Properties

Label 61.2
Level 61
Weight 2
Dimension 126
Nonzero newspaces 8
Newform subspaces 10
Sturm bound 620
Trace bound 3

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Defining parameters

Level: \( N \) = \( 61 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 10 \)
Sturm bound: \(620\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(61))\).

Total New Old
Modular forms 185 185 0
Cusp forms 126 126 0
Eisenstein series 59 59 0

Trace form

\( 126q - 27q^{2} - 26q^{3} - 23q^{4} - 24q^{5} - 18q^{6} - 22q^{7} - 15q^{8} - 17q^{9} + O(q^{10}) \) \( 126q - 27q^{2} - 26q^{3} - 23q^{4} - 24q^{5} - 18q^{6} - 22q^{7} - 15q^{8} - 17q^{9} - 12q^{10} - 18q^{11} - 2q^{12} - 16q^{13} - 6q^{14} - 6q^{15} + q^{16} - 12q^{17} + 9q^{18} - 10q^{19} + 12q^{20} + 2q^{21} + 6q^{22} - 6q^{23} + 30q^{24} + q^{25} + 12q^{26} + 10q^{27} + 26q^{28} + 42q^{30} + 2q^{31} + 33q^{32} + 18q^{33} + 24q^{34} + 18q^{35} + 61q^{36} + 8q^{37} + 30q^{38} + 26q^{39} + 60q^{40} + 12q^{41} + 66q^{42} + 14q^{43} + 54q^{44} + 48q^{45} + 42q^{46} - 12q^{47} - 6q^{48} - 43q^{49} - 12q^{50} - 18q^{51} - 157q^{52} - 6q^{53} - 90q^{54} - 78q^{55} - 90q^{56} - 70q^{57} - 60q^{58} - 30q^{59} - 132q^{60} - 99q^{61} - 144q^{62} - 56q^{63} - 143q^{64} - 6q^{65} - 6q^{66} - 82q^{67} - 84q^{68} - 54q^{69} - 66q^{70} + 12q^{71} - 60q^{72} - 16q^{73} + 9q^{74} + 24q^{75} + 10q^{76} + 36q^{77} + 138q^{78} + 50q^{79} + 156q^{80} + 91q^{81} + 96q^{82} + 54q^{83} + 194q^{84} + 78q^{85} + 102q^{86} + 90q^{87} + 150q^{88} + 60q^{89} + 204q^{90} + 82q^{91} + 138q^{92} + 98q^{93} + 114q^{94} + 90q^{95} + 222q^{96} + 68q^{97} + 141q^{98} + 126q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(61))\)

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
61.2.a \(\chi_{61}(1, \cdot)\) 61.2.a.a 1 1
61.2.a.b 3
61.2.b \(\chi_{61}(60, \cdot)\) 61.2.b.a 4 1
61.2.c \(\chi_{61}(13, \cdot)\) 61.2.c.a 8 2
61.2.e \(\chi_{61}(9, \cdot)\) 61.2.e.a 12 4
61.2.f \(\chi_{61}(14, \cdot)\) 61.2.f.a 2 2
61.2.f.b 8
61.2.g \(\chi_{61}(3, \cdot)\) 61.2.g.a 16 4
61.2.i \(\chi_{61}(12, \cdot)\) 61.2.i.a 32 8
61.2.k \(\chi_{61}(4, \cdot)\) 61.2.k.a 40 8

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 - T + 3 T^{2} - 3 T^{3} + 6 T^{4} - 4 T^{5} + 8 T^{6} \))(\( 1 + 5 T^{4} + 16 T^{8} \))(\( 1 - 4 T^{2} + 2 T^{3} + 7 T^{4} - 6 T^{5} - 3 T^{6} + 7 T^{7} - T^{8} + 14 T^{9} - 12 T^{10} - 48 T^{11} + 112 T^{12} + 64 T^{13} - 256 T^{14} + 256 T^{16} \))(\( 1 + 2 T - 2 T^{2} - 6 T^{3} - 2 T^{4} - 6 T^{5} - 8 T^{6} + 30 T^{7} + 45 T^{8} + 14 T^{9} + 33 T^{10} - 69 T^{11} - 297 T^{12} - 138 T^{13} + 132 T^{14} + 112 T^{15} + 720 T^{16} + 960 T^{17} - 512 T^{18} - 768 T^{19} - 512 T^{20} - 3072 T^{21} - 2048 T^{22} + 4096 T^{23} + 4096 T^{24} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( ( 1 - T + 2 T^{2} - 4 T^{3} + T^{4} - 8 T^{5} + 8 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + T - T^{2} + T^{3} + 7 T^{4} + 2 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} ) \))(\( 1 + 5 T + 15 T^{2} + 35 T^{3} + 65 T^{4} + 100 T^{5} + 120 T^{6} + 110 T^{7} + 69 T^{8} + 5 T^{9} - 40 T^{10} - 85 T^{11} - 170 T^{12} - 565 T^{13} - 1515 T^{14} - 2960 T^{15} - 4739 T^{16} - 5920 T^{17} - 6060 T^{18} - 4520 T^{19} - 2720 T^{20} - 2720 T^{21} - 2560 T^{22} + 640 T^{23} + 17664 T^{24} + 56320 T^{25} + 122880 T^{26} + 204800 T^{27} + 266240 T^{28} + 286720 T^{29} + 245760 T^{30} + 163840 T^{31} + 65536 T^{32} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 - 2 T + 5 T^{2} - 8 T^{3} + 15 T^{4} - 18 T^{5} + 27 T^{6} \))(\( ( 1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4} )^{2} \))(\( ( 1 + 3 T + 10 T^{2} + 23 T^{3} + 41 T^{4} + 69 T^{5} + 90 T^{6} + 81 T^{7} + 81 T^{8} )^{2} \))(\( 1 + T - T^{2} + 9 T^{4} - 12 T^{5} - 22 T^{6} + 48 T^{7} + 14 T^{8} - 120 T^{9} - 36 T^{10} + 119 T^{11} - 857 T^{12} + 357 T^{13} - 324 T^{14} - 3240 T^{15} + 1134 T^{16} + 11664 T^{17} - 16038 T^{18} - 26244 T^{19} + 59049 T^{20} - 59049 T^{22} + 177147 T^{23} + 531441 T^{24} \))(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 + T + 6 T^{2} + 7 T^{3} + 19 T^{4} + 21 T^{5} + 54 T^{6} + 27 T^{7} + 81 T^{8} )^{2} \))(\( 1 + T - 6 T^{2} - T^{3} + 25 T^{4} - 13 T^{5} - 99 T^{6} + 103 T^{7} + 313 T^{8} - 507 T^{9} - 702 T^{10} + 1415 T^{11} + 960 T^{12} - 3374 T^{13} - 109 T^{14} + 5014 T^{15} - 2597 T^{16} + 15042 T^{17} - 981 T^{18} - 91098 T^{19} + 77760 T^{20} + 343845 T^{21} - 511758 T^{22} - 1108809 T^{23} + 2053593 T^{24} + 2027349 T^{25} - 5845851 T^{26} - 2302911 T^{27} + 13286025 T^{28} - 1594323 T^{29} - 28697814 T^{30} + 14348907 T^{31} + 43046721 T^{32} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 6 T^{2} - 3 T^{3} + 30 T^{4} + 25 T^{5} + 125 T^{6} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( 1 + 2 T - 4 T^{2} + 14 T^{3} + 41 T^{4} - 71 T^{5} + 223 T^{6} + 591 T^{7} - 789 T^{8} + 2955 T^{9} + 5575 T^{10} - 8875 T^{11} + 25625 T^{12} + 43750 T^{13} - 62500 T^{14} + 156250 T^{15} + 390625 T^{16} \))(\( 1 + 5 T + 11 T^{2} + T^{3} - 34 T^{4} - 153 T^{5} - 463 T^{6} - 1112 T^{7} - 1071 T^{8} - 745 T^{9} + 4306 T^{10} + 27624 T^{11} + 99061 T^{12} + 138120 T^{13} + 107650 T^{14} - 93125 T^{15} - 669375 T^{16} - 3475000 T^{17} - 7234375 T^{18} - 11953125 T^{19} - 13281250 T^{20} + 1953125 T^{21} + 107421875 T^{22} + 244140625 T^{23} + 244140625 T^{24} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 6 T + 10 T^{2} - 6 T^{3} + 37 T^{4} - 57 T^{5} - 239 T^{6} + 675 T^{7} - 929 T^{8} + 3375 T^{9} - 5975 T^{10} - 7125 T^{11} + 23125 T^{12} - 18750 T^{13} + 156250 T^{14} - 468750 T^{15} + 390625 T^{16} \))(\( 1 - 9 T^{2} + 10 T^{3} + 57 T^{4} - 220 T^{5} - 317 T^{6} + 1615 T^{7} + 285 T^{8} - 10480 T^{9} + 12537 T^{10} + 58535 T^{11} - 100648 T^{12} - 197700 T^{13} + 748344 T^{14} + 215500 T^{15} - 4483684 T^{16} + 1077500 T^{17} + 18708600 T^{18} - 24712500 T^{19} - 62905000 T^{20} + 182921875 T^{21} + 195890625 T^{22} - 818750000 T^{23} + 111328125 T^{24} + 3154296875 T^{25} - 3095703125 T^{26} - 10742187500 T^{27} + 13916015625 T^{28} + 12207031250 T^{29} - 54931640625 T^{30} + 152587890625 T^{32} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 3 T + 20 T^{2} + 41 T^{3} + 140 T^{4} + 147 T^{5} + 343 T^{6} \))(\( 1 - 4 T^{2} + 75 T^{4} - 196 T^{6} + 2401 T^{8} \))(\( 1 + T - 23 T^{2} - 18 T^{3} + 316 T^{4} + 173 T^{5} - 3090 T^{6} - 518 T^{7} + 24095 T^{8} - 3626 T^{9} - 151410 T^{10} + 59339 T^{11} + 758716 T^{12} - 302526 T^{13} - 2705927 T^{14} + 823543 T^{15} + 5764801 T^{16} \))(\( 1 + 10 T + 38 T^{2} + 87 T^{3} + 315 T^{4} + 1182 T^{5} + 1989 T^{6} + 1778 T^{7} + 8482 T^{8} + 10483 T^{9} - 74537 T^{10} - 160780 T^{11} - 38835 T^{12} - 1125460 T^{13} - 3652313 T^{14} + 3595669 T^{15} + 20365282 T^{16} + 29882846 T^{17} + 234003861 T^{18} + 973427826 T^{19} + 1815912315 T^{20} + 3510763809 T^{21} + 10734059462 T^{22} + 19773267430 T^{23} + 13841287201 T^{24} \))(\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 + 3 T + 25 T^{2} + 66 T^{3} + 322 T^{4} + 675 T^{5} + 2758 T^{6} + 5052 T^{7} + 19399 T^{8} + 35364 T^{9} + 135142 T^{10} + 231525 T^{11} + 773122 T^{12} + 1109262 T^{13} + 2941225 T^{14} + 2470629 T^{15} + 5764801 T^{16} \))(\( 1 - 10 T + 79 T^{2} - 445 T^{3} + 2146 T^{4} - 8705 T^{5} + 31515 T^{6} - 101315 T^{7} + 299913 T^{8} - 824040 T^{9} + 2202665 T^{10} - 5904770 T^{11} + 16551554 T^{12} - 48186605 T^{13} + 141963314 T^{14} - 406960070 T^{15} + 1109260303 T^{16} - 2848720490 T^{17} + 6956202386 T^{18} - 16528005515 T^{19} + 39740281154 T^{20} - 99241469390 T^{21} + 259141334585 T^{22} - 678632373720 T^{23} + 1728938762313 T^{24} - 4088425693205 T^{25} + 8902207472235 T^{26} - 17212629297815 T^{27} + 29703402333346 T^{28} - 43115609631115 T^{29} + 53579622755071 T^{30} - 47475615099430 T^{31} + 33232930569601 T^{32} \))
$11$ (\( 1 + 5 T + 11 T^{2} \))(\( 1 - 13 T + 86 T^{2} - 353 T^{3} + 946 T^{4} - 1573 T^{5} + 1331 T^{6} \))(\( 1 - 36 T^{2} + 563 T^{4} - 4356 T^{6} + 14641 T^{8} \))(\( ( 1 + T + 32 T^{2} + 45 T^{3} + 463 T^{4} + 495 T^{5} + 3872 T^{6} + 1331 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 3 T + 44 T^{2} + 78 T^{3} + 898 T^{4} + 1269 T^{5} + 12275 T^{6} + 13959 T^{7} + 108658 T^{8} + 103818 T^{9} + 644204 T^{10} + 483153 T^{11} + 1771561 T^{12} )^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 69 T^{2} + 2224 T^{4} - 43971 T^{6} + 583551 T^{8} - 5320491 T^{10} + 32561584 T^{12} - 122237709 T^{14} + 214358881 T^{16} \))(\( 1 - 89 T^{2} + 3878 T^{4} - 111974 T^{6} + 2446727 T^{8} - 43671170 T^{10} + 666025567 T^{12} - 8872766367 T^{14} + 103997090230 T^{16} - 1073604730407 T^{18} + 9751280326447 T^{20} - 77366141596370 T^{22} + 524477661832487 T^{24} - 2904317182272374 T^{26} + 12170825244924038 T^{28} - 33797735188908449 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 + 9 T + 50 T^{2} + 197 T^{3} + 650 T^{4} + 1521 T^{5} + 2197 T^{6} \))(\( ( 1 - 3 T + 13 T^{2} )^{4} \))(\( 1 - 5 T - 26 T^{2} + 129 T^{3} + 652 T^{4} - 2398 T^{5} - 8817 T^{6} + 10624 T^{7} + 140609 T^{8} + 138112 T^{9} - 1490073 T^{10} - 5268406 T^{11} + 18621772 T^{12} + 47896797 T^{13} - 125497034 T^{14} - 313742585 T^{15} + 815730721 T^{16} \))(\( ( 1 - T + 42 T^{2} - 46 T^{3} + 926 T^{4} - 1009 T^{5} + 13933 T^{6} - 13117 T^{7} + 156494 T^{8} - 101062 T^{9} + 1199562 T^{10} - 371293 T^{11} + 4826809 T^{12} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 + 3 T - 28 T^{2} - 69 T^{3} + 412 T^{4} + 378 T^{5} - 8323 T^{6} + 600 T^{7} + 146293 T^{8} + 7800 T^{9} - 1406587 T^{10} + 830466 T^{11} + 11767132 T^{12} - 25619217 T^{13} - 135150652 T^{14} + 188245551 T^{15} + 815730721 T^{16} \))(\( ( 1 + 6 T + 66 T^{2} + 304 T^{3} + 2198 T^{4} + 8336 T^{5} + 46424 T^{6} + 150617 T^{7} + 707243 T^{8} + 1958021 T^{9} + 7845656 T^{10} + 18314192 T^{11} + 62777078 T^{12} + 112873072 T^{13} + 318569394 T^{14} + 376491102 T^{15} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 43 T^{2} + 72 T^{3} + 731 T^{4} + 578 T^{5} + 4913 T^{6} \))(\( 1 - 48 T^{2} + 1106 T^{4} - 13872 T^{6} + 83521 T^{8} \))(\( 1 - 4 T - 24 T^{2} - 10 T^{3} + 579 T^{4} + 1493 T^{5} - 5195 T^{6} - 17967 T^{7} + 35511 T^{8} - 305439 T^{9} - 1501355 T^{10} + 7335109 T^{11} + 48358659 T^{12} - 14198570 T^{13} - 579301656 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 - 6 T - 32 T^{2} + 149 T^{3} + 902 T^{4} - 5084 T^{5} + 7002 T^{6} + 88100 T^{7} - 362334 T^{8} - 2558599 T^{9} + 11222242 T^{10} + 12415888 T^{11} - 129065177 T^{12} + 211070096 T^{13} + 3243227938 T^{14} - 12570396887 T^{15} - 30262498014 T^{16} + 125089401700 T^{17} + 169011258138 T^{18} - 2086161813532 T^{19} + 6292133211782 T^{20} + 17669593598053 T^{21} - 64511804814368 T^{22} - 205631377845798 T^{23} + 582622237229761 T^{24} \))(\( 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 60 T^{2} - 288 T^{3} + 1723 T^{4} - 8091 T^{5} + 39033 T^{6} - 175173 T^{7} + 735339 T^{8} - 2977941 T^{9} + 11280537 T^{10} - 39751083 T^{11} + 143906683 T^{12} - 408918816 T^{13} + 1448254140 T^{14} - 2462032038 T^{15} + 6975757441 T^{16} \))(\( 1 + 43 T^{2} - 295 T^{3} + 1103 T^{4} - 12685 T^{5} + 64103 T^{6} - 315815 T^{7} + 2239621 T^{8} - 9762110 T^{9} + 51703990 T^{10} - 269739230 T^{11} + 1147692367 T^{12} - 5657629095 T^{13} + 24760716085 T^{14} - 102255926960 T^{15} + 460223486907 T^{16} - 1738350758320 T^{17} + 7155846948565 T^{18} - 27795931743735 T^{19} + 95856414184207 T^{20} - 382991133890110 T^{21} + 1248008626200310 T^{22} - 4005771263080030 T^{23} + 15623052855769861 T^{24} - 37451830215900055 T^{25} + 129231257000482247 T^{26} - 434739004662324605 T^{27} + 642632327664426383 T^{28} - 2921850519707251415 T^{29} + 7240246542054239947 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 9 T^{2} - 20 T^{3} + 171 T^{4} + 6859 T^{6} \))(\( ( 1 - 6 T + 44 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( 1 - T - 36 T^{2} + 31 T^{3} + 392 T^{4} - 18 T^{5} - 7667 T^{6} - 4474 T^{7} + 256473 T^{8} - 85006 T^{9} - 2767787 T^{10} - 123462 T^{11} + 51085832 T^{12} + 76759069 T^{13} - 1693651716 T^{14} - 893871739 T^{15} + 16983563041 T^{16} \))(\( 1 + T - 23 T^{2} - 117 T^{3} + 324 T^{4} + 2161 T^{5} + 580 T^{6} - 64339 T^{7} - 88488 T^{8} + 733724 T^{9} + 5207100 T^{10} - 7823946 T^{11} - 74936717 T^{12} - 148654974 T^{13} + 1879763100 T^{14} + 5032612916 T^{15} - 11531844648 T^{16} - 159309733561 T^{17} + 27286610980 T^{18} + 1931656827979 T^{19} + 5502674425284 T^{20} - 37754460640143 T^{21} - 141014523929423 T^{22} + 116490258898219 T^{23} + 2213314919066161 T^{24} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 34 T^{2} + 327 T^{3} + 274 T^{4} - 8790 T^{5} + 31721 T^{6} + 102258 T^{7} - 868163 T^{8} + 1942902 T^{9} + 11451281 T^{10} - 60290610 T^{11} + 35707954 T^{12} + 809684373 T^{13} - 1599559954 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} \))(\( 1 - 3 T - 40 T^{2} + 148 T^{3} + 492 T^{4} - 1623 T^{5} - 142 T^{6} - 27398 T^{7} - 44916 T^{8} + 1116180 T^{9} + 2675801 T^{10} - 33259049 T^{11} - 85748771 T^{12} + 851900020 T^{13} + 934455131 T^{14} - 8222009647 T^{15} - 3932859421 T^{16} - 156218183293 T^{17} + 337338302291 T^{18} + 5843182237180 T^{19} - 11174865585491 T^{20} - 82352697969851 T^{21} + 125885415425681 T^{22} + 997721757637020 T^{23} - 762833717549556 T^{24} - 8840997543749042 T^{25} - 870611408607742 T^{26} - 189063690191809437 T^{27} + 1088950940180551212 T^{28} + 6223841552414044732 T^{29} - 31960267431315364840 T^{30} - 45543381089624394897 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 9 T + 23 T^{2} \))(\( 1 - 5 T + 74 T^{2} - 229 T^{3} + 1702 T^{4} - 2645 T^{5} + 12167 T^{6} \))(\( 1 - 12 T^{2} + 1067 T^{4} - 6348 T^{6} + 279841 T^{8} \))(\( ( 1 + 4 T + 49 T^{2} + 172 T^{3} + 1685 T^{4} + 3956 T^{5} + 25921 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))(\( 1 + 7 T - 42 T^{2} - 538 T^{3} - 456 T^{4} + 16015 T^{5} + 80472 T^{6} - 49195 T^{7} - 1901339 T^{8} - 8111479 T^{9} - 8042732 T^{10} + 129029774 T^{11} + 1012403101 T^{12} + 2967684802 T^{13} - 4254605228 T^{14} - 98692364993 T^{15} - 532072607099 T^{16} - 316635893885 T^{17} + 11912744059608 T^{18} + 54528279533705 T^{19} - 35709809288136 T^{20} - 969020131867094 T^{21} - 1739913470973258 T^{22} + 6669668305397489 T^{23} + 21914624432020321 T^{24} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 90 T^{2} + 4495 T^{4} - 158652 T^{6} + 4216341 T^{8} - 83926908 T^{10} + 1257885295 T^{12} - 13323230010 T^{14} + 78310985281 T^{16} \))(\( 1 + 15 T + 207 T^{2} + 1925 T^{3} + 16287 T^{4} + 113670 T^{5} + 722203 T^{6} + 3991415 T^{7} + 19845672 T^{8} + 82824520 T^{9} + 275269985 T^{10} + 370268085 T^{11} - 3381524412 T^{12} - 43228698175 T^{13} - 311471986506 T^{14} - 1860890159125 T^{15} - 9431577763665 T^{16} - 42800473659875 T^{17} - 164768680861674 T^{18} - 525963570695225 T^{19} - 946289172978492 T^{20} + 2383172397013155 T^{21} + 40749836944491665 T^{22} + 282003033331560440 T^{23} + 1554134127883553832 T^{24} + 7189147750253340145 T^{25} + 29918350678030948747 T^{26} + \)\(10\!\cdots\!90\)\( T^{27} + \)\(35\!\cdots\!27\)\( T^{28} + \)\(97\!\cdots\!75\)\( T^{29} + \)\(23\!\cdots\!63\)\( T^{30} + \)\(39\!\cdots\!05\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 4 T + 83 T^{2} - 212 T^{3} + 2407 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 - 96 T^{2} + 3938 T^{4} - 80736 T^{6} + 707281 T^{8} \))(\( 1 + 16 T + 102 T^{2} + 290 T^{3} - 77 T^{4} - 5911 T^{5} - 49525 T^{6} - 288749 T^{7} - 1457969 T^{8} - 8373721 T^{9} - 41650525 T^{10} - 144163379 T^{11} - 54460637 T^{12} + 5948233210 T^{13} + 60671978742 T^{14} + 275998020944 T^{15} + 500246412961 T^{16} \))(\( ( 1 + 4 T + 72 T^{2} + 215 T^{3} + 2911 T^{4} + 11489 T^{5} + 105733 T^{6} + 333181 T^{7} + 2448151 T^{8} + 5243635 T^{9} + 50924232 T^{10} + 82044596 T^{11} + 594823321 T^{12} )^{2} \))(\( 1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 66 T^{2} + 2407 T^{4} - 12099 T^{5} + 53265 T^{6} - 637323 T^{7} + 461571 T^{8} - 18482367 T^{9} + 44795865 T^{10} - 295082511 T^{11} + 1702425367 T^{12} + 39258339186 T^{14} + 500246412961 T^{16} \))(\( 1 - 279 T^{2} + 37083 T^{4} - 3118264 T^{6} + 186254757 T^{8} - 8465412925 T^{10} + 310609973592 T^{12} - 9854178730147 T^{14} + 290981028014045 T^{16} - 8287364312053627 T^{18} + 219688532732123352 T^{20} - 5035425029684823925 T^{22} + 93173274086172705477 T^{24} - \)\(13\!\cdots\!64\)\( T^{26} + \)\(13\!\cdots\!03\)\( T^{28} - \)\(83\!\cdots\!99\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 2 T + 17 T^{2} + 240 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6} \))(\( 1 - 64 T^{2} + 2514 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( 1 - 11 T - 20 T^{2} + 277 T^{3} + 3274 T^{4} - 13042 T^{5} - 129249 T^{6} + 182534 T^{7} + 4224793 T^{8} + 5658554 T^{9} - 124208289 T^{10} - 388534222 T^{11} + 3023607754 T^{12} + 7930274827 T^{13} - 17750073620 T^{14} - 302638755221 T^{15} + 852891037441 T^{16} \))(\( 1 - 19 T + 136 T^{2} - 527 T^{3} + 2695 T^{4} - 20193 T^{5} + 98143 T^{6} - 325642 T^{7} + 1787329 T^{8} - 19925018 T^{9} + 158297415 T^{10} - 645589101 T^{11} + 2091689793 T^{12} - 20013262131 T^{13} + 152123815815 T^{14} - 593586211238 T^{15} + 1650635865409 T^{16} - 9322853989942 T^{17} + 87102273764383 T^{18} - 555562216743423 T^{19} + 2298541345903495 T^{20} - 13933680878673617 T^{21} + 111469447029388936 T^{22} - 482761061031691789 T^{23} + 787662783788549761 T^{24} \))(\( ( 1 + 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 3 T + 70 T^{2} + 201 T^{3} + 2350 T^{4} - 4536 T^{5} + 15295 T^{6} - 536172 T^{7} - 1199975 T^{8} - 16621332 T^{9} + 14698495 T^{10} - 135131976 T^{11} + 2170274350 T^{12} + 5754459351 T^{13} + 62125257670 T^{14} + 82537842333 T^{15} + 852891037441 T^{16} \))(\( 1 + 15 T + 229 T^{2} + 2720 T^{3} + 27882 T^{4} + 263495 T^{5} + 2229741 T^{6} + 17793265 T^{7} + 131977326 T^{8} + 925027605 T^{9} + 6198572490 T^{10} + 39477295695 T^{11} + 243905872923 T^{12} + 1454253303145 T^{13} + 8467585214485 T^{14} + 48422784740730 T^{15} + 270849094628077 T^{16} + 1501106326962630 T^{17} + 8137349391120085 T^{18} + 43323660153992695 T^{19} + 225252195667721883 T^{20} + 1130201459523804945 T^{21} + 5501255901820335690 T^{22} + 25449927538387534155 T^{23} + \)\(11\!\cdots\!66\)\( T^{24} + \)\(47\!\cdots\!15\)\( T^{25} + \)\(18\!\cdots\!41\)\( T^{26} + \)\(66\!\cdots\!45\)\( T^{27} + \)\(21\!\cdots\!02\)\( T^{28} + \)\(66\!\cdots\!20\)\( T^{29} + \)\(17\!\cdots\!09\)\( T^{30} + \)\(35\!\cdots\!65\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 75 T^{2} + 336 T^{3} + 2775 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 - 88 T^{2} + 4242 T^{4} - 120472 T^{6} + 1874161 T^{8} \))(\( ( 1 - 2 T + 52 T^{2} + 291 T^{3} + 597 T^{4} + 10767 T^{5} + 71188 T^{6} - 101306 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 - 31 T + 376 T^{2} - 1862 T^{3} - 3382 T^{4} + 103693 T^{5} - 685390 T^{6} + 2034181 T^{7} + 6920601 T^{8} - 124732747 T^{9} + 657905960 T^{10} - 814128670 T^{11} - 5921804427 T^{12} - 30122760790 T^{13} + 900673259240 T^{14} - 6318087833791 T^{15} + 12970320490761 T^{16} + 141058159794217 T^{17} - 1758523223464510 T^{18} + 9843771135552169 T^{19} - 11879205513160822 T^{20} - 241988759498433374 T^{21} + 1808027724029111224 T^{22} - 5515446275163272803 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 71 T^{2} + 1369 T^{4} \))(\( 1 - 176 T^{2} + 14602 T^{4} - 789197 T^{6} + 32518477 T^{8} - 1080410693 T^{10} + 27366498922 T^{12} - 451567847984 T^{14} + 3512479453921 T^{16} \))(\( 1 + 5 T + 33 T^{2} - 115 T^{3} - 1393 T^{4} - 9830 T^{5} - 26883 T^{6} + 712835 T^{7} + 3918312 T^{8} + 36125180 T^{9} - 1458715 T^{10} - 373946925 T^{11} - 6588438982 T^{12} - 31199250245 T^{13} + 128367650526 T^{14} + 957957636515 T^{15} + 15506223176675 T^{16} + 35444432551055 T^{17} + 175735313570094 T^{18} - 1580335622659985 T^{19} - 12347795390944102 T^{20} - 25930959487482225 T^{21} - 3742663598704435 T^{22} + 3429431149167508940 T^{23} + 13762990394052101352 T^{24} + 92641276786823713295 T^{25} - \)\(12\!\cdots\!67\)\( T^{26} - \)\(17\!\cdots\!90\)\( T^{27} - \)\(91\!\cdots\!33\)\( T^{28} - \)\(28\!\cdots\!55\)\( T^{29} + \)\(29\!\cdots\!37\)\( T^{30} + \)\(16\!\cdots\!65\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 5 T + 41 T^{2} \))(\( 1 - 3 T + 62 T^{2} - 55 T^{3} + 2542 T^{4} - 5043 T^{5} + 68921 T^{6} \))(\( ( 1 + 6 T + 79 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 142 T^{2} - 173 T^{3} + 8233 T^{4} - 7093 T^{5} + 238702 T^{6} - 137842 T^{7} + 2825761 T^{8} )^{2} \))(\( 1 - 4 T + 36 T^{2} + 343 T^{3} - 1660 T^{4} + 28812 T^{5} + 37388 T^{6} - 653892 T^{7} + 11898564 T^{8} - 39689643 T^{9} + 121339840 T^{10} + 2201839084 T^{11} - 17037636377 T^{12} + 90275402444 T^{13} + 203972271040 T^{14} - 2735449885203 T^{15} + 33622498107204 T^{16} - 75757442984292 T^{17} + 177596897362508 T^{18} + 5611260139059372 T^{19} - 13254975880340860 T^{20} + 112292003497128623 T^{21} + 483215735165486436 T^{22} - 2201316126864993764 T^{23} + 22563490300366186081 T^{24} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 152 T^{2} + 1107 T^{3} + 8955 T^{4} + 45387 T^{5} + 255512 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2} \))(\( 1 - 12 T - 5 T^{2} + 443 T^{3} + 2319 T^{4} - 42803 T^{5} + 38281 T^{6} + 1134551 T^{7} + 1148287 T^{8} - 52591508 T^{9} - 2634580 T^{10} + 1723605638 T^{11} - 3821894113 T^{12} + 16168497141 T^{13} - 383721286901 T^{14} + 2026154453570 T^{15} - 6329678380097 T^{16} + 83072332596370 T^{17} - 645035483280581 T^{18} + 1114348991454861 T^{19} - 10799759330644993 T^{20} + 199690401240861238 T^{21} - 12514529631253780 T^{22} - 10242420952846802548 T^{23} + 9168985836571665727 T^{24} + \)\(37\!\cdots\!11\)\( T^{25} + \)\(51\!\cdots\!81\)\( T^{26} - \)\(23\!\cdots\!23\)\( T^{27} + \)\(52\!\cdots\!39\)\( T^{28} + \)\(40\!\cdots\!03\)\( T^{29} - \)\(18\!\cdots\!05\)\( T^{30} - \)\(18\!\cdots\!12\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 + 14 T + 185 T^{2} + 1272 T^{3} + 7955 T^{4} + 25886 T^{5} + 79507 T^{6} \))(\( 1 - 40 T^{2} + 210 T^{4} - 73960 T^{6} + 3418801 T^{8} \))(\( 1 - 10 T - 60 T^{2} + 800 T^{3} + 3495 T^{4} - 41500 T^{5} - 101120 T^{6} + 632530 T^{7} + 5339224 T^{8} + 27198790 T^{9} - 186970880 T^{10} - 3299540500 T^{11} + 11948709495 T^{12} + 117606754400 T^{13} - 379281782940 T^{14} - 2718186111070 T^{15} + 11688200277601 T^{16} \))(\( 1 - T + 40 T^{2} + 241 T^{3} + 4316 T^{4} + 17218 T^{5} + 144893 T^{6} + 1614673 T^{7} + 12591792 T^{8} + 87254435 T^{9} + 372981596 T^{10} + 4379781302 T^{11} + 28431710913 T^{12} + 188330595986 T^{13} + 689642971004 T^{14} + 6937338363545 T^{15} + 43048831081392 T^{16} + 237370563684139 T^{17} + 915921256258757 T^{18} + 4680172846040326 T^{19} + 50446272398125916 T^{20} + 121124819476779163 T^{21} + 864459292531369960 T^{22} - 929293739471222707 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 6 T + 55 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 + 82 T^{2} + 3301 T^{4} - 5616 T^{5} - 15638 T^{6} - 492912 T^{7} - 3837332 T^{8} - 21195216 T^{9} - 28914662 T^{10} - 446511312 T^{11} + 11285462101 T^{12} + 518351770018 T^{14} + 11688200277601 T^{16} \))(\( 1 + 25 T + 475 T^{2} + 5760 T^{3} + 54505 T^{4} + 326750 T^{5} + 794110 T^{6} - 11876585 T^{7} - 178232506 T^{8} - 1410378900 T^{9} - 5882739295 T^{10} + 2507100035 T^{11} + 269016911220 T^{12} + 2156454943800 T^{13} + 8159908658200 T^{14} - 7528926419265 T^{15} - 230517908176919 T^{16} - 323743836028395 T^{17} + 15087671109011800 T^{18} + 171453263216706600 T^{19} + 919715285095847220 T^{20} + 368564872590595505 T^{21} - 37186930806313310455 T^{22} - \)\(38\!\cdots\!00\)\( T^{23} - \)\(20\!\cdots\!06\)\( T^{24} - \)\(59\!\cdots\!55\)\( T^{25} + \)\(17\!\cdots\!90\)\( T^{26} + \)\(30\!\cdots\!50\)\( T^{27} + \)\(21\!\cdots\!05\)\( T^{28} + \)\(98\!\cdots\!80\)\( T^{29} + \)\(35\!\cdots\!75\)\( T^{30} + \)\(79\!\cdots\!75\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 53 T^{2} + 392 T^{3} + 2491 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( ( 1 - 12 T + 82 T^{2} - 564 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 57 T^{2} + 208 T^{3} + 552 T^{4} - 10816 T^{5} + 108913 T^{6} + 276744 T^{7} - 5291265 T^{8} + 13006968 T^{9} + 240588817 T^{10} - 1122949568 T^{11} + 2693583912 T^{12} + 47703761456 T^{13} - 614415273753 T^{14} + 23811286661761 T^{16} \))(\( ( 1 + 11 T + 306 T^{2} + 2521 T^{3} + 37470 T^{4} + 233981 T^{5} + 2377977 T^{6} + 10997107 T^{7} + 82771230 T^{8} + 261737783 T^{9} + 1493182386 T^{10} + 2522795077 T^{11} + 10779215329 T^{12} )^{2} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T - 65 T^{2} + 804 T^{3} + 8656 T^{4} - 60264 T^{5} - 435263 T^{6} + 654360 T^{7} + 29253631 T^{8} + 30754920 T^{9} - 961495967 T^{10} - 6256789272 T^{11} + 42238518736 T^{12} + 184393385628 T^{13} - 700648996385 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16} \))(\( ( 1 - 3 T + 252 T^{2} - 271 T^{3} + 29293 T^{4} + 1726 T^{5} + 2226609 T^{6} + 1006952 T^{7} + 122448678 T^{8} + 47326744 T^{9} + 4918579281 T^{10} + 179198498 T^{11} + 142940495533 T^{12} - 62152496897 T^{13} + 2716362262908 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 147 T^{2} + 204 T^{3} + 7791 T^{4} + 5618 T^{5} + 148877 T^{6} \))(\( 1 - 84 T^{2} + 3494 T^{4} - 235956 T^{6} + 7890481 T^{8} \))(\( ( 1 - 19 T + 296 T^{2} - 3017 T^{3} + 25731 T^{4} - 159901 T^{5} + 831464 T^{6} - 2828663 T^{7} + 7890481 T^{8} )^{2} \))(\( 1 + 35 T + 595 T^{2} + 6152 T^{3} + 37765 T^{4} + 40580 T^{5} - 2155638 T^{6} - 30130680 T^{7} - 236238050 T^{8} - 1018545200 T^{9} + 1805599680 T^{10} + 77446824495 T^{11} + 757004919171 T^{12} + 4104681698235 T^{13} + 5071929501120 T^{14} - 151637953740400 T^{15} - 1864031845002050 T^{16} - 12600514577025240 T^{17} - 47778339095395302 T^{18} + 47669778054585460 T^{19} + 2351237208385048165 T^{20} + 20300145616766722216 T^{21} + \)\(10\!\cdots\!55\)\( T^{22} + \)\(32\!\cdots\!95\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 79 T^{2} + 2809 T^{4} \))(\( 1 - 201 T^{2} + 20716 T^{4} - 1413063 T^{6} + 79376943 T^{8} - 3969293967 T^{10} + 163459204396 T^{12} - 4455036586929 T^{14} + 62259690411361 T^{16} \))(\( 1 + 20 T + 199 T^{2} + 2875 T^{3} + 39397 T^{4} + 370045 T^{5} + 3659993 T^{6} + 38740970 T^{7} + 345763277 T^{8} + 3037043755 T^{9} + 27108033320 T^{10} + 222746060060 T^{11} + 1842983363243 T^{12} + 14984596984095 T^{13} + 111847871710691 T^{14} + 855024909525500 T^{15} + 6543454922608830 T^{16} + 45316320204851500 T^{17} + 314180671635331019 T^{18} + 2230861845201111315 T^{19} + 14542025210984989883 T^{20} + 93151398400599309580 T^{21} + \)\(60\!\cdots\!80\)\( T^{22} + \)\(35\!\cdots\!35\)\( T^{23} + \)\(21\!\cdots\!97\)\( T^{24} + \)\(12\!\cdots\!10\)\( T^{25} + \)\(64\!\cdots\!57\)\( T^{26} + \)\(34\!\cdots\!65\)\( T^{27} + \)\(19\!\cdots\!77\)\( T^{28} + \)\(74\!\cdots\!75\)\( T^{29} + \)\(27\!\cdots\!31\)\( T^{30} + \)\(14\!\cdots\!40\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 - 29 T + 408 T^{2} - 3747 T^{3} + 24072 T^{4} - 100949 T^{5} + 205379 T^{6} \))(\( 1 - 156 T^{2} + 11459 T^{4} - 543036 T^{6} + 12117361 T^{8} \))(\( 1 - T - 121 T^{2} + 416 T^{3} + 5312 T^{4} - 30083 T^{5} - 229112 T^{6} + 739956 T^{7} + 18758703 T^{8} + 43657404 T^{9} - 797538872 T^{10} - 6178416457 T^{11} + 64367421632 T^{12} + 297408508384 T^{13} - 5103844570561 T^{14} - 2488651484819 T^{15} + 146830437604321 T^{16} \))(\( 1 + 13 T + 105 T^{2} + 1586 T^{3} + 9191 T^{4} + 15786 T^{5} + 388142 T^{6} - 460008 T^{7} - 17659257 T^{8} + 252362865 T^{9} + 1291282900 T^{10} + 14767301324 T^{11} + 273683744527 T^{12} + 871270778116 T^{13} + 4494955774900 T^{14} + 51830032850835 T^{15} - 213983592060777 T^{16} - 328870896934392 T^{17} + 16372036688485022 T^{18} + 39285852339352734 T^{19} + 1349518552021314311 T^{20} + 13739511368386733254 T^{21} + 53667259096567347105 T^{22} + \)\(39\!\cdots\!67\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 6 T + 71 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 3 T + 75 T^{2} + 216 T^{3} + 2674 T^{4} + 1989 T^{5} - 310752 T^{6} - 960594 T^{7} - 23843193 T^{8} - 56675046 T^{9} - 1081727712 T^{10} + 408498831 T^{11} + 32401823314 T^{12} + 154423648584 T^{13} + 3163540023075 T^{14} + 7465954454457 T^{15} + 146830437604321 T^{16} \))(\( 1 - 5 T + 56 T^{2} + 335 T^{3} + 167 T^{4} + 18855 T^{5} + 14009 T^{6} + 2418195 T^{7} - 14511404 T^{8} + 137830810 T^{9} + 211598925 T^{10} - 447685075 T^{11} + 39793329573 T^{12} + 280053463495 T^{13} + 2697388120680 T^{14} - 3269731262690 T^{15} + 410227757908617 T^{16} - 192914144498710 T^{17} + 9389608048087080 T^{18} + 57517100279139605 T^{19} + 482190139828016853 T^{20} - 320060938417137425 T^{21} + 8925355574361935925 T^{22} + \)\(34\!\cdots\!90\)\( T^{23} - \)\(21\!\cdots\!84\)\( T^{24} + \)\(20\!\cdots\!05\)\( T^{25} + \)\(71\!\cdots\!09\)\( T^{26} + \)\(56\!\cdots\!45\)\( T^{27} + \)\(29\!\cdots\!27\)\( T^{28} + \)\(35\!\cdots\!65\)\( T^{29} + \)\(34\!\cdots\!16\)\( T^{30} - \)\(18\!\cdots\!95\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + T \))(\( ( 1 - T )^{3} \))(\( 1 - 8 T + 30 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 - 10 T - 11 T^{2} + 286 T^{3} - 565 T^{4} + 17446 T^{5} - 40931 T^{6} - 2269810 T^{7} + 13845841 T^{8} \))(\( 1 + 6 T + 47 T^{2} + 1372 T^{3} + 5508 T^{4} + 39787 T^{5} + 831153 T^{6} + 2427007 T^{7} + 20495268 T^{8} + 311417932 T^{9} + 650754527 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 8 T + T^{2} + 500 T^{3} + 8521 T^{4} + 30500 T^{5} + 3721 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 + 53 T + 1264 T^{2} + 17700 T^{3} + 163718 T^{4} + 1199030 T^{5} + 10144001 T^{6} + 106364807 T^{7} + 963246013 T^{8} + 6488253227 T^{9} + 37745827721 T^{10} + 272157028430 T^{11} + 2266813396838 T^{12} + 14949354527700 T^{13} + 65121753192304 T^{14} + 166565370309113 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 7 T + 67 T^{2} \))(\( 1 - 9 T + 116 T^{2} - 647 T^{3} + 7772 T^{4} - 40401 T^{5} + 300763 T^{6} \))(\( 1 - 28 T^{2} + 8931 T^{4} - 125692 T^{6} + 20151121 T^{8} \))(\( 1 + 17 T + 76 T^{2} - 867 T^{3} - 15110 T^{4} - 130934 T^{5} - 364569 T^{6} + 7430156 T^{7} + 111321101 T^{8} + 497820452 T^{9} - 1636550241 T^{10} - 39380102642 T^{11} - 304483438310 T^{12} - 1170558467769 T^{13} + 6874837044844 T^{14} + 103032097290491 T^{15} + 406067677556641 T^{16} \))(\( 1 + 9 T - 110 T^{2} - 582 T^{3} + 14458 T^{4} + 55503 T^{5} - 918949 T^{6} - 2693416 T^{7} + 58965243 T^{8} + 208376463 T^{9} - 3140081710 T^{10} - 5013421313 T^{11} + 218950598879 T^{12} - 335899227971 T^{13} - 14095826796190 T^{14} + 62671930141269 T^{15} + 1188215746487403 T^{16} - 3636448565195512 T^{17} - 83126639835820381 T^{18} + 336387676230242469 T^{19} + 5870926482113915578 T^{20} - 15834203018643659154 T^{21} - \)\(20\!\cdots\!90\)\( T^{22} + \)\(10\!\cdots\!47\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 + 15 T + 280 T^{2} + 3075 T^{3} + 33940 T^{4} + 320436 T^{5} + 3216343 T^{6} + 27416670 T^{7} + 254391505 T^{8} + 1836916890 T^{9} + 14438163727 T^{10} + 96375292668 T^{11} + 683929046740 T^{12} + 4151634704025 T^{13} + 25328347007320 T^{14} + 90910674079845 T^{15} + 406067677556641 T^{16} \))(\( 1 + 55 T + 1623 T^{2} + 33205 T^{3} + 525513 T^{4} + 6842460 T^{5} + 76504738 T^{6} + 754919110 T^{7} + 6632265321 T^{8} + 50595227995 T^{9} + 304083072180 T^{10} + 881347495735 T^{11} - 10332791560373 T^{12} - 238045875765180 T^{13} - 3030756628702715 T^{14} - 30653009199811070 T^{15} - 267409651963011403 T^{16} - 2053751616387341690 T^{17} - 13605066506246487635 T^{18} - 71595391732762832340 T^{19} - \)\(20\!\cdots\!33\)\( T^{20} + \)\(11\!\cdots\!45\)\( T^{21} + \)\(27\!\cdots\!20\)\( T^{22} + \)\(30\!\cdots\!85\)\( T^{23} + \)\(26\!\cdots\!61\)\( T^{24} + \)\(20\!\cdots\!70\)\( T^{25} + \)\(13\!\cdots\!62\)\( T^{26} + \)\(83\!\cdots\!80\)\( T^{27} + \)\(43\!\cdots\!93\)\( T^{28} + \)\(18\!\cdots\!35\)\( T^{29} + \)\(59\!\cdots\!67\)\( T^{30} + \)\(13\!\cdots\!65\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 - 14 T + 201 T^{2} - 1896 T^{3} + 14271 T^{4} - 70574 T^{5} + 357911 T^{6} \))(\( 1 - 240 T^{2} + 24050 T^{4} - 1209840 T^{6} + 25411681 T^{8} \))(\( 1 + 3 T - 148 T^{2} - 305 T^{3} + 10906 T^{4} + 12990 T^{5} - 324129 T^{6} - 321376 T^{7} + 4317725 T^{8} - 22817696 T^{9} - 1633934289 T^{10} + 4649263890 T^{11} + 277139792986 T^{12} - 550289952055 T^{13} - 18958842020308 T^{14} + 27285360475173 T^{15} + 645753531245761 T^{16} \))(\( 1 + 24 T + 275 T^{2} + 3172 T^{3} + 40816 T^{4} + 408678 T^{5} + 3742998 T^{6} + 37344054 T^{7} + 353040948 T^{8} + 3296894425 T^{9} + 32075280670 T^{10} + 284964188583 T^{11} + 2371819319967 T^{12} + 20232457389393 T^{13} + 161691489857470 T^{14} + 1179994780546175 T^{15} + 8971363950513588 T^{16} + 67377238312128954 T^{17} + 479479106515735158 T^{18} + 3716975516090917098 T^{19} + 26357076131326980976 T^{20} + \)\(14\!\cdots\!32\)\( T^{21} + \)\(89\!\cdots\!75\)\( T^{22} + \)\(55\!\cdots\!04\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 18 T + 179 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 15 T + 300 T^{2} - 3375 T^{3} + 45136 T^{4} - 506352 T^{5} + 5101983 T^{6} - 49720512 T^{7} + 402441093 T^{8} - 3530156352 T^{9} + 25719096303 T^{10} - 181228950672 T^{11} + 1146981633616 T^{12} - 6089274059625 T^{13} + 38430085176300 T^{14} - 136426802375865 T^{15} + 645753531245761 T^{16} \))(\( 1 + 50 T + 1310 T^{2} + 25100 T^{3} + 402690 T^{4} + 5710500 T^{5} + 73142420 T^{6} + 857262210 T^{7} + 9291753194 T^{8} + 94226452595 T^{9} + 905415943755 T^{10} + 8315078921580 T^{11} + 73428203558125 T^{12} + 630065735281185 T^{13} + 5326731673336375 T^{14} + 44826640384313440 T^{15} + 377362009506660131 T^{16} + 3182691467286254240 T^{17} + 26852054365288666375 T^{18} + \)\(22\!\cdots\!35\)\( T^{19} + \)\(18\!\cdots\!25\)\( T^{20} + \)\(15\!\cdots\!80\)\( T^{21} + \)\(11\!\cdots\!55\)\( T^{22} + \)\(85\!\cdots\!45\)\( T^{23} + \)\(60\!\cdots\!34\)\( T^{24} + \)\(39\!\cdots\!10\)\( T^{25} + \)\(23\!\cdots\!20\)\( T^{26} + \)\(13\!\cdots\!00\)\( T^{27} + \)\(66\!\cdots\!90\)\( T^{28} + \)\(29\!\cdots\!00\)\( T^{29} + \)\(10\!\cdots\!10\)\( T^{30} + \)\(29\!\cdots\!50\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 11 T + 73 T^{2} \))(\( 1 + T + 174 T^{2} + 121 T^{3} + 12702 T^{4} + 5329 T^{5} + 389017 T^{6} \))(\( ( 1 - 8 T + 135 T^{2} - 584 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 24 T + 117 T^{2} - 68 T^{3} + 21178 T^{4} + 236584 T^{5} - 35003 T^{6} + 9744374 T^{7} + 246610547 T^{8} + 711339302 T^{9} - 186530987 T^{10} + 92035197928 T^{11} + 601417947898 T^{12} - 140968868324 T^{13} + 17706104475813 T^{14} + 265137564458328 T^{15} + 806460091894081 T^{16} \))(\( 1 - 51 T + 1277 T^{2} - 22154 T^{3} + 311135 T^{4} - 3725622 T^{5} + 38662354 T^{6} - 358675454 T^{7} + 3062355737 T^{8} - 24291809463 T^{9} + 183794851578 T^{10} - 1411943103032 T^{11} + 11593980668877 T^{12} - 103071846521336 T^{13} + 979442764059162 T^{14} - 9449926841867871 T^{15} + 86965516247058617 T^{16} - 743559894793778222 T^{17} + 5850937429101424306 T^{18} - 41158430965515203334 T^{19} + \)\(25\!\cdots\!35\)\( T^{20} - \)\(13\!\cdots\!02\)\( T^{21} + \)\(54\!\cdots\!73\)\( T^{22} - \)\(16\!\cdots\!27\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 16 T + 9 T^{2} - 2068 T^{3} - 12562 T^{4} + 150048 T^{5} + 1900429 T^{6} - 4775030 T^{7} - 170006877 T^{8} - 348577190 T^{9} + 10127386141 T^{10} + 58371222816 T^{11} - 356738703442 T^{12} - 4287112054324 T^{13} + 1362008036601 T^{14} + 176758376305552 T^{15} + 806460091894081 T^{16} \))(\( 1 + 11 T - 218 T^{2} - 3889 T^{3} + 6739 T^{4} + 465127 T^{5} + 2249953 T^{6} - 12657591 T^{7} - 198401422 T^{8} - 1622576810 T^{9} - 7165841913 T^{10} + 49748483585 T^{11} + 1050131744229 T^{12} + 13880137357557 T^{13} + 100040852908534 T^{14} - 787338353415830 T^{15} - 17077728803609053 T^{16} - 57475699799355590 T^{17} + 533117705149577686 T^{18} + 5399609394424751469 T^{19} + 29821894354365501189 T^{20} + \)\(10\!\cdots\!05\)\( T^{21} - \)\(10\!\cdots\!57\)\( T^{22} - \)\(17\!\cdots\!70\)\( T^{23} - \)\(16\!\cdots\!82\)\( T^{24} - \)\(74\!\cdots\!83\)\( T^{25} + \)\(96\!\cdots\!97\)\( T^{26} + \)\(14\!\cdots\!79\)\( T^{27} + \)\(15\!\cdots\!19\)\( T^{28} - \)\(65\!\cdots\!37\)\( T^{29} - \)\(26\!\cdots\!62\)\( T^{30} + \)\(98\!\cdots\!27\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 3 T + 79 T^{2} \))(\( 1 - 13 T + 186 T^{2} - 1429 T^{3} + 14694 T^{4} - 81133 T^{5} + 493039 T^{6} \))(\( 1 - 4 T^{2} + 7923 T^{4} - 24964 T^{6} + 38950081 T^{8} \))(\( 1 + 16 T - 81 T^{2} - 1368 T^{3} + 19652 T^{4} + 131296 T^{5} - 2066555 T^{6} - 3782324 T^{7} + 189160727 T^{8} - 298803596 T^{9} - 12897369755 T^{10} + 64734048544 T^{11} + 765446991812 T^{12} - 4209413153832 T^{13} - 19690083897201 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 26 T + 111 T^{2} - 2349 T^{3} - 19024 T^{4} + 63421 T^{5} + 587548 T^{6} + 15440 T^{7} - 4850363 T^{8} - 1051333936 T^{9} - 11216252669 T^{10} + 71065087310 T^{11} + 1709937431683 T^{12} + 5614141897490 T^{13} - 70000632907229 T^{14} - 518348632471504 T^{15} - 188922031729403 T^{16} + 47509750800560 T^{17} + 142825548316452508 T^{18} + 1217931111811189939 T^{19} - 28861477999662416464 T^{20} - \)\(28\!\cdots\!31\)\( T^{21} + \)\(10\!\cdots\!11\)\( T^{22} + \)\(19\!\cdots\!54\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 18 T + 187 T^{2} + 1422 T^{3} + 6241 T^{4} \))(\( 1 - 42 T + 1015 T^{2} - 17934 T^{3} + 257146 T^{4} - 3188376 T^{5} + 35341339 T^{6} - 357084330 T^{7} + 3310809259 T^{8} - 28209662070 T^{9} + 220565296699 T^{10} - 1571993714664 T^{11} + 10015857528826 T^{12} - 55183929459666 T^{13} + 246733767353815 T^{14} - 806564177418678 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 40 T + 1004 T^{2} - 18625 T^{3} + 279368 T^{4} - 3529580 T^{5} + 38355634 T^{6} - 367873670 T^{7} + 3181335467 T^{8} - 26061609010 T^{9} + 212343593095 T^{10} - 1816574334570 T^{11} + 16354414377877 T^{12} - 150158736307295 T^{13} + 1367614192415072 T^{14} - 12190383558121290 T^{15} + 108545585867347675 T^{16} - 963040301091581910 T^{17} + 8535280174862464352 T^{18} - 74034113190212419505 T^{19} + \)\(63\!\cdots\!37\)\( T^{20} - \)\(55\!\cdots\!30\)\( T^{21} + \)\(51\!\cdots\!95\)\( T^{22} - \)\(50\!\cdots\!90\)\( T^{23} + \)\(48\!\cdots\!87\)\( T^{24} - \)\(44\!\cdots\!30\)\( T^{25} + \)\(36\!\cdots\!34\)\( T^{26} - \)\(26\!\cdots\!20\)\( T^{27} + \)\(16\!\cdots\!88\)\( T^{28} - \)\(86\!\cdots\!75\)\( T^{29} + \)\(37\!\cdots\!24\)\( T^{30} - \)\(11\!\cdots\!60\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} \))(\( ( 1 - 12 T + 190 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \))(\( 1 - 16 T - 94 T^{2} + 2326 T^{3} + 14793 T^{4} - 298411 T^{5} - 239845 T^{6} + 6966371 T^{7} + 60249897 T^{8} + 578208793 T^{9} - 1652292205 T^{10} - 170627530457 T^{11} + 702050942553 T^{12} + 9162208535618 T^{13} - 30732395096686 T^{14} - 434176815834032 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 3 T - 91 T^{2} + 1466 T^{3} - 3742 T^{4} - 156341 T^{5} + 1950205 T^{6} - 2165518 T^{7} - 138599549 T^{8} + 1606161909 T^{9} - 3638947395 T^{10} - 84683275245 T^{11} + 1224056573103 T^{12} - 7028711845335 T^{13} - 25068708604155 T^{14} + 918382499461383 T^{15} - 6577701886897229 T^{16} - 8530063415148074 T^{17} + 637600750846090645 T^{18} - 4242477347769274807 T^{19} - 8428077532664291422 T^{20} + \)\(27\!\cdots\!98\)\( T^{21} - \)\(14\!\cdots\!59\)\( T^{22} - \)\(38\!\cdots\!01\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 + 42 T + 778 T^{2} + 11562 T^{3} + 184183 T^{4} + 2417133 T^{5} + 25405465 T^{6} + 271795797 T^{7} + 2757143665 T^{8} + 22559051151 T^{9} + 175018248385 T^{10} + 1382085226671 T^{11} + 8741015936743 T^{12} + 45543187914366 T^{13} + 254359610481082 T^{14} + 1139714141564334 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 31 T + 192 T^{2} + 4279 T^{3} - 59466 T^{4} - 263272 T^{5} + 6923583 T^{6} + 27719176 T^{7} - 773399342 T^{8} - 4251313795 T^{9} + 104799170962 T^{10} + 338765023285 T^{11} - 12069980289866 T^{12} - 6556717409592 T^{13} + 1024908701987209 T^{14} - 197032890173330 T^{15} - 80817926847707993 T^{16} - 16353729884386390 T^{17} + 7060596047989882801 T^{18} - 3749045777478380904 T^{19} - \)\(57\!\cdots\!86\)\( T^{20} + \)\(13\!\cdots\!55\)\( T^{21} + \)\(34\!\cdots\!78\)\( T^{22} - \)\(11\!\cdots\!65\)\( T^{23} - \)\(17\!\cdots\!22\)\( T^{24} + \)\(51\!\cdots\!28\)\( T^{25} + \)\(10\!\cdots\!67\)\( T^{26} - \)\(33\!\cdots\!24\)\( T^{27} - \)\(63\!\cdots\!26\)\( T^{28} + \)\(37\!\cdots\!77\)\( T^{29} + \)\(14\!\cdots\!68\)\( T^{30} - \)\(18\!\cdots\!17\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 + 4 T + 89 T^{2} \))(\( 1 + 4 T + 211 T^{2} + 792 T^{3} + 18779 T^{4} + 31684 T^{5} + 704969 T^{6} \))(\( 1 - 36 T^{2} + 15734 T^{4} - 285156 T^{6} + 62742241 T^{8} \))(\( ( 1 - 4 T + 250 T^{2} - 264 T^{3} + 27211 T^{4} - 23496 T^{5} + 1980250 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2} \))(\( 1 + 17 T + 47 T^{2} + 1310 T^{3} + 48873 T^{4} + 479860 T^{5} + 1504088 T^{6} + 37842912 T^{7} + 895042406 T^{8} + 6353106584 T^{9} + 18278155282 T^{10} + 543520510011 T^{11} + 9100689508467 T^{12} + 48373325390979 T^{13} + 144781267988722 T^{14} + 4478743195415896 T^{15} + 56156966342471846 T^{16} + 211317070331275488 T^{17} + 747503595958948568 T^{18} + 21224848362968545940 T^{19} + \)\(19\!\cdots\!13\)\( T^{20} + \)\(45\!\cdots\!90\)\( T^{21} + \)\(14\!\cdots\!47\)\( T^{22} + \)\(47\!\cdots\!13\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 65 T^{2} + 7921 T^{4} \))(\( 1 - 492 T^{2} + 119866 T^{4} - 18502416 T^{6} + 1965173379 T^{8} - 146557637136 T^{10} + 7520661459706 T^{12} - 244514795152812 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 60 T + 1871 T^{2} - 38495 T^{3} + 562467 T^{4} - 5767705 T^{5} + 35049269 T^{6} + 22683470 T^{7} - 3027868689 T^{8} + 27695647285 T^{9} + 59222262600 T^{10} - 4752582400640 T^{11} + 67318776826303 T^{12} - 507535512149835 T^{13} + 865503028135415 T^{14} + 31691773238870840 T^{15} - 459107547574687058 T^{16} + 2820567818259504760 T^{17} + 6855649485860622215 T^{18} - \)\(35\!\cdots\!15\)\( T^{19} + \)\(42\!\cdots\!23\)\( T^{20} - \)\(26\!\cdots\!60\)\( T^{21} + \)\(29\!\cdots\!00\)\( T^{22} + \)\(12\!\cdots\!65\)\( T^{23} - \)\(11\!\cdots\!09\)\( T^{24} + \)\(79\!\cdots\!30\)\( T^{25} + \)\(10\!\cdots\!69\)\( T^{26} - \)\(16\!\cdots\!45\)\( T^{27} + \)\(13\!\cdots\!07\)\( T^{28} - \)\(84\!\cdots\!55\)\( T^{29} + \)\(36\!\cdots\!11\)\( T^{30} - \)\(10\!\cdots\!40\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 10 T + 175 T^{2} - 844 T^{3} + 16975 T^{4} - 94090 T^{5} + 912673 T^{6} \))(\( ( 1 + 182 T^{2} + 9409 T^{4} )^{2} \))(\( 1 - 13 T - 17 T^{2} + 2976 T^{3} - 28242 T^{4} - 41049 T^{5} + 2206518 T^{6} - 12211732 T^{7} - 39557315 T^{8} - 1184538004 T^{9} + 20761127862 T^{10} - 37464313977 T^{11} - 2500243954002 T^{12} + 25555924604832 T^{13} - 14160524083793 T^{14} - 1050377698215469 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 17 T - 7 T^{2} + 1761 T^{3} + 3648 T^{4} - 119849 T^{5} - 2187618 T^{6} + 18974035 T^{7} + 282217320 T^{8} - 1601697724 T^{9} - 32656751622 T^{10} + 50388883144 T^{11} + 3356668198633 T^{12} + 4887721664968 T^{13} - 307267376011398 T^{14} - 1461826266856252 T^{15} + 24984496425346920 T^{16} + 162936494593226995 T^{17} - 1822224551478769122 T^{18} - 9683593596417364937 T^{19} + 28590957752287153728 T^{20} + \)\(13\!\cdots\!37\)\( T^{21} - \)\(51\!\cdots\!43\)\( T^{22} - \)\(12\!\cdots\!01\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} \))(\( 1 - 3 T - 259 T^{2} + 1206 T^{3} + 34540 T^{4} - 161211 T^{5} - 3423298 T^{6} + 7508760 T^{7} + 328498753 T^{8} + 728349720 T^{9} - 32209810882 T^{10} - 147132927003 T^{11} + 3057801365740 T^{12} + 10356332349942 T^{13} - 215739749276611 T^{14} - 242394853434339 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 45 T + 856 T^{2} - 11860 T^{3} + 210304 T^{4} - 3615225 T^{5} + 44038916 T^{6} - 491524840 T^{7} + 6859409486 T^{8} - 85759087530 T^{9} + 818468817393 T^{10} - 8592168079245 T^{11} + 108625344057871 T^{12} - 1098947552679740 T^{13} + 9418474934221653 T^{14} - 104357219141083475 T^{15} + 1183864542942718737 T^{16} - 10122650256685097075 T^{17} + 88618430656091533077 T^{18} - \)\(10\!\cdots\!20\)\( T^{19} + \)\(96\!\cdots\!51\)\( T^{20} - \)\(73\!\cdots\!65\)\( T^{21} + \)\(68\!\cdots\!97\)\( T^{22} - \)\(69\!\cdots\!90\)\( T^{23} + \)\(53\!\cdots\!46\)\( T^{24} - \)\(37\!\cdots\!80\)\( T^{25} + \)\(32\!\cdots\!84\)\( T^{26} - \)\(25\!\cdots\!25\)\( T^{27} + \)\(14\!\cdots\!64\)\( T^{28} - \)\(79\!\cdots\!20\)\( T^{29} + \)\(55\!\cdots\!64\)\( T^{30} - \)\(28\!\cdots\!85\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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