 Bol identity for modular forms

For integer $k<0$, the operator $D=(2\pi i{)}^{-1}d/dz=qd/dq$ sends a weight-$k$ level-one weak modular form to a weight-$2-k$ one after $1-k$ iterations. In the derivative transformation of a weak modular form, every lower derivative coefficient then contains a zero factor. This also holds on the subspace $M_k^{!}$ of forms meromorphic at the cusp.