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] <javascript:>>
> Date: 28 April 2016 at 15:09
> Subject: [sage-support] Mysterious behaviour of q_eigenform... Bug?
> To: sage-support <[email protected] <javascript:>>
>
>
> 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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