On a non-Archimedean analogue of a question of Atkin and Serre (Q6540604)

Let \(f\) be a non-CM normalized cuspidal Hecke eigenform of even weight \(k\geq 2\) for \(\Gamma_0(N)\) having integer Fourier coefficients \(a_f(n)\) (for \(n=1,2,\ldots\)). Let \(\varepsilon > 0\) be a real number. The authors prove (Theorem 1) the following inequality\N\[\NP(a_f(p)) > (\log p)^{1/8} (\log\log p)^{3/8 - \varepsilon}\N\]\Nfor almost all primes \(p\), where \(P(r)\) denote the largest prime factor of a positive integer \(r\). This improves on earlier bounds. Conditionally on GRH, they prove a stronger result (Theorem 3).\N\NThey also investigate a number field analogue (Theorem 9) of a recent result of \textit{M. A. Bennett} et al. [Math. Ann. 382, No. 1--2, 203--238 (2022; Zbl 1525.11034)] about the largest prime factor of \(a_f(p^m)\) for \(m\geq 2\).