# Properties

 Label 4.33.b Level 4 Weight 33 Character orbit b Rep. character $$\chi_{4}(3,\cdot)$$ Character field $$\Q$$ Dimension 15 Newform subspaces 2 Sturm bound 16 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$33$$ Character orbit: $$[\chi]$$ $$=$$ 4.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$16$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{33}(4, [\chi])$$.

Total New Old
Modular forms 17 17 0
Cusp forms 15 15 0
Eisenstein series 2 2 0

## Trace form

 $$15q + 41756q^{2} + 1372118928q^{4} - 58374617954q^{5} + 1262734959552q^{6} + 90108426597056q^{8} - 9330489743965809q^{9} + O(q^{10})$$ $$15q + 41756q^{2} + 1372118928q^{4} - 58374617954q^{5} + 1262734959552q^{6} + 90108426597056q^{8} - 9330489743965809q^{9} + 18898036649551416q^{10} - 356375853407619840q^{12} - 310330724778787938q^{13} + 7731597686180285568q^{14} - 12124099049477041920q^{16} + 32998092473679242782q^{17} - 235284708498889038564q^{18} + 1178277970470772776736q^{20} - 2117057333603414716416q^{21} + 5453282318362187158080q^{22} - 25676387438412555666432q^{24} + 25580536411347215174061q^{25} - 13318656836733765948872q^{26} + 287406099989137745118720q^{28} + 79528536198734521127326q^{29} - 58105690536229485525120q^{30} - 1840159638716279585895424q^{32} + 300337698474624477849600q^{33} + 3061726194368939324488248q^{34} + 12017787401329470882617232q^{36} + 2620876359380717116432542q^{37} - 23674230631082905997232960q^{38} - 9024227500247159449524864q^{40} - 81178072464964949049401570q^{41} + 115560171253757823362918400q^{42} - 341183497187317069095824640q^{44} + 442973435281896213295744926q^{45} - 292611965791335032977170048q^{46} + 2221472331677427727192412160q^{48} - 3726696106270429609275304689q^{49} + 493922241360801978941421396q^{50} - 3985635270653999132109697248q^{52} + 11697972364361581520631371422q^{53} + 5640914457354711920825943936q^{54} + 6374003242890184756165527552q^{56} - 45021698676819804378731120640q^{57} - 8194344621477086303654741448q^{58} + 47104092918034241778705984000q^{60} + 35013736512382530415793212830q^{61} + 21568431972603200921875622400q^{62} - 117509138728984201932143357952q^{64} - 148178828737840221416914567108q^{65} + 127488965971714418642064552960q^{66} + 360400933872306679913127653152q^{68} - 518263207562337876647092752384q^{69} - 514873634674652910315035354880q^{70} + 635686255013313098011024741056q^{72} + 2039230631915237097214872797982q^{73} - 108331481302464298617628492232q^{74} - 782627770945775078204519980800q^{76} - 3080015547766001613074070589440q^{77} + 87768417039576149684305430400q^{78} + 2654434108669779778173646414336q^{80} + 3391039348733544470993078655759q^{81} - 12257047852236333596334510646728q^{82} + 18798568063225498187446552682496q^{84} + 6477684469910392785583162643772q^{85} - 23458263349697798020331664550848q^{86} + 11102138256776653733497151431680q^{88} - 1065313629081272451230200381154q^{89} - 49358816481571186295303666931144q^{90} + 60798444498044060215697222361600q^{92} + 43431545808036949970541566115840q^{93} - 130643574777689548682138472237312q^{94} + 345792875639857602943412371832832q^{96} - 117697475906931523308761108394978q^{97} - 387626690825356436327499156699364q^{98} + O(q^{100})$$

## Decomposition of $$S_{33}^{\mathrm{new}}(4, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
4.33.b.a $$1$$ $$25.947$$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$65536$$ $$0$$ $$-196496109694$$ $$0$$ $$q+2^{16}q^{2}+2^{32}q^{4}-196496109694q^{5}+\cdots$$
4.33.b.b $$14$$ $$25.947$$ $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ None $$-23780$$ $$0$$ $$138121491740$$ $$0$$ $$q+(-1699+\beta _{1})q^{2}+(9-21\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 65536 T$$)($$1 + 23780 T + 1744168384 T^{2} + 100782736158720 T^{3} + 10654023216358490112 T^{4} +$$$$93\!\cdots\!80$$$$T^{5} +$$$$65\!\cdots\!88$$$$T^{6} -$$$$27\!\cdots\!20$$$$T^{7} +$$$$28\!\cdots\!48$$$$T^{8} +$$$$17\!\cdots\!80$$$$T^{9} +$$$$84\!\cdots\!32$$$$T^{10} +$$$$34\!\cdots\!20$$$$T^{11} +$$$$25\!\cdots\!84$$$$T^{12} +$$$$14\!\cdots\!80$$$$T^{13} +$$$$26\!\cdots\!16$$$$T^{14}$$)
$3$ ($$( 1 - 43046721 T )( 1 + 43046721 T )$$)($$1 - 7379386355554062 T^{2} +$$$$32\!\cdots\!15$$$$T^{4} -$$$$10\!\cdots\!68$$$$T^{6} +$$$$25\!\cdots\!89$$$$T^{8} -$$$$55\!\cdots\!30$$$$T^{10} +$$$$11\!\cdots\!35$$$$T^{12} -$$$$22\!\cdots\!60$$$$T^{14} +$$$$39\!\cdots\!35$$$$T^{16} -$$$$65\!\cdots\!30$$$$T^{18} +$$$$10\!\cdots\!49$$$$T^{20} -$$$$13\!\cdots\!28$$$$T^{22} +$$$$15\!\cdots\!15$$$$T^{24} -$$$$12\!\cdots\!22$$$$T^{26} +$$$$56\!\cdots\!61$$$$T^{28}$$)
$5$ ($$1 + 196496109694 T +$$$$23\!\cdots\!25$$$$T^{2}$$)($$( 1 - 69060745870 T +$$$$81\!\cdots\!75$$$$T^{2} -$$$$60\!\cdots\!00$$$$T^{3} +$$$$35\!\cdots\!25$$$$T^{4} -$$$$26\!\cdots\!50$$$$T^{5} +$$$$10\!\cdots\!75$$$$T^{6} -$$$$77\!\cdots\!00$$$$T^{7} +$$$$25\!\cdots\!75$$$$T^{8} -$$$$14\!\cdots\!50$$$$T^{9} +$$$$45\!\cdots\!25$$$$T^{10} -$$$$17\!\cdots\!00$$$$T^{11} +$$$$55\!\cdots\!75$$$$T^{12} -$$$$11\!\cdots\!50$$$$T^{13} +$$$$37\!\cdots\!25$$$$T^{14} )^{2}$$)
$7$ ($$( 1 - 33232930569601 T )( 1 + 33232930569601 T )$$)($$1 -$$$$53\!\cdots\!62$$$$T^{2} +$$$$15\!\cdots\!35$$$$T^{4} -$$$$35\!\cdots\!08$$$$T^{6} +$$$$65\!\cdots\!89$$$$T^{8} -$$$$10\!\cdots\!10$$$$T^{10} +$$$$13\!\cdots\!75$$$$T^{12} -$$$$15\!\cdots\!40$$$$T^{14} +$$$$16\!\cdots\!75$$$$T^{16} -$$$$14\!\cdots\!10$$$$T^{18} +$$$$11\!\cdots\!89$$$$T^{20} -$$$$78\!\cdots\!08$$$$T^{22} +$$$$42\!\cdots\!35$$$$T^{24} -$$$$17\!\cdots\!62$$$$T^{26} +$$$$40\!\cdots\!01$$$$T^{28}$$)
$11$ ($$( 1 - 45949729863572161 T )( 1 + 45949729863572161 T )$$)($$1 -$$$$13\!\cdots\!94$$$$T^{2} +$$$$94\!\cdots\!91$$$$T^{4} -$$$$47\!\cdots\!04$$$$T^{6} +$$$$18\!\cdots\!81$$$$T^{8} -$$$$59\!\cdots\!02$$$$T^{10} +$$$$16\!\cdots\!63$$$$T^{12} -$$$$36\!\cdots\!32$$$$T^{14} +$$$$71\!\cdots\!83$$$$T^{16} -$$$$11\!\cdots\!62$$$$T^{18} +$$$$16\!\cdots\!01$$$$T^{20} -$$$$18\!\cdots\!44$$$$T^{22} +$$$$16\!\cdots\!91$$$$T^{24} -$$$$10\!\cdots\!54$$$$T^{26} +$$$$34\!\cdots\!81$$$$T^{28}$$)
$13$ ($$1 - 1330087744899070082 T +$$$$44\!\cdots\!81$$$$T^{2}$$)($$( 1 + 820209234838929010 T +$$$$24\!\cdots\!39$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{3} +$$$$27\!\cdots\!57$$$$T^{4} +$$$$16\!\cdots\!30$$$$T^{5} +$$$$19\!\cdots\!43$$$$T^{6} +$$$$91\!\cdots\!80$$$$T^{7} +$$$$85\!\cdots\!83$$$$T^{8} +$$$$31\!\cdots\!30$$$$T^{9} +$$$$24\!\cdots\!37$$$$T^{10} +$$$$65\!\cdots\!00$$$$T^{11} +$$$$41\!\cdots\!39$$$$T^{12} +$$$$61\!\cdots\!10$$$$T^{13} +$$$$33\!\cdots\!61$$$$T^{14} )^{2}$$)
$17$ ($$1 - 1427124567881986562 T +$$$$23\!\cdots\!61$$$$T^{2}$$)($$( 1 - 15785483952898628110 T +$$$$81\!\cdots\!39$$$$T^{2} -$$$$11\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!57$$$$T^{4} -$$$$52\!\cdots\!50$$$$T^{5} +$$$$13\!\cdots\!03$$$$T^{6} -$$$$14\!\cdots\!20$$$$T^{7} +$$$$31\!\cdots\!83$$$$T^{8} -$$$$29\!\cdots\!50$$$$T^{9} +$$$$53\!\cdots\!17$$$$T^{10} -$$$$36\!\cdots\!00$$$$T^{11} +$$$$60\!\cdots\!39$$$$T^{12} -$$$$27\!\cdots\!10$$$$T^{13} +$$$$41\!\cdots\!21$$$$T^{14} )^{2}$$)
$19$ ($$( 1 -$$$$28\!\cdots\!81$$$$T )( 1 +$$$$28\!\cdots\!81$$$$T )$$)($$1 -$$$$56\!\cdots\!14$$$$T^{2} +$$$$15\!\cdots\!91$$$$T^{4} -$$$$29\!\cdots\!44$$$$T^{6} +$$$$41\!\cdots\!41$$$$T^{8} -$$$$49\!\cdots\!02$$$$T^{10} +$$$$51\!\cdots\!03$$$$T^{12} -$$$$45\!\cdots\!32$$$$T^{14} +$$$$35\!\cdots\!63$$$$T^{16} -$$$$23\!\cdots\!82$$$$T^{18} +$$$$13\!\cdots\!01$$$$T^{20} -$$$$67\!\cdots\!64$$$$T^{22} +$$$$25\!\cdots\!91$$$$T^{24} -$$$$62\!\cdots\!94$$$$T^{26} +$$$$76\!\cdots\!41$$$$T^{28}$$)
$23$ ($$( 1 -$$$$61\!\cdots\!61$$$$T )( 1 +$$$$61\!\cdots\!61$$$$T )$$)($$1 -$$$$39\!\cdots\!22$$$$T^{2} +$$$$75\!\cdots\!95$$$$T^{4} -$$$$90\!\cdots\!48$$$$T^{6} +$$$$78\!\cdots\!29$$$$T^{8} -$$$$51\!\cdots\!90$$$$T^{10} +$$$$26\!\cdots\!95$$$$T^{12} -$$$$11\!\cdots\!20$$$$T^{14} +$$$$37\!\cdots\!95$$$$T^{16} -$$$$10\!\cdots\!90$$$$T^{18} +$$$$22\!\cdots\!09$$$$T^{20} -$$$$36\!\cdots\!28$$$$T^{22} +$$$$42\!\cdots\!95$$$$T^{24} -$$$$31\!\cdots\!02$$$$T^{26} +$$$$11\!\cdots\!81$$$$T^{28}$$)
$29$ ($$1 -$$$$46\!\cdots\!42$$$$T +$$$$62\!\cdots\!41$$$$T^{2}$$)($$( 1 +$$$$19\!\cdots\!58$$$$T +$$$$22\!\cdots\!75$$$$T^{2} +$$$$23\!\cdots\!32$$$$T^{3} +$$$$19\!\cdots\!89$$$$T^{4} +$$$$11\!\cdots\!70$$$$T^{5} +$$$$88\!\cdots\!55$$$$T^{6} -$$$$79\!\cdots\!60$$$$T^{7} +$$$$55\!\cdots\!55$$$$T^{8} +$$$$46\!\cdots\!70$$$$T^{9} +$$$$46\!\cdots\!69$$$$T^{10} +$$$$35\!\cdots\!52$$$$T^{11} +$$$$21\!\cdots\!75$$$$T^{12} +$$$$11\!\cdots\!78$$$$T^{13} +$$$$37\!\cdots\!81$$$$T^{14} )^{2}$$)
$31$ ($$( 1 -$$$$72\!\cdots\!81$$$$T )( 1 +$$$$72\!\cdots\!81$$$$T )$$)($$1 -$$$$24\!\cdots\!14$$$$T^{2} +$$$$34\!\cdots\!91$$$$T^{4} -$$$$33\!\cdots\!44$$$$T^{6} +$$$$25\!\cdots\!41$$$$T^{8} -$$$$16\!\cdots\!02$$$$T^{10} +$$$$93\!\cdots\!03$$$$T^{12} -$$$$50\!\cdots\!32$$$$T^{14} +$$$$26\!\cdots\!63$$$$T^{16} -$$$$12\!\cdots\!82$$$$T^{18} +$$$$56\!\cdots\!01$$$$T^{20} -$$$$20\!\cdots\!64$$$$T^{22} +$$$$59\!\cdots\!91$$$$T^{24} -$$$$12\!\cdots\!94$$$$T^{26} +$$$$13\!\cdots\!41$$$$T^{28}$$)
$37$ ($$1 -$$$$13\!\cdots\!82$$$$T +$$$$15\!\cdots\!81$$$$T^{2}$$)($$( 1 +$$$$53\!\cdots\!70$$$$T +$$$$85\!\cdots\!99$$$$T^{2} +$$$$40\!\cdots\!20$$$$T^{3} +$$$$33\!\cdots\!97$$$$T^{4} +$$$$13\!\cdots\!30$$$$T^{5} +$$$$78\!\cdots\!63$$$$T^{6} +$$$$26\!\cdots\!60$$$$T^{7} +$$$$11\!\cdots\!03$$$$T^{8} +$$$$31\!\cdots\!30$$$$T^{9} +$$$$11\!\cdots\!77$$$$T^{10} +$$$$21\!\cdots\!20$$$$T^{11} +$$$$69\!\cdots\!99$$$$T^{12} +$$$$66\!\cdots\!70$$$$T^{13} +$$$$18\!\cdots\!61$$$$T^{14} )^{2}$$)
$41$ ($$1 +$$$$11\!\cdots\!18$$$$T +$$$$40\!\cdots\!81$$$$T^{2}$$)($$( 1 -$$$$18\!\cdots\!74$$$$T +$$$$16\!\cdots\!11$$$$T^{2} -$$$$37\!\cdots\!44$$$$T^{3} +$$$$12\!\cdots\!61$$$$T^{4} -$$$$25\!\cdots\!02$$$$T^{5} +$$$$68\!\cdots\!23$$$$T^{6} -$$$$11\!\cdots\!72$$$$T^{7} +$$$$27\!\cdots\!63$$$$T^{8} -$$$$42\!\cdots\!22$$$$T^{9} +$$$$84\!\cdots\!01$$$$T^{10} -$$$$10\!\cdots\!24$$$$T^{11} +$$$$17\!\cdots\!11$$$$T^{12} -$$$$81\!\cdots\!94$$$$T^{13} +$$$$18\!\cdots\!61$$$$T^{14} )^{2}$$)
$43$ ($$( 1 -$$$$13\!\cdots\!01$$$$T )( 1 +$$$$13\!\cdots\!01$$$$T )$$)($$1 -$$$$10\!\cdots\!42$$$$T^{2} +$$$$66\!\cdots\!55$$$$T^{4} -$$$$29\!\cdots\!68$$$$T^{6} +$$$$10\!\cdots\!29$$$$T^{8} -$$$$29\!\cdots\!50$$$$T^{10} +$$$$71\!\cdots\!15$$$$T^{12} -$$$$14\!\cdots\!80$$$$T^{14} +$$$$24\!\cdots\!15$$$$T^{16} -$$$$36\!\cdots\!50$$$$T^{18} +$$$$43\!\cdots\!29$$$$T^{20} -$$$$43\!\cdots\!68$$$$T^{22} +$$$$33\!\cdots\!55$$$$T^{24} -$$$$18\!\cdots\!42$$$$T^{26} +$$$$62\!\cdots\!01$$$$T^{28}$$)
$47$ ($$( 1 -$$$$56\!\cdots\!21$$$$T )( 1 +$$$$56\!\cdots\!21$$$$T )$$)($$1 -$$$$28\!\cdots\!22$$$$T^{2} +$$$$37\!\cdots\!15$$$$T^{4} -$$$$33\!\cdots\!08$$$$T^{6} +$$$$21\!\cdots\!89$$$$T^{8} -$$$$11\!\cdots\!30$$$$T^{10} +$$$$47\!\cdots\!35$$$$T^{12} -$$$$16\!\cdots\!60$$$$T^{14} +$$$$49\!\cdots\!35$$$$T^{16} -$$$$12\!\cdots\!30$$$$T^{18} +$$$$24\!\cdots\!49$$$$T^{20} -$$$$38\!\cdots\!68$$$$T^{22} +$$$$44\!\cdots\!15$$$$T^{24} -$$$$34\!\cdots\!82$$$$T^{26} +$$$$12\!\cdots\!61$$$$T^{28}$$)
$53$ ($$1 +$$$$67\!\cdots\!58$$$$T +$$$$15\!\cdots\!41$$$$T^{2}$$)($$( 1 -$$$$92\!\cdots\!90$$$$T +$$$$83\!\cdots\!59$$$$T^{2} -$$$$46\!\cdots\!20$$$$T^{3} +$$$$26\!\cdots\!57$$$$T^{4} -$$$$11\!\cdots\!10$$$$T^{5} +$$$$54\!\cdots\!63$$$$T^{6} -$$$$20\!\cdots\!60$$$$T^{7} +$$$$82\!\cdots\!83$$$$T^{8} -$$$$26\!\cdots\!10$$$$T^{9} +$$$$89\!\cdots\!97$$$$T^{10} -$$$$23\!\cdots\!20$$$$T^{11} +$$$$63\!\cdots\!59$$$$T^{12} -$$$$10\!\cdots\!90$$$$T^{13} +$$$$17\!\cdots\!81$$$$T^{14} )^{2}$$)
$59$ ($$( 1 -$$$$21\!\cdots\!41$$$$T )( 1 +$$$$21\!\cdots\!41$$$$T )$$)($$1 -$$$$29\!\cdots\!14$$$$T^{2} +$$$$31\!\cdots\!71$$$$T^{4} -$$$$10\!\cdots\!04$$$$T^{6} -$$$$51\!\cdots\!79$$$$T^{8} +$$$$42\!\cdots\!98$$$$T^{10} +$$$$82\!\cdots\!03$$$$T^{12} -$$$$14\!\cdots\!92$$$$T^{14} +$$$$17\!\cdots\!83$$$$T^{16} +$$$$19\!\cdots\!58$$$$T^{18} -$$$$52\!\cdots\!99$$$$T^{20} -$$$$23\!\cdots\!64$$$$T^{22} +$$$$14\!\cdots\!71$$$$T^{24} -$$$$29\!\cdots\!54$$$$T^{26} +$$$$21\!\cdots\!21$$$$T^{28}$$)
$61$ ($$1 +$$$$71\!\cdots\!78$$$$T +$$$$13\!\cdots\!21$$$$T^{2}$$)($$( 1 -$$$$53\!\cdots\!54$$$$T +$$$$52\!\cdots\!51$$$$T^{2} -$$$$16\!\cdots\!84$$$$T^{3} +$$$$11\!\cdots\!41$$$$T^{4} -$$$$31\!\cdots\!02$$$$T^{5} +$$$$22\!\cdots\!83$$$$T^{6} -$$$$55\!\cdots\!52$$$$T^{7} +$$$$30\!\cdots\!43$$$$T^{8} -$$$$57\!\cdots\!82$$$$T^{9} +$$$$28\!\cdots\!01$$$$T^{10} -$$$$53\!\cdots\!04$$$$T^{11} +$$$$23\!\cdots\!51$$$$T^{12} -$$$$32\!\cdots\!34$$$$T^{13} +$$$$82\!\cdots\!41$$$$T^{14} )^{2}$$)
$67$ ($$( 1 -$$$$16\!\cdots\!81$$$$T )( 1 +$$$$16\!\cdots\!81$$$$T )$$)($$1 -$$$$17\!\cdots\!62$$$$T^{2} +$$$$16\!\cdots\!55$$$$T^{4} -$$$$10\!\cdots\!68$$$$T^{6} +$$$$48\!\cdots\!89$$$$T^{8} -$$$$18\!\cdots\!10$$$$T^{10} +$$$$62\!\cdots\!15$$$$T^{12} -$$$$18\!\cdots\!40$$$$T^{14} +$$$$46\!\cdots\!15$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$19\!\cdots\!29$$$$T^{20} -$$$$30\!\cdots\!08$$$$T^{22} +$$$$35\!\cdots\!55$$$$T^{24} -$$$$29\!\cdots\!02$$$$T^{26} +$$$$12\!\cdots\!41$$$$T^{28}$$)
$71$ ($$( 1 -$$$$41\!\cdots\!21$$$$T )( 1 +$$$$41\!\cdots\!21$$$$T )$$)($$1 -$$$$96\!\cdots\!14$$$$T^{2} +$$$$51\!\cdots\!11$$$$T^{4} -$$$$19\!\cdots\!84$$$$T^{6} +$$$$55\!\cdots\!61$$$$T^{8} -$$$$13\!\cdots\!02$$$$T^{10} +$$$$27\!\cdots\!03$$$$T^{12} -$$$$49\!\cdots\!72$$$$T^{14} +$$$$81\!\cdots\!43$$$$T^{16} -$$$$12\!\cdots\!22$$$$T^{18} +$$$$15\!\cdots\!01$$$$T^{20} -$$$$16\!\cdots\!64$$$$T^{22} +$$$$13\!\cdots\!11$$$$T^{24} -$$$$73\!\cdots\!34$$$$T^{26} +$$$$23\!\cdots\!61$$$$T^{28}$$)
$73$ ($$1 -$$$$60\!\cdots\!22$$$$T +$$$$42\!\cdots\!21$$$$T^{2}$$)($$( 1 -$$$$71\!\cdots\!30$$$$T +$$$$19\!\cdots\!99$$$$T^{2} -$$$$11\!\cdots\!20$$$$T^{3} +$$$$17\!\cdots\!17$$$$T^{4} -$$$$89\!\cdots\!30$$$$T^{5} +$$$$10\!\cdots\!03$$$$T^{6} -$$$$46\!\cdots\!80$$$$T^{7} +$$$$44\!\cdots\!63$$$$T^{8} -$$$$15\!\cdots\!30$$$$T^{9} +$$$$13\!\cdots\!37$$$$T^{10} -$$$$35\!\cdots\!20$$$$T^{11} +$$$$26\!\cdots\!99$$$$T^{12} -$$$$41\!\cdots\!30$$$$T^{13} +$$$$24\!\cdots\!41$$$$T^{14} )^{2}$$)
$79$ ($$( 1 -$$$$23\!\cdots\!21$$$$T )( 1 +$$$$23\!\cdots\!21$$$$T )$$)($$1 -$$$$32\!\cdots\!34$$$$T^{2} +$$$$63\!\cdots\!51$$$$T^{4} -$$$$84\!\cdots\!04$$$$T^{6} +$$$$88\!\cdots\!61$$$$T^{8} -$$$$73\!\cdots\!02$$$$T^{10} +$$$$50\!\cdots\!43$$$$T^{12} -$$$$29\!\cdots\!52$$$$T^{14} +$$$$14\!\cdots\!83$$$$T^{16} -$$$$57\!\cdots\!22$$$$T^{18} +$$$$19\!\cdots\!01$$$$T^{20} -$$$$52\!\cdots\!84$$$$T^{22} +$$$$10\!\cdots\!51$$$$T^{24} -$$$$16\!\cdots\!54$$$$T^{26} +$$$$13\!\cdots\!61$$$$T^{28}$$)
$83$ ($$( 1 -$$$$50\!\cdots\!81$$$$T )( 1 +$$$$50\!\cdots\!81$$$$T )$$)($$1 -$$$$22\!\cdots\!02$$$$T^{2} +$$$$24\!\cdots\!95$$$$T^{4} -$$$$17\!\cdots\!08$$$$T^{6} +$$$$97\!\cdots\!69$$$$T^{8} -$$$$41\!\cdots\!90$$$$T^{10} +$$$$14\!\cdots\!95$$$$T^{12} -$$$$40\!\cdots\!20$$$$T^{14} +$$$$95\!\cdots\!95$$$$T^{16} -$$$$18\!\cdots\!90$$$$T^{18} +$$$$28\!\cdots\!09$$$$T^{20} -$$$$34\!\cdots\!48$$$$T^{22} +$$$$31\!\cdots\!95$$$$T^{24} -$$$$18\!\cdots\!42$$$$T^{26} +$$$$55\!\cdots\!41$$$$T^{28}$$)
$89$ ($$1 -$$$$17\!\cdots\!22$$$$T +$$$$24\!\cdots\!21$$$$T^{2}$$)($$( 1 +$$$$92\!\cdots\!38$$$$T +$$$$73\!\cdots\!55$$$$T^{2} +$$$$12\!\cdots\!92$$$$T^{3} +$$$$36\!\cdots\!89$$$$T^{4} +$$$$55\!\cdots\!50$$$$T^{5} +$$$$12\!\cdots\!15$$$$T^{6} +$$$$16\!\cdots\!20$$$$T^{7} +$$$$30\!\cdots\!15$$$$T^{8} +$$$$32\!\cdots\!50$$$$T^{9} +$$$$50\!\cdots\!29$$$$T^{10} +$$$$40\!\cdots\!52$$$$T^{11} +$$$$58\!\cdots\!55$$$$T^{12} +$$$$17\!\cdots\!98$$$$T^{13} +$$$$46\!\cdots\!41$$$$T^{14} )^{2}$$)
$97$ ($$1 -$$$$84\!\cdots\!42$$$$T +$$$$37\!\cdots\!41$$$$T^{2}$$)($$( 1 +$$$$10\!\cdots\!10$$$$T +$$$$16\!\cdots\!39$$$$T^{2} +$$$$11\!\cdots\!20$$$$T^{3} +$$$$10\!\cdots\!57$$$$T^{4} +$$$$42\!\cdots\!30$$$$T^{5} +$$$$33\!\cdots\!63$$$$T^{6} +$$$$10\!\cdots\!40$$$$T^{7} +$$$$12\!\cdots\!83$$$$T^{8} +$$$$60\!\cdots\!30$$$$T^{9} +$$$$55\!\cdots\!97$$$$T^{10} +$$$$22\!\cdots\!20$$$$T^{11} +$$$$12\!\cdots\!39$$$$T^{12} +$$$$29\!\cdots\!10$$$$T^{13} +$$$$10\!\cdots\!81$$$$T^{14} )^{2}$$)