Eta-quotients and elliptic curves

Abstract

In this paper we list all the weight 2 newforms f(r) that are products and quotients of the Dedekind eta-function 71(r) = 1/24 17 (1 qn) n=1 where q := e2iT,. There are twelve such f (r), and we give a model for the strong Weil curve E whose Hasse-Weil L-function is the Mellin transform for each of them. Five of the f(T) have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Gr6ssencharacter values. These formulae follow easily from well known q-series infinite product identities. In light of the proof of Fermat's Last Theorem by A. Wiles and R. Taylor, there have been many expository articles describing the nature of the ShimuraTaniyama conjecture, the conjecture which asserts that every elliptic curve E over Q is modular. This implies that the Hasse-Weil L-function of an elliptic curve E with conductor N over Q,