Newspace parameters
Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 6.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.61005908336\) |
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 |
|
−32.0000 | 243.000 | 1024.00 | −11730.0 | −7776.00 | −50008.0 | −32768.0 | 59049.0 | 375360. | |||||||||||||||||||||
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 | 6.12.a.b | ✓ | 1 |
3.b | odd | 2 | 1 | 18.12.a.e | 1 | ||
4.b | odd | 2 | 1 | 48.12.a.a | 1 | ||
5.b | even | 2 | 1 | 150.12.a.f | 1 | ||
5.c | odd | 4 | 2 | 150.12.c.b | 2 | ||
8.b | even | 2 | 1 | 192.12.a.j | 1 | ||
8.d | odd | 2 | 1 | 192.12.a.t | 1 | ||
9.c | even | 3 | 2 | 162.12.c.j | 2 | ||
9.d | odd | 6 | 2 | 162.12.c.a | 2 | ||
12.b | even | 2 | 1 | 144.12.a.o | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6.12.a.b | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
18.12.a.e | 1 | 3.b | odd | 2 | 1 | ||
48.12.a.a | 1 | 4.b | odd | 2 | 1 | ||
144.12.a.o | 1 | 12.b | even | 2 | 1 | ||
150.12.a.f | 1 | 5.b | even | 2 | 1 | ||
150.12.c.b | 2 | 5.c | odd | 4 | 2 | ||
162.12.c.a | 2 | 9.d | odd | 6 | 2 | ||
162.12.c.j | 2 | 9.c | even | 3 | 2 | ||
192.12.a.j | 1 | 8.b | even | 2 | 1 | ||
192.12.a.t | 1 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 11730 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 32 \)
$3$
\( T - 243 \)
$5$
\( T + 11730 \)
$7$
\( T + 50008 \)
$11$
\( T + 531420 \)
$13$
\( T - 1332566 \)
$17$
\( T + 5109678 \)
$19$
\( T - 2901404 \)
$23$
\( T - 30597000 \)
$29$
\( T + 77006634 \)
$31$
\( T + 239418352 \)
$37$
\( T + 785041666 \)
$41$
\( T - 411252954 \)
$43$
\( T - 351233348 \)
$47$
\( T - 95821680 \)
$53$
\( T + 1465857378 \)
$59$
\( T - 5621152020 \)
$61$
\( T + 10473587770 \)
$67$
\( T - 4515307532 \)
$71$
\( T + 8509579560 \)
$73$
\( T - 2012496986 \)
$79$
\( T + 22238409568 \)
$83$
\( T - 6328647516 \)
$89$
\( T + 50123706678 \)
$97$
\( T - 94805961314 \)
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