Rationality of the Petersson Inner Product of Cohen’s Kernels

By explicitly calculating and then analytically continuing the Fourier expansion of twisted double Eisenstein series $E_{s,k-s}^{*}(z,w; 1/2)$ Diamantis O'Sullivan, we prove a formula Petersson inner product Cohen's kernel one its twists, obtain rationality result. This extends result Kohnen Zagier.