Newspace parameters
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(110.641418001\) |
Analytic rank: | \(1\) |
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 | 0 | 0 | −1870.00 | 0 | 72312.0 | 0 | 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 | 144.12.a.g | 1 | |
3.b | odd | 2 | 1 | 48.12.a.b | 1 | ||
4.b | odd | 2 | 1 | 72.12.a.a | 1 | ||
12.b | even | 2 | 1 | 24.12.a.c | ✓ | 1 | |
24.f | even | 2 | 1 | 192.12.a.e | 1 | ||
24.h | odd | 2 | 1 | 192.12.a.o | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.12.a.c | ✓ | 1 | 12.b | even | 2 | 1 | |
48.12.a.b | 1 | 3.b | odd | 2 | 1 | ||
72.12.a.a | 1 | 4.b | odd | 2 | 1 | ||
144.12.a.g | 1 | 1.a | even | 1 | 1 | trivial | |
192.12.a.e | 1 | 24.f | even | 2 | 1 | ||
192.12.a.o | 1 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 1870 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(144))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T + 1870 \)
$7$
\( T - 72312 \)
$11$
\( T - 147940 \)
$13$
\( T + 1562858 \)
$17$
\( T - 145774 \)
$19$
\( T + 1096796 \)
$23$
\( T + 60014264 \)
$29$
\( T - 19626954 \)
$31$
\( T - 239950480 \)
$37$
\( T - 488238078 \)
$41$
\( T + 47066010 \)
$43$
\( T + 428866948 \)
$47$
\( T - 450903216 \)
$53$
\( T + 4336685950 \)
$59$
\( T + 8937556460 \)
$61$
\( T - 4673884486 \)
$67$
\( T + 7498937612 \)
$71$
\( T + 27032101480 \)
$73$
\( T - 11676141658 \)
$79$
\( T + 2478876544 \)
$83$
\( T - 42745596956 \)
$89$
\( T - 93270772662 \)
$97$
\( T - 118032786914 \)
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