Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [297,2,Mod(82,297)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(297, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("297.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 297.f (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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 |
|
−0.760284 | − | 2.33991i | 0 | −3.27913 | + | 2.38243i | −0.305860 | + | 0.941339i | 0 | −3.49672 | + | 2.54052i | 4.08684 | + | 2.96926i | 0 | 2.43519 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.2 | −0.386556 | − | 1.18970i | 0 | 0.352078 | − | 0.255800i | 0.507211 | − | 1.56103i | 0 | 2.37869 | − | 1.72822i | −2.46446 | − | 1.79053i | 0 | −2.05323 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.3 | 0.386556 | + | 1.18970i | 0 | 0.352078 | − | 0.255800i | −0.507211 | + | 1.56103i | 0 | 2.37869 | − | 1.72822i | 2.46446 | + | 1.79053i | 0 | −2.05323 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.4 | 0.760284 | + | 2.33991i | 0 | −3.27913 | + | 2.38243i | 0.305860 | − | 0.941339i | 0 | −3.49672 | + | 2.54052i | −4.08684 | − | 2.96926i | 0 | 2.43519 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
136.1 | −1.85934 | − | 1.35089i | 0 | 1.01420 | + | 3.12140i | −1.37287 | + | 0.997447i | 0 | 0.0641710 | + | 0.197498i | 0.910502 | − | 2.80224i | 0 | 3.90006 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
136.2 | −0.255752 | − | 0.185814i | 0 | −0.587152 | − | 1.80707i | 3.28092 | − | 2.38373i | 0 | 1.05386 | + | 3.24346i | −0.380991 | + | 1.17257i | 0 | −1.28203 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
136.3 | 0.255752 | + | 0.185814i | 0 | −0.587152 | − | 1.80707i | −3.28092 | + | 2.38373i | 0 | 1.05386 | + | 3.24346i | 0.380991 | − | 1.17257i | 0 | −1.28203 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
136.4 | 1.85934 | + | 1.35089i | 0 | 1.01420 | + | 3.12140i | 1.37287 | − | 0.997447i | 0 | 0.0641710 | + | 0.197498i | −0.910502 | + | 2.80224i | 0 | 3.90006 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.1 | −0.760284 | + | 2.33991i | 0 | −3.27913 | − | 2.38243i | −0.305860 | − | 0.941339i | 0 | −3.49672 | − | 2.54052i | 4.08684 | − | 2.96926i | 0 | 2.43519 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.2 | −0.386556 | + | 1.18970i | 0 | 0.352078 | + | 0.255800i | 0.507211 | + | 1.56103i | 0 | 2.37869 | + | 1.72822i | −2.46446 | + | 1.79053i | 0 | −2.05323 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.3 | 0.386556 | − | 1.18970i | 0 | 0.352078 | + | 0.255800i | −0.507211 | − | 1.56103i | 0 | 2.37869 | + | 1.72822i | 2.46446 | − | 1.79053i | 0 | −2.05323 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.4 | 0.760284 | − | 2.33991i | 0 | −3.27913 | − | 2.38243i | 0.305860 | + | 0.941339i | 0 | −3.49672 | − | 2.54052i | −4.08684 | + | 2.96926i | 0 | 2.43519 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
190.1 | −1.85934 | + | 1.35089i | 0 | 1.01420 | − | 3.12140i | −1.37287 | − | 0.997447i | 0 | 0.0641710 | − | 0.197498i | 0.910502 | + | 2.80224i | 0 | 3.90006 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
190.2 | −0.255752 | + | 0.185814i | 0 | −0.587152 | + | 1.80707i | 3.28092 | + | 2.38373i | 0 | 1.05386 | − | 3.24346i | −0.380991 | − | 1.17257i | 0 | −1.28203 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
190.3 | 0.255752 | − | 0.185814i | 0 | −0.587152 | + | 1.80707i | −3.28092 | − | 2.38373i | 0 | 1.05386 | − | 3.24346i | 0.380991 | + | 1.17257i | 0 | −1.28203 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
190.4 | 1.85934 | − | 1.35089i | 0 | 1.01420 | − | 3.12140i | 1.37287 | + | 0.997447i | 0 | 0.0641710 | − | 0.197498i | −0.910502 | − | 2.80224i | 0 | 3.90006 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
33.h | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 297.2.f.b | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 297.2.f.b | ✓ | 16 |
9.c | even | 3 | 2 | 891.2.n.h | 32 | ||
9.d | odd | 6 | 2 | 891.2.n.h | 32 | ||
11.c | even | 5 | 1 | inner | 297.2.f.b | ✓ | 16 |
11.c | even | 5 | 1 | 3267.2.a.bj | 8 | ||
11.d | odd | 10 | 1 | 3267.2.a.bi | 8 | ||
33.f | even | 10 | 1 | 3267.2.a.bi | 8 | ||
33.h | odd | 10 | 1 | inner | 297.2.f.b | ✓ | 16 |
33.h | odd | 10 | 1 | 3267.2.a.bj | 8 | ||
99.m | even | 15 | 2 | 891.2.n.h | 32 | ||
99.n | odd | 30 | 2 | 891.2.n.h | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
297.2.f.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
297.2.f.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
297.2.f.b | ✓ | 16 | 11.c | even | 5 | 1 | inner |
297.2.f.b | ✓ | 16 | 33.h | odd | 10 | 1 | inner |
891.2.n.h | 32 | 9.c | even | 3 | 2 | ||
891.2.n.h | 32 | 9.d | odd | 6 | 2 | ||
891.2.n.h | 32 | 99.m | even | 15 | 2 | ||
891.2.n.h | 32 | 99.n | odd | 30 | 2 | ||
3267.2.a.bi | 8 | 11.d | odd | 10 | 1 | ||
3267.2.a.bi | 8 | 33.f | even | 10 | 1 | ||
3267.2.a.bj | 8 | 11.c | even | 5 | 1 | ||
3267.2.a.bj | 8 | 33.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .