  We consider the ring of coinvariants for modular representations of cyclic
groups of prime order. For all cases for which explicit generators for the ring
of invariants are known, we give a reduced Gr\"obner basis for the Hilbert
ideal and the corresponding monomial basis for the coinvariants. We also
describe the decomposition of the coinvariants as a module over the group ring.
For one family of representations, we are able to describe the coinvariants
despite the fact that an explicit generating set for the invariants is not
known. In all cases our results confirm the conjecture of Harm Derksen and
Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an
incidental result, we identify the coefficients of the monomials appearing in
the orbit product of a terminal variable for the three dimensional
indecomposable representation.
