The exponential sums related to cusp forms in the level aspect

Abstract

Let $N$ be a square-free integer. Let $f\in \mathcal{B}^\ast_k(N)$ (or $\mathcal{B}_\lambda^\ast(N)$) be a primitive (either holomorphic or Maaß) cusp form of level $N$, with $\lambda_f(n)$ denoting the $n$-th Hecke eigenvalue. In this paper, we explicitly determine the dependence on the level aspect for the sum \[\sum_{n\le X}\lambda_f(n) e{\left(n^2\alpha+n\beta \right)},\] which is uniform in any $\alpha,\beta\in \R$ and $X\ge 2$. In addition, we also investigate the analog at the prime arguments.