Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(406,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.406");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 585.j (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: | no (minimal twist has level 65) |
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 | ||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
406.1 |
|
−1.15139 | − | 1.99426i | 0 | −1.65139 | + | 2.86029i | 1.00000 | 0 | 0.500000 | − | 0.866025i | 3.00000 | 0 | −1.15139 | − | 1.99426i | ||||||||||||||||||||||
406.2 | 0.651388 | + | 1.12824i | 0 | 0.151388 | − | 0.262211i | 1.00000 | 0 | 0.500000 | − | 0.866025i | 3.00000 | 0 | 0.651388 | + | 1.12824i | |||||||||||||||||||||||
451.1 | −1.15139 | + | 1.99426i | 0 | −1.65139 | − | 2.86029i | 1.00000 | 0 | 0.500000 | + | 0.866025i | 3.00000 | 0 | −1.15139 | + | 1.99426i | |||||||||||||||||||||||
451.2 | 0.651388 | − | 1.12824i | 0 | 0.151388 | + | 0.262211i | 1.00000 | 0 | 0.500000 | + | 0.866025i | 3.00000 | 0 | 0.651388 | − | 1.12824i | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.j.d | 4 | |
3.b | odd | 2 | 1 | 65.2.e.b | ✓ | 4 | |
12.b | even | 2 | 1 | 1040.2.q.o | 4 | ||
13.c | even | 3 | 1 | inner | 585.2.j.d | 4 | |
13.c | even | 3 | 1 | 7605.2.a.bg | 2 | ||
13.e | even | 6 | 1 | 7605.2.a.bb | 2 | ||
15.d | odd | 2 | 1 | 325.2.e.a | 4 | ||
15.e | even | 4 | 2 | 325.2.o.b | 8 | ||
39.d | odd | 2 | 1 | 845.2.e.d | 4 | ||
39.f | even | 4 | 2 | 845.2.m.d | 8 | ||
39.h | odd | 6 | 1 | 845.2.a.f | 2 | ||
39.h | odd | 6 | 1 | 845.2.e.d | 4 | ||
39.i | odd | 6 | 1 | 65.2.e.b | ✓ | 4 | |
39.i | odd | 6 | 1 | 845.2.a.c | 2 | ||
39.k | even | 12 | 2 | 845.2.c.d | 4 | ||
39.k | even | 12 | 2 | 845.2.m.d | 8 | ||
156.p | even | 6 | 1 | 1040.2.q.o | 4 | ||
195.x | odd | 6 | 1 | 325.2.e.a | 4 | ||
195.x | odd | 6 | 1 | 4225.2.a.x | 2 | ||
195.y | odd | 6 | 1 | 4225.2.a.t | 2 | ||
195.bl | even | 12 | 2 | 325.2.o.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
65.2.e.b | ✓ | 4 | 39.i | odd | 6 | 1 | |
325.2.e.a | 4 | 15.d | odd | 2 | 1 | ||
325.2.e.a | 4 | 195.x | odd | 6 | 1 | ||
325.2.o.b | 8 | 15.e | even | 4 | 2 | ||
325.2.o.b | 8 | 195.bl | even | 12 | 2 | ||
585.2.j.d | 4 | 1.a | even | 1 | 1 | trivial | |
585.2.j.d | 4 | 13.c | even | 3 | 1 | inner | |
845.2.a.c | 2 | 39.i | odd | 6 | 1 | ||
845.2.a.f | 2 | 39.h | odd | 6 | 1 | ||
845.2.c.d | 4 | 39.k | even | 12 | 2 | ||
845.2.e.d | 4 | 39.d | odd | 2 | 1 | ||
845.2.e.d | 4 | 39.h | odd | 6 | 1 | ||
845.2.m.d | 8 | 39.f | even | 4 | 2 | ||
845.2.m.d | 8 | 39.k | even | 12 | 2 | ||
1040.2.q.o | 4 | 12.b | even | 2 | 1 | ||
1040.2.q.o | 4 | 156.p | even | 6 | 1 | ||
4225.2.a.t | 2 | 195.y | odd | 6 | 1 | ||
4225.2.a.x | 2 | 195.x | odd | 6 | 1 | ||
7605.2.a.bb | 2 | 13.e | even | 6 | 1 | ||
7605.2.a.bg | 2 | 13.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .