# Properties

 Label 483.2.y.a Level $483$ Weight $2$ Character orbit 483.y Analytic conductor $3.857$ Analytic rank $0$ Dimension $320$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 483.y (of order $$33$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$320$$ Relative dimension: $$16$$ over $$\Q(\zeta_{33})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$320q + 2q^{2} - 16q^{3} + 18q^{4} - 2q^{5} - 18q^{6} + 2q^{7} - 12q^{8} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$320q + 2q^{2} - 16q^{3} + 18q^{4} - 2q^{5} - 18q^{6} + 2q^{7} - 12q^{8} + 16q^{9} + 8q^{10} - 6q^{11} - 18q^{12} - 10q^{14} + 18q^{15} + 8q^{16} + 4q^{17} + 2q^{18} + 18q^{20} + 4q^{21} - 176q^{22} - 18q^{23} - 6q^{24} + 8q^{25} - 14q^{26} + 32q^{27} + 46q^{28} + 34q^{29} - 8q^{30} + 52q^{31} - 8q^{32} - 5q^{33} - 24q^{34} - 12q^{35} - 14q^{36} - 30q^{37} - 157q^{38} + 88q^{40} - 28q^{41} - 45q^{42} + 64q^{43} - 71q^{44} - 2q^{45} + 4q^{46} + 36q^{47} + 60q^{48} + 28q^{49} + 210q^{50} - 26q^{51} - 198q^{52} - 10q^{53} - 2q^{54} - 4q^{55} + 44q^{57} + 31q^{58} + 10q^{59} - 2q^{60} - 34q^{61} - 8q^{62} + 2q^{63} + 84q^{64} + 38q^{65} - 12q^{67} - 22q^{68} + 8q^{69} - 336q^{70} - 144q^{71} + 6q^{72} - 16q^{73} - 68q^{74} - 30q^{75} + 8q^{76} + 98q^{77} + 16q^{78} + 26q^{79} + 225q^{80} + 16q^{81} - 122q^{82} + 44q^{84} - 240q^{85} - 26q^{86} - 16q^{87} + 43q^{88} + 68q^{89} - 16q^{90} + 40q^{91} - 222q^{92} - 8q^{93} + 137q^{94} - 49q^{95} + 30q^{96} - 8q^{97} + 137q^{98} - 10q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −1.59717 + 2.24291i −0.235759 + 0.971812i −1.82557 5.27464i −0.128956 + 2.70712i −1.80314 2.08094i 2.64469 + 0.0748341i 9.46243 + 2.77842i −0.888835 0.458227i −5.86587 4.61297i
4.2 −1.42823 + 2.00566i −0.235759 + 0.971812i −1.32872 3.83909i 0.0678834 1.42505i −1.61241 1.86082i −0.624376 + 2.57102i 4.87268 + 1.43075i −0.888835 0.458227i 2.76121 + 2.17144i
4.3 −1.09070 + 1.53167i −0.235759 + 0.971812i −0.502260 1.45118i −0.0927520 + 1.94710i −1.23135 1.42106i 0.850242 2.50541i −0.837777 0.245993i −0.888835 0.458227i −2.88116 2.26577i
4.4 −0.971087 + 1.36370i −0.235759 + 0.971812i −0.262535 0.758544i −0.194859 + 4.09059i −1.09632 1.26522i −2.20100 + 1.46819i −1.92325 0.564717i −0.888835 0.458227i −5.38912 4.23805i
4.5 −0.957403 + 1.34448i −0.235759 + 0.971812i −0.236882 0.684426i 0.115570 2.42610i −1.08087 1.24739i 2.21964 1.43986i −2.02036 0.593231i −0.888835 0.458227i 3.15121 + 2.47814i
4.6 −0.851765 + 1.19614i −0.235759 + 0.971812i −0.0511029 0.147652i 0.117190 2.46013i −0.961608 1.10975i −2.08452 1.62935i −2.59773 0.762762i −0.888835 0.458227i 2.84283 + 2.23562i
4.7 −0.364327 + 0.511625i −0.235759 + 0.971812i 0.525109 + 1.51720i −0.00931300 + 0.195504i −0.411310 0.474677i 0.694215 + 2.55305i −2.17284 0.638004i −0.888835 0.458227i −0.0966318 0.0759921i
4.8 −0.264798 + 0.371856i −0.235759 + 0.971812i 0.585977 + 1.69307i 0.162988 3.42155i −0.298946 0.345002i −2.64103 0.158003i −1.66077 0.487645i −0.888835 0.458227i 1.22916 + 0.966626i
4.9 0.282147 0.396220i −0.235759 + 0.971812i 0.576752 + 1.66642i −0.0698936 + 1.46725i 0.318533 + 0.367606i −2.64518 + 0.0550715i 1.75642 + 0.515730i −0.888835 0.458227i 0.561633 + 0.441673i
4.10 0.327609 0.460062i −0.235759 + 0.971812i 0.549806 + 1.58856i −0.198810 + 4.17354i 0.369857 + 0.426838i 2.54045 + 0.738979i 1.99478 + 0.585719i −0.888835 0.458227i 1.85495 + 1.45875i
4.11 0.466737 0.655441i −0.235759 + 0.971812i 0.442377 + 1.27816i −0.00881404 + 0.185029i 0.526928 + 0.608107i 1.59903 2.10787i 2.58833 + 0.760002i −0.888835 0.458227i 0.117162 + 0.0921373i
4.12 0.618045 0.867923i −0.235759 + 0.971812i 0.282825 + 0.817169i 0.184708 3.87751i 0.697748 + 0.805244i 0.447567 + 2.60762i 2.92870 + 0.859944i −0.888835 0.458227i −3.25122 2.55679i
4.13 0.910508 1.27863i −0.235759 + 0.971812i −0.151733 0.438405i 0.0858188 1.80156i 1.02793 + 1.18629i −0.649635 2.56476i 2.31350 + 0.679304i −0.888835 0.458227i −2.22539 1.75007i
4.14 1.13327 1.59145i −0.235759 + 0.971812i −0.594283 1.71707i −0.0776676 + 1.63044i 1.27941 + 1.47652i −1.33962 + 2.28154i 0.343047 + 0.100728i −0.888835 0.458227i 2.50675 + 1.97133i
4.15 1.47255 2.06790i −0.235759 + 0.971812i −1.45369 4.20016i 0.132912 2.79016i 1.66245 + 1.91856i 2.51171 0.831465i −5.95457 1.74842i −0.888835 0.458227i −5.57407 4.38350i
4.16 1.55491 2.18356i −0.235759 + 0.971812i −1.69606 4.90045i 0.00530352 0.111335i 1.75543 + 2.02587i −2.64473 0.0735467i −8.19360 2.40586i −0.888835 0.458227i −0.234859 0.184695i
16.1 −0.843018 2.43574i 0.888835 + 0.458227i −3.65004 + 2.87043i −0.918397 0.0876963i 0.366817 2.55126i −0.232163 2.63555i 5.73200 + 3.68373i 0.580057 + 0.814576i 0.560620 + 2.31091i
16.2 −0.783498 2.26377i 0.888835 + 0.458227i −2.93867 + 2.31100i −2.65564 0.253583i 0.340918 2.37114i −1.79847 + 1.94049i 3.50353 + 2.25158i 0.580057 + 0.814576i 1.50664 + 6.21044i
16.3 −0.741922 2.14364i 0.888835 + 0.458227i −2.47265 + 1.94451i 1.61087 + 0.153820i 0.322827 2.24531i 2.64162 0.147800i 2.18625 + 1.40502i 0.580057 + 0.814576i −0.865407 3.56726i
16.4 −0.461078 1.33220i 0.888835 + 0.458227i 0.00994599 0.00782161i 3.91624 + 0.373956i 0.200626 1.39538i −1.00683 2.44669i −2.38689 1.53396i 0.580057 + 0.814576i −1.30751 5.38963i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 478.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.y.a 320
7.c even 3 1 inner 483.2.y.a 320
23.c even 11 1 inner 483.2.y.a 320
161.m even 33 1 inner 483.2.y.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.y.a 320 1.a even 1 1 trivial
483.2.y.a 320 7.c even 3 1 inner
483.2.y.a 320 23.c even 11 1 inner
483.2.y.a 320 161.m even 33 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$21\!\cdots\!30$$$$T_{2}^{293} +$$$$79\!\cdots\!83$$$$T_{2}^{292} -$$$$43\!\cdots\!88$$$$T_{2}^{291} -$$$$91\!\cdots\!55$$$$T_{2}^{290} +$$$$15\!\cdots\!56$$$$T_{2}^{289} +$$$$29\!\cdots\!64$$$$T_{2}^{288} -$$$$89\!\cdots\!29$$$$T_{2}^{287} +$$$$25\!\cdots\!11$$$$T_{2}^{286} -$$$$19\!\cdots\!69$$$$T_{2}^{285} -$$$$35\!\cdots\!23$$$$T_{2}^{284} +$$$$62\!\cdots\!01$$$$T_{2}^{283} +$$$$14\!\cdots\!78$$$$T_{2}^{282} -$$$$35\!\cdots\!24$$$$T_{2}^{281} +$$$$57\!\cdots\!63$$$$T_{2}^{280} -$$$$84\!\cdots\!87$$$$T_{2}^{279} -$$$$10\!\cdots\!78$$$$T_{2}^{278} +$$$$24\!\cdots\!50$$$$T_{2}^{277} +$$$$46\!\cdots\!20$$$$T_{2}^{276} -$$$$13\!\cdots\!84$$$$T_{2}^{275} +$$$$53\!\cdots\!34$$$$T_{2}^{274} -$$$$21\!\cdots\!04$$$$T_{2}^{273} -$$$$19\!\cdots\!33$$$$T_{2}^{272} +$$$$51\!\cdots\!30$$$$T_{2}^{271} +$$$$10\!\cdots\!86$$$$T_{2}^{270} -$$$$32\!\cdots\!58$$$$T_{2}^{269} -$$$$10\!\cdots\!76$$$$T_{2}^{268} +$$$$29\!\cdots\!24$$$$T_{2}^{267} -$$$$30\!\cdots\!18$$$$T_{2}^{266} +$$$$10\!\cdots\!18$$$$T_{2}^{265} +$$$$25\!\cdots\!28$$$$T_{2}^{264} -$$$$82\!\cdots\!12$$$$T_{2}^{263} -$$$$74\!\cdots\!53$$$$T_{2}^{262} +$$$$20\!\cdots\!69$$$$T_{2}^{261} -$$$$40\!\cdots\!99$$$$T_{2}^{260} +$$$$16\!\cdots\!17$$$$T_{2}^{259} +$$$$54\!\cdots\!49$$$$T_{2}^{258} -$$$$19\!\cdots\!79$$$$T_{2}^{257} -$$$$21\!\cdots\!54$$$$T_{2}^{256} +$$$$73\!\cdots\!88$$$$T_{2}^{255} -$$$$59\!\cdots\!07$$$$T_{2}^{254} +$$$$21\!\cdots\!03$$$$T_{2}^{253} +$$$$11\!\cdots\!56$$$$T_{2}^{252} -$$$$38\!\cdots\!93$$$$T_{2}^{251} -$$$$45\!\cdots\!77$$$$T_{2}^{250} +$$$$15\!\cdots\!18$$$$T_{2}^{249} -$$$$81\!\cdots\!00$$$$T_{2}^{248} +$$$$34\!\cdots\!89$$$$T_{2}^{247} +$$$$17\!\cdots\!91$$$$T_{2}^{246} -$$$$69\!\cdots\!47$$$$T_{2}^{245} -$$$$71\!\cdots\!19$$$$T_{2}^{244} +$$$$28\!\cdots\!50$$$$T_{2}^{243} -$$$$11\!\cdots\!22$$$$T_{2}^{242} +$$$$47\!\cdots\!11$$$$T_{2}^{241} +$$$$24\!\cdots\!72$$$$T_{2}^{240} -$$$$10\!\cdots\!15$$$$T_{2}^{239} -$$$$94\!\cdots\!54$$$$T_{2}^{238} +$$$$40\!\cdots\!23$$$$T_{2}^{237} -$$$$12\!\cdots\!72$$$$T_{2}^{236} +$$$$54\!\cdots\!28$$$$T_{2}^{235} +$$$$27\!\cdots\!08$$$$T_{2}^{234} -$$$$12\!\cdots\!93$$$$T_{2}^{233} -$$$$10\!\cdots\!59$$$$T_{2}^{232} +$$$$44\!\cdots\!35$$$$T_{2}^{231} -$$$$59\!\cdots\!27$$$$T_{2}^{230} +$$$$55\!\cdots\!87$$$$T_{2}^{229} +$$$$23\!\cdots\!02$$$$T_{2}^{228} -$$$$12\!\cdots\!85$$$$T_{2}^{227} -$$$$96\!\cdots\!53$$$$T_{2}^{226} +$$$$44\!\cdots\!18$$$$T_{2}^{225} -$$$$11\!\cdots\!90$$$$T_{2}^{224} +$$$$21\!\cdots\!95$$$$T_{2}^{223} +$$$$19\!\cdots\!22$$$$T_{2}^{222} -$$$$91\!\cdots\!78$$$$T_{2}^{221} -$$$$98\!\cdots\!56$$$$T_{2}^{220} +$$$$38\!\cdots\!35$$$$T_{2}^{219} +$$$$11\!\cdots\!60$$$$T_{2}^{218} -$$$$31\!\cdots\!23$$$$T_{2}^{217} +$$$$13\!\cdots\!57$$$$T_{2}^{216} -$$$$42\!\cdots\!27$$$$T_{2}^{215} -$$$$90\!\cdots\!30$$$$T_{2}^{214} +$$$$23\!\cdots\!95$$$$T_{2}^{213} +$$$$24\!\cdots\!40$$$$T_{2}^{212} -$$$$45\!\cdots\!81$$$$T_{2}^{211} +$$$$26\!\cdots\!21$$$$T_{2}^{210} -$$$$11\!\cdots\!26$$$$T_{2}^{209} -$$$$51\!\cdots\!13$$$$T_{2}^{208} +$$$$11\!\cdots\!58$$$$T_{2}^{207} +$$$$20\!\cdots\!45$$$$T_{2}^{206} -$$$$37\!\cdots\!72$$$$T_{2}^{205} -$$$$16\!\cdots\!44$$$$T_{2}^{204} +$$$$28\!\cdots\!19$$$$T_{2}^{203} -$$$$25\!\cdots\!00$$$$T_{2}^{202} +$$$$34\!\cdots\!43$$$$T_{2}^{201} +$$$$15\!\cdots\!43$$$$T_{2}^{200} -$$$$18\!\cdots\!99$$$$T_{2}^{199} -$$$$39\!\cdots\!80$$$$T_{2}^{198} +$$$$41\!\cdots\!95$$$$T_{2}^{197} -$$$$13\!\cdots\!67$$$$T_{2}^{196} -$$$$17\!\cdots\!33$$$$T_{2}^{195} +$$$$62\!\cdots\!34$$$$T_{2}^{194} -$$$$14\!\cdots\!15$$$$T_{2}^{193} -$$$$28\!\cdots\!74$$$$T_{2}^{192} -$$$$38\!\cdots\!24$$$$T_{2}^{191} +$$$$55\!\cdots\!87$$$$T_{2}^{190} +$$$$32\!\cdots\!52$$$$T_{2}^{189} +$$$$72\!\cdots\!36$$$$T_{2}^{188} +$$$$19\!\cdots\!94$$$$T_{2}^{187} -$$$$78\!\cdots\!09$$$$T_{2}^{186} -$$$$80\!\cdots\!78$$$$T_{2}^{185} +$$$$19\!\cdots\!54$$$$T_{2}^{184} +$$$$33\!\cdots\!95$$$$T_{2}^{183} +$$$$58\!\cdots\!43$$$$T_{2}^{182} -$$$$36\!\cdots\!44$$$$T_{2}^{181} -$$$$15\!\cdots\!16$$$$T_{2}^{180} -$$$$20\!\cdots\!28$$$$T_{2}^{179} +$$$$29\!\cdots\!69$$$$T_{2}^{178} +$$$$94\!\cdots\!80$$$$T_{2}^{177} +$$$$57\!\cdots\!37$$$$T_{2}^{176} -$$$$10\!\cdots\!30$$$$T_{2}^{175} -$$$$38\!\cdots\!90$$$$T_{2}^{174} -$$$$45\!\cdots\!83$$$$T_{2}^{173} +$$$$59\!\cdots\!06$$$$T_{2}^{172} +$$$$21\!\cdots\!68$$$$T_{2}^{171} +$$$$12\!\cdots\!27$$$$T_{2}^{170} -$$$$31\!\cdots\!64$$$$T_{2}^{169} -$$$$82\!\cdots\!57$$$$T_{2}^{168} -$$$$54\!\cdots\!09$$$$T_{2}^{167} +$$$$17\!\cdots\!15$$$$T_{2}^{166} +$$$$39\!\cdots\!80$$$$T_{2}^{165} -$$$$90\!\cdots\!15$$$$T_{2}^{164} -$$$$95\!\cdots\!16$$$$T_{2}^{163} -$$$$11\!\cdots\!07$$$$T_{2}^{162} +$$$$63\!\cdots\!04$$$$T_{2}^{161} +$$$$36\!\cdots\!92$$$$T_{2}^{160} +$$$$37\!\cdots\!47$$$$T_{2}^{159} -$$$$48\!\cdots\!98$$$$T_{2}^{158} -$$$$15\!\cdots\!19$$$$T_{2}^{157} -$$$$71\!\cdots\!91$$$$T_{2}^{156} +$$$$23\!\cdots\!54$$$$T_{2}^{155} +$$$$48\!\cdots\!84$$$$T_{2}^{154} +$$$$18\!\cdots\!04$$$$T_{2}^{153} -$$$$91\!\cdots\!49$$$$T_{2}^{152} -$$$$17\!\cdots\!90$$$$T_{2}^{151} -$$$$25\!\cdots\!63$$$$T_{2}^{150} +$$$$34\!\cdots\!76$$$$T_{2}^{149} +$$$$55\!\cdots\!07$$$$T_{2}^{148} +$$$$47\!\cdots\!62$$$$T_{2}^{147} -$$$$12\!\cdots\!50$$$$T_{2}^{146} -$$$$18\!\cdots\!81$$$$T_{2}^{145} +$$$$44\!\cdots\!77$$$$T_{2}^{144} +$$$$43\!\cdots\!86$$$$T_{2}^{143} +$$$$44\!\cdots\!49$$$$T_{2}^{142} -$$$$24\!\cdots\!22$$$$T_{2}^{141} -$$$$11\!\cdots\!28$$$$T_{2}^{140} -$$$$11\!\cdots\!68$$$$T_{2}^{139} +$$$$68\!\cdots\!53$$$$T_{2}^{138} +$$$$28\!\cdots\!40$$$$T_{2}^{137} +$$$$26\!\cdots\!90$$$$T_{2}^{136} -$$$$10\!\cdots\!82$$$$T_{2}^{135} -$$$$67\!\cdots\!78$$$$T_{2}^{134} -$$$$84\!\cdots\!27$$$$T_{2}^{133} +$$$$25\!\cdots\!47$$$$T_{2}^{132} +$$$$19\!\cdots\!54$$$$T_{2}^{131} +$$$$18\!\cdots\!10$$$$T_{2}^{130} -$$$$10\!\cdots\!73$$$$T_{2}^{129} -$$$$43\!\cdots\!55$$$$T_{2}^{128} -$$$$40\!\cdots\!87$$$$T_{2}^{127} +$$$$28\!\cdots\!01$$$$T_{2}^{126} +$$$$95\!\cdots\!27$$$$T_{2}^{125} +$$$$60\!\cdots\!09$$$$T_{2}^{124} -$$$$53\!\cdots\!88$$$$T_{2}^{123} -$$$$15\!\cdots\!26$$$$T_{2}^{122} -$$$$14\!\cdots\!45$$$$T_{2}^{121} +$$$$84\!\cdots\!25$$$$T_{2}^{120} +$$$$33\!\cdots\!82$$$$T_{2}^{119} +$$$$24\!\cdots\!76$$$$T_{2}^{118} -$$$$15\!\cdots\!84$$$$T_{2}^{117} -$$$$60\!\cdots\!60$$$$T_{2}^{116} -$$$$54\!\cdots\!21$$$$T_{2}^{115} +$$$$52\!\cdots\!71$$$$T_{2}^{114} +$$$$13\!\cdots\!80$$$$T_{2}^{113} +$$$$27\!\cdots\!04$$$$T_{2}^{112} -$$$$14\!\cdots\!21$$$$T_{2}^{111} -$$$$15\!\cdots\!89$$$$T_{2}^{110} +$$$$33\!\cdots\!78$$$$T_{2}^{109} +$$$$26\!\cdots\!69$$$$T_{2}^{108} +$$$$15\!\cdots\!05$$$$T_{2}^{107} -$$$$27\!\cdots\!44$$$$T_{2}^{106} -$$$$31\!\cdots\!94$$$$T_{2}^{105} +$$$$11\!\cdots\!10$$$$T_{2}^{104} +$$$$33\!\cdots\!26$$$$T_{2}^{103} +$$$$19\!\cdots\!60$$$$T_{2}^{102} -$$$$22\!\cdots\!20$$$$T_{2}^{101} -$$$$47\!\cdots\!50$$$$T_{2}^{100} +$$$$61\!\cdots\!82$$$$T_{2}^{99} +$$$$50\!\cdots\!28$$$$T_{2}^{98} +$$$$64\!\cdots\!97$$$$T_{2}^{97} -$$$$23\!\cdots\!18$$$$T_{2}^{96} -$$$$11\!\cdots\!65$$$$T_{2}^{95} -$$$$99\!\cdots\!11$$$$T_{2}^{94} +$$$$14\!\cdots\!24$$$$T_{2}^{93} +$$$$19\!\cdots\!87$$$$T_{2}^{92} -$$$$16\!\cdots\!96$$$$T_{2}^{91} +$$$$34\!\cdots\!18$$$$T_{2}^{90} +$$$$11\!\cdots\!73$$$$T_{2}^{89} -$$$$30\!\cdots\!01$$$$T_{2}^{88} +$$$$54\!\cdots\!36$$$$T_{2}^{87} +$$$$30\!\cdots\!22$$$$T_{2}^{86} -$$$$25\!\cdots\!55$$$$T_{2}^{85} -$$$$25\!\cdots\!09$$$$T_{2}^{84} +$$$$32\!\cdots\!84$$$$T_{2}^{83} -$$$$28\!\cdots\!06$$$$T_{2}^{82} -$$$$20\!\cdots\!20$$$$T_{2}^{81} +$$$$39\!\cdots\!23$$$$T_{2}^{80} +$$$$17\!\cdots\!99$$$$T_{2}^{79} -$$$$28\!\cdots\!41$$$$T_{2}^{78} +$$$$10\!\cdots\!48$$$$T_{2}^{77} +$$$$10\!\cdots\!17$$$$T_{2}^{76} -$$$$11\!\cdots\!66$$$$T_{2}^{75} +$$$$21\!\cdots\!44$$$$T_{2}^{74} +$$$$58\!\cdots\!54$$$$T_{2}^{73} -$$$$56\!\cdots\!87$$$$T_{2}^{72} -$$$$15\!\cdots\!72$$$$T_{2}^{71} +$$$$41\!\cdots\!07$$$$T_{2}^{70} +$$$$88\!\cdots\!52$$$$T_{2}^{69} -$$$$18\!\cdots\!38$$$$T_{2}^{68} +$$$$27\!\cdots\!81$$$$T_{2}^{67} +$$$$60\!\cdots\!77$$$$T_{2}^{66} +$$$$40\!\cdots\!90$$$$T_{2}^{65} -$$$$19\!\cdots\!16$$$$T_{2}^{64} -$$$$38\!\cdots\!07$$$$T_{2}^{63} +$$$$79\!\cdots\!19$$$$T_{2}^{62} +$$$$11\!\cdots\!63$$$$T_{2}^{61} -$$$$30\!\cdots\!81$$$$T_{2}^{60} +$$$$80\!\cdots\!82$$$$T_{2}^{59} +$$$$10\!\cdots\!39$$$$T_{2}^{58} -$$$$20\!\cdots\!56$$$$T_{2}^{57} -$$$$33\!\cdots\!55$$$$T_{2}^{56} +$$$$16\!\cdots\!47$$$$T_{2}^{55} +$$$$71\!\cdots\!20$$$$T_{2}^{54} -$$$$91\!\cdots\!64$$$$T_{2}^{53} +$$$$10\!\cdots\!50$$$$T_{2}^{52} +$$$$26\!\cdots\!21$$$$T_{2}^{51} -$$$$17\!\cdots\!75$$$$T_{2}^{50} -$$$$74\!\cdots\!67$$$$T_{2}^{49} +$$$$60\!\cdots\!71$$$$T_{2}^{48} -$$$$22\!\cdots\!01$$$$T_{2}^{47} -$$$$66\!\cdots\!39$$$$T_{2}^{46} +$$$$78\!\cdots\!26$$$$T_{2}^{45} -$$$$14\!\cdots\!78$$$$T_{2}^{44} -$$$$10\!\cdots\!31$$$$T_{2}^{43} +$$$$65\!\cdots\!96$$$$T_{2}^{42} -$$$$10\!\cdots\!86$$$$T_{2}^{41} -$$$$11\!\cdots\!47$$$$T_{2}^{40} +$$$$33\!\cdots\!54$$$$T_{2}^{39} +$$$$18\!\cdots\!40$$$$T_{2}^{38} -$$$$11\!\cdots\!59$$$$T_{2}^{37} -$$$$17\!\cdots\!48$$$$T_{2}^{36} +$$$$27\!\cdots\!09$$$$T_{2}^{35} -$$$$62\!\cdots\!16$$$$T_{2}^{34} -$$$$36\!\cdots\!93$$$$T_{2}^{33} +$$$$23\!\cdots\!59$$$$T_{2}^{32} -$$$$27\!\cdots\!27$$$$T_{2}^{31} -$$$$26\!\cdots\!94$$$$T_{2}^{30} +$$$$13\!\cdots\!39$$$$T_{2}^{29} -$$$$13\!\cdots\!49$$$$T_{2}^{28} -$$$$82\!\cdots\!33$$$$T_{2}^{27} +$$$$46\!\cdots\!59$$$$T_{2}^{26} -$$$$68\!\cdots\!12$$$$T_{2}^{25} -$$$$80\!\cdots\!79$$$$T_{2}^{24} +$$$$85\!\cdots\!91$$$$T_{2}^{23} -$$$$17\!\cdots\!82$$$$T_{2}^{22} +$$$$10\!\cdots\!77$$$$T_{2}^{21} +$$$$81\!\cdots\!18$$$$T_{2}^{20} -$$$$22\!\cdots\!74$$$$T_{2}^{19} +$$$$31\!\cdots\!70$$$$T_{2}^{18} +$$$$27\!\cdots\!36$$$$T_{2}^{17} -$$$$12\!\cdots\!65$$$$T_{2}^{16} +$$$$27\!\cdots\!08$$$$T_{2}^{15} -$$$$23\!\cdots\!73$$$$T_{2}^{14} -$$$$38\!\cdots\!64$$$$T_{2}^{13} +$$$$12\!\cdots\!27$$$$T_{2}^{12} -$$$$22\!\cdots\!24$$$$T_{2}^{11} -$$$$30\!\cdots\!43$$$$T_{2}^{10} +$$$$35\!\cdots\!81$$$$T_{2}^{9} +$$$$53\!\cdots\!89$$$$T_{2}^{8} +$$$$15\!\cdots\!22$$$$T_{2}^{7} +$$$$66\!\cdots\!50$$$$T_{2}^{6} +$$$$40\!\cdots\!55$$$$T_{2}^{5} +$$$$83\!\cdots\!32$$$$T_{2}^{4} +$$$$50\!\cdots\!36$$$$T_{2}^{3} +$$$$14\!\cdots\!53$$$$T_{2}^{2} +$$$$95\!\cdots\!06$$$$T_{2} +$$$$15\!\cdots\!01$$">$$T_{2}^{320} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$.