Average behaviour of Hecke eigenvalues over certain polynomial

Abstract: In the article, we investigate the average behaviour of normalised Hecke eigenvalues over certain polynomials and establish an estimate for the power moments of the normalised Hecke eigenvalues of a normalised Hecke eigenform of weight $k \ge 2$ for the full modular group $SL_2(\mathbb{Z})$ over certain polynomial, given by a sum of triangular numbers with certain positive coefficients. More precisely, for each $r \in \mathbb{N}$, we obtain an asymptotic for the following sum \begin{equation*} \begin{split} \displaystyle{\sideset{}{^{\flat }}\sum_{ \alpha(\underline{x}))+1 \le X \atop \underline{x} \in {\mathbb Z}^{4}} } \lambda_{f}^{r}(\alpha(\underline{x})+1) , \ \end{split} \end{equation*} where $\displaystyle{\sideset{}{^{\flat }}\sum}$ means that the sum runs over the square-free positive integers, and $\lambda_{f} (n)$ is the normalised $n^{\rm th}$-Hecke eigenvalue of a normalised Hecke eigenform $f \in S_{k}(SL_2(\mathbb{Z}))$, and $\alpha(\underline{x}) = \frac{1}{2} \left( x_{1}^{2}+ x_{1} + x_{2}^{2} + x_{2} + 2 ( x_{3}^{2} + x_{3}) + 4 (x_{4}^{2} + x_{4}) \right) \in {\mathbb Q}[x_{1},x_{2},x_{3},x_{4}] $ is a polynomial, and $\underline{x} = (x_{1},x_{2},x_{3},x_{4}) \in {\mathbb Z}^{4}$.