Newspace parameters
Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 8.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.12028668931\) |
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 | −60.0000 | 0 | −2074.00 | 0 | −4344.00 | 0 | −16083.0 | 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 | 8.10.a.a | ✓ | 1 |
3.b | odd | 2 | 1 | 72.10.a.e | 1 | ||
4.b | odd | 2 | 1 | 16.10.a.c | 1 | ||
5.b | even | 2 | 1 | 200.10.a.b | 1 | ||
5.c | odd | 4 | 2 | 200.10.c.b | 2 | ||
7.b | odd | 2 | 1 | 392.10.a.b | 1 | ||
8.b | even | 2 | 1 | 64.10.a.f | 1 | ||
8.d | odd | 2 | 1 | 64.10.a.d | 1 | ||
12.b | even | 2 | 1 | 144.10.a.n | 1 | ||
16.e | even | 4 | 2 | 256.10.b.i | 2 | ||
16.f | odd | 4 | 2 | 256.10.b.c | 2 | ||
20.d | odd | 2 | 1 | 400.10.a.d | 1 | ||
20.e | even | 4 | 2 | 400.10.c.g | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.10.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
16.10.a.c | 1 | 4.b | odd | 2 | 1 | ||
64.10.a.d | 1 | 8.d | odd | 2 | 1 | ||
64.10.a.f | 1 | 8.b | even | 2 | 1 | ||
72.10.a.e | 1 | 3.b | odd | 2 | 1 | ||
144.10.a.n | 1 | 12.b | even | 2 | 1 | ||
200.10.a.b | 1 | 5.b | even | 2 | 1 | ||
200.10.c.b | 2 | 5.c | odd | 4 | 2 | ||
256.10.b.c | 2 | 16.f | odd | 4 | 2 | ||
256.10.b.i | 2 | 16.e | even | 4 | 2 | ||
392.10.a.b | 1 | 7.b | odd | 2 | 1 | ||
400.10.a.d | 1 | 20.d | odd | 2 | 1 | ||
400.10.c.g | 2 | 20.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 60 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(8))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 60 \)
$5$
\( T + 2074 \)
$7$
\( T + 4344 \)
$11$
\( T - 93644 \)
$13$
\( T + 12242 \)
$17$
\( T + 319598 \)
$19$
\( T + 553516 \)
$23$
\( T + 712936 \)
$29$
\( T - 2075838 \)
$31$
\( T + 6420448 \)
$37$
\( T + 18197754 \)
$41$
\( T - 9033834 \)
$43$
\( T - 19594732 \)
$47$
\( T + 18484176 \)
$53$
\( T - 10255766 \)
$59$
\( T - 121666556 \)
$61$
\( T + 45948962 \)
$67$
\( T - 50535428 \)
$71$
\( T - 267044680 \)
$73$
\( T + 176213366 \)
$79$
\( T + 269685680 \)
$83$
\( T + 227032556 \)
$89$
\( T - 72141594 \)
$97$
\( T - 228776546 \)
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