  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
${\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.
