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

In this paper, we study the explicit structure of the graded ring of elliptic modular forms for the congruence subgroup ${\mathrm{\Gamma }}_0(N)$ with N=1,2,3,4,5,6,7,8,9,10,12,16,18,25. More precisely, making use of some relations between Fourier expansions, we determine the ring by a quotient of the polynomial ring in several variables.