Newspace parameters
| Level: | \( N \) | \(=\) | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 550.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(32.4510505032\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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 |
|
−2.00000 | −1.00000 | 4.00000 | 0 | 2.00000 | 10.0000 | −8.00000 | −26.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 550.4.a.e | 1 | |
| 5.b | even | 2 | 1 | 22.4.a.c | ✓ | 1 | |
| 5.c | odd | 4 | 2 | 550.4.b.g | 2 | ||
| 15.d | odd | 2 | 1 | 198.4.a.b | 1 | ||
| 20.d | odd | 2 | 1 | 176.4.a.c | 1 | ||
| 35.c | odd | 2 | 1 | 1078.4.a.f | 1 | ||
| 40.e | odd | 2 | 1 | 704.4.a.g | 1 | ||
| 40.f | even | 2 | 1 | 704.4.a.e | 1 | ||
| 55.d | odd | 2 | 1 | 242.4.a.a | 1 | ||
| 55.h | odd | 10 | 4 | 242.4.c.k | 4 | ||
| 55.j | even | 10 | 4 | 242.4.c.d | 4 | ||
| 60.h | even | 2 | 1 | 1584.4.a.k | 1 | ||
| 165.d | even | 2 | 1 | 2178.4.a.r | 1 | ||
| 220.g | even | 2 | 1 | 1936.4.a.h | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.4.a.c | ✓ | 1 | 5.b | even | 2 | 1 | |
| 176.4.a.c | 1 | 20.d | odd | 2 | 1 | ||
| 198.4.a.b | 1 | 15.d | odd | 2 | 1 | ||
| 242.4.a.a | 1 | 55.d | odd | 2 | 1 | ||
| 242.4.c.d | 4 | 55.j | even | 10 | 4 | ||
| 242.4.c.k | 4 | 55.h | odd | 10 | 4 | ||
| 550.4.a.e | 1 | 1.a | even | 1 | 1 | trivial | |
| 550.4.b.g | 2 | 5.c | odd | 4 | 2 | ||
| 704.4.a.e | 1 | 40.f | even | 2 | 1 | ||
| 704.4.a.g | 1 | 40.e | odd | 2 | 1 | ||
| 1078.4.a.f | 1 | 35.c | odd | 2 | 1 | ||
| 1584.4.a.k | 1 | 60.h | even | 2 | 1 | ||
| 1936.4.a.h | 1 | 220.g | even | 2 | 1 | ||
| 2178.4.a.r | 1 | 165.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(550))\):
|
\( T_{3} + 1 \)
|
|
\( T_{7} - 10 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 2 \)
|
| $3$ |
\( T + 1 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T - 10 \)
|
| $11$ |
\( T - 11 \)
|
| $13$ |
\( T - 16 \)
|
| $17$ |
\( T + 42 \)
|
| $19$ |
\( T - 116 \)
|
| $23$ |
\( T + 189 \)
|
| $29$ |
\( T + 120 \)
|
| $31$ |
\( T + 163 \)
|
| $37$ |
\( T - 409 \)
|
| $41$ |
\( T - 468 \)
|
| $43$ |
\( T + 110 \)
|
| $47$ |
\( T + 144 \)
|
| $53$ |
\( T + 90 \)
|
| $59$ |
\( T + 453 \)
|
| $61$ |
\( T - 20 \)
|
| $67$ |
\( T - 97 \)
|
| $71$ |
\( T + 465 \)
|
| $73$ |
\( T + 848 \)
|
| $79$ |
\( T + 742 \)
|
| $83$ |
\( T + 438 \)
|
| $89$ |
\( T + 273 \)
|
| $97$ |
\( T + 761 \)
|
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