= Serre derivative
{c}
{title2=$D_kf=(2\pi i)^{-1}f'-(k/12)E_2f$}
The Serre derivative sends a level-one weight-$k$ <modular form> to weight $k+2$. Differentiating $f(-1/z)=z^kf(z)$ produces the extra term $kz^{k+1}f/(2\pi i)$; 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 $-ka_0(f)/12$, so it is a <cusp form> exactly when $f$ is.
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