  Let f be a weight two newform for Gamma_1(N) without complex multiplication.
In this article we study the conductor of the absolutely simple factors B of
the variety A_f over certain number fields L. The strategy we follow is to
compute the restriction of scalars Res_{L/\Q}(B), and then to apply Milne's
formula for the conductor of the restriction of scalars. In this way we obtain
an expression for the local exponents of the conductor N_L(B). Under some
hypothesis it is possible to give global formulas relating this conductor with
N. For instance, if N is squarefree we find that N_L(B) belongs to Z and
N_L(B)*f_L^{dim B}=N^{dim B}, where f_L is the conductor of L.
