Hi Kevin and Misja,
This has something to do with the fraught issue of *variable names*.
Kevin's code prints a mixture of True's and False's; but, on the other
hand, if you run the code
sage: G=DirichletGroup(80);
sage: for chi in G:
D=ModularSymbols(chi,2,-1).cuspidal_subspace().new_subspace().decomposition()
for f in D:
elt=f.q_eigenform(10,'alpha' + str(randint(1,1000)))[3];
print(elt.is_integral())
then you get all True's (at least, with rather high probability!). (Or,
more civilised: replace str(randint(1, 1000)) with str(f.factor_number()),
which is just an arbitrary ordering on the decomposition factors of a given
Hecke module.)
There's a natural response that this is absurd -- how can the name given to
a variable affect its behaviour, in any sane environment? if I remember
correctly, the reason for this is that Sage's coercion system has to have a
way of knowing when two "number field" Sage objects with the same defining
polynomial are supposed to represent the same field (so there is a
canonical map between them) and when they're supposed to represent
different, non-canonically-isomorphic fields. Sage's solution is to assume
that if two number field objects have the same defining polynomial *and the
same generator name*, then they're identical. This leads to weird behaviour
when you create a bunch of number fields one after the other and call the
generators "alpha" in each case.
So now you know why the default behaviour of the "Newforms" constructor is
to append an integer to the variable names -- if you do
"Newforms(DirichletGroup(80), names='alpha')" you'll get coefficient fields
with generators alpha0, alpha1, etc (not all alpha). I'll leave it to
other, more expert users to comment on why the collision of names causes
the exact bug above, but hopefully this should give you the means to work
around it.
David
On 3 May 2016 at 17:07, Kevin Buzzard <[email protected]> wrote:
> Here's a shorter version of the code which I think already exhibits a
> problem. I should say that I'm currently only just learning sage so I might
> be victim to a gotcha. But here we go:
>
> G=DirichletGroup(80);
> for chi in G:
>
> D=ModularSymbols(chi,2,-1).cuspidal_subspace().new_subspace().decomposition()
> for f in D:
> elt=f.q_eigenform(10,'alpha')[3];
> print(elt.is_integral())
>
> If I cut and paste this into a sage session (either 6.4.1 or 7.0 on Ubuntu
> 14.04) I get mostly true's but a few false's. My understanding is that I'm
> computing cuspidal new eigenforms here and checking to see that the
> coefficient of q^3 is an algebraic integer in each case. It sometimes
> isn't. What are we doing wrong?
>
> Kevin
>
>
>
>
> On Thursday, 28 April 2016 15:12:15 UTC+1, John Cremona wrote:
>>
>> ---------- Forwarded message ----------
>> From: Misja <[email protected]>
>> Date: 28 April 2016 at 15:09
>> Subject: [sage-support] Mysterious behaviour of q_eigenform... Bug?
>> To: sage-support <[email protected]>
>>
>>
>> When understand the specific reason why my code is not working
>> properly, I managed to pin it down to the following mysterious
>> behaviour of q_eigenform.
>>
>> First run the following code in sage.
>>
>> G=DirichletGroup(80);
>> chi=G[22];
>> D=ModularSymbols(chi,2,-1).cuspidal_subspace().new_subspace().decomposition();
>>
>> for f in D:
>> elt=f.q_eigenform(10,'alpha')[3];
>> N=elt.parent().absolute_field('a');
>> fact=N.factor(2);
>> for P,e in fact:
>> res_field=N.residue_field(P);
>> print res_field(elt);
>>
>>
>> It will print
>>
>> 0
>> 0
>> 0
>> 0
>>
>> which, I think, is the 'right' answer.
>>
>>
>> Now close your sage session entirely. Open a new session and then run
>> the following *silly* code:
>>
>> G=DirichletGroup(80);
>> for chi in G:
>>
>> D=ModularSymbols(chi,2,-1).cuspidal_subspace().new_subspace().decomposition();
>>
>> for f in D:
>> elt=f.q_eigenform(10,'alpha')[3];
>> if not elt.parent()==QQ:
>> K=elt.parent().absolute_field('b');
>> if chi==G[22]:
>> fact=K.factor(2);
>> for P,e in fact:
>> res_field=K.residue_field(P);
>> print res_field(elt);
>>
>>
>> It will print:
>>
>> 0
>> 0
>> 1
>> 0
>>
>> As far as I understand the theory, this cannot happen. If you let sage
>> print the alpha^3 coefficient of you see that in both cases it picks a
>> different q_eigenform in f, the Galois conjugacy class of newforms.
>> Although this can be a bit annoying, in theory it is fine. But I am
>> pretty sure that when your reduce this coefficient modulo some prime
>> P, any two elements of the same Galois conjugacy class can differ at
>> most by some automorphism of the residue field (and obviously 1 and 0
>> do not satisfy this criterion).
>>
>>
>> To make matters worse: if you run a single sage session and you run
>> the 'good' code first and the 'bad' code second, then suddenly the
>> 'bad' code is fixed and printing only 0s. If you run the 'bad' code
>> first and the 'good' code second, then they are both 'bad' and the
>> 'good' code suddenly prints 0,0,1,0 as well.
>>
>> By trying I found out that this is because somehow q_eigenform picks
>> the same q_eigenform as whichever code that ran first and somehow
>> these choices are not compatible! I don't know the inner workings of
>> q_eigenform, but this behaviour seems strange to me.
>>
>> Can anyone explain what is going on here? Is it a bug?
>>
>> Thanks!
>>
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