Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6272,2,Mod(1,6272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6272, 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("6272.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6272 = 2^{7} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6272.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: | \(50.0821721477\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 896) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \)
|
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 | −1.95594 | 0 | 0.703158 | 0 | 0 | 0 | 0.825711 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.816594 | 0 | −0.538445 | 0 | 0 | 0 | −2.33317 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.14243 | 0 | 3.87834 | 0 | 0 | 0 | 1.59002 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.63010 | 0 | −2.04306 | 0 | 0 | 0 | 3.91744 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
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 | 6272.2.a.bt | 4 | |
| 4.b | odd | 2 | 1 | 6272.2.a.bd | 4 | ||
| 7.b | odd | 2 | 1 | 6272.2.a.ba | 4 | ||
| 7.c | even | 3 | 2 | 896.2.i.b | ✓ | 8 | |
| 8.b | even | 2 | 1 | 6272.2.a.z | 4 | ||
| 8.d | odd | 2 | 1 | 6272.2.a.bp | 4 | ||
| 28.d | even | 2 | 1 | 6272.2.a.bq | 4 | ||
| 28.g | odd | 6 | 2 | 896.2.i.e | yes | 8 | |
| 56.e | even | 2 | 1 | 6272.2.a.be | 4 | ||
| 56.h | odd | 2 | 1 | 6272.2.a.bu | 4 | ||
| 56.k | odd | 6 | 2 | 896.2.i.d | yes | 8 | |
| 56.p | even | 6 | 2 | 896.2.i.g | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 896.2.i.b | ✓ | 8 | 7.c | even | 3 | 2 | |
| 896.2.i.d | yes | 8 | 56.k | odd | 6 | 2 | |
| 896.2.i.e | yes | 8 | 28.g | odd | 6 | 2 | |
| 896.2.i.g | yes | 8 | 56.p | even | 6 | 2 | |
| 6272.2.a.z | 4 | 8.b | even | 2 | 1 | ||
| 6272.2.a.ba | 4 | 7.b | odd | 2 | 1 | ||
| 6272.2.a.bd | 4 | 4.b | odd | 2 | 1 | ||
| 6272.2.a.be | 4 | 56.e | even | 2 | 1 | ||
| 6272.2.a.bp | 4 | 8.d | odd | 2 | 1 | ||
| 6272.2.a.bq | 4 | 28.d | even | 2 | 1 | ||
| 6272.2.a.bt | 4 | 1.a | even | 1 | 1 | trivial | |
| 6272.2.a.bu | 4 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6272))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 8T_{3} + 9 \)
|
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\( T_{5}^{4} - 2T_{5}^{3} - 8T_{5}^{2} + 2T_{5} + 3 \)
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\( T_{11}^{4} - 38T_{11}^{2} + 38T_{11} + 197 \)
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\( T_{13}^{4} - 8T_{13}^{3} + 8T_{13}^{2} + 60T_{13} - 116 \)
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\( T_{23}^{4} - 4T_{23}^{3} - 40T_{23}^{2} + 150T_{23} - 125 \)
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\( T_{29}^{4} + 8T_{29}^{3} - 20T_{29}^{2} - 68T_{29} + 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 9 \)
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| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 3 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} - 38 T^{2} + \cdots + 197 \)
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| $13$ |
\( T^{4} - 8 T^{3} + \cdots - 116 \)
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| $17$ |
\( T^{4} + 4 T^{3} + \cdots + 97 \)
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| $19$ |
\( T^{4} - 4 T^{3} + \cdots - 151 \)
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| $23$ |
\( T^{4} - 4 T^{3} + \cdots - 125 \)
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| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 12 \)
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| $31$ |
\( T^{4} - 10 T^{3} + \cdots + 11 \)
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| $37$ |
\( T^{4} + 2 T^{3} + \cdots + 251 \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots - 44 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots - 64 \)
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| $47$ |
\( T^{4} - 14 T^{3} + \cdots - 201 \)
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| $53$ |
\( T^{4} - 10 T^{3} + \cdots - 605 \)
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| $59$ |
\( T^{4} - 6 T^{3} + \cdots + 97 \)
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| $61$ |
\( T^{4} - 6 T^{3} + \cdots - 3557 \)
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| $67$ |
\( T^{4} - 22 T^{3} + \cdots - 11 \)
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| $71$ |
\( (T + 6)^{4} \)
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| $73$ |
\( T^{4} + 16 T^{3} + \cdots - 5563 \)
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| $79$ |
\( T^{4} + 8 T^{3} + \cdots + 3167 \)
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| $83$ |
\( T^{4} - 16 T^{3} + \cdots - 14640 \)
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| $89$ |
\( T^{4} + 8 T^{3} + \cdots - 1227 \)
|
| $97$ |
\( T^{4} + 16 T^{3} + \cdots - 6828 \)
|
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