# Properties

 Label 547.2.a Level $547$ Weight $2$ Character orbit 547.a Rep. character $\chi_{547}(1,\cdot)$ Character field $\Q$ Dimension $45$ Newform subspaces $3$ Sturm bound $91$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 547.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$91$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(547))$$.

Total New Old
Modular forms 46 46 0
Cusp forms 45 45 0
Eisenstein series 1 1 0

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$547$$Dim.
$$+$$$$20$$
$$-$$$$25$$

## Trace form

 $$45q - 2q^{2} - 2q^{3} + 44q^{4} + 2q^{5} - 6q^{6} - 2q^{7} - 12q^{8} + 41q^{9} + O(q^{10})$$ $$45q - 2q^{2} - 2q^{3} + 44q^{4} + 2q^{5} - 6q^{6} - 2q^{7} - 12q^{8} + 41q^{9} - 2q^{10} + 2q^{11} - 10q^{12} - 2q^{14} + 10q^{15} + 30q^{16} - 2q^{17} - 16q^{18} + 2q^{19} - 16q^{20} - 24q^{21} - 20q^{22} - 4q^{23} - 32q^{24} + 61q^{25} - 32q^{26} - 26q^{27} - 20q^{28} - 2q^{29} - 8q^{30} - 6q^{31} - 16q^{32} - 4q^{33} + 6q^{34} + 2q^{35} + 42q^{36} + 4q^{37} - 6q^{38} - 10q^{39} - 14q^{40} - 20q^{42} - 18q^{43} - 6q^{44} + 20q^{45} - 20q^{46} - 2q^{47} + 16q^{48} + 43q^{49} + 10q^{50} + 14q^{51} - 10q^{52} + 6q^{53} + 14q^{54} + 14q^{55} + 2q^{56} - 16q^{57} - 4q^{58} - 22q^{59} + 68q^{60} + 28q^{62} - 14q^{63} + 2q^{64} + 24q^{65} - 14q^{67} - 6q^{68} + 4q^{69} - 18q^{70} - 2q^{71} - 10q^{72} - 2q^{73} + 16q^{74} - 48q^{75} - 6q^{76} - 26q^{77} + 102q^{78} - 24q^{79} - 2q^{80} - 3q^{81} + 12q^{82} - 12q^{83} - 30q^{84} + 62q^{85} - 24q^{86} - 18q^{87} - 38q^{88} + 10q^{89} - 36q^{90} - 30q^{91} + 8q^{92} - 56q^{93} + 6q^{94} - 52q^{95} - 62q^{96} - 18q^{97} + 26q^{98} - 16q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(547))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 547
547.2.a.a $$2$$ $$4.368$$ $$\Q(\sqrt{2})$$ None $$-2$$ $$0$$ $$0$$ $$4$$ $$+$$ $$q+(-1+\beta )q^{2}-\beta q^{3}+(1-2\beta )q^{4}+\cdots$$
547.2.a.b $$18$$ $$4.368$$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$-4$$ $$-10$$ $$-27$$ $$-11$$ $$+$$ $$q-\beta _{1}q^{2}+(-1+\beta _{9})q^{3}+(1+\beta _{2})q^{4}+\cdots$$
547.2.a.c $$25$$ $$4.368$$ None $$4$$ $$8$$ $$29$$ $$5$$ $$-$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$)($$1 + 4 T + 18 T^{2} + 52 T^{3} + 152 T^{4} + 364 T^{5} + 854 T^{6} + 1784 T^{7} + 3639 T^{8} + 6871 T^{9} + 12688 T^{10} + 22114 T^{11} + 37818 T^{12} + 61733 T^{13} + 99096 T^{14} + 152728 T^{15} + 231789 T^{16} + 338874 T^{17} + 488092 T^{18} + 677748 T^{19} + 927156 T^{20} + 1221824 T^{21} + 1585536 T^{22} + 1975456 T^{23} + 2420352 T^{24} + 2830592 T^{25} + 3248128 T^{26} + 3517952 T^{27} + 3726336 T^{28} + 3653632 T^{29} + 3497984 T^{30} + 2981888 T^{31} + 2490368 T^{32} + 1703936 T^{33} + 1179648 T^{34} + 524288 T^{35} + 262144 T^{36}$$)
$3$ ($$1 + 4 T^{2} + 9 T^{4}$$)($$1 + 10 T + 70 T^{2} + 369 T^{3} + 1646 T^{4} + 6380 T^{5} + 22262 T^{6} + 70831 T^{7} + 208802 T^{8} + 574237 T^{9} + 1485851 T^{10} + 3631572 T^{11} + 8425641 T^{12} + 18599141 T^{13} + 39182807 T^{14} + 78875280 T^{15} + 151998095 T^{16} + 280521922 T^{17} + 496314970 T^{18} + 841565766 T^{19} + 1367982855 T^{20} + 2129632560 T^{21} + 3173807367 T^{22} + 4519591263 T^{23} + 6142292289 T^{24} + 7942247964 T^{25} + 9748668411 T^{26} + 11302706871 T^{27} + 12329549298 T^{28} + 12547499157 T^{29} + 11830939542 T^{30} + 10171780740 T^{31} + 7872766974 T^{32} + 5294746683 T^{33} + 3013270470 T^{34} + 1291401630 T^{35} + 387420489 T^{36}$$)
$5$ ($$1 + 8 T^{2} + 25 T^{4}$$)($$1 + 27 T + 394 T^{2} + 4066 T^{3} + 33017 T^{4} + 223247 T^{5} + 1301478 T^{6} + 6696595 T^{7} + 30923803 T^{8} + 129765410 T^{9} + 499584606 T^{10} + 1777920166 T^{11} + 5884067150 T^{12} + 18197020803 T^{13} + 52790609192 T^{14} + 144100415815 T^{15} + 370957947126 T^{16} + 902062261929 T^{17} + 2074042607038 T^{18} + 4510311309645 T^{19} + 9273948678150 T^{20} + 18012551976875 T^{21} + 32994130745000 T^{22} + 56865690009375 T^{23} + 91938549218750 T^{24} + 138900012968750 T^{25} + 195150236718750 T^{26} + 253448066406250 T^{27} + 301990263671875 T^{28} + 326982177734375 T^{29} + 317743652343750 T^{30} + 272518310546875 T^{31} + 201519775390625 T^{32} + 124084472656250 T^{33} + 60119628906250 T^{34} + 20599365234375 T^{35} + 3814697265625 T^{36}$$)
$7$ ($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$1 + 11 T + 109 T^{2} + 718 T^{3} + 4269 T^{4} + 20685 T^{5} + 92163 T^{6} + 360502 T^{7} + 1322737 T^{8} + 4452817 T^{9} + 14363197 T^{10} + 43877097 T^{11} + 130743026 T^{12} + 375593700 T^{13} + 1062872977 T^{14} + 2919179505 T^{15} + 7963303304 T^{16} + 21214026883 T^{17} + 56674382450 T^{18} + 148498188181 T^{19} + 390201861896 T^{20} + 1001278570215 T^{21} + 2551958017777 T^{22} + 6312603315900 T^{23} + 15381786265874 T^{24} + 36134676094671 T^{25} + 82800972428797 T^{26} + 179687227260919 T^{27} + 373640463436513 T^{28} + 712830245504986 T^{29} + 1275654552305763 T^{30} + 2004149180268795 T^{31} + 2895334297992381 T^{32} + 3408749164139074 T^{33} + 3622389432086509 T^{34} + 2558935653859277 T^{35} + 1628413597910449 T^{36}$$)
$11$ ($$1 + 10 T + 45 T^{2} + 110 T^{3} + 121 T^{4}$$)($$1 - 2 T + 85 T^{2} - 150 T^{3} + 3546 T^{4} - 5526 T^{5} + 98861 T^{6} - 141478 T^{7} + 2126155 T^{8} - 2969051 T^{9} + 38313335 T^{10} - 54369337 T^{11} + 603595390 T^{12} - 870619543 T^{13} + 8472071478 T^{14} - 12151342611 T^{15} + 107165383120 T^{16} - 149528311671 T^{17} + 1233316997354 T^{18} - 1644811428381 T^{19} + 12967011357520 T^{20} - 16173437015241 T^{21} + 124039598509398 T^{22} - 140214148019693 T^{23} + 1069306052703790 T^{24} - 1059504567275627 T^{25} + 8212803617978135 T^{26} - 7000866949911241 T^{27} + 55146985002539155 T^{28} - 40365324534703058 T^{29} + 310268167751014781 T^{30} - 190772507307362706 T^{31} + 1346592909886172586 T^{32} - 626587225412347650 T^{33} + 3905727038403633685 T^{34} - 1010894056998587542 T^{35} + 5559917313492231481 T^{36}$$)
$13$ ($$( 1 - 3 T + 13 T^{2} )^{2}$$)($$1 + 25 T + 419 T^{2} + 5077 T^{3} + 50838 T^{4} + 430037 T^{5} + 3215885 T^{6} + 21488134 T^{7} + 131328937 T^{8} + 738626108 T^{9} + 3880155769 T^{10} + 19109203885 T^{11} + 89183382015 T^{12} + 395302024323 T^{13} + 1678118625851 T^{14} + 6828960297885 T^{15} + 26814195655203 T^{16} + 101525943915242 T^{17} + 372390103391781 T^{18} + 1319837270898146 T^{19} + 4531599065729307 T^{20} + 15003225774453345 T^{21} + 47928746072930411 T^{22} + 146772874516959639 T^{23} + 430471150960440135 T^{24} + 1199074204834388545 T^{25} + 3165162263038679449 T^{26} + 7832760099167430284 T^{27} + 18104809190952334513 T^{28} + 38510182696559856958 T^{29} + 74923962474109810685 T^{30} +$$$$13\!\cdots\!61$$$$T^{31} +$$$$20\!\cdots\!82$$$$T^{32} +$$$$25\!\cdots\!89$$$$T^{33} +$$$$27\!\cdots\!79$$$$T^{34} +$$$$21\!\cdots\!25$$$$T^{35} +$$$$11\!\cdots\!29$$$$T^{36}$$)
$17$ ($$1 + 12 T + 68 T^{2} + 204 T^{3} + 289 T^{4}$$)($$1 + 30 T + 584 T^{2} + 8365 T^{3} + 98434 T^{4} + 985682 T^{5} + 8696277 T^{6} + 68768626 T^{7} + 495900647 T^{8} + 3296047078 T^{9} + 20410580082 T^{10} + 118632779583 T^{11} + 651877922855 T^{12} + 3403006356186 T^{13} + 16950930016609 T^{14} + 80772988719250 T^{15} + 368969665979574 T^{16} + 1617067599179214 T^{17} + 6804533522448758 T^{18} + 27490149186046638 T^{19} + 106632233468096886 T^{20} + 396837693577675250 T^{21} + 1415758625917200289 T^{22} + 4831782395875185402 T^{23} + 15734748342489239495 T^{24} + 48679617348389713359 T^{25} +$$$$14\!\cdots\!62$$$$T^{26} +$$$$39\!\cdots\!66$$$$T^{27} +$$$$99\!\cdots\!03$$$$T^{28} +$$$$23\!\cdots\!58$$$$T^{29} +$$$$50\!\cdots\!97$$$$T^{30} +$$$$97\!\cdots\!34$$$$T^{31} +$$$$16\!\cdots\!86$$$$T^{32} +$$$$23\!\cdots\!45$$$$T^{33} +$$$$28\!\cdots\!04$$$$T^{34} +$$$$24\!\cdots\!10$$$$T^{35} +$$$$14\!\cdots\!09$$$$T^{36}$$)
$19$ ($$1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4}$$)($$1 - 4 T + 130 T^{2} - 516 T^{3} + 8958 T^{4} - 36501 T^{5} + 441125 T^{6} - 1838457 T^{7} + 17317003 T^{8} - 72845222 T^{9} + 571934352 T^{10} - 2393239283 T^{11} + 16359891658 T^{12} - 67164262668 T^{13} + 412288983795 T^{14} - 1640265737352 T^{15} + 9248689375227 T^{16} - 35240518894600 T^{17} + 185627099738938 T^{18} - 669569858997400 T^{19} + 3338776864456947 T^{20} - 11250582692497368 T^{21} + 53729912657148195 T^{22} - 166305363627972132 T^{23} + 769665516115160698 T^{24} - 2139248959738323137 T^{25} + 9713483122505484432 T^{26} - 23506256981380161938 T^{27} +$$$$10\!\cdots\!03$$$$T^{28} -$$$$21\!\cdots\!83$$$$T^{29} +$$$$97\!\cdots\!25$$$$T^{30} -$$$$15\!\cdots\!59$$$$T^{31} +$$$$71\!\cdots\!18$$$$T^{32} -$$$$78\!\cdots\!84$$$$T^{33} +$$$$37\!\cdots\!30$$$$T^{34} -$$$$21\!\cdots\!56$$$$T^{35} +$$$$10\!\cdots\!41$$$$T^{36}$$)
$23$ ($$1 + 4 T + 92 T^{3} + 529 T^{4}$$)($$1 + 26 T + 516 T^{2} + 7395 T^{3} + 90928 T^{4} + 951516 T^{5} + 8960211 T^{6} + 75889920 T^{7} + 593605359 T^{8} + 4293455106 T^{9} + 29167458828 T^{10} + 186453853585 T^{11} + 1133632710313 T^{12} + 6568187578396 T^{13} + 36553007673681 T^{14} + 195688621929534 T^{15} + 1013582264018988 T^{16} + 5081312191544094 T^{17} + 24743429994041086 T^{18} + 116870180405514162 T^{19} + 536185017666044652 T^{20} + 2380943463016640178 T^{21} + 10229030220410564721 T^{22} + 42275108142896045828 T^{23} +$$$$16\!\cdots\!57$$$$T^{24} +$$$$63\!\cdots\!95$$$$T^{25} +$$$$22\!\cdots\!68$$$$T^{26} +$$$$77\!\cdots\!78$$$$T^{27} +$$$$24\!\cdots\!91$$$$T^{28} +$$$$72\!\cdots\!40$$$$T^{29} +$$$$19\!\cdots\!31$$$$T^{30} +$$$$47\!\cdots\!28$$$$T^{31} +$$$$10\!\cdots\!52$$$$T^{32} +$$$$19\!\cdots\!65$$$$T^{33} +$$$$31\!\cdots\!76$$$$T^{34} +$$$$36\!\cdots\!78$$$$T^{35} +$$$$32\!\cdots\!69$$$$T^{36}$$)
$29$ ($$1 + 14 T + 99 T^{2} + 406 T^{3} + 841 T^{4}$$)($$1 + 18 T + 531 T^{2} + 7242 T^{3} + 124429 T^{4} + 1388519 T^{5} + 17800604 T^{6} + 169607253 T^{7} + 1778477154 T^{8} + 14850454505 T^{9} + 133353152337 T^{10} + 992168503525 T^{11} + 7832929693192 T^{12} + 52481764441320 T^{13} + 370156555204319 T^{14} + 2247849800307197 T^{15} + 14304279024344391 T^{16} + 78983107272313558 T^{17} + 456054513586518578 T^{18} + 2290510110897093182 T^{19} + 12029898659473632831 T^{20} + 54822808779692227633 T^{21} +$$$$26\!\cdots\!39$$$$T^{22} +$$$$10\!\cdots\!80$$$$T^{23} +$$$$46\!\cdots\!32$$$$T^{24} +$$$$17\!\cdots\!25$$$$T^{25} +$$$$66\!\cdots\!57$$$$T^{26} +$$$$21\!\cdots\!45$$$$T^{27} +$$$$74\!\cdots\!54$$$$T^{28} +$$$$20\!\cdots\!37$$$$T^{29} +$$$$62\!\cdots\!64$$$$T^{30} +$$$$14\!\cdots\!91$$$$T^{31} +$$$$37\!\cdots\!49$$$$T^{32} +$$$$62\!\cdots\!58$$$$T^{33} +$$$$13\!\cdots\!51$$$$T^{34} +$$$$13\!\cdots\!62$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$)
$31$ ($$1 - 4 T + 58 T^{2} - 124 T^{3} + 961 T^{4}$$)($$1 + 5 T + 226 T^{2} + 736 T^{3} + 25466 T^{4} + 48427 T^{5} + 1947515 T^{6} + 1181005 T^{7} + 114803460 T^{8} - 68748799 T^{9} + 5711179330 T^{10} - 8784337491 T^{11} + 254611322915 T^{12} - 519109476169 T^{13} + 10413960551250 T^{14} - 22094105896997 T^{15} + 386758640569635 T^{16} - 776800932680999 T^{17} + 12771866096424756 T^{18} - 24080828913110969 T^{19} + 371675053587419235 T^{20} - 658205508777437627 T^{21} + 9617511262250951250 T^{22} - 14861663578773202519 T^{23} +$$$$22\!\cdots\!15$$$$T^{24} -$$$$24\!\cdots\!01$$$$T^{25} +$$$$48\!\cdots\!30$$$$T^{26} -$$$$18\!\cdots\!29$$$$T^{27} +$$$$94\!\cdots\!60$$$$T^{28} +$$$$30\!\cdots\!55$$$$T^{29} +$$$$15\!\cdots\!15$$$$T^{30} +$$$$11\!\cdots\!57$$$$T^{31} +$$$$19\!\cdots\!86$$$$T^{32} +$$$$17\!\cdots\!36$$$$T^{33} +$$$$16\!\cdots\!06$$$$T^{34} +$$$$11\!\cdots\!55$$$$T^{35} +$$$$69\!\cdots\!41$$$$T^{36}$$)
$37$ ($$1 + 4 T + 76 T^{2} + 148 T^{3} + 1369 T^{4}$$)($$1 + 18 T + 477 T^{2} + 6182 T^{3} + 98542 T^{4} + 1025448 T^{5} + 12472914 T^{6} + 109789868 T^{7} + 1115943998 T^{8} + 8578978218 T^{9} + 76645978734 T^{10} + 526915335411 T^{11} + 4285047049335 T^{12} + 26892539266031 T^{13} + 204473204412967 T^{14} + 1194720790323650 T^{15} + 8656318678483359 T^{16} + 47928410626044380 T^{17} + 333914579074632830 T^{18} + 1773351193163642060 T^{19} + 11850500270843718471 T^{20} + 60516192192263843450 T^{21} +$$$$38\!\cdots\!87$$$$T^{22} +$$$$18\!\cdots\!67$$$$T^{23} +$$$$10\!\cdots\!15$$$$T^{24} +$$$$50\!\cdots\!63$$$$T^{25} +$$$$26\!\cdots\!14$$$$T^{26} +$$$$11\!\cdots\!86$$$$T^{27} +$$$$53\!\cdots\!02$$$$T^{28} +$$$$19\!\cdots\!84$$$$T^{29} +$$$$82\!\cdots\!34$$$$T^{30} +$$$$24\!\cdots\!56$$$$T^{31} +$$$$88\!\cdots\!38$$$$T^{32} +$$$$20\!\cdots\!26$$$$T^{33} +$$$$58\!\cdots\!57$$$$T^{34} +$$$$82\!\cdots\!06$$$$T^{35} +$$$$16\!\cdots\!29$$$$T^{36}$$)
$41$ ($$1 - 8 T + 48 T^{2} - 328 T^{3} + 1681 T^{4}$$)($$1 + 17 T + 500 T^{2} + 6935 T^{3} + 118746 T^{4} + 1411049 T^{5} + 18245075 T^{6} + 191387044 T^{7} + 2060476099 T^{8} + 19451816951 T^{9} + 182957350222 T^{10} + 1574307116444 T^{11} + 13284361461403 T^{12} + 105065275805631 T^{13} + 807813171251088 T^{14} + 5903001862494918 T^{15} + 41738761877644030 T^{16} + 282544368422135749 T^{17} + 1846894667477774252 T^{18} + 11584319105307565709 T^{19} + 70162858716319614430 T^{20} +$$$$40\!\cdots\!78$$$$T^{21} +$$$$22\!\cdots\!68$$$$T^{22} +$$$$12\!\cdots\!31$$$$T^{23} +$$$$63\!\cdots\!23$$$$T^{24} +$$$$30\!\cdots\!64$$$$T^{25} +$$$$14\!\cdots\!62$$$$T^{26} +$$$$63\!\cdots\!11$$$$T^{27} +$$$$27\!\cdots\!99$$$$T^{28} +$$$$10\!\cdots\!04$$$$T^{29} +$$$$41\!\cdots\!75$$$$T^{30} +$$$$13\!\cdots\!29$$$$T^{31} +$$$$45\!\cdots\!06$$$$T^{32} +$$$$10\!\cdots\!35$$$$T^{33} +$$$$31\!\cdots\!00$$$$T^{34} +$$$$44\!\cdots\!77$$$$T^{35} +$$$$10\!\cdots\!21$$$$T^{36}$$)
$43$ ($$( 1 + 8 T + 43 T^{2} )^{2}$$)($$1 - 8 T + 492 T^{2} - 3407 T^{3} + 117668 T^{4} - 715275 T^{5} + 18347951 T^{6} - 99049565 T^{7} + 2107507666 T^{8} - 10209475962 T^{9} + 190684760633 T^{10} - 837060417712 T^{11} + 14161353767533 T^{12} - 56841478497834 T^{13} + 886000874851046 T^{14} - 3276977381712634 T^{15} + 47444588926667291 T^{16} - 162652673998347287 T^{17} + 2192977866661315750 T^{18} - 6994064981928933341 T^{19} + 87725044925407821059 T^{20} -$$$$26\!\cdots\!38$$$$T^{21} +$$$$30\!\cdots\!46$$$$T^{22} -$$$$83\!\cdots\!62$$$$T^{23} +$$$$89\!\cdots\!17$$$$T^{24} -$$$$22\!\cdots\!84$$$$T^{25} +$$$$22\!\cdots\!33$$$$T^{26} -$$$$51\!\cdots\!66$$$$T^{27} +$$$$45\!\cdots\!34$$$$T^{28} -$$$$92\!\cdots\!55$$$$T^{29} +$$$$73\!\cdots\!51$$$$T^{30} -$$$$12\!\cdots\!25$$$$T^{31} +$$$$86\!\cdots\!32$$$$T^{32} -$$$$10\!\cdots\!49$$$$T^{33} +$$$$67\!\cdots\!92$$$$T^{34} -$$$$46\!\cdots\!44$$$$T^{35} +$$$$25\!\cdots\!49$$$$T^{36}$$)
$47$ ($$1 - 22 T + 213 T^{2} - 1034 T^{3} + 2209 T^{4}$$)($$1 + 52 T + 1865 T^{2} + 49131 T^{3} + 1077176 T^{4} + 20125023 T^{5} + 332676442 T^{6} + 4927129277 T^{7} + 66491991801 T^{8} + 823722681768 T^{9} + 9453432637832 T^{10} + 100973005595782 T^{11} + 1009215441829873 T^{12} + 9466179137600663 T^{13} + 83603559634490998 T^{14} + 696394998474434729 T^{15} + 5481817752006743685 T^{16} + 40807789859381398323 T^{17} +$$$$28\!\cdots\!97$$$$T^{18} +$$$$19\!\cdots\!81$$$$T^{19} +$$$$12\!\cdots\!65$$$$T^{20} +$$$$72\!\cdots\!67$$$$T^{21} +$$$$40\!\cdots\!38$$$$T^{22} +$$$$21\!\cdots\!41$$$$T^{23} +$$$$10\!\cdots\!17$$$$T^{24} +$$$$51\!\cdots\!66$$$$T^{25} +$$$$22\!\cdots\!52$$$$T^{26} +$$$$92\!\cdots\!56$$$$T^{27} +$$$$34\!\cdots\!49$$$$T^{28} +$$$$12\!\cdots\!31$$$$T^{29} +$$$$38\!\cdots\!22$$$$T^{30} +$$$$10\!\cdots\!21$$$$T^{31} +$$$$27\!\cdots\!44$$$$T^{32} +$$$$59\!\cdots\!33$$$$T^{33} +$$$$10\!\cdots\!65$$$$T^{34} +$$$$13\!\cdots\!24$$$$T^{35} +$$$$12\!\cdots\!89$$$$T^{36}$$)
$53$ ($$1 + 14 T + 123 T^{2} + 742 T^{3} + 2809 T^{4}$$)($$1 + 60 T + 2292 T^{2} + 64439 T^{3} + 1482802 T^{4} + 29045053 T^{5} + 500611675 T^{6} + 7729740473 T^{7} + 108589762152 T^{8} + 1401743191155 T^{9} + 16770166121672 T^{10} + 187053090421054 T^{11} + 1955182548973064 T^{12} + 19221087500660183 T^{13} + 178277342948315803 T^{14} + 1563393906027613236 T^{15} + 12985863111074957420 T^{16} +$$$$10\!\cdots\!72$$$$T^{17} +$$$$76\!\cdots\!76$$$$T^{18} +$$$$54\!\cdots\!16$$$$T^{19} +$$$$36\!\cdots\!80$$$$T^{20} +$$$$23\!\cdots\!72$$$$T^{21} +$$$$14\!\cdots\!43$$$$T^{22} +$$$$80\!\cdots\!19$$$$T^{23} +$$$$43\!\cdots\!56$$$$T^{24} +$$$$21\!\cdots\!98$$$$T^{25} +$$$$10\!\cdots\!92$$$$T^{26} +$$$$46\!\cdots\!15$$$$T^{27} +$$$$18\!\cdots\!48$$$$T^{28} +$$$$71\!\cdots\!81$$$$T^{29} +$$$$24\!\cdots\!75$$$$T^{30} +$$$$75\!\cdots\!69$$$$T^{31} +$$$$20\!\cdots\!38$$$$T^{32} +$$$$47\!\cdots\!23$$$$T^{33} +$$$$88\!\cdots\!32$$$$T^{34} +$$$$12\!\cdots\!80$$$$T^{35} +$$$$10\!\cdots\!89$$$$T^{36}$$)
$59$ ($$1 + 12 T + 136 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$1 + 8 T + 547 T^{2} + 4886 T^{3} + 151709 T^{4} + 1426288 T^{5} + 28528730 T^{6} + 268965857 T^{7} + 4083691371 T^{8} + 37250203024 T^{9} + 471936791901 T^{10} + 4078707707888 T^{11} + 45548114306989 T^{12} + 370272133366719 T^{13} + 3754286226421140 T^{14} + 28722234855750938 T^{15} + 268673863454487823 T^{16} + 1936169927933708248 T^{17} + 16882310523288642930 T^{18} +$$$$11\!\cdots\!32$$$$T^{19} +$$$$93\!\cdots\!63$$$$T^{20} +$$$$58\!\cdots\!02$$$$T^{21} +$$$$45\!\cdots\!40$$$$T^{22} +$$$$26\!\cdots\!81$$$$T^{23} +$$$$19\!\cdots\!49$$$$T^{24} +$$$$10\!\cdots\!72$$$$T^{25} +$$$$69\!\cdots\!21$$$$T^{26} +$$$$32\!\cdots\!36$$$$T^{27} +$$$$20\!\cdots\!71$$$$T^{28} +$$$$81\!\cdots\!63$$$$T^{29} +$$$$50\!\cdots\!30$$$$T^{30} +$$$$14\!\cdots\!52$$$$T^{31} +$$$$93\!\cdots\!49$$$$T^{32} +$$$$17\!\cdots\!14$$$$T^{33} +$$$$11\!\cdots\!27$$$$T^{34} +$$$$10\!\cdots\!52$$$$T^{35} +$$$$75\!\cdots\!21$$$$T^{36}$$)
$61$ ($$1 - 4 T + 124 T^{2} - 244 T^{3} + 3721 T^{4}$$)($$1 + 26 T + 1092 T^{2} + 21198 T^{3} + 517593 T^{4} + 8124611 T^{5} + 147632317 T^{6} + 1959197923 T^{7} + 29069634275 T^{8} + 335550327593 T^{9} + 4266607342662 T^{10} + 43706374725751 T^{11} + 490749289239130 T^{12} + 4530429185717696 T^{13} + 45852509636769219 T^{14} + 386091138768948475 T^{15} + 3572270707620359676 T^{16} + 27670900627019165169 T^{17} +$$$$23\!\cdots\!26$$$$T^{18} +$$$$16\!\cdots\!09$$$$T^{19} +$$$$13\!\cdots\!96$$$$T^{20} +$$$$87\!\cdots\!75$$$$T^{21} +$$$$63\!\cdots\!79$$$$T^{22} +$$$$38\!\cdots\!96$$$$T^{23} +$$$$25\!\cdots\!30$$$$T^{24} +$$$$13\!\cdots\!71$$$$T^{25} +$$$$81\!\cdots\!22$$$$T^{26} +$$$$39\!\cdots\!13$$$$T^{27} +$$$$20\!\cdots\!75$$$$T^{28} +$$$$85\!\cdots\!03$$$$T^{29} +$$$$39\!\cdots\!57$$$$T^{30} +$$$$13\!\cdots\!91$$$$T^{31} +$$$$51\!\cdots\!13$$$$T^{32} +$$$$12\!\cdots\!98$$$$T^{33} +$$$$40\!\cdots\!12$$$$T^{34} +$$$$58\!\cdots\!46$$$$T^{35} +$$$$13\!\cdots\!81$$$$T^{36}$$)
$67$ ($$1 + 10 T + 61 T^{2} + 670 T^{3} + 4489 T^{4}$$)($$1 - 12 T + 743 T^{2} - 8061 T^{3} + 273924 T^{4} - 2736258 T^{5} + 67007649 T^{6} - 622982958 T^{7} + 12221255375 T^{8} - 106355514112 T^{9} + 1765865096326 T^{10} - 14415408887134 T^{11} + 209411947253484 T^{12} - 1603069573249209 T^{13} + 20828295096643948 T^{14} - 149173768364533151 T^{15} + 1760455572360289685 T^{16} - 11748171421911845627 T^{17} +$$$$12\!\cdots\!21$$$$T^{18} -$$$$78\!\cdots\!09$$$$T^{19} +$$$$79\!\cdots\!65$$$$T^{20} -$$$$44\!\cdots\!13$$$$T^{21} +$$$$41\!\cdots\!08$$$$T^{22} -$$$$21\!\cdots\!63$$$$T^{23} +$$$$18\!\cdots\!96$$$$T^{24} -$$$$87\!\cdots\!82$$$$T^{25} +$$$$71\!\cdots\!66$$$$T^{26} -$$$$28\!\cdots\!64$$$$T^{27} +$$$$22\!\cdots\!75$$$$T^{28} -$$$$76\!\cdots\!14$$$$T^{29} +$$$$54\!\cdots\!89$$$$T^{30} -$$$$15\!\cdots\!46$$$$T^{31} +$$$$10\!\cdots\!96$$$$T^{32} -$$$$19\!\cdots\!23$$$$T^{33} +$$$$12\!\cdots\!83$$$$T^{34} -$$$$13\!\cdots\!24$$$$T^{35} +$$$$74\!\cdots\!09$$$$T^{36}$$)
$71$ ($$1 + 140 T^{2} + 5041 T^{4}$$)($$1 + T + 570 T^{2} - 395 T^{3} + 165367 T^{4} - 402481 T^{5} + 32659200 T^{6} - 135249850 T^{7} + 4976102531 T^{8} - 28212173634 T^{9} + 627081345260 T^{10} - 4301680526387 T^{11} + 68002974751364 T^{12} - 517699284019560 T^{13} + 6477452704212298 T^{14} - 51294377851106202 T^{15} + 547109894053610156 T^{16} - 4280850522490499670 T^{17} + 41143559904963586258 T^{18} -$$$$30\!\cdots\!70$$$$T^{19} +$$$$27\!\cdots\!96$$$$T^{20} -$$$$18\!\cdots\!22$$$$T^{21} +$$$$16\!\cdots\!38$$$$T^{22} -$$$$93\!\cdots\!60$$$$T^{23} +$$$$87\!\cdots\!44$$$$T^{24} -$$$$39\!\cdots\!17$$$$T^{25} +$$$$40\!\cdots\!60$$$$T^{26} -$$$$12\!\cdots\!54$$$$T^{27} +$$$$16\!\cdots\!31$$$$T^{28} -$$$$31\!\cdots\!50$$$$T^{29} +$$$$53\!\cdots\!00$$$$T^{30} -$$$$46\!\cdots\!91$$$$T^{31} +$$$$13\!\cdots\!27$$$$T^{32} -$$$$23\!\cdots\!45$$$$T^{33} +$$$$23\!\cdots\!70$$$$T^{34} +$$$$29\!\cdots\!91$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$)
$73$ ($$1 + 2 T + 115 T^{2} + 146 T^{3} + 5329 T^{4}$$)($$1 + 2 T + 402 T^{2} + 2100 T^{3} + 91054 T^{4} + 664143 T^{5} + 16192289 T^{6} + 128526512 T^{7} + 2389630448 T^{8} + 19163681612 T^{9} + 298745214234 T^{10} + 2361654820993 T^{11} + 32513178733431 T^{12} + 249000864582652 T^{13} + 3122389883437745 T^{14} + 23022354077637360 T^{15} + 267483657127421832 T^{16} + 1885230323459498199 T^{17} + 20590875702168070970 T^{18} +$$$$13\!\cdots\!27$$$$T^{19} +$$$$14\!\cdots\!28$$$$T^{20} +$$$$89\!\cdots\!20$$$$T^{21} +$$$$88\!\cdots\!45$$$$T^{22} +$$$$51\!\cdots\!36$$$$T^{23} +$$$$49\!\cdots\!59$$$$T^{24} +$$$$26\!\cdots\!21$$$$T^{25} +$$$$24\!\cdots\!54$$$$T^{26} +$$$$11\!\cdots\!56$$$$T^{27} +$$$$10\!\cdots\!52$$$$T^{28} +$$$$40\!\cdots\!24$$$$T^{29} +$$$$37\!\cdots\!69$$$$T^{30} +$$$$11\!\cdots\!19$$$$T^{31} +$$$$11\!\cdots\!86$$$$T^{32} +$$$$18\!\cdots\!00$$$$T^{33} +$$$$26\!\cdots\!22$$$$T^{34} +$$$$94\!\cdots\!06$$$$T^{35} +$$$$34\!\cdots\!69$$$$T^{36}$$)
$79$ ($$1 + 8 T + 124 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 - 18 T + 1297 T^{2} - 20031 T^{3} + 786188 T^{4} - 10631617 T^{5} + 298506186 T^{6} - 3585561802 T^{7} + 80086048911 T^{8} - 863338284200 T^{9} + 16209659796490 T^{10} - 157996153097097 T^{11} + 2576626887472372 T^{12} - 22823417890305456 T^{13} + 330077405985329133 T^{14} - 2664890406520132266 T^{15} + 34639279636509263037 T^{16} -$$$$25\!\cdots\!59$$$$T^{17} +$$$$30\!\cdots\!66$$$$T^{18} -$$$$20\!\cdots\!61$$$$T^{19} +$$$$21\!\cdots\!17$$$$T^{20} -$$$$13\!\cdots\!74$$$$T^{21} +$$$$12\!\cdots\!73$$$$T^{22} -$$$$70\!\cdots\!44$$$$T^{23} +$$$$62\!\cdots\!12$$$$T^{24} -$$$$30\!\cdots\!23$$$$T^{25} +$$$$24\!\cdots\!90$$$$T^{26} -$$$$10\!\cdots\!00$$$$T^{27} +$$$$75\!\cdots\!11$$$$T^{28} -$$$$26\!\cdots\!58$$$$T^{29} +$$$$17\!\cdots\!26$$$$T^{30} -$$$$49\!\cdots\!63$$$$T^{31} +$$$$28\!\cdots\!28$$$$T^{32} -$$$$58\!\cdots\!69$$$$T^{33} +$$$$29\!\cdots\!37$$$$T^{34} -$$$$32\!\cdots\!62$$$$T^{35} +$$$$14\!\cdots\!61$$$$T^{36}$$)
$83$ ($$1 - 16 T + 180 T^{2} - 1328 T^{3} + 6889 T^{4}$$)($$1 + 43 T + 1214 T^{2} + 27029 T^{3} + 517233 T^{4} + 8734852 T^{5} + 134157792 T^{6} + 1904363732 T^{7} + 25324885272 T^{8} + 318326987644 T^{9} + 3816043239532 T^{10} + 43881446080752 T^{11} + 486363584266611 T^{12} + 5212562566603327 T^{13} + 54157832160852686 T^{14} + 546073362822746053 T^{15} + 5347456077270647030 T^{16} + 50872297755235833738 T^{17} +$$$$47\!\cdots\!82$$$$T^{18} +$$$$42\!\cdots\!54$$$$T^{19} +$$$$36\!\cdots\!70$$$$T^{20} +$$$$31\!\cdots\!11$$$$T^{21} +$$$$25\!\cdots\!06$$$$T^{22} +$$$$20\!\cdots\!61$$$$T^{23} +$$$$15\!\cdots\!59$$$$T^{24} +$$$$11\!\cdots\!04$$$$T^{25} +$$$$85\!\cdots\!12$$$$T^{26} +$$$$59\!\cdots\!32$$$$T^{27} +$$$$39\!\cdots\!28$$$$T^{28} +$$$$24\!\cdots\!44$$$$T^{29} +$$$$14\!\cdots\!12$$$$T^{30} +$$$$77\!\cdots\!76$$$$T^{31} +$$$$38\!\cdots\!57$$$$T^{32} +$$$$16\!\cdots\!03$$$$T^{33} +$$$$61\!\cdots\!34$$$$T^{34} +$$$$18\!\cdots\!89$$$$T^{35} +$$$$34\!\cdots\!09$$$$T^{36}$$)
$89$ ($$1 + 160 T^{2} + 7921 T^{4}$$)($$1 + 28 T + 1278 T^{2} + 27910 T^{3} + 750883 T^{4} + 13674851 T^{5} + 278327874 T^{6} + 4395094780 T^{7} + 74207320481 T^{8} + 1041079823079 T^{9} + 15262444839605 T^{10} + 193236416956553 T^{11} + 2524149327752443 T^{12} + 29133832029172985 T^{13} + 344418211589959806 T^{14} + 3646576704043225682 T^{15} + 39387762497694988269 T^{16} +$$$$38\!\cdots\!80$$$$T^{17} +$$$$38\!\cdots\!72$$$$T^{18} +$$$$34\!\cdots\!20$$$$T^{19} +$$$$31\!\cdots\!49$$$$T^{20} +$$$$25\!\cdots\!58$$$$T^{21} +$$$$21\!\cdots\!46$$$$T^{22} +$$$$16\!\cdots\!65$$$$T^{23} +$$$$12\!\cdots\!23$$$$T^{24} +$$$$85\!\cdots\!37$$$$T^{25} +$$$$60\!\cdots\!05$$$$T^{26} +$$$$36\!\cdots\!11$$$$T^{27} +$$$$23\!\cdots\!81$$$$T^{28} +$$$$12\!\cdots\!20$$$$T^{29} +$$$$68\!\cdots\!54$$$$T^{30} +$$$$30\!\cdots\!19$$$$T^{31} +$$$$14\!\cdots\!03$$$$T^{32} +$$$$48\!\cdots\!90$$$$T^{33} +$$$$19\!\cdots\!58$$$$T^{34} +$$$$38\!\cdots\!12$$$$T^{35} +$$$$12\!\cdots\!81$$$$T^{36}$$)
$97$ ($$1 - 18 T + 203 T^{2} - 1746 T^{3} + 9409 T^{4}$$)($$1 + 34 T + 1734 T^{2} + 44982 T^{3} + 1350738 T^{4} + 28687000 T^{5} + 647807944 T^{6} + 11728222048 T^{7} + 217762436551 T^{8} + 3446769515305 T^{9} + 54984423618660 T^{10} + 773527117230350 T^{11} + 10868994800485399 T^{12} + 137384482180690541 T^{13} + 1726110187080262341 T^{14} + 19737225141617642237 T^{15} +$$$$22\!\cdots\!37$$$$T^{16} +$$$$23\!\cdots\!18$$$$T^{17} +$$$$23\!\cdots\!52$$$$T^{18} +$$$$22\!\cdots\!46$$$$T^{19} +$$$$21\!\cdots\!33$$$$T^{20} +$$$$18\!\cdots\!01$$$$T^{21} +$$$$15\!\cdots\!21$$$$T^{22} +$$$$11\!\cdots\!37$$$$T^{23} +$$$$90\!\cdots\!71$$$$T^{24} +$$$$62\!\cdots\!50$$$$T^{25} +$$$$43\!\cdots\!60$$$$T^{26} +$$$$26\!\cdots\!85$$$$T^{27} +$$$$16\!\cdots\!99$$$$T^{28} +$$$$83\!\cdots\!44$$$$T^{29} +$$$$44\!\cdots\!04$$$$T^{30} +$$$$19\!\cdots\!00$$$$T^{31} +$$$$88\!\cdots\!22$$$$T^{32} +$$$$28\!\cdots\!26$$$$T^{33} +$$$$10\!\cdots\!14$$$$T^{34} +$$$$20\!\cdots\!58$$$$T^{35} +$$$$57\!\cdots\!89$$$$T^{36}$$)