Newspace parameters
| Level: | \( N \) | \(=\) | \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5082.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(40.5799743072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.28400.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 17x^{2} + 71 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 462) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{4} - 17x^{2} + 71 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 9 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 9\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 9\beta_1 \)
|
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −1.00000 | 1.00000 | −2.10130 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −2.10130 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −1.71698 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −1.71698 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | 3.71698 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 3.71698 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 4.10130 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 4.10130 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 5082.2.a.cd | 4 | |
| 11.b | odd | 2 | 1 | 5082.2.a.by | 4 | ||
| 11.d | odd | 10 | 2 | 462.2.j.h | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.2.j.h | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 5082.2.a.by | 4 | 11.b | odd | 2 | 1 | ||
| 5082.2.a.cd | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5082))\):
|
\( T_{5}^{4} - 4T_{5}^{3} - 11T_{5}^{2} + 30T_{5} + 55 \)
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\( T_{13}^{4} - 10T_{13}^{3} + 12T_{13}^{2} + 100T_{13} - 164 \)
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\( T_{17}^{2} - 3T_{17} - 9 \)
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\( T_{19}^{4} - 4T_{19}^{3} - 17T_{19}^{2} + 42T_{19} + 49 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 55 \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 10 T^{3} + \cdots - 164 \)
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| $17$ |
\( (T^{2} - 3 T - 9)^{2} \)
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| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 49 \)
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| $23$ |
\( T^{4} - 57 T^{2} + \cdots + 131 \)
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| $29$ |
\( T^{4} + 4 T^{3} + \cdots + 20 \)
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| $31$ |
\( T^{4} - 8 T^{3} + \cdots - 181 \)
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| $37$ |
\( T^{4} - 107 T^{2} + \cdots + 1331 \)
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| $41$ |
\( T^{4} + 18 T^{3} + \cdots - 5011 \)
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| $43$ |
\( T^{4} - 10 T^{3} + \cdots - 604 \)
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| $47$ |
\( T^{4} - 12 T^{3} + \cdots - 316 \)
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| $53$ |
\( T^{4} + 18 T^{3} + \cdots - 8020 \)
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| $59$ |
\( T^{4} - 12 T^{3} + \cdots - 2636 \)
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| $61$ |
\( T^{4} - 6 T^{3} + \cdots + 596 \)
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| $67$ |
\( T^{4} + 6 T^{3} + \cdots + 1516 \)
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| $71$ |
\( T^{4} + 28 T^{3} + \cdots - 880 \)
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| $73$ |
\( T^{4} - 12 T^{3} + \cdots - 2156 \)
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| $79$ |
\( T^{4} - 16 T^{3} + \cdots - 19580 \)
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| $83$ |
\( T^{4} - 14 T^{3} + \cdots + 836 \)
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| $89$ |
\( T^{4} - 22 T^{3} + \cdots - 4895 \)
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| $97$ |
\( T^{4} - 6 T^{3} + \cdots + 20 \)
|
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