Newspace parameters
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(146.270043669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 456) |
| 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
\( x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4393\nu^{5} - 288577\nu^{4} + 17990538\nu^{3} + 820304928\nu^{2} - 23681399193\nu - 121791601863 ) / 828150480 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 30899430423 \nu + 62101268487 ) / 3312601920 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 17649022743 \nu + 58788666567 ) / 3312601920 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5543\nu^{5} - 64113\nu^{4} - 28829198\nu^{3} + 639322632\nu^{2} + 11916019503\nu - 53106861087 ) / 552100320 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6041\nu^{5} + 3471\nu^{4} + 28357426\nu^{3} - 539262904\nu^{2} - 9893470161\nu + 97030518849 ) / 184033440 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -36\beta_{5} - 39\beta_{4} - 29\beta_{3} + 143\beta_{2} - 3\beta _1 + 12622 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1044\beta_{5} - 69\beta_{4} + 6971\beta_{3} - 12485\beta_{2} + 759\beta _1 - 335116 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -77850\beta_{5} - 80907\beta_{4} - 148531\beta_{3} + 410857\beta_{2} - 21057\beta _1 + 25100360 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3890574\beta_{5} + 1532274\beta_{4} + 15932809\beta_{3} - 33812113\beta_{2} + 1903236\beta _1 - 1272876977 ) / 4 \)
|
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 | 9.00000 | 0 | −86.8767 | 0 | −2.94760 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −80.1366 | 0 | 221.037 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | −41.5770 | 0 | −98.3255 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 28.9245 | 0 | −210.915 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 9.00000 | 0 | 46.6057 | 0 | 58.6437 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 9.00000 | 0 | 68.0602 | 0 | 181.508 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 912.6.a.y | 6 | |
| 4.b | odd | 2 | 1 | 456.6.a.f | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 456.6.a.f | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 912.6.a.y | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 65T_{5}^{5} - 9557T_{5}^{4} - 445577T_{5}^{3} + 29529208T_{5}^{2} + 602356212T_{5} - 26557430352 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T - 9)^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 26557430352 \)
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| $7$ |
\( T^{6} + \cdots - 143821013760 \)
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| $11$ |
\( T^{6} + \cdots + 1637954874112 \)
|
| $13$ |
\( T^{6} + \cdots - 67\!\cdots\!72 \)
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| $17$ |
\( T^{6} + \cdots + 11\!\cdots\!20 \)
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| $19$ |
\( (T + 361)^{6} \)
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| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $29$ |
\( T^{6} + \cdots + 86\!\cdots\!48 \)
|
| $31$ |
\( T^{6} + \cdots + 17\!\cdots\!16 \)
|
| $37$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 13\!\cdots\!32 \)
|
| $43$ |
\( T^{6} + \cdots - 77\!\cdots\!60 \)
|
| $47$ |
\( T^{6} + \cdots + 60\!\cdots\!28 \)
|
| $53$ |
\( T^{6} + \cdots + 85\!\cdots\!12 \)
|
| $59$ |
\( T^{6} + \cdots - 49\!\cdots\!12 \)
|
| $61$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 85\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots + 11\!\cdots\!80 \)
|
| $73$ |
\( T^{6} + \cdots + 21\!\cdots\!08 \)
|
| $79$ |
\( T^{6} + \cdots + 13\!\cdots\!88 \)
|
| $83$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 37\!\cdots\!84 \)
|
| $97$ |
\( T^{6} + \cdots - 53\!\cdots\!24 \)
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