A <modular function> of weight $k$ and level $\Gamma(1)$ is a <meromorphic function> $f:\mathfrak h\to\mathbb C$ satisfying
$$
f(\gamma\tau)=(c\tau+d)^k f(\tau)
$$
for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma(1)$, and having a meromorphic Fourier expansion at the cusp infinity. Equivalently, $f|_k\gamma=f$ for the <slash operator for modular forms>.

A <modular form> is a modular function that is <holomorphic function>[holomorphic] on $\mathfrak h$ and <holomorphic at a cusp>[holomorphic at infinity]. Its Fourier expansion therefore has the form
$$
f(\tau)=\sum_{n\geq0}a_nq^n,
\qquad q=e^{2\pi i\tau}.
$$
Since $-I\in\Gamma(1)$, a nonzero level-one form must have even weight.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 2 / a / Solution

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A modular function of weight $k$ for a subgroup $\Gamma\leq SL_2(\mathbb Z)$ is a meromorphic function on the <complex upper half-plane> that satisfies
$$
f(\gamma\tau)=(c\tau+d)^k f(\tau)
$$
for $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma$ and is meromorphic at every cusp.

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