Properties

Label 588.2.ba.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9} - 23 q^{10} - 6 q^{11} - 30 q^{14} - 12 q^{16} + 6 q^{19} + 25 q^{20} + 4 q^{21} + 6 q^{22} + 15 q^{24} - 26 q^{25} - 12 q^{26} - 56 q^{27} + 36 q^{28} - 13 q^{30} - 2 q^{31} - 25 q^{32} + 6 q^{33} + 68 q^{34} + 12 q^{35} + 16 q^{37} + 82 q^{38} - 8 q^{39} - 19 q^{40} - 9 q^{42} - 11 q^{44} + 10 q^{46} - 4 q^{47} - 8 q^{48} - 4 q^{49} - 114 q^{50} - 8 q^{52} - 4 q^{53} - 41 q^{56} - 12 q^{57} - 33 q^{58} + 10 q^{59} + 17 q^{60} + 2 q^{61} + 16 q^{62} + 12 q^{63} + 84 q^{64} - 4 q^{65} + 15 q^{66} - 42 q^{67} + 10 q^{68} - 38 q^{70} - 28 q^{71} + 33 q^{72} + 18 q^{73} + 2 q^{74} - 54 q^{75} - 7 q^{76} - 8 q^{77} - 6 q^{78} + 6 q^{79} - 14 q^{80} + 28 q^{81} - 87 q^{82} - 10 q^{83} - 14 q^{84} + 24 q^{85} + 126 q^{86} - 244 q^{88} - 20 q^{90} + 34 q^{91} + 14 q^{92} - 2 q^{93} - 184 q^{94} + 24 q^{95} - 20 q^{96} - 122 q^{98} + O(q^{100}) \)

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 −1.41341 + 0.0477030i 0.826239 + 0.563320i 1.99545 0.134848i −0.793563 0.0594693i −1.19469 0.756787i −0.251406 2.63378i −2.81395 + 0.285784i 0.365341 + 0.930874i 1.12447 + 0.0461991i
103.2 −1.38479 0.286956i 0.826239 + 0.563320i 1.83531 + 0.794750i 0.961592 + 0.0720614i −0.982523 1.01718i −2.50588 + 0.848875i −2.31347 1.62722i 0.365341 + 0.930874i −1.31093 0.375725i
103.3 −1.34609 0.433645i 0.826239 + 0.563320i 1.62390 + 1.16745i 1.33423 + 0.0999870i −0.867909 1.11657i 2.38917 + 1.13661i −1.67966 2.27569i 0.365341 + 0.930874i −1.75264 0.713175i
103.4 −1.31353 + 0.524063i 0.826239 + 0.563320i 1.45072 1.37674i 3.32240 + 0.248980i −1.38050 0.306936i 2.08944 + 1.62303i −1.18406 + 2.56866i 0.365341 + 0.930874i −4.49455 + 1.41411i
103.5 −1.13745 0.840366i 0.826239 + 0.563320i 0.587569 + 1.91174i −4.04500 0.303131i −0.466407 1.33509i 2.59666 0.507311i 0.938236 2.66828i 0.365341 + 0.930874i 4.34623 + 3.74408i
103.6 −1.12991 + 0.850479i 0.826239 + 0.563320i 0.553371 1.92192i −1.47341 0.110417i −1.41266 + 0.0662004i −2.51658 0.816579i 1.00930 + 2.64222i 0.365341 + 0.930874i 1.75872 1.12834i
103.7 −0.984808 + 1.01496i 0.826239 + 0.563320i −0.0603079 1.99909i −2.59884 0.194756i −1.38544 + 0.283842i 2.28001 1.34222i 2.08840 + 1.90751i 0.365341 + 0.930874i 2.75703 2.44594i
103.8 −0.945130 1.05201i 0.826239 + 0.563320i −0.213459 + 1.98858i 4.20665 + 0.315245i −0.188284 1.40162i −2.58503 0.563570i 2.29375 1.65490i 0.365341 + 0.930874i −3.64419 4.72340i
103.9 −0.847839 1.13189i 0.826239 + 0.563320i −0.562339 + 1.91932i −0.786913 0.0589710i −0.0629023 1.41281i −0.621496 + 2.57172i 2.64922 0.990766i 0.365341 + 0.930874i 0.600427 + 0.940695i
103.10 −0.775324 + 1.18274i 0.826239 + 0.563320i −0.797745 1.83401i 2.88553 + 0.216241i −1.30686 + 0.540469i −0.232552 2.63551i 2.78767 + 0.478430i 0.365341 + 0.930874i −2.49298 + 3.24518i
103.11 −0.504390 1.32121i 0.826239 + 0.563320i −1.49118 + 1.33281i −2.91108 0.218156i 0.327516 1.37577i −2.63369 0.252375i 2.51305 + 1.29790i 0.365341 + 0.930874i 1.18009 + 3.95618i
103.12 −0.467447 1.33473i 0.826239 + 0.563320i −1.56299 + 1.24783i 2.71904 + 0.203764i 0.365655 1.36612i 2.62441 0.335390i 2.39612 + 1.50287i 0.365341 + 0.930874i −0.999039 3.72443i
103.13 −0.313470 + 1.37903i 0.826239 + 0.563320i −1.80347 0.864572i −0.481855 0.0361100i −1.03584 + 0.962828i 1.51904 + 2.16623i 1.75761 2.21603i 0.365341 + 0.930874i 0.200844 0.653175i
103.14 −0.281403 + 1.38593i 0.826239 + 0.563320i −1.84162 0.780012i −2.31910 0.173792i −1.01323 + 0.986592i −2.40273 + 1.10766i 1.59928 2.33287i 0.365341 + 0.930874i 0.893467 3.16521i
103.15 0.227338 + 1.39582i 0.826239 + 0.563320i −1.89663 + 0.634646i 1.91992 + 0.143878i −0.598459 + 1.28135i 0.854457 2.50398i −1.31703 2.50308i 0.365341 + 0.930874i 0.235643 + 2.71258i
103.16 0.307661 1.38034i 0.826239 + 0.563320i −1.81069 0.849355i −2.57232 0.192768i 1.03178 0.967181i 0.135490 + 2.64228i −1.72948 + 2.23806i 0.365341 + 0.930874i −1.05749 + 3.49137i
103.17 0.443386 + 1.34291i 0.826239 + 0.563320i −1.60682 + 1.19085i 4.26189 + 0.319384i −0.390146 + 1.35933i −0.230798 + 2.63567i −2.31165 1.62981i 0.365341 + 0.930874i 1.46076 + 5.86495i
103.18 0.471094 1.33344i 0.826239 + 0.563320i −1.55614 1.25635i 0.00499363 0.000374221i 1.14039 0.836366i 2.43918 1.02488i −2.40836 + 1.48317i 0.365341 + 0.930874i 0.00285147 0.00648243i
103.19 0.604797 + 1.27837i 0.826239 + 0.563320i −1.26844 + 1.54630i −1.87993 0.140882i −0.220423 + 1.39693i −1.80315 1.93615i −2.74389 0.686335i 0.365341 + 0.930874i −0.956880 2.48845i
103.20 0.898343 1.09224i 0.826239 + 0.563320i −0.385959 1.96241i −3.46688 0.259807i 1.35752 0.396393i −2.42772 1.05175i −2.49013 1.34135i 0.365341 + 0.930874i −3.39822 + 3.55326i
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.b yes 336
4.b odd 2 1 588.2.ba.a 336
49.h odd 42 1 588.2.ba.a 336
196.p even 42 1 inner 588.2.ba.b yes 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.ba.a 336 4.b odd 2 1
588.2.ba.a 336 49.h odd 42 1
588.2.ba.b yes 336 1.a even 1 1 trivial
588.2.ba.b yes 336 196.p even 42 1 inner

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 \(S_{2}^{\mathrm{new}}(588, [\chi])\).