Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(183.682394985\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{5} - x^{4} - 8082x^{3} + 221404x^{2} + 4885736x - 136273920 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 84) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 8082x^{3} + 221404x^{2} + 4885736x - 136273920 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 473\nu^{4} + 1211\nu^{3} - 3853670\nu^{2} + 66569596\nu + 2170116840 ) / 4161864 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4351\nu^{4} + 187117\nu^{3} - 25646938\nu^{2} - 253593436\nu + 10984196016 ) / 4161864 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 929\nu^{4} + 33803\nu^{3} - 6710318\nu^{2} - 52425668\nu + 5548672488 ) / 594552 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2603\nu^{4} + 86954\nu^{3} - 18139901\nu^{2} + 4965628\nu + 12383024466 ) / 1040466 \)
|
| \(\nu\) | \(=\) |
\( ( 10\beta_{4} - 9\beta_{3} - 7\beta_{2} - 32\beta _1 + 139 ) / 756 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -160\beta_{4} - 108\beta_{3} + 427\beta_{2} + 1079\beta _1 + 1222559 ) / 378 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 532\beta_{4} - 256\beta_{3} - 509\beta_{2} - 3509\beta _1 - 769357 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2093072\beta_{4} - 205290\beta_{3} + 4053581\beta_{2} + 14934715\beta _1 + 8340601993 ) / 378 \)
|
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 |
|
0 | −27.0000 | 0 | −388.554 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | −249.769 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −27.0000 | 0 | 61.0371 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −27.0000 | 0 | 294.614 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −27.0000 | 0 | 480.671 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 588.8.a.k | 5 | |
| 7.b | odd | 2 | 1 | 588.8.a.l | 5 | ||
| 7.c | even | 3 | 2 | 588.8.i.p | 10 | ||
| 7.d | odd | 6 | 2 | 84.8.i.b | ✓ | 10 | |
| 21.g | even | 6 | 2 | 252.8.k.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.8.i.b | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 252.8.k.d | 10 | 21.g | even | 6 | 2 | ||
| 588.8.a.k | 5 | 1.a | even | 1 | 1 | trivial | |
| 588.8.a.l | 5 | 7.b | odd | 2 | 1 | ||
| 588.8.i.p | 10 | 7.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 198T_{5}^{4} - 247861T_{5}^{3} + 30793110T_{5}^{2} + 12818319300T_{5} - 838849851000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 838849851000 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} + \cdots + 16\!\cdots\!32 \)
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| $13$ |
\( T^{5} + \cdots - 13\!\cdots\!00 \)
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| $17$ |
\( T^{5} + \cdots - 47\!\cdots\!92 \)
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| $19$ |
\( T^{5} + \cdots - 12\!\cdots\!84 \)
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| $23$ |
\( T^{5} + \cdots - 34\!\cdots\!56 \)
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| $29$ |
\( T^{5} + \cdots + 31\!\cdots\!32 \)
|
| $31$ |
\( T^{5} + \cdots - 94\!\cdots\!87 \)
|
| $37$ |
\( T^{5} + \cdots - 28\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots - 14\!\cdots\!84 \)
|
| $43$ |
\( T^{5} + \cdots + 33\!\cdots\!48 \)
|
| $47$ |
\( T^{5} + \cdots - 28\!\cdots\!88 \)
|
| $53$ |
\( T^{5} + \cdots + 78\!\cdots\!36 \)
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| $59$ |
\( T^{5} + \cdots + 11\!\cdots\!68 \)
|
| $61$ |
\( T^{5} + \cdots + 93\!\cdots\!76 \)
|
| $67$ |
\( T^{5} + \cdots - 44\!\cdots\!48 \)
|
| $71$ |
\( T^{5} + \cdots - 34\!\cdots\!40 \)
|
| $73$ |
\( T^{5} + \cdots + 12\!\cdots\!88 \)
|
| $79$ |
\( T^{5} + \cdots + 23\!\cdots\!81 \)
|
| $83$ |
\( T^{5} + \cdots + 68\!\cdots\!12 \)
|
| $89$ |
\( T^{5} + \cdots + 58\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots + 21\!\cdots\!40 \)
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