A lower bound for the discrepancy in a Sato-Tate type measure

Let $S_k(N)$ denote the space of cusp forms of even integer weight $k$ and level $N$. We prove an asymptotic for the Petersson trace formula for $S_k(N)$ under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato-Tate distribution for levels not divisible by $8$. This generalizes a result of Jung and Sardari for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda for the distribution of eigenvalues $\lambda_{p^2}(f)$ where $f$ is a Hecke eigenform and $p$ is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights $k_n$ such that discrepancy in the analogue distribution obtained by Omar and Mazhouda has a lower bound.