Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(18.6343807732\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-10})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 10x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 27) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 10x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 10\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
−8.21584 | − | 4.74342i | 0 | 13.0000 | + | 22.5167i | 115.022 | − | 66.4078i | 0 | 201.500 | − | 349.008i | 360.500i | 0 | −1260.00 | ||||||||||||||||||||||
| 26.2 | 8.21584 | + | 4.74342i | 0 | 13.0000 | + | 22.5167i | −115.022 | + | 66.4078i | 0 | 201.500 | − | 349.008i | − | 360.500i | 0 | −1260.00 | ||||||||||||||||||||||
| 53.1 | −8.21584 | + | 4.74342i | 0 | 13.0000 | − | 22.5167i | 115.022 | + | 66.4078i | 0 | 201.500 | + | 349.008i | − | 360.500i | 0 | −1260.00 | ||||||||||||||||||||||
| 53.2 | 8.21584 | − | 4.74342i | 0 | 13.0000 | − | 22.5167i | −115.022 | − | 66.4078i | 0 | 201.500 | + | 349.008i | 360.500i | 0 | −1260.00 | |||||||||||||||||||||||
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 | 81.7.d.c | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.7.d.c | 4 | |
| 9.c | even | 3 | 1 | 27.7.b.a | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.7.d.c | 4 | |
| 9.d | odd | 6 | 1 | 27.7.b.a | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.7.d.c | 4 | |
| 36.f | odd | 6 | 1 | 432.7.e.g | 2 | ||
| 36.h | even | 6 | 1 | 432.7.e.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.7.b.a | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.7.b.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.7.d.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.7.d.c | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.7.d.c | 4 | 9.c | even | 3 | 1 | inner | |
| 81.7.d.c | 4 | 9.d | odd | 6 | 1 | inner | |
| 432.7.e.g | 2 | 36.f | odd | 6 | 1 | ||
| 432.7.e.g | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 90T_{2}^{2} + 8100 \)
acting on \(S_{7}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 90T^{2} + 8100 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 17640 T^{2} + 311169600 \)
|
| $7$ |
\( (T^{2} - 403 T + 162409)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 5047930497600 \)
|
| $13$ |
\( (T^{2} - 961 T + 923521)^{2} \)
|
| $17$ |
\( (T^{2} + 92537640)^{2} \)
|
| $19$ |
\( (T - 8021)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 37015056000000 \)
|
| $31$ |
\( (T^{2} + 48854 T + 2386713316)^{2} \)
|
| $37$ |
\( (T - 24167)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} - 60802 T + 3696883204)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + 18943385760)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 90\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + 272999 T + 74528454001)^{2} \)
|
| $67$ |
\( (T^{2} - 85579 T + 7323765241)^{2} \)
|
| $71$ |
\( (T^{2} + 116795571840)^{2} \)
|
| $73$ |
\( (T + 152737)^{4} \)
|
| $79$ |
\( (T^{2} - 74059 T + 5484735481)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 86\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + 1424896404840)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 1433558419969)^{2} \)
|
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