Properties

Label 1008.4.bt.b
Level $1008$
Weight $4$
Character orbit 1008.bt
Analytic conductor $59.474$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + 1239640536 x^{6} - 1407381792 x^{5} - 1961185792 x^{4} + 4297169408 x^{3} + 2991779296 x^{2} - 11217342336 x + 7375227456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{5} + ( -1 - 2 \beta_{1} + \beta_{6} + \beta_{8} ) q^{7} +O(q^{10})\) \( q + \beta_{9} q^{5} + ( -1 - 2 \beta_{1} + \beta_{6} + \beta_{8} ) q^{7} + ( \beta_{3} + \beta_{5} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{8} ) q^{13} + ( -\beta_{3} + 3 \beta_{5} - 2 \beta_{9} + \beta_{11} - \beta_{14} ) q^{17} + ( 6 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{6} + \beta_{12} - \beta_{13} ) q^{19} + ( \beta_{3} + 2 \beta_{5} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{14} ) q^{23} + ( -26 - 27 \beta_{1} + 5 \beta_{2} + 4 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + \beta_{12} - 2 \beta_{13} ) q^{25} + ( -4 \beta_{3} - 5 \beta_{5} + 12 \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{14} - \beta_{15} ) q^{29} + ( 25 - 34 \beta_{1} - 6 \beta_{2} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 4 \beta_{12} ) q^{31} + ( 5 \beta_{3} - \beta_{4} - 9 \beta_{5} - 4 \beta_{9} - 2 \beta_{11} + \beta_{15} ) q^{35} + ( 7 - 6 \beta_{1} + 4 \beta_{2} - 9 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} ) q^{37} + ( -4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + \beta_{14} + \beta_{15} ) q^{41} + ( -92 - 8 \beta_{1} - 5 \beta_{2} + 5 \beta_{6} - 5 \beta_{7} + 11 \beta_{8} + 6 \beta_{12} - 3 \beta_{13} ) q^{43} + ( 9 \beta_{3} - \beta_{4} + 3 \beta_{5} - 17 \beta_{9} - 3 \beta_{10} + 3 \beta_{14} - \beta_{15} ) q^{47} + ( 25 - 2 \beta_{1} + 4 \beta_{6} - 2 \beta_{8} + 7 \beta_{13} ) q^{49} + ( -5 \beta_{3} + \beta_{4} - 11 \beta_{5} - 5 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{14} ) q^{53} + ( -60 - 111 \beta_{1} - 11 \beta_{2} - 11 \beta_{8} + 2 \beta_{13} ) q^{55} + ( \beta_{4} - 24 \beta_{5} + 25 \beta_{9} - 4 \beta_{11} - 4 \beta_{14} - 2 \beta_{15} ) q^{59} + ( -136 - 70 \beta_{1} - 3 \beta_{2} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} ) q^{61} + ( -\beta_{4} - 22 \beta_{5} + 15 \beta_{9} - 8 \beta_{14} + \beta_{15} ) q^{65} + ( 194 + 188 \beta_{1} - 13 \beta_{2} - 7 \beta_{6} - 7 \beta_{8} - 6 \beta_{12} + 12 \beta_{13} ) q^{67} + ( -9 \beta_{3} - 3 \beta_{5} - 4 \beta_{9} - \beta_{10} - \beta_{11} - 10 \beta_{14} + 9 \beta_{15} ) q^{71} + ( -118 + 107 \beta_{1} - 17 \beta_{2} + 20 \beta_{6} - 26 \beta_{7} - 6 \beta_{8} + 3 \beta_{12} ) q^{73} + ( 11 \beta_{3} + 9 \beta_{4} + 40 \beta_{5} - 7 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 5 \beta_{14} - 8 \beta_{15} ) q^{77} + ( 13 - 39 \beta_{1} - 9 \beta_{2} - 17 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} - 4 \beta_{12} - 4 \beta_{13} ) q^{79} + ( -3 \beta_{3} + 16 \beta_{4} + 32 \beta_{5} - 2 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} - 3 \beta_{14} - 8 \beta_{15} ) q^{83} + ( 452 - 20 \beta_{1} - 21 \beta_{2} - 6 \beta_{6} + 6 \beta_{7} + 19 \beta_{8} - 2 \beta_{12} + \beta_{13} ) q^{85} + ( 18 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 20 \beta_{9} - 10 \beta_{10} + 4 \beta_{14} + 8 \beta_{15} ) q^{89} + ( 30 + 368 \beta_{1} + 7 \beta_{2} + 5 \beta_{6} + 5 \beta_{8} + 7 \beta_{12} ) q^{91} + ( -12 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 31 \beta_{9} - 10 \beta_{10} + 5 \beta_{11} - 5 \beta_{14} ) q^{95} + ( 294 + 611 \beta_{1} - 42 \beta_{2} - 10 \beta_{6} - 10 \beta_{7} - 42 \beta_{8} - \beta_{13} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{7} + O(q^{10}) \) \( 16q + 4q^{7} + 72q^{19} - 212q^{25} + 708q^{31} + 76q^{37} - 1408q^{43} + 400q^{49} - 1632q^{61} + 1528q^{67} - 2700q^{73} + 364q^{79} + 7392q^{85} - 2472q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + 1239640536 x^{6} - 1407381792 x^{5} - 1961185792 x^{4} + 4297169408 x^{3} + 2991779296 x^{2} - 11217342336 x + 7375227456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-1229561931 \nu^{15} + 2498779578 \nu^{14} + 394581872608 \nu^{13} - 1317945813970 \nu^{12} - 48388978127485 \nu^{11} + 216626897426602 \nu^{10} + 2708906003712780 \nu^{9} - 15005503515059110 \nu^{8} - 62911168783150524 \nu^{7} + 432720973755039680 \nu^{6} + 328948851683104464 \nu^{5} - 3693022570101876040 \nu^{4} - 1858925629616754192 \nu^{3} + 13888390311265898368 \nu^{2} + 3151631933173997312 \nu - 24728853666847532128\)\()/ 13574653155388836608 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(61\!\cdots\!81\)\( \nu^{15} - \)\(15\!\cdots\!75\)\( \nu^{14} - \)\(24\!\cdots\!20\)\( \nu^{13} + \)\(44\!\cdots\!00\)\( \nu^{12} + \)\(28\!\cdots\!71\)\( \nu^{11} - \)\(51\!\cdots\!97\)\( \nu^{10} - \)\(90\!\cdots\!44\)\( \nu^{9} + \)\(27\!\cdots\!00\)\( \nu^{8} - \)\(29\!\cdots\!56\)\( \nu^{7} - \)\(55\!\cdots\!72\)\( \nu^{6} + \)\(16\!\cdots\!04\)\( \nu^{5} + \)\(28\!\cdots\!96\)\( \nu^{4} - \)\(43\!\cdots\!48\)\( \nu^{3} + \)\(20\!\cdots\!32\)\( \nu^{2} + \)\(86\!\cdots\!56\)\( \nu - \)\(20\!\cdots\!52\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(11\!\cdots\!52\)\( \nu^{15} - \)\(21\!\cdots\!81\)\( \nu^{14} + \)\(29\!\cdots\!22\)\( \nu^{13} + \)\(56\!\cdots\!88\)\( \nu^{12} - \)\(36\!\cdots\!28\)\( \nu^{11} - \)\(57\!\cdots\!55\)\( \nu^{10} + \)\(28\!\cdots\!54\)\( \nu^{9} + \)\(26\!\cdots\!20\)\( \nu^{8} - \)\(13\!\cdots\!20\)\( \nu^{7} - \)\(49\!\cdots\!12\)\( \nu^{6} + \)\(30\!\cdots\!56\)\( \nu^{5} + \)\(60\!\cdots\!16\)\( \nu^{4} - \)\(13\!\cdots\!12\)\( \nu^{3} - \)\(21\!\cdots\!72\)\( \nu^{2} + \)\(23\!\cdots\!88\)\( \nu - \)\(56\!\cdots\!12\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(24\!\cdots\!91\)\( \nu^{15} + \)\(62\!\cdots\!62\)\( \nu^{14} - \)\(70\!\cdots\!74\)\( \nu^{13} - \)\(47\!\cdots\!02\)\( \nu^{12} + \)\(76\!\cdots\!05\)\( \nu^{11} - \)\(71\!\cdots\!38\)\( \nu^{10} - \)\(36\!\cdots\!50\)\( \nu^{9} + \)\(78\!\cdots\!74\)\( \nu^{8} + \)\(61\!\cdots\!96\)\( \nu^{7} - \)\(13\!\cdots\!44\)\( \nu^{6} + \)\(84\!\cdots\!92\)\( \nu^{5} - \)\(34\!\cdots\!32\)\( \nu^{4} + \)\(84\!\cdots\!76\)\( \nu^{3} + \)\(10\!\cdots\!48\)\( \nu^{2} - \)\(20\!\cdots\!08\)\( \nu - \)\(18\!\cdots\!08\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(13\!\cdots\!96\)\( \nu^{15} + \)\(28\!\cdots\!71\)\( \nu^{14} + \)\(39\!\cdots\!79\)\( \nu^{13} - \)\(16\!\cdots\!25\)\( \nu^{12} - \)\(39\!\cdots\!36\)\( \nu^{11} + \)\(27\!\cdots\!69\)\( \nu^{10} + \)\(14\!\cdots\!13\)\( \nu^{9} - \)\(16\!\cdots\!47\)\( \nu^{8} + \)\(11\!\cdots\!24\)\( \nu^{7} + \)\(35\!\cdots\!20\)\( \nu^{6} - \)\(13\!\cdots\!36\)\( \nu^{5} + \)\(71\!\cdots\!72\)\( \nu^{4} + \)\(32\!\cdots\!64\)\( \nu^{3} - \)\(23\!\cdots\!56\)\( \nu^{2} - \)\(45\!\cdots\!32\)\( \nu + \)\(88\!\cdots\!72\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(39\!\cdots\!72\)\( \nu^{15} + \)\(22\!\cdots\!97\)\( \nu^{14} + \)\(11\!\cdots\!66\)\( \nu^{13} - \)\(30\!\cdots\!92\)\( \nu^{12} - \)\(11\!\cdots\!96\)\( \nu^{11} + \)\(57\!\cdots\!71\)\( \nu^{10} + \)\(47\!\cdots\!98\)\( \nu^{9} - \)\(38\!\cdots\!36\)\( \nu^{8} - \)\(11\!\cdots\!72\)\( \nu^{7} + \)\(85\!\cdots\!80\)\( \nu^{6} - \)\(28\!\cdots\!36\)\( \nu^{5} + \)\(15\!\cdots\!64\)\( \nu^{4} + \)\(82\!\cdots\!44\)\( \nu^{3} - \)\(11\!\cdots\!48\)\( \nu^{2} - \)\(16\!\cdots\!80\)\( \nu + \)\(23\!\cdots\!48\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(59\!\cdots\!29\)\( \nu^{15} + \)\(37\!\cdots\!61\)\( \nu^{14} + \)\(17\!\cdots\!02\)\( \nu^{13} - \)\(14\!\cdots\!86\)\( \nu^{12} - \)\(18\!\cdots\!07\)\( \nu^{11} + \)\(18\!\cdots\!27\)\( \nu^{10} + \)\(62\!\cdots\!94\)\( \nu^{9} - \)\(10\!\cdots\!30\)\( \nu^{8} + \)\(11\!\cdots\!84\)\( \nu^{7} + \)\(21\!\cdots\!16\)\( \nu^{6} - \)\(83\!\cdots\!48\)\( \nu^{5} + \)\(37\!\cdots\!60\)\( \nu^{4} + \)\(23\!\cdots\!28\)\( \nu^{3} - \)\(18\!\cdots\!96\)\( \nu^{2} - \)\(43\!\cdots\!80\)\( \nu + \)\(59\!\cdots\!04\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(10\!\cdots\!68\)\( \nu^{15} - \)\(33\!\cdots\!01\)\( \nu^{14} - \)\(32\!\cdots\!84\)\( \nu^{13} + \)\(15\!\cdots\!72\)\( \nu^{12} + \)\(34\!\cdots\!60\)\( \nu^{11} - \)\(23\!\cdots\!43\)\( \nu^{10} - \)\(14\!\cdots\!56\)\( \nu^{9} + \)\(14\!\cdots\!32\)\( \nu^{8} + \)\(78\!\cdots\!52\)\( \nu^{7} - \)\(31\!\cdots\!08\)\( \nu^{6} + \)\(92\!\cdots\!20\)\( \nu^{5} - \)\(22\!\cdots\!48\)\( \nu^{4} - \)\(25\!\cdots\!28\)\( \nu^{3} + \)\(76\!\cdots\!04\)\( \nu^{2} + \)\(48\!\cdots\!36\)\( \nu - \)\(37\!\cdots\!96\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(12\!\cdots\!66\)\( \nu^{15} - \)\(16\!\cdots\!29\)\( \nu^{14} - \)\(37\!\cdots\!16\)\( \nu^{13} + \)\(76\!\cdots\!36\)\( \nu^{12} + \)\(41\!\cdots\!06\)\( \nu^{11} - \)\(15\!\cdots\!07\)\( \nu^{10} - \)\(19\!\cdots\!88\)\( \nu^{9} + \)\(11\!\cdots\!08\)\( \nu^{8} + \)\(24\!\cdots\!24\)\( \nu^{7} - \)\(28\!\cdots\!24\)\( \nu^{6} + \)\(45\!\cdots\!40\)\( \nu^{5} + \)\(58\!\cdots\!36\)\( \nu^{4} - \)\(14\!\cdots\!72\)\( \nu^{3} - \)\(92\!\cdots\!48\)\( \nu^{2} + \)\(28\!\cdots\!04\)\( \nu - \)\(11\!\cdots\!36\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(13\!\cdots\!89\)\( \nu^{15} - \)\(71\!\cdots\!40\)\( \nu^{14} - \)\(39\!\cdots\!20\)\( \nu^{13} + \)\(28\!\cdots\!00\)\( \nu^{12} + \)\(38\!\cdots\!27\)\( \nu^{11} - \)\(39\!\cdots\!76\)\( \nu^{10} - \)\(10\!\cdots\!48\)\( \nu^{9} + \)\(22\!\cdots\!96\)\( \nu^{8} - \)\(38\!\cdots\!04\)\( \nu^{7} - \)\(41\!\cdots\!24\)\( \nu^{6} + \)\(20\!\cdots\!20\)\( \nu^{5} - \)\(21\!\cdots\!44\)\( \nu^{4} - \)\(39\!\cdots\!96\)\( \nu^{3} + \)\(61\!\cdots\!76\)\( \nu^{2} + \)\(87\!\cdots\!44\)\( \nu - \)\(16\!\cdots\!72\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(14\!\cdots\!19\)\( \nu^{15} - \)\(90\!\cdots\!00\)\( \nu^{14} - \)\(43\!\cdots\!02\)\( \nu^{13} + \)\(34\!\cdots\!32\)\( \nu^{12} + \)\(42\!\cdots\!93\)\( \nu^{11} - \)\(47\!\cdots\!04\)\( \nu^{10} - \)\(11\!\cdots\!74\)\( \nu^{9} + \)\(26\!\cdots\!12\)\( \nu^{8} - \)\(47\!\cdots\!16\)\( \nu^{7} - \)\(52\!\cdots\!00\)\( \nu^{6} + \)\(24\!\cdots\!72\)\( \nu^{5} - \)\(18\!\cdots\!76\)\( \nu^{4} - \)\(60\!\cdots\!32\)\( \nu^{3} + \)\(52\!\cdots\!32\)\( \nu^{2} + \)\(14\!\cdots\!52\)\( \nu - \)\(16\!\cdots\!68\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(15\!\cdots\!47\)\( \nu^{15} - \)\(87\!\cdots\!12\)\( \nu^{14} - \)\(43\!\cdots\!06\)\( \nu^{13} + \)\(11\!\cdots\!78\)\( \nu^{12} + \)\(46\!\cdots\!37\)\( \nu^{11} - \)\(21\!\cdots\!52\)\( \nu^{10} - \)\(19\!\cdots\!58\)\( \nu^{9} + \)\(14\!\cdots\!38\)\( \nu^{8} + \)\(12\!\cdots\!84\)\( \nu^{7} - \)\(33\!\cdots\!92\)\( \nu^{6} + \)\(89\!\cdots\!08\)\( \nu^{5} + \)\(32\!\cdots\!48\)\( \nu^{4} - \)\(24\!\cdots\!52\)\( \nu^{3} - \)\(50\!\cdots\!88\)\( \nu^{2} + \)\(45\!\cdots\!84\)\( \nu - \)\(21\!\cdots\!56\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(10\!\cdots\!21\)\( \nu^{15} + \)\(36\!\cdots\!98\)\( \nu^{14} + \)\(31\!\cdots\!08\)\( \nu^{13} - \)\(16\!\cdots\!34\)\( \nu^{12} - \)\(32\!\cdots\!63\)\( \nu^{11} + \)\(24\!\cdots\!54\)\( \nu^{10} + \)\(12\!\cdots\!72\)\( \nu^{9} - \)\(14\!\cdots\!30\)\( \nu^{8} + \)\(43\!\cdots\!16\)\( \nu^{7} + \)\(31\!\cdots\!00\)\( \nu^{6} - \)\(10\!\cdots\!68\)\( \nu^{5} + \)\(21\!\cdots\!60\)\( \nu^{4} + \)\(28\!\cdots\!20\)\( \nu^{3} - \)\(17\!\cdots\!76\)\( \nu^{2} - \)\(54\!\cdots\!28\)\( \nu + \)\(63\!\cdots\!76\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(22\!\cdots\!43\)\( \nu^{15} + \)\(69\!\cdots\!67\)\( \nu^{14} + \)\(65\!\cdots\!76\)\( \nu^{13} - \)\(32\!\cdots\!98\)\( \nu^{12} - \)\(69\!\cdots\!53\)\( \nu^{11} + \)\(48\!\cdots\!01\)\( \nu^{10} + \)\(26\!\cdots\!44\)\( \nu^{9} - \)\(29\!\cdots\!90\)\( \nu^{8} + \)\(67\!\cdots\!96\)\( \nu^{7} + \)\(64\!\cdots\!32\)\( \nu^{6} - \)\(20\!\cdots\!36\)\( \nu^{5} + \)\(43\!\cdots\!80\)\( \nu^{4} + \)\(55\!\cdots\!36\)\( \nu^{3} - \)\(20\!\cdots\!08\)\( \nu^{2} - \)\(11\!\cdots\!24\)\( \nu + \)\(11\!\cdots\!96\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(28\!\cdots\!32\)\( \nu^{15} - \)\(52\!\cdots\!45\)\( \nu^{14} - \)\(83\!\cdots\!02\)\( \nu^{13} + \)\(31\!\cdots\!26\)\( \nu^{12} + \)\(88\!\cdots\!52\)\( \nu^{11} - \)\(51\!\cdots\!43\)\( \nu^{10} - \)\(36\!\cdots\!42\)\( \nu^{9} + \)\(32\!\cdots\!50\)\( \nu^{8} + \)\(77\!\cdots\!76\)\( \nu^{7} - \)\(71\!\cdots\!60\)\( \nu^{6} + \)\(21\!\cdots\!00\)\( \nu^{5} - \)\(71\!\cdots\!32\)\( \nu^{4} - \)\(51\!\cdots\!44\)\( \nu^{3} + \)\(29\!\cdots\!60\)\( \nu^{2} + \)\(11\!\cdots\!44\)\( \nu - \)\(13\!\cdots\!52\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{15} + 3 \beta_{13} - 6 \beta_{12} + 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - 12 \beta_{2} - 9 \beta_{1} + 9\)\()/54\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{15} + 6 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 6 \beta_{10} + 32 \beta_{9} - 60 \beta_{8} - 15 \beta_{7} + 15 \beta_{6} - 58 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 66 \beta_{2} + 171 \beta_{1} + 2097\)\()/54\)
\(\nu^{3}\)\(=\)\((\)\(262 \beta_{15} + 153 \beta_{14} + 207 \beta_{13} - 360 \beta_{12} - 126 \beta_{11} + 63 \beta_{10} - 8 \beta_{9} + 594 \beta_{8} - 333 \beta_{7} + 279 \beta_{6} - 259 \beta_{5} - 258 \beta_{4} + 63 \beta_{3} - 954 \beta_{2} - 693 \beta_{1} - 5841\)\()/54\)
\(\nu^{4}\)\(=\)\((\)\(-1512 \beta_{15} + 612 \beta_{14} - 747 \beta_{13} + 1386 \beta_{12} + 2016 \beta_{11} - 972 \beta_{10} + 3240 \beta_{9} - 4824 \beta_{8} - 27 \beta_{7} + 1647 \beta_{6} - 4732 \beta_{5} + 1496 \beta_{4} - 972 \beta_{3} + 7290 \beta_{2} + 29763 \beta_{1} + 156969\)\()/54\)
\(\nu^{5}\)\(=\)\((\)\(32630 \beta_{15} + 16515 \beta_{14} + 19083 \beta_{13} - 27726 \beta_{12} - 19350 \beta_{11} + 9045 \beta_{10} - 15340 \beta_{9} + 54780 \beta_{8} - 30477 \beta_{7} + 13197 \beta_{6} - 6157 \beta_{5} - 30902 \beta_{4} + 10485 \beta_{3} - 82146 \beta_{2} - 179703 \beta_{1} - 816273\)\()/54\)
\(\nu^{6}\)\(=\)\((\)\(-247460 \beta_{15} + 30918 \beta_{14} - 112923 \beta_{13} + 164556 \beta_{12} + 254220 \beta_{11} - 112062 \beta_{10} + 359104 \beta_{9} - 403758 \beta_{8} + 144729 \beta_{7} + 186021 \beta_{6} - 406954 \beta_{5} + 232452 \beta_{4} - 116382 \beta_{3} + 768654 \beta_{2} + 4634181 \beta_{1} + 13960017\)\()/54\)
\(\nu^{7}\)\(=\)\((\)\(3810126 \beta_{15} + 1493289 \beta_{14} + 1989039 \beta_{13} - 2371200 \beta_{12} - 2477664 \beta_{11} + 1037421 \beta_{10} - 2684670 \beta_{9} + 4879122 \beta_{8} - 3297981 \beta_{7} - 135645 \beta_{6} + 663401 \beta_{5} - 3370162 \beta_{4} + 1398789 \beta_{3} - 7714254 \beta_{2} - 35715861 \beta_{1} - 96370029\)\()/54\)
\(\nu^{8}\)\(=\)\((\)\(-33101968 \beta_{15} + 310632 \beta_{14} - 15078675 \beta_{13} + 17453526 \beta_{12} + 29246880 \beta_{11} - 11119128 \beta_{10} + 41540144 \beta_{9} - 34834524 \beta_{8} + 27802893 \beta_{7} + 23997219 \beta_{6} - 36285976 \beta_{5} + 28678192 \beta_{4} - 13139160 \beta_{3} + 79332690 \beta_{2} + 646343919 \beta_{1} + 1341758745\)\()/54\)
\(\nu^{9}\)\(=\)\((\)\(433006054 \beta_{15} + 126418959 \beta_{14} + 217504551 \beta_{13} - 213140382 \beta_{12} - 290064474 \beta_{11} + 101356605 \beta_{10} - 374035880 \beta_{9} + 420319500 \beta_{8} - 394012569 \beta_{7} - 152949975 \beta_{6} + 149655719 \beta_{5} - 348832710 \beta_{4} + 165791997 \beta_{3} - 762484458 \beta_{2} - 5729178195 \beta_{1} - 10853159733\)\()/54\)
\(\nu^{10}\)\(=\)\((\)\(-4062381420 \beta_{15} - 138233742 \beta_{14} - 1892854743 \beta_{13} + 1767898236 \beta_{12} + 3216969492 \beta_{11} - 969690954 \beta_{10} + 4874326800 \beta_{9} - 2977985478 \beta_{8} + 4022831661 \beta_{7} + 3184620969 \beta_{6} - 3297732478 \beta_{5} + 3160458860 \beta_{4} - 1462291914 \beta_{3} + 8082357414 \beta_{2} + 84322941993 \beta_{1} + 133822224693\)\()/54\)
\(\nu^{11}\)\(=\)\((\)\(48499514030 \beta_{15} + 10112577321 \beta_{14} + 24136734963 \beta_{13} - 19515342264 \beta_{12} - 32215401768 \beta_{11} + 8414179125 \beta_{10} - 47933990566 \beta_{9} + 34614474882 \beta_{8} - 48169974153 \beta_{7} - 28885566849 \beta_{6} + 20000068937 \beta_{5} - 34702271426 \beta_{4} + 18381347853 \beta_{3} - 76725341526 \beta_{2} - 810563558121 \beta_{1} - 1187441814825\)\()/54\)
\(\nu^{12}\)\(=\)\((\)\(-475514212088 \beta_{15} - 19107195348 \beta_{14} - 227182483671 \beta_{13} + 172645902462 \beta_{12} + 343680414960 \beta_{11} - 70338926292 \beta_{10} + 572400682216 \beta_{9} - 240825302796 \beta_{8} + 521917357929 \beta_{7} + 415038592551 \beta_{6} - 302093990164 \beta_{5} + 323331111720 \beta_{4} - 160686785652 \beta_{3} + 812336246250 \beta_{2} + 10530888691203 \beta_{1} + 13532011977573\)\()/54\)
\(\nu^{13}\)\(=\)\((\)\(5371594977942 \beta_{15} + 739350340911 \beta_{14} + 2677131100059 \beta_{13} - 1777856438046 \beta_{12} - 3450978812034 \beta_{11} + 531828979461 \beta_{10} - 5875603977024 \beta_{9} + 2631892112868 \beta_{8} - 5828570580597 \beta_{7} - 4237713608403 \beta_{6} + 2243982590087 \beta_{5} - 3318605490406 \beta_{4} + 1957956794469 \beta_{3} - 7689064460082 \beta_{2} - 105979054369167 \beta_{1} - 126521367686001\)\()/54\)
\(\nu^{14}\)\(=\)\((\)\(-53952898642708 \beta_{15} - 1366278539058 \beta_{14} - 26310946432011 \beta_{13} + 16173600573612 \beta_{12} + 35853220709676 \beta_{11} - 3169094144310 \beta_{10} + 66778246261040 \beta_{9} - 17055274287342 \beta_{8} + 63894013845801 \beta_{7} + 52531623420501 \beta_{6} - 27514722147010 \beta_{5} + 31070385811348 \beta_{4} - 17346556422198 \beta_{3} + 80162137539438 \beta_{2} + 1274209961062293 \beta_{1} + 1366755753041361\)\()/54\)
\(\nu^{15}\)\(=\)\((\)\(588210720499726 \beta_{15} + 44260971175041 \beta_{14} + 294499803340167 \beta_{13} - 157561454266608 \beta_{12} - 359035155404016 \beta_{11} + 9304639275477 \beta_{10} - 698920301264366 \beta_{9} + 168165221674818 \beta_{8} - 691024482816405 \beta_{7} - 559281785494917 \beta_{6} + 227591471220929 \beta_{5} - 302504235040530 \beta_{4} + 202943484516045 \beta_{3} - 756406138662510 \beta_{2} - 13139670825969885 \beta_{1} - 13121441550661221\)\()/54\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
4.65022 0.707107i
−10.4548 + 0.707107i
−1.35249 + 0.707107i
8.15703 0.707107i
5.70754 + 0.707107i
1.09700 0.707107i
−8.00527 0.707107i
2.20073 + 0.707107i
4.65022 + 0.707107i
−10.4548 0.707107i
−1.35249 0.707107i
8.15703 + 0.707107i
5.70754 0.707107i
1.09700 + 0.707107i
−8.00527 + 0.707107i
2.20073 0.707107i
0 0 0 −10.1259 17.5386i 0 −12.0269 + 14.0838i 0 0 0
17.2 0 0 0 −4.36813 7.56582i 0 −14.4904 11.5338i 0 0 0
17.3 0 0 0 −4.27106 7.39769i 0 12.5787 + 13.5933i 0 0 0
17.4 0 0 0 −3.41226 5.91021i 0 14.9386 10.9471i 0 0 0
17.5 0 0 0 3.41226 + 5.91021i 0 14.9386 10.9471i 0 0 0
17.6 0 0 0 4.27106 + 7.39769i 0 12.5787 + 13.5933i 0 0 0
17.7 0 0 0 4.36813 + 7.56582i 0 −14.4904 11.5338i 0 0 0
17.8 0 0 0 10.1259 + 17.5386i 0 −12.0269 + 14.0838i 0 0 0
593.1 0 0 0 −10.1259 + 17.5386i 0 −12.0269 14.0838i 0 0 0
593.2 0 0 0 −4.36813 + 7.56582i 0 −14.4904 + 11.5338i 0 0 0
593.3 0 0 0 −4.27106 + 7.39769i 0 12.5787 13.5933i 0 0 0
593.4 0 0 0 −3.41226 + 5.91021i 0 14.9386 + 10.9471i 0 0 0
593.5 0 0 0 3.41226 5.91021i 0 14.9386 + 10.9471i 0 0 0
593.6 0 0 0 4.27106 7.39769i 0 12.5787 13.5933i 0 0 0
593.7 0 0 0 4.36813 7.56582i 0 −14.4904 + 11.5338i 0 0 0
593.8 0 0 0 10.1259 17.5386i 0 −12.0269 14.0838i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.bt.b 16
3.b odd 2 1 inner 1008.4.bt.b 16
4.b odd 2 1 252.4.t.a 16
7.d odd 6 1 inner 1008.4.bt.b 16
12.b even 2 1 252.4.t.a 16
21.g even 6 1 inner 1008.4.bt.b 16
28.d even 2 1 1764.4.t.b 16
28.f even 6 1 252.4.t.a 16
28.f even 6 1 1764.4.f.a 16
28.g odd 6 1 1764.4.f.a 16
28.g odd 6 1 1764.4.t.b 16
84.h odd 2 1 1764.4.t.b 16
84.j odd 6 1 252.4.t.a 16
84.j odd 6 1 1764.4.f.a 16
84.n even 6 1 1764.4.f.a 16
84.n even 6 1 1764.4.t.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.t.a 16 4.b odd 2 1
252.4.t.a 16 12.b even 2 1
252.4.t.a 16 28.f even 6 1
252.4.t.a 16 84.j odd 6 1
1008.4.bt.b 16 1.a even 1 1 trivial
1008.4.bt.b 16 3.b odd 2 1 inner
1008.4.bt.b 16 7.d odd 6 1 inner
1008.4.bt.b 16 21.g even 6 1 inner
1764.4.f.a 16 28.f even 6 1
1764.4.f.a 16 28.g odd 6 1
1764.4.f.a 16 84.j odd 6 1
1764.4.f.a 16 84.n even 6 1
1764.4.t.b 16 28.d even 2 1
1764.4.t.b 16 28.g odd 6 1
1764.4.t.b 16 84.h odd 2 1
1764.4.t.b 16 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(19\!\cdots\!12\)\( T_{5}^{4} + \)\(57\!\cdots\!40\)\( T_{5}^{2} + \)\(11\!\cdots\!16\)\( \)">\(T_{5}^{16} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 11316405686931216 + 573927418423440 T^{2} + 19229963845212 T^{4} + 372024076068 T^{6} + 5245846173 T^{8} + 45478638 T^{10} + 274383 T^{12} + 606 T^{14} + T^{16} \)
$7$ \( ( 13841287201 - 80707214 T - 11529602 T^{2} - 192080 T^{3} + 221431 T^{4} - 560 T^{5} - 98 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$11$ \( \)\(70\!\cdots\!96\)\( - \)\(12\!\cdots\!72\)\( T^{2} + 1703868846409635804 T^{4} - 6791181573369132 T^{6} + 19862336359773 T^{8} - 22036449654 T^{10} + 17903511 T^{12} - 4806 T^{14} + T^{16} \)
$13$ \( ( 22573860516 + 426062052 T^{2} + 2267613 T^{4} + 2826 T^{6} + T^{8} )^{2} \)
$17$ \( \)\(49\!\cdots\!76\)\( + \)\(99\!\cdots\!40\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{4} + 9962008776094977792 T^{6} + 5337923687148816 T^{8} + 1682121675600 T^{10} + 365954796 T^{12} + 21396 T^{14} + T^{16} \)
$19$ \( ( 46677071171844 + 4897686621816 T + 67363750602 T^{2} - 10905712884 T^{3} + 216000891 T^{4} + 547668 T^{5} - 14781 T^{6} - 36 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(49\!\cdots\!36\)\( - \)\(23\!\cdots\!88\)\( T^{2} + \)\(75\!\cdots\!72\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{6} + 1477439351941048848 T^{8} - 86217178855824 T^{10} + 3608759628 T^{12} - 71460 T^{14} + T^{16} \)
$29$ \( ( 416548977121475136 + 86086124811216 T^{2} + 5723596161 T^{4} + 133110 T^{6} + T^{8} )^{2} \)
$31$ \( ( 17938763430477561 + 4104504382503798 T + 301548356400690 T^{2} - 2630657447964 T^{3} + 3618762939 T^{4} + 30388068 T^{5} - 44070 T^{6} - 354 T^{7} + T^{8} )^{2} \)
$37$ \( ( 179850423092764516 - 5973918932125064 T + 152069932014238 T^{2} - 1572126338876 T^{3} + 12909581851 T^{4} - 24018986 T^{5} + 110761 T^{6} - 38 T^{7} + T^{8} )^{2} \)
$41$ \( ( 3423608728179335424 - 403281542442816 T^{2} + 16046933220 T^{4} - 240060 T^{6} + T^{8} )^{2} \)
$43$ \( ( -6254809742 - 68765576 T - 146451 T^{2} + 352 T^{3} + T^{4} )^{4} \)
$47$ \( \)\(47\!\cdots\!36\)\( + \)\(30\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(29\!\cdots\!56\)\( T^{6} + \)\(45\!\cdots\!80\)\( T^{8} + 34589931830684928 T^{10} + 190320805800 T^{12} + 522912 T^{14} + T^{16} \)
$53$ \( \)\(39\!\cdots\!76\)\( - \)\(75\!\cdots\!56\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{4} - \)\(60\!\cdots\!24\)\( T^{6} + \)\(24\!\cdots\!21\)\( T^{8} - 3936297096958518 T^{10} + 44910196227 T^{12} - 252054 T^{14} + T^{16} \)
$59$ \( \)\(56\!\cdots\!76\)\( + \)\(11\!\cdots\!12\)\( T^{2} + \)\(22\!\cdots\!84\)\( T^{4} + \)\(63\!\cdots\!48\)\( T^{6} + \)\(14\!\cdots\!69\)\( T^{8} + 85770996108094134 T^{10} + 362612345931 T^{12} + 705414 T^{14} + T^{16} \)
$61$ \( ( 10077652565550144 - 274392800500032 T - 3939444913968 T^{2} + 175070170800 T^{3} + 4946257404 T^{4} + 52264800 T^{5} + 286002 T^{6} + 816 T^{7} + T^{8} )^{2} \)
$67$ \( ( \)\(90\!\cdots\!56\)\( + 5710030600549908304 T + 46396767109353724 T^{2} - 19594109854844 T^{3} + 234348933949 T^{4} - 115322636 T^{5} + 929389 T^{6} - 764 T^{7} + T^{8} )^{2} \)
$71$ \( ( \)\(17\!\cdots\!44\)\( + 42297868577973888 T^{2} + 337035071448 T^{4} + 1006848 T^{6} + T^{8} )^{2} \)
$73$ \( ( 11798164103488007184 + 579941181193088880 T + 11266873327061460 T^{2} + 86734229600700 T^{3} + 191349429597 T^{4} - 693501750 T^{5} + 93795 T^{6} + 1350 T^{7} + T^{8} )^{2} \)
$79$ \( ( 1530560661733206601 + 214369146481110838 T + 30810683064091618 T^{2} - 109679037609848 T^{3} + 434253280891 T^{4} - 230876408 T^{5} + 668698 T^{6} - 182 T^{7} + T^{8} )^{2} \)
$83$ \( ( \)\(10\!\cdots\!84\)\( - 79735840708215204 T^{2} + 734790325365 T^{4} - 1589970 T^{6} + T^{8} )^{2} \)
$89$ \( \)\(10\!\cdots\!96\)\( + \)\(17\!\cdots\!44\)\( T^{2} + \)\(20\!\cdots\!24\)\( T^{4} + \)\(10\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!72\)\( T^{8} + 6291396333799327872 T^{10} + 7072435128816 T^{12} + 3076488 T^{14} + T^{16} \)
$97$ \( ( \)\(47\!\cdots\!36\)\( + 3276863917344744120 T^{2} + 6899032044873 T^{4} + 5135454 T^{6} + T^{8} )^{2} \)
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