Periods of modular forms on ${\Gamma }_0(N)$ and products of Jacobi theta functions

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 ${\Gamma }_0(N)$.