Properties

Label 966.2.be.a
Level $966$
Weight $2$
Character orbit 966.be
Analytic conductor $7.714$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.be (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(16\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320q - 16q^{2} + 16q^{4} + 32q^{8} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 320q - 16q^{2} + 16q^{4} + 32q^{8} - 16q^{9} - 22q^{14} + 16q^{16} + 66q^{17} + 16q^{18} + 40q^{23} - 48q^{25} + 12q^{26} + 44q^{28} - 24q^{29} + 24q^{31} - 16q^{32} + 98q^{35} + 32q^{36} - 22q^{37} - 66q^{38} - 8q^{39} - 88q^{43} + 4q^{46} - 144q^{47} - 24q^{49} + 80q^{50} - 22q^{51} + 12q^{52} + 44q^{53} + 44q^{57} + 10q^{58} + 12q^{59} - 32q^{64} + 108q^{70} - 16q^{71} + 16q^{72} - 180q^{73} - 22q^{74} - 12q^{75} + 18q^{77} - 16q^{78} + 44q^{79} + 16q^{81} + 36q^{82} + 22q^{84} + 68q^{85} - 22q^{86} + 48q^{87} + 22q^{88} + 8q^{92} + 8q^{93} - 12q^{94} + 66q^{95} - 90q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −3.45344 + 1.78037i −0.755750 + 0.654861i −2.24314 + 1.40298i 0.959493 + 0.281733i −0.0475819 0.998867i −3.60703 + 1.44404i
19.2 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −2.25010 + 1.16001i −0.755750 + 0.654861i 1.37319 2.26149i 0.959493 + 0.281733i −0.0475819 0.998867i −2.35018 + 0.940871i
19.3 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −1.41181 + 0.727837i −0.755750 + 0.654861i 1.71541 + 2.01429i 0.959493 + 0.281733i −0.0475819 0.998867i −1.47460 + 0.590341i
19.4 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −0.290919 + 0.149979i −0.755750 + 0.654861i −0.738752 2.54052i 0.959493 + 0.281733i −0.0475819 0.998867i −0.303858 + 0.121646i
19.5 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 0.631353 0.325485i −0.755750 + 0.654861i 2.64138 0.152057i 0.959493 + 0.281733i −0.0475819 0.998867i 0.659433 0.263997i
19.6 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 1.05593 0.544372i −0.755750 + 0.654861i −2.51151 + 0.832044i 0.959493 + 0.281733i −0.0475819 0.998867i 1.10290 0.441534i
19.7 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 3.01447 1.55407i −0.755750 + 0.654861i 1.02282 2.44005i 0.959493 + 0.281733i −0.0475819 0.998867i 3.14854 1.26049i
19.8 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 3.18053 1.63968i −0.755750 + 0.654861i 1.02341 + 2.43980i 0.959493 + 0.281733i −0.0475819 0.998867i 3.32199 1.32992i
19.9 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −3.73421 + 1.92512i 0.755750 0.654861i −0.120285 2.64302i 0.959493 + 0.281733i −0.0475819 0.998867i −3.90030 + 1.56144i
19.10 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −3.26423 + 1.68283i 0.755750 0.654861i 2.33879 + 1.23696i 0.959493 + 0.281733i −0.0475819 0.998867i −3.40941 + 1.36492i
19.11 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −1.59513 + 0.822347i 0.755750 0.654861i −1.53650 + 2.15387i 0.959493 + 0.281733i −0.0475819 0.998867i −1.66608 + 0.666996i
19.12 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −0.867285 + 0.447116i 0.755750 0.654861i 2.46361 0.964688i 0.959493 + 0.281733i −0.0475819 0.998867i −0.905859 + 0.362651i
19.13 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 0.525481 0.270904i 0.755750 0.654861i −0.753581 2.53616i 0.959493 + 0.281733i −0.0475819 0.998867i 0.548853 0.219728i
19.14 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 1.83046 0.943666i 0.755750 0.654861i −2.58469 0.565134i 0.959493 + 0.281733i −0.0475819 0.998867i 1.91187 0.765397i
19.15 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 2.34527 1.20907i 0.755750 0.654861i 2.38412 1.14715i 0.959493 + 0.281733i −0.0475819 0.998867i 2.44958 0.980664i
19.16 0.995472 + 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 2.54871 1.31395i 0.755750 0.654861i −0.0690376 + 2.64485i 0.959493 + 0.281733i −0.0475819 0.998867i 2.66207 1.06573i
61.1 0.786053 + 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −2.60993 + 0.503023i −0.281733 0.959493i −0.258273 2.63312i −0.415415 + 0.909632i 0.327068 + 0.945001i −2.36249 1.21795i
61.2 0.786053 + 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −2.19052 + 0.422188i −0.281733 0.959493i 1.40168 + 2.24395i −0.415415 + 0.909632i 0.327068 + 0.945001i −1.98285 1.02223i
61.3 0.786053 + 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −1.06718 + 0.205682i −0.281733 0.959493i −1.24125 + 2.33651i −0.415415 + 0.909632i 0.327068 + 0.945001i −0.966005 0.498010i
61.4 0.786053 + 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −0.436006 + 0.0840333i −0.281733 0.959493i −2.27972 1.34272i −0.415415 + 0.909632i 0.327068 + 0.945001i −0.394670 0.203466i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 871.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.be.a 320
7.d odd 6 1 inner 966.2.be.a 320
23.d odd 22 1 inner 966.2.be.a 320
161.o even 66 1 inner 966.2.be.a 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.be.a 320 1.a even 1 1 trivial
966.2.be.a 320 7.d odd 6 1 inner
966.2.be.a 320 23.d odd 22 1 inner
966.2.be.a 320 161.o even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(29\!\cdots\!34\)\( T_{5}^{301} - \)\(56\!\cdots\!09\)\( T_{5}^{300} - \)\(19\!\cdots\!44\)\( T_{5}^{299} + \)\(15\!\cdots\!45\)\( T_{5}^{298} - \)\(32\!\cdots\!00\)\( T_{5}^{297} - \)\(20\!\cdots\!45\)\( T_{5}^{296} + \)\(10\!\cdots\!48\)\( T_{5}^{295} - \)\(10\!\cdots\!90\)\( T_{5}^{294} - \)\(17\!\cdots\!40\)\( T_{5}^{293} + \)\(63\!\cdots\!98\)\( T_{5}^{292} + \)\(68\!\cdots\!74\)\( T_{5}^{291} - \)\(93\!\cdots\!82\)\( T_{5}^{290} + \)\(38\!\cdots\!06\)\( T_{5}^{289} + \)\(81\!\cdots\!31\)\( T_{5}^{288} - \)\(86\!\cdots\!62\)\( T_{5}^{287} + \)\(64\!\cdots\!87\)\( T_{5}^{286} + \)\(58\!\cdots\!28\)\( T_{5}^{285} - \)\(40\!\cdots\!17\)\( T_{5}^{284} + \)\(47\!\cdots\!16\)\( T_{5}^{283} + \)\(65\!\cdots\!58\)\( T_{5}^{282} - \)\(19\!\cdots\!98\)\( T_{5}^{281} - \)\(18\!\cdots\!82\)\( T_{5}^{280} + \)\(30\!\cdots\!32\)\( T_{5}^{279} - \)\(92\!\cdots\!39\)\( T_{5}^{278} - \)\(22\!\cdots\!50\)\( T_{5}^{277} + \)\(18\!\cdots\!62\)\( T_{5}^{276} - \)\(22\!\cdots\!16\)\( T_{5}^{275} - \)\(14\!\cdots\!72\)\( T_{5}^{274} + \)\(84\!\cdots\!44\)\( T_{5}^{273} - \)\(27\!\cdots\!75\)\( T_{5}^{272} - \)\(97\!\cdots\!98\)\( T_{5}^{271} + \)\(25\!\cdots\!85\)\( T_{5}^{270} + \)\(11\!\cdots\!52\)\( T_{5}^{269} - \)\(43\!\cdots\!68\)\( T_{5}^{268} + \)\(13\!\cdots\!76\)\( T_{5}^{267} + \)\(39\!\cdots\!48\)\( T_{5}^{266} - \)\(14\!\cdots\!18\)\( T_{5}^{265} + \)\(14\!\cdots\!84\)\( T_{5}^{264} + \)\(55\!\cdots\!56\)\( T_{5}^{263} - \)\(10\!\cdots\!35\)\( T_{5}^{262} + \)\(66\!\cdots\!84\)\( T_{5}^{261} + \)\(10\!\cdots\!67\)\( T_{5}^{260} - \)\(29\!\cdots\!72\)\( T_{5}^{259} + \)\(35\!\cdots\!23\)\( T_{5}^{258} + \)\(70\!\cdots\!00\)\( T_{5}^{257} - \)\(17\!\cdots\!21\)\( T_{5}^{256} - \)\(27\!\cdots\!72\)\( T_{5}^{255} + \)\(11\!\cdots\!58\)\( T_{5}^{254} + \)\(19\!\cdots\!46\)\( T_{5}^{253} + \)\(84\!\cdots\!25\)\( T_{5}^{252} + \)\(29\!\cdots\!96\)\( T_{5}^{251} - \)\(19\!\cdots\!89\)\( T_{5}^{250} - \)\(69\!\cdots\!02\)\( T_{5}^{249} + \)\(69\!\cdots\!53\)\( T_{5}^{248} + \)\(39\!\cdots\!68\)\( T_{5}^{247} + \)\(14\!\cdots\!96\)\( T_{5}^{246} + \)\(51\!\cdots\!40\)\( T_{5}^{245} - \)\(15\!\cdots\!53\)\( T_{5}^{244} - \)\(10\!\cdots\!96\)\( T_{5}^{243} - \)\(66\!\cdots\!42\)\( T_{5}^{242} + \)\(41\!\cdots\!48\)\( T_{5}^{241} + \)\(21\!\cdots\!17\)\( T_{5}^{240} + \)\(84\!\cdots\!42\)\( T_{5}^{239} - \)\(39\!\cdots\!98\)\( T_{5}^{238} - \)\(10\!\cdots\!04\)\( T_{5}^{237} - \)\(18\!\cdots\!07\)\( T_{5}^{236} + \)\(42\!\cdots\!02\)\( T_{5}^{235} + \)\(15\!\cdots\!36\)\( T_{5}^{234} + \)\(84\!\cdots\!16\)\( T_{5}^{233} + \)\(56\!\cdots\!32\)\( T_{5}^{232} - \)\(70\!\cdots\!54\)\( T_{5}^{231} - \)\(16\!\cdots\!63\)\( T_{5}^{230} - \)\(67\!\cdots\!86\)\( T_{5}^{229} + \)\(80\!\cdots\!89\)\( T_{5}^{228} + \)\(57\!\cdots\!86\)\( T_{5}^{227} + \)\(71\!\cdots\!26\)\( T_{5}^{226} - \)\(36\!\cdots\!78\)\( T_{5}^{225} - \)\(11\!\cdots\!34\)\( T_{5}^{224} - \)\(14\!\cdots\!14\)\( T_{5}^{223} + \)\(25\!\cdots\!12\)\( T_{5}^{222} + \)\(32\!\cdots\!40\)\( T_{5}^{221} + \)\(52\!\cdots\!85\)\( T_{5}^{220} - \)\(13\!\cdots\!00\)\( T_{5}^{219} - \)\(39\!\cdots\!83\)\( T_{5}^{218} - \)\(66\!\cdots\!58\)\( T_{5}^{217} - \)\(10\!\cdots\!40\)\( T_{5}^{216} + \)\(10\!\cdots\!24\)\( T_{5}^{215} + \)\(95\!\cdots\!33\)\( T_{5}^{214} - \)\(75\!\cdots\!02\)\( T_{5}^{213} - \)\(78\!\cdots\!01\)\( T_{5}^{212} + \)\(30\!\cdots\!10\)\( T_{5}^{211} + \)\(13\!\cdots\!43\)\( T_{5}^{210} + \)\(37\!\cdots\!44\)\( T_{5}^{209} - \)\(91\!\cdots\!25\)\( T_{5}^{208} - \)\(67\!\cdots\!42\)\( T_{5}^{207} - \)\(74\!\cdots\!34\)\( T_{5}^{206} + \)\(24\!\cdots\!20\)\( T_{5}^{205} + \)\(14\!\cdots\!09\)\( T_{5}^{204} + \)\(34\!\cdots\!18\)\( T_{5}^{203} - \)\(41\!\cdots\!87\)\( T_{5}^{202} - \)\(40\!\cdots\!72\)\( T_{5}^{201} - \)\(71\!\cdots\!85\)\( T_{5}^{200} + \)\(50\!\cdots\!92\)\( T_{5}^{199} + \)\(70\!\cdots\!56\)\( T_{5}^{198} + \)\(19\!\cdots\!14\)\( T_{5}^{197} + \)\(72\!\cdots\!00\)\( T_{5}^{196} - \)\(14\!\cdots\!88\)\( T_{5}^{195} - \)\(36\!\cdots\!70\)\( T_{5}^{194} - \)\(17\!\cdots\!98\)\( T_{5}^{193} + \)\(20\!\cdots\!81\)\( T_{5}^{192} + \)\(72\!\cdots\!46\)\( T_{5}^{191} + \)\(58\!\cdots\!65\)\( T_{5}^{190} - \)\(29\!\cdots\!22\)\( T_{5}^{189} - \)\(11\!\cdots\!36\)\( T_{5}^{188} - \)\(13\!\cdots\!28\)\( T_{5}^{187} + \)\(35\!\cdots\!06\)\( T_{5}^{186} + \)\(18\!\cdots\!00\)\( T_{5}^{185} + \)\(24\!\cdots\!73\)\( T_{5}^{184} - \)\(45\!\cdots\!64\)\( T_{5}^{183} - \)\(24\!\cdots\!55\)\( T_{5}^{182} - \)\(37\!\cdots\!26\)\( T_{5}^{181} + \)\(48\!\cdots\!08\)\( T_{5}^{180} + \)\(33\!\cdots\!22\)\( T_{5}^{179} + \)\(49\!\cdots\!92\)\( T_{5}^{178} - \)\(55\!\cdots\!96\)\( T_{5}^{177} - \)\(35\!\cdots\!25\)\( T_{5}^{176} - \)\(55\!\cdots\!40\)\( T_{5}^{175} + \)\(49\!\cdots\!62\)\( T_{5}^{174} + \)\(37\!\cdots\!10\)\( T_{5}^{173} + \)\(57\!\cdots\!06\)\( T_{5}^{172} - \)\(37\!\cdots\!50\)\( T_{5}^{171} - \)\(32\!\cdots\!37\)\( T_{5}^{170} - \)\(57\!\cdots\!38\)\( T_{5}^{169} + \)\(19\!\cdots\!31\)\( T_{5}^{168} + \)\(30\!\cdots\!12\)\( T_{5}^{167} + \)\(51\!\cdots\!74\)\( T_{5}^{166} - \)\(26\!\cdots\!26\)\( T_{5}^{165} - \)\(23\!\cdots\!65\)\( T_{5}^{164} - \)\(36\!\cdots\!40\)\( T_{5}^{163} + \)\(15\!\cdots\!55\)\( T_{5}^{162} + \)\(14\!\cdots\!44\)\( T_{5}^{161} + \)\(16\!\cdots\!12\)\( T_{5}^{160} - \)\(23\!\cdots\!28\)\( T_{5}^{159} - \)\(78\!\cdots\!45\)\( T_{5}^{158} + \)\(55\!\cdots\!82\)\( T_{5}^{157} + \)\(43\!\cdots\!33\)\( T_{5}^{156} + \)\(83\!\cdots\!82\)\( T_{5}^{155} - \)\(11\!\cdots\!92\)\( T_{5}^{154} - \)\(71\!\cdots\!22\)\( T_{5}^{153} - \)\(84\!\cdots\!53\)\( T_{5}^{152} + \)\(14\!\cdots\!88\)\( T_{5}^{151} + \)\(66\!\cdots\!39\)\( T_{5}^{150} + \)\(10\!\cdots\!30\)\( T_{5}^{149} - \)\(25\!\cdots\!52\)\( T_{5}^{148} - \)\(59\!\cdots\!96\)\( T_{5}^{147} - \)\(12\!\cdots\!66\)\( T_{5}^{146} - \)\(10\!\cdots\!28\)\( T_{5}^{145} + \)\(41\!\cdots\!44\)\( T_{5}^{144} + \)\(85\!\cdots\!12\)\( T_{5}^{143} + \)\(39\!\cdots\!77\)\( T_{5}^{142} - \)\(21\!\cdots\!22\)\( T_{5}^{141} - \)\(57\!\cdots\!12\)\( T_{5}^{140} - \)\(46\!\cdots\!56\)\( T_{5}^{139} + \)\(94\!\cdots\!35\)\( T_{5}^{138} + \)\(33\!\cdots\!00\)\( T_{5}^{137} + \)\(37\!\cdots\!67\)\( T_{5}^{136} - \)\(24\!\cdots\!22\)\( T_{5}^{135} - \)\(14\!\cdots\!14\)\( T_{5}^{134} - \)\(18\!\cdots\!38\)\( T_{5}^{133} + \)\(12\!\cdots\!38\)\( T_{5}^{132} + \)\(34\!\cdots\!26\)\( T_{5}^{131} + \)\(26\!\cdots\!38\)\( T_{5}^{130} - \)\(32\!\cdots\!78\)\( T_{5}^{129} + \)\(91\!\cdots\!99\)\( T_{5}^{128} + \)\(68\!\cdots\!84\)\( T_{5}^{127} + \)\(10\!\cdots\!55\)\( T_{5}^{126} - \)\(94\!\cdots\!80\)\( T_{5}^{125} - \)\(56\!\cdots\!95\)\( T_{5}^{124} - \)\(84\!\cdots\!34\)\( T_{5}^{123} - \)\(12\!\cdots\!75\)\( T_{5}^{122} + \)\(17\!\cdots\!52\)\( T_{5}^{121} + \)\(31\!\cdots\!99\)\( T_{5}^{120} + \)\(17\!\cdots\!10\)\( T_{5}^{119} - \)\(23\!\cdots\!10\)\( T_{5}^{118} - \)\(41\!\cdots\!72\)\( T_{5}^{117} - \)\(25\!\cdots\!25\)\( T_{5}^{116} + \)\(42\!\cdots\!22\)\( T_{5}^{115} + \)\(66\!\cdots\!08\)\( T_{5}^{114} - \)\(10\!\cdots\!92\)\( T_{5}^{113} - \)\(17\!\cdots\!92\)\( T_{5}^{112} + \)\(14\!\cdots\!84\)\( T_{5}^{111} + \)\(51\!\cdots\!99\)\( T_{5}^{110} + \)\(73\!\cdots\!78\)\( T_{5}^{109} - \)\(36\!\cdots\!92\)\( T_{5}^{108} - \)\(24\!\cdots\!66\)\( T_{5}^{107} - \)\(25\!\cdots\!20\)\( T_{5}^{106} + \)\(16\!\cdots\!48\)\( T_{5}^{105} + \)\(67\!\cdots\!35\)\( T_{5}^{104} + \)\(51\!\cdots\!38\)\( T_{5}^{103} - \)\(55\!\cdots\!29\)\( T_{5}^{102} - \)\(14\!\cdots\!56\)\( T_{5}^{101} - \)\(70\!\cdots\!90\)\( T_{5}^{100} + \)\(15\!\cdots\!96\)\( T_{5}^{99} + \)\(28\!\cdots\!91\)\( T_{5}^{98} + \)\(88\!\cdots\!64\)\( T_{5}^{97} - \)\(28\!\cdots\!89\)\( T_{5}^{96} - \)\(37\!\cdots\!06\)\( T_{5}^{95} + \)\(36\!\cdots\!81\)\( T_{5}^{94} + \)\(54\!\cdots\!64\)\( T_{5}^{93} + \)\(50\!\cdots\!45\)\( T_{5}^{92} - \)\(11\!\cdots\!94\)\( T_{5}^{91} - \)\(59\!\cdots\!10\)\( T_{5}^{90} - \)\(31\!\cdots\!54\)\( T_{5}^{89} + \)\(42\!\cdots\!18\)\( T_{5}^{88} + \)\(76\!\cdots\!68\)\( T_{5}^{87} + \)\(32\!\cdots\!09\)\( T_{5}^{86} - \)\(33\!\cdots\!62\)\( T_{5}^{85} - \)\(45\!\cdots\!04\)\( T_{5}^{84} + \)\(27\!\cdots\!20\)\( T_{5}^{83} + \)\(50\!\cdots\!01\)\( T_{5}^{82} + \)\(47\!\cdots\!58\)\( T_{5}^{81} + \)\(84\!\cdots\!24\)\( T_{5}^{80} - \)\(19\!\cdots\!34\)\( T_{5}^{79} - \)\(12\!\cdots\!12\)\( T_{5}^{78} + \)\(11\!\cdots\!54\)\( T_{5}^{77} + \)\(24\!\cdots\!18\)\( T_{5}^{76} + \)\(18\!\cdots\!24\)\( T_{5}^{75} + \)\(45\!\cdots\!30\)\( T_{5}^{74} - \)\(38\!\cdots\!58\)\( T_{5}^{73} - \)\(33\!\cdots\!30\)\( T_{5}^{72} + \)\(13\!\cdots\!22\)\( T_{5}^{71} + \)\(44\!\cdots\!81\)\( T_{5}^{70} + \)\(38\!\cdots\!16\)\( T_{5}^{69} + \)\(15\!\cdots\!02\)\( T_{5}^{68} - \)\(15\!\cdots\!32\)\( T_{5}^{67} - \)\(49\!\cdots\!54\)\( T_{5}^{66} - \)\(77\!\cdots\!18\)\( T_{5}^{65} + \)\(34\!\cdots\!77\)\( T_{5}^{64} + \)\(41\!\cdots\!48\)\( T_{5}^{63} + \)\(23\!\cdots\!97\)\( T_{5}^{62} + \)\(39\!\cdots\!86\)\( T_{5}^{61} - \)\(42\!\cdots\!44\)\( T_{5}^{60} - \)\(32\!\cdots\!68\)\( T_{5}^{59} + \)\(26\!\cdots\!24\)\( T_{5}^{58} + \)\(18\!\cdots\!02\)\( T_{5}^{57} + \)\(17\!\cdots\!06\)\( T_{5}^{56} + \)\(77\!\cdots\!28\)\( T_{5}^{55} + \)\(20\!\cdots\!79\)\( T_{5}^{54} - \)\(23\!\cdots\!00\)\( T_{5}^{53} - \)\(17\!\cdots\!57\)\( T_{5}^{52} - \)\(50\!\cdots\!02\)\( T_{5}^{51} + \)\(21\!\cdots\!65\)\( T_{5}^{50} + \)\(34\!\cdots\!66\)\( T_{5}^{49} + \)\(19\!\cdots\!15\)\( T_{5}^{48} + \)\(47\!\cdots\!12\)\( T_{5}^{47} - \)\(19\!\cdots\!30\)\( T_{5}^{46} - \)\(25\!\cdots\!54\)\( T_{5}^{45} - \)\(12\!\cdots\!23\)\( T_{5}^{44} - \)\(16\!\cdots\!78\)\( T_{5}^{43} + \)\(24\!\cdots\!14\)\( T_{5}^{42} + \)\(24\!\cdots\!92\)\( T_{5}^{41} + \)\(13\!\cdots\!75\)\( T_{5}^{40} + \)\(46\!\cdots\!26\)\( T_{5}^{39} + \)\(61\!\cdots\!27\)\( T_{5}^{38} - \)\(45\!\cdots\!16\)\( T_{5}^{37} - \)\(43\!\cdots\!22\)\( T_{5}^{36} - \)\(22\!\cdots\!66\)\( T_{5}^{35} - \)\(79\!\cdots\!35\)\( T_{5}^{34} - \)\(18\!\cdots\!38\)\( T_{5}^{33} + \)\(86\!\cdots\!23\)\( T_{5}^{32} + \)\(39\!\cdots\!92\)\( T_{5}^{31} + \)\(29\!\cdots\!86\)\( T_{5}^{30} + \)\(15\!\cdots\!46\)\( T_{5}^{29} + \)\(67\!\cdots\!63\)\( T_{5}^{28} + \)\(24\!\cdots\!04\)\( T_{5}^{27} + \)\(72\!\cdots\!75\)\( T_{5}^{26} + \)\(14\!\cdots\!26\)\( T_{5}^{25} - \)\(36\!\cdots\!51\)\( T_{5}^{24} - \)\(20\!\cdots\!70\)\( T_{5}^{23} - \)\(12\!\cdots\!56\)\( T_{5}^{22} - \)\(51\!\cdots\!54\)\( T_{5}^{21} - \)\(16\!\cdots\!58\)\( T_{5}^{20} - \)\(44\!\cdots\!34\)\( T_{5}^{19} - \)\(77\!\cdots\!65\)\( T_{5}^{18} - \)\(13\!\cdots\!76\)\( T_{5}^{17} + \)\(72\!\cdots\!93\)\( T_{5}^{16} + \)\(38\!\cdots\!74\)\( T_{5}^{15} + \)\(14\!\cdots\!23\)\( T_{5}^{14} + \)\(39\!\cdots\!66\)\( T_{5}^{13} + \)\(92\!\cdots\!23\)\( T_{5}^{12} + \)\(17\!\cdots\!92\)\( T_{5}^{11} + \)\(28\!\cdots\!13\)\( T_{5}^{10} + \)\(35\!\cdots\!48\)\( T_{5}^{9} + \)\(36\!\cdots\!32\)\( T_{5}^{8} + \)\(27\!\cdots\!28\)\( T_{5}^{7} + \)\(15\!\cdots\!04\)\( T_{5}^{6} + \)\(53\!\cdots\!12\)\( T_{5}^{5} + \)\(90\!\cdots\!69\)\( T_{5}^{4} + \)\(37\!\cdots\!96\)\( T_{5}^{3} + \)\(69\!\cdots\!73\)\( T_{5}^{2} + \)\(67\!\cdots\!36\)\( T_{5} + \)\(28\!\cdots\!21\)\( \)">\(T_{5}^{320} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).