Properties

Label 1003.2.a
Level 1003
Weight 2
Character orbit a
Rep. character \(\chi_{1003}(1,\cdot)\)
Character field \(\Q\)
Dimension 77
Newform subspaces 10
Sturm bound 180
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1003.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(180\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1003))\).

Total New Old
Modular forms 92 77 15
Cusp forms 89 77 12
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(17\)\(59\)FrickeDim.
\(+\)\(+\)\(+\)\(18\)
\(+\)\(-\)\(-\)\(22\)
\(-\)\(+\)\(-\)\(20\)
\(-\)\(-\)\(+\)\(17\)
Plus space\(+\)\(35\)
Minus space\(-\)\(42\)

Trace form

\( 77q - q^{2} + 75q^{4} + 2q^{5} + 4q^{6} - 8q^{7} + 3q^{8} + 77q^{9} + O(q^{10}) \) \( 77q - q^{2} + 75q^{4} + 2q^{5} + 4q^{6} - 8q^{7} + 3q^{8} + 77q^{9} + 18q^{10} + 4q^{11} + 16q^{12} - 10q^{13} + 12q^{14} + 20q^{15} + 67q^{16} - 3q^{17} - 17q^{18} - 4q^{19} - 18q^{20} - 32q^{21} - 12q^{22} + 8q^{23} - 24q^{24} + 95q^{25} - 26q^{26} + 12q^{27} - 12q^{28} - 34q^{29} - 40q^{30} - 8q^{31} - 5q^{32} + 3q^{34} + 36q^{35} + 11q^{36} - 18q^{37} - 40q^{38} + 20q^{39} + 30q^{40} + 2q^{41} - 88q^{42} - 4q^{43} - 40q^{44} - 14q^{45} - 48q^{46} - 4q^{47} - 16q^{48} + 57q^{49} - 59q^{50} - 4q^{51} - 50q^{52} + 26q^{53} - 12q^{54} + 24q^{55} - 16q^{56} - 12q^{57} + 42q^{58} + q^{59} + 64q^{60} - 22q^{61} - 36q^{62} + 8q^{63} + 67q^{64} + 48q^{65} - 40q^{66} - 20q^{67} - 5q^{68} + 24q^{69} + 20q^{70} - 24q^{71} - 9q^{72} + 10q^{73} + 18q^{74} - 32q^{75} - 8q^{76} - 20q^{77} - 52q^{78} - 48q^{79} - 98q^{80} + 53q^{81} + 38q^{82} - 20q^{83} - 96q^{84} - 2q^{85} + 12q^{86} - 40q^{87} + 28q^{88} + 2q^{89} + 58q^{90} - 4q^{91} - 4q^{92} - 16q^{93} + 56q^{94} - 44q^{95} + 4q^{96} + 6q^{97} - 21q^{98} - 32q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 17 59
1003.2.a.a \(1\) \(8.009\) \(\Q\) None \(-1\) \(3\) \(1\) \(1\) \(-\) \(+\) \(q-q^{2}+3q^{3}-q^{4}+q^{5}-3q^{6}+q^{7}+\cdots\)
1003.2.a.b \(1\) \(8.009\) \(\Q\) None \(0\) \(2\) \(-2\) \(2\) \(+\) \(+\) \(q+2q^{3}-2q^{4}-2q^{5}+2q^{7}+q^{9}+\cdots\)
1003.2.a.c \(1\) \(8.009\) \(\Q\) None \(1\) \(1\) \(1\) \(3\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}+3q^{7}+\cdots\)
1003.2.a.d \(1\) \(8.009\) \(\Q\) None \(2\) \(0\) \(-2\) \(-2\) \(-\) \(-\) \(q+2q^{2}+2q^{4}-2q^{5}-2q^{7}-3q^{9}+\cdots\)
1003.2.a.e \(3\) \(8.009\) 3.3.229.1 None \(-3\) \(2\) \(-4\) \(-2\) \(+\) \(+\) \(q-q^{2}+(1+\beta _{2})q^{3}-q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
1003.2.a.f \(4\) \(8.009\) 4.4.2225.1 None \(-3\) \(-2\) \(1\) \(8\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1003.2.a.g \(10\) \(8.009\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-1\) \(-7\) \(-12\) \(-9\) \(+\) \(+\) \(q-\beta _{8}q^{2}+(-1+\beta _{1})q^{3}+(2-\beta _{5})q^{4}+\cdots\)
1003.2.a.h \(16\) \(8.009\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-6\) \(-7\) \(-21\) \(-11\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{10}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1003.2.a.i \(18\) \(8.009\) \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(5\) \(1\) \(21\) \(-1\) \(-\) \(+\) \(q+\beta _{1}q^{2}+\beta _{10}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
1003.2.a.j \(22\) \(8.009\) None \(5\) \(7\) \(19\) \(3\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1003))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1003)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( ( 1 + T + 2 T^{2} )^{3} \))(\( 1 + 3 T + 6 T^{2} + 11 T^{3} + 17 T^{4} + 22 T^{5} + 24 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 + T + 3 T^{2} + 6 T^{3} + 16 T^{4} + 20 T^{5} + 44 T^{6} + 67 T^{7} + 117 T^{8} + 149 T^{9} + 259 T^{10} + 298 T^{11} + 468 T^{12} + 536 T^{13} + 704 T^{14} + 640 T^{15} + 1024 T^{16} + 768 T^{17} + 768 T^{18} + 512 T^{19} + 1024 T^{20} \))(\( 1 + 6 T + 27 T^{2} + 90 T^{3} + 258 T^{4} + 636 T^{5} + 1415 T^{6} + 2841 T^{7} + 5267 T^{8} + 9017 T^{9} + 14477 T^{10} + 21833 T^{11} + 31441 T^{12} + 43557 T^{13} + 59447 T^{14} + 81092 T^{15} + 113283 T^{16} + 162184 T^{17} + 237788 T^{18} + 348456 T^{19} + 503056 T^{20} + 698656 T^{21} + 926528 T^{22} + 1154176 T^{23} + 1348352 T^{24} + 1454592 T^{25} + 1448960 T^{26} + 1302528 T^{27} + 1056768 T^{28} + 737280 T^{29} + 442368 T^{30} + 196608 T^{31} + 65536 T^{32} \))(\( 1 - 5 T + 20 T^{2} - 58 T^{3} + 152 T^{4} - 344 T^{5} + 735 T^{6} - 1440 T^{7} + 2727 T^{8} - 4866 T^{9} + 8497 T^{10} - 14198 T^{11} + 23403 T^{12} - 37248 T^{13} + 58661 T^{14} - 89451 T^{15} + 134796 T^{16} - 196502 T^{17} + 282612 T^{18} - 393004 T^{19} + 539184 T^{20} - 715608 T^{21} + 938576 T^{22} - 1191936 T^{23} + 1497792 T^{24} - 1817344 T^{25} + 2175232 T^{26} - 2491392 T^{27} + 2792448 T^{28} - 2949120 T^{29} + 3010560 T^{30} - 2818048 T^{31} + 2490368 T^{32} - 1900544 T^{33} + 1310720 T^{34} - 655360 T^{35} + 262144 T^{36} \))
$3$ (\( 1 - 3 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 4 T^{2} - 10 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 7 T^{2} + 12 T^{3} + 28 T^{4} + 36 T^{5} + 63 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + 7 T + 38 T^{2} + 155 T^{3} + 541 T^{4} + 1621 T^{5} + 4342 T^{6} + 10389 T^{7} + 22635 T^{8} + 44766 T^{9} + 81197 T^{10} + 134298 T^{11} + 203715 T^{12} + 280503 T^{13} + 351702 T^{14} + 393903 T^{15} + 394389 T^{16} + 338985 T^{17} + 249318 T^{18} + 137781 T^{19} + 59049 T^{20} \))(\( 1 + 7 T + 40 T^{2} + 161 T^{3} + 587 T^{4} + 1837 T^{5} + 5398 T^{6} + 14467 T^{7} + 36819 T^{8} + 87678 T^{9} + 199691 T^{10} + 431612 T^{11} + 896496 T^{12} + 1779482 T^{13} + 3402090 T^{14} + 6235902 T^{15} + 11024208 T^{16} + 18707706 T^{17} + 30618810 T^{18} + 48046014 T^{19} + 72616176 T^{20} + 104881716 T^{21} + 145574739 T^{22} + 191751786 T^{23} + 241569459 T^{24} + 284753961 T^{25} + 318746502 T^{26} + 325419039 T^{27} + 311955867 T^{28} + 256686003 T^{29} + 191318760 T^{30} + 100442349 T^{31} + 43046721 T^{32} \))(\( 1 - T + 19 T^{2} - 22 T^{3} + 201 T^{4} - 250 T^{5} + 1538 T^{6} - 1963 T^{7} + 9399 T^{8} - 11928 T^{9} + 48218 T^{10} - 59862 T^{11} + 213988 T^{12} - 257547 T^{13} + 838124 T^{14} - 971758 T^{15} + 2935787 T^{16} - 3258875 T^{17} + 9267386 T^{18} - 9776625 T^{19} + 26422083 T^{20} - 26237466 T^{21} + 67888044 T^{22} - 62583921 T^{23} + 155997252 T^{24} - 130918194 T^{25} + 316358298 T^{26} - 234778824 T^{27} + 555001551 T^{28} - 347739561 T^{29} + 817356258 T^{30} - 398580750 T^{31} + 961376769 T^{32} - 315675954 T^{33} + 817887699 T^{34} - 129140163 T^{35} + 387420489 T^{36} \))
$5$ (\( 1 - T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 4 T + 8 T^{2} + 14 T^{3} + 40 T^{4} + 100 T^{5} + 125 T^{6} \))(\( 1 - T + 12 T^{2} - 4 T^{3} + 71 T^{4} - 20 T^{5} + 300 T^{6} - 125 T^{7} + 625 T^{8} \))(\( 1 + 12 T + 96 T^{2} + 570 T^{3} + 2783 T^{4} + 11529 T^{5} + 41780 T^{6} + 134057 T^{7} + 385889 T^{8} + 1000919 T^{9} + 2351889 T^{10} + 5004595 T^{11} + 9647225 T^{12} + 16757125 T^{13} + 26112500 T^{14} + 36028125 T^{15} + 43484375 T^{16} + 44531250 T^{17} + 37500000 T^{18} + 23437500 T^{19} + 9765625 T^{20} \))(\( 1 + 21 T + 249 T^{2} + 2143 T^{3} + 14807 T^{4} + 86535 T^{5} + 441605 T^{6} + 2009432 T^{7} + 8272917 T^{8} + 31143873 T^{9} + 108048711 T^{10} + 347499761 T^{11} + 1040653133 T^{12} + 2911461051 T^{13} + 7628016320 T^{14} + 18746200712 T^{15} + 43253965373 T^{16} + 93731003560 T^{17} + 190700408000 T^{18} + 363932631375 T^{19} + 650408208125 T^{20} + 1085936753125 T^{21} + 1688261109375 T^{22} + 2433115078125 T^{23} + 3231608203125 T^{24} + 3924671875000 T^{25} + 4312548828125 T^{26} + 4225341796875 T^{27} + 3614990234375 T^{28} + 2615966796875 T^{29} + 1519775390625 T^{30} + 640869140625 T^{31} + 152587890625 T^{32} \))(\( 1 - 21 T + 248 T^{2} - 2112 T^{3} + 14390 T^{4} - 82850 T^{5} + 417181 T^{6} - 1880214 T^{7} + 7712027 T^{8} - 29143104 T^{9} + 102409048 T^{10} - 337007088 T^{11} + 1044205383 T^{12} - 3058958706 T^{13} + 8499119894 T^{14} - 22450225214 T^{15} + 56477307355 T^{16} - 135473844787 T^{17} + 310079023954 T^{18} - 677369223935 T^{19} + 1411932683875 T^{20} - 2806278151750 T^{21} + 5311949933750 T^{22} - 9559245956250 T^{23} + 16315709109375 T^{24} - 26328678750000 T^{25} + 40003534375000 T^{26} - 56920125000000 T^{27} + 75312763671875 T^{28} - 91807324218750 T^{29} + 101850830078125 T^{30} - 101135253906250 T^{31} + 87829589843750 T^{32} - 64453125000000 T^{33} + 37841796875000 T^{34} - 16021728515625 T^{35} + 3814697265625 T^{36} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 16 T^{2} + 26 T^{3} + 112 T^{4} + 98 T^{5} + 343 T^{6} \))(\( ( 1 - 4 T + 13 T^{2} - 28 T^{3} + 49 T^{4} )^{2} \))(\( 1 + 9 T + 80 T^{2} + 468 T^{3} + 2555 T^{4} + 11334 T^{5} + 46888 T^{6} + 168019 T^{7} + 564109 T^{8} + 1680606 T^{9} + 4704893 T^{10} + 11764242 T^{11} + 27641341 T^{12} + 57630517 T^{13} + 112578088 T^{14} + 190490538 T^{15} + 300593195 T^{16} + 385418124 T^{17} + 461184080 T^{18} + 363182463 T^{19} + 282475249 T^{20} \))(\( 1 + 11 T + 99 T^{2} + 631 T^{3} + 3529 T^{4} + 16644 T^{5} + 71610 T^{6} + 275534 T^{7} + 992543 T^{8} + 3302021 T^{9} + 10512336 T^{10} + 31647691 T^{11} + 92926579 T^{12} + 262444709 T^{13} + 731975367 T^{14} + 1980226443 T^{15} + 5314535269 T^{16} + 13861585101 T^{17} + 35866792983 T^{18} + 90018535187 T^{19} + 223116716179 T^{20} + 531902742637 T^{21} + 1236765818064 T^{22} + 2719356280403 T^{23} + 5721812878943 T^{24} + 11118790751138 T^{25} + 20228052580890 T^{26} + 32910626310492 T^{27} + 48845902532329 T^{28} + 61136965566817 T^{29} + 67144084212051 T^{30} + 52223176609373 T^{31} + 33232930569601 T^{32} \))(\( 1 + T + 56 T^{2} + 34 T^{3} + 1518 T^{4} + 485 T^{5} + 26882 T^{6} + 3211 T^{7} + 350122 T^{8} - 1782 T^{9} + 3562855 T^{10} - 221519 T^{11} + 29467805 T^{12} - 1805704 T^{13} + 206767815 T^{14} - 3783304 T^{15} + 1329201842 T^{16} + 43203918 T^{17} + 8830674752 T^{18} + 302427426 T^{19} + 65130890258 T^{20} - 1297673272 T^{21} + 496449523815 T^{22} - 30348467128 T^{23} + 3466857790445 T^{24} - 182430421817 T^{25} + 20539150066855 T^{26} - 71910127674 T^{27} + 98900799130378 T^{28} + 6349196171773 T^{29} + 372081482537282 T^{30} + 46991170047395 T^{31} + 1029542624584782 T^{32} + 161417091338062 T^{33} + 1861044111897656 T^{34} + 232630513987207 T^{35} + 1628413597910449 T^{36} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{3} \))(\( 1 + 11 T + 79 T^{2} + 391 T^{3} + 1480 T^{4} + 4301 T^{5} + 9559 T^{6} + 14641 T^{7} + 14641 T^{8} \))(\( 1 - 12 T + 106 T^{2} - 672 T^{3} + 3820 T^{4} - 18679 T^{5} + 86210 T^{6} - 357923 T^{7} + 1411720 T^{8} - 5083546 T^{9} + 17554057 T^{10} - 55919006 T^{11} + 170818120 T^{12} - 476395513 T^{13} + 1262200610 T^{14} - 3008271629 T^{15} + 6767363020 T^{16} - 13095378912 T^{17} + 22722041386 T^{18} - 28295372292 T^{19} + 25937424601 T^{20} \))(\( 1 + 7 T + 99 T^{2} + 573 T^{3} + 4855 T^{4} + 24373 T^{5} + 158595 T^{6} + 708606 T^{7} + 3877077 T^{8} + 15687965 T^{9} + 75519819 T^{10} + 280177541 T^{11} + 1218630412 T^{12} + 4181500767 T^{13} + 16705740691 T^{14} + 53294814856 T^{15} + 197433330166 T^{16} + 586242963416 T^{17} + 2021394623611 T^{18} + 5565577520877 T^{19} + 17841967862092 T^{20} + 45122873155591 T^{21} + 133787966067459 T^{22} + 305714056597015 T^{23} + 831085887270837 T^{24} + 1670855881528746 T^{25} + 4113545854595595 T^{26} + 6953901347801903 T^{27} + 15237069768980455 T^{28} + 19781514058472463 T^{29} + 37595233524740859 T^{30} + 29240737185909557 T^{31} + 45949729863572161 T^{32} \))(\( 1 - 6 T + 127 T^{2} - 689 T^{3} + 7708 T^{4} - 38227 T^{5} + 298469 T^{6} - 1369387 T^{7} + 8327943 T^{8} - 35790520 T^{9} + 179869059 T^{10} - 732797929 T^{11} + 3164067020 T^{12} - 12337375884 T^{13} + 47169541671 T^{14} - 176813586395 T^{15} + 614445550848 T^{16} - 2207242164895 T^{17} + 7134211635492 T^{18} - 24279663813845 T^{19} + 74347911652608 T^{20} - 235338883491745 T^{21} + 690609259605111 T^{22} - 1986946723494084 T^{23} + 5605337734018220 T^{24} - 14280158550868859 T^{25} + 38556530213762979 T^{26} - 84392173993689320 T^{27} + 216005393643925743 T^{28} - 390702092682985457 T^{29} + 936723579171540149 T^{30} - 1319699717126050337 T^{31} + 2927111717259621628 T^{32} - 2878123988727383539 T^{33} + 5835615692673664447 T^{34} - 3032682170995762626 T^{35} + 5559917313492231481 T^{36} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 23 T^{2} + 48 T^{3} + 299 T^{4} + 676 T^{5} + 2197 T^{6} \))(\( 1 + 9 T + 77 T^{2} + 373 T^{3} + 1668 T^{4} + 4849 T^{5} + 13013 T^{6} + 19773 T^{7} + 28561 T^{8} \))(\( 1 + 11 T + 115 T^{2} + 763 T^{3} + 4875 T^{4} + 24307 T^{5} + 120492 T^{6} + 501644 T^{7} + 2136116 T^{8} + 7904440 T^{9} + 30412295 T^{10} + 102757720 T^{11} + 361003604 T^{12} + 1102111868 T^{13} + 3441372012 T^{14} + 9025018951 T^{15} + 23530693875 T^{16} + 47877118471 T^{17} + 93809032915 T^{18} + 116649493103 T^{19} + 137858491849 T^{20} \))(\( 1 + 16 T + 203 T^{2} + 1792 T^{3} + 14275 T^{4} + 95415 T^{5} + 600564 T^{6} + 3371571 T^{7} + 18169721 T^{8} + 89901243 T^{9} + 431538912 T^{10} + 1930538895 T^{11} + 8424397174 T^{12} + 34540053626 T^{13} + 138565115677 T^{14} + 524517883170 T^{15} + 1945586553346 T^{16} + 6818732481210 T^{17} + 23417504549413 T^{18} + 75884497816322 T^{19} + 240609207686614 T^{20} + 716795577941235 T^{21} + 2082955904291808 T^{22} + 5641169674706631 T^{23} + 14821599611698841 T^{24} + 35753822555524983 T^{25} + 82792847298802836 T^{26} + 170998983997040355 T^{27} + 332580165123416275 T^{28} + 542752191013317376 T^{29} + 799287406296955667 T^{30} + 818974288225452112 T^{31} + 665416609183179841 T^{32} \))(\( 1 - 12 T + 195 T^{2} - 1722 T^{3} + 16570 T^{4} - 117747 T^{5} + 854099 T^{6} - 5136447 T^{7} + 30754314 T^{8} - 161864038 T^{9} + 839694908 T^{10} - 3965488172 T^{11} + 18391231959 T^{12} - 79463553929 T^{13} + 336854135767 T^{14} - 1350987506629 T^{15} + 5315299094844 T^{16} - 19964398048608 T^{17} + 73562236154478 T^{18} - 259537174631904 T^{19} + 898285547028636 T^{20} - 2968119552063913 T^{21} + 9620890971641287 T^{22} - 29504261328960197 T^{23} + 88770963940788831 T^{24} - 248828501974040924 T^{25} + 684964932722868668 T^{26} - 1716487089482248174 T^{27} + 4239743345890586586 T^{28} - 9205336879470166539 T^{29} + 19898871205025899619 T^{30} - 35662635175918013991 T^{31} + 65242326711037218730 T^{32} - 88142107770264283554 T^{33} + \)\(12\!\cdots\!95\)\( T^{34} - \)\(10\!\cdots\!96\)\( T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \))
$17$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{10} \))(\( ( 1 - T )^{16} \))(\( ( 1 - T )^{18} \))
$19$ (\( 1 - 5 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 7 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 8 T + 72 T^{2} + 308 T^{3} + 1368 T^{4} + 2888 T^{5} + 6859 T^{6} \))(\( 1 - 10 T + 92 T^{2} - 520 T^{3} + 2733 T^{4} - 9880 T^{5} + 33212 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( 1 + 5 T + 104 T^{2} + 550 T^{3} + 5320 T^{4} + 27780 T^{5} + 187167 T^{6} + 888731 T^{7} + 5101648 T^{8} + 21055340 T^{9} + 109553353 T^{10} + 400051460 T^{11} + 1841694928 T^{12} + 6095805929 T^{13} + 24391790607 T^{14} + 68786030220 T^{15} + 250284086920 T^{16} + 491629456450 T^{17} + 1766290556264 T^{18} + 1613438488895 T^{19} + 6131066257801 T^{20} \))(\( 1 - T + 107 T^{2} - 107 T^{3} + 6348 T^{4} - 4850 T^{5} + 273578 T^{6} - 134840 T^{7} + 9529325 T^{8} - 2760589 T^{9} + 282243356 T^{10} - 46032983 T^{11} + 7281865981 T^{12} - 636489901 T^{13} + 165561919843 T^{14} - 8211582005 T^{15} + 3336711850579 T^{16} - 156020058095 T^{17} + 59767853063323 T^{18} - 4365684230959 T^{19} + 948980056509901 T^{20} - 113982223173317 T^{21} + 13278387339416636 T^{22} - 2467612490094271 T^{23} + 161841891875677325 T^{24} - 43511209168520360 T^{25} + 1677324844676681978 T^{26} - 564977755656362150 T^{27} + 14050123106231990028 T^{28} - 4499669230461505313 T^{29} + 85493715378768600947 T^{30} - 15181127029874798299 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 14 T + 290 T^{2} + 3029 T^{3} + 36817 T^{4} + 314426 T^{5} + 2870012 T^{6} + 21074543 T^{7} + 158491227 T^{8} + 1032221026 T^{9} + 6708312133 T^{10} + 39543233724 T^{11} + 228501358352 T^{12} + 1235145267619 T^{13} + 6463358843799 T^{14} + 32283670501184 T^{15} + 154818187749602 T^{16} + 716919841360198 T^{17} + 3174109718888434 T^{18} + 13621476985843762 T^{19} + 55889365777606322 T^{20} + 221433695967621056 T^{21} + 842311387882729479 T^{22} + 3058341962006138281 T^{23} + 10750047713366548112 T^{24} + 35346579094555326036 T^{25} + \)\(11\!\cdots\!53\)\( T^{26} + \)\(33\!\cdots\!54\)\( T^{27} + \)\(97\!\cdots\!27\)\( T^{28} + \)\(24\!\cdots\!17\)\( T^{29} + \)\(63\!\cdots\!32\)\( T^{30} + \)\(13\!\cdots\!34\)\( T^{31} + \)\(29\!\cdots\!57\)\( T^{32} + \)\(45\!\cdots\!71\)\( T^{33} + \)\(83\!\cdots\!90\)\( T^{34} + \)\(76\!\cdots\!46\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 + 5 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 2 T + 49 T^{2} - 108 T^{3} + 1127 T^{4} - 1058 T^{5} + 12167 T^{6} \))(\( 1 + 3 T + 57 T^{2} + 193 T^{3} + 1568 T^{4} + 4439 T^{5} + 30153 T^{6} + 36501 T^{7} + 279841 T^{8} \))(\( 1 + 7 T + 120 T^{2} + 588 T^{3} + 6289 T^{4} + 22668 T^{5} + 201269 T^{6} + 514368 T^{7} + 4737639 T^{8} + 8908707 T^{9} + 104593011 T^{10} + 204900261 T^{11} + 2506211031 T^{12} + 6258315456 T^{13} + 56323318229 T^{14} + 145899023124 T^{15} + 930997705921 T^{16} + 2002037362836 T^{17} + 9397318233720 T^{18} + 12608068630241 T^{19} + 41426511213649 T^{20} \))(\( 1 + 8 T + 160 T^{2} + 1156 T^{3} + 13275 T^{4} + 89694 T^{5} + 775926 T^{6} + 4914858 T^{7} + 35792094 T^{8} + 211598797 T^{9} + 1371736394 T^{10} + 7558519108 T^{11} + 44803889428 T^{12} + 230539116504 T^{13} + 1264759463636 T^{14} + 6088439654145 T^{15} + 31112640363596 T^{16} + 140034112045335 T^{17} + 669057756263444 T^{18} + 2804969430504168 T^{19} + 12537965221420948 T^{20} + 48649221551142044 T^{21} + 203066216559444266 T^{22} + 720456968580187259 T^{23} + 2802914146410168414 T^{24} + 8852409567412717254 T^{25} + 32143907139961813974 T^{26} + 85461318426331768338 T^{27} + \)\(29\!\cdots\!75\)\( T^{28} + \)\(58\!\cdots\!48\)\( T^{29} + \)\(18\!\cdots\!40\)\( T^{30} + \)\(21\!\cdots\!56\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 7 T + 239 T^{2} - 1576 T^{3} + 28588 T^{4} - 179050 T^{5} + 2269277 T^{6} - 13561935 T^{7} + 133919327 T^{8} - 766089347 T^{9} + 6252142135 T^{10} - 34292124369 T^{11} + 240295781412 T^{12} - 1263196000211 T^{13} + 7818397648251 T^{14} - 39255760559581 T^{15} + 219632672936210 T^{16} - 1045456603448794 T^{17} + 5392265405260800 T^{18} - 24045501879322262 T^{19} + 116185683983255090 T^{20} - 477624838728422027 T^{21} + 2187908216284208091 T^{22} - 8130362733586068373 T^{23} + 35572399624275095268 T^{24} - \)\(11\!\cdots\!43\)\( T^{25} + \)\(48\!\cdots\!35\)\( T^{26} - \)\(13\!\cdots\!61\)\( T^{27} + \)\(55\!\cdots\!23\)\( T^{28} - \)\(12\!\cdots\!45\)\( T^{29} + \)\(49\!\cdots\!17\)\( T^{30} - \)\(90\!\cdots\!50\)\( T^{31} + \)\(33\!\cdots\!92\)\( T^{32} - \)\(42\!\cdots\!32\)\( T^{33} + \)\(14\!\cdots\!79\)\( T^{34} - \)\(98\!\cdots\!21\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))
$29$ (\( 1 + 9 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 44 T^{2} + 106 T^{3} + 1276 T^{4} + 24389 T^{6} \))(\( 1 + 9 T + 126 T^{2} + 740 T^{3} + 5585 T^{4} + 21460 T^{5} + 105966 T^{6} + 219501 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 157 T^{2} - 1430 T^{3} + 14283 T^{4} - 105029 T^{5} + 840303 T^{6} - 5372919 T^{7} + 35889806 T^{8} - 201535046 T^{9} + 1186241053 T^{10} - 5844516334 T^{11} + 30183326846 T^{12} - 131040121491 T^{13} + 594330346143 T^{14} - 2154265468321 T^{15} + 8495861493843 T^{16} - 24667323121870 T^{17} + 78538686834877 T^{18} - 145071459758690 T^{19} + 420707233300201 T^{20} \))(\( 1 + 39 T + 928 T^{2} + 16208 T^{3} + 230056 T^{4} + 2780632 T^{5} + 29604156 T^{6} + 283587370 T^{7} + 2485169966 T^{8} + 20154979173 T^{9} + 152664366097 T^{10} + 1087056903986 T^{11} + 7313935686038 T^{12} + 46666449986350 T^{13} + 283154391474588 T^{14} + 1636759340906011 T^{15} + 9023892718674587 T^{16} + 47466020886274319 T^{17} + 238132843230128508 T^{18} + 1138148048717090150 T^{19} + 5173007745956642678 T^{20} + 22296786129135539914 T^{21} + 90808325240177348137 T^{22} + \)\(34\!\cdots\!57\)\( T^{23} + \)\(12\!\cdots\!26\)\( T^{24} + \)\(41\!\cdots\!30\)\( T^{25} + \)\(12\!\cdots\!56\)\( T^{26} + \)\(33\!\cdots\!28\)\( T^{27} + \)\(81\!\cdots\!96\)\( T^{28} + \)\(16\!\cdots\!12\)\( T^{29} + \)\(27\!\cdots\!68\)\( T^{30} + \)\(33\!\cdots\!11\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 23 T + 563 T^{2} - 8615 T^{3} + 128432 T^{4} - 1522980 T^{5} + 17354219 T^{6} - 171102877 T^{7} + 1620106476 T^{8} - 13815187483 T^{9} + 113412360674 T^{10} - 857384208124 T^{11} + 6258622685406 T^{12} - 42644913205081 T^{13} + 281339577465084 T^{14} - 1747023215055564 T^{15} + 10525426577055857 T^{16} - 59964845786916469 T^{17} + 331827567818956736 T^{18} - 1738980527820577601 T^{19} + 8851883751303975737 T^{20} - 42608149191990150396 T^{21} + \)\(19\!\cdots\!04\)\( T^{22} - \)\(87\!\cdots\!69\)\( T^{23} + \)\(37\!\cdots\!26\)\( T^{24} - \)\(14\!\cdots\!16\)\( T^{25} + \)\(56\!\cdots\!14\)\( T^{26} - \)\(20\!\cdots\!27\)\( T^{27} + \)\(68\!\cdots\!76\)\( T^{28} - \)\(20\!\cdots\!33\)\( T^{29} + \)\(61\!\cdots\!79\)\( T^{30} - \)\(15\!\cdots\!20\)\( T^{31} + \)\(38\!\cdots\!92\)\( T^{32} - \)\(74\!\cdots\!35\)\( T^{33} + \)\(14\!\cdots\!23\)\( T^{34} - \)\(16\!\cdots\!07\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$31$ (\( 1 - 10 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 18 T + 185 T^{2} + 1228 T^{3} + 5735 T^{4} + 17298 T^{5} + 29791 T^{6} \))(\( 1 + 4 T + 109 T^{2} + 374 T^{3} + 4840 T^{4} + 11594 T^{5} + 104749 T^{6} + 119164 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 170 T^{2} - 1670 T^{3} + 14959 T^{4} - 116004 T^{5} + 832653 T^{6} - 5483248 T^{7} + 34521695 T^{8} - 202943201 T^{9} + 1156502631 T^{10} - 6291239231 T^{11} + 33175348895 T^{12} - 163351441168 T^{13} + 768972531213 T^{14} - 3321096032604 T^{15} + 13276167564079 T^{16} - 45946065565370 T^{17} + 144991476364970 T^{18} - 343715088088723 T^{19} + 819628286980801 T^{20} \))(\( 1 + 3 T + 255 T^{2} + 791 T^{3} + 32526 T^{4} + 101121 T^{5} + 2753424 T^{6} + 8468886 T^{7} + 174562090 T^{8} + 528154096 T^{9} + 8901907102 T^{10} + 26393054295 T^{11} + 382768477555 T^{12} + 1104116824605 T^{13} + 14294540542319 T^{14} + 39577703446301 T^{15} + 470796472164880 T^{16} + 1226908806835331 T^{17} + 13737053461168559 T^{18} + 32892744321807555 T^{19} + 353494727160071155 T^{20} + 755610736762753545 T^{21} + 7900475320945042462 T^{22} + 14530899834392048656 T^{23} + \)\(14\!\cdots\!90\)\( T^{24} + \)\(22\!\cdots\!06\)\( T^{25} + \)\(22\!\cdots\!24\)\( T^{26} + \)\(25\!\cdots\!51\)\( T^{27} + \)\(25\!\cdots\!86\)\( T^{28} + \)\(19\!\cdots\!81\)\( T^{29} + \)\(19\!\cdots\!55\)\( T^{30} + \)\(70\!\cdots\!53\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + T + 371 T^{2} + 375 T^{3} + 66747 T^{4} + 74684 T^{5} + 7774251 T^{6} + 10103299 T^{7} + 660259304 T^{8} + 1004583593 T^{9} + 43677801352 T^{10} + 76298900803 T^{11} + 2348382550731 T^{12} + 4538784174677 T^{13} + 105697196910147 T^{14} + 215218702414202 T^{15} + 4064525040452553 T^{16} + 8228165505228620 T^{17} + 135195912490404158 T^{18} + 255073130662087220 T^{19} + 3906008563874903433 T^{20} + 6411580363621491782 T^{21} + 97613580987655867587 T^{22} + \)\(12\!\cdots\!27\)\( T^{23} + \)\(20\!\cdots\!11\)\( T^{24} + \)\(20\!\cdots\!33\)\( T^{25} + \)\(37\!\cdots\!32\)\( T^{26} + \)\(26\!\cdots\!03\)\( T^{27} + \)\(54\!\cdots\!04\)\( T^{28} + \)\(25\!\cdots\!69\)\( T^{29} + \)\(61\!\cdots\!11\)\( T^{30} + \)\(18\!\cdots\!44\)\( T^{31} + \)\(50\!\cdots\!87\)\( T^{32} + \)\(87\!\cdots\!25\)\( T^{33} + \)\(26\!\cdots\!51\)\( T^{34} + \)\(22\!\cdots\!11\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \))
$37$ (\( 1 + 8 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 4 T + 31 T^{2} - 424 T^{3} + 1147 T^{4} - 5476 T^{5} + 50653 T^{6} \))(\( 1 + 32 T + 511 T^{2} + 5224 T^{3} + 37492 T^{4} + 193288 T^{5} + 699559 T^{6} + 1620896 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 270 T^{2} - 2244 T^{3} + 30448 T^{4} - 195442 T^{5} + 2093530 T^{6} - 11100210 T^{7} + 105028311 T^{8} - 486723652 T^{9} + 4252341769 T^{10} - 18008775124 T^{11} + 143783757759 T^{12} - 562258937130 T^{13} + 3923612278330 T^{14} - 13552721643994 T^{15} + 78121237701232 T^{16} - 213027132286452 T^{17} + 948369452558670 T^{18} - 1559540877540924 T^{19} + 4808584372417849 T^{20} \))(\( 1 + 28 T + 659 T^{2} + 11230 T^{3} + 168029 T^{4} + 2160736 T^{5} + 25241316 T^{6} + 267548264 T^{7} + 2629412336 T^{8} + 24030058442 T^{9} + 206141237141 T^{10} + 1665515462176 T^{11} + 12724942102618 T^{12} + 92203962435862 T^{13} + 634461000765648 T^{14} + 4155167012371534 T^{15} + 25893210569519688 T^{16} + 153741179457746758 T^{17} + 868577110048172112 T^{18} + 4670407309263717886 T^{19} + 23848590215984653498 T^{20} + \)\(11\!\cdots\!32\)\( T^{21} + \)\(52\!\cdots\!69\)\( T^{22} + \)\(22\!\cdots\!86\)\( T^{23} + \)\(92\!\cdots\!56\)\( T^{24} + \)\(34\!\cdots\!28\)\( T^{25} + \)\(12\!\cdots\!84\)\( T^{26} + \)\(38\!\cdots\!68\)\( T^{27} + \)\(11\!\cdots\!49\)\( T^{28} + \)\(27\!\cdots\!10\)\( T^{29} + \)\(59\!\cdots\!51\)\( T^{30} + \)\(93\!\cdots\!04\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 36 T + 1035 T^{2} - 21370 T^{3} + 381842 T^{4} - 5803940 T^{5} + 79432215 T^{6} - 976875470 T^{7} + 11061829355 T^{8} - 115471197904 T^{9} + 1125156678811 T^{10} - 10250379469620 T^{11} + 87935059995072 T^{12} - 711273924784364 T^{13} + 5448065421722451 T^{14} - 39546158667594450 T^{15} + 272747091667535146 T^{16} - 1787574464717241302 T^{17} + 11149407736420495504 T^{18} - 66140255194537928174 T^{19} + \)\(37\!\cdots\!74\)\( T^{20} - \)\(20\!\cdots\!50\)\( T^{21} + \)\(10\!\cdots\!11\)\( T^{22} - \)\(49\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!48\)\( T^{24} - \)\(97\!\cdots\!60\)\( T^{25} + \)\(39\!\cdots\!31\)\( T^{26} - \)\(15\!\cdots\!08\)\( T^{27} + \)\(53\!\cdots\!95\)\( T^{28} - \)\(17\!\cdots\!10\)\( T^{29} + \)\(52\!\cdots\!15\)\( T^{30} - \)\(14\!\cdots\!80\)\( T^{31} + \)\(34\!\cdots\!38\)\( T^{32} - \)\(71\!\cdots\!10\)\( T^{33} + \)\(12\!\cdots\!35\)\( T^{34} - \)\(16\!\cdots\!12\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 12 T + 68 T^{2} - 242 T^{3} + 2788 T^{4} - 20172 T^{5} + 68921 T^{6} \))(\( 1 - 4 T + 86 T^{2} - 648 T^{3} + 3671 T^{4} - 26568 T^{5} + 144566 T^{6} - 275684 T^{7} + 2825761 T^{8} \))(\( 1 + 29 T + 588 T^{2} + 9019 T^{3} + 117459 T^{4} + 1312900 T^{5} + 13048499 T^{6} + 115992786 T^{7} + 935973813 T^{8} + 6858355770 T^{9} + 45972909783 T^{10} + 281192586570 T^{11} + 1573371979653 T^{12} + 7994338803906 T^{13} + 36871939582739 T^{14} + 152107606292900 T^{15} + 557942494043619 T^{16} + 1756488796132739 T^{17} + 4695136034723148 T^{18} + 9494076097424869 T^{19} + 13422659310152401 T^{20} \))(\( 1 + 31 T + 845 T^{2} + 16144 T^{3} + 274419 T^{4} + 3937303 T^{5} + 51468151 T^{6} + 602401175 T^{7} + 6533515132 T^{8} + 65168277381 T^{9} + 609165944944 T^{10} + 5312237628797 T^{11} + 43729918253400 T^{12} + 338547209812076 T^{13} + 2484799560505024 T^{14} + 17223431233458923 T^{15} + 113409862522256899 T^{16} + 706160680571815843 T^{17} + 4176948061208945344 T^{18} + 23333012247458089996 T^{19} + \)\(12\!\cdots\!00\)\( T^{20} + \)\(61\!\cdots\!97\)\( T^{21} + \)\(28\!\cdots\!04\)\( T^{22} + \)\(12\!\cdots\!61\)\( T^{23} + \)\(52\!\cdots\!72\)\( T^{24} + \)\(19\!\cdots\!75\)\( T^{25} + \)\(69\!\cdots\!51\)\( T^{26} + \)\(21\!\cdots\!23\)\( T^{27} + \)\(61\!\cdots\!39\)\( T^{28} + \)\(14\!\cdots\!24\)\( T^{29} + \)\(32\!\cdots\!45\)\( T^{30} + \)\(48\!\cdots\!31\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 37 T + 1162 T^{2} - 25639 T^{3} + 497024 T^{4} - 8108783 T^{5} + 119638831 T^{6} - 1574467807 T^{7} + 19095573312 T^{8} - 212257240486 T^{9} + 2201959688464 T^{10} - 21259177815952 T^{11} + 193234881336991 T^{12} - 1650317886320321 T^{13} + 13349672392428094 T^{14} - 102082179475824511 T^{15} + 742232061628877504 T^{16} - 5119197873181398524 T^{17} + 33636669577737855514 T^{18} - \)\(20\!\cdots\!84\)\( T^{19} + \)\(12\!\cdots\!24\)\( T^{20} - \)\(70\!\cdots\!31\)\( T^{21} + \)\(37\!\cdots\!34\)\( T^{22} - \)\(19\!\cdots\!21\)\( T^{23} + \)\(91\!\cdots\!31\)\( T^{24} - \)\(41\!\cdots\!12\)\( T^{25} + \)\(17\!\cdots\!44\)\( T^{26} - \)\(69\!\cdots\!46\)\( T^{27} + \)\(25\!\cdots\!12\)\( T^{28} - \)\(86\!\cdots\!87\)\( T^{29} + \)\(26\!\cdots\!11\)\( T^{30} - \)\(75\!\cdots\!43\)\( T^{31} + \)\(18\!\cdots\!64\)\( T^{32} - \)\(39\!\cdots\!39\)\( T^{33} + \)\(74\!\cdots\!42\)\( T^{34} - \)\(96\!\cdots\!97\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))
$43$ (\( 1 - 10 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 6 T + 125 T^{2} + 500 T^{3} + 5375 T^{4} + 11094 T^{5} + 79507 T^{6} \))(\( 1 - 13 T + 215 T^{2} - 1635 T^{3} + 14508 T^{4} - 70305 T^{5} + 397535 T^{6} - 1033591 T^{7} + 3418801 T^{8} \))(\( 1 + 18 T + 334 T^{2} + 2980 T^{3} + 29029 T^{4} + 136196 T^{5} + 1103813 T^{6} + 3815337 T^{7} + 62800951 T^{8} + 348213533 T^{9} + 3943962421 T^{10} + 14973181919 T^{11} + 116118958399 T^{12} + 303345998859 T^{13} + 3773716988213 T^{14} + 20021961902828 T^{15} + 183502847949421 T^{16} + 810019461098860 T^{17} + 3903858892718734 T^{18} + 9046667014863174 T^{19} + 21611482313284249 T^{20} \))(\( 1 - 5 T + 197 T^{2} - 481 T^{3} + 23722 T^{4} - 45125 T^{5} + 2315260 T^{6} - 3018587 T^{7} + 179938455 T^{8} - 163326585 T^{9} + 12170024380 T^{10} - 8524437672 T^{11} + 711460104377 T^{12} - 359409231654 T^{13} + 36587132230511 T^{14} - 16088275772689 T^{15} + 1675798794529458 T^{16} - 691795858225627 T^{17} + 67649607494214839 T^{18} - 28575549781114578 T^{19} + 2432340516304191977 T^{20} - 1253164309611264696 T^{21} + 76931142421161134620 T^{22} - 44395205491549379595 T^{23} + \)\(21\!\cdots\!55\)\( T^{24} - \)\(15\!\cdots\!41\)\( T^{25} + \)\(50\!\cdots\!40\)\( T^{26} - \)\(41\!\cdots\!75\)\( T^{27} + \)\(94\!\cdots\!22\)\( T^{28} - \)\(82\!\cdots\!83\)\( T^{29} + \)\(14\!\cdots\!53\)\( T^{30} - \)\(15\!\cdots\!35\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 + 15 T + 649 T^{2} + 7805 T^{3} + 190599 T^{4} + 1912780 T^{5} + 34448375 T^{6} + 295984148 T^{7} + 4370757901 T^{8} + 32739693264 T^{9} + 419942836228 T^{10} + 2782160636183 T^{11} + 32141059408382 T^{12} + 190837001390527 T^{13} + 2032298526457431 T^{14} + 10960026906223203 T^{15} + 108958866784162301 T^{16} + 540878278891610325 T^{17} + 5033504906601400058 T^{18} + 23257765992339243975 T^{19} + \)\(20\!\cdots\!49\)\( T^{20} + \)\(87\!\cdots\!21\)\( T^{21} + \)\(69\!\cdots\!31\)\( T^{22} + \)\(28\!\cdots\!61\)\( T^{23} + \)\(20\!\cdots\!18\)\( T^{24} + \)\(75\!\cdots\!81\)\( T^{25} + \)\(49\!\cdots\!28\)\( T^{26} + \)\(16\!\cdots\!52\)\( T^{27} + \)\(94\!\cdots\!49\)\( T^{28} + \)\(27\!\cdots\!36\)\( T^{29} + \)\(13\!\cdots\!75\)\( T^{30} + \)\(32\!\cdots\!40\)\( T^{31} + \)\(14\!\cdots\!51\)\( T^{32} + \)\(24\!\cdots\!35\)\( T^{33} + \)\(88\!\cdots\!49\)\( T^{34} + \)\(88\!\cdots\!45\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 2 T + 33 T^{2} - 324 T^{3} + 1551 T^{4} + 4418 T^{5} + 103823 T^{6} \))(\( 1 + 33 T + 583 T^{2} + 6699 T^{3} + 54268 T^{4} + 314853 T^{5} + 1287847 T^{6} + 3426159 T^{7} + 4879681 T^{8} \))(\( 1 + 18 T + 494 T^{2} + 6270 T^{3} + 99181 T^{4} + 979232 T^{5} + 11419378 T^{6} + 92557225 T^{7} + 873890510 T^{8} + 6001170605 T^{9} + 47926902307 T^{10} + 282055018435 T^{11} + 1930424136590 T^{12} + 9609568771175 T^{13} + 55722921858418 T^{14} + 224581969894624 T^{15} + 1069093355545549 T^{16} + 3176526965303010 T^{17} + 11762775610909934 T^{18} + 20144348515849806 T^{19} + 52599132235830049 T^{20} \))(\( 1 + 47 T + 1521 T^{2} + 35695 T^{3} + 695344 T^{4} + 11443167 T^{5} + 166215515 T^{6} + 2150890354 T^{7} + 25336473439 T^{8} + 273240068043 T^{9} + 2733546016518 T^{10} + 25462180643677 T^{11} + 222818061560779 T^{12} + 1835643848459763 T^{13} + 14325453981515050 T^{14} + 105939294919881198 T^{15} + 745268954307368866 T^{16} + 4979146861234416306 T^{17} + 31644927845166745450 T^{18} + \)\(19\!\cdots\!49\)\( T^{19} + \)\(10\!\cdots\!99\)\( T^{20} + \)\(58\!\cdots\!39\)\( T^{21} + \)\(29\!\cdots\!22\)\( T^{22} + \)\(13\!\cdots\!09\)\( T^{23} + \)\(60\!\cdots\!79\)\( T^{24} + \)\(24\!\cdots\!18\)\( T^{25} + \)\(87\!\cdots\!35\)\( T^{26} + \)\(28\!\cdots\!01\)\( T^{27} + \)\(80\!\cdots\!04\)\( T^{28} + \)\(19\!\cdots\!65\)\( T^{29} + \)\(39\!\cdots\!49\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 23 T + 769 T^{2} - 13551 T^{3} + 269949 T^{4} - 3915438 T^{5} + 59208414 T^{6} - 735695639 T^{7} + 9217604329 T^{8} - 100563658366 T^{9} + 1089466533153 T^{10} - 10604131770061 T^{11} + 101737531943472 T^{12} - 892746822726844 T^{13} + 7694162352201876 T^{14} - 61273031451667859 T^{15} + 478443586728967441 T^{16} - 3470319267236089527 T^{17} + 24664505296042832200 T^{18} - \)\(16\!\cdots\!69\)\( T^{19} + \)\(10\!\cdots\!69\)\( T^{20} - \)\(63\!\cdots\!57\)\( T^{21} + \)\(37\!\cdots\!56\)\( T^{22} - \)\(20\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!88\)\( T^{24} - \)\(53\!\cdots\!43\)\( T^{25} + \)\(25\!\cdots\!33\)\( T^{26} - \)\(11\!\cdots\!22\)\( T^{27} + \)\(48\!\cdots\!21\)\( T^{28} - \)\(18\!\cdots\!17\)\( T^{29} + \)\(68\!\cdots\!74\)\( T^{30} - \)\(21\!\cdots\!26\)\( T^{31} + \)\(69\!\cdots\!81\)\( T^{32} - \)\(16\!\cdots\!93\)\( T^{33} + \)\(43\!\cdots\!49\)\( T^{34} - \)\(61\!\cdots\!01\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \))
$53$ (\( 1 - 3 T + 53 T^{2} \))(\( 1 + 11 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + 28 T + 408 T^{2} + 3654 T^{3} + 21624 T^{4} + 78652 T^{5} + 148877 T^{6} \))(\( 1 - 6 T + 128 T^{2} - 700 T^{3} + 9881 T^{4} - 37100 T^{5} + 359552 T^{6} - 893262 T^{7} + 7890481 T^{8} \))(\( 1 + 280 T^{2} - 453 T^{3} + 40547 T^{4} - 95841 T^{5} + 4116700 T^{6} - 10306599 T^{7} + 316060756 T^{8} - 756016153 T^{9} + 18872290141 T^{10} - 40068856109 T^{11} + 887814663604 T^{12} - 1534415539323 T^{13} + 32482743132700 T^{14} - 40080274244613 T^{15} + 898698350697563 T^{16} - 532144146346161 T^{17} + 17432713315181080 T^{18} + 174887470365513049 T^{20} \))(\( 1 + 36 T + 1175 T^{2} + 26167 T^{3} + 526553 T^{4} + 8814464 T^{5} + 135423948 T^{6} + 1846719868 T^{7} + 23401307600 T^{8} + 271068847869 T^{9} + 2943032349289 T^{10} + 29666205870053 T^{11} + 281889792560510 T^{12} + 2508334485133314 T^{13} + 21112835718481295 T^{14} + 167153704971172239 T^{15} + 1253870241859978341 T^{16} + 8859146363472128667 T^{17} + 59305955533213957655 T^{18} + \)\(37\!\cdots\!78\)\( T^{19} + \)\(22\!\cdots\!10\)\( T^{20} + \)\(12\!\cdots\!29\)\( T^{21} + \)\(65\!\cdots\!81\)\( T^{22} + \)\(31\!\cdots\!53\)\( T^{23} + \)\(14\!\cdots\!00\)\( T^{24} + \)\(60\!\cdots\!44\)\( T^{25} + \)\(23\!\cdots\!52\)\( T^{26} + \)\(81\!\cdots\!08\)\( T^{27} + \)\(25\!\cdots\!73\)\( T^{28} + \)\(68\!\cdots\!91\)\( T^{29} + \)\(16\!\cdots\!75\)\( T^{30} + \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 35 T + 1083 T^{2} - 22489 T^{3} + 425053 T^{4} - 6567248 T^{5} + 94678506 T^{6} - 1195535428 T^{7} + 14340016873 T^{8} - 155936505747 T^{9} + 1632034394994 T^{10} - 15801546478878 T^{11} + 148677841270600 T^{12} - 1310514992350566 T^{13} + 11302408728124255 T^{14} - 92025015377152967 T^{15} + 736342587091688232 T^{16} - 5586084118835796339 T^{17} + 41736989702520768260 T^{18} - \)\(29\!\cdots\!67\)\( T^{19} + \)\(20\!\cdots\!88\)\( T^{20} - \)\(13\!\cdots\!59\)\( T^{21} + \)\(89\!\cdots\!55\)\( T^{22} - \)\(54\!\cdots\!38\)\( T^{23} + \)\(32\!\cdots\!00\)\( T^{24} - \)\(18\!\cdots\!86\)\( T^{25} + \)\(10\!\cdots\!34\)\( T^{26} - \)\(51\!\cdots\!51\)\( T^{27} + \)\(25\!\cdots\!77\)\( T^{28} - \)\(11\!\cdots\!16\)\( T^{29} + \)\(46\!\cdots\!46\)\( T^{30} - \)\(17\!\cdots\!04\)\( T^{31} + \)\(58\!\cdots\!57\)\( T^{32} - \)\(16\!\cdots\!73\)\( T^{33} + \)\(41\!\cdots\!43\)\( T^{34} - \)\(71\!\cdots\!55\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \))
$59$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{10} \))(\( ( 1 - T )^{16} \))(\( ( 1 + T )^{18} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + 2 T + 123 T^{2} + 20 T^{3} + 7503 T^{4} + 7442 T^{5} + 226981 T^{6} \))(\( 1 + 18 T + 307 T^{2} + 3132 T^{3} + 29688 T^{4} + 191052 T^{5} + 1142347 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 8 T + 217 T^{2} - 2336 T^{3} + 31389 T^{4} - 303921 T^{5} + 3563673 T^{6} - 28470976 T^{7} + 295582745 T^{8} - 2240831291 T^{9} + 19394855451 T^{10} - 136690708751 T^{11} + 1099863394145 T^{12} - 6462370603456 T^{13} + 49342049733993 T^{14} - 256690552396221 T^{15} + 1617173030817429 T^{16} - 7341447264945056 T^{17} + 41600486920409977 T^{18} - 93553168742673128 T^{19} + 713342911662882601 T^{20} \))(\( 1 + 22 T + 726 T^{2} + 11788 T^{3} + 229971 T^{4} + 2989207 T^{5} + 44054827 T^{6} + 482781207 T^{7} + 5931009252 T^{8} + 57058208894 T^{9} + 619281836577 T^{10} + 5407165081688 T^{11} + 53761513199754 T^{12} + 435350040127477 T^{13} + 4029639992006474 T^{14} + 30479162475907733 T^{15} + 263314323349656692 T^{16} + 1859228911030371713 T^{17} + 14994290410256089754 T^{18} + 98816187458174856937 T^{19} + \)\(74\!\cdots\!14\)\( T^{20} + \)\(45\!\cdots\!88\)\( T^{21} + \)\(31\!\cdots\!97\)\( T^{22} + \)\(17\!\cdots\!74\)\( T^{23} + \)\(11\!\cdots\!12\)\( T^{24} + \)\(56\!\cdots\!87\)\( T^{25} + \)\(31\!\cdots\!27\)\( T^{26} + \)\(13\!\cdots\!27\)\( T^{27} + \)\(61\!\cdots\!91\)\( T^{28} + \)\(19\!\cdots\!28\)\( T^{29} + \)\(71\!\cdots\!66\)\( T^{30} + \)\(13\!\cdots\!22\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 39 T + 1471 T^{2} - 35818 T^{3} + 817118 T^{4} - 14960561 T^{5} + 258108425 T^{6} - 3854122514 T^{7} + 54695318545 T^{8} - 696771547696 T^{9} + 8499221240849 T^{10} - 95019195234761 T^{11} + 1023357880808331 T^{12} - 10232780871009974 T^{13} + 99036704888231764 T^{14} - 897188364949677630 T^{15} + 7893182830860804533 T^{16} - 65299249552356104475 T^{17} + \)\(52\!\cdots\!42\)\( T^{18} - \)\(39\!\cdots\!75\)\( T^{19} + \)\(29\!\cdots\!93\)\( T^{20} - \)\(20\!\cdots\!30\)\( T^{21} + \)\(13\!\cdots\!24\)\( T^{22} - \)\(86\!\cdots\!74\)\( T^{23} + \)\(52\!\cdots\!91\)\( T^{24} - \)\(29\!\cdots\!81\)\( T^{25} + \)\(16\!\cdots\!69\)\( T^{26} - \)\(81\!\cdots\!36\)\( T^{27} + \)\(39\!\cdots\!45\)\( T^{28} - \)\(16\!\cdots\!54\)\( T^{29} + \)\(68\!\cdots\!25\)\( T^{30} - \)\(24\!\cdots\!41\)\( T^{31} + \)\(80\!\cdots\!38\)\( T^{32} - \)\(21\!\cdots\!18\)\( T^{33} + \)\(54\!\cdots\!31\)\( T^{34} - \)\(87\!\cdots\!19\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 10 T + 67 T^{2} \))(\( 1 + 4 T + 105 T^{2} + 280 T^{3} + 7035 T^{4} + 17956 T^{5} + 300763 T^{6} \))(\( 1 + 8 T + 251 T^{2} + 1596 T^{3} + 24672 T^{4} + 106932 T^{5} + 1126739 T^{6} + 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 6 T + 329 T^{2} + 2024 T^{3} + 62805 T^{4} + 348054 T^{5} + 8142587 T^{6} + 41127531 T^{7} + 790515460 T^{8} + 3576798755 T^{9} + 59745142595 T^{10} + 239645516585 T^{11} + 3548623899940 T^{12} + 12369639606153 T^{13} + 164082255890027 T^{14} + 469916443991778 T^{15} + 5681238692124045 T^{16} + 12266880289173752 T^{17} + 133596265916134889 T^{18} + 163239206377769682 T^{19} + 1822837804551761449 T^{20} \))(\( 1 - 4 T + 508 T^{2} - 486 T^{3} + 121581 T^{4} + 261148 T^{5} + 19273613 T^{6} + 99314231 T^{7} + 2421011347 T^{8} + 17808146326 T^{9} + 267088396674 T^{10} + 2154454341593 T^{11} + 26510155079768 T^{12} + 200163741901303 T^{13} + 2300638246781225 T^{14} + 15443017906683695 T^{15} + 168758840060292406 T^{16} + 1034682199747807565 T^{17} + 10327565089800919025 T^{18} + 60201847505461594189 T^{19} + \)\(53\!\cdots\!28\)\( T^{20} + \)\(29\!\cdots\!51\)\( T^{21} + \)\(24\!\cdots\!06\)\( T^{22} + \)\(10\!\cdots\!98\)\( T^{23} + \)\(98\!\cdots\!27\)\( T^{24} + \)\(27\!\cdots\!57\)\( T^{25} + \)\(35\!\cdots\!37\)\( T^{26} + \)\(31\!\cdots\!84\)\( T^{27} + \)\(99\!\cdots\!41\)\( T^{28} - \)\(26\!\cdots\!82\)\( T^{29} + \)\(18\!\cdots\!32\)\( T^{30} - \)\(98\!\cdots\!72\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 14 T + 584 T^{2} - 7402 T^{3} + 175998 T^{4} - 2019958 T^{5} + 35736509 T^{6} - 377347957 T^{7} + 5499794378 T^{8} - 54163644568 T^{9} + 685649961037 T^{10} - 6367484198644 T^{11} + 72217104791589 T^{12} - 636332394978799 T^{13} + 6595772574396989 T^{14} - 55209996682386672 T^{15} + 529915485627492618 T^{16} - 4201344704663207028 T^{17} + 37698537055203245858 T^{18} - \)\(28\!\cdots\!76\)\( T^{19} + \)\(23\!\cdots\!02\)\( T^{20} - \)\(16\!\cdots\!36\)\( T^{21} + \)\(13\!\cdots\!69\)\( T^{22} - \)\(85\!\cdots\!93\)\( T^{23} + \)\(65\!\cdots\!41\)\( T^{24} - \)\(38\!\cdots\!12\)\( T^{25} + \)\(27\!\cdots\!17\)\( T^{26} - \)\(14\!\cdots\!96\)\( T^{27} + \)\(10\!\cdots\!22\)\( T^{28} - \)\(46\!\cdots\!31\)\( T^{29} + \)\(29\!\cdots\!49\)\( T^{30} - \)\(11\!\cdots\!46\)\( T^{31} + \)\(64\!\cdots\!42\)\( T^{32} - \)\(18\!\cdots\!86\)\( T^{33} + \)\(96\!\cdots\!04\)\( T^{34} - \)\(15\!\cdots\!78\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 22 T + 349 T^{2} + 3348 T^{3} + 24779 T^{4} + 110902 T^{5} + 357911 T^{6} \))(\( 1 + 5 T + 239 T^{2} + 765 T^{3} + 23456 T^{4} + 54315 T^{5} + 1204799 T^{6} + 1789555 T^{7} + 25411681 T^{8} \))(\( 1 - 8 T + 422 T^{2} - 1784 T^{3} + 78397 T^{4} - 137528 T^{5} + 9660861 T^{6} - 1206535 T^{7} + 911777254 T^{8} + 707944132 T^{9} + 70230028469 T^{10} + 50264033372 T^{11} + 4596269137414 T^{12} - 431832148385 T^{13} + 245498717917341 T^{14} - 248132054184328 T^{15} + 10042677958554637 T^{16} - 16225694362569544 T^{17} + 272507990185711142 T^{18} - 366788005747592248 T^{19} + 3255243551009881201 T^{20} \))(\( 1 + 5 T + 707 T^{2} + 3199 T^{3} + 248382 T^{4} + 1037845 T^{5} + 57673238 T^{6} + 225467389 T^{7} + 9923775540 T^{8} + 36556228657 T^{9} + 1343843822580 T^{10} + 4670089592113 T^{11} + 148323449480723 T^{12} + 484195559141675 T^{13} + 13622265401829407 T^{14} + 41396926211404761 T^{15} + 1052601653972946460 T^{16} + 2939181761009738031 T^{17} + 68669839890622040687 T^{18} + \)\(17\!\cdots\!25\)\( T^{19} + \)\(37\!\cdots\!63\)\( T^{20} + \)\(84\!\cdots\!63\)\( T^{21} + \)\(17\!\cdots\!80\)\( T^{22} + \)\(33\!\cdots\!87\)\( T^{23} + \)\(64\!\cdots\!40\)\( T^{24} + \)\(10\!\cdots\!59\)\( T^{25} + \)\(18\!\cdots\!38\)\( T^{26} + \)\(23\!\cdots\!95\)\( T^{27} + \)\(40\!\cdots\!62\)\( T^{28} + \)\(37\!\cdots\!89\)\( T^{29} + \)\(58\!\cdots\!67\)\( T^{30} + \)\(29\!\cdots\!55\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 3 T + 537 T^{2} - 2435 T^{3} + 153205 T^{4} - 910500 T^{5} + 30959641 T^{6} - 215783078 T^{7} + 4973264314 T^{8} - 37289791104 T^{9} + 668654362394 T^{10} - 5085523989212 T^{11} + 76904950958463 T^{12} - 573811536150510 T^{13} + 7644694616954221 T^{14} - 54976086436920544 T^{15} + 661642485133273521 T^{16} - 4525499790997505672 T^{17} + 50144246014068624782 T^{18} - \)\(32\!\cdots\!12\)\( T^{19} + \)\(33\!\cdots\!61\)\( T^{20} - \)\(19\!\cdots\!84\)\( T^{21} + \)\(19\!\cdots\!01\)\( T^{22} - \)\(10\!\cdots\!10\)\( T^{23} + \)\(98\!\cdots\!23\)\( T^{24} - \)\(46\!\cdots\!92\)\( T^{25} + \)\(43\!\cdots\!34\)\( T^{26} - \)\(17\!\cdots\!24\)\( T^{27} + \)\(16\!\cdots\!14\)\( T^{28} - \)\(49\!\cdots\!38\)\( T^{29} + \)\(50\!\cdots\!81\)\( T^{30} - \)\(10\!\cdots\!00\)\( T^{31} + \)\(12\!\cdots\!05\)\( T^{32} - \)\(14\!\cdots\!85\)\( T^{33} + \)\(22\!\cdots\!77\)\( T^{34} - \)\(88\!\cdots\!73\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 13 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 15 T + 73 T^{2} \))(\( 1 + 4 T + 199 T^{2} + 568 T^{3} + 14527 T^{4} + 21316 T^{5} + 389017 T^{6} \))(\( 1 - 18 T + 345 T^{2} - 3510 T^{3} + 38108 T^{4} - 256230 T^{5} + 1838505 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))(\( 1 + 41 T + 1176 T^{2} + 25011 T^{3} + 446720 T^{4} + 6818286 T^{5} + 92020459 T^{6} + 1105799804 T^{7} + 12010240509 T^{8} + 118061925228 T^{9} + 1058028065433 T^{10} + 8618520541644 T^{11} + 64002571672461 T^{12} + 430174922352668 T^{13} + 2613219171612619 T^{14} + 14134795019549598 T^{15} + 67604025567822080 T^{16} + 276306484361135067 T^{17} + 948397068067439256 T^{18} + 2413735055038984433 T^{19} + 4297625829703557649 T^{20} \))(\( 1 + 35 T + 1035 T^{2} + 20128 T^{3} + 344201 T^{4} + 4528714 T^{5} + 52898365 T^{6} + 472574263 T^{7} + 3575301535 T^{8} + 14472242265 T^{9} - 10259759258 T^{10} - 1413670638428 T^{11} - 13344230940186 T^{12} - 98999228688970 T^{13} + 14056056520030 T^{14} + 4087388912498153 T^{15} + 69628091750516746 T^{16} + 298379390612365169 T^{17} + 74904725195239870 T^{18} - 38512382946897042490 T^{19} - \)\(37\!\cdots\!26\)\( T^{20} - \)\(29\!\cdots\!04\)\( T^{21} - \)\(15\!\cdots\!62\)\( T^{22} + \)\(15\!\cdots\!05\)\( T^{23} + \)\(28\!\cdots\!35\)\( T^{24} + \)\(27\!\cdots\!19\)\( T^{25} + \)\(22\!\cdots\!85\)\( T^{26} + \)\(14\!\cdots\!78\)\( T^{27} + \)\(78\!\cdots\!21\)\( T^{28} + \)\(33\!\cdots\!24\)\( T^{29} + \)\(12\!\cdots\!15\)\( T^{30} + \)\(31\!\cdots\!95\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 56 T + 2109 T^{2} - 58148 T^{3} + 1323233 T^{4} - 25545917 T^{5} + 434491514 T^{6} - 6603368704 T^{7} + 91350560633 T^{8} - 1160627321124 T^{9} + 13701049378596 T^{10} - 151273983137480 T^{11} + 1576247493655595 T^{12} - 15595164176234720 T^{13} + 147797237362272105 T^{14} - 1351002409324893588 T^{15} + 12022419854601633312 T^{16} - \)\(10\!\cdots\!47\)\( T^{17} + \)\(90\!\cdots\!48\)\( T^{18} - \)\(76\!\cdots\!31\)\( T^{19} + \)\(64\!\cdots\!48\)\( T^{20} - \)\(52\!\cdots\!96\)\( T^{21} + \)\(41\!\cdots\!05\)\( T^{22} - \)\(32\!\cdots\!60\)\( T^{23} + \)\(23\!\cdots\!55\)\( T^{24} - \)\(16\!\cdots\!60\)\( T^{25} + \)\(11\!\cdots\!76\)\( T^{26} - \)\(68\!\cdots\!12\)\( T^{27} + \)\(39\!\cdots\!17\)\( T^{28} - \)\(20\!\cdots\!08\)\( T^{29} + \)\(99\!\cdots\!94\)\( T^{30} - \)\(42\!\cdots\!61\)\( T^{31} + \)\(16\!\cdots\!97\)\( T^{32} - \)\(51\!\cdots\!36\)\( T^{33} + \)\(13\!\cdots\!49\)\( T^{34} - \)\(26\!\cdots\!68\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \))
$79$ (\( 1 + 9 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 - 6 T + 56 T^{2} + 434 T^{3} + 4424 T^{4} - 37446 T^{5} + 493039 T^{6} \))(\( 1 - 27 T + 324 T^{2} - 1956 T^{3} + 10761 T^{4} - 154524 T^{5} + 2022084 T^{6} - 13312053 T^{7} + 38950081 T^{8} \))(\( 1 - 3 T + 518 T^{2} - 1587 T^{3} + 128745 T^{4} - 390976 T^{5} + 20517279 T^{6} - 60195058 T^{7} + 2367861681 T^{8} - 6490084302 T^{9} + 210779814221 T^{10} - 512716659858 T^{11} + 14777824751121 T^{12} - 29678511201262 T^{13} + 799149678949599 T^{14} - 1203055202655424 T^{15} + 31296294461051145 T^{16} - 30476603561034333 T^{17} + 785862363531598598 T^{18} - 359554787947854957 T^{19} + 9468276082626847201 T^{20} \))(\( 1 + 48 T + 1716 T^{2} + 43733 T^{3} + 949683 T^{4} + 17347139 T^{5} + 284281513 T^{6} + 4151599168 T^{7} + 55961588988 T^{8} + 692772349249 T^{9} + 8070707342734 T^{10} + 88091360119644 T^{11} + 919032606186396 T^{12} + 9123573520975185 T^{13} + 87713721926917761 T^{14} + 811771645641552513 T^{15} + 7336995592941878765 T^{16} + 64129960005682648527 T^{17} + \)\(54\!\cdots\!01\)\( T^{18} + \)\(44\!\cdots\!15\)\( T^{19} + \)\(35\!\cdots\!76\)\( T^{20} + \)\(27\!\cdots\!56\)\( T^{21} + \)\(19\!\cdots\!14\)\( T^{22} + \)\(13\!\cdots\!91\)\( T^{23} + \)\(84\!\cdots\!68\)\( T^{24} + \)\(49\!\cdots\!92\)\( T^{25} + \)\(26\!\cdots\!13\)\( T^{26} + \)\(12\!\cdots\!81\)\( T^{27} + \)\(56\!\cdots\!03\)\( T^{28} + \)\(20\!\cdots\!87\)\( T^{29} + \)\(63\!\cdots\!96\)\( T^{30} + \)\(13\!\cdots\!52\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 + 22 T + 851 T^{2} + 14013 T^{3} + 321543 T^{4} + 4352824 T^{5} + 76480555 T^{6} + 899721359 T^{7} + 13400472097 T^{8} + 142374191076 T^{9} + 1896300817314 T^{10} + 18668917836386 T^{11} + 228746023792580 T^{12} + 2115817543963722 T^{13} + 24174321193637272 T^{14} + 211220187895134527 T^{15} + 2265598352861749407 T^{16} + 18723917741583607029 T^{17} + \)\(18\!\cdots\!20\)\( T^{18} + \)\(14\!\cdots\!91\)\( T^{19} + \)\(14\!\cdots\!87\)\( T^{20} + \)\(10\!\cdots\!53\)\( T^{21} + \)\(94\!\cdots\!32\)\( T^{22} + \)\(65\!\cdots\!78\)\( T^{23} + \)\(55\!\cdots\!80\)\( T^{24} + \)\(35\!\cdots\!74\)\( T^{25} + \)\(28\!\cdots\!54\)\( T^{26} + \)\(17\!\cdots\!44\)\( T^{27} + \)\(12\!\cdots\!97\)\( T^{28} + \)\(67\!\cdots\!61\)\( T^{29} + \)\(45\!\cdots\!55\)\( T^{30} + \)\(20\!\cdots\!36\)\( T^{31} + \)\(11\!\cdots\!83\)\( T^{32} + \)\(40\!\cdots\!87\)\( T^{33} + \)\(19\!\cdots\!71\)\( T^{34} + \)\(40\!\cdots\!98\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \))
$83$ (\( 1 + 16 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 185 T^{2} + 64 T^{3} + 15355 T^{4} + 571787 T^{6} \))(\( 1 + 22 T + 421 T^{2} + 5126 T^{3} + 54364 T^{4} + 425458 T^{5} + 2900269 T^{6} + 12579314 T^{7} + 47458321 T^{8} \))(\( 1 + 369 T^{2} + 611 T^{3} + 68049 T^{4} + 305165 T^{5} + 8173289 T^{6} + 66211883 T^{7} + 744395127 T^{8} + 8505557036 T^{9} + 61617763203 T^{10} + 705961233988 T^{11} + 5128138029903 T^{12} + 37859093944921 T^{13} + 387890572987769 T^{14} + 1202057337821095 T^{15} + 22247965467387081 T^{16} + 16580127154662097 T^{17} + 831095833659306129 T^{18} + 15516041187205853449 T^{20} \))(\( 1 + 42 T + 1656 T^{2} + 45425 T^{3} + 1121429 T^{4} + 23462170 T^{5} + 447315865 T^{6} + 7661927885 T^{7} + 121285518000 T^{8} + 1767394289294 T^{9} + 24044579507563 T^{10} + 304858972200110 T^{11} + 3632307914546706 T^{12} + 40607544102297339 T^{13} + 428241004835045976 T^{14} + 4252453091155187749 T^{15} + 39902891433324960408 T^{16} + \)\(35\!\cdots\!67\)\( T^{17} + \)\(29\!\cdots\!64\)\( T^{18} + \)\(23\!\cdots\!93\)\( T^{19} + \)\(17\!\cdots\!26\)\( T^{20} + \)\(12\!\cdots\!30\)\( T^{21} + \)\(78\!\cdots\!47\)\( T^{22} + \)\(47\!\cdots\!38\)\( T^{23} + \)\(27\!\cdots\!00\)\( T^{24} + \)\(14\!\cdots\!55\)\( T^{25} + \)\(69\!\cdots\!85\)\( T^{26} + \)\(30\!\cdots\!90\)\( T^{27} + \)\(11\!\cdots\!69\)\( T^{28} + \)\(40\!\cdots\!75\)\( T^{29} + \)\(12\!\cdots\!24\)\( T^{30} + \)\(25\!\cdots\!94\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 12 T + 1042 T^{2} - 10723 T^{3} + 506124 T^{4} - 4414628 T^{5} + 152289937 T^{6} - 1096576246 T^{7} + 31815480079 T^{8} - 179334069678 T^{9} + 4910741731628 T^{10} - 19368015579723 T^{11} + 585423077938040 T^{12} - 1209725488576827 T^{13} + 56585014808837265 T^{14} - 6058433211885189 T^{15} + 4782661356230931324 T^{16} + 6581721606983553308 T^{17} + \)\(39\!\cdots\!52\)\( T^{18} + \)\(54\!\cdots\!64\)\( T^{19} + \)\(32\!\cdots\!36\)\( T^{20} - \)\(34\!\cdots\!43\)\( T^{21} + \)\(26\!\cdots\!65\)\( T^{22} - \)\(47\!\cdots\!61\)\( T^{23} + \)\(19\!\cdots\!60\)\( T^{24} - \)\(52\!\cdots\!21\)\( T^{25} + \)\(11\!\cdots\!48\)\( T^{26} - \)\(33\!\cdots\!34\)\( T^{27} + \)\(49\!\cdots\!71\)\( T^{28} - \)\(14\!\cdots\!82\)\( T^{29} + \)\(16\!\cdots\!57\)\( T^{30} - \)\(39\!\cdots\!64\)\( T^{31} + \)\(37\!\cdots\!96\)\( T^{32} - \)\(65\!\cdots\!61\)\( T^{33} + \)\(52\!\cdots\!02\)\( T^{34} - \)\(50\!\cdots\!76\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 20 T + 379 T^{2} + 3744 T^{3} + 33731 T^{4} + 158420 T^{5} + 704969 T^{6} \))(\( 1 + 9 T + 243 T^{2} + 1849 T^{3} + 31448 T^{4} + 164561 T^{5} + 1924803 T^{6} + 6344721 T^{7} + 62742241 T^{8} \))(\( 1 + 45 T + 1573 T^{2} + 37721 T^{3} + 776454 T^{4} + 13065786 T^{5} + 196990215 T^{6} + 2578456481 T^{7} + 30964392375 T^{8} + 331742873437 T^{9} + 3299461837305 T^{10} + 29525115735893 T^{11} + 245268952002375 T^{12} + 1817731886954089 T^{13} + 12359607544171815 T^{14} + 72960125771911914 T^{15} + 385883111291832294 T^{16} + 1668450183594249409 T^{17} + 6192254191369373413 T^{18} + 15766038166836834405 T^{19} + 31181719929966183601 T^{20} \))(\( 1 - 20 T + 597 T^{2} - 8420 T^{3} + 158284 T^{4} - 1939051 T^{5} + 29478345 T^{6} - 339122373 T^{7} + 4443220742 T^{8} - 48716866782 T^{9} + 567385758156 T^{10} - 5969911399957 T^{11} + 63612180884481 T^{12} - 645883424308806 T^{13} + 6449935124831842 T^{14} - 63212766860278727 T^{15} + 599655028685899152 T^{16} - 5625936250564806703 T^{17} + 51089936123793020482 T^{18} - \)\(45\!\cdots\!14\)\( T^{19} + \)\(39\!\cdots\!21\)\( T^{20} - \)\(33\!\cdots\!93\)\( T^{21} + \)\(28\!\cdots\!16\)\( T^{22} - \)\(21\!\cdots\!78\)\( T^{23} + \)\(17\!\cdots\!02\)\( T^{24} - \)\(11\!\cdots\!57\)\( T^{25} + \)\(91\!\cdots\!45\)\( T^{26} - \)\(53\!\cdots\!39\)\( T^{27} + \)\(39\!\cdots\!64\)\( T^{28} - \)\(18\!\cdots\!80\)\( T^{29} + \)\(11\!\cdots\!77\)\( T^{30} - \)\(34\!\cdots\!80\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 34 T + 1437 T^{2} - 34838 T^{3} + 879809 T^{4} - 16884543 T^{5} + 322190914 T^{6} - 5167587423 T^{7} + 81202210588 T^{8} - 1125560659095 T^{9} + 15233128252595 T^{10} - 186980501670944 T^{11} + 2245952507904067 T^{12} - 24938842702007338 T^{13} + 272651064040594252 T^{14} - 2797664833595998600 T^{15} + 28496596163061382239 T^{16} - \)\(27\!\cdots\!65\)\( T^{17} + \)\(26\!\cdots\!56\)\( T^{18} - \)\(24\!\cdots\!85\)\( T^{19} + \)\(22\!\cdots\!19\)\( T^{20} - \)\(19\!\cdots\!00\)\( T^{21} + \)\(17\!\cdots\!32\)\( T^{22} - \)\(13\!\cdots\!62\)\( T^{23} + \)\(11\!\cdots\!87\)\( T^{24} - \)\(82\!\cdots\!76\)\( T^{25} + \)\(59\!\cdots\!95\)\( T^{26} - \)\(39\!\cdots\!55\)\( T^{27} + \)\(25\!\cdots\!88\)\( T^{28} - \)\(14\!\cdots\!47\)\( T^{29} + \)\(79\!\cdots\!94\)\( T^{30} - \)\(37\!\cdots\!67\)\( T^{31} + \)\(17\!\cdots\!69\)\( T^{32} - \)\(60\!\cdots\!62\)\( T^{33} + \)\(22\!\cdots\!57\)\( T^{34} - \)\(46\!\cdots\!86\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 3 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 - 22 T + 375 T^{2} - 4300 T^{3} + 36375 T^{4} - 206998 T^{5} + 912673 T^{6} \))(\( 1 + 31 T + 679 T^{2} + 9567 T^{3} + 110264 T^{4} + 927999 T^{5} + 6388711 T^{6} + 28292863 T^{7} + 88529281 T^{8} \))(\( 1 + 5 T + 559 T^{2} + 3358 T^{3} + 163611 T^{4} + 1065617 T^{5} + 31994112 T^{6} + 210736090 T^{7} + 4604742021 T^{8} + 28659825956 T^{9} + 507252820407 T^{10} + 2780003117732 T^{11} + 43326017675589 T^{12} + 192333139468570 T^{13} + 2832415731593472 T^{14} + 9150815762643569 T^{15} + 136283382698438619 T^{16} + 271320639277503454 T^{17} + 4381125379256721199 T^{18} + 3801155293272826085 T^{19} + 73742412689492826049 T^{20} \))(\( 1 + 2 T + 621 T^{2} - 213 T^{3} + 205664 T^{4} - 437598 T^{5} + 49093119 T^{6} - 163737942 T^{7} + 9352075548 T^{8} - 38413772588 T^{9} + 1489316155695 T^{10} - 6708588826879 T^{11} + 203631678495233 T^{12} - 934785974558696 T^{13} + 24183348030921161 T^{14} - 107966335249417574 T^{15} + 2507492762870435228 T^{16} - 10472734519193504678 T^{17} + \)\(22\!\cdots\!49\)\( T^{18} - \)\(85\!\cdots\!08\)\( T^{19} + \)\(18\!\cdots\!73\)\( T^{20} - \)\(57\!\cdots\!03\)\( T^{21} + \)\(12\!\cdots\!55\)\( T^{22} - \)\(31\!\cdots\!44\)\( T^{23} + \)\(73\!\cdots\!28\)\( T^{24} - \)\(12\!\cdots\!14\)\( T^{25} + \)\(36\!\cdots\!31\)\( T^{26} - \)\(31\!\cdots\!94\)\( T^{27} + \)\(14\!\cdots\!24\)\( T^{28} - \)\(14\!\cdots\!01\)\( T^{29} + \)\(40\!\cdots\!49\)\( T^{30} + \)\(12\!\cdots\!86\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 31 T + 1150 T^{2} - 26640 T^{3} + 621033 T^{4} - 11795524 T^{5} + 217725486 T^{6} - 3569006410 T^{7} + 56489564240 T^{8} - 823771297356 T^{9} + 11600188070637 T^{10} - 153302569826193 T^{11} + 1959177410228710 T^{12} - 23739268881468945 T^{13} + 278562721657343088 T^{14} - 3117371594141937398 T^{15} + 33818200304155728284 T^{16} - \)\(35\!\cdots\!87\)\( T^{17} + \)\(35\!\cdots\!66\)\( T^{18} - \)\(34\!\cdots\!39\)\( T^{19} + \)\(31\!\cdots\!56\)\( T^{20} - \)\(28\!\cdots\!54\)\( T^{21} + \)\(24\!\cdots\!28\)\( T^{22} - \)\(20\!\cdots\!65\)\( T^{23} + \)\(16\!\cdots\!90\)\( T^{24} - \)\(12\!\cdots\!09\)\( T^{25} + \)\(90\!\cdots\!57\)\( T^{26} - \)\(62\!\cdots\!52\)\( T^{27} + \)\(41\!\cdots\!60\)\( T^{28} - \)\(25\!\cdots\!30\)\( T^{29} + \)\(15\!\cdots\!26\)\( T^{30} - \)\(79\!\cdots\!48\)\( T^{31} + \)\(40\!\cdots\!77\)\( T^{32} - \)\(16\!\cdots\!20\)\( T^{33} + \)\(70\!\cdots\!50\)\( T^{34} - \)\(18\!\cdots\!47\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \))
show more
show less