Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 26 / 2 / i

Let $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$ and $j=c\tau +d$. Differentiating the transformation law of a weight-$k$ modular form gives the derivative transformation of a weak modular formThe corresponding anomaly for $Dg$ has coefficient $\ell c/(2\pi i)$. Consequently these anomalies cancel in $\ell gDf-kfDg$, which transforms with weight $k+\ell +2$. The derivatives are holomorphic on the complex upper half-plane. In a Fourier expansion, $D({\sum }_{n\ge 0}{a}_n{q}^n)={\sum }_{n\ge 1}n{a}_n{q}^n$, so each term in the combination has zero constant coefficient and no negative powers. There is just one modular cusp up to the full modular group. ThereforeThis is, up to the chosen sign, the first Rankin-Cohen bracket.