According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian
variety then its L-function must capture substantial part of the arithmetic
properties of A. The smallest number field L where A has all its endomorphisms
defined must also have a role. This article deals with the relationship between
these two objects in the specific case of modular abelian varieties A_f/Q
associated to weight 2 newforms for the modular group Gamma_1(N).