A Theta Relation in Genus 4

April 1999

Eberhard Freitag Mathematisches Institut Im Neuenheimer Feld 288 D69120 Heidelberg

FREITAG@ MATHI.UNI-HEIDELBERG.DE

Manabu Oura**)

Graduate School of Mathematics Kyushu University Japan

OURA@ GAUSS.MATH.UNI-MANNHEIM.DE

Abstract

The 2^{g} theta constants of second kind of genus g generate a graded ring of dimension g(g + 1)/2. In the case g ≥ 3 there must exist algebraic relations. In genus g = 3 it is known that there is one defining relation. In this paper we give a relation in the case g = 4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as linear combination of code polynomials of the 9 self-dual doubly even codes of length 24.

Introduction

The theta constants of genus g of second kind are the 2^{g} functions

f_{a}(Z) := ∑_{n∈ℤ^{g}} exp 2πiZ[n + a/2].

They generate a ring of dimension g(g + 1)/2 + 1. In the cases g ≤ 2 they are algebraically independent, in the case g = 3 there is a defining relation in degree 16, which is related to the so-called Schottky relation [Ru3]. In this paper we describe a similar relation in genus 4. Its degree is 24. This relation has two remarkable properties. It is invariant under the full Siegel modular group and it is mapped to a non trivial relation under Siegels Φ-operator.

*) Classification: 11T71 94B05 11F46 11F27 **) The second author’s work was supported in part by a grant from the Japan Society for the Promotion of Science.

It is a natural question, whether our relation is a consequence of Riemanns theta relations. We could not decide this. One reason is that the Riemann theta relations are quartic relations in the theta constants of first kind

ϑ[m] = ∑_{n∈ℤ^{g}} exp πi(Z[n + a/2] + b′(n + a/2)) m = ( \begin{matrix}a \\ b \end{matrix} ) ∈ ℤ^{2g}.

The functions f_{a}f_{b} generate the same vetor space as the squares ϑ[m]^{2}. We learnt form B. van Geemen and R. Salvati-Manni how to obtain relations between the f_{a} from the Riemann relations using elimination tricks. But it seems very hard to obtain the distinguished relation which we describe below.

Our relation gives rise to a certain Siegel cusp form of weight 12 and genus 5 with respect to the full Siegel modular group.

We use the theory of codes. There are 9 isomorphy classes of binary selfdual doubly-even codes in 𝔽_{2}^{24}. They are associated to 9 of the 24 Niemeier lattices. To any binary code C and any genus g there is associated a certain code polynomial W_{g}, which is a polynomial in 2^{g} variables F_{a}, a ∈ 𝔽_{2}^{g}. If one replaces F_{a} by f_{a} one obtains a function which is nothing else but the usual theta function of the lattice L which is related to the code C. Recall that L is the inverse image of C ⊂ 𝔽_{2}^{n} in ℤ^{n} with the scalar product 2⟨x, y⟩ = ∑ x_{i}y_{i}. Our relation will be constructed as linear combination of the code polynomials of the nine mentioned codes.

With help of computers we determined completely the code polynomials of genus 4 of the nine “Niemeier codes”. The tables are printed at the end of this paper.

Section 1 of the paper contains the results of the paper. In section 2 some of the computational problems are explained. Tables can be found in section 3.

We would like to thank R. Salvati-Manni, B. van Geemen and B. Runge for helpful discussions. This work was done during the second author’s stay in Mannheim. He would like to thank Professor R. Weissauer for his hospitality.

1. Siegel modular forms of genus four and codes

Let 𝔽_{2} be the field of two elements and V = 𝔽_{2}^{g}. Sometimes we will identify the elements of 𝔽_{2} with the integers 0, 1. For a, b ∈ V let a · b denote the standard scalar product. Set

T_{g} := (\frac{1 + i}{2})^{g} ((−1)^{a·b})_{a,b∈V}

and for any symmetric g × g-matrix with integral coefficients

D_{S} := diag(i^{S[a]} with a ∈ V ),

where

S[a] := ∑_{ij} s_{ij}a_{i}a_{j} (∈ ℤ).

The group generated by the matrices T_{g} and D_{S} is

H_{g} :=< T_{g}, D_{S} > .

After a choice of an ordering of V this is a subgroup of GL(2^{g}, ℂ). It is known that H_{g} is a finite group. We introduce for every a ∈ V a variable F_{a} and consider the ring of invariants

R_{g} := ℂ[(F_{a})_{a∈V}]^{H_{g}}.

This ring is connected with the ring A(Γ_{g}) of Siegel modular forms. The link comes from the “theta constants of second kind”:

f_{a}(Z) := ∑_{n∈ℤ^{g}} exp 2πiZ[n + a/2].

Here Z variis in the Siegel half plane of genus g which consists of all symmetric g × g-matrices with positive definite imaginary part. From the classical theta transformation formula one knows that the evaluation F_{a} ↦ f_{a} defines a homomorphism

R_{g} ⟼ A_{g}.

Here

A_{g} = A(Γ_{g}) = ∑_{k=0}^{∞}[Γ_{g}, k],

where [Γ_{g}, k] denotes the vector space of Siegel modular forms of weight k (with respect to the full Siegel modular group Γ_{g} = Sp(g, ℤ)). We use the weight convention that a form of weight k transforms as

f((AZ + B)(CZ + D)^{−1}) = det(CZ + D)^{k}f(Z).

The weight of the f_{a} have to be counted as 1/2.

A variant of “Igusa’s fundamental lemma” states that the subring A_{g}^{(2)} ⊂ A of modular forms of even weight equals the normalization of the image of the ring R_{g}. From this point of view the determination of the ring of Siegel modular forms depends on the determination of the kernel of R_{g} → A_{g}. Along these lines the cases g ≤ 3 have been treated succesfully (reproducing and enlarging known results) [Ru1, Ru2].

This paper is an approach to attack the case g = 4. What we get is a simple element of the kernel. The second author determined the dimension formula for the ring R_{4} [Ou]. The dimension of the space of Siegel modular forms in genus four is known in some cases [PY]:

weight k  4 6 8 10 12 14 16
dim(R_{4})_{k}  1 1 2 3 7 7 19
dim[Γ_{4}, k]  1 1 2 ? 6 ? ?

This table shows that there must be a relation in weight 12, i.e. there must exist an invariant polynomial of degree 24 in the 16 variables F_{a} which is not identically zero but which vanishes if one replaces F_{a} ↦ f_{a}. We use codes to determine this polynomial. Recall that a (binary) code C is nothing else but a subvector space C ⊂ 𝔽_{2}^{n}. The code ploynomial of genus g associated to such a code is

P (C) = P^{(g)}(C) := ∑_{α_{1},...,α_{g}∈C} ∏_{a∈𝔽_{2}^{g}} F_{a}^{a(α_{1},...α_{g})},

where a(α_{1}, … α_{g}) denotes the number of all i ∈ {1 … n} such that a = (α_{1i} …, α_{gi}). The weight of an element α ∈ C is the number of digits 1 in α. A code is called doubly-even if all weights are divisible by four. A code is called self-dual if it agrees with its orthogonal complement with respect to a · b.

It is well-known [Du, Gl, Ru1], that a H_{g}-invariant polynomial P ∈ R_{g} is a linear combination of code polynomials of self-dual doubly-even codes if and only if its degree is divisible by 8. In our case (g = 4 and k = 12) we have to consider codes in 𝔽_{2}^{24}. It is known that upto isomorphism precisely 9 self-dual doubly-even codes exist. We denote representatives by C_{1}, …, C_{9}. The first seven denote the indecomposable ones. We refer to the list [PS]. The 9 types are characterized by their root subcodes which are direct products of the codes a_{n}, d_{n}, n ≥ 4, e_{7}, e_{8}. The correspondence is as follows:

\begin{matrix}C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} & C_{8} & C_{9} \\ d_{12}^{2} & d_{10}e_{7}^{2} & d_{8}^{3} & d_{6}^{4} & d_{24} & d_{4}^{6} & a_{1}^{24} & d_{16}e_{8} & e_{8}^{3} \end{matrix}

First we consider the 9 Siegel modular forms. We know that the dimension of the space [Γ_{4}, 12] is 6.

1.1 Proposition. The Siegel modular forms S_{i} = S(C_{i}) of genus 4 which are attached to the nine codes C_{i} generate the six dimensional space [Γ_{4}, 12]. A basis is given by S_{1}, S_{2}, S_{3}, S_{4}, S_{6}, S_{7}. The expressions of the three remaining terms with respect to this basis are

\begin{aligned}S_{5} & = 66S_{1} − 495S_{3} + 880S_{4} − 594S_{6} + 144S_{7} \\ 3S_{8} & = 17S_{1} + 20S_{2} − 145S_{3} + 208S_{4} − 125S_{6} + 28S_{7} \\ S_{9} & = 3S_{1} + 20S_{2} − 75S_{3} + 96S_{4} − 55S_{6} + 12S_{7}. \end{aligned}

Each of the codes C_{i} corresponds to one of the 24 Niemeier lattices and the modular form S_{i} is the theta series of this Niemeier lattice. The proof of 1.1 uses the knowledge of some Fourier coefficients. Recall that the theta series ϑ(L; Z) of genus g of a definite lattice L admits a Forier expansion whose Fourier coefficients are given by the number of isometric embeddings of an arbitrary lattice M of rank ≤ g into L. In case of the 24 Niemeier lattices L and the ireducible root lattices M of rank ≤ 12 these numbers can be found in [BFW]. For our purpose it is sufficient to take the six root lattices A_{0}, A_{1}, A_{2}, A_{3}, A_{4}, D_{4} (and the 9 Niemeier code lattices) to seperate the theta series. In [BFW] they carry the same name as the corresponding codes but using capital letters. For example the Neimeier lattice corresponding to the code e_{8} is the lattice E_{8}.

As we know from the results [Ou] the space generated by the code polynomials has dimension 7. To get a basis we computed the coefficients of the monomial

F_{0}^{9} ∏_{a≠0} F_{a}

in the code polynomial P_{i} = P (C_{i}). This monomial (to be precise an equivalent one) carries number 127 in our tables.

1.2 Proposition. The coefficients corresponding to the monomial F_{0}^{9} ∏_{a≠0} F_{a} of the nine code polynomials P_{i} = P (C_{i}) are

0, 70, 24, 72, 0, 148, 253, 80, 168

multiplied with 2^{10} · 3^{3} · 5 · 7.

From 1.1 and 1.2 we obtain

1.3 Theorem. The code polynomial

R := 3P_{1} + 20P_{2} − 75P_{3} + 96P_{4} − 55P_{6} + 12P_{7} − P_{9}

does not vanish but the corresponding Siegel modular form is zero.

We can consider the same linear combination of code polynomials in genus 5. Computing one further Fourier coefficient one sees that the corresponding Siegel modular form is not 0.

1.4 Theorem. The linear combination R considered in genus five defines a non-vanishing Siegel cusp form of weight 12 and genus 5 with respect to the full Siegel modular group.

For sake of completeness we give the relations for the 9 code polynomials in genus 4:

1.5 Proposition. A basis for the H_{4}-invariant code polynomials is given by P_{1}, P_{2}, P_{3}, P_{4}, P_{6}, P_{7}, P_{8}. The expressions for the two remaining are

\begin{aligned}P_{5} & = 66P_{1} − 495P_{3} + 880P_{4} − 594P_{6} + 144P_{7}, \\ P_{9} & = −14P_{1} + 70P_{3} − 112P_{4} + 70P_{6} − 16P_{7} + 3P_{8}. \end{aligned}

There is a homomorphism

Φ : R_{g} ⟶ R_{g−1}, F_{(a_{1},a_{2},a_{3},a_{4})} ⟼ \{ \begin{matrix}0 & if a_{4} = 0, \\ F_{(a_{1},a_{2},a_{3})} & elsewhere. \end{matrix}

This induces the Siegel Φ-operator on the level of modular forms. We apply this operator to the relation P (s. 1.3). We get a relation in genus three. In this case there is a defining relation of degree 16 [Ru1]. It is the code polynmial

P^{(3)}(e_{8}^{2}) − P^{(3)}(d_{16}).

The upper index 3 indicates that we consider code polynomials of genus three. Recall that we denote by e_{8} resp. d_{16} the unique irreducible self-dual doublyeven code in dimension 8 resp. 16. So R|Φ must be a product of this polynomial and an invariant polynomial of degree 8. There is only one upto a constant factor, namely P^{(3)}(e_{8}). We obtain

1.6 Proposition. The image of the relation R (s. 1.3) under the Φ-operator is

R|Φ = (P^{(3)}(e_{8}^{2}) − P^{(3)}(d_{16})) · P^{(3)}(e_{8}).

Only the constant has to be verified. To do this we use the coefficients of the monomial

F_{(0,0,0,0)}^{16} ∏_{a_{4}=0} F_{a}.

An equivalent one carries number 40 in our table. The coefficients of the nine C_{i} in our ordering are

0, 336, 0, 0, 0, 0, 0, 1344, 4032.

Hence the coefficient of R is 2688. This is also the coefficient of R|Φ with respect to the degree-3-monomial

F_{(0,0,0)}^{16} ∏_{a∈𝔽^{3}} F_{a}.

On the other side the coefficient of (P^{(3)}(e_{8}^{2})−P^{(3)}(d_{16}))·P^{(3)}(e_{8}) is also 2688. (s. [Ru3]).

§2. Admissible monomials 7

2. Admissible monomials

The number of monomials of degree 24 in 16 variables is 25 140 840 660. But not all of them can occur in a code polynomial of a self-dual doubly-even code.

2.1 Definition. A monomial (α_{a}) is called admissible if

∑_{a∈𝔽_{2}^{g}} α_{a} ≡ 0 mod 8 and ∑_{a∈𝔽_{2}^{g}} α_{a}S[a] ≡ 0 mod 4

for all integral symmetric g × g-matrices S.

Here we use the usual notation S[a] = ∑_{i,j} s_{ij}a_{i}a_{j}. It is of course sufficient to take for S a system of generators, for example all elements in the diagonal or above the diagonal are zero besides one which is 1. It is easy to prove (s. [Ru3]) that in the code polynomial of self-dual doubly-even codes only admissible monomials appear.

There is a certain group AGL(g) which acts on the set of admissible monomials, namely the semidirect product

AGL(g) = GL(g, 𝔽_{2}) · 𝔽_{2}^{g}.

The group AGL(g) acts on 𝔽_{2}^{g} as group of affine transformations,

x ⟼ Ux + a, (U, a) ∈ AGL(g).

The order of AGL(4) is

♯AGL(4) = 2^{4}(2^{4} − 1)(2^{4} − 2)(2^{4} − 2^{2})(2^{4} − 2^{3}) = 322 560 = 2^{10}3^{2}5 · 7.

The group AGL(g) is a subgroup of H_{g}. Hence our code polynomials are invariant under AGL(g). Our goal is to determine a complete system of representatives of admissible monomials of weight wt(a) = ∑_{a} α_{a} = 24.

Therefore it is sufficient to consider a system of representatives of all admissible monomials under AGL(g). We determined a system of representatives in the case g = 4.

2.2 Proposition. There exist 160 admissible monomials of weight 24, upto AGL(4).

An explicit list is given in table I below.

We give some explanations how this table has been computed. For the computations we used the computer algebra system MAGMA. The admissible monomials can be considered as elements of ℤ^{16}. The group AGL(4) acts on this domain permuting the components. A monomial (α_{a}) ∈ ℤ^{16} is called admissible mod N, if

∑_{a∈𝔽_{2}^{g}} α_{a} ≡ 0 mod (8, N) and ∑_{a∈𝔽_{2}^{g}} α_{a}S[a] ≡ 0 mod (4, N)

for all integral symmetric g × g-matrices S. This condition depends only on α mod N, hence this condition can be defined for elements α ∈ (ℤ/N ℤ)^{16}. Let β ∈ ℤ^{16} be the reduced representative mod N of α, i.e. 0 ≤ β_{i} ≤ N for all i. We use the notation

wt(α, N) := ∑_{i=1}^{16} β_{i}.

We consider the set

A(N) := { α ∈ (ℤ/N ℤ)^{16}; wt(α, N) ≤ 24 }.

The idea is to compute step by step a system of representatives of the sets A(2), A(4), A(8), A(16).

The first step is to determine the admissible monomials mod 2. The number of all possibilities 2^{16} = 65 536 is small enough to compute by a silly computer program. We have 3 representatives. In the next step one consideres, for each α ∈ (ℤ/2ℤ)^{16} of the three, the possible expansions β ∈ (ℤ/4ℤ)^{16} to admissibles mod 4 with the property wt(β, 4) ≤ 24. We take a system of representatives under the stabilizer of α in AGL(4). Using this method we found that A(4) has 22 representatives and similarely that A(8), A(16) have 109, 156 representatives respectively. In a last step one expands to A(24) which gives the final result.

The computation of the coefficients of the monomials is more involved. For each member C of the nine codes one needs a system of representatives (α_{1}, α_{2}, α_{3}, α_{4}) of code words with respect to the action of the automorphism group of C. One restricts to tuples such that the corresponding monomial belongs to the list of the 160 representatives. One starts with a list of representatives of code words α_{1}, computes the stabilizer of α_{1} and determines a system of representatives α_{2} with respect to this stabilizer, and so on. Again we used the MAGMA system which provides programs to handle codes and their automorphism groups. The result of the computations is in table 2.

The programs and tables can be found on the WEB-page

http://www.rzusers.uni-heidelberg.de/∼t91

We would like to mention one further result of the calculations:

2.3 Remark. The total number of monomials with non-zero coefficients in the code polynomial R (s. 1.3) is 2 004 480.

§3. Tables 9

3. Tables

The following table gives a system of representatives of admissible monomials. The tuple (α_{1} …, α_{16}) stands for the monomial ∏ F_{i}^{α_{i}}. We use the binary convention, i.e. the characteristic a = (a_{1}, a_{2}, a_{3}, a_{4}) is encoded as digit i = 8a_{1} + 4a_{2} + 2a_{3} + a_{4}. The number behind the monomial gives the length of the orbit of the monomial with respect to the group AGL(4).

Table I

1 (0,0,0,2,0,0,0,2,0,6,4,0,4,6,0,0) 40320  41 (0,4,0,2,0,8,0,2,0,2,0,0,0,2,0,4) 5040
2 (1,2,0,1,6,1,1,0,1,2,0,1,6,1,1,0) 2520  42 (1,4,0,1,0,1,1,0,3,0,0,3,0,3,3,4) 6720
3 (0,8,0,2,0,0,0,2,0,2,0,0,0,6,0,4) 20160  43 (1,1,3,3,1,1,1,1,1,1,1,1,1,1,3,3) 140
4 (0,0,0,2,8,0,0,10,0,2,0,0,0,2,0,0) 6720  44 (1,0,0,1,8,1,1,0,3,0,0,3,0,3,3,0) 3360
5 (0,0,0,0,2,0,2,0,2,14,0,0,2,0,0,2) 2688  45 (0,6,0,2,0,2,2,0,2,0,2,0,0,2,6,0) 20160
6 (1,0,0,1,4,1,1,4,3,0,0,3,0,3,3,0) 5040  46 (1,0,0,1,8,1,1,8,1,0,0,1,0,1,1,0) 840
7 (1,0,2,3,0,1,3,2,3,2,0,1,2,3,1,0) 840  47 (0,0,0,2,0,0,0,2,8,6,0,0,0,6,0,0) 10080
8 (0,2,0,2,4,2,2,4,2,0,2,0,0,2,2,0) 20160  48 (1,4,0,1,2,1,3,0,3,2,0,1,2,1,1,2) 26880
9 (0,0,0,2,0,0,0,10,0,6,0,0,0,6,0,0) 1680  49 (1,2,4,5,2,1,1,0,1,2,0,1,2,1,1,0) 13440
10 (0,0,0,0,2,0,2,4,2,2,4,0,2,4,0,2) 26880  50 (1,0,0,3,0,1,1,6,1,2,0,1,0,3,5,0) 40320
11 (1,0,0,3,0,1,1,2,1,2,0,1,0,11,1,0) 6720  51 (0,0,0,2,0,4,0,6,0,2,0,0,4,2,0,4) 80640
12 (1,0,0,1,0,1,1,0,11,0,0,3,0,3,3,0) 1680  52 (1,4,0,1,8,1,1,0,1,0,0,1,0,1,1,4) 5040
13 (3,0,2,1,0,1,1,0,1,2,0,3,2,3,3,2) 13440  53 (3,0,0,1,0,3,1,0,1,6,0,1,0,3,3,2) 13440
14 (1,2,0,1,2,1,1,4,1,2,4,1,2,1,1,0) 2520  54 (1,4,0,3,0,1,1,2,1,2,0,1,0,3,1,4) 10080
15 (3,0,0,1,0,7,1,0,1,2,0,1,0,3,3,2) 26880  55 (0,0,4,0,4,0,0,0,8,0,0,0,4,4,0,0) 13440
16 (0,0,2,2,0,0,2,2,2,2,0,4,2,2,4,0) 840  56 (1,0,8,9,0,1,1,0,1,0,0,1,0,1,1,0) 1920
17 (1,0,4,1,0,1,5,4,1,4,0,1,0,1,1,0) 13440  57 (0,0,8,0,12,0,0,4,0,0,0,0,0,0,0,0) 3360
18 (1,4,0,1,0,1,1,0,1,4,4,1,4,1,1,0) 1680  58 (1,0,0,7,0,1,1,2,1,2,0,1,0,7,1,0) 3360
19 (3,2,0,1,0,1,3,2,3,0,2,1,2,1,3,0) 5040  59 (1,0,2,1,2,3,1,2,1,2,0,1,2,1,3,2) 3360
20 (1,0,0,1,0,9,1,0,3,0,0,3,0,3,3,0) 1680  60 (0,0,0,0,0,0,0,0,8,0,0,0,0,16,0,0) 240
21 (5,0,0,3,0,1,1,2,5,2,0,1,0,3,1,0) 40320  61 (0,4,0,2,0,0,0,6,0,6,0,0,0,6,0,0) 6720
22 (1,0,0,1,6,1,3,0,3,2,0,1,2,1,1,2) 26880  62 (0,0,0,0,4,0,0,4,0,0,0,0,0,0,16,0) 1680
23 (0,0,0,2,0,0,0,2,0,2,0,0,0,18,0,0) 560  63 (0,0,0,2,0,0,0,2,0,2,0,0,0,14,0,4) 6720
24 (3,0,0,1,0,3,1,0,5,2,0,1,0,3,3,2) 26880  64 (0,0,2,2,0,0,6,2,6,2,0,0,2,2,0,0) 840
25 (2,2,2,0,2,2,0,2,2,2,0,2,0,4,0,2) 2688  65 (0,4,0,6,0,4,0,2,0,2,0,0,0,2,0,4) 6720
26 (1,0,0,3,4,1,1,6,1,2,0,1,0,3,1,0) 40320  66 (0,0,4,2,0,0,4,2,4,2,4,0,0,2,0,0) 13440
27 (0,2,0,2,0,2,2,4,2,0,2,0,4,2,2,0) 3360  67 (0,0,0,10,0,0,0,10,0,2,0,0,0,2,0,0) 840
28 (0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0) 16  68 (9,0,0,1,0,1,1,0,9,0,0,1,0,1,1,0) 840
29 (0,0,2,2,8,0,2,2,2,2,0,0,2,2,0,0) 240  69 (0,0,0,2,0,0,4,6,0,2,0,0,4,6,0,0) 5040
30 (1,0,0,3,0,1,1,2,9,2,0,1,0,3,1,0) 20160  70 (4,0,4,0,2,0,2,0,2,2,0,4,2,0,0,2) 26880
31 (1,0,0,5,4,1,1,0,9,0,0,1,0,1,1,0) 13440  71 (0,4,4,2,0,0,0,2,8,2,0,0,0,2,0,0) 40320
32 (1,0,0,5,0,5,1,0,3,0,0,3,0,3,3,0) 2520  72 (0,4,0,0,6,0,6,0,2,2,0,0,2,0,0,2) 40320
33 (5,6,0,1,2,1,1,0,1,2,0,1,2,1,1,0) 13440  73 (1,2,0,1,6,1,1,4,1,2,0,1,2,1,1,0) 6720
34 (0,4,0,2,0,0,0,2,8,2,0,0,0,2,0,4) 20160  74 (1,0,0,1,0,1,1,0,3,0,0,7,0,7,3,0) 2520
35 (1,2,0,1,10,1,1,0,1,2,0,1,2,1,1,0) 1680  75 (1,2,0,1,2,1,1,0,9,2,0,1,2,1,1,0) 3360
36 (0,2,0,2,0,2,2,0,2,0,2,4,0,6,2,0) 40320  76 (0,4,4,2,0,0,0,2,0,2,0,0,0,10,0,0) 26880
37 (0,0,0,2,4,0,0,2,4,2,0,0,4,2,4,0) 10080  77 (1,4,0,1,4,1,1,0,1,4,0,1,4,1,1,0) 420
38 (1,0,0,1,8,1,1,0,1,0,0,5,0,1,5,0) 6720  78 (1,0,0,1,0,1,1,0,9,0,0,5,0,1,5,0) 5040
39 (0,0,2,2,4,0,2,6,2,2,0,0,2,2,0,0) 1920  79 (0,2,2,2,0,2,2,2,2,2,0,2,2,2,0,2) 140
40 (1,0,0,1,0,1,1,0,1,0,0,1,0,1,17,0) 240  80 (0,0,0,2,0,0,0,2,8,2,0,0,0,6,0,4) 53760

10 A Theta Relation in Genus 4

81 (0,0,8,8,4,0,0,4,0,0,0,0,0,0,0,0) 10080  121 (1,0,0,3,0,1,1,6,1,6,0,1,0,3,1,0) 3360
82 (1,0,0,1,0,1,5,0,7,0,0,3,0,3,3,0) 6720  122 (0,2,4,6,0,2,2,0,2,0,2,0,0,2,2,0) 13440
83 (0,0,0,0,12,0,0,12,0,0,0,0,0,0,0,0) 120  123 (0,0,4,2,0,0,4,6,0,6,0,0,0,2,0,0) 10080
84 (1,0,0,1,2,1,3,0,7,2,0,1,2,1,1,2) 13440  124 (1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2) 30
85 (4,0,0,0,2,4,2,0,2,6,0,0,2,0,0,2) 40320  125 (8,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0) 560
86 (1,0,0,13,4,1,1,0,1,0,0,1,0,1,1,0) 1920  126 (4,4,0,0,0,0,4,4,8,0,0,0,0,0,0,0) 1680
87 (0,2,0,2,8,2,2,0,2,0,2,0,0,2,2,0) 6720  127 (1,1,1,1,1,1,1,1,9,1,1,1,1,1,1,1) 16
88 (4,0,0,0,0,4,0,0,0,4,0,0,0,4,4,4) 448  128 (0,0,4,0,4,0,0,0,0,0,8,0,4,4,0,0) 6720
89 (0,0,4,4,0,4,0,0,0,0,4,0,0,0,4,4) 6720  129 (0,0,0,0,2,0,2,0,10,2,0,4,2,0,0,2) 26880
90 (0,0,4,0,12,0,0,0,0,0,0,0,4,4,0,0) 6720  130 (1,0,0,3,0,1,1,2,1,2,4,1,0,3,1,4) 40320
91 (1,0,0,3,0,1,1,2,1,2,0,5,0,7,1,0) 40320  131 (0,2,0,2,0,2,2,0,10,0,2,0,0,2,2,0) 6720
92 (4,4,0,2,0,0,4,6,0,2,0,0,0,2,0,0) 8960  132 (0,0,0,0,2,0,2,0,2,6,0,8,2,0,0,2) 26880
93 (1,8,0,5,4,1,1,0,1,0,0,1,0,1,1,0) 13440  133 (0,4,4,0,4,0,0,4,0,0,0,0,4,0,0,4) 840
94 (3,2,2,3,0,3,3,0,3,0,0,3,0,1,1,0) 3360  134 (0,0,0,0,6,0,2,0,2,2,0,0,6,0,0,6) 8960
95 (1,4,0,1,0,1,1,0,9,0,0,1,0,1,1,4) 6720  135 (1,2,0,1,2,1,1,0,1,2,0,5,2,5,1,0) 6720
96 (1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,16) 240  136 (1,1,1,1,5,1,1,5,1,1,1,1,1,1,1,1) 120
97 (0,4,0,2,0,0,0,2,0,10,0,0,0,2,0,4) 10080  137 (0,0,0,0,4,0,0,20,0,0,0,0,0,0,0,0) 240
98 (0,0,0,2,0,0,0,2,8,2,0,0,0,2,8,0) 6720  138 (5,4,0,3,0,1,1,2,1,2,0,1,0,3,1,0) 80640
99 (0,0,0,2,0,0,0,2,0,2,0,0,0,6,0,12) 6720  139 (1,0,0,3,8,1,1,2,1,2,0,1,0,3,1,0) 20160
100 (1,0,0,3,0,1,5,2,5,2,0,1,0,3,1,0) 10080  140 (4,0,0,0,2,0,2,0,6,2,0,0,6,0,0,2) 26880
101 (0,4,4,2,8,0,0,2,0,2,0,0,0,2,0,0) 26880  141 (1,0,0,1,2,1,3,0,3,2,0,5,2,1,1,2) 40320
102 (0,0,0,0,2,0,2,0,10,6,0,0,2,0,0,2) 13440  142 (1,0,0,5,12,1,1,0,1,0,0,1,0,1,1,0) 1920
103 (1,0,0,1,0,1,1,0,1,0,0,13,0,1,5,0) 1680  143 (4,4,0,2,0,0,0,2,0,2,0,0,0,2,4,4) 5040
104 (1,0,0,3,0,1,1,2,5,2,0,1,4,3,1,0) 40320  144 (1,0,4,5,0,1,1,0,3,0,0,3,0,3,3,0) 13440
105 (0,0,0,0,2,8,2,0,2,2,0,4,2,0,0,2) 40320  145 (0,0,0,2,0,0,0,2,0,10,0,0,0,6,0,4) 20160
106 (4,4,0,0,0,0,12,4,0,0,0,0,0,0,0,0) 560  146 (5,0,0,5,0,5,5,0,1,0,0,1,0,1,1,0) 420
107 (3,0,0,3,0,3,3,0,3,0,0,3,0,3,3,0) 30  147 (1,2,0,1,2,1,1,0,1,2,8,1,2,1,1,0) 1680
108 (0,0,4,2,0,0,4,2,4,2,0,0,4,2,0,0) 420  148 (8,0,0,8,4,0,0,4,0,0,0,0,0,0,0,0) 840
109 (1,0,0,3,0,1,1,2,1,10,0,1,0,3,1,0) 6720  149 (5,4,0,1,0,1,5,4,1,0,0,1,0,1,1,0) 3360
110 (0,0,0,2,0,0,0,2,0,14,0,0,0,6,0,0) 1680  150 (0,0,0,6,0,0,0,6,0,6,0,0,0,6,0,0) 140
111 (2,2,2,0,2,2,0,2,6,2,0,2,0,0,0,2) 4480  151 (0,0,2,2,0,0,2,2,10,2,0,0,2,2,0,0) 240
112 (1,0,4,5,0,1,5,0,1,0,0,1,0,5,1,0) 13440  152 (0,0,0,2,0,0,8,2,8,2,0,0,0,2,0,0) 2520
113 (0,4,0,0,6,0,2,0,2,2,0,0,2,0,4,2) 80640  153 (0,12,4,2,0,0,0,2,0,2,0,0,0,2,0,0) 13440
114 (0,4,0,2,0,0,0,2,0,6,4,0,4,2,0,0) 26880  154 (1,0,0,5,0,5,5,0,1,0,0,1,0,5,1,0) 1680
115 (4,0,0,2,0,0,0,2,0,2,0,0,4,2,4,4) 40320  155 (1,12,0,1,0,1,1,0,1,0,0,1,0,1,1,4) 1680
116 (0,12,0,2,0,0,0,2,0,2,0,0,0,2,0,4) 5040  156 (5,0,0,1,0,1,1,0,1,0,0,1,4,1,5,4) 20160
117 (1,0,0,7,0,1,1,6,1,2,0,1,0,3,1,0) 13440  157 (0,0,0,2,0,0,0,2,0,2,16,0,0,2,0,0) 1680
118 (1,0,4,1,0,1,1,0,7,0,0,3,0,3,3,0) 13440  158 (0,0,0,0,2,0,2,0,2,2,0,12,2,0,0,2) 4480
119 (0,6,0,2,0,2,2,0,2,0,6,0,0,2,2,0) 3360  159 (1,0,4,7,0,1,1,2,1,2,0,1,0,3,1,0) 40320
120 (3,0,0,1,4,3,1,0,1,2,0,1,0,3,3,2) 40320  160 (1,2,0,5,2,1,1,0,1,2,0,5,2,1,1,0) 5040

§3. Tables 11

The following table gives the coefficients of the 9 Niemeier codes. The number in the first column refers to the 160 monomials as in table I. The second line is the greatest common divisor of the coefficients of the 9 Niemeier codes. The last 9 columns give the coefficients divided by the greatest common divisor. For example the coefficient of the 127th monomial (1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1) (i.e. F_{(1,0,0,0)}^{8} ∏ F_{a}) with respect to the code C_{7} (this is the so-called Golay code) is given by 967 680 · 253.

Table II

MON GCD C_{1} C_{2} C_{3} C_{4} C_{5}  C_{6} C_{7} C_{8}  C_{9}^{1  96 3720 1715 882 297 34650 60  0  8526 6174}^{2  9216 12300 7301 4440 2064 73920 600 0  23030 18522}^{3  72 27660 24143 18996 13335 23100 6960 0  33236 35868}^{4  24 1560 1057 498 171 6930 0  0  3430 4158}^{5  120 12  1  0  0  792  0  0  56  0}^{6  768 186000 133525 100800 70836 1478400 39120 0  354760 321048}^{7  27648 232000 172725 117600 71028 739200 31760 0  378280 386904}^{8  2304 78375 78400 88515 99864 51975 115005 132825 75215 64827}^{9  96 2652 2856 3047 3225 1155 3390 3542 2205 2205}^{10 2160 8220 11515 13296 17010 27720 21856 28336 9800 16464}^{11 192 1920 1561 576 180 0  0  0  4592 7056}^{12 768 1080 1169 1008 810 0  480 0  1148 1764}^{13 27648 51200 55125 60000 62292 0  63000 61985 33320 24696}^{14 9216 410100 395675 401760 412938 1108800 430140 451605 473830 438354}^{15 2304 6000 5355 5136 4140 0  2520 0  3920 0}^{16 1152 339150 188650 182790 184356 2321550 238290 345345 558110 228438}^{17 288 120000 134995 117504 88848 0  47040 0  43120 16464}^{18 4608 7800 7350 16920 22608 0  26700 26565 4655 1029}^{19 9216 206400 184975 177600 180732 739200 192960 212520 288120 271656}^{20 768 6600 7805 9760 11898 0  14520 17710 3500 1764}^{21 384 88320 79135 70848 52668 0  28800 0  69776 49392}^{22 2304 27600 39445 54288 69480 0  86520 106260 3920 0}^{23 24 10  6  3  1  55  0  0  21  21}^{24 2304 58800 54145 41232 28908 0  15000 0  74480 98784}^{25 25920 44940 50225 54048 56022 9240 56672 56672 25480 16464}^{26 384 156000 132055 118368 87444 0  48240 0  121520 49392}^{27 1152 687750 513275 373230 240642 2321550 117990 0  1079470 1043406}^{28 1  1  1  1  1  1  1  1  1  1}^{29 4032 1650 280 702 486 4950 1230 3795 1550 1638}^{30 192 14400 10045 7680 4212 0  1440 0  14000 7056}^{31 48 9600 5929 2304 576 0  0  0  24304 35280}^{32 1536 159720 160475 160416 160314 147840 159960 159390 158564 160524}^{33 3456 27840 28175 22752 17352 0  9840 0  33712 49392}^{34 96 6000 2450 2187 1143 51975 450 0  10465 3969}^{35 4608 540 721 504 324 0  120 0  938  2646}^{36 576 51000 51695 43164 32805 69300 18180 0  51940 61740}^{37 192 36150 20825 23625 27846 155925 44640 79695 49735 27783}^{38 96 2400 3479 1728 792 0  0  0  6328 17640}^{39 4032 6600 8155 5796 4599 9900 2700 0  11900 22932} 40 336 0  1  0  0  0  0  0  4  12

12 A Theta Relation in Genus 4

\begin{matrix}MON & GCD & C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} & C_{8} & C_{9} \\ 41 & 288 & 24000 & 29470 & 35145 & 41229 & 17325 & 47310 & 53130 & 16555 & 17787 \\ 42 & 768 & 61200 & 45325 & 45600 & 53892 & 0 & 74400 & 106260 & 72520 & 24696 \\ 43 & 221184 & 160800 & 144550 & 133800 & 125232 & 369600 & 119180 & 115115 & 199920 & 189336 \\ 44 & 384 & 3600 & 2695 & 1728 & 1836 & 0 & 1680 & 0 & 7000 & 3528 \\ 45 & 1152 & 5610 & 3283 & 2190 & 1047 & 11550 & 330 & 0 & 8526 & 6174 \\ 46 & 384 & 3120 & 500 & 288 & 0 & 63360 & 0 & 0 & 8533 & 2079 \\ 47 & 48 & 2460 & 2107 & 1710 & 1317 & 11550 & 780 & 0 & 4018 & 4410 \\ 48 & 2304 & 973200 & 1037575 & 1074960 & 1110312 & 0 & 1140600 & 1168860 & 827120 & 889056 \\ 49 & 1152 & 472800 & 496615 & 570816 & 641592 & 0 & 733440 & 850080 & 309680 & 148176 \\ 50 & 768 & 10320 & 12299 & 10224 & 8622 & 0 & 5400 & 0 & 13720 & 24696 \\ 51 & 48 & 52500 & 33565 & 21492 & 10575 & 207900 & 3240 & 0 & 83300 & 61740 \\ 52 & 768 & 9300 & 6475 & 4536 & 2340 & 0 & 720 & 0 & 8855 & 2205 \\ 53 & 4608 & 3000 & 2205 & 2792 & 3420 & 0 & 5220 & 8855 & 1960 & 0 \\ 54 & 384 & 3504000 & 2750125 & 1992960 & 1289196 & 8870400 & 623520 & 0 & 5225360 & 5383728 \\ 55 & 72 & 825 & 245 & 117 & 27 & 17325 & 5 & 0 & 2485 & 1421 \\ 56 & 3696 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 72 & 216 \\ 57 & 12 & 585 & 476 & 363 & 246 & 1155 & 125 & 0 & 791 & 791 \\ 58 & 384 & 16320 & 7831 & 2880 & 684 & 253440 & 0 & 0 & 50960 & 49392 \\ 59 & 13824 & 1327200 & 1275225 & 1205760 & 1137588 & 1478400 & 1065920 & 991760 & 1446480 & 1498224 \\ 60 & 3 & 213 & 221 & 229 & 237 & 165 & 245 & 253 & 197 & 197 \\ 61 & 144 & 4580 & 3675 & 2988 & 2075 & 7700 & 1080 & 0 & 5292 & 4116 \\ 62 & 6 & 105 & 68 & 39 & 18 & 495 & 5 & 0 & 203 & 203 \\ 63 & 24 & 180 & 97 & 36 & 9 & 1980 & 0 & 0 & 532 & 588 \\ 64 & 3456 & 36590 & 36505 & 37722 & 39309 & 53130 & 41510 & 44275 & 37338 & 34986 \\ 65 & 144 & 327900 & 354025 & 377820 & 397017 & 69300 & 412680 & 425040 & 257740 & 251076 \\ 66 & 288 & 49500 & 40425 & 36990 & 27378 & 103950 & 15060 & 0 & 47530 & 26754 \\ 67 & 528 & 130 & 102 & 75 & 49 & 319 & 24 & 0 & 189 & 189 \\ 68 & 96 & 4160 & 2309 & 1152 & 368 & 28160 & 0 & 0 & 9044 & 8316 \\ 69 & 96 & 125040 & 96775 & 71082 & 46209 & 330330 & 22560 & 0 & 183358 & 179046 \\ 70 & 432 & 235500 & 197225 & 157680 & 108774 & 138600 & 55680 & 0 & 252840 & 214032 \\ 71 & 144 & 6600 & 3955 & 1818 & 648 & 34650 & 120 & 0 & 15750 & 17934 \\ 72 & 48 & 13980 & 12691 & 7632 & 4374 & 138600 & 1440 & 0 & 33320 & 49392 \\ 73 & 13824 & 7200 & 10780 & 13176 & 17226 & 0 & 21780 & 26565 & 6860 & 12348 \\ 74 & 1536 & 9240 & 7379 & 5616 & 3786 & 21120 & 1920 & 0 & 12740 & 12348 \\ 75 & 1152 & 3600 & 6125 & 5760 & 6048 & 0 & 4560 & 0 & 5320 & 10584 \\ 76 & 48 & 900 & 455 & 126 & 27 & 20790 & 0 & 0 & 3738 & 4410 \\ 77 & 4608 & 183000 & 110250 & 58320 & 24552 & 1108800 & 6240 & 0 & 404495 & 420861 \\ 78 & 192 & 9840 & 8827 & 7392 & 5364 & 0 & 2880 & 0 & 9436 & 8820 \\ 79 & 41472 & 301800 & 249900 & 224430 & 208632 & 1120350 & 203080 & 203665 & 457170 & 446586 \\ 80 & 24 & 4260 & 2107 & 828 & 225 & 69300 & 0 & 0 & 11900 & 8820 \\ 81 & 12 & 2835 & 1568 & 981 & 450 & 17325 & 135 & 0 & 5201 & 3633 \\ 82 & 1536 & 15840 & 18137 & 20184 & 22302 & 0 & 24420 & 26565 & 11564 & 12348 \\ 83 & 4 & 689 & 680 & 671 & 662 & 743 & 653 & 644 & 707 & 707 \\ 84 & 4608 & 4200 & 6335 & 9576 & 14130 & 0 & 19860 & 26565 & 1960 & 0 \\ 85 & 48 & 311100 & 229565 & 131328 & 62676 & 415800 & 17280 & 0 & 554680 & 691488 \\ 86 & 336 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 112 & 336 \\ 87 & 576 & 10950 & 5705 & 8298 & 6984 & 34650 & 4590 & 0 & 11410 & 882 \\ 88 & 720 & 945 & 0 & 379 & 108 & 10395 & 361 & 1771 & 735 & 343 \\ 89 & 48 & 30375 & 12250 & 9405 & 4212 & 155925 & 1515 & 0 & 54145 & 29841 \\ 90 & 24 & 345 & 175 & 81 & 27 & 3465 & 5 & 0 & 889 & 833 \\ 91 & 384 & 32160 & 23653 & 14688 & 7308 & 0 & 2160 & 0 & 46256 & 49392 \\ 92 & 2160 & 1620 & 1421 & 596 & 219 & 4620 & 0 & 0 & 4116 & 6860 \\ 93 & 48 & 59520 & 33019 & 20736 & 7200 & 0 & 0 & 0 & 74480 & 35280 \\ 94 & 4608 & 140000 & 99225 & 74880 & 47732 & 492800 & 23040 & 0 & 199920 & 148176 \\ 95 & 480 & 1440 & 2107 & 960 & 432 & 0 & 0 & 0 & 1624 & 3528 \\ 96 & 1344 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 97 & 48 & 11820 & 10045 & 8886 & 6687 & 20790 & 3780 & 0 & 11914 & 7938 \\ 98 & 48 & 1410 & 245 & 153 & 0 & 17325 & 0 & 0 & 3703 & 2079 \\ 99 & 24 & 780 & 371 & 228 & 75 & 4620 & 0 & 0 & 1316 & 588 \\ 100 & 384 & 307200 & 196735 & 108288 & 48636 & 1774080 & 12960 & 0 & 634256 & 642096 \end{matrix}

§3. Tables 13

\begin{matrix}MON & GCD & C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} & C_{8} & C_{9} \\ 101 & 480 & 2220 & 917 & 675 & 225 & 10395 & 0 & 0 & 3157 & 441 \\ 102 & 72 & 620 & 105 & 64 & 0 & 9240 & 0 & 0 & 1400 & 0 \\ 103 & 576 & 80 & 63 & 32 & 12 & 0 & 0 & 0 & 140 & 196 \\ 104 & 768 & 276000 & 238385 & 188352 & 130446 & 0 & 66960 & 0 & 307720 & 321048 \\ 105 & 72 & 32700 & 26635 & 17424 & 9450 & 138600 & 2880 & 0 & 43400 & 35280 \\ 106 & 24 & 5265 & 5831 & 6473 & 7191 & 3465 & 7985 & 8855 & 4361 & 4361 \\ 107 & 43008 & 83600 & 77175 & 71280 & 65204 & 123200 & 59160 & 53130 & 95340 & 93492 \\ 108 & 576 & 1745850 & 1669675 & 1592955 & 1520154 & 2269575 & 1449660 & 1381380 & 1915165 & 1927317 \\ 109 & 192 & 3840 & 3115 & 2112 & 900 & 0 & 0 & 0 & 4592 & 7056 \\ 110 & 288 & 40 & 34 & 27 & 19 & 55 & 10 & 0 & 49 & 49 \\ 111 & 1728 & 16500 & 25725 & 39408 & 53076 & 46200 & 64720 & 70840 & 17640 & 32928 \\ 112 & 288 & 14400 & 11025 & 6912 & 3456 & 0 & 960 & 0 & 18032 & 16464 \\ 113 & 48 & 102300 & 84035 & 79488 & 64728 & 415800 & 40320 & 0 & 131320 & 98784 \\ 114 & 144 & 7500 & 4655 & 3660 & 3357 & 69300 & 2640 & 0 & 14700 & 4116 \\ 115 & 192 & 68250 & 67375 & 52245 & 40302 & 155925 & 23490 & 0 & 110495 & 163611 \\ 116 & 96 & 1380 & 938 & 597 & 297 & 3465 & 90 & 0 & 2135 & 1911 \\ 117 & 768 & 8160 & 6629 & 3168 & 1170 & 0 & 0 & 0 & 16072 & 24696 \\ 118 & 384 & 14400 & 13685 & 7776 & 4104 & 0 & 1200 & 0 & 27440 & 49392 \\ 119 & 3456 & 4150 & 1960 & 882 & 279 & 53130 & 50 & 0 & 11466 & 10290 \\ 120 & 4608 & 41400 & 47775 & 52200 & 59184 & 0 & 68100 & 79695 & 37240 & 49392 \\ 121 & 384 & 66240 & 27685 & 16704 & 5100 & 591360 & 0 & 0 & 126224 & 49392 \\ 122 & 576 & 18600 & 16415 & 16524 & 23301 & 69300 & 35580 & 53130 & 28420 & 12348 \\ 123 & 96 & 27360 & 29743 & 32826 & 37809 & 34650 & 44520 & 53130 & 28126 & 30870 \\ 124 & 2322432 & 93000 & 80850 & 71880 & 62292 & 184800 & 53140 & 44275 & 112770 & 102606 \\ 125 & 18 & 845 & 833 & 869 & 953 & 1925 & 1085 & 1265 & 1013 & 1013 \\ 126 & 24 & 15075 & 10045 & 10455 & 12177 & 51975 & 17355 & 26565 & 20335 & 7791 \\ 127 & 967680 & 0 & 70 & 24 & 72 & 0 & 148 & 253 & 80 & 168 \\ 128 & 120 & 6975 & 6097 & 4707 & 3321 & 10395 & 1755 & 0 & 9331 & 10731 \\ 129 & 24 & 9300 & 3745 & 1872 & 450 & 83160 & 0 & 0 & 18872 & 7056 \\ 130 & 384 & 566400 & 750925 & 941760 & 1095084 & 0 & 1209600 & 1275120 & 145040 & 49392 \\ 131 & 1728 & 470 & 210 & 126 & 45 & 2310 & 10 & 0 & 798 & 294 \\ 132 & 24 & 22020 & 14147 & 4896 & 1260 & 138600 & 0 & 0 & 68488 & 98784 \\ 133 & 144 & 175325 & 159250 & 156335 & 161348 & 544775 & 174545 & 194810 & 234955 & 223979 \\ 134 & 144 & 1060 & 245 & 288 & 100 & 15400 & 0 & 0 & 1960 & 0 \\ 135 & 2304 & 30480 & 42287 & 48672 & 60840 & 0 & 78720 & 106260 & 32536 & 74088 \\ 136 & 387072 & 6720 & 5082 & 5448 & 5928 & 21120 & 7100 & 8855 & 9072 & 5880 \\ 137 & 6 & 5 & 4 & 3 & 2 & 11 & 1 & 0 & 7 & 7 \\ 138 & 768 & 45600 & 61985 & 80352 & 103662 & 0 & 130320 & 159390 & 25480 & 24696 \\ 139 & 192 & 33600 & 39655 & 41472 & 37332 & 0 & 24480 & 0 & 14000 & 7056 \\ 140 & 48 & 35580 & 21805 & 11664 & 4050 & 138600 & 0 & 0 & 61544 & 49392 \\ 141 & 4608 & 61800 & 79625 & 93672 & 108414 & 0 & 121620 & 132825 & 37240 & 49392 \\ 142 & 336 & 0 & 51 & 0 & 0 & 0 & 0 & 0 & 112 & 336 \\ 143 & 192 & 282750 & 145775 & 86625 & 37386 & 2373525 & 10620 & 0 & 578935 & 398223 \\ 144 & 384 & 50400 & 45815 & 38592 & 28944 & 0 & 16320 & 0 & 50960 & 49392 \\ 145 & 72 & 1460 & 1029 & 596 & 285 & 4620 & 80 & 0 & 2660 & 2940 \\ 146 & 1152 & 64800 & 49735 & 35136 & 22104 & 177408 & 10368 & 0 & 100156 & 102900 \\ 147 & 4608 & 3300 & 3815 & 6336 & 9702 & 0 & 15900 & 26565 & 1330 & 2646 \\ 148 & 36 & 34385 & 34544 & 34727 & 34934 & 33935 & 35165 & 35420 & 34139 & 34139 \\ 149 & 576 & 86400 & 49735 & 23424 & 7368 & 591360 & 0 & 0 & 200312 & 205800 \\ 150 & 192 & 25922 & 24990 & 24057 & 23123 & 31493 & 22188 & 21252 & 27783 & 27783 \\ 151 & 12096 & 370 & 469 & 570 & 741 & 330 & 970 & 1265 & 386 & 546 \\ 152 & 144 & 8550 & 5329 & 2979 & 1332 & 48015 & 360 & 0 & 17437 & 17157 \\ 153 & 48 & 240 & 35 & 18 & 0 & 6930 & 0 & 0 & 798 & 294 \\ 154 & 1152 & 12192 & 14455 & 16992 & 18936 & 0 & 20400 & 21252 & 6076 & 4116 \\ 155 & 768 & 180 & 35 & 24 & 0 & 0 & 0 & 0 & 329 & 147 \\ 156 & 576 & 13440 & 10535 & 10944 & 8712 & 0 & 5280 & 0 & 12152 & 8232 \\ 157 & 24 & 30 & 5 & 3 & 0 & 495 & 0 & 0 & 77 & 21 \\ 158 & 216 & 140 & 63 & 16 & 2 & 3080 & 0 & 0 & 616 & 784 \\ 159 & 768 & 20400 & 23975 & 22896 & 18882 & 0 & 11160 & 0 & 13720 & 24696 \\ 160 & 2304 & 197520 & 139895 & 112896 & 77400 & 887040 & 41040 & 0 & 277144 & 172872 \end{matrix}

4. Literature

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