Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [476,2,Mod(271,476)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(476, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("476.271");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 476.q (of order , degree , minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | |
Analytic rank: | |
Dimension: | |
Relative dimension: | over |
Coefficient field: | |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
|
Coefficient ring: | |
Coefficient ring index: | |
Twist minimal: | yes |
Sato-Tate group: |
-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the -expansion are expressed in terms of a basis for the coefficient ring described below. We also show the integral -expansion of the trace form.
Basis of coefficient ring in terms of a root of
:
Character values
We give the values of on generators for .
Embeddings
For each embedding of the coefficient field, the values are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
271.1 |
|
−0.707107 | + | 1.22474i | −2.78514 | + | 1.60800i | −1.00000 | − | 1.73205i | 0 | − | 4.54811i | 2.58495 | + | 0.563932i | 2.82843 | 3.67132 | − | 6.35892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.2 | −0.707107 | + | 1.22474i | −0.427031 | + | 0.246547i | −1.00000 | − | 1.73205i | 0 | − | 0.697339i | 2.58495 | − | 0.563932i | 2.82843 | −1.37843 | + | 2.38751i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.3 | −0.707107 | + | 1.22474i | 0.427031 | − | 0.246547i | −1.00000 | − | 1.73205i | 0 | 0.697339i | −2.58495 | + | 0.563932i | 2.82843 | −1.37843 | + | 2.38751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.4 | −0.707107 | + | 1.22474i | 2.78514 | − | 1.60800i | −1.00000 | − | 1.73205i | 0 | 4.54811i | −2.58495 | − | 0.563932i | 2.82843 | 3.67132 | − | 6.35892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.5 | 0.707107 | − | 1.22474i | −2.96945 | + | 1.71441i | −1.00000 | − | 1.73205i | 0 | 4.84909i | −0.563932 | − | 2.58495i | −2.82843 | 4.37843 | − | 7.58366i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.6 | 0.707107 | − | 1.22474i | −1.11491 | + | 0.643691i | −1.00000 | − | 1.73205i | 0 | 1.82063i | −0.563932 | + | 2.58495i | −2.82843 | −0.671323 | + | 1.16277i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.7 | 0.707107 | − | 1.22474i | 1.11491 | − | 0.643691i | −1.00000 | − | 1.73205i | 0 | − | 1.82063i | 0.563932 | − | 2.58495i | −2.82843 | −0.671323 | + | 1.16277i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.8 | 0.707107 | − | 1.22474i | 2.96945 | − | 1.71441i | −1.00000 | − | 1.73205i | 0 | − | 4.84909i | 0.563932 | + | 2.58495i | −2.82843 | 4.37843 | − | 7.58366i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.1 | −0.707107 | − | 1.22474i | −2.78514 | − | 1.60800i | −1.00000 | + | 1.73205i | 0 | 4.54811i | 2.58495 | − | 0.563932i | 2.82843 | 3.67132 | + | 6.35892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.2 | −0.707107 | − | 1.22474i | −0.427031 | − | 0.246547i | −1.00000 | + | 1.73205i | 0 | 0.697339i | 2.58495 | + | 0.563932i | 2.82843 | −1.37843 | − | 2.38751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.3 | −0.707107 | − | 1.22474i | 0.427031 | + | 0.246547i | −1.00000 | + | 1.73205i | 0 | − | 0.697339i | −2.58495 | − | 0.563932i | 2.82843 | −1.37843 | − | 2.38751i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.4 | −0.707107 | − | 1.22474i | 2.78514 | + | 1.60800i | −1.00000 | + | 1.73205i | 0 | − | 4.54811i | −2.58495 | + | 0.563932i | 2.82843 | 3.67132 | + | 6.35892i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.5 | 0.707107 | + | 1.22474i | −2.96945 | − | 1.71441i | −1.00000 | + | 1.73205i | 0 | − | 4.84909i | −0.563932 | + | 2.58495i | −2.82843 | 4.37843 | + | 7.58366i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.6 | 0.707107 | + | 1.22474i | −1.11491 | − | 0.643691i | −1.00000 | + | 1.73205i | 0 | − | 1.82063i | −0.563932 | − | 2.58495i | −2.82843 | −0.671323 | − | 1.16277i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.7 | 0.707107 | + | 1.22474i | 1.11491 | + | 0.643691i | −1.00000 | + | 1.73205i | 0 | 1.82063i | 0.563932 | + | 2.58495i | −2.82843 | −0.671323 | − | 1.16277i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
339.8 | 0.707107 | + | 1.22474i | 2.96945 | + | 1.71441i | −1.00000 | + | 1.73205i | 0 | 4.84909i | 0.563932 | − | 2.58495i | −2.82843 | 4.37843 | + | 7.58366i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
68.d | odd | 2 | 1 | CM by |
4.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
17.b | even | 2 | 1 | inner |
28.f | even | 6 | 1 | inner |
119.h | odd | 6 | 1 | inner |
476.q | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 476.2.q.a | ✓ | 16 |
4.b | odd | 2 | 1 | inner | 476.2.q.a | ✓ | 16 |
7.d | odd | 6 | 1 | inner | 476.2.q.a | ✓ | 16 |
17.b | even | 2 | 1 | inner | 476.2.q.a | ✓ | 16 |
28.f | even | 6 | 1 | inner | 476.2.q.a | ✓ | 16 |
68.d | odd | 2 | 1 | CM | 476.2.q.a | ✓ | 16 |
119.h | odd | 6 | 1 | inner | 476.2.q.a | ✓ | 16 |
476.q | even | 6 | 1 | inner | 476.2.q.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
476.2.q.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
476.2.q.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
476.2.q.a | ✓ | 16 | 7.d | odd | 6 | 1 | inner |
476.2.q.a | ✓ | 16 | 17.b | even | 2 | 1 | inner |
476.2.q.a | ✓ | 16 | 28.f | even | 6 | 1 | inner |
476.2.q.a | ✓ | 16 | 68.d | odd | 2 | 1 | CM |
476.2.q.a | ✓ | 16 | 119.h | odd | 6 | 1 | inner |
476.2.q.a | ✓ | 16 | 476.q | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .