Properties

Label 1.74.a.a
Level 1
Weight 74
Character orbit 1.a
Self dual yes
Analytic conductor 33.748
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7483973737\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{15}\cdot 5^{6}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q +(-18417866698 - \beta_{1}) q^{2} +(-25839159645211912 + 123123 \beta_{1} - \beta_{2}) q^{3} +(\)\(17\!\cdots\!30\)\( + 26620058935 \beta_{1} - 3441 \beta_{2} + \beta_{3}) q^{4} +(\)\(46\!\cdots\!55\)\( + 14159858805524 \beta_{1} + 12391606 \beta_{2} + 132 \beta_{3} + \beta_{4}) q^{5} +(-\)\(66\!\cdots\!40\)\( - 2320831478698956 \beta_{1} - 1747442760 \beta_{2} - 29592 \beta_{3} - 144 \beta_{4}) q^{6} +(-\)\(87\!\cdots\!00\)\( - 3945223198957624010 \beta_{1} - 2139829538826 \beta_{2} - 73498544 \beta_{3} - 186732 \beta_{4}) q^{7} +(-\)\(76\!\cdots\!04\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - 37115517216784 \beta_{2} - 41549502192 \beta_{3} - 15102976 \beta_{4}) q^{8} +(\)\(64\!\cdots\!19\)\( - \)\(16\!\cdots\!92\)\( \beta_{1} + 53110578546819540 \beta_{2} - 6698316560904 \beta_{3} + 1614263742 \beta_{4}) q^{9} +O(q^{10})\) \( q +(-18417866698 - \beta_{1}) q^{2} +(-25839159645211912 + 123123 \beta_{1} - \beta_{2}) q^{3} +(\)\(17\!\cdots\!30\)\( + 26620058935 \beta_{1} - 3441 \beta_{2} + \beta_{3}) q^{4} +(\)\(46\!\cdots\!55\)\( + 14159858805524 \beta_{1} + 12391606 \beta_{2} + 132 \beta_{3} + \beta_{4}) q^{5} +(-\)\(66\!\cdots\!40\)\( - 2320831478698956 \beta_{1} - 1747442760 \beta_{2} - 29592 \beta_{3} - 144 \beta_{4}) q^{6} +(-\)\(87\!\cdots\!00\)\( - 3945223198957624010 \beta_{1} - 2139829538826 \beta_{2} - 73498544 \beta_{3} - 186732 \beta_{4}) q^{7} +(-\)\(76\!\cdots\!04\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - 37115517216784 \beta_{2} - 41549502192 \beta_{3} - 15102976 \beta_{4}) q^{8} +(\)\(64\!\cdots\!19\)\( - \)\(16\!\cdots\!92\)\( \beta_{1} + 53110578546819540 \beta_{2} - 6698316560904 \beta_{3} + 1614263742 \beta_{4}) q^{9} +(-\)\(21\!\cdots\!40\)\( - \)\(57\!\cdots\!82\)\( \beta_{1} + 2196470269361916192 \beta_{2} - 184621370870176 \beta_{3} - 52532895168 \beta_{4}) q^{10} +(\)\(10\!\cdots\!32\)\( + \)\(18\!\cdots\!17\)\( \beta_{1} + \)\(20\!\cdots\!33\)\( \beta_{2} + 7662892565307936 \beta_{3} + 649409704328 \beta_{4}) q^{11} +(\)\(27\!\cdots\!44\)\( - \)\(22\!\cdots\!36\)\( \beta_{1} + \)\(91\!\cdots\!20\)\( \beta_{2} + 27025252969283172 \beta_{3} + 7724662972416 \beta_{4}) q^{12} +(\)\(95\!\cdots\!91\)\( - \)\(98\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!34\)\( \beta_{2} - 3198282957929973212 \beta_{3} - 483886195074711 \beta_{4}) q^{13} +(\)\(52\!\cdots\!48\)\( + \)\(15\!\cdots\!60\)\( \beta_{1} - \)\(40\!\cdots\!48\)\( \beta_{2} + 36504584384330910288 \beta_{3} + 10548213915031520 \beta_{4}) q^{14} +(-\)\(10\!\cdots\!20\)\( + \)\(19\!\cdots\!74\)\( \beta_{1} - \)\(27\!\cdots\!94\)\( \beta_{2} + 71581385179388966832 \beta_{3} - 145876262636005524 \beta_{4}) q^{15} +(\)\(11\!\cdots\!32\)\( + \)\(25\!\cdots\!04\)\( \beta_{1} + \)\(81\!\cdots\!44\)\( \beta_{2} - \)\(50\!\cdots\!36\)\( \beta_{3} + 1392819877299142656 \beta_{4}) q^{16} +(\)\(13\!\cdots\!16\)\( + \)\(67\!\cdots\!44\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} + \)\(41\!\cdots\!84\)\( \beta_{3} - 8696329022352154898 \beta_{4}) q^{17} +(\)\(13\!\cdots\!78\)\( + \)\(64\!\cdots\!91\)\( \beta_{1} + \)\(38\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!76\)\( \beta_{3} + 20923570108645529472 \beta_{4}) q^{18} +(\)\(62\!\cdots\!72\)\( + \)\(42\!\cdots\!99\)\( \beta_{1} - \)\(10\!\cdots\!13\)\( \beta_{2} - \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(24\!\cdots\!56\)\( \beta_{4}) q^{19} +(\)\(13\!\cdots\!60\)\( + \)\(20\!\cdots\!18\)\( \beta_{1} - \)\(22\!\cdots\!58\)\( \beta_{2} + \)\(15\!\cdots\!74\)\( \beta_{3} - \)\(36\!\cdots\!68\)\( \beta_{4}) q^{20} +(\)\(17\!\cdots\!88\)\( - \)\(36\!\cdots\!00\)\( \beta_{1} + \)\(46\!\cdots\!88\)\( \beta_{2} - \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(26\!\cdots\!20\)\( \beta_{4}) q^{21} +(-\)\(18\!\cdots\!36\)\( - \)\(84\!\cdots\!88\)\( \beta_{1} + \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3} - \)\(11\!\cdots\!24\)\( \beta_{4}) q^{22} +(-\)\(82\!\cdots\!36\)\( - \)\(26\!\cdots\!54\)\( \beta_{1} - \)\(11\!\cdots\!74\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} + \)\(16\!\cdots\!20\)\( \beta_{4}) q^{23} +(\)\(32\!\cdots\!96\)\( - \)\(28\!\cdots\!36\)\( \beta_{1} + \)\(48\!\cdots\!84\)\( \beta_{2} - \)\(20\!\cdots\!56\)\( \beta_{3} + \)\(18\!\cdots\!96\)\( \beta_{4}) q^{24} +(\)\(64\!\cdots\!75\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(21\!\cdots\!00\)\( \beta_{2} - \)\(45\!\cdots\!00\)\( \beta_{3} - \)\(18\!\cdots\!00\)\( \beta_{4}) q^{25} +(\)\(89\!\cdots\!36\)\( + \)\(36\!\cdots\!82\)\( \beta_{1} - \)\(67\!\cdots\!40\)\( \beta_{2} + \)\(21\!\cdots\!84\)\( \beta_{3} + \)\(92\!\cdots\!68\)\( \beta_{4}) q^{26} +(-\)\(24\!\cdots\!84\)\( + \)\(26\!\cdots\!34\)\( \beta_{1} - \)\(39\!\cdots\!74\)\( \beta_{2} - \)\(87\!\cdots\!52\)\( \beta_{3} - \)\(30\!\cdots\!56\)\( \beta_{4}) q^{27} +(-\)\(75\!\cdots\!44\)\( - \)\(41\!\cdots\!32\)\( \beta_{1} + \)\(42\!\cdots\!00\)\( \beta_{2} - \)\(31\!\cdots\!88\)\( \beta_{3} + \)\(61\!\cdots\!36\)\( \beta_{4}) q^{28} +(-\)\(43\!\cdots\!97\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} + \)\(42\!\cdots\!86\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(71\!\cdots\!63\)\( \beta_{4}) q^{29} +(-\)\(16\!\cdots\!40\)\( - \)\(61\!\cdots\!32\)\( \beta_{1} - \)\(31\!\cdots\!08\)\( \beta_{2} + \)\(64\!\cdots\!24\)\( \beta_{3} + \)\(26\!\cdots\!32\)\( \beta_{4}) q^{30} +(-\)\(79\!\cdots\!08\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} - \)\(52\!\cdots\!56\)\( \beta_{2} - \)\(95\!\cdots\!52\)\( \beta_{3} - \)\(27\!\cdots\!96\)\( \beta_{4}) q^{31} +(-\)\(18\!\cdots\!28\)\( + \)\(49\!\cdots\!52\)\( \beta_{1} + \)\(48\!\cdots\!52\)\( \beta_{2} - \)\(23\!\cdots\!16\)\( \beta_{3} + \)\(14\!\cdots\!52\)\( \beta_{4}) q^{32} +(-\)\(14\!\cdots\!34\)\( + \)\(46\!\cdots\!48\)\( \beta_{1} + \)\(39\!\cdots\!16\)\( \beta_{2} + \)\(17\!\cdots\!16\)\( \beta_{3} - \)\(38\!\cdots\!02\)\( \beta_{4}) q^{33} +(-\)\(64\!\cdots\!60\)\( - \)\(56\!\cdots\!26\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} - \)\(60\!\cdots\!96\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4}) q^{34} +(-\)\(21\!\cdots\!40\)\( - \)\(87\!\cdots\!92\)\( \beta_{1} - \)\(39\!\cdots\!48\)\( \beta_{2} + \)\(77\!\cdots\!44\)\( \beta_{3} + \)\(36\!\cdots\!92\)\( \beta_{4}) q^{35} +(-\)\(68\!\cdots\!78\)\( - \)\(17\!\cdots\!41\)\( \beta_{1} - \)\(22\!\cdots\!81\)\( \beta_{2} - \)\(52\!\cdots\!51\)\( \beta_{3} - \)\(15\!\cdots\!24\)\( \beta_{4}) q^{36} +(-\)\(13\!\cdots\!97\)\( + \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(34\!\cdots\!90\)\( \beta_{2} + \)\(78\!\cdots\!56\)\( \beta_{3} + \)\(29\!\cdots\!93\)\( \beta_{4}) q^{37} +(-\)\(40\!\cdots\!76\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(20\!\cdots\!56\)\( \beta_{2} - \)\(41\!\cdots\!64\)\( \beta_{3} + \)\(48\!\cdots\!08\)\( \beta_{4}) q^{38} +(-\)\(10\!\cdots\!60\)\( + \)\(13\!\cdots\!98\)\( \beta_{1} - \)\(22\!\cdots\!66\)\( \beta_{2} + \)\(81\!\cdots\!72\)\( \beta_{3} - \)\(17\!\cdots\!88\)\( \beta_{4}) q^{39} +(-\)\(17\!\cdots\!00\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!60\)\( \beta_{3} + \)\(42\!\cdots\!80\)\( \beta_{4}) q^{40} +(-\)\(17\!\cdots\!78\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2} - \)\(21\!\cdots\!52\)\( \beta_{3} - \)\(26\!\cdots\!96\)\( \beta_{4}) q^{41} +(\)\(30\!\cdots\!16\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} + \)\(56\!\cdots\!72\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} - \)\(52\!\cdots\!12\)\( \beta_{4}) q^{42} +(\)\(23\!\cdots\!20\)\( + \)\(23\!\cdots\!49\)\( \beta_{1} - \)\(21\!\cdots\!27\)\( \beta_{2} + \)\(73\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!00\)\( \beta_{4}) q^{43} +(\)\(72\!\cdots\!60\)\( + \)\(52\!\cdots\!44\)\( \beta_{1} - \)\(23\!\cdots\!36\)\( \beta_{2} + \)\(38\!\cdots\!24\)\( \beta_{3} + \)\(64\!\cdots\!16\)\( \beta_{4}) q^{44} +(\)\(17\!\cdots\!15\)\( - \)\(78\!\cdots\!28\)\( \beta_{1} + \)\(53\!\cdots\!18\)\( \beta_{2} - \)\(18\!\cdots\!04\)\( \beta_{3} - \)\(28\!\cdots\!47\)\( \beta_{4}) q^{45} +(\)\(26\!\cdots\!52\)\( - \)\(19\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!36\)\( \beta_{2} + \)\(20\!\cdots\!28\)\( \beta_{3} - \)\(64\!\cdots\!96\)\( \beta_{4}) q^{46} +(\)\(52\!\cdots\!24\)\( - \)\(36\!\cdots\!40\)\( \beta_{1} + \)\(17\!\cdots\!84\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4}) q^{47} +(-\)\(61\!\cdots\!16\)\( + \)\(23\!\cdots\!68\)\( \beta_{1} - \)\(82\!\cdots\!36\)\( \beta_{2} - \)\(25\!\cdots\!44\)\( \beta_{3} - \)\(13\!\cdots\!32\)\( \beta_{4}) q^{48} +(-\)\(95\!\cdots\!63\)\( + \)\(18\!\cdots\!24\)\( \beta_{1} + \)\(72\!\cdots\!16\)\( \beta_{2} - \)\(12\!\cdots\!48\)\( \beta_{3} - \)\(70\!\cdots\!84\)\( \beta_{4}) q^{49} +(-\)\(39\!\cdots\!50\)\( + \)\(42\!\cdots\!25\)\( \beta_{1} - \)\(14\!\cdots\!00\)\( \beta_{2} - \)\(58\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!00\)\( \beta_{4}) q^{50} +(-\)\(13\!\cdots\!36\)\( + \)\(18\!\cdots\!74\)\( \beta_{1} + \)\(52\!\cdots\!82\)\( \beta_{2} + \)\(15\!\cdots\!96\)\( \beta_{3} - \)\(98\!\cdots\!44\)\( \beta_{4}) q^{51} +(-\)\(36\!\cdots\!40\)\( - \)\(24\!\cdots\!02\)\( \beta_{1} - \)\(10\!\cdots\!34\)\( \beta_{2} - \)\(24\!\cdots\!50\)\( \beta_{3} - \)\(35\!\cdots\!00\)\( \beta_{4}) q^{52} +(-\)\(45\!\cdots\!97\)\( - \)\(38\!\cdots\!36\)\( \beta_{1} - \)\(19\!\cdots\!62\)\( \beta_{2} - \)\(82\!\cdots\!72\)\( \beta_{3} + \)\(35\!\cdots\!09\)\( \beta_{4}) q^{53} +(-\)\(19\!\cdots\!68\)\( + \)\(12\!\cdots\!24\)\( \beta_{1} - \)\(64\!\cdots\!08\)\( \beta_{2} + \)\(28\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!56\)\( \beta_{4}) q^{54} +(\)\(10\!\cdots\!60\)\( + \)\(14\!\cdots\!18\)\( \beta_{1} + \)\(56\!\cdots\!42\)\( \beta_{2} + \)\(75\!\cdots\!24\)\( \beta_{3} - \)\(78\!\cdots\!68\)\( \beta_{4}) q^{55} +(\)\(34\!\cdots\!12\)\( + \)\(29\!\cdots\!72\)\( \beta_{1} + \)\(42\!\cdots\!84\)\( \beta_{2} + \)\(90\!\cdots\!80\)\( \beta_{3} - \)\(79\!\cdots\!12\)\( \beta_{4}) q^{56} +(\)\(79\!\cdots\!62\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(20\!\cdots\!76\)\( \beta_{2} - \)\(79\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!50\)\( \beta_{4}) q^{57} +(\)\(12\!\cdots\!76\)\( - \)\(37\!\cdots\!10\)\( \beta_{1} - \)\(61\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!16\)\( \beta_{3} - \)\(48\!\cdots\!52\)\( \beta_{4}) q^{58} +(\)\(99\!\cdots\!76\)\( - \)\(22\!\cdots\!75\)\( \beta_{1} - \)\(43\!\cdots\!87\)\( \beta_{2} + \)\(47\!\cdots\!92\)\( \beta_{3} - \)\(53\!\cdots\!80\)\( \beta_{4}) q^{59} +(\)\(18\!\cdots\!60\)\( - \)\(90\!\cdots\!32\)\( \beta_{1} + \)\(25\!\cdots\!92\)\( \beta_{2} - \)\(57\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!32\)\( \beta_{4}) q^{60} +(-\)\(40\!\cdots\!93\)\( + \)\(13\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!30\)\( \beta_{2} - \)\(36\!\cdots\!60\)\( \beta_{3} - \)\(14\!\cdots\!55\)\( \beta_{4}) q^{61} +(-\)\(91\!\cdots\!16\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!32\)\( \beta_{2} - \)\(49\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4}) q^{62} +(-\)\(28\!\cdots\!48\)\( + \)\(14\!\cdots\!82\)\( \beta_{1} - \)\(98\!\cdots\!70\)\( \beta_{2} + \)\(34\!\cdots\!96\)\( \beta_{3} + \)\(12\!\cdots\!88\)\( \beta_{4}) q^{63} +(-\)\(53\!\cdots\!52\)\( - \)\(58\!\cdots\!32\)\( \beta_{1} + \)\(48\!\cdots\!00\)\( \beta_{2} - \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(19\!\cdots\!68\)\( \beta_{4}) q^{64} +(-\)\(44\!\cdots\!20\)\( - \)\(11\!\cdots\!96\)\( \beta_{1} + \)\(33\!\cdots\!76\)\( \beta_{2} - \)\(36\!\cdots\!28\)\( \beta_{3} + \)\(41\!\cdots\!96\)\( \beta_{4}) q^{65} +(-\)\(15\!\cdots\!80\)\( - \)\(20\!\cdots\!68\)\( \beta_{1} - \)\(79\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!52\)\( \beta_{3} + \)\(10\!\cdots\!08\)\( \beta_{4}) q^{66} +(\)\(34\!\cdots\!96\)\( - \)\(12\!\cdots\!61\)\( \beta_{1} - \)\(24\!\cdots\!85\)\( \beta_{2} + \)\(10\!\cdots\!24\)\( \beta_{3} - \)\(13\!\cdots\!28\)\( \beta_{4}) q^{67} +(\)\(51\!\cdots\!48\)\( + \)\(68\!\cdots\!94\)\( \beta_{1} - \)\(18\!\cdots\!06\)\( \beta_{2} + \)\(24\!\cdots\!02\)\( \beta_{3} + \)\(16\!\cdots\!56\)\( \beta_{4}) q^{68} +(\)\(86\!\cdots\!08\)\( - \)\(16\!\cdots\!44\)\( \beta_{1} + \)\(24\!\cdots\!20\)\( \beta_{2} - \)\(68\!\cdots\!68\)\( \beta_{3} + \)\(12\!\cdots\!44\)\( \beta_{4}) q^{69} +(\)\(12\!\cdots\!20\)\( + \)\(12\!\cdots\!56\)\( \beta_{1} + \)\(28\!\cdots\!64\)\( \beta_{2} + \)\(19\!\cdots\!08\)\( \beta_{3} - \)\(35\!\cdots\!56\)\( \beta_{4}) q^{70} +(\)\(59\!\cdots\!12\)\( - \)\(30\!\cdots\!10\)\( \beta_{1} + \)\(57\!\cdots\!10\)\( \beta_{2} - \)\(64\!\cdots\!80\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4}) q^{71} +(\)\(11\!\cdots\!08\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} - \)\(69\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!52\)\( \beta_{3} + \)\(66\!\cdots\!56\)\( \beta_{4}) q^{72} +(-\)\(47\!\cdots\!96\)\( - \)\(72\!\cdots\!92\)\( \beta_{1} - \)\(17\!\cdots\!76\)\( \beta_{2} + \)\(23\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!38\)\( \beta_{4}) q^{73} +(-\)\(77\!\cdots\!16\)\( + \)\(54\!\cdots\!86\)\( \beta_{1} + \)\(98\!\cdots\!24\)\( \beta_{2} - \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(21\!\cdots\!76\)\( \beta_{4}) q^{74} +(-\)\(15\!\cdots\!00\)\( + \)\(32\!\cdots\!25\)\( \beta_{1} + \)\(29\!\cdots\!25\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(47\!\cdots\!00\)\( \beta_{4}) q^{75} +(-\)\(80\!\cdots\!76\)\( + \)\(42\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!88\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!96\)\( \beta_{4}) q^{76} +(-\)\(23\!\cdots\!00\)\( - \)\(29\!\cdots\!76\)\( \beta_{1} - \)\(94\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(11\!\cdots\!52\)\( \beta_{4}) q^{77} +(\)\(67\!\cdots\!40\)\( + \)\(16\!\cdots\!32\)\( \beta_{1} - \)\(47\!\cdots\!48\)\( \beta_{2} + \)\(54\!\cdots\!92\)\( \beta_{3} - \)\(58\!\cdots\!24\)\( \beta_{4}) q^{78} +(\)\(24\!\cdots\!08\)\( - \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(29\!\cdots\!20\)\( \beta_{2} - \)\(32\!\cdots\!72\)\( \beta_{3} - \)\(44\!\cdots\!24\)\( \beta_{4}) q^{79} +(\)\(15\!\cdots\!80\)\( - \)\(22\!\cdots\!16\)\( \beta_{1} + \)\(24\!\cdots\!96\)\( \beta_{2} - \)\(74\!\cdots\!88\)\( \beta_{3} + \)\(81\!\cdots\!16\)\( \beta_{4}) q^{80} +(\)\(25\!\cdots\!79\)\( - \)\(81\!\cdots\!80\)\( \beta_{1} + \)\(75\!\cdots\!84\)\( \beta_{2} + \)\(19\!\cdots\!96\)\( \beta_{3} - \)\(27\!\cdots\!10\)\( \beta_{4}) q^{81} +(\)\(39\!\cdots\!44\)\( + \)\(47\!\cdots\!70\)\( \beta_{1} - \)\(13\!\cdots\!32\)\( \beta_{2} + \)\(45\!\cdots\!56\)\( \beta_{3} + \)\(67\!\cdots\!68\)\( \beta_{4}) q^{82} +(\)\(21\!\cdots\!72\)\( + \)\(78\!\cdots\!35\)\( \beta_{1} + \)\(43\!\cdots\!87\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(34\!\cdots\!48\)\( + \)\(15\!\cdots\!88\)\( \beta_{1} - \)\(44\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!20\)\( \beta_{3} - \)\(20\!\cdots\!48\)\( \beta_{4}) q^{84} +(-\)\(57\!\cdots\!90\)\( + \)\(45\!\cdots\!88\)\( \beta_{1} + \)\(16\!\cdots\!72\)\( \beta_{2} + \)\(20\!\cdots\!84\)\( \beta_{3} + \)\(62\!\cdots\!62\)\( \beta_{4}) q^{85} +(-\)\(26\!\cdots\!96\)\( - \)\(33\!\cdots\!92\)\( \beta_{1} + \)\(58\!\cdots\!28\)\( \beta_{2} - \)\(23\!\cdots\!12\)\( \beta_{3} - \)\(11\!\cdots\!88\)\( \beta_{4}) q^{86} +(-\)\(31\!\cdots\!12\)\( + \)\(53\!\cdots\!86\)\( \beta_{1} + \)\(11\!\cdots\!14\)\( \beta_{2} + \)\(33\!\cdots\!96\)\( \beta_{3} - \)\(79\!\cdots\!12\)\( \beta_{4}) q^{87} +(-\)\(44\!\cdots\!28\)\( - \)\(42\!\cdots\!72\)\( \beta_{1} + \)\(77\!\cdots\!92\)\( \beta_{2} - \)\(26\!\cdots\!84\)\( \beta_{3} - \)\(13\!\cdots\!52\)\( \beta_{4}) q^{88} +(-\)\(88\!\cdots\!56\)\( + \)\(81\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} - \)\(78\!\cdots\!20\)\( \beta_{3} + \)\(88\!\cdots\!66\)\( \beta_{4}) q^{89} +(\)\(41\!\cdots\!80\)\( + \)\(36\!\cdots\!54\)\( \beta_{1} + \)\(85\!\cdots\!76\)\( \beta_{2} + \)\(10\!\cdots\!72\)\( \beta_{3} + \)\(36\!\cdots\!96\)\( \beta_{4}) q^{90} +(\)\(10\!\cdots\!64\)\( + \)\(22\!\cdots\!12\)\( \beta_{1} - \)\(58\!\cdots\!16\)\( \beta_{2} + \)\(79\!\cdots\!60\)\( \beta_{3} - \)\(75\!\cdots\!52\)\( \beta_{4}) q^{91} +(\)\(21\!\cdots\!16\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(89\!\cdots\!56\)\( \beta_{2} + \)\(25\!\cdots\!16\)\( \beta_{3} - \)\(14\!\cdots\!52\)\( \beta_{4}) q^{92} +(\)\(41\!\cdots\!96\)\( - \)\(14\!\cdots\!68\)\( \beta_{1} + \)\(96\!\cdots\!72\)\( \beta_{2} - \)\(38\!\cdots\!12\)\( \beta_{3} + \)\(13\!\cdots\!64\)\( \beta_{4}) q^{93} +(\)\(24\!\cdots\!36\)\( - \)\(67\!\cdots\!16\)\( \beta_{1} + \)\(32\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4}) q^{94} +(\)\(22\!\cdots\!00\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} - \)\(31\!\cdots\!90\)\( \beta_{2} + \)\(46\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{95} +(-\)\(24\!\cdots\!40\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} - \)\(81\!\cdots\!08\)\( \beta_{2} + \)\(30\!\cdots\!72\)\( \beta_{3} - \)\(18\!\cdots\!52\)\( \beta_{4}) q^{96} +(-\)\(94\!\cdots\!96\)\( + \)\(11\!\cdots\!96\)\( \beta_{1} - \)\(84\!\cdots\!72\)\( \beta_{2} - \)\(25\!\cdots\!72\)\( \beta_{3} - \)\(14\!\cdots\!66\)\( \beta_{4}) q^{97} +(-\)\(15\!\cdots\!26\)\( + \)\(23\!\cdots\!51\)\( \beta_{1} - \)\(55\!\cdots\!68\)\( \beta_{2} - \)\(52\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!12\)\( \beta_{4}) q^{98} +(-\)\(31\!\cdots\!92\)\( - \)\(13\!\cdots\!83\)\( \beta_{1} + \)\(25\!\cdots\!69\)\( \beta_{2} - \)\(15\!\cdots\!40\)\( \beta_{3} - \)\(53\!\cdots\!32\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 92089333488q^{2} - 129195798226305804q^{3} + \)\(89\!\cdots\!60\)\(q^{4} + \)\(23\!\cdots\!50\)\(q^{5} - \)\(33\!\cdots\!40\)\(q^{6} - \)\(43\!\cdots\!08\)\(q^{7} - \)\(38\!\cdots\!80\)\(q^{8} + \)\(32\!\cdots\!65\)\(q^{9} + O(q^{10}) \) \( 5q - 92089333488q^{2} - 129195798226305804q^{3} + \)\(89\!\cdots\!60\)\(q^{4} + \)\(23\!\cdots\!50\)\(q^{5} - \)\(33\!\cdots\!40\)\(q^{6} - \)\(43\!\cdots\!08\)\(q^{7} - \)\(38\!\cdots\!80\)\(q^{8} + \)\(32\!\cdots\!65\)\(q^{9} - \)\(10\!\cdots\!00\)\(q^{10} + \)\(50\!\cdots\!60\)\(q^{11} + \)\(13\!\cdots\!92\)\(q^{12} + \)\(47\!\cdots\!86\)\(q^{13} + \)\(26\!\cdots\!20\)\(q^{14} - \)\(50\!\cdots\!00\)\(q^{15} + \)\(57\!\cdots\!80\)\(q^{16} + \)\(66\!\cdots\!02\)\(q^{17} + \)\(69\!\cdots\!16\)\(q^{18} + \)\(31\!\cdots\!00\)\(q^{19} + \)\(68\!\cdots\!00\)\(q^{20} + \)\(87\!\cdots\!60\)\(q^{21} - \)\(94\!\cdots\!16\)\(q^{22} - \)\(41\!\cdots\!24\)\(q^{23} + \)\(16\!\cdots\!00\)\(q^{24} + \)\(32\!\cdots\!75\)\(q^{25} + \)\(44\!\cdots\!60\)\(q^{26} - \)\(12\!\cdots\!80\)\(q^{27} - \)\(37\!\cdots\!16\)\(q^{28} - \)\(21\!\cdots\!50\)\(q^{29} - \)\(80\!\cdots\!00\)\(q^{30} - \)\(39\!\cdots\!40\)\(q^{31} - \)\(94\!\cdots\!68\)\(q^{32} - \)\(73\!\cdots\!28\)\(q^{33} - \)\(32\!\cdots\!80\)\(q^{34} - \)\(10\!\cdots\!00\)\(q^{35} - \)\(34\!\cdots\!20\)\(q^{36} - \)\(67\!\cdots\!78\)\(q^{37} - \)\(20\!\cdots\!20\)\(q^{38} - \)\(53\!\cdots\!20\)\(q^{39} - \)\(87\!\cdots\!00\)\(q^{40} - \)\(89\!\cdots\!90\)\(q^{41} + \)\(15\!\cdots\!84\)\(q^{42} + \)\(11\!\cdots\!56\)\(q^{43} + \)\(36\!\cdots\!20\)\(q^{44} + \)\(86\!\cdots\!50\)\(q^{45} + \)\(13\!\cdots\!60\)\(q^{46} + \)\(26\!\cdots\!32\)\(q^{47} - \)\(30\!\cdots\!24\)\(q^{48} - \)\(47\!\cdots\!15\)\(q^{49} - \)\(19\!\cdots\!00\)\(q^{50} - \)\(66\!\cdots\!40\)\(q^{51} - \)\(18\!\cdots\!28\)\(q^{52} - \)\(22\!\cdots\!54\)\(q^{53} - \)\(98\!\cdots\!00\)\(q^{54} + \)\(52\!\cdots\!00\)\(q^{55} + \)\(17\!\cdots\!00\)\(q^{56} + \)\(39\!\cdots\!40\)\(q^{57} + \)\(63\!\cdots\!20\)\(q^{58} + \)\(49\!\cdots\!00\)\(q^{59} + \)\(91\!\cdots\!00\)\(q^{60} - \)\(20\!\cdots\!90\)\(q^{61} - \)\(45\!\cdots\!96\)\(q^{62} - \)\(14\!\cdots\!44\)\(q^{63} - \)\(26\!\cdots\!40\)\(q^{64} - \)\(22\!\cdots\!00\)\(q^{65} - \)\(79\!\cdots\!80\)\(q^{66} + \)\(17\!\cdots\!52\)\(q^{67} + \)\(25\!\cdots\!04\)\(q^{68} + \)\(43\!\cdots\!80\)\(q^{69} + \)\(60\!\cdots\!00\)\(q^{70} + \)\(29\!\cdots\!60\)\(q^{71} + \)\(59\!\cdots\!60\)\(q^{72} - \)\(23\!\cdots\!74\)\(q^{73} - \)\(38\!\cdots\!80\)\(q^{74} - \)\(78\!\cdots\!00\)\(q^{75} - \)\(40\!\cdots\!00\)\(q^{76} - \)\(11\!\cdots\!56\)\(q^{77} + \)\(33\!\cdots\!72\)\(q^{78} + \)\(12\!\cdots\!00\)\(q^{79} + \)\(76\!\cdots\!00\)\(q^{80} + \)\(12\!\cdots\!05\)\(q^{81} + \)\(19\!\cdots\!64\)\(q^{82} + \)\(10\!\cdots\!16\)\(q^{83} - \)\(17\!\cdots\!80\)\(q^{84} - \)\(28\!\cdots\!00\)\(q^{85} - \)\(13\!\cdots\!40\)\(q^{86} - \)\(15\!\cdots\!40\)\(q^{87} - \)\(22\!\cdots\!60\)\(q^{88} - \)\(44\!\cdots\!50\)\(q^{89} + \)\(20\!\cdots\!00\)\(q^{90} + \)\(50\!\cdots\!60\)\(q^{91} + \)\(10\!\cdots\!52\)\(q^{92} + \)\(20\!\cdots\!32\)\(q^{93} + \)\(12\!\cdots\!20\)\(q^{94} + \)\(11\!\cdots\!00\)\(q^{95} - \)\(12\!\cdots\!40\)\(q^{96} - \)\(47\!\cdots\!18\)\(q^{97} - \)\(76\!\cdots\!16\)\(q^{98} - \)\(15\!\cdots\!20\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 10073499617947743056 x^{3} + 1429272143092482488433869600 x^{2} + 7661214288514935343595600445215756800 x + 1722510836040319301450745177697157900206688000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 10 \)
\(\beta_{2}\)\(=\)\((\)\(-301 \nu^{4} + 116324489073 \nu^{3} + 2946016895675808598652 \nu^{2} - 1572067662351917592485561734992 \nu - 1398107315399150788449578990722768882624\)\()/ \)\(18\!\cdots\!48\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-11137 \nu^{4} + 4304006095701 \nu^{3} + 155411927822539953292268 \nu^{2} - 48289371689549994763869174554256 \nu - 238731607808439142895551358181633844540096\)\()/ 20142926511516942336 \)
\(\beta_{4}\)\(=\)\((\)\(1794556703 \nu^{4} + 6165094803239335989 \nu^{3} - 15605951075985556463410173844 \nu^{2} - 48868520632222511605431163829960893584 \nu + 6327022782245483253661063757607563332086973760\)\()/ \)\(93\!\cdots\!24\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 10\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3441 \beta_{2} - 10215674441 \beta_{1} + 9283737247896553729718\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(943936 \beta_{4} - 856506117 \beta_{3} + 14202697189813 \beta_{2} + 1231550857811227978477 \beta_{1} - 5927477389563705411674836348238\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(121597865910976 \beta_{4} + 9677096401527687305 \beta_{3} - 46188046919629143790841 \beta_{2} - 192031968536100477971908100241 \beta_{1} + 79398573440552352581825579517805287677222\)\()/2304\)

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
2.93794e9
1.10442e9
−2.60629e8
−6.50494e8
−3.13124e9
−1.59439e11 2.55219e16 1.59761e22 2.70839e25 −4.06919e27 −5.83802e30 −1.04135e33 −6.69338e34 −4.31824e36
1.2 −7.14302e10 −1.08968e17 −4.34246e21 −5.37334e25 7.78363e27 1.02465e31 9.84822e32 −5.57111e34 3.83819e36
1.3 −5.90767e9 3.95253e17 −9.40983e21 1.69056e25 −2.33502e27 −2.69951e30 1.11387e32 8.86395e34 −9.98727e34
1.4 1.28059e10 −4.48831e17 −9.28074e21 4.05981e25 −5.74767e27 −7.12492e30 −2.39796e32 1.33864e35 5.19894e35
1.5 1.31882e11 7.82937e15 7.94807e21 −7.75470e24 1.03255e27 1.05974e30 −1.97383e32 −6.75239e34 −1.02270e36
\(n\): 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 1.74.a.a 5
3.b odd 2 1 9.74.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.74.a.a 5 1.a even 1 1 trivial
9.74.a.a 5 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{74}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 92089333488 T + \)\(27\!\cdots\!72\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!88\)\( T^{4} + \)\(23\!\cdots\!64\)\( T^{5} + \)\(32\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!36\)\( T^{8} + \)\(73\!\cdots\!48\)\( T^{9} + \)\(75\!\cdots\!32\)\( T^{10} \)
$3$ \( 1 + 129195798226305804 T + \)\(16\!\cdots\!83\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} + \)\(17\!\cdots\!32\)\( T^{5} + \)\(70\!\cdots\!74\)\( T^{6} + \)\(97\!\cdots\!60\)\( T^{7} + \)\(49\!\cdots\!61\)\( T^{8} + \)\(26\!\cdots\!64\)\( T^{9} + \)\(14\!\cdots\!43\)\( T^{10} \)
$5$ \( 1 - \)\(23\!\cdots\!50\)\( T + \)\(27\!\cdots\!25\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!50\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!50\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!25\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 + \)\(43\!\cdots\!08\)\( T + \)\(15\!\cdots\!07\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(70\!\cdots\!84\)\( T^{5} + \)\(51\!\cdots\!86\)\( T^{6} + \)\(66\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} + \)\(25\!\cdots\!08\)\( T^{9} + \)\(28\!\cdots\!07\)\( T^{10} \)
$11$ \( 1 - \)\(50\!\cdots\!60\)\( T + \)\(30\!\cdots\!95\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!10\)\( T^{4} - \)\(73\!\cdots\!52\)\( T^{5} + \)\(46\!\cdots\!10\)\( T^{6} - \)\(83\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!45\)\( T^{8} - \)\(61\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 - \)\(47\!\cdots\!86\)\( T + \)\(33\!\cdots\!73\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(68\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!34\)\( T^{6} - \)\(32\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!21\)\( T^{8} - \)\(88\!\cdots\!66\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \)
$17$ \( 1 - \)\(66\!\cdots\!02\)\( T + \)\(11\!\cdots\!77\)\( T^{2} - \)\(48\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!18\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{5} + \)\(47\!\cdots\!66\)\( T^{6} - \)\(21\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} - \)\(12\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!57\)\( T^{10} \)
$19$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(41\!\cdots\!95\)\( T^{2} - \)\(41\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!05\)\( T^{8} - \)\(78\!\cdots\!00\)\( T^{9} + \)\(55\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + \)\(41\!\cdots\!24\)\( T + \)\(74\!\cdots\!63\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!18\)\( T^{4} + \)\(65\!\cdots\!72\)\( T^{5} + \)\(72\!\cdots\!94\)\( T^{6} + \)\(14\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} + \)\(17\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!43\)\( T^{10} \)
$29$ \( 1 + \)\(21\!\cdots\!50\)\( T + \)\(20\!\cdots\!45\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!90\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!05\)\( T^{8} + \)\(22\!\cdots\!50\)\( T^{9} + \)\(59\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 + \)\(39\!\cdots\!40\)\( T + \)\(31\!\cdots\!95\)\( T^{2} + \)\(94\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!10\)\( T^{4} + \)\(98\!\cdots\!48\)\( T^{5} + \)\(32\!\cdots\!10\)\( T^{6} + \)\(51\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(22\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + \)\(67\!\cdots\!78\)\( T + \)\(24\!\cdots\!17\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} + \)\(42\!\cdots\!26\)\( T^{6} + \)\(57\!\cdots\!40\)\( T^{7} + \)\(66\!\cdots\!41\)\( T^{8} + \)\(55\!\cdots\!18\)\( T^{9} + \)\(24\!\cdots\!57\)\( T^{10} \)
$41$ \( 1 + \)\(89\!\cdots\!90\)\( T + \)\(21\!\cdots\!45\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!10\)\( T^{6} + \)\(46\!\cdots\!80\)\( T^{7} + \)\(33\!\cdots\!45\)\( T^{8} + \)\(77\!\cdots\!90\)\( T^{9} + \)\(46\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(12\!\cdots\!43\)\( T^{2} - \)\(85\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!88\)\( T^{5} + \)\(90\!\cdots\!14\)\( T^{6} - \)\(26\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!01\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(16\!\cdots\!43\)\( T^{10} \)
$47$ \( 1 - \)\(26\!\cdots\!32\)\( T + \)\(64\!\cdots\!87\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!78\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!06\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(99\!\cdots\!21\)\( T^{8} - \)\(47\!\cdots\!12\)\( T^{9} + \)\(20\!\cdots\!07\)\( T^{10} \)
$53$ \( 1 + \)\(22\!\cdots\!54\)\( T + \)\(53\!\cdots\!33\)\( T^{2} + \)\(71\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!38\)\( T^{4} + \)\(81\!\cdots\!32\)\( T^{5} + \)\(68\!\cdots\!74\)\( T^{6} + \)\(39\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!61\)\( T^{8} + \)\(69\!\cdots\!14\)\( T^{9} + \)\(22\!\cdots\!93\)\( T^{10} \)
$59$ \( 1 - \)\(49\!\cdots\!00\)\( T + \)\(80\!\cdots\!95\)\( T^{2} - \)\(32\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!10\)\( T^{4} - \)\(87\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!90\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!05\)\( T^{8} - \)\(61\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + \)\(20\!\cdots\!90\)\( T + \)\(10\!\cdots\!45\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!10\)\( T^{4} + \)\(49\!\cdots\!48\)\( T^{5} + \)\(99\!\cdots\!10\)\( T^{6} + \)\(72\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!45\)\( T^{8} + \)\(41\!\cdots\!90\)\( T^{9} + \)\(44\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 - \)\(17\!\cdots\!52\)\( T + \)\(45\!\cdots\!27\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(28\!\cdots\!76\)\( T^{5} + \)\(29\!\cdots\!66\)\( T^{6} - \)\(53\!\cdots\!20\)\( T^{7} + \)\(37\!\cdots\!81\)\( T^{8} - \)\(28\!\cdots\!72\)\( T^{9} + \)\(32\!\cdots\!07\)\( T^{10} \)
$71$ \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(44\!\cdots\!95\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!10\)\( T^{4} - \)\(13\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} - \)\(16\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!45\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(51\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 + \)\(23\!\cdots\!74\)\( T + \)\(55\!\cdots\!13\)\( T^{2} + \)\(77\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!94\)\( T^{6} + \)\(85\!\cdots\!80\)\( T^{7} + \)\(64\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!54\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \)
$79$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!10\)\( T^{4} - \)\(42\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!05\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 - \)\(10\!\cdots\!16\)\( T + \)\(23\!\cdots\!03\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!08\)\( T^{5} + \)\(57\!\cdots\!54\)\( T^{6} - \)\(22\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} - \)\(24\!\cdots\!76\)\( T^{9} + \)\(29\!\cdots\!43\)\( T^{10} \)
$89$ \( 1 + \)\(44\!\cdots\!50\)\( T + \)\(62\!\cdots\!45\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!90\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!05\)\( T^{8} + \)\(73\!\cdots\!50\)\( T^{9} + \)\(33\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + \)\(47\!\cdots\!18\)\( T + \)\(59\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} + \)\(32\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!06\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!21\)\( T^{8} + \)\(64\!\cdots\!38\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} \)
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