Newspace parameters
Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 384.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.4632421514\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | yes |
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
\(\beta_{1}\) | \(=\) | \( -\zeta_{12}^{3} + 2\zeta_{12} \) |
\(\beta_{2}\) | \(=\) | \( 8\zeta_{12}^{2} - 4 \) |
\(\beta_{3}\) | \(=\) | \( 4\zeta_{12}^{3} \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{3} + 4\beta_1 ) / 8 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( ( \beta_{2} + 4 ) / 8 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( ( \beta_{3} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(133\) | \(257\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
319.1 |
|
0 | −1.73205 | 0 | − | 4.00000i | 0 | 6.92820i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||
319.2 | 0 | −1.73205 | 0 | 4.00000i | 0 | − | 6.92820i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||
319.3 | 0 | 1.73205 | 0 | − | 4.00000i | 0 | − | 6.92820i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||
319.4 | 0 | 1.73205 | 0 | 4.00000i | 0 | 6.92820i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 384.3.b.a | ✓ | 4 |
3.b | odd | 2 | 1 | 1152.3.b.h | 4 | ||
4.b | odd | 2 | 1 | inner | 384.3.b.a | ✓ | 4 |
8.b | even | 2 | 1 | inner | 384.3.b.a | ✓ | 4 |
8.d | odd | 2 | 1 | inner | 384.3.b.a | ✓ | 4 |
12.b | even | 2 | 1 | 1152.3.b.h | 4 | ||
16.e | even | 4 | 1 | 768.3.g.a | 2 | ||
16.e | even | 4 | 1 | 768.3.g.b | 2 | ||
16.f | odd | 4 | 1 | 768.3.g.a | 2 | ||
16.f | odd | 4 | 1 | 768.3.g.b | 2 | ||
24.f | even | 2 | 1 | 1152.3.b.h | 4 | ||
24.h | odd | 2 | 1 | 1152.3.b.h | 4 | ||
48.i | odd | 4 | 1 | 2304.3.g.g | 2 | ||
48.i | odd | 4 | 1 | 2304.3.g.n | 2 | ||
48.k | even | 4 | 1 | 2304.3.g.g | 2 | ||
48.k | even | 4 | 1 | 2304.3.g.n | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
384.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
384.3.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
384.3.b.a | ✓ | 4 | 8.b | even | 2 | 1 | inner |
384.3.b.a | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
768.3.g.a | 2 | 16.e | even | 4 | 1 | ||
768.3.g.a | 2 | 16.f | odd | 4 | 1 | ||
768.3.g.b | 2 | 16.e | even | 4 | 1 | ||
768.3.g.b | 2 | 16.f | odd | 4 | 1 | ||
1152.3.b.h | 4 | 3.b | odd | 2 | 1 | ||
1152.3.b.h | 4 | 12.b | even | 2 | 1 | ||
1152.3.b.h | 4 | 24.f | even | 2 | 1 | ||
1152.3.b.h | 4 | 24.h | odd | 2 | 1 | ||
2304.3.g.g | 2 | 48.i | odd | 4 | 1 | ||
2304.3.g.g | 2 | 48.k | even | 4 | 1 | ||
2304.3.g.n | 2 | 48.i | odd | 4 | 1 | ||
2304.3.g.n | 2 | 48.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 3)^{2} \)
$5$
\( (T^{2} + 16)^{2} \)
$7$
\( (T^{2} + 48)^{2} \)
$11$
\( (T^{2} - 48)^{2} \)
$13$
\( T^{4} \)
$17$
\( (T + 18)^{4} \)
$19$
\( (T^{2} - 432)^{2} \)
$23$
\( (T^{2} + 1728)^{2} \)
$29$
\( (T^{2} + 16)^{2} \)
$31$
\( (T^{2} + 2352)^{2} \)
$37$
\( (T^{2} + 5184)^{2} \)
$41$
\( (T - 18)^{4} \)
$43$
\( (T^{2} - 3888)^{2} \)
$47$
\( (T^{2} + 1728)^{2} \)
$53$
\( (T^{2} + 1936)^{2} \)
$59$
\( (T^{2} - 3888)^{2} \)
$61$
\( (T^{2} + 5184)^{2} \)
$67$
\( (T^{2} - 432)^{2} \)
$71$
\( (T^{2} + 1728)^{2} \)
$73$
\( (T + 82)^{4} \)
$79$
\( (T^{2} + 3888)^{2} \)
$83$
\( (T^{2} - 17328)^{2} \)
$89$
\( (T - 126)^{4} \)
$97$
\( (T - 110)^{4} \)
show more
show less