For almost all primes p, the largest prime factor of a_f(p) + a_g(p) for twist-inequivalent non-CM newforms f,g exceeds (log p)^{1/14} (log log p)^{3/7-ε}, with a similar density-one result for integers and an exponential lower bound under GRH.
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On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms
For almost all primes p, the largest prime factor of a_f(p) + a_g(p) for twist-inequivalent non-CM newforms f,g exceeds (log p)^{1/14} (log log p)^{3/7-ε}, with a similar density-one result for integers and an exponential lower bound under GRH.