We present a lower bound for the classical Kloosterman sum S(a, b; c) where (ab,c) = 1 and c is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson’s trace formula, subject to a given condition. Consequently, we achieve a modified version of JS20, Theorem 1.7, where weight k and level N are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator T n acting on the space S k (N), of cusp forms of weight k and level N with (n,N) = 1.
Baier et al. (Thu,) studied this question.
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