= Solution
For an integer weight $k$, a <cusp form> for a <congruence subgroup> $\Gamma$ is a <holomorphic function> on $\mathbb H$ with $f|_k\gamma=f$ for every $\gamma\in\Gamma$, whose <Fourier expansion of a modular form> vanishes at every <modular cusp>. More explicitly, choose $\sigma\in SL_2(\mathbb Z)$ representing a <modular cusp> $\sigma\infty$, and a genuine translation period $h>0$ with $\sigma T^h\sigma^{-1}\in\Gamma$. Then the requirement is
$$
(f|_k\sigma)(\tau)=\sum_{n\geq1}a_{\sigma,n}e^{2\pi in\tau/h},
$$
convergent for sufficiently large imaginary part. Thus the slash translate extends holomorphically at $q_h=0$ and has value zero. The <vector> space of these functions is $S_k(\Gamma)$.
A genuine period is convenient when $-I$ is absent and the weight is odd: the smaller effective <cusp width> may supply a negative rather than positive translation in $\Gamma$, but an actual period always exists, possibly twice that width. Using an actual period avoids imposing an incorrect periodicity. The zero-value condition is independent of the chosen representative or period. If $-I\in\Gamma$ and $k$ is odd, its slash relation forces $f=(-1)^kf=-f$, so the space is zero.
Back to article page
