Newspace parameters
Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 42.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(6.73612043215\) |
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 |
|
−4.00000 | 9.00000 | 16.0000 | −72.0000 | −36.0000 | 49.0000 | −64.0000 | 81.0000 | 288.000 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(3\) | \(-1\) |
\(7\) | \(-1\) |
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 | 42.6.a.c | ✓ | 1 |
3.b | odd | 2 | 1 | 126.6.a.l | 1 | ||
4.b | odd | 2 | 1 | 336.6.a.b | 1 | ||
5.b | even | 2 | 1 | 1050.6.a.g | 1 | ||
5.c | odd | 4 | 2 | 1050.6.g.b | 2 | ||
7.b | odd | 2 | 1 | 294.6.a.c | 1 | ||
7.c | even | 3 | 2 | 294.6.e.l | 2 | ||
7.d | odd | 6 | 2 | 294.6.e.n | 2 | ||
12.b | even | 2 | 1 | 1008.6.a.ba | 1 | ||
21.c | even | 2 | 1 | 882.6.a.n | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
42.6.a.c | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
126.6.a.l | 1 | 3.b | odd | 2 | 1 | ||
294.6.a.c | 1 | 7.b | odd | 2 | 1 | ||
294.6.e.l | 2 | 7.c | even | 3 | 2 | ||
294.6.e.n | 2 | 7.d | odd | 6 | 2 | ||
336.6.a.b | 1 | 4.b | odd | 2 | 1 | ||
882.6.a.n | 1 | 21.c | even | 2 | 1 | ||
1008.6.a.ba | 1 | 12.b | even | 2 | 1 | ||
1050.6.a.g | 1 | 5.b | even | 2 | 1 | ||
1050.6.g.b | 2 | 5.c | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 72 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(42))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 4 \)
$3$
\( T - 9 \)
$5$
\( T + 72 \)
$7$
\( T - 49 \)
$11$
\( T + 414 \)
$13$
\( T + 1054 \)
$17$
\( T + 1848 \)
$19$
\( T - 236 \)
$23$
\( T - 2898 \)
$29$
\( T + 6522 \)
$31$
\( T - 6200 \)
$37$
\( T - 9650 \)
$41$
\( T - 8484 \)
$43$
\( T + 10804 \)
$47$
\( T - 60 \)
$53$
\( T - 22506 \)
$59$
\( T + 28176 \)
$61$
\( T + 35194 \)
$67$
\( T + 28216 \)
$71$
\( T + 6642 \)
$73$
\( T + 52090 \)
$79$
\( T - 43340 \)
$83$
\( T - 25716 \)
$89$
\( T - 98724 \)
$97$
\( T + 148954 \)
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