  Let $f \in S_{k}(SL_2(\mathbb{Z}))$ be a normalized Hecke eigenforms of
integral weight $k$ for the full modular group. In the article, we study the
average behaviour of Fourier coefficients of $\ell$-fold product $L$-function.
More precisely, we establish the asymptotics of power moments associated to the
sequence $\{\lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(n)\}_{n- {\rm
squarefree}}$ where ${f \otimes f \otimes \cdots \otimes_{\ell} f}$ denotes the
$\ell$-fold product of $f$. As a consequence, we prove results concerning the
behaviour of sign changes associated to these sequences for odd $\ell$-fold
product $L$-function. A similar result also holds for the sequence
$\{\lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(n)\}_{n \in
\mathbb{N}}$.
