Newspace parameters
Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(34.6931230834\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 84) |
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 | 3.00000 | 0 | −6.00000 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(-1\) |
\(7\) | \(-1\) |
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 | 588.4.a.d | 1 | |
3.b | odd | 2 | 1 | 1764.4.a.j | 1 | ||
4.b | odd | 2 | 1 | 2352.4.a.d | 1 | ||
7.b | odd | 2 | 1 | 84.4.a.a | ✓ | 1 | |
7.c | even | 3 | 2 | 588.4.i.c | 2 | ||
7.d | odd | 6 | 2 | 588.4.i.f | 2 | ||
21.c | even | 2 | 1 | 252.4.a.b | 1 | ||
21.g | even | 6 | 2 | 1764.4.k.l | 2 | ||
21.h | odd | 6 | 2 | 1764.4.k.f | 2 | ||
28.d | even | 2 | 1 | 336.4.a.k | 1 | ||
35.c | odd | 2 | 1 | 2100.4.a.l | 1 | ||
35.f | even | 4 | 2 | 2100.4.k.j | 2 | ||
56.e | even | 2 | 1 | 1344.4.a.d | 1 | ||
56.h | odd | 2 | 1 | 1344.4.a.q | 1 | ||
84.h | odd | 2 | 1 | 1008.4.a.h | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
84.4.a.a | ✓ | 1 | 7.b | odd | 2 | 1 | |
252.4.a.b | 1 | 21.c | even | 2 | 1 | ||
336.4.a.k | 1 | 28.d | even | 2 | 1 | ||
588.4.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
588.4.i.c | 2 | 7.c | even | 3 | 2 | ||
588.4.i.f | 2 | 7.d | odd | 6 | 2 | ||
1008.4.a.h | 1 | 84.h | odd | 2 | 1 | ||
1344.4.a.d | 1 | 56.e | even | 2 | 1 | ||
1344.4.a.q | 1 | 56.h | odd | 2 | 1 | ||
1764.4.a.j | 1 | 3.b | odd | 2 | 1 | ||
1764.4.k.f | 2 | 21.h | odd | 6 | 2 | ||
1764.4.k.l | 2 | 21.g | even | 6 | 2 | ||
2100.4.a.l | 1 | 35.c | odd | 2 | 1 | ||
2100.4.k.j | 2 | 35.f | even | 4 | 2 | ||
2352.4.a.d | 1 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 6 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 3 \)
$5$
\( T + 6 \)
$7$
\( T \)
$11$
\( T - 36 \)
$13$
\( T + 62 \)
$17$
\( T + 114 \)
$19$
\( T - 76 \)
$23$
\( T + 24 \)
$29$
\( T - 54 \)
$31$
\( T - 112 \)
$37$
\( T + 178 \)
$41$
\( T + 378 \)
$43$
\( T + 172 \)
$47$
\( T - 192 \)
$53$
\( T + 402 \)
$59$
\( T + 396 \)
$61$
\( T + 254 \)
$67$
\( T + 1012 \)
$71$
\( T - 840 \)
$73$
\( T + 890 \)
$79$
\( T - 80 \)
$83$
\( T - 108 \)
$89$
\( T - 1638 \)
$97$
\( T + 1010 \)
show more
show less