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Given a Hecke operator which acts on the space of modular forms for $\mathrm{SL}_2(\mathbb{Z})$, are the eigenvalues necessarily distinct?

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I would also like to know if there is a modular form for $\mathrm{SL}_2(\mathbb{Z})$ with a vanishing Hecke eigenvalue. –  GH from MO Aug 28 '12 at 14:13
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Just to clarify, are you asking if a particular Hecke operator $T_\ell$ acting on the space $M_k(\mathrm{SL}_2(\mathbb{Z}))$ of forms of fixed weight $k$ for the full modular group has distinct eigenvalues (i.e. square-free characteristic polynomial)? –  Ramsey Aug 28 '12 at 14:28
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@Will: That bound is right in weight $2$, but in general the bound for weight $k$ is $2p^{(k-1)/2}$. –  Ramsey Aug 28 '12 at 14:45
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It's conjectured (Maeda's conjecture) that for any prime $p$ and any even weight $k \ge 12$, the characteristic polynomial of $T_p$ acting on $S_k(\operatorname{SL}_2(\mathbb{Z}))$ is irreducible over $\mathbb{Q}$; in particular, it has no multiple roots. Various authors (eg Buzzard, Kleinerman) have verified this for lots of values of $p$ and $k$. If I remember correctly, in all cases that have been tested the stronger statement holds that the char polys all have Galois group equal to the full symmetric group of the appropriate degree. –  David Loeffler Aug 28 '12 at 15:22
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The way I understand the question: is it true that for any Hecke eigenform $f\in S_k(\operatorname{SL}_2(\mathbb{Z}))$ the Hecke eigenvalues $\lambda_f(p)$, normalized so that they lie between $\pm 2p^{(k-1)/2}$, are all distinct? –  GH from MO Aug 28 '12 at 15:30
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