Newspace parameters
Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 48.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(7.69842335102\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 24) |
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 | 9.00000 | 0 | −34.0000 | 0 | 240.000 | 0 | 81.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(3\) | \(-1\) |
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 | 48.6.a.d | 1 | |
3.b | odd | 2 | 1 | 144.6.a.i | 1 | ||
4.b | odd | 2 | 1 | 24.6.a.a | ✓ | 1 | |
8.b | even | 2 | 1 | 192.6.a.f | 1 | ||
8.d | odd | 2 | 1 | 192.6.a.n | 1 | ||
12.b | even | 2 | 1 | 72.6.a.e | 1 | ||
16.e | even | 4 | 2 | 768.6.d.a | 2 | ||
16.f | odd | 4 | 2 | 768.6.d.r | 2 | ||
20.d | odd | 2 | 1 | 600.6.a.i | 1 | ||
20.e | even | 4 | 2 | 600.6.f.f | 2 | ||
24.f | even | 2 | 1 | 576.6.a.k | 1 | ||
24.h | odd | 2 | 1 | 576.6.a.l | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.6.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
48.6.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
72.6.a.e | 1 | 12.b | even | 2 | 1 | ||
144.6.a.i | 1 | 3.b | odd | 2 | 1 | ||
192.6.a.f | 1 | 8.b | even | 2 | 1 | ||
192.6.a.n | 1 | 8.d | odd | 2 | 1 | ||
576.6.a.k | 1 | 24.f | even | 2 | 1 | ||
576.6.a.l | 1 | 24.h | odd | 2 | 1 | ||
600.6.a.i | 1 | 20.d | odd | 2 | 1 | ||
600.6.f.f | 2 | 20.e | even | 4 | 2 | ||
768.6.d.a | 2 | 16.e | even | 4 | 2 | ||
768.6.d.r | 2 | 16.f | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 34 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 9 \)
$5$
\( T + 34 \)
$7$
\( T - 240 \)
$11$
\( T - 124 \)
$13$
\( T - 46 \)
$17$
\( T - 1954 \)
$19$
\( T - 1924 \)
$23$
\( T + 2840 \)
$29$
\( T + 8922 \)
$31$
\( T - 4648 \)
$37$
\( T + 4362 \)
$41$
\( T + 2886 \)
$43$
\( T + 11332 \)
$47$
\( T + 7008 \)
$53$
\( T + 22594 \)
$59$
\( T - 28 \)
$61$
\( T + 6386 \)
$67$
\( T - 39076 \)
$71$
\( T - 54872 \)
$73$
\( T - 21034 \)
$79$
\( T + 26632 \)
$83$
\( T + 56188 \)
$89$
\( T - 64410 \)
$97$
\( T + 116158 \)
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