Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,2,Mod(43,110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("110.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 110.f (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
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Defining polynomial: |
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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 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 | ||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 |
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−0.707107 | − | 0.707107i | 1.70711 | + | 1.70711i | 1.00000i | −0.707107 | − | 2.12132i | − | 2.41421i | 2.12132 | + | 2.12132i | 0.707107 | − | 0.707107i | 2.82843i | −1.00000 | + | 2.00000i | |||||||||||||||||
43.2 | 0.707107 | + | 0.707107i | 0.292893 | + | 0.292893i | 1.00000i | 0.707107 | + | 2.12132i | 0.414214i | −2.12132 | − | 2.12132i | −0.707107 | + | 0.707107i | − | 2.82843i | −1.00000 | + | 2.00000i | ||||||||||||||||||
87.1 | −0.707107 | + | 0.707107i | 1.70711 | − | 1.70711i | − | 1.00000i | −0.707107 | + | 2.12132i | 2.41421i | 2.12132 | − | 2.12132i | 0.707107 | + | 0.707107i | − | 2.82843i | −1.00000 | − | 2.00000i | |||||||||||||||||
87.2 | 0.707107 | − | 0.707107i | 0.292893 | − | 0.292893i | − | 1.00000i | 0.707107 | − | 2.12132i | − | 0.414214i | −2.12132 | + | 2.12132i | −0.707107 | − | 0.707107i | 2.82843i | −1.00000 | − | 2.00000i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
55.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 110.2.f.b | ✓ | 4 |
3.b | odd | 2 | 1 | 990.2.m.c | 4 | ||
4.b | odd | 2 | 1 | 880.2.bd.c | 4 | ||
5.b | even | 2 | 1 | 550.2.f.a | 4 | ||
5.c | odd | 4 | 1 | 110.2.f.c | yes | 4 | |
5.c | odd | 4 | 1 | 550.2.f.b | 4 | ||
11.b | odd | 2 | 1 | 110.2.f.c | yes | 4 | |
15.e | even | 4 | 1 | 990.2.m.d | 4 | ||
20.e | even | 4 | 1 | 880.2.bd.b | 4 | ||
33.d | even | 2 | 1 | 990.2.m.d | 4 | ||
44.c | even | 2 | 1 | 880.2.bd.b | 4 | ||
55.d | odd | 2 | 1 | 550.2.f.b | 4 | ||
55.e | even | 4 | 1 | inner | 110.2.f.b | ✓ | 4 |
55.e | even | 4 | 1 | 550.2.f.a | 4 | ||
165.l | odd | 4 | 1 | 990.2.m.c | 4 | ||
220.i | odd | 4 | 1 | 880.2.bd.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
110.2.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
110.2.f.b | ✓ | 4 | 55.e | even | 4 | 1 | inner |
110.2.f.c | yes | 4 | 5.c | odd | 4 | 1 | |
110.2.f.c | yes | 4 | 11.b | odd | 2 | 1 | |
550.2.f.a | 4 | 5.b | even | 2 | 1 | ||
550.2.f.a | 4 | 55.e | even | 4 | 1 | ||
550.2.f.b | 4 | 5.c | odd | 4 | 1 | ||
550.2.f.b | 4 | 55.d | odd | 2 | 1 | ||
880.2.bd.b | 4 | 20.e | even | 4 | 1 | ||
880.2.bd.b | 4 | 44.c | even | 2 | 1 | ||
880.2.bd.c | 4 | 4.b | odd | 2 | 1 | ||
880.2.bd.c | 4 | 220.i | odd | 4 | 1 | ||
990.2.m.c | 4 | 3.b | odd | 2 | 1 | ||
990.2.m.c | 4 | 165.l | odd | 4 | 1 | ||
990.2.m.d | 4 | 15.e | even | 4 | 1 | ||
990.2.m.d | 4 | 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 :
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