Newspace parameters
Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 192.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(19.8470329121\) |
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 | − | 5.19615i | 0 | 42.0000 | 0 | − | 76.2102i | 0 | −27.0000 | 0 | ||||||||||||||||||||||
127.2 | 0 | 5.19615i | 0 | 42.0000 | 0 | 76.2102i | 0 | −27.0000 | 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 | 192.5.g.b | 2 | |
3.b | odd | 2 | 1 | 576.5.g.d | 2 | ||
4.b | odd | 2 | 1 | inner | 192.5.g.b | 2 | |
8.b | even | 2 | 1 | 48.5.g.a | ✓ | 2 | |
8.d | odd | 2 | 1 | 48.5.g.a | ✓ | 2 | |
12.b | even | 2 | 1 | 576.5.g.d | 2 | ||
16.e | even | 4 | 2 | 768.5.b.c | 4 | ||
16.f | odd | 4 | 2 | 768.5.b.c | 4 | ||
24.f | even | 2 | 1 | 144.5.g.f | 2 | ||
24.h | odd | 2 | 1 | 144.5.g.f | 2 | ||
40.e | odd | 2 | 1 | 1200.5.e.b | 2 | ||
40.f | even | 2 | 1 | 1200.5.e.b | 2 | ||
40.i | odd | 4 | 2 | 1200.5.j.b | 4 | ||
40.k | even | 4 | 2 | 1200.5.j.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.5.g.a | ✓ | 2 | 8.b | even | 2 | 1 | |
48.5.g.a | ✓ | 2 | 8.d | odd | 2 | 1 | |
144.5.g.f | 2 | 24.f | even | 2 | 1 | ||
144.5.g.f | 2 | 24.h | odd | 2 | 1 | ||
192.5.g.b | 2 | 1.a | even | 1 | 1 | trivial | |
192.5.g.b | 2 | 4.b | odd | 2 | 1 | inner | |
576.5.g.d | 2 | 3.b | odd | 2 | 1 | ||
576.5.g.d | 2 | 12.b | even | 2 | 1 | ||
768.5.b.c | 4 | 16.e | even | 4 | 2 | ||
768.5.b.c | 4 | 16.f | odd | 4 | 2 | ||
1200.5.e.b | 2 | 40.e | odd | 2 | 1 | ||
1200.5.e.b | 2 | 40.f | even | 2 | 1 | ||
1200.5.j.b | 4 | 40.i | odd | 4 | 2 | ||
1200.5.j.b | 4 | 40.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 42 \)
acting on \(S_{5}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 27 \)
$5$
\( (T - 42)^{2} \)
$7$
\( T^{2} + 5808 \)
$11$
\( T^{2} + 432 \)
$13$
\( (T - 182)^{2} \)
$17$
\( (T + 246)^{2} \)
$19$
\( T^{2} + 13872 \)
$23$
\( T^{2} + 559872 \)
$29$
\( (T + 78)^{2} \)
$31$
\( T^{2} + 2177712 \)
$37$
\( (T + 530)^{2} \)
$41$
\( (T + 918)^{2} \)
$43$
\( T^{2} + 726192 \)
$47$
\( T^{2} + 14309568 \)
$53$
\( (T - 4626)^{2} \)
$59$
\( T^{2} + 52272 \)
$61$
\( (T + 1346)^{2} \)
$67$
\( T^{2} + 1183152 \)
$71$
\( T^{2} + 3345408 \)
$73$
\( (T + 926)^{2} \)
$79$
\( T^{2} + 19354800 \)
$83$
\( T^{2} + 143825328 \)
$89$
\( (T - 11586)^{2} \)
$97$
\( (T + 13118)^{2} \)
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