= Hecke eigenvalues are algebraic integers
{c}
The rational space of level-one <cusp forms> has a full lattice of forms with integral <Fourier coefficients>. It has finite rank by the <valence formula for the modular group>, and is preserved by the <Fourier coefficients of a composite-index Hecke operator> formula. Thus every <Hecke operator> has a monic integral characteristic polynomial on this lattice, and its <eigenvalues> are <algebraic integers>. The common eigenvalue field is totally real by the <Petersson inner product> self-adjointness.
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