Newform invariants
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!56\)\( T_{7}^{221} - \)\(71\!\cdots\!54\)\( T_{7}^{220} + \)\(81\!\cdots\!76\)\( T_{7}^{219} + \)\(30\!\cdots\!44\)\( T_{7}^{218} - \)\(57\!\cdots\!36\)\( T_{7}^{217} + \)\(41\!\cdots\!81\)\( T_{7}^{216} - \)\(69\!\cdots\!40\)\( T_{7}^{215} - \)\(23\!\cdots\!28\)\( T_{7}^{214} + \)\(40\!\cdots\!92\)\( T_{7}^{213} - \)\(24\!\cdots\!56\)\( T_{7}^{212} + \)\(51\!\cdots\!04\)\( T_{7}^{211} + \)\(16\!\cdots\!88\)\( T_{7}^{210} - \)\(25\!\cdots\!44\)\( T_{7}^{209} + \)\(13\!\cdots\!63\)\( T_{7}^{208} - \)\(29\!\cdots\!40\)\( T_{7}^{207} - \)\(79\!\cdots\!16\)\( T_{7}^{206} + \)\(11\!\cdots\!44\)\( T_{7}^{205} - \)\(57\!\cdots\!46\)\( T_{7}^{204} + \)\(10\!\cdots\!76\)\( T_{7}^{203} + \)\(34\!\cdots\!48\)\( T_{7}^{202} - \)\(43\!\cdots\!28\)\( T_{7}^{201} + \)\(21\!\cdots\!20\)\( T_{7}^{200} - \)\(39\!\cdots\!64\)\( T_{7}^{199} - \)\(12\!\cdots\!44\)\( T_{7}^{198} + \)\(14\!\cdots\!16\)\( T_{7}^{197} - \)\(64\!\cdots\!90\)\( T_{7}^{196} + \)\(98\!\cdots\!12\)\( T_{7}^{195} + \)\(45\!\cdots\!28\)\( T_{7}^{194} - \)\(45\!\cdots\!44\)\( T_{7}^{193} + \)\(19\!\cdots\!79\)\( T_{7}^{192} - \)\(25\!\cdots\!00\)\( T_{7}^{191} - \)\(13\!\cdots\!32\)\( T_{7}^{190} + \)\(11\!\cdots\!24\)\( T_{7}^{189} - \)\(43\!\cdots\!84\)\( T_{7}^{188} + \)\(28\!\cdots\!96\)\( T_{7}^{187} + \)\(38\!\cdots\!12\)\( T_{7}^{186} - \)\(26\!\cdots\!72\)\( T_{7}^{185} + \)\(91\!\cdots\!47\)\( T_{7}^{184} - \)\(37\!\cdots\!08\)\( T_{7}^{183} - \)\(79\!\cdots\!16\)\( T_{7}^{182} + \)\(52\!\cdots\!92\)\( T_{7}^{181} - \)\(17\!\cdots\!28\)\( T_{7}^{180} - \)\(67\!\cdots\!08\)\( T_{7}^{179} + \)\(17\!\cdots\!68\)\( T_{7}^{178} - \)\(90\!\cdots\!52\)\( T_{7}^{177} + \)\(24\!\cdots\!58\)\( T_{7}^{176} + \)\(31\!\cdots\!76\)\( T_{7}^{175} - \)\(26\!\cdots\!20\)\( T_{7}^{174} + \)\(14\!\cdots\!44\)\( T_{7}^{173} - \)\(37\!\cdots\!31\)\( T_{7}^{172} - \)\(77\!\cdots\!32\)\( T_{7}^{171} + \)\(46\!\cdots\!36\)\( T_{7}^{170} - \)\(16\!\cdots\!52\)\( T_{7}^{169} + \)\(34\!\cdots\!93\)\( T_{7}^{168} + \)\(17\!\cdots\!36\)\( T_{7}^{167} - \)\(41\!\cdots\!04\)\( T_{7}^{166} + \)\(21\!\cdots\!52\)\( T_{7}^{165} - \)\(15\!\cdots\!17\)\( T_{7}^{164} - \)\(24\!\cdots\!48\)\( T_{7}^{163} + \)\(46\!\cdots\!08\)\( T_{7}^{162} - \)\(37\!\cdots\!52\)\( T_{7}^{161} + \)\(22\!\cdots\!91\)\( T_{7}^{160} + \)\(27\!\cdots\!04\)\( T_{7}^{159} + \)\(15\!\cdots\!20\)\( T_{7}^{158} + \)\(23\!\cdots\!24\)\( T_{7}^{157} + \)\(76\!\cdots\!80\)\( T_{7}^{156} - \)\(15\!\cdots\!28\)\( T_{7}^{155} + \)\(47\!\cdots\!32\)\( T_{7}^{154} + \)\(84\!\cdots\!84\)\( T_{7}^{153} + \)\(21\!\cdots\!66\)\( T_{7}^{152} + \)\(12\!\cdots\!64\)\( T_{7}^{151} + \)\(32\!\cdots\!16\)\( T_{7}^{150} + \)\(14\!\cdots\!92\)\( T_{7}^{149} + \)\(72\!\cdots\!24\)\( T_{7}^{148} + \)\(32\!\cdots\!16\)\( T_{7}^{147} - \)\(11\!\cdots\!20\)\( T_{7}^{146} + \)\(31\!\cdots\!72\)\( T_{7}^{145} + \)\(31\!\cdots\!15\)\( T_{7}^{144} + \)\(76\!\cdots\!48\)\( T_{7}^{143} + \)\(17\!\cdots\!16\)\( T_{7}^{142} - \)\(79\!\cdots\!76\)\( T_{7}^{141} - \)\(35\!\cdots\!78\)\( T_{7}^{140} - \)\(15\!\cdots\!88\)\( T_{7}^{139} - \)\(54\!\cdots\!60\)\( T_{7}^{138} - \)\(12\!\cdots\!12\)\( T_{7}^{137} + \)\(74\!\cdots\!03\)\( T_{7}^{136} + \)\(13\!\cdots\!96\)\( T_{7}^{135} + \)\(62\!\cdots\!00\)\( T_{7}^{134} + \)\(17\!\cdots\!92\)\( T_{7}^{133} + \)\(32\!\cdots\!65\)\( T_{7}^{132} + \)\(23\!\cdots\!28\)\( T_{7}^{131} - \)\(11\!\cdots\!84\)\( T_{7}^{130} - \)\(75\!\cdots\!08\)\( T_{7}^{129} - \)\(27\!\cdots\!45\)\( T_{7}^{128} - \)\(66\!\cdots\!12\)\( T_{7}^{127} - \)\(88\!\cdots\!04\)\( T_{7}^{126} + \)\(30\!\cdots\!24\)\( T_{7}^{125} + \)\(43\!\cdots\!04\)\( T_{7}^{124} + \)\(11\!\cdots\!16\)\( T_{7}^{123} + \)\(24\!\cdots\!80\)\( T_{7}^{122} + \)\(58\!\cdots\!52\)\( T_{7}^{121} + \)\(13\!\cdots\!82\)\( T_{7}^{120} + \)\(18\!\cdots\!48\)\( T_{7}^{119} + \)\(19\!\cdots\!36\)\( T_{7}^{118} + \)\(11\!\cdots\!84\)\( T_{7}^{117} + \)\(59\!\cdots\!30\)\( T_{7}^{116} + \)\(13\!\cdots\!28\)\( T_{7}^{115} + \)\(92\!\cdots\!44\)\( T_{7}^{114} - \)\(29\!\cdots\!84\)\( T_{7}^{113} - \)\(40\!\cdots\!18\)\( T_{7}^{112} + \)\(12\!\cdots\!40\)\( T_{7}^{111} + \)\(40\!\cdots\!52\)\( T_{7}^{110} + \)\(11\!\cdots\!28\)\( T_{7}^{109} - \)\(85\!\cdots\!42\)\( T_{7}^{108} - \)\(99\!\cdots\!20\)\( T_{7}^{107} + \)\(60\!\cdots\!68\)\( T_{7}^{106} + \)\(86\!\cdots\!48\)\( T_{7}^{105} - \)\(12\!\cdots\!75\)\( T_{7}^{104} - \)\(60\!\cdots\!00\)\( T_{7}^{103} - \)\(54\!\cdots\!64\)\( T_{7}^{102} + \)\(89\!\cdots\!76\)\( T_{7}^{101} + \)\(39\!\cdots\!11\)\( T_{7}^{100} + \)\(60\!\cdots\!40\)\( T_{7}^{99} + \)\(44\!\cdots\!56\)\( T_{7}^{98} - \)\(12\!\cdots\!76\)\( T_{7}^{97} - \)\(58\!\cdots\!43\)\( T_{7}^{96} - \)\(11\!\cdots\!76\)\( T_{7}^{95} - \)\(31\!\cdots\!32\)\( T_{7}^{94} + \)\(35\!\cdots\!28\)\( T_{7}^{93} + \)\(85\!\cdots\!54\)\( T_{7}^{92} + \)\(47\!\cdots\!80\)\( T_{7}^{91} - \)\(14\!\cdots\!16\)\( T_{7}^{90} - \)\(29\!\cdots\!72\)\( T_{7}^{89} + \)\(91\!\cdots\!80\)\( T_{7}^{88} + \)\(93\!\cdots\!68\)\( T_{7}^{87} + \)\(60\!\cdots\!84\)\( T_{7}^{86} - \)\(26\!\cdots\!16\)\( T_{7}^{85} - \)\(52\!\cdots\!43\)\( T_{7}^{84} + \)\(30\!\cdots\!48\)\( T_{7}^{83} + \)\(22\!\cdots\!72\)\( T_{7}^{82} + \)\(15\!\cdots\!88\)\( T_{7}^{81} - \)\(79\!\cdots\!93\)\( T_{7}^{80} - \)\(22\!\cdots\!52\)\( T_{7}^{79} - \)\(15\!\cdots\!72\)\( T_{7}^{78} + \)\(49\!\cdots\!52\)\( T_{7}^{77} + \)\(16\!\cdots\!30\)\( T_{7}^{76} + \)\(18\!\cdots\!96\)\( T_{7}^{75} - \)\(53\!\cdots\!76\)\( T_{7}^{74} - \)\(59\!\cdots\!60\)\( T_{7}^{73} - \)\(10\!\cdots\!29\)\( T_{7}^{72} - \)\(62\!\cdots\!84\)\( T_{7}^{71} + \)\(10\!\cdots\!28\)\( T_{7}^{70} + \)\(30\!\cdots\!12\)\( T_{7}^{69} + \)\(31\!\cdots\!00\)\( T_{7}^{68} - \)\(37\!\cdots\!20\)\( T_{7}^{67} - \)\(56\!\cdots\!44\)\( T_{7}^{66} - \)\(76\!\cdots\!24\)\( T_{7}^{65} - \)\(21\!\cdots\!82\)\( T_{7}^{64} + \)\(87\!\cdots\!28\)\( T_{7}^{63} + \)\(17\!\cdots\!40\)\( T_{7}^{62} + \)\(15\!\cdots\!84\)\( T_{7}^{61} + \)\(53\!\cdots\!53\)\( T_{7}^{60} - \)\(55\!\cdots\!40\)\( T_{7}^{59} - \)\(96\!\cdots\!32\)\( T_{7}^{58} - \)\(67\!\cdots\!88\)\( T_{7}^{57} - \)\(20\!\cdots\!36\)\( T_{7}^{56} + \)\(78\!\cdots\!16\)\( T_{7}^{55} - \)\(50\!\cdots\!44\)\( T_{7}^{54} - \)\(11\!\cdots\!20\)\( T_{7}^{53} - \)\(41\!\cdots\!35\)\( T_{7}^{52} + \)\(77\!\cdots\!88\)\( T_{7}^{51} + \)\(11\!\cdots\!16\)\( T_{7}^{50} + \)\(70\!\cdots\!20\)\( T_{7}^{49} + \)\(12\!\cdots\!05\)\( T_{7}^{48} + \)\(17\!\cdots\!32\)\( T_{7}^{47} + \)\(18\!\cdots\!80\)\( T_{7}^{46} + \)\(30\!\cdots\!56\)\( T_{7}^{45} + \)\(10\!\cdots\!58\)\( T_{7}^{44} - \)\(20\!\cdots\!76\)\( T_{7}^{43} - \)\(19\!\cdots\!36\)\( T_{7}^{42} - \)\(18\!\cdots\!32\)\( T_{7}^{41} - \)\(54\!\cdots\!35\)\( T_{7}^{40} + \)\(65\!\cdots\!76\)\( T_{7}^{39} + \)\(22\!\cdots\!20\)\( T_{7}^{38} + \)\(37\!\cdots\!00\)\( T_{7}^{37} + \)\(18\!\cdots\!88\)\( T_{7}^{36} - \)\(45\!\cdots\!68\)\( T_{7}^{35} + \)\(40\!\cdots\!76\)\( T_{7}^{34} - \)\(17\!\cdots\!20\)\( T_{7}^{33} - \)\(16\!\cdots\!76\)\( T_{7}^{32} + \)\(99\!\cdots\!44\)\( T_{7}^{31} - \)\(71\!\cdots\!16\)\( T_{7}^{30} + \)\(43\!\cdots\!64\)\( T_{7}^{29} + \)\(15\!\cdots\!72\)\( T_{7}^{28} - \)\(21\!\cdots\!64\)\( T_{7}^{27} - \)\(71\!\cdots\!60\)\( T_{7}^{26} - \)\(73\!\cdots\!64\)\( T_{7}^{25} - \)\(54\!\cdots\!48\)\( T_{7}^{24} + \)\(17\!\cdots\!44\)\( T_{7}^{23} - \)\(11\!\cdots\!96\)\( T_{7}^{22} + \)\(17\!\cdots\!04\)\( T_{7}^{21} + \)\(58\!\cdots\!48\)\( T_{7}^{20} - \)\(20\!\cdots\!96\)\( T_{7}^{19} + \)\(25\!\cdots\!24\)\( T_{7}^{18} + \)\(21\!\cdots\!40\)\( T_{7}^{17} - \)\(19\!\cdots\!96\)\( T_{7}^{16} - \)\(31\!\cdots\!64\)\( T_{7}^{15} + \)\(16\!\cdots\!76\)\( T_{7}^{14} - \)\(68\!\cdots\!92\)\( T_{7}^{13} + \)\(11\!\cdots\!88\)\( T_{7}^{12} + \)\(98\!\cdots\!32\)\( T_{7}^{11} - \)\(53\!\cdots\!56\)\( T_{7}^{10} + \)\(44\!\cdots\!56\)\( T_{7}^{9} + \)\(16\!\cdots\!08\)\( T_{7}^{8} - \)\(18\!\cdots\!00\)\( T_{7}^{7} + \)\(90\!\cdots\!68\)\( T_{7}^{6} - \)\(35\!\cdots\!44\)\( T_{7}^{5} + \)\(11\!\cdots\!48\)\( T_{7}^{4} - \)\(27\!\cdots\!16\)\( T_{7}^{3} + \)\(51\!\cdots\!28\)\( T_{7}^{2} - \)\(75\!\cdots\!76\)\( T_{7} + \)\(55\!\cdots\!96\)\( \)">\(T_{7}^{240} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).