Properties

Label 210.3.l
Level 210
Weight 3
Character orbit l
Rep. character \(\chi_{210}(43,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 24
Newform subspaces 2
Sturm bound 144
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(144\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(210, [\chi])\).

Total New Old
Modular forms 208 24 184
Cusp forms 176 24 152
Eisenstein series 32 0 32

Trace form

\( 24q - 8q^{2} - 16q^{5} + 16q^{8} + O(q^{10}) \) \( 24q - 8q^{2} - 16q^{5} + 16q^{8} + 24q^{10} - 24q^{13} - 96q^{16} + 24q^{17} - 24q^{18} - 16q^{20} - 16q^{22} - 16q^{23} - 8q^{25} + 80q^{26} + 48q^{30} + 32q^{31} + 32q^{32} + 144q^{33} + 144q^{36} + 8q^{37} - 16q^{40} - 320q^{41} - 32q^{43} + 24q^{45} - 128q^{46} - 64q^{47} - 184q^{50} - 48q^{52} - 152q^{53} + 160q^{55} + 32q^{58} + 96q^{60} + 384q^{61} + 256q^{62} + 40q^{65} + 192q^{66} - 48q^{68} - 112q^{70} + 256q^{71} - 48q^{72} + 312q^{73} - 288q^{75} - 224q^{77} + 64q^{80} - 216q^{81} - 320q^{82} - 576q^{83} + 120q^{85} - 64q^{86} - 192q^{87} + 32q^{88} + 24q^{90} - 32q^{92} + 240q^{93} - 112q^{95} + 664q^{97} + 56q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
210.3.l.a \(8\) \(5.722\) 8.0.\(\cdots\).1 None \(8\) \(0\) \(0\) \(0\) \(q+(1+\beta _{2})q^{2}+\beta _{7}q^{3}+2\beta _{2}q^{4}+(-2\beta _{3}+\cdots)q^{5}+\cdots\)
210.3.l.b \(16\) \(5.722\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-16\) \(0\) \(-16\) \(0\) \(q+(-1-\beta _{2})q^{2}+\beta _{3}q^{3}+2\beta _{2}q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} )^{4} \))(\( ( 1 + 2 T + 2 T^{2} )^{8} \))
$3$ (\( ( 1 + 9 T^{4} )^{2} \))(\( ( 1 + 9 T^{4} )^{4} \))
$5$ (\( 1 + 24 T^{2} + 50 T^{4} + 15000 T^{6} + 390625 T^{8} \))(\( 1 + 16 T + 108 T^{2} + 240 T^{3} - 1736 T^{4} - 15856 T^{5} - 26268 T^{6} + 353520 T^{7} + 2938350 T^{8} + 8838000 T^{9} - 16417500 T^{10} - 247750000 T^{11} - 678125000 T^{12} + 2343750000 T^{13} + 26367187500 T^{14} + 97656250000 T^{15} + 152587890625 T^{16} \))
$7$ (\( ( 1 + 49 T^{4} )^{2} \))(\( ( 1 + 49 T^{4} )^{4} \))
$11$ (\( ( 1 + 4 T + 120 T^{2} - 964 T^{3} + 9758 T^{4} - 116644 T^{5} + 1756920 T^{6} + 7086244 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 - 4 T + 428 T^{2} - 3020 T^{3} + 112132 T^{4} - 750180 T^{5} + 21255892 T^{6} - 132232716 T^{7} + 2897491254 T^{8} - 16000158636 T^{9} + 311207514772 T^{10} - 1328989630980 T^{11} + 24036490044292 T^{12} - 78331022295020 T^{13} + 1343247345236588 T^{14} - 1518999334332964 T^{15} + 45949729863572161 T^{16} )^{2} \))
$13$ (\( 1 - 8 T + 32 T^{2} - 1000 T^{3} + 57188 T^{4} - 379192 T^{5} + 1703520 T^{6} - 57464472 T^{7} + 1905455878 T^{8} - 9711495768 T^{9} + 48654234720 T^{10} - 1830287358328 T^{11} + 46650008472548 T^{12} - 137858491849000 T^{13} + 745538723919392 T^{14} - 31499011085594312 T^{15} + 665416609183179841 T^{16} \))(\( 1 + 32 T + 512 T^{2} + 12624 T^{3} + 210400 T^{4} + 1218608 T^{5} + 10953344 T^{6} - 12361888 T^{7} - 9595251012 T^{8} - 144909422048 T^{9} - 1442470310272 T^{10} - 30698489798640 T^{11} - 199614992290272 T^{12} + 2586969858854768 T^{13} + 29532641854346496 T^{14} + 799237423444942944 T^{15} + 19007129532539001606 T^{16} + \)\(13\!\cdots\!36\)\( T^{17} + \)\(84\!\cdots\!56\)\( T^{18} + \)\(12\!\cdots\!12\)\( T^{19} - \)\(16\!\cdots\!12\)\( T^{20} - \)\(42\!\cdots\!60\)\( T^{21} - \)\(33\!\cdots\!32\)\( T^{22} - \)\(57\!\cdots\!72\)\( T^{23} - \)\(63\!\cdots\!92\)\( T^{24} - \)\(13\!\cdots\!52\)\( T^{25} + \)\(20\!\cdots\!44\)\( T^{26} + \)\(39\!\cdots\!52\)\( T^{27} + \)\(11\!\cdots\!00\)\( T^{28} + \)\(11\!\cdots\!16\)\( T^{29} + \)\(79\!\cdots\!52\)\( T^{30} + \)\(83\!\cdots\!68\)\( T^{31} + \)\(44\!\cdots\!81\)\( T^{32} \))
$17$ (\( 1 + 32 T + 512 T^{2} + 13280 T^{3} + 494308 T^{4} + 9125408 T^{5} + 127106560 T^{6} + 2946229728 T^{7} + 67506528198 T^{8} + 851460391392 T^{9} + 10616066997760 T^{10} + 220265165253152 T^{11} + 3448172709145828 T^{12} + 26772398997962720 T^{13} + 298302585461637632 T^{14} + 5388090449900829728 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 56 T + 1568 T^{2} - 29832 T^{3} + 267808 T^{4} + 3397960 T^{5} - 165234592 T^{6} + 3465012472 T^{7} - 48572233284 T^{8} + 491667950696 T^{9} - 4716183819232 T^{10} + 39445096533336 T^{11} + 280624802792928 T^{12} - 26208629063564312 T^{13} + 761979769352889696 T^{14} - 15987258252632453928 T^{15} + \)\(28\!\cdots\!74\)\( T^{16} - \)\(46\!\cdots\!92\)\( T^{17} + \)\(63\!\cdots\!16\)\( T^{18} - \)\(63\!\cdots\!28\)\( T^{19} + \)\(19\!\cdots\!48\)\( T^{20} + \)\(79\!\cdots\!64\)\( T^{21} - \)\(27\!\cdots\!52\)\( T^{22} + \)\(82\!\cdots\!84\)\( T^{23} - \)\(23\!\cdots\!04\)\( T^{24} + \)\(48\!\cdots\!48\)\( T^{25} - \)\(67\!\cdots\!92\)\( T^{26} + \)\(39\!\cdots\!40\)\( T^{27} + \)\(90\!\cdots\!68\)\( T^{28} - \)\(29\!\cdots\!08\)\( T^{29} + \)\(44\!\cdots\!88\)\( T^{30} - \)\(45\!\cdots\!44\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \))
$19$ (\( 1 - 1632 T^{2} + 1354756 T^{4} - 767600928 T^{6} + 320946269766 T^{8} - 100034520537888 T^{10} + 23008583931172996 T^{12} - 3612129947915974752 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 - 2056 T^{2} + 2230944 T^{4} - 1603740248 T^{6} + 819184531004 T^{8} - 297045269538504 T^{10} + 69987022118810464 T^{12} - 7175141044809308632 T^{14} - \)\(43\!\cdots\!46\)\( T^{16} - \)\(93\!\cdots\!72\)\( T^{18} + \)\(11\!\cdots\!24\)\( T^{20} - \)\(65\!\cdots\!44\)\( T^{22} + \)\(23\!\cdots\!24\)\( T^{24} - \)\(60\!\cdots\!48\)\( T^{26} + \)\(10\!\cdots\!24\)\( T^{28} - \)\(13\!\cdots\!96\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( 1 + 40 T + 800 T^{2} + 23320 T^{3} + 928156 T^{4} + 22803400 T^{5} + 441522400 T^{6} + 12864455160 T^{7} + 373563445830 T^{8} + 6805296779640 T^{9} + 123556069938400 T^{10} + 3375721591222600 T^{11} + 72684810854471836 T^{12} + 966066241502294680 T^{13} + 17531699545616256800 T^{14} + \)\(46\!\cdots\!60\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 24 T + 288 T^{2} - 17192 T^{3} + 221472 T^{4} - 2391768 T^{5} + 141400928 T^{6} - 3172207336 T^{7} + 137674628156 T^{8} - 274022502200 T^{9} - 6993233578720 T^{10} - 103772124341000 T^{11} - 60508914798085408 T^{12} + 933592691352233992 T^{13} - 15427273581773601184 T^{14} + \)\(57\!\cdots\!36\)\( T^{15} - \)\(64\!\cdots\!26\)\( T^{16} + \)\(30\!\cdots\!44\)\( T^{17} - \)\(43\!\cdots\!44\)\( T^{18} + \)\(13\!\cdots\!88\)\( T^{19} - \)\(47\!\cdots\!48\)\( T^{20} - \)\(42\!\cdots\!00\)\( T^{21} - \)\(15\!\cdots\!20\)\( T^{22} - \)\(31\!\cdots\!00\)\( T^{23} + \)\(84\!\cdots\!16\)\( T^{24} - \)\(10\!\cdots\!84\)\( T^{25} + \)\(24\!\cdots\!28\)\( T^{26} - \)\(21\!\cdots\!72\)\( T^{27} + \)\(10\!\cdots\!52\)\( T^{28} - \)\(43\!\cdots\!88\)\( T^{29} + \)\(38\!\cdots\!28\)\( T^{30} - \)\(17\!\cdots\!76\)\( T^{31} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( 1 - 5816 T^{2} + 15494364 T^{4} - 24578727304 T^{6} + 25344087778310 T^{8} - 17384066826300424 T^{10} + 7751000012112051804 T^{12} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \))(\( 1 - 4032 T^{2} + 8105976 T^{4} - 11130637632 T^{6} + 11901856158364 T^{8} - 11208853722734016 T^{10} + 10516466734892486856 T^{12} - \)\(99\!\cdots\!40\)\( T^{14} + \)\(87\!\cdots\!46\)\( T^{16} - \)\(70\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!16\)\( T^{20} - \)\(39\!\cdots\!56\)\( T^{22} + \)\(29\!\cdots\!44\)\( T^{24} - \)\(19\!\cdots\!32\)\( T^{26} + \)\(10\!\cdots\!56\)\( T^{28} - \)\(35\!\cdots\!52\)\( T^{30} + \)\(62\!\cdots\!41\)\( T^{32} \))
$31$ (\( ( 1 - 72 T + 4816 T^{2} - 187848 T^{3} + 7076706 T^{4} - 180521928 T^{5} + 4447677136 T^{6} - 63900265032 T^{7} + 852891037441 T^{8} )^{2} \))(\( ( 1 + 56 T + 4484 T^{2} + 186776 T^{3} + 8992264 T^{4} + 304202984 T^{5} + 11800345036 T^{6} + 348556649544 T^{7} + 12266333612430 T^{8} + 334962940211784 T^{9} + 10897866447991756 T^{10} + 269981268071184104 T^{11} + 7669421371903356424 T^{12} + \)\(15\!\cdots\!76\)\( T^{13} + \)\(35\!\cdots\!24\)\( T^{14} + \)\(42\!\cdots\!76\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \))
$37$ (\( 1 - 160 T + 12800 T^{2} - 692576 T^{3} + 28393916 T^{4} - 973559072 T^{5} + 32158084608 T^{6} - 1118111280864 T^{7} + 40611983255110 T^{8} - 1530694343502816 T^{9} + 60269428007013888 T^{10} - 2497886221751932448 T^{11} + 99733046566358744636 T^{12} - \)\(33\!\cdots\!24\)\( T^{13} + \)\(84\!\cdots\!00\)\( T^{14} - \)\(14\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 152 T + 11552 T^{2} + 598344 T^{3} + 25962168 T^{4} + 1167158904 T^{5} + 56500959840 T^{6} + 2639490056104 T^{7} + 111109511071516 T^{8} + 4242418231122872 T^{9} + 154876079427922976 T^{10} + 5762441784032775208 T^{11} + \)\(22\!\cdots\!72\)\( T^{12} + \)\(89\!\cdots\!08\)\( T^{13} + \)\(32\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!40\)\( T^{15} + \)\(37\!\cdots\!46\)\( T^{16} + \)\(14\!\cdots\!60\)\( T^{17} + \)\(60\!\cdots\!32\)\( T^{18} + \)\(22\!\cdots\!72\)\( T^{19} + \)\(79\!\cdots\!12\)\( T^{20} + \)\(27\!\cdots\!92\)\( T^{21} + \)\(10\!\cdots\!56\)\( T^{22} + \)\(38\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!56\)\( T^{24} + \)\(44\!\cdots\!16\)\( T^{25} + \)\(13\!\cdots\!40\)\( T^{26} + \)\(36\!\cdots\!76\)\( T^{27} + \)\(11\!\cdots\!48\)\( T^{28} + \)\(35\!\cdots\!96\)\( T^{29} + \)\(93\!\cdots\!92\)\( T^{30} + \)\(16\!\cdots\!48\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \))
$41$ (\( ( 1 + 160 T + 11944 T^{2} + 591520 T^{3} + 24800530 T^{4} + 994345120 T^{5} + 33750889384 T^{6} + 760016678560 T^{7} + 7984925229121 T^{8} )^{2} \))(\( ( 1 + 8252 T^{2} - 19936 T^{3} + 34068728 T^{4} - 127251488 T^{5} + 92291784532 T^{6} - 367862933760 T^{7} + 180432835211182 T^{8} - 618377591650560 T^{9} + 260794525350928852 T^{10} - 604457832822360608 T^{11} + \)\(27\!\cdots\!88\)\( T^{12} - \)\(26\!\cdots\!36\)\( T^{13} + \)\(18\!\cdots\!12\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))
$43$ (\( 1 + 32 T + 512 T^{2} - 44896 T^{3} - 2147132 T^{4} + 116117984 T^{5} + 5822932480 T^{6} + 160684182624 T^{7} + 3183857489478 T^{8} + 297105053671776 T^{9} + 19907447385556480 T^{10} + 734023933381973216 T^{11} - 25096108838445990332 T^{12} - \)\(97\!\cdots\!04\)\( T^{13} + \)\(20\!\cdots\!12\)\( T^{14} + \)\(23\!\cdots\!68\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} \))(\( 1 - 158592 T^{3} + 2725896 T^{4} - 61004160 T^{5} + 12575711232 T^{6} - 618250768128 T^{7} + 18017190018460 T^{8} - 968512496926464 T^{9} + 65630298643144704 T^{10} - 3092793950726971008 T^{11} + \)\(10\!\cdots\!80\)\( T^{12} - \)\(37\!\cdots\!52\)\( T^{13} + \)\(33\!\cdots\!64\)\( T^{14} - \)\(12\!\cdots\!48\)\( T^{15} + \)\(20\!\cdots\!26\)\( T^{16} - \)\(22\!\cdots\!52\)\( T^{17} + \)\(11\!\cdots\!64\)\( T^{18} - \)\(23\!\cdots\!48\)\( T^{19} + \)\(11\!\cdots\!80\)\( T^{20} - \)\(66\!\cdots\!92\)\( T^{21} + \)\(26\!\cdots\!04\)\( T^{22} - \)\(71\!\cdots\!36\)\( T^{23} + \)\(24\!\cdots\!60\)\( T^{24} - \)\(15\!\cdots\!72\)\( T^{25} + \)\(58\!\cdots\!32\)\( T^{26} - \)\(52\!\cdots\!40\)\( T^{27} + \)\(43\!\cdots\!96\)\( T^{28} - \)\(46\!\cdots\!08\)\( T^{29} + \)\(18\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 144 T + 10368 T^{2} + 613776 T^{3} + 40380676 T^{4} + 2550145200 T^{5} + 136914549120 T^{6} + 6692132733360 T^{7} + 317421876290310 T^{8} + 14782921207992240 T^{9} + 668099323964430720 T^{10} + 27488564231015770800 T^{11} + \)\(96\!\cdots\!36\)\( T^{12} + \)\(32\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!88\)\( T^{14} + \)\(36\!\cdots\!36\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 80 T + 3200 T^{2} - 185424 T^{3} + 10961416 T^{4} - 416599088 T^{5} + 15442425728 T^{6} - 918151881776 T^{7} + 40326873674268 T^{8} - 452413353633424 T^{9} - 5098985333045632 T^{10} + 1563817770500267376 T^{11} - \)\(31\!\cdots\!76\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} - \)\(66\!\cdots\!16\)\( T^{14} + \)\(36\!\cdots\!52\)\( T^{15} - \)\(19\!\cdots\!38\)\( T^{16} + \)\(79\!\cdots\!68\)\( T^{17} - \)\(32\!\cdots\!96\)\( T^{18} + \)\(19\!\cdots\!40\)\( T^{19} - \)\(75\!\cdots\!36\)\( T^{20} + \)\(82\!\cdots\!24\)\( T^{21} - \)\(59\!\cdots\!12\)\( T^{22} - \)\(11\!\cdots\!56\)\( T^{23} + \)\(22\!\cdots\!28\)\( T^{24} - \)\(11\!\cdots\!64\)\( T^{25} + \)\(42\!\cdots\!28\)\( T^{26} - \)\(25\!\cdots\!92\)\( T^{27} + \)\(14\!\cdots\!96\)\( T^{28} - \)\(55\!\cdots\!96\)\( T^{29} + \)\(21\!\cdots\!00\)\( T^{30} - \)\(11\!\cdots\!20\)\( T^{31} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( 1 + 200 T + 20000 T^{2} + 1719464 T^{3} + 146284516 T^{4} + 10449435128 T^{5} + 642474929248 T^{6} + 39041749119576 T^{7} + 2230315686063750 T^{8} + 109668273276888984 T^{9} + 5069436222207688288 T^{10} + \)\(23\!\cdots\!12\)\( T^{11} + \)\(91\!\cdots\!76\)\( T^{12} + \)\(30\!\cdots\!36\)\( T^{13} + \)\(98\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!00\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 48 T + 1152 T^{2} - 172480 T^{3} + 14460640 T^{4} - 33746752 T^{5} - 164137984 T^{6} + 7272556080 T^{7} - 144052676687940 T^{8} + 6754246786116880 T^{9} - 156566570729557248 T^{10} + 21067810577807322688 T^{11} - \)\(28\!\cdots\!60\)\( T^{12} - \)\(50\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!80\)\( T^{14} - \)\(15\!\cdots\!88\)\( T^{15} + \)\(20\!\cdots\!78\)\( T^{16} - \)\(43\!\cdots\!92\)\( T^{17} + \)\(90\!\cdots\!80\)\( T^{18} - \)\(11\!\cdots\!00\)\( T^{19} - \)\(17\!\cdots\!60\)\( T^{20} + \)\(36\!\cdots\!12\)\( T^{21} - \)\(76\!\cdots\!68\)\( T^{22} + \)\(93\!\cdots\!20\)\( T^{23} - \)\(55\!\cdots\!40\)\( T^{24} + \)\(79\!\cdots\!20\)\( T^{25} - \)\(50\!\cdots\!84\)\( T^{26} - \)\(28\!\cdots\!68\)\( T^{27} + \)\(34\!\cdots\!40\)\( T^{28} - \)\(11\!\cdots\!20\)\( T^{29} + \)\(21\!\cdots\!72\)\( T^{30} - \)\(25\!\cdots\!52\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \))
$59$ (\( 1 - 616 T^{2} + 1419676 T^{4} + 5207600552 T^{6} - 204890323221242 T^{8} + 63102375832383272 T^{10} + \)\(20\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!96\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 - 14320 T^{2} + 157767608 T^{4} - 1244890527696 T^{6} + 8207935216533276 T^{8} - 45520264315988706288 T^{10} + \)\(21\!\cdots\!36\)\( T^{12} - \)\(91\!\cdots\!96\)\( T^{14} + \)\(33\!\cdots\!38\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(32\!\cdots\!56\)\( T^{20} - \)\(80\!\cdots\!28\)\( T^{22} + \)\(17\!\cdots\!16\)\( T^{24} - \)\(32\!\cdots\!96\)\( T^{26} + \)\(49\!\cdots\!88\)\( T^{28} - \)\(54\!\cdots\!20\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( ( 1 - 144 T + 14124 T^{2} - 1141872 T^{3} + 82373126 T^{4} - 4248905712 T^{5} + 195558658284 T^{6} - 7418933907984 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 - 48 T + 22640 T^{2} - 673936 T^{3} + 215932124 T^{4} - 3175993520 T^{5} + 1217381291920 T^{6} - 6110146782096 T^{7} + 5029652273881990 T^{8} - 22735856176179216 T^{9} + 16855667804298904720 T^{10} - \)\(16\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!44\)\( T^{12} - \)\(48\!\cdots\!36\)\( T^{13} + \)\(60\!\cdots\!40\)\( T^{14} - \)\(47\!\cdots\!68\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))
$67$ (\( 1 - 80 T + 3200 T^{2} - 105296 T^{3} - 42100604 T^{4} + 2558251408 T^{5} - 64394556032 T^{6} - 2106545111664 T^{7} + 886628492165190 T^{8} - 9456281006259696 T^{9} - 1297622490342111872 T^{10} + \)\(23\!\cdots\!52\)\( T^{11} - \)\(17\!\cdots\!64\)\( T^{12} - \)\(19\!\cdots\!04\)\( T^{13} + \)\(26\!\cdots\!00\)\( T^{14} - \)\(29\!\cdots\!20\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 80 T + 3200 T^{2} - 609968 T^{3} - 25803832 T^{4} + 2674348464 T^{5} + 482550620032 T^{6} + 12521071726512 T^{7} - 1529742891382500 T^{8} - 116256625005179504 T^{9} + 3984919157383710336 T^{10} + \)\(89\!\cdots\!92\)\( T^{11} + \)\(25\!\cdots\!88\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{13} - \)\(18\!\cdots\!68\)\( T^{14} + \)\(71\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!46\)\( T^{16} + \)\(31\!\cdots\!12\)\( T^{17} - \)\(38\!\cdots\!28\)\( T^{18} - \)\(19\!\cdots\!44\)\( T^{19} + \)\(10\!\cdots\!08\)\( T^{20} + \)\(16\!\cdots\!08\)\( T^{21} + \)\(32\!\cdots\!96\)\( T^{22} - \)\(42\!\cdots\!16\)\( T^{23} - \)\(25\!\cdots\!00\)\( T^{24} + \)\(92\!\cdots\!08\)\( T^{25} + \)\(16\!\cdots\!32\)\( T^{26} + \)\(39\!\cdots\!96\)\( T^{27} - \)\(17\!\cdots\!72\)\( T^{28} - \)\(18\!\cdots\!92\)\( T^{29} + \)\(43\!\cdots\!00\)\( T^{30} + \)\(48\!\cdots\!20\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 + 140 T + 26424 T^{2} + 2214100 T^{3} + 216062270 T^{4} + 11161278100 T^{5} + 671478258744 T^{6} + 17934039748940 T^{7} + 645753531245761 T^{8} )^{2} \))(\( ( 1 - 268 T + 44204 T^{2} - 5071044 T^{3} + 471192708 T^{4} - 37316092140 T^{5} + 2755016710036 T^{6} - 195026045020868 T^{7} + 13901563235868662 T^{8} - 983126292950195588 T^{9} + 70009605785104330516 T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!88\)\( T^{12} - \)\(16\!\cdots\!44\)\( T^{13} + \)\(72\!\cdots\!64\)\( T^{14} - \)\(22\!\cdots\!08\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( 1 - 312 T + 48672 T^{2} - 5493144 T^{3} + 450793636 T^{4} - 23206886280 T^{5} + 386836170336 T^{6} + 68692339480344 T^{7} - 8375186112407610 T^{8} + 366061477090753176 T^{9} + 10985466792718778976 T^{10} - \)\(35\!\cdots\!20\)\( T^{11} + \)\(36\!\cdots\!16\)\( T^{12} - \)\(23\!\cdots\!56\)\( T^{13} + \)\(11\!\cdots\!12\)\( T^{14} - \)\(38\!\cdots\!08\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 - 638960 T^{3} - 86919200 T^{4} - 369113360 T^{5} + 204134940800 T^{6} + 33110468899200 T^{7} + 5206311620706876 T^{8} + 26720496193596800 T^{9} - 3344897125184227200 T^{10} - \)\(13\!\cdots\!20\)\( T^{11} - \)\(22\!\cdots\!00\)\( T^{12} - \)\(53\!\cdots\!20\)\( T^{13} + \)\(75\!\cdots\!00\)\( T^{14} + \)\(36\!\cdots\!00\)\( T^{15} + \)\(78\!\cdots\!66\)\( T^{16} + \)\(19\!\cdots\!00\)\( T^{17} + \)\(21\!\cdots\!00\)\( T^{18} - \)\(80\!\cdots\!80\)\( T^{19} - \)\(17\!\cdots\!00\)\( T^{20} - \)\(59\!\cdots\!80\)\( T^{21} - \)\(76\!\cdots\!00\)\( T^{22} + \)\(32\!\cdots\!00\)\( T^{23} + \)\(33\!\cdots\!36\)\( T^{24} + \)\(11\!\cdots\!00\)\( T^{25} + \)\(37\!\cdots\!00\)\( T^{26} - \)\(36\!\cdots\!40\)\( T^{27} - \)\(45\!\cdots\!00\)\( T^{28} - \)\(17\!\cdots\!40\)\( T^{29} + \)\(42\!\cdots\!21\)\( T^{32} \))
$79$ (\( 1 - 2856 T^{2} - 59627684 T^{4} + 132589015272 T^{6} + 2412904555054278 T^{8} + 5164352884554637032 T^{10} - \)\(90\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!96\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 - 52208 T^{2} + 1391680568 T^{4} - 25119513250000 T^{6} + 343967562165562012 T^{8} - \)\(37\!\cdots\!72\)\( T^{10} + \)\(34\!\cdots\!24\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!30\)\( T^{16} - \)\(10\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!64\)\( T^{20} - \)\(22\!\cdots\!52\)\( T^{22} + \)\(79\!\cdots\!52\)\( T^{24} - \)\(22\!\cdots\!00\)\( T^{26} + \)\(48\!\cdots\!08\)\( T^{28} - \)\(71\!\cdots\!88\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( 1 + 320 T + 51200 T^{2} + 5915840 T^{3} + 553217092 T^{4} + 45528000320 T^{5} + 3742826444800 T^{6} + 317655699460800 T^{7} + 26711748349549254 T^{8} + 2188330113585451200 T^{9} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!80\)\( T^{11} + \)\(12\!\cdots\!72\)\( T^{12} + \)\(91\!\cdots\!60\)\( T^{13} + \)\(54\!\cdots\!00\)\( T^{14} + \)\(23\!\cdots\!80\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( 1 + 256 T + 32768 T^{2} + 2489472 T^{3} + 78882184 T^{4} + 642592896 T^{5} + 678427795456 T^{6} + 150782876556800 T^{7} + 17212084280490396 T^{8} + 1275723698299503616 T^{9} + 84640034952293916672 T^{10} + \)\(62\!\cdots\!76\)\( T^{11} + \)\(45\!\cdots\!48\)\( T^{12} + \)\(40\!\cdots\!04\)\( T^{13} + \)\(47\!\cdots\!44\)\( T^{14} + \)\(59\!\cdots\!24\)\( T^{15} + \)\(59\!\cdots\!02\)\( T^{16} + \)\(41\!\cdots\!36\)\( T^{17} + \)\(22\!\cdots\!24\)\( T^{18} + \)\(13\!\cdots\!76\)\( T^{19} + \)\(10\!\cdots\!68\)\( T^{20} + \)\(96\!\cdots\!24\)\( T^{21} + \)\(90\!\cdots\!92\)\( T^{22} + \)\(93\!\cdots\!64\)\( T^{23} + \)\(87\!\cdots\!76\)\( T^{24} + \)\(52\!\cdots\!00\)\( T^{25} + \)\(16\!\cdots\!56\)\( T^{26} + \)\(10\!\cdots\!44\)\( T^{27} + \)\(90\!\cdots\!64\)\( T^{28} + \)\(19\!\cdots\!68\)\( T^{29} + \)\(17\!\cdots\!88\)\( T^{30} + \)\(95\!\cdots\!44\)\( T^{31} + \)\(25\!\cdots\!61\)\( T^{32} \))
$89$ (\( 1 - 26800 T^{2} + 323387236 T^{4} - 2700009379600 T^{6} + 20962417894146886 T^{8} - \)\(16\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!16\)\( T^{12} - \)\(66\!\cdots\!00\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 - 60568 T^{2} + 1632136448 T^{4} - 25689569431240 T^{6} + 260052896713761532 T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(70\!\cdots\!24\)\( T^{12} - \)\(68\!\cdots\!20\)\( T^{14} - \)\(92\!\cdots\!90\)\( T^{16} - \)\(43\!\cdots\!20\)\( T^{18} + \)\(27\!\cdots\!44\)\( T^{20} - \)\(42\!\cdots\!72\)\( T^{22} + \)\(40\!\cdots\!52\)\( T^{24} - \)\(24\!\cdots\!40\)\( T^{26} + \)\(99\!\cdots\!68\)\( T^{28} - \)\(23\!\cdots\!08\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \))
$97$ (\( 1 + 24 T + 288 T^{2} + 197496 T^{3} - 223582556 T^{4} - 4223145048 T^{5} - 17461370016 T^{6} + 856985885448 T^{7} + 24034841148321606 T^{8} + 8063380196180232 T^{9} - 1545842532791438496 T^{10} - \)\(35\!\cdots\!92\)\( T^{11} - \)\(17\!\cdots\!16\)\( T^{12} + \)\(14\!\cdots\!04\)\( T^{13} + \)\(19\!\cdots\!08\)\( T^{14} + \)\(15\!\cdots\!56\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} \))(\( 1 - 688 T + 236672 T^{2} - 56991072 T^{3} + 10854023776 T^{4} - 1693042723360 T^{5} + 219961022412800 T^{6} - 24020408470599760 T^{7} + 2158083026478171708 T^{8} - \)\(14\!\cdots\!44\)\( T^{9} + \)\(52\!\cdots\!16\)\( T^{10} + \)\(44\!\cdots\!04\)\( T^{11} - \)\(12\!\cdots\!36\)\( T^{12} + \)\(17\!\cdots\!52\)\( T^{13} - \)\(20\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!08\)\( T^{15} - \)\(20\!\cdots\!18\)\( T^{16} + \)\(19\!\cdots\!72\)\( T^{17} - \)\(18\!\cdots\!88\)\( T^{18} + \)\(14\!\cdots\!08\)\( T^{19} - \)\(97\!\cdots\!96\)\( T^{20} + \)\(32\!\cdots\!96\)\( T^{21} + \)\(36\!\cdots\!56\)\( T^{22} - \)\(96\!\cdots\!36\)\( T^{23} + \)\(13\!\cdots\!68\)\( T^{24} - \)\(13\!\cdots\!40\)\( T^{25} + \)\(11\!\cdots\!00\)\( T^{26} - \)\(86\!\cdots\!40\)\( T^{27} + \)\(52\!\cdots\!56\)\( T^{28} - \)\(25\!\cdots\!88\)\( T^{29} + \)\(10\!\cdots\!92\)\( T^{30} - \)\(27\!\cdots\!12\)\( T^{31} + \)\(37\!\cdots\!41\)\( T^{32} \))
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