Hecke Eigenvalues for Real Quadratic Fields

Abstract

We describe an algorithm to compute the trace of Hecke operators acting on the space of Hilbert cusp forms defined relative to a real quadratic field with class number greater than one. Using this algorithm, we obtain numerical data for eigenvalues and characteristic polynomials of the Hecke operators. Within the limit of our computation, the conductors of the orders spanned by the Hecke eigenvalue for any principal split prime ideal contain a nontrivial common factor, which is equal to a Hecke {\small$L$}-value.