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

Abstract

Let g g be a holomorphic Hecke newform of level D D and λ g ( n ) \lambda _{g}(n) be its n n -th Fourier coefficient. We prove that the sum S D ( N , α , β , X ) = ∑ X > n ≤ 2 X n ≡ l \textrm {mod} N λ g ( n ) e ( α n β ) \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 β = 1 / 2 \beta =1/2 , α \alpha close to ± 2 q / c 2 D 2 \pm 2\sqrt {q/c^2D_2} , where l l , q q , c c , D 2 D_2 are positive integers satisfying ( l , N ) = 1 (l,N)=1 , c | N c|N , D 2 = D / ( c , D ) D_2=D/(c, D) and X X is sufficiently large. We obtain upper bounds of S D ( N , α , β , X ) \mathcal {S}_{D}(N,\alpha , \beta ,X) for the case of 0 > β > 1 0>\beta >1 and α ∈ R \alpha \in \mathbb {R} .