I prove this in [25]. Several related problems remain: when it may be possible to compute

I was surprised to find that systematic computations using this formula indicate the following conjectural refinement of a result of Mazur [16].

Theorem 3 also suggests a way to compute Tamagawa numbers of motives attached to eigenforms of higher weight. These numbers appear in the conjectures of Bloch and Kato, which generalize the BSD conjecture to motives (see [5]).