#18086 closed enhancement (fixed)
Twists of newforms
Reported by:  pbruin  Owned by:  

Priority:  major  Milestone:  sage6.6 
Component:  modular forms  Keywords:  twist newform Dirichlet character 
Cc:  wuthrich  Merged in:  
Authors:  Peter Bruin  Reviewers:  Chris Wuthrich, David Loeffler 
Report Upstream:  N/A  Work issues:  
Branch:  751bf48 (Commits)  Commit:  
Dependencies:  #6837, #18436, #18479  Stopgaps: 
Description (last modified by )
As a followup to #6837, twists of newforms by Dirichlet characters should be implemented. This ticket implements a method Newform.twist()
, which returns the twisted form as a Newform
.
Change History (35)
comment:1 Changed 6 years ago by
comment:2 Changed 6 years ago by
 Branch set to u/pbruin/18086Newform_twist
 Commit set to dc9a9cc8b08d912974bed3001873f8487903b3ae
A first version; this is still unable to compute the level in many cases.
New commits:
f415597  trac 6837: implement twist of modular form by a Dirichlet character

57b7688  Trac 6837: improve docstring formatting

20ab273  Trac 6837: slightly simplify ModularForm_abstract.twist()

fd741ea  Trac 6837: take a lower level for the twisted form

0584707  Trac 6837: expand documentation

68354fb  Trac 6837: twist of cusp form should be a cusp form

766ef05  Trac 6837: update/expand doctests

09d70e8  Trac 6837: always use AtkinLi formula for the level of the twisted form

a53ee1f  Trac 18068: move ModularForm_abstract.twist() to ModularFormElement

dc9a9cc  Trac 18086: implement twisting of newforms by Dirichlet characters

comment:3 Changed 6 years ago by
 Commit changed from dc9a9cc8b08d912974bed3001873f8487903b3ae to 2e66ac6a56894901ef8039135e8e3df0e16e445d
comment:4 Changed 6 years ago by
 Commit changed from 2e66ac6a56894901ef8039135e8e3df0e16e445d to 5284dda20fd2c4932e36899f8d3394eebc435030
Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:
e576ad4  Trac 6837: replace Unicode dash by ASCII hyphen

2d3c8a2  Trac 6837: move ModularForm_abstract.twist() to ModularFormElement

d9e849a  Trac 6837: handle forms without character

5f5694a  Trac 6837: use the right base ring for the twisted form

5284dda  Trac 18086: implement twisting of newforms by Dirichlet characters

comment:5 Changed 6 years ago by
 Description modified (diff)
comment:6 Changed 6 years ago by
 Commit changed from 5284dda20fd2c4932e36899f8d3394eebc435030 to f1533dde29e01cbca26cb57e51a680a3853c6440
Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:
f1533dd  Trac 18086: implement twisting of newforms by Dirichlet characters

comment:7 Changed 6 years ago by
 Description modified (diff)
 Status changed from new to needs_review
Calculating the level of the twisted form is still only implemented in the cases where [AtkinLi, Theorem 3.1] applies. However, I currently don't have time to find out if this can easily be done in greater generality, and the current implementation suffices for #18061, which depends on this ticket.
comment:8 Changed 6 years ago by
 Cc wuthrich added
comment:9 followup: ↓ 11 Changed 6 years ago by
 Reviewers set to Chris Wuthrich
 Status changed from needs_review to needs_work
Two issues. One a very small missing second ` after check in the docstring. The other will need more work:
sage: M = CuspForms(Gamma1(13),2) sage: f = M.newforms('a')[0] sage: G = DirichletGroup(3) sage: chi = G.0 sage: f.twist(chi)
returns
TypeError: no common canonical parent for objects with parents: 'Number Field in a0 with defining polynomial x^2 + 3*x + 3' and 'Cyclotomic Field of order 2 and degree 1'
It appears when the form has coefficients outside ZZ.
By the way, this could be another nice example.
sage: D = Newforms(1,12)[0] sage: G = DirichletGroup(3) sage: chi = G.0 sage: D.twist(chi)
comment:10 Changed 6 years ago by
 Commit changed from f1533dde29e01cbca26cb57e51a680a3853c6440 to dd977f21a88f86e7ecc7712c1b921a23340f6879
comment:11 in reply to: ↑ 9 Changed 6 years ago by
 Dependencies changed from #6837 to #6837, #18436
Replying to wuthrich:
Two issues. One a very small missing second ` after check in the docstring.
Done, thanks,
The other will need more work:
sage: M = CuspForms(Gamma1(13),2) sage: f = M.newforms('a')[0] sage: G = DirichletGroup(3) sage: chi = G.0 sage: f.twist(chi)returns
TypeError: no common canonical parent for objects with parents: 'Number Field in a0 with defining polynomial x^2 + 3*x + 3' and 'Cyclotomic Field of order 2 and degree 1'It appears when the form has coefficients outside ZZ.
This is because the field of values of chi
and the field of coefficients of f
are not compatible (this may be a bad example; chi.base_ring()
is CyclotomicField(2)
, which is actually QQ
but not recognised as such). In such cases, one could try to take a compositum, but I think it is better to leave it to the user to specify the correct field.
I added youe example, but with chi
having coefficients in QQ
. Unfortunately, this uncovered a new bug; see #18436.
By the way, this could be another nice example.
sage: D = Newforms(1,12)[0] sage: G = DirichletGroup(3) sage: chi = G.0 sage: D.twist(chi)
This is actually the first doctest in a different guise. I added another one to check that twisting by the trivial character (with values in QQ
) does not change the form.
comment:12 Changed 6 years ago by
 Status changed from needs_work to needs_review
comment:13 followup: ↓ 14 Changed 6 years ago by
Here's a bug I found while testing your AtkinLehner ticket:
sage: f = Newforms(Gamma1(25), names='a')[1] sage: eps = f.character() sage: f.twist(eps)  TypeError Traceback (most recent call last) ... TypeError: No compatible natural embeddings found for Complex Lazy Field and Number Field in a1 with defining polynomial x^8 + 5*x^7 + 11*x^6 + 10*x^5 + x^4 + 10*x^3 + 26*x^2  10*x + 1
comment:14 in reply to: ↑ 13 Changed 6 years ago by
Replying to davidloeffler:
Here's a bug I found while testing your AtkinLehner ticket:
sage: f = Newforms(Gamma1(25), names='a')[1] sage: eps = f.character() sage: f.twist(eps)  TypeError Traceback (most recent call last) ... TypeError: No compatible natural embeddings found for Complex Lazy Field and Number Field in a1 with defining polynomial x^8 + 5*x^7 + 11*x^6 + 10*x^5 + x^4 + 10*x^3 + 26*x^2  10*x + 1
This is in fact caused by a bug in DirichletCharacter.minimize_base_ring()
: in the above setting, typing
sage: eps.minimize_base_ring()
gives the same error. This method is called via the ModularForms
constructor, so e.g. ModularForms(eps, 2)
also fails.
comment:15 Changed 6 years ago by
 Status changed from needs_review to needs_work
 Work issues set to speed
The following is much slower than it should be (about 6.7 seconds on the system I tested this on):
sage: F = Newforms(299,names='a')[0] sage: F.twist(F.character()) q + a0*q^2 + a0*q^3 + (a0  1)*q^4 + (a0  1)*q^5 + O(q^6)
This is caused by a lot of unnecessary computations of roots of unity in various number fields; see David's comment:16:ticket:18061.
comment:16 Changed 6 years ago by
 Dependencies changed from #6837, #18436 to #6837, #18436, #18479
comment:17 Changed 6 years ago by
 Status changed from needs_work to needs_review
 Work issues speed deleted
It turns out that the slowness in comment:15 can be solved by improving the construction of Dirichlet groups so as to avoid computing roots of unity until they are needed. (This basically comes down to allowing Dirichlet groups without a userspecified root of unity.) Since the problem is orthogonal to this ticket, I will create a different ticket to solve it, and I am setting this one back to needs_review. (It probably won't be necessary to make either ticket depend on the other.)
comment:18 Changed 5 years ago by
 Branch changed from u/pbruin/18086Newform_twist to u/davidloeffler/18086Newform_twist
comment:19 Changed 5 years ago by
 Commit changed from dd977f21a88f86e7ecc7712c1b921a23340f6879 to 2e82464519f002200db7d6fb1bd9ee56f49b67b8
comment:20 Changed 5 years ago by
Oops, I seem to have broken something and I can't work out why  this Git branch won't compile. Sorry, I don't know how to fix this! Any offers?
comment:21 Changed 5 years ago by
I won't claim to be an gitexpert, but I guess a "merge" is better than a "rebase" when other people are involved. The commit at which the two branches merge then tells you exactly what went on.
In this case the two involve disjoint set of files so it should be straight forward. So I am surprised this does not work.
Your branch (merged into the 6.8.beta1) compiles fine for me. I will try on it own later. Where does your compilation get a problem ?
comment:22 Changed 5 years ago by
When I try compiling this branch, it raises an error at sage/ext/interrupt/interrupt.pxd and refuses to continue.
comment:23 Changed 5 years ago by
No, I can't reproduce this. It all works fine for me on that branch. That file should not have been touched and should not compile. Probably you best abandon your branch and start a new one pulling from the trac and try compiling again.
comment:24 Changed 5 years ago by
I worked out the problem. The point is that the current "develop" branch doesn't actually build on my machine  as is sometimes the case with unreleased betas  and the branch at #18479 hangs off develop, not master.
comment:25 Changed 5 years ago by
 Branch changed from u/davidloeffler/18086Newform_twist to u/davidloeffler/18086twistsmerged
comment:26 Changed 5 years ago by
 Commit changed from 2e82464519f002200db7d6fb1bd9ee56f49b67b8 to d57edf416adaace77e013e71223b1b5bae7d686a
Branch pushed to git repo; I updated commit sha1. New commits:
d57edf4  Trac 18086: fix two small bugs involving Dirichlet characters revealed in testing

comment:27 Changed 5 years ago by
 Reviewers changed from Chris Wuthrich to Chris Wuthrich, David Loeffler
Right, I created a new branch which has only the relevant commits on top of current "master", using the rather useful "git cherrypick" command, and now I'm testing that.
But I don't think the example from comment 14 is fixed yet. With #18479 in place, calling eps.minimize_base_ring()
works fine, but still f.twist(eps)
returns exactly the same error. I've traced this to sloppy coding in some dimension counting functions. I also spotted another error in minimize_base_ring()
, unrelated to #18479, when the coefficient field is a relative number field.
With these two changes the code looks sound. Peter: if you're happy with these changes, please set the ticket to positive review.
comment:28 Changed 5 years ago by
Aargh! Now it won't merge and I can't work out why! This is giving me a serious headache.
comment:29 Changed 5 years ago by
comment:30 Changed 5 years ago by
 Branch changed from u/davidloeffler/18086twistsmerged to u/davidloeffler/18086rebased
comment:31 Changed 5 years ago by
 Commit changed from d57edf416adaace77e013e71223b1b5bae7d686a to 751bf48f8c2f65fd4a7478e0b22fbe76c6edf3bb
Branch pushed to git repo; I updated commit sha1. New commits:
751bf48  Trac 18086: fixed a small Dirichlet character bug

comment:32 Changed 5 years ago by
OK, so it turns out that:
 the reason I couldn't get the develop branch to compile was that I was using "sage b" to recompile after switching between branches, which is not robust when there are changes to packages; it seems to be safer to use "make";
 two of the three changes in the patch I uploaded earlier had already been fixed by other patches of Peter's, which was the cause of the conflict.
I've now uploaded a patch with the one extra change that Peter hadn't already fixed  minimize_base_ring
returns an error when the base is a relative number field, which is relevant if you create a newform in a space with a specified character, so the Hecke eigenvalue field is a relative field with the coefficient field of the character as its base. Now I think we're now good to go.
comment:33 Changed 5 years ago by
 Status changed from needs_review to positive_review
I saw the gitrelated messages come by but didn't have time to look into it. I think the way you solved it is good, although the branch at #18061 is now less tidy than it would have been if you had used merging instead of rebasing. (The situation might have been clearer if I had merged #18479; I don't always find it easy to decide if and when dependencies should be merged into a branch.)
Thanks to both of you for the review!
comment:34 Changed 5 years ago by
 Branch changed from u/davidloeffler/18086rebased to 751bf48f8c2f65fd4a7478e0b22fbe76c6edf3bb
 Resolution set to fixed
 Status changed from positive_review to closed
comment:35 Changed 5 years ago by
 Commit 751bf48f8c2f65fd4a7478e0b22fbe76c6edf3bb deleted
See #18672 for a followup.
Working on this; the main challenge is determining the level of the twisted form.