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

Downloads

Learn more

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:

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