Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,3,Mod(73,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.73");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 168.z (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: | 8.0.35911766016.9 |
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 |
|
0 | 1.50000 | + | 0.866025i | 0 | −6.80550 | + | 3.92916i | 0 | 6.99187 | − | 0.337312i | 0 | 1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||||||||||||
73.2 | 0 | 1.50000 | + | 0.866025i | 0 | −4.68140 | + | 2.70281i | 0 | −6.12873 | + | 3.38210i | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
73.3 | 0 | 1.50000 | + | 0.866025i | 0 | 3.18140 | − | 1.83678i | 0 | 2.47188 | − | 6.54903i | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
73.4 | 0 | 1.50000 | + | 0.866025i | 0 | 5.30550 | − | 3.06313i | 0 | 0.664986 | + | 6.96834i | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
145.1 | 0 | 1.50000 | − | 0.866025i | 0 | −6.80550 | − | 3.92916i | 0 | 6.99187 | + | 0.337312i | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
145.2 | 0 | 1.50000 | − | 0.866025i | 0 | −4.68140 | − | 2.70281i | 0 | −6.12873 | − | 3.38210i | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
145.3 | 0 | 1.50000 | − | 0.866025i | 0 | 3.18140 | + | 1.83678i | 0 | 2.47188 | + | 6.54903i | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
145.4 | 0 | 1.50000 | − | 0.866025i | 0 | 5.30550 | + | 3.06313i | 0 | 0.664986 | − | 6.96834i | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 168.3.z.b | ✓ | 8 |
3.b | odd | 2 | 1 | 504.3.by.c | 8 | ||
4.b | odd | 2 | 1 | 336.3.bh.g | 8 | ||
7.b | odd | 2 | 1 | 1176.3.z.c | 8 | ||
7.c | even | 3 | 1 | 1176.3.f.c | 8 | ||
7.c | even | 3 | 1 | 1176.3.z.c | 8 | ||
7.d | odd | 6 | 1 | inner | 168.3.z.b | ✓ | 8 |
7.d | odd | 6 | 1 | 1176.3.f.c | 8 | ||
12.b | even | 2 | 1 | 1008.3.cg.p | 8 | ||
21.g | even | 6 | 1 | 504.3.by.c | 8 | ||
21.g | even | 6 | 1 | 3528.3.f.b | 8 | ||
21.h | odd | 6 | 1 | 3528.3.f.b | 8 | ||
28.f | even | 6 | 1 | 336.3.bh.g | 8 | ||
28.f | even | 6 | 1 | 2352.3.f.g | 8 | ||
28.g | odd | 6 | 1 | 2352.3.f.g | 8 | ||
84.j | odd | 6 | 1 | 1008.3.cg.p | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.3.z.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
168.3.z.b | ✓ | 8 | 7.d | odd | 6 | 1 | inner |
336.3.bh.g | 8 | 4.b | odd | 2 | 1 | ||
336.3.bh.g | 8 | 28.f | even | 6 | 1 | ||
504.3.by.c | 8 | 3.b | odd | 2 | 1 | ||
504.3.by.c | 8 | 21.g | even | 6 | 1 | ||
1008.3.cg.p | 8 | 12.b | even | 2 | 1 | ||
1008.3.cg.p | 8 | 84.j | odd | 6 | 1 | ||
1176.3.f.c | 8 | 7.c | even | 3 | 1 | ||
1176.3.f.c | 8 | 7.d | odd | 6 | 1 | ||
1176.3.z.c | 8 | 7.b | odd | 2 | 1 | ||
1176.3.z.c | 8 | 7.c | even | 3 | 1 | ||
2352.3.f.g | 8 | 28.f | even | 6 | 1 | ||
2352.3.f.g | 8 | 28.g | odd | 6 | 1 | ||
3528.3.f.b | 8 | 21.g | even | 6 | 1 | ||
3528.3.f.b | 8 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .