On spaces of modular forms spanned by eta-quotients

An eta-quotient of level N is a modular form of the shape f(z)=∏δ|Nη(δz)rδ. We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ0(N) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N. In addition, we prove that if f(z) is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then f(z) is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of J0(2k)(Q).