We develop an algorithm for determining an explicit set of coset representatives (indexed by lattices) for the action of the Hecke operators T(p), T,(p(2)) on Siegel modular forms of fixed degree and weight. This algorithm associates each coset representative with a particular lattice Omega, pA subset of or equal to Omega subset of or equal to (1)/(p) A where A is a fixed reference lattice. We then evaluate the action of the Hecke operators on Fourier series. Since this evaluation yields incomplete character sums for T-1(p(2)), we complete these sums by replacing this operator with a linear combination of T-1(p(2)), 0 less than or equal to l less than or equal to J. In all cases, this yields a clean and simple description of the action on Fourier coefficients. (C) 2002 Elsevier Science (USA).