Newspace parameters
Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 64.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(10.2645644680\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 8) |
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 | 20.0000 | 0 | 74.0000 | 0 | 24.0000 | 0 | 157.000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
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 | 64.6.a.g | 1 | |
3.b | odd | 2 | 1 | 576.6.a.h | 1 | ||
4.b | odd | 2 | 1 | 64.6.a.a | 1 | ||
8.b | even | 2 | 1 | 16.6.a.a | 1 | ||
8.d | odd | 2 | 1 | 8.6.a.a | ✓ | 1 | |
12.b | even | 2 | 1 | 576.6.a.g | 1 | ||
16.e | even | 4 | 2 | 256.6.b.d | 2 | ||
16.f | odd | 4 | 2 | 256.6.b.f | 2 | ||
24.f | even | 2 | 1 | 72.6.a.f | 1 | ||
24.h | odd | 2 | 1 | 144.6.a.k | 1 | ||
40.e | odd | 2 | 1 | 200.6.a.a | 1 | ||
40.f | even | 2 | 1 | 400.6.a.l | 1 | ||
40.i | odd | 4 | 2 | 400.6.c.d | 2 | ||
40.k | even | 4 | 2 | 200.6.c.a | 2 | ||
56.e | even | 2 | 1 | 392.6.a.b | 1 | ||
56.h | odd | 2 | 1 | 784.6.a.l | 1 | ||
56.k | odd | 6 | 2 | 392.6.i.b | 2 | ||
56.m | even | 6 | 2 | 392.6.i.e | 2 | ||
88.g | even | 2 | 1 | 968.6.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.6.a.a | ✓ | 1 | 8.d | odd | 2 | 1 | |
16.6.a.a | 1 | 8.b | even | 2 | 1 | ||
64.6.a.a | 1 | 4.b | odd | 2 | 1 | ||
64.6.a.g | 1 | 1.a | even | 1 | 1 | trivial | |
72.6.a.f | 1 | 24.f | even | 2 | 1 | ||
144.6.a.k | 1 | 24.h | odd | 2 | 1 | ||
200.6.a.a | 1 | 40.e | odd | 2 | 1 | ||
200.6.c.a | 2 | 40.k | even | 4 | 2 | ||
256.6.b.d | 2 | 16.e | even | 4 | 2 | ||
256.6.b.f | 2 | 16.f | odd | 4 | 2 | ||
392.6.a.b | 1 | 56.e | even | 2 | 1 | ||
392.6.i.b | 2 | 56.k | odd | 6 | 2 | ||
392.6.i.e | 2 | 56.m | even | 6 | 2 | ||
400.6.a.l | 1 | 40.f | even | 2 | 1 | ||
400.6.c.d | 2 | 40.i | odd | 4 | 2 | ||
576.6.a.g | 1 | 12.b | even | 2 | 1 | ||
576.6.a.h | 1 | 3.b | odd | 2 | 1 | ||
784.6.a.l | 1 | 56.h | odd | 2 | 1 | ||
968.6.a.a | 1 | 88.g | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 20 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 20 \)
$5$
\( T - 74 \)
$7$
\( T - 24 \)
$11$
\( T - 124 \)
$13$
\( T + 478 \)
$17$
\( T + 1198 \)
$19$
\( T - 3044 \)
$23$
\( T + 184 \)
$29$
\( T - 3282 \)
$31$
\( T - 5728 \)
$37$
\( T + 10326 \)
$41$
\( T + 8886 \)
$43$
\( T + 9188 \)
$47$
\( T + 23664 \)
$53$
\( T + 11686 \)
$59$
\( T - 16876 \)
$61$
\( T - 18482 \)
$67$
\( T + 15532 \)
$71$
\( T - 31960 \)
$73$
\( T + 4886 \)
$79$
\( T + 44560 \)
$83$
\( T - 67364 \)
$89$
\( T - 71994 \)
$97$
\( T - 48866 \)
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