  We use a relative trace formula on GL(2) to compute a sum of twisted modular
L-functions anywhere in the critical strip, weighted by a Fourier coefficient
and a Hecke eigenvalue. When the weight k or level N is sufficiently large, the
sum is nonzero. Specializing to the central point, we show in some cases that
the resulting bound for the average is as good as that predicted by the
Lindelof hypothesis in the k and N aspects.
