Newspace parameters
Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
Weight: | \( k \) | \(=\) | \( 15 \) |
Character orbit: | \([\chi]\) | \(=\) | 8.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(9.94631745215\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(7\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 |
|
−128.000 | 3022.00 | 16384.0 | 0 | −386816. | 0 | −2.09715e6 | 4.34952e6 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8.15.d.a | ✓ | 1 |
3.b | odd | 2 | 1 | 72.15.b.a | 1 | ||
4.b | odd | 2 | 1 | 32.15.d.a | 1 | ||
8.b | even | 2 | 1 | 32.15.d.a | 1 | ||
8.d | odd | 2 | 1 | CM | 8.15.d.a | ✓ | 1 |
24.f | even | 2 | 1 | 72.15.b.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.15.d.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
8.15.d.a | ✓ | 1 | 8.d | odd | 2 | 1 | CM |
32.15.d.a | 1 | 4.b | odd | 2 | 1 | ||
32.15.d.a | 1 | 8.b | even | 2 | 1 | ||
72.15.b.a | 1 | 3.b | odd | 2 | 1 | ||
72.15.b.a | 1 | 24.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 3022 \)
acting on \(S_{15}^{\mathrm{new}}(8, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 128 \)
$3$
\( T - 3022 \)
$5$
\( T \)
$7$
\( T \)
$11$
\( T - 38712254 \)
$13$
\( T \)
$17$
\( T + 328636222 \)
$19$
\( T - 1778973806 \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T + 333393570766 \)
$43$
\( T + 495012562114 \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T + 3914494552162 \)
$61$
\( T \)
$67$
\( T - 2711103884558 \)
$71$
\( T \)
$73$
\( T - 1579402558802 \)
$79$
\( T \)
$83$
\( T + 31146255762898 \)
$89$
\( T + 38433671549134 \)
$97$
\( T + 62815034524126 \)
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