# Properties

 Label 224.3.s Level 224 Weight 3 Character orbit s Rep. character $$\chi_{224}(33,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 32 Newform subspaces 2 Sturm bound 96 Trace bound 9

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.s (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(224, [\chi])$$.

Total New Old
Modular forms 144 32 112
Cusp forms 112 32 80
Eisenstein series 32 0 32

## Trace form

 $$32q + 48q^{9} + O(q^{10})$$ $$32q + 48q^{9} - 80q^{21} + 96q^{25} + 96q^{29} - 144q^{33} + 80q^{37} + 240q^{45} + 16q^{53} - 160q^{57} - 144q^{61} - 176q^{65} - 240q^{73} + 64q^{77} - 352q^{81} - 352q^{85} + 432q^{89} - 400q^{93} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(224, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
224.3.s.a $$16$$ $$6.104$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+\beta _{10}q^{5}-\beta _{13}q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots$$
224.3.s.b $$16$$ $$6.104$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}-\beta _{5}q^{5}+(\beta _{2}+\beta _{12})q^{7}+(-5\beta _{7}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(224, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(224, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 32 T^{2} + 486 T^{4} + 3776 T^{6} + 6841 T^{8} - 164352 T^{10} - 1549098 T^{12} - 1704672 T^{14} + 53230932 T^{16} - 138078432 T^{18} - 10163631978 T^{20} - 87343391232 T^{22} + 294482618361 T^{24} + 13166097898176 T^{26} + 137260754729766 T^{28} + 732057358558752 T^{30} + 1853020188851841 T^{32}$$)($$1 + 16 T^{2} + 134 T^{4} + 1120 T^{6} + 7897 T^{8} + 73856 T^{10} + 1194422 T^{12} + 13226000 T^{14} + 117489172 T^{16} + 1071306000 T^{18} + 7836602742 T^{20} + 39250106496 T^{22} + 339939955737 T^{24} + 3905198529120 T^{26} + 37845557888454 T^{28} + 366028679279376 T^{30} + 1853020188851841 T^{32}$$)
$5$ ($$( 1 + 46 T^{2} + 1073 T^{4} - 5040 T^{5} + 9678 T^{6} - 250320 T^{7} - 145276 T^{8} - 6258000 T^{9} + 6048750 T^{10} - 78750000 T^{11} + 419140625 T^{12} + 11230468750 T^{14} + 152587890625 T^{16} )^{2}$$)($$( 1 + 30 T^{2} + 321 T^{4} + 1200 T^{5} - 930 T^{6} + 138960 T^{7} - 137884 T^{8} + 3474000 T^{9} - 581250 T^{10} + 18750000 T^{11} + 125390625 T^{12} + 7324218750 T^{14} + 152587890625 T^{16} )^{2}$$)
$7$ ($$1 + 64 T^{2} + 5084 T^{4} + 338880 T^{6} + 15148742 T^{8} + 813650880 T^{10} + 29308248284 T^{12} + 885842380864 T^{14} + 33232930569601 T^{16}$$)($$1 - 64 T^{2} + 2268 T^{4} + 3136 T^{6} - 7054138 T^{8} + 7529536 T^{10} + 13074568668 T^{12} - 885842380864 T^{14} + 33232930569601 T^{16}$$)
$11$ ($$1 - 640 T^{2} + 227030 T^{4} - 52241408 T^{6} + 8348702825 T^{8} - 853428477056 T^{10} + 32116185305766 T^{12} + 6409181104515840 T^{14} - 1320087233250529292 T^{16} + 93836820551216413440 T^{18} +$$$$68\!\cdots\!46$$$$T^{20} -$$$$26\!\cdots\!76$$$$T^{22} +$$$$38\!\cdots\!25$$$$T^{24} -$$$$35\!\cdots\!08$$$$T^{26} +$$$$22\!\cdots\!30$$$$T^{28} -$$$$92\!\cdots\!40$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$)($$1 - 432 T^{2} + 84854 T^{4} - 7780896 T^{6} + 22311049 T^{8} + 70438127616 T^{10} + 144760417670 T^{12} - 2441540782042416 T^{14} + 459119675281580212 T^{16} - 35746598589883012656 T^{18} + 31030681144833827270 T^{20} +$$$$22\!\cdots\!36$$$$T^{22} +$$$$10\!\cdots\!89$$$$T^{24} -$$$$52\!\cdots\!96$$$$T^{26} +$$$$83\!\cdots\!14$$$$T^{28} -$$$$62\!\cdots\!92$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$)
$13$ ($$( 1 - 192 T^{2} - 16036 T^{4} + 1118400 T^{6} + 1003382406 T^{8} + 31942622400 T^{10} - 13081057841956 T^{12} - 4473232343516352 T^{14} + 665416609183179841 T^{16} )^{2}$$)($$( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 4307359811040 T^{10} + 416424007224732 T^{12} - 24602777889339936 T^{14} + 665416609183179841 T^{16} )^{2}$$)
$17$ ($$( 1 - 24 T + 766 T^{2} - 13776 T^{3} + 249905 T^{4} - 2407680 T^{5} - 7157346 T^{6} + 486721512 T^{7} - 15232969756 T^{8} + 140662516968 T^{9} - 597788695266 T^{10} - 58115542129920 T^{11} + 1743276663293105 T^{12} - 27772331972585424 T^{13} + 446288633717996926 T^{14} - 4041067837425622296 T^{15} + 48661191875666868481 T^{16} )^{2}$$)($$( 1 + 24 T + 1022 T^{2} + 19920 T^{3} + 479985 T^{4} + 8829120 T^{5} + 194168350 T^{6} + 3297192024 T^{7} + 67118725988 T^{8} + 952888494936 T^{9} + 16217134760350 T^{10} + 213113493209280 T^{11} + 3348258935318385 T^{12} + 40158598496944080 T^{13} + 595439926448815742 T^{14} + 4041067837425622296 T^{15} + 48661191875666868481 T^{16} )^{2}$$)
$19$ ($$1 + 528 T^{2} + 24262 T^{4} + 11189472 T^{6} + 3728656729 T^{8} - 6420105050688 T^{10} + 940099701135094 T^{12} + 1121910404491838544 T^{14} +$$$$16\!\cdots\!80$$$$T^{16} +$$$$14\!\cdots\!24$$$$T^{18} +$$$$15\!\cdots\!54$$$$T^{20} -$$$$14\!\cdots\!68$$$$T^{22} +$$$$10\!\cdots\!49$$$$T^{24} +$$$$42\!\cdots\!72$$$$T^{26} +$$$$11\!\cdots\!02$$$$T^{28} +$$$$33\!\cdots\!48$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$)($$1 + 1440 T^{2} + 935910 T^{4} + 438945600 T^{6} + 186482707321 T^{8} + 56682302220480 T^{10} + 7723885307690070 T^{12} - 285644590063174560 T^{14} -$$$$25\!\cdots\!40$$$$T^{16} -$$$$37\!\cdots\!60$$$$T^{18} +$$$$13\!\cdots\!70$$$$T^{20} +$$$$12\!\cdots\!80$$$$T^{22} +$$$$53\!\cdots\!01$$$$T^{24} +$$$$16\!\cdots\!00$$$$T^{26} +$$$$45\!\cdots\!10$$$$T^{28} +$$$$91\!\cdots\!40$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$)
$23$ ($$1 - 928 T^{2} - 110362 T^{4} + 76027072 T^{6} + 161836431161 T^{8} - 2090789193920 T^{10} - 54941991557559978 T^{12} + 14303404978544854944 T^{14} -$$$$12\!\cdots\!04$$$$T^{16} +$$$$40\!\cdots\!04$$$$T^{18} -$$$$43\!\cdots\!18$$$$T^{20} -$$$$45\!\cdots\!20$$$$T^{22} +$$$$99\!\cdots\!21$$$$T^{24} +$$$$13\!\cdots\!72$$$$T^{26} -$$$$53\!\cdots\!42$$$$T^{28} -$$$$12\!\cdots\!68$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$)($$1 - 1648 T^{2} + 1017702 T^{4} - 249312032 T^{6} + 42919392953 T^{8} - 81219467468736 T^{10} + 45669118532829142 T^{12} + 12560582613209435984 T^{14} -$$$$19\!\cdots\!76$$$$T^{16} +$$$$35\!\cdots\!44$$$$T^{18} +$$$$35\!\cdots\!02$$$$T^{20} -$$$$17\!\cdots\!56$$$$T^{22} +$$$$26\!\cdots\!33$$$$T^{24} -$$$$42\!\cdots\!32$$$$T^{26} +$$$$48\!\cdots\!82$$$$T^{28} -$$$$22\!\cdots\!88$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$)
$29$ ($$( 1 - 28 T + 1376 T^{2} - 41412 T^{3} + 2055086 T^{4} - 34827492 T^{5} + 973218656 T^{6} - 16655052988 T^{7} + 500246412961 T^{8} )^{4}$$)($$( 1 + 4 T + 2608 T^{2} + 8572 T^{3} + 2977102 T^{4} + 7209052 T^{5} + 1844588848 T^{6} + 2379293284 T^{7} + 500246412961 T^{8} )^{4}$$)
$31$ ($$1 + 4480 T^{2} + 9494774 T^{4} + 14306099840 T^{6} + 19217452630025 T^{8} + 24003699927942080 T^{10} + 26963891194383292806 T^{12} +$$$$27\!\cdots\!40$$$$T^{14} +$$$$26\!\cdots\!04$$$$T^{16} +$$$$25\!\cdots\!40$$$$T^{18} +$$$$22\!\cdots\!46$$$$T^{20} +$$$$18\!\cdots\!80$$$$T^{22} +$$$$13\!\cdots\!25$$$$T^{24} +$$$$96\!\cdots\!40$$$$T^{26} +$$$$58\!\cdots\!54$$$$T^{28} +$$$$25\!\cdots\!80$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$)($$1 + 4944 T^{2} + 12401142 T^{4} + 21706728288 T^{6} + 30278528813833 T^{8} + 35431200182034624 T^{10} + 36428088221729894790 T^{12} +$$$$35\!\cdots\!36$$$$T^{14} +$$$$33\!\cdots\!48$$$$T^{16} +$$$$32\!\cdots\!56$$$$T^{18} +$$$$31\!\cdots\!90$$$$T^{20} +$$$$27\!\cdots\!64$$$$T^{22} +$$$$22\!\cdots\!73$$$$T^{24} +$$$$14\!\cdots\!88$$$$T^{26} +$$$$76\!\cdots\!82$$$$T^{28} +$$$$28\!\cdots\!04$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$)
$37$ ($$( 1 - 4 T - 2770 T^{2} + 121912 T^{3} + 3850313 T^{4} - 247260872 T^{5} + 4348488942 T^{6} + 233812310196 T^{7} - 10747661313644 T^{8} + 320089052658324 T^{9} + 8149768384027662 T^{10} - 634403749202768648 T^{11} + 13524145303664927273 T^{12} +$$$$58\!\cdots\!88$$$$T^{13} -$$$$18\!\cdots\!70$$$$T^{14} -$$$$36\!\cdots\!56$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 36 T - 2658 T^{2} + 179832 T^{3} + 2782297 T^{4} - 349680552 T^{5} + 4944965886 T^{6} + 248707896180 T^{7} - 11947732536204 T^{8} + 340481109870420 T^{9} + 9267662209871646 T^{10} - 897184626980097768 T^{11} + 9772761047206036537 T^{12} +$$$$86\!\cdots\!68$$$$T^{13} -$$$$17\!\cdots\!98$$$$T^{14} -$$$$32\!\cdots\!04$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$)
$41$ ($$( 1 - 4384 T^{2} + 14370524 T^{4} - 32104130784 T^{6} + 62557592438726 T^{8} - 90718600708326624 T^{10} +$$$$11\!\cdots\!04$$$$T^{12} -$$$$98\!\cdots\!04$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 459871021169908160 T^{10} +$$$$42\!\cdots\!48$$$$T^{12} -$$$$24\!\cdots\!96$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$)
$43$ ($$( 1 + 11176 T^{2} + 58293116 T^{4} + 188186249880 T^{6} + 415246366681670 T^{8} + 643371339275993880 T^{10} +$$$$68\!\cdots\!16$$$$T^{12} +$$$$44\!\cdots\!76$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 241961144887582552 T^{10} +$$$$29\!\cdots\!96$$$$T^{12} +$$$$23\!\cdots\!92$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$)
$47$ ($$1 + 7504 T^{2} + 21182918 T^{4} + 37880838368 T^{6} + 100768600793177 T^{8} + 218689095748226816 T^{10} +$$$$22\!\cdots\!70$$$$T^{12} +$$$$76\!\cdots\!12$$$$T^{14} +$$$$29\!\cdots\!24$$$$T^{16} +$$$$37\!\cdots\!72$$$$T^{18} +$$$$52\!\cdots\!70$$$$T^{20} +$$$$25\!\cdots\!56$$$$T^{22} +$$$$57\!\cdots\!17$$$$T^{24} +$$$$10\!\cdots\!68$$$$T^{26} +$$$$28\!\cdots\!58$$$$T^{28} +$$$$49\!\cdots\!44$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$)($$1 + 7200 T^{2} + 21086790 T^{4} + 20987252160 T^{6} - 28679941197479 T^{8} - 70284530432138880 T^{10} +$$$$21\!\cdots\!30$$$$T^{12} +$$$$13\!\cdots\!80$$$$T^{14} +$$$$38\!\cdots\!20$$$$T^{16} +$$$$66\!\cdots\!80$$$$T^{18} +$$$$50\!\cdots\!30$$$$T^{20} -$$$$81\!\cdots\!80$$$$T^{22} -$$$$16\!\cdots\!59$$$$T^{24} +$$$$58\!\cdots\!60$$$$T^{26} +$$$$28\!\cdots\!90$$$$T^{28} +$$$$47\!\cdots\!00$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$)
$53$ ($$( 1 + 12 T - 5314 T^{2} - 123432 T^{3} + 13289689 T^{4} + 412137720 T^{5} + 8001350174 T^{6} - 662204794044 T^{7} - 79868347316108 T^{8} - 1860133266469596 T^{9} + 63134501522293694 T^{10} + 9134769260962685880 T^{11} +$$$$82\!\cdots\!29$$$$T^{12} -$$$$21\!\cdots\!68$$$$T^{13} -$$$$26\!\cdots\!74$$$$T^{14} +$$$$16\!\cdots\!28$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 - 20 T - 6098 T^{2} + 355352 T^{3} + 15866345 T^{4} - 1398217832 T^{5} + 8057716910 T^{6} + 2171403120260 T^{7} - 80650058761388 T^{8} + 6099471364810340 T^{9} + 63579262181733710 T^{10} - 30990604965455452328 T^{11} +$$$$98\!\cdots\!45$$$$T^{12} +$$$$62\!\cdots\!48$$$$T^{13} -$$$$29\!\cdots\!18$$$$T^{14} -$$$$27\!\cdots\!80$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$)
$59$ ($$1 + 12496 T^{2} + 69591254 T^{4} + 231196127072 T^{6} + 548940902299433 T^{8} + 1224313485336173888 T^{10} +$$$$76\!\cdots\!62$$$$T^{12} -$$$$17\!\cdots\!52$$$$T^{14} -$$$$10\!\cdots\!32$$$$T^{16} -$$$$21\!\cdots\!72$$$$T^{18} +$$$$11\!\cdots\!02$$$$T^{20} +$$$$21\!\cdots\!28$$$$T^{22} +$$$$11\!\cdots\!53$$$$T^{24} +$$$$60\!\cdots\!72$$$$T^{26} +$$$$22\!\cdots\!94$$$$T^{28} +$$$$47\!\cdots\!16$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$)($$1 + 13056 T^{2} + 66899190 T^{4} + 216438091392 T^{6} + 1041428149652041 T^{8} + 6415617182341109568 T^{10} +$$$$27\!\cdots\!66$$$$T^{12} +$$$$85\!\cdots\!44$$$$T^{14} +$$$$25\!\cdots\!80$$$$T^{16} +$$$$10\!\cdots\!84$$$$T^{18} +$$$$41\!\cdots\!86$$$$T^{20} +$$$$11\!\cdots\!08$$$$T^{22} +$$$$22\!\cdots\!81$$$$T^{24} +$$$$56\!\cdots\!92$$$$T^{26} +$$$$21\!\cdots\!90$$$$T^{28} +$$$$50\!\cdots\!76$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$)
$61$ ($$( 1 + 180 T + 27726 T^{2} + 3046680 T^{3} + 308297129 T^{4} + 26423205000 T^{5} + 2082233555790 T^{6} + 145145432104140 T^{7} + 9348584465944212 T^{8} + 540086152859504940 T^{9} + 28830274738332969390 T^{10} +$$$$13\!\cdots\!00$$$$T^{11} +$$$$59\!\cdots\!49$$$$T^{12} +$$$$21\!\cdots\!80$$$$T^{13} +$$$$73\!\cdots\!46$$$$T^{14} +$$$$17\!\cdots\!80$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 108 T + 15678 T^{2} - 1273320 T^{3} + 112514745 T^{4} - 8281698840 T^{5} + 635104006110 T^{6} - 41916162290292 T^{7} + 2814046950849908 T^{8} - 155970039882176532 T^{9} + 8793549087062088510 T^{10} -$$$$42\!\cdots\!40$$$$T^{11} +$$$$21\!\cdots\!45$$$$T^{12} -$$$$90\!\cdots\!20$$$$T^{13} +$$$$41\!\cdots\!38$$$$T^{14} -$$$$10\!\cdots\!28$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$)
$67$ ($$1 - 5792 T^{2} - 24227898 T^{4} + 351601426496 T^{6} - 205859799182695 T^{8} - 8707473271916577024 T^{10} +$$$$34\!\cdots\!66$$$$T^{12} +$$$$81\!\cdots\!24$$$$T^{14} -$$$$92\!\cdots\!56$$$$T^{16} +$$$$16\!\cdots\!04$$$$T^{18} +$$$$14\!\cdots\!06$$$$T^{20} -$$$$71\!\cdots\!64$$$$T^{22} -$$$$33\!\cdots\!95$$$$T^{24} +$$$$11\!\cdots\!96$$$$T^{26} -$$$$16\!\cdots\!58$$$$T^{28} -$$$$78\!\cdots\!72$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$)($$1 - 19248 T^{2} + 171825062 T^{4} - 1105279125792 T^{6} + 6322631625605113 T^{8} - 28716244420307792256 T^{10} +$$$$95\!\cdots\!42$$$$T^{12} -$$$$31\!\cdots\!16$$$$T^{14} +$$$$13\!\cdots\!84$$$$T^{16} -$$$$63\!\cdots\!36$$$$T^{18} +$$$$38\!\cdots\!22$$$$T^{20} -$$$$23\!\cdots\!16$$$$T^{22} +$$$$10\!\cdots\!53$$$$T^{24} -$$$$36\!\cdots\!92$$$$T^{26} +$$$$11\!\cdots\!02$$$$T^{28} -$$$$25\!\cdots\!68$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$)
$71$ ($$( 1 + 27624 T^{2} + 370534588 T^{4} + 3151580461272 T^{6} + 18747759294515334 T^{8} + 80086957327676918232 T^{10} +$$$$23\!\cdots\!68$$$$T^{12} +$$$$45\!\cdots\!84$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 67143192723001800472 T^{10} +$$$$20\!\cdots\!60$$$$T^{12} +$$$$39\!\cdots\!12$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$)
$73$ ($$( 1 - 36 T + 13374 T^{2} - 465912 T^{3} + 80365289 T^{4} - 3011641896 T^{5} + 480881406846 T^{6} - 17666763587772 T^{7} + 3025613230783092 T^{8} - 94146183159236988 T^{9} + 13656186084031757886 T^{10} -$$$$45\!\cdots\!44$$$$T^{11} +$$$$64\!\cdots\!09$$$$T^{12} -$$$$20\!\cdots\!88$$$$T^{13} +$$$$30\!\cdots\!54$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 + 156 T + 22270 T^{2} + 2208648 T^{3} + 203639977 T^{4} + 17676576216 T^{5} + 1406019151294 T^{6} + 115719264813828 T^{7} + 8350708700464372 T^{8} + 616667962192889412 T^{9} + 39928470709062473854 T^{10} +$$$$26\!\cdots\!24$$$$T^{11} +$$$$16\!\cdots\!37$$$$T^{12} +$$$$94\!\cdots\!52$$$$T^{13} +$$$$51\!\cdots\!70$$$$T^{14} +$$$$19\!\cdots\!04$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$)
$79$ ($$1 - 10112 T^{2} - 55382202 T^{4} + 565038672512 T^{6} + 4616239476368153 T^{8} - 23803283016229531968 T^{10} -$$$$25\!\cdots\!02$$$$T^{12} +$$$$18\!\cdots\!72$$$$T^{14} +$$$$13\!\cdots\!08$$$$T^{16} +$$$$71\!\cdots\!32$$$$T^{18} -$$$$37\!\cdots\!22$$$$T^{20} -$$$$14\!\cdots\!88$$$$T^{22} +$$$$10\!\cdots\!13$$$$T^{24} +$$$$50\!\cdots\!12$$$$T^{26} -$$$$19\!\cdots\!62$$$$T^{28} -$$$$13\!\cdots\!32$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$)($$1 - 34640 T^{2} + 602988550 T^{4} - 7689396100960 T^{6} + 82922855327705561 T^{8} -$$$$77\!\cdots\!00$$$$T^{10} +$$$$63\!\cdots\!50$$$$T^{12} -$$$$46\!\cdots\!20$$$$T^{14} +$$$$30\!\cdots\!00$$$$T^{16} -$$$$17\!\cdots\!20$$$$T^{18} +$$$$95\!\cdots\!50$$$$T^{20} -$$$$45\!\cdots\!00$$$$T^{22} +$$$$19\!\cdots\!81$$$$T^{24} -$$$$68\!\cdots\!60$$$$T^{26} +$$$$21\!\cdots\!50$$$$T^{28} -$$$$47\!\cdots\!40$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$)
$83$ ($$( 1 - 24232 T^{2} + 235912988 T^{4} - 1035732354840 T^{6} + 3398417338650758 T^{8} - 49154118566082623640 T^{10} +$$$$53\!\cdots\!08$$$$T^{12} -$$$$25\!\cdots\!52$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$)($$( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} -$$$$29\!\cdots\!28$$$$T^{10} +$$$$13\!\cdots\!20$$$$T^{12} -$$$$37\!\cdots\!48$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$)
$89$ ($$( 1 - 204 T + 42982 T^{2} - 5938440 T^{3} + 800796785 T^{4} - 88063687656 T^{5} + 9449600595510 T^{6} - 894257389252164 T^{7} + 83094368751630116 T^{8} - 7083412780266391044 T^{9} +$$$$59\!\cdots\!10$$$$T^{10} -$$$$43\!\cdots\!16$$$$T^{11} +$$$$31\!\cdots\!85$$$$T^{12} -$$$$18\!\cdots\!40$$$$T^{13} +$$$$10\!\cdots\!22$$$$T^{14} -$$$$39\!\cdots\!64$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 12 T + 8678 T^{2} - 103560 T^{3} + 5715441 T^{4} + 150778776 T^{5} - 475047477962 T^{6} + 26903544978684 T^{7} - 3925857799931356 T^{8} + 213102979776155964 T^{9} - 29805543348733992842 T^{10} + 74934230745999443736 T^{11} +$$$$22\!\cdots\!21$$$$T^{12} -$$$$32\!\cdots\!60$$$$T^{13} +$$$$21\!\cdots\!38$$$$T^{14} -$$$$23\!\cdots\!92$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$)
$97$ ($$( 1 - 51968 T^{2} + 1305731356 T^{4} - 20766356196608 T^{6} + 230652504659485894 T^{8} -$$$$18\!\cdots\!48$$$$T^{10} +$$$$10\!\cdots\!16$$$$T^{12} -$$$$36\!\cdots\!88$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} -$$$$10\!\cdots\!56$$$$T^{10} +$$$$61\!\cdots\!40$$$$T^{12} -$$$$27\!\cdots\!20$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$)