Newspace parameters
| Level: | \( N \) | \(=\) | \( 86 = 2 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 86.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.7930085039\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{5} - x^{4} - 390x^{3} - 35x^{2} + 33455x + 98550 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 390x^{3} - 35x^{2} + 33455x + 98550 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 370\nu^{2} - 1075\nu + 22410 ) / 810 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 370\nu^{2} + 1355\nu + 22005 ) / 405 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 208\nu^{2} - 913\nu - 2700 ) / 162 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8\nu^{4} + 103\nu^{3} + 2960\nu^{2} - 26365\nu - 208035 ) / 1215 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{3} - \beta_{2} - 28\beta _1 + 929 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 54\beta_{4} + 259\beta_{2} - 230\beta _1 + 1537 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -216\beta_{4} + 2220\beta_{3} - 331\beta_{2} - 6730\beta _1 + 204197 ) / 6 \)
|
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 | −20.3695 | 16.0000 | 75.3252 | 81.4779 | −228.918 | −64.0000 | 171.915 | −301.301 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −11.4357 | 16.0000 | −86.5921 | 45.7427 | −46.0932 | −64.0000 | −112.225 | 346.369 | ||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −5.81559 | 16.0000 | 65.6537 | 23.2623 | 158.348 | −64.0000 | −209.179 | −262.615 | ||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 9.68048 | 16.0000 | 68.8454 | −38.7219 | 172.009 | −64.0000 | −149.288 | −275.382 | ||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | 26.9403 | 16.0000 | −62.2321 | −107.761 | 66.6552 | −64.0000 | 482.777 | 248.928 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(43\) | \( -1 \) |
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 | 86.6.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 774.6.a.n | 5 | ||
| 4.b | odd | 2 | 1 | 688.6.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 86.6.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 688.6.a.c | 5 | 4.b | odd | 2 | 1 | ||
| 774.6.a.n | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + T_{3}^{4} - 699T_{3}^{3} - 4138T_{3}^{2} + 59378T_{3} + 353292 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(86))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{5} \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 353292 \)
|
| $5$ |
\( T^{5} + \cdots - 1834706376 \)
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| $7$ |
\( T^{5} + \cdots - 19156346880 \)
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| $11$ |
\( T^{5} + \cdots + 1299937408704 \)
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| $13$ |
\( T^{5} + \cdots + 99530462037780 \)
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| $17$ |
\( T^{5} + \cdots + 617471068117833 \)
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| $19$ |
\( T^{5} + \cdots - 442471226103464 \)
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| $23$ |
\( T^{5} + \cdots - 760748967394587 \)
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| $29$ |
\( T^{5} + \cdots - 11\!\cdots\!32 \)
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| $31$ |
\( T^{5} + \cdots + 54\!\cdots\!43 \)
|
| $37$ |
\( T^{5} + \cdots + 32\!\cdots\!40 \)
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| $41$ |
\( T^{5} + \cdots + 44\!\cdots\!25 \)
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| $43$ |
\( (T - 1849)^{5} \)
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| $47$ |
\( T^{5} + \cdots - 23\!\cdots\!96 \)
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| $53$ |
\( T^{5} + \cdots - 57\!\cdots\!00 \)
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| $59$ |
\( T^{5} + \cdots - 54\!\cdots\!72 \)
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| $61$ |
\( T^{5} + \cdots - 80\!\cdots\!60 \)
|
| $67$ |
\( T^{5} + \cdots + 41\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots + 10\!\cdots\!44 \)
|
| $73$ |
\( T^{5} + \cdots - 10\!\cdots\!92 \)
|
| $79$ |
\( T^{5} + \cdots + 14\!\cdots\!20 \)
|
| $83$ |
\( T^{5} + \cdots - 27\!\cdots\!20 \)
|
| $89$ |
\( T^{5} + \cdots + 15\!\cdots\!44 \)
|
| $97$ |
\( T^{5} + \cdots + 77\!\cdots\!15 \)
|
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