# Properties

 Label 108.5 Level 108 Weight 5 Dimension 581 Nonzero newspaces 6 Newform subspaces 9 Sturm bound 3240 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$108\( 108 = 2^{2} \cdot 3^{3}$$ \) Weight: $$k$$ = $$5$$ Nonzero newspaces: $$6$$ Newform subspaces: $$9$$ Sturm bound: $$3240$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(108))$$.

Total New Old
Modular forms 1371 613 758
Cusp forms 1221 581 640
Eisenstein series 150 32 118

## Trace form

 $$581q - 5q^{2} + 7q^{4} + 8q^{5} - 6q^{6} - 16q^{7} - 125q^{8} - 114q^{9} + O(q^{10})$$ $$581q - 5q^{2} + 7q^{4} + 8q^{5} - 6q^{6} - 16q^{7} - 125q^{8} - 114q^{9} - q^{10} + 36q^{11} + 39q^{12} - 554q^{13} - 1113q^{14} - 225q^{15} - 125q^{16} + 50q^{17} + 729q^{18} + 539q^{19} + 2087q^{20} - 294q^{21} + 1083q^{22} + 441q^{23} - 354q^{24} + 2473q^{25} - 5806q^{26} + 54q^{27} + 1626q^{28} - 2383q^{29} + 3951q^{30} - 4234q^{31} + 5065q^{32} - 3777q^{33} - 589q^{34} - 2673q^{35} - 2136q^{36} + 6535q^{37} - 7479q^{38} + 6951q^{39} - 3949q^{40} + 13478q^{41} - 19566q^{42} + 2618q^{43} - 5205q^{44} + 4341q^{45} + 1785q^{46} - 8586q^{47} + 31965q^{48} - 21119q^{49} + 16788q^{50} + 1899q^{51} - 4297q^{52} + 10064q^{53} - 8994q^{54} + 15516q^{55} - 25659q^{56} - 6279q^{57} - 877q^{58} + 9108q^{59} - 39102q^{60} + 4294q^{61} + 5550q^{62} - 11985q^{63} + 14761q^{64} - 1670q^{65} + 47157q^{66} - 7495q^{67} + 67076q^{68} + 12027q^{69} + 8931q^{70} - 19764q^{71} + 6276q^{72} + 18358q^{73} - 7837q^{74} - 3453q^{75} - 40665q^{76} + 6900q^{77} - 43746q^{78} - 4546q^{79} - 109522q^{80} - 35358q^{81} - 46894q^{82} - 57078q^{83} - 39846q^{84} - 10028q^{85} - 24675q^{86} + 40545q^{87} + 8475q^{88} - 13189q^{89} + 107706q^{90} + 28786q^{91} + 97113q^{92} + 30087q^{93} + 62271q^{94} + 85140q^{95} - 6870q^{96} - 34643q^{97} + 33940q^{98} + 12177q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(108))$$

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
108.5.c $$\chi_{108}(53, \cdot)$$ 108.5.c.a 1 1
108.5.c.b 2
108.5.c.c 2
108.5.d $$\chi_{108}(55, \cdot)$$ 108.5.d.a 16 1
108.5.d.b 16
108.5.f $$\chi_{108}(19, \cdot)$$ 108.5.f.a 44 2
108.5.g $$\chi_{108}(17, \cdot)$$ 108.5.g.a 8 2
108.5.j $$\chi_{108}(7, \cdot)$$ 108.5.j.a 420 6
108.5.k $$\chi_{108}(5, \cdot)$$ 108.5.k.a 72 6

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(108))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(108)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 7 T^{2} + 76 T^{4} + 304 T^{6} + 94720 T^{8} + 77824 T^{10} + 4980736 T^{12} + 117440512 T^{14} + 4294967296 T^{16}$$)($$1 - 14 T^{2} + 76 T^{4} - 992 T^{6} - 2816 T^{8} - 253952 T^{10} + 4980736 T^{12} - 234881024 T^{14} + 4294967296 T^{16}$$)
$3$ 1
$5$ ($$( 1 - 25 T )( 1 + 25 T )$$)($$1 + 694 T^{2} + 390625 T^{4}$$)($$1 - 1169 T^{2} + 390625 T^{4}$$)($$( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 633700862500000 T^{10} + 327991027832031250 T^{12} +$$$$12\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} )^{2}$$)($$( 1 + 1816 T^{2} + 1373884 T^{4} + 428961832 T^{6} + 50788147846 T^{8} + 167563215625000 T^{10} + 209638061523437500 T^{12} +$$$$10\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} )^{2}$$)($$1 - 9 T + 1114 T^{2} - 9783 T^{3} + 533599 T^{4} - 10528056 T^{5} + 59806456 T^{6} - 10305069192 T^{7} - 6001445444 T^{8} - 6440668245000 T^{9} + 23361896875000 T^{10} - 2570326171875000 T^{11} + 81420745849609375 T^{12} - 932979583740234375 T^{13} + 66399574279785156250 T^{14} -$$$$33\!\cdots\!25$$$$T^{15} +$$$$23\!\cdots\!25$$$$T^{16}$$)
$7$ ($$1 - 23 T + 2401 T^{2}$$)($$( 1 + 31 T + 2401 T^{2} )^{2}$$)($$( 1 - 5 T + 2401 T^{2} )^{2}$$)($$( 1 - 9380 T^{2} + 36105250 T^{4} - 73168453568 T^{6} + 129372615567787 T^{8} - 421801574297259968 T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$17\!\cdots\!80$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 7628 T^{2} + 28548058 T^{4} - 90794922320 T^{6} + 251781425444899 T^{8} - 523414658985258320 T^{10} +$$$$94\!\cdots\!58$$$$T^{12} -$$$$14\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$)($$1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} +$$$$29\!\cdots\!91$$$$T^{12} +$$$$99\!\cdots\!53$$$$T^{13} -$$$$87\!\cdots\!54$$$$T^{14} -$$$$59\!\cdots\!13$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$)
$11$ ($$( 1 - 121 T )( 1 + 121 T )$$)($$1 + 19318 T^{2} + 214358881 T^{4}$$)($$1 - 15593 T^{2} + 214358881 T^{4}$$)($$( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} -$$$$43\!\cdots\!88$$$$T^{10} +$$$$53\!\cdots\!62$$$$T^{12} -$$$$47\!\cdots\!04$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 - 66488 T^{2} + 2449218172 T^{4} - 58905964133768 T^{6} + 1015821535494509830 T^{8} -$$$$12\!\cdots\!08$$$$T^{10} +$$$$11\!\cdots\!92$$$$T^{12} -$$$$65\!\cdots\!08$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16} )^{2}$$)($$1 - 18 T + 33850 T^{2} - 607356 T^{3} + 452208721 T^{4} - 17443950048 T^{5} + 9074477968558 T^{6} - 445095012639474 T^{7} + 191968264536523468 T^{8} - 6516636080054538834 T^{9} +$$$$19\!\cdots\!98$$$$T^{10} -$$$$54\!\cdots\!08$$$$T^{11} +$$$$20\!\cdots\!81$$$$T^{12} -$$$$40\!\cdots\!56$$$$T^{13} +$$$$33\!\cdots\!50$$$$T^{14} -$$$$25\!\cdots\!58$$$$T^{15} +$$$$21\!\cdots\!21$$$$T^{16}$$)
$13$ ($$1 - 191 T + 28561 T^{2}$$)($$( 1 + 241 T + 28561 T^{2} )^{2}$$)($$( 1 + 34 T + 28561 T^{2} )^{2}$$)($$( 1 + 88 T + 63508 T^{2} + 5077384 T^{3} + 2650180198 T^{4} + 145015164424 T^{5} + 51805426629268 T^{6} + 2050231490778328 T^{7} + 665416609183179841 T^{8} )^{4}$$)($$( 1 - 44 T + 67090 T^{2} - 966368 T^{3} + 2192145307 T^{4} - 27600436448 T^{5} + 54727374071890 T^{6} - 1025115745389164 T^{7} + 665416609183179841 T^{8} )^{4}$$)($$1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} +$$$$18\!\cdots\!84$$$$T^{10} +$$$$58\!\cdots\!40$$$$T^{11} +$$$$23\!\cdots\!75$$$$T^{12} -$$$$13\!\cdots\!65$$$$T^{13} -$$$$22\!\cdots\!94$$$$T^{14} +$$$$77\!\cdots\!05$$$$T^{15} +$$$$44\!\cdots\!81$$$$T^{16}$$)
$17$ ($$( 1 - 289 T )( 1 + 289 T )$$)($$1 - 118442 T^{2} + 6975757441 T^{4}$$)($$1 + 35458 T^{2} + 6975757441 T^{4}$$)($$( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} +$$$$47\!\cdots\!02$$$$T^{8} +$$$$35\!\cdots\!92$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{12} +$$$$81\!\cdots\!44$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 + 422296 T^{2} + 87803746876 T^{4} + 11972376766816936 T^{6} +$$$$11\!\cdots\!30$$$$T^{8} +$$$$83\!\cdots\!76$$$$T^{10} +$$$$42\!\cdots\!56$$$$T^{12} +$$$$14\!\cdots\!16$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16} )^{2}$$)($$1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} +$$$$63\!\cdots\!86$$$$T^{8} -$$$$47\!\cdots\!55$$$$T^{10} +$$$$25\!\cdots\!74$$$$T^{12} -$$$$97\!\cdots\!25$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16}$$)
$19$ ($$1 - 647 T + 130321 T^{2}$$)($$( 1 + 271 T + 130321 T^{2} )^{2}$$)($$( 1 + 64 T + 130321 T^{2} )^{2}$$)($$( 1 - 680120 T^{2} + 233557156252 T^{4} - 51394020863660552 T^{6} +$$$$79\!\cdots\!98$$$$T^{8} -$$$$87\!\cdots\!32$$$$T^{10} +$$$$67\!\cdots\!12$$$$T^{12} -$$$$33\!\cdots\!20$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 298556 T^{2} + 58331731402 T^{4} - 10017587005489136 T^{6} +$$$$14\!\cdots\!39$$$$T^{8} -$$$$17\!\cdots\!76$$$$T^{10} +$$$$16\!\cdots\!62$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$)
$23$ ($$( 1 - 529 T )( 1 + 529 T )$$)($$1 - 511082 T^{2} + 78310985281 T^{4}$$)($$1 - 185138 T^{2} + 78310985281 T^{4}$$)($$( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} +$$$$35\!\cdots\!18$$$$T^{8} -$$$$90\!\cdots\!52$$$$T^{10} +$$$$19\!\cdots\!68$$$$T^{12} -$$$$32\!\cdots\!28$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 - 1184696 T^{2} + 509745640444 T^{4} - 84604026939000968 T^{6} +$$$$71\!\cdots\!22$$$$T^{8} -$$$$66\!\cdots\!08$$$$T^{10} +$$$$31\!\cdots\!84$$$$T^{12} -$$$$56\!\cdots\!36$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16} )^{2}$$)($$1 - 1719 T + 1529458 T^{2} - 935945649 T^{3} + 342642958747 T^{4} + 16333135609500 T^{5} - 118516645434237428 T^{6} +$$$$10\!\cdots\!68$$$$T^{7} -$$$$65\!\cdots\!00$$$$T^{8} +$$$$29\!\cdots\!88$$$$T^{9} -$$$$92\!\cdots\!68$$$$T^{10} +$$$$35\!\cdots\!00$$$$T^{11} +$$$$21\!\cdots\!67$$$$T^{12} -$$$$16\!\cdots\!49$$$$T^{13} +$$$$73\!\cdots\!78$$$$T^{14} -$$$$23\!\cdots\!39$$$$T^{15} +$$$$37\!\cdots\!21$$$$T^{16}$$)
$29$ ($$( 1 - 841 T )( 1 + 841 T )$$)($$1 - 1220162 T^{2} + 500246412961 T^{4}$$)($$1 - 286718 T^{2} + 500246412961 T^{4}$$)($$( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} +$$$$44\!\cdots\!14$$$$T^{8} +$$$$27\!\cdots\!92$$$$T^{10} +$$$$12\!\cdots\!32$$$$T^{12} +$$$$37\!\cdots\!64$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 + 1223560 T^{2} + 603752953756 T^{4} - 126384872678073416 T^{6} -$$$$34\!\cdots\!30$$$$T^{8} -$$$$63\!\cdots\!76$$$$T^{10} +$$$$15\!\cdots\!76$$$$T^{12} +$$$$15\!\cdots\!60$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16} )^{2}$$)($$1 + 2115 T + 4091014 T^{2} + 5498870985 T^{3} + 7048695081595 T^{4} + 8024737206821040 T^{5} + 8297140249856169556 T^{6} +$$$$79\!\cdots\!80$$$$T^{7} +$$$$68\!\cdots\!24$$$$T^{8} +$$$$56\!\cdots\!80$$$$T^{9} +$$$$41\!\cdots\!16$$$$T^{10} +$$$$28\!\cdots\!40$$$$T^{11} +$$$$17\!\cdots\!95$$$$T^{12} +$$$$97\!\cdots\!85$$$$T^{13} +$$$$51\!\cdots\!34$$$$T^{14} +$$$$18\!\cdots\!15$$$$T^{15} +$$$$62\!\cdots\!41$$$$T^{16}$$)
$31$ ($$1 - 194 T + 923521 T^{2}$$)($$( 1 + 778 T + 923521 T^{2} )^{2}$$)($$( 1 + 697 T + 923521 T^{2} )^{2}$$)($$( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} +$$$$29\!\cdots\!59$$$$T^{8} -$$$$20\!\cdots\!72$$$$T^{10} +$$$$19\!\cdots\!54$$$$T^{12} -$$$$82\!\cdots\!76$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 4495112 T^{2} + 10589743228444 T^{4} - 16346503421024387384 T^{6} +$$$$17\!\cdots\!66$$$$T^{8} -$$$$13\!\cdots\!44$$$$T^{10} +$$$$77\!\cdots\!64$$$$T^{12} -$$$$27\!\cdots\!52$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16} )^{2}$$)($$1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} +$$$$31\!\cdots\!92$$$$T^{8} -$$$$12\!\cdots\!14$$$$T^{9} -$$$$29\!\cdots\!86$$$$T^{10} -$$$$23\!\cdots\!56$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} +$$$$12\!\cdots\!37$$$$T^{13} -$$$$15\!\cdots\!84$$$$T^{14} -$$$$10\!\cdots\!67$$$$T^{15} +$$$$52\!\cdots\!61$$$$T^{16}$$)
$37$ ($$1 - 2591 T + 1874161 T^{2}$$)($$( 1 - 1079 T + 1874161 T^{2} )^{2}$$)($$( 1 + 748 T + 1874161 T^{2} )^{2}$$)($$( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 7093653145699808 T^{5} + 6507683083621962172 T^{6} -$$$$52\!\cdots\!00$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{4}$$)($$( 1 - 20 T + 5679442 T^{2} - 534101024 T^{3} + 14604995159131 T^{4} - 1000991309240864 T^{5} + 19948923334735992082 T^{6} -$$$$13\!\cdots\!20$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{4}$$)($$( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$)
$41$ ($$( 1 - 1681 T )( 1 + 1681 T )$$)($$1 - 791522 T^{2} + 7984925229121 T^{4}$$)($$1 - 5183666 T^{2} + 7984925229121 T^{4}$$)($$( 1 + 14045080 T^{2} + 100482710328412 T^{4} +$$$$46\!\cdots\!84$$$$T^{6} +$$$$15\!\cdots\!54$$$$T^{8} +$$$$37\!\cdots\!64$$$$T^{10} +$$$$64\!\cdots\!92$$$$T^{12} +$$$$71\!\cdots\!80$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16} )^{2}$$)($$( 1 + 8498440 T^{2} + 39457953574684 T^{4} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$47\!\cdots\!42$$$$T^{8} +$$$$11\!\cdots\!28$$$$T^{10} +$$$$25\!\cdots\!44$$$$T^{12} +$$$$43\!\cdots\!40$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16} )^{2}$$)($$1 - 7920 T + 37687894 T^{2} - 132890424480 T^{3} + 385083705354505 T^{4} - 963185727644706960 T^{5} +$$$$21\!\cdots\!66$$$$T^{6} -$$$$41\!\cdots\!20$$$$T^{7} +$$$$74\!\cdots\!64$$$$T^{8} -$$$$11\!\cdots\!20$$$$T^{9} +$$$$16\!\cdots\!86$$$$T^{10} -$$$$21\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!05$$$$T^{12} -$$$$23\!\cdots\!80$$$$T^{13} +$$$$19\!\cdots\!34$$$$T^{14} -$$$$11\!\cdots\!20$$$$T^{15} +$$$$40\!\cdots\!81$$$$T^{16}$$)
$43$ ($$1 + 3214 T + 3418801 T^{2}$$)($$( 1 + 298 T + 3418801 T^{2} )^{2}$$)($$( 1 - 2618 T + 3418801 T^{2} )^{2}$$)($$( 1 - 11990648 T^{2} + 69115070774812 T^{4} -$$$$29\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!18$$$$T^{8} -$$$$34\!\cdots\!24$$$$T^{10} +$$$$94\!\cdots\!12$$$$T^{12} -$$$$19\!\cdots\!48$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 16979720 T^{2} + 140301823474588 T^{4} -$$$$77\!\cdots\!28$$$$T^{6} +$$$$30\!\cdots\!14$$$$T^{8} -$$$$90\!\cdots\!28$$$$T^{10} +$$$$19\!\cdots\!88$$$$T^{12} -$$$$27\!\cdots\!20$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16} )^{2}$$)($$1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} -$$$$54\!\cdots\!36$$$$T^{6} +$$$$27\!\cdots\!16$$$$T^{7} +$$$$21\!\cdots\!12$$$$T^{8} +$$$$94\!\cdots\!16$$$$T^{9} -$$$$63\!\cdots\!36$$$$T^{10} +$$$$23\!\cdots\!04$$$$T^{11} +$$$$14\!\cdots\!91$$$$T^{12} -$$$$97\!\cdots\!48$$$$T^{13} -$$$$20\!\cdots\!04$$$$T^{14} +$$$$37\!\cdots\!68$$$$T^{15} +$$$$18\!\cdots\!01$$$$T^{16}$$)
$47$ ($$( 1 - 2209 T )( 1 + 2209 T )$$)($$1 + 1175638 T^{2} + 23811286661761 T^{4}$$)($$1 - 2758046 T^{2} + 23811286661761 T^{4}$$)($$( 1 - 27701336 T^{2} + 374201000701276 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$18\!\cdots\!14$$$$T^{8} -$$$$75\!\cdots\!28$$$$T^{10} +$$$$21\!\cdots\!96$$$$T^{12} -$$$$37\!\cdots\!16$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 11004728 T^{2} + 98757031754236 T^{4} -$$$$67\!\cdots\!20$$$$T^{6} +$$$$36\!\cdots\!06$$$$T^{8} -$$$$16\!\cdots\!20$$$$T^{10} +$$$$55\!\cdots\!56$$$$T^{12} -$$$$14\!\cdots\!68$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16} )^{2}$$)($$1 + 13689 T + 103685338 T^{2} + 564293857959 T^{3} + 2435028217967227 T^{4} + 8736748337842042500 T^{5} +$$$$26\!\cdots\!72$$$$T^{6} +$$$$71\!\cdots\!32$$$$T^{7} +$$$$16\!\cdots\!20$$$$T^{8} +$$$$35\!\cdots\!92$$$$T^{9} +$$$$63\!\cdots\!92$$$$T^{10} +$$$$10\!\cdots\!00$$$$T^{11} +$$$$13\!\cdots\!67$$$$T^{12} +$$$$15\!\cdots\!59$$$$T^{13} +$$$$13\!\cdots\!78$$$$T^{14} +$$$$90\!\cdots\!29$$$$T^{15} +$$$$32\!\cdots\!41$$$$T^{16}$$)
$53$ ($$( 1 - 2809 T )( 1 + 2809 T )$$)($$1 - 6255362 T^{2} + 62259690411361 T^{4}$$)($$1 - 14633921 T^{2} + 62259690411361 T^{4}$$)($$( 1 + 35837884 T^{2} + 582188420839618 T^{4} +$$$$60\!\cdots\!16$$$$T^{6} +$$$$50\!\cdots\!79$$$$T^{8} +$$$$37\!\cdots\!76$$$$T^{10} +$$$$22\!\cdots\!78$$$$T^{12} +$$$$86\!\cdots\!04$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 + 25978312 T^{2} + 440128249758748 T^{4} +$$$$51\!\cdots\!52$$$$T^{6} +$$$$46\!\cdots\!86$$$$T^{8} +$$$$32\!\cdots\!72$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{12} +$$$$62\!\cdots\!72$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$)($$1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} +$$$$67\!\cdots\!26$$$$T^{8} -$$$$52\!\cdots\!60$$$$T^{10} +$$$$44\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16}$$)
$59$ ($$( 1 - 3481 T )( 1 + 3481 T )$$)($$1 - 16021322 T^{2} + 146830437604321 T^{4}$$)($$1 + 9567874 T^{2} + 146830437604321 T^{4}$$)($$( 1 - 70312136 T^{2} + 2419955263154716 T^{4} -$$$$51\!\cdots\!68$$$$T^{6} +$$$$75\!\cdots\!74$$$$T^{8} -$$$$76\!\cdots\!28$$$$T^{10} +$$$$52\!\cdots\!56$$$$T^{12} -$$$$22\!\cdots\!96$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16} )^{2}$$)($$( 1 - 31826744 T^{2} + 538590143907196 T^{4} -$$$$92\!\cdots\!92$$$$T^{6} +$$$$13\!\cdots\!74$$$$T^{8} -$$$$13\!\cdots\!32$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{12} -$$$$10\!\cdots\!84$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16} )^{2}$$)($$1 - 20052 T + 216711700 T^{2} - 1657982214864 T^{3} + 9931594296358591 T^{4} - 49579528565018409012 T^{5} +$$$$21\!\cdots\!48$$$$T^{6} -$$$$84\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!08$$$$T^{8} -$$$$10\!\cdots\!36$$$$T^{9} +$$$$31\!\cdots\!08$$$$T^{10} -$$$$88\!\cdots\!72$$$$T^{11} +$$$$21\!\cdots\!31$$$$T^{12} -$$$$43\!\cdots\!64$$$$T^{13} +$$$$68\!\cdots\!00$$$$T^{14} -$$$$76\!\cdots\!92$$$$T^{15} +$$$$46\!\cdots\!81$$$$T^{16}$$)
$61$ ($$1 + 5233 T + 13845841 T^{2}$$)($$( 1 + 2641 T + 13845841 T^{2} )^{2}$$)($$( 1 - 6404 T + 13845841 T^{2} )^{2}$$)($$( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 401458424080372240 T^{5} +$$$$67\!\cdots\!80$$$$T^{6} +$$$$18\!\cdots\!48$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{4}$$)($$( 1 + 412 T + 9607138 T^{2} + 3267599488 T^{3} + 216368249231659 T^{4} + 45242662962529408 T^{5} +$$$$18\!\cdots\!78$$$$T^{6} +$$$$10\!\cdots\!52$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{4}$$)($$1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} +$$$$12\!\cdots\!24$$$$T^{6} +$$$$40\!\cdots\!44$$$$T^{7} -$$$$98\!\cdots\!68$$$$T^{8} +$$$$55\!\cdots\!04$$$$T^{9} +$$$$23\!\cdots\!44$$$$T^{10} -$$$$35\!\cdots\!44$$$$T^{11} +$$$$15\!\cdots\!91$$$$T^{12} +$$$$76\!\cdots\!23$$$$T^{13} -$$$$74\!\cdots\!94$$$$T^{14} +$$$$18\!\cdots\!97$$$$T^{15} +$$$$13\!\cdots\!21$$$$T^{16}$$)
$67$ ($$1 + 8809 T + 20151121 T^{2}$$)($$( 1 - 5609 T + 20151121 T^{2} )^{2}$$)($$( 1 + 5218 T + 20151121 T^{2} )^{2}$$)($$( 1 - 29579000 T^{2} + 1274223737389084 T^{4} -$$$$30\!\cdots\!28$$$$T^{6} +$$$$73\!\cdots\!58$$$$T^{8} -$$$$12\!\cdots\!48$$$$T^{10} +$$$$21\!\cdots\!04$$$$T^{12} -$$$$19\!\cdots\!00$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 38213180 T^{2} + 1702311028453834 T^{4} -$$$$38\!\cdots\!28$$$$T^{6} +$$$$99\!\cdots\!27$$$$T^{8} -$$$$15\!\cdots\!48$$$$T^{10} +$$$$28\!\cdots\!54$$$$T^{12} -$$$$25\!\cdots\!80$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16} )^{2}$$)($$1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} +$$$$70\!\cdots\!02$$$$T^{6} -$$$$11\!\cdots\!78$$$$T^{7} -$$$$26\!\cdots\!64$$$$T^{8} -$$$$22\!\cdots\!38$$$$T^{9} +$$$$28\!\cdots\!82$$$$T^{10} +$$$$60\!\cdots\!96$$$$T^{11} +$$$$88\!\cdots\!37$$$$T^{12} -$$$$85\!\cdots\!28$$$$T^{13} -$$$$22\!\cdots\!34$$$$T^{14} -$$$$20\!\cdots\!14$$$$T^{15} +$$$$27\!\cdots\!61$$$$T^{16}$$)
$71$ ($$( 1 - 5041 T )( 1 + 5041 T )$$)($$1 - 31383362 T^{2} + 645753531245761 T^{4}$$)($$1 - 7658462 T^{2} + 645753531245761 T^{4}$$)($$( 1 - 86649128 T^{2} + 3567527319446236 T^{4} -$$$$11\!\cdots\!36$$$$T^{6} +$$$$32\!\cdots\!90$$$$T^{8} -$$$$74\!\cdots\!96$$$$T^{10} +$$$$14\!\cdots\!56$$$$T^{12} -$$$$23\!\cdots\!68$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 102214088 T^{2} + 3903098982427036 T^{4} -$$$$62\!\cdots\!96$$$$T^{6} +$$$$63\!\cdots\!58$$$$T^{8} -$$$$40\!\cdots\!56$$$$T^{10} +$$$$16\!\cdots\!56$$$$T^{12} -$$$$27\!\cdots\!28$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16} )^{2}$$)($$1 - 68871716 T^{2} + 3244147638477940 T^{4} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$30\!\cdots\!74$$$$T^{8} -$$$$73\!\cdots\!64$$$$T^{10} +$$$$13\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$)
$73$ ($$1 - 9791 T + 28398241 T^{2}$$)($$( 1 - 7199 T + 28398241 T^{2} )^{2}$$)($$( 1 + 4519 T + 28398241 T^{2} )^{2}$$)($$( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 2874590492512190816 T^{5} +$$$$44\!\cdots\!14$$$$T^{6} -$$$$47\!\cdots\!60$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{4}$$)($$( 1 - 20 T + 67167226 T^{2} + 662878096 T^{3} + 2724353301452611 T^{4} + 18824571923829136 T^{5} +$$$$54\!\cdots\!06$$$$T^{6} -$$$$45\!\cdots\!20$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{4}$$)($$( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} +$$$$47\!\cdots\!70$$$$T^{6} +$$$$89\!\cdots\!21$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$)
$79$ ($$1 + 12361 T + 38950081 T^{2}$$)($$( 1 - 329 T + 38950081 T^{2} )^{2}$$)($$( 1 - 7502 T + 38950081 T^{2} )^{2}$$)($$( 1 - 162885800 T^{2} + 12409200314583772 T^{4} -$$$$63\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!06$$$$T^{8} -$$$$96\!\cdots\!96$$$$T^{10} +$$$$28\!\cdots\!12$$$$T^{12} -$$$$56\!\cdots\!00$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 164533964 T^{2} + 15212503116962266 T^{4} -$$$$95\!\cdots\!24$$$$T^{6} +$$$$43\!\cdots\!71$$$$T^{8} -$$$$14\!\cdots\!64$$$$T^{10} +$$$$35\!\cdots\!86$$$$T^{12} -$$$$57\!\cdots\!84$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$)($$1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} -$$$$16\!\cdots\!36$$$$T^{6} +$$$$76\!\cdots\!80$$$$T^{7} +$$$$88\!\cdots\!64$$$$T^{8} +$$$$29\!\cdots\!80$$$$T^{9} -$$$$24\!\cdots\!96$$$$T^{10} +$$$$21\!\cdots\!60$$$$T^{11} +$$$$72\!\cdots\!75$$$$T^{12} -$$$$16\!\cdots\!15$$$$T^{13} -$$$$30\!\cdots\!34$$$$T^{14} +$$$$29\!\cdots\!95$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16}$$)
$83$ ($$( 1 - 6889 T )( 1 + 6889 T )$$)($$1 - 93167042 T^{2} + 2252292232139041 T^{4}$$)($$1 - 64875281 T^{2} + 2252292232139041 T^{4}$$)($$( 1 - 140155436 T^{2} + 8287556309932090 T^{4} -$$$$42\!\cdots\!92$$$$T^{6} +$$$$22\!\cdots\!03$$$$T^{8} -$$$$95\!\cdots\!72$$$$T^{10} +$$$$42\!\cdots\!90$$$$T^{12} -$$$$16\!\cdots\!56$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 185955656 T^{2} + 18958921421058076 T^{4} -$$$$13\!\cdots\!16$$$$T^{6} +$$$$73\!\cdots\!26$$$$T^{8} -$$$$30\!\cdots\!56$$$$T^{10} +$$$$96\!\cdots\!56$$$$T^{12} -$$$$21\!\cdots\!76$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$)($$1 + 37017 T + 725723290 T^{2} + 9956481997959 T^{3} + 104510585134438411 T^{4} +$$$$87\!\cdots\!72$$$$T^{5} +$$$$62\!\cdots\!28$$$$T^{6} +$$$$40\!\cdots\!76$$$$T^{7} +$$$$26\!\cdots\!48$$$$T^{8} +$$$$19\!\cdots\!96$$$$T^{9} +$$$$14\!\cdots\!48$$$$T^{10} +$$$$93\!\cdots\!92$$$$T^{11} +$$$$53\!\cdots\!91$$$$T^{12} +$$$$23\!\cdots\!59$$$$T^{13} +$$$$82\!\cdots\!90$$$$T^{14} +$$$$20\!\cdots\!97$$$$T^{15} +$$$$25\!\cdots\!61$$$$T^{16}$$)
$89$ ($$( 1 - 7921 T )( 1 + 7921 T )$$)($$1 - 58951082 T^{2} + 3936588805702081 T^{4}$$)($$1 - 46736606 T^{2} + 3936588805702081 T^{4}$$)($$( 1 + 257253400 T^{2} + 38619499284608860 T^{4} +$$$$38\!\cdots\!44$$$$T^{6} +$$$$28\!\cdots\!58$$$$T^{8} +$$$$15\!\cdots\!64$$$$T^{10} +$$$$59\!\cdots\!60$$$$T^{12} +$$$$15\!\cdots\!00$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 + 103821208 T^{2} + 12140769019811644 T^{4} +$$$$81\!\cdots\!28$$$$T^{6} +$$$$66\!\cdots\!06$$$$T^{8} +$$$$32\!\cdots\!68$$$$T^{10} +$$$$18\!\cdots\!84$$$$T^{12} +$$$$63\!\cdots\!28$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16} )^{2}$$)($$1 - 294759296 T^{2} + 46567064448316540 T^{4} -$$$$48\!\cdots\!04$$$$T^{6} +$$$$35\!\cdots\!14$$$$T^{8} -$$$$19\!\cdots\!24$$$$T^{10} +$$$$72\!\cdots\!40$$$$T^{12} -$$$$17\!\cdots\!36$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16}$$)
$97$ ($$1 - 9743 T + 88529281 T^{2}$$)($$( 1 + 15961 T + 88529281 T^{2} )^{2}$$)($$( 1 - 10571 T + 88529281 T^{2} )^{2}$$)($$( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 37912615976467685776 T^{5} +$$$$23\!\cdots\!98$$$$T^{6} +$$$$12\!\cdots\!12$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{4}$$)($$( 1 - 3716 T + 259176202 T^{2} - 799969063952 T^{3} + 30273937496959507 T^{4} - 70820686053913578512 T^{5} +$$$$20\!\cdots\!22$$$$T^{6} -$$$$25\!\cdots\!56$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{4}$$)($$1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} -$$$$75\!\cdots\!16$$$$T^{6} +$$$$33\!\cdots\!26$$$$T^{7} +$$$$75\!\cdots\!52$$$$T^{8} +$$$$29\!\cdots\!06$$$$T^{9} -$$$$58\!\cdots\!76$$$$T^{10} -$$$$67\!\cdots\!96$$$$T^{11} +$$$$35\!\cdots\!91$$$$T^{12} +$$$$54\!\cdots\!52$$$$T^{13} -$$$$13\!\cdots\!84$$$$T^{14} -$$$$31\!\cdots\!02$$$$T^{15} +$$$$37\!\cdots\!41$$$$T^{16}$$)