A unified treatment is given of low-weight modular forms on \Gamma_0(N),N=2,3,4, that have Eisenstein series representations. For each N, certainweight-1 forms are shown to satisfy a coupled system of nonlinear differentialequations, which yields a single nonlinear third-order equation, called ageneralized Chazy equation. As byproducts, a table of divisor function andtheta identities is generated by means of q-expansions, and a transformationlaw under \Gamma_0(4) for the second complete elliptic integral is derived.More generally, it is shown how Picard-Fuchs equations of triangle subgroups ofPSL(2,R) which are hypergeometric equations, yield systems of nonlinearequations for weight-1 forms, and generalized Chazy equations. Each trianglegroup commensurable with \Gamma(1) is treated.
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