Structure of the space of polyharmonic Maass forms with an application to L-values

A polyharmonic Maass form introduced by Lagarias and Rhoades is a generalization of a harmonic Maass form in a sense. In this paper, we construct some bases for the spaces of polyharmonic Maass forms from the structure of the spaces of weakly holomorphic modular forms for prime levels or square-free levels. Furthermore, we relate the polyharmonic Maass forms of depth 1 or depth 3/2 to the L-series of holomorphic cusp forms for square-free levels.