Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(74.0334991994\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-26})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 26x^{2} + 676 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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
\( x^{4} - 26x^{2} + 676 \)
:
| \(\beta_{1}\) | \(=\) |
\( 18\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 26 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{3} ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( 26\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 13\beta_{3} ) / 9 \)
|
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 |
|
−79.4858 | − | 45.8912i | 0 | 2164.00 | + | 3748.16i | 18281.7 | − | 10555.0i | 0 | −20125.0 | + | 34857.5i | − | 21293.5i | 0 | −1.93752e6 | |||||||||||||||||||||
| 26.2 | 79.4858 | + | 45.8912i | 0 | 2164.00 | + | 3748.16i | −18281.7 | + | 10555.0i | 0 | −20125.0 | + | 34857.5i | 21293.5i | 0 | −1.93752e6 | |||||||||||||||||||||||
| 53.1 | −79.4858 | + | 45.8912i | 0 | 2164.00 | − | 3748.16i | 18281.7 | + | 10555.0i | 0 | −20125.0 | − | 34857.5i | 21293.5i | 0 | −1.93752e6 | |||||||||||||||||||||||
| 53.2 | 79.4858 | − | 45.8912i | 0 | 2164.00 | − | 3748.16i | −18281.7 | − | 10555.0i | 0 | −20125.0 | − | 34857.5i | − | 21293.5i | 0 | −1.93752e6 | ||||||||||||||||||||||
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 | 81.13.d.c | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.13.d.c | 4 | |
| 9.c | even | 3 | 1 | 3.13.b.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.13.d.c | 4 | |
| 9.d | odd | 6 | 1 | 3.13.b.b | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.13.d.c | 4 | |
| 36.f | odd | 6 | 1 | 48.13.e.b | 2 | ||
| 36.h | even | 6 | 1 | 48.13.e.b | 2 | ||
| 45.h | odd | 6 | 1 | 75.13.c.c | 2 | ||
| 45.j | even | 6 | 1 | 75.13.c.c | 2 | ||
| 45.k | odd | 12 | 2 | 75.13.d.b | 4 | ||
| 45.l | even | 12 | 2 | 75.13.d.b | 4 | ||
| 72.j | odd | 6 | 1 | 192.13.e.d | 2 | ||
| 72.l | even | 6 | 1 | 192.13.e.c | 2 | ||
| 72.n | even | 6 | 1 | 192.13.e.d | 2 | ||
| 72.p | odd | 6 | 1 | 192.13.e.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.13.b.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 3.13.b.b | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 48.13.e.b | 2 | 36.f | odd | 6 | 1 | ||
| 48.13.e.b | 2 | 36.h | even | 6 | 1 | ||
| 75.13.c.c | 2 | 45.h | odd | 6 | 1 | ||
| 75.13.c.c | 2 | 45.j | even | 6 | 1 | ||
| 75.13.d.b | 4 | 45.k | odd | 12 | 2 | ||
| 75.13.d.b | 4 | 45.l | even | 12 | 2 | ||
| 81.13.d.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.13.d.c | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.13.d.c | 4 | 9.c | even | 3 | 1 | inner | |
| 81.13.d.c | 4 | 9.d | odd | 6 | 1 | inner | |
| 192.13.e.c | 2 | 72.l | even | 6 | 1 | ||
| 192.13.e.c | 2 | 72.p | odd | 6 | 1 | ||
| 192.13.e.d | 2 | 72.j | odd | 6 | 1 | ||
| 192.13.e.d | 2 | 72.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 8424T_{2}^{2} + 70963776 \)
acting on \(S_{13}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 8424 T^{2} + 70963776 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} + 40250 T + 1620062500)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots + 1648784402500)^{2} \)
|
| $17$ |
\( (T^{2} + 220359487855104)^{2} \)
|
| $19$ |
\( (T - 53343578)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots + 44\!\cdots\!04)^{2} \)
|
| $37$ |
\( (T - 2228726450)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots + 80\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!16 \)
|
| $53$ |
\( (T^{2} + 16\!\cdots\!04)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots + 16\!\cdots\!24)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{2} + 20\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T + 60956187550)^{4} \)
|
| $79$ |
\( (T^{2} + \cdots + 63\!\cdots\!04)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} + 12\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 42\!\cdots\!00)^{2} \)
|
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