On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms

Abstract

In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier coefficients, the largest prime factor of $a_f(p)+a_g(p)$ satisfies $P(a_f(p)+a_g(p)) > (\log p)^{1/14} (\log \log p)^{3/7-\epsilon}$ for almost all primes $p$ and for any $\epsilon > 0$. Beyond primes, we apply Brun's sieve to show that a similar phenomenon holds for a set of positive integers with natural density one. The main result is further strengthened under the Generalized Riemann Hypothesis, where we establish exponential growth for the absolute value of $a_f(p)+a_g(p)$ in terms of $p$.Additionally, we derive an interesting result related to the multiplicity one theorem, demonstrating that if the sum $a_f(p)+a_g(p)$ is small for a positive-density subset of primes, then $f$ and $g$ must be twist-equivalent by a quadratic character.