Oscillations of Fourier coefficients over the sparse set of integers

Let f is an element of S-k(Gamma(0)(N))be a normalized Hecke eigenforms of integral weightkand level N >= 1. In the article, we establish the asymptotics of power moment associated to the sequences {lambda(f circle times f circle times f )(Q(x))}Q(is an element of)S(D),(x is an element of)Z(2 )and {lambda(2 )(f circle times sym)(f)(Q(x))}Q(is an element of)S(D),(x is an element of)Z(2 )where S(D )denotes the set of inequivalent primitive integral positive-definite binary quadraticforms (reduced forms) of fixed discriminant D<0.As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences.