Re: Coefficient field of modular forms

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Sep 17, 2019 at 11:08:08AM +0200, Baptiste wrote:
> Hello,
> 
> I recently started to deal with modular forms in Pari/gp and I would like to
> get the coefficient field of a given modular form. I've found two functions
> that should help me:
> 
> -mfparams, which is supposed to give a list [level, weight, character, P],
> where P defines the extension Q(f)/Q(character);
> 
> -f.mod, which is supposed to give a polynomial P defining the extension
> Q(f)/Q.
> 
> I've tried to use those two methods on the Eisenstein series E :=
> mfeisenstein(6,Mod(7,9),Mod(4,9)) and I get the following results:
> 
> ? E = mfeisenstein(6,Mod(7,9),Mod(4,9));
> ? mfparams(E)
> %2 = [81, 6, 1, y]
> ? E.mod
> %3 = t - 1
> ? mfcoefs(E,10)
> %4 = [Mod(0, t^2 + t + 1), Mod(1, t^2 + t + 1), Mod(-31*t - 32, t^2 + t +
> 1), Mod(0, t^2 + t + 1), Mod(1023*t + 31, t^2 + t + 1), Mod(3124*t - 1, t^2
> + t + 1), Mod(0, t^2 + t + 1), Mod(-16806*t - 16807, t^2 + t + 1),
> Mod(-992*t + 31745, t^2 + t + 1), Mod(0, t^2 + t + 1), Mod(-3093*t + 96876,
> t^2 + t + 1)]
> 
> The problem is, mfparams(E)[4] return y, so Q(f) should be equal to Q(chi) =
> Q (but the coefficients of E are in Q(t)/(t^2 + t + 1)), and E.mod return t
> - 1, so Q(f) should be equal Q, which is again not the case.
> 
> Did I do something wrong or is this behavior really strange?

Hello Baptiste,

The issue is as follow: 
The mf package is meant to deal with spaces of modular forms.
space of modular forms are properly C vector spaces. PARI finds a basis
which is defined over the minimal possible number field.
However for practical reasons, only linear combinations of the basis
forms with coefficients over this number field are supported.

Unfortunately some functions like mfeisenstein can return modular forms
that does not follow this rule, in which case mfparams return the
definition field of the underlying spaces of modular forms, which
is smaller than the actual definition field of the field.

In your case, this means that your form can be written as a linear
combination with coef in Q(zeta_3) of rationale forms.

Cheers,
Bill.