Newspace parameters
Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 18.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(13.8301772501\) |
Analytic rank: | \(0\) |
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 |
|
−32.0000 | 0 | 1024.00 | 5280.00 | 0 | −49036.0 | −32768.0 | 0 | −168960. | |||||||||||||||||||||
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 | 18.12.a.b | ✓ | 1 |
3.b | odd | 2 | 1 | 18.12.a.d | yes | 1 | |
4.b | odd | 2 | 1 | 144.12.a.k | 1 | ||
9.c | even | 3 | 2 | 162.12.c.h | 2 | ||
9.d | odd | 6 | 2 | 162.12.c.c | 2 | ||
12.b | even | 2 | 1 | 144.12.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
18.12.a.b | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
18.12.a.d | yes | 1 | 3.b | odd | 2 | 1 | |
144.12.a.c | 1 | 12.b | even | 2 | 1 | ||
144.12.a.k | 1 | 4.b | odd | 2 | 1 | ||
162.12.c.c | 2 | 9.d | odd | 6 | 2 | ||
162.12.c.h | 2 | 9.c | even | 3 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 5280 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(18))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 32 \)
$3$
\( T \)
$5$
\( T - 5280 \)
$7$
\( T + 49036 \)
$11$
\( T - 414336 \)
$13$
\( T + 522982 \)
$17$
\( T - 9499968 \)
$19$
\( T - 13053944 \)
$23$
\( T - 58755840 \)
$29$
\( T + 117142944 \)
$31$
\( T - 142907156 \)
$37$
\( T - 718521806 \)
$41$
\( T + 668055360 \)
$43$
\( T - 141575864 \)
$47$
\( T - 729235200 \)
$53$
\( T - 4917225312 \)
$59$
\( T - 1408015104 \)
$61$
\( T + 3223327018 \)
$67$
\( T + 2358681328 \)
$71$
\( T + 22245092352 \)
$73$
\( T + 28036594330 \)
$79$
\( T + 20685045676 \)
$83$
\( T - 37818604416 \)
$89$
\( T - 11288711808 \)
$97$
\( T + 115724393266 \)
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