Additive twists of Fourier coefficients of modular forms

We study sums of the form ∑n⩽Na(n)e2πiαn, where α is any real number and the a(n) are the Fourier coefficients of either a holomorphic cusp form, a Maass cusp form, or the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform in both α and the form itself. We also improve a bound on a sum of the form ∑n⩽Na(n)e2πi(αn+βnθ), where the a(n) are the Fourier coefficients of a holomorphic cusp form, α and β are any real numbers, and 0⩽θ<1. This last bound is uniform in α, but not with respect to the form.