Properties

Label 4.35.b
Level 4
Weight 35
Character orbit b
Rep. character \(\chi_{4}(3,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 1
Sturm bound 17
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 35 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(17\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{35}(4, [\chi])\).

Total New Old
Modular forms 18 18 0
Cusp forms 16 16 0
Eisenstein series 2 2 0

Trace form

\( 16q - 27372q^{2} - 20032875248q^{4} - 21372255840q^{5} + 20836736461728q^{6} + 2284358011394112q^{8} - 67828545645364080q^{9} + O(q^{10}) \) \( 16q - 27372q^{2} - 20032875248q^{4} - 21372255840q^{5} + 20836736461728q^{6} + 2284358011394112q^{8} - 67828545645364080q^{9} - 45113792455239160q^{10} + 2389205919902175360q^{12} - 5233507036411659616q^{13} - 78584294183795706432q^{14} + 495182112183820124416q^{16} - 603670905919309938144q^{17} - 3049682100893194226508q^{18} - 33058393758449688290400q^{20} + 51436061805005728413696q^{21} + 71660876143050404765280q^{22} + 216144869785928938234368q^{24} + 1224730607199300816400560q^{25} - 2068583291215840445568696q^{26} + 891252510406495618671360q^{28} - 4086289565288128397679456q^{29} + 29601064663371600024444480q^{30} + 20026414012547463259849728q^{32} + 73954448514753271238568960q^{33} + 46288216238217191722231976q^{34} + 569062048300724903070378384q^{36} + 950385175701396416644593056q^{37} - 123498782719680108073563360q^{38} - 3250024485723791435879459200q^{40} + 469238845433510127731349792q^{41} + 2352670327884382399711157760q^{42} + 7327222247557584897899214720q^{44} - 24427097188106605438098250080q^{45} - 19562783684402230406331003072q^{46} + 10576141603633237485164267520q^{48} - 5662649722408510114314091760q^{49} + 20191932196464263485504736220q^{50} - 299155663662273783907276503904q^{52} - 287274278887575172234950512736q^{53} + 6047252470624543429578889536q^{54} + 957626384837555513184792726528q^{56} + 2791286265941381406754467578880q^{57} - 314499391713380785247660833336q^{58} + 1103184540481814815474103512320q^{60} - 7360215904797065436447249778528q^{61} + 6668101684324825273374014288640q^{62} - 4672502276277215477475812569088q^{64} + 7390500022468645317433010287680q^{65} - 12460359580334220374288279205120q^{66} + 3201090312765007118305938045984q^{68} - 20909099675706939600670541666304q^{69} - 13302319794857519494110175324800q^{70} - 776423968147266229824274003392q^{72} - 21322365514881309087265291408096q^{73} + 60200090281780873185690769101576q^{74} - 104909582027053727863743869884800q^{76} + 31510097566037837419459121448960q^{77} + 173718506967281862717403400063040q^{78} + 360511559214548868760655536888320q^{80} - 262819454584631237298859074129648q^{81} - 127799686177535642546403976474264q^{82} - 481860976739335096182940424300544q^{84} + 2066519418228326883252673443837760q^{85} + 680164732953303409567751182232928q^{86} - 1868588957981221169228370505105920q^{88} - 2487774705888249186282098457137376q^{89} + 2710136859758550244207319911170120q^{90} - 5719642193953545223478177490435840q^{92} - 4648987096869411453766406983557120q^{93} - 174301665834847119752293250078592q^{94} - 1356265248520894619800195113541632q^{96} + 7119191713926779638291321673063456q^{97} + 16929214407930779244441537548523732q^{98} + O(q^{100}) \)

Decomposition of \(S_{35}^{\mathrm{new}}(4, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4.35.b.a \(16\) \(29.290\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-27372\) \(0\) \(-21372255840\) \(0\) \(q+(-1711+\beta _{1})q^{2}+(19-76\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 27372 T + 10391050816 T^{2} - 483864769056768 T^{3} - 90697820359441252352 T^{4} - \)\(13\!\cdots\!56\)\( T^{5} - \)\(35\!\cdots\!16\)\( T^{6} + \)\(83\!\cdots\!04\)\( T^{7} + \)\(59\!\cdots\!92\)\( T^{8} + \)\(14\!\cdots\!36\)\( T^{9} - \)\(10\!\cdots\!96\)\( T^{10} - \)\(70\!\cdots\!24\)\( T^{11} - \)\(79\!\cdots\!72\)\( T^{12} - \)\(72\!\cdots\!32\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} + \)\(12\!\cdots\!68\)\( T^{15} + \)\(75\!\cdots\!96\)\( T^{16} \)
$3$ \( 1 - 99503180774650512 T^{2} + \)\(52\!\cdots\!64\)\( T^{4} - \)\(20\!\cdots\!16\)\( T^{6} + \)\(64\!\cdots\!24\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{10} + \)\(39\!\cdots\!92\)\( T^{12} - \)\(78\!\cdots\!52\)\( T^{14} + \)\(13\!\cdots\!18\)\( T^{16} - \)\(21\!\cdots\!72\)\( T^{18} + \)\(30\!\cdots\!32\)\( T^{20} - \)\(36\!\cdots\!40\)\( T^{22} + \)\(38\!\cdots\!84\)\( T^{24} - \)\(34\!\cdots\!16\)\( T^{26} + \)\(24\!\cdots\!04\)\( T^{28} - \)\(12\!\cdots\!52\)\( T^{30} + \)\(35\!\cdots\!81\)\( T^{32} \)
$5$ \( ( 1 + 10686127920 T + \)\(20\!\cdots\!60\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!00\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(63\!\cdots\!00\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(26\!\cdots\!00\)\( T^{12} - \)\(81\!\cdots\!00\)\( T^{13} + \)\(78\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!25\)\( T^{16} )^{2} \)
$7$ \( 1 - \)\(43\!\cdots\!12\)\( T^{2} + \)\(98\!\cdots\!24\)\( T^{4} - \)\(15\!\cdots\!36\)\( T^{6} + \)\(19\!\cdots\!64\)\( T^{8} - \)\(19\!\cdots\!40\)\( T^{10} + \)\(15\!\cdots\!72\)\( T^{12} - \)\(10\!\cdots\!12\)\( T^{14} + \)\(63\!\cdots\!78\)\( T^{16} - \)\(31\!\cdots\!12\)\( T^{18} + \)\(13\!\cdots\!72\)\( T^{20} - \)\(47\!\cdots\!40\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{24} - \)\(34\!\cdots\!36\)\( T^{26} + \)\(62\!\cdots\!24\)\( T^{28} - \)\(79\!\cdots\!12\)\( T^{30} + \)\(54\!\cdots\!01\)\( T^{32} \)
$11$ \( 1 - \)\(18\!\cdots\!56\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(70\!\cdots\!00\)\( T^{8} - \)\(30\!\cdots\!68\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{12} - \)\(35\!\cdots\!00\)\( T^{14} + \)\(98\!\cdots\!90\)\( T^{16} - \)\(23\!\cdots\!00\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(86\!\cdots\!88\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{24} - \)\(15\!\cdots\!20\)\( T^{26} + \)\(14\!\cdots\!00\)\( T^{28} - \)\(95\!\cdots\!16\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$13$ \( ( 1 + 2616753518205829808 T + \)\(29\!\cdots\!16\)\( T^{2} + \)\(19\!\cdots\!48\)\( T^{3} + \)\(45\!\cdots\!08\)\( T^{4} + \)\(37\!\cdots\!76\)\( T^{5} + \)\(55\!\cdots\!84\)\( T^{6} + \)\(38\!\cdots\!56\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} + \)\(31\!\cdots\!64\)\( T^{10} + \)\(15\!\cdots\!44\)\( T^{11} + \)\(14\!\cdots\!28\)\( T^{12} + \)\(45\!\cdots\!52\)\( T^{13} + \)\(51\!\cdots\!76\)\( T^{14} + \)\(34\!\cdots\!32\)\( T^{15} + \)\(98\!\cdots\!81\)\( T^{16} )^{2} \)
$17$ \( ( 1 + \)\(30\!\cdots\!72\)\( T + \)\(28\!\cdots\!76\)\( T^{2} + \)\(47\!\cdots\!92\)\( T^{3} + \)\(44\!\cdots\!88\)\( T^{4} + \)\(62\!\cdots\!04\)\( T^{5} + \)\(48\!\cdots\!04\)\( T^{6} + \)\(54\!\cdots\!24\)\( T^{7} + \)\(37\!\cdots\!82\)\( T^{8} + \)\(37\!\cdots\!96\)\( T^{9} + \)\(22\!\cdots\!64\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(97\!\cdots\!28\)\( T^{12} + \)\(72\!\cdots\!08\)\( T^{13} + \)\(29\!\cdots\!96\)\( T^{14} + \)\(21\!\cdots\!48\)\( T^{15} + \)\(48\!\cdots\!61\)\( T^{16} )^{2} \)
$19$ \( 1 - \)\(21\!\cdots\!16\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{8} - \)\(84\!\cdots\!48\)\( T^{10} + \)\(37\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{14} + \)\(46\!\cdots\!50\)\( T^{16} - \)\(12\!\cdots\!40\)\( T^{18} + \)\(30\!\cdots\!68\)\( T^{20} - \)\(61\!\cdots\!08\)\( T^{22} + \)\(10\!\cdots\!20\)\( T^{24} - \)\(13\!\cdots\!00\)\( T^{26} + \)\(14\!\cdots\!20\)\( T^{28} - \)\(10\!\cdots\!96\)\( T^{30} + \)\(43\!\cdots\!21\)\( T^{32} \)
$23$ \( 1 - \)\(72\!\cdots\!32\)\( T^{2} + \)\(34\!\cdots\!84\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(36\!\cdots\!24\)\( T^{8} - \)\(99\!\cdots\!40\)\( T^{10} + \)\(24\!\cdots\!12\)\( T^{12} - \)\(55\!\cdots\!12\)\( T^{14} + \)\(11\!\cdots\!98\)\( T^{16} - \)\(22\!\cdots\!72\)\( T^{18} + \)\(38\!\cdots\!32\)\( T^{20} - \)\(61\!\cdots\!40\)\( T^{22} + \)\(90\!\cdots\!04\)\( T^{24} - \)\(11\!\cdots\!76\)\( T^{26} + \)\(13\!\cdots\!04\)\( T^{28} - \)\(11\!\cdots\!52\)\( T^{30} + \)\(60\!\cdots\!41\)\( T^{32} \)
$29$ \( ( 1 + \)\(20\!\cdots\!28\)\( T + \)\(21\!\cdots\!84\)\( T^{2} + \)\(36\!\cdots\!44\)\( T^{3} + \)\(22\!\cdots\!64\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!52\)\( T^{6} - \)\(10\!\cdots\!72\)\( T^{7} + \)\(11\!\cdots\!38\)\( T^{8} - \)\(56\!\cdots\!32\)\( T^{9} + \)\(50\!\cdots\!72\)\( T^{10} - \)\(24\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!44\)\( T^{12} + \)\(14\!\cdots\!44\)\( T^{13} + \)\(46\!\cdots\!04\)\( T^{14} + \)\(22\!\cdots\!08\)\( T^{15} + \)\(59\!\cdots\!41\)\( T^{16} )^{2} \)
$31$ \( 1 - \)\(32\!\cdots\!16\)\( T^{2} + \)\(54\!\cdots\!20\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!20\)\( T^{8} - \)\(51\!\cdots\!48\)\( T^{10} + \)\(35\!\cdots\!28\)\( T^{12} - \)\(21\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!50\)\( T^{16} - \)\(56\!\cdots\!40\)\( T^{18} + \)\(24\!\cdots\!68\)\( T^{20} - \)\(88\!\cdots\!08\)\( T^{22} + \)\(28\!\cdots\!20\)\( T^{24} - \)\(75\!\cdots\!00\)\( T^{26} + \)\(16\!\cdots\!20\)\( T^{28} - \)\(24\!\cdots\!96\)\( T^{30} + \)\(19\!\cdots\!21\)\( T^{32} \)
$37$ \( ( 1 - \)\(47\!\cdots\!28\)\( T + \)\(89\!\cdots\!96\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(40\!\cdots\!48\)\( T^{4} - \)\(19\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!84\)\( T^{6} - \)\(59\!\cdots\!56\)\( T^{7} + \)\(30\!\cdots\!02\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(55\!\cdots\!64\)\( T^{10} - \)\(17\!\cdots\!84\)\( T^{11} + \)\(76\!\cdots\!68\)\( T^{12} - \)\(16\!\cdots\!52\)\( T^{13} + \)\(73\!\cdots\!56\)\( T^{14} - \)\(81\!\cdots\!12\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} )^{2} \)
$41$ \( ( 1 - \)\(23\!\cdots\!96\)\( T + \)\(37\!\cdots\!40\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(67\!\cdots\!40\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(77\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!50\)\( T^{8} - \)\(83\!\cdots\!80\)\( T^{9} + \)\(36\!\cdots\!08\)\( T^{10} - \)\(61\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!40\)\( T^{12} - \)\(43\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!40\)\( T^{14} - \)\(16\!\cdots\!16\)\( T^{15} + \)\(47\!\cdots\!81\)\( T^{16} )^{2} \)
$43$ \( 1 - \)\(42\!\cdots\!72\)\( T^{2} + \)\(86\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!64\)\( T^{8} - \)\(76\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!12\)\( T^{14} + \)\(76\!\cdots\!18\)\( T^{16} - \)\(23\!\cdots\!12\)\( T^{18} + \)\(61\!\cdots\!52\)\( T^{20} - \)\(12\!\cdots\!00\)\( T^{22} + \)\(21\!\cdots\!64\)\( T^{24} - \)\(27\!\cdots\!16\)\( T^{26} + \)\(24\!\cdots\!24\)\( T^{28} - \)\(14\!\cdots\!72\)\( T^{30} + \)\(40\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - \)\(48\!\cdots\!32\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{4} - \)\(25\!\cdots\!36\)\( T^{6} + \)\(37\!\cdots\!04\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{12} - \)\(42\!\cdots\!92\)\( T^{14} + \)\(32\!\cdots\!78\)\( T^{16} - \)\(21\!\cdots\!12\)\( T^{18} + \)\(12\!\cdots\!92\)\( T^{20} - \)\(59\!\cdots\!60\)\( T^{22} + \)\(24\!\cdots\!64\)\( T^{24} - \)\(82\!\cdots\!36\)\( T^{26} + \)\(21\!\cdots\!84\)\( T^{28} - \)\(40\!\cdots\!72\)\( T^{30} + \)\(41\!\cdots\!81\)\( T^{32} \)
$53$ \( ( 1 + \)\(14\!\cdots\!68\)\( T + \)\(20\!\cdots\!36\)\( T^{2} + \)\(37\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!88\)\( T^{4} + \)\(45\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} + \)\(30\!\cdots\!76\)\( T^{7} + \)\(59\!\cdots\!22\)\( T^{8} + \)\(13\!\cdots\!44\)\( T^{9} + \)\(22\!\cdots\!44\)\( T^{10} + \)\(33\!\cdots\!44\)\( T^{11} + \)\(63\!\cdots\!48\)\( T^{12} + \)\(50\!\cdots\!72\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} + \)\(34\!\cdots\!52\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} )^{2} \)
$59$ \( 1 - \)\(11\!\cdots\!16\)\( T^{2} + \)\(65\!\cdots\!60\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{6} + \)\(87\!\cdots\!60\)\( T^{8} - \)\(23\!\cdots\!08\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!70\)\( T^{16} - \)\(27\!\cdots\!20\)\( T^{18} + \)\(36\!\cdots\!88\)\( T^{20} - \)\(42\!\cdots\!88\)\( T^{22} + \)\(41\!\cdots\!60\)\( T^{24} - \)\(33\!\cdots\!60\)\( T^{26} + \)\(21\!\cdots\!60\)\( T^{28} - \)\(94\!\cdots\!56\)\( T^{30} + \)\(22\!\cdots\!61\)\( T^{32} \)
$61$ \( ( 1 + \)\(36\!\cdots\!64\)\( T + \)\(19\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!40\)\( T^{4} + \)\(40\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(72\!\cdots\!50\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(32\!\cdots\!08\)\( T^{10} + \)\(51\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!40\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(31\!\cdots\!40\)\( T^{14} + \)\(29\!\cdots\!84\)\( T^{15} + \)\(40\!\cdots\!21\)\( T^{16} )^{2} \)
$67$ \( 1 - \)\(37\!\cdots\!12\)\( T^{2} + \)\(91\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!96\)\( T^{6} + \)\(35\!\cdots\!64\)\( T^{8} - \)\(56\!\cdots\!60\)\( T^{10} + \)\(82\!\cdots\!32\)\( T^{12} - \)\(11\!\cdots\!12\)\( T^{14} + \)\(14\!\cdots\!58\)\( T^{16} - \)\(16\!\cdots\!92\)\( T^{18} + \)\(18\!\cdots\!92\)\( T^{20} - \)\(18\!\cdots\!60\)\( T^{22} + \)\(17\!\cdots\!04\)\( T^{24} - \)\(14\!\cdots\!96\)\( T^{26} + \)\(10\!\cdots\!44\)\( T^{28} - \)\(60\!\cdots\!72\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$71$ \( 1 - \)\(42\!\cdots\!16\)\( T^{2} + \)\(95\!\cdots\!80\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{8} - \)\(31\!\cdots\!88\)\( T^{10} + \)\(35\!\cdots\!88\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(33\!\cdots\!30\)\( T^{16} - \)\(27\!\cdots\!60\)\( T^{18} + \)\(21\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!28\)\( T^{22} + \)\(83\!\cdots\!80\)\( T^{24} - \)\(43\!\cdots\!40\)\( T^{26} + \)\(19\!\cdots\!80\)\( T^{28} - \)\(66\!\cdots\!36\)\( T^{30} + \)\(12\!\cdots\!81\)\( T^{32} \)
$73$ \( ( 1 + \)\(10\!\cdots\!48\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(61\!\cdots\!08\)\( T^{4} - \)\(59\!\cdots\!24\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} - \)\(32\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} - \)\(73\!\cdots\!56\)\( T^{9} + \)\(10\!\cdots\!04\)\( T^{10} - \)\(67\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!88\)\( T^{12} + \)\(23\!\cdots\!52\)\( T^{13} + \)\(15\!\cdots\!96\)\( T^{14} + \)\(31\!\cdots\!12\)\( T^{15} + \)\(66\!\cdots\!21\)\( T^{16} )^{2} \)
$79$ \( 1 - \)\(29\!\cdots\!76\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{8} - \)\(18\!\cdots\!48\)\( T^{10} + \)\(93\!\cdots\!08\)\( T^{12} - \)\(38\!\cdots\!40\)\( T^{14} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(42\!\cdots\!40\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{20} - \)\(24\!\cdots\!88\)\( T^{22} + \)\(45\!\cdots\!20\)\( T^{24} - \)\(67\!\cdots\!00\)\( T^{26} + \)\(74\!\cdots\!20\)\( T^{28} - \)\(55\!\cdots\!96\)\( T^{30} + \)\(20\!\cdots\!81\)\( T^{32} \)
$83$ \( 1 - \)\(17\!\cdots\!92\)\( T^{2} + \)\(14\!\cdots\!24\)\( T^{4} - \)\(80\!\cdots\!16\)\( T^{6} + \)\(32\!\cdots\!04\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{10} + \)\(27\!\cdots\!72\)\( T^{12} - \)\(60\!\cdots\!32\)\( T^{14} + \)\(11\!\cdots\!18\)\( T^{16} - \)\(19\!\cdots\!12\)\( T^{18} + \)\(27\!\cdots\!32\)\( T^{20} - \)\(32\!\cdots\!40\)\( T^{22} + \)\(31\!\cdots\!44\)\( T^{24} - \)\(24\!\cdots\!16\)\( T^{26} + \)\(13\!\cdots\!84\)\( T^{28} - \)\(51\!\cdots\!52\)\( T^{30} + \)\(95\!\cdots\!21\)\( T^{32} \)
$89$ \( ( 1 + \)\(12\!\cdots\!88\)\( T + \)\(96\!\cdots\!84\)\( T^{2} + \)\(72\!\cdots\!24\)\( T^{3} + \)\(41\!\cdots\!84\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!92\)\( T^{6} + \)\(56\!\cdots\!08\)\( T^{7} + \)\(27\!\cdots\!98\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!00\)\( T^{11} + \)\(54\!\cdots\!24\)\( T^{12} + \)\(18\!\cdots\!24\)\( T^{13} + \)\(45\!\cdots\!44\)\( T^{14} + \)\(11\!\cdots\!28\)\( T^{15} + \)\(17\!\cdots\!21\)\( T^{16} )^{2} \)
$97$ \( ( 1 - \)\(35\!\cdots\!28\)\( T + \)\(13\!\cdots\!76\)\( T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(67\!\cdots\!48\)\( T^{4} + \)\(33\!\cdots\!84\)\( T^{5} + \)\(38\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!62\)\( T^{8} + \)\(39\!\cdots\!96\)\( T^{9} + \)\(49\!\cdots\!24\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(10\!\cdots\!08\)\( T^{12} + \)\(93\!\cdots\!28\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} - \)\(25\!\cdots\!92\)\( T^{15} + \)\(25\!\cdots\!41\)\( T^{16} )^{2} \)
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