TITLE: Are the eigenvalues of Hecke operators distinct?
QUESTION [15 upvotes]: Given a Hecke operator which acts on the space of modular forms for $\mathrm{SL}_2(\mathbb{Z})$, are the eigenvalues necessarily distinct?

REPLY [2 votes]: Let $f(z)=\sum_{n\geq1}a_f(n)e^{2\pi i n z}$ be a holomorphic cuspidal Hecke eigenform of even weight $k\geq 12$ on the full modular group with $a_f(1)=1$, with $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$.  A conjecture of Atkin and Serre proposes that for every prime $p$,
$|a_f(p)|\gg_{\epsilon} p^{\frac{k-3}{2}-\epsilon}$,
from which it would follow by multiplicativity that the map $n\mapsto a_f(n)$ is finite-to-1.  In this direction, Murty and Murty (http://www.numdam.org/article/BSMF_1987__115__391_0.pdf) proved that there exists an effective constant $c>0$ such that $|a_f(n)|>(\log n)^c$ when $a_f(n)$ is odd.  Their result also extends to $f$ on congruence subgroups.  Assuming the generalized Riemann hypothesis for Hecke $L$-functions, one can replace $(\log n)^c$ with $n^c$ on a subset of the integers of natural density 1.