A lower bound for classical Kloosterman sums and an application

In this paper, we present a lower bound for the classical Kloosterman sum [Formula: see text] where [Formula: see text] and [Formula: see text] 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 Theorem 1.7 from J. Jung and N. Talebizadeh Sardari [Asymptotic trace formula for the Hecke operators, Math. Ann. 378(1–2) (2020) 513–557], where weight [Formula: see text] and level [Formula: see text] are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator [Formula: see text] acting on the space [Formula: see text], of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text].