Properties

Label 1470.3.f.d
Level 1470
Weight 3
Character orbit 1470.f
Analytic conductor 40.055
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1470.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.0545988610\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{6} q^{3} + 2 q^{4} -\beta_{7} q^{5} -\beta_{8} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{6} q^{3} + 2 q^{4} -\beta_{7} q^{5} -\beta_{8} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} + \beta_{10} q^{10} + ( 1 - \beta_{1} + \beta_{2} - \beta_{12} ) q^{11} + 2 \beta_{6} q^{12} + ( 2 \beta_{6} - \beta_{9} ) q^{13} -\beta_{2} q^{15} + 4 q^{16} + ( \beta_{4} + \beta_{7} + 5 \beta_{10} ) q^{17} -3 \beta_{1} q^{18} + ( -3 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + \beta_{14} ) q^{19} -2 \beta_{7} q^{20} + ( -3 + \beta_{1} + \beta_{3} + \beta_{13} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{23} -2 \beta_{8} q^{24} -5 q^{25} + ( -\beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{26} -3 \beta_{6} q^{27} + ( 4 + 5 \beta_{1} - 6 \beta_{2} + \beta_{5} - \beta_{11} + \beta_{12} ) q^{29} -\beta_{3} q^{30} + ( -\beta_{4} - 6 \beta_{6} - \beta_{7} + \beta_{9} + 7 \beta_{10} ) q^{31} + 4 \beta_{1} q^{32} + ( \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{14} ) q^{33} + ( \beta_{4} - \beta_{6} - 11 \beta_{7} + \beta_{9} - \beta_{10} ) q^{34} -6 q^{36} + ( -6 + 6 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{5} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{37} + ( 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + 5 \beta_{10} - \beta_{15} ) q^{38} + ( -5 - \beta_{2} + \beta_{11} + \beta_{12} ) q^{39} + 2 \beta_{10} q^{40} + ( -2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{14} + \beta_{15} ) q^{41} + ( -3 + 5 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{12} ) q^{44} + 3 \beta_{7} q^{45} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{46} + ( -2 \beta_{6} - 6 \beta_{7} + 2 \beta_{9} + 12 \beta_{10} ) q^{47} + 4 \beta_{6} q^{48} -5 \beta_{1} q^{50} + ( 2 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{5} - \beta_{12} + \beta_{13} ) q^{51} + ( 4 \beta_{6} - 2 \beta_{9} ) q^{52} + ( -4 - 14 \beta_{1} + 2 \beta_{3} - 2 \beta_{11} + 2 \beta_{13} ) q^{53} + 3 \beta_{8} q^{54} + ( 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{15} ) q^{55} + ( 10 - 3 \beta_{1} - 4 \beta_{2} + \beta_{11} - 2 \beta_{12} ) q^{57} + ( 11 + 5 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 2 \beta_{5} - \beta_{13} ) q^{58} + ( -5 \beta_{6} + 17 \beta_{7} - 6 \beta_{8} - \beta_{9} + 14 \beta_{10} - 3 \beta_{14} + 2 \beta_{15} ) q^{59} -2 \beta_{2} q^{60} + ( 3 \beta_{4} + 7 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + 9 \beta_{10} - \beta_{14} ) q^{61} + ( 2 \beta_{6} - 12 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} ) q^{62} + 8 q^{64} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{11} - \beta_{12} ) q^{65} + ( -\beta_{4} - 4 \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{15} ) q^{66} + ( 20 + 29 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} + ( 2 \beta_{4} + 2 \beta_{7} + 10 \beta_{10} ) q^{68} + ( \beta_{4} + \beta_{6} + 2 \beta_{9} - 2 \beta_{10} - \beta_{15} ) q^{69} + ( -9 - 3 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{12} + 2 \beta_{13} ) q^{71} -6 \beta_{1} q^{72} + ( -\beta_{4} + 13 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} + 2 \beta_{9} + 7 \beta_{10} + 3 \beta_{14} ) q^{73} + ( 15 - 4 \beta_{1} - 10 \beta_{2} - \beta_{3} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{74} -5 \beta_{6} q^{75} + ( -6 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} + 2 \beta_{14} ) q^{76} + ( -6 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{11} - \beta_{13} ) q^{78} + ( -34 + \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{79} -4 \beta_{7} q^{80} + 9 q^{81} + ( -\beta_{4} + 12 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{14} + \beta_{15} ) q^{82} + ( 3 \beta_{4} + 13 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( 7 - 26 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} + ( 9 - 6 \beta_{1} - 6 \beta_{2} + \beta_{3} - 2 \beta_{5} + 4 \beta_{12} + \beta_{13} ) q^{86} + ( -2 \beta_{4} + 4 \beta_{6} + 18 \beta_{7} - 6 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{14} - \beta_{15} ) q^{87} + ( -6 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{13} ) q^{88} + ( -2 \beta_{4} - 12 \beta_{6} + 12 \beta_{8} - 3 \beta_{9} + 25 \beta_{10} + 3 \beta_{14} - \beta_{15} ) q^{89} -3 \beta_{10} q^{90} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{92} + ( 15 - 3 \beta_{1} - \beta_{2} - 7 \beta_{3} - \beta_{5} - \beta_{11} - \beta_{13} ) q^{93} + ( 2 \beta_{4} + 2 \beta_{6} - 22 \beta_{7} - 2 \beta_{9} + 4 \beta_{10} ) q^{94} + ( -25 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{11} - 2 \beta_{13} ) q^{95} -4 \beta_{8} q^{96} + ( -\beta_{4} - 13 \beta_{6} + 28 \beta_{8} + 3 \beta_{9} - 17 \beta_{10} + \beta_{14} ) q^{97} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 32q^{4} - 48q^{9} + O(q^{10}) \) \( 16q + 32q^{4} - 48q^{9} + 8q^{11} + 64q^{16} - 48q^{22} + 24q^{23} - 80q^{25} + 72q^{29} - 96q^{36} - 88q^{37} - 72q^{39} - 56q^{43} + 16q^{44} - 16q^{46} + 24q^{51} - 64q^{53} + 144q^{57} + 176q^{58} + 128q^{64} - 40q^{65} + 328q^{67} - 136q^{71} + 224q^{74} - 560q^{79} + 144q^{81} + 120q^{85} + 176q^{86} - 96q^{88} + 48q^{92} + 240q^{93} - 400q^{95} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + 4836403 x^{8} - 6808704 x^{7} + 64376800 x^{6} - 91953512 x^{5} + 595763862 x^{4} - 630430976 x^{3} + 1087013404 x^{2} + 294123256 x + 101626561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(57754008957283349 \nu^{15} + 676562680999836294 \nu^{14} + 8239103585403158159 \nu^{13} + 54078590776251682475 \nu^{12} + 455975368108340251668 \nu^{11} + 2793391292439344623383 \nu^{10} + 16545566601055515429559 \nu^{9} + 76261143494901216401580 \nu^{8} + 306572240997688184788012 \nu^{7} + 1467974756910134628134465 \nu^{6} + 3497231496721962844392765 \nu^{5} + 10591214719874231223793348 \nu^{4} - 4685579583089306764764583 \nu^{3} + 35874681437029057265441973 \nu^{2} + 9955887688266404429946106 \nu - 1202728311133779179554909109\)\()/ \)\(85\!\cdots\!24\)\( \)
\(\beta_{2}\)\(=\)\((\)\(6718070224756 \nu^{15} - 35399436274870 \nu^{14} + 824042900206056 \nu^{13} - 3468795000369922 \nu^{12} + 55447463307528888 \nu^{11} - 237157576567025547 \nu^{10} + 2160971883833844512 \nu^{9} - 7103633821675122284 \nu^{8} + 46638150914308904520 \nu^{7} - 152233236071607452553 \nu^{6} + 595135449699402838792 \nu^{5} - 1018214320678792586196 \nu^{4} + 1279768421429813362272 \nu^{3} - 459537667512276799289 \nu^{2} - 103279705437214175760 \nu + 206657596524136518657064\)\()/ \)\(53\!\cdots\!38\)\( \)
\(\beta_{3}\)\(=\)\((\)\(3871767397361 \nu^{15} + 79093222616710 \nu^{14} + 341679513333957 \nu^{13} + 5895790087314937 \nu^{12} + 14035722749452236 \nu^{11} + 371621017202953791 \nu^{10} + 356604274247547817 \nu^{9} + 10456974871241558252 \nu^{8} + 4802805224198506740 \nu^{7} + 233091302320357629453 \nu^{6} + 23538017273856016799 \nu^{5} + 1467948791527369196388 \nu^{4} - 940733060256762782559 \nu^{3} + 3491995997388483650717 \nu^{2} + 957085277114840834430 \nu - 148193606010062737313275\)\()/ \)\(27\!\cdots\!92\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(24\!\cdots\!58\)\( \nu^{15} + \)\(21\!\cdots\!04\)\( \nu^{14} - \)\(23\!\cdots\!64\)\( \nu^{13} + \)\(33\!\cdots\!95\)\( \nu^{12} - \)\(15\!\cdots\!72\)\( \nu^{11} + \)\(22\!\cdots\!77\)\( \nu^{10} - \)\(56\!\cdots\!72\)\( \nu^{9} + \)\(65\!\cdots\!74\)\( \nu^{8} - \)\(14\!\cdots\!24\)\( \nu^{7} + \)\(15\!\cdots\!19\)\( \nu^{6} - \)\(21\!\cdots\!44\)\( \nu^{5} + \)\(16\!\cdots\!38\)\( \nu^{4} - \)\(19\!\cdots\!74\)\( \nu^{3} + \)\(23\!\cdots\!01\)\( \nu^{2} - \)\(45\!\cdots\!04\)\( \nu - \)\(49\!\cdots\!51\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(15\!\cdots\!15\)\( \nu^{15} + \)\(21\!\cdots\!74\)\( \nu^{14} - \)\(13\!\cdots\!09\)\( \nu^{13} + \)\(34\!\cdots\!07\)\( \nu^{12} - \)\(88\!\cdots\!16\)\( \nu^{11} + \)\(20\!\cdots\!79\)\( \nu^{10} - \)\(29\!\cdots\!65\)\( \nu^{9} + \)\(64\!\cdots\!12\)\( \nu^{8} - \)\(71\!\cdots\!88\)\( \nu^{7} + \)\(11\!\cdots\!81\)\( \nu^{6} - \)\(89\!\cdots\!51\)\( \nu^{5} + \)\(93\!\cdots\!96\)\( \nu^{4} - \)\(86\!\cdots\!83\)\( \nu^{3} - \)\(71\!\cdots\!15\)\( \nu^{2} - \)\(22\!\cdots\!34\)\( \nu - \)\(23\!\cdots\!33\)\(\)\()/ \)\(31\!\cdots\!08\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(34\!\cdots\!04\)\( \nu^{15} - \)\(10\!\cdots\!36\)\( \nu^{14} + \)\(31\!\cdots\!40\)\( \nu^{13} - \)\(48\!\cdots\!52\)\( \nu^{12} + \)\(20\!\cdots\!68\)\( \nu^{11} - \)\(32\!\cdots\!16\)\( \nu^{10} + \)\(68\!\cdots\!40\)\( \nu^{9} - \)\(12\!\cdots\!18\)\( \nu^{8} + \)\(16\!\cdots\!16\)\( \nu^{7} - \)\(28\!\cdots\!40\)\( \nu^{6} + \)\(22\!\cdots\!48\)\( \nu^{5} - \)\(37\!\cdots\!60\)\( \nu^{4} + \)\(20\!\cdots\!48\)\( \nu^{3} - \)\(27\!\cdots\!84\)\( \nu^{2} + \)\(36\!\cdots\!44\)\( \nu + \)\(35\!\cdots\!31\)\(\)\()/ \)\(62\!\cdots\!01\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-536662839109972 \nu^{15} + 184355843936000 \nu^{14} - 49316294058364856 \nu^{13} + 76985389887345138 \nu^{12} - 3154550976828630456 \nu^{11} + 5211291917675260305 \nu^{10} - 106964341125832723328 \nu^{9} + 195705915902593012444 \nu^{8} - 2639716566132467353984 \nu^{7} + 4517022991968883377451 \nu^{6} - 35333929028214918917480 \nu^{5} + 60517159199633559261440 \nu^{4} - 328552596355156909606256 \nu^{3} + 436496862559525516173889 \nu^{2} - 567954327233848070808304 \nu - 55329661987849902130608\)\()/ \)\(75\!\cdots\!34\)\( \)
\(\beta_{8}\)\(=\)\((\)\(2130173036201 \nu^{15} - 540413699312 \nu^{14} + 193661918213181 \nu^{13} - 293621077625441 \nu^{12} + 12285060311199138 \nu^{11} - 19810557668746935 \nu^{10} + 410965646117486839 \nu^{9} - 746713194852137098 \nu^{8} + 10036111646725656126 \nu^{7} - 17105922536607134721 \nu^{6} + 131897626701278295101 \nu^{5} - 228882972939831674094 \nu^{4} + 1220130262910161491531 \nu^{3} - 1620423665127954679375 \nu^{2} + 2091806790152256573456 \nu + 203021156853115775993\)\()/ \)\(25\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(27\!\cdots\!01\)\( \nu^{15} - \)\(90\!\cdots\!76\)\( \nu^{14} + \)\(24\!\cdots\!57\)\( \nu^{13} - \)\(35\!\cdots\!28\)\( \nu^{12} + \)\(15\!\cdots\!94\)\( \nu^{11} - \)\(23\!\cdots\!02\)\( \nu^{10} + \)\(52\!\cdots\!75\)\( \nu^{9} - \)\(85\!\cdots\!28\)\( \nu^{8} + \)\(12\!\cdots\!46\)\( \nu^{7} - \)\(19\!\cdots\!78\)\( \nu^{6} + \)\(16\!\cdots\!77\)\( \nu^{5} - \)\(25\!\cdots\!96\)\( \nu^{4} + \)\(15\!\cdots\!23\)\( \nu^{3} - \)\(18\!\cdots\!22\)\( \nu^{2} + \)\(29\!\cdots\!00\)\( \nu + \)\(31\!\cdots\!96\)\(\)\()/ \)\(31\!\cdots\!48\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-4448842632361015 \nu^{15} + 1527746136142088 \nu^{14} - 406573176070056149 \nu^{13} + 638852366768395737 \nu^{12} - 25956155317369110414 \nu^{11} + 43002282249586977093 \nu^{10} - 874597310627211904019 \nu^{9} + 1616020902079177673194 \nu^{8} - 21496564169263011322522 \nu^{7} + 37198980049890495166591 \nu^{6} - 284874104240500582055717 \nu^{5} + 500982390617084205830630 \nu^{4} - 2644373246682812474960417 \nu^{3} + 3517427264905839032095693 \nu^{2} - 4518691295624378534050984 \nu - 437660350242222703221165\)\()/ \)\(42\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(13\!\cdots\!03\)\( \nu^{15} - \)\(20\!\cdots\!62\)\( \nu^{14} - \)\(12\!\cdots\!73\)\( \nu^{13} - \)\(27\!\cdots\!91\)\( \nu^{12} - \)\(79\!\cdots\!12\)\( \nu^{11} - \)\(10\!\cdots\!97\)\( \nu^{10} - \)\(25\!\cdots\!49\)\( \nu^{9} + \)\(24\!\cdots\!04\)\( \nu^{8} - \)\(60\!\cdots\!56\)\( \nu^{7} - \)\(17\!\cdots\!11\)\( \nu^{6} - \)\(74\!\cdots\!67\)\( \nu^{5} + \)\(59\!\cdots\!92\)\( \nu^{4} - \)\(58\!\cdots\!31\)\( \nu^{3} - \)\(22\!\cdots\!95\)\( \nu^{2} - \)\(64\!\cdots\!58\)\( \nu - \)\(91\!\cdots\!03\)\(\)\()/ \)\(95\!\cdots\!24\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(50\!\cdots\!76\)\( \nu^{15} + \)\(88\!\cdots\!98\)\( \nu^{14} + \)\(46\!\cdots\!52\)\( \nu^{13} + \)\(23\!\cdots\!41\)\( \nu^{12} + \)\(28\!\cdots\!28\)\( \nu^{11} + \)\(10\!\cdots\!43\)\( \nu^{10} + \)\(93\!\cdots\!48\)\( \nu^{9} + \)\(98\!\cdots\!44\)\( \nu^{8} + \)\(22\!\cdots\!24\)\( \nu^{7} + \)\(33\!\cdots\!21\)\( \nu^{6} + \)\(26\!\cdots\!88\)\( \nu^{5} + \)\(45\!\cdots\!72\)\( \nu^{4} + \)\(22\!\cdots\!04\)\( \nu^{3} + \)\(87\!\cdots\!65\)\( \nu^{2} + \)\(24\!\cdots\!32\)\( \nu + \)\(54\!\cdots\!09\)\(\)\()/ \)\(31\!\cdots\!08\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(35\!\cdots\!52\)\( \nu^{15} + \)\(57\!\cdots\!36\)\( \nu^{14} + \)\(32\!\cdots\!91\)\( \nu^{13} + \)\(11\!\cdots\!82\)\( \nu^{12} + \)\(20\!\cdots\!52\)\( \nu^{11} + \)\(50\!\cdots\!62\)\( \nu^{10} + \)\(65\!\cdots\!39\)\( \nu^{9} + \)\(94\!\cdots\!40\)\( \nu^{8} + \)\(15\!\cdots\!28\)\( \nu^{7} + \)\(13\!\cdots\!36\)\( \nu^{6} + \)\(18\!\cdots\!25\)\( \nu^{5} - \)\(51\!\cdots\!08\)\( \nu^{4} + \)\(16\!\cdots\!40\)\( \nu^{3} + \)\(59\!\cdots\!82\)\( \nu^{2} + \)\(16\!\cdots\!14\)\( \nu + \)\(40\!\cdots\!54\)\(\)\()/ \)\(15\!\cdots\!54\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(15\!\cdots\!08\)\( \nu^{15} - \)\(45\!\cdots\!56\)\( \nu^{14} + \)\(14\!\cdots\!19\)\( \nu^{13} - \)\(17\!\cdots\!22\)\( \nu^{12} + \)\(90\!\cdots\!98\)\( \nu^{11} - \)\(11\!\cdots\!64\)\( \nu^{10} + \)\(30\!\cdots\!90\)\( \nu^{9} - \)\(46\!\cdots\!82\)\( \nu^{8} + \)\(76\!\cdots\!56\)\( \nu^{7} - \)\(10\!\cdots\!86\)\( \nu^{6} + \)\(10\!\cdots\!75\)\( \nu^{5} - \)\(14\!\cdots\!36\)\( \nu^{4} + \)\(97\!\cdots\!41\)\( \nu^{3} - \)\(10\!\cdots\!00\)\( \nu^{2} + \)\(21\!\cdots\!10\)\( \nu + \)\(24\!\cdots\!97\)\(\)\()/ \)\(23\!\cdots\!61\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(96\!\cdots\!17\)\( \nu^{15} - \)\(19\!\cdots\!56\)\( \nu^{14} + \)\(89\!\cdots\!11\)\( \nu^{13} - \)\(10\!\cdots\!49\)\( \nu^{12} + \)\(57\!\cdots\!70\)\( \nu^{11} - \)\(75\!\cdots\!51\)\( \nu^{10} + \)\(19\!\cdots\!13\)\( \nu^{9} - \)\(29\!\cdots\!14\)\( \nu^{8} + \)\(47\!\cdots\!18\)\( \nu^{7} - \)\(67\!\cdots\!65\)\( \nu^{6} + \)\(64\!\cdots\!95\)\( \nu^{5} - \)\(92\!\cdots\!50\)\( \nu^{4} + \)\(60\!\cdots\!39\)\( \nu^{3} - \)\(64\!\cdots\!11\)\( \nu^{2} + \)\(13\!\cdots\!12\)\( \nu + \)\(14\!\cdots\!41\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{15} - 5 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - 4 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 10 \beta_{1} - 1\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(-5 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} + \beta_{11} - \beta_{10} + 6 \beta_{9} + 96 \beta_{8} - 127 \beta_{7} - 327 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 39 \beta_{2} - 100 \beta_{1} - 325\)\()/28\)
\(\nu^{3}\)\(=\)\((\)\(17 \beta_{13} - 18 \beta_{12} + 8 \beta_{11} - 3 \beta_{5} + 5 \beta_{3} + 9 \beta_{2} + 88 \beta_{1} + 51\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(198 \beta_{15} - 272 \beta_{14} - 124 \beta_{13} + 232 \beta_{12} + 34 \beta_{11} - 834 \beta_{10} - 176 \beta_{9} - 2228 \beta_{8} + 3100 \beta_{7} + 5749 \beta_{6} + 74 \beta_{5} + 24 \beta_{4} + 212 \beta_{3} + 892 \beta_{2} - 2476 \beta_{1} - 5765\)\()/14\)
\(\nu^{5}\)\(=\)\((\)\(-5990 \beta_{15} + 8667 \beta_{14} - 5413 \beta_{13} + 6530 \beta_{12} - 1560 \beta_{11} + 9155 \beta_{10} - 2003 \beta_{9} + 20200 \beta_{8} - 9948 \beta_{7} - 19409 \beta_{6} + 577 \beta_{5} + 4259 \beta_{4} + 1055 \beta_{3} - 325 \beta_{2} - 33126 \beta_{1} - 25805\)\()/28\)
\(\nu^{6}\)\(=\)\((\)\(2622 \beta_{13} - 4310 \beta_{12} - 373 \beta_{11} - 811 \beta_{5} - 4770 \beta_{3} - 11579 \beta_{2} + 35956 \beta_{1} + 67649\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(279262 \beta_{15} - 400797 \beta_{14} - 259745 \beta_{13} + 334798 \beta_{12} - 46482 \beta_{11} - 699547 \beta_{10} + 17911 \beta_{9} - 1109420 \beta_{8} + 965760 \beta_{7} + 1345299 \beta_{6} + 19517 \beta_{5} - 220711 \beta_{4} + 140095 \beta_{3} + 120721 \beta_{2} - 1730928 \beta_{1} - 1632009\)\()/28\)
\(\nu^{8}\)\(=\)\((\)\(-691284 \beta_{15} + 964464 \beta_{14} - 585096 \beta_{13} + 900336 \beta_{12} + 42060 \beta_{11} + 3656460 \beta_{10} + 399360 \beta_{9} + 5207880 \beta_{8} - 7456656 \beta_{7} - 10260655 \beta_{6} + 106188 \beta_{5} + 372720 \beta_{4} + 988392 \beta_{3} + 1906704 \beta_{2} - 6545064 \beta_{1} - 10697503\)\()/14\)
\(\nu^{9}\)\(=\)\((\)\(1846509 \beta_{13} - 2471702 \beta_{12} + 211658 \beta_{11} - 112891 \beta_{5} - 1436811 \beta_{3} - 1653581 \beta_{2} + 13011958 \beta_{1} + 13889443\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(77787413 \beta_{15} - 109133120 \beta_{14} - 69460730 \beta_{13} + 103071894 \beta_{12} + 2265457 \beta_{11} - 403264199 \beta_{10} - 38142078 \beta_{9} - 527802720 \beta_{8} + 757264471 \beta_{7} + 971458359 \beta_{6} + 8326683 \beta_{5} - 52807364 \beta_{4} + 108492262 \beta_{3} + 185727303 \beta_{2} - 689743204 \beta_{1} - 1030326949\)\()/28\)
\(\nu^{11}\)\(=\)\((\)\(-696502112 \beta_{15} + 990432227 \beta_{14} - 660025487 \beta_{13} + 903385734 \beta_{12} - 50569868 \beta_{11} + 2558370023 \beta_{10} + 142220511 \beta_{9} + 3268336104 \beta_{8} - 4115827654 \beta_{7} - 4912684271 \beta_{6} + 36476625 \beta_{5} + 587072237 \beta_{4} + 620622637 \beta_{3} + 798700893 \beta_{2} - 4845840568 \beta_{1} - 5586803001\)\()/28\)
\(\nu^{12}\)\(=\)\(283303276 \beta_{13} - 411884296 \beta_{12} - 2844970 \beta_{11} - 25153442 \beta_{5} - 415417460 \beta_{3} - 664916932 \beta_{2} + 2616526588 \beta_{1} + 3695031179\)
\(\nu^{13}\)\(=\)\((\)\(36136952786 \beta_{15} - 51274366995 \beta_{14} - 34314451069 \beta_{13} + 47593238282 \beta_{12} - 1858626996 \beta_{11} - 144106197839 \beta_{10} - 9561533221 \beta_{9} - 177814191664 \beta_{8} + 238342892184 \beta_{7} + 279407517869 \beta_{6} + 1822501717 \beta_{5} - 30669447635 \beta_{4} + 35989748351 \beta_{3} + 48945848111 \beta_{2} - 259758006294 \beta_{1} - 313758094217\)\()/28\)
\(\nu^{14}\)\(=\)\((\)\(-237558015007 \beta_{15} + 335044352924 \beta_{14} - 221473349274 \beta_{13} + 318262664810 \beta_{12} - 697022381 \beta_{11} + 1167123223825 \beta_{10} + 95395270774 \beta_{9} + 1430841773632 \beta_{8} - 2050689201961 \beta_{7} - 2461353139957 \beta_{6} + 16084665733 \beta_{5} + 189304017808 \beta_{4} + 309855069606 \beta_{3} + 476981853485 \beta_{2} - 1955190144364 \beta_{1} - 2667438845879\)\()/28\)
\(\nu^{15}\)\(=\)\((\)\(257998257311 \beta_{13} - 360676661674 \beta_{12} + 10562294418 \beta_{11} - 13560958727 \beta_{5} - 289138472617 \beta_{3} - 405091585843 \beta_{2} + 1997234018400 \beta_{1} + 2480567013939\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1
−0.141814 0.245629i
0.848921 + 1.47037i
−2.10711 3.64962i
2.81422 + 4.87437i
−2.10711 + 3.64962i
2.81422 4.87437i
−0.141814 + 0.245629i
0.848921 1.47037i
−2.63284 4.56021i
1.92573 + 3.33546i
2.96377 + 5.13339i
−3.67087 6.35814i
2.96377 5.13339i
−3.67087 + 6.35814i
−2.63284 + 4.56021i
1.92573 3.33546i
−1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.2 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.3 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.4 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.5 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.6 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.7 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.8 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.9 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.10 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.11 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.12 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.13 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.14 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.15 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.16 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.3.f.d 16
7.b odd 2 1 inner 1470.3.f.d 16
7.c even 3 1 210.3.o.b 16
7.d odd 6 1 210.3.o.b 16
21.g even 6 1 630.3.v.c 16
21.h odd 6 1 630.3.v.c 16
35.i odd 6 1 1050.3.p.i 16
35.j even 6 1 1050.3.p.i 16
35.k even 12 2 1050.3.q.e 32
35.l odd 12 2 1050.3.q.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 7.c even 3 1
210.3.o.b 16 7.d odd 6 1
630.3.v.c 16 21.g even 6 1
630.3.v.c 16 21.h odd 6 1
1050.3.p.i 16 35.i odd 6 1
1050.3.p.i 16 35.j even 6 1
1050.3.q.e 32 35.k even 12 2
1050.3.q.e 32 35.l odd 12 2
1470.3.f.d 16 1.a even 1 1 trivial
1470.3.f.d 16 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{11}^{8} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(1470, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{8} \)
$3$ \( ( 1 + 3 T^{2} )^{8} \)
$5$ \( ( 1 + 5 T^{2} )^{8} \)
$7$ 1
$11$ \( ( 1 - 4 T + 380 T^{2} - 2036 T^{3} + 82148 T^{4} - 539228 T^{5} + 12902676 T^{6} - 93038220 T^{7} + 1637770150 T^{8} - 11257624620 T^{9} + 188908079316 T^{10} - 955275294908 T^{11} + 17609153356388 T^{12} - 52808596487636 T^{13} + 1192602783153980 T^{14} - 1518999334332964 T^{15} + 45949729863572161 T^{16} )^{2} \)
$13$ \( 1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} - \)\(21\!\cdots\!12\)\( T^{18} + \)\(12\!\cdots\!16\)\( T^{20} - \)\(31\!\cdots\!76\)\( T^{22} + \)\(51\!\cdots\!70\)\( T^{24} - \)\(58\!\cdots\!64\)\( T^{26} + \)\(44\!\cdots\!28\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(44\!\cdots\!81\)\( T^{32} \)
$17$ \( 1 - 696 T^{2} + 231064 T^{4} - 37149384 T^{6} + 6926154940 T^{8} - 4433534480088 T^{10} + 1832268292772776 T^{12} - 374043336733365672 T^{14} + 58828830910943250694 T^{16} - \)\(31\!\cdots\!12\)\( T^{18} + \)\(12\!\cdots\!16\)\( T^{20} - \)\(25\!\cdots\!68\)\( T^{22} + \)\(33\!\cdots\!40\)\( T^{24} - \)\(15\!\cdots\!84\)\( T^{26} + \)\(78\!\cdots\!44\)\( T^{28} - \)\(19\!\cdots\!36\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( 1 - 2440 T^{2} + 3262484 T^{4} - 3093518480 T^{6} + 2289552719514 T^{8} - 1393085088559960 T^{10} + 717656314443664304 T^{12} - \)\(31\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!75\)\( T^{16} - \)\(41\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!64\)\( T^{20} - \)\(30\!\cdots\!60\)\( T^{22} + \)\(66\!\cdots\!34\)\( T^{24} - \)\(11\!\cdots\!80\)\( T^{26} + \)\(15\!\cdots\!64\)\( T^{28} - \)\(15\!\cdots\!40\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \)
$23$ \( ( 1 - 12 T + 1948 T^{2} - 32316 T^{3} + 2328740 T^{4} - 41047380 T^{5} + 1879055412 T^{6} - 32536827396 T^{7} + 1148877345254 T^{8} - 17211981692484 T^{9} + 525836745549492 T^{10} - 6076485389420820 T^{11} + 182365923863275940 T^{12} - 1338739136380281084 T^{13} + 42689688393575585308 T^{14} - \)\(13\!\cdots\!08\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \)
$29$ \( ( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 25259833173132 T^{9} + 59483856997836 T^{10} + 40924113344941092 T^{11} - 637744142027460460 T^{12} + 8042239471766642316 T^{13} + \)\(51\!\cdots\!04\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \)
$31$ \( 1 - 8776 T^{2} + 39983444 T^{4} - 123311980304 T^{6} + 285704968947930 T^{8} - 524575072927909528 T^{10} + \)\(78\!\cdots\!96\)\( T^{12} - \)\(98\!\cdots\!72\)\( T^{14} + \)\(10\!\cdots\!99\)\( T^{16} - \)\(90\!\cdots\!12\)\( T^{18} + \)\(67\!\cdots\!36\)\( T^{20} - \)\(41\!\cdots\!08\)\( T^{22} + \)\(20\!\cdots\!30\)\( T^{24} - \)\(82\!\cdots\!04\)\( T^{26} + \)\(24\!\cdots\!24\)\( T^{28} - \)\(50\!\cdots\!16\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \)
$37$ \( ( 1 + 44 T + 3892 T^{2} + 79760 T^{3} + 7985418 T^{4} + 149405340 T^{5} + 14878534320 T^{6} + 188674612764 T^{7} + 20151694610467 T^{8} + 258295544873916 T^{9} + 27884768759705520 T^{10} + 383333226483624060 T^{11} + 28048616655970923978 T^{12} + \)\(38\!\cdots\!40\)\( T^{13} + \)\(25\!\cdots\!52\)\( T^{14} + \)\(39\!\cdots\!16\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \)
$41$ \( 1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} + \)\(50\!\cdots\!56\)\( T^{12} - \)\(10\!\cdots\!32\)\( T^{14} + \)\(18\!\cdots\!14\)\( T^{16} - \)\(28\!\cdots\!52\)\( T^{18} + \)\(40\!\cdots\!76\)\( T^{20} - \)\(49\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!40\)\( T^{24} - \)\(47\!\cdots\!64\)\( T^{26} + \)\(33\!\cdots\!04\)\( T^{28} - \)\(16\!\cdots\!36\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \)
$43$ \( ( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 1611056221164 T^{7} + 122468758030675 T^{8} + 2978842952932236 T^{9} + 204841330302006704 T^{10} + 4087494160581305452 T^{11} + \)\(25\!\cdots\!10\)\( T^{12} + \)\(41\!\cdots\!12\)\( T^{13} + \)\(22\!\cdots\!32\)\( T^{14} + \)\(20\!\cdots\!72\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 19504 T^{2} + 194566136 T^{4} - 1312184963216 T^{6} + 6686766048287772 T^{8} - 27242570426384646448 T^{10} + \)\(91\!\cdots\!04\)\( T^{12} - \)\(25\!\cdots\!12\)\( T^{14} + \)\(61\!\cdots\!10\)\( T^{16} - \)\(12\!\cdots\!72\)\( T^{18} + \)\(21\!\cdots\!44\)\( T^{20} - \)\(31\!\cdots\!68\)\( T^{22} + \)\(37\!\cdots\!12\)\( T^{24} - \)\(36\!\cdots\!16\)\( T^{26} + \)\(26\!\cdots\!16\)\( T^{28} - \)\(12\!\cdots\!44\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( ( 1 + 32 T + 13096 T^{2} + 382176 T^{3} + 87745820 T^{4} + 2185324448 T^{5} + 391392084888 T^{6} + 8408381790560 T^{7} + 1270026486053126 T^{8} + 23619144449683040 T^{9} + 3088271809359151128 T^{10} + 48436320249504581792 T^{11} + \)\(54\!\cdots\!20\)\( T^{12} + \)\(66\!\cdots\!24\)\( T^{13} + \)\(64\!\cdots\!36\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} )^{2} \)
$59$ \( 1 - 1896 T^{2} + 10373176 T^{4} - 69423767064 T^{6} + 392354089819132 T^{8} - 1032139977647109192 T^{10} + \)\(50\!\cdots\!84\)\( T^{12} - \)\(17\!\cdots\!48\)\( T^{14} + \)\(83\!\cdots\!30\)\( T^{16} - \)\(21\!\cdots\!28\)\( T^{18} + \)\(73\!\cdots\!64\)\( T^{20} - \)\(18\!\cdots\!52\)\( T^{22} + \)\(84\!\cdots\!12\)\( T^{24} - \)\(18\!\cdots\!64\)\( T^{26} + \)\(32\!\cdots\!36\)\( T^{28} - \)\(72\!\cdots\!16\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( 1 - 26032 T^{2} + 316572152 T^{4} - 2428041481232 T^{6} + 13620146914246812 T^{8} - 63272070928644615856 T^{10} + \)\(27\!\cdots\!52\)\( T^{12} - \)\(11\!\cdots\!52\)\( T^{14} + \)\(44\!\cdots\!86\)\( T^{16} - \)\(15\!\cdots\!32\)\( T^{18} + \)\(52\!\cdots\!12\)\( T^{20} - \)\(16\!\cdots\!76\)\( T^{22} + \)\(50\!\cdots\!32\)\( T^{24} - \)\(12\!\cdots\!32\)\( T^{26} + \)\(22\!\cdots\!32\)\( T^{28} - \)\(25\!\cdots\!92\)\( T^{30} + \)\(13\!\cdots\!21\)\( T^{32} \)
$67$ \( ( 1 - 164 T + 33236 T^{2} - 4024336 T^{3} + 476556074 T^{4} - 45254634868 T^{5} + 4016466262608 T^{6} - 308250966055572 T^{7} + 22055966830747363 T^{8} - 1383738586623462708 T^{9} + 80936297650231583568 T^{10} - \)\(40\!\cdots\!92\)\( T^{11} + \)\(19\!\cdots\!34\)\( T^{12} - \)\(73\!\cdots\!64\)\( T^{13} + \)\(27\!\cdots\!96\)\( T^{14} - \)\(60\!\cdots\!56\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} )^{2} \)
$71$ \( ( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 1096871001959146028 T^{9} + 81023237162072440932 T^{10} + \)\(36\!\cdots\!44\)\( T^{11} + \)\(22\!\cdots\!92\)\( T^{12} + \)\(71\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} + \)\(56\!\cdots\!08\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \)
$73$ \( 1 - 44312 T^{2} + 949379444 T^{4} - 13136848413808 T^{6} + 133216272007717754 T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(73\!\cdots\!20\)\( T^{12} - \)\(44\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!47\)\( T^{16} - \)\(12\!\cdots\!60\)\( T^{18} + \)\(58\!\cdots\!20\)\( T^{20} - \)\(24\!\cdots\!48\)\( T^{22} + \)\(86\!\cdots\!94\)\( T^{24} - \)\(24\!\cdots\!08\)\( T^{26} + \)\(49\!\cdots\!04\)\( T^{28} - \)\(66\!\cdots\!72\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \)
$79$ \( ( 1 + 280 T + 63868 T^{2} + 9752992 T^{3} + 1312957026 T^{4} + 143007366360 T^{5} + 14416274240064 T^{6} + 1263875597787768 T^{7} + 105782687301458731 T^{8} + 7887847605793460088 T^{9} + \)\(56\!\cdots\!84\)\( T^{10} + \)\(34\!\cdots\!60\)\( T^{11} + \)\(19\!\cdots\!86\)\( T^{12} + \)\(92\!\cdots\!92\)\( T^{13} + \)\(37\!\cdots\!88\)\( T^{14} + \)\(10\!\cdots\!80\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( 1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} - \)\(83\!\cdots\!72\)\( T^{10} + \)\(84\!\cdots\!60\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(53\!\cdots\!06\)\( T^{16} - \)\(34\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!60\)\( T^{20} - \)\(88\!\cdots\!92\)\( T^{22} + \)\(34\!\cdots\!16\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(24\!\cdots\!68\)\( T^{28} - \)\(35\!\cdots\!44\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \)
$89$ \( 1 - 19336 T^{2} + 310289528 T^{4} - 4184900827256 T^{6} + 49417825883213052 T^{8} - \)\(50\!\cdots\!60\)\( T^{10} + \)\(49\!\cdots\!52\)\( T^{12} - \)\(43\!\cdots\!88\)\( T^{14} + \)\(35\!\cdots\!50\)\( T^{16} - \)\(26\!\cdots\!08\)\( T^{18} + \)\(19\!\cdots\!12\)\( T^{20} - \)\(12\!\cdots\!60\)\( T^{22} + \)\(76\!\cdots\!72\)\( T^{24} - \)\(40\!\cdots\!56\)\( T^{26} + \)\(18\!\cdots\!48\)\( T^{28} - \)\(74\!\cdots\!16\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$97$ \( 1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} - \)\(21\!\cdots\!72\)\( T^{10} + \)\(29\!\cdots\!68\)\( T^{12} - \)\(34\!\cdots\!84\)\( T^{14} + \)\(34\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!52\)\( T^{22} + \)\(81\!\cdots\!08\)\( T^{24} - \)\(36\!\cdots\!36\)\( T^{26} + \)\(12\!\cdots\!92\)\( T^{28} - \)\(30\!\cdots\!28\)\( T^{30} + \)\(37\!\cdots\!41\)\( T^{32} \)
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