Newspace parameters
Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 64.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(6.61567763737\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 4) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Expression as an eta quotient
\(f(z) = \dfrac{\eta(8z)^{38}}{\eta(4z)^{14}\eta(16z)^{14}}=q\prod_{n=1}^\infty(1 - q^{4n})^{-14}(1 - q^{8n})^{38}(1 - q^{16n})^{-14}\)
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(63\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
63.1 |
|
0 | 0 | 0 | 14.0000 | 0 | 0 | 0 | 81.0000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 64.5.c.a | 1 | |
3.b | odd | 2 | 1 | 576.5.g.b | 1 | ||
4.b | odd | 2 | 1 | CM | 64.5.c.a | 1 | |
8.b | even | 2 | 1 | 4.5.b.a | ✓ | 1 | |
8.d | odd | 2 | 1 | 4.5.b.a | ✓ | 1 | |
12.b | even | 2 | 1 | 576.5.g.b | 1 | ||
16.e | even | 4 | 2 | 256.5.d.c | 2 | ||
16.f | odd | 4 | 2 | 256.5.d.c | 2 | ||
24.f | even | 2 | 1 | 36.5.d.a | 1 | ||
24.h | odd | 2 | 1 | 36.5.d.a | 1 | ||
40.e | odd | 2 | 1 | 100.5.b.a | 1 | ||
40.f | even | 2 | 1 | 100.5.b.a | 1 | ||
40.i | odd | 4 | 2 | 100.5.d.a | 2 | ||
40.k | even | 4 | 2 | 100.5.d.a | 2 | ||
56.e | even | 2 | 1 | 196.5.c.a | 1 | ||
56.h | odd | 2 | 1 | 196.5.c.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.5.b.a | ✓ | 1 | 8.b | even | 2 | 1 | |
4.5.b.a | ✓ | 1 | 8.d | odd | 2 | 1 | |
36.5.d.a | 1 | 24.f | even | 2 | 1 | ||
36.5.d.a | 1 | 24.h | odd | 2 | 1 | ||
64.5.c.a | 1 | 1.a | even | 1 | 1 | trivial | |
64.5.c.a | 1 | 4.b | odd | 2 | 1 | CM | |
100.5.b.a | 1 | 40.e | odd | 2 | 1 | ||
100.5.b.a | 1 | 40.f | even | 2 | 1 | ||
100.5.d.a | 2 | 40.i | odd | 4 | 2 | ||
100.5.d.a | 2 | 40.k | even | 4 | 2 | ||
196.5.c.a | 1 | 56.e | even | 2 | 1 | ||
196.5.c.a | 1 | 56.h | odd | 2 | 1 | ||
256.5.d.c | 2 | 16.e | even | 4 | 2 | ||
256.5.d.c | 2 | 16.f | odd | 4 | 2 | ||
576.5.g.b | 1 | 3.b | odd | 2 | 1 | ||
576.5.g.b | 1 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{5}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T - 14 \)
$7$
\( T \)
$11$
\( T \)
$13$
\( T - 238 \)
$17$
\( T - 322 \)
$19$
\( T \)
$23$
\( T \)
$29$
\( T + 82 \)
$31$
\( T \)
$37$
\( T + 2162 \)
$41$
\( T + 3038 \)
$43$
\( T \)
$47$
\( T \)
$53$
\( T + 2482 \)
$59$
\( T \)
$61$
\( T - 6958 \)
$67$
\( T \)
$71$
\( T \)
$73$
\( T - 1442 \)
$79$
\( T \)
$83$
\( T \)
$89$
\( T + 9758 \)
$97$
\( T + 1918 \)
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