Average behaviour of Hecke eigenvalues over certain polynomial

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 $S{L}_2(\mathbb{Z})$ 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_{ α(\underline{x}))+1 \le X \atop \underline{x} \in {\mathbb Z}^{4}} } λ_{f}^{r}(α(\underline{x})+1) , \ \end{split} \end{equation*} where ${\displaystyle \text{sideset}{}^{\mathrm{♭}}\sum }$ means that the sum runs over the square-free positive integers, and ${\lambda }_f(n)$ is the normalised $n^{\mathrm{t}\mathrm{h}}$-Hecke eigenvalue of a normalised Hecke eigenform $f\in S_k(S{L}_2(\mathbb{Z}))$, and $\alpha (\underset{‾}{x})=\frac{1}{2}(x_1^2+x_1+x_2^2+x_2+2(x_3^2+x_3)+4(x_4^2+x_4))\in \mathbb{Q}[x_1,x_2,x_3,x_4]$ is a polynomial, and $\underset{‾}{x}=(x_1,x_2,x_3,x_4)\in {\mathbb{Z}}^4$.