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; \begin{equation*} \begin{split} S(f, \mathcal{Q}; X ) &:= \sideset{}{^{\flat }}\sum_{n= \mathcal{Q}(\underline{x}) \le X \atop \gcd(n,N) =1 } \lambda_{f}(n), \end{split}\end{equation*} where $\flat$ means that sum runs over the square-free positive integers, $\lambda_{f}(n)$ denotes the normalised $n^{\rm th}$ Fourier coefficients of a Hecke eigenform $f$ of integral weight $k$ for the congruence subgroup $\Gamma_{0}(N)$ and $\mathcal{Q}$ 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 $\mathcal{Q}$. This work is an improvement and generalisation of the previous results.