Newspace parameters
Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 576.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(132.511152165\) |
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: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{8}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 96) |
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
\(\beta_{1}\) | \(=\) | \( 4\zeta_{12}^{3} \) |
\(\beta_{2}\) | \(=\) | \( -12\zeta_{12}^{3} + 24\zeta_{12} \) |
\(\beta_{3}\) | \(=\) | \( 24\zeta_{12}^{2} - 12 \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{2} + 3\beta_1 ) / 24 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( ( \beta_{3} + 12 ) / 24 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( ( \beta_1 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).
\(n\) | \(65\) | \(127\) | \(325\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | 0 | 0 | −153.923 | 0 | − | 395.923i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
127.2 | 0 | 0 | 0 | −153.923 | 0 | 395.923i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
127.3 | 0 | 0 | 0 | 53.9230 | 0 | − | 188.077i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
127.4 | 0 | 0 | 0 | 53.9230 | 0 | 188.077i | 0 | 0 | 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 | 576.7.g.k | 4 | |
3.b | odd | 2 | 1 | 192.7.g.d | 4 | ||
4.b | odd | 2 | 1 | inner | 576.7.g.k | 4 | |
8.b | even | 2 | 1 | 288.7.g.c | 4 | ||
8.d | odd | 2 | 1 | 288.7.g.c | 4 | ||
12.b | even | 2 | 1 | 192.7.g.d | 4 | ||
24.f | even | 2 | 1 | 96.7.g.a | ✓ | 4 | |
24.h | odd | 2 | 1 | 96.7.g.a | ✓ | 4 | |
48.i | odd | 4 | 1 | 768.7.b.a | 4 | ||
48.i | odd | 4 | 1 | 768.7.b.e | 4 | ||
48.k | even | 4 | 1 | 768.7.b.a | 4 | ||
48.k | even | 4 | 1 | 768.7.b.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.7.g.a | ✓ | 4 | 24.f | even | 2 | 1 | |
96.7.g.a | ✓ | 4 | 24.h | odd | 2 | 1 | |
192.7.g.d | 4 | 3.b | odd | 2 | 1 | ||
192.7.g.d | 4 | 12.b | even | 2 | 1 | ||
288.7.g.c | 4 | 8.b | even | 2 | 1 | ||
288.7.g.c | 4 | 8.d | odd | 2 | 1 | ||
576.7.g.k | 4 | 1.a | even | 1 | 1 | trivial | |
576.7.g.k | 4 | 4.b | odd | 2 | 1 | inner | |
768.7.b.a | 4 | 48.i | odd | 4 | 1 | ||
768.7.b.a | 4 | 48.k | even | 4 | 1 | ||
768.7.b.e | 4 | 48.i | odd | 4 | 1 | ||
768.7.b.e | 4 | 48.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 100T_{5} - 8300 \)
acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + 100 T - 8300)^{2} \)
$7$
\( T^{4} + 192128 T^{2} + \cdots + 5544887296 \)
$11$
\( T^{4} + 594656 T^{2} + \cdots + 212459776 \)
$13$
\( (T^{2} + 4916 T + 2846692)^{2} \)
$17$
\( (T^{2} - 396 T - 584604)^{2} \)
$19$
\( T^{4} + 84224736 T^{2} + \cdots + 7440097536 \)
$23$
\( T^{4} + 515524736 T^{2} + \cdots + 63\!\cdots\!24 \)
$29$
\( (T^{2} + 41396 T + 257490292)^{2} \)
$31$
\( T^{4} + 892954496 T^{2} + \cdots + 17\!\cdots\!36 \)
$37$
\( (T^{2} + 1668 T - 4129424892)^{2} \)
$41$
\( (T^{2} + 91924 T - 66725468)^{2} \)
$43$
\( T^{4} + 23532599264 T^{2} + \cdots + 11\!\cdots\!56 \)
$47$
\( T^{4} + 3805630976 T^{2} + \cdots + 27\!\cdots\!76 \)
$53$
\( (T^{2} - 52140 T - 50204602188)^{2} \)
$59$
\( T^{4} + 69156949856 T^{2} + \cdots + 12\!\cdots\!84 \)
$61$
\( (T^{2} + 327428 T - 4029006332)^{2} \)
$67$
\( T^{4} + 15928236128 T^{2} + \cdots + 62\!\cdots\!64 \)
$71$
\( T^{4} + 189100829312 T^{2} + \cdots + 89\!\cdots\!04 \)
$73$
\( (T^{2} - 720132 T - 20837470716)^{2} \)
$79$
\( T^{4} + 459428175488 T^{2} + \cdots + 56\!\cdots\!04 \)
$83$
\( T^{4} + 1024589311712 T^{2} + \cdots + 25\!\cdots\!36 \)
$89$
\( (T^{2} + 36132 T - 1131604920252)^{2} \)
$97$
\( (T^{2} - 977540 T - 554382853628)^{2} \)
show more
show less