Newspace parameters
Level: | \( N \) | \(=\) | \( 512 = 2^{9} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 512.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.9509895352\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 4x^{2} + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{3} \) |
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 \( x^{4} - 4x^{2} + 9 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{3} + 7\nu ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( ( -2\nu^{3} + 2\nu ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( 2\nu^{2} - 4 \) |
\(\nu\) | \(=\) | \( ( -\beta_{2} + 2\beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 4 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -7\beta_{2} + 2\beta_1 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(511\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
255.1 |
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0 | −5.16228 | 0 | − | 4.47214i | 0 | − | 11.7727i | 0 | 17.6491 | 0 | ||||||||||||||||||||||||||||
255.2 | 0 | −5.16228 | 0 | 4.47214i | 0 | 11.7727i | 0 | 17.6491 | 0 | |||||||||||||||||||||||||||||||
255.3 | 0 | 1.16228 | 0 | − | 4.47214i | 0 | − | 6.11584i | 0 | −7.64911 | 0 | |||||||||||||||||||||||||||||
255.4 | 0 | 1.16228 | 0 | 4.47214i | 0 | 6.11584i | 0 | −7.64911 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
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 | 512.3.d.c | 4 | |
4.b | odd | 2 | 1 | 512.3.d.e | 4 | ||
8.b | even | 2 | 1 | 512.3.d.e | 4 | ||
8.d | odd | 2 | 1 | inner | 512.3.d.c | 4 | |
16.e | even | 4 | 2 | 512.3.c.f | ✓ | 8 | |
16.f | odd | 4 | 2 | 512.3.c.f | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
512.3.c.f | ✓ | 8 | 16.e | even | 4 | 2 | |
512.3.c.f | ✓ | 8 | 16.f | odd | 4 | 2 | |
512.3.d.c | 4 | 1.a | even | 1 | 1 | trivial | |
512.3.d.c | 4 | 8.d | odd | 2 | 1 | inner | |
512.3.d.e | 4 | 4.b | odd | 2 | 1 | ||
512.3.d.e | 4 | 8.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 4T_{3} - 6 \)
acting on \(S_{3}^{\mathrm{new}}(512, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} + 4 T - 6)^{2} \)
$5$
\( (T^{2} + 20)^{2} \)
$7$
\( T^{4} + 176T^{2} + 5184 \)
$11$
\( (T^{2} + 12 T + 26)^{2} \)
$13$
\( T^{4} + 296 T^{2} + 11664 \)
$17$
\( (T^{2} - 16 T - 96)^{2} \)
$19$
\( (T^{2} + 28 T - 54)^{2} \)
$23$
\( T^{4} + 560 T^{2} + 14400 \)
$29$
\( T^{4} + 2024T^{2} + 144 \)
$31$
\( T^{4} + 2240 T^{2} + 230400 \)
$37$
\( T^{4} + 5096 T^{2} + \cdots + 5253264 \)
$41$
\( (T^{2} + 28 T - 2364)^{2} \)
$43$
\( (T^{2} + 100 T + 2490)^{2} \)
$47$
\( T^{4} + 6656 T^{2} + 589824 \)
$53$
\( T^{4} + 6440 T^{2} + \cdots + 10112400 \)
$59$
\( (T^{2} + 20 T - 3510)^{2} \)
$61$
\( T^{4} + 5544 T^{2} + 219024 \)
$67$
\( (T^{2} + 44 T - 5766)^{2} \)
$71$
\( T^{4} + 10544 T^{2} + \cdots + 6594624 \)
$73$
\( (T^{2} + 56 T + 624)^{2} \)
$79$
\( T^{4} + 28864 T^{2} + \cdots + 190219264 \)
$83$
\( (T^{2} - 76 T + 234)^{2} \)
$89$
\( (T^{2} - 72 T - 144)^{2} \)
$97$
\( (T^{2} - 160)^{2} \)
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