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Formulas for the coefficients of half-integral weight harmonic Maass forms

Recently, Bruinier and Ono proved that the coefficients of certain weight -1/2 harmonic weak Maaß forms are given as "traces" of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight $3/2+k$, $k$ even, and weight $1/2-k$, $k$ odd, by extending the theta lift of Bruinier-Funke and Bruinier-Ono. Moreover, we generalize their result to include \textit{twisted} traces of singular moduli using earlier work of the author and Ehlen. Employing a duality result between weight $k$ and $2-k$, we are able to cover all half-integral weights. We also show that the non-holomorphic part of the theta lift in weight $1/2-k$, $k$ odd, is connected to the vanishing of the special value of the $L$-function of a certain derivative of the lifted function.