# Properties

 Label 210.3.o Level 210 Weight 3 Character orbit o Rep. character $$\chi_{210}(31,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 24 Newform subspaces 2 Sturm bound 144 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.o (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$144$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(210, [\chi])$$.

Total New Old
Modular forms 208 24 184
Cusp forms 176 24 152
Eisenstein series 32 0 32

## Trace form

 $$24q - 12q^{3} - 24q^{4} + 4q^{7} + 36q^{9} + O(q^{10})$$ $$24q - 12q^{3} - 24q^{4} + 4q^{7} + 36q^{9} - 8q^{11} + 24q^{12} - 32q^{14} - 48q^{16} + 96q^{17} + 36q^{19} - 24q^{21} - 96q^{22} + 60q^{25} - 96q^{26} + 32q^{28} + 144q^{29} - 12q^{31} + 80q^{35} - 144q^{36} - 52q^{37} - 240q^{38} + 60q^{39} - 96q^{42} - 168q^{43} - 16q^{44} + 16q^{46} - 48q^{47} + 116q^{49} + 72q^{51} - 24q^{52} + 64q^{53} + 32q^{56} + 360q^{57} + 16q^{58} + 264q^{59} + 192q^{61} + 60q^{63} + 192q^{64} + 40q^{65} - 284q^{67} - 192q^{68} - 128q^{71} - 324q^{73} - 128q^{74} - 60q^{75} - 120q^{77} - 192q^{78} + 292q^{79} - 108q^{81} + 288q^{82} - 24q^{84} + 240q^{85} - 128q^{86} + 96q^{88} + 192q^{89} - 580q^{91} - 252q^{93} + 480q^{94} + 160q^{95} + 384q^{98} - 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(210, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
210.3.o.a $$8$$ $$5.722$$ 8.0.3317760000.3 None $$0$$ $$12$$ $$0$$ $$0$$ $$q+(\beta _{2}+\beta _{4})q^{2}+(1+\beta _{3})q^{3}+(-2+2\beta _{3}+\cdots)q^{4}+\cdots$$
210.3.o.b $$16$$ $$5.722$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$-24$$ $$0$$ $$4$$ $$q+\beta _{9}q^{2}+(-1+\beta _{5})q^{3}+(-2-2\beta _{5}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(210, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(210, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T^{2} + 4 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 4 T^{4} )^{4}$$)
$3$ ($$( 1 - 3 T + 3 T^{2} )^{4}$$)($$( 1 + 3 T + 3 T^{2} )^{8}$$)
$5$ ($$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 5 T^{2} + 25 T^{4} )^{4}$$)
$7$ ($$1 - 78 T^{2} + 5243 T^{4} - 187278 T^{6} + 5764801 T^{8}$$)($$1 - 4 T + 28 T^{2} - 352 T^{3} + 2898 T^{4} - 16836 T^{5} + 70272 T^{6} - 1158948 T^{7} + 7794283 T^{8} - 56788452 T^{9} + 168723072 T^{10} - 1980738564 T^{11} + 16706393298 T^{12} - 99431287648 T^{13} + 387556041628 T^{14} - 2712892291396 T^{15} + 33232930569601 T^{16}$$)
$11$ ($$1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 3930886828 T^{9} + 34456246528 T^{10} - 605342393700 T^{11} + 3483331816250 T^{12} + 50007354630728 T^{13} - 527255967289128 T^{14} + 1518999334332964 T^{15} + 45949729863572161 T^{16}$$)($$1 + 4 T - 364 T^{2} - 2552 T^{3} + 54108 T^{4} + 655724 T^{5} - 3422504 T^{6} - 97911028 T^{7} - 146055574 T^{8} + 11147160032 T^{9} + 52178728220 T^{10} - 1271915177612 T^{11} - 12776124769680 T^{12} + 140823145942564 T^{13} + 3084355376442156 T^{14} - 7698059096283072 T^{15} - 472018303966569005 T^{16} - 931465150650251712 T^{17} + 45158047066489605996 T^{18} +$$$$24\!\cdots\!04$$$$T^{19} -$$$$27\!\cdots\!80$$$$T^{20} -$$$$32\!\cdots\!12$$$$T^{21} +$$$$16\!\cdots\!20$$$$T^{22} +$$$$42\!\cdots\!12$$$$T^{23} -$$$$67\!\cdots\!14$$$$T^{24} -$$$$54\!\cdots\!68$$$$T^{25} -$$$$23\!\cdots\!04$$$$T^{26} +$$$$53\!\cdots\!04$$$$T^{27} +$$$$53\!\cdots\!28$$$$T^{28} -$$$$30\!\cdots\!72$$$$T^{29} -$$$$52\!\cdots\!84$$$$T^{30} +$$$$69\!\cdots\!04$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$)
$13$ ($$1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 2725827566800 T^{10} + 291622101296058 T^{12} - 20036353205333660 T^{14} + 665416609183179841 T^{16}$$)($$1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} -$$$$21\!\cdots\!12$$$$T^{18} +$$$$12\!\cdots\!16$$$$T^{20} -$$$$31\!\cdots\!76$$$$T^{22} +$$$$51\!\cdots\!70$$$$T^{24} -$$$$58\!\cdots\!64$$$$T^{26} +$$$$44\!\cdots\!28$$$$T^{28} -$$$$20\!\cdots\!56$$$$T^{30} +$$$$44\!\cdots\!81$$$$T^{32}$$)
$17$ ($$1 - 84 T + 4208 T^{2} - 155904 T^{3} + 4746250 T^{4} - 123749124 T^{5} + 2818194464 T^{6} - 56792172564 T^{7} + 1020121281523 T^{8} - 16412937870996 T^{9} + 235378419827744 T^{10} - 2987003019239556 T^{11} + 33108688754346250 T^{12} - 314301513055600896 T^{13} + 2451674374262834288 T^{14} - 14143737430989678036 T^{15} + 48661191875666868481 T^{16}$$)($$1 - 12 T + 420 T^{2} - 4464 T^{3} + 92044 T^{4} - 785292 T^{5} + 23989560 T^{6} + 227912772 T^{7} - 3236850998 T^{8} + 177940653384 T^{9} - 2004860299572 T^{10} + 43692613137348 T^{11} - 466147839356624 T^{12} + 8985915182672820 T^{13} - 29797959923559588 T^{14} - 769960022203743096 T^{15} + 34516652247342574675 T^{16} -$$$$22\!\cdots\!44$$$$T^{17} -$$$$24\!\cdots\!48$$$$T^{18} +$$$$21\!\cdots\!80$$$$T^{19} -$$$$32\!\cdots\!84$$$$T^{20} +$$$$88\!\cdots\!52$$$$T^{21} -$$$$11\!\cdots\!92$$$$T^{22} +$$$$29\!\cdots\!36$$$$T^{23} -$$$$15\!\cdots\!38$$$$T^{24} +$$$$32\!\cdots\!48$$$$T^{25} +$$$$97\!\cdots\!60$$$$T^{26} -$$$$92\!\cdots\!88$$$$T^{27} +$$$$31\!\cdots\!24$$$$T^{28} -$$$$43\!\cdots\!16$$$$T^{29} +$$$$11\!\cdots\!20$$$$T^{30} -$$$$98\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!61$$$$T^{32}$$)
$19$ ($$1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 36186702110772 T^{9} + 607384473849086 T^{10} - 9338250206171448 T^{11} + 128197793162717145 T^{12} - 1492497721269013032 T^{13} + 13594180232904360862 T^{14} - 86292722064551485068 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$)($$1 + 72 T + 3812 T^{2} + 150048 T^{3} + 4883342 T^{4} + 136887912 T^{5} + 3384543112 T^{6} + 75691203624 T^{7} + 1567163213049 T^{8} + 30524546336184 T^{9} + 578634766673960 T^{10} + 10979211701479464 T^{11} + 213751725988675694 T^{12} + 4327693265152902096 T^{13} + 88720456587214840044 T^{14} +$$$$17\!\cdots\!80$$$$T^{15} +$$$$34\!\cdots\!48$$$$T^{16} +$$$$64\!\cdots\!80$$$$T^{17} +$$$$11\!\cdots\!24$$$$T^{18} +$$$$20\!\cdots\!76$$$$T^{19} +$$$$36\!\cdots\!54$$$$T^{20} +$$$$67\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!60$$$$T^{22} +$$$$24\!\cdots\!64$$$$T^{23} +$$$$45\!\cdots\!69$$$$T^{24} +$$$$78\!\cdots\!84$$$$T^{25} +$$$$12\!\cdots\!12$$$$T^{26} +$$$$18\!\cdots\!32$$$$T^{27} +$$$$23\!\cdots\!82$$$$T^{28} +$$$$26\!\cdots\!88$$$$T^{29} +$$$$24\!\cdots\!92$$$$T^{30} +$$$$16\!\cdots\!72$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$)
$23$ ($$1 - 12 T - 1512 T^{2} + 15144 T^{3} + 1340890 T^{4} - 9662580 T^{5} - 850652928 T^{6} + 2479269564 T^{7} + 454065720819 T^{8} + 1311533599356 T^{9} - 238047566024448 T^{10} - 1430408620333620 T^{11} + 105006417053440090 T^{12} + 627363085819500456 T^{13} - 33134912141214725352 T^{14} -$$$$13\!\cdots\!08$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$)($$1 + 12 T - 1804 T^{2} - 41256 T^{3} + 1078172 T^{4} + 48109188 T^{5} - 226519400 T^{6} - 29579538300 T^{7} + 68149425194 T^{8} + 16885870838304 T^{9} + 1241401844668 T^{10} - 10296857200481412 T^{11} - 100210062266615184 T^{12} + 3750005340988049388 T^{13} + 80231845600155444876 T^{14} -$$$$51\!\cdots\!40$$$$T^{15} -$$$$38\!\cdots\!85$$$$T^{16} -$$$$27\!\cdots\!60$$$$T^{17} +$$$$22\!\cdots\!16$$$$T^{18} +$$$$55\!\cdots\!32$$$$T^{19} -$$$$78\!\cdots\!04$$$$T^{20} -$$$$42\!\cdots\!88$$$$T^{21} +$$$$27\!\cdots\!28$$$$T^{22} +$$$$19\!\cdots\!36$$$$T^{23} +$$$$41\!\cdots\!34$$$$T^{24} -$$$$95\!\cdots\!00$$$$T^{25} -$$$$38\!\cdots\!00$$$$T^{26} +$$$$43\!\cdots\!52$$$$T^{27} +$$$$51\!\cdots\!52$$$$T^{28} -$$$$10\!\cdots\!84$$$$T^{29} -$$$$24\!\cdots\!24$$$$T^{30} +$$$$85\!\cdots\!88$$$$T^{31} +$$$$37\!\cdots\!21$$$$T^{32}$$)
$29$ ($$( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 69765996 T^{5} + 2053944024 T^{6} - 21413639556 T^{7} + 500246412961 T^{8} )^{2}$$)($$( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 25259833173132 T^{9} + 59483856997836 T^{10} + 40924113344941092 T^{11} - 637744142027460460 T^{12} + 8042239471766642316 T^{13} +$$$$51\!\cdots\!04$$$$T^{14} -$$$$10\!\cdots\!16$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$)
$31$ ($$1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2534463516281628 T^{9} + 66978143857738190 T^{10} + 1578451886268782760 T^{11} + 32923703244758191593 T^{12} +$$$$59\!\cdots\!40$$$$T^{13} +$$$$88\!\cdots\!58$$$$T^{14} +$$$$99\!\cdots\!72$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$)($$1 - 120 T + 11588 T^{2} - 814560 T^{3} + 49123694 T^{4} - 2488742424 T^{5} + 113293093576 T^{6} - 4558808688792 T^{7} + 170270490089625 T^{8} - 5869377562140936 T^{9} + 196357797692622056 T^{10} - 6377368910762118360 T^{11} +$$$$21\!\cdots\!82$$$$T^{12} -$$$$69\!\cdots\!28$$$$T^{13} +$$$$23\!\cdots\!88$$$$T^{14} -$$$$74\!\cdots\!40$$$$T^{15} +$$$$23\!\cdots\!16$$$$T^{16} -$$$$71\!\cdots\!40$$$$T^{17} +$$$$21\!\cdots\!48$$$$T^{18} -$$$$61\!\cdots\!68$$$$T^{19} +$$$$17\!\cdots\!62$$$$T^{20} -$$$$52\!\cdots\!60$$$$T^{21} +$$$$15\!\cdots\!16$$$$T^{22} -$$$$44\!\cdots\!56$$$$T^{23} +$$$$12\!\cdots\!25$$$$T^{24} -$$$$31\!\cdots\!72$$$$T^{25} +$$$$76\!\cdots\!76$$$$T^{26} -$$$$16\!\cdots\!64$$$$T^{27} +$$$$30\!\cdots\!74$$$$T^{28} -$$$$48\!\cdots\!60$$$$T^{29} +$$$$66\!\cdots\!08$$$$T^{30} -$$$$66\!\cdots\!20$$$$T^{31} +$$$$52\!\cdots\!61$$$$T^{32}$$)
$37$ ($$1 + 96 T + 1110 T^{2} - 46464 T^{3} + 7652401 T^{4} + 360085920 T^{5} - 3532027338 T^{6} + 170035213440 T^{7} + 29111429055396 T^{8} + 232778207199360 T^{9} - 6619587887813418 T^{10} + 923881954453061280 T^{11} + 26878901285664514321 T^{12} -$$$$22\!\cdots\!36$$$$T^{13} +$$$$73\!\cdots\!10$$$$T^{14} +$$$$86\!\cdots\!44$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$)($$1 - 44 T - 1956 T^{2} - 11728 T^{3} + 3652806 T^{4} + 242885524 T^{5} - 1534355576 T^{6} - 317275895964 T^{7} - 9605253544351 T^{8} + 75954637404076 T^{9} + 6072970847630504 T^{10} + 248299820626414612 T^{11} + 4168440621293518230 T^{12} +$$$$29\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!28$$$$T^{14} -$$$$69\!\cdots\!04$$$$T^{15} -$$$$27\!\cdots\!16$$$$T^{16} -$$$$95\!\cdots\!76$$$$T^{17} +$$$$20\!\cdots\!08$$$$T^{18} +$$$$75\!\cdots\!40$$$$T^{19} +$$$$14\!\cdots\!30$$$$T^{20} +$$$$11\!\cdots\!88$$$$T^{21} +$$$$39\!\cdots\!24$$$$T^{22} +$$$$68\!\cdots\!64$$$$T^{23} -$$$$11\!\cdots\!91$$$$T^{24} -$$$$53\!\cdots\!56$$$$T^{25} -$$$$35\!\cdots\!76$$$$T^{26} +$$$$76\!\cdots\!56$$$$T^{27} +$$$$15\!\cdots\!66$$$$T^{28} -$$$$69\!\cdots\!52$$$$T^{29} -$$$$15\!\cdots\!76$$$$T^{30} -$$$$48\!\cdots\!56$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$)
$41$ ($$1 - 5456 T^{2} + 19963884 T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 140783428358562736 T^{10} +$$$$15\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!36$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$)($$1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} +$$$$50\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!32$$$$T^{14} +$$$$18\!\cdots\!14$$$$T^{16} -$$$$28\!\cdots\!52$$$$T^{18} +$$$$40\!\cdots\!76$$$$T^{20} -$$$$49\!\cdots\!28$$$$T^{22} +$$$$53\!\cdots\!40$$$$T^{24} -$$$$47\!\cdots\!64$$$$T^{26} +$$$$33\!\cdots\!04$$$$T^{28} -$$$$16\!\cdots\!36$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$)
$43$ ($$( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 334092112 T^{5} + 17374346682 T^{6} + 353996330744 T^{7} + 11688200277601 T^{8} )^{2}$$)($$( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 1611056221164 T^{7} + 122468758030675 T^{8} + 2978842952932236 T^{9} + 204841330302006704 T^{10} + 4087494160581305452 T^{11} +$$$$25\!\cdots\!10$$$$T^{12} +$$$$41\!\cdots\!12$$$$T^{13} +$$$$22\!\cdots\!32$$$$T^{14} +$$$$20\!\cdots\!72$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$)
$47$ ($$1 + 24 T + 3580 T^{2} + 81312 T^{3} + 3430986 T^{4} + 200307384 T^{5} + 5424417392 T^{6} + 942687331128 T^{7} + 25778360177363 T^{8} + 2082396314461752 T^{9} + 26469426483811952 T^{10} + 2159156424124689336 T^{11} + 81696191178488726346 T^{12} +$$$$42\!\cdots\!88$$$$T^{13} +$$$$41\!\cdots\!80$$$$T^{14} +$$$$61\!\cdots\!56$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$)($$1 + 24 T + 10040 T^{2} + 236352 T^{3} + 48191588 T^{4} + 529473000 T^{5} + 142018937488 T^{6} - 1533668366952 T^{7} + 289429105887690 T^{8} - 11114511768111792 T^{9} + 616929826339364168 T^{10} - 28449623988679606104 T^{11} +$$$$19\!\cdots\!48$$$$T^{12} -$$$$39\!\cdots\!52$$$$T^{13} +$$$$60\!\cdots\!84$$$$T^{14} -$$$$28\!\cdots\!36$$$$T^{15} +$$$$14\!\cdots\!27$$$$T^{16} -$$$$62\!\cdots\!24$$$$T^{17} +$$$$29\!\cdots\!04$$$$T^{18} -$$$$42\!\cdots\!08$$$$T^{19} +$$$$46\!\cdots\!28$$$$T^{20} -$$$$14\!\cdots\!96$$$$T^{21} +$$$$71\!\cdots\!88$$$$T^{22} -$$$$28\!\cdots\!48$$$$T^{23} +$$$$16\!\cdots\!90$$$$T^{24} -$$$$19\!\cdots\!28$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} +$$$$32\!\cdots\!00$$$$T^{27} +$$$$65\!\cdots\!28$$$$T^{28} +$$$$70\!\cdots\!08$$$$T^{29} +$$$$66\!\cdots\!40$$$$T^{30} +$$$$34\!\cdots\!76$$$$T^{31} +$$$$32\!\cdots\!41$$$$T^{32}$$)
$53$ ($$1 - 32 T - 3572 T^{2} + 570944 T^{3} - 366070 T^{4} - 2018388320 T^{5} + 123180800432 T^{6} + 3809526158624 T^{7} - 445561676340941 T^{8} + 10700958979574816 T^{9} + 971955765373487792 T^{10} - 44736287623035613280 T^{11} - 22791404868886921270 T^{12} +$$$$99\!\cdots\!56$$$$T^{13} -$$$$17\!\cdots\!52$$$$T^{14} -$$$$44\!\cdots\!08$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$)($$1 - 32 T - 12072 T^{2} + 345280 T^{3} + 71529764 T^{4} - 1574568864 T^{5} - 290206976304 T^{6} + 2770742930720 T^{7} + 1006589576083018 T^{8} + 3889510628997504 T^{9} - 3360408915215254232 T^{10} - 25194864185334307040 T^{11} +$$$$10\!\cdots\!32$$$$T^{12} +$$$$34\!\cdots\!88$$$$T^{13} -$$$$28\!\cdots\!00$$$$T^{14} -$$$$63\!\cdots\!04$$$$T^{15} +$$$$75\!\cdots\!55$$$$T^{16} -$$$$17\!\cdots\!36$$$$T^{17} -$$$$22\!\cdots\!00$$$$T^{18} +$$$$75\!\cdots\!52$$$$T^{19} +$$$$64\!\cdots\!52$$$$T^{20} -$$$$44\!\cdots\!60$$$$T^{21} -$$$$16\!\cdots\!12$$$$T^{22} +$$$$53\!\cdots\!76$$$$T^{23} +$$$$39\!\cdots\!78$$$$T^{24} +$$$$30\!\cdots\!80$$$$T^{25} -$$$$88\!\cdots\!04$$$$T^{26} -$$$$13\!\cdots\!76$$$$T^{27} +$$$$17\!\cdots\!84$$$$T^{28} +$$$$23\!\cdots\!20$$$$T^{29} -$$$$22\!\cdots\!92$$$$T^{30} -$$$$17\!\cdots\!68$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$)
$59$ ($$1 - 132 T + 15632 T^{2} - 1296768 T^{3} + 88851370 T^{4} - 4629146772 T^{5} + 218386477856 T^{6} - 7743785317668 T^{7} + 393482793133843 T^{8} - 26956116690802308 T^{9} + 2646267789699658016 T^{10} -$$$$19\!\cdots\!52$$$$T^{11} +$$$$13\!\cdots\!70$$$$T^{12} -$$$$66\!\cdots\!68$$$$T^{13} +$$$$27\!\cdots\!92$$$$T^{14} -$$$$81\!\cdots\!52$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$)($$1 - 132 T + 9660 T^{2} - 508464 T^{3} + 17070268 T^{4} - 1368116580 T^{5} + 85370423112 T^{6} - 5735724829716 T^{7} + 274189008275050 T^{8} - 2776853544451464 T^{9} + 245103035882958612 T^{10} + 4667418467579246988 T^{11} +$$$$23\!\cdots\!68$$$$T^{12} -$$$$15\!\cdots\!44$$$$T^{13} +$$$$45\!\cdots\!16$$$$T^{14} -$$$$18\!\cdots\!68$$$$T^{15} -$$$$23\!\cdots\!33$$$$T^{16} -$$$$62\!\cdots\!08$$$$T^{17} +$$$$55\!\cdots\!76$$$$T^{18} -$$$$65\!\cdots\!04$$$$T^{19} +$$$$34\!\cdots\!28$$$$T^{20} +$$$$23\!\cdots\!88$$$$T^{21} +$$$$43\!\cdots\!72$$$$T^{22} -$$$$17\!\cdots\!04$$$$T^{23} +$$$$59\!\cdots\!50$$$$T^{24} -$$$$43\!\cdots\!36$$$$T^{25} +$$$$22\!\cdots\!12$$$$T^{26} -$$$$12\!\cdots\!80$$$$T^{27} +$$$$54\!\cdots\!48$$$$T^{28} -$$$$56\!\cdots\!24$$$$T^{29} +$$$$37\!\cdots\!60$$$$T^{30} -$$$$17\!\cdots\!32$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$)
$61$ ($$1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 50563704436664544 T^{9} + 2739321895348217168 T^{10} -$$$$19\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!78$$$$T^{12} -$$$$56\!\cdots\!68$$$$T^{13} +$$$$30\!\cdots\!80$$$$T^{14} -$$$$94\!\cdots\!36$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$)($$1 - 96 T + 17624 T^{2} - 1396992 T^{3} + 159354980 T^{4} - 12864972960 T^{5} + 1094820209872 T^{6} - 90213896412000 T^{7} + 6336184490820426 T^{8} - 516522392725304640 T^{9} + 33635567216910203432 T^{10} -$$$$25\!\cdots\!96$$$$T^{11} +$$$$16\!\cdots\!12$$$$T^{12} -$$$$10\!\cdots\!72$$$$T^{13} +$$$$72\!\cdots\!12$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$28\!\cdots\!43$$$$T^{16} -$$$$16\!\cdots\!04$$$$T^{17} +$$$$10\!\cdots\!92$$$$T^{18} -$$$$56\!\cdots\!92$$$$T^{19} +$$$$31\!\cdots\!72$$$$T^{20} -$$$$18\!\cdots\!96$$$$T^{21} +$$$$89\!\cdots\!72$$$$T^{22} -$$$$51\!\cdots\!40$$$$T^{23} +$$$$23\!\cdots\!86$$$$T^{24} -$$$$12\!\cdots\!00$$$$T^{25} +$$$$55\!\cdots\!72$$$$T^{26} -$$$$24\!\cdots\!60$$$$T^{27} +$$$$11\!\cdots\!80$$$$T^{28} -$$$$36\!\cdots\!12$$$$T^{29} +$$$$17\!\cdots\!44$$$$T^{30} -$$$$34\!\cdots\!96$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$)
$67$ ($$1 + 120 T - 4522 T^{2} - 749040 T^{3} + 46360561 T^{4} + 4026690720 T^{5} - 240439443082 T^{6} - 2848051223400 T^{7} + 1781423144538724 T^{8} - 12784901941842600 T^{9} - 4845124310717994922 T^{10} +$$$$36\!\cdots\!80$$$$T^{11} +$$$$18\!\cdots\!01$$$$T^{12} -$$$$13\!\cdots\!60$$$$T^{13} -$$$$37\!\cdots\!42$$$$T^{14} +$$$$44\!\cdots\!80$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16}$$)($$1 + 164 T - 6340 T^{2} - 2597968 T^{3} - 31915482 T^{4} + 22697073892 T^{5} + 967634972296 T^{6} - 123392891205548 T^{7} - 9455087268413407 T^{8} + 333882071018564764 T^{9} + 51639925158822741896 T^{10} +$$$$26\!\cdots\!32$$$$T^{11} -$$$$16\!\cdots\!86$$$$T^{12} -$$$$60\!\cdots\!36$$$$T^{13} +$$$$18\!\cdots\!60$$$$T^{14} +$$$$15\!\cdots\!28$$$$T^{15} +$$$$54\!\cdots\!88$$$$T^{16} +$$$$70\!\cdots\!92$$$$T^{17} +$$$$36\!\cdots\!60$$$$T^{18} -$$$$55\!\cdots\!84$$$$T^{19} -$$$$67\!\cdots\!26$$$$T^{20} +$$$$47\!\cdots\!68$$$$T^{21} +$$$$42\!\cdots\!56$$$$T^{22} +$$$$12\!\cdots\!56$$$$T^{23} -$$$$15\!\cdots\!67$$$$T^{24} -$$$$91\!\cdots\!32$$$$T^{25} +$$$$32\!\cdots\!96$$$$T^{26} +$$$$33\!\cdots\!88$$$$T^{27} -$$$$21\!\cdots\!22$$$$T^{28} -$$$$78\!\cdots\!92$$$$T^{29} -$$$$85\!\cdots\!40$$$$T^{30} +$$$$99\!\cdots\!36$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$)
$71$ ($$( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 1708031948 T^{5} + 334824308856 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{2}$$)($$( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 1096871001959146028 T^{9} + 81023237162072440932 T^{10} +$$$$36\!\cdots\!44$$$$T^{11} +$$$$22\!\cdots\!92$$$$T^{12} +$$$$71\!\cdots\!28$$$$T^{13} +$$$$42\!\cdots\!72$$$$T^{14} +$$$$56\!\cdots\!08$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$)
$73$ ($$1 - 24 T + 9630 T^{2} - 226512 T^{3} + 47274361 T^{4} + 2709573216 T^{5} - 18252715698 T^{6} + 34150744827912 T^{7} - 1287593588654412 T^{8} + 181989319187943048 T^{9} - 518345019296287218 T^{10} +$$$$41\!\cdots\!24$$$$T^{11} +$$$$38\!\cdots\!41$$$$T^{12} -$$$$97\!\cdots\!88$$$$T^{13} +$$$$22\!\cdots\!30$$$$T^{14} -$$$$29\!\cdots\!16$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$)($$1 + 348 T + 82708 T^{2} + 14734320 T^{3} + 2214323678 T^{4} + 290251205388 T^{5} + 34312832193656 T^{6} + 3733330194438684 T^{7} + 379285162573239545 T^{8} + 36467064302261366820 T^{9} +$$$$33\!\cdots\!56$$$$T^{10} +$$$$29\!\cdots\!16$$$$T^{11} +$$$$24\!\cdots\!70$$$$T^{12} +$$$$20\!\cdots\!16$$$$T^{13} +$$$$16\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!12$$$$T^{15} +$$$$91\!\cdots\!12$$$$T^{16} +$$$$65\!\cdots\!48$$$$T^{17} +$$$$45\!\cdots\!52$$$$T^{18} +$$$$30\!\cdots\!24$$$$T^{19} +$$$$20\!\cdots\!70$$$$T^{20} +$$$$12\!\cdots\!84$$$$T^{21} +$$$$76\!\cdots\!76$$$$T^{22} +$$$$44\!\cdots\!80$$$$T^{23} +$$$$24\!\cdots\!45$$$$T^{24} +$$$$12\!\cdots\!96$$$$T^{25} +$$$$63\!\cdots\!56$$$$T^{26} +$$$$28\!\cdots\!52$$$$T^{27} +$$$$11\!\cdots\!98$$$$T^{28} +$$$$41\!\cdots\!80$$$$T^{29} +$$$$12\!\cdots\!48$$$$T^{30} +$$$$27\!\cdots\!52$$$$T^{31} +$$$$42\!\cdots\!21$$$$T^{32}$$)
$79$ ($$1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 66037081594965828 T^{9} + 2575578239741296206 T^{10} - 78980823689760123672 T^{11} -$$$$89\!\cdots\!35$$$$T^{12} +$$$$36\!\cdots\!72$$$$T^{13} -$$$$14\!\cdots\!38$$$$T^{14} -$$$$44\!\cdots\!72$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$)($$1 - 280 T + 14532 T^{2} + 1622944 T^{3} + 35326638 T^{4} - 30655524856 T^{5} + 55399514824 T^{6} + 161497193433480 T^{7} + 10355496367899353 T^{8} - 1123575678148193512 T^{9} - 61265347082641448728 T^{10} +$$$$14\!\cdots\!40$$$$T^{11} +$$$$44\!\cdots\!62$$$$T^{12} -$$$$73\!\cdots\!60$$$$T^{13} +$$$$53\!\cdots\!92$$$$T^{14} -$$$$10\!\cdots\!56$$$$T^{15} +$$$$35\!\cdots\!48$$$$T^{16} -$$$$66\!\cdots\!96$$$$T^{17} +$$$$20\!\cdots\!52$$$$T^{18} -$$$$17\!\cdots\!60$$$$T^{19} +$$$$67\!\cdots\!82$$$$T^{20} +$$$$13\!\cdots\!40$$$$T^{21} -$$$$36\!\cdots\!48$$$$T^{22} -$$$$41\!\cdots\!72$$$$T^{23} +$$$$23\!\cdots\!13$$$$T^{24} +$$$$23\!\cdots\!80$$$$T^{25} +$$$$49\!\cdots\!24$$$$T^{26} -$$$$17\!\cdots\!96$$$$T^{27} +$$$$12\!\cdots\!78$$$$T^{28} +$$$$35\!\cdots\!24$$$$T^{29} +$$$$19\!\cdots\!52$$$$T^{30} -$$$$23\!\cdots\!80$$$$T^{31} +$$$$52\!\cdots\!41$$$$T^{32}$$)
$83$ ($$1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 8078059876516133296 T^{10} +$$$$12\!\cdots\!00$$$$T^{12} -$$$$25\!\cdots\!24$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$)($$1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} -$$$$83\!\cdots\!72$$$$T^{10} +$$$$84\!\cdots\!60$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$53\!\cdots\!06$$$$T^{16} -$$$$34\!\cdots\!84$$$$T^{18} +$$$$19\!\cdots\!60$$$$T^{20} -$$$$88\!\cdots\!92$$$$T^{22} +$$$$34\!\cdots\!16$$$$T^{24} -$$$$10\!\cdots\!60$$$$T^{26} +$$$$24\!\cdots\!68$$$$T^{28} -$$$$35\!\cdots\!44$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$)
$89$ ($$1 - 492 T + 137248 T^{2} - 27827520 T^{3} + 4512312618 T^{4} - 615183153660 T^{5} + 72837035721440 T^{6} - 7634393359602348 T^{7} + 716618506839255347 T^{8} - 60472029801410198508 T^{9} +$$$$45\!\cdots\!40$$$$T^{10} -$$$$30\!\cdots\!60$$$$T^{11} +$$$$17\!\cdots\!58$$$$T^{12} -$$$$86\!\cdots\!20$$$$T^{13} +$$$$33\!\cdots\!08$$$$T^{14} -$$$$96\!\cdots\!72$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$)($$1 + 300 T + 54668 T^{2} + 7400400 T^{3} + 757620572 T^{4} + 53632718700 T^{5} + 870152589352 T^{6} - 480156239482500 T^{7} - 98736006237054870 T^{8} - 12632355432189921000 T^{9} -$$$$12\!\cdots\!92$$$$T^{10} -$$$$89\!\cdots\!00$$$$T^{11} -$$$$37\!\cdots\!44$$$$T^{12} +$$$$17\!\cdots\!40$$$$T^{13} +$$$$60\!\cdots\!12$$$$T^{14} +$$$$83\!\cdots\!20$$$$T^{15} +$$$$83\!\cdots\!23$$$$T^{16} +$$$$66\!\cdots\!20$$$$T^{17} +$$$$37\!\cdots\!92$$$$T^{18} +$$$$84\!\cdots\!40$$$$T^{19} -$$$$14\!\cdots\!64$$$$T^{20} -$$$$27\!\cdots\!00$$$$T^{21} -$$$$30\!\cdots\!32$$$$T^{22} -$$$$24\!\cdots\!00$$$$T^{23} -$$$$15\!\cdots\!70$$$$T^{24} -$$$$58\!\cdots\!00$$$$T^{25} +$$$$84\!\cdots\!52$$$$T^{26} +$$$$41\!\cdots\!00$$$$T^{27} +$$$$46\!\cdots\!52$$$$T^{28} +$$$$35\!\cdots\!00$$$$T^{29} +$$$$20\!\cdots\!08$$$$T^{30} +$$$$90\!\cdots\!00$$$$T^{31} +$$$$24\!\cdots\!21$$$$T^{32}$$)
$97$ ($$1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} -$$$$86\!\cdots\!40$$$$T^{10} +$$$$56\!\cdots\!24$$$$T^{12} -$$$$24\!\cdots\!80$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$)($$1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} -$$$$21\!\cdots\!72$$$$T^{10} +$$$$29\!\cdots\!68$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$34\!\cdots\!38$$$$T^{16} -$$$$30\!\cdots\!04$$$$T^{18} +$$$$23\!\cdots\!48$$$$T^{20} -$$$$14\!\cdots\!52$$$$T^{22} +$$$$81\!\cdots\!08$$$$T^{24} -$$$$36\!\cdots\!36$$$$T^{26} +$$$$12\!\cdots\!92$$$$T^{28} -$$$$30\!\cdots\!28$$$$T^{30} +$$$$37\!\cdots\!41$$$$T^{32}$$)