Properties

Label 51.2
Level 51
Weight 2
Dimension 55
Nonzero newspaces 5
Newform subspaces 7
Sturm bound 384
Trace bound 4

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 51 = 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 7 \)
Sturm bound: \(384\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(51))\).

Total New Old
Modular forms 128 87 41
Cusp forms 65 55 10
Eisenstein series 63 32 31

Trace form

\( 55q - 3q^{2} - 9q^{3} - 23q^{4} - 6q^{5} - 11q^{6} - 24q^{7} - 15q^{8} - 9q^{9} + O(q^{10}) \) \( 55q - 3q^{2} - 9q^{3} - 23q^{4} - 6q^{5} - 11q^{6} - 24q^{7} - 15q^{8} - 9q^{9} - 26q^{10} + 4q^{11} + 9q^{12} - 14q^{13} + 8q^{14} + 10q^{15} + 25q^{16} - q^{17} + 13q^{18} - 20q^{19} + 14q^{20} + 8q^{21} - 20q^{22} - 8q^{23} + 17q^{24} + 9q^{25} + 30q^{26} - 9q^{27} + 24q^{28} + 10q^{29} + 38q^{30} + 16q^{31} + 17q^{32} + 20q^{33} + 61q^{34} + 16q^{35} + 17q^{36} + 10q^{37} + 20q^{38} - 6q^{39} + 22q^{40} - 2q^{41} - 32q^{42} - 12q^{43} - 20q^{44} - 38q^{45} - 56q^{46} - 48q^{47} - 95q^{48} - 73q^{49} - 93q^{50} - 57q^{51} - 130q^{52} - 30q^{53} - 27q^{54} - 24q^{55} - 40q^{56} - 12q^{57} + 6q^{58} + 4q^{59} - 10q^{60} + 18q^{61} + 16q^{62} + 56q^{63} + 17q^{64} + 36q^{65} + 36q^{66} - 4q^{67} + 81q^{68} + 72q^{69} + 112q^{70} + 24q^{71} + 73q^{72} + 110q^{73} + 54q^{74} + 89q^{75} + 84q^{76} + 64q^{77} + 118q^{78} + 32q^{79} + 86q^{80} + 47q^{81} + 74q^{82} + 28q^{83} + 40q^{84} + 2q^{85} - 68q^{86} - 38q^{87} - 4q^{88} - 42q^{89} - 98q^{90} - 32q^{91} - 8q^{92} - 72q^{93} - 32q^{94} - 56q^{95} - 111q^{96} - 50q^{97} - 99q^{98} - 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(51))\)

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
51.2.a \(\chi_{51}(1, \cdot)\) 51.2.a.a 1 1
51.2.a.b 2
51.2.d \(\chi_{51}(16, \cdot)\) 51.2.d.a 2 1
51.2.d.b 2
51.2.e \(\chi_{51}(4, \cdot)\) 51.2.e.a 8 2
51.2.h \(\chi_{51}(19, \cdot)\) 51.2.h.a 8 4
51.2.i \(\chi_{51}(5, \cdot)\) 51.2.i.a 32 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(51))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(51)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 5 T^{4} - 12 T^{6} + 12 T^{8} - 48 T^{10} + 80 T^{12} - 128 T^{14} + 256 T^{16} \))(\( 1 - 4 T^{4} + 4 T^{8} - 64 T^{12} + 256 T^{16} \))
$3$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( 1 + T^{8} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 4 T + 8 T^{2} + 4 T^{3} + 13 T^{4} + 72 T^{5} + 192 T^{6} - 176 T^{7} - 1116 T^{8} - 880 T^{9} + 4800 T^{10} + 9000 T^{11} + 8125 T^{12} + 12500 T^{13} + 125000 T^{14} + 312500 T^{15} + 390625 T^{16} \))(\( 1 + 8 T + 40 T^{2} + 160 T^{3} + 544 T^{4} + 1624 T^{5} + 4384 T^{6} + 10904 T^{7} + 25151 T^{8} + 54520 T^{9} + 109600 T^{10} + 203000 T^{11} + 340000 T^{12} + 500000 T^{13} + 625000 T^{14} + 625000 T^{15} + 390625 T^{16} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 - 4 T + 22 T^{2} - 68 T^{3} + 226 T^{4} - 476 T^{5} + 1078 T^{6} - 1372 T^{7} + 2401 T^{8} )( 1 + 8 T + 18 T^{2} - 32 T^{3} - 222 T^{4} - 224 T^{5} + 882 T^{6} + 2744 T^{7} + 2401 T^{8} ) \))(\( ( 1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4} )^{2}( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 616 T^{5} + 1568 T^{6} - 2744 T^{7} + 2401 T^{8} ) \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 3 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 48 T^{3} + 57 T^{4} + 768 T^{5} + 1152 T^{6} + 1872 T^{7} - 34240 T^{8} + 20592 T^{9} + 139392 T^{10} + 1022208 T^{11} + 834537 T^{12} - 7730448 T^{13} + 214358881 T^{16} \))(\( 1 - 8 T + 16 T^{2} + 96 T^{3} - 640 T^{4} + 1240 T^{5} + 1952 T^{6} - 15496 T^{7} + 53087 T^{8} - 170456 T^{9} + 236192 T^{10} + 1650440 T^{11} - 9370240 T^{12} + 15460896 T^{13} + 28344976 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} - 2 T^{3} + 352 T^{4} - 26 T^{5} + 4901 T^{6} + 4394 T^{7} + 28561 T^{8} )^{2} \))(\( 1 - 60 T^{2} + 1946 T^{4} - 41776 T^{6} + 637155 T^{8} - 7060144 T^{10} + 55579706 T^{12} - 289608540 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T - 26 T^{2} - 28 T^{3} + 642 T^{4} - 476 T^{5} - 7514 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 8 T + 40 T^{2} + 88 T^{3} + 240 T^{4} + 1496 T^{5} + 11560 T^{6} + 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - 3 T + 2 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 4969 T^{4} - 154390 T^{6} + 3418820 T^{8} - 55734790 T^{10} + 647565049 T^{12} - 4798679862 T^{14} + 16983563041 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 224 T^{3} + 734 T^{4} - 1320 T^{5} - 8960 T^{6} - 84232 T^{7} - 655901 T^{8} - 1600408 T^{9} - 3234560 T^{10} - 9053880 T^{11} + 95655614 T^{12} + 554646176 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 9 T + 23 T^{2} \))(\( 1 + 9 T + 62 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 45 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 + 16 T + 128 T^{2} + 920 T^{3} + 6297 T^{4} + 34144 T^{5} + 163488 T^{6} + 816776 T^{7} + 4054256 T^{8} + 18785848 T^{9} + 86485152 T^{10} + 415430048 T^{11} + 1762158777 T^{12} + 5921435560 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16} \))(\( 1 - 8 T + 88 T^{2} - 560 T^{3} + 3744 T^{4} - 22424 T^{5} + 123872 T^{6} - 722072 T^{7} + 3312623 T^{8} - 16607656 T^{9} + 65528288 T^{10} - 272832808 T^{11} + 1047724704 T^{12} - 3604352080 T^{13} + 13027158232 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 10 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 4 T + 8 T^{2} + 244 T^{3} + 48 T^{4} - 8644 T^{5} + 63960 T^{6} + 61812 T^{7} - 1448642 T^{8} + 1792548 T^{9} + 53790360 T^{10} - 210818516 T^{11} + 33949488 T^{12} + 5004720356 T^{13} + 4758586568 T^{14} - 68999505236 T^{15} + 500246412961 T^{16} \))(\( 1 - 48 T^{2} + 1152 T^{4} + 8016 T^{6} - 960286 T^{8} + 6741456 T^{10} + 814787712 T^{12} - 28551519408 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 2 T + 46 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 38 T^{2} + 961 T^{4} \))(\( 1 - 46 T^{2} + 961 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 248 T^{3} + 2436 T^{4} + 12056 T^{5} + 49248 T^{6} + 402408 T^{7} + 3280390 T^{8} + 12474648 T^{9} + 47327328 T^{10} + 359160296 T^{11} + 2249697156 T^{12} + 7100029448 T^{13} + 28400117792 T^{14} + 220100912888 T^{15} + 852891037441 T^{16} \))(\( 1 - 8 T + 104 T^{2} - 344 T^{3} + 2528 T^{4} + 10072 T^{5} - 28728 T^{6} + 893640 T^{7} - 2291934 T^{8} + 27702840 T^{9} - 27607608 T^{10} + 300054952 T^{11} + 2334661088 T^{12} - 9848427944 T^{13} + 92300382824 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 4 T + 37 T^{2} \))(\( 1 + 2 T + 58 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 - 8 T + 32 T^{2} - 488 T^{3} + 1868 T^{4} + 12744 T^{5} - 42656 T^{6} + 628648 T^{7} - 8549306 T^{8} + 23259976 T^{9} - 58396064 T^{10} + 645521832 T^{11} + 3500932748 T^{12} - 33839851016 T^{13} + 82103245088 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \))(\( 1 + 8 T - 68 T^{2} - 520 T^{3} + 3528 T^{4} + 21736 T^{5} - 121324 T^{6} - 363944 T^{7} + 4087214 T^{8} - 13465928 T^{9} - 166092556 T^{10} + 1100993608 T^{11} + 6612040008 T^{12} - 36058857640 T^{13} - 174469395812 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 + 3 T + 80 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 57 T^{2} + 1681 T^{4} \))(\( ( 1 - 10 T + 41 T^{2} )( 1 + 10 T + 41 T^{2} ) \))(\( 1 - 28 T + 392 T^{2} - 3836 T^{3} + 29901 T^{4} - 195160 T^{5} + 1100736 T^{6} - 5569536 T^{7} + 30711316 T^{8} - 228350976 T^{9} + 1850337216 T^{10} - 13450622360 T^{11} + 84493079661 T^{12} - 444424387036 T^{13} + 1862040862472 T^{14} - 5453119668668 T^{15} + 7984925229121 T^{16} \))(\( 1 + 24 T + 184 T^{2} - 64 T^{3} - 7648 T^{4} - 17592 T^{5} + 256864 T^{6} + 1952264 T^{7} + 9209503 T^{8} + 80042824 T^{9} + 431788384 T^{10} - 1212458232 T^{11} - 21611420128 T^{12} - 7414796864 T^{13} + 874019180344 T^{14} + 4674102573144 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 + 7 T + 43 T^{2} \))(\( 1 + 3 T + 50 T^{2} + 129 T^{3} + 1849 T^{4} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( 1 - 182 T^{2} + 18905 T^{4} - 1301734 T^{6} + 65505604 T^{8} - 2406906166 T^{10} + 64632432905 T^{12} - 1150488074918 T^{14} + 11688200277601 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 128 T^{3} - 2338 T^{4} - 10376 T^{5} + 492952 T^{7} + 8953251 T^{8} + 21196936 T^{9} - 824964632 T^{11} - 7993156738 T^{12} + 18817080704 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 + 14 T + 126 T^{2} + 658 T^{3} + 2209 T^{4} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 136 T^{2} + 404 T^{3} + 8302 T^{4} + 18988 T^{5} + 300424 T^{6} + 415292 T^{7} + 4879681 T^{8} )^{2} \))(\( 1 - 168 T^{2} + 16604 T^{4} - 1181336 T^{6} + 63379782 T^{8} - 2609571224 T^{10} + 81022223324 T^{12} - 1810908175272 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( 1 - 272 T^{2} + 35708 T^{4} - 3041904 T^{6} + 187058790 T^{8} - 8544708336 T^{10} + 281753295548 T^{12} - 6028706227088 T^{14} + 62259690411361 T^{16} \))(\( 1 + 32 T + 512 T^{2} + 6368 T^{3} + 74084 T^{4} + 768928 T^{5} + 6950400 T^{6} + 57590880 T^{7} + 440626726 T^{8} + 3052316640 T^{9} + 19523673600 T^{10} + 114475693856 T^{11} + 584558394404 T^{12} + 2663068899424 T^{13} + 11348152898048 T^{14} + 37590756474784 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 - 6 T + 110 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( 1 - 148 T^{2} + 18488 T^{4} - 1465532 T^{6} + 102604366 T^{8} - 5101516892 T^{10} + 224025770168 T^{12} - 6242718978868 T^{14} + 146830437604321 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 1024 T^{3} + 7072 T^{4} - 53760 T^{5} + 479232 T^{6} - 5119792 T^{7} + 50495970 T^{8} - 302067728 T^{9} + 1668206592 T^{10} - 11041175040 T^{11} + 85693976992 T^{12} - 732082482176 T^{13} + 5399108306048 T^{14} - 39818423757104 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 - 10 T + 130 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 16 T + 128 T^{2} - 1344 T^{3} + 17612 T^{4} - 146272 T^{5} + 989184 T^{6} - 8987536 T^{7} + 81613830 T^{8} - 548239696 T^{9} + 3680753664 T^{10} - 33200964832 T^{11} + 243852951692 T^{12} - 1135137428544 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \))(\( 1 - 16 T + 324 T^{2} - 2576 T^{3} + 27144 T^{4} - 29808 T^{5} - 166612 T^{6} + 17565840 T^{7} - 104746898 T^{8} + 1071516240 T^{9} - 619963252 T^{10} - 6765849648 T^{11} + 375831508104 T^{12} - 2175680071376 T^{13} + 16692601292964 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))(\( ( 1 - 4 T + 72 T^{2} + 220 T^{3} + 2718 T^{4} + 14740 T^{5} + 323208 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 8 T + 272 T^{2} + 1592 T^{3} + 27474 T^{4} + 106664 T^{5} + 1221008 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 - 4 T + 78 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 5041 T^{4} \))(\( 1 - 24 T + 288 T^{2} - 3560 T^{3} + 51012 T^{4} - 552840 T^{5} + 4913504 T^{6} - 48830968 T^{7} + 466825030 T^{8} - 3466998728 T^{9} + 24768973664 T^{10} - 197867517240 T^{11} + 1296300671172 T^{12} - 6423056489560 T^{13} + 36892881769248 T^{14} - 218282883801384 T^{15} + 645753531245761 T^{16} \))(\( 1 + 16 T - 32 T^{2} - 1280 T^{3} + 4608 T^{4} + 133888 T^{5} + 581984 T^{6} - 1077392 T^{7} - 26627902 T^{8} - 76494832 T^{9} + 2933781344 T^{10} + 47919987968 T^{11} + 117097026048 T^{12} - 2309413569280 T^{13} - 4099209085472 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 20 T + 200 T^{2} - 2140 T^{3} + 26256 T^{4} - 253540 T^{5} + 2109400 T^{6} - 20347820 T^{7} + 193840094 T^{8} - 1485390860 T^{9} + 11240992600 T^{10} - 98631370180 T^{11} + 745624215696 T^{12} - 4436373209020 T^{13} + 30266845257800 T^{14} - 220947970381940 T^{15} + 806460091894081 T^{16} \))(\( 1 - 48 T + 1200 T^{2} - 21344 T^{3} + 306816 T^{4} - 3800544 T^{5} + 41743472 T^{6} - 412310704 T^{7} + 3694140226 T^{8} - 30098681392 T^{9} + 222450962288 T^{10} - 1478476225248 T^{11} + 8713034710656 T^{12} - 44247640080992 T^{13} + 181601071546800 T^{14} - 530275128916656 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 14 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 - 142 T^{2} + 6241 T^{4} \))(\( 1 - 142 T^{2} + 6241 T^{4} \))(\( 1 - 20 T + 200 T^{2} - 2116 T^{3} + 33888 T^{4} - 411652 T^{5} + 3694168 T^{6} - 35764820 T^{7} + 343360958 T^{8} - 2825420780 T^{9} + 23055302488 T^{10} - 202960490428 T^{11} + 1319940344928 T^{12} - 6511051340284 T^{13} + 48617491104200 T^{14} - 384078179723180 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 36 T^{2} - 1488 T^{3} + 648 T^{4} - 122448 T^{5} + 841644 T^{6} - 2436480 T^{7} + 130708238 T^{8} - 192481920 T^{9} + 5252700204 T^{10} - 60371639472 T^{11} + 25239652488 T^{12} - 4578659921712 T^{13} + 8751148398756 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 10 T + 174 T^{2} + 830 T^{3} + 6889 T^{4} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( 1 - 320 T^{2} + 59996 T^{4} - 7489216 T^{6} + 715153510 T^{8} - 51593209024 T^{10} + 2847309426716 T^{12} - 104620919478080 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 32 T + 512 T^{2} - 6304 T^{3} + 73444 T^{4} - 833312 T^{5} + 8932864 T^{6} - 91493280 T^{7} + 877696230 T^{8} - 7593942240 T^{9} + 61538500096 T^{10} - 476476968544 T^{11} + 3485528927524 T^{12} - 24831712213472 T^{13} + 167393471164928 T^{14} - 868353631668064 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 4 T + 246 T^{2} + 716 T^{3} + 29034 T^{4} + 63724 T^{5} + 1948566 T^{6} + 2819876 T^{7} + 62742241 T^{8} )^{2} \))(\( 1 - 416 T^{2} + 87680 T^{4} - 12136864 T^{6} + 1238928322 T^{8} - 96136099744 T^{10} + 5501239690880 T^{12} - 206744217039776 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( 1 + 16 T + 97 T^{2} \))(\( 1 + 14 T + 226 T^{2} + 1358 T^{3} + 9409 T^{4} \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( 1 + 62 T^{2} + 9409 T^{4} \))(\( 1 + 12 T + 72 T^{2} + 84 T^{3} + 10928 T^{4} + 244044 T^{5} + 2145240 T^{6} + 15241620 T^{7} + 71840734 T^{8} + 1478437140 T^{9} + 20184563160 T^{10} + 222732369612 T^{11} + 967447982768 T^{12} + 721336581588 T^{13} + 59973984354888 T^{14} + 969579413737356 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 8 T - 24 T^{2} + 2200 T^{3} - 14752 T^{4} + 58200 T^{5} + 1664072 T^{6} - 6187784 T^{7} + 82745634 T^{8} - 600215048 T^{9} + 15657253448 T^{10} + 53117568600 T^{11} - 1305983953312 T^{12} + 18892148565400 T^{13} - 19991328118296 T^{14} - 646386275824904 T^{15} + 7837433594376961 T^{16} \))
show more
show less