Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,2,Mod(131,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.131");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 156.c (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: | |
Coefficient field: | 8.0.121550625.1 |
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
131.1 |
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−1.11803 | − | 0.866025i | −0.586627 | − | 1.62968i | 0.500000 | + | 1.93649i | 3.82407i | −0.755479 | + | 2.33008i | 3.25937i | 1.11803 | − | 2.59808i | −2.31174 | + | 1.91203i | 3.31174 | − | 4.27543i | ||||||||||||||||||||||||||||
131.2 | −1.11803 | − | 0.866025i | 1.70466 | − | 0.306808i | 0.500000 | + | 1.93649i | − | 2.09201i | −2.17157 | − | 1.13326i | 0.613616i | 1.11803 | − | 2.59808i | 2.81174 | − | 1.04601i | −1.81174 | + | 2.33894i | ||||||||||||||||||||||||||||
131.3 | −1.11803 | + | 0.866025i | −0.586627 | + | 1.62968i | 0.500000 | − | 1.93649i | − | 3.82407i | −0.755479 | − | 2.33008i | − | 3.25937i | 1.11803 | + | 2.59808i | −2.31174 | − | 1.91203i | 3.31174 | + | 4.27543i | |||||||||||||||||||||||||||
131.4 | −1.11803 | + | 0.866025i | 1.70466 | + | 0.306808i | 0.500000 | − | 1.93649i | 2.09201i | −2.17157 | + | 1.13326i | − | 0.613616i | 1.11803 | + | 2.59808i | 2.81174 | + | 1.04601i | −1.81174 | − | 2.33894i | ||||||||||||||||||||||||||||
131.5 | 1.11803 | − | 0.866025i | −1.70466 | + | 0.306808i | 0.500000 | − | 1.93649i | − | 2.09201i | −1.64017 | + | 1.81930i | − | 0.613616i | −1.11803 | − | 2.59808i | 2.81174 | − | 1.04601i | −1.81174 | − | 2.33894i | |||||||||||||||||||||||||||
131.6 | 1.11803 | − | 0.866025i | 0.586627 | + | 1.62968i | 0.500000 | − | 1.93649i | 3.82407i | 2.06722 | + | 1.31401i | − | 3.25937i | −1.11803 | − | 2.59808i | −2.31174 | + | 1.91203i | 3.31174 | + | 4.27543i | ||||||||||||||||||||||||||||
131.7 | 1.11803 | + | 0.866025i | −1.70466 | − | 0.306808i | 0.500000 | + | 1.93649i | 2.09201i | −1.64017 | − | 1.81930i | 0.613616i | −1.11803 | + | 2.59808i | 2.81174 | + | 1.04601i | −1.81174 | + | 2.33894i | |||||||||||||||||||||||||||||
131.8 | 1.11803 | + | 0.866025i | 0.586627 | − | 1.62968i | 0.500000 | + | 1.93649i | − | 3.82407i | 2.06722 | − | 1.31401i | 3.25937i | −1.11803 | + | 2.59808i | −2.31174 | − | 1.91203i | 3.31174 | − | 4.27543i | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 156.2.c.c | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 156.2.c.c | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 156.2.c.c | ✓ | 8 |
8.b | even | 2 | 1 | 2496.2.d.m | 8 | ||
8.d | odd | 2 | 1 | 2496.2.d.m | 8 | ||
12.b | even | 2 | 1 | inner | 156.2.c.c | ✓ | 8 |
24.f | even | 2 | 1 | 2496.2.d.m | 8 | ||
24.h | odd | 2 | 1 | 2496.2.d.m | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
156.2.c.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
156.2.c.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
156.2.c.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
156.2.c.c | ✓ | 8 | 12.b | even | 2 | 1 | inner |
2496.2.d.m | 8 | 8.b | even | 2 | 1 | ||
2496.2.d.m | 8 | 8.d | odd | 2 | 1 | ||
2496.2.d.m | 8 | 24.f | even | 2 | 1 | ||
2496.2.d.m | 8 | 24.h | odd | 2 | 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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