Newspace parameters
| Level: | \( N \) | \(=\) | \( 158 = 2 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 158.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(25.3406435305\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 517x^{4} + 294x^{3} + 75132x^{2} + 2488x - 3309088 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 3x^{5} - 517x^{4} + 294x^{3} + 75132x^{2} + 2488x - 3309088 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -53\nu^{5} + 1825\nu^{4} + 14751\nu^{3} - 583604\nu^{2} - 1326708\nu + 36217712 ) / 316000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 119\nu^{5} + 225\nu^{4} - 48473\nu^{3} - 299008\nu^{2} + 3650484\nu + 28516624 ) / 316000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -119\nu^{5} - 225\nu^{4} + 48473\nu^{3} + 299008\nu^{2} - 3018484\nu - 28832624 ) / 316000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\nu^{5} - 95\nu^{4} - 11569\nu^{3} + 11596\nu^{2} + 950492\nu - 982768 ) / 31600 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -781\nu^{5} + 13925\nu^{4} + 263427\nu^{3} - 4403808\nu^{2} - 19669116\nu + 312579824 ) / 948000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} + 3\beta_{3} + 8\beta _1 + 346 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -69\beta_{5} - 58\beta_{4} + 202\beta_{3} + 245\beta_{2} + 140\beta _1 + 1223 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -1362\beta_{5} - 206\beta_{4} + 1679\beta_{3} + 861\beta_{2} + 3804\beta _1 + 82189 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -33069\beta_{5} - 23236\beta_{4} + 55969\beta_{3} + 72804\beta_{2} + 69936\beta _1 + 702210 ) / 2 \)
|
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 |
|
4.00000 | −30.4434 | 16.0000 | −54.1150 | −121.774 | 158.298 | 64.0000 | 683.800 | −216.460 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −22.7897 | 16.0000 | 26.1459 | −91.1588 | −22.0965 | 64.0000 | 276.371 | 104.584 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −4.18381 | 16.0000 | −19.2215 | −16.7352 | −11.5532 | 64.0000 | −225.496 | −76.8861 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | −2.97218 | 16.0000 | −20.6918 | −11.8887 | 119.379 | 64.0000 | −234.166 | −82.7671 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 2.09312 | 16.0000 | 55.7935 | 8.37248 | −208.498 | 64.0000 | −238.619 | 223.174 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | 22.2960 | 16.0000 | −81.9111 | 89.1839 | −229.529 | 64.0000 | 254.110 | −327.644 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(79\) | \( -1 \) |
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 | 158.6.a.b | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 158.6.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 36T_{3}^{5} - 339T_{3}^{4} - 18070T_{3}^{3} - 77868T_{3}^{2} + 52176T_{3} + 402624 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(158))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{6} \)
|
| $3$ |
\( T^{6} + 36 T^{5} + \cdots + 402624 \)
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| $5$ |
\( T^{6} + \cdots + 2571774399 \)
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| $7$ |
\( T^{6} + \cdots + 230869573776 \)
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| $11$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
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| $13$ |
\( T^{6} + \cdots - 39\!\cdots\!45 \)
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| $17$ |
\( T^{6} + \cdots - 962713750256640 \)
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| $19$ |
\( T^{6} + \cdots + 93\!\cdots\!68 \)
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| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!84 \)
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| $29$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 24\!\cdots\!08 \)
|
| $37$ |
\( T^{6} + \cdots + 31\!\cdots\!12 \)
|
| $41$ |
\( T^{6} + \cdots + 51\!\cdots\!64 \)
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| $43$ |
\( T^{6} + \cdots - 66\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots - 16\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots - 16\!\cdots\!72 \)
|
| $59$ |
\( T^{6} + \cdots + 80\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots + 63\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots + 33\!\cdots\!92 \)
|
| $71$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 57\!\cdots\!04 \)
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| $79$ |
\( (T - 6241)^{6} \)
|
| $83$ |
\( T^{6} + \cdots - 44\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots - 48\!\cdots\!75 \)
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| $97$ |
\( T^{6} + \cdots + 86\!\cdots\!97 \)
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