#8622: Atkin-Lehner operators don't work for odd weight modular forms
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Reporter: davidloeffler | Owner: craigcitro
Type: defect | Status: new
Priority: major | Milestone:
Component: modular forms | Resolution:
Keywords: atkin-lehner | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by robharron):
What's worse is the following:
{{{
sage: f = Newforms(Gamma1(7),3,names='a')[0]
sage: f
q - 3*q^2 + 5*q^4 + O(q^6)
sage: f.atkin_lehner_eigenvalue()
3
}}}
Here, f is a CM form with CM by its own Nebentype and hence has totally
real coefficients. It is therefore an eigenvector of Atkin–Lehner. But the
eigenvalue is certainly not 3, since it must be i or -i. This function
should at least raise an NotImplementedError for odd weights. Note that
the documentation for this function says that it always returns 1 or -1
(which would be wrong even if it was properly implemented).
Would it be silly to simply compute a ratio of L-values? I.e. if you have
an odd weight newform with totally real Hecke eigenvalue field (i.e. with
CM by its own Nebentypus), you could use Dokchitser's code to determine
the sign of the functional equation and hence the Atkin–Lehner eigenvalue.
You don't know the sign, so you could just try 1 and -1 and use
checkfeq(). Then, for the above modular form, you can compute
2*pi*i*L(1,f) / (sqrt(8)*L(2,f) and get i, which is therefore the
Atkin–Lehner eigvenvalue. In general, in the odd weight case (with totally
real coefficients), a ratio of near-central L-values should tell you the
Atkin–Lehner eigenvalue.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8622#comment:3>
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