Returns true if and only if M is an ambient space. Ambient
spaces are those space constructed in
Section Ambient Spaces.
Returns true if M is contained in the cuspidal subspace
of the ambient space.
Returns true if M is contained in the Eisenstein subspace
of the ambient space.
Returns true if f is an Eisenstein newform or was computed
using the intrinsic EisensteinSeries.
(See Section Eisenstein Series.)
Returns true if M is a space of modular forms for Γ0(N).
Returns true if M was created explicitly as a space of modular forms
for Γ1(N), or if the AmbientSpace of M is such a space.
(Note that IsGamma1 will return false for any space
ModularForms(chars,k), even if chars
consists of all mod N Dirichlet characters.)
Returns true if M is contained in the new subspace of its AmbientSpace.
Returns true if f was created using Newforms. (Sometimes
true in other cases in which f is obviously a newform.
In number theory, "newform" means "normalized eigenform
that lies in the new subspace".)
Returns true if and only if M is the ring of all modular forms over
a given ring.
We illustrate each of the above predicates with
some simple computations in M
3(Γ
1(11)).
> M := ModularForms(Gamma1(11),3);
> f := Newform(M,1);
> IsAmbientSpace(M);
true
> IsAmbientSpace(CuspidalSubspace(M));
false
> IsCuspidal(M);
false
> IsCuspidal(CuspidalSubspace(M));
true
> IsEisenstein(CuspidalSubspace(M));
false
> IsEisenstein(EisensteinSubspace(M));
true
> IsGamma1(M);
true
> IsNew(M);
true
> IsNewform(M.1);
false
> IsNewform(f);
true
> IsRingOfAllModularForms(M);
false
> Level(f);
11
> Level(M);
11
> Weight(f);
3
> Weight(M);
3
> Weight(M.1);
3
V2.28, 13 July 2023