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 the article, 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 behaviour 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.