Properties

Label 164.2
Level 164
Weight 2
Dimension 450
Nonzero newspaces 8
Newform subspaces 10
Sturm bound 3360
Trace bound 3

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Defining parameters

Level: \( N \) = \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 10 \)
Sturm bound: \(3360\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(164))\).

Total New Old
Modular forms 940 530 410
Cusp forms 741 450 291
Eisenstein series 199 80 119

Trace form

\( 450q - 20q^{2} - 20q^{4} - 40q^{5} - 20q^{6} - 20q^{8} - 40q^{9} + O(q^{10}) \) \( 450q - 20q^{2} - 20q^{4} - 40q^{5} - 20q^{6} - 20q^{8} - 40q^{9} - 20q^{10} - 20q^{12} - 40q^{13} - 20q^{14} - 20q^{16} - 40q^{17} - 20q^{18} - 20q^{20} - 40q^{21} - 20q^{22} - 20q^{24} - 40q^{25} - 20q^{26} - 20q^{28} - 40q^{29} - 20q^{30} - 20q^{31} - 20q^{32} - 100q^{33} - 20q^{34} - 40q^{35} - 20q^{36} - 100q^{37} - 20q^{38} - 80q^{39} - 60q^{41} - 40q^{42} - 20q^{43} - 20q^{44} - 120q^{45} - 20q^{46} - 60q^{47} - 20q^{48} - 80q^{49} - 20q^{50} - 60q^{51} - 20q^{52} - 60q^{53} - 20q^{54} - 20q^{56} - 40q^{57} - 20q^{58} - 20q^{60} - 40q^{61} - 20q^{62} - 20q^{64} - 30q^{65} + 60q^{66} + 60q^{67} + 80q^{68} + 40q^{69} + 180q^{70} + 80q^{71} + 160q^{72} + 40q^{73} + 100q^{74} + 160q^{75} + 280q^{76} + 40q^{77} + 180q^{78} + 80q^{79} + 180q^{80} + 170q^{81} + 220q^{82} + 80q^{83} + 220q^{84} + 130q^{85} + 180q^{86} + 80q^{87} + 180q^{88} + 40q^{89} + 280q^{90} + 160q^{91} + 100q^{92} + 40q^{93} + 160q^{94} + 80q^{95} + 180q^{96} + 40q^{97} + 80q^{98} + 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(164))\)

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
164.2.a \(\chi_{164}(1, \cdot)\) 164.2.a.a 4 1
164.2.b \(\chi_{164}(81, \cdot)\) 164.2.b.a 4 1
164.2.f \(\chi_{164}(9, \cdot)\) 164.2.f.a 6 2
164.2.g \(\chi_{164}(37, \cdot)\) 164.2.g.a 16 4
164.2.i \(\chi_{164}(3, \cdot)\) 164.2.i.a 4 4
164.2.i.b 72
164.2.k \(\chi_{164}(25, \cdot)\) 164.2.k.a 16 4
164.2.m \(\chi_{164}(5, \cdot)\) 164.2.m.a 24 8
164.2.o \(\chi_{164}(7, \cdot)\) 164.2.o.a 16 16
164.2.o.b 288

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(164))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(164)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(41))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(82))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))(\( \))(\( ( 1 + 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} + 16 T^{7} + 16 T^{8} )^{2} \))
$3$ (\( 1 - 2 T + 2 T^{2} + 4 T^{3} - 8 T^{4} + 12 T^{5} + 18 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 - 6 T^{2} + 20 T^{4} - 54 T^{6} + 81 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} - 5 T^{4} + 18 T^{5} - 18 T^{6} + 54 T^{7} - 45 T^{8} - 108 T^{9} + 162 T^{10} - 486 T^{11} + 729 T^{12} \))(\( ( 1 + T + 8 T^{2} + 5 T^{3} + 37 T^{4} + 7 T^{5} + 112 T^{6} - 21 T^{7} + 340 T^{8} - 63 T^{9} + 1008 T^{10} + 189 T^{11} + 2997 T^{12} + 1215 T^{13} + 5832 T^{14} + 2187 T^{15} + 6561 T^{16} )^{2} \))(\( 1 + 81 T^{8} \))(\( 1 - 9 T^{2} + 36 T^{4} - 135 T^{6} + 653 T^{8} - 2745 T^{10} + 9504 T^{12} - 29799 T^{14} + 89788 T^{16} - 268191 T^{18} + 769824 T^{20} - 2001105 T^{22} + 4284333 T^{24} - 7971615 T^{26} + 19131876 T^{28} - 43046721 T^{30} + 43046721 T^{32} \))(\( ( 1 + 81 T^{8} )^{4} \))
$5$ (\( 1 - 4 T + 12 T^{2} - 16 T^{3} + 34 T^{4} - 80 T^{5} + 300 T^{6} - 500 T^{7} + 625 T^{8} \))(\( ( 1 + 2 T + 4 T^{2} + 10 T^{3} + 25 T^{4} )^{2} \))(\( 1 - 18 T^{2} + 159 T^{4} - 936 T^{6} + 3975 T^{8} - 11250 T^{10} + 15625 T^{12} \))(\( 1 + 4 T + 9 T^{2} + 39 T^{3} + 133 T^{4} + 263 T^{5} + 744 T^{6} + 2047 T^{7} + 3396 T^{8} + 6972 T^{9} + 15659 T^{10} + 14860 T^{11} + 5595 T^{12} - 16339 T^{13} - 248977 T^{14} - 697866 T^{15} - 1345184 T^{16} - 3489330 T^{17} - 6224425 T^{18} - 2042375 T^{19} + 3496875 T^{20} + 46437500 T^{21} + 244671875 T^{22} + 544687500 T^{23} + 1326562500 T^{24} + 3998046875 T^{25} + 7265625000 T^{26} + 12841796875 T^{27} + 32470703125 T^{28} + 47607421875 T^{29} + 54931640625 T^{30} + 122070312500 T^{31} + 152587890625 T^{32} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )( 1 + 8 T^{2} + 25 T^{4} ) \))(\( 1 - 4 T - T^{2} + 41 T^{3} - 69 T^{4} - 37 T^{5} + 96 T^{6} - 317 T^{7} + 3602 T^{8} - 6078 T^{9} - 17079 T^{10} + 49592 T^{11} + 19471 T^{12} - 38671 T^{13} - 342361 T^{14} - 274826 T^{15} + 3399336 T^{16} - 1374130 T^{17} - 8559025 T^{18} - 4833875 T^{19} + 12169375 T^{20} + 154975000 T^{21} - 266859375 T^{22} - 474843750 T^{23} + 1407031250 T^{24} - 619140625 T^{25} + 937500000 T^{26} - 1806640625 T^{27} - 16845703125 T^{28} + 50048828125 T^{29} - 6103515625 T^{30} - 122070312500 T^{31} + 152587890625 T^{32} \))(\( ( 1 - 8 T^{2} + 39 T^{4} - 112 T^{6} - 79 T^{8} - 2800 T^{10} + 24375 T^{12} - 125000 T^{14} + 390625 T^{16} )( 1 + 8 T^{2} + 39 T^{4} + 112 T^{6} - 79 T^{8} + 2800 T^{10} + 24375 T^{12} + 125000 T^{14} + 390625 T^{16} ) \))
$7$ (\( 1 + 6 T^{2} + 26 T^{3} + 24 T^{4} + 182 T^{5} + 294 T^{6} + 2401 T^{8} \))(\( 1 - 18 T^{2} + 172 T^{4} - 882 T^{6} + 2401 T^{8} \))(\( 1 + 18 T^{3} - 45 T^{4} - 162 T^{5} + 162 T^{6} - 1134 T^{7} - 2205 T^{8} + 6174 T^{9} + 117649 T^{12} \))(\( 1 - 6 T^{2} + 4 T^{3} + 42 T^{4} + 40 T^{5} - 336 T^{6} - 296 T^{7} + 3113 T^{8} + 2947 T^{9} - 35514 T^{10} + 3872 T^{11} + 252559 T^{12} - 27043 T^{13} - 1195345 T^{14} + 348148 T^{15} + 12685276 T^{16} + 2437036 T^{17} - 58571905 T^{18} - 9275749 T^{19} + 606394159 T^{20} + 65076704 T^{21} - 4178186586 T^{22} + 2426981221 T^{23} + 17945825513 T^{24} - 11944667672 T^{25} - 94911683664 T^{26} + 79093069720 T^{27} + 581334062442 T^{28} + 387556041628 T^{29} - 4069338437094 T^{30} + 33232930569601 T^{32} \))(\( 1 + 2401 T^{8} \))(\( 1 + 28 T^{2} + 20 T^{3} + 392 T^{4} + 560 T^{5} + 4512 T^{6} + 7740 T^{7} + 47317 T^{8} + 83495 T^{9} + 418400 T^{10} + 832150 T^{11} + 3293403 T^{12} + 7196995 T^{13} + 25128411 T^{14} + 53516060 T^{15} + 182846400 T^{16} + 374612420 T^{17} + 1231292139 T^{18} + 2468569285 T^{19} + 7907460603 T^{20} + 13985945050 T^{21} + 49224341600 T^{22} + 68761722785 T^{23} + 272773088917 T^{24} + 312336918180 T^{25} + 1274528323488 T^{26} + 1107302976080 T^{27} + 5425784582792 T^{28} + 1937780208140 T^{29} + 18990246039772 T^{30} + 33232930569601 T^{32} \))(\( 1 - 2401 T^{8} + 5764801 T^{16} - 13841287201 T^{24} + 33232930569601 T^{32} \))
$11$ (\( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 1254 T^{5} + 3146 T^{6} - 5324 T^{7} + 14641 T^{8} \))(\( 1 - 18 T^{2} + 316 T^{4} - 2178 T^{6} + 14641 T^{8} \))(\( 1 + 4 T + 8 T^{2} + 38 T^{3} + 279 T^{4} + 818 T^{5} + 1762 T^{6} + 8998 T^{7} + 33759 T^{8} + 50578 T^{9} + 117128 T^{10} + 644204 T^{11} + 1771561 T^{12} \))(\( 1 - T - 45 T^{2} - 10 T^{3} + 772 T^{4} + 2167 T^{5} - 3639 T^{6} - 58700 T^{7} - 103421 T^{8} + 640361 T^{9} + 2408448 T^{10} - 66583 T^{11} - 20748662 T^{12} - 74120409 T^{13} + 9338901 T^{14} + 518677995 T^{15} + 1194868466 T^{16} + 5705457945 T^{17} + 1130007021 T^{18} - 98654264379 T^{19} - 303781160342 T^{20} - 10723258733 T^{21} + 4266712547328 T^{22} + 12478824308731 T^{23} - 22169209831901 T^{24} - 138411529461700 T^{25} - 94386288123039 T^{26} + 618270390214037 T^{27} + 2422866706828612 T^{28} - 345227121439310 T^{29} - 17088742511245845 T^{30} - 4177248169415651 T^{31} + 45949729863572161 T^{32} \))(\( 1 + 14641 T^{8} \))(\( 1 + 5 T + 63 T^{2} + 420 T^{3} + 2568 T^{4} + 16205 T^{5} + 82849 T^{6} + 427280 T^{7} + 2047567 T^{8} + 9076645 T^{9} + 39590230 T^{10} + 161775955 T^{11} + 634554422 T^{12} + 2416654295 T^{13} + 8751803303 T^{14} + 30653196935 T^{15} + 104022058530 T^{16} + 337185166285 T^{17} + 1058968199663 T^{18} + 3216566866645 T^{19} + 9290511292502 T^{20} + 26054179328705 T^{21} + 70136507449030 T^{22} + 176878133221295 T^{23} + 438914170892527 T^{24} + 1007503889410480 T^{25} + 2148889690768249 T^{26} + 4623475622251255 T^{27} + 8059484071419528 T^{28} + 14499539100451020 T^{29} + 23924239515744183 T^{30} + 20886240847078255 T^{31} + 45949729863572161 T^{32} \))(\( 1 - 14641 T^{8} + 214358881 T^{16} - 3138428376721 T^{24} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 12 T^{2} - 48 T^{3} + 118 T^{4} - 624 T^{5} + 2028 T^{6} + 28561 T^{8} \))(\( 1 - 12 T^{2} + 262 T^{4} - 2028 T^{6} + 28561 T^{8} \))(\( 1 + 2 T + 2 T^{2} + 18 T^{3} + 87 T^{4} + 68 T^{5} + 124 T^{6} + 884 T^{7} + 14703 T^{8} + 39546 T^{9} + 57122 T^{10} + 742586 T^{11} + 4826809 T^{12} \))(\( 1 + 3 T^{2} + 28 T^{3} + 6 T^{4} + 1542 T^{5} + 1495 T^{6} + 9827 T^{7} + 30745 T^{8} + 29725 T^{9} + 1469860 T^{10} + 2003100 T^{11} + 12293752 T^{12} + 21134827 T^{13} + 47512001 T^{14} + 876437227 T^{15} + 1560334738 T^{16} + 11393683951 T^{17} + 8029528169 T^{18} + 46433214919 T^{19} + 351121850872 T^{20} + 743737008300 T^{21} + 7094733476740 T^{22} + 1865199667825 T^{23} + 25079641017145 T^{24} + 104210415338471 T^{25} + 206098445314255 T^{26} + 2763511327605054 T^{27} + 139788510734886 T^{28} + 8480502984583084 T^{29} + 11812129157097867 T^{30} + 665416609183179841 T^{32} \))(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 - 24 T^{2} + 169 T^{4} ) \))(\( 1 + 27 T^{2} - 150 T^{3} + 542 T^{4} - 4050 T^{5} + 20713 T^{6} - 82305 T^{7} + 440287 T^{8} - 1975395 T^{9} + 7781680 T^{10} - 35114300 T^{11} + 144911468 T^{12} - 531445435 T^{13} + 2179765819 T^{14} - 8325043825 T^{15} + 27709757390 T^{16} - 108225569725 T^{17} + 368380423411 T^{18} - 1167585620695 T^{19} + 4138816437548 T^{20} - 13037693789900 T^{21} + 37560683059120 T^{22} - 123953106739215 T^{23} + 359155631956927 T^{24} - 872803320894765 T^{25} + 2855462941668337 T^{26} - 7258249595849850 T^{27} + 12627562136384702 T^{28} - 45431265988837950 T^{29} + 106309162413880803 T^{30} + 665416609183179841 T^{32} \))(\( ( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 520 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2}( 1 + 24 T^{2} + 407 T^{4} + 5712 T^{6} + 68305 T^{8} + 965328 T^{10} + 11624327 T^{12} + 115843416 T^{14} + 815730721 T^{16} ) \))
$17$ (\( 1 + 4 T + 20 T^{2} + 124 T^{3} + 534 T^{4} + 2108 T^{5} + 5780 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 12 T^{2} + 166 T^{4} + 3468 T^{6} + 83521 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 62 T^{3} - 225 T^{4} - 164 T^{5} + 2044 T^{6} - 2788 T^{7} - 65025 T^{8} - 304606 T^{9} + 167042 T^{10} + 2839714 T^{11} + 24137569 T^{12} \))(\( 1 + T - 29 T^{2} - 93 T^{3} + 334 T^{4} - 3123 T^{5} - 3498 T^{6} + 122958 T^{7} + 421640 T^{8} - 1697151 T^{9} + 1570408 T^{10} + 7160095 T^{11} - 201602448 T^{12} - 882181692 T^{13} + 3644160466 T^{14} + 7639608105 T^{15} - 10203388108 T^{16} + 129873337785 T^{17} + 1053162374674 T^{18} - 4334158652796 T^{19} - 16838038059408 T^{20} + 10166311006415 T^{21} + 37905831458152 T^{22} - 696406689220623 T^{23} + 2941258367423240 T^{24} + 14581328118318126 T^{25} - 7051946663770602 T^{26} - 107031132168737859 T^{27} + 194595827234740174 T^{28} - 921125757060252141 T^{29} - 4882956970222626941 T^{30} + 2862423051509815793 T^{31} + 48661191875666868481 T^{32} \))(\( ( 1 - 2 T + 17 T^{2} )^{2}( 1 - 16 T^{2} + 289 T^{4} ) \))(\( 1 - 5 T + 63 T^{2} - 425 T^{3} + 2568 T^{4} - 16395 T^{5} + 78638 T^{6} - 439920 T^{7} + 2015796 T^{8} - 9574165 T^{9} + 46403810 T^{10} - 202652735 T^{11} + 971098702 T^{12} - 4240819090 T^{13} + 18882484980 T^{14} - 82602641445 T^{15} + 334049922892 T^{16} - 1404244904565 T^{17} + 5457038159220 T^{18} - 20835144189170 T^{19} + 81107134689742 T^{20} - 287737904358895 T^{21} + 1120075165737890 T^{22} - 3928650161183045 T^{23} + 14061703946538036 T^{24} - 52169178628560240 T^{25} + 158533728343508462 T^{26} - 561887739963643035 T^{27} + 1496173905206026248 T^{28} - 4209445663985023225 T^{29} + 10607803073242258527 T^{30} - 14312115257549078965 T^{31} + 48661191875666868481 T^{32} \))(\( ( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 1020 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2}( 1 + 16 T^{2} - 33 T^{4} - 5152 T^{6} - 72895 T^{8} - 1488928 T^{10} - 2756193 T^{12} + 386201104 T^{14} + 6975757441 T^{16} ) \))
$19$ (\( 1 - 6 T + 62 T^{2} - 208 T^{3} + 1448 T^{4} - 3952 T^{5} + 22382 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 6 T^{2} + 388 T^{4} - 2166 T^{6} + 130321 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 56 T^{3} + 39 T^{4} + 1346 T^{5} - 1202 T^{6} + 25574 T^{7} + 14079 T^{8} - 384104 T^{9} + 260642 T^{10} - 4952198 T^{11} + 47045881 T^{12} \))(\( 1 - 9 T + 39 T^{2} - 285 T^{3} + 2307 T^{4} - 12818 T^{5} + 65656 T^{6} - 354482 T^{7} + 1906501 T^{8} - 9433749 T^{9} + 42393273 T^{10} - 198127749 T^{11} + 936880855 T^{12} - 4075764828 T^{13} + 17694491752 T^{14} - 78620604008 T^{15} + 342754896768 T^{16} - 1493791476152 T^{17} + 6387711522472 T^{18} - 27955670955252 T^{19} + 122095249904455 T^{20} - 490583921171151 T^{21} + 1994428876758513 T^{22} - 8432561623919511 T^{23} + 32379179921229541 T^{24} - 114386980484095478 T^{25} + 402541286222182456 T^{26} - 1493172138557371142 T^{27} + 5106117518285633427 T^{28} - 11985100286743261815 T^{29} + 31161260745532480719 T^{30} - \)\(13\!\cdots\!91\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 130321 T^{8} \))(\( 1 - 15 T + 161 T^{2} - 1525 T^{3} + 11343 T^{4} - 75230 T^{5} + 443166 T^{6} - 2254670 T^{7} + 10341187 T^{8} - 39337075 T^{9} + 112992035 T^{10} - 121397655 T^{11} - 1613776893 T^{12} + 16094021990 T^{13} - 105965524802 T^{14} + 571838637400 T^{15} - 2626157417260 T^{16} + 10864934110600 T^{17} - 38253554453522 T^{18} + 110388896829410 T^{19} - 210309018472653 T^{20} - 300592612147845 T^{21} + 5315809832557835 T^{22} - 35162299637423425 T^{23} + 175630201333269667 T^{24} - 727554271551377930 T^{25} + 2717080109204637966 T^{26} - 8763562176913015370 T^{27} + 25105631126967464223 T^{28} - 64130799779942014975 T^{29} + \)\(12\!\cdots\!81\)\( T^{30} - \)\(22\!\cdots\!85\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 - 130321 T^{8} + 16983563041 T^{16} - 2213314919066161 T^{24} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 16100 T^{5} + 57132 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( ( 1 - 4 T + 22 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 49 T^{2} - 68 T^{3} + 1127 T^{4} - 1058 T^{5} + 12167 T^{6} )^{2} \))(\( 1 + 18 T + 101 T^{2} - 141 T^{3} - 3989 T^{4} - 19501 T^{5} - 72466 T^{6} - 228747 T^{7} + 1314644 T^{8} + 19654696 T^{9} + 87356413 T^{10} + 98694086 T^{11} - 789957763 T^{12} - 4700967521 T^{13} - 11829577437 T^{14} - 25737742002 T^{15} - 120195483200 T^{16} - 591968066046 T^{17} - 6257846464173 T^{18} - 57196671828007 T^{19} - 221062570355683 T^{20} + 635228989567498 T^{21} + 12931884258306157 T^{22} + 66920809093849112 T^{23} + 102951066933754964 T^{24} - 412008267851676861 T^{25} - 3002013561608288434 T^{26} - 18580743089079490427 T^{27} - 87417436859329060469 T^{28} - 71069127033041901003 T^{29} + \)\(11\!\cdots\!09\)\( T^{30} + \)\(47\!\cdots\!26\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( 1 - 12 T + 19 T^{2} + 157 T^{3} + 373 T^{4} - 373 T^{5} - 55496 T^{6} + 151147 T^{7} + 538774 T^{8} + 101222 T^{9} - 13084983 T^{10} - 28574544 T^{11} + 649978935 T^{12} - 1623339109 T^{13} - 11361153025 T^{14} + 27845846104 T^{15} + 180405296180 T^{16} + 640454460392 T^{17} - 6010049950225 T^{18} - 19751166939203 T^{19} + 181890755149335 T^{20} - 183915566252592 T^{21} - 1937047090954887 T^{22} + 344643241396234 T^{23} + 42191922783785494 T^{24} + 272238821322148061 T^{25} - 2299005666312664904 T^{26} - 355398039701894771 T^{27} + 8174154913143579733 T^{28} + 79133708824025379131 T^{29} + \)\(22\!\cdots\!71\)\( T^{30} - \)\(31\!\cdots\!84\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 23 T^{2} + 529 T^{4} - 12167 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( 1 + 4 T + 76 T^{2} + 348 T^{3} + 2870 T^{4} + 10092 T^{5} + 63916 T^{6} + 97556 T^{7} + 707281 T^{8} \))(\( 1 - 92 T^{2} + 3686 T^{4} - 77372 T^{6} + 707281 T^{8} \))(\( 1 + 10 T + 50 T^{2} + 338 T^{3} + 2295 T^{4} + 10460 T^{5} + 46972 T^{6} + 303340 T^{7} + 1930095 T^{8} + 8243482 T^{9} + 35364050 T^{10} + 205111490 T^{11} + 594823321 T^{12} \))(\( 1 - 4 T - 10 T^{2} - 29 T^{3} + 913 T^{4} + 8464 T^{5} - 67090 T^{6} - 83005 T^{7} - 78471 T^{8} + 13229031 T^{9} + 20120381 T^{10} - 459248940 T^{11} - 734819136 T^{12} + 5934733814 T^{13} + 72071415514 T^{14} - 88771674011 T^{15} - 2278696929308 T^{16} - 2574378546319 T^{17} + 60612060447274 T^{18} + 144742222989646 T^{19} - 519723613329216 T^{20} - 9419723436432060 T^{21} + 11968071846205301 T^{22} + 228199148437926579 T^{23} - 39254836271462631 T^{24} - 1204165651727006345 T^{25} - 28225248282110485090 T^{26} + \)\(10\!\cdots\!56\)\( T^{27} + \)\(32\!\cdots\!33\)\( T^{28} - \)\(29\!\cdots\!81\)\( T^{29} - \)\(29\!\cdots\!10\)\( T^{30} - \)\(34\!\cdots\!96\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 10 T + 29 T^{2} )^{2}( 1 + 40 T^{2} + 841 T^{4} ) \))(\( 1 - 20 T + 262 T^{2} - 2865 T^{3} + 26553 T^{4} - 218810 T^{5} + 1634626 T^{6} - 11342245 T^{7} + 74202867 T^{8} - 467497335 T^{9} + 2884176435 T^{10} - 17480048660 T^{11} + 105365484402 T^{12} - 627344579160 T^{13} + 3654811098552 T^{14} - 20729749116985 T^{15} + 113615735370380 T^{16} - 601162724392565 T^{17} + 3073696133882232 T^{18} - 15300306941133240 T^{19} + 74523005173330962 T^{20} - 358535882592510340 T^{21} + 1715575405416640635 T^{22} - 8064271203537136515 T^{23} + 37119718048172159187 T^{24} - \)\(16\!\cdots\!05\)\( T^{25} + \)\(68\!\cdots\!26\)\( T^{26} - \)\(26\!\cdots\!90\)\( T^{27} + \)\(93\!\cdots\!73\)\( T^{28} - \)\(29\!\cdots\!85\)\( T^{29} + \)\(77\!\cdots\!22\)\( T^{30} - \)\(17\!\cdots\!80\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 10 T + 71 T^{2} - 420 T^{3} + 2141 T^{4} - 12180 T^{5} + 59711 T^{6} - 243890 T^{7} + 707281 T^{8} )^{2}( 1 - 40 T^{2} + 759 T^{4} + 3280 T^{6} - 769519 T^{8} + 2758480 T^{10} + 536826279 T^{12} - 23792932840 T^{14} + 500246412961 T^{16} ) \))
$31$ (\( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 22072 T^{5} + 88412 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( ( 1 + 4 T + 38 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 4 T + 53 T^{2} - 232 T^{3} + 1643 T^{4} - 3844 T^{5} + 29791 T^{6} )^{2} \))(\( 1 - 3 T - 78 T^{2} + 318 T^{3} + 3477 T^{4} - 25592 T^{5} - 132457 T^{6} + 1485772 T^{7} + 2216173 T^{8} - 63263770 T^{9} + 90511528 T^{10} + 2178444357 T^{11} - 8206034651 T^{12} - 53022475824 T^{13} + 415755833622 T^{14} + 587289334612 T^{15} - 15398084325934 T^{16} + 18205969372972 T^{17} + 399541356110742 T^{18} - 1579592577272784 T^{19} - 7578445326926171 T^{20} + 62367012441650907 T^{21} + 80329314272934568 T^{22} - 1740551691217058470 T^{23} + 1890154089118733293 T^{24} + 39283250296904473012 T^{25} - \)\(10\!\cdots\!57\)\( T^{26} - \)\(65\!\cdots\!52\)\( T^{27} + \)\(27\!\cdots\!97\)\( T^{28} + \)\(77\!\cdots\!38\)\( T^{29} - \)\(59\!\cdots\!38\)\( T^{30} - \)\(70\!\cdots\!53\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( 1 - 3 T - 58 T^{2} + 298 T^{3} + 2045 T^{4} - 9320 T^{5} - 84713 T^{6} + 195904 T^{7} + 4105617 T^{8} - 3872962 T^{9} - 208627856 T^{10} + 219047229 T^{11} + 7596781121 T^{12} - 10652941876 T^{13} - 215328917398 T^{14} + 143234049680 T^{15} + 6349226698338 T^{16} + 4440255540080 T^{17} - 206931089619478 T^{18} - 317361791427916 T^{19} + 7015786897647041 T^{20} + 6271136195172579 T^{21} - 185157990159137936 T^{22} - 106555308972566782 T^{23} + 3501643942465406097 T^{24} + 5179627739764091584 T^{25} - 69433171075004595113 T^{26} - \)\(23\!\cdots\!20\)\( T^{27} + \)\(16\!\cdots\!45\)\( T^{28} + \)\(72\!\cdots\!18\)\( T^{29} - \)\(43\!\cdots\!18\)\( T^{30} - \)\(70\!\cdots\!53\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 31 T^{2} + 961 T^{4} - 29791 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( 1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 64380 T^{5} + 290228 T^{6} - 810448 T^{7} + 1874161 T^{8} \))(\( ( 1 + 6 T + 76 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 18 T + 183 T^{2} + 1250 T^{3} + 6771 T^{4} + 24642 T^{5} + 50653 T^{6} )^{2} \))(\( 1 - 4 T - 76 T^{2} + 372 T^{3} + 2378 T^{4} + 12512 T^{5} - 129251 T^{6} - 1973912 T^{7} + 10452357 T^{8} + 76097384 T^{9} - 79511934 T^{10} - 2512993184 T^{11} - 21506163267 T^{12} + 150699205628 T^{13} + 1019126372585 T^{14} - 3291661887676 T^{15} - 28600810704162 T^{16} - 121791489844012 T^{17} + 1395184004068865 T^{18} + 7633366862675084 T^{19} - 40306012454643987 T^{20} - 174260891292589088 T^{21} - 204005868894465006 T^{22} + 7224067508030720072 T^{23} + 36713689207547341797 T^{24} - \)\(25\!\cdots\!24\)\( T^{25} - \)\(62\!\cdots\!99\)\( T^{26} + \)\(22\!\cdots\!56\)\( T^{27} + \)\(15\!\cdots\!18\)\( T^{28} + \)\(90\!\cdots\!84\)\( T^{29} - \)\(68\!\cdots\!64\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 24 T^{2} + 1369 T^{4} )^{2} \))(\( 1 - 2 T - 74 T^{2} + 678 T^{3} - 780 T^{4} - 29004 T^{5} + 378159 T^{6} - 1092324 T^{7} - 11455407 T^{8} + 127580274 T^{9} - 441840318 T^{10} - 1264690140 T^{11} + 30961324455 T^{12} - 180620947362 T^{13} + 138027863589 T^{14} + 5063066119648 T^{15} - 41354097034082 T^{16} + 187333446426976 T^{17} + 188960145253341 T^{18} - 9148992846727386 T^{19} + 58026506801907255 T^{20} - 87698618686483980 T^{21} - 1133641372453558062 T^{22} + 12111434895962474442 T^{23} - 40236881723802800847 T^{24} - \)\(14\!\cdots\!48\)\( T^{25} + \)\(18\!\cdots\!91\)\( T^{26} - \)\(51\!\cdots\!52\)\( T^{27} - \)\(51\!\cdots\!80\)\( T^{28} + \)\(16\!\cdots\!66\)\( T^{29} - \)\(66\!\cdots\!86\)\( T^{30} - \)\(66\!\cdots\!86\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 24 T^{2} + 1369 T^{4} )^{4}( 1 - 24 T^{2} - 793 T^{4} + 51888 T^{6} - 159695 T^{8} + 71034672 T^{10} - 1486209673 T^{12} - 61577433816 T^{14} + 3512479453921 T^{16} ) \))
$41$ (\( ( 1 + T )^{4} \))(\( 1 - 4 T - 26 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 6 T + 63 T^{2} - 148 T^{3} + 2583 T^{4} - 10086 T^{5} + 68921 T^{6} \))(\( 1 + 36 T + 620 T^{2} + 7035 T^{3} + 61795 T^{4} + 474468 T^{5} + 3495443 T^{6} + 25199865 T^{7} + 169571570 T^{8} + 1033194465 T^{9} + 5875839683 T^{10} + 32700809028 T^{11} + 174617900995 T^{12} + 815048374035 T^{13} + 2945064629420 T^{14} + 7011153859716 T^{15} + 7984925229121 T^{16} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( 1 + 4 T - 8 T^{2} - 39 T^{3} + 781 T^{4} - 1780 T^{5} - 9077 T^{6} + 235635 T^{7} + 4795542 T^{8} + 9661035 T^{9} - 15258437 T^{10} - 122679380 T^{11} + 2206919341 T^{12} - 4518391839 T^{13} - 38000833928 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} \))(\( ( 1 + 10 T + 59 T^{2} + 180 T^{3} - 619 T^{4} + 7380 T^{5} + 99179 T^{6} + 689210 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 4 T + 124 T^{2} - 244 T^{3} + 6678 T^{4} - 10492 T^{5} + 229276 T^{6} - 318028 T^{7} + 3418801 T^{8} \))(\( ( 1 + 4 T + 62 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \))(\( 1 - 54 T^{2} + 5511 T^{4} - 175204 T^{6} + 10189839 T^{8} - 184615254 T^{10} + 6321363049 T^{12} \))(\( 1 + 4 T - 123 T^{2} - 979 T^{3} + 2319 T^{4} + 95741 T^{5} + 543998 T^{6} - 3647641 T^{7} - 55964878 T^{8} - 113362370 T^{9} + 2429111227 T^{10} + 20752429110 T^{11} - 8373305515 T^{12} - 1111261810689 T^{13} - 5521364012955 T^{14} + 21926633034508 T^{15} + 356690581829892 T^{16} + 942845220483844 T^{17} - 10209002059953795 T^{18} - 88353092782450323 T^{19} - 28626665267987515 T^{20} + 3050782291928975730 T^{21} + 15355293952268851123 T^{22} - 30814001965197843590 T^{23} - \)\(65\!\cdots\!78\)\( T^{24} - \)\(18\!\cdots\!63\)\( T^{25} + \)\(11\!\cdots\!02\)\( T^{26} + \)\(88\!\cdots\!87\)\( T^{27} + \)\(92\!\cdots\!19\)\( T^{28} - \)\(16\!\cdots\!97\)\( T^{29} - \)\(90\!\cdots\!27\)\( T^{30} + \)\(12\!\cdots\!28\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( ( 1 + 1849 T^{4} )^{2} \))(\( 1 + 22 T + 79 T^{2} - 1977 T^{3} - 21187 T^{4} + 36273 T^{5} + 1842424 T^{6} + 7386453 T^{7} - 81497976 T^{8} - 823659492 T^{9} + 946182427 T^{10} + 48714667904 T^{11} + 150258950255 T^{12} - 1852630882821 T^{13} - 14387507087295 T^{14} + 31395815128266 T^{15} + 753378404697000 T^{16} + 1350020050515438 T^{17} - 26602500604408455 T^{18} - 147297123600449247 T^{19} + 513705449390744255 T^{20} + 7161467479829113472 T^{21} + 5981162631650939923 T^{22} - \)\(22\!\cdots\!44\)\( T^{23} - \)\(95\!\cdots\!76\)\( T^{24} + \)\(37\!\cdots\!79\)\( T^{25} + \)\(39\!\cdots\!76\)\( T^{26} + \)\(33\!\cdots\!11\)\( T^{27} - \)\(84\!\cdots\!87\)\( T^{28} - \)\(33\!\cdots\!11\)\( T^{29} + \)\(58\!\cdots\!71\)\( T^{30} + \)\(69\!\cdots\!54\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( ( 1 - 1849 T^{4} + 3418801 T^{8} - 6321363049 T^{12} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 + 6 T + 126 T^{2} + 640 T^{3} + 8608 T^{4} + 30080 T^{5} + 278334 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( 1 - 134 T^{2} + 8900 T^{4} - 296006 T^{6} + 4879681 T^{8} \))(\( 1 + 26 T + 338 T^{2} + 3544 T^{3} + 33135 T^{4} + 258406 T^{5} + 1798894 T^{6} + 12145082 T^{7} + 73195215 T^{8} + 367948712 T^{9} + 1649332178 T^{10} + 5962970182 T^{11} + 10779215329 T^{12} \))(\( 1 + 19 T - 35 T^{2} - 2359 T^{3} - 744 T^{4} + 154777 T^{5} - 155832 T^{6} - 6853952 T^{7} + 32073378 T^{8} + 183108955 T^{9} - 2248864848 T^{10} + 1988958467 T^{11} + 89988181938 T^{12} - 460141966288 T^{13} - 2831031561300 T^{14} + 12236111216195 T^{15} + 101479363864500 T^{16} + 575097227161165 T^{17} - 6253748718911700 T^{18} - 47773319365919024 T^{19} + 439113621627401778 T^{20} + 456157693536824269 T^{21} - 24240998442410854992 T^{22} + 92767230166819046165 T^{23} + \)\(76\!\cdots\!58\)\( T^{24} - \)\(76\!\cdots\!84\)\( T^{25} - \)\(81\!\cdots\!68\)\( T^{26} + \)\(38\!\cdots\!31\)\( T^{27} - \)\(86\!\cdots\!04\)\( T^{28} - \)\(12\!\cdots\!93\)\( T^{29} - \)\(89\!\cdots\!15\)\( T^{30} + \)\(22\!\cdots\!17\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 4879681 T^{8} \))(\( 1 - 15 T + 319 T^{2} - 2235 T^{3} + 21716 T^{4} + 65835 T^{5} - 1013420 T^{6} + 23744670 T^{7} - 101995022 T^{8} + 692729475 T^{9} + 5996295850 T^{10} - 52425012705 T^{11} + 657222543244 T^{12} - 1547403488760 T^{13} + 1776164108904 T^{14} + 186789793556565 T^{15} - 1153542114136992 T^{16} + 8779120297158555 T^{17} + 3923546516568936 T^{18} - 160656072413529480 T^{19} + 3207036357039425164 T^{20} - 12023414905803313935 T^{21} + 64635364143539084650 T^{22} + \)\(35\!\cdots\!25\)\( T^{23} - \)\(24\!\cdots\!42\)\( T^{24} + \)\(26\!\cdots\!90\)\( T^{25} - \)\(53\!\cdots\!80\)\( T^{26} + \)\(16\!\cdots\!05\)\( T^{27} + \)\(25\!\cdots\!56\)\( T^{28} - \)\(12\!\cdots\!45\)\( T^{29} + \)\(81\!\cdots\!11\)\( T^{30} - \)\(18\!\cdots\!45\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 4879681 T^{8} + 23811286661761 T^{16} - \)\(11\!\cdots\!41\)\( T^{24} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 16 T + 212 T^{2} + 1824 T^{3} + 15558 T^{4} + 96672 T^{5} + 595508 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 - 108 T^{2} + 8422 T^{4} - 303372 T^{6} + 7890481 T^{8} \))(\( 1 + 14 T + 98 T^{2} + 134 T^{3} - 505 T^{4} + 15012 T^{5} + 268636 T^{6} + 795636 T^{7} - 1418545 T^{8} + 19949518 T^{9} + 773267138 T^{10} + 5854736902 T^{11} + 22164361129 T^{12} \))(\( 1 + 19 T + 89 T^{2} - 721 T^{3} - 5721 T^{4} - 3266 T^{5} - 473026 T^{6} - 6473862 T^{7} + 6702907 T^{8} + 356626567 T^{9} + 619006635 T^{10} + 1774417457 T^{11} + 175428764259 T^{12} + 698594053926 T^{13} - 7876419392802 T^{14} - 46310954757376 T^{15} + 21349633288564 T^{16} - 2454480602140928 T^{17} - 22124862074380818 T^{18} + 104004586966341102 T^{19} + 1384217331239118579 T^{20} + 742053383217921301 T^{21} + 13719886599387090915 T^{22} + \)\(41\!\cdots\!79\)\( T^{23} + \)\(41\!\cdots\!27\)\( T^{24} - \)\(21\!\cdots\!46\)\( T^{25} - \)\(82\!\cdots\!74\)\( T^{26} - \)\(30\!\cdots\!02\)\( T^{27} - \)\(28\!\cdots\!61\)\( T^{28} - \)\(18\!\cdots\!33\)\( T^{29} + \)\(12\!\cdots\!41\)\( T^{30} + \)\(13\!\cdots\!83\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 14 T + 53 T^{2} )^{2}( 1 + 56 T^{2} + 2809 T^{4} ) \))(\( 1 - 25 T + 413 T^{2} - 4205 T^{3} + 32887 T^{4} - 166490 T^{5} + 817862 T^{6} - 4406590 T^{7} + 65399227 T^{8} - 632468245 T^{9} + 5998127695 T^{10} - 35889696115 T^{11} + 268218227003 T^{12} - 1658499250930 T^{13} + 16763296531886 T^{14} - 122805038494960 T^{15} + 1088882590000420 T^{16} - 6508667040232880 T^{17} + 47088099958067774 T^{18} - 246912392980705610 T^{19} + 2116370824020858443 T^{20} - 15008909160432609695 T^{21} + \)\(13\!\cdots\!55\)\( T^{22} - \)\(74\!\cdots\!65\)\( T^{23} + \)\(40\!\cdots\!47\)\( T^{24} - \)\(14\!\cdots\!70\)\( T^{25} + \)\(14\!\cdots\!38\)\( T^{26} - \)\(15\!\cdots\!30\)\( T^{27} + \)\(16\!\cdots\!67\)\( T^{28} - \)\(10\!\cdots\!65\)\( T^{29} + \)\(56\!\cdots\!97\)\( T^{30} - \)\(18\!\cdots\!25\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 14 T + 143 T^{2} - 1260 T^{3} + 10061 T^{4} - 66780 T^{5} + 401687 T^{6} - 2084278 T^{7} + 7890481 T^{8} )^{2}( 1 - 56 T^{2} + 327 T^{4} + 138992 T^{6} - 8702095 T^{8} + 390428528 T^{10} + 2580187287 T^{12} - 1241204223224 T^{14} + 62259690411361 T^{16} ) \))
$59$ (\( 1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 117764 T^{5} + 877212 T^{6} - 2464548 T^{7} + 12117361 T^{8} \))(\( ( 1 + 4 T + 94 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 2 T + 157 T^{2} + 212 T^{3} + 9263 T^{4} + 6962 T^{5} + 205379 T^{6} )^{2} \))(\( 1 + 32 T + 357 T^{2} - 449 T^{3} - 49217 T^{4} - 332901 T^{5} + 2542570 T^{6} + 41717135 T^{7} - 39731852 T^{8} - 3681002918 T^{9} - 11296021029 T^{10} + 241586940986 T^{11} + 1766705223655 T^{12} - 9755069588037 T^{13} - 142398904326723 T^{14} + 180382034079664 T^{15} + 8829112951908052 T^{16} + 10642540010700176 T^{17} - 495690585961322763 T^{18} - 2003486436921451023 T^{19} + 21407804975613374455 T^{20} + \)\(17\!\cdots\!14\)\( T^{21} - \)\(47\!\cdots\!89\)\( T^{22} - \)\(91\!\cdots\!42\)\( T^{23} - \)\(58\!\cdots\!92\)\( T^{24} + \)\(36\!\cdots\!65\)\( T^{25} + \)\(12\!\cdots\!70\)\( T^{26} - \)\(10\!\cdots\!59\)\( T^{27} - \)\(87\!\cdots\!77\)\( T^{28} - \)\(47\!\cdots\!71\)\( T^{29} + \)\(22\!\cdots\!77\)\( T^{30} + \)\(11\!\cdots\!68\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 59 T^{2} )^{4} \))(\( 1 - 8 T + 15 T^{2} - 857 T^{3} + 6687 T^{4} - 41163 T^{5} + 671648 T^{6} - 2868903 T^{7} + 8726994 T^{8} - 173530180 T^{9} + 350103221 T^{10} + 1362124186 T^{11} - 23336798021 T^{12} + 942522742505 T^{13} - 6933616153029 T^{14} + 46988570559288 T^{15} - 517997837841536 T^{16} + 2772325662997992 T^{17} - 24135917828693949 T^{18} + 193574378332934395 T^{19} - 282780406204542581 T^{20} + 973815678826995614 T^{21} + 14767540691212957661 T^{22} - \)\(43\!\cdots\!20\)\( T^{23} + \)\(12\!\cdots\!74\)\( T^{24} - \)\(24\!\cdots\!17\)\( T^{25} + \)\(34\!\cdots\!48\)\( T^{26} - \)\(12\!\cdots\!17\)\( T^{27} + \)\(11\!\cdots\!47\)\( T^{28} - \)\(89\!\cdots\!03\)\( T^{29} + \)\(92\!\cdots\!15\)\( T^{30} - \)\(29\!\cdots\!92\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{4} \))
$61$ (\( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 294264 T^{5} + 1562820 T^{6} - 5447544 T^{7} + 13845841 T^{8} \))(\( ( 1 + 12 T + 130 T^{2} + 732 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 342 T^{2} + 50007 T^{4} - 4009636 T^{6} + 186076047 T^{8} - 4735277622 T^{10} + 51520374361 T^{12} \))(\( 1 - 6 T - 129 T^{2} + 283 T^{3} + 1137 T^{4} - 447 T^{5} + 1485624 T^{6} + 2853839 T^{7} - 108667902 T^{8} - 489408882 T^{9} - 1501966073 T^{10} - 21353827578 T^{11} + 520532127153 T^{12} + 5803442160327 T^{13} - 5692104110227 T^{14} - 239297160857716 T^{15} - 1045242970270992 T^{16} - 14597126812320676 T^{17} - 21180319394154667 T^{18} + 1317271104993182787 T^{19} + 7207205067952220673 T^{20} - 18035363784570588978 T^{21} - 77381854358481054353 T^{22} - \)\(15\!\cdots\!22\)\( T^{23} - \)\(20\!\cdots\!62\)\( T^{24} + \)\(33\!\cdots\!99\)\( T^{25} + \)\(10\!\cdots\!24\)\( T^{26} - \)\(19\!\cdots\!67\)\( T^{27} + \)\(30\!\cdots\!77\)\( T^{28} + \)\(45\!\cdots\!23\)\( T^{29} - \)\(12\!\cdots\!89\)\( T^{30} - \)\(36\!\cdots\!06\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 12 T + 61 T^{2} )^{2}( 1 - 10 T + 61 T^{2} )^{2} \))(\( 1 + 46 T + 1087 T^{2} + 17449 T^{3} + 208527 T^{4} + 1850343 T^{5} + 10881866 T^{6} + 17261585 T^{7} - 460530370 T^{8} - 6071458282 T^{9} - 36969498781 T^{10} - 32534160406 T^{11} + 1562486034213 T^{12} + 13365571910049 T^{13} + 222707996563 T^{14} - 1080062594520004 T^{15} - 12289078565267476 T^{16} - 65883818265720244 T^{17} + 828696455210923 T^{18} + 3033730877714832069 T^{19} + 21633933194433758133 T^{20} - 27478231535048258206 T^{21} - \)\(19\!\cdots\!41\)\( T^{22} - \)\(19\!\cdots\!22\)\( T^{23} - \)\(88\!\cdots\!70\)\( T^{24} + \)\(20\!\cdots\!85\)\( T^{25} + \)\(77\!\cdots\!66\)\( T^{26} + \)\(80\!\cdots\!23\)\( T^{27} + \)\(55\!\cdots\!67\)\( T^{28} + \)\(28\!\cdots\!69\)\( T^{29} + \)\(10\!\cdots\!67\)\( T^{30} + \)\(27\!\cdots\!46\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 10 T + 39 T^{2} - 220 T^{3} - 4579 T^{4} - 13420 T^{5} + 145119 T^{6} + 2269810 T^{7} + 13845841 T^{8} )^{2}( 1 + 12 T + 83 T^{2} + 264 T^{3} - 1895 T^{4} + 16104 T^{5} + 308843 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 28 T + 538 T^{2} - 6638 T^{3} + 64208 T^{4} - 444746 T^{5} + 2415082 T^{6} - 8421364 T^{7} + 20151121 T^{8} \))(\( 1 - 66 T^{2} + 6364 T^{4} - 296274 T^{6} + 20151121 T^{8} \))(\( 1 - 16 T + 128 T^{2} - 706 T^{3} - 4233 T^{4} + 120886 T^{5} - 1143134 T^{6} + 8099362 T^{7} - 19001937 T^{8} - 212338678 T^{9} + 2579343488 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))(\( 1 - 47 T + 811 T^{2} - 4523 T^{3} - 30671 T^{4} + 308708 T^{5} + 3794116 T^{6} - 49770346 T^{7} - 84884803 T^{8} + 3132538219 T^{9} + 1646397175 T^{10} - 174884317579 T^{11} - 212156414611 T^{12} + 15093468962842 T^{13} - 85324088095318 T^{14} + 84898977769442 T^{15} + 870026236538184 T^{16} + 5688231510552614 T^{17} - 383019831459882502 T^{18} + 4539557005671248446 T^{19} - 4275189581752428931 T^{20} - \)\(23\!\cdots\!53\)\( T^{21} + \)\(14\!\cdots\!75\)\( T^{22} + \)\(18\!\cdots\!37\)\( T^{23} - \)\(34\!\cdots\!23\)\( T^{24} - \)\(13\!\cdots\!62\)\( T^{25} + \)\(69\!\cdots\!84\)\( T^{26} + \)\(37\!\cdots\!64\)\( T^{27} - \)\(25\!\cdots\!31\)\( T^{28} - \)\(24\!\cdots\!01\)\( T^{29} + \)\(29\!\cdots\!19\)\( T^{30} - \)\(11\!\cdots\!21\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 20151121 T^{8} \))(\( 1 + 45 T + 971 T^{2} + 15715 T^{3} + 242357 T^{4} + 3491360 T^{5} + 44996702 T^{6} + 544413470 T^{7} + 6325993107 T^{8} + 69410222595 T^{9} + 725476220135 T^{10} + 7307681959045 T^{11} + 70480990733133 T^{12} + 653690636722980 T^{13} + 5867787032194624 T^{14} + 50626271793942110 T^{15} + 420725817995597460 T^{16} + 3391960210194121370 T^{17} + 26340495987521667136 T^{18} + \)\(19\!\cdots\!40\)\( T^{19} + \)\(14\!\cdots\!93\)\( T^{20} + \)\(98\!\cdots\!15\)\( T^{21} + \)\(65\!\cdots\!15\)\( T^{22} + \)\(42\!\cdots\!85\)\( T^{23} + \)\(25\!\cdots\!87\)\( T^{24} + \)\(14\!\cdots\!90\)\( T^{25} + \)\(82\!\cdots\!98\)\( T^{26} + \)\(42\!\cdots\!80\)\( T^{27} + \)\(19\!\cdots\!77\)\( T^{28} + \)\(86\!\cdots\!05\)\( T^{29} + \)\(35\!\cdots\!59\)\( T^{30} + \)\(11\!\cdots\!35\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 20151121 T^{8} + 406067677556641 T^{16} - \)\(81\!\cdots\!61\)\( T^{24} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 2 T + 98 T^{2} - 268 T^{3} + 3408 T^{4} - 19028 T^{5} + 494018 T^{6} + 715822 T^{7} + 25411681 T^{8} \))(\( 1 - 278 T^{2} + 29396 T^{4} - 1401398 T^{6} + 25411681 T^{8} \))(\( 1 - 26 T + 338 T^{2} - 3880 T^{3} + 40755 T^{4} - 348874 T^{5} + 2822734 T^{6} - 24770054 T^{7} + 205445955 T^{8} - 1388694680 T^{9} + 8589148178 T^{10} - 46909963126 T^{11} + 128100283921 T^{12} \))(\( 1 + 3 T - 49 T^{2} - 243 T^{3} - 2176 T^{4} - 635 T^{5} + 457522 T^{6} + 3318384 T^{7} - 2507220 T^{8} - 53196165 T^{9} + 245433972 T^{10} - 10241396805 T^{11} + 75857114136 T^{12} + 931062511948 T^{13} + 1628992425250 T^{14} + 8421955072503 T^{15} + 131763938090504 T^{16} + 597958810147713 T^{17} + 8211750815685250 T^{18} + 333237514713820628 T^{19} + 1927656786004622616 T^{20} - 18477828710818623555 T^{21} + 31440161497058764212 T^{22} - \)\(48\!\cdots\!15\)\( T^{23} - \)\(16\!\cdots\!20\)\( T^{24} + \)\(15\!\cdots\!04\)\( T^{25} + \)\(14\!\cdots\!22\)\( T^{26} - \)\(14\!\cdots\!85\)\( T^{27} - \)\(35\!\cdots\!16\)\( T^{28} - \)\(28\!\cdots\!73\)\( T^{29} - \)\(40\!\cdots\!69\)\( T^{30} + \)\(17\!\cdots\!53\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 + 25411681 T^{8} \))(\( 1 - 15 T + 333 T^{2} - 2115 T^{3} + 23468 T^{4} + 110895 T^{5} - 564886 T^{6} + 27387070 T^{7} - 18538168 T^{8} + 1092956195 T^{9} + 13634357530 T^{10} - 12169056285 T^{11} + 1382715900342 T^{12} + 2624673612340 T^{13} + 23918358394978 T^{14} + 1010473424039345 T^{15} - 3122417250610940 T^{16} + 71743613106793495 T^{17} + 120572444669084098 T^{18} + 939399557266221740 T^{19} + 35137135373118694902 T^{20} - 21955768523368021035 T^{21} + \)\(17\!\cdots\!30\)\( T^{22} + \)\(99\!\cdots\!45\)\( T^{23} - \)\(11\!\cdots\!48\)\( T^{24} + \)\(12\!\cdots\!70\)\( T^{25} - \)\(18\!\cdots\!86\)\( T^{26} + \)\(25\!\cdots\!45\)\( T^{27} + \)\(38\!\cdots\!88\)\( T^{28} - \)\(24\!\cdots\!65\)\( T^{29} + \)\(27\!\cdots\!73\)\( T^{30} - \)\(88\!\cdots\!65\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 25411681 T^{8} + 645753531245761 T^{16} - \)\(16\!\cdots\!41\)\( T^{24} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 77380 T^{5} + 1129748 T^{6} - 3112136 T^{7} + 28398241 T^{8} \))(\( ( 1 + 2 T + 84 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 18 T^{2} + 12063 T^{4} + 179360 T^{6} + 64283727 T^{8} + 511168338 T^{10} + 151334226289 T^{12} \))(\( ( 1 - 21 T + 603 T^{2} - 9152 T^{3} + 147444 T^{4} - 1753420 T^{5} + 20348516 T^{6} - 197328295 T^{7} + 1811783160 T^{8} - 14404965535 T^{9} + 108437241764 T^{10} - 682110188140 T^{11} + 4187150246004 T^{12} - 18972751219136 T^{13} + 91254538452267 T^{14} - 231995368901037 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 96 T^{2} + 5329 T^{4} )( 1 + 96 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 17 T + 485 T^{2} - 5868 T^{3} + 100534 T^{4} - 986446 T^{5} + 12875724 T^{6} - 106236101 T^{7} + 1127476896 T^{8} - 7755235373 T^{9} + 68614733196 T^{10} - 383744263582 T^{11} + 2854988760694 T^{12} - 12164784107724 T^{13} + 73397099750165 T^{14} - 187805774824649 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 96 T^{2} + 3887 T^{4} + 138432 T^{6} - 34003295 T^{8} + 737704128 T^{10} + 110383962767 T^{12} - 14528085723744 T^{14} + 806460091894081 T^{16} )( 1 + 96 T^{2} + 3887 T^{4} - 138432 T^{6} - 34003295 T^{8} - 737704128 T^{10} + 110383962767 T^{12} + 14528085723744 T^{14} + 806460091894081 T^{16} ) \))
$79$ (\( 1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 333696 T^{5} + 2284206 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( 1 - 198 T^{2} + 20260 T^{4} - 1235718 T^{6} + 38950081 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 196 T^{3} - 3937 T^{4} + 24602 T^{5} + 76286 T^{6} + 1943558 T^{7} - 24570817 T^{8} - 96635644 T^{9} + 77900162 T^{10} + 6154112798 T^{11} + 243087455521 T^{12} \))(\( ( 1 - 9 T + 426 T^{2} - 2721 T^{3} + 79731 T^{4} - 348427 T^{5} + 9182750 T^{6} - 28649979 T^{7} + 797866568 T^{8} - 2263348341 T^{9} + 57309542750 T^{10} - 171788099653 T^{11} + 3105528908211 T^{12} - 8372670461679 T^{13} + 103555256051946 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( 1 + 38950081 T^{8} \))(\( 1 - 737 T^{2} + 270304 T^{4} - 65719015 T^{6} + 11895746573 T^{8} - 1706613469745 T^{10} + 201629278078916 T^{12} - 20096005908122503 T^{14} + 1712905283658948508 T^{16} - \)\(12\!\cdots\!23\)\( T^{18} + \)\(78\!\cdots\!96\)\( T^{20} - \)\(41\!\cdots\!45\)\( T^{22} + \)\(18\!\cdots\!53\)\( T^{24} - \)\(62\!\cdots\!15\)\( T^{26} + \)\(15\!\cdots\!64\)\( T^{28} - \)\(27\!\cdots\!97\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 38950081 T^{8} )^{4} \))
$83$ (\( 1 + 12 T + 252 T^{2} + 1644 T^{3} + 24598 T^{4} + 136452 T^{5} + 1736028 T^{6} + 6861444 T^{7} + 47458321 T^{8} \))(\( ( 1 - 12 T + 83 T^{2} )^{4} \))(\( ( 1 - 18 T + 213 T^{2} - 1684 T^{3} + 17679 T^{4} - 124002 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 26 T + 743 T^{2} - 12894 T^{3} + 218641 T^{4} - 2860668 T^{5} + 35902097 T^{6} - 372552968 T^{7} + 3701971660 T^{8} - 30921896344 T^{9} + 247329546233 T^{10} - 1635692773716 T^{11} + 10376334761761 T^{12} - 50789990050842 T^{13} + 242916697413167 T^{14} - 705537325730302 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 6 T + 471 T^{2} - 3054 T^{3} + 107213 T^{4} - 687396 T^{5} + 15437769 T^{6} - 89913372 T^{7} + 1529216388 T^{8} - 7462809876 T^{9} + 106350790641 T^{10} - 393044096652 T^{11} + 5088148969373 T^{12} - 12029830123722 T^{13} + 153988915856799 T^{14} - 162816305937762 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 83 T^{2} )^{16} \))
$89$ (\( 1 - 4 T + 228 T^{2} - 796 T^{3} + 29014 T^{4} - 70844 T^{5} + 1805988 T^{6} - 2819876 T^{7} + 62742241 T^{8} \))(\( 1 - 20 T^{2} - 6010 T^{4} - 158420 T^{6} + 62742241 T^{8} \))(\( 1 - 14 T + 98 T^{2} + 346 T^{3} - 7777 T^{4} + 58980 T^{5} - 3716 T^{6} + 5249220 T^{7} - 61601617 T^{8} + 243919274 T^{9} + 6148739618 T^{10} - 78176832286 T^{11} + 496981290961 T^{12} \))(\( 1 - 16 T + 173 T^{2} - 3834 T^{3} + 54890 T^{4} - 646256 T^{5} + 9147323 T^{6} - 104888533 T^{7} + 1181415911 T^{8} - 14168265619 T^{9} + 146677989448 T^{10} - 1534312883224 T^{11} + 16223424290070 T^{12} - 157433822628107 T^{13} + 1592161339501311 T^{14} - 15445968366892687 T^{15} + 141532658635405802 T^{16} - 1374691184653449143 T^{17} + 12611509970189884431 T^{18} - \)\(11\!\cdots\!83\)\( T^{19} + \)\(10\!\cdots\!70\)\( T^{20} - \)\(85\!\cdots\!76\)\( T^{21} + \)\(72\!\cdots\!28\)\( T^{22} - \)\(62\!\cdots\!51\)\( T^{23} + \)\(46\!\cdots\!91\)\( T^{24} - \)\(36\!\cdots\!97\)\( T^{25} + \)\(28\!\cdots\!23\)\( T^{26} - \)\(17\!\cdots\!84\)\( T^{27} + \)\(13\!\cdots\!90\)\( T^{28} - \)\(84\!\cdots\!46\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} - \)\(27\!\cdots\!84\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 16 T + 89 T^{2} )^{2}( 1 - 160 T^{2} + 7921 T^{4} ) \))(\( 1 + 225 T^{2} + 980 T^{3} + 20690 T^{4} + 220500 T^{5} + 1287265 T^{6} + 31278295 T^{7} + 74234149 T^{8} + 3462955545 T^{9} + 13523503080 T^{10} + 281191192280 T^{11} + 2332233990710 T^{12} + 19198263573455 T^{13} + 290452934135005 T^{14} + 987525869517405 T^{15} + 29870868329825326 T^{16} + 87889802387049045 T^{17} + 2300677691283374605 T^{18} + 13534180673114997895 T^{19} + \)\(14\!\cdots\!10\)\( T^{20} + \)\(15\!\cdots\!20\)\( T^{21} + \)\(67\!\cdots\!80\)\( T^{22} + \)\(15\!\cdots\!05\)\( T^{23} + \)\(29\!\cdots\!69\)\( T^{24} + \)\(10\!\cdots\!55\)\( T^{25} + \)\(40\!\cdots\!65\)\( T^{26} + \)\(61\!\cdots\!00\)\( T^{27} + \)\(51\!\cdots\!90\)\( T^{28} + \)\(21\!\cdots\!20\)\( T^{29} + \)\(44\!\cdots\!25\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 16 T + 167 T^{2} - 1248 T^{3} + 5105 T^{4} - 111072 T^{5} + 1322807 T^{6} - 11279504 T^{7} + 62742241 T^{8} )^{2}( 1 + 160 T^{2} + 17679 T^{4} + 1561280 T^{6} + 109769441 T^{8} + 12366898880 T^{10} + 1109220078639 T^{12} + 79517006553760 T^{14} + 3936588805702081 T^{16} ) \))
$97$ (\( 1 - 16 T + 268 T^{2} - 3376 T^{3} + 38118 T^{4} - 327472 T^{5} + 2521612 T^{6} - 14602768 T^{7} + 88529281 T^{8} \))(\( 1 - 228 T^{2} + 30022 T^{4} - 2145252 T^{6} + 88529281 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 278 T^{3} + 13695 T^{4} + 101412 T^{5} + 400604 T^{6} + 9836964 T^{7} + 128856255 T^{8} + 253723094 T^{9} + 1593527058 T^{10} + 51524041542 T^{11} + 832972004929 T^{12} \))(\( 1 - 34 T + 333 T^{2} - 464 T^{3} + 21920 T^{4} - 684662 T^{5} + 6010820 T^{6} - 42261670 T^{7} + 703102016 T^{8} - 8796288432 T^{9} + 74149855339 T^{10} - 656351717246 T^{11} + 7388770535911 T^{12} - 88304483551232 T^{13} + 906328194719136 T^{14} - 7887931801020420 T^{15} + 71164816564652600 T^{16} - 765129384698980740 T^{17} + 8527641984112350624 T^{18} - 80593117916153563136 T^{19} + \)\(65\!\cdots\!91\)\( T^{20} - \)\(56\!\cdots\!22\)\( T^{21} + \)\(61\!\cdots\!31\)\( T^{22} - \)\(71\!\cdots\!16\)\( T^{23} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(32\!\cdots\!90\)\( T^{25} + \)\(44\!\cdots\!80\)\( T^{26} - \)\(48\!\cdots\!86\)\( T^{27} + \)\(15\!\cdots\!20\)\( T^{28} - \)\(31\!\cdots\!28\)\( T^{29} + \)\(21\!\cdots\!77\)\( T^{30} - \)\(21\!\cdots\!62\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 18 T + 97 T^{2} )^{2}( 1 - 144 T^{2} + 9409 T^{4} ) \))(\( 1 + 40 T + 1103 T^{2} + 23400 T^{3} + 417832 T^{4} + 6531120 T^{5} + 91158672 T^{6} + 1150615280 T^{7} + 13185683112 T^{8} + 138351028200 T^{9} + 1326877723405 T^{10} + 11643088476120 T^{11} + 92843992326243 T^{12} + 669334303943600 T^{13} + 4412489859579696 T^{14} + 28605503736845280 T^{15} + 230828447985637600 T^{16} + 2774733862473992160 T^{17} + 41517117088785359664 T^{18} + \)\(61\!\cdots\!00\)\( T^{19} + \)\(82\!\cdots\!83\)\( T^{20} + \)\(99\!\cdots\!40\)\( T^{21} + \)\(11\!\cdots\!45\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{23} + \)\(10\!\cdots\!32\)\( T^{24} + \)\(87\!\cdots\!60\)\( T^{25} + \)\(67\!\cdots\!28\)\( T^{26} + \)\(46\!\cdots\!60\)\( T^{27} + \)\(28\!\cdots\!12\)\( T^{28} + \)\(15\!\cdots\!00\)\( T^{29} + \)\(72\!\cdots\!07\)\( T^{30} + \)\(25\!\cdots\!20\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 144 T^{2} + 9409 T^{4} )^{4}( 1 - 18 T + 227 T^{2} - 2340 T^{3} + 20101 T^{4} - 226980 T^{5} + 2135843 T^{6} - 16428114 T^{7} + 88529281 T^{8} )^{2} \))
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