# Properties

 Label 4.37.b Level $4$ Weight $37$ Character orbit 4.b Rep. character $\chi_{4}(3,\cdot)$ Character field $\Q$ Dimension $17$ Newform subspaces $2$ Sturm bound $18$ Trace bound $1$

# Learn more about

## Defining parameters

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$37$$ 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: $$18$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{37}(4, [\chi])$$.

Total New Old
Modular forms 19 19 0
Cusp forms 17 17 0
Eisenstein series 2 2 0

## Trace form

 $$17q - 84916q^{2} + 63108769040q^{4} + 1587598983826q^{5} - 217628996575488q^{6} - 6116838281275456q^{8} - 785089825520546847q^{9} + O(q^{10})$$ $$17q - 84916q^{2} + 63108769040q^{4} + 1587598983826q^{5} - 217628996575488q^{6} - 6116838281275456q^{8} - 785089825520546847q^{9} - 1269669491125656104q^{10} - 42882825786868930560q^{12} + 41676149406206500018q^{13} + 360192888461659120128q^{14} - 18131657039655345618688q^{16} + 18678125271466757001058q^{17} + 98858065961248026757644q^{18} - 185987631386800333033184q^{20} - 175281296641396774711296q^{21} - 4320007363229839528400640q^{22} + 5118364442017682383085568q^{24} + 64389827691705544439075331q^{25} - 30067267580294149817323688q^{26} - 303534200373547502486016000q^{28} + 83558829778969622281411570q^{29} + 361206724962468787334008320q^{30} + 3424487663300783641955892224q^{32} - 877278707167997370072668160q^{33} - 2956121337393437394265165928q^{34} + 9386969606899626916903182864q^{36} + 1028936832542213926690666258q^{37} + 78752216997100984306444174080q^{38} - 61948625233183002270749961344q^{40} + 241910655498966263829819397954q^{41} - 193406458925326074258433843200q^{42} + 303084904999663560614745692160q^{44} + 876750889440806776372165555506q^{45} - 919681205391808405630032112128q^{46} - 1511312501851283767715389931520q^{48} - 8690584837344731555055515727247q^{49} + 10185343958170807211600579472036q^{50} + 5584769803431310739681792135968q^{52} + 2864977575306763415468638557778q^{53} - 42689634574575573966970327785984q^{54} - 18346492780981997297263940026368q^{56} - 17591710635767034592583231047680q^{57} - 13872941779387924373159026137512q^{58} - 69704584574466315130190074982400q^{60} + 488044483791787508231294360980594q^{61} + 406820870675021165265665434183680q^{62} + 58407051193644631965991706562560q^{64} - 1419136027205678294994787009303228q^{65} - 816784109200805225218136599695360q^{66} + 1913850173378374444103604070923808q^{68} + 3923547517779842564469076002041856q^{69} + 312094823977274793710809307489280q^{70} - 4579322712415668078348091815468096q^{72} - 7016735948804628395640934715523902q^{73} - 670997304617426449840386839200808q^{74} - 15736875982808342938482824299100160q^{76} + 22639774155764461736245527294259200q^{77} - 18389491597233550087389536189698560q^{78} + 47267506511318977155908021422813696q^{80} - 18188477562036937666899669720078159q^{81} + 5807936682837544119287618687716888q^{82} - 37119022564017857733030089777086464q^{84} - 109102745881421109039130518648221788q^{85} + 102108696026625816655543232864792832q^{86} + 39749581676781586144951142033940480q^{88} + 536580108278993677338741367468283650q^{89} - 195283296567279720904119484240636584q^{90} - 17511543929609412714530045318983680q^{92} - 231978818535002459544922699145134080q^{93} - 696830931593880576023589572144120832q^{94} + 1050073689946737522972596709346639872q^{96} - 467501081650930946766716978739129182q^{97} + 340279939831197597514188700404039884q^{98} + O(q^{100})$$

## Decomposition of $$S_{37}^{\mathrm{new}}(4, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
4.37.b.a $$1$$ $$32.837$$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$-262144$$ $$0$$ $$-4\!\cdots\!34$$ $$0$$ $$q-2^{18}q^{2}+2^{36}q^{4}-4228490555534q^{5}+\cdots$$
4.37.b.b $$16$$ $$32.837$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$177228$$ $$0$$ $$58\!\cdots\!60$$ $$0$$ $$q+(11077+\beta _{1})q^{2}+(50+198\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 262144 T$$)($$1 - 177228 T + 18510235840 T^{2} - 5390822269734912 T^{3} +$$$$65\!\cdots\!36$$$$T^{4} -$$$$20\!\cdots\!76$$$$T^{5} +$$$$32\!\cdots\!20$$$$T^{6} -$$$$12\!\cdots\!24$$$$T^{7} +$$$$23\!\cdots\!76$$$$T^{8} -$$$$83\!\cdots\!64$$$$T^{9} +$$$$15\!\cdots\!20$$$$T^{10} -$$$$65\!\cdots\!56$$$$T^{11} +$$$$14\!\cdots\!76$$$$T^{12} -$$$$82\!\cdots\!12$$$$T^{13} +$$$$19\!\cdots\!40$$$$T^{14} -$$$$12\!\cdots\!88$$$$T^{15} +$$$$49\!\cdots\!56$$$$T^{16}$$)
$3$ ($$( 1 - 387420489 T )( 1 + 387420489 T )$$)($$1 - 733164851967219984 T^{2} +$$$$31\!\cdots\!60$$$$T^{4} -$$$$96\!\cdots\!00$$$$T^{6} +$$$$23\!\cdots\!60$$$$T^{8} -$$$$50\!\cdots\!76$$$$T^{10} +$$$$92\!\cdots\!84$$$$T^{12} -$$$$15\!\cdots\!20$$$$T^{14} +$$$$24\!\cdots\!50$$$$T^{16} -$$$$35\!\cdots\!20$$$$T^{18} +$$$$46\!\cdots\!04$$$$T^{20} -$$$$57\!\cdots\!96$$$$T^{22} +$$$$61\!\cdots\!60$$$$T^{24} -$$$$56\!\cdots\!00$$$$T^{26} +$$$$41\!\cdots\!60$$$$T^{28} -$$$$21\!\cdots\!04$$$$T^{30} +$$$$66\!\cdots\!21$$$$T^{32}$$)
$5$ ($$1 + 4228490555534 T +$$$$14\!\cdots\!25$$$$T^{2}$$)($$( 1 - 2908044769680 T +$$$$47\!\cdots\!00$$$$T^{2} -$$$$86\!\cdots\!00$$$$T^{3} +$$$$72\!\cdots\!00$$$$T^{4} +$$$$60\!\cdots\!00$$$$T^{5} +$$$$90\!\cdots\!00$$$$T^{6} +$$$$56\!\cdots\!00$$$$T^{7} -$$$$80\!\cdots\!50$$$$T^{8} +$$$$82\!\cdots\!00$$$$T^{9} +$$$$19\!\cdots\!00$$$$T^{10} +$$$$18\!\cdots\!00$$$$T^{11} +$$$$32\!\cdots\!00$$$$T^{12} -$$$$56\!\cdots\!00$$$$T^{13} +$$$$44\!\cdots\!00$$$$T^{14} -$$$$40\!\cdots\!00$$$$T^{15} +$$$$20\!\cdots\!25$$$$T^{16} )^{2}$$)
$7$ ($$( 1 - 1628413597910449 T )( 1 + 1628413597910449 T )$$)($$1 -$$$$15\!\cdots\!84$$$$T^{2} +$$$$12\!\cdots\!80$$$$T^{4} -$$$$78\!\cdots\!60$$$$T^{6} +$$$$38\!\cdots\!60$$$$T^{8} -$$$$16\!\cdots\!16$$$$T^{10} +$$$$58\!\cdots\!24$$$$T^{12} -$$$$18\!\cdots\!40$$$$T^{14} +$$$$52\!\cdots\!70$$$$T^{16} -$$$$13\!\cdots\!40$$$$T^{18} +$$$$28\!\cdots\!24$$$$T^{20} -$$$$56\!\cdots\!16$$$$T^{22} +$$$$94\!\cdots\!60$$$$T^{24} -$$$$13\!\cdots\!60$$$$T^{26} +$$$$15\!\cdots\!80$$$$T^{28} -$$$$13\!\cdots\!84$$$$T^{30} +$$$$59\!\cdots\!01$$$$T^{32}$$)
$11$ ($$( 1 - 5559917313492231481 T )( 1 + 5559917313492231481 T )$$)($$1 -$$$$20\!\cdots\!16$$$$T^{2} +$$$$21\!\cdots\!60$$$$T^{4} -$$$$15\!\cdots\!60$$$$T^{6} +$$$$81\!\cdots\!60$$$$T^{8} -$$$$36\!\cdots\!08$$$$T^{10} +$$$$14\!\cdots\!68$$$$T^{12} -$$$$51\!\cdots\!20$$$$T^{14} +$$$$16\!\cdots\!70$$$$T^{16} -$$$$49\!\cdots\!20$$$$T^{18} +$$$$13\!\cdots\!88$$$$T^{20} -$$$$32\!\cdots\!88$$$$T^{22} +$$$$68\!\cdots\!60$$$$T^{24} -$$$$12\!\cdots\!60$$$$T^{26} +$$$$16\!\cdots\!60$$$$T^{28} -$$$$14\!\cdots\!56$$$$T^{30} +$$$$69\!\cdots\!61$$$$T^{32}$$)
$13$ ($$1 +$$$$15\!\cdots\!58$$$$T +$$$$12\!\cdots\!41$$$$T^{2}$$)($$( 1 - 97306007082829587088 T +$$$$49\!\cdots\!60$$$$T^{2} -$$$$40\!\cdots\!72$$$$T^{3} +$$$$13\!\cdots\!56$$$$T^{4} -$$$$92\!\cdots\!76$$$$T^{5} +$$$$24\!\cdots\!80$$$$T^{6} -$$$$15\!\cdots\!24$$$$T^{7} +$$$$35\!\cdots\!86$$$$T^{8} -$$$$19\!\cdots\!84$$$$T^{9} +$$$$39\!\cdots\!80$$$$T^{10} -$$$$18\!\cdots\!96$$$$T^{11} +$$$$34\!\cdots\!16$$$$T^{12} -$$$$13\!\cdots\!72$$$$T^{13} +$$$$20\!\cdots\!60$$$$T^{14} -$$$$50\!\cdots\!28$$$$T^{15} +$$$$65\!\cdots\!21$$$$T^{16} )^{2}$$)
$17$ ($$1 +$$$$23\!\cdots\!18$$$$T +$$$$19\!\cdots\!81$$$$T^{2}$$)($$( 1 -$$$$20\!\cdots\!88$$$$T +$$$$11\!\cdots\!00$$$$T^{2} -$$$$15\!\cdots\!72$$$$T^{3} +$$$$51\!\cdots\!16$$$$T^{4} -$$$$46\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!40$$$$T^{6} -$$$$96\!\cdots\!64$$$$T^{7} +$$$$30\!\cdots\!86$$$$T^{8} -$$$$19\!\cdots\!84$$$$T^{9} +$$$$55\!\cdots\!40$$$$T^{10} -$$$$36\!\cdots\!16$$$$T^{11} +$$$$78\!\cdots\!36$$$$T^{12} -$$$$45\!\cdots\!72$$$$T^{13} +$$$$67\!\cdots\!00$$$$T^{14} -$$$$24\!\cdots\!68$$$$T^{15} +$$$$23\!\cdots\!41$$$$T^{16} )^{2}$$)
$19$ ($$( 1 -$$$$10\!\cdots\!41$$$$T )( 1 +$$$$10\!\cdots\!41$$$$T )$$)($$1 -$$$$96\!\cdots\!36$$$$T^{2} +$$$$44\!\cdots\!60$$$$T^{4} -$$$$13\!\cdots\!60$$$$T^{6} +$$$$28\!\cdots\!60$$$$T^{8} -$$$$47\!\cdots\!08$$$$T^{10} +$$$$67\!\cdots\!28$$$$T^{12} -$$$$83\!\cdots\!20$$$$T^{14} +$$$$93\!\cdots\!70$$$$T^{16} -$$$$97\!\cdots\!20$$$$T^{18} +$$$$93\!\cdots\!88$$$$T^{20} -$$$$77\!\cdots\!48$$$$T^{22} +$$$$54\!\cdots\!60$$$$T^{24} -$$$$29\!\cdots\!60$$$$T^{26} +$$$$11\!\cdots\!60$$$$T^{28} -$$$$29\!\cdots\!56$$$$T^{30} +$$$$36\!\cdots\!81$$$$T^{32}$$)
$23$ ($$( 1 -$$$$32\!\cdots\!69$$$$T )( 1 +$$$$32\!\cdots\!69$$$$T )$$)($$1 -$$$$10\!\cdots\!24$$$$T^{2} +$$$$51\!\cdots\!60$$$$T^{4} -$$$$16\!\cdots\!60$$$$T^{6} +$$$$39\!\cdots\!80$$$$T^{8} -$$$$74\!\cdots\!16$$$$T^{10} +$$$$11\!\cdots\!64$$$$T^{12} -$$$$14\!\cdots\!40$$$$T^{14} +$$$$16\!\cdots\!70$$$$T^{16} -$$$$16\!\cdots\!40$$$$T^{18} +$$$$14\!\cdots\!24$$$$T^{20} -$$$$10\!\cdots\!76$$$$T^{22} +$$$$60\!\cdots\!80$$$$T^{24} -$$$$27\!\cdots\!60$$$$T^{26} +$$$$94\!\cdots\!60$$$$T^{28} -$$$$20\!\cdots\!84$$$$T^{30} +$$$$22\!\cdots\!61$$$$T^{32}$$)
$29$ ($$1 -$$$$17\!\cdots\!22$$$$T +$$$$44\!\cdots\!21$$$$T^{2}$$)($$( 1 +$$$$47\!\cdots\!76$$$$T +$$$$23\!\cdots\!00$$$$T^{2} +$$$$21\!\cdots\!40$$$$T^{3} +$$$$25\!\cdots\!40$$$$T^{4} -$$$$58\!\cdots\!16$$$$T^{5} +$$$$17\!\cdots\!64$$$$T^{6} -$$$$69\!\cdots\!40$$$$T^{7} +$$$$90\!\cdots\!70$$$$T^{8} -$$$$30\!\cdots\!40$$$$T^{9} +$$$$35\!\cdots\!24$$$$T^{10} -$$$$50\!\cdots\!76$$$$T^{11} +$$$$99\!\cdots\!40$$$$T^{12} +$$$$36\!\cdots\!40$$$$T^{13} +$$$$17\!\cdots\!00$$$$T^{14} +$$$$15\!\cdots\!16$$$$T^{15} +$$$$14\!\cdots\!61$$$$T^{16} )^{2}$$)
$31$ ($$( 1 -$$$$69\!\cdots\!41$$$$T )( 1 +$$$$69\!\cdots\!41$$$$T )$$)($$1 -$$$$49\!\cdots\!36$$$$T^{2} +$$$$12\!\cdots\!60$$$$T^{4} -$$$$21\!\cdots\!60$$$$T^{6} +$$$$26\!\cdots\!60$$$$T^{8} -$$$$26\!\cdots\!08$$$$T^{10} +$$$$20\!\cdots\!28$$$$T^{12} -$$$$13\!\cdots\!20$$$$T^{14} +$$$$71\!\cdots\!70$$$$T^{16} -$$$$32\!\cdots\!20$$$$T^{18} +$$$$11\!\cdots\!88$$$$T^{20} -$$$$35\!\cdots\!48$$$$T^{22} +$$$$87\!\cdots\!60$$$$T^{24} -$$$$16\!\cdots\!60$$$$T^{26} +$$$$23\!\cdots\!60$$$$T^{28} -$$$$22\!\cdots\!56$$$$T^{30} +$$$$10\!\cdots\!81$$$$T^{32}$$)
$37$ ($$1 -$$$$31\!\cdots\!42$$$$T +$$$$28\!\cdots\!41$$$$T^{2}$$)($$( 1 +$$$$15\!\cdots\!92$$$$T +$$$$13\!\cdots\!40$$$$T^{2} +$$$$17\!\cdots\!08$$$$T^{3} +$$$$90\!\cdots\!56$$$$T^{4} +$$$$91\!\cdots\!24$$$$T^{5} +$$$$38\!\cdots\!60$$$$T^{6} +$$$$32\!\cdots\!36$$$$T^{7} +$$$$12\!\cdots\!06$$$$T^{8} +$$$$92\!\cdots\!76$$$$T^{9} +$$$$31\!\cdots\!60$$$$T^{10} +$$$$21\!\cdots\!04$$$$T^{11} +$$$$59\!\cdots\!16$$$$T^{12} +$$$$32\!\cdots\!08$$$$T^{13} +$$$$74\!\cdots\!40$$$$T^{14} +$$$$23\!\cdots\!52$$$$T^{15} +$$$$43\!\cdots\!21$$$$T^{16} )^{2}$$)
$41$ ($$1 -$$$$14\!\cdots\!42$$$$T +$$$$11\!\cdots\!41$$$$T^{2}$$)($$( 1 -$$$$49\!\cdots\!56$$$$T +$$$$54\!\cdots\!20$$$$T^{2} -$$$$19\!\cdots\!20$$$$T^{3} +$$$$13\!\cdots\!20$$$$T^{4} -$$$$37\!\cdots\!88$$$$T^{5} +$$$$21\!\cdots\!08$$$$T^{6} -$$$$56\!\cdots\!40$$$$T^{7} +$$$$26\!\cdots\!90$$$$T^{8} -$$$$64\!\cdots\!40$$$$T^{9} +$$$$28\!\cdots\!48$$$$T^{10} -$$$$56\!\cdots\!48$$$$T^{11} +$$$$23\!\cdots\!20$$$$T^{12} -$$$$38\!\cdots\!20$$$$T^{13} +$$$$12\!\cdots\!20$$$$T^{14} -$$$$13\!\cdots\!36$$$$T^{15} +$$$$30\!\cdots\!21$$$$T^{16} )^{2}$$)
$43$ ($$( 1 -$$$$25\!\cdots\!49$$$$T )( 1 +$$$$25\!\cdots\!49$$$$T )$$)($$1 -$$$$46\!\cdots\!24$$$$T^{2} +$$$$11\!\cdots\!20$$$$T^{4} -$$$$20\!\cdots\!00$$$$T^{6} +$$$$27\!\cdots\!00$$$$T^{8} -$$$$29\!\cdots\!56$$$$T^{10} +$$$$27\!\cdots\!04$$$$T^{12} -$$$$21\!\cdots\!20$$$$T^{14} +$$$$14\!\cdots\!50$$$$T^{16} -$$$$88\!\cdots\!20$$$$T^{18} +$$$$45\!\cdots\!04$$$$T^{20} -$$$$20\!\cdots\!56$$$$T^{22} +$$$$74\!\cdots\!00$$$$T^{24} -$$$$22\!\cdots\!00$$$$T^{26} +$$$$52\!\cdots\!20$$$$T^{28} -$$$$85\!\cdots\!24$$$$T^{30} +$$$$75\!\cdots\!01$$$$T^{32}$$)
$47$ ($$( 1 -$$$$12\!\cdots\!89$$$$T )( 1 +$$$$12\!\cdots\!89$$$$T )$$)($$1 -$$$$20\!\cdots\!64$$$$T^{2} +$$$$20\!\cdots\!20$$$$T^{4} -$$$$13\!\cdots\!00$$$$T^{6} +$$$$60\!\cdots\!00$$$$T^{8} -$$$$21\!\cdots\!96$$$$T^{10} +$$$$57\!\cdots\!84$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$21\!\cdots\!50$$$$T^{16} -$$$$30\!\cdots\!20$$$$T^{18} +$$$$34\!\cdots\!04$$$$T^{20} -$$$$31\!\cdots\!16$$$$T^{22} +$$$$22\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!00$$$$T^{26} +$$$$46\!\cdots\!20$$$$T^{28} -$$$$11\!\cdots\!84$$$$T^{30} +$$$$13\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 +$$$$18\!\cdots\!78$$$$T +$$$$11\!\cdots\!21$$$$T^{2}$$)($$( 1 -$$$$10\!\cdots\!28$$$$T +$$$$82\!\cdots\!00$$$$T^{2} -$$$$68\!\cdots\!12$$$$T^{3} +$$$$30\!\cdots\!36$$$$T^{4} -$$$$20\!\cdots\!56$$$$T^{5} +$$$$65\!\cdots\!00$$$$T^{6} -$$$$36\!\cdots\!24$$$$T^{7} +$$$$94\!\cdots\!86$$$$T^{8} -$$$$43\!\cdots\!04$$$$T^{9} +$$$$92\!\cdots\!00$$$$T^{10} -$$$$33\!\cdots\!16$$$$T^{11} +$$$$59\!\cdots\!16$$$$T^{12} -$$$$16\!\cdots\!12$$$$T^{13} +$$$$22\!\cdots\!00$$$$T^{14} -$$$$34\!\cdots\!48$$$$T^{15} +$$$$39\!\cdots\!61$$$$T^{16} )^{2}$$)
$59$ ($$( 1 -$$$$75\!\cdots\!21$$$$T )( 1 +$$$$75\!\cdots\!21$$$$T )$$)($$1 -$$$$53\!\cdots\!36$$$$T^{2} +$$$$14\!\cdots\!20$$$$T^{4} -$$$$27\!\cdots\!00$$$$T^{6} +$$$$38\!\cdots\!20$$$$T^{8} -$$$$41\!\cdots\!48$$$$T^{10} +$$$$37\!\cdots\!88$$$$T^{12} -$$$$27\!\cdots\!40$$$$T^{14} +$$$$16\!\cdots\!50$$$$T^{16} -$$$$86\!\cdots\!40$$$$T^{18} +$$$$37\!\cdots\!68$$$$T^{20} -$$$$13\!\cdots\!68$$$$T^{22} +$$$$38\!\cdots\!20$$$$T^{24} -$$$$88\!\cdots\!00$$$$T^{26} +$$$$15\!\cdots\!20$$$$T^{28} -$$$$17\!\cdots\!96$$$$T^{30} +$$$$10\!\cdots\!41$$$$T^{32}$$)
$61$ ($$1 -$$$$27\!\cdots\!62$$$$T +$$$$18\!\cdots\!61$$$$T^{2}$$)($$( 1 -$$$$10\!\cdots\!16$$$$T +$$$$99\!\cdots\!20$$$$T^{2} -$$$$11\!\cdots\!20$$$$T^{3} +$$$$47\!\cdots\!20$$$$T^{4} -$$$$55\!\cdots\!88$$$$T^{5} +$$$$14\!\cdots\!88$$$$T^{6} -$$$$15\!\cdots\!40$$$$T^{7} +$$$$30\!\cdots\!90$$$$T^{8} -$$$$29\!\cdots\!40$$$$T^{9} +$$$$49\!\cdots\!48$$$$T^{10} -$$$$36\!\cdots\!28$$$$T^{11} +$$$$57\!\cdots\!20$$$$T^{12} -$$$$26\!\cdots\!20$$$$T^{13} +$$$$42\!\cdots\!20$$$$T^{14} -$$$$86\!\cdots\!36$$$$T^{15} +$$$$14\!\cdots\!81$$$$T^{16} )^{2}$$)
$67$ ($$( 1 -$$$$74\!\cdots\!09$$$$T )( 1 +$$$$74\!\cdots\!09$$$$T )$$)($$1 -$$$$44\!\cdots\!04$$$$T^{2} +$$$$10\!\cdots\!40$$$$T^{4} -$$$$17\!\cdots\!80$$$$T^{6} +$$$$22\!\cdots\!40$$$$T^{8} -$$$$22\!\cdots\!56$$$$T^{10} +$$$$18\!\cdots\!84$$$$T^{12} -$$$$13\!\cdots\!80$$$$T^{14} +$$$$77\!\cdots\!10$$$$T^{16} -$$$$39\!\cdots\!80$$$$T^{18} +$$$$16\!\cdots\!64$$$$T^{20} -$$$$60\!\cdots\!36$$$$T^{22} +$$$$17\!\cdots\!40$$$$T^{24} -$$$$42\!\cdots\!80$$$$T^{26} +$$$$77\!\cdots\!40$$$$T^{28} -$$$$97\!\cdots\!84$$$$T^{30} +$$$$65\!\cdots\!81$$$$T^{32}$$)
$71$ ($$( 1 -$$$$21\!\cdots\!61$$$$T )( 1 +$$$$21\!\cdots\!61$$$$T )$$)($$1 -$$$$33\!\cdots\!16$$$$T^{2} +$$$$57\!\cdots\!20$$$$T^{4} -$$$$70\!\cdots\!00$$$$T^{6} +$$$$66\!\cdots\!20$$$$T^{8} -$$$$50\!\cdots\!48$$$$T^{10} +$$$$32\!\cdots\!28$$$$T^{12} -$$$$18\!\cdots\!40$$$$T^{14} +$$$$86\!\cdots\!50$$$$T^{16} -$$$$35\!\cdots\!40$$$$T^{18} +$$$$12\!\cdots\!68$$$$T^{20} -$$$$37\!\cdots\!08$$$$T^{22} +$$$$96\!\cdots\!20$$$$T^{24} -$$$$19\!\cdots\!00$$$$T^{26} +$$$$32\!\cdots\!20$$$$T^{28} -$$$$35\!\cdots\!96$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$)
$73$ ($$1 -$$$$65\!\cdots\!62$$$$T +$$$$12\!\cdots\!61$$$$T^{2}$$)($$( 1 +$$$$67\!\cdots\!32$$$$T +$$$$51\!\cdots\!60$$$$T^{2} +$$$$28\!\cdots\!68$$$$T^{3} +$$$$17\!\cdots\!96$$$$T^{4} +$$$$75\!\cdots\!24$$$$T^{5} +$$$$33\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!96$$$$T^{7} +$$$$48\!\cdots\!66$$$$T^{8} +$$$$15\!\cdots\!56$$$$T^{9} +$$$$48\!\cdots\!80$$$$T^{10} +$$$$13\!\cdots\!44$$$$T^{11} +$$$$35\!\cdots\!36$$$$T^{12} +$$$$72\!\cdots\!68$$$$T^{13} +$$$$15\!\cdots\!60$$$$T^{14} +$$$$24\!\cdots\!72$$$$T^{15} +$$$$43\!\cdots\!81$$$$T^{16} )^{2}$$)
$79$ ($$( 1 -$$$$14\!\cdots\!61$$$$T )( 1 +$$$$14\!\cdots\!61$$$$T )$$)($$1 -$$$$10\!\cdots\!76$$$$T^{2} +$$$$62\!\cdots\!60$$$$T^{4} -$$$$26\!\cdots\!60$$$$T^{6} +$$$$90\!\cdots\!60$$$$T^{8} -$$$$26\!\cdots\!08$$$$T^{10} +$$$$71\!\cdots\!48$$$$T^{12} -$$$$16\!\cdots\!20$$$$T^{14} +$$$$36\!\cdots\!70$$$$T^{16} -$$$$71\!\cdots\!20$$$$T^{18} +$$$$12\!\cdots\!88$$$$T^{20} -$$$$20\!\cdots\!68$$$$T^{22} +$$$$29\!\cdots\!60$$$$T^{24} -$$$$36\!\cdots\!60$$$$T^{26} +$$$$37\!\cdots\!60$$$$T^{28} -$$$$27\!\cdots\!56$$$$T^{30} +$$$$10\!\cdots\!21$$$$T^{32}$$)
$83$ ($$( 1 -$$$$34\!\cdots\!09$$$$T )( 1 +$$$$34\!\cdots\!09$$$$T )$$)($$1 -$$$$73\!\cdots\!24$$$$T^{2} +$$$$27\!\cdots\!00$$$$T^{4} -$$$$66\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!40$$$$T^{8} -$$$$20\!\cdots\!96$$$$T^{10} +$$$$30\!\cdots\!44$$$$T^{12} -$$$$43\!\cdots\!40$$$$T^{14} +$$$$57\!\cdots\!70$$$$T^{16} -$$$$65\!\cdots\!40$$$$T^{18} +$$$$67\!\cdots\!24$$$$T^{20} -$$$$66\!\cdots\!76$$$$T^{22} +$$$$61\!\cdots\!40$$$$T^{24} -$$$$49\!\cdots\!60$$$$T^{26} +$$$$30\!\cdots\!00$$$$T^{28} -$$$$12\!\cdots\!04$$$$T^{30} +$$$$24\!\cdots\!81$$$$T^{32}$$)
$89$ ($$1 -$$$$75\!\cdots\!62$$$$T +$$$$15\!\cdots\!61$$$$T^{2}$$)($$( 1 -$$$$23\!\cdots\!44$$$$T +$$$$10\!\cdots\!20$$$$T^{2} -$$$$16\!\cdots\!00$$$$T^{3} +$$$$42\!\cdots\!00$$$$T^{4} -$$$$55\!\cdots\!56$$$$T^{5} +$$$$10\!\cdots\!24$$$$T^{6} -$$$$11\!\cdots\!20$$$$T^{7} +$$$$19\!\cdots\!50$$$$T^{8} -$$$$17\!\cdots\!20$$$$T^{9} +$$$$24\!\cdots\!04$$$$T^{10} -$$$$19\!\cdots\!36$$$$T^{11} +$$$$22\!\cdots\!00$$$$T^{12} -$$$$12\!\cdots\!00$$$$T^{13} +$$$$11\!\cdots\!20$$$$T^{14} -$$$$40\!\cdots\!24$$$$T^{15} +$$$$26\!\cdots\!81$$$$T^{16} )^{2}$$)
$97$ ($$1 +$$$$91\!\cdots\!78$$$$T +$$$$33\!\cdots\!21$$$$T^{2}$$)($$( 1 -$$$$22\!\cdots\!48$$$$T +$$$$14\!\cdots\!40$$$$T^{2} -$$$$61\!\cdots\!12$$$$T^{3} +$$$$83\!\cdots\!76$$$$T^{4} +$$$$19\!\cdots\!04$$$$T^{5} +$$$$26\!\cdots\!00$$$$T^{6} +$$$$16\!\cdots\!16$$$$T^{7} +$$$$74\!\cdots\!86$$$$T^{8} +$$$$56\!\cdots\!36$$$$T^{9} +$$$$29\!\cdots\!00$$$$T^{10} +$$$$71\!\cdots\!44$$$$T^{11} +$$$$10\!\cdots\!56$$$$T^{12} -$$$$25\!\cdots\!12$$$$T^{13} +$$$$19\!\cdots\!40$$$$T^{14} -$$$$10\!\cdots\!68$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$)
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