  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 $\underline{x} =
(x_{1},x_{2},x_{3},x_{4}) \in {\mathbb Z}^{4}$.
