  Given two distinct newforms with real Fourier coefficients, we show that the
set of primes where the Hecke eigenvalues of one of them dominate the Hecke
eigenvalues of the other has density at least 1/16. Furthermore, if the two
newforms do not have complex multiplication, and neither is a quadratic twist
of the other, we also prove a similar result for the squares of their Hecke
eigenvalues.
