# 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 $2$

# 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.

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

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