  We determine the space of 1-point correlation functions associated with the
Moonshine module: they are precisely those modular forms of non-negative
integral weight which are holomorphic in the upper half plane, have a pole of
order at most 1 at infinity, and whose Fourier expansion has constant 0. There
are Monster-equivariant analogues in which one naturally associates to each
element in the Monster a modular form of fixed weight k, the case k=0
corresponding to the original ``Moonshine'' of Conway and Norton.
