Newspace parameters
| Level: | \( N \) | \(=\) | \( 182 = 2 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 182.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.8540746381\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 10976x^{4} - 73212x^{3} + 26099211x^{2} + 295001154x - 8025269616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 2x^{5} - 10976x^{4} - 73212x^{3} + 26099211x^{2} + 295001154x - 8025269616 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2497 \nu^{5} - 675473 \nu^{4} - 8981633 \nu^{3} + 6088357107 \nu^{2} - 31121733138 \nu - 7842514042800 ) / 14589399456 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3017\nu^{5} + 157655\nu^{4} - 21807253\nu^{3} - 1181735865\nu^{2} - 12202844250\nu + 3424867452 ) / 1823674932 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 39499 \nu^{5} - 622475 \nu^{4} - 457910939 \nu^{3} + 5854317057 \nu^{2} + 1072268733738 \nu - 8709072219360 ) / 7294699728 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 579\nu^{5} - 19787\nu^{4} - 5608675\nu^{3} + 118605345\nu^{2} + 10048267194\nu - 54302973264 ) / 71516664 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 13\beta _1 + 3658 \)
|
| \(\nu^{3}\) | \(=\) |
\( 11\beta_{5} - 47\beta_{4} + 82\beta_{3} + 174\beta_{2} + 6283\beta _1 + 45619 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8644\beta_{5} + 8239\beta_{4} + 11563\beta_{3} + 36462\beta_{2} + 158581\beta _1 + 22851418 \)
|
| \(\nu^{5}\) | \(=\) |
\( 139514\beta_{5} - 378563\beta_{4} + 984634\beta_{3} + 1702506\beta_{2} + 46264267\beta _1 + 567301855 \)
|
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 |
|
8.00000 | −84.1918 | 64.0000 | 28.4152 | −673.534 | 343.000 | 512.000 | 4901.25 | 227.322 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −46.6325 | 64.0000 | 142.436 | −373.060 | 343.000 | 512.000 | −12.4124 | 1139.49 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | −4.29534 | 64.0000 | −487.261 | −34.3627 | 343.000 | 512.000 | −2168.55 | −3898.09 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 37.9408 | 64.0000 | −97.3740 | 303.527 | 343.000 | 512.000 | −747.494 | −778.992 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 8.00000 | 57.8853 | 64.0000 | 393.626 | 463.082 | 343.000 | 512.000 | 1163.70 | 3149.00 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.00000 | 91.2935 | 64.0000 | 141.158 | 730.348 | 343.000 | 512.000 | 6147.50 | 1129.27 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 182.8.a.h | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.8.a.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 52T_{3}^{5} - 9851T_{3}^{4} + 455388T_{3}^{3} + 18871986T_{3}^{2} - 715279104T_{3} - 3381194880 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(182))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 3381194880 \)
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| $5$ |
\( T^{6} + \cdots + 10670006880000 \)
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| $7$ |
\( (T - 343)^{6} \)
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| $11$ |
\( T^{6} + \cdots - 55\!\cdots\!60 \)
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| $13$ |
\( (T + 2197)^{6} \)
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| $17$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
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| $19$ |
\( T^{6} + \cdots + 67\!\cdots\!96 \)
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| $23$ |
\( T^{6} + \cdots + 39\!\cdots\!32 \)
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| $29$ |
\( T^{6} + \cdots - 53\!\cdots\!24 \)
|
| $31$ |
\( T^{6} + \cdots - 90\!\cdots\!60 \)
|
| $37$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 14\!\cdots\!68 \)
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| $43$ |
\( T^{6} + \cdots - 26\!\cdots\!60 \)
|
| $47$ |
\( T^{6} + \cdots - 17\!\cdots\!12 \)
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| $53$ |
\( T^{6} + \cdots - 59\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 84\!\cdots\!40 \)
|
| $61$ |
\( T^{6} + \cdots - 13\!\cdots\!92 \)
|
| $67$ |
\( T^{6} + \cdots - 77\!\cdots\!68 \)
|
| $71$ |
\( T^{6} + \cdots - 93\!\cdots\!24 \)
|
| $73$ |
\( T^{6} + \cdots - 99\!\cdots\!74 \)
|
| $79$ |
\( T^{6} + \cdots - 40\!\cdots\!84 \)
|
| $83$ |
\( T^{6} + \cdots - 41\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots - 23\!\cdots\!72 \)
|
| $97$ |
\( T^{6} + \cdots - 33\!\cdots\!58 \)
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