Almost holomorphic weight-two Eisenstein series ($E_2^{\ast }(\tau )=E_2(\tau )-3/(\pi \mathrm{Im}\tau )$)

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.

The finite Laurent coefficient $C(z)$ in the Kronecker limit formula is invariant under the modular group. Its Wirtinger derivative satisfies ${\partial }_{z}C=-\pi i(E_2(z)-3/(\pi y))/12$, by differentiating the convergent product in that formula. Differentiating $C(\gamma z)=C(z)$ gives weight-two covariance for $E_2^{\ast }$. Since $\Im (\gamma z)=y/|cz+d{|}^2$, substitution gives the displayed transformation law. This proof needs no prior modularity assertion for the eta product.

For a matrix in the Gamma 0 congruence subgroup, argument dilation conjugates its action to an integral determinant-one matrix for each divisor $M$ of $N$. The anomaly of $E_2(M\tau )$ is $6c(c\tau +d)/(M\pi i)$, so the displayed linear condition cancels it. Necessity follows by testing the matrix with rows $(1,0)$ and $(N,1)$. The combinations $E_2(\tau )-M{E}_2(M\tau )$, with $M>1$ dividing $N$, span this space.