On the coefficients of symmetric power L-functions

Abstract

We study the signs of the Fourier coefficients of a newform. Let [Formula: see text] be a normalized newform of weight [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the [Formula: see text]th Fourier coefficient of [Formula: see text]. For any fixed positive integer [Formula: see text], we study the distribution of the signs of [Formula: see text], where [Formula: see text] runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power [Formula: see text]-functions.