Newspace parameters
Level: | \( N \) | \(=\) | \( 704 = 2^{6} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 704.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(112.910209148\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 11) |
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 | −15.0000 | 0 | 19.0000 | 0 | −10.0000 | 0 | −18.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(11\) | \(1\) |
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 | 704.6.a.c | 1 | |
4.b | odd | 2 | 1 | 704.6.a.h | 1 | ||
8.b | even | 2 | 1 | 176.6.a.c | 1 | ||
8.d | odd | 2 | 1 | 11.6.a.a | ✓ | 1 | |
24.f | even | 2 | 1 | 99.6.a.c | 1 | ||
40.e | odd | 2 | 1 | 275.6.a.a | 1 | ||
40.k | even | 4 | 2 | 275.6.b.a | 2 | ||
56.e | even | 2 | 1 | 539.6.a.c | 1 | ||
88.g | even | 2 | 1 | 121.6.a.b | 1 | ||
264.p | odd | 2 | 1 | 1089.6.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.6.a.a | ✓ | 1 | 8.d | odd | 2 | 1 | |
99.6.a.c | 1 | 24.f | even | 2 | 1 | ||
121.6.a.b | 1 | 88.g | even | 2 | 1 | ||
176.6.a.c | 1 | 8.b | even | 2 | 1 | ||
275.6.a.a | 1 | 40.e | odd | 2 | 1 | ||
275.6.b.a | 2 | 40.k | even | 4 | 2 | ||
539.6.a.c | 1 | 56.e | even | 2 | 1 | ||
704.6.a.c | 1 | 1.a | even | 1 | 1 | trivial | |
704.6.a.h | 1 | 4.b | odd | 2 | 1 | ||
1089.6.a.c | 1 | 264.p | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 15 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 15 \)
$5$
\( T - 19 \)
$7$
\( T + 10 \)
$11$
\( T + 121 \)
$13$
\( T - 1148 \)
$17$
\( T - 686 \)
$19$
\( T + 384 \)
$23$
\( T + 3709 \)
$29$
\( T - 5424 \)
$31$
\( T - 6443 \)
$37$
\( T + 12063 \)
$41$
\( T + 1528 \)
$43$
\( T + 4026 \)
$47$
\( T + 7168 \)
$53$
\( T - 29862 \)
$59$
\( T + 6461 \)
$61$
\( T - 16980 \)
$67$
\( T - 29999 \)
$71$
\( T + 31023 \)
$73$
\( T - 1924 \)
$79$
\( T + 65138 \)
$83$
\( T + 102714 \)
$89$
\( T - 17415 \)
$97$
\( T - 66905 \)
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