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The Gamma 1 congruence subgroup isThe space $M_k({\Gamma }_1(N))$ consists of the holomorphic functions $f:\mathfrak{h}\to \mathbb{C}$ satisfyingfor every $\gamma \in {\Gamma }_1(N)$ and which are holomorphic at a cusp for every cusp. Its subspace $S_k({\Gamma }_1(N))$ consists of the forms that vanish at every cusp.

The Gamma 1 congruence subgroup is

The cusps are the ${\Gamma }_1(p)$-orbits of primitive columns $(a,c{)}^T$, with simultaneous negation representing the same point of ${\mathbb{P}}^1(\mathbb{Q})$. For the Gamma 1 congruence subgroup, the standard primitive-vector classification separates the orbits according to $d=\gcd (c,p)$. For each divisor $d\mid p$, the two surviving unit coordinates giveorbits when $p>2$. Hence an odd prime gives

For $p=2$, simultaneous negation is already trivial modulo $2$, so the division by two does not apply. In that case ${\Gamma }_1(2)={\Gamma }_0(2)$ has two cusps, represented by infinity and zero. Thus the Number of cusps of Gamma 1 of prime level is