Ask Sage question: basis of eigenforms for Hecke operators

[Samuel Lelièvre]

Dear sage-nt,

Can someone answer this Ask Sage question about

orthonormal eigenbases for spaces of newforms:

  https://ask.sagemath.org/question/56896

Is the requested functionality part of Sage,

perhaps via some external package?

Modularly yours,    --Samuel Lelièvre

[John Cremona]

I will answer it. The solution is to use modular symbols:

sage: N=120
sage: S=ModularSymbols(N,2,+1)
sage: NS=S.new_submodule()
sage: CNS=NS.cuspidal_submodule()
sage: D=CNS.decomposition()
sage: D
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 32 for Gamma_0(120) of weight 2 with sign 1 over Rational
Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 32 for Gamma_0(120) of weight 2 with sign 1 over Rational
Field
]
sage: [d.q_eigenform(50) for d in D]
[q + q^3 - q^5 + 4*q^7 + q^9 - 6*q^13 - q^15 - 2*q^17 + 4*q^19 +
4*q^21 - 8*q^23 + q^25 + q^27 - 6*q^29 - 4*q^35 - 6*q^37 - 6*q^39 +
10*q^41 - 4*q^43 - q^45 + 8*q^47 + 9*q^49 + O(q^50),
q + q^3 + q^5 + q^9 - 4*q^11 + 6*q^13 + q^15 - 6*q^17 - 4*q^19 + q^25
+ q^27 - 2*q^29 - 8*q^31 - 4*q^33 - 2*q^37 + 6*q^39 - 6*q^41 + 12*q^43
+ q^45 + 8*q^47 - 7*q^49 + O(q^50)]

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