# Properties

 Label 350.2.h Level 350 Weight 2 Character orbit h Rep. character $$\chi_{350}(71,\cdot)$$ Character field $$\Q(\zeta_{5})$$ Dimension 56 Newform subspaces 4 Sturm bound 120 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.h (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$4$$ Sturm bound: $$120$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(350, [\chi])$$.

Total New Old
Modular forms 256 56 200
Cusp forms 224 56 168
Eisenstein series 32 0 32

## Trace form

 $$56q + 2q^{2} + 8q^{3} - 14q^{4} - 14q^{5} + 2q^{8} - 2q^{9} + O(q^{10})$$ $$56q + 2q^{2} + 8q^{3} - 14q^{4} - 14q^{5} + 2q^{8} - 2q^{9} + 6q^{10} + 4q^{11} - 12q^{12} + 28q^{13} + 4q^{14} - 14q^{16} + 4q^{17} - 32q^{18} - 24q^{19} - 4q^{20} + 4q^{21} - 12q^{22} + 4q^{23} - 66q^{25} + 12q^{26} - 28q^{27} + 12q^{29} + 24q^{31} - 8q^{32} + 8q^{33} - 30q^{34} + 4q^{35} - 2q^{36} + 14q^{37} + 24q^{38} - 12q^{39} + 6q^{40} + 12q^{41} + 24q^{43} + 4q^{44} + 2q^{45} - 4q^{46} - 76q^{47} - 12q^{48} + 56q^{49} + 14q^{50} + 8q^{51} + 28q^{52} + 14q^{53} + 24q^{54} - 36q^{55} + 4q^{56} + 88q^{57} + 12q^{58} + 12q^{59} - 28q^{61} + 16q^{62} - 12q^{63} - 14q^{64} - 46q^{65} - 52q^{67} + 4q^{68} - 36q^{69} + 4q^{70} + 32q^{71} + 18q^{72} - 12q^{73} - 68q^{74} + 80q^{75} + 16q^{76} - 24q^{77} - 88q^{78} - 40q^{79} + 6q^{80} - 6q^{81} + 52q^{82} + 28q^{83} + 4q^{84} - 18q^{85} + 20q^{86} - 144q^{87} + 8q^{88} + 46q^{89} + 62q^{90} + 4q^{91} - 16q^{92} + 32q^{93} + 60q^{95} + 56q^{97} + 2q^{98} + 64q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(350, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
350.2.h.a $$8$$ $$2.795$$ $$\Q(\zeta_{15})$$ None $$-2$$ $$3$$ $$0$$ $$8$$ $$q+\zeta_{15}^{4}q^{2}+(-\zeta_{15}^{3}-\zeta_{15}^{4}+\zeta_{15}^{6}+\cdots)q^{3}+\cdots$$
350.2.h.b $$12$$ $$2.795$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$3$$ $$1$$ $$-5$$ $$-12$$ $$q+\beta _{5}q^{2}+(\beta _{1}-\beta _{3}+\beta _{7})q^{3}-\beta _{3}q^{4}+\cdots$$
350.2.h.c $$16$$ $$2.795$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-4$$ $$1$$ $$-4$$ $$-16$$ $$q+\beta _{6}q^{2}-\beta _{10}q^{3}-\beta _{1}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
350.2.h.d $$20$$ $$2.795$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$5$$ $$3$$ $$-5$$ $$20$$ $$q+\beta _{14}q^{2}+\beta _{1}q^{3}+\beta _{6}q^{4}+(\beta _{10}+\beta _{13}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}$$)
$3$ ($$1 - 3 T + 2 T^{2} + T^{3} + 6 T^{4} - 8 T^{5} - 30 T^{6} + 88 T^{7} - 131 T^{8} + 264 T^{9} - 270 T^{10} - 216 T^{11} + 486 T^{12} + 243 T^{13} + 1458 T^{14} - 6561 T^{15} + 6561 T^{16}$$)($$1 - T - 7 T^{2} + 27 T^{4} + 54 T^{5} - 85 T^{6} - 282 T^{7} + 38 T^{8} + 744 T^{9} + 1035 T^{10} - 917 T^{11} - 4667 T^{12} - 2751 T^{13} + 9315 T^{14} + 20088 T^{15} + 3078 T^{16} - 68526 T^{17} - 61965 T^{18} + 118098 T^{19} + 177147 T^{20} - 413343 T^{22} - 177147 T^{23} + 531441 T^{24}$$)($$1 - T - 5 T^{2} + 12 T^{3} - 11 T^{4} - 34 T^{5} + 135 T^{6} - 166 T^{7} - 223 T^{8} + 1011 T^{9} - 1180 T^{10} - 315 T^{11} + 5026 T^{12} - 9385 T^{13} + 3426 T^{14} + 18392 T^{15} - 50450 T^{16} + 55176 T^{17} + 30834 T^{18} - 253395 T^{19} + 407106 T^{20} - 76545 T^{21} - 860220 T^{22} + 2211057 T^{23} - 1463103 T^{24} - 3267378 T^{25} + 7971615 T^{26} - 6022998 T^{27} - 5845851 T^{28} + 19131876 T^{29} - 23914845 T^{30} - 14348907 T^{31} + 43046721 T^{32}$$)($$1 - 3 T + 9 T^{3} - 5 T^{4} - 11 T^{5} - 28 T^{6} + 181 T^{7} - 201 T^{8} - 198 T^{9} + 342 T^{10} + 432 T^{11} + 391 T^{12} - 6101 T^{13} + 12092 T^{14} - 1175 T^{15} - 16887 T^{16} - 11609 T^{17} + 41866 T^{18} + 124287 T^{19} - 435782 T^{20} + 372861 T^{21} + 376794 T^{22} - 313443 T^{23} - 1367847 T^{24} - 285525 T^{25} + 8815068 T^{26} - 13342887 T^{27} + 2565351 T^{28} + 8503056 T^{29} + 20194758 T^{30} - 35075106 T^{31} - 106819641 T^{32} + 288572463 T^{33} - 133923132 T^{34} - 157837977 T^{35} - 215233605 T^{36} + 1162261467 T^{37} - 3486784401 T^{39} + 3486784401 T^{40}$$)
$5$ ($$( 1 + 5 T^{2} + 25 T^{4} )^{2}$$)($$1 + 5 T + 15 T^{2} + 45 T^{3} + 80 T^{4} + 125 T^{5} + 325 T^{6} + 625 T^{7} + 2000 T^{8} + 5625 T^{9} + 9375 T^{10} + 15625 T^{11} + 15625 T^{12}$$)($$1 + 4 T + 11 T^{2} - 6 T^{3} - 84 T^{4} - 280 T^{5} - 130 T^{6} + 1050 T^{7} + 4525 T^{8} + 5250 T^{9} - 3250 T^{10} - 35000 T^{11} - 52500 T^{12} - 18750 T^{13} + 171875 T^{14} + 312500 T^{15} + 390625 T^{16}$$)($$1 + 5 T + 30 T^{2} + 110 T^{3} + 390 T^{4} + 1125 T^{5} + 3005 T^{6} + 7340 T^{7} + 16455 T^{8} + 37775 T^{9} + 80575 T^{10} + 188875 T^{11} + 411375 T^{12} + 917500 T^{13} + 1878125 T^{14} + 3515625 T^{15} + 6093750 T^{16} + 8593750 T^{17} + 11718750 T^{18} + 9765625 T^{19} + 9765625 T^{20}$$)
$7$ ($$( 1 - T )^{8}$$)($$( 1 + T )^{12}$$)($$( 1 + T )^{16}$$)($$( 1 - T )^{20}$$)
$11$ ($$1 + T - 30 T^{2} + 15 T^{3} + 260 T^{4} - 732 T^{5} + 3428 T^{6} + 5530 T^{7} - 83595 T^{8} + 60830 T^{9} + 414788 T^{10} - 974292 T^{11} + 3806660 T^{12} + 2415765 T^{13} - 53146830 T^{14} + 19487171 T^{15} + 214358881 T^{16}$$)($$1 - 7 T + 15 T^{2} - 48 T^{3} + 369 T^{4} - 1218 T^{5} + 3391 T^{6} - 18040 T^{7} + 68194 T^{8} - 176652 T^{9} + 566355 T^{10} - 2220705 T^{11} + 7824965 T^{12} - 24427755 T^{13} + 68528955 T^{14} - 235123812 T^{15} + 998428354 T^{16} - 2905360040 T^{17} + 6007363351 T^{18} - 23735374278 T^{19} + 79098427089 T^{20} - 113181489168 T^{21} + 389061369015 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24}$$)($$1 - 7 T + 4 T^{2} + T^{3} + 367 T^{4} - 671 T^{5} - 874 T^{6} - 12433 T^{7} + 46158 T^{8} + 27843 T^{9} + 284444 T^{10} - 2592183 T^{11} + 1138482 T^{12} + 4725054 T^{13} + 79059510 T^{14} - 177853274 T^{15} - 239193093 T^{16} - 1956386014 T^{17} + 9566200710 T^{18} + 6289046874 T^{19} + 16668514962 T^{20} - 417473664333 T^{21} + 503909897084 T^{22} + 542581302153 T^{23} + 9894377229198 T^{24} - 29316363642203 T^{25} - 22669309101274 T^{26} - 191444130979981 T^{27} + 1151803214256607 T^{28} + 34522712143931 T^{29} + 1518999334332964 T^{30} - 29240737185909557 T^{31} + 45949729863572161 T^{32}$$)($$1 + 9 T + T^{2} - 270 T^{3} - 1106 T^{4} + 1437 T^{5} + 28254 T^{6} + 81600 T^{7} - 185184 T^{8} - 1956121 T^{9} - 3939441 T^{10} + 14683187 T^{11} + 99595113 T^{12} + 123974460 T^{13} - 845289048 T^{14} - 3927803039 T^{15} - 1004769233 T^{16} + 41106907430 T^{17} + 112816425513 T^{18} - 173691256523 T^{19} - 1636513680309 T^{20} - 1910603821753 T^{21} + 13650787487073 T^{22} + 54713293789330 T^{23} - 14710826340353 T^{24} - 632576607233989 T^{25} - 1497481111163928 T^{26} + 2415911501652660 T^{27} + 21349096975748553 T^{28} + 34622186883171217 T^{29} - 102178953907588041 T^{30} - 558104150427259931 T^{31} - 581186720514701664 T^{32} + 2817053310944769600 T^{33} + 10729451798060891214 T^{34} + 6002705619450290487 T^{35} - 50820401229110810066 T^{36} -$$$$13\!\cdots\!70$$$$T^{37} + 5559917313492231481 T^{38} +$$$$55\!\cdots\!19$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$13$ ($$1 - 11 T + 31 T^{2} + 71 T^{3} - 547 T^{4} + 1564 T^{5} - 4156 T^{6} - 10012 T^{7} + 113083 T^{8} - 130156 T^{9} - 702364 T^{10} + 3436108 T^{11} - 15622867 T^{12} + 26361803 T^{13} + 149631079 T^{14} - 690233687 T^{15} + 815730721 T^{16}$$)($$1 - 3 T - 20 T^{2} + 118 T^{3} + 161 T^{4} - 2778 T^{5} + 507 T^{6} + 49992 T^{7} - 117950 T^{8} - 558088 T^{9} + 2845345 T^{10} + 2678925 T^{11} - 39869651 T^{12} + 34826025 T^{13} + 480863305 T^{14} - 1226119336 T^{15} - 3368769950 T^{16} + 18561679656 T^{17} + 2447192163 T^{18} - 174315380226 T^{19} + 131332646081 T^{20} + 1251330926014 T^{21} - 2757169836980 T^{22} - 5376481182111 T^{23} + 23298085122481 T^{24}$$)($$1 - 9 T + 23 T^{2} - 65 T^{3} + 420 T^{4} + 793 T^{5} - 8537 T^{6} + 3579 T^{7} - 93425 T^{8} + 576885 T^{9} + 290947 T^{10} + 84117 T^{11} - 7580374 T^{12} - 72282820 T^{13} + 134961730 T^{14} - 124529416 T^{15} + 2958255499 T^{16} - 1618882408 T^{17} + 22808532370 T^{18} - 158805355540 T^{19} - 216503061814 T^{20} + 31232053281 T^{21} + 1404345598123 T^{22} + 36198678229545 T^{23} - 76209642609425 T^{24} + 37953503255967 T^{25} - 1176897944914913 T^{26} + 1421183192471341 T^{27} + 9785195751442020 T^{28} - 19686881928496445 T^{29} + 90559656871083647 T^{30} - 460673037126816813 T^{31} + 665416609183179841 T^{32}$$)($$1 - 5 T + 180 T^{4} - 1119 T^{5} + 5205 T^{6} - 8500 T^{7} + 26300 T^{8} - 238895 T^{9} + 685648 T^{10} - 3004655 T^{11} + 11645825 T^{12} - 25367850 T^{13} + 167751875 T^{14} - 586109075 T^{15} + 1407193925 T^{16} - 9517504150 T^{17} + 31286861100 T^{18} - 80311296305 T^{19} + 353659054271 T^{20} - 1044046851965 T^{21} + 5287479525900 T^{22} - 20909956617550 T^{23} + 40190865691925 T^{24} - 217618196783975 T^{25} + 809706260016875 T^{26} - 1591794966978450 T^{27} + 9499857223889825 T^{28} - 31862862063581315 T^{29} + 94522399219283152 T^{30} - 428138157333469115 T^{31} + 612739638721250300 T^{32} - 2574438406034150500 T^{33} + 20494044087564799245 T^{34} - 57277014282767557083 T^{35} +$$$$11\!\cdots\!80$$$$T^{36} -$$$$73\!\cdots\!85$$$$T^{39} +$$$$19\!\cdots\!01$$$$T^{40}$$)
$17$ ($$1 - 14 T + 113 T^{2} - 798 T^{3} + 5296 T^{4} - 29904 T^{5} + 149375 T^{6} - 707504 T^{7} + 3101359 T^{8} - 12027568 T^{9} + 43169375 T^{10} - 146918352 T^{11} + 442327216 T^{12} - 1133045886 T^{13} + 2727545297 T^{14} - 5744741422 T^{15} + 6975757441 T^{16}$$)($$1 - 4 T - 52 T^{2} + 212 T^{3} + 1204 T^{4} - 8304 T^{5} - 1499 T^{6} + 234288 T^{7} - 622968 T^{8} - 4319236 T^{9} + 20524788 T^{10} + 33120788 T^{11} - 413458971 T^{12} + 563053396 T^{13} + 5931663732 T^{14} - 21220406468 T^{15} - 52030910328 T^{16} + 332655456816 T^{17} - 36182215931 T^{18} - 3407452340592 T^{19} + 8398811958964 T^{20} + 25140629817364 T^{21} - 104831682823348 T^{22} - 137087585230532 T^{23} + 582622237229761 T^{24}$$)($$1 + 2 T - 81 T^{2} - 164 T^{3} + 2667 T^{4} + 5428 T^{5} - 41998 T^{6} - 90626 T^{7} + 212692 T^{8} + 1088886 T^{9} + 3872547 T^{10} - 26863856 T^{11} - 147904226 T^{12} + 680142794 T^{13} + 4324416006 T^{14} - 6144073958 T^{15} - 92016330155 T^{16} - 104449257286 T^{17} + 1249756225734 T^{18} + 3341541546922 T^{19} - 12353108859746 T^{20} - 38142833988592 T^{21} + 93473870418243 T^{22} + 446812036288278 T^{23} + 1483687801641172 T^{24} - 10747144895417122 T^{25} - 84667711831057102 T^{26} + 186027853157831924 T^{27} + 1553853506691772587 T^{28} - 1624350797396573668 T^{29} - 13638603951311475249 T^{30} + 5724846103019631586 T^{31} + 48661191875666868481 T^{32}$$)($$1 + 12 T + 6 T^{2} - 466 T^{3} - 2255 T^{4} - 2604 T^{5} + 41979 T^{6} + 475288 T^{7} + 1352471 T^{8} - 6403588 T^{9} - 54774001 T^{10} - 170639242 T^{11} + 107637136 T^{12} + 5518940082 T^{13} + 25925948559 T^{14} + 1565494782 T^{15} - 419550373786 T^{16} - 2073571073444 T^{17} - 3941818822220 T^{18} + 22861596658438 T^{19} + 188979842286129 T^{20} + 388647143193446 T^{21} - 1139185639621580 T^{22} - 10187454683830372 T^{23} - 35041266768980506 T^{24} + 2222778724686174 T^{25} + 625789372233313071 T^{26} + 2264634549614391186 T^{27} + 750850552379928976 T^{28} - 20235745355837695274 T^{29} -$$$$11\!\cdots\!49$$$$T^{30} -$$$$21\!\cdots\!04$$$$T^{31} +$$$$78\!\cdots\!31$$$$T^{32} +$$$$47\!\cdots\!56$$$$T^{33} +$$$$70\!\cdots\!91$$$$T^{34} -$$$$74\!\cdots\!72$$$$T^{35} -$$$$10\!\cdots\!55$$$$T^{36} -$$$$38\!\cdots\!82$$$$T^{37} +$$$$84\!\cdots\!54$$$$T^{38} +$$$$28\!\cdots\!36$$$$T^{39} +$$$$40\!\cdots\!01$$$$T^{40}$$)
$19$ ($$1 + 6 T + 24 T^{2} + 106 T^{3} + 681 T^{4} + 2398 T^{5} + 8776 T^{6} + 57768 T^{7} + 306817 T^{8} + 1097592 T^{9} + 3168136 T^{10} + 16447882 T^{11} + 88748601 T^{12} + 262466494 T^{13} + 1129101144 T^{14} + 5363230434 T^{15} + 16983563041 T^{16}$$)($$1 - 4 T - 55 T^{2} + 322 T^{3} + 786 T^{4} - 10430 T^{5} + 19782 T^{6} + 121858 T^{7} - 830761 T^{8} + 935330 T^{9} + 9745914 T^{10} - 23118154 T^{11} - 60429309 T^{12} - 439244926 T^{13} + 3518274954 T^{14} + 6415428470 T^{15} - 108265604281 T^{16} + 301732471942 T^{17} + 930661617942 T^{18} - 9323082237770 T^{19} + 13349080550226 T^{20} + 103905438684838 T^{21} - 337208644179055 T^{22} - 465961035592876 T^{23} + 2213314919066161 T^{24}$$)($$1 + 24 T + 296 T^{2} + 2592 T^{3} + 17982 T^{4} + 98712 T^{5} + 406844 T^{6} + 1030944 T^{7} - 1201257 T^{8} - 34550296 T^{9} - 246931384 T^{10} - 1149072336 T^{11} - 3482962948 T^{12} - 1595520248 T^{13} + 65096881880 T^{14} + 533938597512 T^{15} + 2732959874037 T^{16} + 10144833352728 T^{17} + 23499974358680 T^{18} - 10943673381032 T^{19} - 453903214346308 T^{20} - 2845216862097264 T^{21} - 11617104506829304 T^{22} - 30883533168484744 T^{23} - 20401623987942537 T^{24} + 332672945899073376 T^{25} + 2494387520588790044 T^{26} + 11498986436360993928 T^{27} + 39799828874647707102 T^{28} +$$$$10\!\cdots\!28$$$$T^{29} +$$$$23\!\cdots\!16$$$$T^{30} +$$$$36\!\cdots\!76$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$)($$1 - 2 T - 35 T^{2} - 22 T^{3} + 475 T^{4} + 682 T^{5} + 13747 T^{6} + 12118 T^{7} - 305918 T^{8} - 809522 T^{9} - 2291715 T^{10} - 21808678 T^{11} + 161227877 T^{12} + 967062682 T^{13} + 279777607 T^{14} - 5638086218 T^{15} - 48823661508 T^{16} - 380542806978 T^{17} + 242658755757 T^{18} + 4750008910922 T^{19} + 14190850058469 T^{20} + 90250169307518 T^{21} + 87599810828277 T^{22} - 2610143113062102 T^{23} - 6362748391384068 T^{24} - 13960459646303582 T^{25} + 13162384005386767 T^{26} + 864430001281343998 T^{27} + 2738223812996093957 T^{28} - 7037392095423526162 T^{29} - 14050656508996418715 T^{30} - 94301427363804041318 T^{31} -$$$$67\!\cdots\!98$$$$T^{32} +$$$$50\!\cdots\!62$$$$T^{33} +$$$$10\!\cdots\!87$$$$T^{34} +$$$$10\!\cdots\!18$$$$T^{35} +$$$$13\!\cdots\!75$$$$T^{36} -$$$$12\!\cdots\!58$$$$T^{37} -$$$$36\!\cdots\!35$$$$T^{38} -$$$$39\!\cdots\!58$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$)
$23$ ($$1 - 15 T + 64 T^{2} - 15 T^{3} + 192 T^{4} - 2310 T^{5} - 10768 T^{6} - 66000 T^{7} + 1144505 T^{8} - 1518000 T^{9} - 5696272 T^{10} - 28105770 T^{11} + 53729472 T^{12} - 96545145 T^{13} + 9474296896 T^{14} - 51072381705 T^{15} + 78310985281 T^{16}$$)($$1 + T - 53 T^{2} - 306 T^{3} + 975 T^{4} + 11712 T^{5} + 26649 T^{6} - 153426 T^{7} - 961598 T^{8} - 2533552 T^{9} - 218007 T^{10} + 52959247 T^{11} + 441957817 T^{12} + 1218062681 T^{13} - 115325703 T^{14} - 30825727184 T^{15} - 269094545918 T^{16} - 987502361118 T^{17} + 3945008405961 T^{18} + 39877315635264 T^{19} + 76353210648975 T^{20} - 551152714407678 T^{21} - 2195605094323397 T^{22} + 952809757913927 T^{23} + 21914624432020321 T^{24}$$)($$1 + 5 T - 22 T^{2} - 25 T^{3} + 495 T^{4} - 7425 T^{5} - 19220 T^{6} + 207375 T^{7} - 35370 T^{8} - 1776625 T^{9} + 23723034 T^{10} - 27251505 T^{11} - 556173278 T^{12} + 857611800 T^{13} - 314171630 T^{14} - 4900274400 T^{15} + 245436206455 T^{16} - 112706311200 T^{17} - 166196792270 T^{18} + 10434562770600 T^{19} - 155640086288798 T^{20} - 175400033446215 T^{21} + 3511860427967226 T^{22} - 6049098009776375 T^{23} - 2769859549388970 T^{24} + 373514033170889625 T^{25} - 796217545526333780 T^{26} - 7074612452510907975 T^{27} + 10847739093850058895 T^{28} - 12600909048411684575 T^{29} -$$$$25\!\cdots\!98$$$$T^{30} +$$$$13\!\cdots\!35$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 + 5 T - 35 T^{2} - 120 T^{3} + 1180 T^{4} + 4415 T^{5} - 24560 T^{6} - 152080 T^{7} + 462030 T^{8} + 3027805 T^{9} - 18239153 T^{10} - 62982805 T^{11} + 423352235 T^{12} + 329737570 T^{13} - 11032544730 T^{14} - 9735553715 T^{15} + 254133694985 T^{16} + 440207234730 T^{17} - 6047843923755 T^{18} - 348343474205 T^{19} + 189940842334619 T^{20} - 8011899906715 T^{21} - 3199309435666395 T^{22} + 5356001424959910 T^{23} + 71117027338297385 T^{24} - 62661363004664245 T^{25} - 1633212567037814970 T^{26} + 1122698869167943790 T^{27} + 33153130643763453035 T^{28} -$$$$11\!\cdots\!15$$$$T^{29} -$$$$75\!\cdots\!97$$$$T^{30} +$$$$28\!\cdots\!35$$$$T^{31} +$$$$10\!\cdots\!30$$$$T^{32} -$$$$76\!\cdots\!40$$$$T^{33} -$$$$28\!\cdots\!40$$$$T^{34} +$$$$11\!\cdots\!05$$$$T^{35} +$$$$72\!\cdots\!80$$$$T^{36} -$$$$16\!\cdots\!60$$$$T^{37} -$$$$11\!\cdots\!15$$$$T^{38} +$$$$37\!\cdots\!35$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$29$ ($$1 + 8 T + 35 T^{2} + 120 T^{3} + 1480 T^{4} + 24 T^{5} - 47443 T^{6} - 312460 T^{7} - 647105 T^{8} - 9061340 T^{9} - 39899563 T^{10} + 585336 T^{11} + 1046775880 T^{12} + 2461337880 T^{13} + 20818816235 T^{14} + 137999010472 T^{15} + 500246412961 T^{16}$$)($$1 - 22 T + 156 T^{2} + 68 T^{3} - 6774 T^{4} + 30632 T^{5} + 46573 T^{6} - 1017678 T^{7} + 5929708 T^{8} - 26575028 T^{9} + 50923194 T^{10} + 552546070 T^{11} - 5342883971 T^{12} + 16023836030 T^{13} + 42826406154 T^{14} - 648138357892 T^{15} + 4193969803948 T^{16} - 20873745092022 T^{17} + 27702706528933 T^{18} + 528398211097288 T^{19} - 3388669201397814 T^{20} + 986485926359092 T^{21} + 65630328394831356 T^{22} - 268411214845528238 T^{23} + 353814783205469041 T^{24}$$)($$1 - 20 T + 201 T^{2} - 1320 T^{3} + 7525 T^{4} - 37240 T^{5} + 89750 T^{6} + 615040 T^{7} - 8311490 T^{8} + 61959320 T^{9} - 387993087 T^{10} + 1800586880 T^{11} - 3797715622 T^{12} - 15900561440 T^{13} + 250739424210 T^{14} - 2172262465880 T^{15} + 13894638163425 T^{16} - 62995611510520 T^{17} + 210871855760610 T^{18} - 387798792960160 T^{19} - 2686052102843782 T^{20} + 36932105783125120 T^{21} - 230787336534381927 T^{22} + 1068790606189749880 T^{23} - 4157793058861221890 T^{24} + 8922475060998469760 T^{25} + 37758474188693039750 T^{26} -$$$$45\!\cdots\!60$$$$T^{27} +$$$$26\!\cdots\!25$$$$T^{28} -$$$$13\!\cdots\!80$$$$T^{29} +$$$$59\!\cdots\!81$$$$T^{30} -$$$$17\!\cdots\!80$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$)($$1 + 22 T + 162 T^{2} + 364 T^{3} + 1221 T^{4} + 31882 T^{5} + 241485 T^{6} + 885670 T^{7} + 3609905 T^{8} + 27103550 T^{9} + 174298901 T^{10} + 735666692 T^{11} + 2745459462 T^{12} + 17332002484 T^{13} + 71322017141 T^{14} - 56024592128 T^{15} - 538624284330 T^{16} + 6226688616470 T^{17} + 24729523205740 T^{18} - 193015275164740 T^{19} - 1931410756140335 T^{20} - 5597442979777460 T^{21} + 20797529016027340 T^{22} + 151862708667086830 T^{23} - 380958722445206730 T^{24} - 1149128756801635072 T^{25} + 42423999096228545261 T^{26} +$$$$29\!\cdots\!56$$$$T^{27} +$$$$13\!\cdots\!82$$$$T^{28} +$$$$10\!\cdots\!48$$$$T^{29} +$$$$73\!\cdots\!01$$$$T^{30} +$$$$33\!\cdots\!50$$$$T^{31} +$$$$12\!\cdots\!05$$$$T^{32} +$$$$90\!\cdots\!30$$$$T^{33} +$$$$71\!\cdots\!85$$$$T^{34} +$$$$27\!\cdots\!18$$$$T^{35} +$$$$30\!\cdots\!41$$$$T^{36} +$$$$26\!\cdots\!76$$$$T^{37} +$$$$34\!\cdots\!82$$$$T^{38} +$$$$13\!\cdots\!18$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$31$ ($$1 + 7 T - 4 T^{2} - 2 T^{3} + 526 T^{4} - 2249 T^{5} + 16384 T^{6} + 151404 T^{7} + 78027 T^{8} + 4693524 T^{9} + 15745024 T^{10} - 66999959 T^{11} + 485772046 T^{12} - 57258302 T^{13} - 3550014724 T^{14} + 192588298777 T^{15} + 852891037441 T^{16}$$)($$1 - 31 T + 453 T^{2} - 4225 T^{3} + 29693 T^{4} - 192850 T^{5} + 1377427 T^{6} - 10076394 T^{7} + 65664126 T^{8} - 379719432 T^{9} + 2153666263 T^{10} - 12728315052 T^{11} + 73593545547 T^{12} - 394577766612 T^{13} + 2069673278743 T^{14} - 11312221598712 T^{15} + 60642199307646 T^{16} - 288478605361494 T^{17} + 1222471532808787 T^{18} - 5305807631306350 T^{19} + 25324893574735613 T^{20} - 111707403628834975 T^{21} + 371291614002302853 T^{22} - 787662783788549761 T^{23} + 787662783788549761 T^{24}$$)($$1 - 7 T - 52 T^{2} + 434 T^{3} - 611 T^{4} + 9420 T^{5} + 20954 T^{6} - 1040008 T^{7} + 3045078 T^{8} - 3464302 T^{9} + 23457032 T^{10} + 1136716529 T^{11} - 7254596780 T^{12} + 2110239704 T^{13} - 18287520106 T^{14} - 556338309516 T^{15} + 9102576366561 T^{16} - 17246487594996 T^{17} - 17574306821866 T^{18} + 62866151021864 T^{19} - 6699772472862380 T^{20} + 32543229152936879 T^{21} + 20818202245334792 T^{22} - 95312004089965522 T^{23} + 2597119734508765398 T^{24} - 27497418564075125368 T^{25} + 17174491125395704154 T^{26} +$$$$23\!\cdots\!20$$$$T^{27} -$$$$48\!\cdots\!71$$$$T^{28} +$$$$10\!\cdots\!94$$$$T^{29} -$$$$39\!\cdots\!92$$$$T^{30} -$$$$16\!\cdots\!57$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$)($$1 + 7 T - 79 T^{2} - 579 T^{3} + 3032 T^{4} + 28139 T^{5} + 4936 T^{6} - 743317 T^{7} - 4912332 T^{8} + 9560721 T^{9} + 212484051 T^{10} + 81948331 T^{11} - 2914483577 T^{12} - 1389679507 T^{13} - 55278648174 T^{14} - 279870156423 T^{15} + 759056753083 T^{16} + 13090764459319 T^{17} + 147368113191249 T^{18} - 221840826023487 T^{19} - 7945817595531147 T^{20} - 6877065606728097 T^{21} + 141620756776790289 T^{22} + 389986964007572329 T^{23} + 701004851663965243 T^{24} - 8012444968627686873 T^{25} - 49060003735128928494 T^{26} - 38233716014055723277 T^{27} -$$$$24\!\cdots\!57$$$$T^{28} +$$$$21\!\cdots\!01$$$$T^{29} +$$$$17\!\cdots\!51$$$$T^{30} +$$$$24\!\cdots\!51$$$$T^{31} -$$$$38\!\cdots\!52$$$$T^{32} -$$$$18\!\cdots\!47$$$$T^{33} +$$$$37\!\cdots\!56$$$$T^{34} +$$$$66\!\cdots\!89$$$$T^{35} +$$$$22\!\cdots\!92$$$$T^{36} -$$$$13\!\cdots\!69$$$$T^{37} -$$$$55\!\cdots\!39$$$$T^{38} +$$$$15\!\cdots\!97$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$37$ ($$1 - 14 T + 8 T^{2} + 1162 T^{3} - 7679 T^{4} - 16574 T^{5} + 334840 T^{6} - 260964 T^{7} - 8129271 T^{8} - 9655668 T^{9} + 458395960 T^{10} - 839522822 T^{11} - 14391682319 T^{12} + 80577678034 T^{13} + 20525811272 T^{14} - 1329046279862 T^{15} + 3512479453921 T^{16}$$)($$1 - 9 T - 17 T^{2} + 47 T^{3} + 2009 T^{4} + 17896 T^{5} - 183604 T^{6} - 200352 T^{7} - 1312908 T^{8} + 29143544 T^{9} + 202793238 T^{10} - 1300954802 T^{11} - 1160573366 T^{12} - 48135327674 T^{13} + 277623942822 T^{14} + 1476207934232 T^{15} - 2460600970188 T^{16} - 13893200472864 T^{17} - 471077631598036 T^{18} + 1698900873172168 T^{19} + 7056571222927289 T^{20} + 6108201770368619 T^{21} - 81745934331103433 T^{22} - 1601258596015143717 T^{23} + 6582952005840035281 T^{24}$$)($$1 + 6 T - 86 T^{2} - 202 T^{3} + 4956 T^{4} - 10220 T^{5} - 180710 T^{6} + 1119974 T^{7} + 3483905 T^{8} - 45334964 T^{9} - 27988086 T^{10} + 613887092 T^{11} + 1192594492 T^{12} + 19789376570 T^{13} - 169429782588 T^{14} - 709747280326 T^{15} + 8113479874525 T^{16} - 26260649372062 T^{17} - 231949372362972 T^{18} + 1002391291400210 T^{19} + 2235114085721212 T^{20} + 42569360110503044 T^{21} - 71809771387563174 T^{22} - 4303733232276978212 T^{23} + 12237144731912641505 T^{24} +$$$$14\!\cdots\!98$$$$T^{25} -$$$$86\!\cdots\!90$$$$T^{26} -$$$$18\!\cdots\!60$$$$T^{27} +$$$$32\!\cdots\!36$$$$T^{28} -$$$$49\!\cdots\!94$$$$T^{29} -$$$$77\!\cdots\!54$$$$T^{30} +$$$$20\!\cdots\!58$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$)($$1 + 3 T - 75 T^{2} - 431 T^{3} + 1910 T^{4} + 1531 T^{5} - 52233 T^{6} + 918601 T^{7} + 7816136 T^{8} - 5198737 T^{9} - 134683907 T^{10} - 1561396851 T^{11} - 16230592826 T^{12} - 14283194053 T^{13} + 439735380987 T^{14} + 1622954356537 T^{15} + 13721182465584 T^{16} + 141107352922883 T^{17} - 5174944573003 T^{18} - 4875052887814471 T^{19} - 34960991903139092 T^{20} - 180376956849135427 T^{21} - 7084499120441107 T^{22} + 7147510747602792599 T^{23} + 25715705050881375024 T^{24} +$$$$11\!\cdots\!09$$$$T^{25} +$$$$11\!\cdots\!83$$$$T^{26} -$$$$13\!\cdots\!49$$$$T^{27} -$$$$57\!\cdots\!46$$$$T^{28} -$$$$20\!\cdots\!27$$$$T^{29} -$$$$64\!\cdots\!43$$$$T^{30} -$$$$92\!\cdots\!81$$$$T^{31} +$$$$51\!\cdots\!16$$$$T^{32} +$$$$22\!\cdots\!97$$$$T^{33} -$$$$47\!\cdots\!37$$$$T^{34} +$$$$51\!\cdots\!83$$$$T^{35} +$$$$23\!\cdots\!10$$$$T^{36} -$$$$19\!\cdots\!27$$$$T^{37} -$$$$12\!\cdots\!75$$$$T^{38} +$$$$18\!\cdots\!19$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$)
$41$ ($$1 - 3 T - 19 T^{2} + 18 T^{3} - 459 T^{4} - 19074 T^{5} + 109024 T^{6} + 227799 T^{7} - 525013 T^{8} + 9339759 T^{9} + 183269344 T^{10} - 1314599154 T^{11} - 1297024299 T^{12} + 2085411618 T^{13} - 90251980579 T^{14} - 584262821643 T^{15} + 7984925229121 T^{16}$$)($$1 + 19 T + 68 T^{2} - 655 T^{3} - 1387 T^{4} + 63285 T^{5} + 319397 T^{6} - 2378554 T^{7} - 20541144 T^{8} + 66494623 T^{9} + 976204363 T^{10} - 1644832272 T^{11} - 49442383823 T^{12} - 67438123152 T^{13} + 1640999534203 T^{14} + 4582875911783 T^{15} - 58044363610584 T^{16} - 275570230313354 T^{17} + 1517169044262677 T^{18} + 12325024222559085 T^{19} - 11075091292790827 T^{20} - 214435167028044455 T^{21} + 912740833090363268 T^{22} + 10456251602608720379 T^{23} + 22563490300366186081 T^{24}$$)($$1 - 9 T + 209 T^{2} - 1888 T^{3} + 27988 T^{4} - 215873 T^{5} + 2568095 T^{6} - 17950187 T^{7} + 184170237 T^{8} - 1171290382 T^{9} + 11003655253 T^{10} - 64730717249 T^{11} + 572386438742 T^{12} - 3170831215026 T^{13} + 26747011255536 T^{14} - 140954342581060 T^{15} + 1145771417437051 T^{16} - 5779128045823460 T^{17} + 44961725920556016 T^{18} - 218536858170806946 T^{19} + 1617427275526032662 T^{20} - 7499454988474311049 T^{21} + 52268509483777227973 T^{22} -$$$$22\!\cdots\!42$$$$T^{23} +$$$$14\!\cdots\!77$$$$T^{24} -$$$$58\!\cdots\!07$$$$T^{25} +$$$$34\!\cdots\!95$$$$T^{26} -$$$$11\!\cdots\!93$$$$T^{27} +$$$$63\!\cdots\!28$$$$T^{28} -$$$$17\!\cdots\!48$$$$T^{29} +$$$$79\!\cdots\!49$$$$T^{30} -$$$$13\!\cdots\!09$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$)($$1 - 19 T + 12 T^{2} + 1913 T^{3} - 9016 T^{4} - 84640 T^{5} + 727901 T^{6} + 733147 T^{7} - 19334902 T^{8} + 149819 T^{9} + 159905008 T^{10} + 1763738609 T^{11} + 15953599463 T^{12} - 202733667043 T^{13} - 1098060502629 T^{14} + 11161254620488 T^{15} + 56842341836837 T^{16} - 697945434779521 T^{17} + 859311312315416 T^{18} + 12981930863804423 T^{19} - 102522143793719309 T^{20} + 532259165415981343 T^{21} + 1444502316002214296 T^{22} - 48103097310439366841 T^{23} +$$$$16\!\cdots\!57$$$$T^{24} +$$$$12\!\cdots\!88$$$$T^{25} -$$$$52\!\cdots\!89$$$$T^{26} -$$$$39\!\cdots\!83$$$$T^{27} +$$$$12\!\cdots\!23$$$$T^{28} +$$$$57\!\cdots\!49$$$$T^{29} +$$$$21\!\cdots\!08$$$$T^{30} +$$$$82\!\cdots\!79$$$$T^{31} -$$$$43\!\cdots\!62$$$$T^{32} +$$$$67\!\cdots\!87$$$$T^{33} +$$$$27\!\cdots\!61$$$$T^{34} -$$$$13\!\cdots\!40$$$$T^{35} -$$$$57\!\cdots\!56$$$$T^{36} +$$$$50\!\cdots\!53$$$$T^{37} +$$$$12\!\cdots\!52$$$$T^{38} -$$$$83\!\cdots\!59$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$43$ ($$( 1 + 3 T + 111 T^{2} + 234 T^{3} + 6089 T^{4} + 10062 T^{5} + 205239 T^{6} + 238521 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 25 T + 439 T^{2} - 5258 T^{3} + 52746 T^{4} - 426931 T^{5} + 3057464 T^{6} - 18358033 T^{7} + 97527354 T^{8} - 418047806 T^{9} + 1500853639 T^{10} - 3675211075 T^{11} + 6321363049 T^{12} )^{2}$$)($$( 1 + 11 T + 307 T^{2} + 2726 T^{3} + 41567 T^{4} + 303824 T^{5} + 3314561 T^{6} + 20111097 T^{7} + 173316472 T^{8} + 864777171 T^{9} + 6128623289 T^{10} + 24156134768 T^{11} + 142109301167 T^{12} + 400745015618 T^{13} + 1940658456043 T^{14} + 2990004722177 T^{15} + 11688200277601 T^{16} )^{2}$$)($$( 1 - T + 131 T^{2} - 222 T^{3} + 13120 T^{4} - 17529 T^{5} + 947878 T^{6} - 1296581 T^{7} + 53628143 T^{8} - 65230381 T^{9} + 2588452654 T^{10} - 2804906383 T^{11} + 99158436407 T^{12} - 103087265567 T^{13} + 3240606254278 T^{14} - 2576910997347 T^{15} + 82936283202880 T^{16} - 60343731665754 T^{17} + 1531154236365731 T^{18} - 502592611936843 T^{19} + 21611482313284249 T^{20} )^{2}$$)
$47$ ($$1 - 2 T - 16 T^{2} - 162 T^{3} - 1727 T^{4} - 26682 T^{5} + 136976 T^{6} + 626824 T^{7} + 4715593 T^{8} + 29460728 T^{9} + 302579984 T^{10} - 2770205286 T^{11} - 8427209087 T^{12} - 37153891134 T^{13} - 172467445264 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$)($$1 + 24 T + 115 T^{2} - 2376 T^{3} - 35394 T^{4} - 95696 T^{5} + 1682298 T^{6} + 17182896 T^{7} + 34195735 T^{8} - 445798096 T^{9} - 3285259630 T^{10} - 981842400 T^{11} + 79567059979 T^{12} - 46146592800 T^{13} - 7257138522670 T^{14} - 46284095721008 T^{15} + 166864278360535 T^{16} + 3940811403400272 T^{17} + 18133852389546042 T^{18} - 48481806135827248 T^{19} - 842776680106368834 T^{20} - 2659054004092174392 T^{21} + 6048900207120455635 T^{22} + 59331821162016295272 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$)($$1 + 40 T + 860 T^{2} + 13620 T^{3} + 177582 T^{4} + 1962120 T^{5} + 18728240 T^{6} + 157365740 T^{7} + 1162186863 T^{8} + 7330805360 T^{9} + 36645561260 T^{10} + 98824008780 T^{11} - 685267268196 T^{12} - 14848564608900 T^{13} - 164307010116600 T^{14} - 1429293953116700 T^{15} - 10561394484169275 T^{16} - 67176815796484900 T^{17} - 362954185347569400 T^{18} - 1541622523389824700 T^{19} - 3343885668537925476 T^{20} + 22664792985417161460 T^{21} +$$$$39\!\cdots\!40$$$$T^{22} +$$$$37\!\cdots\!80$$$$T^{23} +$$$$27\!\cdots\!43$$$$T^{24} +$$$$17\!\cdots\!80$$$$T^{25} +$$$$98\!\cdots\!60$$$$T^{26} +$$$$48\!\cdots\!60$$$$T^{27} +$$$$20\!\cdots\!62$$$$T^{28} +$$$$74\!\cdots\!40$$$$T^{29} +$$$$22\!\cdots\!40$$$$T^{30} +$$$$48\!\cdots\!20$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$1 + 14 T - 17 T^{2} - 332 T^{3} + 11875 T^{4} + 85608 T^{5} - 375631 T^{6} + 491530 T^{7} + 67558562 T^{8} + 169238914 T^{9} - 1464748913 T^{10} + 15388021112 T^{11} + 192696067221 T^{12} - 64267686812 T^{13} - 257501703091 T^{14} + 62244145699390 T^{15} + 272409586441108 T^{16} + 72500860159662 T^{17} + 11516861002196303 T^{18} + 93024868043107652 T^{19} + 301659113965774101 T^{20} + 4372168798026059644 T^{21} + 25440745953851633327 T^{22} + 7527256804356587826 T^{23} +$$$$13\!\cdots\!48$$$$T^{24} +$$$$14\!\cdots\!30$$$$T^{25} -$$$$27\!\cdots\!39$$$$T^{26} -$$$$32\!\cdots\!56$$$$T^{27} +$$$$45\!\cdots\!81$$$$T^{28} +$$$$17\!\cdots\!04$$$$T^{29} -$$$$77\!\cdots\!37$$$$T^{30} +$$$$41\!\cdots\!42$$$$T^{31} +$$$$78\!\cdots\!42$$$$T^{32} +$$$$26\!\cdots\!10$$$$T^{33} -$$$$96\!\cdots\!39$$$$T^{34} +$$$$10\!\cdots\!44$$$$T^{35} +$$$$67\!\cdots\!75$$$$T^{36} -$$$$88\!\cdots\!84$$$$T^{37} -$$$$21\!\cdots\!13$$$$T^{38} +$$$$82\!\cdots\!62$$$$T^{39} +$$$$27\!\cdots\!01$$$$T^{40}$$)
$53$ ($$1 - 4 T - 19 T^{2} + 634 T^{3} - 412 T^{4} - 2174 T^{5} + 75189 T^{6} + 261992 T^{7} + 4266723 T^{8} + 13885576 T^{9} + 211205901 T^{10} - 323658598 T^{11} - 3250878172 T^{12} + 265135942562 T^{13} - 421122861451 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16}$$)($$1 - 35 T + 588 T^{2} - 6446 T^{3} + 52523 T^{4} - 302216 T^{5} + 600290 T^{6} + 9127215 T^{7} - 111906848 T^{8} + 601751953 T^{9} - 6692433 T^{10} - 37037958071 T^{11} + 396587073686 T^{12} - 1963011777763 T^{13} - 18799044297 T^{14} + 89587025506781 T^{15} - 882998857913888 T^{16} + 3816960176641995 T^{17} + 13305044342127410 T^{18} - 355016501836978792 T^{19} + 3270065719475913803 T^{20} - 21270276112756549318 T^{21} +$$$$10\!\cdots\!12$$$$T^{22} -$$$$32\!\cdots\!95$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$)($$1 + 24 T + 131 T^{2} - 2082 T^{3} - 29943 T^{4} - 25594 T^{5} + 1376548 T^{6} + 284722 T^{7} - 80364198 T^{8} + 348867682 T^{9} + 9354078043 T^{10} + 4903415212 T^{11} - 484269386646 T^{12} - 422646277832 T^{13} + 23628062037094 T^{14} - 16685214230384 T^{15} - 1553788861344115 T^{16} - 884316354210352 T^{17} + 66371226262197046 T^{18} - 62922309904794664 T^{19} - 3821118394211916726 T^{20} + 2050586141966039516 T^{21} +$$$$20\!\cdots\!47$$$$T^{22} +$$$$40\!\cdots\!34$$$$T^{23} -$$$$50\!\cdots\!78$$$$T^{24} +$$$$93\!\cdots\!26$$$$T^{25} +$$$$24\!\cdots\!52$$$$T^{26} -$$$$23\!\cdots\!18$$$$T^{27} -$$$$14\!\cdots\!63$$$$T^{28} -$$$$54\!\cdots\!86$$$$T^{29} +$$$$18\!\cdots\!39$$$$T^{30} +$$$$17\!\cdots\!68$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$1 + T + 222 T^{2} + 576 T^{3} + 29866 T^{4} + 121159 T^{5} + 2973080 T^{6} + 16226713 T^{7} + 233637046 T^{8} + 1673331728 T^{9} + 15995234209 T^{10} + 136618682878 T^{11} + 1032730160982 T^{12} + 9346217467105 T^{13} + 66824236403364 T^{14} + 556826155800381 T^{15} + 4361081323456902 T^{16} + 29933837836232390 T^{17} + 270757625857435954 T^{18} + 1566157358977656593 T^{19} + 15195332378604934484 T^{20} + 83006340025815799429 T^{21} +$$$$76\!\cdots\!86$$$$T^{22} +$$$$44\!\cdots\!30$$$$T^{23} +$$$$34\!\cdots\!62$$$$T^{24} +$$$$23\!\cdots\!33$$$$T^{25} +$$$$14\!\cdots\!56$$$$T^{26} +$$$$10\!\cdots\!85$$$$T^{27} +$$$$64\!\cdots\!02$$$$T^{28} +$$$$45\!\cdots\!74$$$$T^{29} +$$$$27\!\cdots\!41$$$$T^{30} +$$$$15\!\cdots\!16$$$$T^{31} +$$$$11\!\cdots\!86$$$$T^{32} +$$$$42\!\cdots\!49$$$$T^{33} +$$$$41\!\cdots\!20$$$$T^{34} +$$$$88\!\cdots\!63$$$$T^{35} +$$$$11\!\cdots\!86$$$$T^{36} +$$$$11\!\cdots\!88$$$$T^{37} +$$$$24\!\cdots\!58$$$$T^{38} +$$$$57\!\cdots\!17$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$)
$59$ ($$1 - 11 T - 76 T^{2} + 894 T^{3} + 3886 T^{4} - 58383 T^{5} + 315956 T^{6} + 623272 T^{7} - 26942453 T^{8} + 36773048 T^{9} + 1099842836 T^{10} - 11990642157 T^{11} + 47088064846 T^{12} + 639142323306 T^{13} - 3205720556716 T^{14} - 27375166333009 T^{15} + 146830437604321 T^{16}$$)($$1 - T - 45 T^{2} - 377 T^{3} + 5021 T^{4} - 8190 T^{5} - 34303 T^{6} - 1922028 T^{7} + 23142924 T^{8} - 85445370 T^{9} + 25142239 T^{10} - 5172383726 T^{11} + 106812734191 T^{12} - 305170639834 T^{13} + 87520133959 T^{14} - 17548684645230 T^{15} + 280431164703564 T^{16} - 1374104520558372 T^{17} - 1446918845487223 T^{18} - 20382055660667610 T^{19} + 737235627211295741 T^{20} - 3265949423632912003 T^{21} - 23000253898528863045 T^{22} - 30155888444737842659 T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$)($$1 + 17 T + 231 T^{2} + 2711 T^{3} + 31383 T^{4} + 368824 T^{5} + 3899415 T^{6} + 38857474 T^{7} + 369688167 T^{8} + 3403580961 T^{9} + 31220539482 T^{10} + 272706121363 T^{11} + 2333967246102 T^{12} + 19121516358498 T^{13} + 154947098550224 T^{14} + 1243271597158160 T^{15} + 9651740778329686 T^{16} + 73353024232331440 T^{17} + 539370850053329744 T^{18} + 3927157908191960742 T^{19} + 28281523683193776822 T^{20} +$$$$19\!\cdots\!37$$$$T^{21} +$$$$13\!\cdots\!62$$$$T^{22} +$$$$84\!\cdots\!59$$$$T^{23} +$$$$54\!\cdots\!07$$$$T^{24} +$$$$33\!\cdots\!86$$$$T^{25} +$$$$19\!\cdots\!15$$$$T^{26} +$$$$11\!\cdots\!16$$$$T^{27} +$$$$55\!\cdots\!23$$$$T^{28} +$$$$28\!\cdots\!69$$$$T^{29} +$$$$14\!\cdots\!91$$$$T^{30} +$$$$62\!\cdots\!83$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$)($$1 - 17 T - 24 T^{2} + 1080 T^{3} + 3041 T^{4} + 91298 T^{5} - 1705592 T^{6} - 6176752 T^{7} + 59242869 T^{8} + 435071881 T^{9} + 9725236734 T^{10} - 84667840483 T^{11} - 544988908505 T^{12} + 396013967022 T^{13} + 15351394464494 T^{14} + 534770086845542 T^{15} - 1784279235241195 T^{16} - 18902754975455470 T^{17} - 90766731594831180 T^{18} - 113158777818941145 T^{19} + 18511300884404128250 T^{20} - 6676367891317527555 T^{21} -$$$$31\!\cdots\!80$$$$T^{22} -$$$$38\!\cdots\!30$$$$T^{23} -$$$$21\!\cdots\!95$$$$T^{24} +$$$$38\!\cdots\!58$$$$T^{25} +$$$$64\!\cdots\!54$$$$T^{26} +$$$$98\!\cdots\!18$$$$T^{27} -$$$$80\!\cdots\!05$$$$T^{28} -$$$$73\!\cdots\!37$$$$T^{29} +$$$$49\!\cdots\!34$$$$T^{30} +$$$$13\!\cdots\!79$$$$T^{31} +$$$$10\!\cdots\!89$$$$T^{32} -$$$$64\!\cdots\!08$$$$T^{33} -$$$$10\!\cdots\!12$$$$T^{34} +$$$$33\!\cdots\!02$$$$T^{35} +$$$$65\!\cdots\!81$$$$T^{36} +$$$$13\!\cdots\!20$$$$T^{37} -$$$$18\!\cdots\!04$$$$T^{38} -$$$$75\!\cdots\!63$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$)
$61$ ($$1 + 3 T^{2} + 650 T^{3} + 3438 T^{4} + 16250 T^{5} + 181801 T^{6} + 1901250 T^{7} + 15718855 T^{8} + 115976250 T^{9} + 676481521 T^{10} + 3688441250 T^{11} + 47602001358 T^{12} + 548987595650 T^{13} + 154561123083 T^{14} + 191707312997281 T^{16}$$)($$1 - 8 T - 144 T^{2} + 1210 T^{3} + 15692 T^{4} - 91008 T^{5} - 1749157 T^{6} + 5810908 T^{7} + 166130048 T^{8} - 350338768 T^{9} - 12151497572 T^{10} + 9558414952 T^{11} + 772713424737 T^{12} + 583063312072 T^{13} - 45215722465412 T^{14} - 79520243899408 T^{15} + 2300210229930368 T^{16} + 4907871402251308 T^{17} - 90117223456163677 T^{18} - 286014740020599168 T^{19} + 3008271155553333452 T^{20} + 14149916772329310610 T^{21} -$$$$10\!\cdots\!44$$$$T^{22} -$$$$34\!\cdots\!88$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$)($$1 - 2 T + 178 T^{2} - 346 T^{3} + 11144 T^{4} + 40340 T^{5} + 160139 T^{6} + 8917862 T^{7} - 14033607 T^{8} + 339988058 T^{9} + 800783942 T^{10} - 13436388236 T^{11} + 187167906365 T^{12} - 898375851426 T^{13} + 479169344749 T^{14} + 14288394224174 T^{15} - 732267975983454 T^{16} + 871592047674614 T^{17} + 1782989131811029 T^{18} - 203914249132524906 T^{19} + 2591497071832677965 T^{20} - 11348323802925515036 T^{21} + 41256688474117311062 T^{22} +$$$$10\!\cdots\!18$$$$T^{23} -$$$$26\!\cdots\!67$$$$T^{24} +$$$$10\!\cdots\!42$$$$T^{25} +$$$$11\!\cdots\!39$$$$T^{26} +$$$$17\!\cdots\!40$$$$T^{27} +$$$$29\!\cdots\!24$$$$T^{28} -$$$$56\!\cdots\!26$$$$T^{29} +$$$$17\!\cdots\!98$$$$T^{30} -$$$$12\!\cdots\!02$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$)($$1 + 38 T + 613 T^{2} + 3660 T^{3} - 42463 T^{4} - 1139168 T^{5} - 10432970 T^{6} - 11592012 T^{7} + 876621133 T^{8} + 11435430830 T^{9} + 57987789192 T^{10} - 257674721658 T^{11} - 7267129029675 T^{12} - 60744458842252 T^{13} - 147149444018752 T^{14} + 2537573290658796 T^{15} + 36705196328155185 T^{16} + 228237801345912040 T^{17} + 134377432547021165 T^{18} - 12634742916802888610 T^{19} -$$$$14\!\cdots\!30$$$$T^{20} -$$$$77\!\cdots\!10$$$$T^{21} +$$$$50\!\cdots\!65$$$$T^{22} +$$$$51\!\cdots\!40$$$$T^{23} +$$$$50\!\cdots\!85$$$$T^{24} +$$$$21\!\cdots\!96$$$$T^{25} -$$$$75\!\cdots\!72$$$$T^{26} -$$$$19\!\cdots\!92$$$$T^{27} -$$$$13\!\cdots\!75$$$$T^{28} -$$$$30\!\cdots\!78$$$$T^{29} +$$$$41\!\cdots\!92$$$$T^{30} +$$$$49\!\cdots\!30$$$$T^{31} +$$$$23\!\cdots\!93$$$$T^{32} -$$$$18\!\cdots\!72$$$$T^{33} -$$$$10\!\cdots\!70$$$$T^{34} -$$$$68\!\cdots\!68$$$$T^{35} -$$$$15\!\cdots\!43$$$$T^{36} +$$$$82\!\cdots\!60$$$$T^{37} +$$$$83\!\cdots\!53$$$$T^{38} +$$$$31\!\cdots\!58$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$)
$67$ ($$1 - 14 T + 13 T^{2} + 482 T^{3} + 2336 T^{4} - 29384 T^{5} - 340345 T^{6} + 3464096 T^{7} - 12764441 T^{8} + 232094432 T^{9} - 1527808705 T^{10} - 8837619992 T^{11} + 47073018656 T^{12} + 650760301574 T^{13} + 1175958968197 T^{14} - 84849962474522 T^{15} + 406067677556641 T^{16}$$)($$1 + 36 T + 448 T^{2} + 882 T^{3} - 30116 T^{4} - 288924 T^{5} - 1370359 T^{6} - 12802922 T^{7} - 99303768 T^{8} + 312986874 T^{9} + 9836955068 T^{10} + 57716972648 T^{11} + 232196539349 T^{12} + 3867037167416 T^{13} + 44158091300252 T^{14} + 94134871184862 T^{15} - 2001082244723928 T^{16} - 17285546435162654 T^{17} - 123960458130728671 T^{18} - 1751085039856342452 T^{19} - 12229134177295800356 T^{20} + 23996163337532143254 T^{21} +$$$$81\!\cdots\!52$$$$T^{22} +$$$$43\!\cdots\!88$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$1 + 14 T - 5 T^{2} - 1002 T^{3} - 7675 T^{4} - 114318 T^{5} - 864154 T^{6} + 5397658 T^{7} + 76521194 T^{8} + 18318554 T^{9} + 2123751863 T^{10} + 51556521834 T^{11} + 75655239118 T^{12} - 659111383092 T^{13} + 11489865587542 T^{14} - 146859077325800 T^{15} - 3515784432557183 T^{16} - 9839558180828600 T^{17} + 51578006622476038 T^{18} - 198236316912899196 T^{19} + 1524537877750751278 T^{20} + 69607754557677086238 T^{21} +$$$$19\!\cdots\!47$$$$T^{22} +$$$$11\!\cdots\!42$$$$T^{23} +$$$$31\!\cdots\!54$$$$T^{24} +$$$$14\!\cdots\!26$$$$T^{25} -$$$$15\!\cdots\!46$$$$T^{26} -$$$$13\!\cdots\!94$$$$T^{27} -$$$$62\!\cdots\!75$$$$T^{28} -$$$$54\!\cdots\!74$$$$T^{29} -$$$$18\!\cdots\!45$$$$T^{30} +$$$$34\!\cdots\!02$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)($$1 + 16 T - 58 T^{2} - 2978 T^{3} - 18445 T^{4} + 126976 T^{5} + 2658477 T^{6} + 13876972 T^{7} - 93385065 T^{8} - 1939016994 T^{9} - 8366709641 T^{10} + 75540871444 T^{11} + 1126536292624 T^{12} + 3842182564850 T^{13} - 44052154844443 T^{14} - 587678167507398 T^{15} - 1161816367846488 T^{16} + 27430169634446196 T^{17} + 203676466527797368 T^{18} - 490172917501845106 T^{19} - 13679143434364915411 T^{20} - 32841585472623622102 T^{21} +$$$$91\!\cdots\!52$$$$T^{22} +$$$$82\!\cdots\!48$$$$T^{23} -$$$$23\!\cdots\!48$$$$T^{24} -$$$$79\!\cdots\!86$$$$T^{25} -$$$$39\!\cdots\!67$$$$T^{26} +$$$$23\!\cdots\!50$$$$T^{27} +$$$$45\!\cdots\!84$$$$T^{28} +$$$$20\!\cdots\!68$$$$T^{29} -$$$$15\!\cdots\!09$$$$T^{30} -$$$$23\!\cdots\!02$$$$T^{31} -$$$$76\!\cdots\!65$$$$T^{32} +$$$$76\!\cdots\!64$$$$T^{33} +$$$$97\!\cdots\!33$$$$T^{34} +$$$$31\!\cdots\!68$$$$T^{35} -$$$$30\!\cdots\!45$$$$T^{36} -$$$$32\!\cdots\!06$$$$T^{37} -$$$$42\!\cdots\!22$$$$T^{38} +$$$$79\!\cdots\!48$$$$T^{39} +$$$$33\!\cdots\!01$$$$T^{40}$$)
$71$ ($$1 - 23 T + 251 T^{2} - 2812 T^{3} + 33341 T^{4} - 344824 T^{5} + 3544134 T^{6} - 31567271 T^{7} + 253196517 T^{8} - 2241276241 T^{9} + 17865979494 T^{10} - 123416302664 T^{11} + 847250856221 T^{12} - 5073492935012 T^{13} + 32153171264171 T^{14} - 209187763642993 T^{15} + 645753531245761 T^{16}$$)($$1 - T - 272 T^{2} + 1515 T^{3} + 31793 T^{4} - 390805 T^{5} - 1127933 T^{6} + 51435256 T^{7} - 189880844 T^{8} - 3895432897 T^{9} + 35948452553 T^{10} + 120778460288 T^{11} - 3267845694113 T^{12} + 8575270680448 T^{13} + 181216149319673 T^{14} - 1394218283598167 T^{15} - 4825191435738764 T^{16} + 92800998551398856 T^{17} - 144488537543865293 T^{18} - 3554418433499994755 T^{19} + 20530442018896479473 T^{20} + 69460478588450281965 T^{21} -$$$$88\!\cdots\!72$$$$T^{22} -$$$$23\!\cdots\!71$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$)($$1 - 7 T - 131 T^{2} - 114 T^{3} + 21322 T^{4} - 49561 T^{5} - 1032319 T^{6} + 2461237 T^{7} + 92179373 T^{8} - 675870262 T^{9} - 2152863231 T^{10} + 44390382057 T^{11} - 83384161648 T^{12} - 2913469018956 T^{13} + 14917443172720 T^{14} + 129116899378886 T^{15} - 1673894927111283 T^{16} + 9167299855900906 T^{17} + 75198831033681520 T^{18} - 1042762610043560916 T^{19} - 2118931716251410288 T^{20} + 80090430209343155007 T^{21} -$$$$27\!\cdots\!51$$$$T^{22} -$$$$61\!\cdots\!42$$$$T^{23} +$$$$59\!\cdots\!53$$$$T^{24} +$$$$11\!\cdots\!47$$$$T^{25} -$$$$33\!\cdots\!19$$$$T^{26} -$$$$11\!\cdots\!31$$$$T^{27} +$$$$34\!\cdots\!02$$$$T^{28} -$$$$13\!\cdots\!54$$$$T^{29} -$$$$10\!\cdots\!11$$$$T^{30} -$$$$41\!\cdots\!57$$$$T^{31} +$$$$41\!\cdots\!21$$$$T^{32}$$)($$1 - T - 374 T^{2} + 385 T^{3} + 59424 T^{4} - 61198 T^{5} - 4274911 T^{6} + 3121075 T^{7} - 5229794 T^{8} + 450375939 T^{9} + 21565718974 T^{10} - 89365774783 T^{11} - 422299011827 T^{12} + 5204430801845 T^{13} - 172828405013793 T^{14} + 177634632473756 T^{15} + 14915998344657807 T^{16} - 43484081312487565 T^{17} - 30084719421630102 T^{18} + 1740254535967054397 T^{19} - 48390764100475009279 T^{20} +$$$$12\!\cdots\!87$$$$T^{21} -$$$$15\!\cdots\!82$$$$T^{22} -$$$$15\!\cdots\!15$$$$T^{23} +$$$$37\!\cdots\!67$$$$T^{24} +$$$$32\!\cdots\!56$$$$T^{25} -$$$$22\!\cdots\!53$$$$T^{26} +$$$$47\!\cdots\!95$$$$T^{27} -$$$$27\!\cdots\!47$$$$T^{28} -$$$$40\!\cdots\!73$$$$T^{29} +$$$$70\!\cdots\!74$$$$T^{30} +$$$$10\!\cdots\!69$$$$T^{31} -$$$$85\!\cdots\!54$$$$T^{32} +$$$$36\!\cdots\!25$$$$T^{33} -$$$$35\!\cdots\!91$$$$T^{34} -$$$$35\!\cdots\!98$$$$T^{35} +$$$$24\!\cdots\!04$$$$T^{36} +$$$$11\!\cdots\!35$$$$T^{37} -$$$$78\!\cdots\!14$$$$T^{38} -$$$$14\!\cdots\!31$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$)
$73$ ($$1 + 5 T + 14 T^{2} - 640 T^{3} + 1222 T^{4} - 60055 T^{5} - 28808 T^{6} - 934650 T^{7} + 60949575 T^{8} - 68229450 T^{9} - 153517832 T^{10} - 23362415935 T^{11} + 34702650502 T^{12} - 1326765819520 T^{13} + 2118679168046 T^{14} + 55236992595485 T^{15} + 806460091894081 T^{16}$$)($$1 + 31 T + 157 T^{2} - 4351 T^{3} - 56165 T^{4} + 21012 T^{5} + 4960409 T^{6} + 40392564 T^{7} - 66672508 T^{8} - 3587879652 T^{9} - 16845025467 T^{10} + 109824364092 T^{11} + 1752697474667 T^{12} + 8017178578716 T^{13} - 89767140713643 T^{14} - 1395746178582084 T^{15} - 1893381950258428 T^{16} + 83736676996834452 T^{17} + 750679658091992201 T^{18} + 232127937683266164 T^{19} - 45294831061231059365 T^{20} -$$$$25\!\cdots\!63$$$$T^{21} +$$$$67\!\cdots\!93$$$$T^{22} +$$$$97\!\cdots\!87$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$)($$1 - 5 T - 184 T^{2} - 660 T^{3} + 18729 T^{4} + 101130 T^{5} + 583946 T^{6} + 1517480 T^{7} - 99548154 T^{8} - 2205366530 T^{9} - 3106828272 T^{10} + 113345857735 T^{11} + 1007191618776 T^{12} + 4851951748650 T^{13} + 43053806124718 T^{14} - 510880245844670 T^{15} - 9238219037249483 T^{16} - 37294257946660910 T^{17} + 229433732838622222 T^{18} + 1887491713404577050 T^{19} + 28602470323180973016 T^{20} +$$$$23\!\cdots\!55$$$$T^{21} -$$$$47\!\cdots\!08$$$$T^{22} -$$$$24\!\cdots\!10$$$$T^{23} -$$$$80\!\cdots\!74$$$$T^{24} +$$$$89\!\cdots\!40$$$$T^{25} +$$$$25\!\cdots\!54$$$$T^{26} +$$$$31\!\cdots\!10$$$$T^{27} +$$$$42\!\cdots\!09$$$$T^{28} -$$$$11\!\cdots\!80$$$$T^{29} -$$$$22\!\cdots\!56$$$$T^{30} -$$$$44\!\cdots\!85$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$)($$1 - 19 T - 37 T^{2} + 1911 T^{3} + 14984 T^{4} - 343095 T^{5} + 44886 T^{6} + 20424421 T^{7} + 38131320 T^{8} - 2107575143 T^{9} + 6644560253 T^{10} + 31127895089 T^{11} + 163541353759 T^{12} - 996108478075 T^{13} + 8406248222548 T^{14} - 631161430088471 T^{15} + 5329956394145463 T^{16} + 17871030714682627 T^{17} - 359037128943857193 T^{18} - 2008361662490082115 T^{19} + 51740292756835167037 T^{20} -$$$$14\!\cdots\!95$$$$T^{21} -$$$$19\!\cdots\!97$$$$T^{22} +$$$$69\!\cdots\!59$$$$T^{23} +$$$$15\!\cdots\!83$$$$T^{24} -$$$$13\!\cdots\!03$$$$T^{25} +$$$$12\!\cdots\!72$$$$T^{26} -$$$$11\!\cdots\!75$$$$T^{27} +$$$$13\!\cdots\!79$$$$T^{28} +$$$$18\!\cdots\!57$$$$T^{29} +$$$$28\!\cdots\!97$$$$T^{30} -$$$$66\!\cdots\!11$$$$T^{31} +$$$$87\!\cdots\!20$$$$T^{32} +$$$$34\!\cdots\!93$$$$T^{33} +$$$$54\!\cdots\!74$$$$T^{34} -$$$$30\!\cdots\!15$$$$T^{35} +$$$$97\!\cdots\!24$$$$T^{36} +$$$$90\!\cdots\!83$$$$T^{37} -$$$$12\!\cdots\!53$$$$T^{38} -$$$$48\!\cdots\!03$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$79$ ($$1 - 2 T - 175 T^{2} + 940 T^{3} + 7090 T^{4} - 127586 T^{5} + 1192447 T^{6} + 5744430 T^{7} - 171664185 T^{8} + 453809970 T^{9} + 7442061727 T^{10} - 62904873854 T^{11} + 276156074290 T^{12} + 2892433015060 T^{13} - 42540304716175 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16}$$)($$1 - 2 T + 286 T^{2} - 1942 T^{3} + 45506 T^{4} - 458068 T^{5} + 6694253 T^{6} - 59915278 T^{7} + 860339148 T^{8} - 6564781658 T^{9} + 84094060634 T^{10} - 659242015430 T^{11} + 6879140123889 T^{12} - 52080119218970 T^{13} + 524831032416794 T^{14} - 3236693383878662 T^{15} + 33510279502070988 T^{16} - 184362689567763922 T^{17} + 1627288928383820813 T^{18} - 8796696181471880812 T^{19} + 69037553503607964866 T^{20} -$$$$23\!\cdots\!98$$$$T^{21} +$$$$27\!\cdots\!86$$$$T^{22} -$$$$14\!\cdots\!58$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$)($$1 - 20 T + 72 T^{2} + 240 T^{3} + 3616 T^{4} + 52220 T^{5} - 1218607 T^{6} - 3975630 T^{7} + 89503171 T^{8} + 339499660 T^{9} + 541594996 T^{10} - 86409648250 T^{11} - 192249239601 T^{12} + 5478908409570 T^{13} + 38059098256319 T^{14} + 34437029350830 T^{15} - 7141755789421758 T^{16} + 2720525318715570 T^{17} + 237526832217686879 T^{18} + 2701315523345983230 T^{19} - 7488123454647357681 T^{20} -$$$$26\!\cdots\!50$$$$T^{21} +$$$$13\!\cdots\!16$$$$T^{22} +$$$$65\!\cdots\!40$$$$T^{23} +$$$$13\!\cdots\!31$$$$T^{24} -$$$$47\!\cdots\!70$$$$T^{25} -$$$$11\!\cdots\!07$$$$T^{26} +$$$$39\!\cdots\!80$$$$T^{27} +$$$$21\!\cdots\!56$$$$T^{28} +$$$$11\!\cdots\!60$$$$T^{29} +$$$$26\!\cdots\!32$$$$T^{30} -$$$$58\!\cdots\!80$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$)($$1 + 64 T + 1813 T^{2} + 28300 T^{3} + 220923 T^{4} - 272960 T^{5} - 26038614 T^{6} - 272781482 T^{7} - 1191792493 T^{8} - 1015821708 T^{9} + 10837883272 T^{10} + 850456415848 T^{11} + 23123314759077 T^{12} + 258076348414252 T^{13} + 716331205356126 T^{14} - 13571832494284164 T^{15} - 144937325341815863 T^{16} - 90759777606546922 T^{17} + 6424026371015525703 T^{18} + 21997663317142103344 T^{19} -$$$$14\!\cdots\!34$$$$T^{20} +$$$$17\!\cdots\!76$$$$T^{21} +$$$$40\!\cdots\!23$$$$T^{22} -$$$$44\!\cdots\!58$$$$T^{23} -$$$$56\!\cdots\!03$$$$T^{24} -$$$$41\!\cdots\!36$$$$T^{25} +$$$$17\!\cdots\!46$$$$T^{26} +$$$$49\!\cdots\!68$$$$T^{27} +$$$$35\!\cdots\!97$$$$T^{28} +$$$$10\!\cdots\!12$$$$T^{29} +$$$$10\!\cdots\!72$$$$T^{30} -$$$$75\!\cdots\!32$$$$T^{31} -$$$$70\!\cdots\!13$$$$T^{32} -$$$$12\!\cdots\!98$$$$T^{33} -$$$$96\!\cdots\!34$$$$T^{34} -$$$$79\!\cdots\!40$$$$T^{35} +$$$$50\!\cdots\!83$$$$T^{36} +$$$$51\!\cdots\!00$$$$T^{37} +$$$$26\!\cdots\!93$$$$T^{38} +$$$$72\!\cdots\!16$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$)
$83$ ($$1 + 27 T + 322 T^{2} + 2361 T^{3} + 19056 T^{4} + 273252 T^{5} + 2808440 T^{6} + 13313418 T^{7} + 42193589 T^{8} + 1105013694 T^{9} + 19347343160 T^{10} + 156241941324 T^{11} + 904365764976 T^{12} + 9300074958123 T^{13} + 105274800224818 T^{14} + 732673376719929 T^{15} + 2252292232139041 T^{16}$$)($$1 + 19 T + 3 T^{2} - 2230 T^{3} - 9853 T^{4} + 260324 T^{5} + 3150455 T^{6} - 5556832 T^{7} - 239317072 T^{8} + 68226904 T^{9} + 18813056155 T^{10} + 47128776483 T^{11} - 758831244287 T^{12} + 3911688448089 T^{13} + 129603143851795 T^{14} + 39011256757448 T^{15} - 11357586423756112 T^{16} - 21888587094322976 T^{17} + 1030010933982232895 T^{18} + 7064165337823659148 T^{19} - 22191835363265970973 T^{20} -$$$$41\!\cdots\!90$$$$T^{21} + 46548123561617560347 T^{22} +$$$$24\!\cdots\!73$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$)($$1 - 17 T - 70 T^{2} + 2509 T^{3} - 7605 T^{4} + 46949 T^{5} - 1021496 T^{6} - 21178921 T^{7} + 317998444 T^{8} + 567110603 T^{9} - 13415734458 T^{10} - 13379267583 T^{11} - 1499536740482 T^{12} + 16468540679846 T^{13} + 120096814767848 T^{14} - 1133667594299970 T^{15} - 1727008891553813 T^{16} - 94094410326897510 T^{17} + 827346956935704872 T^{18} + 9416497469707104802 T^{19} - 71165495981088450722 T^{20} - 52701478783009375869 T^{21} -$$$$43\!\cdots\!02$$$$T^{22} +$$$$15\!\cdots\!81$$$$T^{23} +$$$$71\!\cdots\!04$$$$T^{24} -$$$$39\!\cdots\!63$$$$T^{25} -$$$$15\!\cdots\!04$$$$T^{26} +$$$$60\!\cdots\!83$$$$T^{27} -$$$$81\!\cdots\!05$$$$T^{28} +$$$$22\!\cdots\!67$$$$T^{29} -$$$$51\!\cdots\!30$$$$T^{30} -$$$$10\!\cdots\!19$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)($$1 - 57 T + 1531 T^{2} - 25602 T^{3} + 283378 T^{4} - 1661127 T^{5} - 8691182 T^{6} + 349426040 T^{7} - 4444505934 T^{8} + 28616053427 T^{9} + 47050780495 T^{10} - 3583740129277 T^{11} + 45464734870983 T^{12} - 281124959027648 T^{13} - 395656472000900 T^{14} + 30726600590053373 T^{15} - 378959258485740125 T^{16} + 2393776145680167358 T^{17} + 572766742440286311 T^{18} -$$$$20\!\cdots\!17$$$$T^{19} +$$$$26\!\cdots\!27$$$$T^{20} -$$$$17\!\cdots\!11$$$$T^{21} +$$$$39\!\cdots\!79$$$$T^{22} +$$$$13\!\cdots\!46$$$$T^{23} -$$$$17\!\cdots\!25$$$$T^{24} +$$$$12\!\cdots\!39$$$$T^{25} -$$$$12\!\cdots\!00$$$$T^{26} -$$$$76\!\cdots\!96$$$$T^{27} +$$$$10\!\cdots\!03$$$$T^{28} -$$$$66\!\cdots\!31$$$$T^{29} +$$$$73\!\cdots\!55$$$$T^{30} +$$$$36\!\cdots\!09$$$$T^{31} -$$$$47\!\cdots\!74$$$$T^{32} +$$$$31\!\cdots\!20$$$$T^{33} -$$$$63\!\cdots\!78$$$$T^{34} -$$$$10\!\cdots\!89$$$$T^{35} +$$$$14\!\cdots\!18$$$$T^{36} -$$$$10\!\cdots\!46$$$$T^{37} +$$$$53\!\cdots\!79$$$$T^{38} -$$$$16\!\cdots\!79$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$)
$89$ ($$1 + 15 T - 43 T^{2} - 135 T^{3} + 11073 T^{4} - 6060 T^{5} + 887044 T^{6} + 4526550 T^{7} - 143552425 T^{8} + 402862950 T^{9} + 7026275524 T^{10} - 4272112140 T^{11} + 694744834593 T^{12} - 753848025615 T^{13} - 21370195511323 T^{14} + 663470023432935 T^{15} + 3936588805702081 T^{16}$$)($$1 - 40 T + 783 T^{2} - 11380 T^{3} + 136646 T^{4} - 1233320 T^{5} + 8628700 T^{6} - 61036520 T^{7} + 518583425 T^{8} - 6988908095 T^{9} + 118589802798 T^{10} - 1538822973165 T^{11} + 15615590135134 T^{12} - 136955244611685 T^{13} + 939349827962958 T^{14} - 4926963550824055 T^{15} + 32537086229955425 T^{16} - 340831556240077480 T^{17} + 4288302465315180700 T^{18} - 54551389953353826280 T^{19} +$$$$53\!\cdots\!26$$$$T^{20} -$$$$39\!\cdots\!20$$$$T^{21} +$$$$24\!\cdots\!83$$$$T^{22} -$$$$11\!\cdots\!60$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$)($$1 - 27 T - 73 T^{2} + 5791 T^{3} + 20328 T^{4} - 941897 T^{5} - 2350459 T^{6} + 90406805 T^{7} + 102589649 T^{8} - 6634669403 T^{9} + 39515779247 T^{10} + 388019762329 T^{11} - 7800813705650 T^{12} - 29920799249404 T^{13} + 911312691296408 T^{14} + 1205098080379952 T^{15} - 84382434058483177 T^{16} + 107253729153815728 T^{17} + 7218507827758847768 T^{18} - 21093235926053088476 T^{19} -$$$$48\!\cdots\!50$$$$T^{20} +$$$$21\!\cdots\!21$$$$T^{21} +$$$$19\!\cdots\!67$$$$T^{22} -$$$$29\!\cdots\!87$$$$T^{23} +$$$$40\!\cdots\!69$$$$T^{24} +$$$$31\!\cdots\!45$$$$T^{25} -$$$$73\!\cdots\!59$$$$T^{26} -$$$$26\!\cdots\!33$$$$T^{27} +$$$$50\!\cdots\!88$$$$T^{28} +$$$$12\!\cdots\!79$$$$T^{29} -$$$$14\!\cdots\!93$$$$T^{30} -$$$$47\!\cdots\!23$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 + 6 T - 225 T^{2} - 2414 T^{3} + 10475 T^{4} + 275574 T^{5} + 2441593 T^{6} + 4668416 T^{7} - 252701912 T^{8} - 3560779949 T^{9} - 8899915733 T^{10} + 174253060095 T^{11} + 1975857342718 T^{12} + 12822124485500 T^{13} + 98844654551923 T^{14} - 556869091622998 T^{15} - 19095987187955715 T^{16} - 193768608185247050 T^{17} - 611163344428279505 T^{18} + 12505565957375472620 T^{19} +$$$$22\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!80$$$$T^{21} -$$$$48\!\cdots\!05$$$$T^{22} -$$$$13\!\cdots\!50$$$$T^{23} -$$$$11\!\cdots\!15$$$$T^{24} -$$$$31\!\cdots\!02$$$$T^{25} +$$$$49\!\cdots\!03$$$$T^{26} +$$$$56\!\cdots\!00$$$$T^{27} +$$$$77\!\cdots\!58$$$$T^{28} +$$$$61\!\cdots\!55$$$$T^{29} -$$$$27\!\cdots\!33$$$$T^{30} -$$$$98\!\cdots\!61$$$$T^{31} -$$$$62\!\cdots\!52$$$$T^{32} +$$$$10\!\cdots\!04$$$$T^{33} +$$$$47\!\cdots\!13$$$$T^{34} +$$$$47\!\cdots\!26$$$$T^{35} +$$$$16\!\cdots\!75$$$$T^{36} -$$$$33\!\cdots\!06$$$$T^{37} -$$$$27\!\cdots\!25$$$$T^{38} +$$$$65\!\cdots\!54$$$$T^{39} +$$$$97\!\cdots\!01$$$$T^{40}$$)
$97$ ($$1 - 40 T + 1006 T^{2} - 19760 T^{3} + 332807 T^{4} - 4819960 T^{5} + 61842548 T^{6} - 712007600 T^{7} + 7409982325 T^{8} - 69064737200 T^{9} + 581876534132 T^{10} - 4399047353080 T^{11} + 29463164421767 T^{12} - 169685843478320 T^{13} + 837969836958574 T^{14} - 3231931379124520 T^{15} + 7837433594376961 T^{16}$$)($$1 - 28 T + 403 T^{2} - 4628 T^{3} + 55650 T^{4} - 570136 T^{5} + 5049586 T^{6} - 55109928 T^{7} + 751828627 T^{8} - 8579658424 T^{9} + 86105944162 T^{10} - 959504128864 T^{11} + 10485381483807 T^{12} - 93071900499808 T^{13} + 810170828620258 T^{14} - 7830422592807352 T^{15} + 66558847783527187 T^{16} - 473247703274771496 T^{17} + 4206163774481409394 T^{18} - 46066010719213433368 T^{19} +$$$$43\!\cdots\!50$$$$T^{20} -$$$$35\!\cdots\!76$$$$T^{21} +$$$$29\!\cdots\!47$$$$T^{22} -$$$$20\!\cdots\!84$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$)($$1 - 6 T - 226 T^{2} + 3602 T^{3} + 30196 T^{4} - 833820 T^{5} - 1681890 T^{6} + 150728346 T^{7} - 383845095 T^{8} - 20809464676 T^{9} + 138289506334 T^{10} + 2169694559068 T^{11} - 24997900175828 T^{12} - 165816523144770 T^{13} + 3403873171920092 T^{14} + 6407107447256686 T^{15} - 366221833845958995 T^{16} + 621489422383898542 T^{17} + 32027042674596145628 T^{18} -$$$$15\!\cdots\!10$$$$T^{19} -$$$$22\!\cdots\!68$$$$T^{20} +$$$$18\!\cdots\!76$$$$T^{21} +$$$$11\!\cdots\!86$$$$T^{22} -$$$$16\!\cdots\!88$$$$T^{23} -$$$$30\!\cdots\!95$$$$T^{24} +$$$$11\!\cdots\!82$$$$T^{25} -$$$$12\!\cdots\!10$$$$T^{26} -$$$$59\!\cdots\!60$$$$T^{27} +$$$$20\!\cdots\!36$$$$T^{28} +$$$$24\!\cdots\!54$$$$T^{29} -$$$$14\!\cdots\!94$$$$T^{30} -$$$$37\!\cdots\!58$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)($$1 + 18 T - 181 T^{2} - 5224 T^{3} - 12425 T^{4} + 240780 T^{5} + 3449805 T^{6} + 82687874 T^{7} + 326931282 T^{8} - 11449359402 T^{9} - 99823613357 T^{10} - 119817410092 T^{11} + 1031495320729 T^{12} + 126529834617016 T^{13} + 1496866997685305 T^{14} - 6508878292176770 T^{15} - 128528710934989580 T^{16} - 589248127879190526 T^{17} - 8372017197617105093 T^{18} + 50791694907309315928 T^{19} +$$$$20\!\cdots\!29$$$$T^{20} +$$$$49\!\cdots\!16$$$$T^{21} -$$$$78\!\cdots\!37$$$$T^{22} -$$$$53\!\cdots\!98$$$$T^{23} -$$$$11\!\cdots\!80$$$$T^{24} -$$$$55\!\cdots\!90$$$$T^{25} +$$$$12\!\cdots\!45$$$$T^{26} +$$$$10\!\cdots\!08$$$$T^{27} +$$$$80\!\cdots\!69$$$$T^{28} -$$$$91\!\cdots\!64$$$$T^{29} -$$$$73\!\cdots\!93$$$$T^{30} -$$$$81\!\cdots\!06$$$$T^{31} +$$$$22\!\cdots\!62$$$$T^{32} +$$$$55\!\cdots\!98$$$$T^{33} +$$$$22\!\cdots\!45$$$$T^{34} +$$$$15\!\cdots\!40$$$$T^{35} -$$$$76\!\cdots\!25$$$$T^{36} -$$$$31\!\cdots\!88$$$$T^{37} -$$$$10\!\cdots\!09$$$$T^{38} +$$$$10\!\cdots\!94$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$)