# Properties

 Label 89.2 Level 89 Weight 2 Dimension 287 Nonzero newspaces 6 Newform subspaces 8 Sturm bound 1320 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$89$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$8$$ Sturm bound: $$1320$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(89))$$.

Total New Old
Modular forms 374 374 0
Cusp forms 287 287 0
Eisenstein series 87 87 0

## Trace form

 $$287q - 41q^{2} - 40q^{3} - 37q^{4} - 38q^{5} - 32q^{6} - 36q^{7} - 29q^{8} - 31q^{9} + O(q^{10})$$ $$287q - 41q^{2} - 40q^{3} - 37q^{4} - 38q^{5} - 32q^{6} - 36q^{7} - 29q^{8} - 31q^{9} - 26q^{10} - 32q^{11} - 16q^{12} - 30q^{13} - 20q^{14} - 20q^{15} - 13q^{16} - 26q^{17} - 5q^{18} - 24q^{19} - 2q^{20} - 12q^{21} - 8q^{22} - 20q^{23} + 16q^{24} - 13q^{25} - 2q^{26} - 4q^{27} + 12q^{28} - 14q^{29} + 28q^{30} - 12q^{31} + 19q^{32} + 4q^{33} + 10q^{34} + 4q^{35} + 47q^{36} - 6q^{37} + 16q^{38} + 12q^{39} + 46q^{40} - 2q^{41} + 52q^{42} + 40q^{44} + 34q^{45} + 28q^{46} + 4q^{47} + 80q^{48} + 13q^{49} + 49q^{50} + 28q^{51} + 54q^{52} + 10q^{53} + 76q^{54} + 28q^{55} + 76q^{56} + 36q^{57} + 46q^{58} + 16q^{59} + 124q^{60} + 18q^{61} + 52q^{62} + 60q^{63} + 83q^{64} + 40q^{65} + 100q^{66} + 24q^{67} + 82q^{68} + 52q^{69} + 100q^{70} + 28q^{71} + 19q^{72} - 14q^{73} - 40q^{74} - 140q^{75} - 168q^{76} - 36q^{77} - 140q^{78} - 52q^{79} - 188q^{80} - 143q^{81} - 94q^{82} - 92q^{83} - 436q^{84} - 68q^{85} - 88q^{86} - 144q^{87} - 172q^{88} - 131q^{89} - 382q^{90} - 108q^{91} - 140q^{92} - 136q^{93} - 76q^{94} - 56q^{95} - 408q^{96} - 78q^{97} - 49q^{98} - 108q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(89))$$

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
89.2.a $$\chi_{89}(1, \cdot)$$ 89.2.a.a 1 1
89.2.a.b 1
89.2.a.c 5
89.2.b $$\chi_{89}(88, \cdot)$$ 89.2.b.a 6 1
89.2.c $$\chi_{89}(34, \cdot)$$ 89.2.c.a 14 2
89.2.e $$\chi_{89}(2, \cdot)$$ 89.2.e.a 60 10
89.2.f $$\chi_{89}(11, \cdot)$$ 89.2.f.a 60 10
89.2.g $$\chi_{89}(5, \cdot)$$ 89.2.g.a 140 20

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + 2 T^{2}$$)($$1 - T + 2 T^{2}$$)($$1 + T - 2 T^{3} + T^{4} + T^{5} + 2 T^{6} - 8 T^{7} + 16 T^{9} + 32 T^{10}$$)($$( 1 + T + 3 T^{2} + 3 T^{3} + 6 T^{4} + 4 T^{5} + 8 T^{6} )^{2}$$)($$( 1 + T + 5 T^{2} + 5 T^{3} + 13 T^{4} + 13 T^{5} + 29 T^{6} + 27 T^{7} + 58 T^{8} + 52 T^{9} + 104 T^{10} + 80 T^{11} + 160 T^{12} + 64 T^{13} + 128 T^{14} )^{2}$$)
$3$ ($$1 + T + 3 T^{2}$$)($$1 - 2 T + 3 T^{2}$$)($$1 + 3 T + 11 T^{2} + 20 T^{3} + 45 T^{4} + 65 T^{5} + 135 T^{6} + 180 T^{7} + 297 T^{8} + 243 T^{9} + 243 T^{10}$$)($$1 - T^{2} + 14 T^{4} + 5 T^{6} + 126 T^{8} - 81 T^{10} + 729 T^{12}$$)($$1 + 2 T + 2 T^{2} + 2 T^{3} + 18 T^{4} + 32 T^{5} + 30 T^{6} + 62 T^{7} + 299 T^{8} + 440 T^{9} + 378 T^{10} + 1038 T^{11} + 3403 T^{12} + 4370 T^{13} + 4238 T^{14} + 13110 T^{15} + 30627 T^{16} + 28026 T^{17} + 30618 T^{18} + 106920 T^{19} + 217971 T^{20} + 135594 T^{21} + 196830 T^{22} + 629856 T^{23} + 1062882 T^{24} + 354294 T^{25} + 1062882 T^{26} + 3188646 T^{27} + 4782969 T^{28}$$)
$5$ ($$1 + T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 + T + 11 T^{2} + 6 T^{3} + 69 T^{4} + 23 T^{5} + 345 T^{6} + 150 T^{7} + 1375 T^{8} + 625 T^{9} + 3125 T^{10}$$)($$( 1 + T + 10 T^{2} + 9 T^{3} + 50 T^{4} + 25 T^{5} + 125 T^{6} )^{2}$$)($$1 - 32 T^{2} + 546 T^{4} - 6362 T^{6} + 56467 T^{8} - 406132 T^{10} + 2476523 T^{12} - 13179966 T^{14} + 61913075 T^{16} - 253832500 T^{18} + 882296875 T^{20} - 2485156250 T^{22} + 5332031250 T^{24} - 7812500000 T^{26} + 6103515625 T^{28}$$)
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 - 2 T + 7 T^{2}$$)($$1 - 8 T + 45 T^{2} - 188 T^{3} + 632 T^{4} - 1820 T^{5} + 4424 T^{6} - 9212 T^{7} + 15435 T^{8} - 19208 T^{9} + 16807 T^{10}$$)($$1 - 12 T^{2} + 147 T^{4} - 1068 T^{6} + 7203 T^{8} - 28812 T^{10} + 117649 T^{12}$$)($$1 + 2 T + 2 T^{2} + 10 T^{3} + 43 T^{4} - 244 T^{5} - 524 T^{6} - 1796 T^{7} - 5991 T^{8} - 19118 T^{9} + 8086 T^{10} + 5090 T^{11} + 231071 T^{12} + 777888 T^{13} + 3935152 T^{14} + 5445216 T^{15} + 11322479 T^{16} + 1745870 T^{17} + 19414486 T^{18} - 321316226 T^{19} - 704835159 T^{20} - 1479083228 T^{21} - 3020755724 T^{22} - 9846280108 T^{23} + 12146435707 T^{24} + 19773267430 T^{25} + 27682574402 T^{26} + 193778020814 T^{27} + 678223072849 T^{28}$$)
$11$ ($$1 + 2 T + 11 T^{2}$$)($$1 + 4 T + 11 T^{2}$$)($$1 - 6 T + 35 T^{2} - 152 T^{3} + 630 T^{4} - 2004 T^{5} + 6930 T^{6} - 18392 T^{7} + 46585 T^{8} - 87846 T^{9} + 161051 T^{10}$$)($$( 1 + 4 T + 29 T^{2} + 68 T^{3} + 319 T^{4} + 484 T^{5} + 1331 T^{6} )^{2}$$)($$( 1 + 2 T + 49 T^{2} + 104 T^{3} + 1261 T^{4} + 2398 T^{5} + 20577 T^{6} + 33376 T^{7} + 226347 T^{8} + 290158 T^{9} + 1678391 T^{10} + 1522664 T^{11} + 7891499 T^{12} + 3543122 T^{13} + 19487171 T^{14} )^{2}$$)
$13$ ($$1 - 2 T + 13 T^{2}$$)($$1 - 2 T + 13 T^{2}$$)($$1 + 37 T^{2} - 56 T^{3} + 598 T^{4} - 1440 T^{5} + 7774 T^{6} - 9464 T^{7} + 81289 T^{8} + 371293 T^{10}$$)($$1 - 18 T^{2} + 383 T^{4} - 6268 T^{6} + 64727 T^{8} - 514098 T^{10} + 4826809 T^{12}$$)($$1 + 4 T + 8 T^{2} - 24 T^{3} - 149 T^{4} + 252 T^{5} + 2488 T^{6} + 8516 T^{7} + 24321 T^{8} - 20200 T^{9} - 77288 T^{10} - 1779348 T^{11} - 3574765 T^{12} + 15363856 T^{13} + 91902064 T^{14} + 199730128 T^{15} - 604135285 T^{16} - 3909227556 T^{17} - 2207422568 T^{18} - 7500118600 T^{19} + 117392821689 T^{20} + 534366370772 T^{21} + 2029538033848 T^{22} + 2672333841996 T^{23} - 20540915285501 T^{24} - 43011849456888 T^{25} + 186384680979848 T^{26} + 1211500426369012 T^{27} + 3937376385699289 T^{28}$$)
$17$ ($$1 - 3 T + 17 T^{2}$$)($$1 - 6 T + 17 T^{2}$$)($$1 + 13 T + 119 T^{2} + 730 T^{3} + 3833 T^{4} + 16423 T^{5} + 65161 T^{6} + 210970 T^{7} + 584647 T^{8} + 1085773 T^{9} + 1419857 T^{10}$$)($$( 1 - 7 T + 44 T^{2} - 179 T^{3} + 748 T^{4} - 2023 T^{5} + 4913 T^{6} )^{2}$$)($$1 - 124 T^{2} + 8162 T^{4} - 369114 T^{6} + 12672043 T^{8} - 347017224 T^{10} + 7783458323 T^{12} - 144931578006 T^{14} + 2249419455347 T^{16} - 28983225565704 T^{18} + 305872312283467 T^{20} - 2574849732077274 T^{22} + 16454542215464738 T^{24} - 72245157416490364 T^{26} + 168377826559400929 T^{28}$$)
$19$ ($$1 + 5 T + 19 T^{2}$$)($$1 + 2 T + 19 T^{2}$$)($$1 - 13 T + 137 T^{2} - 946 T^{3} + 5707 T^{4} - 26363 T^{5} + 108433 T^{6} - 341506 T^{7} + 939683 T^{8} - 1694173 T^{9} + 2476099 T^{10}$$)($$1 - 75 T^{2} + 2588 T^{4} - 57787 T^{6} + 934268 T^{8} - 9774075 T^{10} + 47045881 T^{12}$$)($$1 + 10 T + 50 T^{2} + 222 T^{3} - 374 T^{4} - 9720 T^{5} - 53858 T^{6} - 255342 T^{7} - 350445 T^{8} + 3384160 T^{9} + 21774234 T^{10} + 102778866 T^{11} + 128302787 T^{12} - 1174613430 T^{13} - 6369335410 T^{14} - 22317655170 T^{15} + 46317306107 T^{16} + 704960241894 T^{17} + 2837639949114 T^{18} + 8379515191840 T^{19} - 16486993767045 T^{20} - 228242997579738 T^{21} - 914700738262178 T^{22} - 3136524422411880 T^{23} - 2293018780417574 T^{24} + 25860837475404618 T^{25} + 110665745953308050 T^{26} + 420529834622570590 T^{27} + 799006685782884121 T^{28}$$)
$23$ ($$1 - 7 T + 23 T^{2}$$)($$1 - 2 T + 23 T^{2}$$)($$1 - T + 53 T^{2} + 58 T^{3} + 1643 T^{4} + 2069 T^{5} + 37789 T^{6} + 30682 T^{7} + 644851 T^{8} - 279841 T^{9} + 6436343 T^{10}$$)($$1 - 83 T^{2} + 3748 T^{4} - 105803 T^{6} + 1982692 T^{8} - 23226803 T^{10} + 148035889 T^{12}$$)($$1 + 14 T + 98 T^{2} + 518 T^{3} + 3006 T^{4} + 17020 T^{5} + 77854 T^{6} + 314690 T^{7} + 1166507 T^{8} + 925424 T^{9} - 27541254 T^{10} - 257057158 T^{11} - 1555354817 T^{12} - 8568279442 T^{13} - 43449359890 T^{14} - 197070427166 T^{15} - 822782698193 T^{16} - 3127614441386 T^{17} - 7707172060614 T^{18} + 5956346284432 T^{19} + 172684900769723 T^{20} + 1071464519916430 T^{21} + 6096823448066974 T^{22} + 30655618298100260 T^{23} + 124528092708228894 T^{24} + 493555454599414186 T^{25} + 2147633194337991458 T^{26} + 7056509067110543362 T^{27} + 11592836324538749809 T^{28}$$)
$29$ ($$1 + 29 T^{2}$$)($$1 + 6 T + 29 T^{2}$$)($$1 - 2 T + 73 T^{2} + 80 T^{3} + 2098 T^{4} + 7220 T^{5} + 60842 T^{6} + 67280 T^{7} + 1780397 T^{8} - 1414562 T^{9} + 20511149 T^{10}$$)($$1 - 46 T^{2} + 2575 T^{4} - 70756 T^{6} + 2165575 T^{8} - 32534926 T^{10} + 594823321 T^{12}$$)($$1 - 16 T + 128 T^{2} - 1408 T^{3} + 15627 T^{4} - 118040 T^{5} + 879616 T^{6} - 7452064 T^{7} + 51846113 T^{8} - 322053496 T^{9} + 2230021152 T^{10} - 14226703288 T^{11} + 78071292883 T^{12} - 455921279944 T^{13} + 2661425742560 T^{14} - 13221717118376 T^{15} + 65657957314603 T^{16} - 346975066491032 T^{17} + 1577251590407712 T^{18} - 6605687242426904 T^{19} + 30839277115601273 T^{20} - 128547182246751776 T^{21} + 440024748783102976 T^{22} - 1712423510991576760 T^{23} + 6574391934782241027 T^{24} - 17178317750113807232 T^{25} + 45288292250300037248 T^{26} -$$$$16\!\cdots\!24$$$$T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$)
$31$ ($$1 + 9 T + 31 T^{2}$$)($$1 - 6 T + 31 T^{2}$$)($$1 - 19 T + 257 T^{2} - 2470 T^{3} + 19109 T^{4} - 116615 T^{5} + 592379 T^{6} - 2373670 T^{7} + 7656287 T^{8} - 17546899 T^{9} + 28629151 T^{10}$$)($$1 - 85 T^{2} + 2118 T^{4} - 27403 T^{6} + 2035398 T^{8} - 78499285 T^{10} + 887503681 T^{12}$$)($$1 - 4 T + 8 T^{2} - 330 T^{3} + 630 T^{4} + 10146 T^{5} + 8826 T^{6} + 173392 T^{7} - 3244481 T^{8} + 1429858 T^{9} + 8927334 T^{10} + 222673008 T^{11} + 1670611051 T^{12} - 11426581072 T^{13} + 11410644226 T^{14} - 354224013232 T^{15} + 1605457220011 T^{16} + 6633651581328 T^{17} + 8244580423014 T^{18} + 40935620590558 T^{19} - 2879488830434561 T^{20} + 4770467185934512 T^{21} + 7527616296454266 T^{22} + 268256406442167966 T^{23} + 516365820797904630 T^{24} - 8384797375813594230 T^{25} + 6301302270308398088 T^{26} - 97670185189780170364 T^{27} +$$$$75\!\cdots\!21$$$$T^{28}$$)
$37$ ($$1 + 2 T + 37 T^{2}$$)($$1 - 10 T + 37 T^{2}$$)($$1 + 14 T + 193 T^{2} + 1736 T^{3} + 14658 T^{4} + 91252 T^{5} + 542346 T^{6} + 2376584 T^{7} + 9776029 T^{8} + 26238254 T^{9} + 69343957 T^{10}$$)($$1 - 70 T^{2} + 3927 T^{4} - 168244 T^{6} + 5376063 T^{8} - 131191270 T^{10} + 2565726409 T^{12}$$)($$1 + 22 T + 242 T^{2} + 1750 T^{3} + 7267 T^{4} + 11748 T^{5} + 31092 T^{6} + 1322580 T^{7} + 16825113 T^{8} + 94345746 T^{9} + 161506254 T^{10} - 1263418014 T^{11} - 4551373229 T^{12} + 71775443224 T^{13} + 755700938840 T^{14} + 2655691399288 T^{15} - 6230829950501 T^{16} - 63995912663142 T^{17} + 302688722502894 T^{18} + 6542307353756922 T^{19} + 43168636758509217 T^{20} + 125555002058563140 T^{21} + 109210011181311732 T^{22} + 1526790519112564596 T^{23} + 34943982634360508683 T^{24} +$$$$31\!\cdots\!50$$$$T^{25} +$$$$15\!\cdots\!02$$$$T^{26} +$$$$53\!\cdots\!34$$$$T^{27} +$$$$90\!\cdots\!89$$$$T^{28}$$)
$41$ ($$1 + 41 T^{2}$$)($$1 + 6 T + 41 T^{2}$$)($$1 + 2 T + 145 T^{2} + 304 T^{3} + 10230 T^{4} + 17132 T^{5} + 419430 T^{6} + 511024 T^{7} + 9993545 T^{8} + 5651522 T^{9} + 115856201 T^{10}$$)($$1 - 186 T^{2} + 16343 T^{4} - 850636 T^{6} + 27472583 T^{8} - 525591546 T^{10} + 4750104241 T^{12}$$)($$1 - 12 T + 72 T^{2} - 556 T^{3} + 7019 T^{4} - 57896 T^{5} + 343952 T^{6} - 2763984 T^{7} + 28448809 T^{8} - 196269132 T^{9} + 1105464760 T^{10} - 8590003548 T^{11} + 70063443675 T^{12} - 400196298120 T^{13} + 2172549144320 T^{14} - 16408048222920 T^{15} + 117776648817675 T^{16} - 592031634531708 T^{17} + 3123779205682360 T^{18} - 22738996007087532 T^{19} + 135134808282298969 T^{20} - 538297696938701904 T^{21} + 2746431002406626192 T^{22} - 18954104473672766056 T^{23} + 94213645697959702619 T^{24} -$$$$30\!\cdots\!96$$$$T^{25} +$$$$16\!\cdots\!32$$$$T^{26} -$$$$11\!\cdots\!52$$$$T^{27} +$$$$37\!\cdots\!61$$$$T^{28}$$)
$43$ ($$1 + 7 T + 43 T^{2}$$)($$1 - 2 T + 43 T^{2}$$)($$1 - T + 147 T^{2} - 228 T^{3} + 10595 T^{4} - 14337 T^{5} + 455585 T^{6} - 421572 T^{7} + 11687529 T^{8} - 3418801 T^{9} + 147008443 T^{10}$$)($$1 - 203 T^{2} + 18728 T^{4} - 1018803 T^{6} + 34628072 T^{8} - 694016603 T^{10} + 6321363049 T^{12}$$)($$1 - 28 T + 392 T^{2} - 4562 T^{3} + 52882 T^{4} - 530218 T^{5} + 4522282 T^{6} - 36812060 T^{7} + 286165135 T^{8} - 1992567554 T^{9} + 12970023350 T^{10} - 83557756760 T^{11} + 521727361623 T^{12} - 3203807206688 T^{13} + 20438812605970 T^{14} - 137763709887584 T^{15} + 964673891640927 T^{16} - 6643426566717320 T^{17} + 44341928799003350 T^{18} - 292924253685858422 T^{19} + 1808953710301096615 T^{20} - 10006203021187550420 T^{21} + 52857337727790005482 T^{22} -$$$$26\!\cdots\!74$$$$T^{23} +$$$$11\!\cdots\!18$$$$T^{24} -$$$$42\!\cdots\!34$$$$T^{25} +$$$$15\!\cdots\!92$$$$T^{26} -$$$$48\!\cdots\!04$$$$T^{27} +$$$$73\!\cdots\!49$$$$T^{28}$$)
$47$ ($$1 + 12 T + 47 T^{2}$$)($$1 - 12 T + 47 T^{2}$$)($$1 + 4 T + 191 T^{2} + 784 T^{3} + 15998 T^{4} + 56008 T^{5} + 751906 T^{6} + 1731856 T^{7} + 19830193 T^{8} + 19518724 T^{9} + 229345007 T^{10}$$)($$( 1 - 6 T + 113 T^{2} - 568 T^{3} + 5311 T^{4} - 13254 T^{5} + 103823 T^{6} )^{2}$$)($$1 - 338 T^{2} + 54419 T^{4} - 5648652 T^{6} + 432542577 T^{8} - 26582620462 T^{10} + 1405536746315 T^{12} - 68054730181256 T^{14} + 3104830672609835 T^{16} - 129714707998632622 T^{18} + 4662469576443562833 T^{20} -$$$$13\!\cdots\!72$$$$T^{22} +$$$$28\!\cdots\!31$$$$T^{24} -$$$$39\!\cdots\!58$$$$T^{26} +$$$$25\!\cdots\!69$$$$T^{28}$$)
$53$ ($$1 + 3 T + 53 T^{2}$$)($$1 + 6 T + 53 T^{2}$$)($$1 + 11 T + 259 T^{2} + 1990 T^{3} + 26589 T^{4} + 150461 T^{5} + 1409217 T^{6} + 5589910 T^{7} + 38559143 T^{8} + 86795291 T^{9} + 418195493 T^{10}$$)($$( 1 - 15 T + 230 T^{2} - 1693 T^{3} + 12190 T^{4} - 42135 T^{5} + 148877 T^{6} )^{2}$$)($$1 - 636 T^{2} + 191570 T^{4} - 36285506 T^{6} + 4836020611 T^{8} - 480392973480 T^{10} + 36716573992427 T^{12} - 2194817183961406 T^{14} + 103136856344727443 T^{16} - 3790531629777443880 T^{18} +$$$$10\!\cdots\!19$$$$T^{20} -$$$$22\!\cdots\!66$$$$T^{22} +$$$$33\!\cdots\!30$$$$T^{24} -$$$$31\!\cdots\!76$$$$T^{26} +$$$$13\!\cdots\!69$$$$T^{28}$$)
$59$ ($$1 - 4 T + 59 T^{2}$$)($$1 + 10 T + 59 T^{2}$$)($$1 + 177 T^{2} + 784 T^{3} + 12024 T^{4} + 94092 T^{5} + 709416 T^{6} + 2729104 T^{7} + 36352083 T^{8} + 714924299 T^{10}$$)($$1 - 104 T^{2} + 10715 T^{4} - 638580 T^{6} + 37298915 T^{8} - 1260205544 T^{10} + 42180533641 T^{12}$$)($$1 + 26 T + 338 T^{2} + 3186 T^{3} + 25511 T^{4} + 196252 T^{5} + 1555132 T^{6} + 12113036 T^{7} + 94970485 T^{8} + 752851954 T^{9} + 5650710870 T^{10} + 39259804354 T^{11} + 301161563639 T^{12} + 2691835042880 T^{13} + 22541436238208 T^{14} + 158818267529920 T^{15} + 1048343403027359 T^{16} + 8063139358420166 T^{17} + 68471703518414070 T^{18} + 538232155464230246 T^{19} + 4005905737444585885 T^{20} + 30145125027066000484 T^{21} +$$$$22\!\cdots\!72$$$$T^{22} +$$$$17\!\cdots\!28$$$$T^{23} +$$$$13\!\cdots\!11$$$$T^{24} +$$$$96\!\cdots\!74$$$$T^{25} +$$$$60\!\cdots\!78$$$$T^{26} +$$$$27\!\cdots\!54$$$$T^{27} +$$$$61\!\cdots\!61$$$$T^{28}$$)
$61$ ($$1 - 6 T + 61 T^{2}$$)($$1 + 6 T + 61 T^{2}$$)($$1 - 4 T + 297 T^{2} - 952 T^{3} + 35762 T^{4} - 86392 T^{5} + 2181482 T^{6} - 3542392 T^{7} + 67413357 T^{8} - 55383364 T^{9} + 844596301 T^{10}$$)($$1 - 254 T^{2} + 30975 T^{4} - 2340644 T^{6} + 115257975 T^{8} - 3516843614 T^{10} + 51520374361 T^{12}$$)($$1 - 4 T + 8 T^{2} + 488 T^{3} - 85 T^{4} - 7948 T^{5} + 151544 T^{6} + 400764 T^{7} - 19829535 T^{8} + 115093352 T^{9} - 61697704 T^{10} - 12243100668 T^{11} + 59167758003 T^{12} - 63846491072 T^{13} - 4027242548368 T^{14} - 3894635955392 T^{15} + 220163227529163 T^{16} - 2778951232723308 T^{17} - 854256599649064 T^{18} + 97207419368890952 T^{19} - 1021625066604552135 T^{20} + 1259498189935120044 T^{21} + 29052093040859951864 T^{22} - 92945073145845752668 T^{23} - 60634147491345021085 T^{24} +$$$$21\!\cdots\!68$$$$T^{25} +$$$$21\!\cdots\!68$$$$T^{26} -$$$$64\!\cdots\!24$$$$T^{27} +$$$$98\!\cdots\!41$$$$T^{28}$$)
$67$ ($$1 - 12 T + 67 T^{2}$$)($$1 - 12 T + 67 T^{2}$$)($$1 - 4 T + 199 T^{2} - 832 T^{3} + 22354 T^{4} - 73576 T^{5} + 1497718 T^{6} - 3734848 T^{7} + 59851837 T^{8} - 80604484 T^{9} + 1350125107 T^{10}$$)($$( 1 + 8 T + 177 T^{2} + 980 T^{3} + 11859 T^{4} + 35912 T^{5} + 300763 T^{6} )^{2}$$)($$( 1 - 12 T + 213 T^{2} - 1184 T^{3} + 12797 T^{4} - 6928 T^{5} + 387605 T^{6} + 2482312 T^{7} + 25969535 T^{8} - 31099792 T^{9} + 3848864111 T^{10} - 23858927264 T^{11} + 287576647791 T^{12} - 1085500586028 T^{13} + 6060711605323 T^{14} )^{2}$$)
$71$ ($$1 + 10 T + 71 T^{2}$$)($$1 - 4 T + 71 T^{2}$$)($$1 + 2 T + 75 T^{2} - 56 T^{3} + 10050 T^{4} + 18892 T^{5} + 713550 T^{6} - 282296 T^{7} + 26843325 T^{8} + 50823362 T^{9} + 1804229351 T^{10}$$)($$( 1 - 18 T + 257 T^{2} - 2260 T^{3} + 18247 T^{4} - 90738 T^{5} + 357911 T^{6} )^{2}$$)($$1 - 510 T^{2} + 132955 T^{4} - 23571692 T^{6} + 3191870969 T^{8} - 349818877474 T^{10} + 31952750326859 T^{12} - 2464386329928296 T^{14} + 161073814397696219 T^{16} - 8889485722147373794 T^{18} +$$$$40\!\cdots\!49$$$$T^{20} -$$$$15\!\cdots\!12$$$$T^{22} +$$$$43\!\cdots\!55$$$$T^{24} -$$$$83\!\cdots\!10$$$$T^{26} +$$$$82\!\cdots\!81$$$$T^{28}$$)
$73$ ($$1 - 7 T + 73 T^{2}$$)($$1 - 10 T + 73 T^{2}$$)($$1 + 25 T + 551 T^{2} + 7534 T^{3} + 92429 T^{4} + 830039 T^{5} + 6747317 T^{6} + 40148686 T^{7} + 214348367 T^{8} + 709956025 T^{9} + 2073071593 T^{10}$$)($$( 1 + 15 T + 210 T^{2} + 1627 T^{3} + 15330 T^{4} + 79935 T^{5} + 389017 T^{6} )^{2}$$)($$( 1 + 10 T + 420 T^{2} + 4150 T^{3} + 80729 T^{4} + 730308 T^{5} + 9208237 T^{6} + 70179122 T^{7} + 672201301 T^{8} + 3891811332 T^{9} + 31404953393 T^{10} + 117852700150 T^{11} + 870690069060 T^{12} + 1513342262890 T^{13} + 11047398519097 T^{14} )^{2}$$)
$79$ ($$1 + 6 T + 79 T^{2}$$)($$1 + 12 T + 79 T^{2}$$)($$1 - 54 T + 1491 T^{2} - 27416 T^{3} + 367554 T^{4} - 3732164 T^{5} + 29036766 T^{6} - 171103256 T^{7} + 735121149 T^{8} - 2103304374 T^{9} + 3077056399 T^{10}$$)($$( 1 + 10 T + 177 T^{2} + 1444 T^{3} + 13983 T^{4} + 62410 T^{5} + 493039 T^{6} )^{2}$$)($$1 - 754 T^{2} + 282523 T^{4} - 69270836 T^{6} + 12376897865 T^{8} - 1701945768494 T^{10} + 185595925800795 T^{12} - 16299300492956760 T^{14} + 1158304172922761595 T^{16} - 66290925540448548014 T^{18} +$$$$30\!\cdots\!65$$$$T^{20} -$$$$10\!\cdots\!96$$$$T^{22} +$$$$26\!\cdots\!23$$$$T^{24} -$$$$44\!\cdots\!14$$$$T^{26} +$$$$36\!\cdots\!81$$$$T^{28}$$)
$83$ ($$1 - 12 T + 83 T^{2}$$)($$1 + 6 T + 83 T^{2}$$)($$1 + 20 T + 493 T^{2} + 6396 T^{3} + 88140 T^{4} + 786372 T^{5} + 7315620 T^{6} + 44062044 T^{7} + 281890991 T^{8} + 949166420 T^{9} + 3939040643 T^{10}$$)($$1 - 332 T^{2} + 56875 T^{4} - 5866028 T^{6} + 391811875 T^{8} - 15756162572 T^{10} + 326940373369 T^{12}$$)($$1 - 18 T + 162 T^{2} + 1130 T^{3} - 25753 T^{4} + 86556 T^{5} + 3252428 T^{6} - 34848668 T^{7} - 8035499 T^{8} + 3552562546 T^{9} - 14517620978 T^{10} - 200342246650 T^{11} + 2669682685911 T^{12} + 6698847332000 T^{13} - 212481562494256 T^{14} + 556004328556000 T^{15} + 18391444023240879 T^{16} - 114553092185263550 T^{17} - 688981916530257938 T^{18} + 13993688255493557078 T^{19} - 2627129043266226131 T^{20} -$$$$94\!\cdots\!36$$$$T^{21} +$$$$73\!\cdots\!48$$$$T^{22} +$$$$16\!\cdots\!68$$$$T^{23} -$$$$39\!\cdots\!97$$$$T^{24} +$$$$14\!\cdots\!10$$$$T^{25} +$$$$17\!\cdots\!82$$$$T^{26} -$$$$15\!\cdots\!34$$$$T^{27} +$$$$73\!\cdots\!29$$$$T^{28}$$)
$89$ ($$1 + T$$)($$1 - T$$)($$( 1 - T )^{5}$$)($$1 + 2 T - 41 T^{2} - 1188 T^{3} - 3649 T^{4} + 15842 T^{5} + 704969 T^{6}$$)($$1 + 8 T - 45 T^{2} - 1608 T^{3} - 20871 T^{4} - 45256 T^{5} + 1587875 T^{6} + 21100752 T^{7} + 141320875 T^{8} - 358472776 T^{9} - 14713407999 T^{10} - 100889523528 T^{11} - 251282675205 T^{12} + 3975850327688 T^{13} + 44231334895529 T^{14}$$)
$97$ ($$1 - 9 T + 97 T^{2}$$)($$1 + 18 T + 97 T^{2}$$)($$1 - 13 T + 355 T^{2} - 2294 T^{3} + 42401 T^{4} - 178803 T^{5} + 4112897 T^{6} - 21584246 T^{7} + 323998915 T^{8} - 1150880653 T^{9} + 8587340257 T^{10}$$)($$( 1 + 9 T + 284 T^{2} + 1609 T^{3} + 27548 T^{4} + 84681 T^{5} + 912673 T^{6} )^{2}$$)($$( 1 + 24 T + 642 T^{2} + 10418 T^{3} + 167995 T^{4} + 2127794 T^{5} + 25696579 T^{6} + 261010122 T^{7} + 2492568163 T^{8} + 20020413746 T^{9} + 153324500635 T^{10} + 922298049458 T^{11} + 5513072444994 T^{12} + 19991328118296 T^{13} + 80798284478113 T^{14} )^{2}$$)