On exponential sums involving Fourier coefficients of cusp forms

Abstract

Let a ( n ) be the normalized Fourier coefficient of a holomorphic cusp form of weight k or a Maass cusp form with Laplacian eigenvalue 1 4 + r 2 for SL 2 ( Z ) . We consider exponential sum of the form ∑ n ⩽ X a ( n ) e ( α n 2 + β n ) and obtain the upper bound X 7 8 + ε which is uniformly in α , β ∈ R .