Properties

Label 109.2
Level 109
Weight 2
Dimension 442
Nonzero newspaces 8
Newform subspaces 12
Sturm bound 1980
Trace bound 2

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Defining parameters

Level: \( N \) = \( 109 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 12 \)
Sturm bound: \(1980\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(109))\).

Total New Old
Modular forms 549 549 0
Cusp forms 442 442 0
Eisenstein series 107 107 0

Trace form

\( 442q - 51q^{2} - 50q^{3} - 47q^{4} - 48q^{5} - 42q^{6} - 46q^{7} - 39q^{8} - 41q^{9} + O(q^{10}) \) \( 442q - 51q^{2} - 50q^{3} - 47q^{4} - 48q^{5} - 42q^{6} - 46q^{7} - 39q^{8} - 41q^{9} - 36q^{10} - 42q^{11} - 26q^{12} - 40q^{13} - 30q^{14} - 30q^{15} - 23q^{16} - 36q^{17} - 15q^{18} - 34q^{19} - 12q^{20} - 22q^{21} - 18q^{22} - 30q^{23} + 6q^{24} - 23q^{25} - 12q^{26} - 14q^{27} + 2q^{28} - 24q^{29} + 18q^{30} - 22q^{31} + 9q^{32} - 6q^{33} - 6q^{35} + 37q^{36} - 16q^{37} + 6q^{38} + 2q^{39} + 36q^{40} - 12q^{41} + 42q^{42} - 10q^{43} + 30q^{44} + 24q^{45} + 18q^{46} - 6q^{47} + 70q^{48} + 3q^{49} + 39q^{50} + 18q^{51} + 44q^{52} + 66q^{54} + 18q^{55} + 66q^{56} + 26q^{57} + 36q^{58} + 6q^{59} + 114q^{60} + 8q^{61} + 42q^{62} + 50q^{63} + 73q^{64} + 30q^{65} + 90q^{66} + 14q^{67} + 72q^{68} + 42q^{69} + 90q^{70} + 18q^{71} + 141q^{72} + 20q^{73} + 60q^{74} + 70q^{75} + 86q^{76} + 42q^{77} + 114q^{78} + 26q^{79} + 132q^{80} + 67q^{81} + 72q^{82} + 30q^{83} + 170q^{84} + 54q^{85} + 78q^{86} + 66q^{87} + 126q^{88} + 36q^{89} + 45q^{90} - 68q^{91} - 102q^{92} - 52q^{93} - 234q^{94} - 96q^{95} - 234q^{96} - 64q^{97} - 99q^{98} - 222q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(109))\)

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
109.2.a \(\chi_{109}(1, \cdot)\) 109.2.a.a 1 1
109.2.a.b 3
109.2.a.c 4
109.2.b \(\chi_{109}(108, \cdot)\) 109.2.b.a 2 1
109.2.b.b 6
109.2.c \(\chi_{109}(45, \cdot)\) 109.2.c.a 14 2
109.2.e \(\chi_{109}(46, \cdot)\) 109.2.e.a 2 2
109.2.e.b 14
109.2.f \(\chi_{109}(16, \cdot)\) 109.2.f.a 42 6
109.2.h \(\chi_{109}(4, \cdot)\) 109.2.h.a 48 6
109.2.i \(\chi_{109}(3, \cdot)\) 109.2.i.a 144 18
109.2.k \(\chi_{109}(12, \cdot)\) 109.2.k.a 162 18

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 8 T^{5} + 8 T^{6} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 7 T^{4} + 4 T^{5} + 12 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 + 3 T^{4} + T^{6} + 12 T^{8} + 64 T^{12} \))(\( ( 1 + 2 T + 7 T^{2} + 12 T^{3} + 26 T^{4} + 40 T^{5} + 73 T^{6} + 95 T^{7} + 146 T^{8} + 160 T^{9} + 208 T^{10} + 192 T^{11} + 224 T^{12} + 128 T^{13} + 128 T^{14} )^{2} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} - 154 T^{6} + 346 T^{8} - 604 T^{10} + 933 T^{12} - 1617 T^{14} + 3732 T^{16} - 9664 T^{18} + 22144 T^{20} - 39424 T^{22} + 50176 T^{24} - 40960 T^{26} + 16384 T^{28} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 23 T^{3} + 36 T^{4} + 36 T^{5} + 27 T^{6} \))(\( 1 - 4 T + 11 T^{2} - 21 T^{3} + 40 T^{4} - 63 T^{5} + 99 T^{6} - 108 T^{7} + 81 T^{8} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 2 T + 6 T^{2} + 9 T^{3} + 18 T^{4} + 18 T^{5} + 27 T^{6} )^{2} \))(\( 1 - T - 7 T^{2} + 16 T^{3} + 11 T^{4} - 84 T^{5} + 54 T^{6} + 213 T^{7} - 331 T^{8} - 464 T^{9} + 1109 T^{10} + 1280 T^{11} - 4316 T^{12} - 1749 T^{13} + 15702 T^{14} - 5247 T^{15} - 38844 T^{16} + 34560 T^{17} + 89829 T^{18} - 112752 T^{19} - 241299 T^{20} + 465831 T^{21} + 354294 T^{22} - 1653372 T^{23} + 649539 T^{24} + 2834352 T^{25} - 3720087 T^{26} - 1594323 T^{27} + 4782969 T^{28} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + 4 T + 2 T^{2} - 2 T^{3} + 26 T^{4} + 37 T^{5} + T^{6} + 216 T^{7} + 285 T^{8} - 504 T^{9} + 501 T^{10} + 2799 T^{11} - 1584 T^{12} - 1548 T^{13} + 12051 T^{14} - 4644 T^{15} - 14256 T^{16} + 75573 T^{17} + 40581 T^{18} - 122472 T^{19} + 207765 T^{20} + 472392 T^{21} + 6561 T^{22} + 728271 T^{23} + 1535274 T^{24} - 354294 T^{25} + 1062882 T^{26} + 6377292 T^{27} + 4782969 T^{28} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( 1 + 6 T + 20 T^{2} + 47 T^{3} + 100 T^{4} + 150 T^{5} + 125 T^{6} \))(\( 1 - T + 15 T^{2} - 11 T^{3} + 103 T^{4} - 55 T^{5} + 375 T^{6} - 125 T^{7} + 625 T^{8} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 16 T^{2} - 37 T^{3} + 80 T^{4} - 100 T^{5} + 125 T^{6} )^{2} \))(\( 1 - 5 T - 10 T^{2} + 73 T^{3} + 92 T^{4} - 602 T^{5} - 1145 T^{6} + 3882 T^{7} + 12291 T^{8} - 22200 T^{9} - 95736 T^{10} + 95372 T^{11} + 593067 T^{12} - 190977 T^{13} - 3158874 T^{14} - 954885 T^{15} + 14826675 T^{16} + 11921500 T^{17} - 59835000 T^{18} - 69375000 T^{19} + 192046875 T^{20} + 303281250 T^{21} - 447265625 T^{22} - 1175781250 T^{23} + 898437500 T^{24} + 3564453125 T^{25} - 2441406250 T^{26} - 6103515625 T^{27} + 6103515625 T^{28} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - 19 T^{2} - 30 T^{3} + 200 T^{4} + 211 T^{5} - 1513 T^{6} - 808 T^{7} + 9195 T^{8} + 1023 T^{9} - 46950 T^{10} + 8913 T^{11} + 225547 T^{12} - 32365 T^{13} - 1100341 T^{14} - 161825 T^{15} + 5638675 T^{16} + 1114125 T^{17} - 29343750 T^{18} + 3196875 T^{19} + 143671875 T^{20} - 63125000 T^{21} - 591015625 T^{22} + 412109375 T^{23} + 1953125000 T^{24} - 1464843750 T^{25} - 4638671875 T^{26} + 2441406250 T^{27} + 6103515625 T^{28} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + T + 5 T^{2} + 27 T^{3} + 35 T^{4} + 49 T^{5} + 343 T^{6} \))(\( 1 + 3 T + 18 T^{2} + 40 T^{3} + 152 T^{4} + 280 T^{5} + 882 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 5 T + 23 T^{2} + 59 T^{3} + 161 T^{4} + 245 T^{5} + 343 T^{6} )^{2} \))(\( 1 + 2 T - 25 T^{2} - 24 T^{3} + 388 T^{4} + 74 T^{5} - 3594 T^{6} + 2544 T^{7} + 20568 T^{8} - 37934 T^{9} - 21402 T^{10} + 289948 T^{11} - 758014 T^{12} - 827533 T^{13} + 8263640 T^{14} - 5792731 T^{15} - 37142686 T^{16} + 99452164 T^{17} - 51386202 T^{18} - 637556738 T^{19} + 2419804632 T^{20} + 2095093392 T^{21} - 20718694794 T^{22} + 2986166918 T^{23} + 109600396612 T^{24} - 47455841832 T^{25} - 346032180025 T^{26} + 193778020814 T^{27} + 678223072849 T^{28} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 + 7 T + 15 T^{2} + 82 T^{3} + 436 T^{4} + 558 T^{5} + 1694 T^{6} + 9506 T^{7} - 180 T^{8} + 2252 T^{9} + 105236 T^{10} - 241578 T^{11} + 570 T^{12} + 996513 T^{13} - 5262717 T^{14} + 6975591 T^{15} + 27930 T^{16} - 82861254 T^{17} + 252671636 T^{18} + 37849364 T^{19} - 21176820 T^{20} + 7828599758 T^{21} + 9765572894 T^{22} + 22517312706 T^{23} + 123159208564 T^{24} + 162140792926 T^{25} + 207619308015 T^{26} + 678223072849 T^{27} + 678223072849 T^{28} \))
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 + 13 T + 87 T^{2} + 357 T^{3} + 957 T^{4} + 1573 T^{5} + 1331 T^{6} \))(\( 1 - 12 T + 77 T^{2} - 349 T^{3} + 1275 T^{4} - 3839 T^{5} + 9317 T^{6} - 15972 T^{7} + 14641 T^{8} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( 1 - 43 T^{2} + 959 T^{4} - 13093 T^{6} + 116039 T^{8} - 629563 T^{10} + 1771561 T^{12} \))(\( 1 - 49 T^{2} + 26 T^{3} + 1153 T^{4} - 982 T^{5} - 20601 T^{6} + 12321 T^{7} + 341559 T^{8} - 80089 T^{9} - 5042978 T^{10} + 1052394 T^{11} + 64153064 T^{12} - 8095195 T^{13} - 733692394 T^{14} - 89047145 T^{15} + 7762520744 T^{16} + 1400736414 T^{17} - 73834240898 T^{18} - 12898413539 T^{19} + 605092603599 T^{20} + 240101433891 T^{21} - 4416007307481 T^{22} - 2315504632562 T^{23} + 29905850564953 T^{24} + 7418103435886 T^{25} - 153782990459329 T^{26} + 379749833583241 T^{28} \))(\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 87 T^{2} + 540 T^{3} + 3221 T^{4} + 15813 T^{5} + 74249 T^{6} + 310059 T^{7} + 1256130 T^{8} + 4670838 T^{9} + 17157496 T^{10} + 59198517 T^{11} + 205279494 T^{12} + 682694604 T^{13} + 2302063791 T^{14} + 7509640644 T^{15} + 24838818774 T^{16} + 78793226127 T^{17} + 251202898936 T^{18} + 752243130738 T^{19} + 2225310918930 T^{20} + 6042172753089 T^{21} + 15915932555369 T^{22} + 37286226837783 T^{23} + 83544444639821 T^{24} + 154068302129940 T^{25} + 273043268774727 T^{26} + 310704409295379 T^{27} + 379749833583241 T^{28} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + T + 23 T^{2} + 39 T^{3} + 299 T^{4} + 169 T^{5} + 2197 T^{6} \))(\( 1 + 7 T + 42 T^{2} + 180 T^{3} + 770 T^{4} + 2340 T^{5} + 7098 T^{6} + 15379 T^{7} + 28561 T^{8} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 + T^{2} + 285 T^{4} + 647 T^{6} + 48165 T^{8} + 28561 T^{10} + 4826809 T^{12} \))(\( 1 - 7 T - 37 T^{2} + 336 T^{3} + 872 T^{4} - 8849 T^{5} - 17680 T^{6} + 151454 T^{7} + 410752 T^{8} - 1814171 T^{9} - 9586356 T^{10} + 14872253 T^{11} + 187101420 T^{12} - 64659977 T^{13} - 2782770618 T^{14} - 840579701 T^{15} + 31620139980 T^{16} + 32674339841 T^{17} - 273795913716 T^{18} - 673588993103 T^{19} + 1982621450368 T^{20} + 9503513893718 T^{21} - 14422119147280 T^{22} - 93839214951677 T^{23} + 120212604892328 T^{24} + 602165892396432 T^{25} - 862029149531797 T^{26} - 2120125746145771 T^{27} + 3937376385699289 T^{28} \))(\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 9 T + 87 T^{2} - 540 T^{3} + 3100 T^{4} - 15069 T^{5} + 67990 T^{6} - 282786 T^{7} + 1155036 T^{8} - 4450989 T^{9} + 17871622 T^{10} - 68392923 T^{11} + 270819788 T^{12} - 1020980421 T^{13} + 3779798850 T^{14} - 13272745473 T^{15} + 45768544172 T^{16} - 150259251831 T^{17} + 510431395942 T^{18} - 1652621058777 T^{19} + 5575138160124 T^{20} - 17744402128362 T^{21} + 55461531720790 T^{22} - 159799201051737 T^{23} + 427361324731900 T^{24} - 967766612779980 T^{25} + 2026933405655847 T^{26} - 2725875959330277 T^{27} + 3937376385699289 T^{28} \))
$17$ (\( 1 + 8 T + 17 T^{2} \))(\( 1 - 3 T + 47 T^{2} - 89 T^{3} + 799 T^{4} - 867 T^{5} + 4913 T^{6} \))(\( 1 - 11 T + 78 T^{2} - 346 T^{3} + 1498 T^{4} - 5882 T^{5} + 22542 T^{6} - 54043 T^{7} + 83521 T^{8} \))(\( 1 - 22 T^{2} + 289 T^{4} \))(\( 1 - 81 T^{2} + 2955 T^{4} - 63449 T^{6} + 853995 T^{8} - 6765201 T^{10} + 24137569 T^{12} \))(\( ( 1 - 3 T + 46 T^{2} - 22 T^{3} + 1089 T^{4} - 189 T^{5} + 28263 T^{6} - 18966 T^{7} + 480471 T^{8} - 54621 T^{9} + 5350257 T^{10} - 1837462 T^{11} + 65313422 T^{12} - 72412707 T^{13} + 410338673 T^{14} )^{2} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 - 154 T^{2} + 11809 T^{4} - 598231 T^{6} + 22319098 T^{8} - 647090269 T^{10} + 15007445187 T^{12} - 282533200959 T^{14} + 4337151659043 T^{16} - 54045626357149 T^{18} + 538728767992762 T^{20} - 4173114349686871 T^{22} + 23806871970402241 T^{24} - 89723824533383194 T^{26} + 168377826559400929 T^{28} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 + 5 T + 49 T^{2} + 149 T^{3} + 931 T^{4} + 1805 T^{5} + 6859 T^{6} \))(\( 1 - 10 T + 103 T^{2} - 567 T^{3} + 3133 T^{4} - 10773 T^{5} + 37183 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 65 T^{2} + 2379 T^{4} - 54841 T^{6} + 858819 T^{8} - 8470865 T^{10} + 47045881 T^{12} \))(\( ( 1 - 7 T + 75 T^{2} - 307 T^{3} + 1898 T^{4} - 4179 T^{5} + 25145 T^{6} - 34306 T^{7} + 477755 T^{8} - 1508619 T^{9} + 13018382 T^{10} - 40008547 T^{11} + 185707425 T^{12} - 329321167 T^{13} + 893871739 T^{14} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 171 T^{2} + 13495 T^{4} - 648401 T^{6} + 21195666 T^{8} - 508180397 T^{10} + 9885798383 T^{12} - 183043667880 T^{14} + 3568773216263 T^{16} - 66226577517437 T^{18} + 997168780351746 T^{20} - 11012159259347441 T^{22} + 82738739149024495 T^{24} - 378476851160313531 T^{26} + 799006685782884121 T^{28} \))
$23$ (\( 1 - 7 T + 23 T^{2} \))(\( 1 - T + 11 T^{2} - 59 T^{3} + 253 T^{4} - 529 T^{5} + 12167 T^{6} \))(\( 1 + 2 T + 61 T^{2} + 95 T^{3} + 1925 T^{4} + 2185 T^{5} + 32269 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 - 43 T^{2} + 529 T^{4} \))(\( 1 - 63 T^{2} + 1641 T^{4} - 33137 T^{6} + 868089 T^{8} - 17629983 T^{10} + 148035889 T^{12} \))(\( ( 1 + 12 T + 186 T^{2} + 1534 T^{3} + 13723 T^{4} + 84457 T^{5} + 540127 T^{6} + 2553922 T^{7} + 12422921 T^{8} + 44677753 T^{9} + 166967741 T^{10} + 429276094 T^{11} + 1197159798 T^{12} + 1776430668 T^{13} + 3404825447 T^{14} )^{2} \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 - 148 T^{2} + 11158 T^{4} - 577126 T^{6} + 23128723 T^{8} - 765546169 T^{10} + 21664425147 T^{12} - 533041733280 T^{14} + 11460480902763 T^{16} - 214231205479129 T^{18} + 3423881070739747 T^{20} - 45195305691282406 T^{22} + 462237012121895542 T^{24} - 3243364415939007508 T^{26} + 11592836324538749809 T^{28} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 6 T + 50 T^{2} + 167 T^{3} + 1450 T^{4} + 5046 T^{5} + 24389 T^{6} \))(\( 1 - T + 57 T^{2} + 67 T^{3} + 1567 T^{4} + 1943 T^{5} + 47937 T^{6} - 24389 T^{7} + 707281 T^{8} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + 54 T^{2} - 9 T^{3} + 1566 T^{4} + 24389 T^{6} )^{2} \))(\( 1 + 5 T - 74 T^{2} - 121 T^{3} + 3914 T^{4} - 2166 T^{5} - 93839 T^{6} + 289968 T^{7} + 166803 T^{8} - 5919018 T^{9} + 63074982 T^{10} - 88345396 T^{11} - 2512842439 T^{12} + 1588318079 T^{13} + 58132597430 T^{14} + 46061224291 T^{15} - 2113300491199 T^{16} - 2154655863044 T^{17} + 44611736343942 T^{18} - 121405860131682 T^{19} + 99218314412763 T^{20} + 5001912133568112 T^{21} - 46942623145847279 T^{22} - 31422478183732254 T^{23} + 1646648111136986714 T^{24} - 1476261681650405309 T^{25} - 26182293957204709034 T^{26} + 51303143564793010945 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 86 T^{2} + 147 T^{3} + 4546 T^{4} - 3726 T^{5} - 144739 T^{6} - 69630 T^{7} + 2280827 T^{8} + 7129752 T^{9} + 61238666 T^{10} - 266400360 T^{11} - 5489372847 T^{12} + 2978004429 T^{13} + 211561430766 T^{14} + 86362128441 T^{15} - 4616562564327 T^{16} - 6497238380040 T^{17} + 43312944927146 T^{18} + 146239405605048 T^{19} + 1356689090766467 T^{20} - 1201108887395670 T^{21} - 72405165565562179 T^{22} - 54053625906087894 T^{23} + 1912535082582713746 T^{24} + 1793474935558756863 T^{25} - 30428071355670337526 T^{26} - 30781886138875806567 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))
$31$ (\( 1 - 6 T + 31 T^{2} \))(\( 1 + 7 T + 65 T^{2} + 441 T^{3} + 2015 T^{4} + 6727 T^{5} + 29791 T^{6} \))(\( 1 + 5 T + 102 T^{2} + 396 T^{3} + 4560 T^{4} + 12276 T^{5} + 98022 T^{6} + 148955 T^{7} + 923521 T^{8} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 - T + 89 T^{2} - 61 T^{3} + 2759 T^{4} - 961 T^{5} + 29791 T^{6} )^{2} \))(\( 1 - 14 T - 4 T^{2} + 620 T^{3} + 777 T^{4} - 16756 T^{5} - 132212 T^{6} + 921899 T^{7} + 2562885 T^{8} - 15563136 T^{9} - 123261055 T^{10} + 289810704 T^{11} + 6237634884 T^{12} - 11613630724 T^{13} - 147711213332 T^{14} - 360022552444 T^{15} + 5994367123524 T^{16} + 8633750682864 T^{17} - 113834172774655 T^{18} - 445559370577536 T^{19} + 2274569871479685 T^{20} + 25363851436316789 T^{21} - 112762429842149492 T^{22} - 443022308924203276 T^{23} + 636851178984082377 T^{24} + 15753255675770995220 T^{25} - 3150651135154199044 T^{26} - \)\(34\!\cdots\!74\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + 3 T - 140 T^{2} - 809 T^{3} + 10183 T^{4} + 84310 T^{5} - 401882 T^{6} - 5525399 T^{7} + 5185096 T^{8} + 241285259 T^{9} + 455895208 T^{10} - 6951682276 T^{11} - 35592385525 T^{12} + 87803532392 T^{13} + 1409152530857 T^{14} + 2721909504152 T^{15} - 34204282489525 T^{16} - 207097566684316 T^{17} + 421028798387368 T^{18} + 6907792113985109 T^{19} + 4601791786338376 T^{20} - 152018170496305289 T^{21} - 342761555908863962 T^{22} + 2229124544366172010 T^{23} + 8346274846325496583 T^{24} - 20555457809191508279 T^{25} - \)\(11\!\cdots\!40\)\( T^{26} + 73252638892335127773 T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 104 T^{2} - 7 T^{3} + 3848 T^{4} + 50653 T^{6} \))(\( 1 + 12 T + 83 T^{2} + 301 T^{3} + 1366 T^{4} + 11137 T^{5} + 113627 T^{6} + 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 34 T^{2} + 1369 T^{4} \))(\( 1 - 122 T^{2} + 6816 T^{4} - 269305 T^{6} + 9331104 T^{8} - 228647642 T^{10} + 2565726409 T^{12} \))(\( 1 - 142 T^{2} + 11067 T^{4} - 1780 T^{5} - 514146 T^{6} + 372989 T^{7} + 13157703 T^{8} - 30138422 T^{9} + 90994613 T^{10} + 1373151654 T^{11} - 25668638664 T^{12} - 22483079694 T^{13} + 1310138880382 T^{14} - 831873948678 T^{15} - 35140366331016 T^{16} + 69554250730062 T^{17} + 170538554894693 T^{18} - 2089917439215854 T^{19} + 33759066068878527 T^{20} + 35408545919960537 T^{21} - 1805927261315666466 T^{22} - 231331896835237060 T^{23} + 53216603249548334883 T^{24} - \)\(93\!\cdots\!02\)\( T^{26} + \)\(90\!\cdots\!89\)\( T^{28} \))(\( 1 - 15 T + 112 T^{2} - 555 T^{3} + 1369 T^{4} \))(\( 1 - 24 T + 336 T^{2} - 3456 T^{3} + 27265 T^{4} - 179094 T^{5} + 1008016 T^{6} - 5261787 T^{7} + 28651383 T^{8} - 164827734 T^{9} + 967493767 T^{10} - 4687804986 T^{11} + 14918087468 T^{12} - 6591759054 T^{13} - 203376109758 T^{14} - 243895084998 T^{15} + 20422861743692 T^{16} - 237451385955858 T^{17} + 1813239085854487 T^{18} - 11429807298903438 T^{19} + 73511610017473647 T^{20} - 499511316984016671 T^{21} + 3540635489223630736 T^{22} - 23275367826859520238 T^{23} + \)\(13\!\cdots\!85\)\( T^{24} - \)\(61\!\cdots\!28\)\( T^{25} + \)\(22\!\cdots\!16\)\( T^{26} - \)\(58\!\cdots\!28\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 72 T^{2} + 563 T^{3} + 2952 T^{4} + 10086 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 211 T^{2} - 1537 T^{3} + 13946 T^{4} - 63017 T^{5} + 354691 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 - 54 T^{2} + 5538 T^{4} - 180701 T^{6} + 9309378 T^{8} - 152591094 T^{10} + 4750104241 T^{12} \))(\( ( 1 - 5 T + 154 T^{2} - 690 T^{3} + 13278 T^{4} - 47984 T^{5} + 744974 T^{6} - 2350140 T^{7} + 30543934 T^{8} - 80661104 T^{9} + 915133038 T^{10} - 1949775090 T^{11} + 17841854954 T^{12} - 23750521205 T^{13} + 194754273881 T^{14} )^{2} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 202 T^{2} + 22423 T^{4} - 1661389 T^{6} + 90408307 T^{8} - 3819006697 T^{10} + 138561193887 T^{12} - 5207189680377 T^{14} + 232921366924047 T^{16} - 10791600183121417 T^{18} + 429448882502329987 T^{20} - 13266066941484109069 T^{22} + \)\(30\!\cdots\!23\)\( T^{24} - \)\(45\!\cdots\!62\)\( T^{26} + \)\(37\!\cdots\!61\)\( T^{28} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 9 T + 93 T^{2} - 423 T^{3} + 3999 T^{4} - 16641 T^{5} + 79507 T^{6} \))(\( 1 - 5 T + 132 T^{2} - 570 T^{3} + 8042 T^{4} - 24510 T^{5} + 244068 T^{6} - 397535 T^{7} + 3418801 T^{8} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 173 T^{2} - 1159 T^{3} + 7439 T^{4} - 24037 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 - 8 T + 72 T^{2} - 642 T^{3} + 6134 T^{4} - 38976 T^{5} + 320008 T^{6} - 2368300 T^{7} + 13760344 T^{8} - 72066624 T^{9} + 487695938 T^{10} - 2194870242 T^{11} + 10584607896 T^{12} - 50570904392 T^{13} + 271818611107 T^{14} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 12 T + 134 T^{2} + 1274 T^{3} + 12864 T^{4} + 94708 T^{5} + 745626 T^{6} + 4959810 T^{7} + 32061918 T^{8} + 175115092 T^{9} + 1022778048 T^{10} + 4355552474 T^{11} + 19699131362 T^{12} + 75856356588 T^{13} + 271818611107 T^{14} )^{2} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 10 T + 116 T^{2} + 815 T^{3} + 5452 T^{4} + 22090 T^{5} + 103823 T^{6} \))(\( 1 + T + 183 T^{2} + 137 T^{3} + 12787 T^{4} + 6439 T^{5} + 404247 T^{6} + 103823 T^{7} + 4879681 T^{8} \))(\( 1 + 53 T^{2} + 2209 T^{4} \))(\( 1 - 150 T^{2} + 13698 T^{4} - 762929 T^{6} + 30258882 T^{8} - 731952150 T^{10} + 10779215329 T^{12} \))(\( 1 + 7 T - 131 T^{2} - 90 T^{3} + 13092 T^{4} - 36737 T^{5} - 525374 T^{6} + 3170310 T^{7} + 321404 T^{8} + 31977 T^{9} + 169445562 T^{10} - 10273810565 T^{11} + 79205198456 T^{12} + 367055157005 T^{13} - 6107815427144 T^{14} + 17251592379235 T^{15} + 174964283389304 T^{16} - 1066657834289995 T^{17} + 826840289425722 T^{18} + 7333765288839 T^{19} + 3464482923601916 T^{20} + 1606152345035053530 T^{21} - 12509830918636023614 T^{22} - 41113496190376351279 T^{23} + \)\(68\!\cdots\!08\)\( T^{24} - \)\(22\!\cdots\!70\)\( T^{25} - \)\(15\!\cdots\!71\)\( T^{26} + \)\(38\!\cdots\!89\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))(\( 1 + 15 T + 122 T^{2} + 705 T^{3} + 2209 T^{4} \))(\( 1 - 30 T + 584 T^{2} - 8520 T^{3} + 103333 T^{4} - 1069185 T^{5} + 9534476 T^{6} - 72742968 T^{7} + 456714346 T^{8} - 2068959987 T^{9} + 1911484325 T^{10} + 90846863325 T^{11} - 1317875483394 T^{12} + 12495584315589 T^{13} - 94346161680183 T^{14} + 587292462832683 T^{15} - 2911186942817346 T^{16} + 9431993890991475 T^{17} + 9327433742500325 T^{18} - 474505642701234909 T^{19} + 4923022279377409834 T^{20} - 36853269439900154184 T^{21} + \)\(22\!\cdots\!36\)\( T^{22} - \)\(11\!\cdots\!95\)\( T^{23} + \)\(54\!\cdots\!17\)\( T^{24} - \)\(21\!\cdots\!60\)\( T^{25} + \)\(67\!\cdots\!44\)\( T^{26} - \)\(16\!\cdots\!10\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 - 9 T + 179 T^{2} - 967 T^{3} + 9487 T^{4} - 25281 T^{5} + 148877 T^{6} \))(\( 1 + 19 T + 188 T^{2} + 1670 T^{3} + 13626 T^{4} + 88510 T^{5} + 528092 T^{6} + 2828663 T^{7} + 7890481 T^{8} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 151 T^{2} + 15011 T^{4} - 925501 T^{6} + 42165899 T^{8} - 1191462631 T^{10} + 22164361129 T^{12} \))(\( 1 + 5 T - 168 T^{2} - 667 T^{3} + 14748 T^{4} + 35404 T^{5} - 717057 T^{6} + 741452 T^{7} + 18064385 T^{8} - 229467306 T^{9} - 68709698 T^{10} + 15189679044 T^{11} - 2447314357 T^{12} - 369768135845 T^{13} - 257611268646 T^{14} - 19597711199785 T^{15} - 6874506028813 T^{16} + 2261393847033588 T^{17} - 542152566584738 T^{18} - 95962193160051858 T^{19} + 400385552713290665 T^{20} + 870991924054423324 T^{21} - 44643746827299284577 T^{22} + \)\(11\!\cdots\!32\)\( T^{23} + \)\(25\!\cdots\!52\)\( T^{24} - \)\(61\!\cdots\!99\)\( T^{25} - \)\(82\!\cdots\!88\)\( T^{26} + \)\(13\!\cdots\!65\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))(\( 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 6 T + 177 T^{2} + 990 T^{3} + 12146 T^{4} + 80799 T^{5} + 541301 T^{6} + 4397820 T^{7} + 32759667 T^{8} + 169473831 T^{9} + 2327816956 T^{10} + 10991900031 T^{11} + 147575394237 T^{12} + 1090984870677 T^{13} + 8401398981261 T^{14} + 57822198145881 T^{15} + 414539282411733 T^{16} + 1636441100915187 T^{17} + 18367595462795836 T^{18} + 70873192305643683 T^{19} + 726097089853784043 T^{20} + 5166168144997955340 T^{21} + 33701232679360120661 T^{22} + \)\(26\!\cdots\!67\)\( T^{23} + \)\(21\!\cdots\!54\)\( T^{24} + \)\(91\!\cdots\!30\)\( T^{25} + \)\(86\!\cdots\!57\)\( T^{26} + \)\(15\!\cdots\!38\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 + 25 T + 369 T^{2} + 3411 T^{3} + 21771 T^{4} + 87025 T^{5} + 205379 T^{6} \))(\( 1 - 27 T + 452 T^{2} - 5292 T^{3} + 46698 T^{4} - 312228 T^{5} + 1573412 T^{6} - 5545233 T^{7} + 12117361 T^{8} \))(\( 1 - 106 T^{2} + 3481 T^{4} \))(\( 1 - 159 T^{2} + 15327 T^{4} - 1011437 T^{6} + 53353287 T^{8} - 1926660399 T^{10} + 42180533641 T^{12} \))(\( 1 - T - 256 T^{2} + 359 T^{3} + 31874 T^{4} - 47930 T^{5} - 2907445 T^{6} + 2728904 T^{7} + 234594707 T^{8} - 87403618 T^{9} - 16751719950 T^{10} + 5972169084 T^{11} + 1053043121365 T^{12} - 244419945183 T^{13} - 62439761526824 T^{14} - 14420776765797 T^{15} + 3665643105471565 T^{16} + 1226558114302836 T^{17} - 202986638005051950 T^{18} - 62486970328713782 T^{19} + 9895329930614038187 T^{20} + 6791290991528508376 T^{21} - \)\(42\!\cdots\!45\)\( T^{22} - \)\(41\!\cdots\!70\)\( T^{23} + \)\(16\!\cdots\!74\)\( T^{24} + \)\(10\!\cdots\!81\)\( T^{25} - \)\(45\!\cdots\!36\)\( T^{26} - \)\(10\!\cdots\!79\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))(\( 1 + 3 T + 62 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 18 T + 449 T^{2} + 6138 T^{3} + 94012 T^{4} + 1089549 T^{5} + 13130579 T^{6} + 135126204 T^{7} + 1402031431 T^{8} + 13133686215 T^{9} + 123059048840 T^{10} + 1062581079861 T^{11} + 9170955443853 T^{12} + 73185751938309 T^{13} + 584575307569053 T^{14} + 4317959364360231 T^{15} + 31924095900052293 T^{16} + 218231839600772319 T^{17} + 1491150919110911240 T^{18} + 9389591410544838285 T^{19} + 59138433941034870271 T^{20} + \)\(33\!\cdots\!76\)\( T^{21} + \)\(19\!\cdots\!59\)\( T^{22} + \)\(94\!\cdots\!11\)\( T^{23} + \)\(48\!\cdots\!12\)\( T^{24} + \)\(18\!\cdots\!42\)\( T^{25} + \)\(79\!\cdots\!69\)\( T^{26} + \)\(18\!\cdots\!22\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))
$61$ (\( 1 + 5 T + 61 T^{2} \))(\( 1 + 10 T + 39 T^{2} - 116 T^{3} + 2379 T^{4} + 37210 T^{5} + 226981 T^{6} \))(\( 1 + 7 T + 142 T^{2} + 1353 T^{3} + 10098 T^{4} + 82533 T^{5} + 528382 T^{6} + 1588867 T^{7} + 13845841 T^{8} \))(\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 159 T^{2} + 676 T^{3} + 9699 T^{4} + 22326 T^{5} + 226981 T^{6} )^{2} \))(\( 1 + 3 T - 195 T^{2} + 1594 T^{3} + 23416 T^{4} - 298808 T^{5} + 316569 T^{6} + 24858977 T^{7} - 187108708 T^{8} + 410552095 T^{9} + 10930306155 T^{10} - 144567948264 T^{11} + 806435215123 T^{12} + 6305606099303 T^{13} - 97736065044098 T^{14} + 384641972057483 T^{15} + 3000745435472683 T^{16} - 32814177464910984 T^{17} + 151339281103451355 T^{18} + 346750780804800595 T^{19} - 9639910682363035588 T^{20} + 78125371877560810517 T^{21} + 60688592368236248889 T^{22} - \)\(34\!\cdots\!28\)\( T^{23} + \)\(16\!\cdots\!16\)\( T^{24} + \)\(69\!\cdots\!34\)\( T^{25} - \)\(51\!\cdots\!95\)\( T^{26} + \)\(48\!\cdots\!43\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 10 T - 216 T^{2} + 732 T^{3} + 41211 T^{4} + 18096 T^{5} - 4221918 T^{6} - 18912534 T^{7} + 303410016 T^{8} + 2187510102 T^{9} - 10161342210 T^{10} - 153708076656 T^{11} - 25088028468 T^{12} + 3682119076470 T^{13} + 27544693118064 T^{14} + 224609263664670 T^{15} - 93352553929428 T^{16} - 34888812947455536 T^{17} - 140692328586248610 T^{18} + 1847562940549332702 T^{19} + 15631797609196999776 T^{20} - 59437230739503587214 T^{21} - \)\(80\!\cdots\!58\)\( T^{22} + \)\(21\!\cdots\!36\)\( T^{23} + \)\(29\!\cdots\!11\)\( T^{24} + \)\(31\!\cdots\!52\)\( T^{25} - \)\(57\!\cdots\!36\)\( T^{26} - \)\(16\!\cdots\!10\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( 1 + 11 T + 176 T^{2} + 1431 T^{3} + 11792 T^{4} + 49379 T^{5} + 300763 T^{6} \))(\( 1 - 7 T + 215 T^{2} - 952 T^{3} + 19060 T^{4} - 63784 T^{5} + 965135 T^{6} - 2105341 T^{7} + 20151121 T^{8} \))(\( 1 - 26 T^{2} + 4489 T^{4} \))(\( 1 + 9 T^{2} + 10818 T^{4} + 125449 T^{6} + 48562002 T^{8} + 181360089 T^{10} + 90458382169 T^{12} \))(\( 1 - 257 T^{2} + 150 T^{3} + 32321 T^{4} - 41434 T^{5} - 2488493 T^{6} + 6547163 T^{7} + 135731623 T^{8} - 679818391 T^{9} - 7634823218 T^{10} + 45030949070 T^{11} + 604186994940 T^{12} - 1312843966751 T^{13} - 46396709910754 T^{14} - 87960545772317 T^{15} + 2712195420285660 T^{16} + 13543643335140410 T^{17} - 153850246479527378 T^{18} - 917839877889442837 T^{19} + 12278063025752630287 T^{20} + 39680466776041348649 T^{21} - \)\(10\!\cdots\!13\)\( T^{22} - \)\(11\!\cdots\!98\)\( T^{23} + \)\(58\!\cdots\!29\)\( T^{24} + \)\(18\!\cdots\!50\)\( T^{25} - \)\(21\!\cdots\!77\)\( T^{26} + \)\(36\!\cdots\!29\)\( T^{28} \))(\( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 421 T^{2} + 3546 T^{3} + 94549 T^{4} + 694539 T^{5} + 14221205 T^{6} + 90266103 T^{7} + 1598714972 T^{8} + 8669730024 T^{9} + 144368891972 T^{10} + 674041358691 T^{11} + 11142609100566 T^{12} + 46613450256372 T^{13} + 777006276980415 T^{14} + 3123101167176924 T^{15} + 50019172252440774 T^{16} + 202726701163981233 T^{17} + 2909195010763700612 T^{18} + 11705220176314112568 T^{19} + \)\(14\!\cdots\!68\)\( T^{20} + \)\(54\!\cdots\!69\)\( T^{21} + \)\(57\!\cdots\!05\)\( T^{22} + \)\(18\!\cdots\!33\)\( T^{23} + \)\(17\!\cdots\!01\)\( T^{24} + \)\(43\!\cdots\!18\)\( T^{25} + \)\(34\!\cdots\!81\)\( T^{26} + \)\(49\!\cdots\!83\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 + 10 T + 202 T^{2} + 1197 T^{3} + 14342 T^{4} + 50410 T^{5} + 357911 T^{6} \))(\( 1 - 32 T + 493 T^{2} - 4973 T^{3} + 42626 T^{4} - 353083 T^{5} + 2485213 T^{6} - 11453152 T^{7} + 25411681 T^{8} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 10 T + 88 T^{2} + 295 T^{3} + 6248 T^{4} + 50410 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 20 T + 579 T^{2} + 8115 T^{3} + 132531 T^{4} + 1398656 T^{5} + 16153348 T^{6} + 131118394 T^{7} + 1146887708 T^{8} + 7050624896 T^{9} + 47434302741 T^{10} + 206215791315 T^{11} + 1044648794229 T^{12} + 2562005678420 T^{13} + 9095120158391 T^{14} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 32 T + 815 T^{2} + 14201 T^{3} + 212941 T^{4} + 2568286 T^{5} + 27359596 T^{6} + 244426652 T^{7} + 1942531316 T^{8} + 12946729726 T^{9} + 76213926251 T^{10} + 360871281881 T^{11} + 1470446921065 T^{12} + 4099209085472 T^{13} + 9095120158391 T^{14} )^{2} \))
$73$ (\( 1 + 5 T + 73 T^{2} \))(\( 1 - 20 T + 350 T^{2} - 3201 T^{3} + 25550 T^{4} - 106580 T^{5} + 389017 T^{6} \))(\( 1 + 9 T + 215 T^{2} + 1261 T^{3} + 19735 T^{4} + 92053 T^{5} + 1145735 T^{6} + 3501153 T^{7} + 28398241 T^{8} \))(\( ( 1 + 5 T + 73 T^{2} )^{2} \))(\( ( 1 - 12 T + 156 T^{2} - 1019 T^{3} + 11388 T^{4} - 63948 T^{5} + 389017 T^{6} )^{2} \))(\( 1 + 7 T - 216 T^{2} - 761 T^{3} + 30236 T^{4} - 5132 T^{5} - 2309705 T^{6} + 17653826 T^{7} + 94942449 T^{8} - 2498427900 T^{9} + 5734974162 T^{10} + 198172952320 T^{11} - 1236199673969 T^{12} - 5916747621401 T^{13} + 120658511795086 T^{14} - 431922576362273 T^{15} - 6587708062580801 T^{16} + 77092647392669440 T^{17} + 162863178381249042 T^{18} - 5179419906648644700 T^{19} + 14368042061397841761 T^{20} + \)\(19\!\cdots\!22\)\( T^{21} - \)\(18\!\cdots\!05\)\( T^{22} - \)\(30\!\cdots\!16\)\( T^{23} + \)\(12\!\cdots\!64\)\( T^{24} - \)\(23\!\cdots\!97\)\( T^{25} - \)\(49\!\cdots\!36\)\( T^{26} + \)\(11\!\cdots\!31\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 5 T - 222 T^{2} + 1469 T^{3} + 33722 T^{4} - 288312 T^{5} - 2803789 T^{6} + 35232284 T^{7} + 135507783 T^{8} - 3287012128 T^{9} + 5566247642 T^{10} + 196460921568 T^{11} - 1425079738759 T^{12} - 6102443606857 T^{13} + 145697415902526 T^{14} - 445478383300561 T^{15} - 7594249927846711 T^{16} + 76426638325618656 T^{17} + 158071642003197722 T^{18} - 6814211468403279904 T^{19} + 20506965496442707287 T^{20} + \)\(38\!\cdots\!48\)\( T^{21} - \)\(22\!\cdots\!09\)\( T^{22} - \)\(16\!\cdots\!56\)\( T^{23} + \)\(14\!\cdots\!78\)\( T^{24} + \)\(46\!\cdots\!13\)\( T^{25} - \)\(50\!\cdots\!62\)\( T^{26} - \)\(83\!\cdots\!65\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 6 T + 158 T^{2} + 487 T^{3} + 12482 T^{4} + 37446 T^{5} + 493039 T^{6} \))(\( 1 + 24 T + 381 T^{2} + 4753 T^{3} + 48980 T^{4} + 375487 T^{5} + 2377821 T^{6} + 11832936 T^{7} + 38950081 T^{8} \))(\( 1 - 50 T^{2} + 6241 T^{4} \))(\( 1 - 416 T^{2} + 75546 T^{4} - 7729573 T^{6} + 471482586 T^{8} - 16203233696 T^{10} + 243087455521 T^{12} \))(\( 1 + 15 T - 209 T^{2} - 5006 T^{3} + 10203 T^{4} + 728204 T^{5} + 1795484 T^{6} - 52184699 T^{7} - 242747615 T^{8} + 635976918 T^{9} - 6456112423 T^{10} + 187012772662 T^{11} + 4353261319184 T^{12} - 9794029551585 T^{13} - 499519213467574 T^{14} - 773728334575215 T^{15} + 27168703893027344 T^{16} + 92204590420499818 T^{17} - 251466101820956263 T^{18} + 1956936845148198282 T^{19} - 59008900064141332415 T^{20} - \)\(10\!\cdots\!41\)\( T^{21} + \)\(27\!\cdots\!24\)\( T^{22} + \)\(87\!\cdots\!76\)\( T^{23} + \)\(96\!\cdots\!03\)\( T^{24} - \)\(37\!\cdots\!74\)\( T^{25} - \)\(12\!\cdots\!69\)\( T^{26} + \)\(70\!\cdots\!85\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))(\( 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( 1 - 30 T + 864 T^{2} - 16920 T^{3} + 312754 T^{4} - 4851879 T^{5} + 71061277 T^{6} - 934017690 T^{7} + 11650092741 T^{8} - 134428874772 T^{9} + 1483468640155 T^{10} - 15379388685633 T^{11} + 153443306278238 T^{12} - 1449071639103042 T^{13} + 13221803703290193 T^{14} - 114476659489140318 T^{15} + 957639674482483358 T^{16} - 7582638418175808687 T^{17} + 57781223694997102555 T^{18} - \)\(41\!\cdots\!28\)\( T^{19} + \)\(28\!\cdots\!61\)\( T^{20} - \)\(17\!\cdots\!10\)\( T^{21} + \)\(10\!\cdots\!97\)\( T^{22} - \)\(58\!\cdots\!01\)\( T^{23} + \)\(29\!\cdots\!54\)\( T^{24} - \)\(12\!\cdots\!80\)\( T^{25} + \)\(51\!\cdots\!24\)\( T^{26} - \)\(14\!\cdots\!70\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))
$83$ (\( 1 + 2 T + 83 T^{2} \))(\( 1 + 13 T + 247 T^{2} + 2019 T^{3} + 20501 T^{4} + 89557 T^{5} + 571787 T^{6} \))(\( 1 - 21 T + 412 T^{2} - 4928 T^{3} + 54080 T^{4} - 409024 T^{5} + 2838268 T^{6} - 12007527 T^{7} + 47458321 T^{8} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 29 T + 517 T^{2} + 5585 T^{3} + 42911 T^{4} + 199781 T^{5} + 571787 T^{6} )^{2} \))(\( 1 - 39 T + 530 T^{2} - 3161 T^{3} + 28225 T^{4} - 370688 T^{5} - 87566 T^{6} + 23065050 T^{7} + 138276576 T^{8} - 1874585286 T^{9} - 12668644086 T^{10} + 87558436288 T^{11} + 75003983467 T^{12} + 22083555186823 T^{13} - 370955487318014 T^{14} + 1832935080506309 T^{15} + 516702442104163 T^{16} + 50064775609806656 T^{17} - 601232577668139606 T^{18} - 7384067630323778898 T^{19} + 45208195385626904544 T^{20} + \)\(62\!\cdots\!50\)\( T^{21} - \)\(19\!\cdots\!06\)\( T^{22} - \)\(69\!\cdots\!64\)\( T^{23} + \)\(43\!\cdots\!25\)\( T^{24} - \)\(40\!\cdots\!87\)\( T^{25} + \)\(56\!\cdots\!30\)\( T^{26} - \)\(34\!\cdots\!57\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 28 T + 41 T^{2} + 4596 T^{3} - 4315 T^{4} - 681152 T^{5} - 523714 T^{6} + 79151132 T^{7} + 277631718 T^{8} - 7893971508 T^{9} - 40028311782 T^{10} + 488680000368 T^{11} + 5194178913307 T^{12} - 14953846613272 T^{13} - 499975501296289 T^{14} - 1241169268901576 T^{15} + 35782698533771923 T^{16} + 279420871370417616 T^{17} - 1899676469638238022 T^{18} - 31094674604695999644 T^{19} + 90769017541996917942 T^{20} + \)\(21\!\cdots\!64\)\( T^{21} - \)\(11\!\cdots\!74\)\( T^{22} - \)\(12\!\cdots\!56\)\( T^{23} - \)\(66\!\cdots\!35\)\( T^{24} + \)\(59\!\cdots\!32\)\( T^{25} + \)\(43\!\cdots\!01\)\( T^{26} - \)\(24\!\cdots\!64\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))
$89$ (\( 1 - T + 89 T^{2} \))(\( 1 + 21 T + 351 T^{2} + 3829 T^{3} + 31239 T^{4} + 166341 T^{5} + 704969 T^{6} \))(\( 1 + 16 T + 327 T^{2} + 3923 T^{3} + 42877 T^{4} + 349147 T^{5} + 2590167 T^{6} + 11279504 T^{7} + 62742241 T^{8} \))(\( ( 1 - 15 T + 89 T^{2} )^{2} \))(\( ( 1 - 17 T + 313 T^{2} - 3035 T^{3} + 27857 T^{4} - 134657 T^{5} + 704969 T^{6} )^{2} \))(\( 1 - 12 T - 194 T^{2} + 22 T^{3} + 49929 T^{4} + 231091 T^{5} - 4806057 T^{6} - 77296472 T^{7} + 239277060 T^{8} + 9675713647 T^{9} + 38702295221 T^{10} - 820673898507 T^{11} - 8419133241029 T^{12} + 27773390115522 T^{13} + 970766944390476 T^{14} + 2471831720281458 T^{15} - 66687954402190709 T^{16} - 578549657556581283 T^{17} + 2428268734009130261 T^{18} + 54029760216348600503 T^{19} + \)\(11\!\cdots\!60\)\( T^{20} - \)\(34\!\cdots\!88\)\( T^{21} - \)\(18\!\cdots\!17\)\( T^{22} + \)\(80\!\cdots\!19\)\( T^{23} + \)\(15\!\cdots\!29\)\( T^{24} + \)\(61\!\cdots\!58\)\( T^{25} - \)\(47\!\cdots\!74\)\( T^{26} - \)\(26\!\cdots\!28\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 28 T + 140 T^{2} - 5730 T^{3} - 93967 T^{4} + 96233 T^{5} + 15970193 T^{6} + 123811324 T^{7} - 1044909120 T^{8} - 22922220183 T^{9} - 51502050867 T^{10} + 2062620559623 T^{11} + 19108855660321 T^{12} - 75971879521166 T^{13} - 2234817993677728 T^{14} - 6761497277383774 T^{15} + 151361245685402641 T^{16} + 1454083553296866687 T^{17} - 3231354087491572947 T^{18} - \)\(12\!\cdots\!67\)\( T^{19} - \)\(51\!\cdots\!20\)\( T^{20} + \)\(54\!\cdots\!96\)\( T^{21} + \)\(62\!\cdots\!33\)\( T^{22} + \)\(33\!\cdots\!97\)\( T^{23} - \)\(29\!\cdots\!67\)\( T^{24} - \)\(15\!\cdots\!70\)\( T^{25} + \)\(34\!\cdots\!40\)\( T^{26} + \)\(61\!\cdots\!32\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))
$97$ (\( 1 - T + 97 T^{2} \))(\( 1 + 20 T + 366 T^{2} + 3755 T^{3} + 35502 T^{4} + 188180 T^{5} + 912673 T^{6} \))(\( 1 + 11 T + 343 T^{2} + 3105 T^{3} + 47701 T^{4} + 301185 T^{5} + 3227287 T^{6} + 10039403 T^{7} + 88529281 T^{8} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 276 T^{2} + 1171 T^{3} + 26772 T^{4} + 56454 T^{5} + 912673 T^{6} )^{2} \))(\( 1 + T - 259 T^{2} + 1278 T^{3} + 30916 T^{4} - 308928 T^{5} + 715001 T^{6} + 30629599 T^{7} - 475159668 T^{8} + 1475361605 T^{9} + 39479667059 T^{10} - 483225071104 T^{11} + 2823068991119 T^{12} + 29743475078269 T^{13} - 618526417728786 T^{14} + 2885117082592093 T^{15} + 26562256137438671 T^{16} - 441026475319700992 T^{17} + 3495106538852654579 T^{18} + 12669432104248632485 T^{19} - \)\(39\!\cdots\!72\)\( T^{20} + \)\(24\!\cdots\!87\)\( T^{21} + \)\(56\!\cdots\!61\)\( T^{22} - \)\(23\!\cdots\!76\)\( T^{23} + \)\(22\!\cdots\!84\)\( T^{24} + \)\(91\!\cdots\!34\)\( T^{25} - \)\(17\!\cdots\!19\)\( T^{26} + \)\(67\!\cdots\!77\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))(\( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} \))(\( 1 + 19 T - 87 T^{2} - 5386 T^{3} - 42748 T^{4} + 192060 T^{5} + 5838221 T^{6} + 65830325 T^{7} + 539930676 T^{8} - 2091581233 T^{9} - 121849354489 T^{10} - 1174130487420 T^{11} - 483818959705 T^{12} + 86355889709651 T^{13} + 1096665730261830 T^{14} + 8376521301836147 T^{15} - 4552252591864345 T^{16} - 1071597194345073660 T^{17} - 10787235743225292409 T^{18} - 17961119722926596881 T^{19} + \)\(44\!\cdots\!04\)\( T^{20} + \)\(53\!\cdots\!25\)\( T^{21} + \)\(45\!\cdots\!81\)\( T^{22} + \)\(14\!\cdots\!20\)\( T^{23} - \)\(31\!\cdots\!52\)\( T^{24} - \)\(38\!\cdots\!58\)\( T^{25} - \)\(60\!\cdots\!67\)\( T^{26} + \)\(12\!\cdots\!63\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))
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