Newspace parameters
Level: | \( N \) | \(=\) | \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 468.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.73699881460\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{2} \) |
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
\(\beta_{1}\) | \(=\) | \( 2\zeta_{8}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{8}^{3} + \zeta_{8} \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{8}^{3} + \zeta_{8} \) |
\(\zeta_{8}\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 2 \) |
\(\zeta_{8}^{2}\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\zeta_{8}^{3}\) | \(=\) | \( ( -\beta_{3} + \beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(235\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
287.1 |
|
−1.41421 | 0 | 2.00000 | − | 2.82843i | 0 | 2.00000i | −2.82843 | 0 | 4.00000i | |||||||||||||||||||||||||||||
287.2 | −1.41421 | 0 | 2.00000 | 2.82843i | 0 | − | 2.00000i | −2.82843 | 0 | − | 4.00000i | |||||||||||||||||||||||||||||
287.3 | 1.41421 | 0 | 2.00000 | − | 2.82843i | 0 | − | 2.00000i | 2.82843 | 0 | − | 4.00000i | ||||||||||||||||||||||||||||
287.4 | 1.41421 | 0 | 2.00000 | 2.82843i | 0 | 2.00000i | 2.82843 | 0 | 4.00000i | |||||||||||||||||||||||||||||||
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 | 468.2.c.a | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 468.2.c.a | ✓ | 4 |
4.b | odd | 2 | 1 | inner | 468.2.c.a | ✓ | 4 |
8.b | even | 2 | 1 | 7488.2.d.a | 4 | ||
8.d | odd | 2 | 1 | 7488.2.d.a | 4 | ||
12.b | even | 2 | 1 | inner | 468.2.c.a | ✓ | 4 |
24.f | even | 2 | 1 | 7488.2.d.a | 4 | ||
24.h | odd | 2 | 1 | 7488.2.d.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
468.2.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
468.2.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
468.2.c.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
468.2.c.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
7488.2.d.a | 4 | 8.b | even | 2 | 1 | ||
7488.2.d.a | 4 | 8.d | odd | 2 | 1 | ||
7488.2.d.a | 4 | 24.f | even | 2 | 1 | ||
7488.2.d.a | 4 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2)^{2} \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + 8)^{2} \)
$7$
\( (T^{2} + 4)^{2} \)
$11$
\( (T^{2} - 18)^{2} \)
$13$
\( (T - 1)^{4} \)
$17$
\( (T^{2} + 2)^{2} \)
$19$
\( (T^{2} + 16)^{2} \)
$23$
\( (T^{2} - 8)^{2} \)
$29$
\( (T^{2} + 18)^{2} \)
$31$
\( (T^{2} + 64)^{2} \)
$37$
\( (T - 10)^{4} \)
$41$
\( (T^{2} + 128)^{2} \)
$43$
\( (T^{2} + 64)^{2} \)
$47$
\( (T^{2} - 2)^{2} \)
$53$
\( (T^{2} + 50)^{2} \)
$59$
\( (T^{2} - 18)^{2} \)
$61$
\( (T + 8)^{4} \)
$67$
\( (T^{2} + 4)^{2} \)
$71$
\( (T^{2} - 242)^{2} \)
$73$
\( (T + 6)^{4} \)
$79$
\( (T^{2} + 16)^{2} \)
$83$
\( (T^{2} - 50)^{2} \)
$89$
\( (T^{2} + 8)^{2} \)
$97$
\( (T + 10)^{4} \)
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