Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank  abelian varieties  that are optimal quotients of attached to newforms. We prove theorems about the ratio , develop tools for computing with , and gather data about certain arithmetic invariants of the nearly abelian varieties of level . Over half of these have analytic rank , and for these we compute upper and lower bounds on the conjectural order of  . We find that there are at least such for which the Birch and Swinnerton-Dyer conjecture implies that is divisible by an odd prime, and we prove for of these that the odd part of the conjectural order of really divides by constructing nontrivial elements of using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.