#6837: Implementation of twisting modular forms by Dirichlet characters
------------------------------------------+---------------------------------
Reporter: ljpk | Owner: craigcitro
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-4.3.2
Component: modular forms | Keywords: modular form twist
character
Author: Lloyd Kilford, Alex Ghitza | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
------------------------------------------+---------------------------------
Changes (by wuthrich):
* status: needs_review => needs_work
Old description:
> In Koblitz's book "Introduction to Elliptic Curves and Modular Forms",
> Proposition III.3.17, it is proved that if f(q)=sum_{n=0}^infinity a_n
> q^n is the Fourier expansion of a modular form in M_k(Gamma_0(M),chi) and
> chi_1 is a primitive Dirichlet character modulo N, then
> f_chi(q)=sum_{n=0}^infinity \chi_1(n) a_n q^n is a modular form in
> M_k(\Gamma_0(MN^2),\chi\chi_1^2). This is a proposed method for objects
> of type 'sage.modular.modform.element.ModularFormElement' which will
> create f_\chi given f and chi.
>
> {{{
> def twist(f,chi):
> r"""
> This function returns the twist of the modular form f by the
> Dirichlet character chi
>
> INPUT:
>
> - ``f`` - a modular form
>
> - ``chi`` - a Dirichlet character
>
> OUTPUT:
>
> f_\chi, the twist of f by chi; if f(q)=sum_{n=0}^infty a_n q^n,
> then f_chi(q)=sum_{n=0}^infty chi(n) a_n q^n ([Koblitz]).
>
> EXAMPLES:
>
> Here is a basic example:
>
> ::
>
> sage: f=CuspForms(11,2).0
> sage: f.q_expansion(6)
> q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
> sage: eps=DirichletGroup(3).0
> sage: f.twist(eps)
> q + 2*q^2 + 2*q^4 - q^5 + O(q^6)
>
> ::
>
> NOTES:
> This function is rather slow when chi is not a quadratic
> character; this may be improved with a new definition of the Sturm bound.
>
> REFERENCES:
>
> - [Koblitz], Neal Koblitz, "Introduction to Elliptic Curves and
> Modular Forms", Springer GTM 97, 1993, Proposition III.3.17.
>
> AUTHORS:
>
> - L. J. P. Kilford (2009-08-28)
>
> """
> DG_MNsq=DirichletGroup(f.level()*chi.modulus()^2)
> basering=DG_MNsq.base_ring()
> M_k_MNsq=ModularForms(DG_MNsq(self.character())*DG_MNsq(chi)^2,self.weight(),base_ring=basering)
> bound=M_k_MNsq.sturm_bound()+1
> PSR.<q>=PowerSeriesRing(DG_MNsq.base_ring())
> f_twist=PSR([self[i]*chi(i) for i in
> xrange(0,bound+2)])+O(q^bound)
> return M_k_MNsq(f_twist)
New description:
In Koblitz's book "Introduction to Elliptic Curves and Modular Forms",
Proposition III.3.17, it is proved that if f(q)=sum,,n=0,,^infinity^ a_n
q^n^ is the Fourier expansion of a modular form in
M,,k,,(Gamma,,0,,(M),chi) and chi_1 is a primitive Dirichlet character
modulo N, then f_chi(q)=sum,,n=0,,^infinity^ \chi_1(n) a_n q^n^ is a
modular form in M_k(\Gamma_0(MN^2^),\chi\chi_1^2^). This is a proposed
method for objects of type
'sage.modular.modform.element.ModularFormElement' which will create f_\chi
given f and chi.
{{{
def twist(f,chi):
r"""
This function returns the twist of the modular form f by the
Dirichlet character chi
INPUT:
- ``f`` - a modular form
- ``chi`` - a Dirichlet character
OUTPUT:
f_\chi, the twist of f by chi; if f(q)=sum_{n=0}^infty a_n q^n,
then f_chi(q)=sum_{n=0}^infty chi(n) a_n q^n ([Koblitz]).
EXAMPLES:
Here is a basic example:
::
sage: f=CuspForms(11,2).0
sage: f.q_expansion(6)
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
sage: eps=DirichletGroup(3).0
sage: f.twist(eps)
q + 2*q^2 + 2*q^4 - q^5 + O(q^6)
::
NOTES:
This function is rather slow when chi is not a quadratic
character; this may be improved with a new definition of the Sturm bound.
REFERENCES:
- [Koblitz], Neal Koblitz, "Introduction to Elliptic Curves and
Modular Forms", Springer GTM 97, 1993, Proposition III.3.17.
AUTHORS:
- L. J. P. Kilford (2009-08-28)
"""
DG_MNsq=DirichletGroup(f.level()*chi.modulus()^2)
basering=DG_MNsq.base_ring()
M_k_MNsq=ModularForms(DG_MNsq(self.character())*DG_MNsq(chi)^2,self.weight(),base_ring=basering)
bound=M_k_MNsq.sturm_bound()+1
PSR.<q>=PowerSeriesRing(DG_MNsq.base_ring())
f_twist=PSR([self[i]*chi(i) for i in
xrange(0,bound+2)])+O(q^bound)
return M_k_MNsq(f_twist)
--
Comment:
* There should be an empty line before {{{... math:}}}.
* I don't know why, but Lloyd's first names are transformed into 12. 10.
16. in the reference page.
* 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].
{{{
sage: M = CuspForms(11,2)
sage: f = M.0
sage: chi = DirichletGroup(3).0
sage: fchi = f.twist(chi)
sage: fchi
q + 2*q^2 + 2*q^4 - q^5 + O(q^6)
sage: fchi.twist(chi)
}}}
* If the form is on Gamma_0(N) and one twists with a character modulo N,
I think it should change the Nebetypus, no ?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6837#comment:3>
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