Hi Misja. Your silly code works fine for me on ubuntu 14.04 with sage
version 6.4.1 (I get zeroes), but I can reproduce your random '1' with sage
7.0.0. I can confirm that after the bad code has run once on 7.0.0, David's
suggestion of clearing the modular symbols cache does not fix the problem
-- I still get a 1.
Kevin
$ sage
┌────────────────────────────────────────────────────────────────────┐
│ Sage Version 6.4.1, Release Date: 2014-11-23 │
│ Type "notebook()" for the browser-based notebook interface. │
│ Type "help()" for help. │
└────────────────────────────────────────────────────────────────────┘
sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
: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);
:<EOF>
0
0
0
0
sage:
On Thursday, 28 April 2016 15:09:59 UTC+1, Misja wrote:
>
> 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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