Properties

Label 108.2
Level 108
Weight 2
Dimension 133
Nonzero newspaces 6
Newform subspaces 7
Sturm bound 1296
Trace bound 2

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Defining parameters

Level: \( N \) = \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 7 \)
Sturm bound: \(1296\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(108))\).

Total New Old
Modular forms 399 165 234
Cusp forms 250 133 117
Eisenstein series 149 32 117

Trace form

\( 133q - 3q^{2} - 5q^{4} - 6q^{6} + 6q^{7} - 9q^{8} - 6q^{9} + O(q^{10}) \) \( 133q - 3q^{2} - 5q^{4} - 6q^{6} + 6q^{7} - 9q^{8} - 6q^{9} - 13q^{10} + 6q^{11} - 15q^{12} - 16q^{13} - 33q^{14} - 9q^{15} - 29q^{16} - 42q^{17} - 27q^{18} - 9q^{19} - 45q^{20} - 42q^{21} - 21q^{22} - 33q^{23} - 12q^{24} - 30q^{25} - 27q^{27} - 6q^{28} - 51q^{29} + 9q^{30} + 57q^{32} - 60q^{33} + 23q^{34} - 15q^{35} + 24q^{36} - 31q^{37} + 45q^{38} + 3q^{39} + 11q^{40} - 42q^{41} + 54q^{42} + 63q^{44} + 21q^{45} + 9q^{46} + 36q^{47} + 69q^{48} - 32q^{49} + 66q^{50} + 63q^{51} - 25q^{52} + 78q^{53} + 78q^{54} + 18q^{55} + 81q^{56} + 30q^{57} - 25q^{58} + 57q^{59} + 102q^{60} - 4q^{61} + 90q^{62} + 57q^{63} + 13q^{64} + 72q^{65} + 87q^{66} - 9q^{67} + 66q^{68} + 3q^{69} + 27q^{70} + 12q^{71} + 12q^{72} + 20q^{73} + 51q^{74} - 33q^{75} + 15q^{76} - 36q^{77} - 24q^{78} - 60q^{79} - 42q^{81} - 58q^{82} - 54q^{83} - 12q^{84} - 80q^{85} - 51q^{86} - 63q^{87} - 21q^{88} - 60q^{89} - 78q^{90} - 24q^{91} - 147q^{92} + 27q^{93} - 33q^{94} - 6q^{95} - 138q^{96} - 67q^{97} - 180q^{98} + 27q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(108))\)

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
108.2.a \(\chi_{108}(1, \cdot)\) 108.2.a.a 1 1
108.2.b \(\chi_{108}(107, \cdot)\) 108.2.b.a 4 1
108.2.b.b 4
108.2.e \(\chi_{108}(37, \cdot)\) 108.2.e.a 2 2
108.2.h \(\chi_{108}(35, \cdot)\) 108.2.h.a 8 2
108.2.i \(\chi_{108}(13, \cdot)\) 108.2.i.a 18 6
108.2.l \(\chi_{108}(11, \cdot)\) 108.2.l.a 96 6

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(108))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(108)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 6 T^{4} - 12 T^{5} + 20 T^{6} - 24 T^{7} + 16 T^{8} \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( 1 - 3 T^{2} + 9 T^{3} + 18 T^{4} - 18 T^{5} + 3 T^{6} + 135 T^{7} + 81 T^{8} - 108 T^{9} + 243 T^{10} + 1215 T^{11} + 81 T^{12} - 1458 T^{13} + 4374 T^{14} + 6561 T^{15} - 6561 T^{16} + 19683 T^{18} \))
$5$ (\( 1 + 5 T^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( ( 1 - 3 T + 11 T^{2} - 24 T^{3} + 54 T^{4} - 120 T^{5} + 275 T^{6} - 375 T^{7} + 625 T^{8} )^{2} \))(\( 1 - 3 T + 18 T^{3} - 36 T^{4} + 42 T^{5} + 12 T^{6} - 66 T^{7} + 207 T^{8} - 306 T^{9} - 1881 T^{10} + 8988 T^{11} + 3777 T^{12} - 51780 T^{13} + 130257 T^{14} - 141489 T^{15} - 166455 T^{16} + 864168 T^{17} - 1249796 T^{18} + 4320840 T^{19} - 4161375 T^{20} - 17686125 T^{21} + 81410625 T^{22} - 161812500 T^{23} + 59015625 T^{24} + 702187500 T^{25} - 734765625 T^{26} - 597656250 T^{27} + 2021484375 T^{28} - 3222656250 T^{29} + 2929687500 T^{30} + 51269531250 T^{31} - 219726562500 T^{32} + 549316406250 T^{33} - 2288818359375 T^{35} + 3814697265625 T^{36} \))
$7$ (\( 1 - 5 T + 7 T^{2} \))(\( ( 1 + T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 76342 T^{10} + 434581 T^{12} + 2235331 T^{14} + 5764801 T^{16} \))(\( 1 + 9 T^{2} + 6 T^{3} + 90 T^{4} - 144 T^{5} + 642 T^{6} - 1305 T^{7} + 3294 T^{8} - 16123 T^{9} + 41769 T^{10} - 164808 T^{11} + 268632 T^{12} - 1051596 T^{13} + 2182869 T^{14} - 10020276 T^{15} + 22667607 T^{16} - 57261276 T^{17} + 121068222 T^{18} - 400828932 T^{19} + 1110712743 T^{20} - 3436954668 T^{21} + 5241068469 T^{22} - 17674173972 T^{23} + 31604286168 T^{24} - 135726474744 T^{25} + 240789972969 T^{26} - 650621205661 T^{27} + 930473470206 T^{28} - 2580411399615 T^{29} + 8886106383042 T^{30} - 13952017498608 T^{31} + 61040076556410 T^{32} + 28485369059658 T^{33} + 299096375126409 T^{34} + 1628413597910449 T^{36} \))
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 + 19 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 863456 T^{10} + 8184319 T^{12} - 56689952 T^{14} + 214358881 T^{16} \))(\( 1 - 3 T + 36 T^{3} - 252 T^{4} - 183 T^{5} + 3414 T^{6} - 14331 T^{7} + 19242 T^{8} + 82413 T^{9} - 362790 T^{10} + 885057 T^{11} + 3067980 T^{12} - 12803547 T^{13} + 53216793 T^{14} - 119847402 T^{15} - 27497151 T^{16} + 185418060 T^{17} - 3750422330 T^{18} + 2039598660 T^{19} - 3327155271 T^{20} - 159516892062 T^{21} + 779147066313 T^{22} - 2062024047897 T^{23} + 5435113716780 T^{24} + 17247257103747 T^{25} - 77767258437990 T^{26} + 194325543058383 T^{27} + 499087924172442 T^{28} - 4088801551526241 T^{29} + 10714594478125494 T^{30} - 6317656322339373 T^{31} - 95696958062976732 T^{32} + 150380934098963436 T^{33} - 1516341085497881313 T^{35} + 5559917313492231481 T^{36} \))
$13$ (\( 1 + 7 T + 13 T^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + T + 13 T^{2} )^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 + T - 17 T^{2} - 8 T^{3} + 142 T^{4} - 104 T^{5} - 2873 T^{6} + 2197 T^{7} + 28561 T^{8} )^{2} \))(\( 1 - 45 T^{2} + 33 T^{3} + 819 T^{4} - 2223 T^{5} - 6621 T^{6} + 67680 T^{7} + 2295 T^{8} - 1105627 T^{9} + 769527 T^{10} + 10632708 T^{11} - 24128055 T^{12} - 87782535 T^{13} + 521321076 T^{14} + 1050449085 T^{15} - 7341713262 T^{16} - 6958867761 T^{17} + 89217898098 T^{18} - 90465280893 T^{19} - 1240749541278 T^{20} + 2307836639745 T^{21} + 14889451251636 T^{22} - 32593040767755 T^{23} - 116461513026495 T^{24} + 667186658694036 T^{25} + 627726814538967 T^{26} - 11724620828271871 T^{27} + 316385238793455 T^{28} + 121293415468424160 T^{29} - 154256621595946701 T^{30} - 673291361954578419 T^{31} + 3224711259887717691 T^{32} + 1689134469464994981 T^{33} - 29943747413243092845 T^{34} + \)\(11\!\cdots\!29\)\( T^{36} \))
$17$ (\( 1 + 17 T^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} + 1500 T^{4} - 17629 T^{6} + 83521 T^{8} )^{2} \))(\( 1 + 12 T - 630 T^{3} - 2493 T^{4} + 9264 T^{5} + 71715 T^{6} - 20382 T^{7} - 663750 T^{8} + 1631844 T^{9} + 5926023 T^{10} - 55869558 T^{11} - 116813226 T^{12} + 478010541 T^{13} - 1415343366 T^{14} - 8092973394 T^{15} + 53937481662 T^{16} + 118392015135 T^{17} - 620668817804 T^{18} + 2012664257295 T^{19} + 15587932200318 T^{20} - 39760778284722 T^{21} - 118210893271686 T^{22} + 678706612712637 T^{23} - 2819587302687594 T^{24} - 22925440290816534 T^{25} + 41338499037787143 T^{26} + 193516914734370468 T^{27} - 1338115951423023750 T^{28} - 698529790542175806 T^{29} + 41782753742932310115 T^{30} + 91756010896840600368 T^{31} - \)\(41\!\cdots\!97\)\( T^{32} - \)\(18\!\cdots\!90\)\( T^{33} + \)\(99\!\cdots\!24\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 49 T^{2} + 1248 T^{4} - 17689 T^{6} + 130321 T^{8} )^{2} \))(\( 1 - 81 T^{2} - 12 T^{3} + 2916 T^{4} + 378 T^{5} - 63789 T^{6} + 26217 T^{7} + 1135107 T^{8} - 2182495 T^{9} - 24267384 T^{10} + 68364540 T^{11} + 564078960 T^{12} - 1060991325 T^{13} - 9783469926 T^{14} + 6143751819 T^{15} + 107000768556 T^{16} + 9549612666 T^{17} - 1139182247424 T^{18} + 181442640654 T^{19} + 38627277448716 T^{20} + 42139993726521 T^{21} - 1274991584226246 T^{22} - 2627119558841175 T^{23} + 26537591626763760 T^{24} + 61109130255735060 T^{25} - 412146646004154744 T^{26} - 704264286964178605 T^{27} + 6959416226693719707 T^{28} + 3054025117534607523 T^{29} - \)\(14\!\cdots\!29\)\( T^{30} + 15896027748733168302 T^{31} + \)\(23\!\cdots\!36\)\( T^{32} - \)\(18\!\cdots\!88\)\( T^{33} - \)\(23\!\cdots\!61\)\( T^{34} + \)\(10\!\cdots\!41\)\( T^{36} \))
$23$ (\( 1 + 23 T^{2} \))(\( ( 1 - 2 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 60040442 T^{10} + 950619877 T^{12} - 11398763453 T^{14} + 78310985281 T^{16} \))(\( 1 + 30 T + 459 T^{2} + 4716 T^{3} + 35739 T^{4} + 203682 T^{5} + 823269 T^{6} + 1638444 T^{7} - 6429924 T^{8} - 76742136 T^{9} - 307889172 T^{10} + 367007538 T^{11} + 14732471886 T^{12} + 121578320496 T^{13} + 628848268830 T^{14} + 2040092984925 T^{15} + 1341049467978 T^{16} - 31755560848485 T^{17} - 226915587058772 T^{18} - 730377899515155 T^{19} + 709415168560362 T^{20} + 24821811347582475 T^{21} + 175977528397656030 T^{22} + 782519772076186128 T^{23} + 2180934572811516654 T^{24} + 1249596604623219486 T^{25} - 24111104416671277332 T^{26} - \)\(13\!\cdots\!68\)\( T^{27} - \)\(26\!\cdots\!76\)\( T^{28} + \)\(15\!\cdots\!88\)\( T^{29} + \)\(18\!\cdots\!49\)\( T^{30} + \)\(10\!\cdots\!06\)\( T^{31} + \)\(41\!\cdots\!51\)\( T^{32} + \)\(12\!\cdots\!12\)\( T^{33} + \)\(28\!\cdots\!99\)\( T^{34} + \)\(42\!\cdots\!90\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))
$29$ (\( 1 + 29 T^{2} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( ( 1 + 3 T + 59 T^{2} + 168 T^{3} + 2382 T^{4} + 4872 T^{5} + 49619 T^{6} + 73167 T^{7} + 707281 T^{8} )^{2} \))(\( 1 + 24 T + 216 T^{2} + 513 T^{3} - 5085 T^{4} - 38622 T^{5} - 39165 T^{6} + 335661 T^{7} - 972927 T^{8} - 15344370 T^{9} + 17037873 T^{10} + 849699627 T^{11} + 5794706544 T^{12} + 21856511310 T^{13} + 65301140610 T^{14} + 312703277274 T^{15} + 367683018633 T^{16} - 19987658397597 T^{17} - 182924253089192 T^{18} - 579642093530313 T^{19} + 309221418670353 T^{20} + 7626520229435586 T^{21} + 46186256031781410 T^{22} + 448302160099595190 T^{23} + 3446826590722512624 T^{24} + 14657213465553436743 T^{25} + 8523134852735071953 T^{26} - \)\(22\!\cdots\!30\)\( T^{27} - \)\(40\!\cdots\!27\)\( T^{28} + \)\(40\!\cdots\!69\)\( T^{29} - \)\(13\!\cdots\!65\)\( T^{30} - \)\(39\!\cdots\!58\)\( T^{31} - \)\(15\!\cdots\!85\)\( T^{32} + \)\(44\!\cdots\!37\)\( T^{33} + \)\(54\!\cdots\!36\)\( T^{34} + \)\(17\!\cdots\!16\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 47 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 12790910 T^{10} + 1242135745 T^{12} + 48812702455 T^{14} + 852891037441 T^{16} \))(\( 1 - 9 T - 45 T^{2} + 816 T^{3} - 1629 T^{4} - 21807 T^{5} + 131943 T^{6} + 1719 T^{7} - 1559916 T^{8} - 2236900 T^{9} - 3018924 T^{10} + 598265892 T^{11} - 3251038833 T^{12} - 8698989627 T^{13} + 152835071832 T^{14} - 400071404862 T^{15} - 1915318217007 T^{16} + 9512206426008 T^{17} - 6801570693948 T^{18} + 294878399206248 T^{19} - 1840620806543727 T^{20} - 11918527222243842 T^{21} + 141146398373360472 T^{22} - 249044687578816677 T^{23} - 2885308931361444273 T^{24} + 16459858622369202012 T^{25} - 2574813222315533484 T^{26} - 59142790811204959900 T^{27} - \)\(12\!\cdots\!16\)\( T^{28} + 43677171784919904489 T^{29} + \)\(10\!\cdots\!23\)\( T^{30} - \)\(53\!\cdots\!37\)\( T^{31} - \)\(12\!\cdots\!09\)\( T^{32} + \)\(19\!\cdots\!16\)\( T^{33} - \)\(32\!\cdots\!45\)\( T^{34} - \)\(20\!\cdots\!99\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \))
$37$ (\( 1 + T + 37 T^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))(\( ( 1 + T + 37 T^{2} )^{4} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 42 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \))(\( 1 - 162 T^{2} - 498 T^{3} + 12069 T^{4} + 69120 T^{5} - 436659 T^{6} - 3993678 T^{7} + 5314518 T^{8} + 92943608 T^{9} - 67375773 T^{10} + 822064248 T^{11} + 22823309772 T^{12} - 67304973087 T^{13} - 1588732176582 T^{14} - 666327415302 T^{15} + 53054144866422 T^{16} + 47616232092741 T^{17} - 1542710676723420 T^{18} + 1761800587431417 T^{19} + 72631124322131718 T^{20} - 33751482567292206 T^{21} - 2977539884795097702 T^{22} - 4667193159631085259 T^{23} + 58558368622808168748 T^{24} + 78040102186568040984 T^{25} - \)\(23\!\cdots\!33\)\( T^{26} + \)\(12\!\cdots\!16\)\( T^{27} + \)\(25\!\cdots\!82\)\( T^{28} - \)\(71\!\cdots\!14\)\( T^{29} - \)\(28\!\cdots\!79\)\( T^{30} + \)\(16\!\cdots\!40\)\( T^{31} + \)\(10\!\cdots\!41\)\( T^{32} - \)\(16\!\cdots\!14\)\( T^{33} - \)\(19\!\cdots\!42\)\( T^{34} + \)\(16\!\cdots\!29\)\( T^{36} \))
$41$ (\( 1 + 41 T^{2} \))(\( ( 1 - 2 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( ( 1 + 12 T + 131 T^{2} + 996 T^{3} + 7176 T^{4} + 40836 T^{5} + 220211 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2} \))(\( 1 - 21 T + 270 T^{2} - 2844 T^{3} + 28449 T^{4} - 225534 T^{5} + 1364187 T^{6} - 5783241 T^{7} + 7120458 T^{8} + 266586381 T^{9} - 3961333476 T^{10} + 35030856111 T^{11} - 226766681787 T^{12} + 1184373682962 T^{13} - 3751533361044 T^{14} - 14056953699150 T^{15} + 350489698231473 T^{16} - 3267580417540392 T^{17} + 22906077409966138 T^{18} - 133970797119156072 T^{19} + 589173182727106113 T^{20} - 968819305899117150 T^{21} - 10600936661837054484 T^{22} + \)\(13\!\cdots\!62\)\( T^{23} - \)\(10\!\cdots\!67\)\( T^{24} + \)\(68\!\cdots\!91\)\( T^{25} - \)\(31\!\cdots\!96\)\( T^{26} + \)\(87\!\cdots\!41\)\( T^{27} + \)\(95\!\cdots\!58\)\( T^{28} - \)\(31\!\cdots\!81\)\( T^{29} + \)\(30\!\cdots\!47\)\( T^{30} - \)\(20\!\cdots\!14\)\( T^{31} + \)\(10\!\cdots\!89\)\( T^{32} - \)\(44\!\cdots\!44\)\( T^{33} + \)\(17\!\cdots\!70\)\( T^{34} - \)\(54\!\cdots\!01\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 117270976 T^{10} - 2027348993 T^{12} + 404567235136 T^{14} + 11688200277601 T^{16} \))(\( 1 + 9 T + 99 T^{2} + 816 T^{3} + 7380 T^{4} + 44028 T^{5} + 425784 T^{6} + 2654973 T^{7} + 17329428 T^{8} + 77745686 T^{9} + 612406224 T^{10} + 1498691268 T^{11} + 14593371894 T^{12} + 17177157162 T^{13} + 121626443133 T^{14} - 4355914982778 T^{15} - 8046633177891 T^{16} - 188327320360398 T^{17} - 573554182033272 T^{18} - 8098074775497114 T^{19} - 14878224745920459 T^{20} - 346325732535730446 T^{21} + 415816605409543533 T^{22} + 2525187129551918766 T^{23} + 92250001851046744806 T^{24} + \)\(40\!\cdots\!76\)\( T^{25} + \)\(71\!\cdots\!24\)\( T^{26} + \)\(39\!\cdots\!98\)\( T^{27} + \)\(37\!\cdots\!72\)\( T^{28} + \)\(24\!\cdots\!11\)\( T^{29} + \)\(17\!\cdots\!84\)\( T^{30} + \)\(75\!\cdots\!04\)\( T^{31} + \)\(54\!\cdots\!20\)\( T^{32} + \)\(25\!\cdots\!12\)\( T^{33} + \)\(13\!\cdots\!99\)\( T^{34} + \)\(52\!\cdots\!87\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 + 47 T^{2} \))(\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 428735974 T^{10} + 10017985093 T^{12} - 571298412437 T^{14} + 23811286661761 T^{16} \))(\( 1 - 45 T + 1026 T^{2} - 15507 T^{3} + 173529 T^{4} - 1550934 T^{5} + 12137826 T^{6} - 94180626 T^{7} + 777562713 T^{8} - 6408653373 T^{9} + 47766713094 T^{10} - 314746073955 T^{11} + 2010959274003 T^{12} - 14589090065835 T^{13} + 120231976929849 T^{14} - 958518905650704 T^{15} + 6721325237600946 T^{16} - 42688545402868929 T^{17} + 277542048358340170 T^{18} - 2006361633934839663 T^{19} + 14847407449860489714 T^{20} - 99516308341373041392 T^{21} + \)\(58\!\cdots\!69\)\( T^{22} - \)\(33\!\cdots\!45\)\( T^{23} + \)\(21\!\cdots\!87\)\( T^{24} - \)\(15\!\cdots\!65\)\( T^{25} + \)\(11\!\cdots\!34\)\( T^{26} - \)\(71\!\cdots\!91\)\( T^{27} + \)\(40\!\cdots\!37\)\( T^{28} - \)\(23\!\cdots\!78\)\( T^{29} + \)\(14\!\cdots\!66\)\( T^{30} - \)\(84\!\cdots\!18\)\( T^{31} + \)\(44\!\cdots\!01\)\( T^{32} - \)\(18\!\cdots\!01\)\( T^{33} + \)\(58\!\cdots\!46\)\( T^{34} - \)\(11\!\cdots\!15\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \))
$53$ (\( 1 + 53 T^{2} \))(\( ( 1 - 101 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 33 T + 711 T^{2} - 11529 T^{3} + 156285 T^{4} - 1818492 T^{5} + 18782340 T^{6} - 173567805 T^{7} + 1455502203 T^{8} - 11084684058 T^{9} + 77141616759 T^{10} - 487551964245 T^{11} + 2796258432180 T^{12} - 14348776574652 T^{13} + 65357682623505 T^{14} - 255532919456241 T^{15} + 835219620424107 T^{16} - 2054569783574913 T^{17} + 3299763591802133 T^{18} )^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 94 T^{2} + 3481 T^{4} )^{2} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 625549624 T^{10} - 7476411737 T^{12} - 2362109883896 T^{14} + 146830437604321 T^{16} \))(\( 1 - 60 T + 1593 T^{2} - 24660 T^{3} + 247662 T^{4} - 1708872 T^{5} + 8490216 T^{6} - 33550125 T^{7} + 148813002 T^{8} - 1136539629 T^{9} + 11730436263 T^{10} - 113179328610 T^{11} + 651096766104 T^{12} + 3048081644244 T^{13} - 120313325069001 T^{14} + 1538872617278898 T^{15} - 13453209049931613 T^{16} + 98005926759519540 T^{17} - 714266619705703766 T^{18} + 5782349678811652860 T^{19} - 46830620702811944853 T^{20} + \)\(31\!\cdots\!42\)\( T^{21} - \)\(14\!\cdots\!61\)\( T^{22} + \)\(21\!\cdots\!56\)\( T^{23} + \)\(27\!\cdots\!64\)\( T^{24} - \)\(28\!\cdots\!90\)\( T^{25} + \)\(17\!\cdots\!23\)\( T^{26} - \)\(98\!\cdots\!31\)\( T^{27} + \)\(76\!\cdots\!02\)\( T^{28} - \)\(10\!\cdots\!75\)\( T^{29} + \)\(15\!\cdots\!96\)\( T^{30} - \)\(17\!\cdots\!88\)\( T^{31} + \)\(15\!\cdots\!82\)\( T^{32} - \)\(90\!\cdots\!40\)\( T^{33} + \)\(34\!\cdots\!13\)\( T^{34} - \)\(76\!\cdots\!40\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \))
$61$ (\( 1 + 13 T + 61 T^{2} \))(\( ( 1 + 4 T + 61 T^{2} )^{4} \))(\( ( 1 - 11 T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 488 T^{5} - 420473 T^{6} + 226981 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 + 18 T + 144 T^{2} + 870 T^{3} + 693 T^{4} + 46719 T^{5} + 1363548 T^{6} + 13321629 T^{7} + 112641282 T^{8} + 631876487 T^{9} + 4072259232 T^{10} + 57229816275 T^{11} + 499398580341 T^{12} + 4512706586946 T^{13} + 40501364633055 T^{14} + 302618732115951 T^{15} + 2515574747971161 T^{16} + 17759529627029685 T^{17} + 116077839022099230 T^{18} + 1083331307248810785 T^{19} + 9360453637200690081 T^{20} + 68688702434410673931 T^{21} + \)\(56\!\cdots\!55\)\( T^{22} + \)\(38\!\cdots\!46\)\( T^{23} + \)\(25\!\cdots\!01\)\( T^{24} + \)\(17\!\cdots\!75\)\( T^{25} + \)\(78\!\cdots\!92\)\( T^{26} + \)\(73\!\cdots\!67\)\( T^{27} + \)\(80\!\cdots\!82\)\( T^{28} + \)\(57\!\cdots\!69\)\( T^{29} + \)\(36\!\cdots\!08\)\( T^{30} + \)\(75\!\cdots\!39\)\( T^{31} + \)\(68\!\cdots\!13\)\( T^{32} + \)\(52\!\cdots\!70\)\( T^{33} + \)\(52\!\cdots\!84\)\( T^{34} + \)\(40\!\cdots\!78\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))
$67$ (\( 1 - 11 T + 67 T^{2} \))(\( ( 1 - 74 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 4573034080 T^{10} + 206649745855 T^{12} + 14473341147040 T^{14} + 406067677556641 T^{16} \))(\( 1 + 27 T + 378 T^{2} + 4110 T^{3} + 42309 T^{4} + 359532 T^{5} + 1903683 T^{6} - 2087343 T^{7} - 177145164 T^{8} - 2600725729 T^{9} - 29272869126 T^{10} - 274304636919 T^{11} - 1972009698093 T^{12} - 9826793958816 T^{13} - 14621223469968 T^{14} + 409049170442514 T^{15} + 7291627922092167 T^{16} + 85616450880794064 T^{17} + 791533857384890682 T^{18} + 5736302209013202288 T^{19} + 32732117742271737663 T^{20} + \)\(12\!\cdots\!82\)\( T^{21} - \)\(29\!\cdots\!28\)\( T^{22} - \)\(13\!\cdots\!12\)\( T^{23} - \)\(17\!\cdots\!17\)\( T^{24} - \)\(16\!\cdots\!37\)\( T^{25} - \)\(11\!\cdots\!66\)\( T^{26} - \)\(70\!\cdots\!63\)\( T^{27} - \)\(32\!\cdots\!36\)\( T^{28} - \)\(25\!\cdots\!69\)\( T^{29} + \)\(15\!\cdots\!63\)\( T^{30} + \)\(19\!\cdots\!84\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} + \)\(10\!\cdots\!30\)\( T^{33} + \)\(62\!\cdots\!18\)\( T^{34} + \)\(29\!\cdots\!29\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 140 T^{2} + 10230 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 + 12 T - 216 T^{2} - 3978 T^{3} + 8901 T^{4} + 529392 T^{5} + 1798959 T^{6} - 37173408 T^{7} - 298164978 T^{8} + 1281013128 T^{9} + 23327409465 T^{10} + 43122203202 T^{11} - 995587443048 T^{12} - 12405448357179 T^{13} - 32462907484212 T^{14} + 1158165262802520 T^{15} + 11716337115395100 T^{16} - 40103807685557901 T^{17} - 1167130398411512264 T^{18} - 2847370345674610971 T^{19} + 59062055398706699100 T^{20} + \)\(41\!\cdots\!20\)\( T^{21} - \)\(82\!\cdots\!72\)\( T^{22} - \)\(22\!\cdots\!29\)\( T^{23} - \)\(12\!\cdots\!08\)\( T^{24} + \)\(39\!\cdots\!82\)\( T^{25} + \)\(15\!\cdots\!65\)\( T^{26} + \)\(58\!\cdots\!68\)\( T^{27} - \)\(97\!\cdots\!78\)\( T^{28} - \)\(85\!\cdots\!68\)\( T^{29} + \)\(29\!\cdots\!19\)\( T^{30} + \)\(61\!\cdots\!12\)\( T^{31} + \)\(73\!\cdots\!81\)\( T^{32} - \)\(23\!\cdots\!78\)\( T^{33} - \)\(90\!\cdots\!36\)\( T^{34} + \)\(35\!\cdots\!92\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$73$ (\( 1 - 17 T + 73 T^{2} \))(\( ( 1 - 5 T + 73 T^{2} )^{4} \))(\( ( 1 + T + 73 T^{2} )^{4} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - T + 138 T^{2} - 73 T^{3} + 5329 T^{4} )^{4} \))(\( 1 - 9 T - 423 T^{2} + 3174 T^{3} + 104238 T^{4} - 633897 T^{5} - 17768040 T^{6} + 86261373 T^{7} + 2286597285 T^{8} - 8821811794 T^{9} - 229412589363 T^{10} + 696472394517 T^{11} + 18405723457890 T^{12} - 43263862214751 T^{13} - 1209322407919761 T^{14} + 2007285699967488 T^{15} + 70978846328739252 T^{16} - 49678641613188693 T^{17} - 4558841577782266638 T^{18} - 3626540837762774589 T^{19} + \)\(37\!\cdots\!08\)\( T^{20} + \)\(78\!\cdots\!96\)\( T^{21} - \)\(34\!\cdots\!01\)\( T^{22} - \)\(89\!\cdots\!43\)\( T^{23} + \)\(27\!\cdots\!10\)\( T^{24} + \)\(76\!\cdots\!49\)\( T^{25} - \)\(18\!\cdots\!03\)\( T^{26} - \)\(51\!\cdots\!22\)\( T^{27} + \)\(98\!\cdots\!65\)\( T^{28} + \)\(27\!\cdots\!21\)\( T^{29} - \)\(40\!\cdots\!40\)\( T^{30} - \)\(10\!\cdots\!01\)\( T^{31} + \)\(12\!\cdots\!42\)\( T^{32} + \)\(28\!\cdots\!18\)\( T^{33} - \)\(27\!\cdots\!03\)\( T^{34} - \)\(42\!\cdots\!77\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \))
$79$ (\( 1 + 13 T + 79 T^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 155 T^{2} + 6241 T^{4} )^{2} \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 2367024070 T^{10} - 99517456955 T^{12} + 27955057384915 T^{14} + 1517108809906561 T^{16} \))(\( 1 + 36 T + 702 T^{2} + 9861 T^{3} + 119196 T^{4} + 1409409 T^{5} + 16913820 T^{6} + 201585240 T^{7} + 2283030162 T^{8} + 24444793301 T^{9} + 255412728198 T^{10} + 2659995762120 T^{11} + 27368820023094 T^{12} + 271885146597855 T^{13} + 2605507077760911 T^{14} + 24385423616176365 T^{15} + 226920837770375571 T^{16} + 2103229216817817381 T^{17} + 19027927100722312626 T^{18} + \)\(16\!\cdots\!99\)\( T^{19} + \)\(14\!\cdots\!11\)\( T^{20} + \)\(12\!\cdots\!35\)\( T^{21} + \)\(10\!\cdots\!91\)\( T^{22} + \)\(83\!\cdots\!45\)\( T^{23} + \)\(66\!\cdots\!74\)\( T^{24} + \)\(51\!\cdots\!80\)\( T^{25} + \)\(38\!\cdots\!78\)\( T^{26} + \)\(29\!\cdots\!19\)\( T^{27} + \)\(21\!\cdots\!62\)\( T^{28} + \)\(15\!\cdots\!60\)\( T^{29} + \)\(99\!\cdots\!20\)\( T^{30} + \)\(65\!\cdots\!51\)\( T^{31} + \)\(43\!\cdots\!76\)\( T^{32} + \)\(28\!\cdots\!39\)\( T^{33} + \)\(16\!\cdots\!42\)\( T^{34} + \)\(65\!\cdots\!24\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \))
$83$ (\( 1 + 83 T^{2} \))(\( ( 1 + 19 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 18577166738 T^{10} + 1084944676381 T^{12} - 72253822514549 T^{14} + 2252292232139041 T^{16} \))(\( 1 + 45 T + 945 T^{2} + 10116 T^{3} + 12339 T^{4} - 1380339 T^{5} - 22826715 T^{6} - 161300349 T^{7} + 246197610 T^{8} + 18368963478 T^{9} + 201770388396 T^{10} + 948024221238 T^{11} - 3270167174595 T^{12} - 95828229532791 T^{13} - 847092215711700 T^{14} - 3777334618062672 T^{15} + 7299207689386761 T^{16} + 335349191198370492 T^{17} + 3919398090577201708 T^{18} + 27833982869464750836 T^{19} + 50284241772185396529 T^{20} - \)\(21\!\cdots\!64\)\( T^{21} - \)\(40\!\cdots\!00\)\( T^{22} - \)\(37\!\cdots\!13\)\( T^{23} - \)\(10\!\cdots\!55\)\( T^{24} + \)\(25\!\cdots\!26\)\( T^{25} + \)\(45\!\cdots\!36\)\( T^{26} + \)\(34\!\cdots\!34\)\( T^{27} + \)\(38\!\cdots\!90\)\( T^{28} - \)\(20\!\cdots\!83\)\( T^{29} - \)\(24\!\cdots\!15\)\( T^{30} - \)\(12\!\cdots\!57\)\( T^{31} + \)\(90\!\cdots\!31\)\( T^{32} + \)\(61\!\cdots\!12\)\( T^{33} + \)\(47\!\cdots\!45\)\( T^{34} + \)\(18\!\cdots\!35\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))
$89$ (\( 1 + 89 T^{2} \))(\( ( 1 - 158 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 170 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 184 T^{2} + 21006 T^{4} - 1457464 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 + 48 T + 729 T^{2} + 540 T^{3} - 60516 T^{4} + 309648 T^{5} + 9164505 T^{6} - 43924695 T^{7} - 296290449 T^{8} + 17649268605 T^{9} + 60514001892 T^{10} - 1616101715796 T^{11} + 6831214397718 T^{12} + 217593105310137 T^{13} - 956051155825812 T^{14} - 10434643262054325 T^{15} + 213339340235303568 T^{16} + 707450617777750446 T^{17} - 16906270535571748808 T^{18} + 62963104982219789694 T^{19} + \)\(16\!\cdots\!28\)\( T^{20} - \)\(73\!\cdots\!25\)\( T^{21} - \)\(59\!\cdots\!92\)\( T^{22} + \)\(12\!\cdots\!13\)\( T^{23} + \)\(33\!\cdots\!98\)\( T^{24} - \)\(71\!\cdots\!84\)\( T^{25} + \)\(23\!\cdots\!52\)\( T^{26} + \)\(61\!\cdots\!45\)\( T^{27} - \)\(92\!\cdots\!49\)\( T^{28} - \)\(12\!\cdots\!55\)\( T^{29} + \)\(22\!\cdots\!05\)\( T^{30} + \)\(68\!\cdots\!12\)\( T^{31} - \)\(11\!\cdots\!56\)\( T^{32} + \)\(94\!\cdots\!60\)\( T^{33} + \)\(11\!\cdots\!69\)\( T^{34} + \)\(66\!\cdots\!92\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))
$97$ (\( 1 - 5 T + 97 T^{2} \))(\( ( 1 - 11 T + 97 T^{2} )^{4} \))(\( ( 1 + 13 T + 97 T^{2} )^{4} \))(\( 1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4} \))(\( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 25414 T^{5} - 555131 T^{6} - 1825346 T^{7} + 88529281 T^{8} )^{2} \))(\( 1 + 27 T + 171 T^{2} - 1506 T^{3} - 5796 T^{4} + 238266 T^{5} - 1047822 T^{6} - 47407473 T^{7} + 141058260 T^{8} + 5136199520 T^{9} - 34758298530 T^{10} - 613734906312 T^{11} + 4119485195664 T^{12} + 37279586300094 T^{13} - 763210465522713 T^{14} - 3540249166293810 T^{15} + 84441145547637873 T^{16} + 225577194346293930 T^{17} - 6812284288441068552 T^{18} + 21880987851590511210 T^{19} + \)\(79\!\cdots\!57\)\( T^{20} - \)\(32\!\cdots\!30\)\( T^{21} - \)\(67\!\cdots\!53\)\( T^{22} + \)\(32\!\cdots\!58\)\( T^{23} + \)\(34\!\cdots\!56\)\( T^{24} - \)\(49\!\cdots\!56\)\( T^{25} - \)\(27\!\cdots\!30\)\( T^{26} + \)\(39\!\cdots\!40\)\( T^{27} + \)\(10\!\cdots\!40\)\( T^{28} - \)\(33\!\cdots\!69\)\( T^{29} - \)\(72\!\cdots\!02\)\( T^{30} + \)\(16\!\cdots\!82\)\( T^{31} - \)\(37\!\cdots\!24\)\( T^{32} - \)\(95\!\cdots\!58\)\( T^{33} + \)\(10\!\cdots\!91\)\( T^{34} + \)\(16\!\cdots\!99\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \))
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