Average Values of Modular L-series Via the Relative Trace Formula

First we reprove, using representation theory and the relative trace formula of Jacquet, an average value result of Duke for modular L-series at the critical center. We also establish a refinement. To be precise, the L-value which appears is L(1/2, f)L(1/2,f,χ) (divided by the Petersson norm of f), and the average is over newforms f of prime level N and coefficients a_p(f), with χ being an odd quadratic Dirichlet character of conductor -D and associated quadratic field K. For any prime p not dividing ND, the asymptotic as N goes to infinity is governed by a measure μ_p, which is the Plancherel measure at p when χ(p)=-1, but is new if χ(p)=1; as p goes to infinity both measures approach the Sato-Tate measure. A particular consequence of our refinement is that for any non-empty interval J in [-2,2], there are infinitely many primes N, which are inert in K, such that for some f of level N, a_p(f) is in J and L(1/2, f)L(1/2,f,χ) is non-zero.