Locally harmonic Maass forms of positive even weight

We twist Zagier's function f(k,D) by a sign function and a genus character. Assuming weight 0 < k equivalent to 2(mod 4), and letting D be a positive non-square discriminant, we prove that the obstruction to modularity caused by the sign function can be corrected obtaining a locally harmonic Maa ss form or a local cusp form of the same weight. In addition, we provide an alternative representation of our new function in terms of a twisted trace of modular cycle integrals of a Poincar & eacute; series due to Petersson.