# Properties

 Label 13.7 Level 13 Weight 7 Dimension 36 Nonzero newspaces 2 Newform subspaces 2 Sturm bound 98 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$98$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(13))$$.

Total New Old
Modular forms 48 48 0
Cusp forms 36 36 0
Eisenstein series 12 12 0

## Trace form

 $$36q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} + 714q^{7} - 2310q^{8} - 6q^{9} + O(q^{10})$$ $$36q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} + 714q^{7} - 2310q^{8} - 6q^{9} + 4674q^{10} + 1914q^{11} - 5478q^{13} - 9132q^{14} - 7782q^{15} - 6q^{16} + 20898q^{17} + 9624q^{18} - 12270q^{19} - 33678q^{20} + 3114q^{21} + 66684q^{22} + 29634q^{23} + 78762q^{24} - 57756q^{26} - 127524q^{27} - 36480q^{28} - 89082q^{29} - 247734q^{30} - 57102q^{31} + 115944q^{32} + 261594q^{33} + 361794q^{34} + 263634q^{35} + 105066q^{36} - 71682q^{37} + 166362q^{39} - 379524q^{40} - 547632q^{41} - 984840q^{42} - 480486q^{43} - 308340q^{44} + 312804q^{45} + 967638q^{46} + 702474q^{47} + 1606410q^{48} + 854562q^{49} + 839136q^{50} - 1350204q^{52} - 887940q^{53} - 2142480q^{54} - 1469526q^{55} - 1461108q^{56} - 543486q^{57} + 113826q^{58} + 573594q^{59} + 3294612q^{60} + 1782372q^{61} + 3741966q^{62} + 2202042q^{63} - 2077032q^{65} - 5495280q^{66} - 2932278q^{67} - 2882184q^{68} - 1475766q^{69} + 332412q^{70} + 1481082q^{71} + 3080544q^{72} + 2914554q^{73} + 5533470q^{74} + 3466008q^{75} + 1769142q^{76} - 1496064q^{78} - 1809660q^{79} - 7899708q^{80} - 4789212q^{81} - 3540426q^{82} - 2747166q^{83} - 268140q^{84} + 3615312q^{85} + 3572508q^{86} + 2640762q^{87} + 4478052q^{88} - 344622q^{89} - 1858038q^{91} + 2026812q^{92} - 511782q^{93} - 4686090q^{94} - 2753718q^{95} - 3082824q^{96} + 3241602q^{97} + 6023610q^{98} + 7231194q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(13))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
13.7.d $$\chi_{13}(5, \cdot)$$ 13.7.d.a 12 2
13.7.f $$\chi_{13}(2, \cdot)$$ 13.7.f.a 24 4

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 6 T + 18 T^{2} + 140 T^{3} - 6655 T^{4} + 38334 T^{5} - 100414 T^{6} + 726352 T^{7} - 18453952 T^{8} + 65125376 T^{9} + 109600768 T^{10} - 13164609536 T^{11} + 244439318528 T^{12} - 842535010304 T^{13} + 448924745728 T^{14} + 17072226566144 T^{15} - 309605938757632 T^{16} + 779914521346048 T^{17} - 6900397536968704 T^{18} + 168594714956660736 T^{19} - 1873215970009415680 T^{20} + 2522015791327477760 T^{21} + 20752587082923245568 T^{22} -$$$$44\!\cdots\!84$$$$T^{23} +$$$$47\!\cdots\!96$$$$T^{24}$$)
$3$ ($$( 1 + 2 T + 1704 T^{2} + 6330 T^{3} + 2136456 T^{4} + 7744698 T^{5} + 1696048146 T^{6} + 5645884842 T^{7} + 1135400313096 T^{8} + 2452371695370 T^{9} + 481259930163624 T^{10} + 411782264189298 T^{11} + 150094635296999121 T^{12} )^{2}$$)
$5$ ($$1 - 108 T + 5832 T^{2} + 31196 T^{3} - 770587876 T^{4} + 59922609780 T^{5} - 1977086768200 T^{6} - 361180635738500 T^{7} + 326413718853155000 T^{8} - 16465642976655137500 T^{9} +$$$$13\!\cdots\!00$$$$T^{10} +$$$$21\!\cdots\!00$$$$T^{11} -$$$$96\!\cdots\!50$$$$T^{12} +$$$$33\!\cdots\!00$$$$T^{13} +$$$$33\!\cdots\!00$$$$T^{14} -$$$$62\!\cdots\!00$$$$T^{15} +$$$$19\!\cdots\!00$$$$T^{16} -$$$$33\!\cdots\!00$$$$T^{17} -$$$$28\!\cdots\!00$$$$T^{18} +$$$$13\!\cdots\!00$$$$T^{19} -$$$$27\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!00$$$$T^{21} +$$$$50\!\cdots\!00$$$$T^{22} -$$$$14\!\cdots\!00$$$$T^{23} +$$$$21\!\cdots\!25$$$$T^{24}$$)
$7$ ($$1 - 398 T + 79202 T^{2} - 28188670 T^{3} - 10722345216 T^{4} + 3939021777462 T^{5} - 321198923447794 T^{6} - 176763204490308034 T^{7} +$$$$55\!\cdots\!08$$$$T^{8} -$$$$12\!\cdots\!14$$$$T^{9} +$$$$16\!\cdots\!54$$$$T^{10} -$$$$90\!\cdots\!14$$$$T^{11} -$$$$37\!\cdots\!10$$$$T^{12} -$$$$10\!\cdots\!86$$$$T^{13} +$$$$23\!\cdots\!54$$$$T^{14} -$$$$20\!\cdots\!86$$$$T^{15} +$$$$10\!\cdots\!08$$$$T^{16} -$$$$39\!\cdots\!66$$$$T^{17} -$$$$85\!\cdots\!94$$$$T^{18} +$$$$12\!\cdots\!38$$$$T^{19} -$$$$39\!\cdots\!16$$$$T^{20} -$$$$12\!\cdots\!30$$$$T^{21} +$$$$40\!\cdots\!02$$$$T^{22} -$$$$23\!\cdots\!02$$$$T^{23} +$$$$70\!\cdots\!01$$$$T^{24}$$)
$11$ ($$1 - 1686 T + 1421298 T^{2} - 9449711854 T^{3} + 21784578331358 T^{4} - 16083795355398150 T^{5} + 40803428417812876874 T^{6} -$$$$11\!\cdots\!94$$$$T^{7} +$$$$99\!\cdots\!75$$$$T^{8} -$$$$10\!\cdots\!04$$$$T^{9} +$$$$34\!\cdots\!28$$$$T^{10} -$$$$39\!\cdots\!32$$$$T^{11} +$$$$22\!\cdots\!72$$$$T^{12} -$$$$69\!\cdots\!52$$$$T^{13} +$$$$10\!\cdots\!88$$$$T^{14} -$$$$57\!\cdots\!24$$$$T^{15} +$$$$97\!\cdots\!75$$$$T^{16} -$$$$19\!\cdots\!94$$$$T^{17} +$$$$12\!\cdots\!14$$$$T^{18} -$$$$88\!\cdots\!50$$$$T^{19} +$$$$21\!\cdots\!98$$$$T^{20} -$$$$16\!\cdots\!14$$$$T^{21} +$$$$43\!\cdots\!98$$$$T^{22} -$$$$90\!\cdots\!46$$$$T^{23} +$$$$95\!\cdots\!21$$$$T^{24}$$)
$13$ ($$1 + 3926 T - 3214718 T^{2} - 41376896314 T^{3} - 59058634955853 T^{4} + 107869141290743820 T^{5} +$$$$51\!\cdots\!12$$$$T^{6} +$$$$52\!\cdots\!80$$$$T^{7} -$$$$13\!\cdots\!93$$$$T^{8} -$$$$46\!\cdots\!06$$$$T^{9} -$$$$17\!\cdots\!98$$$$T^{10} +$$$$10\!\cdots\!74$$$$T^{11} +$$$$12\!\cdots\!41$$$$T^{12}$$)
$17$ ($$1 - 95568036 T^{2} + 4520630311261512 T^{4} -$$$$18\!\cdots\!40$$$$T^{6} +$$$$66\!\cdots\!80$$$$T^{8} -$$$$19\!\cdots\!68$$$$T^{10} +$$$$48\!\cdots\!06$$$$T^{12} -$$$$11\!\cdots\!48$$$$T^{14} +$$$$22\!\cdots\!80$$$$T^{16} -$$$$35\!\cdots\!40$$$$T^{18} +$$$$52\!\cdots\!92$$$$T^{20} -$$$$64\!\cdots\!36$$$$T^{22} +$$$$39\!\cdots\!61$$$$T^{24}$$)
$19$ ($$1 - 1766 T + 1559378 T^{2} - 696664382926 T^{3} + 1154247310387398 T^{4} + 28961508664554671754 T^{5} +$$$$18\!\cdots\!30$$$$T^{6} -$$$$58\!\cdots\!02$$$$T^{7} -$$$$11\!\cdots\!37$$$$T^{8} -$$$$42\!\cdots\!84$$$$T^{9} +$$$$33\!\cdots\!00$$$$T^{10} -$$$$27\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!32$$$$T^{12} -$$$$13\!\cdots\!60$$$$T^{13} +$$$$73\!\cdots\!00$$$$T^{14} -$$$$43\!\cdots\!44$$$$T^{15} -$$$$55\!\cdots\!77$$$$T^{16} -$$$$13\!\cdots\!02$$$$T^{17} +$$$$20\!\cdots\!30$$$$T^{18} +$$$$14\!\cdots\!94$$$$T^{19} +$$$$27\!\cdots\!18$$$$T^{20} -$$$$78\!\cdots\!46$$$$T^{21} +$$$$82\!\cdots\!78$$$$T^{22} -$$$$44\!\cdots\!46$$$$T^{23} +$$$$11\!\cdots\!61$$$$T^{24}$$)
$23$ ($$1 - 814212660 T^{2} + 317916327712063914 T^{4} -$$$$86\!\cdots\!44$$$$T^{6} +$$$$19\!\cdots\!11$$$$T^{8} -$$$$36\!\cdots\!04$$$$T^{10} +$$$$58\!\cdots\!32$$$$T^{12} -$$$$79\!\cdots\!84$$$$T^{14} +$$$$92\!\cdots\!51$$$$T^{16} -$$$$90\!\cdots\!84$$$$T^{18} +$$$$73\!\cdots\!34$$$$T^{20} -$$$$41\!\cdots\!60$$$$T^{22} +$$$$11\!\cdots\!21$$$$T^{24}$$)
$29$ ($$( 1 + 45054 T + 3480794474 T^{2} + 115344513309902 T^{3} + 5022872858956651955 T^{4} +$$$$12\!\cdots\!56$$$$T^{5} +$$$$39\!\cdots\!64$$$$T^{6} +$$$$75\!\cdots\!76$$$$T^{7} +$$$$17\!\cdots\!55$$$$T^{8} +$$$$24\!\cdots\!22$$$$T^{9} +$$$$43\!\cdots\!94$$$$T^{10} +$$$$33\!\cdots\!54$$$$T^{11} +$$$$44\!\cdots\!21$$$$T^{12} )^{2}$$)
$31$ ($$1 - 61014 T + 1861354098 T^{2} - 53209300169710 T^{3} + 2357813398795510374 T^{4} -$$$$94\!\cdots\!82$$$$T^{5} +$$$$27\!\cdots\!46$$$$T^{6} -$$$$72\!\cdots\!38$$$$T^{7} +$$$$25\!\cdots\!35$$$$T^{8} -$$$$10\!\cdots\!80$$$$T^{9} +$$$$34\!\cdots\!28$$$$T^{10} -$$$$96\!\cdots\!40$$$$T^{11} +$$$$27\!\cdots\!24$$$$T^{12} -$$$$85\!\cdots\!40$$$$T^{13} +$$$$26\!\cdots\!08$$$$T^{14} -$$$$73\!\cdots\!80$$$$T^{15} +$$$$15\!\cdots\!35$$$$T^{16} -$$$$39\!\cdots\!38$$$$T^{17} +$$$$13\!\cdots\!26$$$$T^{18} -$$$$41\!\cdots\!02$$$$T^{19} +$$$$90\!\cdots\!34$$$$T^{20} -$$$$18\!\cdots\!10$$$$T^{21} +$$$$56\!\cdots\!98$$$$T^{22} -$$$$16\!\cdots\!34$$$$T^{23} +$$$$23\!\cdots\!61$$$$T^{24}$$)
$37$ ($$1 + 40212 T + 808502472 T^{2} - 94108369185124 T^{3} - 2212353752837064900 T^{4} +$$$$60\!\cdots\!48$$$$T^{5} +$$$$30\!\cdots\!64$$$$T^{6} +$$$$15\!\cdots\!64$$$$T^{7} -$$$$30\!\cdots\!60$$$$T^{8} -$$$$71\!\cdots\!72$$$$T^{9} +$$$$73\!\cdots\!72$$$$T^{10} +$$$$75\!\cdots\!96$$$$T^{11} +$$$$81\!\cdots\!62$$$$T^{12} +$$$$19\!\cdots\!64$$$$T^{13} +$$$$48\!\cdots\!32$$$$T^{14} -$$$$12\!\cdots\!88$$$$T^{15} -$$$$13\!\cdots\!60$$$$T^{16} +$$$$17\!\cdots\!36$$$$T^{17} +$$$$86\!\cdots\!24$$$$T^{18} +$$$$44\!\cdots\!12$$$$T^{19} -$$$$41\!\cdots\!00$$$$T^{20} -$$$$45\!\cdots\!36$$$$T^{21} +$$$$99\!\cdots\!72$$$$T^{22} +$$$$12\!\cdots\!08$$$$T^{23} +$$$$81\!\cdots\!81$$$$T^{24}$$)
$41$ ($$1 + 190416 T + 18129126528 T^{2} + 1488204135776624 T^{3} + 64138724847656792258 T^{4} -$$$$31\!\cdots\!60$$$$T^{5} -$$$$64\!\cdots\!96$$$$T^{6} -$$$$70\!\cdots\!64$$$$T^{7} -$$$$50\!\cdots\!05$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{9} +$$$$57\!\cdots\!56$$$$T^{10} +$$$$14\!\cdots\!24$$$$T^{11} +$$$$14\!\cdots\!88$$$$T^{12} +$$$$68\!\cdots\!84$$$$T^{13} +$$$$12\!\cdots\!36$$$$T^{14} -$$$$14\!\cdots\!96$$$$T^{15} -$$$$25\!\cdots\!05$$$$T^{16} -$$$$17\!\cdots\!64$$$$T^{17} -$$$$74\!\cdots\!36$$$$T^{18} -$$$$16\!\cdots\!60$$$$T^{19} +$$$$16\!\cdots\!18$$$$T^{20} +$$$$18\!\cdots\!64$$$$T^{21} +$$$$10\!\cdots\!28$$$$T^{22} +$$$$52\!\cdots\!56$$$$T^{23} +$$$$13\!\cdots\!81$$$$T^{24}$$)
$43$ ($$1 - 36018404640 T^{2} +$$$$71\!\cdots\!12$$$$T^{4} -$$$$98\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!56$$$$T^{8} -$$$$87\!\cdots\!36$$$$T^{10} +$$$$60\!\cdots\!62$$$$T^{12} -$$$$34\!\cdots\!36$$$$T^{14} +$$$$16\!\cdots\!56$$$$T^{16} -$$$$62\!\cdots\!24$$$$T^{18} +$$$$18\!\cdots\!12$$$$T^{20} -$$$$36\!\cdots\!40$$$$T^{22} +$$$$40\!\cdots\!01$$$$T^{24}$$)
$47$ ($$1 - 562446 T + 158172751458 T^{2} - 33342810078438670 T^{3} +$$$$58\!\cdots\!68$$$$T^{4} -$$$$81\!\cdots\!86$$$$T^{5} +$$$$91\!\cdots\!62$$$$T^{6} -$$$$79\!\cdots\!34$$$$T^{7} +$$$$33\!\cdots\!20$$$$T^{8} +$$$$42\!\cdots\!90$$$$T^{9} -$$$$12\!\cdots\!38$$$$T^{10} +$$$$19\!\cdots\!94$$$$T^{11} -$$$$23\!\cdots\!42$$$$T^{12} +$$$$21\!\cdots\!26$$$$T^{13} -$$$$14\!\cdots\!58$$$$T^{14} +$$$$52\!\cdots\!10$$$$T^{15} +$$$$45\!\cdots\!20$$$$T^{16} -$$$$11\!\cdots\!66$$$$T^{17} +$$$$14\!\cdots\!02$$$$T^{18} -$$$$13\!\cdots\!74$$$$T^{19} +$$$$10\!\cdots\!48$$$$T^{20} -$$$$65\!\cdots\!30$$$$T^{21} +$$$$33\!\cdots\!58$$$$T^{22} -$$$$12\!\cdots\!34$$$$T^{23} +$$$$24\!\cdots\!41$$$$T^{24}$$)
$53$ ($$( 1 - 254568 T + 108652497230 T^{2} - 20683934542944136 T^{3} +$$$$52\!\cdots\!83$$$$T^{4} -$$$$79\!\cdots\!72$$$$T^{5} +$$$$14\!\cdots\!32$$$$T^{6} -$$$$17\!\cdots\!88$$$$T^{7} +$$$$26\!\cdots\!03$$$$T^{8} -$$$$22\!\cdots\!04$$$$T^{9} +$$$$26\!\cdots\!30$$$$T^{10} -$$$$13\!\cdots\!32$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12} )^{2}$$)
$59$ ($$1 + 994458 T + 494473356882 T^{2} + 189903786626655986 T^{3} +$$$$68\!\cdots\!90$$$$T^{4} +$$$$22\!\cdots\!94$$$$T^{5} +$$$$67\!\cdots\!70$$$$T^{6} +$$$$18\!\cdots\!14$$$$T^{7} +$$$$48\!\cdots\!75$$$$T^{8} +$$$$11\!\cdots\!88$$$$T^{9} +$$$$27\!\cdots\!76$$$$T^{10} +$$$$61\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!24$$$$T^{12} +$$$$25\!\cdots\!36$$$$T^{13} +$$$$48\!\cdots\!56$$$$T^{14} +$$$$89\!\cdots\!48$$$$T^{15} +$$$$15\!\cdots\!75$$$$T^{16} +$$$$24\!\cdots\!14$$$$T^{17} +$$$$37\!\cdots\!70$$$$T^{18} +$$$$54\!\cdots\!14$$$$T^{19} +$$$$68\!\cdots\!90$$$$T^{20} +$$$$80\!\cdots\!46$$$$T^{21} +$$$$88\!\cdots\!82$$$$T^{22} +$$$$74\!\cdots\!78$$$$T^{23} +$$$$31\!\cdots\!81$$$$T^{24}$$)
$61$ ($$( 1 - 506848 T + 282902598538 T^{2} - 105182501059034848 T^{3} +$$$$35\!\cdots\!39$$$$T^{4} -$$$$97\!\cdots\!56$$$$T^{5} +$$$$24\!\cdots\!88$$$$T^{6} -$$$$49\!\cdots\!16$$$$T^{7} +$$$$94\!\cdots\!19$$$$T^{8} -$$$$14\!\cdots\!88$$$$T^{9} +$$$$19\!\cdots\!58$$$$T^{10} -$$$$18\!\cdots\!48$$$$T^{11} +$$$$18\!\cdots\!61$$$$T^{12} )^{2}$$)
$67$ ($$1 + 1442386 T + 1040238686498 T^{2} + 574663178423383274 T^{3} +$$$$28\!\cdots\!54$$$$T^{4} +$$$$12\!\cdots\!82$$$$T^{5} +$$$$49\!\cdots\!98$$$$T^{6} +$$$$18\!\cdots\!90$$$$T^{7} +$$$$66\!\cdots\!55$$$$T^{8} +$$$$23\!\cdots\!36$$$$T^{9} +$$$$77\!\cdots\!36$$$$T^{10} +$$$$25\!\cdots\!32$$$$T^{11} +$$$$78\!\cdots\!36$$$$T^{12} +$$$$22\!\cdots\!08$$$$T^{13} +$$$$63\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!24$$$$T^{15} +$$$$44\!\cdots\!55$$$$T^{16} +$$$$11\!\cdots\!10$$$$T^{17} +$$$$26\!\cdots\!38$$$$T^{18} +$$$$61\!\cdots\!98$$$$T^{19} +$$$$12\!\cdots\!14$$$$T^{20} +$$$$23\!\cdots\!46$$$$T^{21} +$$$$38\!\cdots\!98$$$$T^{22} +$$$$47\!\cdots\!34$$$$T^{23} +$$$$30\!\cdots\!61$$$$T^{24}$$)
$71$ ($$1 + 655866 T + 215080104978 T^{2} + 141159966904204586 T^{3} +$$$$93\!\cdots\!48$$$$T^{4} +$$$$25\!\cdots\!74$$$$T^{5} +$$$$62\!\cdots\!38$$$$T^{6} +$$$$33\!\cdots\!78$$$$T^{7} +$$$$73\!\cdots\!60$$$$T^{8} -$$$$16\!\cdots\!30$$$$T^{9} -$$$$63\!\cdots\!38$$$$T^{10} -$$$$36\!\cdots\!50$$$$T^{11} -$$$$20\!\cdots\!42$$$$T^{12} -$$$$46\!\cdots\!50$$$$T^{13} -$$$$10\!\cdots\!58$$$$T^{14} -$$$$33\!\cdots\!30$$$$T^{15} +$$$$19\!\cdots\!60$$$$T^{16} +$$$$11\!\cdots\!78$$$$T^{17} +$$$$27\!\cdots\!98$$$$T^{18} +$$$$14\!\cdots\!34$$$$T^{19} +$$$$68\!\cdots\!28$$$$T^{20} +$$$$13\!\cdots\!66$$$$T^{21} +$$$$25\!\cdots\!78$$$$T^{22} +$$$$99\!\cdots\!86$$$$T^{23} +$$$$19\!\cdots\!41$$$$T^{24}$$)
$73$ ($$1 - 2588228 T + 3349462089992 T^{2} - 2980435039024420492 T^{3} +$$$$20\!\cdots\!26$$$$T^{4} -$$$$11\!\cdots\!48$$$$T^{5} +$$$$49\!\cdots\!84$$$$T^{6} -$$$$16\!\cdots\!44$$$$T^{7} +$$$$27\!\cdots\!95$$$$T^{8} +$$$$10\!\cdots\!08$$$$T^{9} -$$$$12\!\cdots\!48$$$$T^{10} +$$$$77\!\cdots\!00$$$$T^{11} -$$$$34\!\cdots\!20$$$$T^{12} +$$$$11\!\cdots\!00$$$$T^{13} -$$$$29\!\cdots\!08$$$$T^{14} +$$$$36\!\cdots\!52$$$$T^{15} +$$$$14\!\cdots\!95$$$$T^{16} -$$$$12\!\cdots\!56$$$$T^{17} +$$$$59\!\cdots\!24$$$$T^{18} -$$$$20\!\cdots\!92$$$$T^{19} +$$$$56\!\cdots\!06$$$$T^{20} -$$$$12\!\cdots\!28$$$$T^{21} +$$$$21\!\cdots\!92$$$$T^{22} -$$$$24\!\cdots\!92$$$$T^{23} +$$$$14\!\cdots\!21$$$$T^{24}$$)
$79$ ($$( 1 + 37658 T + 781884604234 T^{2} - 44764114305384646 T^{3} +$$$$35\!\cdots\!55$$$$T^{4} -$$$$18\!\cdots\!44$$$$T^{5} +$$$$10\!\cdots\!76$$$$T^{6} -$$$$44\!\cdots\!24$$$$T^{7} +$$$$20\!\cdots\!55$$$$T^{8} -$$$$64\!\cdots\!06$$$$T^{9} +$$$$27\!\cdots\!54$$$$T^{10} +$$$$31\!\cdots\!58$$$$T^{11} +$$$$20\!\cdots\!21$$$$T^{12} )^{2}$$)
$83$ ($$1 - 894966 T + 400482070578 T^{2} - 28674110295070414 T^{3} -$$$$31\!\cdots\!42$$$$T^{4} +$$$$47\!\cdots\!86$$$$T^{5} -$$$$29\!\cdots\!02$$$$T^{6} +$$$$46\!\cdots\!02$$$$T^{7} -$$$$19\!\cdots\!85$$$$T^{8} -$$$$15\!\cdots\!48$$$$T^{9} +$$$$81\!\cdots\!84$$$$T^{10} -$$$$39\!\cdots\!56$$$$T^{11} +$$$$26\!\cdots\!72$$$$T^{12} -$$$$12\!\cdots\!64$$$$T^{13} +$$$$87\!\cdots\!24$$$$T^{14} -$$$$54\!\cdots\!32$$$$T^{15} -$$$$22\!\cdots\!85$$$$T^{16} +$$$$17\!\cdots\!98$$$$T^{17} -$$$$35\!\cdots\!62$$$$T^{18} +$$$$18\!\cdots\!54$$$$T^{19} -$$$$41\!\cdots\!22$$$$T^{20} -$$$$12\!\cdots\!06$$$$T^{21} +$$$$55\!\cdots\!78$$$$T^{22} -$$$$40\!\cdots\!54$$$$T^{23} +$$$$14\!\cdots\!61$$$$T^{24}$$)
$89$ ($$1 + 977376 T + 477631922688 T^{2} + 751481099068671296 T^{3} +$$$$91\!\cdots\!82$$$$T^{4} +$$$$32\!\cdots\!88$$$$T^{5} +$$$$15\!\cdots\!80$$$$T^{6} +$$$$21\!\cdots\!88$$$$T^{7} -$$$$48\!\cdots\!97$$$$T^{8} -$$$$14\!\cdots\!96$$$$T^{9} -$$$$50\!\cdots\!00$$$$T^{10} -$$$$76\!\cdots\!96$$$$T^{11} -$$$$10\!\cdots\!76$$$$T^{12} -$$$$38\!\cdots\!56$$$$T^{13} -$$$$12\!\cdots\!00$$$$T^{14} -$$$$18\!\cdots\!76$$$$T^{15} -$$$$29\!\cdots\!77$$$$T^{16} +$$$$66\!\cdots\!88$$$$T^{17} +$$$$23\!\cdots\!80$$$$T^{18} +$$$$24\!\cdots\!48$$$$T^{19} +$$$$34\!\cdots\!42$$$$T^{20} +$$$$13\!\cdots\!36$$$$T^{21} +$$$$43\!\cdots\!88$$$$T^{22} +$$$$44\!\cdots\!36$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$)
$97$ ($$1 - 983388 T + 483525979272 T^{2} - 532915345861301044 T^{3} +$$$$79\!\cdots\!30$$$$T^{4} -$$$$71\!\cdots\!48$$$$T^{5} +$$$$46\!\cdots\!32$$$$T^{6} -$$$$68\!\cdots\!88$$$$T^{7} +$$$$13\!\cdots\!95$$$$T^{8} -$$$$89\!\cdots\!64$$$$T^{9} +$$$$46\!\cdots\!56$$$$T^{10} -$$$$52\!\cdots\!88$$$$T^{11} +$$$$52\!\cdots\!08$$$$T^{12} -$$$$43\!\cdots\!52$$$$T^{13} +$$$$32\!\cdots\!96$$$$T^{14} -$$$$51\!\cdots\!96$$$$T^{15} +$$$$66\!\cdots\!95$$$$T^{16} -$$$$27\!\cdots\!12$$$$T^{17} +$$$$15\!\cdots\!72$$$$T^{18} -$$$$19\!\cdots\!32$$$$T^{19} +$$$$18\!\cdots\!30$$$$T^{20} -$$$$10\!\cdots\!36$$$$T^{21} +$$$$77\!\cdots\!72$$$$T^{22} -$$$$13\!\cdots\!52$$$$T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$)