Jun 20, 2014 - In this work we will focus on the cases where A is an N = 4 or N = 2 ... group need only fix a 2-plane and preserve an N = 2 superconformal algebra, there .... where k = (k1,...,k12) is an arbitrary 12-tuple of C-numbers. .... Now, we can generalize our formula for the partition function (2.6) to ...... 1A 2A 3A 4A.
SU/ITP-14/17
Mock Modular Mathieu Moonshine Modules Miranda C. N. Cheng1 , Xi Dong2 , John F. R. Duncan3 , Sarah Harrison2 , Shamit Kachru2 , and Timm Wrase2
arXiv:1406.5502v1 [hep-th] 20 Jun 2014
1
Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands∗
2
Stanford Institute for Theoretical Physics, Department of Physics, and Theory Group, SLAC, Stanford University, Stanford, CA 94305, USA
3
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA
Abstract We construct vertex operator super-algebras which lead to modules for moonshine relations connecting the sporadic simple Mathieu groups M22 and M23 , with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co0 that fixes a 3dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N = 4 superconformal algebra. Similarly, any subgroup of Co0 that fixes a 2-dimensional
subspace of the 24-dimensional representation commutes with a certain N = 2 superconformal
algebra. Through the decomposition of the corresponding twined partition functions into characters of the N = 4 (resp. N = 2) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional Z-graded module for the
corresponding group. The two Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding mock modular forms to be regular at all cusps inequivalent to the low temperature cusp at i∞. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.
∗
On leave from CNRS, Paris.
1
Contents 1 Introduction
2
2 The Free Field Theory
5
3 The Superconformal Algebras
7
4 Global Symmetries
11
5 Twining the Module
13
6 The N = 4 Decompositions
16
7 The N = 2 Decompositions
21
8 Mathieu Moonshine
24
9 Discussion
28
A Jacobi Theta Functions
30
B Character Tables
31
C Coefficient Tables
41
D Decomposition Tables
55
References
87
1
Introduction
The investigation of moonshine connecting modular objects, sporadic groups, and 2d conformal field theories has been revitalized in recent years by the discovery of several new classes of examples. While monstrous moonshine [1–6] remains the best understood and prototypical case, a new class of umbral moonshines tying mock modular forms to automorphism groups of Niemeier lattices has recently been uncovered [7, 8]. The best studied example, and the first to be discovered, involves the group M24 and was discovered through the study of the elliptic genus of K3 [9]. The twining functions have been constructed in [10–12] and were proved to be the graded characters of an infinite-dimensional M24 -module in [13]. Steps towards a better and deeper understanding of this Mathieu moonshine can be found in [14–25], and particularly in [26], where the importance of K3 surface geometry for all cases of umbral moonshine is
2
elucidated. Possible connections to space-time physics in string theory have been discussed in [27–31]. In none of these cases, however, has a connection to an underlying conformal field theory (whose Hilbert space furnishes the underlying module) been established. The goal of this paper is to provide first examples of mock modular moonshine for sporadic simple groups G, where the underlying G-module can be explicitly constructed in the state space of a simple and soluble conformal field theory. Our starting point is the Conway module sketched in [2], studied in detail in [32], and revisited recently in [33]. The original construction was in terms of a supersymmetric theory of bosons on the E8 root lattice, but this has the drawback of obscuring the true symmetries of the model. In [32], a different formulation of the same theory, as a Z2 orbifold of the theory of 24 free chiral fermions, is introduced. A priori, the theory has a Spin(24) symmetry. However, one can also view this theory as an N = 1 superconformal field theory. The choice of N = 1
structure breaks the Spin(24) symmetry to a subgroup. In [32] it was shown that the subgroup preserving the natural choice of N = 1 structure is precisely the Conway group Co0 , a double cover of the sporadic group Co1 . In [33] it is shown that this action can be used to attach a
normalized principal modulus (i.e. normalized Hauptmodul) for a genus zero group to every element of Co0 . In this paper, we show that generalizations of the basic strategy of [32, 33] can be used to construct a wide variety of new examples of mock modular moonshine. Instead of choosing an N = 1 superconformal structure, we choose larger extended chiral algebras A. The subgroup GA of Spin(24) that commutes with a given choice can be determined by simple geometric considerations; in the cases of interest to us, it will be a subgroup that preserves point-wise a 2-plane or a 3-plane in the 24 dimensional representation of Co0 , or in plain English, a subgroup that acts trivially on two or three of the free fermions in some basis. In the rest of the paper, we will often use 24 to denote the unique non-trivial 24-dimensional representation of Co0 , and an n-plane to refer to a n-dimensional subspace in 24. It is natural to ask about the role in moonshine, or geometry, of n-planes in 24 for other values of n. One of the inspirations for our analysis here is the recent result of Gaberdiel– Hohenegger–Volpato [15] which indicates the importance of 4-planes in 24 for non-linear K3 sigma models. The relationship between their results and the Co0 -module considered here is studied in [34], where connections to umbral moonshine for various higher n are also established. We refer the reader to §9 for a discussion of the interesting case that n = 1.
In this work we will focus on the cases where A is an N = 4 or N = 2 superconformal
algebra, though other possibilities exist. In the first case, we demonstrate that any subgroup of Co0 that preserves a 3-plane in the 24 dimensional representation can give rise to an N = 4
structure. The groups that arise are discussed in e.g. Chapter 10 of the book by Conway and
3
Sloane [35]. They include in particular the Mathieu group M22 . In the second case, where the group need only fix a 2-plane and preserve an N = 2 superconformal algebra, there are again many possibilities (again, see [35]), including the larger Mathieu group M23 . Note that the
larger the superconformal algebra we wish to preserve, the smaller the global symmetry group is. Corresponding to specific choices of the N = 0, 1, 2, 4 algebras, we have the global symmetry groups Spin(24) ⊃ Co0 ⊃ M23 ⊃ M22 .
We should stress again that there are other Co0 subgroups that preserve N = 4 resp. N = 2
superconformal algebras arising from 3-planes resp. 2-planes in 24. Some examples are: the
group U4 (3) for the former case, and the McLaughlin (McL) and Higman–Sims (HS) sporadic groups, and also U6 (2) for the latter case. However, only for the Mathieu groups do the twined mock modular forms arising from the twined partition function of the module display uniformly a special property, which we regard as an essential feature of the moonshine phenomena. Namely, all the mock modular forms obtained via twining by elements of the Mathieu groups are regular at all cusps inequivalent to the “low-temperature” cusp τ → i∞.
The importance of this property is its predictive power: armed with it we are able to write
down trace functions for the actions of Mathieu group elements with no more information than a certain fixed multiplier system, and the levels of the functions we expect to find. A priori these are just guesses, but the constructions we present here verify their validity. For this reason, we focus on the two Mathieu groups as the two cases of mock modular moonshine arising from the present chiral conformal field theory. This may be compared to the predictive power of the genus zero property of monstrous moonshine: if Γ < SL2 (R) determines a genus zero quotient of the upper-half plane, and if the stabilizer of i∞ in Γ is generated by ± ( 10 11 ), then there is a unique Γ-invariant holomorphic function satisfying TΓ (τ ) = q −1 + O(q) as τ → i∞, for q = e2πiτ . The miracle of monstrous moonshine, and the content of the moonshine conjectures of Conway–Norton [1], is that for
suitable choices of Γ, the function TΓ is the trace of an element of the monster on some graded infinite-dimensional module (namely, the moonshine module of [3]). The optimal growth property formulated in [8] plays the analogous predictive role in umbral moonshine, and is very similar to the special property we formulate for the Mathieu moonshine considered here. We mention here that although the moonshine conjectures have been proven in the monstrous case by Borcherds [4], and verified in [18] for the M24 case of umbral moonshine, conceptual explanations of the genus zero property of monstrous moonshine, and of the analogous properties of umbral moonshine, and the Mathieu moonshine studied here, remain to be determined. An approach to establishing the genus zero property of monstrous moonshine via quantum gravity is discussed in [46]. The organization of the paper is as follows. We begin with a review of the module discussed in [32]. In §3, we describe how one can endow this module with an N = 4 or N = 2 structure.
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In §4, we discuss what this does to the manifest symmetry group of the model - reducing the
symmetry from Co0 to a variety of other possible groups which preserve a 3-plane (respectively
2-plane) in 24 of Co0 . In §4 and §5, we discuss the action of these Co0 -subgroups on the modules
and compute the corresponding twining functions. We identify M23 , M22 , McL, HS, U6 (2) and U4 (3) as some of the most interesting Co0 -subgroups preserving some extended superconformal algebra. In §6 and §7, we discuss in some detail the decomposition of the graded partition
function of our chiral CFT into characters of irreducible representations of the N = 4 and N = 2 superconformal algebra. In §8, we discuss the special property we require from a
moonshine twining function, and show how this property singles out the Mathieu groups in our
setup. We close with a discussion in §9. The appendices contain many tables: character tables
for the various groups we discuss, tables of coefficients of the vector valued mock modular forms that arise in our class functions, and tables describing the decompositions of our modules into irreducible representations of the various groups.
2
The Free Field Theory
The chiral 2d conformal field theory that will play a starring role in this paper has two different constructions. The first is described in [3] and starts with 8 free bosons X i compactified on the 8-dimensional torus given by the E8 root lattice, together with their Fermi superpartners ψ i . One then orbifolds by the Z2 symmetry (X i , ψ i ) → (−X i , −ψ i ) .
(2.1)
This construction has manifest N = 1 supersymmetry; after orbifolding, it is a c = 12 theory with no NS primary fields of dimension 12 . The partition function of this free field theory can
easily be computed; the NS sector partition function is given by ZN S,E8(τ )
(2.2)
= q −1/2 + 0 + 276q 1/2 + 2048q + 11202q 3/2 + · · · ,
(2.3)
=
1 2
E4 (τ )θ3 (τ, 0)4 θ4 (τ, 0)4 θ2 (τ, 0)4 + 16 + 16 12 4 η(τ ) θ2 (τ, 0) θ4 (τ, 0)4
where E4 is the weight 4 Eisenstein series, η(τ ) = q 1/24
Q∞
n=1 (1
− q n ) is the Dedekind eta
function, and θi are the Jacobi theta functions recorded in Appendix A. We have also written q = e(τ ) and we use the shorthand notation e(x) = e2πix throughout this paper. One recognizes representations of the Co1 sporadic group appearing in the q-series: apart
5
from 276, one can also observe 2048
=
1 + 276 + 1771 ,
(2.4)
11202
=
1 + 276 + 299 + 1771 + 8855 ,
(2.5)
··· In fact, this model has a (non-manifest) Co0 ∼ = 2.Co1 symmetry. In this paper, we will sometimes use n or Zn to denote Z/nZ depending on the context. A better representation, for our purposes, was discussed in [32]. The E8 orbifold theory is equivalent to a theory of 24 free chiral fermions λ1 , λ2 , . . . , λ24 , also orbifolded by the Z2 symmetry λα → −λα . This gives an alternative description of the Conway module above. The
partition function from this “free fermion” point of view is more naturally written as 4
ZN S,fermion(τ ) =
1 X θi12 (τ, 0) . 2 i=1 η 12 (τ )
(2.6)
This is equivalent to the answer (2.2) by non-trivial identities on theta functions. Note that θ1 (τ, 0) = 0. The free fermion theory has a manifest Spin(24) symmetry, but not a manifest N = 1
supersymmetry. However, one can construct an N = 1 superalgebra as follows. There is a
unique NS ground state, but there are 212 = 4096 Ramond sector ground states, constructed of products of the Ramond sector fermion zero modes. It will be convenient to label the 4096 ˜ 12 , where F ˜ 2 = {−1/2, 1/2}. There are therefore 4096 spin fields of states by a vector s ∈ F 2
dimension
3 2
which implement the flow from the NS to the R sector. Denoting these as Wa , one
can try to find a linear combination which can serve as an N = 1 supercharge by taking linear combinations
W=
X
˜ 12 s∈F 2
c s Ws .
(2.7)
As discussed in [32], and as we will review in the next section, there exists a set of ca satisfying the conditions for the operator product algebra of W and the stress tensor T to close properly onto an N = 1 superconformal algebra. This solution breaks the Spin(24) symmetry, and is
stabilized precisely by the subgroup Co0 .
In the rest of this paper, we extend this idea as follows. Instead of choosing a chiral N = 1
super-Virasoro algebra and viewing the theory as an N = 1 SCFT, we choose instead various
extended chiral algebras. We will argue that N = 4 and N = 2 super-Virasoro presentations of the theory are in one to one correspondence with choices of subgroups of Co0 which fix a 3-plane (respectively, 2-plane) in the 24 dimensional representation. This leads us naturally to super-modules with various interesting global symmetry groups, and whose twining functions
6
are easily computed in terms of the partition function (or elliptic genus) of the free fermion conformal field theory. These functions in turn are expressed nicely in terms of mock modular forms, and thus we establish moonshine-like relations for subgroups of Co0 via this family of modules.
3
The Superconformal Algebras
We first discuss the largest chiral algebra we will consider, which gives rise to the smallest global symmetry groups. We will construct an N = 4 super-conformal algebra (SCA) in the
free fermion orbifold theory. Our strategy is to first construct the SU (2) R-currents, and act with them on an N = 1 supercurrent to generate the full N = 4 SCA. In this process, we break the Co0 symmetry group down to a proper subgroup as we will discuss in §4.
We start with 24 real free fermions λ1 , λ2 , . . . , λ24 . Picking out the first three fermions, we
obtain the currents Ji : Ji = −iǫijk λj λk ,
i, j, k ∈ {1, 2, 3} .
(3.1)
They form an affine SU (2) current algebra with level 2 as may be seen from their OPE: Ji (z)Jj (0) ∼
i 1 δij + ǫijk Jk (0) . z2 z
(3.2)
The next step is to pick an N = 1 supercurrent and act with Ji on it. As we reviewed in §2,
an N = 1 supercurrent exists in this model and may be written as a linear combination of spin fields. Let us go through this construction in more detail, as we will extend it to find the N = 4
super-algebra. To write the N = 1 supercurrent explicitly, we first group the 24 real fermions
into 12 complex ones and bosonize them: ψa ≡ 2−1/2 (λ2a−1 + iλ2a ) ∼ = eiHa ,
ψ¯a ≡ 2−1/2 (λ2a−1 − iλ2a ) ∼ = e−iHa ,
a = 1, 2, · · · , 12 .
(3.3)
In terms of the bosonic fields H = (H1 , . . . , H12 ), an N = 1 supercurrent W may be written as W =
X
ws eis·H cs (p) ,
(3.4)
˜ 12 s∈F 2
where each component of s = (s1 , s2 , · · · , s12 ) takes the values ±1/2, and the coefficients ws
are C-numbers. We have introduced cocycle operators cs (p) to ensure that the operators with integer spins commute with all other operators, and the operators with half integral spins anticommute among themselves. The cocycle operators depend on the zero-mode operators p which
7
are characterized by the commutation relation
p, eik·H = keik·H ,
(3.5)
where k = (k1 , . . . , k12 ) is an arbitrary 12-tuple of C-numbers. The associativity and closure of the OPE of the “dressed” vertex operators Vk = eik·H ck (p)
(3.6)
ck (p + k′ )ck′ (p) = ǫ(k, k′ )ck+k′ (p)
(3.7)
requires that
and that ǫ(k, k′ ) satisfies the 2-cocycle condition ǫ(k, k′ ) ǫ(k + k′ , k′′ ) = ǫ(k′ , k′′ ) ǫ(k, k′ + k′′ ) .
(3.8)
Moreover, in order for Vk to have the desired (anti)-commutation relation, a further condition is imposed: ′
ǫ(k, k′ ) = (−1)k·k +k
2
k′2
ǫ(k′ , k) .
(3.9)
An explicit description of the cocycle for a general vertex operator eik·H may be chosen as [36] ck (p) = eiπk·M·p .
(3.10)
In our case, M is a 12 × 12 matrix that has the block form
M4 M = I4 I4
0
0
M4
0
I4
M4
0
1 M4 = 1 −1
,
0
0
0
0
1
0
1
−1
0
0 , 0 0
(3.11)
and I4 is a 4 × 4 matrix with all elements being 1. Generically, the OPE of W with itself is
wΓ ¯ αβγδ w T (0) 1 + + λα λβ λγ λδ (0) , W (z)W (0) ∼ ww ¯ z3 4z 96z
(3.12)
where we have defined w ¯s = w−s c−s (s), Γαβγδ ss′
(3.13) = Γα Γβ Γγ Γ ss′ c−s (s′ − s) −2s⌈ α2 ⌉+1 · · · (−2s⌈ β ⌉ ) −2s⌈ γ2 ⌉+1 · · · (−2s⌈ δ ⌉ ) , (3.14) δ
2
8
2
for α < β < γ < δ. Other components of Γαβγδ are defined by the requirement that it is totally antisymmetrized. For W to be an N = 1 supercurrent, the last term must vanish: wΓ ¯ αβγδ w = 0 ,
∀α, β, γ, δ ∈ {1, 2, · · · , 12} ,
(3.15)
and the first two terms must have the correct normalization ww ¯ =
X
w−s ws cs (−s) = 8 .
(3.16)
˜ 12 s∈F 2
We may also write the SU (2) R-currents in (3.1) in the bosonized language: 1 iH1 − e−iH1 eiH2 eiπp1 + e−iH2 e−iπp1 , e 2 i iH1 + e−iH1 eiH2 eiπp1 + e−iH2 e−iπp1 , e J2 = 2 J1 = −
J3 = i∂H1 ,
(3.17) (3.18) (3.19)
where we have included the cocycles e±iπp1 . We may now act with the SU (2) currents Ji on our N = 1 supercurrent W and extract the singular terms of their OPEs: Ji (z)W (0) ∼ −
i Wi (0) , 2z
(3.20)
where Wi are slightly modified combinations of spin fields: W1 = − W2 = i
X
2s2 wRs eis·H cs (p) ,
(3.21)
4s1 s2 wRs eis·H cs (p) ,
(3.22)
2s1 ws eis·H cs (p) ,
(3.23)
s
X s
W3 = i
X s
where Rs ≡ (−s1 , −s2 , s3 , · · · , s12 ).
We claim that all three Wi defined above are valid N = 1 supercurrents. This is because we
may obtain, for instance, W3 from W by rotating the 1-2 plane by π, and the conditions (3.15),
(3.16) for being an N = 1 supercurrent are invariant under SO(24) rotations. We may obtain
W2 and W3 similarly. This shows that each of the Wi is an N = 1 supercurrent.
9
Furthermore, we can check that the OPEs of the Wi are given by 2 2 1 8 Wi (z)Wj (0) ∼ δij 3 + T (0) + 2iǫijk 2 Jk (0) + ∂Jk (0) , z z z z 2 ∂ W (z)Wi (0) ∼ −2i Ji (0) , + z2 z i (δij W + ǫijk Wk ) , Ji (z)Wj (0) ∼ 2z
(3.24) (3.25) (3.26)
upon using the identity (3.15). This shows that W , Wi , Ji , and the stress tensor T defined as T =−
1X 1X λα ∂λα = − ∂Ha ∂Ha 2 α 2 a
(3.27)
form an N = 4 SCA with central charge c = 12. We may recombine the four N = 1 supercurrents W , Wi into the more conventional N = 4 supercurrents W1± ≡ 2−1/2 (W ± iW3 ) ,
W2± ≡ ±2−1/2 i(W1 ± iW2 ) ,
(3.28)
¯ of the SU (2) R-symmetry. In terms of these supercurrents we obtain which transform as 2 + 2 the standard (small) N = 4 SCA with c = 12 characterized by the following set of OPEs: 2 1 6 + 2 T (0) + ∂T (0) , 4 z z z 1 3 ± ± T (z)Wa (0) ∼ 2 Wa (0) + ∂Wa± (0) , 2z z 1 1 T (z)Ji (0) ∼ 2 Ji (0) + ∂Ji (0) , z z 2 8 2 1 − + i J (0) + Wa (z)Wb (0) ∼ δab 3 + T (0) − 2σab ∂J (0) , i i z z z2 z T (z)T (0) ∼
Wa+ (z)Wb+ (0) ∼ Wa− (z)Wb− (0) ∼ 0 , Ji (z)Wa+ (0) ∼ −
1 i σ W + (0) , 2z ab b
1 i∗ − σ W (0) , 2z ab b 1 i Ji (z)Jj (0) ∼ 2 δij + ǫijk Jk (0) . z z
Ji (z)Wa− (0) ∼
(3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36)
Here σ i are the Pauli matrices. Now, we can generalize our formula for the partition function (2.6) to include a grading by the U (1) charge under the Cartan generator of the SU (2). The U (1) current J0 is twice the zero-mode of the J3 current. From this and the definition J3 = −iλ1 λ2 = ψ1 ψ¯1 , we see that under J0 the complex fermion ψ1 has charge 2 while the other 11 complex fermions are neutral.
10
Therefore, the U (1)-graded NS-sector partition function becomes 4
ZN S,fermion(τ, z) =
1 X θi (τ, 2z) θi (τ, 0)11 . 2 i=1 η(τ )12
(3.37)
In the above discussion, we have chosen the first three fermions out of a total of 24 to generate a set of SU (2) currents. Together with an N = 1 supercurrent they generate a full N = 4
SCA. It is clear that we are free to choose any three fermions for this purpose. In fact, we could choose an arbitrary three-dimensional subspace of the 24-dimensional vector space spanned by the fermions, and obtain an N = 4 SCA. For a given N = 1 supercurrent, not all choices of three-spaces are equivalent, as we will see in a moment.
Before moving on, we should also mention that we could instead have chosen to single out only two real fermions, and construct a U (1) current algebra instead of an SU(2) current algebra. Completely analogous manipulations then show that each such choice provides an N = 2 superconformal algebra; as a result we can associate N = 2 SCAs with G symmetry to
subgroups G of Co0 which stabilise 2-planes in 24.
4
Global Symmetries
Enhancing the N = 1 structure of the theory to N = 4 breaks the Co0 symmetry. We now show that for a specific choice of a three-dimensional subspace of the 24-dimensional vector space,
resulting in a specific copy of the N = 4 SCA, the stabilising subgroup of Co0 is the sporadic
group M22 . Similarly, for a specific choice of a two-dimensional subspace of the 24-dimensional vector space, resulting in a specific copy of the N = 2 SCA, the stabilising subgroup of Co0 is
the sporadic group M23 . This amounts to a proof that the model described in §2 results in an infinite-dimensional M22 (resp. M23 )-module underlying the mock modular forms described in §6 (resp. §7) arising from its interpretation as an N = 4 (resp. N = 2) module.
Recall that the theory regarded as an N = 0 theory has a Spin(24) symmetry resulting
from the SO(24) rotations on the 24-dimensional space, and choosing an N = 1 supercurrent
breaks the Spin(24) group down to its subgroup Co0 . The group Co0 is the automorphism group of the Leech lattice ΛLeech , and various interesting subgroups of Co0 can be regarded as the subgroups of Co0 which stabilise certain combinations of lattice vectors in ΛLeech . To study the automorphism group of the module when fixing more structure—more supersymmetries in this case—it will therefore be useful to describe the enhanced supersymmetries in terms of Leech
lattice vectors. In chapter 10 of [35], it is shown that if we choose an appropriate tetrahedron in the Leech lattice whose edges have lengths squared 16 × (2, 2, 2, 2, 3, 3) in the normalisation
described below, the subgroup of Co0 that leaves all vertices of the tetrahedron invariant is M22 . To be more precise, let eγ , γ ∈ {1, 2, . . . , 24} be an orthonormal basis of R24 and let ΛLeech be
11
P the copy of the Leech lattice generated by the vectors {2 γ∈C eγ | C is a special octad of the P24 extended Golay code} and the vector −4e1 + γ=1 eγ . (One can show that all 24 vectors of P the form −4eα + 24 γ=1 eγ are in ΛLeech .) Define the tetrahedron T{α,β} to be that whose four P24 P24 vertices are O = 0, Xα = 4eα + γ=1 eγ , Xβ = 4eβ + γ=1 eγ and Pαβ = 4eα + 4eβ , for any α, β ∈ {1, 2, . . . , 24} with α 6= β. For every such T{α,β} , the subgroup fixing every vertex is a
copy of M22 , a sporadic simple group of order 27 · 32 · 5 · 7 · 11 = 443, 520 and the subgroup of
M24 fixing eα and eβ .
From the above discussion, it is clear that given {α, β}, a copy of M22 stabilises the real P24 span of eα , eβ and γ=1 eγ . Given a choice of the N = 1 superconformal algebra, this copy
of R3 in 24 then determines, up to rotations, the three fermions, denoted λ1,2,3 , from which the SU (2) current algebra was built in §3. By definition then, a copy of M22 leaves the N = 4
superconformal algebra invariant.
A natural question is: what is the symmetry group G that fixes a given choice of N = 2
superconformal structure? Given the above description of the M22 action, we can choose the P24 R2 ⊂ R3 generated by eα and γ=1 eγ and use the two free fermions lying in the R2 to construct the N = 2 sub-algebra of the N = 4 SCA. Specifically, the U (1) R-symmetry is simply the rotation of the R2 . From the above discussion, it is not hard to see that there is a copy of M23
fixing eα and hence stabilising the N = 2 structure. Recall that M23 is a sporadic simple group of order 27 · 32 · 5 · 7 · 11 · 23 = 10, 200, 960. In terms of the Leech lattice, it corresponds to the fact that the stabiliser of the triangle in ΛLeech whose edges have lengths squared 16 × (6, 3, 2), with P24 vertices chosen to be O, Xα and 2 γ=1 eγ , is a copy of M23 inside the copy of Co0 stabilising ΛLeech .
This furnishes a proof that the theory considered in §2 leads to M22 - and M23 -modules
underlying the mock modular forms defined in §6 and §7.
We should mention that by stabilising slightly different choices of geometric structure (in-
stead of the tetrahedron and triangle which result in preservation of M22 or M23 , respectively), one can find instead other global symmetry groups G. We can therefore construct G-modules with N = 4 (N = 2) superconformal symmetry, leading to mock modular forms, for any G
which arises by stabilizing an object contained in a 3-plane (2-plane). We will discuss as further examples, beyond M22 and M23 , the following cases. We stress however that the special properties of the twining functions enjoyed by the M22 and M23 moonshines do not extend to these other groups, as we will show in §8. Another N = 4 super-module
The group U4 (3), which has order 3,265,920, can arise as the stabilizer of a suitably chosen 3-
plane in the 24 of Co0 [35]. The U4 (3) characters are presented in table 20, the coefficients in the associated (twined) vector valued mock modular forms in tables 3 and 4, and the decomposition
12
of the module into irreducible representations of the group in tables 26 and 27. Several additional N = 2 super-modules
In addition to M23 , one can find several other interesting groups which arise as stabilizers
of suitable 2-planes in the 24. These include: • The Higman-Sims group HS. This is a sporadic simple group of order 44,352,000. Its characters
are presented in table 22, the coefficients of the (twined) mock modular forms in tables 9 and 10, and the decomposition of the module into irreducible representations of the group in tables 39-44.
• The McLaughlin group McL. This is a sporadic simple group of order 898,128,000. Its characters are presented in table 21, the coefficients of the (twined) mock modular forms in tables
7 and 8, and the decomposition of the module into irreducible representations of the group in tables 33-38. • The group U6 (2), of order 9,196,830,720. The characters are given in tables 17-19 , the
coefficients of the (twined) mock modular forms in tables 11-14, and the decomposition of the module into irreducible representations of the group in tables 45-54.
5
Twining the Module
In the last sections, we have described how to view the Hilbert space of the orbifolded free fermion CFT as an N = 4 or N = 2 super-module. Let us call the relevant Ramond-sector module V ; and let us choose a R-symmetry generator (twice the Cartan generator of the SU (2)R
in the N = 4 case, or the single U (1) generator of the N = 2 SCA) J0 . Then it is natural to define the Ramond-sector U(1)-graded partition function Z(τ, z) = Tr V (−1)F q L0 −c/24 y J0 = =
1
4 X
(5.1)
(−1)i+1 θi (τ, 2z)θi11 (τ, 0)
(5.2)
4 4 X θi (τ, z) 1 E4 (τ )θ14 (τ, z) , + 8 2 η 12 (τ ) θi (τ, 0) i=2
(5.3)
2η 12 (τ )
i=1
where we have introduced a chemical potential for the J0 charges and written y = e(z). As we have seen in §2, the two different ways of writing this function, (5.2) and (5.3), are intuitively
connected more closely with the free fermion and E8 root lattice descriptions of the chiral CFT, respectively. Of course, the NS sector partition function (cf. (3.37), (2.2) and (2.6)) is related
13
to the above graded Ramond-sector partition function by a spectral flow transformation ZN S (τ, z) = q 1/2 y −2 Z(τ, z −
τ +1 2 ).
(5.4)
There is a natural way in which one can twine the above function under certain subgroups of Co0 . From the previous discussions, we see that the representation 24 plays a central role in the way various subgroups of Co0 act on the model. Let’s denote by {ℓg,k , ℓ¯g,k }, k = 1, . . . 12, the 12
complex conjugate pairs of eigenvalues of g when acting on 24. This information is conveniently encoded in the so-called Frame shape of g, given by Πg =
Y
L n mn ,
n
satisfying
P
n
1 ≤ L1 < L2 < L3 . . . ,
and mn ∈ Z, mn 6= 0 ,
Ln mn = 24, through the fact that the 12 pairs {ℓg,k , ℓ¯g,k } are precisely the 24
roots solving the equation
Y (xLn − 1)mn = 0. n
As discussed in §3 and §4, in order to preserve at least N = 2 superconformal symmetry and
hence be able to twine the graded R-sector partition function (5.1), the subgroup G must leave
at least a 2-dimensional subspace in 24 pointwise invariant. In the graded partition function this corresponds to leaving the factor θi (τ, 2z) in (5.2) invariant. As a result, for every conjugacy class [g] of such a group G we can choose ℓg,1 = ℓ¯g,1 = 1. It is easy to see that when acting on the twisted sector contributing to the terms involving θi with i = 3, 4 in (5.2), the group element g simply replaces θi11 (τ, 0) with 12 Y
θi (τ, ρg,k )
(5.5)
k=2
where e(ρg,k ) = ℓg,k . When trying to do the same for the contribution from the untwisted sector contributing to the term involving θ2 in (5.2), however, we see that the above simple consideration suffers from an ambiguity. This can be seen from the fact that θ2 (τ, ρ) = −θ2 (τ, ρ+
1) and hence the answer cannot be determined simply by looking at the g-eigenvalues on 24. This of course is a reflection of the fact that the ground states in the untwisted sector transforms in the 212 -dimensional representation of Spin(24) (when the theory is treated as a Virasoro module; that is, when we do not keep any superconformal structure fixed), and the 2-fold ambiguity in taking the logarithm of the eigenvalues ℓg,k is precisely the 2-fold covering structure of Spin(24) with respect to SO(24). As a result, to specify the twining of the untwisted sector contribution we also need to know the action of G on its 212 -dimensional representation, henceforth denoted 4096, which corresponds to 4096 = 1 + 276 + 1771 + 24 + 2024 in terms of the decomposition
14
into irreducible representations of Co0 . Finally we will discuss the twining of the vanishing term 0 =
θ1 (τ,2z)θ111 (τ,0) 2η 12 (τ )
in (5.2). It
vanishes because the 212 ground states in the twisted sector come in pairs with opposite eigenvalues for (−1)F . Moreover, exchanging the pair corresponds to complex conjugation ψa ↔ ψ¯a , a = 1, . . . , 12 of the complex fermions. Recall that one of the complex fermions, called ψ1 in (3.19), was used to construct the U (1)-current and we are interested in the graded partition function where we introduce a chemical potential z for the corresponding U (1)-charge. Because exchanging ψ1 ↔ ψ¯1 also induces a flip of U (1)-charges, captured by z ↔ −z, the contribution of the first complex fermion does not vanish, corresponding to the fact that the identity θ1 (τ, z) = θ1 (τ, z + 2) = −θ1 (τ, −z) only forces θ1 (τ, z) to vanish at z ∈ Z. Consequently, the g-twining of 0 =
(5.6) θ1 (τ,2z)θ111 (τ,0) 2η 12 (τ )
in
(5.2) is only non-zero if and only if ρg,k 6∈ Z for all k = 2, . . . , 12; in other words, when the
cyclic group generated by g fixes nothing but a 2-plane. By inspection we find that, among
the groups we consider, such group elements must be in the conjugacy classes 23AB ⊂ M23 ,
6AB, 12AB, 12DE, 18AB ⊂ U6 (2), 15AB, 30AB ⊂ McL, or 20AB ⊂ HS. The pairs of these conjugacy classes corresponding to the letter A and B (D and E) are mutually inverse, and so
their respective traces, on any representation, are related by complex conjugation. In terms of our construction, choosing one over the other is the same as choosing what one labels ψ1 and ψ¯1 , and the same as choosing an orientation on the 2-plane fixed by the group element in 24. As a result, from (5.6) we see that the θ1 term in the partition functions twined by these conjugate “A” (D) and “B” (E) classes come with an opposite sign. Let us work with the principal branch of the logarithm, and choose ρg,k ∈ [0, 1/2] in (5.5). Then, by direct
computation—we must compute directly, for the choice of labels for mutually inverse conjugacy classes is not natural—there exists a copy of Co0 in Spin(24) for which the signs in (5.8) are
ǫg,1 = 1 for g in
23A ⊂ M23 , 20A ⊂ HS,
(5.7)
15A ∪ 30A ⊂ M cL, 12A ∪ 12D ∪ 6B ∪ 18B ⊂ U6 (2),
and ǫg,1 = −1 for the inverse classes, 23B ⊂ M23 , 20B ⊂ HS, &c.
Putting these different contributions together, we conclude that for every [g] ⊂ G where G is
a Co0 subgroup preserving (at least) a 2-plane in 24, the corresponding twined graded R-sector
15
partition function reads Zg (τ, z) = TrV g(−1)F q L0 −c/24 y J0 =
(5.8)
12 4 Y X 1 i+1 (−1) ǫ θ (τ, 2z) θi (τ, ρg,k ), g,i i 2η(τ )12 i=1
(5.9)
k=2
where ǫg,2 =
212
Tr 4096 g ∈ {−1, 1} , Q12 k=1 cos(πρg,k /2)
ǫg,3 = ǫg,4 = 1.
(5.10)
The cycle shapes Πg and the values of Tr4096 g for [g] ∈ G are collected for various G ⊂ Co0 in Appendix B. In §6 and §7 we will see how the above twining leads to the mock modular forms playing the role of the McKay–Thompson series in our mock modular moonshine.
6
The N = 4 Decompositions
From the discussion in §3, it is clear that the theory discussed in §2 is a module of the N = 4
superconformal algebra. In this section we will study the decomposition of the Hilbert space V into irreducible representations of the N = 4 SCA and see how the decomposition leads to mock modular forms relevant for the M22 moonshine which we will discuss in §8.
Recall that the N = 4 superconformal algebra contains subalgebras isomorphic to the affine ˆ 2 and the Virasoro algebra. In a unitary representation the former of these acts Lie algebra sl with level m − 1, for some integer m > 1, and the latter with central charge c = 6(m − 1).
N =4 The unitary irreducible highest weight representations vm;h,j are labeled by the two quantum
numbers h and j which are the eigenvalues of L0 and
1 3 2 J0 ,
respectively, when acting on the
highest weight state [37, 38]. In the Ramond sector of the superconformal algebra there are two types of highest weight representations: the short (or BPS, supersymmetric) ones with h=
c 24
=
with h >
m−1 4 m−1 4
and j ∈ {0, 12 , · · · , m−1 2 }, and the long (or non-BPS, non-supersymmetric) ones
and j ∈ { 12 , 1, · · · , m−1 2 }. Their graded characters, defined as 3 3 =4 chN N =4 (−1)J0 y J0 q L0 −c/24 , m;h,j (τ, z) = trvm;h,j
(6.1)
=4 −1 chN µm;j (τ, z) m;h,j (τ, z) = (Ψ1,1 (τ, z))
(6.2)
2 c =4 −1 h− 24 − jm chN q θm,2j (τ, z) − θm,−2j (τ, z) m;h,j (τ, z) = (Ψ1,1 (τ, z))
(6.3)
are given by
and
in the short and long cases, respectively, [37]. In the above formulas, the function µm;j (τ, z) is
16
given by µm;j (τ, z) = (−1)1+2j
X
2
q mk y 2mk
k∈Z
(yq k )−2j + (yq k )−2j+1 + · · · + (yq k )1+2j 1 − yq k
(6.4)
and Ψ1,1 is a meromorphic Jacobi form of weight 1 and index 1 given by Ψ1,1 (τ, z) = −i
θ1 (τ, 2z) η(τ )3 y+1 = − (y 2 − y −2 )q + · · · . (θ1 (τ, z))2 y−1
(6.5)
Finally, we have used the theta functions X
θm,r (τ, z) = k=r
e( k2 ) q k
2
/4m k
y ,
(6.6)
(mod 2m)
defined for all 2m ∈ Z>0 and r − m ∈ Z, and satisfying θm,r (τ, z) = θm,r+2m (τ, z) = e(m) θm,−r (τ, −z). The vector-valued theta function θm = (θm,r ), r − m ∈ Z/2mZ, is a vector-valued Jacobi form of weight 1/2 and index m satisfying θm (τ, z) =
r
1 2m
r
i 2 1 z e(− m τ z ) Sθ .θm (− τ , τ ) τ
= Tθ .θm (τ + 1, z) = θm (τ, z + 1) = e(m(τ + 2z + 1))θm (τ, z + τ ),
(6.7)
where the Sθ and Tθ matrices are 2m × 2m matrices with entries ′
′
rr ) e( −r+r ) (Sθ )r,r′ = e( 2m 2
,
2
r (Tθ )r,r′ = e(− 4m ) δr,r′ .
(6.8)
We will take m ∈ Z for the rest of this section. When we consider N = 2 decompositions in the
next section, we will use the theta function with half-integral indices.
From the above discussion, it is clear that the graded partition function of a module for the c = 6(m − 1) N = 4 SCA admits the following decomposition Z N =4,m =
X
n≥0,0≤r≤m−1 r6=0 when n>0
c′r (n −
17
N =4 r2 r (τ, z) . 4m ) chm; m−1 4 +n, 2
(6.9)
Furthermore, from the identity µ
m; r2
r
r X
n−1
= (−1) (r + 1)µm;0 + (−1)
n q−
(r−n+1)2 4m
n=1
(θm,r−n+1 − θm,−(r−n+1) )
we arrive at
X
Z N =4,m = (Ψ1,1 (τ, z))−1 c0 µm;0 (τ, z) +
r∈Z/2mZ
Fr(m) (τ ) θm,r (τ, z)
(6.10)
where Fr(m) (τ ) = c0 =
∞ X
2
r n− 4m r2 4m ) q
cr (n −
n=0 m−1 X
1≤r ≤m−1
,
(6.11)
2
r (−1)r (r + 1) c′r (− 4m )
(6.12)
r=0
cr (n −
r2 4m )
=
r ′2 r ′ −r ′ Pm−1 ) , n=0 (r + 1 − r) c′r′ (− 4m r ′ =r (−1) c′r (n −
r2 4m )
(m)
The rest of the components of F (m) = (Fr
(m)
(m)
(m)
(τ, z) + f0
fu(m) (τ, z)
(6.13)
), r ∈ Z/2mZ, are defined by setting (m)
Fr(m) (τ ) = −F−r (τ ) = Fr+2m (τ ). Recall that µm;0 (τ, z) = −f0
.
, n>0
(6.14)
(τ, −z), a specialisation of the Appell–Lerch sum
=
X k∈Z
2
q mk y 2mk , 1 − yq k e(−u)
(6.15)
studied in [39], for instance, has the following relation to the modular group SL2 (Z): let the (non-holomorphic) completion of µm;0 (τ, z) be Z i∞ X 1 (τ ′ + τ )−1/2 Sm,r (−¯ θm,r (τ, z) µ ˆm;0 (τ, τ¯, z) = µm;0 (τ, z) − e(− 18 ) √ τ ′ ) dτ ′ . 2m r∈Z/2mZ −¯ τ (6.16) Then µ ˆm;0 transforms like a Jacobi form of weight 1 and index m under the Jacobi group SL2 (Z) ⋉ Z2 . Here Sm = (Sm,r ) is the vector-valued cusp form under SL2 (Z) with a non-trivial multiplier, whose components are given by the unary theta function Sm,r (τ ) = k=r
X
e( k2 ) k q k
(mod 2m)
18
2
/4m
=
1 ∂ θm,r (τ, z)|z=0 . 2πi ∂z
For later use, note that the cusp form Sm,r (τ ) is defined for all 2m ∈ Z and r − m ∈ Z/2mZ.
The way in which the functions Z (m) and µ ˆm;0 transform under the Jacobi group shows P (m) that the non-holomorphic function r∈Z/2mZ Fˆr (τ ) θm,r (τ, z) transforms as a Jacobi form of
weight 1 and index m under SL2 (Z) ⋉ Z2 , where
1 Fˆr(m) (τ ) = Fr(m) (τ ) + c0 e(− 18 ) √ 2m (m)
In other words, F (m) = (Fr
Z
i∞
τ ′ ) dτ ′ . (τ ′ + τ )−1/2 Sm,r (−¯
−¯ τ
), r ∈ Z/2mZ is a vector-valued mock modular form with a
vector-valued shadow c0 Sm , whose r-th component is given by Sm,r (τ ), with the multiplier for SL2 (Z) given by the inverse of the multiplier system of S (m) (cf. (6.8)). Now we are ready to apply the above discussion to the graded partition function of the
theory discussed in §2. Recall that in this case we have c = 12, or m = 3 in other words. The N = 4 decomposition of (5.1) gives
N =4 N =4 N =4 N =4 =4 Z(τ, z) = 21 chN 3; 12 ,0 + ch3; 21 ,1 + 560 ch3; 32 , 21 + 8470 ch3; 52 , 21 + 70576 ch3; 72 , 21 + . . . =4 N =4 N =4 + 210 chN 3; 32 ,1 + 4444 ch3; 25 ,1 + 42560 ch3; 72 ,1 + . . . X hr (τ )θ3,r (τ, z) = (Ψ1,1 (τ, z))−1 24 µ3;0 (τ, z) +
(6.17) (6.18)
r∈Z/6Z
α β 2 where . . . stand for terms with expansion Ψ−1 1,1 q y with α−β /12 > 3. More Fourier coefficients
of the functions hr (τ ) are recorded in Appendix C, where h = hg for [g] = 1A. Note that all the graded multiplicities c′r (n −
r2 12 )
appear to be non-negative. Of course, this is guaranteed
by the fact that V is a module for the N = 4 SCA as shown in §3. In particular, the Fourier coefficients of hr (τ ) appear to be all non-negative apart from that of the polar term −2q −1/12
in h1 . From the above discussion, we see that h = (hr ), r ∈ Z/6Z is a weight 1/2 vector-
valued mock modular form with 6 components and 2 independent components (h0 = h3 = 0, h−1 = −h1 , h−2 = −h2 ), with the shadow given by 24 S (3) .
This is to be contrasted with the elliptic genus of a generic non-chiral SCFT. For example,
the sigma model of a K3 surface has c = 6, and the elliptic genus is given by ˜
EG(τ, z; K3) = TrHRR (−1)FL +FR y J0 q L0 −c/24 q¯L0 −˜c/24 N =4 N =4 N =4 N =4 =4 = 20 chN 1 + ... 2; 41 ,0 − 2 ch2; 14 , 12 + 90 ch2; 54 , 21 + 462 ch2; 49 , 12 + 1540 ch2; 13 4 ,2 n = (Ψ1,1 (τ, z))−1 24 µ2;0 (τ, z) + (θ2,1 (τ, z) − θ2,−1 (τ, z)) o × (−2q −1/8 + 90q 7/8 + 462q 15/8 + 1540q 23/8 + . . . ) ,
(6.19)
(6.20)
−1 α β where . . . stand for terms with expansion Ψ1,1 q y with α − β 2 /8 > 3. In this case, the
19
=4 coefficient multiplying the massless character chN 2; 1 , 1 is negative, arising from the Witten index 4 2
N =4 of the right-moving massless multiplets paired with the representation v2; 1 1 of the left-moving , 4 2
N = 4 SCA.
In §3 we have shown that the theory under consideration, as a module for the N = 4 SCA,
admits a faithful action via automorphisms by a group G, as long as G is a subgroup of Co0
fixing at least a 3-plane. The graded partition function Zg (τ, z) twined by any such g ∈ G is
given by (5.8), and from the fact that the action of g commutes with the N = 4 SCA, we expect
Zg (τ, z) to admit the decomposition
Zg (τ, z) = (Ψ1,1 (τ, z))−1 (Tr24 g) µ3;0 (τ, z) +
X
r∈Z/6Z
hg,r (τ )θ3,r (τ, z) ,
(6.21)
and the coefficients of hg,r (τ ) = ar q −r
2
/12
+
∞ X
n−r G g) q (TrVr,n
2
/12
(6.22)
n=1
to be given by characters of the G-module VG =
∞ M M
G Vr,n
(6.23)
r=1,2 n=1
arising from the free field theory in §2.
We have explicitly computed the first 30 or so coefficients of each q-series hg,r (τ ) for all
conjugacy classes [g] of G, for G = M22 and G = U4 (3). These can be found in the tables in G in terms of Appendix C. Subsequently, we compute the first 30 or so G-representations Vr,n
their decompositions into the irreducible G-representations. They can be found in the tables in Appendix D. Next we would like to discuss the mock modular property of the functions hg = (hg,r ). The Hecke congruence subgroups are defined as a Γ0 (n) = c
b
∈ SL2 (Z), n|c . d
We expect Zg to be a weak Jacobi form of weight zero, index 2 for the group Γ0 (og ) where og is the order of the group element g ∈ G. This can be verified explicitly from the expression (5.8). Repeating the similar arguments as above, we conclude that the vector-valued functions
hg are vector-valued mock modular forms of weight 1/2 and with shadow (Tr24 g)S (3) for the congruence subgroup Γ0 (og ).
20
The N = 2 Decompositions
7
As discussed in §4, the theory presented in §2 can be regarded as a module for an N = 2 SCA
as well as for an N = 4 SCA. Moreover, for every subgroup G < Co0 fixing a 2-plane there is
a N = 2 SCA commuting with the action of G on the theory. As a result, the decomposition of the partition function (5.8) twined by elements of G into N = 2 characters leads to sets
of vector-valued mock modular forms, now of weight 1/2 and index 3/2, which are the graded characters of an infinite-dimensional G-module inherited from the free fermion module V . ˜ g = (h ˜ g,j ) are, let us start by To see what these (vector-valued) mock modular forms h recalling the characters of the irreducible representations of the N = 2 SCA. For the SCA with N =2 central charge c = 3(2ℓ + 1) = 3ˆ c, the unitary irreducible highest weight representations vℓ;h,Q
are labeled by the two quantum numbers h and Q which are the eigenvalues of L0 and J0 from the U (1) R-symmetry current, respectively, when acting on the highest weight state [40,41]. Just as in the N = 4 case, in the Ramond sector of the superconformal algebra there are two types of highest weight representations: the short (or BPS, supersymmetric) ones with h =
Q∈
h>
c 24
=
cˆ 8
and
{− 2cˆ + 1, − 2cˆ + 2, . . . , 2cˆ − 1, 2cˆ }, and the long (or non-BPS, non-supersymmetric) ones with cˆ cˆ cˆ cˆ cˆ cˆ 8 and Q ∈ {− 2 + 1, − 2 + 2, . . . , 2 − 2, 2 − 1, 2 }, Q 6= 0. From now on we will concentrate
on the case when ℓ is half-integral, and hence cˆ is even. Their graded characters, defined as 3 ˜3 =2 chN N =2 (−1)J0 y J0 q L0 −c/24 , ℓ;h,Q (τ, z) = trvℓ;h,Q
(7.1)
are given by c
2
=2 −1 h− 24 −j ℓ θℓ,j (τ, z) , chN q ℓ;h,Q (τ, z) = e( 2 )(Ψ1,− 21 (τ, z))
j = sgn(Q) (|Q| − 1/2) ,
(7.2)
for the long multiplets, and 1
ℓ+Q+1/2 =2 chN )(Ψ1,− 12 (τ, z))−1 y Q+ 2 fu(ℓ) (τ, z + u) , ℓ;c/24,Q (τ, z) = e( 2
for the short multiplets with Q 6=
cˆ 2,
u=
1 2
+
(1+2Q)τ 4ℓ
, (7.3)
=2 respectively. The character chN ℓ;c/24,Q (τ, z) for Q =
cˆ 2
given in (7.5). In the above formula, we have used the Appell–Lerch sum (6.15) and defined Ψ1,− 21 = −i
1 η(τ )3 = 1/2 + q (y 1/2 − y −1/2 ) + O(q 2 ). θ1 (τ, z) y − y −1/2
Note that the above characters satisfy =2 N =2 chN ℓ;c/24,Q (τ, z) = chℓ;c/24,−Q (τ, −z)
under charge conjugation.
21
is
From the relation between the massless and massive characters |Q|−1 =2 N =2 −n chN ℓ;c/24,Q + chℓ;c/24,−Q = q
X
=2 N =2 (−1)k chN ℓ,n+c/24,Q−k + chℓ,n+c/24,k−Q
k=0 =2 + 2(−1)Q chN ℓ;c/24,0 ,
n > 0, |Q| ≤ ℓ ,
(7.4)
c ˆ 2 −1
X =2 N =2 =2 =2 −n chN chN (−1)k chN ℓ;c/24, cˆ = q ℓ;n+c/24, cˆ + ℓ,n+c/24, cˆ −k + chℓ,n+c/24,k− cˆ 2
2
2
k=1
2
c ˆ
=2 + 2(−1) 2 chN ℓ;c/24,0 ,
(7.5)
as well as the charge conjugation symmetry of the theory, we expect the graded partition function of a module, invariant under charge conjugation, for the N = 2 SCA with even central charge
c = 3(2ℓ + 1), to admit the following decomposition: =2 c Z N =2,ℓ = C0′ chN ,0 (τ, z) + ℓ; 24
+
X
n≥0,j∈{ 12 , 23 ,...,ℓ−1}
X
n≥0
=2 Cℓ′ (n − 4ℓ ) chN ℓ; c +n,ℓ+ 1 (τ, z) 24
Cj′ (n −
2
=2 N =2 chN ℓ; c +n,j+ 1 (τ, z) + chℓ; c +n,−(j+ 1 ) (τ, z)
j2 4ℓ )
24
˜ℓ;0 (τ, z) + = e( 2ℓ )(Ψ1,− 21 )−1 C0 µ
2
X
24
2
j−ℓ∈Z/2ℓZ
(ℓ) F˜j (τ )θℓ,j (τ, z) .
(7.6)
(7.7)
In the last equation, we have defined µ ˜ℓ;0 = e( 14 ) y 1/2 fu(ℓ) (τ, u + z) ,
u=
1 τ + , 2 4ℓ
and (ℓ) (ℓ) (ℓ) F˜j (τ ) = F˜−j (τ ) = F˜j+2ℓ (τ ) =
C0 = C0′ + 2
X
n≥0
X
2
Cj (n −
1
j2 n− j4ℓ 4ℓ ) q
2
(−1)j+ 2 Cj′ (− j4ℓ ) ,
,
(7.8) (7.9)
j∈{ 12 , 32 ,...,ℓ}
Cj (n −
j2 4ℓ )
=
Pℓ−j (−1)k C ′
j+k (−
k=0
Cj′ (n −
2
j 4ℓ )
(j+k)2 4ℓ )
n=0
.
(7.10)
n>0
Similar to the case of short N = 4 characters, through its relation to the Appell–Lerch sum, µ ˜ ℓ;0
admits a completion which transforms as a weight one, half-integral index Jacobi form under the Jacobi group. More precisely, define µ ˜d ˜ ℓ;0 and the integral m with the ℓ;0 by replacing µm;0 with µ
d half-integral ℓ in (6.16). Then µ ˜ℓ;0 transforms like a Jacobi form of weight 1 and index ℓ under
the Jacobi group SL2 (Z) ⋉ Z2 . Following the same computation as in the previous section, we
22
(ℓ) hence conclude that F˜ (ℓ) = (F˜j ), j − 1/2 ∈ Z/2ℓZ is a vector-valued mock modular form with
a vector-valued shadow C0 Sℓ = C0 (Sℓ,j (τ )).
Now we are ready to apply the above discussion to the graded partition function of the VOA discussed in §2. The N = 2 decomposition gives =2 =2 =2 =2 =2 =2 + chN + 770 (chN + chN ) + 13915 (chN + chN )+ ... Z(τ, z) = 23 chN 3 1 3 3 3 3 3 5 3 5 3 1 ; ,0 ; ,2 ; ,1 ; ,−1 ; ,1 ; ,−1 2 2 2 2 2 2 2 2 2 2 2 2 N =2 N =2 (7.11) + 231 ch 3 ; 3 ,2 + 5796 ch 3 ; 5 ,2 + . . . 2 2 2 2 1 23 47 24 µ ˜ 23 ;0 + (−q − 24 + 770 q 24 + 13915 q 24 + . . . ) (θ 32 , 21 + θ 32 ,− 12 ) = e( 34 )Ψ−1 1,− 21 3 5 13 (7.12) + (q − 8 + 231 q 8 + 57962q 8 + . . . ) θ 32 , 23 q α y β with α − β 2 /6 > 2. Again, we observe where . . . denote the terms with expansion Ψ−1 1,− 1 2
=2 that all the multiplicities of the multiplets with characters chN 3 ;h,Q appear to be non-negative, 2
consistent with our construction of V as an N = 2 SCA module.
In general, from the previous sections we have seen that the graded partition function twined
by any element g of a subgroup G of Co0 should admit a decomposition into N = 2 characters. We write
Zg (τ, z) =
e( 34 )Ψ−1 1,− 12
(Tr24 g) µ ˜ 3 ;0 (τ, z) +
X
2
˜hg,j (τ )θ 3 j . , 2 2
j∈{1/2,−1/2,3/2}
(7.13)
Moreover, from the discussion in §5 we have seen that ˜ g,−1/2 (τ ) = h ˜ g,−1/2 (−¯ h τ ),
(7.14)
and the coefficients of these functions ˜ g,j (τ ) = aj q −j 2 /6 + h
∞ X
n−j (TrV˜r,n G g) q
2
/6
(7.15)
n=1
are given by characters of the G-module V˜ G =
M
∞ M
G V˜j,n
(7.16)
j=−1/2,1/2,3/2 n=1
which descends from the free field theory in §2. In particular, for any n, the G-representation G G V˜−1/2,n is the complex conjugate of V˜1/2,n .
23
8
Mathieu Moonshine
In the previous sections we have seen that the free field theory described in §2 leads to infinite-
dimensional G-modules underlying a set of vector-valued mock modular forms from its N = 4
(N = 2) structures for any subgroup G of Co0 fixing at least a 3-plane (2-plane) in 24. In this section we will discuss a natural property of the vector-valued mock modular forms that distinguishes M22 (M23 ) from other 3-plane (2-plane) fixing subgroups. These considerations lead to M22 and M23 versions of moonshine involving distinguished vector-valued mock modular forms of weight 1/2. Recall that the celebrated genus zero condition of monstrous moonshine states, among other things, that the monstrous McKay-Thompson series are Hauptmoduls with only a polar term q −1 at the cusp representative at i∞ and no poles at any other cusps. To be more precise, denote by P Tg (τ ) = n≥−1 q n trVn♮ g the graded character of the FLM moonshine module V ♮ = ⊕n≥−1 Vn♮ [3], then Tg (τ ) is the unique function invariant under a particular Γg < P SL2 (R) (specified in [1]), such that (i)
qTg (τ ) = O(1) as τ → i∞,
(ii)
Tg (τ ) = O(1) as τ → α ∈ Q whenever ∞ ∈ / Γg α.
(8.1)
Similarly, in the 23 instances of umbral moonshine [7, 8] involving mock modular forms, it has X been conjectured that, for any Niemeier root system X, the graded characters HgX = (Hg,r ) of
the (conjectural) umbral GX -modules K X , are all mock modular forms for certain subgroups ΓX g of SL2 (Z), with the following behaviour at the cusps (i) (ii)
X q 1/4m Hg,r (τ ) = O(1) as τ → i∞ for all r, X Hg,r (τ ) = O(1) for all r as τ → α ∈ Q, whenever ∞ ∈ / ΓX g α.
(8.2)
See [8] for more details. A conceptual explanation for such a striking special property of the functions involved in moonshine is still unsettled. However, following the precedent set in the monstrous case by [46], this property can be regarded as a consequence of the (conjectural for umbral cases other than X = A24 1 ) construction of these functions via a uniform sum over modular images, sometimes referred to as the regularised Poincar´e sum, or Rademacher sum. See for instance [49] for a review. In turn, such a sum over modular images has been given a physical interpretation in terms of the gravity dual of the 2d CFT [48]. See [46] for specific conjectures that relate the genus zero property of monstrous moonshine to gravity. A natural question is therefore: among the subgroups of Co0 fixing 2- or 3-planes for which we have constructed a module in this work, is there any group G whose corresponding twined mock modular forms satisfy a condition analogous to the preceding cases of moonshine, monstrous
24
and umbral, described above? We will see that the Mathieu groups M22 and M23 , in the N = 4 and N = 2 cases respectively, indeed render mock modular forms satisfying (i)
q 1/6 hg,r (τ ) = O(1) as τ → i∞ for all r,
(ii)
hg,r (τ ) = O(1) for all r as τ → α ∈ Q, whenever ∞ ∈ / Γ(og )α ,
(8.3)
for all g ∈ M22 , and (i)
hg,j (τ ) = O(1) as τ → i∞ for all j, q 3/8 ˜
(ii)
˜ g,j (τ ) = O(1) for all j as τ → α ∈ Q, whenever ∞ ∈ h / Γ(og )α ,
(8.4)
for all g ∈ M23 . On the other hand, all the other groups mentioned in §4 contain elements g
for which the condition (ii) is not satisfied. This fact singles out the Mathieu groups, and the modules we have constructed in the present paper provide a module underlying mock modular moonshine for M22 and for M23 . To investigate the behaviour of hg,r and ˜hg,j , let us look at the behaviour of the twined
graded partition function Zg (τ, z) (cf. (5.8)) at other cusps. From the SL2 (Z) transformation of the Jacobi theta functions (cf. Appendix A), we obtain
Zg
N 2N z 2 z τ =e , + Nτ + 1 Nτ + 1 Nτ + 1
where Θi,N (τ, z) =
P12
ℓ=1
2
4
ρ2ℓ X
ǫi,N (−1)i+1 Θi,N (τ, z)
i=1
12 θi (τ, 2z) Y ρ2k N 2 τ ) θi (τ, ρk + N ρk τ ) e( η 12 (τ ) 2
(8.5)
k=2
and
ǫi,N = ǫi , 2|N
and
ǫ1,N ǫ2,N ǫ3,N ǫ 4,N
25
= −ǫ1 = −ǫ3 = −1 = −ǫ2 = −ǫ4 = −1
otherwise.
(8.6)
Near the cusp τ → i∞, the different contributions have the following leading behaviour: Θ1,N (τ, z) = e
1 2
Θ2,N (τ, z) = e
12 X
+
12 X
k=1
k=1
Θ3,N (τ, z) = e −
( 21 − ρk )⌊N ρk ⌋ −
−ρk ⌊N ρk ⌋ −
12 X
k=1
ρk 2
ρk 2
q fN,1 (Πg )/2 (y + y −1 ) [1 + O(q 1/og )]
k=2
Θ4,N (τ, z) = e −
k=1
(8.7)
ρk ⌊ 21 + N ρk ⌋ q fN,2 (Πg )/2 1 + q 1/2 (y 2 + y −2 )
12 1 1 Y ⌊ 2 +N ρk ⌋+ 2 −N ρk × 1 + e(−ρk ) q 12 X
q fN,1 (Πg )/2 (y − y −1 ) [1 + O(q 1/og )]
(8.8)
1 1 + e(ρk )q 2 +N ρk −n × [1 + O(q 1/og )]
1 ⌊ 2 +N ρk ⌋
Y
n=1
(8.9)
( 12 + ρk )⌊ 21 + N ρk ⌋ q fN,2 (Πg )/2 1 − q 1/2 (y 2 + y −2 )
12 1 1 Y ⌊ 2 +N ρk ⌋+ 2 −N ρk × 1 − e(−ρk ) q k=2
1 ⌊ 2 +N ρk ⌋
Y
n=1
1 1 − e(ρk ) q 2 +N ρk −n × [1 + O(q 1/og )]
(8.10)
with fN,1 (Πg ) = −1 + fN,2 (Πg ) = −1 +
12 X
(N ρk − 1/2)2 + ⌊N ρk ⌋(1 + ⌊N ρk ⌋ − 2N ρk ) ,
(8.11)
N 2 ρ2k − ⌊ 21 + N ρk ⌋(2N ρk − ⌊ 21 + N ρk ⌋) ,
(8.12)
k=1 12 X
k=1
where ⌊x⌋ denotes the largest integer that is not greater than x.
From the fact that the character of the short, uncharged (in the sense that j = 0 resp.
Q = 0) multiplet of both the N = 2 and N = 4 SCAs approaches a constant at any cusp, ∂ cz 2 aτ + b z =2 lim e(−2 ) chN , ) 3/2;1/2,Q=0 ( ∂z τ →∞ cτ + d cτ + d cτ + d ∂ cz 2 z aτ + b =4 = lim e(−2 ) chN , ) = 0, 3;1/2,j=0 ( τ →∞ ∂z cτ + d cτ + d cτ + d
(8.13)
˜ g ) has a pole at the cusp τ → 1/N whenever fN,1 (Πg ) ≤ 0. we see that hg (resp. h
We see that, among all the groups we have considered, U4 (3), U6 (2) as well as the McL
contain a conjugacy classes with Frame shape Πg = 39 /13 , and the HS group has a conjugacy class with Frame shape Πg = 55 /11 . One can explicitly check that f1,1 (Πg ) = 0 for these classes and hence the corresponding twining function has a pole at the cusp τ → 1.
26
Now we are left to check explicitly the condition (ii) in (8.3)-(8.4) for all M22 and M23 twinings. First, recall a set of representatives of the cusps of the group Γ0 (n) can be chosen to be i∞ ∪ { ni | i|n, i 6= 1}. We can check explicitly that fN,1 (Πg ) > 0,
for all [g] ⊂ M23 , N |og , N < og .
(8.14)
In fact, in all cases fN,1 (Πg )/2 is given by the inverse of the width of the cusp. This verifies that there is no pole in the twining function coming from the contribution of θ1 and θ2 to the twined graded partition function (5.8) at cusps different from i∞ and its images under Γg . To make sure that there is no pole in the twining function coming from the contribution of θ3 and θ4 , we first note that fN,2 (Πg ) = 0 whenever g has at least one eigenvalue −1 and at least one
eigenvalue whose N -th power is −1. In this case, the contribution from limτ →i∞ Θ3,N (τ, z) and limτ →i∞ Θ4,N (τ, z) is a (non-vanishing) constant. For the rest of the cases, namely for N = 1
for the order 3,5,7,11,15,23 classes, N = 2 for the order 6 and 14 classes, as well as N = 3 and N = 5 for the order 15 conjugacy classes, we verify explicitly that Θ3,N − Θ4,N goes to a ˜ g′ with [g] ⊂ M22 and [g ′ ] ⊂ M23 constant as τ → ∞. This finishes the proof that all hg and h satisfy (8.3) and (8.4). We therefore claim that our module leads to two cases of mock modular
Mathieu moonshine, with an explicit construction of the module whose twined graded characters are mock modular forms satisfying the distinguishing conditions (8.3), (8.4), which we may now recognise as the natural analogues of the powerful principal modulus property (a.k.a. genus zero property) (8.1) of monstrous moonshine. Indeed, the multiplicities of the N = 4 multiplets in the decomposition (6.17) are very
suggestive of the following group theoretic interpretation1 : the 21 h = 1/2, j = 0 massless multiplets transform as the 21-dimensional irreducible representation of M22 , similarly, the 560 h = 3/2, j = 1/2 massive multiplets transform as χ10 + χ11 (see Appendix B), or “280 + 280”, under M22 , etc. The the first few Fourier coefficients of the vector-valued mock modular forms are recorded in Appendix C, and the corresponding M22 -representations in terms of the irreducible representations can be found in Appendix D. A similar observation can be made on the coefficients multiplying the characters of low lying N = 2 representations in (7.11). See again Appendix C for the corresponding Fourier coefficients, and Appendix D for decompositions into irreducible representations.
The reader will note that many of the numbers which occur as dimensions of irreducible representations of M23 also occur as dimensions of irreducible representations for M24 . Indeed, looking at the tables in C, one is tempted to guess that there is an alternative construction, or hidden symmetry in our model, which yields an M24 -module with the same graded dimensions. 1 The observation that the decomposition into N = 4 characters of (a multiple of) the function Z(τ, z) returns positive integers that are suggestive of representations of the Mathieu group M22 was first communicated privately by Jeff Harvey to J.D. in 2010.
27
In fact, the procedure we have explained for computing twinings can be carried out for any element of M24 , regarded as a subgroup of Co0 , for any such element fixes a 2-space in 24. However, there is no 2-space that is fixed by every element of a given copy of M24 , and explicit computations reveal that any M24 -module structure on the module we have constructed for M23 would have to involve virtual representations. This indicates that there is no natural extension to M24 of the M23 -module we have considered here. However, there is a certain modification of our method for which M24 can be expected to play a leading role, and we refer the reader to the next, and final, section for a description of this.
9
Discussion
In this paper, we have demonstrated that, starting with the free field Co0 module of [32], one can construct explicit examples of modules for various subgroups G ⊂ Co0 which underlie certain
mock modular forms. In particular, subgroups which preserve a 3-plane (respectively 2-plane) in the 24 give rise to N = 4 (N = 2) super-modules with G symmetry. This gives completely explicit examples of mock modular moonshine for the Mathieu groups M22 and M23 , where the
super-modules are known and where all twining functions are Rademacher sums based on a fixed polar part. Other examples, including modules for the sporadic groups McL and HS, are also described. There are several future directions. We considered here the N = 2 and N = 4 extended
chiral algebras, and the subgroups of Co0 that they preserve. Other extended chiral algebras may also yield interesting results. For instance, supersymmetric sigma models with target a Spin(7) manifold give rise to an extended chiral algebra [42], whose representations were studied in [43]. It is an extension of the N = 1 superconformal algebra where instead of adding a U (1) current (which extends the theory to an N = 2 superconformal theory), one chooses an additional Ising factor. This has a natural implementation in our setup, and studying this algebra and the
resulting moonshine is something we intend to do in the future. In particular, moonshine for the groups M24 , Co2 , and Co3 (involving modular forms, rather than mock modular forms) can be expected to arise in this setting. The motivation that led, eventually, to the present study was actually to find connections between geometrical target manifolds associated to c = 12 conformal field theories, and sporadic groups. The elliptic genera of Calabi-Yau fourfolds were computed in [44], for instance; their structure is reminiscent of some of the modules we have seen here, and we intend to further explore and describe some of these connections in a future publication. Likewise, hyperk¨ahler fourfolds, as well as the Spin(7) manifolds mentioned above, provide a wide class of geometries where an analogue of the connections between M24 and K3 may be sought. Last but not least, there are suggestive connections between the trace functions in moonshine
28
modules, and the path integral for quantum gravity. Both the CFT appearing in monstrous moonshine and the Co0 module that played a starring role in this paper appear to play special roles also in AdS3 quantum gravity, where they are candidates for CFT duals to pure (super)gravity [45]. The genus zero property of the twining functions in monstrous moonshine can be reformulated as a condition that these class functions should be expressed as Rademacher sums based on a fixed polar part [46, 47]; this latter description then applies uniformly to monstrous moonshine and umbral moonshine. In this paper we conjecture and provide evidence that it also applies to our M22 and M23 mock modular moonshine. In particular, we have shown that the mock modular forms relevant for the M22 and M23 moonshine satisfy the specific pole condition which indicates a possible construction via Rademacher sums. It is tempting to associate this property with the existence of a “Farey tail”-like expansion for the partition function of a dual quantum gravity theory [48]. Making this connection more precise, and especially finding examples which extend to asymptotically large central charge (as opposed to the present modules, which are at c = 12), is an enticing direction for the future. Acknowledgements We are grateful to Jeff Harvey, Sander Mack-Crane and Daniel Whalen for conversations about related subjects, and to Daniel Baumann for helpful comments on a draft. We note that the appearance of dimensions of M22 representations in an N = 4 decomposition of the partition function of the model studied in this paper was described in correspondence from J. Harvey to J.
Duncan in 2010. The expression (5.2) for Z(τ, z) first came to our attention during the course of conversations with S. Mack-Crane. We thank the Simons Center for Geometry and Physics, and in particular the organizers of the workshop on “Mock Modular Forms, Moonshine, and String Theory,” for hospitality when this work was initiated. S.K. is grateful to the Aspen Center for Physics for providing the rockies during the completion of this work. X.D., S.H. and S.K. are supported by the U.S. National Science Foundation grant PHY-0756174, the Department of Energy under contract DE-AC02-76SF00515, and the John Templeton Foundation. J.D. is supported by the U.S. National Science Foundation (DMS 1203162). T.W. is supported by a Research Fellowship (Grant number WR 166/1-1) of the German Research Foundation (DFG).
29
A
Jacobi Theta Functions
We define the Jacobi theta functions θi (τ, z) as follows for q = e(τ ) and y = e(z): θ1 (τ, z) = −iq 1/8 y 1/2 θ2 (τ, z) = q 1/8 y 1/2
∞ Y
n=1 ∞ Y
n=1
θ3 (τ, z) = θ4 (τ, z) =
∞ Y
(1 − q n )(1 − yq n )(1 − y −1 q n−1 ) ,
(1 − q n )(1 + yq n )(1 + y −1 q n−1 ) ,
(A.1) (A.2)
(1 − q n )(1 + y q n−1/2 )(1 + y −1 q n−1/2 ) ,
(A.3)
(1 − q n )(1 − y q n−1/2 )(1 − y −1 q n−1/2 ) .
(A.4)
n=1 ∞ Y
n=1
They transform in the following way under the Jacobi group SL2 (Z) ⋉ Z2 1 z θ1 (τ, z) = i α−1 (τ, z)θ1 (− , ) = e(−1/8) θ1 (τ + 1, z) τ τ = (−1)λ+µ e( 21 (λ2 τ + 2λz))θ1 (τ, z + λτ + µ) , 1 z θ2 (τ, z) = α−1 (τ, z)θ4 (− , ) = e(−1/8) θ2(τ + 1, z) τ τ = (−1)µ e( 12 (λ2 τ + 2λz))θ2 (τ, z + λτ + µ) , 1 z θ3 (τ, z) = α−1 (τ, z)θ3 (− , ) = θ4 (τ + 1, z) τ τ
(A.5)
= e( 12 (λ2 τ + 2λz))θ3 (τ, z + λτ + µ) , 1 z θ4 (τ, z) = α−1 (τ, z)θ2 (− , ) = θ3 (τ + 1, z) τ τ = (−1)λ e( 12 (λ2 τ + 2λz))θ4 (τ, z + λτ + µ) , with α(τ, z) =
√ z2 ), and for all λ, µ ∈ Z. −iτ e( 2τ
The weight four Eisenstein series E4 can be written in terms of the Jacobi theta functions as E4 (τ ) =
1 θ2 (τ, 0)8 + θ3 (τ, 0)8 + θ4 (τ, 0)8 . 2
30
(A.6)
B
Character Tables
B1. Frame Shapes and Spinor Representations Table 1: Frame Shapes and Spinor Characters for M22 .
[g]
1A
2A
3A
4A
4B
5A
6A
7A
7B
8A
11A
11B
Πg
124
18 28
16 36
14 24 44
14 24 44
14 54
12 22 33 62
13 73
13 73
12 2.4.82
12 112
12 112
Tr4096g
2048
0
64
0
0
0
0
8
8
0
4
4
Table 2: Frame Shapes and Spinor Characters for U4 (3). [g]
1A
2A
3A 9
3BCD
4AB
5A
Πg
124
18 28
3 13
16 36
14 22 44
14 54
Rg
2048
0
-8
64
0
0
6A 5
6BC 4
1 3.6 24
7AB
8A
9ABCD
12 22 33 62
13 73
12 2.4.82
0
8
0
72
12A 1.22 3.122 42
3 3
1 9 32
4
0
Table 3: Frame Shapes and Spinor Characters for M23 . [g]
1A
Πg
24
Tr4096 g
1
2A
3A
8 8
6 6
4A
5A
4 2 4
6A
4 4
2 2 3 2
7AB
8A
3 3
2
11AB 2
2
2
14AB
15AB
23AB
1 2
1 3
1 2 4
1 5
1 2 3 6
1 7
1 2.4.8
1 11
1.2.7.14
1.3.5.15
1.23
0
64
0
0
0
8
0
4
0
4
2
2048
Table 4: Frame Shapes and Spinor Characters for McL.
[g]
1A
2A
3A 9
3B
4A
5A
5B
5
6A 5
4
6B
7AB
Πg
124
18 28
3 13
16 36
14 22 44
5 1
14 54
1 3.6 24
12 22 33 62
13 73
Tr4096g
2048
0
-8
64
0
-4
0
72
0
8
[g]
8A
9AB 3 3
10A 3
11AB 2
12A 2
14AB 2
15AB 2
2
30AB
Πg
12 2.4.82
1 9 32
1 5.10 22
12 112
1.2 3.12 42
1.2.7.14
1 15 3.5
2.3.5.30 6.10
Tr4096g
0
4
20
4
0
0
2
2
31
Table 5: Frame Shapes and Spinor Characters for HS.
[g]
1A
2A
2B
3A
4A
4BC
6 4
5A 5
5BC
6A
6B
Πg
124
18 28
212
16 36
2 4 14
14 22 44
5 1
14 54
23 63
12 22 33 62
Tr4096 g
2048
0
0
64
0
0
-4
0
0
0
[g]
7A
8ABC
10A
10B
11AB
12A
15A
20AB
Πg
13 73
12 2.4.82
13 5.102 22
22 102
12 112
12 4.62 12 32
1.3.5.15
1.2.10.20 4.5
Tr4096g
8
0
20
0
4
0
4
0
Table 6: Frame Shapes and Spinor Characters for U6 (2).
[g]
1A
2A
Πg
124
16
2 18
18 28
212
16 36
3 13
Tr4096 g
2048
0
0
0
64
-8
[g]
5A
6AB
2B
6C
3A
6D
4 4
6
2C
5
3B
3C
9
6E 4
4A
16 36
8 8
1 4 28
64
256
6F 4
4B
4CDE
4F
4G
4 24
14 22 44
24 44
14 22 44
0
0
0
0
8
6G 5
6H
7A
8A
8BCD
4 4
Πg
14 54
1.6 22 33
2 6 12 32
1 3.6 24
12 22 32 62
1 2.6 34
12 22 32 62
23 63
13 73
1 8 22 42
12 2.4.82
Tr4096g
0
0
0
72
0
0
0
0
8
32
0
[g]
9ABC
10A 2
3
11AB
12AB 3
3
12C 2 2 2
12DE 2
3
3
12FGH 2
2
12I
15A
18AB
Πg
1 9 32
1 2.10 52
12 112
2.3 12 1.4.63
1 3 4 12 22 62
1 12 2.3.4.6
1.2 3.12 42
2.4.6.12
1.3.5.15
1.2.182 6.9
Tr4096g
4
0
4
4
16
12
0
0
4
0
3 3
32
B2. Irreducible Representations √ Table 7: Character table of M22 . bp = (−1 + i p)/2.
[g]
1A
2A
3A
4A
4B
5A
6A
7A
7B
8A
11A
11B
[g 2 ] [g 3 ] [g 5 ] [g 7 ] [g 11 ]
1A 1A 1A 1A 1A
1A 2A 2A 2A 2A
3A 1A 3A 3A 3A
2A 4A 4A 4A 4A
2A 4B 4B 4B 4B
5A 5A 1A 5A 5A
3A 2A 6A 6A 6A
7A 7B 7B 1A 7A
7B 7A 7A 1A 7B
4A 8A 8A 8A 8A
11B 11A 11A 11B 1A
11A 11B 11B 11A 1A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12
1 21 45 45 55 99 154 210 231 280 280 385
1 5 -3 -3 7 3 10 2 7 -8 -8 1
1 3 0 0 1 0 1 3 -3 1 1 -2
1 1 1 1 3 3 -2 -2 -1 0 0 1
1 1 1 1 -1 -1 2 -2 -1 0 0 1
1 1 0 0 0 -1 -1 0 1 0 0 0
1 -1 0 0 1 0 1 -1 1 1 1 -2
1 0 b7 b7 -1 1 0 0 0 0 0 0
1 0 b7 b7 -1 1 0 0 0 0 0 0
1 -1 -1 -1 1 -1 0 0 -1 0 0 1
1 -1 1 1 0 0 0 1 0
1 -1 1 1 0 0 0 1 0
b11 b11 0
b11 b11 0
33
√ Table 8: Character table of U4 (3). bp = (−1 + i p)/2.
34
[g]
1A
2A
3A
3B
3C
3D
4A
4B
5A
6A
6B
6C
7A
7B
8A
9A
9B
9C
9D
12A
[g 2 ] [g 3 ] [g 5 ] [g 7 ]
1A 1A 1A 1A
1A 2A 2A 2A
3A 1A 3A 3A
3B 1A 3B 3B
3C 1A 3C 3C
3D 1A 3D 3D
2A 4A 4A 4A
2A 4B 4B 4B
5A 5A 1A 5A
3A 2A 6A 6A
3B 2A 6B 6B
3C 2A 6C 6C
7A 7B 7B 1A
7B 7A 7A 1A
4A 8A 8A 8A
9B 3A 9B 9A
9A 3A 9A 9B
9D 3A 9D 9C
9C 3A 9C 9D
6A 4A 12A 12A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20
1 21 35 35 90 140 189 210 280 280 280 280 315 315 420 560 640 640 729 896
1 5 3 3 10 12 -3 2 -8 -8 -8 -8 11 11 4 -16 0 0 9 0
1 -6 8 8 9 5 27 21 10 10 10 10 -9 -9 -39 -34 -8 -8 0 32
1 3 8 -1 9 -4 0 3 10 10 1 1 18 -9 6 2 -8 -8 0 -4
1 3 -1 8 9 -4 0 3 1 1 10 10 -9 18 6 2 -8 -8 0 -4
1 3 -1 -1 0 5 0 3 1 1 1 1 0 0 -3 2 1 1 0 -4
1 1 3 3 -2 4 5 -2 0 0 0 0 -1 -1 4 0 0 0 -3 0
1 1 -1 -1 2 0 1 -2 0 0 0 0 -1 -1 0 0 0 0 1 0
1 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 1
1 2 0 0 1 -3 3 5 -2 -2 -2 -2 -1 -1 1 2 0 0 0 0
1 -1 0 3 1 0 0 -1 -2 -2 1 1 2 -1 -2 2 0 0 0 0
1 -1 3 0 1 0 0 -1 1 1 -2 -2 -1 2 -2 2 0 0 0 0
1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 b7 b7 1 0
1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 b7 b7 1 0
1 -1 -1 -1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 -1 0
1 0 2 -1 0 -1 0 0 1 + 3b3 1 + 3b3 1 1 0 0 0 -1 1 1 0 -1
1 0 2 -1 0 -1 0 0 1 + 3b3 1 + 3b3 1 1 0 0 0 -1 1 1 0 -1
1 0 -1 2 0 -1 0 0 1 1 1 + 3b3 1 + 3b3 0 0 0 -1 1 1 0 -1
1 0 -1 2 0 -1 0 0 1 1 1 + 3b3 1 + 3b3 0 0 0 -1 1 1 0 -1
1 -2 0 0 1 1 -1 1 0 0 0 0 -1 -1 1 0 0 0 0 0
√ Table 9: Character table of M23 . bp = (−1 + i p)/2.
35
[g]
1A
2A
3A
4A
5A
6A
7A
7B
8A
11A
11B
14A
14B
15A
15B
23A
23B
2
[g ] [g 3 ] [g 5 ] [g 7 ] [g 11 ] [g 23 ]
1A 1A 1A 1A 1A 1A
1A 2A 2A 2A 2A 2A
3A 1A 3A 3A 3A 3A
2A 4A 4A 4A 4A 4A
5A 5A 1A 5A 5A 5A
3A 2A 6A 6A 6A 6A
7A 7B 7B 1A 7A 7A
7B 7A 7A 1A 7B 7B
4A 8A 8A 8A 8A 8A
11B 11A 11A 11B 1A 11A
11A 11B 11B 11A 1A 11B
7A 14B 14B 2A 14A 14A
7B 14A 14A 2A 14B 14B
15A 5A 3A 15B 15B 15A
15B 5A 3A 15A 15A 15B
23A 23A 23B 23B 23B 1A
23B 23B 23A 23A 23A 1A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17
1 22 45 45 230 231 231 231 253 770 770 896 896 990 990 1035 2024
1 6 -3 -3 22 7 7 7 13 -14 -14 0 0 -18 -18 27 8
1 4 0 0 5 6 -3 -3 1 5 5 -4 -4 0 0 0 -1
1 2 1 1 2 -1 -1 -1 1 -2 -2 0 0 2 2 -1 0
1 2 0 0 0 1 1 1 -2 0 0 1 1 0 0 0 -1
1 0 0 0 1 -2 1 1 1 1 1 0 0 0 0 0 -1
1 1 b7 b7 -1 0 0 0 1 0 0 0 0 b7 b7 -1 1
1 1 b7 b7 -1 0 0 0 1 0 0 0 0 b7 b7 -1 1
1 0 -1 -1 0 -1 -1 -1 -1 0 0 0 0 0 0 1 0
1 0 1 1 -1 0 0 0 0 0 0
1 0 1 1 -1 0 0 0 0 0 0
1 -1 0 0 0 1
b15 b15 1 0 0 1 1 0 0 0 -1
b15 b15 1 0 0 1 1 0 0 0 -1
1 -1 -1 -1 0 1 1 1 0
1 -1 -1 -1 0 1 1 1 0
b11 b11 0 0 1 0
1 -1 -b7 -b7 1 0 0 0 -1 0 0 0 0 b7 b7 -1 1
1 -1 0 0 0 1
b11 b11 0 0 1 0
1 -1 -b7 -b7 1 0 0 0 -1 0 0 0 0 b7 b7 -1 1
b23 b23 -1 -1 1 1 0 0
b23 b23 -1 -1 1 1 0 0
√ Table 10: Character table of McL. bp = (−1 + i p)/2, a = 1 + 3b3 .
36
[g]
1A
2A
3A
3B
4A
5A
5B
6A
6B
7A
7B
8A
9A
9B
10A
11A
11B
12A
14A
14B
15A
15B
30A
30B
[g 2 ] [g 3 ] [g 5 ] [g 7 ] [g 11 ]
1A 1A 1A 1A 1A
1A 2A 2A 2A 2A
3A 1A 3A 3A 3A
3B 1A 3B 3B 3B
2A 4A 4A 4A 4A
5A 5A 1A 5A 5A
5B 5B 1A 5B 5B
3A 2A 6A 6A 6A
3B 2A 6B 6B 6B
7A 7B 7B 1A 7A
7B 7A 7A 1A 7B
4A 8A 8A 8A 8A
9B 3A 9B 9A 9B
9A 3A 9A 9B 9A
5A 10A 2A 10A 10A
11B 11A 11A 11B 1A
11A 11B 11B 11A 1A
6A 4A 12A 12A 12A
7A 14B 14B 2A 14A
7B 14A 14A 2A 14B
15A 5A 3A 15B 15B
15B 5A 3A 15A 15A
15A 10A 6A 30B 30B
15B 10A 6A 30A 30A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24
1 22 231 252 770 770 896 896 1750 3520 3520 4500 4752 5103 5544 8019 8019 8250 8250 9625 9856 9856 10395 10395
1 6 7 28 -14 -14 0 0 70 64 -64 20 -48 63 -56 -45 -45 10 10 105 0 0 -21 -21
1 -5 15 9 -13 -13 32 32 -5 -44 -44 45 54 0 36 0 0 15 15 40 -80 -80 27 27
1 4 6 9 5 5 -4 -4 13 10 10 -9 0 0 9 0 0 6 6 -5 -8 -8 0 0
1 2 -1 4 -2 -2 0 0 2 0 0 4 0 3 0 3 3 -2 -2 -3 0 0 -1 -1
1 -3 6 2 -5 -5 -4 -4 0 -5 -5 0 2 3 19 -6 -6 0 0 0 6 6 -5 -5
1 2 1 2 0 0 1 1 0 0 0 0 2 -2 -1 -1 -1 0 0 0 1 1 0 0
1 3 7 1 7 7 0 0 -5 4 -4 5 -6 0 4 0 0 -5 -5 0 0 0 3 3
1 0 -2 1 1 1 0 0 1 -2 2 -1 0 0 1 0 0 -2 -2 3 0 0 0 0
1 1 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 −b7 −b7 −b7 −b7 0 0 0 0 0
1 1 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 −b7 −b7 −b7 −b7 0 0 0 0 0
1 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 -1 -1 0 0 -1 0 0 1 1
1 1 0 0 -1 -1 -1 -1 -2 1 1 0 0 0 0 0 0 0 0 1 a a 0 0
1 1 0 0 -1 -1 -1 -1 -2 1 1 0 0 0 0 0 0 0 0 1 a a 0 0
1 1 2 -2 1 1 0 0 0 -1 1 0 2 3 -1 0 0 0 0 0 0 0 -1 -1
1 0 0 -1 0 0
1 0 0 -1 0 0
1 -1 0 0 0 0 0 0 0 1 -1 -1 1 0 0 −b7 −b7 b7 b7 0 0 0 0 0
1 0 0 -1
1 -2 2 1
1 -2 2 1
b11 b11 1 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0
1 -1 0 0 0 0 0 0 0 1 -1 -1 1 0 0 −b7 −b7 b7 b7 0 0 0 0 0
1 0 0 -1
b11 b11 1 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0
1 -1 -1 1 1 1 0 0 -1 0 0 1 0 0 0 0 0 1 1 0 0 0 -1 -1
b15 b15 2 2 0 1 1 0 -1 0 1 0 0 0 0 0 0 0 b15 b15
b15 b15 2 2 0 1 1 0 -1 0 1 0 0 0 0 0 0 0 b15 b15
b15 b15 0 0 0 -1 1 0 -1 0 -1 0 0 0 0 0 0 0 -b15 -b15
b15 b15 0 0 0 -1 1 0 -1 0 -1 0 0 0 0 0 0 0 -b15 -b15
√ √ Table 11: Character table of HS. bp = (−1 + i p)/2, ap = i p .
37
[g]
1A
2A
2B
3A
4A
4B
4C
5A
5B
5C
6A
6B
7A
8A
8B
8C
10A
10B
11A
11B
12A
15A
20A
20B
2
[g ] [g 3 ] [g 5 ] [g 7 ] [g 11 ]
1A 1A 1A 1A 1A
1A 2A 2A 2A 2A
1A 2B 2B 2B 2B
3A 1A 3A 3A 3A
2A 4A 4A 4A 4A
2A 4B 4B 4B 4B
2A 4C 4C 4C 4C
5A 5A 1A 5A 5A
5B 5B 1A 5B 5B
5C 5C 1A 5C 5C
3A 2B 6A 6A 6A
3A 2A 6B 6B 6B
7A 7A 7A 1A 7A
4B 8A 8A 8A 8A
4C 8B 8B 8B 8B
4C 8C 8C 8C 8C
5A 10A 2A 10A 10A
5B 10B 2B 10B 10B
11B 11A 11A 11B 1A
11A 11B 11B 11A 1A
6B 4A 12A 12A 12A
15A 5B 3A 15A 15A
10A 20A 4A 20A 20B
10A 20B 4A 20B 20A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24
1 22 77 154 154 154 175 231 693 770 770 770 825 896 896 1056 1386 1408 1750 1925 1925 2520 2750 3200
1 6 13 10 10 10 15 7 21 34 -14 -14 25 0 0 32 -6 0 -10 5 5 24 -50 0
1 -2 1 10 -10 -10 11 -9 9 -10 10 10 9 16 16 0 18 16 10 -19 1 0 -10 -16
1 4 5 1 1 1 4 6 0 5 5 5 6 -4 -4 -6 0 4 -5 -1 -1 0 5 -4
1 -6 5 -2 -10 -10 15 15 21 -14 -10 -10 -15 0 0 0 6 0 -10 5 -35 24 10 0
1 2 5 6 -2 -2 -1 -1 5 2 -2 -2 1 0 0 0 -2 0 6 5 -3 -8 2 0
1 2 1 -2 2 2 3 -1 1 -2 -2 -2 1 0 0 0 -2 0 2 -3 1 0 2 0
1 -3 2 4 4 4 0 6 -7 -5 -5 -5 0 -4 -4 6 11 8 0 0 0 -5 0 0
1 2 -3 4 4 4 5 1 3 0 0 0 -5 1 1 -4 6 -7 0 5 5 0 0 -5
1 2 2 -1 -1 -1 0 1 -2 0 0 0 0 1 1 1 1 -2 0 0 0 0 0 0
1 -2 1 1 -1 -1 2 0 0 -1 1 1 0 -2 -2 0 0 -2 1 -1 1 0 -1 2
1 0 1 1 1 1 0 -2 0 1 1 1 -2 0 0 2 0 0 -1 -1 -1 0 1 0
1 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 1 0 0 0 0 -1 1
1 0 1 0 0 0 -1 -1 1 -2 0 0 1 0 0 0 0 0 -2 1 1 0 0 0
1 0 -1 0 2 -2 1 -1 -1 0 0 0 1 0 0 0 0 0 0 1 -1 0 0 0
1 0 -1 0 -2 2 1 -1 -1 0 0 0 1 0 0 0 0 0 0 1 -1 0 0 0
1 1 -2 0 0 0 0 2 1 -1 1 1 0 0 0 2 -1 0 0 0 0 -1 0 0
1 -2 1 0 0 0 1 1 -1 0 0 0 -1 1 1 0 -2 1 0 1 1 0 0 -1
1 0 0 0 0 0 -1 0 0 0 0 0 0
1 0 0 0 0 0 -1 0 0 0 0 0 0
b11 b11 0 0 0 1 0 0 1 0 -1
b11 b11 0 0 0 1 0 0 1 0 -1
1 0 -1 1 -1 -1 0 0 0 1 -1 -1 0 0 0 0 0 0 -1 -1 1 0 1 0
1 -1 0 1 1 1 -1 1 0 0 0 0 1 1 1 -1 0 -1 0 -1 -1 0 0 1
1 -1 0 -2 0 0 0 0 1 1 a5 a5 0 0 0 0 1 0 0 0 0 -1 0 0
1 -1 0 -2 0 0 0 0 1 1 a5 a5 0 0 0 0 1 0 0 0 0 -1 0 0
Table 12: Character table of U6 (2) - Part I. [g] [g 2 ] [g 3 ] [g 5 ] [g 7 ] [g 11 ]
1A 1A 1A 1A 1A 1A
2A 1A 2A 2A 2A 2A
2B 1A 2B 2B 2B 2B
2C 1A 2C 2C 2C 2C
3A 3A 1A 3A 3A 3A
3B 3B 1A 3B 3B 3B
3C 3C 1A 3C 3C 3C
4A 2A 4A 4A 4A 4A
4B 2A 4B 4B 4B 4B
4C 2B 4C 4C 4C 4C
4D 2B 4D 4D 4D 4D
4E 2B 4E 4E 4E 4E
4F 2B 4F 4F 4F 4F
4G 2B 4G 4G 4G 4G
5A 5A 5A 1A 5A 5A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24 χ25 χ26 χ27 χ28 χ29 χ30 χ31 χ32 χ33 χ34 χ35 χ36 χ37 χ38 χ39 χ40 χ41 χ42 χ43 χ44 χ45 χ46
1 22 231 252 385 440 560 616 770 770 1155 1155 1155 1386 1540 3080 3080 3520 4620 4928 5544 6160 6160 6930 8064 9240 9240 9240 10395 10395 10395 10395 10395 11264 13860 14784 18711 18711 20790 20790 24640 25515 25515 32768 37422 40095
1 -10 39 60 -95 120 -80 -24 -30 -30 195 195 195 -246 260 -440 -440 -320 -180 320 -24 400 400 690 384 -360 -360 -360 315 315 315 315 315 1024 420 -1344 -1161 1431 -810 -810 -960 -405 -405 0 270 1215
1 6 7 28 17 24 -16 40 -14 -14 35 35 35 58 4 40 40 64 44 64 -56 -48 -48 98 128 88 88 88 -21 -21 -21 -21 -21 0 100 64 -57 87 6 6 64 -117 -117 0 30 -81
1 -2 -9 12 -7 8 16 8 10 10 19 19 19 -30 -28 -8 -8 0 -36 64 24 -16 -16 42 0 -8 -8 -8 -45 -45 -45 -45 -45 0 36 0 63 -9 -18 -18 -64 27 27 0 54 -9
1 4 6 9 25 35 20 -14 5 5 30 30 30 36 55 65 65 10 -15 -4 9 40 40 45 -36 -30 -30 -30 0 0 0 0 0 104 -45 114 81 81 0 0 -20 0 0 -64 -81 0
1 -5 15 9 7 8 20 -5 -13 -13 -6 -6 -6 9 1 2 2 -44 57 68 36 -50 -50 -36 -36 33 33 33 27 27 27 27 27 32 9 -12 0 0 54 54 -92 0 0 -64 0 0
1 4 6 9 -2 -1 2 13 5 5 3 3 3 9 1 -7 -7 10 12 5 9 4 4 -9 18 -3 -3 -3 0 0 0 0 0 -4 -18 6 0 0 0 0 -11 0 0 8 0 0
1 6 23 12 1 24 16 8 34 34 -13 -13 -13 -6 -12 40 40 64 28 0 72 48 48 18 0 -8 -8 -8 75 75 -21 -21 -21 0 84 -64 -9 -9 6 6 0 27 27 0 -18 -81
1 -2 7 -4 9 8 -16 8 -6 -6 3 3 3 2 -12 -8 -8 0 -20 0 24 16 16 -6 0 -8 -8 -8 -13 -13 19 19 19 0 20 0 15 -9 14 14 0 -21 -21 0 6 -9
1 2 -1 4 5 0 0 8 -2 -2 11 -5 -5 -2 4 0 0 0 4 0 0 0 0 6 0 24 -8 -8 -1 -1 15 -1 -1 0 4 0 3 -9 -6 -6 0 3 3 0 -6 3
1 2 -1 4 5 0 0 8 -2 -2 -5 11 -5 -2 4 0 0 0 4 0 0 0 0 6 0 -8 24 -8 -1 -1 -1 15 -1 0 4 0 3 -9 -6 -6 0 3 3 0 -6 3
1 2 -1 4 5 0 0 8 -2 -2 -5 -5 11 -2 4 0 0 0 4 0 0 0 0 6 0 -8 -8 24 -1 -1 -1 -1 15 0 4 0 3 -9 -6 -6 0 3 3 0 -6 3
1 -2 -1 4 -7 8 0 0 2 2 3 3 3 -6 4 -8 -8 0 -4 0 -8 0 0 10 0 0 0 0 3 3 3 3 3 0 -4 0 -1 -1 6 6 0 3 3 0 -2 -9
1 2 -1 4 -3 0 0 0 -2 -2 3 3 3 6 -4 0 0 0 -4 0 0 0 0 -2 0 0 0 0 -1 -1 -1 -1 -1 0 4 0 -5 -1 2 2 0 3 3 0 -6 3
1 2 1 2 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -2 -1 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 -1 1 1 0 0 0 0 0 -2 2 0
38
√ √ Table 13: Character table of U6 (2) - Part II. aq = q + 6i 3, c−3 = (1 − 3i 3)/2. [g] [g 2 ] [g 3 ] [g 5 ] [g 7 ] [g 11 ]
6A 3B 2A 6B 6A 6B
6B 3B 2A 6A 6B 6A
6C 3A 2A 6C 6C 6C
6D 3B 2B 6D 6D 6D
6E 3A 2B 6E 6E 6E
6F 3C 2A 6F 6F 6F
6G 3C 2B 6G 6G 6G
6H 3C 2C 6H 6H 6H
7A 7A 7A 7A 1A 7A
8A 4B 8A 8A 8A 8A
8B 4C 8B 8B 8B 8B
8C 4D 8C 8C 8C 8C
8D 4E 8D 8D 8D 8D
9A 9B 3B 9B 9A 9B
9B 9A 3B 9A 9B 9A
9C 9C 3B 9C 9C 9C
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24 χ25 χ26 χ27 χ28 χ29 χ30 χ31 χ32 χ33 χ34 χ35 χ36 χ37 χ38 χ39 χ40 χ41 χ42 χ43 χ44 χ45 χ46
1 -1 3 -3 -5 12 -8 3
1 -1 3 -3 -5 12 -8 3
a−3 a−3 6 6 6 -3 17 a−8 a−8 4 9 -4 12 a4 a4 -12 -12 9 9 9 -9 -9 -9 -9 -9 16 -3 -12 0 √0 18i√3 -18i 3 12 0 0 0 0 0
a−3 a−3 6 6 6 -3 17 a−8 a−8 4 9 -4 12 a4 a4 -12 -12 9 9 9 -9 -9 -9 -9 -9 16 -3 -12 0 √0 -18i√3 18i 3 12 0 0 0 0 0
1 -4 6 9 1 3 4 -6 -3 -3 6 6 6 -12 -1 1 1 -14 9 -4 9 -8 -8 -3 12 -6 -6 -6 0 0 0 0 0 -8 3 -6 9 9 0 0 12 0 0 0 -9 0
1 3 7 1 -1 0 -4 -5 7 7 2 2 2 1 1 -2 -2 4 -7 4 4 -6 -6 -4 -4 1 1 1 3 3 3 3 3 0 1 4 0 0 -6 -6 4 0 0 0 0 0
1 0 -2 1 5 3 -4 -2 1 1 2 2 2 4 -5 1 1 -2 5 4 1 0 0 5 -4 -2 -2 -2 0 0 0 0 0 0 -5 -2 -3 -3 0 0 4 0 0 0 3 0
1 2 0 3 -2 3 -2 3 -3 -3 -3 -3 -3 -3 -1 1 1 4 0 5 -3 -2 -2 -3 0 -3 -3 -3 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 0 0
1 0 -2 1 2 3 2 1 1 1 -1 -1 -1 1 1 1 1 -2 -4 1 1 0 0 -1 2 1 1 1 0 0 0 0 0 0 -2 -2 0 0 0 0 1 0 0 0 0 0
1 -2 0 3 2 -1 -2 -1 1 1 1 1 1 -3 -1 1 1 0 0 1 -3 2 2 -3 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0
1 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 -1
1 2 3 0 1 0 0 0 2 2 -1 -1 -1 -2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 -1 -1 2 2 0 -1 -1 0 -2 3
1 0 -1 0 -1 0 0 0 0 0 3 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -3 1 1 0 0 0 1 -1 0 0 0 -1 -1 0 0 1
1 0 -1 0 -1 0 0 0 0 0 -1 3 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 -3 1 0 0 0 1 -1 0 0 0 -1 -1 0 0 1
1 0 -1 0 -1 0 0 0 0 0 -1 -1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 -3 0 0 0 1 -1 0 0 0 -1 -1 0 0 1
1 1 0 0 1 2 -1 -2 -1 -1 0 0 0 0 1
1 1 0 0 1 2 -1 -2 -1 -1 0 0 0 0 1
c−3 c¯−3 1 0 -1 0 -c−3 -¯ c−3 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -2 0 0 2 0 0
c¯−3 c−3 1 0 -1 0 -¯ c−3 -c−3 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -2 0 0 2 0 0
1 1 0 0 1 -1 2 -2 -1 -1 0 0 0 0 1 -1 -1 1 0 2 0 1 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 -1 0 0
39
√ √ √ Table 14: Character table of U6 (2) - Part III. aq = q + 6i 3, bp = (−1 + i p)/2, dn m = m + in 3. [g] [g 2 ] [g 3 ] [g 5 ] [g 7 ] [g 11 ]
10A 5A 10A 2A 10A 10A
11A 11B 11A 11A 11B 1A
11B 11A 11B 11B 11A 1A
12A 6B 4A 12B 12A 12B
12B 6A 4A 12A 12B 12A
12C 6C 4A 12C 12C 12C
12D 6B 4B 12E 12D 12E
12E 6A 4B 12D 12E 12D
12F 6D 4C 12F 12F 12F
12G 6D 4D 12G 12G 12G
12H 6D 4E 12H 12H 12H
12I 6E 4F 12I 12I 12I
15A 15A 5A 3A 15A 15A
18A 9B 6A 18B 18A 18B
18B 9A 6B 18A 18B 18A
χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24 χ25 χ26 χ27 χ28 χ29 χ30 χ31 χ32 χ33 χ34 χ35 χ36 χ37 χ38 χ39 χ40 χ41 χ42 χ43 χ44 χ45 χ46
1 0 -1 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 1 -1 1 0 0 0 0 0 0 0 0
1 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −¯b11 −b11 -1 0 0
1 -3 5 3 1 6 -2 -1
1 -3 5 3 1 6 -2 -1
1 0 2 -3 1 3 4 2 1 1 2 2 2 0 3 1 1 -2 1 0 -3 0 0 -3 0 -2 -2 -2 0 0 0 0 0 0 3 2 -3 -3 0 0 0 0 0 0 3 0
1 1 1 -1 -3 2 2 -1
1 1 1 -1 -3 2 2 -1 d10
1 -1 -1 1 -1 0 0 -1 1 1 2 -2 -2 1 1 0 0 0 1 0 0 0 0 0 0 -3 1 1 -1 -1 3 -1 -1 0 1 0 0 0 0 0 0 0 0 0 0 0
1 -1 -1 1 -1 0 0 -1 1 1 -2 2 -2 1 1 0 0 0 1 0 0 0 0 0 0 1 -3 1 -1 -1 -1 3 -1 0 1 0 0 0 0 0 0 0 0 0 0 0
1 -1 -1 1 -1 0 0 -1 1 1 -2 -2 2 1 1 0 0 0 1 0 0 0 0 0 0 1 1 -3 -1 -1 -1 -1 3 0 1 0 0 0 0 0 0 0 0 0 0 0
1 -2 2 1 -1 -1 0 0 -1 -1 0 0 0 0 1 1 1 0 -1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 -1 0 0 0 0 0 0 1 0
1 -1 1 -1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 -1 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 -1 1 1 0 0 0 0 0 1 -1 0
1 -1 0 0 1 0 1 0 d10
1 -1 0 0 1 0 1 0
−b11 −¯b11 -1 0 0
d−3 −2 d¯−3 −2 -4 -4 -4 3 -3 d−3 −5 d¯−3 −5
-8 1 0 0 −6b3 −6¯b3 0 0 1 1 1 -¯ a−3 -a−3 -3 -3 -3 0 3 8 0 0 6b3 6¯b3 0 0 0 0 0 0
d¯−3 −2 d−3 −2 -4 -4 -4 3 -3 d¯−3 −5 d−3 −5 -8 1 0 0 −6¯b3 −6b3 0 0 1 1 1 -a−3 -¯ a−3 -3 -3 -3 0 3 8 0 0 6¯b3 6b3 0 0 0 0 0 0
d−1 0 d10 0 0 0 -1 -3 d−1 1 d¯−1 1
0 1 0 0 d−1 1 d¯−1 1
0 0 1 1 1 d−2 −1 d¯−2 −1 1 1 1 0 -1 0 0 0 d1−1 d¯1−1 0 0 0 0 0 0
40
d−1 0 0 0 0 -1 -3 d¯−1 1 d−1 1 0 1 0 0 d¯−1 1 d−1 1 0 0 1 1 1 d¯−2 −1 d−2 −1 1 1 1 0 -1 0 0 0 d¯1−1 d1−1 0 0 0 0 0 0
d−1 0 0 0 0 0 -1 −1/2
d−1/2 −1/2 d¯ −1/2
1 0 -1 0 −1/2
d−1/2 −1/2 d¯ −1/2
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
d−1 0 d10 0 0 0 0 -1 −1/2 d¯−1/2 −1/2 d−1/2 1 0 -1 0 −1/2 d¯−1/2 −1/2 d−1/2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
C
Coefficient Tables
Table 15: The twined series for M22 . The table shows the Fourier coefficients multiplying q −D/12 in the q-expansion of the function hg,1 (τ ).
[g] −D -1 11 23 35 47 59 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239 251 263 275 287
1A
2A
3A
4AB
5A
6A
7AB
8A
11AB
-2 560 8470 70576 435820 2187328 9493330 36792560 130399766 429229920 1327987562 3895785632 10912966810 29351354032 76141761850 191223891936 466389602756 1107626293840 2567229121428 5818567673360 12917927405852 28135083779792 60194978116000 126660856741328 262393981258310
-2 -16 54 -144 332 -704 1394 -2640 4822 -8480 14506 -24288 39770 -63888 101018 -157344 241764 -366960 550676 -817840 1203068 -1753840 2535488 -3637232 5179526
-2 2 -8 16 4 -50 58 38 -172 174 104 -502 466 316 -1232 1098 710 -2876 2472 1658 -6148 5174 3382 -12634 10430
-2 0 -2 0 4 0 -6 0 6 0 -6 0 10 0 -14 0 12 0 -12 0 20 0 -24 0 22
-2 0 0 -4 0 8 0 0 -14 0 22 -8 0 -28 0 56 -14 0 -62 0 112 -48 0 -112 0
-2 2 0 0 -4 -2 2 6 4 -2 -8 -6 2 12 8 -6 -18 -12 8 26 20 -10 -34 -26 14
-2 0 0 2 0 -4 0 0 0 0 6 4 -10 0 0 0 0 10 6 -20 0 -4 0 0 20
-2 0 2 0 0 0 -2 0 -2 0 2 0 2 0 -2 0 0 0 4 0 0 0 -4 0 -2
-2 -1 0 0 0 0 0 2 2 -2 0 2 -2 0 0 0 0 0 4 2 -4 0 2 -3 0
41
Table 16: The twined series for M22 . The table shows the Fourier coefficients multiplying q −D/12 in the q-expansion of the function hg,2 (τ ).
[g] −D -4 8 20 32 44 56 68 80 92 104 116 128 140 152 164 176 188 200 212 224 236 248 260 272 284
1A
2A
3A
4AB
5A
6A
7AB
8A
11AB
1 210 4444 42560 281512 1481964 6649200 26455264 95731405 320626372 1006567156 2990338680 8469129448 23000871960 60186506768 152335803872 374172530930 894352929498 2085157528300 4751675601024 10602363945184 23199658816580 49851590654096 105323108387200 219021850730991
1 2 -4 16 -24 12 -16 64 -83 36 -44 168 -216 88 -112 416 -494 202 -244 864 -1056 420 -496 1792 -2097
1 3 7 -19 19 24 -81 70 70 -248 217 183 -656 558 431 -1582 1340 981 -3545 2946 2077 -7480 6146 4228 -15099
1 -2 4 4 -8 -12 16 24 -27 -36 44 52 -72 -88 112 144 -166 -202 244 280 -352 -420 496 608 -697
1 0 -1 0 7 -6 0 -6 0 22 -14 0 -17 0 48 -28 0 -42 0 104 -51 0 -84 0 196
1 -1 -1 1 3 0 -1 -2 -2 0 1 3 0 -2 -1 2 4 1 -1 -6 -3 0 2 4 -3
1 0 -1 0 0 1 5 -4 0 -4 0 0 3 10 -11 0 -5 0 0 9 18 -20 0 -10 0
1 0 0 -2 0 0 0 4 1 0 0 -6 0 0 0 8 -2 0 0 -12 0 0 0 16 3
1 1 0 1 0 0 -3 0 0 0 1 -1 -1 0 3 -1 0 -5 0 0 0 4 0 0 0
42
Table 17: The twined series for U4 (3). The table shows the Fourier coefficients multiplying q −D/12 in the q-expansion of the function hg,1 (τ ).
[g] −D
43
-1 11 23 35 47 59 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239 251 263 275 287
1A
2A
3A
3BCD
4AB
5A
6A
6BC
7AB
8A
9ABCD
12A
-2 560 8470 70576 435820 2187328 9493330 36792560 130399766 429229920 1327987562 3895785632 10912966810 29351354032 76141761850 191223891936 466389602756 1107626293840 2567229121428 5818567673360 12917927405852 28135083779792 60194978116000 126660856741328 262393981258310
-2 -16 54 -144 332 -704 1394 -2640 4822 -8480 14506 -24288 39770 -63888 101018 -157344 241764 -366960 550676 -817840 1203068 -1753840 2535488 -3637232 5179526
-2 -34 -116 -272 -662 -1454 -2732 -5254 -9802 -16782 -28930 -49066 -79058 -127484 -203264 -313578 -482842 -736772 -1098876 -1634074 -2412226 -3502486 -5067686 -7287046 -10348570
-2 2 -8 16 4 -50 58 38 -172 174 104 -502 466 316 -1232 1098 710 -2876 2472 1658 -6148 5174 3382 -12634 10430
-2 0 -2 0 4 0 -6 0 6 0 -6 0 10 0 -14 0 12 0 -12 0 20 0 -24 0 22
-2 0 0 -4 0 8 0 0 -14 0 22 -8 0 -28 0 56 -14 0 -62 0 112 -48 0 -112 0
-2 2 0 0 2 -2 -4 6 10 -2 -14 -6 14 12 -4 -6 -6 -12 -4 26 38 -10 -58 -26 38
-2 2 0 0 -4 -2 2 6 4 -2 -8 -6 2 12 8 -6 -18 -12 8 26 20 -10 -34 -26 14
-2 0 0 2 0 -4 0 0 0 0 6 4 -10 0 0 0 0 10 6 -20 0 -4 0 0 20
-2 0 2 0 0 0 -2 0 -2 0 2 0 2 0 -2 0 0 0 4 0 0 0 -4 0 -2
-2 -1 1 1 -2 4 1 -7 2 3 -7 5 4 -11 13 3 -22 10 9 -22 17 14 -32 32 11
-2 0 -2 0 -2 0 0 0 0 0 0 0 -2 0 -2 0 0 0 0 0 2 0 0 0 -2
Table 18: The twined series for U4 (3). The table shows the Fourier coefficients multiplying q −D/12 in the q-expansion of the function hg,2 (τ ).
[g] −D
44
-4 8 20 32 44 56 68 80 92 104 116 128 140 152 164 176 188 200 212 224 236 248 260 272 284
1A
2A
3A
3BCD
4AB
5A
6A
6BC
7AB
8A
9ABCD
12A
1 210 4444 42560 281512 1481964 6649200 26455264 95731405 320626372 1006567156 2990338680 8469129448 23000871960 60186506768 152335803872 374172530930 894352929498 2085157528300 4751675601024 10602363945184 23199658816580 49851590654096 105323108387200 219021850730991
1 2 -4 16 -24 12 -16 64 -83 36 -44 168 -216 88 -112 416 -494 202 -244 864 -1056 420 -496 1792 -2097
1 21 97 197 577 1176 2313 4552 8440 14440 25759 42861 69904 114066 181097 280208 436490 662499 993025 1485462 2189455 3186440 4635278 6654706 9475383
1 3 7 -19 19 24 -81 70 70 -248 217 183 -656 558 431 -1582 1340 981 -3545 2946 2077 -7480 6146 4228 -15099
1 -2 4 4 -8 -12 16 24 -27 -36 44 52 -72 -88 112 144 -166 -202 244 280 -352 -420 496 608 -697
1 0 -1 0 7 -6 0 -6 0 22 -14 0 -17 0 48 -28 0 -42 0 104 -51 0 -84 0 196
1 5 -7 13 -15 24 -31 40 -56 72 -89 117 -144 178 -223 272 -326 403 -487 582 -705 840 -994 1186 -1401
1 -1 -1 1 3 0 -1 -2 -2 0 1 3 0 -2 -1 2 4 1 -1 -6 -3 0 2 4 -3
1 0 -1 0 0 1 5 -4 0 -4 0 0 3 10 -11 0 -5 0 0 9 18 -20 0 -10 0
1 0 0 -2 0 0 0 4 1 0 0 -6 0 0 0 8 -2 0 0 -12 0 0 0 16 3
1 0 1 -1 -2 3 -3 -2 7 -5 1 9 -5 -9 11 -10 -4 24 -17 -6 25 -16 -16 34 -30
1 1 1 1 1 0 1 0 0 0 -1 1 0 2 1 0 2 -1 1 -2 -1 0 -2 2 -1
˜ 1 (τ ).bp = Table 19: The twined series for M23 . The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function h g, 2 √ (−1 + i p)/2.
[g] −D
45
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
1A
2A
3A 4A 5A 6A 7AB 8A 11AB 14AB 15AB 23A 23B
-1 -1 -1 770 -14 5 13915 43 10 132825 -119 21 915124 308 31 5069867 -693 59 24053215 1407 85 101268540 -2772 135 387746282 5306 200 1372935090 -9710 300 4552039296 17136 414 14265412315 -29589 610 42568680715 50155 835 121665949240 -83160 1165 334658246604 135148 1581 889413095662 -216482 2158 2291482148835 342259 2865 5739333227670 -533610 3855 14008317423968 821296 5051 33388385201699 -1250717 6656 77853744768906 1886234 8649 177881794535250 -2816798 11250 398808419854845 4167709 14430 878461575586727 -6116665 18581 1903241478167799 8909383 23631
-1 -2 -1 5 4 -13 -9 24 14 -38 -20 63 35 -108 -60 170 87 -250 -124 371 190 -554 -283 799 395
-1 0 0 -5 4 2 0 0 -18 15 11 0 0 -45 34 22 0 0 -107 79 56 0 0 -218 154
-1 -1 -1 1 0 0 -2 -1 1 1 0 -1 -1 0 0 3 -2 -1 -3 4 -1 3 2 2 -4 0 0 4 -5 2 -6 0 2 6 0 -3 -5 -6 -1 9 10 -2 -11 3 -4 10 0 4 -11 -9 1 15 0 2 -17 0 6 16 -11 -5 -19 22 0 22 8 -4 -26 0 -7 29 -22 7 -29 0 1
-1 0 0 0 1 0 -1 -1 0 -1 2 0 0 0 0 3 0 -2 -1 0 -4 3 0 0 0
-1 0 1 0 0 0 0 0 0 -1 0 0 0 0 -1 0 1 0 0 1 0 2 0 -2 0
-1 -1 -1 0 b23 b23 0 0 0 1 0 0 1 0 0 -1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 -1 1 1 0 0 0 0 1 1 1 0 0 1 b23 b23
˜ 3 (τ ). Table 20: The twined series for M23 . The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function h g, 2
[g] −D
46
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1A
2A
3A 4A 5A 6A 7AB 8A 11AB 14AB 15AB 23AB
1 1 1 231 7 6 5796 -28 18 65505 97 -15 494385 -239 60 2922381 525 90 14525511 -1113 -75 63447087 2255 228 250188435 -4333 360 907876585 7945 -260 3073155810 -14174 762 9804660777 24809 1062 29717775186 -42286 -759 86116649220 70308 2070 239806592730 -115046 2880 644418434331 185563 -1926 1676994065901 -294483 5256 4238788584987 460571 6948 10432762525295 -711985 -4645 25058448433770 1088842 12120 58848028309224 -1647128 15912 135349351964727 2466359 -10281 305326880332593 -3660335 26646 676433083819185 5388145 34110 1473468429847035 -7867397 -21840
1 1 1 1 1 -1 1 -2 0 -1 4 1 2 0 0 1 0 1 -1 1 -7 0 4 3 1 -3 -9 -6 0 1 15 6 -3 0 -1 7 7 -4 -3 -1 -29 0 8 0 -1 -15 0 4 0 -3 50 -25 10 -3 2 25 17 -10 9 1 -78 16 -7 1 2 -36 0 -18 0 4 122 0 16 -9 -2 59 -64 10 0 -1 -195 46 24 0 -3 -101 37 -28 -9 -5 295 0 -13 19 3 146 0 -32 3 2 -424 -136 40 0 4 -201 92 23 -16 7 617 68 46 0 -7 305 0 -50 0 -3 -901 0 -32 -15 -5
1 0 -1 0 1 0 0 0 2 0 -1 -1 0 -3 1 1 0 0 0 4 0 -1 -2 0 -5
1 0 0 -1 -1 0 0 1 0 0 1 1 1 0 -1 0 0 -1 -1 -1 0 0 0 0 1
1 1 -2 0 0 0 0 -2 0 0 2 2 1 0 0 -1 1 -2 0 0 2 -1 -4 0 0
1 1 0 1 0 1 -1 0 1 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 1
Table 21: The twined series for McL. The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function ˜ hg, 1 (τ ). bp = 2 √ (−1 + i p)/2.
[g] −D
47
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
1A
2A
3A
-1 -1 -1 770 -14 -13 13915 43 -17 132825 -119 -42 915124 308 -68 5069867 -693 -112 24053215 1407 -167 101268540 -2772 -279 387746282 5306 -394 1372935090 -9710 -600 4552039296 17136 -837 14265412315 -29589 -1208 42568680715 50155 -1667 121665949240 -83160 -2345 334658246604 135148 -3153 889413095662 -216482 -4313 2291482148835 342259 -5748 5739333227670 -533610 -7692 14008317423968 821296 -10096 33388385201699 -1250717 -13333 77853744768906 1886234 -17280 177881794535250 -2816798 -22500 398808419854845 4167709 -28887 878461575586727 -6116665 -37138 1903241478167799 8909383 -47253
3B 4A -1 5 10 21 31 59 85 135 200 300 414 610 835 1165 1581 2158 2865 3855 5051 6656 8649 11250 14430 18581 23631
-1 -2 -1 5 4 -13 -9 24 14 -38 -20 63 35 -108 -60 170 87 -250 -124 371 190 -554 -283 799 395
5A 5B -1 -5 -10 -25 -26 -58 -85 -135 -218 -285 -404 -610 -835 -1210 -1546 -2138 -2865 -3855 -5157 -6576 -8594 -11250 -14430 -18798 -23476
-1 0 0 -5 4 2 0 0 -18 15 11 0 0 -45 34 22 0 0 -107 79 56 0 0 -218 154
6A 6B 7AB 8A 9AB 10A 11AB 12A 14AB 15A, 15B -1 7 7 22 32 60 81 141 194 304 411 612 829 1179 1567 2167 2860 3864 5032 6679 8624 11272 14413 18602 23599
-1 -1 -1 1 0 0 -2 -1 1 1 0 -1 -1 0 0 3 -2 -1 -3 4 -1 3 2 2 -4 0 0 4 -5 2 -6 0 2 6 0 -3 -5 -6 -1 9 10 -2 -11 3 -4 10 0 4 -11 -9 1 15 0 2 -17 0 6 16 -11 -5 -19 22 0 22 8 -4 -26 0 -7 29 -22 7 -29 0 1
-1 -1 1 0 -2 2 1 -3 2 0 -3 4 1 -5 3 1 -6 6 2 -7 6 0 -9 8 3
-1 1 -2 1 -2 2 -3 3 -4 5 -4 6 -5 10 -12 8 -11 15 -19 18 -16 22 -26 30 -32
-1 0 0 0 1 0 -1 -1 0 -1 2 0 0 0 0 3 0 -2 -1 0 -4 3 0 0 0
-1 1 -1 2 -2 2 -3 3 -4 4 -5 6 -7 9 -9 11 -12 14 -16 17 -20 22 -25 28 -31
-1 0 1 0 0 0 0 0 0 -1 0 0 0 0 -1 0 1 0 0 1 0 2 0 -2 0
-1 b15 −b15 −b15 b15 −b15 −b15 (b15 − 1) −2b15 0 (2b15 − 1) 2 −b15 (b15 − 2) (1 − 2b15 ) b15 (4b15 − 1) (2 − 2b15 ) (1 − b15 ) (4b15 − 1) (1 − 3b15 ) 0 (3 − 3b15 ) (2 − 5b15 ) 2
30A, 30B -1 b15 b15 b15 b15 (b15 − 2) (b15 − 1) (b15 − 1) 2b15 2b15 (2b15 − 3) (2b15 − 2) (3b15 − 2) (3b15 − 2) (4b15 − 1) (5b15 − 3) (4b15 − 3) (4b15 − 4) (5b15 − 3) (6b15 − 3) (7b15 − 5) (8b15 − 4) (9b15 − 5) (9b15 − 6) (10b15 − 6)
˜ 3 (τ ). Table 22: The twined series for McL. The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function h g, 2
[g] −D
48
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1A
2A
3A
1 1 1 231 7 15 5796 -28 45 65505 97 30 494385 -239 150 2922381 525 225 14525511 -1113 159 63447087 2255 570 250188435 -4333 900 907876585 7945 505 3073155810 -14174 1905 9804660777 24809 2655 29717775186 -42286 1518 86116649220 70308 5175 239806592730 -115046 7200 644418434331 185563 3879 1676994065901 -294483 13140 4238788584987 460571 17370 10432762525295 -711985 9260 25058448433770 1088842 30300 58848028309224 -1647128 39780 135349351964727 2466359 20562 305326880332593 -3660335 66615 676433083819185 5388145 85275 1473468429847035 -7867397 43725
3B 4A 1 6 18 -15 60 90 -75 228 360 -260 762 1062 -759 2070 2880 -1926 5256 6948 -4645 12120 15912 -10281 26646 34110 -21840
1 -1 4 1 -7 -3 15 7 -29 -15 50 25 -78 -36 122 59 -195 -101 295 146 -424 -201 617 305 -901
5A 5B
6A 6B 7AB 8A 9AB 10A 11AB 12A 14AB 15AB 30AB
1 1 1 1 1 1 1 1 6 1 7 -2 0 -1 0 2 21 1 5 2 0 0 0 -3 30 0 22 1 -1 1 0 2 60 0 22 4 3 1 0 -4 81 -9 57 -6 0 1 0 5 161 6 63 -3 0 -1 3 -3 237 7 122 -4 -3 -1 0 5 360 0 164 8 0 -1 0 -8 510 0 289 4 0 -3 -5 10 735 -25 361 10 -3 2 0 -9 1077 17 551 -10 9 1 0 9 1536 16 710 -7 1 2 0 -16 2070 0 1071 -18 0 4 0 18 2880 0 1408 16 -9 -2 0 -16 3806 -64 1999 10 0 -1 9 18 5301 46 2580 24 0 -3 0 -23 6987 37 3530 -28 -9 -5 0 31 9270 0 4532 -13 19 3 -10 -30 12120 0 6124 -32 3 2 0 32 15774 -136 7876 40 0 4 0 -38 20652 92 10442 23 -16 7 0 44 26718 68 13231 46 0 -7 0 -50 34110 0 17155 -50 0 -3 0 50 43710 0 21637 -32 -15 -5 15 -62
1 0 -1 0 1 0 0 0 2 0 -1 -1 0 -3 1 1 0 0 0 4 0 -1 -2 0 -5
1 -1 1 -2 2 -3 3 -2 4 -3 5 -5 6 -9 8 -13 12 -14 16 -16 20 -18 23 -25 29
1 0 0 -1 -1 0 0 1 0 0 1 1 1 0 -1 0 0 -1 -1 -1 0 0 0 0 1
1 0 0 0 0 0 -1 0 0 0 0 0 3 0 0 -1 0 0 0 0 0 -3 0 0 0
1 2 0 2 2 2 3 2 4 4 6 6 5 6 8 9 10 10 12 14 16 17 16 20 22
√ Table 23: The twined series for HS. The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function ˜ hg, 1 (τ ). ap = i p. 2
[g] −D
49
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
1A
2A
2B
3A
-1 -1 -1 -1 770 -14 10 5 13915 43 -45 10 132825 -119 129 21 915124 308 -300 31 5069867 -693 667 59 24053215 1407 -1425 85 101268540 -2772 2820 135 387746282 5306 -5278 200 1372935090 -9710 9634 300 4552039296 17136 -17176 414 14265412315 -29589 29715 610 42568680715 50155 -50085 835 121665949240 -83160 82944 1165 334658246604 135148 -135268 1581 889413095662 -216482 216822 2158 2291482148835 342259 -342085 2865 5739333227670 -533610 533110 3855 14008317423968 821296 -821544 5051 33388385201699 -1250717 1251459 6656 77853744768906 1886234 -1885854 8649 177881794535250 -2816798 2815690 11250 398808419854845 4167709 -4168275 14430 878461575586727 -6116665 6118263 18581 1903241478167799 8909383 -8908593 23631
4A 4BC -1 -10 -25 -35 -60 -125 -185 -240 -394 -630 -860 -1145 -1645 -2420 -3244 -4126 -5665 -7930 -10260 -12909 -17146 -23010 -29195 -36305 -46925
-1 -2 -1 5 4 -13 -9 24 14 -38 -20 63 35 -108 -60 170 87 -250 -124 371 190 -554 -283 799 395
5A 5BC 6A 6B 7A 8ABC 10A 10B 11AB 12A 15A 20A, 20B -1 -5 -10 -25 -26 -58 -85 -135 -218 -285 -404 -610 -835 -1210 -1546 -2138 -2865 -3855 -5157 -6576 -8594 -11250 -14430 -18798 -23476
-1 0 0 -5 4 2 0 0 -18 15 11 0 0 -45 34 22 0 0 -107 79 56 0 0 -218 154
-1 1 0 3 -3 1 -3 3 -4 4 -4 6 -9 9 -7 12 -13 13 -15 18 -21 22 -24 27 -33
-1 1 -2 1 -1 3 -3 3 -4 4 -6 6 -5 9 -11 10 -11 15 -17 16 -19 22 -26 29 -29
-1 0 -1 0 0 -2 4 2 0 -5 0 0 -6 10 3 0 -9 0 0 -11 22 8 0 -22 0
-1 0 1 -1 0 -1 -1 2 0 2 2 -3 -1 -2 -4 4 1 2 6 -5 0 -4 -7 7 1
-1 1 -2 1 -2 2 -3 3 -4 5 -4 6 -5 10 -12 8 -11 15 -19 18 -16 22 -26 30 -32
-1 0 0 -1 0 2 0 0 2 -1 -1 0 0 -1 2 2 0 0 1 -1 -4 0 0 -2 2
-1 0 0 0 1 0 -1 -1 0 -1 2 0 0 0 0 3 0 -2 -1 0 -4 3 0 0 0
-1 -1 2 1 -3 1 1 -3 2 0 -2 4 -1 -5 5 2 -7 5 3 -6 5 0 -8 7 1
-1 0 0 1 1 -1 0 0 0 0 -1 0 0 0 1 -2 0 0 1 1 -1 0 0 1 1
-1 a5 0 a5 0 0 a5 a5 (a5 + 1) a5 0 0 a5 0 (a5 + 1) (a5 − 1) a5 a5 a5 (a5 + 1) (a5 − 1) 2a5 0 2a5 0
˜ 3 (τ ). Table 24: The twined series for HS. The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function h g, 2
[g] −D
50
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1A
2A
2B
3A
1 1 1 1 231 7 -9 6 5796 -28 36 18 65505 97 -95 -15 494385 -239 225 60 2922381 525 -531 90 14525511 -1113 1143 -75 63447087 2255 -2241 228 250188435 -4333 4275 360 907876585 7945 -7975 -260 3073155810 -14174 14274 762 9804660777 24809 -24759 1062 29717775186 -42286 42130 -759 86116649220 70308 -70380 2070 239806592730 -115046 115290 2880 644418434331 185563 -185445 -1926 1676994065901 -294483 294093 5256 4238788584987 460571 -460773 6948 10432762525295 -711985 712575 -4645 25058448433770 1088842 -1088550 12120 58848028309224 -1647128 1646280 15912 135349351964727 2466359 -2466761 -10281 305326880332593 -3660335 3661569 26646 676433083819185 5388145 -5387535 34110 1473468429847035 -7867397 7865595 -21840
4A 4BC 1 15 36 65 105 189 319 471 675 1025 1554 2169 2930 4140 5850 7835 10269 13851 18775 24450 31320 41015 53817 68625 86395
1 -1 4 1 -7 -3 15 7 -29 -15 50 25 -78 -36 122 59 -195 -101 295 146 -424 -201 617 305 -901
5A 5BC 6A 6B 7A 8ABC 10A 10B 11AB 12A 15A 20AB 1 1 1 1 1 6 1 0 -2 0 21 1 0 2 0 30 0 -5 1 -1 60 0 0 4 3 81 -9 0 -6 0 161 6 9 -3 0 237 7 0 -4 -3 360 0 0 8 0 510 0 -10 4 0 735 -25 0 10 -3 1077 17 0 -10 9 1536 16 19 -7 1 2070 0 0 -18 0 2880 0 0 16 -9 3806 -64 -36 10 0 5301 46 0 24 0 6987 37 0 -28 -9 9270 0 45 -13 19 12120 0 0 -32 3 15774 -136 0 40 0 20652 92 -59 23 -16 26718 68 0 46 0 34110 0 0 -50 0 43710 0 90 -32 -15
1 -1 0 1 1 1 -1 -1 -1 -3 2 1 2 4 -2 -1 -3 -5 3 2 4 7 -7 -3 -5
1 2 -3 2 -4 5 -3 5 -8 10 -9 9 -16 18 -16 18 -23 31 -30 32 -38 44 -50 50 -62
1 1 1 0 0 -1 -2 -1 0 0 -1 1 0 0 0 0 -2 -3 0 0 0 4 4 0 0
1 0 -1 0 1 0 0 0 2 0 -1 -1 0 -3 1 1 0 0 0 4 0 -1 -2 0 -5
1 0 0 -1 0 0 1 0 0 2 0 0 -1 0 0 -4 0 0 1 0 0 5 0 0 -2
1 1 -2 0 0 0 0 -2 0 0 2 2 1 0 0 -1 1 -2 0 0 2 -1 -4 0 0
1 0 1 0 0 -1 -1 1 0 0 -1 -1 0 0 0 0 -1 1 0 0 0 0 2 0 0
˜ 1 (τ ) - part I. Table 25: The twined series for U6 (2). The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function h g, 2 √ ap = i p.
[g] −D
51
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
1A
2A
2B
2C
3A
-1 -1 -1 -1 -1 770 -30 -14 10 5 13915 -5 43 -45 10 132825 -199 -119 129 21 915124 180 308 -300 31 5069867 -917 -693 667 59 24053215 1055 1407 -1425 85 101268540 -3300 -2772 2820 135 387746282 4490 5306 -5278 200 1372935090 -10894 -9710 9634 300 4552039296 15456 17136 -17176 414 14265412315 -32005 -29589 29715 610 42568680715 46795 50155 -50085 835 121665949240 -87784 -83160 82944 1165 334658246604 128780 135148 -135268 1581 889413095662 -225074 -216482 216822 2158 2291482148835 330755 342259 -342085 2865 5739333227670 -548970 -533610 533110 3855 14008317423968 801024 821296 -821544 5051 33388385201699 -1277277 -1250717 1251459 6656 77853744768906 1851562 1886234 -1885854 8649 177881794535250 -2861710 -2816798 2815690 11250 398808419854845 4109885 4167709 -4168275 14430 878461575586727 -6190873 -6116665 6118263 18581 1903241478167799 8814743 8909383 -8908593 23631
3B
3C
-1 -1 -13 5 -17 10 -42 21 -68 31 -112 59 -167 85 -279 135 -394 200 -600 300 -837 414 -1208 610 -1667 835 -2345 1165 -3153 1581 -4313 2158 -5748 2865 -7692 3855 -10096 5051 -13333 6656 -17280 8649 -22500 11250 -28887 14430 -37138 18581 -47253 23631
4A
4B 4CDE
-1 -1 34 -6 139 -13 505 -15 1412 -28 3627 -69 8559 -97 19068 -108 40154 -190 81250 -334 158736 -440 300987 -541 555307 -805 1001016 -1264 1767388 -1652 3062830 -1978 5217427 -2789 8751462 -4090 14473168 -5192 23625763 -6269 38101658 -8478 60762514 -11782 95895325 -14739 149872439 -17753 232094167 -23265
-1 -2 -1 5 4 -13 -9 24 14 -38 -20 63 35 -108 -60 170 87 -250 -124 371 190 -554 -283 799 395
4F 4G 5A -1 2 3 -7 -4 11 7 -20 -14 42 24 -69 -37 104 52 -162 -85 254 136 -381 -190 546 269 -785 -393
-1 -2 -1 5 4 -13 -9 24 14 -38 -20 63 35 -108 -60 170 87 -250 -124 371 190 -554 -283 799 395
6A, 6B
-1 -1 0 -3- 6a3 0 -5 -5 -10- 12a3 4 -18- 18a3 2 -26- 30a3 0 -43-36a3 0 -69- 78a3 -18 -100- 90a3 15 -148- 156a3 11 -213-204a3 0 -298- 306a3 0 -419- 408a3 -45 -583- 606a3 34 -793- 768a3 22 -1073- 1092a3 0 -1444-1428a3 0 -1914- 1938a3 -107 -2532- 2496a3 79 -3327- 3366a3 56 -4328- 4284a3 0 -5614- 5658a3 0 -7237- 7194a3 -218 -9268- 9318a3 154 -11827- 11766a3
˜ 1 (τ ) - part II. Table 26: The twined series for U6 (2). The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function h g, 2 √ ap = i p.
[g] −D
52
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
6C
6D 6E 6F 6G 6H 7A 8A 8BCD 9ABC 10A 11AB
-1 -1 -3 7 -14 7 -19 22 -33 32 -53 60 -91 81 -129 141 -208 194 -292 304 -426 411 -598 612 -845 829 -1147 1179 -1603 1567 -2138 2167 -2887 2860 -3825 3864 -5085 5032 -6624 6679 -8687 8624 -11206 11272 -14482 14413 -18523 18602 -23689 23599
-1 1 -2 1 -1 3 -3 3 -4 4 -6 6 -5 9 -11 10 -11 15 -17 16 -19 22 -26 29 -29
-1 -3 4 -1 -3 1 5 -9 8 -4 0 2 7 -19 17 -8 -1 -3 21 -30 31 -22 8 -13 35
-1 1 -2 1 -1 3 -3 3 -4 4 -6 6 -5 9 -11 10 -11 15 -17 16 -19 22 -26 29 -29
-1 1 0 3 -3 1 -3 3 -4 4 -4 6 -9 9 -7 12 -13 13 -15 18 -21 22 -24 27 -33
-1 0 -1 0 0 -2 4 2 0 -5 0 0 -6 10 3 0 -9 0 0 -11 22 8 0 -22 0
-1 2 -1 1 0 3 -5 4 -2 6 -4 3 -5 8 -16 14 -9 18 -12 11 -18 18 -35 35 -25
-1 0 1 -1 0 -1 -1 2 0 2 2 -3 -1 -2 -4 4 1 2 6 -5 0 -4 -7 7 1
-1 -1 1 0 -2 2 1 -3 2 0 -3 4 1 -5 3 1 -6 6 2 -7 6 0 -9 8 3
-1 0 0 1 0 -2 0 0 0 1 1 0 0 1 0 -4 0 0 -1 3 2 0 0 2 -2
12A, 12B 12C 12D, 12E 12FGH 12I 15A 18A, 18B
-1 -1 0 -2+ 3a3 0 -5+ 6a3 0 -8+ 12a3 1 -19+ 15a3 0 -27+ 27a3 -1 -45+ 42a3 -1 -66+ 69a3 0 -103+ 99a3 -1 -146+ 150a3 2 -213+ 210a3 0 -297+ 303a3 0 -425+ 414a3 0 -576+ 585a3 0 -797+ 792a3 3 -1067+ 1080a3 0 -1448+ 1434a3 -2 -1911+ 1923a3 -1 -2540+ 2526a3 0 -3314+ 3333a3 -4 -4342+ 4320a3 3 -5603+ 5625a3 0 -7244+ 7221a3 0 -9259+ 9285a3 0 -11846+ 11811a3
-1 -1 1 a3 -2 -1-2a3 1 0 -1 -1+a3 3 3+a3 -3 -1-2a3 3 3a3 -4 -1-3a3 4 2+2a3 -6 -5-2a3 6 5+a3 -5 -1-2a3 9 2+7a3 -11 -5-8a3 10 5+4a3 -11 -8-2a3 15 11+5a3 -17 -8-10a3 16 4+11a3 -19 -6-12a3 22 11+11a3 -26 -18-9a3 29 19+11a3 -29 -12-15a3
-1 1 -1 2 -2 2 -3 3 -4 4 -5 6 -7 9 -9 11 -12 14 -16 17 -20 22 -25 28 -31
-1 -1 0 -1 -1 -1 1 1 -2 0 0 0 -1 -1 1 0 -1 -1 1 0 -1 0 2 1 -3
-1 0 0 1 1 -1 0 0 0 0 -1 0 0 0 1 -2 0 0 1 1 -1 0 0 1 1
-1 −a3 1 -1+a3 0 1+a3 -1 −a3 2 -1+a3 −a3 2 -2+a3 -1-2a3 2+a3 -2+a3 -1-a3 3+a3 -3+a3 -3a3 4 -4+2a3 -1-2a3 5+a3 -4+a3
Table 27: The twined series for U6 (2). The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function ˜ hg, 3 (τ ) - part I. 2
[g] −D
53
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1A
2A
2B
2C
3A
3B
3C
1 1 1 1 1 231 39 7 -9 6 5796 36 -28 36 18 65505 225 97 -95 -15 494385 -15 -239 225 60 2922381 909 525 -531 90 14525511 -505 -1113 1143 -75 63447087 3183 2255 -2241 228 250188435 -2925 -4333 4275 360 907876585 10025 7945 -7975 -260 3073155810 -11166 -14174 14274 762 9804660777 29097 24809 -24759 1062 29717775186 -36270 -42286 42130 -759 86116649220 78660 70308 -70380 2070 239806592730 -103590 -115046 115290 2880 644418434331 201115 185563 -185445 -1926 1676994065901 -273555 -294483 294093 5256 4238788584987 488475 460571 -460773 6948 10432762525295 -675025 -711985 712575 -4645 25058448433770 1137450 1088842 -1088550 12120 58848028309224 -1583640 -1647128 1646280 15912 135349351964727 2548791 2466359 -2466761 -10281 305326880332593 -3553935 -3660335 3661569 26646 676433083819185 5524785 5388145 -5387535 34110 1473468429847035 -7692805 -7867397 7865595 -21840
1 15 45 30 150 225 159 570 900 505 1905 2655 1518 5175 7200 3879 13140 17370 9260 30300 39780 20562 66615 85275 43725
1 6 18 -15 60 90 -75 228 360 -260 762 1062 -759 2070 2880 -1926 5256 6948 -4645 12120 15912 -10281 26646 34110 -21840
4A
4B 4CDE
1 1 23 7 84 20 353 33 993 49 2685 93 6487 167 14719 239 31395 323 64553 505 127490 802 243945 1097 453842 1426 824596 2052 1465578 2986 2555051 3947 4376077 5037 7377627 6875 12257551 9535 20093306 12298 32531448 15448 52070711 20407 82458145 27217 129280929 34465 200804347 42747
1 -1 4 1 -7 -3 15 7 -29 -15 50 25 -78 -36 122 59 -195 -101 295 146 -424 -201 617 305 -901
4F 4G 5A 6AB 1 -1 -4 1 9 5 -17 -9 27 9 -46 -23 82 44 -126 -61 189 91 -289 -142 432 215 -631 -311 891
6C
6D
1 1 1 1 1 -1 1 3 6 7 4 1 9 18 5 1 0 18 33 22 -7 0 30 60 22 -3 -9 45 90 57 15 6 71 149 63 7 7 114 228 122 -29 0 180 360 164 -15 0 269 524 289 50 -25 381 762 361 25 17 531 1062 551 -78 16 738 1497 710 -36 0 1035 2070 1071 122 0 1440 2880 1408 59 -64 1963 3898 1999 -195 46 2628 5256 2580 -101 37 3474 6948 3530 295 0 4592 9227 4532 146 0 6060 12120 6124 -424 -136 7956 15912 7876 -201 92 10350 20631 10442 617 68 13323 26646 13231 305 0 17055 34110 17155 -901 0 21761 43616 21637
Table 28: The twined series for U6 (2). The table shows the Fourier coefficients multiplying q −D/24 in the q-expansion of the function ˜ hg, 3 (τ ) - part II. 2
[g] −D
54
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
6E 6F 6G 6H 7A 8A 8BCD 9ABC 10A 11AB 12AB 12C 12DE 12FGH 12I 15A 18AB 1 -2 2 1 4 -6 -3 -4 8 4 10 -10 -7 -18 16 10 24 -28 -13 -32 40 23 46 -50 -32
1 0 0 3 0 0 -7 0 0 14 0 0 -21 0 0 28 0 0 -43 0 0 69 0 0 -94
1 -2 2 1 4 -6 -3 -4 8 4 10 -10 -7 -18 16 10 24 -28 -13 -32 40 23 46 -50 -32
1 0 0 -5 0 0 9 0 0 -10 0 0 19 0 0 -36 0 0 45 0 0 -59 0 0 90
1 0 0 -1 3 0 0 -3 0 0 -3 9 1 0 -9 0 0 -9 19 3 0 -16 0 0 -15
1 3 0 1 -3 9 -5 3 -9 9 -6 9 -14 24 -18 15 -27 27 -25 30 -36 55 -51 45 -69
1 -1 0 1 1 1 -1 -1 -1 -3 2 1 2 4 -2 -1 -3 -5 3 2 4 7 -7 -3 -5
1 0 0 0 0 0 3 0 0 -5 0 0 0 0 0 9 0 0 -10 0 0 0 0 0 15
1 -1 1 0 0 -1 0 3 0 0 -1 -3 0 0 0 0 0 5 0 0 0 -4 0 0 0
1 0 -1 0 1 0 0 0 2 0 -1 -1 0 -3 1 1 0 0 0 4 0 -1 -2 0 -5
1 5 3 20 30 39 79 130 156 275 383 513 764 1069 1368 1985 2644 3414 4654 6152 7788 10400 13357 16917 21907
1 2 -6 5 0 -6 1 16 -24 20 2 -18 5 34 -72 50 16 -60 19 92 -168 119 34 -138 52
1 1 -1 0 -2 3 -1 2 -4 7 -5 5 -8 9 -8 5 -12 14 -14 16 -20 28 -23 25 -33
1 -1 1 -2 2 -3 3 -2 4 -3 5 -5 6 -9 8 -13 12 -14 16 -16 20 -18 23 -25 29
1 2 2 1 0 2 1 0 0 0 2 -2 1 2 0 2 0 4 -1 -4 0 -1 2 -2 0
1 1 -2 0 0 0 0 -2 0 0 2 2 1 0 0 -1 1 -2 0 0 2 -1 -4 0 0
1 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 1 0 0 2 0 0 0 0 0 -1
D
Decomposition Tables
Table 29: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,1 (τ ) into irreducible representations χn of M22 .
55
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
χ11
χ12
-1 11 23 35 47 59 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239
−2 0 0 0 2 2 26 78 300 950 3040 8706 24720 66016 171904 430782 1052210 2496320 5789798 13116978 29128906
0 0 0 2 26 92 472 1710 6220 20228 63080 184100 517270 1388930 3606408 9052216 22086048 52439410 121561770 275489850 611659466
0 0 0 8 42 228 952 3754 13194 43616 134624 395460 1106932 2978518 7724632 19403004 47318486 112383780 260469430 590364314 1310656938
0 0 0 8 42 228 952 3754 13194 43616 134624 395460 1106932 2978518 7724632 19403004 47318486 112383780 260469430 590364314 1310656938
0 0 2 6 60 258 1204 4516 16254 53078 164946 482650 1354036 3638646 9443990 23710442 57840438 137347678 318366892 721533716 1601947284
0 0 2 16 100 480 2130 8192 29148 95744 296536 869408 2436238 6551148 16996718 42682664 104106714 247235148 573046530 1298781032 2883475316
0 0 4 22 160 738 3334 12708 45402 148822 461482 1352058 3790274 10189794 26440708 66393106 161947152 384582826 891413422 2020314748 4485422600
0 0 4 34 208 1028 4508 17410 61752 203204 628868 1844428 5167368 13897114 36052396 90540888 220829728 524442118 1215547168 2754999510 6116448648
0 0 6 32 232 1132 4966 19112 68016 223388 691920 2028652 5684522 15285968 39659114 99592828 242915594 576882244 1337108322 3030488952 6728109656
0 1 4 48 268 1394 5966 23284 82218 271160 838076 2459952 6888684 18531236 48067156 120725462 294432862 699266588 1620714052 3673355264 8155231968
0 1 4 48 268 1394 5966 23284 82218 271160 838076 2459952 6888684 18531236 48067156 120725462 294432862 699266588 1620714052 3673355264 8155231968
0 0 8 60 380 1900 8240 31928 113216 372564 1152800 3381724 9473140 25478420 66095608 165992492 404852680 961480364 2228498796 5050837768 11213482124
Table 30: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,2 (τ ) into irreducible representations χn of M22 .
56
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
χ11
χ12
-4 8 20 32 44 56 68 80 92 104 116 128 140 152 164 176 188 200 212 224 236 248 260
1 0 0 0 2 2 15 62 215 716 2277 6752 19067 51870 135731 343436 843662 2016488 4701307 10713686 23905058 52307782 112400070
0 0 1 1 15 70 310 1260 4535 15162 47678 141611 400933 1089096 2849794 7212750 17716592 42346314 98728774 224984946 502005985 1098468036 2360397820
0 0 1 5 28 149 675 2686 9711 32528 102132 303407 859281 2333682 6106599 15456158 37963923 90741956 211562274 482109963 1075726834 2353861448 5057994232
0 0 1 5 28 149 675 2686 9711 32528 102132 303407 859281 2333682 6106599 15456158 37963923 90741956 211562274 482109963 1075726834 2353861448 5057994232
0 0 1 5 35 184 821 3286 11871 39754 124829 370835 1050214 2852302 7463621 18890818 46400388 110906886 258575987 589245588 1314777297 2876941610 6181993066
0 0 1 10 61 332 1486 5906 21367 71562 224684 667490 1890431 5134124 13434478 34003540 83520644 199632354 465436954 1060641874 2366599091 5178495256 11127587228
0 0 2 15 96 516 2307 9192 33238 111316 349513 1038325 2940645 7986426 20898099 52894358 129921032 310539240 724012941 1649887462 3681376388 8055436866 17309580294
0 1 2 18 136 706 3138 12528 45338 151798 476604 1415886 4009962 10890628 28497414 72128538 177165164 423462681 987290154 2249846600 5020058920 10984686398 23603973264
0 0 1 24 147 770 3468 13770 49855 167022 524228 1557452 4411059 11979584 31347100 79341684 194881422 465808750 1086019814 2474830808 5522064385 12083156304 25964369548
0 0 3 26 179 936 4196 16702 60440 202408 635466 1887842 5346659 14520766 37996546 96171541 236220080 564616772 1316387235 2999795266 6693411661 14646249036 31471963978
0 0 3 26 179 936 4196 16702 60440 202408 635466 1887842 5346659 14520766 37996546 96171541 236220080 564616772 1316387235 2999795266 6693411661 14646249036 31471963978
0 0 4 38 242 1284 5778 22964 83094 278332 873748 2595774 7351704 19965996 52245218 132236044 324802452 776347956 1810032796 4124718268 9203440774 20138593100 43273949924
Table 31: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,1 (τ ) into irreducible representations χn of U4 (3) - part I.
57
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
-1 11 23 35 47 59 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239 251 263 275 287
-2 0 0 0 0 0 4 8 38 124 420 1156 3368 8908 23350 58388 142936 338664 786382 1780628 3955942 8612692 18432618 38777988 80345934
0 0 0 0 6 12 72 232 860 2748 8640 24972 70446 188592 490188 1229276 3000480 7121124 16511002 37411832 83070266 180905952 387072550 814425664 1687237678
0 0 0 0 4 20 102 380 1398 4554 14228 41622 116944 314208 815984 2048450 4998134 11868186 27512230 62351670 138437866 301506846 645093274 1357371466 2812006070
0 0 0 0 4 20 102 380 1398 4554 14228 41622 116944 314208 815984 2048450 4998134 11868186 27512230 62351670 138437866 301506846 645093274 1357371466 2812006070
0 0 0 0 14 52 270 984 3618 11732 36676 107060 300968 808100 2098802 5267788 12853814 30518832 70749080 160334544 355990060 775306952 1658826474 3490390608 7230903886
0 0 0 2 22 82 422 1548 5622 18306 57060 166678 468176 1257446 3264774 8195334 19994842 47475942 110054184 249414368 553762164 1206043980 2580396830 5429518660 11248072954
0 0 0 4 22 120 532 2112 7492 24784 76674 225288 631070 1698156 4405146 11065144 26987278 64096288 148559990 336717232 747550924 1628176080 3483475608 7329886448 15184775484
0 0 0 4 26 132 606 2344 8350 27532 85316 250260 701512 1886748 4895338 12294420 29987754 71217412 165070996 374129064 830621184 1809082032 3870548166 8144312112 16872013154
0 0 0 7 34 189 801 3164 11125 36834 113706 334074 935210 2516637 6526850 16394948 39982901 94962320 220092869 498851309 1107491427 2412136251 5160722937 10859139417 22496001167
0 0 0 7 34 189 801 3164 11125 36834 113706 334074 935210 2516637 6526850 16394948 39982901 94962320 220092869 498851309 1107491427 2412136251 5160722937 10859139417 22496001167
Table 32: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,1 (τ ) into irreducible representations χn of U4 (3) - part II.
58
-D
χ11
χ12
χ13
χ14
χ15
χ16
χ17
χ18
χ19
χ20
-1 11 23 35 47 59 71 83 95 107 119 131 143 155 167 179 191 203 215 227 239 251 263 275 287
0 0 0 7 34 189 801 3164 11125 36834 113706 334074 935210 2516637 6526850 16394948 39982901 94962320 220092869 498851309 1107491427 2412136251 5160722937 10859139417 22496001167
0 0 0 7 34 189 801 3164 11125 36834 113706 334074 935210 2516637 6526850 16394948 39982901 94962320 220092869 498851309 1107491427 2412136251 5160722937 10859139417 22496001167
0 0 2 6 46 206 934 3532 12636 41346 128270 375590 1053068 2830546 7345188 18442654 44986624 106828846 247617812 561199172 1245957224 2713634904 5805874860 12216494216 25308126986
0 0 2 6 46 206 934 3532 12636 41346 128270 375590 1053068 2830546 7345188 18442654 44986624 106828846 247617812 561199172 1245957224 2713634904 5805874860 12216494216 25308126986
0 0 2 10 62 290 1242 4756 16856 55278 171022 501258 1404072 3775230 9793632 24593074 59982140 142445378 330156976 748280636 1661276556 3618212136 7741166340 16288726840 33744168882
0 1 1 16 74 391 1626 6379 22345 73819 227677 668609 1871144 5034454 13055603 32792777 79970261 189931510 440195866 997717729 2215005306 4824304831 10321492731 21718346527 44992097964
0 0 2 14 86 432 1864 7216 25568 84136 260274 763500 2138648 5751956 14921266 37473264 91395836 217054970 503083826 1140227136 2531441198 5513444680 11796004996 24820871652 51419567602
0 0 2 14 86 432 1864 7216 25568 84136 260274 763500 2138648 5751956 14921266 37473264 91395836 217054970 503083826 1140227136 2531441198 5513444680 11796004996 24820871652 51419567602
0 0 2 16 100 480 2130 8192 29148 95744 296536 869408 2436238 6551148 16996718 42682664 104106714 247235148 573046530 1298781032 2883475316 6280138936 13436398852 28272484272 58570125574
0 0 2 16 116 596 2586 10064 35728 117656 364172 1068560 2993494 8051776 20888326 52460200 127950562 303871648 704309074 1596305804 3544000144 7718796324 16514369292 34749166956 71987317446
Table 33: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,2 (τ ) into irreducible representations χn of U4 (3) - part I.
59
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
-4 8 20 32 44 56 68 80 92 104 116 128 140 152 164 176 188 200 212 224 236 248 260 272 284
1 0 0 0 1 0 3 10 30 98 315 932 2585 7068 18479 46678 114644 273958 638587 1455274 3246731 7103972 15265066 32250408 67064204
0 0 0 0 2 8 37 170 606 2040 6453 19200 54343 147802 386845 979182 2405530 5750076 13406477 30552132 68171414 149171154 320543266 677225340 1408309053
0 0 0 1 4 18 74 288 1038 3458 10820 32105 90862 246650 645246 1632930 4010504 9585456 22347452 50924466 113625730 248629024 534252608 1128728962 2347212278
0 0 0 1 4 18 74 288 1038 3458 10820 32105 90862 246650 645246 1632930 4010504 9585456 22347452 50924466 113625730 248629024 534252608 1128728962 2347212278
0 0 1 2 8 42 185 742 2648 8852 27785 82484 233470 634020 1658877 4198392 10311866 24646996 57462835 130945830 292176084 639325160 1373783498 2902432966 6035669290
0 0 1 1 14 64 285 1142 4114 13740 43187 128231 363072 986096 2580201 6530326 16040080 38338768 89385017 203691226 454492898 994500692 2136989558 4514885396 9388804084
0 0 1 4 16 92 396 1558 5574 18616 58374 173260 490425 1331594 3483848 8817060 21655492 51759604 120673444 274988136 613572808 1342587586 2884951508 6095118216 12674919317
0 1 0 3 21 104 428 1720 6189 20668 64812 192443 544792 1479426 3870656 9796194 24061078 57509747 134079840 305540174 681744339 1491758880 3205494608 6772343354 14083228602
0 0 1 3 26 129 573 2276 8226 27505 86349 256447 726201 1972157 5160351 13060772 32080056 76677496 178770283 407382190 908985651 1989002072 4273978516 9029770416 18777609455
0 0 1 3 26 129 573 2276 8226 27505 86349 256447 726201 1972157 5160351 13060772 32080056 76677496 178770283 407382190 908985651 1989002072 4273978516 9029770416 18777609455
Table 34: The table shows the decomposition of the Fourier coefficients multiplying q −D/12 in the function hg,1 (τ ) into irreducible representations χn of U4 (3) - part II.
60
-D
χ11
χ12
χ13
χ14
χ15
χ16
χ17
χ18
χ19
χ20
-4 8 20 32 44 56 68 80 92 104 116 128 140 152 164 176 188 200 212 224 236 248 260 272 284
0 0 1 3 26 129 573 2276 8226 27505 86349 256447 726201 1972157 5160351 13060772 32080056 76677496 178770283 407382190 908985651 1989002072 4273978516 9029770416 18777609455
0 0 1 3 26 129 573 2276 8226 27505 86349 256447 726201 1972157 5160351 13060772 32080056 76677496 178770283 407382190 908985651 1989002072 4273978516 9029770416 18777609455
0 0 0 3 27 142 636 2544 9223 30902 97044 288351 816744 2218282 5804744 14692426 36088522 86259890 201113152 458299758 1022601213 2237616450 4808209568 10158468546 21124777990
0 0 0 3 27 142 636 2544 9223 30902 97044 288351 816744 2218282 5804744 14692426 36088522 86259890 201113152 458299758 1022601213 2237616450 4808209568 10158468546 21124777990
0 0 0 5 31 182 842 3372 12251 41142 129268 384273 1088680 2957154 7738814 19588686 48115930 115010066 268146424 611059410 1363458009 2983474090 6410924444 13544593866 28166327222
0 0 0 6 46 248 1124 4512 16366 54888 172448 512502 1451756 3943264 10318996 26119004 64156020 153348868 357531400 814750620 1817950926 3977974852 8547913920 18059479228 37555131344
0 0 1 8 54 289 1299 5178 18749 62812 197214 585938 1659543 4507162 11794091 29851826 73323427 175259316 408612706 931151309 2077669720 4546274088 9769068796 20639439888 42920200803
0 0 1 8 54 289 1299 5178 18749 62812 197214 585938 1659543 4507162 11794091 29851826 73323427 175259316 408612706 931151309 2077669720 4546274088 9769068796 20639439888 42920200803
0 0 1 10 61 332 1486 5906 21367 71562 224684 667490 1890431 5134124 13434478 34003540 83520644 199632354 465436954 1060641874 2366599091 5178495256 11127587228 23509622436 48888805889
0 0 1 14 81 410 1842 7278 26305 88062 276276 820618 2323911 6310848 16513050 41794712 102655962 245367882 572065334 1303622708 2908753703 6364807716 13676730320 28895264904 60088352145
˜ 1 (τ ) into irreducible representations χn of Table 35: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 M23 - part I.
61
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
-1 0 0 0 1 0 3 11 40 132 455 1393 4196 11898 32865 87129 224778 562443 1373557 3272651 7632762 17436751 39096804 86113417 186578208
0 0 0 0 4 10 57 217 851 2945 9865 30717 91938 262221 722082 1917740 4942779 12376668 30213098 72004775 167908464 383624438 860103679 1894529513 4104664777
0 0 0 1 4 23 104 450 1705 6066 20061 62965 187731 536800 1476142 3923759 10108150 25318792 61794663 147289246 343437978 784701896 1759278672 3875207902 8395853450
0 0 0 1 4 23 104 450 1705 6066 20061 62965 187731 536800 1476142 3923759 10108150 25318792 61794663 147289246 343437978 784701896 1759278672 3875207902 8395853450
0 0 1 3 24 110 555 2266 8792 30883 102784 321420 960228 2742534 7546650 20051805 51668700 129399884 315850951 752794475 1755376093 4010659821 8991927075 19806533653 42912264213
0 0 1 3 23 115 552 2289 8800 31076 103141 322980 964123 2754938 7578723 20140197 51891433 129965529 317219639 756074520 1762997610 4028113401 9030999117 19892683368 43098789349
0 0 0 2 21 113 547 2283 8789 31062 103119 322951 964080 2754882 7578641 20140092 51891287 129965340 317219382 756074191 1762997173 4028112844 9030998389 19892682446 43098788160
0 0 0 2 21 113 547 2283 8789 31062 103119 322951 964080 2754882 7578641 20140092 51891287 129965340 317219382 756074191 1762997173 4028112844 9030998389 19892682446 43098788160
0 0 0 4 24 122 604 2500 9646 34002 112980 353668 1056018 3017118 8300712 22057824 56834066 142342008 347432516 828078940 1930905618 4411737282 9891102068 21787212032 47203452886
˜ 1 (τ ) into irreducible representations χn of Table 36: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 M23 - part II.
62
-D
χ10
χ11
χ12
χ13
χ14
χ15
χ16
χ17
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 1 1 11 68 389 1811 7661 29245 103695 343525 1076965 3212975 9184195 25260382 67136828 172966448 433225520 1057386979 2520265222 5876631765 13427082844 30103271271 66309028068 143662505416
0 0 1 11 68 389 1811 7661 29245 103695 343525 1076965 3212975 9184195 25260382 67136828 172966448 433225520 1057386979 2520265222 5876631765 13427082844 30103271271 66309028068 143662505417
0 0 1 11 80 444 2111 8892 34052 120586 399819 1252987 3738996 10686484 29394631 78121437 201271974 504113505 1230418627 2932664346 6838273397 15624224127 35029285569 77159558254 167170968143
0 0 1 11 80 444 2111 8892 34052 120586 399819 1252987 3738996 10686484 29394631 78121437 201271974 504113505 1230418627 2932664346 6838273397 15624224127 35029285569 77159558254 167170968143
0 0 1 14 87 496 2324 9848 37596 133306 441658 1384656 4130944 11808192 32477570 86318725 222385345 557004065 1359497348 3240340814 7555669131 17263391744 38704205367 85254464108 184708934751
0 0 1 14 87 496 2324 9848 37596 133306 441658 1384656 4130944 11808192 32477570 86318725 222385345 557004065 1359497348 3240340814 7555669131 17263391744 38704205367 85254464108 184708934751
0 0 2 12 96 508 2454 10246 39394 139204 462028 1447084 4319566 12343520 33956132 90238598 232499596 582313360 1421306694 3387607678 7899140768 18048043408 40463558496 89129562686 193104947248
0 0 3 26 182 1003 4777 20084 76950 272375 903231 2830347 8446311 24139819 66400838 176470194 454660159 1138754982 2779430485 6624675836 15447177148 35293998406 79128667059 174297913365 377627303791
˜ 3 (τ ) into irreducible representations χn of Table 37: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 M23 - part I.
63
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1 0 0 0 1 0 1 8 25 90 301 979 2892 8477 23485 63229 164314 415733 1022444 2456955 5768329 13269156 29930003 66312927 144441012
0 0 1 0 2 8 29 148 536 1969 6612 21229 63977 185925 516995 1390158 3616159 9142805 22498248 54045257 126911807 291907776 658478668 1458848820 3177751973
0 0 0 0 2 12 66 278 1108 3996 13574 43224 131140 379810 1058004 2842550 7398130 18698262 46023364 110540364 259601046 597070540 1346907692 2983976756 6499993004
0 0 0 0 2 12 66 278 1108 3996 13574 43224 131140 379810 1058004 2842550 7398130 18698262 46023364 110540364 259601046 597070540 1346907692 2983976756 6499993004
0 0 1 2 10 72 317 1456 5614 20527 69200 221297 669671 1942293 5406043 14531106 37808739 95575487 235220486 564999523 1326827403 3051727864 6884144714 15251512500 33222076673
0 1 0 1 12 71 324 1451 5666 20571 69575 222128 672826 1950361 5430193 14593220 37974823 95988502 236247135 567449989 1332605575 3064982355 6914096414 15317793334 33366564639
0 0 0 2 10 65 327 1439 5650 20585 69539 222072 672862 1950253 5430053 14593319 37974566 95988148 236247364 567449375 1332604789 3064982875 6914095094 15317791616 33366565723
0 0 0 2 10 65 327 1439 5650 20585 69539 222072 672862 1950253 5430053 14593319 37974566 95988148 236247364 567449375 1332604789 3064982875 6914095094 15317791616 33366565723
0 0 0 2 12 76 354 1584 6186 22554 76160 243296 736834 2136178 5947048 15983498 41590710 105130940 258745612 621494632 1459516642 3356890620 7572573738 16776640436 36544317696
˜ 3 (τ ) into irreducible representations χn of Table 38: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 M23 - part II.
64
-D
χ10
χ11
χ12
χ13
χ14
χ15
χ16
χ17
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 1 4 41 220 1099 4783 18920 68482 232064 739984 2243393 6500043 18102016 48641673 126586391 319954666 787500731 1891483817 4442040173 10216574441 23047036703 51059233415 111221993159
0 0 1 4 41 220 1099 4783 18920 68482 232064 739984 2243393 6500043 18102016 48641673 126586391 319954666 787500731 1891483817 4442040173 10216574441 23047036703 51059233415 111221993159
0 0 0 6 42 254 1278 5568 21967 79749 269912 861169 2610275 7563999 21063317 56602448 147298443 372313294 916360448 2201005302 5168908580 11888393029 26818346389 59414411523 129421909124
0 0 0 6 42 254 1278 5568 21967 79749 269912 861169 2610275 7563999 21063317 56602448 147298443 372313294 916360448 2201005302 5168908580 11888393029 26818346389 59414411523 129421909124
0 0 1 6 49 280 1418 6143 24308 88055 298347 951374 2884379 8357121 23273934 62539370 162753713 411370026 1012501148 2431907356 5711193875 13135596047 29631903464 65647584650 142999706146
0 0 1 6 49 280 1418 6143 24308 88055 298347 951374 2884379 8357121 23273934 62539370 162753713 411370026 1012501148 2431907356 5711193875 13135596047 29631903464 65647584650 142999706146
0 0 0 8 48 302 1462 6460 25342 92194 311662 995038 3014774 8738192 24329868 65385226 170146608 430076524 1058511762 2542467146 5970765562 13732710656 30978745714 68631657584 149499558772
0 0 1 13 97 582 2879 12594 49625 180159 609707 1945438 5896262 17086836 47580312 127861367 332736007 841030812 2069990475 4971917548 11676191108 26855037539 60580719175 134212929307 292354823899
˜ 1 (τ ) into irreducible representations χn of Table 39: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 McL - part I.
65
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
-1 0 0 0 0 0 0 1 0 2 6 18 49 135 379 995 2564 6386 15622 37167 86756 198014 444173 977997 2119404
0 0 0 0 1 1 3 5 13 37 122 357 1065 2983 8247 21796 56231 140568 343337 817799 1907498 4357044 9769828 21517663 46622397
0 0 0 0 1 2 8 27 103 357 1181 3673 10971 31296 86127 228754 589479 1476138 3603184 8587440 20024583 45751192 102575149 225940987 489518783
0 0 0 0 1 1 8 29 113 381 1292 3990 11987 34082 94007 249432 643218 1610016 3931104 9367446 21845882 49908836 111902115 246477854 534024793
0 1 0 1 2 7 23 93 336 1192 3911 12260 36504 104381 286919 762672 1964549 4920870 12009753 28625805 66746654 152506325 341912961 753141922 1631720458
0 0 0 1 1 7 22 92 336 1191 3909 12259 36503 104379 286918 762670 1964545 4920869 12009751 28625800 66746652 152506321 341912958 753141920 1631720453
0 0 0 0 1 5 24 101 386 1370 4541 14231 42467 121375 333865 887304 2286050 5725731 13975118 33309272 77669274 177460315 397863480 876380147 1898731972
0 0 0 0 1 5 24 101 386 1370 4541 14231 42467 121375 333865 887304 2286050 5725731 13975118 33309272 77669274 177460315 397863480 876380147 1898731972
˜ 1 (τ ) into irreducible representations χn of Table 40: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 McL - part II.
66
-D
χ9
χ10
χ11
χ12
χ13
χ14
χ15
χ16
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 0 3 8 49 193 765 2657 8900 27745 83033 236919 652314 1732646 4465544 11182141 27296592 65055014 151701090 346597295 777084367 1711669416 3708476391
0 0 1 1 5 21 99 396 1535 5375 17881 55882 166945 476743 1311878 3485567 8981556 22493230 54903746 130856090 305132577 697161444 1563042221 3442912979 7459319124
0 0 0 1 3 22 92 402 1513 5400 17816 55963 166766 476982 1311412 3486207 8980403 22494840 54901023 130859914 305126393 697170138 1563028664 3442931988 7459290309
0 0 0 1 5 25 120 507 1946 6874 22816 71465 213315 609553 1676845 4456239 11481470 28756211 70187986 167289297 390081042 891261397 1998200428 4401459954 9536046829
0 0 0 0 4 27 123 536 2039 7271 24056 75500 225153 643803 1770483 4706100 12123772 30367390 74116978 176659468 411922132 941176611 2110092165 4647951685 10070049592
0 0 0 1 5 27 138 572 2212 7783 25887 81007 241943 691150 1901666 5053141 13020310 32608993 79593979 189704760 442353736 1010687552 2265963373 4991249366 10813885859
0 0 0 1 5 32 146 628 2385 8486 28072 88095 262694 751132 2065592 5490489 14144455 35428696 86469919 206102830 480575987 1098039598 2461774474 5422610680 11748391635
0 0 0 2 8 46 214 909 3459 12269 40626 127410 380029 1086405 2987879 7941451 20459305 51244665 125073387 298112039 695120690 1588233526 3560784919 7843413933 16993217734
˜ 1 (τ ) into irreducible representations χn of Table 41: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 McL - part III.
67
-D
χ17
χ18
χ19
χ20
χ21
χ22
χ23
χ24
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 2 8 46 214 909 3459 12269 40626 127410 380029 1086405 2987879 7941451 20459305 51244665 125073387 298112039 695120690 1588233526 3560784919 7843413933 16993217734
0 0 0 1 8 46 221 928 3561 12607 41815 131025 391030 1117568 3074116 8169871 21049092 52720052 128677387 306697753 715147222 1633981014 3663364558 8069346991 17482746319
0 0 0 1 8 46 221 928 3561 12607 41815 131025 391030 1117568 3074116 8169871 21049092 52720052 128677387 306697753 715147222 1633981014 3663364558 8069346991 17482746319
0 0 0 1 10 53 261 1076 4167 14687 48824 152796 456320 1303640 3586785 9531019 24558072 61505504 150125530 357811165 834342808 1906304694 4273934999 9414224038 20396558046
0 0 0 1 10 53 261 1076 4167 14687 48824 152796 456320 1303640 3586785 9531019 24558072 61505504 150125530 357811165 834342808 1906304694 4273934999 9414224038 20396558046
0 0 0 1 10 55 263 1110 4252 15064 49950 156541 467137 1335140 3672503 9760341 25146550 62983039 153726332 366401965 854361992 1952063468 4376498280 9640181689 20886051640
0 0 0 1 11 59 279 1175 4487 15899 52683 165130 492677 1408229 3873309 10294270 26521644 66427796 162132945 386440375 901084137 2058818682 4615836213 10167382764 22028253515
0 0 1 2 11 60 279 1175 4489 15900 52683 165131 492679 1408230 3873312 10294273 26521644 66427799 162132948 386440376 901084142 2058818686 4615836219 10167382771 22028253520
˜ 3 (τ ) into irreducible representations χn of Table 42: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 McL - part I.
68
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1 0 0 0 1 0 1 1 2 2 6 18 36 105 276 730 1881 4756 11625 27977 65536 150811 339981 753426 1640491
0 0 0 0 0 0 0 3 5 22 73 251 716 2128 5867 15803 41051 103927 255433 614030 1441297 3315763 7478672 16570437 36091901
0 1 0 1 1 4 6 24 72 245 804 2556 7659 22218 61728 165837 431408 1090489 2683334 6445575 15135916 34812830 78530602 173981507 378977806
0 0 1 0 1 1 5 23 72 259 865 2785 8308 24238 67254 180937 470406 1189748 2926781 7031919 16510850 37978515 85667592 189800269 413425592
0 0 0 0 1 3 12 55 219 777 2644 8404 25493 73822 205653 552420 1437895 3633969 8944652 21483285 50453393 116039405 261770033 579930706 1263264431
0 0 0 0 1 3 12 55 219 777 2644 8404 25493 73822 205653 552420 1437895 3633969 8944652 21483285 50453393 116039405 261770033 579930706 1263264431
0 0 0 0 0 2 14 62 247 905 3060 9775 29645 85899 239219 642880 1672987 4228700 10408024 24998994 58708484 135028617 304603271 674830133 1469977138
0 0 0 0 0 2 14 62 247 905 3060 9775 29645 85899 239219 642880 1672987 4228700 10408024 24998994 58708484 135028617 304603271 674830133 1469977138
˜ 3 (τ ) into irreducible representations χn of Table 43: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 McL - part II.
69
-D
χ9
χ10
χ11
χ12
χ13
χ14
χ15
χ16
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 0 1 7 24 129 482 1776 5969 19154 57809 167931 467083 1255917 3267135 8260107 20326853 48828286 114662453 263731843 594922426 1318037615 2871035001
0 0 0 0 1 13 54 254 975 3566 12026 38471 116393 337637 939698 2525910 6572140 16613698 40887536 98212448 230638380 530473486 1196650730 2651128016 5774897738
0 0 0 0 2 10 56 244 986 3536 12064 38380 116510 337390 940034 2525280 6573020 16612158 40889694 98208856 230643438 530465432 1196662060 2651110530 5774922228
0 0 0 1 2 15 75 320 1251 4559 15394 49139 148894 431517 1201486 3228953 8402301 21238355 52272318 125553959 294852626 678158814 1529814670 3389217725 7382694796
0 0 0 0 2 14 78 334 1328 4792 16274 51846 157282 455558 1268952 3409380 8873332 22426902 55200610 132583074 311367160 716131132 1615490472 3579004576 7796138458
0 0 0 1 2 19 81 365 1414 5173 17445 55752 168786 489419 1362371 3661782 9527924 24084770 59275991 142379305 334361281 769034337 1734806216 3843378364 8371967563
0 0 1 1 6 21 95 398 1566 5607 19023 60533 183543 531575 1480572 3977741 10352451 26165020 64401011 154680790 363262412 835486965 1884740065 4175506851 9095496261
0 0 0 0 4 25 130 562 2235 8093 27451 87506 265369 768802 2141239 5753503 14973440 37845752 93150396 223734839 525430528 1208473427 2726136749 6039574829 13155976526
˜ 3 (τ ) into irreducible representations χn of Table 44: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 McL - part III.
70
-D
χ17
χ18
χ19
χ20
χ21
χ22
χ23
χ24
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 0 4 25 130 562 2235 8093 27451 87506 265369 768802 2141239 5753503 14973440 37845752 93150396 223734839 525430528 1208473427 2726136749 6039574829 13155976526
0 0 0 0 5 27 132 583 2298 8336 28225 90070 272959 791067 2202779 5919487 15404421 38936669 95832723 230181494 540564243 1243288847 2804662530 6213562155 13534944287
0 0 0 0 5 27 132 583 2298 8336 28225 90070 272959 791067 2202779 5919487 15404421 38936669 95832723 230181494 540564243 1243288847 2804662530 6213562155 13534944287
0 0 0 1 5 32 153 685 2671 9753 32895 105135 318375 923068 2569637 6906560 17971134 45427164 111803325 268547565 630654439 1450509683 3272097750 7249168236 15790750779
0 0 0 1 5 32 153 685 2671 9753 32895 105135 318375 923068 2569637 6906560 17971134 45427164 111803325 268547565 630654439 1450509683 3272097750 7249168236 15790750779
0 0 0 1 5 31 161 695 2743 9968 33718 107589 326136 945024 2631602 7071840 18403192 46516151 114488565 274989746 645794430 1485315445 3350637733 7423133737 16169749982
0 0 0 1 6 34 168 733 2898 10503 35576 113467 343976 996685 2775597 7458445 19409807 49059813 120749919 290027769 681112299 1566542113 3533877823 7829083804 17054036789
0 0 0 1 6 34 168 733 2898 10503 35576 113467 343976 996685 2775597 7458445 19409807 49059813 120749919 290027769 681112299 1566542113 3533877823 7829083804 17054036789
˜ 1 (τ ) into irreducible representations χn of Table 45: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 HS - part I.
71
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
-1 0 0 0 0 0 0 4 7 33 99 330 948 2759 7516 20108 51591 129518 315659 753088 1754960 4011289 8990981 19807924 42910275
0 0 0 0 2 2 16 50 203 672 2292 7047 21207 60245 166233 440909 1137213 2846173 6949871 16560006 38620903 88231069 197828069 435736187 944081846
0 0 0 0 3 8 44 174 681 2372 7929 24733 73979 211115 581198 1543852 3978753 9963420 24321125 57964332 135165363 308818831 692381433 1525098819 3304250864
0 0 0 1 2 19 80 357 1334 4787 15766 49595 147695 422628 1161707 3088701 7955779 19929384 48638189 115934578 270321371 617651188 1384742351 3050227006 6608457916
0 0 0 0 5 14 90 339 1372 4720 15887 49392 148048 422054 1162651 3087208 7958171 19925689 48643912 115925926 270334509 617631664 1384771352 3050184616 6608519893
0 0 0 0 5 14 90 339 1372 4720 15887 49392 148048 422054 1162651 3087208 7958171 19925689 48643912 115925926 270334509 617631664 1384771352 3050184616 6608519893
0 0 0 1 3 21 92 407 1521 5435 17929 56348 167874 480211 1320213 3509785 9040892 22646724 55271192 131743166 307184590 701874719 1573573454 3466163569 7509616830
0 0 0 0 6 24 131 515 2039 7112 23777 74177 221911 633338 1743556 4631476 11936173 29890200 72963281 173892768 405495839 926456288 2077143971 4575295928 9912751919
˜ 1 (τ ) into irreducible representations χn of Table 46: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 HS - part II.
72
-D
χ9
χ10
χ11
χ12
χ13
χ14
χ15
χ16
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 3 14 79 375 1585 6059 21454 71118 222914 665120 1901066 5228981 13897177 35804287 89677291 218879656 521694040 1216464055 2779404150 6231379962 13725964622 29738144692
0 0 1 2 19 85 432 1739 6780 23769 79177 247447 739461 2111639 5811159 15439550 39785466 99637115 243206606 579649779 1351642677 3088203718 6923790672 15251021256 33042457875
0 1 0 3 15 94 413 1776 6708 23898 78950 247836 738796 2112736 5809370 15442403 39780944 99644150 243195756 579666270 1351617767 3088240865 6923735648 15251101926 33042340266
0 0 0 4 15 94 412 1777 6709 23897 78950 247836 738797 2112736 5809369 15442404 39780943 99644149 243195757 579666271 1351617768 3088240863 6923735648 15251101928 33042340266
0 0 1 3 18 95 449 1887 7219 25544 84686 265365 791857 2263145 6225096 16544170 42624415 106758449 260571613 621063719 1448172795 3308813035 7418312272 16340430775 35402559005
0 0 0 3 17 106 478 2061 7803 27789 91864 288353 859693 2458353 6760018 17969139 46290660 115949075 282991729 674519766 1572792424 3593587213 8056712554 17746732420 38449273233
0 0 0 3 17 106 478 2061 7803 27789 91864 288353 859693 2458353 6760018 17969139 46290660 115949075 282991729 674519766 1572792424 3593587213 8056712554 17746732420 38449273233
0 0 0 2 22 117 575 2396 9247 32642 108441 339513 1013725 2896427 7968570 21175541 54560442 136648460 333534637 794956158 1853668197 4235268766 9495455515 20915725781 45315309735
˜ 1 (τ ) into irreducible representations χn of Table 47: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 HS - part III.
73
-D
χ17
χ18
χ19
χ20
χ21
χ22
χ23
χ24
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 4 26 162 740 3180 12073 42967 142122 445985 1329897 3802615 10457085 27795624 71606337 179357824 437754013 1043395661 2432916045 5558825615 12462733462 27451966739 59476232959
0 0 0 5 27 164 756 3230 12281 43635 144411 453032 1351104 3862871 10623310 28236528 72743550 182203997 444703911 1059955647 2471536933 5647056684 12660561531 27887702980 60420314767
0 0 0 6 35 202 941 4009 15273 54213 179524 563011 1679391 4800963 13203972 35094660 90413520 226459775 552723198 1317413812 3071871642 7018707863 15735791983 34661549588 75096290859
0 0 1 4 42 215 1054 4374 16866 59519 197695 618937 1847957 5280034 14526071 38601412 99459154 249099125 608005819 1449139547 3379082403 7720543541 17309423366 38127628080 82606031374
0 0 1 6 40 221 1046 4396 16834 59591 197583 619159 1847627 5280632 14525165 38602966 99456838 249102903 608000215 1449148385 3379069491 7720563319 17309394714 38127670972 82605970008
0 0 1 7 53 286 1371 5744 22047 77977 258692 810439 2418829 6912578 19015093 50534149 130198910 326096807 795929685 1897063405 4423514093 10106911327 22659582306 49912570342 108138748123
0 0 1 9 56 317 1487 6289 24028 85159 282195 884610 2639284 7544029 20749740 55147824 142079975 355863082 868568813 2070216248 4827235467 11029385777 24727692056 54468122422 118008497342
0 0 1 9 67 361 1743 7290 28003 99000 328520 1029079 3071601 8777736 24146350 64169990 165332469 414090303 1010705622 2408967373 5617163685 12834168544 28774079262 63381031865 137319059194
˜ 3 (τ ) into irreducible representations χn of Table 48: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 HS - part I.
74
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1 0 0 0 1 0 1 2 8 19 76 224 680 1937 5448 14492 37900 95506 235400 564823 1327268 3051197 6885123 15250485 33223874
0 0 0 0 0 2 4 36 117 455 1504 4906 14653 42824 118790 319869 831414 2103267 5173804 12431424 29188063 67140916 151446502 335540316 730874824
0 0 1 0 2 6 25 118 436 1585 5333 17078 51536 149641 416242 1119026 2911159 7359765 18111488 43505965 102164805 234984819 530077046 1174370946 2558094287
0 0 0 0 3 10 54 217 882 3138 10710 34000 103287 298883 832940 2237183 5823587 14717082 36226467 87006255 204337166 469957788 1060170981 2348714622 5116227145
0 0 0 1 1 13 45 232 850 3191 10609 34168 102985 299362 832131 2238459 5821517 14720250 36221486 87013768 204325655 469974829 1060145461 2348751902 5116172310
0 0 0 1 1 13 45 232 850 3191 10609 34168 102985 299362 832131 2238459 5821517 14720250 36221486 87013768 204325655 469974829 1060145461 2348751902 5116172310
0 0 1 0 3 12 61 251 1000 3567 12169 38662 117337 339698 946482 2542349 6617566 16724285 41165993 98871502 232200431 534044327 1204737801 2668997408 5813889540
0 1 0 1 3 21 73 349 1298 4768 15972 51207 154628 448887 1248628 3357136 8733350 22079147 54334528 130517592 306494073 704954744 1590230348 3523111975 7674283501
˜ 3 (τ ) into irreducible representations χn of Table 49: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 HS - part II.
75
-D
χ9
χ10
χ11
χ12
χ13
χ14
χ15
χ16
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 1 7 46 227 990 3908 14181 48025 153185 464346 1345538 3747019 10068963 26203117 66230865 163012176 391537803 919501026 2114832583 4770733874 10569264743 23022947110
0 0 0 1 7 54 239 1118 4310 15813 53238 170410 515565 1495628 4162388 11189191 29112145 73593695 181118490 435051059 1021654112 2349834128 5300784874 11743672359 25580985954
0 0 0 0 10 48 254 1089 4367 15709 53426 170086 516126 1494707 4163910 11186751 29116039 73587635 181127903 435036721 1021675875 2349801630 5300833219 11743601341 25581089850
0 0 0 0 10 48 254 1089 4367 15709 53426 170086 516126 1494707 4163910 11186751 29116039 73587635 181127903 435036721 1021675875 2349801630 5300833219 11743601341 25581089850
0 0 0 1 9 56 268 1183 4655 16879 57171 182391 552755 1601901 4460701 11986913 31194087 78846635 194061566 466117227 1094643301 2517659061 5679443225 12582461650 27408265179
0 0 0 1 10 54 299 1265 5071 18296 62148 197917 600589 1739300 4845163 13017514 33880203 85629401 210766813 506225002 1188857894 2734316595 6168239640 13665284242 29767082573
0 0 0 1 10 54 299 1265 5071 18296 62148 197917 600589 1739300 4845163 13017514 33880203 85629401 210766813 506225002 1188857894 2734316595 6168239640 13665284242 29767082573
0 0 0 3 10 71 344 1520 5937 21661 73107 233542 707419 2050680 5709189 15344148 39927180 100925428 248396323 596634205 1401136588 3222614325 7269672253 16105571907 35082549764
˜ 3 (τ ) into irreducible representations χn of Table 50: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 HS - part III.
76
-D
χ17
χ18
χ19
χ20
χ21
χ22
χ23
χ24
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 1 2 19 89 466 1973 7859 28328 96152 306248 929020 2690703 7494835 20136861 52408277 132459084 326029062 783069157 1839012867 4229650434 9541491409 21138496831 46045944874
0 0 1 2 19 93 468 2007 7976 28783 97662 311150 943669 2733527 7613625 20456746 53239679 134562341 331202866 795500581 1868200964 4296791328 9692937895 21474037147 46776819698
0 0 0 2 19 111 580 2489 9884 35784 121313 386726 1172777 3397533 9462567 25425987 66170567 167247845 411649683 988727552 2321978347 5340477195 12047321308 26690041661 58138785609
0 0 0 4 20 131 622 2770 10828 39464 133278 425729 1289537 3738214 10407432 27970908 72784072 183978541 452805738 1087614155 2554155967 5874555816 13252008175 29359113857 63952566006
0 0 0 2 20 125 628 2750 10850 39398 133362 425535 1289797 3737680 10408176 27969538 72786002 183975195 452810502 1087606345 2554167059 5874538248 13252033063 29359075739 63952619706
0 0 0 4 28 168 822 3612 14204 51610 174568 557162 1688384 4893216 13625020 36615272 95282852 240841718 592768390 1423778908 3343632878 7690311850 17348106814 38433714858 83719772698
0 0 1 4 32 181 904 3932 15532 56267 190608 607872 1842742 5339398 14869371 39955933 103981276 262820443 646874829 1553720168 3648816574 8392189392 18931489848 41941519745 91360913541
0 0 0 5 34 213 1043 4587 18023 65550 221641 707534 2143915 6213691 17301389 46495887 120993550 305831383 752720667 1807974775 4245880168 9765479321 22029335806 48804725354 106310810073
˜ 1 (τ ) into irreducible representations χn of Table 51: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part I.
77
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
-1 0 0 0 0 0 0 1 0 2 3 9 12 33 66 160 334 795 1763 4088 9121 20500 45009 98354 210968
0 0 0 0 1 1 3 4 8 15 36 73 179 404 1029 2473 6120 14678 35220 82414 190679 431992 964995 2117328 4579167
0 0 0 0 1 2 5 10 28 67 187 477 1318 3451 9188 23558 59837 147655 358129 848112 1971916 4492407 10058179 22125220 47903632
0 0 0 0 0 1 2 8 17 57 156 461 1279 3556 9534 25066 63872 159223 386854 920162 2141266 4887625 10947884 24103804 52199413
0 0 0 0 0 0 1 3 18 54 196 592 1803 5080 14074 37214 96109 240240 586899 1397738 3260343 7446731 16697919 36775178 79680703
0 0 0 0 0 2 3 11 33 99 280 807 2265 6244 16751 43863 111842 278362 676374 1607611 3741059 8536684 19121267 42092946 91155986
0 0 0 0 0 2 3 13 32 112 312 971 2723 7741 20811 55156 140844 352289 856699 2040543 4750527 10850420 24309087 53536679 115951616
0 0 0 0 1 0 3 10 32 103 330 998 2936 8287 22680 60008 154261 385653 940408 2239707 5220237 11923091 26726129 58860381 127512423
0 0 0 1 1 4 7 22 56 172 475 1394 3892 10843 29067 76443 194953 486133 1181472 2810456 6541054 14931553 33447564 73643280 159487514
0 1 0 1 2 5 9 27 61 180 487 1411 3914 10878 29109 76503 195033 486240 1181610 2810645 6541292 14931866 33447965 73643797 159488167
˜ 1 (τ ) into irreducible representations χn of Table 52: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part II.
78
-D
χ11
χ12
χ13
χ14
χ15
χ16
χ17
χ18
χ19
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 0 0 0 1 10 39 157 532 1732 5207 15071 41585 111048 286479 718884 1755665 4187951 9767990 22326103 50061479 110289605 238965109
0 0 0 0 0 0 1 10 39 157 532 1732 5207 15071 41585 111048 286479 718884 1755665 4187951 9767990 22326103 50061479 110289605 238965109
0 0 0 0 0 0 1 10 39 157 532 1732 5207 15071 41585 111048 286479 718884 1755665 4187951 9767990 22326103 50061479 110289605 238965109
0 0 0 0 1 0 5 12 62 189 693 2090 6424 18136 50419 133476 345208 863390 2110561 5027797 11731080 26797674 60096749 132364423 286811937
0 0 0 0 0 0 4 11 61 207 744 2302 7052 20097 55773 148102 382864 958742 2343230 5584747 13029827 29770742 66762700 147060137 318652800
0 0 0 0 1 3 14 45 159 510 1642 4975 14671 41417 113397 299955 771309 1928130 4702139 11198147 26101380 59614785 133631223 294300418 637563387
0 0 0 1 2 5 16 49 164 519 1653 4992 14694 41450 113440 300016 771388 1928238 4702278 11198333 26101618 59615100 133631622 294300936 637564041
0 0 1 1 3 8 22 63 204 625 1946 5815 16983 47724 130238 343936 883312 2206704 5378796 12806125 29842923 68152090 152753296 336394392 728720690
0 0 0 0 1 3 17 55 219 715 2383 7273 21717 61506 169159 448042 1154153 2886846 7045333 16782848 39131147 89385449 200393822 441360427 956216022
˜ 1 (τ ) into irreducible representations χn of Table 53: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part III.
79
-D
χ20
χ21
χ22
χ23
χ24
χ25
χ26
χ27
χ28
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 1 0 4 12 59 204 750 2420 7688 22754 65313 179162 476877 1227450 3076064 7505177 17892382 41714567 95319311 213691022 470720019 1019815583
0 0 0 1 2 8 24 89 281 942 2932 9022 26332 74730 203895 540363 1387781 3471701 8462435 20159114 46979165 107312059 240528777 529752466 1147598073
0 0 0 0 1 5 22 81 291 982 3180 9799 28983 82323 225671 598307 1539339 3851620 9395281 22383595 52179270 119196613 267203769 588518852 1274985629
0 0 0 1 2 7 24 85 296 991 3191 9816 29006 82356 225714 598368 1539418 3851728 9395420 22383781 52179508 119196928 267204168 588519370 1274986283
0 0 0 0 1 5 20 83 302 1064 3473 10856 32232 92034 252683 671268 1728219 4327763 10559896 25167016 58675953 134058491 300539689 661990039 1434203044
0 0 0 0 1 4 22 86 345 1194 4006 12481 37369 106605 293554 779662 2009529 5031900 12283559 29274597 68265693 155968449 349688430 770250507 1668815367
0 0 0 0 1 4 24 97 388 1360 4564 14266 42728 122027 336080 892976 2301754 5764588 14072625 33540710 78215160 178705755 400669808 882558528 1912148601
0 0 0 0 1 4 24 97 388 1360 4564 14266 42728 122027 336080 892976 2301754 5764588 14072625 33540710 78215160 178705755 400669808 882558528 1912148601
0 0 0 0 1 4 24 97 388 1360 4564 14266 42728 122027 336080 892976 2301754 5764588 14072625 33540710 78215160 178705755 400669808 882558528 1912148601
˜ 1 (τ ) into irreducible representations χn of Table 54: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part IV.
80
-D
χ29
χ30
χ31
χ32
χ33
χ34
χ35
χ36
χ37
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 1 1 4 11 43 140 504 1659 5391 16513 48944 138812 380846 1009253 2597544 6498438 15853998 37769072 88050558 201135952 450900478 993105129 2151522265
0 0 0 0 3 9 39 135 495 1647 5373 16488 48909 138765 380778 1009164 2597424 6498279 15853785 37768797 88050195 201135486 450899874 993104358 2151521277
0 0 0 0 1 4 26 105 430 1512 5106 15981 47964 137052 377745 1003898 2588431 6483169 15828769 37727937 87984244 201030242 450733710 992844595 2151118874
0 0 0 0 1 4 26 105 430 1512 5106 15981 47964 137052 377745 1003898 2588431 6483169 15828769 37727937 87984244 201030242 450733710 992844595 2151118874
0 0 0 0 1 4 26 105 430 1512 5106 15981 47964 137052 377745 1003898 2588431 6483169 15828769 37727937 87984244 201030242 450733710 992844595 2151118874
0 0 0 0 1 6 30 124 476 1679 5576 17469 52139 149008 409888 1089307 2806561 7029317 17157026 40892977 95353051 217864013 488448796 1075912674 2331033239
0 0 0 1 3 12 47 181 641 2189 7089 21956 64970 184876 506964 1345063 3461147 8662775 21132786 50353789 117386017 268167991 601164912 1324106543 2868612560
0 0 0 0 0 4 30 141 581 2112 7143 22579 67811 194405 535979 1426142 3677682 9215765 22502085 53644469 125107300 285875651 640979164 1411960695 3059214800
0 0 0 0 1 11 44 208 769 2800 9196 29034 86401 247540 680273 1809476 4660418 11676421 28495813 67927686 158383477 361897629 811353013 1787223481 3872097331
˜ 1 (τ ) into irreducible representations χn of Table 55: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part V.
81
-D
χ38
χ39
χ40
χ41
χ42
χ43
χ44
χ45
χ46
-1 23 47 71 95 119 143 167 191 215 239 263 287 311 335 359 383 407 431 455 479 503 527 551 575
0 0 0 0 2 9 49 199 786 2771 9251 28943 86560 247286 680702 1808802 4661495 11674769 28498396 67923794 158389396 361888871 811366088 1787204411 3872125240
0 0 0 0 2 11 55 228 881 3105 10311 32259 96303 275091 756753 2010778 5180729 12974774 31668560 75478392 175998083 402117862 901543092 1985829461 4302423193
0 0 0 0 2 11 55 228 881 3105 10311 32259 96303 275091 756753 2010778 5180729 12974774 31668560 75478392 175998083 402117862 901543092 1985829461 4302423193
0 0 0 0 3 12 67 265 1049 3657 12230 38155 114151 325787 896888 2382438 6140008 15375595 37532560 89450970 208588414 476572073 1068489749 2353547126 5099152355
0 0 0 1 3 17 69 295 1088 3864 12681 39773 118296 338162 929109 2469363 6359343 15927891 38869434 92643752 216007314 493536047 1106464669 2437220934 5280310504
0 0 0 1 3 17 69 295 1088 3864 12681 39773 118296 338162 929109 2469363 6359343 15927891 38869434 92643752 216007314 493536047 1106464669 2437220934 5280310504
0 0 0 1 3 18 87 362 1384 4892 16220 50831 151675 433503 1192384 3168960 8164490 20449076 49911204 118961732 277390339 633786948 1420941137 3129929312 6781185810
0 0 0 0 3 21 94 409 1557 5575 18452 57985 172971 494841 1360994 3618306 9321948 23351239 56994271 135851564 316772980 723786642 1622719280 3574428126 7744222864
0 0 0 0 2 17 91 415 1631 5869 19609 61760 184763 528970 1456375 3873101 9982360 25008800 61050043 145527486 339359186 775414671 1738525374 3829570352 8297132508
˜ 3 (τ ) into irreducible representations χn of Table 56: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part I.
82
-D
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
1 0 0 0 1 0 1 1 2 2 5 10 17 30 64 122 287 608 1400 3111 7070 15666 34864 75899 164118
0 0 0 0 0 0 0 2 2 8 16 54 107 299 716 1817 4447 10973 26176 62208 144100 329562 738943 1632596 3545538
0 1 0 1 1 4 5 14 24 62 140 379 946 2574 6655 17366 43965 109822 267074 638341 1491474 3422339 7702776 17046286 37091514
0 0 1 0 1 1 4 8 19 40 124 334 928 2556 6947 18206 47011 117781 288683 691001 1619959 3719775 8384938 18562586 40418996
0 0 0 0 0 0 1 4 10 42 128 423 1242 3642 10035 27087 70199 177721 436768 1049704 2463620 5667740 12782104 28320987 61684063
0 0 0 1 1 2 5 13 30 79 213 585 1639 4502 12149 31983 82214 206063 504591 1207687 2829836 6498220 14643912 32418588 70582578
0 0 0 0 0 0 2 6 24 66 228 648 1952 5434 15072 39848 103514 259946 639120 1530976 3593518 8255052 18617246 41222644 89782368
0 0 0 0 0 0 0 7 20 66 223 693 2037 5889 16267 43501 112936 284997 700427 1681362 3946210 9073094 20462399 45326175 98720482
0 0 0 0 1 2 5 14 43 113 342 964 2782 7697 21026 55450 143220 359275 881263 2109987 4947675 11363145 25615443 56711601 123491729
0 0 0 0 1 2 5 14 43 113 342 964 2782 7697 21026 55450 143220 359275 881263 2109987 4947675 11363145 25615443 56711601 123491729
˜ 3 (τ ) into irreducible representations χn of Table 57: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part II.
83
-D
χ11
χ12
χ13
χ14
χ15
χ16
χ17
χ18
χ19
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 0 0 0 1 5 26 97 363 1172 3638 10611 29814 80286 209675 530506 1307537 3142048 7383374 16985215 38326695 84919350 185003006
0 0 0 0 0 0 1 5 26 97 363 1172 3638 10611 29814 80286 209675 530506 1307537 3142048 7383374 16985215 38326695 84919350 185003006
0 0 0 0 0 0 1 5 26 97 363 1172 3638 10611 29814 80286 209675 530506 1307537 3142048 7383374 16985215 38326695 84919350 185003006
0 0 0 0 0 0 0 11 30 136 436 1484 4370 12975 35856 97039 251918 638563 1570043 3775558 8863180 20394871 46000805 101933701 222026326
0 0 0 0 0 0 0 9 32 143 475 1607 4842 14297 39741 107493 279626 708547 1743695 4192520 9845654 22654755 51105811 113244297 246680793
0 0 0 0 0 2 8 30 100 350 1106 3458 10248 29442 81284 217714 564586 1425020 3502378 8406902 19730666 45366836 102311008 226631616 493602894
0 0 0 0 0 2 8 30 100 350 1106 3458 10248 29442 81284 217714 564586 1425020 3502378 8406902 19730666 45366836 102311008 226631616 493602894
0 0 0 0 0 4 10 40 127 416 1310 4034 11856 33922 93406 249613 646742 1631017 4006776 9614460 22560343 51864617 116954636 259049862 564184563
0 0 0 0 1 4 9 44 139 497 1596 5088 15102 43783 121084 325236 844320 2133713 5246147 12599677 29576649 68021751 153417371 339876627 740284029
˜ 3 (τ ) into irreducible representations χn of Table 58: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part III.
84
-D
χ20
χ21
χ22
χ23
χ24
χ25
χ26
χ27
χ28
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 0 1 2 9 32 142 477 1672 5228 15977 46068 128664 345074 899003 2270753 5591173 13425927 31535156 72522016 163610220 362451055 789548632
0 0 1 1 3 7 20 60 204 630 2051 6225 18566 52957 146652 391657 1017030 2564491 6305860 15130943 35518958 81656325 184167767 407928442 888501021
0 0 0 0 1 4 15 55 193 662 2165 6779 20286 58400 161869 433851 1126986 2845570 6998318 16801605 39444133 90701880 204577702 453185010 987096186
0 0 0 0 1 4 15 55 193 662 2165 6779 20286 58400 161869 433851 1126986 2845570 6998318 16801605 39444133 90701880 204577702 453185010 987096186
0 0 0 0 0 2 12 48 196 698 2352 7442 22548 65100 181184 486308 1265124 3196124 7865522 18887958 44354258 102004268 230098050 509745480 1110356978
0 0 0 0 1 4 11 59 219 801 2689 8627 26013 75588 210189 565195 1470206 3717138 9147020 21972919 51598009 118679661 267714806 593117732 1291964951
0 0 0 0 0 3 13 63 243 909 3057 9834 29759 86454 240627 647221 1683993 4257972 10479406 25174005 59118234 135979088 306745221 679593708 1480349568
0 0 0 0 0 3 13 63 243 909 3057 9834 29759 86454 240627 647221 1683993 4257972 10479406 25174005 59118234 135979088 306745221 679593708 1480349568
0 0 0 0 0 3 13 63 243 909 3057 9834 29759 86454 240627 647221 1683993 4257972 10479406 25174005 59118234 135979088 306745221 679593708 1480349568
˜ 3 (τ ) into irreducible representations χn of Table 59: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part IV.
85
-D
χ29
χ30
χ31
χ32
χ33
χ34
χ35
χ36
χ37
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 1 2 8 24 95 322 1124 3633 11444 34164 98541 272918 732055 1901107 4801562 11807813 28351578 66557162 153056164 345212969 764740650 1665696713
0 0 0 1 2 8 24 95 322 1124 3633 11444 34164 98541 272918 732055 1901107 4801562 11807813 28351578 66557162 153056164 345212969 764740650 1665696713
0 0 0 0 0 3 13 69 267 1011 3411 11019 33372 97104 270372 727612 1893498 4788741 11786521 28316673 66500652 152965722 345069777 764516151 1665348005
0 0 0 0 0 3 13 69 267 1011 3411 11019 33372 97104 270372 727612 1893498 4788741 11786521 28316673 66500652 152965722 345069777 764516151 1665348005
0 0 0 0 0 3 13 69 267 1011 3411 11019 33372 97104 270372 727612 1893498 4788741 11786521 28316673 66500652 152965722 345069777 764516151 1665348005
0 0 0 0 1 4 17 77 306 1113 3763 12011 36393 105478 293697 789280 2053906 5191571 12777678 30690911 72075158 165771974 373954834 828475182 1804658930
0 0 0 1 2 8 35 118 424 1471 4830 15140 45506 131014 363656 975053 2534016 6399104 15741552 37794342 88736216 204055733 460266342 1019607264 2220884907
0 0 0 0 0 3 16 89 367 1390 4796 15497 47238 137526 383784 1033033 2690675 6805582 16756387 40258951 94560774 217516112 490719721 1087225658 2368380807
0 0 0 0 2 5 32 123 517 1822 6268 19864 60491 174934 487925 1310274 3411830 8621691 21225130 50975995 119725056 275355244 621183917 1376174113 2997763988
˜ 3 (τ ) into irreducible representations χn of Table 60: The table shows the decomposition of the Fourier coefficients multiplying q −D/24 in the function h g, 2 U6 (2) - part V.
86
-D
χ38
χ39
χ40
χ41
χ42
χ43
χ44
χ45
χ46
-9 15 39 63 87 111 135 159 183 207 231 255 279 303 327 351 375 399 423 447 471 495 519 543 567
0 0 0 0 1 6 28 130 503 1846 6220 19937 60355 175147 487555 1310841 3410897 8623112 21222871 50979357 119719865 275362888 621172387 1376190840 2997739281
0 0 0 0 1 7 32 145 566 2060 6951 22198 67201 194787 542191 1457089 3791267 9583033 23584843 56648802 133032283 305972205 690217270 1529134899 3330883649
0 0 0 0 1 7 32 145 566 2060 6951 22198 67201 194787 542191 1457089 3791267 9583033 23584843 56648802 133032283 305972205 690217270 1529134899 3330883649
0 0 0 1 1 9 38 175 662 2457 8205 26322 79550 230873 642249 1726958 4492367 11357471 27949916 67138492 157661257 362631248 818019198 1812299925 3947676846
0 0 0 0 2 8 45 179 715 2536 8608 27278 82728 239200 666188 1788772 4655102 11762638 28950941 69528362 163281960 375524639 847121958 1876700029 4087990850
0 0 0 0 2 8 45 179 715 2536 8608 27278 82728 239200 666188 1788772 4655102 11762638 28950941 69528362 163281960 375524639 847121958 1876700029 4087990850
0 0 0 0 2 11 51 225 891 3233 10951 34934 105878 306831 854421 2296034 5975070 15102657 37171558 89282411 209673562 482245140 1087869446 2410108430 5249918560
0 0 0 0 2 9 58 250 1012 3662 12474 39779 120825 350044 975422 2621064 6822630 17244669 42447872 101955129 239444613 550717850 1242355809 2752364303 5995502658
0 0 0 0 0 9 54 255 1044 3861 13215 42381 128899 374214 1043352 2805657 7304754 18468738 45465426 109216491 256509711 589999521 1330999539 2948823657 6423520020
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