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 $4$

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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:

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 \(S_{2}^{\mathrm{new}}(483, [\chi])\).