Asymptotic trace formula for the Hecke operators

Given integers m, n and k, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the mth and nth Fourier coefficients of an orthonormal basis of S_k(N) ^∗ (the weight k newforms with fixed square-free level N) provided that |4πmn-k|=o(k13). Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator Tn∗ on S_k(N) ^∗ averaged over k in a short interval. By bounding the second moment of the trace of T_n over a larger interval, we show that the trace of T_n is unusually large in the range |4πn-k|=o(n16). As an application, for any fixed prime p coprime to N, we show that there exists a sequence { k_n} of weights such that the error term of Weyl’s law for T_p is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd et al. (J Eur Math Soc 1(1):51–85, 1999) [Theorem 1.4] with an improved exponent.