Properties

Label 77.2.f
Level 77
Weight 2
Character orbit f
Rep. character \(\chi_{77}(15,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 24
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(77, [\chi])\).

Total New Old
Modular forms 40 24 16
Cusp forms 24 24 0
Eisenstein series 16 0 16

Trace form

\( 24q - 4q^{2} - 6q^{3} - 8q^{4} - 2q^{5} + 6q^{6} - 2q^{7} - 2q^{8} - 14q^{9} + O(q^{10}) \) \( 24q - 4q^{2} - 6q^{3} - 8q^{4} - 2q^{5} + 6q^{6} - 2q^{7} - 2q^{8} - 14q^{9} - 16q^{10} + 2q^{11} + 4q^{12} - 2q^{13} + 3q^{14} - 12q^{15} + 14q^{16} - 16q^{17} + 15q^{18} + 10q^{19} + 22q^{20} + 12q^{21} - 34q^{22} + 16q^{23} - 14q^{24} + 12q^{25} - 6q^{26} - 12q^{27} + 11q^{28} - 6q^{29} + 12q^{30} - 18q^{31} + 12q^{32} - 16q^{33} - 48q^{34} - 8q^{35} + 50q^{36} + 10q^{37} + 30q^{38} + 6q^{39} - 26q^{40} - 32q^{41} + 8q^{43} - 9q^{44} + 88q^{45} - 13q^{46} - 16q^{47} - 32q^{48} - 6q^{49} + 18q^{50} - 26q^{51} + 54q^{52} + 6q^{53} + 64q^{54} + 14q^{55} - 18q^{56} - 14q^{57} - 19q^{58} + 42q^{59} - 68q^{60} + 74q^{62} - 10q^{63} + 8q^{64} + 16q^{65} + 26q^{66} - 68q^{67} + 6q^{68} + 18q^{69} - 20q^{70} - 28q^{71} + 3q^{72} - 32q^{73} - 38q^{74} + 14q^{75} - 8q^{77} - 76q^{78} + 6q^{79} - 120q^{80} - 42q^{81} + 50q^{82} + 28q^{83} + 12q^{84} - 4q^{85} - 69q^{86} - 60q^{87} + 48q^{88} + 40q^{89} - 12q^{90} + 18q^{91} - 89q^{92} + 40q^{93} - 54q^{94} + 56q^{95} - 32q^{96} + 50q^{97} + 6q^{98} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(77, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
77.2.f.a \(8\) \(0.615\) 8.0.159390625.1 None \(-1\) \(-4\) \(3\) \(2\) \(q+\beta _{4}q^{2}+(-\beta _{2}-\beta _{3})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
77.2.f.b \(16\) \(0.615\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-3\) \(-2\) \(-5\) \(-4\) \(q+(-\beta _{5}-\beta _{6})q^{2}+(\beta _{9}-\beta _{11}+\beta _{13}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - 3 T^{2} - 5 T^{3} + 19 T^{5} + 21 T^{6} - 20 T^{7} - 51 T^{8} - 40 T^{9} + 84 T^{10} + 152 T^{11} - 160 T^{13} - 192 T^{14} + 128 T^{15} + 256 T^{16} \))(\( 1 + 3 T + 6 T^{2} + 12 T^{3} + 18 T^{4} + 17 T^{5} + 15 T^{6} - T^{7} - 58 T^{8} - 119 T^{9} - 185 T^{10} - 292 T^{11} - 242 T^{12} + 33 T^{13} + 312 T^{14} + 878 T^{15} + 1785 T^{16} + 1756 T^{17} + 1248 T^{18} + 264 T^{19} - 3872 T^{20} - 9344 T^{21} - 11840 T^{22} - 15232 T^{23} - 14848 T^{24} - 512 T^{25} + 15360 T^{26} + 34816 T^{27} + 73728 T^{28} + 98304 T^{29} + 98304 T^{30} + 98304 T^{31} + 65536 T^{32} \))
$3$ (\( ( 1 + 2 T + T^{2} + 6 T^{3} + 19 T^{4} + 18 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} + 4 T^{5} + 42 T^{6} + 104 T^{7} + 161 T^{8} + 8 T^{9} - 278 T^{10} - 122 T^{11} + 1632 T^{12} + 3996 T^{13} + 4420 T^{14} - 908 T^{15} - 3527 T^{16} - 2724 T^{17} + 39780 T^{18} + 107892 T^{19} + 132192 T^{20} - 29646 T^{21} - 202662 T^{22} + 17496 T^{23} + 1056321 T^{24} + 2047032 T^{25} + 2480058 T^{26} + 708588 T^{27} - 4251528 T^{28} - 6377292 T^{29} + 9565938 T^{30} + 28697814 T^{31} + 43046721 T^{32} \))
$5$ (\( 1 - 3 T - 3 T^{2} + 30 T^{3} - 39 T^{4} - 150 T^{5} + 428 T^{6} + 231 T^{7} - 2169 T^{8} + 1155 T^{9} + 10700 T^{10} - 18750 T^{11} - 24375 T^{12} + 93750 T^{13} - 46875 T^{14} - 234375 T^{15} + 390625 T^{16} \))(\( 1 + 5 T - T^{2} - 66 T^{3} - 156 T^{4} + 267 T^{5} + 1789 T^{6} + 1241 T^{7} - 10249 T^{8} - 25094 T^{9} + 25181 T^{10} + 188691 T^{11} + 137316 T^{12} - 851484 T^{13} - 1968914 T^{14} + 1709452 T^{15} + 12417791 T^{16} + 8547260 T^{17} - 49222850 T^{18} - 106435500 T^{19} + 85822500 T^{20} + 589659375 T^{21} + 393453125 T^{22} - 1960468750 T^{23} - 4003515625 T^{24} + 2423828125 T^{25} + 17470703125 T^{26} + 13037109375 T^{27} - 38085937500 T^{28} - 80566406250 T^{29} - 6103515625 T^{30} + 152587890625 T^{31} + 152587890625 T^{32} \))
$7$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))
$11$ (\( 1 - 5 T + 26 T^{2} - 75 T^{3} + 251 T^{4} - 825 T^{5} + 3146 T^{6} - 6655 T^{7} + 14641 T^{8} \))(\( 1 + 3 T + 19 T^{2} - 76 T^{3} - 286 T^{4} - 2245 T^{5} + 1052 T^{6} + 7668 T^{7} + 114073 T^{8} + 84348 T^{9} + 127292 T^{10} - 2988095 T^{11} - 4187326 T^{12} - 12239876 T^{13} + 33659659 T^{14} + 58461513 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 - 5 T + 35 T^{2} - 90 T^{3} + 611 T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 115245 T^{9} + 740220 T^{10} - 1471990 T^{11} + 17450771 T^{12} - 33416370 T^{13} + 168938315 T^{14} - 313742585 T^{15} + 815730721 T^{16} \))(\( 1 + 7 T - 13 T^{2} - 114 T^{3} + 302 T^{4} + 619 T^{5} - 3203 T^{6} + 11659 T^{7} - 57039 T^{8} - 415732 T^{9} + 983197 T^{10} + 802283 T^{11} - 7060468 T^{12} + 40515806 T^{13} - 160505050 T^{14} - 179149458 T^{15} + 5116354083 T^{16} - 2328942954 T^{17} - 27125353450 T^{18} + 89013225782 T^{19} - 201654026548 T^{20} + 297882061919 T^{21} + 4745704128373 T^{22} - 26086566469444 T^{23} - 46528464595119 T^{24} + 123637858189807 T^{25} - 441560749392347 T^{26} + 1109347283908903 T^{27} + 7036021706989262 T^{28} - 34527762151516842 T^{29} - 51185893014090757 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 11 T + 39 T^{2} + 139 T^{3} + 1249 T^{4} + 7484 T^{5} + 34098 T^{6} + 128922 T^{7} + 457683 T^{8} + 2191674 T^{9} + 9854322 T^{10} + 36768892 T^{11} + 104317729 T^{12} + 197360123 T^{13} + 941365191 T^{14} + 4513725403 T^{15} + 6975757441 T^{16} \))(\( 1 + 5 T + 13 T^{2} + 163 T^{3} + 1572 T^{4} + 7139 T^{5} + 24321 T^{6} + 178695 T^{7} + 1262499 T^{8} + 5104293 T^{9} + 17336797 T^{10} + 107294017 T^{11} + 637972982 T^{12} + 2309382396 T^{13} + 7576691718 T^{14} + 42485576390 T^{15} + 218631742615 T^{16} + 722254798630 T^{17} + 2189663906502 T^{18} + 11345995711548 T^{19} + 53284141429622 T^{20} + 152342161095569 T^{21} + 418468133826493 T^{22} + 2094488816223189 T^{23} + 8806886793505059 T^{24} + 21191060590631415 T^{25} + 49030987652820129 T^{26} + 244667067740191987 T^{27} + 915882156925184292 T^{28} + 1614446219363667731 T^{29} + 2188911745272212077 T^{30} + 14312115257549078965 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 9 T + 43 T^{2} + 171 T^{3} + 1023 T^{4} + 1674 T^{5} - 12940 T^{6} - 86580 T^{7} - 253639 T^{8} - 1645020 T^{9} - 4671340 T^{10} + 11481966 T^{11} + 133318383 T^{12} + 423412929 T^{13} + 2022972883 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \))(\( 1 - 19 T + 113 T^{2} + 15 T^{3} - 2596 T^{4} + 2631 T^{5} + 106245 T^{6} - 756527 T^{7} + 1484429 T^{8} + 6206619 T^{9} - 26235105 T^{10} - 133461535 T^{11} + 1444663142 T^{12} - 5677361136 T^{13} + 11109732198 T^{14} - 13150474478 T^{15} + 38443302841 T^{16} - 249859015082 T^{17} + 4010613323478 T^{18} - 38941020031824 T^{19} + 188269945328582 T^{20} - 330463973351965 T^{21} - 1234253627852505 T^{22} + 5547921318840441 T^{23} + 25210893501388589 T^{24} - 244121955937653533 T^{25} + 651395134560067245 T^{26} + 306485871161214189 T^{27} - 5745765529895753956 T^{28} + 630794751933855885 T^{29} + 90287755493465905673 T^{30} - \)\(28\!\cdots\!81\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( ( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 9246 T^{5} + 43907 T^{6} + 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 926670 T^{6} - 5230592 T^{7} + 26816927 T^{8} - 120303616 T^{9} + 490208430 T^{10} - 1724209904 T^{11} + 5475648847 T^{12} - 14378790262 T^{13} + 33160039136 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 5559300 T^{9} + 68911540 T^{10} + 286668306 T^{11} - 538948122 T^{12} - 8799282921 T^{13} - 13086113062 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \))(\( 1 - 3 T - 75 T^{3} + 2441 T^{4} - 473 T^{5} - 37550 T^{6} - 114715 T^{7} + 3177286 T^{8} + 2623287 T^{9} - 24972736 T^{10} - 263064707 T^{11} + 3471423716 T^{12} + 694532172 T^{13} - 11302302466 T^{14} - 87801320252 T^{15} + 2485262854281 T^{16} - 2546238287308 T^{17} - 9505236373906 T^{18} + 16938945142908 T^{19} + 2455272037276196 T^{20} - 5395759401918343 T^{21} - 14854365761976256 T^{22} + 45251376273007683 T^{23} + 1589425924451203846 T^{24} - 1664187250621812335 T^{25} - 15797556610422547550 T^{26} - 5770841119178857117 T^{27} + \)\(86\!\cdots\!81\)\( T^{28} - \)\(76\!\cdots\!75\)\( T^{29} - \)\(25\!\cdots\!47\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 11 T + 34 T^{2} + 183 T^{3} + 3210 T^{4} + 12836 T^{5} - 31430 T^{6} + 25700 T^{7} + 2517721 T^{8} + 796700 T^{9} - 30204230 T^{10} + 382397276 T^{11} + 2964502410 T^{12} + 5239134633 T^{13} + 30175125154 T^{14} + 302638755221 T^{15} + 852891037441 T^{16} \))(\( 1 + 7 T + 13 T^{3} + 77 T^{4} - 8801 T^{5} - 22254 T^{6} + 56119 T^{7} - 947868 T^{8} + 3202093 T^{9} + 57615842 T^{10} - 102279009 T^{11} - 159564504 T^{12} + 5764151836 T^{13} - 36979480986 T^{14} - 181717554252 T^{15} + 841105835367 T^{16} - 5633244181812 T^{17} - 35537281227546 T^{18} + 171719847346276 T^{19} - 147361170298584 T^{20} - 2928161192791359 T^{21} + 51134271858914402 T^{22} + 88097949056534323 T^{23} - 808428121877125788 T^{24} + 1483765156034695849 T^{25} - 18240007898470745454 T^{26} - \)\(22\!\cdots\!31\)\( T^{27} + 60650034351718331597 T^{28} + \)\(31\!\cdots\!83\)\( T^{29} + \)\(16\!\cdots\!57\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 3418430 T^{9} - 27666121 T^{10} - 656158962 T^{11} + 2530117350 T^{12} + 36058857640 T^{13} - 161640763767 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 4 T - 64 T^{2} + 392 T^{3} + 2308 T^{4} - 10820 T^{5} - 47691 T^{6} - 31178 T^{7} + 2748163 T^{8} + 12045860 T^{9} - 124307808 T^{10} - 75116270 T^{11} + 4767089611 T^{12} - 456957494 T^{13} - 122520873497 T^{14} + 217199596514 T^{15} + 4709853143282 T^{16} + 8036385071018 T^{17} - 167731075817393 T^{18} - 23146267943582 T^{19} + 8934293432441371 T^{20} - 5208859396880390 T^{21} - 318939825830501472 T^{22} + 1143536101481319380 T^{23} + 9652866073525897123 T^{24} - 4051947123330910706 T^{25} - \)\(22\!\cdots\!59\)\( T^{26} - \)\(19\!\cdots\!60\)\( T^{27} + \)\(15\!\cdots\!48\)\( T^{28} + \)\(95\!\cdots\!24\)\( T^{29} - \)\(57\!\cdots\!96\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 22 T + 160 T^{2} + 414 T^{3} + 2641 T^{4} + 46342 T^{5} + 334288 T^{6} + 1062556 T^{7} + 2274033 T^{8} + 43564796 T^{9} + 561938128 T^{10} + 3193936982 T^{11} + 7462834801 T^{12} + 47964467214 T^{13} + 760016678560 T^{14} + 4284594025382 T^{15} + 7984925229121 T^{16} \))(\( 1 + 10 T + 32 T^{2} + 192 T^{3} + 5438 T^{4} + 48920 T^{5} + 140400 T^{6} + 769008 T^{7} + 14682039 T^{8} + 96831496 T^{9} + 182717296 T^{10} + 1985070190 T^{11} + 32206741548 T^{12} + 131499578440 T^{13} + 27334729480 T^{14} + 4391234846920 T^{15} + 62202943867805 T^{16} + 180040628723720 T^{17} + 45949680255880 T^{18} + 9063082445663240 T^{19} + 91008554203418028 T^{20} + 229982690931748190 T^{21} + 867926202633652336 T^{22} + 18858347692290955976 T^{23} + \)\(11\!\cdots\!19\)\( T^{24} + \)\(25\!\cdots\!88\)\( T^{25} + \)\(18\!\cdots\!00\)\( T^{26} + \)\(26\!\cdots\!20\)\( T^{27} + \)\(12\!\cdots\!78\)\( T^{28} + \)\(17\!\cdots\!32\)\( T^{29} + \)\(12\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!10\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 - 7 T - 100 T^{2} + 1414 T^{3} + 286 T^{4} - 103523 T^{5} + 579636 T^{6} + 2543800 T^{7} - 42803781 T^{8} + 119558600 T^{9} + 1280415924 T^{10} - 10748068429 T^{11} + 1395588766 T^{12} + 324293839898 T^{13} - 1077921532900 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16} \))(\( 1 + 23 T + 186 T^{2} + 322 T^{3} - 4127 T^{4} - 30138 T^{5} - 70920 T^{6} + 1094344 T^{7} + 18527582 T^{8} + 119631636 T^{9} + 262096090 T^{10} - 2200503287 T^{11} - 30693701772 T^{12} - 88025997822 T^{13} + 1410058928022 T^{14} + 15969608141898 T^{15} + 104939392199145 T^{16} + 750571582669206 T^{17} + 3114820172000598 T^{18} - 9139123171873506 T^{19} - 149775473356494732 T^{20} - 504674441760538009 T^{21} + 2825190190998963610 T^{22} + 60608152736413767468 T^{23} + \)\(44\!\cdots\!02\)\( T^{24} + \)\(12\!\cdots\!48\)\( T^{25} - \)\(37\!\cdots\!80\)\( T^{26} - \)\(74\!\cdots\!14\)\( T^{27} - \)\(47\!\cdots\!07\)\( T^{28} + \)\(17\!\cdots\!94\)\( T^{29} + \)\(47\!\cdots\!34\)\( T^{30} + \)\(27\!\cdots\!89\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 2 T - 173 T^{2} + 964 T^{3} + 9936 T^{4} - 104742 T^{5} + 122655 T^{6} + 3229712 T^{7} - 31796061 T^{8} + 171174736 T^{9} + 344537895 T^{10} - 15593674734 T^{11} + 78399819216 T^{12} + 403140455252 T^{13} - 3834434475317 T^{14} - 2349422279674 T^{15} + 62259690411361 T^{16} \))(\( 1 - 4 T - 22 T^{2} - 440 T^{3} + 660 T^{4} + 49868 T^{5} + 140113 T^{6} - 2285596 T^{7} - 15873165 T^{8} - 76638860 T^{9} + 1323931162 T^{10} + 10376469922 T^{11} - 35381234389 T^{12} - 446229246850 T^{13} - 3132652820265 T^{14} + 9035271835264 T^{15} + 292112805840914 T^{16} + 478869407268992 T^{17} - 8799621772124385 T^{18} - 66433271583287450 T^{19} - 279174957702951109 T^{20} + 4339392954630461546 T^{21} + 29344088384504601898 T^{22} - 90028522586408265820 T^{23} - \)\(98\!\cdots\!65\)\( T^{24} - \)\(75\!\cdots\!68\)\( T^{25} + \)\(24\!\cdots\!37\)\( T^{26} + \)\(46\!\cdots\!96\)\( T^{27} + \)\(32\!\cdots\!60\)\( T^{28} - \)\(11\!\cdots\!20\)\( T^{29} - \)\(30\!\cdots\!18\)\( T^{30} - \)\(29\!\cdots\!28\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 25 T + 206 T^{2} + 25 T^{3} - 15220 T^{4} + 157300 T^{5} - 429276 T^{6} - 8492650 T^{7} + 113408229 T^{8} - 501066350 T^{9} - 1494309756 T^{10} + 32306116700 T^{11} - 184426234420 T^{12} + 17873107475 T^{13} + 8689189930046 T^{14} - 62216287120475 T^{15} + 146830437604321 T^{16} \))(\( 1 - 17 T - 50 T^{2} + 1985 T^{3} + 451 T^{4} - 76307 T^{5} - 426220 T^{6} - 1429235 T^{7} + 60743716 T^{8} + 309308083 T^{9} - 1984484046 T^{10} - 33011892753 T^{11} - 101353409674 T^{12} + 2740413120218 T^{13} + 8464635562844 T^{14} - 86256705224328 T^{15} - 317724345351369 T^{16} - 5089145608235352 T^{17} + 29465396394259964 T^{18} + 562823306217252622 T^{19} - 1228135853600750314 T^{20} - 23601004285101705147 T^{21} - 83706596062330791486 T^{22} + \)\(76\!\cdots\!77\)\( T^{23} + \)\(89\!\cdots\!36\)\( T^{24} - \)\(12\!\cdots\!65\)\( T^{25} - \)\(21\!\cdots\!20\)\( T^{26} - \)\(23\!\cdots\!13\)\( T^{27} + \)\(80\!\cdots\!31\)\( T^{28} + \)\(20\!\cdots\!15\)\( T^{29} - \)\(30\!\cdots\!50\)\( T^{30} - \)\(62\!\cdots\!83\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 7 T - 108 T^{2} + 1783 T^{3} + 318 T^{4} - 164342 T^{5} + 1092660 T^{6} + 5279480 T^{7} - 100824709 T^{8} + 322048280 T^{9} + 4065787860 T^{10} - 37302511502 T^{11} + 4402977438 T^{12} + 1505915204683 T^{13} - 5564200430988 T^{14} - 21999199852147 T^{15} + 191707312997281 T^{16} \))(\( 1 + 7 T - 20 T^{2} + 403 T^{3} + 10067 T^{4} + 48739 T^{5} - 116314 T^{6} + 1172749 T^{7} + 26082242 T^{8} + 24039263 T^{9} - 379016678 T^{10} - 5077729829 T^{11} - 113601259714 T^{12} - 664603591174 T^{13} + 819501759744 T^{14} - 46004665392052 T^{15} - 871290620425703 T^{16} - 2806284588915172 T^{17} + 3049366048007424 T^{18} - 150852387728265694 T^{19} - 1572904979399749474 T^{20} - 4288631831050762529 T^{21} - 19527081139622592758 T^{22} + 75549221576474692523 T^{23} + \)\(50\!\cdots\!02\)\( T^{24} + \)\(13\!\cdots\!09\)\( T^{25} - \)\(82\!\cdots\!14\)\( T^{26} + \)\(21\!\cdots\!79\)\( T^{27} + \)\(26\!\cdots\!07\)\( T^{28} + \)\(65\!\cdots\!43\)\( T^{29} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 287397208 T^{9} + 3671395587 T^{10} + 18632488749 T^{11} + 176026714287 T^{12} - 265221714597 T^{13} - 1152902555289 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} \))(\( 1 + 14 T + 140 T^{2} + 763 T^{3} + 10437 T^{4} + 25566 T^{5} + 32310 T^{6} - 3862236 T^{7} + 10586088 T^{8} - 330657408 T^{9} - 2081383949 T^{10} - 39603959100 T^{11} + 15217861524 T^{12} - 1721196695007 T^{13} + 1593568125167 T^{14} - 22773634065006 T^{15} + 1780879231377332 T^{16} - 1616928018615426 T^{17} + 8033176918966847 T^{18} - 616035230306650377 T^{19} + 386711442550061844 T^{20} - 71454625424023544100 T^{21} - \)\(26\!\cdots\!29\)\( T^{22} - \)\(30\!\cdots\!28\)\( T^{23} + \)\(68\!\cdots\!68\)\( T^{24} - \)\(17\!\cdots\!16\)\( T^{25} + \)\(10\!\cdots\!10\)\( T^{26} + \)\(59\!\cdots\!86\)\( T^{27} + \)\(17\!\cdots\!17\)\( T^{28} + \)\(88\!\cdots\!93\)\( T^{29} + \)\(11\!\cdots\!40\)\( T^{30} + \)\(82\!\cdots\!14\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 3 T - 2 T^{2} - 345 T^{3} + 220 T^{4} - 100292 T^{5} + 620974 T^{6} + 13580 T^{7} + 31898639 T^{8} + 991340 T^{9} + 3309170446 T^{10} - 39015292964 T^{11} + 6247613020 T^{12} - 715209699585 T^{13} - 302668452578 T^{14} - 33142195557291 T^{15} + 806460091894081 T^{16} \))(\( 1 + 35 T + 622 T^{2} + 8881 T^{3} + 123217 T^{4} + 1506367 T^{5} + 15880754 T^{6} + 159131205 T^{7} + 1535260024 T^{8} + 13342640169 T^{9} + 108277262748 T^{10} + 880406179959 T^{11} + 6960680213462 T^{12} + 52467783616198 T^{13} + 405906809006112 T^{14} + 3284917386947140 T^{15} + 27500270697009885 T^{16} + 239798969247141220 T^{17} + 2163077385193570848 T^{18} + 20410859779022497366 T^{19} + \)\(19\!\cdots\!42\)\( T^{20} + \)\(18\!\cdots\!87\)\( T^{21} + \)\(16\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!93\)\( T^{23} + \)\(12\!\cdots\!44\)\( T^{24} + \)\(93\!\cdots\!65\)\( T^{25} + \)\(68\!\cdots\!46\)\( T^{26} + \)\(47\!\cdots\!59\)\( T^{27} + \)\(28\!\cdots\!57\)\( T^{28} + \)\(14\!\cdots\!73\)\( T^{29} + \)\(75\!\cdots\!98\)\( T^{30} + \)\(31\!\cdots\!95\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 116411793 T^{9} - 9693383898 T^{10} + 12111009996 T^{11} + 390708262511 T^{12} - 2363179314432 T^{13} - 18231559164075 T^{14} + 172835180875431 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 15 T + 94 T^{2} - 2495 T^{3} + 38855 T^{4} - 414785 T^{5} + 5676295 T^{6} - 62322940 T^{7} + 690670255 T^{8} - 8423294280 T^{9} + 83337602737 T^{10} - 856819337855 T^{11} + 8989269459608 T^{12} - 82967566755485 T^{13} + 808287092858990 T^{14} - 7591629044633355 T^{15} + 65763672844350810 T^{16} - 599738694526035045 T^{17} + 5044519746532956590 T^{18} - 40906246145557568915 T^{19} + \)\(35\!\cdots\!48\)\( T^{20} - \)\(26\!\cdots\!45\)\( T^{21} + \)\(20\!\cdots\!77\)\( T^{22} - \)\(16\!\cdots\!20\)\( T^{23} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(74\!\cdots\!60\)\( T^{25} + \)\(53\!\cdots\!95\)\( T^{26} - \)\(31\!\cdots\!15\)\( T^{27} + \)\(22\!\cdots\!55\)\( T^{28} - \)\(11\!\cdots\!05\)\( T^{29} + \)\(34\!\cdots\!14\)\( T^{30} - \)\(43\!\cdots\!85\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 23 T + 98 T^{2} + 1745 T^{3} - 15470 T^{4} - 98632 T^{5} + 1750974 T^{6} + 1504700 T^{7} - 137136651 T^{8} + 124890100 T^{9} + 12062459886 T^{10} - 56396495384 T^{11} - 734180225870 T^{12} + 6873625922035 T^{13} + 32040156590162 T^{14} - 624129172761421 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 5 T - 318 T^{2} + 3741 T^{3} + 48517 T^{4} - 994133 T^{5} - 3255786 T^{6} + 170309105 T^{7} - 263285806 T^{8} - 21750909411 T^{9} + 119250152558 T^{10} + 2097202277489 T^{11} - 20999477370338 T^{12} - 141328121959992 T^{13} + 2545394345691772 T^{14} + 4513257769614610 T^{15} - 237031046695965905 T^{16} + 374600394878012630 T^{17} + 17535221647470617308 T^{18} - 80809582871137945704 T^{19} - \)\(99\!\cdots\!98\)\( T^{20} + \)\(82\!\cdots\!27\)\( T^{21} + \)\(38\!\cdots\!02\)\( T^{22} - \)\(59\!\cdots\!97\)\( T^{23} - \)\(59\!\cdots\!46\)\( T^{24} + \)\(31\!\cdots\!15\)\( T^{25} - \)\(50\!\cdots\!14\)\( T^{26} - \)\(12\!\cdots\!11\)\( T^{27} + \)\(51\!\cdots\!37\)\( T^{28} + \)\(33\!\cdots\!83\)\( T^{29} - \)\(23\!\cdots\!22\)\( T^{30} - \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 304291 T^{5} + 2471352 T^{6} + 11984473 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 65606528 T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 65145596911 T^{9} + 519669308288 T^{10} - 3632450763191 T^{11} + 22450616901343 T^{12} - 116712426543549 T^{13} + 512884692271752 T^{14} - 1636559391134573 T^{15} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 3177152550 T^{9} - 5432916553 T^{10} - 76089548010 T^{11} + 890781625422 T^{12} - 18548654955120 T^{13} + 300702893779369 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 20 T + 123 T^{2} + 28 T^{3} + 4997 T^{4} - 239296 T^{5} + 2683386 T^{6} - 13918670 T^{7} + 12108314 T^{8} + 81926108 T^{9} + 9118270907 T^{10} - 123570735168 T^{11} + 816960697702 T^{12} + 3552108922056 T^{13} - 86512314985382 T^{14} - 43450963654390 T^{15} + 5424229096525225 T^{16} - 4214743474475830 T^{17} - 813994371697459238 T^{18} + 3241913906219615688 T^{19} + 72324943172816412262 T^{20} - \)\(10\!\cdots\!76\)\( T^{21} + \)\(75\!\cdots\!03\)\( T^{22} + \)\(66\!\cdots\!04\)\( T^{23} + \)\(94\!\cdots\!54\)\( T^{24} - \)\(10\!\cdots\!90\)\( T^{25} + \)\(19\!\cdots\!14\)\( T^{26} - \)\(17\!\cdots\!88\)\( T^{27} + \)\(34\!\cdots\!77\)\( T^{28} + \)\(18\!\cdots\!56\)\( T^{29} + \)\(80\!\cdots\!87\)\( T^{30} - \)\(12\!\cdots\!60\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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