 Hecke operators are self-adjoint for the Petersson inner product

On level-one cusp forms, determinant-normalized Hecke operators satisfy $⟨T_{n}f,g⟩=⟨f,T_{n}g⟩$. Unfolding their finite correspondences transfers a matrix to its inverse; its adjugate has the same determinant and reverses the correspondence. Since these operators commute, the cusp space has a common orthonormal basis of Hecke eigenforms, with real eigenvalues.