Note: this issue was resolved by David Loeffler in the sage-nt group. See
https://groups.google.com/forum/#!topic/sage-nt/NM7bbCgefdo
On Tuesday, 3 May 2016 17:03:16 UTC+1, Kevin Buzzard wrote:
>
> 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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