The Bases of and the Number of Representation of Integers

Abstract

Following a fundamental theorem of Hecke, some bases of and are determined, and explicit formulas are obtained for the number of representations of positive integers by all possible direct sums (111 different combinations) of seven quadratic forms from the class group of equivalence classes of quadratic forms with discriminant −71 whose representatives are .

Fourıer coeffıcıents of a class of ETA quotıents of weıght 20 wıth level 12

“…Define the rational numbers (12)) and η a1 (z) η a2 (2z) η a3 (3z) η a4 (4z) η a6 (6z) η a12 (12z) = δ (b 1 ) + Proof: It follows from (6)-(11) that a 1 + 2a 2 + 3a 3 + 4a 4 + 6a 6 + 12a 12 = 24b 1 ,…”

“…One can find more information in [3], [6], [14], [16], [17]. I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see [7], [8], [9] [10], [11] and [12]. Recently, Williams, see [18] discovered explicit formulas for the coefficients of Fourier series expansions of a class of 126 eta quotients in terms of σ(n), σ( n 2 ), σ( n 3 ) and σ( n 6 ).…”

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( ).…”