Newspace parameters
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.24813752041\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 | −16.0000 | 0 | −12.0000 | 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 | 72.4.a.a | ✓ | 1 |
3.b | odd | 2 | 1 | 72.4.a.d | yes | 1 | |
4.b | odd | 2 | 1 | 144.4.a.a | 1 | ||
5.b | even | 2 | 1 | 1800.4.a.z | 1 | ||
5.c | odd | 4 | 2 | 1800.4.f.b | 2 | ||
8.b | even | 2 | 1 | 576.4.a.w | 1 | ||
8.d | odd | 2 | 1 | 576.4.a.x | 1 | ||
9.c | even | 3 | 2 | 648.4.i.l | 2 | ||
9.d | odd | 6 | 2 | 648.4.i.a | 2 | ||
12.b | even | 2 | 1 | 144.4.a.f | 1 | ||
15.d | odd | 2 | 1 | 1800.4.a.ba | 1 | ||
15.e | even | 4 | 2 | 1800.4.f.x | 2 | ||
24.f | even | 2 | 1 | 576.4.a.d | 1 | ||
24.h | odd | 2 | 1 | 576.4.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.4.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
72.4.a.d | yes | 1 | 3.b | odd | 2 | 1 | |
144.4.a.a | 1 | 4.b | odd | 2 | 1 | ||
144.4.a.f | 1 | 12.b | even | 2 | 1 | ||
576.4.a.c | 1 | 24.h | odd | 2 | 1 | ||
576.4.a.d | 1 | 24.f | even | 2 | 1 | ||
576.4.a.w | 1 | 8.b | even | 2 | 1 | ||
576.4.a.x | 1 | 8.d | odd | 2 | 1 | ||
648.4.i.a | 2 | 9.d | odd | 6 | 2 | ||
648.4.i.l | 2 | 9.c | even | 3 | 2 | ||
1800.4.a.z | 1 | 5.b | even | 2 | 1 | ||
1800.4.a.ba | 1 | 15.d | odd | 2 | 1 | ||
1800.4.f.b | 2 | 5.c | odd | 4 | 2 | ||
1800.4.f.x | 2 | 15.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T + 16 \)
$7$
\( T + 12 \)
$11$
\( T + 64 \)
$13$
\( T - 58 \)
$17$
\( T + 32 \)
$19$
\( T + 136 \)
$23$
\( T - 128 \)
$29$
\( T - 144 \)
$31$
\( T - 20 \)
$37$
\( T + 18 \)
$41$
\( T - 288 \)
$43$
\( T + 200 \)
$47$
\( T + 384 \)
$53$
\( T + 496 \)
$59$
\( T - 128 \)
$61$
\( T + 458 \)
$67$
\( T + 496 \)
$71$
\( T + 512 \)
$73$
\( T + 602 \)
$79$
\( T - 1108 \)
$83$
\( T + 704 \)
$89$
\( T - 960 \)
$97$
\( T - 206 \)
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