# 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

# Related objects

## 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

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