Properties

Label 588.2.ba.a
Level $588$
Weight $2$
Character orbit 588.ba
Analytic conductor $4.695$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.ba (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9} + 41 q^{10} + 6 q^{11} + 52 q^{14} + 20 q^{16} - 6 q^{19} - 25 q^{20} + 4 q^{21} + 6 q^{22} + 9 q^{24} - 26 q^{25} - 18 q^{26} + 56 q^{27} + 5 q^{30} + 2 q^{31} + 15 q^{32} + 6 q^{33} + 44 q^{34} - 12 q^{35} + 16 q^{37} - 100 q^{38} + 8 q^{39} - 7 q^{40} - 7 q^{42} - 53 q^{44} - 10 q^{46} + 4 q^{47} + 8 q^{48} - 4 q^{49} - 114 q^{50} - 28 q^{52} - 4 q^{53} + q^{56} - 12 q^{57} + 27 q^{58} - 10 q^{59} - 7 q^{60} + 2 q^{61} - 16 q^{62} - 12 q^{63} - 84 q^{64} - 4 q^{65} + 21 q^{66} + 42 q^{67} + 26 q^{68} - 70 q^{70} + 28 q^{71} + 15 q^{72} + 18 q^{73} + 20 q^{74} + 54 q^{75} + 49 q^{76} - 8 q^{77} - 6 q^{78} - 6 q^{79} - 40 q^{80} + 28 q^{81} - 99 q^{82} + 10 q^{83} + 54 q^{84} + 24 q^{85} - 314 q^{86} - 40 q^{88} + 20 q^{90} - 34 q^{91} + 14 q^{92} - 2 q^{93} - 152 q^{94} - 24 q^{95} - 10 q^{96} - 156 q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −1.40964 0.113611i −0.826239 0.563320i 1.97419 + 0.320301i 3.32240 + 0.248980i 1.10070 + 0.887950i −2.08944 1.62303i −2.74651 0.675798i 0.365341 + 0.930874i −4.65512 0.728433i
103.2 −1.36468 + 0.371026i −0.826239 0.563320i 1.72468 1.01266i −0.793563 0.0594693i 1.33655 + 0.462193i 0.251406 + 2.63378i −1.97791 + 2.02185i 0.365341 + 0.930874i 1.10502 0.213276i
103.3 −1.33039 0.479649i −0.826239 0.563320i 1.53987 + 1.27624i −1.47341 0.110417i 0.829024 + 1.14574i 2.51658 + 0.816579i −1.43648 2.43650i 0.365341 + 0.930874i 1.90724 + 0.853615i
103.4 −1.24022 0.679596i −0.826239 0.563320i 1.07630 + 1.68570i −2.59884 0.194756i 0.641889 + 1.26015i −2.28001 + 1.34222i −0.189256 2.82209i 0.365341 + 0.930874i 3.09079 + 2.00770i
103.5 −1.23869 + 0.682383i −0.826239 0.563320i 1.06871 1.69052i 0.961592 + 0.0720614i 1.40785 + 0.133968i 2.50588 0.848875i −0.170216 + 2.82330i 0.365341 + 0.930874i −1.24029 + 0.566912i
103.6 −1.15847 + 0.811146i −0.826239 0.563320i 0.684086 1.87937i 1.33423 + 0.0999870i 1.41410 0.0176130i −2.38917 1.13661i 0.731952 + 2.73208i 0.365341 + 0.930874i −1.62677 + 0.966426i
103.7 −1.08950 0.901663i −0.826239 0.563320i 0.374008 + 1.96472i 2.88553 + 0.216241i 0.392260 + 1.35872i 0.232552 + 2.63551i 1.36403 2.47778i 0.365341 + 0.930874i −2.94880 2.83737i
103.8 −0.839211 + 1.13830i −0.826239 0.563320i −0.591451 1.91055i −4.04500 0.303131i 1.33462 0.467763i −2.59666 + 0.507311i 2.67113 + 0.930102i 0.365341 + 0.930874i 3.73966 4.35003i
103.9 −0.706021 1.22537i −0.826239 0.563320i −1.00307 + 1.73028i −0.481855 0.0361100i −0.106934 + 1.41016i −1.51904 2.16623i 2.82842 + 0.00752107i 0.365341 + 0.930874i 0.295951 + 0.615945i
103.10 −0.677412 1.24142i −0.826239 0.563320i −1.08223 + 1.68190i −2.31910 0.173792i −0.139610 + 1.40731i 2.40273 1.10766i 2.82105 + 0.204152i 0.365341 + 0.930874i 1.35524 + 2.99670i
103.11 −0.593055 + 1.28386i −0.826239 0.563320i −1.29657 1.52279i 4.20665 + 0.315245i 1.21323 0.726692i 2.58503 + 0.563570i 2.72399 0.761513i 0.365341 + 0.930874i −2.89950 + 5.21378i
103.12 −0.476542 + 1.33151i −0.826239 0.563320i −1.54582 1.26904i −0.786913 0.0589710i 1.14380 0.831696i 0.621496 2.57172i 2.42638 1.45351i 0.365341 + 0.930874i 0.453517 1.01968i
103.13 −0.194188 1.40082i −0.826239 0.563320i −1.92458 + 0.544043i 1.91992 + 0.143878i −0.628664 + 1.26680i −0.854457 + 2.50398i 1.13584 + 2.59034i 0.365341 + 0.930874i −0.171278 2.71740i
103.14 −0.0925486 + 1.41118i −0.826239 0.563320i −1.98287 0.261206i −2.91108 0.218156i 0.871414 1.11384i 2.63369 + 0.252375i 0.552121 2.77402i 0.365341 + 0.930874i 0.577274 4.08788i
103.15 −0.0532621 + 1.41321i −0.826239 0.563320i −1.99433 0.150541i 2.71904 + 0.203764i 0.840097 1.13765i −2.62441 + 0.335390i 0.318968 2.81038i 0.365341 + 0.930874i −0.432783 + 3.83173i
103.16 0.0278575 1.41394i −0.826239 0.563320i −1.99845 0.0787776i 4.26189 + 0.319384i −0.819517 + 1.15256i 0.230798 2.63567i −0.167058 + 2.82349i 0.365341 + 0.930874i 0.570316 6.01715i
103.17 0.201122 1.39984i −0.826239 0.563320i −1.91910 0.563077i −1.87993 0.140882i −0.954732 + 1.04331i 1.80315 + 1.93615i −1.17419 + 2.57318i 0.365341 + 0.930874i −0.575308 + 2.60327i
103.18 0.672250 1.24422i −0.826239 0.563320i −1.09616 1.67285i −0.188298 0.0141110i −1.25633 + 0.649330i 1.32214 2.29171i −2.81829 + 0.239290i 0.365341 + 0.930874i −0.144141 + 0.224798i
103.19 0.700855 + 1.22833i −0.826239 0.563320i −1.01760 + 1.72177i −2.57232 0.192768i 0.112871 1.40970i −0.135490 2.64228i −2.82810 0.0432451i 0.365341 + 0.930874i −1.56604 3.29476i
103.20 0.826670 1.14744i −0.826239 0.563320i −0.633232 1.89711i −1.15601 0.0866309i −1.32940 + 0.482379i −2.18569 + 1.49090i −2.70029 0.841687i 0.365341 + 0.930874i −1.05504 + 1.25483i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 535.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.p even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.ba.a 336
4.b odd 2 1 588.2.ba.b yes 336
49.h odd 42 1 588.2.ba.b yes 336
196.p even 42 1 inner 588.2.ba.a 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.ba.a 336 1.a even 1 1 trivial
588.2.ba.a 336 196.p even 42 1 inner
588.2.ba.b yes 336 4.b odd 2 1
588.2.ba.b yes 336 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(28\!\cdots\!59\)\( T_{11}^{322} + \)\(13\!\cdots\!34\)\( T_{11}^{321} + \)\(18\!\cdots\!31\)\( T_{11}^{320} - \)\(62\!\cdots\!30\)\( T_{11}^{319} + \)\(53\!\cdots\!99\)\( T_{11}^{318} - \)\(50\!\cdots\!88\)\( T_{11}^{317} + \)\(22\!\cdots\!13\)\( T_{11}^{316} - \)\(12\!\cdots\!58\)\( T_{11}^{315} + \)\(33\!\cdots\!23\)\( T_{11}^{314} + \)\(13\!\cdots\!80\)\( T_{11}^{313} - \)\(12\!\cdots\!66\)\( T_{11}^{312} + \)\(20\!\cdots\!22\)\( T_{11}^{311} - \)\(85\!\cdots\!78\)\( T_{11}^{310} + \)\(54\!\cdots\!46\)\( T_{11}^{309} - \)\(18\!\cdots\!58\)\( T_{11}^{308} - \)\(29\!\cdots\!22\)\( T_{11}^{307} + \)\(12\!\cdots\!14\)\( T_{11}^{306} - \)\(59\!\cdots\!08\)\( T_{11}^{305} + \)\(20\!\cdots\!79\)\( T_{11}^{304} - \)\(13\!\cdots\!98\)\( T_{11}^{303} + \)\(48\!\cdots\!96\)\( T_{11}^{302} + \)\(17\!\cdots\!94\)\( T_{11}^{301} - \)\(25\!\cdots\!37\)\( T_{11}^{300} + \)\(15\!\cdots\!82\)\( T_{11}^{299} - \)\(43\!\cdots\!71\)\( T_{11}^{298} + \)\(20\!\cdots\!42\)\( T_{11}^{297} - \)\(80\!\cdots\!51\)\( T_{11}^{296} - \)\(81\!\cdots\!78\)\( T_{11}^{295} + \)\(16\!\cdots\!62\)\( T_{11}^{294} - \)\(35\!\cdots\!72\)\( T_{11}^{293} + \)\(10\!\cdots\!67\)\( T_{11}^{292} - \)\(17\!\cdots\!44\)\( T_{11}^{291} + \)\(12\!\cdots\!30\)\( T_{11}^{290} + \)\(22\!\cdots\!94\)\( T_{11}^{289} - \)\(49\!\cdots\!41\)\( T_{11}^{288} + \)\(60\!\cdots\!04\)\( T_{11}^{287} - \)\(20\!\cdots\!22\)\( T_{11}^{286} - \)\(23\!\cdots\!68\)\( T_{11}^{285} - \)\(14\!\cdots\!15\)\( T_{11}^{284} - \)\(42\!\cdots\!12\)\( T_{11}^{283} + \)\(86\!\cdots\!18\)\( T_{11}^{282} - \)\(64\!\cdots\!30\)\( T_{11}^{281} + \)\(25\!\cdots\!65\)\( T_{11}^{280} + \)\(13\!\cdots\!04\)\( T_{11}^{279} - \)\(67\!\cdots\!10\)\( T_{11}^{278} + \)\(63\!\cdots\!14\)\( T_{11}^{277} - \)\(13\!\cdots\!61\)\( T_{11}^{276} + \)\(37\!\cdots\!02\)\( T_{11}^{275} - \)\(22\!\cdots\!00\)\( T_{11}^{274} - \)\(29\!\cdots\!78\)\( T_{11}^{273} + \)\(36\!\cdots\!84\)\( T_{11}^{272} - \)\(74\!\cdots\!78\)\( T_{11}^{271} + \)\(18\!\cdots\!38\)\( T_{11}^{270} + \)\(17\!\cdots\!14\)\( T_{11}^{269} + \)\(18\!\cdots\!17\)\( T_{11}^{268} + \)\(38\!\cdots\!24\)\( T_{11}^{267} - \)\(51\!\cdots\!70\)\( T_{11}^{266} + \)\(54\!\cdots\!66\)\( T_{11}^{265} - \)\(17\!\cdots\!35\)\( T_{11}^{264} - \)\(83\!\cdots\!58\)\( T_{11}^{263} - \)\(93\!\cdots\!74\)\( T_{11}^{262} - \)\(37\!\cdots\!60\)\( T_{11}^{261} + \)\(49\!\cdots\!19\)\( T_{11}^{260} - \)\(21\!\cdots\!10\)\( T_{11}^{259} + \)\(10\!\cdots\!61\)\( T_{11}^{258} + \)\(13\!\cdots\!68\)\( T_{11}^{257} - \)\(44\!\cdots\!38\)\( T_{11}^{256} + \)\(30\!\cdots\!92\)\( T_{11}^{255} - \)\(50\!\cdots\!27\)\( T_{11}^{254} - \)\(11\!\cdots\!74\)\( T_{11}^{253} - \)\(70\!\cdots\!38\)\( T_{11}^{252} - \)\(11\!\cdots\!18\)\( T_{11}^{251} + \)\(69\!\cdots\!20\)\( T_{11}^{250} - \)\(16\!\cdots\!36\)\( T_{11}^{249} + \)\(38\!\cdots\!66\)\( T_{11}^{248} + \)\(11\!\cdots\!90\)\( T_{11}^{247} + \)\(47\!\cdots\!60\)\( T_{11}^{246} + \)\(66\!\cdots\!70\)\( T_{11}^{245} - \)\(24\!\cdots\!16\)\( T_{11}^{244} + \)\(43\!\cdots\!28\)\( T_{11}^{243} - \)\(15\!\cdots\!97\)\( T_{11}^{242} - \)\(16\!\cdots\!44\)\( T_{11}^{241} - \)\(10\!\cdots\!01\)\( T_{11}^{240} - \)\(38\!\cdots\!78\)\( T_{11}^{239} + \)\(40\!\cdots\!90\)\( T_{11}^{238} - \)\(55\!\cdots\!86\)\( T_{11}^{237} + \)\(10\!\cdots\!49\)\( T_{11}^{236} + \)\(97\!\cdots\!56\)\( T_{11}^{235} + \)\(37\!\cdots\!80\)\( T_{11}^{234} + \)\(14\!\cdots\!92\)\( T_{11}^{233} - \)\(24\!\cdots\!83\)\( T_{11}^{232} - \)\(57\!\cdots\!60\)\( T_{11}^{231} - \)\(50\!\cdots\!40\)\( T_{11}^{230} - \)\(40\!\cdots\!60\)\( T_{11}^{229} - \)\(15\!\cdots\!56\)\( T_{11}^{228} - \)\(31\!\cdots\!00\)\( T_{11}^{227} + \)\(94\!\cdots\!99\)\( T_{11}^{226} + \)\(61\!\cdots\!96\)\( T_{11}^{225} + \)\(13\!\cdots\!25\)\( T_{11}^{224} + \)\(14\!\cdots\!16\)\( T_{11}^{223} - \)\(77\!\cdots\!64\)\( T_{11}^{222} + \)\(36\!\cdots\!04\)\( T_{11}^{221} - \)\(43\!\cdots\!44\)\( T_{11}^{220} - \)\(27\!\cdots\!50\)\( T_{11}^{219} - \)\(31\!\cdots\!62\)\( T_{11}^{218} - \)\(39\!\cdots\!32\)\( T_{11}^{217} + \)\(69\!\cdots\!42\)\( T_{11}^{216} + \)\(35\!\cdots\!26\)\( T_{11}^{215} + \)\(16\!\cdots\!45\)\( T_{11}^{214} + \)\(70\!\cdots\!90\)\( T_{11}^{213} + \)\(86\!\cdots\!56\)\( T_{11}^{212} + \)\(61\!\cdots\!98\)\( T_{11}^{211} - \)\(18\!\cdots\!30\)\( T_{11}^{210} - \)\(55\!\cdots\!62\)\( T_{11}^{209} - \)\(32\!\cdots\!20\)\( T_{11}^{208} - \)\(15\!\cdots\!24\)\( T_{11}^{207} - \)\(23\!\cdots\!94\)\( T_{11}^{206} - \)\(62\!\cdots\!64\)\( T_{11}^{205} + \)\(46\!\cdots\!83\)\( T_{11}^{204} + \)\(15\!\cdots\!48\)\( T_{11}^{203} + \)\(42\!\cdots\!29\)\( T_{11}^{202} + \)\(23\!\cdots\!84\)\( T_{11}^{201} - \)\(33\!\cdots\!33\)\( T_{11}^{200} + \)\(32\!\cdots\!84\)\( T_{11}^{199} - \)\(97\!\cdots\!77\)\( T_{11}^{198} - \)\(30\!\cdots\!68\)\( T_{11}^{197} - \)\(55\!\cdots\!61\)\( T_{11}^{196} - \)\(29\!\cdots\!18\)\( T_{11}^{195} + \)\(65\!\cdots\!76\)\( T_{11}^{194} + \)\(89\!\cdots\!56\)\( T_{11}^{193} + \)\(13\!\cdots\!54\)\( T_{11}^{192} + \)\(45\!\cdots\!52\)\( T_{11}^{191} + \)\(58\!\cdots\!71\)\( T_{11}^{190} + \)\(37\!\cdots\!62\)\( T_{11}^{189} - \)\(58\!\cdots\!76\)\( T_{11}^{188} - \)\(85\!\cdots\!08\)\( T_{11}^{187} - \)\(79\!\cdots\!57\)\( T_{11}^{186} - \)\(40\!\cdots\!32\)\( T_{11}^{185} + \)\(10\!\cdots\!78\)\( T_{11}^{184} - \)\(22\!\cdots\!98\)\( T_{11}^{183} + \)\(95\!\cdots\!64\)\( T_{11}^{182} + \)\(22\!\cdots\!22\)\( T_{11}^{181} + \)\(66\!\cdots\!61\)\( T_{11}^{180} + \)\(43\!\cdots\!96\)\( T_{11}^{179} - \)\(37\!\cdots\!37\)\( T_{11}^{178} + \)\(10\!\cdots\!72\)\( T_{11}^{177} - \)\(93\!\cdots\!22\)\( T_{11}^{176} - \)\(37\!\cdots\!52\)\( T_{11}^{175} - \)\(90\!\cdots\!54\)\( T_{11}^{174} - \)\(50\!\cdots\!04\)\( T_{11}^{173} - \)\(27\!\cdots\!32\)\( T_{11}^{172} - \)\(23\!\cdots\!66\)\( T_{11}^{171} + \)\(38\!\cdots\!41\)\( T_{11}^{170} + \)\(20\!\cdots\!12\)\( T_{11}^{169} + \)\(22\!\cdots\!75\)\( T_{11}^{168} + \)\(30\!\cdots\!22\)\( T_{11}^{167} + \)\(46\!\cdots\!03\)\( T_{11}^{166} + \)\(17\!\cdots\!94\)\( T_{11}^{165} + \)\(42\!\cdots\!01\)\( T_{11}^{164} + \)\(88\!\cdots\!38\)\( T_{11}^{163} + \)\(33\!\cdots\!97\)\( T_{11}^{162} - \)\(38\!\cdots\!36\)\( T_{11}^{161} + \)\(48\!\cdots\!31\)\( T_{11}^{160} + \)\(32\!\cdots\!18\)\( T_{11}^{159} - \)\(83\!\cdots\!51\)\( T_{11}^{158} + \)\(40\!\cdots\!28\)\( T_{11}^{157} - \)\(34\!\cdots\!75\)\( T_{11}^{156} + \)\(14\!\cdots\!12\)\( T_{11}^{155} + \)\(23\!\cdots\!49\)\( T_{11}^{154} - \)\(91\!\cdots\!04\)\( T_{11}^{153} + \)\(18\!\cdots\!11\)\( T_{11}^{152} - \)\(19\!\cdots\!54\)\( T_{11}^{151} - \)\(24\!\cdots\!22\)\( T_{11}^{150} - \)\(63\!\cdots\!90\)\( T_{11}^{149} - \)\(43\!\cdots\!16\)\( T_{11}^{148} - \)\(33\!\cdots\!82\)\( T_{11}^{147} - \)\(10\!\cdots\!39\)\( T_{11}^{146} - \)\(20\!\cdots\!66\)\( T_{11}^{145} - \)\(11\!\cdots\!29\)\( T_{11}^{144} + \)\(90\!\cdots\!70\)\( T_{11}^{143} + \)\(38\!\cdots\!31\)\( T_{11}^{142} + \)\(58\!\cdots\!12\)\( T_{11}^{141} + \)\(31\!\cdots\!48\)\( T_{11}^{140} + \)\(57\!\cdots\!40\)\( T_{11}^{139} + \)\(10\!\cdots\!86\)\( T_{11}^{138} + \)\(34\!\cdots\!90\)\( T_{11}^{137} + \)\(46\!\cdots\!06\)\( T_{11}^{136} + \)\(19\!\cdots\!92\)\( T_{11}^{135} + \)\(52\!\cdots\!03\)\( T_{11}^{134} + \)\(11\!\cdots\!30\)\( T_{11}^{133} + \)\(39\!\cdots\!06\)\( T_{11}^{132} + \)\(93\!\cdots\!38\)\( T_{11}^{131} + \)\(19\!\cdots\!69\)\( T_{11}^{130} + \)\(39\!\cdots\!30\)\( T_{11}^{129} + \)\(70\!\cdots\!91\)\( T_{11}^{128} + \)\(72\!\cdots\!64\)\( T_{11}^{127} - \)\(96\!\cdots\!14\)\( T_{11}^{126} - \)\(53\!\cdots\!62\)\( T_{11}^{125} - \)\(19\!\cdots\!05\)\( T_{11}^{124} - \)\(57\!\cdots\!40\)\( T_{11}^{123} - \)\(13\!\cdots\!65\)\( T_{11}^{122} - \)\(27\!\cdots\!16\)\( T_{11}^{121} - \)\(45\!\cdots\!76\)\( T_{11}^{120} - \)\(64\!\cdots\!24\)\( T_{11}^{119} - \)\(69\!\cdots\!16\)\( T_{11}^{118} + \)\(34\!\cdots\!66\)\( T_{11}^{117} + \)\(35\!\cdots\!77\)\( T_{11}^{116} + \)\(10\!\cdots\!62\)\( T_{11}^{115} + \)\(25\!\cdots\!74\)\( T_{11}^{114} + \)\(53\!\cdots\!62\)\( T_{11}^{113} + \)\(97\!\cdots\!35\)\( T_{11}^{112} + \)\(14\!\cdots\!60\)\( T_{11}^{111} + \)\(21\!\cdots\!07\)\( T_{11}^{110} + \)\(29\!\cdots\!28\)\( T_{11}^{109} + \)\(38\!\cdots\!37\)\( T_{11}^{108} + \)\(61\!\cdots\!86\)\( T_{11}^{107} + \)\(11\!\cdots\!89\)\( T_{11}^{106} + \)\(25\!\cdots\!98\)\( T_{11}^{105} + \)\(52\!\cdots\!47\)\( T_{11}^{104} + \)\(10\!\cdots\!78\)\( T_{11}^{103} + \)\(17\!\cdots\!47\)\( T_{11}^{102} + \)\(22\!\cdots\!82\)\( T_{11}^{101} + \)\(22\!\cdots\!58\)\( T_{11}^{100} - \)\(21\!\cdots\!02\)\( T_{11}^{99} - \)\(68\!\cdots\!76\)\( T_{11}^{98} - \)\(17\!\cdots\!02\)\( T_{11}^{97} - \)\(36\!\cdots\!52\)\( T_{11}^{96} - \)\(56\!\cdots\!72\)\( T_{11}^{95} - \)\(72\!\cdots\!64\)\( T_{11}^{94} - \)\(76\!\cdots\!64\)\( T_{11}^{93} - \)\(35\!\cdots\!30\)\( T_{11}^{92} + \)\(35\!\cdots\!22\)\( T_{11}^{91} + \)\(18\!\cdots\!84\)\( T_{11}^{90} + \)\(36\!\cdots\!22\)\( T_{11}^{89} + \)\(54\!\cdots\!71\)\( T_{11}^{88} + \)\(73\!\cdots\!50\)\( T_{11}^{87} + \)\(73\!\cdots\!30\)\( T_{11}^{86} + \)\(59\!\cdots\!48\)\( T_{11}^{85} + \)\(32\!\cdots\!04\)\( T_{11}^{84} - \)\(54\!\cdots\!88\)\( T_{11}^{83} - \)\(65\!\cdots\!28\)\( T_{11}^{82} - \)\(22\!\cdots\!36\)\( T_{11}^{81} - \)\(14\!\cdots\!06\)\( T_{11}^{80} - \)\(27\!\cdots\!84\)\( T_{11}^{79} - \)\(94\!\cdots\!18\)\( T_{11}^{78} - \)\(94\!\cdots\!98\)\( T_{11}^{77} + \)\(10\!\cdots\!38\)\( T_{11}^{76} + \)\(88\!\cdots\!46\)\( T_{11}^{75} + \)\(42\!\cdots\!52\)\( T_{11}^{74} - \)\(94\!\cdots\!76\)\( T_{11}^{73} + \)\(76\!\cdots\!82\)\( T_{11}^{72} - \)\(61\!\cdots\!60\)\( T_{11}^{71} + \)\(82\!\cdots\!13\)\( T_{11}^{70} - \)\(79\!\cdots\!34\)\( T_{11}^{69} + \)\(20\!\cdots\!37\)\( T_{11}^{68} + \)\(62\!\cdots\!70\)\( T_{11}^{67} - \)\(11\!\cdots\!70\)\( T_{11}^{66} + \)\(17\!\cdots\!62\)\( T_{11}^{65} - \)\(24\!\cdots\!90\)\( T_{11}^{64} + \)\(28\!\cdots\!20\)\( T_{11}^{63} - \)\(26\!\cdots\!21\)\( T_{11}^{62} + \)\(23\!\cdots\!70\)\( T_{11}^{61} - \)\(13\!\cdots\!47\)\( T_{11}^{60} + \)\(29\!\cdots\!48\)\( T_{11}^{59} + \)\(86\!\cdots\!92\)\( T_{11}^{58} - \)\(19\!\cdots\!54\)\( T_{11}^{57} + \)\(27\!\cdots\!92\)\( T_{11}^{56} - \)\(31\!\cdots\!54\)\( T_{11}^{55} + \)\(33\!\cdots\!44\)\( T_{11}^{54} - \)\(31\!\cdots\!44\)\( T_{11}^{53} + \)\(27\!\cdots\!47\)\( T_{11}^{52} - \)\(22\!\cdots\!02\)\( T_{11}^{51} + \)\(17\!\cdots\!32\)\( T_{11}^{50} - \)\(13\!\cdots\!88\)\( T_{11}^{49} + \)\(91\!\cdots\!41\)\( T_{11}^{48} - \)\(59\!\cdots\!40\)\( T_{11}^{47} + \)\(37\!\cdots\!57\)\( T_{11}^{46} - \)\(21\!\cdots\!14\)\( T_{11}^{45} + \)\(11\!\cdots\!22\)\( T_{11}^{44} - \)\(59\!\cdots\!76\)\( T_{11}^{43} + \)\(27\!\cdots\!28\)\( T_{11}^{42} - \)\(10\!\cdots\!12\)\( T_{11}^{41} + \)\(33\!\cdots\!16\)\( T_{11}^{40} - \)\(36\!\cdots\!88\)\( T_{11}^{39} - \)\(52\!\cdots\!43\)\( T_{11}^{38} + \)\(59\!\cdots\!08\)\( T_{11}^{37} - \)\(42\!\cdots\!30\)\( T_{11}^{36} + \)\(24\!\cdots\!46\)\( T_{11}^{35} - \)\(12\!\cdots\!95\)\( T_{11}^{34} + \)\(55\!\cdots\!66\)\( T_{11}^{33} - \)\(21\!\cdots\!28\)\( T_{11}^{32} + \)\(65\!\cdots\!60\)\( T_{11}^{31} - \)\(11\!\cdots\!57\)\( T_{11}^{30} - \)\(34\!\cdots\!88\)\( T_{11}^{29} + \)\(50\!\cdots\!63\)\( T_{11}^{28} - \)\(34\!\cdots\!28\)\( T_{11}^{27} + \)\(18\!\cdots\!67\)\( T_{11}^{26} - \)\(84\!\cdots\!14\)\( T_{11}^{25} + \)\(34\!\cdots\!85\)\( T_{11}^{24} - \)\(12\!\cdots\!92\)\( T_{11}^{23} + \)\(42\!\cdots\!56\)\( T_{11}^{22} - \)\(13\!\cdots\!24\)\( T_{11}^{21} + \)\(36\!\cdots\!12\)\( T_{11}^{20} - \)\(93\!\cdots\!16\)\( T_{11}^{19} + \)\(21\!\cdots\!80\)\( T_{11}^{18} - \)\(47\!\cdots\!68\)\( T_{11}^{17} + \)\(96\!\cdots\!28\)\( T_{11}^{16} - \)\(18\!\cdots\!88\)\( T_{11}^{15} + \)\(31\!\cdots\!44\)\( T_{11}^{14} - \)\(50\!\cdots\!00\)\( T_{11}^{13} + \)\(71\!\cdots\!96\)\( T_{11}^{12} - \)\(88\!\cdots\!92\)\( T_{11}^{11} + \)\(92\!\cdots\!96\)\( T_{11}^{10} - \)\(83\!\cdots\!72\)\( T_{11}^{9} + \)\(67\!\cdots\!64\)\( T_{11}^{8} - \)\(46\!\cdots\!28\)\( T_{11}^{7} + \)\(31\!\cdots\!28\)\( T_{11}^{6} - \)\(14\!\cdots\!48\)\( T_{11}^{5} + \)\(63\!\cdots\!88\)\( T_{11}^{4} + \)\(12\!\cdots\!52\)\( T_{11}^{3} - \)\(58\!\cdots\!28\)\( T_{11}^{2} + \)\(83\!\cdots\!80\)\( T_{11} + \)\(41\!\cdots\!16\)\( \)">\(T_{11}^{336} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\).