Properties

Label 3.70.a.b
Level 3
Weight 70
Character orbit 3.a
Self dual yes
Analytic conductor 90.454
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 70 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(90.4544859877\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 8785036246714673132 x^{4} - 489946547045174855374556256 x^{3} + 19187418488273751521495823450187335040 x^{2} + 711105729064303642253694769744270208387385600 x - 4081433259823761741733687714132191037604592129279360000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{33}\cdot 5^{5}\cdot 7^{3}\cdot 11\cdot 17\cdot 23^{2} \)
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.

\(f(q)\) \(=\) \( q +(3283493823 - \beta_{1}) q^{2} +16677181699666569 q^{3} +(\)\(36\!\cdots\!78\)\( - 5061181993 \beta_{1} + \beta_{2}) q^{4} +(\)\(10\!\cdots\!06\)\( + 4262517026085 \beta_{1} + 11 \beta_{2} - \beta_{3}) q^{5} +(\)\(54\!\cdots\!87\)\( - 16677181699666569 \beta_{1}) q^{6} +(\)\(79\!\cdots\!56\)\( + 526765162606293393 \beta_{1} - 9184931 \beta_{2} - 66381 \beta_{3} - \beta_{4}) q^{7} +(\)\(40\!\cdots\!24\)\( - \)\(26\!\cdots\!07\)\( \beta_{1} + 4891590906 \beta_{2} + 5970698 \beta_{3} - 73 \beta_{4} - \beta_{5}) q^{8} +\)\(27\!\cdots\!61\)\( q^{9} +O(q^{10})\) \( q +(3283493823 - \beta_{1}) q^{2} +16677181699666569 q^{3} +(\)\(36\!\cdots\!78\)\( - 5061181993 \beta_{1} + \beta_{2}) q^{4} +(\)\(10\!\cdots\!06\)\( + 4262517026085 \beta_{1} + 11 \beta_{2} - \beta_{3}) q^{5} +(\)\(54\!\cdots\!87\)\( - 16677181699666569 \beta_{1}) q^{6} +(\)\(79\!\cdots\!56\)\( + 526765162606293393 \beta_{1} - 9184931 \beta_{2} - 66381 \beta_{3} - \beta_{4}) q^{7} +(\)\(40\!\cdots\!24\)\( - \)\(26\!\cdots\!07\)\( \beta_{1} + 4891590906 \beta_{2} + 5970698 \beta_{3} - 73 \beta_{4} - \beta_{5}) q^{8} +\)\(27\!\cdots\!61\)\( q^{9} +(-\)\(37\!\cdots\!66\)\( - \)\(10\!\cdots\!46\)\( \beta_{1} - 33003957661792 \beta_{2} - 5837903664 \beta_{3} - 257848 \beta_{4} + 22392 \beta_{5}) q^{10} +(\)\(23\!\cdots\!04\)\( - \)\(64\!\cdots\!22\)\( \beta_{1} - 23595238915834 \beta_{2} + 182284172574 \beta_{3} - 1817624 \beta_{4} - 178688 \beta_{5}) q^{11} +(\)\(61\!\cdots\!82\)\( - \)\(84\!\cdots\!17\)\( \beta_{1} + 16677181699666569 \beta_{2}) q^{12} +(\)\(22\!\cdots\!22\)\( - \)\(39\!\cdots\!06\)\( \beta_{1} - 57930433476915488 \beta_{2} + 72155909361642 \beta_{3} - 1746035729 \beta_{4} - 109587456 \beta_{5}) q^{13} +(-\)\(47\!\cdots\!68\)\( - \)\(26\!\cdots\!80\)\( \beta_{1} - 678969693124927456 \beta_{2} - 1676267483300592 \beta_{3} - 21751713176 \beta_{4} + 1541378776 \beta_{5}) q^{14} +(\)\(17\!\cdots\!14\)\( + \)\(71\!\cdots\!65\)\( \beta_{1} + 183448998696332259 \beta_{2} - 16677181699666569 \beta_{3}) q^{15} +(\)\(49\!\cdots\!80\)\( - \)\(39\!\cdots\!30\)\( \beta_{1} - 15201840963241298172 \beta_{2} + 9525615067635396 \beta_{3} + 799998214966 \beta_{4} - 138910217850 \beta_{5}) q^{16} +(-\)\(45\!\cdots\!98\)\( - \)\(49\!\cdots\!62\)\( \beta_{1} + \)\(36\!\cdots\!62\)\( \beta_{2} - 161380231713275434 \beta_{3} + 71466188898 \beta_{4} + 1072826199552 \beta_{5}) q^{17} +(\)\(91\!\cdots\!03\)\( - \)\(27\!\cdots\!61\)\( \beta_{1}) q^{18} +(\)\(23\!\cdots\!52\)\( + \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(17\!\cdots\!30\)\( \beta_{2} - 52291956889680239418 \beta_{3} + 261165360652514 \beta_{4} - 46550242349568 \beta_{5}) q^{19} +(\)\(23\!\cdots\!04\)\( + \)\(16\!\cdots\!94\)\( \beta_{1} + \)\(19\!\cdots\!18\)\( \beta_{2} - \)\(73\!\cdots\!84\)\( \beta_{3} - 777192358499328 \beta_{4} + 254822040333312 \beta_{5}) q^{20} +(\)\(13\!\cdots\!64\)\( + \)\(87\!\cdots\!17\)\( \beta_{1} - \)\(15\!\cdots\!39\)\( \beta_{2} - \)\(11\!\cdots\!89\)\( \beta_{3} - 16677181699666569 \beta_{4}) q^{21} +(\)\(68\!\cdots\!44\)\( - \)\(23\!\cdots\!88\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(49\!\cdots\!72\)\( \beta_{3} + 41262318672893584 \beta_{4} - 4611928114191888 \beta_{5}) q^{22} +(\)\(36\!\cdots\!68\)\( - \)\(75\!\cdots\!38\)\( \beta_{1} + \)\(11\!\cdots\!98\)\( \beta_{2} + \)\(65\!\cdots\!42\)\( \beta_{3} + 446544990505940582 \beta_{4} + 13806440208128000 \beta_{5}) q^{23} +(\)\(67\!\cdots\!56\)\( - \)\(44\!\cdots\!83\)\( \beta_{1} + \)\(81\!\cdots\!14\)\( \beta_{2} + \)\(99\!\cdots\!62\)\( \beta_{3} - 1217434264075659537 \beta_{4} - 16677181699666569 \beta_{5}) q^{24} +(\)\(52\!\cdots\!91\)\( - \)\(47\!\cdots\!28\)\( \beta_{1} + \)\(18\!\cdots\!28\)\( \beta_{2} + \)\(14\!\cdots\!64\)\( \beta_{3} - 5071357720320646534 \beta_{4} + 286535397163355136 \beta_{5}) q^{25} +(\)\(38\!\cdots\!42\)\( - \)\(15\!\cdots\!10\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(17\!\cdots\!60\)\( \beta_{3} + 14906434176120480624 \beta_{4} - 1868551413515625456 \beta_{5}) q^{26} +\)\(46\!\cdots\!09\)\( q^{27} +(-\)\(37\!\cdots\!08\)\( + \)\(48\!\cdots\!04\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2} - \)\(55\!\cdots\!20\)\( \beta_{3} + 71484758615814773760 \beta_{4} + 44679000501922913280 \beta_{5}) q^{28} +(\)\(64\!\cdots\!34\)\( - \)\(24\!\cdots\!97\)\( \beta_{1} - \)\(10\!\cdots\!71\)\( \beta_{2} + \)\(15\!\cdots\!77\)\( \beta_{3} + \)\(51\!\cdots\!82\)\( \beta_{4} - \)\(20\!\cdots\!60\)\( \beta_{5}) q^{29} +(-\)\(61\!\cdots\!54\)\( - \)\(17\!\cdots\!74\)\( \beta_{1} - \)\(55\!\cdots\!48\)\( \beta_{2} - \)\(97\!\cdots\!16\)\( \beta_{3} - \)\(43\!\cdots\!12\)\( \beta_{4} + \)\(37\!\cdots\!48\)\( \beta_{5}) q^{30} +(-\)\(67\!\cdots\!04\)\( + \)\(21\!\cdots\!61\)\( \beta_{1} - \)\(37\!\cdots\!71\)\( \beta_{2} + \)\(10\!\cdots\!95\)\( \beta_{3} - \)\(31\!\cdots\!97\)\( \beta_{4} - 77148604373737669632 \beta_{5}) q^{31} +(\)\(14\!\cdots\!16\)\( + \)\(10\!\cdots\!56\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(22\!\cdots\!32\)\( \beta_{3} + \)\(52\!\cdots\!56\)\( \beta_{4} - \)\(13\!\cdots\!36\)\( \beta_{5}) q^{32} +(\)\(38\!\cdots\!76\)\( - \)\(10\!\cdots\!18\)\( \beta_{1} - \)\(39\!\cdots\!46\)\( \beta_{2} + \)\(30\!\cdots\!06\)\( \beta_{3} - \)\(30\!\cdots\!56\)\( \beta_{4} - \)\(29\!\cdots\!72\)\( \beta_{5}) q^{33} +(\)\(46\!\cdots\!66\)\( - \)\(21\!\cdots\!82\)\( \beta_{1} + \)\(35\!\cdots\!28\)\( \beta_{2} - \)\(36\!\cdots\!32\)\( \beta_{3} - \)\(82\!\cdots\!56\)\( \beta_{4} + \)\(70\!\cdots\!36\)\( \beta_{5}) q^{34} +(\)\(14\!\cdots\!20\)\( - \)\(32\!\cdots\!14\)\( \beta_{1} + \)\(11\!\cdots\!66\)\( \beta_{2} - \)\(17\!\cdots\!70\)\( \beta_{3} - \)\(66\!\cdots\!52\)\( \beta_{4} + \)\(58\!\cdots\!08\)\( \beta_{5}) q^{35} +(\)\(10\!\cdots\!58\)\( - \)\(14\!\cdots\!73\)\( \beta_{1} + \)\(27\!\cdots\!61\)\( \beta_{2}) q^{36} +(-\)\(17\!\cdots\!54\)\( + \)\(27\!\cdots\!08\)\( \beta_{1} + \)\(23\!\cdots\!46\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!69\)\( \beta_{4} - \)\(25\!\cdots\!76\)\( \beta_{5}) q^{37} +(-\)\(88\!\cdots\!12\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(25\!\cdots\!20\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3} - \)\(12\!\cdots\!76\)\( \beta_{4} + \)\(97\!\cdots\!68\)\( \beta_{5}) q^{38} +(\)\(37\!\cdots\!18\)\( - \)\(66\!\cdots\!14\)\( \beta_{1} - \)\(96\!\cdots\!72\)\( \beta_{2} + \)\(12\!\cdots\!98\)\( \beta_{3} - \)\(29\!\cdots\!01\)\( \beta_{4} - \)\(18\!\cdots\!64\)\( \beta_{5}) q^{39} +(-\)\(13\!\cdots\!52\)\( - \)\(28\!\cdots\!74\)\( \beta_{1} - \)\(19\!\cdots\!56\)\( \beta_{2} - \)\(95\!\cdots\!08\)\( \beta_{3} - \)\(59\!\cdots\!22\)\( \beta_{4} + \)\(40\!\cdots\!38\)\( \beta_{5}) q^{40} +(-\)\(12\!\cdots\!34\)\( - \)\(31\!\cdots\!34\)\( \beta_{1} + \)\(18\!\cdots\!46\)\( \beta_{2} + \)\(53\!\cdots\!70\)\( \beta_{3} + \)\(46\!\cdots\!82\)\( \beta_{4} - \)\(13\!\cdots\!48\)\( \beta_{5}) q^{41} +(-\)\(78\!\cdots\!92\)\( - \)\(43\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!64\)\( \beta_{2} - \)\(27\!\cdots\!48\)\( \beta_{3} - \)\(36\!\cdots\!44\)\( \beta_{4} + \)\(25\!\cdots\!44\)\( \beta_{5}) q^{42} +(\)\(86\!\cdots\!20\)\( - \)\(32\!\cdots\!58\)\( \beta_{1} + \)\(15\!\cdots\!46\)\( \beta_{2} - \)\(16\!\cdots\!46\)\( \beta_{3} - \)\(92\!\cdots\!18\)\( \beta_{4} - \)\(38\!\cdots\!92\)\( \beta_{5}) q^{43} +(\)\(10\!\cdots\!04\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(35\!\cdots\!76\)\( \beta_{2} + \)\(23\!\cdots\!08\)\( \beta_{3} + \)\(14\!\cdots\!92\)\( \beta_{4} - \)\(39\!\cdots\!96\)\( \beta_{5}) q^{44} +(\)\(29\!\cdots\!66\)\( + \)\(11\!\cdots\!85\)\( \beta_{1} + \)\(30\!\cdots\!71\)\( \beta_{2} - \)\(27\!\cdots\!61\)\( \beta_{3}) q^{45} +(\)\(83\!\cdots\!36\)\( - \)\(91\!\cdots\!76\)\( \beta_{1} + \)\(24\!\cdots\!32\)\( \beta_{2} + \)\(77\!\cdots\!32\)\( \beta_{3} - \)\(42\!\cdots\!56\)\( \beta_{4} - \)\(22\!\cdots\!88\)\( \beta_{5}) q^{46} +(\)\(40\!\cdots\!80\)\( - \)\(94\!\cdots\!94\)\( \beta_{1} - \)\(56\!\cdots\!34\)\( \beta_{2} - \)\(12\!\cdots\!42\)\( \beta_{3} + \)\(95\!\cdots\!74\)\( \beta_{4} + \)\(15\!\cdots\!76\)\( \beta_{5}) q^{47} +(\)\(82\!\cdots\!20\)\( - \)\(65\!\cdots\!70\)\( \beta_{1} - \)\(25\!\cdots\!68\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} + \)\(13\!\cdots\!54\)\( \beta_{4} - \)\(23\!\cdots\!50\)\( \beta_{5}) q^{48} +(\)\(40\!\cdots\!05\)\( - \)\(42\!\cdots\!12\)\( \beta_{1} - \)\(53\!\cdots\!88\)\( \beta_{2} - \)\(37\!\cdots\!44\)\( \beta_{3} - \)\(19\!\cdots\!34\)\( \beta_{4} + \)\(44\!\cdots\!40\)\( \beta_{5}) q^{49} +(\)\(62\!\cdots\!49\)\( - \)\(14\!\cdots\!67\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(42\!\cdots\!04\)\( \beta_{3} - \)\(12\!\cdots\!76\)\( \beta_{4} - \)\(38\!\cdots\!96\)\( \beta_{5}) q^{50} +(-\)\(76\!\cdots\!62\)\( - \)\(82\!\cdots\!78\)\( \beta_{1} + \)\(60\!\cdots\!78\)\( \beta_{2} - \)\(26\!\cdots\!46\)\( \beta_{3} + \)\(11\!\cdots\!62\)\( \beta_{4} + \)\(17\!\cdots\!88\)\( \beta_{5}) q^{51} +(\)\(88\!\cdots\!72\)\( - \)\(63\!\cdots\!98\)\( \beta_{1} + \)\(73\!\cdots\!82\)\( \beta_{2} + \)\(11\!\cdots\!28\)\( \beta_{3} + \)\(85\!\cdots\!04\)\( \beta_{4} + \)\(81\!\cdots\!36\)\( \beta_{5}) q^{52} +(\)\(12\!\cdots\!98\)\( - \)\(80\!\cdots\!13\)\( \beta_{1} + \)\(20\!\cdots\!37\)\( \beta_{2} - \)\(62\!\cdots\!95\)\( \beta_{3} - \)\(81\!\cdots\!82\)\( \beta_{4} - \)\(75\!\cdots\!52\)\( \beta_{5}) q^{53} +(\)\(15\!\cdots\!07\)\( - \)\(46\!\cdots\!09\)\( \beta_{1}) q^{54} +(-\)\(40\!\cdots\!44\)\( + \)\(10\!\cdots\!68\)\( \beta_{1} - \)\(75\!\cdots\!76\)\( \beta_{2} - \)\(23\!\cdots\!76\)\( \beta_{3} - \)\(16\!\cdots\!56\)\( \beta_{4} + \)\(77\!\cdots\!24\)\( \beta_{5}) q^{55} +(-\)\(19\!\cdots\!00\)\( + \)\(13\!\cdots\!04\)\( \beta_{1} - \)\(21\!\cdots\!80\)\( \beta_{2} - \)\(89\!\cdots\!64\)\( \beta_{3} - \)\(49\!\cdots\!20\)\( \beta_{4} + \)\(49\!\cdots\!04\)\( \beta_{5}) q^{56} +(\)\(38\!\cdots\!88\)\( + \)\(16\!\cdots\!70\)\( \beta_{1} + \)\(29\!\cdots\!70\)\( \beta_{2} - \)\(87\!\cdots\!42\)\( \beta_{3} + \)\(43\!\cdots\!66\)\( \beta_{4} - \)\(77\!\cdots\!92\)\( \beta_{5}) q^{57} +(\)\(25\!\cdots\!66\)\( - \)\(63\!\cdots\!18\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} + \)\(78\!\cdots\!20\)\( \beta_{3} + \)\(10\!\cdots\!16\)\( \beta_{4} - \)\(10\!\cdots\!84\)\( \beta_{5}) q^{58} +(\)\(19\!\cdots\!88\)\( + \)\(22\!\cdots\!20\)\( \beta_{1} + \)\(14\!\cdots\!36\)\( \beta_{2} + \)\(57\!\cdots\!72\)\( \beta_{3} - \)\(11\!\cdots\!48\)\( \beta_{4} + \)\(94\!\cdots\!40\)\( \beta_{5}) q^{59} +(\)\(38\!\cdots\!76\)\( + \)\(28\!\cdots\!86\)\( \beta_{1} + \)\(33\!\cdots\!42\)\( \beta_{2} - \)\(12\!\cdots\!96\)\( \beta_{3} - \)\(12\!\cdots\!32\)\( \beta_{4} + \)\(42\!\cdots\!28\)\( \beta_{5}) q^{60} +(\)\(55\!\cdots\!10\)\( + \)\(12\!\cdots\!84\)\( \beta_{1} + \)\(17\!\cdots\!98\)\( \beta_{2} + \)\(40\!\cdots\!16\)\( \beta_{3} - \)\(22\!\cdots\!91\)\( \beta_{4} + \)\(18\!\cdots\!08\)\( \beta_{5}) q^{61} +(-\)\(20\!\cdots\!68\)\( + \)\(19\!\cdots\!08\)\( \beta_{1} + \)\(14\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!96\)\( \beta_{3} + \)\(52\!\cdots\!32\)\( \beta_{4} - \)\(19\!\cdots\!92\)\( \beta_{5}) q^{62} +(\)\(22\!\cdots\!16\)\( + \)\(14\!\cdots\!73\)\( \beta_{1} - \)\(25\!\cdots\!91\)\( \beta_{2} - \)\(18\!\cdots\!41\)\( \beta_{3} - \)\(27\!\cdots\!61\)\( \beta_{4}) q^{63} +(-\)\(12\!\cdots\!08\)\( + \)\(52\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} + \)\(83\!\cdots\!08\)\( \beta_{3} + \)\(31\!\cdots\!92\)\( \beta_{4} + \)\(27\!\cdots\!04\)\( \beta_{5}) q^{64} +(-\)\(17\!\cdots\!32\)\( + \)\(29\!\cdots\!54\)\( \beta_{1} - \)\(41\!\cdots\!78\)\( \beta_{2} - \)\(17\!\cdots\!78\)\( \beta_{3} - \)\(15\!\cdots\!18\)\( \beta_{4} + \)\(52\!\cdots\!72\)\( \beta_{5}) q^{65} +(\)\(11\!\cdots\!36\)\( - \)\(38\!\cdots\!72\)\( \beta_{1} + \)\(27\!\cdots\!80\)\( \beta_{2} + \)\(82\!\cdots\!68\)\( \beta_{3} + \)\(68\!\cdots\!96\)\( \beta_{4} - \)\(76\!\cdots\!72\)\( \beta_{5}) q^{66} +(\)\(30\!\cdots\!72\)\( + \)\(70\!\cdots\!52\)\( \beta_{1} + \)\(79\!\cdots\!60\)\( \beta_{2} - \)\(73\!\cdots\!44\)\( \beta_{3} + \)\(61\!\cdots\!68\)\( \beta_{4} - \)\(17\!\cdots\!68\)\( \beta_{5}) q^{67} +(\)\(38\!\cdots\!76\)\( - \)\(35\!\cdots\!54\)\( \beta_{1} - \)\(95\!\cdots\!18\)\( \beta_{2} - \)\(24\!\cdots\!72\)\( \beta_{3} - \)\(12\!\cdots\!40\)\( \beta_{4} + \)\(16\!\cdots\!12\)\( \beta_{5}) q^{68} +(\)\(60\!\cdots\!92\)\( - \)\(12\!\cdots\!22\)\( \beta_{1} + \)\(18\!\cdots\!62\)\( \beta_{2} + \)\(10\!\cdots\!98\)\( \beta_{3} + \)\(74\!\cdots\!58\)\( \beta_{4} + \)\(23\!\cdots\!00\)\( \beta_{5}) q^{69} +(\)\(36\!\cdots\!80\)\( - \)\(20\!\cdots\!28\)\( \beta_{1} + \)\(38\!\cdots\!72\)\( \beta_{2} - \)\(43\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!04\)\( \beta_{4} + \)\(31\!\cdots\!16\)\( \beta_{5}) q^{70} +(-\)\(22\!\cdots\!08\)\( - \)\(37\!\cdots\!22\)\( \beta_{1} - \)\(58\!\cdots\!78\)\( \beta_{2} - \)\(57\!\cdots\!58\)\( \beta_{3} + \)\(56\!\cdots\!74\)\( \beta_{4} - \)\(10\!\cdots\!68\)\( \beta_{5}) q^{71} +(\)\(11\!\cdots\!64\)\( - \)\(74\!\cdots\!27\)\( \beta_{1} + \)\(13\!\cdots\!66\)\( \beta_{2} + \)\(16\!\cdots\!78\)\( \beta_{3} - \)\(20\!\cdots\!53\)\( \beta_{4} - \)\(27\!\cdots\!61\)\( \beta_{5}) q^{72} +(-\)\(17\!\cdots\!02\)\( - \)\(99\!\cdots\!28\)\( \beta_{1} - \)\(42\!\cdots\!96\)\( \beta_{2} - \)\(80\!\cdots\!80\)\( \beta_{3} - \)\(78\!\cdots\!78\)\( \beta_{4} + \)\(56\!\cdots\!92\)\( \beta_{5}) q^{73} +(-\)\(26\!\cdots\!78\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} - \)\(47\!\cdots\!60\)\( \beta_{2} + \)\(23\!\cdots\!84\)\( \beta_{3} + \)\(18\!\cdots\!84\)\( \beta_{4} + \)\(65\!\cdots\!20\)\( \beta_{5}) q^{74} +(\)\(87\!\cdots\!79\)\( - \)\(79\!\cdots\!32\)\( \beta_{1} + \)\(31\!\cdots\!32\)\( \beta_{2} + \)\(24\!\cdots\!16\)\( \beta_{3} - \)\(84\!\cdots\!46\)\( \beta_{4} + \)\(47\!\cdots\!84\)\( \beta_{5}) q^{75} +(\)\(11\!\cdots\!88\)\( + \)\(14\!\cdots\!00\)\( \beta_{1} + \)\(83\!\cdots\!40\)\( \beta_{2} - \)\(23\!\cdots\!48\)\( \beta_{3} + \)\(40\!\cdots\!60\)\( \beta_{4} - \)\(48\!\cdots\!72\)\( \beta_{5}) q^{76} +(-\)\(72\!\cdots\!68\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} - \)\(10\!\cdots\!52\)\( \beta_{2} - \)\(23\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!24\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5}) q^{77} +(\)\(63\!\cdots\!98\)\( - \)\(25\!\cdots\!90\)\( \beta_{1} + \)\(16\!\cdots\!08\)\( \beta_{2} + \)\(29\!\cdots\!40\)\( \beta_{3} + \)\(24\!\cdots\!56\)\( \beta_{4} - \)\(31\!\cdots\!64\)\( \beta_{5}) q^{78} +(\)\(14\!\cdots\!00\)\( + \)\(41\!\cdots\!65\)\( \beta_{1} + \)\(15\!\cdots\!69\)\( \beta_{2} - \)\(11\!\cdots\!41\)\( \beta_{3} - \)\(70\!\cdots\!33\)\( \beta_{4} + \)\(28\!\cdots\!88\)\( \beta_{5}) q^{79} +(-\)\(31\!\cdots\!76\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(28\!\cdots\!20\)\( \beta_{2} + \)\(48\!\cdots\!96\)\( \beta_{3} - \)\(95\!\cdots\!32\)\( \beta_{4} + \)\(99\!\cdots\!28\)\( \beta_{5}) q^{80} +\)\(77\!\cdots\!21\)\( q^{81} +(\)\(25\!\cdots\!42\)\( + \)\(30\!\cdots\!90\)\( \beta_{1} - \)\(23\!\cdots\!68\)\( \beta_{2} + \)\(11\!\cdots\!36\)\( \beta_{3} + \)\(26\!\cdots\!88\)\( \beta_{4} - \)\(20\!\cdots\!28\)\( \beta_{5}) q^{82} +(-\)\(24\!\cdots\!16\)\( + \)\(60\!\cdots\!86\)\( \beta_{1} - \)\(10\!\cdots\!86\)\( \beta_{2} - \)\(86\!\cdots\!70\)\( \beta_{3} + \)\(36\!\cdots\!12\)\( \beta_{4} - \)\(49\!\cdots\!28\)\( \beta_{5}) q^{83} +(-\)\(62\!\cdots\!52\)\( + \)\(80\!\cdots\!76\)\( \beta_{1} - \)\(24\!\cdots\!24\)\( \beta_{2} - \)\(92\!\cdots\!80\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(74\!\cdots\!20\)\( \beta_{5}) q^{84} +(\)\(15\!\cdots\!72\)\( - \)\(87\!\cdots\!10\)\( \beta_{1} - \)\(72\!\cdots\!98\)\( \beta_{2} - \)\(79\!\cdots\!62\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(11\!\cdots\!60\)\( \beta_{5}) q^{85} +(\)\(33\!\cdots\!84\)\( - \)\(16\!\cdots\!28\)\( \beta_{1} + \)\(48\!\cdots\!96\)\( \beta_{2} - \)\(95\!\cdots\!64\)\( \beta_{3} - \)\(16\!\cdots\!68\)\( \beta_{4} - \)\(88\!\cdots\!04\)\( \beta_{5}) q^{86} +(\)\(10\!\cdots\!46\)\( - \)\(40\!\cdots\!93\)\( \beta_{1} - \)\(16\!\cdots\!99\)\( \beta_{2} + \)\(26\!\cdots\!13\)\( \beta_{3} + \)\(85\!\cdots\!58\)\( \beta_{4} - \)\(34\!\cdots\!40\)\( \beta_{5}) q^{87} +(\)\(70\!\cdots\!44\)\( - \)\(15\!\cdots\!80\)\( \beta_{1} + \)\(60\!\cdots\!76\)\( \beta_{2} + \)\(36\!\cdots\!52\)\( \beta_{3} + \)\(17\!\cdots\!28\)\( \beta_{4} - \)\(29\!\cdots\!44\)\( \beta_{5}) q^{88} +(\)\(10\!\cdots\!22\)\( - \)\(11\!\cdots\!12\)\( \beta_{1} + \)\(71\!\cdots\!20\)\( \beta_{2} + \)\(41\!\cdots\!68\)\( \beta_{3} + \)\(77\!\cdots\!68\)\( \beta_{4} + \)\(82\!\cdots\!40\)\( \beta_{5}) q^{89} +(-\)\(10\!\cdots\!26\)\( - \)\(28\!\cdots\!06\)\( \beta_{1} - \)\(91\!\cdots\!12\)\( \beta_{2} - \)\(16\!\cdots\!04\)\( \beta_{3} - \)\(71\!\cdots\!28\)\( \beta_{4} + \)\(62\!\cdots\!12\)\( \beta_{5}) q^{90} +(\)\(74\!\cdots\!76\)\( + \)\(46\!\cdots\!94\)\( \beta_{1} - \)\(64\!\cdots\!54\)\( \beta_{2} - \)\(83\!\cdots\!98\)\( \beta_{3} - \)\(16\!\cdots\!78\)\( \beta_{4} - \)\(21\!\cdots\!20\)\( \beta_{5}) q^{91} +(\)\(67\!\cdots\!16\)\( - \)\(56\!\cdots\!80\)\( \beta_{1} + \)\(57\!\cdots\!36\)\( \beta_{2} + \)\(35\!\cdots\!08\)\( \beta_{3} - \)\(86\!\cdots\!16\)\( \beta_{4} - \)\(26\!\cdots\!64\)\( \beta_{5}) q^{92} +(-\)\(11\!\cdots\!76\)\( + \)\(36\!\cdots\!09\)\( \beta_{1} - \)\(63\!\cdots\!99\)\( \beta_{2} + \)\(17\!\cdots\!55\)\( \beta_{3} - \)\(53\!\cdots\!93\)\( \beta_{4} - \)\(12\!\cdots\!08\)\( \beta_{5}) q^{93} +(\)\(90\!\cdots\!48\)\( + \)\(21\!\cdots\!60\)\( \beta_{1} + \)\(29\!\cdots\!32\)\( \beta_{2} - \)\(82\!\cdots\!36\)\( \beta_{3} + \)\(11\!\cdots\!92\)\( \beta_{4} + \)\(39\!\cdots\!08\)\( \beta_{5}) q^{94} +(\)\(12\!\cdots\!04\)\( + \)\(46\!\cdots\!28\)\( \beta_{1} + \)\(83\!\cdots\!92\)\( \beta_{2} + \)\(21\!\cdots\!16\)\( \beta_{3} - \)\(55\!\cdots\!16\)\( \beta_{4} + \)\(58\!\cdots\!64\)\( \beta_{5}) q^{95} +(\)\(24\!\cdots\!04\)\( + \)\(18\!\cdots\!64\)\( \beta_{1} + \)\(17\!\cdots\!08\)\( \beta_{2} + \)\(37\!\cdots\!08\)\( \beta_{3} + \)\(88\!\cdots\!64\)\( \beta_{4} - \)\(22\!\cdots\!84\)\( \beta_{5}) q^{96} +(\)\(15\!\cdots\!62\)\( + \)\(34\!\cdots\!32\)\( \beta_{1} - \)\(28\!\cdots\!20\)\( \beta_{2} + \)\(14\!\cdots\!24\)\( \beta_{3} - \)\(39\!\cdots\!64\)\( \beta_{4} - \)\(82\!\cdots\!28\)\( \beta_{5}) q^{97} +(\)\(40\!\cdots\!43\)\( + \)\(14\!\cdots\!43\)\( \beta_{1} + \)\(35\!\cdots\!28\)\( \beta_{2} - \)\(79\!\cdots\!60\)\( \beta_{3} - \)\(68\!\cdots\!92\)\( \beta_{4} + \)\(90\!\cdots\!28\)\( \beta_{5}) q^{98} +(\)\(64\!\cdots\!44\)\( - \)\(17\!\cdots\!42\)\( \beta_{1} - \)\(65\!\cdots\!74\)\( \beta_{2} + \)\(50\!\cdots\!14\)\( \beta_{3} - \)\(50\!\cdots\!64\)\( \beta_{4} - \)\(49\!\cdots\!68\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 19700962938q^{2} + 100063090197999414q^{3} + \)\(22\!\cdots\!68\)\(q^{4} + \)\(62\!\cdots\!36\)\(q^{5} + \)\(32\!\cdots\!22\)\(q^{6} + \)\(47\!\cdots\!36\)\(q^{7} + \)\(24\!\cdots\!44\)\(q^{8} + \)\(16\!\cdots\!66\)\(q^{9} + O(q^{10}) \) \( 6q + 19700962938q^{2} + 100063090197999414q^{3} + \)\(22\!\cdots\!68\)\(q^{4} + \)\(62\!\cdots\!36\)\(q^{5} + \)\(32\!\cdots\!22\)\(q^{6} + \)\(47\!\cdots\!36\)\(q^{7} + \)\(24\!\cdots\!44\)\(q^{8} + \)\(16\!\cdots\!66\)\(q^{9} - \)\(22\!\cdots\!96\)\(q^{10} + \)\(13\!\cdots\!24\)\(q^{11} + \)\(36\!\cdots\!92\)\(q^{12} + \)\(13\!\cdots\!32\)\(q^{13} - \)\(28\!\cdots\!08\)\(q^{14} + \)\(10\!\cdots\!84\)\(q^{15} + \)\(29\!\cdots\!80\)\(q^{16} - \)\(27\!\cdots\!88\)\(q^{17} + \)\(54\!\cdots\!18\)\(q^{18} + \)\(13\!\cdots\!12\)\(q^{19} + \)\(13\!\cdots\!24\)\(q^{20} + \)\(79\!\cdots\!84\)\(q^{21} + \)\(41\!\cdots\!64\)\(q^{22} + \)\(21\!\cdots\!08\)\(q^{23} + \)\(40\!\cdots\!36\)\(q^{24} + \)\(31\!\cdots\!46\)\(q^{25} + \)\(23\!\cdots\!52\)\(q^{26} + \)\(27\!\cdots\!54\)\(q^{27} - \)\(22\!\cdots\!48\)\(q^{28} + \)\(38\!\cdots\!04\)\(q^{29} - \)\(37\!\cdots\!24\)\(q^{30} - \)\(40\!\cdots\!24\)\(q^{31} + \)\(88\!\cdots\!96\)\(q^{32} + \)\(23\!\cdots\!56\)\(q^{33} + \)\(28\!\cdots\!96\)\(q^{34} + \)\(89\!\cdots\!20\)\(q^{35} + \)\(61\!\cdots\!48\)\(q^{36} - \)\(10\!\cdots\!24\)\(q^{37} - \)\(52\!\cdots\!72\)\(q^{38} + \)\(22\!\cdots\!08\)\(q^{39} - \)\(82\!\cdots\!12\)\(q^{40} - \)\(75\!\cdots\!04\)\(q^{41} - \)\(47\!\cdots\!52\)\(q^{42} + \)\(51\!\cdots\!20\)\(q^{43} + \)\(63\!\cdots\!24\)\(q^{44} + \)\(17\!\cdots\!96\)\(q^{45} + \)\(50\!\cdots\!16\)\(q^{46} + \)\(24\!\cdots\!80\)\(q^{47} + \)\(49\!\cdots\!20\)\(q^{48} + \)\(24\!\cdots\!30\)\(q^{49} + \)\(37\!\cdots\!94\)\(q^{50} - \)\(45\!\cdots\!72\)\(q^{51} + \)\(53\!\cdots\!32\)\(q^{52} + \)\(74\!\cdots\!88\)\(q^{53} + \)\(91\!\cdots\!42\)\(q^{54} - \)\(24\!\cdots\!64\)\(q^{55} - \)\(11\!\cdots\!00\)\(q^{56} + \)\(23\!\cdots\!28\)\(q^{57} + \)\(15\!\cdots\!96\)\(q^{58} + \)\(11\!\cdots\!28\)\(q^{59} + \)\(23\!\cdots\!56\)\(q^{60} + \)\(33\!\cdots\!60\)\(q^{61} - \)\(12\!\cdots\!08\)\(q^{62} + \)\(13\!\cdots\!96\)\(q^{63} - \)\(76\!\cdots\!48\)\(q^{64} - \)\(10\!\cdots\!92\)\(q^{65} + \)\(68\!\cdots\!16\)\(q^{66} + \)\(18\!\cdots\!32\)\(q^{67} + \)\(23\!\cdots\!56\)\(q^{68} + \)\(36\!\cdots\!52\)\(q^{69} + \)\(21\!\cdots\!80\)\(q^{70} - \)\(13\!\cdots\!48\)\(q^{71} + \)\(68\!\cdots\!84\)\(q^{72} - \)\(10\!\cdots\!12\)\(q^{73} - \)\(16\!\cdots\!68\)\(q^{74} + \)\(52\!\cdots\!74\)\(q^{75} + \)\(71\!\cdots\!28\)\(q^{76} - \)\(43\!\cdots\!08\)\(q^{77} + \)\(38\!\cdots\!88\)\(q^{78} + \)\(84\!\cdots\!00\)\(q^{79} - \)\(18\!\cdots\!56\)\(q^{80} + \)\(46\!\cdots\!26\)\(q^{81} + \)\(15\!\cdots\!52\)\(q^{82} - \)\(14\!\cdots\!96\)\(q^{83} - \)\(37\!\cdots\!12\)\(q^{84} + \)\(90\!\cdots\!32\)\(q^{85} + \)\(20\!\cdots\!04\)\(q^{86} + \)\(64\!\cdots\!76\)\(q^{87} + \)\(42\!\cdots\!64\)\(q^{88} + \)\(63\!\cdots\!32\)\(q^{89} - \)\(61\!\cdots\!56\)\(q^{90} + \)\(44\!\cdots\!56\)\(q^{91} + \)\(40\!\cdots\!96\)\(q^{92} - \)\(67\!\cdots\!56\)\(q^{93} + \)\(54\!\cdots\!88\)\(q^{94} + \)\(72\!\cdots\!24\)\(q^{95} + \)\(14\!\cdots\!24\)\(q^{96} + \)\(94\!\cdots\!72\)\(q^{97} + \)\(24\!\cdots\!58\)\(q^{98} + \)\(38\!\cdots\!64\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 8785036246714673132 x^{4} - 489946547045174855374556256 x^{3} + 19187418488273751521495823450187335040 x^{2} + 711105729064303642253694769744270208387385600 x - 4081433259823761741733687714132191037604592129279360000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 18 \nu - 9 \)
\(\beta_{2}\)\(=\)\( 324 \nu^{2} - 27104502078 \nu - 948783914631632447703 \)
\(\beta_{3}\)\(=\)\((\)\(-90125876775213 \nu^{5} + 53463655026814254751251 \nu^{4} + 463995524025435957735230750531592 \nu^{3} - 169630723465173028211721594061420744807680 \nu^{2} - 192774109436882505235791020666290835385297306979200 \nu - 38569662358280835890776749668815770433982419295276334048000\)\()/ \)\(29\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-844242270277677309 \nu^{5} + 852793370799663688969611843 \nu^{4} + 6805898868230016519173348507543890056 \nu^{3} - 4174347018446025305811891046658812787546154240 \nu^{2} - 12812523921837199366736290192203052903024777016753865600 \nu + 2054497240491824901859748724724492524200853340667726327309856000\)\()/ \)\(36\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-22538453183923580109 \nu^{5} - 89407995202856888571734236557 \nu^{4} + 253602431219869893160139944541657813256 \nu^{3} + 476752262057768969522187796599515568922990037760 \nu^{2} - 594162295978334728640491136206480563520126946584383945600 \nu - 234331745408306749031340280645677075551726492674855377324750816000\)\()/ \)\(14\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 9\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 1505805671 \beta_{1} + 948783914645184698742\)\()/324\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + 73 \beta_{4} - 5970698 \beta_{3} + 4958890590 \beta_{2} + 1431053408173977515375 \beta_{1} + 1428684208035226964532166627476\)\()/5832\)
\(\nu^{4}\)\(=\)\((\)\(-62888121261 \beta_{5} + 879389206955 \beta_{4} - 34446692577651774 \beta_{3} + 878063773153747377570 \beta_{2} + 2981620610645308994107428230485 \beta_{1} + 678880227343919202666191220801731424795564\)\()/52488\)
\(\nu^{5}\)\(=\)\((\)\(81259437629475430781 \beta_{5} + 35136898498255210337957 \beta_{4} - 4211938483175451572482133474 \beta_{3} + 4011634084777833225027188228990 \beta_{2} + 552419430177431820255294359471158638229275 \beta_{1} + 1414456840650861342334087316033052404557801536107796\)\()/472392\)

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
2.38628e9
1.73590e9
4.68489e8
−5.07643e8
−1.88393e9
−2.19909e9
−3.96695e10 1.66772e16 9.83375e20 1.65867e24 −6.61576e26 4.47127e28 −1.55933e31 2.78128e32 −6.57985e34
1.2 −2.79627e10 1.66772e16 1.91618e20 −1.83280e24 −4.66339e26 −3.84348e28 1.11481e31 2.78128e32 5.12500e34
1.3 −5.14931e9 1.66772e16 −5.63780e20 5.32454e23 −8.58759e25 1.60190e29 5.94269e30 2.78128e32 −2.74177e33
1.4 1.24211e10 1.66772e16 −4.36013e20 1.87674e23 2.07148e26 −1.62329e29 −1.27479e31 2.78128e32 2.33111e33
1.5 3.71943e10 1.66772e16 7.93117e20 1.89718e24 6.20295e26 2.07605e29 7.54379e30 2.78128e32 7.05641e34
1.6 4.28672e10 1.66772e16 1.24730e21 −1.81509e24 7.14904e26 −1.63830e29 2.81639e31 2.78128e32 −7.78079e34
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 3.70.a.b 6
3.b odd 2 1 9.70.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.70.a.b 6 1.a even 1 1 trivial
9.70.a.c 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 19700962938 T_{2}^{5} - \)\(26\!\cdots\!48\)\( T_{2}^{4} + \)\(39\!\cdots\!96\)\( T_{2}^{3} + \)\(18\!\cdots\!36\)\( T_{2}^{2} - \)\(14\!\cdots\!12\)\( T_{2} - \)\(11\!\cdots\!68\)\( \) acting on \(S_{70}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 19700962938 T + \)\(85\!\cdots\!24\)\( T^{2} - \)\(18\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!92\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{5} + \)\(51\!\cdots\!84\)\( T^{6} - \)\(75\!\cdots\!12\)\( T^{7} + \)\(24\!\cdots\!48\)\( T^{8} - \)\(38\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!64\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(42\!\cdots\!84\)\( T^{12} \)
$3$ \( ( 1 - 16677181699666569 T )^{6} \)
$5$ \( 1 - \)\(62\!\cdots\!36\)\( T + \)\(37\!\cdots\!50\)\( T^{2} - \)\(98\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!75\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(60\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!75\)\( T^{8} - \)\(47\!\cdots\!00\)\( T^{9} + \)\(30\!\cdots\!50\)\( T^{10} - \)\(87\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(47\!\cdots\!36\)\( T + \)\(61\!\cdots\!54\)\( T^{2} - \)\(33\!\cdots\!08\)\( T^{3} + \)\(22\!\cdots\!91\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{5} + \)\(55\!\cdots\!68\)\( T^{6} - \)\(23\!\cdots\!36\)\( T^{7} + \)\(93\!\cdots\!59\)\( T^{8} - \)\(29\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!54\)\( T^{10} - \)\(17\!\cdots\!52\)\( T^{11} + \)\(74\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 - \)\(13\!\cdots\!24\)\( T + \)\(43\!\cdots\!94\)\( T^{2} - \)\(46\!\cdots\!28\)\( T^{3} + \)\(78\!\cdots\!83\)\( T^{4} - \)\(64\!\cdots\!20\)\( T^{5} + \)\(75\!\cdots\!28\)\( T^{6} - \)\(46\!\cdots\!20\)\( T^{7} + \)\(40\!\cdots\!23\)\( T^{8} - \)\(17\!\cdots\!88\)\( T^{9} + \)\(11\!\cdots\!34\)\( T^{10} - \)\(26\!\cdots\!24\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$13$ \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(13\!\cdots\!90\)\( T^{2} - \)\(47\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(41\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!96\)\( T^{6} - \)\(29\!\cdots\!56\)\( T^{7} + \)\(79\!\cdots\!35\)\( T^{8} - \)\(18\!\cdots\!12\)\( T^{9} + \)\(36\!\cdots\!90\)\( T^{10} - \)\(27\!\cdots\!76\)\( T^{11} + \)\(14\!\cdots\!89\)\( T^{12} \)
$17$ \( 1 + \)\(27\!\cdots\!88\)\( T + \)\(28\!\cdots\!42\)\( T^{2} + \)\(21\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(61\!\cdots\!32\)\( T^{5} + \)\(36\!\cdots\!24\)\( T^{6} - \)\(48\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!95\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{9} + \)\(11\!\cdots\!02\)\( T^{10} + \)\(87\!\cdots\!16\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} \)
$19$ \( 1 - \)\(13\!\cdots\!12\)\( T + \)\(65\!\cdots\!62\)\( T^{2} - \)\(78\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!63\)\( T^{4} - \)\(19\!\cdots\!28\)\( T^{5} + \)\(38\!\cdots\!48\)\( T^{6} - \)\(34\!\cdots\!12\)\( T^{7} + \)\(57\!\cdots\!83\)\( T^{8} - \)\(39\!\cdots\!36\)\( T^{9} + \)\(56\!\cdots\!22\)\( T^{10} - \)\(20\!\cdots\!88\)\( T^{11} + \)\(25\!\cdots\!21\)\( T^{12} \)
$23$ \( 1 - \)\(21\!\cdots\!08\)\( T + \)\(46\!\cdots\!06\)\( T^{2} - \)\(49\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!31\)\( T^{4} - \)\(30\!\cdots\!24\)\( T^{5} + \)\(33\!\cdots\!72\)\( T^{6} - \)\(27\!\cdots\!12\)\( T^{7} + \)\(44\!\cdots\!39\)\( T^{8} - \)\(37\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!66\)\( T^{10} - \)\(13\!\cdots\!44\)\( T^{11} + \)\(56\!\cdots\!09\)\( T^{12} \)
$29$ \( 1 - \)\(38\!\cdots\!04\)\( T + \)\(81\!\cdots\!66\)\( T^{2} - \)\(36\!\cdots\!92\)\( T^{3} + \)\(91\!\cdots\!87\)\( T^{4} - \)\(27\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!72\)\( T^{6} - \)\(22\!\cdots\!56\)\( T^{7} + \)\(59\!\cdots\!07\)\( T^{8} - \)\(19\!\cdots\!28\)\( T^{9} + \)\(33\!\cdots\!86\)\( T^{10} - \)\(13\!\cdots\!96\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \)
$31$ \( 1 + \)\(40\!\cdots\!24\)\( T + \)\(19\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(29\!\cdots\!43\)\( T^{4} + \)\(16\!\cdots\!96\)\( T^{5} + \)\(27\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!16\)\( T^{7} + \)\(19\!\cdots\!63\)\( T^{8} + \)\(69\!\cdots\!52\)\( T^{9} + \)\(82\!\cdots\!58\)\( T^{10} + \)\(13\!\cdots\!24\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \)
$37$ \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(32\!\cdots\!94\)\( T^{2} + \)\(14\!\cdots\!92\)\( T^{3} + \)\(21\!\cdots\!51\)\( T^{4} + \)\(53\!\cdots\!72\)\( T^{5} + \)\(39\!\cdots\!68\)\( T^{6} + \)\(85\!\cdots\!44\)\( T^{7} + \)\(56\!\cdots\!79\)\( T^{8} + \)\(59\!\cdots\!36\)\( T^{9} + \)\(21\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!89\)\( T^{12} \)
$41$ \( 1 + \)\(75\!\cdots\!04\)\( T + \)\(41\!\cdots\!58\)\( T^{2} + \)\(27\!\cdots\!72\)\( T^{3} + \)\(18\!\cdots\!43\)\( T^{4} + \)\(81\!\cdots\!56\)\( T^{5} + \)\(34\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!16\)\( T^{7} + \)\(68\!\cdots\!03\)\( T^{8} + \)\(19\!\cdots\!32\)\( T^{9} + \)\(56\!\cdots\!78\)\( T^{10} + \)\(19\!\cdots\!04\)\( T^{11} + \)\(49\!\cdots\!61\)\( T^{12} \)
$43$ \( 1 - \)\(51\!\cdots\!20\)\( T + \)\(32\!\cdots\!10\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!47\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(55\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{9} + \)\(22\!\cdots\!10\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(24\!\cdots\!80\)\( T + \)\(37\!\cdots\!30\)\( T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!67\)\( T^{4} - \)\(71\!\cdots\!40\)\( T^{5} + \)\(37\!\cdots\!40\)\( T^{6} - \)\(17\!\cdots\!80\)\( T^{7} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(35\!\cdots\!80\)\( T^{9} + \)\(11\!\cdots\!30\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!69\)\( T^{12} \)
$53$ \( 1 - \)\(74\!\cdots\!88\)\( T + \)\(21\!\cdots\!06\)\( T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(30\!\cdots\!51\)\( T^{4} - \)\(13\!\cdots\!64\)\( T^{5} + \)\(33\!\cdots\!72\)\( T^{6} - \)\(12\!\cdots\!12\)\( T^{7} + \)\(26\!\cdots\!39\)\( T^{8} - \)\(12\!\cdots\!12\)\( T^{9} + \)\(17\!\cdots\!26\)\( T^{10} - \)\(55\!\cdots\!84\)\( T^{11} + \)\(70\!\cdots\!69\)\( T^{12} \)
$59$ \( 1 - \)\(11\!\cdots\!28\)\( T + \)\(82\!\cdots\!62\)\( T^{2} - \)\(80\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!83\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{5} + \)\(60\!\cdots\!08\)\( T^{6} - \)\(36\!\cdots\!08\)\( T^{7} + \)\(71\!\cdots\!43\)\( T^{8} - \)\(29\!\cdots\!84\)\( T^{9} + \)\(47\!\cdots\!42\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} + \)\(13\!\cdots\!61\)\( T^{12} \)
$61$ \( 1 - \)\(33\!\cdots\!60\)\( T + \)\(31\!\cdots\!54\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(39\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!15\)\( T^{8} - \)\(46\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(28\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$67$ \( 1 - \)\(18\!\cdots\!32\)\( T + \)\(48\!\cdots\!42\)\( T^{2} - \)\(74\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(13\!\cdots\!32\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} - \)\(13\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(73\!\cdots\!40\)\( T^{9} + \)\(48\!\cdots\!02\)\( T^{10} - \)\(18\!\cdots\!24\)\( T^{11} + \)\(98\!\cdots\!29\)\( T^{12} \)
$71$ \( 1 + \)\(13\!\cdots\!48\)\( T + \)\(23\!\cdots\!46\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(65\!\cdots\!08\)\( T^{5} + \)\(68\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!48\)\( T^{7} + \)\(48\!\cdots\!95\)\( T^{8} + \)\(27\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!66\)\( T^{10} + \)\(64\!\cdots\!48\)\( T^{11} + \)\(26\!\cdots\!81\)\( T^{12} \)
$73$ \( 1 + \)\(10\!\cdots\!12\)\( T + \)\(15\!\cdots\!26\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!31\)\( T^{4} + \)\(52\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!12\)\( T^{6} + \)\(19\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!39\)\( T^{8} + \)\(52\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!86\)\( T^{10} + \)\(73\!\cdots\!16\)\( T^{11} + \)\(26\!\cdots\!09\)\( T^{12} \)
$79$ \( 1 - \)\(84\!\cdots\!00\)\( T + \)\(62\!\cdots\!14\)\( T^{2} - \)\(31\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(44\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(20\!\cdots\!00\)\( T^{9} + \)\(34\!\cdots\!94\)\( T^{10} - \)\(40\!\cdots\!00\)\( T^{11} + \)\(41\!\cdots\!81\)\( T^{12} \)
$83$ \( 1 + \)\(14\!\cdots\!96\)\( T + \)\(61\!\cdots\!06\)\( T^{2} + \)\(59\!\cdots\!32\)\( T^{3} + \)\(20\!\cdots\!27\)\( T^{4} + \)\(29\!\cdots\!32\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} + \)\(77\!\cdots\!96\)\( T^{7} + \)\(13\!\cdots\!43\)\( T^{8} + \)\(10\!\cdots\!64\)\( T^{9} + \)\(28\!\cdots\!86\)\( T^{10} + \)\(17\!\cdots\!28\)\( T^{11} + \)\(31\!\cdots\!29\)\( T^{12} \)
$89$ \( 1 - \)\(63\!\cdots\!32\)\( T + \)\(24\!\cdots\!02\)\( T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(16\!\cdots\!28\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} - \)\(54\!\cdots\!52\)\( T^{7} + \)\(12\!\cdots\!23\)\( T^{8} - \)\(20\!\cdots\!76\)\( T^{9} + \)\(26\!\cdots\!22\)\( T^{10} - \)\(22\!\cdots\!68\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$97$ \( 1 - \)\(94\!\cdots\!72\)\( T + \)\(84\!\cdots\!62\)\( T^{2} - \)\(45\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!55\)\( T^{4} - \)\(94\!\cdots\!32\)\( T^{5} + \)\(37\!\cdots\!44\)\( T^{6} - \)\(11\!\cdots\!44\)\( T^{7} + \)\(36\!\cdots\!95\)\( T^{8} - \)\(83\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!02\)\( T^{10} - \)\(25\!\cdots\!04\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \)
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