LMFDB - Space of modular forms of level 2, weight 44, and Trivial character

Defining parameters

Level:	$ N $	$=$	$ 2 $
Weight:	$ k $	$=$	$ 44 $
Character orbit:	$[\chi]$	$=$	2.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 2 $
Sturm bound:			$11$
Trace bound:			$2$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{44}(\Gamma_0(2))$.

	Total	New	Old
Modular forms	12	4	8
Cusp forms	10	4	6
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.

$2$	Dim.
$+$	$2$
$-$	$2$

Trace form

$ 4q - 35323265040q^{3} + 17592186044416q^{4} - 445983369680520q^{5} - 19629349162450944q^{6} + 952409902196202080q^{7} + 57392344928214127188q^{9} + O(q^{10}) $

\( 4q - 35323265040q^{3} + 17592186044416q^{4} - 445983369680520q^{5} - 19629349162450944q^{6} + 952409902196202080q^{7} + 57392344928214127188q^{9} + 736817688890793000960q^{10} - 53505666230187039126192q^{11} - 155353362569973895004160q^{12} - 3170092445107476877943080q^{13} - 2930413749959980348342272q^{14} - 39896100567148303597839840q^{15} + 77371252455336267181195264q^{16} + 275811531610294198491666120q^{17} + 2155446423566237915236270080q^{18} + 704368595409646764116849840q^{19} - 1961455603033816441112494080q^{20} - 71943252193711849582874093952q^{21} - 63480896917120890827786158080q^{22} + 267622990007463399665328694560q^{23} - 86330790599159598780751282176q^{24} + 2883522820368023310220104607900q^{25} + 309677544891030919403091787776q^{26} - 13660960708029142515089178825120q^{27} + 4188743047494908425223347896320q^{28} - 38644798233978054156037925733480q^{29} + 8988615448994818274005163704320q^{30} + 202517590352351068698084253267328q^{31} + 698603125139732151618674183490240q^{33} - 860904569326544750263603436716032q^{34} - 3325520912837045799202976122441920q^{35} + 252414202375609491412627910295552q^{36} - 1007806155266724596382724999945480q^{37} + 3685618436761385576387510041313280q^{38} + 34627230647677526261318633218835616q^{39} + 3240558465945864657539990122659840q^{40} + 14769056217421573842182424655114728q^{41} - 188529371232675648676832840578498560q^{42} - 50656152082275083325922323570825520q^{43} - 235320408687969219594308666385235968q^{44} + 813851237061199242360170633473196760q^{45} - 401162053228680963277901151835521024q^{46} + 1795871719777727412019789817026927680q^{47} - 683251314239148431991403503566192640q^{48} + 2472588624099381815176131582897177252q^{49} - 14094510821977045744752591014933299200q^{50} + 9366222065014568589261957669399307488q^{51} - 13942214018082087317341913445899960320q^{52} + 34930183704299684660043854347335201720q^{53} - 54991919905881001033766855194737377280q^{54} + 101675970194038744959268516881965777760q^{55} - 12888095969102680990651131959640588288q^{56} + 24725309490521092416268712856158940480q^{57} - 224221408455435359075502788567724195840q^{58} + 74517661892831252610921679030012874640q^{59} - 175464905906000912317031679022973583360q^{60} + 95466819219187169456508921216701652248q^{61} - 256066710492626982167265418768715612160q^{62} + 1302503682143077872333511602225242757600q^{63} + 340282366920938463463374607431768211456q^{64} - 1140879441958849707741225198393647217840q^{65} + 2758089365183515747873923781071955034112q^{66} - 5185823154883915148339698804696290802960q^{67} + 1213031944320905010647284238286040596480q^{68} - 13169724835217980176395670977192312083584q^{69} + 13750686399482936758591003632374812508160q^{70} + 6078131343463252822749802888112407049568q^{71} + 9479753623037087268551680109182262968320q^{72} - 11299472539399729498004578793488644374680q^{73} + 16306213923786421705403162747840986349568q^{74} + 968840833644049096432036826739720166800q^{75} + 3097845843572621900589154698591060623360q^{76} - 122687760481577524751229519037503537559680q^{77} + 5665681380473715909273979796512592363520q^{78} + 39481139269211124080055934235826831819200q^{79} - 8626572971608268796564758456927854264320q^{80} + 303693792362524627388345705941115534711844q^{81} - 276362461442393760174426915844135311114240q^{82} + 450113087170899479515222133277261850695600q^{83} - 316409769308029594425462246614498707243008q^{84} + 339447401278646006771627630621380418317680q^{85} - 1089531960634141917764113636419993530793984q^{86} + 63269631265006087376268943554864452768160q^{87} - 279191937208096203349737387043928219320320q^{88} + 473243058306814632719521085066228455846440q^{89} - 42617612448701427992575832218399125012480q^{90} + 27907556672811409832272913003692321235008q^{91} + 1177018357493545059819073626342986864394240q^{92} + 3376287307445883190884526599190714638543360q^{93} + 3930274486025559007941997987062303820873728q^{94} - 9878276980119994764350873096834160483420000q^{95} - 379686832395483875172155628806131421282304q^{96} - 5934740187101728628006412969339693653467000q^{97} + 7327603468103810611042817664861329078353920q^{98} - 15605924803061079239520118761603262248845424q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{44}^{\mathrm{new}}(\Gamma_0(2))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs	$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	2	$q$-expansion
2.44.a.a	$2$	$23.422$	$\mathbb{Q}[x]/(x^{2} - \cdots)$	None	$-4194304$	$-12981630984$	$-3\!\cdots\!00$	$11\!\cdots\!08$	$+$	$q-2^{21}q^{2}+(-6490815492-\beta )q^{3}+\cdots$
2.44.a.b	$2$	$23.422$	$\mathbb{Q}[x]/(x^{2} - \cdots)$	None	$4194304$	$-22341634056$	$-4\!\cdots\!20$	$-2\!\cdots\!28$	$-$	$q+2^{21}q^{2}+(-11170817028-\beta )q^{3}+\cdots$

Decomposition of $S_{44}^{\mathrm{old}}(\Gamma_0(2))$ into lower level spaces

$ S_{44}^{\mathrm{old}}(\Gamma_0(2)) \cong $ $S_{44}^{\mathrm{new}}(\Gamma_0(1))$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ ( 1 + 2097152 T )^{2} $)($ ( 1 - 2097152 T )^{2} $)
$3$	($ 1 + 12981630984 T + $$65\!\cdots\!18$$ T^{2} + $$42\!\cdots\!68$$ T^{3} + $$10\!\cdots\!29$$ T^{4} $)($ 1 + 22341634056 T + $$30\!\cdots\!38$$ T^{2} + $$73\!\cdots\!12$$ T^{3} + $$10\!\cdots\!29$$ T^{4} $)
$5$	($ 1 + 398662711282500 T - $$11\!\cdots\!50$$ T^{2} + $$45\!\cdots\!00$$ T^{3} + $$12\!\cdots\!25$$ T^{4} $)($ 1 + 47320658398020 T + $$20\!\cdots\!50$$ T^{2} + $$53\!\cdots\!00$$ T^{3} + $$12\!\cdots\!25$$ T^{4} $)
$7$	($ 1 - 1174870033543241008 T + $$31\!\cdots\!02$$ T^{2} - $$25\!\cdots\!44$$ T^{3} + $$47\!\cdots\!49$$ T^{4} $)($ 1 + 222460131347038928 T + $$71\!\cdots\!82$$ T^{2} + $$48\!\cdots\!04$$ T^{3} + $$47\!\cdots\!49$$ T^{4} $)
$11$	($ 1 + $$11\!\cdots\!76$$ T + $$39\!\cdots\!06$$ T^{2} + $$69\!\cdots\!56$$ T^{3} + $$36\!\cdots\!61$$ T^{4} $)($ 1 + $$41\!\cdots\!16$$ T + $$14\!\cdots\!26$$ T^{2} + $$25\!\cdots\!96$$ T^{3} + $$36\!\cdots\!61$$ T^{4} $)
$13$	($ 1 + $$16\!\cdots\!84$$ T + $$21\!\cdots\!58$$ T^{2} + $$13\!\cdots\!48$$ T^{3} + $$62\!\cdots\!09$$ T^{4} $)($ 1 + $$15\!\cdots\!96$$ T + $$21\!\cdots\!98$$ T^{2} + $$11\!\cdots\!12$$ T^{3} + $$62\!\cdots\!09$$ T^{4} $)
$17$	($ 1 - $$34\!\cdots\!68$$ T + $$16\!\cdots\!82$$ T^{2} - $$27\!\cdots\!84$$ T^{3} + $$65\!\cdots\!69$$ T^{4} $)($ 1 + $$67\!\cdots\!48$$ T + $$72\!\cdots\!02$$ T^{2} + $$54\!\cdots\!24$$ T^{3} + $$65\!\cdots\!69$$ T^{4} $)
$19$	($ 1 + $$52\!\cdots\!00$$ T + $$12\!\cdots\!18$$ T^{2} + $$51\!\cdots\!00$$ T^{3} + $$93\!\cdots\!81$$ T^{4} $)($ 1 - $$12\!\cdots\!40$$ T + $$19\!\cdots\!18$$ T^{2} - $$11\!\cdots\!60$$ T^{3} + $$93\!\cdots\!81$$ T^{4} $)
$23$	($ 1 - $$22\!\cdots\!36$$ T + $$72\!\cdots\!58$$ T^{2} - $$82\!\cdots\!12$$ T^{3} + $$12\!\cdots\!89$$ T^{4} $)($ 1 - $$38\!\cdots\!24$$ T + $$21\!\cdots\!78$$ T^{2} - $$13\!\cdots\!08$$ T^{3} + $$12\!\cdots\!89$$ T^{4} $)
$29$	($ 1 - $$34\!\cdots\!20$$ T + $$16\!\cdots\!78$$ T^{2} - $$26\!\cdots\!80$$ T^{3} + $$58\!\cdots\!21$$ T^{4} $)($ 1 + $$72\!\cdots\!00$$ T + $$25\!\cdots\!78$$ T^{2} + $$55\!\cdots\!00$$ T^{3} + $$58\!\cdots\!21$$ T^{4} $)
$31$	($ 1 - $$16\!\cdots\!04$$ T + $$32\!\cdots\!86$$ T^{2} - $$21\!\cdots\!64$$ T^{3} + $$18\!\cdots\!81$$ T^{4} $)($ 1 - $$40\!\cdots\!24$$ T + $$12\!\cdots\!26$$ T^{2} - $$54\!\cdots\!84$$ T^{3} + $$18\!\cdots\!81$$ T^{4} $)
$37$	($ 1 + $$43\!\cdots\!32$$ T + $$28\!\cdots\!62$$ T^{2} + $$11\!\cdots\!96$$ T^{3} + $$73\!\cdots\!09$$ T^{4} $)($ 1 - $$33\!\cdots\!52$$ T + $$13\!\cdots\!82$$ T^{2} - $$91\!\cdots\!56$$ T^{3} + $$73\!\cdots\!09$$ T^{4} $)
$41$	($ 1 - $$73\!\cdots\!24$$ T + $$57\!\cdots\!86$$ T^{2} - $$16\!\cdots\!04$$ T^{3} + $$50\!\cdots\!41$$ T^{4} $)($ 1 + $$58\!\cdots\!96$$ T + $$48\!\cdots\!46$$ T^{2} + $$13\!\cdots\!16$$ T^{3} + $$50\!\cdots\!41$$ T^{4} $)
$43$	($ 1 - $$23\!\cdots\!36$$ T + $$41\!\cdots\!38$$ T^{2} - $$40\!\cdots\!52$$ T^{3} + $$30\!\cdots\!49$$ T^{4} $)($ 1 + $$28\!\cdots\!56$$ T + $$53\!\cdots\!98$$ T^{2} + $$49\!\cdots\!92$$ T^{3} + $$30\!\cdots\!49$$ T^{4} $)
$47$	($ 1 + $$39\!\cdots\!92$$ T + $$27\!\cdots\!62$$ T^{2} + $$31\!\cdots\!16$$ T^{3} + $$63\!\cdots\!29$$ T^{4} $)($ 1 - $$18\!\cdots\!72$$ T + $$20\!\cdots\!42$$ T^{2} - $$14\!\cdots\!56$$ T^{3} + $$63\!\cdots\!29$$ T^{4} $)
$53$	($ 1 - $$21\!\cdots\!56$$ T + $$24\!\cdots\!38$$ T^{2} - $$29\!\cdots\!12$$ T^{3} + $$19\!\cdots\!29$$ T^{4} $)($ 1 - $$13\!\cdots\!64$$ T + $$32\!\cdots\!78$$ T^{2} - $$18\!\cdots\!28$$ T^{3} + $$19\!\cdots\!29$$ T^{4} $)
$59$	($ 1 - $$67\!\cdots\!40$$ T + $$20\!\cdots\!58$$ T^{2} - $$95\!\cdots\!60$$ T^{3} + $$19\!\cdots\!41$$ T^{4} $)($ 1 - $$65\!\cdots\!00$$ T + $$26\!\cdots\!58$$ T^{2} - $$91\!\cdots\!00$$ T^{3} + $$19\!\cdots\!41$$ T^{4} $)
$61$	($ 1 - $$10\!\cdots\!04$$ T + $$94\!\cdots\!66$$ T^{2} - $$62\!\cdots\!24$$ T^{3} + $$34\!\cdots\!61$$ T^{4} $)($ 1 - $$84\!\cdots\!44$$ T + $$10\!\cdots\!46$$ T^{2} - $$49\!\cdots\!64$$ T^{3} + $$34\!\cdots\!61$$ T^{4} $)
$67$	($ 1 + $$12\!\cdots\!52$$ T + $$36\!\cdots\!02$$ T^{2} + $$43\!\cdots\!76$$ T^{3} + $$11\!\cdots\!69$$ T^{4} $)($ 1 + $$38\!\cdots\!08$$ T + $$10\!\cdots\!42$$ T^{2} + $$12\!\cdots\!04$$ T^{3} + $$11\!\cdots\!69$$ T^{4} $)
$71$	($ 1 - $$10\!\cdots\!64$$ T + $$10\!\cdots\!46$$ T^{2} - $$42\!\cdots\!04$$ T^{3} + $$16\!\cdots\!21$$ T^{4} $)($ 1 + $$45\!\cdots\!96$$ T + $$76\!\cdots\!26$$ T^{2} + $$18\!\cdots\!56$$ T^{3} + $$16\!\cdots\!21$$ T^{4} $)
$73$	($ 1 + $$10\!\cdots\!24$$ T + $$26\!\cdots\!78$$ T^{2} + $$14\!\cdots\!08$$ T^{3} + $$17\!\cdots\!89$$ T^{4} $)($ 1 + $$10\!\cdots\!56$$ T + $$20\!\cdots\!18$$ T^{2} + $$13\!\cdots\!52$$ T^{3} + $$17\!\cdots\!89$$ T^{4} $)
$79$	($ 1 + $$66\!\cdots\!20$$ T + $$90\!\cdots\!78$$ T^{2} + $$26\!\cdots\!80$$ T^{3} + $$15\!\cdots\!21$$ T^{4} $)($ 1 - $$10\!\cdots\!20$$ T + $$10\!\cdots\!78$$ T^{2} - $$42\!\cdots\!80$$ T^{3} + $$15\!\cdots\!21$$ T^{4} $)
$83$	($ 1 - $$54\!\cdots\!56$$ T + $$14\!\cdots\!58$$ T^{2} - $$18\!\cdots\!72$$ T^{3} + $$10\!\cdots\!69$$ T^{4} $)($ 1 + $$99\!\cdots\!56$$ T + $$45\!\cdots\!58$$ T^{2} + $$32\!\cdots\!72$$ T^{3} + $$10\!\cdots\!69$$ T^{4} $)
$89$	($ 1 - $$29\!\cdots\!20$$ T + $$85\!\cdots\!38$$ T^{2} - $$19\!\cdots\!80$$ T^{3} + $$44\!\cdots\!61$$ T^{4} $)($ 1 - $$17\!\cdots\!20$$ T + $$11\!\cdots\!38$$ T^{2} - $$11\!\cdots\!80$$ T^{3} + $$44\!\cdots\!61$$ T^{4} $)
$97$	($ 1 + $$22\!\cdots\!12$$ T + $$34\!\cdots\!82$$ T^{2} + $$61\!\cdots\!76$$ T^{3} + $$72\!\cdots\!29$$ T^{4} $)($ 1 + $$36\!\cdots\!88$$ T - $$10\!\cdots\!18$$ T^{2} + $$98\!\cdots\!24$$ T^{3} + $$72\!\cdots\!29$$ T^{4} $)
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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:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 44 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(11\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{44}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Decomposition of \(S_{44}^{\mathrm{old}}(\Gamma_0(2))\) 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	2.44.a

Level	2
Weight	44
Character orbit	a
Rep. character	\(\chi_{2}(1,\cdot)\)
Character field	\(\Q\)
Dimension	4
Newform subspaces	2
Sturm bound	11
Trace bound	2

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs	$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	2	$q$-expansion
2.44.a.a	\(2\)	\(23.422\)	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	\(-4194304\)	\(-12981630984\)	\(-3\!\cdots\!00\)	\(11\!\cdots\!08\)	\(+\)	\(q-2^{21}q^{2}+(-6490815492-\beta )q^{3}+\cdots\)
2.44.a.b	\(2\)	\(23.422\)	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	\(4194304\)	\(-22341634056\)	\(-4\!\cdots\!20\)	\(-2\!\cdots\!28\)	\(-\)	\(q+2^{21}q^{2}+(-11170817028-\beta )q^{3}+\cdots\)

$p$	$F_p(T)$
$2$	(\( ( 1 + 2097152 T )^{2} \))(\( ( 1 - 2097152 T )^{2} \))
$3$	(\( 1 + 12981630984 T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(42\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \))(\( 1 + 22341634056 T + \)\(30\!\cdots\!38\)\( T^{2} + \)\(73\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \))
$5$	(\( 1 + 398662711282500 T - \)\(11\!\cdots\!50\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!25\)\( T^{4} \))(\( 1 + 47320658398020 T + \)\(20\!\cdots\!50\)\( T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!25\)\( T^{4} \))
$7$	(\( 1 - 1174870033543241008 T + \)\(31\!\cdots\!02\)\( T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(47\!\cdots\!49\)\( T^{4} \))(\( 1 + 222460131347038928 T + \)\(71\!\cdots\!82\)\( T^{2} + \)\(48\!\cdots\!04\)\( T^{3} + \)\(47\!\cdots\!49\)\( T^{4} \))
$11$	(\( 1 + \)\(11\!\cdots\!76\)\( T + \)\(39\!\cdots\!06\)\( T^{2} + \)\(69\!\cdots\!56\)\( T^{3} + \)\(36\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(41\!\cdots\!16\)\( T + \)\(14\!\cdots\!26\)\( T^{2} + \)\(25\!\cdots\!96\)\( T^{3} + \)\(36\!\cdots\!61\)\( T^{4} \))
$13$	(\( 1 + \)\(16\!\cdots\!84\)\( T + \)\(21\!\cdots\!58\)\( T^{2} + \)\(13\!\cdots\!48\)\( T^{3} + \)\(62\!\cdots\!09\)\( T^{4} \))(\( 1 + \)\(15\!\cdots\!96\)\( T + \)\(21\!\cdots\!98\)\( T^{2} + \)\(11\!\cdots\!12\)\( T^{3} + \)\(62\!\cdots\!09\)\( T^{4} \))
$17$	(\( 1 - \)\(34\!\cdots\!68\)\( T + \)\(16\!\cdots\!82\)\( T^{2} - \)\(27\!\cdots\!84\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(67\!\cdots\!48\)\( T + \)\(72\!\cdots\!02\)\( T^{2} + \)\(54\!\cdots\!24\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))
$19$	(\( 1 + \)\(52\!\cdots\!00\)\( T + \)\(12\!\cdots\!18\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(12\!\cdots\!40\)\( T + \)\(19\!\cdots\!18\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(93\!\cdots\!81\)\( T^{4} \))
$23$	(\( 1 - \)\(22\!\cdots\!36\)\( T + \)\(72\!\cdots\!58\)\( T^{2} - \)\(82\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(38\!\cdots\!24\)\( T + \)\(21\!\cdots\!78\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))
$29$	(\( 1 - \)\(34\!\cdots\!20\)\( T + \)\(16\!\cdots\!78\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(72\!\cdots\!00\)\( T + \)\(25\!\cdots\!78\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!21\)\( T^{4} \))
$31$	(\( 1 - \)\(16\!\cdots\!04\)\( T + \)\(32\!\cdots\!86\)\( T^{2} - \)\(21\!\cdots\!64\)\( T^{3} + \)\(18\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(40\!\cdots\!24\)\( T + \)\(12\!\cdots\!26\)\( T^{2} - \)\(54\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!81\)\( T^{4} \))
$37$	(\( 1 + \)\(43\!\cdots\!32\)\( T + \)\(28\!\cdots\!62\)\( T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(73\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(33\!\cdots\!52\)\( T + \)\(13\!\cdots\!82\)\( T^{2} - \)\(91\!\cdots\!56\)\( T^{3} + \)\(73\!\cdots\!09\)\( T^{4} \))
$41$	(\( 1 - \)\(73\!\cdots\!24\)\( T + \)\(57\!\cdots\!86\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \))(\( 1 + \)\(58\!\cdots\!96\)\( T + \)\(48\!\cdots\!46\)\( T^{2} + \)\(13\!\cdots\!16\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \))
$43$	(\( 1 - \)\(23\!\cdots\!36\)\( T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(40\!\cdots\!52\)\( T^{3} + \)\(30\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(28\!\cdots\!56\)\( T + \)\(53\!\cdots\!98\)\( T^{2} + \)\(49\!\cdots\!92\)\( T^{3} + \)\(30\!\cdots\!49\)\( T^{4} \))
$47$	(\( 1 + \)\(39\!\cdots\!92\)\( T + \)\(27\!\cdots\!62\)\( T^{2} + \)\(31\!\cdots\!16\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(18\!\cdots\!72\)\( T + \)\(20\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \))
$53$	(\( 1 - \)\(21\!\cdots\!56\)\( T + \)\(24\!\cdots\!38\)\( T^{2} - \)\(29\!\cdots\!12\)\( T^{3} + \)\(19\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(13\!\cdots\!64\)\( T + \)\(32\!\cdots\!78\)\( T^{2} - \)\(18\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!29\)\( T^{4} \))
$59$	(\( 1 - \)\(67\!\cdots\!40\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(95\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(65\!\cdots\!00\)\( T + \)\(26\!\cdots\!58\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))
$61$	(\( 1 - \)\(10\!\cdots\!04\)\( T + \)\(94\!\cdots\!66\)\( T^{2} - \)\(62\!\cdots\!24\)\( T^{3} + \)\(34\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(84\!\cdots\!44\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(49\!\cdots\!64\)\( T^{3} + \)\(34\!\cdots\!61\)\( T^{4} \))
$67$	(\( 1 + \)\(12\!\cdots\!52\)\( T + \)\(36\!\cdots\!02\)\( T^{2} + \)\(43\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(38\!\cdots\!08\)\( T + \)\(10\!\cdots\!42\)\( T^{2} + \)\(12\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!69\)\( T^{4} \))
$71$	(\( 1 - \)\(10\!\cdots\!64\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(42\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(45\!\cdots\!96\)\( T + \)\(76\!\cdots\!26\)\( T^{2} + \)\(18\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \))
$73$	(\( 1 + \)\(10\!\cdots\!24\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(10\!\cdots\!56\)\( T + \)\(20\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!89\)\( T^{4} \))
$79$	(\( 1 + \)\(66\!\cdots\!20\)\( T + \)\(90\!\cdots\!78\)\( T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(10\!\cdots\!20\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(42\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \))
$83$	(\( 1 - \)\(54\!\cdots\!56\)\( T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(18\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(99\!\cdots\!56\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(32\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \))
$89$	(\( 1 - \)\(29\!\cdots\!20\)\( T + \)\(85\!\cdots\!38\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(17\!\cdots\!20\)\( T + \)\(11\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!61\)\( T^{4} \))
$97$	(\( 1 + \)\(22\!\cdots\!12\)\( T + \)\(34\!\cdots\!82\)\( T^{2} + \)\(61\!\cdots\!76\)\( T^{3} + \)\(72\!\cdots\!29\)\( T^{4} \))(\( 1 + \)\(36\!\cdots\!88\)\( T - \)\(10\!\cdots\!18\)\( T^{2} + \)\(98\!\cdots\!24\)\( T^{3} + \)\(72\!\cdots\!29\)\( T^{4} \))
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