Properties

Label 161.2
Level 161
Weight 2
Dimension 901
Nonzero newspaces 8
Newform subspaces 15
Sturm bound 4224
Trace bound 2

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Defining parameters

Level: \( N \) = \( 161 = 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 15 \)
Sturm bound: \(4224\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(161))\).

Total New Old
Modular forms 1188 1113 75
Cusp forms 925 901 24
Eisenstein series 263 212 51

Trace form

\( 901q - 47q^{2} - 48q^{3} - 51q^{4} - 50q^{5} - 56q^{6} - 56q^{7} - 125q^{8} - 57q^{9} + O(q^{10}) \) \( 901q - 47q^{2} - 48q^{3} - 51q^{4} - 50q^{5} - 56q^{6} - 56q^{7} - 125q^{8} - 57q^{9} - 62q^{10} - 56q^{11} - 72q^{12} - 58q^{13} - 58q^{14} - 112q^{15} - 31q^{16} - 40q^{17} + 5q^{18} - 42q^{19} + 2q^{20} - 26q^{21} - 80q^{22} - 23q^{23} - 16q^{24} - 31q^{25} - 42q^{26} - 18q^{27} - 18q^{28} - 118q^{29} - 28q^{30} - 54q^{31} - 63q^{32} - 70q^{33} - 54q^{34} - 39q^{35} - 91q^{36} + 6q^{37} + 6q^{38} - 12q^{39} + 42q^{40} - 42q^{41} + 43q^{42} - 66q^{43} + 70q^{44} + 32q^{45} + 63q^{46} - 48q^{47} + 52q^{48} + 10q^{49} - 49q^{50} - 28q^{51} + 78q^{52} - 54q^{53} + 56q^{54} + 16q^{55} + 51q^{56} - 36q^{57} + 64q^{58} + 28q^{59} + 52q^{60} - 18q^{61} - 8q^{62} - 13q^{63} - 61q^{64} + 26q^{65} + 54q^{66} - 68q^{67} + 116q^{68} - 26q^{69} - 40q^{70} - 72q^{71} + 91q^{72} - 74q^{73} + 40q^{74} + 74q^{75} + 102q^{76} + 32q^{77} + 74q^{78} + 52q^{79} + 188q^{80} + 187q^{81} + 50q^{82} + 26q^{83} + 137q^{84} + 46q^{85} + 88q^{86} + 100q^{87} + 40q^{88} - 2q^{89} + 184q^{90} + 30q^{91} - 73q^{92} + 48q^{93} + 54q^{94} + 34q^{95} + 298q^{96} + 56q^{97} - 3q^{98} + 108q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(161))\)

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
161.2.a \(\chi_{161}(1, \cdot)\) 161.2.a.a 1 1
161.2.a.b 2
161.2.a.c 3
161.2.a.d 5
161.2.c \(\chi_{161}(160, \cdot)\) 161.2.c.a 2 1
161.2.c.b 12
161.2.e \(\chi_{161}(93, \cdot)\) 161.2.e.a 14 2
161.2.e.b 14
161.2.g \(\chi_{161}(45, \cdot)\) 161.2.g.a 28 2
161.2.i \(\chi_{161}(8, \cdot)\) 161.2.i.a 50 10
161.2.i.b 70
161.2.k \(\chi_{161}(20, \cdot)\) 161.2.k.a 20 10
161.2.k.b 120
161.2.m \(\chi_{161}(2, \cdot)\) 161.2.m.a 280 20
161.2.o \(\chi_{161}(5, \cdot)\) 161.2.o.a 280 20

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(161))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(161)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 3 T^{3} + 2 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 - 2 T + T^{2} + T^{3} + 2 T^{4} - 7 T^{5} + 4 T^{6} + 4 T^{7} + 8 T^{8} - 32 T^{9} + 32 T^{10} \))(\( ( 1 + T + 2 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + T^{3} + 4 T^{4} + 8 T^{6} )^{4} \))(\( 1 - 4 T^{2} + 2 T^{3} + 3 T^{4} - 3 T^{5} + 9 T^{6} - 20 T^{7} - 10 T^{8} + 69 T^{9} - 12 T^{10} - 56 T^{11} - 49 T^{12} - 14 T^{13} + 245 T^{14} - 28 T^{15} - 196 T^{16} - 448 T^{17} - 192 T^{18} + 2208 T^{19} - 640 T^{20} - 2560 T^{21} + 2304 T^{22} - 1536 T^{23} + 3072 T^{24} + 4096 T^{25} - 16384 T^{26} + 16384 T^{28} \))(\( 1 - 4 T^{2} + 2 T^{3} + 7 T^{4} - 11 T^{5} + 9 T^{6} + 28 T^{7} - 58 T^{8} - 39 T^{9} + 112 T^{10} + 16 T^{11} - 21 T^{12} + 2 T^{13} - 163 T^{14} + 4 T^{15} - 84 T^{16} + 128 T^{17} + 1792 T^{18} - 1248 T^{19} - 3712 T^{20} + 3584 T^{21} + 2304 T^{22} - 5632 T^{23} + 7168 T^{24} + 4096 T^{25} - 16384 T^{26} + 16384 T^{28} \))(\( ( 1 - T - T^{2} + 3 T^{3} - T^{4} - 5 T^{5} + 7 T^{6} + 3 T^{7} - 17 T^{8} + 11 T^{9} + 23 T^{10} + 22 T^{11} - 68 T^{12} + 24 T^{13} + 112 T^{14} - 160 T^{15} - 64 T^{16} + 384 T^{17} - 256 T^{18} - 512 T^{19} + 1024 T^{20} )^{2} \))
$3$ (\( 1 + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 - 2 T + 7 T^{2} - 10 T^{3} + 21 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T^{2} + 11 T^{4} + 10 T^{5} + 33 T^{6} + 54 T^{8} + 243 T^{10} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 22 T^{4} - 34 T^{6} + 198 T^{8} - 162 T^{10} + 729 T^{12} )^{2} \))(\( 1 + 5 T + 4 T^{2} - 25 T^{3} - 61 T^{4} + 7 T^{5} + 142 T^{6} - 9 T^{7} - 229 T^{8} + 758 T^{9} + 2153 T^{10} - 998 T^{11} - 7446 T^{12} - 1718 T^{13} + 14743 T^{14} - 5154 T^{15} - 67014 T^{16} - 26946 T^{17} + 174393 T^{18} + 184194 T^{19} - 166941 T^{20} - 19683 T^{21} + 931662 T^{22} + 137781 T^{23} - 3601989 T^{24} - 4428675 T^{25} + 2125764 T^{26} + 7971615 T^{27} + 4782969 T^{28} \))(\( 1 - 3 T + 11 T^{3} - 17 T^{4} + 23 T^{5} + 26 T^{6} - 269 T^{7} + 399 T^{8} + 290 T^{9} - 1251 T^{10} + 2098 T^{11} - 2414 T^{12} - 5022 T^{13} + 21255 T^{14} - 15066 T^{15} - 21726 T^{16} + 56646 T^{17} - 101331 T^{18} + 70470 T^{19} + 290871 T^{20} - 588303 T^{21} + 170586 T^{22} + 452709 T^{23} - 1003833 T^{24} + 1948617 T^{25} - 4782969 T^{27} + 4782969 T^{28} \))(\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} + 243 T^{10} + 729 T^{12} + 2187 T^{14} + 6561 T^{16} + 19683 T^{18} + 59049 T^{20} )^{2} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 13 T^{2} - 18 T^{3} + 65 T^{4} - 50 T^{5} + 125 T^{6} \))(\( 1 + 4 T + 11 T^{2} + 26 T^{3} + 92 T^{4} + 228 T^{5} + 460 T^{6} + 650 T^{7} + 1375 T^{8} + 2500 T^{9} + 3125 T^{10} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 83 T^{4} + 468 T^{6} + 2075 T^{8} + 6250 T^{10} + 15625 T^{12} )^{2} \))(\( 1 + 4 T - 8 T^{2} - 28 T^{3} + 92 T^{4} + 65 T^{5} - 720 T^{6} + 771 T^{7} + 4258 T^{8} - 6494 T^{9} - 11273 T^{10} + 30342 T^{11} - 8135 T^{12} - 49280 T^{13} + 232271 T^{14} - 246400 T^{15} - 203375 T^{16} + 3792750 T^{17} - 7045625 T^{18} - 20293750 T^{19} + 66531250 T^{20} + 60234375 T^{21} - 281250000 T^{22} + 126953125 T^{23} + 898437500 T^{24} - 1367187500 T^{25} - 1953125000 T^{26} + 4882812500 T^{27} + 6103515625 T^{28} \))(\( 1 - 4 T - 12 T^{2} + 56 T^{3} + 106 T^{4} - 455 T^{5} - 792 T^{6} + 2899 T^{7} + 4114 T^{8} - 14064 T^{9} - 16209 T^{10} + 51098 T^{11} + 53173 T^{12} - 100008 T^{13} - 178249 T^{14} - 500040 T^{15} + 1329325 T^{16} + 6387250 T^{17} - 10130625 T^{18} - 43950000 T^{19} + 64281250 T^{20} + 226484375 T^{21} - 309375000 T^{22} - 888671875 T^{23} + 1035156250 T^{24} + 2734375000 T^{25} - 2929687500 T^{26} - 4882812500 T^{27} + 6103515625 T^{28} \))(\( ( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2} \))
$7$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))(\( 1 + 7 T^{2} \))(\( 1 + 4 T^{2} + 131 T^{4} + 336 T^{6} + 6419 T^{8} + 9604 T^{10} + 117649 T^{12} \))(\( 1 + 2 T + 12 T^{2} + 24 T^{3} + 45 T^{4} + 172 T^{5} + 213 T^{6} + 1319 T^{7} + 1491 T^{8} + 8428 T^{9} + 15435 T^{10} + 57624 T^{11} + 201684 T^{12} + 235298 T^{13} + 823543 T^{14} \))(\( 1 + 8 T^{2} + 16 T^{3} + 51 T^{4} + 146 T^{5} + 609 T^{6} + 519 T^{7} + 4263 T^{8} + 7154 T^{9} + 17493 T^{10} + 38416 T^{11} + 134456 T^{12} + 823543 T^{14} \))(\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} - 16807 T^{10} + 117649 T^{12} - 823543 T^{14} + 5764801 T^{16} - 40353607 T^{18} + 282475249 T^{20} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T^{2} + 121 T^{4} \))(\( 1 - 4 T + 33 T^{2} - 84 T^{3} + 363 T^{4} - 484 T^{5} + 1331 T^{6} \))(\( 1 + 4 T + 27 T^{2} + 28 T^{3} + 126 T^{4} - 400 T^{5} + 1386 T^{6} + 3388 T^{7} + 35937 T^{8} + 58564 T^{9} + 161051 T^{10} \))(\( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) \))(\( ( 1 - 20 T^{2} + 475 T^{4} - 5024 T^{6} + 57475 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2} \))(\( 1 + 4 T - 18 T^{2} + 32 T^{3} + 591 T^{4} - 1120 T^{5} - 1932 T^{6} + 34335 T^{7} - 46026 T^{8} - 264102 T^{9} + 1042918 T^{10} - 357430 T^{11} - 9574002 T^{12} - 1110479 T^{13} + 2379645 T^{14} - 12215269 T^{15} - 1158454242 T^{16} - 475739330 T^{17} + 15269362438 T^{18} - 42533891202 T^{19} - 81537866586 T^{20} + 669092016285 T^{21} - 414141358092 T^{22} - 2640901413920 T^{23} + 15329017939191 T^{24} + 9129973459552 T^{25} - 56491710780978 T^{26} + 138090848575724 T^{27} + 379749833583241 T^{28} \))(\( 1 - 52 T^{2} + 36 T^{3} + 1441 T^{4} - 1720 T^{5} - 26298 T^{6} + 45909 T^{7} + 350856 T^{8} - 767928 T^{9} - 3622184 T^{10} + 8195738 T^{11} + 32656658 T^{12} - 37624149 T^{13} - 315857827 T^{14} - 413865639 T^{15} + 3951455618 T^{16} + 10908527278 T^{17} - 53032395944 T^{18} - 123675572328 T^{19} + 621562806216 T^{20} + 894636533439 T^{21} - 5637209852538 T^{22} - 4055670028520 T^{23} + 37375828850041 T^{24} + 10271220141996 T^{25} - 163198275589492 T^{26} + 379749833583241 T^{28} \))(\( ( 1 - 4 T + 5 T^{2} + 24 T^{3} - 151 T^{4} + 340 T^{5} + 301 T^{6} - 4944 T^{7} + 16465 T^{8} - 11476 T^{9} - 135211 T^{10} - 126236 T^{11} + 1992265 T^{12} - 6580464 T^{13} + 4406941 T^{14} + 54757340 T^{15} - 267505711 T^{16} + 467692104 T^{17} + 1071794405 T^{18} - 9431790764 T^{19} + 25937424601 T^{20} )( 1 + 4 T + 5 T^{2} - 24 T^{3} - 151 T^{4} - 340 T^{5} + 301 T^{6} + 4944 T^{7} + 16465 T^{8} + 11476 T^{9} - 135211 T^{10} + 126236 T^{11} + 1992265 T^{12} + 6580464 T^{13} + 4406941 T^{14} - 54757340 T^{15} - 267505711 T^{16} - 467692104 T^{17} + 1071794405 T^{18} + 9431790764 T^{19} + 25937424601 T^{20} ) \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 27 T^{2} - 44 T^{3} + 351 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 + 6 T + 56 T^{2} + 266 T^{3} + 1351 T^{4} + 4944 T^{5} + 17563 T^{6} + 44954 T^{7} + 123032 T^{8} + 171366 T^{9} + 371293 T^{10} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 26 T^{2} + 510 T^{4} - 8402 T^{6} + 86190 T^{8} - 742586 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 14 T + 133 T^{2} - 925 T^{3} + 5433 T^{4} - 26920 T^{5} + 117293 T^{6} - 447265 T^{7} + 1524809 T^{8} - 4549480 T^{9} + 11936301 T^{10} - 26418925 T^{11} + 49381969 T^{12} - 67575326 T^{13} + 62748517 T^{14} )^{2} \))(\( ( 1 + 18 T + 213 T^{2} + 1779 T^{3} + 11937 T^{4} + 65092 T^{5} + 300509 T^{6} + 1169999 T^{7} + 3906617 T^{8} + 11000548 T^{9} + 26225589 T^{10} + 50810019 T^{11} + 79085409 T^{12} + 86882562 T^{13} + 62748517 T^{14} )^{2} \))(\( ( 1 + 13 T^{2} + 169 T^{4} + 2197 T^{6} + 28561 T^{8} + 371293 T^{10} + 4826809 T^{12} + 62748517 T^{14} + 815730721 T^{16} + 10604499373 T^{18} + 137858491849 T^{20} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 - 4 T + 53 T^{2} - 134 T^{3} + 901 T^{4} - 1156 T^{5} + 4913 T^{6} \))(\( 1 + 12 T + 91 T^{2} + 430 T^{3} + 1692 T^{4} + 6148 T^{5} + 28764 T^{6} + 124270 T^{7} + 447083 T^{8} + 1002252 T^{9} + 1419857 T^{10} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 939 T^{4} + 11264 T^{6} + 271371 T^{8} + 1670420 T^{10} + 24137569 T^{12} )^{2} \))(\( 1 + 4 T - 77 T^{2} - 240 T^{3} + 3441 T^{4} + 7419 T^{5} - 114250 T^{6} - 153483 T^{7} + 3073148 T^{8} + 2307077 T^{9} - 69904299 T^{10} - 24708984 T^{11} + 1391359660 T^{12} + 133904631 T^{13} - 24872514751 T^{14} + 2276378727 T^{15} + 402102941740 T^{16} - 121395238392 T^{17} - 5838476956779 T^{18} + 3275719427989 T^{19} + 74178321897212 T^{20} - 62980010548059 T^{21} - 796980287634250 T^{22} + 879803455731243 T^{23} + 6937035011445009 T^{24} - 8225255113831920 T^{25} - 44861912266691597 T^{26} + 39618312131623748 T^{27} + 168377826559400929 T^{28} \))(\( 1 - 14 T + 41 T^{2} + 262 T^{3} - 1325 T^{4} - 1987 T^{5} + 6886 T^{6} + 10947 T^{7} + 610452 T^{8} - 2018165 T^{9} - 15692215 T^{10} + 66099152 T^{11} + 124763236 T^{12} - 494866469 T^{13} - 1079368531 T^{14} - 8412729973 T^{15} + 36056575204 T^{16} + 324745133776 T^{17} - 1310629489015 T^{18} - 2865505702405 T^{19} + 14734827271188 T^{20} + 4491977453331 T^{21} + 48035065738726 T^{22} - 235634110599539 T^{23} - 2671191918094925 T^{24} + 8979236832599846 T^{25} + 23887511726420201 T^{26} - 138664092460683118 T^{27} + 168377826559400929 T^{28} \))(\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} - 1419857 T^{10} + 24137569 T^{12} - 410338673 T^{14} + 6975757441 T^{16} - 118587876497 T^{18} + 2015993900449 T^{20} )^{2} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 10 T + 58 T^{2} + 190 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 144 T^{3} + 779 T^{4} - 2888 T^{5} + 6859 T^{6} \))(\( 1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 44346 T^{6} - 129960 T^{7} + 459553 T^{8} - 781926 T^{9} + 2476099 T^{10} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 56 T^{2} + 1339 T^{4} + 23720 T^{6} + 483379 T^{8} + 7297976 T^{10} + 47045881 T^{12} )^{2} \))(\( 1 + 9 T - 29 T^{2} - 556 T^{3} - 513 T^{4} + 14653 T^{5} + 32536 T^{6} - 254940 T^{7} - 602264 T^{8} + 4633279 T^{9} + 10426785 T^{10} - 78118735 T^{11} - 262652449 T^{12} + 611975728 T^{13} + 5791355975 T^{14} + 11627538832 T^{15} - 94817534089 T^{16} - 535816403365 T^{17} + 1358829047985 T^{18} + 11472457498621 T^{19} - 28334040474584 T^{20} - 227883661140660 T^{21} + 552577207101976 T^{22} + 4728342835555687 T^{23} - 3145236990251913 T^{24} - 64768583947409764 T^{25} - 64186132652918669 T^{26} + 378476851160313531 T^{27} + 799006685782884121 T^{28} \))(\( 1 - 11 T - 7 T^{2} + 410 T^{3} - 173 T^{4} - 7563 T^{5} - 15602 T^{6} + 146912 T^{7} + 787356 T^{8} - 2954863 T^{9} - 21499627 T^{10} + 57069141 T^{11} + 393966665 T^{12} - 555745860 T^{13} - 6293183029 T^{14} - 10559171340 T^{15} + 142221966065 T^{16} + 391437238119 T^{17} - 2801852890267 T^{18} - 7316533319437 T^{19} + 37041856680636 T^{20} + 131320484919968 T^{21} - 264977550565682 T^{22} - 2440487058302577 T^{23} - 1060674462599573 T^{24} + 47761006148269790 T^{25} - 15493204433463127 T^{26} - 462582818084827649 T^{27} + 799006685782884121 T^{28} \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} - 2476099 T^{10} + 47045881 T^{12} - 893871739 T^{14} + 16983563041 T^{16} - 322687697779 T^{18} + 6131066257801 T^{20} )^{2} \))
$23$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{5} \))(\( 1 - 8 T + 23 T^{2} \))(\( ( 1 + 4 T + 25 T^{2} + 216 T^{3} + 575 T^{4} + 2116 T^{5} + 12167 T^{6} )^{2} \))(\( ( 1 + T + T^{2} )^{7} \))(\( ( 1 - T + T^{2} )^{7} \))(\( 1 + 8 T + 41 T^{2} + 144 T^{3} + 209 T^{4} - 1640 T^{5} - 17927 T^{6} - 105696 T^{7} - 433247 T^{8} - 1034968 T^{9} + 1684937 T^{10} - 23804264 T^{11} - 229187663 T^{12} - 1286003232 T^{13} - 5016709607 T^{14} - 10555602520 T^{15} + 30939500801 T^{16} + 490294864368 T^{17} + 3210750396521 T^{18} + 14409221291704 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 2 T + 79 T^{2} + 120 T^{3} + 2291 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 + 4 T + 34 T^{2} + 214 T^{3} + 1217 T^{4} + 4232 T^{5} + 35293 T^{6} + 179974 T^{7} + 829226 T^{8} + 2829124 T^{9} + 20511149 T^{10} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 80 T^{2} - 206 T^{3} + 2320 T^{4} - 3364 T^{5} + 24389 T^{6} )^{4} \))(\( ( 1 + 5 T + 75 T^{2} + 424 T^{3} + 3013 T^{4} + 13199 T^{5} + 96131 T^{6} + 299981 T^{7} + 2787799 T^{8} + 11100359 T^{9} + 73484057 T^{10} + 299887144 T^{11} + 1538336175 T^{12} + 2974116605 T^{13} + 17249876309 T^{14} )^{2} \))(\( ( 1 - 3 T + 71 T^{2} - 256 T^{3} + 3961 T^{4} - 11449 T^{5} + 149151 T^{6} - 452943 T^{7} + 4325379 T^{8} - 9628609 T^{9} + 96604829 T^{10} - 181063936 T^{11} + 1456291579 T^{12} - 1784469963 T^{13} + 17249876309 T^{14} )^{2} \))(\( ( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} + 4150 T^{5} - 6461 T^{6} - 133272 T^{7} - 79175 T^{8} + 3706538 T^{9} + 9709151 T^{10} + 107489602 T^{11} - 66586175 T^{12} - 3250370808 T^{13} - 4569742541 T^{14} + 85121268350 T^{15} + 302765070389 T^{16} - 1862986641372 T^{17} - 12506160324025 T^{18} + 29014291951738 T^{19} + 420707233300201 T^{20} )^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 + 9 T + 31 T^{2} )^{2} \))(\( 1 - 16 T + 119 T^{2} - 654 T^{3} + 3689 T^{4} - 15376 T^{5} + 29791 T^{6} \))(\( 1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 1456845 T^{6} - 5425806 T^{7} + 14955082 T^{8} - 27705630 T^{9} + 28629151 T^{10} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 12510 T^{4} - 508346 T^{6} + 12022110 T^{8} - 156998570 T^{10} + 887503681 T^{12} )^{2} \))(\( 1 + 10 T - 107 T^{2} - 1394 T^{3} + 6739 T^{4} + 111131 T^{5} - 251194 T^{6} - 5775167 T^{7} + 6438738 T^{8} + 216282649 T^{9} - 85995919 T^{10} - 5475709110 T^{11} + 246763196 T^{12} + 66827032119 T^{13} + 30777004867 T^{14} + 2071637995689 T^{15} + 237139431356 T^{16} - 163126850096010 T^{17} - 79419037110799 T^{18} + 6191988616900999 T^{19} + 5714403675994578 T^{20} - 158889941097581537 T^{21} - 214241111258954554 T^{22} + 2938261650337528901 T^{23} + 5523475025963617939 T^{24} - 35419416793588334414 T^{25} - 84279917865374824427 T^{26} + \)\(24\!\cdots\!10\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))(\( 1 - 6 T - 87 T^{2} + 654 T^{3} + 2715 T^{4} - 27449 T^{5} - 40930 T^{6} + 572717 T^{7} + 1272266 T^{8} - 1158263 T^{9} - 97958843 T^{10} - 445156934 T^{11} + 6552145616 T^{12} + 10013852583 T^{13} - 269758420421 T^{14} + 310429430073 T^{15} + 6296611936976 T^{16} - 13261670220794 T^{17} - 90467048646203 T^{18} - 33160086324713 T^{19} + 1129140758211146 T^{20} + 15756941815809587 T^{21} - 34908830162460130 T^{22} - 725741188688258279 T^{23} + 2225290799152874715 T^{24} + 16617143890248759474 T^{25} - 68526662189603829207 T^{26} - \)\(14\!\cdots\!46\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))(\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} + 28629151 T^{10} + 887503681 T^{12} + 27512614111 T^{14} + 852891037441 T^{16} + 26439622160671 T^{18} + 819628286980801 T^{20} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 30 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 107 T^{2} + 404 T^{3} + 3959 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 - 4 T + 109 T^{2} - 216 T^{3} + 5126 T^{4} - 5064 T^{5} + 189662 T^{6} - 295704 T^{7} + 5521177 T^{8} - 7496644 T^{9} + 69343957 T^{10} \))(\( ( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} ) \))(\( ( 1 - 90 T^{2} + 5051 T^{4} - 215668 T^{6} + 6914819 T^{8} - 168674490 T^{10} + 2565726409 T^{12} )^{2} \))(\( 1 - 5 T - 94 T^{2} + 829 T^{3} + 1911 T^{4} - 47881 T^{5} + 116847 T^{6} + 1268124 T^{7} - 7660170 T^{8} - 8270520 T^{9} + 192664898 T^{10} - 636988698 T^{11} - 1611700 T^{12} + 17476209359 T^{13} - 140648186317 T^{14} + 646619746283 T^{15} - 2206417300 T^{16} - 32265388519794 T^{17} + 361085037900578 T^{18} - 573510583247640 T^{19} - 19653900466429530 T^{20} + 120385391757408492 T^{21} + 410422686752307087 T^{22} - 6222698063128081837 T^{23} + 9189204735690509439 T^{24} + \)\(14\!\cdots\!77\)\( T^{25} - \)\(61\!\cdots\!14\)\( T^{26} - \)\(12\!\cdots\!85\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))(\( 1 - T - 136 T^{2} - 313 T^{3} + 9531 T^{4} + 48281 T^{5} - 396511 T^{6} - 3417806 T^{7} + 8961444 T^{8} + 162230828 T^{9} + 119557092 T^{10} - 5415877114 T^{11} - 23513831478 T^{12} + 82020758267 T^{13} + 1216114109755 T^{14} + 3034768055879 T^{15} - 32190435293382 T^{16} - 274330423455442 T^{17} + 224069239099812 T^{18} + 11249727560906396 T^{19} + 22992613533574596 T^{20} - 324458739256430198 T^{21} - 1392736740753669631 T^{22} + 6274682759046112637 T^{23} + 45830617653514518819 T^{24} - 55688215616971109269 T^{25} - \)\(89\!\cdots\!16\)\( T^{26} - \)\(24\!\cdots\!97\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))(\( ( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} - 450 T^{5} + 51947 T^{6} - 295032 T^{7} - 151847 T^{8} + 11827266 T^{9} - 65345257 T^{10} + 437608842 T^{11} - 207878543 T^{12} - 14944255896 T^{13} + 97357041467 T^{14} - 31204780650 T^{15} - 3414981850379 T^{16} + 21644467986324 T^{17} - 3512479453921 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} )( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} + 450 T^{5} + 51947 T^{6} + 295032 T^{7} - 151847 T^{8} - 11827266 T^{9} - 65345257 T^{10} - 437608842 T^{11} - 207878543 T^{12} + 14944255896 T^{13} + 97357041467 T^{14} + 31204780650 T^{15} - 3414981850379 T^{16} - 21644467986324 T^{17} - 3512479453921 T^{18} + 779770438770462 T^{19} + 4808584372417849 T^{20} ) \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 77 T^{2} + 1681 T^{4} \))(\( 1 + 14 T + 135 T^{2} + 996 T^{3} + 5535 T^{4} + 23534 T^{5} + 68921 T^{6} \))(\( 1 - 6 T + 176 T^{2} - 838 T^{3} + 13295 T^{4} - 49000 T^{5} + 545095 T^{6} - 1408678 T^{7} + 12130096 T^{8} - 16954566 T^{9} + 115856201 T^{10} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 90 T^{2} + 4070 T^{4} - 167146 T^{6} + 6841670 T^{8} - 254318490 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 23 T + 431 T^{2} - 5502 T^{3} + 61070 T^{4} - 546827 T^{5} + 4377065 T^{6} - 29524413 T^{7} + 179459665 T^{8} - 919216187 T^{9} + 4209005470 T^{10} - 15547337022 T^{11} + 49934022631 T^{12} - 109252397543 T^{13} + 194754273881 T^{14} )^{2} \))(\( ( 1 + 9 T + 163 T^{2} + 866 T^{3} + 9646 T^{4} + 22845 T^{5} + 295945 T^{6} + 182459 T^{7} + 12133745 T^{8} + 38402445 T^{9} + 664811966 T^{10} + 2447109026 T^{11} + 18884560763 T^{12} + 42750938169 T^{13} + 194754273881 T^{14} )^{2} \))(\( ( 1 + 41 T^{2} + 1681 T^{4} + 68921 T^{6} + 2825761 T^{8} + 115856201 T^{10} + 4750104241 T^{12} + 194754273881 T^{14} + 7984925229121 T^{16} + 327381934393961 T^{18} + 13422659310152401 T^{20} )^{2} \))
$43$ (\( 1 - 12 T + 43 T^{2} \))(\( 1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 89 T^{2} + 384 T^{3} + 3827 T^{4} + 14792 T^{5} + 79507 T^{6} \))(\( 1 + 12 T + 191 T^{2} + 1696 T^{3} + 16226 T^{4} + 101736 T^{5} + 697718 T^{6} + 3135904 T^{7} + 15185837 T^{8} + 41025612 T^{9} + 147008443 T^{10} \))(\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 - 86 T^{2} + 6087 T^{4} - 312500 T^{6} + 11254863 T^{8} - 294016886 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 11 T + 246 T^{2} + 2471 T^{3} + 29362 T^{4} + 237647 T^{5} + 2072803 T^{6} + 13048507 T^{7} + 89130529 T^{8} + 439409303 T^{9} + 2334484534 T^{10} + 8447857271 T^{11} + 36164076978 T^{12} + 69534993539 T^{13} + 271818611107 T^{14} )^{2} \))(\( ( 1 - 9 T + 138 T^{2} - 1327 T^{3} + 12682 T^{4} - 96363 T^{5} + 772413 T^{6} - 4889401 T^{7} + 33213759 T^{8} - 178175187 T^{9} + 1008307774 T^{10} - 4536748927 T^{11} + 20287165134 T^{12} - 56892267441 T^{13} + 271818611107 T^{14} )^{2} \))(\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 18180 T^{5} + 45773 T^{6} + 232464 T^{7} - 4757807 T^{8} + 47097732 T^{9} - 360587083 T^{10} + 2025202476 T^{11} - 8797185143 T^{12} + 18482515248 T^{13} + 156488778173 T^{14} - 2672613493740 T^{15} + 25342344463441 T^{16} - 189185753330472 T^{17} + 1180508228037701 T^{18} - 6031111343242116 T^{19} + 21611482313284249 T^{20} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 18180 T^{5} + 45773 T^{6} - 232464 T^{7} - 4757807 T^{8} - 47097732 T^{9} - 360587083 T^{10} - 2025202476 T^{11} - 8797185143 T^{12} - 18482515248 T^{13} + 156488778173 T^{14} + 2672613493740 T^{15} + 25342344463441 T^{16} + 189185753330472 T^{17} + 1180508228037701 T^{18} + 6031111343242116 T^{19} + 21611482313284249 T^{20} ) \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 - 2 T + 75 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 16 T + 203 T^{2} - 1514 T^{3} + 9541 T^{4} - 35344 T^{5} + 103823 T^{6} \))(\( 1 - 10 T + 110 T^{2} + 38 T^{3} - 3609 T^{4} + 58894 T^{5} - 169623 T^{6} + 83942 T^{7} + 11420530 T^{8} - 48796810 T^{9} + 229345007 T^{10} \))(\( ( 1 - 47 T^{2} )^{2} \))(\( ( 1 - 146 T^{2} + 13150 T^{4} - 737354 T^{6} + 29048350 T^{8} - 712433426 T^{10} + 10779215329 T^{12} )^{2} \))(\( 1 + 4 T - 164 T^{2} + 456 T^{3} + 17694 T^{4} - 112113 T^{5} - 869164 T^{6} + 11211837 T^{7} + 10001108 T^{8} - 619647148 T^{9} + 1640363499 T^{10} + 20810538216 T^{11} - 144860495795 T^{12} - 360990177594 T^{13} + 7359717751649 T^{14} - 16966538346918 T^{15} - 319996835211155 T^{16} + 2160612509199768 T^{17} + 8004450599163819 T^{18} - 142112979495590036 T^{19} + 107804096660584532 T^{20} + 5680175847062520531 T^{21} - 20695913160082837804 T^{22} - \)\(12\!\cdots\!71\)\( T^{23} + \)\(93\!\cdots\!06\)\( T^{24} + \)\(11\!\cdots\!68\)\( T^{25} - \)\(19\!\cdots\!24\)\( T^{26} + \)\(21\!\cdots\!08\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))(\( 1 + 12 T - 112 T^{2} - 1008 T^{3} + 16326 T^{4} + 75923 T^{5} - 1105412 T^{6} - 592667 T^{7} + 58749604 T^{8} - 105560376 T^{9} - 725627737 T^{10} + 12330443508 T^{11} - 55906514127 T^{12} - 171905350342 T^{13} + 6197753904793 T^{14} - 8079551466074 T^{15} - 123497489706543 T^{16} + 1280183636331084 T^{17} - 3540831881311897 T^{18} - 24209745172642632 T^{19} + 633274632009479716 T^{20} - 300258804935444821 T^{21} - 26321282011350550532 T^{22} + 84967742909381378941 T^{23} + \)\(85\!\cdots\!74\)\( T^{24} - \)\(24\!\cdots\!24\)\( T^{25} - \)\(13\!\cdots\!92\)\( T^{26} + \)\(65\!\cdots\!24\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))(\( ( 1 - 47 T^{2} )^{20} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4} \))(\( 1 - 6 T + 107 T^{2} - 388 T^{3} + 5671 T^{4} - 16854 T^{5} + 148877 T^{6} \))(\( 1 - 16 T + 317 T^{2} - 3240 T^{3} + 35942 T^{4} - 254032 T^{5} + 1904926 T^{6} - 9101160 T^{7} + 47194009 T^{8} - 126247696 T^{9} + 418195493 T^{10} \))(\( ( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} ) \))(\( ( 1 - 116 T^{2} + 2723 T^{4} + 78288 T^{6} + 7648907 T^{8} - 915295796 T^{10} + 22164361129 T^{12} )^{2} \))(\( 1 + 2 T - 245 T^{2} - 828 T^{3} + 30805 T^{4} + 130448 T^{5} - 2583247 T^{6} - 11994019 T^{7} + 167182393 T^{8} + 722276129 T^{9} - 9584262354 T^{10} - 28775017956 T^{11} + 534148041103 T^{12} + 551780317534 T^{13} - 28996401021379 T^{14} + 29244356829302 T^{15} + 1500421847458327 T^{16} - 4283938348235412 T^{17} - 75624440003252274 T^{18} + 302052621849286597 T^{19} + 3705490932862401697 T^{20} - 14089507730716634903 T^{21} - \)\(16\!\cdots\!67\)\( T^{22} + \)\(43\!\cdots\!84\)\( T^{23} + \)\(53\!\cdots\!45\)\( T^{24} - \)\(76\!\cdots\!16\)\( T^{25} - \)\(12\!\cdots\!45\)\( T^{26} + \)\(52\!\cdots\!46\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))(\( 1 - 12 T - 111 T^{2} + 1770 T^{3} + 4763 T^{4} - 107866 T^{5} - 252317 T^{6} + 3706549 T^{7} + 29889405 T^{8} - 167715329 T^{9} - 2123968506 T^{10} + 12717375734 T^{11} + 59146498711 T^{12} - 393869096868 T^{13} - 434221655775 T^{14} - 20875062134004 T^{15} + 166142514879199 T^{16} + 1893324747150718 T^{17} - 16759133141191386 T^{18} - 70137794694812197 T^{19} + 662479566350938245 T^{20} + 4354124400651692513 T^{21} - 15709178305523373437 T^{22} - \)\(35\!\cdots\!78\)\( T^{23} + \)\(83\!\cdots\!87\)\( T^{24} + \)\(16\!\cdots\!90\)\( T^{25} - \)\(54\!\cdots\!51\)\( T^{26} - \)\(31\!\cdots\!76\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))(\( ( 1 - 10 T + 47 T^{2} + 60 T^{3} - 3091 T^{4} + 27730 T^{5} - 113477 T^{6} - 334920 T^{7} + 9363481 T^{8} - 75884050 T^{9} + 262576007 T^{10} - 4021854650 T^{11} + 26302018129 T^{12} - 49861884840 T^{13} - 895388112437 T^{14} + 11596561020890 T^{15} - 68510040249739 T^{16} + 70482668390220 T^{17} + 2926205449333967 T^{18} - 32997635918021330 T^{19} + 174887470365513049 T^{20} )( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 27730 T^{5} - 113477 T^{6} + 334920 T^{7} + 9363481 T^{8} + 75884050 T^{9} + 262576007 T^{10} + 4021854650 T^{11} + 26302018129 T^{12} + 49861884840 T^{13} - 895388112437 T^{14} - 11596561020890 T^{15} - 68510040249739 T^{16} - 70482668390220 T^{17} + 2926205449333967 T^{18} + 32997635918021330 T^{19} + 174887470365513049 T^{20} ) \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 12 T + 134 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 75 T^{2} + 210 T^{3} + 4425 T^{4} + 34810 T^{5} + 205379 T^{6} \))(\( 1 - 22 T + 413 T^{2} - 5194 T^{3} + 54600 T^{4} - 458288 T^{5} + 3221400 T^{6} - 18080314 T^{7} + 84821527 T^{8} - 266581942 T^{9} + 714924299 T^{10} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 17879 T^{4} - 1245996 T^{6} + 62236799 T^{8} - 2059951370 T^{10} + 42180533641 T^{12} )^{2} \))(\( 1 + 7 T - 115 T^{2} - 442 T^{3} + 9461 T^{4} + 2499 T^{5} - 47206 T^{6} + 4111324 T^{7} - 34690852 T^{8} - 410269739 T^{9} + 2359628105 T^{10} + 18572701115 T^{11} - 40315073743 T^{12} - 563058208084 T^{13} - 3784009218829 T^{14} - 33220434276956 T^{15} - 140336771699383 T^{16} + 3814442782297585 T^{17} + 28592465574030905 T^{18} - 293311805555487961 T^{19} - 1463278649820952132 T^{20} + 10231652577171990356 T^{21} - 6931277637549577126 T^{22} + 21648826550818692561 T^{23} + \)\(48\!\cdots\!61\)\( T^{24} - \)\(13\!\cdots\!78\)\( T^{25} - \)\(20\!\cdots\!15\)\( T^{26} + \)\(73\!\cdots\!53\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))(\( 1 + 7 T - 267 T^{2} - 2162 T^{3} + 36689 T^{4} + 313339 T^{5} - 3578046 T^{6} - 27072380 T^{7} + 309721300 T^{8} + 1505036857 T^{9} - 26509887691 T^{10} - 53622868517 T^{11} + 2098788276497 T^{12} + 948384586584 T^{13} - 138261874028533 T^{14} + 55954690608456 T^{15} + 7305881990486057 T^{16} - 11013011113152943 T^{17} - 321229879221303451 T^{18} + 1075987419959888243 T^{19} + 13064209713984253300 T^{20} - 67373718684584199220 T^{21} - \)\(52\!\cdots\!66\)\( T^{22} + \)\(27\!\cdots\!21\)\( T^{23} + \)\(18\!\cdots\!89\)\( T^{24} - \)\(65\!\cdots\!58\)\( T^{25} - \)\(47\!\cdots\!27\)\( T^{26} + \)\(73\!\cdots\!53\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 10 T + 137 T^{2} - 726 T^{3} + 8357 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 2668384 T^{6} + 14712834 T^{7} + 76946559 T^{8} + 249225138 T^{9} + 844596301 T^{10} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 122 T^{2} + 55 T^{4} - 449044 T^{6} + 204655 T^{8} + 1689192602 T^{10} + 51520374361 T^{12} )^{2} \))(\( 1 + 12 T - 163 T^{2} - 2032 T^{3} + 16685 T^{4} + 166935 T^{5} - 1450434 T^{6} - 7995269 T^{7} + 115046590 T^{8} + 212062833 T^{9} - 7911419927 T^{10} + 781512232 T^{11} + 507494067062 T^{12} - 117238411909 T^{13} - 30876284204843 T^{14} - 7151543126449 T^{15} + 1888385423537702 T^{16} + 177388427931592 T^{17} - 109540262393473607 T^{18} + 179107484331380733 T^{19} + 5927243385756478990 T^{20} - 25127074371810784649 T^{21} - \)\(27\!\cdots\!54\)\( T^{22} + \)\(19\!\cdots\!35\)\( T^{23} + \)\(11\!\cdots\!85\)\( T^{24} - \)\(88\!\cdots\!52\)\( T^{25} - \)\(43\!\cdots\!23\)\( T^{26} + \)\(19\!\cdots\!72\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))(\( 1 - 24 T + 67 T^{2} + 2688 T^{3} - 17035 T^{4} - 157397 T^{5} + 1117646 T^{6} + 6824923 T^{7} + 1198162 T^{8} - 584222045 T^{9} - 3448342057 T^{10} + 43489283118 T^{11} + 219332621356 T^{12} - 1196365340965 T^{13} - 12101653380135 T^{14} - 72978285798865 T^{15} + 816136684065676 T^{16} + 9871240971406758 T^{17} - 47745195834834937 T^{18} - 493431778169655545 T^{19} + 61729754785124482 T^{20} + 21448977864644951383 T^{21} + \)\(21\!\cdots\!26\)\( T^{22} - \)\(18\!\cdots\!77\)\( T^{23} - \)\(12\!\cdots\!35\)\( T^{24} + \)\(11\!\cdots\!68\)\( T^{25} + \)\(17\!\cdots\!07\)\( T^{26} - \)\(38\!\cdots\!44\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} - 844596301 T^{10} + 51520374361 T^{12} - 3142742836021 T^{14} + 191707312997281 T^{16} - 11694146092834141 T^{18} + 713342911662882601 T^{20} )^{2} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 16 T + 193 T^{2} - 1468 T^{3} + 12931 T^{4} - 71824 T^{5} + 300763 T^{6} \))(\( 1 + 2 T + 35 T^{2} - 204 T^{3} + 6790 T^{4} + 16644 T^{5} + 454930 T^{6} - 915756 T^{7} + 10526705 T^{8} + 40302242 T^{9} + 1350125107 T^{10} \))(\( ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) \))(\( ( 1 - 258 T^{2} + 33803 T^{4} - 2811556 T^{6} + 151741667 T^{8} - 5198989218 T^{10} + 90458382169 T^{12} )^{2} \))(\( 1 + 2 T - 160 T^{2} - 1954 T^{3} + 4493 T^{4} + 240091 T^{5} + 1558345 T^{6} - 6198830 T^{7} - 145826397 T^{8} - 849223419 T^{9} + 1791946668 T^{10} + 63989276321 T^{11} + 408449117313 T^{12} - 1513546200047 T^{13} - 32669265939113 T^{14} - 101407595403149 T^{15} + 1833528087618057 T^{16} + 19245606714132923 T^{17} + 36109734132414828 T^{18} - 1146557859444280833 T^{19} - 13191219950154315093 T^{20} - 37569320920424372090 T^{21} + \)\(63\!\cdots\!45\)\( T^{22} + \)\(65\!\cdots\!77\)\( T^{23} + \)\(81\!\cdots\!57\)\( T^{24} - \)\(23\!\cdots\!82\)\( T^{25} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(10\!\cdots\!74\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \))(\( 1 - 12 T - 208 T^{2} + 4498 T^{3} + 11581 T^{4} - 769969 T^{5} + 2330811 T^{6} + 79723168 T^{7} - 563969769 T^{8} - 5413517539 T^{9} + 65475540438 T^{10} + 240195017209 T^{11} - 5444224845133 T^{12} - 5337298937893 T^{13} + 382806778628335 T^{14} - 357599028838831 T^{15} - 24439125329802037 T^{16} + 72241773960830467 T^{17} + 1319405537906530998 T^{18} - 7308925946588751673 T^{19} - 51015792895964648961 T^{20} + \)\(48\!\cdots\!64\)\( T^{21} + \)\(94\!\cdots\!51\)\( T^{22} - \)\(20\!\cdots\!43\)\( T^{23} + \)\(21\!\cdots\!69\)\( T^{24} + \)\(54\!\cdots\!34\)\( T^{25} - \)\(17\!\cdots\!88\)\( T^{26} - \)\(65\!\cdots\!44\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \))(\( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} - 37740 T^{5} + 48517 T^{6} + 2334512 T^{7} - 12588687 T^{8} - 106057556 T^{9} + 1267672253 T^{10} - 7105856252 T^{11} - 56510615943 T^{12} + 702134832656 T^{13} + 977671937557 T^{14} - 50953721538180 T^{15} + 138310866336401 T^{16} + 2860655877712456 T^{17} - 20709451555388691 T^{18} - 108826137585179788 T^{19} + 1822837804551761449 T^{20} )( 1 + 4 T - 51 T^{2} - 472 T^{3} + 1529 T^{4} + 37740 T^{5} + 48517 T^{6} - 2334512 T^{7} - 12588687 T^{8} + 106057556 T^{9} + 1267672253 T^{10} + 7105856252 T^{11} - 56510615943 T^{12} - 702134832656 T^{13} + 977671937557 T^{14} + 50953721538180 T^{15} + 138310866336401 T^{16} - 2860655877712456 T^{17} - 20709451555388691 T^{18} + 108826137585179788 T^{19} + 1822837804551761449 T^{20} ) \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 16 T + 201 T^{2} + 1136 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 133 T^{2} - 504 T^{3} + 9443 T^{4} - 20164 T^{5} + 357911 T^{6} \))(\( 1 - 4 T + 254 T^{2} - 860 T^{3} + 30833 T^{4} - 86976 T^{5} + 2189143 T^{6} - 4335260 T^{7} + 90909394 T^{8} - 101646724 T^{9} + 1804229351 T^{10} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 204 T^{2} + 1488 T^{3} + 14484 T^{4} + 60492 T^{5} + 357911 T^{6} )^{4} \))(\( ( 1 + 28 T + 714 T^{2} + 12158 T^{3} + 180233 T^{4} + 2178780 T^{5} + 22911109 T^{6} + 206850943 T^{7} + 1626688739 T^{8} + 10983229980 T^{9} + 64507373263 T^{10} + 308955217598 T^{11} + 1288219756614 T^{12} + 3586807949788 T^{13} + 9095120158391 T^{14} )^{2} \))(\( ( 1 - 20 T + 478 T^{2} - 6038 T^{3} + 87913 T^{4} - 856632 T^{5} + 9552201 T^{6} - 75571685 T^{7} + 678206271 T^{8} - 4318281912 T^{9} + 31465029743 T^{10} - 153435729878 T^{11} + 862421629778 T^{12} - 2562005678420 T^{13} + 9095120158391 T^{14} )^{2} \))(\( ( 1 + 16 T + 185 T^{2} + 1824 T^{3} + 16049 T^{4} + 127280 T^{5} + 897001 T^{6} + 5315136 T^{7} + 21355105 T^{8} - 35692976 T^{9} - 2087300071 T^{10} - 2534201296 T^{11} + 107651084305 T^{12} + 1902345640896 T^{13} + 22794303268681 T^{14} + 229642311795280 T^{15} + 2055881456648129 T^{16} + 16589499168905184 T^{17} + 119464403280465785 T^{18} + 733576011495184496 T^{19} + 3255243551009881201 T^{20} )^{2} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 + 101 T^{2} + 5329 T^{4} \))(\( 1 + 6 T + 111 T^{2} + 660 T^{3} + 8103 T^{4} + 31974 T^{5} + 389017 T^{6} \))(\( 1 + 2 T + 168 T^{2} + 946 T^{3} + 18175 T^{4} + 89144 T^{5} + 1326775 T^{6} + 5041234 T^{7} + 65354856 T^{8} + 56796482 T^{9} + 2073071593 T^{10} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 330 T^{2} + 50862 T^{4} - 4684890 T^{6} + 271043598 T^{8} - 9371419530 T^{10} + 151334226289 T^{12} )^{2} \))(\( 1 + 59 T + 1558 T^{2} + 26773 T^{3} + 395234 T^{4} + 5614061 T^{5} + 71491948 T^{6} + 818465504 T^{7} + 9139019109 T^{8} + 97770699609 T^{9} + 960356271312 T^{10} + 9153154562004 T^{11} + 86714108301328 T^{12} + 772925330853350 T^{13} + 6595874091137989 T^{14} + 56423549152294550 T^{15} + 462099483137776912 T^{16} + 3560732728247110068 T^{17} + 27272428838579562192 T^{18} + \)\(20\!\cdots\!37\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} + \)\(90\!\cdots\!88\)\( T^{21} + \)\(57\!\cdots\!88\)\( T^{22} + \)\(33\!\cdots\!93\)\( T^{23} + \)\(16\!\cdots\!66\)\( T^{24} + \)\(83\!\cdots\!21\)\( T^{25} + \)\(35\!\cdots\!18\)\( T^{26} + \)\(98\!\cdots\!47\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))(\( 1 - 33 T + 318 T^{2} + 361 T^{3} - 12030 T^{4} - 129355 T^{5} + 1080240 T^{6} + 5668784 T^{7} + 21452721 T^{8} - 566539083 T^{9} - 7087638076 T^{10} + 71569896916 T^{11} + 43985247832 T^{12} + 3384120274258 T^{13} - 75264087406419 T^{14} + 247040780020834 T^{15} + 234397385696728 T^{16} + 27841906588571572 T^{17} - 201276454203024316 T^{18} - 1174476079291569219 T^{19} + 3246530934328782369 T^{20} + 62625315966680768048 T^{21} + \)\(87\!\cdots\!40\)\( T^{22} - \)\(76\!\cdots\!15\)\( T^{23} - \)\(51\!\cdots\!70\)\( T^{24} + \)\(11\!\cdots\!97\)\( T^{25} + \)\(72\!\cdots\!78\)\( T^{26} - \)\(55\!\cdots\!89\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))(\( ( 1 + 73 T^{2} + 5329 T^{4} + 389017 T^{6} + 28398241 T^{8} + 2073071593 T^{10} + 151334226289 T^{12} + 11047398519097 T^{14} + 806460091894081 T^{16} + 58871586708267913 T^{18} + 4297625829703557649 T^{20} )^{2} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 10 T + 178 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 245 T^{2} - 2100 T^{3} + 19355 T^{4} - 99856 T^{5} + 493039 T^{6} \))(\( 1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 10773546 T^{6} - 66628916 T^{7} + 346606417 T^{8} - 1168502430 T^{9} + 3077056399 T^{10} \))(\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 308 T^{2} + 42611 T^{4} - 3870912 T^{6} + 265935251 T^{8} - 11996624948 T^{10} + 243087455521 T^{12} )^{2} \))(\( 1 - 4 T - 521 T^{2} + 1492 T^{3} + 156337 T^{4} - 330004 T^{5} - 32561264 T^{6} + 47905619 T^{7} + 5201467272 T^{8} - 4993760677 T^{9} - 662849060908 T^{10} + 346164223948 T^{11} + 69259823652310 T^{12} - 11162720451616 T^{13} - 5994553376270005 T^{14} - 881854915677664 T^{15} + 432250559414066710 T^{16} + 170672462811097972 T^{17} - 25818024613140533548 T^{18} - 15366083246237422123 T^{19} + \)\(12\!\cdots\!12\)\( T^{20} + \)\(91\!\cdots\!21\)\( T^{21} - \)\(49\!\cdots\!04\)\( T^{22} - \)\(39\!\cdots\!76\)\( T^{23} + \)\(14\!\cdots\!37\)\( T^{24} + \)\(11\!\cdots\!68\)\( T^{25} - \)\(30\!\cdots\!61\)\( T^{26} - \)\(18\!\cdots\!56\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))(\( 1 - 10 T - 199 T^{2} - 410 T^{3} + 40627 T^{4} + 263926 T^{5} - 2826574 T^{6} - 45143369 T^{7} - 10015052 T^{8} + 3146123353 T^{9} + 15703734448 T^{10} - 60810956822 T^{11} - 1034864369576 T^{12} + 644187900550 T^{13} + 9343075967051 T^{14} + 50890844143450 T^{15} - 6458588530523816 T^{16} - 29982173340562058 T^{17} + 611661728752090288 T^{18} + 9680798995391985847 T^{19} - 2434533507590502092 T^{20} - \)\(86\!\cdots\!71\)\( T^{21} - \)\(42\!\cdots\!14\)\( T^{22} + \)\(31\!\cdots\!94\)\( T^{23} + \)\(38\!\cdots\!27\)\( T^{24} - \)\(30\!\cdots\!90\)\( T^{25} - \)\(11\!\cdots\!59\)\( T^{26} - \)\(46\!\cdots\!90\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))(\( ( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} - 20760 T^{5} + 547729 T^{6} - 2741792 T^{7} - 21336255 T^{8} + 387291608 T^{9} - 1412768719 T^{10} + 30596037032 T^{11} - 133159567455 T^{12} - 1351810385888 T^{13} + 21334088916049 T^{14} - 63879690843240 T^{15} - 1174355497621951 T^{16} + 14441339557591568 T^{17} - 22756632148598415 T^{18} - 958812767860946552 T^{19} + 9468276082626847201 T^{20} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} + 20760 T^{5} + 547729 T^{6} + 2741792 T^{7} - 21336255 T^{8} - 387291608 T^{9} - 1412768719 T^{10} - 30596037032 T^{11} - 133159567455 T^{12} + 1351810385888 T^{13} + 21334088916049 T^{14} + 63879690843240 T^{15} - 1174355497621951 T^{16} - 14441339557591568 T^{17} - 22756632148598415 T^{18} + 958812767860946552 T^{19} + 9468276082626847201 T^{20} ) \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 150 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 - 20 T + 145 T^{2} - 648 T^{3} + 12035 T^{4} - 137780 T^{5} + 571787 T^{6} \))(\( 1 - 8 T + 359 T^{2} - 2224 T^{3} + 55778 T^{4} - 264336 T^{5} + 4629574 T^{6} - 15321136 T^{7} + 205271533 T^{8} - 379666568 T^{9} + 3939040643 T^{10} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 420 T^{2} + 77963 T^{4} + 8298224 T^{6} + 537087107 T^{8} + 19932494820 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 + 9 T + 199 T^{2} + 2164 T^{3} + 32668 T^{4} + 299057 T^{5} + 3497361 T^{6} + 30718463 T^{7} + 290280963 T^{8} + 2060203673 T^{9} + 18679137716 T^{10} + 102699806644 T^{11} + 783869087957 T^{12} + 2942463360321 T^{13} + 27136050989627 T^{14} )^{2} \))(\( ( 1 + 25 T + 835 T^{2} + 13794 T^{3} + 254004 T^{4} + 3035699 T^{5} + 38492317 T^{6} + 342462563 T^{7} + 3194862311 T^{8} + 20912930411 T^{9} + 145236185148 T^{10} + 654640079874 T^{11} + 3289098936905 T^{12} + 8173509334225 T^{13} + 27136050989627 T^{14} )^{2} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} - 3939040643 T^{10} + 326940373369 T^{12} - 27136050989627 T^{14} + 2252292232139041 T^{16} - 186940255267540403 T^{18} + 15516041187205853449 T^{20} )^{2} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 98 T^{2} + 7921 T^{4} \))(\( 1 - 4 T + 153 T^{2} - 990 T^{3} + 13617 T^{4} - 31684 T^{5} + 704969 T^{6} \))(\( 1 + 20 T + 511 T^{2} + 6422 T^{3} + 92672 T^{4} + 821572 T^{5} + 8247808 T^{6} + 50868662 T^{7} + 360239159 T^{8} + 1254844820 T^{9} + 5584059449 T^{10} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 258 T^{2} + 41039 T^{4} + 4168892 T^{6} + 325069919 T^{8} + 16187498178 T^{10} + 496981290961 T^{12} )^{2} \))(\( 1 + T - 372 T^{2} - 1225 T^{3} + 68499 T^{4} + 345380 T^{5} - 8137725 T^{6} - 52985316 T^{7} + 738187722 T^{8} + 5132287668 T^{9} - 63317923712 T^{10} - 315741303346 T^{11} + 5949624228375 T^{12} + 9456422184784 T^{13} - 558957419490099 T^{14} + 841621574445776 T^{15} + 47126973512958375 T^{16} - 222587830878526274 T^{17} - 3972708429157918592 T^{18} + 28658999447481574932 T^{19} + \)\(36\!\cdots\!42\)\( T^{20} - \)\(23\!\cdots\!64\)\( T^{21} - \)\(32\!\cdots\!25\)\( T^{22} + \)\(12\!\cdots\!20\)\( T^{23} + \)\(21\!\cdots\!99\)\( T^{24} - \)\(33\!\cdots\!25\)\( T^{25} - \)\(91\!\cdots\!12\)\( T^{26} + \)\(21\!\cdots\!69\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))(\( 1 - 11 T - 266 T^{2} + 4293 T^{3} + 30631 T^{4} - 871406 T^{5} - 697007 T^{6} + 125070564 T^{7} - 467472362 T^{8} - 13062013806 T^{9} + 107966026354 T^{10} + 958397275840 T^{11} - 14887218185097 T^{12} - 33671658148990 T^{13} + 1523631362132685 T^{14} - 2996777575260110 T^{15} - 117921655244153337 T^{16} + 675640369151648960 T^{17} + 6774030445315019314 T^{18} - 72939061616362752894 T^{19} - \)\(23\!\cdots\!82\)\( T^{20} + \)\(55\!\cdots\!56\)\( T^{21} - \)\(27\!\cdots\!67\)\( T^{22} - \)\(30\!\cdots\!54\)\( T^{23} + \)\(95\!\cdots\!31\)\( T^{24} + \)\(11\!\cdots\!77\)\( T^{25} - \)\(65\!\cdots\!86\)\( T^{26} - \)\(24\!\cdots\!59\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} - 704969 T^{6} + 62742241 T^{8} - 5584059449 T^{10} + 496981290961 T^{12} - 44231334895529 T^{14} + 3936588805702081 T^{16} - 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 6 T + 158 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 269 T^{2} - 1166 T^{3} + 26093 T^{4} - 56454 T^{5} + 912673 T^{6} \))(\( 1 + 12 T + 371 T^{2} + 3294 T^{3} + 61432 T^{4} + 417340 T^{5} + 5958904 T^{6} + 30993246 T^{7} + 338601683 T^{8} + 1062351372 T^{9} + 8587340257 T^{10} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 322 T^{2} + 56939 T^{4} + 6497508 T^{6} + 535739051 T^{8} + 28506428482 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 35 T + 813 T^{2} - 13184 T^{3} + 185925 T^{4} - 2280519 T^{5} + 26419621 T^{6} - 271075175 T^{7} + 2562703237 T^{8} - 21457403271 T^{9} + 169688727525 T^{10} - 1167170040704 T^{11} + 6981507628941 T^{12} - 29154020172515 T^{13} + 80798284478113 T^{14} )^{2} \))(\( ( 1 + 51 T + 1585 T^{2} + 35526 T^{3} + 632541 T^{4} + 9292285 T^{5} + 115583147 T^{6} + 1228073393 T^{7} + 11211565259 T^{8} + 87431109565 T^{9} + 577303092093 T^{10} + 3145091236806 T^{11} + 13610934307345 T^{12} + 42481572251379 T^{13} + 80798284478113 T^{14} )^{2} \))(\( ( 1 - 97 T^{2} + 9409 T^{4} - 912673 T^{6} + 88529281 T^{8} - 8587340257 T^{10} + 832972004929 T^{12} - 80798284478113 T^{14} + 7837433594376961 T^{16} - 760231058654565217 T^{18} + 73742412689492826049 T^{20} )^{2} \))
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