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 $m$-th and $n$-th Fourier coefficients of an orthonormal basis of $S_k(N)^*$ (the weight $k$ newforms with fixed square-free level $N$) provided that $|4 \pi \sqrt{ mn}- k|=o(k^{ \frac{ 1}{ 3}})$. Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $\mathcal{ T}_n^*$ on $S_k(N)^*$ averaged over $k$ in a short interval. By bounding the second moment of the trace of $\mathcal{ T}_{ n}$ over a larger interval, we show that the trace of $\mathcal{ T}_n$ is unusually large in the range $|4 \pi \sqrt{ n}- k| = o(n^{ \frac{ 1}{ 6}})$. As an application, for any fixed prime $p$ with $\gcd(p,N)=1$, we show that there exists a sequence $\{ k_n\}$ of weights such that the error term of Weyl's law for $\mathcal{ T}_p$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd, Jakobson and Sarnak~\cite[Theorem 1.4]{ Gamburd} with an improved exponent.
Categories: math.NT math-ph math.MP