Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(406,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.406");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 585.j (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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
406.1 |
|
−1.06141 | − | 1.83842i | 0 | −1.25320 | + | 2.17061i | −1.00000 | 0 | −0.733534 | + | 1.27052i | 1.07500 | 0 | 1.06141 | + | 1.83842i | ||||||||||||||||||||||||||||||||||||||||
406.2 | −0.473183 | − | 0.819577i | 0 | 0.552196 | − | 0.956432i | −1.00000 | 0 | 0.781529 | − | 1.35365i | −2.93789 | 0 | 0.473183 | + | 0.819577i | |||||||||||||||||||||||||||||||||||||||||
406.3 | 0.313396 | + | 0.542817i | 0 | 0.803566 | − | 1.39182i | −1.00000 | 0 | −2.21563 | + | 3.83759i | 2.26092 | 0 | −0.313396 | − | 0.542817i | |||||||||||||||||||||||||||||||||||||||||
406.4 | 0.905157 | + | 1.56778i | 0 | −0.638619 | + | 1.10612i | −1.00000 | 0 | 2.21251 | − | 3.83219i | 1.30843 | 0 | −0.905157 | − | 1.56778i | |||||||||||||||||||||||||||||||||||||||||
406.5 | 1.31604 | + | 2.27945i | 0 | −2.46394 | + | 4.26767i | −1.00000 | 0 | −0.544875 | + | 0.943751i | −7.70645 | 0 | −1.31604 | − | 2.27945i | |||||||||||||||||||||||||||||||||||||||||
451.1 | −1.06141 | + | 1.83842i | 0 | −1.25320 | − | 2.17061i | −1.00000 | 0 | −0.733534 | − | 1.27052i | 1.07500 | 0 | 1.06141 | − | 1.83842i | |||||||||||||||||||||||||||||||||||||||||
451.2 | −0.473183 | + | 0.819577i | 0 | 0.552196 | + | 0.956432i | −1.00000 | 0 | 0.781529 | + | 1.35365i | −2.93789 | 0 | 0.473183 | − | 0.819577i | |||||||||||||||||||||||||||||||||||||||||
451.3 | 0.313396 | − | 0.542817i | 0 | 0.803566 | + | 1.39182i | −1.00000 | 0 | −2.21563 | − | 3.83759i | 2.26092 | 0 | −0.313396 | + | 0.542817i | |||||||||||||||||||||||||||||||||||||||||
451.4 | 0.905157 | − | 1.56778i | 0 | −0.638619 | − | 1.10612i | −1.00000 | 0 | 2.21251 | + | 3.83219i | 1.30843 | 0 | −0.905157 | + | 1.56778i | |||||||||||||||||||||||||||||||||||||||||
451.5 | 1.31604 | − | 2.27945i | 0 | −2.46394 | − | 4.26767i | −1.00000 | 0 | −0.544875 | − | 0.943751i | −7.70645 | 0 | −1.31604 | + | 2.27945i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.j.i | yes | 10 |
3.b | odd | 2 | 1 | 585.2.j.h | ✓ | 10 | |
13.c | even | 3 | 1 | inner | 585.2.j.i | yes | 10 |
13.c | even | 3 | 1 | 7605.2.a.cm | 5 | ||
13.e | even | 6 | 1 | 7605.2.a.cn | 5 | ||
39.h | odd | 6 | 1 | 7605.2.a.cl | 5 | ||
39.i | odd | 6 | 1 | 585.2.j.h | ✓ | 10 | |
39.i | odd | 6 | 1 | 7605.2.a.co | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.j.h | ✓ | 10 | 3.b | odd | 2 | 1 | |
585.2.j.h | ✓ | 10 | 39.i | odd | 6 | 1 | |
585.2.j.i | yes | 10 | 1.a | even | 1 | 1 | trivial |
585.2.j.i | yes | 10 | 13.c | even | 3 | 1 | inner |
7605.2.a.cl | 5 | 39.h | odd | 6 | 1 | ||
7605.2.a.cm | 5 | 13.c | even | 3 | 1 | ||
7605.2.a.cn | 5 | 13.e | even | 6 | 1 | ||
7605.2.a.co | 5 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .