LMFDB - Space of Modular Forms of Level 189, Weight 2, and Character 189.g

Defining parameters

Level:	$ N $	=	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	189.g (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	=	$ 63 $
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$ 2 $
Sturm bound:			$48$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(189, [\chi])$.

	Total	New	Old
Modular forms	60	20	40
Cusp forms	36	12	24
Eisenstein series	24	8	16

Trace form

$ 12q - q^{2} - 3q^{4} + 10q^{5} + 12q^{8} + O(q^{10}) $

$ 12q - q^{2} - 3q^{4} + 10q^{5} + 12q^{8} - 6q^{10} - 2q^{11} - 3q^{13} - 11q^{14} + 3q^{16} - 9q^{17} - 4q^{20} - 6q^{22} - 6q^{25} - 16q^{26} - 6q^{28} - 8q^{29} - 3q^{31} + 7q^{32} - 4q^{35} - 3q^{37} + 38q^{38} + 12q^{40} - 10q^{41} - 6q^{43} + 5q^{44} - 27q^{47} + 12q^{49} - 23q^{50} + 30q^{52} + 12q^{53} - 6q^{55} + 48q^{56} + 18q^{58} - 30q^{59} + 12q^{62} - 36q^{64} + 16q^{65} - 6q^{67} + 60q^{68} - 24q^{70} + 30q^{71} + 12q^{73} - 78q^{74} + 6q^{76} + 26q^{77} - 12q^{79} - 19q^{80} - 18q^{83} - 3q^{85} - 14q^{86} + 6q^{88} - 41q^{89} + 21q^{91} - 30q^{92} - 3q^{94} + 13q^{95} - 3q^{97} - 61q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(189, [\chi])$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	$q$-expansion
189.2.g.a	$2$	$1.509$	$\Q(\sqrt{-3}) $	None	$1$	$0$	$2$	$1$	$q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+q^{5}+(-1+3\zeta_{6})q^{7}+\cdots$
189.2.g.b	$10$	$1.509$	10.0.$\cdots$.1	None	$-2$	$0$	$8$	$-1$	$q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(1+\cdots)q^{5}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(189, [\chi])$ into lower level spaces

$ S_{2}^{\mathrm{old}}(189, [\chi]) \cong $ $S_{2}^{\mathrm{new}}(63, [\chi])$$^{\oplus 2}$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	($ 1 - T - T^{2} - 2 T^{3} + 4 T^{4} $)($ 1 + 2 T - T^{2} - 4 T^{3} - 2 T^{4} - 2 T^{5} + 6 T^{6} + 21 T^{7} + 6 T^{8} - 13 T^{9} - 5 T^{10} - 26 T^{11} + 24 T^{12} + 168 T^{13} + 96 T^{14} - 64 T^{15} - 128 T^{16} - 512 T^{17} - 256 T^{18} + 1024 T^{19} + 1024 T^{20} $)
$3$	1
$5$	($ ( 1 - T + 5 T^{2} )^{2} $)($ ( 1 - 4 T + 20 T^{2} - 62 T^{3} + 193 T^{4} - 423 T^{5} + 965 T^{6} - 1550 T^{7} + 2500 T^{8} - 2500 T^{9} + 3125 T^{10} )^{2} $)
$7$	($ 1 - T + 7 T^{2} $)($ 1 + T - 12 T^{2} + 8 T^{3} + 62 T^{4} - 75 T^{5} + 434 T^{6} + 392 T^{7} - 4116 T^{8} + 2401 T^{9} + 16807 T^{10} $)
$11$	($ ( 1 + 5 T + 11 T^{2} )^{2} $)($ ( 1 - 4 T + 47 T^{2} - 161 T^{3} + 958 T^{4} - 2589 T^{5} + 10538 T^{6} - 19481 T^{7} + 62557 T^{8} - 58564 T^{9} + 161051 T^{10} )^{2} $)
$13$	($ ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) $)($ 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} $)
$17$	($ 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} $)($ 1 + 12 T + 14 T^{2} - 192 T^{3} + 1185 T^{4} + 11847 T^{5} - 6180 T^{6} - 65736 T^{7} + 1002861 T^{8} + 2436261 T^{9} - 7749777 T^{10} + 41416437 T^{11} + 289826829 T^{12} - 322960968 T^{13} - 516159780 T^{14} + 16821045879 T^{15} + 28603019265 T^{16} - 78785025216 T^{17} + 97660604174 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20} $)
$19$	($ ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) $)($ 1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} $)
$23$	($ ( 1 + 3 T + 23 T^{2} )^{2} $)($ ( 1 - 3 T + 52 T^{2} - 225 T^{3} + 2023 T^{4} - 5565 T^{5} + 46529 T^{6} - 119025 T^{7} + 632684 T^{8} - 839523 T^{9} + 6436343 T^{10} )^{2} $)
$29$	($ 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} $)($ 1 + 7 T - 76 T^{2} - 419 T^{3} + 4561 T^{4} + 15146 T^{5} - 199563 T^{6} - 341373 T^{7} + 6918636 T^{8} + 2570041 T^{9} - 219913241 T^{10} + 74531189 T^{11} + 5818572876 T^{12} - 8325746097 T^{13} - 141147118203 T^{14} + 310661862754 T^{15} + 2712989167081 T^{16} - 7227698173471 T^{17} - 38018727385036 T^{18} + 101550021831083 T^{19} + 420707233300201 T^{20} $)
$31$	($ 1 - 31 T^{2} + 961 T^{4} $)($ 1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 3619498 T^{9} - 734820027 T^{10} + 112204438 T^{11} + 20622592954 T^{12} - 7301237862 T^{13} - 465437037101 T^{14} + 240112689437 T^{15} + 8237809167042 T^{16} - 5887699419754 T^{17} - 106611379680125 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20} $)
$37$	($ 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} $)($ 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} $)
$41$	($ 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} $)($ 1 + 5 T - 136 T^{2} - 733 T^{3} + 10507 T^{4} + 54412 T^{5} - 554055 T^{6} - 2345451 T^{7} + 23706084 T^{8} + 41392439 T^{9} - 952045937 T^{10} + 1697089999 T^{11} + 39849927204 T^{12} - 161650828371 T^{13} - 1565627010855 T^{14} + 6303967608812 T^{15} + 49909345260187 T^{16} - 142754882754773 T^{17} - 1085949831160456 T^{18} + 1636909671969805 T^{19} + 13422659310152401 T^{20} $)
$43$	($ 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} $)($ 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} $)
$47$	($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 + 27 T + 281 T^{2} + 1758 T^{3} + 13050 T^{4} + 78783 T^{5} - 25248 T^{6} - 1518381 T^{7} + 9454350 T^{8} + 53043051 T^{9} - 242331903 T^{10} + 2493023397 T^{11} + 20884659150 T^{12} - 157642870563 T^{13} - 123202185888 T^{14} + 18068487686481 T^{15} + 140668760043450 T^{16} + 890643445773954 T^{17} + 6690971551954841 T^{18} + 30216522773774709 T^{19} + 52599132235830049 T^{20} $)
$53$	($ 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} $)($ 1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 18025928898 T^{11} + 311723881722 T^{12} + 563341933134 T^{13} - 4748136394155 T^{14} - 74031893539311 T^{15} + 287383106398614 T^{16} + 1085433093209388 T^{17} + 2552647306865801 T^{18} - 69295035427844793 T^{19} + 174887470365513049 T^{20} $)
$59$	($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 + 30 T + 299 T^{2} + 1644 T^{3} + 26547 T^{4} + 344442 T^{5} + 1635267 T^{6} + 9620487 T^{7} + 170035344 T^{8} + 1056366303 T^{9} + 3109579647 T^{10} + 62325611877 T^{11} + 591893032464 T^{12} + 1975845999573 T^{13} + 19815120570387 T^{14} + 246249955396158 T^{15} + 1119766626567627 T^{16} + 4091343041042436 T^{17} + 43902300843691979 T^{18} + 259889874559648170 T^{19} + 511116753300641401 T^{20} $)
$61$	($ ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) $)($ 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} $)
$67$	($ 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} $)($ 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20} $)
$71$	($ ( 1 - 12 T + 71 T^{2} )^{2} $)($ ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 1113209 T^{6} - 1436685 T^{7} + 66929357 T^{8} - 76235043 T^{9} + 1804229351 T^{10} )^{2} $)
$73$	($ 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} $)($ 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} $)
$79$	($ 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} $)($ 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20} $)
$83$	($ 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} $)($ 1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 37560500118 T^{11} - 259360371831 T^{12} + 12226027968819 T^{13} - 38307122794854 T^{14} - 465299175954375 T^{15} + 8060387965039326 T^{16} + 8059407143919219 T^{17} - 333339250356578068 T^{18} + 1682462297407863627 T^{19} + 15516041187205853449 T^{20} $)
$89$	($ 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} $)($ 1 + 28 T + 104 T^{2} - 1736 T^{3} + 31273 T^{4} + 611939 T^{5} - 1780638 T^{6} - 18973932 T^{7} + 740914101 T^{8} + 3271180573 T^{9} - 40614588329 T^{10} + 291135070997 T^{11} + 5868780594021 T^{12} - 13376033868108 T^{13} - 111721218529758 T^{14} + 3417103755161611 T^{15} + 15542095912223353 T^{16} - 76785597378638344 T^{17} + 409405235793016424 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20} $)
$97$	($ 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} $)($ 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	189.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	=	\( 63 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(48\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
189.2.g.a	\(2\)	\(1.509\)	\(\Q(\sqrt{-3}) \)	None	\(1\)	\(0\)	\(2\)	\(1\)	\(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+q^{5}+(-1+3\zeta_{6})q^{7}+\cdots\)
189.2.g.b	\(10\)	\(1.509\)	10.0.\(\cdots\).1	None	\(-2\)	\(0\)	\(8\)	\(-1\)	\(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(1+\cdots)q^{5}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(189, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(189, [\chi])\) into lower level spaces

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	189.2.g

Level	189
Weight	2
Character orbit	g
Rep. character	\(\chi_{189}(100,\cdot)\)
Character field	\(\Q(\zeta_{3})\)
Dimension	12
Newform subspaces	2
Sturm bound	48
Trace bound	1

$p$	$F_p(T)$
$2$	(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 + 2 T - T^{2} - 4 T^{3} - 2 T^{4} - 2 T^{5} + 6 T^{6} + 21 T^{7} + 6 T^{8} - 13 T^{9} - 5 T^{10} - 26 T^{11} + 24 T^{12} + 168 T^{13} + 96 T^{14} - 64 T^{15} - 128 T^{16} - 512 T^{17} - 256 T^{18} + 1024 T^{19} + 1024 T^{20} \))
$3$	1
$5$	(\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 20 T^{2} - 62 T^{3} + 193 T^{4} - 423 T^{5} + 965 T^{6} - 1550 T^{7} + 2500 T^{8} - 2500 T^{9} + 3125 T^{10} )^{2} \))
$7$	(\( 1 - T + 7 T^{2} \))(\( 1 + T - 12 T^{2} + 8 T^{3} + 62 T^{4} - 75 T^{5} + 434 T^{6} + 392 T^{7} - 4116 T^{8} + 2401 T^{9} + 16807 T^{10} \))
$11$	(\( ( 1 + 5 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} - 161 T^{3} + 958 T^{4} - 2589 T^{5} + 10538 T^{6} - 19481 T^{7} + 62557 T^{8} - 58564 T^{9} + 161051 T^{10} )^{2} \))
$13$	(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \))
$17$	(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 14 T^{2} - 192 T^{3} + 1185 T^{4} + 11847 T^{5} - 6180 T^{6} - 65736 T^{7} + 1002861 T^{8} + 2436261 T^{9} - 7749777 T^{10} + 41416437 T^{11} + 289826829 T^{12} - 322960968 T^{13} - 516159780 T^{14} + 16821045879 T^{15} + 28603019265 T^{16} - 78785025216 T^{17} + 97660604174 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20} \))
$19$	(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \))
$23$	(\( ( 1 + 3 T + 23 T^{2} )^{2} \))(\( ( 1 - 3 T + 52 T^{2} - 225 T^{3} + 2023 T^{4} - 5565 T^{5} + 46529 T^{6} - 119025 T^{7} + 632684 T^{8} - 839523 T^{9} + 6436343 T^{10} )^{2} \))
$29$	(\( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} \))(\( 1 + 7 T - 76 T^{2} - 419 T^{3} + 4561 T^{4} + 15146 T^{5} - 199563 T^{6} - 341373 T^{7} + 6918636 T^{8} + 2570041 T^{9} - 219913241 T^{10} + 74531189 T^{11} + 5818572876 T^{12} - 8325746097 T^{13} - 141147118203 T^{14} + 310661862754 T^{15} + 2712989167081 T^{16} - 7227698173471 T^{17} - 38018727385036 T^{18} + 101550021831083 T^{19} + 420707233300201 T^{20} \))
$31$	(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 3619498 T^{9} - 734820027 T^{10} + 112204438 T^{11} + 20622592954 T^{12} - 7301237862 T^{13} - 465437037101 T^{14} + 240112689437 T^{15} + 8237809167042 T^{16} - 5887699419754 T^{17} - 106611379680125 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20} \))
$37$	(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} \))
$41$	(\( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 + 5 T - 136 T^{2} - 733 T^{3} + 10507 T^{4} + 54412 T^{5} - 554055 T^{6} - 2345451 T^{7} + 23706084 T^{8} + 41392439 T^{9} - 952045937 T^{10} + 1697089999 T^{11} + 39849927204 T^{12} - 161650828371 T^{13} - 1565627010855 T^{14} + 6303967608812 T^{15} + 49909345260187 T^{16} - 142754882754773 T^{17} - 1085949831160456 T^{18} + 1636909671969805 T^{19} + 13422659310152401 T^{20} \))
$43$	(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} \))
$47$	(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 27 T + 281 T^{2} + 1758 T^{3} + 13050 T^{4} + 78783 T^{5} - 25248 T^{6} - 1518381 T^{7} + 9454350 T^{8} + 53043051 T^{9} - 242331903 T^{10} + 2493023397 T^{11} + 20884659150 T^{12} - 157642870563 T^{13} - 123202185888 T^{14} + 18068487686481 T^{15} + 140668760043450 T^{16} + 890643445773954 T^{17} + 6690971551954841 T^{18} + 30216522773774709 T^{19} + 52599132235830049 T^{20} \))
$53$	(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 18025928898 T^{11} + 311723881722 T^{12} + 563341933134 T^{13} - 4748136394155 T^{14} - 74031893539311 T^{15} + 287383106398614 T^{16} + 1085433093209388 T^{17} + 2552647306865801 T^{18} - 69295035427844793 T^{19} + 174887470365513049 T^{20} \))
$59$	(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 30 T + 299 T^{2} + 1644 T^{3} + 26547 T^{4} + 344442 T^{5} + 1635267 T^{6} + 9620487 T^{7} + 170035344 T^{8} + 1056366303 T^{9} + 3109579647 T^{10} + 62325611877 T^{11} + 591893032464 T^{12} + 1975845999573 T^{13} + 19815120570387 T^{14} + 246249955396158 T^{15} + 1119766626567627 T^{16} + 4091343041042436 T^{17} + 43902300843691979 T^{18} + 259889874559648170 T^{19} + 511116753300641401 T^{20} \))
$61$	(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} \))
$67$	(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20} \))
$71$	(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 1113209 T^{6} - 1436685 T^{7} + 66929357 T^{8} - 76235043 T^{9} + 1804229351 T^{10} )^{2} \))
$73$	(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} \))
$79$	(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20} \))
$83$	(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 37560500118 T^{11} - 259360371831 T^{12} + 12226027968819 T^{13} - 38307122794854 T^{14} - 465299175954375 T^{15} + 8060387965039326 T^{16} + 8059407143919219 T^{17} - 333339250356578068 T^{18} + 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \))
$89$	(\( 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} \))(\( 1 + 28 T + 104 T^{2} - 1736 T^{3} + 31273 T^{4} + 611939 T^{5} - 1780638 T^{6} - 18973932 T^{7} + 740914101 T^{8} + 3271180573 T^{9} - 40614588329 T^{10} + 291135070997 T^{11} + 5868780594021 T^{12} - 13376033868108 T^{13} - 111721218529758 T^{14} + 3417103755161611 T^{15} + 15542095912223353 T^{16} - 76785597378638344 T^{17} + 409405235793016424 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))
$97$	(\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \))
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