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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - 182x^{6} + 32932x^{4} - 34944x^{2} + 36864 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{19} \) |
| Twist minimal: | no (minimal twist has level 27) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 182x^{6} + 32932x^{4} - 34944x^{2} + 36864 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 91\nu^{7} - 16466\nu^{5} + 2996812\nu^{3} + 3143040\nu ) / 1053824 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -91\nu^{7} + 16466\nu^{5} - 2996812\nu^{3} + 6341376\nu ) / 1053824 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 91\nu^{6} - 16466\nu^{4} + 2996812\nu^{2} - 18432 ) / 3161472 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\nu^{6} + 79970436 ) / 32932 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 218447\nu^{7} - 39995914\nu^{5} + 7236576476\nu^{3} - 15312627456\nu ) / 1053824 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 222335\nu^{7} - 39995914\nu^{5} + 7236576476\nu^{3} + 7589156736\nu ) / 1053824 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -73665\nu^{6} + 13485654\nu^{4} - 2425935780\nu^{2} + 2574149760 ) / 1053824 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{4} + 2457\beta_{3} ) / 27 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{6} - 2\beta_{5} - 4858\beta_{2} + 9716\beta_1 ) / 243 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 182\beta_{7} + 441990\beta_{3} - 441990 ) / 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -364\beta_{6} - 728\beta_{5} - 1757944\beta_{2} + 878972\beta_1 ) / 243 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 32932\beta_{4} - 79970436 ) / 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 65864\beta_{6} - 65864\beta_{5} - 159040168\beta_{2} - 159040168\beta_1 ) / 243 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{3}\) |
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 |
|
−34.9476 | − | 60.5311i | 0 | −1418.67 | + | 2457.22i | −1037.06 | + | 1796.24i | 0 | −22984.3 | − | 39810.0i | 55171.8 | 0 | 144971. | ||||||||||||||||||||||||||||||||||
| 28.2 | −2.67631 | − | 4.63551i | 0 | 1009.67 | − | 1748.81i | 6419.60 | − | 11119.1i | 0 | 15869.3 | + | 27486.4i | −21771.0 | 0 | −68723.4 | |||||||||||||||||||||||||||||||||||
| 28.3 | 2.67631 | + | 4.63551i | 0 | 1009.67 | − | 1748.81i | −6419.60 | + | 11119.1i | 0 | 15869.3 | + | 27486.4i | 21771.0 | 0 | −68723.4 | |||||||||||||||||||||||||||||||||||
| 28.4 | 34.9476 | + | 60.5311i | 0 | −1418.67 | + | 2457.22i | 1037.06 | − | 1796.24i | 0 | −22984.3 | − | 39810.0i | −55171.8 | 0 | 144971. | |||||||||||||||||||||||||||||||||||
| 55.1 | −34.9476 | + | 60.5311i | 0 | −1418.67 | − | 2457.22i | −1037.06 | − | 1796.24i | 0 | −22984.3 | + | 39810.0i | 55171.8 | 0 | 144971. | |||||||||||||||||||||||||||||||||||
| 55.2 | −2.67631 | + | 4.63551i | 0 | 1009.67 | + | 1748.81i | 6419.60 | + | 11119.1i | 0 | 15869.3 | − | 27486.4i | −21771.0 | 0 | −68723.4 | |||||||||||||||||||||||||||||||||||
| 55.3 | 2.67631 | − | 4.63551i | 0 | 1009.67 | + | 1748.81i | −6419.60 | − | 11119.1i | 0 | 15869.3 | − | 27486.4i | 21771.0 | 0 | −68723.4 | |||||||||||||||||||||||||||||||||||
| 55.4 | 34.9476 | − | 60.5311i | 0 | −1418.67 | − | 2457.22i | 1037.06 | + | 1796.24i | 0 | −22984.3 | + | 39810.0i | −55171.8 | 0 | 144971. | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.12.c.i | 8 | |
| 3.b | odd | 2 | 1 | inner | 81.12.c.i | 8 | |
| 9.c | even | 3 | 1 | 27.12.a.d | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 81.12.c.i | 8 | |
| 9.d | odd | 6 | 1 | 27.12.a.d | ✓ | 4 | |
| 9.d | odd | 6 | 1 | inner | 81.12.c.i | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.12.a.d | ✓ | 4 | 9.c | even | 3 | 1 | |
| 27.12.a.d | ✓ | 4 | 9.d | odd | 6 | 1 | |
| 81.12.c.i | 8 | 1.a | even | 1 | 1 | trivial | |
| 81.12.c.i | 8 | 3.b | odd | 2 | 1 | inner | |
| 81.12.c.i | 8 | 9.c | even | 3 | 1 | inner | |
| 81.12.c.i | 8 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4914T_{2}^{6} + 24007428T_{2}^{4} + 687802752T_{2}^{2} + 19591041024 \)
acting on \(S_{12}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 19591041024 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 50\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots + 21\!\cdots\!21)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 33\!\cdots\!84 \)
|
| $13$ |
\( (T^{4} + \cdots + 62\!\cdots\!25)^{2} \)
|
| $17$ |
\( (T^{4} + \cdots + 57\!\cdots\!72)^{2} \)
|
| $19$ |
\( (T^{2} + \cdots - 66935139322199)^{4} \)
|
| $23$ |
\( T^{8} + \cdots + 35\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 66\!\cdots\!24 \)
|
| $31$ |
\( (T^{4} + \cdots + 49\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 63\!\cdots\!75)^{4} \)
|
| $41$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $43$ |
\( (T^{4} + \cdots + 16\!\cdots\!76)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 15\!\cdots\!44 \)
|
| $53$ |
\( (T^{4} + \cdots + 14\!\cdots\!12)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 42\!\cdots\!44 \)
|
| $61$ |
\( (T^{4} + \cdots + 82\!\cdots\!01)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 86\!\cdots\!89)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 88\!\cdots\!88)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 30\!\cdots\!15)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 12\!\cdots\!25)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 18\!\cdots\!84 \)
|
| $89$ |
\( (T^{4} + \cdots + 61\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 12\!\cdots\!81)^{2} \)
|
| show more | |
| show less |