Newspace parameters
Level: | \( N \) | \(=\) | \( 578 = 2 \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 578.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.61535323683\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 34) |
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 |
|
1.00000 | 2.00000 | 1.00000 | 0 | 2.00000 | 4.00000 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(17\) | \(1\) |
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 | 578.2.a.a | 1 | |
3.b | odd | 2 | 1 | 5202.2.a.d | 1 | ||
4.b | odd | 2 | 1 | 4624.2.a.a | 1 | ||
17.b | even | 2 | 1 | 34.2.a.a | ✓ | 1 | |
17.c | even | 4 | 2 | 578.2.b.a | 2 | ||
17.d | even | 8 | 4 | 578.2.c.e | 4 | ||
17.e | odd | 16 | 8 | 578.2.d.e | 8 | ||
51.c | odd | 2 | 1 | 306.2.a.a | 1 | ||
68.d | odd | 2 | 1 | 272.2.a.d | 1 | ||
85.c | even | 2 | 1 | 850.2.a.e | 1 | ||
85.g | odd | 4 | 2 | 850.2.c.b | 2 | ||
119.d | odd | 2 | 1 | 1666.2.a.m | 1 | ||
136.e | odd | 2 | 1 | 1088.2.a.d | 1 | ||
136.h | even | 2 | 1 | 1088.2.a.l | 1 | ||
187.b | odd | 2 | 1 | 4114.2.a.a | 1 | ||
204.h | even | 2 | 1 | 2448.2.a.k | 1 | ||
221.b | even | 2 | 1 | 5746.2.a.b | 1 | ||
255.h | odd | 2 | 1 | 7650.2.a.ci | 1 | ||
340.d | odd | 2 | 1 | 6800.2.a.b | 1 | ||
408.b | odd | 2 | 1 | 9792.2.a.y | 1 | ||
408.h | even | 2 | 1 | 9792.2.a.bj | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
34.2.a.a | ✓ | 1 | 17.b | even | 2 | 1 | |
272.2.a.d | 1 | 68.d | odd | 2 | 1 | ||
306.2.a.a | 1 | 51.c | odd | 2 | 1 | ||
578.2.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
578.2.b.a | 2 | 17.c | even | 4 | 2 | ||
578.2.c.e | 4 | 17.d | even | 8 | 4 | ||
578.2.d.e | 8 | 17.e | odd | 16 | 8 | ||
850.2.a.e | 1 | 85.c | even | 2 | 1 | ||
850.2.c.b | 2 | 85.g | odd | 4 | 2 | ||
1088.2.a.d | 1 | 136.e | odd | 2 | 1 | ||
1088.2.a.l | 1 | 136.h | even | 2 | 1 | ||
1666.2.a.m | 1 | 119.d | odd | 2 | 1 | ||
2448.2.a.k | 1 | 204.h | even | 2 | 1 | ||
4114.2.a.a | 1 | 187.b | odd | 2 | 1 | ||
4624.2.a.a | 1 | 4.b | odd | 2 | 1 | ||
5202.2.a.d | 1 | 3.b | odd | 2 | 1 | ||
5746.2.a.b | 1 | 221.b | even | 2 | 1 | ||
6800.2.a.b | 1 | 340.d | odd | 2 | 1 | ||
7650.2.a.ci | 1 | 255.h | odd | 2 | 1 | ||
9792.2.a.y | 1 | 408.b | odd | 2 | 1 | ||
9792.2.a.bj | 1 | 408.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(578))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 1 \)
$3$
\( T - 2 \)
$5$
\( T \)
$7$
\( T - 4 \)
$11$
\( T + 6 \)
$13$
\( T - 2 \)
$17$
\( T \)
$19$
\( T + 4 \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T - 4 \)
$37$
\( T - 4 \)
$41$
\( T + 6 \)
$43$
\( T - 8 \)
$47$
\( T \)
$53$
\( T + 6 \)
$59$
\( T \)
$61$
\( T - 4 \)
$67$
\( T - 8 \)
$71$
\( T \)
$73$
\( T + 2 \)
$79$
\( T + 8 \)
$83$
\( T \)
$89$
\( T + 6 \)
$97$
\( T + 14 \)
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