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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-31})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 7x^{2} - 8x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 7x^{2} - 8x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 7\nu^{2} - 7\nu - 64 ) / 56 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} + 3\nu^{2} + 45\nu + 12 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 51\nu^{3} + 21\nu^{2} + 315\nu - 744 ) / 28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 18\beta_1 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 270\beta _1 + 270 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14\beta_{3} - 7\beta_{2} + 414 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 |
|
−28.9310 | − | 50.1099i | 0 | −650.000 | + | 1125.83i | 289.310 | − | 501.099i | 0 | 25035.5 | + | 43362.8i | −43280.7 | 0 | −33480.0 | ||||||||||||||||||||||
| 28.2 | 28.9310 | + | 50.1099i | 0 | −650.000 | + | 1125.83i | −289.310 | + | 501.099i | 0 | 25035.5 | + | 43362.8i | 43280.7 | 0 | −33480.0 | |||||||||||||||||||||||
| 55.1 | −28.9310 | + | 50.1099i | 0 | −650.000 | − | 1125.83i | 289.310 | + | 501.099i | 0 | 25035.5 | − | 43362.8i | −43280.7 | 0 | −33480.0 | |||||||||||||||||||||||
| 55.2 | 28.9310 | − | 50.1099i | 0 | −650.000 | − | 1125.83i | −289.310 | − | 501.099i | 0 | 25035.5 | − | 43362.8i | 43280.7 | 0 | −33480.0 | |||||||||||||||||||||||
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.f | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.12.c.f | 4 | |
| 9.c | even | 3 | 1 | 27.12.a.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.12.c.f | 4 | |
| 9.d | odd | 6 | 1 | 27.12.a.b | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.12.c.f | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.12.a.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.12.a.b | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.12.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.12.c.f | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.12.c.f | 4 | 9.c | even | 3 | 1 | inner | |
| 81.12.c.f | 4 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3348T_{2}^{2} + 11209104 \)
acting on \(S_{12}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3348 T^{2} + 11209104 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 112091040000 \)
|
| $7$ |
\( (T^{2} - 50071 T + 2507105041)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 93\!\cdots\!24 \)
|
| $13$ |
\( (T^{2} + \cdots + 1079772452641)^{2} \)
|
| $17$ |
\( (T^{2} - 34086703146192)^{2} \)
|
| $19$ |
\( (T + 579301)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $31$ |
\( (T^{2} + \cdots + 79\!\cdots\!04)^{2} \)
|
| $37$ |
\( (T + 566931913)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 32\!\cdots\!24 \)
|
| $43$ |
\( (T^{2} + \cdots + 15\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 71\!\cdots\!64 \)
|
| $53$ |
\( (T^{2} - 23\!\cdots\!52)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 14\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} + \cdots + 12\!\cdots\!69)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 16\!\cdots\!41)^{2} \)
|
| $71$ |
\( (T^{2} - 49\!\cdots\!12)^{2} \)
|
| $73$ |
\( (T + 31135508287)^{4} \)
|
| $79$ |
\( (T^{2} + \cdots + 38\!\cdots\!25)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 54\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} - 27\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 10\!\cdots\!41)^{2} \)
|
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