  In this article, we investigate a non-Archimedean analogue of a question of
Atkin and Serre. More precisely, we derive lower bounds for the largest prime
factor of non-zero Fourier coefficients of non-CM normalized cuspidal Hecke
eigenforms of even weight $k \geq 2$, level $N$ with integer Fourier
coefficients. In particular, we show that for such a form $f$ and for any real
number $\epsilon>0$, the largest prime factor of the $p$-th Fourier coefficient
$a_f(p)$ of $f$, denoted by $P(a_f(p))$, satisfies $$ P(a_f(p)) ~>~ (\log
p)^{1/8}(\log\log p)^{3/8 -\epsilon} $$ for almost all primes $p$. This
improves on earlier bounds. We also investigate a number field analogue of a
recent result of Bennett, Gherga, Patel and Siksek about the largest prime
factor of $a_f(p^m)$ for $m \geq 2$.
