Newspace parameters
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(87.4678071356\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 8510x^{4} - 55396x^{3} + 15342657x^{2} + 241384348x + 544656580 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{16} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 8510x^{4} - 55396x^{3} + 15342657x^{2} + 241384348x + 544656580 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 86\nu^{5} - 2045\nu^{4} - 726263\nu^{3} + 11516085\nu^{2} + 1288709949\nu - 586701385 ) / 3700683 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 86\nu^{5} - 2045\nu^{4} - 726263\nu^{3} + 7815402\nu^{2} + 1325716779\nu + 9908435603 ) / 3700683 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5017\nu^{5} - 9794\nu^{4} + 43099687\nu^{3} + 161052570\nu^{2} - 78147732822\nu - 552636868228 ) / 170231418 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 474\nu^{5} - 5807\nu^{4} - 3583508\nu^{3} + 21913801\nu^{2} + 5076307460\nu + 27011721022 ) / 8106258 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 10\beta _1 + 2836 \)
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| \(\nu^{3}\) | \(=\) |
\( 18\beta_{5} + 16\beta_{4} - 14\beta_{3} - 11\beta_{2} + 4919\beta _1 + 27703 \)
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| \(\nu^{4}\) | \(=\) |
\( 102\beta_{5} - 1228\beta_{4} - 6531\beta_{3} + 4717\beta_{2} + 69400\beta _1 + 13907880 \)
|
| \(\nu^{5}\) | \(=\) |
\( 154434\beta_{5} + 105918\beta_{4} - 139622\beta_{3} - 71605\beta_{2} + 26866743\beta _1 + 191725149 \)
|
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 | −76.7045 | 0 | −125.000 | 0 | 343.000 | 0 | 3696.58 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −45.3704 | 0 | −125.000 | 0 | 343.000 | 0 | −128.525 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −15.9612 | 0 | −125.000 | 0 | 343.000 | 0 | −1932.24 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −4.73433 | 0 | −125.000 | 0 | 343.000 | 0 | −2164.59 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 53.6737 | 0 | −125.000 | 0 | 343.000 | 0 | 693.863 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 77.0968 | 0 | −125.000 | 0 | 343.000 | 0 | 3756.91 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 280.8.a.g | ✓ | 6 |
| 4.b | odd | 2 | 1 | 560.8.a.u | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.8.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 560.8.a.u | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 12T_{3}^{5} - 8450T_{3}^{4} - 123316T_{3}^{3} + 14806281T_{3}^{2} + 301818096T_{3} + 1088216640 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 1088216640 \)
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| $5$ |
\( (T + 125)^{6} \)
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| $7$ |
\( (T - 343)^{6} \)
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| $11$ |
\( T^{6} + \cdots - 59\!\cdots\!20 \)
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| $13$ |
\( T^{6} + \cdots + 11\!\cdots\!40 \)
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| $17$ |
\( T^{6} + \cdots + 46\!\cdots\!84 \)
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| $19$ |
\( T^{6} + \cdots - 63\!\cdots\!84 \)
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| $23$ |
\( T^{6} + \cdots - 11\!\cdots\!80 \)
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| $29$ |
\( T^{6} + \cdots + 22\!\cdots\!44 \)
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| $31$ |
\( T^{6} + \cdots + 54\!\cdots\!08 \)
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| $37$ |
\( T^{6} + \cdots + 69\!\cdots\!96 \)
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| $41$ |
\( T^{6} + \cdots - 45\!\cdots\!80 \)
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| $43$ |
\( T^{6} + \cdots + 12\!\cdots\!80 \)
|
| $47$ |
\( T^{6} + \cdots - 74\!\cdots\!84 \)
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| $53$ |
\( T^{6} + \cdots + 11\!\cdots\!60 \)
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| $59$ |
\( T^{6} + \cdots + 42\!\cdots\!40 \)
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| $61$ |
\( T^{6} + \cdots - 10\!\cdots\!16 \)
|
| $67$ |
\( T^{6} + \cdots + 38\!\cdots\!60 \)
|
| $71$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 78\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 98\!\cdots\!92 \)
|
| $83$ |
\( T^{6} + \cdots - 13\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots - 40\!\cdots\!20 \)
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| $97$ |
\( T^{6} + \cdots + 28\!\cdots\!56 \)
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