Newspace parameters
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(51.3228223402\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 40) |
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 | −2.00000 | 0 | 25.0000 | 0 | 62.0000 | 0 | −239.000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(-1\) |
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 | 320.6.a.h | 1 | |
4.b | odd | 2 | 1 | 320.6.a.i | 1 | ||
8.b | even | 2 | 1 | 80.6.a.d | 1 | ||
8.d | odd | 2 | 1 | 40.6.a.c | ✓ | 1 | |
24.f | even | 2 | 1 | 360.6.a.f | 1 | ||
24.h | odd | 2 | 1 | 720.6.a.t | 1 | ||
40.e | odd | 2 | 1 | 200.6.a.b | 1 | ||
40.f | even | 2 | 1 | 400.6.a.h | 1 | ||
40.i | odd | 4 | 2 | 400.6.c.k | 2 | ||
40.k | even | 4 | 2 | 200.6.c.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
40.6.a.c | ✓ | 1 | 8.d | odd | 2 | 1 | |
80.6.a.d | 1 | 8.b | even | 2 | 1 | ||
200.6.a.b | 1 | 40.e | odd | 2 | 1 | ||
200.6.c.d | 2 | 40.k | even | 4 | 2 | ||
320.6.a.h | 1 | 1.a | even | 1 | 1 | trivial | |
320.6.a.i | 1 | 4.b | odd | 2 | 1 | ||
360.6.a.f | 1 | 24.f | even | 2 | 1 | ||
400.6.a.h | 1 | 40.f | even | 2 | 1 | ||
400.6.c.k | 2 | 40.i | odd | 4 | 2 | ||
720.6.a.t | 1 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 2 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 2 \)
$5$
\( T - 25 \)
$7$
\( T - 62 \)
$11$
\( T + 144 \)
$13$
\( T - 654 \)
$17$
\( T + 1190 \)
$19$
\( T - 556 \)
$23$
\( T + 2182 \)
$29$
\( T - 1578 \)
$31$
\( T + 9660 \)
$37$
\( T - 3534 \)
$41$
\( T - 7462 \)
$43$
\( T + 7114 \)
$47$
\( T - 28294 \)
$53$
\( T - 13046 \)
$59$
\( T + 37092 \)
$61$
\( T + 39570 \)
$67$
\( T + 56734 \)
$71$
\( T + 45588 \)
$73$
\( T - 11842 \)
$79$
\( T + 94216 \)
$83$
\( T + 31482 \)
$89$
\( T + 94054 \)
$97$
\( T - 23714 \)
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