A modular framework of generalized Hurwitz class numbers

We discover a non-trivial rather simple relation between the mock modular generating functions of the level 1 and level N Hurwitz class numbers. This relation yields a holomorphic modular form of weight 32 and level 4N, where N > 1 is stipulated to be odd and square-free. We extend this observation to a non-holomorphic framework and obtain a higher level non-holomorphic Zagier Eisenstein series as well as a higher level preimage of it under the Bruinier--Funke operator ₁₂. All of these observations are deduced from a more general inspection of a certain weight 12 Maass--Eisenstein series of level 4N at its spectral point s=34. This idea goes back to Duke, Imamo\=glu and T\'oth in level 4 and relies on the theory of so-called sesquiharmonic Maass forms. We calculate the Fourier expansion of our sesquiharmonic preimage and of its shadow. We conclude by offering an example if N=5 or N=7 and we provide the SAGE code to compute the Fourier coefficients involved.