Newspace parameters
| Level: | \( N \) | \(=\) | \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9196.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(73.4304296988\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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 |
|
0 | 2.00000 | 0 | −1.00000 | 0 | 3.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 9196.2.a.f | 1 | |
| 11.b | odd | 2 | 1 | 76.2.a.a | ✓ | 1 | |
| 33.d | even | 2 | 1 | 684.2.a.b | 1 | ||
| 44.c | even | 2 | 1 | 304.2.a.a | 1 | ||
| 55.d | odd | 2 | 1 | 1900.2.a.b | 1 | ||
| 55.e | even | 4 | 2 | 1900.2.c.b | 2 | ||
| 77.b | even | 2 | 1 | 3724.2.a.a | 1 | ||
| 88.b | odd | 2 | 1 | 1216.2.a.c | 1 | ||
| 88.g | even | 2 | 1 | 1216.2.a.q | 1 | ||
| 132.d | odd | 2 | 1 | 2736.2.a.q | 1 | ||
| 209.d | even | 2 | 1 | 1444.2.a.a | 1 | ||
| 209.g | even | 6 | 2 | 1444.2.e.c | 2 | ||
| 209.h | odd | 6 | 2 | 1444.2.e.a | 2 | ||
| 220.g | even | 2 | 1 | 7600.2.a.p | 1 | ||
| 836.h | odd | 2 | 1 | 5776.2.a.p | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.2.a.a | ✓ | 1 | 11.b | odd | 2 | 1 | |
| 304.2.a.a | 1 | 44.c | even | 2 | 1 | ||
| 684.2.a.b | 1 | 33.d | even | 2 | 1 | ||
| 1216.2.a.c | 1 | 88.b | odd | 2 | 1 | ||
| 1216.2.a.q | 1 | 88.g | even | 2 | 1 | ||
| 1444.2.a.a | 1 | 209.d | even | 2 | 1 | ||
| 1444.2.e.a | 2 | 209.h | odd | 6 | 2 | ||
| 1444.2.e.c | 2 | 209.g | even | 6 | 2 | ||
| 1900.2.a.b | 1 | 55.d | odd | 2 | 1 | ||
| 1900.2.c.b | 2 | 55.e | even | 4 | 2 | ||
| 2736.2.a.q | 1 | 132.d | odd | 2 | 1 | ||
| 3724.2.a.a | 1 | 77.b | even | 2 | 1 | ||
| 5776.2.a.p | 1 | 836.h | odd | 2 | 1 | ||
| 7600.2.a.p | 1 | 220.g | even | 2 | 1 | ||
| 9196.2.a.f | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):
|
\( T_{3} - 2 \)
|
|
\( T_{5} + 1 \)
|
|
\( T_{7} - 3 \)
|
|
\( T_{13} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 2 \)
|
| $5$ |
\( T + 1 \)
|
| $7$ |
\( T - 3 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T - 4 \)
|
| $17$ |
\( T - 3 \)
|
| $19$ |
\( T - 1 \)
|
| $23$ |
\( T - 8 \)
|
| $29$ |
\( T - 2 \)
|
| $31$ |
\( T - 4 \)
|
| $37$ |
\( T - 10 \)
|
| $41$ |
\( T + 10 \)
|
| $43$ |
\( T + 1 \)
|
| $47$ |
\( T + 1 \)
|
| $53$ |
\( T + 4 \)
|
| $59$ |
\( T - 6 \)
|
| $61$ |
\( T - 13 \)
|
| $67$ |
\( T + 12 \)
|
| $71$ |
\( T - 2 \)
|
| $73$ |
\( T + 9 \)
|
| $79$ |
\( T + 8 \)
|
| $83$ |
\( T - 12 \)
|
| $89$ |
\( T - 12 \)
|
| $97$ |
\( T + 8 \)
|
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