  For integers $k\geq 2$, we study two differential operators on harmonic weak
Maass forms of weight $2-k$. The operator $\xi_{2-k}$ (resp. $D^{k-1}$) defines
a map to the space of weight $k$ cusp forms (resp. weakly holomorphic modular
forms). We leverage these operators to study coefficients of harmonic weak
Maass forms. Although generic harmonic weak Maass forms are expected to have
transcendental coefficients, we show that those forms which are "dual" under
$\xi_{2-k}$ to newforms with vanishing Hecke eigenvalues (such as CM forms)
have algebraic coefficients. Using regularized inner products, we also
characterize the image of $D^{k-1}$.
