On comparing Hecke eigenvalues of cusp forms.

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 λ f (n) . Let g(z) be another distinct primitive holomorphic cusp form of even integral weight ℓ for the full modular group. In this paper, we establish that the set { p | λ f (p j) < λ g (p j) } with 1 ≤ j ≤ 8 has analytic density at least 1 16 [ j + 1 2 ] 2 , and the set { p | λ f (p j) 2 < λ g (p j) 2 } with 1 ≤ j ≤ 4 has analytic density at least 1 4 j (j + 1) 2 . [ABSTRACT FROM AUTHOR]