  In this paper we define a new type of modular object and construct explicit
examples of such functions. Our functions are closely related to cusp forms
constructed by Zagier which played an important role in the construction by
Kohnen and Zagier of a kernel function for the Shimura and Shintani lifts
between half-integral and integral weight cusp forms. Although our functions
share many properties in common with harmonic weak Maass forms, they also have
some properties which strikingly contrast those exhibited by harmonic weak
Maass forms. As a first application of the new theory developed in this paper,
one obtains a new perspective on the fact that the even periods of Zagier's
cusp forms are rational as an easy corollary.
