Properties

Label 21.12.a
Level 21
Weight 12
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 5
Sturm bound 32
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(21))\).

Total New Old
Modular forms 32 12 20
Cusp forms 28 12 16
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12q - 110q^{2} + 7298q^{4} + 20864q^{5} + 22356q^{6} + 33614q^{7} - 178530q^{8} + 708588q^{9} + O(q^{10}) \) \( 12q - 110q^{2} + 7298q^{4} + 20864q^{5} + 22356q^{6} + 33614q^{7} - 178530q^{8} + 708588q^{9} - 628628q^{10} + 479768q^{11} + 468504q^{12} + 211816q^{13} + 2252138q^{14} + 1945944q^{15} + 15060050q^{16} - 17677776q^{17} - 6495390q^{18} + 14340400q^{19} + 80062972q^{20} + 8168202q^{21} - 141776528q^{22} - 36696792q^{23} + 124008732q^{24} + 170526692q^{25} + 20750260q^{26} + 64438038q^{28} + 157567448q^{29} + 83372328q^{30} - 145925920q^{31} + 218643110q^{32} - 381578040q^{33} - 231045636q^{34} + 202557964q^{35} + 430939602q^{36} - 549593704q^{37} + 1120674064q^{38} + 1311947280q^{39} - 5178070044q^{40} - 2022176960q^{41} + 261382464q^{42} - 53478240q^{43} + 2698519120q^{44} + 1231998336q^{45} + 2478782472q^{46} - 8003791248q^{47} - 2121428880q^{48} + 3389702988q^{49} - 16753717858q^{50} + 3351877848q^{51} + 7515182916q^{52} + 6139416360q^{53} + 1320099444q^{54} + 613335632q^{55} + 9830582370q^{56} - 3569250096q^{57} + 25290484780q^{58} + 10657359408q^{59} - 7813297584q^{60} - 25651586024q^{61} + 2309408448q^{62} + 1984873086q^{63} + 9391808682q^{64} + 5245084544q^{65} + 28063197144q^{66} + 37188880240q^{67} - 77247208644q^{68} - 4494378312q^{69} + 6367903388q^{70} - 6587245800q^{71} - 10542017970q^{72} - 8394109912q^{73} - 55306181652q^{74} - 13311415584q^{75} + 62495317632q^{76} - 31343038160q^{77} + 25344923328q^{78} + 53406163008q^{79} + 59610226684q^{80} + 41841412812q^{81} - 29032629796q^{82} + 29954364864q^{83} + 25092716544q^{84} + 108643471296q^{85} - 304681471672q^{86} - 25734127680q^{87} - 407148400704q^{88} + 227251464224q^{89} - 37119854772q^{90} + 15560525652q^{91} + 42455083272q^{92} + 13948460496q^{93} - 278600413056q^{94} - 75625756384q^{95} + 324143639892q^{96} + 37409914376q^{97} - 31072277390q^{98} + 28329820632q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(21))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
21.12.a.a \(1\) \(16.135\) \(\Q\) None \(-62\) \(243\) \(-3310\) \(-16807\) \(-\) \(+\) \(q-62q^{2}+3^{5}q^{3}+1796q^{4}-3310q^{5}+\cdots\)
21.12.a.b \(1\) \(16.135\) \(\Q\) None \(8\) \(243\) \(4390\) \(-16807\) \(-\) \(+\) \(q+8q^{2}+3^{5}q^{3}-1984q^{4}+4390q^{5}+\cdots\)
21.12.a.c \(3\) \(16.135\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-68\) \(-729\) \(3326\) \(-50421\) \(+\) \(+\) \(q+(-23-\beta _{1})q^{2}-3^{5}q^{3}+(664+72\beta _{1}+\cdots)q^{4}+\cdots\)
21.12.a.d \(3\) \(16.135\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-33\) \(-729\) \(3102\) \(50421\) \(+\) \(-\) \(q+(-11-\beta _{2})q^{2}-3^{5}q^{3}+(255+13\beta _{1}+\cdots)q^{4}+\cdots\)
21.12.a.e \(4\) \(16.135\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(45\) \(972\) \(13356\) \(67228\) \(-\) \(-\) \(q+(11+\beta _{1})q^{2}+3^{5}q^{3}+(1196+18\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 62 T + 2048 T^{2} \))(\( 1 - 8 T + 2048 T^{2} \))(\( 1 + 68 T + 4424 T^{2} + 271296 T^{3} + 9060352 T^{4} + 285212672 T^{5} + 8589934592 T^{6} \))(\( 1 + 33 T + 3234 T^{2} + 65328 T^{3} + 6623232 T^{4} + 138412032 T^{5} + 8589934592 T^{6} \))(\( 1 - 45 T + 2708 T^{2} - 121608 T^{3} + 6136896 T^{4} - 249053184 T^{5} + 11358175232 T^{6} - 386547056640 T^{7} + 17592186044416 T^{8} \))
$3$ (\( 1 - 243 T \))(\( 1 - 243 T \))(\( ( 1 + 243 T )^{3} \))(\( ( 1 + 243 T )^{3} \))(\( ( 1 - 243 T )^{4} \))
$5$ (\( 1 + 3310 T + 48828125 T^{2} \))(\( 1 - 4390 T + 48828125 T^{2} \))(\( 1 - 3326 T - 8598985 T^{2} + 496386465300 T^{3} - 419872314453125 T^{4} - 7929801940917968750 T^{5} + \)\(11\!\cdots\!25\)\( T^{6} \))(\( 1 - 3102 T + 109099155 T^{2} - 244544745300 T^{3} + 5327107177734375 T^{4} - 7395744323730468750 T^{5} + \)\(11\!\cdots\!25\)\( T^{6} \))(\( 1 - 13356 T + 124196792 T^{2} - 1017460679220 T^{3} + 7734099382859550 T^{4} - 49680697227539062500 T^{5} + \)\(29\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + 16807 T \))(\( 1 + 16807 T \))(\( ( 1 + 16807 T )^{3} \))(\( ( 1 - 16807 T )^{3} \))(\( ( 1 - 16807 T )^{4} \))
$11$ (\( 1 - 628904 T + 285311670611 T^{2} \))(\( 1 + 804836 T + 285311670611 T^{2} \))(\( 1 - 1348256 T + 1328316204845 T^{2} - 818883157804101840 T^{3} + \)\(37\!\cdots\!95\)\( T^{4} - \)\(10\!\cdots\!76\)\( T^{5} + \)\(23\!\cdots\!31\)\( T^{6} \))(\( 1 + 323232 T + 605825713353 T^{2} + 191914722846881088 T^{3} + \)\(17\!\cdots\!83\)\( T^{4} + \)\(26\!\cdots\!72\)\( T^{5} + \)\(23\!\cdots\!31\)\( T^{6} \))(\( 1 + 369324 T + 403074476576 T^{2} - 106613237636417556 T^{3} + \)\(29\!\cdots\!90\)\( T^{4} - \)\(30\!\cdots\!16\)\( T^{5} + \)\(32\!\cdots\!96\)\( T^{6} + \)\(85\!\cdots\!44\)\( T^{7} + \)\(66\!\cdots\!41\)\( T^{8} \))
$13$ (\( 1 - 176854 T + 1792160394037 T^{2} \))(\( 1 - 358294 T + 1792160394037 T^{2} \))(\( 1 + 892158 T + 179565871515 T^{2} - 3016541647757647564 T^{3} + \)\(32\!\cdots\!55\)\( T^{4} + \)\(28\!\cdots\!02\)\( T^{5} + \)\(57\!\cdots\!53\)\( T^{6} \))(\( 1 + 1701414 T + 681707099691 T^{2} - 1196493656783552444 T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(54\!\cdots\!66\)\( T^{5} + \)\(57\!\cdots\!53\)\( T^{6} \))(\( 1 - 2270240 T + 7556638674508 T^{2} - 11382938531649718112 T^{3} + \)\(20\!\cdots\!62\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!52\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$17$ (\( 1 - 566958 T + 34271896307633 T^{2} \))(\( 1 + 5657862 T + 34271896307633 T^{2} \))(\( 1 + 2450550 T + 103338050580915 T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(35\!\cdots\!95\)\( T^{4} + \)\(28\!\cdots\!50\)\( T^{5} + \)\(40\!\cdots\!37\)\( T^{6} \))(\( 1 + 13285206 T + 78572049951279 T^{2} + \)\(40\!\cdots\!84\)\( T^{3} + \)\(26\!\cdots\!07\)\( T^{4} + \)\(15\!\cdots\!34\)\( T^{5} + \)\(40\!\cdots\!37\)\( T^{6} \))(\( 1 - 3148884 T + 72164810000648 T^{2} - 38523070468676402172 T^{3} + \)\(25\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{5} + \)\(84\!\cdots\!72\)\( T^{6} - \)\(12\!\cdots\!08\)\( T^{7} + \)\(13\!\cdots\!21\)\( T^{8} \))
$19$ (\( 1 + 12916124 T + 116490258898219 T^{2} \))(\( 1 + 14602004 T + 116490258898219 T^{2} \))(\( 1 - 11057244 T + 340209044693553 T^{2} - \)\(24\!\cdots\!12\)\( T^{3} + \)\(39\!\cdots\!07\)\( T^{4} - \)\(15\!\cdots\!84\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \))(\( 1 - 3457092 T + 29222966120817 T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!23\)\( T^{4} - \)\(46\!\cdots\!12\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \))(\( 1 - 27344192 T + 627937285363084 T^{2} - \)\(89\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(85\!\cdots\!24\)\( T^{6} - \)\(43\!\cdots\!28\)\( T^{7} + \)\(18\!\cdots\!21\)\( T^{8} \))
$23$ (\( 1 + 25664100 T + 952809757913927 T^{2} \))(\( 1 + 36724800 T + 952809757913927 T^{2} \))(\( 1 - 9675972 T + 1239650213300889 T^{2} + \)\(64\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(87\!\cdots\!88\)\( T^{5} + \)\(86\!\cdots\!83\)\( T^{6} \))(\( 1 + 18776676 T + 1243906160877525 T^{2} + \)\(29\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!83\)\( T^{6} \))(\( 1 - 34792812 T + 3124117532857568 T^{2} - \)\(81\!\cdots\!56\)\( T^{3} + \)\(42\!\cdots\!22\)\( T^{4} - \)\(77\!\cdots\!12\)\( T^{5} + \)\(28\!\cdots\!72\)\( T^{6} - \)\(30\!\cdots\!96\)\( T^{7} + \)\(82\!\cdots\!41\)\( T^{8} \))
$29$ (\( 1 + 47411458 T + 12200509765705829 T^{2} \))(\( 1 - 51126982 T + 12200509765705829 T^{2} \))(\( 1 - 196390898 T + 34048694626865435 T^{2} - \)\(41\!\cdots\!24\)\( T^{3} + \)\(41\!\cdots\!15\)\( T^{4} - \)\(29\!\cdots\!18\)\( T^{5} + \)\(18\!\cdots\!89\)\( T^{6} \))(\( 1 + 64656294 T + 26773497504130299 T^{2} + \)\(17\!\cdots\!44\)\( T^{3} + \)\(32\!\cdots\!71\)\( T^{4} + \)\(96\!\cdots\!54\)\( T^{5} + \)\(18\!\cdots\!89\)\( T^{6} \))(\( 1 - 22117320 T + 24611875536382220 T^{2} + \)\(17\!\cdots\!64\)\( T^{3} + \)\(36\!\cdots\!58\)\( T^{4} + \)\(21\!\cdots\!56\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} - \)\(40\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!81\)\( T^{8} \))
$31$ (\( 1 - 13942680 T + 25408476896404831 T^{2} \))(\( 1 + 208102080 T + 25408476896404831 T^{2} \))(\( 1 + 70964448 T + 24628842392211741 T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(62\!\cdots\!71\)\( T^{4} + \)\(45\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} \))(\( 1 + 30699048 T + 9550670648972829 T^{2} - \)\(19\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!99\)\( T^{4} + \)\(19\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} \))(\( 1 - 149896976 T + 54361211069964220 T^{2} - \)\(37\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!02\)\( T^{4} - \)\(94\!\cdots\!16\)\( T^{5} + \)\(35\!\cdots\!20\)\( T^{6} - \)\(24\!\cdots\!16\)\( T^{7} + \)\(41\!\cdots\!21\)\( T^{8} \))
$37$ (\( 1 + 641657298 T + 177917621779460413 T^{2} \))(\( 1 - 652145982 T + 177917621779460413 T^{2} \))(\( 1 + 874700478 T + 645121507114416819 T^{2} + \)\(30\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!47\)\( T^{4} + \)\(27\!\cdots\!82\)\( T^{5} + \)\(56\!\cdots\!97\)\( T^{6} \))(\( 1 - 1046484186 T + 853645703377279299 T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!87\)\( T^{4} - \)\(33\!\cdots\!34\)\( T^{5} + \)\(56\!\cdots\!97\)\( T^{6} \))(\( 1 + 731866096 T + 375375129139253260 T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(37\!\cdots\!70\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} + \)\(41\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$41$ (\( 1 + 600859298 T + 550329031716248441 T^{2} \))(\( 1 - 951188402 T + 550329031716248441 T^{2} \))(\( 1 + 386189798 T + 505761200017373963 T^{2} + \)\(32\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!83\)\( T^{4} + \)\(11\!\cdots\!38\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} \))(\( 1 + 158666502 T + 1207460495601115719 T^{2} + \)\(22\!\cdots\!56\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} + \)\(48\!\cdots\!62\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} \))(\( 1 + 1827649764 T + 3425289793227810872 T^{2} + \)\(33\!\cdots\!12\)\( T^{3} + \)\(32\!\cdots\!42\)\( T^{4} + \)\(18\!\cdots\!92\)\( T^{5} + \)\(10\!\cdots\!32\)\( T^{6} + \)\(30\!\cdots\!44\)\( T^{7} + \)\(91\!\cdots\!61\)\( T^{8} \))
$43$ (\( 1 + 1417753612 T + 929293739471222707 T^{2} \))(\( 1 - 858607748 T + 929293739471222707 T^{2} \))(\( 1 - 544378572 T + 959573315307790665 T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(89\!\cdots\!55\)\( T^{4} - \)\(47\!\cdots\!28\)\( T^{5} + \)\(80\!\cdots\!43\)\( T^{6} \))(\( 1 - 746774892 T + 2708489779435582281 T^{2} - \)\(12\!\cdots\!28\)\( T^{3} + \)\(25\!\cdots\!67\)\( T^{4} - \)\(64\!\cdots\!08\)\( T^{5} + \)\(80\!\cdots\!43\)\( T^{6} \))(\( 1 + 785485840 T + 1099183924502801068 T^{2} - \)\(96\!\cdots\!92\)\( T^{3} - \)\(56\!\cdots\!78\)\( T^{4} - \)\(89\!\cdots\!44\)\( T^{5} + \)\(94\!\cdots\!32\)\( T^{6} + \)\(63\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - 860414040 T + 2472159215084012303 T^{2} \))(\( 1 + 1336554720 T + 2472159215084012303 T^{2} \))(\( 1 - 20129256 T + 7061746613299470141 T^{2} - \)\(63\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!27\)\( T^{6} \))(\( 1 + 3828844824 T + 12097500436872553773 T^{2} + \)\(20\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!19\)\( T^{4} + \)\(23\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!27\)\( T^{6} \))(\( 1 + 3718935000 T + 8298243945748786508 T^{2} + \)\(10\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!54\)\( T^{4} + \)\(24\!\cdots\!12\)\( T^{5} + \)\(50\!\cdots\!72\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!81\)\( T^{8} \))
$53$ (\( 1 - 3221420478 T + 9269035929372191597 T^{2} \))(\( 1 - 1497595998 T + 9269035929372191597 T^{2} \))(\( 1 - 4746325602 T + 26795018682979651779 T^{2} - \)\(74\!\cdots\!16\)\( T^{3} + \)\(24\!\cdots\!63\)\( T^{4} - \)\(40\!\cdots\!18\)\( T^{5} + \)\(79\!\cdots\!73\)\( T^{6} \))(\( 1 + 3245231286 T + 24451724434264758291 T^{2} + \)\(59\!\cdots\!92\)\( T^{3} + \)\(22\!\cdots\!27\)\( T^{4} + \)\(27\!\cdots\!74\)\( T^{5} + \)\(79\!\cdots\!73\)\( T^{6} \))(\( 1 + 80694432 T + 27187721021168035436 T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!06\)\( T^{4} + \)\(15\!\cdots\!60\)\( T^{5} + \)\(23\!\cdots\!24\)\( T^{6} + \)\(64\!\cdots\!36\)\( T^{7} + \)\(73\!\cdots\!81\)\( T^{8} \))
$59$ (\( 1 + 6082959012 T + 30155888444737842659 T^{2} \))(\( 1 - 7067944068 T + 30155888444737842659 T^{2} \))(\( 1 - 19835761884 T + \)\(21\!\cdots\!13\)\( T^{2} - \)\(14\!\cdots\!92\)\( T^{3} + \)\(65\!\cdots\!67\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(27\!\cdots\!79\)\( T^{6} \))(\( 1 + 12392434452 T + \)\(10\!\cdots\!77\)\( T^{2} + \)\(55\!\cdots\!96\)\( T^{3} + \)\(30\!\cdots\!43\)\( T^{4} + \)\(11\!\cdots\!12\)\( T^{5} + \)\(27\!\cdots\!79\)\( T^{6} \))(\( 1 - 2229046920 T + 37800794755571623196 T^{2} - \)\(23\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} - \)\(71\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!76\)\( T^{6} - \)\(61\!\cdots\!80\)\( T^{7} + \)\(82\!\cdots\!61\)\( T^{8} \))
$61$ (\( 1 + 864141122 T + 43513917611435838661 T^{2} \))(\( 1 + 7643926442 T + 43513917611435838661 T^{2} \))(\( 1 + 6480362406 T + \)\(14\!\cdots\!31\)\( T^{2} + \)\(56\!\cdots\!84\)\( T^{3} + \)\(61\!\cdots\!91\)\( T^{4} + \)\(12\!\cdots\!26\)\( T^{5} + \)\(82\!\cdots\!81\)\( T^{6} \))(\( 1 + 17915258046 T + \)\(21\!\cdots\!55\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(92\!\cdots\!55\)\( T^{4} + \)\(33\!\cdots\!66\)\( T^{5} + \)\(82\!\cdots\!81\)\( T^{6} \))(\( 1 - 7252101992 T + \)\(15\!\cdots\!04\)\( T^{2} - \)\(82\!\cdots\!88\)\( T^{3} + \)\(98\!\cdots\!50\)\( T^{4} - \)\(35\!\cdots\!68\)\( T^{5} + \)\(29\!\cdots\!84\)\( T^{6} - \)\(59\!\cdots\!52\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \))
$67$ (\( 1 - 11897667268 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 + 5086757252 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 25314754956 T + \)\(51\!\cdots\!81\)\( T^{2} - \)\(62\!\cdots\!92\)\( T^{3} + \)\(62\!\cdots\!23\)\( T^{4} - \)\(37\!\cdots\!84\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} \))(\( 1 + 11801453124 T + \)\(12\!\cdots\!41\)\( T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!03\)\( T^{4} + \)\(17\!\cdots\!36\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} \))(\( 1 - 16864668392 T + \)\(17\!\cdots\!76\)\( T^{2} - \)\(18\!\cdots\!48\)\( T^{3} + \)\(28\!\cdots\!30\)\( T^{4} - \)\(22\!\cdots\!84\)\( T^{5} + \)\(26\!\cdots\!64\)\( T^{6} - \)\(30\!\cdots\!04\)\( T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \))
$71$ (\( 1 + 14077803900 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 2801411040 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 41912998956 T + \)\(12\!\cdots\!25\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!75\)\( T^{4} - \)\(22\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!11\)\( T^{6} \))(\( 1 + 26571896460 T + \)\(86\!\cdots\!33\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!43\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!11\)\( T^{6} \))(\( 1 + 10651955436 T + \)\(35\!\cdots\!44\)\( T^{2} + \)\(18\!\cdots\!64\)\( T^{3} + \)\(45\!\cdots\!30\)\( T^{4} + \)\(42\!\cdots\!44\)\( T^{5} + \)\(18\!\cdots\!04\)\( T^{6} + \)\(13\!\cdots\!96\)\( T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \))
$73$ (\( 1 + 18814150398 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 7844280438 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 18034242474 T + \)\(75\!\cdots\!39\)\( T^{2} + \)\(10\!\cdots\!24\)\( T^{3} + \)\(23\!\cdots\!03\)\( T^{4} + \)\(17\!\cdots\!46\)\( T^{5} + \)\(30\!\cdots\!33\)\( T^{6} \))(\( 1 - 15228295590 T + \)\(92\!\cdots\!23\)\( T^{2} - \)\(93\!\cdots\!48\)\( T^{3} + \)\(29\!\cdots\!71\)\( T^{4} - \)\(14\!\cdots\!10\)\( T^{5} + \)\(30\!\cdots\!33\)\( T^{6} \))(\( 1 - 21070267808 T + 79407289323377872732 T^{2} + \)\(77\!\cdots\!52\)\( T^{3} - \)\(16\!\cdots\!78\)\( T^{4} + \)\(24\!\cdots\!04\)\( T^{5} + \)\(78\!\cdots\!28\)\( T^{6} - \)\(65\!\cdots\!64\)\( T^{7} + \)\(96\!\cdots\!41\)\( T^{8} \))
$79$ (\( 1 - 17021361416 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 21156661264 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 33952994712 T + \)\(15\!\cdots\!21\)\( T^{2} + \)\(37\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!92\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \))(\( 1 - 8453153688 T + \)\(13\!\cdots\!65\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(98\!\cdots\!35\)\( T^{4} - \)\(47\!\cdots\!08\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \))(\( 1 - 83041303880 T + \)\(35\!\cdots\!96\)\( T^{2} - \)\(89\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!26\)\( T^{4} - \)\(66\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!36\)\( T^{6} - \)\(34\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!81\)\( T^{8} \))
$83$ (\( 1 - 47613135564 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 + 10894949316 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 25737739644 T + \)\(32\!\cdots\!65\)\( T^{2} - \)\(56\!\cdots\!04\)\( T^{3} + \)\(42\!\cdots\!55\)\( T^{4} - \)\(42\!\cdots\!16\)\( T^{5} + \)\(21\!\cdots\!63\)\( T^{6} \))(\( 1 + 31295925492 T + \)\(10\!\cdots\!09\)\( T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(13\!\cdots\!03\)\( T^{4} + \)\(51\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!63\)\( T^{6} \))(\( 1 + 1205635536 T + \)\(82\!\cdots\!80\)\( T^{2} + \)\(14\!\cdots\!32\)\( T^{3} + \)\(29\!\cdots\!78\)\( T^{4} + \)\(18\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!68\)\( T^{7} + \)\(27\!\cdots\!21\)\( T^{8} \))
$89$ (\( 1 - 61562070254 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 70788775714 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 101517768986 T + \)\(97\!\cdots\!19\)\( T^{2} - \)\(50\!\cdots\!68\)\( T^{3} + \)\(27\!\cdots\!91\)\( T^{4} - \)\(78\!\cdots\!06\)\( T^{5} + \)\(21\!\cdots\!69\)\( T^{6} \))(\( 1 + 16030004742 T + \)\(68\!\cdots\!07\)\( T^{2} + \)\(98\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!23\)\( T^{4} + \)\(12\!\cdots\!82\)\( T^{5} + \)\(21\!\cdots\!69\)\( T^{6} \))(\( 1 - 9412854012 T + \)\(73\!\cdots\!44\)\( T^{2} - \)\(94\!\cdots\!16\)\( T^{3} + \)\(26\!\cdots\!50\)\( T^{4} - \)\(26\!\cdots\!24\)\( T^{5} + \)\(56\!\cdots\!24\)\( T^{6} - \)\(20\!\cdots\!28\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))
$97$ (\( 1 + 166479510534 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 - 82223797746 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 - 3701782446 T + \)\(17\!\cdots\!95\)\( T^{2} + \)\(40\!\cdots\!72\)\( T^{3} + \)\(12\!\cdots\!35\)\( T^{4} - \)\(18\!\cdots\!14\)\( T^{5} + \)\(36\!\cdots\!77\)\( T^{6} \))(\( 1 + 87243902370 T + \)\(22\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} + \)\(44\!\cdots\!30\)\( T^{5} + \)\(36\!\cdots\!77\)\( T^{6} \))(\( 1 - 205207747088 T + \)\(44\!\cdots\!84\)\( T^{2} - \)\(49\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!18\)\( T^{4} - \)\(35\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!56\)\( T^{6} - \)\(75\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} \))
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