The object of this paper is to initiate a study of the Fourier coefficients of a weight harmonic Maass form and relate them to the complex Galois representation associated to a weight newform, which is the form’s image under a certain differential operator. In this paper, our focus will be on weight dihedral newforms of prime level . In this case we give properties of the Fourier coefficients that are similar to (and sometimes reduce to) cases of Stark’s conjectures on derivatives of -functions. We also give a new modular interpretation of certain products of differences of singular moduli studied by Gross and Zagier. Finally, we provide some numerical evidence that the Fourier coefficients of a mock-modular form whose shadow is exotic are similarly related to the associated complex Galois representation.
"Harmonic Maass forms of weight ." Duke Math. J. 164 (1) 39 - 113, 15 January 2015. https://doi.org/10.1215/00127094-2838436