LMFDB - Space of modular forms of level 43, weight 4, and Trivial character

Defining parameters

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	43.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 2 $
Sturm bound:			$14$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(43))$.

	Total	New	Old
Modular forms	12	10	2
Cusp forms	10	10	0
Eisenstein series	2	0	2

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

$43$	Dim.
$+$	$6$
$-$	$4$

Trace form

$ 10q + 2q^{2} - 4q^{3} + 24q^{4} + 16q^{5} - 30q^{6} - 12q^{7} - 12q^{8} + 72q^{9} + O(q^{10}) $

\( 10q + 2q^{2} - 4q^{3} + 24q^{4} + 16q^{5} - 30q^{6} - 12q^{7} - 12q^{8} + 72q^{9} + 54q^{10} - 90q^{11} - 96q^{12} + 54q^{13} - 72q^{14} - 32q^{15} + 148q^{16} - 188q^{17} + 218q^{18} + 24q^{19} + 216q^{20} - 108q^{21} - 302q^{22} + 28q^{23} - 358q^{24} + 4q^{25} - 6q^{26} + 20q^{27} - 484q^{28} + 416q^{29} - 96q^{30} + 368q^{31} + 216q^{32} + 508q^{33} - 54q^{34} - 176q^{35} - 306q^{36} - 180q^{37} - 82q^{38} + 388q^{39} + 746q^{40} - 20q^{41} - 32q^{42} - 86q^{43} - 1192q^{44} - 4q^{45} + 994q^{46} + 434q^{47} - 752q^{48} + 586q^{49} + 1934q^{50} + 408q^{51} + 216q^{52} - 770q^{53} + 80q^{54} - 372q^{55} - 436q^{56} - 768q^{57} - 1826q^{58} + 1172q^{59} - 2888q^{60} - 956q^{61} + 1506q^{62} - 376q^{63} + 900q^{64} - 412q^{65} + 968q^{66} - 522q^{67} - 1126q^{68} - 2368q^{69} - 2620q^{70} + 324q^{71} + 252q^{72} + 1492q^{73} + 1602q^{74} - 2432q^{75} + 2676q^{76} + 616q^{77} + 2132q^{78} - 518q^{79} + 2368q^{80} + 1954q^{81} - 614q^{82} - 1662q^{83} + 3148q^{84} - 68q^{85} - 430q^{86} - 58q^{87} + 1208q^{88} + 2640q^{89} - 1468q^{90} - 3048q^{91} + 974q^{92} + 1844q^{93} + 964q^{94} + 1070q^{95} - 2142q^{96} - 1820q^{97} + 4050q^{98} - 1670q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(43))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs	$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	43	$q$-expansion
43.4.a.a	$4$	$2.537$	4.4.45868.1	None	$-4$	$-11$	$-27$	$-20$	$-$	$q+(-1+\beta _{3})q^{2}+(-3+\beta _{2}-\beta _{3})q^{3}+\cdots$
43.4.a.b	$6$	$2.537$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	$6$	$7$	$43$	$8$	$+$	$q+(1-\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + 4 T + 23 T^{2} + 82 T^{3} + 242 T^{4} + 656 T^{5} + 1472 T^{6} + 2048 T^{7} + 4096 T^{8} $)($ 1 - 6 T + 31 T^{2} - 116 T^{3} + 442 T^{4} - 1472 T^{5} + 4668 T^{6} - 11776 T^{7} + 28288 T^{8} - 59392 T^{9} + 126976 T^{10} - 196608 T^{11} + 262144 T^{12} $)
$3$	($ 1 + 11 T + 119 T^{2} + 839 T^{3} + 4996 T^{4} + 22653 T^{5} + 86751 T^{6} + 216513 T^{7} + 531441 T^{8} $)($ 1 - 7 T + 65 T^{2} - 357 T^{3} + 2599 T^{4} - 15158 T^{5} + 96098 T^{6} - 409266 T^{7} + 1894671 T^{8} - 7026831 T^{9} + 34543665 T^{10} - 100442349 T^{11} + 387420489 T^{12} $)
$5$	($ 1 + 27 T + 665 T^{2} + 9849 T^{3} + 134272 T^{4} + 1231125 T^{5} + 10390625 T^{6} + 52734375 T^{7} + 244140625 T^{8} $)($ 1 - 43 T + 1247 T^{2} - 26367 T^{3} + 452519 T^{4} - 6421366 T^{5} + 77975134 T^{6} - 802670750 T^{7} + 7070609375 T^{8} - 51498046875 T^{9} + 304443359375 T^{10} - 1312255859375 T^{11} + 3814697265625 T^{12} $)
$7$	($ 1 + 20 T + 768 T^{2} + 13108 T^{3} + 314942 T^{4} + 4496044 T^{5} + 90354432 T^{6} + 807072140 T^{7} + 13841287201 T^{8} $)($ 1 - 8 T + 886 T^{2} - 4080 T^{3} + 442155 T^{4} - 3221432 T^{5} + 186242508 T^{6} - 1104951176 T^{7} + 52019093595 T^{8} - 164642716560 T^{9} + 12263380460086 T^{10} - 37980492079544 T^{11} + 1628413597910449 T^{12} $)
$11$	($ 1 + 62 T + 3077 T^{2} + 39962 T^{3} + 1317884 T^{4} + 53189422 T^{5} + 5451093197 T^{6} + 146192756842 T^{7} + 3138428376721 T^{8} $)($ 1 + 28 T + 3144 T^{2} + 41080 T^{3} + 3977536 T^{4} + 3139972 T^{5} + 4111332998 T^{6} + 4179302732 T^{7} + 7046447653696 T^{8} + 96864491146280 T^{9} + 9867218816410824 T^{10} + 116962948743638228 T^{11} + 5559917313492231481 T^{12} $)
$13$	($ 1 + 2 T + 8273 T^{2} + 17238 T^{3} + 26690692 T^{4} + 37871886 T^{5} + 39932190857 T^{6} + 21208998746 T^{7} + 23298085122481 T^{8} $)($ 1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 4725231126824 T^{7} + 180879261248920 T^{8} - 5629759045135732 T^{9} + 204463995034893256 T^{10} - 2866410008789082392 T^{11} + $$11\!\cdots\!29$$ T^{12} $)
$17$	($ 1 + 207 T + 14258 T^{2} - 190032 T^{3} - 71042783 T^{4} - 933627216 T^{5} + 344153458802 T^{6} + 24547690434879 T^{7} + 582622237229761 T^{8} $)($ 1 - 19 T + 23143 T^{2} - 543393 T^{3} + 241535186 T^{4} - 5650313095 T^{5} + 1493450611759 T^{6} - 27759988235735 T^{7} + 5830072218002834 T^{8} - 64439821973334321 T^{9} + 13483626436208358823 T^{10} - 54386037978686500067 T^{11} + $$14\!\cdots\!09$$ T^{12} $)
$19$	($ 1 - 99 T + 22201 T^{2} - 1662075 T^{3} + 222917608 T^{4} - 11400172425 T^{5} + 1044465604081 T^{6} - 31946082080121 T^{7} + 2213314919066161 T^{8} $)($ 1 + 75 T + 38259 T^{2} + 2365781 T^{3} + 629552155 T^{4} + 31151517862 T^{5} + 5681041321490 T^{6} + 213668261015458 T^{7} + 29617835767423555 T^{8} + 763408424339300399 T^{9} + 84679215488552253699 T^{10} + $$11\!\cdots\!25$$ T^{11} + $$10\!\cdots\!41$$ T^{12} $)
$23$	($ 1 + 103 T + 33718 T^{2} + 2793222 T^{3} + 586593953 T^{4} + 33985132074 T^{5} + 4991474105302 T^{6} + 185518724130689 T^{7} + 21914624432020321 T^{8} $)($ 1 - 131 T + 52195 T^{2} - 3677795 T^{3} + 974730114 T^{4} - 33210519163 T^{5} + 11858751245947 T^{6} - 404072386656221 T^{7} + 144295038961061346 T^{8} - 6624270252565314085 T^{9} + $$11\!\cdots\!95$$ T^{10} - $$34\!\cdots\!17$$ T^{11} + $$32\!\cdots\!69$$ T^{12} $)
$29$	($ 1 + 99 T - 13567 T^{2} - 167967 T^{3} + 888213804 T^{4} - 4096547163 T^{5} - 8069967996007 T^{6} + 1436207451611031 T^{7} + 353814783205469041 T^{8} $)($ 1 - 515 T + 204583 T^{2} - 58068807 T^{3} + 13900558631 T^{4} - 2733986934494 T^{5} + 464005745217070 T^{6} - 66679207345374166 T^{7} + 8268376448646633551 T^{8} - $$84\!\cdots\!83$$ T^{9} + $$72\!\cdots\!03$$ T^{10} - $$44\!\cdots\!35$$ T^{11} + $$21\!\cdots\!61$$ T^{12} $)
$31$	($ 1 - 131 T + 64436 T^{2} - 3696474 T^{3} + 1950279047 T^{4} - 110121656934 T^{5} + 57187187188916 T^{6} - 3463590503047901 T^{7} + 787662783788549761 T^{8} $)($ 1 - 237 T + 125373 T^{2} - 21016589 T^{3} + 7321362670 T^{4} - 1031063540341 T^{5} + 275109610824401 T^{6} - 30716413930298731 T^{7} + 6497736319560988270 T^{8} - $$55\!\cdots\!19$$ T^{9} + $$98\!\cdots\!53$$ T^{10} - $$55\!\cdots\!87$$ T^{11} + $$69\!\cdots\!41$$ T^{12} $)
$37$	($ 1 + 449 T + 266817 T^{2} + 71392239 T^{3} + 21942685768 T^{4} + 3616231082067 T^{5} + 684579423270153 T^{6} + 58352821167989573 T^{7} + 6582952005840035281 T^{8} $)($ 1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 65968300092541106 T^{7} + 20981383282418309715 T^{8} - $$39\!\cdots\!61$$ T^{9} + $$83\!\cdots\!91$$ T^{10} - $$89\!\cdots\!17$$ T^{11} + $$16\!\cdots\!29$$ T^{12} $)
$41$	($ 1 + 491 T + 187430 T^{2} + 55413200 T^{3} + 16109516117 T^{4} + 3819133157200 T^{5} + 890312037890630 T^{6} + 160744529787434851 T^{7} + 22563490300366186081 T^{8} $)($ 1 - 471 T + 349763 T^{2} - 112600045 T^{3} + 50459129866 T^{4} - 12922113800443 T^{5} + 4372223136871043 T^{6} - 890605005240332003 T^{7} + $$23\!\cdots\!06$$ T^{8} - $$36\!\cdots\!45$$ T^{9} + $$78\!\cdots\!03$$ T^{10} - $$73\!\cdots\!71$$ T^{11} + $$10\!\cdots\!21$$ T^{12} $)
$43$	($ ( 1 - 43 T )^{4} $)($ ( 1 + 43 T )^{6} $)
$47$	($ 1 - 19 T + 108577 T^{2} - 15073647 T^{3} + 11126438636 T^{4} - 1564991252481 T^{5} + 1170374862776833 T^{6} - 21263478988952573 T^{7} + $$11\!\cdots\!41$$ T^{8} $)($ 1 - 415 T + 421631 T^{2} - 116866317 T^{3} + 77411983523 T^{4} - 16052264514750 T^{5} + 9154052369892234 T^{6} - 1666594258714889250 T^{7} + $$83\!\cdots\!67$$ T^{8} - $$13\!\cdots\!39$$ T^{9} + $$48\!\cdots\!71$$ T^{10} - $$50\!\cdots\!45$$ T^{11} + $$12\!\cdots\!89$$ T^{12} $)
$53$	($ 1 + 1220 T + 1002413 T^{2} + 551938550 T^{3} + 246387730196 T^{4} + 82170955508350 T^{5} + 22217843732404277 T^{6} + 4025711581998602260 T^{7} + $$49\!\cdots\!41$$ T^{8} $)($ 1 - 450 T + 321704 T^{2} - 149982378 T^{3} + 77929548632 T^{4} - 28654270442506 T^{5} + 12746558079363422 T^{6} - 4265961820668965762 T^{7} + $$17\!\cdots\!28$$ T^{8} - $$49\!\cdots\!74$$ T^{9} + $$15\!\cdots\!64$$ T^{10} - $$32\!\cdots\!50$$ T^{11} + $$10\!\cdots\!89$$ T^{12} $)
$59$	($ 1 - 816 T + 987944 T^{2} - 513131184 T^{3} + 321570111646 T^{4} - 105386369438736 T^{5} + 41672005127424104 T^{6} - 7069004588022430224 T^{7} + $$17\!\cdots\!81$$ T^{8} $)($ 1 - 356 T + 1153950 T^{2} - 347607228 T^{3} + 570632518215 T^{4} - 139273869185096 T^{5} + 154351054402988548 T^{6} - 28603927979365831384 T^{7} + $$24\!\cdots\!15$$ T^{8} - $$30\!\cdots\!92$$ T^{9} + $$20\!\cdots\!50$$ T^{10} - $$13\!\cdots\!44$$ T^{11} + $$75\!\cdots\!21$$ T^{12} $)
$61$	($ 1 - 372 T + 843468 T^{2} - 226528412 T^{3} + 278949483798 T^{4} - 51417645484172 T^{5} + 43455787121523948 T^{6} - 4350222346534300452 T^{7} + $$26\!\cdots\!21$$ T^{8} $)($ 1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} + $$14\!\cdots\!08$$ T^{7} + $$58\!\cdots\!39$$ T^{8} + $$17\!\cdots\!84$$ T^{9} + $$47\!\cdots\!74$$ T^{10} + $$80\!\cdots\!28$$ T^{11} + $$13\!\cdots\!81$$ T^{12} $)
$67$	($ 1 - 110 T + 855641 T^{2} - 172379298 T^{3} + 331907899304 T^{4} - 51845314804374 T^{5} + 77399900577465329 T^{6} - 2992718783592444170 T^{7} + $$81\!\cdots\!61$$ T^{8} $)($ 1 + 632 T + 927628 T^{2} + 253029932 T^{3} + 179674044568 T^{4} - 62778009874096 T^{5} - 5212390617577006 T^{6} - 18881302583762735248 T^{7} + $$16\!\cdots\!92$$ T^{8} + $$68\!\cdots\!04$$ T^{9} + $$75\!\cdots\!08$$ T^{10} + $$15\!\cdots\!76$$ T^{11} + $$74\!\cdots\!09$$ T^{12} $)
$71$	($ 1 - 468 T + 925804 T^{2} - 494793108 T^{3} + 407494264950 T^{4} - 177091896077388 T^{5} + 118595755255197484 T^{6} - 21457098336234146508 T^{7} + $$16\!\cdots\!41$$ T^{8} $)($ 1 + 144 T + 863230 T^{2} + 72744496 T^{3} + 475313592223 T^{4} + 62333254020128 T^{5} + 209643801201434276 T^{6} + 22309757279598032608 T^{7} + $$60\!\cdots\!83$$ T^{8} + $$33\!\cdots\!76$$ T^{9} + $$14\!\cdots\!30$$ T^{10} + $$84\!\cdots\!44$$ T^{11} + $$21\!\cdots\!61$$ T^{12} $)
$73$	($ 1 - 628 T + 1127772 T^{2} - 544172556 T^{3} + 627880103782 T^{4} - 211692375217452 T^{5} + 170670503050398108 T^{6} - 36971356452792249364 T^{7} + $$22\!\cdots\!21$$ T^{8} $)($ 1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 81531488761123023064 T^{7} + $$69\!\cdots\!63$$ T^{8} - $$25\!\cdots\!44$$ T^{9} + $$24\!\cdots\!18$$ T^{10} - $$76\!\cdots\!48$$ T^{11} + $$34\!\cdots\!69$$ T^{12} $)
$79$	($ 1 - 1095 T + 1663851 T^{2} - 1399726691 T^{3} + 1195813590120 T^{4} - 690119848003949 T^{5} + 404461305956071371 T^{6} - $$13\!\cdots\!05$$ T^{7} + $$59\!\cdots\!41$$ T^{8} $)($ 1 + 1613 T + 2731081 T^{2} + 3130876751 T^{3} + 3268965374139 T^{4} + 2799662350208018 T^{5} + 2111366737833161318 T^{6} + $$13\!\cdots\!02$$ T^{7} + $$79\!\cdots\!19$$ T^{8} + $$37\!\cdots\!69$$ T^{9} + $$16\!\cdots\!21$$ T^{10} + $$46\!\cdots\!87$$ T^{11} + $$14\!\cdots\!61$$ T^{12} $)
$83$	($ 1 + 980 T + 1568765 T^{2} + 705601262 T^{3} + 896947637720 T^{4} + 403453628795194 T^{5} + 512892614828219285 T^{6} + $$18\!\cdots\!40$$ T^{7} + $$10\!\cdots\!61$$ T^{8} $)($ 1 + 682 T + 1120004 T^{2} + 1783287782 T^{3} + 1291656570448 T^{4} + 1149121488958882 T^{5} + 1227368803599345682 T^{6} + $$65\!\cdots\!34$$ T^{7} + $$42\!\cdots\!12$$ T^{8} + $$33\!\cdots\!46$$ T^{9} + $$11\!\cdots\!44$$ T^{10} + $$41\!\cdots\!74$$ T^{11} + $$34\!\cdots\!09$$ T^{12} $)
$89$	($ 1 + 738 T + 1788008 T^{2} + 1031145174 T^{3} + 1620376904526 T^{4} + 726925382169606 T^{5} + 888606524088595688 T^{6} + $$25\!\cdots\!42$$ T^{7} + $$24\!\cdots\!21$$ T^{8} $)($ 1 - 3378 T + 7851850 T^{2} - 13055260850 T^{3} + 17370332054203 T^{4} - 19017874978895668 T^{5} + 17369747195795800052 T^{6} - $$13\!\cdots\!92$$ T^{7} + $$86\!\cdots\!83$$ T^{8} - $$45\!\cdots\!50$$ T^{9} + $$19\!\cdots\!50$$ T^{10} - $$58\!\cdots\!22$$ T^{11} + $$12\!\cdots\!81$$ T^{12} $)
$97$	($ 1 + 1765 T + 3796458 T^{2} + 4179032168 T^{3} + 5180997869897 T^{4} + 3814089825865064 T^{5} + 3162343231888741482 T^{6} + $$13\!\cdots\!05$$ T^{7} + $$69\!\cdots\!41$$ T^{8} $)($ 1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} - $$15\!\cdots\!33$$ T^{7} + $$27\!\cdots\!30$$ T^{8} - $$71\!\cdots\!87$$ T^{9} + $$17\!\cdots\!11$$ T^{10} + $$34\!\cdots\!15$$ T^{11} + $$57\!\cdots\!89$$ T^{12} $)
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Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(14\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

\(43\)	Dim.
\(+\)	\(6\)
\(-\)	\(4\)

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs	$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	43	$q$-expansion
43.4.a.a	\(4\)	\(2.537\)	4.4.45868.1	None	\(-4\)	\(-11\)	\(-27\)	\(-20\)	\(-\)	\(q+(-1+\beta _{3})q^{2}+(-3+\beta _{2}-\beta _{3})q^{3}+\cdots\)
43.4.a.b	\(6\)	\(2.537\)	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	\(6\)	\(7\)	\(43\)	\(8\)	\(+\)	\(q+(1-\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)

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L-functions

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(43))\) into newform subspaces

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	43.4.a

Level	43
Weight	4
Character orbit	a
Rep. character	\(\chi_{43}(1,\cdot)\)
Character field	\(\Q\)
Dimension	10
Newform subspaces	2
Sturm bound	14
Trace bound	1

$p$	$F_p(T)$
$2$	(\( 1 + 4 T + 23 T^{2} + 82 T^{3} + 242 T^{4} + 656 T^{5} + 1472 T^{6} + 2048 T^{7} + 4096 T^{8} \))(\( 1 - 6 T + 31 T^{2} - 116 T^{3} + 442 T^{4} - 1472 T^{5} + 4668 T^{6} - 11776 T^{7} + 28288 T^{8} - 59392 T^{9} + 126976 T^{10} - 196608 T^{11} + 262144 T^{12} \))
$3$	(\( 1 + 11 T + 119 T^{2} + 839 T^{3} + 4996 T^{4} + 22653 T^{5} + 86751 T^{6} + 216513 T^{7} + 531441 T^{8} \))(\( 1 - 7 T + 65 T^{2} - 357 T^{3} + 2599 T^{4} - 15158 T^{5} + 96098 T^{6} - 409266 T^{7} + 1894671 T^{8} - 7026831 T^{9} + 34543665 T^{10} - 100442349 T^{11} + 387420489 T^{12} \))
$5$	(\( 1 + 27 T + 665 T^{2} + 9849 T^{3} + 134272 T^{4} + 1231125 T^{5} + 10390625 T^{6} + 52734375 T^{7} + 244140625 T^{8} \))(\( 1 - 43 T + 1247 T^{2} - 26367 T^{3} + 452519 T^{4} - 6421366 T^{5} + 77975134 T^{6} - 802670750 T^{7} + 7070609375 T^{8} - 51498046875 T^{9} + 304443359375 T^{10} - 1312255859375 T^{11} + 3814697265625 T^{12} \))
$7$	(\( 1 + 20 T + 768 T^{2} + 13108 T^{3} + 314942 T^{4} + 4496044 T^{5} + 90354432 T^{6} + 807072140 T^{7} + 13841287201 T^{8} \))(\( 1 - 8 T + 886 T^{2} - 4080 T^{3} + 442155 T^{4} - 3221432 T^{5} + 186242508 T^{6} - 1104951176 T^{7} + 52019093595 T^{8} - 164642716560 T^{9} + 12263380460086 T^{10} - 37980492079544 T^{11} + 1628413597910449 T^{12} \))
$11$	(\( 1 + 62 T + 3077 T^{2} + 39962 T^{3} + 1317884 T^{4} + 53189422 T^{5} + 5451093197 T^{6} + 146192756842 T^{7} + 3138428376721 T^{8} \))(\( 1 + 28 T + 3144 T^{2} + 41080 T^{3} + 3977536 T^{4} + 3139972 T^{5} + 4111332998 T^{6} + 4179302732 T^{7} + 7046447653696 T^{8} + 96864491146280 T^{9} + 9867218816410824 T^{10} + 116962948743638228 T^{11} + 5559917313492231481 T^{12} \))
$13$	(\( 1 + 2 T + 8273 T^{2} + 17238 T^{3} + 26690692 T^{4} + 37871886 T^{5} + 39932190857 T^{6} + 21208998746 T^{7} + 23298085122481 T^{8} \))(\( 1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 4725231126824 T^{7} + 180879261248920 T^{8} - 5629759045135732 T^{9} + 204463995034893256 T^{10} - 2866410008789082392 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} \))
$17$	(\( 1 + 207 T + 14258 T^{2} - 190032 T^{3} - 71042783 T^{4} - 933627216 T^{5} + 344153458802 T^{6} + 24547690434879 T^{7} + 582622237229761 T^{8} \))(\( 1 - 19 T + 23143 T^{2} - 543393 T^{3} + 241535186 T^{4} - 5650313095 T^{5} + 1493450611759 T^{6} - 27759988235735 T^{7} + 5830072218002834 T^{8} - 64439821973334321 T^{9} + 13483626436208358823 T^{10} - 54386037978686500067 T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \))
$19$	(\( 1 - 99 T + 22201 T^{2} - 1662075 T^{3} + 222917608 T^{4} - 11400172425 T^{5} + 1044465604081 T^{6} - 31946082080121 T^{7} + 2213314919066161 T^{8} \))(\( 1 + 75 T + 38259 T^{2} + 2365781 T^{3} + 629552155 T^{4} + 31151517862 T^{5} + 5681041321490 T^{6} + 213668261015458 T^{7} + 29617835767423555 T^{8} + 763408424339300399 T^{9} + 84679215488552253699 T^{10} + \)\(11\!\cdots\!25\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \))
$23$	(\( 1 + 103 T + 33718 T^{2} + 2793222 T^{3} + 586593953 T^{4} + 33985132074 T^{5} + 4991474105302 T^{6} + 185518724130689 T^{7} + 21914624432020321 T^{8} \))(\( 1 - 131 T + 52195 T^{2} - 3677795 T^{3} + 974730114 T^{4} - 33210519163 T^{5} + 11858751245947 T^{6} - 404072386656221 T^{7} + 144295038961061346 T^{8} - 6624270252565314085 T^{9} + \)\(11\!\cdots\!95\)\( T^{10} - \)\(34\!\cdots\!17\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \))
$29$	(\( 1 + 99 T - 13567 T^{2} - 167967 T^{3} + 888213804 T^{4} - 4096547163 T^{5} - 8069967996007 T^{6} + 1436207451611031 T^{7} + 353814783205469041 T^{8} \))(\( 1 - 515 T + 204583 T^{2} - 58068807 T^{3} + 13900558631 T^{4} - 2733986934494 T^{5} + 464005745217070 T^{6} - 66679207345374166 T^{7} + 8268376448646633551 T^{8} - \)\(84\!\cdots\!83\)\( T^{9} + \)\(72\!\cdots\!03\)\( T^{10} - \)\(44\!\cdots\!35\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \))
$31$	(\( 1 - 131 T + 64436 T^{2} - 3696474 T^{3} + 1950279047 T^{4} - 110121656934 T^{5} + 57187187188916 T^{6} - 3463590503047901 T^{7} + 787662783788549761 T^{8} \))(\( 1 - 237 T + 125373 T^{2} - 21016589 T^{3} + 7321362670 T^{4} - 1031063540341 T^{5} + 275109610824401 T^{6} - 30716413930298731 T^{7} + 6497736319560988270 T^{8} - \)\(55\!\cdots\!19\)\( T^{9} + \)\(98\!\cdots\!53\)\( T^{10} - \)\(55\!\cdots\!87\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \))
$37$	(\( 1 + 449 T + 266817 T^{2} + 71392239 T^{3} + 21942685768 T^{4} + 3616231082067 T^{5} + 684579423270153 T^{6} + 58352821167989573 T^{7} + 6582952005840035281 T^{8} \))(\( 1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 65968300092541106 T^{7} + 20981383282418309715 T^{8} - \)\(39\!\cdots\!61\)\( T^{9} + \)\(83\!\cdots\!91\)\( T^{10} - \)\(89\!\cdots\!17\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} \))
$41$	(\( 1 + 491 T + 187430 T^{2} + 55413200 T^{3} + 16109516117 T^{4} + 3819133157200 T^{5} + 890312037890630 T^{6} + 160744529787434851 T^{7} + 22563490300366186081 T^{8} \))(\( 1 - 471 T + 349763 T^{2} - 112600045 T^{3} + 50459129866 T^{4} - 12922113800443 T^{5} + 4372223136871043 T^{6} - 890605005240332003 T^{7} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!45\)\( T^{9} + \)\(78\!\cdots\!03\)\( T^{10} - \)\(73\!\cdots\!71\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} \))
$43$	(\( ( 1 - 43 T )^{4} \))(\( ( 1 + 43 T )^{6} \))
$47$	(\( 1 - 19 T + 108577 T^{2} - 15073647 T^{3} + 11126438636 T^{4} - 1564991252481 T^{5} + 1170374862776833 T^{6} - 21263478988952573 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 415 T + 421631 T^{2} - 116866317 T^{3} + 77411983523 T^{4} - 16052264514750 T^{5} + 9154052369892234 T^{6} - 1666594258714889250 T^{7} + \)\(83\!\cdots\!67\)\( T^{8} - \)\(13\!\cdots\!39\)\( T^{9} + \)\(48\!\cdots\!71\)\( T^{10} - \)\(50\!\cdots\!45\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \))
$53$	(\( 1 + 1220 T + 1002413 T^{2} + 551938550 T^{3} + 246387730196 T^{4} + 82170955508350 T^{5} + 22217843732404277 T^{6} + 4025711581998602260 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 450 T + 321704 T^{2} - 149982378 T^{3} + 77929548632 T^{4} - 28654270442506 T^{5} + 12746558079363422 T^{6} - 4265961820668965762 T^{7} + \)\(17\!\cdots\!28\)\( T^{8} - \)\(49\!\cdots\!74\)\( T^{9} + \)\(15\!\cdots\!64\)\( T^{10} - \)\(32\!\cdots\!50\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \))
$59$	(\( 1 - 816 T + 987944 T^{2} - 513131184 T^{3} + 321570111646 T^{4} - 105386369438736 T^{5} + 41672005127424104 T^{6} - 7069004588022430224 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 - 356 T + 1153950 T^{2} - 347607228 T^{3} + 570632518215 T^{4} - 139273869185096 T^{5} + 154351054402988548 T^{6} - 28603927979365831384 T^{7} + \)\(24\!\cdots\!15\)\( T^{8} - \)\(30\!\cdots\!92\)\( T^{9} + \)\(20\!\cdots\!50\)\( T^{10} - \)\(13\!\cdots\!44\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \))
$61$	(\( 1 - 372 T + 843468 T^{2} - 226528412 T^{3} + 278949483798 T^{4} - 51417645484172 T^{5} + 43455787121523948 T^{6} - 4350222346534300452 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} + \)\(14\!\cdots\!08\)\( T^{7} + \)\(58\!\cdots\!39\)\( T^{8} + \)\(17\!\cdots\!84\)\( T^{9} + \)\(47\!\cdots\!74\)\( T^{10} + \)\(80\!\cdots\!28\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \))
$67$	(\( 1 - 110 T + 855641 T^{2} - 172379298 T^{3} + 331907899304 T^{4} - 51845314804374 T^{5} + 77399900577465329 T^{6} - 2992718783592444170 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 + 632 T + 927628 T^{2} + 253029932 T^{3} + 179674044568 T^{4} - 62778009874096 T^{5} - 5212390617577006 T^{6} - 18881302583762735248 T^{7} + \)\(16\!\cdots\!92\)\( T^{8} + \)\(68\!\cdots\!04\)\( T^{9} + \)\(75\!\cdots\!08\)\( T^{10} + \)\(15\!\cdots\!76\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} \))
$71$	(\( 1 - 468 T + 925804 T^{2} - 494793108 T^{3} + 407494264950 T^{4} - 177091896077388 T^{5} + 118595755255197484 T^{6} - 21457098336234146508 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( 1 + 144 T + 863230 T^{2} + 72744496 T^{3} + 475313592223 T^{4} + 62333254020128 T^{5} + 209643801201434276 T^{6} + 22309757279598032608 T^{7} + \)\(60\!\cdots\!83\)\( T^{8} + \)\(33\!\cdots\!76\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} + \)\(84\!\cdots\!44\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \))
$73$	(\( 1 - 628 T + 1127772 T^{2} - 544172556 T^{3} + 627880103782 T^{4} - 211692375217452 T^{5} + 170670503050398108 T^{6} - 36971356452792249364 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 81531488761123023064 T^{7} + \)\(69\!\cdots\!63\)\( T^{8} - \)\(25\!\cdots\!44\)\( T^{9} + \)\(24\!\cdots\!18\)\( T^{10} - \)\(76\!\cdots\!48\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} \))
$79$	(\( 1 - 1095 T + 1663851 T^{2} - 1399726691 T^{3} + 1195813590120 T^{4} - 690119848003949 T^{5} + 404461305956071371 T^{6} - \)\(13\!\cdots\!05\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( 1 + 1613 T + 2731081 T^{2} + 3130876751 T^{3} + 3268965374139 T^{4} + 2799662350208018 T^{5} + 2111366737833161318 T^{6} + \)\(13\!\cdots\!02\)\( T^{7} + \)\(79\!\cdots\!19\)\( T^{8} + \)\(37\!\cdots\!69\)\( T^{9} + \)\(16\!\cdots\!21\)\( T^{10} + \)\(46\!\cdots\!87\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \))
$83$	(\( 1 + 980 T + 1568765 T^{2} + 705601262 T^{3} + 896947637720 T^{4} + 403453628795194 T^{5} + 512892614828219285 T^{6} + \)\(18\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 + 682 T + 1120004 T^{2} + 1783287782 T^{3} + 1291656570448 T^{4} + 1149121488958882 T^{5} + 1227368803599345682 T^{6} + \)\(65\!\cdots\!34\)\( T^{7} + \)\(42\!\cdots\!12\)\( T^{8} + \)\(33\!\cdots\!46\)\( T^{9} + \)\(11\!\cdots\!44\)\( T^{10} + \)\(41\!\cdots\!74\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \))
$89$	(\( 1 + 738 T + 1788008 T^{2} + 1031145174 T^{3} + 1620376904526 T^{4} + 726925382169606 T^{5} + 888606524088595688 T^{6} + \)\(25\!\cdots\!42\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \))(\( 1 - 3378 T + 7851850 T^{2} - 13055260850 T^{3} + 17370332054203 T^{4} - 19017874978895668 T^{5} + 17369747195795800052 T^{6} - \)\(13\!\cdots\!92\)\( T^{7} + \)\(86\!\cdots\!83\)\( T^{8} - \)\(45\!\cdots\!50\)\( T^{9} + \)\(19\!\cdots\!50\)\( T^{10} - \)\(58\!\cdots\!22\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \))
$97$	(\( 1 + 1765 T + 3796458 T^{2} + 4179032168 T^{3} + 5180997869897 T^{4} + 3814089825865064 T^{5} + 3162343231888741482 T^{6} + \)\(13\!\cdots\!05\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} - \)\(15\!\cdots\!33\)\( T^{7} + \)\(27\!\cdots\!30\)\( T^{8} - \)\(71\!\cdots\!87\)\( T^{9} + \)\(17\!\cdots\!11\)\( T^{10} + \)\(34\!\cdots\!15\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} \))
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