# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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