Cusp Forms in and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant

Abstract

A basis of is given and the formulas for the number of representations of positive integers by some direct sum of the quadratic forms , , are determined.

The Bases of and the Number of Representation of Integers

“…Of couse, the work of Hijikata et al [7] for the construction of a general bases of modular forms of Γ 0 ( ) [7] is very important work. Here, we have found bases in special cases 4 (Γ 0 (71)), 6 (Γ 0 (71)) by extending our work on [8,9] to the case of discriminant −71 including weight 6 case.…”

Fourier Coefficients of a Class of Eta Quotients of Weight 4

“…One can find more information in [2], [3], [4], [5] and [6]. The author has also determined the Fourier coefficients of the theta series associated with some quadratic forms, see [8], [9], [10], [11], [12] and [13]. Recently, Williams [14] discovered explicit formulas for the coefficients of Fourier series expansions of a class of one hundred and twenty-six eta quotients in terms of σ(n), σ( ).…”

“…Here, we will express the coefficients of the Fourier series expansions of a class of one hundred and fifty-six eta quotients in M 4 (Γ 0 , χ) in terms of σ 3 (n), σ 3 ( In particular, since 2 a 2 3 a 3 4 a 4 6 a 6 12 a 12 = 2 a 2 +2a 4 +a 6 +2a 12 Since a 2 + a 6 is an even integer, we conclude that it is a meromorphic modular form iff a 3 + a 6 + a 12 is an even integer, and it is a meromorphic modular form with χ 3 iff a 3 + a 6 + a 12 is an odd integer, where χ 3 is the unique primitive Dirichlet character mod12 . On the other hand, the modular forms are holomorphic iff its order at cusps, are nonnegative, see [1] i.e., Theorem 1: Let 0 be the trivial character mod 1, i.e., it sends to 1, 1 be the primitive Dirichlet character mod 4, and 2 is the primitive Dirichlet character mod 3.…”

Representations by Certain Sextenary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6

“…Other methods for representation number have been used in (cf. [7] [10] [12] [18] [19]). Here, we will classify all fourtuples ( ) , , , a b c d for which Q Θ is a modular form of weight 8 with level 24.…”