Properties

Label 930.2.bt.a
Level $930$
Weight $2$
Character orbit 930.bt
Analytic conductor $7.426$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bt (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(16\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256q + 8q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 256q + 8q^{7} + 4q^{10} - 24q^{14} - 8q^{15} + 64q^{16} + 36q^{17} - 40q^{19} + 12q^{20} - 16q^{21} + 8q^{22} - 32q^{24} + 20q^{25} + 4q^{28} - 16q^{29} - 8q^{31} - 4q^{33} - 24q^{35} + 128q^{36} + 4q^{37} + 4q^{38} + 48q^{41} + 32q^{42} - 24q^{43} + 4q^{44} - 12q^{45} - 20q^{46} - 168q^{47} - 40q^{49} + 48q^{50} - 64q^{53} + 64q^{54} + 64q^{55} + 28q^{57} + 68q^{58} - 24q^{59} + 4q^{62} + 4q^{63} - 60q^{65} + 12q^{66} - 76q^{67} - 4q^{68} + 12q^{69} - 16q^{70} - 48q^{71} + 52q^{73} + 68q^{74} + 48q^{75} + 48q^{76} + 84q^{77} - 56q^{79} - 32q^{81} - 32q^{82} + 136q^{83} + 68q^{85} + 24q^{86} + 20q^{87} + 56q^{88} - 224q^{89} + 40q^{91} - 48q^{93} - 4q^{95} + 8q^{97} + 64q^{98} + 4q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i −1.68012 1.47553i 0.866025 + 0.500000i −1.74433 0.0914165i 0.453990 0.891007i −0.743145 0.669131i 1.72019 1.42861i
13.2 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i −1.65687 + 1.50160i 0.866025 + 0.500000i −2.43455 0.127590i 0.453990 0.891007i −0.743145 0.669131i −1.22392 1.87137i
13.3 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i −1.65366 + 1.50512i 0.866025 + 0.500000i 0.140670 + 0.00737222i 0.453990 0.891007i −0.743145 0.669131i −1.22790 1.86876i
13.4 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i −1.12100 1.93477i 0.866025 + 0.500000i 0.00733325 0.000384319i 0.453990 0.891007i −0.743145 0.669131i 2.08632 0.804537i
13.5 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i 1.57500 1.58726i 0.866025 + 0.500000i 4.42218 + 0.231757i 0.453990 0.891007i −0.743145 0.669131i 1.32133 + 1.80391i
13.6 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i 1.66702 + 1.49031i 0.866025 + 0.500000i 2.09913 + 0.110011i 0.453990 0.891007i −0.743145 0.669131i −1.73274 + 1.41336i
13.7 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i 2.05562 + 0.880016i 0.866025 + 0.500000i −0.216795 0.0113618i 0.453990 0.891007i −0.743145 0.669131i −1.19075 + 1.89265i
13.8 −0.156434 + 0.987688i 0.358368 0.933580i −0.951057 0.309017i 2.21938 + 0.272664i 0.866025 + 0.500000i −4.57122 0.239568i 0.453990 0.891007i −0.743145 0.669131i −0.616494 + 2.14940i
13.9 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i −2.21293 0.320863i 0.866025 + 0.500000i −2.17593 0.114036i −0.453990 + 0.891007i −0.743145 0.669131i −0.663091 + 2.13549i
13.10 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i −1.90371 + 1.17298i 0.866025 + 0.500000i 1.91384 + 0.100300i −0.453990 + 0.891007i −0.743145 0.669131i 0.860728 + 2.06377i
13.11 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i −1.16716 1.90728i 0.866025 + 0.500000i 2.78313 + 0.145858i −0.453990 + 0.891007i −0.743145 0.669131i −2.06639 + 0.854429i
13.12 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i −0.443582 + 2.19163i 0.866025 + 0.500000i −2.50424 0.131242i −0.453990 + 0.891007i −0.743145 0.669131i 2.09525 + 0.780967i
13.13 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i 0.904456 2.04498i 0.866025 + 0.500000i −2.34346 0.122815i −0.453990 + 0.891007i −0.743145 0.669131i −1.87832 1.21323i
13.14 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i 1.00881 + 1.99557i 0.866025 + 0.500000i 4.83852 + 0.253576i −0.453990 + 0.891007i −0.743145 0.669131i 2.12882 0.684211i
13.15 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i 1.45393 1.69884i 0.866025 + 0.500000i 3.84003 + 0.201247i −0.453990 + 0.891007i −0.743145 0.669131i −1.45048 1.70179i
13.16 0.156434 0.987688i −0.358368 + 0.933580i −0.951057 0.309017i 1.76830 + 1.36862i 0.866025 + 0.500000i −0.704223 0.0369068i −0.453990 + 0.891007i −0.743145 0.669131i 1.62840 1.53243i
43.1 −0.987688 + 0.156434i 0.933580 0.358368i 0.951057 0.309017i −1.95765 + 1.08055i −0.866025 + 0.500000i −0.179844 3.43163i −0.891007 + 0.453990i 0.743145 0.669131i 1.76452 1.37349i
43.2 −0.987688 + 0.156434i 0.933580 0.358368i 0.951057 0.309017i −1.92766 + 1.13319i −0.866025 + 0.500000i 0.0854612 + 1.63070i −0.891007 + 0.453990i 0.743145 0.669131i 1.72666 1.42079i
43.3 −0.987688 + 0.156434i 0.933580 0.358368i 0.951057 0.309017i −1.71760 1.43173i −0.866025 + 0.500000i 0.0366609 + 0.699531i −0.891007 + 0.453990i 0.743145 0.669131i 1.92043 + 1.14541i
43.4 −0.987688 + 0.156434i 0.933580 0.358368i 0.951057 0.309017i −0.387188 + 2.20229i −0.866025 + 0.500000i −0.0918879 1.75333i −0.891007 + 0.453990i 0.743145 0.669131i 0.0379066 2.23575i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 823.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bt.a 256
5.c odd 4 1 930.2.bt.b yes 256
31.h odd 30 1 930.2.bt.b yes 256
155.x even 60 1 inner 930.2.bt.a 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bt.a 256 1.a even 1 1 trivial
930.2.bt.a 256 155.x even 60 1 inner
930.2.bt.b yes 256 5.c odd 4 1
930.2.bt.b yes 256 31.h odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!70\)\( T_{7}^{240} - \)\(80\!\cdots\!96\)\( T_{7}^{239} + \)\(33\!\cdots\!60\)\( T_{7}^{238} - \)\(10\!\cdots\!40\)\( T_{7}^{237} + \)\(33\!\cdots\!14\)\( T_{7}^{236} - \)\(13\!\cdots\!36\)\( T_{7}^{235} + \)\(36\!\cdots\!40\)\( T_{7}^{234} + \)\(55\!\cdots\!12\)\( T_{7}^{233} - \)\(10\!\cdots\!17\)\( T_{7}^{232} + \)\(71\!\cdots\!36\)\( T_{7}^{231} - \)\(42\!\cdots\!14\)\( T_{7}^{230} + \)\(25\!\cdots\!08\)\( T_{7}^{229} - \)\(14\!\cdots\!89\)\( T_{7}^{228} + \)\(73\!\cdots\!96\)\( T_{7}^{227} - \)\(35\!\cdots\!70\)\( T_{7}^{226} + \)\(16\!\cdots\!72\)\( T_{7}^{225} - \)\(74\!\cdots\!54\)\( T_{7}^{224} + \)\(33\!\cdots\!24\)\( T_{7}^{223} - \)\(14\!\cdots\!42\)\( T_{7}^{222} + \)\(55\!\cdots\!32\)\( T_{7}^{221} - \)\(21\!\cdots\!87\)\( T_{7}^{220} + \)\(80\!\cdots\!68\)\( T_{7}^{219} - \)\(27\!\cdots\!94\)\( T_{7}^{218} + \)\(71\!\cdots\!16\)\( T_{7}^{217} - \)\(96\!\cdots\!09\)\( T_{7}^{216} - \)\(46\!\cdots\!04\)\( T_{7}^{215} + \)\(60\!\cdots\!38\)\( T_{7}^{214} - \)\(44\!\cdots\!00\)\( T_{7}^{213} + \)\(27\!\cdots\!45\)\( T_{7}^{212} - \)\(15\!\cdots\!40\)\( T_{7}^{211} + \)\(77\!\cdots\!26\)\( T_{7}^{210} - \)\(38\!\cdots\!40\)\( T_{7}^{209} + \)\(18\!\cdots\!52\)\( T_{7}^{208} - \)\(86\!\cdots\!84\)\( T_{7}^{207} + \)\(39\!\cdots\!86\)\( T_{7}^{206} - \)\(17\!\cdots\!60\)\( T_{7}^{205} + \)\(76\!\cdots\!95\)\( T_{7}^{204} - \)\(32\!\cdots\!08\)\( T_{7}^{203} + \)\(13\!\cdots\!76\)\( T_{7}^{202} - \)\(57\!\cdots\!56\)\( T_{7}^{201} + \)\(23\!\cdots\!89\)\( T_{7}^{200} - \)\(95\!\cdots\!84\)\( T_{7}^{199} + \)\(37\!\cdots\!58\)\( T_{7}^{198} - \)\(14\!\cdots\!20\)\( T_{7}^{197} + \)\(57\!\cdots\!37\)\( T_{7}^{196} - \)\(21\!\cdots\!04\)\( T_{7}^{195} + \)\(81\!\cdots\!00\)\( T_{7}^{194} - \)\(30\!\cdots\!52\)\( T_{7}^{193} + \)\(11\!\cdots\!90\)\( T_{7}^{192} - \)\(39\!\cdots\!08\)\( T_{7}^{191} + \)\(13\!\cdots\!16\)\( T_{7}^{190} - \)\(48\!\cdots\!20\)\( T_{7}^{189} + \)\(16\!\cdots\!16\)\( T_{7}^{188} - \)\(55\!\cdots\!04\)\( T_{7}^{187} + \)\(17\!\cdots\!80\)\( T_{7}^{186} - \)\(57\!\cdots\!56\)\( T_{7}^{185} + \)\(17\!\cdots\!33\)\( T_{7}^{184} - \)\(52\!\cdots\!24\)\( T_{7}^{183} + \)\(15\!\cdots\!58\)\( T_{7}^{182} - \)\(41\!\cdots\!32\)\( T_{7}^{181} + \)\(10\!\cdots\!47\)\( T_{7}^{180} - \)\(23\!\cdots\!76\)\( T_{7}^{179} + \)\(45\!\cdots\!40\)\( T_{7}^{178} - \)\(43\!\cdots\!16\)\( T_{7}^{177} - \)\(13\!\cdots\!93\)\( T_{7}^{176} + \)\(11\!\cdots\!04\)\( T_{7}^{175} - \)\(56\!\cdots\!98\)\( T_{7}^{174} + \)\(22\!\cdots\!12\)\( T_{7}^{173} - \)\(72\!\cdots\!89\)\( T_{7}^{172} + \)\(23\!\cdots\!68\)\( T_{7}^{171} - \)\(66\!\cdots\!92\)\( T_{7}^{170} + \)\(17\!\cdots\!40\)\( T_{7}^{169} - \)\(39\!\cdots\!57\)\( T_{7}^{168} + \)\(85\!\cdots\!68\)\( T_{7}^{167} - \)\(12\!\cdots\!32\)\( T_{7}^{166} - \)\(17\!\cdots\!36\)\( T_{7}^{165} + \)\(95\!\cdots\!92\)\( T_{7}^{164} - \)\(44\!\cdots\!32\)\( T_{7}^{163} + \)\(18\!\cdots\!32\)\( T_{7}^{162} - \)\(58\!\cdots\!80\)\( T_{7}^{161} + \)\(15\!\cdots\!75\)\( T_{7}^{160} - \)\(39\!\cdots\!56\)\( T_{7}^{159} + \)\(91\!\cdots\!22\)\( T_{7}^{158} - \)\(13\!\cdots\!68\)\( T_{7}^{157} + \)\(85\!\cdots\!85\)\( T_{7}^{156} - \)\(57\!\cdots\!20\)\( T_{7}^{155} - \)\(18\!\cdots\!34\)\( T_{7}^{154} + \)\(10\!\cdots\!00\)\( T_{7}^{153} - \)\(30\!\cdots\!63\)\( T_{7}^{152} + \)\(53\!\cdots\!84\)\( T_{7}^{151} - \)\(13\!\cdots\!98\)\( T_{7}^{150} + \)\(18\!\cdots\!28\)\( T_{7}^{149} - \)\(30\!\cdots\!19\)\( T_{7}^{148} - \)\(34\!\cdots\!12\)\( T_{7}^{147} + \)\(10\!\cdots\!30\)\( T_{7}^{146} - \)\(14\!\cdots\!12\)\( T_{7}^{145} + \)\(40\!\cdots\!58\)\( T_{7}^{144} + \)\(27\!\cdots\!52\)\( T_{7}^{143} - \)\(35\!\cdots\!30\)\( T_{7}^{142} + \)\(62\!\cdots\!32\)\( T_{7}^{141} - \)\(35\!\cdots\!87\)\( T_{7}^{140} + \)\(32\!\cdots\!40\)\( T_{7}^{139} - \)\(47\!\cdots\!84\)\( T_{7}^{138} - \)\(11\!\cdots\!24\)\( T_{7}^{137} + \)\(79\!\cdots\!00\)\( T_{7}^{136} - \)\(14\!\cdots\!48\)\( T_{7}^{135} + \)\(82\!\cdots\!10\)\( T_{7}^{134} - \)\(15\!\cdots\!04\)\( T_{7}^{133} + \)\(12\!\cdots\!81\)\( T_{7}^{132} + \)\(24\!\cdots\!84\)\( T_{7}^{131} + \)\(76\!\cdots\!48\)\( T_{7}^{130} + \)\(46\!\cdots\!04\)\( T_{7}^{129} - \)\(20\!\cdots\!85\)\( T_{7}^{128} + \)\(25\!\cdots\!96\)\( T_{7}^{127} - \)\(18\!\cdots\!60\)\( T_{7}^{126} + \)\(12\!\cdots\!00\)\( T_{7}^{125} - \)\(34\!\cdots\!00\)\( T_{7}^{124} - \)\(36\!\cdots\!24\)\( T_{7}^{123} + \)\(25\!\cdots\!78\)\( T_{7}^{122} - \)\(87\!\cdots\!88\)\( T_{7}^{121} + \)\(12\!\cdots\!23\)\( T_{7}^{120} + \)\(33\!\cdots\!76\)\( T_{7}^{119} - \)\(59\!\cdots\!26\)\( T_{7}^{118} + \)\(56\!\cdots\!40\)\( T_{7}^{117} - \)\(76\!\cdots\!89\)\( T_{7}^{116} + \)\(80\!\cdots\!16\)\( T_{7}^{115} - \)\(17\!\cdots\!92\)\( T_{7}^{114} - \)\(20\!\cdots\!52\)\( T_{7}^{113} + \)\(17\!\cdots\!85\)\( T_{7}^{112} - \)\(26\!\cdots\!52\)\( T_{7}^{111} + \)\(30\!\cdots\!56\)\( T_{7}^{110} + \)\(70\!\cdots\!72\)\( T_{7}^{109} - \)\(19\!\cdots\!64\)\( T_{7}^{108} + \)\(47\!\cdots\!60\)\( T_{7}^{107} - \)\(47\!\cdots\!70\)\( T_{7}^{106} - \)\(13\!\cdots\!32\)\( T_{7}^{105} + \)\(53\!\cdots\!87\)\( T_{7}^{104} - \)\(67\!\cdots\!56\)\( T_{7}^{103} - \)\(62\!\cdots\!58\)\( T_{7}^{102} + \)\(38\!\cdots\!04\)\( T_{7}^{101} - \)\(71\!\cdots\!21\)\( T_{7}^{100} + \)\(29\!\cdots\!96\)\( T_{7}^{99} + \)\(17\!\cdots\!20\)\( T_{7}^{98} - \)\(49\!\cdots\!44\)\( T_{7}^{97} + \)\(54\!\cdots\!08\)\( T_{7}^{96} + \)\(25\!\cdots\!12\)\( T_{7}^{95} - \)\(21\!\cdots\!20\)\( T_{7}^{94} + \)\(42\!\cdots\!72\)\( T_{7}^{93} - \)\(30\!\cdots\!52\)\( T_{7}^{92} - \)\(73\!\cdots\!32\)\( T_{7}^{91} + \)\(28\!\cdots\!94\)\( T_{7}^{90} - \)\(38\!\cdots\!48\)\( T_{7}^{89} + \)\(33\!\cdots\!29\)\( T_{7}^{88} + \)\(75\!\cdots\!84\)\( T_{7}^{87} - \)\(10\!\cdots\!42\)\( T_{7}^{86} - \)\(13\!\cdots\!48\)\( T_{7}^{85} + \)\(32\!\cdots\!91\)\( T_{7}^{84} - \)\(56\!\cdots\!36\)\( T_{7}^{83} + \)\(36\!\cdots\!74\)\( T_{7}^{82} + \)\(34\!\cdots\!92\)\( T_{7}^{81} - \)\(13\!\cdots\!24\)\( T_{7}^{80} + \)\(34\!\cdots\!48\)\( T_{7}^{79} - \)\(81\!\cdots\!76\)\( T_{7}^{78} + \)\(14\!\cdots\!20\)\( T_{7}^{77} - \)\(20\!\cdots\!04\)\( T_{7}^{76} + \)\(25\!\cdots\!92\)\( T_{7}^{75} - \)\(30\!\cdots\!16\)\( T_{7}^{74} + \)\(32\!\cdots\!92\)\( T_{7}^{73} - \)\(18\!\cdots\!43\)\( T_{7}^{72} - \)\(23\!\cdots\!92\)\( T_{7}^{71} + \)\(12\!\cdots\!12\)\( T_{7}^{70} - \)\(33\!\cdots\!32\)\( T_{7}^{69} + \)\(75\!\cdots\!68\)\( T_{7}^{68} - \)\(14\!\cdots\!60\)\( T_{7}^{67} + \)\(25\!\cdots\!86\)\( T_{7}^{66} - \)\(38\!\cdots\!64\)\( T_{7}^{65} + \)\(54\!\cdots\!10\)\( T_{7}^{64} - \)\(70\!\cdots\!36\)\( T_{7}^{63} + \)\(83\!\cdots\!76\)\( T_{7}^{62} - \)\(91\!\cdots\!56\)\( T_{7}^{61} + \)\(91\!\cdots\!66\)\( T_{7}^{60} - \)\(84\!\cdots\!44\)\( T_{7}^{59} + \)\(72\!\cdots\!76\)\( T_{7}^{58} - \)\(59\!\cdots\!28\)\( T_{7}^{57} + \)\(46\!\cdots\!92\)\( T_{7}^{56} - \)\(34\!\cdots\!48\)\( T_{7}^{55} + \)\(25\!\cdots\!54\)\( T_{7}^{54} - \)\(17\!\cdots\!88\)\( T_{7}^{53} + \)\(11\!\cdots\!11\)\( T_{7}^{52} - \)\(61\!\cdots\!24\)\( T_{7}^{51} - \)\(18\!\cdots\!62\)\( T_{7}^{50} + \)\(56\!\cdots\!80\)\( T_{7}^{49} - \)\(75\!\cdots\!45\)\( T_{7}^{48} + \)\(52\!\cdots\!64\)\( T_{7}^{47} - \)\(83\!\cdots\!02\)\( T_{7}^{46} - \)\(24\!\cdots\!08\)\( T_{7}^{45} + \)\(28\!\cdots\!51\)\( T_{7}^{44} - \)\(13\!\cdots\!32\)\( T_{7}^{43} - \)\(41\!\cdots\!24\)\( T_{7}^{42} + \)\(12\!\cdots\!72\)\( T_{7}^{41} - \)\(12\!\cdots\!44\)\( T_{7}^{40} + \)\(81\!\cdots\!80\)\( T_{7}^{39} - \)\(38\!\cdots\!38\)\( T_{7}^{38} + \)\(10\!\cdots\!64\)\( T_{7}^{37} + \)\(43\!\cdots\!41\)\( T_{7}^{36} - \)\(10\!\cdots\!00\)\( T_{7}^{35} + \)\(95\!\cdots\!66\)\( T_{7}^{34} - \)\(62\!\cdots\!68\)\( T_{7}^{33} + \)\(30\!\cdots\!57\)\( T_{7}^{32} - \)\(10\!\cdots\!16\)\( T_{7}^{31} + \)\(31\!\cdots\!88\)\( T_{7}^{30} - \)\(10\!\cdots\!76\)\( T_{7}^{29} + \)\(48\!\cdots\!28\)\( T_{7}^{28} - \)\(21\!\cdots\!84\)\( T_{7}^{27} + \)\(92\!\cdots\!28\)\( T_{7}^{26} - \)\(70\!\cdots\!48\)\( T_{7}^{25} + \)\(65\!\cdots\!20\)\( T_{7}^{24} - \)\(45\!\cdots\!32\)\( T_{7}^{23} + \)\(21\!\cdots\!32\)\( T_{7}^{22} - \)\(62\!\cdots\!80\)\( T_{7}^{21} + \)\(63\!\cdots\!76\)\( T_{7}^{20} + \)\(30\!\cdots\!16\)\( T_{7}^{19} - \)\(14\!\cdots\!04\)\( T_{7}^{18} + \)\(24\!\cdots\!00\)\( T_{7}^{17} + \)\(12\!\cdots\!80\)\( T_{7}^{16} - \)\(11\!\cdots\!48\)\( T_{7}^{15} + \)\(14\!\cdots\!24\)\( T_{7}^{14} + \)\(20\!\cdots\!36\)\( T_{7}^{13} - \)\(72\!\cdots\!56\)\( T_{7}^{12} + \)\(39\!\cdots\!88\)\( T_{7}^{11} + \)\(14\!\cdots\!08\)\( T_{7}^{10} - \)\(37\!\cdots\!52\)\( T_{7}^{9} + \)\(37\!\cdots\!68\)\( T_{7}^{8} - \)\(21\!\cdots\!32\)\( T_{7}^{7} + \)\(16\!\cdots\!32\)\( T_{7}^{6} - \)\(13\!\cdots\!80\)\( T_{7}^{5} + \)\(66\!\cdots\!72\)\( T_{7}^{4} - \)\(18\!\cdots\!68\)\( T_{7}^{3} + \)\(53\!\cdots\!96\)\( T_{7}^{2} - \)\(56\!\cdots\!64\)\( T_{7} + \)\(18\!\cdots\!36\)\( \)">\(T_{7}^{256} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).