Properties

Label 49.2
Level 49
Weight 2
Dimension 69
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 392
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 49\( 49 = 7^{2} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(392\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(49))\).

Total New Old
Modular forms 128 118 10
Cusp forms 69 69 0
Eisenstein series 59 49 10

Trace form

\( 69q - 18q^{2} - 17q^{3} - 14q^{4} - 15q^{5} - 9q^{6} - 14q^{7} - 24q^{8} - 8q^{9} + O(q^{10}) \) \( 69q - 18q^{2} - 17q^{3} - 14q^{4} - 15q^{5} - 9q^{6} - 14q^{7} - 24q^{8} - 8q^{9} - 3q^{10} - 9q^{11} + 7q^{12} - 7q^{13} - 15q^{15} + 10q^{16} - 3q^{17} + 18q^{18} - q^{19} + 21q^{20} + 7q^{21} - 3q^{22} + 3q^{23} + 39q^{24} + 10q^{25} + 21q^{26} + 19q^{27} + 28q^{28} - 9q^{29} + 51q^{30} + 11q^{31} + 42q^{32} + 27q^{33} + 33q^{34} + 21q^{35} - 2q^{36} - 11q^{37} - 3q^{38} - 14q^{39} - 57q^{40} - 21q^{41} - 21q^{42} - 37q^{43} - 63q^{44} - 48q^{45} - 75q^{46} - 15q^{47} - 72q^{48} - 56q^{49} - 51q^{50} - 33q^{51} - 77q^{52} - 9q^{53} - 27q^{54} - 54q^{55} - 42q^{56} - q^{57} - 15q^{58} - 3q^{59} + 21q^{60} - 8q^{61} + 33q^{62} + 42q^{63} + 16q^{64} + 63q^{65} + 123q^{66} + 47q^{67} + 105q^{68} + 75q^{69} + 105q^{70} + 33q^{71} + 174q^{72} + 53q^{73} + 93q^{74} + 103q^{75} + 119q^{76} + 63q^{77} + 21q^{78} + 59q^{79} + 102q^{80} + 16q^{81} - 21q^{83} - 56q^{84} - 15q^{85} - 36q^{86} - 69q^{87} - 72q^{88} - 15q^{89} - 186q^{90} - 7q^{91} - 18q^{92} - 145q^{93} - 66q^{94} - 69q^{95} - 168q^{96} - 28q^{97} - 168q^{98} - 72q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)

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
49.2.a \(\chi_{49}(1, \cdot)\) 49.2.a.a 1 1
49.2.c \(\chi_{49}(18, \cdot)\) 49.2.c.a 2 2
49.2.e \(\chi_{49}(8, \cdot)\) 49.2.e.a 6 6
49.2.e.b 12
49.2.g \(\chi_{49}(2, \cdot)\) 49.2.g.a 48 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 15 T^{3} + 31 T^{4} + 49 T^{5} + 71 T^{6} + 98 T^{7} + 124 T^{8} + 120 T^{9} + 112 T^{10} + 96 T^{11} + 64 T^{12} \))(\( 1 + 2 T - T^{2} - 8 T^{3} - 14 T^{4} + 35 T^{6} + 44 T^{7} + 4 T^{8} - 72 T^{9} - 86 T^{10} + 35 T^{11} + 147 T^{12} + 70 T^{13} - 344 T^{14} - 576 T^{15} + 64 T^{16} + 1408 T^{17} + 2240 T^{18} - 3584 T^{20} - 4096 T^{21} - 1024 T^{22} + 4096 T^{23} + 4096 T^{24} \))
$3$ (\( 1 + 3 T^{2} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + 3 T + 6 T^{2} + 16 T^{3} + 44 T^{4} + 77 T^{5} + 127 T^{6} + 231 T^{7} + 396 T^{8} + 432 T^{9} + 486 T^{10} + 729 T^{11} + 729 T^{12} \))(\( 1 + T^{2} - 7 T^{3} + 6 T^{4} - 7 T^{5} + 67 T^{6} - 14 T^{7} + 27 T^{8} - 441 T^{9} + 96 T^{10} - 245 T^{11} + 2107 T^{12} - 735 T^{13} + 864 T^{14} - 11907 T^{15} + 2187 T^{16} - 3402 T^{17} + 48843 T^{18} - 15309 T^{19} + 39366 T^{20} - 137781 T^{21} + 59049 T^{22} + 531441 T^{24} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 6 T + 10 T^{2} + 5 T^{3} - 3 T^{4} - 161 T^{5} + 561 T^{6} - 805 T^{7} - 75 T^{8} + 625 T^{9} + 6250 T^{10} - 18750 T^{11} + 15625 T^{12} \))(\( 1 + 7 T + 18 T^{2} - 14 T^{3} - 254 T^{4} - 854 T^{5} - 976 T^{6} + 3129 T^{7} + 17440 T^{8} + 36911 T^{9} + 12729 T^{10} - 168609 T^{11} - 575211 T^{12} - 843045 T^{13} + 318225 T^{14} + 4613875 T^{15} + 10900000 T^{16} + 9778125 T^{17} - 15250000 T^{18} - 66718750 T^{19} - 99218750 T^{20} - 27343750 T^{21} + 175781250 T^{22} + 341796875 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 - 7 T + 21 T^{2} - 49 T^{3} + 147 T^{4} - 343 T^{5} + 343 T^{6} \))(\( 1 + 7 T + 21 T^{2} + 35 T^{3} + 98 T^{4} + 588 T^{5} + 2107 T^{6} + 4116 T^{7} + 4802 T^{8} + 12005 T^{9} + 50421 T^{10} + 117649 T^{11} + 117649 T^{12} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 14 T^{2} + 99 T^{3} - 177 T^{4} - 707 T^{5} + 5559 T^{6} - 7777 T^{7} - 21417 T^{8} + 131769 T^{9} - 204974 T^{10} - 322102 T^{11} + 1771561 T^{12} \))(\( 1 + 8 T + 5 T^{2} - 197 T^{3} - 931 T^{4} + 35 T^{5} + 16625 T^{6} + 60537 T^{7} - 46528 T^{8} - 987086 T^{9} - 2323147 T^{10} + 5151853 T^{11} + 40848171 T^{12} + 56670383 T^{13} - 281100787 T^{14} - 1313811466 T^{15} - 681216448 T^{16} + 9749544387 T^{17} + 29452201625 T^{18} + 682050985 T^{19} - 199568118211 T^{20} - 464515695127 T^{21} + 129687123005 T^{22} + 2282493364888 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 + T^{2} - 14 T^{3} + 43 T^{4} + 602 T^{5} - 209 T^{6} + 7826 T^{7} + 7267 T^{8} - 30758 T^{9} + 28561 T^{10} + 4826809 T^{12} \))(\( 1 - 7 T + 2 T^{2} + 105 T^{3} - 74 T^{4} - 1484 T^{5} + 2174 T^{6} + 6181 T^{7} + 17379 T^{8} + 23660 T^{9} - 927256 T^{10} - 444087 T^{11} + 17334093 T^{12} - 5773131 T^{13} - 156706264 T^{14} + 51981020 T^{15} + 496361619 T^{16} + 2294962033 T^{17} + 10493482766 T^{18} - 93118799228 T^{19} - 60364073354 T^{20} + 1113472434165 T^{21} + 275716983698 T^{22} - 12545122758259 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 4 T - 15 T^{2} + 226 T^{3} - 537 T^{4} - 2198 T^{5} + 21911 T^{6} - 37366 T^{7} - 155193 T^{8} + 1110338 T^{9} - 1252815 T^{10} - 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + T^{2} + 21 T^{3} - 351 T^{4} + 805 T^{5} + 1285 T^{6} + 21259 T^{7} + 15840 T^{8} + 89768 T^{9} + 494247 T^{10} - 5620153 T^{11} + 32910115 T^{12} - 95542601 T^{13} + 142837383 T^{14} + 441030184 T^{15} + 1322972640 T^{16} + 30184739963 T^{17} + 31016776165 T^{18} + 330322631765 T^{19} - 2448490861791 T^{20} + 2490345406437 T^{21} + 2015993900449 T^{22} + 582622237229761 T^{24} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( ( 1 + 4 T + 46 T^{2} + 151 T^{3} + 874 T^{4} + 1444 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 7 T + 93 T^{2} + 455 T^{3} + 3770 T^{4} + 14868 T^{5} + 91939 T^{6} + 282492 T^{7} + 1360970 T^{8} + 3120845 T^{9} + 12119853 T^{10} + 17332693 T^{11} + 47045881 T^{12} )^{2} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 + 10 T + 7 T^{2} - 300 T^{3} - 1131 T^{4} + 5110 T^{5} + 53943 T^{6} + 117530 T^{7} - 598299 T^{8} - 3650100 T^{9} + 1958887 T^{10} + 64363430 T^{11} + 148035889 T^{12} \))(\( 1 + 2 T - 36 T^{2} - 253 T^{3} + 1043 T^{4} + 5789 T^{5} - 8190 T^{6} - 151793 T^{7} + 356927 T^{8} + 570344 T^{9} - 4524879 T^{10} - 17763963 T^{11} + 335310423 T^{12} - 408571149 T^{13} - 2393660991 T^{14} + 6939375448 T^{15} + 99882808607 T^{16} - 976991812999 T^{17} - 1212413930910 T^{18} + 19710534512683 T^{19} + 81678357648083 T^{20} - 455691623350139 T^{21} - 1491354403691364 T^{22} + 1905619515827854 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( 1 + 16 T + 129 T^{2} + 914 T^{3} + 5787 T^{4} + 30218 T^{5} + 154847 T^{6} + 876322 T^{7} + 4866867 T^{8} + 22291546 T^{9} + 91239249 T^{10} + 328178384 T^{11} + 594823321 T^{12} \))(\( 1 + 11 T + 17 T^{2} - 449 T^{3} - 1393 T^{4} + 22610 T^{5} + 151697 T^{6} - 280710 T^{7} - 4342483 T^{8} + 10376917 T^{9} + 187164155 T^{10} + 92982673 T^{11} - 3869306847 T^{12} + 2696497517 T^{13} + 157405054355 T^{14} + 253082628713 T^{15} - 3071355718723 T^{16} - 5757684635790 T^{17} + 90232913325737 T^{18} + 390019703346490 T^{19} - 696843253254673 T^{20} - 6513708543165181 T^{21} + 7152022966103417 T^{22} + 134205607422764119 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( ( 1 + 5 T + 99 T^{2} + 311 T^{3} + 3069 T^{4} + 4805 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 7 T + 88 T^{2} + 511 T^{3} + 4076 T^{4} + 21679 T^{5} + 134975 T^{6} + 672049 T^{7} + 3917036 T^{8} + 15223201 T^{9} + 81269848 T^{10} + 200404057 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 35 T^{2} + 134 T^{3} + 1193 T^{4} - 3934 T^{5} + 120891 T^{6} - 145558 T^{7} + 1633217 T^{8} + 6787502 T^{9} + 65595635 T^{10} - 277375828 T^{11} + 2565726409 T^{12} \))(\( 1 + 30 T + 419 T^{2} + 3436 T^{3} + 15204 T^{4} - 19572 T^{5} - 958447 T^{6} - 8542504 T^{7} - 41955147 T^{8} - 80200122 T^{9} + 633349064 T^{10} + 8406169968 T^{11} + 60478612229 T^{12} + 311028288816 T^{13} + 867054868616 T^{14} - 4062376779666 T^{15} - 78630700256667 T^{16} - 592371030048328 T^{17} - 2459112779526823 T^{18} - 1858006699247076 T^{19} + 53403737617414884 T^{20} + 446548537935884572 T^{21} + 2014796852043078731 T^{22} + 5337528653383812390 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 15 T^{2} + 112 T^{3} + 1317 T^{4} + 3178 T^{5} - 18717 T^{6} + 130298 T^{7} + 2213877 T^{8} + 7719152 T^{9} + 42386415 T^{10} + 4750104241 T^{12} \))(\( 1 - 21 T + 114 T^{2} + 1015 T^{3} - 13416 T^{4} - 24332 T^{5} + 1124617 T^{6} - 5080138 T^{7} - 22183975 T^{8} + 257369588 T^{9} - 213592702 T^{10} - 7739726694 T^{11} + 66231437167 T^{12} - 317328794454 T^{13} - 359049332062 T^{14} + 17738169374548 T^{15} - 62686611379975 T^{16} - 588565489235738 T^{17} + 5342047981200697 T^{18} - 4738760992072492 T^{19} - 107125756873887336 T^{20} + 332292663409870415 T^{21} + 1530183161357373714 T^{22} - 11556909666041217261 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( 1 + 12 T + 43 T^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( 1 - 12 T + 31 T^{2} - 136 T^{3} + 2539 T^{4} - 5930 T^{5} - 45857 T^{6} - 254990 T^{7} + 4694611 T^{8} - 10812952 T^{9} + 105982831 T^{10} - 1764101316 T^{11} + 6321363049 T^{12} \))(\( 1 - 17 T - 4 T^{2} + 1665 T^{3} - 8414 T^{4} - 31717 T^{5} + 362285 T^{6} - 1104421 T^{7} + 2730975 T^{8} - 12051566 T^{9} + 39133202 T^{10} + 1793387022 T^{11} - 23518305111 T^{12} + 77115641946 T^{13} + 72357290498 T^{14} - 958183857962 T^{15} + 9336660060975 T^{16} - 162359211626503 T^{17} + 2290135012206965 T^{18} - 8621270888480719 T^{19} - 98344517135734814 T^{20} + 836816698874843595 T^{21} - 86445929253136996 T^{22} - 15797993571010786019 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 15 T + 115 T^{2} + 985 T^{3} + 10854 T^{4} + 81956 T^{5} + 533835 T^{6} + 3851932 T^{7} + 23976486 T^{8} + 102265655 T^{9} + 561163315 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} \))(\( 1 + 21 T + 256 T^{2} + 2793 T^{3} + 30210 T^{4} + 321587 T^{5} + 2992819 T^{6} + 24855299 T^{7} + 206144235 T^{8} + 1692381838 T^{9} + 13105466958 T^{10} + 93235046254 T^{11} + 634794782293 T^{12} + 4382047173938 T^{13} + 28949976510222 T^{14} + 175708159566674 T^{15} + 1005918106789035 T^{16} + 5700438723142093 T^{17} + 32260240441722451 T^{18} + 162923409440334781 T^{19} + 719338970051799810 T^{20} + 3125731411376028231 T^{21} + 13465377852372492544 T^{22} + 51915343516764258363 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 + 26 T + 343 T^{2} + 2892 T^{3} + 17925 T^{4} + 100058 T^{5} + 632955 T^{6} + 5303074 T^{7} + 50351325 T^{8} + 430552284 T^{9} + 2706434983 T^{10} + 10873082818 T^{11} + 22164361129 T^{12} \))(\( 1 - 6 T - 142 T^{2} + 1098 T^{3} + 9891 T^{4} - 144396 T^{5} - 120260 T^{6} + 11267082 T^{7} - 36894535 T^{8} - 561327846 T^{9} + 4842655950 T^{10} + 11314891224 T^{11} - 319975006505 T^{12} + 599689234872 T^{13} + 13603020563550 T^{14} - 83568805728942 T^{15} - 291115627421335 T^{16} + 4711842911661426 T^{17} - 2665486069373540 T^{18} - 169623589747903452 T^{19} + 615810597858771651 T^{20} + 3623140423798742034 T^{21} - 24834020791902852958 T^{22} - 55614215576233149582 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 11 T + 6 T^{2} + 590 T^{3} - 2637 T^{4} - 28819 T^{5} + 423452 T^{6} - 1700321 T^{7} - 9179397 T^{8} + 121173610 T^{9} + 72704166 T^{10} - 7864167289 T^{11} + 42180533641 T^{12} \))(\( 1 - 14 T - 6 T^{2} + 49 T^{3} + 9743 T^{4} - 18893 T^{5} - 282222 T^{6} - 2654281 T^{7} - 4440451 T^{8} + 319262776 T^{9} + 353736969 T^{10} + 2345454825 T^{11} - 184991898189 T^{12} + 138381834675 T^{13} + 1231358389089 T^{14} + 65569869672104 T^{15} - 53806547769811 T^{16} - 1897609983274019 T^{17} - 11904274565230302 T^{18} - 47018092502685367 T^{19} + 1430568953578899503 T^{20} + 424486795114092011 T^{21} - 3066700519803848406 T^{22} - \)\(42\!\cdots\!26\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 61 T^{2} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 8 T - 18 T^{2} - 359 T^{3} + 1229 T^{4} - 16569 T^{5} - 310715 T^{6} - 1010709 T^{7} + 4573109 T^{8} - 81486179 T^{9} - 249225138 T^{10} + 6756770408 T^{11} + 51520374361 T^{12} \))(\( 1 + 7 T + 18 T^{2} + 399 T^{3} + 5241 T^{4} + 65499 T^{5} + 509772 T^{6} + 1220030 T^{7} + 17894495 T^{8} + 360127180 T^{9} + 2886283711 T^{10} + 14727499143 T^{11} + 68550746229 T^{12} + 898377447723 T^{13} + 10739861688631 T^{14} + 81742027443580 T^{15} + 247764332545295 T^{16} + 1030432825109030 T^{17} + 26263644278755692 T^{18} + 205846513016539479 T^{19} + 1004738027418749721 T^{20} + 4665964291040822259 T^{21} + 12840172409931886818 T^{22} + \)\(30\!\cdots\!27\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 + 6 T + 206 T^{2} + 791 T^{3} + 13802 T^{4} + 26934 T^{5} + 300763 T^{6} )^{2} \))(\( ( 1 - 24 T + 544 T^{2} - 7969 T^{3} + 103505 T^{4} - 1061603 T^{5} + 9586871 T^{6} - 71127401 T^{7} + 464633945 T^{8} - 2396780347 T^{9} + 10962209824 T^{10} - 32403002568 T^{11} + 90458382169 T^{12} )^{2} \))
$71$ (\( 1 - 16 T + 71 T^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( 1 - 5 T + 66 T^{2} + 116 T^{3} + 4296 T^{4} + 31079 T^{5} + 6755 T^{6} + 2206609 T^{7} + 21656136 T^{8} + 41517676 T^{9} + 1677170946 T^{10} - 9021146755 T^{11} + 128100283921 T^{12} \))(\( 1 + 39 T + 871 T^{2} + 14307 T^{3} + 190092 T^{4} + 2083466 T^{5} + 18915358 T^{6} + 138847559 T^{7} + 724402569 T^{8} + 765520348 T^{9} - 40173731301 T^{10} - 631486451309 T^{11} - 6236115389021 T^{12} - 44835538042939 T^{13} - 202515779488341 T^{14} + 273988153273028 T^{15} + 18408286999008489 T^{16} + 250512841262504209 T^{17} + 2423062730267358718 T^{18} + 18949373615922263206 T^{19} + \)\(12\!\cdots\!12\)\( T^{20} + \)\(65\!\cdots\!17\)\( T^{21} + \)\(28\!\cdots\!71\)\( T^{22} + \)\(90\!\cdots\!69\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 + 73 T^{2} \))(\( 1 - 73 T^{2} + 5329 T^{4} \))(\( 1 - 4 T + 111 T^{2} - 68 T^{3} + 11531 T^{4} + 24234 T^{5} + 541591 T^{6} + 1769082 T^{7} + 61448699 T^{8} - 26453156 T^{9} + 3152204751 T^{10} - 8292286372 T^{11} + 151334226289 T^{12} \))(\( 1 - 42 T + 715 T^{2} - 5880 T^{3} + 11304 T^{4} + 423150 T^{5} - 8979169 T^{6} + 106207794 T^{7} - 735951847 T^{8} + 1892847096 T^{9} + 28699443498 T^{10} - 614746868148 T^{11} + 6731348891281 T^{12} - 44876521374804 T^{13} + 152939334400842 T^{14} + 736349698744632 T^{15} - 20899737915501127 T^{16} + 220176360696595842 T^{17} - 1358855593333173841 T^{18} + 4674706683355895550 T^{19} + 9116224878770691624 T^{20} - \)\(34\!\cdots\!40\)\( T^{21} + \)\(30\!\cdots\!35\)\( T^{22} - \)\(13\!\cdots\!34\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( ( 1 - 30 T + 530 T^{2} - 5677 T^{3} + 41870 T^{4} - 187230 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 8 T + 377 T^{2} + 2713 T^{3} + 66443 T^{4} + 395274 T^{5} + 6756729 T^{6} + 31226646 T^{7} + 414670763 T^{8} + 1337614807 T^{9} + 14684180537 T^{10} + 24616451192 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 14 T + 169 T^{2} + 2604 T^{3} + 26195 T^{4} + 277228 T^{5} + 3109373 T^{6} + 23009924 T^{7} + 180457355 T^{8} + 1488933348 T^{9} + 8020456249 T^{10} + 55146569002 T^{11} + 326940373369 T^{12} \))(\( 1 + 7 T - 180 T^{2} - 2240 T^{3} + 4210 T^{4} + 197155 T^{5} + 2020379 T^{6} + 9004401 T^{7} - 183343253 T^{8} - 3038332654 T^{9} - 7239918141 T^{10} + 149580689517 T^{11} + 1902505655127 T^{12} + 12415197229911 T^{13} - 49875796073349 T^{14} - 1737279113232698 T^{15} - 8701162954058213 T^{16} + 35468701504869843 T^{17} + 660543464606886851 T^{18} + 5350008132859911185 T^{19} + 9482150297305362610 T^{20} - \)\(41\!\cdots\!20\)\( T^{21} - \)\(27\!\cdots\!20\)\( T^{22} + \)\(90\!\cdots\!69\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 13 T + 192 T^{2} - 2018 T^{3} + 26268 T^{4} - 269759 T^{5} + 3321179 T^{6} - 24008551 T^{7} + 208068828 T^{8} - 1422627442 T^{9} + 12046510272 T^{10} - 72592772837 T^{11} + 496981290961 T^{12} \))(\( 1 + 14 T - 108 T^{2} - 343 T^{3} + 21712 T^{4} - 241381 T^{5} - 2769868 T^{6} + 37209816 T^{7} - 31055819 T^{8} - 3858106259 T^{9} + 34438376829 T^{10} + 162131963175 T^{11} - 4244781250857 T^{12} + 14429744722575 T^{13} + 272786382862509 T^{14} - 2719845311300971 T^{15} - 1948511680150379 T^{16} + 207781824630351384 T^{17} - 1376572574431563148 T^{18} - 10676603848417685549 T^{19} + 85471216149403582672 T^{20} - \)\(12\!\cdots\!87\)\( T^{21} - \)\(33\!\cdots\!08\)\( T^{22} + \)\(38\!\cdots\!46\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 97 T^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 284 T^{2} - 7 T^{3} + 27548 T^{4} + 912673 T^{6} )^{2} \))(\( ( 1 + 14 T + 554 T^{2} + 5824 T^{3} + 128360 T^{4} + 1046052 T^{5} + 16320585 T^{6} + 101467044 T^{7} + 1207739240 T^{8} + 5315407552 T^{9} + 49045221674 T^{10} + 120222763598 T^{11} + 832972004929 T^{12} )^{2} \))
show more
show less