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: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 2) |
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 | −12.0000 | 0 | −210.000 | 0 | −1016.00 | 0 | −2043.00 | 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.b | 1 | |
3.b | odd | 2 | 1 | 144.8.a.i | 1 | ||
4.b | odd | 2 | 1 | 2.8.a.a | ✓ | 1 | |
5.b | even | 2 | 1 | 400.8.a.l | 1 | ||
5.c | odd | 4 | 2 | 400.8.c.j | 2 | ||
8.b | even | 2 | 1 | 64.8.a.e | 1 | ||
8.d | odd | 2 | 1 | 64.8.a.c | 1 | ||
12.b | even | 2 | 1 | 18.8.a.b | 1 | ||
16.e | even | 4 | 2 | 256.8.b.f | 2 | ||
16.f | odd | 4 | 2 | 256.8.b.b | 2 | ||
20.d | odd | 2 | 1 | 50.8.a.g | 1 | ||
20.e | even | 4 | 2 | 50.8.b.c | 2 | ||
24.f | even | 2 | 1 | 576.8.a.g | 1 | ||
24.h | odd | 2 | 1 | 576.8.a.f | 1 | ||
28.d | even | 2 | 1 | 98.8.a.a | 1 | ||
28.f | even | 6 | 2 | 98.8.c.e | 2 | ||
28.g | odd | 6 | 2 | 98.8.c.d | 2 | ||
36.f | odd | 6 | 2 | 162.8.c.l | 2 | ||
36.h | even | 6 | 2 | 162.8.c.a | 2 | ||
44.c | even | 2 | 1 | 242.8.a.e | 1 | ||
52.b | odd | 2 | 1 | 338.8.a.d | 1 | ||
52.f | even | 4 | 2 | 338.8.b.d | 2 | ||
60.h | even | 2 | 1 | 450.8.a.c | 1 | ||
60.l | odd | 4 | 2 | 450.8.c.g | 2 | ||
68.d | odd | 2 | 1 | 578.8.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2.8.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
16.8.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
18.8.a.b | 1 | 12.b | even | 2 | 1 | ||
50.8.a.g | 1 | 20.d | odd | 2 | 1 | ||
50.8.b.c | 2 | 20.e | even | 4 | 2 | ||
64.8.a.c | 1 | 8.d | odd | 2 | 1 | ||
64.8.a.e | 1 | 8.b | even | 2 | 1 | ||
98.8.a.a | 1 | 28.d | even | 2 | 1 | ||
98.8.c.d | 2 | 28.g | odd | 6 | 2 | ||
98.8.c.e | 2 | 28.f | even | 6 | 2 | ||
144.8.a.i | 1 | 3.b | odd | 2 | 1 | ||
162.8.c.a | 2 | 36.h | even | 6 | 2 | ||
162.8.c.l | 2 | 36.f | odd | 6 | 2 | ||
242.8.a.e | 1 | 44.c | even | 2 | 1 | ||
256.8.b.b | 2 | 16.f | odd | 4 | 2 | ||
256.8.b.f | 2 | 16.e | even | 4 | 2 | ||
338.8.a.d | 1 | 52.b | odd | 2 | 1 | ||
338.8.b.d | 2 | 52.f | even | 4 | 2 | ||
400.8.a.l | 1 | 5.b | even | 2 | 1 | ||
400.8.c.j | 2 | 5.c | odd | 4 | 2 | ||
450.8.a.c | 1 | 60.h | even | 2 | 1 | ||
450.8.c.g | 2 | 60.l | odd | 4 | 2 | ||
576.8.a.f | 1 | 24.h | odd | 2 | 1 | ||
576.8.a.g | 1 | 24.f | even | 2 | 1 | ||
578.8.a.b | 1 | 68.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 12 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 12 \)
$5$
\( T + 210 \)
$7$
\( T + 1016 \)
$11$
\( T + 1092 \)
$13$
\( T - 1382 \)
$17$
\( T - 14706 \)
$19$
\( T - 39940 \)
$23$
\( T + 68712 \)
$29$
\( T + 102570 \)
$31$
\( T + 227552 \)
$37$
\( T - 160526 \)
$41$
\( T - 10842 \)
$43$
\( T - 630748 \)
$47$
\( T + 472656 \)
$53$
\( T + 1494018 \)
$59$
\( T + 2640660 \)
$61$
\( T - 827702 \)
$67$
\( T - 126004 \)
$71$
\( T - 1414728 \)
$73$
\( T - 980282 \)
$79$
\( T - 3566800 \)
$83$
\( T + 5672892 \)
$89$
\( T + 11951190 \)
$97$
\( T - 8682146 \)
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