For a prime p, we compute the minimum of \(m_P\) over all possible special fundamental polygons (in the sense of Kulkarni) P for \(\Gamma _0(p)\) , where \(m_P\) denotes the largest denominator in the cusp set of P. This minimum value \(m(\Gamma _0(p))\) is expressed in terms of the solution set to a certain finite system of quadratic Diophantine equations and inequalities, and can be explicitly computed with time complexity \(O(p^2)\) . From this computation, we obtain freely independent generators of \(\Gamma _0(p)\) that have 0 or p in their (2, 1) components, answering a question of Kulkarni. By an analogous argument, we establish that \(\Gamma _0(N)\) admits freely independent generators whose Frobenius norms satisfy O(N) for \(N=p\) or \(N=pq\) , where p and q are odd primes satisfying \(0\leqslant |\sqrt{p}-\sqrt{q}|<\sqrt{2}\) .
