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: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{3}) \) |
Defining polynomial: |
\( x^{2} - 3 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
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 | 0.803848 | 0 | −2.39230 | 8.00000 | 0 | 1.60770 | ||||||||||||||||||||||||
1.2 | 2.00000 | 0 | 4.00000 | 11.1962 | 0 | 18.3923 | 8.00000 | 0 | 22.3923 | |||||||||||||||||||||||||
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.h | yes | 2 |
3.b | odd | 2 | 1 | 162.4.a.e | ✓ | 2 | |
4.b | odd | 2 | 1 | 1296.4.a.s | 2 | ||
9.c | even | 3 | 2 | 162.4.c.i | 4 | ||
9.d | odd | 6 | 2 | 162.4.c.j | 4 | ||
12.b | even | 2 | 1 | 1296.4.a.j | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
162.4.a.e | ✓ | 2 | 3.b | odd | 2 | 1 | |
162.4.a.h | yes | 2 | 1.a | even | 1 | 1 | trivial |
162.4.c.i | 4 | 9.c | even | 3 | 2 | ||
162.4.c.j | 4 | 9.d | odd | 6 | 2 | ||
1296.4.a.j | 2 | 12.b | even | 2 | 1 | ||
1296.4.a.s | 2 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 12T_{5} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 2)^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 12T + 9 \)
$7$
\( T^{2} - 16T - 44 \)
$11$
\( T^{2} - 36T - 1404 \)
$13$
\( T^{2} - 10T - 5267 \)
$17$
\( T^{2} - 120T + 333 \)
$19$
\( T^{2} + 8T - 13052 \)
$23$
\( T^{2} - 180T + 7668 \)
$29$
\( T^{2} - 324T + 24921 \)
$31$
\( T^{2} + 248T - 19616 \)
$37$
\( T^{2} + 434T + 46981 \)
$41$
\( T^{2} - 192T - 43056 \)
$43$
\( T^{2} + 608T + 91444 \)
$47$
\( T^{2} + 384T + 1872 \)
$53$
\( T^{2} + 408T - 114336 \)
$59$
\( T^{2} + 1008 T + 226368 \)
$61$
\( T^{2} - 742T + 128893 \)
$67$
\( T^{2} + 104T - 511484 \)
$71$
\( T^{2} + 1140 T + 323172 \)
$73$
\( T^{2} - 850T - 68207 \)
$79$
\( T^{2} + 440T - 19100 \)
$83$
\( T^{2} - 264T - 79776 \)
$89$
\( T^{2} - 768T - 66411 \)
$97$
\( T^{2} + 764T + 61252 \)
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