  In this paper, we study the explicit structure of the graded ring of elliptic
modular forms for the congruence subgroup $\Gamma_0(N)$ 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.
