OK how about this for a minimal example of unexpected (to me) behaviour. I
get problems with sage 6.4.1 and 7.0.0 with the below code.
G=DirichletGroup(80);
for chi in G:
D=ModularSymbols(chi,2,-1).cuspidal_subspace().new_subspace().decomposition();
for f in D:
e=f.q_eigenform(10,'alpha')[3];
print(e.is_integral())
I believe that this code just computes a bunch of random spaces of cuspidal
new modular symbols and then for each conj class of eigenforms it finds, it
checks to see if the coefficient of q^3 is an algebraic integer.
In sage 7.0 on Ubuntu 14.04 I get
True
True
True
False
True
True
True
True
True
True
True
True
True
True
True
True
False
True
True
True
False
True
False
Have I misunderstood something or is this a bug?
Kevin
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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