# 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

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