Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [170,2,Mod(101,170)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(170, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("170.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 170.b (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: | |
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 . We also show the integral -expansion of the trace form.
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 | ||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 |
|
1.00000 | 0 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | 1.00000 | 3.00000 | − | 1.00000i | |||||||||||||||||||||
101.2 | 1.00000 | 0 | 1.00000 | 1.00000i | 0 | 2.00000i | 1.00000 | 3.00000 | 1.00000i | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 170.2.b.c | ✓ | 2 |
3.b | odd | 2 | 1 | 1530.2.c.a | 2 | ||
4.b | odd | 2 | 1 | 1360.2.c.d | 2 | ||
5.b | even | 2 | 1 | 850.2.b.e | 2 | ||
5.c | odd | 4 | 1 | 850.2.d.d | 2 | ||
5.c | odd | 4 | 1 | 850.2.d.e | 2 | ||
17.b | even | 2 | 1 | inner | 170.2.b.c | ✓ | 2 |
17.c | even | 4 | 1 | 2890.2.a.f | 1 | ||
17.c | even | 4 | 1 | 2890.2.a.g | 1 | ||
51.c | odd | 2 | 1 | 1530.2.c.a | 2 | ||
68.d | odd | 2 | 1 | 1360.2.c.d | 2 | ||
85.c | even | 2 | 1 | 850.2.b.e | 2 | ||
85.g | odd | 4 | 1 | 850.2.d.d | 2 | ||
85.g | odd | 4 | 1 | 850.2.d.e | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
170.2.b.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
170.2.b.c | ✓ | 2 | 17.b | even | 2 | 1 | inner |
850.2.b.e | 2 | 5.b | even | 2 | 1 | ||
850.2.b.e | 2 | 85.c | even | 2 | 1 | ||
850.2.d.d | 2 | 5.c | odd | 4 | 1 | ||
850.2.d.d | 2 | 85.g | odd | 4 | 1 | ||
850.2.d.e | 2 | 5.c | odd | 4 | 1 | ||
850.2.d.e | 2 | 85.g | odd | 4 | 1 | ||
1360.2.c.d | 2 | 4.b | odd | 2 | 1 | ||
1360.2.c.d | 2 | 68.d | odd | 2 | 1 | ||
1530.2.c.a | 2 | 3.b | odd | 2 | 1 | ||
1530.2.c.a | 2 | 51.c | odd | 2 | 1 | ||
2890.2.a.f | 1 | 17.c | even | 4 | 1 | ||
2890.2.a.g | 1 | 17.c | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .