Properties

Label 177.4.e.b
Level $177$
Weight $4$
Character orbit 177.e
Analytic conductor $10.443$
Analytic rank $0$
Dimension $420$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(420\)
Relative dimension: \(15\) over \(\Q(\zeta_{29})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 420q + 2q^{2} - 45q^{3} - 56q^{4} - 14q^{5} + 6q^{6} + 6q^{7} + 66q^{8} - 135q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 420q + 2q^{2} - 45q^{3} - 56q^{4} - 14q^{5} + 6q^{6} + 6q^{7} + 66q^{8} - 135q^{9} - 754q^{10} + 64q^{11} - 168q^{12} + 90q^{13} + 1024q^{14} - 42q^{15} - 172q^{16} + 96q^{17} + 18q^{18} + 16q^{19} - 178q^{20} - 243q^{21} - 604q^{22} + 48q^{23} + 198q^{24} - 173q^{25} + 156q^{26} - 405q^{27} + 84q^{28} - 50q^{29} + 174q^{30} + 222q^{31} - 45q^{32} + 192q^{33} - 64q^{34} - 132q^{35} - 504q^{36} + 30q^{37} - 108q^{38} + 270q^{39} + 6984q^{40} + 400q^{41} + 114q^{42} + 1220q^{43} + 1180q^{44} - 126q^{45} + 4938q^{46} + 370q^{47} - 516q^{48} + 771q^{49} + 1048q^{50} + 288q^{51} - 1964q^{52} - 822q^{53} + 54q^{54} - 2342q^{55} - 13824q^{56} + 48q^{57} - 13150q^{58} - 3423q^{59} - 534q^{60} - 2182q^{61} + 845q^{62} + 54q^{63} - 8018q^{64} + 4140q^{65} - 1812q^{66} + 506q^{67} + 4826q^{68} - 3684q^{69} + 6882q^{70} + 5874q^{71} + 594q^{72} + 206q^{73} + 11124q^{74} - 867q^{75} + 16940q^{76} + 9940q^{77} + 468q^{78} - 1762q^{79} - 28954q^{80} - 1215q^{81} + 3770q^{82} - 1216q^{83} - 8013q^{84} - 420q^{85} - 2256q^{86} + 5940q^{87} + 13458q^{88} - 1580q^{89} + 522q^{90} - 12204q^{91} + 4562q^{92} + 666q^{93} + 176q^{94} - 28456q^{95} + 2040q^{96} + 2574q^{97} + 50690q^{98} + 576q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.48057 0.596048i 1.94216 2.28649i 21.8684 + 4.81360i 0.769914 + 0.585273i −12.0070 + 11.3736i 4.65584 11.6853i −75.1876 25.3337i −1.45604 8.88144i −3.87071 3.66653i
4.2 −4.38291 0.476671i 1.94216 2.28649i 11.1698 + 2.45865i −11.3321 8.61446i −9.60221 + 9.09570i −7.94992 + 19.9528i −14.3603 4.83855i −1.45604 8.88144i 45.5615 + 43.1581i
4.3 −3.97824 0.432660i 1.94216 2.28649i 7.82627 + 1.72269i 6.73681 + 5.12119i −8.71565 + 8.25591i −6.58976 + 16.5390i −0.0516573 0.0174054i −1.45604 8.88144i −24.5850 23.2881i
4.4 −3.05964 0.332755i 1.94216 2.28649i 1.43768 + 0.316458i 12.7861 + 9.71975i −6.70314 + 6.34955i 6.84669 17.1839i 19.0391 + 6.41502i −1.45604 8.88144i −35.8865 33.9935i
4.5 −2.96757 0.322743i 1.94216 2.28649i 0.889340 + 0.195759i −11.1726 8.49315i −6.50144 + 6.15849i 12.7555 32.0138i 20.0545 + 6.75714i −1.45604 8.88144i 30.4142 + 28.8099i
4.6 −1.68795 0.183576i 1.94216 2.28649i −4.99749 1.10003i 3.49616 + 2.65771i −3.69801 + 3.50294i 0.0781424 0.196123i 21.1058 + 7.11136i −1.45604 8.88144i −5.41345 5.12789i
4.7 −0.488701 0.0531494i 1.94216 2.28649i −7.57696 1.66782i −7.57411 5.75769i −1.07066 + 1.01418i −3.71388 + 9.32114i 7.34103 + 2.47348i −1.45604 8.88144i 3.39546 + 3.21635i
4.8 0.253308 + 0.0275489i 1.94216 2.28649i −7.74956 1.70581i 7.78313 + 5.91658i 0.554954 0.525681i −0.440801 + 1.10633i −3.84774 1.29645i −1.45604 8.88144i 1.80853 + 1.71313i
4.9 1.24705 + 0.135624i 1.94216 2.28649i −6.27623 1.38150i 16.4405 + 12.4978i 2.73207 2.58795i −11.4144 + 28.6480i −17.1493 5.77826i −1.45604 8.88144i 18.8071 + 17.8150i
4.10 2.00109 + 0.217632i 1.94216 2.28649i −3.85597 0.848763i −3.87099 2.94265i 4.38404 4.15279i 5.75041 14.4324i −22.7916 7.67938i −1.45604 8.88144i −7.10578 6.73095i
4.11 2.91566 + 0.317097i 1.94216 2.28649i 0.587538 + 0.129327i −2.37530 1.80566i 6.38771 6.05076i 4.97289 12.4810i −20.5625 6.92833i −1.45604 8.88144i −6.35299 6.01788i
4.12 3.13622 + 0.341085i 1.94216 2.28649i 1.90660 + 0.419674i −14.9247 11.3455i 6.87093 6.50849i −10.7338 + 26.9398i −18.0802 6.09194i −1.45604 8.88144i −42.9375 40.6726i
4.13 4.20930 + 0.457789i 1.94216 2.28649i 9.69564 + 2.13417i 11.3095 + 8.59723i 9.22185 8.73540i 5.73956 14.4052i 7.73503 + 2.60623i −1.45604 8.88144i 43.6692 + 41.3657i
4.14 5.06804 + 0.551183i 1.94216 2.28649i 17.5683 + 3.86707i −15.9287 12.1087i 11.1032 10.5175i 8.66226 21.7406i 48.2567 + 16.2596i −1.45604 8.88144i −74.0531 70.1468i
4.15 5.20320 + 0.565882i 1.94216 2.28649i 18.9401 + 4.16902i 3.73106 + 2.83628i 11.3993 10.7980i −7.57844 + 19.0204i 56.5105 + 19.0406i −1.45604 8.88144i 17.8084 + 16.8690i
7.1 −2.86839 4.23056i 0.162417 2.99560i −6.70886 + 16.8380i 19.1429 8.85647i −13.1389 + 7.90544i −6.91419 + 24.9026i 50.5433 11.1254i −8.94724 0.973071i −92.3773 55.5815i
7.2 −2.57971 3.80479i 0.162417 2.99560i −4.86039 + 12.1987i −1.87335 + 0.866706i −11.8166 + 7.10982i 0.688121 2.47839i 23.0364 5.07071i −8.94724 0.973071i 8.13034 + 4.89186i
7.3 −2.23273 3.29303i 0.162417 2.99560i −2.89786 + 7.27309i 3.07784 1.42396i −10.2272 + 6.15353i 3.58463 12.9107i −0.663884 + 0.146132i −8.94724 0.973071i −11.5611 6.95609i
7.4 −2.02689 2.98944i 0.162417 2.99560i −1.86736 + 4.68671i −15.6898 + 7.25888i −9.28436 + 5.58622i −2.35079 + 8.46678i −10.4232 + 2.29432i −8.94724 0.973071i 53.5015 + 32.1908i
7.5 −1.28887 1.90094i 0.162417 2.99560i 1.00873 2.53171i 8.34074 3.85884i −5.90378 + 3.55218i 3.61604 13.0238i −24.0566 + 5.29526i −8.94724 0.973071i −18.0855 10.8817i
See next 80 embeddings (of 420 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.4.e.b 420
59.c even 29 1 inner 177.4.e.b 420
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.e.b 420 1.a even 1 1 trivial
177.4.e.b 420 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!51\)\( T_{2}^{404} - \)\(25\!\cdots\!45\)\( T_{2}^{403} + \)\(28\!\cdots\!94\)\( T_{2}^{402} - \)\(67\!\cdots\!33\)\( T_{2}^{401} + \)\(80\!\cdots\!60\)\( T_{2}^{400} - \)\(18\!\cdots\!71\)\( T_{2}^{399} + \)\(22\!\cdots\!54\)\( T_{2}^{398} - \)\(50\!\cdots\!08\)\( T_{2}^{397} + \)\(66\!\cdots\!65\)\( T_{2}^{396} - \)\(15\!\cdots\!41\)\( T_{2}^{395} + \)\(19\!\cdots\!98\)\( T_{2}^{394} - \)\(52\!\cdots\!45\)\( T_{2}^{393} + \)\(61\!\cdots\!73\)\( T_{2}^{392} - \)\(18\!\cdots\!79\)\( T_{2}^{391} + \)\(19\!\cdots\!41\)\( T_{2}^{390} - \)\(64\!\cdots\!18\)\( T_{2}^{389} + \)\(63\!\cdots\!73\)\( T_{2}^{388} - \)\(22\!\cdots\!52\)\( T_{2}^{387} + \)\(20\!\cdots\!79\)\( T_{2}^{386} - \)\(74\!\cdots\!82\)\( T_{2}^{385} + \)\(62\!\cdots\!97\)\( T_{2}^{384} - \)\(23\!\cdots\!66\)\( T_{2}^{383} + \)\(18\!\cdots\!92\)\( T_{2}^{382} - \)\(74\!\cdots\!25\)\( T_{2}^{381} + \)\(57\!\cdots\!17\)\( T_{2}^{380} - \)\(23\!\cdots\!39\)\( T_{2}^{379} + \)\(17\!\cdots\!92\)\( T_{2}^{378} - \)\(71\!\cdots\!70\)\( T_{2}^{377} + \)\(52\!\cdots\!93\)\( T_{2}^{376} - \)\(22\!\cdots\!94\)\( T_{2}^{375} + \)\(15\!\cdots\!13\)\( T_{2}^{374} - \)\(66\!\cdots\!53\)\( T_{2}^{373} + \)\(45\!\cdots\!37\)\( T_{2}^{372} - \)\(19\!\cdots\!98\)\( T_{2}^{371} + \)\(13\!\cdots\!11\)\( T_{2}^{370} - \)\(55\!\cdots\!22\)\( T_{2}^{369} + \)\(37\!\cdots\!85\)\( T_{2}^{368} - \)\(15\!\cdots\!21\)\( T_{2}^{367} + \)\(10\!\cdots\!45\)\( T_{2}^{366} - \)\(43\!\cdots\!41\)\( T_{2}^{365} + \)\(28\!\cdots\!75\)\( T_{2}^{364} - \)\(11\!\cdots\!58\)\( T_{2}^{363} + \)\(77\!\cdots\!05\)\( T_{2}^{362} - \)\(30\!\cdots\!07\)\( T_{2}^{361} + \)\(20\!\cdots\!51\)\( T_{2}^{360} - \)\(79\!\cdots\!75\)\( T_{2}^{359} + \)\(51\!\cdots\!89\)\( T_{2}^{358} - \)\(19\!\cdots\!17\)\( T_{2}^{357} + \)\(12\!\cdots\!11\)\( T_{2}^{356} - \)\(48\!\cdots\!58\)\( T_{2}^{355} + \)\(30\!\cdots\!33\)\( T_{2}^{354} - \)\(11\!\cdots\!86\)\( T_{2}^{353} + \)\(73\!\cdots\!10\)\( T_{2}^{352} - \)\(27\!\cdots\!32\)\( T_{2}^{351} + \)\(17\!\cdots\!70\)\( T_{2}^{350} - \)\(61\!\cdots\!54\)\( T_{2}^{349} + \)\(39\!\cdots\!99\)\( T_{2}^{348} - \)\(13\!\cdots\!36\)\( T_{2}^{347} + \)\(88\!\cdots\!27\)\( T_{2}^{346} - \)\(29\!\cdots\!50\)\( T_{2}^{345} + \)\(19\!\cdots\!92\)\( T_{2}^{344} - \)\(60\!\cdots\!69\)\( T_{2}^{343} + \)\(43\!\cdots\!14\)\( T_{2}^{342} - \)\(12\!\cdots\!61\)\( T_{2}^{341} + \)\(97\!\cdots\!19\)\( T_{2}^{340} - \)\(24\!\cdots\!84\)\( T_{2}^{339} + \)\(21\!\cdots\!36\)\( T_{2}^{338} - \)\(45\!\cdots\!03\)\( T_{2}^{337} + \)\(49\!\cdots\!32\)\( T_{2}^{336} - \)\(77\!\cdots\!62\)\( T_{2}^{335} + \)\(10\!\cdots\!37\)\( T_{2}^{334} - \)\(11\!\cdots\!05\)\( T_{2}^{333} + \)\(24\!\cdots\!26\)\( T_{2}^{332} - \)\(13\!\cdots\!65\)\( T_{2}^{331} + \)\(53\!\cdots\!37\)\( T_{2}^{330} - \)\(59\!\cdots\!52\)\( T_{2}^{329} + \)\(11\!\cdots\!06\)\( T_{2}^{328} + \)\(31\!\cdots\!69\)\( T_{2}^{327} + \)\(24\!\cdots\!26\)\( T_{2}^{326} + \)\(14\!\cdots\!16\)\( T_{2}^{325} + \)\(52\!\cdots\!02\)\( T_{2}^{324} + \)\(44\!\cdots\!77\)\( T_{2}^{323} + \)\(11\!\cdots\!51\)\( T_{2}^{322} + \)\(11\!\cdots\!19\)\( T_{2}^{321} + \)\(22\!\cdots\!80\)\( T_{2}^{320} + \)\(28\!\cdots\!49\)\( T_{2}^{319} + \)\(47\!\cdots\!00\)\( T_{2}^{318} + \)\(72\!\cdots\!39\)\( T_{2}^{317} + \)\(10\!\cdots\!38\)\( T_{2}^{316} + \)\(18\!\cdots\!71\)\( T_{2}^{315} + \)\(21\!\cdots\!88\)\( T_{2}^{314} + \)\(44\!\cdots\!22\)\( T_{2}^{313} + \)\(45\!\cdots\!85\)\( T_{2}^{312} + \)\(98\!\cdots\!63\)\( T_{2}^{311} + \)\(91\!\cdots\!81\)\( T_{2}^{310} + \)\(19\!\cdots\!30\)\( T_{2}^{309} + \)\(17\!\cdots\!08\)\( T_{2}^{308} + \)\(31\!\cdots\!56\)\( T_{2}^{307} + \)\(29\!\cdots\!40\)\( T_{2}^{306} + \)\(43\!\cdots\!92\)\( T_{2}^{305} + \)\(48\!\cdots\!34\)\( T_{2}^{304} + \)\(48\!\cdots\!52\)\( T_{2}^{303} + \)\(78\!\cdots\!08\)\( T_{2}^{302} + \)\(47\!\cdots\!54\)\( T_{2}^{301} + \)\(13\!\cdots\!42\)\( T_{2}^{300} + \)\(61\!\cdots\!40\)\( T_{2}^{299} + \)\(24\!\cdots\!90\)\( T_{2}^{298} + \)\(14\!\cdots\!86\)\( T_{2}^{297} + \)\(46\!\cdots\!81\)\( T_{2}^{296} + \)\(36\!\cdots\!97\)\( T_{2}^{295} + \)\(87\!\cdots\!65\)\( T_{2}^{294} + \)\(82\!\cdots\!65\)\( T_{2}^{293} + \)\(16\!\cdots\!14\)\( T_{2}^{292} + \)\(17\!\cdots\!14\)\( T_{2}^{291} + \)\(29\!\cdots\!75\)\( T_{2}^{290} + \)\(35\!\cdots\!72\)\( T_{2}^{289} + \)\(54\!\cdots\!31\)\( T_{2}^{288} + \)\(72\!\cdots\!51\)\( T_{2}^{287} + \)\(10\!\cdots\!79\)\( T_{2}^{286} + \)\(14\!\cdots\!08\)\( T_{2}^{285} + \)\(17\!\cdots\!52\)\( T_{2}^{284} + \)\(28\!\cdots\!18\)\( T_{2}^{283} + \)\(31\!\cdots\!30\)\( T_{2}^{282} + \)\(53\!\cdots\!12\)\( T_{2}^{281} + \)\(53\!\cdots\!50\)\( T_{2}^{280} + \)\(99\!\cdots\!70\)\( T_{2}^{279} + \)\(90\!\cdots\!30\)\( T_{2}^{278} + \)\(17\!\cdots\!52\)\( T_{2}^{277} + \)\(14\!\cdots\!69\)\( T_{2}^{276} + \)\(30\!\cdots\!47\)\( T_{2}^{275} + \)\(24\!\cdots\!87\)\( T_{2}^{274} + \)\(50\!\cdots\!16\)\( T_{2}^{273} + \)\(38\!\cdots\!47\)\( T_{2}^{272} + \)\(80\!\cdots\!01\)\( T_{2}^{271} + \)\(61\!\cdots\!58\)\( T_{2}^{270} + \)\(12\!\cdots\!47\)\( T_{2}^{269} + \)\(97\!\cdots\!08\)\( T_{2}^{268} + \)\(19\!\cdots\!43\)\( T_{2}^{267} + \)\(15\!\cdots\!21\)\( T_{2}^{266} + \)\(28\!\cdots\!78\)\( T_{2}^{265} + \)\(22\!\cdots\!53\)\( T_{2}^{264} + \)\(39\!\cdots\!03\)\( T_{2}^{263} + \)\(34\!\cdots\!27\)\( T_{2}^{262} + \)\(53\!\cdots\!78\)\( T_{2}^{261} + \)\(50\!\cdots\!38\)\( T_{2}^{260} + \)\(69\!\cdots\!02\)\( T_{2}^{259} + \)\(73\!\cdots\!47\)\( T_{2}^{258} + \)\(84\!\cdots\!36\)\( T_{2}^{257} + \)\(10\!\cdots\!24\)\( T_{2}^{256} + \)\(91\!\cdots\!12\)\( T_{2}^{255} + \)\(14\!\cdots\!07\)\( T_{2}^{254} + \)\(84\!\cdots\!71\)\( T_{2}^{253} + \)\(20\!\cdots\!45\)\( T_{2}^{252} + \)\(48\!\cdots\!55\)\( T_{2}^{251} + \)\(28\!\cdots\!71\)\( T_{2}^{250} - \)\(30\!\cdots\!56\)\( T_{2}^{249} + \)\(38\!\cdots\!53\)\( T_{2}^{248} - \)\(18\!\cdots\!00\)\( T_{2}^{247} + \)\(52\!\cdots\!91\)\( T_{2}^{246} - \)\(43\!\cdots\!40\)\( T_{2}^{245} + \)\(69\!\cdots\!17\)\( T_{2}^{244} - \)\(83\!\cdots\!59\)\( T_{2}^{243} + \)\(92\!\cdots\!23\)\( T_{2}^{242} - \)\(14\!\cdots\!20\)\( T_{2}^{241} + \)\(12\!\cdots\!89\)\( T_{2}^{240} - \)\(23\!\cdots\!96\)\( T_{2}^{239} + \)\(15\!\cdots\!65\)\( T_{2}^{238} - \)\(34\!\cdots\!13\)\( T_{2}^{237} + \)\(20\!\cdots\!50\)\( T_{2}^{236} - \)\(48\!\cdots\!82\)\( T_{2}^{235} + \)\(25\!\cdots\!28\)\( T_{2}^{234} - \)\(63\!\cdots\!61\)\( T_{2}^{233} + \)\(30\!\cdots\!15\)\( T_{2}^{232} - \)\(79\!\cdots\!50\)\( T_{2}^{231} + \)\(35\!\cdots\!99\)\( T_{2}^{230} - \)\(92\!\cdots\!65\)\( T_{2}^{229} + \)\(39\!\cdots\!36\)\( T_{2}^{228} - \)\(10\!\cdots\!73\)\( T_{2}^{227} + \)\(40\!\cdots\!88\)\( T_{2}^{226} - \)\(10\!\cdots\!48\)\( T_{2}^{225} + \)\(38\!\cdots\!01\)\( T_{2}^{224} - \)\(90\!\cdots\!58\)\( T_{2}^{223} + \)\(33\!\cdots\!50\)\( T_{2}^{222} - \)\(70\!\cdots\!12\)\( T_{2}^{221} + \)\(25\!\cdots\!06\)\( T_{2}^{220} - \)\(42\!\cdots\!82\)\( T_{2}^{219} + \)\(15\!\cdots\!13\)\( T_{2}^{218} - \)\(11\!\cdots\!98\)\( T_{2}^{217} + \)\(53\!\cdots\!94\)\( T_{2}^{216} + \)\(16\!\cdots\!01\)\( T_{2}^{215} - \)\(26\!\cdots\!51\)\( T_{2}^{214} + \)\(36\!\cdots\!09\)\( T_{2}^{213} - \)\(76\!\cdots\!08\)\( T_{2}^{212} + \)\(46\!\cdots\!67\)\( T_{2}^{211} - \)\(99\!\cdots\!96\)\( T_{2}^{210} + \)\(53\!\cdots\!99\)\( T_{2}^{209} - \)\(11\!\cdots\!98\)\( T_{2}^{208} + \)\(55\!\cdots\!75\)\( T_{2}^{207} - \)\(72\!\cdots\!65\)\( T_{2}^{206} + \)\(26\!\cdots\!76\)\( T_{2}^{205} + \)\(94\!\cdots\!88\)\( T_{2}^{204} - \)\(48\!\cdots\!27\)\( T_{2}^{203} + \)\(39\!\cdots\!14\)\( T_{2}^{202} - \)\(15\!\cdots\!16\)\( T_{2}^{201} + \)\(70\!\cdots\!90\)\( T_{2}^{200} - \)\(22\!\cdots\!87\)\( T_{2}^{199} + \)\(80\!\cdots\!86\)\( T_{2}^{198} - \)\(20\!\cdots\!99\)\( T_{2}^{197} + \)\(56\!\cdots\!29\)\( T_{2}^{196} - \)\(86\!\cdots\!71\)\( T_{2}^{195} + \)\(97\!\cdots\!69\)\( T_{2}^{194} + \)\(58\!\cdots\!97\)\( T_{2}^{193} - \)\(29\!\cdots\!44\)\( T_{2}^{192} + \)\(13\!\cdots\!34\)\( T_{2}^{191} - \)\(38\!\cdots\!72\)\( T_{2}^{190} + \)\(11\!\cdots\!03\)\( T_{2}^{189} - \)\(21\!\cdots\!55\)\( T_{2}^{188} + \)\(36\!\cdots\!38\)\( T_{2}^{187} + \)\(86\!\cdots\!90\)\( T_{2}^{186} - \)\(25\!\cdots\!01\)\( T_{2}^{185} + \)\(12\!\cdots\!49\)\( T_{2}^{184} - \)\(43\!\cdots\!09\)\( T_{2}^{183} + \)\(12\!\cdots\!97\)\( T_{2}^{182} - \)\(34\!\cdots\!82\)\( T_{2}^{181} + \)\(83\!\cdots\!81\)\( T_{2}^{180} - \)\(21\!\cdots\!06\)\( T_{2}^{179} + \)\(57\!\cdots\!60\)\( T_{2}^{178} - \)\(13\!\cdots\!59\)\( T_{2}^{177} + \)\(44\!\cdots\!74\)\( T_{2}^{176} - \)\(80\!\cdots\!08\)\( T_{2}^{175} + \)\(35\!\cdots\!05\)\( T_{2}^{174} - \)\(45\!\cdots\!03\)\( T_{2}^{173} + \)\(26\!\cdots\!36\)\( T_{2}^{172} - \)\(30\!\cdots\!78\)\( T_{2}^{171} + \)\(20\!\cdots\!19\)\( T_{2}^{170} - \)\(25\!\cdots\!60\)\( T_{2}^{169} + \)\(15\!\cdots\!70\)\( T_{2}^{168} - \)\(22\!\cdots\!09\)\( T_{2}^{167} + \)\(11\!\cdots\!92\)\( T_{2}^{166} - \)\(18\!\cdots\!81\)\( T_{2}^{165} + \)\(78\!\cdots\!15\)\( T_{2}^{164} - \)\(13\!\cdots\!81\)\( T_{2}^{163} + \)\(52\!\cdots\!22\)\( T_{2}^{162} - \)\(90\!\cdots\!50\)\( T_{2}^{161} + \)\(34\!\cdots\!84\)\( T_{2}^{160} - \)\(54\!\cdots\!88\)\( T_{2}^{159} + \)\(21\!\cdots\!76\)\( T_{2}^{158} - \)\(28\!\cdots\!57\)\( T_{2}^{157} + \)\(13\!\cdots\!10\)\( T_{2}^{156} - \)\(12\!\cdots\!21\)\( T_{2}^{155} + \)\(76\!\cdots\!53\)\( T_{2}^{154} - \)\(49\!\cdots\!32\)\( T_{2}^{153} + \)\(41\!\cdots\!93\)\( T_{2}^{152} - \)\(20\!\cdots\!35\)\( T_{2}^{151} + \)\(20\!\cdots\!35\)\( T_{2}^{150} - \)\(85\!\cdots\!76\)\( T_{2}^{149} + \)\(93\!\cdots\!72\)\( T_{2}^{148} - \)\(34\!\cdots\!30\)\( T_{2}^{147} + \)\(41\!\cdots\!73\)\( T_{2}^{146} - \)\(14\!\cdots\!41\)\( T_{2}^{145} + \)\(16\!\cdots\!55\)\( T_{2}^{144} - \)\(65\!\cdots\!86\)\( T_{2}^{143} + \)\(63\!\cdots\!08\)\( T_{2}^{142} - \)\(31\!\cdots\!94\)\( T_{2}^{141} + \)\(21\!\cdots\!61\)\( T_{2}^{140} - \)\(14\!\cdots\!88\)\( T_{2}^{139} + \)\(58\!\cdots\!36\)\( T_{2}^{138} - \)\(65\!\cdots\!84\)\( T_{2}^{137} + \)\(13\!\cdots\!16\)\( T_{2}^{136} - \)\(20\!\cdots\!72\)\( T_{2}^{135} + \)\(28\!\cdots\!32\)\( T_{2}^{134} - \)\(39\!\cdots\!72\)\( T_{2}^{133} + \)\(79\!\cdots\!76\)\( T_{2}^{132} + \)\(25\!\cdots\!92\)\( T_{2}^{131} + \)\(42\!\cdots\!08\)\( T_{2}^{130} + \)\(75\!\cdots\!32\)\( T_{2}^{129} + \)\(21\!\cdots\!52\)\( T_{2}^{128} + \)\(53\!\cdots\!44\)\( T_{2}^{127} + \)\(13\!\cdots\!88\)\( T_{2}^{126} + \)\(28\!\cdots\!48\)\( T_{2}^{125} + \)\(70\!\cdots\!08\)\( T_{2}^{124} + \)\(14\!\cdots\!72\)\( T_{2}^{123} + \)\(30\!\cdots\!64\)\( T_{2}^{122} + \)\(61\!\cdots\!36\)\( T_{2}^{121} + \)\(13\!\cdots\!72\)\( T_{2}^{120} + \)\(23\!\cdots\!76\)\( T_{2}^{119} + \)\(46\!\cdots\!80\)\( T_{2}^{118} + \)\(88\!\cdots\!04\)\( T_{2}^{117} + \)\(16\!\cdots\!44\)\( T_{2}^{116} + \)\(29\!\cdots\!28\)\( T_{2}^{115} + \)\(60\!\cdots\!12\)\( T_{2}^{114} + \)\(10\!\cdots\!40\)\( T_{2}^{113} + \)\(19\!\cdots\!80\)\( T_{2}^{112} + \)\(34\!\cdots\!72\)\( T_{2}^{111} + \)\(66\!\cdots\!32\)\( T_{2}^{110} + \)\(11\!\cdots\!60\)\( T_{2}^{109} + \)\(22\!\cdots\!04\)\( T_{2}^{108} + \)\(43\!\cdots\!96\)\( T_{2}^{107} + \)\(77\!\cdots\!28\)\( T_{2}^{106} + \)\(13\!\cdots\!24\)\( T_{2}^{105} + \)\(25\!\cdots\!92\)\( T_{2}^{104} + \)\(42\!\cdots\!76\)\( T_{2}^{103} + \)\(73\!\cdots\!32\)\( T_{2}^{102} + \)\(12\!\cdots\!80\)\( T_{2}^{101} + \)\(22\!\cdots\!00\)\( T_{2}^{100} + \)\(38\!\cdots\!24\)\( T_{2}^{99} + \)\(67\!\cdots\!40\)\( T_{2}^{98} + \)\(10\!\cdots\!88\)\( T_{2}^{97} + \)\(17\!\cdots\!68\)\( T_{2}^{96} + \)\(27\!\cdots\!36\)\( T_{2}^{95} + \)\(43\!\cdots\!12\)\( T_{2}^{94} + \)\(69\!\cdots\!44\)\( T_{2}^{93} + \)\(11\!\cdots\!04\)\( T_{2}^{92} + \)\(19\!\cdots\!24\)\( T_{2}^{91} + \)\(32\!\cdots\!04\)\( T_{2}^{90} + \)\(50\!\cdots\!08\)\( T_{2}^{89} + \)\(70\!\cdots\!68\)\( T_{2}^{88} + \)\(92\!\cdots\!44\)\( T_{2}^{87} + \)\(11\!\cdots\!28\)\( T_{2}^{86} + \)\(14\!\cdots\!16\)\( T_{2}^{85} + \)\(16\!\cdots\!72\)\( T_{2}^{84} + \)\(18\!\cdots\!32\)\( T_{2}^{83} + \)\(20\!\cdots\!20\)\( T_{2}^{82} + \)\(15\!\cdots\!44\)\( T_{2}^{81} + \)\(10\!\cdots\!44\)\( T_{2}^{80} - \)\(28\!\cdots\!48\)\( T_{2}^{79} - \)\(70\!\cdots\!20\)\( T_{2}^{78} - \)\(12\!\cdots\!04\)\( T_{2}^{77} - \)\(17\!\cdots\!08\)\( T_{2}^{76} - \)\(22\!\cdots\!48\)\( T_{2}^{75} - \)\(23\!\cdots\!44\)\( T_{2}^{74} - \)\(19\!\cdots\!56\)\( T_{2}^{73} - \)\(72\!\cdots\!32\)\( T_{2}^{72} + \)\(12\!\cdots\!32\)\( T_{2}^{71} + \)\(39\!\cdots\!36\)\( T_{2}^{70} + \)\(70\!\cdots\!16\)\( T_{2}^{69} + \)\(10\!\cdots\!60\)\( T_{2}^{68} + \)\(13\!\cdots\!04\)\( T_{2}^{67} + \)\(15\!\cdots\!20\)\( T_{2}^{66} + \)\(16\!\cdots\!20\)\( T_{2}^{65} + \)\(16\!\cdots\!60\)\( T_{2}^{64} + \)\(15\!\cdots\!72\)\( T_{2}^{63} + \)\(13\!\cdots\!04\)\( T_{2}^{62} + \)\(11\!\cdots\!28\)\( T_{2}^{61} + \)\(86\!\cdots\!40\)\( T_{2}^{60} + \)\(63\!\cdots\!16\)\( T_{2}^{59} + \)\(44\!\cdots\!12\)\( T_{2}^{58} + \)\(28\!\cdots\!28\)\( T_{2}^{57} + \)\(18\!\cdots\!60\)\( T_{2}^{56} + \)\(10\!\cdots\!64\)\( T_{2}^{55} + \)\(65\!\cdots\!24\)\( T_{2}^{54} + \)\(37\!\cdots\!00\)\( T_{2}^{53} + \)\(22\!\cdots\!36\)\( T_{2}^{52} + \)\(13\!\cdots\!68\)\( T_{2}^{51} + \)\(85\!\cdots\!72\)\( T_{2}^{50} + \)\(46\!\cdots\!88\)\( T_{2}^{49} + \)\(32\!\cdots\!08\)\( T_{2}^{48} + \)\(15\!\cdots\!32\)\( T_{2}^{47} + \)\(11\!\cdots\!84\)\( T_{2}^{46} + \)\(47\!\cdots\!28\)\( T_{2}^{45} + \)\(34\!\cdots\!28\)\( T_{2}^{44} + \)\(12\!\cdots\!40\)\( T_{2}^{43} + \)\(83\!\cdots\!12\)\( T_{2}^{42} + \)\(31\!\cdots\!16\)\( T_{2}^{41} + \)\(13\!\cdots\!76\)\( T_{2}^{40} + \)\(75\!\cdots\!56\)\( T_{2}^{39} + \)\(13\!\cdots\!00\)\( T_{2}^{38} + \)\(14\!\cdots\!56\)\( T_{2}^{37} + \)\(18\!\cdots\!60\)\( T_{2}^{36} + \)\(12\!\cdots\!04\)\( T_{2}^{35} + \)\(38\!\cdots\!24\)\( T_{2}^{34} + \)\(96\!\cdots\!48\)\( T_{2}^{33} + \)\(35\!\cdots\!88\)\( T_{2}^{32} + \)\(22\!\cdots\!76\)\( T_{2}^{31} - \)\(61\!\cdots\!28\)\( T_{2}^{30} - \)\(67\!\cdots\!12\)\( T_{2}^{29} + \)\(66\!\cdots\!16\)\( T_{2}^{28} - \)\(23\!\cdots\!32\)\( T_{2}^{27} + \)\(93\!\cdots\!04\)\( T_{2}^{26} - \)\(45\!\cdots\!80\)\( T_{2}^{25} + \)\(16\!\cdots\!24\)\( T_{2}^{24} - \)\(34\!\cdots\!60\)\( T_{2}^{23} + \)\(24\!\cdots\!20\)\( T_{2}^{22} + \)\(18\!\cdots\!48\)\( T_{2}^{21} - \)\(22\!\cdots\!52\)\( T_{2}^{20} + \)\(12\!\cdots\!40\)\( T_{2}^{19} - \)\(28\!\cdots\!96\)\( T_{2}^{18} - \)\(16\!\cdots\!52\)\( T_{2}^{17} + \)\(35\!\cdots\!84\)\( T_{2}^{16} - \)\(15\!\cdots\!84\)\( T_{2}^{15} + \)\(50\!\cdots\!08\)\( T_{2}^{14} - \)\(11\!\cdots\!04\)\( T_{2}^{13} + \)\(18\!\cdots\!08\)\( T_{2}^{12} - \)\(12\!\cdots\!68\)\( T_{2}^{11} - \)\(82\!\cdots\!76\)\( T_{2}^{10} + \)\(19\!\cdots\!40\)\( T_{2}^{9} + \)\(74\!\cdots\!92\)\( T_{2}^{8} - \)\(56\!\cdots\!96\)\( T_{2}^{7} + \)\(84\!\cdots\!04\)\( T_{2}^{6} - \)\(59\!\cdots\!28\)\( T_{2}^{5} + \)\(28\!\cdots\!20\)\( T_{2}^{4} - \)\(94\!\cdots\!80\)\( T_{2}^{3} + \)\(18\!\cdots\!76\)\( T_{2}^{2} - \)\(18\!\cdots\!84\)\( T_{2} + \)\(72\!\cdots\!36\)\( \)">\(T_{2}^{420} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(177, [\chi])\).