Newspace parameters
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.99816040775\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 8) |
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 | −44.0000 | 0 | 430.000 | 0 | 1224.00 | 0 | −251.000 | 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 | 16.8.a.a | 1 | |
3.b | odd | 2 | 1 | 144.8.a.a | 1 | ||
4.b | odd | 2 | 1 | 8.8.a.b | ✓ | 1 | |
5.b | even | 2 | 1 | 400.8.a.p | 1 | ||
5.c | odd | 4 | 2 | 400.8.c.f | 2 | ||
8.b | even | 2 | 1 | 64.8.a.f | 1 | ||
8.d | odd | 2 | 1 | 64.8.a.b | 1 | ||
12.b | even | 2 | 1 | 72.8.a.a | 1 | ||
16.e | even | 4 | 2 | 256.8.b.a | 2 | ||
16.f | odd | 4 | 2 | 256.8.b.g | 2 | ||
20.d | odd | 2 | 1 | 200.8.a.b | 1 | ||
20.e | even | 4 | 2 | 200.8.c.c | 2 | ||
24.f | even | 2 | 1 | 576.8.a.y | 1 | ||
24.h | odd | 2 | 1 | 576.8.a.z | 1 | ||
28.d | even | 2 | 1 | 392.8.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.8.a.b | ✓ | 1 | 4.b | odd | 2 | 1 | |
16.8.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
64.8.a.b | 1 | 8.d | odd | 2 | 1 | ||
64.8.a.f | 1 | 8.b | even | 2 | 1 | ||
72.8.a.a | 1 | 12.b | even | 2 | 1 | ||
144.8.a.a | 1 | 3.b | odd | 2 | 1 | ||
200.8.a.b | 1 | 20.d | odd | 2 | 1 | ||
200.8.c.c | 2 | 20.e | even | 4 | 2 | ||
256.8.b.a | 2 | 16.e | even | 4 | 2 | ||
256.8.b.g | 2 | 16.f | odd | 4 | 2 | ||
392.8.a.b | 1 | 28.d | even | 2 | 1 | ||
400.8.a.p | 1 | 5.b | even | 2 | 1 | ||
400.8.c.f | 2 | 5.c | odd | 4 | 2 | ||
576.8.a.y | 1 | 24.f | even | 2 | 1 | ||
576.8.a.z | 1 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 44 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 44 \)
$5$
\( T - 430 \)
$7$
\( T - 1224 \)
$11$
\( T - 3164 \)
$13$
\( T - 6118 \)
$17$
\( T + 16270 \)
$19$
\( T - 5476 \)
$23$
\( T + 1576 \)
$29$
\( T - 122838 \)
$31$
\( T + 251360 \)
$37$
\( T + 52338 \)
$41$
\( T + 319398 \)
$43$
\( T + 710788 \)
$47$
\( T + 284112 \)
$53$
\( T - 296062 \)
$59$
\( T - 897548 \)
$61$
\( T + 884810 \)
$67$
\( T + 4659692 \)
$71$
\( T - 2710792 \)
$73$
\( T + 5670854 \)
$79$
\( T - 5124176 \)
$83$
\( T - 1563556 \)
$89$
\( T - 11605674 \)
$97$
\( T - 10931618 \)
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