LMFDB - Space of modular forms of level 36 and weight 6

Defining parameters

Level:	$ N $	=	$ 36\( 36 = 2^{2} \cdot 3^{2} $ \)
Weight:	$ k $	=	$ 6 $
Nonzero newspaces:			$ 4 $
Newform subspaces:			$ 6 $
Sturm bound:			$432$
Trace bound:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(\Gamma_1(36))$.

	Total	New	Old
Modular forms	200	86	114
Cusp forms	160	78	82
Eisenstein series	40	8	32

Trace form

$ 78q - 3q^{2} + 12q^{3} - 21q^{4} - 81q^{5} - 27q^{6} + 177q^{7} + 30q^{9} + O(q^{10}) $

\( 78q - 3q^{2} + 12q^{3} - 21q^{4} - 81q^{5} - 27q^{6} + 177q^{7} + 30q^{9} + 600q^{10} - 363q^{11} - 486q^{12} + 369q^{13} - 1518q^{14} + 117q^{15} - 2169q^{16} + 1686q^{17} + 1992q^{18} - 1428q^{19} - 1242q^{20} + 123q^{21} + 10047q^{22} + 4503q^{23} + 2235q^{24} + 864q^{25} - 7128q^{27} - 25284q^{28} - 17385q^{29} - 6882q^{30} - 3309q^{31} - 7233q^{32} + 26643q^{33} + 53997q^{34} + 41898q^{35} + 6399q^{36} - 540q^{37} - 14877q^{38} + 5529q^{39} - 78078q^{40} - 79197q^{41} + 18564q^{42} - 13983q^{43} - 64131q^{45} + 57600q^{46} - 23859q^{47} - 5931q^{48} + 102462q^{49} + 38631q^{50} + 90612q^{51} - 55764q^{52} + 97350q^{53} + 37587q^{54} + 43938q^{55} + 21186q^{56} - 63852q^{57} + 78738q^{58} - 82869q^{59} + 60930q^{60} - 136287q^{61} - 198255q^{63} - 186486q^{64} - 89865q^{65} - 47838q^{66} + 183q^{67} + 31413q^{68} + 212571q^{69} - 17892q^{70} + 276240q^{71} - 130941q^{72} + 286068q^{73} - 20406q^{74} + 44640q^{75} + 76881q^{76} - 138285q^{77} - 96684q^{78} - 147483q^{79} - 418962q^{81} - 263610q^{82} - 296667q^{83} + 141630q^{84} - 393030q^{85} + 279237q^{86} + 397323q^{87} + 226515q^{88} + 568578q^{89} + 235278q^{90} + 445386q^{91} + 435804q^{92} + 321879q^{93} - 223752q^{94} - 439908q^{95} - 37476q^{96} + 277113q^{97} - 697239q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{6}^{\mathrm{new}}(\Gamma_1(36))$

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
36.6.a	$\chi_{36}(1, \cdot)$	36.6.a.a	1	1
36.6.a	$\chi_{36}(1, \cdot)$	36.6.a.b	1	1
36.6.b	$\chi_{36}(35, \cdot)$	36.6.b.a	2	1
36.6.b	$\chi_{36}(35, \cdot)$	36.6.b.b	8	1
36.6.e	$\chi_{36}(13, \cdot)$	36.6.e.a	10	2
36.6.h	$\chi_{36}(11, \cdot)$	36.6.h.a	56	2

Decomposition of $S_{6}^{\mathrm{old}}(\Gamma_1(36))$ into lower level spaces

$ S_{6}^{\mathrm{old}}(\Gamma_1(36)) \cong $ $S_{6}^{\mathrm{new}}(\Gamma_1(3))$$^{\oplus 6}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$^{\oplus 3}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_1(6))$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_1(9))$$^{\oplus 3}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_1(12))$$^{\oplus 2}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_1(18))$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + 32 T^{2} $)($ 1 - 22 T^{2} + 1296 T^{4} - 22528 T^{6} + 1048576 T^{8} $)
$3$	($ 1 - 12 T + 66 T^{2} + 2160 T^{3} + 72981 T^{4} - 1732104 T^{5} + 17734383 T^{6} + 127545840 T^{7} + 947027862 T^{8} - 41841412812 T^{9} + 847288609443 T^{10} $)
$5$	($ 1 + 54 T + 3125 T^{2} $)($ 1 + 3125 T^{2} $)($ 1 + 6232 T^{2} + 9765625 T^{4} $)($ ( 1 - 10048 T^{2} + 43375026 T^{4} - 98125000000 T^{6} + 95367431640625 T^{8} )^{2} $)($ 1 + 21 T - 5203 T^{2} - 519930 T^{3} + 14035794 T^{4} + 2854822770 T^{5} + 76722872007 T^{6} - 9761967315441 T^{7} - 599011867854189 T^{8} + 11924309583255600 T^{9} + 2533145723872694124 T^{10} + 37263467447673750000 T^{11} - $$58\!\cdots\!25$$ T^{12} - $$29\!\cdots\!25$$ T^{13} + $$73\!\cdots\!75$$ T^{14} + $$85\!\cdots\!50$$ T^{15} + $$13\!\cdots\!50$$ T^{16} - $$15\!\cdots\!50$$ T^{17} - $$47\!\cdots\!75$$ T^{18} + $$59\!\cdots\!25$$ T^{19} + $$88\!\cdots\!25$$ T^{20} $)
$7$	($ 1 + 88 T + 16807 T^{2} $)($ 1 - 236 T + 16807 T^{2} $)($ ( 1 - 16807 T^{2} )^{2} $)($ ( 1 - 34780 T^{2} + 609010470 T^{4} - 9824489160220 T^{6} + 79792266297612001 T^{8} )^{2} $)($ 1 - 29 T - 39569 T^{2} + 3762444 T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 2769093584103 T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} - $$69\!\cdots\!60$$ T^{10} + $$26\!\cdots\!24$$ T^{11} + $$67\!\cdots\!43$$ T^{12} - $$26\!\cdots\!39$$ T^{13} - $$22\!\cdots\!03$$ T^{14} - $$10\!\cdots\!72$$ T^{15} + $$99\!\cdots\!64$$ T^{16} + $$14\!\cdots\!92$$ T^{17} - $$25\!\cdots\!69$$ T^{18} - $$31\!\cdots\!03$$ T^{19} + $$17\!\cdots\!49$$ T^{20} $)
$11$	($ 1 + 540 T + 161051 T^{2} $)($ 1 + 161051 T^{2} $)($ ( 1 + 161051 T^{2} )^{2} $)($ ( 1 + 35564 T^{2} + 14255301654 T^{4} + 922438568509964 T^{6} + $$67\!\cdots\!01$$ T^{8} )^{2} $)($ 1 - 177 T - 396232 T^{2} - 71434269 T^{3} + 104816625882 T^{4} + 33726096455301 T^{5} - 6913987980717606 T^{6} - 8552599160812456257 T^{7} - $$10\!\cdots\!51$$ T^{8} + $$50\!\cdots\!82$$ T^{9} + $$47\!\cdots\!16$$ T^{10} + $$81\!\cdots\!82$$ T^{11} - $$27\!\cdots\!51$$ T^{12} - $$35\!\cdots\!07$$ T^{13} - $$46\!\cdots\!06$$ T^{14} + $$36\!\cdots\!51$$ T^{15} + $$18\!\cdots\!82$$ T^{16} - $$20\!\cdots\!19$$ T^{17} - $$17\!\cdots\!32$$ T^{18} - $$12\!\cdots\!27$$ T^{19} + $$11\!\cdots\!01$$ T^{20} $)
$13$	($ 1 + 418 T + 371293 T^{2} $)($ 1 - 1202 T + 371293 T^{2} $)($ ( 1 + 244 T + 371293 T^{2} )^{2} $)($ ( 1 - 64 T + 59178 T^{2} - 23762752 T^{3} + 137858491849 T^{4} )^{4} $)($ 1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} + $$98\!\cdots\!51$$ T^{8} + $$77\!\cdots\!72$$ T^{9} - $$41\!\cdots\!24$$ T^{10} + $$28\!\cdots\!96$$ T^{11} + $$13\!\cdots\!99$$ T^{12} - $$41\!\cdots\!29$$ T^{13} - $$36\!\cdots\!73$$ T^{14} - $$59\!\cdots\!66$$ T^{15} + $$11\!\cdots\!26$$ T^{16} + $$14\!\cdots\!74$$ T^{17} - $$36\!\cdots\!31$$ T^{18} + $$24\!\cdots\!33$$ T^{19} + $$49\!\cdots\!49$$ T^{20} $)
$17$	($ 1 + 594 T + 1419857 T^{2} $)($ 1 + 1419857 T^{2} $)($ 1 + 1811536 T^{2} + 2015993900449 T^{4} $)($ ( 1 - 4370176 T^{2} + 8533712933442 T^{4} - 8810248159888609024 T^{6} + $$40\!\cdots\!01$$ T^{8} )^{2} $)($ ( 1 - 1140 T + 4980550 T^{2} - 3443850354 T^{3} + 10068870522169 T^{4} - 5069379208548852 T^{5} + 14296356292995309833 T^{6} - $$69\!\cdots\!46$$ T^{7} + $$14\!\cdots\!50$$ T^{8} - $$46\!\cdots\!40$$ T^{9} + $$57\!\cdots\!57$$ T^{10} )^{2} $)
$19$	($ 1 - 836 T + 2476099 T^{2} $)($ 1 + 1432 T + 2476099 T^{2} $)($ ( 1 - 2476099 T^{2} )^{2} $)($ ( 1 + 1154804 T^{2} + 7850056485174 T^{4} + 7080179838773626004 T^{6} + $$37\!\cdots\!01$$ T^{8} )^{2} $)($ ( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + $$38\!\cdots\!44$$ T^{7} + $$76\!\cdots\!42$$ T^{8} + $$15\!\cdots\!16$$ T^{9} + $$93\!\cdots\!99$$ T^{10} )^{2} $)
$23$	($ 1 - 4104 T + 6436343 T^{2} $)($ 1 + 6436343 T^{2} $)($ ( 1 + 6436343 T^{2} )^{2} $)($ ( 1 + 2787932 T^{2} - 22939788268314 T^{4} + $$11\!\cdots\!68$$ T^{6} + $$17\!\cdots\!01$$ T^{8} )^{2} $)($ 1 - 399 T - 16077241 T^{2} - 38108825820 T^{3} + 155650662506976 T^{4} + 562944417983120520 T^{5} - 48522958516353490863 T^{6} - $$48\!\cdots\!51$$ T^{7} - $$71\!\cdots\!41$$ T^{8} + $$11\!\cdots\!52$$ T^{9} + $$81\!\cdots\!80$$ T^{10} + $$74\!\cdots\!36$$ T^{11} - $$29\!\cdots\!09$$ T^{12} - $$12\!\cdots\!57$$ T^{13} - $$83\!\cdots\!63$$ T^{14} + $$62\!\cdots\!60$$ T^{15} + $$11\!\cdots\!24$$ T^{16} - $$17\!\cdots\!40$$ T^{17} - $$47\!\cdots\!41$$ T^{18} - $$75\!\cdots\!57$$ T^{19} + $$12\!\cdots\!49$$ T^{20} $)
$29$	($ 1 - 594 T + 20511149 T^{2} $)($ 1 + 20511149 T^{2} $)($ 1 - 25263800 T^{2} + 420707233300201 T^{4} $)($ ( 1 - 38569024 T^{2} + 1205003146164114 T^{4} - $$16\!\cdots\!24$$ T^{6} + $$17\!\cdots\!01$$ T^{8} )^{2} $)($ 1 + 6033 T + 3652157 T^{2} + 31641196734 T^{3} + 283528398607854 T^{4} + 668469168127712358 T^{5} + $$64\!\cdots\!39$$ T^{6} + $$82\!\cdots\!55$$ T^{7} - $$90\!\cdots\!17$$ T^{8} + $$14\!\cdots\!60$$ T^{9} + $$58\!\cdots\!16$$ T^{10} + $$29\!\cdots\!40$$ T^{11} - $$38\!\cdots\!17$$ T^{12} + $$71\!\cdots\!95$$ T^{13} + $$11\!\cdots\!39$$ T^{14} + $$24\!\cdots\!42$$ T^{15} + $$21\!\cdots\!54$$ T^{16} + $$48\!\cdots\!66$$ T^{17} + $$11\!\cdots\!57$$ T^{18} + $$38\!\cdots\!17$$ T^{19} + $$13\!\cdots\!01$$ T^{20} $)
$31$	($ 1 - 4256 T + 28629151 T^{2} $)($ 1 + 10324 T + 28629151 T^{2} $)($ ( 1 - 28629151 T^{2} )^{2} $)($ ( 1 - 75666364 T^{2} + 2821679963807238 T^{4} - $$62\!\cdots\!64$$ T^{6} + $$67\!\cdots\!01$$ T^{8} )^{2} $)($ 1 - 2759 T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} - $$32\!\cdots\!47$$ T^{6} - $$42\!\cdots\!39$$ T^{7} - $$13\!\cdots\!49$$ T^{8} + $$36\!\cdots\!48$$ T^{9} + $$63\!\cdots\!24$$ T^{10} + $$10\!\cdots\!48$$ T^{11} - $$11\!\cdots\!49$$ T^{12} - $$99\!\cdots\!89$$ T^{13} - $$21\!\cdots\!47$$ T^{14} + $$25\!\cdots\!20$$ T^{15} + $$11\!\cdots\!00$$ T^{16} - $$29\!\cdots\!72$$ T^{17} - $$24\!\cdots\!77$$ T^{18} - $$35\!\cdots\!09$$ T^{19} + $$36\!\cdots\!01$$ T^{20} $)
$37$	($ 1 + 298 T + 69343957 T^{2} $)($ 1 - 16550 T + 69343957 T^{2} $)($ ( 1 - 12242 T + 69343957 T^{2} )^{2} $)($ ( 1 + 6524 T + 143741358 T^{2} + 452399975468 T^{3} + 4808584372417849 T^{4} )^{4} $)($ ( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + $$13\!\cdots\!82$$ T^{6} + $$38\!\cdots\!04$$ T^{7} + $$67\!\cdots\!49$$ T^{8} + $$17\!\cdots\!86$$ T^{9} + $$16\!\cdots\!57$$ T^{10} )^{2} $)
$41$	($ 1 + 17226 T + 115856201 T^{2} $)($ 1 + 115856201 T^{2} $)($ 1 - 103744400 T^{2} + 13422659310152401 T^{4} $)($ ( 1 - 314275840 T^{2} + 51537381426570402 T^{4} - $$42\!\cdots\!40$$ T^{6} + $$18\!\cdots\!01$$ T^{8} )^{2} $)($ 1 + 18435 T - 117679042 T^{2} - 4344492069675 T^{3} - 505249106564622 T^{4} + $$52\!\cdots\!97$$ T^{5} + $$16\!\cdots\!40$$ T^{6} - $$39\!\cdots\!43$$ T^{7} - $$31\!\cdots\!03$$ T^{8} + $$10\!\cdots\!42$$ T^{9} + $$29\!\cdots\!40$$ T^{10} + $$11\!\cdots\!42$$ T^{11} - $$41\!\cdots\!03$$ T^{12} - $$62\!\cdots\!43$$ T^{13} + $$30\!\cdots\!40$$ T^{14} + $$10\!\cdots\!97$$ T^{15} - $$12\!\cdots\!22$$ T^{16} - $$12\!\cdots\!75$$ T^{17} - $$38\!\cdots\!42$$ T^{18} + $$69\!\cdots\!35$$ T^{19} + $$43\!\cdots\!01$$ T^{20} $)
$43$	($ 1 + 12100 T + 147008443 T^{2} $)($ 1 + 3352 T + 147008443 T^{2} $)($ ( 1 - 147008443 T^{2} )^{2} $)($ ( 1 - 278633452 T^{2} + 39481856607804822 T^{4} - $$60\!\cdots\!48$$ T^{6} + $$46\!\cdots\!01$$ T^{8} )^{2} $)($ 1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} - $$75\!\cdots\!27$$ T^{5} + $$55\!\cdots\!62$$ T^{6} - $$12\!\cdots\!57$$ T^{7} - $$29\!\cdots\!39$$ T^{8} + $$61\!\cdots\!62$$ T^{9} - $$11\!\cdots\!84$$ T^{10} + $$90\!\cdots\!66$$ T^{11} - $$64\!\cdots\!11$$ T^{12} - $$38\!\cdots\!99$$ T^{13} + $$26\!\cdots\!62$$ T^{14} - $$52\!\cdots\!61$$ T^{15} + $$12\!\cdots\!66$$ T^{16} + $$59\!\cdots\!89$$ T^{17} - $$59\!\cdots\!36$$ T^{18} - $$47\!\cdots\!67$$ T^{19} + $$47\!\cdots\!49$$ T^{20} $)
$47$	($ 1 - 1296 T + 229345007 T^{2} $)($ 1 + 229345007 T^{2} $)($ ( 1 + 229345007 T^{2} )^{2} $)($ ( 1 + 457289276 T^{2} + 104560114643041734 T^{4} + $$24\!\cdots\!24$$ T^{6} + $$27\!\cdots\!01$$ T^{8} )^{2} $)($ 1 + 25155 T - 401246233 T^{2} - 14349179861244 T^{3} + 97557609874842960 T^{4} + $$41\!\cdots\!12$$ T^{5} - $$25\!\cdots\!27$$ T^{6} - $$55\!\cdots\!85$$ T^{7} + $$12\!\cdots\!11$$ T^{8} + $$42\!\cdots\!56$$ T^{9} - $$37\!\cdots\!60$$ T^{10} + $$96\!\cdots\!92$$ T^{11} + $$67\!\cdots\!39$$ T^{12} - $$67\!\cdots\!55$$ T^{13} - $$71\!\cdots\!27$$ T^{14} + $$26\!\cdots\!84$$ T^{15} + $$14\!\cdots\!40$$ T^{16} - $$47\!\cdots\!92$$ T^{17} - $$30\!\cdots\!33$$ T^{18} + $$44\!\cdots\!85$$ T^{19} + $$40\!\cdots\!49$$ T^{20} $)
$53$	($ 1 + 19494 T + 418195493 T^{2} $)($ 1 + 418195493 T^{2} $)($ 1 - 293539736 T^{2} + 174887470365513049 T^{4} $)($ ( 1 - 906737728 T^{2} + 546218780733568626 T^{4} - $$15\!\cdots\!72$$ T^{6} + $$30\!\cdots\!01$$ T^{8} )^{2} $)($ ( 1 - 58422 T + 3354568213 T^{2} - 110313236959296 T^{3} + 3390725554692289246 T^{4} - $$71\!\cdots\!28$$ T^{5} + $$14\!\cdots\!78$$ T^{6} - $$19\!\cdots\!04$$ T^{7} + $$24\!\cdots\!41$$ T^{8} - $$17\!\cdots\!22$$ T^{9} + $$12\!\cdots\!93$$ T^{10} )^{2} $)
$59$	($ 1 - 7668 T + 714924299 T^{2} $)($ 1 + 714924299 T^{2} $)($ ( 1 + 714924299 T^{2} )^{2} $)($ ( 1 + 2486042156 T^{2} + 2533431575735206614 T^{4} + $$12\!\cdots\!56$$ T^{6} + $$26\!\cdots\!01$$ T^{8} )^{2} $)($ 1 + 90537 T + 2831117840 T^{2} + 13805150996349 T^{3} - 966660594685472478 T^{4} - $$10\!\cdots\!09$$ T^{5} + $$69\!\cdots\!78$$ T^{6} + $$39\!\cdots\!93$$ T^{7} + $$84\!\cdots\!01$$ T^{8} - $$17\!\cdots\!74$$ T^{9} - $$12\!\cdots\!20$$ T^{10} - $$12\!\cdots\!26$$ T^{11} + $$42\!\cdots\!01$$ T^{12} + $$14\!\cdots\!07$$ T^{13} + $$18\!\cdots\!78$$ T^{14} - $$19\!\cdots\!91$$ T^{15} - $$12\!\cdots\!78$$ T^{16} + $$13\!\cdots\!51$$ T^{17} + $$19\!\cdots\!40$$ T^{18} + $$44\!\cdots\!63$$ T^{19} + $$34\!\cdots\!01$$ T^{20} $)
$61$	($ 1 + 34738 T + 844596301 T^{2} $)($ 1 + 38626 T + 844596301 T^{2} $)($ ( 1 - 18950 T + 844596301 T^{2} )^{2} $)($ ( 1 + 25556 T + 1664516478 T^{2} + 21584503068356 T^{3} + 713342911662882601 T^{4} )^{4} $)($ 1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} + $$54\!\cdots\!90$$ T^{5} - $$10\!\cdots\!33$$ T^{6} - $$70\!\cdots\!89$$ T^{7} + $$11\!\cdots\!67$$ T^{8} + $$32\!\cdots\!04$$ T^{9} - $$10\!\cdots\!12$$ T^{10} + $$27\!\cdots\!04$$ T^{11} + $$80\!\cdots\!67$$ T^{12} - $$42\!\cdots\!89$$ T^{13} - $$51\!\cdots\!33$$ T^{14} + $$23\!\cdots\!90$$ T^{15} + $$25\!\cdots\!98$$ T^{16} - $$13\!\cdots\!46$$ T^{17} - $$91\!\cdots\!83$$ T^{18} - $$30\!\cdots\!03$$ T^{19} + $$18\!\cdots\!01$$ T^{20} $)
$67$	($ 1 - 21812 T + 1350125107 T^{2} $)($ 1 + 35536 T + 1350125107 T^{2} $)($ ( 1 - 1350125107 T^{2} )^{2} $)($ ( 1 - 511854028 T^{2} + 1402554319081014006 T^{4} - $$93\!\cdots\!72$$ T^{6} + $$33\!\cdots\!01$$ T^{8} )^{2} $)($ 1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} + $$30\!\cdots\!39$$ T^{5} - $$79\!\cdots\!54$$ T^{6} - $$69\!\cdots\!51$$ T^{7} - $$33\!\cdots\!67$$ T^{8} + $$37\!\cdots\!46$$ T^{9} + $$22\!\cdots\!76$$ T^{10} + $$51\!\cdots\!22$$ T^{11} - $$60\!\cdots\!83$$ T^{12} - $$16\!\cdots\!93$$ T^{13} - $$26\!\cdots\!54$$ T^{14} + $$13\!\cdots\!73$$ T^{15} + $$60\!\cdots\!70$$ T^{16} - $$63\!\cdots\!29$$ T^{17} - $$42\!\cdots\!44$$ T^{18} - $$20\!\cdots\!49$$ T^{19} + $$20\!\cdots\!49$$ T^{20} $)
$71$	($ 1 - 46872 T + 1804229351 T^{2} $)($ 1 + 1804229351 T^{2} $)($ ( 1 + 1804229351 T^{2} )^{2} $)($ ( 1 + 5059927580 T^{2} + 12401663987882134950 T^{4} + $$16\!\cdots\!80$$ T^{6} + $$10\!\cdots\!01$$ T^{8} )^{2} $)($ ( 1 - 114684 T + 7758380659 T^{2} - 426246123888336 T^{3} + 19260501229393543450 T^{4} - $$77\!\cdots\!40$$ T^{5} + $$34\!\cdots\!50$$ T^{6} - $$13\!\cdots\!36$$ T^{7} + $$45\!\cdots\!09$$ T^{8} - $$12\!\cdots\!84$$ T^{9} + $$19\!\cdots\!51$$ T^{10} )^{2} $)
$73$	($ 1 - 67562 T + 2073071593 T^{2} $)($ 1 + 1450 T + 2073071593 T^{2} $)($ ( 1 - 20144 T + 2073071593 T^{2} )^{2} $)($ ( 1 - 27712 T + 4150178514 T^{2} - 57448959985216 T^{3} + 4297625829703557649 T^{4} )^{4} $)($ ( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - $$80\!\cdots\!84$$ T^{5} + $$25\!\cdots\!33$$ T^{6} - $$13\!\cdots\!86$$ T^{7} + $$32\!\cdots\!22$$ T^{8} - $$14\!\cdots\!00$$ T^{9} + $$38\!\cdots\!93$$ T^{10} )^{2} $)
$79$	($ 1 + 76912 T + 3077056399 T^{2} $)($ 1 + 100564 T + 3077056399 T^{2} $)($ ( 1 - 3077056399 T^{2} )^{2} $)($ ( 1 - 9285769084 T^{2} + 40487319189030666438 T^{4} - $$87\!\cdots\!84$$ T^{6} + $$89\!\cdots\!01$$ T^{8} )^{2} $)($ 1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} + $$21\!\cdots\!52$$ T^{5} + $$27\!\cdots\!85$$ T^{6} - $$84\!\cdots\!45$$ T^{7} - $$35\!\cdots\!45$$ T^{8} + $$81\!\cdots\!80$$ T^{9} + $$16\!\cdots\!00$$ T^{10} + $$25\!\cdots\!20$$ T^{11} - $$33\!\cdots\!45$$ T^{12} - $$24\!\cdots\!55$$ T^{13} + $$24\!\cdots\!85$$ T^{14} + $$60\!\cdots\!48$$ T^{15} + $$22\!\cdots\!36$$ T^{16} - $$93\!\cdots\!32$$ T^{17} - $$43\!\cdots\!29$$ T^{18} - $$74\!\cdots\!07$$ T^{19} + $$76\!\cdots\!01$$ T^{20} $)
$83$	($ 1 + 67716 T + 3939040643 T^{2} $)($ 1 + 3939040643 T^{2} $)($ ( 1 + 3939040643 T^{2} )^{2} $)($ ( 1 + 8833858700 T^{2} + 50333055349962951990 T^{4} + $$13\!\cdots\!00$$ T^{6} + $$24\!\cdots\!01$$ T^{8} )^{2} $)($ 1 + 228951 T + 21403431983 T^{2} + 1202282302650156 T^{3} + 62567029919071222368 T^{4} + $$36\!\cdots\!68$$ T^{5} + $$11\!\cdots\!01$$ T^{6} - $$11\!\cdots\!41$$ T^{7} - $$18\!\cdots\!73$$ T^{8} - $$14\!\cdots\!84$$ T^{9} - $$88\!\cdots\!72$$ T^{10} - $$55\!\cdots\!12$$ T^{11} - $$28\!\cdots\!77$$ T^{12} - $$67\!\cdots\!87$$ T^{13} + $$28\!\cdots\!01$$ T^{14} + $$34\!\cdots\!24$$ T^{15} + $$23\!\cdots\!32$$ T^{16} + $$17\!\cdots\!92$$ T^{17} + $$12\!\cdots\!83$$ T^{18} + $$52\!\cdots\!93$$ T^{19} + $$89\!\cdots\!49$$ T^{20} $)
$89$	($ 1 + 29754 T + 5584059449 T^{2} $)($ 1 + 5584059449 T^{2} $)($ 1 - 7170687200 T^{2} + 31181719929966183601 T^{4} $)($ ( 1 - 18591086080 T^{2} + $$14\!\cdots\!34$$ T^{4} - $$57\!\cdots\!80$$ T^{6} + $$97\!\cdots\!01$$ T^{8} )^{2} $)($ ( 1 - 299166 T + 52616244181 T^{2} - 6660261403977288 T^{3} + $$67\!\cdots\!10$$ T^{4} - $$55\!\cdots\!64$$ T^{5} + $$37\!\cdots\!90$$ T^{6} - $$20\!\cdots\!88$$ T^{7} + $$91\!\cdots\!69$$ T^{8} - $$29\!\cdots\!66$$ T^{9} + $$54\!\cdots\!49$$ T^{10} )^{2} $)
$97$	($ 1 + 122398 T + 8587340257 T^{2} $)($ 1 + 134386 T + 8587340257 T^{2} $)($ ( 1 - 160808 T + 8587340257 T^{2} )^{2} $)($ ( 1 - 57472 T + 2365161858 T^{2} - 493531619250304 T^{3} + 73742412689492826049 T^{4} )^{4} $)($ 1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} - $$30\!\cdots\!95$$ T^{5} + $$10\!\cdots\!20$$ T^{6} + $$21\!\cdots\!29$$ T^{7} - $$20\!\cdots\!15$$ T^{8} - $$65\!\cdots\!06$$ T^{9} + $$19\!\cdots\!00$$ T^{10} - $$56\!\cdots\!42$$ T^{11} - $$15\!\cdots\!35$$ T^{12} + $$13\!\cdots\!97$$ T^{13} + $$56\!\cdots\!20$$ T^{14} - $$14\!\cdots\!15$$ T^{15} + $$39\!\cdots\!90$$ T^{16} + $$77\!\cdots\!49$$ T^{17} - $$52\!\cdots\!38$$ T^{18} - $$10\!\cdots\!37$$ T^{19} + $$21\!\cdots\!49$$ T^{20} $)
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Overview	Features
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	=	\( 36\( 36 = 2^{2} \cdot 3^{2} \) \)
Weight:	\( k \)	=	\( 6 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 6 \)
Sturm bound:			\(432\)
Trace bound:			\(3\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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Learn more about

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(36))\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(36))\) 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	36.6

Level	36
Weight	6
Dimension	78
Nonzero newspaces	4
Newform subspaces	6
Sturm bound	432
Trace bound	3

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
36.6.a	\(\chi_{36}(1, \cdot)\)	36.6.a.a	1	1
36.6.a	\(\chi_{36}(1, \cdot)\)	36.6.a.b	1	1
36.6.b	\(\chi_{36}(35, \cdot)\)	36.6.b.a	2	1
36.6.b	\(\chi_{36}(35, \cdot)\)	36.6.b.b	8	1
36.6.e	\(\chi_{36}(13, \cdot)\)	36.6.e.a	10	2
36.6.h	\(\chi_{36}(11, \cdot)\)	36.6.h.a	56	2

$p$	$F_p(T)$
$2$	(\( 1 + 32 T^{2} \))(\( 1 - 22 T^{2} + 1296 T^{4} - 22528 T^{6} + 1048576 T^{8} \))
$3$	(\( 1 - 12 T + 66 T^{2} + 2160 T^{3} + 72981 T^{4} - 1732104 T^{5} + 17734383 T^{6} + 127545840 T^{7} + 947027862 T^{8} - 41841412812 T^{9} + 847288609443 T^{10} \))
$5$	(\( 1 + 54 T + 3125 T^{2} \))(\( 1 + 3125 T^{2} \))(\( 1 + 6232 T^{2} + 9765625 T^{4} \))(\( ( 1 - 10048 T^{2} + 43375026 T^{4} - 98125000000 T^{6} + 95367431640625 T^{8} )^{2} \))(\( 1 + 21 T - 5203 T^{2} - 519930 T^{3} + 14035794 T^{4} + 2854822770 T^{5} + 76722872007 T^{6} - 9761967315441 T^{7} - 599011867854189 T^{8} + 11924309583255600 T^{9} + 2533145723872694124 T^{10} + 37263467447673750000 T^{11} - \)\(58\!\cdots\!25\)\( T^{12} - \)\(29\!\cdots\!25\)\( T^{13} + \)\(73\!\cdots\!75\)\( T^{14} + \)\(85\!\cdots\!50\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(15\!\cdots\!50\)\( T^{17} - \)\(47\!\cdots\!75\)\( T^{18} + \)\(59\!\cdots\!25\)\( T^{19} + \)\(88\!\cdots\!25\)\( T^{20} \))
$7$	(\( 1 + 88 T + 16807 T^{2} \))(\( 1 - 236 T + 16807 T^{2} \))(\( ( 1 - 16807 T^{2} )^{2} \))(\( ( 1 - 34780 T^{2} + 609010470 T^{4} - 9824489160220 T^{6} + 79792266297612001 T^{8} )^{2} \))(\( 1 - 29 T - 39569 T^{2} + 3762444 T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 2769093584103 T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} - \)\(69\!\cdots\!60\)\( T^{10} + \)\(26\!\cdots\!24\)\( T^{11} + \)\(67\!\cdots\!43\)\( T^{12} - \)\(26\!\cdots\!39\)\( T^{13} - \)\(22\!\cdots\!03\)\( T^{14} - \)\(10\!\cdots\!72\)\( T^{15} + \)\(99\!\cdots\!64\)\( T^{16} + \)\(14\!\cdots\!92\)\( T^{17} - \)\(25\!\cdots\!69\)\( T^{18} - \)\(31\!\cdots\!03\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \))
$11$	(\( 1 + 540 T + 161051 T^{2} \))(\( 1 + 161051 T^{2} \))(\( ( 1 + 161051 T^{2} )^{2} \))(\( ( 1 + 35564 T^{2} + 14255301654 T^{4} + 922438568509964 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 177 T - 396232 T^{2} - 71434269 T^{3} + 104816625882 T^{4} + 33726096455301 T^{5} - 6913987980717606 T^{6} - 8552599160812456257 T^{7} - \)\(10\!\cdots\!51\)\( T^{8} + \)\(50\!\cdots\!82\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} + \)\(81\!\cdots\!82\)\( T^{11} - \)\(27\!\cdots\!51\)\( T^{12} - \)\(35\!\cdots\!07\)\( T^{13} - \)\(46\!\cdots\!06\)\( T^{14} + \)\(36\!\cdots\!51\)\( T^{15} + \)\(18\!\cdots\!82\)\( T^{16} - \)\(20\!\cdots\!19\)\( T^{17} - \)\(17\!\cdots\!32\)\( T^{18} - \)\(12\!\cdots\!27\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \))
$13$	(\( 1 + 418 T + 371293 T^{2} \))(\( 1 - 1202 T + 371293 T^{2} \))(\( ( 1 + 244 T + 371293 T^{2} )^{2} \))(\( ( 1 - 64 T + 59178 T^{2} - 23762752 T^{3} + 137858491849 T^{4} )^{4} \))(\( 1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} + \)\(98\!\cdots\!51\)\( T^{8} + \)\(77\!\cdots\!72\)\( T^{9} - \)\(41\!\cdots\!24\)\( T^{10} + \)\(28\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!29\)\( T^{13} - \)\(36\!\cdots\!73\)\( T^{14} - \)\(59\!\cdots\!66\)\( T^{15} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(14\!\cdots\!74\)\( T^{17} - \)\(36\!\cdots\!31\)\( T^{18} + \)\(24\!\cdots\!33\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} \))
$17$	(\( 1 + 594 T + 1419857 T^{2} \))(\( 1 + 1419857 T^{2} \))(\( 1 + 1811536 T^{2} + 2015993900449 T^{4} \))(\( ( 1 - 4370176 T^{2} + 8533712933442 T^{4} - 8810248159888609024 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 1140 T + 4980550 T^{2} - 3443850354 T^{3} + 10068870522169 T^{4} - 5069379208548852 T^{5} + 14296356292995309833 T^{6} - \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} - \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} )^{2} \))
$19$	(\( 1 - 836 T + 2476099 T^{2} \))(\( 1 + 1432 T + 2476099 T^{2} \))(\( ( 1 - 2476099 T^{2} )^{2} \))(\( ( 1 + 1154804 T^{2} + 7850056485174 T^{4} + 7080179838773626004 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} )^{2} \))
$23$	(\( 1 - 4104 T + 6436343 T^{2} \))(\( 1 + 6436343 T^{2} \))(\( ( 1 + 6436343 T^{2} )^{2} \))(\( ( 1 + 2787932 T^{2} - 22939788268314 T^{4} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 399 T - 16077241 T^{2} - 38108825820 T^{3} + 155650662506976 T^{4} + 562944417983120520 T^{5} - 48522958516353490863 T^{6} - \)\(48\!\cdots\!51\)\( T^{7} - \)\(71\!\cdots\!41\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(81\!\cdots\!80\)\( T^{10} + \)\(74\!\cdots\!36\)\( T^{11} - \)\(29\!\cdots\!09\)\( T^{12} - \)\(12\!\cdots\!57\)\( T^{13} - \)\(83\!\cdots\!63\)\( T^{14} + \)\(62\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!24\)\( T^{16} - \)\(17\!\cdots\!40\)\( T^{17} - \)\(47\!\cdots\!41\)\( T^{18} - \)\(75\!\cdots\!57\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} \))
$29$	(\( 1 - 594 T + 20511149 T^{2} \))(\( 1 + 20511149 T^{2} \))(\( 1 - 25263800 T^{2} + 420707233300201 T^{4} \))(\( ( 1 - 38569024 T^{2} + 1205003146164114 T^{4} - \)\(16\!\cdots\!24\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 6033 T + 3652157 T^{2} + 31641196734 T^{3} + 283528398607854 T^{4} + 668469168127712358 T^{5} + \)\(64\!\cdots\!39\)\( T^{6} + \)\(82\!\cdots\!55\)\( T^{7} - \)\(90\!\cdots\!17\)\( T^{8} + \)\(14\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!16\)\( T^{10} + \)\(29\!\cdots\!40\)\( T^{11} - \)\(38\!\cdots\!17\)\( T^{12} + \)\(71\!\cdots\!95\)\( T^{13} + \)\(11\!\cdots\!39\)\( T^{14} + \)\(24\!\cdots\!42\)\( T^{15} + \)\(21\!\cdots\!54\)\( T^{16} + \)\(48\!\cdots\!66\)\( T^{17} + \)\(11\!\cdots\!57\)\( T^{18} + \)\(38\!\cdots\!17\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \))
$31$	(\( 1 - 4256 T + 28629151 T^{2} \))(\( 1 + 10324 T + 28629151 T^{2} \))(\( ( 1 - 28629151 T^{2} )^{2} \))(\( ( 1 - 75666364 T^{2} + 2821679963807238 T^{4} - \)\(62\!\cdots\!64\)\( T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 2759 T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} - \)\(32\!\cdots\!47\)\( T^{6} - \)\(42\!\cdots\!39\)\( T^{7} - \)\(13\!\cdots\!49\)\( T^{8} + \)\(36\!\cdots\!48\)\( T^{9} + \)\(63\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{11} - \)\(11\!\cdots\!49\)\( T^{12} - \)\(99\!\cdots\!89\)\( T^{13} - \)\(21\!\cdots\!47\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!00\)\( T^{16} - \)\(29\!\cdots\!72\)\( T^{17} - \)\(24\!\cdots\!77\)\( T^{18} - \)\(35\!\cdots\!09\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \))
$37$	(\( 1 + 298 T + 69343957 T^{2} \))(\( 1 - 16550 T + 69343957 T^{2} \))(\( ( 1 - 12242 T + 69343957 T^{2} )^{2} \))(\( ( 1 + 6524 T + 143741358 T^{2} + 452399975468 T^{3} + 4808584372417849 T^{4} )^{4} \))(\( ( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{2} \))
$41$	(\( 1 + 17226 T + 115856201 T^{2} \))(\( 1 + 115856201 T^{2} \))(\( 1 - 103744400 T^{2} + 13422659310152401 T^{4} \))(\( ( 1 - 314275840 T^{2} + 51537381426570402 T^{4} - \)\(42\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 18435 T - 117679042 T^{2} - 4344492069675 T^{3} - 505249106564622 T^{4} + \)\(52\!\cdots\!97\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(39\!\cdots\!43\)\( T^{7} - \)\(31\!\cdots\!03\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!42\)\( T^{11} - \)\(41\!\cdots\!03\)\( T^{12} - \)\(62\!\cdots\!43\)\( T^{13} + \)\(30\!\cdots\!40\)\( T^{14} + \)\(10\!\cdots\!97\)\( T^{15} - \)\(12\!\cdots\!22\)\( T^{16} - \)\(12\!\cdots\!75\)\( T^{17} - \)\(38\!\cdots\!42\)\( T^{18} + \)\(69\!\cdots\!35\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} \))
$43$	(\( 1 + 12100 T + 147008443 T^{2} \))(\( 1 + 3352 T + 147008443 T^{2} \))(\( ( 1 - 147008443 T^{2} )^{2} \))(\( ( 1 - 278633452 T^{2} + 39481856607804822 T^{4} - \)\(60\!\cdots\!48\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} - \)\(75\!\cdots\!27\)\( T^{5} + \)\(55\!\cdots\!62\)\( T^{6} - \)\(12\!\cdots\!57\)\( T^{7} - \)\(29\!\cdots\!39\)\( T^{8} + \)\(61\!\cdots\!62\)\( T^{9} - \)\(11\!\cdots\!84\)\( T^{10} + \)\(90\!\cdots\!66\)\( T^{11} - \)\(64\!\cdots\!11\)\( T^{12} - \)\(38\!\cdots\!99\)\( T^{13} + \)\(26\!\cdots\!62\)\( T^{14} - \)\(52\!\cdots\!61\)\( T^{15} + \)\(12\!\cdots\!66\)\( T^{16} + \)\(59\!\cdots\!89\)\( T^{17} - \)\(59\!\cdots\!36\)\( T^{18} - \)\(47\!\cdots\!67\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} \))
$47$	(\( 1 - 1296 T + 229345007 T^{2} \))(\( 1 + 229345007 T^{2} \))(\( ( 1 + 229345007 T^{2} )^{2} \))(\( ( 1 + 457289276 T^{2} + 104560114643041734 T^{4} + \)\(24\!\cdots\!24\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 25155 T - 401246233 T^{2} - 14349179861244 T^{3} + 97557609874842960 T^{4} + \)\(41\!\cdots\!12\)\( T^{5} - \)\(25\!\cdots\!27\)\( T^{6} - \)\(55\!\cdots\!85\)\( T^{7} + \)\(12\!\cdots\!11\)\( T^{8} + \)\(42\!\cdots\!56\)\( T^{9} - \)\(37\!\cdots\!60\)\( T^{10} + \)\(96\!\cdots\!92\)\( T^{11} + \)\(67\!\cdots\!39\)\( T^{12} - \)\(67\!\cdots\!55\)\( T^{13} - \)\(71\!\cdots\!27\)\( T^{14} + \)\(26\!\cdots\!84\)\( T^{15} + \)\(14\!\cdots\!40\)\( T^{16} - \)\(47\!\cdots\!92\)\( T^{17} - \)\(30\!\cdots\!33\)\( T^{18} + \)\(44\!\cdots\!85\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} \))
$53$	(\( 1 + 19494 T + 418195493 T^{2} \))(\( 1 + 418195493 T^{2} \))(\( 1 - 293539736 T^{2} + 174887470365513049 T^{4} \))(\( ( 1 - 906737728 T^{2} + 546218780733568626 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 58422 T + 3354568213 T^{2} - 110313236959296 T^{3} + 3390725554692289246 T^{4} - \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} - \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} - \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} )^{2} \))
$59$	(\( 1 - 7668 T + 714924299 T^{2} \))(\( 1 + 714924299 T^{2} \))(\( ( 1 + 714924299 T^{2} )^{2} \))(\( ( 1 + 2486042156 T^{2} + 2533431575735206614 T^{4} + \)\(12\!\cdots\!56\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 90537 T + 2831117840 T^{2} + 13805150996349 T^{3} - 966660594685472478 T^{4} - \)\(10\!\cdots\!09\)\( T^{5} + \)\(69\!\cdots\!78\)\( T^{6} + \)\(39\!\cdots\!93\)\( T^{7} + \)\(84\!\cdots\!01\)\( T^{8} - \)\(17\!\cdots\!74\)\( T^{9} - \)\(12\!\cdots\!20\)\( T^{10} - \)\(12\!\cdots\!26\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} + \)\(14\!\cdots\!07\)\( T^{13} + \)\(18\!\cdots\!78\)\( T^{14} - \)\(19\!\cdots\!91\)\( T^{15} - \)\(12\!\cdots\!78\)\( T^{16} + \)\(13\!\cdots\!51\)\( T^{17} + \)\(19\!\cdots\!40\)\( T^{18} + \)\(44\!\cdots\!63\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \))
$61$	(\( 1 + 34738 T + 844596301 T^{2} \))(\( 1 + 38626 T + 844596301 T^{2} \))(\( ( 1 - 18950 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 25556 T + 1664516478 T^{2} + 21584503068356 T^{3} + 713342911662882601 T^{4} )^{4} \))(\( 1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} + \)\(54\!\cdots\!90\)\( T^{5} - \)\(10\!\cdots\!33\)\( T^{6} - \)\(70\!\cdots\!89\)\( T^{7} + \)\(11\!\cdots\!67\)\( T^{8} + \)\(32\!\cdots\!04\)\( T^{9} - \)\(10\!\cdots\!12\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{11} + \)\(80\!\cdots\!67\)\( T^{12} - \)\(42\!\cdots\!89\)\( T^{13} - \)\(51\!\cdots\!33\)\( T^{14} + \)\(23\!\cdots\!90\)\( T^{15} + \)\(25\!\cdots\!98\)\( T^{16} - \)\(13\!\cdots\!46\)\( T^{17} - \)\(91\!\cdots\!83\)\( T^{18} - \)\(30\!\cdots\!03\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \))
$67$	(\( 1 - 21812 T + 1350125107 T^{2} \))(\( 1 + 35536 T + 1350125107 T^{2} \))(\( ( 1 - 1350125107 T^{2} )^{2} \))(\( ( 1 - 511854028 T^{2} + 1402554319081014006 T^{4} - \)\(93\!\cdots\!72\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} + \)\(30\!\cdots\!39\)\( T^{5} - \)\(79\!\cdots\!54\)\( T^{6} - \)\(69\!\cdots\!51\)\( T^{7} - \)\(33\!\cdots\!67\)\( T^{8} + \)\(37\!\cdots\!46\)\( T^{9} + \)\(22\!\cdots\!76\)\( T^{10} + \)\(51\!\cdots\!22\)\( T^{11} - \)\(60\!\cdots\!83\)\( T^{12} - \)\(16\!\cdots\!93\)\( T^{13} - \)\(26\!\cdots\!54\)\( T^{14} + \)\(13\!\cdots\!73\)\( T^{15} + \)\(60\!\cdots\!70\)\( T^{16} - \)\(63\!\cdots\!29\)\( T^{17} - \)\(42\!\cdots\!44\)\( T^{18} - \)\(20\!\cdots\!49\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} \))
$71$	(\( 1 - 46872 T + 1804229351 T^{2} \))(\( 1 + 1804229351 T^{2} \))(\( ( 1 + 1804229351 T^{2} )^{2} \))(\( ( 1 + 5059927580 T^{2} + 12401663987882134950 T^{4} + \)\(16\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 114684 T + 7758380659 T^{2} - 426246123888336 T^{3} + 19260501229393543450 T^{4} - \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} - \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} )^{2} \))
$73$	(\( 1 - 67562 T + 2073071593 T^{2} \))(\( 1 + 1450 T + 2073071593 T^{2} \))(\( ( 1 - 20144 T + 2073071593 T^{2} )^{2} \))(\( ( 1 - 27712 T + 4150178514 T^{2} - 57448959985216 T^{3} + 4297625829703557649 T^{4} )^{4} \))(\( ( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} )^{2} \))
$79$	(\( 1 + 76912 T + 3077056399 T^{2} \))(\( 1 + 100564 T + 3077056399 T^{2} \))(\( ( 1 - 3077056399 T^{2} )^{2} \))(\( ( 1 - 9285769084 T^{2} + 40487319189030666438 T^{4} - \)\(87\!\cdots\!84\)\( T^{6} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!85\)\( T^{6} - \)\(84\!\cdots\!45\)\( T^{7} - \)\(35\!\cdots\!45\)\( T^{8} + \)\(81\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!20\)\( T^{11} - \)\(33\!\cdots\!45\)\( T^{12} - \)\(24\!\cdots\!55\)\( T^{13} + \)\(24\!\cdots\!85\)\( T^{14} + \)\(60\!\cdots\!48\)\( T^{15} + \)\(22\!\cdots\!36\)\( T^{16} - \)\(93\!\cdots\!32\)\( T^{17} - \)\(43\!\cdots\!29\)\( T^{18} - \)\(74\!\cdots\!07\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \))
$83$	(\( 1 + 67716 T + 3939040643 T^{2} \))(\( 1 + 3939040643 T^{2} \))(\( ( 1 + 3939040643 T^{2} )^{2} \))(\( ( 1 + 8833858700 T^{2} + 50333055349962951990 T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 228951 T + 21403431983 T^{2} + 1202282302650156 T^{3} + 62567029919071222368 T^{4} + \)\(36\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!01\)\( T^{6} - \)\(11\!\cdots\!41\)\( T^{7} - \)\(18\!\cdots\!73\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{9} - \)\(88\!\cdots\!72\)\( T^{10} - \)\(55\!\cdots\!12\)\( T^{11} - \)\(28\!\cdots\!77\)\( T^{12} - \)\(67\!\cdots\!87\)\( T^{13} + \)\(28\!\cdots\!01\)\( T^{14} + \)\(34\!\cdots\!24\)\( T^{15} + \)\(23\!\cdots\!32\)\( T^{16} + \)\(17\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!83\)\( T^{18} + \)\(52\!\cdots\!93\)\( T^{19} + \)\(89\!\cdots\!49\)\( T^{20} \))
$89$	(\( 1 + 29754 T + 5584059449 T^{2} \))(\( 1 + 5584059449 T^{2} \))(\( 1 - 7170687200 T^{2} + 31181719929966183601 T^{4} \))(\( ( 1 - 18591086080 T^{2} + \)\(14\!\cdots\!34\)\( T^{4} - \)\(57\!\cdots\!80\)\( T^{6} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 299166 T + 52616244181 T^{2} - 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} - \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} - \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} - \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} )^{2} \))
$97$	(\( 1 + 122398 T + 8587340257 T^{2} \))(\( 1 + 134386 T + 8587340257 T^{2} \))(\( ( 1 - 160808 T + 8587340257 T^{2} )^{2} \))(\( ( 1 - 57472 T + 2365161858 T^{2} - 493531619250304 T^{3} + 73742412689492826049 T^{4} )^{4} \))(\( 1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} - \)\(30\!\cdots\!95\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!29\)\( T^{7} - \)\(20\!\cdots\!15\)\( T^{8} - \)\(65\!\cdots\!06\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(56\!\cdots\!42\)\( T^{11} - \)\(15\!\cdots\!35\)\( T^{12} + \)\(13\!\cdots\!97\)\( T^{13} + \)\(56\!\cdots\!20\)\( T^{14} - \)\(14\!\cdots\!15\)\( T^{15} + \)\(39\!\cdots\!90\)\( T^{16} + \)\(77\!\cdots\!49\)\( T^{17} - \)\(52\!\cdots\!38\)\( T^{18} - \)\(10\!\cdots\!37\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} \))
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