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: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 6) |
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 | −5766.00 | 0 | −72464.0 | 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.b | 1 | |
3.b | odd | 2 | 1 | 48.12.a.h | 1 | ||
4.b | odd | 2 | 1 | 18.12.a.c | 1 | ||
12.b | even | 2 | 1 | 6.12.a.a | ✓ | 1 | |
24.f | even | 2 | 1 | 192.12.a.l | 1 | ||
24.h | odd | 2 | 1 | 192.12.a.b | 1 | ||
36.f | odd | 6 | 2 | 162.12.c.d | 2 | ||
36.h | even | 6 | 2 | 162.12.c.g | 2 | ||
60.h | even | 2 | 1 | 150.12.a.g | 1 | ||
60.l | odd | 4 | 2 | 150.12.c.f | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6.12.a.a | ✓ | 1 | 12.b | even | 2 | 1 | |
18.12.a.c | 1 | 4.b | odd | 2 | 1 | ||
48.12.a.h | 1 | 3.b | odd | 2 | 1 | ||
144.12.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
150.12.a.g | 1 | 60.h | even | 2 | 1 | ||
150.12.c.f | 2 | 60.l | odd | 4 | 2 | ||
162.12.c.d | 2 | 36.f | odd | 6 | 2 | ||
162.12.c.g | 2 | 36.h | even | 6 | 2 | ||
192.12.a.b | 1 | 24.h | odd | 2 | 1 | ||
192.12.a.l | 1 | 24.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 5766 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(144))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T + 5766 \)
$7$
\( T + 72464 \)
$11$
\( T + 408948 \)
$13$
\( T - 1367558 \)
$17$
\( T + 5422914 \)
$19$
\( T + 15166100 \)
$23$
\( T + 52194072 \)
$29$
\( T + 118581150 \)
$31$
\( T - 57652408 \)
$37$
\( T + 375985186 \)
$41$
\( T + 856316202 \)
$43$
\( T - 1245189172 \)
$47$
\( T + 1306762656 \)
$53$
\( T + 409556358 \)
$59$
\( T + 2882866260 \)
$61$
\( T - 5731767302 \)
$67$
\( T + 3893272244 \)
$71$
\( T + 9075890088 \)
$73$
\( T + 15571822822 \)
$79$
\( T - 30196762600 \)
$83$
\( T - 23135252628 \)
$89$
\( T - 25614819990 \)
$97$
\( T + 61937553406 \)
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