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LetQQbe an integral positive definite quadratic form of levelNNin2k2kvariables. Further, we assume that(−1)kN{\left(-1)}^{k}Nis a fundamental discriminant. We express the average theta function ofQQas an explicit sum of Eisenstein series, which are Hecke eigenforms by applying Atkin-Lehner involution.

On average theta functions of certain quadratic forms as sums of Eisenstein series