Number of Polynomials Vanishing on a Basis of $S_m({\mathrm{\Gamma }}_0(N))$

In this paper we find the number of homogeneous polynomials of degree d such that they vanish on cuspidal modular forms of even weight $m\ge 2$ that form a basis for $S_m({\mathrm{\Gamma }}_0(N))$. We use these cuspidal forms to embedd $X_0(N)$ to projective space and we find the Hilbert polynomial of the graded ideal of the projective curve that is the image of this embedding.