# Properties

 Label 170.2.n Level $170$ Weight $2$ Character orbit 170.n Rep. character $\chi_{170}(9,\cdot)$ Character field $\Q(\zeta_{8})$ Dimension $40$ Newform subspaces $2$ Sturm bound $54$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.n (of order $$8$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$85$$ Character field: $$\Q(\zeta_{8})$$ Newform subspaces: $$2$$ Sturm bound: $$54$$ Trace bound: $$10$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(170, [\chi])$$.

Total New Old
Modular forms 120 40 80
Cusp forms 88 40 48
Eisenstein series 32 0 32

## Trace form

 $$40q - 8q^{5} + O(q^{10})$$ $$40q - 8q^{5} + 12q^{10} - 16q^{11} + 24q^{15} - 40q^{16} + 4q^{20} - 4q^{25} - 24q^{26} - 24q^{29} + 16q^{31} + 16q^{34} - 16q^{35} + 8q^{41} - 16q^{44} - 44q^{45} + 32q^{46} - 112q^{49} + 16q^{50} - 16q^{51} - 48q^{54} + 32q^{59} + 24q^{60} + 16q^{61} - 36q^{65} - 16q^{66} - 32q^{69} - 16q^{70} + 16q^{71} + 56q^{74} + 64q^{75} + 112q^{79} + 8q^{80} + 32q^{84} + 68q^{85} + 96q^{86} + 20q^{90} - 48q^{91} + 64q^{94} + 96q^{95} + 144q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(170, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
170.2.n.a $$20$$ $$1.357$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\beta _{7}q^{2}-\beta _{6}q^{3}-\beta _{10}q^{4}+\beta _{14}q^{5}+\cdots$$
170.2.n.b $$20$$ $$1.357$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\beta _{8}q^{2}+\beta _{5}q^{3}+\beta _{10}q^{4}-\beta _{16}q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(170, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(170, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(85, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{4} )^{5}$$)($$( 1 + T^{4} )^{5}$$)
$3$ ($$1 - 8 T^{3} + 32 T^{5} + 28 T^{6} - 72 T^{7} - 234 T^{8} - 24 T^{9} + 1224 T^{10} + 728 T^{11} - 2488 T^{12} - 6728 T^{13} + 1256 T^{14} + 27240 T^{15} + 14713 T^{16} - 60616 T^{17} - 122316 T^{18} + 66920 T^{19} + 485264 T^{20} + 200760 T^{21} - 1100844 T^{22} - 1636632 T^{23} + 1191753 T^{24} + 6619320 T^{25} + 915624 T^{26} - 14714136 T^{27} - 16323768 T^{28} + 14329224 T^{29} + 72275976 T^{30} - 4251528 T^{31} - 124357194 T^{32} - 114791256 T^{33} + 133923132 T^{34} + 459165024 T^{35} - 1033121304 T^{37} + 3486784401 T^{40}$$)($$1 + 8 T^{3} - 32 T^{5} + 28 T^{6} + 72 T^{7} - 234 T^{8} + 24 T^{9} + 1224 T^{10} - 728 T^{11} - 2488 T^{12} + 6728 T^{13} + 1256 T^{14} - 27240 T^{15} + 14713 T^{16} + 60616 T^{17} - 122316 T^{18} - 66920 T^{19} + 485264 T^{20} - 200760 T^{21} - 1100844 T^{22} + 1636632 T^{23} + 1191753 T^{24} - 6619320 T^{25} + 915624 T^{26} + 14714136 T^{27} - 16323768 T^{28} - 14329224 T^{29} + 72275976 T^{30} + 4251528 T^{31} - 124357194 T^{32} + 114791256 T^{33} + 133923132 T^{34} - 459165024 T^{35} + 1033121304 T^{37} + 3486784401 T^{40}$$)
$5$ ($$1 + 4 T + 14 T^{2} + 28 T^{3} + 41 T^{4} + 32 T^{5} - 72 T^{6} - 416 T^{7} - 1550 T^{8} - 4944 T^{9} - 10916 T^{10} - 24720 T^{11} - 38750 T^{12} - 52000 T^{13} - 45000 T^{14} + 100000 T^{15} + 640625 T^{16} + 2187500 T^{17} + 5468750 T^{18} + 7812500 T^{19} + 9765625 T^{20}$$)($$1 + 4 T + 4 T^{2} - 4 T^{3} + T^{4} + 32 T^{5} + 24 T^{6} + 224 T^{7} + 906 T^{8} - 752 T^{9} - 7288 T^{10} - 3760 T^{11} + 22650 T^{12} + 28000 T^{13} + 15000 T^{14} + 100000 T^{15} + 15625 T^{16} - 312500 T^{17} + 1562500 T^{18} + 7812500 T^{19} + 9765625 T^{20}$$)
$7$ ($$1 + 28 T^{2} + 72 T^{3} + 392 T^{4} + 2152 T^{5} + 5636 T^{6} + 31696 T^{7} + 93057 T^{8} + 321232 T^{9} + 1251096 T^{10} + 3073136 T^{11} + 11857952 T^{12} + 32094064 T^{13} + 87528552 T^{14} + 298482128 T^{15} + 656393822 T^{16} + 2152770768 T^{17} + 5629095584 T^{18} + 13238676192 T^{19} + 44114872624 T^{20} + 92670733344 T^{21} + 275825683616 T^{22} + 738400373424 T^{23} + 1576001566622 T^{24} + 5016589125296 T^{25} + 10297646614248 T^{26} + 26430841748752 T^{27} + 68358733547552 T^{28} + 124012122401552 T^{29} + 353403654122904 T^{30} + 635180624307376 T^{31} + 1288028663063457 T^{32} + 3070994073860272 T^{33} + 3822465238576964 T^{34} + 10216752369397336 T^{35} + 13027308783283592 T^{36} + 16749397007078904 T^{37} + 45595580741492572 T^{38} + 79792266297612001 T^{40}$$)($$1 + 28 T^{2} - 72 T^{3} + 392 T^{4} - 2152 T^{5} + 5636 T^{6} - 31696 T^{7} + 93057 T^{8} - 321232 T^{9} + 1251096 T^{10} - 3073136 T^{11} + 11857952 T^{12} - 32094064 T^{13} + 87528552 T^{14} - 298482128 T^{15} + 656393822 T^{16} - 2152770768 T^{17} + 5629095584 T^{18} - 13238676192 T^{19} + 44114872624 T^{20} - 92670733344 T^{21} + 275825683616 T^{22} - 738400373424 T^{23} + 1576001566622 T^{24} - 5016589125296 T^{25} + 10297646614248 T^{26} - 26430841748752 T^{27} + 68358733547552 T^{28} - 124012122401552 T^{29} + 353403654122904 T^{30} - 635180624307376 T^{31} + 1288028663063457 T^{32} - 3070994073860272 T^{33} + 3822465238576964 T^{34} - 10216752369397336 T^{35} + 13027308783283592 T^{36} - 16749397007078904 T^{37} + 45595580741492572 T^{38} + 79792266297612001 T^{40}$$)
$11$ ($$1 + 8 T + 32 T^{2} + 88 T^{3} + 192 T^{4} + 1336 T^{5} + 8272 T^{6} + 31784 T^{7} + 74909 T^{8} + 75696 T^{9} + 299120 T^{10} + 2946256 T^{11} + 14548096 T^{12} + 38626064 T^{13} + 40847888 T^{14} + 70277616 T^{15} + 1208780434 T^{16} + 7630937792 T^{17} + 25235168784 T^{18} + 46671777024 T^{19} + 75581355904 T^{20} + 513389547264 T^{21} + 3053455422864 T^{22} + 10156778201152 T^{23} + 17697754334194 T^{24} + 11318280334416 T^{25} + 72364525313168 T^{26} + 752712714224944 T^{27} + 3118513579240576 T^{28} + 6947117532294896 T^{29} + 7758402446651120 T^{30} + 21596952218570256 T^{31} + 235096531271793389 T^{32} + 1097269882782702904 T^{33} + 3141290623400569552 T^{34} + 5580803554339309736 T^{35} + 8822348133805854912 T^{36} + 44479338507937851848 T^{37} +$$$$17\!\cdots\!92$$$$T^{38} +$$$$48\!\cdots\!28$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)($$1 + 8 T + 32 T^{2} + 88 T^{3} + 192 T^{4} + 1336 T^{5} + 8272 T^{6} + 31784 T^{7} + 74909 T^{8} + 75696 T^{9} + 299120 T^{10} + 2946256 T^{11} + 14548096 T^{12} + 38626064 T^{13} + 40847888 T^{14} + 70277616 T^{15} + 1208780434 T^{16} + 7630937792 T^{17} + 25235168784 T^{18} + 46671777024 T^{19} + 75581355904 T^{20} + 513389547264 T^{21} + 3053455422864 T^{22} + 10156778201152 T^{23} + 17697754334194 T^{24} + 11318280334416 T^{25} + 72364525313168 T^{26} + 752712714224944 T^{27} + 3118513579240576 T^{28} + 6947117532294896 T^{29} + 7758402446651120 T^{30} + 21596952218570256 T^{31} + 235096531271793389 T^{32} + 1097269882782702904 T^{33} + 3141290623400569552 T^{34} + 5580803554339309736 T^{35} + 8822348133805854912 T^{36} + 44479338507937851848 T^{37} +$$$$17\!\cdots\!92$$$$T^{38} +$$$$48\!\cdots\!28$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$13$ ($$( 1 + 12 T + 126 T^{2} + 944 T^{3} + 6370 T^{4} + 36804 T^{5} + 195036 T^{6} + 930636 T^{7} + 4101389 T^{8} + 16591780 T^{9} + 62248284 T^{10} + 215693140 T^{11} + 693134741 T^{12} + 2044607292 T^{13} + 5570423196 T^{14} + 13665067572 T^{15} + 30746773330 T^{16} + 59234600048 T^{17} + 102782070846 T^{18} + 127253992476 T^{19} + 137858491849 T^{20} )^{2}$$)($$( 1 - 12 T + 126 T^{2} - 944 T^{3} + 6370 T^{4} - 36804 T^{5} + 195036 T^{6} - 930636 T^{7} + 4101389 T^{8} - 16591780 T^{9} + 62248284 T^{10} - 215693140 T^{11} + 693134741 T^{12} - 2044607292 T^{13} + 5570423196 T^{14} - 13665067572 T^{15} + 30746773330 T^{16} - 59234600048 T^{17} + 102782070846 T^{18} - 127253992476 T^{19} + 137858491849 T^{20} )^{2}$$)
$17$ ($$1 + 18 T^{2} - 16 T^{3} + 803 T^{4} - 1696 T^{5} + 11588 T^{6} - 48032 T^{7} + 238596 T^{8} - 1229744 T^{9} + 3632100 T^{10} - 20905648 T^{11} + 68954244 T^{12} - 235981216 T^{13} + 967841348 T^{14} - 2408077472 T^{15} + 19382467907 T^{16} - 6565418768 T^{17} + 125563633938 T^{18} + 2015993900449 T^{20}$$)($$1 + 18 T^{2} + 16 T^{3} + 803 T^{4} + 1696 T^{5} + 11588 T^{6} + 48032 T^{7} + 238596 T^{8} + 1229744 T^{9} + 3632100 T^{10} + 20905648 T^{11} + 68954244 T^{12} + 235981216 T^{13} + 967841348 T^{14} + 2408077472 T^{15} + 19382467907 T^{16} + 6565418768 T^{17} + 125563633938 T^{18} + 2015993900449 T^{20}$$)
$19$ ($$1 - 104 T^{3} - 440 T^{4} - 1768 T^{5} + 5408 T^{6} - 5432 T^{7} + 242710 T^{8} + 738648 T^{9} + 4507360 T^{10} + 7934112 T^{11} + 6690854 T^{12} - 462658112 T^{13} - 1445661280 T^{14} - 8971090888 T^{15} - 40777542279 T^{16} - 43430694488 T^{17} + 214418884704 T^{18} + 3590696280576 T^{19} + 25541206222212 T^{20} + 68223229330944 T^{21} + 77405217378144 T^{22} - 297891133493192 T^{23} - 5314170087341559 T^{24} - 22213309176685912 T^{25} - 68012408545187680 T^{26} - 413557011135896768 T^{27} + 113634540707127014 T^{28} + 2560240335200737248 T^{29} + 27634922807761915360 T^{30} + 86045296754651667912 T^{31} +$$$$53\!\cdots\!10$$$$T^{32} -$$$$22\!\cdots\!88$$$$T^{33} +$$$$43\!\cdots\!68$$$$T^{34} -$$$$26\!\cdots\!32$$$$T^{35} -$$$$12\!\cdots\!40$$$$T^{36} -$$$$56\!\cdots\!56$$$$T^{37} +$$$$37\!\cdots\!01$$$$T^{40}$$)($$1 - 104 T^{3} - 440 T^{4} - 1768 T^{5} + 5408 T^{6} - 5432 T^{7} + 242710 T^{8} + 738648 T^{9} + 4507360 T^{10} + 7934112 T^{11} + 6690854 T^{12} - 462658112 T^{13} - 1445661280 T^{14} - 8971090888 T^{15} - 40777542279 T^{16} - 43430694488 T^{17} + 214418884704 T^{18} + 3590696280576 T^{19} + 25541206222212 T^{20} + 68223229330944 T^{21} + 77405217378144 T^{22} - 297891133493192 T^{23} - 5314170087341559 T^{24} - 22213309176685912 T^{25} - 68012408545187680 T^{26} - 413557011135896768 T^{27} + 113634540707127014 T^{28} + 2560240335200737248 T^{29} + 27634922807761915360 T^{30} + 86045296754651667912 T^{31} +$$$$53\!\cdots\!10$$$$T^{32} -$$$$22\!\cdots\!88$$$$T^{33} +$$$$43\!\cdots\!68$$$$T^{34} -$$$$26\!\cdots\!32$$$$T^{35} -$$$$12\!\cdots\!40$$$$T^{36} -$$$$56\!\cdots\!56$$$$T^{37} +$$$$37\!\cdots\!01$$$$T^{40}$$)
$23$ ($$1 - 16 T + 56 T^{2} + 376 T^{3} - 2400 T^{4} - 7256 T^{5} + 72440 T^{6} + 121072 T^{7} - 2621531 T^{8} + 4314208 T^{9} + 70866208 T^{10} - 388785408 T^{11} - 779461504 T^{12} + 12408165568 T^{13} - 8863476320 T^{14} - 287308681888 T^{15} + 756303717514 T^{16} + 5606916061312 T^{17} - 30507258885872 T^{18} - 66054575985616 T^{19} + 960733835992000 T^{20} - 1519255247669168 T^{21} - 16138339950626288 T^{22} + 68219347717983104 T^{23} + 211644788612835274 T^{24} - 1849217223509055584 T^{25} - 1312112596661648480 T^{26} + 42247637876515608896 T^{27} - 61040398366850122624 T^{28} -$$$$70\!\cdots\!04$$$$T^{29} +$$$$29\!\cdots\!92$$$$T^{30} +$$$$41\!\cdots\!16$$$$T^{31} -$$$$57\!\cdots\!51$$$$T^{32} +$$$$61\!\cdots\!76$$$$T^{33} +$$$$83\!\cdots\!60$$$$T^{34} -$$$$19\!\cdots\!92$$$$T^{35} -$$$$14\!\cdots\!00$$$$T^{36} +$$$$53\!\cdots\!28$$$$T^{37} +$$$$18\!\cdots\!64$$$$T^{38} -$$$$11\!\cdots\!92$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)($$1 + 16 T + 56 T^{2} - 376 T^{3} - 2400 T^{4} + 7256 T^{5} + 72440 T^{6} - 121072 T^{7} - 2621531 T^{8} - 4314208 T^{9} + 70866208 T^{10} + 388785408 T^{11} - 779461504 T^{12} - 12408165568 T^{13} - 8863476320 T^{14} + 287308681888 T^{15} + 756303717514 T^{16} - 5606916061312 T^{17} - 30507258885872 T^{18} + 66054575985616 T^{19} + 960733835992000 T^{20} + 1519255247669168 T^{21} - 16138339950626288 T^{22} - 68219347717983104 T^{23} + 211644788612835274 T^{24} + 1849217223509055584 T^{25} - 1312112596661648480 T^{26} - 42247637876515608896 T^{27} - 61040398366850122624 T^{28} +$$$$70\!\cdots\!04$$$$T^{29} +$$$$29\!\cdots\!92$$$$T^{30} -$$$$41\!\cdots\!16$$$$T^{31} -$$$$57\!\cdots\!51$$$$T^{32} -$$$$61\!\cdots\!76$$$$T^{33} +$$$$83\!\cdots\!60$$$$T^{34} +$$$$19\!\cdots\!92$$$$T^{35} -$$$$14\!\cdots\!00$$$$T^{36} -$$$$53\!\cdots\!28$$$$T^{37} +$$$$18\!\cdots\!64$$$$T^{38} +$$$$11\!\cdots\!92$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$29$ ($$1 + 12 T + 38 T^{2} + 212 T^{3} + 2978 T^{4} + 17140 T^{5} + 92422 T^{6} + 571116 T^{7} + 2331938 T^{8} + 14922092 T^{9} + 130048102 T^{10} + 770925860 T^{11} + 3851687522 T^{12} + 19564235252 T^{13} + 133504284918 T^{14} + 966299457452 T^{15} + 4642708862081 T^{16} + 19643962031712 T^{17} + 123559244554608 T^{18} + 710977606960016 T^{19} + 3560114011124736 T^{20} + 20618350601840464 T^{21} + 103913324670425328 T^{22} + 479096589991423968 T^{23} + 3283699766681511761 T^{24} + 19819912150417132348 T^{25} + 79411462122654972678 T^{26} +$$$$33\!\cdots\!68$$$$T^{27} +$$$$19\!\cdots\!42$$$$T^{28} +$$$$11\!\cdots\!40$$$$T^{29} +$$$$54\!\cdots\!02$$$$T^{30} +$$$$18\!\cdots\!68$$$$T^{31} +$$$$82\!\cdots\!58$$$$T^{32} +$$$$58\!\cdots\!24$$$$T^{33} +$$$$27\!\cdots\!82$$$$T^{34} +$$$$14\!\cdots\!60$$$$T^{35} +$$$$74\!\cdots\!38$$$$T^{36} +$$$$15\!\cdots\!08$$$$T^{37} +$$$$79\!\cdots\!18$$$$T^{38} +$$$$73\!\cdots\!28$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)($$1 + 12 T + 38 T^{2} + 212 T^{3} + 2978 T^{4} + 17140 T^{5} + 92422 T^{6} + 571116 T^{7} + 2331938 T^{8} + 14922092 T^{9} + 130048102 T^{10} + 770925860 T^{11} + 3851687522 T^{12} + 19564235252 T^{13} + 133504284918 T^{14} + 966299457452 T^{15} + 4642708862081 T^{16} + 19643962031712 T^{17} + 123559244554608 T^{18} + 710977606960016 T^{19} + 3560114011124736 T^{20} + 20618350601840464 T^{21} + 103913324670425328 T^{22} + 479096589991423968 T^{23} + 3283699766681511761 T^{24} + 19819912150417132348 T^{25} + 79411462122654972678 T^{26} +$$$$33\!\cdots\!68$$$$T^{27} +$$$$19\!\cdots\!42$$$$T^{28} +$$$$11\!\cdots\!40$$$$T^{29} +$$$$54\!\cdots\!02$$$$T^{30} +$$$$18\!\cdots\!68$$$$T^{31} +$$$$82\!\cdots\!58$$$$T^{32} +$$$$58\!\cdots\!24$$$$T^{33} +$$$$27\!\cdots\!82$$$$T^{34} +$$$$14\!\cdots\!60$$$$T^{35} +$$$$74\!\cdots\!38$$$$T^{36} +$$$$15\!\cdots\!08$$$$T^{37} +$$$$79\!\cdots\!18$$$$T^{38} +$$$$73\!\cdots\!28$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$31$ ($$1 - 8 T + 4 T^{2} + 384 T^{3} - 2296 T^{4} + 6096 T^{5} + 40040 T^{6} - 153440 T^{7} + 1584090 T^{8} - 14476232 T^{9} + 89328068 T^{10} + 315411224 T^{11} - 3760050272 T^{12} + 22889881832 T^{13} + 17254141740 T^{14} - 332815622952 T^{15} + 2990654769257 T^{16} - 6114910723832 T^{17} + 38570348356996 T^{18} + 367744608569728 T^{19} - 1791803809774288 T^{20} + 11400082865661568 T^{21} + 37066104771073156 T^{22} - 182169305373679112 T^{23} + 2761932483158993897 T^{24} - 9528228724651873752 T^{25} + 15313114306745744940 T^{26} +$$$$62\!\cdots\!52$$$$T^{27} -$$$$32\!\cdots\!52$$$$T^{28} +$$$$83\!\cdots\!04$$$$T^{29} +$$$$73\!\cdots\!68$$$$T^{30} -$$$$36\!\cdots\!92$$$$T^{31} +$$$$12\!\cdots\!90$$$$T^{32} -$$$$37\!\cdots\!40$$$$T^{33} +$$$$30\!\cdots\!40$$$$T^{34} +$$$$14\!\cdots\!96$$$$T^{35} -$$$$16\!\cdots\!76$$$$T^{36} +$$$$86\!\cdots\!24$$$$T^{37} +$$$$27\!\cdots\!64$$$$T^{38} -$$$$17\!\cdots\!68$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)($$1 - 8 T + 4 T^{2} + 384 T^{3} - 2296 T^{4} + 6096 T^{5} + 40040 T^{6} - 153440 T^{7} + 1584090 T^{8} - 14476232 T^{9} + 89328068 T^{10} + 315411224 T^{11} - 3760050272 T^{12} + 22889881832 T^{13} + 17254141740 T^{14} - 332815622952 T^{15} + 2990654769257 T^{16} - 6114910723832 T^{17} + 38570348356996 T^{18} + 367744608569728 T^{19} - 1791803809774288 T^{20} + 11400082865661568 T^{21} + 37066104771073156 T^{22} - 182169305373679112 T^{23} + 2761932483158993897 T^{24} - 9528228724651873752 T^{25} + 15313114306745744940 T^{26} +$$$$62\!\cdots\!52$$$$T^{27} -$$$$32\!\cdots\!52$$$$T^{28} +$$$$83\!\cdots\!04$$$$T^{29} +$$$$73\!\cdots\!68$$$$T^{30} -$$$$36\!\cdots\!92$$$$T^{31} +$$$$12\!\cdots\!90$$$$T^{32} -$$$$37\!\cdots\!40$$$$T^{33} +$$$$30\!\cdots\!40$$$$T^{34} +$$$$14\!\cdots\!96$$$$T^{35} -$$$$16\!\cdots\!76$$$$T^{36} +$$$$86\!\cdots\!24$$$$T^{37} +$$$$27\!\cdots\!64$$$$T^{38} -$$$$17\!\cdots\!68$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$37$ ($$1 + 8 T + 118 T^{2} + 960 T^{3} + 8114 T^{4} + 50464 T^{5} + 338982 T^{6} + 1440600 T^{7} + 4335441 T^{8} - 9800336 T^{9} - 346829856 T^{10} - 3402505696 T^{11} - 26710359232 T^{12} - 167815912224 T^{13} - 904985189600 T^{14} - 3602804409392 T^{15} - 5581747209506 T^{16} + 63717621771584 T^{17} + 1054915539916916 T^{18} + 8107624670209056 T^{19} + 59840742028993692 T^{20} + 299982112797735072 T^{21} + 1444179374146258004 T^{22} + 3227488695596044352 T^{23} - 10461092931914974466 T^{24} -$$$$24\!\cdots\!44$$$$T^{25} -$$$$23\!\cdots\!00$$$$T^{26} -$$$$15\!\cdots\!92$$$$T^{27} -$$$$93\!\cdots\!72$$$$T^{28} -$$$$44\!\cdots\!92$$$$T^{29} -$$$$16\!\cdots\!44$$$$T^{30} -$$$$17\!\cdots\!68$$$$T^{31} +$$$$28\!\cdots\!21$$$$T^{32} +$$$$35\!\cdots\!00$$$$T^{33} +$$$$30\!\cdots\!98$$$$T^{34} +$$$$16\!\cdots\!52$$$$T^{35} +$$$$10\!\cdots\!74$$$$T^{36} +$$$$43\!\cdots\!20$$$$T^{37} +$$$$19\!\cdots\!22$$$$T^{38} +$$$$49\!\cdots\!84$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$)($$1 - 8 T + 118 T^{2} - 960 T^{3} + 8114 T^{4} - 50464 T^{5} + 338982 T^{6} - 1440600 T^{7} + 4335441 T^{8} + 9800336 T^{9} - 346829856 T^{10} + 3402505696 T^{11} - 26710359232 T^{12} + 167815912224 T^{13} - 904985189600 T^{14} + 3602804409392 T^{15} - 5581747209506 T^{16} - 63717621771584 T^{17} + 1054915539916916 T^{18} - 8107624670209056 T^{19} + 59840742028993692 T^{20} - 299982112797735072 T^{21} + 1444179374146258004 T^{22} - 3227488695596044352 T^{23} - 10461092931914974466 T^{24} +$$$$24\!\cdots\!44$$$$T^{25} -$$$$23\!\cdots\!00$$$$T^{26} +$$$$15\!\cdots\!92$$$$T^{27} -$$$$93\!\cdots\!72$$$$T^{28} +$$$$44\!\cdots\!92$$$$T^{29} -$$$$16\!\cdots\!44$$$$T^{30} +$$$$17\!\cdots\!68$$$$T^{31} +$$$$28\!\cdots\!21$$$$T^{32} -$$$$35\!\cdots\!00$$$$T^{33} +$$$$30\!\cdots\!98$$$$T^{34} -$$$$16\!\cdots\!52$$$$T^{35} +$$$$10\!\cdots\!74$$$$T^{36} -$$$$43\!\cdots\!20$$$$T^{37} +$$$$19\!\cdots\!22$$$$T^{38} -$$$$49\!\cdots\!84$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$)
$41$ ($$1 - 4 T + 30 T^{2} + 92 T^{3} - 334 T^{4} - 10116 T^{5} + 85854 T^{6} - 842916 T^{7} - 2808099 T^{8} + 21400784 T^{9} - 13842264 T^{10} - 1079831088 T^{11} + 13460410040 T^{12} + 2388153040 T^{13} - 195580560792 T^{14} + 2644046148048 T^{15} + 10500427183938 T^{16} - 192141210610008 T^{17} + 664554433081812 T^{18} - 1101475111191192 T^{19} - 30298126262993364 T^{20} - 45160479558838872 T^{21} + 1117116002010525972 T^{22} - 13242564376452361368 T^{23} + 29671697619711826818 T^{24} +$$$$30\!\cdots\!48$$$$T^{25} -$$$$92\!\cdots\!72$$$$T^{26} +$$$$46\!\cdots\!40$$$$T^{27} +$$$$10\!\cdots\!40$$$$T^{28} -$$$$35\!\cdots\!68$$$$T^{29} -$$$$18\!\cdots\!64$$$$T^{30} +$$$$11\!\cdots\!44$$$$T^{31} -$$$$63\!\cdots\!19$$$$T^{32} -$$$$77\!\cdots\!36$$$$T^{33} +$$$$32\!\cdots\!94$$$$T^{34} -$$$$15\!\cdots\!16$$$$T^{35} -$$$$21\!\cdots\!94$$$$T^{36} +$$$$24\!\cdots\!52$$$$T^{37} +$$$$32\!\cdots\!30$$$$T^{38} -$$$$17\!\cdots\!44$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)($$1 - 4 T + 30 T^{2} + 92 T^{3} - 334 T^{4} - 10116 T^{5} + 85854 T^{6} - 842916 T^{7} - 2808099 T^{8} + 21400784 T^{9} - 13842264 T^{10} - 1079831088 T^{11} + 13460410040 T^{12} + 2388153040 T^{13} - 195580560792 T^{14} + 2644046148048 T^{15} + 10500427183938 T^{16} - 192141210610008 T^{17} + 664554433081812 T^{18} - 1101475111191192 T^{19} - 30298126262993364 T^{20} - 45160479558838872 T^{21} + 1117116002010525972 T^{22} - 13242564376452361368 T^{23} + 29671697619711826818 T^{24} +$$$$30\!\cdots\!48$$$$T^{25} -$$$$92\!\cdots\!72$$$$T^{26} +$$$$46\!\cdots\!40$$$$T^{27} +$$$$10\!\cdots\!40$$$$T^{28} -$$$$35\!\cdots\!68$$$$T^{29} -$$$$18\!\cdots\!64$$$$T^{30} +$$$$11\!\cdots\!44$$$$T^{31} -$$$$63\!\cdots\!19$$$$T^{32} -$$$$77\!\cdots\!36$$$$T^{33} +$$$$32\!\cdots\!94$$$$T^{34} -$$$$15\!\cdots\!16$$$$T^{35} -$$$$21\!\cdots\!94$$$$T^{36} +$$$$24\!\cdots\!52$$$$T^{37} +$$$$32\!\cdots\!30$$$$T^{38} -$$$$17\!\cdots\!44$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$43$ ($$1 - 16 T + 128 T^{2} - 1488 T^{3} + 20694 T^{4} - 176960 T^{5} + 1289600 T^{6} - 12374208 T^{7} + 114689745 T^{8} - 816890480 T^{9} + 5773220224 T^{10} - 47396668080 T^{11} + 376357329952 T^{12} - 2558318885360 T^{13} + 17028142721920 T^{14} - 127218542416432 T^{15} + 963177088145694 T^{16} - 6226032087351344 T^{17} + 38642451157210368 T^{18} - 278587214299949552 T^{19} + 2002378843515302292 T^{20} - 11979250214897830736 T^{21} + 71449892189681970432 T^{22} -$$$$49\!\cdots\!08$$$$T^{23} +$$$$32\!\cdots\!94$$$$T^{24} -$$$$18\!\cdots\!76$$$$T^{25} +$$$$10\!\cdots\!80$$$$T^{26} -$$$$69\!\cdots\!20$$$$T^{27} +$$$$43\!\cdots\!52$$$$T^{28} -$$$$23\!\cdots\!40$$$$T^{29} +$$$$12\!\cdots\!76$$$$T^{30} -$$$$75\!\cdots\!60$$$$T^{31} +$$$$45\!\cdots\!45$$$$T^{32} -$$$$21\!\cdots\!44$$$$T^{33} +$$$$95\!\cdots\!00$$$$T^{34} -$$$$56\!\cdots\!20$$$$T^{35} +$$$$28\!\cdots\!94$$$$T^{36} -$$$$87\!\cdots\!84$$$$T^{37} +$$$$32\!\cdots\!72$$$$T^{38} -$$$$17\!\cdots\!12$$$$T^{39} +$$$$46\!\cdots\!01$$$$T^{40}$$)($$1 + 16 T + 128 T^{2} + 1488 T^{3} + 20694 T^{4} + 176960 T^{5} + 1289600 T^{6} + 12374208 T^{7} + 114689745 T^{8} + 816890480 T^{9} + 5773220224 T^{10} + 47396668080 T^{11} + 376357329952 T^{12} + 2558318885360 T^{13} + 17028142721920 T^{14} + 127218542416432 T^{15} + 963177088145694 T^{16} + 6226032087351344 T^{17} + 38642451157210368 T^{18} + 278587214299949552 T^{19} + 2002378843515302292 T^{20} + 11979250214897830736 T^{21} + 71449892189681970432 T^{22} +$$$$49\!\cdots\!08$$$$T^{23} +$$$$32\!\cdots\!94$$$$T^{24} +$$$$18\!\cdots\!76$$$$T^{25} +$$$$10\!\cdots\!80$$$$T^{26} +$$$$69\!\cdots\!20$$$$T^{27} +$$$$43\!\cdots\!52$$$$T^{28} +$$$$23\!\cdots\!40$$$$T^{29} +$$$$12\!\cdots\!76$$$$T^{30} +$$$$75\!\cdots\!60$$$$T^{31} +$$$$45\!\cdots\!45$$$$T^{32} +$$$$21\!\cdots\!44$$$$T^{33} +$$$$95\!\cdots\!00$$$$T^{34} +$$$$56\!\cdots\!20$$$$T^{35} +$$$$28\!\cdots\!94$$$$T^{36} +$$$$87\!\cdots\!84$$$$T^{37} +$$$$32\!\cdots\!72$$$$T^{38} +$$$$17\!\cdots\!12$$$$T^{39} +$$$$46\!\cdots\!01$$$$T^{40}$$)
$47$ ($$( 1 - 20 T + 388 T^{2} - 4976 T^{3} + 62092 T^{4} - 635100 T^{5} + 6214238 T^{6} - 53431788 T^{7} + 439450571 T^{8} - 3289992044 T^{9} + 23554230924 T^{10} - 154629626068 T^{11} + 970746311339 T^{12} - 5547448525524 T^{13} + 30323499098078 T^{14} - 145657013945700 T^{15} + 669303038208268 T^{16} - 2520956647423888 T^{17} + 9238779224763268 T^{18} - 22382609462055340 T^{19} + 52599132235830049 T^{20} )^{2}$$)($$( 1 + 20 T + 388 T^{2} + 4976 T^{3} + 62092 T^{4} + 635100 T^{5} + 6214238 T^{6} + 53431788 T^{7} + 439450571 T^{8} + 3289992044 T^{9} + 23554230924 T^{10} + 154629626068 T^{11} + 970746311339 T^{12} + 5547448525524 T^{13} + 30323499098078 T^{14} + 145657013945700 T^{15} + 669303038208268 T^{16} + 2520956647423888 T^{17} + 9238779224763268 T^{18} + 22382609462055340 T^{19} + 52599132235830049 T^{20} )^{2}$$)
$53$ ($$1 - 44 T + 968 T^{2} - 14204 T^{3} + 153360 T^{4} - 1253852 T^{5} + 7593816 T^{6} - 29525844 T^{7} + 21121854 T^{8} + 436104500 T^{9} + 1235332696 T^{10} - 80090622156 T^{11} + 970719706210 T^{12} - 6135983063340 T^{13} + 7164913588248 T^{14} + 319649478179492 T^{15} - 4261370408835583 T^{16} + 31329479689545576 T^{17} - 136710820885404048 T^{18} + 162474207458147632 T^{19} + 1532805880285790428 T^{20} + 8611132995281824496 T^{21} -$$$$38\!\cdots\!32$$$$T^{22} +$$$$46\!\cdots\!52$$$$T^{23} -$$$$33\!\cdots\!23$$$$T^{24} +$$$$13\!\cdots\!56$$$$T^{25} +$$$$15\!\cdots\!92$$$$T^{26} -$$$$72\!\cdots\!80$$$$T^{27} +$$$$60\!\cdots\!10$$$$T^{28} -$$$$26\!\cdots\!48$$$$T^{29} +$$$$21\!\cdots\!04$$$$T^{30} +$$$$40\!\cdots\!00$$$$T^{31} +$$$$10\!\cdots\!14$$$$T^{32} -$$$$76\!\cdots\!12$$$$T^{33} +$$$$10\!\cdots\!04$$$$T^{34} -$$$$91\!\cdots\!64$$$$T^{35} +$$$$59\!\cdots\!60$$$$T^{36} -$$$$29\!\cdots\!52$$$$T^{37} +$$$$10\!\cdots\!52$$$$T^{38} -$$$$25\!\cdots\!48$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$)($$1 + 44 T + 968 T^{2} + 14204 T^{3} + 153360 T^{4} + 1253852 T^{5} + 7593816 T^{6} + 29525844 T^{7} + 21121854 T^{8} - 436104500 T^{9} + 1235332696 T^{10} + 80090622156 T^{11} + 970719706210 T^{12} + 6135983063340 T^{13} + 7164913588248 T^{14} - 319649478179492 T^{15} - 4261370408835583 T^{16} - 31329479689545576 T^{17} - 136710820885404048 T^{18} - 162474207458147632 T^{19} + 1532805880285790428 T^{20} - 8611132995281824496 T^{21} -$$$$38\!\cdots\!32$$$$T^{22} -$$$$46\!\cdots\!52$$$$T^{23} -$$$$33\!\cdots\!23$$$$T^{24} -$$$$13\!\cdots\!56$$$$T^{25} +$$$$15\!\cdots\!92$$$$T^{26} +$$$$72\!\cdots\!80$$$$T^{27} +$$$$60\!\cdots\!10$$$$T^{28} +$$$$26\!\cdots\!48$$$$T^{29} +$$$$21\!\cdots\!04$$$$T^{30} -$$$$40\!\cdots\!00$$$$T^{31} +$$$$10\!\cdots\!14$$$$T^{32} +$$$$76\!\cdots\!12$$$$T^{33} +$$$$10\!\cdots\!04$$$$T^{34} +$$$$91\!\cdots\!64$$$$T^{35} +$$$$59\!\cdots\!60$$$$T^{36} +$$$$29\!\cdots\!52$$$$T^{37} +$$$$10\!\cdots\!52$$$$T^{38} +$$$$25\!\cdots\!48$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$)
$59$ ($$1 - 16 T + 128 T^{2} - 640 T^{3} + 6032 T^{4} - 73144 T^{5} + 603008 T^{6} - 4225576 T^{7} + 33076134 T^{8} - 244311232 T^{9} + 1417281312 T^{10} - 16887395168 T^{11} + 211053947230 T^{12} - 1522774288880 T^{13} + 6460849018400 T^{14} - 56720710039568 T^{15} + 808791885028057 T^{16} - 6447237579834824 T^{17} + 35399211288359360 T^{18} - 220142009338344872 T^{19} + 1853897520463942116 T^{20} - 12988378550962347448 T^{21} +$$$$12\!\cdots\!60$$$$T^{22} -$$$$13\!\cdots\!96$$$$T^{23} +$$$$98\!\cdots\!77$$$$T^{24} -$$$$40\!\cdots\!32$$$$T^{25} +$$$$27\!\cdots\!00$$$$T^{26} -$$$$37\!\cdots\!20$$$$T^{27} +$$$$30\!\cdots\!30$$$$T^{28} -$$$$14\!\cdots\!52$$$$T^{29} +$$$$72\!\cdots\!12$$$$T^{30} -$$$$73\!\cdots\!88$$$$T^{31} +$$$$58\!\cdots\!54$$$$T^{32} -$$$$44\!\cdots\!04$$$$T^{33} +$$$$37\!\cdots\!88$$$$T^{34} -$$$$26\!\cdots\!56$$$$T^{35} +$$$$13\!\cdots\!12$$$$T^{36} -$$$$81\!\cdots\!60$$$$T^{37} +$$$$96\!\cdots\!88$$$$T^{38} -$$$$70\!\cdots\!24$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$)($$1 - 16 T + 128 T^{2} - 640 T^{3} + 6032 T^{4} - 73144 T^{5} + 603008 T^{6} - 4225576 T^{7} + 33076134 T^{8} - 244311232 T^{9} + 1417281312 T^{10} - 16887395168 T^{11} + 211053947230 T^{12} - 1522774288880 T^{13} + 6460849018400 T^{14} - 56720710039568 T^{15} + 808791885028057 T^{16} - 6447237579834824 T^{17} + 35399211288359360 T^{18} - 220142009338344872 T^{19} + 1853897520463942116 T^{20} - 12988378550962347448 T^{21} +$$$$12\!\cdots\!60$$$$T^{22} -$$$$13\!\cdots\!96$$$$T^{23} +$$$$98\!\cdots\!77$$$$T^{24} -$$$$40\!\cdots\!32$$$$T^{25} +$$$$27\!\cdots\!00$$$$T^{26} -$$$$37\!\cdots\!20$$$$T^{27} +$$$$30\!\cdots\!30$$$$T^{28} -$$$$14\!\cdots\!52$$$$T^{29} +$$$$72\!\cdots\!12$$$$T^{30} -$$$$73\!\cdots\!88$$$$T^{31} +$$$$58\!\cdots\!54$$$$T^{32} -$$$$44\!\cdots\!04$$$$T^{33} +$$$$37\!\cdots\!88$$$$T^{34} -$$$$26\!\cdots\!56$$$$T^{35} +$$$$13\!\cdots\!12$$$$T^{36} -$$$$81\!\cdots\!60$$$$T^{37} +$$$$96\!\cdots\!88$$$$T^{38} -$$$$70\!\cdots\!24$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$)
$61$ ($$1 - 8 T + 10 T^{2} + 320 T^{3} - 2126 T^{4} - 27296 T^{5} - 72510 T^{6} + 8840 T^{7} - 3520990 T^{8} - 69652728 T^{9} + 659172698 T^{10} + 1929507872 T^{11} + 77537561570 T^{12} + 128446570208 T^{13} + 1861119756730 T^{14} + 37614663351992 T^{15} - 71694040127423 T^{16} - 1493114370031936 T^{17} + 3398743476006728 T^{18} - 110814064642641248 T^{19} - 1163583925604399136 T^{20} - 6759657943201116128 T^{21} + 12646724474221034888 T^{22} -$$$$33\!\cdots\!16$$$$T^{23} -$$$$99\!\cdots\!43$$$$T^{24} +$$$$31\!\cdots\!92$$$$T^{25} +$$$$95\!\cdots\!30$$$$T^{26} +$$$$40\!\cdots\!68$$$$T^{27} +$$$$14\!\cdots\!70$$$$T^{28} +$$$$22\!\cdots\!52$$$$T^{29} +$$$$47\!\cdots\!98$$$$T^{30} -$$$$30\!\cdots\!08$$$$T^{31} -$$$$93\!\cdots\!90$$$$T^{32} +$$$$14\!\cdots\!40$$$$T^{33} -$$$$71\!\cdots\!10$$$$T^{34} -$$$$16\!\cdots\!96$$$$T^{35} -$$$$78\!\cdots\!86$$$$T^{36} +$$$$71\!\cdots\!20$$$$T^{37} +$$$$13\!\cdots\!10$$$$T^{38} -$$$$66\!\cdots\!28$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$)($$1 - 8 T + 10 T^{2} + 320 T^{3} - 2126 T^{4} - 27296 T^{5} - 72510 T^{6} + 8840 T^{7} - 3520990 T^{8} - 69652728 T^{9} + 659172698 T^{10} + 1929507872 T^{11} + 77537561570 T^{12} + 128446570208 T^{13} + 1861119756730 T^{14} + 37614663351992 T^{15} - 71694040127423 T^{16} - 1493114370031936 T^{17} + 3398743476006728 T^{18} - 110814064642641248 T^{19} - 1163583925604399136 T^{20} - 6759657943201116128 T^{21} + 12646724474221034888 T^{22} -$$$$33\!\cdots\!16$$$$T^{23} -$$$$99\!\cdots\!43$$$$T^{24} +$$$$31\!\cdots\!92$$$$T^{25} +$$$$95\!\cdots\!30$$$$T^{26} +$$$$40\!\cdots\!68$$$$T^{27} +$$$$14\!\cdots\!70$$$$T^{28} +$$$$22\!\cdots\!52$$$$T^{29} +$$$$47\!\cdots\!98$$$$T^{30} -$$$$30\!\cdots\!08$$$$T^{31} -$$$$93\!\cdots\!90$$$$T^{32} +$$$$14\!\cdots\!40$$$$T^{33} -$$$$71\!\cdots\!10$$$$T^{34} -$$$$16\!\cdots\!96$$$$T^{35} -$$$$78\!\cdots\!86$$$$T^{36} +$$$$71\!\cdots\!20$$$$T^{37} +$$$$13\!\cdots\!10$$$$T^{38} -$$$$66\!\cdots\!28$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$)
$67$ ($$1 - 532 T^{2} + 148646 T^{4} - 29200180 T^{6} + 4500891213 T^{8} - 575578725168 T^{10} + 63179028972648 T^{12} - 6089051557526448 T^{14} + 523435323978922754 T^{16} - 40558419543281652984 T^{18} +$$$$28\!\cdots\!68$$$$T^{20} -$$$$18\!\cdots\!76$$$$T^{22} +$$$$10\!\cdots\!34$$$$T^{24} -$$$$55\!\cdots\!12$$$$T^{26} +$$$$25\!\cdots\!68$$$$T^{28} -$$$$10\!\cdots\!32$$$$T^{30} +$$$$36\!\cdots\!93$$$$T^{32} -$$$$10\!\cdots\!20$$$$T^{34} +$$$$24\!\cdots\!26$$$$T^{36} -$$$$39\!\cdots\!88$$$$T^{38} +$$$$33\!\cdots\!01$$$$T^{40}$$)($$1 - 532 T^{2} + 148646 T^{4} - 29200180 T^{6} + 4500891213 T^{8} - 575578725168 T^{10} + 63179028972648 T^{12} - 6089051557526448 T^{14} + 523435323978922754 T^{16} - 40558419543281652984 T^{18} +$$$$28\!\cdots\!68$$$$T^{20} -$$$$18\!\cdots\!76$$$$T^{22} +$$$$10\!\cdots\!34$$$$T^{24} -$$$$55\!\cdots\!12$$$$T^{26} +$$$$25\!\cdots\!68$$$$T^{28} -$$$$10\!\cdots\!32$$$$T^{30} +$$$$36\!\cdots\!93$$$$T^{32} -$$$$10\!\cdots\!20$$$$T^{34} +$$$$24\!\cdots\!26$$$$T^{36} -$$$$39\!\cdots\!88$$$$T^{38} +$$$$33\!\cdots\!01$$$$T^{40}$$)
$71$ ($$1 - 8 T + 4 T^{2} + 1064 T^{3} - 7736 T^{4} + 33056 T^{5} + 411864 T^{6} - 3498384 T^{7} + 43320906 T^{8} - 175160784 T^{9} + 1711157444 T^{10} + 23184409480 T^{11} - 282606722400 T^{12} + 2190349172904 T^{13} + 5955655055116 T^{14} - 92105865793280 T^{15} + 1484388456856825 T^{16} - 5521262001586648 T^{17} + 13570667323190324 T^{18} + 605072991295826840 T^{19} - 4474880001515114064 T^{20} + 42960182382003705640 T^{21} + 68409733976202423284 T^{22} -$$$$19\!\cdots\!28$$$$T^{23} +$$$$37\!\cdots\!25$$$$T^{24} -$$$$16\!\cdots\!80$$$$T^{25} +$$$$76\!\cdots\!36$$$$T^{26} +$$$$19\!\cdots\!64$$$$T^{27} -$$$$18\!\cdots\!00$$$$T^{28} +$$$$10\!\cdots\!80$$$$T^{29} +$$$$55\!\cdots\!44$$$$T^{30} -$$$$40\!\cdots\!64$$$$T^{31} +$$$$71\!\cdots\!46$$$$T^{32} -$$$$40\!\cdots\!24$$$$T^{33} +$$$$34\!\cdots\!84$$$$T^{34} +$$$$19\!\cdots\!56$$$$T^{35} -$$$$32\!\cdots\!56$$$$T^{36} +$$$$31\!\cdots\!24$$$$T^{37} +$$$$84\!\cdots\!44$$$$T^{38} -$$$$11\!\cdots\!48$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$)($$1 - 8 T + 4 T^{2} + 1064 T^{3} - 7736 T^{4} + 33056 T^{5} + 411864 T^{6} - 3498384 T^{7} + 43320906 T^{8} - 175160784 T^{9} + 1711157444 T^{10} + 23184409480 T^{11} - 282606722400 T^{12} + 2190349172904 T^{13} + 5955655055116 T^{14} - 92105865793280 T^{15} + 1484388456856825 T^{16} - 5521262001586648 T^{17} + 13570667323190324 T^{18} + 605072991295826840 T^{19} - 4474880001515114064 T^{20} + 42960182382003705640 T^{21} + 68409733976202423284 T^{22} -$$$$19\!\cdots\!28$$$$T^{23} +$$$$37\!\cdots\!25$$$$T^{24} -$$$$16\!\cdots\!80$$$$T^{25} +$$$$76\!\cdots\!36$$$$T^{26} +$$$$19\!\cdots\!64$$$$T^{27} -$$$$18\!\cdots\!00$$$$T^{28} +$$$$10\!\cdots\!80$$$$T^{29} +$$$$55\!\cdots\!44$$$$T^{30} -$$$$40\!\cdots\!64$$$$T^{31} +$$$$71\!\cdots\!46$$$$T^{32} -$$$$40\!\cdots\!24$$$$T^{33} +$$$$34\!\cdots\!84$$$$T^{34} +$$$$19\!\cdots\!56$$$$T^{35} -$$$$32\!\cdots\!56$$$$T^{36} +$$$$31\!\cdots\!24$$$$T^{37} +$$$$84\!\cdots\!44$$$$T^{38} -$$$$11\!\cdots\!48$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$)
$73$ ($$1 + 60 T + 1682 T^{2} + 29988 T^{3} + 398642 T^{4} + 4449180 T^{5} + 44616114 T^{6} + 400755436 T^{7} + 3146402174 T^{8} + 22577269012 T^{9} + 172416257546 T^{10} + 1548661495372 T^{11} + 15112755685578 T^{12} + 149613932704668 T^{13} + 1492299482027146 T^{14} + 14514912884993716 T^{15} + 131595396462860353 T^{16} + 1114942726481802512 T^{17} + 9296763449815475432 T^{18} + 79328198294011301400 T^{19} +$$$$68\!\cdots\!92$$$$T^{20} +$$$$57\!\cdots\!00$$$$T^{21} +$$$$49\!\cdots\!28$$$$T^{22} +$$$$43\!\cdots\!04$$$$T^{23} +$$$$37\!\cdots\!73$$$$T^{24} +$$$$30\!\cdots\!88$$$$T^{25} +$$$$22\!\cdots\!94$$$$T^{26} +$$$$16\!\cdots\!96$$$$T^{27} +$$$$12\!\cdots\!18$$$$T^{28} +$$$$91\!\cdots\!36$$$$T^{29} +$$$$74\!\cdots\!54$$$$T^{30} +$$$$70\!\cdots\!24$$$$T^{31} +$$$$72\!\cdots\!54$$$$T^{32} +$$$$67\!\cdots\!88$$$$T^{33} +$$$$54\!\cdots\!26$$$$T^{34} +$$$$39\!\cdots\!60$$$$T^{35} +$$$$25\!\cdots\!62$$$$T^{36} +$$$$14\!\cdots\!64$$$$T^{37} +$$$$58\!\cdots\!58$$$$T^{38} +$$$$15\!\cdots\!20$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)($$1 - 60 T + 1682 T^{2} - 29988 T^{3} + 398642 T^{4} - 4449180 T^{5} + 44616114 T^{6} - 400755436 T^{7} + 3146402174 T^{8} - 22577269012 T^{9} + 172416257546 T^{10} - 1548661495372 T^{11} + 15112755685578 T^{12} - 149613932704668 T^{13} + 1492299482027146 T^{14} - 14514912884993716 T^{15} + 131595396462860353 T^{16} - 1114942726481802512 T^{17} + 9296763449815475432 T^{18} - 79328198294011301400 T^{19} +$$$$68\!\cdots\!92$$$$T^{20} -$$$$57\!\cdots\!00$$$$T^{21} +$$$$49\!\cdots\!28$$$$T^{22} -$$$$43\!\cdots\!04$$$$T^{23} +$$$$37\!\cdots\!73$$$$T^{24} -$$$$30\!\cdots\!88$$$$T^{25} +$$$$22\!\cdots\!94$$$$T^{26} -$$$$16\!\cdots\!96$$$$T^{27} +$$$$12\!\cdots\!18$$$$T^{28} -$$$$91\!\cdots\!36$$$$T^{29} +$$$$74\!\cdots\!54$$$$T^{30} -$$$$70\!\cdots\!24$$$$T^{31} +$$$$72\!\cdots\!54$$$$T^{32} -$$$$67\!\cdots\!88$$$$T^{33} +$$$$54\!\cdots\!26$$$$T^{34} -$$$$39\!\cdots\!60$$$$T^{35} +$$$$25\!\cdots\!62$$$$T^{36} -$$$$14\!\cdots\!64$$$$T^{37} +$$$$58\!\cdots\!58$$$$T^{38} -$$$$15\!\cdots\!20$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$79$ ($$1 - 56 T + 1608 T^{2} - 32544 T^{3} + 531232 T^{4} - 7502656 T^{5} + 94723432 T^{6} - 1079876968 T^{7} + 11152309445 T^{8} - 104467318464 T^{9} + 885876705344 T^{10} - 6748309500960 T^{11} + 45237704790656 T^{12} - 254352077824672 T^{13} + 1043944393809024 T^{14} - 1173550884529024 T^{15} - 28330582069716470 T^{16} + 326793601144278544 T^{17} - 2173623899474329584 T^{18} + 10801150466444693600 T^{19} - 64674172953547878720 T^{20} +$$$$85\!\cdots\!00$$$$T^{21} -$$$$13\!\cdots\!44$$$$T^{22} +$$$$16\!\cdots\!16$$$$T^{23} -$$$$11\!\cdots\!70$$$$T^{24} -$$$$36\!\cdots\!76$$$$T^{25} +$$$$25\!\cdots\!04$$$$T^{26} -$$$$48\!\cdots\!48$$$$T^{27} +$$$$68\!\cdots\!16$$$$T^{28} -$$$$80\!\cdots\!40$$$$T^{29} +$$$$83\!\cdots\!44$$$$T^{30} -$$$$78\!\cdots\!56$$$$T^{31} +$$$$65\!\cdots\!45$$$$T^{32} -$$$$50\!\cdots\!52$$$$T^{33} +$$$$34\!\cdots\!92$$$$T^{34} -$$$$21\!\cdots\!44$$$$T^{35} +$$$$12\!\cdots\!72$$$$T^{36} -$$$$59\!\cdots\!96$$$$T^{37} +$$$$23\!\cdots\!88$$$$T^{38} -$$$$63\!\cdots\!64$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$)($$1 - 56 T + 1608 T^{2} - 32544 T^{3} + 531232 T^{4} - 7502656 T^{5} + 94723432 T^{6} - 1079876968 T^{7} + 11152309445 T^{8} - 104467318464 T^{9} + 885876705344 T^{10} - 6748309500960 T^{11} + 45237704790656 T^{12} - 254352077824672 T^{13} + 1043944393809024 T^{14} - 1173550884529024 T^{15} - 28330582069716470 T^{16} + 326793601144278544 T^{17} - 2173623899474329584 T^{18} + 10801150466444693600 T^{19} - 64674172953547878720 T^{20} +$$$$85\!\cdots\!00$$$$T^{21} -$$$$13\!\cdots\!44$$$$T^{22} +$$$$16\!\cdots\!16$$$$T^{23} -$$$$11\!\cdots\!70$$$$T^{24} -$$$$36\!\cdots\!76$$$$T^{25} +$$$$25\!\cdots\!04$$$$T^{26} -$$$$48\!\cdots\!48$$$$T^{27} +$$$$68\!\cdots\!16$$$$T^{28} -$$$$80\!\cdots\!40$$$$T^{29} +$$$$83\!\cdots\!44$$$$T^{30} -$$$$78\!\cdots\!56$$$$T^{31} +$$$$65\!\cdots\!45$$$$T^{32} -$$$$50\!\cdots\!52$$$$T^{33} +$$$$34\!\cdots\!92$$$$T^{34} -$$$$21\!\cdots\!44$$$$T^{35} +$$$$12\!\cdots\!72$$$$T^{36} -$$$$59\!\cdots\!96$$$$T^{37} +$$$$23\!\cdots\!88$$$$T^{38} -$$$$63\!\cdots\!64$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$)
$83$ ($$1 - 544 T^{3} - 1354 T^{4} + 59152 T^{5} + 147968 T^{6} + 546160 T^{7} - 31611727 T^{8} + 119552016 T^{9} + 1652717184 T^{10} - 6405201136 T^{11} + 252699874336 T^{12} + 937926276176 T^{13} + 17620618729088 T^{14} - 121733300402992 T^{15} - 1091081305080034 T^{16} + 10833819307749168 T^{17} + 164063275302237440 T^{18} + 183331162121083376 T^{19} - 10465764546854435628 T^{20} + 15216486456049920208 T^{21} +$$$$11\!\cdots\!60$$$$T^{22} +$$$$61\!\cdots\!16$$$$T^{23} -$$$$51\!\cdots\!14$$$$T^{24} -$$$$47\!\cdots\!56$$$$T^{25} +$$$$57\!\cdots\!72$$$$T^{26} +$$$$25\!\cdots\!52$$$$T^{27} +$$$$56\!\cdots\!76$$$$T^{28} -$$$$11\!\cdots\!08$$$$T^{29} +$$$$25\!\cdots\!16$$$$T^{30} +$$$$15\!\cdots\!72$$$$T^{31} -$$$$33\!\cdots\!47$$$$T^{32} +$$$$48\!\cdots\!80$$$$T^{33} +$$$$10\!\cdots\!72$$$$T^{34} +$$$$36\!\cdots\!64$$$$T^{35} -$$$$68\!\cdots\!74$$$$T^{36} -$$$$22\!\cdots\!12$$$$T^{37} +$$$$24\!\cdots\!01$$$$T^{40}$$)($$1 + 544 T^{3} - 1354 T^{4} - 59152 T^{5} + 147968 T^{6} - 546160 T^{7} - 31611727 T^{8} - 119552016 T^{9} + 1652717184 T^{10} + 6405201136 T^{11} + 252699874336 T^{12} - 937926276176 T^{13} + 17620618729088 T^{14} + 121733300402992 T^{15} - 1091081305080034 T^{16} - 10833819307749168 T^{17} + 164063275302237440 T^{18} - 183331162121083376 T^{19} - 10465764546854435628 T^{20} - 15216486456049920208 T^{21} +$$$$11\!\cdots\!60$$$$T^{22} -$$$$61\!\cdots\!16$$$$T^{23} -$$$$51\!\cdots\!14$$$$T^{24} +$$$$47\!\cdots\!56$$$$T^{25} +$$$$57\!\cdots\!72$$$$T^{26} -$$$$25\!\cdots\!52$$$$T^{27} +$$$$56\!\cdots\!76$$$$T^{28} +$$$$11\!\cdots\!08$$$$T^{29} +$$$$25\!\cdots\!16$$$$T^{30} -$$$$15\!\cdots\!72$$$$T^{31} -$$$$33\!\cdots\!47$$$$T^{32} -$$$$48\!\cdots\!80$$$$T^{33} +$$$$10\!\cdots\!72$$$$T^{34} -$$$$36\!\cdots\!64$$$$T^{35} -$$$$68\!\cdots\!74$$$$T^{36} +$$$$22\!\cdots\!12$$$$T^{37} +$$$$24\!\cdots\!01$$$$T^{40}$$)
$89$ ($$1 - 948 T^{2} + 453936 T^{4} - 146353496 T^{6} + 35701988702 T^{8} - 7013810546556 T^{10} + 1152109743195558 T^{12} - 162048052656574324 T^{14} + 19816119477140894897 T^{16} -$$$$21\!\cdots\!68$$$$T^{18} +$$$$20\!\cdots\!84$$$$T^{20} -$$$$16\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!77$$$$T^{24} -$$$$80\!\cdots\!64$$$$T^{26} +$$$$45\!\cdots\!98$$$$T^{28} -$$$$21\!\cdots\!56$$$$T^{30} +$$$$88\!\cdots\!42$$$$T^{32} -$$$$28\!\cdots\!36$$$$T^{34} +$$$$70\!\cdots\!96$$$$T^{36} -$$$$11\!\cdots\!88$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$)($$1 - 948 T^{2} + 453936 T^{4} - 146353496 T^{6} + 35701988702 T^{8} - 7013810546556 T^{10} + 1152109743195558 T^{12} - 162048052656574324 T^{14} + 19816119477140894897 T^{16} -$$$$21\!\cdots\!68$$$$T^{18} +$$$$20\!\cdots\!84$$$$T^{20} -$$$$16\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!77$$$$T^{24} -$$$$80\!\cdots\!64$$$$T^{26} +$$$$45\!\cdots\!98$$$$T^{28} -$$$$21\!\cdots\!56$$$$T^{30} +$$$$88\!\cdots\!42$$$$T^{32} -$$$$28\!\cdots\!36$$$$T^{34} +$$$$70\!\cdots\!96$$$$T^{36} -$$$$11\!\cdots\!88$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$)
$97$ ($$1 - 48 T + 1142 T^{2} - 16936 T^{3} + 160946 T^{4} - 603200 T^{5} - 12193658 T^{6} + 329836064 T^{7} - 4766618146 T^{8} + 48871055912 T^{9} - 342706004482 T^{10} + 607199068352 T^{11} + 28990974681866 T^{12} - 605921738623824 T^{13} + 7806367455647038 T^{14} - 74491245978555832 T^{15} + 484575109311053409 T^{16} - 549564314450980192 T^{17} - 41722742455227040440 T^{18} +$$$$77\!\cdots\!16$$$$T^{19} -$$$$90\!\cdots\!40$$$$T^{20} +$$$$75\!\cdots\!52$$$$T^{21} -$$$$39\!\cdots\!60$$$$T^{22} -$$$$50\!\cdots\!16$$$$T^{23} +$$$$42\!\cdots\!29$$$$T^{24} -$$$$63\!\cdots\!24$$$$T^{25} +$$$$65\!\cdots\!02$$$$T^{26} -$$$$48\!\cdots\!12$$$$T^{27} +$$$$22\!\cdots\!26$$$$T^{28} +$$$$46\!\cdots\!84$$$$T^{29} -$$$$25\!\cdots\!18$$$$T^{30} +$$$$34\!\cdots\!36$$$$T^{31} -$$$$33\!\cdots\!86$$$$T^{32} +$$$$22\!\cdots\!28$$$$T^{33} -$$$$79\!\cdots\!02$$$$T^{34} -$$$$38\!\cdots\!00$$$$T^{35} +$$$$98\!\cdots\!66$$$$T^{36} -$$$$10\!\cdots\!32$$$$T^{37} +$$$$66\!\cdots\!38$$$$T^{38} -$$$$26\!\cdots\!84$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$)($$1 + 48 T + 1142 T^{2} + 16936 T^{3} + 160946 T^{4} + 603200 T^{5} - 12193658 T^{6} - 329836064 T^{7} - 4766618146 T^{8} - 48871055912 T^{9} - 342706004482 T^{10} - 607199068352 T^{11} + 28990974681866 T^{12} + 605921738623824 T^{13} + 7806367455647038 T^{14} + 74491245978555832 T^{15} + 484575109311053409 T^{16} + 549564314450980192 T^{17} - 41722742455227040440 T^{18} -$$$$77\!\cdots\!16$$$$T^{19} -$$$$90\!\cdots\!40$$$$T^{20} -$$$$75\!\cdots\!52$$$$T^{21} -$$$$39\!\cdots\!60$$$$T^{22} +$$$$50\!\cdots\!16$$$$T^{23} +$$$$42\!\cdots\!29$$$$T^{24} +$$$$63\!\cdots\!24$$$$T^{25} +$$$$65\!\cdots\!02$$$$T^{26} +$$$$48\!\cdots\!12$$$$T^{27} +$$$$22\!\cdots\!26$$$$T^{28} -$$$$46\!\cdots\!84$$$$T^{29} -$$$$25\!\cdots\!18$$$$T^{30} -$$$$34\!\cdots\!36$$$$T^{31} -$$$$33\!\cdots\!86$$$$T^{32} -$$$$22\!\cdots\!28$$$$T^{33} -$$$$79\!\cdots\!02$$$$T^{34} +$$$$38\!\cdots\!00$$$$T^{35} +$$$$98\!\cdots\!66$$$$T^{36} +$$$$10\!\cdots\!32$$$$T^{37} +$$$$66\!\cdots\!38$$$$T^{38} +$$$$26\!\cdots\!84$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$)