First moment of Hecke eigenvalues at the integers represented by binary quadratic forms

Abstract

In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum; where means that sum runs over the square-free positive integers, denotes the normalised th Fourier coefficients of a Hecke eigenform of integral weight for the congruence subgroup and is a primitive integral positive-definite binary quadratic forms of fixed discriminant with the class number . As a consequence, we determine the size, in terms of conductor of associated -function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by . This work is an improvement and generalisation of the previous results.