First Moment of Hecke Eigenvalues at the Integers Represented by Binary Quadratic Forms

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; S(f,Q;X)≔∑♭n=Q(x̲)≤Xx̲∈Z2,gcd(n,N)=1λf(n),where ♭ means that sum runs over the square-free positive integers, λf(n) denotes the normalised nth Fourier coefficients of a Hecke eigenform f of integral weight k for the congruence subgroup Γ0(N) and Q is a primitive integral positive-definite binary quadratic forms of fixed discriminant D<0 with the class number h(D)=1. As a consequence, we determine the size, in terms of conductor of associated L-function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by Q. This work is an improvement and generalisation of the previous results.