Newspace parameters
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.1739635558\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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 | 252.000 | 0 | −4830.00 | 0 | 16744.0 | 0 | −113643. | 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 | 64.12.a.f | 1 | |
| 4.b | odd | 2 | 1 | 64.12.a.b | 1 | ||
| 8.b | even | 2 | 1 | 16.12.a.a | 1 | ||
| 8.d | odd | 2 | 1 | 1.12.a.a | ✓ | 1 | |
| 16.e | even | 4 | 2 | 256.12.b.c | 2 | ||
| 16.f | odd | 4 | 2 | 256.12.b.e | 2 | ||
| 24.f | even | 2 | 1 | 9.12.a.b | 1 | ||
| 24.h | odd | 2 | 1 | 144.12.a.d | 1 | ||
| 40.e | odd | 2 | 1 | 25.12.a.b | 1 | ||
| 40.k | even | 4 | 2 | 25.12.b.b | 2 | ||
| 56.e | even | 2 | 1 | 49.12.a.a | 1 | ||
| 56.k | odd | 6 | 2 | 49.12.c.b | 2 | ||
| 56.m | even | 6 | 2 | 49.12.c.c | 2 | ||
| 72.l | even | 6 | 2 | 81.12.c.b | 2 | ||
| 72.p | odd | 6 | 2 | 81.12.c.d | 2 | ||
| 88.g | even | 2 | 1 | 121.12.a.b | 1 | ||
| 104.h | odd | 2 | 1 | 169.12.a.a | 1 | ||
| 120.m | even | 2 | 1 | 225.12.a.b | 1 | ||
| 120.q | odd | 4 | 2 | 225.12.b.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.12.a.a | ✓ | 1 | 8.d | odd | 2 | 1 | |
| 9.12.a.b | 1 | 24.f | even | 2 | 1 | ||
| 16.12.a.a | 1 | 8.b | even | 2 | 1 | ||
| 25.12.a.b | 1 | 40.e | odd | 2 | 1 | ||
| 25.12.b.b | 2 | 40.k | even | 4 | 2 | ||
| 49.12.a.a | 1 | 56.e | even | 2 | 1 | ||
| 49.12.c.b | 2 | 56.k | odd | 6 | 2 | ||
| 49.12.c.c | 2 | 56.m | even | 6 | 2 | ||
| 64.12.a.b | 1 | 4.b | odd | 2 | 1 | ||
| 64.12.a.f | 1 | 1.a | even | 1 | 1 | trivial | |
| 81.12.c.b | 2 | 72.l | even | 6 | 2 | ||
| 81.12.c.d | 2 | 72.p | odd | 6 | 2 | ||
| 121.12.a.b | 1 | 88.g | even | 2 | 1 | ||
| 144.12.a.d | 1 | 24.h | odd | 2 | 1 | ||
| 169.12.a.a | 1 | 104.h | odd | 2 | 1 | ||
| 225.12.a.b | 1 | 120.m | even | 2 | 1 | ||
| 225.12.b.d | 2 | 120.q | odd | 4 | 2 | ||
| 256.12.b.c | 2 | 16.e | even | 4 | 2 | ||
| 256.12.b.e | 2 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 252 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 252 \)
$5$
\( T + 4830 \)
$7$
\( T - 16744 \)
$11$
\( T - 534612 \)
$13$
\( T - 577738 \)
$17$
\( T + 6905934 \)
$19$
\( T - 10661420 \)
$23$
\( T + 18643272 \)
$29$
\( T + 128406630 \)
$31$
\( T - 52843168 \)
$37$
\( T - 182213314 \)
$41$
\( T - 308120442 \)
$43$
\( T + 17125708 \)
$47$
\( T + 2687348496 \)
$53$
\( T - 1596055698 \)
$59$
\( T + 5189203740 \)
$61$
\( T + 6956478662 \)
$67$
\( T + 15481826884 \)
$71$
\( T + 9791485272 \)
$73$
\( T - 1463791322 \)
$79$
\( T + 38116845680 \)
$83$
\( T + 29335099668 \)
$89$
\( T + 24992917110 \)
$97$
\( T - 75013568546 \)
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