Properties

Label 48.3.l
Level 48
Weight 3
Character orbit l
Rep. character \(\chi_{48}(19,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 16
Newform subspaces 1
Sturm bound 24
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(48, [\chi])\).

Total New Old
Modular forms 36 16 20
Cusp forms 28 16 12
Eisenstein series 8 0 8

Trace form

\( 16q + 12q^{4} - 12q^{8} + O(q^{10}) \) \( 16q + 12q^{4} - 12q^{8} - 56q^{10} + 32q^{11} - 24q^{12} - 44q^{14} + 32q^{16} + 12q^{18} - 32q^{19} + 80q^{20} + 32q^{22} - 128q^{23} + 36q^{24} - 100q^{26} - 120q^{28} + 32q^{29} + 72q^{30} + 160q^{32} + 96q^{34} + 96q^{35} + 12q^{36} - 96q^{37} + 168q^{38} + 48q^{40} - 60q^{42} + 160q^{43} + 88q^{44} + 136q^{46} - 144q^{48} + 112q^{49} - 236q^{50} - 96q^{51} - 48q^{52} - 160q^{53} - 36q^{54} - 256q^{55} - 224q^{56} + 144q^{58} - 128q^{59} - 72q^{60} - 32q^{61} - 276q^{62} - 408q^{64} - 32q^{65} + 72q^{66} + 320q^{67} - 448q^{68} + 96q^{69} - 384q^{70} + 512q^{71} + 60q^{72} + 348q^{74} + 192q^{75} + 72q^{76} + 224q^{77} + 396q^{78} + 552q^{80} - 144q^{81} - 40q^{82} - 160q^{83} + 72q^{84} + 160q^{85} + 528q^{86} + 480q^{88} - 24q^{90} - 480q^{91} + 496q^{92} + 312q^{94} - 480q^{96} - 440q^{98} + 96q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(48, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
48.3.l.a \(16\) \(1.308\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}-\beta _{5}q^{3}+(1-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(48, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(48, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T^{2} + 4 T^{3} + 10 T^{4} - 56 T^{5} + 88 T^{6} + 128 T^{7} - 496 T^{8} + 512 T^{9} + 1408 T^{10} - 3584 T^{11} + 2560 T^{12} + 4096 T^{13} - 24576 T^{14} + 65536 T^{16} \)
$3$ \( ( 1 + 9 T^{4} )^{4} \)
$5$ \( 1 + 32 T^{3} - 344 T^{4} + 5664 T^{5} + 512 T^{6} + 145600 T^{7} + 223452 T^{8} - 2255168 T^{9} + 20875776 T^{10} - 67282720 T^{11} + 753060504 T^{12} + 1881828576 T^{13} + 2740220928 T^{14} + 75471830656 T^{15} - 399298967994 T^{16} + 1886795766400 T^{17} + 1712638080000 T^{18} + 29403571500000 T^{19} + 294164259375000 T^{20} - 657057812500000 T^{21} + 5096625000000000 T^{22} - 13764453125000000 T^{23} + 34096069335937500 T^{24} + 555419921875000000 T^{25} + 48828125000000000 T^{26} + 13504028320312500000 T^{27} - 20503997802734375000 T^{28} + 47683715820312500000 T^{29} + \)\(23\!\cdots\!25\)\( T^{32} \)
$7$ \( ( 1 + 168 T^{2} - 448 T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 178833536 T^{9} + 2570011192 T^{10} - 6648580288 T^{11} + 86910139876 T^{12} - 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2} \)
$11$ \( 1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 431768458252384 T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 7491143217161407584 T^{17} + 77497246731528160768 T^{18} - \)\(76\!\cdots\!24\)\( T^{19} + \)\(72\!\cdots\!52\)\( T^{20} - \)\(75\!\cdots\!28\)\( T^{21} + \)\(79\!\cdots\!72\)\( T^{22} - \)\(83\!\cdots\!00\)\( T^{23} + \)\(10\!\cdots\!00\)\( T^{24} - \)\(12\!\cdots\!16\)\( T^{25} + \)\(12\!\cdots\!48\)\( T^{26} - \)\(13\!\cdots\!96\)\( T^{27} + \)\(13\!\cdots\!12\)\( T^{28} - \)\(10\!\cdots\!80\)\( T^{29} + \)\(73\!\cdots\!72\)\( T^{30} - \)\(55\!\cdots\!32\)\( T^{31} + \)\(21\!\cdots\!21\)\( T^{32} \)
$13$ \( 1 + 3200 T^{3} + 7608 T^{4} + 95360 T^{5} + 5120000 T^{6} + 68335872 T^{7} + 2004669468 T^{8} + 7270355200 T^{9} + 184268595200 T^{10} + 4889456013184 T^{11} + 5354592144136 T^{12} + 669839496880000 T^{13} + 7008632866619392 T^{14} + 70586941744778752 T^{15} + 2398056097119178950 T^{16} + 11929193154867609088 T^{17} + \)\(20\!\cdots\!12\)\( T^{18} + \)\(32\!\cdots\!00\)\( T^{19} + \)\(43\!\cdots\!56\)\( T^{20} + \)\(67\!\cdots\!16\)\( T^{21} + \)\(42\!\cdots\!00\)\( T^{22} + \)\(28\!\cdots\!00\)\( T^{23} + \)\(13\!\cdots\!88\)\( T^{24} + \)\(76\!\cdots\!88\)\( T^{25} + \)\(97\!\cdots\!00\)\( T^{26} + \)\(30\!\cdots\!40\)\( T^{27} + \)\(41\!\cdots\!88\)\( T^{28} + \)\(29\!\cdots\!00\)\( T^{29} + \)\(44\!\cdots\!81\)\( T^{32} \)
$17$ \( ( 1 + 968 T^{2} - 2944 T^{3} + 516540 T^{4} - 3209600 T^{5} + 201700088 T^{6} - 1543904000 T^{7} + 63894476806 T^{8} - 446188256000 T^{9} + 16846193049848 T^{10} - 77471941462400 T^{11} + 3603257748574140 T^{12} - 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2} \)
$19$ \( 1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} - \)\(19\!\cdots\!42\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(41\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!36\)\( T^{20} - \)\(42\!\cdots\!12\)\( T^{21} - \)\(67\!\cdots\!08\)\( T^{22} - \)\(56\!\cdots\!96\)\( T^{23} + \)\(31\!\cdots\!64\)\( T^{24} + \)\(46\!\cdots\!64\)\( T^{25} + \)\(73\!\cdots\!88\)\( T^{26} - \)\(11\!\cdots\!88\)\( T^{27} - \)\(25\!\cdots\!08\)\( T^{28} - \)\(46\!\cdots\!36\)\( T^{29} + \)\(32\!\cdots\!92\)\( T^{30} + \)\(73\!\cdots\!32\)\( T^{31} + \)\(83\!\cdots\!61\)\( T^{32} \)
$23$ \( ( 1 + 64 T + 3496 T^{2} + 127936 T^{3} + 4410332 T^{4} + 130001728 T^{5} + 3673719192 T^{6} + 94049622208 T^{7} + 2261818535238 T^{8} + 49752250148032 T^{9} + 1028057252408472 T^{10} + 19244921376016192 T^{11} + 345377444336323292 T^{12} + 5299942138629398464 T^{13} + 76613527014343042216 T^{14} + \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \)
$29$ \( 1 - 32 T + 512 T^{2} + 18368 T^{3} - 1552984 T^{4} + 20596992 T^{5} + 304715776 T^{6} - 25469097376 T^{7} + 491466517980 T^{8} + 9791032230816 T^{9} - 347423504794624 T^{10} + 2649303176415616 T^{11} + 694517140133881240 T^{12} - 20658732330776531008 T^{13} + \)\(19\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!40\)\( T^{15} - \)\(82\!\cdots\!10\)\( T^{16} + \)\(96\!\cdots\!40\)\( T^{17} + \)\(13\!\cdots\!68\)\( T^{18} - \)\(12\!\cdots\!68\)\( T^{19} + \)\(34\!\cdots\!40\)\( T^{20} + \)\(11\!\cdots\!16\)\( T^{21} - \)\(12\!\cdots\!84\)\( T^{22} + \)\(29\!\cdots\!96\)\( T^{23} + \)\(12\!\cdots\!80\)\( T^{24} - \)\(53\!\cdots\!36\)\( T^{25} + \)\(53\!\cdots\!76\)\( T^{26} + \)\(30\!\cdots\!72\)\( T^{27} - \)\(19\!\cdots\!04\)\( T^{28} + \)\(19\!\cdots\!28\)\( T^{29} + \)\(45\!\cdots\!32\)\( T^{30} - \)\(23\!\cdots\!32\)\( T^{31} + \)\(62\!\cdots\!41\)\( T^{32} \)
$31$ \( 1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} + \)\(42\!\cdots\!64\)\( T^{12} - \)\(51\!\cdots\!00\)\( T^{14} + \)\(53\!\cdots\!38\)\( T^{16} - \)\(47\!\cdots\!00\)\( T^{18} + \)\(36\!\cdots\!24\)\( T^{20} - \)\(23\!\cdots\!92\)\( T^{22} + \)\(12\!\cdots\!12\)\( T^{24} - \)\(53\!\cdots\!76\)\( T^{26} + \)\(18\!\cdots\!24\)\( T^{28} - \)\(41\!\cdots\!92\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \)
$37$ \( 1 + 96 T + 4608 T^{2} + 145952 T^{3} + 4040888 T^{4} + 217733344 T^{5} + 12932982272 T^{6} + 602883756192 T^{7} + 21839639792924 T^{8} + 655265530977504 T^{9} + 21703692469355008 T^{10} + 815191556064282016 T^{11} + 35433653736114978312 T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!84\)\( T^{15} + \)\(43\!\cdots\!90\)\( T^{16} + \)\(20\!\cdots\!96\)\( T^{17} + \)\(95\!\cdots\!72\)\( T^{18} + \)\(37\!\cdots\!60\)\( T^{19} + \)\(12\!\cdots\!52\)\( T^{20} + \)\(39\!\cdots\!84\)\( T^{21} + \)\(14\!\cdots\!48\)\( T^{22} + \)\(59\!\cdots\!56\)\( T^{23} + \)\(26\!\cdots\!84\)\( T^{24} + \)\(10\!\cdots\!68\)\( T^{25} + \)\(29\!\cdots\!72\)\( T^{26} + \)\(68\!\cdots\!36\)\( T^{27} + \)\(17\!\cdots\!68\)\( T^{28} + \)\(86\!\cdots\!68\)\( T^{29} + \)\(37\!\cdots\!68\)\( T^{30} + \)\(10\!\cdots\!04\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \)
$41$ \( 1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} + \)\(16\!\cdots\!68\)\( T^{12} - \)\(35\!\cdots\!72\)\( T^{14} + \)\(64\!\cdots\!06\)\( T^{16} - \)\(99\!\cdots\!92\)\( T^{18} + \)\(13\!\cdots\!28\)\( T^{20} - \)\(14\!\cdots\!48\)\( T^{22} + \)\(13\!\cdots\!44\)\( T^{24} - \)\(94\!\cdots\!80\)\( T^{26} + \)\(52\!\cdots\!44\)\( T^{28} - \)\(19\!\cdots\!40\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \)
$43$ \( 1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 9339737479444512 T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} - \)\(12\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!84\)\( T^{13} - \)\(44\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!76\)\( T^{15} - \)\(17\!\cdots\!30\)\( T^{16} + \)\(53\!\cdots\!24\)\( T^{17} - \)\(15\!\cdots\!20\)\( T^{18} + \)\(83\!\cdots\!16\)\( T^{19} - \)\(15\!\cdots\!76\)\( T^{20} - \)\(15\!\cdots\!20\)\( T^{21} + \)\(11\!\cdots\!20\)\( T^{22} - \)\(69\!\cdots\!88\)\( T^{23} + \)\(45\!\cdots\!00\)\( T^{24} - \)\(22\!\cdots\!88\)\( T^{25} + \)\(83\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!36\)\( T^{27} + \)\(11\!\cdots\!32\)\( T^{28} - \)\(28\!\cdots\!36\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(16\!\cdots\!40\)\( T^{31} + \)\(18\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{16} - \)\(21\!\cdots\!24\)\( T^{18} + \)\(37\!\cdots\!76\)\( T^{20} - \)\(54\!\cdots\!48\)\( T^{22} + \)\(64\!\cdots\!32\)\( T^{24} - \)\(58\!\cdots\!84\)\( T^{26} + \)\(37\!\cdots\!72\)\( T^{28} - \)\(15\!\cdots\!84\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 + 160 T + 12800 T^{2} + 602944 T^{3} + 3948712 T^{4} - 1481707264 T^{5} - 105845942272 T^{6} - 3791430241760 T^{7} + 34861972067036 T^{8} + 14471440004155872 T^{9} + 1098860393015073792 T^{10} + 54880211634179791488 T^{11} + \)\(13\!\cdots\!12\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} - \)\(18\!\cdots\!48\)\( T^{14} + \)\(16\!\cdots\!08\)\( T^{15} + \)\(51\!\cdots\!54\)\( T^{16} + \)\(46\!\cdots\!72\)\( T^{17} - \)\(14\!\cdots\!88\)\( T^{18} - \)\(38\!\cdots\!00\)\( T^{19} + \)\(84\!\cdots\!32\)\( T^{20} + \)\(95\!\cdots\!12\)\( T^{21} + \)\(53\!\cdots\!72\)\( T^{22} + \)\(19\!\cdots\!68\)\( T^{23} + \)\(13\!\cdots\!56\)\( T^{24} - \)\(41\!\cdots\!40\)\( T^{25} - \)\(32\!\cdots\!72\)\( T^{26} - \)\(12\!\cdots\!76\)\( T^{27} + \)\(95\!\cdots\!72\)\( T^{28} + \)\(40\!\cdots\!76\)\( T^{29} + \)\(24\!\cdots\!00\)\( T^{30} + \)\(85\!\cdots\!40\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} + \)\(69\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!56\)\( T^{14} + \)\(29\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!38\)\( T^{16} + \)\(10\!\cdots\!44\)\( T^{17} + \)\(62\!\cdots\!16\)\( T^{18} + \)\(29\!\cdots\!56\)\( T^{19} + \)\(16\!\cdots\!40\)\( T^{20} + \)\(98\!\cdots\!80\)\( T^{21} + \)\(45\!\cdots\!92\)\( T^{22} + \)\(19\!\cdots\!32\)\( T^{23} + \)\(11\!\cdots\!64\)\( T^{24} + \)\(53\!\cdots\!08\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} + \)\(84\!\cdots\!48\)\( T^{27} + \)\(43\!\cdots\!24\)\( T^{28} + \)\(12\!\cdots\!28\)\( T^{29} + \)\(31\!\cdots\!32\)\( T^{30} + \)\(17\!\cdots\!28\)\( T^{31} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( 1 + 32 T + 512 T^{2} - 38048 T^{3} - 60439624 T^{4} - 1520787552 T^{5} - 16996289024 T^{6} + 2981900088544 T^{7} + 2018049968078364 T^{8} + 40394929489472928 T^{9} + 275088896591278592 T^{10} - \)\(10\!\cdots\!56\)\( T^{11} - \)\(45\!\cdots\!20\)\( T^{12} - \)\(71\!\cdots\!16\)\( T^{13} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!64\)\( T^{15} + \)\(72\!\cdots\!42\)\( T^{16} + \)\(88\!\cdots\!44\)\( T^{17} - \)\(26\!\cdots\!48\)\( T^{18} - \)\(36\!\cdots\!76\)\( T^{19} - \)\(86\!\cdots\!20\)\( T^{20} - \)\(75\!\cdots\!56\)\( T^{21} + \)\(73\!\cdots\!32\)\( T^{22} + \)\(39\!\cdots\!48\)\( T^{23} + \)\(74\!\cdots\!04\)\( T^{24} + \)\(40\!\cdots\!64\)\( T^{25} - \)\(86\!\cdots\!24\)\( T^{26} - \)\(28\!\cdots\!92\)\( T^{27} - \)\(42\!\cdots\!84\)\( T^{28} - \)\(99\!\cdots\!28\)\( T^{29} + \)\(49\!\cdots\!72\)\( T^{30} + \)\(11\!\cdots\!32\)\( T^{31} + \)\(13\!\cdots\!21\)\( T^{32} \)
$67$ \( 1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 525110406070976 T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} + \)\(26\!\cdots\!12\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{13} + \)\(71\!\cdots\!04\)\( T^{14} - \)\(48\!\cdots\!52\)\( T^{15} + \)\(32\!\cdots\!74\)\( T^{16} - \)\(21\!\cdots\!28\)\( T^{17} + \)\(14\!\cdots\!84\)\( T^{18} - \)\(93\!\cdots\!72\)\( T^{19} + \)\(58\!\cdots\!20\)\( T^{20} - \)\(36\!\cdots\!20\)\( T^{21} + \)\(21\!\cdots\!32\)\( T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(72\!\cdots\!80\)\( T^{24} - \)\(38\!\cdots\!84\)\( T^{25} + \)\(19\!\cdots\!72\)\( T^{26} - \)\(95\!\cdots\!84\)\( T^{27} + \)\(42\!\cdots\!84\)\( T^{28} - \)\(18\!\cdots\!88\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(19\!\cdots\!80\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 - 256 T + 68104 T^{2} - 10692864 T^{3} + 1610923548 T^{4} - 179723087616 T^{5} + 18972832358712 T^{6} - 1588998739085056 T^{7} + 125568612540426694 T^{8} - 8010142643727767296 T^{9} + \)\(48\!\cdots\!72\)\( T^{10} - \)\(23\!\cdots\!36\)\( T^{11} + \)\(10\!\cdots\!28\)\( T^{12} - \)\(34\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!64\)\( T^{14} - \)\(21\!\cdots\!36\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \)
$73$ \( 1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} - \)\(15\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} - \)\(83\!\cdots\!96\)\( T^{14} + \)\(47\!\cdots\!22\)\( T^{16} - \)\(23\!\cdots\!36\)\( T^{18} + \)\(10\!\cdots\!56\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(10\!\cdots\!52\)\( T^{24} - \)\(26\!\cdots\!36\)\( T^{26} + \)\(49\!\cdots\!04\)\( T^{28} - \)\(63\!\cdots\!08\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \)
$79$ \( 1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} - \)\(60\!\cdots\!08\)\( T^{10} + \)\(55\!\cdots\!96\)\( T^{12} - \)\(43\!\cdots\!36\)\( T^{14} + \)\(29\!\cdots\!62\)\( T^{16} - \)\(16\!\cdots\!16\)\( T^{18} + \)\(84\!\cdots\!56\)\( T^{20} - \)\(35\!\cdots\!28\)\( T^{22} + \)\(12\!\cdots\!24\)\( T^{24} - \)\(33\!\cdots\!88\)\( T^{26} + \)\(66\!\cdots\!08\)\( T^{28} - \)\(85\!\cdots\!08\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \)
$83$ \( 1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} + \)\(11\!\cdots\!96\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} + \)\(12\!\cdots\!44\)\( T^{13} + \)\(89\!\cdots\!76\)\( T^{14} + \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!66\)\( T^{16} + \)\(51\!\cdots\!64\)\( T^{17} + \)\(42\!\cdots\!96\)\( T^{18} + \)\(39\!\cdots\!36\)\( T^{19} + \)\(30\!\cdots\!00\)\( T^{20} + \)\(17\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!56\)\( T^{22} + \)\(10\!\cdots\!52\)\( T^{23} + \)\(92\!\cdots\!52\)\( T^{24} + \)\(60\!\cdots\!36\)\( T^{25} + \)\(39\!\cdots\!88\)\( T^{26} + \)\(27\!\cdots\!52\)\( T^{27} + \)\(12\!\cdots\!04\)\( T^{28} + \)\(70\!\cdots\!76\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} + \)\(59\!\cdots\!40\)\( T^{31} + \)\(25\!\cdots\!61\)\( T^{32} \)
$89$ \( 1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} - \)\(23\!\cdots\!04\)\( T^{10} + \)\(28\!\cdots\!72\)\( T^{12} - \)\(29\!\cdots\!36\)\( T^{14} + \)\(25\!\cdots\!10\)\( T^{16} - \)\(18\!\cdots\!76\)\( T^{18} + \)\(11\!\cdots\!32\)\( T^{20} - \)\(57\!\cdots\!84\)\( T^{22} + \)\(24\!\cdots\!96\)\( T^{24} - \)\(80\!\cdots\!12\)\( T^{26} + \)\(19\!\cdots\!76\)\( T^{28} - \)\(31\!\cdots\!48\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$97$ \( ( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} + \)\(96\!\cdots\!84\)\( T^{10} + \)\(29\!\cdots\!72\)\( T^{11} + \)\(60\!\cdots\!04\)\( T^{12} + \)\(85\!\cdots\!76\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \)
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