Newspace parameters
| Level: | \( N \) | \(=\) | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8670.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(69.2302985525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4352.1 |
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|
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| Defining polynomial: |
\( x^{4} - 6x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 510) |
| 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
\( x^{4} - 6x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 2\nu + 3 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} + 4\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 2\beta_{2} + 4\beta _1 + 3 \)
|
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 | −1.00000 | −1.00000 | −3.79793 | 1.00000 | 1.00000 | −1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −1.00000 | −1.00000 | −2.47363 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −1.00000 | −1.00000 | 0.473626 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | −1.00000 | −1.00000 | 1.79793 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 8670.2.a.bx | 4 | |
| 17.b | even | 2 | 1 | 8670.2.a.ca | 4 | ||
| 17.d | even | 8 | 2 | 510.2.p.c | ✓ | 8 | |
| 51.g | odd | 8 | 2 | 1530.2.q.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.p.c | ✓ | 8 | 17.d | even | 8 | 2 | |
| 1530.2.q.j | 8 | 51.g | odd | 8 | 2 | ||
| 8670.2.a.bx | 4 | 1.a | even | 1 | 1 | trivial | |
| 8670.2.a.ca | 4 | 17.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8670))\):
|
\( T_{7}^{4} + 4T_{7}^{3} - 4T_{7}^{2} - 16T_{7} + 8 \)
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\( T_{11}^{4} - 32T_{11}^{2} - 64T_{11} + 16 \)
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\( T_{13}^{4} - 4T_{13}^{3} - 8T_{13}^{2} + 24T_{13} + 4 \)
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\( T_{23}^{2} + 4T_{23} + 2 \)
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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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| $5$ |
\( (T + 1)^{4} \)
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| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 8 \)
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| $11$ |
\( T^{4} - 32 T^{2} + \cdots + 16 \)
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| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 4 \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} - 8 T^{3} + \cdots - 448 \)
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| $23$ |
\( (T^{2} + 4 T + 2)^{2} \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots + 32 \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots - 28 \)
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| $37$ |
\( T^{4} - 56 T^{2} + \cdots + 16 \)
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| $41$ |
\( T^{4} + 12 T^{3} + \cdots - 248 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots - 28 \)
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| $47$ |
\( T^{4} - 16 T^{3} + \cdots + 64 \)
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| $53$ |
\( T^{4} - 8 T^{3} + \cdots + 3728 \)
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| $59$ |
\( T^{4} - 12 T^{3} + \cdots - 604 \)
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| $61$ |
\( T^{4} - 64 T^{2} + \cdots - 112 \)
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| $67$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
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| $71$ |
\( T^{4} - 20 T^{3} + \cdots - 184 \)
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| $73$ |
\( T^{4} + 12 T^{3} + \cdots + 8 \)
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| $79$ |
\( T^{4} - 380 T^{2} + 35972 \)
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| $83$ |
\( T^{4} - 224 T^{2} + \cdots + 256 \)
|
| $89$ |
\( T^{4} - 32 T^{3} + \cdots - 496 \)
|
| $97$ |
\( T^{4} + 4 T^{3} + \cdots + 8 \)
|
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