# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$.