↑

Properties

Label 1.68.a.a
Level 1
Weight 68
Character orbit 1.a
Self dual yes
Analytic conductor 28.429
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 68 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4290351930\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 939384011925257456 x^{3} + 31046449413968483513911200 x^{2} + 156793504704482691874379743265203200 x + 20916736226052669578405116700517591609696000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 \)
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.

\(f(q)\) \(=\) \( q +(1110980251 - \beta_{1}) q^{2} +(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3} +(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4} +(\)\(66\!\cdots\!25\)\( + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5} +(\)\(29\!\cdots\!36\)\( + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6} +(\)\(67\!\cdots\!98\)\( - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7} +(\)\(65\!\cdots\!36\)\( - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +(\)\(55\!\cdots\!99\)\( + \)\(96\!\cdots\!02\)\( \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9} +O(q^{10})\) \( q +(1110980251 - \beta_{1}) q^{2} +(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3} +(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4} +(\)\(66\!\cdots\!25\)\( + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5} +(\)\(29\!\cdots\!36\)\( + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6} +(\)\(67\!\cdots\!98\)\( - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7} +(\)\(65\!\cdots\!36\)\( - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +(\)\(55\!\cdots\!99\)\( + \)\(96\!\cdots\!02\)\( \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9} +(-\)\(10\!\cdots\!50\)\( - \)\(36\!\cdots\!46\)\( \beta_{1} + 387747904434926976 \beta_{2} - 10724715572992 \beta_{3} + 6662909664 \beta_{4}) q^{10} +(\)\(40\!\cdots\!97\)\( - \)\(14\!\cdots\!25\)\( \beta_{1} + 4432914057561345675 \beta_{2} + 337416804371680 \beta_{3} - 154884117080 \beta_{4}) q^{11} +(-\)\(41\!\cdots\!16\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} - 12007852213004039576 \beta_{2} - 2856419893635300 \beta_{3} + 2653671352320 \beta_{4}) q^{12} +(\)\(35\!\cdots\!77\)\( - \)\(19\!\cdots\!87\)\( \beta_{1} - \)\(15\!\cdots\!60\)\( \beta_{2} - 29875083592222700 \beta_{3} - 34622077787505 \beta_{4}) q^{13} +(\)\(58\!\cdots\!44\)\( - \)\(18\!\cdots\!92\)\( \beta_{1} + 33318747846508840128 \beta_{2} + 964772473179561088 \beta_{3} + 344998573654000 \beta_{4}) q^{14} +(-\)\(18\!\cdots\!50\)\( - \)\(17\!\cdots\!22\)\( \beta_{1} + \)\(22\!\cdots\!82\)\( \beta_{2} - 10229182088434956144 \beta_{3} - 2522507816770452 \beta_{4}) q^{15} +(\)\(36\!\cdots\!04\)\( - \)\(18\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} + 43878715027197261888 \beta_{3} + 11266877309667840 \beta_{4}) q^{16} +(\)\(15\!\cdots\!32\)\( - \)\(71\!\cdots\!02\)\( \beta_{1} - \)\(95\!\cdots\!12\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} + 8396275243314890 \beta_{4}) q^{17} +(-\)\(20\!\cdots\!21\)\( + \)\(30\!\cdots\!91\)\( \beta_{1} + \)\(80\!\cdots\!76\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} - 700896140455976640 \beta_{4}) q^{18} +(\)\(79\!\cdots\!79\)\( + \)\(29\!\cdots\!09\)\( \beta_{1} + \)\(63\!\cdots\!69\)\( \beta_{2} + \)\(24\!\cdots\!64\)\( \beta_{3} + 7385036747071057560 \beta_{4}) q^{19} +(-\)\(30\!\cdots\!50\)\( + \)\(24\!\cdots\!02\)\( \beta_{1} - \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(48\!\cdots\!46\)\( \beta_{3} - 47581246658342739968 \beta_{4}) q^{20} +(-\)\(35\!\cdots\!00\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(36\!\cdots\!52\)\( \beta_{3} + \)\(19\!\cdots\!00\)\( \beta_{4}) q^{21} +(\)\(36\!\cdots\!72\)\( - \)\(71\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!60\)\( \beta_{2} + \)\(32\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!80\)\( \beta_{4}) q^{22} +(-\)\(83\!\cdots\!90\)\( - \)\(73\!\cdots\!10\)\( \beta_{1} - \)\(14\!\cdots\!86\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!00\)\( \beta_{4}) q^{23} +(-\)\(26\!\cdots\!48\)\( + \)\(45\!\cdots\!92\)\( \beta_{1} - \)\(78\!\cdots\!28\)\( \beta_{2} - \)\(48\!\cdots\!48\)\( \beta_{3} + \)\(36\!\cdots\!60\)\( \beta_{4}) q^{24} +(\)\(35\!\cdots\!75\)\( + \)\(38\!\cdots\!00\)\( \beta_{1} + \)\(94\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!00\)\( \beta_{4}) q^{25} +(\)\(42\!\cdots\!94\)\( - \)\(62\!\cdots\!18\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} + \)\(74\!\cdots\!20\)\( \beta_{4}) q^{26} +(\)\(37\!\cdots\!34\)\( - \)\(15\!\cdots\!74\)\( \beta_{1} - \)\(25\!\cdots\!18\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!20\)\( \beta_{4}) q^{27} +(\)\(29\!\cdots\!20\)\( - \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} + \)\(48\!\cdots\!80\)\( \beta_{4}) q^{28} +(\)\(37\!\cdots\!01\)\( + \)\(75\!\cdots\!21\)\( \beta_{1} + \)\(75\!\cdots\!36\)\( \beta_{2} - \)\(43\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!95\)\( \beta_{4}) q^{29} +(\)\(36\!\cdots\!00\)\( + \)\(15\!\cdots\!08\)\( \beta_{1} + \)\(17\!\cdots\!52\)\( \beta_{2} + \)\(45\!\cdots\!16\)\( \beta_{3} - \)\(76\!\cdots\!72\)\( \beta_{4}) q^{30} +(\)\(73\!\cdots\!12\)\( + \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(37\!\cdots\!00\)\( \beta_{2} - \)\(46\!\cdots\!60\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4}) q^{31} +(\)\(31\!\cdots\!16\)\( - \)\(66\!\cdots\!36\)\( \beta_{1} - \)\(91\!\cdots\!80\)\( \beta_{2} + \)\(55\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!60\)\( \beta_{4}) q^{32} +(\)\(48\!\cdots\!38\)\( - \)\(23\!\cdots\!58\)\( \beta_{1} + \)\(63\!\cdots\!92\)\( \beta_{2} - \)\(38\!\cdots\!00\)\( \beta_{3} - \)\(80\!\cdots\!10\)\( \beta_{4}) q^{33} +(\)\(15\!\cdots\!18\)\( - \)\(77\!\cdots\!18\)\( \beta_{1} - \)\(51\!\cdots\!88\)\( \beta_{2} + \)\(98\!\cdots\!92\)\( \beta_{3} + \)\(35\!\cdots\!60\)\( \beta_{4}) q^{34} +(\)\(90\!\cdots\!00\)\( + \)\(24\!\cdots\!84\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4}) q^{35} +(-\)\(76\!\cdots\!51\)\( + \)\(78\!\cdots\!23\)\( \beta_{1} - \)\(18\!\cdots\!82\)\( \beta_{2} - \)\(49\!\cdots\!87\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4}) q^{36} +(-\)\(11\!\cdots\!15\)\( - \)\(90\!\cdots\!15\)\( \beta_{1} + \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!35\)\( \beta_{4}) q^{37} +(-\)\(62\!\cdots\!36\)\( - \)\(27\!\cdots\!84\)\( \beta_{1} - \)\(15\!\cdots\!48\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4}) q^{38} +(-\)\(14\!\cdots\!18\)\( - \)\(64\!\cdots\!62\)\( \beta_{1} - \)\(43\!\cdots\!42\)\( \beta_{2} + \)\(20\!\cdots\!48\)\( \beta_{3} - \)\(81\!\cdots\!80\)\( \beta_{4}) q^{39} +(-\)\(37\!\cdots\!00\)\( + \)\(14\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(28\!\cdots\!60\)\( \beta_{4}) q^{40} +(\)\(22\!\cdots\!82\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(78\!\cdots\!00\)\( \beta_{2} + \)\(38\!\cdots\!40\)\( \beta_{3} + \)\(62\!\cdots\!60\)\( \beta_{4}) q^{41} +(\)\(56\!\cdots\!20\)\( + \)\(75\!\cdots\!00\)\( \beta_{1} - \)\(72\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} - \)\(22\!\cdots\!40\)\( \beta_{4}) q^{42} +(\)\(13\!\cdots\!81\)\( - \)\(39\!\cdots\!81\)\( \beta_{1} + \)\(21\!\cdots\!47\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} + \)\(67\!\cdots\!00\)\( \beta_{4}) q^{43} +(\)\(98\!\cdots\!28\)\( - \)\(59\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(45\!\cdots\!12\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{44} +(\)\(19\!\cdots\!25\)\( - \)\(33\!\cdots\!87\)\( \beta_{1} + \)\(14\!\cdots\!72\)\( \beta_{2} - \)\(91\!\cdots\!24\)\( \beta_{3} - \)\(43\!\cdots\!17\)\( \beta_{4}) q^{45} +(\)\(15\!\cdots\!12\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} - \)\(28\!\cdots\!40\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4}) q^{46} +(-\)\(25\!\cdots\!96\)\( + \)\(44\!\cdots\!96\)\( \beta_{1} - \)\(19\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{47} +(-\)\(39\!\cdots\!20\)\( + \)\(36\!\cdots\!00\)\( \beta_{1} - \)\(83\!\cdots\!56\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(35\!\cdots\!60\)\( \beta_{4}) q^{48} +(-\)\(15\!\cdots\!47\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} - \)\(17\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!40\)\( \beta_{4}) q^{49} +(-\)\(78\!\cdots\!75\)\( - \)\(39\!\cdots\!75\)\( \beta_{1} + \)\(89\!\cdots\!00\)\( \beta_{2} - \)\(35\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{50} +(-\)\(67\!\cdots\!42\)\( + \)\(36\!\cdots\!26\)\( \beta_{1} - \)\(51\!\cdots\!34\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} - \)\(57\!\cdots\!60\)\( \beta_{4}) q^{51} +(\)\(13\!\cdots\!18\)\( + \)\(54\!\cdots\!82\)\( \beta_{1} - \)\(32\!\cdots\!04\)\( \beta_{2} + \)\(48\!\cdots\!50\)\( \beta_{3} - \)\(46\!\cdots\!00\)\( \beta_{4}) q^{52} +(\)\(20\!\cdots\!69\)\( + \)\(31\!\cdots\!21\)\( \beta_{1} - \)\(35\!\cdots\!44\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(66\!\cdots\!95\)\( \beta_{4}) q^{53} +(\)\(38\!\cdots\!44\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(58\!\cdots\!84\)\( \beta_{2} - \)\(87\!\cdots\!96\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4}) q^{54} +(\)\(13\!\cdots\!50\)\( + \)\(32\!\cdots\!58\)\( \beta_{1} + \)\(96\!\cdots\!02\)\( \beta_{2} - \)\(67\!\cdots\!84\)\( \beta_{3} - \)\(64\!\cdots\!72\)\( \beta_{4}) q^{55} +(\)\(32\!\cdots\!16\)\( - \)\(30\!\cdots\!64\)\( \beta_{1} - \)\(17\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!76\)\( \beta_{3} + \)\(30\!\cdots\!20\)\( \beta_{4}) q^{56} +(-\)\(18\!\cdots\!22\)\( + \)\(51\!\cdots\!22\)\( \beta_{1} - \)\(62\!\cdots\!36\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!50\)\( \beta_{4}) q^{57} +(-\)\(12\!\cdots\!34\)\( + \)\(34\!\cdots\!14\)\( \beta_{1} - \)\(37\!\cdots\!72\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3} - \)\(42\!\cdots\!60\)\( \beta_{4}) q^{58} +(-\)\(61\!\cdots\!03\)\( + \)\(11\!\cdots\!87\)\( \beta_{1} + \)\(12\!\cdots\!67\)\( \beta_{2} + \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(78\!\cdots\!00\)\( \beta_{4}) q^{59} +(-\)\(26\!\cdots\!00\)\( - \)\(12\!\cdots\!96\)\( \beta_{1} + \)\(14\!\cdots\!76\)\( \beta_{2} + \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(85\!\cdots\!64\)\( \beta_{4}) q^{60} +(-\)\(22\!\cdots\!03\)\( + \)\(14\!\cdots\!25\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!25\)\( \beta_{4}) q^{61} +(-\)\(43\!\cdots\!88\)\( - \)\(76\!\cdots\!92\)\( \beta_{1} - \)\(51\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{62} +(-\)\(41\!\cdots\!02\)\( + \)\(62\!\cdots\!82\)\( \beta_{1} + \)\(27\!\cdots\!98\)\( \beta_{2} - \)\(43\!\cdots\!00\)\( \beta_{3} - \)\(50\!\cdots\!60\)\( \beta_{4}) q^{63} +(\)\(12\!\cdots\!20\)\( - \)\(14\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!92\)\( \beta_{2} + \)\(55\!\cdots\!12\)\( \beta_{3} + \)\(31\!\cdots\!20\)\( \beta_{4}) q^{64} +(\)\(34\!\cdots\!00\)\( + \)\(33\!\cdots\!68\)\( \beta_{1} - \)\(94\!\cdots\!08\)\( \beta_{2} + \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4}) q^{65} +(\)\(55\!\cdots\!92\)\( + \)\(68\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(20\!\cdots\!52\)\( \beta_{3} - \)\(21\!\cdots\!80\)\( \beta_{4}) q^{66} +(-\)\(23\!\cdots\!73\)\( + \)\(58\!\cdots\!53\)\( \beta_{1} - \)\(60\!\cdots\!27\)\( \beta_{2} - \)\(73\!\cdots\!00\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4}) q^{67} +(\)\(16\!\cdots\!02\)\( - \)\(19\!\cdots\!42\)\( \beta_{1} - \)\(62\!\cdots\!28\)\( \beta_{2} + \)\(15\!\cdots\!50\)\( \beta_{3} + \)\(50\!\cdots\!80\)\( \beta_{4}) q^{68} +(\)\(71\!\cdots\!48\)\( - \)\(43\!\cdots\!56\)\( \beta_{1} + \)\(76\!\cdots\!04\)\( \beta_{2} + \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(70\!\cdots\!60\)\( \beta_{4}) q^{69} +(-\)\(52\!\cdots\!00\)\( - \)\(43\!\cdots\!76\)\( \beta_{1} + \)\(38\!\cdots\!56\)\( \beta_{2} - \)\(27\!\cdots\!52\)\( \beta_{3} + \)\(96\!\cdots\!84\)\( \beta_{4}) q^{70} +(-\)\(22\!\cdots\!58\)\( + \)\(16\!\cdots\!50\)\( \beta_{1} - \)\(23\!\cdots\!50\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4}) q^{71} +(-\)\(14\!\cdots\!68\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!24\)\( \beta_{2} - \)\(62\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4}) q^{72} +(-\)\(60\!\cdots\!00\)\( - \)\(28\!\cdots\!70\)\( \beta_{1} + \)\(14\!\cdots\!04\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!90\)\( \beta_{4}) q^{73} +(\)\(18\!\cdots\!26\)\( + \)\(42\!\cdots\!90\)\( \beta_{1} - \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(72\!\cdots\!60\)\( \beta_{4}) q^{74} +(\)\(43\!\cdots\!25\)\( - \)\(41\!\cdots\!25\)\( \beta_{1} + \)\(68\!\cdots\!75\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4}) q^{75} +(\)\(41\!\cdots\!12\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(89\!\cdots\!68\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} - \)\(29\!\cdots\!60\)\( \beta_{4}) q^{76} +(\)\(10\!\cdots\!56\)\( - \)\(47\!\cdots\!76\)\( \beta_{1} - \)\(81\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(54\!\cdots\!40\)\( \beta_{4}) q^{77} +(\)\(12\!\cdots\!52\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(36\!\cdots\!36\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(23\!\cdots\!80\)\( \beta_{4}) q^{78} +(\)\(59\!\cdots\!36\)\( + \)\(85\!\cdots\!56\)\( \beta_{1} - \)\(81\!\cdots\!04\)\( \beta_{2} - \)\(63\!\cdots\!44\)\( \beta_{3} - \)\(77\!\cdots\!40\)\( \beta_{4}) q^{79} +(-\)\(30\!\cdots\!00\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(42\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!92\)\( \beta_{3} + \)\(14\!\cdots\!64\)\( \beta_{4}) q^{80} +(-\)\(22\!\cdots\!25\)\( - \)\(39\!\cdots\!26\)\( \beta_{1} + \)\(13\!\cdots\!84\)\( \beta_{2} + \)\(12\!\cdots\!64\)\( \beta_{3} + \)\(24\!\cdots\!50\)\( \beta_{4}) q^{81} +(-\)\(35\!\cdots\!18\)\( - \)\(22\!\cdots\!62\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!40\)\( \beta_{4}) q^{82} +(-\)\(94\!\cdots\!57\)\( - \)\(35\!\cdots\!43\)\( \beta_{1} - \)\(14\!\cdots\!27\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(10\!\cdots\!68\)\( - \)\(19\!\cdots\!24\)\( \beta_{1} + \)\(99\!\cdots\!16\)\( \beta_{2} - \)\(20\!\cdots\!84\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4}) q^{84} +(-\)\(78\!\cdots\!50\)\( + \)\(15\!\cdots\!94\)\( \beta_{1} + \)\(65\!\cdots\!36\)\( \beta_{2} - \)\(53\!\cdots\!12\)\( \beta_{3} + \)\(64\!\cdots\!54\)\( \beta_{4}) q^{85} +(\)\(99\!\cdots\!28\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(23\!\cdots\!16\)\( \beta_{2} - \)\(18\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{86} +(\)\(71\!\cdots\!82\)\( + \)\(24\!\cdots\!38\)\( \beta_{1} + \)\(71\!\cdots\!46\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4}) q^{87} +(\)\(86\!\cdots\!92\)\( - \)\(88\!\cdots\!12\)\( \beta_{1} - \)\(15\!\cdots\!84\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!40\)\( \beta_{4}) q^{88} +(-\)\(23\!\cdots\!12\)\( + \)\(45\!\cdots\!98\)\( \beta_{1} - \)\(86\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!32\)\( \beta_{3} - \)\(74\!\cdots\!90\)\( \beta_{4}) q^{89} +(\)\(95\!\cdots\!50\)\( - \)\(15\!\cdots\!82\)\( \beta_{1} + \)\(13\!\cdots\!92\)\( \beta_{2} - \)\(84\!\cdots\!64\)\( \beta_{3} - \)\(25\!\cdots\!12\)\( \beta_{4}) q^{90} +(\)\(46\!\cdots\!08\)\( + \)\(82\!\cdots\!96\)\( \beta_{1} - \)\(38\!\cdots\!64\)\( \beta_{2} + \)\(60\!\cdots\!36\)\( \beta_{3} - \)\(37\!\cdots\!80\)\( \beta_{4}) q^{91} +(-\)\(19\!\cdots\!68\)\( + \)\(38\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!48\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4}) q^{92} +(-\)\(36\!\cdots\!52\)\( + \)\(11\!\cdots\!92\)\( \beta_{1} + \)\(34\!\cdots\!72\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(21\!\cdots\!20\)\( \beta_{4}) q^{93} +(-\)\(98\!\cdots\!40\)\( + \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} - \)\(49\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4}) q^{94} +(-\)\(40\!\cdots\!50\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} - \)\(18\!\cdots\!90\)\( \beta_{2} + \)\(52\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!60\)\( \beta_{4}) q^{95} +(-\)\(44\!\cdots\!24\)\( - \)\(60\!\cdots\!36\)\( \beta_{1} + \)\(25\!\cdots\!24\)\( \beta_{2} - \)\(19\!\cdots\!16\)\( \beta_{3} + \)\(50\!\cdots\!20\)\( \beta_{4}) q^{96} +(\)\(57\!\cdots\!24\)\( - \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(76\!\cdots\!16\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(50\!\cdots\!30\)\( \beta_{4}) q^{97} +(\)\(24\!\cdots\!03\)\( - \)\(68\!\cdots\!33\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(79\!\cdots\!60\)\( \beta_{4}) q^{98} +(\)\(32\!\cdots\!53\)\( + \)\(51\!\cdots\!19\)\( \beta_{1} + \)\(62\!\cdots\!79\)\( \beta_{2} - \)\(36\!\cdots\!96\)\( \beta_{3} - \)\(18\!\cdots\!20\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5554901256q^{2} + 3443360269119372q^{3} + \)\(35\!\cdots\!40\)\(q^{4} + \)\(33\!\cdots\!50\)\(q^{5} + \)\(14\!\cdots\!60\)\(q^{6} + \)\(33\!\cdots\!56\)\(q^{7} + \)\(32\!\cdots\!80\)\(q^{8} + \)\(27\!\cdots\!85\)\(q^{9} + O(q^{10}) \) \( 5q + 5554901256q^{2} + 3443360269119372q^{3} + \)\(35\!\cdots\!40\)\(q^{4} + \)\(33\!\cdots\!50\)\(q^{5} + \)\(14\!\cdots\!60\)\(q^{6} + \)\(33\!\cdots\!56\)\(q^{7} + \)\(32\!\cdots\!80\)\(q^{8} + \)\(27\!\cdots\!85\)\(q^{9} - \)\(52\!\cdots\!00\)\(q^{10} + \)\(20\!\cdots\!60\)\(q^{11} - \)\(20\!\cdots\!24\)\(q^{12} + \)\(17\!\cdots\!02\)\(q^{13} + \)\(29\!\cdots\!80\)\(q^{14} - \)\(93\!\cdots\!00\)\(q^{15} + \)\(18\!\cdots\!80\)\(q^{16} + \)\(75\!\cdots\!06\)\(q^{17} - \)\(10\!\cdots\!68\)\(q^{18} + \)\(39\!\cdots\!00\)\(q^{19} - \)\(15\!\cdots\!00\)\(q^{20} - \)\(17\!\cdots\!40\)\(q^{21} + \)\(18\!\cdots\!32\)\(q^{22} - \)\(41\!\cdots\!68\)\(q^{23} - \)\(13\!\cdots\!00\)\(q^{24} + \)\(17\!\cdots\!75\)\(q^{25} + \)\(21\!\cdots\!60\)\(q^{26} + \)\(18\!\cdots\!20\)\(q^{27} + \)\(14\!\cdots\!48\)\(q^{28} + \)\(18\!\cdots\!50\)\(q^{29} + \)\(18\!\cdots\!00\)\(q^{30} + \)\(36\!\cdots\!60\)\(q^{31} + \)\(15\!\cdots\!96\)\(q^{32} + \)\(24\!\cdots\!84\)\(q^{33} + \)\(78\!\cdots\!80\)\(q^{34} + \)\(45\!\cdots\!00\)\(q^{35} - \)\(38\!\cdots\!20\)\(q^{36} - \)\(56\!\cdots\!94\)\(q^{37} - \)\(31\!\cdots\!80\)\(q^{38} - \)\(71\!\cdots\!80\)\(q^{39} - \)\(18\!\cdots\!00\)\(q^{40} + \)\(11\!\cdots\!10\)\(q^{41} + \)\(28\!\cdots\!92\)\(q^{42} + \)\(65\!\cdots\!92\)\(q^{43} + \)\(49\!\cdots\!80\)\(q^{44} + \)\(99\!\cdots\!50\)\(q^{45} + \)\(79\!\cdots\!60\)\(q^{46} - \)\(12\!\cdots\!44\)\(q^{47} - \)\(19\!\cdots\!68\)\(q^{48} - \)\(77\!\cdots\!35\)\(q^{49} - \)\(39\!\cdots\!00\)\(q^{50} - \)\(33\!\cdots\!40\)\(q^{51} + \)\(67\!\cdots\!16\)\(q^{52} + \)\(10\!\cdots\!22\)\(q^{53} + \)\(19\!\cdots\!00\)\(q^{54} + \)\(66\!\cdots\!00\)\(q^{55} + \)\(16\!\cdots\!00\)\(q^{56} - \)\(92\!\cdots\!60\)\(q^{57} - \)\(61\!\cdots\!20\)\(q^{58} - \)\(30\!\cdots\!00\)\(q^{59} - \)\(13\!\cdots\!00\)\(q^{60} - \)\(11\!\cdots\!90\)\(q^{61} - \)\(21\!\cdots\!28\)\(q^{62} - \)\(20\!\cdots\!68\)\(q^{63} + \)\(63\!\cdots\!40\)\(q^{64} + \)\(17\!\cdots\!00\)\(q^{65} + \)\(27\!\cdots\!20\)\(q^{66} - \)\(11\!\cdots\!44\)\(q^{67} + \)\(81\!\cdots\!48\)\(q^{68} + \)\(35\!\cdots\!20\)\(q^{69} - \)\(26\!\cdots\!00\)\(q^{70} - \)\(11\!\cdots\!40\)\(q^{71} - \)\(74\!\cdots\!40\)\(q^{72} - \)\(30\!\cdots\!18\)\(q^{73} + \)\(92\!\cdots\!80\)\(q^{74} + \)\(21\!\cdots\!00\)\(q^{75} + \)\(20\!\cdots\!00\)\(q^{76} + \)\(50\!\cdots\!32\)\(q^{77} + \)\(61\!\cdots\!64\)\(q^{78} + \)\(29\!\cdots\!00\)\(q^{79} - \)\(15\!\cdots\!00\)\(q^{80} - \)\(11\!\cdots\!95\)\(q^{81} - \)\(17\!\cdots\!08\)\(q^{82} - \)\(47\!\cdots\!88\)\(q^{83} - \)\(52\!\cdots\!20\)\(q^{84} - \)\(39\!\cdots\!00\)\(q^{85} + \)\(49\!\cdots\!60\)\(q^{86} + \)\(35\!\cdots\!60\)\(q^{87} + \)\(43\!\cdots\!60\)\(q^{88} - \)\(11\!\cdots\!50\)\(q^{89} + \)\(47\!\cdots\!00\)\(q^{90} + \)\(23\!\cdots\!60\)\(q^{91} - \)\(98\!\cdots\!44\)\(q^{92} - \)\(18\!\cdots\!36\)\(q^{93} - \)\(49\!\cdots\!20\)\(q^{94} - \)\(20\!\cdots\!00\)\(q^{95} - \)\(22\!\cdots\!40\)\(q^{96} + \)\(28\!\cdots\!06\)\(q^{97} + \)\(12\!\cdots\!08\)\(q^{98} + \)\(16\!\cdots\!20\)\(q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 939384011925257456 x^{3} + 31046449413968483513911200 x^{2} + 156793504704482691874379743265203200 x + 20916736226052669578405116700517591609696000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-1463 \nu^{4} + 61316688179 \nu^{3} + 1134677465176013610916 \nu^{2} - 175871625402178878144147591024 \nu - 92322819705534579210406721528027067456\)\()/ \)\(37\!\cdots\!28\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-1461537 \nu^{4} + 61255371490821 \nu^{3} + 1493224099349187733125372 \nu^{2} - 157864696669790310577560760925712 \nu - 227382046305944113582206155994463766078912\)\()/ \)\(62\!\cdots\!88\)\( \)
\(\beta_{4}\)\(=\)\((\)\(4738030953 \nu^{4} - 6783896933686506093 \nu^{3} - 6265292406150844870725632988 \nu^{2} + 4428965898625782506113044897501160080 \nu + 1149735030546567532337013572577831183558493120\)\()/ \)\(15\!\cdots\!20\)\( \)

Display \(\nu^j\) in terms of \(\beta_i\)

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 5994 \beta_{2} - 1189790397 \beta_{1} + 216434076347341357501\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-409640 \beta_{4} - 1180151429 \beta_{3} + 3889870865010 \beta_{2} + 43600466505262789881 \beta_{1} - 32188957769729085286587521881\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-17168672690120 \beta_{4} + 2277286000240633339 \beta_{3} - 18208831024879688328462 \beta_{2} - 9596312969973284759959813095 \beta_{1} + 393192791921864844386864518484726790407\)\()/1728\)

Display \(\beta_i\) in terms of \(\nu^j\)

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
7.63188e8
6.06038e8
−1.64845e8
−3.05046e8
−8.99335e8
−1.72055e10 −1.09955e16 1.48456e20 4.93311e23 1.89183e26 1.79351e28 −1.51842e28 2.81916e31 −8.48767e33
1.2 −1.34339e10 7.87958e15 3.28965e19 −3.01383e23 −1.05854e26 −4.20742e26 1.54057e30 −3.06216e31 4.04876e33
1.3 5.06726e9 −7.82458e15 −1.21897e20 −8.33551e22 −3.96491e25 −1.15070e28 −1.36548e30 −3.14854e31 −4.22381e32
1.4 8.43209e9 1.56682e16 −7.64738e19 3.83370e23 1.32115e26 −1.75533e27 −1.88919e30 1.52782e32 3.23261e33
1.5 2.26950e10 −1.28429e15 3.67490e20 −1.61198e23 −2.91471e25 2.93806e28 4.99099e30 −9.10601e31 −3.65838e33
\(n\): 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 1.68.a.a ✓ 5
3.b odd 2 1 9.68.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.68.a.a ✓ 5 1.a even 1 1 trivial
9.68.a.a 5 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{68}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5554901256 T + \)\(20\!\cdots\!68\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!88\)\( T^{4} - \)\(54\!\cdots\!08\)\( T^{5} + \)\(51\!\cdots\!64\)\( T^{6} - \)\(41\!\cdots\!60\)\( T^{7} + \)\(67\!\cdots\!36\)\( T^{8} - \)\(26\!\cdots\!36\)\( T^{9} + \)\(69\!\cdots\!68\)\( T^{10} \)
$3$ \( 1 - 3443360269119372 T + \)\(22\!\cdots\!67\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!98\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(28\!\cdots\!26\)\( T^{6} - \)\(10\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} - \)\(25\!\cdots\!92\)\( T^{9} + \)\(68\!\cdots\!07\)\( T^{10} \)
$5$ \( 1 - \)\(33\!\cdots\!50\)\( T + \)\(13\!\cdots\!25\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!50\)\( T^{4} - \)\(41\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!50\)\( T^{6} - \)\(26\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!25\)\( T^{8} - \)\(69\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 - \)\(33\!\cdots\!56\)\( T + \)\(19\!\cdots\!43\)\( T^{2} - \)\(50\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!98\)\( T^{4} - \)\(30\!\cdots\!88\)\( T^{5} + \)\(68\!\cdots\!14\)\( T^{6} - \)\(88\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(10\!\cdots\!56\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} \)
$11$ \( 1 - \)\(20\!\cdots\!60\)\( T + \)\(27\!\cdots\!95\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!10\)\( T^{6} - \)\(88\!\cdots\!20\)\( T^{7} + \)\(57\!\cdots\!45\)\( T^{8} - \)\(25\!\cdots\!60\)\( T^{9} + \)\(73\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 - \)\(17\!\cdots\!02\)\( T + \)\(12\!\cdots\!77\)\( T^{2} + \)\(44\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!98\)\( T^{4} + \)\(39\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!66\)\( T^{6} + \)\(82\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} - \)\(61\!\cdots\!42\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} \)
$17$ \( 1 - \)\(75\!\cdots\!06\)\( T + \)\(88\!\cdots\!93\)\( T^{2} - \)\(84\!\cdots\!20\)\( T^{3} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(34\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!14\)\( T^{6} - \)\(64\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!81\)\( T^{8} - \)\(43\!\cdots\!46\)\( T^{9} + \)\(15\!\cdots\!93\)\( T^{10} \)
$19$ \( 1 - \)\(39\!\cdots\!00\)\( T + \)\(14\!\cdots\!95\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!90\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!05\)\( T^{8} - \)\(20\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + \)\(41\!\cdots\!68\)\( T + \)\(73\!\cdots\!87\)\( T^{2} + \)\(30\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!98\)\( T^{4} + \)\(80\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!06\)\( T^{6} + \)\(89\!\cdots\!80\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} + \)\(36\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!07\)\( T^{10} \)
$29$ \( 1 - \)\(18\!\cdots\!50\)\( T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(33\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!10\)\( T^{4} - \)\(48\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!90\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!05\)\( T^{8} - \)\(15\!\cdots\!50\)\( T^{9} + \)\(80\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 - \)\(36\!\cdots\!60\)\( T + \)\(80\!\cdots\!95\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} - \)\(86\!\cdots\!20\)\( T^{7} + \)\(46\!\cdots\!45\)\( T^{8} - \)\(17\!\cdots\!60\)\( T^{9} + \)\(40\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + \)\(56\!\cdots\!94\)\( T + \)\(47\!\cdots\!93\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} + \)\(34\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!14\)\( T^{6} + \)\(29\!\cdots\!60\)\( T^{7} + \)\(77\!\cdots\!41\)\( T^{8} + \)\(10\!\cdots\!74\)\( T^{9} + \)\(22\!\cdots\!93\)\( T^{10} \)
$41$ \( 1 - \)\(11\!\cdots\!10\)\( T + \)\(30\!\cdots\!45\)\( T^{2} + \)\(95\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!10\)\( T^{4} + \)\(48\!\cdots\!48\)\( T^{5} + \)\(49\!\cdots\!10\)\( T^{6} + \)\(12\!\cdots\!80\)\( T^{7} + \)\(45\!\cdots\!45\)\( T^{8} - \)\(19\!\cdots\!10\)\( T^{9} + \)\(19\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(65\!\cdots\!92\)\( T + \)\(10\!\cdots\!07\)\( T^{2} - \)\(56\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!98\)\( T^{4} - \)\(21\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!86\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} - \)\(38\!\cdots\!92\)\( T^{9} + \)\(16\!\cdots\!07\)\( T^{10} \)
$47$ \( 1 + \)\(12\!\cdots\!44\)\( T + \)\(42\!\cdots\!43\)\( T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!78\)\( T^{4} + \)\(62\!\cdots\!72\)\( T^{5} + \)\(88\!\cdots\!14\)\( T^{6} + \)\(49\!\cdots\!80\)\( T^{7} + \)\(52\!\cdots\!21\)\( T^{8} + \)\(16\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!43\)\( T^{10} \)
$53$ \( 1 - \)\(10\!\cdots\!22\)\( T + \)\(74\!\cdots\!17\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(63\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!26\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!01\)\( T^{8} - \)\(13\!\cdots\!42\)\( T^{9} + \)\(42\!\cdots\!57\)\( T^{10} \)
$59$ \( 1 + \)\(30\!\cdots\!00\)\( T + \)\(13\!\cdots\!95\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(87\!\cdots\!10\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!90\)\( T^{6} + \)\(58\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!05\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + \)\(11\!\cdots\!90\)\( T + \)\(21\!\cdots\!45\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(93\!\cdots\!48\)\( T^{5} + \)\(71\!\cdots\!10\)\( T^{6} + \)\(27\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!45\)\( T^{8} + \)\(33\!\cdots\!90\)\( T^{9} + \)\(12\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(30\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!14\)\( T^{6} + \)\(79\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!04\)\( T^{9} + \)\(54\!\cdots\!43\)\( T^{10} \)
$71$ \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!95\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!10\)\( T^{4} + \)\(54\!\cdots\!48\)\( T^{5} + \)\(87\!\cdots\!10\)\( T^{6} + \)\(43\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!45\)\( T^{8} + \)\(15\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 + \)\(30\!\cdots\!18\)\( T + \)\(36\!\cdots\!37\)\( T^{2} + \)\(83\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!98\)\( T^{4} + \)\(86\!\cdots\!24\)\( T^{5} + \)\(36\!\cdots\!06\)\( T^{6} + \)\(40\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} + \)\(71\!\cdots\!58\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} \)
$79$ \( 1 - \)\(29\!\cdots\!00\)\( T + \)\(51\!\cdots\!95\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!90\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(50\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 + \)\(47\!\cdots\!88\)\( T + \)\(21\!\cdots\!47\)\( T^{2} + \)\(64\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!98\)\( T^{4} + \)\(35\!\cdots\!04\)\( T^{5} + \)\(65\!\cdots\!46\)\( T^{6} + \)\(92\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} + \)\(97\!\cdots\!08\)\( T^{9} + \)\(77\!\cdots\!07\)\( T^{10} \)
$89$ \( 1 + \)\(11\!\cdots\!50\)\( T + \)\(98\!\cdots\!45\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!10\)\( T^{4} + \)\(27\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!90\)\( T^{6} + \)\(64\!\cdots\!00\)\( T^{7} + \)\(65\!\cdots\!05\)\( T^{8} + \)\(32\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 - \)\(28\!\cdots\!06\)\( T + \)\(36\!\cdots\!93\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!78\)\( T^{4} - \)\(24\!\cdots\!28\)\( T^{5} + \)\(86\!\cdots\!14\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(79\!\cdots\!21\)\( T^{8} - \)\(82\!\cdots\!66\)\( T^{9} + \)\(37\!\cdots\!93\)\( T^{10} \)
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