# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{4}^{\mathrm{new}}(177, [\chi])$$.