On comparing Hecke eigenvalues of cusp forms

Abstract

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by \(\lambda_{f}(n)\). Let g(z) be another distinct primitive holomorphic cusp form of even integral weight \(\ell\) for the full modular group. In this paper, we establish that the set $$\{p| \lambda_f(p^j)<\lambda_g(p^j)\} \quad \mbox{with } 1\leq j \leq 8$$ has analytic density at least \(\frac{1}{16[\frac{j+1}{2}]^{2}}\), and the set $$\{p| \lambda_f(p^j)^2<\lambda_g(p^j)^2\} \quad \mbox{with } 1\leq j \leq 4$$ has analytic density at least \(\frac{1}{4j(j+1)^{2}}\).