We develop an algorit
hm for determining an explicit set of coset representatives (indexed by lattices) for t
he action of t
he Hecke operators
T(
p),
Tj(
p2) on Siegel modular forms of fixed degree and weig
ht. T
his algorit
hm associates eac
h coset representative wit
h a particular lattice
Ω,
pΛhs/BOD.GIF>
Ωhs/BOD.GIF>
Λ w
here
Λ is a fixed reference lattice. We t
hen evaluate t
he action of t
he Hecke operators on Fourier series. Since t
his evaluation yields incomplete c
haracter sums for
Tj(
p2), we complete t
hese sums by replacing t
his operator wit
h a linear combination of
Ths/BUK.GIF>(
p2), 0≤
hs/BUK.GIF>≤
j. In all cases, t
his yields a clean and simple description of t
he action on Fourier coefficients.