# Properties

 Label 507.2.t.b Level $507$ Weight $2$ Character orbit 507.t 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.t (of order $$78$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$360$$ Relative dimension: $$15$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$360q - 15q^{3} - 14q^{4} + 3q^{7} + 15q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$360q - 15q^{3} - 14q^{4} + 3q^{7} + 15q^{9} + 6q^{11} - 28q^{12} + 6q^{13} + 4q^{14} - 6q^{15} + 20q^{16} - 6q^{19} - 12q^{20} - 28q^{22} + 46q^{23} - 6q^{25} - 39q^{26} + 30q^{27} - 6q^{28} + 43q^{29} + 26q^{31} - 195q^{32} + 19q^{33} + 65q^{34} + 84q^{35} - 14q^{36} - 65q^{38} - 2q^{39} + 12q^{41} - 128q^{42} + 83q^{43} - 39q^{44} - 6q^{45} - 20q^{48} + 72q^{49} - 52q^{50} - 55q^{52} + 49q^{53} - 49q^{55} - 2q^{56} + 26q^{57} + 26q^{58} - 202q^{59} - 182q^{60} - 3q^{61} - 65q^{62} - 3q^{63} - 14q^{64} - 58q^{65} + 48q^{66} - 41q^{67} + 139q^{68} + 6q^{69} - 60q^{71} - 52q^{73} - 269q^{74} + 23q^{75} - 14q^{76} + 70q^{77} - 65q^{78} + 18q^{79} + 492q^{80} + 15q^{81} - 65q^{82} + 78q^{83} - 6q^{84} - 91q^{85} - 169q^{86} + 48q^{87} - 522q^{88} - 12q^{89} + 373q^{91} + 72q^{92} - 3q^{93} - 13q^{94} - 110q^{95} + 65q^{96} - 121q^{97} - 104q^{98} + O(q^{100})$$

## 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 −2.60202 0.104858i −0.278217 + 0.960518i 4.76600 + 0.384751i −0.705070 0.267398i 0.824645 2.47011i 0.861409 + 2.02180i −7.19059 0.873096i −0.845190 0.534466i 1.80657 + 0.769706i
4.2 −2.53957 0.102341i −0.278217 + 0.960518i 4.44542 + 0.358871i 1.66050 + 0.629744i 0.804853 2.41083i −2.05060 4.81293i −6.20652 0.753608i −0.845190 0.534466i −4.15250 1.76922i
4.3 −1.85726 0.0748449i −0.278217 + 0.960518i 1.45029 + 0.117079i −1.06783 0.404975i 0.588611 1.76311i 0.332955 + 0.781475i 1.00563 + 0.122106i −0.845190 0.534466i 1.95292 + 0.832063i
4.4 −1.79660 0.0724007i −0.278217 + 0.960518i 1.22903 + 0.0992178i 1.73729 + 0.658868i 0.569389 1.70553i 1.36244 + 3.19777i 1.36900 + 0.166227i −0.845190 0.534466i −3.07353 1.30951i
4.5 −1.38591 0.0558504i −0.278217 + 0.960518i −0.0758757 0.00612532i −1.08360 0.410955i 0.439231 1.31566i −0.389218 0.913528i 2.85867 + 0.347105i −0.845190 0.534466i 1.47882 + 0.630068i
4.6 −0.772377 0.0311257i −0.278217 + 0.960518i −1.39792 0.112852i 3.22103 + 1.22158i 0.244786 0.733223i −1.25005 2.93398i 2.61094 + 0.317026i −0.845190 0.534466i −2.44983 1.04377i
4.7 −0.218325 0.00879818i −0.278217 + 0.960518i −1.94593 0.157091i −3.83328 1.45377i 0.0691925 0.207257i 1.70758 + 4.00784i 0.857279 + 0.104093i −0.845190 0.534466i 0.824109 + 0.351120i
4.8 −0.208145 0.00838797i −0.278217 + 0.960518i −1.95026 0.157441i −1.53730 0.583019i 0.0659665 0.197594i −0.815402 1.91382i 0.818208 + 0.0993485i −0.845190 0.534466i 0.315091 + 0.134248i
4.9 0.805348 + 0.0324544i −0.278217 + 0.960518i −1.34598 0.108659i 3.20699 + 1.21625i −0.255235 + 0.764522i −0.483860 1.13566i −2.68071 0.325497i −0.845190 0.534466i 2.54327 + 1.08359i
4.10 0.989614 + 0.0398801i −0.278217 + 0.960518i −1.01577 0.0820013i −0.558145 0.211676i −0.313633 + 0.939447i −0.641263 1.50510i −2.96834 0.360422i −0.845190 0.534466i −0.543906 0.231737i
4.11 0.992569 + 0.0399992i −0.278217 + 0.960518i −1.00992 0.0815293i −0.620050 0.235154i −0.314570 + 0.942252i 1.02980 + 2.41704i −2.97142 0.360796i −0.845190 0.534466i −0.606036 0.258208i
4.12 1.81864 + 0.0732885i −0.278217 + 0.960518i 1.30855 + 0.105637i −3.76569 1.42814i −0.576371 + 1.72644i −1.29170 3.03172i −1.24165 0.150763i −0.845190 0.534466i −6.74376 2.87325i
4.13 1.92708 + 0.0776588i −0.278217 + 0.960518i 1.71410 + 0.138377i 3.62781 + 1.37585i −0.610741 + 1.82939i 1.17937 + 2.76808i −0.536697 0.0651668i −0.845190 0.534466i 6.88424 + 2.93310i
4.14 2.27767 + 0.0917871i −0.278217 + 0.960518i 3.18585 + 0.257189i −1.29798 0.492260i −0.721851 + 2.16221i 1.28355 + 3.01261i 2.70692 + 0.328679i −0.845190 0.534466i −2.91120 1.24035i
4.15 2.56929 + 0.103539i −0.278217 + 0.960518i 4.59701 + 0.371109i 1.84810 + 0.700893i −0.814272 + 2.43904i −1.21279 2.84653i 6.66737 + 0.809566i −0.845190 0.534466i 4.67574 + 1.99215i
10.1 −2.52917 0.516334i −0.987050 + 0.160411i 4.29016 + 1.82787i 0.764962 3.10357i 2.57925 + 0.103940i 4.06091 1.92693i −5.65796 3.90541i 0.948536 0.316668i −3.53720 + 7.45450i
10.2 −2.42455 0.494975i −0.987050 + 0.160411i 3.79347 + 1.61625i 0.0746454 0.302848i 2.47255 + 0.0996403i −4.01396 + 1.90465i −4.32440 2.98492i 0.948536 0.316668i −0.330883 + 0.697322i
10.3 −1.83778 0.375186i −0.987050 + 0.160411i 1.39672 + 0.595086i −0.959864 + 3.89432i 1.87417 + 0.0755263i 1.09680 0.520440i 0.743728 + 0.513358i 0.948536 0.316668i 3.22511 6.79678i
10.4 −1.49980 0.306186i −0.987050 + 0.160411i 0.315692 + 0.134504i −0.217058 + 0.880640i 1.52949 + 0.0616365i 0.786387 0.373145i 2.08725 + 1.44072i 0.948536 0.316668i 0.595184 1.25432i
10.5 −1.31845 0.269162i −0.987050 + 0.160411i −0.174110 0.0741814i 0.381569 1.54808i 1.34455 + 0.0541834i 1.87281 0.888658i 2.42447 + 1.67349i 0.948536 0.316668i −0.919764 + 1.93836i
See next 80 embeddings (of 360 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 478.15 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.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.b 360
169.k even 78 1 inner 507.2.t.b 360

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.t.b 360 1.a even 1 1 trivial
507.2.t.b 360 169.k even 78 1 inner

## Hecke kernels

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