#18061: Implement (correct) action of Atkin-Lehner operators on newforms
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Reporter: pbruin | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.8
Component: modular forms | Resolution:
Keywords: newform Atkin- | Merged in:
Lehner operator | Reviewers:
Authors: Peter Bruin | Work issues:
Report Upstream: N/A | Commit:
Branch: | a98bee2a3337431582e1762a0bf30f7577742bee
u/pbruin/18061-atkin_lehner_action | Stopgaps:
Dependencies: #18068, #18072, |
#18086, #18478 |
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Comment (by davidloeffler):
I checked this against a newform of level 21 and character of conductor 7,
and it does indeed appear to be the case that the Atkin--Lehner operator
as currently implemented in Sage differs from Atkin and Li's conventions
(even in weight 2, where the issue of powers of Q does not arise).
Sage's definition of the Atkin--Lehner operator seems to be that W,,Q,,^2^
= e,,Q,,(-1) e,,N/Q,,(Q), while Atkin and Li require that W,,Q,,^2^ =
e,,Q,,(-1) e,,N/Q,,(Q)^-1^.
So there's a choice to be made here: we must have either an inconsistency
between different parts of Sage, an inconsistency between Sage and Atkin--
Li, or a non-backward-compatible change of conventions.
I would actually favour the second option: there are plenty of other
papers in the literature which use the convention W,,Q,,^2^ =
Q^k-2^e,,Q,,(-1) e,,N/Q,,(Q). The great advantage of this convention is
that if you use it, then the pseudo-eigenvalues l,,Q,, satisfy l,,QQ',, =
l,,Q,, l,,Q',,, which is not the case with Atkin and Li's conventions.
--
Ticket URL: <http://trac.sagemath.org/ticket/18061#comment:22>
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