 Almost holomorphic weight-two Eisenstein series

The logarithmic derivative of the modular discriminant gives $E_2={\Delta }^{\prime }/(2\pi i\Delta )$. 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.