# Properties

 Label 128.3 Level 128 Weight 3 Dimension 552 Nonzero newspaces 5 Newform subspaces 9 Sturm bound 3072 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$5$$ Newform subspaces: $$9$$ Sturm bound: $$3072$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(128))$$.

Total New Old
Modular forms 1104 600 504
Cusp forms 944 552 392
Eisenstein series 160 48 112

## Trace form

 $$552q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} + O(q^{10})$$ $$552q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} - 16q^{10} - 12q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 8q^{15} - 16q^{16} - 24q^{17} - 16q^{18} - 12q^{19} - 16q^{20} + 20q^{21} - 16q^{22} + 52q^{23} - 16q^{24} + 76q^{25} - 16q^{26} + 84q^{27} - 16q^{28} + 16q^{29} - 16q^{30} - 16q^{31} - 16q^{32} - 96q^{33} - 16q^{34} - 108q^{35} - 16q^{36} - 112q^{37} - 16q^{38} - 204q^{39} - 16q^{40} - 180q^{41} - 16q^{42} - 108q^{43} - 16q^{44} - 188q^{45} - 16q^{46} - 8q^{47} - 16q^{48} - 220q^{49} - 640q^{50} - 240q^{51} - 1072q^{52} - 336q^{53} - 1168q^{54} - 268q^{55} - 800q^{56} - 404q^{57} - 736q^{58} - 140q^{59} - 592q^{60} - 144q^{61} - 112q^{62} - 32q^{63} + 176q^{64} + 112q^{65} + 560q^{66} + 148q^{67} + 464q^{68} + 404q^{69} + 1328q^{70} + 244q^{71} + 1280q^{72} + 620q^{73} + 1216q^{74} + 472q^{75} + 1648q^{76} + 628q^{77} + 1424q^{78} + 504q^{79} + 800q^{80} + 748q^{81} - 16q^{82} + 468q^{83} - 16q^{84} + 504q^{85} - 16q^{86} + 436q^{87} - 16q^{88} + 364q^{89} - 16q^{90} + 180q^{91} - 16q^{92} + 80q^{93} - 16q^{94} - 16q^{95} - 16q^{96} - 160q^{97} - 16q^{98} - 232q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(128))$$

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
128.3.c $$\chi_{128}(127, \cdot)$$ 128.3.c.a 4 1
128.3.c.b 4
128.3.d $$\chi_{128}(63, \cdot)$$ 128.3.d.a 2 1
128.3.d.b 2
128.3.d.c 4
128.3.f $$\chi_{128}(31, \cdot)$$ 128.3.f.a 6 2
128.3.f.b 6
128.3.h $$\chi_{128}(15, \cdot)$$ 128.3.h.a 28 4
128.3.j $$\chi_{128}(7, \cdot)$$ None 0 8
128.3.l $$\chi_{128}(3, \cdot)$$ 128.3.l.a 496 16

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(128))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(128)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 - 12 T^{2} + 70 T^{4} - 972 T^{6} + 6561 T^{8}$$)($$1 - 12 T^{2} + 70 T^{4} - 972 T^{6} + 6561 T^{8}$$)($$( 1 + 9 T^{2} )^{2}$$)($$1 - 14 T^{2} + 81 T^{4}$$)($$( 1 + 10 T^{2} + 81 T^{4} )^{2}$$)($$1 + 2 T + 2 T^{2} - 14 T^{3} - 65 T^{4} + 124 T^{5} + 476 T^{6} + 1116 T^{7} - 5265 T^{8} - 10206 T^{9} + 13122 T^{10} + 118098 T^{11} + 531441 T^{12}$$)($$1 - 2 T + 2 T^{2} + 14 T^{3} - 65 T^{4} - 124 T^{5} + 476 T^{6} - 1116 T^{7} - 5265 T^{8} + 10206 T^{9} + 13122 T^{10} - 118098 T^{11} + 531441 T^{12}$$)
$5$ ($$( 1 + 4 T + 22 T^{2} + 100 T^{3} + 625 T^{4} )^{2}$$)($$( 1 - 4 T + 22 T^{2} - 100 T^{3} + 625 T^{4} )^{2}$$)($$( 1 - 6 T + 25 T^{2} )( 1 + 6 T + 25 T^{2} )$$)($$( 1 - 5 T )^{2}( 1 + 5 T )^{2}$$)($$( 1 - 34 T^{2} + 625 T^{4} )^{2}$$)($$1 - 2 T + 2 T^{2} + 14 T^{3} - 369 T^{4} - 636 T^{5} + 2108 T^{6} - 15900 T^{7} - 230625 T^{8} + 218750 T^{9} + 781250 T^{10} - 19531250 T^{11} + 244140625 T^{12}$$)($$1 - 2 T + 2 T^{2} + 14 T^{3} - 369 T^{4} - 636 T^{5} + 2108 T^{6} - 15900 T^{7} - 230625 T^{8} + 218750 T^{9} + 781250 T^{10} - 19531250 T^{11} + 244140625 T^{12}$$)
$7$ ($$1 - 100 T^{2} + 5254 T^{4} - 240100 T^{6} + 5764801 T^{8}$$)($$1 - 100 T^{2} + 5254 T^{4} - 240100 T^{6} + 5764801 T^{8}$$)($$( 1 - 7 T )^{2}( 1 + 7 T )^{2}$$)($$( 1 - 7 T )^{2}( 1 + 7 T )^{2}$$)($$( 1 + 30 T^{2} + 2401 T^{4} )^{2}$$)($$( 1 - 2 T + 87 T^{2} - 332 T^{3} + 4263 T^{4} - 4802 T^{5} + 117649 T^{6} )^{2}$$)($$( 1 + 2 T + 87 T^{2} + 332 T^{3} + 4263 T^{4} + 4802 T^{5} + 117649 T^{6} )^{2}$$)
$11$ ($$1 - 140 T^{2} + 5382 T^{4} - 2049740 T^{6} + 214358881 T^{8}$$)($$1 - 140 T^{2} + 5382 T^{4} - 2049740 T^{6} + 214358881 T^{8}$$)($$( 1 + 121 T^{2} )^{2}$$)($$1 - 46 T^{2} + 14641 T^{4}$$)($$( 1 + 42 T^{2} + 14641 T^{4} )^{2}$$)($$1 + 18 T + 162 T^{2} + 2146 T^{3} + 17759 T^{4} + 65756 T^{5} + 609308 T^{6} + 7956476 T^{7} + 260009519 T^{8} + 3801769906 T^{9} + 34726138722 T^{10} + 466873642818 T^{11} + 3138428376721 T^{12}$$)($$1 - 18 T + 162 T^{2} - 2146 T^{3} + 17759 T^{4} - 65756 T^{5} + 609308 T^{6} - 7956476 T^{7} + 260009519 T^{8} - 3801769906 T^{9} + 34726138722 T^{10} - 466873642818 T^{11} + 3138428376721 T^{12}$$)
$13$ ($$( 1 - 12 T + 342 T^{2} - 2028 T^{3} + 28561 T^{4} )^{2}$$)($$( 1 + 12 T + 342 T^{2} + 2028 T^{3} + 28561 T^{4} )^{2}$$)($$( 1 - 10 T + 169 T^{2} )( 1 + 10 T + 169 T^{2} )$$)($$( 1 - 13 T )^{2}( 1 + 13 T )^{2}$$)($$( 1 + 62 T^{2} + 28561 T^{4} )^{2}$$)($$1 - 2 T + 2 T^{2} - 1554 T^{3} - 7825 T^{4} + 453380 T^{5} + 316348 T^{6} + 76621220 T^{7} - 223489825 T^{8} - 7500861186 T^{9} + 1631461442 T^{10} - 275716983698 T^{11} + 23298085122481 T^{12}$$)($$1 - 2 T + 2 T^{2} - 1554 T^{3} - 7825 T^{4} + 453380 T^{5} + 316348 T^{6} + 76621220 T^{7} - 223489825 T^{8} - 7500861186 T^{9} + 1631461442 T^{10} - 275716983698 T^{11} + 23298085122481 T^{12}$$)
$17$ ($$( 1 + 4 T + 454 T^{2} + 1156 T^{3} + 83521 T^{4} )^{2}$$)($$( 1 + 4 T + 454 T^{2} + 1156 T^{3} + 83521 T^{4} )^{2}$$)($$( 1 - 30 T + 289 T^{2} )^{2}$$)($$( 1 + 2 T + 289 T^{2} )^{2}$$)($$( 1 + 10 T + 289 T^{2} )^{4}$$)($$( 1 + 2 T + 607 T^{2} - 388 T^{3} + 175423 T^{4} + 167042 T^{5} + 24137569 T^{6} )^{2}$$)($$( 1 + 2 T + 607 T^{2} - 388 T^{3} + 175423 T^{4} + 167042 T^{5} + 24137569 T^{6} )^{2}$$)
$19$ ($$1 - 780 T^{2} + 402374 T^{4} - 101650380 T^{6} + 16983563041 T^{8}$$)($$1 - 780 T^{2} + 402374 T^{4} - 101650380 T^{6} + 16983563041 T^{8}$$)($$( 1 + 361 T^{2} )^{2}$$)($$1 + 434 T^{2} + 130321 T^{4}$$)($$( 1 + 522 T^{2} + 130321 T^{4} )^{2}$$)($$1 - 30 T + 450 T^{2} - 12014 T^{3} + 441215 T^{4} - 8004292 T^{5} + 113750108 T^{6} - 2889549412 T^{7} + 57499580015 T^{8} - 565209214334 T^{9} + 7642603368450 T^{10} - 183931987734030 T^{11} + 2213314919066161 T^{12}$$)($$1 + 30 T + 450 T^{2} + 12014 T^{3} + 441215 T^{4} + 8004292 T^{5} + 113750108 T^{6} + 2889549412 T^{7} + 57499580015 T^{8} + 565209214334 T^{9} + 7642603368450 T^{10} + 183931987734030 T^{11} + 2213314919066161 T^{12}$$)
$23$ ($$1 - 484 T^{2} + 517894 T^{4} - 135443044 T^{6} + 78310985281 T^{8}$$)($$1 - 484 T^{2} + 517894 T^{4} - 135443044 T^{6} + 78310985281 T^{8}$$)($$( 1 - 23 T )^{2}( 1 + 23 T )^{2}$$)($$( 1 - 23 T )^{2}( 1 + 23 T )^{2}$$)($$( 1 - 930 T^{2} + 279841 T^{4} )^{2}$$)($$( 1 + 30 T + 1751 T^{2} + 30772 T^{3} + 926279 T^{4} + 8395230 T^{5} + 148035889 T^{6} )^{2}$$)($$( 1 - 30 T + 1751 T^{2} - 30772 T^{3} + 926279 T^{4} - 8395230 T^{5} + 148035889 T^{6} )^{2}$$)
$29$ ($$( 1 + 68 T + 2806 T^{2} + 57188 T^{3} + 707281 T^{4} )^{2}$$)($$( 1 - 68 T + 2806 T^{2} - 57188 T^{3} + 707281 T^{4} )^{2}$$)($$( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} )$$)($$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$)($$( 1 - 1282 T^{2} + 707281 T^{4} )^{2}$$)($$1 - 18 T + 162 T^{2} + 4894 T^{3} + 124463 T^{4} - 24625372 T^{5} + 435069308 T^{6} - 20709937852 T^{7} + 88030315103 T^{8} + 2911065332974 T^{9} + 81039918899682 T^{10} - 7572730199403618 T^{11} + 353814783205469041 T^{12}$$)($$1 - 18 T + 162 T^{2} + 4894 T^{3} + 124463 T^{4} - 24625372 T^{5} + 435069308 T^{6} - 20709937852 T^{7} + 88030315103 T^{8} + 2911065332974 T^{9} + 81039918899682 T^{10} - 7572730199403618 T^{11} + 353814783205469041 T^{12}$$)
$31$ ($$( 1 + 126 T^{2} + 923521 T^{4} )^{2}$$)($$( 1 + 126 T^{2} + 923521 T^{4} )^{2}$$)($$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$)($$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$)($$( 1 - 31 T )^{4}( 1 + 31 T )^{4}$$)($$1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 6586358756111 T^{8} - 3280218929998086 T^{10} + 787662783788549761 T^{12}$$)($$1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 6586358756111 T^{8} - 3280218929998086 T^{10} + 787662783788549761 T^{12}$$)
$37$ ($$( 1 + 20 T + 1270 T^{2} + 27380 T^{3} + 1874161 T^{4} )^{2}$$)($$( 1 - 20 T + 1270 T^{2} - 27380 T^{3} + 1874161 T^{4} )^{2}$$)($$( 1 - 70 T + 1369 T^{2} )( 1 + 70 T + 1369 T^{2} )$$)($$( 1 - 37 T )^{2}( 1 + 37 T )^{2}$$)($$( 1 - 2338 T^{2} + 1874161 T^{4} )^{2}$$)($$1 + 46 T + 1058 T^{2} - 6594 T^{3} - 356337 T^{4} + 78343460 T^{5} + 4002544124 T^{6} + 107252196740 T^{7} - 667832908257 T^{8} - 16918399940946 T^{9} + 3716203262248418 T^{10} + 221194881131221054 T^{11} + 6582952005840035281 T^{12}$$)($$1 + 46 T + 1058 T^{2} - 6594 T^{3} - 356337 T^{4} + 78343460 T^{5} + 4002544124 T^{6} + 107252196740 T^{7} - 667832908257 T^{8} - 16918399940946 T^{9} + 3716203262248418 T^{10} + 221194881131221054 T^{11} + 6582952005840035281 T^{12}$$)
$41$ ($$( 1 - 4 T + 2854 T^{2} - 6724 T^{3} + 2825761 T^{4} )^{2}$$)($$( 1 - 4 T + 2854 T^{2} - 6724 T^{3} + 2825761 T^{4} )^{2}$$)($$( 1 - 18 T + 1681 T^{2} )^{2}$$)($$( 1 + 46 T + 1681 T^{2} )^{2}$$)($$( 1 + 30 T + 1681 T^{2} )^{4}$$)($$1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 42528333194703 T^{8} - 40675209117142374 T^{10} + 22563490300366186081 T^{12}$$)($$1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 42528333194703 T^{8} - 40675209117142374 T^{10} + 22563490300366186081 T^{12}$$)
$43$ ($$1 - 2764 T^{2} + 8710534 T^{4} - 9449565964 T^{6} + 11688200277601 T^{8}$$)($$1 - 2764 T^{2} + 8710534 T^{4} - 9449565964 T^{6} + 11688200277601 T^{8}$$)($$( 1 + 1849 T^{2} )^{2}$$)($$1 - 3502 T^{2} + 3418801 T^{4}$$)($$( 1 + 3690 T^{2} + 3418801 T^{4} )^{2}$$)($$1 + 114 T + 6498 T^{2} + 241730 T^{3} + 12357983 T^{4} + 838941724 T^{5} + 44553879452 T^{6} + 1551203247676 T^{7} + 42249484638383 T^{8} + 1528063089834770 T^{9} + 75949925403851298 T^{10} + 2463708983714404386 T^{11} + 39959630797262576401 T^{12}$$)($$1 - 114 T + 6498 T^{2} - 241730 T^{3} + 12357983 T^{4} - 838941724 T^{5} + 44553879452 T^{6} - 1551203247676 T^{7} + 42249484638383 T^{8} - 1528063089834770 T^{9} + 75949925403851298 T^{10} - 2463708983714404386 T^{11} + 39959630797262576401 T^{12}$$)
$47$ ($$1 - 7428 T^{2} + 23258246 T^{4} - 36246270468 T^{6} + 23811286661761 T^{8}$$)($$1 - 7428 T^{2} + 23258246 T^{4} - 36246270468 T^{6} + 23811286661761 T^{8}$$)($$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$)($$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$)($$( 1 + 190 T^{2} + 4879681 T^{4} )^{2}$$)($$1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 60811985090447 T^{8} - 111389199003717958 T^{10} +$$$$11\!\cdots\!41$$$$T^{12}$$)($$1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 60811985090447 T^{8} - 111389199003717958 T^{10} +$$$$11\!\cdots\!41$$$$T^{12}$$)
$53$ ($$( 1 - 44 T + 6070 T^{2} - 123596 T^{3} + 7890481 T^{4} )^{2}$$)($$( 1 + 44 T + 6070 T^{2} + 123596 T^{3} + 7890481 T^{4} )^{2}$$)($$( 1 - 90 T + 2809 T^{2} )( 1 + 90 T + 2809 T^{2} )$$)($$( 1 - 53 T )^{2}( 1 + 53 T )^{2}$$)($$( 1 - 2018 T^{2} + 7890481 T^{4} )^{2}$$)($$1 + 78 T + 3042 T^{2} + 270110 T^{3} + 31648463 T^{4} + 1389102820 T^{5} + 48555101564 T^{6} + 3901989821380 T^{7} + 249721595980703 T^{8} + 5986815584554190 T^{9} + 189393978231360162 T^{10} + 13641222688510017822 T^{11} +$$$$49\!\cdots\!41$$$$T^{12}$$)($$1 + 78 T + 3042 T^{2} + 270110 T^{3} + 31648463 T^{4} + 1389102820 T^{5} + 48555101564 T^{6} + 3901989821380 T^{7} + 249721595980703 T^{8} + 5986815584554190 T^{9} + 189393978231360162 T^{10} + 13641222688510017822 T^{11} +$$$$49\!\cdots\!41$$$$T^{12}$$)
$59$ ($$1 - 11020 T^{2} + 52720774 T^{4} - 133533318220 T^{6} + 146830437604321 T^{8}$$)($$1 - 11020 T^{2} + 52720774 T^{4} - 133533318220 T^{6} + 146830437604321 T^{8}$$)($$( 1 + 3481 T^{2} )^{2}$$)($$1 - 238 T^{2} + 12117361 T^{4}$$)($$( 1 + 5162 T^{2} + 12117361 T^{4} )^{2}$$)($$1 - 206 T + 21218 T^{2} - 1942462 T^{3} + 171214239 T^{4} - 11916831972 T^{5} + 708622973852 T^{6} - 41482492094532 T^{7} + 2074664742303279 T^{8} - 81934083737364142 T^{9} + 3115448225088482978 T^{10} -$$$$10\!\cdots\!06$$$$T^{11} +$$$$17\!\cdots\!81$$$$T^{12}$$)($$1 + 206 T + 21218 T^{2} + 1942462 T^{3} + 171214239 T^{4} + 11916831972 T^{5} + 708622973852 T^{6} + 41482492094532 T^{7} + 2074664742303279 T^{8} + 81934083737364142 T^{9} + 3115448225088482978 T^{10} +$$$$10\!\cdots\!06$$$$T^{11} +$$$$17\!\cdots\!81$$$$T^{12}$$)
$61$ ($$( 1 + 148 T + 11350 T^{2} + 550708 T^{3} + 13845841 T^{4} )^{2}$$)($$( 1 - 148 T + 11350 T^{2} - 550708 T^{3} + 13845841 T^{4} )^{2}$$)($$( 1 - 22 T + 3721 T^{2} )( 1 + 22 T + 3721 T^{2} )$$)($$( 1 - 61 T )^{2}( 1 + 61 T )^{2}$$)($$( 1 - 6658 T^{2} + 13845841 T^{4} )^{2}$$)($$1 + 30 T + 450 T^{2} + 111694 T^{3} + 33268655 T^{4} + 582006980 T^{5} + 8727089468 T^{6} + 2165647972580 T^{7} + 460632507413855 T^{8} + 5754516693877534 T^{9} + 86268290848776450 T^{10} + 21400287349886478030 T^{11} +$$$$26\!\cdots\!21$$$$T^{12}$$)($$1 + 30 T + 450 T^{2} + 111694 T^{3} + 33268655 T^{4} + 582006980 T^{5} + 8727089468 T^{6} + 2165647972580 T^{7} + 460632507413855 T^{8} + 5754516693877534 T^{9} + 86268290848776450 T^{10} + 21400287349886478030 T^{11} +$$$$26\!\cdots\!21$$$$T^{12}$$)
$67$ ($$1 - 11980 T^{2} + 75342534 T^{4} - 241410429580 T^{6} + 406067677556641 T^{8}$$)($$1 - 11980 T^{2} + 75342534 T^{4} - 241410429580 T^{6} + 406067677556641 T^{8}$$)($$( 1 + 4489 T^{2} )^{2}$$)($$1 - 5134 T^{2} + 20151121 T^{4}$$)($$( 1 + 2250 T^{2} + 20151121 T^{4} )^{2}$$)($$1 + 226 T + 25538 T^{2} + 2083538 T^{3} + 120508479 T^{4} + 5203289532 T^{5} + 268963196252 T^{6} + 23357566709148 T^{7} + 2428380941854959 T^{8} + 188473476667633922 T^{9} + 10370156349441497858 T^{10} +$$$$41\!\cdots\!74$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12}$$)($$1 - 226 T + 25538 T^{2} - 2083538 T^{3} + 120508479 T^{4} - 5203289532 T^{5} + 268963196252 T^{6} - 23357566709148 T^{7} + 2428380941854959 T^{8} - 188473476667633922 T^{9} + 10370156349441497858 T^{10} -$$$$41\!\cdots\!74$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12}$$)
$71$ ($$1 - 18276 T^{2} + 133865606 T^{4} - 464423881956 T^{6} + 645753531245761 T^{8}$$)($$1 - 18276 T^{2} + 133865606 T^{4} - 464423881956 T^{6} + 645753531245761 T^{8}$$)($$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$)($$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$)($$( 1 - 6882 T^{2} + 25411681 T^{4} )^{2}$$)($$( 1 - 130 T + 11575 T^{2} - 918796 T^{3} + 58349575 T^{4} - 3303518530 T^{5} + 128100283921 T^{6} )^{2}$$)($$( 1 + 130 T + 11575 T^{2} + 918796 T^{3} + 58349575 T^{4} + 3303518530 T^{5} + 128100283921 T^{6} )^{2}$$)
$73$ ($$( 1 - 44 T + 9990 T^{2} - 234476 T^{3} + 28398241 T^{4} )^{2}$$)($$( 1 - 44 T + 9990 T^{2} - 234476 T^{3} + 28398241 T^{4} )^{2}$$)($$( 1 - 110 T + 5329 T^{2} )^{2}$$)($$( 1 - 142 T + 5329 T^{2} )^{2}$$)($$( 1 + 10 T + 5329 T^{2} )^{4}$$)($$1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 2941261225338191 T^{8} - 10585595166201707206 T^{10} +$$$$22\!\cdots\!21$$$$T^{12}$$)($$1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 2941261225338191 T^{8} - 10585595166201707206 T^{10} +$$$$22\!\cdots\!21$$$$T^{12}$$)
$79$ ($$1 - 1028 T^{2} + 28324230 T^{4} - 40040683268 T^{6} + 1517108809906561 T^{8}$$)($$1 - 1028 T^{2} + 28324230 T^{4} - 40040683268 T^{6} + 1517108809906561 T^{8}$$)($$( 1 - 79 T )^{2}( 1 + 79 T )^{2}$$)($$( 1 - 79 T )^{2}( 1 + 79 T )^{2}$$)($$( 1 + 318 T^{2} + 38950081 T^{4} )^{2}$$)($$1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 3284433446508175 T^{8} - 106197616693459270 T^{10} +$$$$59\!\cdots\!41$$$$T^{12}$$)($$1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 3284433446508175 T^{8} - 106197616693459270 T^{10} +$$$$59\!\cdots\!41$$$$T^{12}$$)
$83$ ($$1 - 5452 T^{2} + 97966918 T^{4} - 258742766092 T^{6} + 2252292232139041 T^{8}$$)($$1 - 5452 T^{2} + 97966918 T^{4} - 258742766092 T^{6} + 2252292232139041 T^{8}$$)($$( 1 + 6889 T^{2} )^{2}$$)($$1 + 11186 T^{2} + 47458321 T^{4}$$)($$( 1 + 13130 T^{2} + 47458321 T^{4} )^{2}$$)($$1 - 318 T + 50562 T^{2} - 6712846 T^{3} + 819490815 T^{4} - 81203275140 T^{5} + 6918697616348 T^{6} - 559409362439460 T^{7} + 38891658154821615 T^{8} - 2194700377608598174 T^{9} +$$$$11\!\cdots\!42$$$$T^{10} -$$$$49\!\cdots\!82$$$$T^{11} +$$$$10\!\cdots\!61$$$$T^{12}$$)($$1 + 318 T + 50562 T^{2} + 6712846 T^{3} + 819490815 T^{4} + 81203275140 T^{5} + 6918697616348 T^{6} + 559409362439460 T^{7} + 38891658154821615 T^{8} + 2194700377608598174 T^{9} +$$$$11\!\cdots\!42$$$$T^{10} +$$$$49\!\cdots\!82$$$$T^{11} +$$$$10\!\cdots\!61$$$$T^{12}$$)
$89$ ($$( 1 - 108 T + 15558 T^{2} - 855468 T^{3} + 62742241 T^{4} )^{2}$$)($$( 1 - 108 T + 15558 T^{2} - 855468 T^{3} + 62742241 T^{4} )^{2}$$)($$( 1 - 78 T + 7921 T^{2} )^{2}$$)($$( 1 + 146 T + 7921 T^{2} )^{2}$$)($$( 1 - 22 T + 7921 T^{2} )^{4}$$)($$1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 29249082478431439 T^{8} -$$$$12\!\cdots\!78$$$$T^{10} +$$$$24\!\cdots\!21$$$$T^{12}$$)($$1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 29249082478431439 T^{8} -$$$$12\!\cdots\!78$$$$T^{10} +$$$$24\!\cdots\!21$$$$T^{12}$$)
$97$ ($$( 1 + 164 T + 22342 T^{2} + 1543076 T^{3} + 88529281 T^{4} )^{2}$$)($$( 1 + 164 T + 22342 T^{2} + 1543076 T^{3} + 88529281 T^{4} )^{2}$$)($$( 1 + 130 T + 9409 T^{2} )^{2}$$)($$( 1 - 94 T + 9409 T^{2} )^{2}$$)($$( 1 - 150 T + 9409 T^{2} )^{4}$$)($$( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 100553983 T^{4} + 177058562 T^{5} + 832972004929 T^{6} )^{2}$$)($$( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 100553983 T^{4} + 177058562 T^{5} + 832972004929 T^{6} )^{2}$$)