Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions

TL;DR

This paper derives a closed-form product formula involving Jacobi theta functions for sums of Hecke eigenforms and their period polynomials on (N), providing explicit Fourier expansions for small squarefree levels.

Contribution

It generalizes a previous level-one result to squarefree levels, linking Hecke eigenforms, period polynomials, and Jacobi theta functions in a novel closed-form expression.

Findings

Provides a closed formula for sums of Hecke eigenforms and their period polynomials.

Determines Fourier expansions for all eigenforms at levels 2, 3, and 5.

Connects modular forms with Jacobi theta series through explicit product formulas.

Abstract

Generalizing a result of~\cite{Z1991} for modular forms of level~one, we give a closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$. We also show that for $N=2$,~3 and~5 this formula completely determines the Fourier expansions of all Hecke eigenforms of all weights on $\Gamma_0(N)$.