Properties

Label 230.3.k.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 240q - 24q^{2} - 4q^{5} - 74q^{7} + 48q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q - 24q^{2} - 4q^{5} - 74q^{7} + 48q^{8} - 16q^{10} + 8q^{11} + 44q^{12} + 24q^{13} + 24q^{15} + 96q^{16} + 12q^{17} + 88q^{18} - 24q^{20} + 24q^{21} + 8q^{22} - 44q^{23} - 128q^{25} + 48q^{26} - 60q^{27} - 116q^{28} + 120q^{30} - 12q^{31} + 96q^{32} - 334q^{33} - 224q^{35} - 176q^{36} + 188q^{37} + 76q^{38} - 16q^{40} - 116q^{41} + 24q^{42} + 120q^{43} + 204q^{45} + 396q^{46} - 144q^{47} - 88q^{48} + 170q^{50} - 176q^{51} + 48q^{52} + 192q^{53} - 312q^{55} + 296q^{56} + 88q^{57} - 28q^{58} - 72q^{60} - 552q^{61} - 12q^{62} - 122q^{63} - 392q^{65} - 8q^{66} - 72q^{67} - 24q^{68} + 100q^{70} + 424q^{71} - 176q^{72} + 452q^{73} + 604q^{75} - 112q^{76} + 356q^{77} + 32q^{78} + 16q^{80} - 704q^{81} + 148q^{82} - 360q^{83} + 428q^{85} - 376q^{86} - 462q^{87} - 104q^{88} - 510q^{90} + 432q^{91} - 192q^{93} - 166q^{95} - 1042q^{97} - 88q^{98} + O(q^{100}) \)

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 −5.47365 + 2.04157i 0.563465 + 1.91899i 1.48407 + 4.77468i 7.92716 + 2.32763i −2.01723 + 9.27304i 0.988434 2.65009i 18.9911 16.4559i 2.36640 6.66334i
3.2 −1.13214 0.847507i −4.48436 + 1.67258i 0.563465 + 1.91899i −1.64232 4.72258i 6.49443 + 1.90694i 1.22526 5.63243i 0.988434 2.65009i 10.5102 9.10714i −2.14308 + 6.73849i
3.3 −1.13214 0.847507i −2.51818 + 0.939231i 0.563465 + 1.91899i 2.35609 4.41008i 3.64692 + 1.07083i −1.24199 + 5.70935i 0.988434 2.65009i −1.34269 + 1.16345i −6.40499 + 2.99601i
3.4 −1.13214 0.847507i −2.44177 + 0.910732i 0.563465 + 1.91899i −4.63287 + 1.88057i 3.53626 + 1.03834i 0.738198 3.39344i 0.988434 2.65009i −1.66895 + 1.44616i 6.83883 + 1.79732i
3.5 −1.13214 0.847507i −1.89358 + 0.706270i 0.563465 + 1.91899i 3.30772 + 3.74953i 2.74236 + 0.805231i 1.94574 8.94444i 0.988434 2.65009i −3.71490 + 3.21898i −0.567036 7.04830i
3.6 −1.13214 0.847507i −1.16785 + 0.435585i 0.563465 + 1.91899i −0.970697 + 4.90487i 1.69133 + 0.496618i −1.47212 + 6.76723i 0.988434 2.65009i −5.62761 + 4.87635i 5.25587 4.73031i
3.7 −1.13214 0.847507i −0.0190211 + 0.00709452i 0.563465 + 1.91899i 4.84193 1.24727i 0.0275472 + 0.00808858i −1.98672 + 9.13280i 0.988434 2.65009i −6.80143 + 5.89348i −6.53880 2.69149i
3.8 −1.13214 0.847507i 1.09130 0.407033i 0.563465 + 1.91899i −4.93104 0.827554i −1.58046 0.464066i 0.600939 2.76247i 0.988434 2.65009i −5.77649 + 5.00536i 4.88125 + 5.11599i
3.9 −1.13214 0.847507i 1.90726 0.711370i 0.563465 + 1.91899i 1.62652 4.72805i −2.76216 0.811045i 2.04047 9.37989i 0.988434 2.65009i −3.67017 + 3.18022i −5.84849 + 3.97431i
3.10 −1.13214 0.847507i 3.55876 1.32735i 0.563465 + 1.91899i −2.88474 + 4.08390i −5.15393 1.51333i −1.27831 + 5.87629i 0.988434 2.65009i 4.10115 3.55367i 6.72706 2.17869i
3.11 −1.13214 0.847507i 4.17989 1.55902i 0.563465 + 1.91899i 4.74689 + 1.57069i −6.05348 1.77746i 0.338789 1.55739i 0.988434 2.65009i 8.23916 7.13928i −4.04295 5.80125i
3.12 −1.13214 0.847507i 4.65466 1.73610i 0.563465 + 1.91899i −3.20030 3.84162i −6.74107 1.97936i −1.76266 + 8.10282i 0.988434 2.65009i 11.8501 10.2682i 0.367373 + 7.06152i
13.1 −1.41061 0.100889i −1.09457 5.03168i 1.97964 + 0.284630i 3.91427 + 3.11103i 1.03638 + 7.20817i −4.48066 + 2.44662i −2.76379 0.601225i −15.9330 + 7.27636i −5.20764 4.78336i
13.2 −1.41061 0.100889i −1.02391 4.70683i 1.97964 + 0.284630i −0.635666 4.95943i 0.969469 + 6.74281i −8.25197 + 4.50592i −2.76379 0.601225i −12.9192 + 5.90000i 0.396327 + 7.05995i
13.3 −1.41061 0.100889i −0.745890 3.42880i 1.97964 + 0.284630i 1.45154 4.78467i 0.706232 + 4.91195i 10.4743 5.71940i −2.76379 0.601225i −3.01364 + 1.37628i −2.53027 + 6.60286i
13.4 −1.41061 0.100889i −0.730822 3.35954i 1.97964 + 0.284630i −1.24754 + 4.84186i 0.691966 + 4.81273i 0.518845 0.283311i −2.76379 0.601225i −2.56569 + 1.17171i 2.24828 6.70412i
13.5 −1.41061 0.100889i −0.160640 0.738452i 1.97964 + 0.284630i −4.82720 1.30312i 0.152100 + 1.05788i 3.79539 2.07244i −2.76379 0.601225i 7.66718 3.50148i 6.67783 + 2.32520i
13.6 −1.41061 0.100889i −0.0934166 0.429429i 1.97964 + 0.284630i 4.81942 1.33161i 0.0884498 + 0.615182i −0.371231 + 0.202708i −2.76379 0.601225i 8.01101 3.65850i −6.93267 + 1.39216i
13.7 −1.41061 0.100889i 0.203005 + 0.933200i 1.97964 + 0.284630i −4.34220 2.47897i −0.192212 1.33686i −8.01935 + 4.37890i −2.76379 0.601225i 7.35704 3.35985i 5.87505 + 3.93494i
13.8 −1.41061 0.100889i 0.313117 + 1.43938i 1.97964 + 0.284630i 0.477701 + 4.97713i −0.296470 2.06199i 7.52566 4.10932i −2.76379 0.601225i 6.21292 2.83735i −0.171714 7.06898i
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.a 240
5.c odd 4 1 inner 230.3.k.a 240
23.c even 11 1 inner 230.3.k.a 240
115.k odd 44 1 inner 230.3.k.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.k.a 240 1.a even 1 1 trivial
230.3.k.a 240 5.c odd 4 1 inner
230.3.k.a 240 23.c even 11 1 inner
230.3.k.a 240 115.k odd 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(20\!\cdots\!64\)\( T_{3}^{224} + \)\(45\!\cdots\!10\)\( T_{3}^{223} - \)\(82\!\cdots\!28\)\( T_{3}^{222} + \)\(23\!\cdots\!90\)\( T_{3}^{221} - \)\(15\!\cdots\!15\)\( T_{3}^{220} - \)\(45\!\cdots\!72\)\( T_{3}^{219} + \)\(38\!\cdots\!70\)\( T_{3}^{218} - \)\(21\!\cdots\!72\)\( T_{3}^{217} + \)\(87\!\cdots\!27\)\( T_{3}^{216} + \)\(42\!\cdots\!02\)\( T_{3}^{215} - \)\(46\!\cdots\!86\)\( T_{3}^{214} + \)\(13\!\cdots\!22\)\( T_{3}^{213} - \)\(31\!\cdots\!77\)\( T_{3}^{212} - \)\(26\!\cdots\!56\)\( T_{3}^{211} + \)\(60\!\cdots\!88\)\( T_{3}^{210} - \)\(99\!\cdots\!28\)\( T_{3}^{209} + \)\(19\!\cdots\!09\)\( T_{3}^{208} + \)\(22\!\cdots\!22\)\( T_{3}^{207} - \)\(43\!\cdots\!30\)\( T_{3}^{206} + \)\(74\!\cdots\!40\)\( T_{3}^{205} - \)\(15\!\cdots\!37\)\( T_{3}^{204} - \)\(13\!\cdots\!20\)\( T_{3}^{203} + \)\(26\!\cdots\!72\)\( T_{3}^{202} - \)\(38\!\cdots\!66\)\( T_{3}^{201} + \)\(18\!\cdots\!03\)\( T_{3}^{200} + \)\(85\!\cdots\!58\)\( T_{3}^{199} - \)\(16\!\cdots\!66\)\( T_{3}^{198} + \)\(23\!\cdots\!62\)\( T_{3}^{197} - \)\(83\!\cdots\!58\)\( T_{3}^{196} - \)\(46\!\cdots\!18\)\( T_{3}^{195} + \)\(10\!\cdots\!94\)\( T_{3}^{194} - \)\(13\!\cdots\!72\)\( T_{3}^{193} + \)\(56\!\cdots\!65\)\( T_{3}^{192} + \)\(21\!\cdots\!38\)\( T_{3}^{191} - \)\(62\!\cdots\!82\)\( T_{3}^{190} + \)\(87\!\cdots\!90\)\( T_{3}^{189} - \)\(21\!\cdots\!77\)\( T_{3}^{188} - \)\(10\!\cdots\!18\)\( T_{3}^{187} + \)\(33\!\cdots\!32\)\( T_{3}^{186} - \)\(44\!\cdots\!58\)\( T_{3}^{185} + \)\(12\!\cdots\!88\)\( T_{3}^{184} + \)\(60\!\cdots\!06\)\( T_{3}^{183} - \)\(16\!\cdots\!36\)\( T_{3}^{182} + \)\(21\!\cdots\!64\)\( T_{3}^{181} - \)\(43\!\cdots\!35\)\( T_{3}^{180} - \)\(38\!\cdots\!82\)\( T_{3}^{179} + \)\(84\!\cdots\!22\)\( T_{3}^{178} - \)\(83\!\cdots\!06\)\( T_{3}^{177} + \)\(24\!\cdots\!22\)\( T_{3}^{176} + \)\(16\!\cdots\!36\)\( T_{3}^{175} - \)\(46\!\cdots\!40\)\( T_{3}^{174} + \)\(37\!\cdots\!00\)\( T_{3}^{173} - \)\(12\!\cdots\!08\)\( T_{3}^{172} - \)\(65\!\cdots\!68\)\( T_{3}^{171} + \)\(24\!\cdots\!28\)\( T_{3}^{170} - \)\(19\!\cdots\!14\)\( T_{3}^{169} + \)\(81\!\cdots\!50\)\( T_{3}^{168} + \)\(15\!\cdots\!24\)\( T_{3}^{167} - \)\(83\!\cdots\!58\)\( T_{3}^{166} + \)\(83\!\cdots\!16\)\( T_{3}^{165} - \)\(44\!\cdots\!74\)\( T_{3}^{164} - \)\(67\!\cdots\!20\)\( T_{3}^{163} + \)\(18\!\cdots\!54\)\( T_{3}^{162} - \)\(23\!\cdots\!90\)\( T_{3}^{161} + \)\(14\!\cdots\!98\)\( T_{3}^{160} - \)\(32\!\cdots\!44\)\( T_{3}^{159} - \)\(46\!\cdots\!48\)\( T_{3}^{158} + \)\(58\!\cdots\!00\)\( T_{3}^{157} - \)\(31\!\cdots\!30\)\( T_{3}^{156} - \)\(87\!\cdots\!58\)\( T_{3}^{155} + \)\(16\!\cdots\!94\)\( T_{3}^{154} - \)\(15\!\cdots\!66\)\( T_{3}^{153} + \)\(58\!\cdots\!86\)\( T_{3}^{152} + \)\(69\!\cdots\!62\)\( T_{3}^{151} - \)\(44\!\cdots\!76\)\( T_{3}^{150} + \)\(36\!\cdots\!66\)\( T_{3}^{149} - \)\(95\!\cdots\!22\)\( T_{3}^{148} - \)\(24\!\cdots\!30\)\( T_{3}^{147} + \)\(10\!\cdots\!64\)\( T_{3}^{146} - \)\(66\!\cdots\!06\)\( T_{3}^{145} + \)\(12\!\cdots\!47\)\( T_{3}^{144} + \)\(55\!\cdots\!08\)\( T_{3}^{143} - \)\(20\!\cdots\!84\)\( T_{3}^{142} + \)\(10\!\cdots\!30\)\( T_{3}^{141} - \)\(77\!\cdots\!51\)\( T_{3}^{140} - \)\(11\!\cdots\!86\)\( T_{3}^{139} + \)\(34\!\cdots\!28\)\( T_{3}^{138} - \)\(16\!\cdots\!10\)\( T_{3}^{137} - \)\(17\!\cdots\!44\)\( T_{3}^{136} + \)\(20\!\cdots\!76\)\( T_{3}^{135} - \)\(52\!\cdots\!86\)\( T_{3}^{134} + \)\(20\!\cdots\!26\)\( T_{3}^{133} + \)\(17\!\cdots\!60\)\( T_{3}^{132} - \)\(28\!\cdots\!84\)\( T_{3}^{131} + \)\(64\!\cdots\!30\)\( T_{3}^{130} - \)\(20\!\cdots\!62\)\( T_{3}^{129} - \)\(43\!\cdots\!05\)\( T_{3}^{128} + \)\(30\!\cdots\!60\)\( T_{3}^{127} - \)\(54\!\cdots\!54\)\( T_{3}^{126} + \)\(14\!\cdots\!30\)\( T_{3}^{125} + \)\(63\!\cdots\!44\)\( T_{3}^{124} - \)\(26\!\cdots\!46\)\( T_{3}^{123} + \)\(29\!\cdots\!20\)\( T_{3}^{122} - \)\(57\!\cdots\!24\)\( T_{3}^{121} - \)\(62\!\cdots\!37\)\( T_{3}^{120} + \)\(18\!\cdots\!36\)\( T_{3}^{119} - \)\(64\!\cdots\!18\)\( T_{3}^{118} + \)\(21\!\cdots\!90\)\( T_{3}^{117} + \)\(49\!\cdots\!52\)\( T_{3}^{116} - \)\(85\!\cdots\!54\)\( T_{3}^{115} + \)\(69\!\cdots\!26\)\( T_{3}^{114} - \)\(69\!\cdots\!96\)\( T_{3}^{113} - \)\(35\!\cdots\!54\)\( T_{3}^{112} + \)\(27\!\cdots\!98\)\( T_{3}^{111} - \)\(34\!\cdots\!66\)\( T_{3}^{110} - \)\(20\!\cdots\!92\)\( T_{3}^{109} + \)\(21\!\cdots\!17\)\( T_{3}^{108} - \)\(73\!\cdots\!66\)\( T_{3}^{107} + \)\(14\!\cdots\!54\)\( T_{3}^{106} + \)\(26\!\cdots\!24\)\( T_{3}^{105} - \)\(84\!\cdots\!45\)\( T_{3}^{104} - \)\(21\!\cdots\!86\)\( T_{3}^{103} + \)\(25\!\cdots\!00\)\( T_{3}^{102} - \)\(91\!\cdots\!90\)\( T_{3}^{101} + \)\(21\!\cdots\!21\)\( T_{3}^{100} + \)\(26\!\cdots\!10\)\( T_{3}^{99} - \)\(50\!\cdots\!62\)\( T_{3}^{98} + \)\(82\!\cdots\!02\)\( T_{3}^{97} - \)\(86\!\cdots\!97\)\( T_{3}^{96} + \)\(85\!\cdots\!30\)\( T_{3}^{95} - \)\(27\!\cdots\!58\)\( T_{3}^{94} + \)\(13\!\cdots\!54\)\( T_{3}^{93} + \)\(58\!\cdots\!36\)\( T_{3}^{92} - \)\(76\!\cdots\!02\)\( T_{3}^{91} + \)\(66\!\cdots\!04\)\( T_{3}^{90} + \)\(14\!\cdots\!48\)\( T_{3}^{89} - \)\(20\!\cdots\!30\)\( T_{3}^{88} + \)\(20\!\cdots\!76\)\( T_{3}^{87} - \)\(74\!\cdots\!46\)\( T_{3}^{86} - \)\(16\!\cdots\!92\)\( T_{3}^{85} + \)\(49\!\cdots\!48\)\( T_{3}^{84} - \)\(39\!\cdots\!32\)\( T_{3}^{83} + \)\(11\!\cdots\!80\)\( T_{3}^{82} - \)\(72\!\cdots\!02\)\( T_{3}^{81} - \)\(89\!\cdots\!15\)\( T_{3}^{80} + \)\(83\!\cdots\!38\)\( T_{3}^{79} + \)\(56\!\cdots\!38\)\( T_{3}^{78} + \)\(20\!\cdots\!86\)\( T_{3}^{77} + \)\(15\!\cdots\!41\)\( T_{3}^{76} + \)\(35\!\cdots\!04\)\( T_{3}^{75} - \)\(52\!\cdots\!30\)\( T_{3}^{74} - \)\(74\!\cdots\!48\)\( T_{3}^{73} - \)\(17\!\cdots\!55\)\( T_{3}^{72} - \)\(17\!\cdots\!38\)\( T_{3}^{71} + \)\(13\!\cdots\!40\)\( T_{3}^{70} + \)\(84\!\cdots\!52\)\( T_{3}^{69} + \)\(26\!\cdots\!46\)\( T_{3}^{68} + \)\(17\!\cdots\!20\)\( T_{3}^{67} - \)\(82\!\cdots\!74\)\( T_{3}^{66} - \)\(90\!\cdots\!76\)\( T_{3}^{65} - \)\(19\!\cdots\!99\)\( T_{3}^{64} - \)\(15\!\cdots\!54\)\( T_{3}^{63} + \)\(78\!\cdots\!50\)\( T_{3}^{62} + \)\(51\!\cdots\!42\)\( T_{3}^{61} + \)\(97\!\cdots\!02\)\( T_{3}^{60} + \)\(73\!\cdots\!08\)\( T_{3}^{59} - \)\(28\!\cdots\!84\)\( T_{3}^{58} - \)\(19\!\cdots\!78\)\( T_{3}^{57} - \)\(31\!\cdots\!47\)\( T_{3}^{56} - \)\(17\!\cdots\!02\)\( T_{3}^{55} + \)\(89\!\cdots\!66\)\( T_{3}^{54} + \)\(42\!\cdots\!34\)\( T_{3}^{53} + \)\(62\!\cdots\!96\)\( T_{3}^{52} + \)\(27\!\cdots\!74\)\( T_{3}^{51} - \)\(29\!\cdots\!72\)\( T_{3}^{50} - \)\(73\!\cdots\!18\)\( T_{3}^{49} - \)\(67\!\cdots\!98\)\( T_{3}^{48} + \)\(14\!\cdots\!82\)\( T_{3}^{47} + \)\(77\!\cdots\!50\)\( T_{3}^{46} + \)\(10\!\cdots\!14\)\( T_{3}^{45} + \)\(45\!\cdots\!41\)\( T_{3}^{44} - \)\(81\!\cdots\!44\)\( T_{3}^{43} - \)\(10\!\cdots\!00\)\( T_{3}^{42} - \)\(43\!\cdots\!26\)\( T_{3}^{41} + \)\(12\!\cdots\!93\)\( T_{3}^{40} + \)\(16\!\cdots\!98\)\( T_{3}^{39} + \)\(12\!\cdots\!94\)\( T_{3}^{38} + \)\(67\!\cdots\!58\)\( T_{3}^{37} - \)\(20\!\cdots\!55\)\( T_{3}^{36} - \)\(82\!\cdots\!24\)\( T_{3}^{35} - \)\(59\!\cdots\!66\)\( T_{3}^{34} - \)\(21\!\cdots\!00\)\( T_{3}^{33} - \)\(11\!\cdots\!58\)\( T_{3}^{32} - \)\(41\!\cdots\!04\)\( T_{3}^{31} + \)\(15\!\cdots\!90\)\( T_{3}^{30} + \)\(37\!\cdots\!64\)\( T_{3}^{29} + \)\(45\!\cdots\!97\)\( T_{3}^{28} + \)\(32\!\cdots\!58\)\( T_{3}^{27} + \)\(18\!\cdots\!14\)\( T_{3}^{26} + \)\(12\!\cdots\!58\)\( T_{3}^{25} + \)\(10\!\cdots\!78\)\( T_{3}^{24} + \)\(64\!\cdots\!48\)\( T_{3}^{23} + \)\(28\!\cdots\!50\)\( T_{3}^{22} + \)\(11\!\cdots\!68\)\( T_{3}^{21} + \)\(63\!\cdots\!22\)\( T_{3}^{20} + \)\(39\!\cdots\!64\)\( T_{3}^{19} + \)\(16\!\cdots\!84\)\( T_{3}^{18} + \)\(27\!\cdots\!20\)\( T_{3}^{17} - \)\(20\!\cdots\!15\)\( T_{3}^{16} - \)\(16\!\cdots\!84\)\( T_{3}^{15} - \)\(60\!\cdots\!86\)\( T_{3}^{14} - \)\(10\!\cdots\!58\)\( T_{3}^{13} + \)\(14\!\cdots\!74\)\( T_{3}^{12} + \)\(14\!\cdots\!80\)\( T_{3}^{11} + \)\(45\!\cdots\!64\)\( T_{3}^{10} + \)\(72\!\cdots\!26\)\( T_{3}^{9} + \)\(53\!\cdots\!51\)\( T_{3}^{8} - \)\(14\!\cdots\!66\)\( T_{3}^{7} + \)\(14\!\cdots\!60\)\( T_{3}^{6} + \)\(54\!\cdots\!30\)\( T_{3}^{5} + \)\(78\!\cdots\!06\)\( T_{3}^{4} + \)\(17\!\cdots\!24\)\( T_{3}^{3} + \)\(24\!\cdots\!18\)\( T_{3}^{2} + \)\(73\!\cdots\!46\)\( T_{3} + \)\(11\!\cdots\!81\)\( \)">\(T_{3}^{240} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(230, [\chi])\).