Newspace parameters
Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 8.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(11.4154804080\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0 | −3444.00 | 0 | 313358. | 0 | −2.32462e6 | 0 | −2.48777e6 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8.16.a.a | ✓ | 1 |
3.b | odd | 2 | 1 | 72.16.a.a | 1 | ||
4.b | odd | 2 | 1 | 16.16.a.e | 1 | ||
8.b | even | 2 | 1 | 64.16.a.j | 1 | ||
8.d | odd | 2 | 1 | 64.16.a.b | 1 | ||
12.b | even | 2 | 1 | 144.16.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.16.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
16.16.a.e | 1 | 4.b | odd | 2 | 1 | ||
64.16.a.b | 1 | 8.d | odd | 2 | 1 | ||
64.16.a.j | 1 | 8.b | even | 2 | 1 | ||
72.16.a.a | 1 | 3.b | odd | 2 | 1 | ||
144.16.a.a | 1 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 3444 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(8))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 3444 \)
$5$
\( T - 313358 \)
$7$
\( T + 2324616 \)
$11$
\( T + 55249084 \)
$13$
\( T + 110259578 \)
$17$
\( T + 2601428750 \)
$19$
\( T - 1952124284 \)
$23$
\( T + 25430340376 \)
$29$
\( T + 2277224202 \)
$31$
\( T + 190667257120 \)
$37$
\( T + 288229450002 \)
$41$
\( T - 756412456602 \)
$43$
\( T + 354186592988 \)
$47$
\( T - 6035922573648 \)
$53$
\( T + 12198920684962 \)
$59$
\( T + 4090911936748 \)
$61$
\( T - 17565907389910 \)
$67$
\( T + 3931246965172 \)
$71$
\( T - 58825436072248 \)
$73$
\( T - 107571519617914 \)
$79$
\( T - 61543860115504 \)
$83$
\( T - 13432070277436 \)
$89$
\( T - 269696339030634 \)
$97$
\( T + 793796744596318 \)
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