Newspace parameters
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 847.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(135.845095382\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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 | −6.00000 | −28.0000 | −74.0000 | −12.0000 | 49.0000 | −120.000 | −207.000 | −148.000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 847.6.a.a | 1 | |
| 11.b | odd | 2 | 1 | 77.6.a.a | ✓ | 1 | |
| 33.d | even | 2 | 1 | 693.6.a.a | 1 | ||
| 77.b | even | 2 | 1 | 539.6.a.d | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.6.a.a | ✓ | 1 | 11.b | odd | 2 | 1 | |
| 539.6.a.d | 1 | 77.b | even | 2 | 1 | ||
| 693.6.a.a | 1 | 33.d | even | 2 | 1 | ||
| 847.6.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 2 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 2 \)
$3$
\( T + 6 \)
$5$
\( T + 74 \)
$7$
\( T - 49 \)
$11$
\( T \)
$13$
\( T + 364 \)
$17$
\( T + 148 \)
$19$
\( T - 1320 \)
$23$
\( T + 436 \)
$29$
\( T + 2970 \)
$31$
\( T - 8842 \)
$37$
\( T - 138 \)
$41$
\( T + 532 \)
$43$
\( T - 20676 \)
$47$
\( T + 11722 \)
$53$
\( T - 5274 \)
$59$
\( T + 27670 \)
$61$
\( T + 19512 \)
$67$
\( T - 64088 \)
$71$
\( T + 3708 \)
$73$
\( T - 24296 \)
$79$
\( T - 2200 \)
$83$
\( T + 74424 \)
$89$
\( T - 34170 \)
$97$
\( T - 151718 \)
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