Average behaviour of Fourier coefficients of j-symmetric power L-functions over some polynomials

Abstract

We establish the asymptotics of the second moment of the coefficient of j-th symmetric poower lift of classical Hecke eigenforms over certain polynomials, given by a sum of triangular numbers with certain positive coefficients. More precisely, for each j∈N, we obtain asymptotics for the sums given by (Formula presented.),where λsym2(n) denotes the coefficient of j-th symmetric power lift of classical Hecke eigenforms f, the polynomials α and β are given by (Formula presented.) and (Formula presented.) © The Author(s), under exclusive licence to Akadémiai Kiadó, Budapest, Hungary 2024.