Newspace parameters
| Level: | \( N \) | \(=\) | \( 9920 = 2^{6} \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9920.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(79.2115988051\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - x^{5} - 19x^{4} + 15x^{3} + 98x^{2} - 44x - 108 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1240) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 19x^{4} + 15x^{3} + 98x^{2} - 44x - 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} - 9\nu^{4} + 10\nu^{3} + 111\nu^{2} + 92\nu - 180 ) / 43 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} - \nu^{4} + 68\nu^{3} - 2\nu^{2} - 157\nu + 66 ) / 43 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{5} - \nu^{4} + 68\nu^{3} + 41\nu^{2} - 157\nu - 235 ) / 43 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 18\nu^{4} - 63\nu^{3} - 222\nu^{2} + 203\nu + 403 ) / 43 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 7 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{2} + 9\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 13\beta_{4} - 11\beta_{3} - \beta_{2} + 65 \)
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| \(\nu^{5}\) | \(=\) |
\( -14\beta_{5} - 3\beta_{4} - 6\beta_{3} - 27\beta_{2} + 91\beta _1 + 11 \)
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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 | −3.31203 | 0 | −1.00000 | 0 | −1.94649 | 0 | 7.96954 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.36730 | 0 | −1.00000 | 0 | 0.985130 | 0 | 2.60412 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.984679 | 0 | −1.00000 | 0 | 4.16564 | 0 | −2.03041 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.45950 | 0 | −1.00000 | 0 | −3.90059 | 0 | −0.869870 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.90399 | 0 | −1.00000 | 0 | −5.00062 | 0 | 5.43318 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.30052 | 0 | −1.00000 | 0 | 3.69693 | 0 | 7.89343 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 9920.2.a.cp | 6 | |
| 4.b | odd | 2 | 1 | 9920.2.a.co | 6 | ||
| 8.b | even | 2 | 1 | 1240.2.a.m | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 2480.2.a.ba | 6 | ||
| 40.f | even | 2 | 1 | 6200.2.a.w | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1240.2.a.m | ✓ | 6 | 8.b | even | 2 | 1 | |
| 2480.2.a.ba | 6 | 8.d | odd | 2 | 1 | ||
| 6200.2.a.w | 6 | 40.f | even | 2 | 1 | ||
| 9920.2.a.co | 6 | 4.b | odd | 2 | 1 | ||
| 9920.2.a.cp | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9920))\):
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\( T_{3}^{6} - T_{3}^{5} - 19T_{3}^{4} + 15T_{3}^{3} + 98T_{3}^{2} - 44T_{3} - 108 \)
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\( T_{7}^{6} + 2T_{7}^{5} - 36T_{7}^{4} - 52T_{7}^{3} + 352T_{7}^{2} + 320T_{7} - 576 \)
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\( T_{11}^{6} + 8T_{11}^{5} - 16T_{11}^{4} - 192T_{11}^{3} + 32T_{11}^{2} + 960T_{11} - 400 \)
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\( T_{13}^{6} + 4T_{13}^{5} - 48T_{13}^{4} - 248T_{13}^{3} + 288T_{13}^{2} + 3008T_{13} + 3856 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - T^{5} + \cdots - 108 \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} + 2 T^{5} + \cdots - 576 \)
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| $11$ |
\( T^{6} + 8 T^{5} + \cdots - 400 \)
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| $13$ |
\( T^{6} + 4 T^{5} + \cdots + 3856 \)
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| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 48 \)
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| $19$ |
\( T^{6} - 5 T^{5} + \cdots + 1376 \)
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| $23$ |
\( T^{6} + 12 T^{5} + \cdots + 1376 \)
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| $29$ |
\( T^{6} - 4 T^{5} + \cdots - 159792 \)
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| $31$ |
\( (T - 1)^{6} \)
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| $37$ |
\( T^{6} + 3 T^{5} + \cdots - 65956 \)
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| $41$ |
\( T^{6} - 17 T^{5} + \cdots + 344 \)
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| $43$ |
\( T^{6} - 17 T^{5} + \cdots - 31988 \)
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| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 33408 \)
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| $53$ |
\( T^{6} + 9 T^{5} + \cdots - 21908 \)
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| $59$ |
\( T^{6} + 39 T^{5} + \cdots - 4128 \)
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| $61$ |
\( T^{6} + 2 T^{5} + \cdots - 23232 \)
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| $67$ |
\( T^{6} + 12 T^{5} + \cdots - 153984 \)
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| $71$ |
\( T^{6} - 21 T^{5} + \cdots + 53888 \)
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| $73$ |
\( T^{6} - 43 T^{5} + \cdots - 44368 \)
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| $79$ |
\( T^{6} + 2 T^{5} + \cdots + 35712 \)
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| $83$ |
\( T^{6} - 5 T^{5} + \cdots - 396 \)
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| $89$ |
\( T^{6} - 6 T^{5} + \cdots - 40288 \)
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| $97$ |
\( T^{6} + 12 T^{5} + \cdots - 129728 \)
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