Properties

Label 3.72.a.b
Level 3
Weight 72
Character orbit 3.a
Self dual yes
Analytic conductor 95.774
Analytic rank 0
Dimension 6
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(95.7738481683\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{49}\cdot 3^{29}\cdot 5^{7}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-12150609471 + \beta_{1}) q^{2} -50031545098999707 q^{3} +(\)\(17\!\cdots\!22\)\( - 17516044621 \beta_{1} + \beta_{2}) q^{4} +(\)\(16\!\cdots\!50\)\( + 28213031926853 \beta_{1} + 1134 \beta_{2} + \beta_{3}) q^{5} +(\)\(60\!\cdots\!97\)\( - 50031545098999707 \beta_{1}) q^{6} +(\)\(99\!\cdots\!48\)\( - 1082693092113113668 \beta_{1} + 104180280 \beta_{2} - 18415 \beta_{3} - 71 \beta_{4} + 38 \beta_{5}) q^{7} +(-\)\(61\!\cdots\!72\)\( + \)\(17\!\cdots\!98\)\( \beta_{1} - 18881241852 \beta_{2} - 6340812 \beta_{3} + 8666 \beta_{4} + 10241 \beta_{5}) q^{8} +\)\(25\!\cdots\!49\)\( q^{9} +O(q^{10})\) \( q +(-12150609471 + \beta_{1}) q^{2} -50031545098999707 q^{3} +(\)\(17\!\cdots\!22\)\( - 17516044621 \beta_{1} + \beta_{2}) q^{4} +(\)\(16\!\cdots\!50\)\( + 28213031926853 \beta_{1} + 1134 \beta_{2} + \beta_{3}) q^{5} +(\)\(60\!\cdots\!97\)\( - 50031545098999707 \beta_{1}) q^{6} +(\)\(99\!\cdots\!48\)\( - 1082693092113113668 \beta_{1} + 104180280 \beta_{2} - 18415 \beta_{3} - 71 \beta_{4} + 38 \beta_{5}) q^{7} +(-\)\(61\!\cdots\!72\)\( + \)\(17\!\cdots\!98\)\( \beta_{1} - 18881241852 \beta_{2} - 6340812 \beta_{3} + 8666 \beta_{4} + 10241 \beta_{5}) q^{8} +\)\(25\!\cdots\!49\)\( q^{9} +(\)\(90\!\cdots\!50\)\( + \)\(41\!\cdots\!78\)\( \beta_{1} - 7427576128556 \beta_{2} - 72797015944 \beta_{3} + 30812860 \beta_{4} + 9467030 \beta_{5}) q^{10} +(\)\(36\!\cdots\!44\)\( + \)\(51\!\cdots\!74\)\( \beta_{1} + 1059967454702844 \beta_{2} - 545209258238 \beta_{3} + 374353744 \beta_{4} + 443137504 \beta_{5}) q^{11} +(-\)\(86\!\cdots\!54\)\( + \)\(87\!\cdots\!47\)\( \beta_{1} - 50031545098999707 \beta_{2}) q^{12} +(-\)\(62\!\cdots\!46\)\( + \)\(13\!\cdots\!81\)\( \beta_{1} + 51014920186356326 \beta_{2} + 144291871839670 \beta_{3} + 17030753453 \beta_{4} + 243171997726 \beta_{5}) q^{13} +(-\)\(54\!\cdots\!72\)\( + \)\(34\!\cdots\!68\)\( \beta_{1} + 3195232706844438996 \beta_{2} - 2437317859969928 \beta_{3} - 55678808900 \beta_{4} + 3269326203286 \beta_{5}) q^{14} +(-\)\(84\!\cdots\!50\)\( - \)\(14\!\cdots\!71\)\( \beta_{1} - 56735772142265667738 \beta_{2} - 50031545098999707 \beta_{3}) q^{15} +(\)\(36\!\cdots\!80\)\( - \)\(73\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2} + 598606103381663032 \beta_{3} - 716962040898532 \beta_{4} - 128965651174778 \beta_{5}) q^{16} +(\)\(49\!\cdots\!66\)\( - \)\(60\!\cdots\!76\)\( \beta_{1} - \)\(21\!\cdots\!08\)\( \beta_{2} + 1119360920456075938 \beta_{3} + 2146481076140310 \beta_{4} - 218736202471836 \beta_{5}) q^{17} +(-\)\(30\!\cdots\!79\)\( + \)\(25\!\cdots\!49\)\( \beta_{1}) q^{18} +(-\)\(56\!\cdots\!12\)\( - \)\(34\!\cdots\!88\)\( \beta_{1} - \)\(35\!\cdots\!56\)\( \beta_{2} - 7564500504646046230 \beta_{3} + 327165097315901566 \beta_{4} + 58305069829914452 \beta_{5}) q^{19} +(\)\(11\!\cdots\!00\)\( - \)\(15\!\cdots\!46\)\( \beta_{1} + \)\(65\!\cdots\!42\)\( \beta_{2} + \)\(32\!\cdots\!08\)\( \beta_{3} - 2198822332160593920 \beta_{4} - 309308960437340160 \beta_{5}) q^{20} +(-\)\(49\!\cdots\!36\)\( + \)\(54\!\cdots\!76\)\( \beta_{1} - \)\(52\!\cdots\!60\)\( \beta_{2} + \)\(92\!\cdots\!05\)\( \beta_{3} + 3552239702028979197 \beta_{4} - 1901198713761988866 \beta_{5}) q^{21} +(\)\(15\!\cdots\!32\)\( + \)\(58\!\cdots\!16\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + \)\(31\!\cdots\!44\)\( \beta_{3} - 20435352797062709896 \beta_{4} + 14235034987616496940 \beta_{5}) q^{22} +(-\)\(40\!\cdots\!04\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(63\!\cdots\!16\)\( \beta_{2} + \)\(85\!\cdots\!98\)\( \beta_{3} + \)\(18\!\cdots\!62\)\( \beta_{4} + 58268384487909764828 \beta_{5}) q^{23} +(\)\(30\!\cdots\!04\)\( - \)\(88\!\cdots\!86\)\( \beta_{1} + \)\(94\!\cdots\!64\)\( \beta_{2} + \)\(31\!\cdots\!84\)\( \beta_{3} - \)\(43\!\cdots\!62\)\( \beta_{4} - \)\(51\!\cdots\!87\)\( \beta_{5}) q^{24} +(\)\(24\!\cdots\!75\)\( - \)\(56\!\cdots\!50\)\( \beta_{1} + \)\(98\!\cdots\!00\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3} + \)\(63\!\cdots\!50\)\( \beta_{4} - \)\(11\!\cdots\!00\)\( \beta_{5}) q^{25} +(\)\(60\!\cdots\!02\)\( - \)\(57\!\cdots\!02\)\( \beta_{1} + \)\(48\!\cdots\!88\)\( \beta_{2} - \)\(14\!\cdots\!68\)\( \beta_{3} - \)\(47\!\cdots\!64\)\( \beta_{4} + \)\(50\!\cdots\!88\)\( \beta_{5}) q^{26} -\)\(12\!\cdots\!43\)\( q^{27} +(\)\(11\!\cdots\!44\)\( + \)\(26\!\cdots\!52\)\( \beta_{1} + \)\(70\!\cdots\!20\)\( \beta_{2} + \)\(11\!\cdots\!68\)\( \beta_{3} + \)\(37\!\cdots\!44\)\( \beta_{4} + \)\(18\!\cdots\!52\)\( \beta_{5}) q^{28} +(\)\(23\!\cdots\!66\)\( + \)\(69\!\cdots\!01\)\( \beta_{1} + \)\(10\!\cdots\!62\)\( \beta_{2} - \)\(30\!\cdots\!65\)\( \beta_{3} + \)\(33\!\cdots\!06\)\( \beta_{4} - \)\(24\!\cdots\!08\)\( \beta_{5}) q^{29} +(-\)\(45\!\cdots\!50\)\( - \)\(20\!\cdots\!46\)\( \beta_{1} + \)\(37\!\cdots\!92\)\( \beta_{2} + \)\(36\!\cdots\!08\)\( \beta_{3} - \)\(15\!\cdots\!20\)\( \beta_{4} - \)\(47\!\cdots\!10\)\( \beta_{5}) q^{30} +(-\)\(45\!\cdots\!04\)\( + \)\(12\!\cdots\!44\)\( \beta_{1} - \)\(83\!\cdots\!44\)\( \beta_{2} + \)\(51\!\cdots\!89\)\( \beta_{3} - \)\(17\!\cdots\!03\)\( \beta_{4} + \)\(43\!\cdots\!26\)\( \beta_{5}) q^{31} +(-\)\(18\!\cdots\!52\)\( + \)\(35\!\cdots\!72\)\( \beta_{1} - \)\(95\!\cdots\!16\)\( \beta_{2} - \)\(54\!\cdots\!80\)\( \beta_{3} + \)\(16\!\cdots\!52\)\( \beta_{4} + \)\(17\!\cdots\!04\)\( \beta_{5}) q^{32} +(-\)\(18\!\cdots\!08\)\( - \)\(25\!\cdots\!18\)\( \beta_{1} - \)\(53\!\cdots\!08\)\( \beta_{2} + \)\(27\!\cdots\!66\)\( \beta_{3} - \)\(18\!\cdots\!08\)\( \beta_{4} - \)\(22\!\cdots\!28\)\( \beta_{5}) q^{33} +(-\)\(24\!\cdots\!86\)\( + \)\(32\!\cdots\!98\)\( \beta_{1} - \)\(60\!\cdots\!96\)\( \beta_{2} + \)\(11\!\cdots\!76\)\( \beta_{3} - \)\(81\!\cdots\!28\)\( \beta_{4} - \)\(69\!\cdots\!28\)\( \beta_{5}) q^{34} +(\)\(22\!\cdots\!00\)\( - \)\(13\!\cdots\!82\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2} + \)\(55\!\cdots\!26\)\( \beta_{3} - \)\(58\!\cdots\!60\)\( \beta_{4} + \)\(21\!\cdots\!20\)\( \beta_{5}) q^{35} +(\)\(43\!\cdots\!78\)\( - \)\(43\!\cdots\!29\)\( \beta_{1} + \)\(25\!\cdots\!49\)\( \beta_{2}) q^{36} +(-\)\(59\!\cdots\!42\)\( + \)\(22\!\cdots\!17\)\( \beta_{1} + \)\(90\!\cdots\!58\)\( \beta_{2} + \)\(83\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!91\)\( \beta_{4} + \)\(23\!\cdots\!78\)\( \beta_{5}) q^{37} +(-\)\(66\!\cdots\!64\)\( - \)\(13\!\cdots\!96\)\( \beta_{1} + \)\(17\!\cdots\!24\)\( \beta_{2} + \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(82\!\cdots\!36\)\( \beta_{4} - \)\(86\!\cdots\!92\)\( \beta_{5}) q^{38} +(\)\(31\!\cdots\!22\)\( - \)\(67\!\cdots\!67\)\( \beta_{1} - \)\(25\!\cdots\!82\)\( \beta_{2} - \)\(72\!\cdots\!90\)\( \beta_{3} - \)\(85\!\cdots\!71\)\( \beta_{4} - \)\(12\!\cdots\!82\)\( \beta_{5}) q^{39} +(-\)\(41\!\cdots\!00\)\( + \)\(16\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!80\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3} + \)\(82\!\cdots\!60\)\( \beta_{4} + \)\(72\!\cdots\!30\)\( \beta_{5}) q^{40} +(\)\(79\!\cdots\!86\)\( - \)\(12\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!44\)\( \beta_{2} + \)\(10\!\cdots\!38\)\( \beta_{3} - \)\(18\!\cdots\!06\)\( \beta_{4} - \)\(66\!\cdots\!48\)\( \beta_{5}) q^{41} +(\)\(27\!\cdots\!04\)\( - \)\(17\!\cdots\!76\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} - \)\(16\!\cdots\!02\)\( \beta_{5}) q^{42} +(\)\(19\!\cdots\!80\)\( + \)\(15\!\cdots\!40\)\( \beta_{1} - \)\(46\!\cdots\!48\)\( \beta_{2} + \)\(13\!\cdots\!74\)\( \beta_{3} + \)\(76\!\cdots\!58\)\( \beta_{4} - \)\(32\!\cdots\!12\)\( \beta_{5}) q^{43} +(\)\(12\!\cdots\!16\)\( + \)\(35\!\cdots\!20\)\( \beta_{1} + \)\(37\!\cdots\!52\)\( \beta_{2} - \)\(26\!\cdots\!68\)\( \beta_{3} + \)\(82\!\cdots\!80\)\( \beta_{4} + \)\(11\!\cdots\!16\)\( \beta_{5}) q^{44} +(\)\(42\!\cdots\!50\)\( + \)\(70\!\cdots\!97\)\( \beta_{1} + \)\(28\!\cdots\!66\)\( \beta_{2} + \)\(25\!\cdots\!49\)\( \beta_{3}) q^{45} +(-\)\(37\!\cdots\!64\)\( + \)\(11\!\cdots\!04\)\( \beta_{1} + \)\(11\!\cdots\!88\)\( \beta_{2} - \)\(13\!\cdots\!36\)\( \beta_{3} - \)\(13\!\cdots\!44\)\( \beta_{4} - \)\(18\!\cdots\!76\)\( \beta_{5}) q^{46} +(-\)\(12\!\cdots\!60\)\( - \)\(29\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!16\)\( \beta_{2} - \)\(80\!\cdots\!86\)\( \beta_{3} - \)\(15\!\cdots\!90\)\( \beta_{4} + \)\(20\!\cdots\!52\)\( \beta_{5}) q^{47} +(-\)\(18\!\cdots\!60\)\( + \)\(36\!\cdots\!96\)\( \beta_{1} - \)\(79\!\cdots\!40\)\( \beta_{2} - \)\(29\!\cdots\!24\)\( \beta_{3} + \)\(35\!\cdots\!24\)\( \beta_{4} + \)\(64\!\cdots\!46\)\( \beta_{5}) q^{48} +(-\)\(11\!\cdots\!15\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} - \)\(19\!\cdots\!68\)\( \beta_{2} + \)\(24\!\cdots\!36\)\( \beta_{3} + \)\(49\!\cdots\!66\)\( \beta_{4} - \)\(56\!\cdots\!40\)\( \beta_{5}) q^{49} +(-\)\(25\!\cdots\!25\)\( + \)\(50\!\cdots\!75\)\( \beta_{1} - \)\(43\!\cdots\!00\)\( \beta_{2} + \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(63\!\cdots\!00\)\( \beta_{4} + \)\(94\!\cdots\!00\)\( \beta_{5}) q^{50} +(-\)\(24\!\cdots\!62\)\( + \)\(30\!\cdots\!32\)\( \beta_{1} + \)\(10\!\cdots\!56\)\( \beta_{2} - \)\(56\!\cdots\!66\)\( \beta_{3} - \)\(10\!\cdots\!70\)\( \beta_{4} + \)\(10\!\cdots\!52\)\( \beta_{5}) q^{51} +(-\)\(15\!\cdots\!24\)\( + \)\(14\!\cdots\!42\)\( \beta_{1} + \)\(37\!\cdots\!66\)\( \beta_{2} + \)\(35\!\cdots\!72\)\( \beta_{3} - \)\(40\!\cdots\!56\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5}) q^{52} +(\)\(90\!\cdots\!86\)\( + \)\(93\!\cdots\!65\)\( \beta_{1} + \)\(20\!\cdots\!42\)\( \beta_{2} - \)\(16\!\cdots\!21\)\( \beta_{3} + \)\(23\!\cdots\!02\)\( \beta_{4} - \)\(85\!\cdots\!84\)\( \beta_{5}) q^{53} +(\)\(15\!\cdots\!53\)\( - \)\(12\!\cdots\!43\)\( \beta_{1}) q^{54} +(-\)\(38\!\cdots\!00\)\( + \)\(10\!\cdots\!36\)\( \beta_{1} + \)\(48\!\cdots\!48\)\( \beta_{2} + \)\(37\!\cdots\!32\)\( \beta_{3} - \)\(76\!\cdots\!60\)\( \beta_{4} + \)\(27\!\cdots\!20\)\( \beta_{5}) q^{55} +(\)\(87\!\cdots\!60\)\( + \)\(19\!\cdots\!04\)\( \beta_{1} - \)\(56\!\cdots\!00\)\( \beta_{2} - \)\(54\!\cdots\!84\)\( \beta_{3} + \)\(76\!\cdots\!80\)\( \beta_{4} - \)\(10\!\cdots\!92\)\( \beta_{5}) q^{56} +(\)\(28\!\cdots\!84\)\( + \)\(17\!\cdots\!16\)\( \beta_{1} + \)\(17\!\cdots\!92\)\( \beta_{2} + \)\(37\!\cdots\!10\)\( \beta_{3} - \)\(16\!\cdots\!62\)\( \beta_{4} - \)\(29\!\cdots\!64\)\( \beta_{5}) q^{57} +(-\)\(11\!\cdots\!18\)\( + \)\(46\!\cdots\!14\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(29\!\cdots\!04\)\( \beta_{3} + \)\(70\!\cdots\!72\)\( \beta_{4} - \)\(64\!\cdots\!74\)\( \beta_{5}) q^{58} +(-\)\(49\!\cdots\!08\)\( - \)\(33\!\cdots\!64\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2} + \)\(43\!\cdots\!16\)\( \beta_{3} - \)\(34\!\cdots\!56\)\( \beta_{4} - \)\(27\!\cdots\!44\)\( \beta_{5}) q^{59} +(-\)\(56\!\cdots\!00\)\( + \)\(79\!\cdots\!22\)\( \beta_{1} - \)\(32\!\cdots\!94\)\( \beta_{2} - \)\(16\!\cdots\!56\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(15\!\cdots\!20\)\( \beta_{5}) q^{60} +(\)\(33\!\cdots\!70\)\( - \)\(15\!\cdots\!71\)\( \beta_{1} + \)\(42\!\cdots\!74\)\( \beta_{2} + \)\(55\!\cdots\!00\)\( \beta_{3} + \)\(25\!\cdots\!27\)\( \beta_{4} - \)\(96\!\cdots\!86\)\( \beta_{5}) q^{61} +(\)\(10\!\cdots\!76\)\( - \)\(65\!\cdots\!28\)\( \beta_{1} + \)\(59\!\cdots\!96\)\( \beta_{2} - \)\(43\!\cdots\!96\)\( \beta_{3} - \)\(88\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!34\)\( \beta_{5}) q^{62} +(\)\(25\!\cdots\!52\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} + \)\(26\!\cdots\!20\)\( \beta_{2} - \)\(46\!\cdots\!35\)\( \beta_{3} - \)\(17\!\cdots\!79\)\( \beta_{4} + \)\(95\!\cdots\!62\)\( \beta_{5}) q^{63} +(\)\(75\!\cdots\!08\)\( - \)\(25\!\cdots\!92\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(32\!\cdots\!52\)\( \beta_{3} - \)\(48\!\cdots\!40\)\( \beta_{4} - \)\(84\!\cdots\!64\)\( \beta_{5}) q^{64} +(\)\(80\!\cdots\!00\)\( - \)\(15\!\cdots\!24\)\( \beta_{1} + \)\(44\!\cdots\!68\)\( \beta_{2} - \)\(24\!\cdots\!38\)\( \beta_{3} + \)\(10\!\cdots\!90\)\( \beta_{4} + \)\(14\!\cdots\!20\)\( \beta_{5}) q^{65} +(-\)\(78\!\cdots\!24\)\( - \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(76\!\cdots\!84\)\( \beta_{2} - \)\(15\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!72\)\( \beta_{4} - \)\(71\!\cdots\!80\)\( \beta_{5}) q^{66} +(-\)\(59\!\cdots\!64\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} - \)\(13\!\cdots\!96\)\( \beta_{2} + \)\(48\!\cdots\!04\)\( \beta_{3} + \)\(26\!\cdots\!88\)\( \beta_{4} + \)\(26\!\cdots\!88\)\( \beta_{5}) q^{67} +(\)\(30\!\cdots\!52\)\( - \)\(24\!\cdots\!54\)\( \beta_{1} - \)\(27\!\cdots\!06\)\( \beta_{2} + \)\(16\!\cdots\!84\)\( \beta_{3} - \)\(76\!\cdots\!36\)\( \beta_{4} - \)\(70\!\cdots\!60\)\( \beta_{5}) q^{68} +(\)\(20\!\cdots\!28\)\( + \)\(54\!\cdots\!60\)\( \beta_{1} - \)\(31\!\cdots\!12\)\( \beta_{2} - \)\(42\!\cdots\!86\)\( \beta_{3} - \)\(93\!\cdots\!34\)\( \beta_{4} - \)\(29\!\cdots\!96\)\( \beta_{5}) q^{69} +(-\)\(57\!\cdots\!00\)\( + \)\(38\!\cdots\!48\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} - \)\(19\!\cdots\!64\)\( \beta_{3} + \)\(78\!\cdots\!40\)\( \beta_{4} + \)\(21\!\cdots\!20\)\( \beta_{5}) q^{70} +(-\)\(85\!\cdots\!88\)\( + \)\(49\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(34\!\cdots\!38\)\( \beta_{3} + \)\(29\!\cdots\!58\)\( \beta_{4} - \)\(12\!\cdots\!20\)\( \beta_{5}) q^{71} +(-\)\(15\!\cdots\!28\)\( + \)\(44\!\cdots\!02\)\( \beta_{1} - \)\(47\!\cdots\!48\)\( \beta_{2} - \)\(15\!\cdots\!88\)\( \beta_{3} + \)\(21\!\cdots\!34\)\( \beta_{4} + \)\(25\!\cdots\!09\)\( \beta_{5}) q^{72} +(-\)\(32\!\cdots\!54\)\( - \)\(39\!\cdots\!86\)\( \beta_{1} - \)\(19\!\cdots\!24\)\( \beta_{2} + \)\(13\!\cdots\!16\)\( \beta_{3} - \)\(83\!\cdots\!42\)\( \beta_{4} - \)\(10\!\cdots\!36\)\( \beta_{5}) q^{73} +(\)\(95\!\cdots\!58\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + \)\(62\!\cdots\!08\)\( \beta_{2} - \)\(91\!\cdots\!48\)\( \beta_{3} - \)\(99\!\cdots\!12\)\( \beta_{4} + \)\(98\!\cdots\!72\)\( \beta_{5}) q^{74} +(-\)\(12\!\cdots\!25\)\( + \)\(28\!\cdots\!50\)\( \beta_{1} - \)\(49\!\cdots\!00\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} - \)\(31\!\cdots\!50\)\( \beta_{4} + \)\(57\!\cdots\!00\)\( \beta_{5}) q^{75} +(-\)\(39\!\cdots\!92\)\( + \)\(49\!\cdots\!56\)\( \beta_{1} - \)\(26\!\cdots\!48\)\( \beta_{2} - \)\(94\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4} - \)\(93\!\cdots\!56\)\( \beta_{5}) q^{76} +(\)\(31\!\cdots\!16\)\( + \)\(99\!\cdots\!16\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} - \)\(11\!\cdots\!72\)\( \beta_{3} + \)\(19\!\cdots\!36\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5}) q^{77} +(-\)\(30\!\cdots\!14\)\( + \)\(28\!\cdots\!14\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} + \)\(70\!\cdots\!76\)\( \beta_{3} + \)\(23\!\cdots\!48\)\( \beta_{4} - \)\(25\!\cdots\!16\)\( \beta_{5}) q^{78} +(\)\(80\!\cdots\!60\)\( - \)\(43\!\cdots\!12\)\( \beta_{1} - \)\(56\!\cdots\!16\)\( \beta_{2} + \)\(21\!\cdots\!85\)\( \beta_{3} + \)\(78\!\cdots\!17\)\( \beta_{4} + \)\(51\!\cdots\!54\)\( \beta_{5}) q^{79} +(\)\(43\!\cdots\!00\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(44\!\cdots\!56\)\( \beta_{3} - \)\(30\!\cdots\!20\)\( \beta_{4} - \)\(14\!\cdots\!60\)\( \beta_{5}) q^{80} +\)\(62\!\cdots\!01\)\( q^{81} +(-\)\(50\!\cdots\!94\)\( - \)\(20\!\cdots\!86\)\( \beta_{1} - \)\(22\!\cdots\!76\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3} + \)\(28\!\cdots\!76\)\( \beta_{4} + \)\(15\!\cdots\!32\)\( \beta_{5}) q^{82} +(\)\(38\!\cdots\!28\)\( - \)\(11\!\cdots\!50\)\( \beta_{1} + \)\(27\!\cdots\!08\)\( \beta_{2} + \)\(16\!\cdots\!78\)\( \beta_{3} + \)\(33\!\cdots\!64\)\( \beta_{4} + \)\(42\!\cdots\!52\)\( \beta_{5}) q^{83} +(-\)\(59\!\cdots\!08\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(35\!\cdots\!40\)\( \beta_{2} - \)\(57\!\cdots\!76\)\( \beta_{3} - \)\(18\!\cdots\!08\)\( \beta_{4} - \)\(92\!\cdots\!64\)\( \beta_{5}) q^{84} +(-\)\(25\!\cdots\!00\)\( - \)\(24\!\cdots\!42\)\( \beta_{1} - \)\(32\!\cdots\!76\)\( \beta_{2} + \)\(41\!\cdots\!86\)\( \beta_{3} - \)\(16\!\cdots\!00\)\( \beta_{4} - \)\(77\!\cdots\!00\)\( \beta_{5}) q^{85} +(\)\(38\!\cdots\!64\)\( + \)\(77\!\cdots\!40\)\( \beta_{1} - \)\(78\!\cdots\!28\)\( \beta_{2} - \)\(74\!\cdots\!40\)\( \beta_{3} + \)\(33\!\cdots\!52\)\( \beta_{4} - \)\(15\!\cdots\!96\)\( \beta_{5}) q^{86} +(-\)\(11\!\cdots\!62\)\( - \)\(34\!\cdots\!07\)\( \beta_{1} - \)\(50\!\cdots\!34\)\( \beta_{2} + \)\(15\!\cdots\!55\)\( \beta_{3} - \)\(16\!\cdots\!42\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5}) q^{87} +(\)\(88\!\cdots\!68\)\( + \)\(52\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!40\)\( \beta_{2} + \)\(80\!\cdots\!84\)\( \beta_{3} - \)\(21\!\cdots\!40\)\( \beta_{4} + \)\(18\!\cdots\!92\)\( \beta_{5}) q^{88} +(\)\(73\!\cdots\!18\)\( + \)\(78\!\cdots\!04\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2} + \)\(12\!\cdots\!88\)\( \beta_{3} - \)\(32\!\cdots\!72\)\( \beta_{4} + \)\(40\!\cdots\!80\)\( \beta_{5}) q^{89} +(\)\(22\!\cdots\!50\)\( + \)\(10\!\cdots\!22\)\( \beta_{1} - \)\(18\!\cdots\!44\)\( \beta_{2} - \)\(18\!\cdots\!56\)\( \beta_{3} + \)\(77\!\cdots\!40\)\( \beta_{4} + \)\(23\!\cdots\!70\)\( \beta_{5}) q^{90} +(\)\(17\!\cdots\!16\)\( + \)\(40\!\cdots\!68\)\( \beta_{1} + \)\(90\!\cdots\!12\)\( \beta_{2} + \)\(17\!\cdots\!06\)\( \beta_{3} + \)\(14\!\cdots\!62\)\( \beta_{4} - \)\(15\!\cdots\!88\)\( \beta_{5}) q^{91} +(\)\(58\!\cdots\!28\)\( + \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(35\!\cdots\!80\)\( \beta_{2} - \)\(18\!\cdots\!52\)\( \beta_{3} - \)\(29\!\cdots\!48\)\( \beta_{4} - \)\(31\!\cdots\!32\)\( \beta_{5}) q^{92} +(\)\(22\!\cdots\!28\)\( - \)\(61\!\cdots\!08\)\( \beta_{1} + \)\(41\!\cdots\!08\)\( \beta_{2} - \)\(25\!\cdots\!23\)\( \beta_{3} + \)\(89\!\cdots\!21\)\( \beta_{4} - \)\(21\!\cdots\!82\)\( \beta_{5}) q^{93} +(\)\(35\!\cdots\!52\)\( + \)\(39\!\cdots\!96\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3} + \)\(10\!\cdots\!56\)\( \beta_{4} + \)\(59\!\cdots\!76\)\( \beta_{5}) q^{94} +(-\)\(53\!\cdots\!00\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} - \)\(61\!\cdots\!60\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!60\)\( \beta_{4} + \)\(19\!\cdots\!80\)\( \beta_{5}) q^{95} +(\)\(93\!\cdots\!64\)\( - \)\(17\!\cdots\!04\)\( \beta_{1} + \)\(47\!\cdots\!12\)\( \beta_{2} + \)\(27\!\cdots\!60\)\( \beta_{3} - \)\(84\!\cdots\!64\)\( \beta_{4} - \)\(88\!\cdots\!28\)\( \beta_{5}) q^{96} +(\)\(16\!\cdots\!66\)\( - \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(19\!\cdots\!04\)\( \beta_{2} - \)\(96\!\cdots\!48\)\( \beta_{3} - \)\(50\!\cdots\!20\)\( \beta_{4} - \)\(20\!\cdots\!24\)\( \beta_{5}) q^{97} +(\)\(43\!\cdots\!21\)\( - \)\(21\!\cdots\!19\)\( \beta_{1} + \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(21\!\cdots\!72\)\( \beta_{4} - \)\(15\!\cdots\!64\)\( \beta_{5}) q^{98} +(\)\(92\!\cdots\!56\)\( + \)\(12\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!56\)\( \beta_{2} - \)\(13\!\cdots\!62\)\( \beta_{3} + \)\(93\!\cdots\!56\)\( \beta_{4} + \)\(11\!\cdots\!96\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 72903656826q^{2} - 300189270593998242q^{3} + \)\(10\!\cdots\!32\)\(q^{4} + \)\(10\!\cdots\!00\)\(q^{5} + \)\(36\!\cdots\!82\)\(q^{6} + \)\(59\!\cdots\!88\)\(q^{7} - \)\(36\!\cdots\!32\)\(q^{8} + \)\(15\!\cdots\!94\)\(q^{9} + O(q^{10}) \) \( 6q - 72903656826q^{2} - 300189270593998242q^{3} + \)\(10\!\cdots\!32\)\(q^{4} + \)\(10\!\cdots\!00\)\(q^{5} + \)\(36\!\cdots\!82\)\(q^{6} + \)\(59\!\cdots\!88\)\(q^{7} - \)\(36\!\cdots\!32\)\(q^{8} + \)\(15\!\cdots\!94\)\(q^{9} + \)\(54\!\cdots\!00\)\(q^{10} + \)\(22\!\cdots\!64\)\(q^{11} - \)\(51\!\cdots\!24\)\(q^{12} - \)\(37\!\cdots\!76\)\(q^{13} - \)\(32\!\cdots\!32\)\(q^{14} - \)\(50\!\cdots\!00\)\(q^{15} + \)\(21\!\cdots\!80\)\(q^{16} + \)\(29\!\cdots\!96\)\(q^{17} - \)\(18\!\cdots\!74\)\(q^{18} - \)\(34\!\cdots\!72\)\(q^{19} + \)\(67\!\cdots\!00\)\(q^{20} - \)\(29\!\cdots\!16\)\(q^{21} + \)\(94\!\cdots\!92\)\(q^{22} - \)\(24\!\cdots\!24\)\(q^{23} + \)\(18\!\cdots\!24\)\(q^{24} + \)\(14\!\cdots\!50\)\(q^{25} + \)\(36\!\cdots\!12\)\(q^{26} - \)\(75\!\cdots\!58\)\(q^{27} + \)\(71\!\cdots\!64\)\(q^{28} + \)\(14\!\cdots\!96\)\(q^{29} - \)\(27\!\cdots\!00\)\(q^{30} - \)\(27\!\cdots\!24\)\(q^{31} - \)\(11\!\cdots\!12\)\(q^{32} - \)\(11\!\cdots\!48\)\(q^{33} - \)\(14\!\cdots\!16\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} + \)\(25\!\cdots\!68\)\(q^{36} - \)\(35\!\cdots\!52\)\(q^{37} - \)\(40\!\cdots\!84\)\(q^{38} + \)\(18\!\cdots\!32\)\(q^{39} - \)\(24\!\cdots\!00\)\(q^{40} + \)\(47\!\cdots\!16\)\(q^{41} + \)\(16\!\cdots\!24\)\(q^{42} + \)\(11\!\cdots\!80\)\(q^{43} + \)\(74\!\cdots\!96\)\(q^{44} + \)\(25\!\cdots\!00\)\(q^{45} - \)\(22\!\cdots\!84\)\(q^{46} - \)\(74\!\cdots\!60\)\(q^{47} - \)\(10\!\cdots\!60\)\(q^{48} - \)\(68\!\cdots\!90\)\(q^{49} - \)\(15\!\cdots\!50\)\(q^{50} - \)\(14\!\cdots\!72\)\(q^{51} - \)\(92\!\cdots\!44\)\(q^{52} + \)\(54\!\cdots\!16\)\(q^{53} + \)\(91\!\cdots\!18\)\(q^{54} - \)\(23\!\cdots\!00\)\(q^{55} + \)\(52\!\cdots\!60\)\(q^{56} + \)\(17\!\cdots\!04\)\(q^{57} - \)\(68\!\cdots\!08\)\(q^{58} - \)\(29\!\cdots\!48\)\(q^{59} - \)\(33\!\cdots\!00\)\(q^{60} + \)\(20\!\cdots\!20\)\(q^{61} + \)\(61\!\cdots\!56\)\(q^{62} + \)\(15\!\cdots\!12\)\(q^{63} + \)\(45\!\cdots\!48\)\(q^{64} + \)\(48\!\cdots\!00\)\(q^{65} - \)\(47\!\cdots\!44\)\(q^{66} - \)\(35\!\cdots\!84\)\(q^{67} + \)\(18\!\cdots\!12\)\(q^{68} + \)\(12\!\cdots\!68\)\(q^{69} - \)\(34\!\cdots\!00\)\(q^{70} - \)\(51\!\cdots\!28\)\(q^{71} - \)\(91\!\cdots\!68\)\(q^{72} - \)\(19\!\cdots\!24\)\(q^{73} + \)\(57\!\cdots\!48\)\(q^{74} - \)\(73\!\cdots\!50\)\(q^{75} - \)\(23\!\cdots\!52\)\(q^{76} + \)\(18\!\cdots\!96\)\(q^{77} - \)\(18\!\cdots\!84\)\(q^{78} + \)\(48\!\cdots\!60\)\(q^{79} + \)\(26\!\cdots\!00\)\(q^{80} + \)\(37\!\cdots\!06\)\(q^{81} - \)\(30\!\cdots\!64\)\(q^{82} + \)\(23\!\cdots\!68\)\(q^{83} - \)\(35\!\cdots\!48\)\(q^{84} - \)\(15\!\cdots\!00\)\(q^{85} + \)\(23\!\cdots\!84\)\(q^{86} - \)\(70\!\cdots\!72\)\(q^{87} + \)\(52\!\cdots\!08\)\(q^{88} + \)\(43\!\cdots\!08\)\(q^{89} + \)\(13\!\cdots\!00\)\(q^{90} + \)\(10\!\cdots\!96\)\(q^{91} + \)\(34\!\cdots\!68\)\(q^{92} + \)\(13\!\cdots\!68\)\(q^{93} + \)\(21\!\cdots\!12\)\(q^{94} - \)\(32\!\cdots\!00\)\(q^{95} + \)\(56\!\cdots\!84\)\(q^{96} + \)\(97\!\cdots\!96\)\(q^{97} + \)\(26\!\cdots\!26\)\(q^{98} + \)\(55\!\cdots\!36\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 327712249929578892882 x^{4} - 247064970776889197907293537206 x^{3} + 25021443441555202497465421340907628615965 x^{2} + 14077016837018585361473224499724917623325562275625 x - 279259064416593965218586682187488478036140359478345093745500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 3 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} - 40711045962 \nu - 3932546999134591191657 \)
\(\beta_{3}\)\(=\)\((\)\(-2647338101397 \nu^{5} + 15585983560336005406296 \nu^{4} + 665943825059194354529114527395714 \nu^{3} - 2709584662600088683935949720665253390974588 \nu^{2} - 28336308661367042176479914042787806437245868632255285 \nu + 63322920809782082624626193833747499036625134066418078569730780\)\()/ \)\(13\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-104778897869878551 \nu^{5} - 723744240429543687924148152 \nu^{4} + 34403186840616316215985349942118946422 \nu^{3} + 190818785626362696757958152258341409915806056556 \nu^{2} - 2182591086580538223721931001219022522622317728806750482935 \nu - 5417817678162782131119470382015236091722504014174630329062729721420\)\()/ \)\(62\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(25300363094553231 \nu^{5} + 2119930948658032782322797432 \nu^{4} + 6541929024458501170586700888486179418 \nu^{3} - 556218784724563435725073421322083729718007982796 \nu^{2} - 2487365207004615067701905635488661412782409232072541917585 \nu + 18013763051749487640773267896799037138147661392064065749480539920300\)\()/ \)\(12\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 6785174327 \beta_{1} + 3932546999154946714638\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(10241 \beta_{5} + 8666 \beta_{4} - 6340812 \beta_{3} + 17570586570 \beta_{2} + 6300306283451987631960 \beta_{1} + 26683016950082802351171263312988\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(184385957659079 \beta_{5} - 147886657045898 \beta_{4} + 145213590970725740 \beta_{3} + 4318306282598743033064 \beta_{2} + 55109579314199972427421588418346 \beta_{1} + 12388125284412213438358600535730522935095644\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(13220994683194238297297297 \beta_{5} + 8503776992325990821009018 \beta_{4} - 10873323761890330947413991308 \beta_{3} + 30390299029534496433664184974350 \beta_{2} + 5697025877958147537301573458895105056666306 \beta_{1} + 54180252698368277670536905870375644096963350849575086\)\()/972\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.38952e10
−1.03374e10
−3.99343e9
3.35353e9
9.35614e9
1.55163e10
−9.55218e10 −5.00315e16 6.76322e21 1.07436e25 4.77910e27 6.00629e29 −4.20491e32 2.50316e33 −1.02625e36
1.2 −7.41748e10 −5.00315e16 3.14071e21 −1.28474e25 3.71108e27 −1.80308e28 −5.78216e31 2.50316e33 9.52956e35
1.3 −3.61112e10 −5.00315e16 −1.05717e21 4.57686e24 1.80670e27 −2.20036e29 1.23441e32 2.50316e33 −1.65276e35
1.4 7.97054e9 −5.00315e16 −2.29765e21 −4.57364e24 −3.98779e26 9.29524e29 −3.71335e31 2.50316e33 −3.64544e34
1.5 4.39862e10 −5.00315e16 −4.26395e20 4.61068e24 −2.20070e27 −1.72926e30 −1.22615e32 2.50316e33 2.02806e35
1.6 8.09473e10 −5.00315e16 4.19128e21 7.59856e24 −4.04992e27 1.03665e30 1.48142e32 2.50316e33 6.15083e35
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 3.72.a.b 6
3.b odd 2 1 9.72.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.72.a.b 6 1.a even 1 1 trivial
9.72.a.c 6 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 72903656826 T_{2}^{5} - \)\(95\!\cdots\!72\)\( T_{2}^{4} - \)\(59\!\cdots\!88\)\( T_{2}^{3} + \)\(20\!\cdots\!36\)\( T_{2}^{2} + \)\(79\!\cdots\!44\)\( T_{2} - \)\(72\!\cdots\!72\)\( \) acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 72903656826 T + \)\(45\!\cdots\!16\)\( T^{2} + \)\(26\!\cdots\!52\)\( T^{3} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(66\!\cdots\!12\)\( T^{5} + \)\(31\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!76\)\( T^{7} + \)\(75\!\cdots\!88\)\( T^{8} + \)\(35\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!64\)\( T^{12} \)
$3$ \( ( 1 + 50031545098999707 T )^{6} \)
$5$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} - \)\(58\!\cdots\!25\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{5} - \)\(17\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} - \)\(10\!\cdots\!25\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!50\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(57\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(59\!\cdots\!88\)\( T + \)\(35\!\cdots\!46\)\( T^{2} - \)\(43\!\cdots\!36\)\( T^{3} + \)\(48\!\cdots\!11\)\( T^{4} + \)\(15\!\cdots\!16\)\( T^{5} + \)\(46\!\cdots\!52\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} + \)\(49\!\cdots\!39\)\( T^{8} - \)\(43\!\cdots\!52\)\( T^{9} + \)\(36\!\cdots\!46\)\( T^{10} - \)\(61\!\cdots\!84\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 - \)\(22\!\cdots\!64\)\( T + \)\(52\!\cdots\!54\)\( T^{2} - \)\(83\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!63\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(85\!\cdots\!23\)\( T^{8} - \)\(54\!\cdots\!88\)\( T^{9} + \)\(29\!\cdots\!14\)\( T^{10} - \)\(10\!\cdots\!64\)\( T^{11} + \)\(42\!\cdots\!61\)\( T^{12} \)
$13$ \( 1 + \)\(37\!\cdots\!76\)\( T + \)\(45\!\cdots\!90\)\( T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} + \)\(28\!\cdots\!96\)\( T^{5} + \)\(15\!\cdots\!44\)\( T^{6} + \)\(34\!\cdots\!52\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} + \)\(25\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!90\)\( T^{10} + \)\(10\!\cdots\!32\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \)
$17$ \( 1 - \)\(29\!\cdots\!96\)\( T + \)\(84\!\cdots\!38\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!55\)\( T^{4} - \)\(38\!\cdots\!76\)\( T^{5} + \)\(96\!\cdots\!56\)\( T^{6} - \)\(89\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!95\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(23\!\cdots\!98\)\( T^{10} - \)\(19\!\cdots\!28\)\( T^{11} + \)\(14\!\cdots\!69\)\( T^{12} \)
$19$ \( 1 + \)\(34\!\cdots\!72\)\( T + \)\(17\!\cdots\!82\)\( T^{2} + \)\(49\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!23\)\( T^{4} + \)\(41\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!88\)\( T^{6} + \)\(25\!\cdots\!32\)\( T^{7} + \)\(73\!\cdots\!03\)\( T^{8} + \)\(11\!\cdots\!56\)\( T^{9} + \)\(25\!\cdots\!22\)\( T^{10} + \)\(30\!\cdots\!28\)\( T^{11} + \)\(56\!\cdots\!81\)\( T^{12} \)
$23$ \( 1 + \)\(24\!\cdots\!24\)\( T + \)\(20\!\cdots\!74\)\( T^{2} + \)\(88\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} + \)\(99\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!04\)\( T^{7} + \)\(44\!\cdots\!19\)\( T^{8} + \)\(99\!\cdots\!76\)\( T^{9} + \)\(11\!\cdots\!34\)\( T^{10} + \)\(62\!\cdots\!68\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$29$ \( 1 - \)\(14\!\cdots\!96\)\( T + \)\(37\!\cdots\!46\)\( T^{2} - \)\(38\!\cdots\!88\)\( T^{3} + \)\(60\!\cdots\!67\)\( T^{4} - \)\(47\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!92\)\( T^{6} - \)\(32\!\cdots\!04\)\( T^{7} + \)\(27\!\cdots\!47\)\( T^{8} - \)\(11\!\cdots\!32\)\( T^{9} + \)\(79\!\cdots\!26\)\( T^{10} - \)\(19\!\cdots\!04\)\( T^{11} + \)\(95\!\cdots\!21\)\( T^{12} \)
$31$ \( 1 + \)\(27\!\cdots\!24\)\( T + \)\(59\!\cdots\!78\)\( T^{2} + \)\(90\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(12\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!92\)\( T^{6} + \)\(99\!\cdots\!36\)\( T^{7} + \)\(71\!\cdots\!03\)\( T^{8} + \)\(41\!\cdots\!12\)\( T^{9} + \)\(20\!\cdots\!38\)\( T^{10} + \)\(73\!\cdots\!24\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} \)
$37$ \( 1 + \)\(35\!\cdots\!52\)\( T + \)\(80\!\cdots\!06\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{5} + \)\(58\!\cdots\!92\)\( T^{6} + \)\(31\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!19\)\( T^{8} + \)\(15\!\cdots\!08\)\( T^{9} + \)\(18\!\cdots\!66\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{11} + \)\(11\!\cdots\!09\)\( T^{12} \)
$41$ \( 1 - \)\(47\!\cdots\!16\)\( T + \)\(86\!\cdots\!78\)\( T^{2} - \)\(93\!\cdots\!48\)\( T^{3} + \)\(50\!\cdots\!83\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(19\!\cdots\!12\)\( T^{6} - \)\(14\!\cdots\!84\)\( T^{7} + \)\(51\!\cdots\!23\)\( T^{8} - \)\(31\!\cdots\!08\)\( T^{9} + \)\(92\!\cdots\!58\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$43$ \( 1 - \)\(11\!\cdots\!80\)\( T + \)\(30\!\cdots\!90\)\( T^{2} - \)\(23\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!03\)\( T^{8} - \)\(19\!\cdots\!80\)\( T^{9} + \)\(24\!\cdots\!90\)\( T^{10} - \)\(88\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + \)\(74\!\cdots\!60\)\( T + \)\(23\!\cdots\!70\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!27\)\( T^{4} + \)\(19\!\cdots\!80\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(70\!\cdots\!43\)\( T^{8} + \)\(28\!\cdots\!60\)\( T^{9} + \)\(17\!\cdots\!70\)\( T^{10} + \)\(29\!\cdots\!80\)\( T^{11} + \)\(20\!\cdots\!29\)\( T^{12} \)
$53$ \( 1 - \)\(54\!\cdots\!16\)\( T + \)\(17\!\cdots\!34\)\( T^{2} - \)\(44\!\cdots\!08\)\( T^{3} + \)\(89\!\cdots\!11\)\( T^{4} - \)\(15\!\cdots\!08\)\( T^{5} + \)\(25\!\cdots\!68\)\( T^{6} - \)\(41\!\cdots\!76\)\( T^{7} + \)\(62\!\cdots\!99\)\( T^{8} - \)\(82\!\cdots\!84\)\( T^{9} + \)\(88\!\cdots\!54\)\( T^{10} - \)\(71\!\cdots\!12\)\( T^{11} + \)\(34\!\cdots\!29\)\( T^{12} \)
$59$ \( 1 + \)\(29\!\cdots\!48\)\( T + \)\(58\!\cdots\!62\)\( T^{2} + \)\(83\!\cdots\!36\)\( T^{3} + \)\(95\!\cdots\!43\)\( T^{4} + \)\(90\!\cdots\!32\)\( T^{5} + \)\(72\!\cdots\!88\)\( T^{6} + \)\(48\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!83\)\( T^{8} + \)\(12\!\cdots\!44\)\( T^{9} + \)\(48\!\cdots\!82\)\( T^{10} + \)\(13\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!41\)\( T^{12} \)
$61$ \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(27\!\cdots\!14\)\( T^{2} - \)\(55\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!95\)\( T^{4} - \)\(62\!\cdots\!40\)\( T^{5} + \)\(25\!\cdots\!60\)\( T^{6} - \)\(35\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{9} + \)\(30\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(35\!\cdots\!61\)\( T^{12} \)
$67$ \( 1 + \)\(35\!\cdots\!84\)\( T + \)\(16\!\cdots\!38\)\( T^{2} + \)\(66\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!55\)\( T^{4} + \)\(53\!\cdots\!24\)\( T^{5} + \)\(62\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!92\)\( T^{7} + \)\(25\!\cdots\!95\)\( T^{8} + \)\(59\!\cdots\!80\)\( T^{9} + \)\(67\!\cdots\!98\)\( T^{10} + \)\(64\!\cdots\!12\)\( T^{11} + \)\(80\!\cdots\!69\)\( T^{12} \)
$71$ \( 1 + \)\(51\!\cdots\!28\)\( T + \)\(36\!\cdots\!86\)\( T^{2} + \)\(67\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(54\!\cdots\!08\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!68\)\( T^{7} + \)\(90\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!66\)\( T^{10} + \)\(80\!\cdots\!28\)\( T^{11} + \)\(43\!\cdots\!21\)\( T^{12} \)
$73$ \( 1 + \)\(19\!\cdots\!24\)\( T + \)\(50\!\cdots\!34\)\( T^{2} + \)\(76\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!31\)\( T^{4} + \)\(24\!\cdots\!12\)\( T^{5} + \)\(41\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!24\)\( T^{7} + \)\(70\!\cdots\!99\)\( T^{8} + \)\(59\!\cdots\!56\)\( T^{9} + \)\(77\!\cdots\!94\)\( T^{10} + \)\(59\!\cdots\!68\)\( T^{11} + \)\(59\!\cdots\!89\)\( T^{12} \)
$79$ \( 1 - \)\(48\!\cdots\!60\)\( T + \)\(39\!\cdots\!74\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!80\)\( T^{6} - \)\(79\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!15\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!94\)\( T^{10} - \)\(22\!\cdots\!40\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \)
$83$ \( 1 - \)\(23\!\cdots\!68\)\( T + \)\(83\!\cdots\!14\)\( T^{2} - \)\(13\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!67\)\( T^{4} - \)\(41\!\cdots\!16\)\( T^{5} + \)\(70\!\cdots\!44\)\( T^{6} - \)\(74\!\cdots\!72\)\( T^{7} + \)\(99\!\cdots\!63\)\( T^{8} - \)\(78\!\cdots\!72\)\( T^{9} + \)\(87\!\cdots\!94\)\( T^{10} - \)\(43\!\cdots\!76\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \)
$89$ \( 1 - \)\(43\!\cdots\!08\)\( T + \)\(18\!\cdots\!82\)\( T^{2} - \)\(46\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!63\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!08\)\( T^{6} - \)\(54\!\cdots\!08\)\( T^{7} + \)\(73\!\cdots\!23\)\( T^{8} - \)\(77\!\cdots\!84\)\( T^{9} + \)\(76\!\cdots\!62\)\( T^{10} - \)\(47\!\cdots\!92\)\( T^{11} + \)\(27\!\cdots\!61\)\( T^{12} \)
$97$ \( 1 - \)\(97\!\cdots\!96\)\( T + \)\(10\!\cdots\!58\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!56\)\( T^{6} - \)\(16\!\cdots\!28\)\( T^{7} + \)\(47\!\cdots\!95\)\( T^{8} - \)\(93\!\cdots\!20\)\( T^{9} + \)\(18\!\cdots\!98\)\( T^{10} - \)\(19\!\cdots\!28\)\( T^{11} + \)\(23\!\cdots\!29\)\( T^{12} \)
show more
show less