A modular framework for generalized Hurwitz class numbers III

In $2003$, Pei and Wang introduced higher level analogs of the classical Cohen--Eisenstein series. In recent joint work with Beckwith, we found a weight $\frac{1}{2}$ sesquiharmonic preimage of their weight $\frac{3}{2}$ Eisenstein series under $\xi_{\frac{1}{2}}$ utilizing a construction from seminal work by Duke, Imamo\={g}lu and T\'{o}th. In further joint work with Beckwith, when restricting to prime level, we realized our preimage as a regularized Siegel theta lift and evaluated its (regularized) Fourier coefficients explicitly. This relied crucially on work by Bruinier, Funke and Imamo\={g}lu. In this paper, we extend both works to higher weights. That is, we provide a harmonic preimage of Pei and Wang's generalized Cohen--Eisenstein series under $\xi_{\frac{3}{2}-k}$, where $k > 1$. Furthermore, when restricting to prime level, we realize them as outputs of a regularized Shintani theta lift of a higher level holomorphic Eisenstein series, which builds on recent work by Alfes and Schwagenscheidt. Lastly, we evaluate the regularized Millson theta lift of a higher level Maass--Eisenstein series, which is known to be connected to the Shintani theta lift by a differential equation by earlier work of Alfes and Schwagenscheidt.