Properties

Label 966.2.be.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 320q + 16q^{2} + 16q^{4} - 32q^{8} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 320q + 16q^{2} + 16q^{4} - 32q^{8} - 16q^{9} + 22q^{14} + 16q^{16} - 66q^{17} - 16q^{18} + 36q^{23} + 24q^{25} + 12q^{26} + 44q^{28} + 8q^{29} - 48q^{31} + 16q^{32} - 46q^{35} + 32q^{36} - 22q^{37} + 66q^{38} + 8q^{39} + 176q^{43} - 8q^{46} + 120q^{47} - 24q^{49} - 48q^{50} - 22q^{51} - 12q^{52} - 44q^{53} + 44q^{57} + 18q^{58} + 12q^{59} - 32q^{64} - 108q^{70} - 48q^{71} - 16q^{72} + 252q^{73} + 22q^{74} - 36q^{75} - 42q^{77} - 16q^{78} + 44q^{79} + 16q^{81} + 12q^{82} - 22q^{84} - 76q^{85} + 22q^{86} + 24q^{87} - 22q^{88} + 16q^{92} + 12q^{94} + 26q^{95} + 2q^{98} + 88q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −2.88028 + 1.48489i 0.755750 0.654861i −2.06769 + 1.65065i −0.959493 0.281733i −0.0475819 0.998867i 3.00838 1.20438i
19.2 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −2.11320 + 1.08943i 0.755750 0.654861i 0.852549 2.50463i −0.959493 0.281733i −0.0475819 0.998867i 2.20719 0.883624i
19.3 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −1.73698 + 0.895474i 0.755750 0.654861i 0.815740 + 2.51686i −0.959493 0.281733i −0.0475819 0.998867i 1.81423 0.726309i
19.4 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −1.35389 + 0.697977i 0.755750 0.654861i −2.16100 + 1.52646i −0.959493 0.281733i −0.0475819 0.998867i 1.41410 0.566121i
19.5 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 0.157815 0.0813595i 0.755750 0.654861i −1.34398 2.27897i −0.959493 0.281733i −0.0475819 0.998867i −0.164835 + 0.0659898i
19.6 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 0.505673 0.260692i 0.755750 0.654861i 2.07574 1.64052i −0.959493 0.281733i −0.0475819 0.998867i −0.528163 + 0.211445i
19.7 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 2.27295 1.17179i 0.755750 0.654861i 2.41464 + 1.08144i −0.959493 0.281733i −0.0475819 0.998867i −2.37404 + 0.950422i
19.8 −0.995472 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 2.93697 1.51411i 0.755750 0.654861i −2.06506 1.65394i −0.959493 0.281733i −0.0475819 0.998867i −3.06760 + 1.22808i
19.9 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −3.62091 + 1.86671i −0.755750 + 0.654861i −2.61175 0.422774i −0.959493 0.281733i −0.0475819 0.998867i 3.78196 1.51407i
19.10 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −2.00286 + 1.03255i −0.755750 + 0.654861i 1.87452 1.86713i −0.959493 0.281733i −0.0475819 0.998867i 2.09194 0.837488i
19.11 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −1.84849 + 0.952964i −0.755750 + 0.654861i 1.16229 + 2.37678i −0.959493 0.281733i −0.0475819 0.998867i 1.93071 0.772938i
19.12 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i −0.0836058 + 0.0431018i −0.755750 + 0.654861i −1.37599 2.25979i −0.959493 0.281733i −0.0475819 0.998867i 0.0873243 0.0349594i
19.13 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 0.0713145 0.0367652i −0.755750 + 0.654861i −2.59984 0.490747i −0.959493 0.281733i −0.0475819 0.998867i −0.0744863 + 0.0298198i
19.14 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 0.951099 0.490326i −0.755750 + 0.654861i 1.44395 + 2.21698i −0.959493 0.281733i −0.0475819 0.998867i −0.993401 + 0.397698i
19.15 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 3.08964 1.59282i −0.755750 + 0.654861i 1.21125 2.35221i −0.959493 0.281733i −0.0475819 0.998867i −3.22705 + 1.29192i
19.16 −0.995472 0.0950560i 0.690079 0.723734i 0.981929 + 0.189251i 3.91984 2.02082i −0.755750 + 0.654861i −0.423103 + 2.61170i −0.959493 0.281733i −0.0475819 0.998867i −4.09418 + 1.63906i
61.1 −0.786053 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −2.69832 + 0.520058i 0.281733 + 0.959493i 1.79038 1.94796i 0.415415 0.909632i 0.327068 + 0.945001i 2.44250 + 1.25920i
61.2 −0.786053 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −2.30347 + 0.443957i 0.281733 + 0.959493i −2.21609 + 1.44532i 0.415415 0.909632i 0.327068 + 0.945001i 2.08508 + 1.07494i
61.3 −0.786053 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i −2.22749 + 0.429313i 0.281733 + 0.959493i −1.35585 2.27193i 0.415415 0.909632i 0.327068 + 0.945001i 2.01631 + 1.03948i
61.4 −0.786053 0.618159i −0.814576 0.580057i 0.235759 + 0.971812i 1.21776 0.234705i 0.281733 + 0.959493i −1.70399 2.02396i 0.415415 0.909632i 0.327068 + 0.945001i −1.10231 0.568281i
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.b 320
7.d odd 6 1 inner 966.2.be.b 320
23.d odd 22 1 inner 966.2.be.b 320
161.o even 66 1 inner 966.2.be.b 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.be.b 320 1.a even 1 1 trivial
966.2.be.b 320 7.d odd 6 1 inner
966.2.be.b 320 23.d odd 22 1 inner
966.2.be.b 320 161.o even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(17\!\cdots\!04\)\( T_{5}^{302} - \)\(57\!\cdots\!70\)\( T_{5}^{301} + \)\(39\!\cdots\!63\)\( T_{5}^{300} - \)\(11\!\cdots\!72\)\( T_{5}^{299} - \)\(41\!\cdots\!67\)\( T_{5}^{298} + \)\(20\!\cdots\!08\)\( T_{5}^{297} + \)\(15\!\cdots\!79\)\( T_{5}^{296} - \)\(39\!\cdots\!72\)\( T_{5}^{295} + \)\(19\!\cdots\!02\)\( T_{5}^{294} + \)\(16\!\cdots\!40\)\( T_{5}^{293} - \)\(56\!\cdots\!82\)\( T_{5}^{292} + \)\(53\!\cdots\!54\)\( T_{5}^{291} + \)\(73\!\cdots\!10\)\( T_{5}^{290} - \)\(62\!\cdots\!78\)\( T_{5}^{289} + \)\(90\!\cdots\!03\)\( T_{5}^{288} - \)\(14\!\cdots\!78\)\( T_{5}^{287} - \)\(47\!\cdots\!65\)\( T_{5}^{286} + \)\(50\!\cdots\!92\)\( T_{5}^{285} + \)\(20\!\cdots\!35\)\( T_{5}^{284} - \)\(63\!\cdots\!92\)\( T_{5}^{283} + \)\(24\!\cdots\!50\)\( T_{5}^{282} + \)\(10\!\cdots\!62\)\( T_{5}^{281} - \)\(71\!\cdots\!46\)\( T_{5}^{280} + \)\(12\!\cdots\!12\)\( T_{5}^{279} + \)\(84\!\cdots\!65\)\( T_{5}^{278} - \)\(35\!\cdots\!94\)\( T_{5}^{277} + \)\(39\!\cdots\!98\)\( T_{5}^{276} + \)\(54\!\cdots\!32\)\( T_{5}^{275} - \)\(18\!\cdots\!72\)\( T_{5}^{274} - \)\(42\!\cdots\!32\)\( T_{5}^{273} + \)\(34\!\cdots\!61\)\( T_{5}^{272} - \)\(32\!\cdots\!42\)\( T_{5}^{271} - \)\(32\!\cdots\!03\)\( T_{5}^{270} + \)\(14\!\cdots\!48\)\( T_{5}^{269} + \)\(25\!\cdots\!36\)\( T_{5}^{268} - \)\(19\!\cdots\!88\)\( T_{5}^{267} + \)\(44\!\cdots\!00\)\( T_{5}^{266} + \)\(10\!\cdots\!10\)\( T_{5}^{265} - \)\(76\!\cdots\!24\)\( T_{5}^{264} + \)\(94\!\cdots\!68\)\( T_{5}^{263} + \)\(59\!\cdots\!37\)\( T_{5}^{262} - \)\(26\!\cdots\!68\)\( T_{5}^{261} + \)\(10\!\cdots\!03\)\( T_{5}^{260} + \)\(25\!\cdots\!64\)\( T_{5}^{259} - \)\(80\!\cdots\!25\)\( T_{5}^{258} - \)\(50\!\cdots\!76\)\( T_{5}^{257} + \)\(88\!\cdots\!51\)\( T_{5}^{256} - \)\(20\!\cdots\!36\)\( T_{5}^{255} - \)\(20\!\cdots\!82\)\( T_{5}^{254} + \)\(29\!\cdots\!18\)\( T_{5}^{253} - \)\(60\!\cdots\!91\)\( T_{5}^{252} - \)\(13\!\cdots\!84\)\( T_{5}^{251} + \)\(82\!\cdots\!47\)\( T_{5}^{250} - \)\(14\!\cdots\!10\)\( T_{5}^{249} - \)\(34\!\cdots\!51\)\( T_{5}^{248} + \)\(27\!\cdots\!20\)\( T_{5}^{247} - \)\(36\!\cdots\!80\)\( T_{5}^{246} - \)\(14\!\cdots\!12\)\( T_{5}^{245} + \)\(70\!\cdots\!67\)\( T_{5}^{244} - \)\(69\!\cdots\!72\)\( T_{5}^{243} - \)\(41\!\cdots\!38\)\( T_{5}^{242} + \)\(17\!\cdots\!48\)\( T_{5}^{241} - \)\(18\!\cdots\!19\)\( T_{5}^{240} - \)\(12\!\cdots\!70\)\( T_{5}^{239} + \)\(45\!\cdots\!06\)\( T_{5}^{238} - \)\(35\!\cdots\!96\)\( T_{5}^{237} - \)\(22\!\cdots\!91\)\( T_{5}^{236} + \)\(87\!\cdots\!50\)\( T_{5}^{235} - \)\(11\!\cdots\!48\)\( T_{5}^{234} - \)\(67\!\cdots\!36\)\( T_{5}^{233} + \)\(22\!\cdots\!20\)\( T_{5}^{232} - \)\(63\!\cdots\!46\)\( T_{5}^{231} - \)\(11\!\cdots\!11\)\( T_{5}^{230} + \)\(41\!\cdots\!06\)\( T_{5}^{229} - \)\(55\!\cdots\!71\)\( T_{5}^{228} - \)\(28\!\cdots\!30\)\( T_{5}^{227} + \)\(14\!\cdots\!54\)\( T_{5}^{226} + \)\(11\!\cdots\!10\)\( T_{5}^{225} - \)\(74\!\cdots\!50\)\( T_{5}^{224} + \)\(13\!\cdots\!02\)\( T_{5}^{223} - \)\(24\!\cdots\!76\)\( T_{5}^{222} - \)\(10\!\cdots\!92\)\( T_{5}^{221} + \)\(29\!\cdots\!25\)\( T_{5}^{220} + \)\(24\!\cdots\!36\)\( T_{5}^{219} - \)\(68\!\cdots\!71\)\( T_{5}^{218} + \)\(50\!\cdots\!22\)\( T_{5}^{217} - \)\(44\!\cdots\!64\)\( T_{5}^{216} - \)\(12\!\cdots\!92\)\( T_{5}^{215} + \)\(29\!\cdots\!17\)\( T_{5}^{214} + \)\(35\!\cdots\!62\)\( T_{5}^{213} - \)\(26\!\cdots\!49\)\( T_{5}^{212} - \)\(71\!\cdots\!66\)\( T_{5}^{211} - \)\(23\!\cdots\!89\)\( T_{5}^{210} + \)\(28\!\cdots\!92\)\( T_{5}^{209} + \)\(10\!\cdots\!55\)\( T_{5}^{208} - \)\(16\!\cdots\!22\)\( T_{5}^{207} - \)\(83\!\cdots\!38\)\( T_{5}^{206} - \)\(29\!\cdots\!84\)\( T_{5}^{205} + \)\(65\!\cdots\!13\)\( T_{5}^{204} + \)\(17\!\cdots\!30\)\( T_{5}^{203} + \)\(13\!\cdots\!81\)\( T_{5}^{202} - \)\(10\!\cdots\!00\)\( T_{5}^{201} - \)\(50\!\cdots\!17\)\( T_{5}^{200} - \)\(60\!\cdots\!88\)\( T_{5}^{199} + \)\(14\!\cdots\!92\)\( T_{5}^{198} + \)\(60\!\cdots\!46\)\( T_{5}^{197} + \)\(97\!\cdots\!72\)\( T_{5}^{196} - \)\(92\!\cdots\!72\)\( T_{5}^{195} - \)\(96\!\cdots\!82\)\( T_{5}^{194} - \)\(76\!\cdots\!86\)\( T_{5}^{193} + \)\(74\!\cdots\!33\)\( T_{5}^{192} + \)\(20\!\cdots\!34\)\( T_{5}^{191} - \)\(16\!\cdots\!79\)\( T_{5}^{190} - \)\(22\!\cdots\!66\)\( T_{5}^{189} - \)\(51\!\cdots\!92\)\( T_{5}^{188} + \)\(40\!\cdots\!16\)\( T_{5}^{187} + \)\(49\!\cdots\!82\)\( T_{5}^{186} + \)\(97\!\cdots\!32\)\( T_{5}^{185} - \)\(80\!\cdots\!19\)\( T_{5}^{184} - \)\(82\!\cdots\!44\)\( T_{5}^{183} - \)\(14\!\cdots\!99\)\( T_{5}^{182} + \)\(21\!\cdots\!46\)\( T_{5}^{181} + \)\(15\!\cdots\!96\)\( T_{5}^{180} + \)\(23\!\cdots\!54\)\( T_{5}^{179} - \)\(54\!\cdots\!68\)\( T_{5}^{178} - \)\(33\!\cdots\!96\)\( T_{5}^{177} - \)\(56\!\cdots\!61\)\( T_{5}^{176} + \)\(82\!\cdots\!28\)\( T_{5}^{175} + \)\(66\!\cdots\!14\)\( T_{5}^{174} + \)\(13\!\cdots\!18\)\( T_{5}^{173} - \)\(70\!\cdots\!86\)\( T_{5}^{172} - \)\(10\!\cdots\!74\)\( T_{5}^{171} - \)\(25\!\cdots\!13\)\( T_{5}^{170} - \)\(17\!\cdots\!26\)\( T_{5}^{169} + \)\(15\!\cdots\!07\)\( T_{5}^{168} + \)\(41\!\cdots\!56\)\( T_{5}^{167} + \)\(21\!\cdots\!70\)\( T_{5}^{166} - \)\(19\!\cdots\!06\)\( T_{5}^{165} - \)\(65\!\cdots\!97\)\( T_{5}^{164} - \)\(64\!\cdots\!56\)\( T_{5}^{163} + \)\(20\!\cdots\!27\)\( T_{5}^{162} + \)\(95\!\cdots\!32\)\( T_{5}^{161} + \)\(14\!\cdots\!96\)\( T_{5}^{160} - \)\(12\!\cdots\!84\)\( T_{5}^{159} - \)\(11\!\cdots\!49\)\( T_{5}^{158} - \)\(23\!\cdots\!38\)\( T_{5}^{157} - \)\(71\!\cdots\!07\)\( T_{5}^{156} + \)\(90\!\cdots\!06\)\( T_{5}^{155} + \)\(25\!\cdots\!04\)\( T_{5}^{154} + \)\(25\!\cdots\!30\)\( T_{5}^{153} - \)\(45\!\cdots\!41\)\( T_{5}^{152} - \)\(20\!\cdots\!44\)\( T_{5}^{151} - \)\(29\!\cdots\!01\)\( T_{5}^{150} + \)\(11\!\cdots\!78\)\( T_{5}^{149} + \)\(13\!\cdots\!60\)\( T_{5}^{148} + \)\(25\!\cdots\!00\)\( T_{5}^{147} + \)\(59\!\cdots\!02\)\( T_{5}^{146} - \)\(80\!\cdots\!76\)\( T_{5}^{145} - \)\(19\!\cdots\!64\)\( T_{5}^{144} - \)\(11\!\cdots\!12\)\( T_{5}^{143} + \)\(44\!\cdots\!09\)\( T_{5}^{142} + \)\(12\!\cdots\!90\)\( T_{5}^{141} + \)\(10\!\cdots\!92\)\( T_{5}^{140} - \)\(21\!\cdots\!60\)\( T_{5}^{139} - \)\(72\!\cdots\!73\)\( T_{5}^{138} - \)\(66\!\cdots\!80\)\( T_{5}^{137} + \)\(99\!\cdots\!79\)\( T_{5}^{136} + \)\(36\!\cdots\!94\)\( T_{5}^{135} + \)\(33\!\cdots\!90\)\( T_{5}^{134} - \)\(38\!\cdots\!86\)\( T_{5}^{133} - \)\(13\!\cdots\!22\)\( T_{5}^{132} - \)\(10\!\cdots\!50\)\( T_{5}^{131} + \)\(17\!\cdots\!66\)\( T_{5}^{130} + \)\(51\!\cdots\!06\)\( T_{5}^{129} + \)\(40\!\cdots\!71\)\( T_{5}^{128} - \)\(43\!\cdots\!56\)\( T_{5}^{127} - \)\(13\!\cdots\!09\)\( T_{5}^{126} - \)\(11\!\cdots\!92\)\( T_{5}^{125} + \)\(10\!\cdots\!09\)\( T_{5}^{124} + \)\(38\!\cdots\!50\)\( T_{5}^{123} + \)\(41\!\cdots\!73\)\( T_{5}^{122} - \)\(43\!\cdots\!88\)\( T_{5}^{121} - \)\(77\!\cdots\!97\)\( T_{5}^{120} - \)\(95\!\cdots\!26\)\( T_{5}^{119} + \)\(27\!\cdots\!46\)\( T_{5}^{118} + \)\(23\!\cdots\!20\)\( T_{5}^{117} + \)\(26\!\cdots\!79\)\( T_{5}^{116} - \)\(16\!\cdots\!58\)\( T_{5}^{115} - \)\(82\!\cdots\!08\)\( T_{5}^{114} - \)\(97\!\cdots\!88\)\( T_{5}^{113} - \)\(60\!\cdots\!56\)\( T_{5}^{112} + \)\(13\!\cdots\!76\)\( T_{5}^{111} + \)\(19\!\cdots\!03\)\( T_{5}^{110} + \)\(13\!\cdots\!74\)\( T_{5}^{109} + \)\(50\!\cdots\!12\)\( T_{5}^{108} + \)\(56\!\cdots\!70\)\( T_{5}^{107} + \)\(18\!\cdots\!44\)\( T_{5}^{106} + \)\(28\!\cdots\!36\)\( T_{5}^{105} + \)\(92\!\cdots\!11\)\( T_{5}^{104} - \)\(92\!\cdots\!18\)\( T_{5}^{103} - \)\(20\!\cdots\!65\)\( T_{5}^{102} - \)\(18\!\cdots\!44\)\( T_{5}^{101} + \)\(18\!\cdots\!38\)\( T_{5}^{100} + \)\(22\!\cdots\!16\)\( T_{5}^{99} + \)\(15\!\cdots\!75\)\( T_{5}^{98} - \)\(10\!\cdots\!68\)\( T_{5}^{97} - \)\(90\!\cdots\!61\)\( T_{5}^{96} + \)\(41\!\cdots\!02\)\( T_{5}^{95} + \)\(95\!\cdots\!45\)\( T_{5}^{94} + \)\(80\!\cdots\!72\)\( T_{5}^{93} - \)\(18\!\cdots\!95\)\( T_{5}^{92} - \)\(13\!\cdots\!90\)\( T_{5}^{91} - \)\(17\!\cdots\!82\)\( T_{5}^{90} - \)\(59\!\cdots\!74\)\( T_{5}^{89} + \)\(17\!\cdots\!46\)\( T_{5}^{88} + \)\(31\!\cdots\!76\)\( T_{5}^{87} + \)\(11\!\cdots\!73\)\( T_{5}^{86} - \)\(36\!\cdots\!58\)\( T_{5}^{85} - \)\(63\!\cdots\!68\)\( T_{5}^{84} - \)\(24\!\cdots\!56\)\( T_{5}^{83} + \)\(58\!\cdots\!29\)\( T_{5}^{82} + \)\(94\!\cdots\!14\)\( T_{5}^{81} + \)\(24\!\cdots\!64\)\( T_{5}^{80} - \)\(87\!\cdots\!86\)\( T_{5}^{79} - \)\(10\!\cdots\!84\)\( T_{5}^{78} + \)\(22\!\cdots\!42\)\( T_{5}^{77} + \)\(16\!\cdots\!26\)\( T_{5}^{76} + \)\(16\!\cdots\!88\)\( T_{5}^{75} + \)\(19\!\cdots\!26\)\( T_{5}^{74} - \)\(10\!\cdots\!42\)\( T_{5}^{73} - \)\(74\!\cdots\!86\)\( T_{5}^{72} + \)\(72\!\cdots\!22\)\( T_{5}^{71} + \)\(16\!\cdots\!09\)\( T_{5}^{70} + \)\(10\!\cdots\!56\)\( T_{5}^{69} - \)\(41\!\cdots\!18\)\( T_{5}^{68} - \)\(51\!\cdots\!32\)\( T_{5}^{67} - \)\(10\!\cdots\!38\)\( T_{5}^{66} + \)\(48\!\cdots\!46\)\( T_{5}^{65} + \)\(64\!\cdots\!89\)\( T_{5}^{64} + \)\(40\!\cdots\!28\)\( T_{5}^{63} + \)\(98\!\cdots\!57\)\( T_{5}^{62} - \)\(38\!\cdots\!78\)\( T_{5}^{61} - \)\(29\!\cdots\!32\)\( T_{5}^{60} + \)\(12\!\cdots\!00\)\( T_{5}^{59} + \)\(28\!\cdots\!32\)\( T_{5}^{58} + \)\(18\!\cdots\!62\)\( T_{5}^{57} + \)\(56\!\cdots\!30\)\( T_{5}^{56} - \)\(22\!\cdots\!00\)\( T_{5}^{55} - \)\(73\!\cdots\!65\)\( T_{5}^{54} + \)\(75\!\cdots\!20\)\( T_{5}^{53} + \)\(38\!\cdots\!31\)\( T_{5}^{52} + \)\(24\!\cdots\!98\)\( T_{5}^{51} + \)\(68\!\cdots\!21\)\( T_{5}^{50} + \)\(34\!\cdots\!82\)\( T_{5}^{49} - \)\(61\!\cdots\!37\)\( T_{5}^{48} - \)\(11\!\cdots\!80\)\( T_{5}^{47} + \)\(96\!\cdots\!70\)\( T_{5}^{46} + \)\(95\!\cdots\!10\)\( T_{5}^{45} + \)\(51\!\cdots\!77\)\( T_{5}^{44} + \)\(16\!\cdots\!22\)\( T_{5}^{43} - \)\(83\!\cdots\!58\)\( T_{5}^{42} - \)\(42\!\cdots\!88\)\( T_{5}^{41} - \)\(30\!\cdots\!41\)\( T_{5}^{40} - \)\(15\!\cdots\!78\)\( T_{5}^{39} - \)\(67\!\cdots\!77\)\( T_{5}^{38} - \)\(18\!\cdots\!68\)\( T_{5}^{37} + \)\(60\!\cdots\!38\)\( T_{5}^{36} + \)\(31\!\cdots\!46\)\( T_{5}^{35} + \)\(19\!\cdots\!57\)\( T_{5}^{34} + \)\(76\!\cdots\!14\)\( T_{5}^{33} + \)\(22\!\cdots\!55\)\( T_{5}^{32} + \)\(41\!\cdots\!64\)\( T_{5}^{31} - \)\(17\!\cdots\!94\)\( T_{5}^{30} - \)\(46\!\cdots\!94\)\( T_{5}^{29} - \)\(21\!\cdots\!41\)\( T_{5}^{28} - \)\(70\!\cdots\!36\)\( T_{5}^{27} - \)\(19\!\cdots\!37\)\( T_{5}^{26} - \)\(37\!\cdots\!90\)\( T_{5}^{25} - \)\(12\!\cdots\!47\)\( T_{5}^{24} + \)\(23\!\cdots\!02\)\( T_{5}^{23} + \)\(10\!\cdots\!40\)\( T_{5}^{22} + \)\(26\!\cdots\!78\)\( T_{5}^{21} + \)\(54\!\cdots\!66\)\( T_{5}^{20} + \)\(87\!\cdots\!42\)\( T_{5}^{19} + \)\(85\!\cdots\!67\)\( T_{5}^{18} + \)\(10\!\cdots\!72\)\( T_{5}^{17} + \)\(34\!\cdots\!65\)\( T_{5}^{16} + \)\(84\!\cdots\!38\)\( T_{5}^{15} + \)\(19\!\cdots\!31\)\( T_{5}^{14} + \)\(33\!\cdots\!54\)\( T_{5}^{13} + \)\(53\!\cdots\!27\)\( T_{5}^{12} + \)\(18\!\cdots\!20\)\( T_{5}^{11} + \)\(92\!\cdots\!09\)\( T_{5}^{10} + \)\(50\!\cdots\!52\)\( T_{5}^{9} + \)\(12\!\cdots\!08\)\( T_{5}^{8} + \)\(33\!\cdots\!00\)\( T_{5}^{7} + \)\(59\!\cdots\!32\)\( T_{5}^{6} + \)\(78\!\cdots\!92\)\( T_{5}^{5} + \)\(25\!\cdots\!17\)\( T_{5}^{4} - \)\(24\!\cdots\!00\)\( T_{5}^{3} + \)\(55\!\cdots\!81\)\( T_{5}^{2} - \)\(12\!\cdots\!00\)\( T_{5} + \)\(27\!\cdots\!41\)\( \)">\(T_{5}^{320} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).