We give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on GL(2) over Q. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. We include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, we show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed relative to the Sato-Tate measure in the limit as the level goes to infinity.

Using a simple relative trace formula, we compute averages of twisted modular L-values for newforms of cubic level. In the case of Maass forms, we obtain an exact formula. For holomorphic forms of weight k > 2, we obtain an asymptotic formula, which agrees with the estimate predicted by the Lindelöf hypothesis in the weight and level aspects.

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