Properties

Label 1.64.a.a
Level $1$
Weight $64$
Character orbit 1.a
Self dual yes
Analytic conductor $25.136$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 64 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1360966918\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5096287552528786 x^{3} + 574038763744383494840 x^{2} + 3502610791787684740809332695881 x - 35880030333954415007358004861309901934\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{37}\cdot 3^{17}\cdot 5^{3}\cdot 7^{2}\cdot 13 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(101463019 - \beta_{1}) q^{2} +(190649070219572 - 18394 \beta_{1} - \beta_{2}) q^{3} +(1354584576291079283 - 215090888 \beta_{1} + 669 \beta_{2} + \beta_{3}) q^{4} +(-\)\(10\!\cdots\!40\)\( - 659683719897 \beta_{1} + 145059 \beta_{2} + 245 \beta_{3} + \beta_{4}) q^{5} +(\)\(21\!\cdots\!92\)\( - 114690955086372 \beta_{1} + 99399096 \beta_{2} + 192264 \beta_{3} + 120 \beta_{4}) q^{6} +(\)\(75\!\cdots\!48\)\( + 1747180379931712 \beta_{1} - 9692344086 \beta_{2} + 43833100 \beta_{3} - 19620 \beta_{4}) q^{7} +(\)\(14\!\cdots\!52\)\( - 881208717933160576 \beta_{1} - 12849034917768 \beta_{2} - 315813800 \beta_{3} + 1165760 \beta_{4}) q^{8} +(\)\(12\!\cdots\!77\)\( + 17846178675598245834 \beta_{1} - 1133383860802062 \beta_{2} - 72422061858 \beta_{3} - 42220890 \beta_{4}) q^{9} +O(q^{10})\) \( q +(101463019 - \beta_{1}) q^{2} +(190649070219572 - 18394 \beta_{1} - \beta_{2}) q^{3} +(1354584576291079283 - 215090888 \beta_{1} + 669 \beta_{2} + \beta_{3}) q^{4} +(-\)\(10\!\cdots\!40\)\( - 659683719897 \beta_{1} + 145059 \beta_{2} + 245 \beta_{3} + \beta_{4}) q^{5} +(\)\(21\!\cdots\!92\)\( - 114690955086372 \beta_{1} + 99399096 \beta_{2} + 192264 \beta_{3} + 120 \beta_{4}) q^{6} +(\)\(75\!\cdots\!48\)\( + 1747180379931712 \beta_{1} - 9692344086 \beta_{2} + 43833100 \beta_{3} - 19620 \beta_{4}) q^{7} +(\)\(14\!\cdots\!52\)\( - 881208717933160576 \beta_{1} - 12849034917768 \beta_{2} - 315813800 \beta_{3} + 1165760 \beta_{4}) q^{8} +(\)\(12\!\cdots\!77\)\( + 17846178675598245834 \beta_{1} - 1133383860802062 \beta_{2} - 72422061858 \beta_{3} - 42220890 \beta_{4}) q^{9} +(\)\(69\!\cdots\!10\)\( - \)\(21\!\cdots\!62\)\( \beta_{1} - 18004502499599136 \beta_{2} + 2663271065120 \beta_{3} + 1081776096 \beta_{4}) q^{10} +(-\)\(10\!\cdots\!68\)\( - \)\(10\!\cdots\!70\)\( \beta_{1} + 307238139008440365 \beta_{2} - 42229248861800 \beta_{3} - 21038877640 \beta_{4}) q^{11} +(-\)\(52\!\cdots\!00\)\( - \)\(17\!\cdots\!44\)\( \beta_{1} + 4972868297144221436 \beta_{2} + 273344406086700 \beta_{3} + 322909148160 \beta_{4}) q^{12} +(\)\(21\!\cdots\!88\)\( - \)\(34\!\cdots\!77\)\( \beta_{1} - 71084452195560207165 \beta_{2} + 1862677836522325 \beta_{3} - 3996043453215 \beta_{4}) q^{13} +(-\)\(10\!\cdots\!24\)\( - \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(26\!\cdots\!88\)\( \beta_{2} - 60032646663528752 \beta_{3} + 40204492977200 \beta_{4}) q^{14} +(-\)\(69\!\cdots\!80\)\( - \)\(22\!\cdots\!44\)\( \beta_{1} + \)\(74\!\cdots\!18\)\( \beta_{2} + 639401304412529940 \beta_{3} - 326717880482748 \beta_{4}) q^{15} +(-\)\(30\!\cdots\!64\)\( + \)\(41\!\cdots\!56\)\( \beta_{1} - \)\(18\!\cdots\!68\)\( \beta_{2} - 3493655345864697792 \beta_{3} + 2073868362385920 \beta_{4}) q^{16} +(\)\(46\!\cdots\!66\)\( + \)\(14\!\cdots\!98\)\( \beta_{1} - \)\(27\!\cdots\!78\)\( \beta_{2} + 2768074475558717150 \beta_{3} - 9153503316227930 \beta_{4}) q^{17} +(-\)\(17\!\cdots\!53\)\( + \)\(59\!\cdots\!11\)\( \beta_{1} + \)\(16\!\cdots\!84\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} + 12245970229980480 \beta_{4}) q^{18} +(-\)\(15\!\cdots\!00\)\( - \)\(18\!\cdots\!82\)\( \beta_{1} + \)\(28\!\cdots\!31\)\( \beta_{2} - \)\(10\!\cdots\!56\)\( \beta_{3} + 237205341825686280 \beta_{4}) q^{19} +(\)\(23\!\cdots\!30\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} - \)\(49\!\cdots\!78\)\( \beta_{2} + \)\(41\!\cdots\!10\)\( \beta_{3} - 2885463764230254592 \beta_{4}) q^{20} +(\)\(25\!\cdots\!12\)\( - \)\(39\!\cdots\!24\)\( \beta_{1} + \)\(51\!\cdots\!12\)\( \beta_{2} - \)\(21\!\cdots\!52\)\( \beta_{3} + 20454693249964733100 \beta_{4}) q^{21} +(\)\(10\!\cdots\!08\)\( + \)\(44\!\cdots\!48\)\( \beta_{1} + \)\(84\!\cdots\!40\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!40\)\( \beta_{4}) q^{22} +(\)\(31\!\cdots\!44\)\( + \)\(99\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!94\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3} + \)\(41\!\cdots\!00\)\( \beta_{4}) q^{23} +(\)\(16\!\cdots\!00\)\( - \)\(13\!\cdots\!56\)\( \beta_{1} - \)\(86\!\cdots\!32\)\( \beta_{2} - \)\(33\!\cdots\!08\)\( \beta_{3} - \)\(11\!\cdots\!20\)\( \beta_{4}) q^{24} +(\)\(58\!\cdots\!75\)\( - \)\(21\!\cdots\!40\)\( \beta_{1} + \)\(45\!\cdots\!80\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!20\)\( \beta_{4}) q^{25} +(\)\(36\!\cdots\!02\)\( - \)\(36\!\cdots\!86\)\( \beta_{1} + \)\(66\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!32\)\( \beta_{3} + \)\(76\!\cdots\!60\)\( \beta_{4}) q^{26} +(\)\(11\!\cdots\!08\)\( + \)\(24\!\cdots\!96\)\( \beta_{1} - \)\(70\!\cdots\!22\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(51\!\cdots\!60\)\( \beta_{4}) q^{27} +(\)\(41\!\cdots\!56\)\( + \)\(48\!\cdots\!80\)\( \beta_{1} + \)\(57\!\cdots\!24\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!40\)\( \beta_{4}) q^{28} +(\)\(10\!\cdots\!00\)\( - \)\(36\!\cdots\!73\)\( \beta_{1} + \)\(43\!\cdots\!79\)\( \beta_{2} - \)\(89\!\cdots\!19\)\( \beta_{3} - \)\(27\!\cdots\!15\)\( \beta_{4}) q^{29} +(\)\(23\!\cdots\!20\)\( - \)\(63\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!72\)\( \beta_{2} + \)\(15\!\cdots\!40\)\( \beta_{3} - \)\(35\!\cdots\!08\)\( \beta_{4}) q^{30} +(\)\(31\!\cdots\!92\)\( - \)\(94\!\cdots\!60\)\( \beta_{1} - \)\(55\!\cdots\!80\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(36\!\cdots\!80\)\( \beta_{4}) q^{31} +(-\)\(57\!\cdots\!16\)\( + \)\(43\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!80\)\( \beta_{2} + \)\(77\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4}) q^{32} +(-\)\(39\!\cdots\!96\)\( + \)\(10\!\cdots\!02\)\( \beta_{1} + \)\(43\!\cdots\!98\)\( \beta_{2} + \)\(81\!\cdots\!50\)\( \beta_{3} + \)\(13\!\cdots\!70\)\( \beta_{4}) q^{33} +(-\)\(15\!\cdots\!54\)\( - \)\(32\!\cdots\!46\)\( \beta_{1} + \)\(28\!\cdots\!88\)\( \beta_{2} - \)\(12\!\cdots\!28\)\( \beta_{3} + \)\(28\!\cdots\!80\)\( \beta_{4}) q^{34} +(-\)\(21\!\cdots\!40\)\( - \)\(78\!\cdots\!32\)\( \beta_{1} - \)\(29\!\cdots\!96\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(17\!\cdots\!44\)\( \beta_{4}) q^{35} +(-\)\(74\!\cdots\!09\)\( - \)\(10\!\cdots\!84\)\( \beta_{1} + \)\(81\!\cdots\!77\)\( \beta_{2} - \)\(23\!\cdots\!87\)\( \beta_{3} + \)\(34\!\cdots\!20\)\( \beta_{4}) q^{36} +(-\)\(34\!\cdots\!08\)\( + \)\(26\!\cdots\!15\)\( \beta_{1} - \)\(55\!\cdots\!77\)\( \beta_{2} - \)\(94\!\cdots\!75\)\( \beta_{3} - \)\(52\!\cdots\!95\)\( \beta_{4}) q^{37} +(\)\(19\!\cdots\!88\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(10\!\cdots\!28\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!80\)\( \beta_{4}) q^{38} +(\)\(98\!\cdots\!24\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(94\!\cdots\!78\)\( \beta_{2} - \)\(23\!\cdots\!72\)\( \beta_{3} + \)\(34\!\cdots\!60\)\( \beta_{4}) q^{39} +(\)\(17\!\cdots\!00\)\( - \)\(39\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!00\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{40} +(\)\(31\!\cdots\!22\)\( - \)\(71\!\cdots\!60\)\( \beta_{1} + \)\(73\!\cdots\!20\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3} - \)\(84\!\cdots\!20\)\( \beta_{4}) q^{41} +(\)\(41\!\cdots\!24\)\( - \)\(16\!\cdots\!60\)\( \beta_{1} - \)\(26\!\cdots\!64\)\( \beta_{2} + \)\(74\!\cdots\!00\)\( \beta_{3} + \)\(73\!\cdots\!80\)\( \beta_{4}) q^{42} +(-\)\(59\!\cdots\!24\)\( + \)\(48\!\cdots\!34\)\( \beta_{1} - \)\(60\!\cdots\!27\)\( \beta_{2} + \)\(51\!\cdots\!00\)\( \beta_{3} + \)\(41\!\cdots\!00\)\( \beta_{4}) q^{43} +(-\)\(37\!\cdots\!44\)\( + \)\(58\!\cdots\!24\)\( \beta_{1} - \)\(68\!\cdots\!72\)\( \beta_{2} - \)\(40\!\cdots\!68\)\( \beta_{3} - \)\(78\!\cdots\!20\)\( \beta_{4}) q^{44} +(-\)\(85\!\cdots\!80\)\( - \)\(15\!\cdots\!69\)\( \beta_{1} + \)\(91\!\cdots\!43\)\( \beta_{2} + \)\(68\!\cdots\!65\)\( \beta_{3} - \)\(21\!\cdots\!23\)\( \beta_{4}) q^{45} +(-\)\(10\!\cdots\!88\)\( - \)\(60\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4}) q^{46} +(-\)\(97\!\cdots\!00\)\( + \)\(29\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!84\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{47} +(\)\(21\!\cdots\!84\)\( + \)\(30\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!24\)\( \beta_{2} - \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(32\!\cdots\!80\)\( \beta_{4}) q^{48} +(\)\(67\!\cdots\!93\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(88\!\cdots\!60\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} + \)\(12\!\cdots\!80\)\( \beta_{4}) q^{49} +(\)\(22\!\cdots\!25\)\( - \)\(14\!\cdots\!15\)\( \beta_{1} + \)\(67\!\cdots\!80\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!80\)\( \beta_{4}) q^{50} +(\)\(31\!\cdots\!52\)\( + \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(36\!\cdots\!26\)\( \beta_{2} - \)\(52\!\cdots\!24\)\( \beta_{3} - \)\(24\!\cdots\!80\)\( \beta_{4}) q^{51} +(\)\(22\!\cdots\!98\)\( - \)\(38\!\cdots\!88\)\( \beta_{1} + \)\(69\!\cdots\!34\)\( \beta_{2} + \)\(82\!\cdots\!50\)\( \beta_{3} + \)\(81\!\cdots\!00\)\( \beta_{4}) q^{52} +(-\)\(13\!\cdots\!48\)\( + \)\(24\!\cdots\!71\)\( \beta_{1} - \)\(57\!\cdots\!81\)\( \beta_{2} + \)\(15\!\cdots\!25\)\( \beta_{3} - \)\(61\!\cdots\!15\)\( \beta_{4}) q^{53} +(-\)\(24\!\cdots\!00\)\( - \)\(58\!\cdots\!92\)\( \beta_{1} + \)\(22\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{54} +(-\)\(41\!\cdots\!80\)\( + \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(23\!\cdots\!62\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} + \)\(23\!\cdots\!32\)\( \beta_{4}) q^{55} +(-\)\(46\!\cdots\!00\)\( - \)\(17\!\cdots\!08\)\( \beta_{1} - \)\(31\!\cdots\!16\)\( \beta_{2} + \)\(24\!\cdots\!76\)\( \beta_{3} - \)\(46\!\cdots\!40\)\( \beta_{4}) q^{56} +(-\)\(34\!\cdots\!44\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} + \)\(29\!\cdots\!26\)\( \beta_{2} + \)\(48\!\cdots\!50\)\( \beta_{3} - \)\(46\!\cdots\!50\)\( \beta_{4}) q^{57} +(\)\(48\!\cdots\!62\)\( - \)\(97\!\cdots\!26\)\( \beta_{1} + \)\(52\!\cdots\!32\)\( \beta_{2} - \)\(93\!\cdots\!00\)\( \beta_{3} - \)\(84\!\cdots\!80\)\( \beta_{4}) q^{58} +(\)\(21\!\cdots\!00\)\( + \)\(55\!\cdots\!74\)\( \beta_{1} + \)\(31\!\cdots\!13\)\( \beta_{2} + \)\(29\!\cdots\!52\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4}) q^{59} +(\)\(69\!\cdots\!60\)\( - \)\(36\!\cdots\!52\)\( \beta_{1} - \)\(58\!\cdots\!56\)\( \beta_{2} + \)\(78\!\cdots\!20\)\( \beta_{3} + \)\(30\!\cdots\!16\)\( \beta_{4}) q^{60} +(\)\(78\!\cdots\!32\)\( + \)\(46\!\cdots\!75\)\( \beta_{1} - \)\(12\!\cdots\!25\)\( \beta_{2} - \)\(61\!\cdots\!75\)\( \beta_{3} - \)\(10\!\cdots\!75\)\( \beta_{4}) q^{61} +(\)\(10\!\cdots\!48\)\( - \)\(87\!\cdots\!52\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(67\!\cdots\!80\)\( \beta_{4}) q^{62} +(-\)\(13\!\cdots\!80\)\( - \)\(41\!\cdots\!68\)\( \beta_{1} + \)\(14\!\cdots\!22\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!20\)\( \beta_{4}) q^{63} +(-\)\(43\!\cdots\!12\)\( - \)\(53\!\cdots\!56\)\( \beta_{1} + \)\(19\!\cdots\!08\)\( \beta_{2} - \)\(61\!\cdots\!28\)\( \beta_{3} - \)\(34\!\cdots\!40\)\( \beta_{4}) q^{64} +(-\)\(35\!\cdots\!80\)\( - \)\(21\!\cdots\!64\)\( \beta_{1} - \)\(41\!\cdots\!92\)\( \beta_{2} + \)\(17\!\cdots\!40\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4}) q^{65} +(-\)\(11\!\cdots\!56\)\( + \)\(33\!\cdots\!56\)\( \beta_{1} - \)\(38\!\cdots\!48\)\( \beta_{2} - \)\(83\!\cdots\!52\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{66} +(-\)\(95\!\cdots\!44\)\( + \)\(77\!\cdots\!78\)\( \beta_{1} - \)\(67\!\cdots\!13\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(60\!\cdots\!20\)\( \beta_{4}) q^{67} +(-\)\(24\!\cdots\!10\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(36\!\cdots\!98\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} - \)\(47\!\cdots\!60\)\( \beta_{4}) q^{68} +(\)\(34\!\cdots\!44\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(12\!\cdots\!96\)\( \beta_{3} - \)\(33\!\cdots\!80\)\( \beta_{4}) q^{69} +(\)\(80\!\cdots\!60\)\( - \)\(69\!\cdots\!72\)\( \beta_{1} - \)\(96\!\cdots\!16\)\( \beta_{2} + \)\(79\!\cdots\!20\)\( \beta_{3} + \)\(64\!\cdots\!76\)\( \beta_{4}) q^{70} +(\)\(10\!\cdots\!12\)\( - \)\(44\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!50\)\( \beta_{2} - \)\(92\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{71} +(\)\(12\!\cdots\!04\)\( + \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!76\)\( \beta_{2} - \)\(59\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4}) q^{72} +(-\)\(59\!\cdots\!26\)\( + \)\(28\!\cdots\!30\)\( \beta_{1} + \)\(39\!\cdots\!66\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} + \)\(92\!\cdots\!70\)\( \beta_{4}) q^{73} +(-\)\(28\!\cdots\!14\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!60\)\( \beta_{3} - \)\(55\!\cdots\!80\)\( \beta_{4}) q^{74} +(-\)\(40\!\cdots\!00\)\( + \)\(30\!\cdots\!70\)\( \beta_{1} - \)\(37\!\cdots\!15\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3} + \)\(25\!\cdots\!40\)\( \beta_{4}) q^{75} +(-\)\(91\!\cdots\!00\)\( - \)\(21\!\cdots\!16\)\( \beta_{1} - \)\(41\!\cdots\!32\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!80\)\( \beta_{4}) q^{76} +(-\)\(11\!\cdots\!64\)\( - \)\(60\!\cdots\!36\)\( \beta_{1} + \)\(13\!\cdots\!88\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(79\!\cdots\!80\)\( \beta_{4}) q^{77} +(-\)\(44\!\cdots\!88\)\( - \)\(88\!\cdots\!72\)\( \beta_{1} - \)\(41\!\cdots\!64\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!40\)\( \beta_{4}) q^{78} +(\)\(11\!\cdots\!00\)\( + \)\(89\!\cdots\!32\)\( \beta_{1} + \)\(23\!\cdots\!24\)\( \beta_{2} + \)\(27\!\cdots\!16\)\( \beta_{3} - \)\(25\!\cdots\!20\)\( \beta_{4}) q^{79} +(\)\(21\!\cdots\!60\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(49\!\cdots\!24\)\( \beta_{2} - \)\(31\!\cdots\!80\)\( \beta_{3} + \)\(33\!\cdots\!36\)\( \beta_{4}) q^{80} +(\)\(90\!\cdots\!21\)\( + \)\(29\!\cdots\!58\)\( \beta_{1} - \)\(16\!\cdots\!54\)\( \beta_{2} - \)\(63\!\cdots\!66\)\( \beta_{3} - \)\(20\!\cdots\!50\)\( \beta_{4}) q^{81} +(\)\(78\!\cdots\!18\)\( - \)\(11\!\cdots\!82\)\( \beta_{1} + \)\(48\!\cdots\!20\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3} - \)\(45\!\cdots\!20\)\( \beta_{4}) q^{82} +(\)\(55\!\cdots\!20\)\( - \)\(17\!\cdots\!58\)\( \beta_{1} + \)\(10\!\cdots\!87\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(25\!\cdots\!04\)\( - \)\(68\!\cdots\!68\)\( \beta_{1} - \)\(15\!\cdots\!36\)\( \beta_{2} + \)\(66\!\cdots\!96\)\( \beta_{3} - \)\(58\!\cdots\!40\)\( \beta_{4}) q^{84} +(-\)\(24\!\cdots\!40\)\( - \)\(22\!\cdots\!22\)\( \beta_{1} + \)\(40\!\cdots\!34\)\( \beta_{2} - \)\(14\!\cdots\!30\)\( \beta_{3} - \)\(27\!\cdots\!74\)\( \beta_{4}) q^{85} +(-\)\(51\!\cdots\!68\)\( + \)\(65\!\cdots\!92\)\( \beta_{1} - \)\(96\!\cdots\!76\)\( \beta_{2} - \)\(29\!\cdots\!44\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4}) q^{86} +(-\)\(33\!\cdots\!56\)\( + \)\(30\!\cdots\!08\)\( \beta_{1} - \)\(80\!\cdots\!06\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(34\!\cdots\!80\)\( \beta_{4}) q^{87} +(-\)\(75\!\cdots\!36\)\( + \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!76\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} + \)\(49\!\cdots\!20\)\( \beta_{4}) q^{88} +(\)\(65\!\cdots\!50\)\( - \)\(79\!\cdots\!54\)\( \beta_{1} + \)\(16\!\cdots\!42\)\( \beta_{2} + \)\(13\!\cdots\!38\)\( \beta_{3} - \)\(17\!\cdots\!70\)\( \beta_{4}) q^{89} +(\)\(79\!\cdots\!70\)\( + \)\(19\!\cdots\!26\)\( \beta_{1} - \)\(62\!\cdots\!72\)\( \beta_{2} - \)\(21\!\cdots\!60\)\( \beta_{3} - \)\(41\!\cdots\!08\)\( \beta_{4}) q^{90} +(\)\(21\!\cdots\!72\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} + \)\(79\!\cdots\!44\)\( \beta_{2} - \)\(30\!\cdots\!84\)\( \beta_{3} + \)\(40\!\cdots\!60\)\( \beta_{4}) q^{91} +(\)\(33\!\cdots\!16\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} + \)\(93\!\cdots\!12\)\( \beta_{2} + \)\(68\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4}) q^{92} +(\)\(75\!\cdots\!24\)\( - \)\(31\!\cdots\!68\)\( \beta_{1} - \)\(74\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!40\)\( \beta_{4}) q^{93} +(-\)\(40\!\cdots\!44\)\( + \)\(19\!\cdots\!48\)\( \beta_{1} + \)\(31\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!16\)\( \beta_{3} - \)\(17\!\cdots\!20\)\( \beta_{4}) q^{94} +(\)\(27\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} + \)\(32\!\cdots\!50\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(71\!\cdots\!00\)\( \beta_{4}) q^{95} +(-\)\(18\!\cdots\!88\)\( + \)\(99\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!96\)\( \beta_{3} + \)\(80\!\cdots\!60\)\( \beta_{4}) q^{96} +(-\)\(34\!\cdots\!10\)\( + \)\(37\!\cdots\!46\)\( \beta_{1} - \)\(27\!\cdots\!34\)\( \beta_{2} + \)\(13\!\cdots\!50\)\( \beta_{3} - \)\(21\!\cdots\!10\)\( \beta_{4}) q^{97} +(-\)\(38\!\cdots\!13\)\( - \)\(99\!\cdots\!13\)\( \beta_{1} - \)\(26\!\cdots\!60\)\( \beta_{2} - \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(33\!\cdots\!80\)\( \beta_{4}) q^{98} +(-\)\(25\!\cdots\!36\)\( - \)\(78\!\cdots\!02\)\( \beta_{1} + \)\(18\!\cdots\!21\)\( \beta_{2} + \)\(39\!\cdots\!44\)\( \beta_{3} + \)\(94\!\cdots\!40\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 507315096q^{2} + 953245351116252q^{3} + 6772922881670488640q^{4} - \)\(50\!\cdots\!30\)\(q^{5} + \)\(10\!\cdots\!60\)\(q^{6} + \)\(37\!\cdots\!56\)\(q^{7} + \)\(73\!\cdots\!00\)\(q^{8} + \)\(63\!\cdots\!85\)\(q^{9} + O(q^{10}) \) \( 5q + 507315096q^{2} + 953245351116252q^{3} + 6772922881670488640q^{4} - \)\(50\!\cdots\!30\)\(q^{5} + \)\(10\!\cdots\!60\)\(q^{6} + \)\(37\!\cdots\!56\)\(q^{7} + \)\(73\!\cdots\!00\)\(q^{8} + \)\(63\!\cdots\!85\)\(q^{9} + \)\(34\!\cdots\!20\)\(q^{10} - \)\(54\!\cdots\!40\)\(q^{11} - \)\(26\!\cdots\!84\)\(q^{12} + \)\(10\!\cdots\!62\)\(q^{13} - \)\(54\!\cdots\!20\)\(q^{14} - \)\(34\!\cdots\!60\)\(q^{15} - \)\(15\!\cdots\!20\)\(q^{16} + \)\(23\!\cdots\!26\)\(q^{17} - \)\(87\!\cdots\!08\)\(q^{18} - \)\(78\!\cdots\!00\)\(q^{19} + \)\(11\!\cdots\!60\)\(q^{20} + \)\(12\!\cdots\!60\)\(q^{21} + \)\(52\!\cdots\!72\)\(q^{22} + \)\(15\!\cdots\!72\)\(q^{23} + \)\(81\!\cdots\!00\)\(q^{24} + \)\(29\!\cdots\!75\)\(q^{25} + \)\(18\!\cdots\!60\)\(q^{26} + \)\(59\!\cdots\!00\)\(q^{27} + \)\(20\!\cdots\!48\)\(q^{28} + \)\(50\!\cdots\!50\)\(q^{29} + \)\(11\!\cdots\!40\)\(q^{30} + \)\(15\!\cdots\!60\)\(q^{31} - \)\(28\!\cdots\!44\)\(q^{32} - \)\(19\!\cdots\!36\)\(q^{33} - \)\(77\!\cdots\!20\)\(q^{34} - \)\(10\!\cdots\!80\)\(q^{35} - \)\(37\!\cdots\!20\)\(q^{36} - \)\(17\!\cdots\!34\)\(q^{37} + \)\(96\!\cdots\!00\)\(q^{38} + \)\(49\!\cdots\!20\)\(q^{39} + \)\(89\!\cdots\!00\)\(q^{40} + \)\(15\!\cdots\!10\)\(q^{41} + \)\(20\!\cdots\!52\)\(q^{42} - \)\(29\!\cdots\!08\)\(q^{43} - \)\(18\!\cdots\!20\)\(q^{44} - \)\(42\!\cdots\!10\)\(q^{45} - \)\(51\!\cdots\!40\)\(q^{46} - \)\(48\!\cdots\!64\)\(q^{47} + \)\(10\!\cdots\!72\)\(q^{48} + \)\(33\!\cdots\!65\)\(q^{49} + \)\(11\!\cdots\!00\)\(q^{50} + \)\(15\!\cdots\!60\)\(q^{51} + \)\(11\!\cdots\!96\)\(q^{52} - \)\(69\!\cdots\!98\)\(q^{53} - \)\(12\!\cdots\!00\)\(q^{54} - \)\(20\!\cdots\!60\)\(q^{55} - \)\(23\!\cdots\!00\)\(q^{56} - \)\(17\!\cdots\!00\)\(q^{57} + \)\(24\!\cdots\!00\)\(q^{58} + \)\(10\!\cdots\!00\)\(q^{59} + \)\(34\!\cdots\!20\)\(q^{60} + \)\(39\!\cdots\!10\)\(q^{61} + \)\(51\!\cdots\!32\)\(q^{62} - \)\(68\!\cdots\!88\)\(q^{63} - \)\(21\!\cdots\!60\)\(q^{64} - \)\(17\!\cdots\!60\)\(q^{65} - \)\(55\!\cdots\!80\)\(q^{66} - \)\(47\!\cdots\!24\)\(q^{67} - \)\(12\!\cdots\!92\)\(q^{68} + \)\(17\!\cdots\!20\)\(q^{69} + \)\(40\!\cdots\!20\)\(q^{70} + \)\(50\!\cdots\!60\)\(q^{71} + \)\(61\!\cdots\!00\)\(q^{72} - \)\(29\!\cdots\!78\)\(q^{73} - \)\(14\!\cdots\!20\)\(q^{74} - \)\(20\!\cdots\!00\)\(q^{75} - \)\(45\!\cdots\!00\)\(q^{76} - \)\(58\!\cdots\!08\)\(q^{77} - \)\(22\!\cdots\!96\)\(q^{78} + \)\(58\!\cdots\!00\)\(q^{79} + \)\(10\!\cdots\!20\)\(q^{80} + \)\(45\!\cdots\!05\)\(q^{81} + \)\(39\!\cdots\!12\)\(q^{82} + \)\(27\!\cdots\!32\)\(q^{83} - \)\(12\!\cdots\!20\)\(q^{84} - \)\(12\!\cdots\!80\)\(q^{85} - \)\(25\!\cdots\!40\)\(q^{86} - \)\(16\!\cdots\!00\)\(q^{87} - \)\(37\!\cdots\!00\)\(q^{88} + \)\(32\!\cdots\!50\)\(q^{89} + \)\(39\!\cdots\!40\)\(q^{90} + \)\(10\!\cdots\!60\)\(q^{91} + \)\(16\!\cdots\!76\)\(q^{92} + \)\(37\!\cdots\!84\)\(q^{93} - \)\(20\!\cdots\!20\)\(q^{94} + \)\(13\!\cdots\!00\)\(q^{95} - \)\(90\!\cdots\!40\)\(q^{96} - \)\(17\!\cdots\!14\)\(q^{97} - \)\(19\!\cdots\!72\)\(q^{98} - \)\(12\!\cdots\!80\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5096287552528786 x^{3} + 574038763744383494840 x^{2} + 3502610791787684740809332695881 x - 35880030333954415007358004861309901934\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu - 29 \)
\(\beta_{2}\)\(=\)\((\)\(98361 \nu^{4} + 1875869661375 \nu^{3} - 478485724569513183099 \nu^{2} - 8652395815905765079896446187 \nu + 229804128899029201484728877488049518\)\()/ 24333542524587606016 \)
\(\beta_{3}\)\(=\)\((\)\(-65803509 \nu^{4} - 1254956803459875 \nu^{3} + 446252034184466469080175 \nu^{2} + 5809765699648259782584028883487 \nu - 410887611714893351947779231248849410166\)\()/ 24333542524587606016 \)
\(\beta_{4}\)\(=\)\((\)\(2665776547737 \nu^{4} + 31362235844074290783 \nu^{3} - 12800126178842342648877747099 \nu^{2} - 161978831504363254446653476179804939 \nu + 5879417456202971889545707633040694674748078\)\()/ 60833856311469015040 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 29\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 669 \beta_{2} - 12164792 \beta_{1} + 10567661868921261571\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(-18215 \beta_{4} + 9690671 \beta_{3} + 203947988361 \beta_{2} + 301458839229539738 \beta_{1} - 2008640715943852839028117\)\()/5832\)
\(\nu^{4}\)\(=\)\((\)\(2779066165000 \beta_{4} + 42302782965885731 \beta_{3} + 9715533155837233575 \beta_{2} + 10475558906127233571198464 \beta_{1} + 353968345455287156763638000502126737\)\()/46656\)

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
6.64596e7
1.73795e7
1.42670e7
−3.39250e7
−6.41810e7
−4.68363e9 −3.45751e14 1.27130e19 −3.68153e21 1.61937e24 6.76929e26 −1.63440e28 −1.02502e30 1.72429e31
1.2 −1.14986e9 2.06926e15 −7.90120e18 −9.56939e21 −2.37936e24 −1.15073e26 1.96908e28 3.13728e30 1.10034e31
1.3 −9.25761e8 −5.88048e14 −8.36634e18 1.60235e22 5.44392e23 −7.44490e26 1.62839e28 −7.98761e29 −1.48340e31
1.4 2.54406e9 −9.84533e14 −2.75111e18 −1.69176e22 −2.50471e24 1.59356e26 −3.04638e28 −1.75256e29 −4.30395e31
1.5 4.72250e9 8.02316e14 1.30786e19 1.36438e22 3.78894e24 4.00095e26 1.82063e28 −5.00850e29 6.44329e31
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 1.64.a.a 5
3.b odd 2 1 9.64.a.c 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.64.a.a 5 1.a even 1 1 trivial
9.64.a.c 5 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{64}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(59\!\cdots\!24\)\( + \)\(93\!\cdots\!56\)\( T + \)\(78\!\cdots\!72\)\( T^{2} - 26316207229657439232 T^{3} - 507315096 T^{4} + T^{5} \)
$3$ \( \)\(33\!\cdots\!68\)\( + \)\(12\!\cdots\!16\)\( T + \)\(16\!\cdots\!16\)\( T^{2} - \)\(27\!\cdots\!08\)\( T^{3} - 953245351116252 T^{4} + T^{5} \)
$5$ \( \)\(13\!\cdots\!00\)\( + \)\(39\!\cdots\!00\)\( T - \)\(50\!\cdots\!00\)\( T^{2} - \)\(41\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!30\)\( T^{4} + T^{5} \)
$7$ \( -\)\(36\!\cdots\!76\)\( + \)\(80\!\cdots\!96\)\( T + \)\(23\!\cdots\!32\)\( T^{2} - \)\(53\!\cdots\!72\)\( T^{3} - \)\(37\!\cdots\!56\)\( T^{4} + T^{5} \)
$11$ \( \)\(11\!\cdots\!68\)\( - \)\(13\!\cdots\!20\)\( T - \)\(44\!\cdots\!80\)\( T^{2} - \)\(75\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!40\)\( T^{4} + T^{5} \)
$13$ \( -\)\(46\!\cdots\!32\)\( + \)\(56\!\cdots\!36\)\( T + \)\(43\!\cdots\!56\)\( T^{2} - \)\(49\!\cdots\!88\)\( T^{3} - \)\(10\!\cdots\!62\)\( T^{4} + T^{5} \)
$17$ \( \)\(31\!\cdots\!24\)\( + \)\(18\!\cdots\!76\)\( T + \)\(17\!\cdots\!52\)\( T^{2} - \)\(81\!\cdots\!52\)\( T^{3} - \)\(23\!\cdots\!26\)\( T^{4} + T^{5} \)
$19$ \( \)\(28\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T - \)\(15\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(78\!\cdots\!00\)\( T^{4} + T^{5} \)
$23$ \( \)\(19\!\cdots\!68\)\( - \)\(11\!\cdots\!44\)\( T + \)\(19\!\cdots\!96\)\( T^{2} - \)\(43\!\cdots\!68\)\( T^{3} - \)\(15\!\cdots\!72\)\( T^{4} + T^{5} \)
$29$ \( -\)\(48\!\cdots\!00\)\( + \)\(32\!\cdots\!00\)\( T - \)\(80\!\cdots\!00\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} - \)\(50\!\cdots\!50\)\( T^{4} + T^{5} \)
$31$ \( -\)\(48\!\cdots\!32\)\( + \)\(13\!\cdots\!80\)\( T + \)\(17\!\cdots\!20\)\( T^{2} - \)\(81\!\cdots\!60\)\( T^{3} - \)\(15\!\cdots\!60\)\( T^{4} + T^{5} \)
$37$ \( \)\(71\!\cdots\!24\)\( + \)\(16\!\cdots\!36\)\( T - \)\(51\!\cdots\!08\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!34\)\( T^{4} + T^{5} \)
$41$ \( \)\(72\!\cdots\!68\)\( - \)\(49\!\cdots\!20\)\( T + \)\(47\!\cdots\!20\)\( T^{2} + \)\(65\!\cdots\!40\)\( T^{3} - \)\(15\!\cdots\!10\)\( T^{4} + T^{5} \)
$43$ \( -\)\(13\!\cdots\!32\)\( + \)\(96\!\cdots\!96\)\( T - \)\(83\!\cdots\!24\)\( T^{2} - \)\(44\!\cdots\!28\)\( T^{3} + \)\(29\!\cdots\!08\)\( T^{4} + T^{5} \)
$47$ \( \)\(18\!\cdots\!24\)\( - \)\(74\!\cdots\!84\)\( T - \)\(17\!\cdots\!88\)\( T^{2} + \)\(13\!\cdots\!08\)\( T^{3} + \)\(48\!\cdots\!64\)\( T^{4} + T^{5} \)
$53$ \( -\)\(42\!\cdots\!32\)\( - \)\(26\!\cdots\!84\)\( T - \)\(93\!\cdots\!84\)\( T^{2} - \)\(83\!\cdots\!08\)\( T^{3} + \)\(69\!\cdots\!98\)\( T^{4} + T^{5} \)
$59$ \( -\)\(31\!\cdots\!00\)\( - \)\(25\!\cdots\!00\)\( T + \)\(23\!\cdots\!00\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} - \)\(10\!\cdots\!00\)\( T^{4} + T^{5} \)
$61$ \( -\)\(11\!\cdots\!32\)\( - \)\(24\!\cdots\!20\)\( T + \)\(33\!\cdots\!20\)\( T^{2} - \)\(52\!\cdots\!60\)\( T^{3} - \)\(39\!\cdots\!10\)\( T^{4} + T^{5} \)
$67$ \( \)\(17\!\cdots\!24\)\( - \)\(74\!\cdots\!24\)\( T - \)\(16\!\cdots\!48\)\( T^{2} + \)\(14\!\cdots\!48\)\( T^{3} + \)\(47\!\cdots\!24\)\( T^{4} + T^{5} \)
$71$ \( \)\(24\!\cdots\!68\)\( - \)\(45\!\cdots\!20\)\( T + \)\(23\!\cdots\!20\)\( T^{2} + \)\(26\!\cdots\!40\)\( T^{3} - \)\(50\!\cdots\!60\)\( T^{4} + T^{5} \)
$73$ \( -\)\(26\!\cdots\!32\)\( + \)\(19\!\cdots\!56\)\( T + \)\(68\!\cdots\!96\)\( T^{2} - \)\(52\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!78\)\( T^{4} + T^{5} \)
$79$ \( -\)\(37\!\cdots\!00\)\( + \)\(63\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} - \)\(58\!\cdots\!00\)\( T^{4} + T^{5} \)
$83$ \( -\)\(15\!\cdots\!32\)\( + \)\(31\!\cdots\!76\)\( T + \)\(30\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!48\)\( T^{3} - \)\(27\!\cdots\!32\)\( T^{4} + T^{5} \)
$89$ \( \)\(42\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T - \)\(56\!\cdots\!00\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} - \)\(32\!\cdots\!50\)\( T^{4} + T^{5} \)
$97$ \( -\)\(82\!\cdots\!76\)\( + \)\(73\!\cdots\!16\)\( T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(20\!\cdots\!92\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} + T^{5} \)
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