Properties

Label 1.96.a.a
Level 1
Weight 96
Character orbit 1.a
Self dual yes
Analytic conductor 57.154
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 96 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.1535908815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{38}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-729457392285 + \beta_{1}) q^{2} +(-\)\(11\!\cdots\!60\)\( + 20242501 \beta_{1} - \beta_{2}) q^{3} +(\)\(26\!\cdots\!48\)\( - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4} +(\)\(24\!\cdots\!70\)\( - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5} +(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6} +(\)\(39\!\cdots\!00\)\( + \)\(16\!\cdots\!42\)\( \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7} +(-\)\(18\!\cdots\!20\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8} +(\)\(11\!\cdots\!17\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(34\!\cdots\!86\)\( \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9} +O(q^{10})\) \( q +(-729457392285 + \beta_{1}) q^{2} +(-\)\(11\!\cdots\!60\)\( + 20242501 \beta_{1} - \beta_{2}) q^{3} +(\)\(26\!\cdots\!48\)\( - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4} +(\)\(24\!\cdots\!70\)\( - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5} +(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6} +(\)\(39\!\cdots\!00\)\( + \)\(16\!\cdots\!42\)\( \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7} +(-\)\(18\!\cdots\!20\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8} +(\)\(11\!\cdots\!17\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(34\!\cdots\!86\)\( \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9} +(-\)\(44\!\cdots\!30\)\( + \)\(40\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!56\)\( \beta_{2} + 1823205551639504480 \beta_{3} + 31201730362844 \beta_{4} + 32408570756 \beta_{5} - 8596448 \beta_{6} + 18144 \beta_{7}) q^{10} +(\)\(66\!\cdots\!52\)\( - \)\(38\!\cdots\!81\)\( \beta_{1} + \)\(40\!\cdots\!65\)\( \beta_{2} + 1971098222936635454 \beta_{3} - 2281046769584042 \beta_{4} + 1373746304736 \beta_{5} - 186517774 \beta_{6} - 1124608 \beta_{7}) q^{11} +(\)\(13\!\cdots\!80\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!36\)\( \beta_{2} + \)\(34\!\cdots\!28\)\( \beta_{3} - 1437017081453628624 \beta_{4} - 130671188070048 \beta_{5} + 46052813520 \beta_{6} + 51404976 \beta_{7}) q^{12} +(\)\(14\!\cdots\!30\)\( - \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(76\!\cdots\!69\)\( \beta_{2} + \)\(43\!\cdots\!38\)\( \beta_{3} - 22836391586997725369 \beta_{4} + 1277203065211264 \beta_{5} - 2768620169896 \beta_{6} - 1847131776 \beta_{7}) q^{13} +(\)\(11\!\cdots\!76\)\( + \)\(29\!\cdots\!16\)\( \beta_{1} + \)\(68\!\cdots\!26\)\( \beta_{2} + \)\(36\!\cdots\!08\)\( \beta_{3} + \)\(67\!\cdots\!10\)\( \beta_{4} + 113268416831067546 \beta_{5} + 101942184647552 \beta_{6} + 54313261184 \beta_{7}) q^{14} +(\)\(23\!\cdots\!40\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} - \)\(15\!\cdots\!68\)\( \beta_{2} + \)\(13\!\cdots\!85\)\( \beta_{3} + \)\(21\!\cdots\!93\)\( \beta_{4} - 4760217497661248368 \beta_{5} - 2699459604617481 \beta_{6} - 1343169464832 \beta_{7}) q^{15} +(\)\(11\!\cdots\!16\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} + \)\(35\!\cdots\!16\)\( \beta_{3} - \)\(51\!\cdots\!44\)\( \beta_{4} + 86123790987775929904 \beta_{5} + 54813613926115496 \beta_{6} + 28494357719832 \beta_{7}) q^{16} +(-\)\(17\!\cdots\!30\)\( - \)\(31\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!74\)\( \beta_{2} - \)\(76\!\cdots\!44\)\( \beta_{3} - \)\(11\!\cdots\!38\)\( \beta_{4} - \)\(63\!\cdots\!64\)\( \beta_{5} - 878472313378704156 \beta_{6} - 526281973855680 \beta_{7}) q^{17} +(-\)\(22\!\cdots\!85\)\( + \)\(79\!\cdots\!33\)\( \beta_{1} - \)\(21\!\cdots\!60\)\( \beta_{2} - \)\(75\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!28\)\( \beta_{4} - \)\(62\!\cdots\!56\)\( \beta_{5} + 11177988501349285056 \beta_{6} + 8559395719398720 \beta_{7}) q^{18} +(-\)\(65\!\cdots\!80\)\( - \)\(31\!\cdots\!75\)\( \beta_{1} - \)\(57\!\cdots\!09\)\( \beta_{2} - \)\(39\!\cdots\!34\)\( \beta_{3} - \)\(25\!\cdots\!86\)\( \beta_{4} + \)\(23\!\cdots\!48\)\( \beta_{5} - \)\(11\!\cdots\!22\)\( \beta_{6} - 123672945181895424 \beta_{7}) q^{19} +(\)\(17\!\cdots\!60\)\( + \)\(47\!\cdots\!28\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} + \)\(88\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!88\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} + \)\(76\!\cdots\!00\)\( \beta_{6} + 1598525547088209600 \beta_{7}) q^{20} +(\)\(16\!\cdots\!92\)\( - \)\(29\!\cdots\!24\)\( \beta_{1} - \)\(93\!\cdots\!80\)\( \beta_{2} + \)\(48\!\cdots\!08\)\( \beta_{3} + \)\(30\!\cdots\!80\)\( \beta_{4} + \)\(24\!\cdots\!52\)\( \beta_{5} - \)\(20\!\cdots\!96\)\( \beta_{6} - 18582876201795196032 \beta_{7}) q^{21} +(-\)\(25\!\cdots\!20\)\( + \)\(76\!\cdots\!86\)\( \beta_{1} - \)\(63\!\cdots\!89\)\( \beta_{2} - \)\(60\!\cdots\!52\)\( \beta_{3} - \)\(13\!\cdots\!99\)\( \beta_{4} - \)\(42\!\cdots\!61\)\( \beta_{5} - \)\(37\!\cdots\!76\)\( \beta_{6} + \)\(19\!\cdots\!84\)\( \beta_{7}) q^{22} +(-\)\(89\!\cdots\!80\)\( - \)\(36\!\cdots\!22\)\( \beta_{1} + \)\(49\!\cdots\!56\)\( \beta_{2} - \)\(50\!\cdots\!35\)\( \beta_{3} - \)\(86\!\cdots\!95\)\( \beta_{4} - \)\(13\!\cdots\!60\)\( \beta_{5} + \)\(75\!\cdots\!35\)\( \beta_{6} - \)\(18\!\cdots\!00\)\( \beta_{7}) q^{23} +(\)\(36\!\cdots\!60\)\( + \)\(25\!\cdots\!36\)\( \beta_{1} + \)\(77\!\cdots\!24\)\( \beta_{2} + \)\(55\!\cdots\!04\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4} + \)\(18\!\cdots\!32\)\( \beta_{5} - \)\(83\!\cdots\!04\)\( \beta_{6} + \)\(16\!\cdots\!32\)\( \beta_{7}) q^{24} +(\)\(10\!\cdots\!75\)\( - \)\(26\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(63\!\cdots\!00\)\( \beta_{3} - \)\(54\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!20\)\( \beta_{5} + \)\(68\!\cdots\!60\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7}) q^{25} +(-\)\(86\!\cdots\!38\)\( + \)\(38\!\cdots\!38\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(24\!\cdots\!92\)\( \beta_{4} + \)\(54\!\cdots\!04\)\( \beta_{5} - \)\(43\!\cdots\!48\)\( \beta_{6} + \)\(89\!\cdots\!84\)\( \beta_{7}) q^{26} +(\)\(80\!\cdots\!20\)\( + \)\(84\!\cdots\!74\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} - \)\(77\!\cdots\!58\)\( \beta_{3} + \)\(35\!\cdots\!14\)\( \beta_{4} + \)\(11\!\cdots\!48\)\( \beta_{5} + \)\(21\!\cdots\!70\)\( \beta_{6} - \)\(57\!\cdots\!56\)\( \beta_{7}) q^{27} +(\)\(37\!\cdots\!80\)\( + \)\(20\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!44\)\( \beta_{2} + \)\(85\!\cdots\!72\)\( \beta_{3} - \)\(13\!\cdots\!76\)\( \beta_{4} - \)\(16\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!00\)\( \beta_{6} + \)\(33\!\cdots\!64\)\( \beta_{7}) q^{28} +(\)\(96\!\cdots\!30\)\( + \)\(45\!\cdots\!36\)\( \beta_{1} + \)\(22\!\cdots\!31\)\( \beta_{2} - \)\(10\!\cdots\!70\)\( \beta_{3} - \)\(36\!\cdots\!37\)\( \beta_{4} + \)\(11\!\cdots\!04\)\( \beta_{5} - \)\(82\!\cdots\!08\)\( \beta_{6} - \)\(17\!\cdots\!36\)\( \beta_{7}) q^{29} +(-\)\(78\!\cdots\!60\)\( + \)\(92\!\cdots\!92\)\( \beta_{1} + \)\(20\!\cdots\!94\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(18\!\cdots\!18\)\( \beta_{4} - \)\(26\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6} + \)\(77\!\cdots\!00\)\( \beta_{7}) q^{30} +(\)\(60\!\cdots\!52\)\( + \)\(20\!\cdots\!72\)\( \beta_{1} + \)\(75\!\cdots\!92\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!64\)\( \beta_{4} - \)\(12\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!56\)\( \beta_{6} - \)\(29\!\cdots\!52\)\( \beta_{7}) q^{31} +(\)\(11\!\cdots\!40\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - \)\(31\!\cdots\!56\)\( \beta_{2} + \)\(47\!\cdots\!76\)\( \beta_{3} - \)\(65\!\cdots\!48\)\( \beta_{4} + \)\(14\!\cdots\!56\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(91\!\cdots\!20\)\( \beta_{7}) q^{32} +(-\)\(19\!\cdots\!20\)\( + \)\(33\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(68\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!42\)\( \beta_{4} - \)\(63\!\cdots\!48\)\( \beta_{5} + \)\(67\!\cdots\!96\)\( \beta_{6} - \)\(17\!\cdots\!96\)\( \beta_{7}) q^{33} +(-\)\(20\!\cdots\!54\)\( - \)\(87\!\cdots\!54\)\( \beta_{1} + \)\(43\!\cdots\!88\)\( \beta_{2} - \)\(61\!\cdots\!76\)\( \beta_{3} + \)\(79\!\cdots\!24\)\( \beta_{4} + \)\(10\!\cdots\!48\)\( \beta_{5} - \)\(14\!\cdots\!92\)\( \beta_{6} - \)\(23\!\cdots\!64\)\( \beta_{7}) q^{34} +(\)\(73\!\cdots\!20\)\( - \)\(29\!\cdots\!84\)\( \beta_{1} + \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} - \)\(77\!\cdots\!96\)\( \beta_{4} + \)\(35\!\cdots\!96\)\( \beta_{5} + \)\(10\!\cdots\!32\)\( \beta_{6} + \)\(40\!\cdots\!04\)\( \beta_{7}) q^{35} +(\)\(64\!\cdots\!16\)\( - \)\(45\!\cdots\!56\)\( \beta_{1} + \)\(35\!\cdots\!13\)\( \beta_{2} + \)\(76\!\cdots\!01\)\( \beta_{3} + \)\(20\!\cdots\!56\)\( \beta_{4} - \)\(23\!\cdots\!24\)\( \beta_{5} - \)\(46\!\cdots\!00\)\( \beta_{6} - \)\(21\!\cdots\!00\)\( \beta_{7}) q^{36} +(-\)\(22\!\cdots\!90\)\( - \)\(39\!\cdots\!24\)\( \beta_{1} - \)\(29\!\cdots\!93\)\( \beta_{2} - \)\(15\!\cdots\!06\)\( \beta_{3} + \)\(74\!\cdots\!23\)\( \beta_{4} + \)\(90\!\cdots\!56\)\( \beta_{5} + \)\(14\!\cdots\!80\)\( \beta_{6} + \)\(76\!\cdots\!88\)\( \beta_{7}) q^{37} +(-\)\(20\!\cdots\!80\)\( - \)\(22\!\cdots\!54\)\( \beta_{1} - \)\(67\!\cdots\!87\)\( \beta_{2} - \)\(95\!\cdots\!64\)\( \beta_{3} - \)\(85\!\cdots\!33\)\( \beta_{4} + \)\(50\!\cdots\!45\)\( \beta_{5} - \)\(25\!\cdots\!48\)\( \beta_{6} - \)\(16\!\cdots\!84\)\( \beta_{7}) q^{38} +(\)\(58\!\cdots\!04\)\( + \)\(11\!\cdots\!30\)\( \beta_{1} - \)\(90\!\cdots\!28\)\( \beta_{2} + \)\(15\!\cdots\!57\)\( \beta_{3} - \)\(58\!\cdots\!07\)\( \beta_{4} - \)\(30\!\cdots\!44\)\( \beta_{5} - \)\(37\!\cdots\!29\)\( \beta_{6} - \)\(15\!\cdots\!68\)\( \beta_{7}) q^{39} +(\)\(48\!\cdots\!00\)\( + \)\(45\!\cdots\!20\)\( \beta_{1} + \)\(20\!\cdots\!20\)\( \beta_{2} + \)\(13\!\cdots\!50\)\( \beta_{3} + \)\(41\!\cdots\!30\)\( \beta_{4} + \)\(98\!\cdots\!20\)\( \beta_{5} + \)\(46\!\cdots\!90\)\( \beta_{6} + \)\(20\!\cdots\!30\)\( \beta_{7}) q^{40} +(-\)\(10\!\cdots\!98\)\( + \)\(61\!\cdots\!32\)\( \beta_{1} + \)\(62\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!68\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4} - \)\(17\!\cdots\!76\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(10\!\cdots\!00\)\( \beta_{7}) q^{41} +(-\)\(19\!\cdots\!80\)\( + \)\(43\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!28\)\( \beta_{2} - \)\(99\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!72\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5} + \)\(41\!\cdots\!76\)\( \beta_{6} + \)\(29\!\cdots\!60\)\( \beta_{7}) q^{42} +(\)\(45\!\cdots\!00\)\( - \)\(29\!\cdots\!45\)\( \beta_{1} - \)\(34\!\cdots\!31\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(55\!\cdots\!20\)\( \beta_{4} - \)\(44\!\cdots\!56\)\( \beta_{5} - \)\(18\!\cdots\!32\)\( \beta_{6} - \)\(34\!\cdots\!44\)\( \beta_{7}) q^{43} +(\)\(24\!\cdots\!96\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} + \)\(73\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(33\!\cdots\!44\)\( \beta_{4} + \)\(41\!\cdots\!36\)\( \beta_{5} - \)\(24\!\cdots\!16\)\( \beta_{6} - \)\(13\!\cdots\!72\)\( \beta_{7}) q^{44} +(-\)\(20\!\cdots\!10\)\( - \)\(66\!\cdots\!68\)\( \beta_{1} + \)\(10\!\cdots\!99\)\( \beta_{2} - \)\(16\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!97\)\( \beta_{4} - \)\(14\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(92\!\cdots\!00\)\( \beta_{7}) q^{45} +(-\)\(24\!\cdots\!48\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(83\!\cdots\!74\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(38\!\cdots\!14\)\( \beta_{4} + \)\(91\!\cdots\!54\)\( \beta_{5} - \)\(19\!\cdots\!80\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7}) q^{46} +(\)\(48\!\cdots\!80\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(17\!\cdots\!50\)\( \beta_{3} + \)\(75\!\cdots\!90\)\( \beta_{4} + \)\(12\!\cdots\!48\)\( \beta_{5} + \)\(49\!\cdots\!06\)\( \beta_{6} + \)\(36\!\cdots\!52\)\( \beta_{7}) q^{47} +(\)\(11\!\cdots\!20\)\( + \)\(20\!\cdots\!96\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!76\)\( \beta_{3} - \)\(69\!\cdots\!48\)\( \beta_{4} - \)\(54\!\cdots\!44\)\( \beta_{5} + \)\(76\!\cdots\!24\)\( \beta_{6} + \)\(92\!\cdots\!20\)\( \beta_{7}) q^{48} +(\)\(33\!\cdots\!93\)\( + \)\(27\!\cdots\!12\)\( \beta_{1} + \)\(25\!\cdots\!04\)\( \beta_{2} - \)\(85\!\cdots\!48\)\( \beta_{3} - \)\(47\!\cdots\!36\)\( \beta_{4} + \)\(90\!\cdots\!84\)\( \beta_{5} - \)\(17\!\cdots\!20\)\( \beta_{6} - \)\(67\!\cdots\!40\)\( \beta_{7}) q^{49} +(-\)\(17\!\cdots\!75\)\( + \)\(13\!\cdots\!35\)\( \beta_{1} + \)\(72\!\cdots\!40\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!40\)\( \beta_{4} + \)\(26\!\cdots\!80\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6} + \)\(18\!\cdots\!20\)\( \beta_{7}) q^{50} +(\)\(30\!\cdots\!32\)\( - \)\(21\!\cdots\!10\)\( \beta_{1} + \)\(29\!\cdots\!74\)\( \beta_{2} + \)\(49\!\cdots\!34\)\( \beta_{3} - \)\(30\!\cdots\!94\)\( \beta_{4} - \)\(33\!\cdots\!88\)\( \beta_{5} + \)\(32\!\cdots\!02\)\( \beta_{6} - \)\(17\!\cdots\!16\)\( \beta_{7}) q^{51} +(\)\(19\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(20\!\cdots\!74\)\( \beta_{2} + \)\(36\!\cdots\!50\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!68\)\( \beta_{5} + \)\(20\!\cdots\!96\)\( \beta_{6} - \)\(74\!\cdots\!68\)\( \beta_{7}) q^{52} +(-\)\(36\!\cdots\!10\)\( - \)\(85\!\cdots\!60\)\( \beta_{1} - \)\(62\!\cdots\!33\)\( \beta_{2} - \)\(59\!\cdots\!82\)\( \beta_{3} - \)\(10\!\cdots\!69\)\( \beta_{4} + \)\(14\!\cdots\!12\)\( \beta_{5} - \)\(13\!\cdots\!80\)\( \beta_{6} + \)\(38\!\cdots\!56\)\( \beta_{7}) q^{53} +(\)\(55\!\cdots\!20\)\( - \)\(26\!\cdots\!20\)\( \beta_{1} + \)\(42\!\cdots\!38\)\( \beta_{2} - \)\(17\!\cdots\!24\)\( \beta_{3} + \)\(65\!\cdots\!14\)\( \beta_{4} + \)\(12\!\cdots\!66\)\( \beta_{5} + \)\(25\!\cdots\!04\)\( \beta_{6} - \)\(81\!\cdots\!32\)\( \beta_{7}) q^{54} +(\)\(97\!\cdots\!40\)\( + \)\(68\!\cdots\!62\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} - \)\(33\!\cdots\!45\)\( \beta_{3} - \)\(71\!\cdots\!77\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!25\)\( \beta_{6} + \)\(11\!\cdots\!00\)\( \beta_{7}) q^{55} +(\)\(90\!\cdots\!80\)\( + \)\(27\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!92\)\( \beta_{4} + \)\(67\!\cdots\!32\)\( \beta_{5} - \)\(33\!\cdots\!52\)\( \beta_{6} + \)\(48\!\cdots\!16\)\( \beta_{7}) q^{56} +(\)\(18\!\cdots\!40\)\( + \)\(29\!\cdots\!56\)\( \beta_{1} - \)\(45\!\cdots\!58\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(88\!\cdots\!70\)\( \beta_{4} - \)\(23\!\cdots\!36\)\( \beta_{5} + \)\(76\!\cdots\!08\)\( \beta_{6} - \)\(15\!\cdots\!64\)\( \beta_{7}) q^{57} +(\)\(29\!\cdots\!30\)\( + \)\(76\!\cdots\!30\)\( \beta_{1} - \)\(72\!\cdots\!76\)\( \beta_{2} + \)\(48\!\cdots\!56\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(28\!\cdots\!52\)\( \beta_{5} - \)\(14\!\cdots\!92\)\( \beta_{6} + \)\(19\!\cdots\!08\)\( \beta_{7}) q^{58} +(\)\(30\!\cdots\!60\)\( - \)\(99\!\cdots\!21\)\( \beta_{1} + \)\(18\!\cdots\!25\)\( \beta_{2} - \)\(16\!\cdots\!68\)\( \beta_{3} + \)\(23\!\cdots\!20\)\( \beta_{4} + \)\(68\!\cdots\!88\)\( \beta_{5} - \)\(38\!\cdots\!24\)\( \beta_{6} + \)\(33\!\cdots\!92\)\( \beta_{7}) q^{59} +(\)\(51\!\cdots\!20\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(42\!\cdots\!84\)\( \beta_{2} + \)\(41\!\cdots\!80\)\( \beta_{3} - \)\(85\!\cdots\!16\)\( \beta_{4} - \)\(45\!\cdots\!84\)\( \beta_{5} + \)\(10\!\cdots\!72\)\( \beta_{6} - \)\(20\!\cdots\!16\)\( \beta_{7}) q^{60} +(\)\(32\!\cdots\!02\)\( - \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(75\!\cdots\!19\)\( \beta_{2} - \)\(71\!\cdots\!90\)\( \beta_{3} + \)\(97\!\cdots\!75\)\( \beta_{4} - \)\(52\!\cdots\!04\)\( \beta_{5} - \)\(77\!\cdots\!28\)\( \beta_{6} + \)\(39\!\cdots\!24\)\( \beta_{7}) q^{61} +(\)\(13\!\cdots\!80\)\( - \)\(46\!\cdots\!36\)\( \beta_{1} + \)\(38\!\cdots\!96\)\( \beta_{2} + \)\(90\!\cdots\!44\)\( \beta_{3} + \)\(21\!\cdots\!88\)\( \beta_{4} - \)\(22\!\cdots\!96\)\( \beta_{5} - \)\(22\!\cdots\!64\)\( \beta_{6} - \)\(22\!\cdots\!60\)\( \beta_{7}) q^{62} +(\)\(21\!\cdots\!60\)\( + \)\(29\!\cdots\!22\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} - \)\(66\!\cdots\!29\)\( \beta_{3} - \)\(51\!\cdots\!93\)\( \beta_{4} + \)\(43\!\cdots\!04\)\( \beta_{5} + \)\(67\!\cdots\!45\)\( \beta_{6} - \)\(18\!\cdots\!08\)\( \beta_{7}) q^{63} +(\)\(50\!\cdots\!28\)\( + \)\(27\!\cdots\!68\)\( \beta_{1} - \)\(30\!\cdots\!64\)\( \beta_{2} + \)\(98\!\cdots\!76\)\( \beta_{3} - \)\(14\!\cdots\!92\)\( \beta_{4} + \)\(19\!\cdots\!88\)\( \beta_{5} - \)\(53\!\cdots\!60\)\( \beta_{6} + \)\(49\!\cdots\!80\)\( \beta_{7}) q^{64} +(\)\(88\!\cdots\!40\)\( + \)\(49\!\cdots\!32\)\( \beta_{1} - \)\(69\!\cdots\!08\)\( \beta_{2} - \)\(23\!\cdots\!40\)\( \beta_{3} - \)\(13\!\cdots\!92\)\( \beta_{4} - \)\(88\!\cdots\!08\)\( \beta_{5} - \)\(25\!\cdots\!36\)\( \beta_{6} - \)\(37\!\cdots\!92\)\( \beta_{7}) q^{65} +(\)\(21\!\cdots\!44\)\( - \)\(55\!\cdots\!20\)\( \beta_{1} + \)\(35\!\cdots\!12\)\( \beta_{2} + \)\(16\!\cdots\!72\)\( \beta_{3} + \)\(12\!\cdots\!28\)\( \beta_{4} + \)\(95\!\cdots\!76\)\( \beta_{5} - \)\(92\!\cdots\!84\)\( \beta_{6} - \)\(12\!\cdots\!28\)\( \beta_{7}) q^{66} +(-\)\(80\!\cdots\!80\)\( + \)\(21\!\cdots\!01\)\( \beta_{1} + \)\(33\!\cdots\!87\)\( \beta_{2} - \)\(34\!\cdots\!34\)\( \beta_{3} - \)\(21\!\cdots\!98\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5} + \)\(44\!\cdots\!02\)\( \beta_{6} + \)\(43\!\cdots\!76\)\( \beta_{7}) q^{67} +(\)\(11\!\cdots\!40\)\( - \)\(37\!\cdots\!28\)\( \beta_{1} - \)\(54\!\cdots\!50\)\( \beta_{2} - \)\(97\!\cdots\!78\)\( \beta_{3} - \)\(11\!\cdots\!76\)\( \beta_{4} - \)\(48\!\cdots\!72\)\( \beta_{5} + \)\(79\!\cdots\!40\)\( \beta_{6} - \)\(50\!\cdots\!56\)\( \beta_{7}) q^{68} +(-\)\(20\!\cdots\!16\)\( - \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!56\)\( \beta_{2} + \)\(15\!\cdots\!12\)\( \beta_{3} + \)\(71\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5} - \)\(64\!\cdots\!04\)\( \beta_{6} - \)\(50\!\cdots\!68\)\( \beta_{7}) q^{69} +(-\)\(19\!\cdots\!80\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} - \)\(13\!\cdots\!68\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(56\!\cdots\!96\)\( \beta_{4} + \)\(43\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(27\!\cdots\!00\)\( \beta_{7}) q^{70} +(-\)\(18\!\cdots\!48\)\( + \)\(85\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!55\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} + \)\(22\!\cdots\!68\)\( \beta_{5} + \)\(12\!\cdots\!01\)\( \beta_{6} - \)\(38\!\cdots\!08\)\( \beta_{7}) q^{71} +(-\)\(21\!\cdots\!40\)\( + \)\(21\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(29\!\cdots\!37\)\( \beta_{3} - \)\(55\!\cdots\!99\)\( \beta_{4} - \)\(66\!\cdots\!82\)\( \beta_{5} - \)\(69\!\cdots\!33\)\( \beta_{6} - \)\(60\!\cdots\!55\)\( \beta_{7}) q^{72} +(-\)\(20\!\cdots\!30\)\( + \)\(66\!\cdots\!76\)\( \beta_{1} + \)\(65\!\cdots\!26\)\( \beta_{2} + \)\(21\!\cdots\!16\)\( \beta_{3} + \)\(55\!\cdots\!02\)\( \beta_{4} - \)\(27\!\cdots\!00\)\( \beta_{5} + \)\(65\!\cdots\!72\)\( \beta_{6} + \)\(10\!\cdots\!16\)\( \beta_{7}) q^{73} +(-\)\(26\!\cdots\!14\)\( - \)\(74\!\cdots\!62\)\( \beta_{1} + \)\(17\!\cdots\!12\)\( \beta_{2} - \)\(68\!\cdots\!72\)\( \beta_{3} + \)\(12\!\cdots\!16\)\( \beta_{4} + \)\(31\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!88\)\( \beta_{6} - \)\(15\!\cdots\!04\)\( \beta_{7}) q^{74} +(-\)\(39\!\cdots\!00\)\( - \)\(32\!\cdots\!05\)\( \beta_{1} - \)\(18\!\cdots\!95\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(16\!\cdots\!20\)\( \beta_{4} - \)\(29\!\cdots\!40\)\( \beta_{5} - \)\(40\!\cdots\!80\)\( \beta_{6} + \)\(72\!\cdots\!40\)\( \beta_{7}) q^{75} +(-\)\(11\!\cdots\!40\)\( - \)\(44\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3} - \)\(20\!\cdots\!44\)\( \beta_{4} - \)\(32\!\cdots\!96\)\( \beta_{5} - \)\(16\!\cdots\!04\)\( \beta_{6} - \)\(66\!\cdots\!68\)\( \beta_{7}) q^{76} +(-\)\(44\!\cdots\!00\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(32\!\cdots\!64\)\( \beta_{3} - \)\(67\!\cdots\!28\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} + \)\(17\!\cdots\!84\)\( \beta_{6} + \)\(12\!\cdots\!60\)\( \beta_{7}) q^{77} +(\)\(73\!\cdots\!00\)\( + \)\(79\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!86\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3} + \)\(34\!\cdots\!94\)\( \beta_{4} + \)\(28\!\cdots\!90\)\( \beta_{5} - \)\(11\!\cdots\!36\)\( \beta_{6} + \)\(51\!\cdots\!12\)\( \beta_{7}) q^{78} +(\)\(58\!\cdots\!80\)\( + \)\(80\!\cdots\!64\)\( \beta_{1} - \)\(92\!\cdots\!44\)\( \beta_{2} + \)\(43\!\cdots\!18\)\( \beta_{3} - \)\(10\!\cdots\!06\)\( \beta_{4} - \)\(45\!\cdots\!24\)\( \beta_{5} - \)\(58\!\cdots\!86\)\( \beta_{6} - \)\(37\!\cdots\!12\)\( \beta_{7}) q^{79} +(\)\(23\!\cdots\!20\)\( + \)\(73\!\cdots\!16\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(29\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!36\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(69\!\cdots\!00\)\( \beta_{7}) q^{80} +(\)\(15\!\cdots\!21\)\( + \)\(63\!\cdots\!88\)\( \beta_{1} - \)\(57\!\cdots\!22\)\( \beta_{2} - \)\(51\!\cdots\!56\)\( \beta_{3} + \)\(55\!\cdots\!30\)\( \beta_{4} + \)\(28\!\cdots\!68\)\( \beta_{5} + \)\(17\!\cdots\!16\)\( \beta_{6} + \)\(18\!\cdots\!72\)\( \beta_{7}) q^{81} +(\)\(40\!\cdots\!30\)\( - \)\(26\!\cdots\!06\)\( \beta_{1} + \)\(91\!\cdots\!44\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(22\!\cdots\!28\)\( \beta_{4} + \)\(14\!\cdots\!56\)\( \beta_{5} - \)\(43\!\cdots\!40\)\( \beta_{6} - \)\(30\!\cdots\!72\)\( \beta_{7}) q^{82} +(\)\(42\!\cdots\!60\)\( - \)\(26\!\cdots\!51\)\( \beta_{1} - \)\(88\!\cdots\!73\)\( \beta_{2} - \)\(99\!\cdots\!00\)\( \beta_{3} - \)\(21\!\cdots\!00\)\( \beta_{4} - \)\(54\!\cdots\!20\)\( \beta_{5} + \)\(57\!\cdots\!60\)\( \beta_{6} + \)\(47\!\cdots\!20\)\( \beta_{7}) q^{83} +(\)\(21\!\cdots\!16\)\( - \)\(62\!\cdots\!64\)\( \beta_{1} + \)\(39\!\cdots\!32\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3} - \)\(78\!\cdots\!72\)\( \beta_{4} + \)\(75\!\cdots\!08\)\( \beta_{5} + \)\(19\!\cdots\!40\)\( \beta_{6} + \)\(10\!\cdots\!80\)\( \beta_{7}) q^{84} +(\)\(73\!\cdots\!20\)\( - \)\(69\!\cdots\!04\)\( \beta_{1} - \)\(36\!\cdots\!74\)\( \beta_{2} + \)\(71\!\cdots\!80\)\( \beta_{3} + \)\(24\!\cdots\!74\)\( \beta_{4} - \)\(13\!\cdots\!24\)\( \beta_{5} - \)\(10\!\cdots\!08\)\( \beta_{6} - \)\(12\!\cdots\!76\)\( \beta_{7}) q^{85} +(-\)\(19\!\cdots\!68\)\( + \)\(15\!\cdots\!58\)\( \beta_{1} + \)\(36\!\cdots\!55\)\( \beta_{2} - \)\(39\!\cdots\!88\)\( \beta_{3} + \)\(73\!\cdots\!77\)\( \beta_{4} + \)\(21\!\cdots\!63\)\( \beta_{5} + \)\(73\!\cdots\!72\)\( \beta_{6} + \)\(10\!\cdots\!24\)\( \beta_{7}) q^{86} +(-\)\(68\!\cdots\!40\)\( + \)\(36\!\cdots\!86\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} - \)\(50\!\cdots\!71\)\( \beta_{3} + \)\(28\!\cdots\!53\)\( \beta_{4} + \)\(31\!\cdots\!08\)\( \beta_{5} - \)\(90\!\cdots\!81\)\( \beta_{6} + \)\(88\!\cdots\!96\)\( \beta_{7}) q^{87} +(-\)\(15\!\cdots\!40\)\( + \)\(51\!\cdots\!12\)\( \beta_{1} - \)\(51\!\cdots\!32\)\( \beta_{2} - \)\(19\!\cdots\!84\)\( \beta_{3} - \)\(48\!\cdots\!68\)\( \beta_{4} - \)\(14\!\cdots\!44\)\( \beta_{5} + \)\(52\!\cdots\!04\)\( \beta_{6} + \)\(21\!\cdots\!60\)\( \beta_{7}) q^{88} +(-\)\(42\!\cdots\!10\)\( - \)\(43\!\cdots\!92\)\( \beta_{1} - \)\(85\!\cdots\!98\)\( \beta_{2} + \)\(60\!\cdots\!20\)\( \beta_{3} - \)\(94\!\cdots\!26\)\( \beta_{4} + \)\(19\!\cdots\!48\)\( \beta_{5} - \)\(68\!\cdots\!92\)\( \beta_{6} - \)\(72\!\cdots\!64\)\( \beta_{7}) q^{89} +(-\)\(43\!\cdots\!10\)\( - \)\(83\!\cdots\!78\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(95\!\cdots\!60\)\( \beta_{3} + \)\(25\!\cdots\!68\)\( \beta_{4} + \)\(44\!\cdots\!32\)\( \beta_{5} - \)\(12\!\cdots\!56\)\( \beta_{6} + \)\(31\!\cdots\!68\)\( \beta_{7}) q^{90} +(-\)\(43\!\cdots\!68\)\( - \)\(71\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(19\!\cdots\!52\)\( \beta_{4} - \)\(24\!\cdots\!28\)\( \beta_{5} + \)\(48\!\cdots\!48\)\( \beta_{6} + \)\(36\!\cdots\!16\)\( \beta_{7}) q^{91} +(-\)\(15\!\cdots\!80\)\( - \)\(52\!\cdots\!80\)\( \beta_{1} - \)\(18\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!56\)\( \beta_{3} + \)\(21\!\cdots\!48\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(51\!\cdots\!40\)\( \beta_{6} - \)\(69\!\cdots\!52\)\( \beta_{7}) q^{92} +(\)\(24\!\cdots\!80\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(89\!\cdots\!88\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} + \)\(61\!\cdots\!96\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5} - \)\(78\!\cdots\!24\)\( \beta_{6} - \)\(22\!\cdots\!92\)\( \beta_{7}) q^{93} +(\)\(19\!\cdots\!56\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(57\!\cdots\!32\)\( \beta_{2} + \)\(59\!\cdots\!36\)\( \beta_{3} - \)\(18\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(98\!\cdots\!24\)\( \beta_{6} + \)\(21\!\cdots\!08\)\( \beta_{7}) q^{94} +(\)\(19\!\cdots\!00\)\( + \)\(63\!\cdots\!30\)\( \beta_{1} + \)\(82\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!25\)\( \beta_{3} + \)\(23\!\cdots\!45\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} - \)\(23\!\cdots\!25\)\( \beta_{6} - \)\(16\!\cdots\!00\)\( \beta_{7}) q^{95} +(\)\(12\!\cdots\!72\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} + \)\(33\!\cdots\!64\)\( \beta_{2} + \)\(64\!\cdots\!08\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4} + \)\(65\!\cdots\!48\)\( \beta_{5} + \)\(44\!\cdots\!40\)\( \beta_{6} - \)\(28\!\cdots\!20\)\( \beta_{7}) q^{96} +(\)\(12\!\cdots\!30\)\( + \)\(19\!\cdots\!28\)\( \beta_{1} + \)\(26\!\cdots\!18\)\( \beta_{2} - \)\(35\!\cdots\!68\)\( \beta_{3} - \)\(26\!\cdots\!06\)\( \beta_{4} - \)\(80\!\cdots\!12\)\( \beta_{5} - \)\(28\!\cdots\!20\)\( \beta_{6} + \)\(29\!\cdots\!44\)\( \beta_{7}) q^{97} +(\)\(18\!\cdots\!95\)\( - \)\(14\!\cdots\!95\)\( \beta_{1} - \)\(38\!\cdots\!96\)\( \beta_{2} + \)\(15\!\cdots\!84\)\( \beta_{3} - \)\(68\!\cdots\!72\)\( \beta_{4} + \)\(44\!\cdots\!16\)\( \beta_{5} - \)\(85\!\cdots\!20\)\( \beta_{6} + \)\(85\!\cdots\!68\)\( \beta_{7}) q^{98} +(\)\(38\!\cdots\!84\)\( - \)\(10\!\cdots\!21\)\( \beta_{1} + \)\(97\!\cdots\!57\)\( \beta_{2} - \)\(83\!\cdots\!20\)\( \beta_{3} + \)\(72\!\cdots\!52\)\( \beta_{4} + \)\(31\!\cdots\!48\)\( \beta_{5} + \)\(20\!\cdots\!92\)\( \beta_{6} - \)\(79\!\cdots\!36\)\( \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 5835659138280q^{2} - \)\(95\!\cdots\!80\)\(q^{3} + \)\(20\!\cdots\!84\)\(q^{4} + \)\(19\!\cdots\!60\)\(q^{5} + \)\(10\!\cdots\!76\)\(q^{6} + \)\(31\!\cdots\!00\)\(q^{7} - \)\(14\!\cdots\!60\)\(q^{8} + \)\(92\!\cdots\!36\)\(q^{9} + O(q^{10}) \) \( 8q - 5835659138280q^{2} - \)\(95\!\cdots\!80\)\(q^{3} + \)\(20\!\cdots\!84\)\(q^{4} + \)\(19\!\cdots\!60\)\(q^{5} + \)\(10\!\cdots\!76\)\(q^{6} + \)\(31\!\cdots\!00\)\(q^{7} - \)\(14\!\cdots\!60\)\(q^{8} + \)\(92\!\cdots\!36\)\(q^{9} - \)\(35\!\cdots\!40\)\(q^{10} + \)\(53\!\cdots\!16\)\(q^{11} + \)\(10\!\cdots\!40\)\(q^{12} + \)\(11\!\cdots\!40\)\(q^{13} + \)\(88\!\cdots\!08\)\(q^{14} + \)\(18\!\cdots\!20\)\(q^{15} + \)\(95\!\cdots\!28\)\(q^{16} - \)\(13\!\cdots\!40\)\(q^{17} - \)\(17\!\cdots\!80\)\(q^{18} - \)\(52\!\cdots\!40\)\(q^{19} + \)\(13\!\cdots\!80\)\(q^{20} + \)\(13\!\cdots\!36\)\(q^{21} - \)\(20\!\cdots\!60\)\(q^{22} - \)\(71\!\cdots\!40\)\(q^{23} + \)\(28\!\cdots\!80\)\(q^{24} + \)\(81\!\cdots\!00\)\(q^{25} - \)\(69\!\cdots\!04\)\(q^{26} + \)\(64\!\cdots\!60\)\(q^{27} + \)\(30\!\cdots\!40\)\(q^{28} + \)\(77\!\cdots\!40\)\(q^{29} - \)\(62\!\cdots\!80\)\(q^{30} + \)\(48\!\cdots\!16\)\(q^{31} + \)\(91\!\cdots\!20\)\(q^{32} - \)\(15\!\cdots\!60\)\(q^{33} - \)\(16\!\cdots\!32\)\(q^{34} + \)\(58\!\cdots\!60\)\(q^{35} + \)\(51\!\cdots\!28\)\(q^{36} - \)\(18\!\cdots\!20\)\(q^{37} - \)\(16\!\cdots\!40\)\(q^{38} + \)\(46\!\cdots\!32\)\(q^{39} + \)\(39\!\cdots\!00\)\(q^{40} - \)\(87\!\cdots\!84\)\(q^{41} - \)\(15\!\cdots\!40\)\(q^{42} + \)\(36\!\cdots\!00\)\(q^{43} + \)\(19\!\cdots\!68\)\(q^{44} - \)\(16\!\cdots\!80\)\(q^{45} - \)\(19\!\cdots\!84\)\(q^{46} + \)\(38\!\cdots\!40\)\(q^{47} + \)\(93\!\cdots\!60\)\(q^{48} + \)\(26\!\cdots\!44\)\(q^{49} - \)\(14\!\cdots\!00\)\(q^{50} + \)\(24\!\cdots\!56\)\(q^{51} + \)\(15\!\cdots\!00\)\(q^{52} - \)\(29\!\cdots\!80\)\(q^{53} + \)\(44\!\cdots\!60\)\(q^{54} + \)\(77\!\cdots\!20\)\(q^{55} + \)\(72\!\cdots\!40\)\(q^{56} + \)\(14\!\cdots\!20\)\(q^{57} + \)\(23\!\cdots\!40\)\(q^{58} + \)\(24\!\cdots\!80\)\(q^{59} + \)\(41\!\cdots\!60\)\(q^{60} + \)\(26\!\cdots\!16\)\(q^{61} + \)\(10\!\cdots\!40\)\(q^{62} + \)\(17\!\cdots\!80\)\(q^{63} + \)\(40\!\cdots\!24\)\(q^{64} + \)\(71\!\cdots\!20\)\(q^{65} + \)\(17\!\cdots\!52\)\(q^{66} - \)\(64\!\cdots\!40\)\(q^{67} + \)\(91\!\cdots\!20\)\(q^{68} - \)\(16\!\cdots\!28\)\(q^{69} - \)\(15\!\cdots\!40\)\(q^{70} - \)\(14\!\cdots\!84\)\(q^{71} - \)\(16\!\cdots\!20\)\(q^{72} - \)\(16\!\cdots\!40\)\(q^{73} - \)\(20\!\cdots\!12\)\(q^{74} - \)\(31\!\cdots\!00\)\(q^{75} - \)\(95\!\cdots\!20\)\(q^{76} - \)\(35\!\cdots\!00\)\(q^{77} + \)\(59\!\cdots\!00\)\(q^{78} + \)\(46\!\cdots\!40\)\(q^{79} + \)\(18\!\cdots\!60\)\(q^{80} + \)\(12\!\cdots\!68\)\(q^{81} + \)\(32\!\cdots\!40\)\(q^{82} + \)\(34\!\cdots\!80\)\(q^{83} + \)\(17\!\cdots\!28\)\(q^{84} + \)\(58\!\cdots\!60\)\(q^{85} - \)\(15\!\cdots\!44\)\(q^{86} - \)\(54\!\cdots\!20\)\(q^{87} - \)\(12\!\cdots\!20\)\(q^{88} - \)\(34\!\cdots\!80\)\(q^{89} - \)\(35\!\cdots\!80\)\(q^{90} - \)\(34\!\cdots\!44\)\(q^{91} - \)\(12\!\cdots\!40\)\(q^{92} + \)\(19\!\cdots\!40\)\(q^{93} + \)\(15\!\cdots\!48\)\(q^{94} + \)\(15\!\cdots\!00\)\(q^{95} + \)\(97\!\cdots\!76\)\(q^{96} + \)\(10\!\cdots\!40\)\(q^{97} + \)\(14\!\cdots\!60\)\(q^{98} + \)\(30\!\cdots\!72\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - \)\(45\!\cdots\!12\)\( x^{6} + \)\(33\!\cdots\!60\)\( x^{5} + \)\(59\!\cdots\!16\)\( x^{4} - \)\(12\!\cdots\!20\)\( x^{3} - \)\(20\!\cdots\!88\)\( x^{2} + \)\(46\!\cdots\!68\)\( x + \)\(12\!\cdots\!76\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 3 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(76\!\cdots\!51\)\( \nu^{7} + \)\(14\!\cdots\!81\)\( \nu^{6} - \)\(52\!\cdots\!16\)\( \nu^{5} - \)\(40\!\cdots\!00\)\( \nu^{4} + \)\(90\!\cdots\!96\)\( \nu^{3} + \)\(14\!\cdots\!32\)\( \nu^{2} - \)\(29\!\cdots\!00\)\( \nu - \)\(69\!\cdots\!36\)\(\)\()/ \)\(58\!\cdots\!72\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(52\!\cdots\!49\)\( \nu^{7} + \)\(10\!\cdots\!19\)\( \nu^{6} - \)\(36\!\cdots\!84\)\( \nu^{5} - \)\(27\!\cdots\!00\)\( \nu^{4} + \)\(62\!\cdots\!04\)\( \nu^{3} + \)\(47\!\cdots\!76\)\( \nu^{2} - \)\(16\!\cdots\!16\)\( \nu - \)\(43\!\cdots\!12\)\(\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(13\!\cdots\!97\)\( \nu^{7} - \)\(63\!\cdots\!15\)\( \nu^{6} + \)\(51\!\cdots\!24\)\( \nu^{5} + \)\(41\!\cdots\!84\)\( \nu^{4} - \)\(50\!\cdots\!88\)\( \nu^{3} - \)\(67\!\cdots\!08\)\( \nu^{2} + \)\(12\!\cdots\!68\)\( \nu - \)\(31\!\cdots\!68\)\(\)\()/ \)\(73\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(60\!\cdots\!17\)\( \nu^{7} + \)\(10\!\cdots\!85\)\( \nu^{6} + \)\(17\!\cdots\!64\)\( \nu^{5} - \)\(26\!\cdots\!76\)\( \nu^{4} - \)\(11\!\cdots\!68\)\( \nu^{3} + \)\(76\!\cdots\!12\)\( \nu^{2} + \)\(33\!\cdots\!48\)\( \nu + \)\(10\!\cdots\!52\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(14\!\cdots\!57\)\( \nu^{7} - \)\(12\!\cdots\!15\)\( \nu^{6} + \)\(58\!\cdots\!44\)\( \nu^{5} + \)\(43\!\cdots\!04\)\( \nu^{4} - \)\(62\!\cdots\!28\)\( \nu^{3} - \)\(32\!\cdots\!48\)\( \nu^{2} + \)\(15\!\cdots\!08\)\( \nu + \)\(49\!\cdots\!92\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(37\!\cdots\!87\)\( \nu^{7} + \)\(30\!\cdots\!65\)\( \nu^{6} - \)\(21\!\cdots\!04\)\( \nu^{5} - \)\(12\!\cdots\!64\)\( \nu^{4} + \)\(51\!\cdots\!48\)\( \nu^{3} + \)\(11\!\cdots\!68\)\( \nu^{2} - \)\(54\!\cdots\!28\)\( \nu - \)\(15\!\cdots\!72\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 3\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 62091 \beta_{2} - 26761022262152 \beta_{1} + 65733123937758428360847400200\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 295 \beta_{6} + 160002 \beta_{5} - 2004356999 \beta_{4} + 51851570017067 \beta_{3} + 3276903991940428836 \beta_{2} + 112986517961279136356684832712 \beta_{1} - 1759085591639276897686588409592076569213504\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(3926523411123 \beta_{7} + 6959296706126917 \beta_{6} + 10823831194312423526 \beta_{5} - 64844235599751336866149 \beta_{4} + 19321453573813038341957137905 \beta_{3} - 1394410694473382624668011668515956 \beta_{2} - 298710901026801058381966898320637168298536 \beta_{1} + 928369598546200014171379235704963029862290826589385463872\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(2621067272289636342635857733 \beta_{7} + 781805958213198104016871956611 \beta_{6} + 626720668829570339117854559945610 \beta_{5} - 6015663869305187979638343702232227939 \beta_{4} + 205850449531108259889513399496641754951863 \beta_{3} - 41887807299957782555913661895122244965673155244 \beta_{2} + 229515919805726057201751531896339304137050604018359815144 \beta_{1} - 2454400080858624658714224866616934605524010680291289642149797035267648\)\()/124416\)
\(\nu^{6}\)\(=\)\((\)\(2456698967573154102027892526992622040587 \beta_{7} + 2287336922187280642817740919038569124333229 \beta_{6} + 4173765778599191392471236993532318130284922646 \beta_{5} - 22719176458145430265227683260813234396944017761485 \beta_{4} + 4820608307462255271214311913264579460921736185013012921 \beta_{3} - 421824107166754602266817224392645212904577134947783234195476 \beta_{2} + 14896020318557401322538306359430870429741697951273163415269325560472 \beta_{1} + 209538866711613197149097879153829515723664322115869508558284660196534474158205620800\)\()/41472\)
\(\nu^{7}\)\(=\)\((\)\(\)\(22\!\cdots\!91\)\( \beta_{7} + \)\(67\!\cdots\!33\)\( \beta_{6} + \)\(63\!\cdots\!18\)\( \beta_{5} - \)\(57\!\cdots\!61\)\( \beta_{4} + \)\(23\!\cdots\!85\)\( \beta_{3} - \)\(59\!\cdots\!32\)\( \beta_{2} + \)\(18\!\cdots\!88\)\( \beta_{1} + \)\(40\!\cdots\!88\)\(\)\()/41472\)

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
−1.48943e13
−1.46162e13
−6.56924e12
−1.67307e12
4.79736e12
6.86315e12
9.80521e12
1.62871e13
−3.58194e14 −7.12911e22 8.86885e28 −1.64420e33 2.55360e37 −1.35904e40 −1.75781e43 2.96152e45 5.88943e47
1.2 −3.51519e14 6.56090e22 8.39514e28 2.79946e33 −2.30628e37 5.53054e39 −1.55854e43 2.18364e45 −9.84062e47
1.3 −1.58391e14 9.80327e21 −1.45263e28 −7.31691e32 −1.55275e36 −1.37055e39 8.57536e42 −2.02479e45 1.15893e47
1.4 −4.08831e13 −7.46060e22 −3.79427e28 2.52173e33 3.05013e36 2.11876e40 3.17076e42 3.44515e45 −1.03096e47
1.5 1.14407e14 8.75958e22 −2.65251e28 −1.07741e33 1.00216e37 1.86215e40 −7.56680e42 5.55214e45 −1.23264e47
1.6 1.63986e14 6.46739e21 −1.27226e28 1.20020e33 1.06056e36 −2.14893e40 −8.58249e42 −2.07907e45 1.96817e47
1.7 2.34596e14 −5.48368e22 1.54210e28 −2.52565e33 −1.28645e37 4.80687e39 −5.67559e42 8.86182e44 −5.92506e47
1.8 3.90162e14 2.16924e22 1.12612e29 1.39914e33 8.46353e36 1.75361e40 2.84812e43 −1.65034e45 5.45891e47
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 1.96.a.a 8
3.b odd 2 1 9.96.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.96.a.a 8 1.a even 1 1 trivial
9.96.a.c 8 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{96}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5835659138280 T + \)\(53\!\cdots\!80\)\( T^{2} + \)\(51\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!96\)\( T^{4} - \)\(31\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!60\)\( T^{6} - \)\(20\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!56\)\( T^{8} - \)\(83\!\cdots\!60\)\( T^{9} + \)\(24\!\cdots\!40\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(28\!\cdots\!96\)\( T^{12} + \)\(50\!\cdots\!20\)\( T^{13} + \)\(20\!\cdots\!20\)\( T^{14} + \)\(89\!\cdots\!60\)\( T^{15} + \)\(60\!\cdots\!76\)\( T^{16} \)
$3$ \( 1 + \)\(95\!\cdots\!80\)\( T + \)\(38\!\cdots\!60\)\( T^{2} + \)\(88\!\cdots\!60\)\( T^{3} + \)\(90\!\cdots\!96\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!20\)\( T^{6} - \)\(64\!\cdots\!80\)\( T^{7} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(13\!\cdots\!60\)\( T^{9} + \)\(10\!\cdots\!80\)\( T^{10} - \)\(26\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!96\)\( T^{12} + \)\(37\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!40\)\( T^{15} + \)\(40\!\cdots\!01\)\( T^{16} \)
$5$ \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(78\!\cdots\!00\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!00\)\( T^{4} - \)\(64\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!50\)\( T^{8} - \)\(50\!\cdots\!00\)\( T^{9} + \)\(88\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!00\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{13} + \)\(20\!\cdots\!00\)\( T^{14} - \)\(12\!\cdots\!00\)\( T^{15} + \)\(16\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(11\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!96\)\( T^{4} - \)\(75\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!06\)\( T^{8} - \)\(35\!\cdots\!00\)\( T^{9} + \)\(51\!\cdots\!00\)\( T^{10} - \)\(53\!\cdots\!00\)\( T^{11} + \)\(67\!\cdots\!96\)\( T^{12} - \)\(58\!\cdots\!00\)\( T^{13} + \)\(57\!\cdots\!00\)\( T^{14} - \)\(30\!\cdots\!00\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 - \)\(53\!\cdots\!16\)\( T + \)\(59\!\cdots\!20\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!20\)\( T^{4} - \)\(53\!\cdots\!68\)\( T^{5} + \)\(24\!\cdots\!08\)\( T^{6} - \)\(69\!\cdots\!40\)\( T^{7} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(59\!\cdots\!40\)\( T^{9} + \)\(17\!\cdots\!08\)\( T^{10} - \)\(33\!\cdots\!68\)\( T^{11} + \)\(83\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!60\)\( T^{13} + \)\(23\!\cdots\!20\)\( T^{14} - \)\(17\!\cdots\!16\)\( T^{15} + \)\(28\!\cdots\!01\)\( T^{16} \)
$13$ \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(38\!\cdots\!20\)\( T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!96\)\( T^{4} - \)\(56\!\cdots\!80\)\( T^{5} + \)\(80\!\cdots\!40\)\( T^{6} - \)\(54\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!20\)\( T^{9} + \)\(35\!\cdots\!60\)\( T^{10} - \)\(16\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!96\)\( T^{12} - \)\(48\!\cdots\!60\)\( T^{13} + \)\(33\!\cdots\!80\)\( T^{14} - \)\(69\!\cdots\!20\)\( T^{15} + \)\(39\!\cdots\!01\)\( T^{16} \)
$17$ \( 1 + \)\(13\!\cdots\!40\)\( T + \)\(50\!\cdots\!40\)\( T^{2} + \)\(52\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!96\)\( T^{4} + \)\(94\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!06\)\( T^{8} + \)\(84\!\cdots\!20\)\( T^{9} + \)\(99\!\cdots\!20\)\( T^{10} + \)\(45\!\cdots\!60\)\( T^{11} + \)\(43\!\cdots\!96\)\( T^{12} + \)\(15\!\cdots\!60\)\( T^{13} + \)\(11\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!80\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} \)
$19$ \( 1 + \)\(52\!\cdots\!40\)\( T + \)\(14\!\cdots\!92\)\( T^{2} + \)\(69\!\cdots\!20\)\( T^{3} + \)\(86\!\cdots\!28\)\( T^{4} + \)\(46\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!44\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!70\)\( T^{8} + \)\(62\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!44\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(73\!\cdots\!28\)\( T^{12} + \)\(17\!\cdots\!80\)\( T^{13} + \)\(10\!\cdots\!92\)\( T^{14} + \)\(12\!\cdots\!60\)\( T^{15} + \)\(71\!\cdots\!01\)\( T^{16} \)
$23$ \( 1 + \)\(71\!\cdots\!40\)\( T + \)\(11\!\cdots\!80\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!96\)\( T^{4} + \)\(65\!\cdots\!80\)\( T^{5} + \)\(21\!\cdots\!60\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(56\!\cdots\!06\)\( T^{8} + \)\(52\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!40\)\( T^{10} + \)\(80\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!96\)\( T^{12} + \)\(72\!\cdots\!60\)\( T^{13} + \)\(17\!\cdots\!20\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(81\!\cdots\!01\)\( T^{16} \)
$29$ \( 1 - \)\(77\!\cdots\!40\)\( T + \)\(64\!\cdots\!92\)\( T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!28\)\( T^{4} - \)\(45\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!44\)\( T^{6} - \)\(54\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!70\)\( T^{8} - \)\(46\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} - \)\(27\!\cdots\!60\)\( T^{11} + \)\(71\!\cdots\!28\)\( T^{12} - \)\(12\!\cdots\!80\)\( T^{13} + \)\(23\!\cdots\!92\)\( T^{14} - \)\(24\!\cdots\!60\)\( T^{15} + \)\(26\!\cdots\!01\)\( T^{16} \)
$31$ \( 1 - \)\(48\!\cdots\!16\)\( T + \)\(22\!\cdots\!20\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!20\)\( T^{4} - \)\(12\!\cdots\!68\)\( T^{5} + \)\(21\!\cdots\!08\)\( T^{6} - \)\(88\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(42\!\cdots\!40\)\( T^{9} + \)\(49\!\cdots\!08\)\( T^{10} - \)\(13\!\cdots\!68\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} - \)\(27\!\cdots\!60\)\( T^{13} + \)\(27\!\cdots\!20\)\( T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(27\!\cdots\!01\)\( T^{16} \)
$37$ \( 1 + \)\(18\!\cdots\!20\)\( T + \)\(42\!\cdots\!20\)\( T^{2} + \)\(81\!\cdots\!60\)\( T^{3} + \)\(93\!\cdots\!96\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!40\)\( T^{6} + \)\(26\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!06\)\( T^{8} + \)\(25\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!60\)\( T^{10} + \)\(16\!\cdots\!80\)\( T^{11} + \)\(77\!\cdots\!96\)\( T^{12} + \)\(63\!\cdots\!80\)\( T^{13} + \)\(31\!\cdots\!80\)\( T^{14} + \)\(12\!\cdots\!40\)\( T^{15} + \)\(68\!\cdots\!01\)\( T^{16} \)
$41$ \( 1 + \)\(87\!\cdots\!84\)\( T + \)\(90\!\cdots\!20\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!20\)\( T^{4} + \)\(82\!\cdots\!32\)\( T^{5} + \)\(32\!\cdots\!08\)\( T^{6} + \)\(72\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(87\!\cdots\!08\)\( T^{10} + \)\(36\!\cdots\!32\)\( T^{11} + \)\(18\!\cdots\!20\)\( T^{12} + \)\(54\!\cdots\!40\)\( T^{13} + \)\(17\!\cdots\!20\)\( T^{14} + \)\(27\!\cdots\!84\)\( T^{15} + \)\(51\!\cdots\!01\)\( T^{16} \)
$43$ \( 1 - \)\(36\!\cdots\!00\)\( T + \)\(97\!\cdots\!00\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!96\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!06\)\( T^{8} - \)\(30\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!00\)\( T^{10} - \)\(33\!\cdots\!00\)\( T^{11} + \)\(22\!\cdots\!96\)\( T^{12} - \)\(22\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} - \)\(65\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - \)\(38\!\cdots\!40\)\( T + \)\(44\!\cdots\!60\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(92\!\cdots\!96\)\( T^{4} - \)\(29\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(23\!\cdots\!20\)\( T^{9} + \)\(59\!\cdots\!80\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(23\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!60\)\( T^{13} + \)\(55\!\cdots\!40\)\( T^{14} - \)\(34\!\cdots\!80\)\( T^{15} + \)\(62\!\cdots\!01\)\( T^{16} \)
$53$ \( 1 + \)\(29\!\cdots\!80\)\( T + \)\(35\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(56\!\cdots\!96\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(56\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!20\)\( T^{7} + \)\(41\!\cdots\!06\)\( T^{8} + \)\(21\!\cdots\!40\)\( T^{9} + \)\(23\!\cdots\!80\)\( T^{10} + \)\(80\!\cdots\!80\)\( T^{11} + \)\(95\!\cdots\!96\)\( T^{12} + \)\(16\!\cdots\!20\)\( T^{13} + \)\(24\!\cdots\!40\)\( T^{14} + \)\(12\!\cdots\!40\)\( T^{15} + \)\(28\!\cdots\!01\)\( T^{16} \)
$59$ \( 1 - \)\(24\!\cdots\!80\)\( T + \)\(73\!\cdots\!92\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!28\)\( T^{4} - \)\(36\!\cdots\!80\)\( T^{5} + \)\(68\!\cdots\!44\)\( T^{6} - \)\(80\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!44\)\( T^{10} - \)\(18\!\cdots\!20\)\( T^{11} + \)\(21\!\cdots\!28\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{13} + \)\(17\!\cdots\!92\)\( T^{14} - \)\(10\!\cdots\!20\)\( T^{15} + \)\(70\!\cdots\!01\)\( T^{16} \)
$61$ \( 1 - \)\(26\!\cdots\!16\)\( T + \)\(48\!\cdots\!20\)\( T^{2} - \)\(64\!\cdots\!60\)\( T^{3} + \)\(71\!\cdots\!20\)\( T^{4} - \)\(67\!\cdots\!68\)\( T^{5} + \)\(56\!\cdots\!08\)\( T^{6} - \)\(42\!\cdots\!40\)\( T^{7} + \)\(28\!\cdots\!70\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} + \)\(92\!\cdots\!08\)\( T^{10} - \)\(44\!\cdots\!68\)\( T^{11} + \)\(18\!\cdots\!20\)\( T^{12} - \)\(68\!\cdots\!60\)\( T^{13} + \)\(20\!\cdots\!20\)\( T^{14} - \)\(46\!\cdots\!16\)\( T^{15} + \)\(70\!\cdots\!01\)\( T^{16} \)
$67$ \( 1 + \)\(64\!\cdots\!40\)\( T + \)\(84\!\cdots\!40\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!96\)\( T^{4} - \)\(23\!\cdots\!20\)\( T^{5} + \)\(33\!\cdots\!80\)\( T^{6} - \)\(41\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!06\)\( T^{8} - \)\(12\!\cdots\!80\)\( T^{9} + \)\(29\!\cdots\!20\)\( T^{10} - \)\(62\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!96\)\( T^{12} + \)\(55\!\cdots\!60\)\( T^{13} + \)\(61\!\cdots\!60\)\( T^{14} + \)\(14\!\cdots\!80\)\( T^{15} + \)\(65\!\cdots\!01\)\( T^{16} \)
$71$ \( 1 + \)\(14\!\cdots\!84\)\( T + \)\(54\!\cdots\!20\)\( T^{2} + \)\(66\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(10\!\cdots\!08\)\( T^{10} + \)\(55\!\cdots\!32\)\( T^{11} + \)\(40\!\cdots\!20\)\( T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(89\!\cdots\!20\)\( T^{14} + \)\(17\!\cdots\!84\)\( T^{15} + \)\(90\!\cdots\!01\)\( T^{16} \)
$73$ \( 1 + \)\(16\!\cdots\!40\)\( T + \)\(17\!\cdots\!80\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!96\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{5} + \)\(18\!\cdots\!60\)\( T^{6} + \)\(70\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!06\)\( T^{8} + \)\(73\!\cdots\!20\)\( T^{9} + \)\(19\!\cdots\!40\)\( T^{10} + \)\(46\!\cdots\!40\)\( T^{11} + \)\(94\!\cdots\!96\)\( T^{12} + \)\(16\!\cdots\!60\)\( T^{13} + \)\(22\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!20\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} \)
$79$ \( 1 - \)\(46\!\cdots\!40\)\( T + \)\(20\!\cdots\!92\)\( T^{2} - \)\(58\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!28\)\( T^{4} - \)\(31\!\cdots\!40\)\( T^{5} + \)\(59\!\cdots\!44\)\( T^{6} - \)\(98\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!44\)\( T^{10} - \)\(21\!\cdots\!60\)\( T^{11} + \)\(18\!\cdots\!28\)\( T^{12} - \)\(13\!\cdots\!80\)\( T^{13} + \)\(89\!\cdots\!92\)\( T^{14} - \)\(39\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!01\)\( T^{16} \)
$83$ \( 1 - \)\(34\!\cdots\!80\)\( T + \)\(14\!\cdots\!40\)\( T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(82\!\cdots\!96\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(69\!\cdots\!06\)\( T^{8} - \)\(85\!\cdots\!40\)\( T^{9} + \)\(12\!\cdots\!20\)\( T^{10} - \)\(12\!\cdots\!80\)\( T^{11} + \)\(14\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!60\)\( T^{14} - \)\(52\!\cdots\!40\)\( T^{15} + \)\(31\!\cdots\!01\)\( T^{16} \)
$89$ \( 1 + \)\(34\!\cdots\!80\)\( T + \)\(62\!\cdots\!92\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!28\)\( T^{4} + \)\(61\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(98\!\cdots\!70\)\( T^{8} + \)\(20\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(13\!\cdots\!28\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(88\!\cdots\!92\)\( T^{14} + \)\(75\!\cdots\!20\)\( T^{15} + \)\(34\!\cdots\!01\)\( T^{16} \)
$97$ \( 1 - \)\(10\!\cdots\!40\)\( T + \)\(13\!\cdots\!60\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!96\)\( T^{4} - \)\(25\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} - \)\(19\!\cdots\!40\)\( T^{7} + \)\(84\!\cdots\!06\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{9} + \)\(38\!\cdots\!80\)\( T^{10} - \)\(43\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!60\)\( T^{13} + \)\(39\!\cdots\!40\)\( T^{14} - \)\(16\!\cdots\!80\)\( T^{15} + \)\(88\!\cdots\!01\)\( T^{16} \)
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