  We complete several generating functions to non-holomorphic modular forms in
two variables. For instance, we consider the generating function of a natural
family of meromorphic modular forms of weight two. We then show that this
generating series can be completed to a smooth, non-holomorphic modular form of
weights 3/2 and two. Moreover, it turns out that the same function is also a
modular completion of the generating function of weakly holomorphic modular
forms of weight 3/2, which prominently appear in work of Zagier on traces of
singular moduli.
