Properties

Label 98.2
Level 98
Weight 2
Dimension 95
Nonzero newspaces 4
Newform subspaces 9
Sturm bound 1176
Trace bound 1

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Defining parameters

Level: \( N \) = \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 9 \)
Sturm bound: \(1176\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(98))\).

Total New Old
Modular forms 354 95 259
Cusp forms 235 95 140
Eisenstein series 119 0 119

Trace form

\( 95q + q^{2} - 3q^{4} - 6q^{5} - 8q^{6} - 8q^{7} + q^{8} - 15q^{9} + O(q^{10}) \) \( 95q + q^{2} - 3q^{4} - 6q^{5} - 8q^{6} - 8q^{7} + q^{8} - 15q^{9} - 6q^{10} - 12q^{11} - 6q^{13} - 6q^{14} - 24q^{15} - 3q^{16} - 30q^{17} - 11q^{18} - 24q^{19} - 6q^{20} - 26q^{21} - 12q^{22} - 24q^{23} - 8q^{24} - 21q^{25} - 22q^{26} - 48q^{27} - 8q^{28} - 18q^{29} - 24q^{30} - 24q^{31} + q^{32} - 48q^{33} - 6q^{34} - 42q^{35} - 15q^{36} + 14q^{37} + 26q^{38} + 26q^{39} + 36q^{40} + 30q^{41} + 54q^{42} + 24q^{43} + 30q^{44} + 132q^{45} + 102q^{46} + 60q^{47} + 14q^{48} + 118q^{49} + 43q^{50} + 120q^{51} + 8q^{52} + 18q^{53} + 94q^{54} + 138q^{55} + 36q^{56} + 12q^{57} + 42q^{58} + 36q^{59} + 18q^{60} + 20q^{61} + 2q^{62} - 36q^{63} - 3q^{64} - 84q^{65} - 48q^{66} - 60q^{67} - 30q^{68} - 96q^{69} - 42q^{70} - 72q^{71} - 11q^{72} - 78q^{73} - 34q^{74} - 144q^{75} - 24q^{76} - 84q^{77} - 40q^{78} - 96q^{79} - 6q^{80} - 15q^{81} - 30q^{82} + 12q^{83} - 26q^{84} - 24q^{85} - 28q^{86} + 24q^{87} - 12q^{88} + 6q^{89} - 78q^{90} - 10q^{91} - 24q^{92} + 108q^{93} - 72q^{94} + 48q^{95} - 8q^{96} + 6q^{97} - 48q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(98))\)

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
98.2.a \(\chi_{98}(1, \cdot)\) 98.2.a.a 1 1
98.2.a.b 2
98.2.c \(\chi_{98}(67, \cdot)\) 98.2.c.a 2 2
98.2.c.b 2
98.2.c.c 4
98.2.e \(\chi_{98}(15, \cdot)\) 98.2.e.a 18 6
98.2.e.b 18
98.2.g \(\chi_{98}(9, \cdot)\) 98.2.g.a 24 12
98.2.g.b 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(98))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(98)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{3} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{3} \))
$3$ (\( 1 - 2 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} \))(\( 1 - 3 T - 5 T^{2} + 24 T^{3} + 6 T^{4} - 125 T^{5} + 64 T^{6} + 557 T^{7} - 523 T^{8} - 2172 T^{9} + 2989 T^{10} + 6859 T^{11} - 14717 T^{12} - 17891 T^{13} + 60020 T^{14} + 39396 T^{15} - 208781 T^{16} - 43613 T^{17} + 644245 T^{18} - 130839 T^{19} - 1879029 T^{20} + 1063692 T^{21} + 4861620 T^{22} - 4347513 T^{23} - 10728693 T^{24} + 15000633 T^{25} + 19610829 T^{26} - 42751476 T^{27} - 30882627 T^{28} + 98670879 T^{29} + 34012224 T^{30} - 199290375 T^{31} + 28697814 T^{32} + 344373768 T^{33} - 215233605 T^{34} - 387420489 T^{35} + 387420489 T^{36} \))(\( 1 + 5 T + 13 T^{2} + 22 T^{3} + 24 T^{4} + 19 T^{5} + 58 T^{6} + 165 T^{7} + 255 T^{8} + 216 T^{9} - 153 T^{10} - 639 T^{11} - 399 T^{12} - 897 T^{13} - 4872 T^{14} - 13942 T^{15} - 39597 T^{16} - 81705 T^{17} - 139571 T^{18} - 245115 T^{19} - 356373 T^{20} - 376434 T^{21} - 394632 T^{22} - 217971 T^{23} - 290871 T^{24} - 1397493 T^{25} - 1003833 T^{26} + 4251528 T^{27} + 15057495 T^{28} + 29229255 T^{29} + 30823578 T^{30} + 30292137 T^{31} + 114791256 T^{32} + 315675954 T^{33} + 559607373 T^{34} + 645700815 T^{35} + 387420489 T^{36} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))(\( 1 + 6 T + 22 T^{2} + 81 T^{3} + 250 T^{4} + 598 T^{5} + 1128 T^{6} + 1616 T^{7} + 1198 T^{8} - 3883 T^{9} - 19180 T^{10} - 40456 T^{11} - 40365 T^{12} + 30812 T^{13} + 284754 T^{14} + 1259888 T^{15} + 3746500 T^{16} + 8025442 T^{17} + 16755249 T^{18} + 40127210 T^{19} + 93662500 T^{20} + 157486000 T^{21} + 177971250 T^{22} + 96287500 T^{23} - 630703125 T^{24} - 3160625000 T^{25} - 7492187500 T^{26} - 7583984375 T^{27} + 11699218750 T^{28} + 78906250000 T^{29} + 275390625000 T^{30} + 729980468750 T^{31} + 1525878906250 T^{32} + 2471923828125 T^{33} + 3356933593750 T^{34} + 4577636718750 T^{35} + 3814697265625 T^{36} \))(\( 1 - 8 T^{2} + 7 T^{3} + 52 T^{4} - 28 T^{5} - 340 T^{6} - 28 T^{7} + 1486 T^{8} - 203 T^{9} - 7364 T^{10} - 1386 T^{11} + 28399 T^{12} + 2128 T^{13} - 66336 T^{14} + 17486 T^{15} + 213602 T^{16} + 122752 T^{17} - 875271 T^{18} + 613760 T^{19} + 5340050 T^{20} + 2185750 T^{21} - 41460000 T^{22} + 6650000 T^{23} + 443734375 T^{24} - 108281250 T^{25} - 2876562500 T^{26} - 396484375 T^{27} + 14511718750 T^{28} - 1367187500 T^{29} - 83007812500 T^{30} - 34179687500 T^{31} + 317382812500 T^{32} + 213623046875 T^{33} - 1220703125000 T^{34} + 3814697265625 T^{36} \))
$7$ (\( 1 + 7 T - 154 T^{3} - 490 T^{4} + 630 T^{5} + 7259 T^{6} + 13034 T^{7} - 29008 T^{8} - 172186 T^{9} - 203056 T^{10} + 638666 T^{11} + 2489837 T^{12} + 1512630 T^{13} - 8235430 T^{14} - 18117946 T^{15} + 40353607 T^{17} + 40353607 T^{18} \))(\( 1 + T + 22 T^{2} + 8 T^{3} + 274 T^{4} + 36 T^{5} + 2633 T^{6} + 21728 T^{8} + 294 T^{9} + 152096 T^{10} + 903119 T^{12} + 86436 T^{13} + 4605118 T^{14} + 941192 T^{15} + 18117946 T^{16} + 5764801 T^{17} + 40353607 T^{18} \))
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( 1 + T + 5 T^{2} + 79 T^{3} - 95 T^{4} - 76 T^{5} + 1326 T^{6} - 16865 T^{7} - 30267 T^{8} + 2498 T^{9} - 691459 T^{10} - 433967 T^{11} + 4314741 T^{12} - 10810333 T^{13} + 25601767 T^{14} + 238148631 T^{15} - 110120853 T^{16} + 326664502 T^{17} + 7013995437 T^{18} + 3593309522 T^{19} - 13324623213 T^{20} + 316975827861 T^{21} + 374835470647 T^{22} - 1741014939983 T^{23} + 7643826880701 T^{24} - 8456789137357 T^{25} - 148220377497379 T^{26} + 5890153332118 T^{27} - 785048030398467 T^{28} - 4811781324854515 T^{29} + 4161556027532046 T^{30} - 2623726122938756 T^{31} - 36076234190407895 T^{32} + 330002605383836429 T^{33} + 229748649317860805 T^{34} + 505447028499293771 T^{35} + 5559917313492231481 T^{36} \))(\( 1 - 7 T - 5 T^{2} + 147 T^{3} - 345 T^{4} - 126 T^{5} + 4274 T^{6} - 23611 T^{7} + 3343 T^{8} + 334222 T^{9} - 618065 T^{10} - 1389661 T^{11} + 9199197 T^{12} - 32851721 T^{13} - 24177817 T^{14} + 557023425 T^{15} - 789016793 T^{16} - 2614468038 T^{17} + 12008470549 T^{18} - 28759148418 T^{19} - 95471031953 T^{20} + 741398178675 T^{21} - 353987418697 T^{22} - 5290802518771 T^{23} + 16296938636517 T^{24} - 27080561539031 T^{25} - 132487721785265 T^{26} + 788077993181402 T^{27} + 86708810441143 T^{28} - 6736493854796321 T^{29} + 13413642882105554 T^{30} - 4349861730135306 T^{31} - 131013692586218145 T^{32} + 614055480904100697 T^{33} - 229748649317860805 T^{34} - 3538129199495056397 T^{35} + 5559917313492231481 T^{36} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( 1 - 25 T^{2} + 6 T^{4} + 315 T^{5} + 6611 T^{6} - 4879 T^{7} - 86232 T^{8} - 133112 T^{9} - 58583 T^{10} + 3498768 T^{11} + 9691052 T^{12} + 4709460 T^{13} - 66963514 T^{14} - 992799661 T^{15} + 112996493 T^{16} + 8511389887 T^{17} - 292743113 T^{18} + 110648068531 T^{19} + 19096407317 T^{20} - 2181180855217 T^{21} - 1912544923354 T^{22} + 1748589531780 T^{23} + 46776857013068 T^{24} + 219542503327056 T^{25} - 47787952828343 T^{26} - 1411586120538776 T^{27} - 11887813469122968 T^{28} - 8743950562506523 T^{29} + 154023640744721891 T^{30} + 95405658576559695 T^{31} + 23624258314195734 T^{32} - 16635415229579496025 T^{34} + \)\(11\!\cdots\!29\)\( T^{36} \))(\( 1 + 10 T + 21 T^{2} - 144 T^{3} - 718 T^{4} - 863 T^{5} - 3629 T^{6} - 21483 T^{7} + 27436 T^{8} + 348240 T^{9} + 1418295 T^{10} + 7817900 T^{11} + 14373674 T^{12} - 64688436 T^{13} - 129713172 T^{14} + 176127391 T^{15} - 2181811557 T^{16} - 6112204835 T^{17} + 14386902143 T^{18} - 79458662855 T^{19} - 368726153133 T^{20} + 386951878027 T^{21} - 3704737905492 T^{22} - 24018363467748 T^{23} + 69378979026266 T^{24} + 490561631054300 T^{25} + 1156946802940695 T^{26} + 3692910861653520 T^{27} + 3782285582369164 T^{28} - 38500981745096871 T^{29} - 84548750909483549 T^{30} - 261381216989114339 T^{31} - 2827036244932089502 T^{32} - 7370768594029069008 T^{33} + 13973748792846776661 T^{34} + 86504159193813379330 T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( 1 + 11 T + 2 T^{2} - 487 T^{3} - 2500 T^{4} + 722 T^{5} + 58773 T^{6} + 207960 T^{7} - 361793 T^{8} - 2998886 T^{9} + 8869448 T^{10} + 84079050 T^{11} - 81902649 T^{12} - 2573938814 T^{13} - 8082484317 T^{14} + 17290780045 T^{15} + 157286643438 T^{16} + 97701633159 T^{17} - 1376017537517 T^{18} + 1660927763703 T^{19} + 45455839953582 T^{20} + 84949602361085 T^{21} - 675057172640157 T^{22} - 3654625042629598 T^{23} - 1976930841520281 T^{24} + 34500885804100650 T^{25} + 61871117883562568 T^{26} - 355631522596582342 T^{27} - 729372481225145057 T^{28} + 7127183556135358680 T^{29} + 34242456748704743253 T^{30} + 7151105339758086514 T^{31} - \)\(42\!\cdots\!00\)\( T^{32} - \)\(13\!\cdots\!91\)\( T^{33} + 97322383751333736962 T^{34} + \)\(90\!\cdots\!47\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \))(\( 1 - T - 42 T^{2} + 73 T^{3} + 1084 T^{4} - 2150 T^{5} - 24363 T^{6} + 65268 T^{7} + 544031 T^{8} - 1310830 T^{9} - 10606560 T^{10} + 18351910 T^{11} + 199091291 T^{12} - 197323282 T^{13} - 3719811841 T^{14} + 2892650137 T^{15} + 71883549566 T^{16} - 22562148225 T^{17} - 1265451733317 T^{18} - 383556519825 T^{19} + 20774345824574 T^{20} + 14211590123081 T^{21} - 310682404772161 T^{22} - 280170843210674 T^{23} + 4805579773811579 T^{24} + 7530498396415430 T^{25} - 73988789843412960 T^{26} - 155448546148562510 T^{27} + 1096763177655169919 T^{28} + 2236858128206590644 T^{29} - 14194425565628667243 T^{30} - 21294842770747764550 T^{31} + \)\(18\!\cdots\!36\)\( T^{32} + \)\(20\!\cdots\!89\)\( T^{33} - \)\(20\!\cdots\!02\)\( T^{34} - \)\(82\!\cdots\!77\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 - 12 T^{2} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 12 T^{2} - 217 T^{4} + 4332 T^{6} + 130321 T^{8} \))(\( ( 1 - 18 T + 220 T^{2} - 1974 T^{3} + 15076 T^{4} - 99865 T^{5} + 596888 T^{6} - 3212338 T^{7} + 15876577 T^{8} - 71853703 T^{9} + 301654963 T^{10} - 1159654018 T^{11} + 4094054792 T^{12} - 13014506665 T^{13} + 37329668524 T^{14} - 92868569094 T^{15} + 196651782580 T^{16} - 305704134738 T^{17} + 322687697779 T^{18} )^{2} \))(\( ( 1 + 22 T + 360 T^{2} + 4122 T^{3} + 39514 T^{4} + 310897 T^{5} + 2136276 T^{6} + 12646550 T^{7} + 66557763 T^{8} + 306914335 T^{9} + 1264597497 T^{10} + 4565404550 T^{11} + 14652717084 T^{12} + 40516407937 T^{13} + 97840575886 T^{14} + 193923121482 T^{15} + 321793826040 T^{16} + 373638386902 T^{17} + 322687697779 T^{18} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( ( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))(\( 1 - 5 T + 20 T^{2} + 37 T^{3} + 258 T^{4} - 129 T^{5} - 3189 T^{6} + 120795 T^{7} - 136839 T^{8} - 714190 T^{9} + 5712919 T^{10} - 5435984 T^{11} + 107367096 T^{12} - 1641952414 T^{13} + 5170633749 T^{14} - 4076640736 T^{15} - 118253194072 T^{16} - 72635970262 T^{17} - 597214939565 T^{18} - 1670627316026 T^{19} - 62555939664088 T^{20} - 49600487834912 T^{21} + 1446955318953909 T^{22} - 10568168926182002 T^{23} + 15894183505708344 T^{24} - 18508576652684848 T^{25} + 447384315720545239 T^{26} - 1286365219290259970 T^{27} - 5668762367964515511 T^{28} + \)\(11\!\cdots\!65\)\( T^{29} - 69885737313712803669 T^{30} - 65020690689804292407 T^{31} + \)\(29\!\cdots\!22\)\( T^{32} + \)\(98\!\cdots\!59\)\( T^{33} + \)\(12\!\cdots\!20\)\( T^{34} - \)\(70\!\cdots\!15\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))(\( 1 - 21 T + 120 T^{2} + 497 T^{3} - 7494 T^{4} + 5915 T^{5} + 218621 T^{6} - 556115 T^{7} - 5752283 T^{8} + 30788058 T^{9} + 79489305 T^{10} - 991337102 T^{11} + 169834658 T^{12} + 24643342290 T^{13} - 54685074195 T^{14} - 501219381110 T^{15} + 2617597122438 T^{16} + 4593175083882 T^{17} - 73195291611409 T^{18} + 105643026929286 T^{19} + 1384708877769702 T^{20} - 6098336209965370 T^{21} - 15303125847802995 T^{22} + 158613003644845470 T^{23} + 25141624580040962 T^{24} - 3375329791444834594 T^{25} + 6224885793851919705 T^{26} + 55453992607977208854 T^{27} - \)\(23\!\cdots\!67\)\( T^{28} - \)\(52\!\cdots\!05\)\( T^{29} + \)\(47\!\cdots\!41\)\( T^{30} + \)\(29\!\cdots\!45\)\( T^{31} - \)\(86\!\cdots\!46\)\( T^{32} + \)\(13\!\cdots\!79\)\( T^{33} + \)\(73\!\cdots\!20\)\( T^{34} - \)\(29\!\cdots\!63\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( 1 + 13 T + 14 T^{2} - 105 T^{3} + 3094 T^{4} + 16940 T^{5} + 42693 T^{6} + 803658 T^{7} + 2075225 T^{8} - 2407524 T^{9} + 204229046 T^{10} + 936885852 T^{11} - 280158823 T^{12} + 27808933488 T^{13} + 140261496763 T^{14} + 49426230347 T^{15} + 6405725290328 T^{16} + 27514135369805 T^{17} - 40336836526861 T^{18} + 797909925724345 T^{19} + 5387214969165848 T^{20} + 1205456331932983 T^{21} + 99204291692031403 T^{22} + 570393178303457712 T^{23} - 166645001504311183 T^{24} + 16161165062652080268 T^{25} + \)\(10\!\cdots\!06\)\( T^{26} - 34926302108408038356 T^{27} + \)\(87\!\cdots\!25\)\( T^{28} + \)\(98\!\cdots\!82\)\( T^{29} + \)\(15\!\cdots\!13\)\( T^{30} + \)\(17\!\cdots\!60\)\( T^{31} + \)\(92\!\cdots\!14\)\( T^{32} - \)\(90\!\cdots\!45\)\( T^{33} + \)\(35\!\cdots\!94\)\( T^{34} + \)\(94\!\cdots\!17\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))(\( 1 - 11 T + 52 T^{2} - 275 T^{3} + 2056 T^{4} - 15052 T^{5} + 112661 T^{6} - 430604 T^{7} + 938513 T^{8} - 7030078 T^{9} + 58085220 T^{10} - 311723870 T^{11} + 813220433 T^{12} + 3825214814 T^{13} - 1019693503 T^{14} - 241500803649 T^{15} + 1283371104458 T^{16} - 843424901591 T^{17} - 10037528634909 T^{18} - 24459322146139 T^{19} + 1079315098849178 T^{20} - 5889963100195461 T^{21} - 721209840495343 T^{22} + 78459551006961286 T^{23} + 483722478662117993 T^{24} - 5377198200062795830 T^{25} + 29056922951050536420 T^{26} - \)\(10\!\cdots\!82\)\( T^{27} + \)\(39\!\cdots\!13\)\( T^{28} - \)\(52\!\cdots\!16\)\( T^{29} + \)\(39\!\cdots\!01\)\( T^{30} - \)\(15\!\cdots\!28\)\( T^{31} + \)\(61\!\cdots\!36\)\( T^{32} - \)\(23\!\cdots\!75\)\( T^{33} + \)\(13\!\cdots\!92\)\( T^{34} - \)\(79\!\cdots\!99\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 10 T^{2} - 861 T^{4} + 9610 T^{6} + 923521 T^{8} \))(\( ( 1 + 2 T + 237 T^{2} + 434 T^{3} + 25997 T^{4} + 42566 T^{5} + 1732533 T^{6} + 2478803 T^{7} + 77321509 T^{8} + 94038425 T^{9} + 2396966779 T^{10} + 2382129683 T^{11} + 51613890603 T^{12} + 39310594886 T^{13} + 744272038547 T^{14} + 385176597554 T^{15} + 6520489544307 T^{16} + 1705782074882 T^{17} + 26439622160671 T^{18} )^{2} \))(\( ( 1 + 12 T + 223 T^{2} + 1736 T^{3} + 19067 T^{4} + 109656 T^{5} + 947861 T^{6} + 4432661 T^{7} + 34725039 T^{8} + 145311651 T^{9} + 1076476209 T^{10} + 4259787221 T^{11} + 28237727051 T^{12} + 101269618776 T^{13} + 545872022117 T^{14} + 1540706390216 T^{15} + 6135312946753 T^{16} + 10234692449292 T^{17} + 26439622160671 T^{18} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2} \))(\( 1 - 33 T + 594 T^{2} - 7684 T^{3} + 79211 T^{4} - 675181 T^{5} + 4760822 T^{6} - 27260293 T^{7} + 119538780 T^{8} - 315292636 T^{9} - 431460111 T^{10} + 9839885600 T^{11} - 40620061618 T^{12} - 188056914546 T^{13} + 4611267559928 T^{14} - 47372345261297 T^{15} + 369141656659028 T^{16} - 2485377313954698 T^{17} + 15489636837749194 T^{18} - 91958960616323826 T^{19} + 505354927966209332 T^{20} - 2399551404520476941 T^{21} + 8642257821382220408 T^{22} - 13040610595830498522 T^{23} - \)\(10\!\cdots\!62\)\( T^{24} + \)\(93\!\cdots\!00\)\( T^{25} - \)\(15\!\cdots\!31\)\( T^{26} - \)\(40\!\cdots\!72\)\( T^{27} + \)\(57\!\cdots\!20\)\( T^{28} - \)\(48\!\cdots\!09\)\( T^{29} + \)\(31\!\cdots\!82\)\( T^{30} - \)\(16\!\cdots\!57\)\( T^{31} + \)\(71\!\cdots\!79\)\( T^{32} - \)\(25\!\cdots\!12\)\( T^{33} + \)\(73\!\cdots\!54\)\( T^{34} - \)\(15\!\cdots\!61\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \))(\( 1 + 13 T - 10 T^{2} - 696 T^{3} - 2769 T^{4} - 4353 T^{5} + 97274 T^{6} + 1501747 T^{7} + 2267810 T^{8} - 65690928 T^{9} - 358869961 T^{10} + 856933268 T^{11} + 15269913396 T^{12} + 64821248008 T^{13} - 68625840838 T^{14} - 3756011589619 T^{15} - 20117162507060 T^{16} + 65066560986760 T^{17} + 1090658022068282 T^{18} + 2407462756510120 T^{19} - 27540395472165140 T^{20} - 190253255048971207 T^{21} - 128615874490786918 T^{22} + 4494961834553087656 T^{23} + 39178420063260074964 T^{24} + 81350283708956160644 T^{25} - \)\(12\!\cdots\!81\)\( T^{26} - \)\(85\!\cdots\!56\)\( T^{27} + \)\(10\!\cdots\!90\)\( T^{28} + \)\(26\!\cdots\!11\)\( T^{29} + \)\(64\!\cdots\!94\)\( T^{30} - \)\(10\!\cdots\!41\)\( T^{31} - \)\(24\!\cdots\!41\)\( T^{32} - \)\(23\!\cdots\!28\)\( T^{33} - \)\(12\!\cdots\!10\)\( T^{34} + \)\(59\!\cdots\!21\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 16 T^{2} + 1681 T^{4} )^{2} \))(\( 1 + 28 T + 339 T^{2} + 2576 T^{3} + 13887 T^{4} + 20839 T^{5} - 600800 T^{6} - 8454299 T^{7} - 69339479 T^{8} - 349092128 T^{9} - 426409074 T^{10} + 11047502900 T^{11} + 146817087557 T^{12} + 1146851972791 T^{13} + 5817185018496 T^{14} + 12785477699127 T^{15} - 112878494729510 T^{16} - 1709486039383024 T^{17} - 13192203880991413 T^{18} - 70088927614703984 T^{19} - 189748749640306310 T^{20} + 881187908501531967 T^{21} + 16437974555050275456 T^{22} + \)\(13\!\cdots\!91\)\( T^{23} + \)\(69\!\cdots\!37\)\( T^{24} + \)\(21\!\cdots\!00\)\( T^{25} - \)\(34\!\cdots\!54\)\( T^{26} - \)\(11\!\cdots\!08\)\( T^{27} - \)\(93\!\cdots\!79\)\( T^{28} - \)\(46\!\cdots\!59\)\( T^{29} - \)\(13\!\cdots\!00\)\( T^{30} + \)\(19\!\cdots\!19\)\( T^{31} + \)\(52\!\cdots\!07\)\( T^{32} + \)\(40\!\cdots\!76\)\( T^{33} + \)\(21\!\cdots\!99\)\( T^{34} + \)\(73\!\cdots\!68\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))(\( 1 - 8 T - 149 T^{2} + 596 T^{3} + 12507 T^{4} + 29851 T^{5} - 495696 T^{6} - 7065091 T^{7} - 12469507 T^{8} + 398376684 T^{9} + 3079741494 T^{10} - 2018662440 T^{11} - 174188053823 T^{12} - 1078066935369 T^{13} + 2504003552428 T^{14} + 67457726392891 T^{15} + 265011896196846 T^{16} - 1361150930125912 T^{17} - 18508827294469517 T^{18} - 55807188135162392 T^{19} + 445484997506898126 T^{20} + 4649253960724440611 T^{21} + 7075715582312497708 T^{22} - \)\(12\!\cdots\!69\)\( T^{23} - \)\(82\!\cdots\!43\)\( T^{24} - \)\(39\!\cdots\!40\)\( T^{25} + \)\(24\!\cdots\!74\)\( T^{26} + \)\(13\!\cdots\!24\)\( T^{27} - \)\(16\!\cdots\!07\)\( T^{28} - \)\(38\!\cdots\!31\)\( T^{29} - \)\(11\!\cdots\!76\)\( T^{30} + \)\(27\!\cdots\!71\)\( T^{31} + \)\(47\!\cdots\!27\)\( T^{32} + \)\(92\!\cdots\!96\)\( T^{33} - \)\(95\!\cdots\!09\)\( T^{34} - \)\(20\!\cdots\!48\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{4} \))(\( 1 + 20 T + 77 T^{2} - 63 T^{3} + 7637 T^{4} + 39578 T^{5} - 214578 T^{6} + 1909476 T^{7} + 4295128 T^{8} - 209128990 T^{9} + 554833132 T^{10} + 1652084175 T^{11} - 93064174280 T^{12} + 245197794716 T^{13} + 1580145249728 T^{14} - 29314076973548 T^{15} + 85205967739165 T^{16} + 646689692482344 T^{17} - 7627678256625063 T^{18} + 27807656776740792 T^{19} + 157545834349716085 T^{20} - 2330674317935880836 T^{21} + 5402202159915336128 T^{22} + 36046146028232787188 T^{23} - \)\(58\!\cdots\!20\)\( T^{24} + \)\(44\!\cdots\!25\)\( T^{25} + \)\(64\!\cdots\!32\)\( T^{26} - \)\(10\!\cdots\!70\)\( T^{27} + \)\(92\!\cdots\!72\)\( T^{28} + \)\(17\!\cdots\!32\)\( T^{29} - \)\(85\!\cdots\!78\)\( T^{30} + \)\(68\!\cdots\!54\)\( T^{31} + \)\(56\!\cdots\!13\)\( T^{32} - \)\(20\!\cdots\!41\)\( T^{33} + \)\(10\!\cdots\!77\)\( T^{34} + \)\(11\!\cdots\!60\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \))(\( 1 + 24 T + 171 T^{2} - 149 T^{3} - 4405 T^{4} + 7320 T^{5} - 95904 T^{6} - 2232040 T^{7} + 6929106 T^{8} + 169418082 T^{9} + 605030542 T^{10} + 1781115991 T^{11} + 3875158060 T^{12} - 134197178254 T^{13} - 582144907018 T^{14} - 1410611851348 T^{15} - 38236880858251 T^{16} + 200382281040208 T^{17} + 4744814362180117 T^{18} + 8616438084728944 T^{19} - 70699992706906099 T^{20} - 112153516465125436 T^{21} - 1990237590258045418 T^{22} - 19728118230113998522 T^{23} + 24496280969518524940 T^{24} + \)\(48\!\cdots\!37\)\( T^{25} + \)\(70\!\cdots\!42\)\( T^{26} + \)\(85\!\cdots\!26\)\( T^{27} + \)\(14\!\cdots\!94\)\( T^{28} - \)\(20\!\cdots\!80\)\( T^{29} - \)\(38\!\cdots\!04\)\( T^{30} + \)\(12\!\cdots\!60\)\( T^{31} - \)\(32\!\cdots\!45\)\( T^{32} - \)\(47\!\cdots\!43\)\( T^{33} + \)\(23\!\cdots\!71\)\( T^{34} + \)\(14\!\cdots\!32\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))(\( 1 - 36 T + 484 T^{2} - 2418 T^{3} - 2438 T^{4} + 458 T^{5} + 971769 T^{6} - 6200468 T^{7} + 5796904 T^{8} - 66847814 T^{9} + 881180006 T^{10} + 1965825838 T^{11} - 35807762427 T^{12} + 111410889196 T^{13} - 888932098200 T^{14} - 3661950587498 T^{15} + 66341502588244 T^{16} + 1221587255706956 T^{17} - 18008846763710985 T^{18} + 57414601018226932 T^{19} + 146548379217430996 T^{20} - 380194695845804854 T^{21} - 4337705069876674200 T^{22} + 25551531162532844372 T^{23} - \)\(38\!\cdots\!83\)\( T^{24} + \)\(99\!\cdots\!94\)\( T^{25} + \)\(20\!\cdots\!66\)\( T^{26} - \)\(74\!\cdots\!38\)\( T^{27} + \)\(30\!\cdots\!96\)\( T^{28} - \)\(15\!\cdots\!04\)\( T^{29} + \)\(11\!\cdots\!29\)\( T^{30} + \)\(25\!\cdots\!66\)\( T^{31} - \)\(62\!\cdots\!22\)\( T^{32} - \)\(29\!\cdots\!74\)\( T^{33} + \)\(27\!\cdots\!64\)\( T^{34} - \)\(95\!\cdots\!32\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \))(\( 1 - 40 T + 644 T^{2} - 4224 T^{3} - 19070 T^{4} + 663772 T^{5} - 6327009 T^{6} + 21960512 T^{7} + 194007162 T^{8} - 3443103360 T^{9} + 24102576626 T^{10} - 55039347228 T^{11} - 674402516025 T^{12} + 9009506539674 T^{13} - 55050670205162 T^{14} + 144537339124326 T^{15} + 784541744620374 T^{16} - 13394574409393166 T^{17} + 109693552770385659 T^{18} - 629544997241478802 T^{19} + 1733052713866406166 T^{20} + 15006300159904898298 T^{21} - \)\(26\!\cdots\!22\)\( T^{22} + \)\(20\!\cdots\!18\)\( T^{23} - \)\(72\!\cdots\!25\)\( T^{24} - \)\(27\!\cdots\!64\)\( T^{25} + \)\(57\!\cdots\!86\)\( T^{26} - \)\(38\!\cdots\!20\)\( T^{27} + \)\(10\!\cdots\!38\)\( T^{28} + \)\(54\!\cdots\!36\)\( T^{29} - \)\(73\!\cdots\!69\)\( T^{30} + \)\(36\!\cdots\!44\)\( T^{31} - \)\(48\!\cdots\!30\)\( T^{32} - \)\(50\!\cdots\!32\)\( T^{33} + \)\(36\!\cdots\!24\)\( T^{34} - \)\(10\!\cdots\!80\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( ( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 48 T + 1027 T^{2} - 13032 T^{3} + 113452 T^{4} - 745996 T^{5} + 3093303 T^{6} + 11125840 T^{7} - 385128365 T^{8} + 4672252360 T^{9} - 45099050107 T^{10} + 381474795658 T^{11} - 2560106414904 T^{12} + 12125964585208 T^{13} - 21254028685230 T^{14} - 434572667544596 T^{15} + 7486489637890813 T^{16} - 74556033723244882 T^{17} + 584471007089854893 T^{18} - 3951469787331978746 T^{19} + 21029549392835293717 T^{20} - 64697875026036818692 T^{21} - \)\(16\!\cdots\!30\)\( T^{22} + \)\(50\!\cdots\!44\)\( T^{23} - \)\(56\!\cdots\!16\)\( T^{24} + \)\(44\!\cdots\!46\)\( T^{25} - \)\(28\!\cdots\!27\)\( T^{26} + \)\(15\!\cdots\!80\)\( T^{27} - \)\(67\!\cdots\!85\)\( T^{28} + \)\(10\!\cdots\!80\)\( T^{29} + \)\(15\!\cdots\!23\)\( T^{30} - \)\(19\!\cdots\!08\)\( T^{31} + \)\(15\!\cdots\!88\)\( T^{32} - \)\(95\!\cdots\!24\)\( T^{33} + \)\(39\!\cdots\!67\)\( T^{34} - \)\(98\!\cdots\!24\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \))(\( 1 - 10 T + 29 T^{2} - 228 T^{3} + 4730 T^{4} - 74818 T^{5} + 558777 T^{6} - 5522994 T^{7} + 43799453 T^{8} - 246953110 T^{9} + 2677514965 T^{10} - 28039517216 T^{11} + 262827614136 T^{12} - 1682090426992 T^{13} + 12197246331238 T^{14} - 90813330316490 T^{15} + 661834313235587 T^{16} - 6314501407437490 T^{17} + 48794270880957465 T^{18} - 334668574594186970 T^{19} + 1859092585878763883 T^{20} - 13520016177528081730 T^{21} + 96242140428953145478 T^{22} - \)\(70\!\cdots\!56\)\( T^{23} + \)\(58\!\cdots\!44\)\( T^{24} - \)\(32\!\cdots\!92\)\( T^{25} + \)\(16\!\cdots\!65\)\( T^{26} - \)\(81\!\cdots\!30\)\( T^{27} + \)\(76\!\cdots\!97\)\( T^{28} - \)\(51\!\cdots\!18\)\( T^{29} + \)\(27\!\cdots\!57\)\( T^{30} - \)\(19\!\cdots\!14\)\( T^{31} + \)\(65\!\cdots\!70\)\( T^{32} - \)\(16\!\cdots\!96\)\( T^{33} + \)\(11\!\cdots\!09\)\( T^{34} - \)\(20\!\cdots\!30\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))(\( 1 + 25 T + 282 T^{2} + 3370 T^{3} + 54501 T^{4} + 679351 T^{5} + 6708661 T^{6} + 72121396 T^{7} + 815515241 T^{8} + 8027067641 T^{9} + 73518733596 T^{10} + 702948989693 T^{11} + 6585717674890 T^{12} + 57072728770037 T^{13} + 482396086609999 T^{14} + 4107164871912858 T^{15} + 33936817067694696 T^{16} + 267212451488138097 T^{17} + 2056595036207731957 T^{18} + 15765534637800147723 T^{19} + \)\(11\!\cdots\!76\)\( T^{20} + \)\(84\!\cdots\!82\)\( T^{21} + \)\(58\!\cdots\!39\)\( T^{22} + \)\(40\!\cdots\!63\)\( T^{23} + \)\(27\!\cdots\!90\)\( T^{24} + \)\(17\!\cdots\!67\)\( T^{25} + \)\(10\!\cdots\!16\)\( T^{26} + \)\(69\!\cdots\!99\)\( T^{27} + \)\(41\!\cdots\!41\)\( T^{28} + \)\(21\!\cdots\!64\)\( T^{29} + \)\(11\!\cdots\!41\)\( T^{30} + \)\(71\!\cdots\!29\)\( T^{31} + \)\(33\!\cdots\!61\)\( T^{32} + \)\(12\!\cdots\!30\)\( T^{33} + \)\(60\!\cdots\!62\)\( T^{34} + \)\(31\!\cdots\!75\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \))(\( 1 - 13 T - 168 T^{2} + 3308 T^{3} + 4913 T^{4} - 358091 T^{5} + 1198033 T^{6} + 19363316 T^{7} - 175646011 T^{8} - 99604901 T^{9} + 12471969322 T^{10} - 84370925691 T^{11} - 470930351430 T^{12} + 9329956580485 T^{13} - 10559791467425 T^{14} - 570785200662522 T^{15} + 3369504363792318 T^{16} + 14671224179906015 T^{17} - 275264924921688627 T^{18} + 865602226614454885 T^{19} + 11729244690361058958 T^{20} - \)\(11\!\cdots\!38\)\( T^{21} - \)\(12\!\cdots\!25\)\( T^{22} + \)\(66\!\cdots\!15\)\( T^{23} - \)\(19\!\cdots\!30\)\( T^{24} - \)\(20\!\cdots\!29\)\( T^{25} + \)\(18\!\cdots\!62\)\( T^{26} - \)\(86\!\cdots\!39\)\( T^{27} - \)\(89\!\cdots\!11\)\( T^{28} + \)\(58\!\cdots\!44\)\( T^{29} + \)\(21\!\cdots\!73\)\( T^{30} - \)\(37\!\cdots\!89\)\( T^{31} + \)\(30\!\cdots\!93\)\( T^{32} + \)\(12\!\cdots\!92\)\( T^{33} - \)\(36\!\cdots\!88\)\( T^{34} - \)\(16\!\cdots\!47\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))(\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 114 T^{2} + 9275 T^{4} - 424194 T^{6} + 13845841 T^{8} \))(\( 1 - T - 104 T^{2} - 1039 T^{3} + 1965 T^{4} + 113382 T^{5} + 553582 T^{6} - 1719789 T^{7} - 33834660 T^{8} - 61300405 T^{9} - 1432537967 T^{10} - 14153643444 T^{11} - 32883927380 T^{12} + 1868993049229 T^{13} + 16611121509004 T^{14} - 24347779588048 T^{15} - 705952959985019 T^{16} - 2332913121718237 T^{17} + 21649393309133541 T^{18} - 142307700424812457 T^{19} - 2626850964104255699 T^{20} - 5526483358674723088 T^{21} + \)\(22\!\cdots\!64\)\( T^{22} + \)\(15\!\cdots\!29\)\( T^{23} - \)\(16\!\cdots\!80\)\( T^{24} - \)\(44\!\cdots\!24\)\( T^{25} - \)\(27\!\cdots\!27\)\( T^{26} - \)\(71\!\cdots\!05\)\( T^{27} - \)\(24\!\cdots\!60\)\( T^{28} - \)\(74\!\cdots\!29\)\( T^{29} + \)\(14\!\cdots\!22\)\( T^{30} + \)\(18\!\cdots\!42\)\( T^{31} + \)\(19\!\cdots\!65\)\( T^{32} - \)\(62\!\cdots\!39\)\( T^{33} - \)\(38\!\cdots\!44\)\( T^{34} - \)\(22\!\cdots\!21\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))(\( 1 - 27 T + 358 T^{2} - 2517 T^{3} + 5437 T^{4} + 61198 T^{5} - 270612 T^{6} - 9382085 T^{7} + 167668082 T^{8} - 1277269275 T^{9} + 1433214041 T^{10} + 72295776332 T^{11} - 785635747116 T^{12} + 2617750316813 T^{13} + 22530391569330 T^{14} - 277442038509454 T^{15} + 19041168727695 T^{16} + 25927828100752377 T^{17} - 297028576346892747 T^{18} + 1581597514145894997 T^{19} + 70852188835753095 T^{20} - 62974071342914378374 T^{21} + \)\(31\!\cdots\!30\)\( T^{22} + \)\(22\!\cdots\!13\)\( T^{23} - \)\(40\!\cdots\!76\)\( T^{24} + \)\(22\!\cdots\!72\)\( T^{25} + \)\(27\!\cdots\!21\)\( T^{26} - \)\(14\!\cdots\!75\)\( T^{27} + \)\(11\!\cdots\!82\)\( T^{28} - \)\(40\!\cdots\!85\)\( T^{29} - \)\(71\!\cdots\!52\)\( T^{30} + \)\(99\!\cdots\!38\)\( T^{31} + \)\(53\!\cdots\!17\)\( T^{32} - \)\(15\!\cdots\!17\)\( T^{33} + \)\(13\!\cdots\!38\)\( T^{34} - \)\(60\!\cdots\!67\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( ( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 17 T + 306 T^{2} - 3687 T^{3} + 48666 T^{4} - 513254 T^{5} + 5520487 T^{6} - 50739092 T^{7} + 475083883 T^{8} - 3873616303 T^{9} + 31830620161 T^{10} - 227767783988 T^{11} + 1660358231581 T^{12} - 10342643457734 T^{13} + 65705188457262 T^{14} - 333520055057103 T^{15} + 1854577751228838 T^{16} - 6903150518462897 T^{17} + 27206534396294947 T^{18} )^{2} \))(\( ( 1 + 43 T + 1256 T^{2} + 26303 T^{3} + 453672 T^{4} + 6514284 T^{5} + 81520579 T^{6} + 890095728 T^{7} + 8666196067 T^{8} + 74876454483 T^{9} + 580635136489 T^{10} + 3995639722992 T^{11} + 24518373901777 T^{12} + 131270125112364 T^{13} + 612513957542904 T^{14} + 2379326826191207 T^{15} + 7612253776285688 T^{16} + 17460910134935563 T^{17} + 27206534396294947 T^{18} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{4} \))(\( 1 + 6 T - 210 T^{2} - 952 T^{3} + 16583 T^{4} + 16324 T^{5} - 643363 T^{6} - 258984 T^{7} - 737382 T^{8} + 912139347 T^{9} + 4305537152 T^{10} - 93236145114 T^{11} - 280821494149 T^{12} + 2783232492873 T^{13} - 967685856540 T^{14} - 71237200972668 T^{15} - 1759207347557296 T^{16} + 4294552003410998 T^{17} + 288918929416291839 T^{18} + 304913192242180858 T^{19} - 8868164239036329136 T^{20} - 25496577837328576548 T^{21} - 24590524294606243740 T^{22} + \)\(50\!\cdots\!23\)\( T^{23} - \)\(35\!\cdots\!29\)\( T^{24} - \)\(84\!\cdots\!74\)\( T^{25} + \)\(27\!\cdots\!72\)\( T^{26} + \)\(41\!\cdots\!57\)\( T^{27} - \)\(24\!\cdots\!82\)\( T^{28} - \)\(59\!\cdots\!64\)\( T^{29} - \)\(10\!\cdots\!83\)\( T^{30} + \)\(19\!\cdots\!64\)\( T^{31} + \)\(13\!\cdots\!23\)\( T^{32} - \)\(55\!\cdots\!52\)\( T^{33} - \)\(87\!\cdots\!10\)\( T^{34} + \)\(17\!\cdots\!46\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))(\( 1 - 290 T^{2} - 546 T^{3} + 28629 T^{4} + 156996 T^{5} - 263847 T^{6} - 11002320 T^{7} - 142131800 T^{8} - 1094265249 T^{9} + 4427899630 T^{10} + 198330459348 T^{11} + 927057356295 T^{12} - 5330573304693 T^{13} - 65291921040190 T^{14} - 740831378518098 T^{15} - 4239573868089846 T^{16} + 41003906532639582 T^{17} + 714841627101682855 T^{18} + 2911277363817410322 T^{19} - 21371691869040913686 T^{20} - \)\(26\!\cdots\!78\)\( T^{21} - \)\(16\!\cdots\!90\)\( T^{22} - \)\(96\!\cdots\!43\)\( T^{23} + \)\(11\!\cdots\!95\)\( T^{24} + \)\(18\!\cdots\!68\)\( T^{25} + \)\(28\!\cdots\!30\)\( T^{26} - \)\(50\!\cdots\!19\)\( T^{27} - \)\(46\!\cdots\!00\)\( T^{28} - \)\(25\!\cdots\!20\)\( T^{29} - \)\(43\!\cdots\!27\)\( T^{30} + \)\(18\!\cdots\!56\)\( T^{31} + \)\(23\!\cdots\!49\)\( T^{32} - \)\(32\!\cdots\!46\)\( T^{33} - \)\(12\!\cdots\!90\)\( T^{34} + \)\(21\!\cdots\!61\)\( T^{36} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 144 T^{2} + 5329 T^{4} \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 144 T^{2} + 15407 T^{4} - 767376 T^{6} + 28398241 T^{8} \))(\( 1 + 39 T + 597 T^{2} + 3307 T^{3} - 24123 T^{4} - 505273 T^{5} - 1772091 T^{6} + 38400548 T^{7} + 603319642 T^{8} + 3158577623 T^{9} - 11348234520 T^{10} - 270795790210 T^{11} - 1070092084582 T^{12} + 12896721225256 T^{13} + 217613983286184 T^{14} + 1524753240729012 T^{15} + 4162169837040916 T^{16} - 37984817936832784 T^{17} - 587124083192237451 T^{18} - 2772891709388793232 T^{19} + 22180203061591041364 T^{20} + \)\(59\!\cdots\!04\)\( T^{21} + \)\(61\!\cdots\!44\)\( T^{22} + \)\(26\!\cdots\!08\)\( T^{23} - \)\(16\!\cdots\!98\)\( T^{24} - \)\(29\!\cdots\!70\)\( T^{25} - \)\(91\!\cdots\!20\)\( T^{26} + \)\(18\!\cdots\!99\)\( T^{27} + \)\(25\!\cdots\!58\)\( T^{28} + \)\(12\!\cdots\!96\)\( T^{29} - \)\(40\!\cdots\!11\)\( T^{30} - \)\(84\!\cdots\!09\)\( T^{31} - \)\(29\!\cdots\!07\)\( T^{32} + \)\(29\!\cdots\!99\)\( T^{33} + \)\(38\!\cdots\!17\)\( T^{34} + \)\(18\!\cdots\!67\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \))(\( 1 - 5 T - 323 T^{2} + 2687 T^{3} + 46301 T^{4} - 649773 T^{5} - 2405583 T^{6} + 89955512 T^{7} - 300984170 T^{8} - 7032453301 T^{9} + 72740162048 T^{10} + 142552817514 T^{11} - 6755413388822 T^{12} + 33325150144392 T^{13} + 248027092784912 T^{14} - 3946390059049020 T^{15} + 13804938358862212 T^{16} + 140951235631318316 T^{17} - 2177817924362339187 T^{18} + 10289440201086237068 T^{19} + 73566516514376727748 T^{20} - \)\(15\!\cdots\!40\)\( T^{21} + \)\(70\!\cdots\!92\)\( T^{22} + \)\(69\!\cdots\!56\)\( T^{23} - \)\(10\!\cdots\!58\)\( T^{24} + \)\(15\!\cdots\!58\)\( T^{25} + \)\(58\!\cdots\!88\)\( T^{26} - \)\(41\!\cdots\!13\)\( T^{27} - \)\(12\!\cdots\!30\)\( T^{28} + \)\(28\!\cdots\!24\)\( T^{29} - \)\(55\!\cdots\!43\)\( T^{30} - \)\(10\!\cdots\!09\)\( T^{31} + \)\(56\!\cdots\!09\)\( T^{32} + \)\(23\!\cdots\!59\)\( T^{33} - \)\(21\!\cdots\!03\)\( T^{34} - \)\(23\!\cdots\!65\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 + T + 468 T^{2} - 110 T^{3} + 108679 T^{4} - 88683 T^{5} + 16473653 T^{6} - 16388550 T^{7} + 1776040263 T^{8} - 1663289107 T^{9} + 140307180777 T^{10} - 102280940550 T^{11} + 8122153401467 T^{12} - 3454210033323 T^{13} + 334411412386921 T^{14} - 26739620107310 T^{15} + 8987429405522412 T^{16} + 1517108809906561 T^{17} + 119851595982618319 T^{18} )^{2} \))(\( ( 1 + 33 T + 936 T^{2} + 17622 T^{3} + 300801 T^{4} + 4140381 T^{5} + 53239123 T^{6} + 587509404 T^{7} + 6128037799 T^{8} + 55962904279 T^{9} + 484114986121 T^{10} + 3666646190364 T^{11} + 26248963964797 T^{12} + 161268175320861 T^{13} + 925581641875599 T^{14} + 4283687141191062 T^{15} + 17974858811044824 T^{16} + 50064590726916513 T^{17} + 119851595982618319 T^{18} )^{2} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 68 T^{2} + 6889 T^{4} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 68 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 35 T + 479 T^{2} - 3409 T^{3} + 19375 T^{4} - 31353 T^{5} - 2406555 T^{6} + 24156524 T^{7} + 6883395 T^{8} + 39695271 T^{9} - 10338428590 T^{10} - 5145435974 T^{11} + 135323329869 T^{12} + 3436306060467 T^{13} + 132406605120235 T^{14} - 1752923865830333 T^{15} - 3875663600835921 T^{16} + 115332104316949660 T^{17} - 705090040524994983 T^{18} + 9572564658306821780 T^{19} - 26699446546158659769 T^{20} - \)\(10\!\cdots\!71\)\( T^{21} + \)\(62\!\cdots\!35\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{23} + \)\(44\!\cdots\!61\)\( T^{24} - \)\(13\!\cdots\!98\)\( T^{25} - \)\(23\!\cdots\!90\)\( T^{26} + \)\(74\!\cdots\!13\)\( T^{27} + \)\(10\!\cdots\!55\)\( T^{28} + \)\(31\!\cdots\!08\)\( T^{29} - \)\(25\!\cdots\!55\)\( T^{30} - \)\(27\!\cdots\!39\)\( T^{31} + \)\(14\!\cdots\!75\)\( T^{32} - \)\(20\!\cdots\!63\)\( T^{33} + \)\(24\!\cdots\!99\)\( T^{34} - \)\(14\!\cdots\!05\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))(\( 1 - 55 T + 1307 T^{2} - 15891 T^{3} + 73869 T^{4} + 502107 T^{5} - 7798313 T^{6} + 12490378 T^{7} + 230007623 T^{8} + 1633765029 T^{9} - 52447109858 T^{10} + 284785335188 T^{11} + 1796954314057 T^{12} - 33848843373411 T^{13} + 246384424666331 T^{14} - 1991874247898131 T^{15} + 19360416122489765 T^{16} - 111868014345678078 T^{17} + 496289589992636057 T^{18} - 9285045190691280474 T^{19} + \)\(13\!\cdots\!85\)\( T^{20} - \)\(11\!\cdots\!97\)\( T^{21} + \)\(11\!\cdots\!51\)\( T^{22} - \)\(13\!\cdots\!73\)\( T^{23} + \)\(58\!\cdots\!33\)\( T^{24} + \)\(77\!\cdots\!76\)\( T^{25} - \)\(11\!\cdots\!78\)\( T^{26} + \)\(30\!\cdots\!87\)\( T^{27} + \)\(35\!\cdots\!27\)\( T^{28} + \)\(16\!\cdots\!26\)\( T^{29} - \)\(83\!\cdots\!93\)\( T^{30} + \)\(44\!\cdots\!41\)\( T^{31} + \)\(54\!\cdots\!01\)\( T^{32} - \)\(97\!\cdots\!37\)\( T^{33} + \)\(66\!\cdots\!67\)\( T^{34} - \)\(23\!\cdots\!65\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 128 T^{2} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 128 T^{2} + 8463 T^{4} - 1013888 T^{6} + 62742241 T^{8} \))(\( 1 + 6 T - 237 T^{2} - 192 T^{3} + 39940 T^{4} - 65671 T^{5} - 3663806 T^{6} + 14387533 T^{7} + 201443985 T^{8} - 921675767 T^{9} - 6410096406 T^{10} - 27660200711 T^{11} + 758551978909 T^{12} + 10643035282150 T^{13} - 193758610734576 T^{14} - 1060042884622694 T^{15} + 28600180947878871 T^{16} + 31300409832430604 T^{17} - 3133414544394990373 T^{18} + 2785736475086323756 T^{19} + \)\(22\!\cdots\!91\)\( T^{20} - \)\(74\!\cdots\!86\)\( T^{21} - \)\(12\!\cdots\!16\)\( T^{22} + \)\(59\!\cdots\!50\)\( T^{23} + \)\(37\!\cdots\!49\)\( T^{24} - \)\(12\!\cdots\!19\)\( T^{25} - \)\(25\!\cdots\!86\)\( T^{26} - \)\(32\!\cdots\!03\)\( T^{27} + \)\(62\!\cdots\!85\)\( T^{28} + \)\(39\!\cdots\!37\)\( T^{29} - \)\(90\!\cdots\!26\)\( T^{30} - \)\(14\!\cdots\!99\)\( T^{31} + \)\(78\!\cdots\!40\)\( T^{32} - \)\(33\!\cdots\!08\)\( T^{33} - \)\(36\!\cdots\!57\)\( T^{34} + \)\(82\!\cdots\!74\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))(\( 1 - 62 T + 1695 T^{2} - 28340 T^{3} + 341752 T^{4} - 2937679 T^{5} + 9490566 T^{6} + 195057689 T^{7} - 4216745147 T^{8} + 46871376137 T^{9} - 284049882914 T^{10} - 789011574747 T^{11} + 44618098414089 T^{12} - 646673709169234 T^{13} + 5850815394273200 T^{14} - 28480105599660806 T^{15} - 103171040914897793 T^{16} + 4052328622366548400 T^{17} - 49934342778322290085 T^{18} + \)\(36\!\cdots\!00\)\( T^{19} - \)\(81\!\cdots\!53\)\( T^{20} - \)\(20\!\cdots\!14\)\( T^{21} + \)\(36\!\cdots\!00\)\( T^{22} - \)\(36\!\cdots\!66\)\( T^{23} + \)\(22\!\cdots\!29\)\( T^{24} - \)\(34\!\cdots\!63\)\( T^{25} - \)\(11\!\cdots\!34\)\( T^{26} + \)\(16\!\cdots\!33\)\( T^{27} - \)\(13\!\cdots\!47\)\( T^{28} + \)\(54\!\cdots\!21\)\( T^{29} + \)\(23\!\cdots\!86\)\( T^{30} - \)\(64\!\cdots\!51\)\( T^{31} + \)\(66\!\cdots\!32\)\( T^{32} - \)\(49\!\cdots\!60\)\( T^{33} + \)\(26\!\cdots\!95\)\( T^{34} - \)\(85\!\cdots\!98\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 + 96 T^{2} + 9409 T^{4} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 96 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 28 T + 1111 T^{2} - 21952 T^{3} + 495167 T^{4} - 7490854 T^{5} + 121190909 T^{6} - 1456448147 T^{7} + 18272521659 T^{8} - 176700396733 T^{9} + 1772434600923 T^{10} - 13703720615123 T^{11} + 110607670489757 T^{12} - 663159918695974 T^{13} + 4252167513037919 T^{14} - 18285401452201408 T^{15} + 89766894055183543 T^{16} - 219448140642554908 T^{17} + 760231058654565217 T^{18} )^{2} \))(\( ( 1 + 16 T + 635 T^{2} + 7580 T^{3} + 168999 T^{4} + 1529370 T^{5} + 25950241 T^{6} + 184196649 T^{7} + 2859586075 T^{8} + 17958910263 T^{9} + 277379849275 T^{10} + 1733106270441 T^{11} + 23684084304193 T^{12} + 135394026482970 T^{13} + 1451251916092743 T^{14} + 6313927797361820 T^{15} + 51306910643601755 T^{16} + 125398937510031376 T^{17} + 760231058654565217 T^{18} )^{2} \))
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