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.
W. Duke. Y. Li. "Harmonic Maass forms of weight ." Duke Math. J. 164 (1) 39 - 113, 15 January 2015. https://doi.org/10.1215/00127094-2838436