Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [120,2,Mod(109,120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 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("120.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 120.d (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: | 6.0.839056.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
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 | ||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 |
|
−1.34067 | − | 0.450129i | 1.00000 | 1.59477 | + | 1.20695i | −0.254102 | + | 2.22158i | −1.34067 | − | 0.450129i | 2.64265i | −1.59477 | − | 2.33596i | 1.00000 | 1.34067 | − | 2.86402i | ||||||||||||||||||||||||
109.2 | −1.34067 | + | 0.450129i | 1.00000 | 1.59477 | − | 1.20695i | −0.254102 | − | 2.22158i | −1.34067 | + | 0.450129i | − | 2.64265i | −1.59477 | + | 2.33596i | 1.00000 | 1.34067 | + | 2.86402i | ||||||||||||||||||||||||
109.3 | −0.321037 | − | 1.37729i | 1.00000 | −1.79387 | + | 0.884323i | 2.11491 | + | 0.726062i | −0.321037 | − | 1.37729i | − | 4.05705i | 1.79387 | + | 2.18678i | 1.00000 | 0.321037 | − | 3.14594i | ||||||||||||||||||||||||
109.4 | −0.321037 | + | 1.37729i | 1.00000 | −1.79387 | − | 0.884323i | 2.11491 | − | 0.726062i | −0.321037 | + | 1.37729i | 4.05705i | 1.79387 | − | 2.18678i | 1.00000 | 0.321037 | + | 3.14594i | |||||||||||||||||||||||||
109.5 | 1.16170 | − | 0.806504i | 1.00000 | 0.699104 | − | 1.87383i | −1.86081 | + | 1.23992i | 1.16170 | − | 0.806504i | − | 0.746175i | −0.699104 | − | 2.74067i | 1.00000 | −1.16170 | + | 2.94116i | ||||||||||||||||||||||||
109.6 | 1.16170 | + | 0.806504i | 1.00000 | 0.699104 | + | 1.87383i | −1.86081 | − | 1.23992i | 1.16170 | + | 0.806504i | 0.746175i | −0.699104 | + | 2.74067i | 1.00000 | −1.16170 | − | 2.94116i | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 120.2.d.a | ✓ | 6 |
3.b | odd | 2 | 1 | 360.2.d.f | 6 | ||
4.b | odd | 2 | 1 | 480.2.d.a | 6 | ||
5.b | even | 2 | 1 | 120.2.d.b | yes | 6 | |
5.c | odd | 4 | 2 | 600.2.k.f | 12 | ||
8.b | even | 2 | 1 | 120.2.d.b | yes | 6 | |
8.d | odd | 2 | 1 | 480.2.d.b | 6 | ||
12.b | even | 2 | 1 | 1440.2.d.e | 6 | ||
15.d | odd | 2 | 1 | 360.2.d.e | 6 | ||
15.e | even | 4 | 2 | 1800.2.k.u | 12 | ||
16.e | even | 4 | 2 | 3840.2.f.l | 12 | ||
16.f | odd | 4 | 2 | 3840.2.f.m | 12 | ||
20.d | odd | 2 | 1 | 480.2.d.b | 6 | ||
20.e | even | 4 | 2 | 2400.2.k.f | 12 | ||
24.f | even | 2 | 1 | 1440.2.d.f | 6 | ||
24.h | odd | 2 | 1 | 360.2.d.e | 6 | ||
40.e | odd | 2 | 1 | 480.2.d.a | 6 | ||
40.f | even | 2 | 1 | inner | 120.2.d.a | ✓ | 6 |
40.i | odd | 4 | 2 | 600.2.k.f | 12 | ||
40.k | even | 4 | 2 | 2400.2.k.f | 12 | ||
60.h | even | 2 | 1 | 1440.2.d.f | 6 | ||
60.l | odd | 4 | 2 | 7200.2.k.u | 12 | ||
80.k | odd | 4 | 2 | 3840.2.f.m | 12 | ||
80.q | even | 4 | 2 | 3840.2.f.l | 12 | ||
120.i | odd | 2 | 1 | 360.2.d.f | 6 | ||
120.m | even | 2 | 1 | 1440.2.d.e | 6 | ||
120.q | odd | 4 | 2 | 7200.2.k.u | 12 | ||
120.w | even | 4 | 2 | 1800.2.k.u | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
120.2.d.a | ✓ | 6 | 40.f | even | 2 | 1 | inner |
120.2.d.b | yes | 6 | 5.b | even | 2 | 1 | |
120.2.d.b | yes | 6 | 8.b | even | 2 | 1 | |
360.2.d.e | 6 | 15.d | odd | 2 | 1 | ||
360.2.d.e | 6 | 24.h | odd | 2 | 1 | ||
360.2.d.f | 6 | 3.b | odd | 2 | 1 | ||
360.2.d.f | 6 | 120.i | odd | 2 | 1 | ||
480.2.d.a | 6 | 4.b | odd | 2 | 1 | ||
480.2.d.a | 6 | 40.e | odd | 2 | 1 | ||
480.2.d.b | 6 | 8.d | odd | 2 | 1 | ||
480.2.d.b | 6 | 20.d | odd | 2 | 1 | ||
600.2.k.f | 12 | 5.c | odd | 4 | 2 | ||
600.2.k.f | 12 | 40.i | odd | 4 | 2 | ||
1440.2.d.e | 6 | 12.b | even | 2 | 1 | ||
1440.2.d.e | 6 | 120.m | even | 2 | 1 | ||
1440.2.d.f | 6 | 24.f | even | 2 | 1 | ||
1440.2.d.f | 6 | 60.h | even | 2 | 1 | ||
1800.2.k.u | 12 | 15.e | even | 4 | 2 | ||
1800.2.k.u | 12 | 120.w | even | 4 | 2 | ||
2400.2.k.f | 12 | 20.e | even | 4 | 2 | ||
2400.2.k.f | 12 | 40.k | even | 4 | 2 | ||
3840.2.f.l | 12 | 16.e | even | 4 | 2 | ||
3840.2.f.l | 12 | 80.q | even | 4 | 2 | ||
3840.2.f.m | 12 | 16.f | odd | 4 | 2 | ||
3840.2.f.m | 12 | 80.k | odd | 4 | 2 | ||
7200.2.k.u | 12 | 60.l | odd | 4 | 2 | ||
7200.2.k.u | 12 | 120.q | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .