Properties

Label 507.2.t.b
Level $507$
Weight $2$
Character orbit 507.t
Analytic conductor $4.048$
Analytic rank $0$
Dimension $360$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.t (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(15\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360q - 15q^{3} - 14q^{4} + 3q^{7} + 15q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 360q - 15q^{3} - 14q^{4} + 3q^{7} + 15q^{9} + 6q^{11} - 28q^{12} + 6q^{13} + 4q^{14} - 6q^{15} + 20q^{16} - 6q^{19} - 12q^{20} - 28q^{22} + 46q^{23} - 6q^{25} - 39q^{26} + 30q^{27} - 6q^{28} + 43q^{29} + 26q^{31} - 195q^{32} + 19q^{33} + 65q^{34} + 84q^{35} - 14q^{36} - 65q^{38} - 2q^{39} + 12q^{41} - 128q^{42} + 83q^{43} - 39q^{44} - 6q^{45} - 20q^{48} + 72q^{49} - 52q^{50} - 55q^{52} + 49q^{53} - 49q^{55} - 2q^{56} + 26q^{57} + 26q^{58} - 202q^{59} - 182q^{60} - 3q^{61} - 65q^{62} - 3q^{63} - 14q^{64} - 58q^{65} + 48q^{66} - 41q^{67} + 139q^{68} + 6q^{69} - 60q^{71} - 52q^{73} - 269q^{74} + 23q^{75} - 14q^{76} + 70q^{77} - 65q^{78} + 18q^{79} + 492q^{80} + 15q^{81} - 65q^{82} + 78q^{83} - 6q^{84} - 91q^{85} - 169q^{86} + 48q^{87} - 522q^{88} - 12q^{89} + 373q^{91} + 72q^{92} - 3q^{93} - 13q^{94} - 110q^{95} + 65q^{96} - 121q^{97} - 104q^{98} + O(q^{100}) \)

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 −2.60202 0.104858i −0.278217 + 0.960518i 4.76600 + 0.384751i −0.705070 0.267398i 0.824645 2.47011i 0.861409 + 2.02180i −7.19059 0.873096i −0.845190 0.534466i 1.80657 + 0.769706i
4.2 −2.53957 0.102341i −0.278217 + 0.960518i 4.44542 + 0.358871i 1.66050 + 0.629744i 0.804853 2.41083i −2.05060 4.81293i −6.20652 0.753608i −0.845190 0.534466i −4.15250 1.76922i
4.3 −1.85726 0.0748449i −0.278217 + 0.960518i 1.45029 + 0.117079i −1.06783 0.404975i 0.588611 1.76311i 0.332955 + 0.781475i 1.00563 + 0.122106i −0.845190 0.534466i 1.95292 + 0.832063i
4.4 −1.79660 0.0724007i −0.278217 + 0.960518i 1.22903 + 0.0992178i 1.73729 + 0.658868i 0.569389 1.70553i 1.36244 + 3.19777i 1.36900 + 0.166227i −0.845190 0.534466i −3.07353 1.30951i
4.5 −1.38591 0.0558504i −0.278217 + 0.960518i −0.0758757 0.00612532i −1.08360 0.410955i 0.439231 1.31566i −0.389218 0.913528i 2.85867 + 0.347105i −0.845190 0.534466i 1.47882 + 0.630068i
4.6 −0.772377 0.0311257i −0.278217 + 0.960518i −1.39792 0.112852i 3.22103 + 1.22158i 0.244786 0.733223i −1.25005 2.93398i 2.61094 + 0.317026i −0.845190 0.534466i −2.44983 1.04377i
4.7 −0.218325 0.00879818i −0.278217 + 0.960518i −1.94593 0.157091i −3.83328 1.45377i 0.0691925 0.207257i 1.70758 + 4.00784i 0.857279 + 0.104093i −0.845190 0.534466i 0.824109 + 0.351120i
4.8 −0.208145 0.00838797i −0.278217 + 0.960518i −1.95026 0.157441i −1.53730 0.583019i 0.0659665 0.197594i −0.815402 1.91382i 0.818208 + 0.0993485i −0.845190 0.534466i 0.315091 + 0.134248i
4.9 0.805348 + 0.0324544i −0.278217 + 0.960518i −1.34598 0.108659i 3.20699 + 1.21625i −0.255235 + 0.764522i −0.483860 1.13566i −2.68071 0.325497i −0.845190 0.534466i 2.54327 + 1.08359i
4.10 0.989614 + 0.0398801i −0.278217 + 0.960518i −1.01577 0.0820013i −0.558145 0.211676i −0.313633 + 0.939447i −0.641263 1.50510i −2.96834 0.360422i −0.845190 0.534466i −0.543906 0.231737i
4.11 0.992569 + 0.0399992i −0.278217 + 0.960518i −1.00992 0.0815293i −0.620050 0.235154i −0.314570 + 0.942252i 1.02980 + 2.41704i −2.97142 0.360796i −0.845190 0.534466i −0.606036 0.258208i
4.12 1.81864 + 0.0732885i −0.278217 + 0.960518i 1.30855 + 0.105637i −3.76569 1.42814i −0.576371 + 1.72644i −1.29170 3.03172i −1.24165 0.150763i −0.845190 0.534466i −6.74376 2.87325i
4.13 1.92708 + 0.0776588i −0.278217 + 0.960518i 1.71410 + 0.138377i 3.62781 + 1.37585i −0.610741 + 1.82939i 1.17937 + 2.76808i −0.536697 0.0651668i −0.845190 0.534466i 6.88424 + 2.93310i
4.14 2.27767 + 0.0917871i −0.278217 + 0.960518i 3.18585 + 0.257189i −1.29798 0.492260i −0.721851 + 2.16221i 1.28355 + 3.01261i 2.70692 + 0.328679i −0.845190 0.534466i −2.91120 1.24035i
4.15 2.56929 + 0.103539i −0.278217 + 0.960518i 4.59701 + 0.371109i 1.84810 + 0.700893i −0.814272 + 2.43904i −1.21279 2.84653i 6.66737 + 0.809566i −0.845190 0.534466i 4.67574 + 1.99215i
10.1 −2.52917 0.516334i −0.987050 + 0.160411i 4.29016 + 1.82787i 0.764962 3.10357i 2.57925 + 0.103940i 4.06091 1.92693i −5.65796 3.90541i 0.948536 0.316668i −3.53720 + 7.45450i
10.2 −2.42455 0.494975i −0.987050 + 0.160411i 3.79347 + 1.61625i 0.0746454 0.302848i 2.47255 + 0.0996403i −4.01396 + 1.90465i −4.32440 2.98492i 0.948536 0.316668i −0.330883 + 0.697322i
10.3 −1.83778 0.375186i −0.987050 + 0.160411i 1.39672 + 0.595086i −0.959864 + 3.89432i 1.87417 + 0.0755263i 1.09680 0.520440i 0.743728 + 0.513358i 0.948536 0.316668i 3.22511 6.79678i
10.4 −1.49980 0.306186i −0.987050 + 0.160411i 0.315692 + 0.134504i −0.217058 + 0.880640i 1.52949 + 0.0616365i 0.786387 0.373145i 2.08725 + 1.44072i 0.948536 0.316668i 0.595184 1.25432i
10.5 −1.31845 0.269162i −0.987050 + 0.160411i −0.174110 0.0741814i 0.381569 1.54808i 1.34455 + 0.0541834i 1.87281 0.888658i 2.42447 + 1.67349i 0.948536 0.316668i −0.919764 + 1.93836i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 478.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.k even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.t.b 360
169.k even 78 1 inner 507.2.t.b 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.t.b 360 1.a even 1 1 trivial
507.2.t.b 360 169.k even 78 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(35\!\cdots\!99\)\( T_{2}^{332} + \)\(35\!\cdots\!90\)\( T_{2}^{331} - \)\(82\!\cdots\!09\)\( T_{2}^{330} + \)\(37\!\cdots\!76\)\( T_{2}^{329} - \)\(23\!\cdots\!57\)\( T_{2}^{328} - \)\(27\!\cdots\!21\)\( T_{2}^{327} - \)\(97\!\cdots\!02\)\( T_{2}^{326} - \)\(18\!\cdots\!96\)\( T_{2}^{325} + \)\(57\!\cdots\!93\)\( T_{2}^{324} + \)\(60\!\cdots\!44\)\( T_{2}^{323} + \)\(94\!\cdots\!66\)\( T_{2}^{322} + \)\(15\!\cdots\!67\)\( T_{2}^{321} + \)\(38\!\cdots\!62\)\( T_{2}^{320} + \)\(64\!\cdots\!18\)\( T_{2}^{319} - \)\(20\!\cdots\!79\)\( T_{2}^{318} - \)\(49\!\cdots\!71\)\( T_{2}^{317} - \)\(35\!\cdots\!82\)\( T_{2}^{316} - \)\(70\!\cdots\!82\)\( T_{2}^{315} - \)\(15\!\cdots\!00\)\( T_{2}^{314} - \)\(22\!\cdots\!14\)\( T_{2}^{313} + \)\(61\!\cdots\!84\)\( T_{2}^{312} + \)\(18\!\cdots\!10\)\( T_{2}^{311} + \)\(11\!\cdots\!92\)\( T_{2}^{310} + \)\(22\!\cdots\!26\)\( T_{2}^{309} + \)\(51\!\cdots\!70\)\( T_{2}^{308} + \)\(73\!\cdots\!72\)\( T_{2}^{307} - \)\(10\!\cdots\!82\)\( T_{2}^{306} - \)\(48\!\cdots\!89\)\( T_{2}^{305} - \)\(26\!\cdots\!79\)\( T_{2}^{304} - \)\(62\!\cdots\!37\)\( T_{2}^{303} - \)\(13\!\cdots\!62\)\( T_{2}^{302} - \)\(20\!\cdots\!52\)\( T_{2}^{301} + \)\(13\!\cdots\!19\)\( T_{2}^{300} + \)\(12\!\cdots\!67\)\( T_{2}^{299} + \)\(56\!\cdots\!47\)\( T_{2}^{298} + \)\(16\!\cdots\!63\)\( T_{2}^{297} + \)\(30\!\cdots\!68\)\( T_{2}^{296} + \)\(58\!\cdots\!46\)\( T_{2}^{295} - \)\(89\!\cdots\!32\)\( T_{2}^{294} - \)\(22\!\cdots\!87\)\( T_{2}^{293} - \)\(10\!\cdots\!40\)\( T_{2}^{292} - \)\(32\!\cdots\!52\)\( T_{2}^{291} - \)\(58\!\cdots\!93\)\( T_{2}^{290} - \)\(10\!\cdots\!83\)\( T_{2}^{289} + \)\(29\!\cdots\!82\)\( T_{2}^{288} + \)\(43\!\cdots\!78\)\( T_{2}^{287} + \)\(19\!\cdots\!07\)\( T_{2}^{286} + \)\(57\!\cdots\!95\)\( T_{2}^{285} + \)\(90\!\cdots\!43\)\( T_{2}^{284} + \)\(17\!\cdots\!09\)\( T_{2}^{283} - \)\(11\!\cdots\!20\)\( T_{2}^{282} - \)\(64\!\cdots\!30\)\( T_{2}^{281} - \)\(31\!\cdots\!54\)\( T_{2}^{280} - \)\(78\!\cdots\!91\)\( T_{2}^{279} - \)\(12\!\cdots\!17\)\( T_{2}^{278} - \)\(20\!\cdots\!21\)\( T_{2}^{277} + \)\(25\!\cdots\!55\)\( T_{2}^{276} + \)\(91\!\cdots\!43\)\( T_{2}^{275} + \)\(43\!\cdots\!25\)\( T_{2}^{274} + \)\(81\!\cdots\!32\)\( T_{2}^{273} + \)\(13\!\cdots\!71\)\( T_{2}^{272} + \)\(76\!\cdots\!98\)\( T_{2}^{271} - \)\(39\!\cdots\!81\)\( T_{2}^{270} - \)\(17\!\cdots\!74\)\( T_{2}^{269} - \)\(42\!\cdots\!64\)\( T_{2}^{268} - \)\(97\!\cdots\!61\)\( T_{2}^{267} - \)\(48\!\cdots\!08\)\( T_{2}^{266} - \)\(22\!\cdots\!51\)\( T_{2}^{265} + \)\(93\!\cdots\!38\)\( T_{2}^{264} + \)\(17\!\cdots\!18\)\( T_{2}^{263} + \)\(52\!\cdots\!09\)\( T_{2}^{262} + \)\(49\!\cdots\!19\)\( T_{2}^{261} + \)\(19\!\cdots\!72\)\( T_{2}^{260} - \)\(32\!\cdots\!19\)\( T_{2}^{259} - \)\(93\!\cdots\!74\)\( T_{2}^{258} - \)\(28\!\cdots\!67\)\( T_{2}^{257} - \)\(31\!\cdots\!94\)\( T_{2}^{256} - \)\(55\!\cdots\!23\)\( T_{2}^{255} + \)\(12\!\cdots\!67\)\( T_{2}^{254} + \)\(36\!\cdots\!34\)\( T_{2}^{253} + \)\(13\!\cdots\!92\)\( T_{2}^{252} + \)\(27\!\cdots\!92\)\( T_{2}^{251} + \)\(27\!\cdots\!96\)\( T_{2}^{250} + \)\(50\!\cdots\!14\)\( T_{2}^{249} - \)\(16\!\cdots\!97\)\( T_{2}^{248} - \)\(24\!\cdots\!07\)\( T_{2}^{247} - \)\(13\!\cdots\!80\)\( T_{2}^{246} - \)\(15\!\cdots\!40\)\( T_{2}^{245} - \)\(24\!\cdots\!59\)\( T_{2}^{244} - \)\(13\!\cdots\!74\)\( T_{2}^{243} + \)\(13\!\cdots\!28\)\( T_{2}^{242} + \)\(18\!\cdots\!49\)\( T_{2}^{241} + \)\(96\!\cdots\!61\)\( T_{2}^{240} + \)\(77\!\cdots\!83\)\( T_{2}^{239} + \)\(19\!\cdots\!91\)\( T_{2}^{238} + \)\(30\!\cdots\!97\)\( T_{2}^{237} - \)\(48\!\cdots\!24\)\( T_{2}^{236} - \)\(41\!\cdots\!00\)\( T_{2}^{235} - \)\(34\!\cdots\!40\)\( T_{2}^{234} + \)\(14\!\cdots\!02\)\( T_{2}^{233} - \)\(23\!\cdots\!60\)\( T_{2}^{232} + \)\(18\!\cdots\!98\)\( T_{2}^{231} + \)\(35\!\cdots\!11\)\( T_{2}^{230} + \)\(26\!\cdots\!05\)\( T_{2}^{229} + \)\(72\!\cdots\!69\)\( T_{2}^{228} - \)\(40\!\cdots\!19\)\( T_{2}^{227} - \)\(76\!\cdots\!83\)\( T_{2}^{226} - \)\(31\!\cdots\!94\)\( T_{2}^{225} - \)\(59\!\cdots\!55\)\( T_{2}^{224} - \)\(10\!\cdots\!82\)\( T_{2}^{223} - \)\(17\!\cdots\!34\)\( T_{2}^{222} - \)\(32\!\cdots\!27\)\( T_{2}^{221} + \)\(15\!\cdots\!20\)\( T_{2}^{220} + \)\(14\!\cdots\!32\)\( T_{2}^{219} + \)\(35\!\cdots\!04\)\( T_{2}^{218} + \)\(76\!\cdots\!57\)\( T_{2}^{217} + \)\(15\!\cdots\!83\)\( T_{2}^{216} + \)\(18\!\cdots\!65\)\( T_{2}^{215} + \)\(25\!\cdots\!87\)\( T_{2}^{214} - \)\(10\!\cdots\!17\)\( T_{2}^{213} - \)\(52\!\cdots\!77\)\( T_{2}^{212} - \)\(18\!\cdots\!96\)\( T_{2}^{211} - \)\(45\!\cdots\!55\)\( T_{2}^{210} - \)\(69\!\cdots\!11\)\( T_{2}^{209} - \)\(12\!\cdots\!04\)\( T_{2}^{208} - \)\(12\!\cdots\!51\)\( T_{2}^{207} - \)\(10\!\cdots\!70\)\( T_{2}^{206} + \)\(22\!\cdots\!52\)\( T_{2}^{205} + \)\(47\!\cdots\!27\)\( T_{2}^{204} + \)\(71\!\cdots\!57\)\( T_{2}^{203} + \)\(19\!\cdots\!74\)\( T_{2}^{202} + \)\(19\!\cdots\!83\)\( T_{2}^{201} + \)\(22\!\cdots\!95\)\( T_{2}^{200} + \)\(24\!\cdots\!96\)\( T_{2}^{199} - \)\(47\!\cdots\!36\)\( T_{2}^{198} - \)\(99\!\cdots\!88\)\( T_{2}^{197} - \)\(21\!\cdots\!27\)\( T_{2}^{196} - \)\(12\!\cdots\!89\)\( T_{2}^{195} - \)\(51\!\cdots\!29\)\( T_{2}^{194} - \)\(33\!\cdots\!36\)\( T_{2}^{193} + \)\(16\!\cdots\!23\)\( T_{2}^{192} + \)\(11\!\cdots\!27\)\( T_{2}^{191} + \)\(50\!\cdots\!96\)\( T_{2}^{190} + \)\(51\!\cdots\!97\)\( T_{2}^{189} + \)\(11\!\cdots\!75\)\( T_{2}^{188} + \)\(23\!\cdots\!02\)\( T_{2}^{187} - \)\(23\!\cdots\!10\)\( T_{2}^{186} + \)\(38\!\cdots\!78\)\( T_{2}^{185} - \)\(15\!\cdots\!54\)\( T_{2}^{184} - \)\(61\!\cdots\!43\)\( T_{2}^{183} + \)\(22\!\cdots\!25\)\( T_{2}^{182} - \)\(38\!\cdots\!40\)\( T_{2}^{181} + \)\(68\!\cdots\!35\)\( T_{2}^{180} - \)\(22\!\cdots\!80\)\( T_{2}^{179} - \)\(10\!\cdots\!07\)\( T_{2}^{178} + \)\(26\!\cdots\!35\)\( T_{2}^{177} - \)\(41\!\cdots\!14\)\( T_{2}^{176} + \)\(72\!\cdots\!36\)\( T_{2}^{175} - \)\(51\!\cdots\!43\)\( T_{2}^{174} - \)\(13\!\cdots\!09\)\( T_{2}^{173} + \)\(18\!\cdots\!43\)\( T_{2}^{172} - \)\(40\!\cdots\!19\)\( T_{2}^{171} + \)\(65\!\cdots\!02\)\( T_{2}^{170} - \)\(56\!\cdots\!77\)\( T_{2}^{169} + \)\(28\!\cdots\!71\)\( T_{2}^{168} + \)\(97\!\cdots\!38\)\( T_{2}^{167} - \)\(21\!\cdots\!68\)\( T_{2}^{166} + \)\(40\!\cdots\!05\)\( T_{2}^{165} - \)\(42\!\cdots\!18\)\( T_{2}^{164} + \)\(30\!\cdots\!24\)\( T_{2}^{163} + \)\(13\!\cdots\!24\)\( T_{2}^{162} - \)\(75\!\cdots\!65\)\( T_{2}^{161} + \)\(16\!\cdots\!09\)\( T_{2}^{160} - \)\(21\!\cdots\!33\)\( T_{2}^{159} + \)\(26\!\cdots\!31\)\( T_{2}^{158} - \)\(18\!\cdots\!45\)\( T_{2}^{157} + \)\(11\!\cdots\!86\)\( T_{2}^{156} + \)\(19\!\cdots\!17\)\( T_{2}^{155} - \)\(54\!\cdots\!37\)\( T_{2}^{154} + \)\(11\!\cdots\!40\)\( T_{2}^{153} - \)\(15\!\cdots\!83\)\( T_{2}^{152} + \)\(16\!\cdots\!50\)\( T_{2}^{151} - \)\(63\!\cdots\!27\)\( T_{2}^{150} - \)\(13\!\cdots\!85\)\( T_{2}^{149} + \)\(46\!\cdots\!34\)\( T_{2}^{148} - \)\(66\!\cdots\!73\)\( T_{2}^{147} + \)\(55\!\cdots\!90\)\( T_{2}^{146} + \)\(27\!\cdots\!45\)\( T_{2}^{145} - \)\(16\!\cdots\!16\)\( T_{2}^{144} + \)\(35\!\cdots\!31\)\( T_{2}^{143} - \)\(45\!\cdots\!56\)\( T_{2}^{142} + \)\(37\!\cdots\!71\)\( T_{2}^{141} + \)\(38\!\cdots\!02\)\( T_{2}^{140} - \)\(76\!\cdots\!27\)\( T_{2}^{139} + \)\(16\!\cdots\!79\)\( T_{2}^{138} - \)\(24\!\cdots\!54\)\( T_{2}^{137} + \)\(26\!\cdots\!33\)\( T_{2}^{136} - \)\(17\!\cdots\!23\)\( T_{2}^{135} - \)\(41\!\cdots\!97\)\( T_{2}^{134} + \)\(38\!\cdots\!33\)\( T_{2}^{133} - \)\(76\!\cdots\!76\)\( T_{2}^{132} + \)\(10\!\cdots\!78\)\( T_{2}^{131} - \)\(12\!\cdots\!46\)\( T_{2}^{130} + \)\(10\!\cdots\!44\)\( T_{2}^{129} - \)\(43\!\cdots\!79\)\( T_{2}^{128} - \)\(53\!\cdots\!98\)\( T_{2}^{127} + \)\(17\!\cdots\!19\)\( T_{2}^{126} - \)\(30\!\cdots\!02\)\( T_{2}^{125} + \)\(40\!\cdots\!83\)\( T_{2}^{124} - \)\(46\!\cdots\!49\)\( T_{2}^{123} + \)\(44\!\cdots\!24\)\( T_{2}^{122} - \)\(37\!\cdots\!39\)\( T_{2}^{121} + \)\(24\!\cdots\!71\)\( T_{2}^{120} - \)\(12\!\cdots\!16\)\( T_{2}^{119} + \)\(32\!\cdots\!23\)\( T_{2}^{118} - \)\(15\!\cdots\!93\)\( T_{2}^{117} + \)\(11\!\cdots\!79\)\( T_{2}^{116} - \)\(29\!\cdots\!74\)\( T_{2}^{115} + \)\(58\!\cdots\!47\)\( T_{2}^{114} - \)\(88\!\cdots\!44\)\( T_{2}^{113} + \)\(11\!\cdots\!23\)\( T_{2}^{112} - \)\(14\!\cdots\!46\)\( T_{2}^{111} + \)\(14\!\cdots\!11\)\( T_{2}^{110} - \)\(14\!\cdots\!65\)\( T_{2}^{109} + \)\(12\!\cdots\!50\)\( T_{2}^{108} - \)\(10\!\cdots\!75\)\( T_{2}^{107} + \)\(72\!\cdots\!06\)\( T_{2}^{106} - \)\(39\!\cdots\!53\)\( T_{2}^{105} + \)\(12\!\cdots\!46\)\( T_{2}^{104} + \)\(10\!\cdots\!26\)\( T_{2}^{103} - \)\(22\!\cdots\!89\)\( T_{2}^{102} + \)\(29\!\cdots\!69\)\( T_{2}^{101} - \)\(27\!\cdots\!90\)\( T_{2}^{100} + \)\(21\!\cdots\!82\)\( T_{2}^{99} - \)\(14\!\cdots\!14\)\( T_{2}^{98} + \)\(50\!\cdots\!81\)\( T_{2}^{97} + \)\(34\!\cdots\!95\)\( T_{2}^{96} - \)\(67\!\cdots\!24\)\( T_{2}^{95} + \)\(77\!\cdots\!35\)\( T_{2}^{94} - \)\(91\!\cdots\!83\)\( T_{2}^{93} + \)\(75\!\cdots\!41\)\( T_{2}^{92} - \)\(56\!\cdots\!55\)\( T_{2}^{91} + \)\(38\!\cdots\!51\)\( T_{2}^{90} - \)\(12\!\cdots\!72\)\( T_{2}^{89} + \)\(48\!\cdots\!58\)\( T_{2}^{88} + \)\(11\!\cdots\!40\)\( T_{2}^{87} - \)\(10\!\cdots\!24\)\( T_{2}^{86} + \)\(15\!\cdots\!67\)\( T_{2}^{85} - \)\(12\!\cdots\!71\)\( T_{2}^{84} + \)\(93\!\cdots\!67\)\( T_{2}^{83} - \)\(74\!\cdots\!35\)\( T_{2}^{82} + \)\(34\!\cdots\!40\)\( T_{2}^{81} - \)\(29\!\cdots\!37\)\( T_{2}^{80} + \)\(37\!\cdots\!64\)\( T_{2}^{79} - \)\(46\!\cdots\!17\)\( T_{2}^{78} - \)\(40\!\cdots\!70\)\( T_{2}^{77} + \)\(39\!\cdots\!21\)\( T_{2}^{76} - \)\(36\!\cdots\!24\)\( T_{2}^{75} + \)\(42\!\cdots\!97\)\( T_{2}^{74} - \)\(21\!\cdots\!99\)\( T_{2}^{73} + \)\(24\!\cdots\!09\)\( T_{2}^{72} - \)\(11\!\cdots\!27\)\( T_{2}^{71} + \)\(11\!\cdots\!43\)\( T_{2}^{70} - \)\(57\!\cdots\!82\)\( T_{2}^{69} + \)\(43\!\cdots\!40\)\( T_{2}^{68} - \)\(25\!\cdots\!74\)\( T_{2}^{67} + \)\(15\!\cdots\!26\)\( T_{2}^{66} - \)\(90\!\cdots\!68\)\( T_{2}^{65} + \)\(48\!\cdots\!23\)\( T_{2}^{64} - \)\(24\!\cdots\!14\)\( T_{2}^{63} + \)\(11\!\cdots\!72\)\( T_{2}^{62} - \)\(48\!\cdots\!17\)\( T_{2}^{61} + \)\(17\!\cdots\!53\)\( T_{2}^{60} - \)\(53\!\cdots\!76\)\( T_{2}^{59} + \)\(13\!\cdots\!06\)\( T_{2}^{58} - \)\(16\!\cdots\!80\)\( T_{2}^{57} + \)\(35\!\cdots\!40\)\( T_{2}^{56} - \)\(86\!\cdots\!34\)\( T_{2}^{55} + \)\(17\!\cdots\!54\)\( T_{2}^{54} - \)\(20\!\cdots\!74\)\( T_{2}^{53} + \)\(17\!\cdots\!20\)\( T_{2}^{52} - \)\(12\!\cdots\!01\)\( T_{2}^{51} + \)\(73\!\cdots\!55\)\( T_{2}^{50} - \)\(37\!\cdots\!67\)\( T_{2}^{49} + \)\(16\!\cdots\!02\)\( T_{2}^{48} - \)\(64\!\cdots\!21\)\( T_{2}^{47} + \)\(20\!\cdots\!69\)\( T_{2}^{46} - \)\(49\!\cdots\!38\)\( T_{2}^{45} + \)\(64\!\cdots\!72\)\( T_{2}^{44} + \)\(10\!\cdots\!61\)\( T_{2}^{43} - \)\(54\!\cdots\!52\)\( T_{2}^{42} - \)\(26\!\cdots\!02\)\( T_{2}^{41} + \)\(43\!\cdots\!73\)\( T_{2}^{40} - \)\(31\!\cdots\!23\)\( T_{2}^{39} + \)\(16\!\cdots\!72\)\( T_{2}^{38} - \)\(68\!\cdots\!40\)\( T_{2}^{37} + \)\(23\!\cdots\!79\)\( T_{2}^{36} - \)\(66\!\cdots\!45\)\( T_{2}^{35} + \)\(14\!\cdots\!62\)\( T_{2}^{34} - \)\(21\!\cdots\!98\)\( T_{2}^{33} + \)\(28\!\cdots\!61\)\( T_{2}^{32} + \)\(97\!\cdots\!98\)\( T_{2}^{31} - \)\(18\!\cdots\!63\)\( T_{2}^{30} - \)\(97\!\cdots\!61\)\( T_{2}^{29} + \)\(73\!\cdots\!72\)\( T_{2}^{28} - \)\(21\!\cdots\!51\)\( T_{2}^{27} + \)\(15\!\cdots\!11\)\( T_{2}^{26} + \)\(11\!\cdots\!31\)\( T_{2}^{25} - \)\(46\!\cdots\!31\)\( T_{2}^{24} + \)\(75\!\cdots\!84\)\( T_{2}^{23} - \)\(32\!\cdots\!80\)\( T_{2}^{22} + \)\(11\!\cdots\!92\)\( T_{2}^{21} - \)\(14\!\cdots\!84\)\( T_{2}^{20} + \)\(58\!\cdots\!76\)\( T_{2}^{19} - \)\(87\!\cdots\!32\)\( T_{2}^{18} - \)\(15\!\cdots\!80\)\( T_{2}^{17} + \)\(12\!\cdots\!88\)\( T_{2}^{16} - \)\(32\!\cdots\!76\)\( T_{2}^{15} + \)\(42\!\cdots\!08\)\( T_{2}^{14} + \)\(18\!\cdots\!12\)\( T_{2}^{13} - \)\(22\!\cdots\!12\)\( T_{2}^{12} + \)\(53\!\cdots\!60\)\( T_{2}^{11} - \)\(68\!\cdots\!32\)\( T_{2}^{10} + \)\(20\!\cdots\!52\)\( T_{2}^{9} + \)\(11\!\cdots\!48\)\( T_{2}^{8} - \)\(28\!\cdots\!20\)\( T_{2}^{7} + \)\(38\!\cdots\!28\)\( T_{2}^{6} - \)\(36\!\cdots\!32\)\( T_{2}^{5} + \)\(26\!\cdots\!56\)\( T_{2}^{4} - \)\(14\!\cdots\!60\)\( T_{2}^{3} + \)\(64\!\cdots\!68\)\( T_{2}^{2} - \)\(17\!\cdots\!20\)\( T_{2} + \)\(20\!\cdots\!64\)\( \)">\(T_{2}^{360} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).