Newspace parameters
Level: | \( N \) | \(=\) | \( 96 = 2^{5} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 96.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.61581053786\) |
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
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Defining polynomial: | \( x^{4} + 4x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{3}\cdot 3 \) |
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 in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{3} + \nu^{2} + 5\nu + 2 \) |
\(\beta_{2}\) | \(=\) | \( -2\nu^{3} - 6\nu \) |
\(\beta_{3}\) | \(=\) | \( 2\nu^{3} - 4\nu^{2} + 10\nu - 8 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + 3\beta_{2} + 4\beta_1 ) / 12 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + 2\beta _1 - 12 ) / 6 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{3} - 5\beta_{2} - 4\beta_1 ) / 4 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
|
0 | −1.73205 | − | 2.44949i | 0 | − | 2.82843i | 0 | −10.3923 | 0 | −3.00000 | + | 8.48528i | 0 | |||||||||||||||||||||||||
65.2 | 0 | −1.73205 | + | 2.44949i | 0 | 2.82843i | 0 | −10.3923 | 0 | −3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||||||
65.3 | 0 | 1.73205 | − | 2.44949i | 0 | 2.82843i | 0 | 10.3923 | 0 | −3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||||||
65.4 | 0 | 1.73205 | + | 2.44949i | 0 | − | 2.82843i | 0 | 10.3923 | 0 | −3.00000 | + | 8.48528i | 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 | 96.3.e.a | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 96.3.e.a | ✓ | 4 |
4.b | odd | 2 | 1 | inner | 96.3.e.a | ✓ | 4 |
8.b | even | 2 | 1 | 192.3.e.e | 4 | ||
8.d | odd | 2 | 1 | 192.3.e.e | 4 | ||
12.b | even | 2 | 1 | inner | 96.3.e.a | ✓ | 4 |
16.e | even | 4 | 2 | 768.3.h.f | 8 | ||
16.f | odd | 4 | 2 | 768.3.h.f | 8 | ||
24.f | even | 2 | 1 | 192.3.e.e | 4 | ||
24.h | odd | 2 | 1 | 192.3.e.e | 4 | ||
48.i | odd | 4 | 2 | 768.3.h.f | 8 | ||
48.k | even | 4 | 2 | 768.3.h.f | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.3.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
96.3.e.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
96.3.e.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
96.3.e.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
192.3.e.e | 4 | 8.b | even | 2 | 1 | ||
192.3.e.e | 4 | 8.d | odd | 2 | 1 | ||
192.3.e.e | 4 | 24.f | even | 2 | 1 | ||
192.3.e.e | 4 | 24.h | odd | 2 | 1 | ||
768.3.h.f | 8 | 16.e | even | 4 | 2 | ||
768.3.h.f | 8 | 16.f | odd | 4 | 2 | ||
768.3.h.f | 8 | 48.i | odd | 4 | 2 | ||
768.3.h.f | 8 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 8 \)
acting on \(S_{3}^{\mathrm{new}}(96, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 6T^{2} + 81 \)
$5$
\( (T^{2} + 8)^{2} \)
$7$
\( (T^{2} - 108)^{2} \)
$11$
\( (T^{2} + 216)^{2} \)
$13$
\( (T + 6)^{4} \)
$17$
\( (T^{2} + 512)^{2} \)
$19$
\( (T^{2} - 108)^{2} \)
$23$
\( (T^{2} + 864)^{2} \)
$29$
\( (T^{2} + 968)^{2} \)
$31$
\( (T^{2} - 972)^{2} \)
$37$
\( (T + 38)^{4} \)
$41$
\( (T^{2} + 32)^{2} \)
$43$
\( (T^{2} - 108)^{2} \)
$47$
\( (T^{2} + 3456)^{2} \)
$53$
\( (T^{2} + 200)^{2} \)
$59$
\( (T^{2} + 216)^{2} \)
$61$
\( (T + 22)^{4} \)
$67$
\( (T^{2} - 13068)^{2} \)
$71$
\( (T^{2} + 864)^{2} \)
$73$
\( (T + 30)^{4} \)
$79$
\( (T^{2} - 972)^{2} \)
$83$
\( (T^{2} + 5400)^{2} \)
$89$
\( (T^{2} + 32)^{2} \)
$97$
\( (T - 90)^{4} \)
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