Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.2357976253\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 10488x^{4} - 31048x^{3} + 25151424x^{2} + 164507280x - 10237193792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{9} \) |
| 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
\( x^{6} - 3x^{5} - 10488x^{4} - 31048x^{3} + 25151424x^{2} + 164507280x - 10237193792 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 8\nu - 3494 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 9\nu^{2} - 5645\nu + 3039 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 359\nu^{4} - 13138\nu^{3} - 2797884\nu^{2} + 25972968\nu + 2648031904 ) / 265216 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -43\nu^{5} + 1139\nu^{4} + 382598\nu^{3} - 7326188\nu^{2} - 588726264\nu + 6065568544 ) / 132608 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 8\beta _1 + 3494 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 9\beta_{2} + 5717\beta _1 + 28407 \)
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| \(\nu^{4}\) | \(=\) |
\( 8\beta_{5} + 688\beta_{4} + 11\beta_{3} + 7799\beta_{2} + 92627\beta _1 + 19981061 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2872\beta_{5} + 18224\beta_{4} + 9189\beta_{3} + 116285\beta_{2} + 38266957\beta _1 + 327785059 \)
|
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 |
|
−86.4079 | 0 | 5418.33 | 11308.6 | 0 | 7656.19 | −291223. | 0 | −977155. | ||||||||||||||||||||||||||||||||||||
| 1.2 | −52.9496 | 0 | 755.665 | −9078.67 | 0 | 9377.64 | 68428.7 | 0 | 480712. | |||||||||||||||||||||||||||||||||||||
| 1.3 | −16.4719 | 0 | −1776.68 | 6657.94 | 0 | −23707.4 | 62999.6 | 0 | −109669. | |||||||||||||||||||||||||||||||||||||
| 1.4 | 30.7124 | 0 | −1104.75 | −553.176 | 0 | 72114.9 | −96828.5 | 0 | −16989.4 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 50.9535 | 0 | 548.260 | −7102.15 | 0 | −78935.2 | −76417.0 | 0 | −361880. | |||||||||||||||||||||||||||||||||||||
| 1.6 | 83.1635 | 0 | 4868.17 | 9045.43 | 0 | 36437.9 | 234535. | 0 | 752249. | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
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 | 81.12.a.b | yes | 6 |
| 3.b | odd | 2 | 1 | 81.12.a.a | ✓ | 6 | |
| 9.c | even | 3 | 2 | 81.12.c.k | 12 | ||
| 9.d | odd | 6 | 2 | 81.12.c.l | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.12.a.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 81.12.a.b | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 81.12.c.k | 12 | 9.c | even | 3 | 2 | ||
| 81.12.c.l | 12 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 9T_{2}^{5} - 10458T_{2}^{4} + 114912T_{2}^{3} + 24713424T_{2}^{2} - 264404736T_{2} - 9807989760 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 9807989760 \)
|
| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + \cdots - 24\!\cdots\!75 \)
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| $7$ |
\( T^{6} + \cdots + 35\!\cdots\!40 \)
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| $11$ |
\( T^{6} + \cdots + 24\!\cdots\!40 \)
|
| $13$ |
\( T^{6} + \cdots - 28\!\cdots\!39 \)
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| $17$ |
\( T^{6} + \cdots + 49\!\cdots\!69 \)
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| $19$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots - 38\!\cdots\!76 \)
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| $29$ |
\( T^{6} + \cdots + 11\!\cdots\!85 \)
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| $31$ |
\( T^{6} + \cdots + 17\!\cdots\!04 \)
|
| $37$ |
\( T^{6} + \cdots - 23\!\cdots\!55 \)
|
| $41$ |
\( T^{6} + \cdots + 42\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots - 66\!\cdots\!12 \)
|
| $47$ |
\( T^{6} + \cdots + 51\!\cdots\!76 \)
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| $53$ |
\( T^{6} + \cdots - 66\!\cdots\!68 \)
|
| $59$ |
\( T^{6} + \cdots + 32\!\cdots\!68 \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!55 \)
|
| $67$ |
\( T^{6} + \cdots + 23\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots - 12\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + \cdots - 11\!\cdots\!91 \)
|
| $79$ |
\( T^{6} + \cdots - 36\!\cdots\!88 \)
|
| $83$ |
\( T^{6} + \cdots - 70\!\cdots\!28 \)
|
| $89$ |
\( T^{6} + \cdots + 30\!\cdots\!05 \)
|
| $97$ |
\( T^{6} + \cdots + 38\!\cdots\!56 \)
|
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