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 3) |
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 | 6.00000 | 0 | 40.0000 | 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.a | 1 | |
3.b | odd | 2 | 1 | 144.6.a.f | 1 | ||
4.b | odd | 2 | 1 | 3.6.a.a | ✓ | 1 | |
8.b | even | 2 | 1 | 192.6.a.l | 1 | ||
8.d | odd | 2 | 1 | 192.6.a.d | 1 | ||
12.b | even | 2 | 1 | 9.6.a.a | 1 | ||
16.e | even | 4 | 2 | 768.6.d.h | 2 | ||
16.f | odd | 4 | 2 | 768.6.d.k | 2 | ||
20.d | odd | 2 | 1 | 75.6.a.e | 1 | ||
20.e | even | 4 | 2 | 75.6.b.b | 2 | ||
24.f | even | 2 | 1 | 576.6.a.s | 1 | ||
24.h | odd | 2 | 1 | 576.6.a.t | 1 | ||
28.d | even | 2 | 1 | 147.6.a.a | 1 | ||
28.f | even | 6 | 2 | 147.6.e.k | 2 | ||
28.g | odd | 6 | 2 | 147.6.e.h | 2 | ||
36.f | odd | 6 | 2 | 81.6.c.c | 2 | ||
36.h | even | 6 | 2 | 81.6.c.a | 2 | ||
44.c | even | 2 | 1 | 363.6.a.d | 1 | ||
52.b | odd | 2 | 1 | 507.6.a.b | 1 | ||
60.h | even | 2 | 1 | 225.6.a.a | 1 | ||
60.l | odd | 4 | 2 | 225.6.b.b | 2 | ||
68.d | odd | 2 | 1 | 867.6.a.a | 1 | ||
76.d | even | 2 | 1 | 1083.6.a.c | 1 | ||
84.h | odd | 2 | 1 | 441.6.a.i | 1 | ||
132.d | odd | 2 | 1 | 1089.6.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3.6.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
9.6.a.a | 1 | 12.b | even | 2 | 1 | ||
48.6.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
75.6.a.e | 1 | 20.d | odd | 2 | 1 | ||
75.6.b.b | 2 | 20.e | even | 4 | 2 | ||
81.6.c.a | 2 | 36.h | even | 6 | 2 | ||
81.6.c.c | 2 | 36.f | odd | 6 | 2 | ||
144.6.a.f | 1 | 3.b | odd | 2 | 1 | ||
147.6.a.a | 1 | 28.d | even | 2 | 1 | ||
147.6.e.h | 2 | 28.g | odd | 6 | 2 | ||
147.6.e.k | 2 | 28.f | even | 6 | 2 | ||
192.6.a.d | 1 | 8.d | odd | 2 | 1 | ||
192.6.a.l | 1 | 8.b | even | 2 | 1 | ||
225.6.a.a | 1 | 60.h | even | 2 | 1 | ||
225.6.b.b | 2 | 60.l | odd | 4 | 2 | ||
363.6.a.d | 1 | 44.c | even | 2 | 1 | ||
441.6.a.i | 1 | 84.h | odd | 2 | 1 | ||
507.6.a.b | 1 | 52.b | odd | 2 | 1 | ||
576.6.a.s | 1 | 24.f | even | 2 | 1 | ||
576.6.a.t | 1 | 24.h | odd | 2 | 1 | ||
768.6.d.h | 2 | 16.e | even | 4 | 2 | ||
768.6.d.k | 2 | 16.f | odd | 4 | 2 | ||
867.6.a.a | 1 | 68.d | odd | 2 | 1 | ||
1083.6.a.c | 1 | 76.d | even | 2 | 1 | ||
1089.6.a.b | 1 | 132.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 6 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 9 \)
$5$
\( T - 6 \)
$7$
\( T - 40 \)
$11$
\( T - 564 \)
$13$
\( T - 638 \)
$17$
\( T - 882 \)
$19$
\( T - 556 \)
$23$
\( T - 840 \)
$29$
\( T - 4638 \)
$31$
\( T + 4400 \)
$37$
\( T + 2410 \)
$41$
\( T + 6870 \)
$43$
\( T + 9644 \)
$47$
\( T - 18672 \)
$53$
\( T - 33750 \)
$59$
\( T - 18084 \)
$61$
\( T - 39758 \)
$67$
\( T - 23068 \)
$71$
\( T - 4248 \)
$73$
\( T + 41110 \)
$79$
\( T + 21920 \)
$83$
\( T + 82452 \)
$89$
\( T + 94086 \)
$97$
\( T - 49442 \)
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