Serre derivative ($D_{k}f=(2\pi i{)}^{-1}{f}^{\prime }-(k/12)E_{2}f$)

The Serre derivative sends a level-one weight-$k$ modular form to weight $k+2$. Differentiating $f(-1/z)=z^{k}f(z)$ produces the extra term $k{z}^{k+1}f/(2\pi i)$; the anomalous transformation of the Eisenstein series of weight two cancels it. Translation invariance and the Fourier expansion of a modular form verify the remaining holomorphy conditions. For $k>0$, its constant coefficient is $-k{a}_0(f)/12$, so it is a cusp form exactly when $f$ is.

For modular forms $f,g$ of weights $k,\ell$, respectively, the displayed combination cancels the anomaly in derivative transformation of a weak modular form and is a weight-$k+\ell +2$ cusp form. Equivalently, for weights $k,\ell$ in the order $f,g$, a conventional first bracket is $kfDg-\ell gDf$; its negative is equally modular. Differentiation kills each constant Fourier coefficient, proving cusp vanishing.