Newspace parameters
Level: | \( N \) | \(=\) | \( 96 = 2^{5} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 96.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(29.9889624465\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{6}, \sqrt{-26})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 10x^{2} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{8} \) |
Twist minimal: | no (minimal twist has level 24) |
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 \( x^{4} + 10x^{2} + 64 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 18\nu ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( -\nu^{3} - 2\nu \) |
\(\beta_{3}\) | \(=\) | \( 8\nu^{2} + 40 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + 8\beta_1 ) / 16 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} - 40 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( -9\beta_{2} - 8\beta_1 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(37\) | \(65\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 |
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0 | 9.00000 | − | 45.8912i | 0 | −97.9796 | 0 | − | 1548.76i | 0 | −2025.00 | − | 826.041i | 0 | |||||||||||||||||||||||||
47.2 | 0 | 9.00000 | − | 45.8912i | 0 | 97.9796 | 0 | 1548.76i | 0 | −2025.00 | − | 826.041i | 0 | |||||||||||||||||||||||||||
47.3 | 0 | 9.00000 | + | 45.8912i | 0 | −97.9796 | 0 | 1548.76i | 0 | −2025.00 | + | 826.041i | 0 | |||||||||||||||||||||||||||
47.4 | 0 | 9.00000 | + | 45.8912i | 0 | 97.9796 | 0 | − | 1548.76i | 0 | −2025.00 | + | 826.041i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 96.8.f.b | 4 | |
3.b | odd | 2 | 1 | inner | 96.8.f.b | 4 | |
4.b | odd | 2 | 1 | 24.8.f.b | ✓ | 4 | |
8.b | even | 2 | 1 | 24.8.f.b | ✓ | 4 | |
8.d | odd | 2 | 1 | inner | 96.8.f.b | 4 | |
12.b | even | 2 | 1 | 24.8.f.b | ✓ | 4 | |
24.f | even | 2 | 1 | inner | 96.8.f.b | 4 | |
24.h | odd | 2 | 1 | 24.8.f.b | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.8.f.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
24.8.f.b | ✓ | 4 | 8.b | even | 2 | 1 | |
24.8.f.b | ✓ | 4 | 12.b | even | 2 | 1 | |
24.8.f.b | ✓ | 4 | 24.h | odd | 2 | 1 | |
96.8.f.b | 4 | 1.a | even | 1 | 1 | trivial | |
96.8.f.b | 4 | 3.b | odd | 2 | 1 | inner | |
96.8.f.b | 4 | 8.d | odd | 2 | 1 | inner | |
96.8.f.b | 4 | 24.f | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 9600 \)
acting on \(S_{8}^{\mathrm{new}}(96, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 18 T + 2187)^{2} \)
$5$
\( (T^{2} - 9600)^{2} \)
$7$
\( (T^{2} + 2398656)^{2} \)
$11$
\( (T^{2} + 3185000)^{2} \)
$13$
\( (T^{2} + 102857664)^{2} \)
$17$
\( (T^{2} + 759283616)^{2} \)
$19$
\( (T + 11570)^{4} \)
$23$
\( (T^{2} - 3084117504)^{2} \)
$29$
\( (T^{2} - 2999817600)^{2} \)
$31$
\( (T^{2} + 5139825600)^{2} \)
$37$
\( (T^{2} + 79989114816)^{2} \)
$41$
\( (T^{2} + 162923945600)^{2} \)
$43$
\( (T + 495062)^{4} \)
$47$
\( (T^{2} - 1290350985216)^{2} \)
$53$
\( (T^{2} - 327660488064)^{2} \)
$59$
\( (T^{2} + 2075050640936)^{2} \)
$61$
\( (T^{2} + 3190936382400)^{2} \)
$67$
\( (T + 1400126)^{4} \)
$71$
\( (T^{2} - 12741150610944)^{2} \)
$73$
\( (T + 2223598)^{4} \)
$79$
\( (T^{2} + 31254497167296)^{2} \)
$83$
\( (T^{2} + 9080138221544)^{2} \)
$89$
\( (T^{2} + 34167910932896)^{2} \)
$97$
\( (T - 6867926)^{4} \)
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