Properties

Label 43.3
Level 43
Weight 3
Dimension 133
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 462
Trace bound 1

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Defining parameters

Level: \( N \) = \( 43\( 43 \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(462\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(43))\).

Total New Old
Modular forms 175 175 0
Cusp forms 133 133 0
Eisenstein series 42 42 0

Trace form

\( 133q - 21q^{2} - 21q^{3} - 21q^{4} - 21q^{5} - 21q^{6} - 21q^{7} - 21q^{8} - 21q^{9} + O(q^{10}) \) \( 133q - 21q^{2} - 21q^{3} - 21q^{4} - 21q^{5} - 21q^{6} - 21q^{7} - 21q^{8} - 21q^{9} - 21q^{10} - 21q^{11} - 21q^{12} - 21q^{13} - 21q^{14} - 21q^{15} - 21q^{16} - 21q^{17} - 21q^{18} - 21q^{19} - 21q^{20} - 21q^{21} - 21q^{22} - 21q^{23} - 21q^{24} - 21q^{25} - 21q^{26} - 21q^{27} - 21q^{28} - 21q^{29} - 21q^{30} + 56q^{31} + 399q^{32} + 357q^{33} + 483q^{34} + 273q^{35} + 651q^{36} + 189q^{37} + 315q^{38} + 126q^{39} + 315q^{40} - 126q^{43} - 210q^{44} - 336q^{45} - 357q^{46} - 126q^{47} - 1029q^{48} - 364q^{49} - 693q^{50} - 399q^{51} - 1141q^{52} - 483q^{53} - 777q^{54} - 567q^{55} - 609q^{56} - 126q^{57} - 21q^{58} - 21q^{59} - 21q^{60} - 21q^{61} - 21q^{62} - 21q^{63} - 21q^{64} - 21q^{65} - 21q^{66} - 21q^{67} - 21q^{68} + 651q^{69} + 1239q^{70} + 567q^{71} + 2709q^{72} + 483q^{73} + 1596q^{74} + 1449q^{75} + 1302q^{76} + 987q^{77} + 1890q^{78} + 483q^{79} + 819q^{80} + 651q^{81} + 609q^{82} + 147q^{83} + 441q^{84} - 189q^{86} - 462q^{87} - 609q^{88} - 357q^{89} - 1911q^{90} - 525q^{91} - 1491q^{92} - 1365q^{93} - 1533q^{94} - 861q^{95} - 3024q^{96} - 1533q^{97} - 1932q^{98} - 2079q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(43))\)

We only show spaces with odd 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
43.3.b \(\chi_{43}(42, \cdot)\) 43.3.b.a 1 1
43.3.b.b 6
43.3.d \(\chi_{43}(7, \cdot)\) 43.3.d.a 12 2
43.3.f \(\chi_{43}(2, \cdot)\) 43.3.f.a 42 6
43.3.h \(\chi_{43}(3, \cdot)\) 43.3.h.a 72 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T )( 1 + 2 T ) \))(\( 1 - 4 T^{2} + 41 T^{4} - 114 T^{6} + 656 T^{8} - 1024 T^{10} + 4096 T^{12} \))(\( 1 - 11 T^{2} + 59 T^{4} - 178 T^{6} + 635 T^{8} - 4155 T^{10} + 21681 T^{12} - 66480 T^{14} + 162560 T^{16} - 729088 T^{18} + 3866624 T^{20} - 11534336 T^{22} + 16777216 T^{24} \))
$3$ (\( ( 1 - 3 T )( 1 + 3 T ) \))(\( 1 - 9 T^{2} + 26 T^{4} - 254 T^{6} + 2106 T^{8} - 59049 T^{10} + 531441 T^{12} \))(\( 1 + 33 T^{2} + 544 T^{4} - 3 T^{5} + 6224 T^{6} + 2205 T^{7} + 59278 T^{8} + 74400 T^{9} + 537073 T^{10} + 1154382 T^{11} + 4828798 T^{12} + 10389438 T^{13} + 43502913 T^{14} + 54237600 T^{15} + 388922958 T^{16} + 130203045 T^{17} + 3307688784 T^{18} - 14348907 T^{19} + 23417416224 T^{20} + 115063885233 T^{22} + 282429536481 T^{24} \))
$5$ (\( ( 1 - 5 T )( 1 + 5 T ) \))(\( 1 - 33 T^{2} + 1538 T^{4} - 41006 T^{6} + 961250 T^{8} - 12890625 T^{10} + 244140625 T^{12} \))(\( 1 + 3 T + 90 T^{2} + 261 T^{3} + 3868 T^{4} + 16263 T^{5} + 135026 T^{6} + 750009 T^{7} + 4440152 T^{8} + 24002067 T^{9} + 135845122 T^{10} + 636745773 T^{11} + 3708044426 T^{12} + 15918644325 T^{13} + 84903201250 T^{14} + 375032296875 T^{15} + 1734434375000 T^{16} + 7324306640625 T^{17} + 32965332031250 T^{18} + 99261474609375 T^{19} + 590209960937500 T^{20} + 995635986328125 T^{21} + 8583068847656250 T^{22} + 7152557373046875 T^{23} + 59604644775390625 T^{24} \))
$7$ (\( ( 1 - 7 T )( 1 + 7 T ) \))(\( 1 - 144 T^{2} + 11511 T^{4} - 668464 T^{6} + 27637911 T^{8} - 830131344 T^{10} + 13841287201 T^{12} \))(\( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 29841 T^{4} - 211941 T^{5} + 2317384 T^{6} - 15083235 T^{7} + 123904335 T^{8} - 778414590 T^{9} + 4927316019 T^{10} - 34388074575 T^{11} + 204998231790 T^{12} - 1685015654175 T^{13} + 11830485761619 T^{14} - 91579698098910 T^{15} + 714283834312335 T^{16} - 4260640562350515 T^{17} + 32075577499002184 T^{18} - 143743276282689909 T^{19} + 991703881127463441 T^{20} - 3077701700050748610 T^{21} + 18910767112534044237 T^{22} - 35188389437246892441 T^{23} + \)\(19\!\cdots\!01\)\( T^{24} \))
$11$ (\( 1 + 21 T + 121 T^{2} \))(\( ( 1 - 19 T + 411 T^{2} - 4354 T^{3} + 49731 T^{4} - 278179 T^{5} + 1771561 T^{6} )^{2} \))(\( ( 1 - 14 T + 430 T^{2} - 6047 T^{3} + 101321 T^{4} - 1177595 T^{5} + 15292736 T^{6} - 142488995 T^{7} + 1483440761 T^{8} - 10712629367 T^{9} + 92174318830 T^{10} - 363123944414 T^{11} + 3138428376721 T^{12} )^{2} \))
$13$ (\( 1 + 17 T + 169 T^{2} \))(\( ( 1 - 15 T + 257 T^{2} - 1570 T^{3} + 43433 T^{4} - 428415 T^{5} + 4826809 T^{6} )^{2} \))(\( 1 - 24 T - 211 T^{2} + 7466 T^{3} + 36644 T^{4} - 1139421 T^{5} - 11023890 T^{6} + 164450517 T^{7} + 1885659940 T^{8} - 21544092258 T^{9} - 166029500453 T^{10} + 1118804167120 T^{11} + 20761074178322 T^{12} + 189077904243280 T^{13} - 4741968562438133 T^{14} - 103989218407744722 T^{15} + 1538190742417016740 T^{16} + 22670900257408335933 T^{17} - \)\(25\!\cdots\!90\)\( T^{18} - \)\(44\!\cdots\!69\)\( T^{19} + \)\(24\!\cdots\!04\)\( T^{20} + \)\(83\!\cdots\!14\)\( T^{21} - \)\(40\!\cdots\!11\)\( T^{22} - \)\(77\!\cdots\!56\)\( T^{23} + \)\(54\!\cdots\!61\)\( T^{24} \))
$17$ (\( 1 + 9 T + 289 T^{2} \))(\( ( 1 + 10 T + 767 T^{2} + 5655 T^{3} + 221663 T^{4} + 835210 T^{5} + 24137569 T^{6} )^{2} \))(\( 1 + 7 T - 910 T^{2} - 8453 T^{3} + 343548 T^{4} + 3893616 T^{5} - 110794242 T^{6} - 1195653416 T^{7} + 42885403526 T^{8} + 359068799653 T^{9} - 11658350660948 T^{10} - 53405218008175 T^{11} + 2646840415582786 T^{12} - 15434108004362575 T^{13} - 973717105553037908 T^{14} + 8667047927371463557 T^{15} + \)\(29\!\cdots\!66\)\( T^{16} - \)\(24\!\cdots\!84\)\( T^{17} - \)\(64\!\cdots\!62\)\( T^{18} + \)\(65\!\cdots\!64\)\( T^{19} + \)\(16\!\cdots\!88\)\( T^{20} - \)\(11\!\cdots\!77\)\( T^{21} - \)\(36\!\cdots\!10\)\( T^{22} + \)\(82\!\cdots\!23\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \))
$19$ (\( ( 1 - 19 T )( 1 + 19 T ) \))(\( 1 - 691 T^{2} + 378040 T^{4} - 133433020 T^{6} + 49266550840 T^{8} - 11735642061331 T^{10} + 2213314919066161 T^{12} \))(\( 1 - 66 T + 2608 T^{2} - 76296 T^{3} + 1836036 T^{4} - 38733966 T^{5} + 734123972 T^{6} - 14185527162 T^{7} + 292458472820 T^{8} - 6360444714264 T^{9} + 137883926411448 T^{10} - 2871978163512918 T^{11} + 56971930639346710 T^{12} - 1036784117028163398 T^{13} + 17969171173866314808 T^{14} - \)\(29\!\cdots\!84\)\( T^{15} + \)\(49\!\cdots\!20\)\( T^{16} - \)\(86\!\cdots\!62\)\( T^{17} + \)\(16\!\cdots\!92\)\( T^{18} - \)\(30\!\cdots\!86\)\( T^{19} + \)\(52\!\cdots\!16\)\( T^{20} - \)\(79\!\cdots\!36\)\( T^{21} + \)\(98\!\cdots\!08\)\( T^{22} - \)\(89\!\cdots\!26\)\( T^{23} + \)\(48\!\cdots\!21\)\( T^{24} \))
$23$ (\( 1 - 3 T + 529 T^{2} \))(\( ( 1 + 40 T + 1387 T^{2} + 28195 T^{3} + 733723 T^{4} + 11193640 T^{5} + 148035889 T^{6} )^{2} \))(\( 1 + 16 T - 995 T^{2} - 970 T^{3} + 448208 T^{4} - 9378727 T^{5} + 41040474 T^{6} + 4400241755 T^{7} - 92523779140 T^{8} + 750281486226 T^{9} - 30397457804233 T^{10} - 745325850079516 T^{11} + 45354493455383514 T^{12} - 394277374692063964 T^{13} - 8506454989394366953 T^{14} + \)\(11\!\cdots\!14\)\( T^{15} - \)\(72\!\cdots\!40\)\( T^{16} + \)\(18\!\cdots\!95\)\( T^{17} + \)\(89\!\cdots\!54\)\( T^{18} - \)\(10\!\cdots\!43\)\( T^{19} + \)\(27\!\cdots\!88\)\( T^{20} - \)\(31\!\cdots\!30\)\( T^{21} - \)\(17\!\cdots\!95\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( 1 - 1621 T^{2} + 1946890 T^{4} - 2007466870 T^{6} + 1376998306090 T^{8} - 810899435409781 T^{10} + 353814783205469041 T^{12} \))(\( 1 + 111 T + 7375 T^{2} + 362748 T^{3} + 14016537 T^{4} + 490234341 T^{5} + 16991400860 T^{6} + 618781423887 T^{7} + 22589952693419 T^{8} + 755597311945764 T^{9} + 22852874993878941 T^{10} + 643686573544215837 T^{11} + 18176844555197488294 T^{12} + \)\(54\!\cdots\!17\)\( T^{13} + \)\(16\!\cdots\!21\)\( T^{14} + \)\(44\!\cdots\!44\)\( T^{15} + \)\(11\!\cdots\!59\)\( T^{16} + \)\(26\!\cdots\!87\)\( T^{17} + \)\(60\!\cdots\!60\)\( T^{18} + \)\(14\!\cdots\!21\)\( T^{19} + \)\(35\!\cdots\!77\)\( T^{20} + \)\(76\!\cdots\!28\)\( T^{21} + \)\(13\!\cdots\!75\)\( T^{22} + \)\(16\!\cdots\!51\)\( T^{23} + \)\(12\!\cdots\!81\)\( T^{24} \))
$31$ (\( 1 - 19 T + 961 T^{2} \))(\( ( 1 + 56 T + 2591 T^{2} + 96591 T^{3} + 2489951 T^{4} + 51717176 T^{5} + 887503681 T^{6} )^{2} \))(\( 1 + 29 T - 2854 T^{2} - 122293 T^{3} + 3710916 T^{4} + 236687409 T^{5} - 1692705294 T^{6} - 282476528401 T^{7} - 2178393607912 T^{8} + 217078922116565 T^{9} + 5420523596702458 T^{10} - 78231274595814749 T^{11} - 6486151840040562614 T^{12} - 75180254886577973789 T^{13} + \)\(50\!\cdots\!18\)\( T^{14} + \)\(19\!\cdots\!65\)\( T^{15} - \)\(18\!\cdots\!92\)\( T^{16} - \)\(23\!\cdots\!01\)\( T^{17} - \)\(13\!\cdots\!34\)\( T^{18} + \)\(17\!\cdots\!89\)\( T^{19} + \)\(26\!\cdots\!96\)\( T^{20} - \)\(85\!\cdots\!13\)\( T^{21} - \)\(19\!\cdots\!54\)\( T^{22} + \)\(18\!\cdots\!69\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \))
$37$ (\( ( 1 - 37 T )( 1 + 37 T ) \))(\( 1 - 5259 T^{2} + 14237096 T^{4} - 24083763884 T^{6} + 26682610076456 T^{8} - 18472129448170539 T^{10} + 6582952005840035281 T^{12} \))(\( 1 - 120 T + 13269 T^{2} - 1016280 T^{3} + 71816674 T^{4} - 4159036083 T^{5} + 226460485550 T^{6} - 10769149077249 T^{7} + 489905206154870 T^{8} - 20187174180639120 T^{9} + 817471660147259059 T^{10} - 30887683233450226884 T^{11} + \)\(11\!\cdots\!42\)\( T^{12} - \)\(42\!\cdots\!96\)\( T^{13} + \)\(15\!\cdots\!99\)\( T^{14} - \)\(51\!\cdots\!80\)\( T^{15} + \)\(17\!\cdots\!70\)\( T^{16} - \)\(51\!\cdots\!01\)\( T^{17} + \)\(14\!\cdots\!50\)\( T^{18} - \)\(37\!\cdots\!87\)\( T^{19} + \)\(88\!\cdots\!34\)\( T^{20} - \)\(17\!\cdots\!20\)\( T^{21} + \)\(30\!\cdots\!69\)\( T^{22} - \)\(37\!\cdots\!80\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \))
$41$ (\( 1 - 39 T + 1681 T^{2} \))(\( ( 1 + 86 T + 6451 T^{2} + 292121 T^{3} + 10844131 T^{4} + 243015446 T^{5} + 4750104241 T^{6} )^{2} \))(\( ( 1 - 47 T + 6771 T^{2} - 336496 T^{3} + 24096797 T^{4} - 992715921 T^{5} + 52240608766 T^{6} - 1668755463201 T^{7} + 68091789187517 T^{8} - 1598391076679536 T^{9} + 54065928726378291 T^{10} - 630864987577162847 T^{11} + 22563490300366186081 T^{12} )^{2} \))
$43$ (\( 1 + 43 T \))(\( 1 - 10 T + 147 T^{2} + 135020 T^{3} + 271803 T^{4} - 34188010 T^{5} + 6321363049 T^{6} \))(\( 1 - 5 T + 235 T^{2} + 31132 T^{3} + 5312177 T^{4} + 54939337 T^{5} + 786324230 T^{6} + 101582834113 T^{7} + 18161276039777 T^{8} + 196796674441468 T^{9} + 2746727065236235 T^{10} - 108057411566421245 T^{11} + 39959630797262576401 T^{12} \))
$47$ (\( 1 + 78 T + 2209 T^{2} \))(\( ( 1 - 15 T + 5052 T^{2} - 79770 T^{3} + 11159868 T^{4} - 73195215 T^{5} + 10779215329 T^{6} )^{2} \))(\( ( 1 + 9 T + 7161 T^{2} - 54567 T^{3} + 24318411 T^{4} - 403881570 T^{5} + 60415020470 T^{6} - 892174388130 T^{7} + 118666088106891 T^{8} - 588189442857543 T^{9} + 170512623784870521 T^{10} + 473392190122470441 T^{11} + \)\(11\!\cdots\!41\)\( T^{12} )^{2} \))
$53$ (\( 1 - 63 T + 2809 T^{2} \))(\( ( 1 + 55 T + 4677 T^{2} + 168490 T^{3} + 13137693 T^{4} + 433976455 T^{5} + 22164361129 T^{6} )^{2} \))(\( 1 + 58 T - 10011 T^{2} - 489700 T^{3} + 62208382 T^{4} + 2298288439 T^{5} - 284627569714 T^{6} - 7541357242599 T^{7} + 1029571029628954 T^{8} + 16839043965386300 T^{9} - 3214742139372951389 T^{10} - 18444106212599549754 T^{11} + \)\(92\!\cdots\!42\)\( T^{12} - \)\(51\!\cdots\!86\)\( T^{13} - \)\(25\!\cdots\!09\)\( T^{14} + \)\(37\!\cdots\!00\)\( T^{15} + \)\(64\!\cdots\!94\)\( T^{16} - \)\(13\!\cdots\!51\)\( T^{17} - \)\(13\!\cdots\!74\)\( T^{18} + \)\(31\!\cdots\!91\)\( T^{19} + \)\(24\!\cdots\!22\)\( T^{20} - \)\(53\!\cdots\!00\)\( T^{21} - \)\(30\!\cdots\!11\)\( T^{22} + \)\(49\!\cdots\!22\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \))
$59$ (\( 1 + 54 T + 3481 T^{2} \))(\( ( 1 + 6 T + 4271 T^{2} + 147196 T^{3} + 14867351 T^{4} + 72704166 T^{5} + 42180533641 T^{6} )^{2} \))(\( ( 1 - 168 T + 29086 T^{2} - 3029785 T^{3} + 294619625 T^{4} - 21440752319 T^{5} + 1430753245424 T^{6} - 74635258822439 T^{7} + 3570012353809625 T^{8} - 127797948117497185 T^{9} + 4270710108159280606 T^{10} - 85867614554507755368 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \))
$61$ (\( ( 1 - 61 T )( 1 + 61 T ) \))(\( 1 - 6176 T^{2} + 53251015 T^{4} - 178029474320 T^{6} + 737305086778615 T^{8} - 1183984365071207456 T^{10} + \)\(26\!\cdots\!21\)\( T^{12} \))(\( 1 - 204 T + 33131 T^{2} - 3928836 T^{3} + 396166698 T^{4} - 36077954385 T^{5} + 3074787876040 T^{6} - 249044419787289 T^{7} + 19373960824471256 T^{8} - 1402697797539370524 T^{9} + 96292769362924730019 T^{10} - \)\(62\!\cdots\!62\)\( T^{11} + \)\(38\!\cdots\!74\)\( T^{12} - \)\(23\!\cdots\!02\)\( T^{13} + \)\(13\!\cdots\!79\)\( T^{14} - \)\(72\!\cdots\!64\)\( T^{15} + \)\(37\!\cdots\!36\)\( T^{16} - \)\(17\!\cdots\!89\)\( T^{17} + \)\(81\!\cdots\!40\)\( T^{18} - \)\(35\!\cdots\!85\)\( T^{19} + \)\(14\!\cdots\!78\)\( T^{20} - \)\(53\!\cdots\!16\)\( T^{21} + \)\(16\!\cdots\!31\)\( T^{22} - \)\(38\!\cdots\!84\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \))
$67$ (\( 1 - 91 T + 4489 T^{2} \))(\( ( 1 + 35 T + 9017 T^{2} + 350730 T^{3} + 40477313 T^{4} + 705289235 T^{5} + 90458382169 T^{6} )^{2} \))(\( 1 - 115 T - 7960 T^{2} + 1601411 T^{3} + 12878544 T^{4} - 11631792126 T^{5} + 264032039310 T^{6} + 51372624149438 T^{7} - 2607440286917038 T^{8} - 136422464608902859 T^{9} + 13476394220383696006 T^{10} + \)\(18\!\cdots\!27\)\( T^{11} - \)\(61\!\cdots\!22\)\( T^{12} + \)\(84\!\cdots\!03\)\( T^{13} + \)\(27\!\cdots\!26\)\( T^{14} - \)\(12\!\cdots\!71\)\( T^{15} - \)\(10\!\cdots\!58\)\( T^{16} + \)\(93\!\cdots\!62\)\( T^{17} + \)\(21\!\cdots\!10\)\( T^{18} - \)\(42\!\cdots\!54\)\( T^{19} + \)\(21\!\cdots\!64\)\( T^{20} + \)\(11\!\cdots\!99\)\( T^{21} - \)\(26\!\cdots\!60\)\( T^{22} - \)\(17\!\cdots\!35\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( 1 - 22246 T^{2} + 239223215 T^{4} - 1523736510420 T^{6} + 6079064027374415 T^{8} - 14365433056093199206 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} \))(\( 1 + 66 T + 8383 T^{2} + 457446 T^{3} + 22253522 T^{4} + 2686038483 T^{5} - 142137724894 T^{6} + 6765531085149 T^{7} - 469326810763006 T^{8} + 20521634467403010 T^{9} + 5562036689541880365 T^{10} + \)\(10\!\cdots\!58\)\( T^{11} + \)\(64\!\cdots\!18\)\( T^{12} + \)\(54\!\cdots\!78\)\( T^{13} + \)\(14\!\cdots\!65\)\( T^{14} + \)\(26\!\cdots\!10\)\( T^{15} - \)\(30\!\cdots\!66\)\( T^{16} + \)\(22\!\cdots\!49\)\( T^{17} - \)\(23\!\cdots\!54\)\( T^{18} + \)\(22\!\cdots\!23\)\( T^{19} + \)\(92\!\cdots\!62\)\( T^{20} + \)\(96\!\cdots\!06\)\( T^{21} + \)\(88\!\cdots\!83\)\( T^{22} + \)\(35\!\cdots\!06\)\( T^{23} + \)\(26\!\cdots\!81\)\( T^{24} \))
$73$ (\( ( 1 - 73 T )( 1 + 73 T ) \))(\( 1 - 24604 T^{2} + 277197791 T^{4} - 1859020745064 T^{6} + 7871929673485631 T^{8} - 19842144100961968924 T^{10} + \)\(22\!\cdots\!21\)\( T^{12} \))(\( 1 - 249 T + 47836 T^{2} - 6765081 T^{3} + 836510852 T^{4} - 87998072508 T^{5} + 8327492192822 T^{6} - 712665936231156 T^{7} + 56395869792302342 T^{8} - 4210361193162399351 T^{9} + \)\(30\!\cdots\!94\)\( T^{10} - \)\(21\!\cdots\!91\)\( T^{11} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(11\!\cdots\!39\)\( T^{13} + \)\(85\!\cdots\!54\)\( T^{14} - \)\(63\!\cdots\!39\)\( T^{15} + \)\(45\!\cdots\!02\)\( T^{16} - \)\(30\!\cdots\!44\)\( T^{17} + \)\(19\!\cdots\!62\)\( T^{18} - \)\(10\!\cdots\!72\)\( T^{19} + \)\(54\!\cdots\!72\)\( T^{20} - \)\(23\!\cdots\!89\)\( T^{21} + \)\(88\!\cdots\!36\)\( T^{22} - \)\(24\!\cdots\!21\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \))
$79$ (\( 1 + 14 T + 6241 T^{2} \))(\( ( 1 - 89 T + 20896 T^{2} - 1122134 T^{3} + 130411936 T^{4} - 3466557209 T^{5} + 243087455521 T^{6} )^{2} \))(\( 1 - 236 T + 6211 T^{2} + 1816934 T^{3} + 15618410 T^{4} - 18039764431 T^{5} - 862854134850 T^{6} + 144012352794143 T^{7} + 12282532151020286 T^{8} - 938070438536680854 T^{9} - 93564616453312180819 T^{10} + \)\(13\!\cdots\!72\)\( T^{11} + \)\(78\!\cdots\!82\)\( T^{12} + \)\(84\!\cdots\!52\)\( T^{13} - \)\(36\!\cdots\!39\)\( T^{14} - \)\(22\!\cdots\!34\)\( T^{15} + \)\(18\!\cdots\!46\)\( T^{16} + \)\(13\!\cdots\!43\)\( T^{17} - \)\(50\!\cdots\!50\)\( T^{18} - \)\(66\!\cdots\!11\)\( T^{19} + \)\(35\!\cdots\!10\)\( T^{20} + \)\(26\!\cdots\!74\)\( T^{21} + \)\(55\!\cdots\!11\)\( T^{22} - \)\(13\!\cdots\!76\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \))
$83$ (\( 1 - 123 T + 6889 T^{2} \))(\( ( 1 - 5 T + 14767 T^{2} - 128390 T^{3} + 101729863 T^{4} - 237291605 T^{5} + 326940373369 T^{6} )^{2} \))(\( 1 + 4 T - 21815 T^{2} + 1032278 T^{3} + 256215014 T^{4} - 22099127125 T^{5} - 1301303064984 T^{6} + 269522473356119 T^{7} - 2312375630136352 T^{8} - 1824448248468292326 T^{9} + \)\(11\!\cdots\!01\)\( T^{10} + \)\(55\!\cdots\!10\)\( T^{11} - \)\(10\!\cdots\!42\)\( T^{12} + \)\(37\!\cdots\!90\)\( T^{13} + \)\(54\!\cdots\!21\)\( T^{14} - \)\(59\!\cdots\!94\)\( T^{15} - \)\(52\!\cdots\!32\)\( T^{16} + \)\(41\!\cdots\!31\)\( T^{17} - \)\(13\!\cdots\!24\)\( T^{18} - \)\(16\!\cdots\!25\)\( T^{19} + \)\(12\!\cdots\!34\)\( T^{20} + \)\(36\!\cdots\!02\)\( T^{21} - \)\(52\!\cdots\!15\)\( T^{22} + \)\(66\!\cdots\!56\)\( T^{23} + \)\(11\!\cdots\!21\)\( T^{24} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( 1 - 38876 T^{2} + 668902015 T^{4} - 6712320081320 T^{6} + 41968411430515615 T^{8} - \)\(15\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!21\)\( T^{12} \))(\( 1 + 45 T + 19568 T^{2} + 850185 T^{3} + 135178944 T^{4} + 15768566256 T^{5} + 877991398306 T^{6} + 230067075716784 T^{7} + 11948725877253974 T^{8} + 1772692905496764327 T^{9} + \)\(15\!\cdots\!42\)\( T^{10} + \)\(84\!\cdots\!51\)\( T^{11} + \)\(14\!\cdots\!90\)\( T^{12} + \)\(67\!\cdots\!71\)\( T^{13} + \)\(96\!\cdots\!22\)\( T^{14} + \)\(88\!\cdots\!47\)\( T^{15} + \)\(47\!\cdots\!94\)\( T^{16} + \)\(71\!\cdots\!84\)\( T^{17} + \)\(21\!\cdots\!26\)\( T^{18} + \)\(30\!\cdots\!96\)\( T^{19} + \)\(20\!\cdots\!84\)\( T^{20} + \)\(10\!\cdots\!85\)\( T^{21} + \)\(19\!\cdots\!68\)\( T^{22} + \)\(34\!\cdots\!45\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \))
$97$ (\( 1 + 193 T + 9409 T^{2} \))(\( ( 1 + 190 T + 26827 T^{2} + 2529295 T^{3} + 252415243 T^{4} + 16820563390 T^{5} + 832972004929 T^{6} )^{2} \))(\( ( 1 + 185 T + 45810 T^{2} + 6039385 T^{3} + 922201559 T^{4} + 96415612878 T^{5} + 11078943083116 T^{6} + 907174501569102 T^{7} + 81641840955349079 T^{8} + 5030638631988128665 T^{9} + \)\(35\!\cdots\!10\)\( T^{10} + \)\(13\!\cdots\!65\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))
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