Properties

Label 17.12.a
Level 17
Weight 12
Character orbit a
Rep. character \(\chi_{17}(1,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 2
Sturm bound 18
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 17.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(17))\).

Total New Old
Modular forms 18 14 4
Cusp forms 16 14 2
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(17\)Dim.
\(+\)\(8\)
\(-\)\(6\)

Trace form

\( 14q + 46q^{2} + 20q^{3} + 15370q^{4} - 4292q^{5} + 17746q^{6} + 71852q^{7} - 173058q^{8} + 797046q^{9} + O(q^{10}) \) \( 14q + 46q^{2} + 20q^{3} + 15370q^{4} - 4292q^{5} + 17746q^{6} + 71852q^{7} - 173058q^{8} + 797046q^{9} - 673214q^{10} - 526804q^{11} + 1942838q^{12} + 3758516q^{13} - 3613816q^{14} + 912360q^{15} + 8226226q^{16} - 2839714q^{17} + 8686466q^{18} - 11240288q^{19} + 44409558q^{20} + 41986976q^{21} + 29992986q^{22} - 76798340q^{23} - 151613346q^{24} + 200012338q^{25} + 3898736q^{26} + 174116456q^{27} + 210859792q^{28} - 258972468q^{29} - 159989456q^{30} - 371456620q^{31} - 74491658q^{32} - 246365216q^{33} - 90870848q^{34} + 687191560q^{35} - 1184787162q^{36} - 1048571284q^{37} + 1499726344q^{38} + 324651000q^{39} - 2917454058q^{40} - 1116864084q^{41} - 383375592q^{42} + 1108450592q^{43} + 3258533550q^{44} + 2275669548q^{45} + 4189620772q^{46} - 5567527520q^{47} + 4501089838q^{48} + 10091243646q^{49} - 2038342106q^{50} - 1380101004q^{51} - 3084898544q^{52} - 487871740q^{53} - 2810024660q^{54} - 6299451048q^{55} - 6178133760q^{56} - 12155866720q^{57} + 15131715978q^{58} + 14529638592q^{59} + 26595432800q^{60} - 25675611524q^{61} - 43307851592q^{62} + 15170417228q^{63} - 20861279734q^{64} + 42496345080q^{65} + 12924702860q^{66} + 70124663480q^{67} - 8723601408q^{68} - 10532027568q^{69} - 76100421800q^{70} - 26116702620q^{71} - 13980317574q^{72} - 12455392836q^{73} - 144742795930q^{74} + 43318267804q^{75} + 10119859400q^{76} + 100619066928q^{77} - 52631742492q^{78} - 71714375700q^{79} + 122289744734q^{80} - 84963717282q^{81} + 99296045040q^{82} + 143601653440q^{83} + 120528571208q^{84} - 30492848932q^{85} + 117618276260q^{86} - 159891378936q^{87} - 280242226026q^{88} + 254963977324q^{89} - 14714658934q^{90} - 189744657976q^{91} - 12966101244q^{92} + 43609152976q^{93} + 337819101216q^{94} - 170603129056q^{95} + 14769913038q^{96} + 57157923628q^{97} + 21076786366q^{98} - 11267044468q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(17))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 17
17.12.a.a \(6\) \(13.062\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-9\) \(-476\) \(-12884\) \(-23436\) \(-\) \(q+(-2+\beta _{1})q^{2}+(-79+\beta _{2})q^{3}+(767+\cdots)q^{4}+\cdots\)
17.12.a.b \(8\) \(13.062\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(55\) \(496\) \(8592\) \(95288\) \(+\) \(q+(7-\beta _{1})q^{2}+(62-\beta _{1}+\beta _{3})q^{3}+(1344+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(17))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(17)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 9 T + 3878 T^{2} + 3580 T^{3} + 12725208 T^{4} - 66927744 T^{5} + 24540943872 T^{6} - 137068019712 T^{7} + 53373390815232 T^{8} + 30751965839360 T^{9} + 68222497480245248 T^{10} + 324259173170675712 T^{11} + 73786976294838206464 T^{12} \))(\( 1 - 55 T + 4326 T^{2} - 62304 T^{3} + 2312224 T^{4} + 208500928 T^{5} + 8737082240 T^{6} - 368451463168 T^{7} + 55692574646272 T^{8} - 754588596568064 T^{9} + 36645978987560960 T^{10} + 1791009333891301376 T^{11} + 40677074784363741184 T^{12} - \)\(22\!\cdots\!72\)\( T^{13} + \)\(31\!\cdots\!64\)\( T^{14} - \)\(83\!\cdots\!60\)\( T^{15} + \)\(30\!\cdots\!56\)\( T^{16} \))
$3$ (\( 1 + 476 T + 496530 T^{2} + 137229660 T^{3} + 140129689527 T^{4} + 38091063715632 T^{5} + 30949857022988124 T^{6} + 6747717664033061904 T^{7} + \)\(43\!\cdots\!43\)\( T^{8} + \)\(76\!\cdots\!80\)\( T^{9} + \)\(48\!\cdots\!30\)\( T^{10} + \)\(83\!\cdots\!32\)\( T^{11} + \)\(30\!\cdots\!29\)\( T^{12} \))(\( 1 - 496 T + 581272 T^{2} - 241325544 T^{3} + 166099339536 T^{4} - 74872836303048 T^{5} + 38276807178843288 T^{6} - 19169581990566729456 T^{7} + \)\(77\!\cdots\!82\)\( T^{8} - \)\(33\!\cdots\!32\)\( T^{9} + \)\(12\!\cdots\!92\)\( T^{10} - \)\(41\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!16\)\( T^{12} - \)\(42\!\cdots\!08\)\( T^{13} + \)\(17\!\cdots\!88\)\( T^{14} - \)\(27\!\cdots\!48\)\( T^{15} + \)\(96\!\cdots\!61\)\( T^{16} \))
$5$ (\( 1 + 12884 T + 196440170 T^{2} + 1147423568500 T^{3} + 9843234375782375 T^{4} + 23131636996256705000 T^{5} + \)\(30\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!75\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!50\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!25\)\( T^{12} \))(\( 1 - 8592 T + 165260496 T^{2} - 1195282192048 T^{3} + 17061279605799740 T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} - \)\(65\!\cdots\!00\)\( T^{7} + \)\(64\!\cdots\!50\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!00\)\( T^{10} - \)\(12\!\cdots\!00\)\( T^{11} + \)\(96\!\cdots\!00\)\( T^{12} - \)\(33\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(56\!\cdots\!00\)\( T^{15} + \)\(32\!\cdots\!25\)\( T^{16} \))
$7$ (\( 1 + 23436 T + 7191004114 T^{2} + 271828319992572 T^{3} + 22584363793540867199 T^{4} + \)\(11\!\cdots\!04\)\( T^{5} + \)\(48\!\cdots\!08\)\( T^{6} + \)\(22\!\cdots\!72\)\( T^{7} + \)\(88\!\cdots\!51\)\( T^{8} + \)\(21\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!14\)\( T^{10} + \)\(70\!\cdots\!48\)\( T^{11} + \)\(59\!\cdots\!49\)\( T^{12} \))(\( 1 - 95288 T + 6419185784 T^{2} - 288646872408528 T^{3} + 18205274691115961280 T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(62\!\cdots\!32\)\( T^{6} - \)\(24\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(49\!\cdots\!28\)\( T^{9} + \)\(24\!\cdots\!68\)\( T^{10} - \)\(85\!\cdots\!16\)\( T^{11} + \)\(27\!\cdots\!80\)\( T^{12} - \)\(87\!\cdots\!04\)\( T^{13} + \)\(38\!\cdots\!16\)\( T^{14} - \)\(11\!\cdots\!16\)\( T^{15} + \)\(23\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 + 962060 T + 1085456228018 T^{2} + 645518017104724412 T^{3} + \)\(53\!\cdots\!47\)\( T^{4} + \)\(28\!\cdots\!12\)\( T^{5} + \)\(19\!\cdots\!40\)\( T^{6} + \)\(82\!\cdots\!32\)\( T^{7} + \)\(43\!\cdots\!87\)\( T^{8} + \)\(14\!\cdots\!72\)\( T^{9} + \)\(71\!\cdots\!38\)\( T^{10} + \)\(18\!\cdots\!60\)\( T^{11} + \)\(53\!\cdots\!61\)\( T^{12} \))(\( 1 - 435256 T + 943309828440 T^{2} - 511030849687403248 T^{3} + \)\(50\!\cdots\!44\)\( T^{4} - \)\(29\!\cdots\!52\)\( T^{5} + \)\(20\!\cdots\!96\)\( T^{6} - \)\(11\!\cdots\!84\)\( T^{7} + \)\(63\!\cdots\!30\)\( T^{8} - \)\(32\!\cdots\!24\)\( T^{9} + \)\(16\!\cdots\!16\)\( T^{10} - \)\(68\!\cdots\!12\)\( T^{11} + \)\(33\!\cdots\!04\)\( T^{12} - \)\(96\!\cdots\!48\)\( T^{13} + \)\(50\!\cdots\!40\)\( T^{14} - \)\(66\!\cdots\!76\)\( T^{15} + \)\(43\!\cdots\!81\)\( T^{16} \))
$13$ (\( 1 + 435268 T + 5418179409834 T^{2} + 625424148619622148 T^{3} + \)\(16\!\cdots\!19\)\( T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(37\!\cdots\!64\)\( T^{6} + \)\(38\!\cdots\!24\)\( T^{7} + \)\(53\!\cdots\!11\)\( T^{8} + \)\(36\!\cdots\!44\)\( T^{9} + \)\(55\!\cdots\!74\)\( T^{10} + \)\(80\!\cdots\!76\)\( T^{11} + \)\(33\!\cdots\!09\)\( T^{12} \))(\( 1 - 4193784 T + 14286591469104 T^{2} - 31272710568678623464 T^{3} + \)\(63\!\cdots\!92\)\( T^{4} - \)\(10\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!36\)\( T^{6} - \)\(25\!\cdots\!48\)\( T^{7} + \)\(36\!\cdots\!82\)\( T^{8} - \)\(45\!\cdots\!76\)\( T^{9} + \)\(55\!\cdots\!84\)\( T^{10} - \)\(59\!\cdots\!32\)\( T^{11} + \)\(65\!\cdots\!12\)\( T^{12} - \)\(57\!\cdots\!48\)\( T^{13} + \)\(47\!\cdots\!36\)\( T^{14} - \)\(24\!\cdots\!72\)\( T^{15} + \)\(10\!\cdots\!21\)\( T^{16} \))
$17$ (\( ( 1 - 1419857 T )^{6} \))(\( ( 1 + 1419857 T )^{8} \))
$19$ (\( 1 + 26398480 T + 581968147006866 T^{2} + \)\(96\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!39\)\( T^{4} + \)\(17\!\cdots\!68\)\( T^{5} + \)\(18\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!79\)\( T^{8} + \)\(15\!\cdots\!72\)\( T^{9} + \)\(10\!\cdots\!86\)\( T^{10} + \)\(56\!\cdots\!20\)\( T^{11} + \)\(24\!\cdots\!81\)\( T^{12} \))(\( 1 - 15158192 T + 617876369865560 T^{2} - \)\(77\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!40\)\( T^{4} - \)\(20\!\cdots\!04\)\( T^{5} + \)\(39\!\cdots\!44\)\( T^{6} - \)\(34\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!98\)\( T^{8} - \)\(40\!\cdots\!12\)\( T^{9} + \)\(52\!\cdots\!84\)\( T^{10} - \)\(32\!\cdots\!36\)\( T^{11} + \)\(36\!\cdots\!40\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(44\!\cdots\!88\)\( T^{15} + \)\(33\!\cdots\!41\)\( T^{16} \))
$23$ (\( 1 + 99172772 T + 8610439921468418 T^{2} + \)\(48\!\cdots\!88\)\( T^{3} + \)\(24\!\cdots\!43\)\( T^{4} + \)\(93\!\cdots\!24\)\( T^{5} + \)\(32\!\cdots\!84\)\( T^{6} + \)\(88\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!47\)\( T^{8} + \)\(41\!\cdots\!04\)\( T^{9} + \)\(70\!\cdots\!38\)\( T^{10} + \)\(77\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!89\)\( T^{12} \))(\( 1 - 22374432 T + 4869578611371768 T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!00\)\( T^{4} - \)\(24\!\cdots\!08\)\( T^{5} + \)\(18\!\cdots\!20\)\( T^{6} - \)\(34\!\cdots\!44\)\( T^{7} + \)\(20\!\cdots\!46\)\( T^{8} - \)\(33\!\cdots\!88\)\( T^{9} + \)\(16\!\cdots\!80\)\( T^{10} - \)\(21\!\cdots\!64\)\( T^{11} + \)\(97\!\cdots\!00\)\( T^{12} - \)\(86\!\cdots\!76\)\( T^{13} + \)\(36\!\cdots\!52\)\( T^{14} - \)\(15\!\cdots\!96\)\( T^{15} + \)\(67\!\cdots\!81\)\( T^{16} \))
$29$ (\( 1 - 165683964 T + 54906859084506106 T^{2} - \)\(67\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!39\)\( T^{4} - \)\(12\!\cdots\!68\)\( T^{5} + \)\(19\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(19\!\cdots\!99\)\( T^{8} - \)\(12\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!86\)\( T^{10} - \)\(44\!\cdots\!36\)\( T^{11} + \)\(32\!\cdots\!21\)\( T^{12} \))(\( 1 + 424656432 T + 147902612117139600 T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + \)\(73\!\cdots\!72\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!68\)\( T^{7} + \)\(29\!\cdots\!10\)\( T^{8} + \)\(30\!\cdots\!72\)\( T^{9} + \)\(28\!\cdots\!32\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!32\)\( T^{12} + \)\(96\!\cdots\!40\)\( T^{13} + \)\(48\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!88\)\( T^{15} + \)\(49\!\cdots\!61\)\( T^{16} \))
$31$ (\( 1 + 199133468 T + 93261571180926402 T^{2} + \)\(65\!\cdots\!64\)\( T^{3} + \)\(20\!\cdots\!83\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(18\!\cdots\!84\)\( T^{6} - \)\(47\!\cdots\!64\)\( T^{7} + \)\(13\!\cdots\!63\)\( T^{8} + \)\(10\!\cdots\!24\)\( T^{9} + \)\(38\!\cdots\!42\)\( T^{10} + \)\(21\!\cdots\!68\)\( T^{11} + \)\(26\!\cdots\!81\)\( T^{12} \))(\( 1 + 172323152 T + 100207919100701576 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(60\!\cdots\!80\)\( T^{4} + \)\(84\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!24\)\( T^{6} + \)\(29\!\cdots\!84\)\( T^{7} + \)\(70\!\cdots\!14\)\( T^{8} + \)\(74\!\cdots\!04\)\( T^{9} + \)\(15\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!80\)\( T^{12} + \)\(17\!\cdots\!32\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} + \)\(11\!\cdots\!72\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} \))
$37$ (\( 1 + 785778644 T + 611639175101561226 T^{2} + \)\(30\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} + \)\(43\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} + \)\(77\!\cdots\!76\)\( T^{7} + \)\(39\!\cdots\!79\)\( T^{8} + \)\(17\!\cdots\!44\)\( T^{9} + \)\(61\!\cdots\!86\)\( T^{10} + \)\(14\!\cdots\!92\)\( T^{11} + \)\(31\!\cdots\!09\)\( T^{12} \))(\( 1 + 262792640 T + 494840550851399024 T^{2} + \)\(75\!\cdots\!28\)\( T^{3} + \)\(15\!\cdots\!84\)\( T^{4} + \)\(31\!\cdots\!16\)\( T^{5} + \)\(37\!\cdots\!40\)\( T^{6} + \)\(68\!\cdots\!40\)\( T^{7} + \)\(67\!\cdots\!74\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!60\)\( T^{10} + \)\(17\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!24\)\( T^{12} + \)\(13\!\cdots\!04\)\( T^{13} + \)\(15\!\cdots\!16\)\( T^{14} + \)\(14\!\cdots\!80\)\( T^{15} + \)\(10\!\cdots\!21\)\( T^{16} \))
$41$ (\( 1 - 166444428 T + 1309299570092551186 T^{2} - \)\(83\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} - \)\(71\!\cdots\!72\)\( T^{5} + \)\(76\!\cdots\!80\)\( T^{6} - \)\(39\!\cdots\!52\)\( T^{7} + \)\(32\!\cdots\!55\)\( T^{8} - \)\(13\!\cdots\!48\)\( T^{9} + \)\(12\!\cdots\!46\)\( T^{10} - \)\(84\!\cdots\!28\)\( T^{11} + \)\(27\!\cdots\!41\)\( T^{12} \))(\( 1 + 1283308512 T + 2920355332799766648 T^{2} + \)\(24\!\cdots\!84\)\( T^{3} + \)\(35\!\cdots\!52\)\( T^{4} + \)\(25\!\cdots\!12\)\( T^{5} + \)\(30\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!64\)\( T^{7} + \)\(20\!\cdots\!62\)\( T^{8} + \)\(10\!\cdots\!24\)\( T^{9} + \)\(93\!\cdots\!00\)\( T^{10} + \)\(41\!\cdots\!52\)\( T^{11} + \)\(32\!\cdots\!72\)\( T^{12} + \)\(12\!\cdots\!84\)\( T^{13} + \)\(81\!\cdots\!68\)\( T^{14} + \)\(19\!\cdots\!72\)\( T^{15} + \)\(84\!\cdots\!21\)\( T^{16} \))
$43$ (\( 1 + 1110947880 T + 4408636810310648674 T^{2} + \)\(43\!\cdots\!56\)\( T^{3} + \)\(91\!\cdots\!47\)\( T^{4} + \)\(72\!\cdots\!24\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} + \)\(67\!\cdots\!68\)\( T^{7} + \)\(79\!\cdots\!03\)\( T^{8} + \)\(34\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!74\)\( T^{10} + \)\(76\!\cdots\!60\)\( T^{11} + \)\(64\!\cdots\!49\)\( T^{12} \))(\( 1 - 2219398472 T + 5852627735858357528 T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!40\)\( T^{4} - \)\(15\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!16\)\( T^{6} - \)\(20\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!62\)\( T^{8} - \)\(18\!\cdots\!36\)\( T^{9} + \)\(17\!\cdots\!84\)\( T^{10} - \)\(12\!\cdots\!24\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} - \)\(59\!\cdots\!40\)\( T^{13} + \)\(37\!\cdots\!72\)\( T^{14} - \)\(13\!\cdots\!96\)\( T^{15} + \)\(55\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 + 5828211928 T + 26899437797946093338 T^{2} + \)\(82\!\cdots\!32\)\( T^{3} + \)\(21\!\cdots\!35\)\( T^{4} + \)\(43\!\cdots\!44\)\( T^{5} + \)\(76\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!32\)\( T^{7} + \)\(13\!\cdots\!15\)\( T^{8} + \)\(12\!\cdots\!64\)\( T^{9} + \)\(10\!\cdots\!78\)\( T^{10} + \)\(53\!\cdots\!04\)\( T^{11} + \)\(22\!\cdots\!29\)\( T^{12} \))(\( 1 - 260684408 T + 15142626249200842008 T^{2} - \)\(49\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!64\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!20\)\( T^{6} - \)\(15\!\cdots\!44\)\( T^{7} + \)\(14\!\cdots\!46\)\( T^{8} - \)\(37\!\cdots\!32\)\( T^{9} + \)\(29\!\cdots\!80\)\( T^{10} - \)\(56\!\cdots\!60\)\( T^{11} + \)\(40\!\cdots\!84\)\( T^{12} - \)\(46\!\cdots\!04\)\( T^{13} + \)\(34\!\cdots\!32\)\( T^{14} - \)\(14\!\cdots\!96\)\( T^{15} + \)\(13\!\cdots\!61\)\( T^{16} \))
$53$ (\( 1 + 9889898636 T + 71408594115749981834 T^{2} + \)\(37\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!43\)\( T^{4} + \)\(61\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!76\)\( T^{6} + \)\(56\!\cdots\!96\)\( T^{7} + \)\(14\!\cdots\!87\)\( T^{8} + \)\(29\!\cdots\!48\)\( T^{9} + \)\(52\!\cdots\!54\)\( T^{10} + \)\(67\!\cdots\!52\)\( T^{11} + \)\(63\!\cdots\!29\)\( T^{12} \))(\( 1 - 9402026896 T + 74821093305090747864 T^{2} - \)\(44\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!28\)\( T^{4} - \)\(96\!\cdots\!20\)\( T^{5} + \)\(37\!\cdots\!20\)\( T^{6} - \)\(13\!\cdots\!28\)\( T^{7} + \)\(42\!\cdots\!22\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{9} + \)\(32\!\cdots\!80\)\( T^{10} - \)\(77\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!68\)\( T^{12} - \)\(30\!\cdots\!28\)\( T^{13} + \)\(47\!\cdots\!56\)\( T^{14} - \)\(55\!\cdots\!48\)\( T^{15} + \)\(54\!\cdots\!61\)\( T^{16} \))
$59$ (\( 1 - 204095112 T + 93492548327993904514 T^{2} - \)\(93\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!15\)\( T^{4} - \)\(65\!\cdots\!32\)\( T^{5} + \)\(15\!\cdots\!24\)\( T^{6} - \)\(19\!\cdots\!88\)\( T^{7} + \)\(41\!\cdots\!15\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{9} + \)\(77\!\cdots\!54\)\( T^{10} - \)\(50\!\cdots\!88\)\( T^{11} + \)\(75\!\cdots\!41\)\( T^{12} \))(\( 1 - 14325543480 T + \)\(24\!\cdots\!04\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!60\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(82\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!06\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{9} + \)\(12\!\cdots\!24\)\( T^{10} - \)\(50\!\cdots\!16\)\( T^{11} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(60\!\cdots\!20\)\( T^{13} + \)\(18\!\cdots\!64\)\( T^{14} - \)\(32\!\cdots\!20\)\( T^{15} + \)\(68\!\cdots\!21\)\( T^{16} \))
$61$ (\( 1 + 15864546948 T + \)\(31\!\cdots\!70\)\( T^{2} + \)\(33\!\cdots\!88\)\( T^{3} + \)\(37\!\cdots\!75\)\( T^{4} + \)\(28\!\cdots\!04\)\( T^{5} + \)\(22\!\cdots\!72\)\( T^{6} + \)\(12\!\cdots\!44\)\( T^{7} + \)\(70\!\cdots\!75\)\( T^{8} + \)\(27\!\cdots\!28\)\( T^{9} + \)\(11\!\cdots\!70\)\( T^{10} + \)\(24\!\cdots\!48\)\( T^{11} + \)\(67\!\cdots\!61\)\( T^{12} \))(\( 1 + 9811064576 T + \)\(29\!\cdots\!08\)\( T^{2} + \)\(23\!\cdots\!68\)\( T^{3} + \)\(39\!\cdots\!36\)\( T^{4} + \)\(26\!\cdots\!72\)\( T^{5} + \)\(30\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!40\)\( T^{7} + \)\(16\!\cdots\!66\)\( T^{8} + \)\(74\!\cdots\!40\)\( T^{9} + \)\(58\!\cdots\!72\)\( T^{10} + \)\(21\!\cdots\!32\)\( T^{11} + \)\(14\!\cdots\!76\)\( T^{12} + \)\(37\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!88\)\( T^{14} + \)\(28\!\cdots\!96\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \))
$67$ (\( 1 - 17196640232 T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(74\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!55\)\( T^{4} - \)\(15\!\cdots\!36\)\( T^{5} + \)\(22\!\cdots\!88\)\( T^{6} - \)\(18\!\cdots\!88\)\( T^{7} + \)\(21\!\cdots\!95\)\( T^{8} - \)\(13\!\cdots\!80\)\( T^{9} + \)\(12\!\cdots\!70\)\( T^{10} - \)\(46\!\cdots\!76\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \))(\( 1 - 52928023248 T + \)\(19\!\cdots\!60\)\( T^{2} - \)\(51\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!84\)\( T^{4} - \)\(20\!\cdots\!04\)\( T^{5} + \)\(31\!\cdots\!16\)\( T^{6} - \)\(42\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!02\)\( T^{8} - \)\(51\!\cdots\!24\)\( T^{9} + \)\(46\!\cdots\!24\)\( T^{10} - \)\(36\!\cdots\!48\)\( T^{11} + \)\(24\!\cdots\!64\)\( T^{12} - \)\(13\!\cdots\!88\)\( T^{13} + \)\(64\!\cdots\!40\)\( T^{14} - \)\(21\!\cdots\!96\)\( T^{15} + \)\(49\!\cdots\!41\)\( T^{16} \))
$71$ (\( 1 - 8751653884 T + \)\(16\!\cdots\!66\)\( T^{2} - \)\(49\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(22\!\cdots\!56\)\( T^{6} - \)\(30\!\cdots\!80\)\( T^{7} + \)\(73\!\cdots\!15\)\( T^{8} - \)\(60\!\cdots\!56\)\( T^{9} + \)\(47\!\cdots\!46\)\( T^{10} - \)\(57\!\cdots\!84\)\( T^{11} + \)\(15\!\cdots\!21\)\( T^{12} \))(\( 1 + 34868356504 T + \)\(15\!\cdots\!56\)\( T^{2} + \)\(41\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(22\!\cdots\!60\)\( T^{5} + \)\(42\!\cdots\!00\)\( T^{6} + \)\(75\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!94\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!00\)\( T^{10} + \)\(27\!\cdots\!60\)\( T^{11} + \)\(30\!\cdots\!36\)\( T^{12} + \)\(27\!\cdots\!96\)\( T^{13} + \)\(23\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!64\)\( T^{15} + \)\(81\!\cdots\!61\)\( T^{16} \))
$73$ (\( 1 + 13704156916 T + \)\(17\!\cdots\!54\)\( T^{2} + \)\(20\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!03\)\( T^{4} + \)\(12\!\cdots\!32\)\( T^{5} + \)\(56\!\cdots\!12\)\( T^{6} + \)\(40\!\cdots\!64\)\( T^{7} + \)\(13\!\cdots\!87\)\( T^{8} + \)\(64\!\cdots\!32\)\( T^{9} + \)\(17\!\cdots\!14\)\( T^{10} + \)\(41\!\cdots\!12\)\( T^{11} + \)\(95\!\cdots\!89\)\( T^{12} \))(\( 1 - 1248764080 T + \)\(10\!\cdots\!00\)\( T^{2} - \)\(30\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!40\)\( T^{4} - \)\(47\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!40\)\( T^{6} - \)\(30\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!46\)\( T^{8} - \)\(94\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!60\)\( T^{10} - \)\(14\!\cdots\!32\)\( T^{11} + \)\(45\!\cdots\!40\)\( T^{12} - \)\(92\!\cdots\!72\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} - \)\(37\!\cdots\!40\)\( T^{15} + \)\(93\!\cdots\!81\)\( T^{16} \))
$79$ (\( 1 + 89923384436 T + \)\(53\!\cdots\!82\)\( T^{2} + \)\(21\!\cdots\!52\)\( T^{3} + \)\(68\!\cdots\!91\)\( T^{4} + \)\(18\!\cdots\!36\)\( T^{5} + \)\(48\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!44\)\( T^{7} + \)\(38\!\cdots\!31\)\( T^{8} + \)\(90\!\cdots\!28\)\( T^{9} + \)\(16\!\cdots\!42\)\( T^{10} + \)\(21\!\cdots\!64\)\( T^{11} + \)\(17\!\cdots\!21\)\( T^{12} \))(\( 1 - 18209008736 T + \)\(46\!\cdots\!68\)\( T^{2} - \)\(66\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} - \)\(12\!\cdots\!32\)\( T^{7} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(92\!\cdots\!28\)\( T^{9} + \)\(74\!\cdots\!08\)\( T^{10} - \)\(47\!\cdots\!56\)\( T^{11} + \)\(31\!\cdots\!28\)\( T^{12} - \)\(15\!\cdots\!88\)\( T^{13} + \)\(81\!\cdots\!28\)\( T^{14} - \)\(23\!\cdots\!24\)\( T^{15} + \)\(97\!\cdots\!61\)\( T^{16} \))
$83$ (\( 1 + 26042106648 T + \)\(51\!\cdots\!62\)\( T^{2} + \)\(83\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} + \)\(15\!\cdots\!64\)\( T^{5} + \)\(19\!\cdots\!36\)\( T^{6} + \)\(19\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!07\)\( T^{8} + \)\(17\!\cdots\!88\)\( T^{9} + \)\(14\!\cdots\!02\)\( T^{10} + \)\(92\!\cdots\!36\)\( T^{11} + \)\(45\!\cdots\!69\)\( T^{12} \))(\( 1 - 169643760088 T + \)\(16\!\cdots\!88\)\( T^{2} - \)\(11\!\cdots\!16\)\( T^{3} + \)\(59\!\cdots\!20\)\( T^{4} - \)\(28\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!36\)\( T^{6} - \)\(52\!\cdots\!92\)\( T^{7} + \)\(19\!\cdots\!98\)\( T^{8} - \)\(67\!\cdots\!64\)\( T^{9} + \)\(20\!\cdots\!04\)\( T^{10} - \)\(60\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!20\)\( T^{12} - \)\(39\!\cdots\!12\)\( T^{13} + \)\(74\!\cdots\!72\)\( T^{14} - \)\(99\!\cdots\!24\)\( T^{15} + \)\(75\!\cdots\!41\)\( T^{16} \))
$89$ (\( 1 - 53269579420 T + \)\(27\!\cdots\!54\)\( T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!99\)\( T^{4} - \)\(75\!\cdots\!36\)\( T^{5} + \)\(39\!\cdots\!32\)\( T^{6} - \)\(20\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!79\)\( T^{8} - \)\(34\!\cdots\!64\)\( T^{9} + \)\(16\!\cdots\!14\)\( T^{10} - \)\(87\!\cdots\!80\)\( T^{11} + \)\(45\!\cdots\!61\)\( T^{12} \))(\( 1 - 201694397904 T + \)\(33\!\cdots\!88\)\( T^{2} - \)\(37\!\cdots\!16\)\( T^{3} + \)\(37\!\cdots\!04\)\( T^{4} - \)\(29\!\cdots\!84\)\( T^{5} + \)\(21\!\cdots\!68\)\( T^{6} - \)\(13\!\cdots\!44\)\( T^{7} + \)\(74\!\cdots\!54\)\( T^{8} - \)\(36\!\cdots\!16\)\( T^{9} + \)\(16\!\cdots\!28\)\( T^{10} - \)\(63\!\cdots\!96\)\( T^{11} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(62\!\cdots\!84\)\( T^{13} + \)\(15\!\cdots\!68\)\( T^{14} - \)\(25\!\cdots\!16\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} \))
$97$ (\( 1 + 106272517044 T + \)\(32\!\cdots\!18\)\( T^{2} + \)\(26\!\cdots\!12\)\( T^{3} + \)\(47\!\cdots\!55\)\( T^{4} + \)\(30\!\cdots\!28\)\( T^{5} + \)\(41\!\cdots\!60\)\( T^{6} + \)\(21\!\cdots\!84\)\( T^{7} + \)\(24\!\cdots\!95\)\( T^{8} + \)\(97\!\cdots\!24\)\( T^{9} + \)\(86\!\cdots\!58\)\( T^{10} + \)\(19\!\cdots\!92\)\( T^{11} + \)\(13\!\cdots\!29\)\( T^{12} \))(\( 1 - 163430440672 T + \)\(17\!\cdots\!52\)\( T^{2} - \)\(95\!\cdots\!80\)\( T^{3} - \)\(86\!\cdots\!68\)\( T^{4} + \)\(56\!\cdots\!08\)\( T^{5} - \)\(33\!\cdots\!56\)\( T^{6} - \)\(18\!\cdots\!36\)\( T^{7} + \)\(36\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!08\)\( T^{9} - \)\(16\!\cdots\!04\)\( T^{10} + \)\(20\!\cdots\!16\)\( T^{11} - \)\(22\!\cdots\!08\)\( T^{12} - \)\(17\!\cdots\!40\)\( T^{13} + \)\(23\!\cdots\!08\)\( T^{14} - \)\(15\!\cdots\!64\)\( T^{15} + \)\(68\!\cdots\!61\)\( T^{16} \))
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