Newspace parameters
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 192.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(78.2166931317\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 48) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).
| \(n\) | \(65\) | \(127\) | \(133\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | − | 46.7654i | 0 | 90.0000 | 0 | 921.451i | 0 | −2187.00 | 0 | |||||||||||||||||||||||
| 127.2 | 0 | 46.7654i | 0 | 90.0000 | 0 | − | 921.451i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 192.9.g.b | 2 | |
| 4.b | odd | 2 | 1 | inner | 192.9.g.b | 2 | |
| 8.b | even | 2 | 1 | 48.9.g.a | ✓ | 2 | |
| 8.d | odd | 2 | 1 | 48.9.g.a | ✓ | 2 | |
| 24.f | even | 2 | 1 | 144.9.g.f | 2 | ||
| 24.h | odd | 2 | 1 | 144.9.g.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.9.g.a | ✓ | 2 | 8.b | even | 2 | 1 | |
| 48.9.g.a | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 144.9.g.f | 2 | 24.f | even | 2 | 1 | ||
| 144.9.g.f | 2 | 24.h | odd | 2 | 1 | ||
| 192.9.g.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 192.9.g.b | 2 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 90 \)
acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 2187 \)
$5$
\( (T - 90)^{2} \)
$7$
\( T^{2} + 849072 \)
$11$
\( T^{2} + 26784432 \)
$13$
\( (T - 16358)^{2} \)
$17$
\( (T + 42678)^{2} \)
$19$
\( T^{2} + 13098113328 \)
$23$
\( T^{2} + 62831635200 \)
$29$
\( (T - 1270530)^{2} \)
$31$
\( T^{2} + 263110913712 \)
$37$
\( (T - 2262142)^{2} \)
$41$
\( (T + 872694)^{2} \)
$43$
\( T^{2} + 2762519510448 \)
$47$
\( T^{2} + 635219589312 \)
$53$
\( (T + 1061694)^{2} \)
$59$
\( T^{2} + \cdots + 546263136432432 \)
$61$
\( (T + 15301010)^{2} \)
$67$
\( T^{2} + 84276487324848 \)
$71$
\( T^{2} + \cdots + 521253612260352 \)
$73$
\( (T - 18916354)^{2} \)
$79$
\( T^{2} + 29\!\cdots\!68 \)
$83$
\( T^{2} + 42\!\cdots\!48 \)
$89$
\( (T + 89813214)^{2} \)
$97$
\( (T + 75778238)^{2} \)
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