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: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 138) |
| 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 | 2.00000 | 0 | −2.00000 | 1.00000 | 0 | 2.00000 | |||||||||||||||||||||
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.d | 1 | |
| 3.b | odd | 2 | 1 | 138.2.a.a | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 3312.2.a.n | 1 | ||
| 12.b | even | 2 | 1 | 1104.2.a.e | 1 | ||
| 15.d | odd | 2 | 1 | 3450.2.a.y | 1 | ||
| 15.e | even | 4 | 2 | 3450.2.d.j | 2 | ||
| 21.c | even | 2 | 1 | 6762.2.a.q | 1 | ||
| 23.b | odd | 2 | 1 | 9522.2.a.i | 1 | ||
| 24.f | even | 2 | 1 | 4416.2.a.m | 1 | ||
| 24.h | odd | 2 | 1 | 4416.2.a.z | 1 | ||
| 69.c | even | 2 | 1 | 3174.2.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.2.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 414.2.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
| 1104.2.a.e | 1 | 12.b | even | 2 | 1 | ||
| 3174.2.a.b | 1 | 69.c | even | 2 | 1 | ||
| 3312.2.a.n | 1 | 4.b | odd | 2 | 1 | ||
| 3450.2.a.y | 1 | 15.d | odd | 2 | 1 | ||
| 3450.2.d.j | 2 | 15.e | even | 4 | 2 | ||
| 4416.2.a.m | 1 | 24.f | even | 2 | 1 | ||
| 4416.2.a.z | 1 | 24.h | odd | 2 | 1 | ||
| 6762.2.a.q | 1 | 21.c | even | 2 | 1 | ||
| 9522.2.a.i | 1 | 23.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(414))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 1 \)
$3$
\( T \)
$5$
\( T - 2 \)
$7$
\( T + 2 \)
$11$
\( T - 6 \)
$13$
\( T + 2 \)
$17$
\( T \)
$19$
\( T \)
$23$
\( T - 1 \)
$29$
\( T + 6 \)
$31$
\( T - 8 \)
$37$
\( T \)
$41$
\( T + 10 \)
$43$
\( T + 12 \)
$47$
\( T - 8 \)
$53$
\( T + 2 \)
$59$
\( T - 12 \)
$61$
\( T - 4 \)
$67$
\( T + 12 \)
$71$
\( T \)
$73$
\( T + 10 \)
$79$
\( T + 6 \)
$83$
\( T + 14 \)
$89$
\( T \)
$97$
\( T + 6 \)
show more
show less