Newspace parameters
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(32.9976674150\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-14})\) |
| Defining polynomial: |
\( x^{4} - 14x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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
\( x^{4} - 14x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 14\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} ) / 3 \)
|
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 |
|
−19.4422 | − | 11.2250i | 0 | 124.000 | + | 214.774i | −194.422 | + | 112.250i | 0 | 875.000 | − | 1515.54i | 179.600i | 0 | 5040.00 | ||||||||||||||||||||||
| 26.2 | 19.4422 | + | 11.2250i | 0 | 124.000 | + | 214.774i | 194.422 | − | 112.250i | 0 | 875.000 | − | 1515.54i | − | 179.600i | 0 | 5040.00 | ||||||||||||||||||||||
| 53.1 | −19.4422 | + | 11.2250i | 0 | 124.000 | − | 214.774i | −194.422 | − | 112.250i | 0 | 875.000 | + | 1515.54i | − | 179.600i | 0 | 5040.00 | ||||||||||||||||||||||
| 53.2 | 19.4422 | − | 11.2250i | 0 | 124.000 | − | 214.774i | 194.422 | + | 112.250i | 0 | 875.000 | + | 1515.54i | 179.600i | 0 | 5040.00 | |||||||||||||||||||||||
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 | 81.9.d.d | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.9.d.d | 4 | |
| 9.c | even | 3 | 1 | 3.9.b.a | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.9.d.d | 4 | |
| 9.d | odd | 6 | 1 | 3.9.b.a | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.9.d.d | 4 | |
| 36.f | odd | 6 | 1 | 48.9.e.b | 2 | ||
| 36.h | even | 6 | 1 | 48.9.e.b | 2 | ||
| 45.h | odd | 6 | 1 | 75.9.c.c | 2 | ||
| 45.j | even | 6 | 1 | 75.9.c.c | 2 | ||
| 45.k | odd | 12 | 2 | 75.9.d.b | 4 | ||
| 45.l | even | 12 | 2 | 75.9.d.b | 4 | ||
| 72.j | odd | 6 | 1 | 192.9.e.e | 2 | ||
| 72.l | even | 6 | 1 | 192.9.e.f | 2 | ||
| 72.n | even | 6 | 1 | 192.9.e.e | 2 | ||
| 72.p | odd | 6 | 1 | 192.9.e.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.9.b.a | ✓ | 2 | 9.c | even | 3 | 1 | |
| 3.9.b.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 48.9.e.b | 2 | 36.f | odd | 6 | 1 | ||
| 48.9.e.b | 2 | 36.h | even | 6 | 1 | ||
| 75.9.c.c | 2 | 45.h | odd | 6 | 1 | ||
| 75.9.c.c | 2 | 45.j | even | 6 | 1 | ||
| 75.9.d.b | 4 | 45.k | odd | 12 | 2 | ||
| 75.9.d.b | 4 | 45.l | even | 12 | 2 | ||
| 81.9.d.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.9.d.d | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.9.d.d | 4 | 9.c | even | 3 | 1 | inner | |
| 81.9.d.d | 4 | 9.d | odd | 6 | 1 | inner | |
| 192.9.e.e | 2 | 72.j | odd | 6 | 1 | ||
| 192.9.e.e | 2 | 72.n | even | 6 | 1 | ||
| 192.9.e.f | 2 | 72.l | even | 6 | 1 | ||
| 192.9.e.f | 2 | 72.p | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 504T_{2}^{2} + 254016 \)
acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 504 T^{2} + 254016 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 50400 T^{2} + \cdots + 2540160000 \)
$7$
\( (T^{2} - 1750 T + 3062500)^{2} \)
$11$
\( T^{4} - 48434400 T^{2} + \cdots + 23\!\cdots\!00 \)
$13$
\( (T^{2} + 25730 T + 662032900)^{2} \)
$17$
\( (T^{2} + 5608963584)^{2} \)
$19$
\( (T - 18938)^{4} \)
$23$
\( T^{4} - 221333583744 T^{2} + \cdots + 48\!\cdots\!36 \)
$29$
\( T^{4} - 212426373600 T^{2} + \cdots + 45\!\cdots\!00 \)
$31$
\( (T^{2} - 351478 T + 123536784484)^{2} \)
$37$
\( (T - 1335170)^{4} \)
$41$
\( T^{4} - 3517381526400 T^{2} + \cdots + 12\!\cdots\!00 \)
$43$
\( (T^{2} - 3526150 T + 12433733822500)^{2} \)
$47$
\( T^{4} - 16654893018624 T^{2} + \cdots + 27\!\cdots\!76 \)
$53$
\( (T^{2} + 43583305427424)^{2} \)
$59$
\( T^{4} - 188098358162400 T^{2} + \cdots + 35\!\cdots\!00 \)
$61$
\( (T^{2} + 753602 T + 567915974404)^{2} \)
$67$
\( (T^{2} + 2268890 T + 5147861832100)^{2} \)
$71$
\( (T^{2} + 289748374473600)^{2} \)
$73$
\( (T - 27672770)^{4} \)
$79$
\( (T^{2} - 22980982 T + 528125533684324)^{2} \)
$83$
\( T^{4} + \cdots + 46\!\cdots\!36 \)
$89$
\( (T^{2} + 52\!\cdots\!00)^{2} \)
$97$
\( (T^{2} + 147271010 T + 21\!\cdots\!00)^{2} \)
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