On Higher Order Weierstrass Points on \(X_0(N)\)

Let \(\Gamma\) be the Fuchsian group of the first kind. For an even integer \(m\ge 4\), we describe the space \(H^{m/2}\left(\mathfrak R_\Gamma\right)\) of \(m/2\)--holomorphic differentials in terms of a subspace \(S_m^H(\Gamma)\) of the space of (holomorphic) cuspidal modular forms \(S_m(\Gamma)\). This generalizes classical isomorphism \(S_2(\Gamma)\simeq H^{1}\left(\mathfrak R_\Gamma\right)\). We study the properties of \(S_m^H(\Gamma)\). As an application, we describe the algorithm implemented in SAGE for testing if a cusp at \(\infty\) for non-hyperelliptic \(X_0(N)\) is a \(\frac{m}{2}\)-Weierstrass point.