# Properties

 Label 637.2.bl.a Level $637$ Weight $2$ Character orbit 637.bl Analytic conductor $5.086$ Analytic rank $0$ Dimension $324$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.bl (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$324$$ Relative dimension: $$27$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$324 q - 3 q^{2} + 25 q^{4} + q^{5} - 24 q^{6} - 21 q^{7} + 24 q^{8} + 11 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$324 q - 3 q^{2} + 25 q^{4} + q^{5} - 24 q^{6} - 21 q^{7} + 24 q^{8} + 11 q^{9} - 5 q^{10} - 18 q^{11} - 40 q^{12} + 54 q^{13} - 15 q^{14} + 6 q^{15} + 29 q^{16} - 6 q^{17} + 49 q^{18} - 24 q^{19} + 11 q^{20} - 6 q^{22} - 42 q^{23} + 20 q^{24} + 14 q^{25} + 3 q^{26} - 33 q^{27} + 7 q^{28} - 22 q^{29} + 57 q^{30} - 31 q^{31} - 139 q^{32} + 6 q^{33} + 50 q^{34} + 42 q^{35} - 78 q^{36} - 4 q^{37} - 90 q^{38} + 40 q^{40} + 8 q^{41} + 20 q^{42} + 34 q^{43} - 256 q^{44} + 19 q^{45} + 85 q^{46} + 34 q^{47} - 10 q^{48} + 51 q^{49} - 54 q^{50} + 74 q^{51} - 25 q^{52} + 10 q^{53} + 111 q^{54} - 10 q^{55} - 196 q^{56} - 5 q^{57} - 21 q^{58} + 65 q^{59} + 87 q^{60} + 3 q^{61} - 54 q^{62} - 35 q^{63} - 28 q^{64} - q^{65} - 110 q^{66} + 135 q^{67} - 158 q^{68} + 42 q^{69} + 44 q^{70} + 9 q^{71} - 133 q^{72} + 31 q^{73} - 97 q^{74} - 315 q^{75} - 177 q^{76} + 6 q^{77} - 25 q^{78} + 43 q^{79} - 20 q^{80} - 259 q^{81} + 96 q^{82} + 59 q^{83} + 285 q^{84} + 6 q^{85} + 95 q^{86} - 206 q^{87} + 228 q^{88} + 43 q^{89} - 61 q^{90} - 7 q^{91} + 53 q^{92} - 10 q^{93} - 36 q^{94} - 17 q^{95} + 277 q^{96} + 66 q^{97} + 260 q^{98} - 206 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1 −2.27310 1.54978i 2.32870 + 2.16072i 2.03451 + 5.18386i 1.52353 0.469945i −1.94475 8.52050i −2.61577 0.397162i 2.18478 9.57215i 0.529951 + 7.07171i −4.19144 1.29289i
53.2 −2.04606 1.39498i −2.21248 2.05288i 1.50972 + 3.84670i 1.22875 0.379019i 1.66314 + 7.28669i 2.38208 1.15138i 1.17502 5.14808i 0.456553 + 6.09228i −3.04283 0.938588i
53.3 −1.99966 1.36335i 0.0697391 + 0.0647085i 1.40925 + 3.59072i 4.18596 1.29120i −0.0512347 0.224474i 0.780361 2.52805i 1.00027 4.38249i −0.223514 2.98259i −10.1309 3.12496i
53.4 −1.98190 1.35124i 0.272118 + 0.252488i 1.37140 + 3.49428i −2.27380 + 0.701373i −0.198139 0.868102i −1.25655 + 2.32832i 0.936090 4.10128i −0.213893 2.85420i 5.45416 + 1.68238i
53.5 −1.79391 1.22307i 1.34252 + 1.24567i 0.991536 + 2.52639i 1.96924 0.607429i −0.884812 3.87661i 1.60270 + 2.10508i 0.344956 1.51135i 0.0264586 + 0.353066i −4.27556 1.31883i
53.6 −1.44647 0.986187i −2.38493 2.21289i 0.389031 + 0.991234i −0.723247 + 0.223092i 1.26741 + 5.55286i −0.372756 + 2.61936i −0.364300 + 1.59610i 0.566810 + 7.56355i 1.26617 + 0.390561i
53.7 −1.40283 0.956431i −0.499322 0.463303i 0.322480 + 0.821666i −2.80314 + 0.864654i 0.257345 + 1.12750i 2.49640 0.876348i −0.422130 + 1.84947i −0.189518 2.52894i 4.75930 + 1.46805i
53.8 −1.31734 0.898146i −1.50703 1.39832i 0.198032 + 0.504577i 0.675664 0.208415i 0.729374 + 3.19559i −1.94868 1.78960i −0.517257 + 2.26625i 0.0916520 + 1.22301i −1.07726 0.332292i
53.9 −1.15236 0.785668i 1.86576 + 1.73118i −0.0200159 0.0509996i −1.72793 + 0.532996i −0.789908 3.46081i −0.964248 2.46378i −0.637707 + 2.79398i 0.259914 + 3.46832i 2.40996 + 0.743375i
53.10 −0.971314 0.662231i 1.44449 + 1.34029i −0.225781 0.575279i −0.154277 + 0.0475881i −0.515472 2.25843i 2.49647 + 0.876147i −0.684849 + 3.00052i 0.0659823 + 0.880472i 0.181366 + 0.0559439i
53.11 −0.639784 0.436198i 1.84518 + 1.71208i −0.511626 1.30360i −1.40912 + 0.434655i −0.433714 1.90023i −2.31520 + 1.28057i −0.585909 + 2.56703i 0.249293 + 3.32659i 1.09113 + 0.336568i
53.12 −0.483489 0.329637i −1.12540 1.04422i −0.605581 1.54300i −2.74195 + 0.845781i 0.199905 + 0.875842i 1.46937 + 2.20022i −0.476262 + 2.08664i −0.0480579 0.641288i 1.60451 + 0.494925i
53.13 −0.434073 0.295946i 0.0114514 + 0.0106254i −0.629847 1.60482i 3.15579 0.973431i −0.00182622 0.00800119i 0.290350 + 2.62977i −0.435349 + 1.90739i −0.224172 2.99137i −1.65792 0.511402i
53.14 −0.178924 0.121988i −0.267794 0.248477i −0.713549 1.81809i 0.350054 0.107977i 0.0176035 + 0.0771261i 1.05341 2.42700i −0.190490 + 0.834589i −0.214217 2.85853i −0.0758047 0.0233827i
53.15 −0.0762337 0.0519753i −1.17809 1.09311i −0.727572 1.85382i 0.443733 0.136874i 0.0329956 + 0.144563i −2.48207 + 0.916143i −0.0819496 + 0.359045i −0.0311795 0.416062i −0.0409415 0.0126288i
53.16 0.395693 + 0.269779i 1.67664 + 1.55569i −0.646890 1.64825i 2.78605 0.859383i 0.243741 + 1.06790i −1.76663 1.96952i 0.401828 1.76052i 0.166745 + 2.22506i 1.33427 + 0.411566i
53.17 0.500003 + 0.340896i −1.93585 1.79621i −0.596889 1.52085i −4.16664 + 1.28524i −0.355611 1.55803i −2.59900 0.495162i 0.489325 2.14387i 0.296968 + 3.96277i −2.52146 0.777768i
53.18 0.726317 + 0.495195i −1.98880 1.84534i −0.448363 1.14241i 1.60817 0.496054i −0.530699 2.32514i 0.921177 2.48021i 0.631282 2.76583i 0.325867 + 4.34840i 1.41368 + 0.436064i
53.19 1.01838 + 0.694320i 0.670270 + 0.621920i −0.175663 0.447582i 0.228303 0.0704221i 0.250779 + 1.09873i 2.28590 + 1.33217i 0.680410 2.98107i −0.161712 2.15790i 0.281395 + 0.0867987i
53.20 1.14097 + 0.777904i 2.47075 + 2.29252i −0.0339921 0.0866106i −0.390328 + 0.120400i 1.03571 + 4.53772i 2.64566 0.0219120i 0.643161 2.81787i 0.624765 + 8.33691i −0.539015 0.166264i
See next 80 embeddings (of 324 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 625.27 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bl.a 324
49.g even 21 1 inner 637.2.bl.a 324

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bl.a 324 1.a even 1 1 trivial
637.2.bl.a 324 49.g even 21 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$30\!\cdots\!47$$$$T_{2}^{300} -$$$$40\!\cdots\!10$$$$T_{2}^{299} +$$$$35\!\cdots\!69$$$$T_{2}^{298} +$$$$42\!\cdots\!30$$$$T_{2}^{297} +$$$$13\!\cdots\!67$$$$T_{2}^{296} -$$$$18\!\cdots\!93$$$$T_{2}^{295} -$$$$96\!\cdots\!63$$$$T_{2}^{294} -$$$$24\!\cdots\!34$$$$T_{2}^{293} -$$$$21\!\cdots\!60$$$$T_{2}^{292} +$$$$17\!\cdots\!08$$$$T_{2}^{291} +$$$$74\!\cdots\!65$$$$T_{2}^{290} +$$$$25\!\cdots\!66$$$$T_{2}^{289} -$$$$42\!\cdots\!68$$$$T_{2}^{288} -$$$$12\!\cdots\!32$$$$T_{2}^{287} -$$$$97\!\cdots\!10$$$$T_{2}^{286} +$$$$85\!\cdots\!24$$$$T_{2}^{285} +$$$$28\!\cdots\!88$$$$T_{2}^{284} -$$$$31\!\cdots\!69$$$$T_{2}^{283} -$$$$15\!\cdots\!50$$$$T_{2}^{282} -$$$$36\!\cdots\!45$$$$T_{2}^{281} -$$$$15\!\cdots\!14$$$$T_{2}^{280} +$$$$25\!\cdots\!12$$$$T_{2}^{279} +$$$$70\!\cdots\!45$$$$T_{2}^{278} -$$$$23\!\cdots\!81$$$$T_{2}^{277} -$$$$34\!\cdots\!41$$$$T_{2}^{276} -$$$$78\!\cdots\!57$$$$T_{2}^{275} -$$$$66\!\cdots\!39$$$$T_{2}^{274} +$$$$49\!\cdots\!74$$$$T_{2}^{273} +$$$$15\!\cdots\!11$$$$T_{2}^{272} +$$$$37\!\cdots\!31$$$$T_{2}^{271} -$$$$51\!\cdots\!92$$$$T_{2}^{270} -$$$$20\!\cdots\!22$$$$T_{2}^{269} -$$$$32\!\cdots\!06$$$$T_{2}^{268} +$$$$97\!\cdots\!61$$$$T_{2}^{267} +$$$$37\!\cdots\!10$$$$T_{2}^{266} +$$$$25\!\cdots\!62$$$$T_{2}^{265} -$$$$85\!\cdots\!05$$$$T_{2}^{264} -$$$$51\!\cdots\!13$$$$T_{2}^{263} -$$$$86\!\cdots\!21$$$$T_{2}^{262} +$$$$21\!\cdots\!43$$$$T_{2}^{261} +$$$$77\!\cdots\!34$$$$T_{2}^{260} +$$$$41\!\cdots\!69$$$$T_{2}^{259} -$$$$15\!\cdots\!81$$$$T_{2}^{258} -$$$$85\!\cdots\!98$$$$T_{2}^{257} -$$$$14\!\cdots\!42$$$$T_{2}^{256} +$$$$32\!\cdots\!00$$$$T_{2}^{255} +$$$$11\!\cdots\!94$$$$T_{2}^{254} +$$$$73\!\cdots\!29$$$$T_{2}^{253} -$$$$14\!\cdots\!20$$$$T_{2}^{252} -$$$$11\!\cdots\!51$$$$T_{2}^{251} -$$$$23\!\cdots\!19$$$$T_{2}^{250} +$$$$29\!\cdots\!05$$$$T_{2}^{249} +$$$$13\!\cdots\!68$$$$T_{2}^{248} +$$$$19\!\cdots\!74$$$$T_{2}^{247} +$$$$43\!\cdots\!92$$$$T_{2}^{246} -$$$$17\!\cdots\!83$$$$T_{2}^{245} -$$$$41\!\cdots\!69$$$$T_{2}^{244} +$$$$31\!\cdots\!81$$$$T_{2}^{243} +$$$$18\!\cdots\!15$$$$T_{2}^{242} +$$$$31\!\cdots\!99$$$$T_{2}^{241} +$$$$16\!\cdots\!26$$$$T_{2}^{240} -$$$$24\!\cdots\!91$$$$T_{2}^{239} -$$$$55\!\cdots\!09$$$$T_{2}^{238} +$$$$49\!\cdots\!69$$$$T_{2}^{237} +$$$$23\!\cdots\!12$$$$T_{2}^{236} +$$$$28\!\cdots\!82$$$$T_{2}^{235} +$$$$60\!\cdots\!84$$$$T_{2}^{234} -$$$$22\!\cdots\!40$$$$T_{2}^{233} -$$$$49\!\cdots\!30$$$$T_{2}^{232} +$$$$43\!\cdots\!18$$$$T_{2}^{231} +$$$$19\!\cdots\!60$$$$T_{2}^{230} +$$$$24\!\cdots\!53$$$$T_{2}^{229} +$$$$11\!\cdots\!22$$$$T_{2}^{228} -$$$$16\!\cdots\!12$$$$T_{2}^{227} -$$$$39\!\cdots\!85$$$$T_{2}^{226} +$$$$14\!\cdots\!88$$$$T_{2}^{225} +$$$$11\!\cdots\!31$$$$T_{2}^{224} +$$$$28\!\cdots\!77$$$$T_{2}^{223} +$$$$38\!\cdots\!46$$$$T_{2}^{222} -$$$$15\!\cdots\!18$$$$T_{2}^{221} -$$$$40\!\cdots\!70$$$$T_{2}^{220} +$$$$13\!\cdots\!29$$$$T_{2}^{219} +$$$$10\!\cdots\!82$$$$T_{2}^{218} +$$$$23\!\cdots\!19$$$$T_{2}^{217} +$$$$33\!\cdots\!79$$$$T_{2}^{216} -$$$$13\!\cdots\!98$$$$T_{2}^{215} -$$$$35\!\cdots\!84$$$$T_{2}^{214} +$$$$26\!\cdots\!96$$$$T_{2}^{213} +$$$$11\!\cdots\!61$$$$T_{2}^{212} +$$$$73\!\cdots\!06$$$$T_{2}^{211} +$$$$73\!\cdots\!85$$$$T_{2}^{210} -$$$$64\!\cdots\!21$$$$T_{2}^{209} -$$$$15\!\cdots\!37$$$$T_{2}^{208} +$$$$17\!\cdots\!60$$$$T_{2}^{207} +$$$$66\!\cdots\!41$$$$T_{2}^{206} +$$$$92\!\cdots\!42$$$$T_{2}^{205} -$$$$94\!\cdots\!93$$$$T_{2}^{204} -$$$$15\!\cdots\!74$$$$T_{2}^{203} -$$$$25\!\cdots\!08$$$$T_{2}^{202} +$$$$40\!\cdots\!47$$$$T_{2}^{201} +$$$$14\!\cdots\!39$$$$T_{2}^{200} +$$$$34\!\cdots\!76$$$$T_{2}^{199} -$$$$13\!\cdots\!99$$$$T_{2}^{198} -$$$$44\!\cdots\!00$$$$T_{2}^{197} -$$$$75\!\cdots\!34$$$$T_{2}^{196} +$$$$63\!\cdots\!50$$$$T_{2}^{195} +$$$$14\!\cdots\!66$$$$T_{2}^{194} +$$$$36\!\cdots\!35$$$$T_{2}^{193} +$$$$12\!\cdots\!77$$$$T_{2}^{192} -$$$$21\!\cdots\!01$$$$T_{2}^{191} -$$$$82\!\cdots\!98$$$$T_{2}^{190} +$$$$52\!\cdots\!21$$$$T_{2}^{189} +$$$$20\!\cdots\!49$$$$T_{2}^{188} -$$$$41\!\cdots\!68$$$$T_{2}^{187} -$$$$11\!\cdots\!58$$$$T_{2}^{186} -$$$$70\!\cdots\!93$$$$T_{2}^{185} -$$$$15\!\cdots\!54$$$$T_{2}^{184} +$$$$11\!\cdots\!16$$$$T_{2}^{183} +$$$$51\!\cdots\!69$$$$T_{2}^{182} +$$$$10\!\cdots\!58$$$$T_{2}^{181} -$$$$61\!\cdots\!95$$$$T_{2}^{180} -$$$$21\!\cdots\!20$$$$T_{2}^{179} -$$$$35\!\cdots\!72$$$$T_{2}^{178} -$$$$64\!\cdots\!95$$$$T_{2}^{177} -$$$$51\!\cdots\!07$$$$T_{2}^{176} +$$$$36\!\cdots\!77$$$$T_{2}^{175} +$$$$19\!\cdots\!37$$$$T_{2}^{174} -$$$$31\!\cdots\!46$$$$T_{2}^{173} -$$$$76\!\cdots\!40$$$$T_{2}^{172} -$$$$27\!\cdots\!54$$$$T_{2}^{171} +$$$$11\!\cdots\!39$$$$T_{2}^{170} +$$$$13\!\cdots\!78$$$$T_{2}^{169} +$$$$12\!\cdots\!97$$$$T_{2}^{168} -$$$$29\!\cdots\!49$$$$T_{2}^{167} -$$$$99\!\cdots\!37$$$$T_{2}^{166} +$$$$41\!\cdots\!81$$$$T_{2}^{165} +$$$$19\!\cdots\!08$$$$T_{2}^{164} -$$$$69\!\cdots\!63$$$$T_{2}^{163} -$$$$42\!\cdots\!83$$$$T_{2}^{162} +$$$$26\!\cdots\!72$$$$T_{2}^{161} -$$$$39\!\cdots\!18$$$$T_{2}^{160} -$$$$84\!\cdots\!24$$$$T_{2}^{159} -$$$$26\!\cdots\!77$$$$T_{2}^{158} +$$$$12\!\cdots\!03$$$$T_{2}^{157} +$$$$51\!\cdots\!34$$$$T_{2}^{156} +$$$$10\!\cdots\!10$$$$T_{2}^{155} -$$$$13\!\cdots\!62$$$$T_{2}^{154} -$$$$98\!\cdots\!70$$$$T_{2}^{153} +$$$$80\!\cdots\!02$$$$T_{2}^{152} +$$$$22\!\cdots\!14$$$$T_{2}^{151} +$$$$38\!\cdots\!97$$$$T_{2}^{150} -$$$$24\!\cdots\!61$$$$T_{2}^{149} -$$$$11\!\cdots\!17$$$$T_{2}^{148} +$$$$90\!\cdots\!96$$$$T_{2}^{147} +$$$$72\!\cdots\!12$$$$T_{2}^{146} -$$$$40\!\cdots\!90$$$$T_{2}^{145} +$$$$22\!\cdots\!88$$$$T_{2}^{144} +$$$$24\!\cdots\!04$$$$T_{2}^{143} -$$$$49\!\cdots\!15$$$$T_{2}^{142} -$$$$50\!\cdots\!20$$$$T_{2}^{141} -$$$$21\!\cdots\!11$$$$T_{2}^{140} +$$$$41\!\cdots\!79$$$$T_{2}^{139} +$$$$23\!\cdots\!55$$$$T_{2}^{138} +$$$$23\!\cdots\!01$$$$T_{2}^{137} -$$$$35\!\cdots\!26$$$$T_{2}^{136} -$$$$62\!\cdots\!27$$$$T_{2}^{135} -$$$$20\!\cdots\!59$$$$T_{2}^{134} +$$$$66\!\cdots\!75$$$$T_{2}^{133} +$$$$16\!\cdots\!39$$$$T_{2}^{132} +$$$$37\!\cdots\!57$$$$T_{2}^{131} -$$$$24\!\cdots\!13$$$$T_{2}^{130} -$$$$24\!\cdots\!64$$$$T_{2}^{129} +$$$$34\!\cdots\!30$$$$T_{2}^{128} +$$$$38\!\cdots\!67$$$$T_{2}^{127} +$$$$49\!\cdots\!47$$$$T_{2}^{126} -$$$$16\!\cdots\!22$$$$T_{2}^{125} -$$$$84\!\cdots\!88$$$$T_{2}^{124} -$$$$44\!\cdots\!74$$$$T_{2}^{123} +$$$$33\!\cdots\!74$$$$T_{2}^{122} +$$$$93\!\cdots\!48$$$$T_{2}^{121} +$$$$88\!\cdots\!90$$$$T_{2}^{120} -$$$$57\!\cdots\!93$$$$T_{2}^{119} -$$$$15\!\cdots\!44$$$$T_{2}^{118} -$$$$61\!\cdots\!01$$$$T_{2}^{117} +$$$$41\!\cdots\!07$$$$T_{2}^{116} +$$$$14\!\cdots\!41$$$$T_{2}^{115} +$$$$14\!\cdots\!58$$$$T_{2}^{114} -$$$$60\!\cdots\!46$$$$T_{2}^{113} -$$$$16\!\cdots\!80$$$$T_{2}^{112} -$$$$12\!\cdots\!96$$$$T_{2}^{111} -$$$$53\!\cdots\!48$$$$T_{2}^{110} +$$$$18\!\cdots\!22$$$$T_{2}^{109} +$$$$26\!\cdots\!38$$$$T_{2}^{108} -$$$$74\!\cdots\!08$$$$T_{2}^{107} -$$$$17\!\cdots\!49$$$$T_{2}^{106} -$$$$22\!\cdots\!27$$$$T_{2}^{105} -$$$$15\!\cdots\!05$$$$T_{2}^{104} +$$$$22\!\cdots\!82$$$$T_{2}^{103} +$$$$34\!\cdots\!97$$$$T_{2}^{102} +$$$$61\!\cdots\!91$$$$T_{2}^{101} -$$$$19\!\cdots\!83$$$$T_{2}^{100} -$$$$24\!\cdots\!17$$$$T_{2}^{99} -$$$$11\!\cdots\!45$$$$T_{2}^{98} +$$$$24\!\cdots\!07$$$$T_{2}^{97} +$$$$24\!\cdots\!99$$$$T_{2}^{96} -$$$$35\!\cdots\!75$$$$T_{2}^{95} -$$$$11\!\cdots\!25$$$$T_{2}^{94} -$$$$15\!\cdots\!54$$$$T_{2}^{93} -$$$$66\!\cdots\!11$$$$T_{2}^{92} +$$$$17\!\cdots\!64$$$$T_{2}^{91} +$$$$88\!\cdots\!99$$$$T_{2}^{90} -$$$$68\!\cdots\!14$$$$T_{2}^{89} +$$$$95\!\cdots\!76$$$$T_{2}^{88} -$$$$35\!\cdots\!70$$$$T_{2}^{87} -$$$$74\!\cdots\!74$$$$T_{2}^{86} +$$$$71\!\cdots\!45$$$$T_{2}^{85} +$$$$41\!\cdots\!17$$$$T_{2}^{84} -$$$$53\!\cdots\!94$$$$T_{2}^{83} +$$$$26\!\cdots\!18$$$$T_{2}^{82} +$$$$20\!\cdots\!25$$$$T_{2}^{81} -$$$$54\!\cdots\!36$$$$T_{2}^{80} +$$$$51\!\cdots\!25$$$$T_{2}^{79} +$$$$36\!\cdots\!26$$$$T_{2}^{78} -$$$$16\!\cdots\!03$$$$T_{2}^{77} -$$$$62\!\cdots\!71$$$$T_{2}^{76} +$$$$14\!\cdots\!70$$$$T_{2}^{75} -$$$$10\!\cdots\!70$$$$T_{2}^{74} -$$$$74\!\cdots\!68$$$$T_{2}^{73} +$$$$11\!\cdots\!38$$$$T_{2}^{72} +$$$$17\!\cdots\!25$$$$T_{2}^{71} -$$$$69\!\cdots\!64$$$$T_{2}^{70} +$$$$65\!\cdots\!72$$$$T_{2}^{69} +$$$$26\!\cdots\!70$$$$T_{2}^{68} -$$$$88\!\cdots\!79$$$$T_{2}^{67} -$$$$55\!\cdots\!11$$$$T_{2}^{66} +$$$$45\!\cdots\!81$$$$T_{2}^{65} -$$$$71\!\cdots\!96$$$$T_{2}^{64} -$$$$11\!\cdots\!24$$$$T_{2}^{63} +$$$$52\!\cdots\!98$$$$T_{2}^{62} +$$$$15\!\cdots\!38$$$$T_{2}^{61} -$$$$16\!\cdots\!74$$$$T_{2}^{60} +$$$$13\!\cdots\!30$$$$T_{2}^{59} +$$$$96\!\cdots\!97$$$$T_{2}^{58} -$$$$57\!\cdots\!02$$$$T_{2}^{57} +$$$$10\!\cdots\!43$$$$T_{2}^{56} +$$$$13\!\cdots\!19$$$$T_{2}^{55} -$$$$27\!\cdots\!39$$$$T_{2}^{54} -$$$$86\!\cdots\!75$$$$T_{2}^{53} +$$$$18\!\cdots\!23$$$$T_{2}^{52} -$$$$16\!\cdots\!47$$$$T_{2}^{51} +$$$$24\!\cdots\!09$$$$T_{2}^{50} +$$$$14\!\cdots\!15$$$$T_{2}^{49} -$$$$62\!\cdots\!52$$$$T_{2}^{48} +$$$$58\!\cdots\!60$$$$T_{2}^{47} +$$$$71\!\cdots\!03$$$$T_{2}^{46} +$$$$50\!\cdots\!19$$$$T_{2}^{45} +$$$$48\!\cdots\!99$$$$T_{2}^{44} +$$$$16\!\cdots\!37$$$$T_{2}^{43} +$$$$39\!\cdots\!21$$$$T_{2}^{42} +$$$$21\!\cdots\!66$$$$T_{2}^{41} +$$$$71\!\cdots\!67$$$$T_{2}^{40} +$$$$48\!\cdots\!36$$$$T_{2}^{39} +$$$$27\!\cdots\!44$$$$T_{2}^{38} +$$$$64\!\cdots\!28$$$$T_{2}^{37} +$$$$21\!\cdots\!69$$$$T_{2}^{36} +$$$$94\!\cdots\!85$$$$T_{2}^{35} -$$$$10\!\cdots\!79$$$$T_{2}^{34} -$$$$42\!\cdots\!92$$$$T_{2}^{33} -$$$$92\!\cdots\!39$$$$T_{2}^{32} -$$$$61\!\cdots\!28$$$$T_{2}^{31} +$$$$28\!\cdots\!37$$$$T_{2}^{30} +$$$$67\!\cdots\!67$$$$T_{2}^{29} +$$$$27\!\cdots\!08$$$$T_{2}^{28} -$$$$11\!\cdots\!52$$$$T_{2}^{27} +$$$$11\!\cdots\!82$$$$T_{2}^{26} +$$$$21\!\cdots\!34$$$$T_{2}^{25} +$$$$40\!\cdots\!03$$$$T_{2}^{24} +$$$$11\!\cdots\!56$$$$T_{2}^{23} -$$$$40\!\cdots\!61$$$$T_{2}^{22} -$$$$66\!\cdots\!48$$$$T_{2}^{21} -$$$$46\!\cdots\!16$$$$T_{2}^{20} -$$$$56\!\cdots\!20$$$$T_{2}^{19} -$$$$74\!\cdots\!55$$$$T_{2}^{18} +$$$$15\!\cdots\!67$$$$T_{2}^{17} +$$$$43\!\cdots\!16$$$$T_{2}^{16} +$$$$90\!\cdots\!27$$$$T_{2}^{15} -$$$$49\!\cdots\!29$$$$T_{2}^{14} -$$$$35\!\cdots\!52$$$$T_{2}^{13} +$$$$20\!\cdots\!55$$$$T_{2}^{12} +$$$$25\!\cdots\!92$$$$T_{2}^{11} -$$$$84\!\cdots\!10$$$$T_{2}^{10} -$$$$66\!\cdots\!30$$$$T_{2}^{9} +$$$$47\!\cdots\!99$$$$T_{2}^{8} +$$$$74\!\cdots\!40$$$$T_{2}^{7} -$$$$68\!\cdots\!63$$$$T_{2}^{6} -$$$$29\!\cdots\!88$$$$T_{2}^{5} +$$$$16\!\cdots\!77$$$$T_{2}^{4} -$$$$34\!\cdots\!12$$$$T_{2}^{3} +$$$$84\!\cdots\!47$$$$T_{2}^{2} -$$$$91\!\cdots\!12$$$$T_{2} +$$$$57\!\cdots\!81$$">$$T_{2}^{324} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.