Properties

Label 507.2.q.b
Level $507$
Weight $2$
Character orbit 507.q
Analytic conductor $4.048$
Analytic rank $0$
Dimension $384$
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.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384q + q^{2} + 16q^{3} + 17q^{4} + 6q^{5} + q^{6} - 3q^{7} - 18q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 384q + q^{2} + 16q^{3} + 17q^{4} + 6q^{5} + q^{6} - 3q^{7} - 18q^{8} + 16q^{9} + 3q^{10} - 4q^{11} - 34q^{12} - 15q^{13} + 24q^{14} - 3q^{15} + 11q^{16} - 3q^{17} - 2q^{18} + 6q^{19} + q^{20} + 6q^{21} - 86q^{22} - 52q^{23} + 9q^{24} - 74q^{25} - 64q^{26} - 32q^{27} - 16q^{28} - 44q^{29} + 3q^{30} - 28q^{31} + 74q^{32} - 17q^{33} - 83q^{34} + 70q^{35} + 17q^{36} + 11q^{37} - 29q^{38} + q^{39} - 5q^{40} + q^{41} + 118q^{42} - 79q^{43} - 19q^{44} - 3q^{45} + 314q^{46} + 11q^{48} - 69q^{49} - 76q^{50} + 6q^{51} - 37q^{52} - 102q^{53} + q^{54} - 73q^{55} - 36q^{56} + 14q^{57} - 51q^{58} + 194q^{59} + 76q^{60} + 4q^{61} - 89q^{62} - 3q^{63} - 184q^{64} - 49q^{65} - 36q^{66} - 73q^{67} + 4q^{68} - 100q^{70} + 106q^{71} + 9q^{72} - 132q^{73} - 45q^{74} + 11q^{75} + 24q^{76} - 54q^{77} - 39q^{78} - 2q^{79} + 177q^{80} + 16q^{81} - 76q^{82} - 58q^{83} - 16q^{84} - 123q^{85} - 157q^{86} - 57q^{87} + 476q^{88} + 18q^{89} - 6q^{90} + 233q^{91} - 12q^{92} + q^{93} - 111q^{94} + 64q^{95} - 83q^{96} - 143q^{97} - 79q^{98} - 18q^{99} + O(q^{100}) \)

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 −2.67365 0.215839i −0.845190 0.534466i 5.12771 + 0.833333i −1.64492 1.45728i 2.14438 + 1.61140i 1.09807 1.14322i −8.32101 2.05094i 0.428693 + 0.903450i 4.08341 + 4.25128i
16.2 −2.48128 0.200310i −0.845190 0.534466i 4.14255 + 0.673230i 1.36223 + 1.20683i 1.99010 + 1.49546i −0.288513 + 0.300373i −5.30995 1.30878i 0.428693 + 0.903450i −3.13834 3.26736i
16.3 −2.00855 0.162147i −0.845190 0.534466i 2.03388 + 0.330538i −1.08961 0.965307i 1.61095 + 1.21055i −3.04871 + 3.17404i −0.118504 0.0292085i 0.428693 + 0.903450i 2.03201 + 2.11554i
16.4 −1.82353 0.147211i −0.845190 0.534466i 1.32950 + 0.216064i 1.37382 + 1.21710i 1.46255 + 1.09904i −0.320915 + 0.334108i 1.16003 + 0.285923i 0.428693 + 0.903450i −2.32603 2.42166i
16.5 −1.48832 0.120150i −0.845190 0.534466i 0.226569 + 0.0368210i 1.90213 + 1.68514i 1.19370 + 0.897007i 3.62894 3.77813i 2.56677 + 0.632652i 0.428693 + 0.903450i −2.62851 2.73657i
16.6 −1.39808 0.112865i −0.845190 0.534466i −0.0322011 0.00523319i −2.59137 2.29575i 1.12132 + 0.842620i 2.75817 2.87156i 2.76818 + 0.682294i 0.428693 + 0.903450i 3.36384 + 3.50213i
16.7 −0.529051 0.0427094i −0.845190 0.534466i −1.69603 0.275632i 0.694311 + 0.615105i 0.424322 + 0.318857i 0.272511 0.283714i 1.91621 + 0.472304i 0.428693 + 0.903450i −0.341055 0.355076i
16.8 −0.488381 0.0394262i −0.845190 0.534466i −1.73714 0.282313i −1.47540 1.30709i 0.391703 + 0.294346i −2.51230 + 2.61558i 1.78872 + 0.440880i 0.428693 + 0.903450i 0.669022 + 0.696526i
16.9 0.0277086 + 0.00223687i −0.845190 0.534466i −1.97334 0.320699i 0.676752 + 0.599550i −0.0222235 0.0166999i −0.402242 + 0.418779i −0.107943 0.0266056i 0.428693 + 0.903450i 0.0174108 + 0.0181265i
16.10 0.655764 + 0.0529387i −0.845190 0.534466i −1.54688 0.251392i −1.61643 1.43203i −0.525951 0.395227i 1.45558 1.51542i −2.27864 0.561634i 0.428693 + 0.903450i −0.984187 1.02465i
16.11 0.910598 + 0.0735111i −0.845190 0.534466i −1.15032 0.186944i 3.23408 + 2.86515i −0.730339 0.548814i −1.27940 + 1.33200i −2.80776 0.692051i 0.428693 + 0.903450i 2.73433 + 2.84674i
16.12 1.45946 + 0.117820i −0.845190 0.534466i 0.142046 + 0.0230847i −0.575893 0.510197i −1.17055 0.879613i −2.84784 + 2.96491i −2.63873 0.650390i 0.428693 + 0.903450i −0.780383 0.812465i
16.13 1.80906 + 0.146043i −0.845190 0.534466i 1.27728 + 0.207579i −0.0943821 0.0836152i −1.45095 1.09032i 1.37395 1.43043i −1.24405 0.306631i 0.428693 + 0.903450i −0.158532 0.165049i
16.14 2.08178 + 0.168058i −0.845190 0.534466i 2.33145 + 0.378897i −3.16521 2.80413i −1.66968 1.25468i 1.01537 1.05711i 0.734161 + 0.180955i 0.428693 + 0.903450i −6.11801 6.36952i
16.15 2.27058 + 0.183300i −0.845190 0.534466i 3.14784 + 0.511574i 2.05738 + 1.82268i −1.82111 1.36847i 0.613173 0.638380i 2.63011 + 0.648264i 0.428693 + 0.903450i 4.33734 + 4.51565i
16.16 2.67914 + 0.216283i −0.845190 0.534466i 5.15693 + 0.838083i 1.04622 + 0.926874i −2.14879 1.61471i −2.79839 + 2.91343i 8.41538 + 2.07420i 0.428693 + 0.903450i 2.60252 + 2.70951i
55.1 −1.73567 + 2.12581i −0.996757 0.0804666i −1.10646 5.41982i 0.355841 + 2.93061i 1.90110 1.97925i 0.0396979 0.0251034i 8.58190 + 4.50413i 0.987050 + 0.160411i −6.84755 4.33013i
55.2 −1.55963 + 1.91020i −0.996757 0.0804666i −0.816373 3.99886i −0.349402 2.87759i 1.70828 1.77851i −1.82814 + 1.15605i 4.54475 + 2.38527i 0.987050 + 0.160411i 6.04172 + 3.82055i
55.3 −1.21429 + 1.48724i −0.996757 0.0804666i −0.337321 1.65231i −0.0596193 0.491009i 1.33003 1.38471i 0.351566 0.222317i −0.533160 0.279824i 0.987050 + 0.160411i 0.802643 + 0.507561i
55.4 −1.17775 + 1.44248i −0.996757 0.0804666i −0.293610 1.43820i 0.180273 + 1.48468i 1.29000 1.34304i 3.26448 2.06433i −0.877451 0.460522i 0.987050 + 0.160411i −2.35394 1.48854i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 490.16
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.b 384
169.i even 39 1 inner 507.2.q.b 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.q.b 384 1.a even 1 1 trivial
507.2.q.b 384 169.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(12\!\cdots\!68\)\( T_{2}^{358} + \)\(16\!\cdots\!41\)\( T_{2}^{357} + \)\(11\!\cdots\!61\)\( T_{2}^{356} - \)\(23\!\cdots\!19\)\( T_{2}^{355} + \)\(12\!\cdots\!63\)\( T_{2}^{354} + \)\(42\!\cdots\!25\)\( T_{2}^{353} - \)\(73\!\cdots\!64\)\( T_{2}^{352} + \)\(14\!\cdots\!66\)\( T_{2}^{351} + \)\(34\!\cdots\!61\)\( T_{2}^{350} - \)\(14\!\cdots\!75\)\( T_{2}^{349} + \)\(25\!\cdots\!71\)\( T_{2}^{348} + \)\(10\!\cdots\!72\)\( T_{2}^{347} - \)\(38\!\cdots\!90\)\( T_{2}^{346} + \)\(84\!\cdots\!65\)\( T_{2}^{345} + \)\(12\!\cdots\!16\)\( T_{2}^{344} - \)\(74\!\cdots\!79\)\( T_{2}^{343} + \)\(13\!\cdots\!39\)\( T_{2}^{342} + \)\(12\!\cdots\!22\)\( T_{2}^{341} - \)\(15\!\cdots\!87\)\( T_{2}^{340} + \)\(28\!\cdots\!56\)\( T_{2}^{339} + \)\(37\!\cdots\!22\)\( T_{2}^{338} - \)\(26\!\cdots\!65\)\( T_{2}^{337} + \)\(58\!\cdots\!14\)\( T_{2}^{336} + \)\(45\!\cdots\!93\)\( T_{2}^{335} - \)\(62\!\cdots\!21\)\( T_{2}^{334} + \)\(96\!\cdots\!13\)\( T_{2}^{333} + \)\(16\!\cdots\!10\)\( T_{2}^{332} - \)\(86\!\cdots\!00\)\( T_{2}^{331} + \)\(17\!\cdots\!67\)\( T_{2}^{330} + \)\(14\!\cdots\!17\)\( T_{2}^{329} - \)\(18\!\cdots\!01\)\( T_{2}^{328} + \)\(25\!\cdots\!01\)\( T_{2}^{327} + \)\(46\!\cdots\!25\)\( T_{2}^{326} - \)\(21\!\cdots\!93\)\( T_{2}^{325} + \)\(40\!\cdots\!53\)\( T_{2}^{324} + \)\(31\!\cdots\!05\)\( T_{2}^{323} - \)\(41\!\cdots\!56\)\( T_{2}^{322} + \)\(56\!\cdots\!44\)\( T_{2}^{321} + \)\(11\!\cdots\!34\)\( T_{2}^{320} - \)\(43\!\cdots\!23\)\( T_{2}^{319} + \)\(74\!\cdots\!91\)\( T_{2}^{318} + \)\(64\!\cdots\!97\)\( T_{2}^{317} - \)\(79\!\cdots\!76\)\( T_{2}^{316} + \)\(11\!\cdots\!65\)\( T_{2}^{315} + \)\(19\!\cdots\!01\)\( T_{2}^{314} - \)\(89\!\cdots\!75\)\( T_{2}^{313} + \)\(15\!\cdots\!46\)\( T_{2}^{312} + \)\(16\!\cdots\!96\)\( T_{2}^{311} - \)\(14\!\cdots\!12\)\( T_{2}^{310} + \)\(17\!\cdots\!68\)\( T_{2}^{309} + \)\(16\!\cdots\!18\)\( T_{2}^{308} - \)\(13\!\cdots\!23\)\( T_{2}^{307} + \)\(39\!\cdots\!67\)\( T_{2}^{306} + \)\(13\!\cdots\!06\)\( T_{2}^{305} - \)\(24\!\cdots\!16\)\( T_{2}^{304} + \)\(27\!\cdots\!98\)\( T_{2}^{303} - \)\(10\!\cdots\!58\)\( T_{2}^{302} - \)\(13\!\cdots\!21\)\( T_{2}^{301} + \)\(82\!\cdots\!63\)\( T_{2}^{300} - \)\(30\!\cdots\!91\)\( T_{2}^{299} - \)\(36\!\cdots\!00\)\( T_{2}^{298} + \)\(41\!\cdots\!77\)\( T_{2}^{297} - \)\(67\!\cdots\!70\)\( T_{2}^{296} - \)\(18\!\cdots\!43\)\( T_{2}^{295} + \)\(13\!\cdots\!59\)\( T_{2}^{294} - \)\(12\!\cdots\!26\)\( T_{2}^{293} - \)\(45\!\cdots\!83\)\( T_{2}^{292} + \)\(60\!\cdots\!13\)\( T_{2}^{291} - \)\(14\!\cdots\!27\)\( T_{2}^{290} + \)\(11\!\cdots\!56\)\( T_{2}^{289} + \)\(17\!\cdots\!71\)\( T_{2}^{288} - \)\(23\!\cdots\!31\)\( T_{2}^{287} - \)\(35\!\cdots\!96\)\( T_{2}^{286} + \)\(95\!\cdots\!03\)\( T_{2}^{285} - \)\(28\!\cdots\!20\)\( T_{2}^{284} - \)\(11\!\cdots\!52\)\( T_{2}^{283} + \)\(19\!\cdots\!35\)\( T_{2}^{282} - \)\(15\!\cdots\!74\)\( T_{2}^{281} - \)\(25\!\cdots\!15\)\( T_{2}^{280} + \)\(58\!\cdots\!44\)\( T_{2}^{279} - \)\(55\!\cdots\!51\)\( T_{2}^{278} - \)\(10\!\cdots\!02\)\( T_{2}^{277} + \)\(28\!\cdots\!63\)\( T_{2}^{276} + \)\(10\!\cdots\!48\)\( T_{2}^{275} + \)\(57\!\cdots\!40\)\( T_{2}^{274} - \)\(90\!\cdots\!62\)\( T_{2}^{273} - \)\(76\!\cdots\!61\)\( T_{2}^{272} + \)\(10\!\cdots\!04\)\( T_{2}^{271} + \)\(37\!\cdots\!45\)\( T_{2}^{270} + \)\(29\!\cdots\!69\)\( T_{2}^{269} - \)\(14\!\cdots\!47\)\( T_{2}^{268} - \)\(21\!\cdots\!24\)\( T_{2}^{267} - \)\(73\!\cdots\!16\)\( T_{2}^{266} + \)\(34\!\cdots\!01\)\( T_{2}^{265} + \)\(37\!\cdots\!16\)\( T_{2}^{264} + \)\(39\!\cdots\!83\)\( T_{2}^{263} - \)\(29\!\cdots\!54\)\( T_{2}^{262} - \)\(30\!\cdots\!04\)\( T_{2}^{261} - \)\(63\!\cdots\!01\)\( T_{2}^{260} + \)\(68\!\cdots\!59\)\( T_{2}^{259} + \)\(37\!\cdots\!67\)\( T_{2}^{258} + \)\(34\!\cdots\!21\)\( T_{2}^{257} - \)\(54\!\cdots\!50\)\( T_{2}^{256} - \)\(32\!\cdots\!15\)\( T_{2}^{255} - \)\(44\!\cdots\!52\)\( T_{2}^{254} + \)\(96\!\cdots\!73\)\( T_{2}^{253} + \)\(30\!\cdots\!04\)\( T_{2}^{252} + \)\(15\!\cdots\!07\)\( T_{2}^{251} - \)\(57\!\cdots\!63\)\( T_{2}^{250} - \)\(23\!\cdots\!25\)\( T_{2}^{249} - \)\(24\!\cdots\!44\)\( T_{2}^{248} + \)\(83\!\cdots\!71\)\( T_{2}^{247} + \)\(18\!\cdots\!02\)\( T_{2}^{246} - \)\(17\!\cdots\!83\)\( T_{2}^{245} - \)\(30\!\cdots\!57\)\( T_{2}^{244} - \)\(11\!\cdots\!40\)\( T_{2}^{243} - \)\(16\!\cdots\!82\)\( T_{2}^{242} + \)\(45\!\cdots\!12\)\( T_{2}^{241} + \)\(11\!\cdots\!62\)\( T_{2}^{240} - \)\(32\!\cdots\!61\)\( T_{2}^{239} - \)\(19\!\cdots\!74\)\( T_{2}^{238} - \)\(36\!\cdots\!14\)\( T_{2}^{237} - \)\(84\!\cdots\!98\)\( T_{2}^{236} + \)\(12\!\cdots\!50\)\( T_{2}^{235} + \)\(55\!\cdots\!21\)\( T_{2}^{234} + \)\(42\!\cdots\!66\)\( T_{2}^{233} - \)\(80\!\cdots\!46\)\( T_{2}^{232} - \)\(74\!\cdots\!51\)\( T_{2}^{231} - \)\(38\!\cdots\!03\)\( T_{2}^{230} - \)\(16\!\cdots\!26\)\( T_{2}^{229} + \)\(20\!\cdots\!60\)\( T_{2}^{228} + \)\(22\!\cdots\!53\)\( T_{2}^{227} - \)\(11\!\cdots\!29\)\( T_{2}^{226} - \)\(45\!\cdots\!79\)\( T_{2}^{225} - \)\(18\!\cdots\!77\)\( T_{2}^{224} - \)\(18\!\cdots\!77\)\( T_{2}^{223} + \)\(65\!\cdots\!31\)\( T_{2}^{222} + \)\(11\!\cdots\!77\)\( T_{2}^{221} - \)\(81\!\cdots\!92\)\( T_{2}^{220} - \)\(11\!\cdots\!69\)\( T_{2}^{219} - \)\(48\!\cdots\!59\)\( T_{2}^{218} - \)\(11\!\cdots\!94\)\( T_{2}^{217} + \)\(72\!\cdots\!60\)\( T_{2}^{216} + \)\(58\!\cdots\!39\)\( T_{2}^{215} + \)\(49\!\cdots\!67\)\( T_{2}^{214} - \)\(10\!\cdots\!74\)\( T_{2}^{213} - \)\(27\!\cdots\!48\)\( T_{2}^{212} - \)\(75\!\cdots\!44\)\( T_{2}^{211} + \)\(51\!\cdots\!55\)\( T_{2}^{210} + \)\(69\!\cdots\!59\)\( T_{2}^{209} + \)\(46\!\cdots\!97\)\( T_{2}^{208} - \)\(24\!\cdots\!73\)\( T_{2}^{207} - \)\(46\!\cdots\!26\)\( T_{2}^{206} - \)\(80\!\cdots\!11\)\( T_{2}^{205} + \)\(74\!\cdots\!84\)\( T_{2}^{204} + \)\(35\!\cdots\!98\)\( T_{2}^{203} + \)\(21\!\cdots\!47\)\( T_{2}^{202} - \)\(68\!\cdots\!95\)\( T_{2}^{201} - \)\(14\!\cdots\!63\)\( T_{2}^{200} + \)\(16\!\cdots\!34\)\( T_{2}^{199} + \)\(34\!\cdots\!23\)\( T_{2}^{198} + \)\(24\!\cdots\!18\)\( T_{2}^{197} - \)\(24\!\cdots\!46\)\( T_{2}^{196} - \)\(47\!\cdots\!21\)\( T_{2}^{195} - \)\(10\!\cdots\!39\)\( T_{2}^{194} - \)\(35\!\cdots\!31\)\( T_{2}^{193} + \)\(30\!\cdots\!12\)\( T_{2}^{192} + \)\(28\!\cdots\!84\)\( T_{2}^{191} - \)\(26\!\cdots\!83\)\( T_{2}^{190} - \)\(22\!\cdots\!48\)\( T_{2}^{189} - \)\(13\!\cdots\!30\)\( T_{2}^{188} - \)\(14\!\cdots\!43\)\( T_{2}^{187} + \)\(25\!\cdots\!96\)\( T_{2}^{186} + \)\(26\!\cdots\!55\)\( T_{2}^{185} + \)\(24\!\cdots\!97\)\( T_{2}^{184} + \)\(12\!\cdots\!62\)\( T_{2}^{183} - \)\(15\!\cdots\!69\)\( T_{2}^{182} - \)\(59\!\cdots\!13\)\( T_{2}^{181} - \)\(11\!\cdots\!04\)\( T_{2}^{180} + \)\(67\!\cdots\!63\)\( T_{2}^{179} + \)\(21\!\cdots\!77\)\( T_{2}^{178} + \)\(24\!\cdots\!82\)\( T_{2}^{177} - \)\(57\!\cdots\!80\)\( T_{2}^{176} - \)\(12\!\cdots\!13\)\( T_{2}^{175} + \)\(98\!\cdots\!48\)\( T_{2}^{174} + \)\(24\!\cdots\!54\)\( T_{2}^{173} + \)\(34\!\cdots\!72\)\( T_{2}^{172} - \)\(55\!\cdots\!75\)\( T_{2}^{171} - \)\(10\!\cdots\!81\)\( T_{2}^{170} - \)\(10\!\cdots\!93\)\( T_{2}^{169} + \)\(12\!\cdots\!81\)\( T_{2}^{168} + \)\(30\!\cdots\!80\)\( T_{2}^{167} + \)\(18\!\cdots\!79\)\( T_{2}^{166} - \)\(30\!\cdots\!21\)\( T_{2}^{165} - \)\(10\!\cdots\!56\)\( T_{2}^{164} - \)\(37\!\cdots\!82\)\( T_{2}^{163} + \)\(15\!\cdots\!41\)\( T_{2}^{162} + \)\(20\!\cdots\!18\)\( T_{2}^{161} + \)\(87\!\cdots\!55\)\( T_{2}^{160} - \)\(35\!\cdots\!47\)\( T_{2}^{159} - \)\(79\!\cdots\!19\)\( T_{2}^{158} + \)\(76\!\cdots\!18\)\( T_{2}^{157} + \)\(13\!\cdots\!69\)\( T_{2}^{156} + \)\(12\!\cdots\!80\)\( T_{2}^{155} + \)\(15\!\cdots\!40\)\( T_{2}^{154} - \)\(35\!\cdots\!13\)\( T_{2}^{153} - \)\(48\!\cdots\!82\)\( T_{2}^{152} + \)\(37\!\cdots\!59\)\( T_{2}^{151} + \)\(10\!\cdots\!90\)\( T_{2}^{150} + \)\(44\!\cdots\!72\)\( T_{2}^{149} - \)\(79\!\cdots\!40\)\( T_{2}^{148} - \)\(23\!\cdots\!03\)\( T_{2}^{147} - \)\(11\!\cdots\!66\)\( T_{2}^{146} + \)\(38\!\cdots\!64\)\( T_{2}^{145} + \)\(46\!\cdots\!50\)\( T_{2}^{144} - \)\(16\!\cdots\!93\)\( T_{2}^{143} - \)\(65\!\cdots\!34\)\( T_{2}^{142} - \)\(63\!\cdots\!37\)\( T_{2}^{141} + \)\(22\!\cdots\!12\)\( T_{2}^{140} + \)\(16\!\cdots\!19\)\( T_{2}^{139} + \)\(10\!\cdots\!18\)\( T_{2}^{138} - \)\(16\!\cdots\!19\)\( T_{2}^{137} - \)\(23\!\cdots\!63\)\( T_{2}^{136} - \)\(10\!\cdots\!39\)\( T_{2}^{135} + \)\(20\!\cdots\!33\)\( T_{2}^{134} + \)\(27\!\cdots\!14\)\( T_{2}^{133} + \)\(96\!\cdots\!95\)\( T_{2}^{132} - \)\(36\!\cdots\!12\)\( T_{2}^{131} - \)\(51\!\cdots\!87\)\( T_{2}^{130} + \)\(79\!\cdots\!59\)\( T_{2}^{129} + \)\(63\!\cdots\!07\)\( T_{2}^{128} + \)\(33\!\cdots\!95\)\( T_{2}^{127} - \)\(20\!\cdots\!30\)\( T_{2}^{126} - \)\(31\!\cdots\!63\)\( T_{2}^{125} - \)\(44\!\cdots\!66\)\( T_{2}^{124} - \)\(26\!\cdots\!72\)\( T_{2}^{123} + \)\(76\!\cdots\!55\)\( T_{2}^{122} + \)\(77\!\cdots\!90\)\( T_{2}^{121} - \)\(79\!\cdots\!11\)\( T_{2}^{120} - \)\(67\!\cdots\!62\)\( T_{2}^{119} + \)\(87\!\cdots\!10\)\( T_{2}^{118} + \)\(95\!\cdots\!23\)\( T_{2}^{117} - \)\(98\!\cdots\!82\)\( T_{2}^{116} + \)\(54\!\cdots\!25\)\( T_{2}^{115} + \)\(74\!\cdots\!41\)\( T_{2}^{114} - \)\(10\!\cdots\!69\)\( T_{2}^{113} - \)\(11\!\cdots\!14\)\( T_{2}^{112} + \)\(11\!\cdots\!09\)\( T_{2}^{111} - \)\(61\!\cdots\!12\)\( T_{2}^{110} - \)\(99\!\cdots\!72\)\( T_{2}^{109} + \)\(11\!\cdots\!99\)\( T_{2}^{108} + \)\(50\!\cdots\!68\)\( T_{2}^{107} - \)\(11\!\cdots\!92\)\( T_{2}^{106} + \)\(29\!\cdots\!12\)\( T_{2}^{105} + \)\(89\!\cdots\!58\)\( T_{2}^{104} - \)\(36\!\cdots\!70\)\( T_{2}^{103} - \)\(47\!\cdots\!96\)\( T_{2}^{102} + \)\(44\!\cdots\!25\)\( T_{2}^{101} + \)\(12\!\cdots\!97\)\( T_{2}^{100} - \)\(33\!\cdots\!09\)\( T_{2}^{99} + \)\(52\!\cdots\!93\)\( T_{2}^{98} + \)\(16\!\cdots\!91\)\( T_{2}^{97} - \)\(87\!\cdots\!88\)\( T_{2}^{96} - \)\(47\!\cdots\!34\)\( T_{2}^{95} + \)\(57\!\cdots\!89\)\( T_{2}^{94} - \)\(41\!\cdots\!98\)\( T_{2}^{93} - \)\(22\!\cdots\!34\)\( T_{2}^{92} + \)\(95\!\cdots\!46\)\( T_{2}^{91} + \)\(59\!\cdots\!77\)\( T_{2}^{90} - \)\(17\!\cdots\!37\)\( T_{2}^{89} - \)\(29\!\cdots\!80\)\( T_{2}^{88} - \)\(25\!\cdots\!48\)\( T_{2}^{87} + \)\(41\!\cdots\!08\)\( T_{2}^{86} + \)\(18\!\cdots\!44\)\( T_{2}^{85} - \)\(41\!\cdots\!42\)\( T_{2}^{84} + \)\(31\!\cdots\!50\)\( T_{2}^{83} + \)\(27\!\cdots\!76\)\( T_{2}^{82} - \)\(93\!\cdots\!86\)\( T_{2}^{81} - \)\(12\!\cdots\!99\)\( T_{2}^{80} + \)\(84\!\cdots\!17\)\( T_{2}^{79} + \)\(35\!\cdots\!17\)\( T_{2}^{78} - \)\(45\!\cdots\!46\)\( T_{2}^{77} - \)\(42\!\cdots\!76\)\( T_{2}^{76} + \)\(17\!\cdots\!02\)\( T_{2}^{75} - \)\(14\!\cdots\!02\)\( T_{2}^{74} - \)\(51\!\cdots\!31\)\( T_{2}^{73} + \)\(79\!\cdots\!51\)\( T_{2}^{72} + \)\(12\!\cdots\!75\)\( T_{2}^{71} - \)\(10\!\cdots\!60\)\( T_{2}^{70} - \)\(34\!\cdots\!32\)\( T_{2}^{69} - \)\(61\!\cdots\!89\)\( T_{2}^{68} + \)\(11\!\cdots\!12\)\( T_{2}^{67} + \)\(47\!\cdots\!95\)\( T_{2}^{66} - \)\(30\!\cdots\!71\)\( T_{2}^{65} - \)\(19\!\cdots\!63\)\( T_{2}^{64} + \)\(55\!\cdots\!61\)\( T_{2}^{63} + \)\(56\!\cdots\!40\)\( T_{2}^{62} - \)\(22\!\cdots\!30\)\( T_{2}^{61} - \)\(10\!\cdots\!44\)\( T_{2}^{60} - \)\(19\!\cdots\!68\)\( T_{2}^{59} + \)\(11\!\cdots\!88\)\( T_{2}^{58} + \)\(66\!\cdots\!91\)\( T_{2}^{57} + \)\(56\!\cdots\!86\)\( T_{2}^{56} - \)\(10\!\cdots\!31\)\( T_{2}^{55} - \)\(23\!\cdots\!41\)\( T_{2}^{54} + \)\(27\!\cdots\!29\)\( T_{2}^{53} + \)\(45\!\cdots\!38\)\( T_{2}^{52} + \)\(17\!\cdots\!47\)\( T_{2}^{51} + \)\(81\!\cdots\!53\)\( T_{2}^{50} - \)\(37\!\cdots\!04\)\( T_{2}^{49} - \)\(15\!\cdots\!03\)\( T_{2}^{48} + \)\(72\!\cdots\!83\)\( T_{2}^{47} + \)\(37\!\cdots\!87\)\( T_{2}^{46} + \)\(10\!\cdots\!61\)\( T_{2}^{45} + \)\(61\!\cdots\!35\)\( T_{2}^{44} + \)\(70\!\cdots\!09\)\( T_{2}^{43} - \)\(75\!\cdots\!56\)\( T_{2}^{42} - \)\(72\!\cdots\!09\)\( T_{2}^{41} + \)\(51\!\cdots\!14\)\( T_{2}^{40} - \)\(11\!\cdots\!77\)\( T_{2}^{39} + \)\(10\!\cdots\!25\)\( T_{2}^{38} + \)\(42\!\cdots\!76\)\( T_{2}^{37} - \)\(44\!\cdots\!80\)\( T_{2}^{36} + \)\(58\!\cdots\!25\)\( T_{2}^{35} - \)\(15\!\cdots\!96\)\( T_{2}^{34} - \)\(19\!\cdots\!18\)\( T_{2}^{33} + \)\(82\!\cdots\!42\)\( T_{2}^{32} - \)\(99\!\cdots\!74\)\( T_{2}^{31} + \)\(55\!\cdots\!59\)\( T_{2}^{30} + \)\(19\!\cdots\!93\)\( T_{2}^{29} - \)\(47\!\cdots\!27\)\( T_{2}^{28} - \)\(48\!\cdots\!57\)\( T_{2}^{27} - \)\(14\!\cdots\!72\)\( T_{2}^{26} - \)\(55\!\cdots\!36\)\( T_{2}^{25} + \)\(58\!\cdots\!97\)\( T_{2}^{24} - \)\(10\!\cdots\!60\)\( T_{2}^{23} + \)\(34\!\cdots\!76\)\( T_{2}^{22} - \)\(84\!\cdots\!76\)\( T_{2}^{21} + \)\(12\!\cdots\!72\)\( T_{2}^{20} - \)\(25\!\cdots\!68\)\( T_{2}^{19} + \)\(43\!\cdots\!04\)\( T_{2}^{18} - \)\(56\!\cdots\!88\)\( T_{2}^{17} + \)\(74\!\cdots\!24\)\( T_{2}^{16} - \)\(83\!\cdots\!00\)\( T_{2}^{15} + \)\(88\!\cdots\!16\)\( T_{2}^{14} - \)\(48\!\cdots\!76\)\( T_{2}^{13} + \)\(74\!\cdots\!32\)\( T_{2}^{12} - \)\(32\!\cdots\!20\)\( T_{2}^{11} + \)\(36\!\cdots\!80\)\( T_{2}^{10} - \)\(28\!\cdots\!04\)\( T_{2}^{9} + \)\(31\!\cdots\!52\)\( T_{2}^{8} - \)\(67\!\cdots\!44\)\( T_{2}^{7} - \)\(10\!\cdots\!48\)\( T_{2}^{6} - \)\(34\!\cdots\!88\)\( T_{2}^{5} + \)\(57\!\cdots\!84\)\( T_{2}^{4} + \)\(96\!\cdots\!68\)\( T_{2}^{3} + \)\(41\!\cdots\!20\)\( T_{2}^{2} + \)\(52\!\cdots\!76\)\( T_{2} + \)\(17\!\cdots\!36\)\( \)">\(T_{2}^{384} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).