Newform orbit 483.2.y.b

• 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: $4$

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	−1.48158	+	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

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