# Properties

 Label 637.2.bl.b Level $637$ Weight $2$ Character orbit 637.bl Analytic conductor $5.086$ Analytic rank $0$ Dimension $348$ 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: $$348$$ Relative dimension: $$29$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$348 q + 3 q^{2} + 31 q^{4} + q^{5} - 4 q^{6} + 27 q^{7} - 12 q^{8} + 13 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$348 q + 3 q^{2} + 31 q^{4} + q^{5} - 4 q^{6} + 27 q^{7} - 12 q^{8} + 13 q^{9} - 5 q^{10} + 2 q^{11} - 40 q^{12} - 58 q^{13} - 45 q^{14} - 14 q^{15} + 27 q^{16} - 12 q^{17} - 59 q^{18} - 24 q^{19} - 99 q^{20} - 2 q^{21} - 10 q^{22} - 14 q^{23} + 28 q^{25} + 3 q^{26} - 9 q^{27} - 37 q^{28} - 26 q^{29} - 83 q^{30} - 49 q^{31} + 33 q^{32} + 2 q^{33} - 130 q^{34} - 38 q^{35} - 74 q^{36} - 4 q^{37} + 92 q^{38} + 24 q^{40} + 20 q^{41} - 52 q^{42} + 2 q^{43} + 226 q^{44} - 23 q^{45} + 71 q^{46} + 24 q^{47} + 86 q^{48} + 25 q^{49} + 114 q^{50} - 54 q^{51} + 31 q^{52} + 36 q^{53} + 75 q^{54} + 38 q^{55} + 126 q^{56} + 27 q^{57} - 33 q^{58} + q^{59} - 179 q^{60} - 15 q^{61} + 158 q^{62} - 5 q^{63} - 24 q^{64} + q^{65} - 150 q^{66} - 151 q^{67} + 140 q^{68} - 98 q^{69} - 126 q^{70} - 31 q^{71} - 149 q^{72} - 9 q^{73} + 39 q^{74} + 85 q^{75} - 81 q^{76} + 46 q^{77} - 25 q^{78} - 69 q^{79} - 64 q^{80} + 71 q^{81} - 52 q^{82} - q^{83} - 47 q^{84} + 62 q^{85} + 97 q^{86} + 150 q^{87} - 12 q^{88} + 23 q^{89} + 107 q^{90} - q^{91} - 43 q^{92} + 316 q^{93} - 86 q^{94} - 29 q^{95} + 177 q^{96} + 114 q^{97} - 336 q^{98} + 102 q^{99} + O(q^{100})$$

## 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 −2.30989 1.57486i 0.887307 + 0.823300i 2.12474 + 5.41374i −0.535995 + 0.165332i −0.753001 3.29911i 2.63410 0.247988i 2.37377 10.4002i −0.114700 1.53057i 1.49846 + 0.462215i
53.2 −2.18557 1.49010i −1.35107 1.25361i 1.82564 + 4.65166i 2.14986 0.663145i 1.08486 + 4.75308i −1.64665 + 2.07088i 1.76413 7.72914i 0.0296636 + 0.395833i −5.68682 1.75415i
53.3 −1.81899 1.24017i 0.714120 + 0.662606i 1.04004 + 2.64997i −1.77156 + 0.546453i −0.477236 2.09091i 0.521590 2.59383i 0.414817 1.81743i −0.153270 2.04525i 3.90015 + 1.20304i
53.4 −1.75157 1.19420i 2.41317 + 2.23910i 0.911203 + 2.32171i −3.83737 + 1.18367i −1.55291 6.80375i 1.44247 + 2.21795i 0.233090 1.02123i 0.585657 + 7.81504i 8.13496 + 2.50930i
53.5 −1.69736 1.15724i 1.28266 + 1.19014i 0.811150 + 2.06678i 0.653901 0.201702i −0.799869 3.50445i −2.34964 1.21622i 0.100685 0.441128i 0.00460791 + 0.0614883i −1.34333 0.414361i
53.6 −1.50446 1.02572i −0.944416 0.876290i 0.480605 + 1.22456i 1.77403 0.547215i 0.522004 + 2.28705i 2.64020 0.171310i −0.277345 + 1.21513i −0.100153 1.33645i −3.23024 0.996396i
53.7 −1.37622 0.938291i −0.821341 0.762093i 0.282909 + 0.720840i 1.09295 0.337131i 0.415281 + 1.81946i 1.05356 + 2.42693i −0.454268 + 1.99028i −0.130375 1.73974i −1.82047 0.561540i
53.8 −1.17499 0.801092i 2.33099 + 2.16284i 0.00816248 + 0.0207977i 3.42079 1.05518i −1.00624 4.40864i 1.74821 1.98590i −0.625820 + 2.74190i 0.531434 + 7.09150i −4.86468 1.50055i
53.9 −1.01819 0.694189i 0.299706 + 0.278087i −0.175873 0.448118i −3.01602 + 0.930319i −0.112113 0.491197i −1.93482 + 1.80457i −0.680439 + 2.98120i −0.211699 2.82492i 3.71669 + 1.14645i
53.10 −0.929618 0.633803i −1.91082 1.77298i −0.268199 0.683359i 3.66253 1.12974i 0.652612 + 2.85928i −1.33043 2.28691i −0.684518 + 2.99907i 0.283578 + 3.78409i −4.12078 1.27109i
53.11 −0.610286 0.416086i −2.18416 2.02661i −0.531361 1.35388i −2.47946 + 0.764813i 0.489722 + 2.14561i 1.83554 1.90546i −0.567772 + 2.48757i 0.439242 + 5.86128i 1.83141 + 0.564915i
53.12 −0.496066 0.338212i −0.453403 0.420696i −0.598988 1.52620i −1.06776 + 0.329360i 0.0826330 + 0.362039i −1.07420 2.41787i −0.486239 + 2.13035i −0.195602 2.61012i 0.641072 + 0.197745i
53.13 −0.296505 0.202154i 1.23401 + 1.14499i −0.683633 1.74187i 1.06645 0.328957i −0.134425 0.588954i 2.51862 0.810284i −0.309133 + 1.35440i −0.0124224 0.165766i −0.382709 0.118050i
53.14 0.00745244 + 0.00508099i −2.32094 2.15352i −0.730652 1.86167i 2.27283 0.701076i −0.00635467 0.0278416i −1.04784 + 2.42941i 0.00802814 0.0351736i 0.524933 + 7.00475i 0.0205003 + 0.00632351i
53.15 0.168293 + 0.114740i 2.17291 + 2.01616i −0.715525 1.82313i 0.954892 0.294545i 0.134350 + 0.588628i −0.383890 + 2.61775i 0.179417 0.786079i 0.432422 + 5.77027i 0.194498 + 0.0599948i
53.16 0.371774 + 0.253471i −0.182364 0.169209i −0.656714 1.67328i 3.18926 0.983757i −0.0249085 0.109131i 2.51832 0.811213i 0.380230 1.66590i −0.219565 2.92990i 1.43504 + 0.442650i
53.17 0.414790 + 0.282799i −0.401269 0.372324i −0.638607 1.62714i −2.35807 + 0.727368i −0.0611497 0.267915i 0.971142 + 2.46107i 0.418687 1.83439i −0.201798 2.69281i −1.18380 0.365154i
53.18 0.445939 + 0.304036i 0.759470 + 0.704685i −0.624259 1.59058i −1.39712 + 0.430954i 0.124428 + 0.545153i −2.52342 0.795220i 0.445412 1.95148i −0.143977 1.92124i −0.754055 0.232595i
53.19 0.655312 + 0.446784i −1.79642 1.66683i −0.500865 1.27618i −0.752346 + 0.232068i −0.432499 1.89490i 2.24469 + 1.40049i 0.594930 2.60656i 0.224596 + 2.99702i −0.596706 0.184059i
53.20 1.12577 + 0.767535i −0.0507932 0.0471292i −0.0524406 0.133616i 0.822051 0.253569i −0.0210080 0.0920421i −2.32385 + 1.26480i 0.649898 2.84739i −0.223831 2.98682i 1.12006 + 0.345493i
See next 80 embeddings (of 348 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 625.29 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.b 348
49.g even 21 1 inner 637.2.bl.b 348

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bl.b 348 1.a even 1 1 trivial
637.2.bl.b 348 49.g even 21 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$39\!\cdots\!45$$$$T_{2}^{324} +$$$$89\!\cdots\!80$$$$T_{2}^{323} -$$$$22\!\cdots\!69$$$$T_{2}^{322} -$$$$45\!\cdots\!47$$$$T_{2}^{321} +$$$$27\!\cdots\!96$$$$T_{2}^{320} -$$$$30\!\cdots\!47$$$$T_{2}^{319} -$$$$17\!\cdots\!71$$$$T_{2}^{318} +$$$$54\!\cdots\!23$$$$T_{2}^{317} -$$$$56\!\cdots\!04$$$$T_{2}^{316} -$$$$21\!\cdots\!12$$$$T_{2}^{315} +$$$$15\!\cdots\!19$$$$T_{2}^{314} -$$$$17\!\cdots\!84$$$$T_{2}^{313} -$$$$77\!\cdots\!33$$$$T_{2}^{312} +$$$$26\!\cdots\!22$$$$T_{2}^{311} -$$$$29\!\cdots\!54$$$$T_{2}^{310} -$$$$10\!\cdots\!52$$$$T_{2}^{309} +$$$$62\!\cdots\!64$$$$T_{2}^{308} -$$$$57\!\cdots\!42$$$$T_{2}^{307} -$$$$27\!\cdots\!67$$$$T_{2}^{306} +$$$$84\!\cdots\!06$$$$T_{2}^{305} -$$$$11\!\cdots\!54$$$$T_{2}^{304} -$$$$27\!\cdots\!37$$$$T_{2}^{303} +$$$$19\!\cdots\!48$$$$T_{2}^{302} -$$$$19\!\cdots\!27$$$$T_{2}^{301} -$$$$67\!\cdots\!09$$$$T_{2}^{300} +$$$$22\!\cdots\!28$$$$T_{2}^{299} -$$$$47\!\cdots\!21$$$$T_{2}^{298} -$$$$40\!\cdots\!65$$$$T_{2}^{297} +$$$$55\!\cdots\!44$$$$T_{2}^{296} -$$$$75\!\cdots\!94$$$$T_{2}^{295} -$$$$10\!\cdots\!16$$$$T_{2}^{294} +$$$$59\!\cdots\!16$$$$T_{2}^{293} -$$$$17\!\cdots\!51$$$$T_{2}^{292} -$$$$27\!\cdots\!65$$$$T_{2}^{291} +$$$$14\!\cdots\!22$$$$T_{2}^{290} -$$$$22\!\cdots\!91$$$$T_{2}^{289} -$$$$10\!\cdots\!35$$$$T_{2}^{288} +$$$$13\!\cdots\!54$$$$T_{2}^{287} -$$$$47\!\cdots\!96$$$$T_{2}^{286} +$$$$70\!\cdots\!77$$$$T_{2}^{285} +$$$$29\!\cdots\!12$$$$T_{2}^{284} -$$$$50\!\cdots\!24$$$$T_{2}^{283} +$$$$13\!\cdots\!94$$$$T_{2}^{282} +$$$$22\!\cdots\!40$$$$T_{2}^{281} -$$$$10\!\cdots\!66$$$$T_{2}^{280} +$$$$58\!\cdots\!25$$$$T_{2}^{279} +$$$$50\!\cdots\!74$$$$T_{2}^{278} -$$$$10\!\cdots\!46$$$$T_{2}^{277} +$$$$10\!\cdots\!16$$$$T_{2}^{276} +$$$$26\!\cdots\!40$$$$T_{2}^{275} -$$$$21\!\cdots\!57$$$$T_{2}^{274} +$$$$20\!\cdots\!03$$$$T_{2}^{273} +$$$$68\!\cdots\!47$$$$T_{2}^{272} -$$$$18\!\cdots\!55$$$$T_{2}^{271} +$$$$32\!\cdots\!21$$$$T_{2}^{270} +$$$$25\!\cdots\!50$$$$T_{2}^{269} -$$$$35\!\cdots\!71$$$$T_{2}^{268} +$$$$45\!\cdots\!93$$$$T_{2}^{267} +$$$$69\!\cdots\!35$$$$T_{2}^{266} -$$$$29\!\cdots\!42$$$$T_{2}^{265} +$$$$68\!\cdots\!44$$$$T_{2}^{264} +$$$$78\!\cdots\!08$$$$T_{2}^{263} -$$$$49\!\cdots\!56$$$$T_{2}^{262} +$$$$74\!\cdots\!20$$$$T_{2}^{261} +$$$$35\!\cdots\!00$$$$T_{2}^{260} -$$$$34\!\cdots\!14$$$$T_{2}^{259} +$$$$11\!\cdots\!39$$$$T_{2}^{258} -$$$$47\!\cdots\!99$$$$T_{2}^{257} -$$$$56\!\cdots\!89$$$$T_{2}^{256} +$$$$10\!\cdots\!55$$$$T_{2}^{255} -$$$$45\!\cdots\!24$$$$T_{2}^{254} -$$$$31\!\cdots\!70$$$$T_{2}^{253} +$$$$15\!\cdots\!28$$$$T_{2}^{252} -$$$$13\!\cdots\!68$$$$T_{2}^{251} -$$$$50\!\cdots\!71$$$$T_{2}^{250} +$$$$13\!\cdots\!46$$$$T_{2}^{249} -$$$$15\!\cdots\!71$$$$T_{2}^{248} -$$$$23\!\cdots\!17$$$$T_{2}^{247} +$$$$17\!\cdots\!83$$$$T_{2}^{246} -$$$$19\!\cdots\!20$$$$T_{2}^{245} -$$$$35\!\cdots\!00$$$$T_{2}^{244} +$$$$13\!\cdots\!24$$$$T_{2}^{243} -$$$$23\!\cdots\!91$$$$T_{2}^{242} -$$$$99\!\cdots\!28$$$$T_{2}^{241} +$$$$16\!\cdots\!53$$$$T_{2}^{240} -$$$$22\!\cdots\!74$$$$T_{2}^{239} -$$$$12\!\cdots\!99$$$$T_{2}^{238} +$$$$10\!\cdots\!67$$$$T_{2}^{237} -$$$$26\!\cdots\!06$$$$T_{2}^{236} +$$$$62\!\cdots\!64$$$$T_{2}^{235} +$$$$11\!\cdots\!96$$$$T_{2}^{234} -$$$$21\!\cdots\!18$$$$T_{2}^{233} +$$$$73\!\cdots\!61$$$$T_{2}^{232} +$$$$66\!\cdots\!70$$$$T_{2}^{231} -$$$$23\!\cdots\!00$$$$T_{2}^{230} +$$$$15\!\cdots\!15$$$$T_{2}^{229} +$$$$69\!\cdots\!70$$$$T_{2}^{228} -$$$$17\!\cdots\!04$$$$T_{2}^{227} +$$$$16\!\cdots\!35$$$$T_{2}^{226} +$$$$35\!\cdots\!99$$$$T_{2}^{225} -$$$$17\!\cdots\!98$$$$T_{2}^{224} +$$$$15\!\cdots\!07$$$$T_{2}^{223} +$$$$31\!\cdots\!56$$$$T_{2}^{222} -$$$$10\!\cdots\!38$$$$T_{2}^{221} +$$$$15\!\cdots\!78$$$$T_{2}^{220} +$$$$13\!\cdots\!77$$$$T_{2}^{219} -$$$$96\!\cdots\!41$$$$T_{2}^{218} +$$$$11\!\cdots\!56$$$$T_{2}^{217} +$$$$89\!\cdots\!55$$$$T_{2}^{216} -$$$$53\!\cdots\!85$$$$T_{2}^{215} +$$$$10\!\cdots\!11$$$$T_{2}^{214} +$$$$15\!\cdots\!95$$$$T_{2}^{213} -$$$$41\!\cdots\!32$$$$T_{2}^{212} +$$$$63\!\cdots\!75$$$$T_{2}^{211} -$$$$58\!\cdots\!42$$$$T_{2}^{210} -$$$$20\!\cdots\!07$$$$T_{2}^{209} +$$$$53\!\cdots\!30$$$$T_{2}^{208} -$$$$16\!\cdots\!70$$$$T_{2}^{207} -$$$$15\!\cdots\!10$$$$T_{2}^{206} +$$$$30\!\cdots\!20$$$$T_{2}^{205} -$$$$19\!\cdots\!94$$$$T_{2}^{204} -$$$$73\!\cdots\!15$$$$T_{2}^{203} +$$$$22\!\cdots\!98$$$$T_{2}^{202} -$$$$11\!\cdots\!53$$$$T_{2}^{201} -$$$$46\!\cdots\!82$$$$T_{2}^{200} +$$$$11\!\cdots\!92$$$$T_{2}^{199} -$$$$97\!\cdots\!20$$$$T_{2}^{198} -$$$$22\!\cdots\!30$$$$T_{2}^{197} +$$$$72\!\cdots\!15$$$$T_{2}^{196} -$$$$39\!\cdots\!11$$$$T_{2}^{195} -$$$$11\!\cdots\!99$$$$T_{2}^{194} +$$$$30\!\cdots\!99$$$$T_{2}^{193} -$$$$31\!\cdots\!54$$$$T_{2}^{192} -$$$$48\!\cdots\!27$$$$T_{2}^{191} +$$$$16\!\cdots\!85$$$$T_{2}^{190} -$$$$11\!\cdots\!64$$$$T_{2}^{189} -$$$$15\!\cdots\!08$$$$T_{2}^{188} +$$$$62\!\cdots\!74$$$$T_{2}^{187} -$$$$90\!\cdots\!55$$$$T_{2}^{186} -$$$$59\!\cdots\!86$$$$T_{2}^{185} +$$$$30\!\cdots\!47$$$$T_{2}^{184} -$$$$32\!\cdots\!24$$$$T_{2}^{183} +$$$$47\!\cdots\!49$$$$T_{2}^{182} +$$$$11\!\cdots\!98$$$$T_{2}^{181} -$$$$23\!\cdots\!73$$$$T_{2}^{180} -$$$$37\!\cdots\!66$$$$T_{2}^{179} +$$$$53\!\cdots\!14$$$$T_{2}^{178} -$$$$76\!\cdots\!57$$$$T_{2}^{177} +$$$$32\!\cdots\!71$$$$T_{2}^{176} +$$$$21\!\cdots\!10$$$$T_{2}^{175} -$$$$41\!\cdots\!88$$$$T_{2}^{174} -$$$$40\!\cdots\!88$$$$T_{2}^{173} +$$$$75\!\cdots\!92$$$$T_{2}^{172} -$$$$11\!\cdots\!21$$$$T_{2}^{171} +$$$$67\!\cdots\!81$$$$T_{2}^{170} +$$$$28\!\cdots\!99$$$$T_{2}^{169} -$$$$52\!\cdots\!78$$$$T_{2}^{168} +$$$$12\!\cdots\!37$$$$T_{2}^{167} +$$$$62\!\cdots\!33$$$$T_{2}^{166} -$$$$14\!\cdots\!76$$$$T_{2}^{165} +$$$$16\!\cdots\!35$$$$T_{2}^{164} +$$$$25\!\cdots\!69$$$$T_{2}^{163} -$$$$64\!\cdots\!76$$$$T_{2}^{162} +$$$$25\!\cdots\!17$$$$T_{2}^{161} +$$$$39\!\cdots\!12$$$$T_{2}^{160} -$$$$19\!\cdots\!35$$$$T_{2}^{159} +$$$$26\!\cdots\!49$$$$T_{2}^{158} +$$$$27\!\cdots\!75$$$$T_{2}^{157} -$$$$80\!\cdots\!42$$$$T_{2}^{156} +$$$$32\!\cdots\!91$$$$T_{2}^{155} +$$$$57\!\cdots\!55$$$$T_{2}^{154} -$$$$19\!\cdots\!23$$$$T_{2}^{153} +$$$$22\!\cdots\!44$$$$T_{2}^{152} +$$$$28\!\cdots\!81$$$$T_{2}^{151} -$$$$70\!\cdots\!89$$$$T_{2}^{150} +$$$$14\!\cdots\!19$$$$T_{2}^{149} +$$$$67\!\cdots\!86$$$$T_{2}^{148} -$$$$12\!\cdots\!43$$$$T_{2}^{147} +$$$$98\!\cdots\!32$$$$T_{2}^{146} +$$$$18\!\cdots\!02$$$$T_{2}^{145} -$$$$38\!\cdots\!52$$$$T_{2}^{144} +$$$$32\!\cdots\!89$$$$T_{2}^{143} +$$$$39\!\cdots\!21$$$$T_{2}^{142} -$$$$54\!\cdots\!37$$$$T_{2}^{141} +$$$$34\!\cdots\!63$$$$T_{2}^{140} +$$$$68\!\cdots\!23$$$$T_{2}^{139} -$$$$14\!\cdots\!19$$$$T_{2}^{138} +$$$$42\!\cdots\!97$$$$T_{2}^{137} +$$$$10\!\cdots\!74$$$$T_{2}^{136} -$$$$22\!\cdots\!21$$$$T_{2}^{135} +$$$$20\!\cdots\!36$$$$T_{2}^{134} +$$$$16\!\cdots\!39$$$$T_{2}^{133} -$$$$54\!\cdots\!30$$$$T_{2}^{132} +$$$$40\!\cdots\!59$$$$T_{2}^{131} +$$$$22\!\cdots\!84$$$$T_{2}^{130} -$$$$10\!\cdots\!33$$$$T_{2}^{129} +$$$$10\!\cdots\!18$$$$T_{2}^{128} +$$$$65\!\cdots\!92$$$$T_{2}^{127} -$$$$22\!\cdots\!79$$$$T_{2}^{126} +$$$$14\!\cdots\!41$$$$T_{2}^{125} +$$$$12\!\cdots\!42$$$$T_{2}^{124} -$$$$38\!\cdots\!67$$$$T_{2}^{123} +$$$$25\!\cdots\!77$$$$T_{2}^{122} +$$$$31\!\cdots\!13$$$$T_{2}^{121} -$$$$62\!\cdots\!84$$$$T_{2}^{120} +$$$$15\!\cdots\!05$$$$T_{2}^{119} +$$$$53\!\cdots\!82$$$$T_{2}^{118} -$$$$69\!\cdots\!32$$$$T_{2}^{117} +$$$$83\!\cdots\!10$$$$T_{2}^{116} +$$$$75\!\cdots\!41$$$$T_{2}^{115} -$$$$73\!\cdots\!94$$$$T_{2}^{114} -$$$$19\!\cdots\!06$$$$T_{2}^{113} +$$$$83\!\cdots\!00$$$$T_{2}^{112} -$$$$45\!\cdots\!57$$$$T_{2}^{111} -$$$$35\!\cdots\!57$$$$T_{2}^{110} +$$$$61\!\cdots\!99$$$$T_{2}^{109} -$$$$15\!\cdots\!03$$$$T_{2}^{108} -$$$$27\!\cdots\!25$$$$T_{2}^{107} +$$$$24\!\cdots\!95$$$$T_{2}^{106} -$$$$13\!\cdots\!64$$$$T_{2}^{105} -$$$$30\!\cdots\!35$$$$T_{2}^{104} +$$$$22\!\cdots\!62$$$$T_{2}^{103} -$$$$10\!\cdots\!16$$$$T_{2}^{102} -$$$$79\!\cdots\!86$$$$T_{2}^{101} -$$$$16\!\cdots\!38$$$$T_{2}^{100} -$$$$18\!\cdots\!28$$$$T_{2}^{99} +$$$$22\!\cdots\!33$$$$T_{2}^{98} -$$$$62\!\cdots\!67$$$$T_{2}^{97} -$$$$28\!\cdots\!70$$$$T_{2}^{96} +$$$$20\!\cdots\!18$$$$T_{2}^{95} +$$$$18\!\cdots\!89$$$$T_{2}^{94} -$$$$26\!\cdots\!16$$$$T_{2}^{93} -$$$$72\!\cdots\!50$$$$T_{2}^{92} +$$$$19\!\cdots\!21$$$$T_{2}^{91} +$$$$32\!\cdots\!45$$$$T_{2}^{90} -$$$$10\!\cdots\!27$$$$T_{2}^{89} -$$$$42\!\cdots\!46$$$$T_{2}^{88} +$$$$32\!\cdots\!39$$$$T_{2}^{87} +$$$$48\!\cdots\!65$$$$T_{2}^{86} -$$$$36\!\cdots\!64$$$$T_{2}^{85} -$$$$35\!\cdots\!47$$$$T_{2}^{84} -$$$$99\!\cdots\!21$$$$T_{2}^{83} +$$$$17\!\cdots\!85$$$$T_{2}^{82} +$$$$10\!\cdots\!89$$$$T_{2}^{81} -$$$$72\!\cdots\!40$$$$T_{2}^{80} -$$$$91\!\cdots\!05$$$$T_{2}^{79} +$$$$39\!\cdots\!16$$$$T_{2}^{78} +$$$$73\!\cdots\!63$$$$T_{2}^{77} -$$$$28\!\cdots\!08$$$$T_{2}^{76} -$$$$54\!\cdots\!53$$$$T_{2}^{75} +$$$$16\!\cdots\!15$$$$T_{2}^{74} +$$$$35\!\cdots\!42$$$$T_{2}^{73} -$$$$50\!\cdots\!12$$$$T_{2}^{72} -$$$$18\!\cdots\!25$$$$T_{2}^{71} -$$$$85\!\cdots\!58$$$$T_{2}^{70} +$$$$76\!\cdots\!36$$$$T_{2}^{69} +$$$$18\!\cdots\!20$$$$T_{2}^{68} -$$$$22\!\cdots\!95$$$$T_{2}^{67} -$$$$10\!\cdots\!28$$$$T_{2}^{66} +$$$$44\!\cdots\!08$$$$T_{2}^{65} +$$$$37\!\cdots\!30$$$$T_{2}^{64} -$$$$34\!\cdots\!29$$$$T_{2}^{63} -$$$$85\!\cdots\!04$$$$T_{2}^{62} -$$$$97\!\cdots\!69$$$$T_{2}^{61} +$$$$98\!\cdots\!98$$$$T_{2}^{60} +$$$$34\!\cdots\!78$$$$T_{2}^{59} +$$$$85\!\cdots\!09$$$$T_{2}^{58} -$$$$29\!\cdots\!91$$$$T_{2}^{57} -$$$$57\!\cdots\!93$$$$T_{2}^{56} -$$$$86\!\cdots\!02$$$$T_{2}^{55} +$$$$10\!\cdots\!67$$$$T_{2}^{54} +$$$$22\!\cdots\!97$$$$T_{2}^{53} -$$$$19\!\cdots\!70$$$$T_{2}^{52} +$$$$26\!\cdots\!22$$$$T_{2}^{51} -$$$$80\!\cdots\!21$$$$T_{2}^{50} -$$$$12\!\cdots\!08$$$$T_{2}^{49} -$$$$27\!\cdots\!77$$$$T_{2}^{48} +$$$$11\!\cdots\!51$$$$T_{2}^{47} +$$$$21\!\cdots\!49$$$$T_{2}^{46} +$$$$28\!\cdots\!11$$$$T_{2}^{45} -$$$$32\!\cdots\!81$$$$T_{2}^{44} -$$$$12\!\cdots\!08$$$$T_{2}^{43} -$$$$12\!\cdots\!03$$$$T_{2}^{42} +$$$$68\!\cdots\!47$$$$T_{2}^{41} +$$$$18\!\cdots\!54$$$$T_{2}^{40} +$$$$51\!\cdots\!54$$$$T_{2}^{39} -$$$$19\!\cdots\!43$$$$T_{2}^{38} -$$$$48\!\cdots\!07$$$$T_{2}^{37} +$$$$12\!\cdots\!83$$$$T_{2}^{36} -$$$$81\!\cdots\!19$$$$T_{2}^{35} +$$$$20\!\cdots\!89$$$$T_{2}^{34} +$$$$19\!\cdots\!88$$$$T_{2}^{33} +$$$$36\!\cdots\!01$$$$T_{2}^{32} -$$$$21\!\cdots\!10$$$$T_{2}^{31} -$$$$11\!\cdots\!62$$$$T_{2}^{30} -$$$$25\!\cdots\!84$$$$T_{2}^{29} +$$$$90\!\cdots\!48$$$$T_{2}^{28} +$$$$65\!\cdots\!83$$$$T_{2}^{27} +$$$$12\!\cdots\!90$$$$T_{2}^{26} +$$$$58\!\cdots\!47$$$$T_{2}^{25} -$$$$97\!\cdots\!96$$$$T_{2}^{24} -$$$$26\!\cdots\!86$$$$T_{2}^{23} +$$$$30\!\cdots\!75$$$$T_{2}^{22} +$$$$77\!\cdots\!67$$$$T_{2}^{21} +$$$$17\!\cdots\!36$$$$T_{2}^{20} +$$$$23\!\cdots\!53$$$$T_{2}^{19} -$$$$95\!\cdots\!47$$$$T_{2}^{18} -$$$$25\!\cdots\!08$$$$T_{2}^{17} +$$$$14\!\cdots\!44$$$$T_{2}^{16} +$$$$82\!\cdots\!39$$$$T_{2}^{15} +$$$$66\!\cdots\!91$$$$T_{2}^{14} +$$$$17\!\cdots\!26$$$$T_{2}^{13} -$$$$43\!\cdots\!26$$$$T_{2}^{12} -$$$$19\!\cdots\!60$$$$T_{2}^{11} -$$$$15\!\cdots\!89$$$$T_{2}^{10} +$$$$22\!\cdots\!10$$$$T_{2}^{9} +$$$$26\!\cdots\!16$$$$T_{2}^{8} -$$$$12\!\cdots\!40$$$$T_{2}^{7} -$$$$11\!\cdots\!67$$$$T_{2}^{6} +$$$$26\!\cdots\!72$$$$T_{2}^{5} +$$$$17\!\cdots\!48$$$$T_{2}^{4} -$$$$85\!\cdots\!70$$$$T_{2}^{3} +$$$$13\!\cdots\!40$$$$T_{2}^{2} -$$$$20\!\cdots\!12$$$$T_{2} +$$$$12\!\cdots\!41$$">$$T_{2}^{348} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.