On Hecke algebras for Hilbert modular forms

We use twisting operators defined by characters of order two of Cl+(F)Cl+(F) to represent the Hecke algebra acting on spaces of Hilbert modular forms as an extension of a canonically defined subalgebra. The extension is generated by quadratic subextensions which are explicitly parametrized by cosets in Cl+(F)Cl+(F) relative to canonically defined subgroups. Here Cl+(F)Cl+(F) is the ideal class group of F in the narrow sense. In addition, we give numerical examples of characteristic polynomials of Hecke operators for real quadratic fields F and for totally real non-Galois cubic fields F   with nontrivial Cl+(F)/Cl+(F)2Cl+(F)/Cl+(F)2.