Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [832,2,Mod(321,832)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(832, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("832.321");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 832.i (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: | |
Coefficient field: | |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
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Coefficient ring: | |
Coefficient ring index: | |
Twist minimal: | no (minimal twist has level 104) |
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 primitive root of unity . We also show the integral -expansion of the trace form.
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 | ||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
321.1 |
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0 | 0.500000 | − | 0.866025i | 0 | −2.00000 | 0 | 0.500000 | + | 0.866025i | 0 | 1.00000 | + | 1.73205i | 0 | ||||||||||||||||||
705.1 | 0 | 0.500000 | + | 0.866025i | 0 | −2.00000 | 0 | 0.500000 | − | 0.866025i | 0 | 1.00000 | − | 1.73205i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
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Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 832.2.i.g | 2 | |
4.b | odd | 2 | 1 | 832.2.i.d | 2 | ||
8.b | even | 2 | 1 | 104.2.i.a | ✓ | 2 | |
8.d | odd | 2 | 1 | 208.2.i.c | 2 | ||
13.c | even | 3 | 1 | inner | 832.2.i.g | 2 | |
24.f | even | 2 | 1 | 1872.2.t.d | 2 | ||
24.h | odd | 2 | 1 | 936.2.t.c | 2 | ||
52.j | odd | 6 | 1 | 832.2.i.d | 2 | ||
104.e | even | 2 | 1 | 1352.2.i.a | 2 | ||
104.j | odd | 4 | 2 | 1352.2.o.b | 4 | ||
104.n | odd | 6 | 1 | 208.2.i.c | 2 | ||
104.n | odd | 6 | 1 | 2704.2.a.e | 1 | ||
104.p | odd | 6 | 1 | 2704.2.a.c | 1 | ||
104.r | even | 6 | 1 | 104.2.i.a | ✓ | 2 | |
104.r | even | 6 | 1 | 1352.2.a.c | 1 | ||
104.s | even | 6 | 1 | 1352.2.a.a | 1 | ||
104.s | even | 6 | 1 | 1352.2.i.a | 2 | ||
104.u | even | 12 | 2 | 2704.2.f.c | 2 | ||
104.x | odd | 12 | 2 | 1352.2.f.a | 2 | ||
104.x | odd | 12 | 2 | 1352.2.o.b | 4 | ||
312.bh | odd | 6 | 1 | 936.2.t.c | 2 | ||
312.bn | even | 6 | 1 | 1872.2.t.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
104.2.i.a | ✓ | 2 | 8.b | even | 2 | 1 | |
104.2.i.a | ✓ | 2 | 104.r | even | 6 | 1 | |
208.2.i.c | 2 | 8.d | odd | 2 | 1 | ||
208.2.i.c | 2 | 104.n | odd | 6 | 1 | ||
832.2.i.d | 2 | 4.b | odd | 2 | 1 | ||
832.2.i.d | 2 | 52.j | odd | 6 | 1 | ||
832.2.i.g | 2 | 1.a | even | 1 | 1 | trivial | |
832.2.i.g | 2 | 13.c | even | 3 | 1 | inner | |
936.2.t.c | 2 | 24.h | odd | 2 | 1 | ||
936.2.t.c | 2 | 312.bh | odd | 6 | 1 | ||
1352.2.a.a | 1 | 104.s | even | 6 | 1 | ||
1352.2.a.c | 1 | 104.r | even | 6 | 1 | ||
1352.2.f.a | 2 | 104.x | odd | 12 | 2 | ||
1352.2.i.a | 2 | 104.e | even | 2 | 1 | ||
1352.2.i.a | 2 | 104.s | even | 6 | 1 | ||
1352.2.o.b | 4 | 104.j | odd | 4 | 2 | ||
1352.2.o.b | 4 | 104.x | odd | 12 | 2 | ||
1872.2.t.d | 2 | 24.f | even | 2 | 1 | ||
1872.2.t.d | 2 | 312.bn | even | 6 | 1 | ||
2704.2.a.c | 1 | 104.p | odd | 6 | 1 | ||
2704.2.a.e | 1 | 104.n | odd | 6 | 1 | ||
2704.2.f.c | 2 | 104.u | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on :
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