#8622: Atkin-Lehner operators don't work for odd weight modular forms
-----------------------------+----------------------------------------------
Reporter: davidloeffler | Owner: craigcitro
Type: defect | Status: new
Priority: major | Milestone:
Component: modular forms | Keywords: atkin-lehner
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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In ticket #5262 I implemented a method to find the Atkin-Lehner eigenvalue
of a modular form. Sadly this does not work if the form has odd weight:
{{{
sage: f = Newforms(Gamma1(13),3,names='a')[0]
sage: f
q + a0*q^2 + (1/7*a0^3 + 2/7*a0^2 - 3/7*a0 - 27/7)*q^3 + (-8/21*a0^3 -
23/21*a0^2 - 88/21*a0 + 16/7)*q^4 + (2/7*a0^3 + 11/7*a0^2 + 29/7*a0 +
9/7)*q^5 + O(q^6)
sage: f.atkin_lehner_eigenvalue()
---------------------------------------------------------------------------
ArithmeticError Traceback (most recent call
last)
...
ArithmeticError: subspace is not invariant under matrix
}}}
This comes up because for modular symbols of any odd weight, the Atkin-
Lehner involution doesn't commute with the star involution and hence
doesn't restrict to an operator on the plus submodule of the modular forms
(hence "subspace not invariant under matrix"). In fact they ''anti-
commute'':
{{{
sage: N = f.modular_symbols(sign=0)
sage: a = N.atkin_lehner_operator(13).matrix()
sage: b = N.star_involution().matrix()
sage: a * b * ~a * ~b
[-1 0 0 0 0 0 0 0]
[ 0 -1 0 0 0 0 0 0]
[ 0 0 -1 0 0 0 0 0]
[ 0 0 0 -1 0 0 0 0]
[ 0 0 0 0 -1 0 0 0]
[ 0 0 0 0 0 -1 0 0]
[ 0 0 0 0 0 0 -1 0]
[ 0 0 0 0 0 0 0 -1]
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8622>
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