Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(25.3031870642\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 156x^{4} + 388x^{3} + 5992x^{2} - 18174x + 6597 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{9} \) |
| 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
\( x^{6} - 2x^{5} - 156x^{4} + 388x^{3} + 5992x^{2} - 18174x + 6597 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} - 70\nu^{4} - 123\nu^{3} + 6438\nu^{2} - 10453\nu - 43878 ) / 2385 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -102\nu^{5} - 5\nu^{4} + 11337\nu^{3} - 4242\nu^{2} - 262313\nu + 239727 ) / 4770 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 3\nu^{3} - 78\nu^{2} - 221\nu + 273 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} + 5\nu^{4} + 1489\nu^{3} - 1694\nu^{2} - 77841\nu + 129789 ) / 530 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{4} + 6\nu^{3} + 69\nu^{2} - 526\nu + 780 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} + 3\beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 36 ) / 108 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -10\beta_{5} - 3\beta_{4} + 8\beta_{3} + 13\beta_{2} + 203\beta _1 + 5688 ) / 108 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -70\beta_{5} + 123\beta_{4} - 61\beta_{3} - 35\beta_{2} + 143\beta _1 - 1980 ) / 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -802\beta_{5} - 309\beta_{4} + 872\beta_{3} + 1003\beta_{2} + 15197\beta _1 + 434016 ) / 108 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -9962\beta_{5} + 19767\beta_{4} - 8792\beta_{3} - 10849\beta_{2} + 20029\beta _1 - 536724 ) / 108 \)
|
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 |
|
−19.7742 | 0 | 263.020 | 473.605 | 0 | −1026.85 | −2669.91 | 0 | −9365.17 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −13.3832 | 0 | 51.1095 | −73.4090 | 0 | −484.171 | 1029.04 | 0 | 982.446 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −5.73332 | 0 | −95.1290 | −160.767 | 0 | 545.019 | 1279.27 | 0 | 921.729 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 5.73332 | 0 | −95.1290 | 160.767 | 0 | 545.019 | −1279.27 | 0 | 921.729 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 13.3832 | 0 | 51.1095 | 73.4090 | 0 | −484.171 | −1029.04 | 0 | 982.446 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 19.7742 | 0 | 263.020 | −473.605 | 0 | −1026.85 | 2669.91 | 0 | −9365.17 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.8.a.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 81.8.a.d | ✓ | 6 |
| 9.c | even | 3 | 2 | 81.8.c.k | 12 | ||
| 9.d | odd | 6 | 2 | 81.8.c.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.8.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 81.8.a.d | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 81.8.c.k | 12 | 9.c | even | 3 | 2 | ||
| 81.8.c.k | 12 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 603T_{2}^{4} + 88776T_{2}^{2} - 2302128 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 603 T^{4} + \cdots - 2302128 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + \cdots - 31241071051875 \)
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| $7$ |
\( (T^{3} + 966 T^{2} + \cdots - 270966824)^{2} \)
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| $11$ |
\( T^{6} + \cdots - 44\!\cdots\!08 \)
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| $13$ |
\( (T^{3} + 5943 T^{2} + \cdots - 66238651043)^{2} \)
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| $17$ |
\( T^{6} + \cdots - 10\!\cdots\!87 \)
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| $19$ |
\( (T^{3} + \cdots - 91889736662600)^{2} \)
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| $23$ |
\( T^{6} + \cdots - 18\!\cdots\!32 \)
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| $29$ |
\( T^{6} + \cdots - 35\!\cdots\!87 \)
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| $31$ |
\( (T^{3} + \cdots - 244699463601152)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 22\!\cdots\!69)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 31\!\cdots\!52 \)
|
| $43$ |
\( (T^{3} + \cdots - 43\!\cdots\!00)^{2} \)
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| $47$ |
\( T^{6} + \cdots - 44\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
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| $59$ |
\( T^{6} + \cdots - 12\!\cdots\!68 \)
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| $61$ |
\( (T^{3} + \cdots - 61\!\cdots\!13)^{2} \)
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| $67$ |
\( (T^{3} + \cdots + 14\!\cdots\!00)^{2} \)
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| $71$ |
\( T^{6} + \cdots - 88\!\cdots\!52 \)
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| $73$ |
\( (T^{3} + \cdots - 63\!\cdots\!81)^{2} \)
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| $79$ |
\( (T^{3} + \cdots - 12\!\cdots\!28)^{2} \)
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| $83$ |
\( T^{6} + \cdots - 22\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots - 12\!\cdots\!03 \)
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| $97$ |
\( (T^{3} + \cdots + 65\!\cdots\!36)^{2} \)
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