Newspace parameters
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 847.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.9746177749\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.522072.1 |
| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 77) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 12x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 12\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 11\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 13\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 2\beta _1 + 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} + 6\beta_{2} + \beta _1 + 9 \)
|
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 |
|
−4.60395 | 2.77399 | 13.1964 | 1.84418 | −12.7713 | 7.00000 | −23.9238 | −19.3050 | −8.49053 | ||||||||||||||||||||||||||||||
| 1.2 | −1.53253 | 10.1459 | −5.65135 | 8.69995 | −15.5490 | 7.00000 | 20.9211 | 75.9402 | −13.3330 | |||||||||||||||||||||||||||||||
| 1.3 | 3.24550 | −5.49244 | 2.53327 | −16.0955 | −17.8257 | 7.00000 | −17.7423 | 3.16692 | −52.2379 | |||||||||||||||||||||||||||||||
| 1.4 | 4.89098 | 6.57251 | 15.9217 | 15.5514 | 32.1460 | 7.00000 | 38.7449 | 16.1978 | 76.0614 | |||||||||||||||||||||||||||||||
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 | 847.4.a.d | 4 | |
| 11.b | odd | 2 | 1 | 77.4.a.d | ✓ | 4 | |
| 33.d | even | 2 | 1 | 693.4.a.l | 4 | ||
| 44.c | even | 2 | 1 | 1232.4.a.s | 4 | ||
| 55.d | odd | 2 | 1 | 1925.4.a.p | 4 | ||
| 77.b | even | 2 | 1 | 539.4.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.a.d | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 539.4.a.g | 4 | 77.b | even | 2 | 1 | ||
| 693.4.a.l | 4 | 33.d | even | 2 | 1 | ||
| 847.4.a.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 1232.4.a.s | 4 | 44.c | even | 2 | 1 | ||
| 1925.4.a.p | 4 | 55.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} - 27T_{2}^{2} + 40T_{2} + 112 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} - 27 T^{2} + 40 T + 112 \)
$3$
\( T^{4} - 14 T^{3} + 6 T^{2} + \cdots - 1016 \)
$5$
\( T^{4} - 10 T^{3} - 240 T^{2} + \cdots - 4016 \)
$7$
\( (T - 7)^{4} \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 58 T^{3} - 4926 T^{2} + \cdots - 4947656 \)
$17$
\( T^{4} + 4 T^{3} - 6186 T^{2} + \cdots - 2705024 \)
$19$
\( T^{4} + 258 T^{3} + \cdots - 14423904 \)
$23$
\( T^{4} - 8 T^{3} - 22392 T^{2} + \cdots - 17449856 \)
$29$
\( T^{4} - 396 T^{3} + \cdots + 22336464 \)
$31$
\( T^{4} + 56 T^{3} - 31890 T^{2} + \cdots - 11250248 \)
$37$
\( T^{4} - 84 T^{3} - 63516 T^{2} + \cdots + 11157312 \)
$41$
\( T^{4} + 52 T^{3} + \cdots - 659233664 \)
$43$
\( T^{4} + 408 T^{3} + \cdots + 1210397376 \)
$47$
\( T^{4} - 8 T^{3} - 121650 T^{2} + \cdots - 318931592 \)
$53$
\( T^{4} - 624 T^{3} + \cdots + 403923072 \)
$59$
\( T^{4} + 238 T^{3} + \cdots + 17599820728 \)
$61$
\( T^{4} - 162 T^{3} + \cdots + 6668930664 \)
$67$
\( T^{4} - 1340 T^{3} + \cdots - 140865466496 \)
$71$
\( T^{4} - 1788 T^{3} + \cdots - 72982082688 \)
$73$
\( T^{4} + 1456 T^{3} + \cdots - 322052228384 \)
$79$
\( T^{4} - 1324 T^{3} + \cdots - 59537293568 \)
$83$
\( T^{4} + 450 T^{3} + \cdots + 21951092064 \)
$89$
\( T^{4} + 3072 T^{3} + \cdots - 109303561968 \)
$97$
\( T^{4} + 652 T^{3} + \cdots + 868634650768 \)
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