Properties

Label 8048.2
Level 8048
Weight 2
Dimension 1136273
Nonzero newspaces 8
Sturm bound 8096256

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Defining parameters

Level: \( N \) = \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Sturm bound: \(8096256\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(8048))\).

Total New Old
Modular forms 2031092 1140781 890311
Cusp forms 2017037 1136273 880764
Eisenstein series 14055 4508 9547

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(8048))\)

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
8048.2.a \(\chi_{8048}(1, \cdot)\) 8048.2.a.a 1 1
8048.2.a.b 1
8048.2.a.c 1
8048.2.a.d 1
8048.2.a.e 1
8048.2.a.f 1
8048.2.a.g 1
8048.2.a.h 1
8048.2.a.i 1
8048.2.a.j 1
8048.2.a.k 1
8048.2.a.l 2
8048.2.a.m 3
8048.2.a.n 5
8048.2.a.o 5
8048.2.a.p 10
8048.2.a.q 12
8048.2.a.r 12
8048.2.a.s 21
8048.2.a.t 21
8048.2.a.u 26
8048.2.a.v 28
8048.2.a.w 29
8048.2.a.x 33
8048.2.a.y 33
8048.2.b \(\chi_{8048}(8047, \cdot)\) n/a 252 1
8048.2.c \(\chi_{8048}(4025, \cdot)\) None 0 1
8048.2.h \(\chi_{8048}(4023, \cdot)\) None 0 1
8048.2.j \(\chi_{8048}(2013, \cdot)\) n/a 2008 2
8048.2.l \(\chi_{8048}(2011, \cdot)\) n/a 2012 2
8048.2.m \(\chi_{8048}(33, \cdot)\) n/a 62750 250
8048.2.n \(\chi_{8048}(55, \cdot)\) None 0 250
8048.2.s \(\chi_{8048}(9, \cdot)\) None 0 250
8048.2.t \(\chi_{8048}(15, \cdot)\) n/a 63000 250
8048.2.u \(\chi_{8048}(19, \cdot)\) n/a 503000 500
8048.2.w \(\chi_{8048}(13, \cdot)\) n/a 503000 500

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(8048))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(8048)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(503))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1006))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2012))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4024))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 15 T^{4} + 9 T^{5} + 27 T^{6} \))(\( 1 + 9 T^{2} + T^{3} + 40 T^{4} + 7 T^{5} + 120 T^{6} + 9 T^{7} + 243 T^{8} + 243 T^{10} \))(\( 1 - 4 T + 15 T^{2} - 37 T^{3} + 86 T^{4} - 151 T^{5} + 258 T^{6} - 333 T^{7} + 405 T^{8} - 324 T^{9} + 243 T^{10} \))(\( 1 - 8 T + 48 T^{2} - 206 T^{3} + 752 T^{4} - 2307 T^{5} + 6300 T^{6} - 15215 T^{7} + 33404 T^{8} - 66208 T^{9} + 120343 T^{10} - 198624 T^{11} + 300636 T^{12} - 410805 T^{13} + 510300 T^{14} - 560601 T^{15} + 548208 T^{16} - 450522 T^{17} + 314928 T^{18} - 157464 T^{19} + 59049 T^{20} \))(\( 1 + 5 T + 23 T^{2} + 71 T^{3} + 219 T^{4} + 553 T^{5} + 1403 T^{6} + 3112 T^{7} + 6916 T^{8} + 13793 T^{9} + 27414 T^{10} + 49644 T^{11} + 89792 T^{12} + 148932 T^{13} + 246726 T^{14} + 372411 T^{15} + 560196 T^{16} + 756216 T^{17} + 1022787 T^{18} + 1209411 T^{19} + 1436859 T^{20} + 1397493 T^{21} + 1358127 T^{22} + 885735 T^{23} + 531441 T^{24} \))(\( 1 - 3 T + 9 T^{2} - 21 T^{3} + 51 T^{4} - 125 T^{5} + 293 T^{6} - 592 T^{7} + 1250 T^{8} - 2291 T^{9} + 4498 T^{10} - 8234 T^{11} + 14628 T^{12} - 24702 T^{13} + 40482 T^{14} - 61857 T^{15} + 101250 T^{16} - 143856 T^{17} + 213597 T^{18} - 273375 T^{19} + 334611 T^{20} - 413343 T^{21} + 531441 T^{22} - 531441 T^{23} + 531441 T^{24} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{3} \))(\( 1 + T + 16 T^{2} + 16 T^{3} + 122 T^{4} + 111 T^{5} + 610 T^{6} + 400 T^{7} + 2000 T^{8} + 625 T^{9} + 3125 T^{10} \))(\( 1 + 3 T + 14 T^{2} + 10 T^{3} + 30 T^{4} - 65 T^{5} + 150 T^{6} + 250 T^{7} + 1750 T^{8} + 1875 T^{9} + 3125 T^{10} \))(\( 1 + T + 39 T^{2} + 38 T^{3} + 726 T^{4} + 662 T^{5} + 8471 T^{6} + 7016 T^{7} + 68463 T^{8} + 50106 T^{9} + 400081 T^{10} + 250530 T^{11} + 1711575 T^{12} + 877000 T^{13} + 5294375 T^{14} + 2068750 T^{15} + 11343750 T^{16} + 2968750 T^{17} + 15234375 T^{18} + 1953125 T^{19} + 9765625 T^{20} \))(\( 1 - 5 T + 42 T^{2} - 154 T^{3} + 791 T^{4} - 2412 T^{5} + 9636 T^{6} - 25656 T^{7} + 86256 T^{8} - 204402 T^{9} + 600308 T^{10} - 1277475 T^{11} + 3342092 T^{12} - 6387375 T^{13} + 15007700 T^{14} - 25550250 T^{15} + 53910000 T^{16} - 80175000 T^{17} + 150562500 T^{18} - 188437500 T^{19} + 308984375 T^{20} - 300781250 T^{21} + 410156250 T^{22} - 244140625 T^{23} + 244140625 T^{24} \))(\( 1 - 7 T + 42 T^{2} - 190 T^{3} + 781 T^{4} - 2772 T^{5} + 9074 T^{6} - 27190 T^{7} + 76806 T^{8} - 202490 T^{9} + 509798 T^{10} - 1215619 T^{11} + 2791764 T^{12} - 6078095 T^{13} + 12744950 T^{14} - 25311250 T^{15} + 48003750 T^{16} - 84968750 T^{17} + 141781250 T^{18} - 216562500 T^{19} + 305078125 T^{20} - 371093750 T^{21} + 410156250 T^{22} - 341796875 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 - 4 T + 13 T^{2} - 28 T^{3} + 49 T^{4} \))(\( 1 + T + 13 T^{2} + 11 T^{3} + 91 T^{4} + 49 T^{5} + 343 T^{6} \))(\( 1 + 3 T + 24 T^{2} + 60 T^{3} + 252 T^{4} + 547 T^{5} + 1764 T^{6} + 2940 T^{7} + 8232 T^{8} + 7203 T^{9} + 16807 T^{10} \))(\( 1 - 9 T + 40 T^{2} - 96 T^{3} + 116 T^{4} - 83 T^{5} + 812 T^{6} - 4704 T^{7} + 13720 T^{8} - 21609 T^{9} + 16807 T^{10} \))(\( 1 - 5 T + 61 T^{2} - 251 T^{3} + 1709 T^{4} - 5912 T^{5} + 29171 T^{6} - 85970 T^{7} + 337948 T^{8} - 851532 T^{9} + 2785579 T^{10} - 5960724 T^{11} + 16559452 T^{12} - 29487710 T^{13} + 70039571 T^{14} - 99362984 T^{15} + 201062141 T^{16} - 206709293 T^{17} + 351652861 T^{18} - 201768035 T^{19} + 282475249 T^{20} \))(\( 1 + 8 T + 72 T^{2} + 399 T^{3} + 2199 T^{4} + 9669 T^{5} + 41150 T^{6} + 152258 T^{7} + 542867 T^{8} + 1744965 T^{9} + 5421535 T^{10} + 15427129 T^{11} + 42558007 T^{12} + 107989903 T^{13} + 265655215 T^{14} + 598522995 T^{15} + 1303423667 T^{16} + 2559000206 T^{17} + 4841256350 T^{18} + 7962837267 T^{19} + 12676797399 T^{20} + 16101089193 T^{21} + 20338217928 T^{22} + 15818613944 T^{23} + 13841287201 T^{24} \))(\( 1 - 2 T + 26 T^{2} - 41 T^{3} + 387 T^{4} - 495 T^{5} + 4456 T^{6} - 4830 T^{7} + 43901 T^{8} - 43875 T^{9} + 380095 T^{10} - 361109 T^{11} + 2860403 T^{12} - 2527763 T^{13} + 18624655 T^{14} - 15049125 T^{15} + 105406301 T^{16} - 81177810 T^{17} + 524243944 T^{18} - 407653785 T^{19} + 2230977987 T^{20} - 1654497887 T^{21} + 7344356474 T^{22} - 3954653486 T^{23} + 13841287201 T^{24} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 11 T + 69 T^{2} + 277 T^{3} + 759 T^{4} + 1331 T^{5} + 1331 T^{6} \))(\( 1 - 11 T + 89 T^{2} - 507 T^{3} + 2301 T^{4} - 8463 T^{5} + 25311 T^{6} - 61347 T^{7} + 118459 T^{8} - 161051 T^{9} + 161051 T^{10} \))(\( 1 - 15 T + 115 T^{2} - 625 T^{3} + 2765 T^{4} - 10129 T^{5} + 30415 T^{6} - 75625 T^{7} + 153065 T^{8} - 219615 T^{9} + 161051 T^{10} \))(\( 1 - 3 T + 69 T^{2} - 217 T^{3} + 2352 T^{4} - 7445 T^{5} + 52841 T^{6} - 159886 T^{7} + 869048 T^{8} - 2400415 T^{9} + 10894513 T^{10} - 26404565 T^{11} + 105154808 T^{12} - 212808266 T^{13} + 773645081 T^{14} - 1199024695 T^{15} + 4166711472 T^{16} - 4228716107 T^{17} + 14790762789 T^{18} - 7073843073 T^{19} + 25937424601 T^{20} \))(\( 1 + 18 T + 236 T^{2} + 2219 T^{3} + 17491 T^{4} + 115780 T^{5} + 677529 T^{6} + 3514943 T^{7} + 16587530 T^{8} + 71330085 T^{9} + 283931035 T^{10} + 1045560197 T^{11} + 3596181460 T^{12} + 11501162167 T^{13} + 34355655235 T^{14} + 94940343135 T^{15} + 242858026730 T^{16} + 566085085093 T^{17} + 1200283952769 T^{18} + 2256224658380 T^{19} + 3749351187571 T^{20} + 5232285926329 T^{21} + 6121232205836 T^{22} + 5135610070998 T^{23} + 3138428376721 T^{24} \))(\( 1 + 18 T + 206 T^{2} + 1751 T^{3} + 12433 T^{4} + 76100 T^{5} + 416305 T^{6} + 2060807 T^{7} + 9385744 T^{8} + 39508541 T^{9} + 155009377 T^{10} + 567389449 T^{11} + 1945839084 T^{12} + 6241283939 T^{13} + 18756134617 T^{14} + 52585868071 T^{15} + 137416677904 T^{16} + 331895028157 T^{17} + 737509702105 T^{18} + 1482973713100 T^{19} + 2665123967473 T^{20} + 4128766406941 T^{21} + 5343109467806 T^{22} + 5135610070998 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 - 5 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - 5 T + 37 T^{2} - 105 T^{3} + 481 T^{4} - 845 T^{5} + 2197 T^{6} \))(\( 1 - 5 T + 44 T^{2} - 179 T^{3} + 885 T^{4} - 3033 T^{5} + 11505 T^{6} - 30251 T^{7} + 96668 T^{8} - 142805 T^{9} + 371293 T^{10} \))(\( 1 + 5 T + 38 T^{2} + 175 T^{3} + 647 T^{4} + 2875 T^{5} + 8411 T^{6} + 29575 T^{7} + 83486 T^{8} + 142805 T^{9} + 371293 T^{10} \))(\( 1 + 18 T + 250 T^{2} + 2413 T^{3} + 19873 T^{4} + 134672 T^{5} + 810062 T^{6} + 4230421 T^{7} + 19976351 T^{8} + 83675331 T^{9} + 319368199 T^{10} + 1087779303 T^{11} + 3376003319 T^{12} + 9294234937 T^{13} + 23136180782 T^{14} + 50002770896 T^{15} + 95923175257 T^{16} + 151412171521 T^{17} + 203932680250 T^{18} + 190880988714 T^{19} + 137858491849 T^{20} \))(\( 1 - 4 T + 95 T^{2} - 278 T^{3} + 4083 T^{4} - 8607 T^{5} + 110337 T^{6} - 159728 T^{7} + 2185482 T^{8} - 2023466 T^{9} + 34878800 T^{10} - 21192537 T^{11} + 480640372 T^{12} - 275502981 T^{13} + 5894517200 T^{14} - 4445554802 T^{15} + 62419551402 T^{16} - 59305888304 T^{17} + 532575624633 T^{18} - 540076485819 T^{19} + 3330628533843 T^{20} - 2948050825694 T^{21} + 13096556725655 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \))(\( 1 + 4 T + 65 T^{2} + 248 T^{3} + 2225 T^{4} + 7917 T^{5} + 54689 T^{6} + 179986 T^{7} + 1073280 T^{8} + 3265684 T^{9} + 17414630 T^{10} + 49451901 T^{11} + 242072156 T^{12} + 642874713 T^{13} + 2943072470 T^{14} + 7174707748 T^{15} + 30653950080 T^{16} + 66827541898 T^{17} + 263973357401 T^{18} + 496780009089 T^{19} + 1815000854225 T^{20} + 2629915844504 T^{21} + 8960801970185 T^{22} + 7168641576148 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 8 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( 1 - 12 T + 79 T^{2} - 368 T^{3} + 1343 T^{4} - 3468 T^{5} + 4913 T^{6} \))(\( 1 + 20 T + 239 T^{2} + 1929 T^{3} + 11748 T^{4} + 54699 T^{5} + 199716 T^{6} + 557481 T^{7} + 1174207 T^{8} + 1670420 T^{9} + 1419857 T^{10} \))(\( 1 + 6 T + 63 T^{2} + 331 T^{3} + 2054 T^{4} + 7567 T^{5} + 34918 T^{6} + 95659 T^{7} + 309519 T^{8} + 501126 T^{9} + 1419857 T^{10} \))(\( 1 + 11 T + 172 T^{2} + 1316 T^{3} + 12110 T^{4} + 72481 T^{5} + 497736 T^{6} + 2460829 T^{7} + 13750697 T^{8} + 57744215 T^{9} + 272963791 T^{10} + 981651655 T^{11} + 3973951433 T^{12} + 12090052877 T^{13} + 41571408456 T^{14} + 102912655217 T^{15} + 292305960590 T^{16} + 540005693668 T^{17} + 1199830279852 T^{18} + 1304466641467 T^{19} + 2015993900449 T^{20} \))(\( 1 + 2 T + 86 T^{2} + 211 T^{3} + 3520 T^{4} + 11022 T^{5} + 97722 T^{6} + 347875 T^{7} + 2230039 T^{8} + 7425255 T^{9} + 46049968 T^{10} + 127773955 T^{11} + 844910160 T^{12} + 2172157235 T^{13} + 13308440752 T^{14} + 36480277815 T^{15} + 186255087319 T^{16} + 493932753875 T^{17} + 2358771517818 T^{18} + 4522752853806 T^{19} + 24554666192320 T^{20} + 25022041940867 T^{21} + 173375475438614 T^{22} + 68543792615266 T^{23} + 582622237229761 T^{24} \))(\( 1 - 12 T + 166 T^{2} - 1221 T^{3} + 9798 T^{4} - 51318 T^{5} + 298418 T^{6} - 1123183 T^{7} + 5047911 T^{8} - 11448777 T^{9} + 42672488 T^{10} + 1695375 T^{11} + 253046308 T^{12} + 28821375 T^{13} + 12332349032 T^{14} - 56247841401 T^{15} + 421606574631 T^{16} - 1594759244831 T^{17} + 7203085065842 T^{18} - 21057760021014 T^{19} + 68348471406918 T^{20} - 144795797202837 T^{21} + 334654987474534 T^{22} - 411262755691596 T^{23} + 582622237229761 T^{24} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 + 4 T + 19 T^{2} )^{3} \))(\( 1 - 4 T + 79 T^{2} - 267 T^{3} + 2678 T^{4} - 7255 T^{5} + 50882 T^{6} - 96387 T^{7} + 541861 T^{8} - 521284 T^{9} + 2476099 T^{10} \))(\( 1 + 10 T + 97 T^{2} + 617 T^{3} + 3658 T^{4} + 16559 T^{5} + 69502 T^{6} + 222737 T^{7} + 665323 T^{8} + 1303210 T^{9} + 2476099 T^{10} \))(\( 1 + 117 T^{2} - 144 T^{3} + 6545 T^{4} - 14584 T^{5} + 240551 T^{6} - 681477 T^{7} + 6591561 T^{8} - 19407794 T^{9} + 141051753 T^{10} - 368748086 T^{11} + 2379553521 T^{12} - 4674250743 T^{13} + 31348846871 T^{14} - 36111427816 T^{15} + 307915291145 T^{16} - 128717530416 T^{17} + 1987076875797 T^{18} + 6131066257801 T^{20} \))(\( 1 + 6 T + 179 T^{2} + 1019 T^{3} + 15567 T^{4} + 81493 T^{5} + 860771 T^{6} + 4072782 T^{7} + 33490409 T^{8} + 141658777 T^{9} + 962981822 T^{10} + 3604155843 T^{11} + 20956227758 T^{12} + 68478961017 T^{13} + 347636437742 T^{14} + 971637551443 T^{15} + 4364503591289 T^{16} + 10084611437418 T^{17} + 40495730034251 T^{18} + 72844289626327 T^{19} + 264383125859247 T^{20} + 328818764036801 T^{21} + 1097460860146379 T^{22} + 698941553389314 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 59 T^{2} + 67 T^{3} + 2447 T^{4} + 3079 T^{5} + 74847 T^{6} + 136062 T^{7} + 1860811 T^{8} + 3815153 T^{9} + 41316958 T^{10} + 91606659 T^{11} + 798996362 T^{12} + 1740526521 T^{13} + 14915421838 T^{14} + 26168134427 T^{15} + 242502750331 T^{16} + 336902982138 T^{17} + 3521243055207 T^{18} + 2752231084381 T^{19} + 41558778761327 T^{20} + 21620075751193 T^{21} + 361732909210259 T^{22} + 2213314919066161 T^{24} \))
$23$ (\( 1 - 5 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 + 41 T^{2} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{3} \))(\( 1 - 4 T + 96 T^{2} - 310 T^{3} + 4041 T^{4} - 10201 T^{5} + 92943 T^{6} - 163990 T^{7} + 1168032 T^{8} - 1119364 T^{9} + 6436343 T^{10} \))(\( 1 - 12 T + 124 T^{2} - 730 T^{3} + 4391 T^{4} - 19387 T^{5} + 100993 T^{6} - 386170 T^{7} + 1508708 T^{8} - 3358092 T^{9} + 6436343 T^{10} \))(\( 1 - 2 T + 128 T^{2} - 139 T^{3} + 7117 T^{4} + 755 T^{5} + 224173 T^{6} + 407616 T^{7} + 4690668 T^{8} + 19318174 T^{9} + 93011881 T^{10} + 444318002 T^{11} + 2481363372 T^{12} + 4959463872 T^{13} + 62732796493 T^{14} + 4859438965 T^{15} + 1053571422013 T^{16} - 473270737133 T^{17} + 10023806115968 T^{18} - 3602305322926 T^{19} + 41426511213649 T^{20} \))(\( 1 + 13 T + 237 T^{2} + 2002 T^{3} + 21372 T^{4} + 133225 T^{5} + 1071017 T^{6} + 5181303 T^{7} + 35330634 T^{8} + 137041097 T^{9} + 885747404 T^{10} + 2981518951 T^{11} + 20222903567 T^{12} + 68574935873 T^{13} + 468560376716 T^{14} + 1667379027199 T^{15} + 9886959949194 T^{16} + 33348643294929 T^{17} + 158548953729113 T^{18} + 453607870176575 T^{19} + 1673662377425532 T^{20} + 3605907628248926 T^{21} + 9818083157634813 T^{22} + 12386526852881051 T^{23} + 21914624432020321 T^{24} \))(\( 1 - 9 T + 157 T^{2} - 1198 T^{3} + 12814 T^{4} - 84379 T^{5} + 700261 T^{6} - 4074957 T^{7} + 28422478 T^{8} - 148320623 T^{9} + 904244314 T^{10} - 4251116175 T^{11} + 23114338603 T^{12} - 97775672025 T^{13} + 478345242106 T^{14} - 1804617020041 T^{15} + 7953774665998 T^{16} - 26227820962251 T^{17} + 103663759667029 T^{18} - 287295766392413 T^{19} + 1003476965390734 T^{20} - 2157780888432674 T^{21} + 6503962260542893 T^{22} - 8575287821225343 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 10 T + 78 T^{2} + 290 T^{3} + 841 T^{4} \))(\( 1 + 2 T + 47 T^{2} + 188 T^{3} + 1363 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 + 6 T + 94 T^{2} + 323 T^{3} + 3466 T^{4} + 8479 T^{5} + 100514 T^{6} + 271643 T^{7} + 2292566 T^{8} + 4243686 T^{9} + 20511149 T^{10} \))(\( 1 + 16 T + 168 T^{2} + 1239 T^{3} + 7332 T^{4} + 40831 T^{5} + 212628 T^{6} + 1041999 T^{7} + 4097352 T^{8} + 11316496 T^{9} + 20511149 T^{10} \))(\( 1 + 9 T + 225 T^{2} + 1647 T^{3} + 23252 T^{4} + 143611 T^{5} + 1501329 T^{6} + 8016660 T^{7} + 68262730 T^{8} + 317447419 T^{9} + 2290933469 T^{10} + 9205975151 T^{11} + 57408955930 T^{12} + 195518320740 T^{13} + 1061861476449 T^{14} + 2945626619039 T^{15} + 13830831859892 T^{16} + 28410546280923 T^{17} + 112555442916225 T^{18} + 130564313782821 T^{19} + 420707233300201 T^{20} \))(\( 1 - 20 T + 399 T^{2} - 4927 T^{3} + 59113 T^{4} - 549200 T^{5} + 4994143 T^{6} - 38035759 T^{7} + 285978915 T^{8} - 1866713053 T^{9} + 12107702506 T^{10} - 69364622861 T^{11} + 396392131942 T^{12} - 2011574062969 T^{13} + 10182577807546 T^{14} - 45527264649617 T^{15} + 202267452980115 T^{16} - 780157120177091 T^{17} + 2970632724808903 T^{18} - 9473632068902800 T^{19} + 29571066209363593 T^{20} - 71476708223106563 T^{21} + 167862186086780199 T^{22} - 244010195314116580 T^{23} + 353814783205469041 T^{24} \))(\( 1 - 34 T + 717 T^{2} - 11125 T^{3} + 141257 T^{4} - 1531538 T^{5} + 14628117 T^{6} - 125171225 T^{7} + 971974667 T^{8} - 6905339085 T^{9} + 45156961894 T^{10} - 272841031757 T^{11} + 1526411184342 T^{12} - 7912389920953 T^{13} + 37977004952854 T^{14} - 168414314944065 T^{15} + 687459214450427 T^{16} - 2567405646487525 T^{17} + 8701145133916557 T^{18} - 26418841062533242 T^{19} + 70663307555631977 T^{20} - 161391998981542625 T^{21} + 301647086276244117 T^{22} - 414817332033998186 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 2 T + 18 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 - 10 T + 93 T^{2} - 548 T^{3} + 2883 T^{4} - 9610 T^{5} + 29791 T^{6} \))(\( 1 - 3 T + 127 T^{2} - 249 T^{3} + 6921 T^{4} - 9755 T^{5} + 214551 T^{6} - 239289 T^{7} + 3783457 T^{8} - 2770563 T^{9} + 28629151 T^{10} \))(\( 1 - 33 T + 579 T^{2} - 6751 T^{3} + 57197 T^{4} - 364933 T^{5} + 1773107 T^{6} - 6487711 T^{7} + 17248989 T^{8} - 30476193 T^{9} + 28629151 T^{10} \))(\( 1 - 22 T + 437 T^{2} - 5905 T^{3} + 71196 T^{4} - 707620 T^{5} + 6360319 T^{6} - 49929045 T^{7} + 358058017 T^{8} - 2291835411 T^{9} + 13467173163 T^{10} - 71046897741 T^{11} + 344093754337 T^{12} - 1487436179595 T^{13} + 5873888163199 T^{14} - 20258559830620 T^{15} + 63186712072476 T^{16} - 162461986325455 T^{17} + 372713383361717 T^{18} - 581671687534762 T^{19} + 819628286980801 T^{20} \))(\( 1 + 7 T + 282 T^{2} + 1830 T^{3} + 38139 T^{4} + 230297 T^{5} + 3280516 T^{6} + 18364168 T^{7} + 200366767 T^{8} + 1028922293 T^{9} + 9171046402 T^{10} + 42467891207 T^{11} + 322887365354 T^{12} + 1316504627417 T^{13} + 8813375592322 T^{14} + 30652624030763 T^{15} + 185042917026607 T^{16} + 525750538661368 T^{17} + 2911470025579396 T^{18} + 6336072491920967 T^{19} + 32528411276962299 T^{20} + 48384508554027930 T^{21} + 231135176928585882 T^{22} + 177859338274833817 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 11 T + 246 T^{2} - 2242 T^{3} + 28987 T^{4} - 229197 T^{5} + 2208928 T^{6} - 15528844 T^{7} + 122865191 T^{8} - 777995717 T^{9} + 5304569738 T^{10} - 30338429999 T^{11} + 183095285754 T^{12} - 940491329969 T^{13} + 5097691518218 T^{14} - 23177270405147 T^{15} + 113468584057511 T^{16} - 444577619731444 T^{17} + 1960431731063968 T^{18} - 6305808616398867 T^{19} + 24722752502302267 T^{20} - 59277632884224382 T^{21} + 201628558597277046 T^{22} - 279493245860453141 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 - 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 - 2 T + 43 T^{2} - 204 T^{3} + 1591 T^{4} - 2738 T^{5} + 50653 T^{6} \))(\( 1 + 10 T + 161 T^{2} + 1149 T^{3} + 11234 T^{4} + 60063 T^{5} + 415658 T^{6} + 1572981 T^{7} + 8155133 T^{8} + 18741610 T^{9} + 69343957 T^{10} \))(\( 1 - 2 T + 79 T^{2} + 19 T^{3} + 3444 T^{4} + 4987 T^{5} + 127428 T^{6} + 26011 T^{7} + 4001587 T^{8} - 3748322 T^{9} + 69343957 T^{10} \))(\( 1 + 35 T + 807 T^{2} + 13514 T^{3} + 185245 T^{4} + 2135413 T^{5} + 21450226 T^{6} + 189788329 T^{7} + 1500972503 T^{8} + 10649984634 T^{9} + 68217693835 T^{10} + 394049431458 T^{11} + 2054831356607 T^{12} + 9613348228837 T^{13} + 40201177010386 T^{14} + 148077987249241 T^{15} + 475287988635205 T^{16} + 1282909387575362 T^{17} + 2834570919314247 T^{18} + 4548660892827695 T^{19} + 4808584372417849 T^{20} \))(\( 1 - 10 T + 243 T^{2} - 1891 T^{3} + 28373 T^{4} - 204733 T^{5} + 2343397 T^{6} - 15984122 T^{7} + 146808103 T^{8} - 936951483 T^{9} + 7321030232 T^{10} - 43550538569 T^{11} + 299802706494 T^{12} - 1611369927053 T^{13} + 10022490387608 T^{14} - 47459403468399 T^{15} + 275142021126583 T^{16} - 1108402268650754 T^{17} + 6012515569671373 T^{18} - 19435688001070489 T^{19} + 99659579546100533 T^{20} - 245757649952490607 T^{21} + 1168486002497537307 T^{22} - 1779176217794604130 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 22 T + 455 T^{2} + 6559 T^{3} + 85475 T^{4} + 942209 T^{5} + 9511859 T^{6} + 86045010 T^{7} + 722793987 T^{8} + 5570654269 T^{9} + 40242008154 T^{10} + 269557457559 T^{11} + 1700095666794 T^{12} + 9973625929683 T^{13} + 55091309162826 T^{14} + 282170350687657 T^{15} + 1354632301469907 T^{16} + 5966701473504570 T^{17} + 24404827834984331 T^{18} + 89445669021606797 T^{19} + 300229181323897475 T^{20} + 852419051315910043 T^{21} + 2187905889450121295 T^{22} + 3914187679148129086 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 86 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 2 T + 19 T^{2} - 284 T^{3} + 779 T^{4} - 3362 T^{5} + 68921 T^{6} \))(\( 1 + 6 T + 79 T^{2} + 31 T^{3} + 194 T^{4} - 17503 T^{5} + 7954 T^{6} + 52111 T^{7} + 5444759 T^{8} + 16954566 T^{9} + 115856201 T^{10} \))(\( 1 + 12 T + 207 T^{2} + 1687 T^{3} + 16586 T^{4} + 98677 T^{5} + 680026 T^{6} + 2835847 T^{7} + 14266647 T^{8} + 33909132 T^{9} + 115856201 T^{10} \))(\( 1 + 4 T + 260 T^{2} + 1417 T^{3} + 32755 T^{4} + 218127 T^{5} + 2678340 T^{6} + 19788650 T^{7} + 159907986 T^{8} + 1182434807 T^{9} + 7385189807 T^{10} + 48479827087 T^{11} + 268805324466 T^{12} + 1363853546650 T^{13} + 7568348716740 T^{14} + 25271365555527 T^{15} + 155589664413955 T^{16} + 275966806089377 T^{17} + 2076080559571460 T^{18} + 1309527737575844 T^{19} + 13422659310152401 T^{20} \))(\( 1 - 2 T + 310 T^{2} - 785 T^{3} + 47098 T^{4} - 138456 T^{5} + 4673492 T^{6} - 14922727 T^{7} + 340248619 T^{8} - 1111165915 T^{9} + 19293798486 T^{10} - 60634376507 T^{11} + 878545992372 T^{12} - 2486009436787 T^{13} + 32432875254966 T^{14} - 76582666027715 T^{15} + 961461277874059 T^{16} - 1728890458780127 T^{17} + 22199574169479572 T^{18} - 26964897744467736 T^{19} + 376074008441140858 T^{20} - 256994818499259385 T^{21} + 4161024386147244310 T^{22} - 1100658063432496882 T^{23} + 22563490300366186081 T^{24} \))(\( 1 - 32 T + 770 T^{2} - 13419 T^{3} + 199542 T^{4} - 2512554 T^{5} + 28288052 T^{6} - 283801421 T^{7} + 2605368715 T^{8} - 21811184839 T^{9} + 169130978058 T^{10} - 1209112436159 T^{11} + 8049974165820 T^{12} - 49573609882519 T^{13} + 284309174115498 T^{14} - 1503248670288719 T^{15} + 7362149305467115 T^{16} - 32880154475461621 T^{17} + 134371195774828532 T^{18} - 489330629856802074 T^{19} + 1593327950069262582 T^{20} - 4393138177632562659 T^{21} + 10335447668817348770 T^{22} - 17610529014919950112 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( 1 - T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 16 T + 145 T^{2} + 688 T^{3} + 1849 T^{4} \))(\( 1 + 5 T + 81 T^{2} + 435 T^{3} + 3483 T^{4} + 9245 T^{5} + 79507 T^{6} \))(\( 1 + 11 T + 250 T^{2} + 1923 T^{3} + 23015 T^{4} + 124701 T^{5} + 989645 T^{6} + 3555627 T^{7} + 19876750 T^{8} + 37606811 T^{9} + 147008443 T^{10} \))(\( 1 - 17 T + 270 T^{2} - 2417 T^{3} + 21761 T^{4} - 137929 T^{5} + 935723 T^{6} - 4469033 T^{7} + 21466890 T^{8} - 58119617 T^{9} + 147008443 T^{10} \))(\( 1 - 20 T + 435 T^{2} - 5094 T^{3} + 60693 T^{4} - 456684 T^{5} + 3529418 T^{6} - 14396035 T^{7} + 65477767 T^{8} + 145261781 T^{9} - 450150713 T^{10} + 6246256583 T^{11} + 121068391183 T^{12} - 1144585554745 T^{13} + 12066377787818 T^{14} - 67136403783012 T^{15} + 383662487532957 T^{16} - 1384644004979058 T^{17} + 5084367120756435 T^{18} - 10051852238736860 T^{19} + 21611482313284249 T^{20} \))(\( 1 + 8 T + 333 T^{2} + 1866 T^{3} + 49961 T^{4} + 201111 T^{5} + 4798493 T^{6} + 14405750 T^{7} + 347174076 T^{8} + 838886304 T^{9} + 20293592746 T^{10} + 42545902371 T^{11} + 968100841052 T^{12} + 1829473801953 T^{13} + 37522852987354 T^{14} + 66697333372128 T^{15} + 1186919078202876 T^{16} + 2117766877747250 T^{17} + 30333016341085157 T^{18} + 54665712698339877 T^{19} + 583954174069223561 T^{20} + 937837813874149038 T^{21} + 7196623610323654917 T^{22} + 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 8 T + 213 T^{2} - 1770 T^{3} + 23231 T^{4} - 166041 T^{5} + 1560689 T^{6} - 8765026 T^{7} + 63390606 T^{8} - 259839552 T^{9} + 1507551598 T^{10} - 3483878321 T^{11} + 34544523436 T^{12} - 149806767803 T^{13} + 2787462904702 T^{14} - 20659063260864 T^{15} + 216719867183406 T^{16} - 1288532825114518 T^{17} + 9865681775580761 T^{18} - 45133034006817387 T^{19} + 271528580648948831 T^{20} - 889588923128212110 T^{21} + 4603245732729545037 T^{22} - 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 - 3 T + 47 T^{2} \))(\( 1 + 11 T + 47 T^{2} \))(\( 1 - T + 47 T^{2} \))(\( 1 - T + 47 T^{2} \))(\( 1 - 5 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 - T + 47 T^{2} \))(\( 1 + T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 + 7 T + 47 T^{2} \))(\( 1 + 4 T + 93 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 + 19 T + 237 T^{2} + 1849 T^{3} + 11139 T^{4} + 41971 T^{5} + 103823 T^{6} \))(\( 1 - 9 T + 120 T^{2} - 610 T^{3} + 8412 T^{4} - 44117 T^{5} + 395364 T^{6} - 1347490 T^{7} + 12458760 T^{8} - 43917129 T^{9} + 229345007 T^{10} \))(\( 1 - 7 T + 42 T^{2} + 124 T^{3} + 3434 T^{4} - 23499 T^{5} + 161398 T^{6} + 273916 T^{7} + 4360566 T^{8} - 34157767 T^{9} + 229345007 T^{10} \))(\( 1 + 7 T + 322 T^{2} + 2204 T^{3} + 50553 T^{4} + 325252 T^{5} + 5105346 T^{6} + 30078097 T^{7} + 367899299 T^{8} + 1942791244 T^{9} + 19840453349 T^{10} + 91311188468 T^{11} + 812689551491 T^{12} + 3122798264831 T^{13} + 24912459874626 T^{14} + 74594922216764 T^{15} + 544921672526937 T^{16} + 1116597357500452 T^{17} + 7667234305087042 T^{18} + 7833913311719369 T^{19} + 52599132235830049 T^{20} \))(\( 1 + 12 T + 302 T^{2} + 3187 T^{3} + 47279 T^{4} + 415791 T^{5} + 4717830 T^{6} + 35652736 T^{7} + 339864289 T^{8} + 2268004051 T^{9} + 19491144185 T^{10} + 119470683497 T^{11} + 965417437421 T^{12} + 5615122124359 T^{13} + 43055937504665 T^{14} + 235470984586973 T^{15} + 1658429313611809 T^{16} + 8176776987489152 T^{17} + 50854505455616070 T^{18} + 210649333880431233 T^{19} + 1125773822081398319 T^{20} + 3566668817778518429 T^{21} + 15884937935220674798 T^{22} + 29665910581008147636 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 24 T + 624 T^{2} + 9813 T^{3} + 154353 T^{4} + 1867001 T^{5} + 22286272 T^{6} + 222278140 T^{7} + 2182329723 T^{8} + 18581477729 T^{9} + 155728331067 T^{10} + 1151472566375 T^{11} + 8382970589145 T^{12} + 54119210619625 T^{13} + 344003883327003 T^{14} + 1929184762257967 T^{15} + 10649072885058363 T^{16} + 50978381574246980 T^{17} + 240228524768663488 T^{18} + 945865872527541463 T^{19} + 3675343530102795633 T^{20} + 10982027332557452571 T^{21} + 32821858515157950576 T^{22} + 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - 14 T + 127 T^{2} - 836 T^{3} + 6731 T^{4} - 39326 T^{5} + 148877 T^{6} \))(\( 1 + 22 T + 345 T^{2} + 3931 T^{3} + 36038 T^{4} + 286901 T^{5} + 1910014 T^{6} + 11042179 T^{7} + 51362565 T^{8} + 173590582 T^{9} + 418195493 T^{10} \))(\( 1 + 177 T^{2} - 51 T^{3} + 15884 T^{4} - 2059 T^{5} + 841852 T^{6} - 143259 T^{7} + 26351229 T^{8} + 418195493 T^{10} \))(\( 1 + 24 T + 570 T^{2} + 8461 T^{3} + 123312 T^{4} + 1397686 T^{5} + 15606645 T^{6} + 145668337 T^{7} + 1341977182 T^{8} + 10633420505 T^{9} + 83159919093 T^{10} + 563571286765 T^{11} + 3769613904238 T^{12} + 21686665007549 T^{13} + 123143935846245 T^{14} + 584505985829198 T^{15} + 2733131699539248 T^{16} + 9939230954160857 T^{17} + 35488023534475770 T^{18} + 79194326203251192 T^{19} + 174887470365513049 T^{20} \))(\( 1 - 12 T + 545 T^{2} - 5561 T^{3} + 137317 T^{4} - 1216285 T^{5} + 21437635 T^{6} - 166788154 T^{7} + 2329758205 T^{8} - 15995580255 T^{9} + 186586431716 T^{10} - 1128590015573 T^{11} + 11317232997794 T^{12} - 59815270825369 T^{13} + 524121286690244 T^{14} - 2381374001623635 T^{15} + 18382912851146605 T^{16} - 69750054288589922 T^{17} + 475151483891689915 T^{18} - 1428783538716645545 T^{19} + 8549313908216858437 T^{20} - 18349985334011661613 T^{21} + 95313671349204611705 T^{22} - \)\(11\!\cdots\!64\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 2 T + 395 T^{2} + 573 T^{3} + 79159 T^{4} + 85111 T^{5} + 10523539 T^{6} + 8593086 T^{7} + 1028184511 T^{8} + 664497415 T^{9} + 77483123342 T^{10} + 41921174393 T^{11} + 4607531907962 T^{12} + 2221822242829 T^{13} + 217650093467678 T^{14} + 98928381652955 T^{15} + 8112870348539791 T^{16} + 3593589836161398 T^{17} + 233247518751115531 T^{18} + 99980839822666907 T^{19} + 4928414833272925399 T^{20} + 1890764538102622209 T^{21} + 69080550794377654355 T^{22} + 18538071858744383194 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 - 5 T + 161 T^{2} - 591 T^{3} + 9499 T^{4} - 17405 T^{5} + 205379 T^{6} \))(\( 1 - 10 T + 284 T^{2} - 2204 T^{3} + 33061 T^{4} - 190573 T^{5} + 1950599 T^{6} - 7672124 T^{7} + 58327636 T^{8} - 121173610 T^{9} + 714924299 T^{10} \))(\( 1 + 18 T + 380 T^{2} + 4190 T^{3} + 48801 T^{4} + 367541 T^{5} + 2879259 T^{6} + 14585390 T^{7} + 78044020 T^{8} + 218112498 T^{9} + 714924299 T^{10} \))(\( 1 + 17 T + 467 T^{2} + 5784 T^{3} + 84135 T^{4} + 790560 T^{5} + 7647144 T^{6} + 56238779 T^{7} + 400513752 T^{8} + 2655972904 T^{9} + 18692560589 T^{10} + 156702401336 T^{11} + 1394188370712 T^{12} + 11550264192241 T^{13} + 92663204466984 T^{14} + 565190553817440 T^{15} + 3548859197885535 T^{16} + 14394360188193096 T^{17} + 68569814361217907 T^{18} + 147270928917133963 T^{19} + 511116753300641401 T^{20} \))(\( 1 + 6 T + 277 T^{2} + 1124 T^{3} + 34762 T^{4} + 84929 T^{5} + 3002529 T^{6} + 4076375 T^{7} + 229711951 T^{8} + 229401713 T^{9} + 16358116930 T^{10} + 15943411277 T^{11} + 1033672281260 T^{12} + 940661265343 T^{13} + 56942605033330 T^{14} + 47114294414227 T^{15} + 2783502636281311 T^{16} + 2914299539336125 T^{17} + 126648275492578089 T^{18} + 211358681954192851 T^{19} + 5104119672001406602 T^{20} + 9737207300168151436 T^{21} + \)\(14\!\cdots\!77\)\( T^{22} + \)\(18\!\cdots\!54\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 26 T + 437 T^{2} + 5246 T^{3} + 56014 T^{4} + 561929 T^{5} + 5627753 T^{6} + 53428741 T^{7} + 487722895 T^{8} + 4192620027 T^{9} + 35209729306 T^{10} + 283290463699 T^{11} + 2227511236260 T^{12} + 16714137358241 T^{13} + 122565067714186 T^{14} + 861076108525233 T^{15} + 5909914386680095 T^{16} + 38197505205877559 T^{17} + 237381624739738673 T^{18} + 1398445440212855851 T^{19} + 8224560131968436494 T^{20} + 45446076064663809994 T^{21} + \)\(22\!\cdots\!37\)\( T^{22} + \)\(78\!\cdots\!34\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 11 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 6 T + 111 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 - 2 T + 61 T^{2} )^{3} \))(\( 1 + 11 T + 272 T^{2} + 2418 T^{3} + 31232 T^{4} + 213643 T^{5} + 1905152 T^{6} + 8997378 T^{7} + 61738832 T^{8} + 152304251 T^{9} + 844596301 T^{10} \))(\( 1 - T + 194 T^{2} + 4 T^{3} + 19824 T^{4} - 933 T^{5} + 1209264 T^{6} + 14884 T^{7} + 44034314 T^{8} - 13845841 T^{9} + 844596301 T^{10} \))(\( 1 + 4 T + 252 T^{2} + 859 T^{3} + 35750 T^{4} + 141968 T^{5} + 3623379 T^{6} + 15927450 T^{7} + 286566685 T^{8} + 1331963236 T^{9} + 19096375159 T^{10} + 81249757396 T^{11} + 1066314634885 T^{12} + 3615228528450 T^{13} + 50168729516739 T^{14} + 119905647660368 T^{15} + 1841853383405750 T^{16} + 2699616096142039 T^{17} + 48310242875314812 T^{18} + 46776584371336564 T^{19} + 713342911662882601 T^{20} \))(\( 1 + 10 T + 365 T^{2} + 3179 T^{3} + 61581 T^{4} + 449878 T^{5} + 6491041 T^{6} + 40615545 T^{7} + 512777084 T^{8} + 2961728657 T^{9} + 35403589926 T^{10} + 197962455527 T^{11} + 2258050315556 T^{12} + 12075709787147 T^{13} + 131736758114646 T^{14} + 672256132294517 T^{15} + 7099829973507644 T^{16} + 34303739070099045 T^{17} + 334420862312599801 T^{18} + 1413850861583455438 T^{19} + 11805528041685561261 T^{20} + 37175690429119734239 T^{21} + \)\(26\!\cdots\!65\)\( T^{22} + \)\(43\!\cdots\!10\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 - 12 T + 461 T^{2} - 4767 T^{3} + 106261 T^{4} - 975590 T^{5} + 16121299 T^{6} - 133141815 T^{7} + 1794097610 T^{8} - 13394248577 T^{9} + 154375149996 T^{10} - 1039460652715 T^{11} + 10534585951880 T^{12} - 63407099815615 T^{13} + 574429933135116 T^{14} - 3040239936256037 T^{15} + 24840790246540010 T^{16} - 112451084457426315 T^{17} + 830575359665614939 T^{18} - 3066028483393727390 T^{19} + 20371010786404076341 T^{20} - 55745994424540350147 T^{21} + \)\(32\!\cdots\!61\)\( T^{22} - \)\(52\!\cdots\!32\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 + 7 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 - 4 T + 93 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 35 T + 589 T^{2} + 6009 T^{3} + 39463 T^{4} + 157115 T^{5} + 300763 T^{6} \))(\( 1 + 14 T + 324 T^{2} + 3022 T^{3} + 39869 T^{4} + 276469 T^{5} + 2671223 T^{6} + 13565758 T^{7} + 97447212 T^{8} + 282115694 T^{9} + 1350125107 T^{10} \))(\( 1 + 6 T + 128 T^{2} + 1136 T^{3} + 14413 T^{4} + 78367 T^{5} + 965671 T^{6} + 5099504 T^{7} + 38497664 T^{8} + 120906726 T^{9} + 1350125107 T^{10} \))(\( 1 - 6 T + 318 T^{2} - 2402 T^{3} + 61276 T^{4} - 443521 T^{5} + 8166786 T^{6} - 55393313 T^{7} + 807832572 T^{8} - 4959723492 T^{9} + 61619242847 T^{10} - 332301473964 T^{11} + 3626360415708 T^{12} - 16660258997819 T^{13} + 164569892867106 T^{14} - 598808837581747 T^{15} + 5542927825787644 T^{16} - 14557829275985846 T^{17} + 129129521463011838 T^{18} - 163239206377769682 T^{19} + 1822837804551761449 T^{20} \))(\( 1 + 7 T + 554 T^{2} + 3317 T^{3} + 145357 T^{4} + 743955 T^{5} + 24141915 T^{6} + 105743608 T^{7} + 2874222370 T^{8} + 10870454891 T^{9} + 263904792235 T^{10} + 880117558364 T^{11} + 19537385096368 T^{12} + 58967876410388 T^{13} + 1184668612342915 T^{14} + 3269430624381833 T^{15} + 57918802758776770 T^{16} + 142767100065566056 T^{17} + 2183838573361513635 T^{18} + 4508896702338072465 T^{19} + 59024779406600665837 T^{20} + 90244074592510339199 T^{21} + \)\(10\!\cdots\!46\)\( T^{22} + \)\(85\!\cdots\!81\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 21 T + 458 T^{2} - 6321 T^{3} + 87583 T^{4} - 965771 T^{5} + 10922727 T^{6} - 107004290 T^{7} + 1094708152 T^{8} - 9997299639 T^{9} + 94636033095 T^{10} - 802402093488 T^{11} + 6948793830000 T^{12} - 53760940263696 T^{13} + 424821152563455 T^{14} - 3006817831324557 T^{15} + 22059596430638392 T^{16} - 144469178485709030 T^{17} + 988052213293654863 T^{18} - 5853259507784399033 T^{19} + 35564625403443288703 T^{20} - \)\(17\!\cdots\!87\)\( T^{21} + \)\(83\!\cdots\!42\)\( T^{22} - \)\(25\!\cdots\!43\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 + 6 T + 146 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 + 12 T + 241 T^{2} + 1712 T^{3} + 17111 T^{4} + 60492 T^{5} + 357911 T^{6} \))(\( 1 - 26 T + 478 T^{2} - 6623 T^{3} + 75008 T^{4} - 682865 T^{5} + 5325568 T^{6} - 33386543 T^{7} + 171081458 T^{8} - 660703706 T^{9} + 1804229351 T^{10} \))(\( 1 - 26 T + 402 T^{2} - 4177 T^{3} + 35480 T^{4} - 279111 T^{5} + 2519080 T^{6} - 21056257 T^{7} + 143880222 T^{8} - 660703706 T^{9} + 1804229351 T^{10} \))(\( 1 - T + 262 T^{2} - 1233 T^{3} + 39245 T^{4} - 293739 T^{5} + 4405465 T^{6} - 40998321 T^{7} + 397241676 T^{8} - 3994326482 T^{9} + 30391775917 T^{10} - 283597180222 T^{11} + 2002495288716 T^{12} - 14673750067431 T^{13} + 111950271236665 T^{14} - 529972525333389 T^{15} + 5027295642479645 T^{16} - 11214283155296103 T^{17} + 169187425186389382 T^{18} - 45848500718449031 T^{19} + 3255243551009881201 T^{20} \))(\( 1 + 22 T + 767 T^{2} + 12443 T^{3} + 245151 T^{4} + 3126102 T^{5} + 44600111 T^{6} + 466232143 T^{7} + 5338265595 T^{8} + 47751004997 T^{9} + 474162762402 T^{10} + 3842492624367 T^{11} + 35488548426314 T^{12} + 272816976330057 T^{13} + 2390254485268482 T^{14} + 17090609949481267 T^{15} + 135654302393415195 T^{16} + 841189716780229193 T^{17} + 5713286882008115231 T^{18} + 28432273317386421882 T^{19} + \)\(15\!\cdots\!11\)\( T^{20} + \)\(57\!\cdots\!33\)\( T^{21} + \)\(24\!\cdots\!67\)\( T^{22} + \)\(50\!\cdots\!62\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 50 T + 1711 T^{2} + 43305 T^{3} + 908171 T^{4} + 16189750 T^{5} + 253701007 T^{6} + 3535881037 T^{7} + 44474886435 T^{8} + 507739217863 T^{9} + 5299361943762 T^{10} + 50676244010449 T^{11} + 445479921732642 T^{12} + 3598013324741879 T^{13} + 26714083558504242 T^{14} + 181725451204564193 T^{15} + 1130181626597447235 T^{16} + 6379540348599716987 T^{17} + 32499171027743608447 T^{18} + \)\(14\!\cdots\!50\)\( T^{19} + \)\(58\!\cdots\!31\)\( T^{20} + \)\(19\!\cdots\!55\)\( T^{21} + \)\(55\!\cdots\!11\)\( T^{22} + \)\(11\!\cdots\!50\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 24 T + 270 T^{2} + 1752 T^{3} + 5329 T^{4} \))(\( 1 - 35 T + 605 T^{2} - 6403 T^{3} + 44165 T^{4} - 186515 T^{5} + 389017 T^{6} \))(\( 1 + 7 T + 140 T^{2} + 324 T^{3} + 13742 T^{4} + 47981 T^{5} + 1003166 T^{6} + 1726596 T^{7} + 54462380 T^{8} + 198787687 T^{9} + 2073071593 T^{10} \))(\( 1 - 9 T + 372 T^{2} - 2620 T^{3} + 54822 T^{4} - 286599 T^{5} + 4002006 T^{6} - 13961980 T^{7} + 144714324 T^{8} - 255584169 T^{9} + 2073071593 T^{10} \))(\( 1 + 31 T + 857 T^{2} + 16625 T^{3} + 280173 T^{4} + 4020157 T^{5} + 51396432 T^{6} + 589991993 T^{7} + 6192642640 T^{8} + 59433187850 T^{9} + 529794375619 T^{10} + 4338622713050 T^{11} + 33000592628560 T^{12} + 229516915140881 T^{13} + 1459568262476112 T^{14} + 8334073276100101 T^{15} + 42399764182067997 T^{16} + 183663000379987625 T^{17} + 691136298753227417 T^{18} + 1825019187956305303 T^{19} + 4297625829703557649 T^{20} \))(\( 1 + 23 T + 914 T^{2} + 15862 T^{3} + 359607 T^{4} + 5033434 T^{5} + 83581420 T^{6} + 980185074 T^{7} + 13078758902 T^{8} + 131435921598 T^{9} + 1474371678576 T^{10} + 12844486325361 T^{11} + 123956533139464 T^{12} + 937647501751353 T^{13} + 7856926675131504 T^{14} + 51130807912289166 T^{15} + 371413747279891382 T^{16} + 2031993832792002882 T^{17} + 12648729527835950380 T^{18} + 55606351317572489098 T^{19} + \)\(29\!\cdots\!67\)\( T^{20} + \)\(93\!\cdots\!06\)\( T^{21} + \)\(39\!\cdots\!86\)\( T^{22} + \)\(72\!\cdots\!71\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 - 17 T + 610 T^{2} - 8290 T^{3} + 174647 T^{4} - 2005546 T^{5} + 31885540 T^{6} - 318801846 T^{7} + 4202591870 T^{8} - 37297650102 T^{9} + 427206769000 T^{10} - 3402453895951 T^{11} + 34693832766584 T^{12} - 248379134404423 T^{13} + 2276584872001000 T^{14} - 14509419949729734 T^{15} + 119346216748900670 T^{16} - 660899050738560678 T^{17} + 4825373525706961060 T^{18} - 22156065910380911962 T^{19} + \)\(14\!\cdots\!07\)\( T^{20} - \)\(48\!\cdots\!70\)\( T^{21} + \)\(26\!\cdots\!90\)\( T^{22} - \)\(53\!\cdots\!09\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 4 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 7 T + 85 T^{2} + 39 T^{3} + 6715 T^{4} - 43687 T^{5} + 493039 T^{6} \))(\( 1 - 15 T + 218 T^{2} - 1565 T^{3} + 10815 T^{4} - 54601 T^{5} + 854385 T^{6} - 9767165 T^{7} + 107482502 T^{8} - 584251215 T^{9} + 3077056399 T^{10} \))(\( 1 - 21 T + 494 T^{2} - 6501 T^{3} + 84577 T^{4} - 764307 T^{5} + 6681583 T^{6} - 40572741 T^{7} + 243561266 T^{8} - 817951701 T^{9} + 3077056399 T^{10} \))(\( 1 - 10 T + 248 T^{2} - 2937 T^{3} + 29662 T^{4} - 273591 T^{5} + 1835094 T^{6} - 1901304 T^{7} - 54353511 T^{8} + 1412662644 T^{9} - 15400667207 T^{10} + 111600348876 T^{11} - 339220262151 T^{12} - 937417022856 T^{13} + 71477059942614 T^{14} - 841854937258809 T^{15} + 7210460105663902 T^{16} - 56401880692348983 T^{17} + 376242984856827128 T^{18} - 1198515959826183190 T^{19} + 9468276082626847201 T^{20} \))(\( 1 + 13 T + 740 T^{2} + 9277 T^{3} + 264468 T^{4} + 3122234 T^{5} + 60133963 T^{6} + 655134600 T^{7} + 9654177811 T^{8} + 95352923720 T^{9} + 1149494991777 T^{10} + 10118852614358 T^{11} + 103879296399120 T^{12} + 799389356534282 T^{13} + 7173998243680257 T^{14} + 47012710157985080 T^{15} + 376031007726852691 T^{16} + 2015886113136305400 T^{17} + 14617812056063959723 T^{18} + 59959097569491159206 T^{19} + \)\(40\!\cdots\!48\)\( T^{20} + \)\(11\!\cdots\!63\)\( T^{21} + \)\(70\!\cdots\!40\)\( T^{22} + \)\(97\!\cdots\!27\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 - 9 T + 648 T^{2} - 6279 T^{3} + 204926 T^{4} - 2056730 T^{5} + 42044625 T^{6} - 420858912 T^{7} + 6250252261 T^{8} - 60111649704 T^{9} + 709782021291 T^{10} - 6315539448032 T^{11} + 63127054087280 T^{12} - 498927616394528 T^{13} + 4429749594877131 T^{14} - 29637387658410456 T^{15} + 243447831836383141 T^{16} - 1295006608245777888 T^{17} + 10220520909584624625 T^{18} - 39497255729102800070 T^{19} + \)\(31\!\cdots\!86\)\( T^{20} - \)\(75\!\cdots\!01\)\( T^{21} + \)\(61\!\cdots\!48\)\( T^{22} - \)\(67\!\cdots\!11\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 + 15 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - T + 83 T^{2} \))(\( 1 - 17 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 + 20 T + 221 T^{2} + 1660 T^{3} + 6889 T^{4} \))(\( 1 - T + 241 T^{2} - 163 T^{3} + 20003 T^{4} - 6889 T^{5} + 571787 T^{6} \))(\( 1 - 12 T + 345 T^{2} - 2417 T^{3} + 44256 T^{4} - 226225 T^{5} + 3673248 T^{6} - 16650713 T^{7} + 197266515 T^{8} - 569499852 T^{9} + 3939040643 T^{10} \))(\( 1 + 103 T^{2} + 223 T^{3} + 10958 T^{4} + 36289 T^{5} + 909514 T^{6} + 1536247 T^{7} + 58894061 T^{8} + 3939040643 T^{10} \))(\( 1 + 22 T + 669 T^{2} + 9824 T^{3} + 169480 T^{4} + 1798297 T^{5} + 22307432 T^{6} + 175385507 T^{7} + 1816492439 T^{8} + 11655093474 T^{9} + 132417751709 T^{10} + 967372758342 T^{11} + 12513816412271 T^{12} + 100283152891009 T^{13} + 1058673268541672 T^{14} + 7083564971184971 T^{15} + 55409854478578120 T^{16} + 266584564922095648 T^{17} + 1506783503301018429 T^{18} + 4112685615885888866 T^{19} + 15516041187205853449 T^{20} \))(\( 1 + T + 591 T^{2} - 693 T^{3} + 167307 T^{4} - 517795 T^{5} + 30927255 T^{6} - 144179528 T^{7} + 4250890052 T^{8} - 24232595755 T^{9} + 465398797158 T^{10} - 2800671733828 T^{11} + 42176704430232 T^{12} - 232455753907724 T^{13} + 3206132313621462 T^{14} - 13855883228964185 T^{15} + 201740104623522692 T^{16} - 567929020680556504 T^{17} + 10111368296978272095 T^{18} - 14050911522173912465 T^{19} + \)\(37\!\cdots\!87\)\( T^{20} - \)\(12\!\cdots\!79\)\( T^{21} + \)\(91\!\cdots\!59\)\( T^{22} + \)\(12\!\cdots\!67\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 25 T + 697 T^{2} + 12233 T^{3} + 208949 T^{4} + 2854575 T^{5} + 37114827 T^{6} + 419442950 T^{7} + 4525988754 T^{8} + 44654078439 T^{9} + 428913306760 T^{10} + 3945221485244 T^{11} + 36292751560520 T^{12} + 327453383275252 T^{13} + 2954783770269640 T^{14} + 25532621548400493 T^{15} + 214795827129722034 T^{16} + 1652202827469816850 T^{17} + 12134335396905842163 T^{18} + 77461892753714493525 T^{19} + \)\(47\!\cdots\!09\)\( T^{20} + \)\(22\!\cdots\!99\)\( T^{21} + \)\(10\!\cdots\!53\)\( T^{22} + \)\(32\!\cdots\!75\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} \))(\( 1 - 4 T + 175 T^{2} - 880 T^{3} + 15575 T^{4} - 31684 T^{5} + 704969 T^{6} \))(\( 1 + 5 T + 106 T^{2} - 715 T^{3} + 6727 T^{4} - 64699 T^{5} + 598703 T^{6} - 5663515 T^{7} + 74726714 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \))(\( 1 - 15 T + 292 T^{2} - 2249 T^{3} + 32285 T^{4} - 207719 T^{5} + 2873365 T^{6} - 17814329 T^{7} + 205850948 T^{8} - 941133615 T^{9} + 5584059449 T^{10} \))(\( 1 - T + 510 T^{2} - 62 T^{3} + 113775 T^{4} + 104240 T^{5} + 14626103 T^{6} + 33572815 T^{7} + 1283471757 T^{8} + 5086267491 T^{9} + 104337188501 T^{10} + 452677806699 T^{11} + 10166379787197 T^{12} + 23667793817735 T^{13} + 917674479316823 T^{14} + 582082356963760 T^{15} + 56544046379087775 T^{16} - 2742342763522798 T^{17} + 2007660290908061310 T^{18} - 350356403707485209 T^{19} + 31181719929966183601 T^{20} \))(\( 1 + 3 T + 559 T^{2} + 2334 T^{3} + 165630 T^{4} + 762205 T^{5} + 33875739 T^{6} + 156779452 T^{7} + 5245514383 T^{8} + 23317169957 T^{9} + 641955466630 T^{10} + 2655079688665 T^{11} + 63398187090724 T^{12} + 236302092291185 T^{13} + 5084929251176230 T^{14} + 16437881987416333 T^{15} + 329115327587152303 T^{16} + 875465780349641948 T^{17} + 16835608500477895179 T^{18} + 33713344614046681445 T^{19} + \)\(65\!\cdots\!30\)\( T^{20} + \)\(81\!\cdots\!06\)\( T^{21} + \)\(17\!\cdots\!59\)\( T^{22} + \)\(83\!\cdots\!67\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 21 T + 781 T^{2} - 13992 T^{3} + 297410 T^{4} - 4532053 T^{5} + 72309901 T^{6} - 948480268 T^{7} + 12500242567 T^{8} - 143122686371 T^{9} + 1625460925398 T^{10} - 16402737491831 T^{11} + 163663794940716 T^{12} - 1459843636772959 T^{13} + 12875275990077558 T^{14} - 100897057088277499 T^{15} + 784293231697172647 T^{16} - 5296370202715452332 T^{17} + 35936667948242104861 T^{18} - \)\(20\!\cdots\!37\)\( T^{19} + \)\(11\!\cdots\!10\)\( T^{20} - \)\(49\!\cdots\!28\)\( T^{21} + \)\(24\!\cdots\!81\)\( T^{22} - \)\(58\!\cdots\!69\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 6 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( 1 + 23 T + 329 T^{2} + 3379 T^{3} + 31913 T^{4} + 216407 T^{5} + 912673 T^{6} \))(\( 1 - 6 T + 405 T^{2} - 2139 T^{3} + 71686 T^{4} - 303715 T^{5} + 6953542 T^{6} - 20125851 T^{7} + 369632565 T^{8} - 531175686 T^{9} + 8587340257 T^{10} \))(\( 1 - 26 T + 557 T^{2} - 8411 T^{3} + 110746 T^{4} - 1172915 T^{5} + 10742362 T^{6} - 79139099 T^{7} + 508358861 T^{8} - 2301761306 T^{9} + 8587340257 T^{10} \))(\( 1 + 57 T + 1999 T^{2} + 49485 T^{3} + 981253 T^{4} + 16180839 T^{5} + 233385980 T^{6} + 3001658219 T^{7} + 35423192688 T^{8} + 386010871872 T^{9} + 3937832201969 T^{10} + 37443054571584 T^{11} + 333296820001392 T^{12} + 2739532411709387 T^{13} + 20661493004880380 T^{14} + 138950370136735623 T^{15} + 817356278752596037 T^{16} + 3998303107399421805 T^{17} + 15667029755159545039 T^{18} + 43333170343310217369 T^{19} + 73742412689492826049 T^{20} \))(\( 1 + 70 T + 2935 T^{2} + 87469 T^{3} + 2060013 T^{4} + 39911519 T^{5} + 658581287 T^{6} + 9431186506 T^{7} + 119768221401 T^{8} + 1373580747687 T^{9} + 14608461788988 T^{10} + 147941064750717 T^{11} + 1466383547078142 T^{12} + 14350283280819549 T^{13} + 137451016972588092 T^{14} + 1253630061733737351 T^{15} + 10602994527279342681 T^{16} + 80988807554248972042 T^{17} + \)\(54\!\cdots\!23\)\( T^{18} + \)\(32\!\cdots\!47\)\( T^{19} + \)\(16\!\cdots\!93\)\( T^{20} + \)\(66\!\cdots\!73\)\( T^{21} + \)\(21\!\cdots\!15\)\( T^{22} + \)\(50\!\cdots\!10\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 18 T + 591 T^{2} - 8155 T^{3} + 166085 T^{4} - 1930089 T^{5} + 31869251 T^{6} - 331651870 T^{7} + 4795815957 T^{8} - 45773961233 T^{9} + 596253361156 T^{10} - 5240536538331 T^{11} + 62648823242718 T^{12} - 508332044218107 T^{13} + 5610147875116804 T^{14} - 41776658520405809 T^{15} + 424570138481536917 T^{16} - 2848007454560330590 T^{17} + 26546193901055538179 T^{18} - \)\(15\!\cdots\!57\)\( T^{19} + \)\(13\!\cdots\!85\)\( T^{20} - \)\(61\!\cdots\!35\)\( T^{21} + \)\(43\!\cdots\!59\)\( T^{22} - \)\(12\!\cdots\!54\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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