Number of Polynomials Vanishing on a Basis of Sₘ(Γ₀(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≥ 2 that form a basis for Sₘ(Γ₀(N)). We use these cuspidal forms to embedd X₀(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.