Properties

Label 507.2.t.a
Level $507$
Weight $2$
Character orbit 507.t
Analytic conductor $4.048$
Analytic rank $0$
Dimension $336$
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: \(336\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 336q + 14q^{3} - 12q^{4} + 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 336q + 14q^{3} - 12q^{4} + 14q^{9} + 4q^{10} + 24q^{12} + 26q^{13} + 4q^{14} + 8q^{16} + 4q^{17} + 72q^{22} - 48q^{23} + 68q^{25} + 39q^{26} - 28q^{27} - 45q^{29} + 4q^{30} + 26q^{31} + 65q^{32} - 13q^{33} + 65q^{34} - 82q^{35} - 12q^{36} - 73q^{38} + 24q^{40} - 132q^{42} - 72q^{43} + 39q^{44} + 8q^{48} + 68q^{49} + 52q^{50} - 8q^{51} - 65q^{52} + 37q^{53} - 53q^{55} + 14q^{56} - 26q^{57} + 26q^{58} - 208q^{59} + 78q^{60} - 12q^{61} - 49q^{62} + 14q^{64} + 52q^{65} + 64q^{66} - 26q^{67} - 33q^{68} + 4q^{69} - 78q^{71} + 52q^{73} + 205q^{74} - 8q^{75} - 26q^{76} - 114q^{77} - 65q^{78} + 28q^{79} - 468q^{80} + 14q^{81} - 45q^{82} - 78q^{83} + 13q^{85} + 13q^{86} + 46q^{87} + 26q^{88} - 8q^{90} - 260q^{91} + 8q^{92} - 25q^{94} - 90q^{95} - 65q^{96} - 26q^{97} + 104q^{98} + O(q^{100}) \)

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 −2.42393 0.0976810i 0.278217 0.960518i 3.87238 + 0.312611i −2.76576 1.04892i −0.768204 + 2.30105i −0.673721 1.58128i −4.53942 0.551185i −0.845190 0.534466i 6.60156 + 2.81266i
4.2 −2.28165 0.0919475i 0.278217 0.960518i 3.20398 + 0.258652i −1.50502 0.570779i −0.723113 + 2.16599i 1.60130 + 3.75840i −2.75287 0.334259i −0.845190 0.534466i 3.38146 + 1.44070i
4.3 −2.22635 0.0897189i 0.278217 0.960518i 2.95507 + 0.238558i 1.89329 + 0.718032i −0.705586 + 2.11349i −0.669684 1.57181i −2.13381 0.259091i −0.845190 0.534466i −4.15072 1.76846i
4.4 −1.11129 0.0447836i 0.278217 0.960518i −0.760548 0.0613978i 3.40515 + 1.29140i −0.352197 + 1.05496i 1.25015 + 2.93422i 3.05061 + 0.370412i −0.845190 0.534466i −3.72629 1.58762i
4.5 −1.05386 0.0424689i 0.278217 0.960518i −0.884707 0.0714209i −0.307541 0.116635i −0.333993 + 1.00043i 0.927898 + 2.17786i 3.02336 + 0.367103i −0.845190 0.534466i 0.319151 + 0.135977i
4.6 −1.04226 0.0420016i 0.278217 0.960518i −0.908975 0.0733801i 0.954453 + 0.361976i −0.330318 + 0.989423i −1.88928 4.43431i 3.01530 + 0.366124i −0.845190 0.534466i −0.979583 0.417361i
4.7 −0.0608806 0.00245341i 0.278217 0.960518i −1.98981 0.160634i 0.970084 + 0.367904i −0.0192946 + 0.0577944i 0.630770 + 1.48047i 0.241719 + 0.0293500i −0.845190 0.534466i −0.0581567 0.0247783i
4.8 0.0547950 + 0.00220816i 0.278217 0.960518i −1.99052 0.160691i −2.83821 1.07639i 0.0173659 0.0520173i 0.272711 + 0.640076i −0.217595 0.0264208i −0.845190 0.534466i −0.153143 0.0652481i
4.9 0.756241 + 0.0304755i 0.278217 0.960518i −1.42254 0.114840i −1.80244 0.683575i 0.239672 0.717904i 0.988668 + 2.32049i −2.57496 0.312656i −0.845190 0.534466i −1.34225 0.571877i
4.10 0.819680 + 0.0330320i 0.278217 0.960518i −1.32273 0.106782i 1.33494 + 0.506274i 0.259777 0.778127i −1.47285 3.45690i −2.70942 0.328983i −0.845190 0.534466i 1.07750 + 0.459078i
4.11 1.51468 + 0.0610396i 0.278217 0.960518i 0.297021 + 0.0239780i −3.09424 1.17349i 0.480041 1.43790i −0.201466 0.472857i −2.56129 0.310997i −0.845190 0.534466i −4.61516 1.96633i
4.12 1.90487 + 0.0767637i 0.278217 0.960518i 1.62913 + 0.131517i 2.99420 + 1.13555i 0.603702 1.80831i −0.0821119 0.192724i −0.691848 0.0840056i −0.845190 0.534466i 5.61640 + 2.39293i
4.13 2.45059 + 0.0987553i 0.278217 0.960518i 4.00211 + 0.323084i 0.599077 + 0.227200i 0.776653 2.32636i 0.599988 + 1.40822i 4.90623 + 0.595725i −0.845190 0.534466i 1.44565 + 0.615935i
4.14 2.69937 + 0.108781i 0.278217 0.960518i 5.28123 + 0.426345i −2.66969 1.01248i 0.855497 2.56253i −1.46999 3.45019i 8.84587 + 1.07408i −0.845190 0.534466i −7.09634 3.02347i
10.1 −2.70648 0.552531i 0.987050 0.160411i 5.17976 + 2.20689i −0.330566 + 1.34116i −2.76006 0.111227i −2.19257 + 1.04039i −8.25286 5.69654i 0.948536 0.316668i 1.63570 3.44716i
10.2 −2.16315 0.441610i 0.987050 0.160411i 2.64424 + 1.12661i −0.634667 + 2.57494i −2.20598 0.0888979i 4.23396 2.00904i −1.58846 1.09644i 0.948536 0.316668i 2.51000 5.28972i
10.3 −1.86580 0.380906i 0.987050 0.160411i 1.49616 + 0.637456i 0.0804692 0.326476i −1.90274 0.0766778i 2.39714 1.13746i 0.585658 + 0.404250i 0.948536 0.316668i −0.274496 + 0.578489i
10.4 −1.78276 0.363953i 0.987050 0.160411i 1.20581 + 0.513748i 0.718046 2.91323i −1.81806 0.0732651i −2.45740 + 1.16605i 1.03220 + 0.712476i 0.948536 0.316668i −2.34038 + 4.93225i
10.5 −0.886049 0.180888i 0.987050 0.160411i −1.08760 0.463382i 0.108486 0.440146i −0.903591 0.0364135i −1.78657 + 0.847738i 2.36833 + 1.63474i 0.948536 0.316668i −0.175741 + 0.370367i
10.6 −0.561687 0.114669i 0.987050 0.160411i −1.53762 0.655117i −0.327138 + 1.32725i −0.572808 0.0230834i −1.99021 + 0.944367i 1.73212 + 1.19560i 0.948536 0.316668i 0.335944 0.707987i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 478.14
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.a 336
169.k even 78 1 inner 507.2.t.a 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.t.a 336 1.a even 1 1 trivial
507.2.t.a 336 169.k even 78 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(50\!\cdots\!25\)\( T_{2}^{308} - \)\(68\!\cdots\!28\)\( T_{2}^{307} + \)\(17\!\cdots\!17\)\( T_{2}^{306} - \)\(32\!\cdots\!94\)\( T_{2}^{305} - \)\(91\!\cdots\!21\)\( T_{2}^{304} + \)\(13\!\cdots\!81\)\( T_{2}^{303} - \)\(11\!\cdots\!50\)\( T_{2}^{302} + \)\(25\!\cdots\!14\)\( T_{2}^{301} - \)\(32\!\cdots\!29\)\( T_{2}^{300} + \)\(13\!\cdots\!78\)\( T_{2}^{299} + \)\(22\!\cdots\!64\)\( T_{2}^{298} - \)\(30\!\cdots\!67\)\( T_{2}^{297} + \)\(21\!\cdots\!70\)\( T_{2}^{296} - \)\(41\!\cdots\!44\)\( T_{2}^{295} + \)\(98\!\cdots\!75\)\( T_{2}^{294} - \)\(24\!\cdots\!69\)\( T_{2}^{293} - \)\(82\!\cdots\!28\)\( T_{2}^{292} - \)\(32\!\cdots\!28\)\( T_{2}^{291} - \)\(39\!\cdots\!78\)\( T_{2}^{290} + \)\(49\!\cdots\!04\)\( T_{2}^{289} + \)\(22\!\cdots\!82\)\( T_{2}^{288} + \)\(27\!\cdots\!78\)\( T_{2}^{287} + \)\(31\!\cdots\!74\)\( T_{2}^{286} - \)\(99\!\cdots\!54\)\( T_{2}^{285} + \)\(93\!\cdots\!96\)\( T_{2}^{284} - \)\(14\!\cdots\!80\)\( T_{2}^{283} - \)\(58\!\cdots\!46\)\( T_{2}^{282} - \)\(26\!\cdots\!79\)\( T_{2}^{281} - \)\(65\!\cdots\!81\)\( T_{2}^{280} + \)\(39\!\cdots\!99\)\( T_{2}^{279} - \)\(19\!\cdots\!18\)\( T_{2}^{278} + \)\(27\!\cdots\!24\)\( T_{2}^{277} + \)\(82\!\cdots\!49\)\( T_{2}^{276} + \)\(43\!\cdots\!23\)\( T_{2}^{275} + \)\(89\!\cdots\!81\)\( T_{2}^{274} - \)\(76\!\cdots\!57\)\( T_{2}^{273} + \)\(12\!\cdots\!38\)\( T_{2}^{272} - \)\(64\!\cdots\!32\)\( T_{2}^{271} - \)\(20\!\cdots\!98\)\( T_{2}^{270} - \)\(12\!\cdots\!53\)\( T_{2}^{269} - \)\(11\!\cdots\!10\)\( T_{2}^{268} + \)\(14\!\cdots\!16\)\( T_{2}^{267} + \)\(18\!\cdots\!11\)\( T_{2}^{266} + \)\(10\!\cdots\!39\)\( T_{2}^{265} + \)\(41\!\cdots\!64\)\( T_{2}^{264} + \)\(99\!\cdots\!40\)\( T_{2}^{263} + \)\(12\!\cdots\!89\)\( T_{2}^{262} - \)\(22\!\cdots\!91\)\( T_{2}^{261} - \)\(69\!\cdots\!83\)\( T_{2}^{260} - \)\(12\!\cdots\!49\)\( T_{2}^{259} - \)\(70\!\cdots\!68\)\( T_{2}^{258} + \)\(14\!\cdots\!94\)\( T_{2}^{257} - \)\(13\!\cdots\!40\)\( T_{2}^{256} + \)\(34\!\cdots\!19\)\( T_{2}^{255} + \)\(13\!\cdots\!59\)\( T_{2}^{254} + \)\(19\!\cdots\!69\)\( T_{2}^{253} + \)\(11\!\cdots\!11\)\( T_{2}^{252} + \)\(41\!\cdots\!77\)\( T_{2}^{251} + \)\(32\!\cdots\!65\)\( T_{2}^{250} - \)\(19\!\cdots\!16\)\( T_{2}^{249} - \)\(11\!\cdots\!49\)\( T_{2}^{248} - \)\(16\!\cdots\!18\)\( T_{2}^{247} - \)\(13\!\cdots\!99\)\( T_{2}^{246} - \)\(18\!\cdots\!54\)\( T_{2}^{245} - \)\(34\!\cdots\!34\)\( T_{2}^{244} + \)\(33\!\cdots\!35\)\( T_{2}^{243} + \)\(17\!\cdots\!34\)\( T_{2}^{242} + \)\(24\!\cdots\!95\)\( T_{2}^{241} + \)\(17\!\cdots\!74\)\( T_{2}^{240} + \)\(85\!\cdots\!06\)\( T_{2}^{239} + \)\(48\!\cdots\!83\)\( T_{2}^{238} + \)\(18\!\cdots\!43\)\( T_{2}^{237} - \)\(73\!\cdots\!28\)\( T_{2}^{236} + \)\(14\!\cdots\!63\)\( T_{2}^{235} - \)\(10\!\cdots\!70\)\( T_{2}^{234} - \)\(15\!\cdots\!35\)\( T_{2}^{233} - \)\(26\!\cdots\!18\)\( T_{2}^{232} - \)\(93\!\cdots\!95\)\( T_{2}^{231} + \)\(53\!\cdots\!09\)\( T_{2}^{230} + \)\(20\!\cdots\!56\)\( T_{2}^{229} + \)\(60\!\cdots\!80\)\( T_{2}^{228} + \)\(25\!\cdots\!24\)\( T_{2}^{227} + \)\(17\!\cdots\!22\)\( T_{2}^{226} + \)\(12\!\cdots\!50\)\( T_{2}^{225} - \)\(77\!\cdots\!79\)\( T_{2}^{224} + \)\(25\!\cdots\!87\)\( T_{2}^{223} - \)\(25\!\cdots\!56\)\( T_{2}^{222} - \)\(49\!\cdots\!16\)\( T_{2}^{221} - \)\(93\!\cdots\!43\)\( T_{2}^{220} - \)\(67\!\cdots\!46\)\( T_{2}^{219} - \)\(62\!\cdots\!72\)\( T_{2}^{218} - \)\(31\!\cdots\!03\)\( T_{2}^{217} + \)\(69\!\cdots\!57\)\( T_{2}^{216} - \)\(78\!\cdots\!87\)\( T_{2}^{215} + \)\(24\!\cdots\!11\)\( T_{2}^{214} - \)\(58\!\cdots\!37\)\( T_{2}^{213} + \)\(52\!\cdots\!94\)\( T_{2}^{212} + \)\(40\!\cdots\!02\)\( T_{2}^{211} - \)\(18\!\cdots\!80\)\( T_{2}^{210} + \)\(17\!\cdots\!02\)\( T_{2}^{209} - \)\(31\!\cdots\!50\)\( T_{2}^{208} + \)\(50\!\cdots\!12\)\( T_{2}^{207} + \)\(19\!\cdots\!37\)\( T_{2}^{206} - \)\(28\!\cdots\!59\)\( T_{2}^{205} + \)\(11\!\cdots\!89\)\( T_{2}^{204} - \)\(19\!\cdots\!01\)\( T_{2}^{203} + \)\(17\!\cdots\!01\)\( T_{2}^{202} - \)\(89\!\cdots\!40\)\( T_{2}^{201} - \)\(54\!\cdots\!73\)\( T_{2}^{200} - \)\(31\!\cdots\!36\)\( T_{2}^{199} - \)\(21\!\cdots\!58\)\( T_{2}^{198} - \)\(84\!\cdots\!91\)\( T_{2}^{197} + \)\(80\!\cdots\!24\)\( T_{2}^{196} - \)\(17\!\cdots\!64\)\( T_{2}^{195} + \)\(94\!\cdots\!80\)\( T_{2}^{194} - \)\(18\!\cdots\!17\)\( T_{2}^{193} + \)\(41\!\cdots\!47\)\( T_{2}^{192} + \)\(19\!\cdots\!87\)\( T_{2}^{191} + \)\(11\!\cdots\!37\)\( T_{2}^{190} + \)\(11\!\cdots\!63\)\( T_{2}^{189} + \)\(27\!\cdots\!91\)\( T_{2}^{188} - \)\(39\!\cdots\!82\)\( T_{2}^{187} + \)\(74\!\cdots\!59\)\( T_{2}^{186} - \)\(71\!\cdots\!15\)\( T_{2}^{185} + \)\(30\!\cdots\!50\)\( T_{2}^{184} - \)\(46\!\cdots\!41\)\( T_{2}^{183} + \)\(12\!\cdots\!72\)\( T_{2}^{182} - \)\(20\!\cdots\!02\)\( T_{2}^{181} + \)\(38\!\cdots\!47\)\( T_{2}^{180} - \)\(68\!\cdots\!51\)\( T_{2}^{179} + \)\(91\!\cdots\!04\)\( T_{2}^{178} - \)\(18\!\cdots\!49\)\( T_{2}^{177} + \)\(17\!\cdots\!29\)\( T_{2}^{176} - \)\(42\!\cdots\!18\)\( T_{2}^{175} + \)\(41\!\cdots\!02\)\( T_{2}^{174} - \)\(10\!\cdots\!82\)\( T_{2}^{173} + \)\(18\!\cdots\!07\)\( T_{2}^{172} - \)\(31\!\cdots\!39\)\( T_{2}^{171} + \)\(82\!\cdots\!67\)\( T_{2}^{170} - \)\(12\!\cdots\!94\)\( T_{2}^{169} + \)\(28\!\cdots\!91\)\( T_{2}^{168} - \)\(45\!\cdots\!71\)\( T_{2}^{167} + \)\(82\!\cdots\!20\)\( T_{2}^{166} - \)\(14\!\cdots\!55\)\( T_{2}^{165} + \)\(20\!\cdots\!87\)\( T_{2}^{164} - \)\(37\!\cdots\!36\)\( T_{2}^{163} + \)\(48\!\cdots\!44\)\( T_{2}^{162} - \)\(80\!\cdots\!64\)\( T_{2}^{161} + \)\(11\!\cdots\!12\)\( T_{2}^{160} - \)\(15\!\cdots\!73\)\( T_{2}^{159} + \)\(22\!\cdots\!11\)\( T_{2}^{158} - \)\(27\!\cdots\!42\)\( T_{2}^{157} + \)\(38\!\cdots\!55\)\( T_{2}^{156} - \)\(43\!\cdots\!64\)\( T_{2}^{155} + \)\(44\!\cdots\!81\)\( T_{2}^{154} - \)\(48\!\cdots\!47\)\( T_{2}^{153} + \)\(12\!\cdots\!72\)\( T_{2}^{152} + \)\(29\!\cdots\!08\)\( T_{2}^{151} - \)\(76\!\cdots\!53\)\( T_{2}^{150} + \)\(18\!\cdots\!43\)\( T_{2}^{149} - \)\(19\!\cdots\!17\)\( T_{2}^{148} + \)\(48\!\cdots\!27\)\( T_{2}^{147} - \)\(33\!\cdots\!20\)\( T_{2}^{146} + \)\(63\!\cdots\!37\)\( T_{2}^{145} - \)\(58\!\cdots\!41\)\( T_{2}^{144} + \)\(27\!\cdots\!28\)\( T_{2}^{143} - \)\(78\!\cdots\!62\)\( T_{2}^{142} - \)\(15\!\cdots\!11\)\( T_{2}^{141} - \)\(15\!\cdots\!08\)\( T_{2}^{140} + \)\(57\!\cdots\!40\)\( T_{2}^{139} + \)\(24\!\cdots\!64\)\( T_{2}^{138} + \)\(19\!\cdots\!71\)\( T_{2}^{137} + \)\(49\!\cdots\!79\)\( T_{2}^{136} + \)\(45\!\cdots\!03\)\( T_{2}^{135} + \)\(44\!\cdots\!51\)\( T_{2}^{134} - \)\(54\!\cdots\!93\)\( T_{2}^{133} + \)\(17\!\cdots\!34\)\( T_{2}^{132} - \)\(92\!\cdots\!29\)\( T_{2}^{131} + \)\(10\!\cdots\!65\)\( T_{2}^{130} - \)\(10\!\cdots\!04\)\( T_{2}^{129} + \)\(28\!\cdots\!35\)\( T_{2}^{128} + \)\(20\!\cdots\!54\)\( T_{2}^{127} + \)\(32\!\cdots\!53\)\( T_{2}^{126} + \)\(40\!\cdots\!61\)\( T_{2}^{125} + \)\(17\!\cdots\!78\)\( T_{2}^{124} + \)\(37\!\cdots\!35\)\( T_{2}^{123} + \)\(16\!\cdots\!44\)\( T_{2}^{122} - \)\(37\!\cdots\!93\)\( T_{2}^{121} + \)\(39\!\cdots\!70\)\( T_{2}^{120} - \)\(61\!\cdots\!99\)\( T_{2}^{119} + \)\(59\!\cdots\!38\)\( T_{2}^{118} - \)\(82\!\cdots\!23\)\( T_{2}^{117} + \)\(29\!\cdots\!04\)\( T_{2}^{116} - \)\(25\!\cdots\!67\)\( T_{2}^{115} - \)\(98\!\cdots\!81\)\( T_{2}^{114} + \)\(75\!\cdots\!54\)\( T_{2}^{113} - \)\(30\!\cdots\!29\)\( T_{2}^{112} + \)\(12\!\cdots\!51\)\( T_{2}^{111} - \)\(39\!\cdots\!65\)\( T_{2}^{110} + \)\(66\!\cdots\!23\)\( T_{2}^{109} - \)\(70\!\cdots\!66\)\( T_{2}^{108} - \)\(65\!\cdots\!00\)\( T_{2}^{107} + \)\(64\!\cdots\!34\)\( T_{2}^{106} - \)\(15\!\cdots\!88\)\( T_{2}^{105} + \)\(10\!\cdots\!33\)\( T_{2}^{104} - \)\(10\!\cdots\!10\)\( T_{2}^{103} + \)\(44\!\cdots\!73\)\( T_{2}^{102} + \)\(25\!\cdots\!74\)\( T_{2}^{101} - \)\(57\!\cdots\!31\)\( T_{2}^{100} + \)\(12\!\cdots\!93\)\( T_{2}^{99} - \)\(11\!\cdots\!60\)\( T_{2}^{98} + \)\(11\!\cdots\!71\)\( T_{2}^{97} - \)\(73\!\cdots\!21\)\( T_{2}^{96} + \)\(19\!\cdots\!28\)\( T_{2}^{95} + \)\(27\!\cdots\!63\)\( T_{2}^{94} - \)\(80\!\cdots\!17\)\( T_{2}^{93} + \)\(90\!\cdots\!35\)\( T_{2}^{92} - \)\(82\!\cdots\!26\)\( T_{2}^{91} + \)\(41\!\cdots\!39\)\( T_{2}^{90} + \)\(68\!\cdots\!00\)\( T_{2}^{89} - \)\(40\!\cdots\!81\)\( T_{2}^{88} + \)\(55\!\cdots\!00\)\( T_{2}^{87} - \)\(40\!\cdots\!03\)\( T_{2}^{86} + \)\(12\!\cdots\!91\)\( T_{2}^{85} + \)\(14\!\cdots\!56\)\( T_{2}^{84} - \)\(28\!\cdots\!65\)\( T_{2}^{83} + \)\(25\!\cdots\!00\)\( T_{2}^{82} - \)\(11\!\cdots\!11\)\( T_{2}^{81} - \)\(40\!\cdots\!84\)\( T_{2}^{80} + \)\(13\!\cdots\!32\)\( T_{2}^{79} - \)\(14\!\cdots\!41\)\( T_{2}^{78} + \)\(94\!\cdots\!91\)\( T_{2}^{77} - \)\(20\!\cdots\!52\)\( T_{2}^{76} - \)\(35\!\cdots\!32\)\( T_{2}^{75} + \)\(57\!\cdots\!02\)\( T_{2}^{74} - \)\(52\!\cdots\!91\)\( T_{2}^{73} + \)\(33\!\cdots\!79\)\( T_{2}^{72} - \)\(15\!\cdots\!82\)\( T_{2}^{71} + \)\(39\!\cdots\!45\)\( T_{2}^{70} + \)\(55\!\cdots\!35\)\( T_{2}^{69} - \)\(99\!\cdots\!15\)\( T_{2}^{68} + \)\(24\!\cdots\!75\)\( T_{2}^{67} + \)\(24\!\cdots\!17\)\( T_{2}^{66} - \)\(29\!\cdots\!16\)\( T_{2}^{65} + \)\(16\!\cdots\!72\)\( T_{2}^{64} - \)\(51\!\cdots\!68\)\( T_{2}^{63} + \)\(15\!\cdots\!66\)\( T_{2}^{62} - \)\(15\!\cdots\!75\)\( T_{2}^{61} + \)\(15\!\cdots\!23\)\( T_{2}^{60} - \)\(91\!\cdots\!59\)\( T_{2}^{59} + \)\(48\!\cdots\!03\)\( T_{2}^{58} - \)\(33\!\cdots\!20\)\( T_{2}^{57} + \)\(22\!\cdots\!85\)\( T_{2}^{56} - \)\(95\!\cdots\!60\)\( T_{2}^{55} + \)\(20\!\cdots\!77\)\( T_{2}^{54} - \)\(82\!\cdots\!52\)\( T_{2}^{53} + \)\(15\!\cdots\!43\)\( T_{2}^{52} - \)\(15\!\cdots\!46\)\( T_{2}^{51} + \)\(93\!\cdots\!71\)\( T_{2}^{50} - \)\(38\!\cdots\!85\)\( T_{2}^{49} + \)\(96\!\cdots\!27\)\( T_{2}^{48} + \)\(15\!\cdots\!25\)\( T_{2}^{47} - \)\(19\!\cdots\!07\)\( T_{2}^{46} + \)\(12\!\cdots\!22\)\( T_{2}^{45} - \)\(37\!\cdots\!62\)\( T_{2}^{44} - \)\(44\!\cdots\!46\)\( T_{2}^{43} + \)\(11\!\cdots\!14\)\( T_{2}^{42} - \)\(64\!\cdots\!24\)\( T_{2}^{41} + \)\(22\!\cdots\!45\)\( T_{2}^{40} - \)\(52\!\cdots\!94\)\( T_{2}^{39} + \)\(47\!\cdots\!44\)\( T_{2}^{38} + \)\(22\!\cdots\!47\)\( T_{2}^{37} - \)\(16\!\cdots\!37\)\( T_{2}^{36} + \)\(67\!\cdots\!38\)\( T_{2}^{35} - \)\(20\!\cdots\!24\)\( T_{2}^{34} + \)\(40\!\cdots\!90\)\( T_{2}^{33} + \)\(17\!\cdots\!34\)\( T_{2}^{32} - \)\(53\!\cdots\!86\)\( T_{2}^{31} + \)\(25\!\cdots\!88\)\( T_{2}^{30} - \)\(73\!\cdots\!20\)\( T_{2}^{29} + \)\(13\!\cdots\!88\)\( T_{2}^{28} - \)\(11\!\cdots\!35\)\( T_{2}^{27} - \)\(25\!\cdots\!07\)\( T_{2}^{26} + \)\(12\!\cdots\!29\)\( T_{2}^{25} - \)\(23\!\cdots\!32\)\( T_{2}^{24} + \)\(15\!\cdots\!73\)\( T_{2}^{23} + \)\(54\!\cdots\!53\)\( T_{2}^{22} - \)\(22\!\cdots\!46\)\( T_{2}^{21} + \)\(48\!\cdots\!92\)\( T_{2}^{20} - \)\(68\!\cdots\!89\)\( T_{2}^{19} + \)\(67\!\cdots\!94\)\( T_{2}^{18} - \)\(34\!\cdots\!78\)\( T_{2}^{17} - \)\(73\!\cdots\!59\)\( T_{2}^{16} + \)\(29\!\cdots\!61\)\( T_{2}^{15} - \)\(85\!\cdots\!78\)\( T_{2}^{14} - \)\(70\!\cdots\!10\)\( T_{2}^{13} + \)\(10\!\cdots\!07\)\( T_{2}^{12} + \)\(20\!\cdots\!25\)\( T_{2}^{11} - \)\(11\!\cdots\!98\)\( T_{2}^{10} - \)\(82\!\cdots\!16\)\( T_{2}^{9} - \)\(30\!\cdots\!07\)\( T_{2}^{8} + \)\(25\!\cdots\!58\)\( T_{2}^{7} + \)\(50\!\cdots\!63\)\( T_{2}^{6} + \)\(13\!\cdots\!13\)\( T_{2}^{5} + \)\(58\!\cdots\!42\)\( T_{2}^{4} - \)\(93\!\cdots\!55\)\( T_{2}^{3} + \)\(12\!\cdots\!27\)\( T_{2}^{2} - \)\(94\!\cdots\!45\)\( T_{2} + \)\(14\!\cdots\!01\)\( \)">\(T_{2}^{336} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).