Harmonic Maass forms associated with CM newforms

In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight $k\ge 2$ under the $\xi$-operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maaass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades.