Properties

Label 22.4
Level 22
Weight 4
Dimension 15
Nonzero newspaces 2
Newform subspaces 5
Sturm bound 120
Trace bound 1

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Defining parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 5 \)
Sturm bound: \(120\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(22))\).

Total New Old
Modular forms 55 15 40
Cusp forms 35 15 20
Eisenstein series 20 0 20

Trace form

\( 15q + 50q^{6} + 20q^{7} - 160q^{9} + O(q^{10}) \) \( 15q + 50q^{6} + 20q^{7} - 160q^{9} - 100q^{10} - 100q^{11} - 80q^{12} - 40q^{13} + 20q^{14} + 410q^{15} + 310q^{17} + 230q^{18} - 225q^{19} - 420q^{21} + 390q^{23} + 200q^{24} + 300q^{25} + 200q^{26} + 75q^{27} - 120q^{28} - 250q^{29} - 780q^{30} + 90q^{31} - 160q^{32} - 1265q^{33} - 600q^{34} - 1010q^{35} - 100q^{36} - 240q^{37} - 60q^{38} - 160q^{39} + 240q^{40} + 830q^{41} + 2180q^{42} + 2690q^{43} + 1060q^{44} + 1910q^{45} + 360q^{46} - 250q^{47} - 1330q^{49} - 680q^{50} - 695q^{51} - 960q^{52} + 260q^{53} - 2160q^{54} - 370q^{55} - 945q^{57} - 1680q^{58} - 1825q^{59} - 480q^{60} + 260q^{61} + 1380q^{62} + 1720q^{63} + 760q^{65} + 2920q^{66} + 820q^{67} + 1240q^{68} + 840q^{70} - 3240q^{71} - 1120q^{72} - 870q^{73} - 1400q^{74} - 5045q^{75} - 760q^{76} + 630q^{77} - 2960q^{78} + 240q^{79} - 640q^{80} + 2505q^{81} + 1690q^{82} + 4445q^{83} + 1840q^{84} + 2420q^{85} + 2450q^{86} + 3400q^{87} + 560q^{88} + 350q^{89} + 2940q^{90} + 3830q^{91} - 1680q^{92} + 6230q^{93} - 1600q^{94} - 2200q^{95} - 3345q^{97} - 4410q^{98} - 7790q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(22))\)

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
22.4.a \(\chi_{22}(1, \cdot)\) 22.4.a.a 1 1
22.4.a.b 1
22.4.a.c 1
22.4.c \(\chi_{22}(3, \cdot)\) 22.4.c.a 4 4
22.4.c.b 8

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(22))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(22)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 - 2 T \))(\( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} \))(\( ( 1 - 2 T + 4 T^{2} - 8 T^{3} + 16 T^{4} )^{2} \))
$3$ (\( 1 + 7 T + 27 T^{2} \))(\( 1 - 4 T + 27 T^{2} \))(\( 1 - T + 27 T^{2} \))(\( 1 + T + 49 T^{2} - 83 T^{3} + 1204 T^{4} - 2241 T^{5} + 35721 T^{6} + 19683 T^{7} + 531441 T^{8} \))(\( 1 - 3 T - 12 T^{2} + 158 T^{3} - 1228 T^{4} + 6473 T^{5} + 959 T^{6} - 225344 T^{7} + 1236232 T^{8} - 6084288 T^{9} + 699111 T^{10} + 127408059 T^{11} - 652609548 T^{12} + 2267127306 T^{13} - 4649045868 T^{14} - 31381059609 T^{15} + 282429536481 T^{16} \))
$5$ (\( 1 + 19 T + 125 T^{2} \))(\( 1 - 14 T + 125 T^{2} \))(\( 1 + 3 T + 125 T^{2} \))(\( 1 - 3 T - 121 T^{2} + 123 T^{3} + 15376 T^{4} + 15375 T^{5} - 1890625 T^{6} - 5859375 T^{7} + 244140625 T^{8} \))(\( 1 - 5 T - 104 T^{2} - 1630 T^{3} + 10006 T^{4} + 283885 T^{5} + 2835571 T^{6} - 31638350 T^{7} - 387953844 T^{8} - 3954793750 T^{9} + 44305796875 T^{10} + 554462890625 T^{11} + 2442871093750 T^{12} - 49743652343750 T^{13} - 396728515625000 T^{14} - 2384185791015625 T^{15} + 59604644775390625 T^{16} \))
$7$ (\( 1 - 14 T + 343 T^{2} \))(\( 1 + 8 T + 343 T^{2} \))(\( 1 + 10 T + 343 T^{2} \))(\( 1 - 25 T + 117 T^{2} - 6065 T^{3} + 231284 T^{4} - 2080295 T^{5} + 13764933 T^{6} - 1008840175 T^{7} + 13841287201 T^{8} \))(\( 1 + T + 12 T^{2} - 1104 T^{3} - 87888 T^{4} - 922401 T^{5} + 28062801 T^{6} + 211635072 T^{7} + 13215739812 T^{8} + 72590829696 T^{9} + 3301560474849 T^{10} - 37222207450407 T^{11} - 1216483049521488 T^{12} - 5241307906977072 T^{13} + 19540963174925388 T^{14} + 558545864083284007 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 - 11 T \))(\( 1 + 11 T \))(\( 1 - 11 T \))(\( 1 - 44 T + 726 T^{2} - 58564 T^{3} + 1771561 T^{4} \))(\( 1 + 155 T + 13111 T^{2} + 755095 T^{3} + 31999176 T^{4} + 1005031445 T^{5} + 23226936271 T^{6} + 365481892105 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 + 72 T + 2197 T^{2} \))(\( 1 + 50 T + 2197 T^{2} \))(\( 1 + 16 T + 2197 T^{2} \))(\( 1 - 91 T + 1299 T^{2} + 43003 T^{3} - 510136 T^{4} + 94477591 T^{5} + 6270024891 T^{6} - 965009442943 T^{7} + 23298085122481 T^{8} \))(\( 1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 10009625561 T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 2714446510819810 T^{9} + 48314550744464849 T^{10} + 3465761392820424581 T^{11} + \)\(23\!\cdots\!76\)\( T^{12} + \)\(40\!\cdots\!88\)\( T^{13} + \)\(21\!\cdots\!70\)\( T^{14} - \)\(17\!\cdots\!91\)\( T^{15} + \)\(54\!\cdots\!61\)\( T^{16} \))
$17$ (\( 1 + 46 T + 4913 T^{2} \))(\( 1 - 130 T + 4913 T^{2} \))(\( 1 - 42 T + 4913 T^{2} \))(\( 1 - 23 T + 6011 T^{2} - 260829 T^{3} + 20906864 T^{4} - 1281452877 T^{5} + 145090927259 T^{6} - 2727521159431 T^{7} + 582622237229761 T^{8} \))(\( 1 - 161 T + 5020 T^{2} + 211188 T^{3} + 21130156 T^{4} - 1323457999 T^{5} - 265862283281 T^{6} + 13237803594360 T^{7} + 277440196391544 T^{8} + 65037329059090680 T^{9} - 6417269207192683889 T^{10} - \)\(15\!\cdots\!03\)\( T^{11} + \)\(12\!\cdots\!16\)\( T^{12} + \)\(60\!\cdots\!84\)\( T^{13} + \)\(70\!\cdots\!80\)\( T^{14} - \)\(11\!\cdots\!37\)\( T^{15} + \)\(33\!\cdots\!21\)\( T^{16} \))
$19$ (\( 1 + 20 T + 6859 T^{2} \))(\( 1 + 108 T + 6859 T^{2} \))(\( 1 - 116 T + 6859 T^{2} \))(\( 1 - 59 T - 5223 T^{2} + 483473 T^{3} + 10576100 T^{4} + 3316141307 T^{5} - 245720636463 T^{6} - 19038574168961 T^{7} + 2213314919066161 T^{8} \))(\( 1 + 272 T + 33201 T^{2} + 3039584 T^{3} + 292957015 T^{4} + 21884364992 T^{5} + 909384182595 T^{6} + 32108296725040 T^{7} + 2715838419478696 T^{8} + 220230807237049360 T^{9} + 42782780037646641195 T^{10} + \)\(70\!\cdots\!68\)\( T^{11} + \)\(64\!\cdots\!15\)\( T^{12} + \)\(46\!\cdots\!16\)\( T^{13} + \)\(34\!\cdots\!41\)\( T^{14} + \)\(19\!\cdots\!68\)\( T^{15} + \)\(48\!\cdots\!21\)\( T^{16} \))
$23$ (\( 1 + 107 T + 12167 T^{2} \))(\( 1 + 96 T + 12167 T^{2} \))(\( 1 - 189 T + 12167 T^{2} \))(\( ( 1 + 112 T + 26750 T^{2} + 1362704 T^{3} + 148035889 T^{4} )^{2} \))(\( ( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 97855798574 T^{5} + 9150986514424 T^{6} - 565561935699382 T^{7} + 21914624432020321 T^{8} )^{2} \))
$29$ (\( 1 - 120 T + 24389 T^{2} \))(\( 1 - 142 T + 24389 T^{2} \))(\( 1 + 120 T + 24389 T^{2} \))(\( 1 + 425 T + 71171 T^{2} + 10071255 T^{3} + 1728423176 T^{4} + 245627838195 T^{5} + 42334170578891 T^{6} + 6165537039744325 T^{7} + 353814783205469041 T^{8} \))(\( 1 - 33 T + 14176 T^{2} + 2091144 T^{3} + 419909220 T^{4} + 81926257017 T^{5} + 7198320043835 T^{6} + 2719607629962300 T^{7} + 63828590693142176 T^{8} + 66328510487150534700 T^{9} + \)\(42\!\cdots\!35\)\( T^{10} + \)\(11\!\cdots\!73\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(18\!\cdots\!56\)\( T^{13} + \)\(29\!\cdots\!36\)\( T^{14} - \)\(16\!\cdots\!57\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \))
$31$ (\( 1 - 117 T + 29791 T^{2} \))(\( 1 - 40 T + 29791 T^{2} \))(\( 1 + 163 T + 29791 T^{2} \))(\( 1 + 227 T + 63923 T^{2} + 11938579 T^{3} + 2979456040 T^{4} + 355662206989 T^{5} + 56731897800563 T^{6} + 6001794230472317 T^{7} + 787662783788549761 T^{8} \))(\( 1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 36115996851965 T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!65\)\( T^{10} - \)\(24\!\cdots\!83\)\( T^{11} + \)\(24\!\cdots\!78\)\( T^{12} + \)\(13\!\cdots\!92\)\( T^{13} - \)\(28\!\cdots\!18\)\( T^{14} - \)\(67\!\cdots\!13\)\( T^{15} + \)\(62\!\cdots\!21\)\( T^{16} \))
$37$ (\( 1 + 201 T + 50653 T^{2} \))(\( 1 - 382 T + 50653 T^{2} \))(\( 1 + 409 T + 50653 T^{2} \))(\( 1 + 61 T - 8617 T^{2} + 10210715 T^{3} + 2977938736 T^{4} + 517203346895 T^{5} - 22108864466353 T^{6} + 7927666127499697 T^{7} + 6582952005840035281 T^{8} \))(\( 1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - \)\(73\!\cdots\!14\)\( T^{9} + \)\(23\!\cdots\!39\)\( T^{10} + \)\(13\!\cdots\!53\)\( T^{11} - \)\(37\!\cdots\!78\)\( T^{12} + \)\(14\!\cdots\!78\)\( T^{13} - \)\(13\!\cdots\!72\)\( T^{14} - \)\(41\!\cdots\!13\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} \))
$41$ (\( 1 + 228 T + 68921 T^{2} \))(\( 1 + 118 T + 68921 T^{2} \))(\( 1 - 468 T + 68921 T^{2} \))(\( 1 - 347 T - 5917 T^{2} - 2642769 T^{3} + 5378594000 T^{4} - 182142282249 T^{5} - 28106366793997 T^{6} - 113601531234704467 T^{7} + 22563490300366186081 T^{8} \))(\( 1 - 361 T - 19140 T^{2} + 12862580 T^{3} + 3853225260 T^{4} - 605378813703 T^{5} - 283363657590617 T^{6} + 99783171614693080 T^{7} - 22290482644766813160 T^{8} + \)\(68\!\cdots\!80\)\( T^{9} - \)\(13\!\cdots\!97\)\( T^{10} - \)\(19\!\cdots\!83\)\( T^{11} + \)\(86\!\cdots\!60\)\( T^{12} + \)\(20\!\cdots\!80\)\( T^{13} - \)\(20\!\cdots\!40\)\( T^{14} - \)\(26\!\cdots\!01\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \))
$43$ (\( 1 + 242 T + 79507 T^{2} \))(\( 1 - 220 T + 79507 T^{2} \))(\( 1 - 110 T + 79507 T^{2} \))(\( ( 1 - 580 T + 237334 T^{2} - 46114060 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 12280589284047 T^{5} + 2655712080056733 T^{6} - 362369273206463803 T^{7} + 39959630797262576401 T^{8} )^{2} \))
$47$ (\( 1 + 96 T + 103823 T^{2} \))(\( 1 - 520 T + 103823 T^{2} \))(\( 1 - 144 T + 103823 T^{2} \))(\( 1 - 251 T - 62767 T^{2} + 35545485 T^{3} - 2090662324 T^{4} + 3690438889155 T^{5} - 676579008555343 T^{6} - 280901748748794517 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 1069 T + 301070 T^{2} - 43368042 T^{3} - 24642745634 T^{4} + 6223420821981 T^{5} + 4635059307315839 T^{6} + 746453682627128500 T^{7} + 29698413240018564184 T^{8} + \)\(77\!\cdots\!00\)\( T^{9} + \)\(49\!\cdots\!31\)\( T^{10} + \)\(69\!\cdots\!27\)\( T^{11} - \)\(28\!\cdots\!94\)\( T^{12} - \)\(52\!\cdots\!06\)\( T^{13} + \)\(37\!\cdots\!30\)\( T^{14} + \)\(13\!\cdots\!43\)\( T^{15} + \)\(13\!\cdots\!81\)\( T^{16} \))
$53$ (\( 1 - 458 T + 148877 T^{2} \))(\( 1 - 238 T + 148877 T^{2} \))(\( 1 - 90 T + 148877 T^{2} \))(\( 1 + 245 T + 135263 T^{2} + 62167475 T^{3} + 40936657664 T^{4} + 9255307175575 T^{5} + 2998017979391927 T^{6} + 808442079991522585 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 + 281 T - 343802 T^{2} - 60864958 T^{3} + 33213391926 T^{4} + 4932564267181 T^{5} + 8874412003121445 T^{6} + 107096647673733916 T^{7} - \)\(25\!\cdots\!56\)\( T^{8} + \)\(15\!\cdots\!32\)\( T^{9} + \)\(19\!\cdots\!05\)\( T^{10} + \)\(16\!\cdots\!73\)\( T^{11} + \)\(16\!\cdots\!66\)\( T^{12} - \)\(44\!\cdots\!06\)\( T^{13} - \)\(37\!\cdots\!78\)\( T^{14} + \)\(45\!\cdots\!93\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} \))
$59$ (\( 1 - 435 T + 205379 T^{2} \))(\( 1 + 852 T + 205379 T^{2} \))(\( 1 + 453 T + 205379 T^{2} \))(\( 1 + 827 T + 139925 T^{2} + 42419807 T^{3} + 50592146204 T^{4} + 8712137541853 T^{5} + 5902111169716925 T^{6} + 7164297542027634553 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 + 128 T - 87119 T^{2} - 50674984 T^{3} + 39669673335 T^{4} + 496570131448 T^{5} - 4049042036547885 T^{6} - 22790010341236160 T^{7} + \)\(27\!\cdots\!36\)\( T^{8} - \)\(46\!\cdots\!40\)\( T^{9} - \)\(17\!\cdots\!85\)\( T^{10} + \)\(43\!\cdots\!72\)\( T^{11} + \)\(70\!\cdots\!35\)\( T^{12} - \)\(18\!\cdots\!16\)\( T^{13} - \)\(65\!\cdots\!99\)\( T^{14} + \)\(19\!\cdots\!52\)\( T^{15} + \)\(31\!\cdots\!61\)\( T^{16} \))
$61$ (\( 1 + 668 T + 226981 T^{2} \))(\( 1 - 190 T + 226981 T^{2} \))(\( 1 - 20 T + 226981 T^{2} \))(\( 1 - 1335 T + 583559 T^{2} - 108492345 T^{3} + 24155367616 T^{4} - 24625700960445 T^{5} + 30065178141730799 T^{6} - 15611685033933578235 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 7295748837377 T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} + \)\(68\!\cdots\!36\)\( T^{8} + \)\(77\!\cdots\!00\)\( T^{9} - \)\(68\!\cdots\!15\)\( T^{10} + \)\(85\!\cdots\!57\)\( T^{11} + \)\(59\!\cdots\!98\)\( T^{12} - \)\(68\!\cdots\!18\)\( T^{13} - \)\(83\!\cdots\!38\)\( T^{14} + \)\(19\!\cdots\!37\)\( T^{15} + \)\(70\!\cdots\!41\)\( T^{16} \))
$67$ (\( 1 - 439 T + 300763 T^{2} \))(\( 1 + 12 T + 300763 T^{2} \))(\( 1 + 97 T + 300763 T^{2} \))(\( ( 1 + 44 T + 585190 T^{2} + 13233572 T^{3} + 90458382169 T^{4} )^{2} \))(\( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))
$71$ (\( 1 + 1113 T + 357911 T^{2} \))(\( 1 + 112 T + 357911 T^{2} \))(\( 1 + 465 T + 357911 T^{2} \))(\( 1 + 1665 T + 687799 T^{2} - 442234575 T^{3} - 563174800784 T^{4} - 158280618972825 T^{5} + 88107247180579879 T^{6} + 76337753696217636615 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( 1 - 115 T - 138428 T^{2} + 42960000 T^{3} + 77925723838 T^{4} - 71653925274035 T^{5} + 12191505393571699 T^{6} + 3285417296443548450 T^{7} + \)\(11\!\cdots\!80\)\( T^{8} + \)\(11\!\cdots\!50\)\( T^{9} + \)\(15\!\cdots\!79\)\( T^{10} - \)\(32\!\cdots\!85\)\( T^{11} + \)\(12\!\cdots\!58\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{13} - \)\(29\!\cdots\!08\)\( T^{14} - \)\(86\!\cdots\!65\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} \))
$73$ (\( 1 + 72 T + 389017 T^{2} \))(\( 1 + 6 T + 389017 T^{2} \))(\( 1 - 848 T + 389017 T^{2} \))(\( 1 + 153 T - 378613 T^{2} - 19574565 T^{3} + 149317325056 T^{4} - 7614838552605 T^{5} - 57297105417957157 T^{6} + 9007352766364990689 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 + 1487 T + 142728 T^{2} - 1045087272 T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} - \)\(19\!\cdots\!28\)\( T^{8} - \)\(26\!\cdots\!28\)\( T^{9} + \)\(59\!\cdots\!71\)\( T^{10} + \)\(21\!\cdots\!89\)\( T^{11} - \)\(13\!\cdots\!28\)\( T^{12} - \)\(93\!\cdots\!04\)\( T^{13} + \)\(49\!\cdots\!32\)\( T^{14} + \)\(20\!\cdots\!51\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \))
$79$ (\( 1 + 70 T + 493039 T^{2} \))(\( 1 - 304 T + 493039 T^{2} \))(\( 1 + 742 T + 493039 T^{2} \))(\( 1 - 677 T + 1031235 T^{2} - 490717477 T^{3} + 653512317224 T^{4} - 241942854142603 T^{5} + 250680292194198435 T^{6} - 81139530480232601963 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( 1 - 71 T - 500222 T^{2} - 73013624 T^{3} + 350790961998 T^{4} + 97550067497959 T^{5} - 161731343227399015 T^{6} + 8534351363848547400 T^{7} + \)\(59\!\cdots\!76\)\( T^{8} + \)\(42\!\cdots\!00\)\( T^{9} - \)\(39\!\cdots\!15\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{11} + \)\(20\!\cdots\!18\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{13} - \)\(71\!\cdots\!42\)\( T^{14} - \)\(50\!\cdots\!09\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \))
$83$ (\( 1 - 358 T + 571787 T^{2} \))(\( 1 - 820 T + 571787 T^{2} \))(\( 1 - 438 T + 571787 T^{2} \))(\( 1 - 887 T + 364607 T^{2} - 628939715 T^{3} + 778142659496 T^{4} - 359619552820705 T^{5} + 119204748712950983 T^{6} - \)\(16\!\cdots\!61\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 1942 T + 875395 T^{2} + 857035884 T^{3} - 790010610449 T^{4} - 618864428088728 T^{5} + 845503237658086801 T^{6} + \)\(17\!\cdots\!30\)\( T^{7} - \)\(59\!\cdots\!96\)\( T^{8} + \)\(10\!\cdots\!10\)\( T^{9} + \)\(27\!\cdots\!69\)\( T^{10} - \)\(11\!\cdots\!84\)\( T^{11} - \)\(84\!\cdots\!89\)\( T^{12} + \)\(52\!\cdots\!88\)\( T^{13} + \)\(30\!\cdots\!55\)\( T^{14} - \)\(38\!\cdots\!86\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$ (\( 1 - 895 T + 704969 T^{2} \))(\( 1 - 202 T + 704969 T^{2} \))(\( 1 + 273 T + 704969 T^{2} \))(\( ( 1 - 864 T + 492062 T^{2} - 609093216 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1349696810654843 T^{5} + 1196181784307576095 T^{6} + \)\(38\!\cdots\!09\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$ (\( 1 - 409 T + 912673 T^{2} \))(\( 1 + 1406 T + 912673 T^{2} \))(\( 1 - 761 T + 912673 T^{2} \))(\( 1 - 2019 T + 872763 T^{2} + 94786015 T^{3} + 88862307216 T^{4} + 86508636668095 T^{5} + 726987145937848827 T^{6} - \)\(15\!\cdots\!23\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} + \)\(96\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!64\)\( T^{8} + \)\(88\!\cdots\!76\)\( T^{9} - \)\(12\!\cdots\!25\)\( T^{10} - \)\(37\!\cdots\!04\)\( T^{11} + \)\(10\!\cdots\!91\)\( T^{12} + \)\(60\!\cdots\!52\)\( T^{13} + \)\(59\!\cdots\!53\)\( T^{14} + \)\(27\!\cdots\!16\)\( T^{15} + \)\(48\!\cdots\!81\)\( T^{16} \))
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