Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8112,2,Mod(1,8112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8112.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8112.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: | \(64.7746461197\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 312) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 1.00000 | 0 | −1.73205 | 0 | 0.732051 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | 1.73205 | 0 | −2.73205 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(13\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8112.2.a.br | 2 | |
| 4.b | odd | 2 | 1 | 4056.2.a.u | 2 | ||
| 13.b | even | 2 | 1 | 8112.2.a.bw | 2 | ||
| 13.f | odd | 12 | 2 | 624.2.bv.c | 4 | ||
| 39.k | even | 12 | 2 | 1872.2.by.g | 4 | ||
| 52.b | odd | 2 | 1 | 4056.2.a.t | 2 | ||
| 52.f | even | 4 | 2 | 4056.2.c.k | 4 | ||
| 52.l | even | 12 | 2 | 312.2.bf.a | ✓ | 4 | |
| 156.v | odd | 12 | 2 | 936.2.bi.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.2.bf.a | ✓ | 4 | 52.l | even | 12 | 2 | |
| 624.2.bv.c | 4 | 13.f | odd | 12 | 2 | ||
| 936.2.bi.a | 4 | 156.v | odd | 12 | 2 | ||
| 1872.2.by.g | 4 | 39.k | even | 12 | 2 | ||
| 4056.2.a.t | 2 | 52.b | odd | 2 | 1 | ||
| 4056.2.a.u | 2 | 4.b | odd | 2 | 1 | ||
| 4056.2.c.k | 4 | 52.f | even | 4 | 2 | ||
| 8112.2.a.br | 2 | 1.a | even | 1 | 1 | trivial | |
| 8112.2.a.bw | 2 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):
|
\( T_{5}^{2} - 3 \)
|
|
\( T_{7}^{2} + 2T_{7} - 2 \)
|
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\( T_{11}^{2} + 2T_{11} - 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 1)^{2} \)
|
| $5$ |
\( T^{2} - 3 \)
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| $7$ |
\( T^{2} + 2T - 2 \)
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| $11$ |
\( T^{2} + 2T - 2 \)
|
| $13$ |
\( T^{2} \)
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| $17$ |
\( T^{2} + 4T + 1 \)
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| $19$ |
\( T^{2} - 6T + 6 \)
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| $23$ |
\( T^{2} + 6T - 18 \)
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| $29$ |
\( T^{2} - 2T - 47 \)
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| $31$ |
\( T^{2} + 4T - 8 \)
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| $37$ |
\( T^{2} - 10T + 13 \)
|
| $41$ |
\( (T - 5)^{2} \)
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| $43$ |
\( T^{2} + 14T + 22 \)
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| $47$ |
\( T^{2} - 10T + 22 \)
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| $53$ |
\( T^{2} + 2T - 47 \)
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| $59$ |
\( T^{2} \)
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| $61$ |
\( T^{2} + 16T + 61 \)
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| $67$ |
\( T^{2} + 14T + 46 \)
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| $71$ |
\( T^{2} - 6T - 18 \)
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| $73$ |
\( T^{2} - 8T - 11 \)
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| $79$ |
\( T^{2} + 4T - 104 \)
|
| $83$ |
\( T^{2} + 14T - 26 \)
|
| $89$ |
\( T^{2} + 12T + 24 \)
|
| $97$ |
\( (T + 10)^{2} \)
|
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