Niebur-Poincaré Series and Regularized Inner Products

Zagier introduced weight 2k cusp forms f_{k,D} associated to quadratic forms of positive
discriminant D. We determine the Fourier coefficients of analogues of these functions of
weight 2, higher level, and negative discriminant and relate them to traces of singular moduli
of Niebur-Poincaré series. This allows us to compute regularized inner products of these
functions, which in the higher weight case have been related to evaluations of higher
Green's functions at CM-points.