Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,3,Mod(193,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.193");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 600.u (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 | ||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
193.1 |
|
0 | −1.22474 | − | 1.22474i | 0 | 0 | 0 | 3.22474 | − | 3.22474i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||
193.2 | 0 | 1.22474 | + | 1.22474i | 0 | 0 | 0 | 0.775255 | − | 0.775255i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||
457.1 | 0 | −1.22474 | + | 1.22474i | 0 | 0 | 0 | 3.22474 | + | 3.22474i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
457.2 | 0 | 1.22474 | − | 1.22474i | 0 | 0 | 0 | 0.775255 | + | 0.775255i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.3.u.d | yes | 4 |
3.b | odd | 2 | 1 | 1800.3.v.m | 4 | ||
4.b | odd | 2 | 1 | 1200.3.bg.f | 4 | ||
5.b | even | 2 | 1 | 600.3.u.c | ✓ | 4 | |
5.c | odd | 4 | 1 | 600.3.u.c | ✓ | 4 | |
5.c | odd | 4 | 1 | inner | 600.3.u.d | yes | 4 |
15.d | odd | 2 | 1 | 1800.3.v.l | 4 | ||
15.e | even | 4 | 1 | 1800.3.v.l | 4 | ||
15.e | even | 4 | 1 | 1800.3.v.m | 4 | ||
20.d | odd | 2 | 1 | 1200.3.bg.l | 4 | ||
20.e | even | 4 | 1 | 1200.3.bg.f | 4 | ||
20.e | even | 4 | 1 | 1200.3.bg.l | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
600.3.u.c | ✓ | 4 | 5.b | even | 2 | 1 | |
600.3.u.c | ✓ | 4 | 5.c | odd | 4 | 1 | |
600.3.u.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
600.3.u.d | yes | 4 | 5.c | odd | 4 | 1 | inner |
1200.3.bg.f | 4 | 4.b | odd | 2 | 1 | ||
1200.3.bg.f | 4 | 20.e | even | 4 | 1 | ||
1200.3.bg.l | 4 | 20.d | odd | 2 | 1 | ||
1200.3.bg.l | 4 | 20.e | even | 4 | 1 | ||
1800.3.v.l | 4 | 15.d | odd | 2 | 1 | ||
1800.3.v.l | 4 | 15.e | even | 4 | 1 | ||
1800.3.v.m | 4 | 3.b | odd | 2 | 1 | ||
1800.3.v.m | 4 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .