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: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 12) |
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 | 18.0000 | 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.c | 1 | |
3.b | odd | 2 | 1 | 1764.4.a.b | 1 | ||
4.b | odd | 2 | 1 | 2352.4.a.bk | 1 | ||
7.b | odd | 2 | 1 | 12.4.a.a | ✓ | 1 | |
7.c | even | 3 | 2 | 588.4.i.e | 2 | ||
7.d | odd | 6 | 2 | 588.4.i.d | 2 | ||
21.c | even | 2 | 1 | 36.4.a.a | 1 | ||
21.g | even | 6 | 2 | 1764.4.k.b | 2 | ||
21.h | odd | 6 | 2 | 1764.4.k.o | 2 | ||
28.d | even | 2 | 1 | 48.4.a.a | 1 | ||
35.c | odd | 2 | 1 | 300.4.a.b | 1 | ||
35.f | even | 4 | 2 | 300.4.d.e | 2 | ||
56.e | even | 2 | 1 | 192.4.a.l | 1 | ||
56.h | odd | 2 | 1 | 192.4.a.f | 1 | ||
63.l | odd | 6 | 2 | 324.4.e.h | 2 | ||
63.o | even | 6 | 2 | 324.4.e.a | 2 | ||
77.b | even | 2 | 1 | 1452.4.a.d | 1 | ||
84.h | odd | 2 | 1 | 144.4.a.g | 1 | ||
91.b | odd | 2 | 1 | 2028.4.a.c | 1 | ||
91.i | even | 4 | 2 | 2028.4.b.c | 2 | ||
105.g | even | 2 | 1 | 900.4.a.g | 1 | ||
105.k | odd | 4 | 2 | 900.4.d.c | 2 | ||
112.j | even | 4 | 2 | 768.4.d.j | 2 | ||
112.l | odd | 4 | 2 | 768.4.d.g | 2 | ||
140.c | even | 2 | 1 | 1200.4.a.be | 1 | ||
140.j | odd | 4 | 2 | 1200.4.f.d | 2 | ||
168.e | odd | 2 | 1 | 576.4.a.a | 1 | ||
168.i | even | 2 | 1 | 576.4.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
12.4.a.a | ✓ | 1 | 7.b | odd | 2 | 1 | |
36.4.a.a | 1 | 21.c | even | 2 | 1 | ||
48.4.a.a | 1 | 28.d | even | 2 | 1 | ||
144.4.a.g | 1 | 84.h | odd | 2 | 1 | ||
192.4.a.f | 1 | 56.h | odd | 2 | 1 | ||
192.4.a.l | 1 | 56.e | even | 2 | 1 | ||
300.4.a.b | 1 | 35.c | odd | 2 | 1 | ||
300.4.d.e | 2 | 35.f | even | 4 | 2 | ||
324.4.e.a | 2 | 63.o | even | 6 | 2 | ||
324.4.e.h | 2 | 63.l | odd | 6 | 2 | ||
576.4.a.a | 1 | 168.e | odd | 2 | 1 | ||
576.4.a.b | 1 | 168.i | even | 2 | 1 | ||
588.4.a.c | 1 | 1.a | even | 1 | 1 | trivial | |
588.4.i.d | 2 | 7.d | odd | 6 | 2 | ||
588.4.i.e | 2 | 7.c | even | 3 | 2 | ||
768.4.d.g | 2 | 112.l | odd | 4 | 2 | ||
768.4.d.j | 2 | 112.j | even | 4 | 2 | ||
900.4.a.g | 1 | 105.g | even | 2 | 1 | ||
900.4.d.c | 2 | 105.k | odd | 4 | 2 | ||
1200.4.a.be | 1 | 140.c | even | 2 | 1 | ||
1200.4.f.d | 2 | 140.j | odd | 4 | 2 | ||
1452.4.a.d | 1 | 77.b | even | 2 | 1 | ||
1764.4.a.b | 1 | 3.b | odd | 2 | 1 | ||
1764.4.k.b | 2 | 21.g | even | 6 | 2 | ||
1764.4.k.o | 2 | 21.h | odd | 6 | 2 | ||
2028.4.a.c | 1 | 91.b | odd | 2 | 1 | ||
2028.4.b.c | 2 | 91.i | even | 4 | 2 | ||
2352.4.a.bk | 1 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 18 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 3 \)
$5$
\( T - 18 \)
$7$
\( T \)
$11$
\( T - 36 \)
$13$
\( T - 10 \)
$17$
\( T + 18 \)
$19$
\( T - 100 \)
$23$
\( T - 72 \)
$29$
\( T + 234 \)
$31$
\( T - 16 \)
$37$
\( T + 226 \)
$41$
\( T + 90 \)
$43$
\( T - 452 \)
$47$
\( T + 432 \)
$53$
\( T - 414 \)
$59$
\( T - 684 \)
$61$
\( T + 422 \)
$67$
\( T - 332 \)
$71$
\( T + 360 \)
$73$
\( T + 26 \)
$79$
\( T - 512 \)
$83$
\( T - 1188 \)
$89$
\( T - 630 \)
$97$
\( T - 1054 \)
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