  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 $\Gamma_0(N)$.
