A burgess-like subconvex bound for twisted L-functions

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, X a primitive character of conductor q, and s a point on the critical line Rs = 1/2. It is proved that L(g circle times chi, s) << epsilon,g,s q(1/2-(1/8)(1-20)+epsilon), where epsilon > 0 is arbitrary and theta = 7/64 is the current known approximation towards the RamannJan-Petersson conjecture (which would allow theta = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L-functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch-Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms.