Newspace parameters
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.8059464526\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 735x^{5} + 872x^{4} + 157439x^{3} - 32262x^{2} - 9241409x + 5869696 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 735x^{5} + 872x^{4} + 157439x^{3} - 32262x^{2} - 9241409x + 5869696 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 210 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 503 \nu^{6} + 6221 \nu^{5} + 266180 \nu^{4} - 3483212 \nu^{3} - 26131789 \nu^{2} + \cdots - 365317888 ) / 1308704 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 413 \nu^{6} - 3807 \nu^{5} - 274492 \nu^{4} + 2273268 \nu^{3} + 45282079 \nu^{2} + \cdots - 1086396576 ) / 1308704 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 421 \nu^{6} - 3249 \nu^{5} + 259212 \nu^{4} + 1879740 \nu^{3} - 40308071 \nu^{2} + \cdots + 1097357536 ) / 1308704 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 227 \nu^{6} + 4271 \nu^{5} + 138988 \nu^{4} - 2150196 \nu^{3} - 19096969 \nu^{2} + \cdots + 121909512 ) / 327176 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 210 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 308\beta _1 + 170 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} - \beta_{5} - 20\beta_{4} - 21\beta_{3} + 417\beta_{2} + 476\beta _1 + 64826 \)
|
| \(\nu^{5}\) | \(=\) |
\( 580\beta_{6} + 408\beta_{5} + 1048\beta_{4} - 528\beta_{3} - 384\beta_{2} + 106885\beta _1 + 83724 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1836\beta_{6} - 2408\beta_{5} - 11472\beta_{4} - 13320\beta_{3} + 163969\beta_{2} + 145477\beta _1 + 22527022 \)
|
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 |
|
−19.0345 | 27.0000 | 234.313 | −271.311 | −513.932 | 1572.43 | −2023.61 | 729.000 | 5164.28 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −11.9775 | 27.0000 | 15.4595 | 501.304 | −323.391 | 220.275 | 1347.95 | 729.000 | −6004.35 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −10.6045 | 27.0000 | −15.5449 | −508.968 | −286.321 | −1484.51 | 1522.22 | 729.000 | 5397.34 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.63816 | 27.0000 | −125.316 | −104.560 | 44.2304 | −219.622 | −414.974 | 729.000 | −171.287 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 9.90889 | 27.0000 | −29.8138 | 247.633 | 267.540 | 1451.54 | −1563.76 | 729.000 | 2453.76 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 18.3741 | 27.0000 | 209.607 | 371.149 | 496.100 | −291.441 | 1499.44 | 729.000 | 6819.51 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 20.6953 | 27.0000 | 300.296 | −262.246 | 558.774 | 640.329 | 3565.73 | 729.000 | −5427.26 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 57.8.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 171.8.a.i | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.8.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 171.8.a.i | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 9T_{2}^{6} - 702T_{2}^{5} + 4482T_{2}^{4} + 146676T_{2}^{3} - 492048T_{2}^{2} - 8711712T_{2} + 14923008 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(57))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 9 T^{6} + \cdots + 14923008 \)
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| $3$ |
\( (T - 27)^{7} \)
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| $5$ |
\( T^{7} + \cdots - 17\!\cdots\!00 \)
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| $7$ |
\( T^{7} + \cdots + 30\!\cdots\!24 \)
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| $11$ |
\( T^{7} + \cdots + 74\!\cdots\!72 \)
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| $13$ |
\( T^{7} + \cdots + 19\!\cdots\!80 \)
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| $17$ |
\( T^{7} + \cdots - 13\!\cdots\!00 \)
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| $19$ |
\( (T - 6859)^{7} \)
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| $23$ |
\( T^{7} + \cdots - 18\!\cdots\!16 \)
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| $29$ |
\( T^{7} + \cdots + 24\!\cdots\!00 \)
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| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!48 \)
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| $37$ |
\( T^{7} + \cdots - 17\!\cdots\!72 \)
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| $41$ |
\( T^{7} + \cdots + 27\!\cdots\!08 \)
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| $43$ |
\( T^{7} + \cdots + 11\!\cdots\!52 \)
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| $47$ |
\( T^{7} + \cdots + 14\!\cdots\!68 \)
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| $53$ |
\( T^{7} + \cdots - 66\!\cdots\!08 \)
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| $59$ |
\( T^{7} + \cdots - 70\!\cdots\!40 \)
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| $61$ |
\( T^{7} + \cdots - 24\!\cdots\!20 \)
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| $67$ |
\( T^{7} + \cdots + 24\!\cdots\!00 \)
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| $71$ |
\( T^{7} + \cdots - 18\!\cdots\!80 \)
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| $73$ |
\( T^{7} + \cdots + 40\!\cdots\!04 \)
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| $79$ |
\( T^{7} + \cdots - 17\!\cdots\!20 \)
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| $83$ |
\( T^{7} + \cdots - 10\!\cdots\!92 \)
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| $89$ |
\( T^{7} + \cdots - 27\!\cdots\!00 \)
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| $97$ |
\( T^{7} + \cdots - 57\!\cdots\!56 \)
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