Additive twists of Fourier coefficients of modular forms

We study sums of the form E(n <= N)a(n)e(2 pi i alpha n), where alpha is any real number and the a(n) are the Fourier coefficients of either a holomorphic cusp form, a Maass cusp form, or the symmetricsquare lift of a holomorphic cusp form. We obtain bounds that are uniform in both a and the form itself. We also improve a bound on a sum of the form Sigma(n <= N)a(n)e(2 pi i(alpha n+beta n theta)), where the a(n) are the Fourier coefficients of a holomorphic cusp form, alpha and beta are any real numbers, and 0 <= 0 < 1. This last bound is uniform in alpha, but not with respect to the form. (C) 2012 Elsevier Inc. All rights reserved.