Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,3,Mod(65,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("144.65");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 144.q (of order , degree , not 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: | no (minimal twist has level 18) |
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 | ||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
|
0 | −2.44949 | + | 1.73205i | 0 | −4.50000 | + | 2.59808i | 0 | 3.17423 | − | 5.49794i | 0 | 3.00000 | − | 8.48528i | 0 | ||||||||||||||||||||||
65.2 | 0 | 2.44949 | + | 1.73205i | 0 | −4.50000 | + | 2.59808i | 0 | −4.17423 | + | 7.22999i | 0 | 3.00000 | + | 8.48528i | 0 | |||||||||||||||||||||||
113.1 | 0 | −2.44949 | − | 1.73205i | 0 | −4.50000 | − | 2.59808i | 0 | 3.17423 | + | 5.49794i | 0 | 3.00000 | + | 8.48528i | 0 | |||||||||||||||||||||||
113.2 | 0 | 2.44949 | − | 1.73205i | 0 | −4.50000 | − | 2.59808i | 0 | −4.17423 | − | 7.22999i | 0 | 3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 144.3.q.c | 4 | |
3.b | odd | 2 | 1 | 432.3.q.d | 4 | ||
4.b | odd | 2 | 1 | 18.3.d.a | ✓ | 4 | |
8.b | even | 2 | 1 | 576.3.q.e | 4 | ||
8.d | odd | 2 | 1 | 576.3.q.f | 4 | ||
9.c | even | 3 | 1 | 432.3.q.d | 4 | ||
9.c | even | 3 | 1 | 1296.3.e.g | 4 | ||
9.d | odd | 6 | 1 | inner | 144.3.q.c | 4 | |
9.d | odd | 6 | 1 | 1296.3.e.g | 4 | ||
12.b | even | 2 | 1 | 54.3.d.a | 4 | ||
20.d | odd | 2 | 1 | 450.3.i.b | 4 | ||
20.e | even | 4 | 2 | 450.3.k.a | 8 | ||
24.f | even | 2 | 1 | 1728.3.q.d | 4 | ||
24.h | odd | 2 | 1 | 1728.3.q.c | 4 | ||
36.f | odd | 6 | 1 | 54.3.d.a | 4 | ||
36.f | odd | 6 | 1 | 162.3.b.a | 4 | ||
36.h | even | 6 | 1 | 18.3.d.a | ✓ | 4 | |
36.h | even | 6 | 1 | 162.3.b.a | 4 | ||
60.h | even | 2 | 1 | 1350.3.i.b | 4 | ||
60.l | odd | 4 | 2 | 1350.3.k.a | 8 | ||
72.j | odd | 6 | 1 | 576.3.q.e | 4 | ||
72.l | even | 6 | 1 | 576.3.q.f | 4 | ||
72.n | even | 6 | 1 | 1728.3.q.c | 4 | ||
72.p | odd | 6 | 1 | 1728.3.q.d | 4 | ||
180.n | even | 6 | 1 | 450.3.i.b | 4 | ||
180.p | odd | 6 | 1 | 1350.3.i.b | 4 | ||
180.v | odd | 12 | 2 | 450.3.k.a | 8 | ||
180.x | even | 12 | 2 | 1350.3.k.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
18.3.d.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
18.3.d.a | ✓ | 4 | 36.h | even | 6 | 1 | |
54.3.d.a | 4 | 12.b | even | 2 | 1 | ||
54.3.d.a | 4 | 36.f | odd | 6 | 1 | ||
144.3.q.c | 4 | 1.a | even | 1 | 1 | trivial | |
144.3.q.c | 4 | 9.d | odd | 6 | 1 | inner | |
162.3.b.a | 4 | 36.f | odd | 6 | 1 | ||
162.3.b.a | 4 | 36.h | even | 6 | 1 | ||
432.3.q.d | 4 | 3.b | odd | 2 | 1 | ||
432.3.q.d | 4 | 9.c | even | 3 | 1 | ||
450.3.i.b | 4 | 20.d | odd | 2 | 1 | ||
450.3.i.b | 4 | 180.n | even | 6 | 1 | ||
450.3.k.a | 8 | 20.e | even | 4 | 2 | ||
450.3.k.a | 8 | 180.v | odd | 12 | 2 | ||
576.3.q.e | 4 | 8.b | even | 2 | 1 | ||
576.3.q.e | 4 | 72.j | odd | 6 | 1 | ||
576.3.q.f | 4 | 8.d | odd | 2 | 1 | ||
576.3.q.f | 4 | 72.l | even | 6 | 1 | ||
1296.3.e.g | 4 | 9.c | even | 3 | 1 | ||
1296.3.e.g | 4 | 9.d | odd | 6 | 1 | ||
1350.3.i.b | 4 | 60.h | even | 2 | 1 | ||
1350.3.i.b | 4 | 180.p | odd | 6 | 1 | ||
1350.3.k.a | 8 | 60.l | odd | 4 | 2 | ||
1350.3.k.a | 8 | 180.x | even | 12 | 2 | ||
1728.3.q.c | 4 | 24.h | odd | 2 | 1 | ||
1728.3.q.c | 4 | 72.n | even | 6 | 1 | ||
1728.3.q.d | 4 | 24.f | even | 2 | 1 | ||
1728.3.q.d | 4 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .