I am considering the Hecke operators $T\_n$ acting on the space $M\_k(\text{SL}\_2(\mathbb{Z}))$ of weight $k$ modular forms of level 1. Are their eigenvalues always real?


I have read somewhere that the Fourier coefficients of a normalized eigenform are real. The coefficients are precisely the eigenvalues right? Is this correct?



Are the eigenvalues of the Hecke operator always real?
More simply, the eigenvalues are real because the Hecke operators are Hermitian for the Peterson scalar product (a fact which can be checked by a straightforward computation). See for example the introduction by Serre on modular forms in Cours d'arithmétique.


