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 $2$

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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:

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 \(S_{2}^{\mathrm{new}}(637, [\chi])\).