Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,4,Mod(289,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416.289");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 416.i (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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
|
0 | −4.25724 | − | 7.37376i | 0 | −9.83216 | 0 | −12.0572 | + | 20.8837i | 0 | −22.7482 | + | 39.4011i | 0 | ||||||||||||||||||||||||||||||||||||
289.2 | 0 | −0.613091 | − | 1.06190i | 0 | 13.8322 | 0 | 12.7720 | − | 22.1218i | 0 | 12.7482 | − | 22.0806i | 0 | |||||||||||||||||||||||||||||||||||||
289.3 | 0 | 0.613091 | + | 1.06190i | 0 | 13.8322 | 0 | −12.7720 | + | 22.1218i | 0 | 12.7482 | − | 22.0806i | 0 | |||||||||||||||||||||||||||||||||||||
289.4 | 0 | 4.25724 | + | 7.37376i | 0 | −9.83216 | 0 | 12.0572 | − | 20.8837i | 0 | −22.7482 | + | 39.4011i | 0 | |||||||||||||||||||||||||||||||||||||
321.1 | 0 | −4.25724 | + | 7.37376i | 0 | −9.83216 | 0 | −12.0572 | − | 20.8837i | 0 | −22.7482 | − | 39.4011i | 0 | |||||||||||||||||||||||||||||||||||||
321.2 | 0 | −0.613091 | + | 1.06190i | 0 | 13.8322 | 0 | 12.7720 | + | 22.1218i | 0 | 12.7482 | + | 22.0806i | 0 | |||||||||||||||||||||||||||||||||||||
321.3 | 0 | 0.613091 | − | 1.06190i | 0 | 13.8322 | 0 | −12.7720 | − | 22.1218i | 0 | 12.7482 | + | 22.0806i | 0 | |||||||||||||||||||||||||||||||||||||
321.4 | 0 | 4.25724 | − | 7.37376i | 0 | −9.83216 | 0 | 12.0572 | + | 20.8837i | 0 | −22.7482 | − | 39.4011i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
52.j | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 416.4.i.d | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 416.4.i.d | ✓ | 8 |
13.c | even | 3 | 1 | inner | 416.4.i.d | ✓ | 8 |
52.j | odd | 6 | 1 | inner | 416.4.i.d | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
416.4.i.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
416.4.i.d | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
416.4.i.d | ✓ | 8 | 13.c | even | 3 | 1 | inner |
416.4.i.d | ✓ | 8 | 52.j | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .