Simultaneous computation of Hecke operators

We present a new method for computing Hecke operators acting on spaces of algebraic modular forms based on an idea of Eichler's. The method significantly improves the complexity of computing operators that can be obtained this way and in addition computes two operators simultaneously. We show that in certain cases the approach can be used to attain the action of the full Hecke algebra with respect to a hyperspecial subgroup and use it to compute Hecke eigenforms for compact forms of symplectic groups.