# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{3}^{\mathrm{new}}(230, [\chi])$$.