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We present an elementary uniform lower bound for the classical Kloosterman sum $S(a,b;c)$ under the condition of its non-vanishing and $(ab,c)=1$, with $c$ being an odd integer. We then apply this lower bound for Kloosterman sums to derive an explicit lower bound in the Petersson's trace formula, subject to a pertinent condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, wherein the parameters $k$ and $N$ are permitted to vary independently., Comment: 11 pages

In \cite{BS}, Balasubramanyam and the second named author derived the first moment of the pair correlation function for Hecke angles lying in small subintervals of $[0,1]$ upon averaging over large families of Hecke newforms of weight $k$ with respect to $\Gamma_0(N)$. The goal of this article is to study higher moments of this pair correlation function. For an integer $r \geq 2$, we present bounds for its $r$-th power moments. We apply these bounds to record lower order error terms in the computation of the second and third moments. As a result, one can obtain the convergence of the second and third moments of this pair correlation function for suitably small intervals, and under appropriate growth conditions for the size of the families of Hecke newforms., Comment: This is a significantly revised version incorporating a study of the $r$-th power moments of the above-mentioned pair correlation function for all integers $r \geq 2$. The previous version only discussed the second moment. 47 pages