Newspace parameters
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(3.30580664368\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 46) |
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 |
|
1.00000 | 0 | 1.00000 | −4.00000 | 0 | −4.00000 | 1.00000 | 0 | −4.00000 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(-1\) |
\(23\) | \(1\) |
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 | 414.2.a.b | 1 | |
3.b | odd | 2 | 1 | 46.2.a.a | ✓ | 1 | |
4.b | odd | 2 | 1 | 3312.2.a.b | 1 | ||
12.b | even | 2 | 1 | 368.2.a.d | 1 | ||
15.d | odd | 2 | 1 | 1150.2.a.h | 1 | ||
15.e | even | 4 | 2 | 1150.2.b.d | 2 | ||
21.c | even | 2 | 1 | 2254.2.a.c | 1 | ||
23.b | odd | 2 | 1 | 9522.2.a.p | 1 | ||
24.f | even | 2 | 1 | 1472.2.a.g | 1 | ||
24.h | odd | 2 | 1 | 1472.2.a.f | 1 | ||
33.d | even | 2 | 1 | 5566.2.a.h | 1 | ||
39.d | odd | 2 | 1 | 7774.2.a.d | 1 | ||
60.h | even | 2 | 1 | 9200.2.a.p | 1 | ||
69.c | even | 2 | 1 | 1058.2.a.c | 1 | ||
276.h | odd | 2 | 1 | 8464.2.a.g | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
46.2.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
368.2.a.d | 1 | 12.b | even | 2 | 1 | ||
414.2.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
1058.2.a.c | 1 | 69.c | even | 2 | 1 | ||
1150.2.a.h | 1 | 15.d | odd | 2 | 1 | ||
1150.2.b.d | 2 | 15.e | even | 4 | 2 | ||
1472.2.a.f | 1 | 24.h | odd | 2 | 1 | ||
1472.2.a.g | 1 | 24.f | even | 2 | 1 | ||
2254.2.a.c | 1 | 21.c | even | 2 | 1 | ||
3312.2.a.b | 1 | 4.b | odd | 2 | 1 | ||
5566.2.a.h | 1 | 33.d | even | 2 | 1 | ||
7774.2.a.d | 1 | 39.d | odd | 2 | 1 | ||
8464.2.a.g | 1 | 276.h | odd | 2 | 1 | ||
9200.2.a.p | 1 | 60.h | even | 2 | 1 | ||
9522.2.a.p | 1 | 23.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(414))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 1 \)
$3$
\( T \)
$5$
\( T + 4 \)
$7$
\( T + 4 \)
$11$
\( T + 2 \)
$13$
\( T + 2 \)
$17$
\( T - 2 \)
$19$
\( T + 2 \)
$23$
\( T + 1 \)
$29$
\( T + 2 \)
$31$
\( T \)
$37$
\( T + 4 \)
$41$
\( T + 6 \)
$43$
\( T - 10 \)
$47$
\( T \)
$53$
\( T - 4 \)
$59$
\( T + 12 \)
$61$
\( T + 8 \)
$67$
\( T + 10 \)
$71$
\( T \)
$73$
\( T - 6 \)
$79$
\( T + 12 \)
$83$
\( T + 14 \)
$89$
\( T - 6 \)
$97$
\( T - 6 \)
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