Newspace parameters
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(170.562223914\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
| Defining polynomial: |
\( x^{6} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2024389 \nu^{5} + 1548086629 \nu^{4} + 626570347621 \nu^{3} - 319874939700201 \nu^{2} + \cdots + 94\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 27588299 \nu^{5} + 2827655399 \nu^{4} + 5325442304951 \nu^{3} + \cdots + 89\!\cdots\!00 ) / 74\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1384243 \nu^{5} + 593013523 \nu^{4} + 248055256927 \nu^{3} - 92775859244187 \nu^{2} + \cdots + 18\!\cdots\!00 ) / 117804753346800 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 83321227 \nu^{5} + 14438167357 \nu^{4} + 20223862688443 \nu^{3} + \cdots + 12\!\cdots\!00 ) / 37\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 7\beta_{3} - 5\beta_{2} + 73\beta _1 + 103303 \)
|
| \(\nu^{3}\) | \(=\) |
\( 425\beta_{5} - 771\beta_{4} - 1055\beta_{3} + 2101\beta_{2} + 147337\beta _1 + 7404281 \)
|
| \(\nu^{4}\) | \(=\) |
\( 105869\beta_{5} + 316233\beta_{4} - 1485023\beta_{3} - 349969\beta_{2} + 22069185\beta _1 + 15214288683 \)
|
| \(\nu^{5}\) | \(=\) |
\( 54491425 \beta_{5} - 154814415 \beta_{4} - 356083435 \beta_{3} + 561683925 \beta_{2} + 24495440641 \beta _1 + 2260450105145 \)
|
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 |
|
−8.00000 | −27.0000 | 64.0000 | −487.447 | 216.000 | 343.000 | −512.000 | 729.000 | 3899.58 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −27.0000 | 64.0000 | −378.888 | 216.000 | 343.000 | −512.000 | 729.000 | 3031.11 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −27.0000 | 64.0000 | −113.468 | 216.000 | 343.000 | −512.000 | 729.000 | 907.742 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | −27.0000 | 64.0000 | 163.695 | 216.000 | 343.000 | −512.000 | 729.000 | −1309.56 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | −27.0000 | 64.0000 | 279.836 | 216.000 | 343.000 | −512.000 | 729.000 | −2238.69 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −8.00000 | −27.0000 | 64.0000 | 355.273 | 216.000 | 343.000 | −512.000 | 729.000 | −2842.18 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 546.8.a.n | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.8.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 181T_{5}^{5} - 296299T_{5}^{4} - 22096449T_{5}^{3} + 24869553210T_{5}^{2} - 343324745100T_{5} - 341044841259000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 8)^{6} \)
$3$
\( (T + 27)^{6} \)
$5$
\( T^{6} + \cdots - 341044841259000 \)
$7$
\( (T - 343)^{6} \)
$11$
\( T^{6} + 6130 T^{5} + \cdots - 31\!\cdots\!36 \)
$13$
\( (T - 2197)^{6} \)
$17$
\( T^{6} + 34610 T^{5} + \cdots + 35\!\cdots\!08 \)
$19$
\( T^{6} + 4085 T^{5} + \cdots - 37\!\cdots\!40 \)
$23$
\( T^{6} - 1515 T^{5} + \cdots - 16\!\cdots\!00 \)
$29$
\( T^{6} + 59395 T^{5} + \cdots - 21\!\cdots\!16 \)
$31$
\( T^{6} - 478241 T^{5} + \cdots - 24\!\cdots\!60 \)
$37$
\( T^{6} - 574310 T^{5} + \cdots + 38\!\cdots\!56 \)
$41$
\( T^{6} - 201552 T^{5} + \cdots + 41\!\cdots\!08 \)
$43$
\( T^{6} - 728605 T^{5} + \cdots + 64\!\cdots\!16 \)
$47$
\( T^{6} - 227615 T^{5} + \cdots + 11\!\cdots\!76 \)
$53$
\( T^{6} - 26321 T^{5} + \cdots - 48\!\cdots\!60 \)
$59$
\( T^{6} - 478280 T^{5} + \cdots - 16\!\cdots\!00 \)
$61$
\( T^{6} + 501406 T^{5} + \cdots - 13\!\cdots\!20 \)
$67$
\( T^{6} + 3156366 T^{5} + \cdots + 10\!\cdots\!32 \)
$71$
\( T^{6} + 2003644 T^{5} + \cdots - 16\!\cdots\!20 \)
$73$
\( T^{6} - 3659111 T^{5} + \cdots + 39\!\cdots\!28 \)
$79$
\( T^{6} - 1131065 T^{5} + \cdots - 28\!\cdots\!00 \)
$83$
\( T^{6} + 9629297 T^{5} + \cdots + 29\!\cdots\!28 \)
$89$
\( T^{6} + 21977377 T^{5} + \cdots - 15\!\cdots\!00 \)
$97$
\( T^{6} + 26386649 T^{5} + \cdots - 27\!\cdots\!80 \)
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