Newspace parameters
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.83209168028\) |
| 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 | −3.00000 | 0 | 14.0000 | 0 | 24.0000 | 0 | 9.00000 | 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 | 48.4.a.b | 1 | |
| 3.b | odd | 2 | 1 | 144.4.a.b | 1 | ||
| 4.b | odd | 2 | 1 | 24.4.a.a | ✓ | 1 | |
| 5.b | even | 2 | 1 | 1200.4.a.u | 1 | ||
| 5.c | odd | 4 | 2 | 1200.4.f.p | 2 | ||
| 7.b | odd | 2 | 1 | 2352.4.a.w | 1 | ||
| 8.b | even | 2 | 1 | 192.4.a.g | 1 | ||
| 8.d | odd | 2 | 1 | 192.4.a.a | 1 | ||
| 12.b | even | 2 | 1 | 72.4.a.b | 1 | ||
| 16.e | even | 4 | 2 | 768.4.d.b | 2 | ||
| 16.f | odd | 4 | 2 | 768.4.d.o | 2 | ||
| 20.d | odd | 2 | 1 | 600.4.a.h | 1 | ||
| 20.e | even | 4 | 2 | 600.4.f.b | 2 | ||
| 24.f | even | 2 | 1 | 576.4.a.u | 1 | ||
| 24.h | odd | 2 | 1 | 576.4.a.v | 1 | ||
| 28.d | even | 2 | 1 | 1176.4.a.a | 1 | ||
| 36.f | odd | 6 | 2 | 648.4.i.b | 2 | ||
| 36.h | even | 6 | 2 | 648.4.i.k | 2 | ||
| 60.h | even | 2 | 1 | 1800.4.a.bg | 1 | ||
| 60.l | odd | 4 | 2 | 1800.4.f.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.4.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
| 48.4.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
| 72.4.a.b | 1 | 12.b | even | 2 | 1 | ||
| 144.4.a.b | 1 | 3.b | odd | 2 | 1 | ||
| 192.4.a.a | 1 | 8.d | odd | 2 | 1 | ||
| 192.4.a.g | 1 | 8.b | even | 2 | 1 | ||
| 576.4.a.u | 1 | 24.f | even | 2 | 1 | ||
| 576.4.a.v | 1 | 24.h | odd | 2 | 1 | ||
| 600.4.a.h | 1 | 20.d | odd | 2 | 1 | ||
| 600.4.f.b | 2 | 20.e | even | 4 | 2 | ||
| 648.4.i.b | 2 | 36.f | odd | 6 | 2 | ||
| 648.4.i.k | 2 | 36.h | even | 6 | 2 | ||
| 768.4.d.b | 2 | 16.e | even | 4 | 2 | ||
| 768.4.d.o | 2 | 16.f | odd | 4 | 2 | ||
| 1176.4.a.a | 1 | 28.d | even | 2 | 1 | ||
| 1200.4.a.u | 1 | 5.b | even | 2 | 1 | ||
| 1200.4.f.p | 2 | 5.c | odd | 4 | 2 | ||
| 1800.4.a.bg | 1 | 60.h | even | 2 | 1 | ||
| 1800.4.f.q | 2 | 60.l | odd | 4 | 2 | ||
| 2352.4.a.w | 1 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 14 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 3 \)
$5$
\( T - 14 \)
$7$
\( T - 24 \)
$11$
\( T - 28 \)
$13$
\( T + 74 \)
$17$
\( T - 82 \)
$19$
\( T + 92 \)
$23$
\( T + 8 \)
$29$
\( T + 138 \)
$31$
\( T + 80 \)
$37$
\( T - 30 \)
$41$
\( T - 282 \)
$43$
\( T + 4 \)
$47$
\( T + 240 \)
$53$
\( T + 130 \)
$59$
\( T + 596 \)
$61$
\( T + 218 \)
$67$
\( T - 436 \)
$71$
\( T + 856 \)
$73$
\( T + 998 \)
$79$
\( T - 32 \)
$83$
\( T - 1508 \)
$89$
\( T + 246 \)
$97$
\( T - 866 \)
show more
show less