Properties

Label 8.12
Level 8
Weight 12
Dimension 13
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8\( 8 = 2^{3} \) \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(48\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(8))\).

Total New Old
Modular forms 25 15 10
Cusp forms 19 13 6
Eisenstein series 6 2 4

Trace form

\( 13q + 22q^{2} + 20q^{3} + 436q^{4} + 4378q^{5} - 24308q^{6} + 1976q^{7} - 208472q^{8} - 108043q^{9} + O(q^{10}) \) \( 13q + 22q^{2} + 20q^{3} + 436q^{4} + 4378q^{5} - 24308q^{6} + 1976q^{7} - 208472q^{8} - 108043q^{9} + 29864q^{10} - 437924q^{11} - 476216q^{12} + 2424354q^{13} + 2264272q^{14} - 6977400q^{15} + 3757072q^{16} + 14211466q^{17} + 2329130q^{18} - 29163996q^{19} + 8296432q^{20} + 42049248q^{21} - 24201892q^{22} - 41608536q^{23} + 10668944q^{24} - 16272585q^{25} - 101931080q^{26} + 67917320q^{27} - 95722144q^{28} - 242433102q^{29} + 261362768q^{30} + 739918304q^{31} + 343908512q^{32} - 647584936q^{33} - 187103796q^{34} + 101747952q^{35} - 431195700q^{36} - 182227398q^{37} + 375100844q^{38} + 394990152q^{39} + 1519138784q^{40} + 324867634q^{41} - 3697737056q^{42} - 1743278724q^{43} - 4127012952q^{44} + 2138827042q^{45} + 6028867440q^{46} - 1261796400q^{47} + 9357606048q^{48} + 4008659349q^{49} - 9626584226q^{50} - 9990058520q^{51} - 11595427248q^{52} - 531170102q^{53} + 20050494008q^{54} + 15767737912q^{55} + 18698733760q^{56} - 5229281640q^{57} - 22828679464q^{58} - 9014835572q^{59} - 43681325472q^{60} - 1840230702q^{61} + 40774631744q^{62} + 21964258552q^{63} + 55266728512q^{64} - 10900501924q^{65} - 83128448712q^{66} - 2485905324q^{67} - 83303223768q^{68} - 19932595552q^{69} + 100120831168q^{70} + 18720070520q^{71} + 112862051672q^{72} + 36321508754q^{73} - 102103996184q^{74} - 81254774516q^{75} - 97501928568q^{76} + 14365261728q^{77} + 144445560944q^{78} + 105049383344q^{79} + 128322038976q^{80} - 2226964707q^{81} - 123309929604q^{82} - 147967912348q^{83} - 199990865984q^{84} + 86218163220q^{85} + 130201908124q^{86} - 26582565432q^{87} + 169216119632q^{88} - 26522388350q^{89} - 228506434984q^{90} + 7178123184q^{91} - 141139403232q^{92} - 32147884160q^{93} + 145336130016q^{94} + 58741871848q^{95} + 109508068928q^{96} - 10556170438q^{97} - 6194726778q^{98} + 113298667660q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.12.a \(\chi_{8}(1, \cdot)\) 8.12.a.a 1 1
8.12.a.b 2
8.12.b \(\chi_{8}(5, \cdot)\) 8.12.b.a 10 1

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 22 T + 24 T^{2} + 72512 T^{3} - 2487296 T^{4} - 46628864 T^{5} - 5093982208 T^{6} + 304137371648 T^{7} + 206158430208 T^{8} - 387028092977152 T^{9} + 36028797018963968 T^{10} \))
$3$ (\( 1 + 36 T + 177147 T^{2} \))(\( 1 - 56 T - 91386 T^{2} - 9920232 T^{3} + 31381059609 T^{4} \))(\( 1 - 649538 T^{2} + 237663850725 T^{4} - 64838819239431768 T^{6} + \)\(14\!\cdots\!22\)\( T^{8} - \)\(28\!\cdots\!88\)\( T^{10} + \)\(46\!\cdots\!98\)\( T^{12} - \)\(63\!\cdots\!08\)\( T^{14} + \)\(73\!\cdots\!25\)\( T^{16} - \)\(62\!\cdots\!18\)\( T^{18} + \)\(30\!\cdots\!49\)\( T^{20} \))
$5$ (\( 1 + 3490 T + 48828125 T^{2} \))(\( 1 - 7868 T + 48841790 T^{2} - 384179687500 T^{3} + 2384185791015625 T^{4} \))(\( 1 - 223389298 T^{2} + 28218482100794325 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!50\)\( T^{8} - \)\(93\!\cdots\!00\)\( T^{10} + \)\(41\!\cdots\!50\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{14} + \)\(38\!\cdots\!25\)\( T^{16} - \)\(72\!\cdots\!50\)\( T^{18} + \)\(77\!\cdots\!25\)\( T^{20} \))
$7$ (\( 1 + 55464 T + 1977326743 T^{2} \))(\( 1 - 91056 T + 5239889774 T^{2} - 180047463910608 T^{3} + 3909821048582988049 T^{4} \))(\( ( 1 + 16808 T + 4798655859 T^{2} + 133282572983392 T^{3} + 14656889013644202794 T^{4} + \)\(34\!\cdots\!36\)\( T^{5} + \)\(28\!\cdots\!42\)\( T^{6} + \)\(52\!\cdots\!08\)\( T^{7} + \)\(37\!\cdots\!13\)\( T^{8} + \)\(25\!\cdots\!08\)\( T^{9} + \)\(30\!\cdots\!43\)\( T^{10} )^{2} \))
$11$ (\( 1 + 597004 T + 285311670611 T^{2} \))(\( 1 - 159080 T + 362536063286 T^{2} - 45387380560797880 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))(\( 1 - 1158438441298 T^{2} + \)\(67\!\cdots\!81\)\( T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(78\!\cdots\!58\)\( T^{8} - \)\(22\!\cdots\!64\)\( T^{10} + \)\(64\!\cdots\!18\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(36\!\cdots\!41\)\( T^{16} - \)\(50\!\cdots\!38\)\( T^{18} + \)\(35\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 - 1373878 T + 1792160394037 T^{2} \))(\( 1 - 1050476 T + 3458956762062 T^{2} - 1882621482086411612 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 9406612074850 T^{2} + \)\(48\!\cdots\!13\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!42\)\( T^{8} - \)\(86\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!98\)\( T^{12} - \)\(17\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!17\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(34\!\cdots\!49\)\( T^{20} \))
$17$ (\( 1 - 10140850 T + 34271896307633 T^{2} \))(\( 1 - 1430884 T + 16748384809766 T^{2} - 49039108076251137572 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( ( 1 - 1319866 T + 80316996009501 T^{2} + 82971742437589927112 T^{3} + \)\(33\!\cdots\!42\)\( T^{4} + \)\(68\!\cdots\!32\)\( T^{5} + \)\(11\!\cdots\!86\)\( T^{6} + \)\(97\!\cdots\!68\)\( T^{7} + \)\(32\!\cdots\!37\)\( T^{8} - \)\(18\!\cdots\!86\)\( T^{9} + \)\(47\!\cdots\!93\)\( T^{10} )^{2} \))
$19$ (\( 1 + 7297396 T + 116490258898219 T^{2} \))(\( 1 + 21866600 T + 343123710088422 T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))(\( 1 - 709131993597154 T^{2} + \)\(24\!\cdots\!77\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{6} + \)\(93\!\cdots\!66\)\( T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(12\!\cdots\!26\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{14} + \)\(61\!\cdots\!37\)\( T^{16} - \)\(24\!\cdots\!14\)\( T^{18} + \)\(46\!\cdots\!01\)\( T^{20} \))
$23$ (\( 1 + 32057464 T + 952809757913927 T^{2} \))(\( 1 - 35806736 T + 1952928132896462 T^{2} - \)\(34\!\cdots\!72\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))(\( ( 1 + 22678904 T + 2721031019872995 T^{2} + \)\(79\!\cdots\!84\)\( T^{3} + \)\(42\!\cdots\!74\)\( T^{4} + \)\(10\!\cdots\!72\)\( T^{5} + \)\(40\!\cdots\!98\)\( T^{6} + \)\(71\!\cdots\!36\)\( T^{7} + \)\(23\!\cdots\!85\)\( T^{8} + \)\(18\!\cdots\!64\)\( T^{9} + \)\(78\!\cdots\!07\)\( T^{10} )^{2} \))
$29$ (\( 1 + 13605402 T + 12200509765705829 T^{2} \))(\( 1 + 228827700 T + 35907977791027054 T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 - 59089589213276098 T^{2} + \)\(16\!\cdots\!61\)\( T^{4} - \)\(32\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!18\)\( T^{8} - \)\(66\!\cdots\!24\)\( T^{10} + \)\(74\!\cdots\!38\)\( T^{12} - \)\(71\!\cdots\!00\)\( T^{14} + \)\(55\!\cdots\!81\)\( T^{16} - \)\(29\!\cdots\!78\)\( T^{18} + \)\(73\!\cdots\!01\)\( T^{20} \))
$31$ (\( 1 - 233160800 T + 25408476896404831 T^{2} \))(\( 1 - 64722112 T + 51498184379920062 T^{2} - \)\(16\!\cdots\!72\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 221017696 T + 43703396794606299 T^{2} - \)\(67\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!78\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{5} + \)\(29\!\cdots\!18\)\( T^{6} - \)\(43\!\cdots\!08\)\( T^{7} + \)\(71\!\cdots\!09\)\( T^{8} - \)\(92\!\cdots\!16\)\( T^{9} + \)\(10\!\cdots\!51\)\( T^{10} )^{2} \))
$37$ (\( 1 + 257786178 T + 177917621779460413 T^{2} \))(\( 1 - 75558780 T + 354292341022528382 T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))(\( 1 - 1197799649319754546 T^{2} + \)\(69\!\cdots\!49\)\( T^{4} - \)\(25\!\cdots\!96\)\( T^{6} + \)\(69\!\cdots\!38\)\( T^{8} - \)\(14\!\cdots\!36\)\( T^{10} + \)\(21\!\cdots\!22\)\( T^{12} - \)\(25\!\cdots\!56\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} - \)\(12\!\cdots\!66\)\( T^{18} + \)\(31\!\cdots\!49\)\( T^{20} \))
$41$ (\( 1 + 221438598 T + 550329031716248441 T^{2} \))(\( 1 - 1201214196 T + 1274442257818453270 T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 327453982 T + 1187101734449808069 T^{2} + \)\(54\!\cdots\!24\)\( T^{3} + \)\(94\!\cdots\!82\)\( T^{4} + \)\(39\!\cdots\!84\)\( T^{5} + \)\(52\!\cdots\!62\)\( T^{6} + \)\(16\!\cdots\!44\)\( T^{7} + \)\(19\!\cdots\!49\)\( T^{8} + \)\(30\!\cdots\!02\)\( T^{9} + \)\(50\!\cdots\!01\)\( T^{10} )^{2} \))
$43$ (\( 1 + 1697758892 T + 929293739471222707 T^{2} \))(\( 1 + 45519832 T + 1812222731951247606 T^{2} + \)\(42\!\cdots\!24\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))(\( 1 - 4457304455199526546 T^{2} + \)\(11\!\cdots\!65\)\( T^{4} - \)\(19\!\cdots\!56\)\( T^{6} + \)\(26\!\cdots\!02\)\( T^{8} - \)\(27\!\cdots\!96\)\( T^{10} + \)\(22\!\cdots\!98\)\( T^{12} - \)\(14\!\cdots\!56\)\( T^{14} + \)\(72\!\cdots\!85\)\( T^{16} - \)\(24\!\cdots\!46\)\( T^{18} + \)\(48\!\cdots\!49\)\( T^{20} \))
$47$ (\( 1 - 527509392 T + 2472159215084012303 T^{2} \))(\( 1 + 1229079264 T + 5024390553242310430 T^{2} + \)\(30\!\cdots\!92\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))(\( ( 1 + 280113264 T + 9772169230935669067 T^{2} + \)\(30\!\cdots\!56\)\( T^{3} + \)\(42\!\cdots\!50\)\( T^{4} + \)\(11\!\cdots\!12\)\( T^{5} + \)\(10\!\cdots\!50\)\( T^{6} + \)\(18\!\cdots\!04\)\( T^{7} + \)\(14\!\cdots\!09\)\( T^{8} + \)\(10\!\cdots\!84\)\( T^{9} + \)\(92\!\cdots\!43\)\( T^{10} )^{2} \))
$53$ (\( 1 - 3277379822 T + 9269035929372191597 T^{2} \))(\( 1 + 3808549924 T + 16648234587904299038 T^{2} + \)\(35\!\cdots\!28\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))(\( 1 - 67267282814146198162 T^{2} + \)\(21\!\cdots\!09\)\( T^{4} - \)\(44\!\cdots\!08\)\( T^{6} + \)\(64\!\cdots\!66\)\( T^{8} - \)\(68\!\cdots\!80\)\( T^{10} + \)\(55\!\cdots\!94\)\( T^{12} - \)\(32\!\cdots\!48\)\( T^{14} + \)\(13\!\cdots\!61\)\( T^{16} - \)\(36\!\cdots\!82\)\( T^{18} + \)\(46\!\cdots\!49\)\( T^{20} \))
$59$ (\( 1 + 3001908988 T + 30155888444737842659 T^{2} \))(\( 1 + 6012926584 T + 61066799326040273366 T^{2} + \)\(18\!\cdots\!56\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))(\( 1 - 88134178321575180082 T^{2} + \)\(36\!\cdots\!37\)\( T^{4} - \)\(14\!\cdots\!32\)\( T^{6} + \)\(55\!\cdots\!66\)\( T^{8} - \)\(18\!\cdots\!80\)\( T^{10} + \)\(50\!\cdots\!46\)\( T^{12} - \)\(11\!\cdots\!52\)\( T^{14} + \)\(27\!\cdots\!17\)\( T^{16} - \)\(60\!\cdots\!22\)\( T^{18} + \)\(62\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 + 11630023610 T + 43513917611435838661 T^{2} \))(\( 1 - 9789792908 T + \)\(10\!\cdots\!42\)\( T^{2} - \)\(42\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(20\!\cdots\!22\)\( T^{2} + \)\(22\!\cdots\!17\)\( T^{4} - \)\(16\!\cdots\!52\)\( T^{6} + \)\(99\!\cdots\!46\)\( T^{8} - \)\(47\!\cdots\!80\)\( T^{10} + \)\(18\!\cdots\!66\)\( T^{12} - \)\(60\!\cdots\!32\)\( T^{14} + \)\(15\!\cdots\!37\)\( T^{16} - \)\(26\!\cdots\!82\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \))
$67$ (\( 1 + 17189000548 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 14703095224 T + 95414710392374126214 T^{2} - \)\(17\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(72\!\cdots\!38\)\( T^{2} + \)\(26\!\cdots\!49\)\( T^{4} - \)\(65\!\cdots\!52\)\( T^{6} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(16\!\cdots\!80\)\( T^{10} + \)\(17\!\cdots\!74\)\( T^{12} - \)\(14\!\cdots\!92\)\( T^{14} + \)\(88\!\cdots\!81\)\( T^{16} - \)\(35\!\cdots\!58\)\( T^{18} + \)\(73\!\cdots\!49\)\( T^{20} \))
$71$ (\( 1 - 26169539608 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 4319991088 T + \)\(28\!\cdots\!82\)\( T^{2} - \)\(99\!\cdots\!48\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 5884730088 T + \)\(95\!\cdots\!75\)\( T^{2} + \)\(45\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!34\)\( T^{4} + \)\(14\!\cdots\!96\)\( T^{5} + \)\(91\!\cdots\!14\)\( T^{6} + \)\(24\!\cdots\!24\)\( T^{7} + \)\(11\!\cdots\!25\)\( T^{8} + \)\(16\!\cdots\!28\)\( T^{9} + \)\(65\!\cdots\!51\)\( T^{10} )^{2} \))
$73$ (\( 1 + 7039021094 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 - 11055639476 T + \)\(65\!\cdots\!62\)\( T^{2} - \)\(34\!\cdots\!52\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))(\( ( 1 - 16152445186 T + \)\(11\!\cdots\!73\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!54\)\( T^{4} - \)\(60\!\cdots\!16\)\( T^{5} + \)\(18\!\cdots\!58\)\( T^{6} - \)\(14\!\cdots\!36\)\( T^{7} + \)\(36\!\cdots\!09\)\( T^{8} - \)\(15\!\cdots\!26\)\( T^{9} + \)\(30\!\cdots\!57\)\( T^{10} )^{2} \))
$79$ (\( 1 + 4199910416 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 - 51957623264 T + \)\(14\!\cdots\!78\)\( T^{2} - \)\(38\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 28645835248 T + \)\(23\!\cdots\!19\)\( T^{2} - \)\(74\!\cdots\!96\)\( T^{3} + \)\(27\!\cdots\!54\)\( T^{4} - \)\(79\!\cdots\!12\)\( T^{5} + \)\(20\!\cdots\!66\)\( T^{6} - \)\(41\!\cdots\!36\)\( T^{7} + \)\(98\!\cdots\!41\)\( T^{8} - \)\(89\!\cdots\!88\)\( T^{9} + \)\(23\!\cdots\!99\)\( T^{10} )^{2} \))
$83$ (\( 1 + 39739936436 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 + 108227975912 T + \)\(54\!\cdots\!06\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(82\!\cdots\!06\)\( T^{2} + \)\(34\!\cdots\!45\)\( T^{4} - \)\(91\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!02\)\( T^{8} - \)\(26\!\cdots\!16\)\( T^{10} + \)\(29\!\cdots\!78\)\( T^{12} - \)\(25\!\cdots\!16\)\( T^{14} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(62\!\cdots\!46\)\( T^{18} + \)\(12\!\cdots\!49\)\( T^{20} \))
$89$ (\( 1 - 10565331594 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 71188291860 T + \)\(31\!\cdots\!82\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 54138005902 T + \)\(98\!\cdots\!69\)\( T^{2} + \)\(52\!\cdots\!04\)\( T^{3} + \)\(44\!\cdots\!14\)\( T^{4} + \)\(20\!\cdots\!88\)\( T^{5} + \)\(12\!\cdots\!46\)\( T^{6} + \)\(40\!\cdots\!84\)\( T^{7} + \)\(21\!\cdots\!61\)\( T^{8} + \)\(32\!\cdots\!82\)\( T^{9} + \)\(16\!\cdots\!49\)\( T^{10} )^{2} \))
$97$ (\( 1 + 69851645662 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 1699807676 T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))(\( ( 1 - 30497641450 T + \)\(25\!\cdots\!65\)\( T^{2} - \)\(10\!\cdots\!32\)\( T^{3} + \)\(30\!\cdots\!14\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!42\)\( T^{6} - \)\(54\!\cdots\!88\)\( T^{7} + \)\(94\!\cdots\!05\)\( T^{8} - \)\(79\!\cdots\!50\)\( T^{9} + \)\(18\!\cdots\!93\)\( T^{10} )^{2} \))
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