Newspace parameters
Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 162.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(9.55830942093\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 18) |
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 |
|
−2.00000 | 0 | 4.00000 | 9.00000 | 0 | −31.0000 | −8.00000 | 0 | −18.0000 | |||||||||||||||||||||
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 | 162.4.a.a | 1 | |
3.b | odd | 2 | 1 | 162.4.a.d | 1 | ||
4.b | odd | 2 | 1 | 1296.4.a.g | 1 | ||
9.c | even | 3 | 2 | 54.4.c.a | 2 | ||
9.d | odd | 6 | 2 | 18.4.c.a | ✓ | 2 | |
12.b | even | 2 | 1 | 1296.4.a.b | 1 | ||
36.f | odd | 6 | 2 | 432.4.i.a | 2 | ||
36.h | even | 6 | 2 | 144.4.i.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
18.4.c.a | ✓ | 2 | 9.d | odd | 6 | 2 | |
54.4.c.a | 2 | 9.c | even | 3 | 2 | ||
144.4.i.a | 2 | 36.h | even | 6 | 2 | ||
162.4.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
162.4.a.d | 1 | 3.b | odd | 2 | 1 | ||
432.4.i.a | 2 | 36.f | odd | 6 | 2 | ||
1296.4.a.b | 1 | 12.b | even | 2 | 1 | ||
1296.4.a.g | 1 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 9 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 2 \)
$3$
\( T \)
$5$
\( T - 9 \)
$7$
\( T + 31 \)
$11$
\( T - 15 \)
$13$
\( T + 37 \)
$17$
\( T - 42 \)
$19$
\( T + 28 \)
$23$
\( T + 195 \)
$29$
\( T + 111 \)
$31$
\( T + 205 \)
$37$
\( T + 166 \)
$41$
\( T - 261 \)
$43$
\( T + 43 \)
$47$
\( T + 177 \)
$53$
\( T + 114 \)
$59$
\( T + 159 \)
$61$
\( T - 191 \)
$67$
\( T + 421 \)
$71$
\( T + 156 \)
$73$
\( T - 182 \)
$79$
\( T - 1133 \)
$83$
\( T - 1083 \)
$89$
\( T - 1050 \)
$97$
\( T + 901 \)
show more
show less