Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9801,2,Mod(1,9801)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9801, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9801.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 9801.a (trivial) |
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: | yes |
Analytic conductor: | |
Analytic rank: | |
Dimension: | |
Coefficient field: | 5.5.5144904.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
|
Coefficient ring: | |
Coefficient ring index: | |
Twist minimal: | yes |
Fricke sign: | |
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
:
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 | |||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−2.17399 | 0 | 2.72621 | −2.40479 | 0 | 4.30500 | −1.57878 | 0 | 5.22799 | |||||||||||||||||||||||||||||||||
1.2 | −1.52027 | 0 | 0.311228 | 1.08767 | 0 | −2.25617 | 2.56739 | 0 | −1.65355 | ||||||||||||||||||||||||||||||||||
1.3 | −0.156598 | 0 | −1.97548 | −2.22085 | 0 | −2.59803 | 0.622551 | 0 | 0.347781 | ||||||||||||||||||||||||||||||||||
1.4 | 2.13163 | 0 | 2.54385 | −3.97234 | 0 | 1.38456 | 1.15930 | 0 | −8.46756 | ||||||||||||||||||||||||||||||||||
1.5 | 2.71922 | 0 | 5.39418 | 3.51032 | 0 | −3.83536 | 9.22954 | 0 | 9.54534 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
Sign | |
---|---|
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 9801.2.a.bp | yes | 5 |
3.b | odd | 2 | 1 | 9801.2.a.bo | yes | 5 | |
11.b | odd | 2 | 1 | 9801.2.a.bn | ✓ | 5 | |
33.d | even | 2 | 1 | 9801.2.a.bq | yes | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
9801.2.a.bn | ✓ | 5 | 11.b | odd | 2 | 1 | |
9801.2.a.bo | yes | 5 | 3.b | odd | 2 | 1 | |
9801.2.a.bp | yes | 5 | 1.a | even | 1 | 1 | trivial |
9801.2.a.bq | yes | 5 | 33.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on :
|
|
|
|