Newspace parameters
Level: | \( N \) | \(=\) | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 528.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(164.939293456\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 33) |
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 | −27.0000 | 0 | −410.000 | 0 | 1028.00 | 0 | 729.000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(1\) |
\(11\) | \(-1\) |
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 | 528.8.a.a | 1 | |
4.b | odd | 2 | 1 | 33.8.a.a | ✓ | 1 | |
12.b | even | 2 | 1 | 99.8.a.a | 1 | ||
44.c | even | 2 | 1 | 363.8.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.8.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
99.8.a.a | 1 | 12.b | even | 2 | 1 | ||
363.8.a.a | 1 | 44.c | even | 2 | 1 | ||
528.8.a.a | 1 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 410 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(528))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 27 \)
$5$
\( T + 410 \)
$7$
\( T - 1028 \)
$11$
\( T - 1331 \)
$13$
\( T - 12958 \)
$17$
\( T - 17062 \)
$19$
\( T - 54168 \)
$23$
\( T - 11488 \)
$29$
\( T + 186654 \)
$31$
\( T - 188672 \)
$37$
\( T - 395886 \)
$41$
\( T + 47546 \)
$43$
\( T + 602088 \)
$47$
\( T - 647200 \)
$53$
\( T + 1312722 \)
$59$
\( T - 2681140 \)
$61$
\( T - 551190 \)
$67$
\( T + 459260 \)
$71$
\( T - 18072 \)
$73$
\( T + 426062 \)
$79$
\( T + 297764 \)
$83$
\( T + 5684028 \)
$89$
\( T + 6342966 \)
$97$
\( T - 16651586 \)
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