Properties

 Label 48.3 Level 48 Weight 3 Dimension 49 Nonzero newspaces 4 Newform subspaces 6 Sturm bound 384 Trace bound 1

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Defining parameters

 Level: $$N$$ = $$48\( 48 = 2^{4} \cdot 3$$ \) Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$384$$ Trace bound: $$1$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(48))$$.

Total New Old
Modular forms 156 59 97
Cusp forms 100 49 51
Eisenstein series 56 10 46

Trace form

 $$49q - q^{3} + 8q^{4} + 12q^{5} + 4q^{6} + 10q^{7} - 12q^{8} - 11q^{9} + O(q^{10})$$ $$49q - q^{3} + 8q^{4} + 12q^{5} + 4q^{6} + 10q^{7} - 12q^{8} - 11q^{9} - 80q^{10} + 32q^{11} - 56q^{12} - 34q^{13} - 44q^{14} - 36q^{15} + 16q^{16} - 12q^{17} - 48q^{18} - 98q^{19} + 80q^{20} - 26q^{21} + 72q^{22} - 128q^{23} - 20q^{24} + 33q^{25} - 100q^{26} + 23q^{27} + 92q^{29} + 156q^{30} + 82q^{31} + 160q^{32} + 100q^{33} + 232q^{34} + 96q^{35} + 208q^{36} - 34q^{37} + 168q^{38} - 86q^{39} + 136q^{40} - 108q^{41} + 220q^{42} - 50q^{43} + 88q^{44} - 24q^{45} - 64q^{46} - 32q^{48} + 7q^{49} - 236q^{50} + 128q^{51} - 368q^{52} - 196q^{53} - 164q^{54} - 192q^{55} - 224q^{56} - 50q^{57} - 424q^{58} - 128q^{59} - 648q^{60} - 306q^{61} - 276q^{62} + 90q^{63} - 640q^{64} - 200q^{65} - 500q^{66} + 318q^{67} - 448q^{68} + 12q^{69} - 72q^{70} + 512q^{71} - 252q^{72} + 282q^{73} + 348q^{74} + 459q^{75} + 848q^{76} + 512q^{77} + 632q^{78} + 370q^{79} + 552q^{80} - 15q^{81} + 696q^{82} - 160q^{83} + 952q^{84} + 280q^{85} + 528q^{86} - 96q^{87} + 832q^{88} + 228q^{89} + 984q^{90} - 28q^{91} + 496q^{92} - 46q^{93} - 24q^{94} + 8q^{96} - 126q^{97} - 440q^{98} - 260q^{99} + O(q^{100})$$

Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$

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
48.3.b $$\chi_{48}(7, \cdot)$$ None 0 1
48.3.e $$\chi_{48}(17, \cdot)$$ 48.3.e.a 1 1
48.3.e.b 2
48.3.g $$\chi_{48}(31, \cdot)$$ 48.3.g.a 2 1
48.3.h $$\chi_{48}(41, \cdot)$$ None 0 1
48.3.i $$\chi_{48}(5, \cdot)$$ 48.3.i.a 8 2
48.3.i.b 20
48.3.l $$\chi_{48}(19, \cdot)$$ 48.3.l.a 16 2

Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(48))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(48)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 4 T^{2} + 8 T^{4} + 64 T^{6} + 256 T^{8}$$)($$1 - 2 T^{2} + 6 T^{4} - 24 T^{6} - 24 T^{8} + 1216 T^{10} - 384 T^{12} - 6144 T^{14} + 24576 T^{16} - 131072 T^{18} + 1048576 T^{20}$$)($$1 - 6 T^{2} + 4 T^{3} + 10 T^{4} - 56 T^{5} + 88 T^{6} + 128 T^{7} - 496 T^{8} + 512 T^{9} + 1408 T^{10} - 3584 T^{11} + 2560 T^{12} + 4096 T^{13} - 24576 T^{14} + 65536 T^{16}$$)
$3$ ($$1 - 3 T$$)($$1 + 2 T + 9 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 4 T + 8 T^{2} + 12 T^{3} - 126 T^{4} + 108 T^{5} + 648 T^{6} - 2916 T^{7} + 6561 T^{8}$$)($$1 + 6 T + 18 T^{2} + 30 T^{3} + 65 T^{4} + 168 T^{5} + 288 T^{6} + 1560 T^{7} + 11278 T^{8} + 64332 T^{9} + 225180 T^{10} + 578988 T^{11} + 913518 T^{12} + 1137240 T^{13} + 1889568 T^{14} + 9920232 T^{15} + 34543665 T^{16} + 143489070 T^{17} + 774840978 T^{18} + 2324522934 T^{19} + 3486784401 T^{20}$$)($$( 1 + 9 T^{4} )^{4}$$)
$5$ ($$( 1 - 5 T )( 1 + 5 T )$$)($$1 - 18 T^{2} + 625 T^{4}$$)($$( 1 - 6 T + 25 T^{2} )^{2}$$)($$1 + 372 T^{4} + 464614 T^{8} + 145312500 T^{12} + 152587890625 T^{16}$$)($$1 - 946 T^{4} + 556509 T^{8} + 27762120 T^{12} - 359980854270 T^{16} + 279323158314196 T^{20} - 140617521199218750 T^{24} + 4236163330078125000 T^{28} +$$$$33\!\cdots\!25$$$$T^{32} -$$$$22\!\cdots\!50$$$$T^{36} +$$$$90\!\cdots\!25$$$$T^{40}$$)($$1 + 32 T^{3} - 344 T^{4} + 5664 T^{5} + 512 T^{6} + 145600 T^{7} + 223452 T^{8} - 2255168 T^{9} + 20875776 T^{10} - 67282720 T^{11} + 753060504 T^{12} + 1881828576 T^{13} + 2740220928 T^{14} + 75471830656 T^{15} - 399298967994 T^{16} + 1886795766400 T^{17} + 1712638080000 T^{18} + 29403571500000 T^{19} + 294164259375000 T^{20} - 657057812500000 T^{21} + 5096625000000000 T^{22} - 13764453125000000 T^{23} + 34096069335937500 T^{24} + 555419921875000000 T^{25} + 48828125000000000 T^{26} + 13504028320312500000 T^{27} - 20503997802734375000 T^{28} + 47683715820312500000 T^{29} +$$$$23\!\cdots\!25$$$$T^{32}$$)
$7$ ($$1 + 2 T + 49 T^{2}$$)($$( 1 - 6 T + 49 T^{2} )^{2}$$)($$1 - 50 T^{2} + 2401 T^{4}$$)($$( 1 - 180 T^{2} + 12874 T^{4} - 432180 T^{6} + 5764801 T^{8} )^{2}$$)($$( 1 - 154 T^{2} + 17009 T^{4} - 1179536 T^{6} + 71996590 T^{8} - 3527749420 T^{10} + 172863812590 T^{12} - 6799790312336 T^{14} + 235426454001809 T^{16} - 5117871307718554 T^{18} + 79792266297612001 T^{20} )^{2}$$)($$( 1 + 168 T^{2} - 448 T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 178833536 T^{9} + 2570011192 T^{10} - 6648580288 T^{11} + 86910139876 T^{12} - 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2}$$)
$11$ ($$( 1 - 11 T )( 1 + 11 T )$$)($$1 - 210 T^{2} + 14641 T^{4}$$)($$1 + 190 T^{2} + 14641 T^{4}$$)($$1 + 37380 T^{4} + 692870854 T^{8} + 8012734971780 T^{12} + 45949729863572161 T^{16}$$)($$1 - 17026 T^{4} - 443639331 T^{8} + 11991832514824 T^{12} + 51739802427787650 T^{16} -$$$$36\!\cdots\!16$$$$T^{20} +$$$$11\!\cdots\!50$$$$T^{24} +$$$$55\!\cdots\!64$$$$T^{28} -$$$$43\!\cdots\!71$$$$T^{32} -$$$$35\!\cdots\!46$$$$T^{36} +$$$$45\!\cdots\!01$$$$T^{40}$$)($$1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 431768458252384 T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 7491143217161407584 T^{17} + 77497246731528160768 T^{18} -$$$$76\!\cdots\!24$$$$T^{19} +$$$$72\!\cdots\!52$$$$T^{20} -$$$$75\!\cdots\!28$$$$T^{21} +$$$$79\!\cdots\!72$$$$T^{22} -$$$$83\!\cdots\!00$$$$T^{23} +$$$$10\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!16$$$$T^{25} +$$$$12\!\cdots\!48$$$$T^{26} -$$$$13\!\cdots\!96$$$$T^{27} +$$$$13\!\cdots\!12$$$$T^{28} -$$$$10\!\cdots\!80$$$$T^{29} +$$$$73\!\cdots\!72$$$$T^{30} -$$$$55\!\cdots\!32$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$)
$13$ ($$1 + 22 T + 169 T^{2}$$)($$( 1 - 10 T + 169 T^{2} )^{2}$$)($$( 1 + 14 T + 169 T^{2} )^{2}$$)($$( 1 + 48 T + 1152 T^{2} + 21264 T^{3} + 317422 T^{4} + 3593616 T^{5} + 32902272 T^{6} + 231686832 T^{7} + 815730721 T^{8} )^{2}$$)($$( 1 - 46 T + 1058 T^{2} - 20510 T^{3} + 411997 T^{4} - 7447784 T^{5} + 117035288 T^{6} - 1803425192 T^{7} + 27622285106 T^{8} - 386182026516 T^{9} + 5044801970700 T^{10} - 65264762481204 T^{11} + 788920084912466 T^{12} - 8704788947572328 T^{13} + 95469279862682648 T^{14} - 1026740269857112616 T^{15} + 9598741176206804557 T^{16} - 80755589670692417390 T^{17} +$$$$70\!\cdots\!78$$$$T^{18} -$$$$51\!\cdots\!34$$$$T^{19} +$$$$19\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 + 3200 T^{3} + 7608 T^{4} + 95360 T^{5} + 5120000 T^{6} + 68335872 T^{7} + 2004669468 T^{8} + 7270355200 T^{9} + 184268595200 T^{10} + 4889456013184 T^{11} + 5354592144136 T^{12} + 669839496880000 T^{13} + 7008632866619392 T^{14} + 70586941744778752 T^{15} + 2398056097119178950 T^{16} + 11929193154867609088 T^{17} +$$$$20\!\cdots\!12$$$$T^{18} +$$$$32\!\cdots\!00$$$$T^{19} +$$$$43\!\cdots\!56$$$$T^{20} +$$$$67\!\cdots\!16$$$$T^{21} +$$$$42\!\cdots\!00$$$$T^{22} +$$$$28\!\cdots\!00$$$$T^{23} +$$$$13\!\cdots\!88$$$$T^{24} +$$$$76\!\cdots\!88$$$$T^{25} +$$$$97\!\cdots\!00$$$$T^{26} +$$$$30\!\cdots\!40$$$$T^{27} +$$$$41\!\cdots\!88$$$$T^{28} +$$$$29\!\cdots\!00$$$$T^{29} +$$$$44\!\cdots\!81$$$$T^{32}$$)
$17$ ($$( 1 - 17 T )( 1 + 17 T )$$)($$1 - 66 T^{2} + 83521 T^{4}$$)($$( 1 + 6 T + 289 T^{2} )^{2}$$)($$( 1 - 236 T^{2} + 73334 T^{4} - 19710956 T^{6} + 6975757441 T^{8} )^{2}$$)($$( 1 - 1938 T^{2} + 1855181 T^{4} - 1152381976 T^{6} + 513055082610 T^{8} - 170587207926956 T^{10} + 42850873554669810 T^{12} - 8038737143956283416 T^{14} +$$$$10\!\cdots\!41$$$$T^{16} -$$$$94\!\cdots\!78$$$$T^{18} +$$$$40\!\cdots\!01$$$$T^{20} )^{2}$$)($$( 1 + 968 T^{2} - 2944 T^{3} + 516540 T^{4} - 3209600 T^{5} + 201700088 T^{6} - 1543904000 T^{7} + 63894476806 T^{8} - 446188256000 T^{9} + 16846193049848 T^{10} - 77471941462400 T^{11} + 3603257748574140 T^{12} - 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2}$$)
$19$ ($$1 + 26 T + 361 T^{2}$$)($$( 1 + 2 T + 361 T^{2} )^{2}$$)($$1 - 674 T^{2} + 130321 T^{4}$$)($$( 1 - 8 T + 32 T^{2} - 1160 T^{3} - 4606 T^{4} - 418760 T^{5} + 4170272 T^{6} - 376367048 T^{7} + 16983563041 T^{8} )^{2}$$)($$( 1 + 26 T + 338 T^{2} + 13338 T^{3} + 349281 T^{4} + 2446864 T^{5} + 34512608 T^{6} + 926396720 T^{7} - 22197634642 T^{8} - 577965275108 T^{9} - 4229043710052 T^{10} - 208645464313988 T^{11} - 2892817944180082 T^{12} + 43583149847910320 T^{13} + 586147053677320928 T^{14} + 15001885307827986064 T^{15} +$$$$77\!\cdots\!41$$$$T^{16} +$$$$10\!\cdots\!98$$$$T^{17} +$$$$97\!\cdots\!78$$$$T^{18} +$$$$27\!\cdots\!66$$$$T^{19} +$$$$37\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} -$$$$19\!\cdots\!42$$$$T^{16} -$$$$11\!\cdots\!28$$$$T^{17} +$$$$41\!\cdots\!00$$$$T^{18} +$$$$11\!\cdots\!40$$$$T^{19} +$$$$16\!\cdots\!36$$$$T^{20} -$$$$42\!\cdots\!12$$$$T^{21} -$$$$67\!\cdots\!08$$$$T^{22} -$$$$56\!\cdots\!96$$$$T^{23} +$$$$31\!\cdots\!64$$$$T^{24} +$$$$46\!\cdots\!64$$$$T^{25} +$$$$73\!\cdots\!88$$$$T^{26} -$$$$11\!\cdots\!88$$$$T^{27} -$$$$25\!\cdots\!08$$$$T^{28} -$$$$46\!\cdots\!36$$$$T^{29} +$$$$32\!\cdots\!92$$$$T^{30} +$$$$73\!\cdots\!32$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$)
$23$ ($$( 1 - 23 T )( 1 + 23 T )$$)($$1 - 930 T^{2} + 279841 T^{4}$$)($$( 1 - 23 T )^{2}( 1 + 23 T )^{2}$$)($$( 1 + 996 T^{2} + 719878 T^{4} + 278721636 T^{6} + 78310985281 T^{8} )^{2}$$)($$( 1 + 3054 T^{2} + 4762717 T^{4} + 4930394824 T^{6} + 3773391227074 T^{8} + 2245272418513300 T^{10} + 1055949574375615234 T^{12} +$$$$38\!\cdots\!44$$$$T^{14} +$$$$10\!\cdots\!57$$$$T^{16} +$$$$18\!\cdots\!94$$$$T^{18} +$$$$17\!\cdots\!01$$$$T^{20} )^{2}$$)($$( 1 + 64 T + 3496 T^{2} + 127936 T^{3} + 4410332 T^{4} + 130001728 T^{5} + 3673719192 T^{6} + 94049622208 T^{7} + 2261818535238 T^{8} + 49752250148032 T^{9} + 1028057252408472 T^{10} + 19244921376016192 T^{11} + 345377444336323292 T^{12} + 5299942138629398464 T^{13} + 76613527014343042216 T^{14} +$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$)
$29$ ($$( 1 - 29 T )( 1 + 29 T )$$)($$1 - 1394 T^{2} + 707281 T^{4}$$)($$( 1 - 30 T + 841 T^{2} )^{2}$$)($$1 + 147060 T^{4} + 623812106854 T^{8} + 73566237490044660 T^{12} +$$$$25\!\cdots\!21$$$$T^{16}$$)($$1 + 865038 T^{4} - 341999726179 T^{8} - 621091035708977976 T^{12} -$$$$34\!\cdots\!02$$$$T^{16} +$$$$22\!\cdots\!76$$$$T^{20} -$$$$17\!\cdots\!22$$$$T^{24} -$$$$15\!\cdots\!96$$$$T^{28} -$$$$42\!\cdots\!99$$$$T^{32} +$$$$54\!\cdots\!58$$$$T^{36} +$$$$31\!\cdots\!01$$$$T^{40}$$)($$1 - 32 T + 512 T^{2} + 18368 T^{3} - 1552984 T^{4} + 20596992 T^{5} + 304715776 T^{6} - 25469097376 T^{7} + 491466517980 T^{8} + 9791032230816 T^{9} - 347423504794624 T^{10} + 2649303176415616 T^{11} + 694517140133881240 T^{12} - 20658732330776531008 T^{13} +$$$$19\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!40$$$$T^{15} -$$$$82\!\cdots\!10$$$$T^{16} +$$$$96\!\cdots\!40$$$$T^{17} +$$$$13\!\cdots\!68$$$$T^{18} -$$$$12\!\cdots\!68$$$$T^{19} +$$$$34\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!16$$$$T^{21} -$$$$12\!\cdots\!84$$$$T^{22} +$$$$29\!\cdots\!96$$$$T^{23} +$$$$12\!\cdots\!80$$$$T^{24} -$$$$53\!\cdots\!36$$$$T^{25} +$$$$53\!\cdots\!76$$$$T^{26} +$$$$30\!\cdots\!72$$$$T^{27} -$$$$19\!\cdots\!04$$$$T^{28} +$$$$19\!\cdots\!28$$$$T^{29} +$$$$45\!\cdots\!32$$$$T^{30} -$$$$23\!\cdots\!32$$$$T^{31} +$$$$62\!\cdots\!41$$$$T^{32}$$)
$31$ ($$1 - 46 T + 961 T^{2}$$)($$( 1 - 22 T + 961 T^{2} )^{2}$$)($$1 - 1490 T^{2} + 923521 T^{4}$$)($$( 1 - 18 T + 1996 T^{2} - 17298 T^{3} + 923521 T^{4} )^{4}$$)($$( 1 + 20 T + 2055 T^{2} + 69364 T^{3} + 2243976 T^{4} + 102850448 T^{5} + 2156460936 T^{6} + 64059110644 T^{7} + 1823820064455 T^{8} + 17057820748820 T^{9} + 819628286980801 T^{10} )^{4}$$)($$1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} +$$$$42\!\cdots\!64$$$$T^{12} -$$$$51\!\cdots\!00$$$$T^{14} +$$$$53\!\cdots\!38$$$$T^{16} -$$$$47\!\cdots\!00$$$$T^{18} +$$$$36\!\cdots\!24$$$$T^{20} -$$$$23\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!12$$$$T^{24} -$$$$53\!\cdots\!76$$$$T^{26} +$$$$18\!\cdots\!24$$$$T^{28} -$$$$41\!\cdots\!92$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$)
$37$ ($$1 - 26 T + 1369 T^{2}$$)($$( 1 + 6 T + 1369 T^{2} )^{2}$$)($$( 1 - 26 T + 1369 T^{2} )^{2}$$)($$( 1 - 56 T + 1568 T^{2} - 79016 T^{3} + 3980078 T^{4} - 108172904 T^{5} + 2938684448 T^{6} - 143680678904 T^{7} + 3512479453921 T^{8} )^{2}$$)($$( 1 + 58 T + 1682 T^{2} - 50742 T^{3} - 603459 T^{4} + 139637784 T^{5} + 10401384792 T^{6} + 114472119576 T^{7} + 338413283634 T^{8} + 111361866221948 T^{9} + 16107397431686060 T^{10} + 152454394857846812 T^{11} + 634240978068781074 T^{12} +$$$$29\!\cdots\!84$$$$T^{13} +$$$$36\!\cdots\!32$$$$T^{14} +$$$$67\!\cdots\!16$$$$T^{15} -$$$$39\!\cdots\!79$$$$T^{16} -$$$$45\!\cdots\!38$$$$T^{17} +$$$$20\!\cdots\!62$$$$T^{18} +$$$$97\!\cdots\!82$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 + 96 T + 4608 T^{2} + 145952 T^{3} + 4040888 T^{4} + 217733344 T^{5} + 12932982272 T^{6} + 602883756192 T^{7} + 21839639792924 T^{8} + 655265530977504 T^{9} + 21703692469355008 T^{10} + 815191556064282016 T^{11} + 35433653736114978312 T^{12} +$$$$14\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!52$$$$T^{14} +$$$$14\!\cdots\!84$$$$T^{15} +$$$$43\!\cdots\!90$$$$T^{16} +$$$$20\!\cdots\!96$$$$T^{17} +$$$$95\!\cdots\!72$$$$T^{18} +$$$$37\!\cdots\!60$$$$T^{19} +$$$$12\!\cdots\!52$$$$T^{20} +$$$$39\!\cdots\!84$$$$T^{21} +$$$$14\!\cdots\!48$$$$T^{22} +$$$$59\!\cdots\!56$$$$T^{23} +$$$$26\!\cdots\!84$$$$T^{24} +$$$$10\!\cdots\!68$$$$T^{25} +$$$$29\!\cdots\!72$$$$T^{26} +$$$$68\!\cdots\!36$$$$T^{27} +$$$$17\!\cdots\!68$$$$T^{28} +$$$$86\!\cdots\!68$$$$T^{29} +$$$$37\!\cdots\!68$$$$T^{30} +$$$$10\!\cdots\!04$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$)
$41$ ($$( 1 - 41 T )( 1 + 41 T )$$)($$1 - 2210 T^{2} + 2825761 T^{4}$$)($$( 1 + 54 T + 1681 T^{2} )^{2}$$)($$( 1 + 6060 T^{2} + 14724790 T^{4} + 17124111660 T^{6} + 7984925229121 T^{8} )^{2}$$)($$( 1 + 8166 T^{2} + 35165469 T^{4} + 106382596584 T^{6} + 246760101929730 T^{8} + 458809512185363300 T^{10} +$$$$69\!\cdots\!30$$$$T^{12} +$$$$84\!\cdots\!64$$$$T^{14} +$$$$79\!\cdots\!89$$$$T^{16} +$$$$52\!\cdots\!06$$$$T^{18} +$$$$18\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} +$$$$16\!\cdots\!68$$$$T^{12} -$$$$35\!\cdots\!72$$$$T^{14} +$$$$64\!\cdots\!06$$$$T^{16} -$$$$99\!\cdots\!92$$$$T^{18} +$$$$13\!\cdots\!28$$$$T^{20} -$$$$14\!\cdots\!48$$$$T^{22} +$$$$13\!\cdots\!44$$$$T^{24} -$$$$94\!\cdots\!80$$$$T^{26} +$$$$52\!\cdots\!44$$$$T^{28} -$$$$19\!\cdots\!40$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$)
$43$ ($$1 - 22 T + 1849 T^{2}$$)($$( 1 + 82 T + 1849 T^{2} )^{2}$$)($$1 - 3266 T^{2} + 3418801 T^{4}$$)($$( 1 + 120 T + 7200 T^{2} + 411000 T^{3} + 20977474 T^{4} + 759939000 T^{5} + 24615367200 T^{6} + 758563565880 T^{7} + 11688200277601 T^{8} )^{2}$$)($$( 1 - 86 T + 3698 T^{2} - 270774 T^{3} + 16192641 T^{4} - 356953072 T^{5} + 7476857312 T^{6} - 199821444560 T^{7} - 33578430535506 T^{8} + 2320167893729020 T^{9} - 78617269367427492 T^{10} + 4289990435504957980 T^{11} -$$$$11\!\cdots\!06$$$$T^{12} -$$$$12\!\cdots\!40$$$$T^{13} +$$$$87\!\cdots\!12$$$$T^{14} -$$$$77\!\cdots\!28$$$$T^{15} +$$$$64\!\cdots\!41$$$$T^{16} -$$$$20\!\cdots\!26$$$$T^{17} +$$$$50\!\cdots\!98$$$$T^{18} -$$$$21\!\cdots\!14$$$$T^{19} +$$$$46\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 9339737479444512 T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} -$$$$12\!\cdots\!76$$$$T^{12} +$$$$13\!\cdots\!84$$$$T^{13} -$$$$44\!\cdots\!20$$$$T^{14} +$$$$28\!\cdots\!76$$$$T^{15} -$$$$17\!\cdots\!30$$$$T^{16} +$$$$53\!\cdots\!24$$$$T^{17} -$$$$15\!\cdots\!20$$$$T^{18} +$$$$83\!\cdots\!16$$$$T^{19} -$$$$15\!\cdots\!76$$$$T^{20} -$$$$15\!\cdots\!20$$$$T^{21} +$$$$11\!\cdots\!20$$$$T^{22} -$$$$69\!\cdots\!88$$$$T^{23} +$$$$45\!\cdots\!00$$$$T^{24} -$$$$22\!\cdots\!88$$$$T^{25} +$$$$83\!\cdots\!88$$$$T^{26} -$$$$32\!\cdots\!36$$$$T^{27} +$$$$11\!\cdots\!32$$$$T^{28} -$$$$28\!\cdots\!36$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} -$$$$16\!\cdots\!40$$$$T^{31} +$$$$18\!\cdots\!01$$$$T^{32}$$)
$47$ ($$( 1 - 47 T )( 1 + 47 T )$$)($$1 + 190 T^{2} + 4879681 T^{4}$$)($$1 - 2690 T^{2} + 4879681 T^{4}$$)($$( 1 - 692 T^{2} + 7491686 T^{4} - 3376739252 T^{6} + 23811286661761 T^{8} )^{2}$$)($$( 1 - 17146 T^{2} + 138875501 T^{4} - 703714777016 T^{6} + 2481001995058130 T^{8} - 6375856842165200540 T^{10} +$$$$12\!\cdots\!30$$$$T^{12} -$$$$16\!\cdots\!76$$$$T^{14} +$$$$16\!\cdots\!41$$$$T^{16} -$$$$97\!\cdots\!66$$$$T^{18} +$$$$27\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} +$$$$15\!\cdots\!16$$$$T^{12} -$$$$44\!\cdots\!04$$$$T^{14} +$$$$10\!\cdots\!58$$$$T^{16} -$$$$21\!\cdots\!24$$$$T^{18} +$$$$37\!\cdots\!76$$$$T^{20} -$$$$54\!\cdots\!48$$$$T^{22} +$$$$64\!\cdots\!32$$$$T^{24} -$$$$58\!\cdots\!84$$$$T^{26} +$$$$37\!\cdots\!72$$$$T^{28} -$$$$15\!\cdots\!84$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$)
$53$ ($$( 1 - 53 T )( 1 + 53 T )$$)($$1 - 1746 T^{2} + 7890481 T^{4}$$)($$( 1 + 18 T + 2809 T^{2} )^{2}$$)($$1 + 1441524 T^{4} - 91064986479386 T^{8} + 89748837960546754164 T^{12} +$$$$38\!\cdots\!21$$$$T^{16}$$)($$1 + 1437518 T^{4} + 119553426803037 T^{8} -$$$$89\!\cdots\!68$$$$T^{12} +$$$$36\!\cdots\!22$$$$T^{16} -$$$$10\!\cdots\!60$$$$T^{20} +$$$$22\!\cdots\!42$$$$T^{24} -$$$$34\!\cdots\!28$$$$T^{28} +$$$$28\!\cdots\!97$$$$T^{32} +$$$$21\!\cdots\!38$$$$T^{36} +$$$$93\!\cdots\!01$$$$T^{40}$$)($$1 + 160 T + 12800 T^{2} + 602944 T^{3} + 3948712 T^{4} - 1481707264 T^{5} - 105845942272 T^{6} - 3791430241760 T^{7} + 34861972067036 T^{8} + 14471440004155872 T^{9} + 1098860393015073792 T^{10} + 54880211634179791488 T^{11} +$$$$13\!\cdots\!12$$$$T^{12} -$$$$17\!\cdots\!00$$$$T^{13} -$$$$18\!\cdots\!48$$$$T^{14} +$$$$16\!\cdots\!08$$$$T^{15} +$$$$51\!\cdots\!54$$$$T^{16} +$$$$46\!\cdots\!72$$$$T^{17} -$$$$14\!\cdots\!88$$$$T^{18} -$$$$38\!\cdots\!00$$$$T^{19} +$$$$84\!\cdots\!32$$$$T^{20} +$$$$95\!\cdots\!12$$$$T^{21} +$$$$53\!\cdots\!72$$$$T^{22} +$$$$19\!\cdots\!68$$$$T^{23} +$$$$13\!\cdots\!56$$$$T^{24} -$$$$41\!\cdots\!40$$$$T^{25} -$$$$32\!\cdots\!72$$$$T^{26} -$$$$12\!\cdots\!76$$$$T^{27} +$$$$95\!\cdots\!72$$$$T^{28} +$$$$40\!\cdots\!76$$$$T^{29} +$$$$24\!\cdots\!00$$$$T^{30} +$$$$85\!\cdots\!40$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$)
$59$ ($$( 1 - 59 T )( 1 + 59 T )$$)($$1 - 1554 T^{2} + 12117361 T^{4}$$)($$1 - 6530 T^{2} + 12117361 T^{4}$$)($$1 + 15787044 T^{4} + 346992256453126 T^{8} +$$$$23\!\cdots\!24$$$$T^{12} +$$$$21\!\cdots\!41$$$$T^{16}$$)($$1 - 19699682 T^{4} + 5882421086685 T^{8} +$$$$56\!\cdots\!08$$$$T^{12} +$$$$29\!\cdots\!74$$$$T^{16} -$$$$57\!\cdots\!12$$$$T^{20} +$$$$43\!\cdots\!54$$$$T^{24} +$$$$12\!\cdots\!28$$$$T^{28} +$$$$18\!\cdots\!85$$$$T^{32} -$$$$91\!\cdots\!42$$$$T^{36} +$$$$68\!\cdots\!01$$$$T^{40}$$)($$1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} +$$$$19\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!40$$$$T^{12} +$$$$69\!\cdots\!16$$$$T^{13} +$$$$51\!\cdots\!56$$$$T^{14} +$$$$29\!\cdots\!24$$$$T^{15} +$$$$15\!\cdots\!38$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{17} +$$$$62\!\cdots\!16$$$$T^{18} +$$$$29\!\cdots\!56$$$$T^{19} +$$$$16\!\cdots\!40$$$$T^{20} +$$$$98\!\cdots\!80$$$$T^{21} +$$$$45\!\cdots\!92$$$$T^{22} +$$$$19\!\cdots\!32$$$$T^{23} +$$$$11\!\cdots\!64$$$$T^{24} +$$$$53\!\cdots\!08$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} +$$$$84\!\cdots\!48$$$$T^{27} +$$$$43\!\cdots\!24$$$$T^{28} +$$$$12\!\cdots\!28$$$$T^{29} +$$$$31\!\cdots\!32$$$$T^{30} +$$$$17\!\cdots\!28$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$)
$61$ ($$1 - 74 T + 3721 T^{2}$$)($$( 1 + 86 T + 3721 T^{2} )^{2}$$)($$( 1 + 70 T + 3721 T^{2} )^{2}$$)($$( 1 - 104 T + 5408 T^{2} - 491192 T^{3} + 43609454 T^{4} - 1827725432 T^{5} + 74878308128 T^{6} - 5358118933544 T^{7} + 191707312997281 T^{8} )^{2}$$)($$( 1 + 122 T + 7442 T^{2} + 699306 T^{3} + 60909597 T^{4} + 2415315288 T^{5} + 85893685080 T^{6} + 3780099265368 T^{7} - 289655325423054 T^{8} - 34035914575403972 T^{9} - 1668196322377933396 T^{10} -$$$$12\!\cdots\!12$$$$T^{11} -$$$$40\!\cdots\!14$$$$T^{12} +$$$$19\!\cdots\!48$$$$T^{13} +$$$$16\!\cdots\!80$$$$T^{14} +$$$$17\!\cdots\!88$$$$T^{15} +$$$$16\!\cdots\!37$$$$T^{16} +$$$$69\!\cdots\!46$$$$T^{17} +$$$$27\!\cdots\!62$$$$T^{18} +$$$$16\!\cdots\!82$$$$T^{19} +$$$$50\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 + 32 T + 512 T^{2} - 38048 T^{3} - 60439624 T^{4} - 1520787552 T^{5} - 16996289024 T^{6} + 2981900088544 T^{7} + 2018049968078364 T^{8} + 40394929489472928 T^{9} + 275088896591278592 T^{10} -$$$$10\!\cdots\!56$$$$T^{11} -$$$$45\!\cdots\!20$$$$T^{12} -$$$$71\!\cdots\!16$$$$T^{13} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$23\!\cdots\!64$$$$T^{15} +$$$$72\!\cdots\!42$$$$T^{16} +$$$$88\!\cdots\!44$$$$T^{17} -$$$$26\!\cdots\!48$$$$T^{18} -$$$$36\!\cdots\!76$$$$T^{19} -$$$$86\!\cdots\!20$$$$T^{20} -$$$$75\!\cdots\!56$$$$T^{21} +$$$$73\!\cdots\!32$$$$T^{22} +$$$$39\!\cdots\!48$$$$T^{23} +$$$$74\!\cdots\!04$$$$T^{24} +$$$$40\!\cdots\!64$$$$T^{25} -$$$$86\!\cdots\!24$$$$T^{26} -$$$$28\!\cdots\!92$$$$T^{27} -$$$$42\!\cdots\!84$$$$T^{28} -$$$$99\!\cdots\!28$$$$T^{29} +$$$$49\!\cdots\!72$$$$T^{30} +$$$$11\!\cdots\!32$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$)
$67$ ($$1 + 122 T + 4489 T^{2}$$)($$( 1 + 2 T + 4489 T^{2} )^{2}$$)($$1 + 4894 T^{2} + 20151121 T^{4}$$)($$( 1 + 116 T + 6728 T^{2} + 350436 T^{3} + 16097858 T^{4} + 1573107204 T^{5} + 135576742088 T^{6} + 10493172331604 T^{7} + 406067677556641 T^{8} )^{2}$$)($$( 1 - 178 T + 15842 T^{2} - 1501482 T^{3} + 163953249 T^{4} - 13424405736 T^{5} + 919420948512 T^{6} - 73920183453528 T^{7} + 6211456955376366 T^{8} - 413849456030337652 T^{9} + 25618409953668575740 T^{10} -$$$$18\!\cdots\!28$$$$T^{11} +$$$$12\!\cdots\!86$$$$T^{12} -$$$$66\!\cdots\!32$$$$T^{13} +$$$$37\!\cdots\!92$$$$T^{14} -$$$$24\!\cdots\!64$$$$T^{15} +$$$$13\!\cdots\!89$$$$T^{16} -$$$$55\!\cdots\!78$$$$T^{17} +$$$$26\!\cdots\!02$$$$T^{18} -$$$$13\!\cdots\!02$$$$T^{19} +$$$$33\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 525110406070976 T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} +$$$$26\!\cdots\!12$$$$T^{10} -$$$$19\!\cdots\!80$$$$T^{11} +$$$$14\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!88$$$$T^{13} +$$$$71\!\cdots\!04$$$$T^{14} -$$$$48\!\cdots\!52$$$$T^{15} +$$$$32\!\cdots\!74$$$$T^{16} -$$$$21\!\cdots\!28$$$$T^{17} +$$$$14\!\cdots\!84$$$$T^{18} -$$$$93\!\cdots\!72$$$$T^{19} +$$$$58\!\cdots\!20$$$$T^{20} -$$$$36\!\cdots\!20$$$$T^{21} +$$$$21\!\cdots\!32$$$$T^{22} -$$$$12\!\cdots\!36$$$$T^{23} +$$$$72\!\cdots\!80$$$$T^{24} -$$$$38\!\cdots\!84$$$$T^{25} +$$$$19\!\cdots\!72$$$$T^{26} -$$$$95\!\cdots\!84$$$$T^{27} +$$$$42\!\cdots\!84$$$$T^{28} -$$$$18\!\cdots\!88$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} -$$$$19\!\cdots\!80$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$)
$71$ ($$( 1 - 71 T )( 1 + 71 T )$$)($$1 + 5406 T^{2} + 25411681 T^{4}$$)($$1 - 3170 T^{2} + 25411681 T^{4}$$)($$( 1 + 3604 T^{2} + 10180454 T^{4} + 91583698324 T^{6} + 645753531245761 T^{8} )^{2}$$)($$( 1 + 37534 T^{2} + 679811933 T^{4} + 7799021001352 T^{6} + 62618275556164866 T^{8} +$$$$36\!\cdots\!28$$$$T^{10} +$$$$15\!\cdots\!46$$$$T^{12} +$$$$50\!\cdots\!72$$$$T^{14} +$$$$11\!\cdots\!53$$$$T^{16} +$$$$15\!\cdots\!14$$$$T^{18} +$$$$10\!\cdots\!01$$$$T^{20} )^{2}$$)($$( 1 - 256 T + 68104 T^{2} - 10692864 T^{3} + 1610923548 T^{4} - 179723087616 T^{5} + 18972832358712 T^{6} - 1588998739085056 T^{7} + 125568612540426694 T^{8} - 8010142643727767296 T^{9} +$$$$48\!\cdots\!72$$$$T^{10} -$$$$23\!\cdots\!36$$$$T^{11} +$$$$10\!\cdots\!28$$$$T^{12} -$$$$34\!\cdots\!64$$$$T^{13} +$$$$11\!\cdots\!64$$$$T^{14} -$$$$21\!\cdots\!36$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$)
$73$ ($$1 + 46 T + 5329 T^{2}$$)($$( 1 - 82 T + 5329 T^{2} )^{2}$$)($$( 1 - 82 T + 5329 T^{2} )^{2}$$)($$( 1 - 17604 T^{2} + 131615494 T^{4} - 499922634564 T^{6} + 806460091894081 T^{8} )^{2}$$)($$( 1 - 37130 T^{2} + 665990797 T^{4} - 7634966781176 T^{6} + 62349742704798482 T^{8} -$$$$38\!\cdots\!28$$$$T^{10} +$$$$17\!\cdots\!62$$$$T^{12} -$$$$61\!\cdots\!56$$$$T^{14} +$$$$15\!\cdots\!37$$$$T^{16} -$$$$24\!\cdots\!30$$$$T^{18} +$$$$18\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} -$$$$15\!\cdots\!24$$$$T^{10} +$$$$12\!\cdots\!76$$$$T^{12} -$$$$83\!\cdots\!96$$$$T^{14} +$$$$47\!\cdots\!22$$$$T^{16} -$$$$23\!\cdots\!36$$$$T^{18} +$$$$10\!\cdots\!56$$$$T^{20} -$$$$36\!\cdots\!04$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{24} -$$$$26\!\cdots\!36$$$$T^{26} +$$$$49\!\cdots\!04$$$$T^{28} -$$$$63\!\cdots\!08$$$$T^{30} +$$$$42\!\cdots\!21$$$$T^{32}$$)
$79$ ($$1 - 142 T + 6241 T^{2}$$)($$( 1 + 10 T + 6241 T^{2} )^{2}$$)($$1 - 6674 T^{2} + 38950081 T^{4}$$)($$( 1 + 34 T + 11588 T^{2} + 212194 T^{3} + 38950081 T^{4} )^{4}$$)($$( 1 - 96 T + 27671 T^{2} - 2133500 T^{3} + 327107352 T^{4} - 19299334696 T^{5} + 2041476983832 T^{6} - 83099997813500 T^{7} + 6726472981721591 T^{8} - 145642445751029856 T^{9} + 9468276082626847201 T^{10} )^{4}$$)($$1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} -$$$$60\!\cdots\!08$$$$T^{10} +$$$$55\!\cdots\!96$$$$T^{12} -$$$$43\!\cdots\!36$$$$T^{14} +$$$$29\!\cdots\!62$$$$T^{16} -$$$$16\!\cdots\!16$$$$T^{18} +$$$$84\!\cdots\!56$$$$T^{20} -$$$$35\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!24$$$$T^{24} -$$$$33\!\cdots\!88$$$$T^{26} +$$$$66\!\cdots\!08$$$$T^{28} -$$$$85\!\cdots\!08$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$)
$83$ ($$( 1 - 83 T )( 1 + 83 T )$$)($$1 - 8370 T^{2} + 47458321 T^{4}$$)($$1 - 13346 T^{2} + 47458321 T^{4}$$)($$1 - 64624380 T^{4} + 2364945173919814 T^{8} -$$$$14\!\cdots\!80$$$$T^{12} +$$$$50\!\cdots\!81$$$$T^{16}$$)($$1 + 11186750 T^{4} - 558936874100067 T^{8} -$$$$49\!\cdots\!36$$$$T^{12} +$$$$20\!\cdots\!66$$$$T^{16} +$$$$44\!\cdots\!52$$$$T^{20} +$$$$46\!\cdots\!06$$$$T^{24} -$$$$24\!\cdots\!16$$$$T^{28} -$$$$63\!\cdots\!07$$$$T^{32} +$$$$28\!\cdots\!50$$$$T^{36} +$$$$57\!\cdots\!01$$$$T^{40}$$)($$1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} +$$$$11\!\cdots\!96$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} +$$$$12\!\cdots\!44$$$$T^{13} +$$$$89\!\cdots\!76$$$$T^{14} +$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!66$$$$T^{16} +$$$$51\!\cdots\!64$$$$T^{17} +$$$$42\!\cdots\!96$$$$T^{18} +$$$$39\!\cdots\!36$$$$T^{19} +$$$$30\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!56$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{23} +$$$$92\!\cdots\!52$$$$T^{24} +$$$$60\!\cdots\!36$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} +$$$$27\!\cdots\!52$$$$T^{27} +$$$$12\!\cdots\!04$$$$T^{28} +$$$$70\!\cdots\!76$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} +$$$$59\!\cdots\!40$$$$T^{31} +$$$$25\!\cdots\!61$$$$T^{32}$$)
$89$ ($$( 1 - 89 T )( 1 + 89 T )$$)($$1 - 14690 T^{2} + 62742241 T^{4}$$)($$( 1 - 114 T + 7921 T^{2} )^{2}$$)($$( 1 + 25444 T^{2} + 277784198 T^{4} + 1596413580004 T^{6} + 3936588805702081 T^{8} )^{2}$$)($$( 1 + 29470 T^{2} + 451584989 T^{4} + 4107082352008 T^{6} + 26252370181550850 T^{8} +$$$$16\!\cdots\!64$$$$T^{10} +$$$$16\!\cdots\!50$$$$T^{12} +$$$$16\!\cdots\!48$$$$T^{14} +$$$$11\!\cdots\!69$$$$T^{16} +$$$$45\!\cdots\!70$$$$T^{18} +$$$$97\!\cdots\!01$$$$T^{20} )^{2}$$)($$1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} -$$$$23\!\cdots\!04$$$$T^{10} +$$$$28\!\cdots\!72$$$$T^{12} -$$$$29\!\cdots\!36$$$$T^{14} +$$$$25\!\cdots\!10$$$$T^{16} -$$$$18\!\cdots\!76$$$$T^{18} +$$$$11\!\cdots\!32$$$$T^{20} -$$$$57\!\cdots\!84$$$$T^{22} +$$$$24\!\cdots\!96$$$$T^{24} -$$$$80\!\cdots\!12$$$$T^{26} +$$$$19\!\cdots\!76$$$$T^{28} -$$$$31\!\cdots\!48$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$)
$97$ ($$1 - 2 T + 9409 T^{2}$$)($$( 1 + 94 T + 9409 T^{2} )^{2}$$)($$( 1 - 34 T + 9409 T^{2} )^{2}$$)($$( 1 + 120 T + 21970 T^{2} + 1129080 T^{3} + 88529281 T^{4} )^{4}$$)($$( 1 - 118 T + 35265 T^{2} - 3292640 T^{3} + 583062270 T^{4} - 43725541204 T^{5} + 5486032898430 T^{6} - 291495051791840 T^{7} + 29374757753821185 T^{8} - 924817164136481398 T^{9} + 73742412689492826049 T^{10} )^{4}$$)($$( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} +$$$$96\!\cdots\!84$$$$T^{10} +$$$$29\!\cdots\!72$$$$T^{11} +$$$$60\!\cdots\!04$$$$T^{12} +$$$$85\!\cdots\!76$$$$T^{13} +$$$$26\!\cdots\!56$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$)
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