Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(89,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.89");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 630.bo (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: | |
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
89.1 |
|
−0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.11423 | − | 0.728019i | 0 | 2.30608 | − | 1.29693i | 1.00000 | 0 | 1.68760 | − | 1.46697i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.98669 | + | 1.02619i | 0 | 1.39924 | − | 2.24547i | 1.00000 | 0 | 0.104634 | − | 2.23362i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.3 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.68760 | − | 1.46697i | 0 | −2.30608 | + | 1.29693i | 1.00000 | 0 | 2.11423 | − | 0.728019i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.4 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.948234 | + | 2.02506i | 0 | 0.732536 | + | 2.54232i | 1.00000 | 0 | −1.27963 | − | 1.83372i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.5 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.104634 | − | 2.23362i | 0 | −1.39924 | + | 2.24547i | 1.00000 | 0 | 1.98669 | + | 1.02619i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.6 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0.442358 | + | 2.19188i | 0 | −1.63937 | − | 2.07665i | 1.00000 | 0 | −2.11940 | − | 0.712845i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.7 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.27963 | − | 1.83372i | 0 | −0.732536 | − | 2.54232i | 1.00000 | 0 | 0.948234 | + | 2.02506i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
89.8 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.11940 | − | 0.712845i | 0 | 1.63937 | + | 2.07665i | 1.00000 | 0 | −0.442358 | + | 2.19188i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.1 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.11423 | + | 0.728019i | 0 | 2.30608 | + | 1.29693i | 1.00000 | 0 | 1.68760 | + | 1.46697i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.98669 | − | 1.02619i | 0 | 1.39924 | + | 2.24547i | 1.00000 | 0 | 0.104634 | + | 2.23362i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.3 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.68760 | + | 1.46697i | 0 | −2.30608 | − | 1.29693i | 1.00000 | 0 | 2.11423 | + | 0.728019i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.4 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.948234 | − | 2.02506i | 0 | 0.732536 | − | 2.54232i | 1.00000 | 0 | −1.27963 | + | 1.83372i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.5 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.104634 | + | 2.23362i | 0 | −1.39924 | − | 2.24547i | 1.00000 | 0 | 1.98669 | − | 1.02619i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.6 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.442358 | − | 2.19188i | 0 | −1.63937 | + | 2.07665i | 1.00000 | 0 | −2.11940 | + | 0.712845i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.7 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.27963 | + | 1.83372i | 0 | −0.732536 | + | 2.54232i | 1.00000 | 0 | 0.948234 | − | 2.02506i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
269.8 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 2.11940 | + | 0.712845i | 0 | 1.63937 | − | 2.07665i | 1.00000 | 0 | −0.442358 | − | 2.19188i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
15.d | odd | 2 | 1 | inner |
105.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 630.2.bo.a | ✓ | 16 |
3.b | odd | 2 | 1 | 630.2.bo.b | yes | 16 | |
5.b | even | 2 | 1 | 630.2.bo.b | yes | 16 | |
5.c | odd | 4 | 2 | 3150.2.bf.f | 32 | ||
7.c | even | 3 | 1 | 4410.2.d.b | 16 | ||
7.d | odd | 6 | 1 | inner | 630.2.bo.a | ✓ | 16 |
7.d | odd | 6 | 1 | 4410.2.d.b | 16 | ||
15.d | odd | 2 | 1 | inner | 630.2.bo.a | ✓ | 16 |
15.e | even | 4 | 2 | 3150.2.bf.f | 32 | ||
21.g | even | 6 | 1 | 630.2.bo.b | yes | 16 | |
21.g | even | 6 | 1 | 4410.2.d.a | 16 | ||
21.h | odd | 6 | 1 | 4410.2.d.a | 16 | ||
35.i | odd | 6 | 1 | 630.2.bo.b | yes | 16 | |
35.i | odd | 6 | 1 | 4410.2.d.a | 16 | ||
35.j | even | 6 | 1 | 4410.2.d.a | 16 | ||
35.k | even | 12 | 2 | 3150.2.bf.f | 32 | ||
105.o | odd | 6 | 1 | 4410.2.d.b | 16 | ||
105.p | even | 6 | 1 | inner | 630.2.bo.a | ✓ | 16 |
105.p | even | 6 | 1 | 4410.2.d.b | 16 | ||
105.w | odd | 12 | 2 | 3150.2.bf.f | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bo.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
630.2.bo.a | ✓ | 16 | 7.d | odd | 6 | 1 | inner |
630.2.bo.a | ✓ | 16 | 15.d | odd | 2 | 1 | inner |
630.2.bo.a | ✓ | 16 | 105.p | even | 6 | 1 | inner |
630.2.bo.b | yes | 16 | 3.b | odd | 2 | 1 | |
630.2.bo.b | yes | 16 | 5.b | even | 2 | 1 | |
630.2.bo.b | yes | 16 | 21.g | even | 6 | 1 | |
630.2.bo.b | yes | 16 | 35.i | odd | 6 | 1 | |
3150.2.bf.f | 32 | 5.c | odd | 4 | 2 | ||
3150.2.bf.f | 32 | 15.e | even | 4 | 2 | ||
3150.2.bf.f | 32 | 35.k | even | 12 | 2 | ||
3150.2.bf.f | 32 | 105.w | odd | 12 | 2 | ||
4410.2.d.a | 16 | 21.g | even | 6 | 1 | ||
4410.2.d.a | 16 | 21.h | odd | 6 | 1 | ||
4410.2.d.a | 16 | 35.i | odd | 6 | 1 | ||
4410.2.d.a | 16 | 35.j | even | 6 | 1 | ||
4410.2.d.b | 16 | 7.c | even | 3 | 1 | ||
4410.2.d.b | 16 | 7.d | odd | 6 | 1 | ||
4410.2.d.b | 16 | 105.o | odd | 6 | 1 | ||
4410.2.d.b | 16 | 105.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
acting on .