Properties

Label 4012.2
Level 4012
Weight 2
Dimension 288414
Nonzero newspaces 20
Sturm bound 2004480

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Defining parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 20 \)
Sturm bound: \(2004480\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(4012))\).

Total New Old
Modular forms 505760 291830 213930
Cusp forms 496481 288414 208067
Eisenstein series 9279 3416 5863

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(4012))\)

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
4012.2.a \(\chi_{4012}(1, \cdot)\) 4012.2.a.a 1 1
4012.2.a.b 1
4012.2.a.c 1
4012.2.a.d 2
4012.2.a.e 2
4012.2.a.f 3
4012.2.a.g 12
4012.2.a.h 15
4012.2.a.i 18
4012.2.a.j 21
4012.2.b \(\chi_{4012}(237, \cdot)\) 4012.2.b.a 40 1
4012.2.b.b 46
4012.2.e \(\chi_{4012}(3775, \cdot)\) n/a 480 1
4012.2.f \(\chi_{4012}(4011, \cdot)\) n/a 536 1
4012.2.j \(\chi_{4012}(1415, \cdot)\) n/a 1072 2
4012.2.k \(\chi_{4012}(1653, \cdot)\) n/a 172 2
4012.2.n \(\chi_{4012}(1181, \cdot)\) n/a 352 4
4012.2.o \(\chi_{4012}(943, \cdot)\) n/a 2144 4
4012.2.q \(\chi_{4012}(827, \cdot)\) n/a 4176 8
4012.2.t \(\chi_{4012}(589, \cdot)\) n/a 720 8
4012.2.u \(\chi_{4012}(137, \cdot)\) n/a 2240 28
4012.2.x \(\chi_{4012}(67, \cdot)\) n/a 15008 28
4012.2.y \(\chi_{4012}(103, \cdot)\) n/a 13440 28
4012.2.bb \(\chi_{4012}(169, \cdot)\) n/a 2520 28
4012.2.bd \(\chi_{4012}(21, \cdot)\) n/a 5040 56
4012.2.be \(\chi_{4012}(47, \cdot)\) n/a 30016 56
4012.2.bh \(\chi_{4012}(43, \cdot)\) n/a 60032 112
4012.2.bi \(\chi_{4012}(9, \cdot)\) n/a 10080 112
4012.2.bk \(\chi_{4012}(37, \cdot)\) n/a 20160 224
4012.2.bn \(\chi_{4012}(3, \cdot)\) n/a 120064 224

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(4012))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(4012)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(59))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(118))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(236))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1003))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2006))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( ( 1 + T + 3 T^{2} )^{3} \))(\( 1 + 4 T + 23 T^{2} + 72 T^{3} + 252 T^{4} + 649 T^{5} + 1738 T^{6} + 3847 T^{7} + 8630 T^{8} + 16986 T^{9} + 33622 T^{10} + 60443 T^{11} + 109222 T^{12} + 181329 T^{13} + 302598 T^{14} + 458622 T^{15} + 699030 T^{16} + 934821 T^{17} + 1267002 T^{18} + 1419363 T^{19} + 1653372 T^{20} + 1417176 T^{21} + 1358127 T^{22} + 708588 T^{23} + 531441 T^{24} \))(\( 1 + T + 15 T^{2} + 26 T^{3} + 141 T^{4} + 282 T^{5} + 1017 T^{6} + 2096 T^{7} + 5832 T^{8} + 12052 T^{9} + 27823 T^{10} + 55768 T^{11} + 113005 T^{12} + 215053 T^{13} + 392472 T^{14} + 701652 T^{15} + 1177416 T^{16} + 1935477 T^{17} + 3051135 T^{18} + 4517208 T^{19} + 6760989 T^{20} + 8785908 T^{21} + 12754584 T^{22} + 13751856 T^{23} + 20017611 T^{24} + 16651818 T^{25} + 24977727 T^{26} + 13817466 T^{27} + 23914845 T^{28} + 4782969 T^{29} + 14348907 T^{30} \))(\( 1 - 8 T + 50 T^{2} - 230 T^{3} + 920 T^{4} - 3187 T^{5} + 10085 T^{6} - 29170 T^{7} + 79079 T^{8} - 200978 T^{9} + 485598 T^{10} - 1115098 T^{11} + 2454896 T^{12} - 5177071 T^{13} + 10519717 T^{14} - 20568850 T^{15} + 38860500 T^{16} - 70812848 T^{17} + 124841396 T^{18} - 212438544 T^{19} + 349744500 T^{20} - 555358950 T^{21} + 852097077 T^{22} - 1258028253 T^{23} + 1789619184 T^{24} - 2438719326 T^{25} + 3186008478 T^{26} - 3955849974 T^{27} + 4669535871 T^{28} - 5167377990 T^{29} + 5359582485 T^{30} - 5081107401 T^{31} + 4400331480 T^{32} - 3300248610 T^{33} + 2152336050 T^{34} - 1033121304 T^{35} + 387420489 T^{36} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - 3 T + 11 T^{2} - 15 T^{3} + 25 T^{4} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( 1 - T - 2 T^{2} - T^{3} - 10 T^{4} - 25 T^{5} + 125 T^{6} \))(\( 1 - 3 T + 32 T^{2} - 73 T^{3} + 462 T^{4} - 824 T^{5} + 4263 T^{6} - 6398 T^{7} + 31078 T^{8} - 43703 T^{9} + 198957 T^{10} - 268946 T^{11} + 1092688 T^{12} - 1344730 T^{13} + 4973925 T^{14} - 5462875 T^{15} + 19423750 T^{16} - 19993750 T^{17} + 66609375 T^{18} - 64375000 T^{19} + 180468750 T^{20} - 142578125 T^{21} + 312500000 T^{22} - 146484375 T^{23} + 244140625 T^{24} \))(\( 1 - T + 36 T^{2} - 34 T^{3} + 652 T^{4} - 602 T^{5} + 7984 T^{6} - 7465 T^{7} + 75179 T^{8} - 71730 T^{9} + 582334 T^{10} - 556052 T^{11} + 3842951 T^{12} - 3564841 T^{13} + 21988347 T^{14} - 19305638 T^{15} + 109941735 T^{16} - 89121025 T^{17} + 480368875 T^{18} - 347532500 T^{19} + 1819793750 T^{20} - 1120781250 T^{21} + 5873359375 T^{22} - 2916015625 T^{23} + 15593750000 T^{24} - 5878906250 T^{25} + 31835937500 T^{26} - 8300781250 T^{27} + 43945312500 T^{28} - 6103515625 T^{29} + 30517578125 T^{30} \))(\( 1 - 4 T + 45 T^{2} - 145 T^{3} + 937 T^{4} - 2417 T^{5} + 11954 T^{6} - 24260 T^{7} + 106142 T^{8} - 163121 T^{9} + 731321 T^{10} - 818737 T^{11} + 4488760 T^{12} - 3939650 T^{13} + 27672889 T^{14} - 23572750 T^{15} + 168467946 T^{16} - 146318404 T^{17} + 914054586 T^{18} - 731592020 T^{19} + 4211698650 T^{20} - 2946593750 T^{21} + 17295555625 T^{22} - 12311406250 T^{23} + 70136875000 T^{24} - 63963828125 T^{25} + 285672265625 T^{26} - 318595703125 T^{27} + 1036542968750 T^{28} - 1184570312500 T^{29} + 2918457031250 T^{30} - 2950439453125 T^{31} + 5718994140625 T^{32} - 4425048828125 T^{33} + 6866455078125 T^{34} - 3051757812500 T^{35} + 3814697265625 T^{36} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( 1 + 9 T^{2} + 49 T^{4} \))(\( ( 1 - T + 7 T^{2} )^{3} \))(\( 1 - 2 T + 39 T^{2} - 89 T^{3} + 869 T^{4} - 2026 T^{5} + 13586 T^{6} - 31006 T^{7} + 163043 T^{8} - 352767 T^{9} + 1557036 T^{10} - 3119131 T^{11} + 12073898 T^{12} - 21833917 T^{13} + 76294764 T^{14} - 120999081 T^{15} + 391466243 T^{16} - 521117842 T^{17} + 1598379314 T^{18} - 1668498118 T^{19} + 5009612069 T^{20} - 3591471023 T^{21} + 11016534711 T^{22} - 3954653486 T^{23} + 13841287201 T^{24} \))(\( 1 + 11 T + 104 T^{2} + 688 T^{3} + 4116 T^{4} + 20763 T^{5} + 97477 T^{6} + 410898 T^{7} + 1637282 T^{8} + 6025558 T^{9} + 21140109 T^{10} + 69476752 T^{11} + 218679986 T^{12} + 649484690 T^{13} + 1851901833 T^{14} + 4998625284 T^{15} + 12963312831 T^{16} + 31824749810 T^{17} + 75007235198 T^{18} + 166813681552 T^{19} + 355301811963 T^{20} + 708900873142 T^{21} + 1348372130126 T^{22} + 2368745201298 T^{23} + 3933548549539 T^{24} + 5865033594987 T^{25} + 8138676874188 T^{26} + 9522805594288 T^{27} + 10076457082328 T^{28} + 7460453801339 T^{29} + 4747561509943 T^{30} \))(\( 1 - 2 T + 51 T^{2} - 65 T^{3} + 1205 T^{4} - 802 T^{5} + 18544 T^{6} - 3658 T^{7} + 224973 T^{8} + 12111 T^{9} + 2416962 T^{10} + 256165 T^{11} + 23664095 T^{12} + 2634873 T^{13} + 208225621 T^{14} + 33579812 T^{15} + 1648391180 T^{16} + 356324386 T^{17} + 11975041192 T^{18} + 2494270702 T^{19} + 80771167820 T^{20} + 11517875516 T^{21} + 499949716021 T^{22} + 44284310511 T^{23} + 2784057112655 T^{24} + 210962892595 T^{25} + 13933304954562 T^{26} + 488722534377 T^{27} + 63549304193277 T^{28} - 7233061225894 T^{29} + 256672829855344 T^{30} - 77704986346414 T^{31} + 817258802783045 T^{32} - 308591498146295 T^{33} + 1694879459049651 T^{34} - 465261027974414 T^{35} + 1628413597910449 T^{36} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - T + 21 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 2 T^{2} + 121 T^{4} \))(\( ( 1 + 2 T + 11 T^{2} )^{3} \))(\( 1 + 5 T + 74 T^{2} + 367 T^{3} + 2889 T^{4} + 13297 T^{5} + 76373 T^{6} + 319559 T^{7} + 1507262 T^{8} + 5679553 T^{9} + 23251272 T^{10} + 78597244 T^{11} + 285742952 T^{12} + 864569684 T^{13} + 2813403912 T^{14} + 7559485043 T^{15} + 22067822942 T^{16} + 51465296509 T^{17} + 135299428253 T^{18} + 259120912787 T^{19} + 619282807209 T^{20} + 865366802597 T^{21} + 1919369420474 T^{22} + 1426558353055 T^{23} + 3138428376721 T^{24} \))(\( 1 + 12 T + 133 T^{2} + 1047 T^{3} + 7530 T^{4} + 46565 T^{5} + 266674 T^{6} + 1388283 T^{7} + 6785876 T^{8} + 30896141 T^{9} + 133432541 T^{10} + 543813670 T^{11} + 2116387758 T^{12} + 7831681555 T^{13} + 27783012053 T^{14} + 94052272884 T^{15} + 305613132583 T^{16} + 947633468155 T^{17} + 2816912105898 T^{18} + 7961975942470 T^{19} + 21489444160591 T^{20} + 54734398446101 T^{21} + 132237525996796 T^{22} + 297590790391323 T^{23} + 628803342549734 T^{24} + 1207776176545565 T^{25} + 2148396879700830 T^{26} + 3285934510426887 T^{27} + 4591520715142823 T^{28} + 4556998002998892 T^{29} + 4177248169415651 T^{30} \))(\( 1 - 12 T + 161 T^{2} - 1301 T^{3} + 10576 T^{4} - 66635 T^{5} + 413157 T^{6} - 2178965 T^{7} + 11262499 T^{8} - 52010930 T^{9} + 235750895 T^{10} - 985721475 T^{11} + 4055381848 T^{12} - 15731416070 T^{13} + 60079018449 T^{14} - 219244348355 T^{15} + 786132762890 T^{16} - 2710197223729 T^{17} + 9159661384640 T^{18} - 29812169461019 T^{19} + 95122064309690 T^{20} - 291814227660505 T^{21} + 879616909111809 T^{22} - 2533560289489570 T^{23} + 7184356322024728 T^{24} - 19208922941697225 T^{25} + 50535298046948495 T^{26} - 122639052300262630 T^{27} + 292120218631337899 T^{28} - 621684144352897615 T^{29} + 1296663652840918197 T^{30} - 2300420923710842185 T^{31} + 4016234239976356816 T^{32} - 5434599868409761951 T^{33} + 7397906508035117921 T^{34} - 6065364341991525252 T^{35} + 5559917313492231481 T^{36} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 5 T + 31 T^{2} + 65 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 23 T^{2} - 44 T^{3} + 299 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 - 9 T + 103 T^{2} - 583 T^{3} + 4242 T^{4} - 19416 T^{5} + 117521 T^{6} - 478145 T^{7} + 2549845 T^{8} - 9334119 T^{9} + 44278587 T^{10} - 146176544 T^{11} + 629059858 T^{12} - 1900295072 T^{13} + 7483081203 T^{14} - 20507059443 T^{15} + 72826123045 T^{16} - 177531891485 T^{17} + 567251420489 T^{18} - 1218325206072 T^{19} + 3460329718482 T^{20} - 6182423134459 T^{21} + 14199424660447 T^{22} - 16129443546333 T^{23} + 23298085122481 T^{24} \))(\( 1 + 10 T + 140 T^{2} + 994 T^{3} + 8547 T^{4} + 48921 T^{5} + 326579 T^{6} + 1601994 T^{7} + 9073914 T^{8} + 39552951 T^{9} + 198844678 T^{10} + 787385768 T^{11} + 3597109908 T^{12} + 13089054362 T^{13} + 54935615381 T^{14} + 184451327824 T^{15} + 714162999953 T^{16} + 2212050187178 T^{17} + 7902850467876 T^{18} + 22488524919848 T^{19} + 73829637028654 T^{20} + 190914539863359 T^{21} + 569374646885538 T^{22} + 1306795720657674 T^{23} + 3463206800734967 T^{24} + 6744175279744929 T^{25} + 15317594887834239 T^{26} + 23158296611746114 T^{27} + 42402514922915420 T^{28} + 39373763856992890 T^{29} + 51185893014090757 T^{30} \))(\( 1 - 2 T + 87 T^{2} - 164 T^{3} + 3586 T^{4} - 5935 T^{5} + 94897 T^{6} - 129843 T^{7} + 1884148 T^{8} - 2115714 T^{9} + 31478368 T^{10} - 31635536 T^{11} + 472356619 T^{12} - 455403655 T^{13} + 6445148473 T^{14} - 5614565331 T^{15} + 81572613926 T^{16} - 61294787796 T^{17} + 1032214875838 T^{18} - 796832241348 T^{19} + 13785771753494 T^{20} - 12335200032207 T^{21} + 184079885537353 T^{22} - 169088189275915 T^{23} + 2279975179798771 T^{24} - 1985082968500112 T^{25} + 25677871824543328 T^{26} - 22436087786447322 T^{27} + 259745801700309652 T^{28} - 232699482042946191 T^{29} + 2210918383868079457 T^{30} - 1797563757625021555 T^{31} + 14119431719117650354 T^{32} - 8394486454310884148 T^{33} + 57891244998936646167 T^{34} - 17300831838762675866 T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \))
$17$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{12} \))(\( ( 1 - T )^{15} \))(\( ( 1 + T )^{18} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 33 T^{2} + 361 T^{4} \))(\( 1 - 7 T^{2} + 361 T^{4} \))(\( 1 + 7 T + 56 T^{2} + 235 T^{3} + 1064 T^{4} + 2527 T^{5} + 6859 T^{6} \))(\( 1 + 15 T + 227 T^{2} + 2177 T^{3} + 20285 T^{4} + 148441 T^{5} + 1059503 T^{6} + 6406841 T^{7} + 38109495 T^{8} + 199026663 T^{9} + 1030099984 T^{10} + 4762169337 T^{11} + 21905988513 T^{12} + 90481217403 T^{13} + 371866094224 T^{14} + 1365123881517 T^{15} + 4966467497895 T^{16} + 15863972593259 T^{17} + 49845252057143 T^{18} + 132687214808899 T^{19} + 344511576286685 T^{20} + 702491118064883 T^{21} + 1391752040520827 T^{22} + 1747353883473285 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 83 T^{2} + 97 T^{3} + 4154 T^{4} + 9003 T^{5} + 152790 T^{6} + 455226 T^{7} + 4711130 T^{8} + 16151252 T^{9} + 126895345 T^{10} + 448984987 T^{11} + 3060918579 T^{12} + 10411079176 T^{13} + 65732842410 T^{14} + 210170897854 T^{15} + 1248924005790 T^{16} + 3758399582536 T^{17} + 20994840533361 T^{18} + 58512172490827 T^{19} + 314205436859155 T^{20} + 759849879593012 T^{21} + 4211145965755070 T^{22} + 7731359468902266 T^{23} + 49303453343653410 T^{24} + 55197989518982403 T^{25} + 483900535463201726 T^{26} + 214691547149417617 T^{27} + 3490397627367335897 T^{28} + 15181127029874798299 T^{30} \))(\( 1 - 5 T + 178 T^{2} - 882 T^{3} + 16008 T^{4} - 77838 T^{5} + 966687 T^{6} - 4563465 T^{7} + 43947265 T^{8} - 199555023 T^{9} + 1600591839 T^{10} - 6942109298 T^{11} + 48599184196 T^{12} - 200399364677 T^{13} + 1265081257684 T^{14} - 4945951234609 T^{15} + 28797125449257 T^{16} - 106526874277791 T^{17} + 580299754253631 T^{18} - 2024010611278029 T^{19} + 10395762287181777 T^{20} - 33924279518183131 T^{21} + 164866654582636564 T^{22} - 496208666477355023 T^{23} + 2286391436382096676 T^{24} - 6205355310531329222 T^{25} + 27183752400566622399 T^{26} - 64393950952105393917 T^{27} + \)\(26\!\cdots\!65\)\( T^{28} - \)\(53\!\cdots\!35\)\( T^{29} + \)\(21\!\cdots\!07\)\( T^{30} - \)\(32\!\cdots\!42\)\( T^{31} + \)\(12\!\cdots\!68\)\( T^{32} - \)\(13\!\cdots\!18\)\( T^{33} + \)\(51\!\cdots\!18\)\( T^{34} - \)\(27\!\cdots\!95\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - 7 T + 57 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 + 26 T^{2} + 529 T^{4} \))(\( 1 + 33 T^{2} + 8 T^{3} + 759 T^{4} + 12167 T^{6} \))(\( 1 + 22 T + 367 T^{2} + 4259 T^{3} + 41017 T^{4} + 318344 T^{5} + 2094237 T^{6} + 11260048 T^{7} + 49236660 T^{8} + 153000611 T^{9} + 207740031 T^{10} - 1281777529 T^{11} - 10103437610 T^{12} - 29480883167 T^{13} + 109894476399 T^{14} + 1861558434037 T^{15} + 13778436171060 T^{16} + 72473531124464 T^{17} + 310022236071693 T^{18} + 1083905752099768 T^{19} + 3212081683270777 T^{20} + 7671109185170917 T^{21} + 15203529615409183 T^{22} + 20961814674106394 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 21 T + 347 T^{2} + 4336 T^{3} + 46198 T^{4} + 434470 T^{5} + 3675866 T^{6} + 28621248 T^{7} + 206719642 T^{8} + 1397159765 T^{9} + 8896862239 T^{10} + 53499099763 T^{11} + 305250702792 T^{12} + 1654375544921 T^{13} + 8532394903305 T^{14} + 41930966491950 T^{15} + 196245082776015 T^{16} + 875164663263209 T^{17} + 3713985300870264 T^{18} + 14971241576777683 T^{19} + 57263256993951977 T^{20} + 206829787886806085 T^{21} + 703844297476329974 T^{22} + 2241358130851850688 T^{23} + 6620795829081351958 T^{24} + 17998576326994081030 T^{25} + 44017905196107599546 T^{26} + 95021811537240111856 T^{27} + \)\(17\!\cdots\!01\)\( T^{28} + \)\(24\!\cdots\!89\)\( T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \))(\( 1 - 21 T + 479 T^{2} - 6876 T^{3} + 96142 T^{4} - 1076064 T^{5} + 11524829 T^{6} - 107178981 T^{7} + 952298573 T^{8} - 7627853519 T^{9} + 58483357007 T^{10} - 412453392385 T^{11} + 2790697724302 T^{12} - 17572144271241 T^{13} + 106354141132161 T^{14} - 603129985677831 T^{15} + 3291879942209978 T^{16} - 16895230001530308 T^{17} + 83510289963295008 T^{18} - 388590290035197084 T^{19} + 1741404489429078362 T^{20} - 7338282535742169777 T^{21} + 29762249208565066401 T^{22} - \)\(11\!\cdots\!63\)\( T^{23} + \)\(41\!\cdots\!78\)\( T^{24} - \)\(14\!\cdots\!95\)\( T^{25} + \)\(45\!\cdots\!67\)\( T^{26} - \)\(13\!\cdots\!97\)\( T^{27} + \)\(39\!\cdots\!77\)\( T^{28} - \)\(10\!\cdots\!87\)\( T^{29} + \)\(25\!\cdots\!09\)\( T^{30} - \)\(54\!\cdots\!12\)\( T^{31} + \)\(11\!\cdots\!78\)\( T^{32} - \)\(18\!\cdots\!32\)\( T^{33} + \)\(29\!\cdots\!19\)\( T^{34} - \)\(29\!\cdots\!63\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))
$29$ (\( 1 + 9 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 - 5 T + 53 T^{2} - 145 T^{3} + 841 T^{4} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( 1 + 5 T + 78 T^{2} + 269 T^{3} + 2262 T^{4} + 4205 T^{5} + 24389 T^{6} \))(\( 1 + 17 T + 267 T^{2} + 2896 T^{3} + 29860 T^{4} + 259073 T^{5} + 2154919 T^{6} + 15978321 T^{7} + 113636417 T^{8} + 740007036 T^{9} + 4634565777 T^{10} + 26911758984 T^{11} + 150329209016 T^{12} + 780441010536 T^{13} + 3897669818457 T^{14} + 18048031601004 T^{15} + 80372878652177 T^{16} + 327733722800829 T^{17} + 1281796076065999 T^{18} + 4468977205001557 T^{19} + 14937357891015460 T^{20} + 42012694746116624 T^{21} + 112328831291153667 T^{22} + 207408666016999093 T^{23} + 353814783205469041 T^{24} \))(\( 1 - 23 T + 427 T^{2} - 5461 T^{3} + 61354 T^{4} - 569838 T^{5} + 4861036 T^{6} - 36739478 T^{7} + 263280045 T^{8} - 1739117478 T^{9} + 11146165520 T^{10} - 67433048860 T^{11} + 401224859086 T^{12} - 2281520103087 T^{13} + 12841014957963 T^{14} - 69439016441342 T^{15} + 372389433780927 T^{16} - 1918758406696167 T^{17} + 9785473088248454 T^{18} - 47694114230749660 T^{19} + 228620661759382480 T^{20} - 1034467633873104438 T^{21} + 4541548210877953905 T^{22} - 18378792083559574358 T^{23} + 70519758845954340284 T^{24} - \)\(23\!\cdots\!38\)\( T^{25} + \)\(74\!\cdots\!66\)\( T^{26} - \)\(19\!\cdots\!01\)\( T^{27} + \)\(43\!\cdots\!03\)\( T^{28} - \)\(68\!\cdots\!63\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \))(\( 1 - 14 T + 360 T^{2} - 3587 T^{3} + 54348 T^{4} - 423754 T^{5} + 4945609 T^{6} - 31908922 T^{7} + 322094293 T^{8} - 1787565388 T^{9} + 16675773046 T^{10} - 82069849851 T^{11} + 733625056412 T^{12} - 3284386344346 T^{13} + 28442479738949 T^{14} - 117993109212365 T^{15} + 983185311718844 T^{16} - 3818632484106849 T^{17} + 30253694091956980 T^{18} - 110740342039098621 T^{19} + 826858847155547804 T^{20} - 2877733940580369985 T^{21} + 20116825512243587669 T^{22} - 67366537682446113554 T^{23} + \)\(43\!\cdots\!52\)\( T^{24} - \)\(14\!\cdots\!59\)\( T^{25} + \)\(83\!\cdots\!06\)\( T^{26} - \)\(25\!\cdots\!72\)\( T^{27} + \)\(13\!\cdots\!93\)\( T^{28} - \)\(38\!\cdots\!38\)\( T^{29} + \)\(17\!\cdots\!69\)\( T^{30} - \)\(43\!\cdots\!06\)\( T^{31} + \)\(16\!\cdots\!88\)\( T^{32} - \)\(30\!\cdots\!63\)\( T^{33} + \)\(90\!\cdots\!60\)\( T^{34} - \)\(10\!\cdots\!26\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( ( 1 + 9 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( 1 - 4 T + 49 T^{2} - 80 T^{3} + 1519 T^{4} - 3844 T^{5} + 29791 T^{6} \))(\( 1 - 5 T + 174 T^{2} - 612 T^{3} + 14514 T^{4} - 35717 T^{5} + 843245 T^{6} - 1558202 T^{7} + 39777604 T^{8} - 59702625 T^{9} + 1569775628 T^{10} - 2000648013 T^{11} + 52495876228 T^{12} - 62020088403 T^{13} + 1508554378508 T^{14} - 1778600901375 T^{15} + 36735452623684 T^{16} - 44610000346502 T^{17} + 748383041484845 T^{18} - 982668038202587 T^{19} + 12378860517418674 T^{20} - 16181048762330652 T^{21} + 142615321934659374 T^{22} - 127042384482024155 T^{23} + 787662783788549761 T^{24} \))(\( 1 + 31 T + 692 T^{2} + 11612 T^{3} + 165720 T^{4} + 2039159 T^{5} + 22496680 T^{6} + 224081837 T^{7} + 2050852122 T^{8} + 17313998075 T^{9} + 136131839684 T^{10} + 998684238486 T^{11} + 6875238587605 T^{12} + 44437839105999 T^{13} + 270594191528358 T^{14} + 1551242144539090 T^{15} + 8388419937379098 T^{16} + 42704763380865039 T^{17} + 204820232763340555 T^{18} + 922305866610829206 T^{19} + 3897338994221028284 T^{20} + 15366237024389414075 T^{21} + 56424303031311493542 T^{22} + \)\(19\!\cdots\!17\)\( T^{23} + \)\(59\!\cdots\!80\)\( T^{24} + \)\(16\!\cdots\!59\)\( T^{25} + \)\(42\!\cdots\!20\)\( T^{26} + \)\(91\!\cdots\!32\)\( T^{27} + \)\(16\!\cdots\!72\)\( T^{28} + \)\(23\!\cdots\!51\)\( T^{29} + \)\(23\!\cdots\!51\)\( T^{30} \))(\( 1 - 15 T + 383 T^{2} - 4367 T^{3} + 62771 T^{4} - 573832 T^{5} + 5953341 T^{6} - 45137537 T^{7} + 371609344 T^{8} - 2416476089 T^{9} + 16816630942 T^{10} - 99316478823 T^{11} + 629101175373 T^{12} - 3696057736585 T^{13} + 22910045878845 T^{14} - 140226112843424 T^{15} + 847117926362703 T^{16} - 5086489323981278 T^{17} + 28474335222419506 T^{18} - 157681169043419618 T^{19} + 814080327234557583 T^{20} - 4177476127718444384 T^{21} + 21157908480076813245 T^{22} - \)\(10\!\cdots\!35\)\( T^{23} + \)\(55\!\cdots\!13\)\( T^{24} - \)\(27\!\cdots\!53\)\( T^{25} + \)\(14\!\cdots\!22\)\( T^{26} - \)\(63\!\cdots\!19\)\( T^{27} + \)\(30\!\cdots\!44\)\( T^{28} - \)\(11\!\cdots\!47\)\( T^{29} + \)\(46\!\cdots\!01\)\( T^{30} - \)\(14\!\cdots\!12\)\( T^{31} + \)\(47\!\cdots\!91\)\( T^{32} - \)\(10\!\cdots\!17\)\( T^{33} + \)\(27\!\cdots\!23\)\( T^{34} - \)\(33\!\cdots\!65\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( 1 + 10 T + 54 T^{2} + 370 T^{3} + 1369 T^{4} \))(\( 1 - 6 T + 15 T^{2} + 188 T^{3} + 555 T^{4} - 8214 T^{5} + 50653 T^{6} \))(\( 1 + 2 T + 276 T^{2} + 652 T^{3} + 38837 T^{4} + 96256 T^{5} + 3631670 T^{6} + 8930412 T^{7} + 249282483 T^{8} + 584164766 T^{9} + 13169887841 T^{10} + 28427662304 T^{11} + 547481700896 T^{12} + 1051823505248 T^{13} + 18029576454329 T^{14} + 29589697892198 T^{15} + 467195507621763 T^{16} + 619270105720284 T^{17} + 9317871627773030 T^{18} + 9137762765314048 T^{19} + 136414164551929877 T^{20} + 84735054346390204 T^{21} + 1327169286787326324 T^{22} + 355835243558920826 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 10 T + 312 T^{2} + 3108 T^{3} + 52581 T^{4} + 483244 T^{5} + 6034705 T^{6} + 50418088 T^{7} + 516444781 T^{8} + 3928176048 T^{9} + 34696477720 T^{10} + 240745761242 T^{11} + 1885028766162 T^{12} + 11943747411556 T^{13} + 84189154828618 T^{14} + 486619808224408 T^{15} + 3114998728658866 T^{16} + 16350990206420164 T^{17} + 95482362092403786 T^{18} + 451196316635067962 T^{19} + 2405991059067138040 T^{20} + 10078625025554851632 T^{21} + 49027072495871092873 T^{22} + \)\(17\!\cdots\!48\)\( T^{23} + \)\(78\!\cdots\!85\)\( T^{24} + \)\(23\!\cdots\!56\)\( T^{25} + \)\(93\!\cdots\!53\)\( T^{26} + \)\(20\!\cdots\!48\)\( T^{27} + \)\(75\!\cdots\!64\)\( T^{28} + \)\(90\!\cdots\!90\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \))(\( 1 - 2 T + 453 T^{2} - 976 T^{3} + 99954 T^{4} - 230136 T^{5} + 14341649 T^{6} - 34760092 T^{7} + 1506935873 T^{8} - 3769215366 T^{9} + 123701015387 T^{10} - 312196087532 T^{11} + 8251610947472 T^{12} - 20532487852350 T^{13} + 458470700045013 T^{14} - 1099967812490778 T^{15} + 21536513634648780 T^{16} - 48778927118631752 T^{17} + 862131251088710164 T^{18} - 1804820303389374824 T^{19} + 29483487165834179820 T^{20} - 55716669606095378034 T^{21} + \)\(85\!\cdots\!93\)\( T^{22} - \)\(14\!\cdots\!50\)\( T^{23} + \)\(21\!\cdots\!48\)\( T^{24} - \)\(29\!\cdots\!56\)\( T^{25} + \)\(43\!\cdots\!27\)\( T^{26} - \)\(48\!\cdots\!82\)\( T^{27} + \)\(72\!\cdots\!77\)\( T^{28} - \)\(61\!\cdots\!96\)\( T^{29} + \)\(94\!\cdots\!69\)\( T^{30} - \)\(56\!\cdots\!92\)\( T^{31} + \)\(90\!\cdots\!06\)\( T^{32} - \)\(32\!\cdots\!68\)\( T^{33} + \)\(55\!\cdots\!73\)\( T^{34} - \)\(91\!\cdots\!34\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 + 7 T + 41 T^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( 1 + 6 T + 71 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 7 T + 122 T^{2} + 543 T^{3} + 5002 T^{4} + 11767 T^{5} + 68921 T^{6} \))(\( 1 + 18 T + 513 T^{2} + 6752 T^{3} + 112032 T^{4} + 1180612 T^{5} + 14535318 T^{6} + 128286403 T^{7} + 1278488494 T^{8} + 9670533526 T^{9} + 81232337198 T^{10} + 532034629908 T^{11} + 3842570017322 T^{12} + 21813419826228 T^{13} + 136551558829838 T^{14} + 666502841145446 T^{15} + 3612702925293934 T^{16} + 14862775291535003 T^{17} + 69044275676083638 T^{18} + 229929232795195172 T^{19} + 894567143268883872 T^{20} + 2210482821028024672 T^{21} + 6885824226108181713 T^{22} + 9905922570892471938 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 15 T + 442 T^{2} + 5223 T^{3} + 90894 T^{4} + 911629 T^{5} + 12025392 T^{6} + 106214266 T^{7} + 1162087852 T^{8} + 9210320951 T^{9} + 87305614458 T^{10} + 627124168079 T^{11} + 5277791151653 T^{12} + 34520123652249 T^{13} + 261648125984996 T^{14} + 1559014176372088 T^{15} + 10727573165384836 T^{16} + 58028327859430569 T^{17} + 363750643963076413 T^{18} + 1772103016315083119 T^{19} + 10114896817074554058 T^{20} + 43749984610316253191 T^{21} + \)\(22\!\cdots\!12\)\( T^{22} + \)\(84\!\cdots\!86\)\( T^{23} + \)\(39\!\cdots\!12\)\( T^{24} + \)\(12\!\cdots\!29\)\( T^{25} + \)\(50\!\cdots\!54\)\( T^{26} + \)\(11\!\cdots\!63\)\( T^{27} + \)\(40\!\cdots\!82\)\( T^{28} + \)\(56\!\cdots\!15\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \))(\( 1 - 34 T + 1005 T^{2} - 20528 T^{3} + 374948 T^{4} - 5718724 T^{5} + 79935632 T^{6} - 993553553 T^{7} + 11504795328 T^{8} - 122037807818 T^{9} + 1219519671986 T^{10} - 11352249769588 T^{11} + 100326858628779 T^{12} - 834454910428352 T^{13} + 6624290155120839 T^{14} - 49804911709306802 T^{15} + 358608059604054218 T^{16} - 2454039700753445925 T^{17} + 16109159095203839388 T^{18} - \)\(10\!\cdots\!25\)\( T^{19} + \)\(60\!\cdots\!58\)\( T^{20} - \)\(34\!\cdots\!42\)\( T^{21} + \)\(18\!\cdots\!79\)\( T^{22} - \)\(96\!\cdots\!52\)\( T^{23} + \)\(47\!\cdots\!39\)\( T^{24} - \)\(22\!\cdots\!28\)\( T^{25} + \)\(97\!\cdots\!06\)\( T^{26} - \)\(39\!\cdots\!98\)\( T^{27} + \)\(15\!\cdots\!28\)\( T^{28} - \)\(54\!\cdots\!73\)\( T^{29} + \)\(18\!\cdots\!92\)\( T^{30} - \)\(52\!\cdots\!04\)\( T^{31} + \)\(14\!\cdots\!28\)\( T^{32} - \)\(31\!\cdots\!28\)\( T^{33} + \)\(64\!\cdots\!05\)\( T^{34} - \)\(88\!\cdots\!54\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 5 T + 91 T^{2} + 215 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 82 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 101 T^{2} + 680 T^{3} + 4343 T^{4} + 14792 T^{5} + 79507 T^{6} \))(\( 1 + 16 T + 452 T^{2} + 5166 T^{3} + 84594 T^{4} + 768632 T^{5} + 9494587 T^{6} + 73309881 T^{7} + 755743740 T^{8} + 5146333609 T^{9} + 46182597778 T^{10} + 280738811836 T^{11} + 2228528119624 T^{12} + 12071768908948 T^{13} + 85391623291522 T^{14} + 409169546250763 T^{15} + 2583737454055740 T^{16} + 10777171462325283 T^{17} + 60018731427315763 T^{18} + 208928482692395624 T^{19} + 988751614283378994 T^{20} + 2596393433265730938 T^{21} + 9768390005604480548 T^{22} + 14868699831539563312 T^{23} + 39959630797262576401 T^{24} \))(\( 1 + 3 T + 278 T^{2} + 514 T^{3} + 38864 T^{4} + 25385 T^{5} + 3718122 T^{6} - 1286068 T^{7} + 282546143 T^{8} - 278051405 T^{9} + 18321423239 T^{10} - 23233496517 T^{11} + 1033949685168 T^{12} - 1384563262343 T^{13} + 50876286219341 T^{14} - 65935465846962 T^{15} + 2187680307431663 T^{16} - 2560057472072207 T^{17} + 82206237618652176 T^{18} - 79430701125816117 T^{19} + 2693403903909406877 T^{20} - 1757663877289533845 T^{21} + 76801300163899810301 T^{22} - 15031820354613762868 T^{23} + \)\(18\!\cdots\!46\)\( T^{24} + \)\(54\!\cdots\!65\)\( T^{25} + \)\(36\!\cdots\!48\)\( T^{26} + \)\(20\!\cdots\!14\)\( T^{27} + \)\(47\!\cdots\!54\)\( T^{28} + \)\(22\!\cdots\!47\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \))(\( 1 - 21 T + 519 T^{2} - 7267 T^{3} + 109313 T^{4} - 1198356 T^{5} + 13957551 T^{6} - 129619412 T^{7} + 1285896403 T^{8} - 10602139616 T^{9} + 94268250552 T^{10} - 710895850963 T^{11} + 5838466152734 T^{12} - 41062097173583 T^{13} + 317400515100395 T^{14} - 2110223160672891 T^{15} + 15538521143036709 T^{16} - 98608169677782665 T^{17} + 696306034517542542 T^{18} - 4240151296144654595 T^{19} + 28730725593474874941 T^{20} - \)\(16\!\cdots\!37\)\( T^{21} + \)\(10\!\cdots\!95\)\( T^{22} - \)\(60\!\cdots\!69\)\( T^{23} + \)\(36\!\cdots\!66\)\( T^{24} - \)\(19\!\cdots\!41\)\( T^{25} + \)\(11\!\cdots\!52\)\( T^{26} - \)\(53\!\cdots\!88\)\( T^{27} + \)\(27\!\cdots\!47\)\( T^{28} - \)\(12\!\cdots\!84\)\( T^{29} + \)\(55\!\cdots\!51\)\( T^{30} - \)\(20\!\cdots\!08\)\( T^{31} + \)\(80\!\cdots\!37\)\( T^{32} - \)\(23\!\cdots\!69\)\( T^{33} + \)\(70\!\cdots\!19\)\( T^{34} - \)\(12\!\cdots\!03\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - T + 83 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( 1 + 12 T + 153 T^{2} + 1040 T^{3} + 7191 T^{4} + 26508 T^{5} + 103823 T^{6} \))(\( 1 + 60 T + 2010 T^{2} + 47866 T^{3} + 899112 T^{4} + 14051168 T^{5} + 188897608 T^{6} + 2231635363 T^{7} + 23513775589 T^{8} + 223192326289 T^{9} + 1921928283489 T^{10} + 15081365904050 T^{11} + 108125238818798 T^{12} + 708824197490350 T^{13} + 4245539578227201 T^{14} + 23172496892302847 T^{15} + 114739723979907109 T^{16} + 511814427948682541 T^{17} + 2036167991765033032 T^{18} + 7118646578309850784 T^{19} + 21409013573029256232 T^{20} + 53568299225537045222 T^{21} + \)\(10\!\cdots\!90\)\( T^{22} + \)\(14\!\cdots\!80\)\( T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 47 T + 1360 T^{2} + 29242 T^{3} + 515710 T^{4} + 7806435 T^{5} + 104570527 T^{6} + 1263761173 T^{7} + 13972610942 T^{8} + 142749487480 T^{9} + 1357652693882 T^{10} + 12087146093867 T^{11} + 101151189464855 T^{12} + 798016923155622 T^{13} + 5947378574242063 T^{14} + 41918381294391180 T^{15} + 279526792989376961 T^{16} + 1762819383250768998 T^{17} + 10501819943809640665 T^{18} + 58981417138467016427 T^{19} + \)\(31\!\cdots\!74\)\( T^{20} + \)\(15\!\cdots\!20\)\( T^{21} + \)\(70\!\cdots\!46\)\( T^{22} + \)\(30\!\cdots\!53\)\( T^{23} + \)\(11\!\cdots\!09\)\( T^{24} + \)\(41\!\cdots\!15\)\( T^{25} + \)\(12\!\cdots\!30\)\( T^{26} + \)\(33\!\cdots\!22\)\( T^{27} + \)\(74\!\cdots\!20\)\( T^{28} + \)\(12\!\cdots\!43\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \))(\( 1 - 69 T + 2813 T^{2} - 83259 T^{3} + 1977829 T^{4} - 39575026 T^{5} + 688923872 T^{6} - 10652518865 T^{7} + 148590583135 T^{8} - 1890844183696 T^{9} + 22145465576619 T^{10} - 240338605574193 T^{11} + 2430280974430484 T^{12} - 22995879886692348 T^{13} + 204320195163910158 T^{14} - 1709199322459761645 T^{15} + 13488922699121350055 T^{16} - \)\(10\!\cdots\!15\)\( T^{17} + \)\(70\!\cdots\!44\)\( T^{18} - \)\(47\!\cdots\!05\)\( T^{19} + \)\(29\!\cdots\!95\)\( T^{20} - \)\(17\!\cdots\!35\)\( T^{21} + \)\(99\!\cdots\!98\)\( T^{22} - \)\(52\!\cdots\!36\)\( T^{23} + \)\(26\!\cdots\!36\)\( T^{24} - \)\(12\!\cdots\!59\)\( T^{25} + \)\(52\!\cdots\!59\)\( T^{26} - \)\(21\!\cdots\!32\)\( T^{27} + \)\(78\!\cdots\!15\)\( T^{28} - \)\(26\!\cdots\!95\)\( T^{29} + \)\(80\!\cdots\!52\)\( T^{30} - \)\(21\!\cdots\!02\)\( T^{31} + \)\(50\!\cdots\!01\)\( T^{32} - \)\(10\!\cdots\!37\)\( T^{33} + \)\(15\!\cdots\!73\)\( T^{34} - \)\(18\!\cdots\!03\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \))
$53$ (\( 1 + T + 53 T^{2} \))(\( 1 - 7 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 + 4 T + 65 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 10 T + 111 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 - 17 T + 154 T^{2} - 1085 T^{3} + 8162 T^{4} - 47753 T^{5} + 148877 T^{6} \))(\( 1 + 24 T + 585 T^{2} + 10035 T^{3} + 156536 T^{4} + 2090218 T^{5} + 25483383 T^{6} + 280685847 T^{7} + 2849940379 T^{8} + 26691947343 T^{9} + 232136355273 T^{10} + 1878007036003 T^{11} + 14158182047281 T^{12} + 99534372908159 T^{13} + 652071021961857 T^{14} + 3973817044583811 T^{15} + 22487400411632299 T^{16} + 117381556164287571 T^{17} + 564822903600619407 T^{18} + 2455402369287814466 T^{19} + 9745882898232805496 T^{20} + 33113127643734404655 T^{21} + \)\(10\!\cdots\!65\)\( T^{22} + \)\(22\!\cdots\!28\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 7 T + 196 T^{2} + 608 T^{3} + 21087 T^{4} + 23332 T^{5} + 1863719 T^{6} - 1169126 T^{7} + 147182643 T^{8} - 208121852 T^{9} + 10894186148 T^{10} - 18900938535 T^{11} + 715640854992 T^{12} - 1358638531946 T^{13} + 41942713891806 T^{14} - 80813786672112 T^{15} + 2222963836265718 T^{16} - 3816415636236314 T^{17} + 106542463568643984 T^{18} - 149137496392585335 T^{19} + 4555899546996630964 T^{20} - 4612887886564290908 T^{21} + \)\(17\!\cdots\!91\)\( T^{22} - 72789422811872840486 T^{23} + \)\(61\!\cdots\!27\)\( T^{24} + \)\(40\!\cdots\!68\)\( T^{25} + \)\(19\!\cdots\!39\)\( T^{26} + \)\(29\!\cdots\!28\)\( T^{27} + \)\(51\!\cdots\!08\)\( T^{28} + \)\(96\!\cdots\!83\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \))(\( 1 + 4 T + 484 T^{2} + 2303 T^{3} + 119041 T^{4} + 615075 T^{5} + 19730544 T^{6} + 104020135 T^{7} + 2461306477 T^{8} + 12696580177 T^{9} + 245084819493 T^{10} + 1204914660498 T^{11} + 20257898789527 T^{12} + 93642219626884 T^{13} + 1433925290865274 T^{14} + 6216668515947622 T^{15} + 89238053661872786 T^{16} + 364921034572944308 T^{17} + 4977627118689745467 T^{18} + 19340814832366048324 T^{19} + \)\(25\!\cdots\!74\)\( T^{20} + \)\(92\!\cdots\!94\)\( T^{21} + \)\(11\!\cdots\!94\)\( T^{22} + \)\(39\!\cdots\!12\)\( T^{23} + \)\(44\!\cdots\!83\)\( T^{24} + \)\(14\!\cdots\!26\)\( T^{25} + \)\(15\!\cdots\!73\)\( T^{26} + \)\(41\!\cdots\!41\)\( T^{27} + \)\(43\!\cdots\!73\)\( T^{28} + \)\(96\!\cdots\!95\)\( T^{29} + \)\(96\!\cdots\!04\)\( T^{30} + \)\(16\!\cdots\!75\)\( T^{31} + \)\(16\!\cdots\!29\)\( T^{32} + \)\(16\!\cdots\!71\)\( T^{33} + \)\(18\!\cdots\!64\)\( T^{34} + \)\(82\!\cdots\!52\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \))
$59$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{12} \))(\( ( 1 + T )^{15} \))(\( ( 1 + T )^{18} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 6 T + 111 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 14 T + 126 T^{2} - 854 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 187 T^{2} + 952 T^{3} + 11407 T^{4} + 29768 T^{5} + 226981 T^{6} \))(\( 1 + 17 T + 523 T^{2} + 7084 T^{3} + 125632 T^{4} + 1423665 T^{5} + 19094435 T^{6} + 187175044 T^{7} + 2096912119 T^{8} + 18193464886 T^{9} + 177833498575 T^{10} + 1384227028687 T^{11} + 12060055117058 T^{12} + 84437848749907 T^{13} + 661718448197575 T^{14} + 4129570853289166 T^{15} + 29033511790647079 T^{16} + 158087349801912244 T^{17} + 983752439411781035 T^{18} + 4474212979643836965 T^{19} + 24084573146474406592 T^{20} + 82841330921637054844 T^{21} + \)\(37\!\cdots\!23\)\( T^{22} + \)\(73\!\cdots\!37\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 - T + 535 T^{2} + 896 T^{3} + 129742 T^{4} + 619919 T^{5} + 19695042 T^{6} + 159104993 T^{7} + 2271075096 T^{8} + 23702591530 T^{9} + 229839047737 T^{10} + 2401962078682 T^{11} + 21008186472825 T^{12} + 185309068314783 T^{13} + 1616722061410274 T^{14} + 12035047966225248 T^{15} + 98620045746026714 T^{16} + 689535043199307543 T^{17} + 4768459173788291325 T^{18} + 33257185029460461562 T^{19} + \)\(19\!\cdots\!37\)\( T^{20} + \)\(12\!\cdots\!30\)\( T^{21} + \)\(71\!\cdots\!16\)\( T^{22} + \)\(30\!\cdots\!33\)\( T^{23} + \)\(23\!\cdots\!22\)\( T^{24} + \)\(44\!\cdots\!19\)\( T^{25} + \)\(56\!\cdots\!62\)\( T^{26} + \)\(23\!\cdots\!16\)\( T^{27} + \)\(86\!\cdots\!35\)\( T^{28} - \)\(98\!\cdots\!41\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \))(\( 1 - 11 T + 481 T^{2} - 3908 T^{3} + 106662 T^{4} - 665939 T^{5} + 15677275 T^{6} - 78271338 T^{7} + 1827212675 T^{8} - 7663617944 T^{9} + 183977297525 T^{10} - 675835228611 T^{11} + 16397805671619 T^{12} - 53865870554654 T^{13} + 1298473949014948 T^{14} - 3855780004224404 T^{15} + 91930862643429725 T^{16} - 251775224817831323 T^{17} + 5882200161266945018 T^{18} - 15358288713887710703 T^{19} + \)\(34\!\cdots\!25\)\( T^{20} - \)\(87\!\cdots\!24\)\( T^{21} + \)\(17\!\cdots\!68\)\( T^{22} - \)\(45\!\cdots\!54\)\( T^{23} + \)\(84\!\cdots\!59\)\( T^{24} - \)\(21\!\cdots\!31\)\( T^{25} + \)\(35\!\cdots\!25\)\( T^{26} - \)\(89\!\cdots\!04\)\( T^{27} + \)\(13\!\cdots\!75\)\( T^{28} - \)\(34\!\cdots\!18\)\( T^{29} + \)\(41\!\cdots\!75\)\( T^{30} - \)\(10\!\cdots\!59\)\( T^{31} + \)\(10\!\cdots\!42\)\( T^{32} - \)\(23\!\cdots\!08\)\( T^{33} + \)\(17\!\cdots\!41\)\( T^{34} - \)\(24\!\cdots\!31\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))
$67$ (\( 1 + 6 T + 67 T^{2} \))(\( 1 - 10 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 8 T + 105 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 90 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 6 T + 69 T^{2} + 460 T^{3} + 4623 T^{4} + 26934 T^{5} + 300763 T^{6} \))(\( 1 + 22 T + 741 T^{2} + 12614 T^{3} + 247452 T^{4} + 3454128 T^{5} + 50461200 T^{6} + 596546933 T^{7} + 7086482547 T^{8} + 72239490641 T^{9} + 726632463270 T^{10} + 6441622270486 T^{11} + 55947235078290 T^{12} + 431588692122562 T^{13} + 3261853127619030 T^{14} + 21726965923659083 T^{15} + 142800567268985187 T^{16} + 805412991747146831 T^{17} + 4564638514306342800 T^{18} + 20934473655871123344 T^{19} + \)\(10\!\cdots\!32\)\( T^{20} + \)\(34\!\cdots\!58\)\( T^{21} + \)\(13\!\cdots\!09\)\( T^{22} + \)\(26\!\cdots\!26\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 20 T + 715 T^{2} + 10412 T^{3} + 216322 T^{4} + 2479018 T^{5} + 38923425 T^{6} + 370303811 T^{7} + 4952459444 T^{8} + 41220861511 T^{9} + 506297442528 T^{10} + 3870640116092 T^{11} + 44987121763246 T^{12} + 322484235897585 T^{13} + 3508819206645825 T^{14} + 23386315081882382 T^{15} + 235090886845270275 T^{16} + 1447631734944259065 T^{17} + 13530461702879156698 T^{18} + 77997737326823939132 T^{19} + \)\(68\!\cdots\!96\)\( T^{20} + \)\(37\!\cdots\!59\)\( T^{21} + \)\(30\!\cdots\!12\)\( T^{22} + \)\(15\!\cdots\!51\)\( T^{23} + \)\(10\!\cdots\!75\)\( T^{24} + \)\(45\!\cdots\!82\)\( T^{25} + \)\(26\!\cdots\!26\)\( T^{26} + \)\(85\!\cdots\!32\)\( T^{27} + \)\(39\!\cdots\!05\)\( T^{28} + \)\(73\!\cdots\!80\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \))(\( 1 - 34 T + 1256 T^{2} - 29610 T^{3} + 670616 T^{4} - 12426584 T^{5} + 216557435 T^{6} - 3352890195 T^{7} + 48781974726 T^{8} - 653572817240 T^{9} + 8263250492199 T^{10} - 97921163094028 T^{11} + 1100466341178743 T^{12} - 11699198515220201 T^{13} + 118457844867384795 T^{14} - 1140148189649309110 T^{15} + 10483428760965178506 T^{16} - 91845721016617428512 T^{17} + \)\(76\!\cdots\!42\)\( T^{18} - \)\(61\!\cdots\!04\)\( T^{19} + \)\(47\!\cdots\!34\)\( T^{20} - \)\(34\!\cdots\!30\)\( T^{21} + \)\(23\!\cdots\!95\)\( T^{22} - \)\(15\!\cdots\!07\)\( T^{23} + \)\(99\!\cdots\!67\)\( T^{24} - \)\(59\!\cdots\!44\)\( T^{25} + \)\(33\!\cdots\!59\)\( T^{26} - \)\(17\!\cdots\!80\)\( T^{27} + \)\(88\!\cdots\!74\)\( T^{28} - \)\(40\!\cdots\!85\)\( T^{29} + \)\(17\!\cdots\!35\)\( T^{30} - \)\(68\!\cdots\!08\)\( T^{31} + \)\(24\!\cdots\!64\)\( T^{32} - \)\(72\!\cdots\!30\)\( T^{33} + \)\(20\!\cdots\!36\)\( T^{34} - \)\(37\!\cdots\!18\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 15 T + 167 T^{2} - 1065 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 126 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 117 T^{2} + 760 T^{3} + 8307 T^{4} + 20164 T^{5} + 357911 T^{6} \))(\( 1 + 10 T + 574 T^{2} + 6474 T^{3} + 164602 T^{4} + 1912514 T^{5} + 31427999 T^{6} + 346712137 T^{7} + 4403217165 T^{8} + 43628604814 T^{9} + 467550791406 T^{10} + 4063454064165 T^{11} + 37945230477130 T^{12} + 288505238555715 T^{13} + 2356923539477646 T^{14} + 15615157577583554 T^{15} + 111893149970704365 T^{16} + 625548213923333087 T^{17} + 4025935594968904079 T^{18} + 17394544634605004974 T^{19} + \)\(10\!\cdots\!22\)\( T^{20} + \)\(29\!\cdots\!94\)\( T^{21} + \)\(18\!\cdots\!74\)\( T^{22} + \)\(23\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 13 T + 734 T^{2} + 8444 T^{3} + 256616 T^{4} + 2655901 T^{5} + 57716900 T^{6} + 543971994 T^{7} + 9469919098 T^{8} + 81876459729 T^{9} + 1210704385834 T^{10} + 9631650796678 T^{11} + 125115095370877 T^{12} + 915416527813805 T^{13} + 10666306815567054 T^{14} + 71502153804051136 T^{15} + 757307783905260834 T^{16} + 4614614716709391005 T^{17} + 44780068899285957947 T^{18} + \)\(24\!\cdots\!18\)\( T^{19} + \)\(21\!\cdots\!34\)\( T^{20} + \)\(10\!\cdots\!09\)\( T^{21} + \)\(86\!\cdots\!18\)\( T^{22} + \)\(35\!\cdots\!34\)\( T^{23} + \)\(26\!\cdots\!00\)\( T^{24} + \)\(86\!\cdots\!01\)\( T^{25} + \)\(59\!\cdots\!36\)\( T^{26} + \)\(13\!\cdots\!04\)\( T^{27} + \)\(85\!\cdots\!74\)\( T^{28} + \)\(10\!\cdots\!53\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \))(\( 1 - 37 T + 1155 T^{2} - 24903 T^{3} + 485171 T^{4} - 7896284 T^{5} + 120218449 T^{6} - 1635116928 T^{7} + 21236636002 T^{8} - 254709648948 T^{9} + 2953507339142 T^{10} - 32202301017074 T^{11} + 341744420320871 T^{12} - 3443858987512768 T^{13} + 33892860537735277 T^{14} - 318418018400167788 T^{15} + 2926118537495009523 T^{16} - 25737442969373370088 T^{17} + \)\(22\!\cdots\!46\)\( T^{18} - \)\(18\!\cdots\!48\)\( T^{19} + \)\(14\!\cdots\!43\)\( T^{20} - \)\(11\!\cdots\!68\)\( T^{21} + \)\(86\!\cdots\!37\)\( T^{22} - \)\(62\!\cdots\!68\)\( T^{23} + \)\(43\!\cdots\!91\)\( T^{24} - \)\(29\!\cdots\!34\)\( T^{25} + \)\(19\!\cdots\!62\)\( T^{26} - \)\(11\!\cdots\!88\)\( T^{27} + \)\(69\!\cdots\!02\)\( T^{28} - \)\(37\!\cdots\!88\)\( T^{29} + \)\(19\!\cdots\!09\)\( T^{30} - \)\(91\!\cdots\!24\)\( T^{31} + \)\(40\!\cdots\!51\)\( T^{32} - \)\(14\!\cdots\!53\)\( T^{33} + \)\(48\!\cdots\!55\)\( T^{34} - \)\(10\!\cdots\!67\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 4 T + 25 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 34 T^{2} + 5329 T^{4} \))(\( ( 1 - 2 T + 73 T^{2} )^{3} \))(\( 1 - 4 T + 437 T^{2} - 576 T^{3} + 88919 T^{4} + 116331 T^{5} + 11939372 T^{6} + 45208590 T^{7} + 1236816657 T^{8} + 7281945960 T^{9} + 106422782444 T^{10} + 762970891832 T^{11} + 8103868987534 T^{12} + 55696875103736 T^{13} + 567127007644076 T^{14} + 2832800771521320 T^{15} + 35123417498300337 T^{16} + 93720643688583870 T^{17} + 1806835623996550508 T^{18} + 1285154917125073107 T^{19} + 71709624911129788439 T^{20} - 33910033943962317888 T^{21} + \)\(18\!\cdots\!13\)\( T^{22} - \)\(12\!\cdots\!08\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 24 T + 819 T^{2} + 15192 T^{3} + 309951 T^{4} + 4704095 T^{5} + 72983727 T^{6} + 942396132 T^{7} + 12143149592 T^{8} + 137237813352 T^{9} + 1539029575583 T^{10} + 15575909116591 T^{11} + 156785812756212 T^{12} + 1448205613996866 T^{13} + 13364742955882919 T^{14} + 114116433932227924 T^{15} + 975626235779453087 T^{16} + 7717487716989298914 T^{17} + 60992346520983323604 T^{18} + \)\(44\!\cdots\!31\)\( T^{19} + \)\(31\!\cdots\!19\)\( T^{20} + \)\(20\!\cdots\!28\)\( T^{21} + \)\(13\!\cdots\!24\)\( T^{22} + \)\(76\!\cdots\!92\)\( T^{23} + \)\(42\!\cdots\!51\)\( T^{24} + \)\(20\!\cdots\!55\)\( T^{25} + \)\(97\!\cdots\!27\)\( T^{26} + \)\(34\!\cdots\!32\)\( T^{27} + \)\(13\!\cdots\!27\)\( T^{28} + \)\(29\!\cdots\!16\)\( T^{29} + \)\(89\!\cdots\!57\)\( T^{30} \))(\( 1 - 18 T + 693 T^{2} - 9358 T^{3} + 223991 T^{4} - 2507127 T^{5} + 47434428 T^{6} - 454465578 T^{7} + 7438072501 T^{8} - 61763079622 T^{9} + 923573050490 T^{10} - 6677623153886 T^{11} + 95143153197465 T^{12} - 601588878507954 T^{13} + 8475520214804649 T^{14} - 47608376979440768 T^{15} + 680525784224168184 T^{16} - 3525807261135811289 T^{17} + 51025527266299755452 T^{18} - \)\(25\!\cdots\!97\)\( T^{19} + \)\(36\!\cdots\!36\)\( T^{20} - \)\(18\!\cdots\!56\)\( T^{21} + \)\(24\!\cdots\!09\)\( T^{22} - \)\(12\!\cdots\!22\)\( T^{23} + \)\(14\!\cdots\!85\)\( T^{24} - \)\(73\!\cdots\!42\)\( T^{25} + \)\(74\!\cdots\!90\)\( T^{26} - \)\(36\!\cdots\!86\)\( T^{27} + \)\(31\!\cdots\!49\)\( T^{28} - \)\(14\!\cdots\!06\)\( T^{29} + \)\(10\!\cdots\!88\)\( T^{30} - \)\(41\!\cdots\!91\)\( T^{31} + \)\(27\!\cdots\!19\)\( T^{32} - \)\(83\!\cdots\!06\)\( T^{33} + \)\(45\!\cdots\!73\)\( T^{34} - \)\(85\!\cdots\!54\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \))
$79$ (\( 1 + 11 T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( 1 + 5 T + 79 T^{2} \))(\( 1 + 11 T + 157 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + 4 T + 117 T^{2} + 316 T^{3} + 6241 T^{4} \))(\( ( 1 + 5 T + 79 T^{2} )^{3} \))(\( 1 - 48 T + 1625 T^{2} - 40426 T^{3} + 839442 T^{4} - 14827026 T^{5} + 231784702 T^{6} - 3237232009 T^{7} + 41158789322 T^{8} - 478465878064 T^{9} + 5137583259438 T^{10} - 51004021734028 T^{11} + 470673531133144 T^{12} - 4029317716988212 T^{13} + 32063657122152558 T^{14} - 235902338054796496 T^{15} + 1603138177953835082 T^{16} - 9961145468341075591 T^{17} + 56343953437873239742 T^{18} - \)\(28\!\cdots\!34\)\( T^{19} + \)\(12\!\cdots\!62\)\( T^{20} - \)\(48\!\cdots\!94\)\( T^{21} + \)\(15\!\cdots\!25\)\( T^{22} - \)\(35\!\cdots\!92\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 34 T + 1173 T^{2} + 26305 T^{3} + 562443 T^{4} + 9836766 T^{5} + 162873092 T^{6} + 2382471793 T^{7} + 33050635472 T^{8} + 419254986829 T^{9} + 5061673727393 T^{10} + 56862498191110 T^{11} + 609952841001422 T^{12} + 6142635892248079 T^{13} + 59197444488140998 T^{14} + 537655417992347836 T^{15} + 4676598114563138842 T^{16} + 38336190603520261039 T^{17} + \)\(30\!\cdots\!58\)\( T^{18} + \)\(22\!\cdots\!10\)\( T^{19} + \)\(15\!\cdots\!07\)\( T^{20} + \)\(10\!\cdots\!09\)\( T^{21} + \)\(63\!\cdots\!48\)\( T^{22} + \)\(36\!\cdots\!73\)\( T^{23} + \)\(19\!\cdots\!48\)\( T^{24} + \)\(93\!\cdots\!66\)\( T^{25} + \)\(42\!\cdots\!97\)\( T^{26} + \)\(15\!\cdots\!05\)\( T^{27} + \)\(54\!\cdots\!47\)\( T^{28} + \)\(12\!\cdots\!54\)\( T^{29} + \)\(29\!\cdots\!99\)\( T^{30} \))(\( 1 - 11 T + 685 T^{2} - 6481 T^{3} + 225790 T^{4} - 1851053 T^{5} + 48873268 T^{6} - 351387947 T^{7} + 8014493255 T^{8} - 51360170663 T^{9} + 1080970410860 T^{10} - 6278269508005 T^{11} + 125611304688396 T^{12} - 670828362784983 T^{13} + 12898492496169612 T^{14} - 64144515705391504 T^{15} + 1187177355536698586 T^{16} - 5560084148732057379 T^{17} + 98624258252882862522 T^{18} - \)\(43\!\cdots\!41\)\( T^{19} + \)\(74\!\cdots\!26\)\( T^{20} - \)\(31\!\cdots\!56\)\( T^{21} + \)\(50\!\cdots\!72\)\( T^{22} - \)\(20\!\cdots\!17\)\( T^{23} + \)\(30\!\cdots\!16\)\( T^{24} - \)\(12\!\cdots\!95\)\( T^{25} + \)\(16\!\cdots\!60\)\( T^{26} - \)\(61\!\cdots\!97\)\( T^{27} + \)\(75\!\cdots\!55\)\( T^{28} - \)\(26\!\cdots\!13\)\( T^{29} + \)\(28\!\cdots\!88\)\( T^{30} - \)\(86\!\cdots\!67\)\( T^{31} + \)\(83\!\cdots\!90\)\( T^{32} - \)\(18\!\cdots\!19\)\( T^{33} + \)\(15\!\cdots\!85\)\( T^{34} - \)\(20\!\cdots\!49\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \))
$83$ (\( 1 - 14 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 2 T + 147 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 + 2 T + 162 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 + 6 T + 153 T^{2} + 1212 T^{3} + 12699 T^{4} + 41334 T^{5} + 571787 T^{6} \))(\( 1 + 20 T + 745 T^{2} + 12329 T^{3} + 264212 T^{4} + 3753535 T^{5} + 59783574 T^{6} + 742649018 T^{7} + 9656633440 T^{8} + 105856008277 T^{9} + 1175051350486 T^{10} + 11401113299177 T^{11} + 110512481376744 T^{12} + 946292403831691 T^{13} + 8094928753498054 T^{14} + 60527089404680999 T^{15} + 458287609574854240 T^{16} + 2925324665386038574 T^{17} + 19545664004893240806 T^{18} + \)\(10\!\cdots\!45\)\( T^{19} + \)\(59\!\cdots\!92\)\( T^{20} + \)\(23\!\cdots\!87\)\( T^{21} + \)\(11\!\cdots\!05\)\( T^{22} + \)\(25\!\cdots\!40\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 40 T + 1031 T^{2} + 17725 T^{3} + 256332 T^{4} + 3200681 T^{5} + 40303961 T^{6} + 474910242 T^{7} + 5415040583 T^{8} + 55001306684 T^{9} + 557287568498 T^{10} + 5451190530076 T^{11} + 55382564648974 T^{12} + 518249610734501 T^{13} + 4822234874235804 T^{14} + 42594893289671942 T^{15} + 400245494561571732 T^{16} + 3570221568349977389 T^{17} + 31667030492942896538 T^{18} + \)\(25\!\cdots\!96\)\( T^{19} + \)\(21\!\cdots\!14\)\( T^{20} + \)\(17\!\cdots\!96\)\( T^{21} + \)\(14\!\cdots\!41\)\( T^{22} + \)\(10\!\cdots\!22\)\( T^{23} + \)\(75\!\cdots\!83\)\( T^{24} + \)\(49\!\cdots\!69\)\( T^{25} + \)\(33\!\cdots\!44\)\( T^{26} + \)\(18\!\cdots\!25\)\( T^{27} + \)\(91\!\cdots\!53\)\( T^{28} + \)\(29\!\cdots\!60\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \))(\( 1 - 28 T + 1084 T^{2} - 21171 T^{3} + 488774 T^{4} - 7444810 T^{5} + 129435425 T^{6} - 1607344142 T^{7} + 22867356485 T^{8} - 233351638512 T^{9} + 2823584090192 T^{10} - 22867332415791 T^{11} + 240239654651824 T^{12} - 1319571611951593 T^{13} + 12381897567975973 T^{14} - 4231523957468201 T^{15} + 140617753201218772 T^{16} + 6994569251833345554 T^{17} - 22994441467512066308 T^{18} + \)\(58\!\cdots\!82\)\( T^{19} + \)\(96\!\cdots\!08\)\( T^{20} - \)\(24\!\cdots\!87\)\( T^{21} + \)\(58\!\cdots\!33\)\( T^{22} - \)\(51\!\cdots\!99\)\( T^{23} + \)\(78\!\cdots\!56\)\( T^{24} - \)\(62\!\cdots\!57\)\( T^{25} + \)\(63\!\cdots\!72\)\( T^{26} - \)\(43\!\cdots\!36\)\( T^{27} + \)\(35\!\cdots\!65\)\( T^{28} - \)\(20\!\cdots\!14\)\( T^{29} + \)\(13\!\cdots\!25\)\( T^{30} - \)\(66\!\cdots\!30\)\( T^{31} + \)\(35\!\cdots\!46\)\( T^{32} - \)\(12\!\cdots\!97\)\( T^{33} + \)\(54\!\cdots\!04\)\( T^{34} - \)\(11\!\cdots\!44\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))
$89$ (\( 1 + 10 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 - 9 T + 137 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 - 4 T + 162 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 - 10 T + 251 T^{2} - 1548 T^{3} + 22339 T^{4} - 79210 T^{5} + 704969 T^{6} \))(\( 1 + 29 T + 765 T^{2} + 13253 T^{3} + 201887 T^{4} + 2435499 T^{5} + 24592778 T^{6} + 189383588 T^{7} + 775452159 T^{8} - 5841612153 T^{9} - 191711192048 T^{10} - 2742591163352 T^{11} - 29526628703276 T^{12} - 244090613538328 T^{13} - 1518544352212208 T^{14} - 4118155477888257 T^{15} + 48653606243948319 T^{16} + 1057529214056923012 T^{17} + 12222150558757279658 T^{18} + \)\(10\!\cdots\!71\)\( T^{19} + \)\(79\!\cdots\!47\)\( T^{20} + \)\(46\!\cdots\!77\)\( T^{21} + \)\(23\!\cdots\!65\)\( T^{22} + \)\(80\!\cdots\!81\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 46 T + 1724 T^{2} + 45608 T^{3} + 1055696 T^{4} + 20603015 T^{5} + 365472591 T^{6} + 5803698716 T^{7} + 85671363923 T^{8} + 1164978384166 T^{9} + 14926872643907 T^{10} + 178875659973247 T^{11} + 2035217727608960 T^{12} + 21823550842118395 T^{13} + 222987948881550762 T^{14} + 2153948747947025494 T^{15} + 19845927450458017818 T^{16} + \)\(17\!\cdots\!95\)\( T^{17} + \)\(14\!\cdots\!40\)\( T^{18} + \)\(11\!\cdots\!27\)\( T^{19} + \)\(83\!\cdots\!43\)\( T^{20} + \)\(57\!\cdots\!26\)\( T^{21} + \)\(37\!\cdots\!67\)\( T^{22} + \)\(22\!\cdots\!96\)\( T^{23} + \)\(12\!\cdots\!19\)\( T^{24} + \)\(64\!\cdots\!15\)\( T^{25} + \)\(29\!\cdots\!44\)\( T^{26} + \)\(11\!\cdots\!68\)\( T^{27} + \)\(37\!\cdots\!56\)\( T^{28} + \)\(89\!\cdots\!86\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \))(\( 1 - 44 T + 1927 T^{2} - 54350 T^{3} + 1458139 T^{4} - 31487709 T^{5} + 646528570 T^{6} - 11538721311 T^{7} + 196722571112 T^{8} - 3023385309595 T^{9} + 44592288644567 T^{10} - 604941790914728 T^{11} + 7905187195967763 T^{12} - 96162134560435884 T^{13} + 1129892147851445700 T^{14} - 12447443779589877500 T^{15} + \)\(13\!\cdots\!37\)\( T^{16} - \)\(13\!\cdots\!67\)\( T^{17} + \)\(12\!\cdots\!52\)\( T^{18} - \)\(11\!\cdots\!63\)\( T^{19} + \)\(10\!\cdots\!77\)\( T^{20} - \)\(87\!\cdots\!00\)\( T^{21} + \)\(70\!\cdots\!00\)\( T^{22} - \)\(53\!\cdots\!16\)\( T^{23} + \)\(39\!\cdots\!43\)\( T^{24} - \)\(26\!\cdots\!12\)\( T^{25} + \)\(17\!\cdots\!27\)\( T^{26} - \)\(10\!\cdots\!55\)\( T^{27} + \)\(61\!\cdots\!12\)\( T^{28} - \)\(32\!\cdots\!79\)\( T^{29} + \)\(15\!\cdots\!70\)\( T^{30} - \)\(69\!\cdots\!21\)\( T^{31} + \)\(28\!\cdots\!99\)\( T^{32} - \)\(94\!\cdots\!50\)\( T^{33} + \)\(29\!\cdots\!47\)\( T^{34} - \)\(60\!\cdots\!76\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))
$97$ (\( 1 - 8 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 21 T + 273 T^{2} + 2037 T^{3} + 9409 T^{4} \))(\( 1 + 2 T + 150 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 207 T^{2} + 1936 T^{3} + 20079 T^{4} + 112908 T^{5} + 912673 T^{6} \))(\( 1 + 26 T + 994 T^{2} + 18412 T^{3} + 419067 T^{4} + 6226726 T^{5} + 107948306 T^{6} + 1365044803 T^{7} + 19716930881 T^{8} + 218636377270 T^{9} + 2739861393601 T^{10} + 26981021125004 T^{11} + 298758142387224 T^{12} + 2617159049125388 T^{13} + 25779355852391809 T^{14} + 199543518352142710 T^{15} + 1745525714421626561 T^{16} + 11722104189410534371 T^{17} + 89917916877509200274 T^{18} + \)\(50\!\cdots\!38\)\( T^{19} + \)\(32\!\cdots\!87\)\( T^{20} + \)\(13\!\cdots\!04\)\( T^{21} + \)\(73\!\cdots\!06\)\( T^{22} + \)\(18\!\cdots\!78\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 5 T + 860 T^{2} + 3654 T^{3} + 357585 T^{4} + 1321128 T^{5} + 96179967 T^{6} + 315767895 T^{7} + 18923279992 T^{8} + 56408221506 T^{9} + 2929154946806 T^{10} + 8083074985785 T^{11} + 375649080319275 T^{12} + 972628985193628 T^{13} + 41452416754446442 T^{14} + 101048739140589122 T^{15} + 4020884425181304874 T^{16} + 9151466121686845852 T^{17} + \)\(34\!\cdots\!75\)\( T^{18} + \)\(71\!\cdots\!85\)\( T^{19} + \)\(25\!\cdots\!42\)\( T^{20} + \)\(46\!\cdots\!74\)\( T^{21} + \)\(15\!\cdots\!96\)\( T^{22} + \)\(24\!\cdots\!95\)\( T^{23} + \)\(73\!\cdots\!39\)\( T^{24} + \)\(97\!\cdots\!72\)\( T^{25} + \)\(25\!\cdots\!05\)\( T^{26} + \)\(25\!\cdots\!14\)\( T^{27} + \)\(57\!\cdots\!20\)\( T^{28} + \)\(32\!\cdots\!45\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \))(\( 1 - 11 T + 946 T^{2} - 9538 T^{3} + 445273 T^{4} - 4196116 T^{5} + 140154954 T^{6} - 1253005376 T^{7} + 33279732560 T^{8} - 284872829736 T^{9} + 6352378052523 T^{10} - 52263727924621 T^{11} + 1012224836827266 T^{12} - 7998398413761741 T^{13} + 137908328989793822 T^{14} - 1041611363192867202 T^{15} + 16309719612276148602 T^{16} - \)\(11\!\cdots\!75\)\( T^{17} + \)\(16\!\cdots\!98\)\( T^{18} - \)\(11\!\cdots\!75\)\( T^{19} + \)\(15\!\cdots\!18\)\( T^{20} - \)\(95\!\cdots\!46\)\( T^{21} + \)\(12\!\cdots\!82\)\( T^{22} - \)\(68\!\cdots\!37\)\( T^{23} + \)\(84\!\cdots\!14\)\( T^{24} - \)\(42\!\cdots\!73\)\( T^{25} + \)\(49\!\cdots\!03\)\( T^{26} - \)\(21\!\cdots\!12\)\( T^{27} + \)\(24\!\cdots\!40\)\( T^{28} - \)\(89\!\cdots\!28\)\( T^{29} + \)\(97\!\cdots\!14\)\( T^{30} - \)\(28\!\cdots\!32\)\( T^{31} + \)\(29\!\cdots\!37\)\( T^{32} - \)\(60\!\cdots\!34\)\( T^{33} + \)\(58\!\cdots\!66\)\( T^{34} - \)\(65\!\cdots\!07\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \))
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