Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [155,2,Mod(36,155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(155, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("155.36");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 155.e (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: | |
Relative dimension: | over |
Coefficient field: | 8.0.42575625.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
|
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
36.1 |
|
−1.35567 | −0.440197 | + | 0.762443i | −0.162147 | −0.500000 | − | 0.866025i | 0.596764 | − | 1.03362i | 1.77460 | − | 3.07370i | 2.93117 | 1.11245 | + | 1.92683i | 0.677837 | + | 1.17405i | ||||||||||||||||||||||||||||||
36.2 | −0.477260 | 1.35666 | − | 2.34981i | −1.77222 | −0.500000 | − | 0.866025i | −0.647481 | + | 1.12147i | 0.0911485 | − | 0.157874i | 1.80033 | −2.18107 | − | 3.77773i | 0.238630 | + | 0.413319i | |||||||||||||||||||||||||||||||
36.3 | 0.737640 | −1.48685 | + | 2.57531i | −1.45589 | −0.500000 | − | 0.866025i | −1.09676 | + | 1.89965i | −0.965584 | + | 1.67244i | −2.54920 | −2.92147 | − | 5.06014i | −0.368820 | − | 0.638815i | |||||||||||||||||||||||||||||||
36.4 | 2.09529 | 0.0703870 | − | 0.121914i | 2.39026 | −0.500000 | − | 0.866025i | 0.147481 | − | 0.255445i | −0.400166 | + | 0.693107i | 0.817703 | 1.49009 | + | 2.58091i | −1.04765 | − | 1.81458i | |||||||||||||||||||||||||||||||
56.1 | −1.35567 | −0.440197 | − | 0.762443i | −0.162147 | −0.500000 | + | 0.866025i | 0.596764 | + | 1.03362i | 1.77460 | + | 3.07370i | 2.93117 | 1.11245 | − | 1.92683i | 0.677837 | − | 1.17405i | |||||||||||||||||||||||||||||||
56.2 | −0.477260 | 1.35666 | + | 2.34981i | −1.77222 | −0.500000 | + | 0.866025i | −0.647481 | − | 1.12147i | 0.0911485 | + | 0.157874i | 1.80033 | −2.18107 | + | 3.77773i | 0.238630 | − | 0.413319i | |||||||||||||||||||||||||||||||
56.3 | 0.737640 | −1.48685 | − | 2.57531i | −1.45589 | −0.500000 | + | 0.866025i | −1.09676 | − | 1.89965i | −0.965584 | − | 1.67244i | −2.54920 | −2.92147 | + | 5.06014i | −0.368820 | + | 0.638815i | |||||||||||||||||||||||||||||||
56.4 | 2.09529 | 0.0703870 | + | 0.121914i | 2.39026 | −0.500000 | + | 0.866025i | 0.147481 | + | 0.255445i | −0.400166 | − | 0.693107i | 0.817703 | 1.49009 | − | 2.58091i | −1.04765 | + | 1.81458i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 155.2.e.d | ✓ | 8 |
5.b | even | 2 | 1 | 775.2.e.f | 8 | ||
5.c | odd | 4 | 2 | 775.2.o.f | 16 | ||
31.c | even | 3 | 1 | inner | 155.2.e.d | ✓ | 8 |
31.c | even | 3 | 1 | 4805.2.a.o | 4 | ||
31.e | odd | 6 | 1 | 4805.2.a.m | 4 | ||
155.j | even | 6 | 1 | 775.2.e.f | 8 | ||
155.o | odd | 12 | 2 | 775.2.o.f | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
155.2.e.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
155.2.e.d | ✓ | 8 | 31.c | even | 3 | 1 | inner |
775.2.e.f | 8 | 5.b | even | 2 | 1 | ||
775.2.e.f | 8 | 155.j | even | 6 | 1 | ||
775.2.o.f | 16 | 5.c | odd | 4 | 2 | ||
775.2.o.f | 16 | 155.o | odd | 12 | 2 | ||
4805.2.a.m | 4 | 31.e | odd | 6 | 1 | ||
4805.2.a.o | 4 | 31.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .