Asymptotic of Fourier coefficients of Hecke eigenforms at integers represented by a binary quadratic form of a fixed discriminant

In this article, we establish the general asymptotic behaviour of the Fourier coefficients of the Hecke eigenforms supported at the integers represented by a primitive integral positive-definite binary quadratic form of the fixed discriminant $D ＜ 0$ under the assumption that the class number $h(D) = 1$.
As a consequence, we also obtain a quantitative result for the number of sign changes of the sequence of normalised Fourier coefficients $\lambda_{f}(n)$ of Hecke eigenforms $f$ over the indices, represented by a primitive integral positive definite binary quadratic form of the fixed discriminant $D ＜ 0$ when class number $h(D) = 1$; in the interval $(X,2X]$ for sufficiently large $X$. Moreover, under the assumption of the Lindel{öf hypothesis, the above-said sequence has at least $X^{\frac{1}{2}-\epsilon}$ many sign changes.