Let $\rho=e^{2\pi i/3}$. For a nonzero <meromorphic modular form> $f$ of weight $k$, the <valence formula for the modular group> is
$$
v_\infty(f)+\frac12v_i(f)+\frac13v_\rho(f)
+\sum_{z\in\Gamma(1)\backslash\mathfrak h\atop z\ne i,\rho}v_z(f)
=\frac{k}{12}.
$$
Here $v_z(f)$ 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 $k/12-v_\infty(f)$.

Solution

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

For a nonzero meromorphic modular form $f$ of weight $k$,
$$
v_\infty(f)+\frac12v_i(f)+\frac13v_\rho(f)
+\sum_{z\ne i,\rho}v_z(f)=\frac{k}{12},
\qquad \rho=e^{2\pi i/3},
$$
where the sum takes one representative from each modular-group orbit.


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