Projections of modular forms on Eisenstein series and its application to Siegel’s formula

Abstract

Let \(k \ge 2\) and N be positive integers and let \(\chi \) be a Dirichlet character modulo N. Let f(z) be a modular form in \(M_k(\Gamma _0(N),\chi )\). Then we have a unique decomposition \(f(z)=E_f(z)+S_f(z)\), where \(E_f(z) \in E_k(\Gamma _0(N),\chi )\) and \(S_f(z) \in S_k(\Gamma _0(N),\chi )\). In this paper, we give an explicit formula for \(E_f(z)\) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel’s formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities.