Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / b / Solution

Let $\rho =e^{2\pi i/3}$. For a nonzero meromorphic modular form $f$ of weight $k$, the valence formula for the modular group isHere $v_z(f)$ is positive for a zero and negative for a pole, and the factors $1/2$ and $1/3$ account for the stabilizers of the two elliptic points. Equivalently, the weighted number of zeros minus poles in $\Gamma (1)\backslash \mathfrak{h}$ is $k/12-v_{\infty }(f)$.