# 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 $2$

# 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: $$336$$ 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: 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.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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$.