Diagonalizing Hilbert cusp forms (Q1902038)

It is well known that the space of Hilbert cusp forms \(S_k ({\mathcal N}, \psi)\) of Hecke character \(\psi\) decomposes into a direct sum of common eigenspaces for the Hecke operators \(\{T_p \mid p \nmid {\mathcal N}\}\) which are invariant under the Hecke operators \(\{T_q \mid q |{\mathcal N}\}\). Unfortunately in general there is no basis consisting of eigenforms for \(\{T_q \mid q |{\mathcal N}\}\). In the present paper a certain modification \(C_q (\psi_Q)\) of the Hecke operator \(T_q\) is defined which turns out to coincide with this operator on the subspace of new-forms and allows one to decompose the space \(S_k({\mathcal N}, \psi)\) of cusp forms into a direct sum of common eigenspaces for \(\{T_p,C_q (\psi_Q) \mid p \nmid {\mathcal N}, q |{\mathcal N}\}\) each of which is of dimension one. Furthermore there is a generator whose \(p\)th resp. \(q\)th Fourier coefficient is an eigenvalue of \(T_p\) resp. \(C_q (\psi_Q)\), the latter being bounded by \(2N(q)^{k/2}\).