Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.20709014132\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 4x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 5\nu^{3} + 19\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 6 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{2} + 19\beta_1 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 80.1 |
|
− | 3.34607i | 0 | −7.19615 | − | 4.00240i | 0 | −6.19615 | 10.6945i | 0 | −13.3923 | ||||||||||||||||||||||||||||
| 80.2 | − | 0.896575i | 0 | 3.19615 | 8.24504i | 0 | 4.19615 | − | 6.45189i | 0 | 7.39230 | |||||||||||||||||||||||||||||
| 80.3 | 0.896575i | 0 | 3.19615 | − | 8.24504i | 0 | 4.19615 | 6.45189i | 0 | 7.39230 | ||||||||||||||||||||||||||||||
| 80.4 | 3.34607i | 0 | −7.19615 | 4.00240i | 0 | −6.19615 | − | 10.6945i | 0 | −13.3923 | ||||||||||||||||||||||||||||||
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 | 81.3.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 81.3.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1296.3.e.h | 4 | ||
| 9.c | even | 3 | 2 | 81.3.d.c | 8 | ||
| 9.d | odd | 6 | 2 | 81.3.d.c | 8 | ||
| 12.b | even | 2 | 1 | 1296.3.e.h | 4 | ||
| 36.f | odd | 6 | 2 | 1296.3.q.m | 8 | ||
| 36.h | even | 6 | 2 | 1296.3.q.m | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.3.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 81.3.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 81.3.d.c | 8 | 9.c | even | 3 | 2 | ||
| 81.3.d.c | 8 | 9.d | odd | 6 | 2 | ||
| 1296.3.e.h | 4 | 4.b | odd | 2 | 1 | ||
| 1296.3.e.h | 4 | 12.b | even | 2 | 1 | ||
| 1296.3.q.m | 8 | 36.f | odd | 6 | 2 | ||
| 1296.3.q.m | 8 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 12T_{2}^{2} + 9 \)
acting on \(S_{3}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 12T^{2} + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 84T^{2} + 1089 \)
|
| $7$ |
\( (T^{2} + 2 T - 26)^{2} \)
|
| $11$ |
\( T^{4} + 156T^{2} + 4356 \)
|
| $13$ |
\( (T^{2} - 22 T + 13)^{2} \)
|
| $17$ |
\( T^{4} + 108T^{2} + 729 \)
|
| $19$ |
\( (T^{2} + 14 T - 194)^{2} \)
|
| $23$ |
\( T^{4} + 1956 T^{2} + 617796 \)
|
| $29$ |
\( T^{4} + 12T^{2} + 9 \)
|
| $31$ |
\( (T^{2} + 32 T + 148)^{2} \)
|
| $37$ |
\( (T^{2} - 4 T - 239)^{2} \)
|
| $41$ |
\( T^{4} + 2316 T^{2} + 1313316 \)
|
| $43$ |
\( (T^{2} - 70 T + 1198)^{2} \)
|
| $47$ |
\( T^{4} + 7536 T^{2} + 13927824 \)
|
| $53$ |
\( (T^{2} + 4374)^{2} \)
|
| $59$ |
\( T^{4} + 1488 T^{2} + 24336 \)
|
| $61$ |
\( (T^{2} - 88 T + 1261)^{2} \)
|
| $67$ |
\( (T^{2} - 58 T - 8906)^{2} \)
|
| $71$ |
\( T^{4} + 7236 T^{2} + 10850436 \)
|
| $73$ |
\( (T^{2} + 32 T - 1931)^{2} \)
|
| $79$ |
\( (T^{2} + 182 T + 8254)^{2} \)
|
| $83$ |
\( T^{4} + 17616 T^{2} + 74235456 \)
|
| $89$ |
\( T^{4} + 29268 T^{2} + 213364449 \)
|
| $97$ |
\( (T^{2} - 16 T - 2636)^{2} \)
|
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