Properties

 Label 128.5.f Level 128 Weight 5 Character orbit f Rep. character $$\chi_{128}(31,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 28 Newform subspaces 2 Sturm bound 80 Trace bound 3

Related objects

Defining parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 128.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$80$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(128, [\chi])$$.

Total New Old
Modular forms 144 36 108
Cusp forms 112 28 84
Eisenstein series 32 8 24

Trace form

 $$28q + 4q^{5} + O(q^{10})$$ $$28q + 4q^{5} + 4q^{13} - 8q^{17} + 328q^{21} - 1724q^{29} - 8q^{33} + 3652q^{37} - 2820q^{45} + 1364q^{49} + 964q^{53} + 7556q^{61} - 4040q^{65} - 19256q^{69} + 19016q^{77} + 2908q^{81} - 19896q^{85} - 17792q^{93} - 8q^{97} + O(q^{100})$$

Decomposition of $$S_{5}^{\mathrm{new}}(128, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
128.5.f.a $$14$$ $$13.231$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$-2$$ $$2$$ $$4$$ $$q+\beta _{5}q^{3}+\beta _{8}q^{5}+\beta _{9}q^{7}+(-19\beta _{1}+\cdots)q^{9}+\cdots$$
128.5.f.b $$14$$ $$13.231$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$2$$ $$2$$ $$-4$$ $$q+\beta _{2}q^{3}-\beta _{3}q^{5}-\beta _{9}q^{7}+(19\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots$$

Decomposition of $$S_{5}^{\mathrm{old}}(128, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(128, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 2 T + 2 T^{2} + 610 T^{3} - 4613 T^{4} + 31732 T^{5} + 258740 T^{6} + 2114868 T^{7} + 70347753 T^{8} + 127844190 T^{9} + 3102089886 T^{10} + 22201539966 T^{11} + 75486949083 T^{12} + 2442289052376 T^{13} + 20677791784536 T^{14} + 197825413242456 T^{15} + 495269872933563 T^{16} + 11798808601071006 T^{17} + 133534797839563806 T^{18} + 445765127450480190 T^{19} + 19868283272269877193 T^{20} + 48381396305638460148 T^{21} +$$$$47\!\cdots\!40$$$$T^{22} +$$$$47\!\cdots\!72$$$$T^{23} -$$$$56\!\cdots\!13$$$$T^{24} +$$$$60\!\cdots\!10$$$$T^{25} +$$$$15\!\cdots\!22$$$$T^{26} +$$$$12\!\cdots\!82$$$$T^{27} +$$$$52\!\cdots\!21$$$$T^{28}$$)($$1 - 2 T + 2 T^{2} - 610 T^{3} - 4613 T^{4} - 31732 T^{5} + 258740 T^{6} - 2114868 T^{7} + 70347753 T^{8} - 127844190 T^{9} + 3102089886 T^{10} - 22201539966 T^{11} + 75486949083 T^{12} - 2442289052376 T^{13} + 20677791784536 T^{14} - 197825413242456 T^{15} + 495269872933563 T^{16} - 11798808601071006 T^{17} + 133534797839563806 T^{18} - 445765127450480190 T^{19} + 19868283272269877193 T^{20} - 48381396305638460148 T^{21} +$$$$47\!\cdots\!40$$$$T^{22} -$$$$47\!\cdots\!72$$$$T^{23} -$$$$56\!\cdots\!13$$$$T^{24} -$$$$60\!\cdots\!10$$$$T^{25} +$$$$15\!\cdots\!22$$$$T^{26} -$$$$12\!\cdots\!82$$$$T^{27} +$$$$52\!\cdots\!21$$$$T^{28}$$)
$5$ ($$1 - 2 T + 2 T^{2} - 3938 T^{3} - 94565 T^{4} + 9371916 T^{5} - 10800780 T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 70375486014886875 T^{12} - 1339233603531363800 T^{13} + 14585682673141755608 T^{14} -$$$$83\!\cdots\!00$$$$T^{15} +$$$$27\!\cdots\!75$$$$T^{16} +$$$$28\!\cdots\!50$$$$T^{17} +$$$$27\!\cdots\!50$$$$T^{18} -$$$$53\!\cdots\!50$$$$T^{19} -$$$$90\!\cdots\!75$$$$T^{20} +$$$$24\!\cdots\!00$$$$T^{21} -$$$$25\!\cdots\!00$$$$T^{22} +$$$$13\!\cdots\!00$$$$T^{23} -$$$$86\!\cdots\!25$$$$T^{24} -$$$$22\!\cdots\!50$$$$T^{25} +$$$$71\!\cdots\!50$$$$T^{26} -$$$$44\!\cdots\!50$$$$T^{27} +$$$$13\!\cdots\!25$$$$T^{28}$$)($$1 - 2 T + 2 T^{2} - 3938 T^{3} - 94565 T^{4} + 9371916 T^{5} - 10800780 T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 70375486014886875 T^{12} - 1339233603531363800 T^{13} + 14585682673141755608 T^{14} -$$$$83\!\cdots\!00$$$$T^{15} +$$$$27\!\cdots\!75$$$$T^{16} +$$$$28\!\cdots\!50$$$$T^{17} +$$$$27\!\cdots\!50$$$$T^{18} -$$$$53\!\cdots\!50$$$$T^{19} -$$$$90\!\cdots\!75$$$$T^{20} +$$$$24\!\cdots\!00$$$$T^{21} -$$$$25\!\cdots\!00$$$$T^{22} +$$$$13\!\cdots\!00$$$$T^{23} -$$$$86\!\cdots\!25$$$$T^{24} -$$$$22\!\cdots\!50$$$$T^{25} +$$$$71\!\cdots\!50$$$$T^{26} -$$$$44\!\cdots\!50$$$$T^{27} +$$$$13\!\cdots\!25$$$$T^{28}$$)
$7$ ($$( 1 - 2 T + 8235 T^{2} - 64404 T^{3} + 38860249 T^{4} - 447351454 T^{5} + 125587217723 T^{6} - 1378109878936 T^{7} + 301534909752923 T^{8} - 2578892109370654 T^{9} + 537875867111373049 T^{10} - 2140333660404582804 T^{11} +$$$$65\!\cdots\!35$$$$T^{12} -$$$$38\!\cdots\!02$$$$T^{13} +$$$$45\!\cdots\!01$$$$T^{14} )^{2}$$)($$( 1 + 2 T + 8235 T^{2} + 64404 T^{3} + 38860249 T^{4} + 447351454 T^{5} + 125587217723 T^{6} + 1378109878936 T^{7} + 301534909752923 T^{8} + 2578892109370654 T^{9} + 537875867111373049 T^{10} + 2140333660404582804 T^{11} +$$$$65\!\cdots\!35$$$$T^{12} +$$$$38\!\cdots\!02$$$$T^{13} +$$$$45\!\cdots\!01$$$$T^{14} )^{2}$$)
$11$ ($$1 - 94 T + 4418 T^{2} + 965570 T^{3} - 470088133 T^{4} + 3120961844 T^{5} + 2249641670708 T^{6} - 796542815073292 T^{7} + 97369777194709097 T^{8} + 5317394377195027646 T^{9} -$$$$63\!\cdots\!78$$$$T^{10} +$$$$14\!\cdots\!86$$$$T^{11} -$$$$23\!\cdots\!21$$$$T^{12} -$$$$22\!\cdots\!52$$$$T^{13} +$$$$18\!\cdots\!92$$$$T^{14} -$$$$33\!\cdots\!32$$$$T^{15} -$$$$50\!\cdots\!01$$$$T^{16} +$$$$45\!\cdots\!06$$$$T^{17} -$$$$29\!\cdots\!58$$$$T^{18} +$$$$35\!\cdots\!46$$$$T^{19} +$$$$95\!\cdots\!77$$$$T^{20} -$$$$11\!\cdots\!52$$$$T^{21} +$$$$47\!\cdots\!68$$$$T^{22} +$$$$96\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!33$$$$T^{24} +$$$$63\!\cdots\!70$$$$T^{25} +$$$$42\!\cdots\!58$$$$T^{26} -$$$$13\!\cdots\!74$$$$T^{27} +$$$$20\!\cdots\!61$$$$T^{28}$$)($$1 + 94 T + 4418 T^{2} - 965570 T^{3} - 470088133 T^{4} - 3120961844 T^{5} + 2249641670708 T^{6} + 796542815073292 T^{7} + 97369777194709097 T^{8} - 5317394377195027646 T^{9} -$$$$63\!\cdots\!78$$$$T^{10} -$$$$14\!\cdots\!86$$$$T^{11} -$$$$23\!\cdots\!21$$$$T^{12} +$$$$22\!\cdots\!52$$$$T^{13} +$$$$18\!\cdots\!92$$$$T^{14} +$$$$33\!\cdots\!32$$$$T^{15} -$$$$50\!\cdots\!01$$$$T^{16} -$$$$45\!\cdots\!06$$$$T^{17} -$$$$29\!\cdots\!58$$$$T^{18} -$$$$35\!\cdots\!46$$$$T^{19} +$$$$95\!\cdots\!77$$$$T^{20} +$$$$11\!\cdots\!52$$$$T^{21} +$$$$47\!\cdots\!68$$$$T^{22} -$$$$96\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!33$$$$T^{24} -$$$$63\!\cdots\!70$$$$T^{25} +$$$$42\!\cdots\!58$$$$T^{26} +$$$$13\!\cdots\!74$$$$T^{27} +$$$$20\!\cdots\!61$$$$T^{28}$$)
$13$ ($$1 - 2 T + 2 T^{2} - 6883234 T^{3} + 1464853339 T^{4} - 65707775476 T^{5} + 23817940993652 T^{6} - 16199073445624116 T^{7} + 1142495439970904649 T^{8} -$$$$10\!\cdots\!70$$$$T^{9} +$$$$78\!\cdots\!46$$$$T^{10} -$$$$14\!\cdots\!90$$$$T^{11} +$$$$74\!\cdots\!35$$$$T^{12} -$$$$22\!\cdots\!32$$$$T^{13} +$$$$94\!\cdots\!60$$$$T^{14} -$$$$65\!\cdots\!52$$$$T^{15} +$$$$60\!\cdots\!35$$$$T^{16} -$$$$33\!\cdots\!90$$$$T^{17} +$$$$52\!\cdots\!86$$$$T^{18} -$$$$19\!\cdots\!70$$$$T^{19} +$$$$62\!\cdots\!89$$$$T^{20} -$$$$25\!\cdots\!36$$$$T^{21} +$$$$10\!\cdots\!12$$$$T^{22} -$$$$83\!\cdots\!16$$$$T^{23} +$$$$52\!\cdots\!39$$$$T^{24} -$$$$71\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} -$$$$16\!\cdots\!62$$$$T^{27} +$$$$24\!\cdots\!41$$$$T^{28}$$)($$1 - 2 T + 2 T^{2} - 6883234 T^{3} + 1464853339 T^{4} - 65707775476 T^{5} + 23817940993652 T^{6} - 16199073445624116 T^{7} + 1142495439970904649 T^{8} -$$$$10\!\cdots\!70$$$$T^{9} +$$$$78\!\cdots\!46$$$$T^{10} -$$$$14\!\cdots\!90$$$$T^{11} +$$$$74\!\cdots\!35$$$$T^{12} -$$$$22\!\cdots\!32$$$$T^{13} +$$$$94\!\cdots\!60$$$$T^{14} -$$$$65\!\cdots\!52$$$$T^{15} +$$$$60\!\cdots\!35$$$$T^{16} -$$$$33\!\cdots\!90$$$$T^{17} +$$$$52\!\cdots\!86$$$$T^{18} -$$$$19\!\cdots\!70$$$$T^{19} +$$$$62\!\cdots\!89$$$$T^{20} -$$$$25\!\cdots\!36$$$$T^{21} +$$$$10\!\cdots\!12$$$$T^{22} -$$$$83\!\cdots\!16$$$$T^{23} +$$$$52\!\cdots\!39$$$$T^{24} -$$$$71\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} -$$$$16\!\cdots\!62$$$$T^{27} +$$$$24\!\cdots\!41$$$$T^{28}$$)
$17$ ($$( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} +$$$$53\!\cdots\!11$$$$T^{8} -$$$$20\!\cdots\!62$$$$T^{9} +$$$$32\!\cdots\!45$$$$T^{10} -$$$$62\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!55$$$$T^{12} +$$$$67\!\cdots\!42$$$$T^{13} +$$$$28\!\cdots\!41$$$$T^{14} )^{2}$$)($$( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} +$$$$53\!\cdots\!11$$$$T^{8} -$$$$20\!\cdots\!62$$$$T^{9} +$$$$32\!\cdots\!45$$$$T^{10} -$$$$62\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!55$$$$T^{12} +$$$$67\!\cdots\!42$$$$T^{13} +$$$$28\!\cdots\!41$$$$T^{14} )^{2}$$)
$19$ ($$1 + 706 T + 249218 T^{2} + 101381538 T^{3} + 32599242619 T^{4} + 3617133340788 T^{5} - 431513784802892 T^{6} - 960761925731564364 T^{7} -$$$$71\!\cdots\!35$$$$T^{8} -$$$$22\!\cdots\!14$$$$T^{9} -$$$$57\!\cdots\!66$$$$T^{10} -$$$$18\!\cdots\!22$$$$T^{11} -$$$$19\!\cdots\!61$$$$T^{12} +$$$$38\!\cdots\!56$$$$T^{13} +$$$$11\!\cdots\!48$$$$T^{14} +$$$$50\!\cdots\!76$$$$T^{15} -$$$$32\!\cdots\!01$$$$T^{16} -$$$$40\!\cdots\!42$$$$T^{17} -$$$$16\!\cdots\!46$$$$T^{18} -$$$$84\!\cdots\!14$$$$T^{19} -$$$$35\!\cdots\!35$$$$T^{20} -$$$$61\!\cdots\!24$$$$T^{21} -$$$$35\!\cdots\!12$$$$T^{22} +$$$$39\!\cdots\!28$$$$T^{23} +$$$$46\!\cdots\!19$$$$T^{24} +$$$$18\!\cdots\!98$$$$T^{25} +$$$$59\!\cdots\!38$$$$T^{26} +$$$$22\!\cdots\!66$$$$T^{27} +$$$$40\!\cdots\!81$$$$T^{28}$$)($$1 - 706 T + 249218 T^{2} - 101381538 T^{3} + 32599242619 T^{4} - 3617133340788 T^{5} - 431513784802892 T^{6} + 960761925731564364 T^{7} -$$$$71\!\cdots\!35$$$$T^{8} +$$$$22\!\cdots\!14$$$$T^{9} -$$$$57\!\cdots\!66$$$$T^{10} +$$$$18\!\cdots\!22$$$$T^{11} -$$$$19\!\cdots\!61$$$$T^{12} -$$$$38\!\cdots\!56$$$$T^{13} +$$$$11\!\cdots\!48$$$$T^{14} -$$$$50\!\cdots\!76$$$$T^{15} -$$$$32\!\cdots\!01$$$$T^{16} +$$$$40\!\cdots\!42$$$$T^{17} -$$$$16\!\cdots\!46$$$$T^{18} +$$$$84\!\cdots\!14$$$$T^{19} -$$$$35\!\cdots\!35$$$$T^{20} +$$$$61\!\cdots\!24$$$$T^{21} -$$$$35\!\cdots\!12$$$$T^{22} -$$$$39\!\cdots\!28$$$$T^{23} +$$$$46\!\cdots\!19$$$$T^{24} -$$$$18\!\cdots\!98$$$$T^{25} +$$$$59\!\cdots\!38$$$$T^{26} -$$$$22\!\cdots\!66$$$$T^{27} +$$$$40\!\cdots\!81$$$$T^{28}$$)
$23$ ($$( 1 + 574 T + 1024043 T^{2} + 635922028 T^{3} + 551773439769 T^{4} + 314763003369506 T^{5} + 213700110666561659 T^{6} +$$$$10\!\cdots\!08$$$$T^{7} +$$$$59\!\cdots\!19$$$$T^{8} +$$$$24\!\cdots\!86$$$$T^{9} +$$$$12\!\cdots\!49$$$$T^{10} +$$$$38\!\cdots\!08$$$$T^{11} +$$$$17\!\cdots\!43$$$$T^{12} +$$$$27\!\cdots\!34$$$$T^{13} +$$$$13\!\cdots\!81$$$$T^{14} )^{2}$$)($$( 1 - 574 T + 1024043 T^{2} - 635922028 T^{3} + 551773439769 T^{4} - 314763003369506 T^{5} + 213700110666561659 T^{6} -$$$$10\!\cdots\!08$$$$T^{7} +$$$$59\!\cdots\!19$$$$T^{8} -$$$$24\!\cdots\!86$$$$T^{9} +$$$$12\!\cdots\!49$$$$T^{10} -$$$$38\!\cdots\!08$$$$T^{11} +$$$$17\!\cdots\!43$$$$T^{12} -$$$$27\!\cdots\!34$$$$T^{13} +$$$$13\!\cdots\!81$$$$T^{14} )^{2}$$)
$29$ ($$1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} +$$$$98\!\cdots\!28$$$$T^{7} +$$$$75\!\cdots\!49$$$$T^{8} +$$$$31\!\cdots\!74$$$$T^{9} +$$$$39\!\cdots\!98$$$$T^{10} +$$$$45\!\cdots\!94$$$$T^{11} +$$$$34\!\cdots\!87$$$$T^{12} +$$$$29\!\cdots\!64$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} +$$$$20\!\cdots\!84$$$$T^{15} +$$$$17\!\cdots\!07$$$$T^{16} +$$$$15\!\cdots\!54$$$$T^{17} +$$$$98\!\cdots\!58$$$$T^{18} +$$$$54\!\cdots\!74$$$$T^{19} +$$$$94\!\cdots\!69$$$$T^{20} +$$$$87\!\cdots\!08$$$$T^{21} +$$$$31\!\cdots\!56$$$$T^{22} +$$$$30\!\cdots\!16$$$$T^{23} +$$$$53\!\cdots\!27$$$$T^{24} +$$$$23\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} +$$$$95\!\cdots\!42$$$$T^{27} +$$$$78\!\cdots\!21$$$$T^{28}$$)($$1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} +$$$$98\!\cdots\!28$$$$T^{7} +$$$$75\!\cdots\!49$$$$T^{8} +$$$$31\!\cdots\!74$$$$T^{9} +$$$$39\!\cdots\!98$$$$T^{10} +$$$$45\!\cdots\!94$$$$T^{11} +$$$$34\!\cdots\!87$$$$T^{12} +$$$$29\!\cdots\!64$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} +$$$$20\!\cdots\!84$$$$T^{15} +$$$$17\!\cdots\!07$$$$T^{16} +$$$$15\!\cdots\!54$$$$T^{17} +$$$$98\!\cdots\!58$$$$T^{18} +$$$$54\!\cdots\!74$$$$T^{19} +$$$$94\!\cdots\!69$$$$T^{20} +$$$$87\!\cdots\!08$$$$T^{21} +$$$$31\!\cdots\!56$$$$T^{22} +$$$$30\!\cdots\!16$$$$T^{23} +$$$$53\!\cdots\!27$$$$T^{24} +$$$$23\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} +$$$$95\!\cdots\!42$$$$T^{27} +$$$$78\!\cdots\!21$$$$T^{28}$$)
$31$ ($$1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} +$$$$10\!\cdots\!69$$$$T^{8} -$$$$14\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!83$$$$T^{12} -$$$$17\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!03$$$$T^{16} -$$$$10\!\cdots\!50$$$$T^{18} +$$$$64\!\cdots\!49$$$$T^{20} -$$$$30\!\cdots\!92$$$$T^{22} +$$$$10\!\cdots\!11$$$$T^{24} -$$$$26\!\cdots\!94$$$$T^{26} +$$$$32\!\cdots\!81$$$$T^{28}$$)($$1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} +$$$$10\!\cdots\!69$$$$T^{8} -$$$$14\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!83$$$$T^{12} -$$$$17\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!03$$$$T^{16} -$$$$10\!\cdots\!50$$$$T^{18} +$$$$64\!\cdots\!49$$$$T^{20} -$$$$30\!\cdots\!92$$$$T^{22} +$$$$10\!\cdots\!11$$$$T^{24} -$$$$26\!\cdots\!94$$$$T^{26} +$$$$32\!\cdots\!81$$$$T^{28}$$)
$37$ ($$1 - 1826 T + 1667138 T^{2} - 4976934274 T^{3} + 7030206539163 T^{4} - 1716272691299380 T^{5} + 3798526848883495924 T^{6} -$$$$11\!\cdots\!60$$$$T^{7} -$$$$10\!\cdots\!63$$$$T^{8} +$$$$16\!\cdots\!22$$$$T^{9} +$$$$18\!\cdots\!50$$$$T^{10} +$$$$21\!\cdots\!18$$$$T^{11} -$$$$10\!\cdots\!97$$$$T^{12} +$$$$14\!\cdots\!60$$$$T^{13} -$$$$67\!\cdots\!04$$$$T^{14} +$$$$26\!\cdots\!60$$$$T^{15} -$$$$37\!\cdots\!37$$$$T^{16} +$$$$14\!\cdots\!58$$$$T^{17} +$$$$22\!\cdots\!50$$$$T^{18} +$$$$38\!\cdots\!22$$$$T^{19} -$$$$44\!\cdots\!43$$$$T^{20} -$$$$91\!\cdots\!60$$$$T^{21} +$$$$57\!\cdots\!44$$$$T^{22} -$$$$48\!\cdots\!80$$$$T^{23} +$$$$37\!\cdots\!63$$$$T^{24} -$$$$49\!\cdots\!14$$$$T^{25} +$$$$31\!\cdots\!98$$$$T^{26} -$$$$64\!\cdots\!06$$$$T^{27} +$$$$65\!\cdots\!41$$$$T^{28}$$)($$1 - 1826 T + 1667138 T^{2} - 4976934274 T^{3} + 7030206539163 T^{4} - 1716272691299380 T^{5} + 3798526848883495924 T^{6} -$$$$11\!\cdots\!60$$$$T^{7} -$$$$10\!\cdots\!63$$$$T^{8} +$$$$16\!\cdots\!22$$$$T^{9} +$$$$18\!\cdots\!50$$$$T^{10} +$$$$21\!\cdots\!18$$$$T^{11} -$$$$10\!\cdots\!97$$$$T^{12} +$$$$14\!\cdots\!60$$$$T^{13} -$$$$67\!\cdots\!04$$$$T^{14} +$$$$26\!\cdots\!60$$$$T^{15} -$$$$37\!\cdots\!37$$$$T^{16} +$$$$14\!\cdots\!58$$$$T^{17} +$$$$22\!\cdots\!50$$$$T^{18} +$$$$38\!\cdots\!22$$$$T^{19} -$$$$44\!\cdots\!43$$$$T^{20} -$$$$91\!\cdots\!60$$$$T^{21} +$$$$57\!\cdots\!44$$$$T^{22} -$$$$48\!\cdots\!80$$$$T^{23} +$$$$37\!\cdots\!63$$$$T^{24} -$$$$49\!\cdots\!14$$$$T^{25} +$$$$31\!\cdots\!98$$$$T^{26} -$$$$64\!\cdots\!06$$$$T^{27} +$$$$65\!\cdots\!41$$$$T^{28}$$)
$41$ ($$1 - 24523982 T^{2} + 302442312166171 T^{4} -$$$$24\!\cdots\!76$$$$T^{6} +$$$$14\!\cdots\!01$$$$T^{8} -$$$$70\!\cdots\!70$$$$T^{10} +$$$$27\!\cdots\!51$$$$T^{12} -$$$$84\!\cdots\!88$$$$T^{14} +$$$$21\!\cdots\!71$$$$T^{16} -$$$$45\!\cdots\!70$$$$T^{18} +$$$$76\!\cdots\!61$$$$T^{20} -$$$$10\!\cdots\!56$$$$T^{22} +$$$$98\!\cdots\!71$$$$T^{24} -$$$$63\!\cdots\!22$$$$T^{26} +$$$$20\!\cdots\!41$$$$T^{28}$$)($$1 - 24523982 T^{2} + 302442312166171 T^{4} -$$$$24\!\cdots\!76$$$$T^{6} +$$$$14\!\cdots\!01$$$$T^{8} -$$$$70\!\cdots\!70$$$$T^{10} +$$$$27\!\cdots\!51$$$$T^{12} -$$$$84\!\cdots\!88$$$$T^{14} +$$$$21\!\cdots\!71$$$$T^{16} -$$$$45\!\cdots\!70$$$$T^{18} +$$$$76\!\cdots\!61$$$$T^{20} -$$$$10\!\cdots\!56$$$$T^{22} +$$$$98\!\cdots\!71$$$$T^{24} -$$$$63\!\cdots\!22$$$$T^{26} +$$$$20\!\cdots\!41$$$$T^{28}$$)
$43$ ($$1 - 1694 T + 1434818 T^{2} - 14278395262 T^{3} + 44454402050619 T^{4} - 13474016894363980 T^{5} + 60977294006539554100 T^{6} -$$$$38\!\cdots\!44$$$$T^{7} +$$$$23\!\cdots\!81$$$$T^{8} +$$$$78\!\cdots\!82$$$$T^{9} +$$$$12\!\cdots\!62$$$$T^{10} -$$$$18\!\cdots\!22$$$$T^{11} -$$$$91\!\cdots\!45$$$$T^{12} +$$$$10\!\cdots\!08$$$$T^{13} +$$$$16\!\cdots\!52$$$$T^{14} +$$$$36\!\cdots\!08$$$$T^{15} -$$$$10\!\cdots\!45$$$$T^{16} -$$$$73\!\cdots\!22$$$$T^{17} +$$$$16\!\cdots\!62$$$$T^{18} +$$$$36\!\cdots\!82$$$$T^{19} +$$$$37\!\cdots\!81$$$$T^{20} -$$$$21\!\cdots\!44$$$$T^{21} +$$$$11\!\cdots\!00$$$$T^{22} -$$$$85\!\cdots\!80$$$$T^{23} +$$$$96\!\cdots\!19$$$$T^{24} -$$$$10\!\cdots\!62$$$$T^{25} +$$$$36\!\cdots\!18$$$$T^{26} -$$$$14\!\cdots\!94$$$$T^{27} +$$$$29\!\cdots\!01$$$$T^{28}$$)($$1 + 1694 T + 1434818 T^{2} + 14278395262 T^{3} + 44454402050619 T^{4} + 13474016894363980 T^{5} + 60977294006539554100 T^{6} +$$$$38\!\cdots\!44$$$$T^{7} +$$$$23\!\cdots\!81$$$$T^{8} -$$$$78\!\cdots\!82$$$$T^{9} +$$$$12\!\cdots\!62$$$$T^{10} +$$$$18\!\cdots\!22$$$$T^{11} -$$$$91\!\cdots\!45$$$$T^{12} -$$$$10\!\cdots\!08$$$$T^{13} +$$$$16\!\cdots\!52$$$$T^{14} -$$$$36\!\cdots\!08$$$$T^{15} -$$$$10\!\cdots\!45$$$$T^{16} +$$$$73\!\cdots\!22$$$$T^{17} +$$$$16\!\cdots\!62$$$$T^{18} -$$$$36\!\cdots\!82$$$$T^{19} +$$$$37\!\cdots\!81$$$$T^{20} +$$$$21\!\cdots\!44$$$$T^{21} +$$$$11\!\cdots\!00$$$$T^{22} +$$$$85\!\cdots\!80$$$$T^{23} +$$$$96\!\cdots\!19$$$$T^{24} +$$$$10\!\cdots\!62$$$$T^{25} +$$$$36\!\cdots\!18$$$$T^{26} +$$$$14\!\cdots\!94$$$$T^{27} +$$$$29\!\cdots\!01$$$$T^{28}$$)
$47$ ($$1 - 51887758 T^{2} + 1309844227745755 T^{4} -$$$$21\!\cdots\!56$$$$T^{6} +$$$$24\!\cdots\!09$$$$T^{8} -$$$$21\!\cdots\!30$$$$T^{10} +$$$$15\!\cdots\!39$$$$T^{12} -$$$$82\!\cdots\!04$$$$T^{14} +$$$$35\!\cdots\!79$$$$T^{16} -$$$$12\!\cdots\!30$$$$T^{18} +$$$$33\!\cdots\!29$$$$T^{20} -$$$$68\!\cdots\!96$$$$T^{22} +$$$$10\!\cdots\!55$$$$T^{24} -$$$$94\!\cdots\!38$$$$T^{26} +$$$$43\!\cdots\!21$$$$T^{28}$$)($$1 - 51887758 T^{2} + 1309844227745755 T^{4} -$$$$21\!\cdots\!56$$$$T^{6} +$$$$24\!\cdots\!09$$$$T^{8} -$$$$21\!\cdots\!30$$$$T^{10} +$$$$15\!\cdots\!39$$$$T^{12} -$$$$82\!\cdots\!04$$$$T^{14} +$$$$35\!\cdots\!79$$$$T^{16} -$$$$12\!\cdots\!30$$$$T^{18} +$$$$33\!\cdots\!29$$$$T^{20} -$$$$68\!\cdots\!96$$$$T^{22} +$$$$10\!\cdots\!55$$$$T^{24} -$$$$94\!\cdots\!38$$$$T^{26} +$$$$43\!\cdots\!21$$$$T^{28}$$)
$53$ ($$1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} -$$$$11\!\cdots\!76$$$$T^{7} +$$$$35\!\cdots\!13$$$$T^{8} +$$$$92\!\cdots\!94$$$$T^{9} +$$$$12\!\cdots\!18$$$$T^{10} -$$$$13\!\cdots\!10$$$$T^{11} +$$$$28\!\cdots\!31$$$$T^{12} -$$$$18\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!68$$$$T^{14} -$$$$14\!\cdots\!44$$$$T^{15} +$$$$17\!\cdots\!91$$$$T^{16} -$$$$67\!\cdots\!10$$$$T^{17} +$$$$47\!\cdots\!78$$$$T^{18} +$$$$28\!\cdots\!94$$$$T^{19} +$$$$84\!\cdots\!53$$$$T^{20} -$$$$21\!\cdots\!36$$$$T^{21} +$$$$12\!\cdots\!92$$$$T^{22} -$$$$18\!\cdots\!80$$$$T^{23} +$$$$40\!\cdots\!95$$$$T^{24} +$$$$41\!\cdots\!18$$$$T^{25} +$$$$67\!\cdots\!82$$$$T^{26} -$$$$22\!\cdots\!62$$$$T^{27} +$$$$36\!\cdots\!21$$$$T^{28}$$)($$1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} -$$$$11\!\cdots\!76$$$$T^{7} +$$$$35\!\cdots\!13$$$$T^{8} +$$$$92\!\cdots\!94$$$$T^{9} +$$$$12\!\cdots\!18$$$$T^{10} -$$$$13\!\cdots\!10$$$$T^{11} +$$$$28\!\cdots\!31$$$$T^{12} -$$$$18\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!68$$$$T^{14} -$$$$14\!\cdots\!44$$$$T^{15} +$$$$17\!\cdots\!91$$$$T^{16} -$$$$67\!\cdots\!10$$$$T^{17} +$$$$47\!\cdots\!78$$$$T^{18} +$$$$28\!\cdots\!94$$$$T^{19} +$$$$84\!\cdots\!53$$$$T^{20} -$$$$21\!\cdots\!36$$$$T^{21} +$$$$12\!\cdots\!92$$$$T^{22} -$$$$18\!\cdots\!80$$$$T^{23} +$$$$40\!\cdots\!95$$$$T^{24} +$$$$41\!\cdots\!18$$$$T^{25} +$$$$67\!\cdots\!82$$$$T^{26} -$$$$22\!\cdots\!62$$$$T^{27} +$$$$36\!\cdots\!21$$$$T^{28}$$)
$59$ ($$1 + 2786 T + 3880898 T^{2} + 95235375746 T^{3} + 282835430943931 T^{4} - 951817240129082700 T^{5} +$$$$78\!\cdots\!20$$$$T^{6} -$$$$10\!\cdots\!24$$$$T^{7} -$$$$10\!\cdots\!59$$$$T^{8} -$$$$16\!\cdots\!78$$$$T^{9} +$$$$95\!\cdots\!50$$$$T^{10} -$$$$27\!\cdots\!94$$$$T^{11} +$$$$66\!\cdots\!23$$$$T^{12} +$$$$40\!\cdots\!76$$$$T^{13} +$$$$34\!\cdots\!76$$$$T^{14} +$$$$49\!\cdots\!36$$$$T^{15} +$$$$97\!\cdots\!83$$$$T^{16} -$$$$49\!\cdots\!14$$$$T^{17} +$$$$20\!\cdots\!50$$$$T^{18} -$$$$42\!\cdots\!78$$$$T^{19} -$$$$34\!\cdots\!99$$$$T^{20} -$$$$39\!\cdots\!04$$$$T^{21} +$$$$36\!\cdots\!20$$$$T^{22} -$$$$53\!\cdots\!00$$$$T^{23} +$$$$19\!\cdots\!31$$$$T^{24} +$$$$78\!\cdots\!06$$$$T^{25} +$$$$38\!\cdots\!58$$$$T^{26} +$$$$33\!\cdots\!66$$$$T^{27} +$$$$14\!\cdots\!41$$$$T^{28}$$)($$1 - 2786 T + 3880898 T^{2} - 95235375746 T^{3} + 282835430943931 T^{4} + 951817240129082700 T^{5} +$$$$78\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!24$$$$T^{7} -$$$$10\!\cdots\!59$$$$T^{8} +$$$$16\!\cdots\!78$$$$T^{9} +$$$$95\!\cdots\!50$$$$T^{10} +$$$$27\!\cdots\!94$$$$T^{11} +$$$$66\!\cdots\!23$$$$T^{12} -$$$$40\!\cdots\!76$$$$T^{13} +$$$$34\!\cdots\!76$$$$T^{14} -$$$$49\!\cdots\!36$$$$T^{15} +$$$$97\!\cdots\!83$$$$T^{16} +$$$$49\!\cdots\!14$$$$T^{17} +$$$$20\!\cdots\!50$$$$T^{18} +$$$$42\!\cdots\!78$$$$T^{19} -$$$$34\!\cdots\!99$$$$T^{20} +$$$$39\!\cdots\!04$$$$T^{21} +$$$$36\!\cdots\!20$$$$T^{22} +$$$$53\!\cdots\!00$$$$T^{23} +$$$$19\!\cdots\!31$$$$T^{24} -$$$$78\!\cdots\!06$$$$T^{25} +$$$$38\!\cdots\!58$$$$T^{26} -$$$$33\!\cdots\!66$$$$T^{27} +$$$$14\!\cdots\!41$$$$T^{28}$$)
$61$ ($$1 - 3778 T + 7136642 T^{2} - 13584988130 T^{3} + 15885658886619 T^{4} + 322904339201283724 T^{5} -$$$$12\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!80$$$$T^{7} -$$$$34\!\cdots\!59$$$$T^{8} +$$$$57\!\cdots\!82$$$$T^{9} -$$$$60\!\cdots\!50$$$$T^{10} +$$$$13\!\cdots\!70$$$$T^{11} +$$$$15\!\cdots\!55$$$$T^{12} -$$$$19\!\cdots\!60$$$$T^{13} +$$$$92\!\cdots\!08$$$$T^{14} -$$$$26\!\cdots\!60$$$$T^{15} +$$$$29\!\cdots\!55$$$$T^{16} +$$$$35\!\cdots\!70$$$$T^{17} -$$$$22\!\cdots\!50$$$$T^{18} +$$$$29\!\cdots\!82$$$$T^{19} -$$$$24\!\cdots\!19$$$$T^{20} +$$$$11\!\cdots\!80$$$$T^{21} -$$$$16\!\cdots\!20$$$$T^{22} +$$$$60\!\cdots\!64$$$$T^{23} +$$$$41\!\cdots\!19$$$$T^{24} -$$$$48\!\cdots\!30$$$$T^{25} +$$$$35\!\cdots\!02$$$$T^{26} -$$$$25\!\cdots\!38$$$$T^{27} +$$$$95\!\cdots\!61$$$$T^{28}$$)($$1 - 3778 T + 7136642 T^{2} - 13584988130 T^{3} + 15885658886619 T^{4} + 322904339201283724 T^{5} -$$$$12\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!80$$$$T^{7} -$$$$34\!\cdots\!59$$$$T^{8} +$$$$57\!\cdots\!82$$$$T^{9} -$$$$60\!\cdots\!50$$$$T^{10} +$$$$13\!\cdots\!70$$$$T^{11} +$$$$15\!\cdots\!55$$$$T^{12} -$$$$19\!\cdots\!60$$$$T^{13} +$$$$92\!\cdots\!08$$$$T^{14} -$$$$26\!\cdots\!60$$$$T^{15} +$$$$29\!\cdots\!55$$$$T^{16} +$$$$35\!\cdots\!70$$$$T^{17} -$$$$22\!\cdots\!50$$$$T^{18} +$$$$29\!\cdots\!82$$$$T^{19} -$$$$24\!\cdots\!19$$$$T^{20} +$$$$11\!\cdots\!80$$$$T^{21} -$$$$16\!\cdots\!20$$$$T^{22} +$$$$60\!\cdots\!64$$$$T^{23} +$$$$41\!\cdots\!19$$$$T^{24} -$$$$48\!\cdots\!30$$$$T^{25} +$$$$35\!\cdots\!02$$$$T^{26} -$$$$25\!\cdots\!38$$$$T^{27} +$$$$95\!\cdots\!61$$$$T^{28}$$)
$67$ ($$1 - 7998 T + 31984002 T^{2} - 47670849246 T^{3} - 439076236005637 T^{4} + 648765759780793716 T^{5} +$$$$99\!\cdots\!64$$$$T^{6} -$$$$93\!\cdots\!52$$$$T^{7} +$$$$28\!\cdots\!21$$$$T^{8} +$$$$69\!\cdots\!70$$$$T^{9} -$$$$55\!\cdots\!14$$$$T^{10} +$$$$21\!\cdots\!74$$$$T^{11} -$$$$39\!\cdots\!49$$$$T^{12} +$$$$17\!\cdots\!60$$$$T^{13} -$$$$39\!\cdots\!76$$$$T^{14} +$$$$35\!\cdots\!60$$$$T^{15} -$$$$15\!\cdots\!09$$$$T^{16} +$$$$17\!\cdots\!14$$$$T^{17} -$$$$92\!\cdots\!34$$$$T^{18} +$$$$23\!\cdots\!70$$$$T^{19} +$$$$18\!\cdots\!41$$$$T^{20} -$$$$12\!\cdots\!32$$$$T^{21} +$$$$27\!\cdots\!04$$$$T^{22} +$$$$35\!\cdots\!96$$$$T^{23} -$$$$48\!\cdots\!37$$$$T^{24} -$$$$10\!\cdots\!66$$$$T^{25} +$$$$14\!\cdots\!82$$$$T^{26} -$$$$72\!\cdots\!78$$$$T^{27} +$$$$18\!\cdots\!81$$$$T^{28}$$)($$1 + 7998 T + 31984002 T^{2} + 47670849246 T^{3} - 439076236005637 T^{4} - 648765759780793716 T^{5} +$$$$99\!\cdots\!64$$$$T^{6} +$$$$93\!\cdots\!52$$$$T^{7} +$$$$28\!\cdots\!21$$$$T^{8} -$$$$69\!\cdots\!70$$$$T^{9} -$$$$55\!\cdots\!14$$$$T^{10} -$$$$21\!\cdots\!74$$$$T^{11} -$$$$39\!\cdots\!49$$$$T^{12} -$$$$17\!\cdots\!60$$$$T^{13} -$$$$39\!\cdots\!76$$$$T^{14} -$$$$35\!\cdots\!60$$$$T^{15} -$$$$15\!\cdots\!09$$$$T^{16} -$$$$17\!\cdots\!14$$$$T^{17} -$$$$92\!\cdots\!34$$$$T^{18} -$$$$23\!\cdots\!70$$$$T^{19} +$$$$18\!\cdots\!41$$$$T^{20} +$$$$12\!\cdots\!32$$$$T^{21} +$$$$27\!\cdots\!04$$$$T^{22} -$$$$35\!\cdots\!96$$$$T^{23} -$$$$48\!\cdots\!37$$$$T^{24} +$$$$10\!\cdots\!66$$$$T^{25} +$$$$14\!\cdots\!82$$$$T^{26} +$$$$72\!\cdots\!78$$$$T^{27} +$$$$18\!\cdots\!81$$$$T^{28}$$)
$71$ ($$( 1 + 9982 T + 145017323 T^{2} + 1165310044396 T^{3} + 10014081489420185 T^{4} + 63641985337531169890 T^{5} +$$$$40\!\cdots\!59$$$$T^{6} +$$$$20\!\cdots\!72$$$$T^{7} +$$$$10\!\cdots\!79$$$$T^{8} +$$$$41\!\cdots\!90$$$$T^{9} +$$$$16\!\cdots\!85$$$$T^{10} +$$$$48\!\cdots\!16$$$$T^{11} +$$$$15\!\cdots\!23$$$$T^{12} +$$$$26\!\cdots\!42$$$$T^{13} +$$$$68\!\cdots\!61$$$$T^{14} )^{2}$$)($$( 1 - 9982 T + 145017323 T^{2} - 1165310044396 T^{3} + 10014081489420185 T^{4} - 63641985337531169890 T^{5} +$$$$40\!\cdots\!59$$$$T^{6} -$$$$20\!\cdots\!72$$$$T^{7} +$$$$10\!\cdots\!79$$$$T^{8} -$$$$41\!\cdots\!90$$$$T^{9} +$$$$16\!\cdots\!85$$$$T^{10} -$$$$48\!\cdots\!16$$$$T^{11} +$$$$15\!\cdots\!23$$$$T^{12} -$$$$26\!\cdots\!42$$$$T^{13} +$$$$68\!\cdots\!61$$$$T^{14} )^{2}$$)
$73$ ($$1 - 168573838 T^{2} + 13353714116727067 T^{4} -$$$$70\!\cdots\!96$$$$T^{6} +$$$$30\!\cdots\!57$$$$T^{8} -$$$$11\!\cdots\!98$$$$T^{10} +$$$$39\!\cdots\!07$$$$T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$31\!\cdots\!67$$$$T^{16} -$$$$74\!\cdots\!78$$$$T^{18} +$$$$15\!\cdots\!37$$$$T^{20} -$$$$29\!\cdots\!16$$$$T^{22} +$$$$45\!\cdots\!67$$$$T^{24} -$$$$46\!\cdots\!78$$$$T^{26} +$$$$22\!\cdots\!61$$$$T^{28}$$)($$1 - 168573838 T^{2} + 13353714116727067 T^{4} -$$$$70\!\cdots\!96$$$$T^{6} +$$$$30\!\cdots\!57$$$$T^{8} -$$$$11\!\cdots\!98$$$$T^{10} +$$$$39\!\cdots\!07$$$$T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$31\!\cdots\!67$$$$T^{16} -$$$$74\!\cdots\!78$$$$T^{18} +$$$$15\!\cdots\!37$$$$T^{20} -$$$$29\!\cdots\!16$$$$T^{22} +$$$$45\!\cdots\!67$$$$T^{24} -$$$$46\!\cdots\!78$$$$T^{26} +$$$$22\!\cdots\!61$$$$T^{28}$$)
$79$ ($$1 - 364033678 T^{2} + 65454647116587227 T^{4} -$$$$76\!\cdots\!56$$$$T^{6} +$$$$65\!\cdots\!81$$$$T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$23\!\cdots\!67$$$$T^{12} -$$$$99\!\cdots\!88$$$$T^{14} +$$$$35\!\cdots\!87$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$23\!\cdots\!61$$$$T^{20} -$$$$40\!\cdots\!96$$$$T^{22} +$$$$52\!\cdots\!27$$$$T^{24} -$$$$44\!\cdots\!58$$$$T^{26} +$$$$18\!\cdots\!21$$$$T^{28}$$)($$1 - 364033678 T^{2} + 65454647116587227 T^{4} -$$$$76\!\cdots\!56$$$$T^{6} +$$$$65\!\cdots\!81$$$$T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$23\!\cdots\!67$$$$T^{12} -$$$$99\!\cdots\!88$$$$T^{14} +$$$$35\!\cdots\!87$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$23\!\cdots\!61$$$$T^{20} -$$$$40\!\cdots\!96$$$$T^{22} +$$$$52\!\cdots\!27$$$$T^{24} -$$$$44\!\cdots\!58$$$$T^{26} +$$$$18\!\cdots\!21$$$$T^{28}$$)
$83$ ($$1 + 17282 T + 149333762 T^{2} + 1088719641698 T^{3} + 16964332297412731 T^{4} +$$$$21\!\cdots\!96$$$$T^{5} +$$$$17\!\cdots\!52$$$$T^{6} +$$$$12\!\cdots\!12$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8} +$$$$11\!\cdots\!62$$$$T^{9} +$$$$90\!\cdots\!74$$$$T^{10} +$$$$61\!\cdots\!82$$$$T^{11} +$$$$44\!\cdots\!67$$$$T^{12} +$$$$35\!\cdots\!48$$$$T^{13} +$$$$26\!\cdots\!76$$$$T^{14} +$$$$16\!\cdots\!08$$$$T^{15} +$$$$10\!\cdots\!47$$$$T^{16} +$$$$65\!\cdots\!02$$$$T^{17} +$$$$45\!\cdots\!94$$$$T^{18} +$$$$28\!\cdots\!62$$$$T^{19} +$$$$13\!\cdots\!81$$$$T^{20} +$$$$66\!\cdots\!92$$$$T^{21} +$$$$44\!\cdots\!72$$$$T^{22} +$$$$26\!\cdots\!76$$$$T^{23} +$$$$98\!\cdots\!31$$$$T^{24} +$$$$29\!\cdots\!58$$$$T^{25} +$$$$19\!\cdots\!42$$$$T^{26} +$$$$10\!\cdots\!02$$$$T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$)($$1 - 17282 T + 149333762 T^{2} - 1088719641698 T^{3} + 16964332297412731 T^{4} -$$$$21\!\cdots\!96$$$$T^{5} +$$$$17\!\cdots\!52$$$$T^{6} -$$$$12\!\cdots\!12$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8} -$$$$11\!\cdots\!62$$$$T^{9} +$$$$90\!\cdots\!74$$$$T^{10} -$$$$61\!\cdots\!82$$$$T^{11} +$$$$44\!\cdots\!67$$$$T^{12} -$$$$35\!\cdots\!48$$$$T^{13} +$$$$26\!\cdots\!76$$$$T^{14} -$$$$16\!\cdots\!08$$$$T^{15} +$$$$10\!\cdots\!47$$$$T^{16} -$$$$65\!\cdots\!02$$$$T^{17} +$$$$45\!\cdots\!94$$$$T^{18} -$$$$28\!\cdots\!62$$$$T^{19} +$$$$13\!\cdots\!81$$$$T^{20} -$$$$66\!\cdots\!92$$$$T^{21} +$$$$44\!\cdots\!72$$$$T^{22} -$$$$26\!\cdots\!76$$$$T^{23} +$$$$98\!\cdots\!31$$$$T^{24} -$$$$29\!\cdots\!58$$$$T^{25} +$$$$19\!\cdots\!42$$$$T^{26} -$$$$10\!\cdots\!02$$$$T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$)
$89$ ($$1 - 548528910 T^{2} + 149200943223060123 T^{4} -$$$$26\!\cdots\!16$$$$T^{6} +$$$$35\!\cdots\!49$$$$T^{8} -$$$$37\!\cdots\!50$$$$T^{10} +$$$$31\!\cdots\!11$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!91$$$$T^{16} -$$$$57\!\cdots\!50$$$$T^{18} +$$$$21\!\cdots\!09$$$$T^{20} -$$$$64\!\cdots\!36$$$$T^{22} +$$$$14\!\cdots\!23$$$$T^{24} -$$$$20\!\cdots\!10$$$$T^{26} +$$$$14\!\cdots\!61$$$$T^{28}$$)($$1 - 548528910 T^{2} + 149200943223060123 T^{4} -$$$$26\!\cdots\!16$$$$T^{6} +$$$$35\!\cdots\!49$$$$T^{8} -$$$$37\!\cdots\!50$$$$T^{10} +$$$$31\!\cdots\!11$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!91$$$$T^{16} -$$$$57\!\cdots\!50$$$$T^{18} +$$$$21\!\cdots\!09$$$$T^{20} -$$$$64\!\cdots\!36$$$$T^{22} +$$$$14\!\cdots\!23$$$$T^{24} -$$$$20\!\cdots\!10$$$$T^{26} +$$$$14\!\cdots\!61$$$$T^{28}$$)
$97$ ($$( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} +$$$$94\!\cdots\!67$$$$T^{6} +$$$$89\!\cdots\!44$$$$T^{7} +$$$$83\!\cdots\!27$$$$T^{8} +$$$$57\!\cdots\!54$$$$T^{9} +$$$$52\!\cdots\!45$$$$T^{10} +$$$$15\!\cdots\!56$$$$T^{11} +$$$$21\!\cdots\!19$$$$T^{12} +$$$$96\!\cdots\!62$$$$T^{13} +$$$$42\!\cdots\!61$$$$T^{14} )^{2}$$)($$( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} +$$$$94\!\cdots\!67$$$$T^{6} +$$$$89\!\cdots\!44$$$$T^{7} +$$$$83\!\cdots\!27$$$$T^{8} +$$$$57\!\cdots\!54$$$$T^{9} +$$$$52\!\cdots\!45$$$$T^{10} +$$$$15\!\cdots\!56$$$$T^{11} +$$$$21\!\cdots\!19$$$$T^{12} +$$$$96\!\cdots\!62$$$$T^{13} +$$$$42\!\cdots\!61$$$$T^{14} )^{2}$$)