Representations of positive integers by a direct sum of quadratic forms

Abstract

The number of representation of positive integers by quadratic forms F1=x12+3x1x2+8x22 and G1=2x12+3x1x2+4x22 of discriminant —23 are given. Moreover, a basis for the cusp form space S4(Γ0(23), 1) are constructed. Furthermore, formulas for the representation of positive integers by direct sum of copies of F1 and G1, i.e. formulas for r(n; F4), r(n; G4), r(n; F3 ⊕ G1), r(n; F2 ⊕ G2), and r(n; F1 ⊕ G3), are derived using the elements of the space S4(Γ(23), 1).