Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(25.3031870642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-14})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 14x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| 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
\( x^{4} - 14x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 42\nu ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{3} + 84\nu ) / 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( 14\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -14\beta_{3} + 28\beta_{2} ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 |
|
−9.72111 | − | 16.8375i | 0 | −125.000 | + | 216.506i | −194.422 | + | 336.749i | 0 | 630.500 | + | 1092.06i | 2371.95 | 0 | 7560.00 | ||||||||||||||||||||||
| 28.2 | 9.72111 | + | 16.8375i | 0 | −125.000 | + | 216.506i | 194.422 | − | 336.749i | 0 | 630.500 | + | 1092.06i | −2371.95 | 0 | 7560.00 | |||||||||||||||||||||||
| 55.1 | −9.72111 | + | 16.8375i | 0 | −125.000 | − | 216.506i | −194.422 | − | 336.749i | 0 | 630.500 | − | 1092.06i | 2371.95 | 0 | 7560.00 | |||||||||||||||||||||||
| 55.2 | 9.72111 | − | 16.8375i | 0 | −125.000 | − | 216.506i | 194.422 | + | 336.749i | 0 | 630.500 | − | 1092.06i | −2371.95 | 0 | 7560.00 | |||||||||||||||||||||||
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 | 81.8.c.e | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.8.c.e | 4 | |
| 9.c | even | 3 | 1 | 27.8.a.d | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.8.c.e | 4 | |
| 9.d | odd | 6 | 1 | 27.8.a.d | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.8.c.e | 4 | |
| 36.f | odd | 6 | 1 | 432.8.a.o | 2 | ||
| 36.h | even | 6 | 1 | 432.8.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.8.a.d | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.8.a.d | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.8.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.8.c.e | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.8.c.e | 4 | 9.c | even | 3 | 1 | inner | |
| 81.8.c.e | 4 | 9.d | odd | 6 | 1 | inner | |
| 432.8.a.o | 2 | 36.f | odd | 6 | 1 | ||
| 432.8.a.o | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 378T_{2}^{2} + 142884 \)
acting on \(S_{8}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 378 T^{2} + 142884 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 22861440000 \)
|
| $7$ |
\( (T^{2} - 1261 T + 1590121)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 4766921155584 \)
|
| $13$ |
\( (T^{2} + 9581 T + 91795561)^{2} \)
|
| $17$ |
\( (T^{2} - 450751392)^{2} \)
|
| $19$ |
\( (T + 21931)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 54\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 10\!\cdots\!44 \)
|
| $31$ |
\( (T^{2} - 50908 T + 2591624464)^{2} \)
|
| $37$ |
\( (T - 246467)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $43$ |
\( (T^{2} + 315512 T + 99547822144)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 32\!\cdots\!84 \)
|
| $53$ |
\( (T^{2} - 16227050112)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 86\!\cdots\!24 \)
|
| $61$ |
\( (T^{2} - 497953 T + 247957190209)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 1785860722321)^{2} \)
|
| $71$ |
\( (T^{2} - 813538276992)^{2} \)
|
| $73$ |
\( (T - 3250793)^{4} \)
|
| $79$ |
\( (T^{2} + \cdots + 36911517985225)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 45\!\cdots\!44 \)
|
| $89$ |
\( (T^{2} - 170059231792800)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 43173165455641)^{2} \)
|
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