Periods of modular forms on Γ0 (N) and products of Jacobi theta functions

Generalizing a result of [15] 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).