#6837: Implementation of twisting modular forms by Dirichlet characters
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Reporter: ljpk | Owner: craigcitro
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-4.4
Component: modular forms | Keywords: modular form twist
character
Author: Lloyd Kilford, Alex Ghitza | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by davidloeffler):
Replying to [comment:3 wuthrich]:
> * I think one should catch the case when the conductor of the character
is not coprime to the level of the modular form. Note that the if you
twist a form that has been twisted by chi with chi again, the level should
go down not up. I believe that the correct assumption are in [Mazur-Tate-
Teitelbaum].
I don't agree: twisting by chi and then by chi^-1^ shouldn't give the
original answer back again, it should give some oldform at possibly higher
level, whose q-expansion has zeros at coefficients that aren't prime to
the conductor of chi. Looking for some associated minimal-level form is
tempting, but it's not going to make sense when the original form f is not
an eigenform.
There is a question, however, as to what level the answer should be
returned at when M and N are not coprime. I would contend that the answer
should be returned as an element of the modular forms space of level
LCM(N, N'M, M^2^) where N is the level of f, N' is the conductor of the
character of f, and M is the conductor of the character we're twisting by.
This is more or less the best bound you can get without knowing more about
f; see Atkin and Li, "Twists of newforms and pseudo-eigenvalues of
W-operators". (Perhaps there should be an optional argument for what level
to look in, so the user can override the default.)
> * If the form is on Gamma_0(N) and one twists with a character modulo
N, I think it should change the Nebetypus, no ?
It does, see line 460 of the patch. This is just a notational issue, one
man's M_k(Gamma_0(N), chi) is another man's M_k(Gamma_1(N), chi).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6837#comment:4>
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