# Properties

 Label 5.34.b Level 5 Weight 34 Character orbit b Rep. character $$\chi_{5}(4,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 1 Sturm bound 17 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 5.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$17$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{34}(5, [\chi])$$.

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

## Trace form

 $$16q - 72851326872q^{4} - 232168160280q^{5} + 11001777346872q^{6} - 27535156574590368q^{9} + O(q^{10})$$ $$16q - 72851326872q^{4} - 232168160280q^{5} + 11001777346872q^{6} - 27535156574590368q^{9} - 22271193818331880q^{10} + 219974590742466912q^{11} - 10416043581549356184q^{14} + 9232415497377901440q^{15} + 133171456389985196576q^{16} - 2949217257058889591200q^{19} + 5571772909456424216760q^{20} + 20126160533851210892592q^{21} - 236696360036140740492000q^{24} + 269473151678676722148400q^{25} + 23043234216449373353232q^{26} + 2210824656972934579370400q^{29} - 7438800885032297852420760q^{30} + 4805180075928582021889472q^{31} - 106706080442140703440064q^{34} + 34840114703893102459924320q^{35} + 150609707516636111776170456q^{36} - 950668737676648885834065216q^{39} + 967967034953888345833396000q^{40} + 960625982433026021733974352q^{41} - 4300041994750366563328828704q^{44} + 5959973976670208219568382440q^{45} + 4030532465737707969346868392q^{46} - 48902999941413820155855454112q^{49} + 64132734499011351066825776400q^{50} + 37677263492556888574173469632q^{51} - 416785644990759180210455480400q^{54} + 208827421951347317583761567040q^{55} + 209868986171565551575474447200q^{56} - 204679212804521237512362904800q^{59} - 320542648209593925139588560480q^{60} - 62902839261738400132019069488q^{61} + 3273202940251902417009873735808q^{64} - 1155813780125630532662955267360q^{65} - 2991219624550267460630717491296q^{66} + 6899695112224183044828868241904q^{69} - 5880904834264312779374002982280q^{70} - 87525095316001019795902439808q^{71} + 43140282179597200538055251968176q^{74} - 27726088944293536006405507003200q^{75} - 38605994956626944020944749752800q^{76} - 10590745643822309766800662264000q^{79} + 24913342429758171577982759437920q^{80} + 126385893984816788804508532470336q^{81} - 401968339664232685914350670917664q^{84} + 174616747052458036927895334806720q^{85} - 18805496634715830052189508032488q^{86} - 816457769414638729740610145056800q^{89} + 814730708455223899676092744179240q^{90} + 572310345418666359618454810906752q^{91} + 1084618018208246129370832376900296q^{94} - 960346549067575215463879922304000q^{95} - 1376269400022374885061468958478208q^{96} + 3432482025669259323040953857901024q^{99} + O(q^{100})$$

## Decomposition of $$S_{34}^{\mathrm{new}}(5, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
5.34.b.a $$16$$ $$34.491$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$-232168160280$$ $$0$$ $$q+\beta _{1}q^{2}+(-52\beta _{1}+\beta _{3})q^{3}+(-4553207930+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 32293813300 T^{2} +$$$$64\!\cdots\!12$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{6} +$$$$14\!\cdots\!88$$$$T^{8} -$$$$17\!\cdots\!00$$$$T^{10} +$$$$19\!\cdots\!64$$$$T^{12} -$$$$19\!\cdots\!00$$$$T^{14} +$$$$17\!\cdots\!20$$$$T^{16} -$$$$14\!\cdots\!00$$$$T^{18} +$$$$10\!\cdots\!44$$$$T^{20} -$$$$71\!\cdots\!00$$$$T^{22} +$$$$42\!\cdots\!08$$$$T^{24} -$$$$22\!\cdots\!00$$$$T^{26} +$$$$10\!\cdots\!32$$$$T^{28} -$$$$38\!\cdots\!00$$$$T^{30} +$$$$87\!\cdots\!56$$$$T^{32}$$
$3$ $$1 - 30704906245149000 T^{2} +$$$$47\!\cdots\!32$$$$T^{4} -$$$$49\!\cdots\!00$$$$T^{6} +$$$$37\!\cdots\!48$$$$T^{8} -$$$$24\!\cdots\!00$$$$T^{10} +$$$$15\!\cdots\!84$$$$T^{12} -$$$$96\!\cdots\!00$$$$T^{14} +$$$$55\!\cdots\!70$$$$T^{16} -$$$$29\!\cdots\!00$$$$T^{18} +$$$$14\!\cdots\!44$$$$T^{20} -$$$$73\!\cdots\!00$$$$T^{22} +$$$$34\!\cdots\!88$$$$T^{24} -$$$$13\!\cdots\!00$$$$T^{26} +$$$$41\!\cdots\!72$$$$T^{28} -$$$$82\!\cdots\!00$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$
$5$ $$1 + 232168160280 T -$$$$10\!\cdots\!00$$$$T^{2} -$$$$70\!\cdots\!00$$$$T^{3} +$$$$72\!\cdots\!00$$$$T^{4} -$$$$21\!\cdots\!00$$$$T^{5} -$$$$22\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!00$$$$T^{7} +$$$$70\!\cdots\!50$$$$T^{8} +$$$$30\!\cdots\!00$$$$T^{9} -$$$$30\!\cdots\!00$$$$T^{10} -$$$$34\!\cdots\!00$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} -$$$$15\!\cdots\!00$$$$T^{13} -$$$$26\!\cdots\!00$$$$T^{14} +$$$$67\!\cdots\!00$$$$T^{15} +$$$$33\!\cdots\!25$$$$T^{16}$$
$7$ $$1 -$$$$37\!\cdots\!00$$$$T^{2} +$$$$76\!\cdots\!92$$$$T^{4} -$$$$11\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!28$$$$T^{8} -$$$$17\!\cdots\!00$$$$T^{10} +$$$$17\!\cdots\!44$$$$T^{12} -$$$$15\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!70$$$$T^{16} -$$$$94\!\cdots\!00$$$$T^{18} +$$$$63\!\cdots\!44$$$$T^{20} -$$$$37\!\cdots\!00$$$$T^{22} +$$$$19\!\cdots\!28$$$$T^{24} -$$$$90\!\cdots\!00$$$$T^{26} +$$$$34\!\cdots\!92$$$$T^{28} -$$$$10\!\cdots\!00$$$$T^{30} +$$$$16\!\cdots\!01$$$$T^{32}$$
$11$ $$( 1 - 109987295371233456 T +$$$$91\!\cdots\!20$$$$T^{2} -$$$$12\!\cdots\!60$$$$T^{3} +$$$$43\!\cdots\!20$$$$T^{4} -$$$$64\!\cdots\!68$$$$T^{5} +$$$$14\!\cdots\!28$$$$T^{6} -$$$$20\!\cdots\!40$$$$T^{7} +$$$$37\!\cdots\!70$$$$T^{8} -$$$$48\!\cdots\!40$$$$T^{9} +$$$$78\!\cdots\!08$$$$T^{10} -$$$$81\!\cdots\!88$$$$T^{11} +$$$$12\!\cdots\!20$$$$T^{12} -$$$$85\!\cdots\!60$$$$T^{13} +$$$$14\!\cdots\!20$$$$T^{14} -$$$$40\!\cdots\!16$$$$T^{15} +$$$$84\!\cdots\!41$$$$T^{16} )^{2}$$
$13$ $$1 -$$$$40\!\cdots\!00$$$$T^{2} +$$$$85\!\cdots\!72$$$$T^{4} -$$$$11\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!68$$$$T^{8} -$$$$93\!\cdots\!00$$$$T^{10} +$$$$59\!\cdots\!24$$$$T^{12} -$$$$32\!\cdots\!00$$$$T^{14} +$$$$18\!\cdots\!70$$$$T^{16} -$$$$10\!\cdots\!00$$$$T^{18} +$$$$65\!\cdots\!44$$$$T^{20} -$$$$34\!\cdots\!00$$$$T^{22} +$$$$14\!\cdots\!48$$$$T^{24} -$$$$47\!\cdots\!00$$$$T^{26} +$$$$11\!\cdots\!52$$$$T^{28} -$$$$17\!\cdots\!00$$$$T^{30} +$$$$14\!\cdots\!21$$$$T^{32}$$
$17$ $$1 -$$$$38\!\cdots\!00$$$$T^{2} +$$$$71\!\cdots\!52$$$$T^{4} -$$$$88\!\cdots\!00$$$$T^{6} +$$$$82\!\cdots\!08$$$$T^{8} -$$$$60\!\cdots\!00$$$$T^{10} +$$$$37\!\cdots\!04$$$$T^{12} -$$$$19\!\cdots\!00$$$$T^{14} +$$$$83\!\cdots\!70$$$$T^{16} -$$$$30\!\cdots\!00$$$$T^{18} +$$$$97\!\cdots\!44$$$$T^{20} -$$$$25\!\cdots\!00$$$$T^{22} +$$$$56\!\cdots\!68$$$$T^{24} -$$$$99\!\cdots\!00$$$$T^{26} +$$$$12\!\cdots\!12$$$$T^{28} -$$$$11\!\cdots\!00$$$$T^{30} +$$$$47\!\cdots\!41$$$$T^{32}$$
$19$ $$( 1 +$$$$14\!\cdots\!00$$$$T +$$$$67\!\cdots\!72$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$21\!\cdots\!68$$$$T^{4} +$$$$30\!\cdots\!00$$$$T^{5} +$$$$47\!\cdots\!24$$$$T^{6} +$$$$55\!\cdots\!00$$$$T^{7} +$$$$81\!\cdots\!70$$$$T^{8} +$$$$88\!\cdots\!00$$$$T^{9} +$$$$11\!\cdots\!44$$$$T^{10} +$$$$11\!\cdots\!00$$$$T^{11} +$$$$13\!\cdots\!48$$$$T^{12} +$$$$10\!\cdots\!00$$$$T^{13} +$$$$10\!\cdots\!52$$$$T^{14} +$$$$36\!\cdots\!00$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$23$ $$1 -$$$$64\!\cdots\!00$$$$T^{2} +$$$$21\!\cdots\!12$$$$T^{4} -$$$$49\!\cdots\!00$$$$T^{6} +$$$$87\!\cdots\!88$$$$T^{8} -$$$$12\!\cdots\!00$$$$T^{10} +$$$$15\!\cdots\!64$$$$T^{12} -$$$$16\!\cdots\!00$$$$T^{14} +$$$$15\!\cdots\!70$$$$T^{16} -$$$$12\!\cdots\!00$$$$T^{18} +$$$$86\!\cdots\!44$$$$T^{20} -$$$$52\!\cdots\!00$$$$T^{22} +$$$$27\!\cdots\!08$$$$T^{24} -$$$$11\!\cdots\!00$$$$T^{26} +$$$$37\!\cdots\!32$$$$T^{28} -$$$$84\!\cdots\!00$$$$T^{30} +$$$$98\!\cdots\!81$$$$T^{32}$$
$29$ $$( 1 -$$$$11\!\cdots\!00$$$$T +$$$$95\!\cdots\!12$$$$T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$43\!\cdots\!88$$$$T^{4} -$$$$49\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!64$$$$T^{6} -$$$$13\!\cdots\!00$$$$T^{7} +$$$$26\!\cdots\!70$$$$T^{8} -$$$$25\!\cdots\!00$$$$T^{9} +$$$$41\!\cdots\!44$$$$T^{10} -$$$$29\!\cdots\!00$$$$T^{11} +$$$$47\!\cdots\!08$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{13} +$$$$34\!\cdots\!32$$$$T^{14} -$$$$72\!\cdots\!00$$$$T^{15} +$$$$11\!\cdots\!81$$$$T^{16} )^{2}$$
$31$ $$( 1 -$$$$24\!\cdots\!36$$$$T +$$$$36\!\cdots\!20$$$$T^{2} -$$$$84\!\cdots\!60$$$$T^{3} +$$$$68\!\cdots\!20$$$$T^{4} -$$$$17\!\cdots\!68$$$$T^{5} +$$$$96\!\cdots\!68$$$$T^{6} -$$$$26\!\cdots\!40$$$$T^{7} +$$$$15\!\cdots\!70$$$$T^{8} -$$$$44\!\cdots\!40$$$$T^{9} +$$$$26\!\cdots\!08$$$$T^{10} -$$$$75\!\cdots\!28$$$$T^{11} +$$$$49\!\cdots\!20$$$$T^{12} -$$$$99\!\cdots\!60$$$$T^{13} +$$$$70\!\cdots\!20$$$$T^{14} -$$$$76\!\cdots\!16$$$$T^{15} +$$$$52\!\cdots\!21$$$$T^{16} )^{2}$$
$37$ $$1 -$$$$54\!\cdots\!00$$$$T^{2} +$$$$14\!\cdots\!72$$$$T^{4} -$$$$26\!\cdots\!00$$$$T^{6} +$$$$34\!\cdots\!68$$$$T^{8} -$$$$35\!\cdots\!00$$$$T^{10} +$$$$30\!\cdots\!24$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$13\!\cdots\!70$$$$T^{16} -$$$$68\!\cdots\!00$$$$T^{18} +$$$$30\!\cdots\!44$$$$T^{20} -$$$$11\!\cdots\!00$$$$T^{22} +$$$$34\!\cdots\!48$$$$T^{24} -$$$$83\!\cdots\!00$$$$T^{26} +$$$$14\!\cdots\!52$$$$T^{28} -$$$$17\!\cdots\!00$$$$T^{30} +$$$$10\!\cdots\!21$$$$T^{32}$$
$41$ $$( 1 -$$$$48\!\cdots\!76$$$$T +$$$$93\!\cdots\!20$$$$T^{2} -$$$$33\!\cdots\!60$$$$T^{3} +$$$$40\!\cdots\!20$$$$T^{4} -$$$$12\!\cdots\!68$$$$T^{5} +$$$$11\!\cdots\!88$$$$T^{6} -$$$$28\!\cdots\!40$$$$T^{7} +$$$$22\!\cdots\!70$$$$T^{8} -$$$$47\!\cdots\!40$$$$T^{9} +$$$$31\!\cdots\!08$$$$T^{10} -$$$$55\!\cdots\!48$$$$T^{11} +$$$$31\!\cdots\!20$$$$T^{12} -$$$$43\!\cdots\!60$$$$T^{13} +$$$$19\!\cdots\!20$$$$T^{14} -$$$$17\!\cdots\!16$$$$T^{15} +$$$$59\!\cdots\!61$$$$T^{16} )^{2}$$
$43$ $$1 -$$$$73\!\cdots\!00$$$$T^{2} +$$$$26\!\cdots\!92$$$$T^{4} -$$$$65\!\cdots\!00$$$$T^{6} +$$$$11\!\cdots\!28$$$$T^{8} -$$$$17\!\cdots\!00$$$$T^{10} +$$$$20\!\cdots\!44$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$18\!\cdots\!70$$$$T^{16} -$$$$13\!\cdots\!00$$$$T^{18} +$$$$86\!\cdots\!44$$$$T^{20} -$$$$46\!\cdots\!00$$$$T^{22} +$$$$20\!\cdots\!28$$$$T^{24} -$$$$72\!\cdots\!00$$$$T^{26} +$$$$19\!\cdots\!92$$$$T^{28} -$$$$33\!\cdots\!00$$$$T^{30} +$$$$29\!\cdots\!01$$$$T^{32}$$
$47$ $$1 -$$$$12\!\cdots\!00$$$$T^{2} +$$$$87\!\cdots\!32$$$$T^{4} -$$$$40\!\cdots\!00$$$$T^{6} +$$$$14\!\cdots\!48$$$$T^{8} -$$$$39\!\cdots\!00$$$$T^{10} +$$$$91\!\cdots\!84$$$$T^{12} -$$$$17\!\cdots\!00$$$$T^{14} +$$$$28\!\cdots\!70$$$$T^{16} -$$$$40\!\cdots\!00$$$$T^{18} +$$$$47\!\cdots\!44$$$$T^{20} -$$$$47\!\cdots\!00$$$$T^{22} +$$$$38\!\cdots\!88$$$$T^{24} -$$$$25\!\cdots\!00$$$$T^{26} +$$$$12\!\cdots\!72$$$$T^{28} -$$$$41\!\cdots\!00$$$$T^{30} +$$$$73\!\cdots\!61$$$$T^{32}$$
$53$ $$1 -$$$$95\!\cdots\!00$$$$T^{2} +$$$$44\!\cdots\!32$$$$T^{4} -$$$$13\!\cdots\!00$$$$T^{6} +$$$$29\!\cdots\!48$$$$T^{8} -$$$$49\!\cdots\!00$$$$T^{10} +$$$$65\!\cdots\!84$$$$T^{12} -$$$$70\!\cdots\!00$$$$T^{14} +$$$$61\!\cdots\!70$$$$T^{16} -$$$$44\!\cdots\!00$$$$T^{18} +$$$$26\!\cdots\!44$$$$T^{20} -$$$$12\!\cdots\!00$$$$T^{22} +$$$$47\!\cdots\!88$$$$T^{24} -$$$$13\!\cdots\!00$$$$T^{26} +$$$$28\!\cdots\!72$$$$T^{28} -$$$$39\!\cdots\!00$$$$T^{30} +$$$$26\!\cdots\!61$$$$T^{32}$$
$59$ $$( 1 +$$$$10\!\cdots\!00$$$$T +$$$$14\!\cdots\!32$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!48$$$$T^{4} +$$$$49\!\cdots\!00$$$$T^{5} +$$$$45\!\cdots\!84$$$$T^{6} +$$$$14\!\cdots\!00$$$$T^{7} +$$$$14\!\cdots\!70$$$$T^{8} +$$$$40\!\cdots\!00$$$$T^{9} +$$$$34\!\cdots\!44$$$$T^{10} +$$$$10\!\cdots\!00$$$$T^{11} +$$$$58\!\cdots\!88$$$$T^{12} +$$$$16\!\cdots\!00$$$$T^{13} +$$$$63\!\cdots\!72$$$$T^{14} +$$$$11\!\cdots\!00$$$$T^{15} +$$$$31\!\cdots\!61$$$$T^{16} )^{2}$$
$61$ $$( 1 +$$$$31\!\cdots\!44$$$$T +$$$$43\!\cdots\!20$$$$T^{2} +$$$$22\!\cdots\!40$$$$T^{3} +$$$$94\!\cdots\!20$$$$T^{4} +$$$$56\!\cdots\!32$$$$T^{5} +$$$$13\!\cdots\!28$$$$T^{6} +$$$$76\!\cdots\!60$$$$T^{7} +$$$$12\!\cdots\!70$$$$T^{8} +$$$$62\!\cdots\!60$$$$T^{9} +$$$$89\!\cdots\!08$$$$T^{10} +$$$$31\!\cdots\!12$$$$T^{11} +$$$$43\!\cdots\!20$$$$T^{12} +$$$$86\!\cdots\!40$$$$T^{13} +$$$$13\!\cdots\!20$$$$T^{14} +$$$$81\!\cdots\!84$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$67$ $$1 -$$$$12\!\cdots\!00$$$$T^{2} +$$$$78\!\cdots\!52$$$$T^{4} -$$$$32\!\cdots\!00$$$$T^{6} +$$$$10\!\cdots\!08$$$$T^{8} -$$$$26\!\cdots\!00$$$$T^{10} +$$$$62\!\cdots\!04$$$$T^{12} -$$$$13\!\cdots\!00$$$$T^{14} +$$$$26\!\cdots\!70$$$$T^{16} -$$$$44\!\cdots\!00$$$$T^{18} +$$$$69\!\cdots\!44$$$$T^{20} -$$$$96\!\cdots\!00$$$$T^{22} +$$$$12\!\cdots\!68$$$$T^{24} -$$$$12\!\cdots\!00$$$$T^{26} +$$$$10\!\cdots\!12$$$$T^{28} -$$$$56\!\cdots\!00$$$$T^{30} +$$$$14\!\cdots\!41$$$$T^{32}$$
$71$ $$( 1 +$$$$43\!\cdots\!04$$$$T +$$$$59\!\cdots\!20$$$$T^{2} -$$$$12\!\cdots\!60$$$$T^{3} +$$$$18\!\cdots\!20$$$$T^{4} -$$$$57\!\cdots\!68$$$$T^{5} +$$$$37\!\cdots\!48$$$$T^{6} -$$$$12\!\cdots\!40$$$$T^{7} +$$$$54\!\cdots\!70$$$$T^{8} -$$$$15\!\cdots\!40$$$$T^{9} +$$$$56\!\cdots\!08$$$$T^{10} -$$$$10\!\cdots\!08$$$$T^{11} +$$$$42\!\cdots\!20$$$$T^{12} -$$$$35\!\cdots\!60$$$$T^{13} +$$$$20\!\cdots\!20$$$$T^{14} +$$$$19\!\cdots\!84$$$$T^{15} +$$$$53\!\cdots\!81$$$$T^{16} )^{2}$$
$73$ $$1 -$$$$33\!\cdots\!00$$$$T^{2} +$$$$54\!\cdots\!12$$$$T^{4} -$$$$56\!\cdots\!00$$$$T^{6} +$$$$42\!\cdots\!88$$$$T^{8} -$$$$25\!\cdots\!00$$$$T^{10} +$$$$11\!\cdots\!64$$$$T^{12} -$$$$47\!\cdots\!00$$$$T^{14} +$$$$15\!\cdots\!70$$$$T^{16} -$$$$44\!\cdots\!00$$$$T^{18} +$$$$10\!\cdots\!44$$$$T^{20} -$$$$21\!\cdots\!00$$$$T^{22} +$$$$35\!\cdots\!08$$$$T^{24} -$$$$44\!\cdots\!00$$$$T^{26} +$$$$40\!\cdots\!32$$$$T^{28} -$$$$24\!\cdots\!00$$$$T^{30} +$$$$68\!\cdots\!81$$$$T^{32}$$
$79$ $$( 1 +$$$$52\!\cdots\!00$$$$T +$$$$16\!\cdots\!12$$$$T^{2} +$$$$93\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!88$$$$T^{4} -$$$$12\!\cdots\!00$$$$T^{5} +$$$$89\!\cdots\!64$$$$T^{6} -$$$$12\!\cdots\!00$$$$T^{7} +$$$$42\!\cdots\!70$$$$T^{8} -$$$$51\!\cdots\!00$$$$T^{9} +$$$$15\!\cdots\!44$$$$T^{10} -$$$$90\!\cdots\!00$$$$T^{11} +$$$$43\!\cdots\!08$$$$T^{12} +$$$$12\!\cdots\!00$$$$T^{13} +$$$$86\!\cdots\!32$$$$T^{14} +$$$$11\!\cdots\!00$$$$T^{15} +$$$$94\!\cdots\!81$$$$T^{16} )^{2}$$
$83$ $$1 -$$$$23\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!52$$$$T^{4} -$$$$17\!\cdots\!00$$$$T^{6} +$$$$86\!\cdots\!08$$$$T^{8} -$$$$33\!\cdots\!00$$$$T^{10} +$$$$10\!\cdots\!04$$$$T^{12} -$$$$27\!\cdots\!00$$$$T^{14} +$$$$63\!\cdots\!70$$$$T^{16} -$$$$12\!\cdots\!00$$$$T^{18} +$$$$21\!\cdots\!44$$$$T^{20} -$$$$31\!\cdots\!00$$$$T^{22} +$$$$37\!\cdots\!68$$$$T^{24} -$$$$34\!\cdots\!00$$$$T^{26} +$$$$22\!\cdots\!12$$$$T^{28} -$$$$94\!\cdots\!00$$$$T^{30} +$$$$18\!\cdots\!41$$$$T^{32}$$
$89$ $$( 1 +$$$$40\!\cdots\!00$$$$T +$$$$20\!\cdots\!52$$$$T^{2} +$$$$55\!\cdots\!00$$$$T^{3} +$$$$16\!\cdots\!08$$$$T^{4} +$$$$32\!\cdots\!00$$$$T^{5} +$$$$69\!\cdots\!04$$$$T^{6} +$$$$11\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!70$$$$T^{8} +$$$$23\!\cdots\!00$$$$T^{9} +$$$$31\!\cdots\!44$$$$T^{10} +$$$$31\!\cdots\!00$$$$T^{11} +$$$$33\!\cdots\!68$$$$T^{12} +$$$$24\!\cdots\!00$$$$T^{13} +$$$$19\!\cdots\!12$$$$T^{14} +$$$$83\!\cdots\!00$$$$T^{15} +$$$$43\!\cdots\!41$$$$T^{16} )^{2}$$
$97$ $$1 -$$$$30\!\cdots\!00$$$$T^{2} +$$$$48\!\cdots\!32$$$$T^{4} -$$$$52\!\cdots\!00$$$$T^{6} +$$$$43\!\cdots\!48$$$$T^{8} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$16\!\cdots\!84$$$$T^{12} -$$$$75\!\cdots\!00$$$$T^{14} +$$$$30\!\cdots\!70$$$$T^{16} -$$$$10\!\cdots\!00$$$$T^{18} +$$$$29\!\cdots\!44$$$$T^{20} -$$$$70\!\cdots\!00$$$$T^{22} +$$$$14\!\cdots\!88$$$$T^{24} -$$$$22\!\cdots\!00$$$$T^{26} +$$$$27\!\cdots\!72$$$$T^{28} -$$$$23\!\cdots\!00$$$$T^{30} +$$$$10\!\cdots\!61$$$$T^{32}$$