Properties

Label 36.5
Level 36
Weight 5
Dimension 63
Nonzero newspaces 4
Newform subspaces 6
Sturm bound 360
Trace bound 1

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Defining parameters

Level: \( N \) = \( 36\( 36 = 2^{2} \cdot 3^{2} \) \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 6 \)
Sturm bound: \(360\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(36))\).

Total New Old
Modular forms 164 73 91
Cusp forms 124 63 61
Eisenstein series 40 10 30

Trace form

\( 63q - 3q^{2} - 9q^{3} - 13q^{4} - 21q^{5} + 15q^{6} + 149q^{7} + 186q^{8} - 39q^{9} + O(q^{10}) \) \( 63q - 3q^{2} - 9q^{3} - 13q^{4} - 21q^{5} + 15q^{6} + 149q^{7} + 186q^{8} - 39q^{9} + 136q^{10} - 18q^{11} - 228q^{12} + 75q^{13} - 348q^{14} + 225q^{15} - 625q^{16} + 222q^{17} + 72q^{18} + 146q^{19} + 1452q^{20} - 1029q^{21} + 1143q^{22} - 1719q^{23} - 951q^{24} - 3230q^{25} - 1548q^{26} + 648q^{27} - 3060q^{28} + 1671q^{29} - 1980q^{30} + 3491q^{31} + 3687q^{32} + 6252q^{33} + 3541q^{34} - 1005q^{36} - 234q^{37} - 3285q^{38} - 8265q^{39} - 4196q^{40} - 8196q^{41} + 3330q^{42} + 2252q^{43} + 10410q^{44} + 9789q^{45} + 7800q^{46} + 13689q^{47} + 2163q^{48} + 2852q^{49} + 5697q^{50} + 10449q^{51} + 3526q^{52} - 17682q^{53} - 4983q^{54} - 13482q^{55} - 9234q^{56} - 20109q^{57} - 11012q^{58} - 20052q^{59} - 16392q^{60} + 3q^{61} - 11100q^{62} + 5559q^{63} + 4418q^{64} + 38811q^{65} - 17358q^{66} + 12938q^{67} - 22053q^{68} + 24999q^{69} - 6366q^{70} + 4083q^{72} - 30864q^{73} + 27384q^{74} - 30297q^{75} + 23901q^{76} - 50511q^{77} + 34566q^{78} - 16315q^{79} + 23472q^{80} + 18537q^{81} - 6482q^{82} + 37017q^{83} + 51078q^{84} + 58274q^{85} + 41673q^{86} + 22455q^{87} + 21003q^{88} + 47886q^{89} - 4692q^{90} + 13654q^{91} - 49110q^{92} - 24399q^{93} - 29964q^{94} - 37116q^{95} - 76164q^{96} + 3834q^{97} - 77484q^{98} - 10035q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(36))\)

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
36.5.c \(\chi_{36}(17, \cdot)\) 36.5.c.a 2 1
36.5.d \(\chi_{36}(19, \cdot)\) 36.5.d.a 1 1
36.5.d.b 4
36.5.d.c 4
36.5.f \(\chi_{36}(7, \cdot)\) 36.5.f.a 44 2
36.5.g \(\chi_{36}(5, \cdot)\) 36.5.g.a 8 2

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(36))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(36)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 4 T \))(\( 1 + 6 T + 28 T^{2} + 96 T^{3} + 256 T^{4} \))(\( 1 + 4 T^{2} + 256 T^{4} \))
$3$ (\( 1 + 9 T + 30 T^{2} - 189 T^{3} - 6966 T^{4} - 15309 T^{5} + 196830 T^{6} + 4782969 T^{7} + 43046721 T^{8} \))
$5$ (\( 1 + 208 T^{2} + 390625 T^{4} \))(\( 1 - 14 T + 625 T^{2} \))(\( ( 1 + 12 T + 454 T^{2} + 7500 T^{3} + 390625 T^{4} )^{2} \))(\( ( 1 + 802 T^{2} + 390625 T^{4} )^{2} \))(\( 1 + 9 T + 1114 T^{2} + 9783 T^{3} + 533599 T^{4} + 10528056 T^{5} + 59806456 T^{6} + 10305069192 T^{7} - 6001445444 T^{8} + 6440668245000 T^{9} + 23361896875000 T^{10} + 2570326171875000 T^{11} + 81420745849609375 T^{12} + 932979583740234375 T^{13} + 66399574279785156250 T^{14} + \)\(33\!\cdots\!25\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} \))
$7$ (\( ( 1 - 68 T + 2401 T^{2} )^{2} \))(\( ( 1 - 49 T )( 1 + 49 T ) \))(\( 1 - 4516 T^{2} + 16148934 T^{4} - 26033841316 T^{6} + 33232930569601 T^{8} \))(\( ( 1 - 770 T^{2} + 5764801 T^{4} )^{2} \))(\( 1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} + \)\(29\!\cdots\!91\)\( T^{12} + \)\(99\!\cdots\!53\)\( T^{13} - \)\(87\!\cdots\!54\)\( T^{14} - \)\(59\!\cdots\!13\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 - 5954 T^{2} + 214358881 T^{4} \))(\( ( 1 - 121 T )( 1 + 121 T ) \))(\( 1 - 36196 T^{2} + 708332166 T^{4} - 7758934056676 T^{6} + 45949729863572161 T^{8} \))(\( ( 1 + 7582 T^{2} + 214358881 T^{4} )^{2} \))(\( 1 + 18 T + 33850 T^{2} + 607356 T^{3} + 452208721 T^{4} + 17443950048 T^{5} + 9074477968558 T^{6} + 445095012639474 T^{7} + 191968264536523468 T^{8} + 6516636080054538834 T^{9} + \)\(19\!\cdots\!98\)\( T^{10} + \)\(54\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} + \)\(40\!\cdots\!56\)\( T^{13} + \)\(33\!\cdots\!50\)\( T^{14} + \)\(25\!\cdots\!58\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} \))
$13$ (\( ( 1 + 16 T + 28561 T^{2} )^{2} \))(\( 1 + 238 T + 28561 T^{2} \))(\( ( 1 - 148 T + 32646 T^{2} - 4227028 T^{3} + 815730721 T^{4} )^{2} \))(\( ( 1 - 14 T + 28561 T^{2} )^{4} \))(\( 1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} + \)\(18\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!75\)\( T^{12} - \)\(13\!\cdots\!65\)\( T^{13} - \)\(22\!\cdots\!94\)\( T^{14} + \)\(77\!\cdots\!05\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \))
$17$ (\( 1 + 79360 T^{2} + 6975757441 T^{4} \))(\( 1 + 322 T + 83521 T^{2} \))(\( ( 1 - 300 T + 186214 T^{2} - 25056300 T^{3} + 6975757441 T^{4} )^{2} \))(\( ( 1 - 49790 T^{2} + 6975757441 T^{4} )^{2} \))(\( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!55\)\( T^{10} + \)\(25\!\cdots\!74\)\( T^{12} - \)\(97\!\cdots\!25\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \))
$19$ (\( ( 1 + 208 T + 130321 T^{2} )^{2} \))(\( ( 1 - 361 T )( 1 + 361 T ) \))(\( 1 - 195556 T^{2} + 17286835974 T^{4} - 3321237654045796 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} \))(\( ( 1 - 115490 T^{2} + 16983563041 T^{4} )^{2} \))(\( ( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \))
$23$ (\( 1 - 536354 T^{2} + 78310985281 T^{4} \))(\( ( 1 - 529 T )( 1 + 529 T ) \))(\( 1 - 655492 T^{2} + 255175536006 T^{4} - 51332224363813252 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} \))(\( ( 1 - 412226 T^{2} + 78310985281 T^{4} )^{2} \))(\( 1 + 1719 T + 1529458 T^{2} + 935945649 T^{3} + 342642958747 T^{4} - 16333135609500 T^{5} - 118516645434237428 T^{6} - \)\(10\!\cdots\!68\)\( T^{7} - \)\(65\!\cdots\!00\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{9} - \)\(92\!\cdots\!68\)\( T^{10} - \)\(35\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!67\)\( T^{12} + \)\(16\!\cdots\!49\)\( T^{13} + \)\(73\!\cdots\!78\)\( T^{14} + \)\(23\!\cdots\!39\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \))
$29$ (\( 1 - 993200 T^{2} + 500246412961 T^{4} \))(\( 1 + 82 T + 707281 T^{2} \))(\( ( 1 + 444 T + 1423078 T^{2} + 314032764 T^{3} + 500246412961 T^{4} )^{2} \))(\( ( 1 + 1037794 T^{2} + 500246412961 T^{4} )^{2} \))(\( 1 - 2115 T + 4091014 T^{2} - 5498870985 T^{3} + 7048695081595 T^{4} - 8024737206821040 T^{5} + 8297140249856169556 T^{6} - \)\(79\!\cdots\!80\)\( T^{7} + \)\(68\!\cdots\!24\)\( T^{8} - \)\(56\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!16\)\( T^{10} - \)\(28\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!95\)\( T^{12} - \)\(97\!\cdots\!85\)\( T^{13} + \)\(51\!\cdots\!34\)\( T^{14} - \)\(18\!\cdots\!15\)\( T^{15} + \)\(62\!\cdots\!41\)\( T^{16} \))
$31$ (\( ( 1 - 1652 T + 923521 T^{2} )^{2} \))(\( ( 1 - 961 T )( 1 + 961 T ) \))(\( 1 - 2090020 T^{2} + 2465294161734 T^{4} - 1782559326072438820 T^{6} + \)\(72\!\cdots\!81\)\( T^{8} \))(\( ( 1 - 1746242 T^{2} + 852891037441 T^{4} )^{2} \))(\( 1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} + \)\(31\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!14\)\( T^{9} - \)\(29\!\cdots\!86\)\( T^{10} - \)\(23\!\cdots\!56\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} + \)\(12\!\cdots\!37\)\( T^{13} - \)\(15\!\cdots\!84\)\( T^{14} - \)\(10\!\cdots\!67\)\( T^{15} + \)\(52\!\cdots\!61\)\( T^{16} \))
$37$ (\( ( 1 + 442 T + 1874161 T^{2} )^{2} \))(\( 1 - 2162 T + 1874161 T^{2} \))(\( ( 1 + 2204 T + 4842918 T^{2} + 4130650844 T^{3} + 3512479453921 T^{4} )^{2} \))(\( ( 1 - 722 T + 1874161 T^{2} )^{4} \))(\( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))
$41$ (\( 1 - 5405120 T^{2} + 7984925229121 T^{4} \))(\( 1 - 3038 T + 2825761 T^{2} \))(\( ( 1 + 276 T + 5507494 T^{2} + 779910036 T^{3} + 7984925229121 T^{4} )^{2} \))(\( ( 1 + 3929410 T^{2} + 7984925229121 T^{4} )^{2} \))(\( 1 + 7920 T + 37687894 T^{2} + 132890424480 T^{3} + 385083705354505 T^{4} + 963185727644706960 T^{5} + \)\(21\!\cdots\!66\)\( T^{6} + \)\(41\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!64\)\( T^{8} + \)\(11\!\cdots\!20\)\( T^{9} + \)\(16\!\cdots\!86\)\( T^{10} + \)\(21\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!05\)\( T^{12} + \)\(23\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!34\)\( T^{14} + \)\(11\!\cdots\!20\)\( T^{15} + \)\(40\!\cdots\!81\)\( T^{16} \))
$43$ (\( ( 1 - 1160 T + 3418801 T^{2} )^{2} \))(\( ( 1 - 1849 T )( 1 + 1849 T ) \))(\( 1 - 3593572 T^{2} + 5893643026566 T^{4} - 42002389247979180772 T^{6} + \)\(13\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 2176610 T^{2} + 11688200277601 T^{4} )^{2} \))(\( 1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} - \)\(54\!\cdots\!36\)\( T^{6} + \)\(27\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} + \)\(94\!\cdots\!16\)\( T^{9} - \)\(63\!\cdots\!36\)\( T^{10} + \)\(23\!\cdots\!04\)\( T^{11} + \)\(14\!\cdots\!91\)\( T^{12} - \)\(97\!\cdots\!48\)\( T^{13} - \)\(20\!\cdots\!04\)\( T^{14} + \)\(37\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 9549410 T^{2} + 23811286661761 T^{4} \))(\( ( 1 - 2209 T )( 1 + 2209 T ) \))(\( 1 - 16853380 T^{2} + 118578854811654 T^{4} - \)\(40\!\cdots\!80\)\( T^{6} + \)\(56\!\cdots\!21\)\( T^{8} \))(\( ( 1 - 2534018 T^{2} + 23811286661761 T^{4} )^{2} \))(\( 1 - 13689 T + 103685338 T^{2} - 564293857959 T^{3} + 2435028217967227 T^{4} - 8736748337842042500 T^{5} + \)\(26\!\cdots\!72\)\( T^{6} - \)\(71\!\cdots\!32\)\( T^{7} + \)\(16\!\cdots\!20\)\( T^{8} - \)\(35\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!92\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!67\)\( T^{12} - \)\(15\!\cdots\!59\)\( T^{13} + \)\(13\!\cdots\!78\)\( T^{14} - \)\(90\!\cdots\!29\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \))
$53$ (\( 1 - 9620912 T^{2} + 62259690411361 T^{4} \))(\( 1 + 2482 T + 7890481 T^{2} \))(\( ( 1 + 2556 T + 17173798 T^{2} + 20168069436 T^{3} + 62259690411361 T^{4} )^{2} \))(\( ( 1 + 15027874 T^{2} + 62259690411361 T^{4} )^{2} \))(\( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - \)\(52\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \))
$59$ (\( 1 - 12943970 T^{2} + 146830437604321 T^{4} \))(\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( 1 - 32722276 T^{2} + 559903452302214 T^{4} - \)\(48\!\cdots\!96\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} \))(\( ( 1 - 17009378 T^{2} + 146830437604321 T^{4} )^{2} \))(\( 1 + 20052 T + 216711700 T^{2} + 1657982214864 T^{3} + 9931594296358591 T^{4} + 49579528565018409012 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} + \)\(84\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!08\)\( T^{8} + \)\(10\!\cdots\!36\)\( T^{9} + \)\(31\!\cdots\!08\)\( T^{10} + \)\(88\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!31\)\( T^{12} + \)\(43\!\cdots\!64\)\( T^{13} + \)\(68\!\cdots\!00\)\( T^{14} + \)\(76\!\cdots\!92\)\( T^{15} + \)\(46\!\cdots\!81\)\( T^{16} \))
$61$ (\( ( 1 + 3910 T + 13845841 T^{2} )^{2} \))(\( 1 + 6958 T + 13845841 T^{2} \))(\( ( 1 - 2116 T + 16830246 T^{2} - 29297799556 T^{3} + 191707312997281 T^{4} )^{2} \))(\( ( 1 - 3122 T + 13845841 T^{2} )^{4} \))(\( 1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!44\)\( T^{7} - \)\(98\!\cdots\!68\)\( T^{8} + \)\(55\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!44\)\( T^{10} - \)\(35\!\cdots\!44\)\( T^{11} + \)\(15\!\cdots\!91\)\( T^{12} + \)\(76\!\cdots\!23\)\( T^{13} - \)\(74\!\cdots\!94\)\( T^{14} + \)\(18\!\cdots\!97\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \))
$67$ (\( ( 1 - 6392 T + 20151121 T^{2} )^{2} \))(\( ( 1 - 4489 T )( 1 + 4489 T ) \))(\( 1 - 53256676 T^{2} + 1338152481850374 T^{4} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} \))(\( ( 1 - 29399714 T^{2} + 406067677556641 T^{4} )^{2} \))(\( 1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} + \)\(70\!\cdots\!02\)\( T^{6} - \)\(11\!\cdots\!78\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} - \)\(22\!\cdots\!38\)\( T^{9} + \)\(28\!\cdots\!82\)\( T^{10} + \)\(60\!\cdots\!96\)\( T^{11} + \)\(88\!\cdots\!37\)\( T^{12} - \)\(85\!\cdots\!28\)\( T^{13} - \)\(22\!\cdots\!34\)\( T^{14} - \)\(20\!\cdots\!14\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \))
$71$ (\( 1 - 7689890 T^{2} + 645753531245761 T^{4} \))(\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( 1 - 79127428 T^{2} + 2740885222137990 T^{4} - \)\(51\!\cdots\!08\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} \))(\( ( 1 - 29589698 T^{2} + 645753531245761 T^{4} )^{2} \))(\( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(73\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \))
$73$ (\( ( 1 + 2224 T + 28398241 T^{2} )^{2} \))(\( 1 - 1442 T + 28398241 T^{2} \))(\( ( 1 - 4420 T + 3693510 T^{2} - 125520225220 T^{3} + 806460091894081 T^{4} )^{2} \))(\( ( 1 + 6370 T + 28398241 T^{2} )^{4} \))(\( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(89\!\cdots\!21\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))
$79$ (\( ( 1 + 7060 T + 38950081 T^{2} )^{2} \))(\( ( 1 - 6241 T )( 1 + 6241 T ) \))(\( 1 - 86136100 T^{2} + 4459690111983174 T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!21\)\( T^{8} \))(\( ( 1 + 21484606 T^{2} + 1517108809906561 T^{4} )^{2} \))(\( 1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} - \)\(16\!\cdots\!36\)\( T^{6} + \)\(76\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!64\)\( T^{8} + \)\(29\!\cdots\!80\)\( T^{9} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(21\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!15\)\( T^{13} - \)\(30\!\cdots\!34\)\( T^{14} + \)\(29\!\cdots\!95\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \))
$83$ (\( 1 - 59434754 T^{2} + 2252292232139041 T^{4} \))(\( ( 1 - 6889 T )( 1 + 6889 T ) \))(\( 1 - 855268 T^{2} + 4298268871421190 T^{4} - \)\(19\!\cdots\!88\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} \))(\( ( 1 - 93110306 T^{2} + 2252292232139041 T^{4} )^{2} \))(\( 1 - 37017 T + 725723290 T^{2} - 9956481997959 T^{3} + 104510585134438411 T^{4} - \)\(87\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!28\)\( T^{6} - \)\(40\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} - \)\(19\!\cdots\!96\)\( T^{9} + \)\(14\!\cdots\!48\)\( T^{10} - \)\(93\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!91\)\( T^{12} - \)\(23\!\cdots\!59\)\( T^{13} + \)\(82\!\cdots\!90\)\( T^{14} - \)\(20\!\cdots\!97\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \))
$89$ (\( 1 - 54274304 T^{2} + 3936588805702081 T^{4} \))(\( 1 - 9758 T + 62742241 T^{2} \))(\( ( 1 - 12540 T + 132835270 T^{2} - 786787702140 T^{3} + 3936588805702081 T^{4} )^{2} \))(\( ( 1 + 123412930 T^{2} + 3936588805702081 T^{4} )^{2} \))(\( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!24\)\( T^{10} + \)\(72\!\cdots\!40\)\( T^{12} - \)\(17\!\cdots\!36\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \))
$97$ (\( ( 1 - 4352 T + 88529281 T^{2} )^{2} \))(\( 1 + 1918 T + 88529281 T^{2} \))(\( ( 1 - 11524 T + 146880774 T^{2} - 1020211434244 T^{3} + 7837433594376961 T^{4} )^{2} \))(\( ( 1 + 9730 T + 88529281 T^{2} )^{4} \))(\( 1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{7} + \)\(75\!\cdots\!52\)\( T^{8} + \)\(29\!\cdots\!06\)\( T^{9} - \)\(58\!\cdots\!76\)\( T^{10} - \)\(67\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!91\)\( T^{12} + \)\(54\!\cdots\!52\)\( T^{13} - \)\(13\!\cdots\!84\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \))
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