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: | \(x^{6} - x^{5} - 309949 x^{4} - 14548431 x^{3} + 25221499020 x^{2} + 1862570808000 x - 308009568384000\) |
| 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 \(x^{6} - x^{5} - 309949 x^{4} - 14548431 x^{3} + 25221499020 x^{2} + 1862570808000 x - 308009568384000\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\(-2024389 \nu^{5} + 1548086629 \nu^{4} + 626570347621 \nu^{3} - 319874939700201 \nu^{2} - 53542834336954620 \nu + 9427731423802360200\)\()/ 1236949910141400 \) |
| \(\beta_{3}\) | \(=\) | \((\)\(-27588299 \nu^{5} + 2827655399 \nu^{4} + 5325442304951 \nu^{3} - 1059348303204831 \nu^{2} - 93918776326732980 \nu + 89343994329715910400\)\()/ 7421699460848400 \) |
| \(\beta_{4}\) | \(=\) | \((\)\( -1384243 \nu^{5} + 593013523 \nu^{4} + 248055256927 \nu^{3} - 92775859244187 \nu^{2} - 8954762572396740 \nu + 1854087064738915200 \)\()/ 117804753346800 \) |
| \(\beta_{5}\) | \(=\) | \((\)\(-83321227 \nu^{5} + 14438167357 \nu^{4} + 20223862688443 \nu^{3} - 1872553860103833 \nu^{2} - 1120675241488354020 \nu + 12374299269754128000\)\()/ 3710849730424200 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{5} + \beta_{4} - 7 \beta_{3} - 5 \beta_{2} + 73 \beta_{1} + 103303\) |
| \(\nu^{3}\) | \(=\) | \(425 \beta_{5} - 771 \beta_{4} - 1055 \beta_{3} + 2101 \beta_{2} + 147337 \beta_{1} + 7404281\) |
| \(\nu^{4}\) | \(=\) | \(105869 \beta_{5} + 316233 \beta_{4} - 1485023 \beta_{3} - 349969 \beta_{2} + 22069185 \beta_{1} + 15214288683\) |
| \(\nu^{5}\) | \(=\) | \(54491425 \beta_{5} - 154814415 \beta_{4} - 356083435 \beta_{3} + 561683925 \beta_{2} + 24495440641 \beta_{1} + 2260450105145\) |
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 \( T_{5}^{6} + 181 T_{5}^{5} - 296299 T_{5}^{4} - 22096449 T_{5}^{3} + 24869553210 T_{5}^{2} - 343324745100 T_{5} - \)\(34\!\cdots\!00\)\( \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( ( 8 + T )^{6} \) |
| $3$ | \( ( 27 + T )^{6} \) |
| $5$ | \( -341044841259000 - 343324745100 T + 24869553210 T^{2} - 22096449 T^{3} - 296299 T^{4} + 181 T^{5} + T^{6} \) |
| $7$ | \( ( -343 + T )^{6} \) |
| $11$ | \( -\)\(31\!\cdots\!36\)\( + 1736123897206401984 T + 531250914534696 T^{2} - 233699671890 T^{3} - 43044145 T^{4} + 6130 T^{5} + T^{6} \) |
| $13$ | \( ( -2197 + T )^{6} \) |
| $17$ | \( \)\(35\!\cdots\!08\)\( + \)\(22\!\cdots\!68\)\( T - 5845188076735428 T^{2} - 29198969866242 T^{3} - 829139125 T^{4} + 34610 T^{5} + T^{6} \) |
| $19$ | \( -\)\(37\!\cdots\!40\)\( + \)\(87\!\cdots\!88\)\( T + 5343265149887319716 T^{2} - 9492878171013 T^{3} - 4734740997 T^{4} + 4085 T^{5} + T^{6} \) |
| $23$ | \( -\)\(16\!\cdots\!00\)\( + \)\(31\!\cdots\!72\)\( T + \)\(11\!\cdots\!24\)\( T^{2} + 4727534295759 T^{3} - 20521250801 T^{4} - 1515 T^{5} + T^{6} \) |
| $29$ | \( -\)\(21\!\cdots\!16\)\( - \)\(10\!\cdots\!52\)\( T + \)\(91\!\cdots\!98\)\( T^{2} - 216235904101143 T^{3} - 63710108383 T^{4} + 59395 T^{5} + T^{6} \) |
| $31$ | \( -\)\(24\!\cdots\!60\)\( + \)\(17\!\cdots\!96\)\( T - \)\(21\!\cdots\!40\)\( T^{2} - 2857508603742272 T^{3} + 72784985460 T^{4} - 478241 T^{5} + T^{6} \) |
| $37$ | \( \)\(38\!\cdots\!56\)\( - \)\(36\!\cdots\!04\)\( T - \)\(33\!\cdots\!92\)\( T^{2} + 99163911793545626 T^{3} - 117782223737 T^{4} - 574310 T^{5} + T^{6} \) |
| $41$ | \( \)\(41\!\cdots\!08\)\( - \)\(34\!\cdots\!20\)\( T + \)\(47\!\cdots\!16\)\( T^{2} + 245803952696366640 T^{3} - 622802708432 T^{4} - 201552 T^{5} + T^{6} \) |
| $43$ | \( \)\(64\!\cdots\!16\)\( - \)\(27\!\cdots\!80\)\( T - \)\(68\!\cdots\!04\)\( T^{2} + 467557746167504117 T^{3} - 432298077205 T^{4} - 728605 T^{5} + T^{6} \) |
| $47$ | \( \)\(11\!\cdots\!76\)\( + \)\(12\!\cdots\!92\)\( T + \)\(36\!\cdots\!12\)\( T^{2} - 151915489918348752 T^{3} - 1287424320764 T^{4} - 227615 T^{5} + T^{6} \) |
| $53$ | \( -\)\(48\!\cdots\!60\)\( + \)\(24\!\cdots\!68\)\( T + \)\(13\!\cdots\!28\)\( T^{2} - 1189810742698797576 T^{3} - 4243282064810 T^{4} - 26321 T^{5} + T^{6} \) |
| $59$ | \( -\)\(16\!\cdots\!00\)\( + \)\(43\!\cdots\!92\)\( T + \)\(11\!\cdots\!84\)\( T^{2} - 1202809625885360064 T^{3} - 7301321480608 T^{4} - 478280 T^{5} + T^{6} \) |
| $61$ | \( -\)\(13\!\cdots\!20\)\( + \)\(85\!\cdots\!72\)\( T + \)\(18\!\cdots\!92\)\( T^{2} - 2581773418155138100 T^{3} - 9282662906231 T^{4} + 501406 T^{5} + T^{6} \) |
| $67$ | \( \)\(10\!\cdots\!32\)\( + \)\(68\!\cdots\!36\)\( T + \)\(19\!\cdots\!52\)\( T^{2} - 30812333776301144088 T^{3} - 9791651241644 T^{4} + 3156366 T^{5} + T^{6} \) |
| $71$ | \( -\)\(16\!\cdots\!20\)\( - \)\(33\!\cdots\!04\)\( T + \)\(38\!\cdots\!92\)\( T^{2} - 20665613994075126720 T^{3} - 37121399529968 T^{4} + 2003644 T^{5} + T^{6} \) |
| $73$ | \( \)\(39\!\cdots\!28\)\( - \)\(25\!\cdots\!16\)\( T + \)\(67\!\cdots\!22\)\( T^{2} + 36272125819456405523 T^{3} - 9855730520255 T^{4} - 3659111 T^{5} + T^{6} \) |
| $79$ | \( -\)\(28\!\cdots\!00\)\( - \)\(27\!\cdots\!64\)\( T + \)\(67\!\cdots\!56\)\( T^{2} + 20776819780345680956 T^{3} - 47075544906248 T^{4} - 1131065 T^{5} + T^{6} \) |
| $83$ | \( \)\(29\!\cdots\!28\)\( + \)\(14\!\cdots\!08\)\( T - \)\(42\!\cdots\!52\)\( T^{2} - \)\(26\!\cdots\!08\)\( T^{3} - 6623928686524 T^{4} + 9629297 T^{5} + T^{6} \) |
| $89$ | \( -\)\(15\!\cdots\!00\)\( + \)\(40\!\cdots\!40\)\( T - \)\(21\!\cdots\!04\)\( T^{2} - \)\(47\!\cdots\!36\)\( T^{3} + 94721604221654 T^{4} + 21977377 T^{5} + T^{6} \) |
| $97$ | \( -\)\(27\!\cdots\!80\)\( - \)\(23\!\cdots\!96\)\( T - \)\(12\!\cdots\!48\)\( T^{2} - \)\(16\!\cdots\!48\)\( T^{3} + 108642148800462 T^{4} + 26386649 T^{5} + T^{6} \) |
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