  Duke, Imamoglu, and Toth constructed a polyharmonic Maass form of level 4
whose Fourier coefficients encode real quadratic class numbers. A more general
construction of such forms was subsequently given by Bruinier, Funke, and
Imamoglu. Here we give a direct construction of such a form for the full
modular group and study the properties of its coefficients. We give
interpretations of the coefficients of the holomorphic parts of each of these
polyharmonic Maass forms as inner products of certain weakly holomorphic
modular forms and harmonic Maass forms. The coefficients of square index are
particularly intractable; in order to address these, we develop various
extensions of the usual normalized Peterson inner product using a strategy of
Bringmann, Ehlen and Diamantis.
