Moments of Fourier coefficients of ℓ-fold product L-function

Let [Formula: see text] be a normalized Hecke eigenform of integral weight [Formula: see text] for the full modular group. In this paper, we study the average behaviour of Fourier coefficients of [Formula: see text]-fold product [Formula: see text]-function. More precisely, we establish the asymptotics of power moment associated to the sequence [Formula: see text], where [Formula: see text] denotes the [Formula: see text]-fold product of [Formula: see text], and [Formula: see text] denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant [Formula: see text] As a consequence, we prove results concerning the behavior of sign changes associated to the sequence [Formula: see text] of Fourier coefficients of [Formula: see text]-fold product [Formula: see text]-function for an odd positive integer [Formula: see text].