# Properties

 Label 115.2 Level 115 Weight 2 Dimension 417 Nonzero newspaces 6 Newform subspaces 11 Sturm bound 2112 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$115 = 5 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$11$$ Sturm bound: $$2112$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(115))$$.

Total New Old
Modular forms 616 545 71
Cusp forms 441 417 24
Eisenstein series 175 128 47

## Trace form

 $$417q - 25q^{2} - 26q^{3} - 29q^{4} - 34q^{5} - 78q^{6} - 30q^{7} - 37q^{8} - 35q^{9} + O(q^{10})$$ $$417q - 25q^{2} - 26q^{3} - 29q^{4} - 34q^{5} - 78q^{6} - 30q^{7} - 37q^{8} - 35q^{9} - 36q^{10} - 78q^{11} - 50q^{12} - 36q^{13} - 46q^{14} - 26q^{15} - 53q^{16} - 18q^{17} + 27q^{18} - 20q^{19} + 4q^{20} - 32q^{21} + 8q^{22} - q^{23} + 28q^{24} - 12q^{25} - 64q^{26} + 4q^{27} + 10q^{28} - 30q^{29} - q^{30} - 76q^{31} - 41q^{32} - 48q^{33} - 32q^{34} - 19q^{35} - 47q^{36} + 28q^{37} + 28q^{38} + 10q^{39} + 40q^{40} - 64q^{41} + 102q^{42} + 22q^{43} + 92q^{44} + 31q^{45} + 41q^{46} - 4q^{47} + 74q^{48} + 53q^{49} - 14q^{50} - 50q^{51} + 100q^{52} - 32q^{53} + 78q^{54} + 21q^{55} + 56q^{56} + 52q^{57} + 86q^{58} + 50q^{59} + 71q^{60} - 40q^{61} + 14q^{62} - 16q^{63} + 27q^{64} + 30q^{65} + 32q^{66} - 46q^{67} + 138q^{68} - 4q^{69} - 2q^{70} - 28q^{71} + 113q^{72} - 52q^{73} + 62q^{74} + 84q^{75} + 80q^{76} + 80q^{77} + 162q^{78} + 74q^{79} + 145q^{80} + 165q^{81} + 72q^{82} + 48q^{83} + 194q^{84} + 81q^{85} + 66q^{86} + 122q^{87} - 114q^{88} + 20q^{89} + 159q^{90} + 20q^{91} + 15q^{92} + 92q^{93} + 76q^{94} + 2q^{95} + 100q^{96} - 54q^{97} - 105q^{98} - 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(115))$$

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
115.2.a $$\chi_{115}(1, \cdot)$$ 115.2.a.a 1 1
115.2.a.b 2
115.2.a.c 4
115.2.b $$\chi_{115}(24, \cdot)$$ 115.2.b.a 2 1
115.2.b.b 8
115.2.e $$\chi_{115}(22, \cdot)$$ 115.2.e.a 20 2
115.2.g $$\chi_{115}(6, \cdot)$$ 115.2.g.a 10 10
115.2.g.b 20
115.2.g.c 50
115.2.j $$\chi_{115}(4, \cdot)$$ 115.2.j.a 100 10
115.2.l $$\chi_{115}(7, \cdot)$$ 115.2.l.a 200 20

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(115))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(115)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T + 2 T^{2}$$)($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 - 2 T + 4 T^{2} - 7 T^{3} + 10 T^{4} - 14 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8}$$)($$( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$)($$1 - 6 T^{2} + 19 T^{4} - 50 T^{6} + 113 T^{8} - 200 T^{10} + 304 T^{12} - 384 T^{14} + 256 T^{16}$$)($$( 1 + 2 T + 2 T^{2} + 2 T^{3} - T^{4} - 6 T^{5} - 8 T^{6} - 16 T^{7} - 11 T^{8} + 4 T^{9} + 8 T^{10} + 8 T^{11} - 44 T^{12} - 128 T^{13} - 128 T^{14} - 192 T^{15} - 64 T^{16} + 256 T^{17} + 512 T^{18} + 1024 T^{19} + 1024 T^{20} )^{2}$$)($$1 - 5 T + 12 T^{2} - 17 T^{3} + 6 T^{4} + 37 T^{5} - 98 T^{6} + 119 T^{7} - 25 T^{8} - 212 T^{9} + 461 T^{10} - 424 T^{11} - 100 T^{12} + 952 T^{13} - 1568 T^{14} + 1184 T^{15} + 384 T^{16} - 2176 T^{17} + 3072 T^{18} - 2560 T^{19} + 1024 T^{20}$$)($$( 1 + 2 T + 2 T^{2} + 7 T^{4} + 25 T^{5} + 36 T^{6} + 22 T^{7} + 16 T^{8} + 109 T^{9} + 219 T^{10} + 218 T^{11} + 64 T^{12} + 176 T^{13} + 576 T^{14} + 800 T^{15} + 448 T^{16} + 512 T^{18} + 1024 T^{19} + 1024 T^{20} )^{2}$$)
$3$ ($$1 + 3 T^{2}$$)($$( 1 + T + 3 T^{2} )^{2}$$)($$( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} )^{2}$$)($$1 - 2 T^{2} + 9 T^{4}$$)($$1 - 8 T^{2} + 34 T^{4} - 96 T^{6} + 259 T^{8} - 864 T^{10} + 2754 T^{12} - 5832 T^{14} + 6561 T^{16}$$)($$( 1 + 4 T + 8 T^{2} + 14 T^{3} + 11 T^{4} - 8 T^{5} - 22 T^{6} - 38 T^{7} - 11 T^{8} + 82 T^{9} + 158 T^{10} + 246 T^{11} - 99 T^{12} - 1026 T^{13} - 1782 T^{14} - 1944 T^{15} + 8019 T^{16} + 30618 T^{17} + 52488 T^{18} + 78732 T^{19} + 59049 T^{20} )^{2}$$)($$( 1 - 6 T + 14 T^{2} - 7 T^{3} - 46 T^{4} + 133 T^{5} - 138 T^{6} - 63 T^{7} + 378 T^{8} - 486 T^{9} + 243 T^{10} )( 1 + 5 T + 14 T^{2} + 26 T^{3} + 42 T^{4} + 67 T^{5} + 126 T^{6} + 234 T^{7} + 378 T^{8} + 405 T^{9} + 243 T^{10} )$$)($$1 - T + 6 T^{2} - 14 T^{3} + 27 T^{4} - 40 T^{5} + 108 T^{6} - 142 T^{7} + 185 T^{8} - 485 T^{9} + 1007 T^{10} - 1619 T^{11} + 3309 T^{12} - 9356 T^{13} + 14699 T^{14} - 23307 T^{15} + 48108 T^{16} - 86905 T^{17} + 82057 T^{18} - 156475 T^{19} + 360997 T^{20} - 469425 T^{21} + 738513 T^{22} - 2346435 T^{23} + 3896748 T^{24} - 5663601 T^{25} + 10715571 T^{26} - 20461572 T^{27} + 21710349 T^{28} - 31866777 T^{29} + 59462343 T^{30} - 85916295 T^{31} + 98316585 T^{32} - 226393866 T^{33} + 516560652 T^{34} - 573956280 T^{35} + 1162261467 T^{36} - 1807962282 T^{37} + 2324522934 T^{38} - 1162261467 T^{39} + 3486784401 T^{40}$$)
$5$ ($$1 + T$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{4}$$)($$1 - 4 T + 5 T^{2}$$)($$1 + 6 T + 26 T^{2} + 82 T^{3} + 206 T^{4} + 410 T^{5} + 650 T^{6} + 750 T^{7} + 625 T^{8}$$)($$1 + 6 T^{2} + 3 T^{4} - 80 T^{6} - 600 T^{8} - 3500 T^{10} - 15000 T^{12} - 50000 T^{14} + 46875 T^{16} + 2343750 T^{18} + 9765625 T^{20}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$)
$7$ ($$1 - T + 7 T^{2}$$)($$1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 + 3 T + 14 T^{2} + 11 T^{3} + 66 T^{4} + 77 T^{5} + 686 T^{6} + 1029 T^{7} + 2401 T^{8}$$)($$1 - 13 T^{2} + 49 T^{4}$$)($$1 - 28 T^{2} + 348 T^{4} - 2708 T^{6} + 18230 T^{8} - 132692 T^{10} + 835548 T^{12} - 3294172 T^{14} + 5764801 T^{16}$$)($$1 - 237 T^{4} + 23161 T^{8} - 1134076 T^{12} + 24086022 T^{16} - 196683742 T^{20} + 57830538822 T^{24} - 6537722458876 T^{28} + 320578052862361 T^{32} - 7876204544995437 T^{36} + 79792266297612001 T^{40}$$)($$1 - 5 T + 7 T^{2} - 22 T^{3} + 61 T^{4} - 63 T^{5} + 262 T^{6} + 704 T^{7} - 3165 T^{8} + 7828 T^{9} - 38149 T^{10} + 54796 T^{11} - 155085 T^{12} + 241472 T^{13} + 629062 T^{14} - 1058841 T^{15} + 7176589 T^{16} - 18117946 T^{17} + 40353607 T^{18} - 201768035 T^{19} + 282475249 T^{20}$$)($$1 + 4 T - T^{2} + 11 T^{3} + 245 T^{4} + 349 T^{5} - 1301 T^{6} + 1861 T^{7} + 18638 T^{8} - 28841 T^{9} - 155478 T^{10} + 268498 T^{11} + 305619 T^{12} - 4521142 T^{13} - 1974595 T^{14} + 24747851 T^{15} - 38560597 T^{16} - 137651474 T^{17} + 569072322 T^{18} + 697342953 T^{19} - 3108876707 T^{20} + 4881400671 T^{21} + 27884543778 T^{22} - 47214455582 T^{23} - 92583993397 T^{24} + 415937131757 T^{25} - 232309127155 T^{26} - 3723354846106 T^{27} + 1761832716819 T^{28} + 10834862772286 T^{29} - 43918686764022 T^{30} - 57028080594863 T^{31} + 257973910852238 T^{32} + 180310448367427 T^{33} - 882368217776549 T^{34} + 1656898966970107 T^{35} + 8142067989552245 T^{36} + 2558935653859277 T^{37} - 1628413597910449 T^{38} + 45595580741492572 T^{39} + 79792266297612001 T^{40}$$)
$11$ ($$1 - 2 T + 11 T^{2}$$)($$1 + 2 T + 18 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 - 4 T + 28 T^{2} - 92 T^{3} + 406 T^{4} - 1012 T^{5} + 3388 T^{6} - 5324 T^{7} + 14641 T^{8}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 28 T^{2} - 22 T^{3} + 346 T^{4} - 242 T^{5} + 3388 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 - 42 T^{2} + 1081 T^{4} - 20956 T^{6} + 316762 T^{8} - 3842292 T^{10} + 38328202 T^{12} - 306816796 T^{14} + 1915057441 T^{16} - 9003073002 T^{18} + 25937424601 T^{20} )^{2}$$)($$1 - 8 T + 20 T^{2} - 116 T^{3} + 785 T^{4} - 1924 T^{5} + 7648 T^{6} - 42781 T^{7} + 84254 T^{8} - 306621 T^{9} + 1799073 T^{10} - 3372831 T^{11} + 10194734 T^{12} - 56941511 T^{13} + 111974368 T^{14} - 309862124 T^{15} + 1390675385 T^{16} - 2260511836 T^{17} + 4287177620 T^{18} - 18863581528 T^{19} + 25937424601 T^{20}$$)($$1 + 8 T + 8 T^{2} - 120 T^{3} - 561 T^{4} - 859 T^{5} + 2258 T^{6} + 21321 T^{7} + 77329 T^{8} + 120978 T^{9} - 353594 T^{10} - 3097611 T^{11} - 8842197 T^{12} - 10206633 T^{13} + 28541538 T^{14} + 311515082 T^{15} + 1139840185 T^{16} + 880981278 T^{17} - 5235886142 T^{18} - 18114167536 T^{19} - 48380815625 T^{20} - 199255842896 T^{21} - 633542223182 T^{22} + 1172586081018 T^{23} + 16688400148585 T^{24} + 50169815471182 T^{25} + 50563075600818 T^{26} - 198898402605243 T^{27} - 1895403454501557 T^{28} - 7304004705066201 T^{29} - 9171317714365994 T^{30} + 34516435287177558 T^{31} + 242691527943458209 T^{32} + 736058745620752851 T^{33} + 857475124230958178 T^{34} - 3588256177528044209 T^{35} - 25777798453463982321 T^{36} - 60653643419915252520 T^{37} + 44479338507937851848 T^{38} +$$$$48\!\cdots\!28$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$13$ ($$1 + 2 T + 13 T^{2}$$)($$1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4}$$)($$1 + 11 T^{2} + 160 T^{4} + 1859 T^{6} + 28561 T^{8}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 40 T^{2} + 914 T^{4} - 15968 T^{6} + 228595 T^{8} - 2698592 T^{10} + 26104754 T^{12} - 193072360 T^{14} + 815730721 T^{16}$$)($$( 1 - 2 T + 2 T^{2} - 60 T^{3} + 339 T^{4} + 508 T^{5} + 106 T^{6} - 3702 T^{7} - 35227 T^{8} + 234016 T^{9} - 82360 T^{10} + 3042208 T^{11} - 5953363 T^{12} - 8133294 T^{13} + 3027466 T^{14} + 188616844 T^{15} + 1636288251 T^{16} - 3764911020 T^{17} + 1631461442 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} )^{2}$$)($$1 - 2 T^{2} + 11 T^{3} + 169 T^{4} + 407 T^{5} + 1785 T^{6} - 2068 T^{7} + 30475 T^{8} + 77154 T^{9} + 232145 T^{10} + 1003002 T^{11} + 5150275 T^{12} - 4543396 T^{13} + 50981385 T^{14} + 151116251 T^{15} + 815730721 T^{16} + 690233687 T^{17} - 1631461442 T^{18} + 137858491849 T^{20}$$)($$1 - 10 T + 58 T^{2} - 323 T^{3} + 1705 T^{4} - 8003 T^{5} + 37572 T^{6} - 172909 T^{7} + 767395 T^{8} - 3344202 T^{9} + 13988859 T^{10} - 56848352 T^{11} + 227655708 T^{12} - 894452242 T^{13} + 3496893949 T^{14} - 13511663886 T^{15} + 51241017455 T^{16} - 193291836963 T^{17} + 714868254694 T^{18} - 2598123737952 T^{19} + 9385591752033 T^{20} - 33775608593376 T^{21} + 120812735043286 T^{22} - 424662165807711 T^{23} + 1463494699532255 T^{24} - 5016786219224598 T^{25} + 16878839185078741 T^{26} - 56125551712825114 T^{27} + 185705754826605468 T^{28} - 602848313140083296 T^{29} + 1928483004428310291 T^{30} - 5993346374059323474 T^{31} + 17878834032566306995 T^{32} - 52369831805759873977 T^{33} +$$$$14\!\cdots\!08$$$$T^{34} -$$$$40\!\cdots\!71$$$$T^{35} +$$$$11\!\cdots\!05$$$$T^{36} -$$$$27\!\cdots\!59$$$$T^{37} +$$$$65\!\cdots\!82$$$$T^{38} -$$$$14\!\cdots\!70$$$$T^{39} +$$$$19\!\cdots\!01$$$$T^{40}$$)
$17$ ($$1 - 3 T + 17 T^{2}$$)($$1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4}$$)($$1 + T + 50 T^{2} + 27 T^{3} + 1154 T^{4} + 459 T^{5} + 14450 T^{6} + 4913 T^{7} + 83521 T^{8}$$)($$1 - 9 T^{2} + 289 T^{4}$$)($$1 - 56 T^{2} + 1544 T^{4} - 36168 T^{6} + 720894 T^{8} - 10452552 T^{10} + 128956424 T^{12} - 1351703864 T^{14} + 6975757441 T^{16}$$)($$1 + 519 T^{4} + 89433 T^{8} - 14268808 T^{12} - 7240575434 T^{16} - 2204663931022 T^{20} - 604740100823114 T^{24} - 99535743580200328 T^{28} + 52105654542169215513 T^{32} +$$$$25\!\cdots\!39$$$$T^{36} +$$$$40\!\cdots\!01$$$$T^{40}$$)($$1 + 23 T + 248 T^{2} + 1793 T^{3} + 10843 T^{4} + 61388 T^{5} + 327661 T^{6} + 1644533 T^{7} + 7815982 T^{8} + 35008785 T^{9} + 148068887 T^{10} + 595149345 T^{11} + 2258818798 T^{12} + 8079590629 T^{13} + 27366574381 T^{14} + 87162181516 T^{15} + 261723660667 T^{16} + 735737240689 T^{17} + 1729987845368 T^{18} + 2727521159431 T^{19} + 2015993900449 T^{20}$$)($$1 + 9 T + 40 T^{2} + 100 T^{3} - 246 T^{4} - 2278 T^{5} - 8909 T^{6} - 24479 T^{7} + 1113 T^{8} - 224535 T^{9} - 2290561 T^{10} - 16349752 T^{11} - 91534141 T^{12} - 280206614 T^{13} - 628445950 T^{14} + 1744414036 T^{15} + 25418244278 T^{16} + 123431240367 T^{17} + 377484951692 T^{18} + 330981245066 T^{19} - 1445197923018 T^{20} + 5626681166122 T^{21} + 109093151038988 T^{22} + 606417683923071 T^{23} + 2122957180342838 T^{24} + 2476818479912852 T^{25} - 15169157480895550 T^{26} - 114979610154583222 T^{27} - 638519965186293181 T^{28} - 1938882370932578744 T^{29} - 4617757004606361889 T^{30} - 7695240237434375655 T^{31} + 648458550036723993 T^{32} -$$$$24\!\cdots\!23$$$$T^{33} -$$$$15\!\cdots\!61$$$$T^{34} -$$$$65\!\cdots\!54$$$$T^{35} -$$$$11\!\cdots\!26$$$$T^{36} +$$$$82\!\cdots\!00$$$$T^{37} +$$$$56\!\cdots\!60$$$$T^{38} +$$$$21\!\cdots\!77$$$$T^{39} +$$$$40\!\cdots\!01$$$$T^{40}$$)
$19$ ($$1 + 2 T + 19 T^{2}$$)($$1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4}$$)($$1 + 4 T + 60 T^{2} + 188 T^{3} + 1590 T^{4} + 3572 T^{5} + 21660 T^{6} + 27436 T^{7} + 130321 T^{8}$$)($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 70 T^{2} - 200 T^{3} + 1918 T^{4} - 3800 T^{5} + 25270 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 + 72 T^{2} + 2599 T^{4} + 72796 T^{6} + 1835388 T^{8} + 39000088 T^{10} + 662575068 T^{12} + 9486847516 T^{14} + 122272244719 T^{16} + 1222816538952 T^{18} + 6131066257801 T^{20} )^{2}$$)($$1 + 13 T + 29 T^{2} - 409 T^{3} - 2337 T^{4} + 5165 T^{5} + 66426 T^{6} + 2641 T^{7} - 1149892 T^{8} - 297132 T^{9} + 19829679 T^{10} - 5645508 T^{11} - 415111012 T^{12} + 18114619 T^{13} + 8656702746 T^{14} + 12789051335 T^{15} - 109946223897 T^{16} - 365593541251 T^{17} + 492523328189 T^{18} + 4194940071127 T^{19} + 6131066257801 T^{20}$$)($$1 - 24 T + 252 T^{2} - 1489 T^{3} + 4733 T^{4} + 3733 T^{5} - 165357 T^{6} + 1237434 T^{7} - 5578089 T^{8} + 16320194 T^{9} - 18257942 T^{10} - 187138908 T^{11} + 1933541143 T^{12} - 11304701304 T^{13} + 48272065375 T^{14} - 142867417584 T^{15} + 120166092742 T^{16} + 1544637215496 T^{17} - 11207167930805 T^{18} + 52159457753326 T^{19} - 225110036080269 T^{20} + 991029697313194 T^{21} - 4045787623020605 T^{22} + 10594666661087064 T^{23} + 15660165372230182 T^{24} - 353753869812324816 T^{25} + 2271001843256470375 T^{26} - 10104953013482047656 T^{27} + 32838417894507695863 T^{28} - 60387423387396085332 T^{29} -$$$$11\!\cdots\!42$$$$T^{30} +$$$$19\!\cdots\!86$$$$T^{31} -$$$$12\!\cdots\!29$$$$T^{32} +$$$$52\!\cdots\!06$$$$T^{33} -$$$$13\!\cdots\!97$$$$T^{34} +$$$$56\!\cdots\!67$$$$T^{35} +$$$$13\!\cdots\!73$$$$T^{36} -$$$$81\!\cdots\!71$$$$T^{37} +$$$$26\!\cdots\!32$$$$T^{38} -$$$$47\!\cdots\!96$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$)
$23$ ($$1 - T$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{4}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{4}$$)($$1 + 112 T^{3} + 1429 T^{4} + 1424 T^{5} + 6272 T^{6} + 100048 T^{7} + 1214218 T^{8} + 2254896 T^{9} + 3256576 T^{10} + 51862608 T^{11} + 642321322 T^{12} + 1217284016 T^{13} + 1755162752 T^{14} + 9165352432 T^{15} + 211543285381 T^{16} + 381340450064 T^{17} + 41426511213649 T^{20}$$)($$1 + 21 T + 243 T^{2} + 1913 T^{3} + 11771 T^{4} + 60543 T^{5} + 270733 T^{6} + 1011977 T^{7} + 2956581 T^{8} + 5876661 T^{9} + 6436343 T^{10}$$)($$1 - 11 T + 12 T^{2} + 429 T^{3} - 1627 T^{4} - 7370 T^{5} + 59423 T^{6} - 30723 T^{7} - 510938 T^{8} + 866525 T^{9} + 4291915 T^{10} + 19930075 T^{11} - 270286202 T^{12} - 373806741 T^{13} + 16628991743 T^{14} - 47435847910 T^{15} - 240854391403 T^{16} + 1460670116763 T^{17} + 939731823372 T^{18} - 19812679276093 T^{19} + 41426511213649 T^{20}$$)
$29$ ($$1 - 7 T + 29 T^{2}$$)($$1 + 10 T + 63 T^{2} + 290 T^{3} + 841 T^{4}$$)($$1 - 19 T + 233 T^{2} - 1922 T^{3} + 12034 T^{4} - 55738 T^{5} + 195953 T^{6} - 463391 T^{7} + 707281 T^{8}$$)($$( 1 - 5 T + 29 T^{2} )^{2}$$)($$( 1 + 4 T + 94 T^{2} + 344 T^{3} + 3775 T^{4} + 9976 T^{5} + 79054 T^{6} + 97556 T^{7} + 707281 T^{8} )^{2}$$)($$( 1 - 193 T^{2} + 18883 T^{4} - 1197882 T^{6} + 54156853 T^{8} - 1813801707 T^{10} + 45545913373 T^{12} - 847239178842 T^{14} + 11232048770443 T^{16} - 96547557701473 T^{18} + 420707233300201 T^{20} )^{2}$$)($$1 - 2 T + 41 T^{2} - 222 T^{3} + 531 T^{4} - 9408 T^{5} + 47813 T^{6} - 301118 T^{7} + 2010297 T^{8} - 10226854 T^{9} + 44968155 T^{10} - 296578766 T^{11} + 1690659777 T^{12} - 7343966902 T^{13} + 33817226453 T^{14} - 192968889792 T^{15} + 315851183451 T^{16} - 3829472540598 T^{17} + 20510102931401 T^{18} - 29014291951738 T^{19} + 420707233300201 T^{20}$$)($$1 - 12 T - 67 T^{2} + 1195 T^{3} + 1042 T^{4} - 48597 T^{5} + 5197 T^{6} + 1150766 T^{7} + 1427500 T^{8} - 23690871 T^{9} - 186659295 T^{10} + 749599372 T^{11} + 10720023417 T^{12} - 31499319861 T^{13} - 376737927236 T^{14} + 1065343656837 T^{15} + 10045093559315 T^{16} - 25490540704614 T^{17} - 266261175986743 T^{18} + 297289837349525 T^{19} + 7599931488502746 T^{20} + 8621405283136225 T^{21} - 223925649004850863 T^{22} - 621688797244830846 T^{23} + 7104703817725872515 T^{24} + 21851422481588575713 T^{25} -$$$$22\!\cdots\!56$$$$T^{26} -$$$$54\!\cdots\!49$$$$T^{27} +$$$$53\!\cdots\!37$$$$T^{28} +$$$$10\!\cdots\!68$$$$T^{29} -$$$$78\!\cdots\!95$$$$T^{30} -$$$$28\!\cdots\!59$$$$T^{31} +$$$$50\!\cdots\!00$$$$T^{32} +$$$$11\!\cdots\!74$$$$T^{33} +$$$$15\!\cdots\!57$$$$T^{34} -$$$$41\!\cdots\!53$$$$T^{35} +$$$$26\!\cdots\!82$$$$T^{36} +$$$$86\!\cdots\!55$$$$T^{37} -$$$$14\!\cdots\!87$$$$T^{38} -$$$$73\!\cdots\!28$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$31$ ($$1 + 5 T + 31 T^{2}$$)($$1 - 4 T + 61 T^{2} - 124 T^{3} + 961 T^{4}$$)($$1 + T + 23 T^{2} + 104 T^{3} + 1648 T^{4} + 3224 T^{5} + 22103 T^{6} + 29791 T^{7} + 923521 T^{8}$$)($$( 1 + 5 T + 31 T^{2} )^{2}$$)($$( 1 + 50 T^{2} - 256 T^{3} + 1011 T^{4} - 7936 T^{5} + 48050 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 + T + 97 T^{2} + 62 T^{3} + 4277 T^{4} + 1947 T^{5} + 132587 T^{6} + 59582 T^{7} + 2889727 T^{8} + 923521 T^{9} + 28629151 T^{10} )^{4}$$)($$1 + 20 T + 127 T^{2} + 127 T^{3} - 1089 T^{4} - 3794 T^{5} - 73933 T^{6} - 716061 T^{7} - 2339386 T^{8} + 4043930 T^{9} + 65396035 T^{10} + 125361830 T^{11} - 2248149946 T^{12} - 21332173251 T^{13} - 68278678093 T^{14} - 108618998894 T^{15} - 966491508609 T^{16} + 3494101992097 T^{17} + 108317161755007 T^{18} + 528792443213420 T^{19} + 819628286980801 T^{20}$$)($$1 - 10 T + 134 T^{2} - 758 T^{3} + 7675 T^{4} - 42437 T^{5} + 404282 T^{6} - 1932071 T^{7} + 15085785 T^{8} - 64870146 T^{9} + 524687482 T^{10} - 2365711093 T^{11} + 16855502897 T^{12} - 79312877915 T^{13} + 567068279230 T^{14} - 3353689420274 T^{15} + 21041878154409 T^{16} - 125443438747332 T^{17} + 716718317074566 T^{18} - 4602431108233134 T^{19} + 24926288490663857 T^{20} - 142675364355227154 T^{21} + 688766302708657926 T^{22} - 3737085483721767612 T^{23} + 19432616355037954089 T^{24} - 96013280820126807374 T^{25} +$$$$50\!\cdots\!30$$$$T^{26} -$$$$21\!\cdots\!65$$$$T^{27} +$$$$14\!\cdots\!77$$$$T^{28} -$$$$62\!\cdots\!03$$$$T^{29} +$$$$43\!\cdots\!82$$$$T^{30} -$$$$16\!\cdots\!26$$$$T^{31} +$$$$11\!\cdots\!85$$$$T^{32} -$$$$47\!\cdots\!61$$$$T^{33} +$$$$30\!\cdots\!22$$$$T^{34} -$$$$99\!\cdots\!87$$$$T^{35} +$$$$55\!\cdots\!75$$$$T^{36} -$$$$17\!\cdots\!38$$$$T^{37} +$$$$93\!\cdots\!94$$$$T^{38} -$$$$21\!\cdots\!10$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$37$ ($$1 - 11 T + 37 T^{2}$$)($$1 + 6 T + 38 T^{2} + 222 T^{3} + 1369 T^{4}$$)($$1 + 3 T + 32 T^{2} + 349 T^{3} + 1638 T^{4} + 12913 T^{5} + 43808 T^{6} + 151959 T^{7} + 1874161 T^{8}$$)($$1 - 25 T^{2} + 1369 T^{4}$$)($$1 - 148 T^{2} + 11228 T^{4} - 572828 T^{6} + 23118550 T^{8} - 784201532 T^{10} + 21043079708 T^{12} - 379727508532 T^{14} + 3512479453921 T^{16}$$)($$1 - 9781 T^{4} + 44061273 T^{8} - 123293552508 T^{12} + 245670541820726 T^{16} - 377506395088955022 T^{20} +$$$$46\!\cdots\!86$$$$T^{24} -$$$$43\!\cdots\!68$$$$T^{28} +$$$$29\!\cdots\!13$$$$T^{32} -$$$$12\!\cdots\!21$$$$T^{36} +$$$$23\!\cdots\!01$$$$T^{40}$$)($$1 - 13 T + 132 T^{2} - 1235 T^{3} + 11171 T^{4} - 85668 T^{5} + 658381 T^{6} - 4704949 T^{7} + 33057222 T^{8} - 212975933 T^{9} + 1343462603 T^{10} - 7880109521 T^{11} + 45255336918 T^{12} - 238319781697 T^{13} + 1233911993341 T^{14} - 5940558108276 T^{15} + 28661729714939 T^{16} - 117240868259255 T^{17} + 463647287917572 T^{18} - 1689502617336001 T^{19} + 4808584372417849 T^{20}$$)($$1 + 18 T + 85 T^{2} - 390 T^{3} - 6629 T^{4} - 30356 T^{5} + 95098 T^{6} + 1788792 T^{7} + 4202378 T^{8} - 48185338 T^{9} - 404071013 T^{10} - 1298535796 T^{11} + 380299965 T^{12} + 51711042538 T^{13} + 458631920536 T^{14} + 1459019174670 T^{15} - 10051521013052 T^{16} - 164054790080866 T^{17} - 618726163001666 T^{18} + 4578533633789962 T^{19} + 57065299869074243 T^{20} + 169405744450228594 T^{21} - 847036117149280754 T^{22} - 8309867281966105498 T^{23} - 18838168673342549372 T^{24} +$$$$10\!\cdots\!90$$$$T^{25} +$$$$11\!\cdots\!24$$$$T^{26} +$$$$49\!\cdots\!54$$$$T^{27} +$$$$13\!\cdots\!65$$$$T^{28} -$$$$16\!\cdots\!92$$$$T^{29} -$$$$19\!\cdots\!37$$$$T^{30} -$$$$85\!\cdots\!94$$$$T^{31} +$$$$27\!\cdots\!18$$$$T^{32} +$$$$43\!\cdots\!24$$$$T^{33} +$$$$85\!\cdots\!22$$$$T^{34} -$$$$10\!\cdots\!08$$$$T^{35} -$$$$81\!\cdots\!89$$$$T^{36} -$$$$17\!\cdots\!30$$$$T^{37} +$$$$14\!\cdots\!65$$$$T^{38} +$$$$11\!\cdots\!14$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$)
$41$ ($$1 - T + 41 T^{2}$$)($$1 + 6 T + 71 T^{2} + 246 T^{3} + 1681 T^{4}$$)($$1 - 13 T + 209 T^{2} - 1602 T^{3} + 13682 T^{4} - 65682 T^{5} + 351329 T^{6} - 895973 T^{7} + 2825761 T^{8}$$)($$( 1 + 7 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 25256 T^{5} + 117670 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 + 9 T + 195 T^{2} + 1370 T^{3} + 15765 T^{4} + 82035 T^{5} + 646365 T^{6} + 2302970 T^{7} + 13439595 T^{8} + 25431849 T^{9} + 115856201 T^{10} )^{4}$$)($$1 + 5 T + 50 T^{2} - 681 T^{3} - 2573 T^{4} - 37436 T^{5} + 238919 T^{6} + 364515 T^{7} + 12937390 T^{8} - 71985111 T^{9} + 8447451 T^{10} - 2951389551 T^{11} + 21747752590 T^{12} + 25122738315 T^{13} + 675127992359 T^{14} - 4337192740636 T^{15} - 12222018212093 T^{16} - 132627660512961 T^{17} + 399246261456050 T^{18} + 1636909671969805 T^{19} + 13422659310152401 T^{20}$$)($$1 - 10 T - 114 T^{2} + 1506 T^{3} + 2757 T^{4} - 92209 T^{5} + 243933 T^{6} + 2920729 T^{7} - 26804829 T^{8} - 21977155 T^{9} + 1492380725 T^{10} - 4237439576 T^{11} - 52174293824 T^{12} + 353017933581 T^{13} + 774069433040 T^{14} - 16950354155947 T^{15} + 38704074299184 T^{16} + 551121880563897 T^{17} - 4002933543559192 T^{18} - 8577951456589504 T^{19} + 203910150115021758 T^{20} - 351696009720169664 T^{21} - 6728931286723001752 T^{22} + 37983871130344345137 T^{23} +$$$$10\!\cdots\!24$$$$T^{24} -$$$$19\!\cdots\!47$$$$T^{25} +$$$$36\!\cdots\!40$$$$T^{26} +$$$$68\!\cdots\!61$$$$T^{27} -$$$$41\!\cdots\!04$$$$T^{28} -$$$$13\!\cdots\!36$$$$T^{29} +$$$$20\!\cdots\!25$$$$T^{30} -$$$$12\!\cdots\!55$$$$T^{31} -$$$$60\!\cdots\!49$$$$T^{32} +$$$$27\!\cdots\!09$$$$T^{33} +$$$$92\!\cdots\!13$$$$T^{34} -$$$$14\!\cdots\!09$$$$T^{35} +$$$$17\!\cdots\!37$$$$T^{36} +$$$$39\!\cdots\!86$$$$T^{37} -$$$$12\!\cdots\!94$$$$T^{38} -$$$$43\!\cdots\!10$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$43$ ($$1 + 43 T^{2}$$)($$1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4}$$)($$1 + 6 T + 136 T^{2} + 758 T^{3} + 8126 T^{4} + 32594 T^{5} + 251464 T^{6} + 477042 T^{7} + 3418801 T^{8}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 92 T^{2} + 8096 T^{4} - 480052 T^{6} + 22994446 T^{8} - 887616148 T^{10} + 27678612896 T^{12} - 581565400508 T^{14} + 11688200277601 T^{16}$$)($$1 - 2666 T^{4} - 1105447 T^{8} + 8937665552 T^{12} - 1848148296154 T^{16} - 20946707502982572 T^{20} - 6318451243039591354 T^{24} +$$$$10\!\cdots\!52$$$$T^{28} -$$$$44\!\cdots\!47$$$$T^{32} -$$$$36\!\cdots\!66$$$$T^{36} +$$$$46\!\cdots\!01$$$$T^{40}$$)($$1 - 15 T + 204 T^{2} - 2129 T^{3} + 19643 T^{4} - 172738 T^{5} + 1385269 T^{6} - 10775981 T^{7} + 77916158 T^{8} - 539582197 T^{9} + 3599056165 T^{10} - 23202034471 T^{11} + 144066976142 T^{12} - 856765921367 T^{13} + 4735959042469 T^{14} - 25393944426934 T^{15} + 124170534371507 T^{16} - 578701823046803 T^{17} + 2384392856630604 T^{18} - 7538889179052645 T^{19} + 21611482313284249 T^{20}$$)($$1 - 35 T + 567 T^{2} - 6482 T^{3} + 66662 T^{4} - 645520 T^{5} + 5810987 T^{6} - 48597733 T^{7} + 371510041 T^{8} - 2643204420 T^{9} + 17674945398 T^{10} - 105727723373 T^{11} + 540972251826 T^{12} - 2059075592526 T^{13} + 649816071717 T^{14} + 88212600828558 T^{15} - 1205960437326518 T^{16} + 11953353936135423 T^{17} - 98903689285419458 T^{18} + 732724305899372468 T^{19} - 5012641069620623889 T^{20} + 31507145153673016124 T^{21} -$$$$18\!\cdots\!42$$$$T^{22} +$$$$95\!\cdots\!61$$$$T^{23} -$$$$41\!\cdots\!18$$$$T^{24} +$$$$12\!\cdots\!94$$$$T^{25} +$$$$41\!\cdots\!33$$$$T^{26} -$$$$55\!\cdots\!82$$$$T^{27} +$$$$63\!\cdots\!26$$$$T^{28} -$$$$53\!\cdots\!39$$$$T^{29} +$$$$38\!\cdots\!02$$$$T^{30} -$$$$24\!\cdots\!40$$$$T^{31} +$$$$14\!\cdots\!41$$$$T^{32} -$$$$83\!\cdots\!19$$$$T^{33} +$$$$42\!\cdots\!63$$$$T^{34} -$$$$20\!\cdots\!40$$$$T^{35} +$$$$91\!\cdots\!62$$$$T^{36} -$$$$38\!\cdots\!26$$$$T^{37} +$$$$14\!\cdots\!83$$$$T^{38} -$$$$38\!\cdots\!45$$$$T^{39} +$$$$46\!\cdots\!01$$$$T^{40}$$)
$47$ ($$1 + 47 T^{2}$$)($$1 - 10 T + 99 T^{2} - 470 T^{3} + 2209 T^{4}$$)($$1 - 6 T + 105 T^{2} - 298 T^{3} + 5324 T^{4} - 14006 T^{5} + 231945 T^{6} - 622938 T^{7} + 4879681 T^{8}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 96 T^{2} + 11626 T^{4} - 665136 T^{6} + 41700939 T^{8} - 1469285424 T^{10} + 56731171306 T^{12} - 1034804671584 T^{14} + 23811286661761 T^{16}$$)($$( 1 + 4 T + 8 T^{2} - 330 T^{3} + 435 T^{4} + 17576 T^{5} + 121274 T^{6} - 601990 T^{7} - 5812707 T^{8} + 11707642 T^{9} + 393692622 T^{10} + 550259174 T^{11} - 12840269763 T^{12} - 62500407770 T^{13} + 591778433594 T^{14} + 4030967843032 T^{15} + 4688958668115 T^{16} - 167185629752790 T^{17} + 190490293294088 T^{18} + 4476521892411068 T^{19} + 52599132235830049 T^{20} )^{2}$$)($$( 1 + 26 T + 424 T^{2} + 4806 T^{3} + 43985 T^{4} + 327085 T^{5} + 2067295 T^{6} + 10616454 T^{7} + 44020952 T^{8} + 126871706 T^{9} + 229345007 T^{10} )^{2}$$)($$( 1 + T + 158 T^{2} + 376 T^{3} + 13303 T^{4} + 53535 T^{5} + 946350 T^{6} + 4112977 T^{7} + 59396726 T^{8} + 219980016 T^{9} + 3078120533 T^{10} + 10339060752 T^{11} + 131207367734 T^{12} + 427021611071 T^{13} + 4617886114350 T^{14} + 12277984949745 T^{15} + 143395901521687 T^{16} + 190490293294088 T^{17} + 3762183292558238 T^{18} + 1119130473102767 T^{19} + 52599132235830049 T^{20} )^{2}$$)
$53$ ($$1 - 11 T + 53 T^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$1 - 19 T + 178 T^{2} - 929 T^{3} + 4474 T^{4} - 49237 T^{5} + 500002 T^{6} - 2828663 T^{7} + 7890481 T^{8}$$)($$1 - 105 T^{2} + 2809 T^{4}$$)($$1 - 200 T^{2} + 23516 T^{4} - 1842744 T^{6} + 111792678 T^{8} - 5176267896 T^{10} + 185552551196 T^{12} - 4432872225800 T^{14} + 62259690411361 T^{16}$$)($$1 - 6865 T^{4} + 5634885 T^{8} + 55524537540 T^{12} - 66605380961110 T^{16} - 238481873986046502 T^{20} -$$$$52\!\cdots\!10$$$$T^{24} +$$$$34\!\cdots\!40$$$$T^{28} +$$$$27\!\cdots\!85$$$$T^{32} -$$$$26\!\cdots\!65$$$$T^{36} +$$$$30\!\cdots\!01$$$$T^{40}$$)($$1 - 6 T + 126 T^{2} - 1670 T^{3} + 18434 T^{4} - 163092 T^{5} + 1868723 T^{6} - 15540916 T^{7} + 130470814 T^{8} - 1013135108 T^{9} + 8246457637 T^{10} - 53696160724 T^{11} + 366492516526 T^{12} - 2313684951332 T^{13} + 14745123325763 T^{14} - 68204339344356 T^{15} + 408577833051986 T^{16} - 1961767603527790 T^{17} + 7844720991831486 T^{18} - 19798581550812798 T^{19} + 174887470365513049 T^{20}$$)($$1 + 5 T + 51 T^{2} + 595 T^{3} + 2839 T^{4} + 26462 T^{5} + 145564 T^{6} + 743850 T^{7} + 5187559 T^{8} - 9941890 T^{9} - 35840051 T^{10} - 4581262764 T^{11} - 19433102310 T^{12} - 98940938897 T^{13} - 1096107884097 T^{14} + 4668708827649 T^{15} + 43678935628705 T^{16} + 633329500748410 T^{17} + 8062327369739114 T^{18} + 51635096968093848 T^{19} + 472566844582501503 T^{20} + 2736660139308973944 T^{21} + 22647077581597171226 T^{22} + 94288196082921035570 T^{23} +$$$$34\!\cdots\!05$$$$T^{24} +$$$$19\!\cdots\!57$$$$T^{25} -$$$$24\!\cdots\!13$$$$T^{26} -$$$$11\!\cdots\!89$$$$T^{27} -$$$$12\!\cdots\!10$$$$T^{28} -$$$$15\!\cdots\!12$$$$T^{29} -$$$$62\!\cdots\!99$$$$T^{30} -$$$$92\!\cdots\!30$$$$T^{31} +$$$$25\!\cdots\!19$$$$T^{32} +$$$$19\!\cdots\!50$$$$T^{33} +$$$$20\!\cdots\!16$$$$T^{34} +$$$$19\!\cdots\!34$$$$T^{35} +$$$$11\!\cdots\!19$$$$T^{36} +$$$$12\!\cdots\!35$$$$T^{37} +$$$$55\!\cdots\!39$$$$T^{38} +$$$$28\!\cdots\!85$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$)
$59$ ($$1 + 13 T + 59 T^{2}$$)($$1 + 38 T^{2} + 3481 T^{4}$$)($$1 - 23 T + 336 T^{2} - 3511 T^{3} + 29550 T^{4} - 207149 T^{5} + 1169616 T^{6} - 4723717 T^{7} + 12117361 T^{8}$$)($$( 1 + 3 T + 59 T^{2} )^{2}$$)($$( 1 + 216 T^{2} - 16 T^{3} + 18542 T^{4} - 944 T^{5} + 751896 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 277 T^{2} + 38049 T^{4} - 3370504 T^{6} + 226355270 T^{8} - 13536944758 T^{10} + 787942694870 T^{12} - 40841613719944 T^{14} + 1604927124506409 T^{16} - 40672031216396917 T^{18} + 511116753300641401 T^{20} )^{2}$$)($$1 - 5 T + 142 T^{2} - 646 T^{3} + 8536 T^{4} - 26742 T^{5} + 287886 T^{6} + 702648 T^{7} + 10262887 T^{8} + 139977013 T^{9} + 674338952 T^{10} + 8258643767 T^{11} + 35725109647 T^{12} + 144309143592 T^{13} + 3488418588846 T^{14} - 19118505603858 T^{15} + 360053035159576 T^{16} - 1607668859193074 T^{17} + 20849922139813582 T^{18} - 43314979093274695 T^{19} + 511116753300641401 T^{20}$$)($$1 - 41 T + 913 T^{2} - 14997 T^{3} + 203431 T^{4} - 2350391 T^{5} + 23374344 T^{6} - 203007532 T^{7} + 1564347264 T^{8} - 10971437923 T^{9} + 75041826526 T^{10} - 567907579491 T^{11} + 5167877252932 T^{12} - 52599927939636 T^{13} + 526749214679551 T^{14} - 4821318377051737 T^{15} + 39545367549341390 T^{16} - 292455010641582662 T^{17} + 1999568112398060742 T^{18} - 13406494422249449930 T^{19} + 96691071763912858131 T^{20} -$$$$79\!\cdots\!70$$$$T^{21} +$$$$69\!\cdots\!02$$$$T^{22} -$$$$60\!\cdots\!98$$$$T^{23} +$$$$47\!\cdots\!90$$$$T^{24} -$$$$34\!\cdots\!63$$$$T^{25} +$$$$22\!\cdots\!91$$$$T^{26} -$$$$13\!\cdots\!84$$$$T^{27} +$$$$75\!\cdots\!72$$$$T^{28} -$$$$49\!\cdots\!49$$$$T^{29} +$$$$38\!\cdots\!26$$$$T^{30} -$$$$33\!\cdots\!57$$$$T^{31} +$$$$27\!\cdots\!84$$$$T^{32} -$$$$21\!\cdots\!28$$$$T^{33} +$$$$14\!\cdots\!84$$$$T^{34} -$$$$85\!\cdots\!09$$$$T^{35} +$$$$43\!\cdots\!71$$$$T^{36} -$$$$19\!\cdots\!43$$$$T^{37} +$$$$68\!\cdots\!73$$$$T^{38} -$$$$18\!\cdots\!99$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$)
$61$ ($$1 + 8 T + 61 T^{2}$$)($$1 - 2 T - 2 T^{2} - 122 T^{3} + 3721 T^{4}$$)($$1 + 188 T^{2} + 136 T^{3} + 15462 T^{4} + 8296 T^{5} + 699548 T^{6} + 13845841 T^{8}$$)($$( 1 + 6 T + 61 T^{2} )^{2}$$)($$( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 19276 T^{5} + 528382 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 - 288 T^{2} + 45599 T^{4} - 4912324 T^{6} + 406368068 T^{8} - 27198365512 T^{10} + 1512095581028 T^{12} - 68015257044484 T^{14} + 2349277550487239 T^{16} - 55211706143216928 T^{18} + 713342911662882601 T^{20} )^{2}$$)($$1 + 3 T + 14 T^{2} - 174 T^{3} + 3200 T^{4} + 46207 T^{5} + 124734 T^{6} - 489978 T^{7} + 801426 T^{8} + 45141464 T^{9} + 1300242855 T^{10} + 2753629304 T^{11} + 2982106146 T^{12} - 111215696418 T^{13} + 1727047131294 T^{14} + 39026261280307 T^{15} + 164865197955200 T^{16} - 546837253467654 T^{17} + 2683902381961934 T^{18} + 35082438278502423 T^{19} + 713342911662882601 T^{20}$$)($$1 + 13 T - 60 T^{2} - 2612 T^{3} - 15003 T^{4} + 100279 T^{5} + 1903557 T^{6} + 10954221 T^{7} + 12672790 T^{8} - 800102006 T^{9} - 12112961179 T^{10} - 59157771400 T^{11} + 479177951078 T^{12} + 7550071333662 T^{13} + 31890998032060 T^{14} - 103532444522825 T^{15} - 2309029969616480 T^{16} - 22681070722503046 T^{17} - 90474577182005101 T^{18} + 916305827480323236 T^{19} + 15097916789735610599 T^{20} + 55894655476299717396 T^{21} -$$$$33\!\cdots\!21$$$$T^{22} -$$$$51\!\cdots\!26$$$$T^{23} -$$$$31\!\cdots\!80$$$$T^{24} -$$$$87\!\cdots\!25$$$$T^{25} +$$$$16\!\cdots\!60$$$$T^{26} +$$$$23\!\cdots\!02$$$$T^{27} +$$$$91\!\cdots\!18$$$$T^{28} -$$$$69\!\cdots\!00$$$$T^{29} -$$$$86\!\cdots\!79$$$$T^{30} -$$$$34\!\cdots\!66$$$$T^{31} +$$$$33\!\cdots\!90$$$$T^{32} +$$$$17\!\cdots\!01$$$$T^{33} +$$$$18\!\cdots\!37$$$$T^{34} +$$$$60\!\cdots\!79$$$$T^{35} -$$$$55\!\cdots\!83$$$$T^{36} -$$$$58\!\cdots\!52$$$$T^{37} -$$$$82\!\cdots\!60$$$$T^{38} +$$$$10\!\cdots\!33$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$)
$67$ ($$1 - 5 T + 67 T^{2}$$)($$1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4}$$)($$1 + 3 T + 170 T^{2} + 391 T^{3} + 15834 T^{4} + 26197 T^{5} + 763130 T^{6} + 902289 T^{7} + 20151121 T^{8}$$)($$1 + 35 T^{2} + 4489 T^{4}$$)($$1 - 516 T^{2} + 117772 T^{4} - 15527980 T^{6} + 1293477494 T^{8} - 69705102220 T^{10} + 2373237822412 T^{12} - 46676525199204 T^{14} + 406067677556641 T^{16}$$)($$1 + 12179 T^{4} + 21394393 T^{8} - 187118931868 T^{12} + 373223367469206 T^{16} + 8514442786702047778 T^{20} +$$$$75\!\cdots\!26$$$$T^{24} -$$$$75\!\cdots\!88$$$$T^{28} +$$$$17\!\cdots\!73$$$$T^{32} +$$$$20\!\cdots\!99$$$$T^{36} +$$$$33\!\cdots\!01$$$$T^{40}$$)($$1 - 20 T + 223 T^{2} - 1822 T^{3} + 4262 T^{4} + 78106 T^{5} - 1241112 T^{6} + 7658230 T^{7} + 22056296 T^{8} - 902388884 T^{9} + 10137910717 T^{10} - 60460055228 T^{11} + 99010712744 T^{12} + 2303312229490 T^{13} - 25009798086552 T^{14} + 105452871607342 T^{15} + 385533624804278 T^{16} - 11042616544898506 T^{17} + 90553092095130943 T^{18} - 544130687925898940 T^{19} + 1822837804551761449 T^{20}$$)($$1 + 9 T - 337 T^{2} - 4839 T^{3} + 44298 T^{4} + 1132981 T^{5} - 440595 T^{6} - 154086318 T^{7} - 773767208 T^{8} + 12301905801 T^{9} + 141655410926 T^{10} - 326567723604 T^{11} - 13755036511502 T^{12} - 54542970952515 T^{13} + 727217719747252 T^{14} + 8656605590327801 T^{15} + 1622077958659168 T^{16} - 623781141371941467 T^{17} - 4122992200747057095 T^{18} + 18074669471187244023 T^{19} +$$$$39\!\cdots\!75$$$$T^{20} +$$$$12\!\cdots\!41$$$$T^{21} -$$$$18\!\cdots\!55$$$$T^{22} -$$$$18\!\cdots\!21$$$$T^{23} +$$$$32\!\cdots\!28$$$$T^{24} +$$$$11\!\cdots\!07$$$$T^{25} +$$$$65\!\cdots\!88$$$$T^{26} -$$$$33\!\cdots\!45$$$$T^{27} -$$$$55\!\cdots\!82$$$$T^{28} -$$$$88\!\cdots\!88$$$$T^{29} +$$$$25\!\cdots\!74$$$$T^{30} +$$$$15\!\cdots\!83$$$$T^{31} -$$$$63\!\cdots\!88$$$$T^{32} -$$$$84\!\cdots\!66$$$$T^{33} -$$$$16\!\cdots\!55$$$$T^{34} +$$$$27\!\cdots\!83$$$$T^{35} +$$$$73\!\cdots\!38$$$$T^{36} -$$$$53\!\cdots\!53$$$$T^{37} -$$$$24\!\cdots\!33$$$$T^{38} +$$$$44\!\cdots\!27$$$$T^{39} +$$$$33\!\cdots\!01$$$$T^{40}$$)
$71$ ($$1 - 5 T + 71 T^{2}$$)($$1 + 8 T + 153 T^{2} + 568 T^{3} + 5041 T^{4}$$)($$1 + 3 T + 135 T^{2} + 104 T^{3} + 9080 T^{4} + 7384 T^{5} + 680535 T^{6} + 1073733 T^{7} + 25411681 T^{8}$$)($$( 1 - 13 T + 71 T^{2} )^{2}$$)($$( 1 + 24 T + 350 T^{2} + 3544 T^{3} + 32183 T^{4} + 251624 T^{5} + 1764350 T^{6} + 8589864 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 - 11 T + 235 T^{2} - 1020 T^{3} + 15325 T^{4} - 20035 T^{5} + 1088075 T^{6} - 5141820 T^{7} + 84109085 T^{8} - 279528491 T^{9} + 1804229351 T^{10} )^{4}$$)($$1 + 22 T + 204 T^{2} + 209 T^{3} - 1361 T^{4} + 99759 T^{5} + 1632143 T^{6} + 4544078 T^{7} - 8138253 T^{8} + 377062070 T^{9} + 8288280417 T^{10} + 26771406970 T^{11} - 41024933373 T^{12} + 1626375501058 T^{13} + 41475497262383 T^{14} + 179988115826409 T^{15} - 174344486416481 T^{16} + 1900880113103719 T^{17} + 131733720374135244 T^{18} + 1008667015805878682 T^{19} + 3255243551009881201 T^{20}$$)($$1 + 19 T + 83 T^{2} - 816 T^{3} - 6625 T^{4} + 14575 T^{5} + 69541 T^{6} - 1592636 T^{7} - 12108235 T^{8} - 196163533 T^{9} - 3746523573 T^{10} - 48113457043 T^{11} - 279634239359 T^{12} + 1030580907940 T^{13} + 17532711356992 T^{14} - 64779962713440 T^{15} - 898237358003269 T^{16} + 4646236140350778 T^{17} + 86466675434941796 T^{18} + 900300579683224304 T^{19} + 9271746160980739561 T^{20} + 63921341157508925584 T^{21} +$$$$43\!\cdots\!36$$$$T^{22} +$$$$16\!\cdots\!58$$$$T^{23} -$$$$22\!\cdots\!89$$$$T^{24} -$$$$11\!\cdots\!40$$$$T^{25} +$$$$22\!\cdots\!32$$$$T^{26} +$$$$93\!\cdots\!40$$$$T^{27} -$$$$18\!\cdots\!99$$$$T^{28} -$$$$22\!\cdots\!33$$$$T^{29} -$$$$12\!\cdots\!73$$$$T^{30} -$$$$45\!\cdots\!43$$$$T^{31} -$$$$19\!\cdots\!35$$$$T^{32} -$$$$18\!\cdots\!96$$$$T^{33} +$$$$57\!\cdots\!21$$$$T^{34} +$$$$85\!\cdots\!25$$$$T^{35} -$$$$27\!\cdots\!25$$$$T^{36} -$$$$24\!\cdots\!56$$$$T^{37} +$$$$17\!\cdots\!63$$$$T^{38} +$$$$28\!\cdots\!89$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$)
$73$ ($$1 - 6 T + 73 T^{2}$$)($$1 + 101 T^{2} + 5329 T^{4}$$)($$1 + 32 T + 635 T^{2} + 8400 T^{3} + 83736 T^{4} + 613200 T^{5} + 3383915 T^{6} + 12448544 T^{7} + 28398241 T^{8}$$)($$1 - 82 T^{2} + 5329 T^{4}$$)($$1 - 216 T^{2} + 38578 T^{4} - 4059072 T^{6} + 362257971 T^{8} - 21630794688 T^{10} + 1095547341298 T^{12} - 32688192878424 T^{14} + 806460091894081 T^{16}$$)($$( 1 + 28 T + 392 T^{2} + 4930 T^{3} + 65419 T^{4} + 742868 T^{5} + 7308506 T^{6} + 74556498 T^{7} + 765085933 T^{8} + 6852665726 T^{9} + 57335768430 T^{10} + 500244597998 T^{11} + 4077142936957 T^{12} + 29003745182466 T^{13} + 207548714737946 T^{14} + 1540018548148724 T^{15} + 9900133749600091 T^{16} + 54463674699148210 T^{17} + 316132356022479752 T^{18} + 1648404427831501564 T^{19} + 4297625829703557649 T^{20} )^{2}$$)($$1 - 43 T + 797 T^{2} - 8021 T^{3} + 47021 T^{4} - 205129 T^{5} + 1802674 T^{6} - 24182707 T^{7} + 240208550 T^{8} - 1756457458 T^{9} + 12854913015 T^{10} - 128221394434 T^{11} + 1280071362950 T^{12} - 9407484129019 T^{13} + 51192770696434 T^{14} - 425247102800497 T^{15} + 7115886654335069 T^{16} - 88611183521677037 T^{17} + 642748693239582557 T^{18} - 2531478228455520259 T^{19} + 4297625829703557649 T^{20}$$)($$1 + 27 T - 49 T^{2} - 8346 T^{3} - 56095 T^{4} + 1063391 T^{5} + 14512055 T^{6} - 49029069 T^{7} - 1832046917 T^{8} - 4811768775 T^{9} + 142698735069 T^{10} + 1167290774645 T^{11} - 6314222788672 T^{12} - 130217378630554 T^{13} - 71752041870944 T^{14} + 10325321569056809 T^{15} + 48322092625455436 T^{16} - 592140752562355010 T^{17} - 6063599567403657515 T^{18} + 16569795153193735530 T^{19} +$$$$51\!\cdots\!14$$$$T^{20} +$$$$12\!\cdots\!90$$$$T^{21} -$$$$32\!\cdots\!35$$$$T^{22} -$$$$23\!\cdots\!70$$$$T^{23} +$$$$13\!\cdots\!76$$$$T^{24} +$$$$21\!\cdots\!37$$$$T^{25} -$$$$10\!\cdots\!16$$$$T^{26} -$$$$14\!\cdots\!38$$$$T^{27} -$$$$50\!\cdots\!32$$$$T^{28} +$$$$68\!\cdots\!85$$$$T^{29} +$$$$61\!\cdots\!81$$$$T^{30} -$$$$15\!\cdots\!75$$$$T^{31} -$$$$41\!\cdots\!57$$$$T^{32} -$$$$81\!\cdots\!77$$$$T^{33} +$$$$17\!\cdots\!95$$$$T^{34} +$$$$94\!\cdots\!87$$$$T^{35} -$$$$36\!\cdots\!95$$$$T^{36} -$$$$39\!\cdots\!38$$$$T^{37} -$$$$16\!\cdots\!81$$$$T^{38} +$$$$68\!\cdots\!99$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$79$ ($$1 + 12 T + 79 T^{2}$$)($$1 - 22 T + 274 T^{2} - 1738 T^{3} + 6241 T^{4}$$)($$1 - 2 T + 176 T^{2} - 826 T^{3} + 15838 T^{4} - 65254 T^{5} + 1098416 T^{6} - 986078 T^{7} + 38950081 T^{8}$$)($$( 1 - 14 T + 79 T^{2} )^{2}$$)($$( 1 + 24 T + 494 T^{2} + 6100 T^{3} + 65598 T^{4} + 481900 T^{5} + 3083054 T^{6} + 11832936 T^{7} + 38950081 T^{8} )^{2}$$)($$( 1 + 332 T^{2} + 60799 T^{4} + 7894836 T^{6} + 804461788 T^{8} + 68672209888 T^{10} + 5020646018908 T^{12} + 307504501681716 T^{14} + 14779474208221279 T^{16} + 503680124888978252 T^{18} + 9468276082626847201 T^{20} )^{2}$$)($$1 - 51 T + 1114 T^{2} - 13295 T^{3} + 92135 T^{4} - 385584 T^{5} + 1188383 T^{6} - 2740799 T^{7} + 64216836 T^{8} - 2455180103 T^{9} + 33339862885 T^{10} - 193959228137 T^{11} + 400777273476 T^{12} - 1351320798161 T^{13} + 46287614109023 T^{14} - 1186463714552016 T^{15} + 22396862714427335 T^{16} - 255315969970983905 T^{17} + 1690059214235908954 T^{18} - 6112431395113534269 T^{19} + 9468276082626847201 T^{20}$$)($$1 + 83 T + 3229 T^{2} + 78771 T^{3} + 1360907 T^{4} + 17747029 T^{5} + 178442571 T^{6} + 1318577173 T^{7} + 5245494785 T^{8} - 26916412207 T^{9} - 688944277333 T^{10} - 5280206060532 T^{11} + 11833858144127 T^{12} + 923189234494892 T^{13} + 15263056131282191 T^{14} + 169446648447312438 T^{15} + 1393749417833100974 T^{16} + 7898356705887080574 T^{17} + 15891588591444807484 T^{18} -$$$$26\!\cdots\!23$$$$T^{19} -$$$$36\!\cdots\!29$$$$T^{20} -$$$$20\!\cdots\!17$$$$T^{21} +$$$$99\!\cdots\!44$$$$T^{22} +$$$$38\!\cdots\!86$$$$T^{23} +$$$$54\!\cdots\!94$$$$T^{24} +$$$$52\!\cdots\!62$$$$T^{25} +$$$$37\!\cdots\!11$$$$T^{26} +$$$$17\!\cdots\!28$$$$T^{27} +$$$$17\!\cdots\!47$$$$T^{28} -$$$$63\!\cdots\!08$$$$T^{29} -$$$$65\!\cdots\!33$$$$T^{30} -$$$$20\!\cdots\!53$$$$T^{31} +$$$$30\!\cdots\!85$$$$T^{32} +$$$$61\!\cdots\!47$$$$T^{33} +$$$$65\!\cdots\!51$$$$T^{34} +$$$$51\!\cdots\!71$$$$T^{35} +$$$$31\!\cdots\!47$$$$T^{36} +$$$$14\!\cdots\!89$$$$T^{37} +$$$$46\!\cdots\!69$$$$T^{38} +$$$$94\!\cdots\!77$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$)
$83$ ($$1 - 9 T + 83 T^{2}$$)($$1 + 4 T + 150 T^{2} + 332 T^{3} + 6889 T^{4}$$)($$1 + 21 T + 428 T^{2} + 5005 T^{3} + 56054 T^{4} + 415415 T^{5} + 2948492 T^{6} + 12007527 T^{7} + 47458321 T^{8}$$)($$1 - 157 T^{2} + 6889 T^{4}$$)($$1 - 88 T^{2} + 20552 T^{4} - 1491272 T^{6} + 193546270 T^{8} - 10273372808 T^{10} + 975363413192 T^{12} - 28770752856472 T^{14} + 2252292232139041 T^{16}$$)($$1 + 14383 T^{4} + 96015001 T^{8} + 196390156024 T^{12} - 4395513412321338 T^{16} - 50701641519546426142 T^{20} -$$$$20\!\cdots\!98$$$$T^{24} +$$$$44\!\cdots\!84$$$$T^{28} +$$$$10\!\cdots\!61$$$$T^{32} +$$$$72\!\cdots\!23$$$$T^{36} +$$$$24\!\cdots\!01$$$$T^{40}$$)($$1 + 17 T - 47 T^{2} - 2793 T^{3} - 1329 T^{4} + 469277 T^{5} + 3476090 T^{6} - 30330137 T^{7} - 432941786 T^{8} + 1652517680 T^{9} + 51926903887 T^{10} + 137158967440 T^{11} - 2982535963754 T^{12} - 17342378044819 T^{13} + 164969395044890 T^{14} + 1848501175825111 T^{15} - 434503756207401 T^{16} - 75790990414028211 T^{17} - 105857734910534927 T^{18} + 3177984339548186851 T^{19} + 15516041187205853449 T^{20}$$)($$1 - 67 T + 2213 T^{2} - 49993 T^{3} + 896668 T^{4} - 13779193 T^{5} + 188412483 T^{6} - 2340658030 T^{7} + 26777861702 T^{8} - 285062809625 T^{9} + 2847595768468 T^{10} - 26869752855934 T^{11} + 241233089991778 T^{12} - 2079971489872049 T^{13} + 17436911144135816 T^{14} - 144599134630260603 T^{15} + 1208646255676931134 T^{16} - 10315114326872771811 T^{17} + 90233346610071163341 T^{18} -$$$$80\!\cdots\!31$$$$T^{19} +$$$$72\!\cdots\!67$$$$T^{20} -$$$$66\!\cdots\!73$$$$T^{21} +$$$$62\!\cdots\!49$$$$T^{22} -$$$$58\!\cdots\!57$$$$T^{23} +$$$$57\!\cdots\!14$$$$T^{24} -$$$$56\!\cdots\!29$$$$T^{25} +$$$$57\!\cdots\!04$$$$T^{26} -$$$$56\!\cdots\!23$$$$T^{27} +$$$$54\!\cdots\!98$$$$T^{28} -$$$$50\!\cdots\!02$$$$T^{29} +$$$$44\!\cdots\!32$$$$T^{30} -$$$$36\!\cdots\!75$$$$T^{31} +$$$$28\!\cdots\!22$$$$T^{32} -$$$$20\!\cdots\!90$$$$T^{33} +$$$$13\!\cdots\!07$$$$T^{34} -$$$$84\!\cdots\!51$$$$T^{35} +$$$$45\!\cdots\!08$$$$T^{36} -$$$$21\!\cdots\!39$$$$T^{37} +$$$$77\!\cdots\!17$$$$T^{38} -$$$$19\!\cdots\!49$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$)
$89$ ($$1 - 4 T + 89 T^{2}$$)($$1 - 10 T + 198 T^{2} - 890 T^{3} + 7921 T^{4}$$)($$1 + 140 T^{2} - 1496 T^{3} + 6326 T^{4} - 133144 T^{5} + 1108940 T^{6} + 62742241 T^{8}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 - 8 T + 270 T^{2} - 1764 T^{3} + 34598 T^{4} - 156996 T^{5} + 2138670 T^{6} - 5639752 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 468 T^{2} + 124191 T^{4} + 22228084 T^{6} + 2951501052 T^{8} + 298597247008 T^{10} + 23378839832892 T^{12} + 1394639803296244 T^{14} + 61720603505737551 T^{16} + 1842323561068573908 T^{18} + 31181719929966183601 T^{20} )^{2}$$)($$1 - 57 T + 1455 T^{2} - 22697 T^{3} + 269934 T^{4} - 3046104 T^{5} + 33584052 T^{6} - 328175911 T^{7} + 3078564586 T^{8} - 32141776413 T^{9} + 327813403533 T^{10} - 2860618100757 T^{11} + 24385310085706 T^{12} - 231353843801759 T^{13} + 2107138684340532 T^{14} - 17009625823836696 T^{15} + 134152147794266574 T^{16} - 1003918608123821713 T^{17} + 5727736712296527855 T^{18} - 19970315011326656913 T^{19} + 31181719929966183601 T^{20}$$)($$1 + 5 T - 25 T^{2} - 1096 T^{3} - 8331 T^{4} - 8721 T^{5} + 899098 T^{6} + 2262003 T^{7} - 53415589 T^{8} - 1115445180 T^{9} + 273194454 T^{10} + 154552038615 T^{11} + 1491838219341 T^{12} - 1199327424269 T^{13} - 125456650170894 T^{14} - 1124099813937917 T^{15} + 622354984158761 T^{16} + 119069351123803568 T^{17} + 721825449679471296 T^{18} - 4533691496925569566 T^{19} -$$$$11\!\cdots\!23$$$$T^{20} -$$$$40\!\cdots\!74$$$$T^{21} +$$$$57\!\cdots\!16$$$$T^{22} +$$$$83\!\cdots\!92$$$$T^{23} +$$$$39\!\cdots\!01$$$$T^{24} -$$$$62\!\cdots\!33$$$$T^{25} -$$$$62\!\cdots\!34$$$$T^{26} -$$$$53\!\cdots\!01$$$$T^{27} +$$$$58\!\cdots\!21$$$$T^{28} +$$$$54\!\cdots\!35$$$$T^{29} +$$$$85\!\cdots\!54$$$$T^{30} -$$$$30\!\cdots\!20$$$$T^{31} -$$$$13\!\cdots\!69$$$$T^{32} +$$$$49\!\cdots\!07$$$$T^{33} +$$$$17\!\cdots\!18$$$$T^{34} -$$$$15\!\cdots\!29$$$$T^{35} -$$$$12\!\cdots\!91$$$$T^{36} -$$$$15\!\cdots\!84$$$$T^{37} -$$$$30\!\cdots\!25$$$$T^{38} +$$$$54\!\cdots\!45$$$$T^{39} +$$$$97\!\cdots\!01$$$$T^{40}$$)
$97$ ($$1 + 14 T + 97 T^{2}$$)($$1 - 10 T + 94 T^{2} - 970 T^{3} + 9409 T^{4}$$)($$1 + 18 T + 460 T^{2} + 5038 T^{3} + 69350 T^{4} + 488686 T^{5} + 4328140 T^{6} + 16428114 T^{7} + 88529281 T^{8}$$)($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 316 T^{2} + 65324 T^{4} - 9550388 T^{6} + 1045123990 T^{8} - 89859600692 T^{10} + 5783086752044 T^{12} - 263219153557564 T^{14} + 7837433594376961 T^{16}$$)($$1 - 5386 T^{4} - 21562987 T^{8} - 1116224538328 T^{12} + 10456340019940986 T^{16} - 23761540774597810972 T^{20} +$$$$92\!\cdots\!66$$$$T^{24} -$$$$87\!\cdots\!08$$$$T^{28} -$$$$14\!\cdots\!67$$$$T^{32} -$$$$33\!\cdots\!06$$$$T^{36} +$$$$54\!\cdots\!01$$$$T^{40}$$)($$1 - 52 T + 1221 T^{2} - 17946 T^{3} + 192969 T^{4} - 1571526 T^{5} + 7814029 T^{6} + 34957602 T^{7} - 1703623889 T^{8} + 28959932998 T^{9} - 334806911165 T^{10} + 2809113500806 T^{11} - 16029397171601 T^{12} + 31904859490146 T^{13} + 691770369083149 T^{14} - 13495228484722182 T^{15} + 160737774819144201 T^{16} - 1450006013244215898 T^{17} + 9569506418734269381 T^{18} - 39532015050037391284 T^{19} + 73742412689492826049 T^{20}$$)($$1 - 18 T + 217 T^{2} - 459 T^{3} - 165 T^{4} - 58184 T^{5} + 2022945 T^{6} - 8627114 T^{7} - 72720422 T^{8} + 1290108908 T^{9} + 5885657778 T^{10} - 93480419036 T^{11} - 259130726070 T^{12} + 13697917648969 T^{13} - 218945380462 T^{14} - 1990820824616916 T^{15} + 15584041952796128 T^{16} - 15011258222155937 T^{17} - 70947882815197672 T^{18} - 20273662042860585271 T^{19} +$$$$30\!\cdots\!33$$$$T^{20} -$$$$19\!\cdots\!87$$$$T^{21} -$$$$66\!\cdots\!48$$$$T^{22} -$$$$13\!\cdots\!01$$$$T^{23} +$$$$13\!\cdots\!68$$$$T^{24} -$$$$17\!\cdots\!12$$$$T^{25} -$$$$18\!\cdots\!98$$$$T^{26} +$$$$11\!\cdots\!97$$$$T^{27} -$$$$20\!\cdots\!70$$$$T^{28} -$$$$71\!\cdots\!12$$$$T^{29} +$$$$43\!\cdots\!22$$$$T^{30} +$$$$92\!\cdots\!24$$$$T^{31} -$$$$50\!\cdots\!02$$$$T^{32} -$$$$58\!\cdots\!78$$$$T^{33} +$$$$13\!\cdots\!05$$$$T^{34} -$$$$36\!\cdots\!12$$$$T^{35} -$$$$10\!\cdots\!65$$$$T^{36} -$$$$27\!\cdots\!83$$$$T^{37} +$$$$12\!\cdots\!13$$$$T^{38} -$$$$10\!\cdots\!94$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$)