A note on the Fourier coefficients of Hecke–Maass forms

Abstract

To a Hecke–Maass form f ( z ) with Laplace eigenvalue 1 / 4 + ν 2 , we have an automorphic L -function L ( s , sym 2 f ) which is called the symmetric square L -function associated to f . Suppose that λ sym 2 f ( n ) is the n th Fourier coefficient of L ( s , sym 2 f ) . In this paper, the estimate ∑ n ⩽ x λ sym 2 f ( n ) ≪ ν 3 / 4 + ε x 1 / 2 + ε + x 39 64 + ε is established.