The main result of this paper is a characterization of the abelian varieties B=K defined over\udGalois number fields with the property that the L-function L(B=K; s) is a product of L-functions of\udnon-CM newforms over Q for congruence subgroups of the form T1(N). The characterization involves\udthe structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology\udclass attached to B=K.\udWe call the varieties having this property strongly modular. The last section is devoted to the study\udof a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which\udthe general results of the paper can be applied, we prove the strong modularity of some particular\udabelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly\udmodular varieties by twisting.
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