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In this paper, we study the explicit structure of the graded ring of ellipticmodular forms for the congruence subgroup $\Gamma_0(N)$ withN=1,2,3,4,5,6,7,8,9,10,12,16,18,25. More precisely, making use of somerelations between Fourier expansions, we determine the ring by a quotient ofthe polynomial ring in several variables.

An explicit structure of the graded ring of modular forms of small level