Petrow, Ian N.2015-12-022015-12-022015-12-02201510.1007/s00208-014-1166-8https://infoscience.epfl.ch/handle/20.500.14299/121208WOS:000360853500007We 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.text::journal::journal article::research article