Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [297,2,Mod(100,297)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(297, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("297.100");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 297.e (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: |
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Coefficient ring: | |
Coefficient ring index: | |
Twist minimal: | no (minimal twist has level 99) |
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
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 | ||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
100.1 |
|
−0.939693 | + | 1.62760i | 0 | −0.766044 | − | 1.32683i | 1.43969 | + | 2.49362i | 0 | 0.326352 | − | 0.565258i | −0.879385 | 0 | −5.41147 | ||||||||||||||||||||||||||||
100.2 | 0.173648 | − | 0.300767i | 0 | 0.939693 | + | 1.62760i | 0.326352 | + | 0.565258i | 0 | −0.266044 | + | 0.460802i | 1.34730 | 0 | 0.226682 | |||||||||||||||||||||||||||||
100.3 | 0.766044 | − | 1.32683i | 0 | −0.173648 | − | 0.300767i | −0.266044 | − | 0.460802i | 0 | 1.43969 | − | 2.49362i | 2.53209 | 0 | −0.815207 | |||||||||||||||||||||||||||||
199.1 | −0.939693 | − | 1.62760i | 0 | −0.766044 | + | 1.32683i | 1.43969 | − | 2.49362i | 0 | 0.326352 | + | 0.565258i | −0.879385 | 0 | −5.41147 | |||||||||||||||||||||||||||||
199.2 | 0.173648 | + | 0.300767i | 0 | 0.939693 | − | 1.62760i | 0.326352 | − | 0.565258i | 0 | −0.266044 | − | 0.460802i | 1.34730 | 0 | 0.226682 | |||||||||||||||||||||||||||||
199.3 | 0.766044 | + | 1.32683i | 0 | −0.173648 | + | 0.300767i | −0.266044 | + | 0.460802i | 0 | 1.43969 | + | 2.49362i | 2.53209 | 0 | −0.815207 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 297.2.e.d | 6 | |
3.b | odd | 2 | 1 | 99.2.e.d | ✓ | 6 | |
9.c | even | 3 | 1 | inner | 297.2.e.d | 6 | |
9.c | even | 3 | 1 | 891.2.a.k | 3 | ||
9.d | odd | 6 | 1 | 99.2.e.d | ✓ | 6 | |
9.d | odd | 6 | 1 | 891.2.a.l | 3 | ||
33.d | even | 2 | 1 | 1089.2.e.h | 6 | ||
99.g | even | 6 | 1 | 1089.2.e.h | 6 | ||
99.g | even | 6 | 1 | 9801.2.a.be | 3 | ||
99.h | odd | 6 | 1 | 9801.2.a.bd | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.2.e.d | ✓ | 6 | 3.b | odd | 2 | 1 | |
99.2.e.d | ✓ | 6 | 9.d | odd | 6 | 1 | |
297.2.e.d | 6 | 1.a | even | 1 | 1 | trivial | |
297.2.e.d | 6 | 9.c | even | 3 | 1 | inner | |
891.2.a.k | 3 | 9.c | even | 3 | 1 | ||
891.2.a.l | 3 | 9.d | odd | 6 | 1 | ||
1089.2.e.h | 6 | 33.d | even | 2 | 1 | ||
1089.2.e.h | 6 | 99.g | even | 6 | 1 | ||
9801.2.a.bd | 3 | 99.h | odd | 6 | 1 | ||
9801.2.a.be | 3 | 99.g | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .