Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,2,Mod(1,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, 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("1216.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 1216.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: | 3.3.961.1 |
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 152) |
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
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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 | |||||||||||||||||||||||||||||||||||||
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1.1 |
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0 | −3.29707 | 0 | 3.08387 | 0 | 1.78680 | 0 | 7.87067 | 0 | |||||||||||||||||||||||||||
1.2 | 0 | −0.786802 | 0 | −3.29707 | 0 | −2.08387 | 0 | −2.38094 | 0 | ||||||||||||||||||||||||||||
1.3 | 0 | 3.08387 | 0 | −0.786802 | 0 | 4.29707 | 0 | 6.51027 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
Sign | |
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Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
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Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1216.2.a.u | 3 | |
4.b | odd | 2 | 1 | 1216.2.a.v | 3 | ||
8.b | even | 2 | 1 | 152.2.a.c | ✓ | 3 | |
8.d | odd | 2 | 1 | 304.2.a.g | 3 | ||
24.f | even | 2 | 1 | 2736.2.a.bd | 3 | ||
24.h | odd | 2 | 1 | 1368.2.a.n | 3 | ||
40.e | odd | 2 | 1 | 7600.2.a.bv | 3 | ||
40.f | even | 2 | 1 | 3800.2.a.r | 3 | ||
40.i | odd | 4 | 2 | 3800.2.d.j | 6 | ||
56.h | odd | 2 | 1 | 7448.2.a.bf | 3 | ||
152.b | even | 2 | 1 | 5776.2.a.bp | 3 | ||
152.g | odd | 2 | 1 | 2888.2.a.o | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
152.2.a.c | ✓ | 3 | 8.b | even | 2 | 1 | |
304.2.a.g | 3 | 8.d | odd | 2 | 1 | ||
1216.2.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
1216.2.a.v | 3 | 4.b | odd | 2 | 1 | ||
1368.2.a.n | 3 | 24.h | odd | 2 | 1 | ||
2736.2.a.bd | 3 | 24.f | even | 2 | 1 | ||
2888.2.a.o | 3 | 152.g | odd | 2 | 1 | ||
3800.2.a.r | 3 | 40.f | even | 2 | 1 | ||
3800.2.d.j | 6 | 40.i | odd | 4 | 2 | ||
5776.2.a.bp | 3 | 152.b | even | 2 | 1 | ||
7448.2.a.bf | 3 | 56.h | odd | 2 | 1 | ||
7600.2.a.bv | 3 | 40.e | odd | 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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