Properties

Label 1.92.a.a
Level 1
Weight 92
Character orbit 1.a
Self dual yes
Analytic conductor 52.442
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 92 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.4421558310\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{31}\cdot 5^{8}\cdot 7^{6}\cdot 11\cdot 13^{3}\cdot 23 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(548816691151 + \beta_{1}) q^{2} +(\)\(88\!\cdots\!63\)\( + 19467112 \beta_{1} + \beta_{2}) q^{3} +(\)\(81\!\cdots\!85\)\( - 173326932865 \beta_{1} + 49571 \beta_{2} + \beta_{3}) q^{4} +(\)\(33\!\cdots\!67\)\( + 203684509503647605 \beta_{1} + 3408688857 \beta_{2} + 3464 \beta_{3} - \beta_{4}) q^{5} +(\)\(64\!\cdots\!93\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} + 7916444742738 \beta_{2} - 24868 \beta_{3} - 1212 \beta_{4} + \beta_{5}) q^{6} +(-\)\(24\!\cdots\!13\)\( + \)\(77\!\cdots\!83\)\( \beta_{1} - 15418806458979859 \beta_{2} - 20904077593 \beta_{3} + 79668 \beta_{4} + 29 \beta_{5} + \beta_{6}) q^{7} +(-\)\(14\!\cdots\!32\)\( + \)\(36\!\cdots\!44\)\( \beta_{1} - 4021839698115592272 \beta_{2} - 6483639273776 \beta_{3} - 101411744 \beta_{4} + 103728 \beta_{5} + 312 \beta_{6}) q^{8} +(\)\(54\!\cdots\!55\)\( + \)\(10\!\cdots\!58\)\( \beta_{1} + \)\(81\!\cdots\!14\)\( \beta_{2} + 204820844789196 \beta_{3} - 127519054326 \beta_{4} + 80849268 \beta_{5} - 4860 \beta_{6}) q^{9} +O(q^{10})\) \( q +(548816691151 + \beta_{1}) q^{2} +(\)\(88\!\cdots\!63\)\( + 19467112 \beta_{1} + \beta_{2}) q^{3} +(\)\(81\!\cdots\!85\)\( - 173326932865 \beta_{1} + 49571 \beta_{2} + \beta_{3}) q^{4} +(\)\(33\!\cdots\!67\)\( + 203684509503647605 \beta_{1} + 3408688857 \beta_{2} + 3464 \beta_{3} - \beta_{4}) q^{5} +(\)\(64\!\cdots\!93\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} + 7916444742738 \beta_{2} - 24868 \beta_{3} - 1212 \beta_{4} + \beta_{5}) q^{6} +(-\)\(24\!\cdots\!13\)\( + \)\(77\!\cdots\!83\)\( \beta_{1} - 15418806458979859 \beta_{2} - 20904077593 \beta_{3} + 79668 \beta_{4} + 29 \beta_{5} + \beta_{6}) q^{7} +(-\)\(14\!\cdots\!32\)\( + \)\(36\!\cdots\!44\)\( \beta_{1} - 4021839698115592272 \beta_{2} - 6483639273776 \beta_{3} - 101411744 \beta_{4} + 103728 \beta_{5} + 312 \beta_{6}) q^{8} +(\)\(54\!\cdots\!55\)\( + \)\(10\!\cdots\!58\)\( \beta_{1} + \)\(81\!\cdots\!14\)\( \beta_{2} + 204820844789196 \beta_{3} - 127519054326 \beta_{4} + 80849268 \beta_{5} - 4860 \beta_{6}) q^{9} +(\)\(67\!\cdots\!02\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} + \)\(17\!\cdots\!12\)\( \beta_{2} + 143330903119276464 \beta_{3} - 28411298625456 \beta_{4} + 7949474900 \beta_{5} - 3447680 \beta_{6}) q^{10} +(-\)\(34\!\cdots\!69\)\( + \)\(12\!\cdots\!98\)\( \beta_{1} + \)\(99\!\cdots\!77\)\( \beta_{2} + 8437080972595758094 \beta_{3} - 2522095344612056 \beta_{4} - 615036062982 \beta_{5} + 360284610 \beta_{6}) q^{11} +(\)\(21\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} + \)\(84\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!44\)\( \beta_{3} - 137230898165028864 \beta_{4} + 5600704186368 \beta_{5} - 19730262528 \beta_{6}) q^{12} +(\)\(10\!\cdots\!67\)\( + \)\(61\!\cdots\!01\)\( \beta_{1} - \)\(13\!\cdots\!35\)\( \beta_{2} + \)\(83\!\cdots\!84\)\( \beta_{3} + 3130478306245636911 \beta_{4} + 660012374861848 \beta_{5} + 736959274232 \beta_{6}) q^{13} +(\)\(25\!\cdots\!22\)\( - \)\(73\!\cdots\!40\)\( \beta_{1} + \)\(95\!\cdots\!48\)\( \beta_{2} + \)\(53\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!80\)\( \beta_{4} - 31314710120571030 \beta_{5} - 20712747194880 \beta_{6}) q^{14} +(\)\(11\!\cdots\!29\)\( + \)\(20\!\cdots\!05\)\( \beta_{1} + \)\(18\!\cdots\!99\)\( \beta_{2} + \)\(86\!\cdots\!53\)\( \beta_{3} - \)\(40\!\cdots\!12\)\( \beta_{4} + 709496409196973675 \beta_{5} + 460952841048615 \beta_{6}) q^{15} +(-\)\(79\!\cdots\!20\)\( - \)\(16\!\cdots\!04\)\( \beta_{1} + \)\(22\!\cdots\!72\)\( \beta_{2} - \)\(61\!\cdots\!64\)\( \beta_{3} + \)\(24\!\cdots\!28\)\( \beta_{4} - 9148275738935924864 \beta_{5} - 8369862307627840 \beta_{6}) q^{16} +(-\)\(16\!\cdots\!96\)\( + \)\(19\!\cdots\!94\)\( \beta_{1} + \)\(36\!\cdots\!38\)\( \beta_{2} - \)\(31\!\cdots\!92\)\( \beta_{3} + \)\(61\!\cdots\!42\)\( \beta_{4} + 43604599257025439076 \beta_{5} + 126235309562590644 \beta_{6}) q^{17} +(\)\(36\!\cdots\!43\)\( + \)\(61\!\cdots\!49\)\( \beta_{1} + \)\(27\!\cdots\!96\)\( \beta_{2} + \)\(13\!\cdots\!72\)\( \beta_{3} - \)\(12\!\cdots\!72\)\( \beta_{4} + \)\(83\!\cdots\!84\)\( \beta_{5} - 1595655892723739904 \beta_{6}) q^{18} +(\)\(35\!\cdots\!45\)\( + \)\(27\!\cdots\!14\)\( \beta_{1} + \)\(23\!\cdots\!35\)\( \beta_{2} + \)\(33\!\cdots\!26\)\( \beta_{3} + \)\(71\!\cdots\!72\)\( \beta_{4} - \)\(22\!\cdots\!66\)\( \beta_{5} + 16907326006883839630 \beta_{6}) q^{19} +(\)\(30\!\cdots\!26\)\( + \)\(52\!\cdots\!90\)\( \beta_{1} + \)\(21\!\cdots\!46\)\( \beta_{2} - \)\(90\!\cdots\!58\)\( \beta_{3} + \)\(43\!\cdots\!72\)\( \beta_{4} + \)\(29\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!00\)\( \beta_{6}) q^{20} +(-\)\(42\!\cdots\!32\)\( + \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(26\!\cdots\!32\)\( \beta_{2} - \)\(30\!\cdots\!52\)\( \beta_{3} - \)\(88\!\cdots\!60\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!40\)\( \beta_{6}) q^{21} +(\)\(39\!\cdots\!87\)\( - \)\(17\!\cdots\!84\)\( \beta_{1} - \)\(14\!\cdots\!90\)\( \beta_{2} + \)\(11\!\cdots\!24\)\( \beta_{3} + \)\(34\!\cdots\!96\)\( \beta_{4} + \)\(11\!\cdots\!03\)\( \beta_{5} - \)\(51\!\cdots\!48\)\( \beta_{6}) q^{22} +(\)\(40\!\cdots\!21\)\( - \)\(60\!\cdots\!43\)\( \beta_{1} - \)\(10\!\cdots\!41\)\( \beta_{2} + \)\(15\!\cdots\!45\)\( \beta_{3} + \)\(29\!\cdots\!80\)\( \beta_{4} + \)\(60\!\cdots\!15\)\( \beta_{5} + \)\(75\!\cdots\!35\)\( \beta_{6}) q^{23} +(-\)\(96\!\cdots\!84\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} + \)\(17\!\cdots\!28\)\( \beta_{2} - \)\(82\!\cdots\!76\)\( \beta_{3} - \)\(39\!\cdots\!28\)\( \beta_{4} - \)\(69\!\cdots\!76\)\( \beta_{5} + \)\(17\!\cdots\!60\)\( \beta_{6}) q^{24} +(\)\(46\!\cdots\!35\)\( + \)\(42\!\cdots\!00\)\( \beta_{1} + \)\(49\!\cdots\!60\)\( \beta_{2} - \)\(55\!\cdots\!80\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4} + \)\(68\!\cdots\!00\)\( \beta_{5} - \)\(24\!\cdots\!00\)\( \beta_{6}) q^{25} +(\)\(20\!\cdots\!10\)\( + \)\(26\!\cdots\!38\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} + \)\(39\!\cdots\!36\)\( \beta_{3} + \)\(98\!\cdots\!24\)\( \beta_{4} - \)\(38\!\cdots\!92\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6}) q^{26} +(\)\(12\!\cdots\!12\)\( + \)\(21\!\cdots\!06\)\( \beta_{1} - \)\(33\!\cdots\!48\)\( \beta_{2} + \)\(14\!\cdots\!26\)\( \beta_{3} - \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!22\)\( \beta_{5} - \)\(12\!\cdots\!62\)\( \beta_{6}) q^{27} +(-\)\(18\!\cdots\!16\)\( + \)\(11\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!64\)\( \beta_{2} - \)\(16\!\cdots\!04\)\( \beta_{3} + \)\(33\!\cdots\!24\)\( \beta_{4} + \)\(32\!\cdots\!12\)\( \beta_{5} + \)\(49\!\cdots\!48\)\( \beta_{6}) q^{28} +(-\)\(28\!\cdots\!29\)\( + \)\(53\!\cdots\!13\)\( \beta_{1} - \)\(35\!\cdots\!07\)\( \beta_{2} - \)\(86\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!79\)\( \beta_{4} - \)\(50\!\cdots\!32\)\( \beta_{5} - \)\(74\!\cdots\!80\)\( \beta_{6}) q^{29} +(\)\(66\!\cdots\!34\)\( + \)\(29\!\cdots\!60\)\( \beta_{1} + \)\(26\!\cdots\!64\)\( \beta_{2} + \)\(37\!\cdots\!28\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4} + \)\(28\!\cdots\!50\)\( \beta_{5} - \)\(81\!\cdots\!00\)\( \beta_{6}) q^{30} +(-\)\(85\!\cdots\!76\)\( + \)\(25\!\cdots\!84\)\( \beta_{1} + \)\(54\!\cdots\!36\)\( \beta_{2} - \)\(72\!\cdots\!88\)\( \beta_{3} + \)\(45\!\cdots\!72\)\( \beta_{4} - \)\(83\!\cdots\!36\)\( \beta_{5} + \)\(94\!\cdots\!40\)\( \beta_{6}) q^{31} +(-\)\(49\!\cdots\!24\)\( - \)\(28\!\cdots\!16\)\( \beta_{1} - \)\(18\!\cdots\!40\)\( \beta_{2} - \)\(43\!\cdots\!52\)\( \beta_{3} + \)\(63\!\cdots\!52\)\( \beta_{4} + \)\(15\!\cdots\!56\)\( \beta_{5} - \)\(60\!\cdots\!36\)\( \beta_{6}) q^{32} +(\)\(69\!\cdots\!46\)\( - \)\(13\!\cdots\!18\)\( \beta_{1} - \)\(11\!\cdots\!98\)\( \beta_{2} + \)\(98\!\cdots\!88\)\( \beta_{3} - \)\(93\!\cdots\!18\)\( \beta_{4} + \)\(94\!\cdots\!36\)\( \beta_{5} + \)\(28\!\cdots\!04\)\( \beta_{6}) q^{33} +(\)\(63\!\cdots\!02\)\( - \)\(79\!\cdots\!26\)\( \beta_{1} + \)\(20\!\cdots\!68\)\( \beta_{2} + \)\(83\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4} - \)\(28\!\cdots\!36\)\( \beta_{5} - \)\(95\!\cdots\!20\)\( \beta_{6}) q^{34} +(-\)\(19\!\cdots\!88\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} + \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(30\!\cdots\!16\)\( \beta_{3} + \)\(78\!\cdots\!64\)\( \beta_{4} - \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6}) q^{35} +(\)\(68\!\cdots\!65\)\( + \)\(47\!\cdots\!71\)\( \beta_{1} + \)\(41\!\cdots\!47\)\( \beta_{2} - \)\(61\!\cdots\!79\)\( \beta_{3} - \)\(56\!\cdots\!92\)\( \beta_{4} + \)\(15\!\cdots\!16\)\( \beta_{5} + \)\(76\!\cdots\!00\)\( \beta_{6}) q^{36} +(-\)\(27\!\cdots\!97\)\( + \)\(83\!\cdots\!73\)\( \beta_{1} - \)\(18\!\cdots\!43\)\( \beta_{2} + \)\(40\!\cdots\!12\)\( \beta_{3} + \)\(97\!\cdots\!03\)\( \beta_{4} - \)\(74\!\cdots\!36\)\( \beta_{5} - \)\(29\!\cdots\!44\)\( \beta_{6}) q^{37} +(\)\(92\!\cdots\!97\)\( + \)\(11\!\cdots\!28\)\( \beta_{1} - \)\(77\!\cdots\!58\)\( \beta_{2} + \)\(15\!\cdots\!08\)\( \beta_{3} + \)\(26\!\cdots\!72\)\( \beta_{4} + \)\(21\!\cdots\!01\)\( \beta_{5} + \)\(14\!\cdots\!24\)\( \beta_{6}) q^{38} +(-\)\(26\!\cdots\!71\)\( + \)\(15\!\cdots\!13\)\( \beta_{1} + \)\(18\!\cdots\!35\)\( \beta_{2} - \)\(25\!\cdots\!43\)\( \beta_{3} - \)\(14\!\cdots\!76\)\( \beta_{4} - \)\(33\!\cdots\!97\)\( \beta_{5} - \)\(42\!\cdots\!65\)\( \beta_{6}) q^{39} +(\)\(87\!\cdots\!60\)\( - \)\(76\!\cdots\!00\)\( \beta_{1} + \)\(59\!\cdots\!60\)\( \beta_{2} - \)\(60\!\cdots\!80\)\( \beta_{3} + \)\(10\!\cdots\!20\)\( \beta_{4} - \)\(99\!\cdots\!00\)\( \beta_{5} + \)\(54\!\cdots\!00\)\( \beta_{6}) q^{40} +(\)\(39\!\cdots\!54\)\( - \)\(23\!\cdots\!56\)\( \beta_{1} + \)\(17\!\cdots\!96\)\( \beta_{2} + \)\(34\!\cdots\!52\)\( \beta_{3} + \)\(43\!\cdots\!72\)\( \beta_{4} + \)\(13\!\cdots\!44\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{41} +(\)\(35\!\cdots\!16\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} - \)\(49\!\cdots\!16\)\( \beta_{2} + \)\(50\!\cdots\!52\)\( \beta_{3} + \)\(18\!\cdots\!48\)\( \beta_{4} - \)\(90\!\cdots\!56\)\( \beta_{5} - \)\(11\!\cdots\!64\)\( \beta_{6}) q^{42} +(-\)\(43\!\cdots\!31\)\( - \)\(59\!\cdots\!28\)\( \beta_{1} - \)\(13\!\cdots\!13\)\( \beta_{2} - \)\(36\!\cdots\!80\)\( \beta_{3} - \)\(21\!\cdots\!40\)\( \beta_{4} - \)\(10\!\cdots\!60\)\( \beta_{5} + \)\(34\!\cdots\!40\)\( \beta_{6}) q^{43} +(\)\(29\!\cdots\!52\)\( + \)\(18\!\cdots\!84\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} - \)\(29\!\cdots\!36\)\( \beta_{3} + \)\(71\!\cdots\!12\)\( \beta_{4} - \)\(47\!\cdots\!56\)\( \beta_{5} - \)\(46\!\cdots\!60\)\( \beta_{6}) q^{44} +(\)\(68\!\cdots\!19\)\( + \)\(25\!\cdots\!85\)\( \beta_{1} + \)\(22\!\cdots\!49\)\( \beta_{2} + \)\(30\!\cdots\!48\)\( \beta_{3} - \)\(11\!\cdots\!57\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} - \)\(82\!\cdots\!00\)\( \beta_{6}) q^{45} +(-\)\(19\!\cdots\!86\)\( + \)\(31\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!64\)\( \beta_{2} + \)\(88\!\cdots\!12\)\( \beta_{3} + \)\(19\!\cdots\!12\)\( \beta_{4} - \)\(11\!\cdots\!26\)\( \beta_{5} + \)\(61\!\cdots\!00\)\( \beta_{6}) q^{46} +(\)\(85\!\cdots\!50\)\( + \)\(22\!\cdots\!06\)\( \beta_{1} - \)\(51\!\cdots\!26\)\( \beta_{2} - \)\(17\!\cdots\!10\)\( \beta_{3} - \)\(92\!\cdots\!80\)\( \beta_{4} + \)\(19\!\cdots\!30\)\( \beta_{5} - \)\(15\!\cdots\!70\)\( \beta_{6}) q^{47} +(\)\(48\!\cdots\!36\)\( - \)\(30\!\cdots\!60\)\( \beta_{1} - \)\(28\!\cdots\!96\)\( \beta_{2} - \)\(11\!\cdots\!12\)\( \beta_{3} + \)\(31\!\cdots\!12\)\( \beta_{4} + \)\(19\!\cdots\!36\)\( \beta_{5} + \)\(12\!\cdots\!84\)\( \beta_{6}) q^{48} +(\)\(21\!\cdots\!45\)\( - \)\(90\!\cdots\!16\)\( \beta_{1} + \)\(39\!\cdots\!16\)\( \beta_{2} + \)\(91\!\cdots\!32\)\( \beta_{3} - \)\(28\!\cdots\!08\)\( \beta_{4} - \)\(18\!\cdots\!16\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6}) q^{49} +(\)\(14\!\cdots\!85\)\( - \)\(45\!\cdots\!25\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(30\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4} + \)\(39\!\cdots\!00\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6}) q^{50} +(\)\(10\!\cdots\!72\)\( + \)\(46\!\cdots\!66\)\( \beta_{1} - \)\(19\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!86\)\( \beta_{3} + \)\(36\!\cdots\!68\)\( \beta_{4} - \)\(18\!\cdots\!54\)\( \beta_{5} + \)\(25\!\cdots\!70\)\( \beta_{6}) q^{51} +(\)\(61\!\cdots\!02\)\( + \)\(87\!\cdots\!66\)\( \beta_{1} - \)\(93\!\cdots\!34\)\( \beta_{2} + \)\(70\!\cdots\!90\)\( \beta_{3} - \)\(95\!\cdots\!80\)\( \beta_{4} - \)\(80\!\cdots\!20\)\( \beta_{5} + \)\(96\!\cdots\!80\)\( \beta_{6}) q^{52} +(\)\(19\!\cdots\!63\)\( + \)\(25\!\cdots\!21\)\( \beta_{1} - \)\(14\!\cdots\!79\)\( \beta_{2} + \)\(36\!\cdots\!24\)\( \beta_{3} - \)\(18\!\cdots\!69\)\( \beta_{4} + \)\(68\!\cdots\!28\)\( \beta_{5} - \)\(77\!\cdots\!88\)\( \beta_{6}) q^{53} +(\)\(71\!\cdots\!50\)\( + \)\(39\!\cdots\!04\)\( \beta_{1} + \)\(49\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!16\)\( \beta_{3} - \)\(45\!\cdots\!88\)\( \beta_{4} + \)\(31\!\cdots\!94\)\( \beta_{5} - \)\(23\!\cdots\!60\)\( \beta_{6}) q^{54} +(\)\(10\!\cdots\!79\)\( - \)\(83\!\cdots\!65\)\( \beta_{1} + \)\(48\!\cdots\!09\)\( \beta_{2} - \)\(53\!\cdots\!57\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4} + \)\(19\!\cdots\!25\)\( \beta_{5} + \)\(33\!\cdots\!25\)\( \beta_{6}) q^{55} +(\)\(31\!\cdots\!16\)\( - \)\(38\!\cdots\!92\)\( \beta_{1} + \)\(77\!\cdots\!28\)\( \beta_{2} + \)\(32\!\cdots\!80\)\( \beta_{3} - \)\(69\!\cdots\!16\)\( \beta_{4} - \)\(57\!\cdots\!92\)\( \beta_{5} + \)\(33\!\cdots\!80\)\( \beta_{6}) q^{56} +(\)\(75\!\cdots\!74\)\( - \)\(46\!\cdots\!10\)\( \beta_{1} - \)\(20\!\cdots\!26\)\( \beta_{2} - \)\(39\!\cdots\!80\)\( \beta_{3} - \)\(57\!\cdots\!90\)\( \beta_{4} + \)\(15\!\cdots\!40\)\( \beta_{5} - \)\(54\!\cdots\!60\)\( \beta_{6}) q^{57} +(\)\(17\!\cdots\!78\)\( - \)\(35\!\cdots\!62\)\( \beta_{1} - \)\(14\!\cdots\!12\)\( \beta_{2} + \)\(47\!\cdots\!68\)\( \beta_{3} + \)\(75\!\cdots\!72\)\( \beta_{4} - \)\(39\!\cdots\!04\)\( \beta_{5} + \)\(13\!\cdots\!64\)\( \beta_{6}) q^{58} +(\)\(18\!\cdots\!49\)\( + \)\(20\!\cdots\!00\)\( \beta_{1} - \)\(36\!\cdots\!33\)\( \beta_{2} - \)\(90\!\cdots\!88\)\( \beta_{3} + \)\(12\!\cdots\!60\)\( \beta_{4} - \)\(70\!\cdots\!00\)\( \beta_{5} + \)\(22\!\cdots\!60\)\( \beta_{6}) q^{59} +(\)\(68\!\cdots\!12\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} + \)\(80\!\cdots\!72\)\( \beta_{2} + \)\(27\!\cdots\!84\)\( \beta_{3} - \)\(47\!\cdots\!36\)\( \beta_{4} + \)\(16\!\cdots\!00\)\( \beta_{5} - \)\(92\!\cdots\!80\)\( \beta_{6}) q^{60} +(\)\(72\!\cdots\!27\)\( + \)\(14\!\cdots\!45\)\( \beta_{1} + \)\(10\!\cdots\!45\)\( \beta_{2} - \)\(80\!\cdots\!20\)\( \beta_{3} - \)\(14\!\cdots\!25\)\( \beta_{4} - \)\(57\!\cdots\!40\)\( \beta_{5} + \)\(21\!\cdots\!20\)\( \beta_{6}) q^{61} +(\)\(84\!\cdots\!52\)\( - \)\(23\!\cdots\!56\)\( \beta_{1} - \)\(27\!\cdots\!60\)\( \beta_{2} + \)\(60\!\cdots\!52\)\( \beta_{3} + \)\(28\!\cdots\!48\)\( \beta_{4} - \)\(25\!\cdots\!56\)\( \beta_{5} - \)\(36\!\cdots\!64\)\( \beta_{6}) q^{62} +(-\)\(51\!\cdots\!45\)\( - \)\(85\!\cdots\!25\)\( \beta_{1} - \)\(38\!\cdots\!27\)\( \beta_{2} + \)\(27\!\cdots\!83\)\( \beta_{3} + \)\(18\!\cdots\!52\)\( \beta_{4} + \)\(46\!\cdots\!01\)\( \beta_{5} - \)\(89\!\cdots\!71\)\( \beta_{6}) q^{63} +(-\)\(73\!\cdots\!64\)\( - \)\(10\!\cdots\!92\)\( \beta_{1} - \)\(68\!\cdots\!76\)\( \beta_{2} - \)\(32\!\cdots\!04\)\( \beta_{3} - \)\(84\!\cdots\!56\)\( \beta_{4} + \)\(79\!\cdots\!88\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6}) q^{64} +(-\)\(11\!\cdots\!76\)\( - \)\(98\!\cdots\!20\)\( \beta_{1} + \)\(29\!\cdots\!44\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{3} - \)\(33\!\cdots\!72\)\( \beta_{4} + \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(27\!\cdots\!40\)\( \beta_{6}) q^{65} +(-\)\(44\!\cdots\!96\)\( + \)\(16\!\cdots\!68\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!64\)\( \beta_{4} - \)\(11\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!40\)\( \beta_{6}) q^{66} +(-\)\(53\!\cdots\!71\)\( + \)\(12\!\cdots\!34\)\( \beta_{1} + \)\(22\!\cdots\!43\)\( \beta_{2} + \)\(25\!\cdots\!58\)\( \beta_{3} + \)\(81\!\cdots\!72\)\( \beta_{4} + \)\(15\!\cdots\!26\)\( \beta_{5} + \)\(52\!\cdots\!74\)\( \beta_{6}) q^{67} +(-\)\(21\!\cdots\!10\)\( + \)\(19\!\cdots\!94\)\( \beta_{1} - \)\(27\!\cdots\!98\)\( \beta_{2} - \)\(51\!\cdots\!14\)\( \beta_{3} - \)\(20\!\cdots\!16\)\( \beta_{4} + \)\(21\!\cdots\!92\)\( \beta_{5} + \)\(21\!\cdots\!68\)\( \beta_{6}) q^{68} +(-\)\(31\!\cdots\!20\)\( - \)\(96\!\cdots\!84\)\( \beta_{1} + \)\(34\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!32\)\( \beta_{3} - \)\(17\!\cdots\!92\)\( \beta_{4} - \)\(77\!\cdots\!04\)\( \beta_{5} - \)\(42\!\cdots\!40\)\( \beta_{6}) q^{69} +(-\)\(38\!\cdots\!48\)\( - \)\(74\!\cdots\!20\)\( \beta_{1} - \)\(33\!\cdots\!08\)\( \beta_{2} - \)\(47\!\cdots\!16\)\( \beta_{3} + \)\(39\!\cdots\!44\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5} - \)\(71\!\cdots\!00\)\( \beta_{6}) q^{70} +(\)\(13\!\cdots\!67\)\( - \)\(80\!\cdots\!65\)\( \beta_{1} + \)\(20\!\cdots\!85\)\( \beta_{2} + \)\(22\!\cdots\!15\)\( \beta_{3} + \)\(97\!\cdots\!00\)\( \beta_{4} + \)\(33\!\cdots\!05\)\( \beta_{5} + \)\(35\!\cdots\!85\)\( \beta_{6}) q^{71} +(\)\(67\!\cdots\!76\)\( - \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(51\!\cdots\!04\)\( \beta_{2} + \)\(25\!\cdots\!64\)\( \beta_{3} - \)\(16\!\cdots\!64\)\( \beta_{4} + \)\(29\!\cdots\!08\)\( \beta_{5} - \)\(28\!\cdots\!48\)\( \beta_{6}) q^{72} +(\)\(13\!\cdots\!36\)\( + \)\(63\!\cdots\!70\)\( \beta_{1} + \)\(47\!\cdots\!34\)\( \beta_{2} - \)\(78\!\cdots\!72\)\( \beta_{3} - \)\(42\!\cdots\!98\)\( \beta_{4} - \)\(77\!\cdots\!84\)\( \beta_{5} - \)\(11\!\cdots\!16\)\( \beta_{6}) q^{73} +(\)\(27\!\cdots\!34\)\( + \)\(45\!\cdots\!26\)\( \beta_{1} - \)\(26\!\cdots\!16\)\( \beta_{2} - \)\(80\!\cdots\!72\)\( \beta_{3} + \)\(87\!\cdots\!48\)\( \beta_{4} - \)\(10\!\cdots\!64\)\( \beta_{5} + \)\(31\!\cdots\!80\)\( \beta_{6}) q^{74} +(\)\(17\!\cdots\!45\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!95\)\( \beta_{2} - \)\(98\!\cdots\!60\)\( \beta_{3} - \)\(87\!\cdots\!60\)\( \beta_{4} + \)\(64\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!00\)\( \beta_{6}) q^{75} +(\)\(28\!\cdots\!52\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!16\)\( \beta_{2} + \)\(12\!\cdots\!20\)\( \beta_{3} - \)\(85\!\cdots\!32\)\( \beta_{4} - \)\(68\!\cdots\!84\)\( \beta_{5} - \)\(76\!\cdots\!40\)\( \beta_{6}) q^{76} +(\)\(29\!\cdots\!84\)\( + \)\(68\!\cdots\!00\)\( \beta_{1} + \)\(76\!\cdots\!72\)\( \beta_{2} - \)\(10\!\cdots\!32\)\( \beta_{3} + \)\(57\!\cdots\!32\)\( \beta_{4} - \)\(10\!\cdots\!04\)\( \beta_{5} + \)\(15\!\cdots\!24\)\( \beta_{6}) q^{77} +(\)\(49\!\cdots\!78\)\( - \)\(71\!\cdots\!80\)\( \beta_{1} - \)\(46\!\cdots\!16\)\( \beta_{2} - \)\(68\!\cdots\!04\)\( \beta_{3} - \)\(65\!\cdots\!36\)\( \beta_{4} + \)\(31\!\cdots\!62\)\( \beta_{5} - \)\(37\!\cdots\!12\)\( \beta_{6}) q^{78} +(-\)\(83\!\cdots\!86\)\( - \)\(24\!\cdots\!46\)\( \beta_{1} - \)\(12\!\cdots\!38\)\( \beta_{2} - \)\(21\!\cdots\!42\)\( \beta_{3} + \)\(20\!\cdots\!52\)\( \beta_{4} - \)\(15\!\cdots\!26\)\( \beta_{5} - \)\(22\!\cdots\!10\)\( \beta_{6}) q^{79} +(-\)\(10\!\cdots\!88\)\( - \)\(21\!\cdots\!20\)\( \beta_{1} - \)\(50\!\cdots\!48\)\( \beta_{2} + \)\(28\!\cdots\!04\)\( \beta_{3} - \)\(85\!\cdots\!36\)\( \beta_{4} - \)\(29\!\cdots\!00\)\( \beta_{5} + \)\(16\!\cdots\!00\)\( \beta_{6}) q^{80} +(-\)\(92\!\cdots\!21\)\( + \)\(43\!\cdots\!30\)\( \beta_{1} + \)\(44\!\cdots\!54\)\( \beta_{2} + \)\(14\!\cdots\!24\)\( \beta_{3} - \)\(33\!\cdots\!10\)\( \beta_{4} + \)\(33\!\cdots\!60\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6}) q^{81} +(-\)\(76\!\cdots\!18\)\( + \)\(11\!\cdots\!14\)\( \beta_{1} + \)\(92\!\cdots\!00\)\( \beta_{2} - \)\(18\!\cdots\!08\)\( \beta_{3} + \)\(89\!\cdots\!48\)\( \beta_{4} - \)\(15\!\cdots\!76\)\( \beta_{5} + \)\(86\!\cdots\!96\)\( \beta_{6}) q^{82} +(-\)\(10\!\cdots\!05\)\( + \)\(15\!\cdots\!12\)\( \beta_{1} + \)\(20\!\cdots\!33\)\( \beta_{2} + \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(37\!\cdots\!00\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} - \)\(34\!\cdots\!00\)\( \beta_{6}) q^{83} +(-\)\(24\!\cdots\!88\)\( + \)\(87\!\cdots\!88\)\( \beta_{1} - \)\(75\!\cdots\!52\)\( \beta_{2} - \)\(71\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!64\)\( \beta_{4} + \)\(23\!\cdots\!28\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{84} +(-\)\(27\!\cdots\!18\)\( - \)\(27\!\cdots\!10\)\( \beta_{1} - \)\(49\!\cdots\!58\)\( \beta_{2} + \)\(11\!\cdots\!24\)\( \beta_{3} - \)\(51\!\cdots\!46\)\( \beta_{4} - \)\(66\!\cdots\!00\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6}) q^{85} +(-\)\(19\!\cdots\!25\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} + \)\(64\!\cdots\!54\)\( \beta_{2} + \)\(12\!\cdots\!72\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4} - \)\(99\!\cdots\!93\)\( \beta_{5} - \)\(24\!\cdots\!80\)\( \beta_{6}) q^{86} +(\)\(21\!\cdots\!61\)\( - \)\(92\!\cdots\!03\)\( \beta_{1} - \)\(97\!\cdots\!09\)\( \beta_{2} - \)\(96\!\cdots\!23\)\( \beta_{3} + \)\(17\!\cdots\!28\)\( \beta_{4} + \)\(56\!\cdots\!19\)\( \beta_{5} + \)\(45\!\cdots\!91\)\( \beta_{6}) q^{87} +(\)\(50\!\cdots\!36\)\( + \)\(16\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2} - \)\(33\!\cdots\!72\)\( \beta_{3} + \)\(29\!\cdots\!72\)\( \beta_{4} - \)\(43\!\cdots\!84\)\( \beta_{5} + \)\(42\!\cdots\!04\)\( \beta_{6}) q^{88} +(\)\(84\!\cdots\!48\)\( - \)\(37\!\cdots\!26\)\( \beta_{1} - \)\(12\!\cdots\!66\)\( \beta_{2} - \)\(68\!\cdots\!00\)\( \beta_{3} + \)\(30\!\cdots\!82\)\( \beta_{4} - \)\(13\!\cdots\!96\)\( \beta_{5} - \)\(52\!\cdots\!20\)\( \beta_{6}) q^{89} +(\)\(85\!\cdots\!14\)\( + \)\(13\!\cdots\!30\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(18\!\cdots\!48\)\( \beta_{3} - \)\(90\!\cdots\!92\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6}) q^{90} +(\)\(61\!\cdots\!08\)\( + \)\(40\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2} + \)\(30\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!76\)\( \beta_{4} - \)\(53\!\cdots\!12\)\( \beta_{5} - \)\(62\!\cdots\!20\)\( \beta_{6}) q^{91} +(\)\(92\!\cdots\!64\)\( + \)\(30\!\cdots\!28\)\( \beta_{1} - \)\(10\!\cdots\!72\)\( \beta_{2} + \)\(11\!\cdots\!52\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4} - \)\(27\!\cdots\!56\)\( \beta_{5} + \)\(16\!\cdots\!76\)\( \beta_{6}) q^{92} +(\)\(25\!\cdots\!76\)\( - \)\(26\!\cdots\!52\)\( \beta_{1} - \)\(30\!\cdots\!28\)\( \beta_{2} - \)\(77\!\cdots\!76\)\( \beta_{3} + \)\(13\!\cdots\!16\)\( \beta_{4} + \)\(44\!\cdots\!28\)\( \beta_{5} + \)\(13\!\cdots\!72\)\( \beta_{6}) q^{93} +(\)\(75\!\cdots\!96\)\( - \)\(29\!\cdots\!76\)\( \beta_{1} + \)\(77\!\cdots\!60\)\( \beta_{2} - \)\(50\!\cdots\!44\)\( \beta_{3} - \)\(49\!\cdots\!28\)\( \beta_{4} - \)\(13\!\cdots\!36\)\( \beta_{5} - \)\(10\!\cdots\!60\)\( \beta_{6}) q^{94} +(\)\(65\!\cdots\!65\)\( - \)\(30\!\cdots\!75\)\( \beta_{1} + \)\(68\!\cdots\!15\)\( \beta_{2} + \)\(88\!\cdots\!05\)\( \beta_{3} - \)\(53\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!75\)\( \beta_{5} + \)\(13\!\cdots\!75\)\( \beta_{6}) q^{95} +(-\)\(75\!\cdots\!56\)\( - \)\(63\!\cdots\!72\)\( \beta_{1} - \)\(65\!\cdots\!24\)\( \beta_{2} + \)\(21\!\cdots\!08\)\( \beta_{3} + \)\(20\!\cdots\!44\)\( \beta_{4} - \)\(29\!\cdots\!12\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{96} +(-\)\(10\!\cdots\!00\)\( + \)\(42\!\cdots\!02\)\( \beta_{1} + \)\(55\!\cdots\!94\)\( \beta_{2} + \)\(42\!\cdots\!76\)\( \beta_{3} - \)\(15\!\cdots\!06\)\( \beta_{4} - \)\(11\!\cdots\!28\)\( \beta_{5} - \)\(22\!\cdots\!12\)\( \beta_{6}) q^{97} +(-\)\(29\!\cdots\!37\)\( + \)\(41\!\cdots\!25\)\( \beta_{1} - \)\(56\!\cdots\!20\)\( \beta_{2} - \)\(96\!\cdots\!48\)\( \beta_{3} + \)\(10\!\cdots\!88\)\( \beta_{4} + \)\(64\!\cdots\!44\)\( \beta_{5} + \)\(35\!\cdots\!76\)\( \beta_{6}) q^{98} +(-\)\(34\!\cdots\!07\)\( + \)\(10\!\cdots\!72\)\( \beta_{1} + \)\(17\!\cdots\!07\)\( \beta_{2} - \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!56\)\( \beta_{4} - \)\(90\!\cdots\!28\)\( \beta_{5} - \)\(59\!\cdots\!80\)\( \beta_{6}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3841716838056q^{2} + \)\(62\!\cdots\!32\)\(q^{3} + \)\(56\!\cdots\!76\)\(q^{4} + \)\(23\!\cdots\!30\)\(q^{5} + \)\(45\!\cdots\!84\)\(q^{6} - \)\(17\!\cdots\!44\)\(q^{7} - \)\(10\!\cdots\!60\)\(q^{8} + \)\(38\!\cdots\!59\)\(q^{9} + O(q^{10}) \) \( 7q + 3841716838056q^{2} + \)\(62\!\cdots\!32\)\(q^{3} + \)\(56\!\cdots\!76\)\(q^{4} + \)\(23\!\cdots\!30\)\(q^{5} + \)\(45\!\cdots\!84\)\(q^{6} - \)\(17\!\cdots\!44\)\(q^{7} - \)\(10\!\cdots\!60\)\(q^{8} + \)\(38\!\cdots\!59\)\(q^{9} + \)\(46\!\cdots\!80\)\(q^{10} - \)\(24\!\cdots\!16\)\(q^{11} + \)\(15\!\cdots\!16\)\(q^{12} + \)\(71\!\cdots\!22\)\(q^{13} + \)\(17\!\cdots\!32\)\(q^{14} + \)\(81\!\cdots\!60\)\(q^{15} - \)\(55\!\cdots\!08\)\(q^{16} - \)\(11\!\cdots\!94\)\(q^{17} + \)\(25\!\cdots\!12\)\(q^{18} + \)\(24\!\cdots\!60\)\(q^{19} + \)\(21\!\cdots\!40\)\(q^{20} - \)\(29\!\cdots\!76\)\(q^{21} + \)\(27\!\cdots\!72\)\(q^{22} + \)\(28\!\cdots\!12\)\(q^{23} - \)\(67\!\cdots\!20\)\(q^{24} + \)\(32\!\cdots\!25\)\(q^{25} + \)\(14\!\cdots\!44\)\(q^{26} + \)\(88\!\cdots\!60\)\(q^{27} - \)\(12\!\cdots\!72\)\(q^{28} - \)\(19\!\cdots\!10\)\(q^{29} + \)\(46\!\cdots\!60\)\(q^{30} - \)\(59\!\cdots\!76\)\(q^{31} - \)\(34\!\cdots\!24\)\(q^{32} + \)\(48\!\cdots\!84\)\(q^{33} + \)\(44\!\cdots\!72\)\(q^{34} - \)\(13\!\cdots\!20\)\(q^{35} + \)\(48\!\cdots\!12\)\(q^{36} - \)\(19\!\cdots\!94\)\(q^{37} + \)\(64\!\cdots\!60\)\(q^{38} - \)\(18\!\cdots\!92\)\(q^{39} + \)\(61\!\cdots\!00\)\(q^{40} + \)\(27\!\cdots\!94\)\(q^{41} + \)\(24\!\cdots\!52\)\(q^{42} - \)\(30\!\cdots\!08\)\(q^{43} + \)\(20\!\cdots\!12\)\(q^{44} + \)\(47\!\cdots\!10\)\(q^{45} - \)\(13\!\cdots\!96\)\(q^{46} + \)\(60\!\cdots\!56\)\(q^{47} + \)\(34\!\cdots\!12\)\(q^{48} + \)\(14\!\cdots\!51\)\(q^{49} + \)\(98\!\cdots\!00\)\(q^{50} + \)\(74\!\cdots\!04\)\(q^{51} + \)\(43\!\cdots\!36\)\(q^{52} + \)\(13\!\cdots\!82\)\(q^{53} + \)\(49\!\cdots\!60\)\(q^{54} + \)\(71\!\cdots\!60\)\(q^{55} + \)\(22\!\cdots\!40\)\(q^{56} + \)\(52\!\cdots\!20\)\(q^{57} + \)\(12\!\cdots\!40\)\(q^{58} + \)\(13\!\cdots\!80\)\(q^{59} + \)\(48\!\cdots\!80\)\(q^{60} + \)\(50\!\cdots\!34\)\(q^{61} + \)\(59\!\cdots\!92\)\(q^{62} - \)\(36\!\cdots\!88\)\(q^{63} - \)\(51\!\cdots\!64\)\(q^{64} - \)\(77\!\cdots\!40\)\(q^{65} - \)\(30\!\cdots\!92\)\(q^{66} - \)\(37\!\cdots\!44\)\(q^{67} - \)\(15\!\cdots\!72\)\(q^{68} - \)\(21\!\cdots\!72\)\(q^{69} - \)\(26\!\cdots\!20\)\(q^{70} + \)\(96\!\cdots\!04\)\(q^{71} + \)\(47\!\cdots\!80\)\(q^{72} + \)\(95\!\cdots\!62\)\(q^{73} + \)\(19\!\cdots\!52\)\(q^{74} + \)\(12\!\cdots\!00\)\(q^{75} + \)\(20\!\cdots\!80\)\(q^{76} + \)\(20\!\cdots\!72\)\(q^{77} + \)\(34\!\cdots\!24\)\(q^{78} - \)\(58\!\cdots\!60\)\(q^{79} - \)\(70\!\cdots\!20\)\(q^{80} - \)\(64\!\cdots\!53\)\(q^{81} - \)\(53\!\cdots\!48\)\(q^{82} - \)\(71\!\cdots\!48\)\(q^{83} - \)\(17\!\cdots\!68\)\(q^{84} - \)\(18\!\cdots\!20\)\(q^{85} - \)\(13\!\cdots\!76\)\(q^{86} + \)\(14\!\cdots\!80\)\(q^{87} + \)\(35\!\cdots\!80\)\(q^{88} + \)\(58\!\cdots\!70\)\(q^{89} + \)\(59\!\cdots\!60\)\(q^{90} + \)\(43\!\cdots\!84\)\(q^{91} + \)\(64\!\cdots\!56\)\(q^{92} + \)\(17\!\cdots\!24\)\(q^{93} + \)\(52\!\cdots\!92\)\(q^{94} + \)\(46\!\cdots\!00\)\(q^{95} - \)\(53\!\cdots\!36\)\(q^{96} - \)\(70\!\cdots\!94\)\(q^{97} - \)\(20\!\cdots\!92\)\(q^{98} - \)\(23\!\cdots\!92\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 19965898505076685758763041 x^{5} + 704885126882155619610489099544084432 x^{4} + 107067825809672902244360085998174063583690648295951 x^{3} + 2760974505633618159712492393750116169743299337348578140163910 x^{2} - 117290449585568076678331590341364312042793082454528443905095325102023447375 x + 37970201148992037926462243830462339624462239310525820322458780504727338931040585255500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 7 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(34\!\cdots\!59\)\( \nu^{6} - \)\(93\!\cdots\!77\)\( \nu^{5} + \)\(59\!\cdots\!42\)\( \nu^{4} + \)\(12\!\cdots\!50\)\( \nu^{3} - \)\(27\!\cdots\!39\)\( \nu^{2} - \)\(35\!\cdots\!53\)\( \nu + \)\(26\!\cdots\!08\)\(\)\()/ \)\(24\!\cdots\!72\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(90\!\cdots\!31\)\( \nu^{6} + \)\(24\!\cdots\!93\)\( \nu^{5} - \)\(15\!\cdots\!78\)\( \nu^{4} - \)\(33\!\cdots\!50\)\( \nu^{3} + \)\(14\!\cdots\!39\)\( \nu^{2} + \)\(96\!\cdots\!13\)\( \nu - \)\(49\!\cdots\!48\)\(\)\()/ \)\(12\!\cdots\!88\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(65\!\cdots\!79\)\( \nu^{6} + \)\(30\!\cdots\!11\)\( \nu^{5} + \)\(14\!\cdots\!62\)\( \nu^{4} - \)\(11\!\cdots\!62\)\( \nu^{3} - \)\(79\!\cdots\!95\)\( \nu^{2} - \)\(26\!\cdots\!25\)\( \nu + \)\(55\!\cdots\!00\)\(\)\()/ \)\(30\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(16\!\cdots\!49\)\( \nu^{6} - \)\(89\!\cdots\!59\)\( \nu^{5} + \)\(25\!\cdots\!22\)\( \nu^{4} + \)\(76\!\cdots\!78\)\( \nu^{3} - \)\(85\!\cdots\!45\)\( \nu^{2} - \)\(22\!\cdots\!75\)\( \nu + \)\(31\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(15\!\cdots\!61\)\( \nu^{6} - \)\(28\!\cdots\!49\)\( \nu^{5} - \)\(14\!\cdots\!58\)\( \nu^{4} + \)\(11\!\cdots\!58\)\( \nu^{3} - \)\(22\!\cdots\!95\)\( \nu^{2} - \)\(71\!\cdots\!25\)\( \nu + \)\(61\!\cdots\!00\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 7\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 49571 \beta_{2} - 1270960315153 \beta_{1} + 3285816439692620103304937381\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(39 \beta_{6} + 12966 \beta_{5} - 12676468 \beta_{4} - 1016261168401 \beta_{3} - 512931984338211243 \beta_{2} + 665339679315744541705428191 \beta_{1} - 522017785364203279525612267220887567696\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-141481024033993 \beta_{6} - 146499787029560378 \beta_{5} + 383626286810111142348 \beta_{4} + 106762309451665755675362048 \beta_{3} + 9384661690135860719271389741736 \beta_{2} - 455311605135943378511136613941824967370 \beta_{1} + 273273007015307998096220902015072945718157332183285693\)\()/5184\)
\(\nu^{5}\)\(=\)\((\)\(272294119230702643338278143 \beta_{6} + 115955420708801071468255024662 \beta_{5} - 46965447327553276381417610283156 \beta_{4} - 13713050651072167295090703870908934459 \beta_{3} - 6454898015889767005127124500408305904301609 \beta_{2} + 3423404559609344185123436271357033475350202101321557 \beta_{1} - 10389377462060720364413531410820663002534052824647214875962006902\)\()/864\)
\(\nu^{6}\)\(=\)\((\)\(-845081535489787599813217876804493212586 \beta_{6} - 983551911903246004064930950812150880066660 \beta_{5} + 1971838849993081965086934612786280371118306744 \beta_{4} + 404479468114429999630179233938341116865447981180449 \beta_{3} + 45311208431768090051123574350547561815149654442811609363 \beta_{2} - 3769018111755297590653218515154387604191873885598880668897146997 \beta_{1} + 937389914979775814832843332260231511597800305402118168526359478476802655371255\)\()/1728\)

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
−3.52994e12
−2.36516e12
−1.48073e12
3.73440e11
8.78141e11
2.95949e12
3.16475e12
−8.41697e13 2.94143e21 4.60866e27 2.26211e30 −2.47580e35 −5.00335e38 −1.79515e41 −1.75319e43 −1.90401e44
1.2 −5.62150e13 −7.69269e21 6.84242e26 −3.78165e31 4.32444e35 1.92974e38 1.00717e41 3.29936e43 2.12585e45
1.3 −3.49886e13 3.81854e21 −1.25168e27 4.98385e31 −1.33605e35 4.56727e38 1.30422e41 −1.16026e43 −1.74378e45
1.4 9.51137e12 5.28795e21 −2.38541e27 −1.05575e32 5.02957e34 −1.59557e38 −4.62376e40 1.77856e42 −1.00417e45
1.5 2.16242e13 −4.57928e21 −2.00827e27 4.97855e31 −9.90233e34 −2.82571e38 −9.69662e40 −5.21406e42 1.07657e45
1.6 7.15766e13 9.10839e21 2.64733e27 1.09575e32 6.51948e35 −1.62560e38 1.22718e40 5.67789e43 7.84298e45
1.7 7.65028e13 −2.65748e21 3.37680e27 −4.45639e31 −2.03305e35 2.84237e38 6.89228e40 −1.91217e43 −3.40927e45
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 1.92.a.a 7
3.b odd 2 1 9.92.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.92.a.a 7 1.a even 1 1 trivial
9.92.a.b 7 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{92}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3841716838056 T + \)\(58\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!12\)\( T^{4} + \)\(19\!\cdots\!28\)\( T^{5} + \)\(67\!\cdots\!76\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!48\)\( T^{8} + \)\(11\!\cdots\!12\)\( T^{9} + \)\(33\!\cdots\!04\)\( T^{10} - \)\(59\!\cdots\!00\)\( T^{11} + \)\(54\!\cdots\!64\)\( T^{12} - \)\(88\!\cdots\!84\)\( T^{13} + \)\(57\!\cdots\!72\)\( T^{14} \)
$3$ \( 1 - \)\(62\!\cdots\!32\)\( T + \)\(91\!\cdots\!97\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!77\)\( T^{4} - \)\(20\!\cdots\!64\)\( T^{5} + \)\(14\!\cdots\!69\)\( T^{6} - \)\(62\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!43\)\( T^{8} - \)\(14\!\cdots\!76\)\( T^{9} + \)\(78\!\cdots\!71\)\( T^{10} - \)\(21\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!79\)\( T^{12} - \)\(20\!\cdots\!28\)\( T^{13} + \)\(84\!\cdots\!63\)\( T^{14} \)
$5$ \( 1 - \)\(23\!\cdots\!30\)\( T + \)\(12\!\cdots\!75\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!25\)\( T^{4} - \)\(16\!\cdots\!50\)\( T^{5} + \)\(33\!\cdots\!75\)\( T^{6} - \)\(72\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!75\)\( T^{8} - \)\(27\!\cdots\!50\)\( T^{9} + \)\(52\!\cdots\!25\)\( T^{10} - \)\(75\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!75\)\( T^{12} - \)\(10\!\cdots\!50\)\( T^{13} + \)\(17\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 + \)\(17\!\cdots\!44\)\( T + \)\(22\!\cdots\!93\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!97\)\( T^{4} + \)\(46\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!21\)\( T^{6} + \)\(40\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!03\)\( T^{8} + \)\(29\!\cdots\!32\)\( T^{9} + \)\(14\!\cdots\!79\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(73\!\cdots\!99\)\( T^{12} + \)\(45\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!07\)\( T^{14} \)
$11$ \( 1 + \)\(24\!\cdots\!16\)\( T + \)\(18\!\cdots\!01\)\( T^{2} + \)\(44\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} + \)\(32\!\cdots\!48\)\( T^{5} + \)\(90\!\cdots\!93\)\( T^{6} + \)\(17\!\cdots\!48\)\( T^{7} + \)\(53\!\cdots\!23\)\( T^{8} + \)\(11\!\cdots\!08\)\( T^{9} + \)\(31\!\cdots\!51\)\( T^{10} + \)\(51\!\cdots\!16\)\( T^{11} + \)\(12\!\cdots\!51\)\( T^{12} + \)\(95\!\cdots\!76\)\( T^{13} + \)\(23\!\cdots\!71\)\( T^{14} \)
$13$ \( 1 - \)\(71\!\cdots\!22\)\( T + \)\(12\!\cdots\!87\)\( T^{2} - \)\(65\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!57\)\( T^{4} - \)\(26\!\cdots\!54\)\( T^{5} + \)\(20\!\cdots\!59\)\( T^{6} - \)\(69\!\cdots\!00\)\( T^{7} + \)\(47\!\cdots\!83\)\( T^{8} - \)\(14\!\cdots\!26\)\( T^{9} + \)\(81\!\cdots\!21\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} + \)\(85\!\cdots\!59\)\( T^{12} - \)\(11\!\cdots\!98\)\( T^{13} + \)\(38\!\cdots\!33\)\( T^{14} \)
$17$ \( 1 + \)\(11\!\cdots\!94\)\( T + \)\(35\!\cdots\!83\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!17\)\( T^{4} + \)\(32\!\cdots\!98\)\( T^{5} + \)\(45\!\cdots\!11\)\( T^{6} + \)\(32\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!63\)\( T^{8} + \)\(28\!\cdots\!22\)\( T^{9} + \)\(38\!\cdots\!29\)\( T^{10} + \)\(21\!\cdots\!00\)\( T^{11} + \)\(25\!\cdots\!19\)\( T^{12} + \)\(78\!\cdots\!86\)\( T^{13} + \)\(62\!\cdots\!77\)\( T^{14} \)
$19$ \( 1 - \)\(24\!\cdots\!60\)\( T + \)\(70\!\cdots\!33\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} - \)\(73\!\cdots\!00\)\( T^{5} + \)\(62\!\cdots\!65\)\( T^{6} - \)\(75\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!35\)\( T^{8} - \)\(39\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!79\)\( T^{10} - \)\(13\!\cdots\!40\)\( T^{11} + \)\(48\!\cdots\!67\)\( T^{12} - \)\(39\!\cdots\!60\)\( T^{13} + \)\(36\!\cdots\!39\)\( T^{14} \)
$23$ \( 1 - \)\(28\!\cdots\!12\)\( T + \)\(39\!\cdots\!77\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(78\!\cdots\!37\)\( T^{4} - \)\(25\!\cdots\!44\)\( T^{5} + \)\(96\!\cdots\!49\)\( T^{6} - \)\(26\!\cdots\!00\)\( T^{7} + \)\(79\!\cdots\!23\)\( T^{8} - \)\(17\!\cdots\!76\)\( T^{9} + \)\(44\!\cdots\!71\)\( T^{10} - \)\(61\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(89\!\cdots\!68\)\( T^{13} + \)\(26\!\cdots\!03\)\( T^{14} \)
$29$ \( 1 + \)\(19\!\cdots\!10\)\( T + \)\(41\!\cdots\!03\)\( T^{2} + \)\(70\!\cdots\!40\)\( T^{3} + \)\(80\!\cdots\!61\)\( T^{4} + \)\(95\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!15\)\( T^{6} + \)\(98\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!35\)\( T^{8} + \)\(13\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!29\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!47\)\( T^{12} + \)\(58\!\cdots\!10\)\( T^{13} + \)\(35\!\cdots\!09\)\( T^{14} \)
$31$ \( 1 + \)\(59\!\cdots\!76\)\( T + \)\(28\!\cdots\!21\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(37\!\cdots\!61\)\( T^{4} + \)\(15\!\cdots\!48\)\( T^{5} + \)\(29\!\cdots\!73\)\( T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + \)\(15\!\cdots\!63\)\( T^{8} + \)\(41\!\cdots\!28\)\( T^{9} + \)\(51\!\cdots\!51\)\( T^{10} + \)\(98\!\cdots\!76\)\( T^{11} + \)\(10\!\cdots\!71\)\( T^{12} + \)\(11\!\cdots\!56\)\( T^{13} + \)\(99\!\cdots\!11\)\( T^{14} \)
$37$ \( 1 + \)\(19\!\cdots\!94\)\( T + \)\(77\!\cdots\!63\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!57\)\( T^{4} + \)\(54\!\cdots\!58\)\( T^{5} + \)\(15\!\cdots\!91\)\( T^{6} - \)\(40\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!83\)\( T^{8} + \)\(14\!\cdots\!02\)\( T^{9} + \)\(63\!\cdots\!29\)\( T^{10} + \)\(16\!\cdots\!00\)\( T^{11} + \)\(26\!\cdots\!59\)\( T^{12} + \)\(33\!\cdots\!46\)\( T^{13} + \)\(88\!\cdots\!17\)\( T^{14} \)
$41$ \( 1 - \)\(27\!\cdots\!94\)\( T + \)\(33\!\cdots\!31\)\( T^{2} - \)\(78\!\cdots\!04\)\( T^{3} + \)\(50\!\cdots\!81\)\( T^{4} - \)\(10\!\cdots\!02\)\( T^{5} + \)\(45\!\cdots\!63\)\( T^{6} - \)\(74\!\cdots\!12\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(33\!\cdots\!62\)\( T^{9} + \)\(97\!\cdots\!01\)\( T^{10} - \)\(89\!\cdots\!44\)\( T^{11} + \)\(21\!\cdots\!31\)\( T^{12} - \)\(10\!\cdots\!54\)\( T^{13} + \)\(22\!\cdots\!81\)\( T^{14} \)
$43$ \( 1 + \)\(30\!\cdots\!08\)\( T + \)\(19\!\cdots\!57\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!97\)\( T^{4} + \)\(26\!\cdots\!76\)\( T^{5} + \)\(92\!\cdots\!29\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!03\)\( T^{8} + \)\(51\!\cdots\!24\)\( T^{9} + \)\(14\!\cdots\!71\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(32\!\cdots\!99\)\( T^{12} + \)\(22\!\cdots\!92\)\( T^{13} + \)\(33\!\cdots\!43\)\( T^{14} \)
$47$ \( 1 - \)\(60\!\cdots\!56\)\( T + \)\(61\!\cdots\!53\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!77\)\( T^{4} - \)\(11\!\cdots\!12\)\( T^{5} + \)\(41\!\cdots\!81\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(60\!\cdots\!43\)\( T^{8} - \)\(24\!\cdots\!08\)\( T^{9} + \)\(60\!\cdots\!79\)\( T^{10} - \)\(17\!\cdots\!00\)\( T^{11} + \)\(39\!\cdots\!79\)\( T^{12} - \)\(55\!\cdots\!24\)\( T^{13} + \)\(13\!\cdots\!87\)\( T^{14} \)
$53$ \( 1 - \)\(13\!\cdots\!82\)\( T + \)\(12\!\cdots\!47\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!77\)\( T^{4} - \)\(17\!\cdots\!14\)\( T^{5} + \)\(64\!\cdots\!19\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!43\)\( T^{8} - \)\(11\!\cdots\!26\)\( T^{9} + \)\(22\!\cdots\!21\)\( T^{10} - \)\(34\!\cdots\!00\)\( T^{11} + \)\(43\!\cdots\!79\)\( T^{12} - \)\(37\!\cdots\!78\)\( T^{13} + \)\(23\!\cdots\!13\)\( T^{14} \)
$59$ \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(13\!\cdots\!13\)\( T^{2} - \)\(93\!\cdots\!20\)\( T^{3} + \)\(58\!\cdots\!01\)\( T^{4} - \)\(29\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!65\)\( T^{6} - \)\(54\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!35\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!79\)\( T^{10} - \)\(36\!\cdots\!20\)\( T^{11} + \)\(71\!\cdots\!87\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{13} + \)\(10\!\cdots\!19\)\( T^{14} \)
$61$ \( 1 - \)\(50\!\cdots\!34\)\( T + \)\(22\!\cdots\!51\)\( T^{2} - \)\(73\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!21\)\( T^{4} - \)\(48\!\cdots\!02\)\( T^{5} + \)\(10\!\cdots\!43\)\( T^{6} - \)\(18\!\cdots\!52\)\( T^{7} + \)\(29\!\cdots\!23\)\( T^{8} - \)\(41\!\cdots\!42\)\( T^{9} + \)\(51\!\cdots\!01\)\( T^{10} - \)\(53\!\cdots\!84\)\( T^{11} + \)\(48\!\cdots\!51\)\( T^{12} - \)\(31\!\cdots\!74\)\( T^{13} + \)\(17\!\cdots\!21\)\( T^{14} \)
$67$ \( 1 + \)\(37\!\cdots\!44\)\( T + \)\(11\!\cdots\!33\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!17\)\( T^{4} + \)\(77\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!61\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!63\)\( T^{8} + \)\(17\!\cdots\!72\)\( T^{9} + \)\(15\!\cdots\!79\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(83\!\cdots\!19\)\( T^{12} + \)\(40\!\cdots\!36\)\( T^{13} + \)\(16\!\cdots\!27\)\( T^{14} \)
$71$ \( 1 - \)\(96\!\cdots\!04\)\( T + \)\(10\!\cdots\!61\)\( T^{2} + \)\(86\!\cdots\!16\)\( T^{3} + \)\(31\!\cdots\!41\)\( T^{4} + \)\(69\!\cdots\!48\)\( T^{5} + \)\(80\!\cdots\!33\)\( T^{6} + \)\(35\!\cdots\!28\)\( T^{7} + \)\(23\!\cdots\!43\)\( T^{8} + \)\(58\!\cdots\!68\)\( T^{9} + \)\(78\!\cdots\!51\)\( T^{10} + \)\(62\!\cdots\!96\)\( T^{11} + \)\(21\!\cdots\!11\)\( T^{12} - \)\(59\!\cdots\!84\)\( T^{13} + \)\(17\!\cdots\!91\)\( T^{14} \)
$73$ \( 1 - \)\(95\!\cdots\!62\)\( T + \)\(17\!\cdots\!27\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!37\)\( T^{4} - \)\(73\!\cdots\!94\)\( T^{5} + \)\(64\!\cdots\!99\)\( T^{6} - \)\(32\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!23\)\( T^{8} - \)\(98\!\cdots\!26\)\( T^{9} + \)\(62\!\cdots\!21\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!39\)\( T^{12} - \)\(22\!\cdots\!18\)\( T^{13} + \)\(86\!\cdots\!53\)\( T^{14} \)
$79$ \( 1 + \)\(58\!\cdots\!60\)\( T + \)\(17\!\cdots\!53\)\( T^{2} + \)\(78\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(80\!\cdots\!65\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!35\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!79\)\( T^{10} + \)\(42\!\cdots\!40\)\( T^{11} + \)\(46\!\cdots\!47\)\( T^{12} + \)\(74\!\cdots\!60\)\( T^{13} + \)\(61\!\cdots\!59\)\( T^{14} \)
$83$ \( 1 + \)\(71\!\cdots\!48\)\( T + \)\(41\!\cdots\!17\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!17\)\( T^{4} + \)\(16\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!89\)\( T^{6} + \)\(95\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!63\)\( T^{8} + \)\(31\!\cdots\!24\)\( T^{9} + \)\(47\!\cdots\!71\)\( T^{10} + \)\(58\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!19\)\( T^{12} + \)\(47\!\cdots\!12\)\( T^{13} + \)\(28\!\cdots\!23\)\( T^{14} \)
$89$ \( 1 - \)\(58\!\cdots\!70\)\( T + \)\(99\!\cdots\!23\)\( T^{2} - \)\(33\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!41\)\( T^{4} - \)\(10\!\cdots\!50\)\( T^{5} + \)\(14\!\cdots\!15\)\( T^{6} - \)\(32\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!35\)\( T^{8} - \)\(67\!\cdots\!50\)\( T^{9} + \)\(65\!\cdots\!29\)\( T^{10} - \)\(12\!\cdots\!80\)\( T^{11} + \)\(93\!\cdots\!27\)\( T^{12} - \)\(13\!\cdots\!70\)\( T^{13} + \)\(57\!\cdots\!29\)\( T^{14} \)
$97$ \( 1 + \)\(70\!\cdots\!94\)\( T + \)\(54\!\cdots\!03\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!77\)\( T^{4} + \)\(37\!\cdots\!38\)\( T^{5} + \)\(12\!\cdots\!31\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{7} + \)\(76\!\cdots\!43\)\( T^{8} + \)\(14\!\cdots\!42\)\( T^{9} + \)\(27\!\cdots\!29\)\( T^{10} + \)\(38\!\cdots\!00\)\( T^{11} + \)\(52\!\cdots\!79\)\( T^{12} + \)\(42\!\cdots\!26\)\( T^{13} + \)\(37\!\cdots\!37\)\( T^{14} \)
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