Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

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For integers k≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2-k. The operator ξ₂₋ₖ (resp. Dᵏ⁻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 ξ₂₋ₖ to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of Dᵏ⁻¹.