= Almost holomorphic weight-two Eisenstein series
{title2=$E_2^*(\tau)=E_2(\tau)-3/(\pi\operatorname{Im}\tau)$}
The <logarithmic derivative> of the <modular discriminant> gives $E_2=\Delta'/(2\pi i\Delta)$. Differentiating its weight-twelve transformation yields the anomalous term $6c(c\tau+d)/(\pi i)$. Subtracting the displayed inverse-height term cancels that anomaly. The result transforms with weight two under the <modular group> but is not holomorphic, so it is not a holomorphic <modular form>.
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