GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 1, 1998, 55-70

ON THE REPRESENTATION OF NUMBERS BY THE DIRECT SUMS OF SOME BINARY QUADRATIC FORMS

N. KACHAKHIDZE

Abstract. The systems of bases are constructed for the spaces of cusp forms S_{k}(Γ_{0}(3), χ) (k ≥6), S_{k}(Γ_{0}(7), χ) (k ≥3) and S_{k}(Γ_{0}(11), χ) (k ≥ 3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind x_{1}^{2} +x_{1}x_{2} +x_{2}^{2} (6 ≤ k ≤ 17), of the kind x_{1}^{2} +x_{1}x_{2} +2x_{2}^{2} (3 ≤ k ≤ 11) and of the kind x_{1}^{2} + x_{1}x_{2} + 3x_{1}^{2} (3 ≤ k ≤ 7).

Let F_{k} denote a direct sum of k binary quadratic forms F_{1} with the same negative discriminant −q (q is prime ≡ 3 (mod 4)). The question of the representation of natural numbers by the direct sum of some binary quadratic forms was for the first time considered by Petersson [1]. In particular, he constructed the basis for the space of cusp forms S_{k}(Γ_{0}(3), χ) for an arbitrary integer k ≥ 6 using the Dedekind η-function and obtained formulas for the number of representation r(n, F_{k}) of the positive integer n by F_{k} when F_{1} = x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} and 2 ≤ k ≤ 6. Lomadze [2, 3] obtained formulas for r(n, F_{k}) when F_{1} = x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} (2 ≤ k ≤ 17) and F_{1} = x_{1}^{2} + x_{1}x_{2} + 2x_{2}^{2} (2 ≤ k ≤ 11) and showed that they obey a fairly definite law. To this end, he constructed the bases of the spaces S_{k}(Γ_{0}(3), χ) and S_{k}(Γ(7), χ) apart for each k using a generalized multiple theta series. Merzlyakov [4] constructed the basis of the space S_{k}(Γ_{0}(3), χ) for an arbitrary integer k ≥ 6 and obtained formulas for r(n, F_{k}) when F_{1} = x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} (2 ≤ k ≤ 11). But these formulas do not obey any law.

In the present paper, using the generalized multiple theta series, the systems of bases for the space of cusp forms S_{k}(Γ_{0}(3), χ) with an integer k ≥ 6 and for the spaces S_{k}(Γ_{0}(7), χ) and S_{k}(Γ_{0}(11), χ) with an integer k ≥ 3 are constructed. After that, these bases are used to obtain the formulas for r(n, F_{k}) when F_{1} = x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} (6 ≤ k ≤ 17), F_{1} = x_{1}^{2} + x_{1}x_{2} + 2x_{2}^{2}

55 (3 ≤ k ≤ 11) and F_{1} = x_{1}^{2} + x_{1}x_{2} + 3x_{2}^{2} (3 ≤ k ≤ 7). These formulas obey a fairly definite law.

1991 Mathematics Subject Classification. 11F11, 11F13, 11F27, 11E25.

Key words and phrases. Representation of numbers by quadratic forms, direct sum of quadratic forms, space of cusp forms, generalized multiple theta-series.

1072-947X/98/0100-0055$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

1.1. Let

Q(X) = Q(x_{1}, …, x_{f}) = ∑_{1≤r≤s≤f} b_{rs}x_{r}x_{s}

be a positive quadratic form with f (f is even) variables and integral coefficients b_{rs}. Let D denote a determinant of the quadratic form

2Q(X) = ∑_{r,s=1}^{f} a_{rs}x_{r}x_{s} (a_{rr} = 2b_{rr}; a_{sr} = a_{rs} = b_{rs}, r < s).

A_{rs} is a cofactor of a_{rs} in the matrix A = (a_{ij}). One calls Δ = (−1)^{f/2}D the discriminant of Q(X). N = Δ/δ is the level of Q(X), where δ = g.c.d.(\frac{A_{rs}}{2} , A_{rr})_{(r,s=1,...,f)} · χ(d) is the character of Q(X), i.e., if Δ is a perfect square, then χ(d) = 1; if Δ is not a perfect square and 2 ∤ Δ, then χ(d) = (\frac{d}{|Δ|}) for d > 0, and χ(d) = (−1)^{f/2}χ(−d) for d < 0 ((\frac{Δ}{d}) is the Jacobi symbol). A positive quadratic form with f variables of level N with the character χ is called a quadratic form of type (\frac{f}{2} , N, χ). Further, q denotes an odd prime number, z = exp(2πiτ), z_{N} = exp(\frac{2πiτ}{N}), Im τ > 0.

Let Γ(1) denote a full modular group and Γ any subgroup of a finite index in Γ(1). In particular, if N ∈ ℕ, then

Γ_{0}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1) |c ≡ 0 (mod N)\} ,
Γ_{1}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N) |a ≡ d ≡ 1 (mod N)\} ,
Γ(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{1}(N) |b ≡ 0 (mod N)\} .

Let G_{k}(Γ, χ) and S_{k}(Γ, χ) denote the spaces of entire modular and cusp forms, respectively, of type (k, Γ, χ). If Γ = Γ(N) or Γ = Γ_{1}(N), then χ(d) = 1; hence we shall write G_{k}(Γ) instead of G_{k}(Γ, χ). As is known ([5], p. 109), if F (τ) ∈ G_{k}(Γ, χ), then in the neighborhood of the cusp ζ = i∞

F (τ) = ∑_{m=m_{0}≥0}^{∞} a_{m}z^{m}, a_{m_{0}} ≠ 0.

The order of an entire modular form F (τ) ≠ 0 of type (k, Γ, χ) at the cusp ζ = i∞ with respect to Γ is called the number ([5], p. 41)

ord (F (τ), i∞, Γ) = m_{0}.  (1.1)

ON THE REPRESENTATION OF NUMBERS 57

Let F (τ) be any function on ℍ = {τ ∈ ℂ| Im τ > 0} and k ∈ ℤ. Then, for any matrix L = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1), let

F (τ)|_{k}L = (cτ + d)^{−k}F (Lτ) (τ ∈ ℍ).

Let

θ(τ; Q(X), P_{ν}(X), h) = ∑_{n_{i}≡h_{i} (mod N)} P_{ν}(n_{1}, …, n_{f})z_{N}^{\frac{1}{N} Q(n_{1},...,n_{f})},

θ(τ; Q(X), P_{ν}(X)) = ∑_{X∈ℤ^{f}} P_{ν}(X)z^{Q(X)} = ∑_{n=1}^{∞} ( ∑_{Q(X)=n} P_{ν}(X))z^{n}.

Here Q(X) = \frac{1}{2} ∑_{r,s=1}^{f} a_{rs}x_{r}x_{s} is a quadratic form of type (\frac{f}{2} , N, χ); P_{ν}(X) is a spherical function of order ν with respect to Q(X); n_{1}, …, n_{f} are integers; h = (h_{1}, …, h_{f}), where h_{i} are integers satisfying the conditions

∑_{s=1}^{f} a_{rs}h_{s} ≡ 0 (mod N) (r = 1, …, f).

Lemma 1 ([5], p. 217). If θ(τ; Q(X), P_{ν}(X), h) is not identically equal to zero, then

θ(τ; Q(X), P_{ν}(X), h) ∈ G_{ν+f/2}(Γ(N)).

Lemma 2 ([5], p. 218). If L = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N), then

θ(τ; Q(X), P_{ν}(X), h)|_{ν+f/2}L =

= χ(d) exp (\frac{2πiab}{N^{2}} Q(h))θ(τ; Q(X), P_{ν}(X), ah).

Note ([5], p. 210; [6], p. 856) that

θ(τ; Q(X), P_{ν}(X), h) = θ(τ; Q(X), P_{ν}(X), g) for h ≡ g (mod N), (1.2)
θ(τ; Q(X), P_{ν}(X), −h) = (−1)^{ν}θ(τ; Q(X), P_{ν}(X), h),  (1.3)
θ(τ; Q(X), P_{ν}(X), 0) = N^{ν}θ(τ; Q(X), P_{ν}(X)).  (1.4)

Lemma 3 ([6], p. 855).

θ(τ; Q(X), P_{ν}(X)) ∈ G_{ν+f/2}(Γ_{0}(N), χ)

and when ν > 0

θ(τ; Q(X), P_{ν}(X)) ∈ S_{ν+f/2}(Γ_{0}(N), χ).

Lemma 4 ([5], p. 114; [7], Ch. 3, §1). Let F (τ) ∈ G_{k}(Γ_{1}(N)) and

F (τ) = ∑_{m=0}^{∞} a_{m}z^{m}.

If all a_{m} = 0 when m ≤ \frac{kN^{2}}{24} ∏_{p|N}(1 − \frac{1}{p^{2}} ), then F (τ) is identically equal to

zero.

Lemma 5 ([6], p. 846). If Q_{1}(X) and Q_{2}(X) are quadratic forms of types (k_{1}, N, χ_{1}) and (k_{2}, N, χ_{2}), respectively, then Q_{1}(X) ⊕ Q_{2}(X) is a quadratic form of type (k_{1} + k_{2}, N, χ_{1}χ_{2}).

As is known ([6], pp. 874, 875, 817), if Q(X) is a quadratic form of type (k, q, 1), 2|k, k > 2, then

Δ = q^{2l}, 1 ≤ l ≤ k − 1,  (1.5)

and the Eisenstein series

E(τ, Q(X)) = 1 + ∑_{n=1}^{∞} (ασ_{k−1}(n)z^{n} + βσ_{k−1}(n)z^{qn})  (1.6)

corresponds to it, where

α = \frac{i^{k}}{ρ_{k}} · \frac{q^{k−l} − i^{k}}{q^{k} − 1} , β = \frac{1}{ρ_{k}} · \frac{q^{k} − i^{k}q^{k−l}}{q^{k} − 1} ,  (1.7)

σ_{k−1}(n) = ∑_{d|n} d^{k−1}, ρ_{k} = (−1)^{k/2} · \frac{(k−1)!}{(2π)^{k}} ζ(k) (ζ(k) is the zeta function of

Riemann).

If Q(X) is a quadratic form of type (k, q, χ), k ≥ 2, χ(−1) = (−1)^{k}, then ([6], pp. 877, 818)

Δ = (−1)^{\frac{q−1}{2}} q^{2l+1}, 0 ≤ l ≤ k − 1,  (1.8)

and the Eisenstein series

E(τ, Q(X)) = 1 + \frac{1}{A_{k}(q)} ∑_{n=1}^{∞} ((−1)^{k/2}q^{k−l−1} ∑_{δd=n} χ(δ)d^{k−1} +

+ ∑_{d|n} χ(d)d^{k−1})z^{n}  (1.9)

corresponds to it, where

A_{k}(q) = (−1)^{k/2} · \frac{q^{k− \frac{1}{2}}}{(2π)^{k}} (k − 1)! ∑_{n=1}^{∞} χ(n)n^{−k}.

To any positive quadratic form Q(X), as is well-known, corresponds the theta series

θ(τ, Q(X)) = 1 + ∑_{n=1}^{∞} r(n, Q(X))z^{n} = ∑_{X∈ℤ^{f}} z^{Q(X)},  (1.10)

where r(n, Q(X)) denotes the number of representations of a positive integer n by the form Q(X).

Lemma 6 ([6], pp. 874, 875). If Q(X) is a quadratic form of type (k, q, 1) or of type (k, q, χ), then if k > 2,

θ(τ, Q(X)) − E(τ, Q(X))

is the cusp form of type (k, Γ_{0}(q), 1) or of type (k, Γ_{0}(q), χ), respectively.

Lemma 7 ([8], p.76). Let η(τ) denote the η-function of Dedekind. Then if q > 3

f_{q}(τ) = \frac{η^{q}(qτ)}{η(τ)} = z^{\frac{q^{2}−1}{24}} ∏_{n=1}^{∞} \frac{(1 − z^{qn})^{q}}{1 − z^{n}} ∈ G_{\frac{q−1}{2}} (Γ_{0}(q), χ),

where χ = χ(d) = (\frac{d}{q}).

1.2. Lemma 8. Let

Q(q)_{\frac{q−1}{2}} = Q(q)_{\frac{q−1}{2}} (X) = q ∑_{1≤i≤j≤q−2} x_{i}x_{j} + q ∑_{1≤i≤q−2} x_{i}x_{q−1} + \frac{q − 1}{2} x_{q−1}^{2},

h^{(q)} = (1, 2, …, \frac{q − 1}{2}, \frac{q + 3}{2}, \frac{q + 5}{2}, …, q − 1, \frac{q(1 − q)}{2}).

Then a) if q = 3

θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}) ∈ G_{3}(Γ_{0}(3), χ), ord(θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}), i∞, Γ_{0}(3)) = 1;

b) if q = 5, 7, 11 and 13

θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}) ∈ G_{\frac{q−1}{2}} (Γ_{0}(q), χ),

ord (θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}), i∞, Γ_{0}(q)) = \frac{q^{2} − 1}{24}.

Proof. In [9] it is shown that a quadratic form Q(q)_{\frac{q−1}{2}} is of type (\frac{q−1}{2}, q, χ). It is easy to verify that if Q(q)_{\frac{q−1}{2}} = \frac{1}{2} ∑_{r,s=1}^{q−1} a_{rs}x_{r}x_{s}, then

∑_{s=1}^{q−1} a_{rs}h^{(q)} ≡ 0 (mod q) and Q(q)_{\frac{q−1}{2}} (h^{(q)}) = q^{2} \frac{q^{2} − 1}{24}.

a) By Lemma 1, θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}) ∈ G_{3}(Γ(3)). Let L = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(3). Then (a, c) = 1, i.e., a ≡ 1 or −1 (mod 3) and, as ν = 0, from Lemma 2 and (1.2), (1.3) we get

θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)})|_{3}L = χ^{3}(d) exp (\frac{6πiab}{9} · 3)θ^{3}(τ; Q_{1}^{(3)}, 1, ah^{(3)}) =

= χ(d)θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}).

Hence,by definition, θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}) ∈ G_{3}(Γ_{0}(3), χ). Using the computer we get

θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}) = 27z + 81z^{2} + 243z^{3} + · · · .

Consequently, by (1.1),

ord(θ^{3}(τ; Q_{1}^{(3)}, 1, h^{(3)}), i∞, Γ_{0}(3)) = 1.

b) By Lemma 1, θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}) ∈ G_{\frac{q−1}{2}} (Γ(q)). If L ∈ Γ_{1}(q) ⊂ Γ_{0}(q),

then a ≡ d ≡ 1 (mod q), hence by Lemma 2 and (1.2)

θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)})|_{\frac{q−1}{2}} L =

= χ(d) exp (\frac{2πiab}{q^{2}} · Q(q)_{\frac{q−1}{2}} (h^{(q)}))θ(τ; Q(q)_{\frac{q−1}{2}} , 1, ah^{(q)}) =

= θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}).

That is, by definition, θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}) ∈ G_{\frac{q−1}{2}} (Γ_{1}(q)). By Lemma 7,

f_{q}(τ) = \frac{η^{q}(qτ)}{η(τ)} ∈ G_{\frac{q−1}{2}} (Γ_{0}(q), χ) ⊂ G_{\frac{q−1}{2}} (Γ_{1}(q)).

Hence

ψ_{q}(τ) = f_{q}(τ) − \frac{1}{q} θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}) = ∑_{m=0}^{∞} c_{m}^{(q)}z^{m} ∈ G_{\frac{q−1}{2}} (Γ_{1}(q)).

If q = 5, 7, 11 and 13, using the computer we get that all c_{m}^{(q)} = 0, when m ≤ \frac{q−1}{2} · \frac{q^{2}−1}{24}. Hence,by Lemma 4, ψ_{q}(τ) is identically equal to zero, i.e., θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}) = qf_{q}(τ) ∈ G_{\frac{q−1}{2}} (Γ_{0}(q), χ). Using the computer we get

θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}) = \begin{cases}5z + 5z^{2} + 10z^{3} + ⋯ & when q = 5, \\ 7z^{2} + 7z^{3} + ⋯ & when q = 7, \\ 11z^{5} + ⋯ & when q = 11, \\ 13z^{7} + ⋯ & when q = 13. \end{cases} \begin{matrix}(1.11) \\ (1.12) \\ (1.13) \\ (1.14) \end{matrix}

By (1.11)–(1.14) and (1.1)

ord(θ(τ; Q(q)_{\frac{q−1}{2}} , 1, h^{(q)}), i∞, Γ_{0}(q)) = \frac{q^{2} − 1}{24}.

Further Φ_{km} denotes the direct sum of m quadratic forms Φ_{k}.

Lemma 9. Let l_{1}, l_{2}, t_{1}, t_{2} ∈ ℕ ∪ {0}, X ∈ ℤ^{l_{1}}, Y ∈ ℤ^{l_{2}+t_{1}}, V ∈ ℤ^{t_{2}}, W = (X, Y, Z) ∈ ℤ^{l_{1}+l_{2}+t_{1}+t_{2}}, F_{1} and Φ_{1} are positive quadratic forms. If θ(τ; F_{l_{1}}, P_{1}(X)) ∈ S_{k_{1}}(Γ_{0}(q), χ), θ(τ; F_{l_{2}} ⊕ Φ_{t_{1}}, P_{2}(Y)) ∈ S_{k_{2}}(Γ_{0}(q), χ), θ(τ; Φ_{t_{2}}, P_{3}(V)) ∈ S_{k_{3}}(Γ_{0}(q), χ), then

θ(τ; F_{l_{1}}, P_{1}(X))θ(τ; F_{l_{2}} ⊕ Φ_{t_{1}}, P_{2}(Y))θ(τ; Φ_{t_{2}}, P_{3}(V)) =
= θ(τ; F_{l_{1}+l_{2}} ⊕ Φ_{t_{1}+t_{2}}, P (W)) ∈ S_{k_{1}+k_{2}+k_{3}}(Γ_{0}(q), χ),

where

P (W) = P_{1}(X)P_{2}(Y)P_{3}(V).

Proof directly follows from (1.4) and Lemma 3.

In the sequel let

σ_{t}^{∗}(n) = \begin{cases}σ_{t}(n) & if q ∤ n, \\ σ_{t}(n) + (−q)^{\frac{t+1}{2}} σ_{t}(\frac{n}{q}) & if q|n; \end{cases}

ρ_{t}^{∗}(n) = q^{\frac{t}{2}} ∑_{δd=n} (\frac{δ}{q} )d^{t} + (−1)^{\frac{t}{2}} ∑_{d|n} (\frac{d}{q})d^{t}.

If F_{k} is a quadratic form of type (k, q, χ), then by Lemma 3,

θ(τ, F_{k}) = θ(τ; F_{k}, 1) ∈ G_{k}(Γ_{0}(q), χ).  (1.15)

It follows from (1.10) and (1.1) that

ord(θ(τ, F_{k}), i∞, Γ_{0}(q)) = 0.  (1.16)

Using the formula ([6], pp. 824, 825)

A_{k}(q) = \frac{χ(2)(k − 1)!}{1 − χ(2)2^{k}} · q^{k−1} ∑_{1≤l≤ \frac{q−1}{2}} χ(l)\{2^{k}g_{k}(\frac{l}{q} ) − g_{k}(\frac{2l}{q} )\},

where

g_{k}(x) = \frac{x^{k}}{k!} − \frac{x^{k−1}}{2(k − 1)!} − 4 ∑_{1≤r≤ \frac{k}{2}} \frac{rρ_{2r}x^{k−2r}}{(k − 2r)!(2r)!}

are the Bernoulli polynomials defined on interval 0 < x < 1, we get

A_{3}(11) = −3, A_{5}(11) = \frac{3 · 25 · 17}{11}, A_{7}(11) = −9 · 17 · 71. (1.17)

2. Let

F_{1} = x_{1}^{2} + x_{1}x_{2} + x_{2}^{2}.

In [2] it is shown, that F_{k} is a quadratic form of type (k, 3, χ) (χ = χ(d) = (\frac{d}{3}) when 2 ∤ k and χ = 1 when 2|k) and

θ(τ, F_{1}) = 1 + 6z + 0 · z^{2} + · · · .  (2.1)

Applyng the Jacobi method it is easy to show that the linear transformation

\begin{matrix}x_{1} = y_{1} − \frac{1}{\sqrt{3}} y_{2}, x_{2} = \frac{2}{\sqrt{3}} y_{2}, \\ x_{3} = y_{3} − \frac{1}{\sqrt{3}} y_{4}, x_{4} = \frac{2}{\sqrt{3}} y_{4} \end{matrix}  (2.2)

reduces the quadratic form F_{2} into the normal form.

Lemma 10. a) φ = φ(x_{1}, …, x_{4}) = x_{2}^{4} + x_{4}^{4} − 6x_{2}^{2}x_{4}^{2} is a spherical function of order 4 with respect to F_{2};

b) θ(τ; F_{2}, φ) = 8z − 48z^{2} + 72z^{3} + 32z^{4} + ⋯ ∈ S_{6}(Γ_{0}(3), 1);

c) ord(θ(τ; F_{2}, φ), i∞, Γ_{0}(3)) = 1.

Proof. Using transformation (2.2) we get

φ(x_{1}, …, x_{4}) = \frac{16}{9} y_{2}^{4} + \frac{16}{9} y_{4}^{4} − \frac{32}{3} y_{2}^{2}y_{4}^{2} = φ(y_{1}, …, y_{4}),

∑_{i=1}^{4} \frac{∂^{2}φ}{∂y_{i}^{2}} = (\frac{16 · 4 · 3}{9} − \frac{32 · 2}{3})(y_{2}^{2} + y_{4}^{2}) = 0.

Consequently, by definition, φ is a spherical function of order 4 with respect to F_{2} and, by Lemma 3, θ(τ; F_{2}, φ) ∈ S_{6}(Γ_{0}(3), 1).

Using the computer we get

θ(τ; F_{2}, φ) = ∑_{n=1}^{∞} ( ∑_{F_{2}=n} x_{2}^{4} + x_{4}^{4} − 6x_{2}^{2}x_{4}^{2})z^{n} =

= 8z − 48z^{2} + 72z^{3} + 32z^{4} + · · · .  (2.3)

It follows from (2.3) and (1.1) that

ord(θ(τ; F_{2}, φ), i∞, Γ_{0}(3)) = 1.

Theorem 11. Let C = (c_{sr}) (s = 1, 2, 3; r = 1, …, [\frac{k}{3} ]−1) be the matrix whose elements are non-negative integers satisfying the conditions

\begin{matrix}6c_{1r} + 3c_{2r} + c_{3r} = k \\ c_{1r} + c_{2r} = r, c_{1r} > 0 \end{matrix} \} (r = 1, …, [\frac{k}{3} ] − 1). \begin{matrix}(2.4) \\ (2.5) \end{matrix}

Then for k ≥ 6 the system of functions

θ^{c_{1r}}(τ; F_{2}, φ)θ^{3c_{2r}}(τ; Q_{1}^{(3)}, 1, h^{(3)})θ^{c_{3r}}(τ, F_{1}) (r = 1, …, [\frac{k}{3} ] − 1), (2.6)

where θ(τ; Q_{1}^{(3)}, 1, h^{(3)}) is defined by Lemma 8, is the basis of the space S_{k}(Γ_{0}(3), χ).

Proof. By Lemmas 8–10 and (1.15), (2.4), functions (2.6) are cusp forms of type (k, Γ_{0}(3), χ). By Lemmas 10, 8 and (1.16), (2.5), we have

ord(θ^{c_{1r}}(τ; F_{2}, φ)θ^{3c_{2r}}(τ; Q_{1}^{3)}, 1, h^{(3)})θ^{c_{3r}}(τ, F_{1}), i∞, Γ_{0}(3)) = r

(r = 1, …, [\frac{k}{3} ] − 1).

Functions (2.6) are linearly independent because their orders at the cusp i∞ are different. Hence the theorem is proved as it is known ([6], pp. 815,

816) that dim S_{k}(Γ_{0}(3), χ) = [\frac{k}{3} ] − 1.

Corollary 12. The system of functions

θ(τ; F_{2}, φ)θ^{k−6}(τ, F_{1}) when 6 ≤ k ≤ 8 and

θ^{r}(τ; F_{2}, φ)θ^{k−6r}(τ, F_{1}) (r = 1, … [\frac{k}{6} ] ),

θ^{[ \frac{k}{3} ]−r}(τ; F_{2}, φ)θ^{3(2r−[ \frac{k}{3} ])}(τ; Q_{1}^{(3)}, 1, h^{(3)})θ^{3\{ \frac{k}{3} \}}(τ, F_{1})

(r = [\frac{k}{6} ] + 1, …, [\frac{k}{3} ] − 1), when k ≥ 9,

is the basis of the space S_{k}(Γ_{0}(3), χ).

By Lemma 6, θ(τ, F_{k}) − E(τ, F_{k}) ∈ S_{k}(Γ_{0}(3), χ). Hence,by Corollary 12, there are constants α_{r}^{(k)} such that

θ(τ, F_{k}) = E(τ, F_{k}) + ∑_{r=1}^{[ \frac{k}{6} ]} α_{r}^{(k)}θ^{r}(τ; F_{2}, φ)θ^{k−6r}(τ, F_{1}) +

+ ∑_{r=[ \frac{k}{6} ]+1}^{[ \frac{k}{3} ]−1} α_{r}^{(k)}θ^{)[ \frac{k}{3} ]−r}(τ; F_{2}, φ)θ^{3(2r−[ \frac{k}{3} ])}(τ; Q_{1}^{(3)}, 1, h^{(3)})θ^{3\{ \frac{k}{3} \}}(τ, F_{1}). (2.7)

Using the expansions of θ(τ, F_{k}), E(τ, F_{k}) [2] and the expansions of θ(τ; F_{2}, φ) and θ(τ; Q_{1}^{(3)}, 1, h^{(3)}) and equating, in both parts of (2.7), the coefficients of z, z^{2}, …, z^{l}, where l = dim S_{k}(Γ_{0}(3), χ), we get the systems of linear equations. By solving these systems we find the constants α_{r}^{(k)}. Equating, in both parts of (2.7), the coefficients of z^{n}, we obtain formulas for the arithmetical function r(n, F_{k}).

Let G_{i}(x_{1}, …, x_{4i}) = ∏_{t=1}^{i}(x_{4t−2}^{4} + x_{4t}^{4} − 6x_{4t−2}^{2}x_{4t}^{2}). Then formulas for r(n, F_{k}) when k = 6, 7, …, 17 have the following form:

r(n, F_{6}) = \frac{252}{13} σ_{5}^{∗}(n) + \frac{27}{13} ∑_{F_{2}=n} G_{1}(x_{1}, …, x_{4}),

r(n, F_{7}) = \frac{3}{7} ρ_{6}^{∗}(n) + \frac{27}{7} ∑_{F_{3}=n} G_{1}(x_{1}, …, x_{4}),

r(n, F_{8}) = \frac{240}{41} σ_{7}^{∗}(n) + \frac{216}{41} ∑_{F_{4}=n} G_{1}(x_{1}, …, x_{4}),

r(n, F_{9}) = \frac{27}{809} ρ_{8}^{∗}(n) + \frac{5184}{809} ∑_{F_{5}=n} G_{1}(x_{1}, …, x_{4}),

r(n, F_{10}) = \frac{12}{11} σ_{9}^{∗}(n) + \frac{81}{11} ∑_{F_{6}=n} G_{1}(x_{1}, …, x_{4}),

r(n, F_{11}) = \frac{3}{1847} ρ_{10}^{∗}(n) + \frac{15147}{1847} ∑_{F_{7}=n} G_{1}(x_{1}, …, x_{4}),

r(n, F_{12}) = \frac{6552}{73 · 691} σ_{11}^{∗}(n) + \frac{453168}{73 · 691} ∑_{F_{8}=n} G_{1}(x_{1}, …, x_{4}) −

− \frac{36450}{73 · 691} ∑_{F_{4}=n} G_{2}(x_{1}, …, x_{8}),

r(n, F_{13}) = \frac{3}{7 · 13^{2} · 47} ρ_{12}^{∗}(n) + \frac{541836}{7 · 13^{2} · 47} ∑_{F_{9}=n} G_{1}(x_{1}, …, x_{4}) −

− \frac{138510}{7 · 13^{2} · 47} ∑_{F_{5}=n} G_{2}(x_{1}, …, x_{8}),

r(n, F_{14}) = \frac{12}{1093} σ_{13}^{∗}(n) + \frac{11475}{1093} ∑_{F_{10}=n} G_{1}(x_{1}, …, x_{4}) −

− \frac{5832}{1093} ∑_{F_{6}=n} G_{2}(x_{1}, …, x_{8}),

r(n, F_{15}) = \frac{9}{419 · 16519} ρ_{14}^{∗}(n) + \frac{77863977}{419 · 16519} ∑_{F_{11}=n} G_{1}(x_{1}, …, x_{4}) −

− \frac{63425916}{419 · 16519} ∑_{F_{7}=n} G_{2}(x_{1}, …, x_{8}),

r(n, F_{16}) = \frac{480}{193 · 3617} σ_{15}^{∗}(n) + \frac{8376912}{193 · 3617} ∑_{F_{12}=n} G_{1}(x_{1}, …, x_{4}) −

− \frac{9669456}{193 · 3617} ∑_{F_{8}=n} G_{2}(x_{1}, …, x_{8}),

r(n, F_{17}) = \frac{3}{23 · 401 · 13687} ρ_{16}^{∗}(n)+ \frac{1609496352}{23 · 401 · 13687} ∑_{F_{13}=n}G_{1}(x_{1}, …, x_{4})−

− \frac{2434381776}{23 · 401 · 13687} ∑_{F_{9}=n} G_{2}(x_{1}, …, x_{8}).

3. Let

F_{1} = x_{1}^{2} + x_{1}x_{2} + 2x_{2}^{2}.

In [3] it is shown that the quadratic form F_{k} is of type (k, 7, χ) (χ = χ(d) = (\frac{d}{7}) when 2 ∤ k and χ = 1 when 2|k).

The following Lemma is proved exactly as Lemma 10.

Lemma 13. a) φ = φ(x_{1}, x_{2}) = x_{1}^{2} −2x_{2}^{2} is a spherical function of order 2 with respect to F_{1};

b) θ(τ; F_{1}, φ) = 2z − 6z^{2} + 0 · z^{3} + 10z^{4} + 0 · z^{5} + ⋯ ∈ S_{3}(Γ_{0}(7), χ);

c) ord(θ(τ; F_{1}, φ), i∞, Γ_{0}(7)) = 1.

Theorem 14. Let C = (c_{sr}) (s = 1, 2, 3; r = 1, …, 2[\frac{k}{3} ] − 1) be the matrix whose elements are non-negative integers satisfying the conditions

\begin{matrix}3c_{1r} + 3c_{2r} + c_{3r} = k \\ c_{1r} + 2c_{2r} = r, c_{1r} > 0 \end{matrix} \} (r = 1, …, 2[\frac{k}{3} ] − 1). \begin{matrix}(3.1) \\ (3.2) \end{matrix}

Then for k ≥ 3 the system of functions

θ^{c_{1r}}(τ; F_{1}, φ)θ^{c_{2r}}(τ; Q_{3}^{(7)}, 1, h^{(7)})θ^{c_{3r}}(τ, F_{1}) (r = 1, …, 2[\frac{k}{3} ] − 1), (3.3)

where θ(τ; Q_{3}^{(7)}, 1, h^{(7)}) is defined by Lemma 8, is the basis of the space S_{k}(Γ_{0}(7), χ).

Proof. By Lemmas 8, 9, 13 and (1.15),(3.1), functions (3.3) are cusp forms of type (k, Γ_{0}(7), χ). By Lemmas 8, 13 and (1.16), (3.2), we have

ord(θ^{c_{1r}}(τ; F_{1}, φ)θ^{c_{2r}}(τ; Q_{3}^{(7)}, 1, h^{(7)})θ^{c_{3r}}(τ, F_{1}), i∞, Γ_{0}(7)) = r

(r = 1, …, 2[\frac{k}{3} ] − 1).

Functions (3.3) are linearly independent because their orders at the cusp i∞ are different. Hence the theorem is proved as it is known ([6], pp. 815,

816) that dim S_{k}(Γ_{0}(7), χ) = 2[\frac{k}{3} ] − 1.

Corollary 15. The system of functions

θ(τ; F_{1}, φ)θ^{k−3}(τ, F_{1}) when 3 ≤ k ≤ 5 and

θ^{r}(τ; F_{1}, φ)θ^{k−3r}(τ, F_{1}) (r = 1, …, [\frac{k}{3} ] ),

θ^{2[ \frac{k}{3} ]−r}(τ; F_{1}, φ)θ^{r−[ \frac{k}{3} ]}(τ; Q_{3}^{(7)}, 1, h^{(7)})θ^{3\{ \frac{k}{3} \}}(τ, F_{1})

(r = [\frac{k}{3} ] + 1, …, 2[\frac{k}{3} ] − 1), when k ≥ 6,

is the basis of the space S_{k}(Γ_{0}(7), χ).

By Lemma 6, θ(τ, F_{k}) − E(τ, F_{k}) ∈ S_{k}(Γ_{0}(7), χ). Hence by Corollary 15 there are constants α_{r}^{(k)} such that

θ(τ, F_{k}) = E(τ, F_{k}) + ∑_{r=1}^{[ \frac{k}{3} ]} α_{r}^{(k)}θ^{r}(τ; F_{1}, φ)θ^{k−3r}(τ, F_{1}) +

+ ∑_{r=[ \frac{k}{3} ]+1}^{2[ \frac{k}{3} ]−1} α_{r}^{(k)}θ^{2[ \frac{k}{3} ]−r}(τ; F_{1}, φ)θ^{r−[ \frac{k}{3} ]}(τ; Q_{3}^{(7)}, 1, h^{(7)})θ^{3\{ \frac{k}{3} \}}(τ, F_{1}).

Then, using the expansions of θ(τ, F_{k}), E(τ, F_{k}) [3], the expansions of θ(τ; F_{1}, φ) and θ(τ; Q_{3}^{(7)}, 1, h^{(7)}), exactly as above, we get formulas for r(n, F_{k}).

Let G_{i}(x_{1}, …, x_{2i}) = ∏_{t=1}^{i}(x_{2t−1}^{2} − x_{2t}^{2}). Then formulas for r(n, F_{k}) when k = 3, 4, …, 11 have the form

r(n, F_{3}) = \frac{7}{8} ρ_{2}^{∗}(n) + \frac{3}{8} ∑_{F_{1}=n} G_{1}(x_{1}, x_{2}),

r(n, F_{4}) = \frac{24}{5} σ_{3}^{∗}(n) + \frac{8}{5} ∑_{F_{2}=n} G_{1}(x_{1}, x_{2}),

r(n, F_{5}) = \frac{1}{16} ρ_{4}^{∗}(n) + \frac{55}{16} ∑_{F_{3}=n} G_{1}(x_{1}, x_{2}),

r(n, F_{6}) = \frac{28}{19} σ_{5}^{∗}(n) + \frac{100}{19} ∑_{F_{4}=n} G_{1}(x_{1}, x_{2}) + \frac{18}{19} ∑_{F_{2}=n} G_{2}(x_{1}, …, x_{4}),

r(n, F_{7}) = \frac{1}{584} ρ_{6}^{∗}(n) + \frac{3917}{584} ∑_{F_{5}=n} G_{1}(x_{1}, x_{2}) + \frac{501}{292} ∑_{F_{3}=n} G_{2}(x_{1}, …, x_{4}),

r(n, F_{8}) = \frac{240}{1201} σ_{7}^{∗}(n) + \frac{9488}{1201} ∑_{F_{6}=n} G_{1}(x_{1}, x_{2}) +

+ \frac{2288}{1201} ∑_{F_{4}=n} G_{2}(x_{1}, …, x_{4}),

r(n, F_{9}) = \frac{7}{2^{5} · 8831} ρ_{8}^{∗}(n) + \frac{2534921}{2^{5} · 8831} ∑_{F_{7}=n} G_{1}(x_{1}, x_{2}) +

+ \frac{57299}{2^{3} · 8831} ∑_{F_{5}=n} G_{2}(x_{1}, …, x_{4}) − \frac{27675}{2^{3} · 8831} ∑_{F_{3}=n} G_{3}(x_{1}, …, x_{6}),

r(n, F_{10}) = \frac{44}{2801} σ_{9}^{∗}(n) + \frac{27988}{2801} ∑_{F_{8}=n} G_{1}(x_{1}, x_{2}) −

− \frac{5522}{2801} ∑_{F_{6}=n} G_{2}(x_{1}, …, x_{4}) − \frac{4320}{2801} ∑_{F_{4}=n} G_{3}(x_{1}, …, x_{6}),

r(n, F_{11}) = \frac{1}{2^{3} · 11 · 73 · 701} ρ_{10}^{∗}(n) + \frac{49527061}{2^{3} · 11 · 73 · 701} ∑_{F_{9}=n} G_{1}(x_{1}, x_{2}) −

− \frac{1319075}{2^{2} · 73 · 701} ∑_{F_{7}=n} G_{2}(x_{1}, …, x_{4}) −

− \frac{4070853}{2 · 11 · 73 · 701} ∑_{F_{5}=n} G_{3}(x_{1}, …, x_{6}).

4. Let

F_{1} = x_{1}^{2} + x_{1}x_{2} + 3x_{2}^{2},

F̃_{2} = 2(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}) + 2x_{1}x_{3} + x_{1}x_{4} + x_{2}x_{3} − 2x_{2}x_{4}.

It is easy to verify that the quadratic form F_{k} is of type (k, 11, χ) (χ = χ(d) = (\frac{d}{11}) when 2 ∤ k and χ = 1 when 2|k) and the quadratic form F̃_{2} is of type (2, 11, 1).

The following Lemmas are proved exactly as Lemma 10.

Lemma 16. a) φ_{1} = φ_{1}(x_{1}, x_{2}) = x_{1}^{2} − 3x_{2}^{2} is a spherical function of order 2 with respect to F_{1};

b) θ(τ; F_{1}, φ_{1}) = 2z − 10z^{3} + 8z^{4} − 2z^{5} + 0 · z^{6} + ⋯ ∈ S_{3}(Γ_{0}(11), χ);

c) ord(θ(τ; F_{1}, φ_{1}), i∞, Γ_{0}(11)) = 1.

Lemma 17. a) φ_{2} = φ_{2}(x_{1}, …, x_{4}) = x_{2}^{2} +4x_{2}x_{3} is a spherical function of order 2 with respect to F̃_{2};

b) θ(τ; F̃_{2}, φ_{2}) = 4z^{2} − 16z^{3} + 8z^{4} + 32z^{5} − 20z^{6} + ⋯ ∈ S_{4}(Γ_{0}(11), 1);

c) ord(θ(τ; F̃_{2}, φ_{2}), i∞, Γ_{0}(11)) = 2.

Lemma 18. a) φ_{3} = φ_{3}(x_{1}, …, x_{6}) = x_{6}^{2} +2x_{2}x_{3} is a spherical function of order 2 with respect to F̃_{2} ⊕ F_{1};

b) θ(τ; F̃_{2} ⊕ F_{1}, φ_{3}) = −8z^{3} − 24z^{4} + 64z^{5} + 0 · z^{6} + ⋯ ∈ S_{5}(Γ_{0}(11), χ);

c) ord(θ(τ; F̃ ⊕ F_{1}, φ_{3}), i∞, Γ_{0}(11)) = 3.

Lemma 19. a) φ_{4} = φ_{4}(x_{1}, …, x_{4}) = 2x_{2}^{4} − 3x_{4}^{4} − 6x_{2}^{2}x_{3}^{2} + 4x_{3}x_{4}^{3} + 6x_{3}^{2}x_{4}^{2} + 6x_{2}^{2}x_{4}^{2} is a spherical function of order 4 with respect to F_{2};

b) θ(τ; F_{2}, φ_{4}) = −48z^{4} + 48z^{5} + 96z^{6} + ⋯ ∈ S_{6}(Γ_{0}(11), 1);

c) ord(θ(τ; F_{2}, φ_{4}), i∞, Γ_{0}(11)) = 4.

Lemma 20. a) φ_{5} = φ_{5}(x_{1}, …, x_{6}) = 4x_{2}x_{3}^{3} + 3x_{3}^{2}x_{5}^{2} − 24x_{2}x_{3}x_{6}^{2} − −3x_{3}^{2}x_{6}^{2} − 6x_{5}^{2}x_{6}^{2} − 4x_{5}x_{6}^{3} + x_{6}^{4} is a spherical function of order 4 with respect to F̃_{2} ⊕ F_{1};

b) θ(τ; F̃_{2} ⊕ F_{1}, φ_{5}) = 528z^{6} + ⋯ ∈ S_{7}(Γ_{0}(11), χ);

c) ord(θ(τ; F̃_{2} ⊕ F_{1}, φ_{5}), i∞, Γ_{0}(11)) = 6.

Theorem 21. Let C = (c_{sr}) (s = 1, …, 7; r = 1, …, k−2) be the matrix whose elements are non-negative integers satisfying the following conditions:

\begin{matrix}7c_{1r}+4c_{2r}+5c_{3r}+6c_{4r}+3c_{5r}+5c_{6r}+c_{7r} =k \\ 6c_{1r}+2c_{2r}+3c_{3r}+4c_{4r}+c_{5r}+5c_{6r} =m_{r}, \\ ∑_{s=1}^{5} c_{sr} > 0; c_{1r} · c_{3r} = 0; c_{1r}, c_{3r} = 0 or 1 \end{matrix} \} (r =1, …, k−2), \begin{matrix}(4.1) \\ (4.2) \end{matrix}

m_{r} = r when r = 1, …, k − 3; m_{k−2} = k − 2 if k ≠ 5l + 2 and m_{k−2} = k − 1 if k = 5l + 2, l ∈ ℕ. Further, let φ^{(r)} = φ_{5}^{c_{1r}}φ_{2}^{c_{2r}}φ_{3}^{c_{3r}}φ_{4}^{c_{4r}}φ_{1}^{c_{5r}} be the direct product of functions φ_{s} defined by Lemmas 16–20 (φ_{s}^{0} ≡ 1). Then for each k ≥ 3, k ≠ 8, the system of functions

θ(τ; F̃_{2v_{r}} ⊕ F_{t_{r}}, φ^{(r)}))θ^{c_{6r}}(τ; Q_{5}^{(11)}, 1, h^{(11)})θ^{c_{7r}}(τ, F_{1})
(r = 1, …, k − 2),  (4.3)

where θ(τ; Q_{5}^{(11)}, 1, h^{(11)}) is defined by Lemma 8 and

v_{r} = c_{1r} + c_{2r} + c_{3r}, t_{r} = c_{1r} + c_{3r} + c_{4r} + c_{5r}  (4.4)

is the basis of the space S_{k}(Γ_{0}(11), χ).

Proof. It follows from the definition of φ^{(r)}, (4.4) and Lemma 9 that

θ(τ; F̃_{2v_{r}} ⊕ F_{t_{r}}, φ^{(r)}) = θ^{c_{1r}}(τ; F̃_{2} ⊕ F_{1}, φ_{5})^{c_{2r}}θ(τ; F̃_{2}, φ_{2}) ×
×θ^{c_{3r}}(τ; F̃_{2} ⊕ F_{1}, φ_{3})θ^{c_{4r}}(τ; F_{2}, φ_{4})θ^{c_{5r}}(τ; F_{1}, φ_{1}).  (4.5)

Therefore (4.5), (1.15), (4.1) and Lemmas 8, 16–20 imply that functions (4.3) are cusp forms of type (k, Γ_{0}(11), χ). By Lemmas 16–20, 8 and (1.16), (4.2), we have

ord(θ(τ; F̃_{2v_{r}} ⊕F_{t_{r}}, φ^{(r)}))θ^{c_{6r}}(τ; Q_{5}^{(11)}, 1, h^{(11)})θ^{c_{7r}}(τ, F_{1}), i∞, Γ_{0}(11))=m,

m = 1, 2, …, k − 2 when k ≠ 5l + 2 and m = 1, 2, …, k − 3, k − 1 when k = 5l + 2, l ∈ ℕ. Functions (4.3) are linearly independent because their orders at the cusp i∞ are different. Hence the theorem is proved as it is known ([6], p. 815, 816) that dim S_{k}(Γ_{0}(11), χ) = k − 2.

By Lemma 6, θ(τ, F_{k}) − E(τ, F_{k}) ∈ S_{k}(Γ_{0}(11), χ). Hence by Theorem 21 there are constants α_{r}^{(k)} such that

θ(τ, F_{k}) = E(τ, F_{k}) +

+ ∑_{r=1}^{k−2} α_{r}^{(k)}θ(τ; F̃_{2v_{r}} ⊕ F_{t_{r}}, φ^{(r)})θ(τ; Q_{5}^{(11)}, 1, h^{(11)})θ^{c_{7r}}(τ, F_{1}).

From (1.5)–(1.10) and (1.17) when q = 11, using the computer we get the expansions of θ(τ, F_{k}) and E(τ, F_{k}). Then exactly as above we get formulas for r(n, F_{k}). When k = 3, …, 7 they have the form

r(n, F_{3}) = \frac{1}{3} ρ_{2}^{∗}(n) + \frac{4}{3} ∑_{F_{1}=n} x_{1}^{2} − 3x_{2}^{2},

r(n, F_{4}) = \frac{120}{61} σ_{3}^{∗}(n) + \frac{184}{61} ∑_{F_{2}=n} x_{1}^{2} − 3x_{2}^{2} − \frac{88}{61} ∑_{_{F_{2}}=n} x_{2}^{2} + 4x_{2}x_{3},

r(n, F_{5}) = \frac{11}{3 · 5^{2} · 17} ρ_{4}^{∗}(n) + \frac{5704}{3 · 5^{2} · 17} ∑_{F_{3}=n} x_{1}^{2} − 3x_{2}^{2} −

− \frac{3608}{3 · 5^{2} · 17} ∑_{_{F_{2}}⊕F_{1}=n} x_{2}^{2} + 4x_{2}x_{3},

r(n, F_{6}) = \frac{36}{5 · 19} σ_{5}^{∗}(n) + \frac{552}{5 · 19} ∑_{F_{4}=n} x_{1}^{2} − 3x_{2}^{2} −

− \frac{528}{5 · 19} ∑_{_{F_{2}}⊕F_{2}=n} x_{2}^{2} + 4x_{2}x_{3} − \frac{121}{5 · 19} ∑_{_{F_{2}}⊕F_{2}=n} x_{6}^{2} + 2x_{2}x_{3} +

+ \frac{1331}{2 · 3 · 5 · 19} ∑_{F_{2}=n} 2x_{2}^{4} − 3x_{4}^{4} − 6x_{2}^{2}x_{3}^{2} + 4x_{3}x_{4}^{3} + 6x_{3}^{2}x_{4}^{2} + 6x_{2}^{2}x_{4}^{2},

r(n, F_{7}) = \frac{1}{3^{2} · 17 · 71} ρ_{6}^{∗}(n) + \frac{75376}{3^{2} · 17 · 71} ∑_{F_{5}=n} x_{1}^{2} − 3x_{2}^{2} −

− \frac{94360}{3^{2} · 17 · 71} ∑_{_{F_{2}}⊕F_{3}=n} x_{2}^{2} + 4x_{2}x_{3} −

− \frac{33187}{3^{2} · 17 · 71} ∑_{_{F_{2}}⊕F_{3}=n} x_{6}^{2} + 2x_{2}x_{3} +

+ \frac{365057}{2 · 3^{3} · 17 · 71} ∑_{F_{3}=n} 2x_{2}^{4} − 3x_{4}^{4} − 6x_{2}^{2}x_{3}^{2} + 4x_{3}x_{4}^{3} + 6x_{3}^{2}x_{4}^{2} + 6x_{2}^{2}x_{4}^{2}.

References

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(Received 3.07.1995)

Author’s address: Georgian Technical University Department of Mathematics No. 3 77, Kostava St., Tbilisi 380075 Georgia