Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [507,2,Mod(316,507)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(6))
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("507.316");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 507.j (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: | 12.0.17213603549184.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
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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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
316.1 |
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−2.33136 | − | 1.34601i | 0.500000 | − | 0.866025i | 2.62349 | + | 4.54402i | − | 1.04892i | −2.33136 | + | 1.34601i | −0.480608 | + | 0.277479i | − | 8.74094i | −0.500000 | − | 0.866025i | −1.41185 | + | 2.44540i | ||||||||||||||||||||||||||||||||||||||
316.2 | −2.04113 | − | 1.17845i | 0.500000 | − | 0.866025i | 1.77748 | + | 3.07868i | 3.69202i | −2.04113 | + | 1.17845i | 0.694498 | − | 0.400969i | − | 3.66487i | −0.500000 | − | 0.866025i | 4.35086 | − | 7.53590i | ||||||||||||||||||||||||||||||||||||||||
316.3 | −1.77441 | − | 1.02446i | 0.500000 | − | 0.866025i | 1.09903 | + | 1.90358i | − | 3.35690i | −1.77441 | + | 1.02446i | 1.94594 | − | 1.12349i | − | 0.405813i | −0.500000 | − | 0.866025i | −3.43900 | + | 5.95652i | |||||||||||||||||||||||||||||||||||||||
316.4 | 1.77441 | + | 1.02446i | 0.500000 | − | 0.866025i | 1.09903 | + | 1.90358i | 3.35690i | 1.77441 | − | 1.02446i | −1.94594 | + | 1.12349i | 0.405813i | −0.500000 | − | 0.866025i | −3.43900 | + | 5.95652i | |||||||||||||||||||||||||||||||||||||||||
316.5 | 2.04113 | + | 1.17845i | 0.500000 | − | 0.866025i | 1.77748 | + | 3.07868i | − | 3.69202i | 2.04113 | − | 1.17845i | −0.694498 | + | 0.400969i | 3.66487i | −0.500000 | − | 0.866025i | 4.35086 | − | 7.53590i | ||||||||||||||||||||||||||||||||||||||||
316.6 | 2.33136 | + | 1.34601i | 0.500000 | − | 0.866025i | 2.62349 | + | 4.54402i | 1.04892i | 2.33136 | − | 1.34601i | 0.480608 | − | 0.277479i | 8.74094i | −0.500000 | − | 0.866025i | −1.41185 | + | 2.44540i | |||||||||||||||||||||||||||||||||||||||||
361.1 | −2.33136 | + | 1.34601i | 0.500000 | + | 0.866025i | 2.62349 | − | 4.54402i | 1.04892i | −2.33136 | − | 1.34601i | −0.480608 | − | 0.277479i | 8.74094i | −0.500000 | + | 0.866025i | −1.41185 | − | 2.44540i | |||||||||||||||||||||||||||||||||||||||||
361.2 | −2.04113 | + | 1.17845i | 0.500000 | + | 0.866025i | 1.77748 | − | 3.07868i | − | 3.69202i | −2.04113 | − | 1.17845i | 0.694498 | + | 0.400969i | 3.66487i | −0.500000 | + | 0.866025i | 4.35086 | + | 7.53590i | ||||||||||||||||||||||||||||||||||||||||
361.3 | −1.77441 | + | 1.02446i | 0.500000 | + | 0.866025i | 1.09903 | − | 1.90358i | 3.35690i | −1.77441 | − | 1.02446i | 1.94594 | + | 1.12349i | 0.405813i | −0.500000 | + | 0.866025i | −3.43900 | − | 5.95652i | |||||||||||||||||||||||||||||||||||||||||
361.4 | 1.77441 | − | 1.02446i | 0.500000 | + | 0.866025i | 1.09903 | − | 1.90358i | − | 3.35690i | 1.77441 | + | 1.02446i | −1.94594 | − | 1.12349i | − | 0.405813i | −0.500000 | + | 0.866025i | −3.43900 | − | 5.95652i | |||||||||||||||||||||||||||||||||||||||
361.5 | 2.04113 | − | 1.17845i | 0.500000 | + | 0.866025i | 1.77748 | − | 3.07868i | 3.69202i | 2.04113 | + | 1.17845i | −0.694498 | − | 0.400969i | − | 3.66487i | −0.500000 | + | 0.866025i | 4.35086 | + | 7.53590i | ||||||||||||||||||||||||||||||||||||||||
361.6 | 2.33136 | − | 1.34601i | 0.500000 | + | 0.866025i | 2.62349 | − | 4.54402i | − | 1.04892i | 2.33136 | + | 1.34601i | 0.480608 | + | 0.277479i | − | 8.74094i | −0.500000 | + | 0.866025i | −1.41185 | − | 2.44540i | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 507.2.j.i | 12 | |
13.b | even | 2 | 1 | inner | 507.2.j.i | 12 | |
13.c | even | 3 | 1 | 507.2.b.f | 6 | ||
13.c | even | 3 | 1 | inner | 507.2.j.i | 12 | |
13.d | odd | 4 | 1 | 507.2.e.i | 6 | ||
13.d | odd | 4 | 1 | 507.2.e.l | 6 | ||
13.e | even | 6 | 1 | 507.2.b.f | 6 | ||
13.e | even | 6 | 1 | inner | 507.2.j.i | 12 | |
13.f | odd | 12 | 1 | 507.2.a.i | ✓ | 3 | |
13.f | odd | 12 | 1 | 507.2.a.l | yes | 3 | |
13.f | odd | 12 | 1 | 507.2.e.i | 6 | ||
13.f | odd | 12 | 1 | 507.2.e.l | 6 | ||
39.h | odd | 6 | 1 | 1521.2.b.k | 6 | ||
39.i | odd | 6 | 1 | 1521.2.b.k | 6 | ||
39.k | even | 12 | 1 | 1521.2.a.n | 3 | ||
39.k | even | 12 | 1 | 1521.2.a.s | 3 | ||
52.l | even | 12 | 1 | 8112.2.a.cg | 3 | ||
52.l | even | 12 | 1 | 8112.2.a.cp | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
507.2.a.i | ✓ | 3 | 13.f | odd | 12 | 1 | |
507.2.a.l | yes | 3 | 13.f | odd | 12 | 1 | |
507.2.b.f | 6 | 13.c | even | 3 | 1 | ||
507.2.b.f | 6 | 13.e | even | 6 | 1 | ||
507.2.e.i | 6 | 13.d | odd | 4 | 1 | ||
507.2.e.i | 6 | 13.f | odd | 12 | 1 | ||
507.2.e.l | 6 | 13.d | odd | 4 | 1 | ||
507.2.e.l | 6 | 13.f | odd | 12 | 1 | ||
507.2.j.i | 12 | 1.a | even | 1 | 1 | trivial | |
507.2.j.i | 12 | 13.b | even | 2 | 1 | inner | |
507.2.j.i | 12 | 13.c | even | 3 | 1 | inner | |
507.2.j.i | 12 | 13.e | even | 6 | 1 | inner | |
1521.2.a.n | 3 | 39.k | even | 12 | 1 | ||
1521.2.a.s | 3 | 39.k | even | 12 | 1 | ||
1521.2.b.k | 6 | 39.h | odd | 6 | 1 | ||
1521.2.b.k | 6 | 39.i | odd | 6 | 1 | ||
8112.2.a.cg | 3 | 52.l | even | 12 | 1 | ||
8112.2.a.cp | 3 | 52.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on :
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