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In this paper, we investigate functions introduced by Knopp and complete themto non-holomorphic bimodular forms of positive integral weight related toindefinite binary quadratic forms. We study further properties of ourcompletions, which in turn motivates certain local cusp forms. We then definemodular analogues of negative weight of our local cusp forms, which are locallyharmonic Maass forms with continuously removable singularities. They admitlocal splittings in terms of Eichler integrals, and a realization as outputs ofa certain theta lift.

A modular framework of functions of Knopp and indefinite binary quadratic forms