LMFDB - Space of Modular Forms of Level 16, Weight 4, and Character 16.e

Defining parameters

Level:	$ N $	=	$ 16 = 2^{4} $
Weight:	$ k $	=	$ 4 $
Character orbit:	$[\chi]$	=	16.e (of order $4$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	=	$ 16 $
Character field:			$\Q(i)$
Newform subspaces:			$ 1 $
Sturm bound:			$8$
Trace bound:			$0$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(16, [\chi])$.

	Total	New
Modular forms	14	14
Cusp forms	10	10
Eisenstein series	4	4

Trace form

$ 10q - 2q^{2} - 2q^{3} + 8q^{4} - 2q^{5} - 32q^{6} - 44q^{8} + O(q^{10}) $

\( 10q - 2q^{2} - 2q^{3} + 8q^{4} - 2q^{5} - 32q^{6} - 44q^{8} - 68q^{10} + 18q^{11} + 100q^{12} - 2q^{13} + 188q^{14} - 124q^{15} + 280q^{16} - 4q^{17} + 174q^{18} - 26q^{19} - 196q^{20} + 52q^{21} - 588q^{22} - 848q^{24} - 264q^{26} + 184q^{27} + 280q^{28} - 202q^{29} + 1236q^{30} + 368q^{31} + 968q^{32} - 4q^{33} + 436q^{34} + 476q^{35} - 596q^{36} - 10q^{37} - 1232q^{38} - 1336q^{40} - 680q^{42} - 838q^{43} + 868q^{44} + 194q^{45} + 1132q^{46} - 944q^{47} + 1768q^{48} + 94q^{49} + 726q^{50} - 1500q^{51} - 236q^{52} - 378q^{53} - 1376q^{54} - 488q^{56} + 8q^{58} + 1706q^{59} - 192q^{60} + 910q^{61} - 80q^{62} + 2628q^{63} + 512q^{64} - 492q^{65} - 428q^{66} + 1942q^{67} - 880q^{68} + 580q^{69} + 160q^{70} + 1092q^{72} - 452q^{74} - 2954q^{75} - 1228q^{76} - 268q^{77} - 772q^{78} - 4416q^{79} - 2648q^{80} + 482q^{81} - 704q^{82} - 2562q^{83} + 1960q^{84} - 12q^{85} + 3764q^{86} + 1528q^{88} + 1896q^{90} + 3332q^{91} + 632q^{92} - 2192q^{93} - 3248q^{94} + 6900q^{95} - 4432q^{96} - 4q^{97} + 314q^{98} + 4958q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{4}^{\mathrm{new}}(16, [\chi])$ 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
16.4.e.a	$10$	$0.944$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$-2$	$-2$	$-2$	$0$	$q-\beta _{4}q^{2}-\beta _{5}q^{3}+(1-\beta _{2}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	$ 1 + 2 T - 2 T^{2} + 8 T^{3} - 40 T^{4} - 352 T^{5} - 320 T^{6} + 512 T^{7} - 1024 T^{8} + 8192 T^{9} + 32768 T^{10} $
$3$	$ 1 + 2 T + 2 T^{2} - 42 T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 66792 T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 82248156 T^{11} + 42190146 T^{12} + 1314666936 T^{13} + 1883426904 T^{14} + 10905169320 T^{15} - 221217099219 T^{16} - 439334834526 T^{17} + 564859072962 T^{18} + 15251194969974 T^{19} + 205891132094649 T^{20} $
$5$	$ 1 + 2 T + 2 T^{2} - 966 T^{3} - 13723 T^{4} + 18040 T^{5} + 530104 T^{6} - 11981288 T^{7} - 28535006 T^{8} + 2301854348 T^{9} + 23672040908 T^{10} + 287731793500 T^{11} - 445859468750 T^{12} - 23400953125000 T^{13} + 129419921875000 T^{14} + 550537109375000 T^{15} - 52349090576171875 T^{16} - 460624694824218750 T^{17} + 119209289550781250 T^{18} + 14901161193847656250 T^{19} + $$93\!\cdots\!25$$ T^{20} $
$7$	$ 1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} + $$25\!\cdots\!85$$ T^{16} - $$33\!\cdots\!62$$ T^{18} + $$22\!\cdots\!49$$ T^{20} $
$11$	$ 1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 6887886337838 T^{8} + 101615185776500 T^{9} + 18755914132083020 T^{10} + 135249812268521500 T^{11} - 12202310808546625118 T^{12} - $$73\!\cdots\!28$$ T^{13} + $$17\!\cdots\!44$$ T^{14} + $$33\!\cdots\!00$$ T^{15} + $$22\!\cdots\!57$$ T^{16} - $$90\!\cdots\!74$$ T^{17} + $$15\!\cdots\!42$$ T^{18} - $$23\!\cdots\!78$$ T^{19} + $$17\!\cdots\!01$$ T^{20} $
$13$	$ 1 + 2 T + 2 T^{2} - 45206 T^{3} - 2401451 T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 384228102440 T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 3364607868872346876 T^{11} - 45372173460921944494 T^{12} - $$40\!\cdots\!20$$ T^{13} + $$25\!\cdots\!40$$ T^{14} + $$15\!\cdots\!20$$ T^{15} - $$27\!\cdots\!79$$ T^{16} - $$11\!\cdots\!78$$ T^{17} + $$10\!\cdots\!22$$ T^{18} + $$23\!\cdots\!34$$ T^{19} + $$26\!\cdots\!49$$ T^{20} $
$17$	$ ( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} $
$19$	$ 1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} - $$17\!\cdots\!76$$ T^{11} + $$47\!\cdots\!74$$ T^{12} + $$11\!\cdots\!40$$ T^{13} + $$31\!\cdots\!00$$ T^{14} + $$37\!\cdots\!16$$ T^{15} - $$66\!\cdots\!11$$ T^{16} - $$24\!\cdots\!14$$ T^{17} + $$16\!\cdots\!98$$ T^{18} + $$87\!\cdots\!14$$ T^{19} + $$23\!\cdots\!01$$ T^{20} $
$23$	$ 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - $$18\!\cdots\!84$$ T^{10} + $$19\!\cdots\!94$$ T^{12} - $$16\!\cdots\!40$$ T^{14} + $$94\!\cdots\!93$$ T^{16} - $$36\!\cdots\!26$$ T^{18} + $$71\!\cdots\!49$$ T^{20} $
$29$	$ 1 + 202 T + 20402 T^{2} - 1177934 T^{3} + 398569397 T^{4} + 239164019416 T^{5} + 40873283338616 T^{6} + 2529271278095288 T^{7} + 194871598558001506 T^{8} + $$12\!\cdots\!32$$ T^{9} + $$30\!\cdots\!36$$ T^{10} + $$30\!\cdots\!48$$ T^{11} + $$11\!\cdots\!26$$ T^{12} + $$36\!\cdots\!72$$ T^{13} + $$14\!\cdots\!56$$ T^{14} + $$20\!\cdots\!84$$ T^{15} + $$83\!\cdots\!17$$ T^{16} - $$60\!\cdots\!86$$ T^{17} + $$25\!\cdots\!62$$ T^{18} + $$61\!\cdots\!18$$ T^{19} + $$74\!\cdots\!01$$ T^{20} $
$31$	$ ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} - $$14\!\cdots\!24$$ T^{9} + $$23\!\cdots\!51$$ T^{10} )^{2} $
$37$	$ 1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} + $$10\!\cdots\!24$$ T^{10} + $$14\!\cdots\!16$$ T^{11} + $$89\!\cdots\!58$$ T^{12} - $$68\!\cdots\!00$$ T^{13} + $$21\!\cdots\!92$$ T^{14} - $$51\!\cdots\!52$$ T^{15} + $$27\!\cdots\!93$$ T^{16} + $$16\!\cdots\!94$$ T^{17} + $$21\!\cdots\!50$$ T^{18} + $$21\!\cdots\!30$$ T^{19} + $$11\!\cdots\!49$$ T^{20} $
$41$	$ 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + $$14\!\cdots\!14$$ T^{8} - $$11\!\cdots\!88$$ T^{10} + $$68\!\cdots\!74$$ T^{12} - $$31\!\cdots\!24$$ T^{14} + $$10\!\cdots\!81$$ T^{16} - $$22\!\cdots\!98$$ T^{18} + $$24\!\cdots\!01$$ T^{20} $
$43$	$ 1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} + $$59\!\cdots\!10$$ T^{8} + $$17\!\cdots\!68$$ T^{9} + $$49\!\cdots\!52$$ T^{10} + $$13\!\cdots\!76$$ T^{11} + $$37\!\cdots\!90$$ T^{12} + $$98\!\cdots\!60$$ T^{13} + $$25\!\cdots\!48$$ T^{14} + $$66\!\cdots\!12$$ T^{15} + $$14\!\cdots\!45$$ T^{16} + $$26\!\cdots\!50$$ T^{17} + $$56\!\cdots\!22$$ T^{18} + $$10\!\cdots\!66$$ T^{19} + $$10\!\cdots\!49$$ T^{20} $
$47$	$ ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} + $$51\!\cdots\!73$$ T^{8} + $$54\!\cdots\!52$$ T^{9} + $$12\!\cdots\!43$$ T^{10} )^{2} $
$53$	$ 1 + 378 T + 71442 T^{2} - 52753550 T^{3} + 3016286341 T^{4} - 526686651752 T^{5} + 976891435665272 T^{6} - 1674349213754452168 T^{7} - 1581505854923305054 T^{8} - $$14\!\cdots\!20$$ T^{9} + $$29\!\cdots\!08$$ T^{10} - $$22\!\cdots\!40$$ T^{11} - $$35\!\cdots\!66$$ T^{12} - $$55\!\cdots\!44$$ T^{13} + $$47\!\cdots\!52$$ T^{14} - $$38\!\cdots\!64$$ T^{15} + $$32\!\cdots\!49$$ T^{16} - $$85\!\cdots\!50$$ T^{17} + $$17\!\cdots\!02$$ T^{18} + $$13\!\cdots\!86$$ T^{19} + $$53\!\cdots\!49$$ T^{20} $
$59$	$ 1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} + $$57\!\cdots\!98$$ T^{8} + $$22\!\cdots\!08$$ T^{9} - $$12\!\cdots\!80$$ T^{10} + $$45\!\cdots\!32$$ T^{11} + $$24\!\cdots\!18$$ T^{12} - $$17\!\cdots\!28$$ T^{13} + $$13\!\cdots\!96$$ T^{14} - $$85\!\cdots\!68$$ T^{15} + $$41\!\cdots\!01$$ T^{16} - $$15\!\cdots\!22$$ T^{17} + $$46\!\cdots\!98$$ T^{18} - $$11\!\cdots\!14$$ T^{19} + $$13\!\cdots\!01$$ T^{20} $
$61$	$ 1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} + $$59\!\cdots\!50$$ T^{8} - $$32\!\cdots\!00$$ T^{9} + $$38\!\cdots\!00$$ T^{10} - $$73\!\cdots\!00$$ T^{11} + $$30\!\cdots\!50$$ T^{12} - $$70\!\cdots\!80$$ T^{13} + $$15\!\cdots\!00$$ T^{14} - $$43\!\cdots\!20$$ T^{15} + $$27\!\cdots\!85$$ T^{16} + $$14\!\cdots\!10$$ T^{17} + $$29\!\cdots\!50$$ T^{18} - $$14\!\cdots\!10$$ T^{19} + $$36\!\cdots\!01$$ T^{20} $
$67$	$ 1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} - $$46\!\cdots\!12$$ T^{7} + $$29\!\cdots\!06$$ T^{8} - $$17\!\cdots\!64$$ T^{9} + $$97\!\cdots\!68$$ T^{10} - $$52\!\cdots\!32$$ T^{11} + $$26\!\cdots\!14$$ T^{12} - $$12\!\cdots\!64$$ T^{13} + $$59\!\cdots\!16$$ T^{14} - $$26\!\cdots\!24$$ T^{15} + $$99\!\cdots\!49$$ T^{16} - $$34\!\cdots\!66$$ T^{17} + $$12\!\cdots\!22$$ T^{18} - $$39\!\cdots\!66$$ T^{19} + $$60\!\cdots\!49$$ T^{20} $
$71$	$ 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + $$13\!\cdots\!66$$ T^{8} - $$55\!\cdots\!32$$ T^{10} + $$17\!\cdots\!86$$ T^{12} - $$39\!\cdots\!68$$ T^{14} + $$64\!\cdots\!53$$ T^{16} - $$67\!\cdots\!58$$ T^{18} + $$34\!\cdots\!01$$ T^{20} $
$73$	$ 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + $$26\!\cdots\!74$$ T^{8} - $$12\!\cdots\!92$$ T^{10} + $$40\!\cdots\!86$$ T^{12} - $$97\!\cdots\!12$$ T^{14} + $$16\!\cdots\!49$$ T^{16} - $$16\!\cdots\!62$$ T^{18} + $$79\!\cdots\!49$$ T^{20} $
$79$	$ ( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} + $$10\!\cdots\!52$$ T^{7} + $$45\!\cdots\!33$$ T^{8} + $$13\!\cdots\!28$$ T^{9} + $$29\!\cdots\!99$$ T^{10} )^{2} $
$83$	$ 1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} + $$79\!\cdots\!28$$ T^{7} + $$89\!\cdots\!74$$ T^{8} + $$97\!\cdots\!88$$ T^{9} + $$82\!\cdots\!76$$ T^{10} + $$55\!\cdots\!56$$ T^{11} + $$29\!\cdots\!06$$ T^{12} + $$14\!\cdots\!84$$ T^{13} + $$94\!\cdots\!96$$ T^{14} + $$72\!\cdots\!32$$ T^{15} + $$67\!\cdots\!61$$ T^{16} + $$57\!\cdots\!94$$ T^{17} + $$37\!\cdots\!62$$ T^{18} + $$16\!\cdots\!74$$ T^{19} + $$37\!\cdots\!49$$ T^{20} $
$89$	$ 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + $$69\!\cdots\!98$$ T^{8} - $$52\!\cdots\!28$$ T^{10} + $$34\!\cdots\!78$$ T^{12} - $$19\!\cdots\!88$$ T^{14} + $$79\!\cdots\!73$$ T^{16} - $$22\!\cdots\!98$$ T^{18} + $$30\!\cdots\!01$$ T^{20} $
$97$	$ ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - $$60\!\cdots\!36$$ T^{7} + $$19\!\cdots\!13$$ T^{8} + $$13\!\cdots\!82$$ T^{9} + $$63\!\cdots\!93$$ T^{10} )^{2} $
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 16 = 2^{4} \)
Weight:	\( k \)	=	\( 4 \)
Character orbit:	\([\chi]\)	=	16.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	=	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(8\)
Trace bound:			\(0\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
16.4.e.a	\(10\)	\(0.944\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(-2\)	\(-2\)	\(-2\)	\(0\)	\(q-\beta _{4}q^{2}-\beta _{5}q^{3}+(1-\beta _{2}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(16, [\chi])\) 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	16.4.e

Level	16
Weight	4
Character orbit	e
Rep. character	\(\chi_{16}(5,\cdot)\)
Character field	\(\Q(\zeta_{4})\)
Dimension	10
Newform subspaces	1
Sturm bound	8
Trace bound	0

$p$	$F_p(T)$
$2$	\( 1 + 2 T - 2 T^{2} + 8 T^{3} - 40 T^{4} - 352 T^{5} - 320 T^{6} + 512 T^{7} - 1024 T^{8} + 8192 T^{9} + 32768 T^{10} \)
$3$	\( 1 + 2 T + 2 T^{2} - 42 T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 66792 T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 82248156 T^{11} + 42190146 T^{12} + 1314666936 T^{13} + 1883426904 T^{14} + 10905169320 T^{15} - 221217099219 T^{16} - 439334834526 T^{17} + 564859072962 T^{18} + 15251194969974 T^{19} + 205891132094649 T^{20} \)
$5$	\( 1 + 2 T + 2 T^{2} - 966 T^{3} - 13723 T^{4} + 18040 T^{5} + 530104 T^{6} - 11981288 T^{7} - 28535006 T^{8} + 2301854348 T^{9} + 23672040908 T^{10} + 287731793500 T^{11} - 445859468750 T^{12} - 23400953125000 T^{13} + 129419921875000 T^{14} + 550537109375000 T^{15} - 52349090576171875 T^{16} - 460624694824218750 T^{17} + 119209289550781250 T^{18} + 14901161193847656250 T^{19} + \)\(93\!\cdots\!25\)\( T^{20} \)
$7$	\( 1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} + \)\(25\!\cdots\!85\)\( T^{16} - \)\(33\!\cdots\!62\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \)
$11$	\( 1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 6887886337838 T^{8} + 101615185776500 T^{9} + 18755914132083020 T^{10} + 135249812268521500 T^{11} - 12202310808546625118 T^{12} - \)\(73\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!44\)\( T^{14} + \)\(33\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!57\)\( T^{16} - \)\(90\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!42\)\( T^{18} - \)\(23\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} \)
$13$	\( 1 + 2 T + 2 T^{2} - 45206 T^{3} - 2401451 T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 384228102440 T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 3364607868872346876 T^{11} - 45372173460921944494 T^{12} - \)\(40\!\cdots\!20\)\( T^{13} + \)\(25\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!20\)\( T^{15} - \)\(27\!\cdots\!79\)\( T^{16} - \)\(11\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!22\)\( T^{18} + \)\(23\!\cdots\!34\)\( T^{19} + \)\(26\!\cdots\!49\)\( T^{20} \)
$17$	\( ( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \)
$19$	\( 1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} - \)\(17\!\cdots\!76\)\( T^{11} + \)\(47\!\cdots\!74\)\( T^{12} + \)\(11\!\cdots\!40\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(37\!\cdots\!16\)\( T^{15} - \)\(66\!\cdots\!11\)\( T^{16} - \)\(24\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!98\)\( T^{18} + \)\(87\!\cdots\!14\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \)
$23$	\( 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(19\!\cdots\!94\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(94\!\cdots\!93\)\( T^{16} - \)\(36\!\cdots\!26\)\( T^{18} + \)\(71\!\cdots\!49\)\( T^{20} \)
$29$	\( 1 + 202 T + 20402 T^{2} - 1177934 T^{3} + 398569397 T^{4} + 239164019416 T^{5} + 40873283338616 T^{6} + 2529271278095288 T^{7} + 194871598558001506 T^{8} + \)\(12\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} + \)\(30\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!26\)\( T^{12} + \)\(36\!\cdots\!72\)\( T^{13} + \)\(14\!\cdots\!56\)\( T^{14} + \)\(20\!\cdots\!84\)\( T^{15} + \)\(83\!\cdots\!17\)\( T^{16} - \)\(60\!\cdots\!86\)\( T^{17} + \)\(25\!\cdots\!62\)\( T^{18} + \)\(61\!\cdots\!18\)\( T^{19} + \)\(74\!\cdots\!01\)\( T^{20} \)
$31$	\( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \)
$37$	\( 1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} + \)\(10\!\cdots\!24\)\( T^{10} + \)\(14\!\cdots\!16\)\( T^{11} + \)\(89\!\cdots\!58\)\( T^{12} - \)\(68\!\cdots\!00\)\( T^{13} + \)\(21\!\cdots\!92\)\( T^{14} - \)\(51\!\cdots\!52\)\( T^{15} + \)\(27\!\cdots\!93\)\( T^{16} + \)\(16\!\cdots\!94\)\( T^{17} + \)\(21\!\cdots\!50\)\( T^{18} + \)\(21\!\cdots\!30\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \)
$41$	\( 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(68\!\cdots\!74\)\( T^{12} - \)\(31\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!81\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \)
$43$	\( 1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} + \)\(59\!\cdots\!10\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!76\)\( T^{11} + \)\(37\!\cdots\!90\)\( T^{12} + \)\(98\!\cdots\!60\)\( T^{13} + \)\(25\!\cdots\!48\)\( T^{14} + \)\(66\!\cdots\!12\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(26\!\cdots\!50\)\( T^{17} + \)\(56\!\cdots\!22\)\( T^{18} + \)\(10\!\cdots\!66\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} \)
$47$	\( ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \)
$53$	\( 1 + 378 T + 71442 T^{2} - 52753550 T^{3} + 3016286341 T^{4} - 526686651752 T^{5} + 976891435665272 T^{6} - 1674349213754452168 T^{7} - 1581505854923305054 T^{8} - \)\(14\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} - \)\(22\!\cdots\!40\)\( T^{11} - \)\(35\!\cdots\!66\)\( T^{12} - \)\(55\!\cdots\!44\)\( T^{13} + \)\(47\!\cdots\!52\)\( T^{14} - \)\(38\!\cdots\!64\)\( T^{15} + \)\(32\!\cdots\!49\)\( T^{16} - \)\(85\!\cdots\!50\)\( T^{17} + \)\(17\!\cdots\!02\)\( T^{18} + \)\(13\!\cdots\!86\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \)
$59$	\( 1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} + \)\(57\!\cdots\!98\)\( T^{8} + \)\(22\!\cdots\!08\)\( T^{9} - \)\(12\!\cdots\!80\)\( T^{10} + \)\(45\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!18\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!96\)\( T^{14} - \)\(85\!\cdots\!68\)\( T^{15} + \)\(41\!\cdots\!01\)\( T^{16} - \)\(15\!\cdots\!22\)\( T^{17} + \)\(46\!\cdots\!98\)\( T^{18} - \)\(11\!\cdots\!14\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$61$	\( 1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} + \)\(59\!\cdots\!50\)\( T^{8} - \)\(32\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!00\)\( T^{10} - \)\(73\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!50\)\( T^{12} - \)\(70\!\cdots\!80\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(43\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!85\)\( T^{16} + \)\(14\!\cdots\!10\)\( T^{17} + \)\(29\!\cdots\!50\)\( T^{18} - \)\(14\!\cdots\!10\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$67$	\( 1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} - \)\(46\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} - \)\(17\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!68\)\( T^{10} - \)\(52\!\cdots\!32\)\( T^{11} + \)\(26\!\cdots\!14\)\( T^{12} - \)\(12\!\cdots\!64\)\( T^{13} + \)\(59\!\cdots\!16\)\( T^{14} - \)\(26\!\cdots\!24\)\( T^{15} + \)\(99\!\cdots\!49\)\( T^{16} - \)\(34\!\cdots\!66\)\( T^{17} + \)\(12\!\cdots\!22\)\( T^{18} - \)\(39\!\cdots\!66\)\( T^{19} + \)\(60\!\cdots\!49\)\( T^{20} \)
$71$	\( 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!32\)\( T^{10} + \)\(17\!\cdots\!86\)\( T^{12} - \)\(39\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!53\)\( T^{16} - \)\(67\!\cdots\!58\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} \)
$73$	\( 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + \)\(26\!\cdots\!74\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(40\!\cdots\!86\)\( T^{12} - \)\(97\!\cdots\!12\)\( T^{14} + \)\(16\!\cdots\!49\)\( T^{16} - \)\(16\!\cdots\!62\)\( T^{18} + \)\(79\!\cdots\!49\)\( T^{20} \)
$79$	\( ( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \)
$83$	\( 1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} + \)\(79\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!74\)\( T^{8} + \)\(97\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!76\)\( T^{10} + \)\(55\!\cdots\!56\)\( T^{11} + \)\(29\!\cdots\!06\)\( T^{12} + \)\(14\!\cdots\!84\)\( T^{13} + \)\(94\!\cdots\!96\)\( T^{14} + \)\(72\!\cdots\!32\)\( T^{15} + \)\(67\!\cdots\!61\)\( T^{16} + \)\(57\!\cdots\!94\)\( T^{17} + \)\(37\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!74\)\( T^{19} + \)\(37\!\cdots\!49\)\( T^{20} \)
$89$	\( 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + \)\(69\!\cdots\!98\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{10} + \)\(34\!\cdots\!78\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{14} + \)\(79\!\cdots\!73\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \)
$97$	\( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \)
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