Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(49,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 624.bv (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: | 8.0.649638144.4 |
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: | no (minimal twist has level 312) |
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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49.1 |
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0 | 0.500000 | + | 0.866025i | 0 | − | 1.55452i | 0 | 2.56383 | + | 1.48023i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||
49.2 | 0 | 0.500000 | + | 0.866025i | 0 | − | 0.475353i | 0 | −3.94508 | − | 2.27769i | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||
49.3 | 0 | 0.500000 | + | 0.866025i | 0 | 1.28657i | 0 | −1.69781 | − | 0.980228i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
49.4 | 0 | 0.500000 | + | 0.866025i | 0 | 4.20740i | 0 | 3.07905 | + | 1.77769i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
433.1 | 0 | 0.500000 | − | 0.866025i | 0 | − | 4.20740i | 0 | 3.07905 | − | 1.77769i | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||
433.2 | 0 | 0.500000 | − | 0.866025i | 0 | − | 1.28657i | 0 | −1.69781 | + | 0.980228i | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||
433.3 | 0 | 0.500000 | − | 0.866025i | 0 | 0.475353i | 0 | −3.94508 | + | 2.27769i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
433.4 | 0 | 0.500000 | − | 0.866025i | 0 | 1.55452i | 0 | 2.56383 | − | 1.48023i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
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 | 624.2.bv.g | 8 | |
3.b | odd | 2 | 1 | 1872.2.by.m | 8 | ||
4.b | odd | 2 | 1 | 312.2.bf.b | ✓ | 8 | |
12.b | even | 2 | 1 | 936.2.bi.c | 8 | ||
13.e | even | 6 | 1 | inner | 624.2.bv.g | 8 | |
13.f | odd | 12 | 1 | 8112.2.a.cq | 4 | ||
13.f | odd | 12 | 1 | 8112.2.a.cs | 4 | ||
39.h | odd | 6 | 1 | 1872.2.by.m | 8 | ||
52.i | odd | 6 | 1 | 312.2.bf.b | ✓ | 8 | |
52.i | odd | 6 | 1 | 4056.2.c.p | 8 | ||
52.j | odd | 6 | 1 | 4056.2.c.p | 8 | ||
52.l | even | 12 | 1 | 4056.2.a.bd | 4 | ||
52.l | even | 12 | 1 | 4056.2.a.be | 4 | ||
156.r | even | 6 | 1 | 936.2.bi.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
312.2.bf.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
312.2.bf.b | ✓ | 8 | 52.i | odd | 6 | 1 | |
624.2.bv.g | 8 | 1.a | even | 1 | 1 | trivial | |
624.2.bv.g | 8 | 13.e | even | 6 | 1 | inner | |
936.2.bi.c | 8 | 12.b | even | 2 | 1 | ||
936.2.bi.c | 8 | 156.r | even | 6 | 1 | ||
1872.2.by.m | 8 | 3.b | odd | 2 | 1 | ||
1872.2.by.m | 8 | 39.h | odd | 6 | 1 | ||
4056.2.a.bd | 4 | 52.l | even | 12 | 1 | ||
4056.2.a.be | 4 | 52.l | even | 12 | 1 | ||
4056.2.c.p | 8 | 52.i | odd | 6 | 1 | ||
4056.2.c.p | 8 | 52.j | odd | 6 | 1 | ||
8112.2.a.cq | 4 | 13.f | odd | 12 | 1 | ||
8112.2.a.cs | 4 | 13.f | odd | 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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