  Let $n$ be a prime or its square. We prove that the congruence subgroup
$\Gamma_0(n)$ admits a free product decomposition into cyclic factors in such a
way that the $(2,1)$-component of each cyclic generator is either $n$ or $0$,
answering a conjecture of Kulkarni. We can also require that the Frobenius norm
of each generator is less than $2n-1$. A crucial observation is that if $P$
denotes the convex hull of the extended Farey sequence of order $\lfloor
\sqrt{n} \rfloor$ in the hyperbolic plane $\mathbb{H}^2$, then the projection
$\pi: \mathbb{H}^2\to \mathbb{H}^2/\Gamma_0(n)$ is injective on the interior of
$P$ and each connected component of $\pi(\mathbb{H}^2)\setminus\pi(P)$ is
either an order-three cone of area $\pi/3$ or an ideal triangle. Denoting by
$m(\Gamma_0(n))$ the minimum of the largest denominator in the cusp set of $Q$
where $Q$ ranges over all possible special (fundamental) polygons for
$\Gamma_0(n)$, we establish the inequality $ \lfloor \sqrt{n} \rfloor \le
m(\Gamma_0(n))\le \lfloor \sqrt{4n/3} \rfloor$, and completely characterize the
cases in which the bounds are achieved. We also prove analogous results when
$n$ is the multiplication of two sufficiently close odd primes.
