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 χ_D, and let a_g(n) be its n-th Fourier coefficient. We consider the sum and prove that S1 has an asymptotic formula when β = 1/2 and αis close to \pm \mathrm{2}\sqrt{q/D} for positive integer q\le X/\mathrm{4} and X sufficiently large. And when 0<β<1 and α, β fail to meet the above condition, we obtain upper bounds of S₁. We also consider the sum {{\displaystyle S}}_{\mathrm{2}}={\displaystyle {\sum }_{n>\mathrm{0}}{{\displaystyle a}}_g}(n)e({an}^{\beta })\varphi (n/X) with \varphi (x)\in {{\displaystyle C}}_c^{\infty }(\mathrm{0},+\infty ) and prove that S₂ has better upper bounds than S₁ at some special α and β.

Keywords

exponential sums / cusp form / Fourier coefficients

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