Newspace parameters
| Level: | \( N \) | \(=\) | \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6930.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(55.3363286007\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 770) |
| 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 | 1.00000 | 0 | 1.00000 | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 6930.2.a.o | 1 | |
| 3.b | odd | 2 | 1 | 770.2.a.f | ✓ | 1 | |
| 12.b | even | 2 | 1 | 6160.2.a.j | 1 | ||
| 15.d | odd | 2 | 1 | 3850.2.a.k | 1 | ||
| 15.e | even | 4 | 2 | 3850.2.c.b | 2 | ||
| 21.c | even | 2 | 1 | 5390.2.a.bj | 1 | ||
| 33.d | even | 2 | 1 | 8470.2.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.a.f | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 3850.2.a.k | 1 | 15.d | odd | 2 | 1 | ||
| 3850.2.c.b | 2 | 15.e | even | 4 | 2 | ||
| 5390.2.a.bj | 1 | 21.c | even | 2 | 1 | ||
| 6160.2.a.j | 1 | 12.b | even | 2 | 1 | ||
| 6930.2.a.o | 1 | 1.a | even | 1 | 1 | trivial | |
| 8470.2.a.c | 1 | 33.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):
| \( T_{13} + 4 \) |
| \( T_{17} \) |
| \( T_{19} + 4 \) |
| \( T_{23} \) |
| \( T_{29} - 6 \) |
| \( T_{31} + 10 \) |
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 1 + T \) |
| $3$ | \( T \) |
| $5$ | \( -1 + T \) |
| $7$ | \( -1 + T \) |
| $11$ | \( -1 + T \) |
| $13$ | \( 4 + T \) |
| $17$ | \( T \) |
| $19$ | \( 4 + T \) |
| $23$ | \( T \) |
| $29$ | \( -6 + T \) |
| $31$ | \( 10 + T \) |
| $37$ | \( -2 + T \) |
| $41$ | \( -12 + T \) |
| $43$ | \( 4 + T \) |
| $47$ | \( 6 + T \) |
| $53$ | \( -6 + T \) |
| $59$ | \( -6 + T \) |
| $61$ | \( 4 + T \) |
| $67$ | \( 4 + T \) |
| $71$ | \( 12 + T \) |
| $73$ | \( 4 + T \) |
| $79$ | \( -8 + T \) |
| $83$ | \( 12 + T \) |
| $89$ | \( 18 + T \) |
| $97$ | \( 10 + T \) |
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