Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(62.2357976253\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + 7011 x^{10} - 35000 x^{9} + 17884836 x^{8} - 71329410 x^{7} + 20475172451 x^{6} + \cdots + 14\!\cdots\!75 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{21} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} + 7011 x^{10} - 35000 x^{9} + 17884836 x^{8} - 71329410 x^{7} + 20475172451 x^{6} + \cdots + 14\!\cdots\!75 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3713329 \nu^{10} + 18566645 \nu^{9} - 21503329409 \nu^{8} + 85901917766 \nu^{7} + \cdots - 17\!\cdots\!35 ) / 76\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 25993303 \nu^{10} + 129966515 \nu^{9} - 150523305863 \nu^{8} + 601313424362 \nu^{7} + \cdots + 26\!\cdots\!75 ) / 76\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 56788284859 \nu^{10} - 283941424295 \nu^{9} + 349614115343159 \nu^{8} + \cdots + 58\!\cdots\!45 ) / 35\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4552616241979 \nu^{10} + 22763081209895 \nu^{9} + \cdots - 15\!\cdots\!45 ) / 53\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7458401471 \nu^{10} - 37292007355 \nu^{9} + 43190480481391 \nu^{8} - 172538169881434 \nu^{7} + \cdots - 14\!\cdots\!95 ) / 76\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 176482897265 \nu^{11} + \cdots - 35\!\cdots\!35 ) / 36\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 352965794530 \nu^{11} + 1941311869915 \nu^{10} + \cdots - 22\!\cdots\!65 ) / 36\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 25\!\cdots\!31 \nu^{11} + \cdots - 79\!\cdots\!45 ) / 36\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 68\!\cdots\!13 \nu^{11} + \cdots - 12\!\cdots\!20 ) / 17\!\cdots\!80 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 20\!\cdots\!45 \nu^{11} + \cdots - 36\!\cdots\!15 ) / 36\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 40\!\cdots\!19 \nu^{11} + \cdots + 38\!\cdots\!20 ) / 25\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + \beta_{2} - 6\beta _1 - 3493 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{10} + 15\beta_{8} - 57726\beta_{7} + 11407\beta_{6} - \beta_{5} + 12\beta_{2} - 5735\beta _1 - 38877 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4 \beta_{10} + 30 \beta_{8} - 115455 \beta_{7} + 22820 \beta_{6} + 7 \beta_{5} + 8 \beta_{4} + \cdots + 19859662 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5784 \beta_{11} - 18313 \beta_{10} - 33008 \beta_{9} - 193505 \beta_{8} + 631270881 \beta_{7} + \cdots + 426992828 ) / 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5784 \beta_{11} - 18323 \beta_{10} - 33008 \beta_{9} - 193580 \beta_{8} + 631559520 \beta_{7} + \cdots - 43920644365 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 6640080 \beta_{11} + 15862495 \beta_{10} + 46375968 \beta_{9} + 231396834 \beta_{8} + \cdots - 454665140795 ) / 9 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 79761936 \beta_{11} + 190606490 \beta_{10} + 556973728 \beta_{9} + 2779472338 \beta_{8} + \cdots + 308536451093351 ) / 27 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 17611827672 \beta_{11} - 39603167573 \beta_{10} - 149455947120 \beta_{9} - 744255081033 \beta_{8} + \cdots + 13\!\cdots\!37 ) / 9 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 88258583688 \beta_{11} - 198492482371 \beta_{10} - 748672400976 \beta_{9} - 3728225441220 \beta_{8} + \cdots - 24\!\cdots\!56 ) / 9 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 130374864366576 \beta_{11} + 293431744566487 \beta_{10} + \cdots - 12\!\cdots\!76 ) / 27 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 - \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−43.2040 | − | 74.8315i | 0 | −2709.16 | + | 4692.41i | 5654.31 | − | 9793.56i | 0 | −3828.09 | − | 6630.45i | 291223. | 0 | −977155. | ||||||||||||||||||||||||||||||||||||||||||||||
| 28.2 | −26.4748 | − | 45.8557i | 0 | −377.832 | + | 654.425i | −4539.33 | + | 7862.36i | 0 | −4688.82 | − | 8121.28i | −68428.7 | 0 | 480712. | |||||||||||||||||||||||||||||||||||||||||||||||
| 28.3 | −8.23593 | − | 14.2651i | 0 | 888.339 | − | 1538.65i | 3328.97 | − | 5765.95i | 0 | 11853.7 | + | 20531.2i | −62999.6 | 0 | −109669. | |||||||||||||||||||||||||||||||||||||||||||||||
| 28.4 | 15.3562 | + | 26.5977i | 0 | 552.374 | − | 956.740i | −276.588 | + | 479.064i | 0 | −36057.4 | − | 62453.3i | 96828.5 | 0 | −16989.4 | |||||||||||||||||||||||||||||||||||||||||||||||
| 28.5 | 25.4768 | + | 44.1270i | 0 | −274.130 | + | 474.807i | −3551.08 | + | 6150.64i | 0 | 39467.6 | + | 68359.8i | 76417.0 | 0 | −361880. | |||||||||||||||||||||||||||||||||||||||||||||||
| 28.6 | 41.5818 | + | 72.0217i | 0 | −2434.09 | + | 4215.96i | 4522.71 | − | 7833.57i | 0 | −18218.9 | − | 31556.1i | −234535. | 0 | 752249. | |||||||||||||||||||||||||||||||||||||||||||||||
| 55.1 | −43.2040 | + | 74.8315i | 0 | −2709.16 | − | 4692.41i | 5654.31 | + | 9793.56i | 0 | −3828.09 | + | 6630.45i | 291223. | 0 | −977155. | |||||||||||||||||||||||||||||||||||||||||||||||
| 55.2 | −26.4748 | + | 45.8557i | 0 | −377.832 | − | 654.425i | −4539.33 | − | 7862.36i | 0 | −4688.82 | + | 8121.28i | −68428.7 | 0 | 480712. | |||||||||||||||||||||||||||||||||||||||||||||||
| 55.3 | −8.23593 | + | 14.2651i | 0 | 888.339 | + | 1538.65i | 3328.97 | + | 5765.95i | 0 | 11853.7 | − | 20531.2i | −62999.6 | 0 | −109669. | |||||||||||||||||||||||||||||||||||||||||||||||
| 55.4 | 15.3562 | − | 26.5977i | 0 | 552.374 | + | 956.740i | −276.588 | − | 479.064i | 0 | −36057.4 | + | 62453.3i | 96828.5 | 0 | −16989.4 | |||||||||||||||||||||||||||||||||||||||||||||||
| 55.5 | 25.4768 | − | 44.1270i | 0 | −274.130 | − | 474.807i | −3551.08 | − | 6150.64i | 0 | 39467.6 | − | 68359.8i | 76417.0 | 0 | −361880. | |||||||||||||||||||||||||||||||||||||||||||||||
| 55.6 | 41.5818 | − | 72.0217i | 0 | −2434.09 | − | 4215.96i | 4522.71 | + | 7833.57i | 0 | −18218.9 | + | 31556.1i | −234535. | 0 | 752249. | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.12.c.l | 12 | |
| 3.b | odd | 2 | 1 | 81.12.c.k | 12 | ||
| 9.c | even | 3 | 1 | 81.12.a.a | ✓ | 6 | |
| 9.c | even | 3 | 1 | inner | 81.12.c.l | 12 | |
| 9.d | odd | 6 | 1 | 81.12.a.b | yes | 6 | |
| 9.d | odd | 6 | 1 | 81.12.c.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.12.a.a | ✓ | 6 | 9.c | even | 3 | 1 | |
| 81.12.a.b | yes | 6 | 9.d | odd | 6 | 1 | |
| 81.12.c.k | 12 | 3.b | odd | 2 | 1 | ||
| 81.12.c.k | 12 | 9.d | odd | 6 | 1 | ||
| 81.12.c.l | 12 | 1.a | even | 1 | 1 | trivial | |
| 81.12.c.l | 12 | 9.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 9 T_{2}^{11} + 10539 T_{2}^{10} - 135702 T_{2}^{9} + 85690548 T_{2}^{8} + \cdots + 96\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 96\!\cdots\!00 \)
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| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 59\!\cdots\!25 \)
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| $7$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
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| $11$ |
\( T^{12} + \cdots + 58\!\cdots\!00 \)
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| $13$ |
\( T^{12} + \cdots + 80\!\cdots\!21 \)
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| $17$ |
\( (T^{6} + \cdots + 49\!\cdots\!69)^{2} \)
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| $19$ |
\( (T^{6} + \cdots + 40\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{12} + \cdots + 14\!\cdots\!76 \)
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| $29$ |
\( T^{12} + \cdots + 13\!\cdots\!25 \)
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| $31$ |
\( T^{12} + \cdots + 31\!\cdots\!16 \)
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| $37$ |
\( (T^{6} + \cdots - 23\!\cdots\!55)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 17\!\cdots\!56 \)
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| $43$ |
\( T^{12} + \cdots + 44\!\cdots\!44 \)
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| $47$ |
\( T^{12} + \cdots + 26\!\cdots\!76 \)
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| $53$ |
\( (T^{6} + \cdots - 66\!\cdots\!68)^{2} \)
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| $59$ |
\( T^{12} + \cdots + 10\!\cdots\!24 \)
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| $61$ |
\( T^{12} + \cdots + 14\!\cdots\!25 \)
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| $67$ |
\( T^{12} + \cdots + 56\!\cdots\!16 \)
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| $71$ |
\( (T^{6} + \cdots - 12\!\cdots\!48)^{2} \)
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| $73$ |
\( (T^{6} + \cdots - 11\!\cdots\!91)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 13\!\cdots\!44 \)
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| $83$ |
\( T^{12} + \cdots + 49\!\cdots\!84 \)
|
| $89$ |
\( (T^{6} + \cdots + 30\!\cdots\!05)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 14\!\cdots\!36 \)
|
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