Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1045,2,Mod(1,1045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 1045.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: | 6.6.7281497.1 |
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 |
|
−2.51246 | 0.895533 | 4.31247 | −1.00000 | −2.24999 | −0.393051 | −5.80998 | −2.19802 | 2.51246 | ||||||||||||||||||||||||||||||||||||
1.2 | −2.23198 | −1.34246 | 2.98176 | −1.00000 | 2.99635 | 4.13295 | −2.19127 | −1.19780 | 2.23198 | |||||||||||||||||||||||||||||||||||||
1.3 | −0.412130 | 1.67805 | −1.83015 | −1.00000 | −0.691575 | 0.0703171 | 1.57852 | −0.184142 | 0.412130 | |||||||||||||||||||||||||||||||||||||
1.4 | 0.205229 | −3.10246 | −1.95788 | −1.00000 | −0.636714 | 2.33231 | −0.812271 | 6.62527 | −0.205229 | |||||||||||||||||||||||||||||||||||||
1.5 | 1.21244 | 1.77266 | −0.529980 | −1.00000 | 2.14925 | −3.37010 | −3.06746 | 0.142317 | −1.21244 | |||||||||||||||||||||||||||||||||||||
1.6 | 1.73890 | −0.901323 | 1.02379 | −1.00000 | −1.56731 | 2.22757 | −1.69754 | −2.18762 | −1.73890 | |||||||||||||||||||||||||||||||||||||
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 | 1045.2.a.f | ✓ | 6 |
3.b | odd | 2 | 1 | 9405.2.a.z | 6 | ||
5.b | even | 2 | 1 | 5225.2.a.l | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1045.2.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
5225.2.a.l | 6 | 5.b | even | 2 | 1 | ||
9405.2.a.z | 6 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .