Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,3,Mod(449,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.449");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 960.c (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: | |
Coefficient field: | 8.0.40960000.1 |
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: | no (minimal twist has level 480) |
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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449.1 |
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0 | −2.99535 | − | 0.166925i | 0 | − | 5.00000i | 0 | 6.65841i | 0 | 8.94427 | + | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
449.2 | 0 | −2.99535 | + | 0.166925i | 0 | 5.00000i | 0 | − | 6.65841i | 0 | 8.94427 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
449.3 | 0 | −0.166925 | − | 2.99535i | 0 | − | 5.00000i | 0 | 12.3153i | 0 | −8.94427 | + | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
449.4 | 0 | −0.166925 | + | 2.99535i | 0 | 5.00000i | 0 | − | 12.3153i | 0 | −8.94427 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
449.5 | 0 | 0.166925 | − | 2.99535i | 0 | 5.00000i | 0 | 12.3153i | 0 | −8.94427 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
449.6 | 0 | 0.166925 | + | 2.99535i | 0 | − | 5.00000i | 0 | − | 12.3153i | 0 | −8.94427 | + | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
449.7 | 0 | 2.99535 | − | 0.166925i | 0 | 5.00000i | 0 | 6.65841i | 0 | 8.94427 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
449.8 | 0 | 2.99535 | + | 0.166925i | 0 | − | 5.00000i | 0 | − | 6.65841i | 0 | 8.94427 | + | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
60.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.3.c.i | 8 | |
3.b | odd | 2 | 1 | inner | 960.3.c.i | 8 | |
4.b | odd | 2 | 1 | inner | 960.3.c.i | 8 | |
5.b | even | 2 | 1 | inner | 960.3.c.i | 8 | |
8.b | even | 2 | 1 | 480.3.c.a | ✓ | 8 | |
8.d | odd | 2 | 1 | 480.3.c.a | ✓ | 8 | |
12.b | even | 2 | 1 | inner | 960.3.c.i | 8 | |
15.d | odd | 2 | 1 | inner | 960.3.c.i | 8 | |
20.d | odd | 2 | 1 | CM | 960.3.c.i | 8 | |
24.f | even | 2 | 1 | 480.3.c.a | ✓ | 8 | |
24.h | odd | 2 | 1 | 480.3.c.a | ✓ | 8 | |
40.e | odd | 2 | 1 | 480.3.c.a | ✓ | 8 | |
40.f | even | 2 | 1 | 480.3.c.a | ✓ | 8 | |
60.h | even | 2 | 1 | inner | 960.3.c.i | 8 | |
120.i | odd | 2 | 1 | 480.3.c.a | ✓ | 8 | |
120.m | even | 2 | 1 | 480.3.c.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
480.3.c.a | ✓ | 8 | 8.b | even | 2 | 1 | |
480.3.c.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
480.3.c.a | ✓ | 8 | 24.f | even | 2 | 1 | |
480.3.c.a | ✓ | 8 | 24.h | odd | 2 | 1 | |
480.3.c.a | ✓ | 8 | 40.e | odd | 2 | 1 | |
480.3.c.a | ✓ | 8 | 40.f | even | 2 | 1 | |
480.3.c.a | ✓ | 8 | 120.i | odd | 2 | 1 | |
480.3.c.a | ✓ | 8 | 120.m | even | 2 | 1 | |
960.3.c.i | 8 | 1.a | even | 1 | 1 | trivial | |
960.3.c.i | 8 | 3.b | odd | 2 | 1 | inner | |
960.3.c.i | 8 | 4.b | odd | 2 | 1 | inner | |
960.3.c.i | 8 | 5.b | even | 2 | 1 | inner | |
960.3.c.i | 8 | 12.b | even | 2 | 1 | inner | |
960.3.c.i | 8 | 15.d | odd | 2 | 1 | inner | |
960.3.c.i | 8 | 20.d | odd | 2 | 1 | CM | |
960.3.c.i | 8 | 60.h | even | 2 | 1 | inner |
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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