Properties

 Label 36.5 Level 36 Weight 5 Dimension 63 Nonzero newspaces 4 Newform subspaces 6 Sturm bound 360 Trace bound 1

Defining parameters

 Level: $$N$$ = $$36\( 36 = 2^{2} \cdot 3^{2}$$ \) Weight: $$k$$ = $$5$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$360$$ Trace bound: $$1$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(36))$$.

Total New Old
Modular forms 164 73 91
Cusp forms 124 63 61
Eisenstein series 40 10 30

Trace form

 $$63q - 3q^{2} - 9q^{3} - 13q^{4} - 21q^{5} + 15q^{6} + 149q^{7} + 186q^{8} - 39q^{9} + O(q^{10})$$ $$63q - 3q^{2} - 9q^{3} - 13q^{4} - 21q^{5} + 15q^{6} + 149q^{7} + 186q^{8} - 39q^{9} + 136q^{10} - 18q^{11} - 228q^{12} + 75q^{13} - 348q^{14} + 225q^{15} - 625q^{16} + 222q^{17} + 72q^{18} + 146q^{19} + 1452q^{20} - 1029q^{21} + 1143q^{22} - 1719q^{23} - 951q^{24} - 3230q^{25} - 1548q^{26} + 648q^{27} - 3060q^{28} + 1671q^{29} - 1980q^{30} + 3491q^{31} + 3687q^{32} + 6252q^{33} + 3541q^{34} - 1005q^{36} - 234q^{37} - 3285q^{38} - 8265q^{39} - 4196q^{40} - 8196q^{41} + 3330q^{42} + 2252q^{43} + 10410q^{44} + 9789q^{45} + 7800q^{46} + 13689q^{47} + 2163q^{48} + 2852q^{49} + 5697q^{50} + 10449q^{51} + 3526q^{52} - 17682q^{53} - 4983q^{54} - 13482q^{55} - 9234q^{56} - 20109q^{57} - 11012q^{58} - 20052q^{59} - 16392q^{60} + 3q^{61} - 11100q^{62} + 5559q^{63} + 4418q^{64} + 38811q^{65} - 17358q^{66} + 12938q^{67} - 22053q^{68} + 24999q^{69} - 6366q^{70} + 4083q^{72} - 30864q^{73} + 27384q^{74} - 30297q^{75} + 23901q^{76} - 50511q^{77} + 34566q^{78} - 16315q^{79} + 23472q^{80} + 18537q^{81} - 6482q^{82} + 37017q^{83} + 51078q^{84} + 58274q^{85} + 41673q^{86} + 22455q^{87} + 21003q^{88} + 47886q^{89} - 4692q^{90} + 13654q^{91} - 49110q^{92} - 24399q^{93} - 29964q^{94} - 37116q^{95} - 76164q^{96} + 3834q^{97} - 77484q^{98} - 10035q^{99} + O(q^{100})$$

Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(36))$$

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
36.5.c $$\chi_{36}(17, \cdot)$$ 36.5.c.a 2 1
36.5.d $$\chi_{36}(19, \cdot)$$ 36.5.d.a 1 1
36.5.d.b 4
36.5.d.c 4
36.5.f $$\chi_{36}(7, \cdot)$$ 36.5.f.a 44 2
36.5.g $$\chi_{36}(5, \cdot)$$ 36.5.g.a 8 2

Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(36))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(36)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 4 T$$)($$1 + 6 T + 28 T^{2} + 96 T^{3} + 256 T^{4}$$)($$1 + 4 T^{2} + 256 T^{4}$$)
$3$ ($$1 + 9 T + 30 T^{2} - 189 T^{3} - 6966 T^{4} - 15309 T^{5} + 196830 T^{6} + 4782969 T^{7} + 43046721 T^{8}$$)
$5$ ($$1 + 208 T^{2} + 390625 T^{4}$$)($$1 - 14 T + 625 T^{2}$$)($$( 1 + 12 T + 454 T^{2} + 7500 T^{3} + 390625 T^{4} )^{2}$$)($$( 1 + 802 T^{2} + 390625 T^{4} )^{2}$$)($$1 + 9 T + 1114 T^{2} + 9783 T^{3} + 533599 T^{4} + 10528056 T^{5} + 59806456 T^{6} + 10305069192 T^{7} - 6001445444 T^{8} + 6440668245000 T^{9} + 23361896875000 T^{10} + 2570326171875000 T^{11} + 81420745849609375 T^{12} + 932979583740234375 T^{13} + 66399574279785156250 T^{14} +$$$$33\!\cdots\!25$$$$T^{15} +$$$$23\!\cdots\!25$$$$T^{16}$$)
$7$ ($$( 1 - 68 T + 2401 T^{2} )^{2}$$)($$( 1 - 49 T )( 1 + 49 T )$$)($$1 - 4516 T^{2} + 16148934 T^{4} - 26033841316 T^{6} + 33232930569601 T^{8}$$)($$( 1 - 770 T^{2} + 5764801 T^{4} )^{2}$$)($$1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} +$$$$29\!\cdots\!91$$$$T^{12} +$$$$99\!\cdots\!53$$$$T^{13} -$$$$87\!\cdots\!54$$$$T^{14} -$$$$59\!\cdots\!13$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$)
$11$ ($$1 - 5954 T^{2} + 214358881 T^{4}$$)($$( 1 - 121 T )( 1 + 121 T )$$)($$1 - 36196 T^{2} + 708332166 T^{4} - 7758934056676 T^{6} + 45949729863572161 T^{8}$$)($$( 1 + 7582 T^{2} + 214358881 T^{4} )^{2}$$)($$1 + 18 T + 33850 T^{2} + 607356 T^{3} + 452208721 T^{4} + 17443950048 T^{5} + 9074477968558 T^{6} + 445095012639474 T^{7} + 191968264536523468 T^{8} + 6516636080054538834 T^{9} +$$$$19\!\cdots\!98$$$$T^{10} +$$$$54\!\cdots\!08$$$$T^{11} +$$$$20\!\cdots\!81$$$$T^{12} +$$$$40\!\cdots\!56$$$$T^{13} +$$$$33\!\cdots\!50$$$$T^{14} +$$$$25\!\cdots\!58$$$$T^{15} +$$$$21\!\cdots\!21$$$$T^{16}$$)
$13$ ($$( 1 + 16 T + 28561 T^{2} )^{2}$$)($$1 + 238 T + 28561 T^{2}$$)($$( 1 - 148 T + 32646 T^{2} - 4227028 T^{3} + 815730721 T^{4} )^{2}$$)($$( 1 - 14 T + 28561 T^{2} )^{4}$$)($$1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} +$$$$18\!\cdots\!84$$$$T^{10} +$$$$58\!\cdots\!40$$$$T^{11} +$$$$23\!\cdots\!75$$$$T^{12} -$$$$13\!\cdots\!65$$$$T^{13} -$$$$22\!\cdots\!94$$$$T^{14} +$$$$77\!\cdots\!05$$$$T^{15} +$$$$44\!\cdots\!81$$$$T^{16}$$)
$17$ ($$1 + 79360 T^{2} + 6975757441 T^{4}$$)($$1 + 322 T + 83521 T^{2}$$)($$( 1 - 300 T + 186214 T^{2} - 25056300 T^{3} + 6975757441 T^{4} )^{2}$$)($$( 1 - 49790 T^{2} + 6975757441 T^{4} )^{2}$$)($$1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} +$$$$63\!\cdots\!86$$$$T^{8} -$$$$47\!\cdots\!55$$$$T^{10} +$$$$25\!\cdots\!74$$$$T^{12} -$$$$97\!\cdots\!25$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16}$$)
$19$ ($$( 1 + 208 T + 130321 T^{2} )^{2}$$)($$( 1 - 361 T )( 1 + 361 T )$$)($$1 - 195556 T^{2} + 17286835974 T^{4} - 3321237654045796 T^{6} +$$$$28\!\cdots\!81$$$$T^{8}$$)($$( 1 - 115490 T^{2} + 16983563041 T^{4} )^{2}$$)($$( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$)
$23$ ($$1 - 536354 T^{2} + 78310985281 T^{4}$$)($$( 1 - 529 T )( 1 + 529 T )$$)($$1 - 655492 T^{2} + 255175536006 T^{4} - 51332224363813252 T^{6} +$$$$61\!\cdots\!61$$$$T^{8}$$)($$( 1 - 412226 T^{2} + 78310985281 T^{4} )^{2}$$)($$1 + 1719 T + 1529458 T^{2} + 935945649 T^{3} + 342642958747 T^{4} - 16333135609500 T^{5} - 118516645434237428 T^{6} -$$$$10\!\cdots\!68$$$$T^{7} -$$$$65\!\cdots\!00$$$$T^{8} -$$$$29\!\cdots\!88$$$$T^{9} -$$$$92\!\cdots\!68$$$$T^{10} -$$$$35\!\cdots\!00$$$$T^{11} +$$$$21\!\cdots\!67$$$$T^{12} +$$$$16\!\cdots\!49$$$$T^{13} +$$$$73\!\cdots\!78$$$$T^{14} +$$$$23\!\cdots\!39$$$$T^{15} +$$$$37\!\cdots\!21$$$$T^{16}$$)
$29$ ($$1 - 993200 T^{2} + 500246412961 T^{4}$$)($$1 + 82 T + 707281 T^{2}$$)($$( 1 + 444 T + 1423078 T^{2} + 314032764 T^{3} + 500246412961 T^{4} )^{2}$$)($$( 1 + 1037794 T^{2} + 500246412961 T^{4} )^{2}$$)($$1 - 2115 T + 4091014 T^{2} - 5498870985 T^{3} + 7048695081595 T^{4} - 8024737206821040 T^{5} + 8297140249856169556 T^{6} -$$$$79\!\cdots\!80$$$$T^{7} +$$$$68\!\cdots\!24$$$$T^{8} -$$$$56\!\cdots\!80$$$$T^{9} +$$$$41\!\cdots\!16$$$$T^{10} -$$$$28\!\cdots\!40$$$$T^{11} +$$$$17\!\cdots\!95$$$$T^{12} -$$$$97\!\cdots\!85$$$$T^{13} +$$$$51\!\cdots\!34$$$$T^{14} -$$$$18\!\cdots\!15$$$$T^{15} +$$$$62\!\cdots\!41$$$$T^{16}$$)
$31$ ($$( 1 - 1652 T + 923521 T^{2} )^{2}$$)($$( 1 - 961 T )( 1 + 961 T )$$)($$1 - 2090020 T^{2} + 2465294161734 T^{4} - 1782559326072438820 T^{6} +$$$$72\!\cdots\!81$$$$T^{8}$$)($$( 1 - 1746242 T^{2} + 852891037441 T^{4} )^{2}$$)($$1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} +$$$$31\!\cdots\!92$$$$T^{8} -$$$$12\!\cdots\!14$$$$T^{9} -$$$$29\!\cdots\!86$$$$T^{10} -$$$$23\!\cdots\!56$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} +$$$$12\!\cdots\!37$$$$T^{13} -$$$$15\!\cdots\!84$$$$T^{14} -$$$$10\!\cdots\!67$$$$T^{15} +$$$$52\!\cdots\!61$$$$T^{16}$$)
$37$ ($$( 1 + 442 T + 1874161 T^{2} )^{2}$$)($$1 - 2162 T + 1874161 T^{2}$$)($$( 1 + 2204 T + 4842918 T^{2} + 4130650844 T^{3} + 3512479453921 T^{4} )^{2}$$)($$( 1 - 722 T + 1874161 T^{2} )^{4}$$)($$( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$)
$41$ ($$1 - 5405120 T^{2} + 7984925229121 T^{4}$$)($$1 - 3038 T + 2825761 T^{2}$$)($$( 1 + 276 T + 5507494 T^{2} + 779910036 T^{3} + 7984925229121 T^{4} )^{2}$$)($$( 1 + 3929410 T^{2} + 7984925229121 T^{4} )^{2}$$)($$1 + 7920 T + 37687894 T^{2} + 132890424480 T^{3} + 385083705354505 T^{4} + 963185727644706960 T^{5} +$$$$21\!\cdots\!66$$$$T^{6} +$$$$41\!\cdots\!20$$$$T^{7} +$$$$74\!\cdots\!64$$$$T^{8} +$$$$11\!\cdots\!20$$$$T^{9} +$$$$16\!\cdots\!86$$$$T^{10} +$$$$21\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!05$$$$T^{12} +$$$$23\!\cdots\!80$$$$T^{13} +$$$$19\!\cdots\!34$$$$T^{14} +$$$$11\!\cdots\!20$$$$T^{15} +$$$$40\!\cdots\!81$$$$T^{16}$$)
$43$ ($$( 1 - 1160 T + 3418801 T^{2} )^{2}$$)($$( 1 - 1849 T )( 1 + 1849 T )$$)($$1 - 3593572 T^{2} + 5893643026566 T^{4} - 42002389247979180772 T^{6} +$$$$13\!\cdots\!01$$$$T^{8}$$)($$( 1 - 2176610 T^{2} + 11688200277601 T^{4} )^{2}$$)($$1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} -$$$$54\!\cdots\!36$$$$T^{6} +$$$$27\!\cdots\!16$$$$T^{7} +$$$$21\!\cdots\!12$$$$T^{8} +$$$$94\!\cdots\!16$$$$T^{9} -$$$$63\!\cdots\!36$$$$T^{10} +$$$$23\!\cdots\!04$$$$T^{11} +$$$$14\!\cdots\!91$$$$T^{12} -$$$$97\!\cdots\!48$$$$T^{13} -$$$$20\!\cdots\!04$$$$T^{14} +$$$$37\!\cdots\!68$$$$T^{15} +$$$$18\!\cdots\!01$$$$T^{16}$$)
$47$ ($$1 - 9549410 T^{2} + 23811286661761 T^{4}$$)($$( 1 - 2209 T )( 1 + 2209 T )$$)($$1 - 16853380 T^{2} + 118578854811654 T^{4} -$$$$40\!\cdots\!80$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8}$$)($$( 1 - 2534018 T^{2} + 23811286661761 T^{4} )^{2}$$)($$1 - 13689 T + 103685338 T^{2} - 564293857959 T^{3} + 2435028217967227 T^{4} - 8736748337842042500 T^{5} +$$$$26\!\cdots\!72$$$$T^{6} -$$$$71\!\cdots\!32$$$$T^{7} +$$$$16\!\cdots\!20$$$$T^{8} -$$$$35\!\cdots\!92$$$$T^{9} +$$$$63\!\cdots\!92$$$$T^{10} -$$$$10\!\cdots\!00$$$$T^{11} +$$$$13\!\cdots\!67$$$$T^{12} -$$$$15\!\cdots\!59$$$$T^{13} +$$$$13\!\cdots\!78$$$$T^{14} -$$$$90\!\cdots\!29$$$$T^{15} +$$$$32\!\cdots\!41$$$$T^{16}$$)
$53$ ($$1 - 9620912 T^{2} + 62259690411361 T^{4}$$)($$1 + 2482 T + 7890481 T^{2}$$)($$( 1 + 2556 T + 17173798 T^{2} + 20168069436 T^{3} + 62259690411361 T^{4} )^{2}$$)($$( 1 + 15027874 T^{2} + 62259690411361 T^{4} )^{2}$$)($$1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} +$$$$67\!\cdots\!26$$$$T^{8} -$$$$52\!\cdots\!60$$$$T^{10} +$$$$44\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16}$$)
$59$ ($$1 - 12943970 T^{2} + 146830437604321 T^{4}$$)($$( 1 - 3481 T )( 1 + 3481 T )$$)($$1 - 32722276 T^{2} + 559903452302214 T^{4} -$$$$48\!\cdots\!96$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8}$$)($$( 1 - 17009378 T^{2} + 146830437604321 T^{4} )^{2}$$)($$1 + 20052 T + 216711700 T^{2} + 1657982214864 T^{3} + 9931594296358591 T^{4} + 49579528565018409012 T^{5} +$$$$21\!\cdots\!48$$$$T^{6} +$$$$84\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!08$$$$T^{8} +$$$$10\!\cdots\!36$$$$T^{9} +$$$$31\!\cdots\!08$$$$T^{10} +$$$$88\!\cdots\!72$$$$T^{11} +$$$$21\!\cdots\!31$$$$T^{12} +$$$$43\!\cdots\!64$$$$T^{13} +$$$$68\!\cdots\!00$$$$T^{14} +$$$$76\!\cdots\!92$$$$T^{15} +$$$$46\!\cdots\!81$$$$T^{16}$$)
$61$ ($$( 1 + 3910 T + 13845841 T^{2} )^{2}$$)($$1 + 6958 T + 13845841 T^{2}$$)($$( 1 - 2116 T + 16830246 T^{2} - 29297799556 T^{3} + 191707312997281 T^{4} )^{2}$$)($$( 1 - 3122 T + 13845841 T^{2} )^{4}$$)($$1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} +$$$$12\!\cdots\!24$$$$T^{6} +$$$$40\!\cdots\!44$$$$T^{7} -$$$$98\!\cdots\!68$$$$T^{8} +$$$$55\!\cdots\!04$$$$T^{9} +$$$$23\!\cdots\!44$$$$T^{10} -$$$$35\!\cdots\!44$$$$T^{11} +$$$$15\!\cdots\!91$$$$T^{12} +$$$$76\!\cdots\!23$$$$T^{13} -$$$$74\!\cdots\!94$$$$T^{14} +$$$$18\!\cdots\!97$$$$T^{15} +$$$$13\!\cdots\!21$$$$T^{16}$$)
$67$ ($$( 1 - 6392 T + 20151121 T^{2} )^{2}$$)($$( 1 - 4489 T )( 1 + 4489 T )$$)($$1 - 53256676 T^{2} + 1338152481850374 T^{4} -$$$$21\!\cdots\!16$$$$T^{6} +$$$$16\!\cdots\!81$$$$T^{8}$$)($$( 1 - 29399714 T^{2} + 406067677556641 T^{4} )^{2}$$)($$1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} +$$$$70\!\cdots\!02$$$$T^{6} -$$$$11\!\cdots\!78$$$$T^{7} -$$$$26\!\cdots\!64$$$$T^{8} -$$$$22\!\cdots\!38$$$$T^{9} +$$$$28\!\cdots\!82$$$$T^{10} +$$$$60\!\cdots\!96$$$$T^{11} +$$$$88\!\cdots\!37$$$$T^{12} -$$$$85\!\cdots\!28$$$$T^{13} -$$$$22\!\cdots\!34$$$$T^{14} -$$$$20\!\cdots\!14$$$$T^{15} +$$$$27\!\cdots\!61$$$$T^{16}$$)
$71$ ($$1 - 7689890 T^{2} + 645753531245761 T^{4}$$)($$( 1 - 5041 T )( 1 + 5041 T )$$)($$1 - 79127428 T^{2} + 2740885222137990 T^{4} -$$$$51\!\cdots\!08$$$$T^{6} +$$$$41\!\cdots\!21$$$$T^{8}$$)($$( 1 - 29589698 T^{2} + 645753531245761 T^{4} )^{2}$$)($$1 - 68871716 T^{2} + 3244147638477940 T^{4} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$30\!\cdots\!74$$$$T^{8} -$$$$73\!\cdots\!64$$$$T^{10} +$$$$13\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$)
$73$ ($$( 1 + 2224 T + 28398241 T^{2} )^{2}$$)($$1 - 1442 T + 28398241 T^{2}$$)($$( 1 - 4420 T + 3693510 T^{2} - 125520225220 T^{3} + 806460091894081 T^{4} )^{2}$$)($$( 1 + 6370 T + 28398241 T^{2} )^{4}$$)($$( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} +$$$$47\!\cdots\!70$$$$T^{6} +$$$$89\!\cdots\!21$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$)
$79$ ($$( 1 + 7060 T + 38950081 T^{2} )^{2}$$)($$( 1 - 6241 T )( 1 + 6241 T )$$)($$1 - 86136100 T^{2} + 4459690111983174 T^{4} -$$$$13\!\cdots\!00$$$$T^{6} +$$$$23\!\cdots\!21$$$$T^{8}$$)($$( 1 + 21484606 T^{2} + 1517108809906561 T^{4} )^{2}$$)($$1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} -$$$$16\!\cdots\!36$$$$T^{6} +$$$$76\!\cdots\!80$$$$T^{7} +$$$$88\!\cdots\!64$$$$T^{8} +$$$$29\!\cdots\!80$$$$T^{9} -$$$$24\!\cdots\!96$$$$T^{10} +$$$$21\!\cdots\!60$$$$T^{11} +$$$$72\!\cdots\!75$$$$T^{12} -$$$$16\!\cdots\!15$$$$T^{13} -$$$$30\!\cdots\!34$$$$T^{14} +$$$$29\!\cdots\!95$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16}$$)
$83$ ($$1 - 59434754 T^{2} + 2252292232139041 T^{4}$$)($$( 1 - 6889 T )( 1 + 6889 T )$$)($$1 - 855268 T^{2} + 4298268871421190 T^{4} -$$$$19\!\cdots\!88$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8}$$)($$( 1 - 93110306 T^{2} + 2252292232139041 T^{4} )^{2}$$)($$1 - 37017 T + 725723290 T^{2} - 9956481997959 T^{3} + 104510585134438411 T^{4} -$$$$87\!\cdots\!72$$$$T^{5} +$$$$62\!\cdots\!28$$$$T^{6} -$$$$40\!\cdots\!76$$$$T^{7} +$$$$26\!\cdots\!48$$$$T^{8} -$$$$19\!\cdots\!96$$$$T^{9} +$$$$14\!\cdots\!48$$$$T^{10} -$$$$93\!\cdots\!92$$$$T^{11} +$$$$53\!\cdots\!91$$$$T^{12} -$$$$23\!\cdots\!59$$$$T^{13} +$$$$82\!\cdots\!90$$$$T^{14} -$$$$20\!\cdots\!97$$$$T^{15} +$$$$25\!\cdots\!61$$$$T^{16}$$)
$89$ ($$1 - 54274304 T^{2} + 3936588805702081 T^{4}$$)($$1 - 9758 T + 62742241 T^{2}$$)($$( 1 - 12540 T + 132835270 T^{2} - 786787702140 T^{3} + 3936588805702081 T^{4} )^{2}$$)($$( 1 + 123412930 T^{2} + 3936588805702081 T^{4} )^{2}$$)($$1 - 294759296 T^{2} + 46567064448316540 T^{4} -$$$$48\!\cdots\!04$$$$T^{6} +$$$$35\!\cdots\!14$$$$T^{8} -$$$$19\!\cdots\!24$$$$T^{10} +$$$$72\!\cdots\!40$$$$T^{12} -$$$$17\!\cdots\!36$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16}$$)
$97$ ($$( 1 - 4352 T + 88529281 T^{2} )^{2}$$)($$1 + 1918 T + 88529281 T^{2}$$)($$( 1 - 11524 T + 146880774 T^{2} - 1020211434244 T^{3} + 7837433594376961 T^{4} )^{2}$$)($$( 1 + 9730 T + 88529281 T^{2} )^{4}$$)($$1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} -$$$$75\!\cdots\!16$$$$T^{6} +$$$$33\!\cdots\!26$$$$T^{7} +$$$$75\!\cdots\!52$$$$T^{8} +$$$$29\!\cdots\!06$$$$T^{9} -$$$$58\!\cdots\!76$$$$T^{10} -$$$$67\!\cdots\!96$$$$T^{11} +$$$$35\!\cdots\!91$$$$T^{12} +$$$$54\!\cdots\!52$$$$T^{13} -$$$$13\!\cdots\!84$$$$T^{14} -$$$$31\!\cdots\!02$$$$T^{15} +$$$$37\!\cdots\!41$$$$T^{16}$$)