Newspace parameters
| Level: | \( N \) | \(=\) | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5202.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.5381791315\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 102) |
| 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 |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 5202.2.a.n | 1 | |
| 3.b | odd | 2 | 1 | 1734.2.a.e | 1 | ||
| 17.b | even | 2 | 1 | 5202.2.a.h | 1 | ||
| 17.c | even | 4 | 2 | 306.2.b.a | 2 | ||
| 51.c | odd | 2 | 1 | 1734.2.a.d | 1 | ||
| 51.f | odd | 4 | 2 | 102.2.b.a | ✓ | 2 | |
| 51.g | odd | 8 | 4 | 1734.2.f.h | 4 | ||
| 68.f | odd | 4 | 2 | 2448.2.c.a | 2 | ||
| 204.l | even | 4 | 2 | 816.2.c.a | 2 | ||
| 255.i | odd | 4 | 2 | 2550.2.c.f | 2 | ||
| 255.k | even | 4 | 2 | 2550.2.f.e | 2 | ||
| 255.r | even | 4 | 2 | 2550.2.f.j | 2 | ||
| 408.q | even | 4 | 2 | 3264.2.c.i | 2 | ||
| 408.t | odd | 4 | 2 | 3264.2.c.j | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.2.b.a | ✓ | 2 | 51.f | odd | 4 | 2 | |
| 306.2.b.a | 2 | 17.c | even | 4 | 2 | ||
| 816.2.c.a | 2 | 204.l | even | 4 | 2 | ||
| 1734.2.a.d | 1 | 51.c | odd | 2 | 1 | ||
| 1734.2.a.e | 1 | 3.b | odd | 2 | 1 | ||
| 1734.2.f.h | 4 | 51.g | odd | 8 | 4 | ||
| 2448.2.c.a | 2 | 68.f | odd | 4 | 2 | ||
| 2550.2.c.f | 2 | 255.i | odd | 4 | 2 | ||
| 2550.2.f.e | 2 | 255.k | even | 4 | 2 | ||
| 2550.2.f.j | 2 | 255.r | even | 4 | 2 | ||
| 3264.2.c.i | 2 | 408.q | even | 4 | 2 | ||
| 3264.2.c.j | 2 | 408.t | odd | 4 | 2 | ||
| 5202.2.a.h | 1 | 17.b | even | 2 | 1 | ||
| 5202.2.a.n | 1 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5202))\):
|
\( T_{5} - 2 \)
|
|
\( T_{7} + 2 \)
|
|
\( T_{23} + 6 \)
|
|
\( T_{47} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 1 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T - 2 \)
|
| $7$ |
\( T + 2 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T + 6 \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T + 6 \)
|
| $29$ |
\( T + 6 \)
|
| $31$ |
\( T - 10 \)
|
| $37$ |
\( T - 2 \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T + 4 \)
|
| $47$ |
\( T + 8 \)
|
| $53$ |
\( T + 6 \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T - 10 \)
|
| $67$ |
\( T - 8 \)
|
| $71$ |
\( T + 10 \)
|
| $73$ |
\( T + 16 \)
|
| $79$ |
\( T + 6 \)
|
| $83$ |
\( T + 16 \)
|
| $89$ |
\( T + 10 \)
|
| $97$ |
\( T - 12 \)
|
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