The average of the exponential sums related to cusp forms

Abstract

In this paper, we investigate the average [Formula: see text] where [Formula: see text] for any [Formula: see text] and [Formula: see text], with [Formula: see text] running over primitive holomorphic cusp forms of weight [Formula: see text] and prime level P. As a result, we prove pointwise uniform bounds with respect to [Formula: see text], [Formula: see text] for the frequency of the P-values, and present that there exist strong oscillations in the exponential sum [Formula: see text], so long as the level of f varies. Concerning the applications of the main result, we consider the quadratic Waring problem associated to the coefficients of cusp forms exhibiting considerable cancellations.