Newspace parameters
Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 192.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.3283667211\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3 \) |
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 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
\( x^{4} - 2x^{2} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 2 \)
|
\(\beta_{2}\) | \(=\) |
\( -\nu^{3} - \nu^{2} + 4\nu + 1 \)
|
\(\beta_{3}\) | \(=\) |
\( 6\nu^{3} \)
|
\(\nu\) | \(=\) |
\( ( \beta_{3} + 6\beta_{2} + 3\beta_1 ) / 24 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta _1 + 2 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( \beta_{3} ) / 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
191.1 |
|
0 | −4.89898 | − | 1.73205i | 0 | − | 16.9706i | 0 | − | 17.3205i | 0 | 21.0000 | + | 16.9706i | 0 | ||||||||||||||||||||||||
191.2 | 0 | −4.89898 | + | 1.73205i | 0 | 16.9706i | 0 | 17.3205i | 0 | 21.0000 | − | 16.9706i | 0 | |||||||||||||||||||||||||||
191.3 | 0 | 4.89898 | − | 1.73205i | 0 | 16.9706i | 0 | − | 17.3205i | 0 | 21.0000 | − | 16.9706i | 0 | ||||||||||||||||||||||||||
191.4 | 0 | 4.89898 | + | 1.73205i | 0 | − | 16.9706i | 0 | 17.3205i | 0 | 21.0000 | + | 16.9706i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 192.4.c.c | 4 | |
3.b | odd | 2 | 1 | inner | 192.4.c.c | 4 | |
4.b | odd | 2 | 1 | inner | 192.4.c.c | 4 | |
8.b | even | 2 | 1 | 48.4.c.b | ✓ | 4 | |
8.d | odd | 2 | 1 | 48.4.c.b | ✓ | 4 | |
12.b | even | 2 | 1 | inner | 192.4.c.c | 4 | |
16.e | even | 4 | 2 | 768.4.f.b | 8 | ||
16.f | odd | 4 | 2 | 768.4.f.b | 8 | ||
24.f | even | 2 | 1 | 48.4.c.b | ✓ | 4 | |
24.h | odd | 2 | 1 | 48.4.c.b | ✓ | 4 | |
48.i | odd | 4 | 2 | 768.4.f.b | 8 | ||
48.k | even | 4 | 2 | 768.4.f.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.4.c.b | ✓ | 4 | 8.b | even | 2 | 1 | |
48.4.c.b | ✓ | 4 | 8.d | odd | 2 | 1 | |
48.4.c.b | ✓ | 4 | 24.f | even | 2 | 1 | |
48.4.c.b | ✓ | 4 | 24.h | odd | 2 | 1 | |
192.4.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
192.4.c.c | 4 | 3.b | odd | 2 | 1 | inner | |
192.4.c.c | 4 | 4.b | odd | 2 | 1 | inner | |
192.4.c.c | 4 | 12.b | even | 2 | 1 | inner | |
768.4.f.b | 8 | 16.e | even | 4 | 2 | ||
768.4.f.b | 8 | 16.f | odd | 4 | 2 | ||
768.4.f.b | 8 | 48.i | odd | 4 | 2 | ||
768.4.f.b | 8 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 288 \)
acting on \(S_{4}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 42T^{2} + 729 \)
$5$
\( (T^{2} + 288)^{2} \)
$7$
\( (T^{2} + 300)^{2} \)
$11$
\( (T^{2} - 864)^{2} \)
$13$
\( (T - 26)^{4} \)
$17$
\( (T^{2} + 4608)^{2} \)
$19$
\( (T^{2} + 11532)^{2} \)
$23$
\( (T^{2} - 31104)^{2} \)
$29$
\( (T^{2} + 288)^{2} \)
$31$
\( (T^{2} + 972)^{2} \)
$37$
\( (T + 206)^{4} \)
$41$
\( (T^{2} + 93312)^{2} \)
$43$
\( (T^{2} + 8748)^{2} \)
$47$
\( (T^{2} - 13824)^{2} \)
$53$
\( (T^{2} + 2592)^{2} \)
$59$
\( (T^{2} - 311904)^{2} \)
$61$
\( (T + 278)^{4} \)
$67$
\( (T^{2} + 792588)^{2} \)
$71$
\( (T^{2} - 3456)^{2} \)
$73$
\( (T + 422)^{4} \)
$79$
\( (T^{2} + 446988)^{2} \)
$83$
\( (T^{2} - 864)^{2} \)
$89$
\( (T^{2} + 139392)^{2} \)
$97$
\( (T + 1070)^{4} \)
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