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; $S(f,Q;X)≔{{\sum }^{♭}}_{\frac{n=Q\left(\underline{x}\right)\le X}{\underline{x}\in Z^2,gcd(n,N)=1}}{\lambda }_f\left(n\right),$where $♭$ means that sum runs over the square-free positive integers, ${\lambda }_f\left(n\right)$ denotes the normalised $n$th Fourier coefficients of a Hecke eigenform $f$ of integral weight $k$ for the congruence subgroup ${\Gamma }_0\left(N\right)$ and $Q$ is a primitive integral positive-definite binary quadratic forms of fixed discriminant $D<0$ with the class number $h\left(D\right)=1$. 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.