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

Abstract

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let ag(n) be its n-th Fourier coefficient. We consider the sum \({S_1} = \sum {_{X 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}\) with o(x) ∈ Cc∞ (0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.