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 $\{ \lambda_{ f \otimes f \otimes \cdots \otimes_{ \ell} f}(n)\}_{ n \in \mathbb{ N}}$.
Categories: math.NT