Newform orbit 177.4.e.a

• 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: $2$

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.

\(\operatorname{Tr}(f)(q) = \) \( 420q - 2q^{2} + 45q^{3} - 64q^{4} + 14q^{5} + 6q^{6} + 6q^{7} + 18q^{8} - 135q^{9} + O(q^{10}) \)

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

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