= Cancellation of weight-two Eisenstein anomalies
{title2=$\sum_{M\mid N}c(M)E_2(M\tau)\text{ has weight }2\iff\sum_{M\mid N}c(M)/M=0$}
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 $E_2(\tau)-ME_2(M\tau)$, with $M>1$ dividing $N$, span this space.
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