Sums of Fourier coefficients of cusp forms of level $D$ twisted by exponential functions over arithmetic progressions

Abstract:

Let $g$ be a holomorphic Hecke newform of level $D$ and $\lambda _{g}(n)$ be its $n$-th Fourier coefficient. We prove that the sum $\mathcal {S}_{D}(N,\alpha , \beta ,X)=\sum _{\substack {X<n\leq 2X\\ n\equiv l \text {\textrm {mod}} N}}\lambda _{g}(n)e(\alpha n^\beta )$ has an asymptotic formula for the case of $\beta =1/2$, $\alpha$ close to $\pm 2\sqrt {q/c^2D_2}$, where $l$, $q$, $c$, $D_2$ are positive integers satisfying $(l,N)=1$, $c|N$, $D_2=D/(c, D)$ and $X$ is sufficiently large. We obtain upper bounds of $\mathcal {S}_{D}(N,\alpha , \beta ,X)$ for the case of $0<\beta <1$ and $\alpha \in \mathbb {R}$.