Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,2,Mod(255,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 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("1216.255");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 1216.n (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: | |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
|
Coefficient ring: | |
Coefficient ring index: | |
Twist minimal: | no (minimal twist has level 76) |
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
255.1 |
|
0 | −1.42689 | + | 2.47144i | 0 | 0.139977 | − | 0.242447i | 0 | 1.55280i | 0 | −2.57201 | − | 4.45486i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.2 | 0 | −0.982349 | + | 1.70148i | 0 | 0.349646 | − | 0.605604i | 0 | − | 3.80025i | 0 | −0.430019 | − | 0.744815i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.3 | 0 | −0.637123 | + | 1.10353i | 0 | 1.60333 | − | 2.77705i | 0 | 1.25044i | 0 | 0.688149 | + | 1.19191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.4 | 0 | −0.305055 | + | 0.528371i | 0 | −1.59295 | + | 2.75907i | 0 | 2.36291i | 0 | 1.31388 | + | 2.27571i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.5 | 0 | 0.305055 | − | 0.528371i | 0 | −1.59295 | + | 2.75907i | 0 | − | 2.36291i | 0 | 1.31388 | + | 2.27571i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.6 | 0 | 0.637123 | − | 1.10353i | 0 | 1.60333 | − | 2.77705i | 0 | − | 1.25044i | 0 | 0.688149 | + | 1.19191i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.7 | 0 | 0.982349 | − | 1.70148i | 0 | 0.349646 | − | 0.605604i | 0 | 3.80025i | 0 | −0.430019 | − | 0.744815i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
255.8 | 0 | 1.42689 | − | 2.47144i | 0 | 0.139977 | − | 0.242447i | 0 | − | 1.55280i | 0 | −2.57201 | − | 4.45486i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.1 | 0 | −1.42689 | − | 2.47144i | 0 | 0.139977 | + | 0.242447i | 0 | − | 1.55280i | 0 | −2.57201 | + | 4.45486i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.2 | 0 | −0.982349 | − | 1.70148i | 0 | 0.349646 | + | 0.605604i | 0 | 3.80025i | 0 | −0.430019 | + | 0.744815i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.3 | 0 | −0.637123 | − | 1.10353i | 0 | 1.60333 | + | 2.77705i | 0 | − | 1.25044i | 0 | 0.688149 | − | 1.19191i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.4 | 0 | −0.305055 | − | 0.528371i | 0 | −1.59295 | − | 2.75907i | 0 | − | 2.36291i | 0 | 1.31388 | − | 2.27571i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.5 | 0 | 0.305055 | + | 0.528371i | 0 | −1.59295 | − | 2.75907i | 0 | 2.36291i | 0 | 1.31388 | − | 2.27571i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.6 | 0 | 0.637123 | + | 1.10353i | 0 | 1.60333 | + | 2.77705i | 0 | 1.25044i | 0 | 0.688149 | − | 1.19191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.7 | 0 | 0.982349 | + | 1.70148i | 0 | 0.349646 | + | 0.605604i | 0 | − | 3.80025i | 0 | −0.430019 | + | 0.744815i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
639.8 | 0 | 1.42689 | + | 2.47144i | 0 | 0.139977 | + | 0.242447i | 0 | 1.55280i | 0 | −2.57201 | + | 4.45486i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
76.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1216.2.n.f | 16 | |
4.b | odd | 2 | 1 | inner | 1216.2.n.f | 16 | |
8.b | even | 2 | 1 | 76.2.f.a | ✓ | 16 | |
8.d | odd | 2 | 1 | 76.2.f.a | ✓ | 16 | |
19.d | odd | 6 | 1 | inner | 1216.2.n.f | 16 | |
24.f | even | 2 | 1 | 684.2.r.a | 16 | ||
24.h | odd | 2 | 1 | 684.2.r.a | 16 | ||
76.f | even | 6 | 1 | inner | 1216.2.n.f | 16 | |
152.l | odd | 6 | 1 | 76.2.f.a | ✓ | 16 | |
152.o | even | 6 | 1 | 76.2.f.a | ✓ | 16 | |
456.s | odd | 6 | 1 | 684.2.r.a | 16 | ||
456.v | even | 6 | 1 | 684.2.r.a | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.2.f.a | ✓ | 16 | 8.b | even | 2 | 1 | |
76.2.f.a | ✓ | 16 | 8.d | odd | 2 | 1 | |
76.2.f.a | ✓ | 16 | 152.l | odd | 6 | 1 | |
76.2.f.a | ✓ | 16 | 152.o | even | 6 | 1 | |
684.2.r.a | 16 | 24.f | even | 2 | 1 | ||
684.2.r.a | 16 | 24.h | odd | 2 | 1 | ||
684.2.r.a | 16 | 456.s | odd | 6 | 1 | ||
684.2.r.a | 16 | 456.v | even | 6 | 1 | ||
1216.2.n.f | 16 | 1.a | even | 1 | 1 | trivial | |
1216.2.n.f | 16 | 4.b | odd | 2 | 1 | inner | |
1216.2.n.f | 16 | 19.d | odd | 6 | 1 | inner | |
1216.2.n.f | 16 | 76.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .