Newspace parameters
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(8.49627504083\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 9) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-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 | 0 | 0 | −20.0000 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 144.4.a.d | 1 | |
3.b | odd | 2 | 1 | CM | 144.4.a.d | 1 | |
4.b | odd | 2 | 1 | 9.4.a.a | ✓ | 1 | |
8.b | even | 2 | 1 | 576.4.a.l | 1 | ||
8.d | odd | 2 | 1 | 576.4.a.m | 1 | ||
12.b | even | 2 | 1 | 9.4.a.a | ✓ | 1 | |
20.d | odd | 2 | 1 | 225.4.a.d | 1 | ||
20.e | even | 4 | 2 | 225.4.b.g | 2 | ||
24.f | even | 2 | 1 | 576.4.a.m | 1 | ||
24.h | odd | 2 | 1 | 576.4.a.l | 1 | ||
28.d | even | 2 | 1 | 441.4.a.f | 1 | ||
28.f | even | 6 | 2 | 441.4.e.j | 2 | ||
28.g | odd | 6 | 2 | 441.4.e.i | 2 | ||
36.f | odd | 6 | 2 | 81.4.c.b | 2 | ||
36.h | even | 6 | 2 | 81.4.c.b | 2 | ||
44.c | even | 2 | 1 | 1089.4.a.g | 1 | ||
52.b | odd | 2 | 1 | 1521.4.a.g | 1 | ||
60.h | even | 2 | 1 | 225.4.a.d | 1 | ||
60.l | odd | 4 | 2 | 225.4.b.g | 2 | ||
84.h | odd | 2 | 1 | 441.4.a.f | 1 | ||
84.j | odd | 6 | 2 | 441.4.e.j | 2 | ||
84.n | even | 6 | 2 | 441.4.e.i | 2 | ||
132.d | odd | 2 | 1 | 1089.4.a.g | 1 | ||
156.h | even | 2 | 1 | 1521.4.a.g | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
9.4.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
9.4.a.a | ✓ | 1 | 12.b | even | 2 | 1 | |
81.4.c.b | 2 | 36.f | odd | 6 | 2 | ||
81.4.c.b | 2 | 36.h | even | 6 | 2 | ||
144.4.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
144.4.a.d | 1 | 3.b | odd | 2 | 1 | CM | |
225.4.a.d | 1 | 20.d | odd | 2 | 1 | ||
225.4.a.d | 1 | 60.h | even | 2 | 1 | ||
225.4.b.g | 2 | 20.e | even | 4 | 2 | ||
225.4.b.g | 2 | 60.l | odd | 4 | 2 | ||
441.4.a.f | 1 | 28.d | even | 2 | 1 | ||
441.4.a.f | 1 | 84.h | odd | 2 | 1 | ||
441.4.e.i | 2 | 28.g | odd | 6 | 2 | ||
441.4.e.i | 2 | 84.n | even | 6 | 2 | ||
441.4.e.j | 2 | 28.f | even | 6 | 2 | ||
441.4.e.j | 2 | 84.j | odd | 6 | 2 | ||
576.4.a.l | 1 | 8.b | even | 2 | 1 | ||
576.4.a.l | 1 | 24.h | odd | 2 | 1 | ||
576.4.a.m | 1 | 8.d | odd | 2 | 1 | ||
576.4.a.m | 1 | 24.f | even | 2 | 1 | ||
1089.4.a.g | 1 | 44.c | even | 2 | 1 | ||
1089.4.a.g | 1 | 132.d | odd | 2 | 1 | ||
1521.4.a.g | 1 | 52.b | odd | 2 | 1 | ||
1521.4.a.g | 1 | 156.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(144))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T \)
$7$
\( T + 20 \)
$11$
\( T \)
$13$
\( T + 70 \)
$17$
\( T \)
$19$
\( T + 56 \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T + 308 \)
$37$
\( T - 110 \)
$41$
\( T \)
$43$
\( T - 520 \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T - 182 \)
$67$
\( T - 880 \)
$71$
\( T \)
$73$
\( T - 1190 \)
$79$
\( T + 884 \)
$83$
\( T \)
$89$
\( T \)
$97$
\( T + 1330 \)
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