# Properties

 Label 336.2.w Level 336 Weight 2 Character orbit w Rep. character $$\chi_{336}(85,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 48 Newform subspaces 2 Sturm bound 128 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.w (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$128$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(336, [\chi])$$.

Total New Old
Modular forms 136 48 88
Cusp forms 120 48 72
Eisenstein series 16 0 16

## Trace form

 $$48q + 4q^{4} + O(q^{10})$$ $$48q + 4q^{4} - 8q^{10} + 8q^{11} - 16q^{12} + 4q^{14} + 16q^{15} + 4q^{16} - 4q^{18} + 16q^{19} - 20q^{22} - 8q^{24} - 40q^{26} + 16q^{29} + 16q^{30} - 32q^{34} + 8q^{36} + 16q^{37} + 56q^{38} - 40q^{40} + 24q^{43} + 52q^{44} - 24q^{46} - 48q^{49} + 68q^{50} - 16q^{51} - 16q^{52} - 16q^{53} - 8q^{54} - 28q^{56} + 4q^{58} - 24q^{60} - 32q^{61} - 48q^{62} - 8q^{63} + 28q^{64} + 32q^{65} - 8q^{67} - 104q^{68} - 32q^{69} - 24q^{70} + 4q^{72} + 12q^{74} - 32q^{75} - 40q^{76} - 16q^{77} + 24q^{78} + 48q^{79} - 80q^{80} - 48q^{81} - 40q^{82} + 80q^{83} + 32q^{85} - 60q^{86} + 20q^{88} + 16q^{90} + 48q^{92} - 40q^{94} + 40q^{96} + 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
336.2.w.a $$20$$ $$2.683$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-\beta _{7}q^{3}+(\beta _{1}-\beta _{3}-\beta _{6}-\beta _{16}+\cdots)q^{4}+\cdots$$
336.2.w.b $$28$$ $$2.683$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(336, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T^{2} + 2 T^{4} + 4 T^{6} - 8 T^{7} - 4 T^{8} - 16 T^{12} - 64 T^{13} + 64 T^{14} + 128 T^{16} - 512 T^{18} + 1024 T^{20}$$)
$3$ ($$( 1 + T^{4} )^{5}$$)
$5$ ($$1 - 24 T^{3} + 10 T^{4} + 24 T^{5} + 288 T^{6} - 320 T^{7} + 485 T^{8} - 2720 T^{9} + 5088 T^{10} - 20208 T^{11} + 28392 T^{12} - 21904 T^{13} + 280352 T^{14} - 374560 T^{15} + 85402 T^{16} - 2818304 T^{17} + 5181408 T^{18} - 7099744 T^{19} + 30439356 T^{20} - 35498720 T^{21} + 129535200 T^{22} - 352288000 T^{23} + 53376250 T^{24} - 1170500000 T^{25} + 4380500000 T^{26} - 1711250000 T^{27} + 11090625000 T^{28} - 39468750000 T^{29} + 49687500000 T^{30} - 132812500000 T^{31} + 118408203125 T^{32} - 390625000000 T^{33} + 1757812500000 T^{34} + 732421875000 T^{35} + 1525878906250 T^{36} - 18310546875000 T^{37} + 95367431640625 T^{40}$$)
$7$ ($$( 1 + T^{2} )^{10}$$)
$11$ ($$1 - 12 T + 72 T^{2} - 404 T^{3} + 2214 T^{4} - 9404 T^{5} + 35048 T^{6} - 135156 T^{7} + 456709 T^{8} - 1318720 T^{9} + 4165952 T^{10} - 13629648 T^{11} + 41111256 T^{12} - 141076784 T^{13} + 543949056 T^{14} - 1968344544 T^{15} + 6732047034 T^{16} - 23311377560 T^{17} + 80768058832 T^{18} - 261514302488 T^{19} + 834968255588 T^{20} - 2876657327368 T^{21} + 9772935118672 T^{22} - 31027443532360 T^{23} + 98563900624794 T^{24} - 317003857155744 T^{25} + 963638933596416 T^{26} - 2749187413938064 T^{27} + 8812562832664536 T^{28} - 32137997030742768 T^{29} + 108054065891385152 T^{30} - 376246206268137920 T^{31} + 1433348485503871189 T^{32} - 4665951682525138236 T^{33} + 13309472167425430568 T^{34} - 39282841785184782004 T^{35} +$$$$10\!\cdots\!54$$$$T^{36} -$$$$20\!\cdots\!84$$$$T^{37} +$$$$40\!\cdots\!32$$$$T^{38} -$$$$73\!\cdots\!92$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$13$ ($$1 - 80 T^{3} - 118 T^{4} + 1616 T^{5} + 3200 T^{6} + 22560 T^{7} - 156451 T^{8} - 297120 T^{9} - 121472 T^{10} + 6082400 T^{11} + 43092280 T^{12} - 92564192 T^{13} - 225951616 T^{14} - 2287891296 T^{15} + 3788315570 T^{16} + 30959844064 T^{17} + 31244658048 T^{18} + 7310121920 T^{19} - 2235843760900 T^{20} + 95031584960 T^{21} + 5280347210112 T^{22} + 68018777408608 T^{23} + 108198080994770 T^{24} - 849478022965728 T^{25} - 1090625293673344 T^{26} - 5808265775303264 T^{27} + 35151696633933880 T^{28} + 64500806986335200 T^{29} - 16745946721881728 T^{30} - 532486696276273440 T^{31} - 3645008715497274931 T^{32} + 6832862404721227680 T^{33} + 12599604434237724800 T^{34} + 82716403110770663312 T^{35} - 78519159883615221238 T^{36} -$$$$69\!\cdots\!40$$$$T^{37} +$$$$19\!\cdots\!01$$$$T^{40}$$)
$17$ ($$( 1 + 94 T^{2} + 88 T^{3} + 4293 T^{4} + 6432 T^{5} + 134312 T^{6} + 219984 T^{7} + 3234206 T^{8} + 4965576 T^{9} + 61650188 T^{10} + 84414792 T^{11} + 934685534 T^{12} + 1080781392 T^{13} + 11217872552 T^{14} + 9132520224 T^{15} + 103622583717 T^{16} + 36109803224 T^{17} + 655721199454 T^{18} + 2015993900449 T^{20} )^{2}$$)
$19$ ($$1 - 8 T + 32 T^{2} - 120 T^{3} + 170 T^{4} + 664 T^{5} - 3552 T^{6} + 30376 T^{7} - 137251 T^{8} + 141472 T^{9} + 439424 T^{10} - 1977376 T^{11} + 92609976 T^{12} - 734762848 T^{13} + 3144939136 T^{14} - 11850068256 T^{15} + 18020441650 T^{16} + 30523407824 T^{17} - 82150980672 T^{18} + 1163681013680 T^{19} - 8113217288772 T^{20} + 22109939259920 T^{21} - 29656504022592 T^{22} + 209360054264816 T^{23} + 2348441976269650 T^{24} - 29341942158613344 T^{25} + 147956432344498816 T^{26} - 656783744694352672 T^{27} + 1572847365621497016 T^{28} - 638074909083447904 T^{29} + 2694137659267946624 T^{30} + 16480109906848838368 T^{31} -$$$$30\!\cdots\!11$$$$T^{32} +$$$$12\!\cdots\!84$$$$T^{33} -$$$$28\!\cdots\!92$$$$T^{34} +$$$$10\!\cdots\!36$$$$T^{35} +$$$$49\!\cdots\!70$$$$T^{36} -$$$$65\!\cdots\!80$$$$T^{37} +$$$$33\!\cdots\!12$$$$T^{38} -$$$$15\!\cdots\!32$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$)
$23$ ($$1 - 228 T^{2} + 24654 T^{4} - 1671172 T^{6} + 79227605 T^{8} - 2786970288 T^{10} + 76106467832 T^{12} - 1708777858416 T^{14} + 34629617977562 T^{16} - 710121266972024 T^{18} + 15689370725592020 T^{20} - 375654150228200696 T^{22} + 9690786924458927642 T^{24} -$$$$25\!\cdots\!24$$$$T^{26} +$$$$59\!\cdots\!92$$$$T^{28} -$$$$11\!\cdots\!12$$$$T^{30} +$$$$17\!\cdots\!05$$$$T^{32} -$$$$19\!\cdots\!48$$$$T^{34} +$$$$15\!\cdots\!94$$$$T^{36} -$$$$73\!\cdots\!32$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$29$ ($$1 - 12 T + 72 T^{2} - 660 T^{3} + 5062 T^{4} - 19812 T^{5} + 91080 T^{6} - 552988 T^{7} + 898845 T^{8} + 3230928 T^{9} - 3305184 T^{10} + 180759600 T^{11} - 2401479032 T^{12} + 11590609840 T^{13} - 61731406432 T^{14} + 542524821712 T^{15} - 2180038368078 T^{16} + 2872412612888 T^{17} - 25762237441424 T^{18} + 2546561760168 T^{19} + 1284480158865124 T^{20} + 73850291044872 T^{21} - 21666041688237584 T^{22} + 70055271215725432 T^{23} - 1541899717012575918 T^{24} + 11127807454333267088 T^{25} - 36719280183883000672 T^{26} +$$$$19\!\cdots\!60$$$$T^{27} -$$$$12\!\cdots\!52$$$$T^{28} +$$$$26\!\cdots\!00$$$$T^{29} -$$$$13\!\cdots\!84$$$$T^{30} +$$$$39\!\cdots\!12$$$$T^{31} +$$$$31\!\cdots\!45$$$$T^{32} -$$$$56\!\cdots\!32$$$$T^{33} +$$$$27\!\cdots\!80$$$$T^{34} -$$$$17\!\cdots\!88$$$$T^{35} +$$$$12\!\cdots\!02$$$$T^{36} -$$$$47\!\cdots\!40$$$$T^{37} +$$$$15\!\cdots\!92$$$$T^{38} -$$$$73\!\cdots\!28$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$31$ ($$( 1 + 134 T^{2} - 64 T^{3} + 7637 T^{4} + 64 T^{5} + 250776 T^{6} + 647232 T^{7} + 5787898 T^{8} + 46731200 T^{9} + 141742500 T^{10} + 1448667200 T^{11} + 5562169978 T^{12} + 19281688512 T^{13} + 231596902296 T^{14} + 1832265664 T^{15} + 6777865611797 T^{16} - 1760807303104 T^{17} + 114287399017094 T^{18} + 819628286980801 T^{20} )^{2}$$)
$37$ ($$1 - 12 T + 72 T^{2} - 692 T^{3} + 4646 T^{4} - 25508 T^{5} + 211016 T^{6} - 1455420 T^{7} + 12729021 T^{8} - 102763568 T^{9} + 668772640 T^{10} - 4907491152 T^{11} + 30685175816 T^{12} - 181058288208 T^{13} + 1246622676896 T^{14} - 7968453274800 T^{15} + 52782260071730 T^{16} - 346974533424104 T^{17} + 2138919388551792 T^{18} - 13427579452955416 T^{19} + 82729716000045604 T^{20} - 496820439759350392 T^{21} + 2928180642927403248 T^{22} - 17575301041531139912 T^{23} + 98922453318293568530 T^{24} -$$$$55\!\cdots\!00$$$$T^{25} +$$$$31\!\cdots\!64$$$$T^{26} -$$$$17\!\cdots\!64$$$$T^{27} +$$$$10\!\cdots\!36$$$$T^{28} -$$$$63\!\cdots\!04$$$$T^{29} +$$$$32\!\cdots\!60$$$$T^{30} -$$$$18\!\cdots\!84$$$$T^{31} +$$$$83\!\cdots\!01$$$$T^{32} -$$$$35\!\cdots\!40$$$$T^{33} +$$$$19\!\cdots\!24$$$$T^{34} -$$$$85\!\cdots\!44$$$$T^{35} +$$$$57\!\cdots\!86$$$$T^{36} -$$$$31\!\cdots\!64$$$$T^{37} +$$$$12\!\cdots\!88$$$$T^{38} -$$$$74\!\cdots\!76$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$)
$41$ ($$1 - 444 T^{2} + 98574 T^{4} - 14512124 T^{6} + 1588488149 T^{8} - 137675000112 T^{10} + 9846636632440 T^{12} - 599501624736304 T^{14} + 31880388630283994 T^{16} - 1512585137889846120 T^{18} + 64990434418739739988 T^{20} -$$$$25\!\cdots\!20$$$$T^{22} +$$$$90\!\cdots\!34$$$$T^{24} -$$$$28\!\cdots\!64$$$$T^{26} +$$$$78\!\cdots\!40$$$$T^{28} -$$$$18\!\cdots\!12$$$$T^{30} +$$$$35\!\cdots\!69$$$$T^{32} -$$$$55\!\cdots\!64$$$$T^{34} +$$$$62\!\cdots\!34$$$$T^{36} -$$$$47\!\cdots\!24$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$43$ ($$1 - 4 T + 8 T^{2} + 300 T^{3} + 7078 T^{4} - 44284 T^{5} + 165512 T^{6} + 1626836 T^{7} + 19994909 T^{8} - 198352400 T^{9} + 930543520 T^{10} + 1629993648 T^{11} + 32879446344 T^{12} - 523812434096 T^{13} + 2532563214112 T^{14} - 8844397053040 T^{15} + 52614175288210 T^{16} - 1034528298704760 T^{17} + 4610401791613808 T^{18} - 37116758465801176 T^{19} + 100269344691342500 T^{20} - 1596020614029450568 T^{21} + 8524632912693930992 T^{22} - 82252241445119353320 T^{23} +$$$$17\!\cdots\!10$$$$T^{24} -$$$$13\!\cdots\!20$$$$T^{25} +$$$$16\!\cdots\!88$$$$T^{26} -$$$$14\!\cdots\!72$$$$T^{27} +$$$$38\!\cdots\!44$$$$T^{28} +$$$$81\!\cdots\!64$$$$T^{29} +$$$$20\!\cdots\!80$$$$T^{30} -$$$$18\!\cdots\!00$$$$T^{31} +$$$$79\!\cdots\!09$$$$T^{32} +$$$$27\!\cdots\!48$$$$T^{33} +$$$$12\!\cdots\!88$$$$T^{34} -$$$$14\!\cdots\!88$$$$T^{35} +$$$$96\!\cdots\!78$$$$T^{36} +$$$$17\!\cdots\!00$$$$T^{37} +$$$$20\!\cdots\!92$$$$T^{38} -$$$$43\!\cdots\!28$$$$T^{39} +$$$$46\!\cdots\!01$$$$T^{40}$$)
$47$ ($$( 1 + 254 T^{2} - 128 T^{3} + 33997 T^{4} - 26752 T^{5} + 3110696 T^{6} - 2769024 T^{7} + 212743186 T^{8} - 186077312 T^{9} + 11293892148 T^{10} - 8745633664 T^{11} + 469949697874 T^{12} - 287488378752 T^{13} + 15179204167976 T^{14} - 6135437627264 T^{15} + 366460983540013 T^{16} - 64847759419264 T^{17} + 6048066812087294 T^{18} + 52599132235830049 T^{20} )^{2}$$)
$53$ ($$1 + 36 T + 648 T^{2} + 8284 T^{3} + 81222 T^{4} + 595308 T^{5} + 3111560 T^{6} + 8277492 T^{7} - 42534915 T^{8} - 754029808 T^{9} - 6543023328 T^{10} - 50633847440 T^{11} - 420252241016 T^{12} - 3590853938064 T^{13} - 27232240956512 T^{14} - 163759336672624 T^{15} - 644108178966222 T^{16} - 225724213191624 T^{17} + 21359012626695920 T^{18} + 241077022660459848 T^{19} + 1941342482920284900 T^{20} + 12777082201004371944 T^{21} + 59997466468388839280 T^{22} - 33605143687329406248 T^{23} -$$$$50\!\cdots\!82$$$$T^{24} -$$$$68\!\cdots\!32$$$$T^{25} -$$$$60\!\cdots\!48$$$$T^{26} -$$$$42\!\cdots\!68$$$$T^{27} -$$$$26\!\cdots\!76$$$$T^{28} -$$$$16\!\cdots\!20$$$$T^{29} -$$$$11\!\cdots\!72$$$$T^{30} -$$$$69\!\cdots\!76$$$$T^{31} -$$$$20\!\cdots\!15$$$$T^{32} +$$$$21\!\cdots\!16$$$$T^{33} +$$$$42\!\cdots\!40$$$$T^{34} +$$$$43\!\cdots\!56$$$$T^{35} +$$$$31\!\cdots\!62$$$$T^{36} +$$$$17\!\cdots\!92$$$$T^{37} +$$$$70\!\cdots\!72$$$$T^{38} +$$$$20\!\cdots\!12$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$)
$59$ ($$1 + 64 T^{3} + 17498 T^{4} + 16320 T^{5} + 2048 T^{6} + 964352 T^{7} + 139305821 T^{8} + 265011968 T^{9} + 159053824 T^{10} + 7117236224 T^{11} + 675196772280 T^{12} + 1865453307392 T^{13} + 2177063692288 T^{14} + 46993822767872 T^{15} + 2378495821379602 T^{16} + 7365185499789056 T^{17} + 13796808199690240 T^{18} + 308926250030895744 T^{19} + 7830691115365441372 T^{20} + 18226648751822848896 T^{21} + 48026689343121725440 T^{22} +$$$$15\!\cdots\!24$$$$T^{23} +$$$$28\!\cdots\!22$$$$T^{24} +$$$$33\!\cdots\!28$$$$T^{25} +$$$$91\!\cdots\!08$$$$T^{26} +$$$$46\!\cdots\!48$$$$T^{27} +$$$$99\!\cdots\!80$$$$T^{28} +$$$$61\!\cdots\!36$$$$T^{29} +$$$$81\!\cdots\!24$$$$T^{30} +$$$$79\!\cdots\!12$$$$T^{31} +$$$$24\!\cdots\!01$$$$T^{32} +$$$$10\!\cdots\!08$$$$T^{33} +$$$$12\!\cdots\!28$$$$T^{34} +$$$$59\!\cdots\!80$$$$T^{35} +$$$$37\!\cdots\!18$$$$T^{36} +$$$$81\!\cdots\!16$$$$T^{37} +$$$$26\!\cdots\!01$$$$T^{40}$$)
$61$ ($$1 - 8 T + 32 T^{2} - 24 T^{3} - 22 T^{4} + 28776 T^{5} - 229216 T^{6} + 1701688 T^{7} - 7327587 T^{8} + 193356480 T^{9} - 944223232 T^{10} + 7441249856 T^{11} + 25030907320 T^{12} - 252322194176 T^{13} + 6350522522880 T^{14} - 30700611174144 T^{15} + 72405460654514 T^{16} + 1378721064691056 T^{17} + 10443615781913408 T^{18} + 4859962566983248 T^{19} + 374225284282585404 T^{20} + 296457716585978128 T^{21} + 38860694324499791168 T^{22} +$$$$31\!\cdots\!36$$$$T^{23} +$$$$10\!\cdots\!74$$$$T^{24} -$$$$25\!\cdots\!44$$$$T^{25} +$$$$32\!\cdots\!80$$$$T^{26} -$$$$79\!\cdots\!96$$$$T^{27} +$$$$47\!\cdots\!20$$$$T^{28} +$$$$87\!\cdots\!96$$$$T^{29} -$$$$67\!\cdots\!32$$$$T^{30} +$$$$84\!\cdots\!80$$$$T^{31} -$$$$19\!\cdots\!27$$$$T^{32} +$$$$27\!\cdots\!28$$$$T^{33} -$$$$22\!\cdots\!56$$$$T^{34} +$$$$17\!\cdots\!76$$$$T^{35} -$$$$80\!\cdots\!42$$$$T^{36} -$$$$53\!\cdots\!04$$$$T^{37} +$$$$43\!\cdots\!92$$$$T^{38} -$$$$66\!\cdots\!28$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$)
$67$ ($$1 + 12 T + 72 T^{2} + 444 T^{3} + 4102 T^{4} + 54484 T^{5} + 457032 T^{6} + 4150628 T^{7} - 7647939 T^{8} - 292442192 T^{9} - 1506268000 T^{10} - 8570912144 T^{11} - 68934819128 T^{12} - 1058425781488 T^{13} - 9188787191264 T^{14} - 93290480178608 T^{15} + 105617726671378 T^{16} + 7399255991086056 T^{17} + 46962879527876336 T^{18} + 365051212101087880 T^{19} + 2665710348810285028 T^{20} + 24458431210772887960 T^{21} +$$$$21\!\cdots\!04$$$$T^{22} +$$$$22\!\cdots\!28$$$$T^{23} +$$$$21\!\cdots\!38$$$$T^{24} -$$$$12\!\cdots\!56$$$$T^{25} -$$$$83\!\cdots\!16$$$$T^{26} -$$$$64\!\cdots\!24$$$$T^{27} -$$$$27\!\cdots\!48$$$$T^{28} -$$$$23\!\cdots\!68$$$$T^{29} -$$$$27\!\cdots\!00$$$$T^{30} -$$$$35\!\cdots\!36$$$$T^{31} -$$$$62\!\cdots\!79$$$$T^{32} +$$$$22\!\cdots\!36$$$$T^{33} +$$$$16\!\cdots\!28$$$$T^{34} +$$$$13\!\cdots\!12$$$$T^{35} +$$$$67\!\cdots\!62$$$$T^{36} +$$$$49\!\cdots\!88$$$$T^{37} +$$$$53\!\cdots\!48$$$$T^{38} +$$$$59\!\cdots\!36$$$$T^{39} +$$$$33\!\cdots\!01$$$$T^{40}$$)
$71$ ($$1 - 588 T^{2} + 187310 T^{4} - 41853292 T^{6} + 7286062485 T^{8} - 1044227363664 T^{10} + 127464606182264 T^{12} - 13544991508155920 T^{14} + 1271322412181875034 T^{16} -$$$$10\!\cdots\!40$$$$T^{18} +$$$$79\!\cdots\!36$$$$T^{20} -$$$$53\!\cdots\!40$$$$T^{22} +$$$$32\!\cdots\!54$$$$T^{24} -$$$$17\!\cdots\!20$$$$T^{26} +$$$$82\!\cdots\!04$$$$T^{28} -$$$$33\!\cdots\!64$$$$T^{30} +$$$$11\!\cdots\!85$$$$T^{32} -$$$$34\!\cdots\!52$$$$T^{34} +$$$$78\!\cdots\!10$$$$T^{36} -$$$$12\!\cdots\!68$$$$T^{38} +$$$$10\!\cdots\!01$$$$T^{40}$$)
$73$ ($$1 - 588 T^{2} + 185438 T^{4} - 41247404 T^{6} + 7184467213 T^{8} - 1034846312688 T^{10} + 127434000595944 T^{12} - 13711360420506544 T^{14} + 1308031080629694098 T^{16} -$$$$11\!\cdots\!16$$$$T^{18} +$$$$85\!\cdots\!04$$$$T^{20} -$$$$59\!\cdots\!64$$$$T^{22} +$$$$37\!\cdots\!18$$$$T^{24} -$$$$20\!\cdots\!16$$$$T^{26} +$$$$10\!\cdots\!64$$$$T^{28} -$$$$44\!\cdots\!12$$$$T^{30} +$$$$16\!\cdots\!73$$$$T^{32} -$$$$50\!\cdots\!36$$$$T^{34} +$$$$12\!\cdots\!18$$$$T^{36} -$$$$20\!\cdots\!72$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$79$ ($$( 1 - 12 T + 410 T^{2} - 3636 T^{3} + 71661 T^{4} - 457104 T^{5} + 6962296 T^{6} - 26461552 T^{7} + 422059634 T^{8} - 425195336 T^{9} + 24696310492 T^{10} - 33590431544 T^{11} + 2634074175794 T^{12} - 13046577136528 T^{13} + 271181993145976 T^{14} - 1406534788208496 T^{15} + 17419890150090381 T^{16} - 69825413073674124 T^{17} + 622014612061690010 T^{18} - 1438219151791419828 T^{19} + 9468276082626847201 T^{20} )^{2}$$)
$83$ ($$1 - 40 T + 800 T^{2} - 11352 T^{3} + 130714 T^{4} - 1275720 T^{5} + 10891552 T^{6} - 81158136 T^{7} + 533032573 T^{8} - 3591219744 T^{9} + 28582322304 T^{10} - 250649734880 T^{11} + 2277897381816 T^{12} - 18560960241440 T^{13} + 98547001707136 T^{14} + 326327140527904 T^{15} - 18693002705618542 T^{16} + 320247589539147216 T^{17} - 4017303386407089728 T^{18} + 43457553116792444976 T^{19} -$$$$41\!\cdots\!72$$$$T^{20} +$$$$36\!\cdots\!08$$$$T^{21} -$$$$27\!\cdots\!92$$$$T^{22} +$$$$18\!\cdots\!92$$$$T^{23} -$$$$88\!\cdots\!82$$$$T^{24} +$$$$12\!\cdots\!72$$$$T^{25} +$$$$32\!\cdots\!84$$$$T^{26} -$$$$50\!\cdots\!80$$$$T^{27} +$$$$51\!\cdots\!56$$$$T^{28} -$$$$46\!\cdots\!40$$$$T^{29} +$$$$44\!\cdots\!96$$$$T^{30} -$$$$46\!\cdots\!48$$$$T^{31} +$$$$56\!\cdots\!53$$$$T^{32} -$$$$72\!\cdots\!68$$$$T^{33} +$$$$80\!\cdots\!08$$$$T^{34} -$$$$77\!\cdots\!40$$$$T^{35} +$$$$66\!\cdots\!34$$$$T^{36} -$$$$47\!\cdots\!96$$$$T^{37} +$$$$27\!\cdots\!00$$$$T^{38} -$$$$11\!\cdots\!80$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$)
$89$ ($$1 - 700 T^{2} + 256046 T^{4} - 64249660 T^{6} + 12305003221 T^{8} - 1904804480368 T^{10} + 247648665045240 T^{12} - 27922948227608560 T^{14} + 2822449388027483546 T^{16} -$$$$26\!\cdots\!68$$$$T^{18} +$$$$23\!\cdots\!96$$$$T^{20} -$$$$21\!\cdots\!28$$$$T^{22} +$$$$17\!\cdots\!86$$$$T^{24} -$$$$13\!\cdots\!60$$$$T^{26} +$$$$97\!\cdots\!40$$$$T^{28} -$$$$59\!\cdots\!68$$$$T^{30} +$$$$30\!\cdots\!41$$$$T^{32} -$$$$12\!\cdots\!60$$$$T^{34} +$$$$39\!\cdots\!06$$$$T^{36} -$$$$85\!\cdots\!00$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$)
$97$ ($$( 1 + 36 T + 1006 T^{2} + 17860 T^{3} + 282157 T^{4} + 3451280 T^{5} + 41738920 T^{6} + 425104976 T^{7} + 4638925202 T^{8} + 43768470968 T^{9} + 461725986516 T^{10} + 4245541683896 T^{11} + 43647647225618 T^{12} + 387981833760848 T^{13} + 3695116577316520 T^{14} + 29637315682178960 T^{15} + 235028881994751853 T^{16} + 1443057360779098180 T^{17} + 7884458195943222766 T^{18} + 27368318111564347812 T^{19} + 73742412689492826049 T^{20} )^{2}$$)