# Properties

 Label 13.5.f Level 13 Weight 5 Character orbit f Rep. character $$\chi_{13}(2,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 16 Newform subspaces 1 Sturm bound 5 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 13.f (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$1$$ Sturm bound: $$5$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(13, [\chi])$$.

Total New Old
Modular forms 24 24 0
Cusp forms 16 16 0
Eisenstein series 8 8 0

## Trace form

 $$16q - 4q^{2} - 2q^{3} - 6q^{4} + 8q^{5} - 38q^{6} + 56q^{7} + 90q^{8} - 164q^{9} + O(q^{10})$$ $$16q - 4q^{2} - 2q^{3} - 6q^{4} + 8q^{5} - 38q^{6} + 56q^{7} + 90q^{8} - 164q^{9} - 486q^{10} - 100q^{11} + 294q^{13} + 808q^{14} + 346q^{15} + 230q^{16} + 984q^{17} + 2434q^{18} - 1498q^{19} - 3962q^{20} - 1076q^{21} - 1524q^{22} - 1014q^{23} - 2142q^{24} + 614q^{26} + 3352q^{27} + 5764q^{28} + 814q^{29} + 9162q^{30} + 4060q^{31} - 4996q^{32} - 5636q^{33} - 2502q^{34} - 4892q^{35} - 15750q^{36} - 1790q^{37} + 6982q^{39} + 18816q^{40} + 4280q^{41} - 1204q^{42} - 1368q^{43} + 10736q^{44} - 6806q^{45} - 15246q^{46} + 1484q^{47} - 3002q^{48} - 11820q^{49} - 13574q^{50} + 1432q^{52} + 7204q^{53} + 13240q^{54} + 6936q^{55} + 8124q^{56} + 12736q^{57} + 3030q^{58} - 2380q^{59} - 6472q^{60} - 162q^{61} + 19614q^{62} + 12004q^{63} - 5248q^{65} - 23872q^{66} - 14854q^{67} - 6444q^{68} + 2412q^{69} - 34524q^{70} - 8050q^{71} + 15420q^{72} - 15448q^{73} - 2882q^{74} + 8280q^{75} + 10622q^{76} - 11672q^{78} - 17064q^{79} + 2564q^{80} + 2128q^{81} - 5346q^{82} + 12788q^{83} + 25948q^{84} + 35382q^{85} + 67260q^{86} + 29342q^{87} + 40836q^{88} + 20492q^{89} + 8996q^{91} - 49884q^{92} - 78920q^{93} - 30606q^{94} - 98574q^{95} - 94664q^{96} + 50944q^{97} + 61484q^{98} - 21632q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
13.5.f.a $$16$$ $$1.344$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-4$$ $$-2$$ $$8$$ $$56$$ $$q-\beta _{12}q^{2}+(-1-\beta _{8}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 11 T^{2} - 18 T^{3} + 75 T^{4} + 502 T^{5} + 2554 T^{6} + 9868 T^{7} + 22616 T^{8} + 29288 T^{9} - 27048 T^{10} - 898544 T^{11} - 17177376 T^{12} - 108914720 T^{13} - 426373536 T^{14} - 382913728 T^{15} - 329617344 T^{16} - 6126619648 T^{17} - 109151625216 T^{18} - 446114693120 T^{19} - 1125736513536 T^{20} - 942191673344 T^{21} - 453790138368 T^{22} + 7861937635328 T^{23} + 97134980366336 T^{24} + 678123796430848 T^{25} + 2808152697339904 T^{26} + 8831277394296832 T^{27} + 21110623253299200 T^{28} - 81064793292668928 T^{29} + 792633534417207296 T^{30} + 4611686018427387904 T^{31} + 18446744073709551616 T^{32}$$
$3$ $$1 + 2 T - 240 T^{2} - 1492 T^{3} + 22927 T^{4} + 271668 T^{5} - 890196 T^{6} - 21609198 T^{7} - 25045623 T^{8} + 1172175192 T^{9} + 5748348060 T^{10} - 93679542360 T^{11} - 1026060864642 T^{12} + 8900356926348 T^{13} + 169584054967116 T^{14} - 370754464597920 T^{15} - 17423631492664230 T^{16} - 30031111632431520 T^{17} + 1112640984639248076 T^{18} + 4730014585295307468 T^{19} - 44168555769262938882 T^{20} -$$$$32\!\cdots\!60$$$$T^{21} +$$$$16\!\cdots\!60$$$$T^{22} +$$$$26\!\cdots\!12$$$$T^{23} -$$$$46\!\cdots\!43$$$$T^{24} -$$$$32\!\cdots\!58$$$$T^{25} -$$$$10\!\cdots\!96$$$$T^{26} +$$$$26\!\cdots\!08$$$$T^{27} +$$$$18\!\cdots\!47$$$$T^{28} -$$$$96\!\cdots\!72$$$$T^{29} -$$$$12\!\cdots\!40$$$$T^{30} +$$$$84\!\cdots\!02$$$$T^{31} +$$$$34\!\cdots\!81$$$$T^{32}$$
$5$ $$1 - 8 T + 32 T^{2} - 12096 T^{3} - 527442 T^{4} + 6817816 T^{5} + 35492224 T^{6} + 3499118776 T^{7} + 214598054801 T^{8} - 1770375811288 T^{9} + 6931925138112 T^{10} - 204666718110056 T^{11} - 91059199232161746 T^{12} - 449348487028373168 T^{13} - 924223306284337248 T^{14} -$$$$13\!\cdots\!88$$$$T^{15} +$$$$53\!\cdots\!36$$$$T^{16} -$$$$84\!\cdots\!00$$$$T^{17} -$$$$36\!\cdots\!00$$$$T^{18} -$$$$10\!\cdots\!00$$$$T^{19} -$$$$13\!\cdots\!50$$$$T^{20} -$$$$19\!\cdots\!00$$$$T^{21} +$$$$41\!\cdots\!00$$$$T^{22} -$$$$65\!\cdots\!00$$$$T^{23} +$$$$49\!\cdots\!25$$$$T^{24} +$$$$50\!\cdots\!00$$$$T^{25} +$$$$32\!\cdots\!00$$$$T^{26} +$$$$38\!\cdots\!00$$$$T^{27} -$$$$18\!\cdots\!50$$$$T^{28} -$$$$26\!\cdots\!00$$$$T^{29} +$$$$44\!\cdots\!00$$$$T^{30} -$$$$69\!\cdots\!00$$$$T^{31} +$$$$54\!\cdots\!25$$$$T^{32}$$
$7$ $$1 - 56 T + 7478 T^{2} - 202660 T^{3} + 15425611 T^{4} + 147757036 T^{5} + 351071986 T^{6} + 612647989816 T^{7} + 16654656981341 T^{8} - 3342938469481784 T^{9} + 73292174911510116 T^{10} - 4996161548264166712 T^{11} -$$$$32\!\cdots\!86$$$$T^{12} +$$$$46\!\cdots\!48$$$$T^{13} -$$$$17\!\cdots\!68$$$$T^{14} -$$$$47\!\cdots\!88$$$$T^{15} +$$$$27\!\cdots\!66$$$$T^{16} -$$$$11\!\cdots\!88$$$$T^{17} -$$$$10\!\cdots\!68$$$$T^{18} +$$$$64\!\cdots\!48$$$$T^{19} -$$$$10\!\cdots\!86$$$$T^{20} -$$$$39\!\cdots\!12$$$$T^{21} +$$$$14\!\cdots\!16$$$$T^{22} -$$$$15\!\cdots\!84$$$$T^{23} +$$$$18\!\cdots\!41$$$$T^{24} +$$$$16\!\cdots\!16$$$$T^{25} +$$$$22\!\cdots\!86$$$$T^{26} +$$$$22\!\cdots\!36$$$$T^{27} +$$$$56\!\cdots\!11$$$$T^{28} -$$$$17\!\cdots\!60$$$$T^{29} +$$$$15\!\cdots\!78$$$$T^{30} -$$$$28\!\cdots\!56$$$$T^{31} +$$$$12\!\cdots\!01$$$$T^{32}$$
$11$ $$1 + 100 T + 27170 T^{2} - 106032 T^{3} + 1057947 T^{4} - 44543342240 T^{5} - 2123313764498 T^{6} - 578703081018404 T^{7} - 42234317976348187 T^{8} - 16300704176801189728 T^{9} -$$$$18\!\cdots\!52$$$$T^{10} -$$$$15\!\cdots\!56$$$$T^{11} +$$$$32\!\cdots\!42$$$$T^{12} +$$$$33\!\cdots\!12$$$$T^{13} +$$$$38\!\cdots\!24$$$$T^{14} +$$$$42\!\cdots\!68$$$$T^{15} +$$$$31\!\cdots\!70$$$$T^{16} +$$$$62\!\cdots\!88$$$$T^{17} +$$$$81\!\cdots\!44$$$$T^{18} +$$$$10\!\cdots\!52$$$$T^{19} +$$$$14\!\cdots\!62$$$$T^{20} -$$$$10\!\cdots\!56$$$$T^{21} -$$$$18\!\cdots\!32$$$$T^{22} -$$$$23\!\cdots\!68$$$$T^{23} -$$$$89\!\cdots\!27$$$$T^{24} -$$$$17\!\cdots\!44$$$$T^{25} -$$$$96\!\cdots\!98$$$$T^{26} -$$$$29\!\cdots\!40$$$$T^{27} +$$$$10\!\cdots\!07$$$$T^{28} -$$$$15\!\cdots\!72$$$$T^{29} +$$$$56\!\cdots\!70$$$$T^{30} +$$$$30\!\cdots\!00$$$$T^{31} +$$$$44\!\cdots\!41$$$$T^{32}$$
$13$ $$1 - 294 T - 66586 T^{2} + 42455504 T^{3} - 2316760667 T^{4} - 2128791278900 T^{5} + 410094472234278 T^{6} + 33452215036870058 T^{7} - 17726001609919307172 T^{8} +$$$$95\!\cdots\!38$$$$T^{9} +$$$$33\!\cdots\!38$$$$T^{10} -$$$$49\!\cdots\!00$$$$T^{11} -$$$$15\!\cdots\!47$$$$T^{12} +$$$$80\!\cdots\!04$$$$T^{13} -$$$$36\!\cdots\!46$$$$T^{14} -$$$$45\!\cdots\!74$$$$T^{15} +$$$$44\!\cdots\!81$$$$T^{16}$$
$17$ $$1 - 984 T + 934532 T^{2} - 601991520 T^{3} + 366227956458 T^{4} - 186786892356648 T^{5} + 90290052706451368 T^{6} - 39430381111636884696 T^{7} +$$$$16\!\cdots\!09$$$$T^{8} -$$$$64\!\cdots\!32$$$$T^{9} +$$$$23\!\cdots\!64$$$$T^{10} -$$$$85\!\cdots\!84$$$$T^{11} +$$$$29\!\cdots\!86$$$$T^{12} -$$$$95\!\cdots\!24$$$$T^{13} +$$$$30\!\cdots\!88$$$$T^{14} -$$$$92\!\cdots\!00$$$$T^{15} +$$$$27\!\cdots\!68$$$$T^{16} -$$$$77\!\cdots\!00$$$$T^{17} +$$$$21\!\cdots\!08$$$$T^{18} -$$$$55\!\cdots\!64$$$$T^{19} +$$$$14\!\cdots\!66$$$$T^{20} -$$$$34\!\cdots\!84$$$$T^{21} +$$$$81\!\cdots\!44$$$$T^{22} -$$$$18\!\cdots\!12$$$$T^{23} +$$$$38\!\cdots\!49$$$$T^{24} -$$$$77\!\cdots\!76$$$$T^{25} +$$$$14\!\cdots\!68$$$$T^{26} -$$$$25\!\cdots\!08$$$$T^{27} +$$$$42\!\cdots\!78$$$$T^{28} -$$$$57\!\cdots\!20$$$$T^{29} +$$$$75\!\cdots\!92$$$$T^{30} -$$$$66\!\cdots\!84$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$19$ $$1 + 1498 T + 1242140 T^{2} + 605628956 T^{3} + 144809054215 T^{4} - 30840378908480 T^{5} - 42820390632894416 T^{6} - 17457142236019195214 T^{7} -$$$$21\!\cdots\!35$$$$T^{8} +$$$$14\!\cdots\!12$$$$T^{9} +$$$$88\!\cdots\!60$$$$T^{10} +$$$$16\!\cdots\!32$$$$T^{11} -$$$$48\!\cdots\!18$$$$T^{12} -$$$$39\!\cdots\!12$$$$T^{13} -$$$$82\!\cdots\!76$$$$T^{14} +$$$$18\!\cdots\!60$$$$T^{15} +$$$$15\!\cdots\!38$$$$T^{16} +$$$$23\!\cdots\!60$$$$T^{17} -$$$$14\!\cdots\!16$$$$T^{18} -$$$$87\!\cdots\!32$$$$T^{19} -$$$$14\!\cdots\!58$$$$T^{20} +$$$$62\!\cdots\!32$$$$T^{21} +$$$$43\!\cdots\!60$$$$T^{22} +$$$$92\!\cdots\!92$$$$T^{23} -$$$$17\!\cdots\!35$$$$T^{24} -$$$$18\!\cdots\!34$$$$T^{25} -$$$$60\!\cdots\!16$$$$T^{26} -$$$$56\!\cdots\!80$$$$T^{27} +$$$$34\!\cdots\!15$$$$T^{28} +$$$$18\!\cdots\!16$$$$T^{29} +$$$$50\!\cdots\!40$$$$T^{30} +$$$$79\!\cdots\!98$$$$T^{31} +$$$$69\!\cdots\!21$$$$T^{32}$$
$23$ $$1 + 1014 T + 1541624 T^{2} + 1215676488 T^{3} + 1070749656711 T^{4} + 761258748687096 T^{5} + 523855300773466804 T^{6} +$$$$37\!\cdots\!10$$$$T^{7} +$$$$23\!\cdots\!53$$$$T^{8} +$$$$16\!\cdots\!08$$$$T^{9} +$$$$96\!\cdots\!40$$$$T^{10} +$$$$61\!\cdots\!96$$$$T^{11} +$$$$35\!\cdots\!06$$$$T^{12} +$$$$20\!\cdots\!24$$$$T^{13} +$$$$11\!\cdots\!00$$$$T^{14} +$$$$60\!\cdots\!24$$$$T^{15} +$$$$34\!\cdots\!06$$$$T^{16} +$$$$16\!\cdots\!84$$$$T^{17} +$$$$90\!\cdots\!00$$$$T^{18} +$$$$44\!\cdots\!04$$$$T^{19} +$$$$21\!\cdots\!66$$$$T^{20} +$$$$10\!\cdots\!96$$$$T^{21} +$$$$46\!\cdots\!40$$$$T^{22} +$$$$21\!\cdots\!48$$$$T^{23} +$$$$87\!\cdots\!13$$$$T^{24} +$$$$39\!\cdots\!10$$$$T^{25} +$$$$15\!\cdots\!04$$$$T^{26} +$$$$62\!\cdots\!36$$$$T^{27} +$$$$24\!\cdots\!91$$$$T^{28} +$$$$78\!\cdots\!48$$$$T^{29} +$$$$27\!\cdots\!64$$$$T^{30} +$$$$51\!\cdots\!14$$$$T^{31} +$$$$14\!\cdots\!41$$$$T^{32}$$
$29$ $$1 - 814 T - 3744612 T^{2} + 2089659432 T^{3} + 8177655703406 T^{4} - 2736264841720610 T^{5} - 12804081872369783884 T^{6} +$$$$25\!\cdots\!10$$$$T^{7} +$$$$15\!\cdots\!13$$$$T^{8} -$$$$18\!\cdots\!86$$$$T^{9} -$$$$16\!\cdots\!00$$$$T^{10} +$$$$13\!\cdots\!06$$$$T^{11} +$$$$14\!\cdots\!38$$$$T^{12} -$$$$78\!\cdots\!60$$$$T^{13} -$$$$12\!\cdots\!92$$$$T^{14} +$$$$21\!\cdots\!10$$$$T^{15} +$$$$93\!\cdots\!80$$$$T^{16} +$$$$15\!\cdots\!10$$$$T^{17} -$$$$62\!\cdots\!12$$$$T^{18} -$$$$27\!\cdots\!60$$$$T^{19} +$$$$37\!\cdots\!98$$$$T^{20} +$$$$23\!\cdots\!06$$$$T^{21} -$$$$20\!\cdots\!00$$$$T^{22} -$$$$16\!\cdots\!46$$$$T^{23} +$$$$97\!\cdots\!33$$$$T^{24} +$$$$11\!\cdots\!10$$$$T^{25} -$$$$40\!\cdots\!84$$$$T^{26} -$$$$60\!\cdots\!10$$$$T^{27} +$$$$12\!\cdots\!66$$$$T^{28} +$$$$23\!\cdots\!12$$$$T^{29} -$$$$29\!\cdots\!52$$$$T^{30} -$$$$45\!\cdots\!14$$$$T^{31} +$$$$39\!\cdots\!81$$$$T^{32}$$
$31$ $$1 - 4060 T + 8241800 T^{2} - 11888153764 T^{3} + 12102195192780 T^{4} - 6844355067784460 T^{5} - 1291690806395612552 T^{6} +$$$$91\!\cdots\!16$$$$T^{7} -$$$$12\!\cdots\!64$$$$T^{8} +$$$$80\!\cdots\!00$$$$T^{9} +$$$$12\!\cdots\!36$$$$T^{10} -$$$$10\!\cdots\!52$$$$T^{11} +$$$$15\!\cdots\!04$$$$T^{12} -$$$$11\!\cdots\!48$$$$T^{13} +$$$$36\!\cdots\!28$$$$T^{14} +$$$$41\!\cdots\!28$$$$T^{15} -$$$$74\!\cdots\!42$$$$T^{16} +$$$$38\!\cdots\!88$$$$T^{17} +$$$$31\!\cdots\!48$$$$T^{18} -$$$$93\!\cdots\!28$$$$T^{19} +$$$$11\!\cdots\!24$$$$T^{20} -$$$$72\!\cdots\!52$$$$T^{21} +$$$$78\!\cdots\!56$$$$T^{22} +$$$$46\!\cdots\!00$$$$T^{23} -$$$$66\!\cdots\!04$$$$T^{24} +$$$$44\!\cdots\!96$$$$T^{25} -$$$$58\!\cdots\!52$$$$T^{26} -$$$$28\!\cdots\!60$$$$T^{27} +$$$$46\!\cdots\!80$$$$T^{28} -$$$$42\!\cdots\!04$$$$T^{29} +$$$$27\!\cdots\!00$$$$T^{30} -$$$$12\!\cdots\!60$$$$T^{31} +$$$$27\!\cdots\!21$$$$T^{32}$$
$37$ $$1 + 1790 T + 8613908 T^{2} + 6266080760 T^{3} + 17905874153442 T^{4} - 23706724104957758 T^{5} - 22615506964230414788 T^{6} -$$$$14\!\cdots\!38$$$$T^{7} -$$$$62\!\cdots\!27$$$$T^{8} -$$$$13\!\cdots\!82$$$$T^{9} +$$$$30\!\cdots\!92$$$$T^{10} +$$$$33\!\cdots\!18$$$$T^{11} +$$$$99\!\cdots\!54$$$$T^{12} +$$$$24\!\cdots\!76$$$$T^{13} +$$$$15\!\cdots\!00$$$$T^{14} -$$$$24\!\cdots\!14$$$$T^{15} -$$$$23\!\cdots\!64$$$$T^{16} -$$$$45\!\cdots\!54$$$$T^{17} +$$$$53\!\cdots\!00$$$$T^{18} +$$$$16\!\cdots\!56$$$$T^{19} +$$$$12\!\cdots\!14$$$$T^{20} +$$$$77\!\cdots\!18$$$$T^{21} +$$$$13\!\cdots\!12$$$$T^{22} -$$$$11\!\cdots\!22$$$$T^{23} -$$$$94\!\cdots\!87$$$$T^{24} -$$$$40\!\cdots\!58$$$$T^{25} -$$$$12\!\cdots\!88$$$$T^{26} -$$$$23\!\cdots\!38$$$$T^{27} +$$$$33\!\cdots\!82$$$$T^{28} +$$$$22\!\cdots\!60$$$$T^{29} +$$$$56\!\cdots\!28$$$$T^{30} +$$$$22\!\cdots\!90$$$$T^{31} +$$$$23\!\cdots\!61$$$$T^{32}$$
$41$ $$1 - 4280 T + 16206692 T^{2} - 41605604352 T^{3} + 89007507297078 T^{4} - 156507905015338472 T^{5} +$$$$22\!\cdots\!52$$$$T^{6} -$$$$29\!\cdots\!84$$$$T^{7} +$$$$29\!\cdots\!57$$$$T^{8} -$$$$44\!\cdots\!84$$$$T^{9} +$$$$61\!\cdots\!76$$$$T^{10} -$$$$12\!\cdots\!68$$$$T^{11} +$$$$17\!\cdots\!90$$$$T^{12} -$$$$89\!\cdots\!44$$$$T^{13} -$$$$29\!\cdots\!48$$$$T^{14} +$$$$15\!\cdots\!64$$$$T^{15} -$$$$27\!\cdots\!92$$$$T^{16} +$$$$44\!\cdots\!04$$$$T^{17} -$$$$23\!\cdots\!08$$$$T^{18} -$$$$20\!\cdots\!64$$$$T^{19} +$$$$11\!\cdots\!90$$$$T^{20} -$$$$23\!\cdots\!68$$$$T^{21} +$$$$31\!\cdots\!36$$$$T^{22} -$$$$64\!\cdots\!64$$$$T^{23} +$$$$12\!\cdots\!17$$$$T^{24} -$$$$34\!\cdots\!44$$$$T^{25} +$$$$72\!\cdots\!52$$$$T^{26} -$$$$14\!\cdots\!92$$$$T^{27} +$$$$23\!\cdots\!38$$$$T^{28} -$$$$30\!\cdots\!12$$$$T^{29} +$$$$33\!\cdots\!72$$$$T^{30} -$$$$25\!\cdots\!80$$$$T^{31} +$$$$16\!\cdots\!61$$$$T^{32}$$
$43$ $$1 + 1368 T + 13386518 T^{2} + 17459387280 T^{3} + 89338317662175 T^{4} + 129060665859360744 T^{5} +$$$$40\!\cdots\!42$$$$T^{6} +$$$$73\!\cdots\!56$$$$T^{7} +$$$$14\!\cdots\!61$$$$T^{8} +$$$$35\!\cdots\!04$$$$T^{9} +$$$$55\!\cdots\!44$$$$T^{10} +$$$$14\!\cdots\!84$$$$T^{11} +$$$$24\!\cdots\!50$$$$T^{12} +$$$$52\!\cdots\!76$$$$T^{13} +$$$$10\!\cdots\!68$$$$T^{14} +$$$$17\!\cdots\!08$$$$T^{15} +$$$$39\!\cdots\!58$$$$T^{16} +$$$$61\!\cdots\!08$$$$T^{17} +$$$$12\!\cdots\!68$$$$T^{18} +$$$$21\!\cdots\!76$$$$T^{19} +$$$$33\!\cdots\!50$$$$T^{20} +$$$$67\!\cdots\!84$$$$T^{21} +$$$$89\!\cdots\!44$$$$T^{22} +$$$$19\!\cdots\!04$$$$T^{23} +$$$$27\!\cdots\!61$$$$T^{24} +$$$$46\!\cdots\!56$$$$T^{25} +$$$$87\!\cdots\!42$$$$T^{26} +$$$$96\!\cdots\!44$$$$T^{27} +$$$$22\!\cdots\!75$$$$T^{28} +$$$$15\!\cdots\!80$$$$T^{29} +$$$$39\!\cdots\!18$$$$T^{30} +$$$$13\!\cdots\!68$$$$T^{31} +$$$$34\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 1484 T + 1101128 T^{2} + 8571170172 T^{3} + 67218351180012 T^{4} - 197396893555353260 T^{5} +$$$$25\!\cdots\!96$$$$T^{6} +$$$$27\!\cdots\!20$$$$T^{7} +$$$$18\!\cdots\!48$$$$T^{8} -$$$$96\!\cdots\!72$$$$T^{9} +$$$$16\!\cdots\!92$$$$T^{10} -$$$$79\!\cdots\!80$$$$T^{11} +$$$$29\!\cdots\!72$$$$T^{12} -$$$$27\!\cdots\!40$$$$T^{13} +$$$$63\!\cdots\!92$$$$T^{14} -$$$$86\!\cdots\!68$$$$T^{15} +$$$$34\!\cdots\!58$$$$T^{16} -$$$$42\!\cdots\!08$$$$T^{17} +$$$$15\!\cdots\!12$$$$T^{18} -$$$$31\!\cdots\!40$$$$T^{19} +$$$$16\!\cdots\!12$$$$T^{20} -$$$$21\!\cdots\!80$$$$T^{21} +$$$$22\!\cdots\!52$$$$T^{22} -$$$$63\!\cdots\!92$$$$T^{23} +$$$$59\!\cdots\!68$$$$T^{24} +$$$$42\!\cdots\!20$$$$T^{25} +$$$$19\!\cdots\!96$$$$T^{26} -$$$$73\!\cdots\!60$$$$T^{27} +$$$$12\!\cdots\!32$$$$T^{28} +$$$$76\!\cdots\!52$$$$T^{29} +$$$$47\!\cdots\!88$$$$T^{30} -$$$$31\!\cdots\!84$$$$T^{31} +$$$$10\!\cdots\!81$$$$T^{32}$$
$53$ $$( 1 - 3602 T + 54510142 T^{2} - 169091095000 T^{3} + 1347824194996053 T^{4} - 3593979789946993116 T^{5} +$$$$19\!\cdots\!86$$$$T^{6} -$$$$44\!\cdots\!26$$$$T^{7} +$$$$19\!\cdots\!96$$$$T^{8} -$$$$35\!\cdots\!06$$$$T^{9} +$$$$12\!\cdots\!46$$$$T^{10} -$$$$17\!\cdots\!56$$$$T^{11} +$$$$52\!\cdots\!13$$$$T^{12} -$$$$51\!\cdots\!00$$$$T^{13} +$$$$13\!\cdots\!02$$$$T^{14} -$$$$68\!\cdots\!22$$$$T^{15} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$
$59$ $$1 + 2380 T + 26675390 T^{2} + 58910752356 T^{3} + 184713311551875 T^{4} - 411423908773382660 T^{5} -$$$$31\!\cdots\!82$$$$T^{6} -$$$$24\!\cdots\!80$$$$T^{7} -$$$$72\!\cdots\!87$$$$T^{8} -$$$$10\!\cdots\!96$$$$T^{9} -$$$$55\!\cdots\!12$$$$T^{10} +$$$$30\!\cdots\!12$$$$T^{11} +$$$$10\!\cdots\!74$$$$T^{12} +$$$$31\!\cdots\!40$$$$T^{13} +$$$$19\!\cdots\!96$$$$T^{14} -$$$$82\!\cdots\!24$$$$T^{15} -$$$$14\!\cdots\!10$$$$T^{16} -$$$$99\!\cdots\!64$$$$T^{17} +$$$$27\!\cdots\!16$$$$T^{18} +$$$$55\!\cdots\!40$$$$T^{19} +$$$$23\!\cdots\!34$$$$T^{20} +$$$$78\!\cdots\!12$$$$T^{21} -$$$$17\!\cdots\!32$$$$T^{22} -$$$$41\!\cdots\!16$$$$T^{23} -$$$$33\!\cdots\!47$$$$T^{24} -$$$$13\!\cdots\!80$$$$T^{25} -$$$$21\!\cdots\!82$$$$T^{26} -$$$$34\!\cdots\!60$$$$T^{27} +$$$$18\!\cdots\!75$$$$T^{28} +$$$$71\!\cdots\!36$$$$T^{29} +$$$$39\!\cdots\!90$$$$T^{30} +$$$$42\!\cdots\!80$$$$T^{31} +$$$$21\!\cdots\!61$$$$T^{32}$$
$61$ $$1 + 162 T - 50302892 T^{2} - 60770764376 T^{3} + 1076406400246734 T^{4} + 2212979358132074374 T^{5} -$$$$12\!\cdots\!00$$$$T^{6} -$$$$23\!\cdots\!38$$$$T^{7} +$$$$12\!\cdots\!45$$$$T^{8} -$$$$19\!\cdots\!06$$$$T^{9} -$$$$19\!\cdots\!64$$$$T^{10} +$$$$60\!\cdots\!86$$$$T^{11} +$$$$31\!\cdots\!06$$$$T^{12} -$$$$33\!\cdots\!96$$$$T^{13} -$$$$25\!\cdots\!92$$$$T^{14} -$$$$37\!\cdots\!34$$$$T^{15} +$$$$15\!\cdots\!84$$$$T^{16} -$$$$52\!\cdots\!94$$$$T^{17} -$$$$49\!\cdots\!52$$$$T^{18} -$$$$89\!\cdots\!16$$$$T^{19} +$$$$11\!\cdots\!66$$$$T^{20} +$$$$30\!\cdots\!86$$$$T^{21} -$$$$14\!\cdots\!24$$$$T^{22} -$$$$19\!\cdots\!86$$$$T^{23} +$$$$16\!\cdots\!45$$$$T^{24} -$$$$43\!\cdots\!18$$$$T^{25} -$$$$32\!\cdots\!00$$$$T^{26} +$$$$79\!\cdots\!34$$$$T^{27} +$$$$53\!\cdots\!54$$$$T^{28} -$$$$41\!\cdots\!96$$$$T^{29} -$$$$47\!\cdots\!12$$$$T^{30} +$$$$21\!\cdots\!62$$$$T^{31} +$$$$18\!\cdots\!41$$$$T^{32}$$
$67$ $$1 + 14854 T + 152246780 T^{2} + 1156949917028 T^{3} + 7271671748877823 T^{4} + 42419416555472380768 T^{5} +$$$$23\!\cdots\!44$$$$T^{6} +$$$$13\!\cdots\!62$$$$T^{7} +$$$$73\!\cdots\!33$$$$T^{8} +$$$$39\!\cdots\!52$$$$T^{9} +$$$$21\!\cdots\!32$$$$T^{10} +$$$$10\!\cdots\!60$$$$T^{11} +$$$$53\!\cdots\!70$$$$T^{12} +$$$$25\!\cdots\!60$$$$T^{13} +$$$$12\!\cdots\!72$$$$T^{14} +$$$$55\!\cdots\!96$$$$T^{15} +$$$$24\!\cdots\!74$$$$T^{16} +$$$$11\!\cdots\!16$$$$T^{17} +$$$$49\!\cdots\!52$$$$T^{18} +$$$$21\!\cdots\!60$$$$T^{19} +$$$$88\!\cdots\!70$$$$T^{20} +$$$$35\!\cdots\!60$$$$T^{21} +$$$$14\!\cdots\!72$$$$T^{22} +$$$$53\!\cdots\!32$$$$T^{23} +$$$$20\!\cdots\!13$$$$T^{24} +$$$$72\!\cdots\!22$$$$T^{25} +$$$$25\!\cdots\!44$$$$T^{26} +$$$$94\!\cdots\!28$$$$T^{27} +$$$$32\!\cdots\!43$$$$T^{28} +$$$$10\!\cdots\!08$$$$T^{29} +$$$$27\!\cdots\!80$$$$T^{30} +$$$$54\!\cdots\!54$$$$T^{31} +$$$$73\!\cdots\!21$$$$T^{32}$$
$71$ $$1 + 8050 T + 23381036 T^{2} + 11907331644 T^{3} - 640051132997169 T^{4} - 10630217014363370528 T^{5} -$$$$61\!\cdots\!68$$$$T^{6} -$$$$22\!\cdots\!54$$$$T^{7} -$$$$28\!\cdots\!59$$$$T^{8} +$$$$44\!\cdots\!28$$$$T^{9} +$$$$49\!\cdots\!12$$$$T^{10} +$$$$30\!\cdots\!36$$$$T^{11} +$$$$12\!\cdots\!98$$$$T^{12} +$$$$11\!\cdots\!48$$$$T^{13} -$$$$13\!\cdots\!32$$$$T^{14} -$$$$16\!\cdots\!32$$$$T^{15} -$$$$11\!\cdots\!78$$$$T^{16} -$$$$43\!\cdots\!92$$$$T^{17} -$$$$87\!\cdots\!52$$$$T^{18} +$$$$19\!\cdots\!68$$$$T^{19} +$$$$52\!\cdots\!58$$$$T^{20} +$$$$32\!\cdots\!36$$$$T^{21} +$$$$13\!\cdots\!72$$$$T^{22} +$$$$30\!\cdots\!08$$$$T^{23} -$$$$49\!\cdots\!19$$$$T^{24} -$$$$10\!\cdots\!34$$$$T^{25} -$$$$68\!\cdots\!68$$$$T^{26} -$$$$30\!\cdots\!68$$$$T^{27} -$$$$46\!\cdots\!09$$$$T^{28} +$$$$21\!\cdots\!04$$$$T^{29} +$$$$10\!\cdots\!56$$$$T^{30} +$$$$95\!\cdots\!50$$$$T^{31} +$$$$30\!\cdots\!81$$$$T^{32}$$
$73$ $$1 + 15448 T + 119320352 T^{2} + 673174693736 T^{3} + 2488100657363446 T^{4} + 439794287857885936 T^{5} -$$$$63\!\cdots\!16$$$$T^{6} -$$$$62\!\cdots\!20$$$$T^{7} -$$$$47\!\cdots\!31$$$$T^{8} -$$$$27\!\cdots\!12$$$$T^{9} -$$$$13\!\cdots\!00$$$$T^{10} -$$$$44\!\cdots\!16$$$$T^{11} -$$$$23\!\cdots\!62$$$$T^{12} +$$$$97\!\cdots\!12$$$$T^{13} +$$$$86\!\cdots\!96$$$$T^{14} +$$$$55\!\cdots\!96$$$$T^{15} +$$$$31\!\cdots\!12$$$$T^{16} +$$$$15\!\cdots\!36$$$$T^{17} +$$$$70\!\cdots\!76$$$$T^{18} +$$$$22\!\cdots\!52$$$$T^{19} -$$$$15\!\cdots\!82$$$$T^{20} -$$$$82\!\cdots\!16$$$$T^{21} -$$$$68\!\cdots\!00$$$$T^{22} -$$$$41\!\cdots\!72$$$$T^{23} -$$$$19\!\cdots\!51$$$$T^{24} -$$$$75\!\cdots\!20$$$$T^{25} -$$$$21\!\cdots\!16$$$$T^{26} +$$$$42\!\cdots\!76$$$$T^{27} +$$$$68\!\cdots\!26$$$$T^{28} +$$$$52\!\cdots\!56$$$$T^{29} +$$$$26\!\cdots\!72$$$$T^{30} +$$$$97\!\cdots\!48$$$$T^{31} +$$$$17\!\cdots\!41$$$$T^{32}$$
$79$ $$( 1 + 8532 T + 181789616 T^{2} + 1264697206412 T^{3} + 17222120260071052 T^{4} + 98573552246045188900 T^{5} +$$$$10\!\cdots\!52$$$$T^{6} +$$$$51\!\cdots\!68$$$$T^{7} +$$$$46\!\cdots\!78$$$$T^{8} +$$$$20\!\cdots\!08$$$$T^{9} +$$$$15\!\cdots\!72$$$$T^{10} +$$$$58\!\cdots\!00$$$$T^{11} +$$$$39\!\cdots\!92$$$$T^{12} +$$$$11\!\cdots\!12$$$$T^{13} +$$$$63\!\cdots\!96$$$$T^{14} +$$$$11\!\cdots\!52$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$
$83$ $$1 - 12788 T + 81766472 T^{2} - 219288483636 T^{3} + 2030465335614048 T^{4} - 45830993951572452908 T^{5} +$$$$44\!\cdots\!96$$$$T^{6} -$$$$25\!\cdots\!88$$$$T^{7} +$$$$16\!\cdots\!16$$$$T^{8} -$$$$16\!\cdots\!32$$$$T^{9} +$$$$14\!\cdots\!56$$$$T^{10} -$$$$91\!\cdots\!60$$$$T^{11} +$$$$55\!\cdots\!64$$$$T^{12} -$$$$46\!\cdots\!64$$$$T^{13} +$$$$38\!\cdots\!88$$$$T^{14} -$$$$25\!\cdots\!16$$$$T^{15} +$$$$16\!\cdots\!58$$$$T^{16} -$$$$12\!\cdots\!36$$$$T^{17} +$$$$87\!\cdots\!08$$$$T^{18} -$$$$49\!\cdots\!04$$$$T^{19} +$$$$28\!\cdots\!84$$$$T^{20} -$$$$21\!\cdots\!60$$$$T^{21} +$$$$16\!\cdots\!76$$$$T^{22} -$$$$88\!\cdots\!12$$$$T^{23} +$$$$41\!\cdots\!76$$$$T^{24} -$$$$31\!\cdots\!28$$$$T^{25} +$$$$25\!\cdots\!96$$$$T^{26} -$$$$12\!\cdots\!68$$$$T^{27} +$$$$26\!\cdots\!68$$$$T^{28} -$$$$13\!\cdots\!96$$$$T^{29} +$$$$24\!\cdots\!32$$$$T^{30} -$$$$17\!\cdots\!88$$$$T^{31} +$$$$66\!\cdots\!21$$$$T^{32}$$
$89$ $$1 - 20492 T + 140290034 T^{2} + 294316631388 T^{3} - 7791818672057541 T^{4} + 30934354648294655428 T^{5} -$$$$17\!\cdots\!22$$$$T^{6} +$$$$68\!\cdots\!36$$$$T^{7} -$$$$79\!\cdots\!23$$$$T^{8} +$$$$35\!\cdots\!60$$$$T^{9} +$$$$95\!\cdots\!64$$$$T^{10} -$$$$14\!\cdots\!56$$$$T^{11} +$$$$75\!\cdots\!86$$$$T^{12} -$$$$44\!\cdots\!80$$$$T^{13} +$$$$74\!\cdots\!96$$$$T^{14} -$$$$60\!\cdots\!12$$$$T^{15} +$$$$48\!\cdots\!30$$$$T^{16} -$$$$37\!\cdots\!92$$$$T^{17} +$$$$29\!\cdots\!76$$$$T^{18} -$$$$10\!\cdots\!80$$$$T^{19} +$$$$11\!\cdots\!46$$$$T^{20} -$$$$13\!\cdots\!56$$$$T^{21} +$$$$58\!\cdots\!24$$$$T^{22} +$$$$13\!\cdots\!60$$$$T^{23} -$$$$19\!\cdots\!83$$$$T^{24} +$$$$10\!\cdots\!96$$$$T^{25} -$$$$16\!\cdots\!22$$$$T^{26} +$$$$18\!\cdots\!48$$$$T^{27} -$$$$28\!\cdots\!21$$$$T^{28} +$$$$68\!\cdots\!48$$$$T^{29} +$$$$20\!\cdots\!74$$$$T^{30} -$$$$18\!\cdots\!92$$$$T^{31} +$$$$57\!\cdots\!41$$$$T^{32}$$
$97$ $$1 - 50944 T + 891925202 T^{2} + 15221697404 T^{3} - 236517605790318237 T^{4} +$$$$33\!\cdots\!96$$$$T^{5} -$$$$12\!\cdots\!26$$$$T^{6} -$$$$46\!\cdots\!40$$$$T^{7} +$$$$51\!\cdots\!85$$$$T^{8} -$$$$82\!\cdots\!12$$$$T^{9} -$$$$48\!\cdots\!08$$$$T^{10} +$$$$44\!\cdots\!76$$$$T^{11} -$$$$17\!\cdots\!98$$$$T^{12} -$$$$33\!\cdots\!24$$$$T^{13} +$$$$25\!\cdots\!04$$$$T^{14} +$$$$77\!\cdots\!12$$$$T^{15} -$$$$23\!\cdots\!26$$$$T^{16} +$$$$68\!\cdots\!72$$$$T^{17} +$$$$19\!\cdots\!44$$$$T^{18} -$$$$23\!\cdots\!84$$$$T^{19} -$$$$10\!\cdots\!58$$$$T^{20} +$$$$24\!\cdots\!76$$$$T^{21} -$$$$23\!\cdots\!48$$$$T^{22} -$$$$35\!\cdots\!32$$$$T^{23} +$$$$19\!\cdots\!85$$$$T^{24} -$$$$15\!\cdots\!40$$$$T^{25} -$$$$37\!\cdots\!26$$$$T^{26} +$$$$87\!\cdots\!76$$$$T^{27} -$$$$54\!\cdots\!57$$$$T^{28} +$$$$31\!\cdots\!64$$$$T^{29} +$$$$16\!\cdots\!42$$$$T^{30} -$$$$81\!\cdots\!44$$$$T^{31} +$$$$14\!\cdots\!81$$$$T^{32}$$