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

A polyharmonic Maass form of depth 3/2 for SL_2(Z)