#18061: Fix Newform.atkin_lehner_eigenvalue() for coefficient fields other than
QQ
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Reporter: pbruin | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.6
Component: modular | Keywords: newform Atkin-Lehner operator
forms | Authors:
Merged in: | Report Upstream: N/A
Reviewers: | Branch:
Work issues: | Dependencies:
Commit: |
Stopgaps: |
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Consider the following newform of weight 2 for Γ,,1,,(30) with
coefficients in '''Q'''(''i''):
{{{
sage: f = Newforms(Gamma1(30), 2, names='a')[1]; f
q + a1*q^2 - a1*q^3 - q^4 + (a1 - 2)*q^5 + O(q^6)
sage: f.base_ring()
Number Field in a1 with defining polynomial x^2 + 1
sage: f.character()
Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> -1
}}}
The method `atkin_lehner_eigenvalue()` returns a nonsensical result:
{{{
sage: f.atkin_lehner_eigenvalue()
-2
}}}
This is just the upper left entry of the matrix of the Atkin-Lehner
operator ''W'',,30,, with respect to some basis of the space of modular
symbols attached to ''f'':
{{{
sage: f.modular_symbols(sign=1).atkin_lehner_operator()
Hecke module morphism Atkin-Lehner operator W_30 defined by the matrix
[-2 -3]
[ 1 2]
Domain: Modular Symbols subspace of dimension 2 of Modular Symbols space
...
Codomain: Modular Symbols subspace of dimension 2 of Modular Symbols space
...
}}}
The correct result is in fact only a ''pseudo''-eigenvalue; it is a
complex number ''η'' of absolute value 1 such that ''W'',,30,,''f'' =
''ηf''^*^, where ''f''^*^ is the form whose coefficients are the complex
conjugates of those of ''f''.
--
Ticket URL: <http://trac.sagemath.org/ticket/18061>
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