  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.
