Properties

Label 242.2.g.b
Level $242$
Weight $2$
Character orbit 242.g
Analytic conductor $1.932$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 242.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.93237972891\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(6\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240q + 6q^{2} - 8q^{3} + 6q^{4} + 5q^{5} - 2q^{6} - 2q^{7} + 6q^{8} - 80q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q + 6q^{2} - 8q^{3} + 6q^{4} + 5q^{5} - 2q^{6} - 2q^{7} + 6q^{8} - 80q^{9} - 22q^{10} - 10q^{11} - 13q^{12} - 30q^{13} - 13q^{14} - 30q^{15} + 6q^{16} - 4q^{17} + 12q^{18} + 4q^{19} + 5q^{20} + 20q^{21} - 11q^{22} - 21q^{23} - 2q^{24} - 7q^{25} - 7q^{26} + 46q^{27} - 2q^{28} - 13q^{29} - 19q^{30} - 68q^{31} - 24q^{32} - 11q^{33} + 6q^{34} - 16q^{35} + 12q^{36} - 75q^{37} - 44q^{38} + 178q^{39} - 16q^{40} - 4q^{41} - 44q^{42} + 14q^{43} - 15q^{44} + 4q^{45} + 13q^{46} - 7q^{47} + 3q^{48} - 46q^{49} + 13q^{50} + 21q^{51} - 7q^{52} + 12q^{53} + 16q^{54} - 26q^{55} - 3q^{56} - 57q^{57} + 97q^{58} - 41q^{59} - 31q^{60} - 48q^{61} - 2q^{62} - 81q^{63} + 6q^{64} - 75q^{65} - 82q^{66} - 41q^{67} - 4q^{68} - 22q^{69} + 26q^{70} - 69q^{71} - 3q^{72} - 64q^{73} + 13q^{74} - 2q^{75} - 30q^{76} + 162q^{77} + 32q^{78} - 33q^{79} - 5q^{80} - 131q^{81} - 21q^{82} + 10q^{83} + 166q^{84} - 122q^{85} - 26q^{86} - 17q^{87} - 21q^{88} - 13q^{89} - 160q^{90} - 44q^{91} - 52q^{92} - 49q^{93} - 170q^{94} - 295q^{95} + 3q^{96} - 88q^{97} - 36q^{98} - 119q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0.516397 0.856349i −2.22762 1.61846i −0.466667 0.884433i 2.61490 2.13819i −2.53630 + 1.07185i 0.238950 0.309876i −0.998369 0.0570888i 1.41582 + 4.35745i −0.480712 3.34343i
5.2 0.516397 0.856349i −0.659289 0.479001i −0.466667 0.884433i −2.95672 + 2.41770i −0.750647 + 0.317226i −2.54441 + 3.29965i −0.998369 0.0570888i −0.721832 2.22157i 0.543551 + 3.78048i
5.3 0.516397 0.856349i −0.321298 0.233437i −0.466667 0.884433i 0.747785 0.611460i −0.365821 + 0.154597i 0.646547 0.838457i −0.998369 0.0570888i −0.878311 2.70316i −0.137469 0.956121i
5.4 0.516397 0.856349i −0.107640 0.0782047i −0.466667 0.884433i −0.862795 + 0.705504i −0.122555 + 0.0517923i 2.67441 3.46824i −0.998369 0.0570888i −0.921581 2.83633i 0.158612 + 1.10317i
5.5 0.516397 0.856349i 1.68484 + 1.22411i −0.466667 0.884433i 2.72636 2.22933i 1.91832 0.810687i −0.998224 + 1.29452i −0.998369 0.0570888i 0.413202 + 1.27170i −0.501202 3.48594i
5.6 0.516397 0.856349i 2.41632 + 1.75556i −0.466667 0.884433i −1.13886 + 0.931243i 2.75115 1.16265i 0.346651 0.449545i −0.998369 0.0570888i 1.82956 + 5.63081i 0.209364 + 1.45616i
15.1 0.0855750 + 0.996332i −1.04353 + 3.21167i −0.985354 + 0.170522i −1.10788 + 0.0633506i −3.28919 0.764868i 2.72097 + 3.97928i −0.254218 0.967147i −6.79881 4.93962i −0.157925 1.09839i
15.2 0.0855750 + 0.996332i −0.398506 + 1.22647i −0.985354 + 0.170522i 3.76248 0.215146i −1.25608 0.292088i −0.343343 0.502123i −0.254218 0.967147i 1.08162 + 0.785842i 0.536331 + 3.73026i
15.3 0.0855750 + 0.996332i 0.176884 0.544392i −0.985354 + 0.170522i −4.00461 + 0.228992i 0.557532 + 0.129648i 2.14856 + 3.14217i −0.254218 0.967147i 2.16198 + 1.57077i −0.570847 3.97033i
15.4 0.0855750 + 0.996332i 0.416363 1.28143i −0.985354 + 0.170522i 0.325006 0.0185845i 1.31236 + 0.305177i 0.546419 + 0.799112i −0.254218 0.967147i 0.958336 + 0.696272i 0.0463288 + 0.322224i
15.5 0.0855750 + 0.996332i 0.567015 1.74509i −0.985354 + 0.170522i −2.46010 + 0.140674i 1.78721 + 0.415599i −2.92983 4.28473i −0.254218 0.967147i −0.296792 0.215632i −0.350681 2.43904i
15.6 0.0855750 + 0.996332i 1.02153 3.14394i −0.985354 + 0.170522i 2.66237 0.152240i 3.21982 + 0.748737i 1.76910 + 2.58723i −0.254218 0.967147i −6.41377 4.65988i 0.379513 + 2.63957i
25.1 −0.466667 0.884433i −0.720100 2.21624i −0.564443 + 0.825472i 0.496453 2.45009i −1.62407 + 1.67113i −0.475415 1.80867i 0.993482 + 0.113991i −1.96613 + 1.42848i −2.39862 + 0.704299i
25.2 −0.466667 0.884433i −0.456517 1.40501i −0.564443 + 0.825472i −0.588814 + 2.90591i −1.02960 + 1.05943i 0.913321 + 3.47463i 0.993482 + 0.113991i 0.661395 0.480532i 2.84487 0.835328i
25.3 −0.466667 0.884433i −0.0505515 0.155581i −0.564443 + 0.825472i −0.0911070 + 0.449631i −0.114011 + 0.117314i 0.380264 + 1.44667i 0.993482 + 0.113991i 2.40540 1.74763i 0.440185 0.129250i
25.4 −0.466667 0.884433i 0.340662 + 1.04845i −0.564443 + 0.825472i 0.456414 2.25249i 0.768307 0.790569i −0.473473 1.80128i 0.993482 + 0.113991i 1.44386 1.04902i −2.20517 + 0.647496i
25.5 −0.466667 0.884433i 0.845646 + 2.60263i −0.564443 + 0.825472i −0.614800 + 3.03416i 1.90722 1.96248i −0.909018 3.45826i 0.993482 + 0.113991i −3.63151 + 2.63845i 2.97041 0.872192i
25.6 −0.466667 0.884433i 0.931970 + 2.86831i −0.564443 + 0.825472i 0.304917 1.50482i 2.10191 2.16281i 1.11312 + 4.23475i 0.993482 + 0.113991i −4.93157 + 3.58300i −1.47321 + 0.432573i
31.1 0.774142 0.633012i −2.45131 + 1.78098i 0.198590 0.980083i −1.82825 + 0.898891i −0.770279 + 2.93045i −0.378745 4.40965i −0.466667 0.884433i 1.90999 5.87834i −0.846316 + 1.85317i
31.2 0.774142 0.633012i −2.15202 + 1.56354i 0.198590 0.980083i 2.23021 1.09652i −0.676233 + 2.57266i 0.232047 + 2.70167i −0.466667 0.884433i 1.25951 3.87637i 1.03239 2.26062i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.2.g.b 240
121.g even 55 1 inner 242.2.g.b 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.2.g.b 240 1.a even 1 1 trivial
242.2.g.b 240 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(36\!\cdots\!72\)\( T_{3}^{225} + \)\(18\!\cdots\!17\)\( T_{3}^{224} + \)\(79\!\cdots\!37\)\( T_{3}^{223} + \)\(39\!\cdots\!25\)\( T_{3}^{222} + \)\(15\!\cdots\!37\)\( T_{3}^{221} + \)\(75\!\cdots\!15\)\( T_{3}^{220} + \)\(29\!\cdots\!08\)\( T_{3}^{219} + \)\(13\!\cdots\!52\)\( T_{3}^{218} + \)\(50\!\cdots\!56\)\( T_{3}^{217} + \)\(22\!\cdots\!65\)\( T_{3}^{216} + \)\(80\!\cdots\!26\)\( T_{3}^{215} + \)\(34\!\cdots\!35\)\( T_{3}^{214} + \)\(12\!\cdots\!68\)\( T_{3}^{213} + \)\(50\!\cdots\!72\)\( T_{3}^{212} + \)\(17\!\cdots\!31\)\( T_{3}^{211} + \)\(69\!\cdots\!52\)\( T_{3}^{210} + \)\(22\!\cdots\!23\)\( T_{3}^{209} + \)\(91\!\cdots\!35\)\( T_{3}^{208} + \)\(29\!\cdots\!72\)\( T_{3}^{207} + \)\(11\!\cdots\!60\)\( T_{3}^{206} + \)\(35\!\cdots\!18\)\( T_{3}^{205} + \)\(13\!\cdots\!29\)\( T_{3}^{204} + \)\(41\!\cdots\!94\)\( T_{3}^{203} + \)\(15\!\cdots\!70\)\( T_{3}^{202} + \)\(46\!\cdots\!36\)\( T_{3}^{201} + \)\(17\!\cdots\!74\)\( T_{3}^{200} + \)\(49\!\cdots\!61\)\( T_{3}^{199} + \)\(18\!\cdots\!34\)\( T_{3}^{198} + \)\(50\!\cdots\!24\)\( T_{3}^{197} + \)\(18\!\cdots\!43\)\( T_{3}^{196} + \)\(49\!\cdots\!34\)\( T_{3}^{195} + \)\(17\!\cdots\!38\)\( T_{3}^{194} + \)\(46\!\cdots\!08\)\( T_{3}^{193} + \)\(16\!\cdots\!40\)\( T_{3}^{192} + \)\(42\!\cdots\!31\)\( T_{3}^{191} + \)\(15\!\cdots\!65\)\( T_{3}^{190} + \)\(36\!\cdots\!27\)\( T_{3}^{189} + \)\(13\!\cdots\!86\)\( T_{3}^{188} + \)\(31\!\cdots\!95\)\( T_{3}^{187} + \)\(11\!\cdots\!77\)\( T_{3}^{186} + \)\(25\!\cdots\!70\)\( T_{3}^{185} + \)\(91\!\cdots\!63\)\( T_{3}^{184} + \)\(19\!\cdots\!25\)\( T_{3}^{183} + \)\(72\!\cdots\!63\)\( T_{3}^{182} + \)\(14\!\cdots\!17\)\( T_{3}^{181} + \)\(55\!\cdots\!38\)\( T_{3}^{180} + \)\(10\!\cdots\!97\)\( T_{3}^{179} + \)\(40\!\cdots\!42\)\( T_{3}^{178} + \)\(76\!\cdots\!55\)\( T_{3}^{177} + \)\(29\!\cdots\!81\)\( T_{3}^{176} + \)\(52\!\cdots\!91\)\( T_{3}^{175} + \)\(20\!\cdots\!00\)\( T_{3}^{174} + \)\(34\!\cdots\!82\)\( T_{3}^{173} + \)\(13\!\cdots\!23\)\( T_{3}^{172} + \)\(21\!\cdots\!37\)\( T_{3}^{171} + \)\(88\!\cdots\!21\)\( T_{3}^{170} + \)\(13\!\cdots\!59\)\( T_{3}^{169} + \)\(55\!\cdots\!68\)\( T_{3}^{168} + \)\(77\!\cdots\!75\)\( T_{3}^{167} + \)\(34\!\cdots\!39\)\( T_{3}^{166} + \)\(43\!\cdots\!95\)\( T_{3}^{165} + \)\(20\!\cdots\!70\)\( T_{3}^{164} + \)\(23\!\cdots\!08\)\( T_{3}^{163} + \)\(11\!\cdots\!58\)\( T_{3}^{162} + \)\(12\!\cdots\!07\)\( T_{3}^{161} + \)\(64\!\cdots\!95\)\( T_{3}^{160} + \)\(63\!\cdots\!17\)\( T_{3}^{159} + \)\(35\!\cdots\!00\)\( T_{3}^{158} + \)\(31\!\cdots\!94\)\( T_{3}^{157} + \)\(18\!\cdots\!09\)\( T_{3}^{156} + \)\(14\!\cdots\!56\)\( T_{3}^{155} + \)\(93\!\cdots\!41\)\( T_{3}^{154} + \)\(66\!\cdots\!20\)\( T_{3}^{153} + \)\(45\!\cdots\!80\)\( T_{3}^{152} + \)\(28\!\cdots\!76\)\( T_{3}^{151} + \)\(21\!\cdots\!88\)\( T_{3}^{150} + \)\(12\!\cdots\!84\)\( T_{3}^{149} + \)\(99\!\cdots\!30\)\( T_{3}^{148} + \)\(48\!\cdots\!59\)\( T_{3}^{147} + \)\(44\!\cdots\!85\)\( T_{3}^{146} + \)\(19\!\cdots\!22\)\( T_{3}^{145} + \)\(19\!\cdots\!92\)\( T_{3}^{144} + \)\(72\!\cdots\!21\)\( T_{3}^{143} + \)\(79\!\cdots\!84\)\( T_{3}^{142} + \)\(26\!\cdots\!17\)\( T_{3}^{141} + \)\(31\!\cdots\!56\)\( T_{3}^{140} + \)\(95\!\cdots\!44\)\( T_{3}^{139} + \)\(12\!\cdots\!73\)\( T_{3}^{138} + \)\(34\!\cdots\!18\)\( T_{3}^{137} + \)\(45\!\cdots\!57\)\( T_{3}^{136} + \)\(11\!\cdots\!20\)\( T_{3}^{135} + \)\(16\!\cdots\!16\)\( T_{3}^{134} + \)\(41\!\cdots\!32\)\( T_{3}^{133} + \)\(56\!\cdots\!58\)\( T_{3}^{132} + \)\(14\!\cdots\!30\)\( T_{3}^{131} + \)\(18\!\cdots\!59\)\( T_{3}^{130} + \)\(50\!\cdots\!27\)\( T_{3}^{129} + \)\(61\!\cdots\!63\)\( T_{3}^{128} + \)\(17\!\cdots\!23\)\( T_{3}^{127} + \)\(18\!\cdots\!53\)\( T_{3}^{126} + \)\(58\!\cdots\!76\)\( T_{3}^{125} + \)\(56\!\cdots\!30\)\( T_{3}^{124} + \)\(19\!\cdots\!48\)\( T_{3}^{123} + \)\(16\!\cdots\!81\)\( T_{3}^{122} + \)\(60\!\cdots\!86\)\( T_{3}^{121} + \)\(44\!\cdots\!66\)\( T_{3}^{120} + \)\(18\!\cdots\!39\)\( T_{3}^{119} + \)\(11\!\cdots\!22\)\( T_{3}^{118} + \)\(54\!\cdots\!65\)\( T_{3}^{117} + \)\(30\!\cdots\!35\)\( T_{3}^{116} + \)\(15\!\cdots\!08\)\( T_{3}^{115} + \)\(76\!\cdots\!77\)\( T_{3}^{114} + \)\(41\!\cdots\!98\)\( T_{3}^{113} + \)\(18\!\cdots\!26\)\( T_{3}^{112} + \)\(10\!\cdots\!01\)\( T_{3}^{111} + \)\(41\!\cdots\!30\)\( T_{3}^{110} + \)\(26\!\cdots\!51\)\( T_{3}^{109} + \)\(91\!\cdots\!72\)\( T_{3}^{108} + \)\(62\!\cdots\!23\)\( T_{3}^{107} + \)\(19\!\cdots\!57\)\( T_{3}^{106} + \)\(14\!\cdots\!25\)\( T_{3}^{105} + \)\(39\!\cdots\!27\)\( T_{3}^{104} + \)\(30\!\cdots\!65\)\( T_{3}^{103} + \)\(78\!\cdots\!62\)\( T_{3}^{102} + \)\(62\!\cdots\!59\)\( T_{3}^{101} + \)\(14\!\cdots\!85\)\( T_{3}^{100} + \)\(12\!\cdots\!51\)\( T_{3}^{99} + \)\(26\!\cdots\!39\)\( T_{3}^{98} + \)\(22\!\cdots\!98\)\( T_{3}^{97} + \)\(46\!\cdots\!94\)\( T_{3}^{96} + \)\(40\!\cdots\!58\)\( T_{3}^{95} + \)\(77\!\cdots\!04\)\( T_{3}^{94} + \)\(68\!\cdots\!62\)\( T_{3}^{93} + \)\(12\!\cdots\!97\)\( T_{3}^{92} + \)\(11\!\cdots\!97\)\( T_{3}^{91} + \)\(18\!\cdots\!15\)\( T_{3}^{90} + \)\(16\!\cdots\!67\)\( T_{3}^{89} + \)\(27\!\cdots\!18\)\( T_{3}^{88} + \)\(24\!\cdots\!30\)\( T_{3}^{87} + \)\(37\!\cdots\!66\)\( T_{3}^{86} + \)\(34\!\cdots\!10\)\( T_{3}^{85} + \)\(50\!\cdots\!94\)\( T_{3}^{84} + \)\(44\!\cdots\!34\)\( T_{3}^{83} + \)\(63\!\cdots\!26\)\( T_{3}^{82} + \)\(55\!\cdots\!35\)\( T_{3}^{81} + \)\(75\!\cdots\!93\)\( T_{3}^{80} + \)\(66\!\cdots\!45\)\( T_{3}^{79} + \)\(85\!\cdots\!75\)\( T_{3}^{78} + \)\(74\!\cdots\!82\)\( T_{3}^{77} + \)\(92\!\cdots\!82\)\( T_{3}^{76} + \)\(78\!\cdots\!99\)\( T_{3}^{75} + \)\(94\!\cdots\!83\)\( T_{3}^{74} + \)\(79\!\cdots\!90\)\( T_{3}^{73} + \)\(91\!\cdots\!35\)\( T_{3}^{72} + \)\(75\!\cdots\!43\)\( T_{3}^{71} + \)\(83\!\cdots\!23\)\( T_{3}^{70} + \)\(67\!\cdots\!16\)\( T_{3}^{69} + \)\(72\!\cdots\!82\)\( T_{3}^{68} + \)\(57\!\cdots\!53\)\( T_{3}^{67} + \)\(59\!\cdots\!02\)\( T_{3}^{66} + \)\(46\!\cdots\!77\)\( T_{3}^{65} + \)\(45\!\cdots\!83\)\( T_{3}^{64} + \)\(34\!\cdots\!51\)\( T_{3}^{63} + \)\(33\!\cdots\!48\)\( T_{3}^{62} + \)\(24\!\cdots\!54\)\( T_{3}^{61} + \)\(22\!\cdots\!61\)\( T_{3}^{60} + \)\(16\!\cdots\!52\)\( T_{3}^{59} + \)\(14\!\cdots\!61\)\( T_{3}^{58} + \)\(10\!\cdots\!87\)\( T_{3}^{57} + \)\(87\!\cdots\!14\)\( T_{3}^{56} + \)\(60\!\cdots\!45\)\( T_{3}^{55} + \)\(49\!\cdots\!14\)\( T_{3}^{54} + \)\(32\!\cdots\!30\)\( T_{3}^{53} + \)\(25\!\cdots\!83\)\( T_{3}^{52} + \)\(16\!\cdots\!67\)\( T_{3}^{51} + \)\(12\!\cdots\!09\)\( T_{3}^{50} + \)\(81\!\cdots\!43\)\( T_{3}^{49} + \)\(58\!\cdots\!01\)\( T_{3}^{48} + \)\(36\!\cdots\!55\)\( T_{3}^{47} + \)\(25\!\cdots\!60\)\( T_{3}^{46} + \)\(15\!\cdots\!07\)\( T_{3}^{45} + \)\(99\!\cdots\!54\)\( T_{3}^{44} + \)\(57\!\cdots\!44\)\( T_{3}^{43} + \)\(36\!\cdots\!49\)\( T_{3}^{42} + \)\(20\!\cdots\!51\)\( T_{3}^{41} + \)\(12\!\cdots\!63\)\( T_{3}^{40} + \)\(68\!\cdots\!51\)\( T_{3}^{39} + \)\(39\!\cdots\!57\)\( T_{3}^{38} + \)\(20\!\cdots\!91\)\( T_{3}^{37} + \)\(11\!\cdots\!51\)\( T_{3}^{36} + \)\(57\!\cdots\!70\)\( T_{3}^{35} + \)\(29\!\cdots\!89\)\( T_{3}^{34} + \)\(14\!\cdots\!01\)\( T_{3}^{33} + \)\(71\!\cdots\!80\)\( T_{3}^{32} + \)\(33\!\cdots\!19\)\( T_{3}^{31} + \)\(15\!\cdots\!56\)\( T_{3}^{30} + \)\(69\!\cdots\!74\)\( T_{3}^{29} + \)\(30\!\cdots\!54\)\( T_{3}^{28} + \)\(12\!\cdots\!90\)\( T_{3}^{27} + \)\(51\!\cdots\!86\)\( T_{3}^{26} + \)\(19\!\cdots\!11\)\( T_{3}^{25} + \)\(73\!\cdots\!68\)\( T_{3}^{24} + \)\(26\!\cdots\!91\)\( T_{3}^{23} + \)\(88\!\cdots\!03\)\( T_{3}^{22} + \)\(28\!\cdots\!39\)\( T_{3}^{21} + \)\(86\!\cdots\!49\)\( T_{3}^{20} + \)\(25\!\cdots\!25\)\( T_{3}^{19} + \)\(70\!\cdots\!87\)\( T_{3}^{18} + \)\(19\!\cdots\!27\)\( T_{3}^{17} + \)\(49\!\cdots\!52\)\( T_{3}^{16} + \)\(12\!\cdots\!68\)\( T_{3}^{15} + \)\(28\!\cdots\!19\)\( T_{3}^{14} + \)\(64\!\cdots\!48\)\( T_{3}^{13} + \)\(13\!\cdots\!55\)\( T_{3}^{12} + \)\(26\!\cdots\!79\)\( T_{3}^{11} + \)\(49\!\cdots\!83\)\( T_{3}^{10} + \)\(82\!\cdots\!95\)\( T_{3}^{9} + \)\(12\!\cdots\!42\)\( T_{3}^{8} + \)\(17\!\cdots\!36\)\( T_{3}^{7} + \)\(21\!\cdots\!68\)\( T_{3}^{6} + \)\(24\!\cdots\!54\)\( T_{3}^{5} + \)\(23\!\cdots\!42\)\( T_{3}^{4} + \)\(19\!\cdots\!50\)\( T_{3}^{3} + \)\(12\!\cdots\!29\)\( T_{3}^{2} + \)\(59\!\cdots\!60\)\( T_{3} + \)\(16\!\cdots\!61\)\( \)">\(T_{3}^{240} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(242, [\chi])\).