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 .
Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field
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Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 48 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field
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Modular Symbols subspace of dimension 48 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field
2^10 * 6947
CPU time: 0.81 s, Wall time: 0.81 s
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Simple abelian subvariety 431a(1,431) of dimension 1 of J0(431),
Simple abelian subvariety 431b(1,431) of dimension 1 of J0(431),
Simple abelian subvariety 431c(1,431) of dimension 3 of J0(431),
Simple abelian subvariety 431d(1,431) of dimension 3 of J0(431),
Simple abelian subvariety 431e(1,431) of dimension 4 of J0(431),
Simple abelian subvariety 431f(1,431) of dimension 24 of J0(431)
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Simple abelian subvariety 431f(1,431) of dimension 24 of J0(431)
2^11 * 6947