A polyharmonic Maass form of depth 3/2 for SL2(Z)

Scott Ahlgren, Nickolas Andersen, Detchat Samart

Research output: Contribution to journalArticlepeer-review


Duke, Imamoḡlu, and Tóth 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 Imamoḡlu. 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.

Original languageEnglish (US)
Pages (from-to)1018-1042
Number of pages25
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - Dec 15 2018


  • Coefficients of real analytic modular forms
  • Inner products
  • Polyharmonic Maass forms

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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