Properties

Label 605.2.k.b
Level $605$
Weight $2$
Character orbit 605.k
Analytic conductor $4.831$
Analytic rank $0$
Dimension $220$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.k (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220q - 2q^{2} - 24q^{4} + 22q^{5} + 8q^{6} + 4q^{7} - 6q^{8} + 212q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 220q - 2q^{2} - 24q^{4} + 22q^{5} + 8q^{6} + 4q^{7} - 6q^{8} + 212q^{9} + 2q^{10} - 2q^{11} + 49q^{12} + 8q^{13} - 40q^{14} + 11q^{15} - 28q^{16} - 8q^{17} - 10q^{18} + 24q^{20} - 22q^{21} - 79q^{22} - 31q^{23} - 36q^{24} - 22q^{25} - 6q^{26} - 6q^{27} + 4q^{28} - 4q^{29} - 19q^{30} + 20q^{31} - 104q^{32} - 12q^{34} - 4q^{35} - 30q^{36} - 93q^{37} + 8q^{38} + 16q^{39} + 6q^{40} - 12q^{41} - 8q^{42} - 43q^{43} + 9q^{44} + 30q^{45} - 124q^{46} - 42q^{47} - 158q^{48} - 38q^{49} - 2q^{50} + 27q^{51} + 146q^{52} + 74q^{53} + 93q^{54} + 2q^{55} + 25q^{56} - 55q^{57} + 26q^{58} + 10q^{59} - 16q^{60} - 4q^{61} - 33q^{62} + 20q^{63} + 32q^{64} - 8q^{65} - 69q^{66} - 47q^{67} - 24q^{68} - 82q^{69} - 15q^{70} + 2q^{71} - 294q^{72} + 30q^{73} - 112q^{74} + 132q^{76} + 136q^{77} - 115q^{78} + 58q^{79} + 28q^{80} + 220q^{81} + 32q^{82} - 164q^{83} - 32q^{84} + 41q^{85} - 34q^{86} - 76q^{87} + 115q^{88} - 44q^{89} + 54q^{90} - 60q^{91} + 140q^{92} - 68q^{93} - 74q^{94} - 44q^{95} + 140q^{96} - 39q^{97} + 182q^{98} - 274q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −2.41398 0.708808i 3.07614 3.64238 + 2.34081i 0.654861 0.755750i −7.42572 2.18039i 0.721882 1.58070i −3.83832 4.42965i 6.46261 −2.11650 + 1.36019i
56.2 −2.31434 0.679550i −0.0870913 3.21185 + 2.06413i 0.654861 0.755750i 0.201559 + 0.0591829i −1.78239 + 3.90290i −2.87152 3.31392i −2.99242 −2.02914 + 1.30405i
56.3 −2.28582 0.671178i 0.714225 3.09200 + 1.98711i 0.654861 0.755750i −1.63259 0.479373i −0.0293477 + 0.0642624i −2.61389 3.01659i −2.48988 −2.00414 + 1.28798i
56.4 −2.21684 0.650922i −2.65600 2.80815 + 1.80469i 0.654861 0.755750i 5.88791 + 1.72885i 1.70360 3.73037i −2.02449 2.33638i 4.05433 −1.94365 + 1.24911i
56.5 −1.87250 0.549815i −1.56075 1.52144 + 0.977773i 0.654861 0.755750i 2.92251 + 0.858125i −0.922489 + 2.01997i 0.244679 + 0.282374i −0.564050 −1.64175 + 1.05509i
56.6 −1.47955 0.434434i 1.25158 0.317819 + 0.204250i 0.654861 0.755750i −1.85177 0.543727i 1.94824 4.26605i 1.63811 + 1.89048i −1.43356 −1.29722 + 0.833673i
56.7 −1.20161 0.352823i 3.05910 −0.363136 0.233373i 0.654861 0.755750i −3.67583 1.07932i −1.77893 + 3.89532i 1.99421 + 2.30145i 6.35811 −1.05353 + 0.677063i
56.8 −0.960803 0.282117i −1.66611 −0.838954 0.539163i 0.654861 0.755750i 1.60080 + 0.470037i −0.0635449 + 0.139144i 1.96547 + 2.26828i −0.224089 −0.842402 + 0.541379i
56.9 −0.926188 0.271953i 0.536603 −0.898642 0.577522i 0.654861 0.755750i −0.496995 0.145931i 0.519083 1.13663i 1.93951 + 2.23832i −2.71206 −0.812053 + 0.521875i
56.10 −0.309263 0.0908078i 1.15033 −1.59511 1.02511i 0.654861 0.755750i −0.355754 0.104459i −1.76485 + 3.86448i 0.822368 + 0.949063i −1.67674 −0.271152 + 0.174259i
56.11 −0.127482 0.0374322i −3.30973 −1.66766 1.07174i 0.654861 0.755750i 0.421933 + 0.123891i 1.07269 2.34886i 0.346495 + 0.399876i 7.95431 −0.111773 + 0.0718319i
56.12 −0.0357503 0.0104972i 2.77455 −1.68134 1.08053i 0.654861 0.755750i −0.0991912 0.0291252i 0.491662 1.07659i 0.0975655 + 0.112597i 4.69815 −0.0313448 + 0.0201441i
56.13 0.105289 + 0.0309156i −0.0677895 −1.67238 1.07477i 0.654861 0.755750i −0.00713748 0.00209575i −0.157614 + 0.345127i −0.286576 0.330727i −2.99540 0.0923140 0.0593266i
56.14 0.685415 + 0.201256i −1.63608 −1.25322 0.805394i 0.654861 0.755750i −1.12140 0.329271i −1.36643 + 2.99207i −1.63249 1.88399i −0.323235 0.600951 0.386208i
56.15 0.701517 + 0.205984i 2.32552 −1.23281 0.792279i 0.654861 0.755750i 1.63139 + 0.479020i 1.16427 2.54940i −1.65922 1.91484i 2.40803 0.615068 0.395280i
56.16 1.59296 + 0.467736i −2.26440 0.636241 + 0.408888i 0.654861 0.755750i −3.60711 1.05914i −0.226450 + 0.495857i −1.35216 1.56047i 2.12752 1.39666 0.897578i
56.17 1.60588 + 0.471530i −1.30135 0.674012 + 0.433161i 0.654861 0.755750i −2.08982 0.613626i 1.11835 2.44885i −1.31392 1.51634i −1.30648 1.40799 0.904859i
56.18 1.76280 + 0.517603i 1.27989 1.15703 + 0.743575i 0.654861 0.755750i 2.25619 + 0.662476i 0.990961 2.16990i −0.751518 0.867297i −1.36188 1.54556 0.993273i
56.19 1.92400 + 0.564938i 2.93193 1.70012 + 1.09260i 0.654861 0.755750i 5.64104 + 1.65636i −0.651570 + 1.42674i 0.0274874 + 0.0317221i 5.59624 1.68690 1.08411i
56.20 2.11159 + 0.620020i −3.15222 2.39189 + 1.53718i 0.654861 0.755750i −6.65621 1.95444i −0.373094 + 0.816963i 1.21527 + 1.40250i 6.93651 1.85138 1.18981i
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.k.b 220
121.e even 11 1 inner 605.2.k.b 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.k.b 220 1.a even 1 1 trivial
605.2.k.b 220 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!61\)\( T_{2}^{197} + \)\(60\!\cdots\!19\)\( T_{2}^{196} + \)\(12\!\cdots\!88\)\( T_{2}^{195} + \)\(45\!\cdots\!99\)\( T_{2}^{194} + \)\(94\!\cdots\!13\)\( T_{2}^{193} + \)\(32\!\cdots\!95\)\( T_{2}^{192} + \)\(67\!\cdots\!05\)\( T_{2}^{191} + \)\(22\!\cdots\!62\)\( T_{2}^{190} + \)\(46\!\cdots\!03\)\( T_{2}^{189} + \)\(15\!\cdots\!68\)\( T_{2}^{188} + \)\(30\!\cdots\!49\)\( T_{2}^{187} + \)\(99\!\cdots\!36\)\( T_{2}^{186} + \)\(19\!\cdots\!40\)\( T_{2}^{185} + \)\(63\!\cdots\!45\)\( T_{2}^{184} + \)\(12\!\cdots\!79\)\( T_{2}^{183} + \)\(39\!\cdots\!53\)\( T_{2}^{182} + \)\(74\!\cdots\!49\)\( T_{2}^{181} + \)\(23\!\cdots\!09\)\( T_{2}^{180} + \)\(44\!\cdots\!58\)\( T_{2}^{179} + \)\(13\!\cdots\!04\)\( T_{2}^{178} + \)\(25\!\cdots\!39\)\( T_{2}^{177} + \)\(80\!\cdots\!39\)\( T_{2}^{176} + \)\(14\!\cdots\!19\)\( T_{2}^{175} + \)\(44\!\cdots\!37\)\( T_{2}^{174} + \)\(79\!\cdots\!23\)\( T_{2}^{173} + \)\(24\!\cdots\!28\)\( T_{2}^{172} + \)\(42\!\cdots\!47\)\( T_{2}^{171} + \)\(13\!\cdots\!79\)\( T_{2}^{170} + \)\(21\!\cdots\!02\)\( T_{2}^{169} + \)\(68\!\cdots\!69\)\( T_{2}^{168} + \)\(11\!\cdots\!27\)\( T_{2}^{167} + \)\(34\!\cdots\!60\)\( T_{2}^{166} + \)\(54\!\cdots\!82\)\( T_{2}^{165} + \)\(17\!\cdots\!96\)\( T_{2}^{164} + \)\(26\!\cdots\!22\)\( T_{2}^{163} + \)\(84\!\cdots\!90\)\( T_{2}^{162} + \)\(12\!\cdots\!34\)\( T_{2}^{161} + \)\(40\!\cdots\!62\)\( T_{2}^{160} + \)\(57\!\cdots\!79\)\( T_{2}^{159} + \)\(18\!\cdots\!97\)\( T_{2}^{158} + \)\(25\!\cdots\!92\)\( T_{2}^{157} + \)\(85\!\cdots\!98\)\( T_{2}^{156} + \)\(11\!\cdots\!86\)\( T_{2}^{155} + \)\(38\!\cdots\!17\)\( T_{2}^{154} + \)\(47\!\cdots\!26\)\( T_{2}^{153} + \)\(16\!\cdots\!30\)\( T_{2}^{152} + \)\(19\!\cdots\!67\)\( T_{2}^{151} + \)\(71\!\cdots\!31\)\( T_{2}^{150} + \)\(78\!\cdots\!69\)\( T_{2}^{149} + \)\(29\!\cdots\!58\)\( T_{2}^{148} + \)\(31\!\cdots\!10\)\( T_{2}^{147} + \)\(12\!\cdots\!19\)\( T_{2}^{146} + \)\(12\!\cdots\!26\)\( T_{2}^{145} + \)\(48\!\cdots\!53\)\( T_{2}^{144} + \)\(45\!\cdots\!95\)\( T_{2}^{143} + \)\(19\!\cdots\!84\)\( T_{2}^{142} + \)\(16\!\cdots\!05\)\( T_{2}^{141} + \)\(72\!\cdots\!07\)\( T_{2}^{140} + \)\(60\!\cdots\!51\)\( T_{2}^{139} + \)\(27\!\cdots\!45\)\( T_{2}^{138} + \)\(21\!\cdots\!79\)\( T_{2}^{137} + \)\(99\!\cdots\!74\)\( T_{2}^{136} + \)\(73\!\cdots\!91\)\( T_{2}^{135} + \)\(35\!\cdots\!44\)\( T_{2}^{134} + \)\(24\!\cdots\!76\)\( T_{2}^{133} + \)\(12\!\cdots\!69\)\( T_{2}^{132} + \)\(83\!\cdots\!81\)\( T_{2}^{131} + \)\(41\!\cdots\!42\)\( T_{2}^{130} + \)\(27\!\cdots\!40\)\( T_{2}^{129} + \)\(13\!\cdots\!01\)\( T_{2}^{128} + \)\(89\!\cdots\!15\)\( T_{2}^{127} + \)\(44\!\cdots\!90\)\( T_{2}^{126} + \)\(28\!\cdots\!60\)\( T_{2}^{125} + \)\(14\!\cdots\!71\)\( T_{2}^{124} + \)\(91\!\cdots\!15\)\( T_{2}^{123} + \)\(43\!\cdots\!77\)\( T_{2}^{122} + \)\(28\!\cdots\!84\)\( T_{2}^{121} + \)\(12\!\cdots\!88\)\( T_{2}^{120} + \)\(90\!\cdots\!46\)\( T_{2}^{119} + \)\(37\!\cdots\!13\)\( T_{2}^{118} + \)\(27\!\cdots\!48\)\( T_{2}^{117} + \)\(10\!\cdots\!72\)\( T_{2}^{116} + \)\(85\!\cdots\!00\)\( T_{2}^{115} + \)\(29\!\cdots\!21\)\( T_{2}^{114} + \)\(25\!\cdots\!26\)\( T_{2}^{113} + \)\(78\!\cdots\!79\)\( T_{2}^{112} + \)\(73\!\cdots\!66\)\( T_{2}^{111} + \)\(20\!\cdots\!85\)\( T_{2}^{110} + \)\(20\!\cdots\!98\)\( T_{2}^{109} + \)\(52\!\cdots\!20\)\( T_{2}^{108} + \)\(56\!\cdots\!34\)\( T_{2}^{107} + \)\(13\!\cdots\!03\)\( T_{2}^{106} + \)\(14\!\cdots\!99\)\( T_{2}^{105} + \)\(32\!\cdots\!94\)\( T_{2}^{104} + \)\(37\!\cdots\!20\)\( T_{2}^{103} + \)\(78\!\cdots\!97\)\( T_{2}^{102} + \)\(94\!\cdots\!36\)\( T_{2}^{101} + \)\(18\!\cdots\!37\)\( T_{2}^{100} + \)\(22\!\cdots\!51\)\( T_{2}^{99} + \)\(41\!\cdots\!83\)\( T_{2}^{98} + \)\(53\!\cdots\!46\)\( T_{2}^{97} + \)\(93\!\cdots\!38\)\( T_{2}^{96} + \)\(12\!\cdots\!67\)\( T_{2}^{95} + \)\(20\!\cdots\!89\)\( T_{2}^{94} + \)\(26\!\cdots\!90\)\( T_{2}^{93} + \)\(41\!\cdots\!56\)\( T_{2}^{92} + \)\(56\!\cdots\!38\)\( T_{2}^{91} + \)\(84\!\cdots\!44\)\( T_{2}^{90} + \)\(11\!\cdots\!63\)\( T_{2}^{89} + \)\(16\!\cdots\!87\)\( T_{2}^{88} + \)\(21\!\cdots\!08\)\( T_{2}^{87} + \)\(29\!\cdots\!56\)\( T_{2}^{86} + \)\(38\!\cdots\!28\)\( T_{2}^{85} + \)\(51\!\cdots\!33\)\( T_{2}^{84} + \)\(65\!\cdots\!88\)\( T_{2}^{83} + \)\(85\!\cdots\!59\)\( T_{2}^{82} + \)\(10\!\cdots\!99\)\( T_{2}^{81} + \)\(13\!\cdots\!47\)\( T_{2}^{80} + \)\(15\!\cdots\!36\)\( T_{2}^{79} + \)\(19\!\cdots\!17\)\( T_{2}^{78} + \)\(22\!\cdots\!61\)\( T_{2}^{77} + \)\(25\!\cdots\!61\)\( T_{2}^{76} + \)\(28\!\cdots\!42\)\( T_{2}^{75} + \)\(32\!\cdots\!54\)\( T_{2}^{74} + \)\(34\!\cdots\!51\)\( T_{2}^{73} + \)\(37\!\cdots\!76\)\( T_{2}^{72} + \)\(38\!\cdots\!86\)\( T_{2}^{71} + \)\(39\!\cdots\!02\)\( T_{2}^{70} + \)\(39\!\cdots\!41\)\( T_{2}^{69} + \)\(38\!\cdots\!35\)\( T_{2}^{68} + \)\(35\!\cdots\!91\)\( T_{2}^{67} + \)\(33\!\cdots\!31\)\( T_{2}^{66} + \)\(29\!\cdots\!95\)\( T_{2}^{65} + \)\(26\!\cdots\!12\)\( T_{2}^{64} + \)\(22\!\cdots\!79\)\( T_{2}^{63} + \)\(18\!\cdots\!17\)\( T_{2}^{62} + \)\(14\!\cdots\!61\)\( T_{2}^{61} + \)\(12\!\cdots\!54\)\( T_{2}^{60} + \)\(92\!\cdots\!60\)\( T_{2}^{59} + \)\(73\!\cdots\!67\)\( T_{2}^{58} + \)\(55\!\cdots\!21\)\( T_{2}^{57} + \)\(43\!\cdots\!26\)\( T_{2}^{56} + \)\(32\!\cdots\!67\)\( T_{2}^{55} + \)\(25\!\cdots\!39\)\( T_{2}^{54} + \)\(18\!\cdots\!51\)\( T_{2}^{53} + \)\(14\!\cdots\!38\)\( T_{2}^{52} + \)\(10\!\cdots\!26\)\( T_{2}^{51} + \)\(83\!\cdots\!87\)\( T_{2}^{50} + \)\(57\!\cdots\!95\)\( T_{2}^{49} + \)\(45\!\cdots\!87\)\( T_{2}^{48} + \)\(30\!\cdots\!73\)\( T_{2}^{47} + \)\(23\!\cdots\!03\)\( T_{2}^{46} + \)\(15\!\cdots\!36\)\( T_{2}^{45} + \)\(11\!\cdots\!85\)\( T_{2}^{44} + \)\(69\!\cdots\!54\)\( T_{2}^{43} + \)\(48\!\cdots\!75\)\( T_{2}^{42} + \)\(29\!\cdots\!53\)\( T_{2}^{41} + \)\(19\!\cdots\!52\)\( T_{2}^{40} + \)\(12\!\cdots\!11\)\( T_{2}^{39} + \)\(85\!\cdots\!14\)\( T_{2}^{38} + \)\(57\!\cdots\!89\)\( T_{2}^{37} + \)\(40\!\cdots\!30\)\( T_{2}^{36} + \)\(28\!\cdots\!37\)\( T_{2}^{35} + \)\(18\!\cdots\!91\)\( T_{2}^{34} + \)\(11\!\cdots\!37\)\( T_{2}^{33} + \)\(69\!\cdots\!46\)\( T_{2}^{32} + \)\(37\!\cdots\!19\)\( T_{2}^{31} + \)\(18\!\cdots\!12\)\( T_{2}^{30} + \)\(85\!\cdots\!65\)\( T_{2}^{29} + \)\(35\!\cdots\!29\)\( T_{2}^{28} + \)\(13\!\cdots\!49\)\( T_{2}^{27} + \)\(50\!\cdots\!43\)\( T_{2}^{26} + \)\(18\!\cdots\!98\)\( T_{2}^{25} + \)\(68\!\cdots\!15\)\( T_{2}^{24} + \)\(25\!\cdots\!87\)\( T_{2}^{23} + \)\(86\!\cdots\!50\)\( T_{2}^{22} + \)\(25\!\cdots\!53\)\( T_{2}^{21} + \)\(63\!\cdots\!18\)\( T_{2}^{20} + \)\(12\!\cdots\!78\)\( T_{2}^{19} + \)\(18\!\cdots\!25\)\( T_{2}^{18} + \)\(19\!\cdots\!08\)\( T_{2}^{17} + \)\(11\!\cdots\!70\)\( T_{2}^{16} - \)\(27\!\cdots\!25\)\( T_{2}^{15} - \)\(17\!\cdots\!81\)\( T_{2}^{14} - \)\(11\!\cdots\!90\)\( T_{2}^{13} + \)\(19\!\cdots\!31\)\( T_{2}^{12} + \)\(17\!\cdots\!31\)\( T_{2}^{11} - \)\(12\!\cdots\!62\)\( T_{2}^{10} + \)\(18\!\cdots\!62\)\( T_{2}^{9} + \)\(25\!\cdots\!68\)\( T_{2}^{8} + \)\(49\!\cdots\!57\)\( T_{2}^{7} + \)\(69\!\cdots\!43\)\( T_{2}^{6} - \)\(56\!\cdots\!46\)\( T_{2}^{5} - \)\(85\!\cdots\!56\)\( T_{2}^{4} + \)\(24\!\cdots\!03\)\( T_{2}^{3} + \)\(64\!\cdots\!43\)\( T_{2}^{2} - \)\(13\!\cdots\!29\)\( T_{2} + \)\(33\!\cdots\!61\)\( \)">\(T_{2}^{220} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).