### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 1018-1042 |

Number of pages | 25 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 468 |

Issue number | 2 |

DOIs | |

State | Published - Dec 15 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

## Fingerprint Dive into the research topics of 'A polyharmonic Maass form of depth 3/2 for SL<sub>2</sub>(Z)'. Together they form a unique fingerprint.

## Cite this

_{2}(Z).

*Journal of Mathematical Analysis and Applications*,

*468*(2), 1018-1042. https://doi.org/10.1016/j.jmaa.2018.08.055