Properties

Label 230.3.k.b
Level $230$
Weight $3$
Character orbit 230.k
Analytic conductor $6.267$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{2} + 8q^{3} - 4q^{5} - 16q^{6} + 50q^{7} - 48q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{2} + 8q^{3} - 4q^{5} - 16q^{6} + 50q^{7} - 48q^{8} + 16q^{10} - 24q^{11} - 28q^{12} - 8q^{13} + 8q^{15} + 96q^{16} + 44q^{17} + 200q^{18} - 24q^{20} + 24q^{21} + 24q^{22} - 40q^{23} + 240q^{25} + 16q^{26} - 76q^{27} - 100q^{28} - 216q^{30} + 4q^{31} - 96q^{32} - 206q^{33} + 136q^{35} - 48q^{36} + 556q^{37} - 140q^{38} + 16q^{40} + 44q^{41} - 24q^{42} + 48q^{43} + 12q^{45} - 404q^{46} - 24q^{47} + 56q^{48} - 138q^{50} + 48q^{51} - 16q^{52} + 32q^{53} + 64q^{55} + 200q^{56} - 920q^{57} + 28q^{58} + 152q^{60} + 1800q^{61} - 4q^{62} - 406q^{63} + 392q^{65} + 104q^{66} - 304q^{67} - 88q^{68} + 108q^{70} - 1512q^{71} + 48q^{72} - 44q^{73} - 252q^{75} + 16q^{76} - 492q^{77} + 160q^{78} + 16q^{80} - 1344q^{81} - 308q^{82} - 516q^{85} - 272q^{86} + 814q^{87} + 40q^{88} + 670q^{90} + 144q^{91} + 8q^{92} + 160q^{93} - 670q^{95} + 64q^{96} - 242q^{97} - 776q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 1.13214 + 0.847507i −5.13373 + 1.91478i 0.563465 + 1.91899i 4.33360 2.49398i −7.43487 2.18307i 2.35485 10.8251i −0.988434 + 2.65009i 15.8870 13.7662i 7.01989 + 0.849235i
3.2 1.13214 + 0.847507i −4.87737 + 1.81916i 0.563465 + 1.91899i −2.81617 4.13149i −7.06360 2.07406i −2.62119 + 12.0494i −0.988434 + 2.65009i 13.6776 11.8517i 0.313179 7.06413i
3.3 1.13214 + 0.847507i −3.78586 + 1.41205i 0.563465 + 1.91899i −0.916589 + 4.91527i −5.48284 1.60991i 0.413787 1.90215i −0.988434 + 2.65009i 5.53710 4.79793i −5.20343 + 4.78794i
3.4 1.13214 + 0.847507i −2.59640 + 0.968406i 0.563465 + 1.91899i 4.72347 + 1.63977i −3.76021 1.10410i −1.02169 + 4.69661i −0.988434 + 2.65009i −0.998275 + 0.865010i 3.95789 + 5.85962i
3.5 1.13214 + 0.847507i −1.46393 + 0.546016i 0.563465 + 1.91899i −4.98248 0.418219i −2.12011 0.622522i 2.78627 12.8083i −0.988434 + 2.65009i −4.95680 + 4.29509i −5.28640 4.69616i
3.6 1.13214 + 0.847507i −1.33310 + 0.497219i 0.563465 + 1.91899i −1.71782 4.69565i −1.93064 0.566888i −0.622339 + 2.86085i −0.988434 + 2.65009i −5.27183 + 4.56806i 2.03479 6.77197i
3.7 1.13214 + 0.847507i 0.583215 0.217528i 0.563465 + 1.91899i 0.599174 + 4.96397i 0.844635 + 0.248007i 0.414739 1.90653i −0.988434 + 2.65009i −6.50892 + 5.64002i −3.52865 + 6.12769i
3.8 1.13214 + 0.847507i 1.00233 0.373849i 0.563465 + 1.91899i −4.85970 + 1.17614i 1.45161 + 0.426232i −1.84838 + 8.49688i −0.988434 + 2.65009i −5.93685 + 5.14431i −6.49863 2.78707i
3.9 1.13214 + 0.847507i 3.11913 1.16338i 0.563465 + 1.91899i 4.81990 1.32988i 4.51725 + 1.32639i −2.04259 + 9.38963i −0.988434 + 2.65009i 1.57380 1.36371i 6.58386 + 2.57929i
3.10 1.13214 + 0.847507i 3.21185 1.19796i 0.563465 + 1.91899i 4.41997 + 2.33748i 4.65153 + 1.36581i 2.26142 10.3956i −0.988434 + 2.65009i 2.07914 1.80158i 3.02298 + 6.39231i
3.11 1.13214 + 0.847507i 3.86243 1.44061i 0.563465 + 1.91899i −0.740945 4.94480i 5.59372 + 1.64247i 1.21592 5.58950i −0.988434 + 2.65009i 6.04125 5.23477i 3.35190 6.22614i
3.12 1.13214 + 0.847507i 5.55917 2.07346i 0.563465 + 1.91899i −2.76117 + 4.16845i 8.05101 + 2.36399i −0.783489 + 3.60164i −0.988434 + 2.65009i 19.8034 17.1597i −6.65881 + 2.37914i
13.1 1.41061 + 0.100889i −0.933043 4.28913i 1.97964 + 0.284630i 4.55004 2.07296i −0.883435 6.14443i 1.77102 0.967050i 2.76379 + 0.601225i −9.33939 + 4.26516i 6.62747 2.46510i
13.2 1.41061 + 0.100889i −0.875157 4.02303i 1.97964 + 0.284630i −4.99252 + 0.273452i −0.828626 5.76322i −11.1890 + 6.10964i 2.76379 + 0.601225i −7.23219 + 3.30283i −7.07008 0.117955i
13.3 1.41061 + 0.100889i −0.800118 3.67808i 1.97964 + 0.284630i −2.59942 4.27118i −0.757577 5.26906i 6.22281 3.39791i 2.76379 + 0.601225i −4.70140 + 2.14706i −3.23586 6.28723i
13.4 1.41061 + 0.100889i −0.526675 2.42109i 1.97964 + 0.284630i 1.87475 + 4.63522i −0.498673 3.46835i 1.72156 0.940042i 2.76379 + 0.601225i 2.60241 1.18848i 2.17690 + 6.72764i
13.5 1.41061 + 0.100889i −0.110736 0.509045i 1.97964 + 0.284630i −4.91212 + 0.933335i −0.104848 0.729237i 5.72192 3.12441i 2.76379 + 0.601225i 7.93982 3.62600i −7.02324 + 0.820995i
13.6 1.41061 + 0.100889i 0.0578351 + 0.265863i 1.97964 + 0.284630i −0.330642 4.98906i 0.0547601 + 0.380865i −4.77713 + 2.60851i 2.76379 + 0.601225i 8.11935 3.70798i 0.0369324 7.07097i
13.7 1.41061 + 0.100889i 0.161368 + 0.741796i 1.97964 + 0.284630i 4.91560 0.914803i 0.152788 + 1.06267i 2.14811 1.17296i 2.76379 + 0.601225i 7.66247 3.49933i 7.02629 0.794501i
13.8 1.41061 + 0.100889i 0.302791 + 1.39191i 1.97964 + 0.284630i −2.63202 + 4.25117i 0.286693 + 1.99399i −5.20021 + 2.83953i 2.76379 + 0.601225i 6.34096 2.89582i −4.14165 + 5.73121i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.c even 11 1 inner
115.k odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.k.b 240
5.c odd 4 1 inner 230.3.k.b 240
23.c even 11 1 inner 230.3.k.b 240
115.k odd 44 1 inner 230.3.k.b 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.k.b 240 1.a even 1 1 trivial
230.3.k.b 240 5.c odd 4 1 inner
230.3.k.b 240 23.c even 11 1 inner
230.3.k.b 240 115.k odd 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(97\!\cdots\!64\)\( T_{3}^{222} - \)\(14\!\cdots\!98\)\( T_{3}^{221} - \)\(43\!\cdots\!35\)\( T_{3}^{220} + \)\(94\!\cdots\!72\)\( T_{3}^{219} - \)\(45\!\cdots\!66\)\( T_{3}^{218} + \)\(80\!\cdots\!52\)\( T_{3}^{217} + \)\(20\!\cdots\!43\)\( T_{3}^{216} - \)\(12\!\cdots\!42\)\( T_{3}^{215} + \)\(31\!\cdots\!26\)\( T_{3}^{214} + \)\(22\!\cdots\!82\)\( T_{3}^{213} - \)\(18\!\cdots\!45\)\( T_{3}^{212} + \)\(79\!\cdots\!36\)\( T_{3}^{211} - \)\(12\!\cdots\!44\)\( T_{3}^{210} - \)\(41\!\cdots\!64\)\( T_{3}^{209} + \)\(13\!\cdots\!41\)\( T_{3}^{208} - \)\(53\!\cdots\!86\)\( T_{3}^{207} + \)\(99\!\cdots\!94\)\( T_{3}^{206} + \)\(77\!\cdots\!16\)\( T_{3}^{205} - \)\(80\!\cdots\!97\)\( T_{3}^{204} + \)\(28\!\cdots\!08\)\( T_{3}^{203} - \)\(30\!\cdots\!68\)\( T_{3}^{202} - \)\(13\!\cdots\!18\)\( T_{3}^{201} + \)\(54\!\cdots\!23\)\( T_{3}^{200} - \)\(13\!\cdots\!46\)\( T_{3}^{199} + \)\(16\!\cdots\!98\)\( T_{3}^{198} + \)\(16\!\cdots\!90\)\( T_{3}^{197} - \)\(33\!\cdots\!94\)\( T_{3}^{196} + \)\(69\!\cdots\!70\)\( T_{3}^{195} - \)\(90\!\cdots\!02\)\( T_{3}^{194} - \)\(79\!\cdots\!76\)\( T_{3}^{193} + \)\(17\!\cdots\!65\)\( T_{3}^{192} - \)\(36\!\cdots\!78\)\( T_{3}^{191} + \)\(20\!\cdots\!34\)\( T_{3}^{190} + \)\(33\!\cdots\!66\)\( T_{3}^{189} - \)\(90\!\cdots\!33\)\( T_{3}^{188} + \)\(18\!\cdots\!58\)\( T_{3}^{187} - \)\(20\!\cdots\!32\)\( T_{3}^{186} - \)\(14\!\cdots\!58\)\( T_{3}^{185} + \)\(47\!\cdots\!28\)\( T_{3}^{184} - \)\(98\!\cdots\!26\)\( T_{3}^{183} + \)\(18\!\cdots\!20\)\( T_{3}^{182} + \)\(27\!\cdots\!16\)\( T_{3}^{181} - \)\(22\!\cdots\!39\)\( T_{3}^{180} + \)\(53\!\cdots\!94\)\( T_{3}^{179} - \)\(12\!\cdots\!34\)\( T_{3}^{178} + \)\(25\!\cdots\!54\)\( T_{3}^{177} + \)\(93\!\cdots\!02\)\( T_{3}^{176} - \)\(26\!\cdots\!00\)\( T_{3}^{175} + \)\(63\!\cdots\!12\)\( T_{3}^{174} - \)\(28\!\cdots\!48\)\( T_{3}^{173} - \)\(35\!\cdots\!00\)\( T_{3}^{172} + \)\(10\!\cdots\!12\)\( T_{3}^{171} - \)\(26\!\cdots\!60\)\( T_{3}^{170} + \)\(18\!\cdots\!06\)\( T_{3}^{169} + \)\(12\!\cdots\!94\)\( T_{3}^{168} - \)\(42\!\cdots\!72\)\( T_{3}^{167} + \)\(88\!\cdots\!34\)\( T_{3}^{166} - \)\(97\!\cdots\!04\)\( T_{3}^{165} - \)\(36\!\cdots\!70\)\( T_{3}^{164} + \)\(16\!\cdots\!68\)\( T_{3}^{163} - \)\(21\!\cdots\!18\)\( T_{3}^{162} + \)\(41\!\cdots\!78\)\( T_{3}^{161} + \)\(72\!\cdots\!30\)\( T_{3}^{160} - \)\(56\!\cdots\!00\)\( T_{3}^{159} + \)\(35\!\cdots\!84\)\( T_{3}^{158} - \)\(14\!\cdots\!20\)\( T_{3}^{157} - \)\(45\!\cdots\!78\)\( T_{3}^{156} + \)\(17\!\cdots\!34\)\( T_{3}^{155} + \)\(29\!\cdots\!58\)\( T_{3}^{154} + \)\(42\!\cdots\!50\)\( T_{3}^{153} - \)\(36\!\cdots\!62\)\( T_{3}^{152} - \)\(47\!\cdots\!78\)\( T_{3}^{151} - \)\(46\!\cdots\!84\)\( T_{3}^{150} - \)\(10\!\cdots\!38\)\( T_{3}^{149} + \)\(20\!\cdots\!10\)\( T_{3}^{148} + \)\(12\!\cdots\!74\)\( T_{3}^{147} + \)\(20\!\cdots\!68\)\( T_{3}^{146} + \)\(20\!\cdots\!22\)\( T_{3}^{145} - \)\(69\!\cdots\!93\)\( T_{3}^{144} - \)\(31\!\cdots\!52\)\( T_{3}^{143} - \)\(58\!\cdots\!84\)\( T_{3}^{142} - \)\(33\!\cdots\!02\)\( T_{3}^{141} + \)\(18\!\cdots\!93\)\( T_{3}^{140} + \)\(69\!\cdots\!82\)\( T_{3}^{139} + \)\(11\!\cdots\!60\)\( T_{3}^{138} + \)\(37\!\cdots\!62\)\( T_{3}^{137} - \)\(41\!\cdots\!80\)\( T_{3}^{136} - \)\(12\!\cdots\!08\)\( T_{3}^{135} - \)\(17\!\cdots\!98\)\( T_{3}^{134} - \)\(22\!\cdots\!30\)\( T_{3}^{133} + \)\(71\!\cdots\!84\)\( T_{3}^{132} + \)\(18\!\cdots\!04\)\( T_{3}^{131} + \)\(18\!\cdots\!02\)\( T_{3}^{130} - \)\(66\!\cdots\!54\)\( T_{3}^{129} - \)\(86\!\cdots\!61\)\( T_{3}^{128} - \)\(19\!\cdots\!16\)\( T_{3}^{127} - \)\(17\!\cdots\!50\)\( T_{3}^{126} - \)\(56\!\cdots\!42\)\( T_{3}^{125} + \)\(66\!\cdots\!24\)\( T_{3}^{124} + \)\(18\!\cdots\!30\)\( T_{3}^{123} + \)\(25\!\cdots\!00\)\( T_{3}^{122} + \)\(94\!\cdots\!96\)\( T_{3}^{121} - \)\(28\!\cdots\!69\)\( T_{3}^{120} - \)\(19\!\cdots\!68\)\( T_{3}^{119} - \)\(36\!\cdots\!22\)\( T_{3}^{118} - \)\(18\!\cdots\!22\)\( T_{3}^{117} + \)\(72\!\cdots\!08\)\( T_{3}^{116} + \)\(17\!\cdots\!38\)\( T_{3}^{115} + \)\(43\!\cdots\!86\)\( T_{3}^{114} + \)\(16\!\cdots\!16\)\( T_{3}^{113} + \)\(18\!\cdots\!86\)\( T_{3}^{112} - \)\(11\!\cdots\!82\)\( T_{3}^{111} - \)\(25\!\cdots\!14\)\( T_{3}^{110} - \)\(92\!\cdots\!84\)\( T_{3}^{109} - \)\(35\!\cdots\!23\)\( T_{3}^{108} + \)\(46\!\cdots\!18\)\( T_{3}^{107} + \)\(64\!\cdots\!14\)\( T_{3}^{106} + \)\(30\!\cdots\!24\)\( T_{3}^{105} + \)\(13\!\cdots\!11\)\( T_{3}^{104} - \)\(80\!\cdots\!42\)\( T_{3}^{103} + \)\(16\!\cdots\!48\)\( T_{3}^{102} + \)\(49\!\cdots\!42\)\( T_{3}^{101} + \)\(22\!\cdots\!49\)\( T_{3}^{100} + \)\(25\!\cdots\!26\)\( T_{3}^{99} - \)\(27\!\cdots\!22\)\( T_{3}^{98} - \)\(40\!\cdots\!86\)\( T_{3}^{97} - \)\(13\!\cdots\!29\)\( T_{3}^{96} - \)\(32\!\cdots\!30\)\( T_{3}^{95} - \)\(16\!\cdots\!02\)\( T_{3}^{94} - \)\(79\!\cdots\!02\)\( T_{3}^{93} + \)\(12\!\cdots\!40\)\( T_{3}^{92} + \)\(60\!\cdots\!78\)\( T_{3}^{91} + \)\(43\!\cdots\!88\)\( T_{3}^{90} + \)\(11\!\cdots\!20\)\( T_{3}^{89} + \)\(12\!\cdots\!78\)\( T_{3}^{88} + \)\(51\!\cdots\!72\)\( T_{3}^{87} + \)\(11\!\cdots\!86\)\( T_{3}^{86} + \)\(21\!\cdots\!88\)\( T_{3}^{85} + \)\(63\!\cdots\!20\)\( T_{3}^{84} + \)\(64\!\cdots\!76\)\( T_{3}^{83} + \)\(35\!\cdots\!68\)\( T_{3}^{82} + \)\(72\!\cdots\!14\)\( T_{3}^{81} + \)\(11\!\cdots\!09\)\( T_{3}^{80} - \)\(21\!\cdots\!66\)\( T_{3}^{79} + \)\(40\!\cdots\!58\)\( T_{3}^{78} + \)\(10\!\cdots\!14\)\( T_{3}^{77} - \)\(79\!\cdots\!23\)\( T_{3}^{76} - \)\(18\!\cdots\!04\)\( T_{3}^{75} + \)\(79\!\cdots\!38\)\( T_{3}^{74} - \)\(87\!\cdots\!92\)\( T_{3}^{73} - \)\(13\!\cdots\!95\)\( T_{3}^{72} + \)\(10\!\cdots\!86\)\( T_{3}^{71} + \)\(43\!\cdots\!76\)\( T_{3}^{70} - \)\(68\!\cdots\!60\)\( T_{3}^{69} - \)\(24\!\cdots\!98\)\( T_{3}^{68} + \)\(14\!\cdots\!36\)\( T_{3}^{67} + \)\(18\!\cdots\!22\)\( T_{3}^{66} - \)\(41\!\cdots\!80\)\( T_{3}^{65} + \)\(52\!\cdots\!09\)\( T_{3}^{64} + \)\(20\!\cdots\!22\)\( T_{3}^{63} - \)\(57\!\cdots\!70\)\( T_{3}^{62} - \)\(90\!\cdots\!86\)\( T_{3}^{61} + \)\(13\!\cdots\!62\)\( T_{3}^{60} - \)\(65\!\cdots\!84\)\( T_{3}^{59} - \)\(61\!\cdots\!84\)\( T_{3}^{58} - \)\(74\!\cdots\!10\)\( T_{3}^{57} + \)\(58\!\cdots\!09\)\( T_{3}^{56} - \)\(60\!\cdots\!46\)\( T_{3}^{55} - \)\(17\!\cdots\!78\)\( T_{3}^{54} + \)\(11\!\cdots\!62\)\( T_{3}^{53} - \)\(71\!\cdots\!24\)\( T_{3}^{52} - \)\(10\!\cdots\!14\)\( T_{3}^{51} + \)\(26\!\cdots\!60\)\( T_{3}^{50} - \)\(22\!\cdots\!66\)\( T_{3}^{49} + \)\(56\!\cdots\!78\)\( T_{3}^{48} + \)\(26\!\cdots\!66\)\( T_{3}^{47} - \)\(38\!\cdots\!22\)\( T_{3}^{46} + \)\(26\!\cdots\!74\)\( T_{3}^{45} + \)\(59\!\cdots\!41\)\( T_{3}^{44} - \)\(33\!\cdots\!40\)\( T_{3}^{43} + \)\(26\!\cdots\!76\)\( T_{3}^{42} + \)\(25\!\cdots\!58\)\( T_{3}^{41} - \)\(87\!\cdots\!43\)\( T_{3}^{40} + \)\(95\!\cdots\!94\)\( T_{3}^{39} - \)\(29\!\cdots\!38\)\( T_{3}^{38} - \)\(67\!\cdots\!42\)\( T_{3}^{37} + \)\(13\!\cdots\!73\)\( T_{3}^{36} - \)\(12\!\cdots\!60\)\( T_{3}^{35} + \)\(63\!\cdots\!06\)\( T_{3}^{34} + \)\(60\!\cdots\!04\)\( T_{3}^{33} - \)\(49\!\cdots\!10\)\( T_{3}^{32} + \)\(54\!\cdots\!52\)\( T_{3}^{31} - \)\(38\!\cdots\!74\)\( T_{3}^{30} + \)\(18\!\cdots\!20\)\( T_{3}^{29} - \)\(21\!\cdots\!39\)\( T_{3}^{28} - \)\(51\!\cdots\!66\)\( T_{3}^{27} + \)\(66\!\cdots\!54\)\( T_{3}^{26} - \)\(58\!\cdots\!86\)\( T_{3}^{25} + \)\(32\!\cdots\!46\)\( T_{3}^{24} - \)\(13\!\cdots\!84\)\( T_{3}^{23} + \)\(41\!\cdots\!22\)\( T_{3}^{22} + \)\(15\!\cdots\!72\)\( T_{3}^{21} - \)\(22\!\cdots\!50\)\( T_{3}^{20} + \)\(12\!\cdots\!96\)\( T_{3}^{19} - \)\(82\!\cdots\!28\)\( T_{3}^{18} + \)\(19\!\cdots\!60\)\( T_{3}^{17} + \)\(15\!\cdots\!21\)\( T_{3}^{16} - \)\(72\!\cdots\!08\)\( T_{3}^{15} + \)\(33\!\cdots\!46\)\( T_{3}^{14} - \)\(30\!\cdots\!66\)\( T_{3}^{13} + \)\(31\!\cdots\!06\)\( T_{3}^{12} + \)\(23\!\cdots\!40\)\( T_{3}^{11} + \)\(10\!\cdots\!80\)\( T_{3}^{10} + \)\(57\!\cdots\!58\)\( T_{3}^{9} - \)\(41\!\cdots\!25\)\( T_{3}^{8} - \)\(18\!\cdots\!82\)\( T_{3}^{7} - \)\(40\!\cdots\!44\)\( T_{3}^{6} - \)\(12\!\cdots\!98\)\( T_{3}^{5} + \)\(19\!\cdots\!74\)\( T_{3}^{4} + \)\(85\!\cdots\!40\)\( T_{3}^{3} + \)\(10\!\cdots\!18\)\( T_{3}^{2} + \)\(17\!\cdots\!94\)\( T_{3} + \)\(14\!\cdots\!01\)\( \)">\(T_{3}^{240} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(230, [\chi])\).