Cancellation of weight-two Eisenstein anomalies (${\sum }_{M\mid N}c(M)E_2(M\tau )\text{ has weight }2⟺{\sum }_{M\mid N}c(M)/M=0$)

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.