A basis for S (k) (I" (0)(4)) and representations of integers as sums of squares

Abstract

In this paper, we find a basis for the space S (k) (I" (0)(4)) of cusp forms of even weight k for the congruence subgroup I" (0)(4) in terms of Eisenstein series. As an application, we obtain formulas for r (4s) (n), the number of ways to represent a nonnegative integer n as a sum of 4s integral squares.