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

TL;DR

This paper derives explicit formulas for projections of modular forms onto Eisenstein series and applies these to eta quotients and quadratic form representations, providing an alternative to Siegel's classical formula.

Contribution

It offers a new explicit Eisenstein projection formula for modular forms and applies it to quadratic forms and eta quotients, simplifying calculations of representation numbers.

Findings

Explicit Eisenstein projection formula derived

Alternative Siegel's formula for quadratic forms provided

Simplified computation using divisor functions

Abstract

Let $k \geq 2$ 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. 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 generalized divisor functions and does not involve computation of local densities.