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
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
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
for all optimal
quotients of level
;
this table continues to be
of value to number theorists.