Properties

Label 100.2
Level 100
Weight 2
Dimension 141
Nonzero newspaces 6
Newform subspaces 10
Sturm bound 1200
Trace bound 1

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Defining parameters

Level: \( N \) = \( 100\( 100 = 2^{2} \cdot 5^{2} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 10 \)
Sturm bound: \(1200\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(100))\).

Total New Old
Modular forms 370 181 189
Cusp forms 231 141 90
Eisenstein series 139 40 99

Trace form

\( 141q - 6q^{2} + 4q^{3} - 10q^{4} - 15q^{5} - 18q^{6} - 4q^{7} - 18q^{8} - 22q^{9} + O(q^{10}) \) \( 141q - 6q^{2} + 4q^{3} - 10q^{4} - 15q^{5} - 18q^{6} - 4q^{7} - 18q^{8} - 22q^{9} - 16q^{10} - 10q^{12} - 20q^{13} - 10q^{14} - 2q^{15} - 2q^{16} - 30q^{17} + 2q^{18} - 12q^{19} - 6q^{20} - 48q^{21} - 10q^{22} - 32q^{23} - 57q^{25} - 36q^{26} - 38q^{27} - 10q^{28} - 52q^{29} - 10q^{30} - 12q^{31} - 26q^{32} - 40q^{33} - 10q^{34} - 8q^{35} - 10q^{36} + 9q^{37} + 20q^{38} + 48q^{39} + 44q^{40} + 4q^{41} + 90q^{42} + 60q^{43} + 60q^{44} + 55q^{45} + 42q^{46} + 52q^{47} + 120q^{48} + 51q^{49} + 94q^{50} + 36q^{51} + 88q^{52} + q^{53} + 120q^{54} + 40q^{55} + 42q^{56} + 44q^{57} + 44q^{58} + 6q^{59} + 90q^{60} - 88q^{61} + 40q^{62} - 14q^{63} + 20q^{64} - 65q^{65} + 30q^{66} - 44q^{67} + 14q^{68} - 66q^{69} - 10q^{70} - 16q^{71} - 16q^{72} - 20q^{73} - 78q^{75} - 60q^{76} - 100q^{77} + 20q^{78} - 56q^{79} - 26q^{80} + 54q^{81} - 82q^{82} - 62q^{83} - 90q^{84} + 27q^{85} - 78q^{86} + 14q^{87} - 130q^{88} + 77q^{89} - 166q^{90} + 12q^{91} - 110q^{92} + 74q^{93} - 170q^{94} + 36q^{95} - 118q^{96} + 144q^{97} - 158q^{98} + 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(100))\)

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
100.2.a \(\chi_{100}(1, \cdot)\) 100.2.a.a 1 1
100.2.c \(\chi_{100}(49, \cdot)\) 100.2.c.a 2 1
100.2.e \(\chi_{100}(7, \cdot)\) 100.2.e.a 2 2
100.2.e.b 2
100.2.e.c 2
100.2.e.d 8
100.2.g \(\chi_{100}(21, \cdot)\) 100.2.g.a 12 4
100.2.i \(\chi_{100}(9, \cdot)\) 100.2.i.a 8 4
100.2.l \(\chi_{100}(3, \cdot)\) 100.2.l.a 8 8
100.2.l.b 96

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(100))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(100)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 7 T^{4} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 - 17 T^{4} + 81 T^{8} )^{2} \))(\( 1 - 2 T - T^{2} + 13 T^{3} - 28 T^{4} + 29 T^{5} + 19 T^{6} - 194 T^{7} + 455 T^{8} - 583 T^{9} - 49 T^{10} + 2153 T^{11} - 5330 T^{12} + 6459 T^{13} - 441 T^{14} - 15741 T^{15} + 36855 T^{16} - 47142 T^{17} + 13851 T^{18} + 63423 T^{19} - 183708 T^{20} + 255879 T^{21} - 59049 T^{22} - 354294 T^{23} + 531441 T^{24} \))(\( 1 + 2 T^{2} - 5 T^{3} + 5 T^{4} - 10 T^{5} + 27 T^{6} - 85 T^{7} + 34 T^{8} - 255 T^{9} + 243 T^{10} - 270 T^{11} + 405 T^{12} - 1215 T^{13} + 1458 T^{14} + 6561 T^{16} \))(\( 1 - 9 T^{4} + 81 T^{8} - 729 T^{12} + 6561 T^{16} \))
$5$ (\( 1 + 4 T + 21 T^{2} + 74 T^{3} + 226 T^{4} + 630 T^{5} + 1440 T^{6} + 3150 T^{7} + 5650 T^{8} + 9250 T^{9} + 13125 T^{10} + 12500 T^{11} + 15625 T^{12} \))(\( 1 - 5 T + 15 T^{2} - 35 T^{3} + 80 T^{4} - 175 T^{5} + 375 T^{6} - 625 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 11 T^{2} - 24 T^{3} + 41 T^{4} - 120 T^{5} + 275 T^{6} - 500 T^{7} + 625 T^{8} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 + 49 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} \))(\( ( 1 + 49 T^{4} )^{4} \))(\( ( 1 + T + 13 T^{2} + 17 T^{3} + 167 T^{4} + 208 T^{5} + 1174 T^{6} + 1456 T^{7} + 8183 T^{8} + 5831 T^{9} + 31213 T^{10} + 16807 T^{11} + 117649 T^{12} )^{2} \))(\( 1 - 35 T^{2} + 551 T^{4} - 5425 T^{6} + 41176 T^{8} - 265825 T^{10} + 1322951 T^{12} - 4117715 T^{14} + 5764801 T^{16} \))(\( ( 1 + 49 T^{4} )^{4} \))
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} + 121 T^{4} )^{4} \))(\( 1 + 5 T + 7 T^{2} + 65 T^{3} + 351 T^{4} + 1490 T^{5} + 6090 T^{6} + 16160 T^{7} + 72390 T^{8} + 311790 T^{9} + 1151332 T^{10} + 3608920 T^{11} + 9338114 T^{12} + 39698120 T^{13} + 139311172 T^{14} + 414992490 T^{15} + 1059861990 T^{16} + 2602584160 T^{17} + 10788806490 T^{18} + 29035884790 T^{19} + 75239967231 T^{20} + 153266599915 T^{21} + 181561972207 T^{22} + 1426558353055 T^{23} + 3138428376721 T^{24} \))(\( 1 - 5 T + 8 T^{2} - 20 T^{3} + 188 T^{4} - 655 T^{5} + 681 T^{6} - 3600 T^{7} + 23500 T^{8} - 39600 T^{9} + 82401 T^{10} - 871805 T^{11} + 2752508 T^{12} - 3221020 T^{13} + 14172488 T^{14} - 97435855 T^{15} + 214358881 T^{16} \))(\( ( 1 + 11 T^{2} + 121 T^{4} + 1331 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 + 169 T^{4} \))(\( ( 1 - 142 T^{4} + 28561 T^{8} )^{2} \))(\( 1 + 2 T - 16 T^{2} + 2 T^{3} - 53 T^{4} + 6 T^{5} + 7544 T^{6} + 1784 T^{7} - 56280 T^{8} + 53458 T^{9} - 477164 T^{10} + 428612 T^{11} + 19743920 T^{12} + 5571956 T^{13} - 80640716 T^{14} + 117447226 T^{15} - 1607413080 T^{16} + 662386712 T^{17} + 36413447096 T^{18} + 376491102 T^{19} - 43233728213 T^{20} + 21208998746 T^{21} - 2205735869584 T^{22} + 3584320788074 T^{23} + 23298085122481 T^{24} \))(\( 1 + 7 T^{2} + 50 T^{3} + 105 T^{4} + 350 T^{5} + 2317 T^{6} + 13880 T^{7} + 12484 T^{8} + 180440 T^{9} + 391573 T^{10} + 768950 T^{11} + 2998905 T^{12} + 18564650 T^{13} + 33787663 T^{14} + 815730721 T^{16} \))(\( ( 1 - 6 T + 23 T^{2} - 60 T^{3} + 61 T^{4} - 780 T^{5} + 3887 T^{6} - 13182 T^{7} + 28561 T^{8} )( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 520 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} ) \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 + 289 T^{4} \))(\( ( 1 - 2 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 + 289 T^{4} \))(\( ( 1 + 383 T^{4} + 83521 T^{8} )^{2} \))(\( 1 - T - 9 T^{2} + 51 T^{3} + 507 T^{4} - 718 T^{5} - 6864 T^{6} + 9252 T^{7} + 172520 T^{8} - 281314 T^{9} - 3362156 T^{10} - 273914 T^{11} + 40433450 T^{12} - 4656538 T^{13} - 971663084 T^{14} - 1382095682 T^{15} + 14409042920 T^{16} + 13136516964 T^{17} - 165680273616 T^{18} - 294623167214 T^{19} + 3536709022587 T^{20} + 6047981701347 T^{21} - 18143945104041 T^{22} - 34271896307633 T^{23} + 582622237229761 T^{24} \))(\( 1 + 5 T + 18 T^{2} + 60 T^{3} + 20 T^{4} - 315 T^{5} - 5207 T^{6} - 17930 T^{7} - 24376 T^{8} - 304810 T^{9} - 1504823 T^{10} - 1547595 T^{11} + 1670420 T^{12} + 85191420 T^{13} + 434476242 T^{14} + 2051693365 T^{15} + 6975757441 T^{16} \))(\( ( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 1020 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} )( 1 + 8 T + 47 T^{2} + 240 T^{3} + 1121 T^{4} + 4080 T^{5} + 13583 T^{6} + 39304 T^{7} + 83521 T^{8} ) \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} + 361 T^{4} )^{4} \))(\( 1 + 8 T - 6 T^{2} - 258 T^{3} - 939 T^{4} - 1272 T^{5} + 8744 T^{6} + 96932 T^{7} + 222296 T^{8} - 392992 T^{9} + 160794 T^{10} - 5760978 T^{11} - 96272964 T^{12} - 109458582 T^{13} + 58046634 T^{14} - 2695532128 T^{15} + 28969837016 T^{16} + 240013228268 T^{17} + 411369183464 T^{18} - 1137004852008 T^{19} - 15947565695499 T^{20} - 83253426026982 T^{21} - 36786397546806 T^{22} + 931922071185752 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 8 T + 55 T^{2} + 330 T^{3} + 1905 T^{4} + 10214 T^{5} + 50917 T^{6} + 225500 T^{7} + 1025380 T^{8} + 4284500 T^{9} + 18381037 T^{10} + 70057826 T^{11} + 248261505 T^{12} + 817112670 T^{13} + 2587523455 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 529 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} \))(\( ( 1 - 12 T + 72 T^{2} - 276 T^{3} + 529 T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 276 T^{3} + 529 T^{4} )^{2} \))(\( 1 + 6 T + 13 T^{2} - 155 T^{3} - 428 T^{4} - 3269 T^{5} - 9845 T^{6} - 111568 T^{7} + 95053 T^{8} - 1251629 T^{9} + 5100635 T^{10} + 36361407 T^{11} + 637218038 T^{12} + 836312361 T^{13} + 2698235915 T^{14} - 15228570043 T^{15} + 26599726573 T^{16} - 718089915824 T^{17} - 1457413327205 T^{18} - 11130374386243 T^{19} - 33517101700268 T^{20} - 279178662526765 T^{21} + 538544645777437 T^{22} + 5716858547483562 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 20 T + 180 T^{2} + 695 T^{3} - 1459 T^{4} - 30080 T^{5} - 120345 T^{6} + 100285 T^{7} + 2412686 T^{8} + 2306555 T^{9} - 63662505 T^{10} - 365983360 T^{11} - 408288019 T^{12} + 4473258385 T^{13} + 26646460020 T^{14} + 68096508940 T^{15} + 78310985281 T^{16} \))(\( 1 - 529 T^{4} + 279841 T^{8} - 148035889 T^{12} + 78310985281 T^{16} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{4} \))(\( 1 + 18 T + 154 T^{2} + 1212 T^{3} + 10291 T^{4} + 77478 T^{5} + 547974 T^{6} + 3752112 T^{7} + 23821306 T^{8} + 144955698 T^{9} + 863261644 T^{10} + 4979066562 T^{11} + 27518493276 T^{12} + 144392930298 T^{13} + 726003042604 T^{14} + 3535324518522 T^{15} + 16848357128986 T^{16} + 76960128296688 T^{17} + 325947714501654 T^{18} + 1336485916668702 T^{19} + 5148035835781651 T^{20} + 17582660922753228 T^{21} + 64788913928230954 T^{22} + 219609175782704922 T^{23} + 353814783205469041 T^{24} \))(\( 1 + 8 T + 5 T^{2} - 200 T^{3} - 845 T^{4} + 8104 T^{5} + 60447 T^{6} - 21800 T^{7} - 1138800 T^{8} - 632200 T^{9} + 50835927 T^{10} + 197648456 T^{11} - 597652445 T^{12} - 4102229800 T^{13} + 2974116605 T^{14} + 137999010472 T^{15} + 500246412961 T^{16} \))(\( ( 1 - 10 T + 71 T^{2} - 420 T^{3} + 2141 T^{4} - 12180 T^{5} + 59711 T^{6} - 243890 T^{7} + 707281 T^{8} )( 1 + 10 T + 71 T^{2} + 420 T^{3} + 2141 T^{4} + 12180 T^{5} + 59711 T^{6} + 243890 T^{7} + 707281 T^{8} ) \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{4}( 1 + 8 T + 31 T^{2} )^{4} \))(\( 1 - 12 T + 13 T^{2} + 223 T^{3} + 326 T^{4} - 4537 T^{5} - 5551 T^{6} - 52236 T^{7} - 541651 T^{8} + 6074353 T^{9} + 33540303 T^{10} + 62852409 T^{11} - 3029364626 T^{12} + 1948424679 T^{13} + 32232231183 T^{14} + 180961050223 T^{15} - 500226073171 T^{16} - 1495472331636 T^{17} - 4926532933231 T^{18} - 124824730221607 T^{19} + 278042478205766 T^{20} + 5896035741829633 T^{21} + 10655167730750413 T^{22} - 304901722756857972 T^{23} + 787662783788549761 T^{24} \))(\( 1 + 12 T - 4 T^{2} - 527 T^{3} - 929 T^{4} + 16076 T^{5} + 97629 T^{6} - 73041 T^{7} - 2777218 T^{8} - 2264271 T^{9} + 93821469 T^{10} + 478920116 T^{11} - 857951009 T^{12} - 15087562577 T^{13} - 3550014724 T^{14} + 330151369332 T^{15} + 852891037441 T^{16} \))(\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 - 2 T + 37 T^{2} ) \))(\( 1 + 1369 T^{4} \))(\( ( 1 - 2062 T^{4} + 1874161 T^{8} )^{2} \))(\( 1 - 13 T - 33 T^{2} + 1330 T^{3} - 5308 T^{4} - 45413 T^{5} + 538870 T^{6} - 1058624 T^{7} - 16353842 T^{8} + 128132797 T^{9} - 180178425 T^{10} - 2773787169 T^{11} + 25187757138 T^{12} - 102630125253 T^{13} - 246664263825 T^{14} + 6490310566441 T^{15} - 30649732876562 T^{16} - 73409177135168 T^{17} + 1382592990017830 T^{18} - 4311141336240929 T^{19} - 18644240941412668 T^{20} + 172849113927452410 T^{21} - 158683284289789017 T^{22} - 2312929083132985369 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 10 T + 140 T^{2} + 1685 T^{3} + 12811 T^{4} + 119510 T^{5} + 861615 T^{6} + 5562535 T^{7} + 38923586 T^{8} + 205813795 T^{9} + 1179550935 T^{10} + 6053540030 T^{11} + 24009876571 T^{12} + 116844567545 T^{13} + 359201697260 T^{14} + 949318771330 T^{15} + 3512479453921 T^{16} \))(\( ( 1 + 12 T + 37 T^{2} )^{4}( 1 - 2 T - 33 T^{2} + 140 T^{3} + 941 T^{4} + 5180 T^{5} - 45177 T^{6} - 101306 T^{7} + 1874161 T^{8} ) \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{8} \))(\( 1 + 23 T + 183 T^{2} + 523 T^{3} + 2171 T^{4} + 47618 T^{5} + 413784 T^{6} + 2192484 T^{7} + 17878464 T^{8} + 165471518 T^{9} + 998460948 T^{10} + 4101545294 T^{11} + 18493818874 T^{12} + 168163357054 T^{13} + 1678412853588 T^{14} + 11404462492078 T^{15} + 50520266311104 T^{16} + 254012866993284 T^{17} + 1965517133257944 T^{18} + 9273809013665458 T^{19} + 17335272672421691 T^{20} + 171220751688041603 T^{21} + 2456346653757889383 T^{22} + 12657567729473714143 T^{23} + 22563490300366186081 T^{24} \))(\( 1 - 13 T + 36 T^{2} + 388 T^{3} - 4124 T^{4} + 25421 T^{5} - 46041 T^{6} - 1100136 T^{7} + 12326712 T^{8} - 45105576 T^{9} - 77394921 T^{10} + 1752040741 T^{11} - 11653438364 T^{12} + 44952205988 T^{13} + 171003752676 T^{14} - 2531805560453 T^{15} + 7984925229121 T^{16} \))(\( ( 1 - 8 T + 23 T^{2} + 144 T^{3} - 2095 T^{4} + 5904 T^{5} + 38663 T^{6} - 551368 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 10 T + 43 T^{2} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 18 T + 162 T^{2} - 774 T^{3} + 1849 T^{4} \))(\( 1 + 1849 T^{4} \))(\( 1 + 18 T + 162 T^{2} + 774 T^{3} + 1849 T^{4} \))(\( ( 1 + 1849 T^{4} )^{4} \))(\( ( 1 - 25 T + 443 T^{2} - 5515 T^{3} + 56835 T^{4} - 475510 T^{5} + 3415010 T^{6} - 20446930 T^{7} + 105087915 T^{8} - 438481105 T^{9} + 1514528843 T^{10} - 3675211075 T^{11} + 6321363049 T^{12} )^{2} \))(\( 1 - 219 T^{2} + 21607 T^{4} - 1334737 T^{6} + 62871560 T^{8} - 2467928713 T^{10} + 73870033207 T^{12} - 1384378507731 T^{14} + 11688200277601 T^{16} \))(\( ( 1 + 1849 T^{4} )^{4} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( 1 + 14 T + 98 T^{2} + 658 T^{3} + 2209 T^{4} \))(\( 1 + 2209 T^{4} \))(\( 1 - 14 T + 98 T^{2} - 658 T^{3} + 2209 T^{4} \))(\( ( 1 + 1058 T^{4} + 4879681 T^{8} )^{2} \))(\( 1 - T - 69 T^{2} - 239 T^{3} + 6477 T^{4} + 4432 T^{5} - 387564 T^{6} - 766988 T^{7} + 20113500 T^{8} + 34196416 T^{9} - 1070828686 T^{10} + 468011586 T^{11} + 48081764570 T^{12} + 21996544542 T^{13} - 2365460567374 T^{14} + 3550374498368 T^{15} + 98147463793500 T^{16} - 175904868228916 T^{17} - 4177635809768556 T^{18} + 2245353669892016 T^{19} + 154225703708225997 T^{20} - 267472183071561313 T^{21} - 3629340124272273381 T^{22} - 2472159215084012303 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 45 T + 938 T^{2} - 11970 T^{3} + 107730 T^{4} - 813105 T^{5} + 6411283 T^{6} - 54084780 T^{7} + 409062224 T^{8} - 2541984660 T^{9} + 14162524147 T^{10} - 84419000415 T^{11} + 525688034130 T^{12} - 2745259733790 T^{13} + 10110903978602 T^{14} - 22798040420835 T^{15} + 23811286661761 T^{16} \))(\( 1 - 2209 T^{4} + 4879681 T^{8} - 10779215329 T^{12} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 + 2809 T^{4} \))(\( ( 1 + 4 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))(\( 1 + 2809 T^{4} \))(\( ( 1 - 20 T + 200 T^{2} - 1060 T^{3} + 2809 T^{4} )^{2}( 1 + 20 T + 200 T^{2} + 1060 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 21 T + 113 T^{2} + 480 T^{3} - 3818 T^{4} - 66081 T^{5} + 764610 T^{6} - 1231332 T^{7} - 1322382 T^{8} - 297972771 T^{9} + 3254242025 T^{10} - 4778141607 T^{11} - 77922774362 T^{12} - 253241505171 T^{13} + 9141165848225 T^{14} - 44361292228167 T^{15} - 10434230045742 T^{16} - 514937492786676 T^{17} + 16947092162844690 T^{18} - 77626086831568797 T^{19} - 237707497990576298 T^{20} + 1583886524065023840 T^{21} + 19762284151302974537 T^{22} - \)\(19\!\cdots\!37\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 30 T + 600 T^{2} - 8895 T^{3} + 108851 T^{4} - 1137030 T^{5} + 10519575 T^{6} - 87403425 T^{7} + 665807146 T^{8} - 4632381525 T^{9} + 29549486175 T^{10} - 169277615310 T^{11} + 858886747331 T^{12} - 3719848910235 T^{13} + 13298616677400 T^{14} - 35241334195110 T^{15} + 62259690411361 T^{16} \))(\( ( 1 - 4 T + 53 T^{2} )^{4}( 1 + 14 T + 143 T^{2} + 1260 T^{3} + 10061 T^{4} + 66780 T^{5} + 401687 T^{6} + 2084278 T^{7} + 7890481 T^{8} ) \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 58 T^{2} + 3481 T^{4} )^{4} \))(\( 1 - 9 T + 7 T^{2} + 529 T^{3} - 4049 T^{4} - 12814 T^{5} + 446296 T^{6} - 2350008 T^{7} - 10129056 T^{8} + 281765226 T^{9} - 1540417248 T^{10} - 6251689178 T^{11} + 130056196874 T^{12} - 368849661502 T^{13} - 5362192440288 T^{14} + 57868660350654 T^{15} - 122737428141216 T^{16} - 1680077822044392 T^{17} + 18825003441843736 T^{18} - 31889580126470666 T^{19} - 594516441859895729 T^{20} + 4582724788068462731 T^{21} + 3577817273104489807 T^{22} - \)\(27\!\cdots\!31\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 9 T - 16 T^{2} - 534 T^{3} + 11816 T^{4} - 58317 T^{5} + 169591 T^{6} - 4043232 T^{7} + 51996952 T^{8} - 238550688 T^{9} + 590346271 T^{10} - 11977087143 T^{11} + 143178737576 T^{12} - 381769575666 T^{13} - 674888538256 T^{14} - 22397863363371 T^{15} + 146830437604321 T^{16} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} - 205379 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{8} \))(\( 1 + 26 T + 211 T^{2} - 939 T^{3} - 25344 T^{4} - 117039 T^{5} + 38221 T^{6} - 4961444 T^{7} - 59685069 T^{8} + 405621811 T^{9} + 7923270111 T^{10} + 10091674181 T^{11} - 277516312934 T^{12} + 615592125041 T^{13} + 29482488083031 T^{14} + 92068444282591 T^{15} - 826389975448029 T^{16} - 4190417250018644 T^{17} + 1969160228451781 T^{18} - 367823478785061819 T^{19} - 4858630140603089664 T^{20} - 10980803181171258399 T^{21} + \)\(15\!\cdots\!11\)\( T^{22} + \)\(11\!\cdots\!86\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 - 16 T + 1485 T^{3} - 8915 T^{4} + 31232 T^{5} - 252357 T^{6} - 3665185 T^{7} + 71554830 T^{8} - 223576285 T^{9} - 939020397 T^{10} + 7089070592 T^{11} - 123435672515 T^{12} + 1254225506985 T^{13} - 50283885376336 T^{15} + 191707312997281 T^{16} \))(\( ( 1 + 12 T + 83 T^{2} + 264 T^{3} - 1895 T^{4} + 16104 T^{5} + 308843 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 2 T + 67 T^{2} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 + 4489 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 402 T^{3} + 4489 T^{4} \))(\( ( 1 - 1057 T^{4} + 20151121 T^{8} )^{2} \))(\( 1 + 37 T + 737 T^{2} + 11665 T^{3} + 164147 T^{4} + 2062122 T^{5} + 23810210 T^{6} + 259823016 T^{7} + 2675533158 T^{8} + 25966581222 T^{9} + 239313744080 T^{10} + 2101508568736 T^{11} + 17602530247678 T^{12} + 140801074105312 T^{13} + 1074279397175120 T^{14} + 7809786868072386 T^{15} + 53914992406370118 T^{16} + 350793577278062712 T^{17} + 2153833075704145490 T^{18} + 12497926736991875406 T^{19} + 66654791067889950227 T^{20} + \)\(31\!\cdots\!55\)\( T^{21} + \)\(13\!\cdots\!13\)\( T^{22} + \)\(45\!\cdots\!71\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 5 T + 180 T^{2} + 990 T^{3} + 12196 T^{4} + 74305 T^{5} + 246935 T^{6} + 2417280 T^{7} - 7902424 T^{8} + 161957760 T^{9} + 1108491215 T^{10} + 22348194715 T^{11} + 245763071716 T^{12} + 1336623855930 T^{13} + 16282508790420 T^{14} + 30303558026615 T^{15} + 406067677556641 T^{16} \))(\( 1 - 4489 T^{4} + 20151121 T^{8} - 90458382169 T^{12} + 406067677556641 T^{16} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 82 T^{2} + 5041 T^{4} )^{4} \))(\( 1 - 21 T + T^{2} + 1349 T^{3} + 10771 T^{4} - 67866 T^{5} - 264144 T^{6} - 20842296 T^{7} + 46063856 T^{8} + 1792845654 T^{9} + 7389179196 T^{10} - 96304225006 T^{11} - 413283203394 T^{12} - 6837599975426 T^{13} + 37248852327036 T^{14} + 641679180868794 T^{15} + 1170560014301936 T^{16} - 37604282185429896 T^{17} - 33836921396028624 T^{18} - 617249424669363606 T^{19} + 6955411285048091731 T^{20} + 61849627469187742819 T^{21} + 3255243551009881201 T^{22} - \)\(48\!\cdots\!91\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + T + 170 T^{2} + 410 T^{3} + 25030 T^{4} + 53273 T^{5} + 2381343 T^{6} + 5547800 T^{7} + 194260320 T^{8} + 393893800 T^{9} + 12004350063 T^{10} + 19066992703 T^{11} + 636054375430 T^{12} + 739734033910 T^{13} + 21777048266570 T^{14} + 9095120158391 T^{15} + 645753531245761 T^{16} \))(\( ( 1 + 71 T^{2} + 5041 T^{4} + 357911 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 - 6 T + 73 T^{2} ) \))(\( 1 + 5329 T^{4} \))(\( ( 1 + 3503 T^{4} + 28398241 T^{8} )^{2} \))(\( 1 - 18 T - 11 T^{2} + 1697 T^{3} - 6318 T^{4} - 96309 T^{5} + 2090249 T^{6} - 15143916 T^{7} - 53445245 T^{8} + 1395629633 T^{9} - 5747246199 T^{10} - 79066630723 T^{11} + 1328147128830 T^{12} - 5771864042779 T^{13} - 30627074994471 T^{14} + 542923652940761 T^{15} - 1517750947814045 T^{16} - 31394422066378188 T^{17} + 316326215166355961 T^{18} - 1063963903975712973 T^{19} - 5095214860586803758 T^{20} + 99905082643930648361 T^{21} - 47273884126739134139 T^{22} - \)\(56\!\cdots\!86\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 60 T + 1772 T^{2} + 33525 T^{3} + 445815 T^{4} + 4298160 T^{5} + 30003487 T^{6} + 155927835 T^{7} + 910902194 T^{8} + 11382731955 T^{9} + 159888582223 T^{10} + 1672057308720 T^{11} + 12660361811415 T^{12} + 69499725155325 T^{13} + 268164248984108 T^{14} + 662843911145820 T^{15} + 806460091894081 T^{16} \))(\( ( 1 - 16 T + 183 T^{2} - 1760 T^{3} + 14801 T^{4} - 128480 T^{5} + 975207 T^{6} - 6224272 T^{7} + 28398241 T^{8} )( 1 - 6 T - 37 T^{2} + 660 T^{3} - 1259 T^{4} + 48180 T^{5} - 197173 T^{6} - 2334102 T^{7} + 28398241 T^{8} ) \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 82 T^{2} + 6241 T^{4} )^{4} \))(\( 1 + 24 T + 252 T^{2} + 2216 T^{3} + 27111 T^{4} + 297424 T^{5} + 2287436 T^{6} + 13926068 T^{7} + 93806084 T^{8} + 810763724 T^{9} + 6428940772 T^{10} + 17289470648 T^{11} - 104116056176 T^{12} + 1365868181192 T^{13} + 40123019358052 T^{14} + 399738135717236 T^{15} + 3653754570092804 T^{16} + 42851296652309132 T^{17} + 556046996907134156 T^{18} + 5711703426299354416 T^{19} + 41130336945376775271 T^{20} + \)\(26\!\cdots\!04\)\( T^{21} + \)\(23\!\cdots\!52\)\( T^{22} + \)\(17\!\cdots\!96\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 24 T + 99 T^{2} - 1326 T^{3} - 979 T^{4} + 289302 T^{5} + 3270321 T^{6} + 4493532 T^{7} - 130612868 T^{8} + 354989028 T^{9} + 20410073361 T^{10} + 142637168778 T^{11} - 38132129299 T^{12} - 4080176785074 T^{13} + 24065658096579 T^{14} + 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 + 22 T + 242 T^{2} + 1826 T^{3} + 6889 T^{4} \))(\( 1 + 6889 T^{4} \))(\( 1 - 22 T + 242 T^{2} - 1826 T^{3} + 6889 T^{4} \))(\( ( 1 + 12143 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 + 46 T + 913 T^{2} + 11375 T^{3} + 136602 T^{4} + 1946041 T^{5} + 23993235 T^{6} + 220675522 T^{7} + 1922016543 T^{8} + 21103229401 T^{9} + 221198443925 T^{10} + 1760498099887 T^{11} + 13654984203298 T^{12} + 146121342290621 T^{13} + 1523836080199325 T^{14} + 12066552229509587 T^{15} + 91215678065004303 T^{16} + 869249850073240646 T^{17} + 7844357209230158715 T^{18} + 52807867803904716707 T^{19} + \)\(30\!\cdots\!82\)\( T^{20} + \)\(21\!\cdots\!25\)\( T^{21} + \)\(14\!\cdots\!37\)\( T^{22} + \)\(59\!\cdots\!82\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 10 T + 180 T^{2} - 815 T^{3} - 7979 T^{4} - 308990 T^{5} + 290165 T^{6} + 231275 T^{7} + 274435626 T^{8} + 19195825 T^{9} + 1998946685 T^{10} - 176676465130 T^{11} - 378669943259 T^{12} - 3210318124045 T^{13} + 58849267206420 T^{14} + 271360509896270 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 6889 T^{4} + 47458321 T^{8} - 326940373369 T^{12} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( 1 - 142 T^{2} + 7921 T^{4} \))(\( ( 1 - 10 T + 89 T^{2} )( 1 + 10 T + 89 T^{2} ) \))(\( 1 - 142 T^{2} + 7921 T^{4} \))(\( ( 1 - 97 T^{2} + 7921 T^{4} )^{4} \))(\( 1 + 2 T + 34 T^{2} + 98 T^{3} + 14261 T^{4} + 18542 T^{5} + 156284 T^{6} + 5557948 T^{7} + 96398796 T^{8} + 548901022 T^{9} + 814939374 T^{10} + 68326526948 T^{11} + 536156406596 T^{12} + 6081060898372 T^{13} + 6455134781454 T^{14} + 386958204578318 T^{15} + 6048276490741836 T^{16} + 31035912046450652 T^{17} + 77670224076548924 T^{18} + 820137411632898718 T^{19} + 56139692958117377141 T^{20} + 34334927563333550482 T^{21} + \)\(10\!\cdots\!34\)\( T^{22} + \)\(55\!\cdots\!78\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 37 T + 590 T^{2} + 5480 T^{3} + 34820 T^{4} + 246401 T^{5} + 3372057 T^{6} + 45787390 T^{7} + 479697120 T^{8} + 4075077710 T^{9} + 26710063497 T^{10} + 173705066569 T^{11} + 2184684831620 T^{12} + 30600645780520 T^{13} + 293218961666990 T^{14} + 1636559391134573 T^{15} + 3936588805702081 T^{16} \))(\( ( 1 + 10 T + 89 T^{2} )^{4}( 1 + 10 T + 11 T^{2} - 780 T^{3} - 8779 T^{4} - 69420 T^{5} + 87131 T^{6} + 7049690 T^{7} + 62742241 T^{8} ) \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 + 9409 T^{4} \))(\( ( 1 + 8 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( 1 + 9409 T^{4} \))(\( ( 1 + 2498 T^{4} + 88529281 T^{8} )^{2} \))(\( 1 + 7 T - 123 T^{2} - 1605 T^{3} + 657 T^{4} - 51348 T^{5} + 95680 T^{6} + 18416476 T^{7} + 105394968 T^{8} - 476041588 T^{9} + 10556842230 T^{10} - 65948957114 T^{11} - 2540582876162 T^{12} - 6397048840058 T^{13} + 99329328542070 T^{14} - 434470304244724 T^{15} + 9330540738058008 T^{16} + 158148545746874332 T^{17} + 79698761431606720 T^{18} - 4148830311382146324 T^{19} + 5149193871505663377 T^{20} - \)\(12\!\cdots\!85\)\( T^{21} - \)\(90\!\cdots\!27\)\( T^{22} + \)\(50\!\cdots\!71\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 25 T + 240 T^{2} + 1020 T^{3} - 53264 T^{4} + 582025 T^{5} - 1401985 T^{6} - 44565840 T^{7} + 691186376 T^{8} - 4322886480 T^{9} - 13191276865 T^{10} + 531198502825 T^{11} - 4715423623184 T^{12} + 8759087062140 T^{13} + 199913281182960 T^{14} - 2019957111952825 T^{15} + 7837433594376961 T^{16} \))(\( ( 1 + 8 T - 33 T^{2} - 1040 T^{3} - 5119 T^{4} - 100880 T^{5} - 310497 T^{6} + 7301384 T^{7} + 88529281 T^{8} )( 1 + 18 T + 227 T^{2} + 2340 T^{3} + 20101 T^{4} + 226980 T^{5} + 2135843 T^{6} + 16428114 T^{7} + 88529281 T^{8} ) \))
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