Properties

Label 177.4.e.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 420q - 2q^{2} + 45q^{3} - 64q^{4} + 14q^{5} + 6q^{6} + 6q^{7} + 18q^{8} - 135q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 420q - 2q^{2} + 45q^{3} - 64q^{4} + 14q^{5} + 6q^{6} + 6q^{7} + 18q^{8} - 135q^{9} + 854q^{10} + 32q^{11} + 192q^{12} - 22q^{13} - 884q^{14} - 42q^{15} - 268q^{16} - 56q^{17} - 18q^{18} - 112q^{19} - 250q^{20} + 243q^{21} + 108q^{22} + 16q^{23} - 54q^{24} - 197q^{25} - 328q^{26} + 405q^{27} + 236q^{28} - 54q^{29} - 126q^{30} + 66q^{31} + 857q^{32} - 96q^{33} + 112q^{34} + 532q^{35} - 576q^{36} + 110q^{37} + 56q^{38} + 66q^{39} - 6044q^{40} + 304q^{41} - 306q^{42} - 324q^{43} - 568q^{44} + 126q^{45} + 5842q^{46} + 4082q^{47} + 804q^{48} + 739q^{49} + 576q^{50} + 168q^{51} - 1372q^{52} - 1498q^{53} + 54q^{54} - 4422q^{55} - 12036q^{56} + 336q^{57} + 6298q^{58} - 3305q^{59} + 750q^{60} - 1718q^{61} - 4333q^{62} + 54q^{63} - 6250q^{64} - 5884q^{65} - 324q^{66} + 382q^{67} + 734q^{68} + 3780q^{69} + 8082q^{70} + 8638q^{71} + 162q^{72} + 8894q^{73} + 12428q^{74} + 243q^{75} - 15468q^{76} - 9916q^{77} + 984q^{78} - 578q^{79} + 24570q^{80} - 1215q^{81} - 1382q^{82} - 32q^{83} + 7557q^{84} + 708q^{85} + 2296q^{86} - 5928q^{87} - 17646q^{88} - 2284q^{89} + 378q^{90} + 11124q^{91} + 762q^{92} - 198q^{93} - 1232q^{94} + 24184q^{95} - 396q^{96} - 658q^{97} - 4356q^{98} + 288q^{99} + O(q^{100}) \)

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 −5.26239 0.572319i −1.94216 + 2.28649i 19.5522 + 4.30376i −16.6552 12.6610i 11.5290 10.9208i −1.92301 + 4.82639i −60.2975 20.3166i −1.45604 8.88144i 80.4000 + 76.1589i
4.2 −4.95061 0.538411i −1.94216 + 2.28649i 16.4057 + 3.61116i 10.7888 + 8.20142i 10.8459 10.2738i 11.1308 27.9361i −41.5207 13.9900i −1.45604 8.88144i −48.9953 46.4108i
4.3 −4.04316 0.439720i −1.94216 + 2.28649i 8.34080 + 1.83595i 2.28067 + 1.73372i 8.85787 8.39062i 2.30882 5.79471i −2.08305 0.701862i −1.45604 8.88144i −8.45878 8.01258i
4.4 −3.43911 0.374026i −1.94216 + 2.28649i 3.87463 + 0.852870i −6.75733 5.13679i 7.53451 7.13706i 2.82453 7.08903i 13.2201 + 4.45438i −1.45604 8.88144i 21.3179 + 20.1934i
4.5 −2.99643 0.325881i −1.94216 + 2.28649i 1.05942 + 0.233196i −3.11311 2.36653i 6.56466 6.21838i −9.33995 + 23.4415i 19.7521 + 6.65525i −1.45604 8.88144i 8.55701 + 8.10563i
4.6 −2.03313 0.221116i −1.94216 + 2.28649i −3.72826 0.820651i 16.1269 + 12.2594i 4.45423 4.21927i −2.36180 + 5.92767i 22.9030 + 7.71693i −1.45604 8.88144i −30.0773 28.4907i
4.7 −0.689793 0.0750196i −1.94216 + 2.28649i −7.34278 1.61627i −13.2529 10.0746i 1.51122 1.43150i 3.00216 7.53485i 10.2041 + 3.43815i −1.45604 8.88144i 8.38595 + 7.94360i
4.8 0.115030 + 0.0125102i −1.94216 + 2.28649i −7.79989 1.71689i −3.58691 2.72670i −0.252010 + 0.238717i −13.1064 + 32.8946i −1.75295 0.590637i −1.45604 8.88144i −0.378490 0.358525i
4.9 0.857263 + 0.0932330i −1.94216 + 2.28649i −7.08676 1.55991i 6.24712 + 4.74894i −1.87812 + 1.77905i 0.937924 2.35401i −12.4672 4.20069i −1.45604 8.88144i 4.91267 + 4.65353i
4.10 1.07661 + 0.117088i −1.94216 + 2.28649i −6.66759 1.46765i 6.87670 + 5.22753i −2.35866 + 2.23424i 7.63111 19.1526i −15.2167 5.12709i −1.45604 8.88144i 6.79142 + 6.43318i
4.11 2.45810 + 0.267335i −1.94216 + 2.28649i −1.84216 0.405491i 7.30295 + 5.55156i −5.38528 + 5.10121i −2.79914 + 7.02531i −23.1651 7.80525i −1.45604 8.88144i 16.4673 + 15.5986i
4.12 3.41158 + 0.371031i −1.94216 + 2.28649i 3.68824 + 0.811842i −7.61387 5.78791i −7.47419 + 7.07992i 7.80642 19.5926i −13.7350 4.62785i −1.45604 8.88144i −23.8278 22.5709i
4.13 3.96175 + 0.430866i −1.94216 + 2.28649i 7.69682 + 1.69420i −13.8169 10.5033i −8.67951 + 8.22167i −7.10268 + 17.8264i −0.449126 0.151328i −1.45604 8.88144i −50.2135 47.5647i
4.14 4.44507 + 0.483430i −1.94216 + 2.28649i 11.7120 + 2.57800i 8.85655 + 6.73258i −9.73839 + 9.22469i −9.70013 + 24.3455i 16.9165 + 5.69982i −1.45604 8.88144i 36.1133 + 34.2083i
4.15 5.10094 + 0.554760i −1.94216 + 2.28649i 17.8988 + 3.93983i 10.4418 + 7.93765i −11.1753 + 10.5858i 11.7316 29.4442i 50.2158 + 16.9197i −1.45604 8.88144i 48.8595 + 46.2821i
7.1 −2.99760 4.42113i −0.162417 + 2.99560i −7.59966 + 19.0737i −18.0173 + 8.33567i 13.7308 8.26154i −1.41944 + 5.11237i 65.3749 14.3901i −8.94724 0.973071i 90.8616 + 54.6696i
7.2 −2.77836 4.09778i −0.162417 + 2.99560i −6.11139 + 15.3384i 11.4034 5.27577i 12.7266 7.65732i −0.571992 + 2.06013i 41.1522 9.05829i −8.94724 0.973071i −53.3017 32.0706i
7.3 −2.23303 3.29348i −0.162417 + 2.99560i −2.89945 + 7.27707i −6.22504 + 2.88001i 10.2286 6.15436i −3.74407 + 13.4849i −0.647313 + 0.142484i −8.94724 0.973071i 23.3860 + 14.0709i
7.4 −1.93104 2.84807i −0.162417 + 2.99560i −1.42147 + 3.56762i −12.2161 + 5.65175i 8.84530 5.32204i 8.68646 31.2858i −13.9786 + 3.07691i −8.94724 0.973071i 39.6862 + 23.8784i
7.5 −1.59837 2.35742i −0.162417 + 2.99560i −0.0415309 + 0.104235i 14.5110 6.71353i 7.32148 4.40519i 0.937125 3.37522i −21.9407 + 4.82952i −8.94724 0.973071i −39.0206 23.4779i
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.a 420
59.c even 29 1 inner 177.4.e.a 420
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.e.a 420 1.a even 1 1 trivial
177.4.e.a 420 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(12\!\cdots\!10\)\( T_{2}^{405} + \)\(15\!\cdots\!41\)\( T_{2}^{404} + \)\(45\!\cdots\!51\)\( T_{2}^{403} + \)\(52\!\cdots\!58\)\( T_{2}^{402} + \)\(16\!\cdots\!47\)\( T_{2}^{401} + \)\(16\!\cdots\!20\)\( T_{2}^{400} + \)\(58\!\cdots\!65\)\( T_{2}^{399} + \)\(52\!\cdots\!82\)\( T_{2}^{398} + \)\(19\!\cdots\!40\)\( T_{2}^{397} + \)\(16\!\cdots\!47\)\( T_{2}^{396} + \)\(65\!\cdots\!37\)\( T_{2}^{395} + \)\(49\!\cdots\!90\)\( T_{2}^{394} + \)\(21\!\cdots\!59\)\( T_{2}^{393} + \)\(14\!\cdots\!35\)\( T_{2}^{392} + \)\(65\!\cdots\!27\)\( T_{2}^{391} + \)\(43\!\cdots\!59\)\( T_{2}^{390} + \)\(19\!\cdots\!84\)\( T_{2}^{389} + \)\(12\!\cdots\!75\)\( T_{2}^{388} + \)\(55\!\cdots\!48\)\( T_{2}^{387} + \)\(36\!\cdots\!77\)\( T_{2}^{386} + \)\(15\!\cdots\!42\)\( T_{2}^{385} + \)\(10\!\cdots\!81\)\( T_{2}^{384} + \)\(41\!\cdots\!84\)\( T_{2}^{383} + \)\(29\!\cdots\!24\)\( T_{2}^{382} + \)\(10\!\cdots\!41\)\( T_{2}^{381} + \)\(79\!\cdots\!23\)\( T_{2}^{380} + \)\(27\!\cdots\!93\)\( T_{2}^{379} + \)\(21\!\cdots\!60\)\( T_{2}^{378} + \)\(68\!\cdots\!54\)\( T_{2}^{377} + \)\(54\!\cdots\!31\)\( T_{2}^{376} + \)\(16\!\cdots\!34\)\( T_{2}^{375} + \)\(13\!\cdots\!09\)\( T_{2}^{374} + \)\(42\!\cdots\!03\)\( T_{2}^{373} + \)\(35\!\cdots\!37\)\( T_{2}^{372} + \)\(10\!\cdots\!46\)\( T_{2}^{371} + \)\(89\!\cdots\!75\)\( T_{2}^{370} + \)\(28\!\cdots\!54\)\( T_{2}^{369} + \)\(23\!\cdots\!75\)\( T_{2}^{368} + \)\(75\!\cdots\!87\)\( T_{2}^{367} + \)\(63\!\cdots\!37\)\( T_{2}^{366} + \)\(20\!\cdots\!59\)\( T_{2}^{365} + \)\(18\!\cdots\!93\)\( T_{2}^{364} + \)\(58\!\cdots\!38\)\( T_{2}^{363} + \)\(51\!\cdots\!13\)\( T_{2}^{362} + \)\(16\!\cdots\!93\)\( T_{2}^{361} + \)\(14\!\cdots\!31\)\( T_{2}^{360} + \)\(43\!\cdots\!15\)\( T_{2}^{359} + \)\(36\!\cdots\!89\)\( T_{2}^{358} + \)\(11\!\cdots\!13\)\( T_{2}^{357} + \)\(92\!\cdots\!59\)\( T_{2}^{356} + \)\(31\!\cdots\!82\)\( T_{2}^{355} + \)\(23\!\cdots\!57\)\( T_{2}^{354} + \)\(83\!\cdots\!50\)\( T_{2}^{353} + \)\(57\!\cdots\!80\)\( T_{2}^{352} + \)\(21\!\cdots\!48\)\( T_{2}^{351} + \)\(13\!\cdots\!56\)\( T_{2}^{350} + \)\(53\!\cdots\!26\)\( T_{2}^{349} + \)\(33\!\cdots\!91\)\( T_{2}^{348} + \)\(12\!\cdots\!34\)\( T_{2}^{347} + \)\(81\!\cdots\!87\)\( T_{2}^{346} + \)\(28\!\cdots\!10\)\( T_{2}^{345} + \)\(19\!\cdots\!48\)\( T_{2}^{344} + \)\(64\!\cdots\!19\)\( T_{2}^{343} + \)\(46\!\cdots\!70\)\( T_{2}^{342} + \)\(14\!\cdots\!57\)\( T_{2}^{341} + \)\(10\!\cdots\!05\)\( T_{2}^{340} + \)\(31\!\cdots\!06\)\( T_{2}^{339} + \)\(24\!\cdots\!08\)\( T_{2}^{338} + \)\(66\!\cdots\!13\)\( T_{2}^{337} + \)\(54\!\cdots\!78\)\( T_{2}^{336} + \)\(13\!\cdots\!20\)\( T_{2}^{335} + \)\(12\!\cdots\!69\)\( T_{2}^{334} + \)\(28\!\cdots\!33\)\( T_{2}^{333} + \)\(26\!\cdots\!40\)\( T_{2}^{332} + \)\(55\!\cdots\!43\)\( T_{2}^{331} + \)\(58\!\cdots\!31\)\( T_{2}^{330} + \)\(10\!\cdots\!86\)\( T_{2}^{329} + \)\(12\!\cdots\!84\)\( T_{2}^{328} + \)\(19\!\cdots\!65\)\( T_{2}^{327} + \)\(25\!\cdots\!44\)\( T_{2}^{326} + \)\(35\!\cdots\!02\)\( T_{2}^{325} + \)\(52\!\cdots\!46\)\( T_{2}^{324} + \)\(62\!\cdots\!81\)\( T_{2}^{323} + \)\(10\!\cdots\!65\)\( T_{2}^{322} + \)\(10\!\cdots\!23\)\( T_{2}^{321} + \)\(21\!\cdots\!54\)\( T_{2}^{320} + \)\(17\!\cdots\!61\)\( T_{2}^{319} + \)\(43\!\cdots\!16\)\( T_{2}^{318} + \)\(29\!\cdots\!19\)\( T_{2}^{317} + \)\(86\!\cdots\!96\)\( T_{2}^{316} + \)\(56\!\cdots\!23\)\( T_{2}^{315} + \)\(17\!\cdots\!32\)\( T_{2}^{314} + \)\(11\!\cdots\!50\)\( T_{2}^{313} + \)\(34\!\cdots\!61\)\( T_{2}^{312} + \)\(27\!\cdots\!07\)\( T_{2}^{311} + \)\(70\!\cdots\!55\)\( T_{2}^{310} + \)\(63\!\cdots\!20\)\( T_{2}^{309} + \)\(13\!\cdots\!02\)\( T_{2}^{308} + \)\(14\!\cdots\!46\)\( T_{2}^{307} + \)\(27\!\cdots\!02\)\( T_{2}^{306} + \)\(33\!\cdots\!52\)\( T_{2}^{305} + \)\(52\!\cdots\!32\)\( T_{2}^{304} + \)\(73\!\cdots\!88\)\( T_{2}^{303} + \)\(99\!\cdots\!62\)\( T_{2}^{302} + \)\(15\!\cdots\!32\)\( T_{2}^{301} + \)\(18\!\cdots\!80\)\( T_{2}^{300} + \)\(31\!\cdots\!94\)\( T_{2}^{299} + \)\(34\!\cdots\!88\)\( T_{2}^{298} + \)\(63\!\cdots\!84\)\( T_{2}^{297} + \)\(62\!\cdots\!13\)\( T_{2}^{296} + \)\(12\!\cdots\!51\)\( T_{2}^{295} + \)\(11\!\cdots\!67\)\( T_{2}^{294} + \)\(23\!\cdots\!75\)\( T_{2}^{293} + \)\(20\!\cdots\!96\)\( T_{2}^{292} + \)\(44\!\cdots\!08\)\( T_{2}^{291} + \)\(35\!\cdots\!69\)\( T_{2}^{290} + \)\(81\!\cdots\!10\)\( T_{2}^{289} + \)\(62\!\cdots\!13\)\( T_{2}^{288} + \)\(14\!\cdots\!67\)\( T_{2}^{287} + \)\(10\!\cdots\!65\)\( T_{2}^{286} + \)\(25\!\cdots\!28\)\( T_{2}^{285} + \)\(17\!\cdots\!92\)\( T_{2}^{284} + \)\(42\!\cdots\!84\)\( T_{2}^{283} + \)\(28\!\cdots\!72\)\( T_{2}^{282} + \)\(68\!\cdots\!52\)\( T_{2}^{281} + \)\(46\!\cdots\!06\)\( T_{2}^{280} + \)\(10\!\cdots\!24\)\( T_{2}^{279} + \)\(72\!\cdots\!74\)\( T_{2}^{278} + \)\(15\!\cdots\!92\)\( T_{2}^{277} + \)\(10\!\cdots\!81\)\( T_{2}^{276} + \)\(22\!\cdots\!33\)\( T_{2}^{275} + \)\(16\!\cdots\!73\)\( T_{2}^{274} + \)\(29\!\cdots\!78\)\( T_{2}^{273} + \)\(23\!\cdots\!27\)\( T_{2}^{272} + \)\(33\!\cdots\!45\)\( T_{2}^{271} + \)\(32\!\cdots\!16\)\( T_{2}^{270} + \)\(31\!\cdots\!03\)\( T_{2}^{269} + \)\(44\!\cdots\!52\)\( T_{2}^{268} + \)\(21\!\cdots\!09\)\( T_{2}^{267} + \)\(63\!\cdots\!83\)\( T_{2}^{266} + \)\(10\!\cdots\!46\)\( T_{2}^{265} + \)\(10\!\cdots\!03\)\( T_{2}^{264} + \)\(44\!\cdots\!69\)\( T_{2}^{263} + \)\(18\!\cdots\!97\)\( T_{2}^{262} + \)\(22\!\cdots\!96\)\( T_{2}^{261} + \)\(37\!\cdots\!46\)\( T_{2}^{260} + \)\(72\!\cdots\!40\)\( T_{2}^{259} + \)\(72\!\cdots\!61\)\( T_{2}^{258} + \)\(17\!\cdots\!70\)\( T_{2}^{257} + \)\(13\!\cdots\!86\)\( T_{2}^{256} + \)\(33\!\cdots\!98\)\( T_{2}^{255} + \)\(21\!\cdots\!55\)\( T_{2}^{254} + \)\(56\!\cdots\!73\)\( T_{2}^{253} + \)\(31\!\cdots\!13\)\( T_{2}^{252} + \)\(80\!\cdots\!97\)\( T_{2}^{251} + \)\(41\!\cdots\!03\)\( T_{2}^{250} + \)\(10\!\cdots\!34\)\( T_{2}^{249} + \)\(48\!\cdots\!65\)\( T_{2}^{248} + \)\(10\!\cdots\!22\)\( T_{2}^{247} + \)\(50\!\cdots\!67\)\( T_{2}^{246} + \)\(91\!\cdots\!78\)\( T_{2}^{245} + \)\(44\!\cdots\!31\)\( T_{2}^{244} + \)\(63\!\cdots\!05\)\( T_{2}^{243} + \)\(35\!\cdots\!15\)\( T_{2}^{242} + \)\(36\!\cdots\!76\)\( T_{2}^{241} + \)\(29\!\cdots\!47\)\( T_{2}^{240} + \)\(30\!\cdots\!70\)\( T_{2}^{239} + \)\(33\!\cdots\!17\)\( T_{2}^{238} + \)\(68\!\cdots\!33\)\( T_{2}^{237} + \)\(51\!\cdots\!10\)\( T_{2}^{236} + \)\(15\!\cdots\!84\)\( T_{2}^{235} + \)\(84\!\cdots\!22\)\( T_{2}^{234} + \)\(26\!\cdots\!77\)\( T_{2}^{233} + \)\(12\!\cdots\!49\)\( T_{2}^{232} + \)\(37\!\cdots\!86\)\( T_{2}^{231} + \)\(16\!\cdots\!35\)\( T_{2}^{230} + \)\(45\!\cdots\!57\)\( T_{2}^{229} + \)\(20\!\cdots\!74\)\( T_{2}^{228} + \)\(49\!\cdots\!37\)\( T_{2}^{227} + \)\(23\!\cdots\!48\)\( T_{2}^{226} + \)\(48\!\cdots\!60\)\( T_{2}^{225} + \)\(26\!\cdots\!73\)\( T_{2}^{224} + \)\(45\!\cdots\!90\)\( T_{2}^{223} + \)\(30\!\cdots\!94\)\( T_{2}^{222} + \)\(46\!\cdots\!26\)\( T_{2}^{221} + \)\(36\!\cdots\!74\)\( T_{2}^{220} + \)\(52\!\cdots\!02\)\( T_{2}^{219} + \)\(43\!\cdots\!97\)\( T_{2}^{218} + \)\(67\!\cdots\!62\)\( T_{2}^{217} + \)\(51\!\cdots\!54\)\( T_{2}^{216} + \)\(88\!\cdots\!75\)\( T_{2}^{215} + \)\(57\!\cdots\!75\)\( T_{2}^{214} + \)\(11\!\cdots\!57\)\( T_{2}^{213} + \)\(60\!\cdots\!80\)\( T_{2}^{212} + \)\(13\!\cdots\!65\)\( T_{2}^{211} + \)\(61\!\cdots\!02\)\( T_{2}^{210} + \)\(15\!\cdots\!39\)\( T_{2}^{209} + \)\(60\!\cdots\!36\)\( T_{2}^{208} + \)\(17\!\cdots\!53\)\( T_{2}^{207} + \)\(58\!\cdots\!21\)\( T_{2}^{206} + \)\(17\!\cdots\!30\)\( T_{2}^{205} + \)\(53\!\cdots\!56\)\( T_{2}^{204} + \)\(15\!\cdots\!79\)\( T_{2}^{203} + \)\(47\!\cdots\!58\)\( T_{2}^{202} + \)\(12\!\cdots\!78\)\( T_{2}^{201} + \)\(41\!\cdots\!54\)\( T_{2}^{200} + \)\(99\!\cdots\!35\)\( T_{2}^{199} + \)\(35\!\cdots\!06\)\( T_{2}^{198} + \)\(79\!\cdots\!23\)\( T_{2}^{197} + \)\(30\!\cdots\!15\)\( T_{2}^{196} + \)\(62\!\cdots\!35\)\( T_{2}^{195} + \)\(24\!\cdots\!59\)\( T_{2}^{194} + \)\(45\!\cdots\!77\)\( T_{2}^{193} + \)\(18\!\cdots\!84\)\( T_{2}^{192} + \)\(32\!\cdots\!66\)\( T_{2}^{191} + \)\(13\!\cdots\!98\)\( T_{2}^{190} + \)\(21\!\cdots\!97\)\( T_{2}^{189} + \)\(10\!\cdots\!73\)\( T_{2}^{188} + \)\(14\!\cdots\!84\)\( T_{2}^{187} + \)\(80\!\cdots\!40\)\( T_{2}^{186} + \)\(85\!\cdots\!03\)\( T_{2}^{185} + \)\(62\!\cdots\!73\)\( T_{2}^{184} + \)\(41\!\cdots\!37\)\( T_{2}^{183} + \)\(46\!\cdots\!19\)\( T_{2}^{182} + \)\(11\!\cdots\!48\)\( T_{2}^{181} + \)\(33\!\cdots\!35\)\( T_{2}^{180} + \)\(12\!\cdots\!18\)\( T_{2}^{179} + \)\(25\!\cdots\!94\)\( T_{2}^{178} - \)\(22\!\cdots\!31\)\( T_{2}^{177} + \)\(17\!\cdots\!78\)\( T_{2}^{176} - \)\(45\!\cdots\!84\)\( T_{2}^{175} + \)\(12\!\cdots\!93\)\( T_{2}^{174} - \)\(57\!\cdots\!51\)\( T_{2}^{173} + \)\(88\!\cdots\!06\)\( T_{2}^{172} - \)\(85\!\cdots\!36\)\( T_{2}^{171} + \)\(64\!\cdots\!33\)\( T_{2}^{170} - \)\(70\!\cdots\!94\)\( T_{2}^{169} + \)\(48\!\cdots\!20\)\( T_{2}^{168} - \)\(60\!\cdots\!73\)\( T_{2}^{167} + \)\(31\!\cdots\!06\)\( T_{2}^{166} - \)\(37\!\cdots\!71\)\( T_{2}^{165} + \)\(20\!\cdots\!99\)\( T_{2}^{164} - \)\(21\!\cdots\!87\)\( T_{2}^{163} + \)\(99\!\cdots\!34\)\( T_{2}^{162} - \)\(11\!\cdots\!96\)\( T_{2}^{161} + \)\(52\!\cdots\!32\)\( T_{2}^{160} - \)\(22\!\cdots\!78\)\( T_{2}^{159} + \)\(25\!\cdots\!28\)\( T_{2}^{158} - \)\(32\!\cdots\!99\)\( T_{2}^{157} + \)\(14\!\cdots\!78\)\( T_{2}^{156} + \)\(11\!\cdots\!77\)\( T_{2}^{155} + \)\(85\!\cdots\!13\)\( T_{2}^{154} + \)\(17\!\cdots\!38\)\( T_{2}^{153} + \)\(32\!\cdots\!99\)\( T_{2}^{152} + \)\(10\!\cdots\!87\)\( T_{2}^{151} + \)\(22\!\cdots\!69\)\( T_{2}^{150} + \)\(12\!\cdots\!64\)\( T_{2}^{149} + \)\(12\!\cdots\!50\)\( T_{2}^{148} + \)\(11\!\cdots\!66\)\( T_{2}^{147} + \)\(82\!\cdots\!95\)\( T_{2}^{146} + \)\(13\!\cdots\!63\)\( T_{2}^{145} + \)\(52\!\cdots\!15\)\( T_{2}^{144} + \)\(72\!\cdots\!42\)\( T_{2}^{143} + \)\(24\!\cdots\!50\)\( T_{2}^{142} + \)\(33\!\cdots\!78\)\( T_{2}^{141} + \)\(12\!\cdots\!13\)\( T_{2}^{140} + \)\(22\!\cdots\!36\)\( T_{2}^{139} + \)\(62\!\cdots\!52\)\( T_{2}^{138} + \)\(12\!\cdots\!64\)\( T_{2}^{137} + \)\(28\!\cdots\!32\)\( T_{2}^{136} + \)\(81\!\cdots\!28\)\( T_{2}^{135} + \)\(18\!\cdots\!60\)\( T_{2}^{134} + \)\(53\!\cdots\!88\)\( T_{2}^{133} + \)\(11\!\cdots\!28\)\( T_{2}^{132} + \)\(30\!\cdots\!16\)\( T_{2}^{131} + \)\(72\!\cdots\!96\)\( T_{2}^{130} + \)\(16\!\cdots\!40\)\( T_{2}^{129} + \)\(40\!\cdots\!04\)\( T_{2}^{128} + \)\(80\!\cdots\!08\)\( T_{2}^{127} + \)\(18\!\cdots\!44\)\( T_{2}^{126} + \)\(36\!\cdots\!04\)\( T_{2}^{125} + \)\(83\!\cdots\!36\)\( T_{2}^{124} + \)\(17\!\cdots\!40\)\( T_{2}^{123} + \)\(37\!\cdots\!76\)\( T_{2}^{122} + \)\(78\!\cdots\!36\)\( T_{2}^{121} + \)\(14\!\cdots\!32\)\( T_{2}^{120} + \)\(27\!\cdots\!04\)\( T_{2}^{119} + \)\(53\!\cdots\!68\)\( T_{2}^{118} + \)\(97\!\cdots\!56\)\( T_{2}^{117} + \)\(20\!\cdots\!08\)\( T_{2}^{116} + \)\(35\!\cdots\!16\)\( T_{2}^{115} + \)\(66\!\cdots\!24\)\( T_{2}^{114} + \)\(10\!\cdots\!12\)\( T_{2}^{113} + \)\(16\!\cdots\!60\)\( T_{2}^{112} + \)\(28\!\cdots\!96\)\( T_{2}^{111} + \)\(48\!\cdots\!80\)\( T_{2}^{110} + \)\(95\!\cdots\!92\)\( T_{2}^{109} + \)\(18\!\cdots\!04\)\( T_{2}^{108} + \)\(30\!\cdots\!36\)\( T_{2}^{107} + \)\(52\!\cdots\!20\)\( T_{2}^{106} + \)\(69\!\cdots\!08\)\( T_{2}^{105} + \)\(98\!\cdots\!56\)\( T_{2}^{104} + \)\(11\!\cdots\!48\)\( T_{2}^{103} + \)\(13\!\cdots\!24\)\( T_{2}^{102} + \)\(15\!\cdots\!36\)\( T_{2}^{101} + \)\(23\!\cdots\!68\)\( T_{2}^{100} + \)\(19\!\cdots\!84\)\( T_{2}^{99} + \)\(41\!\cdots\!60\)\( T_{2}^{98} + \)\(67\!\cdots\!72\)\( T_{2}^{97} + \)\(16\!\cdots\!52\)\( T_{2}^{96} - \)\(36\!\cdots\!20\)\( T_{2}^{95} - \)\(90\!\cdots\!20\)\( T_{2}^{94} - \)\(10\!\cdots\!04\)\( T_{2}^{93} - \)\(73\!\cdots\!28\)\( T_{2}^{92} - \)\(23\!\cdots\!24\)\( T_{2}^{91} + \)\(28\!\cdots\!76\)\( T_{2}^{90} - \)\(40\!\cdots\!96\)\( T_{2}^{89} + \)\(66\!\cdots\!68\)\( T_{2}^{88} - \)\(59\!\cdots\!80\)\( T_{2}^{87} + \)\(11\!\cdots\!24\)\( T_{2}^{86} - \)\(15\!\cdots\!60\)\( T_{2}^{85} + \)\(26\!\cdots\!16\)\( T_{2}^{84} - \)\(38\!\cdots\!48\)\( T_{2}^{83} + \)\(52\!\cdots\!24\)\( T_{2}^{82} - \)\(69\!\cdots\!92\)\( T_{2}^{81} + \)\(90\!\cdots\!60\)\( T_{2}^{80} - \)\(11\!\cdots\!08\)\( T_{2}^{79} + \)\(15\!\cdots\!92\)\( T_{2}^{78} - \)\(19\!\cdots\!68\)\( T_{2}^{77} + \)\(24\!\cdots\!20\)\( T_{2}^{76} - \)\(30\!\cdots\!52\)\( T_{2}^{75} + \)\(35\!\cdots\!68\)\( T_{2}^{74} - \)\(39\!\cdots\!92\)\( T_{2}^{73} + \)\(44\!\cdots\!24\)\( T_{2}^{72} - \)\(51\!\cdots\!36\)\( T_{2}^{71} + \)\(60\!\cdots\!04\)\( T_{2}^{70} - \)\(69\!\cdots\!84\)\( T_{2}^{69} + \)\(77\!\cdots\!20\)\( T_{2}^{68} - \)\(84\!\cdots\!76\)\( T_{2}^{67} + \)\(91\!\cdots\!76\)\( T_{2}^{66} - \)\(99\!\cdots\!52\)\( T_{2}^{65} + \)\(10\!\cdots\!08\)\( T_{2}^{64} - \)\(11\!\cdots\!76\)\( T_{2}^{63} + \)\(11\!\cdots\!84\)\( T_{2}^{62} - \)\(10\!\cdots\!20\)\( T_{2}^{61} + \)\(94\!\cdots\!40\)\( T_{2}^{60} - \)\(83\!\cdots\!36\)\( T_{2}^{59} + \)\(74\!\cdots\!72\)\( T_{2}^{58} - \)\(66\!\cdots\!28\)\( T_{2}^{57} + \)\(59\!\cdots\!36\)\( T_{2}^{56} - \)\(50\!\cdots\!16\)\( T_{2}^{55} + \)\(41\!\cdots\!40\)\( T_{2}^{54} - \)\(32\!\cdots\!76\)\( T_{2}^{53} + \)\(24\!\cdots\!80\)\( T_{2}^{52} - \)\(17\!\cdots\!20\)\( T_{2}^{51} + \)\(11\!\cdots\!44\)\( T_{2}^{50} - \)\(69\!\cdots\!04\)\( T_{2}^{49} + \)\(37\!\cdots\!24\)\( T_{2}^{48} - \)\(17\!\cdots\!40\)\( T_{2}^{47} + \)\(54\!\cdots\!32\)\( T_{2}^{46} + \)\(20\!\cdots\!24\)\( T_{2}^{45} - \)\(62\!\cdots\!04\)\( T_{2}^{44} + \)\(79\!\cdots\!56\)\( T_{2}^{43} - \)\(81\!\cdots\!12\)\( T_{2}^{42} + \)\(72\!\cdots\!88\)\( T_{2}^{41} - \)\(56\!\cdots\!48\)\( T_{2}^{40} + \)\(37\!\cdots\!80\)\( T_{2}^{39} - \)\(18\!\cdots\!76\)\( T_{2}^{38} + \)\(41\!\cdots\!96\)\( T_{2}^{37} + \)\(45\!\cdots\!68\)\( T_{2}^{36} - \)\(75\!\cdots\!08\)\( T_{2}^{35} + \)\(69\!\cdots\!40\)\( T_{2}^{34} - \)\(49\!\cdots\!52\)\( T_{2}^{33} + \)\(30\!\cdots\!52\)\( T_{2}^{32} - \)\(15\!\cdots\!36\)\( T_{2}^{31} + \)\(76\!\cdots\!92\)\( T_{2}^{30} - \)\(33\!\cdots\!32\)\( T_{2}^{29} + \)\(13\!\cdots\!32\)\( T_{2}^{28} - \)\(51\!\cdots\!36\)\( T_{2}^{27} + \)\(18\!\cdots\!00\)\( T_{2}^{26} - \)\(62\!\cdots\!32\)\( T_{2}^{25} + \)\(19\!\cdots\!56\)\( T_{2}^{24} - \)\(60\!\cdots\!20\)\( T_{2}^{23} + \)\(17\!\cdots\!00\)\( T_{2}^{22} - \)\(48\!\cdots\!44\)\( T_{2}^{21} + \)\(12\!\cdots\!56\)\( T_{2}^{20} - \)\(29\!\cdots\!28\)\( T_{2}^{19} + \)\(60\!\cdots\!16\)\( T_{2}^{18} - \)\(10\!\cdots\!36\)\( T_{2}^{17} + \)\(14\!\cdots\!84\)\( T_{2}^{16} - \)\(15\!\cdots\!32\)\( T_{2}^{15} + \)\(86\!\cdots\!24\)\( T_{2}^{14} + \)\(16\!\cdots\!68\)\( T_{2}^{13} - \)\(25\!\cdots\!92\)\( T_{2}^{12} - \)\(31\!\cdots\!60\)\( T_{2}^{11} + \)\(98\!\cdots\!00\)\( T_{2}^{10} - \)\(18\!\cdots\!84\)\( T_{2}^{9} + \)\(25\!\cdots\!48\)\( T_{2}^{8} - \)\(29\!\cdots\!08\)\( T_{2}^{7} + \)\(30\!\cdots\!48\)\( T_{2}^{6} - \)\(32\!\cdots\!08\)\( T_{2}^{5} + \)\(37\!\cdots\!36\)\( T_{2}^{4} - \)\(38\!\cdots\!36\)\( T_{2}^{3} + \)\(28\!\cdots\!52\)\( T_{2}^{2} - \)\(12\!\cdots\!08\)\( T_{2} + \)\(25\!\cdots\!36\)\( \)">\(T_{2}^{420} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(177, [\chi])\).