# Properties

 Label 128.4 Level 128 Weight 4 Dimension 840 Nonzero newspaces 5 Newform subspaces 17 Sturm bound 4096 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$5$$ Newform subspaces: $$17$$ Sturm bound: $$4096$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(128))$$.

Total New Old
Modular forms 1616 888 728
Cusp forms 1456 840 616
Eisenstein series 160 48 112

## Trace form

 $$840q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} + O(q^{10})$$ $$840q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} - 16q^{10} - 12q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 16q^{15} - 16q^{16} - 24q^{17} - 16q^{18} - 12q^{19} - 16q^{20} - 124q^{21} - 16q^{22} - 340q^{23} - 16q^{24} - 196q^{25} - 16q^{26} + 252q^{27} - 16q^{28} + 384q^{29} - 16q^{30} + 736q^{31} - 16q^{32} + 896q^{33} - 16q^{34} + 444q^{35} - 16q^{36} - 16q^{38} - 612q^{39} - 16q^{40} - 964q^{41} - 16q^{42} - 820q^{43} - 16q^{44} - 300q^{45} - 16q^{46} - 16q^{47} - 16q^{48} + 1348q^{49} + 5696q^{50} + 1368q^{51} + 6608q^{52} + 1488q^{53} + 3440q^{54} + 276q^{55} - 800q^{56} - 1364q^{57} - 4768q^{58} - 1388q^{59} - 9808q^{60} - 3664q^{61} - 5872q^{62} - 2512q^{63} - 12112q^{64} - 3920q^{65} - 10960q^{66} - 2052q^{67} - 4144q^{68} - 2236q^{69} - 4048q^{70} - 236q^{71} + 1280q^{72} + 1708q^{73} + 5248q^{74} + 1696q^{75} + 11888q^{76} + 2420q^{77} + 14096q^{78} - 16q^{79} + 10016q^{80} - 788q^{81} - 16q^{82} - 2452q^{83} - 16q^{84} + 504q^{85} - 16q^{86} + 1276q^{87} - 16q^{88} + 3372q^{89} - 16q^{90} + 3588q^{91} - 16q^{92} + 4256q^{93} - 16q^{94} + 6072q^{95} - 16q^{96} + 5920q^{97} - 16q^{98} + 5408q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(128))$$

We only show spaces with even 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
128.4.a $$\chi_{128}(1, \cdot)$$ 128.4.a.a 1 1
128.4.a.b 1
128.4.a.c 1
128.4.a.d 1
128.4.a.e 2
128.4.a.f 2
128.4.a.g 2
128.4.a.h 2
128.4.b $$\chi_{128}(65, \cdot)$$ 128.4.b.a 2 1
128.4.b.b 2
128.4.b.c 2
128.4.b.d 2
128.4.b.e 4
128.4.e $$\chi_{128}(33, \cdot)$$ 128.4.e.a 10 2
128.4.e.b 10
128.4.g $$\chi_{128}(17, \cdot)$$ 128.4.g.a 44 4
128.4.i $$\chi_{128}(9, \cdot)$$ None 0 8
128.4.k $$\chi_{128}(5, \cdot)$$ 128.4.k.a 752 16

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(128))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(128)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 2 T + 27 T^{2}$$)($$1 + 2 T + 27 T^{2}$$)($$1 - 2 T + 27 T^{2}$$)($$1 - 2 T + 27 T^{2}$$)($$1 + 4 T + 10 T^{2} + 108 T^{3} + 729 T^{4}$$)($$1 + 4 T + 10 T^{2} + 108 T^{3} + 729 T^{4}$$)($$1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4}$$)($$1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4}$$)($$1 + 10 T^{2} + 729 T^{4}$$)($$( 1 - 10 T + 27 T^{2} )( 1 + 10 T + 27 T^{2} )$$)($$( 1 - 27 T^{2} )^{2}$$)($$1 + 10 T^{2} + 729 T^{4}$$)($$( 1 - 14 T^{2} + 729 T^{4} )^{2}$$)($$1 + 2 T + 2 T^{2} - 42 T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 66792 T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 82248156 T^{11} + 42190146 T^{12} + 1314666936 T^{13} + 1883426904 T^{14} + 10905169320 T^{15} - 221217099219 T^{16} - 439334834526 T^{17} + 564859072962 T^{18} + 15251194969974 T^{19} + 205891132094649 T^{20}$$)($$1 - 2 T + 2 T^{2} + 42 T^{3} - 571 T^{4} - 760 T^{5} + 3544 T^{6} - 66792 T^{7} + 57874 T^{8} + 3046228 T^{9} - 6701044 T^{10} + 82248156 T^{11} + 42190146 T^{12} - 1314666936 T^{13} + 1883426904 T^{14} - 10905169320 T^{15} - 221217099219 T^{16} + 439334834526 T^{17} + 564859072962 T^{18} - 15251194969974 T^{19} + 205891132094649 T^{20}$$)
$5$ ($$1 + 6 T + 125 T^{2}$$)($$1 - 6 T + 125 T^{2}$$)($$1 + 6 T + 125 T^{2}$$)($$1 - 6 T + 125 T^{2}$$)($$1 + 4 T + 62 T^{2} + 500 T^{3} + 15625 T^{4}$$)($$1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4}$$)($$1 + 4 T + 62 T^{2} + 500 T^{3} + 15625 T^{4}$$)($$1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4}$$)($$1 - 106 T^{2} + 15625 T^{4}$$)($$( 1 - 125 T^{2} )^{2}$$)($$( 1 - 22 T + 125 T^{2} )( 1 + 22 T + 125 T^{2} )$$)($$1 - 106 T^{2} + 15625 T^{4}$$)($$( 1 + 70 T^{2} + 15625 T^{4} )^{2}$$)($$1 - 2 T + 2 T^{2} + 966 T^{3} - 13723 T^{4} - 18040 T^{5} + 530104 T^{6} + 11981288 T^{7} - 28535006 T^{8} - 2301854348 T^{9} + 23672040908 T^{10} - 287731793500 T^{11} - 445859468750 T^{12} + 23400953125000 T^{13} + 129419921875000 T^{14} - 550537109375000 T^{15} - 52349090576171875 T^{16} + 460624694824218750 T^{17} + 119209289550781250 T^{18} - 14901161193847656250 T^{19} +$$$$93\!\cdots\!25$$$$T^{20}$$)($$1 - 2 T + 2 T^{2} + 966 T^{3} - 13723 T^{4} - 18040 T^{5} + 530104 T^{6} + 11981288 T^{7} - 28535006 T^{8} - 2301854348 T^{9} + 23672040908 T^{10} - 287731793500 T^{11} - 445859468750 T^{12} + 23400953125000 T^{13} + 129419921875000 T^{14} - 550537109375000 T^{15} - 52349090576171875 T^{16} + 460624694824218750 T^{17} + 119209289550781250 T^{18} - 14901161193847656250 T^{19} +$$$$93\!\cdots\!25$$$$T^{20}$$)
$7$ ($$1 - 20 T + 343 T^{2}$$)($$1 + 20 T + 343 T^{2}$$)($$1 + 20 T + 343 T^{2}$$)($$1 - 20 T + 343 T^{2}$$)($$1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4}$$)($$1 - 8 T + 510 T^{2} - 2744 T^{3} + 117649 T^{4}$$)($$1 - 8 T + 510 T^{2} - 2744 T^{3} + 117649 T^{4}$$)($$1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4}$$)($$( 1 + 32 T + 343 T^{2} )^{2}$$)($$( 1 + 343 T^{2} )^{2}$$)($$( 1 + 343 T^{2} )^{2}$$)($$( 1 - 32 T + 343 T^{2} )^{2}$$)($$( 1 + 174 T^{2} + 117649 T^{4} )^{2}$$)($$1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} +$$$$25\!\cdots\!85$$$$T^{16} -$$$$33\!\cdots\!62$$$$T^{18} +$$$$22\!\cdots\!49$$$$T^{20}$$)($$1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} +$$$$25\!\cdots\!85$$$$T^{16} -$$$$33\!\cdots\!62$$$$T^{18} +$$$$22\!\cdots\!49$$$$T^{20}$$)
$11$ ($$1 + 14 T + 1331 T^{2}$$)($$1 + 14 T + 1331 T^{2}$$)($$1 - 14 T + 1331 T^{2}$$)($$1 - 14 T + 1331 T^{2}$$)($$1 + 92 T + 4730 T^{2} + 122452 T^{3} + 1771561 T^{4}$$)($$1 + 92 T + 4730 T^{2} + 122452 T^{3} + 1771561 T^{4}$$)($$1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4}$$)($$1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4}$$)($$1 - 2598 T^{2} + 1771561 T^{4}$$)($$( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} )$$)($$( 1 - 1331 T^{2} )^{2}$$)($$1 - 2598 T^{2} + 1771561 T^{4}$$)($$( 1 - 58 T + 1331 T^{2} )^{2}( 1 + 58 T + 1331 T^{2} )^{2}$$)($$1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 6887886337838 T^{8} + 101615185776500 T^{9} + 18755914132083020 T^{10} + 135249812268521500 T^{11} - 12202310808546625118 T^{12} -$$$$73\!\cdots\!28$$$$T^{13} +$$$$17\!\cdots\!44$$$$T^{14} +$$$$33\!\cdots\!00$$$$T^{15} +$$$$22\!\cdots\!57$$$$T^{16} -$$$$90\!\cdots\!74$$$$T^{17} +$$$$15\!\cdots\!42$$$$T^{18} -$$$$23\!\cdots\!78$$$$T^{19} +$$$$17\!\cdots\!01$$$$T^{20}$$)($$1 + 18 T + 162 T^{2} + 122934 T^{3} + 4077397 T^{4} - 79597000 T^{5} + 5463099864 T^{6} + 313798751208 T^{7} - 6887886337838 T^{8} - 101615185776500 T^{9} + 18755914132083020 T^{10} - 135249812268521500 T^{11} - 12202310808546625118 T^{12} +$$$$73\!\cdots\!28$$$$T^{13} +$$$$17\!\cdots\!44$$$$T^{14} -$$$$33\!\cdots\!00$$$$T^{15} +$$$$22\!\cdots\!57$$$$T^{16} +$$$$90\!\cdots\!74$$$$T^{17} +$$$$15\!\cdots\!42$$$$T^{18} +$$$$23\!\cdots\!78$$$$T^{19} +$$$$17\!\cdots\!01$$$$T^{20}$$)
$13$ ($$1 + 54 T + 2197 T^{2}$$)($$1 - 54 T + 2197 T^{2}$$)($$1 + 54 T + 2197 T^{2}$$)($$1 - 54 T + 2197 T^{2}$$)($$1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4}$$)($$1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4}$$)($$1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4}$$)($$1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4}$$)($$1 - 3994 T^{2} + 4826809 T^{4}$$)($$( 1 - 2197 T^{2} )^{2}$$)($$( 1 - 18 T + 2197 T^{2} )( 1 + 18 T + 2197 T^{2} )$$)($$1 - 3994 T^{2} + 4826809 T^{4}$$)($$( 1 - 4074 T^{2} + 4826809 T^{4} )^{2}$$)($$1 - 2 T + 2 T^{2} + 45206 T^{3} - 2401451 T^{4} - 29395960 T^{5} + 1085386040 T^{6} + 384228102440 T^{7} - 9400034983966 T^{8} - 1531455561616908 T^{9} + 23489689415409228 T^{10} - 3364607868872346876 T^{11} - 45372173460921944494 T^{12} +$$$$40\!\cdots\!20$$$$T^{13} +$$$$25\!\cdots\!40$$$$T^{14} -$$$$15\!\cdots\!20$$$$T^{15} -$$$$27\!\cdots\!79$$$$T^{16} +$$$$11\!\cdots\!78$$$$T^{17} +$$$$10\!\cdots\!22$$$$T^{18} -$$$$23\!\cdots\!34$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20}$$)($$1 - 2 T + 2 T^{2} + 45206 T^{3} - 2401451 T^{4} - 29395960 T^{5} + 1085386040 T^{6} + 384228102440 T^{7} - 9400034983966 T^{8} - 1531455561616908 T^{9} + 23489689415409228 T^{10} - 3364607868872346876 T^{11} - 45372173460921944494 T^{12} +$$$$40\!\cdots\!20$$$$T^{13} +$$$$25\!\cdots\!40$$$$T^{14} -$$$$15\!\cdots\!20$$$$T^{15} -$$$$27\!\cdots\!79$$$$T^{16} +$$$$11\!\cdots\!78$$$$T^{17} +$$$$10\!\cdots\!22$$$$T^{18} -$$$$23\!\cdots\!34$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20}$$)
$17$ ($$1 + 66 T + 4913 T^{2}$$)($$1 + 66 T + 4913 T^{2}$$)($$1 + 66 T + 4913 T^{2}$$)($$1 + 66 T + 4913 T^{2}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$( 1 + 98 T + 4913 T^{2} )^{2}$$)($$( 1 + 90 T + 4913 T^{2} )^{2}$$)($$( 1 - 94 T + 4913 T^{2} )^{2}$$)($$( 1 + 98 T + 4913 T^{2} )^{2}$$)($$( 1 - 70 T + 4913 T^{2} )^{4}$$)($$( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2}$$)($$( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2}$$)
$19$ ($$1 + 162 T + 6859 T^{2}$$)($$1 + 162 T + 6859 T^{2}$$)($$1 - 162 T + 6859 T^{2}$$)($$1 - 162 T + 6859 T^{2}$$)($$1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4}$$)($$1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4}$$)($$1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4}$$)($$1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4}$$)($$1 - 5974 T^{2} + 47045881 T^{4}$$)($$( 1 - 106 T + 6859 T^{2} )( 1 + 106 T + 6859 T^{2} )$$)($$( 1 - 6859 T^{2} )^{2}$$)($$1 - 5974 T^{2} + 47045881 T^{4}$$)($$( 1 - 6958 T^{2} + 47045881 T^{4} )^{2}$$)($$1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} -$$$$17\!\cdots\!76$$$$T^{11} +$$$$47\!\cdots\!74$$$$T^{12} +$$$$11\!\cdots\!40$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} +$$$$37\!\cdots\!16$$$$T^{15} -$$$$66\!\cdots\!11$$$$T^{16} -$$$$24\!\cdots\!14$$$$T^{17} +$$$$16\!\cdots\!98$$$$T^{18} +$$$$87\!\cdots\!14$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20}$$)($$1 - 26 T + 338 T^{2} + 339906 T^{3} - 64153371 T^{4} - 2461461784 T^{5} + 143449890200 T^{6} - 36783398837960 T^{7} + 1011857007777554 T^{8} + 259766590630759364 T^{9} - 7366645907488092948 T^{10} +$$$$17\!\cdots\!76$$$$T^{11} +$$$$47\!\cdots\!74$$$$T^{12} -$$$$11\!\cdots\!40$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} -$$$$37\!\cdots\!16$$$$T^{15} -$$$$66\!\cdots\!11$$$$T^{16} +$$$$24\!\cdots\!14$$$$T^{17} +$$$$16\!\cdots\!98$$$$T^{18} -$$$$87\!\cdots\!14$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20}$$)
$23$ ($$1 - 172 T + 12167 T^{2}$$)($$1 + 172 T + 12167 T^{2}$$)($$1 + 172 T + 12167 T^{2}$$)($$1 - 172 T + 12167 T^{2}$$)($$1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4}$$)($$1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4}$$)($$1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4}$$)($$1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4}$$)($$( 1 - 32 T + 12167 T^{2} )^{2}$$)($$( 1 + 12167 T^{2} )^{2}$$)($$( 1 + 12167 T^{2} )^{2}$$)($$( 1 + 32 T + 12167 T^{2} )^{2}$$)($$( 1 - 754 T^{2} + 148035889 T^{4} )^{2}$$)($$1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} -$$$$18\!\cdots\!84$$$$T^{10} +$$$$19\!\cdots\!94$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{14} +$$$$94\!\cdots\!93$$$$T^{16} -$$$$36\!\cdots\!26$$$$T^{18} +$$$$71\!\cdots\!49$$$$T^{20}$$)($$1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} -$$$$18\!\cdots\!84$$$$T^{10} +$$$$19\!\cdots\!94$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{14} +$$$$94\!\cdots\!93$$$$T^{16} -$$$$36\!\cdots\!26$$$$T^{18} +$$$$71\!\cdots\!49$$$$T^{20}$$)
$29$ ($$1 - 2 T + 24389 T^{2}$$)($$1 + 2 T + 24389 T^{2}$$)($$1 - 2 T + 24389 T^{2}$$)($$1 + 2 T + 24389 T^{2}$$)($$1 + 84 T + 45742 T^{2} + 2048676 T^{3} + 594823321 T^{4}$$)($$1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4}$$)($$1 + 84 T + 45742 T^{2} + 2048676 T^{3} + 594823321 T^{4}$$)($$1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4}$$)($$1 - 19194 T^{2} + 594823321 T^{4}$$)($$( 1 - 24389 T^{2} )^{2}$$)($$( 1 - 130 T + 24389 T^{2} )( 1 + 130 T + 24389 T^{2} )$$)($$1 - 19194 T^{2} + 594823321 T^{4}$$)($$( 1 - 33098 T^{2} + 594823321 T^{4} )^{2}$$)($$1 - 202 T + 20402 T^{2} + 1177934 T^{3} + 398569397 T^{4} - 239164019416 T^{5} + 40873283338616 T^{6} - 2529271278095288 T^{7} + 194871598558001506 T^{8} -$$$$12\!\cdots\!32$$$$T^{9} +$$$$30\!\cdots\!36$$$$T^{10} -$$$$30\!\cdots\!48$$$$T^{11} +$$$$11\!\cdots\!26$$$$T^{12} -$$$$36\!\cdots\!72$$$$T^{13} +$$$$14\!\cdots\!56$$$$T^{14} -$$$$20\!\cdots\!84$$$$T^{15} +$$$$83\!\cdots\!17$$$$T^{16} +$$$$60\!\cdots\!86$$$$T^{17} +$$$$25\!\cdots\!62$$$$T^{18} -$$$$61\!\cdots\!18$$$$T^{19} +$$$$74\!\cdots\!01$$$$T^{20}$$)($$1 - 202 T + 20402 T^{2} + 1177934 T^{3} + 398569397 T^{4} - 239164019416 T^{5} + 40873283338616 T^{6} - 2529271278095288 T^{7} + 194871598558001506 T^{8} -$$$$12\!\cdots\!32$$$$T^{9} +$$$$30\!\cdots\!36$$$$T^{10} -$$$$30\!\cdots\!48$$$$T^{11} +$$$$11\!\cdots\!26$$$$T^{12} -$$$$36\!\cdots\!72$$$$T^{13} +$$$$14\!\cdots\!56$$$$T^{14} -$$$$20\!\cdots\!84$$$$T^{15} +$$$$83\!\cdots\!17$$$$T^{16} +$$$$60\!\cdots\!86$$$$T^{17} +$$$$25\!\cdots\!62$$$$T^{18} -$$$$61\!\cdots\!18$$$$T^{19} +$$$$74\!\cdots\!01$$$$T^{20}$$)
$31$ ($$1 + 128 T + 29791 T^{2}$$)($$1 - 128 T + 29791 T^{2}$$)($$1 - 128 T + 29791 T^{2}$$)($$1 + 128 T + 29791 T^{2}$$)($$1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4}$$)($$1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4}$$)($$1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4}$$)($$1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4}$$)($$( 1 + 256 T + 29791 T^{2} )^{2}$$)($$( 1 + 29791 T^{2} )^{2}$$)($$( 1 + 29791 T^{2} )^{2}$$)($$( 1 - 256 T + 29791 T^{2} )^{2}$$)($$( 1 + 29791 T^{2} )^{4}$$)($$( 1 + 184 T + 134043 T^{2} + 19809056 T^{3} + 7638677322 T^{4} + 852982867024 T^{5} + 227563836099702 T^{6} + 17580610117135136 T^{7} + 3544046273282822853 T^{8} +$$$$14\!\cdots\!24$$$$T^{9} +$$$$23\!\cdots\!51$$$$T^{10} )^{2}$$)($$( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} -$$$$14\!\cdots\!24$$$$T^{9} +$$$$23\!\cdots\!51$$$$T^{10} )^{2}$$)
$37$ ($$1 + 158 T + 50653 T^{2}$$)($$1 - 158 T + 50653 T^{2}$$)($$1 + 158 T + 50653 T^{2}$$)($$1 - 158 T + 50653 T^{2}$$)($$1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4}$$)($$1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4}$$)($$1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4}$$)($$1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4}$$)($$1 - 92842 T^{2} + 2565726409 T^{4}$$)($$( 1 - 50653 T^{2} )^{2}$$)($$( 1 - 214 T + 50653 T^{2} )( 1 + 214 T + 50653 T^{2} )$$)($$1 - 92842 T^{2} + 2565726409 T^{4}$$)($$( 1 + 39814 T^{2} + 2565726409 T^{4} )^{2}$$)($$1 - 10 T + 50 T^{2} - 1972962 T^{3} + 1630465317 T^{4} + 153991562664 T^{5} + 324850634232 T^{6} + 5252842710654600 T^{7} + 3474549392106364962 T^{8} - 27985624577691139772 T^{9} +$$$$10\!\cdots\!24$$$$T^{10} -$$$$14\!\cdots\!16$$$$T^{11} +$$$$89\!\cdots\!58$$$$T^{12} +$$$$68\!\cdots\!00$$$$T^{13} +$$$$21\!\cdots\!92$$$$T^{14} +$$$$51\!\cdots\!52$$$$T^{15} +$$$$27\!\cdots\!93$$$$T^{16} -$$$$16\!\cdots\!94$$$$T^{17} +$$$$21\!\cdots\!50$$$$T^{18} -$$$$21\!\cdots\!30$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20}$$)($$1 - 10 T + 50 T^{2} - 1972962 T^{3} + 1630465317 T^{4} + 153991562664 T^{5} + 324850634232 T^{6} + 5252842710654600 T^{7} + 3474549392106364962 T^{8} - 27985624577691139772 T^{9} +$$$$10\!\cdots\!24$$$$T^{10} -$$$$14\!\cdots\!16$$$$T^{11} +$$$$89\!\cdots\!58$$$$T^{12} +$$$$68\!\cdots\!00$$$$T^{13} +$$$$21\!\cdots\!92$$$$T^{14} +$$$$51\!\cdots\!52$$$$T^{15} +$$$$27\!\cdots\!93$$$$T^{16} -$$$$16\!\cdots\!94$$$$T^{17} +$$$$21\!\cdots\!50$$$$T^{18} -$$$$21\!\cdots\!30$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20}$$)
$41$ ($$1 - 202 T + 68921 T^{2}$$)($$1 - 202 T + 68921 T^{2}$$)($$1 - 202 T + 68921 T^{2}$$)($$1 - 202 T + 68921 T^{2}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$( 1 + 102 T + 68921 T^{2} )^{2}$$)($$( 1 - 522 T + 68921 T^{2} )^{2}$$)($$( 1 + 230 T + 68921 T^{2} )^{2}$$)($$( 1 + 102 T + 68921 T^{2} )^{2}$$)($$( 1 + 182 T + 68921 T^{2} )^{4}$$)($$1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} +$$$$14\!\cdots\!14$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$68\!\cdots\!74$$$$T^{12} -$$$$31\!\cdots\!24$$$$T^{14} +$$$$10\!\cdots\!81$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20}$$)($$1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} +$$$$14\!\cdots\!14$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$68\!\cdots\!74$$$$T^{12} -$$$$31\!\cdots\!24$$$$T^{14} +$$$$10\!\cdots\!81$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20}$$)
$43$ ($$1 - 298 T + 79507 T^{2}$$)($$1 - 298 T + 79507 T^{2}$$)($$1 + 298 T + 79507 T^{2}$$)($$1 + 298 T + 79507 T^{2}$$)($$1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4}$$)($$1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4}$$)($$1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4}$$)($$1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4}$$)($$1 - 71398 T^{2} + 6321363049 T^{4}$$)($$( 1 - 290 T + 79507 T^{2} )( 1 + 290 T + 79507 T^{2} )$$)($$( 1 - 79507 T^{2} )^{2}$$)($$1 - 71398 T^{2} + 6321363049 T^{4}$$)($$( 1 - 141374 T^{2} + 6321363049 T^{4} )^{2}$$)($$1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} +$$$$59\!\cdots\!10$$$$T^{8} +$$$$17\!\cdots\!68$$$$T^{9} +$$$$49\!\cdots\!52$$$$T^{10} +$$$$13\!\cdots\!76$$$$T^{11} +$$$$37\!\cdots\!90$$$$T^{12} +$$$$98\!\cdots\!60$$$$T^{13} +$$$$25\!\cdots\!48$$$$T^{14} +$$$$66\!\cdots\!12$$$$T^{15} +$$$$14\!\cdots\!45$$$$T^{16} +$$$$26\!\cdots\!50$$$$T^{17} +$$$$56\!\cdots\!22$$$$T^{18} +$$$$10\!\cdots\!66$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20}$$)($$1 - 838 T + 351122 T^{2} - 132133650 T^{3} + 56398378005 T^{4} - 20936462157416 T^{5} + 6471694737204248 T^{6} - 1952595983380873720 T^{7} +$$$$59\!\cdots\!10$$$$T^{8} -$$$$17\!\cdots\!68$$$$T^{9} +$$$$49\!\cdots\!52$$$$T^{10} -$$$$13\!\cdots\!76$$$$T^{11} +$$$$37\!\cdots\!90$$$$T^{12} -$$$$98\!\cdots\!60$$$$T^{13} +$$$$25\!\cdots\!48$$$$T^{14} -$$$$66\!\cdots\!12$$$$T^{15} +$$$$14\!\cdots\!45$$$$T^{16} -$$$$26\!\cdots\!50$$$$T^{17} +$$$$56\!\cdots\!22$$$$T^{18} -$$$$10\!\cdots\!66$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20}$$)
$47$ ($$1 + 408 T + 103823 T^{2}$$)($$1 - 408 T + 103823 T^{2}$$)($$1 - 408 T + 103823 T^{2}$$)($$1 + 408 T + 103823 T^{2}$$)($$1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4}$$)($$1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4}$$)($$1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4}$$)($$1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4}$$)($$( 1 + 320 T + 103823 T^{2} )^{2}$$)($$( 1 + 103823 T^{2} )^{2}$$)($$( 1 + 103823 T^{2} )^{2}$$)($$( 1 - 320 T + 103823 T^{2} )^{2}$$)($$( 1 + 107294 T^{2} + 10779215329 T^{4} )^{2}$$)($$( 1 - 472 T + 462219 T^{2} - 171516064 T^{3} + 90105579914 T^{4} - 25593405310224 T^{5} + 9355031623411222 T^{6} - 1848808586238545056 T^{7} +$$$$51\!\cdots\!73$$$$T^{8} -$$$$54\!\cdots\!52$$$$T^{9} +$$$$12\!\cdots\!43$$$$T^{10} )^{2}$$)($$( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} +$$$$51\!\cdots\!73$$$$T^{8} +$$$$54\!\cdots\!52$$$$T^{9} +$$$$12\!\cdots\!43$$$$T^{10} )^{2}$$)
$53$ ($$1 - 690 T + 148877 T^{2}$$)($$1 + 690 T + 148877 T^{2}$$)($$1 - 690 T + 148877 T^{2}$$)($$1 + 690 T + 148877 T^{2}$$)($$1 - 12 T + 288382 T^{2} - 1786524 T^{3} + 22164361129 T^{4}$$)($$1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4}$$)($$1 - 12 T + 288382 T^{2} - 1786524 T^{3} + 22164361129 T^{4}$$)($$1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4}$$)($$1 - 291978 T^{2} + 22164361129 T^{4}$$)($$( 1 - 148877 T^{2} )^{2}$$)($$( 1 - 518 T + 148877 T^{2} )( 1 + 518 T + 148877 T^{2} )$$)($$1 - 291978 T^{2} + 22164361129 T^{4}$$)($$( 1 - 282074 T^{2} + 22164361129 T^{4} )^{2}$$)($$1 - 378 T + 71442 T^{2} + 52753550 T^{3} + 3016286341 T^{4} + 526686651752 T^{5} + 976891435665272 T^{6} + 1674349213754452168 T^{7} - 1581505854923305054 T^{8} +$$$$14\!\cdots\!20$$$$T^{9} +$$$$29\!\cdots\!08$$$$T^{10} +$$$$22\!\cdots\!40$$$$T^{11} -$$$$35\!\cdots\!66$$$$T^{12} +$$$$55\!\cdots\!44$$$$T^{13} +$$$$47\!\cdots\!52$$$$T^{14} +$$$$38\!\cdots\!64$$$$T^{15} +$$$$32\!\cdots\!49$$$$T^{16} +$$$$85\!\cdots\!50$$$$T^{17} +$$$$17\!\cdots\!02$$$$T^{18} -$$$$13\!\cdots\!86$$$$T^{19} +$$$$53\!\cdots\!49$$$$T^{20}$$)($$1 - 378 T + 71442 T^{2} + 52753550 T^{3} + 3016286341 T^{4} + 526686651752 T^{5} + 976891435665272 T^{6} + 1674349213754452168 T^{7} - 1581505854923305054 T^{8} +$$$$14\!\cdots\!20$$$$T^{9} +$$$$29\!\cdots\!08$$$$T^{10} +$$$$22\!\cdots\!40$$$$T^{11} -$$$$35\!\cdots\!66$$$$T^{12} +$$$$55\!\cdots\!44$$$$T^{13} +$$$$47\!\cdots\!52$$$$T^{14} +$$$$38\!\cdots\!64$$$$T^{15} +$$$$32\!\cdots\!49$$$$T^{16} +$$$$85\!\cdots\!50$$$$T^{17} +$$$$17\!\cdots\!02$$$$T^{18} -$$$$13\!\cdots\!86$$$$T^{19} +$$$$53\!\cdots\!49$$$$T^{20}$$)
$59$ ($$1 - 322 T + 205379 T^{2}$$)($$1 - 322 T + 205379 T^{2}$$)($$1 + 322 T + 205379 T^{2}$$)($$1 + 322 T + 205379 T^{2}$$)($$1 - 644 T + 462170 T^{2} - 132264076 T^{3} + 42180533641 T^{4}$$)($$1 - 644 T + 462170 T^{2} - 132264076 T^{3} + 42180533641 T^{4}$$)($$1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4}$$)($$1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4}$$)($$1 - 244294 T^{2} + 42180533641 T^{4}$$)($$( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} )$$)($$( 1 - 205379 T^{2} )^{2}$$)($$1 - 244294 T^{2} + 42180533641 T^{4}$$)($$( 1 - 403998 T^{2} + 42180533641 T^{4} )^{2}$$)($$1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} +$$$$57\!\cdots\!98$$$$T^{8} +$$$$22\!\cdots\!08$$$$T^{9} -$$$$12\!\cdots\!80$$$$T^{10} +$$$$45\!\cdots\!32$$$$T^{11} +$$$$24\!\cdots\!18$$$$T^{12} -$$$$17\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!96$$$$T^{14} -$$$$85\!\cdots\!68$$$$T^{15} +$$$$41\!\cdots\!01$$$$T^{16} -$$$$15\!\cdots\!22$$$$T^{17} +$$$$46\!\cdots\!98$$$$T^{18} -$$$$11\!\cdots\!14$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$)($$1 + 1706 T + 1455218 T^{2} + 989315358 T^{3} + 555128806581 T^{4} + 232658117164632 T^{5} + 78453755015006616 T^{6} + 20092577710244830152 T^{7} +$$$$57\!\cdots\!98$$$$T^{8} -$$$$22\!\cdots\!08$$$$T^{9} -$$$$12\!\cdots\!80$$$$T^{10} -$$$$45\!\cdots\!32$$$$T^{11} +$$$$24\!\cdots\!18$$$$T^{12} +$$$$17\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!96$$$$T^{14} +$$$$85\!\cdots\!68$$$$T^{15} +$$$$41\!\cdots\!01$$$$T^{16} +$$$$15\!\cdots\!22$$$$T^{17} +$$$$46\!\cdots\!98$$$$T^{18} +$$$$11\!\cdots\!14$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$)
$61$ ($$1 - 298 T + 226981 T^{2}$$)($$1 + 298 T + 226981 T^{2}$$)($$1 - 298 T + 226981 T^{2}$$)($$1 + 298 T + 226981 T^{2}$$)($$1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4}$$)($$1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4}$$)($$1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4}$$)($$1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4}$$)($$1 - 49466 T^{2} + 51520374361 T^{4}$$)($$( 1 - 226981 T^{2} )^{2}$$)($$( 1 - 830 T + 226981 T^{2} )( 1 + 830 T + 226981 T^{2} )$$)($$1 - 49466 T^{2} + 51520374361 T^{4}$$)($$( 1 - 399882 T^{2} + 51520374361 T^{4} )^{2}$$)($$1 + 910 T + 414050 T^{2} - 45940410 T^{3} + 20471098485 T^{4} + 72823214590920 T^{5} + 58848327585507000 T^{6} + 6001380717052735080 T^{7} +$$$$59\!\cdots\!50$$$$T^{8} +$$$$32\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!00$$$$T^{10} +$$$$73\!\cdots\!00$$$$T^{11} +$$$$30\!\cdots\!50$$$$T^{12} +$$$$70\!\cdots\!80$$$$T^{13} +$$$$15\!\cdots\!00$$$$T^{14} +$$$$43\!\cdots\!20$$$$T^{15} +$$$$27\!\cdots\!85$$$$T^{16} -$$$$14\!\cdots\!10$$$$T^{17} +$$$$29\!\cdots\!50$$$$T^{18} +$$$$14\!\cdots\!10$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$)($$1 + 910 T + 414050 T^{2} - 45940410 T^{3} + 20471098485 T^{4} + 72823214590920 T^{5} + 58848327585507000 T^{6} + 6001380717052735080 T^{7} +$$$$59\!\cdots\!50$$$$T^{8} +$$$$32\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!00$$$$T^{10} +$$$$73\!\cdots\!00$$$$T^{11} +$$$$30\!\cdots\!50$$$$T^{12} +$$$$70\!\cdots\!80$$$$T^{13} +$$$$15\!\cdots\!00$$$$T^{14} +$$$$43\!\cdots\!20$$$$T^{15} +$$$$27\!\cdots\!85$$$$T^{16} -$$$$14\!\cdots\!10$$$$T^{17} +$$$$29\!\cdots\!50$$$$T^{18} +$$$$14\!\cdots\!10$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$)
$67$ ($$1 + 202 T + 300763 T^{2}$$)($$1 + 202 T + 300763 T^{2}$$)($$1 - 202 T + 300763 T^{2}$$)($$1 - 202 T + 300763 T^{2}$$)($$1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4}$$)($$1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4}$$)($$1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4}$$)($$1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4}$$)($$1 - 296822 T^{2} + 90458382169 T^{4}$$)($$( 1 - 70 T + 300763 T^{2} )( 1 + 70 T + 300763 T^{2} )$$)($$( 1 - 300763 T^{2} )^{2}$$)($$1 - 296822 T^{2} + 90458382169 T^{4}$$)($$( 1 - 552526 T^{2} + 90458382169 T^{4} )^{2}$$)($$1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} -$$$$46\!\cdots\!12$$$$T^{7} +$$$$29\!\cdots\!06$$$$T^{8} -$$$$17\!\cdots\!64$$$$T^{9} +$$$$97\!\cdots\!68$$$$T^{10} -$$$$52\!\cdots\!32$$$$T^{11} +$$$$26\!\cdots\!14$$$$T^{12} -$$$$12\!\cdots\!64$$$$T^{13} +$$$$59\!\cdots\!16$$$$T^{14} -$$$$26\!\cdots\!24$$$$T^{15} +$$$$99\!\cdots\!49$$$$T^{16} -$$$$34\!\cdots\!66$$$$T^{17} +$$$$12\!\cdots\!22$$$$T^{18} -$$$$39\!\cdots\!66$$$$T^{19} +$$$$60\!\cdots\!49$$$$T^{20}$$)($$1 + 1942 T + 1885682 T^{2} + 1530965298 T^{3} + 1338066168261 T^{4} + 1068751594464168 T^{5} + 724275679990789656 T^{6} +$$$$46\!\cdots\!12$$$$T^{7} +$$$$29\!\cdots\!06$$$$T^{8} +$$$$17\!\cdots\!64$$$$T^{9} +$$$$97\!\cdots\!68$$$$T^{10} +$$$$52\!\cdots\!32$$$$T^{11} +$$$$26\!\cdots\!14$$$$T^{12} +$$$$12\!\cdots\!64$$$$T^{13} +$$$$59\!\cdots\!16$$$$T^{14} +$$$$26\!\cdots\!24$$$$T^{15} +$$$$99\!\cdots\!49$$$$T^{16} +$$$$34\!\cdots\!66$$$$T^{17} +$$$$12\!\cdots\!22$$$$T^{18} +$$$$39\!\cdots\!66$$$$T^{19} +$$$$60\!\cdots\!49$$$$T^{20}$$)
$71$ ($$1 + 700 T + 357911 T^{2}$$)($$1 - 700 T + 357911 T^{2}$$)($$1 - 700 T + 357911 T^{2}$$)($$1 + 700 T + 357911 T^{2}$$)($$1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4}$$)($$1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4}$$)($$1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4}$$)($$1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4}$$)($$( 1 + 416 T + 357911 T^{2} )^{2}$$)($$( 1 + 357911 T^{2} )^{2}$$)($$( 1 + 357911 T^{2} )^{2}$$)($$( 1 - 416 T + 357911 T^{2} )^{2}$$)($$( 1 + 703022 T^{2} + 128100283921 T^{4} )^{2}$$)($$1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} +$$$$13\!\cdots\!66$$$$T^{8} -$$$$55\!\cdots\!32$$$$T^{10} +$$$$17\!\cdots\!86$$$$T^{12} -$$$$39\!\cdots\!68$$$$T^{14} +$$$$64\!\cdots\!53$$$$T^{16} -$$$$67\!\cdots\!58$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20}$$)($$1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} +$$$$13\!\cdots\!66$$$$T^{8} -$$$$55\!\cdots\!32$$$$T^{10} +$$$$17\!\cdots\!86$$$$T^{12} -$$$$39\!\cdots\!68$$$$T^{14} +$$$$64\!\cdots\!53$$$$T^{16} -$$$$67\!\cdots\!58$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20}$$)
$73$ ($$1 + 418 T + 389017 T^{2}$$)($$1 + 418 T + 389017 T^{2}$$)($$1 + 418 T + 389017 T^{2}$$)($$1 + 418 T + 389017 T^{2}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$( 1 + 138 T + 389017 T^{2} )^{2}$$)($$( 1 - 430 T + 389017 T^{2} )^{2}$$)($$( 1 + 1098 T + 389017 T^{2} )^{2}$$)($$( 1 + 138 T + 389017 T^{2} )^{2}$$)($$( 1 - 910 T + 389017 T^{2} )^{4}$$)($$1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} +$$$$26\!\cdots\!74$$$$T^{8} -$$$$12\!\cdots\!92$$$$T^{10} +$$$$40\!\cdots\!86$$$$T^{12} -$$$$97\!\cdots\!12$$$$T^{14} +$$$$16\!\cdots\!49$$$$T^{16} -$$$$16\!\cdots\!62$$$$T^{18} +$$$$79\!\cdots\!49$$$$T^{20}$$)($$1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} +$$$$26\!\cdots\!74$$$$T^{8} -$$$$12\!\cdots\!92$$$$T^{10} +$$$$40\!\cdots\!86$$$$T^{12} -$$$$97\!\cdots\!12$$$$T^{14} +$$$$16\!\cdots\!49$$$$T^{16} -$$$$16\!\cdots\!62$$$$T^{18} +$$$$79\!\cdots\!49$$$$T^{20}$$)
$79$ ($$1 - 744 T + 493039 T^{2}$$)($$1 + 744 T + 493039 T^{2}$$)($$1 + 744 T + 493039 T^{2}$$)($$1 - 744 T + 493039 T^{2}$$)($$1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4}$$)($$1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4}$$)($$1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4}$$)($$1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4}$$)($$( 1 + 64 T + 493039 T^{2} )^{2}$$)($$( 1 + 493039 T^{2} )^{2}$$)($$( 1 + 493039 T^{2} )^{2}$$)($$( 1 - 64 T + 493039 T^{2} )^{2}$$)($$( 1 + 525278 T^{2} + 243087455521 T^{4} )^{2}$$)($$( 1 - 2208 T + 3816107 T^{2} - 4320867712 T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 2093287800654474006 T^{6} -$$$$10\!\cdots\!52$$$$T^{7} +$$$$45\!\cdots\!33$$$$T^{8} -$$$$13\!\cdots\!28$$$$T^{9} +$$$$29\!\cdots\!99$$$$T^{10} )^{2}$$)($$( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} +$$$$10\!\cdots\!52$$$$T^{7} +$$$$45\!\cdots\!33$$$$T^{8} +$$$$13\!\cdots\!28$$$$T^{9} +$$$$29\!\cdots\!99$$$$T^{10} )^{2}$$)
$83$ ($$1 - 678 T + 571787 T^{2}$$)($$1 - 678 T + 571787 T^{2}$$)($$1 + 678 T + 571787 T^{2}$$)($$1 + 678 T + 571787 T^{2}$$)($$1 + 948 T + 1360138 T^{2} + 542054076 T^{3} + 326940373369 T^{4}$$)($$1 + 948 T + 1360138 T^{2} + 542054076 T^{3} + 326940373369 T^{4}$$)($$1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4}$$)($$1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4}$$)($$1 - 989910 T^{2} + 326940373369 T^{4}$$)($$( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} )$$)($$( 1 - 571787 T^{2} )^{2}$$)($$1 - 989910 T^{2} + 326940373369 T^{4}$$)($$( 1 - 632814 T^{2} + 326940373369 T^{4} )^{2}$$)($$1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} +$$$$79\!\cdots\!28$$$$T^{7} +$$$$89\!\cdots\!74$$$$T^{8} +$$$$97\!\cdots\!88$$$$T^{9} +$$$$82\!\cdots\!76$$$$T^{10} +$$$$55\!\cdots\!56$$$$T^{11} +$$$$29\!\cdots\!06$$$$T^{12} +$$$$14\!\cdots\!84$$$$T^{13} +$$$$94\!\cdots\!96$$$$T^{14} +$$$$72\!\cdots\!32$$$$T^{15} +$$$$67\!\cdots\!61$$$$T^{16} +$$$$57\!\cdots\!94$$$$T^{17} +$$$$37\!\cdots\!62$$$$T^{18} +$$$$16\!\cdots\!74$$$$T^{19} +$$$$37\!\cdots\!49$$$$T^{20}$$)($$1 - 2562 T + 3281922 T^{2} - 2891460918 T^{3} + 1934799974629 T^{4} - 1191348439341176 T^{5} + 882645219418437336 T^{6} -$$$$79\!\cdots\!28$$$$T^{7} +$$$$89\!\cdots\!74$$$$T^{8} -$$$$97\!\cdots\!88$$$$T^{9} +$$$$82\!\cdots\!76$$$$T^{10} -$$$$55\!\cdots\!56$$$$T^{11} +$$$$29\!\cdots\!06$$$$T^{12} -$$$$14\!\cdots\!84$$$$T^{13} +$$$$94\!\cdots\!96$$$$T^{14} -$$$$72\!\cdots\!32$$$$T^{15} +$$$$67\!\cdots\!61$$$$T^{16} -$$$$57\!\cdots\!94$$$$T^{17} +$$$$37\!\cdots\!62$$$$T^{18} -$$$$16\!\cdots\!74$$$$T^{19} +$$$$37\!\cdots\!49$$$$T^{20}$$)
$89$ ($$1 + 82 T + 704969 T^{2}$$)($$1 + 82 T + 704969 T^{2}$$)($$1 + 82 T + 704969 T^{2}$$)($$1 + 82 T + 704969 T^{2}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$( 1 - 582 T + 704969 T^{2} )^{2}$$)($$( 1 + 1026 T + 704969 T^{2} )^{2}$$)($$( 1 - 1670 T + 704969 T^{2} )^{2}$$)($$( 1 - 582 T + 704969 T^{2} )^{2}$$)($$( 1 + 546 T + 704969 T^{2} )^{4}$$)($$1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} +$$$$69\!\cdots\!98$$$$T^{8} -$$$$52\!\cdots\!28$$$$T^{10} +$$$$34\!\cdots\!78$$$$T^{12} -$$$$19\!\cdots\!88$$$$T^{14} +$$$$79\!\cdots\!73$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20}$$)($$1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} +$$$$69\!\cdots\!98$$$$T^{8} -$$$$52\!\cdots\!28$$$$T^{10} +$$$$34\!\cdots\!78$$$$T^{12} -$$$$19\!\cdots\!88$$$$T^{14} +$$$$79\!\cdots\!73$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20}$$)
$97$ ($$1 + 1122 T + 912673 T^{2}$$)($$1 + 1122 T + 912673 T^{2}$$)($$1 + 1122 T + 912673 T^{2}$$)($$1 + 1122 T + 912673 T^{2}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$( 1 - 238 T + 912673 T^{2} )^{2}$$)($$( 1 - 1910 T + 912673 T^{2} )^{2}$$)($$( 1 + 594 T + 912673 T^{2} )^{2}$$)($$( 1 - 238 T + 912673 T^{2} )^{2}$$)($$( 1 + 490 T + 912673 T^{2} )^{4}$$)($$( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} -$$$$60\!\cdots\!36$$$$T^{7} +$$$$19\!\cdots\!13$$$$T^{8} +$$$$13\!\cdots\!82$$$$T^{9} +$$$$63\!\cdots\!93$$$$T^{10} )^{2}$$)($$( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} -$$$$60\!\cdots\!36$$$$T^{7} +$$$$19\!\cdots\!13$$$$T^{8} +$$$$13\!\cdots\!82$$$$T^{9} +$$$$63\!\cdots\!93$$$$T^{10} )^{2}$$)