Properties

Label 21.4
Level 21
Weight 4
Dimension 30
Nonzero newspaces 4
Newform subspaces 8
Sturm bound 128
Trace bound 1

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(21))\).

Total New Old
Modular forms 60 42 18
Cusp forms 36 30 6
Eisenstein series 24 12 12

Trace form

\( 30q + 3q^{3} - 6q^{4} - 24q^{5} - 36q^{6} - 54q^{7} + 54q^{8} + 39q^{9} + O(q^{10}) \) \( 30q + 3q^{3} - 6q^{4} - 24q^{5} - 36q^{6} - 54q^{7} + 54q^{8} + 39q^{9} + 48q^{10} - 96q^{12} - 84q^{13} - 312q^{14} - 198q^{15} - 126q^{16} - 12q^{17} + 270q^{18} + 558q^{19} + 1044q^{20} + 585q^{21} + 432q^{22} - 168q^{23} + 144q^{24} - 426q^{25} - 294q^{26} - 108q^{27} - 1110q^{28} - 648q^{29} - 1530q^{30} - 750q^{31} - 1218q^{32} - 801q^{33} - 276q^{34} + 84q^{35} + 258q^{36} + 1422q^{37} + 1710q^{38} + 1212q^{39} + 1416q^{40} + 192q^{41} + 2322q^{42} + 1152q^{43} + 2184q^{44} + 2151q^{45} + 708q^{46} + 792q^{47} - 1056q^{48} - 2346q^{49} - 3258q^{50} - 3069q^{51} - 4680q^{52} - 2100q^{53} - 4158q^{54} - 1020q^{55} + 690q^{56} - 150q^{57} + 3528q^{58} - 336q^{59} + 3060q^{60} + 3138q^{61} + 1812q^{62} + 2283q^{63} + 4278q^{64} + 504q^{65} + 882q^{66} + 918q^{67} - 1536q^{68} - 1656q^{69} - 3504q^{70} - 492q^{71} - 4050q^{72} - 4326q^{73} - 2730q^{74} - 2304q^{75} - 5544q^{76} - 180q^{77} + 504q^{78} - 510q^{79} + 3096q^{80} + 3987q^{81} + 7488q^{82} + 4200q^{83} + 7956q^{84} + 4668q^{85} + 5166q^{86} + 1656q^{87} - 1812q^{88} - 1752q^{89} - 1080q^{90} + 804q^{91} - 4704q^{92} - 633q^{93} + 504q^{94} + 1344q^{95} - 2700q^{96} + 300q^{97} - 5982q^{98} - 4446q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(21))\)

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
21.4.a \(\chi_{21}(1, \cdot)\) 21.4.a.a 1 1
21.4.a.b 1
21.4.a.c 2
21.4.c \(\chi_{21}(20, \cdot)\) 21.4.c.a 2 1
21.4.c.b 4
21.4.e \(\chi_{21}(4, \cdot)\) 21.4.e.a 2 2
21.4.e.b 6
21.4.g \(\chi_{21}(5, \cdot)\) 21.4.g.a 12 2

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(21))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(21)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 8 T^{2} \))(\( 1 - 4 T + 8 T^{2} \))(\( 1 + 3 T + 4 T^{2} + 24 T^{3} + 64 T^{4} \))(\( ( 1 - 8 T^{2} )^{2} \))(\( ( 1 + T^{2} + 64 T^{4} )^{2} \))(\( 1 - 3 T + T^{2} - 24 T^{3} + 64 T^{4} \))(\( 1 + T + T^{2} - 20 T^{3} - 10 T^{4} + 64 T^{5} + 1060 T^{6} + 512 T^{7} - 640 T^{8} - 10240 T^{9} + 4096 T^{10} + 32768 T^{11} + 262144 T^{12} \))(\( 1 + 17 T^{2} + 83 T^{4} + 222 T^{6} + 1372 T^{8} - 40784 T^{10} - 685760 T^{12} - 2610176 T^{14} + 5619712 T^{16} + 58195968 T^{18} + 1392508928 T^{20} + 18253611008 T^{22} + 68719476736 T^{24} \))
$3$ (\( 1 + 3 T \))(\( 1 + 3 T \))(\( ( 1 - 3 T )^{2} \))(\( 1 + 27 T^{2} \))(\( 1 - 48 T^{2} + 729 T^{4} \))(\( 1 + 3 T + 9 T^{2} \))(\( ( 1 - 3 T + 9 T^{2} )^{3} \))(\( 1 + 3 T + 6 T^{2} + 9 T^{3} - 198 T^{4} - 2565 T^{5} - 36018 T^{6} - 69255 T^{7} - 144342 T^{8} + 177147 T^{9} + 3188646 T^{10} + 43046721 T^{11} + 387420489 T^{12} \))
$5$ (\( 1 + 18 T + 125 T^{2} \))(\( 1 + 4 T + 125 T^{2} \))(\( 1 - 6 T + 202 T^{2} - 750 T^{3} + 15625 T^{4} \))(\( ( 1 + 125 T^{2} )^{2} \))(\( ( 1 + 148 T^{2} + 15625 T^{4} )^{2} \))(\( 1 - 3 T - 116 T^{2} - 375 T^{3} + 15625 T^{4} \))(\( 1 + 11 T - 62 T^{2} - 1015 T^{3} - 6040 T^{4} - 54313 T^{5} + 121696 T^{6} - 6789125 T^{7} - 94375000 T^{8} - 1982421875 T^{9} - 15136718750 T^{10} + 335693359375 T^{11} + 3814697265625 T^{12} \))(\( 1 - 354 T^{2} + 53346 T^{4} - 3913744 T^{6} + 238386078 T^{8} - 66225097710 T^{10} + 12349753291374 T^{12} - 1034767151718750 T^{14} + 58199726074218750 T^{16} - 14929748535156250000 T^{18} + \)\(31\!\cdots\!50\)\( T^{20} - \)\(32\!\cdots\!50\)\( T^{22} + \)\(14\!\cdots\!25\)\( T^{24} \))
$7$ (\( 1 - 7 T \))(\( 1 + 7 T \))(\( ( 1 - 7 T )^{2} \))(\( 1 + 20 T + 343 T^{2} \))(\( ( 1 - 14 T + 343 T^{2} )^{2} \))(\( 1 + 7 T + 343 T^{2} \))(\( 1 + 13 T + 236 T^{2} + 12145 T^{3} + 80948 T^{4} + 1529437 T^{5} + 40353607 T^{6} \))(\( ( 1 + 28 T + 476 T^{2} + 10780 T^{3} + 163268 T^{4} + 3294172 T^{5} + 40353607 T^{6} )^{2} \))
$11$ (\( 1 + 36 T + 1331 T^{2} \))(\( 1 - 62 T + 1331 T^{2} \))(\( 1 + 6 T + 1246 T^{2} + 7986 T^{3} + 1771561 T^{4} \))(\( ( 1 - 1331 T^{2} )^{2} \))(\( ( 1 - 1574 T^{2} + 1771561 T^{4} )^{2} \))(\( 1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4} \))(\( 1 + 35 T - 1400 T^{2} - 113593 T^{3} - 198940 T^{4} + 87110135 T^{5} + 3928586038 T^{6} + 115943589685 T^{7} - 352434345340 T^{8} - 267846352063763 T^{9} - 4393799727409400 T^{10} + 146203685929547785 T^{11} + 5559917313492231481 T^{12} \))(\( 1 + 4742 T^{2} + 10147682 T^{4} + 18336975648 T^{6} + 35136366245806 T^{8} + 55422032379705850 T^{10} + 74318354148055569358 T^{12} + \)\(98\!\cdots\!50\)\( T^{14} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(10\!\cdots\!88\)\( T^{18} + \)\(99\!\cdots\!62\)\( T^{20} + \)\(82\!\cdots\!42\)\( T^{22} + \)\(30\!\cdots\!61\)\( T^{24} \))
$13$ (\( 1 + 34 T + 2197 T^{2} \))(\( 1 + 62 T + 2197 T^{2} \))(\( 1 - 16 T + 2406 T^{2} - 35152 T^{3} + 4826809 T^{4} \))(\( ( 1 - 70 T + 2197 T^{2} )( 1 + 70 T + 2197 T^{2} ) \))(\( ( 1 - 1220 T^{2} + 4826809 T^{4} )^{2} \))(\( ( 1 + 64 T + 2197 T^{2} )^{2} \))(\( ( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 15414152 T^{4} - 299262158 T^{5} + 10604499373 T^{6} )^{2} \))(\( ( 1 - 8847 T^{2} + 36037359 T^{4} - 94069474274 T^{6} + 173945448757431 T^{8} - 206118159078589407 T^{10} + \)\(11\!\cdots\!29\)\( T^{12} )^{2} \))
$17$ (\( 1 - 42 T + 4913 T^{2} \))(\( 1 - 84 T + 4913 T^{2} \))(\( 1 + 6 T + 9778 T^{2} + 29478 T^{3} + 24137569 T^{4} \))(\( ( 1 + 4913 T^{2} )^{2} \))(\( ( 1 + 6154 T^{2} + 24137569 T^{4} )^{2} \))(\( 1 + 84 T + 2143 T^{2} + 412692 T^{3} + 24137569 T^{4} \))(\( 1 + 48 T - 10035 T^{2} - 125232 T^{3} + 74409318 T^{4} - 234420432 T^{5} - 437742983351 T^{6} - 1151707582416 T^{7} + 1796060047467942 T^{8} - 14850996949472304 T^{9} - 5846614150600651635 T^{10} + \)\(13\!\cdots\!64\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \))(\( 1 - 17217 T^{2} + 160729722 T^{4} - 865949822635 T^{6} + 2480649769543932 T^{8} + 2994987661331912811 T^{10} - \)\(48\!\cdots\!48\)\( T^{12} + \)\(72\!\cdots\!59\)\( T^{14} + \)\(14\!\cdots\!52\)\( T^{16} - \)\(12\!\cdots\!15\)\( T^{18} + \)\(54\!\cdots\!62\)\( T^{20} - \)\(14\!\cdots\!33\)\( T^{22} + \)\(19\!\cdots\!81\)\( T^{24} \))
$19$ (\( 1 + 124 T + 6859 T^{2} \))(\( 1 - 100 T + 6859 T^{2} \))(\( 1 - 64 T + 6534 T^{2} - 438976 T^{3} + 47045881 T^{4} \))(\( ( 1 - 56 T + 6859 T^{2} )( 1 + 56 T + 6859 T^{2} ) \))(\( ( 1 - 12368 T^{2} + 47045881 T^{4} )^{2} \))(\( 1 - 16 T - 6603 T^{2} - 109744 T^{3} + 47045881 T^{4} \))(\( 1 - 202 T + 7946 T^{2} - 627636 T^{3} + 247297462 T^{4} - 17185599794 T^{5} + 349471935958 T^{6} - 117876028987046 T^{7} + 11634326968854022 T^{8} - 202530415883220444 T^{9} + 17587000346899715306 T^{10} - \)\(30\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \))(\( ( 1 - 150 T + 30330 T^{2} - 3424500 T^{3} + 468855630 T^{4} - 40538051670 T^{5} + 4025485422214 T^{6} - 278050496404530 T^{7} + 22057726175160030 T^{8} - 1105044021044185500 T^{9} + 67129841495276663130 T^{10} - \)\(22\!\cdots\!50\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \))
$23$ (\( 1 + 12167 T^{2} \))(\( 1 + 42 T + 12167 T^{2} \))(\( 1 - 6 T + 7870 T^{2} - 73002 T^{3} + 148035889 T^{4} \))(\( ( 1 - 12167 T^{2} )^{2} \))(\( ( 1 - 16106 T^{2} + 148035889 T^{4} )^{2} \))(\( 1 - 84 T - 5111 T^{2} - 1022028 T^{3} + 148035889 T^{4} \))(\( 1 + 216 T + 10827 T^{2} + 387864 T^{3} + 53856198 T^{4} - 24653558952 T^{5} - 5413409425505 T^{6} - 299959851768984 T^{7} + 7972650149090022 T^{8} + 698602275885685032 T^{9} + \)\(23\!\cdots\!67\)\( T^{10} + \)\(57\!\cdots\!12\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \))(\( 1 + 58691 T^{2} + 1855192226 T^{4} + 43669490183457 T^{6} + 835010896549582396 T^{8} + \)\(13\!\cdots\!91\)\( T^{10} + \)\(17\!\cdots\!88\)\( T^{12} + \)\(19\!\cdots\!99\)\( T^{14} + \)\(18\!\cdots\!16\)\( T^{16} + \)\(14\!\cdots\!33\)\( T^{18} + \)\(89\!\cdots\!66\)\( T^{20} + \)\(41\!\cdots\!59\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))
$29$ (\( 1 - 102 T + 24389 T^{2} \))(\( 1 + 10 T + 24389 T^{2} \))(\( 1 + 252 T + 56446 T^{2} + 6146028 T^{3} + 594823321 T^{4} \))(\( ( 1 - 24389 T^{2} )^{2} \))(\( ( 1 - 45446 T^{2} + 594823321 T^{4} )^{2} \))(\( ( 1 + 297 T + 24389 T^{2} )^{2} \))(\( ( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 1285178355 T^{4} - 31525636013 T^{5} + 14507145975869 T^{6} )^{2} \))(\( ( 1 - 26333 T^{2} + 912806819 T^{4} - 27528483699758 T^{6} + 542958783509025899 T^{8} - \)\(93\!\cdots\!53\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$31$ (\( 1 + 160 T + 29791 T^{2} \))(\( 1 + 48 T + 29791 T^{2} \))(\( 1 - 40 T - 13890 T^{2} - 1191640 T^{3} + 887503681 T^{4} \))(\( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \))(\( ( 1 + 5314 T^{2} + 887503681 T^{4} )^{2} \))(\( 1 - 253 T + 34218 T^{2} - 7537123 T^{3} + 887503681 T^{4} \))(\( 1 - 95 T - 70347 T^{2} + 3756594 T^{3} + 3398738767 T^{4} - 83374434539 T^{5} - 110906046363338 T^{6} - 2483807779351349 T^{7} + 3016393166469901327 T^{8} + 99322925971043714574 T^{9} - \)\(55\!\cdots\!67\)\( T^{10} - \)\(22\!\cdots\!45\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \))(\( ( 1 + 465 T + 176469 T^{2} + 48543210 T^{3} + 12033403527 T^{4} + 2446560037821 T^{5} + 457872227799046 T^{6} + 72885470086725411 T^{7} + 10679689925170882887 T^{8} + \)\(12\!\cdots\!10\)\( T^{9} + \)\(13\!\cdots\!09\)\( T^{10} + \)\(10\!\cdots\!15\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))
$37$ (\( 1 - 398 T + 50653 T^{2} \))(\( 1 + 246 T + 50653 T^{2} \))(\( 1 + 248 T + 98214 T^{2} + 12561944 T^{3} + 2565726409 T^{4} \))(\( ( 1 + 110 T + 50653 T^{2} )^{2} \))(\( ( 1 - 230 T + 50653 T^{2} )^{4} \))(\( 1 - 316 T + 49203 T^{2} - 16006348 T^{3} + 2565726409 T^{4} \))(\( 1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 34463384910226462 T^{7} + 31286934978284739448 T^{8} - \)\(12\!\cdots\!44\)\( T^{9} - \)\(64\!\cdots\!24\)\( T^{10} + \)\(87\!\cdots\!66\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} \))(\( ( 1 - 382 T - 32782 T^{2} + 5434056 T^{3} + 8807187550 T^{4} - 649676877530 T^{5} - 374210188787594 T^{6} - 32908082877527090 T^{7} + 22596833686051007950 T^{8} + \)\(70\!\cdots\!12\)\( T^{9} - \)\(21\!\cdots\!42\)\( T^{10} - \)\(12\!\cdots\!26\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} )^{2} \))
$41$ (\( 1 + 318 T + 68921 T^{2} \))(\( 1 + 248 T + 68921 T^{2} \))(\( 1 + 450 T + 175642 T^{2} + 31014450 T^{3} + 4750104241 T^{4} \))(\( ( 1 + 68921 T^{2} )^{2} \))(\( ( 1 + 117850 T^{2} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 360 T + 68921 T^{2} )^{2} \))(\( ( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 12955976343 T^{4} - 1159025434804 T^{5} + 327381934393961 T^{6} )^{2} \))(\( ( 1 + 240738 T^{2} + 28558186959 T^{4} + 2299629172416892 T^{6} + \)\(13\!\cdots\!19\)\( T^{8} + \)\(54\!\cdots\!78\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))
$43$ (\( 1 + 268 T + 79507 T^{2} \))(\( 1 - 68 T + 79507 T^{2} \))(\( 1 - 376 T + 161526 T^{2} - 29894632 T^{3} + 6321363049 T^{4} \))(\( ( 1 - 520 T + 79507 T^{2} )^{2} \))(\( ( 1 - 44 T + 79507 T^{2} )^{4} \))(\( ( 1 - 26 T + 79507 T^{2} )^{2} \))(\( ( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 13210724106 T^{4} - 2275690697640 T^{5} + 502592611936843 T^{6} )^{2} \))(\( ( 1 + 253 T + 215237 T^{2} + 33567598 T^{3} + 17112848159 T^{4} + 1599304851397 T^{5} + 502592611936843 T^{6} )^{4} \))
$47$ (\( 1 - 240 T + 103823 T^{2} \))(\( 1 - 324 T + 103823 T^{2} \))(\( 1 + 12 T + 141790 T^{2} + 1245876 T^{3} + 10779215329 T^{4} \))(\( ( 1 + 103823 T^{2} )^{2} \))(\( ( 1 + 89734 T^{2} + 10779215329 T^{4} )^{2} \))(\( 1 - 30 T - 102923 T^{2} - 3114690 T^{3} + 10779215329 T^{4} \))(\( 1 - 210 T - 20853 T^{2} + 83809446 T^{3} - 12756928590 T^{4} - 2596137940074 T^{5} + 3698984470026571 T^{6} - 269538829352302902 T^{7} - \)\(13\!\cdots\!10\)\( T^{8} + \)\(93\!\cdots\!82\)\( T^{9} - \)\(24\!\cdots\!73\)\( T^{10} - \)\(25\!\cdots\!30\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \))(\( 1 - 437385 T^{2} + 98505882894 T^{4} - 16642251751296271 T^{6} + \)\(23\!\cdots\!92\)\( T^{8} - \)\(28\!\cdots\!97\)\( T^{10} + \)\(30\!\cdots\!08\)\( T^{12} - \)\(30\!\cdots\!13\)\( T^{14} + \)\(27\!\cdots\!72\)\( T^{16} - \)\(20\!\cdots\!19\)\( T^{18} + \)\(13\!\cdots\!14\)\( T^{20} - \)\(63\!\cdots\!65\)\( T^{22} + \)\(15\!\cdots\!21\)\( T^{24} \))
$53$ (\( 1 + 498 T + 148877 T^{2} \))(\( 1 - 258 T + 148877 T^{2} \))(\( 1 + 1104 T + 602230 T^{2} + 164360208 T^{3} + 22164361129 T^{4} \))(\( ( 1 - 148877 T^{2} )^{2} \))(\( ( 1 - 255254 T^{2} + 22164361129 T^{4} )^{2} \))(\( 1 + 363 T - 17108 T^{2} + 54042351 T^{3} + 22164361129 T^{4} \))(\( 1 + 393 T - 211446 T^{2} - 23899125 T^{3} + 46453564620 T^{4} - 3425920762143 T^{5} - 9724787230272680 T^{6} - 510040805305563411 T^{7} + \)\(10\!\cdots\!80\)\( T^{8} - \)\(78\!\cdots\!25\)\( T^{9} - \)\(10\!\cdots\!86\)\( T^{10} + \)\(28\!\cdots\!01\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \))(\( 1 + 362162 T^{2} + 31342470242 T^{4} + 2991238457561520 T^{6} + \)\(17\!\cdots\!22\)\( T^{8} + \)\(23\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!62\)\( T^{12} + \)\(51\!\cdots\!66\)\( T^{14} + \)\(85\!\cdots\!02\)\( T^{16} + \)\(32\!\cdots\!80\)\( T^{18} + \)\(75\!\cdots\!02\)\( T^{20} + \)\(19\!\cdots\!38\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))
$59$ (\( 1 + 132 T + 205379 T^{2} \))(\( 1 - 120 T + 205379 T^{2} \))(\( 1 - 804 T + 380614 T^{2} - 165124716 T^{3} + 42180533641 T^{4} \))(\( ( 1 + 205379 T^{2} )^{2} \))(\( ( 1 + 393520 T^{2} + 42180533641 T^{4} )^{2} \))(\( 1 - 15 T - 205154 T^{2} - 3080685 T^{3} + 42180533641 T^{4} \))(\( 1 + 1143 T + 557208 T^{2} + 118327563 T^{3} - 14314666608 T^{4} - 27063102119841 T^{5} - 16891447327378130 T^{6} - 5558192850270824739 T^{7} - \)\(60\!\cdots\!28\)\( T^{8} + \)\(10\!\cdots\!57\)\( T^{9} + \)\(99\!\cdots\!48\)\( T^{10} + \)\(41\!\cdots\!57\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \))(\( 1 - 661854 T^{2} + 202275596298 T^{4} - 40619516236130848 T^{6} + \)\(74\!\cdots\!38\)\( T^{8} - \)\(16\!\cdots\!86\)\( T^{10} + \)\(35\!\cdots\!06\)\( T^{12} - \)\(68\!\cdots\!26\)\( T^{14} + \)\(13\!\cdots\!78\)\( T^{16} - \)\(30\!\cdots\!08\)\( T^{18} + \)\(64\!\cdots\!78\)\( T^{20} - \)\(88\!\cdots\!54\)\( T^{22} + \)\(56\!\cdots\!41\)\( T^{24} \))
$61$ (\( 1 - 398 T + 226981 T^{2} \))(\( 1 - 622 T + 226981 T^{2} \))(\( 1 + 428 T + 425886 T^{2} + 97147868 T^{3} + 51520374361 T^{4} \))(\( ( 1 - 182 T + 226981 T^{2} )( 1 + 182 T + 226981 T^{2} ) \))(\( ( 1 - 448916 T^{2} + 51520374361 T^{4} )^{2} \))(\( 1 - 118 T - 213057 T^{2} - 26783758 T^{3} + 51520374361 T^{4} \))(\( 1 - 70 T - 335143 T^{2} - 129510330 T^{3} + 42145697866 T^{4} + 25171752927730 T^{5} - 316289217432887 T^{6} + 5713509651289083130 T^{7} + \)\(21\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!30\)\( T^{9} - \)\(88\!\cdots\!03\)\( T^{10} - \)\(42\!\cdots\!70\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \))(\( ( 1 - 1179 T + 941982 T^{2} - 564310665 T^{3} + 276007099296 T^{4} - 128710086478551 T^{5} + 56963412393227764 T^{6} - 29214744138987984531 T^{7} + \)\(14\!\cdots\!56\)\( T^{8} - \)\(65\!\cdots\!65\)\( T^{9} + \)\(25\!\cdots\!22\)\( T^{10} - \)\(71\!\cdots\!79\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} )^{2} \))
$67$ (\( 1 - 92 T + 300763 T^{2} \))(\( 1 - 904 T + 300763 T^{2} \))(\( 1 - 148 T + 440790 T^{2} - 44512924 T^{3} + 90458382169 T^{4} \))(\( ( 1 + 880 T + 300763 T^{2} )^{2} \))(\( ( 1 + 64 T + 300763 T^{2} )^{4} \))(\( 1 - 370 T - 163863 T^{2} - 111282310 T^{3} + 90458382169 T^{4} \))(\( 1 - 628 T - 202942 T^{2} + 436381932 T^{3} - 77667044702 T^{4} - 73528811914784 T^{5} + 76060129771959310 T^{6} - 22114746057926180192 T^{7} - \)\(70\!\cdots\!38\)\( T^{8} + \)\(11\!\cdots\!04\)\( T^{9} - \)\(16\!\cdots\!62\)\( T^{10} - \)\(15\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} \))(\( ( 1 - 396 T - 208818 T^{2} + 528240452 T^{3} - 109541196954 T^{4} - 71767532435832 T^{5} + 113262131816538126 T^{6} - 21585018357998139816 T^{7} - \)\(99\!\cdots\!26\)\( T^{8} + \)\(14\!\cdots\!44\)\( T^{9} - \)\(17\!\cdots\!98\)\( T^{10} - \)\(97\!\cdots\!28\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \))
$71$ (\( 1 + 720 T + 357911 T^{2} \))(\( 1 + 678 T + 357911 T^{2} \))(\( 1 - 954 T + 930526 T^{2} - 341447094 T^{3} + 128100283921 T^{4} \))(\( ( 1 - 357911 T^{2} )^{2} \))(\( ( 1 - 502574 T^{2} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 342 T + 357911 T^{2} )^{2} \))(\( ( 1 - 318 T + 742929 T^{2} - 256167372 T^{3} + 265902461319 T^{4} - 40735890286878 T^{5} + 45848500718449031 T^{6} )^{2} \))(\( ( 1 - 1922318 T^{2} + 1602890240687 T^{4} - 746575415906526884 T^{6} + \)\(20\!\cdots\!27\)\( T^{8} - \)\(31\!\cdots\!38\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$73$ (\( 1 + 502 T + 389017 T^{2} \))(\( 1 + 642 T + 389017 T^{2} \))(\( 1 - 1072 T + 1063278 T^{2} - 417026224 T^{3} + 151334226289 T^{4} \))(\( ( 1 - 1190 T + 389017 T^{2} )( 1 + 1190 T + 389017 T^{2} ) \))(\( ( 1 - 770258 T^{2} + 151334226289 T^{4} )^{2} \))(\( 1 + 362 T - 257973 T^{2} + 140824154 T^{3} + 151334226289 T^{4} \))(\( 1 + 988 T - 186552 T^{2} - 102237300 T^{3} + 281568890272 T^{4} - 16988127696596 T^{5} - 164639785652996186 T^{6} - 6608670472146686132 T^{7} + \)\(42\!\cdots\!08\)\( T^{8} - \)\(60\!\cdots\!00\)\( T^{9} - \)\(42\!\cdots\!92\)\( T^{10} + \)\(88\!\cdots\!16\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} \))(\( ( 1 + 1452 T + 1911888 T^{2} + 1755642240 T^{3} + 1485047372688 T^{4} + 1012976615051412 T^{5} + 691104732915920110 T^{6} + \)\(39\!\cdots\!04\)\( T^{7} + \)\(22\!\cdots\!32\)\( T^{8} + \)\(10\!\cdots\!20\)\( T^{9} + \)\(43\!\cdots\!48\)\( T^{10} + \)\(12\!\cdots\!64\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} )^{2} \))
$79$ (\( 1 + 1024 T + 493039 T^{2} \))(\( 1 - 740 T + 493039 T^{2} \))(\( 1 + 572 T + 901662 T^{2} + 282018308 T^{3} + 243087455521 T^{4} \))(\( ( 1 - 884 T + 493039 T^{2} )^{2} \))(\( ( 1 + 442 T + 493039 T^{2} )^{4} \))(\( 1 + 467 T - 274950 T^{2} + 230249213 T^{3} + 243087455521 T^{4} \))(\( 1 + 861 T - 479895 T^{2} - 258646666 T^{3} + 325257480351 T^{4} - 27564282842211 T^{5} - 246706047980056146 T^{6} - 13590266448240869229 T^{7} + \)\(79\!\cdots\!71\)\( T^{8} - \)\(30\!\cdots\!54\)\( T^{9} - \)\(28\!\cdots\!95\)\( T^{10} + \)\(25\!\cdots\!39\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \))(\( ( 1 - 837 T - 518715 T^{2} + 592038158 T^{3} + 210634653159 T^{4} - 180227265033825 T^{5} - 13471835399034906 T^{6} - 88859070525012044175 T^{7} + \)\(51\!\cdots\!39\)\( T^{8} + \)\(70\!\cdots\!02\)\( T^{9} - \)\(30\!\cdots\!15\)\( T^{10} - \)\(24\!\cdots\!63\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))
$83$ (\( 1 + 204 T + 571787 T^{2} \))(\( 1 - 468 T + 571787 T^{2} \))(\( 1 - 1944 T + 1957030 T^{2} - 1111553928 T^{3} + 326940373369 T^{4} \))(\( ( 1 + 571787 T^{2} )^{2} \))(\( ( 1 + 898672 T^{2} + 326940373369 T^{4} )^{2} \))(\( ( 1 - 477 T + 571787 T^{2} )^{2} \))(\( ( 1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 905564802315 T^{4} - 169682053778511 T^{5} + 186940255267540403 T^{6} )^{2} \))(\( ( 1 + 2862735 T^{2} + 3680795850915 T^{4} + 2710866692924348218 T^{6} + \)\(12\!\cdots\!35\)\( T^{8} + \)\(30\!\cdots\!35\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \))
$89$ (\( 1 - 354 T + 704969 T^{2} \))(\( 1 - 200 T + 704969 T^{2} \))(\( 1 - 366 T + 1156090 T^{2} - 258018654 T^{3} + 496981290961 T^{4} \))(\( ( 1 + 704969 T^{2} )^{2} \))(\( ( 1 + 1174930 T^{2} + 496981290961 T^{4} )^{2} \))(\( 1 + 906 T + 115867 T^{2} + 638701914 T^{3} + 496981290961 T^{4} \))(\( 1 + 1766 T + 725929 T^{2} - 728159446 T^{3} - 335534377858 T^{4} + 846551335831238 T^{5} + 1249625385561159997 T^{6} + \)\(59\!\cdots\!22\)\( T^{7} - \)\(16\!\cdots\!38\)\( T^{8} - \)\(25\!\cdots\!14\)\( T^{9} + \)\(17\!\cdots\!09\)\( T^{10} + \)\(30\!\cdots\!34\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \))(\( 1 - 1635561 T^{2} + 1787180462010 T^{4} - 817237014874197619 T^{6} - \)\(17\!\cdots\!36\)\( T^{8} + \)\(75\!\cdots\!95\)\( T^{10} - \)\(70\!\cdots\!84\)\( T^{12} + \)\(37\!\cdots\!95\)\( T^{14} - \)\(42\!\cdots\!56\)\( T^{16} - \)\(10\!\cdots\!39\)\( T^{18} + \)\(10\!\cdots\!10\)\( T^{20} - \)\(49\!\cdots\!61\)\( T^{22} + \)\(15\!\cdots\!61\)\( T^{24} \))
$97$ (\( 1 + 286 T + 912673 T^{2} \))(\( 1 + 1266 T + 912673 T^{2} \))(\( 1 - 808 T + 903054 T^{2} - 737439784 T^{3} + 832972004929 T^{4} \))(\( ( 1 - 1330 T + 912673 T^{2} )( 1 + 1330 T + 912673 T^{2} ) \))(\( ( 1 - 631850 T^{2} + 832972004929 T^{4} )^{2} \))(\( ( 1 - 503 T + 912673 T^{2} )^{2} \))(\( ( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 1979057473987 T^{4} - 15826468093651 T^{5} + 760231058654565217 T^{6} )^{2} \))(\( ( 1 - 3316347 T^{2} + 4922505333747 T^{4} - 4971002050523297522 T^{6} + \)\(41\!\cdots\!63\)\( T^{8} - \)\(23\!\cdots\!27\)\( T^{10} + \)\(57\!\cdots\!89\)\( T^{12} )^{2} \))
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