 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 $\sigma \in S{L}_2(\mathbb{Z})$ representing a modular cusp $\sigma \infty$, and a genuine translation period $h>0$ with $\sigma T^h{\sigma }^{-1}\in \Gamma$. Then the requirement isconvergent 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 $f=(-1{)}^{k}f=-f$, so the space is zero.