Remark 3.7 Let be a newform.
When is an elliptic curve, Theorem~3.6 implies
that the modular degree divides the congruence number (since for an
elliptic curve, the modular degree and modular exponent are the
same), and that
(since for an elliptic curve, the modular number is the square of the
modular exponent). In general, for a higher dimensional
newform quotient, the divisibility
need not hold. For example, there is a newform
of degree in such that
Note that is prime and mod~ multiplicity one fails for .