Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(15\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360q + q^{2} - 15q^{3} + 15q^{4} + 2q^{5} - q^{6} + 2q^{7} + 6q^{8} + 15q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 360q + q^{2} - 15q^{3} + 15q^{4} + 2q^{5} - q^{6} + 2q^{7} + 6q^{8} + 15q^{9} - q^{10} - 2q^{11} + 30q^{12} - 19q^{13} + q^{15} + 15q^{16} - 7q^{17} - 2q^{18} - 6q^{19} + q^{20} + 4q^{21} + 22q^{22} + 46q^{23} + 3q^{24} + 28q^{25} + 34q^{26} + 30q^{27} - 2q^{28} + 36q^{29} + q^{30} - 34q^{31} - 190q^{32} + 15q^{33} - 51q^{34} - 80q^{35} + 15q^{36} + q^{37} - 53q^{38} + 5q^{39} - 45q^{40} + 9q^{41} + 130q^{42} + 78q^{43} + 35q^{44} - q^{45} - 318q^{46} - 12q^{47} - 15q^{48} - 71q^{49} + 48q^{50} - 14q^{51} - 63q^{52} - 26q^{53} - q^{54} - 61q^{55} - 4q^{56} - 38q^{57} - 27q^{58} + 208q^{59} - 180q^{60} - 3q^{61} - 61q^{62} + 2q^{63} + 140q^{64} + 45q^{65} - 60q^{66} - 80q^{67} - 140q^{68} + 6q^{69} - 100q^{70} + 84q^{71} - 3q^{72} + 134q^{73} - 151q^{74} - 12q^{75} + 32q^{76} + 82q^{77} - 63q^{78} + 4q^{79} - 155q^{80} + 15q^{81} - 56q^{82} + 106q^{83} + 2q^{84} - 19q^{85} + q^{86} - 49q^{87} + 8q^{88} - 14q^{89} + 2q^{90} - 368q^{91} + 36q^{92} - 4q^{93} - 59q^{94} + 104q^{95} + 75q^{96} - 28q^{97} + 101q^{98} + 30q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.74572 0.221657i 0.845190 + 0.534466i 5.51573 + 0.896393i 0.154238 + 0.136643i −2.20218 1.65483i −1.20996 + 1.25970i −9.59673 2.36538i 0.428693 + 0.903450i −0.393207 0.409372i
16.2 −2.25005 0.181643i 0.845190 + 0.534466i 3.05563 + 0.496588i −2.62979 2.32979i −1.80464 1.35610i 0.735447 0.765681i −2.40157 0.591933i 0.428693 + 0.903450i 5.49396 + 5.71982i
16.3 −2.21324 0.178671i 0.845190 + 0.534466i 2.89239 + 0.470059i −0.0344706 0.0305383i −1.77511 1.33391i 3.31537 3.45167i −2.00573 0.494367i 0.428693 + 0.903450i 0.0708354 + 0.0737474i
16.4 −1.75204 0.141439i 0.845190 + 0.534466i 1.07555 + 0.174793i 0.0960286 + 0.0850739i −1.40521 1.05595i −1.50147 + 1.56320i 1.55365 + 0.382940i 0.428693 + 0.903450i −0.156213 0.162635i
16.5 −1.46442 0.118220i 0.845190 + 0.534466i 0.156450 + 0.0254257i 3.08910 + 2.73671i −1.17453 0.882601i −0.144485 + 0.150425i 2.62688 + 0.647468i 0.428693 + 0.903450i −4.20021 4.37289i
16.6 −0.779114 0.0628966i 0.845190 + 0.534466i −1.37104 0.222815i −2.45133 2.17169i −0.624884 0.469570i −1.20443 + 1.25394i 2.57205 + 0.633954i 0.428693 + 0.903450i 1.77327 + 1.84617i
16.7 −0.676006 0.0545728i 0.845190 + 0.534466i −1.52010 0.247040i −1.01786 0.901745i −0.542186 0.407426i −0.418694 + 0.435907i 2.33111 + 0.574566i 0.428693 + 0.903450i 0.638868 + 0.665132i
16.8 −0.164145 0.0132511i 0.845190 + 0.534466i −1.94733 0.316472i 2.22303 + 1.96943i −0.131651 0.0989295i −0.158039 + 0.164536i 0.635238 + 0.156572i 0.428693 + 0.903450i −0.338802 0.352730i
16.9 0.350694 + 0.0283109i 0.845190 + 0.534466i −1.85192 0.300966i 0.194020 + 0.171887i 0.281272 + 0.211362i 2.30124 2.39584i −1.32416 0.326375i 0.428693 + 0.903450i 0.0631753 + 0.0657724i
16.10 1.03364 + 0.0834440i 0.845190 + 0.534466i −0.912654 0.148321i −1.98811 1.76131i 0.829023 + 0.622971i −2.70082 + 2.81186i −2.94471 0.725807i 0.428693 + 0.903450i −1.90802 1.98646i
16.11 1.26947 + 0.102482i 0.845190 + 0.534466i −0.373055 0.0606273i −2.62216 2.32303i 1.01817 + 0.765104i 1.77636 1.84939i −2.94055 0.724780i 0.428693 + 0.903450i −3.09068 3.21774i
16.12 1.66608 + 0.134500i 0.845190 + 0.534466i 0.783635 + 0.127353i 2.48154 + 2.19845i 1.33627 + 1.00414i 2.05375 2.13818i −1.95739 0.482453i 0.428693 + 0.903450i 3.83875 + 3.99656i
16.13 1.84825 + 0.149206i 0.845190 + 0.534466i 1.41967 + 0.230719i 1.63990 + 1.45283i 1.48238 + 1.11393i −1.92076 + 1.99973i −1.01128 0.249259i 0.428693 + 0.903450i 2.81418 + 2.92987i
16.14 2.40692 + 0.194306i 0.845190 + 0.534466i 3.78139 + 0.614535i −1.78205 1.57876i 1.93045 + 1.45064i 1.67721 1.74616i 4.29291 + 1.05811i 0.428693 + 0.903450i −3.98249 4.14621i
16.15 2.47293 + 0.199635i 0.845190 + 0.534466i 4.10141 + 0.666543i −0.0128422 0.0113772i 1.98339 + 1.49042i −1.37396 + 1.43044i 5.19165 + 1.27963i 0.428693 + 0.903450i −0.0294866 0.0306988i
55.1 −1.73581 + 2.12599i 0.996757 + 0.0804666i −1.10672 5.42106i −0.134154 1.10485i −1.90126 + 1.97942i −3.44326 + 2.17739i 8.58570 + 4.50612i 0.987050 + 0.160411i 2.58177 + 1.63261i
55.2 −1.58863 + 1.94572i 0.996757 + 0.0804666i −0.862027 4.22249i 0.458750 + 3.77814i −1.74005 + 1.81158i 2.22525 1.40716i 5.13689 + 2.69605i 0.987050 + 0.160411i −8.07999 5.10948i
55.3 −1.28348 + 1.57198i 0.996757 + 0.0804666i −0.423741 2.07562i −0.317374 2.61381i −1.40581 + 1.46360i 0.313349 0.198150i 0.212820 + 0.111697i 0.987050 + 0.160411i 4.51619 + 2.85586i
55.4 −1.18444 + 1.45068i 0.996757 + 0.0804666i −0.301514 1.47691i 0.266003 + 2.19074i −1.29733 + 1.35067i −0.690655 + 0.436744i −0.816908 0.428746i 0.987050 + 0.160411i −3.49312 2.20892i
55.5 −0.826374 + 1.01212i 0.996757 + 0.0804666i 0.0585493 + 0.286793i −0.0644718 0.530973i −0.905137 + 0.942347i 0.386707 0.244538i −2.65259 1.39219i 0.987050 + 0.160411i 0.590688 + 0.373529i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 490.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.q.a 360
169.i even 39 1 inner 507.2.q.a 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.q.a 360 1.a even 1 1 trivial
507.2.q.a 360 169.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(29\!\cdots\!73\)\( T_{2}^{332} - \)\(35\!\cdots\!71\)\( T_{2}^{331} + \)\(29\!\cdots\!47\)\( T_{2}^{330} + \)\(31\!\cdots\!05\)\( T_{2}^{329} - \)\(35\!\cdots\!36\)\( T_{2}^{328} - \)\(23\!\cdots\!84\)\( T_{2}^{327} + \)\(78\!\cdots\!73\)\( T_{2}^{326} - \)\(15\!\cdots\!09\)\( T_{2}^{325} + \)\(12\!\cdots\!71\)\( T_{2}^{324} + \)\(72\!\cdots\!72\)\( T_{2}^{323} - \)\(11\!\cdots\!74\)\( T_{2}^{322} + \)\(95\!\cdots\!15\)\( T_{2}^{321} + \)\(10\!\cdots\!84\)\( T_{2}^{320} - \)\(11\!\cdots\!43\)\( T_{2}^{319} + \)\(66\!\cdots\!83\)\( T_{2}^{318} - \)\(30\!\cdots\!46\)\( T_{2}^{317} - \)\(45\!\cdots\!89\)\( T_{2}^{316} + \)\(81\!\cdots\!66\)\( T_{2}^{315} - \)\(24\!\cdots\!28\)\( T_{2}^{314} - \)\(61\!\cdots\!61\)\( T_{2}^{313} + \)\(24\!\cdots\!88\)\( T_{2}^{312} + \)\(38\!\cdots\!09\)\( T_{2}^{311} - \)\(16\!\cdots\!43\)\( T_{2}^{310} + \)\(24\!\cdots\!43\)\( T_{2}^{309} + \)\(94\!\cdots\!38\)\( T_{2}^{308} - \)\(19\!\cdots\!68\)\( T_{2}^{307} + \)\(52\!\cdots\!83\)\( T_{2}^{306} + \)\(33\!\cdots\!39\)\( T_{2}^{305} - \)\(32\!\cdots\!05\)\( T_{2}^{304} + \)\(45\!\cdots\!07\)\( T_{2}^{303} - \)\(18\!\cdots\!59\)\( T_{2}^{302} - \)\(35\!\cdots\!91\)\( T_{2}^{301} + \)\(12\!\cdots\!75\)\( T_{2}^{300} + \)\(10\!\cdots\!33\)\( T_{2}^{299} - \)\(47\!\cdots\!62\)\( T_{2}^{298} + \)\(12\!\cdots\!86\)\( T_{2}^{297} - \)\(28\!\cdots\!94\)\( T_{2}^{296} - \)\(58\!\cdots\!27\)\( T_{2}^{295} + \)\(35\!\cdots\!57\)\( T_{2}^{294} - \)\(34\!\cdots\!57\)\( T_{2}^{293} - \)\(69\!\cdots\!02\)\( T_{2}^{292} + \)\(47\!\cdots\!69\)\( T_{2}^{291} - \)\(11\!\cdots\!95\)\( T_{2}^{290} - \)\(10\!\cdots\!23\)\( T_{2}^{289} + \)\(10\!\cdots\!78\)\( T_{2}^{288} - \)\(15\!\cdots\!54\)\( T_{2}^{287} - \)\(11\!\cdots\!42\)\( T_{2}^{286} + \)\(14\!\cdots\!04\)\( T_{2}^{285} - \)\(35\!\cdots\!72\)\( T_{2}^{284} - \)\(23\!\cdots\!41\)\( T_{2}^{283} + \)\(27\!\cdots\!03\)\( T_{2}^{282} - \)\(45\!\cdots\!94\)\( T_{2}^{281} - \)\(40\!\cdots\!04\)\( T_{2}^{280} + \)\(39\!\cdots\!08\)\( T_{2}^{279} - \)\(77\!\cdots\!10\)\( T_{2}^{278} - \)\(82\!\cdots\!13\)\( T_{2}^{277} + \)\(64\!\cdots\!49\)\( T_{2}^{276} - \)\(89\!\cdots\!29\)\( T_{2}^{275} - \)\(14\!\cdots\!72\)\( T_{2}^{274} + \)\(85\!\cdots\!61\)\( T_{2}^{273} - \)\(11\!\cdots\!10\)\( T_{2}^{272} - \)\(24\!\cdots\!13\)\( T_{2}^{271} + \)\(11\!\cdots\!11\)\( T_{2}^{270} - \)\(11\!\cdots\!86\)\( T_{2}^{269} - \)\(36\!\cdots\!15\)\( T_{2}^{268} + \)\(14\!\cdots\!73\)\( T_{2}^{267} - \)\(11\!\cdots\!57\)\( T_{2}^{266} - \)\(52\!\cdots\!80\)\( T_{2}^{265} + \)\(16\!\cdots\!99\)\( T_{2}^{264} - \)\(71\!\cdots\!85\)\( T_{2}^{263} - \)\(66\!\cdots\!00\)\( T_{2}^{262} + \)\(17\!\cdots\!67\)\( T_{2}^{261} - \)\(33\!\cdots\!42\)\( T_{2}^{260} - \)\(83\!\cdots\!86\)\( T_{2}^{259} + \)\(17\!\cdots\!29\)\( T_{2}^{258} + \)\(53\!\cdots\!22\)\( T_{2}^{257} - \)\(89\!\cdots\!41\)\( T_{2}^{256} + \)\(15\!\cdots\!68\)\( T_{2}^{255} + \)\(10\!\cdots\!35\)\( T_{2}^{254} - \)\(96\!\cdots\!74\)\( T_{2}^{253} + \)\(13\!\cdots\!55\)\( T_{2}^{252} + \)\(18\!\cdots\!42\)\( T_{2}^{251} - \)\(87\!\cdots\!84\)\( T_{2}^{250} + \)\(93\!\cdots\!76\)\( T_{2}^{249} + \)\(19\!\cdots\!19\)\( T_{2}^{248} - \)\(82\!\cdots\!56\)\( T_{2}^{247} + \)\(60\!\cdots\!27\)\( T_{2}^{246} + \)\(24\!\cdots\!03\)\( T_{2}^{245} - \)\(64\!\cdots\!45\)\( T_{2}^{244} + \)\(26\!\cdots\!68\)\( T_{2}^{243} + \)\(20\!\cdots\!96\)\( T_{2}^{242} - \)\(53\!\cdots\!49\)\( T_{2}^{241} + \)\(65\!\cdots\!94\)\( T_{2}^{240} + \)\(21\!\cdots\!03\)\( T_{2}^{239} - \)\(34\!\cdots\!92\)\( T_{2}^{238} - \)\(13\!\cdots\!62\)\( T_{2}^{237} + \)\(14\!\cdots\!87\)\( T_{2}^{236} - \)\(25\!\cdots\!91\)\( T_{2}^{235} - \)\(15\!\cdots\!53\)\( T_{2}^{234} + \)\(13\!\cdots\!53\)\( T_{2}^{233} - \)\(13\!\cdots\!40\)\( T_{2}^{232} - \)\(20\!\cdots\!27\)\( T_{2}^{231} + \)\(76\!\cdots\!04\)\( T_{2}^{230} - \)\(93\!\cdots\!31\)\( T_{2}^{229} - \)\(13\!\cdots\!38\)\( T_{2}^{228} + \)\(64\!\cdots\!25\)\( T_{2}^{227} - \)\(38\!\cdots\!01\)\( T_{2}^{226} - \)\(13\!\cdots\!07\)\( T_{2}^{225} + \)\(30\!\cdots\!62\)\( T_{2}^{224} - \)\(25\!\cdots\!05\)\( T_{2}^{223} - \)\(74\!\cdots\!08\)\( T_{2}^{222} + \)\(26\!\cdots\!57\)\( T_{2}^{221} - \)\(49\!\cdots\!91\)\( T_{2}^{220} - \)\(69\!\cdots\!68\)\( T_{2}^{219} + \)\(94\!\cdots\!18\)\( T_{2}^{218} - \)\(28\!\cdots\!24\)\( T_{2}^{217} - \)\(29\!\cdots\!70\)\( T_{2}^{216} + \)\(80\!\cdots\!61\)\( T_{2}^{215} + \)\(19\!\cdots\!60\)\( T_{2}^{214} - \)\(23\!\cdots\!68\)\( T_{2}^{213} + \)\(20\!\cdots\!04\)\( T_{2}^{212} - \)\(87\!\cdots\!32\)\( T_{2}^{211} - \)\(85\!\cdots\!85\)\( T_{2}^{210} + \)\(21\!\cdots\!92\)\( T_{2}^{209} + \)\(11\!\cdots\!44\)\( T_{2}^{208} - \)\(60\!\cdots\!05\)\( T_{2}^{207} + \)\(35\!\cdots\!25\)\( T_{2}^{206} - \)\(12\!\cdots\!88\)\( T_{2}^{205} - \)\(19\!\cdots\!42\)\( T_{2}^{204} + \)\(45\!\cdots\!65\)\( T_{2}^{203} + \)\(26\!\cdots\!69\)\( T_{2}^{202} - \)\(73\!\cdots\!33\)\( T_{2}^{201} + \)\(67\!\cdots\!57\)\( T_{2}^{200} - \)\(25\!\cdots\!53\)\( T_{2}^{199} - \)\(34\!\cdots\!85\)\( T_{2}^{198} + \)\(10\!\cdots\!71\)\( T_{2}^{197} + \)\(41\!\cdots\!72\)\( T_{2}^{196} + \)\(99\!\cdots\!17\)\( T_{2}^{195} + \)\(91\!\cdots\!93\)\( T_{2}^{194} - \)\(16\!\cdots\!36\)\( T_{2}^{193} - \)\(41\!\cdots\!60\)\( T_{2}^{192} + \)\(39\!\cdots\!01\)\( T_{2}^{191} + \)\(26\!\cdots\!37\)\( T_{2}^{190} + \)\(31\!\cdots\!70\)\( T_{2}^{189} + \)\(46\!\cdots\!44\)\( T_{2}^{188} - \)\(44\!\cdots\!72\)\( T_{2}^{187} - \)\(35\!\cdots\!01\)\( T_{2}^{186} + \)\(98\!\cdots\!53\)\( T_{2}^{185} + \)\(16\!\cdots\!97\)\( T_{2}^{184} + \)\(89\!\cdots\!57\)\( T_{2}^{183} - \)\(45\!\cdots\!42\)\( T_{2}^{182} - \)\(11\!\cdots\!13\)\( T_{2}^{181} + \)\(49\!\cdots\!14\)\( T_{2}^{180} + \)\(34\!\cdots\!12\)\( T_{2}^{179} + \)\(23\!\cdots\!69\)\( T_{2}^{178} - \)\(37\!\cdots\!05\)\( T_{2}^{177} - \)\(14\!\cdots\!99\)\( T_{2}^{176} - \)\(11\!\cdots\!60\)\( T_{2}^{175} + \)\(40\!\cdots\!75\)\( T_{2}^{174} + \)\(67\!\cdots\!28\)\( T_{2}^{173} - \)\(52\!\cdots\!24\)\( T_{2}^{172} - \)\(19\!\cdots\!59\)\( T_{2}^{171} - \)\(55\!\cdots\!93\)\( T_{2}^{170} + \)\(36\!\cdots\!11\)\( T_{2}^{169} + \)\(45\!\cdots\!98\)\( T_{2}^{168} - \)\(46\!\cdots\!36\)\( T_{2}^{167} - \)\(11\!\cdots\!47\)\( T_{2}^{166} + \)\(16\!\cdots\!18\)\( T_{2}^{165} + \)\(17\!\cdots\!20\)\( T_{2}^{164} + \)\(92\!\cdots\!87\)\( T_{2}^{163} - \)\(10\!\cdots\!88\)\( T_{2}^{162} - \)\(25\!\cdots\!55\)\( T_{2}^{161} - \)\(10\!\cdots\!93\)\( T_{2}^{160} + \)\(24\!\cdots\!48\)\( T_{2}^{159} + \)\(26\!\cdots\!55\)\( T_{2}^{158} + \)\(28\!\cdots\!33\)\( T_{2}^{157} - \)\(94\!\cdots\!74\)\( T_{2}^{156} - \)\(11\!\cdots\!49\)\( T_{2}^{155} - \)\(13\!\cdots\!13\)\( T_{2}^{154} + \)\(12\!\cdots\!98\)\( T_{2}^{153} - \)\(55\!\cdots\!24\)\( T_{2}^{152} + \)\(95\!\cdots\!71\)\( T_{2}^{151} + \)\(34\!\cdots\!66\)\( T_{2}^{150} - \)\(57\!\cdots\!02\)\( T_{2}^{149} - \)\(84\!\cdots\!16\)\( T_{2}^{148} + \)\(92\!\cdots\!03\)\( T_{2}^{147} + \)\(11\!\cdots\!31\)\( T_{2}^{146} - \)\(20\!\cdots\!99\)\( T_{2}^{145} - \)\(49\!\cdots\!41\)\( T_{2}^{144} - \)\(19\!\cdots\!00\)\( T_{2}^{143} - \)\(10\!\cdots\!39\)\( T_{2}^{142} + \)\(39\!\cdots\!79\)\( T_{2}^{141} - \)\(15\!\cdots\!10\)\( T_{2}^{140} - \)\(36\!\cdots\!22\)\( T_{2}^{139} + \)\(77\!\cdots\!01\)\( T_{2}^{138} + \)\(25\!\cdots\!74\)\( T_{2}^{137} - \)\(14\!\cdots\!49\)\( T_{2}^{136} - \)\(48\!\cdots\!97\)\( T_{2}^{135} + \)\(76\!\cdots\!33\)\( T_{2}^{134} + \)\(83\!\cdots\!00\)\( T_{2}^{133} + \)\(11\!\cdots\!63\)\( T_{2}^{132} - \)\(16\!\cdots\!36\)\( T_{2}^{131} - \)\(15\!\cdots\!12\)\( T_{2}^{130} - \)\(20\!\cdots\!89\)\( T_{2}^{129} - \)\(40\!\cdots\!74\)\( T_{2}^{128} + \)\(35\!\cdots\!33\)\( T_{2}^{127} + \)\(99\!\cdots\!12\)\( T_{2}^{126} - \)\(21\!\cdots\!16\)\( T_{2}^{125} + \)\(37\!\cdots\!80\)\( T_{2}^{124} - \)\(15\!\cdots\!01\)\( T_{2}^{123} - \)\(10\!\cdots\!74\)\( T_{2}^{122} + \)\(44\!\cdots\!88\)\( T_{2}^{121} + \)\(10\!\cdots\!34\)\( T_{2}^{120} - \)\(56\!\cdots\!21\)\( T_{2}^{119} - \)\(23\!\cdots\!02\)\( T_{2}^{118} + \)\(64\!\cdots\!61\)\( T_{2}^{117} - \)\(59\!\cdots\!24\)\( T_{2}^{116} - \)\(61\!\cdots\!09\)\( T_{2}^{115} + \)\(72\!\cdots\!96\)\( T_{2}^{114} + \)\(23\!\cdots\!21\)\( T_{2}^{113} - \)\(31\!\cdots\!05\)\( T_{2}^{112} + \)\(41\!\cdots\!31\)\( T_{2}^{111} - \)\(57\!\cdots\!09\)\( T_{2}^{110} - \)\(66\!\cdots\!20\)\( T_{2}^{109} + \)\(20\!\cdots\!87\)\( T_{2}^{108} + \)\(22\!\cdots\!02\)\( T_{2}^{107} - \)\(16\!\cdots\!09\)\( T_{2}^{106} + \)\(24\!\cdots\!43\)\( T_{2}^{105} + \)\(78\!\cdots\!55\)\( T_{2}^{104} - \)\(34\!\cdots\!23\)\( T_{2}^{103} - \)\(47\!\cdots\!28\)\( T_{2}^{102} + \)\(13\!\cdots\!99\)\( T_{2}^{101} - \)\(32\!\cdots\!94\)\( T_{2}^{100} + \)\(65\!\cdots\!16\)\( T_{2}^{99} + \)\(99\!\cdots\!75\)\( T_{2}^{98} - \)\(75\!\cdots\!10\)\( T_{2}^{97} - \)\(18\!\cdots\!99\)\( T_{2}^{96} + \)\(54\!\cdots\!00\)\( T_{2}^{95} - \)\(14\!\cdots\!60\)\( T_{2}^{94} - \)\(17\!\cdots\!17\)\( T_{2}^{93} - \)\(20\!\cdots\!90\)\( T_{2}^{92} - \)\(28\!\cdots\!55\)\( T_{2}^{91} - \)\(16\!\cdots\!59\)\( T_{2}^{90} + \)\(12\!\cdots\!03\)\( T_{2}^{89} + \)\(48\!\cdots\!38\)\( T_{2}^{88} + \)\(54\!\cdots\!47\)\( T_{2}^{87} + \)\(48\!\cdots\!66\)\( T_{2}^{86} + \)\(96\!\cdots\!80\)\( T_{2}^{85} + \)\(19\!\cdots\!17\)\( T_{2}^{84} + \)\(43\!\cdots\!52\)\( T_{2}^{83} + \)\(23\!\cdots\!52\)\( T_{2}^{82} + \)\(27\!\cdots\!64\)\( T_{2}^{81} + \)\(74\!\cdots\!32\)\( T_{2}^{80} + \)\(51\!\cdots\!74\)\( T_{2}^{79} + \)\(44\!\cdots\!36\)\( T_{2}^{78} + \)\(69\!\cdots\!03\)\( T_{2}^{77} + \)\(20\!\cdots\!53\)\( T_{2}^{76} + \)\(11\!\cdots\!55\)\( T_{2}^{75} + \)\(70\!\cdots\!85\)\( T_{2}^{74} + \)\(88\!\cdots\!37\)\( T_{2}^{73} + \)\(55\!\cdots\!30\)\( T_{2}^{72} + \)\(36\!\cdots\!20\)\( T_{2}^{71} + \)\(22\!\cdots\!69\)\( T_{2}^{70} - \)\(70\!\cdots\!84\)\( T_{2}^{69} + \)\(15\!\cdots\!08\)\( T_{2}^{68} - \)\(30\!\cdots\!10\)\( T_{2}^{67} + \)\(56\!\cdots\!67\)\( T_{2}^{66} + \)\(81\!\cdots\!81\)\( T_{2}^{65} + \)\(51\!\cdots\!06\)\( T_{2}^{64} + \)\(87\!\cdots\!22\)\( T_{2}^{63} + \)\(91\!\cdots\!72\)\( T_{2}^{62} + \)\(37\!\cdots\!24\)\( T_{2}^{61} - \)\(13\!\cdots\!84\)\( T_{2}^{60} + \)\(27\!\cdots\!94\)\( T_{2}^{59} - \)\(12\!\cdots\!06\)\( T_{2}^{58} + \)\(85\!\cdots\!22\)\( T_{2}^{57} - \)\(75\!\cdots\!73\)\( T_{2}^{56} - \)\(54\!\cdots\!39\)\( T_{2}^{55} + \)\(15\!\cdots\!15\)\( T_{2}^{54} - \)\(89\!\cdots\!66\)\( T_{2}^{53} + \)\(56\!\cdots\!48\)\( T_{2}^{52} - \)\(20\!\cdots\!56\)\( T_{2}^{51} + \)\(10\!\cdots\!76\)\( T_{2}^{50} - \)\(48\!\cdots\!97\)\( T_{2}^{49} + \)\(41\!\cdots\!31\)\( T_{2}^{48} - \)\(26\!\cdots\!01\)\( T_{2}^{47} + \)\(15\!\cdots\!96\)\( T_{2}^{46} - \)\(63\!\cdots\!26\)\( T_{2}^{45} + \)\(19\!\cdots\!41\)\( T_{2}^{44} - \)\(26\!\cdots\!08\)\( T_{2}^{43} + \)\(11\!\cdots\!29\)\( T_{2}^{42} - \)\(27\!\cdots\!63\)\( T_{2}^{41} + \)\(51\!\cdots\!65\)\( T_{2}^{40} - \)\(35\!\cdots\!07\)\( T_{2}^{39} + \)\(16\!\cdots\!90\)\( T_{2}^{38} - \)\(50\!\cdots\!40\)\( T_{2}^{37} + \)\(77\!\cdots\!12\)\( T_{2}^{36} + \)\(12\!\cdots\!86\)\( T_{2}^{35} - \)\(10\!\cdots\!46\)\( T_{2}^{34} + \)\(25\!\cdots\!71\)\( T_{2}^{33} - \)\(29\!\cdots\!98\)\( T_{2}^{32} + \)\(93\!\cdots\!31\)\( T_{2}^{31} - \)\(10\!\cdots\!21\)\( T_{2}^{30} + \)\(27\!\cdots\!59\)\( T_{2}^{29} + \)\(44\!\cdots\!36\)\( T_{2}^{28} - \)\(55\!\cdots\!27\)\( T_{2}^{27} + \)\(47\!\cdots\!57\)\( T_{2}^{26} + \)\(26\!\cdots\!74\)\( T_{2}^{25} + \)\(97\!\cdots\!39\)\( T_{2}^{24} - \)\(35\!\cdots\!85\)\( T_{2}^{23} - \)\(40\!\cdots\!03\)\( T_{2}^{22} + \)\(22\!\cdots\!01\)\( T_{2}^{21} + \)\(93\!\cdots\!19\)\( T_{2}^{20} + \)\(17\!\cdots\!13\)\( T_{2}^{19} + \)\(66\!\cdots\!20\)\( T_{2}^{18} - \)\(28\!\cdots\!21\)\( T_{2}^{17} - \)\(57\!\cdots\!34\)\( T_{2}^{16} - \)\(29\!\cdots\!55\)\( T_{2}^{15} + \)\(75\!\cdots\!27\)\( T_{2}^{14} + \)\(12\!\cdots\!48\)\( T_{2}^{13} + \)\(41\!\cdots\!60\)\( T_{2}^{12} - \)\(83\!\cdots\!21\)\( T_{2}^{11} - \)\(10\!\cdots\!66\)\( T_{2}^{10} + \)\(37\!\cdots\!36\)\( T_{2}^{9} + \)\(83\!\cdots\!18\)\( T_{2}^{8} - \)\(22\!\cdots\!14\)\( T_{2}^{7} - \)\(28\!\cdots\!85\)\( T_{2}^{6} + \)\(10\!\cdots\!65\)\( T_{2}^{5} + \)\(25\!\cdots\!95\)\( T_{2}^{4} - \)\(11\!\cdots\!21\)\( T_{2}^{3} + \)\(28\!\cdots\!36\)\( T_{2}^{2} - \)\(29\!\cdots\!32\)\( T_{2} + \)\(90\!\cdots\!89\)\( \)">\(T_{2}^{360} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).