# Properties

 Label 43.4.c Level 43 Weight 4 Character orbit c Rep. character $$\chi_{43}(6,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 20 Newform subspaces 1 Sturm bound 14 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 43.c (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$43$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$1$$ Sturm bound: $$14$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(43, [\chi])$$.

Total New Old
Modular forms 24 24 0
Cusp forms 20 20 0
Eisenstein series 4 4 0

## Trace form

 $$20q - 2q^{2} - 5q^{3} + 78q^{4} - 19q^{5} + 15q^{6} - 51q^{7} - 72q^{8} - 117q^{9} + O(q^{10})$$ $$20q - 2q^{2} - 5q^{3} + 78q^{4} - 19q^{5} + 15q^{6} - 51q^{7} - 72q^{8} - 117q^{9} + 27q^{10} + 54q^{11} - 72q^{12} - 15q^{13} + 96q^{14} + 65q^{15} + 134q^{16} - 82q^{17} + 247q^{18} + 78q^{19} - 495q^{20} - 18q^{21} + 380q^{22} - 61q^{23} + 202q^{24} - 151q^{25} - 21q^{26} - 194q^{27} - 794q^{28} - 53q^{29} + 627q^{30} + 253q^{31} - 798q^{32} - 424q^{33} - 231q^{34} + 710q^{35} - 1092q^{36} - 129q^{37} - 854q^{38} + 1382q^{39} + 1345q^{40} + 782q^{41} + 62q^{42} + 1025q^{43} + 754q^{44} + 1888q^{45} - 40q^{46} - 668q^{47} - 2401q^{48} - 115q^{49} + 424q^{50} + 1590q^{51} - 564q^{52} + 773q^{53} + 364q^{54} - 1242q^{55} - 923q^{56} - 765q^{57} + 1328q^{58} - 2966q^{59} - 1075q^{60} + 437q^{61} + 1509q^{62} - 2222q^{63} - 1476q^{64} - 2126q^{65} + 1483q^{66} - 642q^{67} - 1052q^{68} - 3503q^{69} - 170q^{70} - 1545q^{71} + 3834q^{72} + 1292q^{73} - 2232q^{74} + 164q^{75} - 252q^{76} + 1448q^{77} + 5644q^{78} - 1405q^{79} - 3157q^{80} + 974q^{81} + 6608q^{82} + 543q^{83} + 7304q^{84} + 1946q^{85} + 2776q^{86} + 2818q^{87} - 5372q^{88} - 2196q^{89} - 1484q^{90} - 3513q^{91} + 2629q^{92} - 983q^{93} + 9878q^{94} - 149q^{95} + 3540q^{96} - 850q^{97} - 213q^{98} - 3181q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(43, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
43.4.c.a $$20$$ $$2.537$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-2$$ $$-5$$ $$-19$$ $$-51$$ $$q+\beta _{5}q^{2}+(-\beta _{4}-\beta _{6})q^{3}+(4+\beta _{2}+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + 21 T^{2} + 30 T^{3} + 291 T^{4} + 493 T^{5} + 3299 T^{6} + 6020 T^{7} + 30260 T^{8} + 57696 T^{9} + 252608 T^{10} + 461568 T^{11} + 1936640 T^{12} + 3082240 T^{13} + 13512704 T^{14} + 16154624 T^{15} + 76283904 T^{16} + 62914560 T^{17} + 352321536 T^{18} + 134217728 T^{19} + 1073741824 T^{20} )^{2}$$
$3$ $$1 + 5 T - 64 T^{2} - 207 T^{3} + 1736 T^{4} - 2493 T^{5} + 14940 T^{6} + 502453 T^{7} - 2029707 T^{8} - 12348180 T^{9} + 49889273 T^{10} - 155791671 T^{11} + 111657687 T^{12} + 13413963364 T^{13} - 31402447776 T^{14} - 104284718676 T^{15} + 597172338392 T^{16} - 7718563954902 T^{17} + 6795103553147 T^{18} + 149929321126415 T^{19} - 522136902551354 T^{20} + 4048091670413205 T^{21} + 4953630490244163 T^{22} - 151924494324336066 T^{23} + 317361864687382872 T^{24} - 1496371729803087132 T^{25} - 12165951673174882464 T^{26} +$$$$14\!\cdots\!92$$$$T^{27} + 31535428783950579447 T^{28} -$$$$11\!\cdots\!77$$$$T^{29} +$$$$10\!\cdots\!77$$$$T^{30} -$$$$68\!\cdots\!40$$$$T^{31} -$$$$30\!\cdots\!47$$$$T^{32} +$$$$20\!\cdots\!51$$$$T^{33} +$$$$16\!\cdots\!60$$$$T^{34} -$$$$73\!\cdots\!99$$$$T^{35} +$$$$13\!\cdots\!96$$$$T^{36} -$$$$44\!\cdots\!29$$$$T^{37} -$$$$37\!\cdots\!16$$$$T^{38} +$$$$78\!\cdots\!15$$$$T^{39} +$$$$42\!\cdots\!01$$$$T^{40}$$
$5$ $$1 + 19 T - 369 T^{2} - 9420 T^{3} + 46555 T^{4} + 2017871 T^{5} + 1496842 T^{6} - 191282201 T^{7} - 899992568 T^{8} - 8445628937 T^{9} - 69732522494 T^{10} + 4381874197151 T^{11} + 59385793792057 T^{12} - 382392294295462 T^{13} - 10614686309241807 T^{14} - 29318878428412277 T^{15} + 668588555926177131 T^{16} + 9697248609861628750 T^{17} + 75906644519801998580 T^{18} -$$$$63\!\cdots\!54$$$$T^{19} -$$$$19\!\cdots\!04$$$$T^{20} -$$$$79\!\cdots\!50$$$$T^{21} +$$$$11\!\cdots\!00$$$$T^{22} +$$$$18\!\cdots\!50$$$$T^{23} +$$$$16\!\cdots\!75$$$$T^{24} -$$$$89\!\cdots\!25$$$$T^{25} -$$$$40\!\cdots\!75$$$$T^{26} -$$$$18\!\cdots\!50$$$$T^{27} +$$$$35\!\cdots\!25$$$$T^{28} +$$$$32\!\cdots\!75$$$$T^{29} -$$$$64\!\cdots\!50$$$$T^{30} -$$$$98\!\cdots\!25$$$$T^{31} -$$$$13\!\cdots\!00$$$$T^{32} -$$$$34\!\cdots\!25$$$$T^{33} +$$$$34\!\cdots\!50$$$$T^{34} +$$$$57\!\cdots\!75$$$$T^{35} +$$$$16\!\cdots\!75$$$$T^{36} -$$$$41\!\cdots\!00$$$$T^{37} -$$$$20\!\cdots\!25$$$$T^{38} +$$$$13\!\cdots\!75$$$$T^{39} +$$$$86\!\cdots\!25$$$$T^{40}$$
$7$ $$1 + 51 T - 357 T^{2} - 45708 T^{3} + 199158 T^{4} + 23478106 T^{5} - 285493259 T^{6} - 11658162295 T^{7} + 146829104946 T^{8} + 4094897966023 T^{9} - 71194687182651 T^{10} - 1461672246803082 T^{11} + 35394258687371444 T^{12} + 643809247546209966 T^{13} - 13649568920422493291 T^{14} -$$$$16\!\cdots\!91$$$$T^{15} +$$$$61\!\cdots\!77$$$$T^{16} +$$$$32\!\cdots\!32$$$$T^{17} -$$$$23\!\cdots\!26$$$$T^{18} -$$$$58\!\cdots\!98$$$$T^{19} +$$$$71\!\cdots\!44$$$$T^{20} -$$$$20\!\cdots\!14$$$$T^{21} -$$$$27\!\cdots\!74$$$$T^{22} +$$$$13\!\cdots\!24$$$$T^{23} +$$$$84\!\cdots\!77$$$$T^{24} -$$$$77\!\cdots\!13$$$$T^{25} -$$$$22\!\cdots\!59$$$$T^{26} +$$$$35\!\cdots\!62$$$$T^{27} +$$$$67\!\cdots\!44$$$$T^{28} -$$$$96\!\cdots\!26$$$$T^{29} -$$$$16\!\cdots\!99$$$$T^{30} +$$$$31\!\cdots\!61$$$$T^{31} +$$$$38\!\cdots\!46$$$$T^{32} -$$$$10\!\cdots\!85$$$$T^{33} -$$$$89\!\cdots\!91$$$$T^{34} +$$$$25\!\cdots\!42$$$$T^{35} +$$$$73\!\cdots\!58$$$$T^{36} -$$$$57\!\cdots\!44$$$$T^{37} -$$$$15\!\cdots\!93$$$$T^{38} +$$$$75\!\cdots\!57$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$
$11$ $$( 1 - 27 T + 5077 T^{2} - 61984 T^{3} + 15043654 T^{4} - 162363748 T^{5} + 35128969454 T^{6} - 264446154846 T^{7} + 61397630693497 T^{8} - 405185649214743 T^{9} + 91897922249805306 T^{10} - 539302099104822933 T^{11} +$$$$10\!\cdots\!17$$$$T^{12} -$$$$62\!\cdots\!86$$$$T^{13} +$$$$11\!\cdots\!34$$$$T^{14} -$$$$67\!\cdots\!48$$$$T^{15} +$$$$83\!\cdots\!74$$$$T^{16} -$$$$45\!\cdots\!24$$$$T^{17} +$$$$50\!\cdots\!57$$$$T^{18} -$$$$35\!\cdots\!17$$$$T^{19} +$$$$17\!\cdots\!01$$$$T^{20} )^{2}$$
$13$ $$1 + 15 T - 15836 T^{2} - 84609 T^{3} + 136428704 T^{4} - 211873241 T^{5} - 798195901414 T^{6} + 5549492881873 T^{7} + 3492359466060033 T^{8} - 39363659442734576 T^{9} - 11971166560261837535 T^{10} +$$$$17\!\cdots\!97$$$$T^{11} +$$$$33\!\cdots\!29$$$$T^{12} -$$$$59\!\cdots\!52$$$$T^{13} -$$$$74\!\cdots\!72$$$$T^{14} +$$$$14\!\cdots\!74$$$$T^{15} +$$$$13\!\cdots\!50$$$$T^{16} -$$$$26\!\cdots\!76$$$$T^{17} -$$$$23\!\cdots\!67$$$$T^{18} +$$$$22\!\cdots\!07$$$$T^{19} +$$$$44\!\cdots\!78$$$$T^{20} +$$$$48\!\cdots\!79$$$$T^{21} -$$$$11\!\cdots\!03$$$$T^{22} -$$$$27\!\cdots\!48$$$$T^{23} +$$$$32\!\cdots\!50$$$$T^{24} +$$$$75\!\cdots\!18$$$$T^{25} -$$$$83\!\cdots\!88$$$$T^{26} -$$$$14\!\cdots\!76$$$$T^{27} +$$$$17\!\cdots\!69$$$$T^{28} +$$$$21\!\cdots\!49$$$$T^{29} -$$$$31\!\cdots\!15$$$$T^{30} -$$$$22\!\cdots\!28$$$$T^{31} +$$$$44\!\cdots\!53$$$$T^{32} +$$$$15\!\cdots\!21$$$$T^{33} -$$$$48\!\cdots\!66$$$$T^{34} -$$$$28\!\cdots\!13$$$$T^{35} +$$$$40\!\cdots\!84$$$$T^{36} -$$$$54\!\cdots\!33$$$$T^{37} -$$$$22\!\cdots\!04$$$$T^{38} +$$$$46\!\cdots\!95$$$$T^{39} +$$$$68\!\cdots\!01$$$$T^{40}$$
$17$ $$1 + 82 T - 24818 T^{2} - 2873008 T^{3} + 275626616 T^{4} + 48442716340 T^{5} - 1590978438438 T^{6} - 545228525674854 T^{7} + 42616778692923 T^{8} + 4700974927774535486 T^{9} +$$$$11\!\cdots\!15$$$$T^{10} -$$$$32\!\cdots\!40$$$$T^{11} -$$$$15\!\cdots\!83$$$$T^{12} +$$$$18\!\cdots\!16$$$$T^{13} +$$$$13\!\cdots\!50$$$$T^{14} -$$$$88\!\cdots\!82$$$$T^{15} -$$$$99\!\cdots\!98$$$$T^{16} +$$$$31\!\cdots\!96$$$$T^{17} +$$$$60\!\cdots\!95$$$$T^{18} -$$$$56\!\cdots\!84$$$$T^{19} -$$$$31\!\cdots\!82$$$$T^{20} -$$$$27\!\cdots\!92$$$$T^{21} +$$$$14\!\cdots\!55$$$$T^{22} +$$$$37\!\cdots\!12$$$$T^{23} -$$$$57\!\cdots\!78$$$$T^{24} -$$$$25\!\cdots\!26$$$$T^{25} +$$$$19\!\cdots\!50$$$$T^{26} +$$$$13\!\cdots\!72$$$$T^{27} -$$$$52\!\cdots\!43$$$$T^{28} -$$$$54\!\cdots\!20$$$$T^{29} +$$$$92\!\cdots\!35$$$$T^{30} +$$$$18\!\cdots\!82$$$$T^{31} +$$$$84\!\cdots\!63$$$$T^{32} -$$$$52\!\cdots\!62$$$$T^{33} -$$$$75\!\cdots\!82$$$$T^{34} +$$$$11\!\cdots\!80$$$$T^{35} +$$$$31\!\cdots\!56$$$$T^{36} -$$$$16\!\cdots\!64$$$$T^{37} -$$$$69\!\cdots\!22$$$$T^{38} +$$$$11\!\cdots\!14$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$19$ $$1 - 78 T - 30307 T^{2} + 1643314 T^{3} + 557392853 T^{4} - 19116976304 T^{5} - 6385440198566 T^{6} + 80081143266136 T^{7} + 48128038162948702 T^{8} + 395871623097331272 T^{9} -$$$$15\!\cdots\!30$$$$T^{10} -$$$$76\!\cdots\!40$$$$T^{11} -$$$$95\!\cdots\!37$$$$T^{12} +$$$$90\!\cdots\!74$$$$T^{13} +$$$$17\!\cdots\!19$$$$T^{14} +$$$$54\!\cdots\!06$$$$T^{15} -$$$$11\!\cdots\!65$$$$T^{16} -$$$$60\!\cdots\!84$$$$T^{17} +$$$$38\!\cdots\!88$$$$T^{18} +$$$$19\!\cdots\!44$$$$T^{19} -$$$$94\!\cdots\!24$$$$T^{20} +$$$$13\!\cdots\!96$$$$T^{21} +$$$$18\!\cdots\!28$$$$T^{22} -$$$$19\!\cdots\!36$$$$T^{23} -$$$$25\!\cdots\!65$$$$T^{24} +$$$$82\!\cdots\!94$$$$T^{25} +$$$$18\!\cdots\!79$$$$T^{26} +$$$$64\!\cdots\!06$$$$T^{27} -$$$$46\!\cdots\!77$$$$T^{28} -$$$$25\!\cdots\!60$$$$T^{29} -$$$$36\!\cdots\!30$$$$T^{30} +$$$$62\!\cdots\!48$$$$T^{31} +$$$$52\!\cdots\!62$$$$T^{32} +$$$$59\!\cdots\!44$$$$T^{33} -$$$$32\!\cdots\!26$$$$T^{34} -$$$$66\!\cdots\!96$$$$T^{35} +$$$$13\!\cdots\!73$$$$T^{36} +$$$$27\!\cdots\!66$$$$T^{37} -$$$$34\!\cdots\!47$$$$T^{38} -$$$$60\!\cdots\!42$$$$T^{39} +$$$$53\!\cdots\!01$$$$T^{40}$$
$23$ $$1 + 61 T - 81824 T^{2} - 6253131 T^{3} + 3389445188 T^{4} + 298260670973 T^{5} - 95613835472806 T^{6} - 9095183914407171 T^{7} + 2101707189884822683 T^{8} +$$$$20\!\cdots\!68$$$$T^{9} -$$$$39\!\cdots\!71$$$$T^{10} -$$$$34\!\cdots\!69$$$$T^{11} +$$$$65\!\cdots\!49$$$$T^{12} +$$$$47\!\cdots\!72$$$$T^{13} -$$$$10\!\cdots\!88$$$$T^{14} -$$$$51\!\cdots\!20$$$$T^{15} +$$$$15\!\cdots\!62$$$$T^{16} +$$$$43\!\cdots\!14$$$$T^{17} -$$$$21\!\cdots\!19$$$$T^{18} -$$$$18\!\cdots\!05$$$$T^{19} +$$$$27\!\cdots\!22$$$$T^{20} -$$$$22\!\cdots\!35$$$$T^{21} -$$$$31\!\cdots\!91$$$$T^{22} +$$$$77\!\cdots\!82$$$$T^{23} +$$$$33\!\cdots\!02$$$$T^{24} -$$$$13\!\cdots\!40$$$$T^{25} -$$$$33\!\cdots\!72$$$$T^{26} +$$$$18\!\cdots\!56$$$$T^{27} +$$$$31\!\cdots\!09$$$$T^{28} -$$$$20\!\cdots\!43$$$$T^{29} -$$$$27\!\cdots\!79$$$$T^{30} +$$$$17\!\cdots\!44$$$$T^{31} +$$$$22\!\cdots\!63$$$$T^{32} -$$$$11\!\cdots\!77$$$$T^{33} -$$$$14\!\cdots\!74$$$$T^{34} +$$$$56\!\cdots\!39$$$$T^{35} +$$$$78\!\cdots\!28$$$$T^{36} -$$$$17\!\cdots\!37$$$$T^{37} -$$$$27\!\cdots\!16$$$$T^{38} +$$$$25\!\cdots\!83$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$
$29$ $$1 + 53 T - 138329 T^{2} - 8566904 T^{3} + 9770438424 T^{4} + 711713671522 T^{5} - 454180068094483 T^{6} - 40428004977832863 T^{7} + 15194742238715729146 T^{8} +$$$$18\!\cdots\!33$$$$T^{9} -$$$$37\!\cdots\!31$$$$T^{10} -$$$$67\!\cdots\!46$$$$T^{11} +$$$$61\!\cdots\!62$$$$T^{12} +$$$$21\!\cdots\!70$$$$T^{13} -$$$$27\!\cdots\!19$$$$T^{14} -$$$$55\!\cdots\!63$$$$T^{15} -$$$$25\!\cdots\!79$$$$T^{16} +$$$$10\!\cdots\!00$$$$T^{17} +$$$$12\!\cdots\!66$$$$T^{18} -$$$$10\!\cdots\!10$$$$T^{19} -$$$$34\!\cdots\!60$$$$T^{20} -$$$$24\!\cdots\!90$$$$T^{21} +$$$$72\!\cdots\!86$$$$T^{22} +$$$$15\!\cdots\!00$$$$T^{23} -$$$$89\!\cdots\!39$$$$T^{24} -$$$$47\!\cdots\!87$$$$T^{25} -$$$$57\!\cdots\!59$$$$T^{26} +$$$$10\!\cdots\!30$$$$T^{27} +$$$$77\!\cdots\!22$$$$T^{28} -$$$$20\!\cdots\!14$$$$T^{29} -$$$$27\!\cdots\!31$$$$T^{30} +$$$$32\!\cdots\!37$$$$T^{31} +$$$$67\!\cdots\!66$$$$T^{32} -$$$$43\!\cdots\!47$$$$T^{33} -$$$$11\!\cdots\!03$$$$T^{34} +$$$$45\!\cdots\!78$$$$T^{35} +$$$$15\!\cdots\!64$$$$T^{36} -$$$$32\!\cdots\!16$$$$T^{37} -$$$$12\!\cdots\!49$$$$T^{38} +$$$$12\!\cdots\!77$$$$T^{39} +$$$$55\!\cdots\!01$$$$T^{40}$$
$31$ $$1 - 253 T - 41875 T^{2} + 7234722 T^{3} + 561602045 T^{4} + 650618176597 T^{5} - 97721636388434 T^{6} - 29167394163008097 T^{7} + 4500921270624132994 T^{8} +$$$$12\!\cdots\!49$$$$T^{9} +$$$$95\!\cdots\!38$$$$T^{10} -$$$$12\!\cdots\!41$$$$T^{11} -$$$$81\!\cdots\!49$$$$T^{12} +$$$$10\!\cdots\!88$$$$T^{13} +$$$$81\!\cdots\!75$$$$T^{14} -$$$$39\!\cdots\!15$$$$T^{15} +$$$$17\!\cdots\!87$$$$T^{16} -$$$$94\!\cdots\!70$$$$T^{17} +$$$$34\!\cdots\!84$$$$T^{18} +$$$$16\!\cdots\!10$$$$T^{19} -$$$$33\!\cdots\!96$$$$T^{20} +$$$$49\!\cdots\!10$$$$T^{21} +$$$$31\!\cdots\!04$$$$T^{22} -$$$$25\!\cdots\!70$$$$T^{23} +$$$$13\!\cdots\!07$$$$T^{24} -$$$$93\!\cdots\!65$$$$T^{25} +$$$$56\!\cdots\!75$$$$T^{26} +$$$$22\!\cdots\!28$$$$T^{27} -$$$$50\!\cdots\!29$$$$T^{28} -$$$$23\!\cdots\!51$$$$T^{29} +$$$$52\!\cdots\!38$$$$T^{30} +$$$$19\!\cdots\!59$$$$T^{31} +$$$$21\!\cdots\!14$$$$T^{32} -$$$$42\!\cdots\!87$$$$T^{33} -$$$$42\!\cdots\!74$$$$T^{34} +$$$$84\!\cdots\!47$$$$T^{35} +$$$$21\!\cdots\!45$$$$T^{36} +$$$$82\!\cdots\!82$$$$T^{37} -$$$$14\!\cdots\!75$$$$T^{38} -$$$$25\!\cdots\!83$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$
$37$ $$1 + 129 T - 268978 T^{2} - 43370241 T^{3} + 34974292754 T^{4} + 6620200735309 T^{5} - 2909843471321326 T^{6} - 614683056380244321 T^{7} +$$$$17\!\cdots\!39$$$$T^{8} +$$$$39\!\cdots\!18$$$$T^{9} -$$$$91\!\cdots\!59$$$$T^{10} -$$$$18\!\cdots\!75$$$$T^{11} +$$$$44\!\cdots\!41$$$$T^{12} +$$$$78\!\cdots\!22$$$$T^{13} -$$$$19\!\cdots\!38$$$$T^{14} -$$$$35\!\cdots\!54$$$$T^{15} +$$$$53\!\cdots\!04$$$$T^{16} +$$$$14\!\cdots\!10$$$$T^{17} +$$$$31\!\cdots\!45$$$$T^{18} -$$$$31\!\cdots\!85$$$$T^{19} -$$$$90\!\cdots\!42$$$$T^{20} -$$$$15\!\cdots\!05$$$$T^{21} +$$$$81\!\cdots\!05$$$$T^{22} +$$$$19\!\cdots\!70$$$$T^{23} +$$$$35\!\cdots\!24$$$$T^{24} -$$$$11\!\cdots\!22$$$$T^{25} -$$$$32\!\cdots\!02$$$$T^{26} +$$$$67\!\cdots\!14$$$$T^{27} +$$$$19\!\cdots\!01$$$$T^{28} -$$$$40\!\cdots\!75$$$$T^{29} -$$$$10\!\cdots\!91$$$$T^{30} +$$$$21\!\cdots\!46$$$$T^{31} +$$$$50\!\cdots\!99$$$$T^{32} -$$$$88\!\cdots\!33$$$$T^{33} -$$$$21\!\cdots\!94$$$$T^{34} +$$$$24\!\cdots\!13$$$$T^{35} +$$$$65\!\cdots\!34$$$$T^{36} -$$$$41\!\cdots\!33$$$$T^{37} -$$$$12\!\cdots\!42$$$$T^{38} +$$$$31\!\cdots\!93$$$$T^{39} +$$$$12\!\cdots\!01$$$$T^{40}$$
$41$ $$( 1 - 391 T + 466858 T^{2} - 159315937 T^{3} + 107799629054 T^{4} - 32107569594125 T^{5} + 16028988094146372 T^{6} - 4181844238111783617 T^{7} +$$$$16\!\cdots\!81$$$$T^{8} -$$$$38\!\cdots\!70$$$$T^{9} +$$$$13\!\cdots\!96$$$$T^{10} -$$$$26\!\cdots\!70$$$$T^{11} +$$$$80\!\cdots\!21$$$$T^{12} -$$$$13\!\cdots\!37$$$$T^{13} +$$$$36\!\cdots\!32$$$$T^{14} -$$$$49\!\cdots\!25$$$$T^{15} +$$$$11\!\cdots\!34$$$$T^{16} -$$$$11\!\cdots\!17$$$$T^{17} +$$$$23\!\cdots\!38$$$$T^{18} -$$$$13\!\cdots\!71$$$$T^{19} +$$$$24\!\cdots\!01$$$$T^{20} )^{2}$$
$43$ $$1 - 1025 T + 585952 T^{2} - 190558261 T^{3} + 20265475934 T^{4} + 14357877818201 T^{5} - 8972906466293482 T^{6} + 2452051846789185023 T^{7} -$$$$15\!\cdots\!87$$$$T^{8} -$$$$14\!\cdots\!18$$$$T^{9} +$$$$66\!\cdots\!40$$$$T^{10} -$$$$11\!\cdots\!26$$$$T^{11} -$$$$10\!\cdots\!63$$$$T^{12} +$$$$12\!\cdots\!89$$$$T^{13} -$$$$35\!\cdots\!82$$$$T^{14} +$$$$45\!\cdots\!07$$$$T^{15} +$$$$51\!\cdots\!66$$$$T^{16} -$$$$38\!\cdots\!23$$$$T^{17} +$$$$93\!\cdots\!52$$$$T^{18} -$$$$13\!\cdots\!75$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20}$$
$47$ $$( 1 + 334 T + 469033 T^{2} + 125686554 T^{3} + 97290890628 T^{4} + 20127660597266 T^{5} + 12996286592743642 T^{6} + 2230581953572455798 T^{7} +$$$$14\!\cdots\!19$$$$T^{8} +$$$$24\!\cdots\!68$$$$T^{9} +$$$$16\!\cdots\!34$$$$T^{10} +$$$$25\!\cdots\!64$$$$T^{11} +$$$$15\!\cdots\!51$$$$T^{12} +$$$$24\!\cdots\!66$$$$T^{13} +$$$$15\!\cdots\!22$$$$T^{14} +$$$$24\!\cdots\!38$$$$T^{15} +$$$$12\!\cdots\!92$$$$T^{16} +$$$$16\!\cdots\!38$$$$T^{17} +$$$$63\!\cdots\!73$$$$T^{18} +$$$$46\!\cdots\!42$$$$T^{19} +$$$$14\!\cdots\!49$$$$T^{20} )^{2}$$
$53$ $$1 - 773 T - 541704 T^{2} + 559887799 T^{3} + 134971478324 T^{4} - 208241369369157 T^{5} - 17263563843724798 T^{6} + 53174144863295438605 T^{7} -$$$$60\!\cdots\!27$$$$T^{8} -$$$$10\!\cdots\!04$$$$T^{9} +$$$$82\!\cdots\!89$$$$T^{10} +$$$$14\!\cdots\!17$$$$T^{11} -$$$$18\!\cdots\!03$$$$T^{12} -$$$$14\!\cdots\!72$$$$T^{13} +$$$$16\!\cdots\!32$$$$T^{14} +$$$$81\!\cdots\!34$$$$T^{15} +$$$$21\!\cdots\!18$$$$T^{16} +$$$$38\!\cdots\!40$$$$T^{17} -$$$$10\!\cdots\!15$$$$T^{18} -$$$$28\!\cdots\!09$$$$T^{19} +$$$$21\!\cdots\!62$$$$T^{20} -$$$$42\!\cdots\!93$$$$T^{21} -$$$$23\!\cdots\!35$$$$T^{22} +$$$$12\!\cdots\!20$$$$T^{23} +$$$$10\!\cdots\!38$$$$T^{24} +$$$$59\!\cdots\!38$$$$T^{25} +$$$$17\!\cdots\!48$$$$T^{26} -$$$$22\!\cdots\!16$$$$T^{27} -$$$$44\!\cdots\!43$$$$T^{28} +$$$$51\!\cdots\!29$$$$T^{29} +$$$$44\!\cdots\!61$$$$T^{30} -$$$$80\!\cdots\!92$$$$T^{31} -$$$$71\!\cdots\!67$$$$T^{32} +$$$$93\!\cdots\!85$$$$T^{33} -$$$$45\!\cdots\!82$$$$T^{34} -$$$$81\!\cdots\!01$$$$T^{35} +$$$$78\!\cdots\!64$$$$T^{36} +$$$$48\!\cdots\!03$$$$T^{37} -$$$$69\!\cdots\!76$$$$T^{38} -$$$$14\!\cdots\!49$$$$T^{39} +$$$$28\!\cdots\!01$$$$T^{40}$$
$59$ $$( 1 + 1483 T + 1378157 T^{2} + 969130766 T^{3} + 686264212828 T^{4} + 451249739385536 T^{5} + 271319879713111662 T^{6} +$$$$14\!\cdots\!74$$$$T^{7} +$$$$73\!\cdots\!15$$$$T^{8} +$$$$36\!\cdots\!33$$$$T^{9} +$$$$17\!\cdots\!86$$$$T^{10} +$$$$74\!\cdots\!07$$$$T^{11} +$$$$30\!\cdots\!15$$$$T^{12} +$$$$12\!\cdots\!86$$$$T^{13} +$$$$48\!\cdots\!22$$$$T^{14} +$$$$16\!\cdots\!64$$$$T^{15} +$$$$51\!\cdots\!88$$$$T^{16} +$$$$14\!\cdots\!94$$$$T^{17} +$$$$43\!\cdots\!77$$$$T^{18} +$$$$96\!\cdots\!77$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20} )^{2}$$
$61$ $$1 - 437 T - 744916 T^{2} + 424927067 T^{3} + 139480796216 T^{4} - 122519571726057 T^{5} + 19509236640919974 T^{6} - 8885281027720171051 T^{7} +$$$$17\!\cdots\!49$$$$T^{8} +$$$$92\!\cdots\!28$$$$T^{9} -$$$$74\!\cdots\!75$$$$T^{10} +$$$$52\!\cdots\!53$$$$T^{11} +$$$$19\!\cdots\!81$$$$T^{12} -$$$$90\!\cdots\!44$$$$T^{13} +$$$$54\!\cdots\!48$$$$T^{14} +$$$$67\!\cdots\!10$$$$T^{15} -$$$$60\!\cdots\!46$$$$T^{16} +$$$$50\!\cdots\!76$$$$T^{17} -$$$$13\!\cdots\!43$$$$T^{18} -$$$$85\!\cdots\!77$$$$T^{19} +$$$$79\!\cdots\!02$$$$T^{20} -$$$$19\!\cdots\!37$$$$T^{21} -$$$$70\!\cdots\!23$$$$T^{22} +$$$$58\!\cdots\!16$$$$T^{23} -$$$$16\!\cdots\!66$$$$T^{24} +$$$$40\!\cdots\!10$$$$T^{25} +$$$$74\!\cdots\!88$$$$T^{26} -$$$$28\!\cdots\!84$$$$T^{27} +$$$$13\!\cdots\!21$$$$T^{28} +$$$$83\!\cdots\!13$$$$T^{29} -$$$$26\!\cdots\!75$$$$T^{30} +$$$$76\!\cdots\!68$$$$T^{31} +$$$$32\!\cdots\!89$$$$T^{32} -$$$$37\!\cdots\!91$$$$T^{33} +$$$$18\!\cdots\!54$$$$T^{34} -$$$$26\!\cdots\!57$$$$T^{35} +$$$$69\!\cdots\!96$$$$T^{36} +$$$$47\!\cdots\!87$$$$T^{37} -$$$$19\!\cdots\!56$$$$T^{38} -$$$$25\!\cdots\!77$$$$T^{39} +$$$$13\!\cdots\!01$$$$T^{40}$$
$67$ $$1 + 642 T - 1655222 T^{2} - 1351264924 T^{3} + 1395720548382 T^{4} + 1507860208990244 T^{5} - 730390077743505158 T^{6} -$$$$11\!\cdots\!40$$$$T^{7} +$$$$20\!\cdots\!15$$$$T^{8} +$$$$69\!\cdots\!46$$$$T^{9} +$$$$32\!\cdots\!85$$$$T^{10} -$$$$33\!\cdots\!96$$$$T^{11} -$$$$78\!\cdots\!53$$$$T^{12} +$$$$12\!\cdots\!70$$$$T^{13} +$$$$56\!\cdots\!78$$$$T^{14} -$$$$39\!\cdots\!80$$$$T^{15} -$$$$28\!\cdots\!94$$$$T^{16} +$$$$89\!\cdots\!68$$$$T^{17} +$$$$11\!\cdots\!05$$$$T^{18} -$$$$10\!\cdots\!22$$$$T^{19} -$$$$36\!\cdots\!78$$$$T^{20} -$$$$30\!\cdots\!86$$$$T^{21} +$$$$10\!\cdots\!45$$$$T^{22} +$$$$24\!\cdots\!96$$$$T^{23} -$$$$22\!\cdots\!34$$$$T^{24} -$$$$96\!\cdots\!40$$$$T^{25} +$$$$41\!\cdots\!02$$$$T^{26} +$$$$28\!\cdots\!90$$$$T^{27} -$$$$52\!\cdots\!13$$$$T^{28} -$$$$66\!\cdots\!08$$$$T^{29} +$$$$19\!\cdots\!65$$$$T^{30} +$$$$12\!\cdots\!02$$$$T^{31} +$$$$11\!\cdots\!15$$$$T^{32} -$$$$19\!\cdots\!20$$$$T^{33} -$$$$36\!\cdots\!62$$$$T^{34} +$$$$22\!\cdots\!08$$$$T^{35} +$$$$62\!\cdots\!62$$$$T^{36} -$$$$18\!\cdots\!92$$$$T^{37} -$$$$67\!\cdots\!38$$$$T^{38} +$$$$78\!\cdots\!34$$$$T^{39} +$$$$36\!\cdots\!01$$$$T^{40}$$
$71$ $$1 + 1545 T - 164612 T^{2} - 823498495 T^{3} + 510816804584 T^{4} + 421399428301951 T^{5} - 410556697843186024 T^{6} + 14010431784563891953 T^{7} +$$$$29\!\cdots\!89$$$$T^{8} -$$$$96\!\cdots\!56$$$$T^{9} -$$$$10\!\cdots\!59$$$$T^{10} +$$$$85\!\cdots\!53$$$$T^{11} +$$$$18\!\cdots\!67$$$$T^{12} -$$$$42\!\cdots\!72$$$$T^{13} +$$$$64\!\cdots\!76$$$$T^{14} +$$$$14\!\cdots\!32$$$$T^{15} -$$$$71\!\cdots\!48$$$$T^{16} -$$$$37\!\cdots\!38$$$$T^{17} +$$$$33\!\cdots\!67$$$$T^{18} +$$$$36\!\cdots\!55$$$$T^{19} -$$$$13\!\cdots\!22$$$$T^{20} +$$$$13\!\cdots\!05$$$$T^{21} +$$$$42\!\cdots\!07$$$$T^{22} -$$$$16\!\cdots\!78$$$$T^{23} -$$$$11\!\cdots\!68$$$$T^{24} +$$$$83\!\cdots\!32$$$$T^{25} +$$$$13\!\cdots\!36$$$$T^{26} -$$$$31\!\cdots\!12$$$$T^{27} +$$$$49\!\cdots\!27$$$$T^{28} +$$$$82\!\cdots\!23$$$$T^{29} -$$$$37\!\cdots\!59$$$$T^{30} -$$$$11\!\cdots\!16$$$$T^{31} +$$$$13\!\cdots\!69$$$$T^{32} +$$$$22\!\cdots\!43$$$$T^{33} -$$$$23\!\cdots\!84$$$$T^{34} +$$$$85\!\cdots\!01$$$$T^{35} +$$$$37\!\cdots\!24$$$$T^{36} -$$$$21\!\cdots\!45$$$$T^{37} -$$$$15\!\cdots\!72$$$$T^{38} +$$$$51\!\cdots\!95$$$$T^{39} +$$$$11\!\cdots\!01$$$$T^{40}$$
$73$ $$1 - 1292 T - 800760 T^{2} + 1888773084 T^{3} - 294654454086 T^{4} - 930615313063966 T^{5} + 430214695631874962 T^{6} +$$$$18\!\cdots\!12$$$$T^{7} -$$$$10\!\cdots\!67$$$$T^{8} -$$$$52\!\cdots\!64$$$$T^{9} -$$$$10\!\cdots\!33$$$$T^{10} +$$$$39\!\cdots\!20$$$$T^{11} +$$$$82\!\cdots\!53$$$$T^{12} -$$$$16\!\cdots\!46$$$$T^{13} -$$$$29\!\cdots\!00$$$$T^{14} +$$$$27\!\cdots\!50$$$$T^{15} +$$$$21\!\cdots\!44$$$$T^{16} +$$$$12\!\cdots\!62$$$$T^{17} -$$$$27\!\cdots\!53$$$$T^{18} -$$$$51\!\cdots\!40$$$$T^{19} +$$$$16\!\cdots\!58$$$$T^{20} -$$$$19\!\cdots\!80$$$$T^{21} -$$$$41\!\cdots\!17$$$$T^{22} +$$$$75\!\cdots\!06$$$$T^{23} +$$$$48\!\cdots\!24$$$$T^{24} +$$$$24\!\cdots\!50$$$$T^{25} -$$$$10\!\cdots\!00$$$$T^{26} -$$$$21\!\cdots\!58$$$$T^{27} +$$$$43\!\cdots\!73$$$$T^{28} +$$$$80\!\cdots\!40$$$$T^{29} -$$$$81\!\cdots\!17$$$$T^{30} -$$$$16\!\cdots\!12$$$$T^{31} -$$$$12\!\cdots\!87$$$$T^{32} +$$$$84\!\cdots\!44$$$$T^{33} +$$$$78\!\cdots\!98$$$$T^{34} -$$$$65\!\cdots\!38$$$$T^{35} -$$$$81\!\cdots\!66$$$$T^{36} +$$$$20\!\cdots\!68$$$$T^{37} -$$$$33\!\cdots\!40$$$$T^{38} -$$$$20\!\cdots\!76$$$$T^{39} +$$$$63\!\cdots\!01$$$$T^{40}$$
$79$ $$1 + 1405 T - 2150268 T^{2} - 3116848211 T^{3} + 3360157724444 T^{4} + 4151188953835363 T^{5} - 3817398869736906676 T^{6} -$$$$37\!\cdots\!15$$$$T^{7} +$$$$34\!\cdots\!89$$$$T^{8} +$$$$24\!\cdots\!40$$$$T^{9} -$$$$24\!\cdots\!47$$$$T^{10} -$$$$12\!\cdots\!11$$$$T^{11} +$$$$14\!\cdots\!03$$$$T^{12} +$$$$47\!\cdots\!60$$$$T^{13} -$$$$67\!\cdots\!72$$$$T^{14} -$$$$13\!\cdots\!52$$$$T^{15} +$$$$28\!\cdots\!56$$$$T^{16} +$$$$29\!\cdots\!90$$$$T^{17} -$$$$11\!\cdots\!77$$$$T^{18} -$$$$35\!\cdots\!53$$$$T^{19} +$$$$54\!\cdots\!18$$$$T^{20} -$$$$17\!\cdots\!67$$$$T^{21} -$$$$29\!\cdots\!17$$$$T^{22} +$$$$34\!\cdots\!10$$$$T^{23} +$$$$17\!\cdots\!96$$$$T^{24} -$$$$40\!\cdots\!48$$$$T^{25} -$$$$97\!\cdots\!92$$$$T^{26} +$$$$33\!\cdots\!40$$$$T^{27} +$$$$49\!\cdots\!43$$$$T^{28} -$$$$21\!\cdots\!49$$$$T^{29} -$$$$20\!\cdots\!47$$$$T^{30} +$$$$10\!\cdots\!60$$$$T^{31} +$$$$70\!\cdots\!69$$$$T^{32} -$$$$38\!\cdots\!85$$$$T^{33} -$$$$19\!\cdots\!16$$$$T^{34} +$$$$10\!\cdots\!37$$$$T^{35} +$$$$40\!\cdots\!84$$$$T^{36} -$$$$18\!\cdots\!69$$$$T^{37} -$$$$63\!\cdots\!08$$$$T^{38} +$$$$20\!\cdots\!95$$$$T^{39} +$$$$72\!\cdots\!01$$$$T^{40}$$
$83$ $$1 - 543 T - 3210972 T^{2} + 2149693073 T^{3} + 5106187705928 T^{4} - 4006405925115629 T^{5} - 5191885555542057060 T^{6} +$$$$46\!\cdots\!33$$$$T^{7} +$$$$37\!\cdots\!57$$$$T^{8} -$$$$37\!\cdots\!40$$$$T^{9} -$$$$22\!\cdots\!35$$$$T^{10} +$$$$23\!\cdots\!85$$$$T^{11} +$$$$11\!\cdots\!11$$$$T^{12} -$$$$13\!\cdots\!48$$$$T^{13} -$$$$56\!\cdots\!52$$$$T^{14} +$$$$70\!\cdots\!28$$$$T^{15} +$$$$18\!\cdots\!76$$$$T^{16} -$$$$31\!\cdots\!22$$$$T^{17} -$$$$49\!\cdots\!25$$$$T^{18} +$$$$72\!\cdots\!87$$$$T^{19} -$$$$25\!\cdots\!94$$$$T^{20} +$$$$41\!\cdots\!69$$$$T^{21} -$$$$16\!\cdots\!25$$$$T^{22} -$$$$59\!\cdots\!66$$$$T^{23} +$$$$19\!\cdots\!36$$$$T^{24} +$$$$43\!\cdots\!96$$$$T^{25} -$$$$19\!\cdots\!68$$$$T^{26} -$$$$26\!\cdots\!84$$$$T^{27} +$$$$13\!\cdots\!31$$$$T^{28} +$$$$15\!\cdots\!95$$$$T^{29} -$$$$83\!\cdots\!15$$$$T^{30} -$$$$80\!\cdots\!20$$$$T^{31} +$$$$46\!\cdots\!17$$$$T^{32} +$$$$32\!\cdots\!51$$$$T^{33} -$$$$20\!\cdots\!40$$$$T^{34} -$$$$91\!\cdots\!47$$$$T^{35} +$$$$66\!\cdots\!48$$$$T^{36} +$$$$16\!\cdots\!91$$$$T^{37} -$$$$13\!\cdots\!88$$$$T^{38} -$$$$13\!\cdots\!89$$$$T^{39} +$$$$13\!\cdots\!01$$$$T^{40}$$
$89$ $$1 + 2196 T - 2279256 T^{2} - 6744108292 T^{3} + 4590931714074 T^{4} + 12293000410375634 T^{5} - 9448115145172476374 T^{6} -$$$$17\!\cdots\!88$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} +$$$$20\!\cdots\!52$$$$T^{9} -$$$$21\!\cdots\!85$$$$T^{10} -$$$$19\!\cdots\!24$$$$T^{11} +$$$$25\!\cdots\!53$$$$T^{12} +$$$$15\!\cdots\!82$$$$T^{13} -$$$$26\!\cdots\!20$$$$T^{14} -$$$$10\!\cdots\!82$$$$T^{15} +$$$$24\!\cdots\!12$$$$T^{16} +$$$$53\!\cdots\!26$$$$T^{17} -$$$$20\!\cdots\!41$$$$T^{18} -$$$$13\!\cdots\!08$$$$T^{19} +$$$$15\!\cdots\!66$$$$T^{20} -$$$$96\!\cdots\!52$$$$T^{21} -$$$$10\!\cdots\!01$$$$T^{22} +$$$$18\!\cdots\!34$$$$T^{23} +$$$$61\!\cdots\!52$$$$T^{24} -$$$$18\!\cdots\!18$$$$T^{25} -$$$$33\!\cdots\!20$$$$T^{26} +$$$$13\!\cdots\!98$$$$T^{27} +$$$$15\!\cdots\!73$$$$T^{28} -$$$$83\!\cdots\!96$$$$T^{29} -$$$$66\!\cdots\!85$$$$T^{30} +$$$$43\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32} -$$$$18\!\cdots\!92$$$$T^{33} -$$$$70\!\cdots\!54$$$$T^{34} +$$$$64\!\cdots\!66$$$$T^{35} +$$$$17\!\cdots\!94$$$$T^{36} -$$$$17\!\cdots\!88$$$$T^{37} -$$$$42\!\cdots\!96$$$$T^{38} +$$$$28\!\cdots\!84$$$$T^{39} +$$$$91\!\cdots\!01$$$$T^{40}$$
$97$ $$( 1 + 425 T + 4855100 T^{2} + 2674736785 T^{3} + 12413935569789 T^{4} + 7844309405033348 T^{5} + 21709474743257501488 T^{6} +$$$$14\!\cdots\!84$$$$T^{7} +$$$$28\!\cdots\!94$$$$T^{8} +$$$$18\!\cdots\!26$$$$T^{9} +$$$$29\!\cdots\!88$$$$T^{10} +$$$$16\!\cdots\!98$$$$T^{11} +$$$$23\!\cdots\!26$$$$T^{12} +$$$$10\!\cdots\!28$$$$T^{13} +$$$$15\!\cdots\!08$$$$T^{14} +$$$$49\!\cdots\!64$$$$T^{15} +$$$$71\!\cdots\!21$$$$T^{16} +$$$$14\!\cdots\!45$$$$T^{17} +$$$$23\!\cdots\!00$$$$T^{18} +$$$$18\!\cdots\!25$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20} )^{2}$$