# Properties

 Label 135.2 Level 135 Weight 2 Dimension 422 Nonzero newspaces 9 Newform subspaces 16 Sturm bound 2592 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$16$$ Sturm bound: $$2592$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(135))$$.

Total New Old
Modular forms 768 518 250
Cusp forms 529 422 107
Eisenstein series 239 96 143

## Trace form

 $$422q - 4q^{2} - 12q^{3} - 12q^{4} - 13q^{5} - 48q^{6} - 14q^{7} - 36q^{8} - 24q^{9} + O(q^{10})$$ $$422q - 4q^{2} - 12q^{3} - 12q^{4} - 13q^{5} - 48q^{6} - 14q^{7} - 36q^{8} - 24q^{9} - 33q^{10} - 46q^{11} - 48q^{12} - 34q^{13} - 66q^{14} - 33q^{15} - 80q^{16} - 46q^{17} - 42q^{18} - 16q^{19} - 45q^{20} - 24q^{21} - 58q^{22} - 12q^{23} + 12q^{24} - 37q^{25} - 4q^{26} - 6q^{27} - 60q^{28} - 32q^{29} - 15q^{30} - 70q^{31} + 16q^{32} - 12q^{33} - 46q^{34} + 8q^{35} - 12q^{36} - 36q^{37} - 16q^{38} - 30q^{39} + 31q^{40} + 10q^{41} + 36q^{42} + 14q^{43} + 166q^{44} + 45q^{45} - 10q^{46} + 98q^{47} + 126q^{48} + 152q^{50} + 36q^{51} + 30q^{52} + 80q^{53} + 156q^{54} - 60q^{55} + 66q^{56} + 18q^{57} - 70q^{58} - 28q^{59} + 48q^{60} - 126q^{61} - 84q^{62} - 42q^{63} - 84q^{64} - 87q^{65} - 30q^{66} - 56q^{67} - 62q^{68} - 42q^{69} - 45q^{70} - 158q^{71} + 36q^{72} - 34q^{73} - 14q^{74} - 21q^{75} + 28q^{76} + 54q^{77} + 96q^{78} + 94q^{79} + 150q^{80} - 24q^{81} + 120q^{82} + 78q^{83} + 192q^{84} + 63q^{85} + 170q^{86} + 78q^{87} + 168q^{88} + 186q^{89} + 186q^{90} + 38q^{91} + 402q^{92} + 198q^{93} + 146q^{94} + 165q^{95} + 168q^{96} + 60q^{97} + 370q^{98} + 150q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$

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
135.2.a $$\chi_{135}(1, \cdot)$$ 135.2.a.a 1 1
135.2.a.b 1
135.2.a.c 2
135.2.a.d 2
135.2.b $$\chi_{135}(109, \cdot)$$ 135.2.b.a 4 1
135.2.b.b 4
135.2.e $$\chi_{135}(46, \cdot)$$ 135.2.e.a 2 2
135.2.e.b 6
135.2.f $$\chi_{135}(53, \cdot)$$ 135.2.f.a 8 2
135.2.f.b 8
135.2.j $$\chi_{135}(19, \cdot)$$ 135.2.j.a 8 2
135.2.k $$\chi_{135}(16, \cdot)$$ 135.2.k.a 30 6
135.2.k.b 42
135.2.m $$\chi_{135}(8, \cdot)$$ 135.2.m.a 16 4
135.2.p $$\chi_{135}(4, \cdot)$$ 135.2.p.a 96 6
135.2.q $$\chi_{135}(2, \cdot)$$ 135.2.q.a 192 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(135))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(135)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T + 2 T^{2}$$)($$1 - 2 T + 2 T^{2}$$)($$1 + T + T^{2} + 2 T^{3} + 4 T^{4}$$)($$1 - T + T^{2} - 2 T^{3} + 4 T^{4}$$)($$1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8}$$)($$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$)($$1 - T - T^{2} - 2 T^{3} + 4 T^{4}$$)($$1 - T + T^{3} - 2 T^{4} - T^{5} + 11 T^{6} - 2 T^{7} - 8 T^{8} + 8 T^{9} - 32 T^{11} + 64 T^{12}$$)($$1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16}$$)($$( 1 - 2 T^{4} + 16 T^{8} )^{2}$$)($$1 + 4 T^{2} + 7 T^{4} + 4 T^{6} + T^{8} + 16 T^{10} + 112 T^{12} + 256 T^{14} + 256 T^{16}$$)($$1 - 6 T + 18 T^{2} - 36 T^{3} + 58 T^{4} - 78 T^{5} + 72 T^{6} - 149 T^{8} + 354 T^{9} - 576 T^{10} + 732 T^{11} - 530 T^{12} - 480 T^{13} + 2592 T^{14} - 5496 T^{15} + 8641 T^{16} - 10992 T^{17} + 10368 T^{18} - 3840 T^{19} - 8480 T^{20} + 23424 T^{21} - 36864 T^{22} + 45312 T^{23} - 38144 T^{24} + 73728 T^{26} - 159744 T^{27} + 237568 T^{28} - 294912 T^{29} + 294912 T^{30} - 196608 T^{31} + 65536 T^{32}$$)
$3$ ()()()()()()()()()()()()
$5$ ($$1 + T$$)($$1 - T$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + 5 T^{2} )^{2}$$)($$1 - 8 T^{2} + 25 T^{4}$$)($$1 + T + T^{2}$$)($$( 1 - T + T^{2} )^{3}$$)($$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$)($$1 - 4 T^{2} - 100 T^{6} + 625 T^{8}$$)($$1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8}$$)($$1 - 6 T + 16 T^{2} - 24 T^{3} + 22 T^{4} - 30 T^{5} + 136 T^{6} - 630 T^{7} + 1831 T^{8} - 3150 T^{9} + 3400 T^{10} - 3750 T^{11} + 13750 T^{12} - 75000 T^{13} + 250000 T^{14} - 468750 T^{15} + 390625 T^{16}$$)
$7$ ($$1 + 3 T + 7 T^{2}$$)($$1 + 3 T + 7 T^{2}$$)($$1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}$$)($$1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{4}$$)($$( 1 - 5 T^{2} + 49 T^{4} )^{2}$$)($$1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4}$$)($$1 + 5 T + T^{2} - 14 T^{3} + 73 T^{4} + 173 T^{5} - 82 T^{6} + 1211 T^{7} + 3577 T^{8} - 4802 T^{9} + 2401 T^{10} + 84035 T^{11} + 117649 T^{12}$$)($$( 1 + 2 T + 2 T^{2} - 6 T^{3} - 82 T^{4} - 42 T^{5} + 98 T^{6} + 686 T^{7} + 2401 T^{8} )^{2}$$)($$( 1 - 4 T + 8 T^{2} - 24 T^{3} + 71 T^{4} - 168 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2}$$)($$1 + 16 T^{2} + 121 T^{4} + 592 T^{6} + 3280 T^{8} + 29008 T^{10} + 290521 T^{12} + 1882384 T^{14} + 5764801 T^{16}$$)($$1 + 2 T + 2 T^{2} - 16 T^{3} - 66 T^{4} - 104 T^{5} + 52 T^{6} + 1072 T^{7} - 235 T^{8} + 2614 T^{9} - 2614 T^{10} + 626 T^{11} + 8430 T^{12} - 20146 T^{13} + 75264 T^{14} - 1168356 T^{15} + 4373152 T^{16} - 8178492 T^{17} + 3687936 T^{18} - 6910078 T^{19} + 20240430 T^{20} + 10521182 T^{21} - 307534486 T^{22} + 2152741402 T^{23} - 1354728235 T^{24} + 43259066704 T^{25} + 14688712948 T^{26} - 205641981272 T^{27} - 913524955266 T^{28} - 1550224166512 T^{29} + 1356446145698 T^{30} + 9495123019886 T^{31} + 33232930569601 T^{32}$$)
$11$ ($$1 + 2 T + 11 T^{2}$$)($$1 - 2 T + 11 T^{2}$$)($$1 - 2 T + 10 T^{2} - 22 T^{3} + 121 T^{4}$$)($$1 + 2 T + 10 T^{2} + 22 T^{3} + 121 T^{4}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 + 4 T^{2} + 121 T^{4} )^{2}$$)($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 + 2 T - 21 T^{2} - 14 T^{3} + 286 T^{4} - 58 T^{5} - 3673 T^{6} - 638 T^{7} + 34606 T^{8} - 18634 T^{9} - 307461 T^{10} + 322102 T^{11} + 1771561 T^{12}$$)($$( 1 - 10 T^{2} + 78 T^{4} - 1210 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 28 T^{2} + 384 T^{4} - 3388 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 396 T^{5} + 968 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 34 T^{2} + 652 T^{4} + 414 T^{5} + 9016 T^{6} + 9072 T^{7} + 100687 T^{8} + 99792 T^{9} + 1090936 T^{10} + 551034 T^{11} + 9545932 T^{12} + 60233074 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 + 5 T + 13 T^{2}$$)($$1 + 5 T + 13 T^{2}$$)($$1 - 6 T + 22 T^{2} - 78 T^{3} + 169 T^{4}$$)($$1 - 6 T + 22 T^{2} - 78 T^{3} + 169 T^{4}$$)($$( 1 - 13 T^{2} )^{4}$$)($$( 1 - 17 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)($$1 + 4 T - 19 T^{2} - 60 T^{3} + 370 T^{4} + 536 T^{5} - 4235 T^{6} + 6968 T^{7} + 62530 T^{8} - 131820 T^{9} - 542659 T^{10} + 1485172 T^{11} + 4826809 T^{12}$$)($$( 1 + 2 T + 2 T^{2} + 6 T^{3} - 82 T^{4} + 78 T^{5} + 338 T^{6} + 4394 T^{7} + 28561 T^{8} )^{2}$$)($$( 1 + 8 T + 32 T^{2} + 144 T^{3} + 623 T^{4} + 1872 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2}$$)($$( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2}$$)($$1 + 2 T + 2 T^{2} + 56 T^{3} + 144 T^{4} + 658 T^{5} + 2596 T^{6} + 4150 T^{7} + 35702 T^{8} - 23108 T^{9} - 42562 T^{10} + 1025978 T^{11} - 7929096 T^{12} - 10433170 T^{13} - 40700658 T^{14} - 417980868 T^{15} - 885935405 T^{16} - 5433751284 T^{17} - 6878411202 T^{18} - 22921674490 T^{19} - 226462910856 T^{20} + 380938449554 T^{21} - 205438644658 T^{22} - 1449992730836 T^{23} + 29123218201142 T^{24} + 44008672397950 T^{25} + 357880644840004 T^{26} + 1179241539276346 T^{27} + 3354924257637264 T^{28} + 16961005969166168 T^{29} + 7874752771398578 T^{30} + 102371786028181514 T^{31} + 665416609183179841 T^{32}$$)
$17$ ($$1 + 8 T + 17 T^{2}$$)($$1 - 8 T + 17 T^{2}$$)($$1 + 4 T + 25 T^{2} + 68 T^{3} + 289 T^{4}$$)($$1 - 4 T + 25 T^{2} - 68 T^{3} + 289 T^{4}$$)($$1 + 14 T^{2} - 93 T^{4} + 4046 T^{6} + 83521 T^{8}$$)($$( 1 - 26 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 4 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 43 T^{2} - 56 T^{3} + 731 T^{4} - 578 T^{5} + 4913 T^{6} )^{2}$$)($$( 1 + 151 T^{4} + 83521 T^{8} )^{2}$$)($$1 - 76 T^{4} - 111450 T^{8} - 6347596 T^{12} + 6975757441 T^{16}$$)($$( 1 - 40 T^{2} + 786 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2}$$)($$1 + 964 T^{4} + 237772 T^{8} - 70157828 T^{12} - 47638889354 T^{16} - 5859651952388 T^{20} + 1658639798261452 T^{24} + 561647836689489604 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$1 - T + 19 T^{2}$$)($$1 - T + 19 T^{2}$$)($$1 + 25 T^{2} + 361 T^{4}$$)($$1 + 25 T^{2} + 361 T^{4}$$)($$( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + T + 19 T^{2} )^{4}$$)($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 53 T^{2} - 148 T^{3} + 1007 T^{4} - 1444 T^{5} + 6859 T^{6} )^{2}$$)($$( 1 - 29 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 29 T^{2} + 361 T^{4} )^{4}$$)($$( 1 + 2 T + 36 T^{2} + 38 T^{3} + 361 T^{4} )^{4}$$)($$( 1 - 92 T^{2} + 4132 T^{4} - 123032 T^{6} + 2693650 T^{8} - 44414552 T^{10} + 538486372 T^{12} - 4328221052 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 - 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$( 1 + 3 T + 23 T^{2} )^{2}$$)($$( 1 - 3 T + 23 T^{2} )^{2}$$)($$1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8}$$)($$( 1 + 4 T^{2} + 529 T^{4} )^{2}$$)($$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 - 3 T - 27 T^{2} - 66 T^{3} + 405 T^{4} + 2625 T^{5} - 9794 T^{6} + 60375 T^{7} + 214245 T^{8} - 803022 T^{9} - 7555707 T^{10} - 19309029 T^{11} + 148035889 T^{12}$$)($$1 + 14 T^{4} - 420045 T^{8} + 3917774 T^{12} + 78310985281 T^{16}$$)($$1 + 1424 T^{4} + 1024290 T^{8} + 398493584 T^{12} + 78310985281 T^{16}$$)($$1 + 88 T^{2} + 4753 T^{4} + 170104 T^{6} + 4569664 T^{8} + 89985016 T^{10} + 1330084273 T^{12} + 13027158232 T^{14} + 78310985281 T^{16}$$)($$1 + 18 T + 162 T^{2} + 972 T^{3} + 4138 T^{4} + 11184 T^{5} + 3348 T^{6} - 119100 T^{7} - 338219 T^{8} + 2163498 T^{9} + 26656146 T^{10} + 182453766 T^{11} + 892508410 T^{12} + 1883089950 T^{13} - 11227491072 T^{14} - 141260851668 T^{15} - 837597200144 T^{16} - 3248999588364 T^{17} - 5939342777088 T^{18} + 22911555421650 T^{19} + 249760445962810 T^{20} + 1174335019617738 T^{21} + 3946066270423794 T^{22} + 7366333044933606 T^{23} - 26486263130754539 T^{24} - 214517281980243300 T^{25} + 138695959543296852 T^{26} + 10656224332509359568 T^{27} + 90682715899700088298 T^{28} +$$$$48\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!58$$$$T^{30} +$$$$47\!\cdots\!26$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$1 - 2 T + 29 T^{2}$$)($$1 + 2 T + 29 T^{2}$$)($$1 - 10 T + 70 T^{2} - 290 T^{3} + 841 T^{4}$$)($$1 + 10 T + 70 T^{2} + 290 T^{3} + 841 T^{4}$$)($$( 1 + 29 T^{2} )^{4}$$)($$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$)($$1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4}$$)($$1 + 7 T - 9 T^{2} - 304 T^{3} - 803 T^{4} + 2929 T^{5} + 36038 T^{6} + 84941 T^{7} - 675323 T^{8} - 7414256 T^{9} - 6365529 T^{10} + 143578043 T^{11} + 594823321 T^{12}$$)($$( 1 + 50 T^{2} + 2118 T^{4} + 42050 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 + 20 T^{2} + 1728 T^{4} + 16820 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 + 6 T - 19 T^{2} - 18 T^{3} + 1140 T^{4} - 522 T^{5} - 15979 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2}$$)($$1 - 148 T^{2} + 10806 T^{4} - 562376 T^{6} + 24743345 T^{8} - 973959144 T^{10} + 34914297094 T^{12} - 1155748123996 T^{14} + 35134519733316 T^{16} - 971984172280636 T^{18} + 24694218962941414 T^{20} - 579333612552397224 T^{22} + 12377769580906494545 T^{24} -$$$$23\!\cdots\!76$$$$T^{26} +$$$$38\!\cdots\!46$$$$T^{28} -$$$$44\!\cdots\!88$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$1 + 31 T^{2}$$)($$1 + 31 T^{2}$$)($$1 + 4 T + 53 T^{2} + 124 T^{3} + 961 T^{4}$$)($$1 + 4 T + 53 T^{2} + 124 T^{3} + 961 T^{4}$$)($$( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 2 T + 31 T^{2} )^{4}$$)($$1 - 31 T^{2} + 961 T^{4}$$)($$1 + 8 T + 31 T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 240436 T^{7} - 151838 T^{8} + 6196528 T^{9} + 28629151 T^{10} + 229033208 T^{11} + 887503681 T^{12}$$)($$( 1 + T + 31 T^{2} )^{8}$$)($$( 1 - 4 T + 12 T^{2} - 124 T^{3} + 961 T^{4} )^{4}$$)($$( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 124 T^{5} - 53816 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 2 T - 78 T^{2} + 76 T^{3} + 3500 T^{4} - 7572 T^{5} - 100340 T^{6} + 131474 T^{7} + 2627523 T^{8} + 4075694 T^{9} - 96426740 T^{10} - 225577452 T^{11} + 3232323500 T^{12} + 2175815476 T^{13} - 69225287118 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$1 - 5 T + 37 T^{2}$$)($$1 - 5 T + 37 T^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 37 T^{2} )^{4}$$)($$( 1 + 7 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 - 6 T + 99 T^{2} - 440 T^{3} + 3663 T^{4} - 8214 T^{5} + 50653 T^{6} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{4}( 1 + 12 T + 37 T^{2} )^{4}$$)($$( 1 + 8 T + 32 T^{2} + 144 T^{3} + 287 T^{4} + 5328 T^{5} + 43808 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 - 56 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 - 2 T + 2 T^{2} + 50 T^{3} + 680 T^{4} - 6430 T^{5} + 12750 T^{6} - 67506 T^{7} - 1916078 T^{8} - 2497722 T^{9} + 17454750 T^{10} - 325698790 T^{11} + 1274429480 T^{12} + 3467197850 T^{13} + 5131452818 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$1 + 10 T + 41 T^{2}$$)($$1 - 10 T + 41 T^{2}$$)($$1 + 2 T + 70 T^{2} + 82 T^{3} + 1681 T^{4}$$)($$1 - 2 T + 70 T^{2} - 82 T^{3} + 1681 T^{4}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 64 T^{2} + 1681 T^{4} )^{2}$$)($$1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4}$$)($$1 + 13 T + 27 T^{2} - 292 T^{3} + 445 T^{4} + 22279 T^{5} + 169790 T^{6} + 913439 T^{7} + 748045 T^{8} - 20124932 T^{9} + 76295547 T^{10} + 1506130613 T^{11} + 4750104241 T^{12}$$)($$( 1 - 130 T^{2} + 7398 T^{4} - 218530 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 28 T^{2} - 816 T^{4} - 47068 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 12 T + 29 T^{2} - 396 T^{3} + 5640 T^{4} - 16236 T^{5} + 48749 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 - 12 T + 190 T^{2} - 1704 T^{3} + 16861 T^{4} - 130290 T^{5} + 1067482 T^{6} - 7225794 T^{7} + 51027208 T^{8} - 296257554 T^{9} + 1794437242 T^{10} - 8979717090 T^{11} + 47645156221 T^{12} - 197418966504 T^{13} + 902519805790 T^{14} - 2337051286572 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$1 - 4 T + 43 T^{2}$$)($$1 - 4 T + 43 T^{2}$$)($$1 + 6 T + 82 T^{2} + 258 T^{3} + 1849 T^{4}$$)($$1 + 6 T + 82 T^{2} + 258 T^{3} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 50 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$1 + 10 T - 25 T^{2} - 462 T^{3} + 1690 T^{4} + 17990 T^{5} + 34975 T^{6} + 773570 T^{7} + 3124810 T^{8} - 36732234 T^{9} - 85470025 T^{10} + 1470084430 T^{11} + 6321363049 T^{12}$$)($$( 1 - 22 T + 242 T^{2} - 2046 T^{3} + 14798 T^{4} - 87978 T^{5} + 447458 T^{6} - 1749154 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 4 T + 8 T^{2} - 132 T^{3} + 2078 T^{4} - 5676 T^{5} + 14792 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$)($$1 + 88 T^{2} + 3838 T^{4} + 18304 T^{6} - 2480621 T^{8} + 33844096 T^{10} + 13121358238 T^{12} + 556279948312 T^{14} + 11688200277601 T^{16}$$)($$1 + 2 T + 2 T^{2} - 112 T^{3} - 3840 T^{4} - 7982 T^{5} - 2012 T^{6} + 464698 T^{7} + 6098798 T^{8} + 19506388 T^{9} - 1100914 T^{10} - 890327938 T^{11} - 7456003800 T^{12} - 38444523010 T^{13} - 1944910962 T^{14} + 1611174599916 T^{15} + 14665036034491 T^{16} + 69280507796388 T^{17} - 3596140368738 T^{18} - 3056608690956070 T^{19} - 25490593247443800 T^{20} - 130885723924780534 T^{21} - 6959277079726786 T^{22} + 5302199293874251516 T^{23} + 71283972476632423598 T^{24} +$$$$23\!\cdots\!14$$$$T^{25} - 43482302414327908988 T^{26} -$$$$74\!\cdots\!74$$$$T^{27} -$$$$15\!\cdots\!40$$$$T^{28} -$$$$19\!\cdots\!16$$$$T^{29} +$$$$14\!\cdots\!98$$$$T^{30} +$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$1 - 4 T + 47 T^{2}$$)($$1 + 4 T + 47 T^{2}$$)($$1 + 4 T + 46 T^{2} + 188 T^{3} + 2209 T^{4}$$)($$1 - 4 T + 46 T^{2} - 188 T^{3} + 2209 T^{4}$$)($$( 1 - 14 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 86 T^{2} + 2209 T^{4} )^{2}$$)($$1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4}$$)($$1 - 13 T + 39 T^{2} + 16 T^{3} - 299 T^{4} + 19253 T^{5} - 232366 T^{6} + 904891 T^{7} - 660491 T^{8} + 1661168 T^{9} + 190307559 T^{10} - 2981485091 T^{11} + 10779215329 T^{12}$$)($$1 + 5444 T^{4} + 14797830 T^{8} + 26564983364 T^{12} + 23811286661761 T^{16}$$)($$1 - 3964 T^{4} + 8158086 T^{8} - 19343055484 T^{12} + 23811286661761 T^{16}$$)($$1 + 160 T^{2} + 14809 T^{4} + 1019680 T^{6} + 55015600 T^{8} + 2252473120 T^{10} + 72263195929 T^{12} + 1724674452640 T^{14} + 23811286661761 T^{16}$$)($$1 - 12 T + 72 T^{2} - 288 T^{3} - 134 T^{4} + 24762 T^{5} - 246024 T^{6} + 2286006 T^{7} - 18983651 T^{8} + 98916168 T^{9} - 204929766 T^{10} - 1977304884 T^{11} + 21327903874 T^{12} - 124668663210 T^{13} + 894258211950 T^{14} - 7685415723390 T^{15} + 46309407650224 T^{16} - 361214538999330 T^{17} + 1975416390197550 T^{18} - 12943474620451830 T^{19} + 104073367303784194 T^{20} - 453485002462114188 T^{21} - 2208982075035583014 T^{22} + 50113217696402345784 T^{23} -$$$$45\!\cdots\!11$$$$T^{24} +$$$$25\!\cdots\!02$$$$T^{25} -$$$$12\!\cdots\!76$$$$T^{26} +$$$$61\!\cdots\!86$$$$T^{27} -$$$$15\!\cdots\!94$$$$T^{28} -$$$$15\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!68$$$$T^{30} -$$$$14\!\cdots\!16$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 2 T + 53 T^{2}$$)($$1 - 2 T + 53 T^{2}$$)($$1 + 4 T + 97 T^{2} + 212 T^{3} + 2809 T^{4}$$)($$1 - 4 T + 97 T^{2} - 212 T^{3} + 2809 T^{4}$$)($$1 + 86 T^{2} + 4587 T^{4} + 241574 T^{6} + 7890481 T^{8}$$)($$( 1 - 8 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 2 T + 53 T^{2} )^{2}$$)($$( 1 - 2 T + 139 T^{2} - 188 T^{3} + 7367 T^{4} - 5618 T^{5} + 148877 T^{6} )^{2}$$)($$1 + 14 T^{4} - 15680685 T^{8} + 110466734 T^{12} + 62259690411361 T^{16}$$)($$1 - 8464 T^{4} + 31941186 T^{8} - 66785031184 T^{12} + 62259690411361 T^{16}$$)($$( 1 - 196 T^{2} + 15174 T^{4} - 550564 T^{6} + 7890481 T^{8} )^{2}$$)($$1 - 6680 T^{4} + 12266524 T^{8} + 25145244376 T^{12} - 161780081477882 T^{16} + 198408072989184856 T^{20} +$$$$76\!\cdots\!64$$$$T^{24} -$$$$32\!\cdots\!80$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$1 + 8 T + 59 T^{2}$$)($$1 - 8 T + 59 T^{2}$$)($$1 + 10 T + 130 T^{2} + 590 T^{3} + 3481 T^{4}$$)($$1 - 10 T + 130 T^{2} - 590 T^{3} + 3481 T^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 46 T^{2} + 3481 T^{4} )^{2}$$)($$1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4}$$)($$1 + 2 T - 153 T^{2} - 110 T^{3} + 14962 T^{4} + 3194 T^{5} - 1012513 T^{6} + 188446 T^{7} + 52082722 T^{8} - 22591690 T^{9} - 1853956233 T^{10} + 1429848598 T^{11} + 42180533641 T^{12}$$)($$( 1 + 170 T^{2} + 13998 T^{4} + 591770 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 4 T^{2} + 1566 T^{4} - 13924 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 12 T + 2 T^{2} + 288 T^{3} + 8187 T^{4} + 16992 T^{5} + 6962 T^{6} + 2464548 T^{7} + 12117361 T^{8} )^{2}$$)($$1 - 328 T^{2} + 54732 T^{4} - 6528848 T^{6} + 640897706 T^{8} - 55004293416 T^{10} + 4215408852016 T^{12} - 289798546717912 T^{14} + 17966936709291123 T^{16} - 1008788741125051672 T^{18} + 51079630822473449776 T^{20} -$$$$23\!\cdots\!56$$$$T^{22} +$$$$94\!\cdots\!26$$$$T^{24} -$$$$33\!\cdots\!48$$$$T^{26} +$$$$97\!\cdots\!92$$$$T^{28} -$$$$20\!\cdots\!08$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$1 - 7 T + 61 T^{2}$$)($$1 - 7 T + 61 T^{2}$$)($$1 - 6 T + 79 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$1 - 6 T + 79 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 13 T + 61 T^{2} )^{4}$$)($$1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4}$$)($$1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 859307 T^{7} + 45459457 T^{8} - 54475440 T^{9} - 2007646945 T^{10} + 844596301 T^{11} + 51520374361 T^{12}$$)($$( 1 + T + 61 T^{2} )^{8}$$)($$( 1 + T + 61 T^{2} )^{8}$$)($$( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 17324 T^{5} - 130235 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 - 4 T - 126 T^{2} + 1444 T^{3} + 6365 T^{4} - 126018 T^{5} + 435286 T^{6} + 4572842 T^{7} - 47796264 T^{8} + 278943362 T^{9} + 1619699206 T^{10} - 28603691658 T^{11} + 88128777965 T^{12} + 1219597058644 T^{13} - 6491567169486 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 + 9 T + 67 T^{2}$$)($$1 + 9 T + 67 T^{2}$$)($$1 + 16 T + 146 T^{2} + 1072 T^{3} + 4489 T^{4}$$)($$1 + 16 T + 146 T^{2} + 1072 T^{3} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 125 T^{2} + 4489 T^{4} )^{2}$$)($$1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4}$$)($$1 + 11 T - 41 T^{2} - 152 T^{3} + 4753 T^{4} - 32755 T^{5} - 764902 T^{6} - 2194585 T^{7} + 21336217 T^{8} - 45715976 T^{9} - 826195961 T^{10} + 14851376177 T^{11} + 90458382169 T^{12}$$)($$( 1 - 16 T + 128 T^{2} - 912 T^{3} + 6398 T^{4} - 61104 T^{5} + 574592 T^{6} - 4812208 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 20 T + 200 T^{2} + 2280 T^{3} + 23783 T^{4} + 152760 T^{5} + 897800 T^{6} + 6015260 T^{7} + 20151121 T^{8} )^{2}$$)($$1 + 184 T^{2} + 16657 T^{4} + 1512664 T^{6} + 123674896 T^{8} + 6790348696 T^{10} + 335657222497 T^{12} + 16644342319096 T^{14} + 406067677556641 T^{16}$$)($$1 - 4 T + 8 T^{2} + 1436 T^{3} - 5070 T^{4} + 45274 T^{5} + 890512 T^{6} - 2479154 T^{7} + 72952373 T^{8} + 410747644 T^{9} - 246911326 T^{10} + 51154442144 T^{11} + 220045950810 T^{12} + 1378484916350 T^{13} + 21635958751422 T^{14} + 133914238479582 T^{15} + 1373516219103616 T^{16} + 8972253978131994 T^{17} + 97123818835133358 T^{18} + 414597258896175050 T^{19} + 4434172580332358010 T^{20} + 69064896673193309408 T^{21} - 22335199089162546094 T^{22} +$$$$24\!\cdots\!12$$$$T^{23} +$$$$29\!\cdots\!93$$$$T^{24} -$$$$67\!\cdots\!38$$$$T^{25} +$$$$16\!\cdots\!88$$$$T^{26} +$$$$55\!\cdots\!42$$$$T^{27} -$$$$41\!\cdots\!70$$$$T^{28} +$$$$78\!\cdots\!32$$$$T^{29} +$$$$29\!\cdots\!32$$$$T^{30} -$$$$98\!\cdots\!72$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$1 - 2 T + 71 T^{2}$$)($$1 + 2 T + 71 T^{2}$$)($$1 - 22 T + 250 T^{2} - 1562 T^{3} + 5041 T^{4}$$)($$1 + 22 T + 250 T^{2} + 1562 T^{3} + 5041 T^{4}$$)($$( 1 + 71 T^{2} )^{4}$$)($$( 1 - 20 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 - 10 T + 121 T^{2} - 712 T^{3} + 8591 T^{4} - 50410 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 - 250 T^{2} + 25518 T^{4} - 1260250 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 100 T^{2} + 12528 T^{4} - 504100 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 + 18 T + 196 T^{2} + 1278 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 452 T^{2} + 95704 T^{4} - 12356144 T^{6} + 1062541066 T^{8} - 62287321904 T^{10} + 2431999518424 T^{12} - 57901328332292 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$1 + 5 T + 73 T^{2}$$)($$1 + 5 T + 73 T^{2}$$)($$1 - 18 T + 214 T^{2} - 1314 T^{3} + 5329 T^{4}$$)($$1 - 18 T + 214 T^{2} - 1314 T^{3} + 5329 T^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 65 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 4 T + 73 T^{2} )^{2}$$)($$( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 11315 T^{4} + 42632 T^{5} + 389017 T^{6} )^{2}$$)($$( 1 + 14 T + 98 T^{2} + 42 T^{3} - 4882 T^{4} + 3066 T^{5} + 522242 T^{6} + 5446238 T^{7} + 28398241 T^{8} )^{2}$$)($$( 1 - 4 T + 8 T^{2} - 192 T^{3} + 3983 T^{4} - 14016 T^{5} + 42632 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2}$$)($$( 1 - 74 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 + 4 T + 8 T^{2} - 292 T^{3} - 844 T^{4} + 28196 T^{5} + 162168 T^{6} - 417348 T^{7} - 41227034 T^{8} - 30466404 T^{9} + 864193272 T^{10} + 10968723332 T^{11} - 23968115404 T^{12} - 605336905156 T^{13} + 1210673810312 T^{14} + 44189594076388 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$1 + 3 T + 79 T^{2}$$)($$1 + 3 T + 79 T^{2}$$)($$1 + 16 T + 209 T^{2} + 1264 T^{3} + 6241 T^{4}$$)($$1 + 16 T + 209 T^{2} + 1264 T^{3} + 6241 T^{4}$$)($$( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 5 T + 79 T^{2} )^{4}$$)($$1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4}$$)($$1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 1276166 T^{7} + 66416722 T^{8} - 184396586 T^{9} - 5803562069 T^{10} + 6154112798 T^{11} + 243087455521 T^{12}$$)($$( 1 - 202 T^{2} + 19659 T^{4} - 1260682 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 286 T^{2} + 32715 T^{4} - 1784926 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 8 T - 98 T^{2} - 32 T^{3} + 14947 T^{4} - 2528 T^{5} - 611618 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2}$$)($$1 + 260 T^{2} + 23724 T^{4} + 1706392 T^{6} + 225017642 T^{8} + 14869625148 T^{10} - 181450880816 T^{12} - 19987311435172 T^{14} + 2160188956785123 T^{16} - 124740810666908452 T^{18} - 7067526505304546096 T^{20} +$$$$36\!\cdots\!08$$$$T^{22} +$$$$34\!\cdots\!62$$$$T^{24} +$$$$16\!\cdots\!92$$$$T^{26} +$$$$14\!\cdots\!84$$$$T^{28} +$$$$95\!\cdots\!60$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$1 - 6 T + 83 T^{2}$$)($$1 + 6 T + 83 T^{2}$$)($$( 1 - 3 T + 83 T^{2} )^{2}$$)($$( 1 + 3 T + 83 T^{2} )^{2}$$)($$1 - 154 T^{2} + 16827 T^{4} - 1060906 T^{6} + 47458321 T^{8}$$)($$( 1 - 164 T^{2} + 6889 T^{4} )^{2}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 15 T - 51 T^{2} - 678 T^{3} + 17133 T^{4} + 80979 T^{5} - 925778 T^{6} + 6721257 T^{7} + 118029237 T^{8} - 387671586 T^{9} - 2420374371 T^{10} + 59085609645 T^{11} + 326940373369 T^{12}$$)($$1 - 19666 T^{4} + 177621555 T^{8} - 933315340786 T^{12} + 2252292232139041 T^{16}$$)($$1 + 17456 T^{4} + 170554626 T^{8} + 828432451376 T^{12} + 2252292232139041 T^{16}$$)($$1 + 160 T^{2} + 5929 T^{4} + 942880 T^{6} + 163131520 T^{8} + 6495500320 T^{10} + 281380385209 T^{12} + 52310459739040 T^{14} + 2252292232139041 T^{16}$$)($$1 - 66 T + 2178 T^{2} - 47916 T^{3} + 791830 T^{4} - 10596156 T^{5} + 122712084 T^{6} - 1311937572 T^{7} + 13564902253 T^{8} - 136924099746 T^{9} + 1327585690602 T^{10} - 12258029803398 T^{11} + 109901521003222 T^{12} - 994600086849090 T^{13} + 9293464416138048 T^{14} - 88105947208965780 T^{15} + 817933709355268336 T^{16} - 7312793618344159740 T^{17} + 64022676362775012672 T^{18} -$$$$56\!\cdots\!30$$$$T^{19} +$$$$52\!\cdots\!62$$$$T^{20} -$$$$48\!\cdots\!14$$$$T^{21} +$$$$43\!\cdots\!38$$$$T^{22} -$$$$37\!\cdots\!42$$$$T^{23} +$$$$30\!\cdots\!73$$$$T^{24} -$$$$24\!\cdots\!16$$$$T^{25} +$$$$19\!\cdots\!16$$$$T^{26} -$$$$13\!\cdots\!52$$$$T^{27} +$$$$84\!\cdots\!30$$$$T^{28} -$$$$42\!\cdots\!08$$$$T^{29} +$$$$16\!\cdots\!62$$$$T^{30} -$$$$40\!\cdots\!62$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$1 + 12 T + 89 T^{2}$$)($$1 - 12 T + 89 T^{2}$$)($$1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4}$$)($$1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 - 15 T + 89 T^{2} )^{2}$$)($$( 1 - 3 T + 89 T^{2} )^{6}$$)($$( 1 + 50 T^{2} + 1158 T^{4} + 396050 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 212 T^{2} + 25134 T^{4} + 1679252 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 6 T + 79 T^{2} - 534 T^{3} + 7921 T^{4} )^{4}$$)($$( 1 + 412 T^{2} + 69850 T^{4} + 6725680 T^{6} + 546021283 T^{8} + 53274111280 T^{10} + 4382545533850 T^{12} + 204756291875932 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$1 + 13 T + 97 T^{2}$$)($$1 + 13 T + 97 T^{2}$$)($$( 1 - 8 T + 97 T^{2} )^{2}$$)($$( 1 - 8 T + 97 T^{2} )^{2}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 185 T^{2} + 9409 T^{4} )^{2}$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 - 18 T + 69 T^{2} - 214 T^{3} + 906 T^{4} + 163158 T^{5} - 2743251 T^{6} + 15826326 T^{7} + 8524554 T^{8} - 195312022 T^{9} + 6108520389 T^{10} - 154572124626 T^{11} + 832972004929 T^{12}$$)($$( 1 - 4 T + 8 T^{2} - 228 T^{3} + 4898 T^{4} - 22116 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8} )^{2}$$)($$( 1 - 16 T + 128 T^{2} - 864 T^{3} + 3983 T^{4} - 83808 T^{5} + 1204352 T^{6} - 14602768 T^{7} + 88529281 T^{8} )^{2}$$)($$1 + 52 T^{2} - 12902 T^{4} - 167024 T^{6} + 129576019 T^{8} - 1571528816 T^{10} - 1142204783462 T^{12} + 43314544256308 T^{14} + 7837433594376961 T^{16}$$)($$1 - 28 T + 392 T^{2} + 872 T^{3} - 117924 T^{4} + 2250436 T^{5} - 16405808 T^{6} - 121301108 T^{7} + 5089087514 T^{8} - 63345173552 T^{9} + 270560580248 T^{10} + 4797083097932 T^{11} - 105427115995440 T^{12} + 964302703147436 T^{13} - 1291090828127400 T^{14} - 85355358908186064 T^{15} + 1288991316725561923 T^{16} - 8279469814094048208 T^{17} - 12147873601850706600 T^{18} +$$$$88\!\cdots\!28$$$$T^{19} -$$$$93\!\cdots\!40$$$$T^{20} +$$$$41\!\cdots\!24$$$$T^{21} +$$$$22\!\cdots\!92$$$$T^{22} -$$$$51\!\cdots\!76$$$$T^{23} +$$$$39\!\cdots\!54$$$$T^{24} -$$$$92\!\cdots\!36$$$$T^{25} -$$$$12\!\cdots\!92$$$$T^{26} +$$$$16\!\cdots\!08$$$$T^{27} -$$$$81\!\cdots\!84$$$$T^{28} +$$$$58\!\cdots\!44$$$$T^{29} +$$$$25\!\cdots\!48$$$$T^{30} -$$$$17\!\cdots\!04$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)