Properties

Label 4017.2
Level 4017
Weight 2
Dimension 437931
Nonzero newspaces 60
Sturm bound 2376192

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Defining parameters

Level: \( N \) = \( 4017\( 4017 = 3 \cdot 13 \cdot 103 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 60 \)
Sturm bound: \(2376192\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(4017))\).

Total New Old
Modular forms 598944 442371 156573
Cusp forms 589153 437931 151222
Eisenstein series 9791 4440 5351

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(4017))\)

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
4017.2.a \(\chi_{4017}(1, \cdot)\) 4017.2.a.a 1 1
4017.2.a.b 1
4017.2.a.c 1
4017.2.a.d 2
4017.2.a.e 16
4017.2.a.f 19
4017.2.a.g 24
4017.2.a.h 25
4017.2.a.i 25
4017.2.a.j 25
4017.2.a.k 32
4017.2.a.l 32
4017.2.b \(\chi_{4017}(1546, \cdot)\) n/a 240 1
4017.2.e \(\chi_{4017}(2471, \cdot)\) n/a 416 1
4017.2.f \(\chi_{4017}(4016, \cdot)\) n/a 480 1
4017.2.i \(\chi_{4017}(880, \cdot)\) n/a 486 2
4017.2.j \(\chi_{4017}(664, \cdot)\) n/a 416 2
4017.2.k \(\chi_{4017}(1855, \cdot)\) n/a 472 2
4017.2.l \(\chi_{4017}(1498, \cdot)\) n/a 486 2
4017.2.m \(\chi_{4017}(1958, \cdot)\) n/a 952 2
4017.2.p \(\chi_{4017}(1750, \cdot)\) n/a 488 2
4017.2.r \(\chi_{4017}(1901, \cdot)\) n/a 962 2
4017.2.s \(\chi_{4017}(355, \cdot)\) n/a 486 2
4017.2.v \(\chi_{4017}(1544, \cdot)\) n/a 964 2
4017.2.ba \(\chi_{4017}(974, \cdot)\) n/a 964 2
4017.2.bb \(\chi_{4017}(881, \cdot)\) n/a 962 2
4017.2.be \(\chi_{4017}(3091, \cdot)\) n/a 480 2
4017.2.bf \(\chi_{4017}(1808, \cdot)\) n/a 832 2
4017.2.bg \(\chi_{4017}(263, \cdot)\) n/a 962 2
4017.2.bl \(\chi_{4017}(2734, \cdot)\) n/a 486 2
4017.2.bm \(\chi_{4017}(571, \cdot)\) n/a 484 2
4017.2.bn \(\chi_{4017}(308, \cdot)\) n/a 964 2
4017.2.bq \(\chi_{4017}(1499, \cdot)\) n/a 962 2
4017.2.bt \(\chi_{4017}(2416, \cdot)\) n/a 972 4
4017.2.bu \(\chi_{4017}(149, \cdot)\) n/a 1924 4
4017.2.bw \(\chi_{4017}(983, \cdot)\) n/a 1928 4
4017.2.ca \(\chi_{4017}(514, \cdot)\) n/a 968 4
4017.2.cb \(\chi_{4017}(253, \cdot)\) n/a 972 4
4017.2.cc \(\chi_{4017}(722, \cdot)\) n/a 1904 4
4017.2.cd \(\chi_{4017}(2312, \cdot)\) n/a 1924 4
4017.2.ch \(\chi_{4017}(1087, \cdot)\) n/a 968 4
4017.2.ci \(\chi_{4017}(79, \cdot)\) n/a 3328 16
4017.2.cl \(\chi_{4017}(233, \cdot)\) n/a 7680 16
4017.2.cm \(\chi_{4017}(209, \cdot)\) n/a 6656 16
4017.2.cp \(\chi_{4017}(64, \cdot)\) n/a 3904 16
4017.2.cq \(\chi_{4017}(334, \cdot)\) n/a 7776 32
4017.2.cr \(\chi_{4017}(61, \cdot)\) n/a 7744 32
4017.2.cs \(\chi_{4017}(118, \cdot)\) n/a 6656 32
4017.2.ct \(\chi_{4017}(16, \cdot)\) n/a 7776 32
4017.2.cu \(\chi_{4017}(31, \cdot)\) n/a 7808 32
4017.2.cx \(\chi_{4017}(8, \cdot)\) n/a 15360 32
4017.2.cz \(\chi_{4017}(212, \cdot)\) n/a 15392 32
4017.2.dc \(\chi_{4017}(113, \cdot)\) n/a 15424 32
4017.2.dd \(\chi_{4017}(25, \cdot)\) n/a 7744 32
4017.2.de \(\chi_{4017}(4, \cdot)\) n/a 7776 32
4017.2.dj \(\chi_{4017}(146, \cdot)\) n/a 15392 32
4017.2.dk \(\chi_{4017}(53, \cdot)\) n/a 13312 32
4017.2.dl \(\chi_{4017}(322, \cdot)\) n/a 7744 32
4017.2.do \(\chi_{4017}(62, \cdot)\) n/a 15392 32
4017.2.dp \(\chi_{4017}(77, \cdot)\) n/a 15424 32
4017.2.du \(\chi_{4017}(95, \cdot)\) n/a 15424 32
4017.2.dx \(\chi_{4017}(121, \cdot)\) n/a 7776 32
4017.2.dy \(\chi_{4017}(35, \cdot)\) n/a 15392 32
4017.2.ea \(\chi_{4017}(70, \cdot)\) n/a 15488 64
4017.2.ee \(\chi_{4017}(2, \cdot)\) n/a 30784 64
4017.2.ef \(\chi_{4017}(137, \cdot)\) n/a 30848 64
4017.2.eg \(\chi_{4017}(124, \cdot)\) n/a 15552 64
4017.2.eh \(\chi_{4017}(37, \cdot)\) n/a 15488 64
4017.2.el \(\chi_{4017}(83, \cdot)\) n/a 30848 64
4017.2.en \(\chi_{4017}(50, \cdot)\) n/a 30784 64
4017.2.eo \(\chi_{4017}(67, \cdot)\) n/a 15552 64

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(4017))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(4017)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(103))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(309))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1339))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 11 T^{2} + 3 T^{3} + 69 T^{4} + 33 T^{5} + 321 T^{6} + 197 T^{7} + 1207 T^{8} + 843 T^{9} + 3838 T^{10} + 2828 T^{11} + 10577 T^{12} + 7805 T^{13} + 25545 T^{14} + 18240 T^{15} + 54371 T^{16} + 36480 T^{17} + 102180 T^{18} + 62440 T^{19} + 169232 T^{20} + 90496 T^{21} + 245632 T^{22} + 107904 T^{23} + 308992 T^{24} + 100864 T^{25} + 328704 T^{26} + 67584 T^{27} + 282624 T^{28} + 24576 T^{29} + 180224 T^{30} + 65536 T^{32} \))(\( 1 + 4 T + 22 T^{2} + 67 T^{3} + 228 T^{4} + 578 T^{5} + 1534 T^{6} + 3396 T^{7} + 7654 T^{8} + 15244 T^{9} + 30463 T^{10} + 55675 T^{11} + 101215 T^{12} + 172075 T^{13} + 289380 T^{14} + 461887 T^{15} + 726640 T^{16} + 1095078 T^{17} + 1623283 T^{18} + 2315500 T^{19} + 3246566 T^{20} + 4380312 T^{21} + 5813120 T^{22} + 7390192 T^{23} + 9260160 T^{24} + 11012800 T^{25} + 12955520 T^{26} + 14252800 T^{27} + 15597056 T^{28} + 15609856 T^{29} + 15675392 T^{30} + 13910016 T^{31} + 12566528 T^{32} + 9469952 T^{33} + 7471104 T^{34} + 4390912 T^{35} + 2883584 T^{36} + 1048576 T^{37} + 524288 T^{38} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{16} \))(\( ( 1 - T )^{19} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 6 T + 63 T^{2} + 297 T^{3} + 1808 T^{4} + 7115 T^{5} + 32405 T^{6} + 110545 T^{7} + 414963 T^{8} + 1258263 T^{9} + 4089810 T^{10} + 11204141 T^{11} + 32424912 T^{12} + 81010514 T^{13} + 212055774 T^{14} + 485086372 T^{15} + 1158154701 T^{16} + 2425431860 T^{17} + 5301394350 T^{18} + 10126314250 T^{19} + 20265570000 T^{20} + 35012940625 T^{21} + 63903281250 T^{22} + 98301796875 T^{23} + 162094921875 T^{24} + 215908203125 T^{25} + 316455078125 T^{26} + 347412109375 T^{27} + 441406250000 T^{28} + 362548828125 T^{29} + 384521484375 T^{30} + 183105468750 T^{31} + 152587890625 T^{32} \))(\( 1 + 3 T + 56 T^{2} + 150 T^{3} + 1521 T^{4} + 3665 T^{5} + 26816 T^{6} + 58367 T^{7} + 346475 T^{8} + 682923 T^{9} + 3515090 T^{10} + 6290519 T^{11} + 29339974 T^{12} + 47880380 T^{13} + 208649182 T^{14} + 312931684 T^{15} + 1298515446 T^{16} + 1811263614 T^{17} + 7207414795 T^{18} + 9484485698 T^{19} + 36037073975 T^{20} + 45281590350 T^{21} + 162314430750 T^{22} + 195582302500 T^{23} + 652028693750 T^{24} + 748130937500 T^{25} + 2292185468750 T^{26} + 2457233984375 T^{27} + 6865410156250 T^{28} + 6669169921875 T^{29} + 16917724609375 T^{30} + 14249755859375 T^{31} + 32734375000000 T^{32} + 22369384765625 T^{33} + 46417236328125 T^{34} + 22888183593750 T^{35} + 42724609375000 T^{36} + 11444091796875 T^{37} + 19073486328125 T^{38} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 6 T^{2} + 49 T^{4} \))(\( 1 + 13 T + 149 T^{2} + 1172 T^{3} + 8306 T^{4} + 48983 T^{5} + 265941 T^{6} + 1280626 T^{7} + 5759546 T^{8} + 23663083 T^{9} + 91657373 T^{10} + 329497626 T^{11} + 1123505752 T^{12} + 3587011075 T^{13} + 10901801474 T^{14} + 31166268813 T^{15} + 84960123175 T^{16} + 218163881691 T^{17} + 534188272226 T^{18} + 1230344798725 T^{19} + 2697537310552 T^{20} + 5537866600182 T^{21} + 10783398276077 T^{22} + 19487566363069 T^{23} + 33202636540346 T^{24} + 51677878317982 T^{25} + 75121750194309 T^{26} + 96855395852369 T^{27} + 114965731491506 T^{28} + 113553920197004 T^{29} + 101055237854501 T^{30} + 61718299629259 T^{31} + 33232930569601 T^{32} \))(\( 1 + 23 T + 334 T^{2} + 3603 T^{3} + 31822 T^{4} + 239968 T^{5} + 1592143 T^{6} + 9467382 T^{7} + 51173242 T^{8} + 253973142 T^{9} + 1166694790 T^{10} + 4991258134 T^{11} + 19985833094 T^{12} + 75196757106 T^{13} + 266710304191 T^{14} + 893993228835 T^{15} + 2837602665044 T^{16} + 8541380572189 T^{17} + 24407317995249 T^{18} + 66249905589668 T^{19} + 170851225966743 T^{20} + 418527648037261 T^{21} + 973297714110092 T^{22} + 2146477742432835 T^{23} + 4482600082538137 T^{24} + 8846823276763794 T^{25} + 16459192943732042 T^{26} + 28773609882141334 T^{27} + 47080343044607530 T^{28} + 71741126525762358 T^{29} + 101186219932610806 T^{30} + 131040753303577782 T^{31} + 154261159696432201 T^{32} + 162751834345428832 T^{33} + 151076902369406146 T^{34} + 119738248842272403 T^{35} + 77698591671727138 T^{36} + 37453512751940327 T^{37} + 11398895185373143 T^{38} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 20 T^{2} + 121 T^{4} \))(\( 1 + 5 T + 103 T^{2} + 468 T^{3} + 5155 T^{4} + 21737 T^{5} + 168960 T^{6} + 674495 T^{7} + 4125967 T^{8} + 15786120 T^{9} + 80534912 T^{10} + 295906003 T^{11} + 1306828751 T^{12} + 4581629419 T^{13} + 18014110345 T^{14} + 59563908141 T^{15} + 213368671275 T^{16} + 655202989551 T^{17} + 2179707351745 T^{18} + 6098148756689 T^{19} + 19133279743391 T^{20} + 47655957689153 T^{21} + 142672509237632 T^{22} + 307626819866520 T^{23} + 884437669162927 T^{24} + 1590423927841045 T^{25} + 4382387260584960 T^{26} + 6201819784071307 T^{27} + 16178598281996755 T^{28} + 16156629283359708 T^{29} + 39114232859073823 T^{30} + 20886240847078255 T^{31} + 45949729863572161 T^{32} \))(\( 1 + 15 T + 216 T^{2} + 2045 T^{3} + 18055 T^{4} + 128958 T^{5} + 863939 T^{6} + 5011533 T^{7} + 27480905 T^{8} + 134437474 T^{9} + 626080020 T^{10} + 2639271791 T^{11} + 10671965252 T^{12} + 39371145209 T^{13} + 141062994741 T^{14} + 465291096429 T^{15} + 1531558432651 T^{16} + 4747783690792 T^{17} + 15453700607668 T^{18} + 49139797418868 T^{19} + 169990706684348 T^{20} + 574481826585832 T^{21} + 2038504273858481 T^{22} + 6812326942816989 T^{23} + 22718336366032791 T^{24} + 69748385377601249 T^{25} + 207966411771782092 T^{26} + 565751347773625871 T^{27} + 1476263937540233820 T^{28} + 3486961845423897874 T^{29} + 7840622915452182955 T^{30} + 15728337378073723293 T^{31} + 29825517406915604209 T^{32} + 48971779039227592878 T^{33} + 75420215698799578805 T^{34} + 93967197571005069245 T^{35} + \)\(10\!\cdots\!36\)\( T^{36} + 83398759702383472215 T^{37} + 61159090448414546291 T^{38} \))
$13$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{16} \))(\( ( 1 - T )^{19} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 8 T + 42 T^{2} - 136 T^{3} + 289 T^{4} \))(\( 1 + T + 185 T^{2} + 128 T^{3} + 16210 T^{4} + 5248 T^{5} + 896252 T^{6} - 97966 T^{7} + 35234374 T^{8} - 21118488 T^{9} + 1057224025 T^{10} - 1172403634 T^{11} + 25563421147 T^{12} - 39596372033 T^{13} + 522806390149 T^{14} - 929982009080 T^{15} + 9402855969331 T^{16} - 15809694154360 T^{17} + 151091046753061 T^{18} - 194536975798129 T^{19} + 2135082497618587 T^{20} - 1664645506560338 T^{21} + 25518817851895225 T^{22} - 8665732341686424 T^{23} + 245786446609476934 T^{24} - 11617579908905102 T^{25} + 1806838565265217148 T^{26} + 179858911822457984 T^{27} + 9444306465494425810 T^{28} + 1267785988211959936 T^{29} + 31149897913489171865 T^{30} + 2862423051509815793 T^{31} + 48661191875666868481 T^{32} \))(\( 1 + 161 T^{2} + 150 T^{3} + 12634 T^{4} + 24558 T^{5} + 659787 T^{6} + 1928082 T^{7} + 26503567 T^{8} + 97783430 T^{9} + 888518315 T^{10} + 3650313947 T^{11} + 25800369848 T^{12} + 108316958592 T^{13} + 654618405646 T^{14} + 2679078381485 T^{15} + 14509412268976 T^{16} + 56661713830115 T^{17} + 281301834590869 T^{18} + 1035217676004762 T^{19} + 4782131188044773 T^{20} + 16375235296903235 T^{21} + 71284742477479088 T^{22} + 223759305500008685 T^{23} + 929464525585312622 T^{24} + 2614508061884542848 T^{25} + 10586889526337531704 T^{26} + 25463704677771329627 T^{27} + \)\(10\!\cdots\!55\)\( T^{28} + \)\(19\!\cdots\!70\)\( T^{29} + \)\(90\!\cdots\!11\)\( T^{30} + \)\(11\!\cdots\!02\)\( T^{31} + \)\(65\!\cdots\!19\)\( T^{32} + \)\(41\!\cdots\!82\)\( T^{33} + \)\(36\!\cdots\!62\)\( T^{34} + \)\(72\!\cdots\!50\)\( T^{35} + \)\(13\!\cdots\!97\)\( T^{36} + \)\(23\!\cdots\!53\)\( T^{38} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 6 T^{2} + 361 T^{4} \))(\( 1 - 6 T + 189 T^{2} - 896 T^{3} + 16481 T^{4} - 59325 T^{5} + 870526 T^{6} - 2078906 T^{7} + 30498830 T^{8} - 25915099 T^{9} + 721784287 T^{10} + 1149666723 T^{11} + 11008302177 T^{12} + 75193606493 T^{13} + 93098661367 T^{14} + 2256718908251 T^{15} + 606067442715 T^{16} + 42877659256769 T^{17} + 33608616753487 T^{18} + 515752946935487 T^{19} + 1434612948008817 T^{20} + 2846688623153577 T^{21} + 33956977673871847 T^{22} - 23164774609487161 T^{23} + 517978801981742030 T^{24} - 670837391038949774 T^{25} + 5337252585138473326 T^{26} - 6910784609136842175 T^{27} + 36477643181129399441 T^{28} - 37679473182182324864 T^{29} + \)\(15\!\cdots\!69\)\( T^{30} - 91086762179248789794 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 32 T + 676 T^{2} + 10546 T^{3} + 136367 T^{4} + 1508247 T^{5} + 14791519 T^{6} + 130556099 T^{7} + 1054866226 T^{8} + 7869431210 T^{9} + 54737438274 T^{10} + 356949412621 T^{11} + 2196096767396 T^{12} + 12794340593121 T^{13} + 70887947112409 T^{14} + 374425848975018 T^{15} + 1890848135805420 T^{16} + 9141927011693352 T^{17} + 42390272726005716 T^{18} + 188580040508177724 T^{19} + 805415181794108604 T^{20} + 3300235651221300072 T^{21} + 12969327363489375780 T^{22} + 48795551064273320778 T^{23} + \)\(17\!\cdots\!91\)\( T^{24} + \)\(60\!\cdots\!01\)\( T^{25} + \)\(19\!\cdots\!44\)\( T^{26} + \)\(60\!\cdots\!61\)\( T^{27} + \)\(17\!\cdots\!46\)\( T^{28} + \)\(48\!\cdots\!10\)\( T^{29} + \)\(12\!\cdots\!94\)\( T^{30} + \)\(28\!\cdots\!39\)\( T^{31} + \)\(62\!\cdots\!21\)\( T^{32} + \)\(12\!\cdots\!87\)\( T^{33} + \)\(20\!\cdots\!33\)\( T^{34} + \)\(30\!\cdots\!26\)\( T^{35} + \)\(37\!\cdots\!64\)\( T^{36} + \)\(33\!\cdots\!12\)\( T^{37} + \)\(19\!\cdots\!79\)\( T^{38} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 21 T + 420 T^{2} + 5588 T^{3} + 68980 T^{4} + 699998 T^{5} + 6639302 T^{6} + 55581914 T^{7} + 438829098 T^{8} + 3160467956 T^{9} + 21623399241 T^{10} + 137363499366 T^{11} + 833411377896 T^{12} + 4741844098829 T^{13} + 25858406038707 T^{14} + 132926304470906 T^{15} + 656062497467189 T^{16} + 3057305002830838 T^{17} + 13679096794476003 T^{18} + 57694017150452443 T^{19} + 233222673401794536 T^{20} + 884118597599858538 T^{21} + 3201039129843360249 T^{22} + 10760841721016876332 T^{23} + 34365139034352506538 T^{24} + \)\(10\!\cdots\!82\)\( T^{25} + \)\(27\!\cdots\!98\)\( T^{26} + \)\(66\!\cdots\!46\)\( T^{27} + \)\(15\!\cdots\!80\)\( T^{28} + \)\(28\!\cdots\!04\)\( T^{29} + \)\(48\!\cdots\!80\)\( T^{30} + \)\(55\!\cdots\!47\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 23 T + 463 T^{2} + 6645 T^{3} + 85003 T^{4} + 927933 T^{5} + 9272977 T^{6} + 83703727 T^{7} + 704480046 T^{8} + 5498886128 T^{9} + 40515524299 T^{10} + 280993200928 T^{11} + 1855003272971 T^{12} + 11633165379971 T^{13} + 69838183305282 T^{14} + 400592440468203 T^{15} + 2207790064849525 T^{16} + 11666141667866817 T^{17} + 59353797866407781 T^{18} + 289994439146464066 T^{19} + 1365137350927378963 T^{20} + 6171388942301546193 T^{21} + 26862181719024170675 T^{22} + \)\(11\!\cdots\!23\)\( T^{23} + \)\(44\!\cdots\!26\)\( T^{24} + \)\(17\!\cdots\!19\)\( T^{25} + \)\(63\!\cdots\!37\)\( T^{26} + \)\(22\!\cdots\!68\)\( T^{27} + \)\(72\!\cdots\!37\)\( T^{28} + \)\(22\!\cdots\!72\)\( T^{29} + \)\(67\!\cdots\!42\)\( T^{30} + \)\(18\!\cdots\!67\)\( T^{31} + \)\(46\!\cdots\!91\)\( T^{32} + \)\(10\!\cdots\!97\)\( T^{33} + \)\(22\!\cdots\!21\)\( T^{34} + \)\(40\!\cdots\!45\)\( T^{35} + \)\(65\!\cdots\!89\)\( T^{36} + \)\(74\!\cdots\!87\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 5 T + 29 T^{2} \))(\( 1 + 4 T + 30 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 + 17 T + 382 T^{2} + 4493 T^{3} + 61304 T^{4} + 572658 T^{5} + 6003778 T^{6} + 47507896 T^{7} + 418744825 T^{8} + 2918415745 T^{9} + 22669608551 T^{10} + 142653322489 T^{11} + 1002250783092 T^{12} + 5775139652754 T^{13} + 37169837531505 T^{14} + 197275016475520 T^{15} + 1169024019773341 T^{16} + 5720975477790080 T^{17} + 31259833363995705 T^{18} + 140849880991017306 T^{19} + 708872936116092852 T^{20} + 2925983552916929861 T^{21} + 13484411844075817871 T^{22} + 50342310619488085205 T^{23} + \)\(20\!\cdots\!25\)\( T^{24} + \)\(68\!\cdots\!24\)\( T^{25} + \)\(25\!\cdots\!78\)\( T^{26} + \)\(69\!\cdots\!82\)\( T^{27} + \)\(21\!\cdots\!64\)\( T^{28} + \)\(46\!\cdots\!77\)\( T^{29} + \)\(11\!\cdots\!42\)\( T^{30} + \)\(14\!\cdots\!33\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 4 T + 230 T^{2} - 498 T^{3} + 24812 T^{4} - 12555 T^{5} + 1764888 T^{6} + 1715140 T^{7} + 98134336 T^{8} + 211222301 T^{9} + 4629673399 T^{10} + 14097997175 T^{11} + 189546437831 T^{12} + 719536341873 T^{13} + 6850956810332 T^{14} + 30284677018272 T^{15} + 226124668385788 T^{16} + 1077783463993004 T^{17} + 6988062519001243 T^{18} + 33301105221734120 T^{19} + 202653813051036047 T^{20} + 906415893218116364 T^{21} + 5514954537260983532 T^{22} + 21419776646160438432 T^{23} + \)\(14\!\cdots\!68\)\( T^{24} + \)\(42\!\cdots\!33\)\( T^{25} + \)\(32\!\cdots\!79\)\( T^{26} + \)\(70\!\cdots\!75\)\( T^{27} + \)\(67\!\cdots\!31\)\( T^{28} + \)\(88\!\cdots\!01\)\( T^{29} + \)\(11\!\cdots\!44\)\( T^{30} + \)\(60\!\cdots\!40\)\( T^{31} + \)\(18\!\cdots\!32\)\( T^{32} - \)\(37\!\cdots\!55\)\( T^{33} + \)\(21\!\cdots\!88\)\( T^{34} - \)\(12\!\cdots\!58\)\( T^{35} + \)\(16\!\cdots\!70\)\( T^{36} - \)\(84\!\cdots\!44\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 12 T + 96 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 + 33 T + 798 T^{2} + 13776 T^{3} + 200919 T^{4} + 2457514 T^{5} + 26725983 T^{6} + 257317564 T^{7} + 2261797203 T^{8} + 18088811728 T^{9} + 134494073707 T^{10} + 927661553610 T^{11} + 6053361007013 T^{12} + 37359222213316 T^{13} + 222554232877347 T^{14} + 1279029067003387 T^{15} + 7215455552667321 T^{16} + 39649901077104997 T^{17} + 213874617795130467 T^{18} + 1112968588956896956 T^{19} + 5590406010557652773 T^{20} + 26558162695195285110 T^{21} + \)\(11\!\cdots\!67\)\( T^{22} + \)\(49\!\cdots\!08\)\( T^{23} + \)\(19\!\cdots\!23\)\( T^{24} + \)\(68\!\cdots\!44\)\( T^{25} + \)\(21\!\cdots\!83\)\( T^{26} + \)\(62\!\cdots\!34\)\( T^{27} + \)\(15\!\cdots\!59\)\( T^{28} + \)\(33\!\cdots\!16\)\( T^{29} + \)\(60\!\cdots\!58\)\( T^{30} + \)\(77\!\cdots\!83\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 50 T + 1500 T^{2} + 32605 T^{3} + 567786 T^{4} + 8290847 T^{5} + 104907104 T^{6} + 1174610464 T^{7} + 11841177493 T^{8} + 108920805454 T^{9} + 925489451444 T^{10} + 7341160323854 T^{11} + 54904024825514 T^{12} + 390393952760208 T^{13} + 2657493298730252 T^{14} + 17398450743096251 T^{15} + 109849826427509735 T^{16} + 669350367850818674 T^{17} + 3935651883726109129 T^{18} + 22315832316500223946 T^{19} + \)\(12\!\cdots\!99\)\( T^{20} + \)\(64\!\cdots\!14\)\( T^{21} + \)\(32\!\cdots\!85\)\( T^{22} + \)\(16\!\cdots\!71\)\( T^{23} + \)\(76\!\cdots\!52\)\( T^{24} + \)\(34\!\cdots\!48\)\( T^{25} + \)\(15\!\cdots\!54\)\( T^{26} + \)\(62\!\cdots\!14\)\( T^{27} + \)\(24\!\cdots\!24\)\( T^{28} + \)\(89\!\cdots\!54\)\( T^{29} + \)\(30\!\cdots\!83\)\( T^{30} + \)\(92\!\cdots\!04\)\( T^{31} + \)\(25\!\cdots\!64\)\( T^{32} + \)\(62\!\cdots\!87\)\( T^{33} + \)\(13\!\cdots\!86\)\( T^{34} + \)\(23\!\cdots\!05\)\( T^{35} + \)\(33\!\cdots\!00\)\( T^{36} + \)\(34\!\cdots\!50\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \))
$37$ (\( 1 + 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 8 T + 40 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 23 T + 546 T^{2} + 8362 T^{3} + 124217 T^{4} + 1497794 T^{5} + 17395660 T^{6} + 177027635 T^{7} + 1735971971 T^{8} + 15439423027 T^{9} + 132540944311 T^{10} + 1050736287635 T^{11} + 8052582550557 T^{12} + 57556305197255 T^{13} + 398120263677646 T^{14} + 2581256279955595 T^{15} + 16205026090197957 T^{16} + 95506482358357015 T^{17} + 545026640974697374 T^{18} + 2915399527156557515 T^{19} + 15091836165534457677 T^{20} + 72862211948101071695 T^{21} + \)\(34\!\cdots\!99\)\( T^{22} + \)\(14\!\cdots\!91\)\( T^{23} + \)\(60\!\cdots\!91\)\( T^{24} + \)\(23\!\cdots\!95\)\( T^{25} + \)\(83\!\cdots\!40\)\( T^{26} + \)\(26\!\cdots\!22\)\( T^{27} + \)\(81\!\cdots\!77\)\( T^{28} + \)\(20\!\cdots\!14\)\( T^{29} + \)\(49\!\cdots\!94\)\( T^{30} + \)\(76\!\cdots\!39\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 + 38 T + 1050 T^{2} + 21553 T^{3} + 376486 T^{4} + 5665885 T^{5} + 76554475 T^{6} + 935956836 T^{7} + 10548520743 T^{8} + 110148266848 T^{9} + 1076609228882 T^{10} + 9882737954944 T^{11} + 85738840276959 T^{12} + 704512181956948 T^{13} + 5505620842008961 T^{14} + 40970573601743475 T^{15} + 291123759158286697 T^{16} + 1976231259377669664 T^{17} + 12838303108481261318 T^{18} + 79798938937090207718 T^{19} + \)\(47\!\cdots\!66\)\( T^{20} + \)\(27\!\cdots\!16\)\( T^{21} + \)\(14\!\cdots\!41\)\( T^{22} + \)\(76\!\cdots\!75\)\( T^{23} + \)\(38\!\cdots\!77\)\( T^{24} + \)\(18\!\cdots\!32\)\( T^{25} + \)\(81\!\cdots\!47\)\( T^{26} + \)\(34\!\cdots\!24\)\( T^{27} + \)\(13\!\cdots\!14\)\( T^{28} + \)\(52\!\cdots\!52\)\( T^{29} + \)\(18\!\cdots\!59\)\( T^{30} + \)\(61\!\cdots\!16\)\( T^{31} + \)\(18\!\cdots\!75\)\( T^{32} + \)\(51\!\cdots\!65\)\( T^{33} + \)\(12\!\cdots\!98\)\( T^{34} + \)\(26\!\cdots\!73\)\( T^{35} + \)\(47\!\cdots\!50\)\( T^{36} + \)\(64\!\cdots\!02\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 8 T + 90 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 - 7 T + 433 T^{2} - 2795 T^{3} + 92046 T^{4} - 561342 T^{5} + 12862008 T^{6} - 74802669 T^{7} + 1325543074 T^{8} - 7350118872 T^{9} + 106865779223 T^{10} - 561760441455 T^{11} + 6970177220808 T^{12} - 34413366371858 T^{13} + 375022839134302 T^{14} - 1718157991944314 T^{15} + 16815911734423615 T^{16} - 70444477669716874 T^{17} + 630413392584761662 T^{18} - 2371803623714825218 T^{19} + 19696054953647634888 T^{20} - 65083430619059212455 T^{21} + \)\(50\!\cdots\!43\)\( T^{22} - \)\(14\!\cdots\!32\)\( T^{23} + \)\(10\!\cdots\!54\)\( T^{24} - \)\(24\!\cdots\!09\)\( T^{25} + \)\(17\!\cdots\!08\)\( T^{26} - \)\(30\!\cdots\!22\)\( T^{27} + \)\(20\!\cdots\!26\)\( T^{28} - \)\(25\!\cdots\!95\)\( T^{29} + \)\(16\!\cdots\!13\)\( T^{30} - \)\(10\!\cdots\!07\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 + 11 T + 446 T^{2} + 4400 T^{3} + 95118 T^{4} + 853912 T^{5} + 13103612 T^{6} + 108385973 T^{7} + 1330006259 T^{8} + 10240472650 T^{9} + 107377461957 T^{10} + 775641499850 T^{11} + 7239423922717 T^{12} + 49337987959514 T^{13} + 420563202084647 T^{14} + 2715283812448549 T^{15} + 21493423996286668 T^{16} + 131806368937077125 T^{17} + 980054993372572395 T^{18} + 5709326279755589648 T^{19} + 40182254728275468195 T^{20} + \)\(22\!\cdots\!25\)\( T^{21} + \)\(14\!\cdots\!28\)\( T^{22} + \)\(76\!\cdots\!89\)\( T^{23} + \)\(48\!\cdots\!47\)\( T^{24} + \)\(23\!\cdots\!74\)\( T^{25} + \)\(14\!\cdots\!77\)\( T^{26} + \)\(61\!\cdots\!50\)\( T^{27} + \)\(35\!\cdots\!77\)\( T^{28} + \)\(13\!\cdots\!50\)\( T^{29} + \)\(73\!\cdots\!19\)\( T^{30} + \)\(24\!\cdots\!13\)\( T^{31} + \)\(12\!\cdots\!52\)\( T^{32} + \)\(32\!\cdots\!32\)\( T^{33} + \)\(14\!\cdots\!18\)\( T^{34} + \)\(28\!\cdots\!00\)\( T^{35} + \)\(11\!\cdots\!26\)\( T^{36} + \)\(11\!\cdots\!31\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \))
$43$ (\( 1 + 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 78 T^{2} + 1849 T^{4} \))(\( 1 + 33 T + 807 T^{2} + 14013 T^{3} + 210617 T^{4} + 2673429 T^{5} + 31041288 T^{6} + 323566690 T^{7} + 3174452934 T^{8} + 28826234693 T^{9} + 250512987576 T^{10} + 2048975124703 T^{11} + 16184599779161 T^{12} + 121327338875479 T^{13} + 881781458787489 T^{14} + 6102466570046400 T^{15} + 40991254409033571 T^{16} + 262406062511995200 T^{17} + 1630413917298067161 T^{18} + 9646372731972708853 T^{19} + 55331925909595405961 T^{20} + \)\(30\!\cdots\!29\)\( T^{21} + \)\(15\!\cdots\!24\)\( T^{22} + \)\(78\!\cdots\!51\)\( T^{23} + \)\(37\!\cdots\!34\)\( T^{24} + \)\(16\!\cdots\!70\)\( T^{25} + \)\(67\!\cdots\!12\)\( T^{26} + \)\(24\!\cdots\!03\)\( T^{27} + \)\(84\!\cdots\!17\)\( T^{28} + \)\(24\!\cdots\!59\)\( T^{29} + \)\(59\!\cdots\!43\)\( T^{30} + \)\(10\!\cdots\!31\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 + 17 T + 676 T^{2} + 9368 T^{3} + 209007 T^{4} + 2461191 T^{5} + 40133229 T^{6} + 413148783 T^{7} + 5455892312 T^{8} + 50140105129 T^{9} + 566071868179 T^{10} + 4718692011715 T^{11} + 47073968184889 T^{12} + 360114640244637 T^{13} + 3242786609224706 T^{14} + 22940882977523533 T^{15} + 189026616829371247 T^{16} + 1241430508025184462 T^{17} + 9438293347761412414 T^{18} + 57581173645198944090 T^{19} + \)\(40\!\cdots\!02\)\( T^{20} + \)\(22\!\cdots\!38\)\( T^{21} + \)\(15\!\cdots\!29\)\( T^{22} + \)\(78\!\cdots\!33\)\( T^{23} + \)\(47\!\cdots\!58\)\( T^{24} + \)\(22\!\cdots\!13\)\( T^{25} + \)\(12\!\cdots\!23\)\( T^{26} + \)\(55\!\cdots\!15\)\( T^{27} + \)\(28\!\cdots\!97\)\( T^{28} + \)\(10\!\cdots\!21\)\( T^{29} + \)\(50\!\cdots\!84\)\( T^{30} + \)\(16\!\cdots\!83\)\( T^{31} + \)\(68\!\cdots\!47\)\( T^{32} + \)\(18\!\cdots\!59\)\( T^{33} + \)\(66\!\cdots\!49\)\( T^{34} + \)\(12\!\cdots\!68\)\( T^{35} + \)\(39\!\cdots\!68\)\( T^{36} + \)\(42\!\cdots\!33\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 44 T^{2} + 2209 T^{4} \))(\( 1 + 13 T + 506 T^{2} + 5531 T^{3} + 121933 T^{4} + 1151310 T^{5} + 18765873 T^{6} + 156130578 T^{7} + 2088051163 T^{8} + 15543390385 T^{9} + 180307471207 T^{10} + 1216379986349 T^{11} + 12664105586572 T^{12} + 78296720053043 T^{13} + 747345536131286 T^{14} + 4272703074590882 T^{15} + 37806653876981405 T^{16} + 200817044505771454 T^{17} + 1650886289314010774 T^{18} + 8129000366067083389 T^{19} + 61796795412789243532 T^{20} + \)\(27\!\cdots\!43\)\( T^{21} + \)\(19\!\cdots\!03\)\( T^{22} + \)\(78\!\cdots\!55\)\( T^{23} + \)\(49\!\cdots\!43\)\( T^{24} + \)\(17\!\cdots\!26\)\( T^{25} + \)\(98\!\cdots\!77\)\( T^{26} + \)\(28\!\cdots\!30\)\( T^{27} + \)\(14\!\cdots\!53\)\( T^{28} + \)\(30\!\cdots\!37\)\( T^{29} + \)\(12\!\cdots\!14\)\( T^{30} + \)\(15\!\cdots\!59\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 38 T + 1184 T^{2} + 26310 T^{3} + 509391 T^{4} + 8386228 T^{5} + 124803580 T^{6} + 1672888287 T^{7} + 20735239238 T^{8} + 237883859520 T^{9} + 2561046117903 T^{10} + 25915537319490 T^{11} + 248514138696738 T^{12} + 2261122284217522 T^{13} + 19622634338884856 T^{14} + 162527075194515100 T^{15} + 1289225740802462879 T^{16} + 9794244602586546037 T^{17} + 71420512046319014688 T^{18} + \)\(49\!\cdots\!96\)\( T^{19} + \)\(33\!\cdots\!36\)\( T^{20} + \)\(21\!\cdots\!33\)\( T^{21} + \)\(13\!\cdots\!17\)\( T^{22} + \)\(79\!\cdots\!00\)\( T^{23} + \)\(45\!\cdots\!92\)\( T^{24} + \)\(24\!\cdots\!38\)\( T^{25} + \)\(12\!\cdots\!94\)\( T^{26} + \)\(61\!\cdots\!90\)\( T^{27} + \)\(28\!\cdots\!01\)\( T^{28} + \)\(12\!\cdots\!80\)\( T^{29} + \)\(51\!\cdots\!14\)\( T^{30} + \)\(19\!\cdots\!67\)\( T^{31} + \)\(68\!\cdots\!60\)\( T^{32} + \)\(21\!\cdots\!32\)\( T^{33} + \)\(61\!\cdots\!13\)\( T^{34} + \)\(14\!\cdots\!10\)\( T^{35} + \)\(31\!\cdots\!08\)\( T^{36} + \)\(47\!\cdots\!82\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 4 T + 78 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 + 20 T + 657 T^{2} + 11019 T^{3} + 208449 T^{4} + 2968289 T^{5} + 42361540 T^{6} + 520856655 T^{7} + 6178485362 T^{8} + 66720949994 T^{9} + 687555638024 T^{10} + 6613477092993 T^{11} + 60572154748597 T^{12} + 524121427328877 T^{13} + 4320082712516532 T^{14} + 33825250778665414 T^{15} + 252463984338875259 T^{16} + 1792738291269266942 T^{17} + 12135112339458938388 T^{18} + 78029625736441221129 T^{19} + \)\(47\!\cdots\!57\)\( T^{20} + \)\(27\!\cdots\!49\)\( T^{21} + \)\(15\!\cdots\!96\)\( T^{22} + \)\(78\!\cdots\!78\)\( T^{23} + \)\(38\!\cdots\!82\)\( T^{24} + \)\(17\!\cdots\!15\)\( T^{25} + \)\(74\!\cdots\!60\)\( T^{26} + \)\(27\!\cdots\!33\)\( T^{27} + \)\(10\!\cdots\!09\)\( T^{28} + \)\(28\!\cdots\!87\)\( T^{29} + \)\(90\!\cdots\!33\)\( T^{30} + \)\(14\!\cdots\!40\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 12 T + 498 T^{2} + 5681 T^{3} + 124449 T^{4} + 1331062 T^{5} + 20665589 T^{6} + 205663693 T^{7} + 2559635404 T^{8} + 23661924533 T^{9} + 252816233095 T^{10} + 2178433687467 T^{11} + 20871463031289 T^{12} + 168853538722947 T^{13} + 1493519480228299 T^{14} + 11443665225955430 T^{15} + 95229402270526716 T^{16} + 696089475303278220 T^{17} + 5511017661440595772 T^{18} + 38566804465414974990 T^{19} + \)\(29\!\cdots\!16\)\( T^{20} + \)\(19\!\cdots\!80\)\( T^{21} + \)\(14\!\cdots\!32\)\( T^{22} + \)\(90\!\cdots\!30\)\( T^{23} + \)\(62\!\cdots\!07\)\( T^{24} + \)\(37\!\cdots\!63\)\( T^{25} + \)\(24\!\cdots\!93\)\( T^{26} + \)\(13\!\cdots\!87\)\( T^{27} + \)\(83\!\cdots\!35\)\( T^{28} + \)\(41\!\cdots\!17\)\( T^{29} + \)\(23\!\cdots\!88\)\( T^{30} + \)\(10\!\cdots\!13\)\( T^{31} + \)\(53\!\cdots\!97\)\( T^{32} + \)\(18\!\cdots\!78\)\( T^{33} + \)\(91\!\cdots\!93\)\( T^{34} + \)\(22\!\cdots\!01\)\( T^{35} + \)\(10\!\cdots\!74\)\( T^{36} + \)\(13\!\cdots\!68\)\( T^{37} + \)\(57\!\cdots\!17\)\( T^{38} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( 1 - 6 T + 450 T^{2} - 3104 T^{3} + 107556 T^{4} - 793021 T^{5} + 17956435 T^{6} - 134051810 T^{7} + 2320209526 T^{8} - 16928358836 T^{9} + 243556826818 T^{10} - 1701915137751 T^{11} + 21337965113765 T^{12} - 141090837953809 T^{13} + 1586076706524103 T^{14} - 9827215165558520 T^{15} + 100965940994665475 T^{16} - 579805694767952680 T^{17} + 5521133015410402543 T^{18} - 28977095208115338611 T^{19} + \)\(25\!\cdots\!65\)\( T^{20} - \)\(12\!\cdots\!49\)\( T^{21} + \)\(10\!\cdots\!38\)\( T^{22} - \)\(42\!\cdots\!84\)\( T^{23} + \)\(34\!\cdots\!46\)\( T^{24} - \)\(11\!\cdots\!90\)\( T^{25} + \)\(91\!\cdots\!35\)\( T^{26} - \)\(23\!\cdots\!39\)\( T^{27} + \)\(19\!\cdots\!36\)\( T^{28} - \)\(32\!\cdots\!16\)\( T^{29} + \)\(27\!\cdots\!50\)\( T^{30} - \)\(21\!\cdots\!94\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 8 T + 623 T^{2} + 4134 T^{3} + 190621 T^{4} + 1050085 T^{5} + 38427010 T^{6} + 175133655 T^{7} + 5779442895 T^{8} + 21722980813 T^{9} + 695588266125 T^{10} + 2159194946108 T^{11} + 69983480585344 T^{12} + 181207160345950 T^{13} + 6049854134885154 T^{14} + 13349726064923830 T^{15} + 456926455532493367 T^{16} + 888599919831587398 T^{17} + 30436547739405903524 T^{18} + 54456112835383097214 T^{19} + \)\(17\!\cdots\!16\)\( T^{20} + \)\(30\!\cdots\!38\)\( T^{21} + \)\(93\!\cdots\!93\)\( T^{22} + \)\(16\!\cdots\!30\)\( T^{23} + \)\(43\!\cdots\!46\)\( T^{24} + \)\(76\!\cdots\!50\)\( T^{25} + \)\(17\!\cdots\!36\)\( T^{26} + \)\(31\!\cdots\!68\)\( T^{27} + \)\(60\!\cdots\!75\)\( T^{28} + \)\(11\!\cdots\!13\)\( T^{29} + \)\(17\!\cdots\!05\)\( T^{30} + \)\(31\!\cdots\!55\)\( T^{31} + \)\(40\!\cdots\!90\)\( T^{32} + \)\(65\!\cdots\!85\)\( T^{33} + \)\(69\!\cdots\!79\)\( T^{34} + \)\(89\!\cdots\!94\)\( T^{35} + \)\(79\!\cdots\!37\)\( T^{36} + \)\(60\!\cdots\!68\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( ( 1 - 4 T + 61 T^{2} )^{2} \))(\( 1 + 49 T + 1445 T^{2} + 30643 T^{3} + 516127 T^{4} + 7207564 T^{5} + 85963998 T^{6} + 890252739 T^{7} + 8093675561 T^{8} + 64740574591 T^{9} + 452368767302 T^{10} + 2670323232598 T^{11} + 11946108049152 T^{12} + 19411519103057 T^{13} - 353341507738210 T^{14} - 5456325917254617 T^{15} - 50050491269126031 T^{16} - 332835880952531637 T^{17} - 1314783750293879410 T^{18} + 4406046017530980917 T^{19} + \)\(16\!\cdots\!32\)\( T^{20} + \)\(22\!\cdots\!98\)\( T^{21} + \)\(23\!\cdots\!22\)\( T^{22} + \)\(20\!\cdots\!11\)\( T^{23} + \)\(15\!\cdots\!41\)\( T^{24} + \)\(10\!\cdots\!99\)\( T^{25} + \)\(61\!\cdots\!98\)\( T^{26} + \)\(31\!\cdots\!04\)\( T^{27} + \)\(13\!\cdots\!67\)\( T^{28} + \)\(49\!\cdots\!83\)\( T^{29} + \)\(14\!\cdots\!45\)\( T^{30} + \)\(29\!\cdots\!49\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 31 T + 1122 T^{2} + 24944 T^{3} + 558039 T^{4} + 9922652 T^{5} + 171701709 T^{6} + 2579144347 T^{7} + 37371094549 T^{8} + 489412722267 T^{9} + 6171689935936 T^{10} + 71865812776928 T^{11} + 806096120003341 T^{12} + 8452965514454037 T^{13} + 85459573955135789 T^{14} + 813688455316642359 T^{15} + 7475528587401161751 T^{16} + 64951419522973826223 T^{17} + \)\(54\!\cdots\!71\)\( T^{18} + \)\(43\!\cdots\!32\)\( T^{19} + \)\(33\!\cdots\!31\)\( T^{20} + \)\(24\!\cdots\!83\)\( T^{21} + \)\(16\!\cdots\!31\)\( T^{22} + \)\(11\!\cdots\!19\)\( T^{23} + \)\(72\!\cdots\!89\)\( T^{24} + \)\(43\!\cdots\!57\)\( T^{25} + \)\(25\!\cdots\!61\)\( T^{26} + \)\(13\!\cdots\!68\)\( T^{27} + \)\(72\!\cdots\!76\)\( T^{28} + \)\(34\!\cdots\!67\)\( T^{29} + \)\(16\!\cdots\!89\)\( T^{30} + \)\(68\!\cdots\!87\)\( T^{31} + \)\(27\!\cdots\!29\)\( T^{32} + \)\(98\!\cdots\!32\)\( T^{33} + \)\(33\!\cdots\!39\)\( T^{34} + \)\(91\!\cdots\!84\)\( T^{35} + \)\(25\!\cdots\!62\)\( T^{36} + \)\(42\!\cdots\!11\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 12 T + 168 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 554 T^{2} + 1713 T^{3} + 147685 T^{4} + 287141 T^{5} + 25021566 T^{6} + 12599179 T^{7} + 3018876326 T^{8} - 3981375073 T^{9} + 278089867837 T^{10} - 1003383262359 T^{11} + 20871040870445 T^{12} - 130529192959041 T^{13} + 1390286487767268 T^{14} - 11769338625737108 T^{15} + 91279376974770109 T^{16} - 788545687924386236 T^{17} + 6240996043587266052 T^{18} - 39258351661940048283 T^{19} + \)\(42\!\cdots\!45\)\( T^{20} - \)\(13\!\cdots\!13\)\( T^{21} + \)\(25\!\cdots\!53\)\( T^{22} - \)\(24\!\cdots\!79\)\( T^{23} + \)\(12\!\cdots\!66\)\( T^{24} + \)\(34\!\cdots\!13\)\( T^{25} + \)\(45\!\cdots\!34\)\( T^{26} + \)\(35\!\cdots\!03\)\( T^{27} + \)\(12\!\cdots\!85\)\( T^{28} + \)\(93\!\cdots\!31\)\( T^{29} + \)\(20\!\cdots\!66\)\( T^{30} + \)\(98\!\cdots\!72\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 48 T + 1717 T^{2} + 43117 T^{3} + 911422 T^{4} + 15936542 T^{5} + 246468456 T^{6} + 3344821831 T^{7} + 41489567667 T^{8} + 469152360816 T^{9} + 5006605610104 T^{10} + 50423971923394 T^{11} + 493847225561446 T^{12} + 4675309865809282 T^{13} + 43588850508562878 T^{14} + 394233330201463681 T^{15} + 3502839888446862097 T^{16} + 30150280176851212568 T^{17} + \)\(25\!\cdots\!94\)\( T^{18} + \)\(21\!\cdots\!42\)\( T^{19} + \)\(17\!\cdots\!98\)\( T^{20} + \)\(13\!\cdots\!52\)\( T^{21} + \)\(10\!\cdots\!11\)\( T^{22} + \)\(79\!\cdots\!01\)\( T^{23} + \)\(58\!\cdots\!46\)\( T^{24} + \)\(42\!\cdots\!58\)\( T^{25} + \)\(29\!\cdots\!58\)\( T^{26} + \)\(20\!\cdots\!54\)\( T^{27} + \)\(13\!\cdots\!88\)\( T^{28} + \)\(85\!\cdots\!84\)\( T^{29} + \)\(50\!\cdots\!61\)\( T^{30} + \)\(27\!\cdots\!91\)\( T^{31} + \)\(13\!\cdots\!72\)\( T^{32} + \)\(58\!\cdots\!18\)\( T^{33} + \)\(22\!\cdots\!46\)\( T^{34} + \)\(71\!\cdots\!77\)\( T^{35} + \)\(18\!\cdots\!59\)\( T^{36} + \)\(35\!\cdots\!32\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + T + 71 T^{2} \))(\( 1 + 124 T^{2} + 5041 T^{4} \))(\( 1 + 29 T + 1067 T^{2} + 22461 T^{3} + 496204 T^{4} + 8372386 T^{5} + 140578765 T^{6} + 2003758465 T^{7} + 27871656891 T^{8} + 346350355143 T^{9} + 4166030041368 T^{10} + 46038827837200 T^{11} + 490677379082246 T^{12} + 4881574615448695 T^{13} + 46758782506118940 T^{14} + 421549755242357416 T^{15} + 3656604511028036343 T^{16} + 29930032622207376536 T^{17} + \)\(23\!\cdots\!40\)\( T^{18} + \)\(17\!\cdots\!45\)\( T^{19} + \)\(12\!\cdots\!26\)\( T^{20} + \)\(83\!\cdots\!00\)\( T^{21} + \)\(53\!\cdots\!28\)\( T^{22} + \)\(31\!\cdots\!13\)\( T^{23} + \)\(17\!\cdots\!51\)\( T^{24} + \)\(91\!\cdots\!15\)\( T^{25} + \)\(45\!\cdots\!65\)\( T^{26} + \)\(19\!\cdots\!06\)\( T^{27} + \)\(81\!\cdots\!64\)\( T^{28} + \)\(26\!\cdots\!71\)\( T^{29} + \)\(88\!\cdots\!27\)\( T^{30} + \)\(17\!\cdots\!79\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 + 14 T + 779 T^{2} + 8545 T^{3} + 291081 T^{4} + 2661015 T^{5} + 72027972 T^{6} + 567475811 T^{7} + 13419574683 T^{8} + 92940837218 T^{9} + 2008722438501 T^{10} + 12400654036369 T^{11} + 250831483501791 T^{12} + 1396667359982226 T^{13} + 26750504765591477 T^{14} + 135880285433004786 T^{15} + 2472988641415101069 T^{16} + 11589377862718551888 T^{17} + \)\(19\!\cdots\!24\)\( T^{18} + \)\(87\!\cdots\!08\)\( T^{19} + \)\(14\!\cdots\!04\)\( T^{20} + \)\(58\!\cdots\!08\)\( T^{21} + \)\(88\!\cdots\!59\)\( T^{22} + \)\(34\!\cdots\!66\)\( T^{23} + \)\(48\!\cdots\!27\)\( T^{24} + \)\(17\!\cdots\!46\)\( T^{25} + \)\(22\!\cdots\!81\)\( T^{26} + \)\(80\!\cdots\!09\)\( T^{27} + \)\(92\!\cdots\!31\)\( T^{28} + \)\(30\!\cdots\!18\)\( T^{29} + \)\(31\!\cdots\!93\)\( T^{30} + \)\(93\!\cdots\!51\)\( T^{31} + \)\(83\!\cdots\!92\)\( T^{32} + \)\(22\!\cdots\!15\)\( T^{33} + \)\(17\!\cdots\!31\)\( T^{34} + \)\(35\!\cdots\!45\)\( T^{35} + \)\(23\!\cdots\!89\)\( T^{36} + \)\(29\!\cdots\!54\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \))
$73$ (\( 1 + 4 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 + 24 T + 272 T^{2} + 1752 T^{3} + 5329 T^{4} \))(\( 1 + 21 T + 843 T^{2} + 12706 T^{3} + 301429 T^{4} + 3562537 T^{5} + 64991253 T^{6} + 633326442 T^{7} + 9968974435 T^{8} + 83364066381 T^{9} + 1212228806064 T^{10} + 9033863092764 T^{11} + 125272833116271 T^{12} + 856091131617299 T^{13} + 11300665713265806 T^{14} + 71642851343601187 T^{15} + 886933815975108711 T^{16} + 5229928148082886651 T^{17} + 60221247585993480174 T^{18} + \)\(33\!\cdots\!83\)\( T^{19} + \)\(35\!\cdots\!11\)\( T^{20} + \)\(18\!\cdots\!52\)\( T^{21} + \)\(18\!\cdots\!96\)\( T^{22} + \)\(92\!\cdots\!57\)\( T^{23} + \)\(80\!\cdots\!35\)\( T^{24} + \)\(37\!\cdots\!46\)\( T^{25} + \)\(27\!\cdots\!97\)\( T^{26} + \)\(11\!\cdots\!49\)\( T^{27} + \)\(69\!\cdots\!09\)\( T^{28} + \)\(21\!\cdots\!98\)\( T^{29} + \)\(10\!\cdots\!87\)\( T^{30} + \)\(18\!\cdots\!97\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 50 T + 2241 T^{2} + 68742 T^{3} + 1893273 T^{4} + 43395218 T^{5} + 909552358 T^{6} + 16911002226 T^{7} + 291602390134 T^{8} + 4593236943696 T^{9} + 67766568645023 T^{10} + 928031111930849 T^{11} + 11985009346299019 T^{12} + 145040527345410188 T^{13} + 1662944381085034554 T^{14} + 17970372965720692341 T^{15} + \)\(18\!\cdots\!69\)\( T^{16} + \)\(17\!\cdots\!83\)\( T^{17} + \)\(16\!\cdots\!48\)\( T^{18} + \)\(14\!\cdots\!46\)\( T^{19} + \)\(12\!\cdots\!04\)\( T^{20} + \)\(95\!\cdots\!07\)\( T^{21} + \)\(71\!\cdots\!73\)\( T^{22} + \)\(51\!\cdots\!81\)\( T^{23} + \)\(34\!\cdots\!22\)\( T^{24} + \)\(21\!\cdots\!32\)\( T^{25} + \)\(13\!\cdots\!43\)\( T^{26} + \)\(74\!\cdots\!69\)\( T^{27} + \)\(39\!\cdots\!99\)\( T^{28} + \)\(19\!\cdots\!04\)\( T^{29} + \)\(91\!\cdots\!18\)\( T^{30} + \)\(38\!\cdots\!46\)\( T^{31} + \)\(15\!\cdots\!14\)\( T^{32} + \)\(52\!\cdots\!62\)\( T^{33} + \)\(16\!\cdots\!61\)\( T^{34} + \)\(44\!\cdots\!62\)\( T^{35} + \)\(10\!\cdots\!73\)\( T^{36} + \)\(17\!\cdots\!50\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( 1 - 4 T + 90 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 + 70 T + 3181 T^{2} + 106206 T^{3} + 2903742 T^{4} + 67363691 T^{5} + 1370556324 T^{6} + 24863746244 T^{7} + 408341937164 T^{8} + 6127085067015 T^{9} + 84697311482740 T^{10} + 1084494938901919 T^{11} + 12925303174447985 T^{12} + 143837961451830280 T^{13} + 1498705028538343730 T^{14} + 14642810470765089212 T^{15} + \)\(13\!\cdots\!83\)\( T^{16} + \)\(11\!\cdots\!48\)\( T^{17} + \)\(93\!\cdots\!30\)\( T^{18} + \)\(70\!\cdots\!20\)\( T^{19} + \)\(50\!\cdots\!85\)\( T^{20} + \)\(33\!\cdots\!81\)\( T^{21} + \)\(20\!\cdots\!40\)\( T^{22} + \)\(11\!\cdots\!85\)\( T^{23} + \)\(61\!\cdots\!04\)\( T^{24} + \)\(29\!\cdots\!36\)\( T^{25} + \)\(12\!\cdots\!24\)\( T^{26} + \)\(50\!\cdots\!89\)\( T^{27} + \)\(17\!\cdots\!22\)\( T^{28} + \)\(49\!\cdots\!34\)\( T^{29} + \)\(11\!\cdots\!61\)\( T^{30} + \)\(20\!\cdots\!30\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 + 21 T + 944 T^{2} + 18089 T^{3} + 452292 T^{4} + 7763774 T^{5} + 144851463 T^{6} + 2217699998 T^{7} + 34486539329 T^{8} + 473807892376 T^{9} + 6456799768996 T^{10} + 80426375979059 T^{11} + 984733126454588 T^{12} + 11226131368696991 T^{13} + 125250028977848806 T^{14} + 1315405987489981678 T^{15} + 13491883597159234565 T^{16} + \)\(13\!\cdots\!13\)\( T^{17} + \)\(12\!\cdots\!90\)\( T^{18} + \)\(11\!\cdots\!58\)\( T^{19} + \)\(98\!\cdots\!10\)\( T^{20} + \)\(81\!\cdots\!33\)\( T^{21} + \)\(66\!\cdots\!35\)\( T^{22} + \)\(51\!\cdots\!18\)\( T^{23} + \)\(38\!\cdots\!94\)\( T^{24} + \)\(27\!\cdots\!11\)\( T^{25} + \)\(18\!\cdots\!92\)\( T^{26} + \)\(12\!\cdots\!99\)\( T^{27} + \)\(77\!\cdots\!24\)\( T^{28} + \)\(44\!\cdots\!76\)\( T^{29} + \)\(25\!\cdots\!91\)\( T^{30} + \)\(13\!\cdots\!18\)\( T^{31} + \)\(67\!\cdots\!57\)\( T^{32} + \)\(28\!\cdots\!94\)\( T^{33} + \)\(13\!\cdots\!08\)\( T^{34} + \)\(41\!\cdots\!69\)\( T^{35} + \)\(17\!\cdots\!96\)\( T^{36} + \)\(30\!\cdots\!81\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 4 T + 162 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 - 5 T + 788 T^{2} - 4064 T^{3} + 314344 T^{4} - 1642590 T^{5} + 83715189 T^{6} - 437000895 T^{7} + 16622508813 T^{8} - 85505455581 T^{9} + 2607632114640 T^{10} - 13029710377055 T^{11} + 334369739375426 T^{12} - 1597484401188633 T^{13} + 35766121959269287 T^{14} - 160481327234523664 T^{15} + 3226751470146674609 T^{16} - 13319950160465464112 T^{17} + \)\(24\!\cdots\!43\)\( T^{18} - \)\(91\!\cdots\!71\)\( T^{19} + \)\(15\!\cdots\!46\)\( T^{20} - \)\(51\!\cdots\!65\)\( T^{21} + \)\(85\!\cdots\!60\)\( T^{22} - \)\(23\!\cdots\!87\)\( T^{23} + \)\(37\!\cdots\!33\)\( T^{24} - \)\(81\!\cdots\!85\)\( T^{25} + \)\(12\!\cdots\!61\)\( T^{26} - \)\(21\!\cdots\!30\)\( T^{27} + \)\(33\!\cdots\!84\)\( T^{28} - \)\(36\!\cdots\!32\)\( T^{29} + \)\(58\!\cdots\!52\)\( T^{30} - \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 15 T + 961 T^{2} + 12799 T^{3} + 440593 T^{4} + 5316495 T^{5} + 129795752 T^{6} + 1442336500 T^{7} + 27937659990 T^{8} + 289445505106 T^{9} + 4735223949496 T^{10} + 46100502808821 T^{11} + 663851101393194 T^{12} + 6096229613201484 T^{13} + 79588921240804102 T^{14} + 690292219769847759 T^{15} + 8343587301880265488 T^{16} + 68396654622218142114 T^{17} + \)\(77\!\cdots\!47\)\( T^{18} + \)\(60\!\cdots\!26\)\( T^{19} + \)\(64\!\cdots\!01\)\( T^{20} + \)\(47\!\cdots\!46\)\( T^{21} + \)\(47\!\cdots\!56\)\( T^{22} + \)\(32\!\cdots\!39\)\( T^{23} + \)\(31\!\cdots\!86\)\( T^{24} + \)\(19\!\cdots\!96\)\( T^{25} + \)\(18\!\cdots\!38\)\( T^{26} + \)\(10\!\cdots\!61\)\( T^{27} + \)\(88\!\cdots\!88\)\( T^{28} + \)\(44\!\cdots\!94\)\( T^{29} + \)\(35\!\cdots\!30\)\( T^{30} + \)\(15\!\cdots\!00\)\( T^{31} + \)\(11\!\cdots\!76\)\( T^{32} + \)\(39\!\cdots\!55\)\( T^{33} + \)\(26\!\cdots\!51\)\( T^{34} + \)\(64\!\cdots\!19\)\( T^{35} + \)\(40\!\cdots\!03\)\( T^{36} + \)\(52\!\cdots\!35\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \))(\( 1 + 8 T + 788 T^{2} + 5622 T^{3} + 314928 T^{4} + 1982365 T^{5} + 84101576 T^{6} + 462998681 T^{7} + 16774707768 T^{8} + 80263731785 T^{9} + 2656947249836 T^{10} + 11051808048320 T^{11} + 347589138735910 T^{12} + 1271156624445054 T^{13} + 38565010417995184 T^{14} + 127083106064549923 T^{15} + 3687295841422531449 T^{16} + 11310396439744943147 T^{17} + \)\(30\!\cdots\!64\)\( T^{18} + \)\(89\!\cdots\!26\)\( T^{19} + \)\(21\!\cdots\!10\)\( T^{20} + \)\(61\!\cdots\!80\)\( T^{21} + \)\(13\!\cdots\!96\)\( T^{22} + \)\(35\!\cdots\!65\)\( T^{23} + \)\(66\!\cdots\!08\)\( T^{24} + \)\(16\!\cdots\!29\)\( T^{25} + \)\(26\!\cdots\!76\)\( T^{26} + \)\(55\!\cdots\!85\)\( T^{27} + \)\(77\!\cdots\!88\)\( T^{28} + \)\(12\!\cdots\!18\)\( T^{29} + \)\(15\!\cdots\!08\)\( T^{30} + \)\(13\!\cdots\!92\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 + 10 T + 841 T^{2} + 10802 T^{3} + 380943 T^{4} + 5398895 T^{5} + 123188463 T^{6} + 1745339034 T^{7} + 31053428600 T^{8} + 418547777381 T^{9} + 6331918577836 T^{10} + 79681045158127 T^{11} + 1068423636074732 T^{12} + 12490209809083642 T^{13} + 151640732674117136 T^{14} + 1646257209054592284 T^{15} + 18309998252800948513 T^{16} + \)\(18\!\cdots\!66\)\( T^{17} + \)\(18\!\cdots\!55\)\( T^{18} + \)\(17\!\cdots\!94\)\( T^{19} + \)\(16\!\cdots\!95\)\( T^{20} + \)\(14\!\cdots\!86\)\( T^{21} + \)\(12\!\cdots\!97\)\( T^{22} + \)\(10\!\cdots\!44\)\( T^{23} + \)\(84\!\cdots\!64\)\( T^{24} + \)\(62\!\cdots\!62\)\( T^{25} + \)\(47\!\cdots\!28\)\( T^{26} + \)\(31\!\cdots\!87\)\( T^{27} + \)\(22\!\cdots\!24\)\( T^{28} + \)\(13\!\cdots\!81\)\( T^{29} + \)\(86\!\cdots\!00\)\( T^{30} + \)\(43\!\cdots\!14\)\( T^{31} + \)\(27\!\cdots\!47\)\( T^{32} + \)\(10\!\cdots\!95\)\( T^{33} + \)\(66\!\cdots\!07\)\( T^{34} + \)\(16\!\cdots\!22\)\( T^{35} + \)\(11\!\cdots\!89\)\( T^{36} + \)\(12\!\cdots\!10\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4} \))(\( 1 + 30 T + 1384 T^{2} + 30095 T^{3} + 821624 T^{4} + 14435275 T^{5} + 298548880 T^{6} + 4482639379 T^{7} + 76950885206 T^{8} + 1018756644687 T^{9} + 15174072985234 T^{10} + 180306557628117 T^{11} + 2387553095005840 T^{12} + 25718042088085683 T^{13} + 307027229025665160 T^{14} + 3012399828928057707 T^{15} + 32674113638507094703 T^{16} + \)\(29\!\cdots\!79\)\( T^{17} + \)\(28\!\cdots\!40\)\( T^{18} + \)\(23\!\cdots\!59\)\( T^{19} + \)\(21\!\cdots\!40\)\( T^{20} + \)\(15\!\cdots\!69\)\( T^{21} + \)\(12\!\cdots\!86\)\( T^{22} + \)\(82\!\cdots\!31\)\( T^{23} + \)\(60\!\cdots\!66\)\( T^{24} + \)\(34\!\cdots\!43\)\( T^{25} + \)\(22\!\cdots\!20\)\( T^{26} + \)\(10\!\cdots\!75\)\( T^{27} + \)\(57\!\cdots\!84\)\( T^{28} + \)\(20\!\cdots\!15\)\( T^{29} + \)\(90\!\cdots\!96\)\( T^{30} + \)\(18\!\cdots\!90\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 42 T + 2073 T^{2} + 61101 T^{3} + 1819039 T^{4} + 42176082 T^{5} + 953979593 T^{6} + 18442895655 T^{7} + 344683597024 T^{8} + 5751894330776 T^{9} + 92654276073252 T^{10} + 1364998464761074 T^{11} + 19431073990884714 T^{12} + 256637619724453180 T^{13} + 3280668004994934464 T^{14} + 39259304460589677610 T^{15} + \)\(45\!\cdots\!61\)\( T^{16} + \)\(49\!\cdots\!18\)\( T^{17} + \)\(52\!\cdots\!87\)\( T^{18} + \)\(52\!\cdots\!84\)\( T^{19} + \)\(51\!\cdots\!39\)\( T^{20} + \)\(46\!\cdots\!62\)\( T^{21} + \)\(41\!\cdots\!53\)\( T^{22} + \)\(34\!\cdots\!10\)\( T^{23} + \)\(28\!\cdots\!48\)\( T^{24} + \)\(21\!\cdots\!20\)\( T^{25} + \)\(15\!\cdots\!82\)\( T^{26} + \)\(10\!\cdots\!14\)\( T^{27} + \)\(70\!\cdots\!84\)\( T^{28} + \)\(42\!\cdots\!24\)\( T^{29} + \)\(24\!\cdots\!72\)\( T^{30} + \)\(12\!\cdots\!55\)\( T^{31} + \)\(64\!\cdots\!61\)\( T^{32} + \)\(27\!\cdots\!58\)\( T^{33} + \)\(11\!\cdots\!27\)\( T^{34} + \)\(37\!\cdots\!21\)\( T^{35} + \)\(12\!\cdots\!01\)\( T^{36} + \)\(24\!\cdots\!38\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \))
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