# Properties

 Label 23.10 Level 23 Weight 10 Dimension 187 Nonzero newspaces 2 Newform subspaces 3 Sturm bound 440 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$23\( 23$$ \) Weight: $$k$$ = $$10$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ Sturm bound: $$440$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_1(23))$$.

Total New Old
Modular forms 209 207 2
Cusp forms 187 187 0
Eisenstein series 22 20 2

## Trace form

 $$187q - 11q^{2} - 11q^{3} - 11q^{4} - 11q^{5} - 11q^{6} - 11q^{7} - 11q^{8} - 11q^{9} + O(q^{10})$$ $$187q - 11q^{2} - 11q^{3} - 11q^{4} - 11q^{5} - 11q^{6} - 11q^{7} - 11q^{8} - 11q^{9} - 11q^{10} - 11q^{11} - 11q^{12} - 11q^{13} - 11q^{14} + 969023q^{15} - 478731q^{16} - 1054867q^{17} + 1519925q^{18} + 1526459q^{19} + 3198965q^{20} - 3575231q^{21} - 5153654q^{22} - 3650823q^{23} + 6167018q^{24} + 8299797q^{25} + 9664149q^{26} + 3875311q^{27} - 21469195q^{28} - 10200751q^{29} - 24330955q^{30} + 9969839q^{31} + 46627317q^{32} - 21902947q^{33} - 8377589q^{34} + 57736239q^{35} - 30672686q^{36} - 107778935q^{37} - 92052576q^{38} + 23347753q^{39} + 213344989q^{40} + 62844639q^{41} + 178280179q^{42} - 29184023q^{43} - 225229796q^{44} - 270641272q^{45} - 311125331q^{46} - 54389346q^{47} + 183376413q^{48} + 341786995q^{49} + 511328114q^{50} + 237700969q^{51} + 138729679q^{52} - 103750977q^{53} - 1104475251q^{54} - 96640643q^{55} + 210507990q^{56} + 59318281q^{57} + 513967300q^{58} - 118047545q^{59} - 1153719721q^{60} + 344711653q^{61} + 980989141q^{62} + 882717759q^{63} + 986185717q^{64} + 158418557q^{65} - 747426636q^{66} - 378627887q^{67} - 2822341654q^{68} - 1069913031q^{69} - 1006270870q^{70} + 56718519q^{71} + 1971614260q^{72} + 582788437q^{73} + 699926392q^{74} + 3594630963q^{75} + 4934994042q^{76} + 832730437q^{77} - 4315286437q^{78} - 4249645807q^{79} - 10615143550q^{80} - 5321334799q^{81} + 416976439q^{82} + 3425248739q^{83} + 14370912435q^{84} + 7253042687q^{85} + 779636011q^{86} + 2829378673q^{87} - 671992739q^{88} - 2402723741q^{89} - 12214147682q^{90} - 6034596590q^{91} - 6333894578q^{92} - 5895816058q^{93} - 1344916188q^{94} - 1504040945q^{95} + 17191683300q^{96} + 11525413603q^{97} + 12597404941q^{98} + 10210410199q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_1(23))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
23.10.a $$\chi_{23}(1, \cdot)$$ 23.10.a.a 7 1
23.10.a.b 10
23.10.c $$\chi_{23}(2, \cdot)$$ 23.10.c.a 170 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 1024 T^{2} - 11640 T^{3} + 784256 T^{4} - 13549312 T^{5} + 434521088 T^{6} - 9484298240 T^{7} + 222474797056 T^{8} - 3551870844928 T^{9} + 105261058490368 T^{10} - 799894709207040 T^{11} + 36028797018963968 T^{12} + 9223372036854775808 T^{14}$$)($$1 - 32 T + 1536 T^{2} - 30595 T^{3} + 820124 T^{4} - 5615752 T^{5} + 236635672 T^{6} + 690646848 T^{7} + 180060561408 T^{8} - 2463941171200 T^{9} + 137592836921344 T^{10} - 1261537879654400 T^{11} + 47201795809738752 T^{12} + 92697050788921344 T^{13} + 16261479556911726592 T^{14} -$$$$19\!\cdots\!64$$$$T^{15} +$$$$14\!\cdots\!16$$$$T^{16} -$$$$28\!\cdots\!60$$$$T^{17} +$$$$72\!\cdots\!56$$$$T^{18} -$$$$77\!\cdots\!64$$$$T^{19} +$$$$12\!\cdots\!24$$$$T^{20}$$)
$3$ ($$1 + 89 T + 56319 T^{2} + 3200832 T^{3} + 1662515172 T^{4} + 32804985744 T^{5} + 34020755420418 T^{6} - 54407515840506 T^{7} + 669630528940087494 T^{8} + 12709323618578508816 T^{9} +$$$$12\!\cdots\!64$$$$T^{10} +$$$$48\!\cdots\!72$$$$T^{11} +$$$$16\!\cdots\!17$$$$T^{12} +$$$$51\!\cdots\!41$$$$T^{13} +$$$$11\!\cdots\!27$$$$T^{14}$$)($$1 - 235 T + 100187 T^{2} - 18356860 T^{3} + 4781484148 T^{4} - 729750871812 T^{5} + 149647969596249 T^{6} - 20073584417672475 T^{7} + 3697845052481213139 T^{8} -$$$$45\!\cdots\!60$$$$T^{9} +$$$$77\!\cdots\!36$$$$T^{10} -$$$$88\!\cdots\!80$$$$T^{11} +$$$$14\!\cdots\!71$$$$T^{12} -$$$$15\!\cdots\!25$$$$T^{13} +$$$$22\!\cdots\!29$$$$T^{14} -$$$$21\!\cdots\!16$$$$T^{15} +$$$$27\!\cdots\!12$$$$T^{16} -$$$$21\!\cdots\!20$$$$T^{17} +$$$$22\!\cdots\!67$$$$T^{18} -$$$$10\!\cdots\!05$$$$T^{19} +$$$$87\!\cdots\!49$$$$T^{20}$$)
$5$ ($$1 + 2388 T + 10347659 T^{2} + 15897713104 T^{3} + 41634187705269 T^{4} + 46393192014942188 T^{5} +$$$$10\!\cdots\!71$$$$T^{6} +$$$$95\!\cdots\!20$$$$T^{7} +$$$$19\!\cdots\!75$$$$T^{8} +$$$$17\!\cdots\!00$$$$T^{9} +$$$$31\!\cdots\!25$$$$T^{10} +$$$$23\!\cdots\!00$$$$T^{11} +$$$$29\!\cdots\!75$$$$T^{12} +$$$$13\!\cdots\!00$$$$T^{13} +$$$$10\!\cdots\!25$$$$T^{14}$$)($$1 - 112 T + 5234278 T^{2} + 2769747776 T^{3} + 16578605745765 T^{4} + 13520682696304400 T^{5} + 50153124455236573000 T^{6} +$$$$35\!\cdots\!00$$$$T^{7} +$$$$12\!\cdots\!50$$$$T^{8} +$$$$93\!\cdots\!00$$$$T^{9} +$$$$25\!\cdots\!00$$$$T^{10} +$$$$18\!\cdots\!00$$$$T^{11} +$$$$48\!\cdots\!50$$$$T^{12} +$$$$26\!\cdots\!00$$$$T^{13} +$$$$72\!\cdots\!00$$$$T^{14} +$$$$38\!\cdots\!00$$$$T^{15} +$$$$92\!\cdots\!25$$$$T^{16} +$$$$30\!\cdots\!00$$$$T^{17} +$$$$11\!\cdots\!50$$$$T^{18} -$$$$46\!\cdots\!00$$$$T^{19} +$$$$80\!\cdots\!25$$$$T^{20}$$)
$7$ ($$1 + 9896 T + 175593517 T^{2} + 1562683196248 T^{3} + 15909705469565177 T^{4} +$$$$11\!\cdots\!52$$$$T^{5} +$$$$91\!\cdots\!25$$$$T^{6} +$$$$57\!\cdots\!08$$$$T^{7} +$$$$37\!\cdots\!75$$$$T^{8} +$$$$19\!\cdots\!48$$$$T^{9} +$$$$10\!\cdots\!11$$$$T^{10} +$$$$41\!\cdots\!48$$$$T^{11} +$$$$18\!\cdots\!19$$$$T^{12} +$$$$42\!\cdots\!04$$$$T^{13} +$$$$17\!\cdots\!43$$$$T^{14}$$)($$1 - 1280 T + 143880674 T^{2} + 306761723856 T^{3} + 11170478766250573 T^{4} + 50758417860990663376 T^{5} +$$$$71\!\cdots\!96$$$$T^{6} +$$$$35\!\cdots\!40$$$$T^{7} +$$$$40\!\cdots\!06$$$$T^{8} +$$$$17\!\cdots\!92$$$$T^{9} +$$$$18\!\cdots\!72$$$$T^{10} +$$$$71\!\cdots\!44$$$$T^{11} +$$$$66\!\cdots\!94$$$$T^{12} +$$$$23\!\cdots\!20$$$$T^{13} +$$$$19\!\cdots\!96$$$$T^{14} +$$$$54\!\cdots\!32$$$$T^{15} +$$$$48\!\cdots\!77$$$$T^{16} +$$$$53\!\cdots\!08$$$$T^{17} +$$$$10\!\cdots\!74$$$$T^{18} -$$$$36\!\cdots\!60$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20}$$)
$11$ ($$1 + 78484 T + 10901228757 T^{2} + 733893101136640 T^{3} + 58334822935416071301 T^{4} +$$$$32\!\cdots\!84$$$$T^{5} +$$$$19\!\cdots\!61$$$$T^{6} +$$$$92\!\cdots\!88$$$$T^{7} +$$$$46\!\cdots\!51$$$$T^{8} +$$$$18\!\cdots\!04$$$$T^{9} +$$$$76\!\cdots\!71$$$$T^{10} +$$$$22\!\cdots\!40$$$$T^{11} +$$$$79\!\cdots\!07$$$$T^{12} +$$$$13\!\cdots\!44$$$$T^{13} +$$$$40\!\cdots\!31$$$$T^{14}$$)($$1 + 2690 T + 8732444578 T^{2} + 88815080094718 T^{3} + 41288912483657217125 T^{4} +$$$$54\!\cdots\!84$$$$T^{5} +$$$$13\!\cdots\!60$$$$T^{6} +$$$$17\!\cdots\!52$$$$T^{7} +$$$$36\!\cdots\!02$$$$T^{8} +$$$$42\!\cdots\!00$$$$T^{9} +$$$$87\!\cdots\!80$$$$T^{10} +$$$$10\!\cdots\!00$$$$T^{11} +$$$$20\!\cdots\!62$$$$T^{12} +$$$$23\!\cdots\!92$$$$T^{13} +$$$$41\!\cdots\!60$$$$T^{14} +$$$$39\!\cdots\!84$$$$T^{15} +$$$$70\!\cdots\!25$$$$T^{16} +$$$$35\!\cdots\!58$$$$T^{17} +$$$$83\!\cdots\!38$$$$T^{18} +$$$$60\!\cdots\!90$$$$T^{19} +$$$$53\!\cdots\!01$$$$T^{20}$$)
$13$ ($$1 + 296769 T + 92822557229 T^{2} + 17981406282795684 T^{3} +$$$$32\!\cdots\!52$$$$T^{4} +$$$$46\!\cdots\!76$$$$T^{5} +$$$$60\!\cdots\!30$$$$T^{6} +$$$$65\!\cdots\!82$$$$T^{7} +$$$$64\!\cdots\!90$$$$T^{8} +$$$$52\!\cdots\!04$$$$T^{9} +$$$$39\!\cdots\!84$$$$T^{10} +$$$$22\!\cdots\!44$$$$T^{11} +$$$$12\!\cdots\!97$$$$T^{12} +$$$$42\!\cdots\!41$$$$T^{13} +$$$$15\!\cdots\!97$$$$T^{14}$$)($$1 - 307847 T + 96488911133 T^{2} - 18728358098838002 T^{3} +$$$$34\!\cdots\!86$$$$T^{4} -$$$$50\!\cdots\!06$$$$T^{5} +$$$$69\!\cdots\!19$$$$T^{6} -$$$$82\!\cdots\!17$$$$T^{7} +$$$$95\!\cdots\!67$$$$T^{8} -$$$$10\!\cdots\!00$$$$T^{9} +$$$$10\!\cdots\!40$$$$T^{10} -$$$$10\!\cdots\!00$$$$T^{11} +$$$$10\!\cdots\!43$$$$T^{12} -$$$$98\!\cdots\!89$$$$T^{13} +$$$$88\!\cdots\!79$$$$T^{14} -$$$$67\!\cdots\!58$$$$T^{15} +$$$$49\!\cdots\!54$$$$T^{16} -$$$$28\!\cdots\!94$$$$T^{17} +$$$$15\!\cdots\!73$$$$T^{18} -$$$$52\!\cdots\!11$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20}$$)
$17$ ($$1 + 1128820 T + 950945462119 T^{2} + 561809900824145328 T^{3} +$$$$27\!\cdots\!53$$$$T^{4} +$$$$11\!\cdots\!16$$$$T^{5} +$$$$43\!\cdots\!27$$$$T^{6} +$$$$15\!\cdots\!20$$$$T^{7} +$$$$51\!\cdots\!19$$$$T^{8} +$$$$16\!\cdots\!44$$$$T^{9} +$$$$46\!\cdots\!69$$$$T^{10} +$$$$11\!\cdots\!68$$$$T^{11} +$$$$22\!\cdots\!83$$$$T^{12} +$$$$31\!\cdots\!80$$$$T^{13} +$$$$32\!\cdots\!13$$$$T^{14}$$)($$1 - 827614 T + 922959160226 T^{2} - 502323774007556782 T^{3} +$$$$32\!\cdots\!69$$$$T^{4} -$$$$13\!\cdots\!64$$$$T^{5} +$$$$69\!\cdots\!20$$$$T^{6} -$$$$25\!\cdots\!16$$$$T^{7} +$$$$11\!\cdots\!54$$$$T^{8} -$$$$37\!\cdots\!16$$$$T^{9} +$$$$15\!\cdots\!52$$$$T^{10} -$$$$44\!\cdots\!52$$$$T^{11} +$$$$16\!\cdots\!86$$$$T^{12} -$$$$42\!\cdots\!68$$$$T^{13} +$$$$13\!\cdots\!20$$$$T^{14} -$$$$32\!\cdots\!48$$$$T^{15} +$$$$90\!\cdots\!01$$$$T^{16} -$$$$16\!\cdots\!66$$$$T^{17} +$$$$36\!\cdots\!86$$$$T^{18} -$$$$38\!\cdots\!38$$$$T^{19} +$$$$55\!\cdots\!49$$$$T^{20}$$)
$19$ ($$1 + 1301252 T + 1939820316785 T^{2} + 1879363496695414344 T^{3} +$$$$16\!\cdots\!37$$$$T^{4} +$$$$12\!\cdots\!92$$$$T^{5} +$$$$85\!\cdots\!09$$$$T^{6} +$$$$49\!\cdots\!44$$$$T^{7} +$$$$27\!\cdots\!11$$$$T^{8} +$$$$12\!\cdots\!72$$$$T^{9} +$$$$56\!\cdots\!43$$$$T^{10} +$$$$20\!\cdots\!64$$$$T^{11} +$$$$67\!\cdots\!15$$$$T^{12} +$$$$14\!\cdots\!92$$$$T^{13} +$$$$36\!\cdots\!59$$$$T^{14}$$)($$1 - 475454 T + 1893976423862 T^{2} - 702088360072893194 T^{3} +$$$$17\!\cdots\!73$$$$T^{4} -$$$$53\!\cdots\!92$$$$T^{5} +$$$$11\!\cdots\!84$$$$T^{6} -$$$$28\!\cdots\!40$$$$T^{7} +$$$$51\!\cdots\!10$$$$T^{8} -$$$$11\!\cdots\!28$$$$T^{9} +$$$$18\!\cdots\!40$$$$T^{10} -$$$$38\!\cdots\!12$$$$T^{11} +$$$$53\!\cdots\!10$$$$T^{12} -$$$$97\!\cdots\!60$$$$T^{13} +$$$$12\!\cdots\!04$$$$T^{14} -$$$$18\!\cdots\!08$$$$T^{15} +$$$$19\!\cdots\!33$$$$T^{16} -$$$$25\!\cdots\!46$$$$T^{17} +$$$$22\!\cdots\!82$$$$T^{18} -$$$$18\!\cdots\!26$$$$T^{19} +$$$$12\!\cdots\!01$$$$T^{20}$$)
$23$ ($$( 1 + 279841 T )^{7}$$)($$( 1 - 279841 T )^{10}$$)
$29$ ($$1 - 2813849 T + 96073558072185 T^{2} -$$$$22\!\cdots\!60$$$$T^{3} +$$$$40\!\cdots\!88$$$$T^{4} -$$$$79\!\cdots\!84$$$$T^{5} +$$$$95\!\cdots\!94$$$$T^{6} -$$$$15\!\cdots\!54$$$$T^{7} +$$$$13\!\cdots\!86$$$$T^{8} -$$$$16\!\cdots\!24$$$$T^{9} +$$$$12\!\cdots\!92$$$$T^{10} -$$$$10\!\cdots\!60$$$$T^{11} +$$$$61\!\cdots\!65$$$$T^{12} -$$$$26\!\cdots\!69$$$$T^{13} +$$$$13\!\cdots\!89$$$$T^{14}$$)($$1 + 6012071 T + 55254712746153 T^{2} +$$$$17\!\cdots\!10$$$$T^{3} +$$$$12\!\cdots\!86$$$$T^{4} +$$$$33\!\cdots\!66$$$$T^{5} +$$$$22\!\cdots\!07$$$$T^{6} +$$$$37\!\cdots\!09$$$$T^{7} +$$$$27\!\cdots\!67$$$$T^{8} +$$$$31\!\cdots\!04$$$$T^{9} +$$$$38\!\cdots\!12$$$$T^{10} +$$$$46\!\cdots\!76$$$$T^{11} +$$$$57\!\cdots\!87$$$$T^{12} +$$$$11\!\cdots\!81$$$$T^{13} +$$$$10\!\cdots\!47$$$$T^{14} +$$$$21\!\cdots\!34$$$$T^{15} +$$$$11\!\cdots\!66$$$$T^{16} +$$$$23\!\cdots\!90$$$$T^{17} +$$$$10\!\cdots\!73$$$$T^{18} +$$$$17\!\cdots\!59$$$$T^{19} +$$$$41\!\cdots\!01$$$$T^{20}$$)
$31$ ($$1 - 7334751 T + 133000085478695 T^{2} -$$$$64\!\cdots\!24$$$$T^{3} +$$$$67\!\cdots\!60$$$$T^{4} -$$$$22\!\cdots\!12$$$$T^{5} +$$$$20\!\cdots\!34$$$$T^{6} -$$$$57\!\cdots\!86$$$$T^{7} +$$$$53\!\cdots\!14$$$$T^{8} -$$$$15\!\cdots\!92$$$$T^{9} +$$$$12\!\cdots\!60$$$$T^{10} -$$$$31\!\cdots\!44$$$$T^{11} +$$$$17\!\cdots\!45$$$$T^{12} -$$$$25\!\cdots\!71$$$$T^{13} +$$$$90\!\cdots\!91$$$$T^{14}$$)($$1 + 8628841 T + 143609550396119 T^{2} +$$$$11\!\cdots\!68$$$$T^{3} +$$$$11\!\cdots\!04$$$$T^{4} +$$$$76\!\cdots\!24$$$$T^{5} +$$$$58\!\cdots\!89$$$$T^{6} +$$$$35\!\cdots\!73$$$$T^{7} +$$$$22\!\cdots\!19$$$$T^{8} +$$$$12\!\cdots\!76$$$$T^{9} +$$$$68\!\cdots\!12$$$$T^{10} +$$$$32\!\cdots\!96$$$$T^{11} +$$$$15\!\cdots\!79$$$$T^{12} +$$$$64\!\cdots\!03$$$$T^{13} +$$$$28\!\cdots\!09$$$$T^{14} +$$$$98\!\cdots\!24$$$$T^{15} +$$$$37\!\cdots\!84$$$$T^{16} +$$$$10\!\cdots\!88$$$$T^{17} +$$$$34\!\cdots\!59$$$$T^{18} +$$$$54\!\cdots\!71$$$$T^{19} +$$$$16\!\cdots\!01$$$$T^{20}$$)
$37$ ($$1 + 13324320 T + 509958403128099 T^{2} +$$$$69\!\cdots\!88$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4} +$$$$17\!\cdots\!16$$$$T^{5} +$$$$28\!\cdots\!91$$$$T^{6} +$$$$27\!\cdots\!72$$$$T^{7} +$$$$36\!\cdots\!07$$$$T^{8} +$$$$28\!\cdots\!64$$$$T^{9} +$$$$32\!\cdots\!53$$$$T^{10} +$$$$19\!\cdots\!08$$$$T^{11} +$$$$18\!\cdots\!43$$$$T^{12} +$$$$64\!\cdots\!80$$$$T^{13} +$$$$62\!\cdots\!53$$$$T^{14}$$)($$1 + 4928252 T + 633532097918862 T^{2} +$$$$99\!\cdots\!88$$$$T^{3} +$$$$17\!\cdots\!05$$$$T^{4} -$$$$45\!\cdots\!16$$$$T^{5} +$$$$27\!\cdots\!08$$$$T^{6} -$$$$22\!\cdots\!20$$$$T^{7} +$$$$34\!\cdots\!18$$$$T^{8} -$$$$45\!\cdots\!88$$$$T^{9} +$$$$43\!\cdots\!84$$$$T^{10} -$$$$59\!\cdots\!76$$$$T^{11} +$$$$58\!\cdots\!22$$$$T^{12} -$$$$48\!\cdots\!60$$$$T^{13} +$$$$78\!\cdots\!28$$$$T^{14} -$$$$16\!\cdots\!12$$$$T^{15} +$$$$82\!\cdots\!45$$$$T^{16} +$$$$62\!\cdots\!64$$$$T^{17} +$$$$51\!\cdots\!22$$$$T^{18} +$$$$52\!\cdots\!24$$$$T^{19} +$$$$13\!\cdots\!49$$$$T^{20}$$)
$41$ ($$1 + 15691573 T + 1375318214425493 T^{2} +$$$$12\!\cdots\!28$$$$T^{3} +$$$$89\!\cdots\!84$$$$T^{4} +$$$$61\!\cdots\!04$$$$T^{5} +$$$$41\!\cdots\!78$$$$T^{6} +$$$$24\!\cdots\!58$$$$T^{7} +$$$$13\!\cdots\!58$$$$T^{8} +$$$$66\!\cdots\!84$$$$T^{9} +$$$$31\!\cdots\!04$$$$T^{10} +$$$$14\!\cdots\!48$$$$T^{11} +$$$$51\!\cdots\!93$$$$T^{12} +$$$$19\!\cdots\!53$$$$T^{13} +$$$$40\!\cdots\!21$$$$T^{14}$$)($$1 - 26206047 T + 1100068384422657 T^{2} -$$$$22\!\cdots\!58$$$$T^{3} +$$$$33\!\cdots\!02$$$$T^{4} -$$$$55\!\cdots\!26$$$$T^{5} +$$$$26\!\cdots\!99$$$$T^{6} -$$$$11\!\cdots\!25$$$$T^{7} +$$$$37\!\cdots\!15$$$$T^{8} -$$$$87\!\cdots\!44$$$$T^{9} +$$$$24\!\cdots\!12$$$$T^{10} -$$$$28\!\cdots\!84$$$$T^{11} +$$$$40\!\cdots\!15$$$$T^{12} -$$$$40\!\cdots\!25$$$$T^{13} +$$$$30\!\cdots\!59$$$$T^{14} -$$$$20\!\cdots\!26$$$$T^{15} +$$$$41\!\cdots\!22$$$$T^{16} -$$$$88\!\cdots\!18$$$$T^{17} +$$$$14\!\cdots\!17$$$$T^{18} -$$$$11\!\cdots\!27$$$$T^{19} +$$$$14\!\cdots\!01$$$$T^{20}$$)
$43$ ($$1 + 46474818 T + 2683578315052717 T^{2} +$$$$93\!\cdots\!60$$$$T^{3} +$$$$35\!\cdots\!89$$$$T^{4} +$$$$96\!\cdots\!94$$$$T^{5} +$$$$27\!\cdots\!01$$$$T^{6} +$$$$60\!\cdots\!88$$$$T^{7} +$$$$13\!\cdots\!43$$$$T^{8} +$$$$24\!\cdots\!06$$$$T^{9} +$$$$44\!\cdots\!23$$$$T^{10} +$$$$59\!\cdots\!60$$$$T^{11} +$$$$86\!\cdots\!31$$$$T^{12} +$$$$74\!\cdots\!82$$$$T^{13} +$$$$81\!\cdots\!07$$$$T^{14}$$)($$1 - 39105260 T + 4463894678087386 T^{2} -$$$$14\!\cdots\!24$$$$T^{3} +$$$$91\!\cdots\!77$$$$T^{4} -$$$$25\!\cdots\!80$$$$T^{5} +$$$$11\!\cdots\!56$$$$T^{6} -$$$$27\!\cdots\!36$$$$T^{7} +$$$$96\!\cdots\!82$$$$T^{8} -$$$$19\!\cdots\!80$$$$T^{9} +$$$$57\!\cdots\!56$$$$T^{10} -$$$$98\!\cdots\!40$$$$T^{11} +$$$$24\!\cdots\!18$$$$T^{12} -$$$$34\!\cdots\!52$$$$T^{13} +$$$$72\!\cdots\!56$$$$T^{14} -$$$$81\!\cdots\!40$$$$T^{15} +$$$$14\!\cdots\!73$$$$T^{16} -$$$$11\!\cdots\!68$$$$T^{17} +$$$$18\!\cdots\!86$$$$T^{18} -$$$$80\!\cdots\!80$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20}$$)
$47$ ($$1 - 8232227 T + 3021791284626767 T^{2} -$$$$59\!\cdots\!68$$$$T^{3} +$$$$55\!\cdots\!28$$$$T^{4} -$$$$12\!\cdots\!04$$$$T^{5} +$$$$70\!\cdots\!94$$$$T^{6} -$$$$18\!\cdots\!74$$$$T^{7} +$$$$79\!\cdots\!98$$$$T^{8} -$$$$15\!\cdots\!56$$$$T^{9} +$$$$77\!\cdots\!64$$$$T^{10} -$$$$93\!\cdots\!28$$$$T^{11} +$$$$53\!\cdots\!69$$$$T^{12} -$$$$16\!\cdots\!63$$$$T^{13} +$$$$21\!\cdots\!23$$$$T^{14}$$)($$1 - 90595459 T + 9931963379089527 T^{2} -$$$$62\!\cdots\!28$$$$T^{3} +$$$$41\!\cdots\!88$$$$T^{4} -$$$$20\!\cdots\!12$$$$T^{5} +$$$$10\!\cdots\!01$$$$T^{6} -$$$$43\!\cdots\!67$$$$T^{7} +$$$$18\!\cdots\!11$$$$T^{8} -$$$$66\!\cdots\!24$$$$T^{9} +$$$$23\!\cdots\!64$$$$T^{10} -$$$$74\!\cdots\!08$$$$T^{11} +$$$$22\!\cdots\!79$$$$T^{12} -$$$$61\!\cdots\!21$$$$T^{13} +$$$$16\!\cdots\!21$$$$T^{14} -$$$$36\!\cdots\!84$$$$T^{15} +$$$$81\!\cdots\!72$$$$T^{16} -$$$$13\!\cdots\!44$$$$T^{17} +$$$$24\!\cdots\!07$$$$T^{18} -$$$$24\!\cdots\!73$$$$T^{19} +$$$$30\!\cdots\!49$$$$T^{20}$$)
$53$ ($$1 + 53545400 T + 14071117691005871 T^{2} +$$$$70\!\cdots\!08$$$$T^{3} +$$$$97\!\cdots\!53$$$$T^{4} +$$$$45\!\cdots\!64$$$$T^{5} +$$$$44\!\cdots\!55$$$$T^{6} +$$$$18\!\cdots\!28$$$$T^{7} +$$$$14\!\cdots\!15$$$$T^{8} +$$$$49\!\cdots\!96$$$$T^{9} +$$$$35\!\cdots\!61$$$$T^{10} +$$$$83\!\cdots\!68$$$$T^{11} +$$$$55\!\cdots\!03$$$$T^{12} +$$$$69\!\cdots\!00$$$$T^{13} +$$$$42\!\cdots\!77$$$$T^{14}$$)($$1 - 174250464 T + 31943485744432362 T^{2} -$$$$36\!\cdots\!60$$$$T^{3} +$$$$41\!\cdots\!05$$$$T^{4} -$$$$36\!\cdots\!68$$$$T^{5} +$$$$32\!\cdots\!84$$$$T^{6} -$$$$24\!\cdots\!92$$$$T^{7} +$$$$17\!\cdots\!14$$$$T^{8} -$$$$11\!\cdots\!24$$$$T^{9} +$$$$68\!\cdots\!48$$$$T^{10} -$$$$36\!\cdots\!92$$$$T^{11} +$$$$19\!\cdots\!46$$$$T^{12} -$$$$86\!\cdots\!04$$$$T^{13} +$$$$38\!\cdots\!64$$$$T^{14} -$$$$14\!\cdots\!24$$$$T^{15} +$$$$53\!\cdots\!45$$$$T^{16} -$$$$15\!\cdots\!20$$$$T^{17} +$$$$44\!\cdots\!42$$$$T^{18} -$$$$80\!\cdots\!92$$$$T^{19} +$$$$15\!\cdots\!49$$$$T^{20}$$)
$59$ ($$1 + 341275144 T + 87172213344057277 T^{2} +$$$$16\!\cdots\!56$$$$T^{3} +$$$$25\!\cdots\!45$$$$T^{4} +$$$$33\!\cdots\!48$$$$T^{5} +$$$$38\!\cdots\!73$$$$T^{6} +$$$$38\!\cdots\!04$$$$T^{7} +$$$$33\!\cdots\!47$$$$T^{8} +$$$$25\!\cdots\!08$$$$T^{9} +$$$$16\!\cdots\!55$$$$T^{10} +$$$$92\!\cdots\!96$$$$T^{11} +$$$$42\!\cdots\!23$$$$T^{12} +$$$$14\!\cdots\!84$$$$T^{13} +$$$$36\!\cdots\!79$$$$T^{14}$$)($$1 - 94151060 T + 24726293886487630 T^{2} -$$$$11\!\cdots\!04$$$$T^{3} +$$$$22\!\cdots\!61$$$$T^{4} -$$$$15\!\cdots\!84$$$$T^{5} +$$$$32\!\cdots\!04$$$$T^{6} -$$$$31\!\cdots\!04$$$$T^{7} +$$$$35\!\cdots\!18$$$$T^{8} -$$$$23\!\cdots\!92$$$$T^{9} +$$$$24\!\cdots\!72$$$$T^{10} -$$$$20\!\cdots\!88$$$$T^{11} +$$$$26\!\cdots\!78$$$$T^{12} -$$$$20\!\cdots\!76$$$$T^{13} +$$$$18\!\cdots\!64$$$$T^{14} -$$$$73\!\cdots\!16$$$$T^{15} +$$$$94\!\cdots\!21$$$$T^{16} -$$$$41\!\cdots\!16$$$$T^{17} +$$$$78\!\cdots\!30$$$$T^{18} -$$$$25\!\cdots\!40$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20}$$)
$61$ ($$1 + 277157656 T + 81383943047092111 T^{2} +$$$$16\!\cdots\!44$$$$T^{3} +$$$$28\!\cdots\!05$$$$T^{4} +$$$$43\!\cdots\!12$$$$T^{5} +$$$$56\!\cdots\!75$$$$T^{6} +$$$$64\!\cdots\!12$$$$T^{7} +$$$$65\!\cdots\!75$$$$T^{8} +$$$$58\!\cdots\!72$$$$T^{9} +$$$$46\!\cdots\!05$$$$T^{10} +$$$$30\!\cdots\!84$$$$T^{11} +$$$$17\!\cdots\!11$$$$T^{12} +$$$$70\!\cdots\!96$$$$T^{13} +$$$$29\!\cdots\!81$$$$T^{14}$$)($$1 - 174839396 T + 59435114322456626 T^{2} -$$$$77\!\cdots\!04$$$$T^{3} +$$$$17\!\cdots\!29$$$$T^{4} -$$$$19\!\cdots\!64$$$$T^{5} +$$$$35\!\cdots\!96$$$$T^{6} -$$$$34\!\cdots\!76$$$$T^{7} +$$$$57\!\cdots\!54$$$$T^{8} -$$$$49\!\cdots\!04$$$$T^{9} +$$$$73\!\cdots\!76$$$$T^{10} -$$$$57\!\cdots\!64$$$$T^{11} +$$$$77\!\cdots\!74$$$$T^{12} -$$$$54\!\cdots\!96$$$$T^{13} +$$$$66\!\cdots\!56$$$$T^{14} -$$$$41\!\cdots\!64$$$$T^{15} +$$$$45\!\cdots\!89$$$$T^{16} -$$$$23\!\cdots\!24$$$$T^{17} +$$$$20\!\cdots\!46$$$$T^{18} -$$$$71\!\cdots\!56$$$$T^{19} +$$$$47\!\cdots\!01$$$$T^{20}$$)
$67$ ($$1 - 89654580 T + 59646420871281213 T^{2} +$$$$50\!\cdots\!32$$$$T^{3} +$$$$13\!\cdots\!25$$$$T^{4} +$$$$24\!\cdots\!52$$$$T^{5} +$$$$89\!\cdots\!09$$$$T^{6} +$$$$95\!\cdots\!76$$$$T^{7} +$$$$24\!\cdots\!23$$$$T^{8} +$$$$18\!\cdots\!68$$$$T^{9} +$$$$27\!\cdots\!75$$$$T^{10} +$$$$27\!\cdots\!92$$$$T^{11} +$$$$88\!\cdots\!91$$$$T^{12} -$$$$36\!\cdots\!20$$$$T^{13} +$$$$11\!\cdots\!63$$$$T^{14}$$)($$1 + 148457006 T + 179852493781505586 T^{2} +$$$$17\!\cdots\!66$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} +$$$$77\!\cdots\!08$$$$T^{5} +$$$$71\!\cdots\!12$$$$T^{6} +$$$$15\!\cdots\!44$$$$T^{7} +$$$$26\!\cdots\!78$$$$T^{8} +$$$$13\!\cdots\!96$$$$T^{9} +$$$$78\!\cdots\!04$$$$T^{10} +$$$$36\!\cdots\!12$$$$T^{11} +$$$$19\!\cdots\!02$$$$T^{12} +$$$$32\!\cdots\!12$$$$T^{13} +$$$$39\!\cdots\!72$$$$T^{14} +$$$$11\!\cdots\!56$$$$T^{15} +$$$$58\!\cdots\!69$$$$T^{16} +$$$$18\!\cdots\!58$$$$T^{17} +$$$$53\!\cdots\!46$$$$T^{18} +$$$$12\!\cdots\!02$$$$T^{19} +$$$$22\!\cdots\!49$$$$T^{20}$$)
$71$ ($$1 + 286098961 T + 51012935713112263 T^{2} -$$$$12\!\cdots\!28$$$$T^{3} -$$$$39\!\cdots\!76$$$$T^{4} -$$$$62\!\cdots\!80$$$$T^{5} +$$$$24\!\cdots\!66$$$$T^{6} +$$$$69\!\cdots\!78$$$$T^{7} +$$$$11\!\cdots\!46$$$$T^{8} -$$$$13\!\cdots\!80$$$$T^{9} -$$$$38\!\cdots\!16$$$$T^{10} -$$$$56\!\cdots\!88$$$$T^{11} +$$$$10\!\cdots\!13$$$$T^{12} +$$$$26\!\cdots\!41$$$$T^{13} +$$$$42\!\cdots\!11$$$$T^{14}$$)($$1 - 245537391 T + 266924822057793527 T^{2} -$$$$63\!\cdots\!48$$$$T^{3} +$$$$39\!\cdots\!12$$$$T^{4} -$$$$83\!\cdots\!72$$$$T^{5} +$$$$37\!\cdots\!37$$$$T^{6} -$$$$73\!\cdots\!03$$$$T^{7} +$$$$26\!\cdots\!27$$$$T^{8} -$$$$45\!\cdots\!96$$$$T^{9} +$$$$14\!\cdots\!12$$$$T^{10} -$$$$20\!\cdots\!76$$$$T^{11} +$$$$56\!\cdots\!47$$$$T^{12} -$$$$70\!\cdots\!73$$$$T^{13} +$$$$16\!\cdots\!77$$$$T^{14} -$$$$16\!\cdots\!72$$$$T^{15} +$$$$36\!\cdots\!72$$$$T^{16} -$$$$27\!\cdots\!28$$$$T^{17} +$$$$52\!\cdots\!07$$$$T^{18} -$$$$21\!\cdots\!61$$$$T^{19} +$$$$41\!\cdots\!01$$$$T^{20}$$)
$73$ ($$1 + 637495039 T + 410258525574225253 T^{2} +$$$$16\!\cdots\!20$$$$T^{3} +$$$$63\!\cdots\!32$$$$T^{4} +$$$$19\!\cdots\!04$$$$T^{5} +$$$$58\!\cdots\!42$$$$T^{6} +$$$$14\!\cdots\!70$$$$T^{7} +$$$$34\!\cdots\!46$$$$T^{8} +$$$$66\!\cdots\!76$$$$T^{9} +$$$$13\!\cdots\!04$$$$T^{10} +$$$$19\!\cdots\!20$$$$T^{11} +$$$$29\!\cdots\!29$$$$T^{12} +$$$$26\!\cdots\!51$$$$T^{13} +$$$$24\!\cdots\!17$$$$T^{14}$$)($$1 - 380918201 T + 501766842869198261 T^{2} -$$$$17\!\cdots\!98$$$$T^{3} +$$$$11\!\cdots\!46$$$$T^{4} -$$$$36\!\cdots\!50$$$$T^{5} +$$$$16\!\cdots\!39$$$$T^{6} -$$$$46\!\cdots\!63$$$$T^{7} +$$$$15\!\cdots\!75$$$$T^{8} -$$$$39\!\cdots\!52$$$$T^{9} +$$$$10\!\cdots\!40$$$$T^{10} -$$$$23\!\cdots\!76$$$$T^{11} +$$$$53\!\cdots\!75$$$$T^{12} -$$$$94\!\cdots\!11$$$$T^{13} +$$$$19\!\cdots\!79$$$$T^{14} -$$$$25\!\cdots\!50$$$$T^{15} +$$$$47\!\cdots\!14$$$$T^{16} -$$$$42\!\cdots\!66$$$$T^{17} +$$$$72\!\cdots\!81$$$$T^{18} -$$$$32\!\cdots\!73$$$$T^{19} +$$$$50\!\cdots\!49$$$$T^{20}$$)
$79$ ($$1 - 274469546 T + 711394885574565961 T^{2} -$$$$16\!\cdots\!88$$$$T^{3} +$$$$22\!\cdots\!85$$$$T^{4} -$$$$42\!\cdots\!62$$$$T^{5} +$$$$42\!\cdots\!97$$$$T^{6} -$$$$65\!\cdots\!92$$$$T^{7} +$$$$51\!\cdots\!43$$$$T^{8} -$$$$61\!\cdots\!82$$$$T^{9} +$$$$39\!\cdots\!15$$$$T^{10} -$$$$33\!\cdots\!48$$$$T^{11} +$$$$17\!\cdots\!39$$$$T^{12} -$$$$81\!\cdots\!26$$$$T^{13} +$$$$35\!\cdots\!39$$$$T^{14}$$)($$1 - 388717158 T + 700814288807930586 T^{2} -$$$$22\!\cdots\!26$$$$T^{3} +$$$$24\!\cdots\!81$$$$T^{4} -$$$$65\!\cdots\!60$$$$T^{5} +$$$$55\!\cdots\!28$$$$T^{6} -$$$$13\!\cdots\!04$$$$T^{7} +$$$$97\!\cdots\!42$$$$T^{8} -$$$$21\!\cdots\!56$$$$T^{9} +$$$$13\!\cdots\!84$$$$T^{10} -$$$$25\!\cdots\!64$$$$T^{11} +$$$$13\!\cdots\!62$$$$T^{12} -$$$$23\!\cdots\!36$$$$T^{13} +$$$$11\!\cdots\!88$$$$T^{14} -$$$$16\!\cdots\!40$$$$T^{15} +$$$$71\!\cdots\!61$$$$T^{16} -$$$$78\!\cdots\!14$$$$T^{17} +$$$$29\!\cdots\!26$$$$T^{18} -$$$$19\!\cdots\!82$$$$T^{19} +$$$$61\!\cdots\!01$$$$T^{20}$$)
$83$ ($$1 - 1164579762 T + 1265100440875063585 T^{2} -$$$$75\!\cdots\!80$$$$T^{3} +$$$$42\!\cdots\!33$$$$T^{4} -$$$$15\!\cdots\!62$$$$T^{5} +$$$$61\!\cdots\!57$$$$T^{6} -$$$$18\!\cdots\!72$$$$T^{7} +$$$$11\!\cdots\!71$$$$T^{8} -$$$$52\!\cdots\!58$$$$T^{9} +$$$$27\!\cdots\!91$$$$T^{10} -$$$$92\!\cdots\!80$$$$T^{11} +$$$$28\!\cdots\!55$$$$T^{12} -$$$$49\!\cdots\!98$$$$T^{13} +$$$$79\!\cdots\!87$$$$T^{14}$$)($$1 - 1453704440 T + 1625395358352804054 T^{2} -$$$$12\!\cdots\!64$$$$T^{3} +$$$$76\!\cdots\!65$$$$T^{4} -$$$$39\!\cdots\!80$$$$T^{5} +$$$$20\!\cdots\!20$$$$T^{6} -$$$$99\!\cdots\!72$$$$T^{7} +$$$$51\!\cdots\!34$$$$T^{8} -$$$$25\!\cdots\!00$$$$T^{9} +$$$$11\!\cdots\!00$$$$T^{10} -$$$$46\!\cdots\!00$$$$T^{11} +$$$$17\!\cdots\!06$$$$T^{12} -$$$$64\!\cdots\!44$$$$T^{13} +$$$$24\!\cdots\!20$$$$T^{14} -$$$$91\!\cdots\!40$$$$T^{15} +$$$$32\!\cdots\!85$$$$T^{16} -$$$$96\!\cdots\!68$$$$T^{17} +$$$$24\!\cdots\!94$$$$T^{18} -$$$$40\!\cdots\!20$$$$T^{19} +$$$$52\!\cdots\!49$$$$T^{20}$$)
$89$ ($$1 + 504153000 T + 1586076374292019667 T^{2} +$$$$88\!\cdots\!24$$$$T^{3} +$$$$13\!\cdots\!09$$$$T^{4} +$$$$65\!\cdots\!24$$$$T^{5} +$$$$69\!\cdots\!51$$$$T^{6} +$$$$28\!\cdots\!44$$$$T^{7} +$$$$24\!\cdots\!59$$$$T^{8} +$$$$80\!\cdots\!44$$$$T^{9} +$$$$56\!\cdots\!61$$$$T^{10} +$$$$13\!\cdots\!64$$$$T^{11} +$$$$83\!\cdots\!83$$$$T^{12} +$$$$93\!\cdots\!00$$$$T^{13} +$$$$64\!\cdots\!69$$$$T^{14}$$)($$1 + 509637702 T + 1644014166102711726 T^{2} +$$$$97\!\cdots\!66$$$$T^{3} +$$$$15\!\cdots\!89$$$$T^{4} +$$$$89\!\cdots\!20$$$$T^{5} +$$$$10\!\cdots\!88$$$$T^{6} +$$$$55\!\cdots\!72$$$$T^{7} +$$$$51\!\cdots\!30$$$$T^{8} +$$$$25\!\cdots\!16$$$$T^{9} +$$$$20\!\cdots\!92$$$$T^{10} +$$$$88\!\cdots\!44$$$$T^{11} +$$$$63\!\cdots\!30$$$$T^{12} +$$$$23\!\cdots\!88$$$$T^{13} +$$$$15\!\cdots\!68$$$$T^{14} +$$$$47\!\cdots\!80$$$$T^{15} +$$$$28\!\cdots\!49$$$$T^{16} +$$$$63\!\cdots\!54$$$$T^{17} +$$$$37\!\cdots\!46$$$$T^{18} +$$$$40\!\cdots\!78$$$$T^{19} +$$$$27\!\cdots\!01$$$$T^{20}$$)
$97$ ($$1 + 3519929016 T + 9443697783553902375 T^{2} +$$$$17\!\cdots\!08$$$$T^{3} +$$$$27\!\cdots\!97$$$$T^{4} +$$$$35\!\cdots\!32$$$$T^{5} +$$$$39\!\cdots\!83$$$$T^{6} +$$$$36\!\cdots\!80$$$$T^{7} +$$$$29\!\cdots\!11$$$$T^{8} +$$$$20\!\cdots\!48$$$$T^{9} +$$$$12\!\cdots\!61$$$$T^{10} +$$$$59\!\cdots\!68$$$$T^{11} +$$$$23\!\cdots\!75$$$$T^{12} +$$$$67\!\cdots\!04$$$$T^{13} +$$$$14\!\cdots\!73$$$$T^{14}$$)($$1 - 4762815042 T + 13084826530517859114 T^{2} -$$$$25\!\cdots\!74$$$$T^{3} +$$$$41\!\cdots\!53$$$$T^{4} -$$$$56\!\cdots\!80$$$$T^{5} +$$$$69\!\cdots\!08$$$$T^{6} -$$$$78\!\cdots\!40$$$$T^{7} +$$$$81\!\cdots\!46$$$$T^{8} -$$$$79\!\cdots\!00$$$$T^{9} +$$$$72\!\cdots\!88$$$$T^{10} -$$$$60\!\cdots\!00$$$$T^{11} +$$$$47\!\cdots\!94$$$$T^{12} -$$$$34\!\cdots\!20$$$$T^{13} +$$$$23\!\cdots\!68$$$$T^{14} -$$$$14\!\cdots\!60$$$$T^{15} +$$$$79\!\cdots\!57$$$$T^{16} -$$$$38\!\cdots\!02$$$$T^{17} +$$$$14\!\cdots\!74$$$$T^{18} -$$$$40\!\cdots\!74$$$$T^{19} +$$$$64\!\cdots\!49$$$$T^{20}$$)