Newform orbit 483.2.y.a

• 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: $4$

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.

\(\operatorname{Tr}(f)(q) = \) \( 320q + 2q^{2} - 16q^{3} + 18q^{4} - 2q^{5} - 18q^{6} + 2q^{7} - 12q^{8} + 16q^{9} + O(q^{10}) \)

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

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