Georgian Mathematical Journal Volume 8 (2001), Number 1, 87-95

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

N. KACHAKHIDZE

abstract. The systems of bases are constructed for the spaces of cusp forms S_{2m}(Γ_{0}(5), χ^{m}) and S_{2m}(Γ_{0}(13), χ^{m}) for an arbitrary integer m ≥ 2. Formulas are obtained for the number of representations of a positive integers by the direct sums of some quaternary quadratic forms.

2000 Mathematics Subject Classification: 11E25, 11E20, 11F11, 11E45, 11F27. Key words and phrases: Modular form, representation of integers by quadratic forms, space of cusp forms.

Let F_{2m} denote a direct sum of m quaternary quadratic forms F_{2} with the same positive discriminant q (q is prime ≡ 1 (mod 4)). We shall use the notions, notation and some results from [1].

In what follows let

Q_{\frac{q−1}{2} m}^{(q)} = Q_{\frac{q−1}{2} m}^{(q)}(y) = Q_{\frac{q−1}{2} m}^{(q)}(y_{1}, y_{2}, …, y_{(q−1)m}),

where

Q_{\frac{q−1}{2} m}^{(q)} = ∑_{k=1}^{m} Q(q)_{\frac{q−1}{2}} (y_{(k−1)(q−1)+1}, …, y_{k(q−1)})

with

Q(q)_{\frac{q−1}{2}} (u_{1}, …, u_{q−1}) = q ∑_{1≤i≤j≤q−2} u_{i}u_{j} + q ∑_{1≤i≤q−2} u_{i}u_{q−1} + \frac{q − 1}{2} u_{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}),

g^{(q,m)} = (g_{1}, g_{2}, …, g_{(q−1)m}) = (h^{(q)}, h^{(q)}, …, h^{(q)});

y ≡ g^{(q,m)} (mod q) is to be understood elementwise;

σ_{t}^{∗}(n) = \begin{cases}(q^{(t+1)/2} + q^{(t+1)/4} + 1)σ_{t}(n), & if q ∤ n, \\ (q^{(t+1)/2} + q^{(t+1)/4} + 1)σ_{t}(n) + q^{3(t+1)/4}σ_{t}(\frac{n}{q}), & if q|n, \end{cases}

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

where

σ_{t}(n) = ∑_{d|n} d^{t};

ρ_{t}^{∗}(n) = q^{(3t+1)/4} ∑_{δd=n} (\frac{δ}{q} )d^{t} + (−1)^{(t+1)/2} ∑_{d|n} (\frac{d}{q})d^{t},

where (\frac{·}{q}) is the Jacobi symbol.

1. Let

F_{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{4},
Φ_{2} = x_{1}^{2} + x_{2}^{2} + 2x_{3}^{2} + 2x_{4}^{2} + x_{1}x_{2} + x_{1}x_{3} + x_{1}x_{4} + x_{2}x_{4} + 2x_{3}x_{4}.

F_{2} and Φ_{2} are the only reduced positive quaternary quadratic forms with determinants 5 and 25 respectively [2, pp. 146, 147]. In [3] it is shown, that F_{2m} is a quadratic form of type (2m, 5, χ^{m}) (χ = χ(d) = (\frac{d}{5})).

To construct a spherical function of order ν, we find a linear transformation which reduces the quadratic form to a sum of squares. Further, all possible monomials of order ν are constructed of the variables of the quadratic form. The found linear transformation are next applied to these monomials. The obtained polynomial is subjected to the action of the Laplace operator and for the resulting polynomials we choose identically vanishing linear combinations. A linear combination of the corresponding monomials of order ν is a spherical function of the same order with respect to the initial quadratic form.

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

\begin{matrix}x_{1} = y_{1} − \frac{1}{\sqrt{3}} y_{2} − \frac{2}{\sqrt{15}} y_{3}, x_{2} = \frac{2}{\sqrt{3}} y_{2} + \frac{1}{\sqrt{15}} y_{3} − \frac{1}{\sqrt{5}} y_{4}, \\ x_{3} = \sqrt{\frac{3}{5}} y_{3} − \frac{1}{\sqrt{5}} y_{4}, x_{4} = \frac{2}{\sqrt{5}} y_{4} \end{matrix}  (1.1)

takes Φ_{2} into a sum of squares.

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

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

(c) ord(ϑ(τ; Φ_{2}, φ), i∞, Γ_{0}(5)) = 1.

Proof. Using transformation (1.1) we get

φ(x_{1}, …, x_{4}) = y_{1}^{2} + \frac{1}{3} y_{2}^{2} + \frac{4}{15} y_{3}^{2} − \frac{8}{5} y_{4}^{2} − \frac{2}{\sqrt{3}} y_{1}y_{2} − \frac{4}{\sqrt{15}} y_{1}y_{3}

+ \frac{4}{3\sqrt{5}} y_{2}y_{3} = φ(y_{1}, …, y_{4}),

∑_{i=1}^{4} \frac{∂^{2}φ}{∂y_{i}^{2}} = 2(1 + \frac{1}{3} + \frac{4}{15} − \frac{8}{5}) = 0.

ON THE REPRESENTATION OF NUMBERS BY THE QUADRATIC FORMS 89

Consequently, by definition (see [4], pp. 849, 853), φ is a spherical function of order 2 with respect to Φ_{2} and, by Lemma 3 from [1], ϑ(τ; Φ_{2}, φ) ∈ S_{4}(Γ_{0}(5), 1).

Having performed suitable calculations we get

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

= 4z − 16z^{2} + 8z^{3} + 32z^{4} − 20z^{5} + · · · .  (1.2)

It follows from (1.2) and formula (1.1) of [1] that

ord (ϑ(τ; Φ_{2}, φ), i∞, Γ_{0}(5)) = 1.

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

\begin{matrix}4c_{1r} + 2c_{2r} + 2c_{3r} = 2m \\ c_{1r} + c_{2r} = r, c_{1r} > 0 \end{matrix} \} (r = 1, 2, …, m−1). \begin{matrix}(1.3) \\ (1.4) \end{matrix}

Then for m ≥ 2 the system of functions

ϑ^{c_{1r}}(τ; Φ_{2}, φ)ϑ^{c_{2r}}(τ; Q_{2}^{(5)}, 1, h^{(5)})ϑ^{c_{3r}}(τ, F_{2}) (r = 1, 2, …, m − 1) (1.5)

is the basis of the space S_{2m}(Γ_{0}(5), χ^{m}).

Proof. By Lemma 1, (1.3) and Lemmas 3, 8, 9 from [1], functions (1.5) are cusp forms of type (2m, Γ_{0}(5), χ^{m}). By Lemma 1, (1.4) and Lemma 8 from [1], we have

ord (ϑ^{c_{1r}}(τ; Φ_{2}, φ)ϑ^{c_{2r}}(τ; Q_{2}^{(5)}, 1, h^{(5)})ϑ^{c_{3r}}(τ, F_{2}), i∞, Γ_{0}(5)) = r
(r = 1, 2, …, m − 1).

Functions (1.5) are linearly independent because their orders at the cusp i∞ are different. Hence the theorem is proved as it is known (see [4], pp. 815, 816) that dim S_{2m}(Γ_{0}(5), χ^{m}) = m − 1.

Corollary 3. For m = 2 the function ϑ(τ; Φ_{2}, φ) and for m ≥ 3 the system of functions

ϑ^{r}(τ; Φ_{2}, φ)ϑ^{m−2r}(τ, F_{2}) (r = 1, 2, …, [m/2]),
ϑ^{m−r}(τ; Φ_{2}, φ)ϑ^{2r−m}(τ; Q_{2}^{(5)}, 1, h^{(5)}) (r = [m/2] + 1, …, m − 1)

is the basis of the space S_{2m}(Γ_{0}(5), χ^{m}).

By Lemma 6 from [1], ϑ(τ, F_{2m})−E(τ, F_{2m}) ∈ S_{2m}(Γ_{0}(5), χ^{m}), where E(τ, F_{2m}) is the Eisenstein series (see [4], pp. 875, 877). Hence, by Corollary 3, there are constants α and α_{r}^{(2m)} such that

ϑ(τ, F_{4}) = E(τ, F_{4}) + αϑ(τ; Φ_{2}, φ)  (1.6)

and for m ≥ 3

ϑ(τ, F_{2m}) = E(τ, F_{2m}) + ∑_{r=1}^{[m/2]} α_{r}^{(2m)}ϑ^{r}(τ; Φ_{2}, φ)ϑ^{m−2r}(τ, F_{2})

+ ∑_{r=[m/2]+1}^{m−1} α_{r}^{(2m)}ϑ^{m−r}(τ; Φ_{2}, φ)ϑ^{2r−m}(τ; Q_{2}^{(5)}, 1, h^{(5)}).  (1.7)

Using the expansions of ϑ(τ, F_{2m}), E(τ, F_{2m}) from [3], that of ϑ(τ; Q_{2}^{(5)}, 1, h^{(5)}) from [1], Lemma 1, and equating, in both parts of (1.6) and (1.7), the coefficients of z and z, z^{2}, …, z^{m−1} we get the systems of linear equations, whence we find the constants α and α_{r}^{(2m)}. Equating, in both parts of (1.6) and (1.7), the coefficients of z^{n}, we obtain formulas for the arithmetical function r(n, F_{2m}).

The formulas for r(n, F_{2m}) when m = 2, 3, …, 6 have the form

r(n, F_{4}) = \frac{20}{13} σ_{3}^{∗}(n) − \frac{25}{13} ∑_{Φ_{2}=n} x_{1}^{2} − 2x_{4}^{2},
r(n, F_{6}) = \frac{5}{67} ρ_{5}^{∗}(n) + \frac{225}{67} ∑_{Φ_{2}⊕F_{2}=n} x_{1}^{2} − 2x_{4}^{2}

− \frac{1250}{67} ∑_{\begin{matrix}Φ_{2}⊕Q_{2}^{(5)}=n \\ y≡h^{(5)} (mod 5) \end{matrix}} x_{1}^{2} − 2x_{4}^{2},

r(n, F_{8}) = \frac{120}{13 · 313} σ_{7}^{∗}(n) + \frac{61850}{13 · 313} ∑_{Φ_{2}⊕F_{4}=n} x_{1}^{2} − 2x_{4}^{2}

− \frac{545625}{13 · 313} ∑_{Φ_{4}=n}(x_{1}^{2} − 2x_{4}^{2})(x_{5}^{2} − 2x_{8}^{2})

− \frac{1550000}{13 · 313} ∑_{\begin{matrix}Φ_{2}⊕Q_{4}^{(5)}=n \\ y≡g^{(5,2)} (mod 5) \end{matrix}} x_{1}^{2} − 2x_{4}^{2},

r(n, F_{10}) = \frac{25}{191 · 2161} ρ_{9}^{∗}(n) + \frac{9830500}{191 · 2161} ∑_{Φ_{2}⊕F_{6}=n} x_{1}^{2} − 2x_{4}^{2}

− \frac{185896875}{2 · 191 · 2161} ∑_{Φ_{4}⊕F_{2}=n}(x_{1}^{2} − 2x_{4}^{2})(x_{5}^{2} − 2x_{8}^{2})

− \frac{596640625}{2 · 191 · 2161} ∑_{\begin{matrix}Φ_{4}⊕Q_{2}^{(5)}=n \\ y≡h^{(5)} (mod 5) \end{matrix}}(x_{1}^{2} − 2x_{4}^{2})(x_{5}^{2} − 2x_{8}^{2})

− \frac{1228906250}{191 · 2161} ∑_{\begin{matrix}Φ_{2}⊕Q_{6}^{(5)}=n \\ y≡g^{(5,3)} (mod 5) \end{matrix}} x_{1}^{2} − 2x_{4}^{2},

r(n, F_{12}) = \frac{20}{601 · 691} σ_{11}^{∗}(n) + \frac{12379975}{601 · 691} ∑_{Φ_{2}⊕F_{8}=n} x_{1}^{2} − 2x_{4}^{2}

− \frac{115155625}{601 · 691} ∑_{Φ_{4}⊕F_{4}=n}(x_{1}^{2} − 2x_{4}^{2})(x_{5}^{2} − 2x_{8}^{2})

− \frac{497140625}{601 · 691} ∑_{Φ_{6}=n}(x_{1}^{2} − 2x_{4}^{2})(x_{5}^{2} − 2x_{8}^{2})(x_{9}^{2} − 2x_{12}^{2})

− \frac{3886875000}{601 · 691} ∑_{\begin{matrix}Φ_{4}⊕Q_{4}^{(5)}=n \\ y≡g^{(5,2)} (mod 5) \end{matrix}}(x_{1}^{2} − 2x_{4}^{2})(x_{5}^{2} − 2x_{8}^{2})

− \frac{7737500000}{601 · 691} ∑_{\begin{matrix}Φ_{2}⊕Q_{8}^{(5)}=n \\ y≡g^{(5,4)} (mod 5) \end{matrix}} x_{1}^{2} − 2x_{4}^{2}.

2. Let

F_{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 2x_{4}^{2} + x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{4},
F_{2}^{∗} = 7x_{1}^{2} + 6x_{2}^{2} + 5x_{3}^{2} + 2x_{4}^{2} − 8x_{1}x_{2} − 7x_{1}x_{3}
+ 2x_{1}x_{4} + 4x_{2}x_{3} − 3x_{2}x_{4} − x_{3}x_{4},
Φ_{2} = x_{1}^{2} + 2x_{2}^{2} + 2x_{3}^{2} + 4x_{4}^{2} + x_{1}x_{2} + x_{1}x_{4}
+ x_{2}x_{3} + x_{2}x_{4} + 2x_{3}x_{4}.

F_{2} is the only reduced positive quaternary quadratic form with determinant 13 ([2], p. 141) and F_{2}^{∗} is adjoint to F_{2}. F_{2} and F_{2}^{∗} are quadratic forms of type (2, 13, χ) [5], and Φ_{2} is a quadratic form of type (2, 13, 1) [6], where χ = χ(d) = (\frac{d}{13}).

Using Lemma 5 from [1], it is easy to show that F_{2m} is a quadratic form of type (2m, 13, χ^{m}).

The following Lemmas are proved exactly as Lemma 1.

Lemma 4. (a) φ_{1} = φ_{1}(x_{1}, …, x_{4}) = x_{1}^{2} − 2x_{2}^{2}, φ_{2} = φ_{2}(x_{1}, …, x_{4}) = x_{2}^{2} − x_{3}^{2}, φ_{3} = φ_{3}(x_{1}, …, x_{4}) = 2x_{1}x_{3} − x_{4}^{2} are spherical functions of order 2 with respect to Φ_{2};

(b) ϑ(τ; Φ_{2}, φ_{1}) = 2z − 6z^{2} − 2z^{3} + 14z^{4} − 10z^{5} + 26z^{6}

− 30z^{7} − 38z^{8} + 56z^{9} + 42z^{10} + ⋯,
ϑ(τ; Φ_{2}, φ_{2}) = 2z^{2} − 4z^{3} + 2z^{5} − 2z^{6} + 18z^{7} − 14z^{8} − 24z^{9} − 4z^{10} + ⋯,
ϑ(τ; Φ_{2}, φ_{3}) = 4z^{3} − 4z^{4} − 8z^{6} − 8z^{7} + 16z^{8} + 12z^{9} − 4z^{10} + ⋯;

(c) ϑ(τ; Φ_{2}, φ_{r}) ∈ S_{4}(Γ_{0}(13), 1);

(d) ord(ϑ(τ; Φ_{2}, φ_{r}), i∞, Γ_{0}(13)) = r (r = 1, 2, 3).

Lemma 5. (a) φ_{4} = φ_{4}(x_{1}, …, x_{8}) = x_{4}^{2} −x_{5}^{2} is a spherical function of order 2 with respect to Φ_{2} ⊕ F_{2}^{∗};

(b) ϑ(τ; Φ_{2} ⊕ F_{2}^{∗}, φ_{4}) = 8z^{4} + 24z^{6} − 24z^{7} − 24z^{8}

− 48z^{9} − 40z^{10} + ⋯ ∈ S_{6}(Γ_{0}(13), χ);

(c) ord(ϑ(τ; Φ_{2} ⊕ F_{2}^{∗}, φ_{4}), i∞, Γ_{0}(13)) = 4.

Lemma 6. (a) φ_{5} = φ_{5}(x_{1}, …, x_{4}) = x_{2}^{4} + x_{3}^{4} − 6x_{2}^{2}x_{3}^{2} and φ_{6} = φ_{6}(x_{1}, …, x_{4}) = x_{1}x_{4}^{3} + x_{3}x_{4}^{3} − 6x_{1}^{2}x_{3}x_{4} are spherical functions of order 4 with respect to F_{2}^{∗};

(b) ϑ(τ; F_{2}^{∗}, φ_{5}) = 8z^{5} + 8z^{6} − 32z^{7} + 8z^{8} + 0 · z^{10} + ⋯,

ϑ(τ; F_{2}^{∗}, φ_{6}) = 12z^{6} − 12z^{7} − 12z^{8} + 0 · z^{10} + ⋯;

(c) ϑ(τ; F_{2}^{∗}, φ_{r}) ∈ S_{6}(Γ_{0}(13), χ);

(d) ord(ϑ(τ; F_{2}^{∗}, φ_{r}), i∞, Γ_{0}(13)) = r (r = 5, 6).

Lemma 7. (a) φ_{7} = φ_{7}(x_{1}, …, x_{8}) = x_{7}^{4} − x_{3}^{4} + 6x_{2}^{2}x_{3}^{2} − 6x_{2}^{2}x_{7}^{2} is a spherical function of order 4 with respect to F_{4}^{∗};

(b) ϑ(τ; F_{4}^{∗}, φ_{7}) = 48z^{7} + 96z^{9} − 96z^{10} + ⋯ ∈ S_{8}(Γ_{0}(13), 1);

(c) ord(ϑ(τ; F_{4}^{∗}, φ_{7}), i∞, Γ_{0}(13)) = 7.

Theorem 8. Let

λ = \begin{cases}2m + [\frac{m}{3}] − 1, & if m ≡ 0 or 2 (mod 3), \\ \frac{7}{3} (m − 1), & if m ≡ 1 (mod 3) \end{cases}

and let C = (c_{sr}) (s = 1, 2, …, 9; r = 1, 2, …, λ) be the matrix whose elements are non-negative integers satisfying the conditions

\begin{matrix}4 ∑_{s=1}^{8} c_{sr} + 2 ∑_{s=4}^{9} c_{sr} + 2c_{7r} = 2m \\ ∑_{s=1}^{7} s · c_{sr} + 7c_{8r} = r \\ c_{4r} = 0 or 1, ∑_{s=1}^{7} c_{sr} > 0 \end{matrix} \} (r = 1, 2, …, λ). \begin{matrix}(2.1) \\ (2.2) \end{matrix}

Further, let φ^{(r)} = ∏_{s=1}^{7} φ_{s}^{c_{sr}} denote the direct product of functions φ_{s}, defined

by Lemmas 4–7 (φ_{s}^{0} ≡ 1). Then for each m ≥ 2, the system of functions

ϑ(τ; Φ_{2v_{r}} ⊕ F_{2t_{r}}^{∗}, φ^{(r)})ϑ^{c_{8r}}(τ; Q_{6}^{(13)}, 1, h^{(13)})ϑ^{c_{9r}}(τ, F_{2}^{∗})  (2.3)
(r = 1, 2, …, λ),

where

v_{r} = ∑_{s=1}^{4} c_{sr}, t_{r} = ∑_{s=4}^{7} c_{sr} + c_{7r}  (2.4)

is the basis of the space S_{2m}(Γ_{0}(13), χ^{m}).

Proof. It follows from the definition of φ^{(r)}, (2.4) and ([1], Lemma 9) that

ϑ(τ; Φ_{2v_{r}} ⊕ F_{2t_{r}}^{∗}, φ^{(r)}) = ϑ^{c_{1r}}(τ; Φ_{2}, φ_{1})
× ϑ^{c_{2r}}(τ; Φ_{2}, φ_{2})ϑ^{c_{3r}}(τ; Φ_{2}, φ_{3})
× ϑ^{c_{4r}}(τ; Φ_{2} ⊕ F_{2}^{∗}, φ_{4})ϑ^{c_{5r}}(τ; F_{2}^{∗}, φ_{5})
× ϑ^{c_{6r}}(τ; F_{2}^{∗}, φ_{6})ϑ^{c_{7r}}(τ; F_{4}^{∗}, φ_{7}).  (2.5)

Therefore (2.5), (2.1), above Lemmas 4–7 and Lemmas 3, 8, 9 from [1] imply that functions (2.3) are cusp forms of type (2m, Γ_{0}(13), χ^{m}). By Lemmas 4–7, (2.2) and Lemma 8 from [1], we have

ord (ϑ(τ; Φ_{2v_{r}} ⊕ F_{2t_{r}}^{∗}, φ^{(r)})ϑ^{c_{8r}}(τ; Q_{6}^{(13)}, 1, h^{(13)})ϑ^{c_{9r}}(τ, F_{2}^{∗}), i∞, Γ_{0}(13)) = r

(r = 1, 2, …, λ).

Functions (2.3) are linearly independent because their orders at the cusp i∞ are different. Hence the theorem is proved as it is known that dim S_{2m}(Γ_{0}(13), χ^{m}) = λ (see [4], pp. 815, 816)).

By ([1], Lemma 6) ϑ(τ, F_{2m}) − E(τ, F_{2m}) ∈ S_{2m}(Γ_{0}(13), χ^{m}). Hence by Theorem 8 there are constants α_{r}^{(2m)} such that

ϑ(τ, F_{2m}) = E(τ, F_{2m})

+ ∑_{r=1}^{λ} α_{r}^{(2m)}ϑ(τ; Φ_{2v_{r}} ⊕ F_{2t_{r}}^{∗}, φ^{(r)})ϑ^{c_{8r}}(τ; Q_{6}^{(13)}, 1, h^{(13)})ϑ^{c_{9r}}(τ, F_{2}^{∗}).

Using the expansion of ϑ(τ, F_{2}) from [5] we get the expansions of ϑ(τ, F_{2m}) = ϑ^{m}(τ, F_{2}). From formulas (1.5)–(1.9) of [1], when q = 13, we get the expansions of E(τ, F_{2m}). Then using the expansions of ϑ(τ, F_{2}^{∗}) from [5] and ϑ(τ; Q_{6}^{(13)}, 1, h^{(13)}) from [1], Lemmas 4-7, exactly as above we get formulas for r(n, F_{2m}). When m = 2, …, 5 they have the form

r(n, F_{4}) = \frac{12}{7 · 17} σ_{3}^{∗}(n) + \frac{330}{7 · 17} ∑_{Φ_{2}=n} x_{1}^{2} − 2x_{2}^{2} + \frac{1342}{7 · 17} ∑_{Φ_{2}=n} x_{2}^{2} − x_{3}^{2}

− \frac{1013}{7 · 17} ∑_{Φ_{2}=n} 2x_{1}x_{3} − x_{4}^{2},

r(n, F_{6}) = \frac{13}{109 · 307} ρ_{5}^{∗}(n) + \frac{416694}{109 · 307} ∑_{Φ_{2}⊕F_{2}^{∗}=n} x_{1}^{2} − 2x_{2}^{2}

+ \frac{3425570}{109 · 307} ∑_{Φ_{2}⊕F_{2}^{∗}=n} x_{2}^{2} − x_{3}^{2} + \frac{4662503}{109 · 307} ∑_{Φ_{2}⊕F_{2}^{∗}=n} 2x_{1}x_{3} − x_{4}^{2}

− \frac{2905512}{109 · 307} ∑_{Φ_{2}⊕F_{2}^{∗}=n} x_{4}^{2} − x_{5}^{2} − \frac{1698555}{109 · 307} ∑_{F_{2}^{∗}=n} x_{2}^{4} + x_{3}^{4} − 6x_{2}^{2}x_{3}^{2}

+ \frac{2615880}{109 · 307} ∑_{F_{2}^{∗}=n} x_{1}x_{4}^{3} + x_{3}x_{4}^{3} − 6x_{1}^{2}x_{3}x_{4},

r(n, F_{8}) = \frac{24}{17 · 14281} σ_{7}^{∗}(n) + \frac{5481876}{17 · 14281} ∑_{Φ_{2}⊕F_{4}^{∗}=n} x_{1}^{2} − 2x_{2}^{2}

+ \frac{83647460}{17 · 14281} ∑_{Φ_{2}⊕F_{4}^{∗}=n} x_{2}^{2} − x_{3}^{2}

+ \frac{251775638}{17 · 14281} ∑_{Φ_{2}⊕F_{4}^{∗}=n} 2x_{1}x_{3} − x_{4}^{2}

+ \frac{239040860}{17 · 14281} ∑_{Φ_{4}=n}(2x_{1}x_{3} − x_{4}^{2})(x_{5}^{2} − 2x_{6}^{2})

+ \frac{591029498}{17 · 14281} ∑_{Φ_{4}=n}(2x_{1}x_{3} − x_{4}^{2})(x_{6}^{2} − x_{7}^{2})

+ \frac{877045926}{17 · 14281} ∑_{Φ_{4}=n}(2x_{1}x_{3} − x_{4}^{2})(2x_{5}x_{7} − x_{8}^{2})

− \frac{315282279}{17 · 14281} ∑_{F_{4}^{∗}=n} x_{7}^{4} − x_{3}^{4} + 6x_{2}^{2}x_{3}^{2} − 6x_{2}^{2}x_{7}^{2},

r(n, F_{10}) = \frac{1}{144547171} ρ_{9}^{∗}(n) + \frac{4305040872}{144547171} ∑_{Φ_{2}⊕F_{6}^{∗}=n} x_{1}^{2} − 2x_{2}^{2}

+ \frac{106015990372}{144547171} ∑_{Φ_{2}⊕F_{6}^{∗}=n} x_{2}^{2} − x_{3}^{2}

+ \frac{541004173576}{144547171} ∑_{Φ_{2}⊕F_{6}^{∗}=n} 2x_{1}x_{3} − x_{4}^{2}

+ \frac{1284103896202}{144547171} ∑_{Φ_{4}⊕F_{2}^{∗}=n}(x_{1}^{2} − 2x_{2}^{2})(2x_{5}x_{7} − x_{8}^{2})

+ \frac{7148202390955}{144547171} ∑_{Φ_{4}⊕F_{2}^{∗}=n}(x_{2}^{2} − x_{3}^{2})(2x_{5}x_{7} − x_{8}^{2})

+ \frac{11816421944265}{144547171} ∑_{Φ_{4}⊕F_{2}^{∗}=n}(2x_{1}x_{3} − x_{4}^{2})(2x_{5}x_{7} − x_{8}^{2})

+ \frac{8100018167088}{144547171} ∑_{Φ_{2}⊕F_{2}^{∗}=n}(x_{1}^{2} − 2x_{2}^{2})(x_{5}x_{8}^{3} + x_{7}x_{8}^{3} − 6x_{5}^{2}x_{7}x_{8})

+ \frac{22777304972068}{144547171} ∑_{Φ_{2}⊕F_{2}^{∗}=n}(x_{2}^{2} − x_{3}^{2})(x_{5}x_{8}^{3} + x_{7}x_{8}^{3} − 6x_{5}^{2}x_{7}x_{8})

+ \frac{41871730633632}{144547171} ∑_{Φ_{2}⊕F_{2}^{∗}=n}(2x_{1}x_{3} − x_{4}^{2})(x_{5}x_{8}^{3} + x_{7}x_{8}^{3} − 6x_{5}^{2}x_{7}x_{8})

− \frac{7694378154216}{144547171} ∑_{\begin{matrix}Φ_{2}⊕Q_{6}^{(13)}=n \\ y≡h^{(13)} (mod 13) \end{matrix}} 2x_{1}x_{3} − x_{4}^{2}.

References

1. N. Kachakhidze, On the representation of numbers by the direct sums of some binary quadratic forms. Georgian Math. J. 5(1998), No. 1, 55–70.

2. K. Germann, Tabellen reduzierter, positiver quaternärer quadratischer formen. Studien zur Theorie der quadratischen Formen. Birkhäuser Verlag, Basel und Stuttgart, 1968.

3. N. D. Kachakhidze, On the representation of numbers by a direct sum of quadratic forms of the kind x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{4}. (Russian) Proc. Tbilisi Univ. Math. Mekh. Astron. 299(1990), 60–85.

4. E. Hecke, Mathematische Werke. Zweite Aufl. Vandenhoeck und Ruprecht, Göttingen, 1970.

5. G. A. Lomadze, On the cusp forms with prime level and of adjacent type. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 63(1980), 36–53.

6. K. Sh. Shavgulidze, On cusp and quadratic forms of type (−4, q, 1). (Russian) Trudy Tbiliss. Univ. Math. Mekh. Astron. 214(1980), 194–219.

(Received 20.11.2000)

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