On the number of dominating Fourier coefficients of two newforms

Abstract

Let f = ∑ n ≥ 1 λ f ( n ) n ( k 1 − 1 ) / 2 q n f\!=\!\sum _{n\geq 1} \lambda _f(n)n^{(k_1-1)/2}q^n and g = ∑ n ≥ 1 λ g ( n ) n ( k 2 − 1 ) / 2 q n g\!=\!\sum _{n\geq 1} \lambda _g(n)n^{(k_2-1)/2}q^n be two newforms with real Fourier coeffcients. If f f and g g do not have complex multiplication and are not related by a character twist, we prove that # { n ≤ x   |   λ f ( n ) > λ g ( n ) } ≫ x . \begin{equation*} \#\{n\leq x~|~\lambda _f(n)>\lambda _g(n)\}\gg x. \end{equation*}