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; \begin{equation*} \begin{split} S(f, $\mathcal{Q}$; X ) &:= \sideset{}{^{\flat }}\sum_{n= $\mathcal{Q}$(\underline{x}) \le X \atop \gcd(n,N) =1 } λ_{f}(n), \end{split}\end{equation*} where $\mathrm{♭}$ means that sum runs over the square-free positive integers, ${\lambda }_f(n)$ denotes the normalised $n^{\mathrm{t}\mathrm{h}}$ Fourier coefficients of a Hecke eigenform $f$ of integral weight $k$ for the congruence subgroup ${\mathrm{\Gamma }}_0(N)$ 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.