We derive a Motohashi-type formula for the cubic moment of central values of -functions of level cusp forms twisted by quadratic characters of conductor , previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for -functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.