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Author: William A. Stein
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Compute Environment: Ubuntu 18.04 (Deprecated)

Remark 3.7 Let fS2(Γ,C)f \in S_2(\Gamma,\CC) be a newform. When AfA_f 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 nAfrAf2n_{A_f} \mid r^2_{A_f} (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 nAfrAf2n_{A_f}\mid r^2_{A_f} need not hold. For example, there is a newform of degree 2424 in S2(Γ0(431))S_2(\Gamma_0(431)) such that nAf=(2116947)2rAf2=(2106947)2.n_{A_f} = (2^{11}\cdot 6947)^2 \,\,\nmid\,\, r^2_{A_f} = (2^{10}\cdot 6947)^2.
Note that 431431 is prime and mod~22 multiplicity one fails for J0(431)J_0(431).

M = ModularSymbols(431,sign=0); M
Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field
D = M.decomposition(); D
[ 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 ]
A = D[-1]; A
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
%time factor(A.congruence_number(A.complement().cuspidal_submodule()))
2^10 * 6947 CPU time: 0.81 s, Wall time: 0.81 s
J = J0(431)
D = J.decomposition(); D
[ 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) ]
A = D[-1]; A
Simple abelian subvariety 431f(1,431) of dimension 24 of J0(431)
factor(A.modular_degree())
2^11 * 6947