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The ratio $L(A,1)/\Omega _A$

Extending classical work on elliptic curves, A. Agashé and I proved the following theorem in [2].

Theorem 1   Let m be the largest square dividing N. The ratio $L(A,1)/\Omega _A$ is a rational number that can be explicitly computed, up to a unit (conjecturally 1) in Z[1/(2m)].

The proof uses modular symbols combined with an extension of the argument used by Mazur in [17] to bound the Manin constant. The ratio $L(A,1)/\Omega _A$ is expressed as the lattice index of two modules over the Hecke algebra. I expect the method to give similar results for special values of twists, and of L-functions attached to eigenforms of higher weight. I have computed $L(A,1)/\Omega _A$ for all optimal quotients of level $N\leq 1500$; this table continues to be of value to number theorists.