Properties

Label 4028.2
Level 4028
Weight 2
Dimension 291260
Nonzero newspaces 36
Sturm bound 2021760

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Defining parameters

Level: \( N \) = \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 36 \)
Sturm bound: \(2021760\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(4028))\).

Total New Old
Modular forms 510120 294724 215396
Cusp forms 500761 291260 209501
Eisenstein series 9359 3464 5895

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(4028))\)

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
4028.2.a \(\chi_{4028}(1, \cdot)\) 4028.2.a.a 1 1
4028.2.a.b 1
4028.2.a.c 19
4028.2.a.d 19
4028.2.a.e 19
4028.2.a.f 19
4028.2.b \(\chi_{4028}(4027, \cdot)\) n/a 536 1
4028.2.c \(\chi_{4028}(3497, \cdot)\) 4028.2.c.a 82 1
4028.2.h \(\chi_{4028}(531, \cdot)\) n/a 520 1
4028.2.i \(\chi_{4028}(425, \cdot)\) n/a 176 2
4028.2.j \(\chi_{4028}(189, \cdot)\) n/a 180 2
4028.2.m \(\chi_{4028}(3687, \cdot)\) n/a 972 2
4028.2.n \(\chi_{4028}(107, \cdot)\) n/a 1040 2
4028.2.s \(\chi_{4028}(635, \cdot)\) n/a 1072 2
4028.2.t \(\chi_{4028}(2861, \cdot)\) n/a 180 2
4028.2.u \(\chi_{4028}(213, \cdot)\) n/a 516 6
4028.2.w \(\chi_{4028}(825, \cdot)\) n/a 360 4
4028.2.x \(\chi_{4028}(83, \cdot)\) n/a 2144 4
4028.2.z \(\chi_{4028}(77, \cdot)\) n/a 960 12
4028.2.bc \(\chi_{4028}(529, \cdot)\) n/a 540 6
4028.2.bd \(\chi_{4028}(319, \cdot)\) n/a 3120 6
4028.2.bf \(\chi_{4028}(211, \cdot)\) n/a 3216 6
4028.2.bh \(\chi_{4028}(227, \cdot)\) n/a 6432 12
4028.2.bm \(\chi_{4028}(229, \cdot)\) n/a 984 12
4028.2.bn \(\chi_{4028}(303, \cdot)\) n/a 6432 12
4028.2.bo \(\chi_{4028}(23, \cdot)\) n/a 6432 12
4028.2.br \(\chi_{4028}(129, \cdot)\) n/a 1080 12
4028.2.bs \(\chi_{4028}(49, \cdot)\) n/a 2160 24
4028.2.bt \(\chi_{4028}(39, \cdot)\) n/a 11664 24
4028.2.bw \(\chi_{4028}(569, \cdot)\) n/a 2160 24
4028.2.bx \(\chi_{4028}(197, \cdot)\) n/a 2160 24
4028.2.by \(\chi_{4028}(255, \cdot)\) n/a 12864 24
4028.2.cd \(\chi_{4028}(183, \cdot)\) n/a 12864 24
4028.2.ce \(\chi_{4028}(81, \cdot)\) n/a 6480 72
4028.2.cg \(\chi_{4028}(87, \cdot)\) n/a 25728 48
4028.2.ch \(\chi_{4028}(65, \cdot)\) n/a 4320 48
4028.2.ck \(\chi_{4028}(59, \cdot)\) n/a 38592 72
4028.2.cm \(\chi_{4028}(15, \cdot)\) n/a 38592 72
4028.2.cn \(\chi_{4028}(9, \cdot)\) n/a 6480 72
4028.2.cq \(\chi_{4028}(21, \cdot)\) n/a 12960 144
4028.2.ct \(\chi_{4028}(35, \cdot)\) n/a 77184 144

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(4028))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(4028)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(53))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(76))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(106))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(212))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1007))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2014))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + 5 T + 30 T^{2} + 109 T^{3} + 415 T^{4} + 1260 T^{5} + 3817 T^{6} + 10164 T^{7} + 26540 T^{8} + 63750 T^{9} + 149676 T^{10} + 330753 T^{11} + 715428 T^{12} + 1474910 T^{13} + 2982881 T^{14} + 5793760 T^{15} + 11050805 T^{16} + 20338180 T^{17} + 36756438 T^{18} + 64249802 T^{19} + 110269314 T^{20} + 183043620 T^{21} + 298371735 T^{22} + 469294560 T^{23} + 724840083 T^{24} + 1075209390 T^{25} + 1564641036 T^{26} + 2170070433 T^{27} + 2946072708 T^{28} + 3764373750 T^{29} + 4701481380 T^{30} + 5401566324 T^{31} + 6085530891 T^{32} + 6026540940 T^{33} + 5954796405 T^{34} + 4692092589 T^{35} + 3874204890 T^{36} + 1937102445 T^{37} + 1162261467 T^{38} \))(\( 1 + 4 T + 27 T^{2} + 84 T^{3} + 341 T^{4} + 886 T^{5} + 2785 T^{6} + 6377 T^{7} + 17118 T^{8} + 35819 T^{9} + 86396 T^{10} + 169078 T^{11} + 377103 T^{12} + 699873 T^{13} + 1468056 T^{14} + 2603456 T^{15} + 5190084 T^{16} + 8831093 T^{17} + 16849145 T^{18} + 27567853 T^{19} + 50547435 T^{20} + 79479837 T^{21} + 140132268 T^{22} + 210879936 T^{23} + 356737608 T^{24} + 510207417 T^{25} + 824724261 T^{26} + 1109320758 T^{27} + 1700532468 T^{28} + 2115076131 T^{29} + 3032402346 T^{30} + 3388999257 T^{31} + 4440189555 T^{32} + 4237710534 T^{33} + 4892977287 T^{34} + 3615924564 T^{35} + 3486784401 T^{36} + 1549681956 T^{37} + 1162261467 T^{38} \))(\( 1 - 3 T + 22 T^{2} - 59 T^{3} + 255 T^{4} - 624 T^{5} + 2093 T^{6} - 4760 T^{7} + 13652 T^{8} - 29178 T^{9} + 74846 T^{10} - 151091 T^{11} + 355018 T^{12} - 678984 T^{13} + 1482735 T^{14} - 2692426 T^{15} + 5516491 T^{16} - 9517900 T^{17} + 18403604 T^{18} - 30145834 T^{19} + 55210812 T^{20} - 85661100 T^{21} + 148945257 T^{22} - 218086506 T^{23} + 360304605 T^{24} - 494979336 T^{25} + 776424366 T^{26} - 991308051 T^{27} + 1473193818 T^{28} - 1722931722 T^{29} + 2418410844 T^{30} - 2529659160 T^{31} + 3336918039 T^{32} - 2984572656 T^{33} + 3658971285 T^{34} - 2539756539 T^{35} + 2841083586 T^{36} - 1162261467 T^{37} + 1162261467 T^{38} \))(\( 1 - 4 T + 27 T^{2} - 92 T^{3} + 373 T^{4} - 1110 T^{5} + 3513 T^{6} - 9363 T^{7} + 25414 T^{8} - 61785 T^{9} + 150394 T^{10} - 337748 T^{11} + 754529 T^{12} - 1579321 T^{13} + 3278928 T^{14} - 6435752 T^{15} + 12508810 T^{16} - 23109339 T^{17} + 42233495 T^{18} - 73562153 T^{19} + 126700485 T^{20} - 207984051 T^{21} + 337737870 T^{22} - 521295912 T^{23} + 796779504 T^{24} - 1151325009 T^{25} + 1650154923 T^{26} - 2215964628 T^{27} + 2960205102 T^{28} - 3648342465 T^{29} + 4502013858 T^{30} - 4975882083 T^{31} + 5600856699 T^{32} - 5309095590 T^{33} + 5352142311 T^{34} - 3960298332 T^{35} + 3486784401 T^{36} - 1549681956 T^{37} + 1162261467 T^{38} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T + 51 T^{2} + 212 T^{3} + 1265 T^{4} + 4564 T^{5} + 20793 T^{6} + 67283 T^{7} + 258185 T^{8} + 765977 T^{9} + 2595001 T^{10} + 7150935 T^{11} + 21953718 T^{12} + 56622550 T^{13} + 160090022 T^{14} + 388285071 T^{15} + 1021768667 T^{16} + 2335933608 T^{17} + 5760303945 T^{18} + 12413847014 T^{19} + 28801519725 T^{20} + 58398340200 T^{21} + 127721083375 T^{22} + 242678169375 T^{23} + 500281318750 T^{24} + 884727343750 T^{25} + 1715134218750 T^{26} + 2793333984375 T^{27} + 5068361328125 T^{28} + 7480244140625 T^{29} + 12606689453125 T^{30} + 16426513671875 T^{31} + 25382080078125 T^{32} + 27856445312500 T^{33} + 38604736328125 T^{34} + 32348632812500 T^{35} + 38909912109375 T^{36} + 19073486328125 T^{37} + 19073486328125 T^{38} \))(\( 1 + 2 T + 39 T^{2} + 82 T^{3} + 749 T^{4} + 1641 T^{5} + 9540 T^{6} + 21342 T^{7} + 91512 T^{8} + 202596 T^{9} + 708864 T^{10} + 1499102 T^{11} + 4624545 T^{12} + 9095878 T^{13} + 26247629 T^{14} + 47825634 T^{15} + 134993734 T^{16} + 233914675 T^{17} + 666823791 T^{18} + 1147399136 T^{19} + 3334118955 T^{20} + 5847866875 T^{21} + 16874216750 T^{22} + 29891021250 T^{23} + 82023840625 T^{24} + 142123093750 T^{25} + 361292578125 T^{26} + 585586718750 T^{27} + 1384500000000 T^{28} + 1978476562500 T^{29} + 4468359375000 T^{30} + 5210449218750 T^{31} + 11645507812500 T^{32} + 10015869140625 T^{33} + 22857666015625 T^{34} + 12512207031250 T^{35} + 29754638671875 T^{36} + 7629394531250 T^{37} + 19073486328125 T^{38} \))(\( 1 - 3 T + 41 T^{2} - 108 T^{3} + 871 T^{4} - 1994 T^{5} + 12383 T^{6} - 24481 T^{7} + 131199 T^{8} - 219639 T^{9} + 1098135 T^{10} - 1501351 T^{11} + 7561038 T^{12} - 7936486 T^{13} + 44493720 T^{14} - 32715313 T^{15} + 234601057 T^{16} - 112311302 T^{17} + 1175893979 T^{18} - 441349142 T^{19} + 5879469895 T^{20} - 2807782550 T^{21} + 29325132125 T^{22} - 20447070625 T^{23} + 139042875000 T^{24} - 124007593750 T^{25} + 590706093750 T^{26} - 586465234375 T^{27} + 2144794921875 T^{28} - 2144912109375 T^{29} + 6406201171875 T^{30} - 5976806640625 T^{31} + 15115966796875 T^{32} - 12170410156250 T^{33} + 26580810546875 T^{34} - 16479492187500 T^{35} + 31280517578125 T^{36} - 11444091796875 T^{37} + 19073486328125 T^{38} \))(\( 1 - 4 T + 45 T^{2} - 156 T^{3} + 1025 T^{4} - 3257 T^{5} + 16178 T^{6} - 47882 T^{7} + 197744 T^{8} - 548118 T^{9} + 1977988 T^{10} - 5155812 T^{11} + 16737235 T^{12} - 41128080 T^{13} + 122310767 T^{14} - 283704750 T^{15} + 782380246 T^{16} - 1713041077 T^{17} + 4416234463 T^{18} - 9114484432 T^{19} + 22081172315 T^{20} - 42826026925 T^{21} + 97797530750 T^{22} - 177315468750 T^{23} + 382221146875 T^{24} - 642626250000 T^{25} + 1307596484375 T^{26} - 2013989062500 T^{27} + 3863257812500 T^{28} - 5352714843750 T^{29} + 9655468750000 T^{30} - 11689941406250 T^{31} + 19748535156250 T^{32} - 19879150390625 T^{33} + 31280517578125 T^{34} - 23803710937500 T^{35} + 34332275390625 T^{36} - 15258789062500 T^{37} + 19073486328125 T^{38} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 10 T + 105 T^{2} + 710 T^{3} + 4663 T^{4} + 24837 T^{5} + 127056 T^{6} + 571129 T^{7} + 2466753 T^{8} + 9728082 T^{9} + 36978479 T^{10} + 131193922 T^{11} + 450365864 T^{12} + 1463600791 T^{13} + 4620576233 T^{14} + 13946174019 T^{15} + 41041968937 T^{16} + 116246943600 T^{17} + 321902673455 T^{18} + 860912771825 T^{19} + 2253318714185 T^{20} + 5696100236400 T^{21} + 14077395345391 T^{22} + 33484763819619 T^{23} + 77658024748031 T^{24} + 172191169460359 T^{25} + 370895654736152 T^{26} + 756306852739522 T^{27} + 1492215009023753 T^{28} + 2747942385242418 T^{29} + 4877576675275479 T^{30} + 7905160517819929 T^{31} + 12310330106271792 T^{32} + 16845026460350613 T^{33} + 22137879320864209 T^{34} + 23595380704416710 T^{35} + 24426203968656735 T^{36} + 16284135979104490 T^{37} + 11398895185373143 T^{38} \))(\( 1 + 11 T + 120 T^{2} + 895 T^{3} + 6137 T^{4} + 35639 T^{5} + 191013 T^{6} + 925604 T^{7} + 4190306 T^{8} + 17643355 T^{9} + 70149327 T^{10} + 263386198 T^{11} + 941475306 T^{12} + 3207913105 T^{13} + 10468086720 T^{14} + 32760289701 T^{15} + 98594518373 T^{16} + 285642829543 T^{17} + 797795552662 T^{18} + 2148879348566 T^{19} + 5584568868634 T^{20} + 13996498647607 T^{21} + 33817919801939 T^{22} + 78657455572101 T^{23} + 175937133503040 T^{24} + 377407768890145 T^{25} + 775345397929158 T^{26} + 1518369017616598 T^{27} + 2830778373072489 T^{28} + 4983811096820395 T^{29} + 8285604115153358 T^{30} + 12811550798394404 T^{31} + 18507060544872291 T^{32} + 24171192093265511 T^{33} + 29135784986520191 T^{34} + 29743472859792895 T^{35} + 27915661678464840 T^{36} + 17912549577014939 T^{37} + 11398895185373143 T^{38} \))(\( 1 - 6 T + 81 T^{2} - 390 T^{3} + 3087 T^{4} - 12659 T^{5} + 76100 T^{6} - 276155 T^{7} + 1391433 T^{8} - 4583234 T^{9} + 20329179 T^{10} - 61771222 T^{11} + 247809480 T^{12} - 701298649 T^{13} + 2585379565 T^{14} - 6850769845 T^{15} + 23440756317 T^{16} - 58307509248 T^{17} + 186351994547 T^{18} - 435315231219 T^{19} + 1304463961829 T^{20} - 2857067953152 T^{21} + 8040179416731 T^{22} - 16448698397845 T^{23} + 43452474348955 T^{24} - 82507084756201 T^{25} + 204081762587640 T^{26} - 356098802356822 T^{27} + 820355699998653 T^{28} - 1294650165375266 T^{29} + 2751317681992719 T^{30} - 3822340666992155 T^{31} + 7373253691972700 T^{32} - 8585625879195491 T^{33} + 14655722381194041 T^{34} - 12960842922144390 T^{35} + 18843071632963767 T^{36} - 9770481587462694 T^{37} + 11398895185373143 T^{38} \))(\( 1 - 13 T + 136 T^{2} - 1037 T^{3} + 6901 T^{4} - 39597 T^{5} + 206553 T^{6} - 980480 T^{7} + 4332310 T^{8} - 17869701 T^{9} + 69645391 T^{10} - 257114014 T^{11} + 905754306 T^{12} - 3049762635 T^{13} + 9861011136 T^{14} - 30642732759 T^{15} + 91783435161 T^{16} - 265014142129 T^{17} + 738974023970 T^{18} - 1988966652946 T^{19} + 5172818167790 T^{20} - 12985692964321 T^{21} + 31481718260223 T^{22} - 73573201354359 T^{23} + 165734014162752 T^{24} - 358801524245115 T^{25} + 745927618426158 T^{26} - 1482211125021214 T^{27} + 2810442737775337 T^{28} - 5047748239530549 T^{29} + 8566392421966330 T^{30} - 13571105274836480 T^{31} + 20012715766597071 T^{32} - 26855599015601853 T^{33} + 32762921980116643 T^{34} - 34462549000676237 T^{35} + 31637749902260152 T^{36} - 21169376772835837 T^{37} + 11398895185373143 T^{38} \))
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 86 T^{2} + 170 T^{3} + 3502 T^{4} + 4201 T^{5} + 94757 T^{6} + 55164 T^{7} + 1972609 T^{8} + 138544 T^{9} + 33901025 T^{10} - 12786731 T^{11} + 502272217 T^{12} - 383328306 T^{13} + 6666711634 T^{14} - 7081445587 T^{15} + 81664493483 T^{16} - 102102205669 T^{17} + 941668653638 T^{18} - 1224307648818 T^{19} + 10358355190018 T^{20} - 12354366885949 T^{21} + 108695440825873 T^{22} - 103679444839267 T^{23} + 1073680575367334 T^{24} - 679089477105666 T^{25} + 9787864581228107 T^{26} - 2740949348808011 T^{27} + 79936843621283275 T^{28} + 3593474553920944 T^{29} + 562808369252294099 T^{30} + 173128262973437244 T^{31} + 3271268634622469767 T^{32} + 1595329050883195441 T^{33} + 14628723089293609802 T^{34} + 7811454076807267370 T^{35} + 43468444450939264306 T^{36} + 16679751940476694443 T^{37} + 61159090448414546291 T^{38} \))(\( 1 + 11 T + 170 T^{2} + 1466 T^{3} + 13698 T^{4} + 97805 T^{5} + 703713 T^{6} + 4316820 T^{7} + 26040453 T^{8} + 140683756 T^{9} + 740890277 T^{10} + 3584559765 T^{11} + 16853652949 T^{12} + 73850586590 T^{13} + 314179777450 T^{14} + 1256137321885 T^{15} + 4875103100411 T^{16} + 17865248014363 T^{17} + 63553609216286 T^{18} + 213932167946862 T^{19} + 699089701379146 T^{20} + 2161695009737923 T^{21} + 6488762226647041 T^{22} + 18391106529718285 T^{23} + 50598967338099950 T^{24} + 130830819029966990 T^{25} + 328430016991817279 T^{26} + 768382220103022965 T^{27} + 1746980517936500407 T^{28} + 3648974313835481356 T^{29} + 7429645148897226783 T^{30} + 13548030385196747220 T^{31} + 24294081330942115803 T^{32} + 37141432473608886005 T^{33} + 57219945424655587398 T^{34} + 67362303979996788026 T^{35} + 85925994844879941070 T^{36} + 61159090448414546291 T^{37} + 61159090448414546291 T^{38} \))(\( 1 - 5 T + 98 T^{2} - 406 T^{3} + 4578 T^{4} - 16227 T^{5} + 141093 T^{6} - 442400 T^{7} + 3330881 T^{8} - 9508984 T^{9} + 65156805 T^{10} - 172480855 T^{11} + 1098648625 T^{12} - 2724423282 T^{13} + 16337846294 T^{14} - 38247791303 T^{15} + 217838871003 T^{16} - 484544772673 T^{17} + 2632051949662 T^{18} - 5583192535930 T^{19} + 28952571446282 T^{20} - 58629917493433 T^{21} + 289943537304993 T^{22} - 559985912467223 T^{23} + 2631226483494994 T^{24} - 4826482033883202 T^{25} + 21409553624289875 T^{26} - 36972803071723255 T^{27} + 153636337902687255 T^{28} - 246638555532115384 T^{29} + 950339222716438291 T^{30} - 1388440713861370400 T^{31} + 4870913024523656583 T^{32} - 6162200549555251707 T^{33} + 19123442119584850278 T^{34} - 18655590324610297366 T^{35} + 49533808792930789558 T^{36} - 27799586567461157405 T^{37} + 61159090448414546291 T^{38} \))(\( 1 - T + 102 T^{2} - 66 T^{3} + 5186 T^{4} - 2255 T^{5} + 176433 T^{6} - 61916 T^{7} + 4520873 T^{8} - 1693296 T^{9} + 92898281 T^{10} - 44531007 T^{11} + 1594490761 T^{12} - 1010104746 T^{13} + 23604860498 T^{14} - 18767430139 T^{15} + 310055076623 T^{16} - 282489964469 T^{17} + 3700344158690 T^{18} - 3448348716218 T^{19} + 40703785745590 T^{20} - 34181285700749 T^{21} + 412683306985213 T^{22} - 274773944665099 T^{23} + 3801586388063398 T^{24} - 1789462173928506 T^{25} + 31072114117527131 T^{26} - 9545616830323167 T^{27} + 219049287181819171 T^{28} - 43919737327174896 T^{29} + 1289857828250163403 T^{30} - 194318931373057436 T^{31} + 6090945671690178123 T^{32} - 856335874730208455 T^{33} + 21663209006589566086 T^{34} - 3032682170995762626 T^{35} + 51555596906927964642 T^{36} - 5559917313492231481 T^{37} + 61159090448414546291 T^{38} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 23 T + 373 T^{2} + 4458 T^{3} + 44688 T^{4} + 384191 T^{5} + 2944945 T^{6} + 20401508 T^{7} + 130015901 T^{8} + 768912828 T^{9} + 4261757890 T^{10} + 22265416213 T^{11} + 110307438793 T^{12} + 520137854792 T^{13} + 2342959118292 T^{14} + 10104255975680 T^{15} + 41807969094757 T^{16} + 166157109111486 T^{17} + 634961388988616 T^{18} + 2333952062602714 T^{19} + 8254498056852008 T^{20} + 28080551439841134 T^{21} + 91852108101181129 T^{22} + 288587654921396480 T^{23} + 869924319907991556 T^{24} + 2510606078750718728 T^{25} + 6921628198329019981 T^{26} + 18162584020795579573 T^{27} + 45193808872382802970 T^{28} + \)\(10\!\cdots\!72\)\( T^{29} + \)\(23\!\cdots\!37\)\( T^{30} + \)\(47\!\cdots\!48\)\( T^{31} + \)\(89\!\cdots\!85\)\( T^{32} + \)\(15\!\cdots\!99\)\( T^{33} + \)\(22\!\cdots\!16\)\( T^{34} + \)\(29\!\cdots\!78\)\( T^{35} + \)\(32\!\cdots\!09\)\( T^{36} + \)\(25\!\cdots\!67\)\( T^{37} + \)\(14\!\cdots\!77\)\( T^{38} \))(\( 1 - T + 127 T^{2} - 68 T^{3} + 8138 T^{4} - 1707 T^{5} + 353317 T^{6} + 1062 T^{7} + 11684909 T^{8} + 1560084 T^{9} + 312600446 T^{10} + 65083687 T^{11} + 7010909859 T^{12} + 1722230180 T^{13} + 134899780238 T^{14} + 34925506310 T^{15} + 2260526879991 T^{16} + 581289655388 T^{17} + 33291077083366 T^{18} + 8182233606402 T^{19} + 432784002083758 T^{20} + 98237951760572 T^{21} + 4966377555340227 T^{22} + 997507385719910 T^{23} + 50087344103907734 T^{24} + 8312876132895620 T^{25} + 439924196472929103 T^{26} + 53090762921848327 T^{27} + 3314971233606520358 T^{28} + 215070827397755316 T^{29} + 20941231117726487633 T^{30} + 24742566400074822 T^{31} + \)\(10\!\cdots\!01\)\( T^{32} - 6721101490388686323 T^{33} + \)\(41\!\cdots\!66\)\( T^{34} - 45248329424456229188 T^{35} + \)\(10\!\cdots\!91\)\( T^{36} - \)\(11\!\cdots\!29\)\( T^{37} + \)\(14\!\cdots\!77\)\( T^{38} \))(\( 1 - 25 T + 439 T^{2} - 5744 T^{3} + 63212 T^{4} - 599371 T^{5} + 5070237 T^{6} - 38764470 T^{7} + 271919319 T^{8} - 1763619620 T^{9} + 10660141728 T^{10} - 60343755965 T^{11} + 321378712051 T^{12} - 1615420207144 T^{13} + 7685199367394 T^{14} - 34673347008766 T^{15} + 148603144630913 T^{16} - 605698464639848 T^{17} + 2349873414300098 T^{18} - 8681569384007902 T^{19} + 30548354385901274 T^{20} - 102363040524134312 T^{21} + 326481108754115861 T^{22} - 990305463917365726 T^{23} + 2853460728717820442 T^{24} - 7797324794624523496 T^{25} + 20166037576570278367 T^{26} - 49224255561177500765 T^{27} + \)\(11\!\cdots\!44\)\( T^{28} - \)\(24\!\cdots\!80\)\( T^{29} + \)\(48\!\cdots\!03\)\( T^{30} - \)\(90\!\cdots\!70\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} - \)\(23\!\cdots\!19\)\( T^{33} + \)\(32\!\cdots\!84\)\( T^{34} - \)\(38\!\cdots\!04\)\( T^{35} + \)\(37\!\cdots\!87\)\( T^{36} - \)\(28\!\cdots\!25\)\( T^{37} + \)\(14\!\cdots\!77\)\( T^{38} \))(\( 1 + T + 119 T^{2} + 90 T^{3} + 7256 T^{4} + 3847 T^{5} + 300173 T^{6} + 97854 T^{7} + 9447095 T^{8} + 1426764 T^{9} + 240938160 T^{10} + 3011801 T^{11} + 5181695515 T^{12} - 445165586 T^{13} + 96461855242 T^{14} - 14054721574 T^{15} + 1580593263593 T^{16} - 272984334266 T^{17} + 23018479806670 T^{18} - 4003200236086 T^{19} + 299240237486710 T^{20} - 46134352490954 T^{21} + 3472563400113821 T^{22} - 401416902875014 T^{23} + 35815611618367906 T^{24} - 2148729256995074 T^{25} + 325143709111801255 T^{26} + 2456818601238521 T^{27} + 2555028566651773680 T^{28} + 196691533264446636 T^{29} + 16930709497704972515 T^{30} + 2279810821575255774 T^{31} + 90914929371116359769 T^{32} + 15147086955785164783 T^{33} + \)\(37\!\cdots\!92\)\( T^{34} + 59887494826486185690 T^{35} + \)\(10\!\cdots\!27\)\( T^{36} + \)\(11\!\cdots\!29\)\( T^{37} + \)\(14\!\cdots\!77\)\( T^{38} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 - 7 T + 17 T^{2} \))(\( 1 + 7 T + 174 T^{2} + 1137 T^{3} + 14790 T^{4} + 90012 T^{5} + 817093 T^{6} + 4624904 T^{7} + 32961151 T^{8} + 173405714 T^{9} + 1035549061 T^{10} + 5068120842 T^{11} + 26488468169 T^{12} + 121088547632 T^{13} + 573736299265 T^{14} + 2473777418155 T^{15} + 10985508205846 T^{16} + 45430258164096 T^{17} + 194825115658374 T^{18} + 786835379404329 T^{19} + 3312026966192358 T^{20} + 13129344609423744 T^{21} + 53971801815321398 T^{22} + 206612363741723755 T^{23} + 814623500665505105 T^{24} + 2922783173577186608 T^{25} + 10869242878270199737 T^{26} + 35353981675468685322 T^{27} + \)\(12\!\cdots\!17\)\( T^{28} + \)\(34\!\cdots\!86\)\( T^{29} + \)\(11\!\cdots\!83\)\( T^{30} + \)\(26\!\cdots\!44\)\( T^{31} + \)\(80\!\cdots\!41\)\( T^{32} + \)\(15\!\cdots\!48\)\( T^{33} + \)\(42\!\cdots\!70\)\( T^{34} + \)\(55\!\cdots\!97\)\( T^{35} + \)\(14\!\cdots\!98\)\( T^{36} + \)\(98\!\cdots\!63\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \))(\( 1 + T + 150 T^{2} + 175 T^{3} + 11058 T^{4} + 13274 T^{5} + 540367 T^{6} + 599326 T^{7} + 19853003 T^{8} + 18456246 T^{9} + 588688177 T^{10} + 417341732 T^{11} + 14782761103 T^{12} + 7212484674 T^{13} + 326065477437 T^{14} + 97948236591 T^{15} + 6493166151252 T^{16} + 1135467731678 T^{17} + 118909992017306 T^{18} + 15315368153951 T^{19} + 2021469864294202 T^{20} + 328150174454942 T^{21} + 31900925301101076 T^{22} + 8180734668316911 T^{23} + 462966350597266509 T^{24} + 174091846480117506 T^{25} + 6065938574281036319 T^{26} + 2911274692438827812 T^{27} + 69811280829320075969 T^{28} + 37207679361186254454 T^{29} + \)\(68\!\cdots\!99\)\( T^{30} + \)\(34\!\cdots\!86\)\( T^{31} + \)\(53\!\cdots\!79\)\( T^{32} + \)\(22\!\cdots\!46\)\( T^{33} + \)\(31\!\cdots\!94\)\( T^{34} + \)\(85\!\cdots\!75\)\( T^{35} + \)\(12\!\cdots\!50\)\( T^{36} + \)\(14\!\cdots\!09\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \))(\( 1 + 7 T + 202 T^{2} + 1265 T^{3} + 19086 T^{4} + 106856 T^{5} + 1113061 T^{6} + 5542296 T^{7} + 44307035 T^{8} + 193478686 T^{9} + 1244913445 T^{10} + 4602398282 T^{11} + 24075684713 T^{12} + 67550989176 T^{13} + 269711787697 T^{14} + 240870621991 T^{15} - 288294148986 T^{16} - 15347486460092 T^{17} - 73722562405782 T^{18} - 412500661787123 T^{19} - 1253283560898294 T^{20} - 4435423586966588 T^{21} - 1416389153968218 T^{22} + 20117755219310311 T^{23} + 382952169744099329 T^{24} + 1630516662253953144 T^{25} + 9879184516698805849 T^{26} + 32105214062107116362 T^{27} + \)\(14\!\cdots\!65\)\( T^{28} + \)\(39\!\cdots\!14\)\( T^{29} + \)\(15\!\cdots\!55\)\( T^{30} + \)\(32\!\cdots\!56\)\( T^{31} + \)\(11\!\cdots\!57\)\( T^{32} + \)\(17\!\cdots\!24\)\( T^{33} + \)\(54\!\cdots\!98\)\( T^{34} + \)\(61\!\cdots\!65\)\( T^{35} + \)\(16\!\cdots\!54\)\( T^{36} + \)\(98\!\cdots\!63\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \))(\( 1 - 3 T + 138 T^{2} - 421 T^{3} + 9546 T^{4} - 29302 T^{5} + 442843 T^{6} - 1367798 T^{7} + 15617967 T^{8} - 48773398 T^{9} + 451294009 T^{10} - 1430425648 T^{11} + 11237311739 T^{12} - 36080850250 T^{13} + 249264860397 T^{14} - 803103452789 T^{15} + 5021797727200 T^{16} - 15985420026762 T^{17} + 92766305155654 T^{18} - 286257199960153 T^{19} + 1577027187646118 T^{20} - 4619786387734218 T^{21} + 24672092233733600 T^{22} - 67076003480390069 T^{23} + 353920456888703229 T^{24} - 870904012488042250 T^{25} + 4611103587068582347 T^{26} - 9978302357833246768 T^{27} + 53517998203128006473 T^{28} - 98326872872171455702 T^{29} + \)\(53\!\cdots\!11\)\( T^{30} - \)\(79\!\cdots\!78\)\( T^{31} + \)\(43\!\cdots\!91\)\( T^{32} - \)\(49\!\cdots\!58\)\( T^{33} + \)\(27\!\cdots\!78\)\( T^{34} - \)\(20\!\cdots\!01\)\( T^{35} + \)\(11\!\cdots\!26\)\( T^{36} - \)\(42\!\cdots\!27\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \))
$19$ (\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{19} \))(\( ( 1 + T )^{19} \))(\( ( 1 - T )^{19} \))(\( ( 1 + T )^{19} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 6 T + 213 T^{2} + 1374 T^{3} + 22302 T^{4} + 151437 T^{5} + 1562932 T^{6} + 10723499 T^{7} + 84049075 T^{8} + 554260699 T^{9} + 3728332167 T^{10} + 22749090660 T^{11} + 141154980638 T^{12} + 791048160563 T^{13} + 4629454378290 T^{14} + 24299410847649 T^{15} + 133089418970398 T^{16} + 667939893364767 T^{17} + 3402749310767452 T^{18} + 16331114712281892 T^{19} + 78263234147651396 T^{20} + 353340203589961743 T^{21} + 1619298960612832466 T^{22} + 6799971431016943809 T^{23} + 29796756281526193470 T^{24} + \)\(11\!\cdots\!07\)\( T^{25} + \)\(48\!\cdots\!86\)\( T^{26} + \)\(17\!\cdots\!60\)\( T^{27} + \)\(67\!\cdots\!21\)\( T^{28} + \)\(22\!\cdots\!51\)\( T^{29} + \)\(80\!\cdots\!25\)\( T^{30} + \)\(23\!\cdots\!79\)\( T^{31} + \)\(78\!\cdots\!56\)\( T^{32} + \)\(17\!\cdots\!33\)\( T^{33} + \)\(59\!\cdots\!14\)\( T^{34} + \)\(84\!\cdots\!14\)\( T^{35} + \)\(30\!\cdots\!39\)\( T^{36} + \)\(19\!\cdots\!14\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \))(\( 1 + 16 T + 319 T^{2} + 3618 T^{3} + 43748 T^{4} + 394447 T^{5} + 3653016 T^{6} + 27822565 T^{7} + 215894329 T^{8} + 1446039217 T^{9} + 9889502217 T^{10} + 60060497332 T^{11} + 374474498134 T^{12} + 2110058255001 T^{13} + 12257266225486 T^{14} + 64967583526007 T^{15} + 355248137995664 T^{16} + 1779048859027991 T^{17} + 9176892138573302 T^{18} + 43385874507698012 T^{19} + 211068519187185946 T^{20} + 941116846425807239 T^{21} + 4322304094993243888 T^{22} + 18180593541501324887 T^{23} + 78891969669543237698 T^{24} + \)\(31\!\cdots\!89\)\( T^{25} + \)\(12\!\cdots\!98\)\( T^{26} + \)\(47\!\cdots\!92\)\( T^{27} + \)\(17\!\cdots\!71\)\( T^{28} + \)\(59\!\cdots\!33\)\( T^{29} + \)\(20\!\cdots\!83\)\( T^{30} + \)\(60\!\cdots\!65\)\( T^{31} + \)\(18\!\cdots\!28\)\( T^{32} + \)\(45\!\cdots\!23\)\( T^{33} + \)\(11\!\cdots\!36\)\( T^{34} + \)\(22\!\cdots\!98\)\( T^{35} + \)\(44\!\cdots\!57\)\( T^{36} + \)\(51\!\cdots\!04\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \))(\( 1 - 18 T + 345 T^{2} - 4066 T^{3} + 48506 T^{4} - 449811 T^{5} + 4186504 T^{6} - 32871465 T^{7} + 258756073 T^{8} - 1786927775 T^{9} + 12398366325 T^{10} - 77148517096 T^{11} + 484227526330 T^{12} - 2763233935095 T^{13} + 15987114783514 T^{14} - 84867895836243 T^{15} + 459259714274516 T^{16} - 2294146567163813 T^{17} + 11732655048434354 T^{18} - 55563582053421876 T^{19} + 269851066113990142 T^{20} - 1213603534029657077 T^{21} + 5587812943578036172 T^{22} - 23749516838710077363 T^{23} + \)\(10\!\cdots\!02\)\( T^{24} - \)\(40\!\cdots\!55\)\( T^{25} + \)\(16\!\cdots\!10\)\( T^{26} - \)\(60\!\cdots\!76\)\( T^{27} + \)\(22\!\cdots\!75\)\( T^{28} - \)\(74\!\cdots\!75\)\( T^{29} + \)\(24\!\cdots\!71\)\( T^{30} - \)\(72\!\cdots\!65\)\( T^{31} + \)\(21\!\cdots\!32\)\( T^{32} - \)\(52\!\cdots\!99\)\( T^{33} + \)\(12\!\cdots\!42\)\( T^{34} - \)\(24\!\cdots\!26\)\( T^{35} + \)\(48\!\cdots\!35\)\( T^{36} - \)\(58\!\cdots\!42\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \))(\( 1 - 10 T + 307 T^{2} - 2674 T^{3} + 45272 T^{4} - 349589 T^{5} + 4289654 T^{6} - 29769251 T^{7} + 294628567 T^{8} - 1857749595 T^{9} + 15687819577 T^{10} - 90687955688 T^{11} + 676197688296 T^{12} - 3610227380985 T^{13} + 24312402394198 T^{14} - 120568838734045 T^{15} + 744649738710880 T^{16} - 3442792302232973 T^{17} + 19696758718839744 T^{18} - 84999658277064268 T^{19} + 453025450533314112 T^{20} - 1821237127881242717 T^{21} + 9060153370895276960 T^{22} - 33740104400173886845 T^{23} + \)\(15\!\cdots\!14\)\( T^{24} - \)\(53\!\cdots\!65\)\( T^{25} + \)\(23\!\cdots\!12\)\( T^{26} - \)\(71\!\cdots\!28\)\( T^{27} + \)\(28\!\cdots\!51\)\( T^{28} - \)\(76\!\cdots\!55\)\( T^{29} + \)\(28\!\cdots\!09\)\( T^{30} - \)\(65\!\cdots\!71\)\( T^{31} + \)\(21\!\cdots\!82\)\( T^{32} - \)\(40\!\cdots\!01\)\( T^{33} + \)\(12\!\cdots\!04\)\( T^{34} - \)\(16\!\cdots\!14\)\( T^{35} + \)\(43\!\cdots\!21\)\( T^{36} - \)\(32\!\cdots\!90\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \))
$29$ (\( 1 - 4 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 4 T + 245 T^{2} + 749 T^{3} + 30209 T^{4} + 63774 T^{5} + 2515227 T^{6} + 3097326 T^{7} + 160966802 T^{8} + 72857701 T^{9} + 8508715798 T^{10} - 1699428275 T^{11} + 386741413141 T^{12} - 274144023987 T^{13} + 15461584833599 T^{14} - 16427413649482 T^{15} + 551381926860055 T^{16} - 688943493024412 T^{17} + 17694022288314267 T^{18} - 22425123788172060 T^{19} + 513126646361113743 T^{20} - 579401477633530492 T^{21} + 13447653814189881395 T^{22} - 11618797553419278442 T^{23} + \)\(31\!\cdots\!51\)\( T^{24} - \)\(16\!\cdots\!27\)\( T^{25} + \)\(66\!\cdots\!69\)\( T^{26} - \)\(85\!\cdots\!75\)\( T^{27} + \)\(12\!\cdots\!62\)\( T^{28} + \)\(30\!\cdots\!01\)\( T^{29} + \)\(19\!\cdots\!58\)\( T^{30} + \)\(10\!\cdots\!66\)\( T^{31} + \)\(25\!\cdots\!03\)\( T^{32} + \)\(18\!\cdots\!94\)\( T^{33} + \)\(26\!\cdots\!41\)\( T^{34} + \)\(18\!\cdots\!29\)\( T^{35} + \)\(17\!\cdots\!05\)\( T^{36} + \)\(84\!\cdots\!44\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \))(\( 1 + 263 T^{2} + 227 T^{3} + 32667 T^{4} + 57816 T^{5} + 2602535 T^{6} + 6904872 T^{7} + 153336076 T^{8} + 520219805 T^{9} + 7310658090 T^{10} + 28193282237 T^{11} + 299486963727 T^{12} + 1195035541881 T^{13} + 10947633789555 T^{14} + 42383215737484 T^{15} + 365629929365909 T^{16} + 1338432920697978 T^{17} + 11331129770587833 T^{18} + 39644618388421176 T^{19} + 328602763347047157 T^{20} + 1125622086306999498 T^{21} + 8917348347305154601 T^{22} + 29976843210023421004 T^{23} + \)\(22\!\cdots\!95\)\( T^{24} + \)\(71\!\cdots\!01\)\( T^{25} + \)\(51\!\cdots\!43\)\( T^{26} + \)\(14\!\cdots\!57\)\( T^{27} + \)\(10\!\cdots\!10\)\( T^{28} + \)\(21\!\cdots\!05\)\( T^{29} + \)\(18\!\cdots\!04\)\( T^{30} + \)\(24\!\cdots\!52\)\( T^{31} + \)\(26\!\cdots\!15\)\( T^{32} + \)\(17\!\cdots\!96\)\( T^{33} + \)\(28\!\cdots\!83\)\( T^{34} + \)\(56\!\cdots\!67\)\( T^{35} + \)\(19\!\cdots\!67\)\( T^{36} + \)\(61\!\cdots\!69\)\( T^{38} \))(\( 1 - 4 T + 353 T^{2} - 1493 T^{3} + 61659 T^{4} - 272442 T^{5} + 7099851 T^{6} - 32357928 T^{7} + 605148920 T^{8} - 2807714907 T^{9} + 40617959290 T^{10} - 189254645211 T^{11} + 2229092073599 T^{12} - 10279664051121 T^{13} + 102470942129199 T^{14} - 460355078589312 T^{15} + 4008264114073569 T^{16} - 17238662183690350 T^{17} + 134693219450077543 T^{18} - 543940108322037232 T^{19} + 3906103364052248747 T^{20} - 14497714896483584350 T^{21} + 97757553478140274341 T^{22} - \)\(32\!\cdots\!72\)\( T^{23} + \)\(21\!\cdots\!51\)\( T^{24} - \)\(61\!\cdots\!41\)\( T^{25} + \)\(38\!\cdots\!91\)\( T^{26} - \)\(94\!\cdots\!71\)\( T^{27} + \)\(58\!\cdots\!10\)\( T^{28} - \)\(11\!\cdots\!07\)\( T^{29} + \)\(73\!\cdots\!80\)\( T^{30} - \)\(11\!\cdots\!48\)\( T^{31} + \)\(72\!\cdots\!39\)\( T^{32} - \)\(81\!\cdots\!02\)\( T^{33} + \)\(53\!\cdots\!91\)\( T^{34} - \)\(37\!\cdots\!53\)\( T^{35} + \)\(25\!\cdots\!77\)\( T^{36} - \)\(84\!\cdots\!44\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \))(\( 1 - 2 T + 287 T^{2} - 411 T^{3} + 41039 T^{4} - 45786 T^{5} + 3934785 T^{6} - 3985478 T^{7} + 284865260 T^{8} - 303271903 T^{9} + 16561718102 T^{10} - 20024267343 T^{11} + 803472028407 T^{12} - 1115838363159 T^{13} + 33433836739095 T^{14} - 51728261983814 T^{15} + 1218307083826775 T^{16} - 1987279360411350 T^{17} + 39449720364149505 T^{18} - 63284093598091124 T^{19} + 1144041890560335645 T^{20} - 1671301942105945350 T^{21} + 29713291467451215475 T^{22} - 36586416864173949734 T^{23} + \)\(68\!\cdots\!55\)\( T^{24} - \)\(66\!\cdots\!39\)\( T^{25} + \)\(13\!\cdots\!63\)\( T^{26} - \)\(10\!\cdots\!23\)\( T^{27} + \)\(24\!\cdots\!38\)\( T^{28} - \)\(12\!\cdots\!03\)\( T^{29} + \)\(34\!\cdots\!40\)\( T^{30} - \)\(14\!\cdots\!98\)\( T^{31} + \)\(40\!\cdots\!65\)\( T^{32} - \)\(13\!\cdots\!66\)\( T^{33} + \)\(35\!\cdots\!11\)\( T^{34} - \)\(10\!\cdots\!31\)\( T^{35} + \)\(20\!\cdots\!83\)\( T^{36} - \)\(42\!\cdots\!22\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \))
$31$ (\( 1 - 9 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 30 T + 706 T^{2} + 12199 T^{3} + 183148 T^{4} + 2373100 T^{5} + 27911693 T^{6} + 298362653 T^{7} + 2962478759 T^{8} + 27377251711 T^{9} + 238186603659 T^{10} + 1953979888991 T^{11} + 15217200234396 T^{12} + 112613660206921 T^{13} + 795338517459528 T^{14} + 5363017385825135 T^{15} + 34624850307121213 T^{16} + 214021384968133322 T^{17} + 1268888108608911174 T^{18} + 7211608209745661246 T^{19} + 39335531366876246394 T^{20} + \)\(20\!\cdots\!42\)\( T^{21} + \)\(10\!\cdots\!83\)\( T^{22} + \)\(49\!\cdots\!35\)\( T^{23} + \)\(22\!\cdots\!28\)\( T^{24} + \)\(99\!\cdots\!01\)\( T^{25} + \)\(41\!\cdots\!56\)\( T^{26} + \)\(16\!\cdots\!31\)\( T^{27} + \)\(62\!\cdots\!89\)\( T^{28} + \)\(22\!\cdots\!11\)\( T^{29} + \)\(75\!\cdots\!29\)\( T^{30} + \)\(23\!\cdots\!33\)\( T^{31} + \)\(68\!\cdots\!63\)\( T^{32} + \)\(17\!\cdots\!00\)\( T^{33} + \)\(42\!\cdots\!48\)\( T^{34} + \)\(88\!\cdots\!19\)\( T^{35} + \)\(15\!\cdots\!66\)\( T^{36} + \)\(20\!\cdots\!30\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \))(\( 1 + 9 T + 268 T^{2} + 2260 T^{3} + 36981 T^{4} + 291295 T^{5} + 3491376 T^{6} + 25906909 T^{7} + 254051465 T^{8} + 1790063164 T^{9} + 15199900498 T^{10} + 102054137753 T^{11} + 775688214172 T^{12} + 4962816000933 T^{13} + 34475251192351 T^{14} + 209822120776297 T^{15} + 1350769017614591 T^{16} + 7801487741761741 T^{17} + 46992591773508464 T^{18} + 256770621749748689 T^{19} + 1456770344978762384 T^{20} + 7497229719833033101 T^{21} + 40240759803756280481 T^{22} + \)\(19\!\cdots\!37\)\( T^{23} + \)\(98\!\cdots\!01\)\( T^{24} + \)\(44\!\cdots\!73\)\( T^{25} + \)\(21\!\cdots\!92\)\( T^{26} + \)\(87\!\cdots\!73\)\( T^{27} + \)\(40\!\cdots\!58\)\( T^{28} + \)\(14\!\cdots\!64\)\( T^{29} + \)\(64\!\cdots\!15\)\( T^{30} + \)\(20\!\cdots\!49\)\( T^{31} + \)\(85\!\cdots\!16\)\( T^{32} + \)\(22\!\cdots\!95\)\( T^{33} + \)\(86\!\cdots\!31\)\( T^{34} + \)\(16\!\cdots\!60\)\( T^{35} + \)\(60\!\cdots\!48\)\( T^{36} + \)\(62\!\cdots\!69\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \))(\( 1 - 12 T + 400 T^{2} - 3899 T^{3} + 72138 T^{4} - 600856 T^{5} + 8049221 T^{6} - 59750051 T^{7} + 642273943 T^{8} - 4411170927 T^{9} + 40083682487 T^{10} - 262404822871 T^{11} + 2075554693522 T^{12} - 13171655503173 T^{13} + 92431251994926 T^{14} - 570417651985029 T^{15} + 3611919987359935 T^{16} - 21529650907675796 T^{17} + 125313565347419826 T^{18} - 712181799768067490 T^{19} + 3884720525770014606 T^{20} - 20689994522276439956 T^{21} + \)\(10\!\cdots\!85\)\( T^{22} - \)\(52\!\cdots\!09\)\( T^{23} + \)\(26\!\cdots\!26\)\( T^{24} - \)\(11\!\cdots\!13\)\( T^{25} + \)\(57\!\cdots\!42\)\( T^{26} - \)\(22\!\cdots\!11\)\( T^{27} + \)\(10\!\cdots\!77\)\( T^{28} - \)\(36\!\cdots\!27\)\( T^{29} + \)\(16\!\cdots\!33\)\( T^{30} - \)\(47\!\cdots\!11\)\( T^{31} + \)\(19\!\cdots\!11\)\( T^{32} - \)\(45\!\cdots\!76\)\( T^{33} + \)\(16\!\cdots\!38\)\( T^{34} - \)\(28\!\cdots\!19\)\( T^{35} + \)\(90\!\cdots\!00\)\( T^{36} - \)\(83\!\cdots\!92\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \))(\( 1 - 25 T + 636 T^{2} - 10752 T^{3} + 170367 T^{4} - 2242021 T^{5} + 27533340 T^{6} - 301639543 T^{7} + 3102582029 T^{8} - 29383843830 T^{9} + 262993394040 T^{10} - 2203608331833 T^{11} + 17541600703490 T^{12} - 131978568070883 T^{13} + 947022697438839 T^{14} - 6459315066811787 T^{15} + 42127055677687887 T^{16} - 262010068672560107 T^{17} + 1560515079538383152 T^{18} - 8876616325334004229 T^{19} + 48375967465689877712 T^{20} - \)\(25\!\cdots\!27\)\( T^{21} + \)\(12\!\cdots\!17\)\( T^{22} - \)\(59\!\cdots\!27\)\( T^{23} + \)\(27\!\cdots\!89\)\( T^{24} - \)\(11\!\cdots\!23\)\( T^{25} + \)\(48\!\cdots\!90\)\( T^{26} - \)\(18\!\cdots\!53\)\( T^{27} + \)\(69\!\cdots\!40\)\( T^{28} - \)\(24\!\cdots\!30\)\( T^{29} + \)\(78\!\cdots\!99\)\( T^{30} - \)\(23\!\cdots\!23\)\( T^{31} + \)\(67\!\cdots\!40\)\( T^{32} - \)\(16\!\cdots\!41\)\( T^{33} + \)\(39\!\cdots\!17\)\( T^{34} - \)\(78\!\cdots\!12\)\( T^{35} + \)\(14\!\cdots\!96\)\( T^{36} - \)\(17\!\cdots\!25\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \))
$37$ (\( 1 - 10 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 31 T + 765 T^{2} + 13225 T^{3} + 201281 T^{4} + 2583789 T^{5} + 30570159 T^{6} + 325949412 T^{7} + 3280422752 T^{8} + 30686775874 T^{9} + 274193476394 T^{10} + 2313324784611 T^{11} + 18756512995405 T^{12} + 144885553768312 T^{13} + 1079571506183625 T^{14} + 7706572284622052 T^{15} + 53193211571852979 T^{16} + 352893773501694034 T^{17} + 2266378283907833479 T^{18} + 14009243302058283256 T^{19} + 83855996504589838723 T^{20} + \)\(48\!\cdots\!46\)\( T^{21} + \)\(26\!\cdots\!87\)\( T^{22} + \)\(14\!\cdots\!72\)\( T^{23} + \)\(74\!\cdots\!25\)\( T^{24} + \)\(37\!\cdots\!08\)\( T^{25} + \)\(17\!\cdots\!65\)\( T^{26} + \)\(81\!\cdots\!31\)\( T^{27} + \)\(35\!\cdots\!38\)\( T^{28} + \)\(14\!\cdots\!26\)\( T^{29} + \)\(58\!\cdots\!76\)\( T^{30} + \)\(21\!\cdots\!72\)\( T^{31} + \)\(74\!\cdots\!23\)\( T^{32} + \)\(23\!\cdots\!21\)\( T^{33} + \)\(67\!\cdots\!33\)\( T^{34} + \)\(16\!\cdots\!25\)\( T^{35} + \)\(34\!\cdots\!05\)\( T^{36} + \)\(52\!\cdots\!99\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \))(\( 1 + 25 T + 751 T^{2} + 13385 T^{3} + 243505 T^{4} + 3446883 T^{5} + 47858705 T^{6} + 568669912 T^{7} + 6550582652 T^{8} + 67507394756 T^{9} + 672343075530 T^{10} + 6136000248927 T^{11} + 54107266908095 T^{12} + 443364156625774 T^{13} + 3512654394037347 T^{14} + 26079561106234904 T^{15} + 187358642326732771 T^{16} + 1267484363772225738 T^{17} + 8301592548220343267 T^{18} + 51315013186076839280 T^{19} + \)\(30\!\cdots\!79\)\( T^{20} + \)\(17\!\cdots\!22\)\( T^{21} + \)\(94\!\cdots\!63\)\( T^{22} + \)\(48\!\cdots\!44\)\( T^{23} + \)\(24\!\cdots\!79\)\( T^{24} + \)\(11\!\cdots\!66\)\( T^{25} + \)\(51\!\cdots\!35\)\( T^{26} + \)\(21\!\cdots\!67\)\( T^{27} + \)\(87\!\cdots\!10\)\( T^{28} + \)\(32\!\cdots\!44\)\( T^{29} + \)\(11\!\cdots\!76\)\( T^{30} + \)\(37\!\cdots\!72\)\( T^{31} + \)\(11\!\cdots\!85\)\( T^{32} + \)\(31\!\cdots\!87\)\( T^{33} + \)\(81\!\cdots\!65\)\( T^{34} + \)\(16\!\cdots\!85\)\( T^{35} + \)\(34\!\cdots\!67\)\( T^{36} + \)\(42\!\cdots\!25\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \))(\( 1 - 19 T + 571 T^{2} - 7997 T^{3} + 139917 T^{4} - 1580345 T^{5} + 20694903 T^{6} - 198719404 T^{7} + 2148472926 T^{8} - 18172628760 T^{9} + 171046542852 T^{10} - 1307103033465 T^{11} + 11068341331633 T^{12} - 77803956330044 T^{13} + 605399572806229 T^{14} - 3962504599069592 T^{15} + 28711335897173841 T^{16} - 176285587145231192 T^{17} + 1198557617336613823 T^{18} - 6927775369066964908 T^{19} + 44346631841454711451 T^{20} - \)\(24\!\cdots\!48\)\( T^{21} + \)\(14\!\cdots\!73\)\( T^{22} - \)\(74\!\cdots\!12\)\( T^{23} + \)\(41\!\cdots\!53\)\( T^{24} - \)\(19\!\cdots\!96\)\( T^{25} + \)\(10\!\cdots\!89\)\( T^{26} - \)\(45\!\cdots\!65\)\( T^{27} + \)\(22\!\cdots\!04\)\( T^{28} - \)\(87\!\cdots\!40\)\( T^{29} + \)\(38\!\cdots\!38\)\( T^{30} - \)\(13\!\cdots\!24\)\( T^{31} + \)\(50\!\cdots\!91\)\( T^{32} - \)\(14\!\cdots\!05\)\( T^{33} + \)\(46\!\cdots\!81\)\( T^{34} - \)\(98\!\cdots\!77\)\( T^{35} + \)\(26\!\cdots\!07\)\( T^{36} - \)\(32\!\cdots\!51\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \))(\( 1 - 19 T + 587 T^{2} - 8625 T^{3} + 154149 T^{4} - 1858369 T^{5} + 24702931 T^{6} - 253266460 T^{7} + 2749534680 T^{8} - 24561040982 T^{9} + 228537069322 T^{10} - 1811987902329 T^{11} + 14908216724367 T^{12} - 106697627933544 T^{13} + 795930350933783 T^{14} - 5236056639515756 T^{15} + 36259174299721281 T^{16} - 223664617536867558 T^{17} + 1468079325472823371 T^{18} - 8635934231125656828 T^{19} + 54318935042494464727 T^{20} - \)\(30\!\cdots\!02\)\( T^{21} + \)\(18\!\cdots\!93\)\( T^{22} - \)\(98\!\cdots\!16\)\( T^{23} + \)\(55\!\cdots\!31\)\( T^{24} - \)\(27\!\cdots\!96\)\( T^{25} + \)\(14\!\cdots\!11\)\( T^{26} - \)\(63\!\cdots\!09\)\( T^{27} + \)\(29\!\cdots\!94\)\( T^{28} - \)\(11\!\cdots\!18\)\( T^{29} + \)\(48\!\cdots\!40\)\( T^{30} - \)\(16\!\cdots\!60\)\( T^{31} + \)\(60\!\cdots\!07\)\( T^{32} - \)\(16\!\cdots\!41\)\( T^{33} + \)\(51\!\cdots\!57\)\( T^{34} - \)\(10\!\cdots\!25\)\( T^{35} + \)\(26\!\cdots\!79\)\( T^{36} - \)\(32\!\cdots\!51\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 15 T + 394 T^{2} + 4415 T^{3} + 68670 T^{4} + 628586 T^{5} + 7510437 T^{6} + 59786976 T^{7} + 612601940 T^{8} + 4450061109 T^{9} + 41451115420 T^{10} + 283084807992 T^{11} + 2464701251908 T^{12} + 16037317438809 T^{13} + 132268581790761 T^{14} + 824964866123264 T^{15} + 6491216614172799 T^{16} + 38854867847970715 T^{17} + 291897254922029097 T^{18} + 1670613898843367188 T^{19} + 11967787451803192977 T^{20} + 65315032852438771915 T^{21} + \)\(44\!\cdots\!79\)\( T^{22} + \)\(23\!\cdots\!04\)\( T^{23} + \)\(15\!\cdots\!61\)\( T^{24} + \)\(76\!\cdots\!69\)\( T^{25} + \)\(48\!\cdots\!48\)\( T^{26} + \)\(22\!\cdots\!32\)\( T^{27} + \)\(13\!\cdots\!20\)\( T^{28} + \)\(59\!\cdots\!09\)\( T^{29} + \)\(33\!\cdots\!40\)\( T^{30} + \)\(13\!\cdots\!56\)\( T^{31} + \)\(69\!\cdots\!77\)\( T^{32} + \)\(23\!\cdots\!46\)\( T^{33} + \)\(10\!\cdots\!70\)\( T^{34} + \)\(28\!\cdots\!15\)\( T^{35} + \)\(10\!\cdots\!14\)\( T^{36} + \)\(16\!\cdots\!15\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \))(\( 1 - T + 550 T^{2} - 945 T^{3} + 146374 T^{4} - 349942 T^{5} + 25219659 T^{6} - 75244426 T^{7} + 3174543670 T^{8} - 10979362017 T^{9} + 311959036558 T^{10} - 1180422294000 T^{11} + 24917954256776 T^{12} - 98297563717089 T^{13} + 1658534291568757 T^{14} - 6548776085629442 T^{15} + 93342246610226873 T^{16} - 356435642476819079 T^{17} + 4476526560391372765 T^{18} - 16040172963957118964 T^{19} + \)\(18\!\cdots\!65\)\( T^{20} - \)\(59\!\cdots\!99\)\( T^{21} + \)\(64\!\cdots\!33\)\( T^{22} - \)\(18\!\cdots\!62\)\( T^{23} + \)\(19\!\cdots\!57\)\( T^{24} - \)\(46\!\cdots\!49\)\( T^{25} + \)\(48\!\cdots\!56\)\( T^{26} - \)\(94\!\cdots\!00\)\( T^{27} + \)\(10\!\cdots\!38\)\( T^{28} - \)\(14\!\cdots\!17\)\( T^{29} + \)\(17\!\cdots\!70\)\( T^{30} - \)\(16\!\cdots\!06\)\( T^{31} + \)\(23\!\cdots\!39\)\( T^{32} - \)\(13\!\cdots\!62\)\( T^{33} + \)\(22\!\cdots\!74\)\( T^{34} - \)\(60\!\cdots\!45\)\( T^{35} + \)\(14\!\cdots\!50\)\( T^{36} - \)\(10\!\cdots\!21\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \))(\( 1 + 9 T + 526 T^{2} + 4105 T^{3} + 132806 T^{4} + 909156 T^{5} + 21519893 T^{6} + 129898512 T^{7} + 2521575692 T^{8} + 13439994141 T^{9} + 228353407676 T^{10} + 1074771558998 T^{11} + 16708339570044 T^{12} + 69564450356501 T^{13} + 1021788631218899 T^{14} + 3792067254951512 T^{15} + 53693910825660731 T^{16} + 180710184640157347 T^{17} + 2477103331244412857 T^{18} + 7758724151411719448 T^{19} + \)\(10\!\cdots\!37\)\( T^{20} + \)\(30\!\cdots\!07\)\( T^{21} + \)\(37\!\cdots\!51\)\( T^{22} + \)\(10\!\cdots\!32\)\( T^{23} + \)\(11\!\cdots\!99\)\( T^{24} + \)\(33\!\cdots\!41\)\( T^{25} + \)\(32\!\cdots\!64\)\( T^{26} + \)\(85\!\cdots\!58\)\( T^{27} + \)\(74\!\cdots\!36\)\( T^{28} + \)\(18\!\cdots\!41\)\( T^{29} + \)\(13\!\cdots\!72\)\( T^{30} + \)\(29\!\cdots\!72\)\( T^{31} + \)\(19\!\cdots\!53\)\( T^{32} + \)\(34\!\cdots\!16\)\( T^{33} + \)\(20\!\cdots\!06\)\( T^{34} + \)\(26\!\cdots\!05\)\( T^{35} + \)\(13\!\cdots\!06\)\( T^{36} + \)\(96\!\cdots\!89\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \))(\( 1 + 9 T + 370 T^{2} + 1937 T^{3} + 59678 T^{4} + 147240 T^{5} + 6303827 T^{6} + 1236526 T^{7} + 538409058 T^{8} - 792733321 T^{9} + 40634941390 T^{10} - 99724739142 T^{11} + 2730767902588 T^{12} - 8061289576889 T^{13} + 161614966035311 T^{14} - 516651586394262 T^{15} + 8405310147836305 T^{16} - 27569139474843983 T^{17} + 386708973768399065 T^{18} - 1234365416460788556 T^{19} + 15855067924504361665 T^{20} - 46343723457212735423 T^{21} + \)\(57\!\cdots\!05\)\( T^{22} - \)\(14\!\cdots\!82\)\( T^{23} + \)\(18\!\cdots\!11\)\( T^{24} - \)\(38\!\cdots\!49\)\( T^{25} + \)\(53\!\cdots\!28\)\( T^{26} - \)\(79\!\cdots\!82\)\( T^{27} + \)\(13\!\cdots\!90\)\( T^{28} - \)\(10\!\cdots\!21\)\( T^{29} + \)\(29\!\cdots\!78\)\( T^{30} + \)\(27\!\cdots\!06\)\( T^{31} + \)\(58\!\cdots\!67\)\( T^{32} + \)\(55\!\cdots\!40\)\( T^{33} + \)\(92\!\cdots\!78\)\( T^{34} + \)\(12\!\cdots\!17\)\( T^{35} + \)\(96\!\cdots\!70\)\( T^{36} + \)\(96\!\cdots\!89\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \))
$43$ (\( 1 - 5 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 + 29 T + 858 T^{2} + 15986 T^{3} + 287374 T^{4} + 4090703 T^{5} + 55989636 T^{6} + 657847429 T^{7} + 7465160080 T^{8} + 75675883101 T^{9} + 745353385415 T^{10} + 6714334612785 T^{11} + 59119252710980 T^{12} + 483502155510094 T^{13} + 3885664317905650 T^{14} + 29308046091830080 T^{15} + 218118688184052547 T^{16} + 1533475608923910719 T^{17} + 10664691064970894514 T^{18} + 70283734691579405075 T^{19} + \)\(45\!\cdots\!02\)\( T^{20} + \)\(28\!\cdots\!31\)\( T^{21} + \)\(17\!\cdots\!29\)\( T^{22} + \)\(10\!\cdots\!80\)\( T^{23} + \)\(57\!\cdots\!50\)\( T^{24} + \)\(30\!\cdots\!06\)\( T^{25} + \)\(16\!\cdots\!60\)\( T^{26} + \)\(78\!\cdots\!85\)\( T^{27} + \)\(37\!\cdots\!45\)\( T^{28} + \)\(16\!\cdots\!49\)\( T^{29} + \)\(69\!\cdots\!60\)\( T^{30} + \)\(26\!\cdots\!29\)\( T^{31} + \)\(96\!\cdots\!48\)\( T^{32} + \)\(30\!\cdots\!47\)\( T^{33} + \)\(91\!\cdots\!18\)\( T^{34} + \)\(21\!\cdots\!86\)\( T^{35} + \)\(50\!\cdots\!94\)\( T^{36} + \)\(73\!\cdots\!21\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \))(\( 1 + 41 T + 1210 T^{2} + 26774 T^{3} + 501650 T^{4} + 8133183 T^{5} + 118151216 T^{6} + 1556708821 T^{7} + 18890009404 T^{8} + 212668330533 T^{9} + 2239932400711 T^{10} + 22175457407977 T^{11} + 207400542113396 T^{12} + 1838180140236470 T^{13} + 15488364055406778 T^{14} + 124314080562234228 T^{15} + 952381712403182927 T^{16} + 6972065687450028683 T^{17} + 48826921600417729150 T^{18} + \)\(32\!\cdots\!51\)\( T^{19} + \)\(20\!\cdots\!50\)\( T^{20} + \)\(12\!\cdots\!67\)\( T^{21} + \)\(75\!\cdots\!89\)\( T^{22} + \)\(42\!\cdots\!28\)\( T^{23} + \)\(22\!\cdots\!54\)\( T^{24} + \)\(11\!\cdots\!30\)\( T^{25} + \)\(56\!\cdots\!72\)\( T^{26} + \)\(25\!\cdots\!77\)\( T^{27} + \)\(11\!\cdots\!73\)\( T^{28} + \)\(45\!\cdots\!17\)\( T^{29} + \)\(17\!\cdots\!28\)\( T^{30} + \)\(62\!\cdots\!21\)\( T^{31} + \)\(20\!\cdots\!88\)\( T^{32} + \)\(60\!\cdots\!67\)\( T^{33} + \)\(15\!\cdots\!50\)\( T^{34} + \)\(36\!\cdots\!74\)\( T^{35} + \)\(71\!\cdots\!30\)\( T^{36} + \)\(10\!\cdots\!09\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \))(\( 1 - 31 T + 770 T^{2} - 14250 T^{3} + 232846 T^{4} - 3313929 T^{5} + 43236724 T^{6} - 516157019 T^{7} + 5761624152 T^{8} - 60184860147 T^{9} + 594844351815 T^{10} - 5569434951687 T^{11} + 49708064519100 T^{12} - 423344548101154 T^{13} + 3453444337482902 T^{14} - 27001656035570220 T^{15} + 202824684948195107 T^{16} - 1464043494695603541 T^{17} + 10169425300698959022 T^{18} - 67961201927745961273 T^{19} + \)\(43\!\cdots\!46\)\( T^{20} - \)\(27\!\cdots\!09\)\( T^{21} + \)\(16\!\cdots\!49\)\( T^{22} - \)\(92\!\cdots\!20\)\( T^{23} + \)\(50\!\cdots\!86\)\( T^{24} - \)\(26\!\cdots\!46\)\( T^{25} + \)\(13\!\cdots\!00\)\( T^{26} - \)\(65\!\cdots\!87\)\( T^{27} + \)\(29\!\cdots\!45\)\( T^{28} - \)\(13\!\cdots\!03\)\( T^{29} + \)\(53\!\cdots\!64\)\( T^{30} - \)\(20\!\cdots\!19\)\( T^{31} + \)\(74\!\cdots\!32\)\( T^{32} - \)\(24\!\cdots\!21\)\( T^{33} + \)\(73\!\cdots\!22\)\( T^{34} - \)\(19\!\cdots\!50\)\( T^{35} + \)\(45\!\cdots\!10\)\( T^{36} - \)\(78\!\cdots\!19\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \))(\( 1 - 35 T + 954 T^{2} - 18894 T^{3} + 323442 T^{4} - 4737025 T^{5} + 62766952 T^{6} - 751441507 T^{7} + 8345837428 T^{8} - 86011941579 T^{9} + 835298551487 T^{10} - 7646518450759 T^{11} + 66663821014436 T^{12} - 553566865926354 T^{13} + 4411465085385462 T^{14} - 33729694227154960 T^{15} + 248874620474012383 T^{16} - 1770516510180817145 T^{17} + 12197741015410559706 T^{18} - 81230467280486884041 T^{19} + \)\(52\!\cdots\!58\)\( T^{20} - \)\(32\!\cdots\!05\)\( T^{21} + \)\(19\!\cdots\!81\)\( T^{22} - \)\(11\!\cdots\!60\)\( T^{23} + \)\(64\!\cdots\!66\)\( T^{24} - \)\(34\!\cdots\!46\)\( T^{25} + \)\(18\!\cdots\!52\)\( T^{26} - \)\(89\!\cdots\!59\)\( T^{27} + \)\(41\!\cdots\!41\)\( T^{28} - \)\(18\!\cdots\!71\)\( T^{29} + \)\(77\!\cdots\!96\)\( T^{30} - \)\(30\!\cdots\!07\)\( T^{31} + \)\(10\!\cdots\!36\)\( T^{32} - \)\(34\!\cdots\!25\)\( T^{33} + \)\(10\!\cdots\!94\)\( T^{34} - \)\(25\!\cdots\!94\)\( T^{35} + \)\(56\!\cdots\!22\)\( T^{36} - \)\(88\!\cdots\!15\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 18 T + 666 T^{2} + 9650 T^{3} + 206558 T^{4} + 2518144 T^{5} + 40179900 T^{6} + 423574679 T^{7} + 5542510762 T^{8} + 51469199134 T^{9} + 580686584864 T^{10} + 4817669222555 T^{11} + 48420184578036 T^{12} + 363577064334165 T^{13} + 3338296852934765 T^{14} + 23009345399163790 T^{15} + 197102840745367302 T^{16} + 1266959817037630889 T^{17} + 10281425740961315411 T^{18} + 62525911803075996605 T^{19} + \)\(48\!\cdots\!17\)\( T^{20} + \)\(27\!\cdots\!01\)\( T^{21} + \)\(20\!\cdots\!46\)\( T^{22} + \)\(11\!\cdots\!90\)\( T^{23} + \)\(76\!\cdots\!55\)\( T^{24} + \)\(39\!\cdots\!85\)\( T^{25} + \)\(24\!\cdots\!68\)\( T^{26} + \)\(11\!\cdots\!55\)\( T^{27} + \)\(64\!\cdots\!88\)\( T^{28} + \)\(27\!\cdots\!66\)\( T^{29} + \)\(13\!\cdots\!86\)\( T^{30} + \)\(49\!\cdots\!39\)\( T^{31} + \)\(21\!\cdots\!00\)\( T^{32} + \)\(64\!\cdots\!36\)\( T^{33} + \)\(24\!\cdots\!94\)\( T^{34} + \)\(54\!\cdots\!50\)\( T^{35} + \)\(17\!\cdots\!42\)\( T^{36} + \)\(22\!\cdots\!02\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \))(\( 1 + 29 T + 909 T^{2} + 18088 T^{3} + 351005 T^{4} + 5488261 T^{5} + 82311862 T^{6} + 1078604038 T^{7} + 13540700542 T^{8} + 154198818287 T^{9} + 1685805736685 T^{10} + 17060978901367 T^{11} + 166174382371186 T^{12} + 1516221634325367 T^{13} + 13342302214277328 T^{14} + 110776535216503272 T^{15} + 888349952423380237 T^{16} + 6749274790948988982 T^{17} + 49571711840322407256 T^{18} + \)\(34\!\cdots\!10\)\( T^{19} + \)\(23\!\cdots\!32\)\( T^{20} + \)\(14\!\cdots\!38\)\( T^{21} + \)\(92\!\cdots\!51\)\( T^{22} + \)\(54\!\cdots\!32\)\( T^{23} + \)\(30\!\cdots\!96\)\( T^{24} + \)\(16\!\cdots\!43\)\( T^{25} + \)\(84\!\cdots\!18\)\( T^{26} + \)\(40\!\cdots\!87\)\( T^{27} + \)\(18\!\cdots\!95\)\( T^{28} + \)\(81\!\cdots\!63\)\( T^{29} + \)\(33\!\cdots\!26\)\( T^{30} + \)\(12\!\cdots\!58\)\( T^{31} + \)\(44\!\cdots\!74\)\( T^{32} + \)\(14\!\cdots\!09\)\( T^{33} + \)\(42\!\cdots\!15\)\( T^{34} + \)\(10\!\cdots\!48\)\( T^{35} + \)\(24\!\cdots\!83\)\( T^{36} + \)\(36\!\cdots\!81\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \))(\( 1 + 2 T + 366 T^{2} + 1026 T^{3} + 71918 T^{4} + 243560 T^{5} + 9990472 T^{6} + 38031931 T^{7} + 1091450730 T^{8} + 4453020166 T^{9} + 98973626512 T^{10} + 417811200027 T^{11} + 7680996145608 T^{12} + 32645312897749 T^{13} + 519536883528085 T^{14} + 2175175782699062 T^{15} + 30966478728434906 T^{16} + 125391924046243693 T^{17} + 1636684022058435491 T^{18} + 6303276874556039293 T^{19} + 76924149036746468077 T^{20} + \)\(27\!\cdots\!37\)\( T^{21} + \)\(32\!\cdots\!38\)\( T^{22} + \)\(10\!\cdots\!22\)\( T^{23} + \)\(11\!\cdots\!95\)\( T^{24} + \)\(35\!\cdots\!21\)\( T^{25} + \)\(38\!\cdots\!04\)\( T^{26} + \)\(99\!\cdots\!47\)\( T^{27} + \)\(11\!\cdots\!04\)\( T^{28} + \)\(23\!\cdots\!34\)\( T^{29} + \)\(26\!\cdots\!90\)\( T^{30} + \)\(44\!\cdots\!71\)\( T^{31} + \)\(54\!\cdots\!44\)\( T^{32} + \)\(62\!\cdots\!40\)\( T^{33} + \)\(86\!\cdots\!74\)\( T^{34} + \)\(58\!\cdots\!46\)\( T^{35} + \)\(97\!\cdots\!42\)\( T^{36} + \)\(25\!\cdots\!78\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \))(\( 1 - 23 T + 781 T^{2} - 14004 T^{3} + 283685 T^{4} - 4210783 T^{5} + 64965782 T^{6} - 828398730 T^{7} + 10626383294 T^{8} - 119248768445 T^{9} + 1327003503921 T^{10} - 13319016061017 T^{11} + 131672869848630 T^{12} - 1194963676863053 T^{13} + 10646823612570704 T^{14} - 87992845313829104 T^{15} + 712844481276829297 T^{16} - 5388563177263744362 T^{17} + 39898913142691893764 T^{18} - \)\(27\!\cdots\!78\)\( T^{19} + \)\(18\!\cdots\!08\)\( T^{20} - \)\(11\!\cdots\!58\)\( T^{21} + \)\(74\!\cdots\!31\)\( T^{22} - \)\(42\!\cdots\!24\)\( T^{23} + \)\(24\!\cdots\!28\)\( T^{24} - \)\(12\!\cdots\!37\)\( T^{25} + \)\(66\!\cdots\!90\)\( T^{26} - \)\(31\!\cdots\!37\)\( T^{27} + \)\(14\!\cdots\!07\)\( T^{28} - \)\(62\!\cdots\!05\)\( T^{29} + \)\(26\!\cdots\!82\)\( T^{30} - \)\(96\!\cdots\!30\)\( T^{31} + \)\(35\!\cdots\!14\)\( T^{32} - \)\(10\!\cdots\!27\)\( T^{33} + \)\(34\!\cdots\!55\)\( T^{34} - \)\(79\!\cdots\!84\)\( T^{35} + \)\(20\!\cdots\!47\)\( T^{36} - \)\(28\!\cdots\!47\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \))
$53$ (\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{19} \))(\( ( 1 - T )^{19} \))(\( ( 1 - T )^{19} \))(\( ( 1 + T )^{19} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 8 T + 439 T^{2} - 3376 T^{3} + 106905 T^{4} - 810403 T^{5} + 18516456 T^{6} - 137898318 T^{7} + 2528074250 T^{8} - 18417331250 T^{9} + 286624364276 T^{10} - 2027437902702 T^{11} + 27822335584309 T^{12} - 189840659111492 T^{13} + 2354967380396877 T^{14} - 15398813121369530 T^{15} + 175791498160172450 T^{16} - 1095273047560524135 T^{17} + 11648424653595690799 T^{18} - 68720820813605597428 T^{19} + \)\(68\!\cdots\!41\)\( T^{20} - \)\(38\!\cdots\!35\)\( T^{21} + \)\(36\!\cdots\!50\)\( T^{22} - \)\(18\!\cdots\!30\)\( T^{23} + \)\(16\!\cdots\!23\)\( T^{24} - \)\(80\!\cdots\!72\)\( T^{25} + \)\(69\!\cdots\!71\)\( T^{26} - \)\(29\!\cdots\!42\)\( T^{27} + \)\(24\!\cdots\!64\)\( T^{28} - \)\(94\!\cdots\!50\)\( T^{29} + \)\(76\!\cdots\!50\)\( T^{30} - \)\(24\!\cdots\!58\)\( T^{31} + \)\(19\!\cdots\!24\)\( T^{32} - \)\(50\!\cdots\!83\)\( T^{33} + \)\(39\!\cdots\!95\)\( T^{34} - \)\(72\!\cdots\!16\)\( T^{35} + \)\(55\!\cdots\!41\)\( T^{36} - \)\(60\!\cdots\!68\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \))(\( 1 + 42 T + 1359 T^{2} + 31896 T^{3} + 654747 T^{4} + 11517935 T^{5} + 185139300 T^{6} + 2692451456 T^{7} + 36557880014 T^{8} + 460650413958 T^{9} + 5487089427876 T^{10} + 61508874932730 T^{11} + 656823640582863 T^{12} + 6655930253502458 T^{13} + 64566281722336627 T^{14} + 597399001750638236 T^{15} + 5307312423594061832 T^{16} + 45105897961474589951 T^{17} + \)\(36\!\cdots\!47\)\( T^{18} + \)\(28\!\cdots\!68\)\( T^{19} + \)\(21\!\cdots\!73\)\( T^{20} + \)\(15\!\cdots\!31\)\( T^{21} + \)\(10\!\cdots\!28\)\( T^{22} + \)\(72\!\cdots\!96\)\( T^{23} + \)\(46\!\cdots\!73\)\( T^{24} + \)\(28\!\cdots\!78\)\( T^{25} + \)\(16\!\cdots\!97\)\( T^{26} + \)\(90\!\cdots\!30\)\( T^{27} + \)\(47\!\cdots\!64\)\( T^{28} + \)\(23\!\cdots\!58\)\( T^{29} + \)\(11\!\cdots\!26\)\( T^{30} + \)\(47\!\cdots\!36\)\( T^{31} + \)\(19\!\cdots\!00\)\( T^{32} + \)\(71\!\cdots\!35\)\( T^{33} + \)\(23\!\cdots\!53\)\( T^{34} + \)\(68\!\cdots\!36\)\( T^{35} + \)\(17\!\cdots\!21\)\( T^{36} + \)\(31\!\cdots\!82\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \))(\( 1 - 2 T + 661 T^{2} - 1142 T^{3} + 214939 T^{4} - 305339 T^{5} + 45981274 T^{6} - 50637978 T^{7} + 7298014794 T^{8} - 5766491364 T^{9} + 918215271432 T^{10} - 464291165456 T^{11} + 95465058154245 T^{12} - 25310096763064 T^{13} + 8432301805711733 T^{14} - 712099692206170 T^{15} + 644715878874295300 T^{16} + 15761554485197905 T^{17} + 43176417246101662375 T^{18} + 2480752384418781108 T^{19} + \)\(25\!\cdots\!25\)\( T^{20} + 54865971162973907305 T^{21} + \)\(13\!\cdots\!00\)\( T^{22} - \)\(86\!\cdots\!70\)\( T^{23} + \)\(60\!\cdots\!67\)\( T^{24} - \)\(10\!\cdots\!24\)\( T^{25} + \)\(23\!\cdots\!55\)\( T^{26} - \)\(68\!\cdots\!76\)\( T^{27} + \)\(79\!\cdots\!48\)\( T^{28} - \)\(29\!\cdots\!64\)\( T^{29} + \)\(22\!\cdots\!46\)\( T^{30} - \)\(90\!\cdots\!18\)\( T^{31} + \)\(48\!\cdots\!46\)\( T^{32} - \)\(18\!\cdots\!79\)\( T^{33} + \)\(78\!\cdots\!61\)\( T^{34} - \)\(24\!\cdots\!22\)\( T^{35} + \)\(84\!\cdots\!59\)\( T^{36} - \)\(15\!\cdots\!42\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \))(\( 1 - 16 T + 905 T^{2} - 13210 T^{3} + 397277 T^{4} - 5300777 T^{5} + 112327826 T^{6} - 1373778932 T^{7} + 22930111098 T^{8} - 257730507212 T^{9} + 3591829759940 T^{10} - 37178526571584 T^{11} + 447967953220387 T^{12} - 4275009274140354 T^{13} + 45548754458809699 T^{14} - 400768896903050404 T^{15} + 3833735193234708250 T^{16} - 31060520529758926913 T^{17} + \)\(26\!\cdots\!87\)\( T^{18} - \)\(20\!\cdots\!08\)\( T^{19} + \)\(15\!\cdots\!33\)\( T^{20} - \)\(10\!\cdots\!53\)\( T^{21} + \)\(78\!\cdots\!50\)\( T^{22} - \)\(48\!\cdots\!44\)\( T^{23} + \)\(32\!\cdots\!01\)\( T^{24} - \)\(18\!\cdots\!14\)\( T^{25} + \)\(11\!\cdots\!53\)\( T^{26} - \)\(54\!\cdots\!64\)\( T^{27} + \)\(31\!\cdots\!60\)\( T^{28} - \)\(13\!\cdots\!12\)\( T^{29} + \)\(69\!\cdots\!82\)\( T^{30} - \)\(24\!\cdots\!92\)\( T^{31} + \)\(11\!\cdots\!54\)\( T^{32} - \)\(32\!\cdots\!97\)\( T^{33} + \)\(14\!\cdots\!23\)\( T^{34} - \)\(28\!\cdots\!10\)\( T^{35} + \)\(11\!\cdots\!95\)\( T^{36} - \)\(12\!\cdots\!36\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \))
$61$ (\( 1 - 12 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 4 T + 424 T^{2} + 2555 T^{3} + 98895 T^{4} + 767607 T^{5} + 16880261 T^{6} + 150245826 T^{7} + 2337424867 T^{8} + 22021569456 T^{9} + 274315262265 T^{10} + 2598534111093 T^{11} + 27843166747504 T^{12} + 257051602313097 T^{13} + 2471417773598938 T^{14} + 21834134575167748 T^{15} + 193073306268690653 T^{16} + 1615834622789416828 T^{17} + 13319194916164150992 T^{18} + \)\(10\!\cdots\!68\)\( T^{19} + \)\(81\!\cdots\!12\)\( T^{20} + \)\(60\!\cdots\!88\)\( T^{21} + \)\(43\!\cdots\!93\)\( T^{22} + \)\(30\!\cdots\!68\)\( T^{23} + \)\(20\!\cdots\!38\)\( T^{24} + \)\(13\!\cdots\!17\)\( T^{25} + \)\(87\!\cdots\!84\)\( T^{26} + \)\(49\!\cdots\!33\)\( T^{27} + \)\(32\!\cdots\!65\)\( T^{28} + \)\(15\!\cdots\!56\)\( T^{29} + \)\(10\!\cdots\!87\)\( T^{30} + \)\(39\!\cdots\!46\)\( T^{31} + \)\(27\!\cdots\!41\)\( T^{32} + \)\(75\!\cdots\!87\)\( T^{33} + \)\(59\!\cdots\!95\)\( T^{34} + \)\(93\!\cdots\!55\)\( T^{35} + \)\(95\!\cdots\!04\)\( T^{36} + \)\(54\!\cdots\!24\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \))(\( 1 - T + 517 T^{2} + 241 T^{3} + 133255 T^{4} + 245250 T^{5} + 23188299 T^{6} + 70944406 T^{7} + 3102873365 T^{8} + 12517695983 T^{9} + 343125143088 T^{10} + 1609447575084 T^{11} + 32729710941875 T^{12} + 163931903376839 T^{13} + 2760204624527518 T^{14} + 13916645083482646 T^{15} + 208309095297321825 T^{16} + 1019676457461419921 T^{17} + 14128281306980348077 T^{18} + 65943742910952541230 T^{19} + \)\(86\!\cdots\!97\)\( T^{20} + \)\(37\!\cdots\!41\)\( T^{21} + \)\(47\!\cdots\!25\)\( T^{22} + \)\(19\!\cdots\!86\)\( T^{23} + \)\(23\!\cdots\!18\)\( T^{24} + \)\(84\!\cdots\!79\)\( T^{25} + \)\(10\!\cdots\!75\)\( T^{26} + \)\(30\!\cdots\!04\)\( T^{27} + \)\(40\!\cdots\!08\)\( T^{28} + \)\(89\!\cdots\!83\)\( T^{29} + \)\(13\!\cdots\!65\)\( T^{30} + \)\(18\!\cdots\!26\)\( T^{31} + \)\(37\!\cdots\!19\)\( T^{32} + \)\(24\!\cdots\!50\)\( T^{33} + \)\(80\!\cdots\!55\)\( T^{34} + \)\(88\!\cdots\!01\)\( T^{35} + \)\(11\!\cdots\!57\)\( T^{36} - \)\(13\!\cdots\!81\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \))(\( 1 - 6 T + 730 T^{2} - 3103 T^{3} + 258891 T^{4} - 752107 T^{5} + 60439941 T^{6} - 112145228 T^{7} + 10512133767 T^{8} - 10968674890 T^{9} + 1452133794129 T^{10} - 631445441533 T^{11} + 165313031593244 T^{12} - 961733898689 T^{13} + 15861258225658508 T^{14} + 4207868867472626 T^{15} + 1300234935910377787 T^{16} + 520350373823388140 T^{17} + 91781172393823906530 T^{18} + 38566348795081569132 T^{19} + \)\(55\!\cdots\!30\)\( T^{20} + \)\(19\!\cdots\!40\)\( T^{21} + \)\(29\!\cdots\!47\)\( T^{22} + \)\(58\!\cdots\!66\)\( T^{23} + \)\(13\!\cdots\!08\)\( T^{24} - \)\(49\!\cdots\!29\)\( T^{25} + \)\(51\!\cdots\!24\)\( T^{26} - \)\(12\!\cdots\!73\)\( T^{27} + \)\(16\!\cdots\!89\)\( T^{28} - \)\(78\!\cdots\!90\)\( T^{29} + \)\(45\!\cdots\!87\)\( T^{30} - \)\(29\!\cdots\!88\)\( T^{31} + \)\(97\!\cdots\!21\)\( T^{32} - \)\(74\!\cdots\!87\)\( T^{33} + \)\(15\!\cdots\!91\)\( T^{34} - \)\(11\!\cdots\!83\)\( T^{35} + \)\(16\!\cdots\!30\)\( T^{36} - \)\(82\!\cdots\!86\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \))(\( 1 - 21 T + 839 T^{2} - 13585 T^{3} + 316479 T^{4} - 4234046 T^{5} + 74370149 T^{6} - 859179658 T^{7} + 12577990259 T^{8} - 129431723489 T^{9} + 1664058662278 T^{10} - 15584266891504 T^{11} + 181229523565747 T^{12} - 1565787424356325 T^{13} + 16744777586200170 T^{14} - 134517790407016746 T^{15} + 1335568797477828429 T^{16} - 10021974942455962271 T^{17} + 92912594859447132645 T^{18} - \)\(65\!\cdots\!02\)\( T^{19} + \)\(56\!\cdots\!45\)\( T^{20} - \)\(37\!\cdots\!91\)\( T^{21} + \)\(30\!\cdots\!49\)\( T^{22} - \)\(18\!\cdots\!86\)\( T^{23} + \)\(14\!\cdots\!70\)\( T^{24} - \)\(80\!\cdots\!25\)\( T^{25} + \)\(56\!\cdots\!87\)\( T^{26} - \)\(29\!\cdots\!24\)\( T^{27} + \)\(19\!\cdots\!98\)\( T^{28} - \)\(92\!\cdots\!89\)\( T^{29} + \)\(54\!\cdots\!99\)\( T^{30} - \)\(22\!\cdots\!18\)\( T^{31} + \)\(12\!\cdots\!69\)\( T^{32} - \)\(41\!\cdots\!86\)\( T^{33} + \)\(19\!\cdots\!79\)\( T^{34} - \)\(49\!\cdots\!85\)\( T^{35} + \)\(18\!\cdots\!19\)\( T^{36} - \)\(28\!\cdots\!01\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \))
$67$ (\( 1 + T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 62 T + 2348 T^{2} + 67216 T^{3} + 1595195 T^{4} + 32768119 T^{5} + 599600687 T^{6} + 9951131772 T^{7} + 151787041838 T^{8} + 2148077790292 T^{9} + 28413860236575 T^{10} + 353258562257633 T^{11} + 4146511451436474 T^{12} + 46111085809937628 T^{13} + 487163318592836631 T^{14} + 4900277504170737741 T^{15} + 47008614950398723604 T^{16} + \)\(43\!\cdots\!35\)\( T^{17} + \)\(37\!\cdots\!99\)\( T^{18} + \)\(31\!\cdots\!12\)\( T^{19} + \)\(25\!\cdots\!33\)\( T^{20} + \)\(19\!\cdots\!15\)\( T^{21} + \)\(14\!\cdots\!52\)\( T^{22} + \)\(98\!\cdots\!61\)\( T^{23} + \)\(65\!\cdots\!17\)\( T^{24} + \)\(41\!\cdots\!32\)\( T^{25} + \)\(25\!\cdots\!02\)\( T^{26} + \)\(14\!\cdots\!53\)\( T^{27} + \)\(77\!\cdots\!25\)\( T^{28} + \)\(39\!\cdots\!08\)\( T^{29} + \)\(18\!\cdots\!54\)\( T^{30} + \)\(81\!\cdots\!92\)\( T^{31} + \)\(32\!\cdots\!69\)\( T^{32} + \)\(12\!\cdots\!51\)\( T^{33} + \)\(39\!\cdots\!85\)\( T^{34} + \)\(11\!\cdots\!96\)\( T^{35} + \)\(25\!\cdots\!96\)\( T^{36} + \)\(45\!\cdots\!58\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \))(\( 1 + 41 T + 1396 T^{2} + 32847 T^{3} + 682311 T^{4} + 11742555 T^{5} + 184509800 T^{6} + 2556216527 T^{7} + 33066576689 T^{8} + 389631495081 T^{9} + 4366480554048 T^{10} + 45585580889627 T^{11} + 460355414139920 T^{12} + 4406910850541114 T^{13} + 41392844964425483 T^{14} + 372892643057838574 T^{15} + 3324013680677058857 T^{16} + 28558548940799366476 T^{17} + \)\(24\!\cdots\!83\)\( T^{18} + \)\(20\!\cdots\!64\)\( T^{19} + \)\(16\!\cdots\!61\)\( T^{20} + \)\(12\!\cdots\!64\)\( T^{21} + \)\(99\!\cdots\!91\)\( T^{22} + \)\(75\!\cdots\!54\)\( T^{23} + \)\(55\!\cdots\!81\)\( T^{24} + \)\(39\!\cdots\!66\)\( T^{25} + \)\(27\!\cdots\!60\)\( T^{26} + \)\(18\!\cdots\!07\)\( T^{27} + \)\(11\!\cdots\!56\)\( T^{28} + \)\(71\!\cdots\!69\)\( T^{29} + \)\(40\!\cdots\!87\)\( T^{30} + \)\(20\!\cdots\!47\)\( T^{31} + \)\(10\!\cdots\!00\)\( T^{32} + \)\(43\!\cdots\!95\)\( T^{33} + \)\(16\!\cdots\!73\)\( T^{34} + \)\(54\!\cdots\!07\)\( T^{35} + \)\(15\!\cdots\!92\)\( T^{36} + \)\(30\!\cdots\!69\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \))(\( 1 - 50 T + 1930 T^{2} - 54358 T^{3} + 1318333 T^{4} - 27382915 T^{5} + 512360843 T^{6} - 8648460352 T^{7} + 134663135678 T^{8} - 1938311033424 T^{9} + 26100099298707 T^{10} - 329330075379945 T^{11} + 3921942568126972 T^{12} - 44128155709736096 T^{13} + 471276051201080851 T^{14} - 4779576808254686207 T^{15} + 46172962226234329022 T^{16} - \)\(42\!\cdots\!11\)\( T^{17} + \)\(37\!\cdots\!63\)\( T^{18} - \)\(31\!\cdots\!12\)\( T^{19} + \)\(24\!\cdots\!21\)\( T^{20} - \)\(19\!\cdots\!79\)\( T^{21} + \)\(13\!\cdots\!86\)\( T^{22} - \)\(96\!\cdots\!47\)\( T^{23} + \)\(63\!\cdots\!57\)\( T^{24} - \)\(39\!\cdots\!24\)\( T^{25} + \)\(23\!\cdots\!56\)\( T^{26} - \)\(13\!\cdots\!45\)\( T^{27} + \)\(71\!\cdots\!29\)\( T^{28} - \)\(35\!\cdots\!76\)\( T^{29} + \)\(16\!\cdots\!74\)\( T^{30} - \)\(70\!\cdots\!72\)\( T^{31} + \)\(28\!\cdots\!41\)\( T^{32} - \)\(10\!\cdots\!35\)\( T^{33} + \)\(32\!\cdots\!19\)\( T^{34} - \)\(89\!\cdots\!98\)\( T^{35} + \)\(21\!\cdots\!10\)\( T^{36} - \)\(37\!\cdots\!50\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \))(\( 1 - 67 T + 2846 T^{2} - 90129 T^{3} + 2349623 T^{4} - 52516691 T^{5} + 1037575644 T^{6} - 18445245285 T^{7} + 299267743445 T^{8} - 4474285702675 T^{9} + 62133476280632 T^{10} - 806084748639787 T^{11} + 9817509036040056 T^{12} - 112661261577607220 T^{13} + 1221899920120837581 T^{14} - 12553918213477095012 T^{15} + \)\(12\!\cdots\!01\)\( T^{16} - \)\(11\!\cdots\!98\)\( T^{17} + \)\(99\!\cdots\!47\)\( T^{18} - \)\(83\!\cdots\!88\)\( T^{19} + \)\(66\!\cdots\!49\)\( T^{20} - \)\(50\!\cdots\!22\)\( T^{21} + \)\(36\!\cdots\!63\)\( T^{22} - \)\(25\!\cdots\!52\)\( T^{23} + \)\(16\!\cdots\!67\)\( T^{24} - \)\(10\!\cdots\!80\)\( T^{25} + \)\(59\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!67\)\( T^{27} + \)\(16\!\cdots\!04\)\( T^{28} - \)\(81\!\cdots\!75\)\( T^{29} + \)\(36\!\cdots\!35\)\( T^{30} - \)\(15\!\cdots\!85\)\( T^{31} + \)\(56\!\cdots\!28\)\( T^{32} - \)\(19\!\cdots\!39\)\( T^{33} + \)\(57\!\cdots\!89\)\( T^{34} - \)\(14\!\cdots\!49\)\( T^{35} + \)\(31\!\cdots\!42\)\( T^{36} - \)\(49\!\cdots\!03\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \))
$71$ (\( 1 + 15 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 17 T + 665 T^{2} + 9693 T^{3} + 215758 T^{4} + 2758548 T^{5} + 45171032 T^{6} + 514544303 T^{7} + 6846102533 T^{8} + 70404798631 T^{9} + 803558128769 T^{10} + 7558948749571 T^{11} + 76947037580518 T^{12} + 672410090262301 T^{13} + 6325266603277196 T^{14} + 52379619563293124 T^{15} + 471772899456128691 T^{16} + 3792843265491037097 T^{17} + 33715761129197145779 T^{18} + \)\(26\!\cdots\!82\)\( T^{19} + \)\(23\!\cdots\!09\)\( T^{20} + \)\(19\!\cdots\!77\)\( T^{21} + \)\(16\!\cdots\!01\)\( T^{22} + \)\(13\!\cdots\!44\)\( T^{23} + \)\(11\!\cdots\!96\)\( T^{24} + \)\(86\!\cdots\!21\)\( T^{25} + \)\(69\!\cdots\!38\)\( T^{26} + \)\(48\!\cdots\!31\)\( T^{27} + \)\(36\!\cdots\!39\)\( T^{28} + \)\(22\!\cdots\!31\)\( T^{29} + \)\(15\!\cdots\!43\)\( T^{30} + \)\(84\!\cdots\!23\)\( T^{31} + \)\(52\!\cdots\!52\)\( T^{32} + \)\(22\!\cdots\!88\)\( T^{33} + \)\(12\!\cdots\!58\)\( T^{34} + \)\(40\!\cdots\!53\)\( T^{35} + \)\(19\!\cdots\!15\)\( T^{36} + \)\(35\!\cdots\!37\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \))(\( 1 + 902 T^{2} + 1085 T^{3} + 392410 T^{4} + 949536 T^{5} + 110022826 T^{6} + 400348787 T^{7} + 22430153668 T^{8} + 108208741832 T^{9} + 3558817249441 T^{10} + 21038930940961 T^{11} + 459400689288747 T^{12} + 3127977931159775 T^{13} + 49778932803940384 T^{14} + 368691409348339198 T^{15} + 4628758875972824499 T^{16} + 35206980919949657196 T^{17} + \)\(37\!\cdots\!96\)\( T^{18} + \)\(27\!\cdots\!60\)\( T^{19} + \)\(26\!\cdots\!16\)\( T^{20} + \)\(17\!\cdots\!36\)\( T^{21} + \)\(16\!\cdots\!89\)\( T^{22} + \)\(93\!\cdots\!38\)\( T^{23} + \)\(89\!\cdots\!84\)\( T^{24} + \)\(40\!\cdots\!75\)\( T^{25} + \)\(41\!\cdots\!77\)\( T^{26} + \)\(13\!\cdots\!21\)\( T^{27} + \)\(16\!\cdots\!71\)\( T^{28} + \)\(35\!\cdots\!32\)\( T^{29} + \)\(51\!\cdots\!28\)\( T^{30} + \)\(65\!\cdots\!67\)\( T^{31} + \)\(12\!\cdots\!86\)\( T^{32} + \)\(78\!\cdots\!16\)\( T^{33} + \)\(23\!\cdots\!10\)\( T^{34} + \)\(45\!\cdots\!85\)\( T^{35} + \)\(26\!\cdots\!82\)\( T^{36} + \)\(14\!\cdots\!31\)\( T^{38} \))(\( 1 - 25 T + 1151 T^{2} - 23543 T^{3} + 616778 T^{4} - 10733796 T^{5} + 207883638 T^{6} - 3159143413 T^{7} + 49947509223 T^{8} - 674990799697 T^{9} + 9161629089779 T^{10} - 111551162045159 T^{11} + 1338337717662228 T^{12} - 14819991866581577 T^{13} + 160029696550631304 T^{14} - 1621983887344868718 T^{15} + 15942925977999490817 T^{16} - \)\(14\!\cdots\!15\)\( T^{17} + \)\(13\!\cdots\!23\)\( T^{18} - \)\(11\!\cdots\!78\)\( T^{19} + \)\(94\!\cdots\!33\)\( T^{20} - \)\(74\!\cdots\!15\)\( T^{21} + \)\(57\!\cdots\!87\)\( T^{22} - \)\(41\!\cdots\!58\)\( T^{23} + \)\(28\!\cdots\!04\)\( T^{24} - \)\(18\!\cdots\!17\)\( T^{25} + \)\(12\!\cdots\!48\)\( T^{26} - \)\(72\!\cdots\!99\)\( T^{27} + \)\(42\!\cdots\!49\)\( T^{28} - \)\(21\!\cdots\!97\)\( T^{29} + \)\(11\!\cdots\!33\)\( T^{30} - \)\(51\!\cdots\!33\)\( T^{31} + \)\(24\!\cdots\!18\)\( T^{32} - \)\(88\!\cdots\!76\)\( T^{33} + \)\(36\!\cdots\!78\)\( T^{34} - \)\(98\!\cdots\!03\)\( T^{35} + \)\(34\!\cdots\!41\)\( T^{36} - \)\(52\!\cdots\!25\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \))(\( 1 - 18 T + 906 T^{2} - 14889 T^{3} + 406256 T^{4} - 6014082 T^{5} + 118541242 T^{6} - 1578367353 T^{7} + 25126376316 T^{8} - 302260603422 T^{9} + 4114015579565 T^{10} - 45037195327141 T^{11} + 542266907609805 T^{12} - 5447684281749189 T^{13} + 59373043179021998 T^{14} - 552057134956144870 T^{15} + 5537214361279434613 T^{16} - 48007184454316797644 T^{17} + \)\(44\!\cdots\!12\)\( T^{18} - \)\(36\!\cdots\!72\)\( T^{19} + \)\(31\!\cdots\!52\)\( T^{20} - \)\(24\!\cdots\!04\)\( T^{21} + \)\(19\!\cdots\!43\)\( T^{22} - \)\(14\!\cdots\!70\)\( T^{23} + \)\(10\!\cdots\!98\)\( T^{24} - \)\(69\!\cdots\!69\)\( T^{25} + \)\(49\!\cdots\!55\)\( T^{26} - \)\(29\!\cdots\!01\)\( T^{27} + \)\(18\!\cdots\!15\)\( T^{28} - \)\(98\!\cdots\!22\)\( T^{29} + \)\(58\!\cdots\!36\)\( T^{30} - \)\(25\!\cdots\!73\)\( T^{31} + \)\(13\!\cdots\!62\)\( T^{32} - \)\(49\!\cdots\!42\)\( T^{33} + \)\(23\!\cdots\!56\)\( T^{34} - \)\(62\!\cdots\!69\)\( T^{35} + \)\(26\!\cdots\!46\)\( T^{36} - \)\(37\!\cdots\!98\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - T + 249 T^{2} + 1794 T^{3} + 42898 T^{4} + 402745 T^{5} + 8030505 T^{6} + 64123645 T^{7} + 1109240971 T^{8} + 9504993966 T^{9} + 131267935795 T^{10} + 1120605908016 T^{11} + 14271910376379 T^{12} + 114526691802619 T^{13} + 1367827149743843 T^{14} + 10702326924832173 T^{15} + 118120519508558708 T^{16} + 896546024321723199 T^{17} + 9437512670921213859 T^{18} + 68066337291903400968 T^{19} + \)\(68\!\cdots\!07\)\( T^{20} + \)\(47\!\cdots\!71\)\( T^{21} + \)\(45\!\cdots\!36\)\( T^{22} + \)\(30\!\cdots\!93\)\( T^{23} + \)\(28\!\cdots\!99\)\( T^{24} + \)\(17\!\cdots\!91\)\( T^{25} + \)\(15\!\cdots\!63\)\( T^{26} + \)\(90\!\cdots\!96\)\( T^{27} + \)\(77\!\cdots\!35\)\( T^{28} + \)\(40\!\cdots\!34\)\( T^{29} + \)\(34\!\cdots\!67\)\( T^{30} + \)\(14\!\cdots\!45\)\( T^{31} + \)\(13\!\cdots\!65\)\( T^{32} + \)\(49\!\cdots\!05\)\( T^{33} + \)\(38\!\cdots\!86\)\( T^{34} + \)\(11\!\cdots\!34\)\( T^{35} + \)\(11\!\cdots\!97\)\( T^{36} - \)\(34\!\cdots\!69\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \))(\( 1 + 20 T + 1035 T^{2} + 17620 T^{3} + 517098 T^{4} + 7690145 T^{5} + 166572834 T^{6} + 2205603797 T^{7} + 38957981169 T^{8} + 465597466491 T^{9} + 7053722838370 T^{10} + 76835565092557 T^{11} + 1028265501509793 T^{12} + 10278623906882763 T^{13} + 123784113041917842 T^{14} + 1140483585764195683 T^{15} + 12509408888531126726 T^{16} + \)\(10\!\cdots\!53\)\( T^{17} + \)\(10\!\cdots\!84\)\( T^{18} + \)\(84\!\cdots\!70\)\( T^{19} + \)\(78\!\cdots\!32\)\( T^{20} + \)\(56\!\cdots\!37\)\( T^{21} + \)\(48\!\cdots\!42\)\( T^{22} + \)\(32\!\cdots\!03\)\( T^{23} + \)\(25\!\cdots\!06\)\( T^{24} + \)\(15\!\cdots\!07\)\( T^{25} + \)\(11\!\cdots\!21\)\( T^{26} + \)\(61\!\cdots\!17\)\( T^{27} + \)\(41\!\cdots\!10\)\( T^{28} + \)\(20\!\cdots\!59\)\( T^{29} + \)\(12\!\cdots\!13\)\( T^{30} + \)\(50\!\cdots\!37\)\( T^{31} + \)\(27\!\cdots\!22\)\( T^{32} + \)\(93\!\cdots\!05\)\( T^{33} + \)\(46\!\cdots\!86\)\( T^{34} + \)\(11\!\cdots\!20\)\( T^{35} + \)\(49\!\cdots\!55\)\( T^{36} + \)\(69\!\cdots\!80\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \))(\( 1 + 5 T + 887 T^{2} + 5088 T^{3} + 384860 T^{4} + 2449143 T^{5} + 109228233 T^{6} + 749665533 T^{7} + 22860465591 T^{8} + 165029417436 T^{9} + 3767338577767 T^{10} + 27951156094122 T^{11} + 509053989714855 T^{12} + 3796956951404607 T^{13} + 57891161784002931 T^{14} + 424884406086319905 T^{15} + 5633787223837469216 T^{16} + 39829785192139786425 T^{17} + \)\(47\!\cdots\!91\)\( T^{18} + \)\(31\!\cdots\!24\)\( T^{19} + \)\(34\!\cdots\!43\)\( T^{20} + \)\(21\!\cdots\!25\)\( T^{21} + \)\(21\!\cdots\!72\)\( T^{22} + \)\(12\!\cdots\!05\)\( T^{23} + \)\(12\!\cdots\!83\)\( T^{24} + \)\(57\!\cdots\!23\)\( T^{25} + \)\(56\!\cdots\!35\)\( T^{26} + \)\(22\!\cdots\!82\)\( T^{27} + \)\(22\!\cdots\!71\)\( T^{28} + \)\(70\!\cdots\!64\)\( T^{29} + \)\(71\!\cdots\!07\)\( T^{30} + \)\(17\!\cdots\!93\)\( T^{31} + \)\(18\!\cdots\!89\)\( T^{32} + \)\(29\!\cdots\!87\)\( T^{33} + \)\(34\!\cdots\!20\)\( T^{34} + \)\(33\!\cdots\!68\)\( T^{35} + \)\(42\!\cdots\!11\)\( T^{36} + \)\(17\!\cdots\!45\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \))(\( 1 + 20 T + 943 T^{2} + 15234 T^{3} + 409076 T^{4} + 5679105 T^{5} + 112473200 T^{6} + 1395113399 T^{7} + 22439085615 T^{8} + 255101250047 T^{9} + 3497073802404 T^{10} + 37072240763833 T^{11} + 446016235651199 T^{12} + 4460399752203137 T^{13} + 48097348246279552 T^{14} + 456740405150050021 T^{15} + 4491469549272609188 T^{16} + 40549980487509398251 T^{17} + \)\(36\!\cdots\!66\)\( T^{18} + \)\(31\!\cdots\!94\)\( T^{19} + \)\(26\!\cdots\!18\)\( T^{20} + \)\(21\!\cdots\!79\)\( T^{21} + \)\(17\!\cdots\!96\)\( T^{22} + \)\(12\!\cdots\!61\)\( T^{23} + \)\(99\!\cdots\!36\)\( T^{24} + \)\(67\!\cdots\!93\)\( T^{25} + \)\(49\!\cdots\!03\)\( T^{26} + \)\(29\!\cdots\!73\)\( T^{27} + \)\(20\!\cdots\!52\)\( T^{28} + \)\(10\!\cdots\!03\)\( T^{29} + \)\(70\!\cdots\!55\)\( T^{30} + \)\(31\!\cdots\!79\)\( T^{31} + \)\(18\!\cdots\!00\)\( T^{32} + \)\(69\!\cdots\!45\)\( T^{33} + \)\(36\!\cdots\!32\)\( T^{34} + \)\(99\!\cdots\!74\)\( T^{35} + \)\(44\!\cdots\!79\)\( T^{36} + \)\(69\!\cdots\!80\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 28 T + 647 T^{2} + 12445 T^{3} + 225138 T^{4} + 3649718 T^{5} + 55305593 T^{6} + 786079505 T^{7} + 10641987542 T^{8} + 136875771508 T^{9} + 1685200947619 T^{10} + 19914329339471 T^{11} + 226641208931544 T^{12} + 2486101418783668 T^{13} + 26334694194328281 T^{14} + 269835700835124903 T^{15} + 2676481014424375878 T^{16} + 25716191293856836690 T^{17} + \)\(23\!\cdots\!82\)\( T^{18} + \)\(21\!\cdots\!94\)\( T^{19} + \)\(18\!\cdots\!78\)\( T^{20} + \)\(16\!\cdots\!90\)\( T^{21} + \)\(13\!\cdots\!42\)\( T^{22} + \)\(10\!\cdots\!43\)\( T^{23} + \)\(81\!\cdots\!19\)\( T^{24} + \)\(60\!\cdots\!28\)\( T^{25} + \)\(43\!\cdots\!96\)\( T^{26} + \)\(30\!\cdots\!31\)\( T^{27} + \)\(20\!\cdots\!61\)\( T^{28} + \)\(12\!\cdots\!08\)\( T^{29} + \)\(79\!\cdots\!18\)\( T^{30} + \)\(46\!\cdots\!05\)\( T^{31} + \)\(25\!\cdots\!27\)\( T^{32} + \)\(13\!\cdots\!58\)\( T^{33} + \)\(65\!\cdots\!62\)\( T^{34} + \)\(28\!\cdots\!45\)\( T^{35} + \)\(11\!\cdots\!73\)\( T^{36} + \)\(40\!\cdots\!08\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \))(\( 1 + 38 T + 1823 T^{2} + 49357 T^{3} + 1409588 T^{4} + 30186522 T^{5} + 649197903 T^{6} + 11607780631 T^{7} + 204826709148 T^{8} + 3157990204334 T^{9} + 47727241601601 T^{10} + 647838665381407 T^{11} + 8597864411050786 T^{12} + 104185240649215988 T^{13} + 1233381949003359879 T^{14} + 13466476286165832589 T^{15} + \)\(14\!\cdots\!78\)\( T^{16} + \)\(14\!\cdots\!96\)\( T^{17} + \)\(13\!\cdots\!94\)\( T^{18} + \)\(12\!\cdots\!36\)\( T^{19} + \)\(10\!\cdots\!26\)\( T^{20} + \)\(88\!\cdots\!36\)\( T^{21} + \)\(70\!\cdots\!42\)\( T^{22} + \)\(52\!\cdots\!09\)\( T^{23} + \)\(37\!\cdots\!21\)\( T^{24} + \)\(25\!\cdots\!48\)\( T^{25} + \)\(16\!\cdots\!74\)\( T^{26} + \)\(98\!\cdots\!27\)\( T^{27} + \)\(57\!\cdots\!19\)\( T^{28} + \)\(29\!\cdots\!34\)\( T^{29} + \)\(15\!\cdots\!92\)\( T^{30} + \)\(68\!\cdots\!71\)\( T^{31} + \)\(30\!\cdots\!17\)\( T^{32} + \)\(11\!\cdots\!82\)\( T^{33} + \)\(41\!\cdots\!12\)\( T^{34} + \)\(11\!\cdots\!97\)\( T^{35} + \)\(33\!\cdots\!57\)\( T^{36} + \)\(54\!\cdots\!18\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \))(\( 1 - 36 T + 1383 T^{2} - 34797 T^{3} + 827558 T^{4} - 16337032 T^{5} + 301461143 T^{6} - 4969915953 T^{7} + 77080156936 T^{8} - 1102564902374 T^{9} + 14956852929515 T^{10} - 190359953020107 T^{11} + 2313225094645024 T^{12} - 26652254411132986 T^{13} + 294724135527805651 T^{14} - 3111174973747133291 T^{15} + 31639113352367117696 T^{16} - \)\(30\!\cdots\!42\)\( T^{17} + \)\(29\!\cdots\!04\)\( T^{18} - \)\(26\!\cdots\!34\)\( T^{19} + \)\(22\!\cdots\!16\)\( T^{20} - \)\(19\!\cdots\!22\)\( T^{21} + \)\(15\!\cdots\!44\)\( T^{22} - \)\(12\!\cdots\!71\)\( T^{23} + \)\(90\!\cdots\!49\)\( T^{24} - \)\(64\!\cdots\!06\)\( T^{25} + \)\(44\!\cdots\!16\)\( T^{26} - \)\(28\!\cdots\!27\)\( T^{27} + \)\(17\!\cdots\!85\)\( T^{28} - \)\(10\!\cdots\!74\)\( T^{29} + \)\(57\!\cdots\!44\)\( T^{30} - \)\(29\!\cdots\!73\)\( T^{31} + \)\(14\!\cdots\!77\)\( T^{32} - \)\(60\!\cdots\!92\)\( T^{33} + \)\(24\!\cdots\!42\)\( T^{34} - \)\(80\!\cdots\!37\)\( T^{35} + \)\(25\!\cdots\!97\)\( T^{36} - \)\(51\!\cdots\!96\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \))(\( 1 - 2 T + 775 T^{2} - 1857 T^{3} + 305340 T^{4} - 832976 T^{5} + 81432501 T^{6} - 243666707 T^{7} + 16496653450 T^{8} - 52566238716 T^{9} + 2697737117585 T^{10} - 8932775659003 T^{11} + 369339134809314 T^{12} - 1243340561727782 T^{13} + 43329568648050313 T^{14} - 145167774471731117 T^{15} + 4422035364034371284 T^{16} - 14422868094603165828 T^{17} + \)\(39\!\cdots\!92\)\( T^{18} - \)\(12\!\cdots\!08\)\( T^{19} + \)\(31\!\cdots\!68\)\( T^{20} - \)\(90\!\cdots\!48\)\( T^{21} + \)\(21\!\cdots\!76\)\( T^{22} - \)\(56\!\cdots\!77\)\( T^{23} + \)\(13\!\cdots\!87\)\( T^{24} - \)\(30\!\cdots\!22\)\( T^{25} + \)\(70\!\cdots\!26\)\( T^{26} - \)\(13\!\cdots\!83\)\( T^{27} + \)\(32\!\cdots\!15\)\( T^{28} - \)\(49\!\cdots\!16\)\( T^{29} + \)\(12\!\cdots\!50\)\( T^{30} - \)\(14\!\cdots\!87\)\( T^{31} + \)\(38\!\cdots\!39\)\( T^{32} - \)\(30\!\cdots\!56\)\( T^{33} + \)\(88\!\cdots\!60\)\( T^{34} - \)\(42\!\cdots\!97\)\( T^{35} + \)\(14\!\cdots\!25\)\( T^{36} - \)\(28\!\cdots\!22\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 804 T^{2} - 4322 T^{3} + 340016 T^{4} - 2116335 T^{5} + 99563315 T^{6} - 666494144 T^{7} + 22383096083 T^{8} - 154305917895 T^{9} + 4070142997588 T^{10} - 28081098862659 T^{11} + 616918332678019 T^{12} - 4172884757288833 T^{13} + 79403801537398328 T^{14} - 518088264160671436 T^{15} + 8779967093170503223 T^{16} - 54486510390147467109 T^{17} + \)\(83\!\cdots\!51\)\( T^{18} - \)\(48\!\cdots\!22\)\( T^{19} + \)\(69\!\cdots\!33\)\( T^{20} - \)\(37\!\cdots\!01\)\( T^{21} + \)\(50\!\cdots\!01\)\( T^{22} - \)\(24\!\cdots\!56\)\( T^{23} + \)\(31\!\cdots\!04\)\( T^{24} - \)\(13\!\cdots\!77\)\( T^{25} + \)\(16\!\cdots\!13\)\( T^{26} - \)\(63\!\cdots\!19\)\( T^{27} + \)\(76\!\cdots\!64\)\( T^{28} - \)\(23\!\cdots\!55\)\( T^{29} + \)\(28\!\cdots\!61\)\( T^{30} - \)\(71\!\cdots\!84\)\( T^{31} + \)\(88\!\cdots\!45\)\( T^{32} - \)\(15\!\cdots\!15\)\( T^{33} + \)\(20\!\cdots\!12\)\( T^{34} - \)\(21\!\cdots\!82\)\( T^{35} + \)\(33\!\cdots\!92\)\( T^{36} - \)\(13\!\cdots\!36\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \))(\( 1 + 36 T + 1320 T^{2} + 31176 T^{3} + 708266 T^{4} + 13095311 T^{5} + 231851717 T^{6} + 3632606314 T^{7} + 54604228491 T^{8} + 756432311401 T^{9} + 10071111067524 T^{10} + 126365511530507 T^{11} + 1525307869152585 T^{12} + 17581725050697915 T^{13} + 195062126290088254 T^{14} + 2083294213860253602 T^{15} + 21423143737594143167 T^{16} + \)\(21\!\cdots\!09\)\( T^{17} + \)\(20\!\cdots\!11\)\( T^{18} + \)\(18\!\cdots\!78\)\( T^{19} + \)\(16\!\cdots\!13\)\( T^{20} + \)\(14\!\cdots\!01\)\( T^{21} + \)\(12\!\cdots\!29\)\( T^{22} + \)\(98\!\cdots\!42\)\( T^{23} + \)\(76\!\cdots\!22\)\( T^{24} + \)\(57\!\cdots\!35\)\( T^{25} + \)\(41\!\cdots\!95\)\( T^{26} + \)\(28\!\cdots\!87\)\( T^{27} + \)\(18\!\cdots\!72\)\( T^{28} + \)\(11\!\cdots\!49\)\( T^{29} + \)\(70\!\cdots\!97\)\( T^{30} + \)\(38\!\cdots\!54\)\( T^{31} + \)\(20\!\cdots\!71\)\( T^{32} + \)\(96\!\cdots\!19\)\( T^{33} + \)\(43\!\cdots\!62\)\( T^{34} + \)\(15\!\cdots\!56\)\( T^{35} + \)\(55\!\cdots\!60\)\( T^{36} + \)\(12\!\cdots\!24\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \))(\( 1 - 20 T + 918 T^{2} - 15392 T^{3} + 385150 T^{4} - 5501135 T^{5} + 99178525 T^{6} - 1224368980 T^{7} + 17869514163 T^{8} - 194267986369 T^{9} + 2465404566106 T^{10} - 24279766079311 T^{11} + 282853941073475 T^{12} - 2616615985347671 T^{13} + 29083705759559030 T^{14} - 260001256419118512 T^{15} + 2798816272612116063 T^{16} - 24298776074494011285 T^{17} + \)\(25\!\cdots\!65\)\( T^{18} - \)\(21\!\cdots\!22\)\( T^{19} + \)\(20\!\cdots\!95\)\( T^{20} - \)\(16\!\cdots\!65\)\( T^{21} + \)\(16\!\cdots\!81\)\( T^{22} - \)\(12\!\cdots\!52\)\( T^{23} + \)\(11\!\cdots\!90\)\( T^{24} - \)\(85\!\cdots\!99\)\( T^{25} + \)\(76\!\cdots\!25\)\( T^{26} - \)\(54\!\cdots\!51\)\( T^{27} + \)\(46\!\cdots\!18\)\( T^{28} - \)\(30\!\cdots\!81\)\( T^{29} + \)\(23\!\cdots\!21\)\( T^{30} - \)\(13\!\cdots\!80\)\( T^{31} + \)\(87\!\cdots\!75\)\( T^{32} - \)\(40\!\cdots\!15\)\( T^{33} + \)\(23\!\cdots\!50\)\( T^{34} - \)\(78\!\cdots\!52\)\( T^{35} + \)\(38\!\cdots\!14\)\( T^{36} - \)\(69\!\cdots\!80\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \))(\( 1 - 38 T + 1506 T^{2} - 38248 T^{3} + 945168 T^{4} - 18869855 T^{5} + 362815037 T^{6} - 6100901814 T^{7} + 98791331069 T^{8} - 1451824044391 T^{9} + 20589074826162 T^{10} - 270384216238247 T^{11} + 3434442163841103 T^{12} - 40881612158802879 T^{13} + 471607930273373812 T^{14} - 5135445086087776514 T^{15} + 54272696275543842603 T^{16} - \)\(54\!\cdots\!77\)\( T^{17} + \)\(52\!\cdots\!27\)\( T^{18} - \)\(48\!\cdots\!86\)\( T^{19} + \)\(43\!\cdots\!41\)\( T^{20} - \)\(37\!\cdots\!53\)\( T^{21} + \)\(31\!\cdots\!61\)\( T^{22} - \)\(24\!\cdots\!94\)\( T^{23} + \)\(18\!\cdots\!16\)\( T^{24} - \)\(13\!\cdots\!51\)\( T^{25} + \)\(93\!\cdots\!81\)\( T^{26} - \)\(60\!\cdots\!27\)\( T^{27} + \)\(38\!\cdots\!86\)\( T^{28} - \)\(22\!\cdots\!59\)\( T^{29} + \)\(12\!\cdots\!23\)\( T^{30} - \)\(65\!\cdots\!54\)\( T^{31} + \)\(32\!\cdots\!31\)\( T^{32} - \)\(13\!\cdots\!95\)\( T^{33} + \)\(57\!\cdots\!76\)\( T^{34} - \)\(19\!\cdots\!88\)\( T^{35} + \)\(63\!\cdots\!38\)\( T^{36} - \)\(13\!\cdots\!42\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \))
$89$ (\( 1 - 8 T + 89 T^{2} \))(\( 1 - 8 T + 89 T^{2} \))(\( 1 - 33 T + 1224 T^{2} - 26348 T^{3} + 598618 T^{4} - 9990413 T^{5} + 175451155 T^{6} - 2454164402 T^{7} + 36533570333 T^{8} - 449000172249 T^{9} + 5974764652946 T^{10} - 66763132550128 T^{11} + 822734153998475 T^{12} - 8572970387216681 T^{13} + 99920823118676988 T^{14} - 985357725194651078 T^{15} + 10940451629455197735 T^{16} - \)\(10\!\cdots\!72\)\( T^{17} + \)\(10\!\cdots\!89\)\( T^{18} - \)\(96\!\cdots\!28\)\( T^{19} + \)\(96\!\cdots\!21\)\( T^{20} - \)\(81\!\cdots\!12\)\( T^{21} + \)\(77\!\cdots\!15\)\( T^{22} - \)\(61\!\cdots\!98\)\( T^{23} + \)\(55\!\cdots\!12\)\( T^{24} - \)\(42\!\cdots\!41\)\( T^{25} + \)\(36\!\cdots\!75\)\( T^{26} - \)\(26\!\cdots\!68\)\( T^{27} + \)\(20\!\cdots\!14\)\( T^{28} - \)\(14\!\cdots\!49\)\( T^{29} + \)\(10\!\cdots\!37\)\( T^{30} - \)\(60\!\cdots\!42\)\( T^{31} + \)\(38\!\cdots\!95\)\( T^{32} - \)\(19\!\cdots\!33\)\( T^{33} + \)\(10\!\cdots\!82\)\( T^{34} - \)\(40\!\cdots\!28\)\( T^{35} + \)\(16\!\cdots\!96\)\( T^{36} - \)\(40\!\cdots\!73\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \))(\( 1 + 25 T + 1294 T^{2} + 26250 T^{3} + 782226 T^{4} + 13481997 T^{5} + 299641351 T^{6} + 4524428234 T^{7} + 82708614115 T^{8} + 1117468413781 T^{9} + 17657786024838 T^{10} + 216628121395716 T^{11} + 3045563398391669 T^{12} + 34266852135290057 T^{13} + 436381940160735136 T^{14} + 4532093373333428262 T^{15} + 52879632718988757665 T^{16} + \)\(50\!\cdots\!48\)\( T^{17} + \)\(54\!\cdots\!73\)\( T^{18} + \)\(48\!\cdots\!40\)\( T^{19} + \)\(48\!\cdots\!97\)\( T^{20} + \)\(40\!\cdots\!08\)\( T^{21} + \)\(37\!\cdots\!85\)\( T^{22} + \)\(28\!\cdots\!42\)\( T^{23} + \)\(24\!\cdots\!64\)\( T^{24} + \)\(17\!\cdots\!77\)\( T^{25} + \)\(13\!\cdots\!01\)\( T^{26} + \)\(85\!\cdots\!96\)\( T^{27} + \)\(61\!\cdots\!42\)\( T^{28} + \)\(34\!\cdots\!81\)\( T^{29} + \)\(22\!\cdots\!35\)\( T^{30} + \)\(11\!\cdots\!14\)\( T^{31} + \)\(65\!\cdots\!19\)\( T^{32} + \)\(26\!\cdots\!77\)\( T^{33} + \)\(13\!\cdots\!74\)\( T^{34} + \)\(40\!\cdots\!50\)\( T^{35} + \)\(17\!\cdots\!26\)\( T^{36} + \)\(30\!\cdots\!25\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \))(\( 1 - 9 T + 1216 T^{2} - 10206 T^{3} + 722284 T^{4} - 5655923 T^{5} + 278644421 T^{6} - 2035842968 T^{7} + 78355998607 T^{8} - 534075515785 T^{9} + 17094308188542 T^{10} - 108666426373214 T^{11} + 3007075820086831 T^{12} - 17819405728937051 T^{13} + 437553643920847190 T^{14} - 2414915882406884548 T^{15} + 53569379415572077849 T^{16} - \)\(27\!\cdots\!12\)\( T^{17} + \)\(55\!\cdots\!87\)\( T^{18} - \)\(26\!\cdots\!68\)\( T^{19} + \)\(49\!\cdots\!43\)\( T^{20} - \)\(21\!\cdots\!52\)\( T^{21} + \)\(37\!\cdots\!81\)\( T^{22} - \)\(15\!\cdots\!68\)\( T^{23} + \)\(24\!\cdots\!10\)\( T^{24} - \)\(88\!\cdots\!11\)\( T^{25} + \)\(13\!\cdots\!99\)\( T^{26} - \)\(42\!\cdots\!34\)\( T^{27} + \)\(59\!\cdots\!78\)\( T^{28} - \)\(16\!\cdots\!85\)\( T^{29} + \)\(21\!\cdots\!23\)\( T^{30} - \)\(50\!\cdots\!28\)\( T^{31} + \)\(61\!\cdots\!49\)\( T^{32} - \)\(11\!\cdots\!43\)\( T^{33} + \)\(12\!\cdots\!16\)\( T^{34} - \)\(15\!\cdots\!66\)\( T^{35} + \)\(16\!\cdots\!64\)\( T^{36} - \)\(11\!\cdots\!29\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \))(\( 1 + T + 822 T^{2} + 1016 T^{3} + 354280 T^{4} + 499771 T^{5} + 105558417 T^{6} + 163147680 T^{7} + 24191601297 T^{8} + 39689155053 T^{9} + 4506146051134 T^{10} + 7635787634954 T^{11} + 704619726684365 T^{12} + 1203989265061723 T^{13} + 94369768879055054 T^{14} + 159081593533786532 T^{15} + 10962290397855609723 T^{16} + 17856824421640813596 T^{17} + \)\(11\!\cdots\!35\)\( T^{18} + \)\(17\!\cdots\!64\)\( T^{19} + \)\(99\!\cdots\!15\)\( T^{20} + \)\(14\!\cdots\!16\)\( T^{21} + \)\(77\!\cdots\!87\)\( T^{22} + \)\(99\!\cdots\!12\)\( T^{23} + \)\(52\!\cdots\!46\)\( T^{24} + \)\(59\!\cdots\!03\)\( T^{25} + \)\(31\!\cdots\!85\)\( T^{26} + \)\(30\!\cdots\!74\)\( T^{27} + \)\(15\!\cdots\!06\)\( T^{28} + \)\(12\!\cdots\!53\)\( T^{29} + \)\(67\!\cdots\!33\)\( T^{30} + \)\(40\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!73\)\( T^{32} + \)\(97\!\cdots\!11\)\( T^{33} + \)\(61\!\cdots\!20\)\( T^{34} + \)\(15\!\cdots\!76\)\( T^{35} + \)\(11\!\cdots\!38\)\( T^{36} + \)\(12\!\cdots\!81\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 - 4 T + 97 T^{2} \))(\( 1 - 5 T + 1108 T^{2} - 6130 T^{3} + 609288 T^{4} - 3665122 T^{5} + 222167659 T^{6} - 1424037937 T^{7} + 60466865645 T^{8} - 403918846781 T^{9} + 13087076763205 T^{10} - 89037646363900 T^{11} + 2339033958336190 T^{12} - 15848426279185775 T^{13} + 353332008731605184 T^{14} - 2335054391587469791 T^{15} + 45750298510426686739 T^{16} - \)\(28\!\cdots\!05\)\( T^{17} + \)\(51\!\cdots\!53\)\( T^{18} - \)\(30\!\cdots\!04\)\( T^{19} + \)\(49\!\cdots\!41\)\( T^{20} - \)\(27\!\cdots\!45\)\( T^{21} + \)\(41\!\cdots\!47\)\( T^{22} - \)\(20\!\cdots\!71\)\( T^{23} + \)\(30\!\cdots\!88\)\( T^{24} - \)\(13\!\cdots\!75\)\( T^{25} + \)\(18\!\cdots\!70\)\( T^{26} - \)\(69\!\cdots\!00\)\( T^{27} + \)\(99\!\cdots\!85\)\( T^{28} - \)\(29\!\cdots\!69\)\( T^{29} + \)\(43\!\cdots\!85\)\( T^{30} - \)\(98\!\cdots\!17\)\( T^{31} + \)\(14\!\cdots\!43\)\( T^{32} - \)\(23\!\cdots\!18\)\( T^{33} + \)\(38\!\cdots\!84\)\( T^{34} - \)\(37\!\cdots\!30\)\( T^{35} + \)\(66\!\cdots\!96\)\( T^{36} - \)\(28\!\cdots\!45\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \))(\( 1 + 13 T + 922 T^{2} + 10182 T^{3} + 416924 T^{4} + 4067466 T^{5} + 125743955 T^{6} + 1118163431 T^{7} + 28782848457 T^{8} + 239023088765 T^{9} + 5358815293129 T^{10} + 42274589977300 T^{11} + 845335550817640 T^{12} + 6396442512504985 T^{13} + 115884929956805182 T^{14} + 843456646689364365 T^{15} + 14033880049339959377 T^{16} + 97976223132614554835 T^{17} + \)\(15\!\cdots\!17\)\( T^{18} + \)\(10\!\cdots\!80\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} + \)\(92\!\cdots\!15\)\( T^{21} + \)\(12\!\cdots\!21\)\( T^{22} + \)\(74\!\cdots\!65\)\( T^{23} + \)\(99\!\cdots\!74\)\( T^{24} + \)\(53\!\cdots\!65\)\( T^{25} + \)\(68\!\cdots\!20\)\( T^{26} + \)\(33\!\cdots\!00\)\( T^{27} + \)\(40\!\cdots\!93\)\( T^{28} + \)\(17\!\cdots\!85\)\( T^{29} + \)\(20\!\cdots\!21\)\( T^{30} + \)\(77\!\cdots\!71\)\( T^{31} + \)\(84\!\cdots\!35\)\( T^{32} + \)\(26\!\cdots\!54\)\( T^{33} + \)\(26\!\cdots\!32\)\( T^{34} + \)\(62\!\cdots\!22\)\( T^{35} + \)\(54\!\cdots\!14\)\( T^{36} + \)\(75\!\cdots\!57\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \))(\( 1 - 7 T + 790 T^{2} - 2814 T^{3} + 283962 T^{4} - 31890 T^{5} + 66005997 T^{6} + 203750501 T^{7} + 12311844201 T^{8} + 63618254067 T^{9} + 2069869061505 T^{10} + 12016304285232 T^{11} + 312408200206454 T^{12} + 1839756827127625 T^{13} + 41284428580631336 T^{14} + 247364849649607011 T^{15} + 4849666685063790501 T^{16} + 29036205252796449405 T^{17} + \)\(51\!\cdots\!65\)\( T^{18} + \)\(29\!\cdots\!80\)\( T^{19} + \)\(50\!\cdots\!05\)\( T^{20} + \)\(27\!\cdots\!45\)\( T^{21} + \)\(44\!\cdots\!73\)\( T^{22} + \)\(21\!\cdots\!91\)\( T^{23} + \)\(35\!\cdots\!52\)\( T^{24} + \)\(15\!\cdots\!25\)\( T^{25} + \)\(25\!\cdots\!02\)\( T^{26} + \)\(94\!\cdots\!52\)\( T^{27} + \)\(15\!\cdots\!85\)\( T^{28} + \)\(46\!\cdots\!83\)\( T^{29} + \)\(88\!\cdots\!53\)\( T^{30} + \)\(14\!\cdots\!41\)\( T^{31} + \)\(44\!\cdots\!69\)\( T^{32} - \)\(20\!\cdots\!10\)\( T^{33} + \)\(17\!\cdots\!66\)\( T^{34} - \)\(17\!\cdots\!94\)\( T^{35} + \)\(47\!\cdots\!30\)\( T^{36} - \)\(40\!\cdots\!23\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \))(\( 1 + 21 T + 1262 T^{2} + 23964 T^{3} + 771772 T^{4} + 13302056 T^{5} + 305356671 T^{6} + 4795628461 T^{7} + 88045272649 T^{8} + 1265019756527 T^{9} + 19758460473275 T^{10} + 260795101805896 T^{11} + 3598661704620890 T^{12} + 43820032998343225 T^{13} + 547381830580611238 T^{14} + 6174353665422467803 T^{15} + 70918121770833424349 T^{16} + \)\(74\!\cdots\!51\)\( T^{17} + \)\(79\!\cdots\!05\)\( T^{18} + \)\(77\!\cdots\!84\)\( T^{19} + \)\(76\!\cdots\!85\)\( T^{20} + \)\(69\!\cdots\!59\)\( T^{21} + \)\(64\!\cdots\!77\)\( T^{22} + \)\(54\!\cdots\!43\)\( T^{23} + \)\(47\!\cdots\!66\)\( T^{24} + \)\(36\!\cdots\!25\)\( T^{25} + \)\(29\!\cdots\!70\)\( T^{26} + \)\(20\!\cdots\!56\)\( T^{27} + \)\(15\!\cdots\!75\)\( T^{28} + \)\(93\!\cdots\!23\)\( T^{29} + \)\(62\!\cdots\!97\)\( T^{30} + \)\(33\!\cdots\!01\)\( T^{31} + \)\(20\!\cdots\!67\)\( T^{32} + \)\(86\!\cdots\!64\)\( T^{33} + \)\(48\!\cdots\!96\)\( T^{34} + \)\(14\!\cdots\!44\)\( T^{35} + \)\(75\!\cdots\!94\)\( T^{36} + \)\(12\!\cdots\!69\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \))
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