Twisted Kronecker series and periods of modular forms on ${\mathrm{\Gamma }}_0(N)$

We introduce an infinite family of Kronecker series twisted by characters. As an application, we give a closed formula for the sum of all Hecke eigenforms on ${\mathrm{\Gamma }}_0(N)$ multiplied by their twisted period polynomials in terms of the product of those twisted Kronecker series, when N is square free. This extends an identity of Zagier among period polynomials, Hecke eigenforms and a quotient of Jacobi theta series.