Exponential sums of squares of Fourier coefficients of cusp forms

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms f, we show that if vertical bar alpha - a/q vertical bar <= 1/q(2) with (a, q) = 1, then for any epsilon > 0, Sigma(n <= X) (lambda)f(()n)(2)e(n alpha) <<(f,epsilon) X4/5+epsilon for X-1/5 << q << X-4/5. Moreover, for any epsilon > 0, there exists a set S subset of (0, 1) with mu(S) = 1 such that for every alpha epsilon S, there exists X-0 = X-0(alpha) such that the above inequality holds true for any alpha epsilon S and X >= X-0(alpha). A weaker bound for Maass cusp forms is also established.