Newspace parameters
Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 48.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(36.8804726669\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 12) |
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 | 243.000 | 0 | 9990.00 | 0 | 86128.0 | 0 | 59049.0 | 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 | 48.12.a.i | 1 | |
3.b | odd | 2 | 1 | 144.12.a.a | 1 | ||
4.b | odd | 2 | 1 | 12.12.a.a | ✓ | 1 | |
8.b | even | 2 | 1 | 192.12.a.a | 1 | ||
8.d | odd | 2 | 1 | 192.12.a.k | 1 | ||
12.b | even | 2 | 1 | 36.12.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
12.12.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
36.12.a.a | 1 | 12.b | even | 2 | 1 | ||
48.12.a.i | 1 | 1.a | even | 1 | 1 | trivial | |
144.12.a.a | 1 | 3.b | odd | 2 | 1 | ||
192.12.a.a | 1 | 8.b | even | 2 | 1 | ||
192.12.a.k | 1 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 9990 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 243 \)
$5$
\( T - 9990 \)
$7$
\( T - 86128 \)
$11$
\( T - 806004 \)
$13$
\( T + 960250 \)
$17$
\( T + 4306878 \)
$19$
\( T + 401300 \)
$23$
\( T + 17751528 \)
$29$
\( T + 84704994 \)
$31$
\( T + 140930504 \)
$37$
\( T + 413506594 \)
$41$
\( T - 150094890 \)
$43$
\( T + 706702028 \)
$47$
\( T - 2475725472 \)
$53$
\( T - 1600124166 \)
$59$
\( T + 3945492396 \)
$61$
\( T + 885973498 \)
$67$
\( T - 4881597772 \)
$71$
\( T + 12631469400 \)
$73$
\( T - 1423335194 \)
$79$
\( T + 667407512 \)
$83$
\( T + 5716071828 \)
$89$
\( T + 85738736790 \)
$97$
\( T + 52302647806 \)
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