Properties

Label 483.2.y.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 320q + 2q^{2} + 16q^{3} + 18q^{4} - 2q^{5} + 18q^{6} + 2q^{7} - 12q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 320q + 2q^{2} + 16q^{3} + 18q^{4} - 2q^{5} + 18q^{6} + 2q^{7} - 12q^{8} + 16q^{9} + 2q^{11} + 18q^{12} + 18q^{14} - 18q^{15} - 8q^{16} + 4q^{17} + 2q^{18} + 8q^{19} - 162q^{20} - 4q^{21} + 144q^{22} - 26q^{23} + 6q^{24} - 8q^{25} - 14q^{26} - 32q^{27} + 86q^{28} - 74q^{29} - 56q^{31} - 28q^{32} + 13q^{33} + 40q^{34} - 32q^{35} - 14q^{36} + 2q^{37} - 39q^{38} - 52q^{40} + 60q^{41} - 61q^{42} + 16q^{43} - 75q^{44} - 2q^{45} - 4q^{46} - 40q^{47} - 28q^{48} - 100q^{49} + 146q^{50} - 18q^{51} - 18q^{52} + 34q^{53} + 2q^{54} + 36q^{55} - 102q^{56} + 28q^{57} - 17q^{58} - 102q^{59} - 18q^{60} - 18q^{61} - 88q^{62} + 2q^{63} - 252q^{64} - 78q^{65} + 16q^{66} + 12q^{67} + 34q^{68} + 8q^{69} + 264q^{70} + 160q^{71} + 6q^{72} - 8q^{73} + 70q^{74} + 14q^{75} - 40q^{76} - 90q^{77} - 16q^{78} + 26q^{79} - 103q^{80} + 16q^{81} - 30q^{82} - 80q^{83} - 52q^{84} - 128q^{85} + 90q^{86} + 4q^{87} - 293q^{88} - 36q^{89} + 16q^{91} - 174q^{92} + 32q^{93} + 57q^{94} - 85q^{95} - 50q^{96} - 8q^{97} - 193q^{98} - 26q^{99} + O(q^{100}) \)

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 −1.48158 + 2.08058i 0.235759 0.971812i −1.47962 4.27509i 0.193923 4.07094i 1.67264 + 1.93033i −2.46944 0.949661i 6.18541 + 1.81620i −0.888835 0.458227i 8.18263 + 6.43489i
4.2 −1.37000 + 1.92390i 0.235759 0.971812i −1.17034 3.38148i 0.00886507 0.186101i 1.54668 + 1.78496i 0.229059 + 2.63582i 3.57665 + 1.05020i −0.888835 0.458227i 0.345893 + 0.272014i
4.3 −1.32550 + 1.86141i 0.235759 0.971812i −1.05374 3.04459i −0.0837421 + 1.75796i 1.49644 + 1.72698i 0.00745284 2.64574i 2.67883 + 0.786576i −0.888835 0.458227i −3.16128 2.48606i
4.4 −1.09595 + 1.53904i 0.235759 0.971812i −0.513414 1.48341i −0.133721 + 2.80715i 1.23728 + 1.42790i 2.23077 + 1.42255i −0.779983 0.229024i −0.888835 0.458227i −4.17377 3.28229i
4.5 −0.848286 + 1.19125i 0.235759 0.971812i −0.0453539 0.131042i 0.176217 3.69925i 0.957681 + 1.10522i 2.34512 + 1.22490i −2.61178 0.766889i −0.888835 0.458227i 4.25725 + 3.34794i
4.6 −0.765883 + 1.07553i 0.235759 0.971812i 0.0839437 + 0.242539i −0.0355806 + 0.746929i 0.864651 + 0.997860i −2.58720 + 0.553515i −2.85890 0.839448i −0.888835 0.458227i −0.776095 0.610328i
4.7 −0.342635 + 0.481163i 0.235759 0.971812i 0.540016 + 1.56028i 0.0918733 1.92866i 0.386821 + 0.446415i −2.12937 + 1.57028i −2.06930 0.607603i −0.888835 0.458227i 0.896520 + 0.705031i
4.8 −0.111942 + 0.157200i 0.235759 0.971812i 0.641955 + 1.85481i −0.171106 + 3.59197i 0.126378 + 0.145848i −2.30883 1.29202i −0.733771 0.215455i −0.888835 0.458227i −0.545504 0.428989i
4.9 −0.0919807 + 0.129169i 0.235759 0.971812i 0.645912 + 1.86624i 0.0626824 1.31586i 0.103842 + 0.119841i 0.568266 2.58400i −0.604769 0.177576i −0.888835 0.458227i 0.164203 + 0.129131i
4.10 0.199911 0.280735i 0.235759 0.971812i 0.615288 + 1.77776i −0.0359296 + 0.754254i −0.225691 0.260461i 2.50738 0.844407i 1.28344 + 0.376852i −0.888835 0.458227i 0.204563 + 0.160870i
4.11 0.733120 1.02952i 0.235759 0.971812i 0.131683 + 0.380473i 0.00357914 0.0751355i −0.827663 0.955174i 1.84836 + 1.89303i 2.91361 + 0.855512i −0.888835 0.458227i −0.0747298 0.0587682i
4.12 0.746582 1.04843i 0.235759 0.971812i 0.112320 + 0.324527i 0.116881 2.45363i −0.842861 0.972713i −2.38903 1.13690i 2.89400 + 0.849754i −0.888835 0.458227i −2.48519 1.95438i
4.13 0.900476 1.26454i 0.235759 0.971812i −0.134072 0.387376i −0.164040 + 3.44363i −1.01660 1.17322i −1.20985 + 2.35293i 2.36844 + 0.695436i −0.888835 0.458227i 4.20690 + 3.30834i
4.14 1.20076 1.68624i 0.235759 0.971812i −0.747425 2.15954i −0.0741223 + 1.55602i −1.35561 1.56446i 2.38094 1.15374i −0.566529 0.166348i −0.888835 0.458227i 2.53481 + 1.99340i
4.15 1.38638 1.94690i 0.235759 0.971812i −1.21423 3.50828i 0.129724 2.72324i −1.56517 1.80630i −0.222669 + 2.63636i −3.92712 1.15311i −0.888835 0.458227i −5.12202 4.02800i
4.16 1.50681 2.11601i 0.235759 0.971812i −1.55291 4.48684i 0.00580718 0.121908i −1.70112 1.96320i −0.123513 2.64287i −6.84923 2.01112i −0.888835 0.458227i −0.249208 0.195979i
16.1 −0.860562 2.48643i −0.888835 0.458227i −3.86967 + 3.04314i 0.317869 + 0.0303528i −0.374451 + 2.60436i 1.99020 + 1.74330i 6.46974 + 4.15785i 0.580057 + 0.814576i −0.198076 0.816481i
16.2 −0.740363 2.13914i −0.888835 0.458227i −2.45566 + 1.93116i 2.48488 + 0.237277i −0.322149 + 2.24059i −2.63923 + 0.185589i 2.14051 + 1.37562i 0.580057 + 0.814576i −1.33214 5.49116i
16.3 −0.695372 2.00914i −0.888835 0.458227i −1.98101 + 1.55789i 1.53680 + 0.146746i −0.302572 + 2.10444i 0.278867 2.63101i 0.930423 + 0.597946i 0.580057 + 0.814576i −0.773810 3.18969i
16.4 −0.620909 1.79400i −0.888835 0.458227i −1.26080 + 0.991503i −3.16803 0.302510i −0.270172 + 1.87909i 1.40046 2.24471i −0.632491 0.406478i 0.580057 + 0.814576i 1.42436 + 5.87128i
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.b 320
7.c even 3 1 inner 483.2.y.b 320
23.c even 11 1 inner 483.2.y.b 320
161.m even 33 1 inner 483.2.y.b 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.y.b 320 1.a even 1 1 trivial
483.2.y.b 320 7.c even 3 1 inner
483.2.y.b 320 23.c even 11 1 inner
483.2.y.b 320 161.m even 33 1 inner

Hecke kernels

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