Properties

Label 605.2.k.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 220q - 4q^{2} - 4q^{3} - 24q^{4} - 22q^{5} - 12q^{6} - 8q^{7} - 12q^{8} + 212q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 220q - 4q^{2} - 4q^{3} - 24q^{4} - 22q^{5} - 12q^{6} - 8q^{7} - 12q^{8} + 212q^{9} - 4q^{10} - 10q^{11} + 5q^{12} + 6q^{13} + 68q^{14} + 7q^{15} - 44q^{16} - 24q^{17} - 36q^{18} - 16q^{19} - 24q^{20} - 10q^{21} + 45q^{22} + 51q^{23} + 72q^{24} - 22q^{25} - 30q^{26} - 34q^{27} - 56q^{28} - 36q^{29} - q^{30} + 4q^{31} + 42q^{32} - 44q^{33} - 52q^{34} - 8q^{35} - 94q^{36} + 69q^{37} - 40q^{38} - 56q^{39} + 54q^{40} - 44q^{41} - 76q^{42} + 7q^{43} - 67q^{44} - 30q^{45} + 56q^{46} + 14q^{47} + 54q^{48} + 10q^{49} - 4q^{50} + 27q^{51} + 58q^{52} + 86q^{53} - 43q^{54} - 10q^{55} - 79q^{56} + 129q^{57} + 100q^{58} - 54q^{59} - 28q^{60} - 52q^{61} + 55q^{62} - 104q^{63} + 36q^{64} - 16q^{65} - 33q^{66} + 13q^{67} - 120q^{68} - 22q^{69} - 9q^{70} - 70q^{71} + 60q^{72} - 66q^{73} + 20q^{74} - 4q^{75} - 12q^{76} + 44q^{77} + 187q^{78} - 22q^{79} - 44q^{80} + 108q^{81} - 76q^{82} + 96q^{83} - 224q^{84} + 9q^{85} - 66q^{86} - 76q^{87} - 47q^{88} + 52q^{89} + 8q^{90} + 88q^{91} + 8q^{92} + 38q^{93} + 22q^{94} + 28q^{95} - 120q^{96} + 37q^{97} + 12q^{98} + 118q^{99} + O(q^{100}) \)

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 −2.62928 0.772025i 2.30100 4.63457 + 2.97846i −0.654861 + 0.755750i −6.04998 1.77643i −1.45531 + 3.18669i −6.29712 7.26726i 2.29462 2.30527 1.48151i
56.2 −2.60010 0.763459i −2.07957 4.49516 + 2.88886i −0.654861 + 0.755750i 5.40710 + 1.58767i −0.775443 + 1.69798i −5.93317 6.84725i 1.32462 2.27969 1.46507i
56.3 −2.17968 0.640012i 2.32772 2.65889 + 1.70876i −0.654861 + 0.755750i −5.07369 1.48977i 1.54290 3.37848i −1.72660 1.99261i 2.41829 1.91108 1.22817i
56.4 −2.10632 0.618472i −0.247785 2.37157 + 1.52412i −0.654861 + 0.755750i 0.521915 + 0.153248i −0.111662 + 0.244505i −1.17751 1.35892i −2.93860 1.84676 1.18684i
56.5 −1.92738 0.565928i −3.27440 1.71199 + 1.10023i −0.654861 + 0.755750i 6.31101 + 1.85308i −0.190974 + 0.418175i −0.0461080 0.0532114i 7.72172 1.68986 1.08601i
56.6 −1.35839 0.398861i 1.82303 0.00363840 + 0.00233826i −0.654861 + 0.755750i −2.47639 0.727134i −0.403444 + 0.883419i 1.85022 + 2.13527i 0.323433 1.19100 0.765408i
56.7 −1.23909 0.363831i −1.31010 −0.279524 0.179639i −0.654861 + 0.755750i 1.62334 + 0.476654i 0.915617 2.00492i 1.97238 + 2.27625i −1.28364 1.08640 0.698187i
56.8 −1.12455 0.330198i −2.26851 −0.526923 0.338633i −0.654861 + 0.755750i 2.55105 + 0.749057i 1.97574 4.32626i 2.01576 + 2.32632i 2.14613 0.985971 0.633645i
56.9 −1.03518 0.303957i 3.20573 −0.703293 0.451979i −0.654861 + 0.755750i −3.31852 0.974405i 0.791166 1.73241i 2.00369 + 2.31239i 7.27670 0.907616 0.583290i
56.10 −0.606624 0.178121i −1.58204 −1.34624 0.865177i −0.654861 + 0.755750i 0.959706 + 0.281795i −1.84880 + 4.04831i 1.49061 + 1.72025i −0.497135 0.531869 0.341811i
56.11 −0.345583 0.101472i 0.573313 −1.57338 1.01115i −0.654861 + 0.755750i −0.198127 0.0581754i 0.206429 0.452017i 0.912855 + 1.05349i −2.67131 0.302997 0.194724i
56.12 −0.0639308 0.0187718i 2.05010 −1.67877 1.07888i −0.654861 + 0.755750i −0.131065 0.0384841i −1.53344 + 3.35777i 0.174339 + 0.201198i 1.20292 0.0560525 0.0360228i
56.13 0.228264 + 0.0670242i −2.04300 −1.63490 1.05068i −0.654861 + 0.755750i −0.466343 0.136931i −0.615760 + 1.34833i −0.614349 0.708996i 1.17386 −0.200134 + 0.128619i
56.14 0.449807 + 0.132075i 0.974961 −1.49762 0.962465i −0.654861 + 0.755750i 0.438545 + 0.128768i 1.61594 3.53842i −1.16052 1.33931i −2.04945 −0.394377 + 0.253451i
56.15 0.797436 + 0.234148i 0.459960 −1.10143 0.707845i −0.654861 + 0.755750i 0.366789 + 0.107699i 0.797120 1.74545i −1.80109 2.07857i −2.78844 −0.699167 + 0.449328i
56.16 1.05521 + 0.309837i −3.06769 −0.665042 0.427397i −0.654861 + 0.755750i −3.23705 0.950483i 0.923710 2.02264i −2.00971 2.31933i 6.41070 −0.925173 + 0.594573i
56.17 1.44492 + 0.424267i 2.86532 0.225288 + 0.144784i −0.654861 + 0.755750i 4.14017 + 1.21566i 0.844449 1.84908i −1.70824 1.97142i 5.21008 −1.26686 + 0.814163i
56.18 1.63524 + 0.480149i −0.780476 0.760954 + 0.489036i −0.654861 + 0.755750i −1.27626 0.374745i −1.25961 + 2.75817i −1.22259 1.41095i −2.39086 −1.43373 + 0.921400i
56.19 1.90451 + 0.559214i −2.09497 1.63192 + 1.04877i −0.654861 + 0.755750i −3.98988 1.17153i 1.65156 3.61642i −0.0781658 0.0902081i 1.38889 −1.66981 + 1.07312i
56.20 2.05201 + 0.602524i 2.34072 2.16519 + 1.39148i −0.654861 + 0.755750i 4.80316 + 1.41034i −1.19051 + 2.60686i 0.803566 + 0.927365i 2.47895 −1.79914 + 1.15623i
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.a 220
121.e even 11 1 inner 605.2.k.a 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.k.a 220 1.a even 1 1 trivial
605.2.k.a 220 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(19\!\cdots\!88\)\( T_{2}^{198} + \)\(51\!\cdots\!77\)\( T_{2}^{197} + \)\(15\!\cdots\!43\)\( T_{2}^{196} + \)\(40\!\cdots\!28\)\( T_{2}^{195} + \)\(11\!\cdots\!45\)\( T_{2}^{194} + \)\(30\!\cdots\!01\)\( T_{2}^{193} + \)\(89\!\cdots\!07\)\( T_{2}^{192} + \)\(22\!\cdots\!17\)\( T_{2}^{191} + \)\(64\!\cdots\!78\)\( T_{2}^{190} + \)\(16\!\cdots\!87\)\( T_{2}^{189} + \)\(45\!\cdots\!96\)\( T_{2}^{188} + \)\(11\!\cdots\!49\)\( T_{2}^{187} + \)\(31\!\cdots\!40\)\( T_{2}^{186} + \)\(74\!\cdots\!24\)\( T_{2}^{185} + \)\(20\!\cdots\!95\)\( T_{2}^{184} + \)\(48\!\cdots\!15\)\( T_{2}^{183} + \)\(13\!\cdots\!17\)\( T_{2}^{182} + \)\(31\!\cdots\!73\)\( T_{2}^{181} + \)\(83\!\cdots\!07\)\( T_{2}^{180} + \)\(19\!\cdots\!62\)\( T_{2}^{179} + \)\(51\!\cdots\!80\)\( T_{2}^{178} + \)\(11\!\cdots\!83\)\( T_{2}^{177} + \)\(30\!\cdots\!75\)\( T_{2}^{176} + \)\(69\!\cdots\!41\)\( T_{2}^{175} + \)\(17\!\cdots\!17\)\( T_{2}^{174} + \)\(39\!\cdots\!37\)\( T_{2}^{173} + \)\(10\!\cdots\!60\)\( T_{2}^{172} + \)\(22\!\cdots\!83\)\( T_{2}^{171} + \)\(55\!\cdots\!37\)\( T_{2}^{170} + \)\(12\!\cdots\!60\)\( T_{2}^{169} + \)\(29\!\cdots\!47\)\( T_{2}^{168} + \)\(63\!\cdots\!41\)\( T_{2}^{167} + \)\(15\!\cdots\!06\)\( T_{2}^{166} + \)\(32\!\cdots\!86\)\( T_{2}^{165} + \)\(78\!\cdots\!42\)\( T_{2}^{164} + \)\(16\!\cdots\!36\)\( T_{2}^{163} + \)\(38\!\cdots\!32\)\( T_{2}^{162} + \)\(78\!\cdots\!96\)\( T_{2}^{161} + \)\(18\!\cdots\!52\)\( T_{2}^{160} + \)\(37\!\cdots\!63\)\( T_{2}^{159} + \)\(86\!\cdots\!71\)\( T_{2}^{158} + \)\(17\!\cdots\!68\)\( T_{2}^{157} + \)\(39\!\cdots\!60\)\( T_{2}^{156} + \)\(77\!\cdots\!42\)\( T_{2}^{155} + \)\(17\!\cdots\!33\)\( T_{2}^{154} + \)\(34\!\cdots\!22\)\( T_{2}^{153} + \)\(76\!\cdots\!52\)\( T_{2}^{152} + \)\(14\!\cdots\!93\)\( T_{2}^{151} + \)\(32\!\cdots\!29\)\( T_{2}^{150} + \)\(60\!\cdots\!17\)\( T_{2}^{149} + \)\(13\!\cdots\!64\)\( T_{2}^{148} + \)\(24\!\cdots\!92\)\( T_{2}^{147} + \)\(53\!\cdots\!59\)\( T_{2}^{146} + \)\(98\!\cdots\!84\)\( T_{2}^{145} + \)\(20\!\cdots\!77\)\( T_{2}^{144} + \)\(37\!\cdots\!33\)\( T_{2}^{143} + \)\(80\!\cdots\!88\)\( T_{2}^{142} + \)\(14\!\cdots\!59\)\( T_{2}^{141} + \)\(29\!\cdots\!81\)\( T_{2}^{140} + \)\(52\!\cdots\!07\)\( T_{2}^{139} + \)\(10\!\cdots\!03\)\( T_{2}^{138} + \)\(18\!\cdots\!85\)\( T_{2}^{137} + \)\(38\!\cdots\!82\)\( T_{2}^{136} + \)\(65\!\cdots\!33\)\( T_{2}^{135} + \)\(13\!\cdots\!04\)\( T_{2}^{134} + \)\(22\!\cdots\!06\)\( T_{2}^{133} + \)\(43\!\cdots\!61\)\( T_{2}^{132} + \)\(72\!\cdots\!97\)\( T_{2}^{131} + \)\(14\!\cdots\!78\)\( T_{2}^{130} + \)\(23\!\cdots\!82\)\( T_{2}^{129} + \)\(43\!\cdots\!47\)\( T_{2}^{128} + \)\(71\!\cdots\!97\)\( T_{2}^{127} + \)\(13\!\cdots\!96\)\( T_{2}^{126} + \)\(21\!\cdots\!58\)\( T_{2}^{125} + \)\(39\!\cdots\!19\)\( T_{2}^{124} + \)\(62\!\cdots\!31\)\( T_{2}^{123} + \)\(11\!\cdots\!27\)\( T_{2}^{122} + \)\(17\!\cdots\!26\)\( T_{2}^{121} + \)\(31\!\cdots\!64\)\( T_{2}^{120} + \)\(49\!\cdots\!24\)\( T_{2}^{119} + \)\(84\!\cdots\!47\)\( T_{2}^{118} + \)\(13\!\cdots\!86\)\( T_{2}^{117} + \)\(22\!\cdots\!90\)\( T_{2}^{116} + \)\(34\!\cdots\!58\)\( T_{2}^{115} + \)\(57\!\cdots\!71\)\( T_{2}^{114} + \)\(88\!\cdots\!18\)\( T_{2}^{113} + \)\(14\!\cdots\!03\)\( T_{2}^{112} + \)\(21\!\cdots\!52\)\( T_{2}^{111} + \)\(33\!\cdots\!45\)\( T_{2}^{110} + \)\(50\!\cdots\!44\)\( T_{2}^{109} + \)\(76\!\cdots\!92\)\( T_{2}^{108} + \)\(11\!\cdots\!02\)\( T_{2}^{107} + \)\(17\!\cdots\!93\)\( T_{2}^{106} + \)\(24\!\cdots\!29\)\( T_{2}^{105} + \)\(36\!\cdots\!32\)\( T_{2}^{104} + \)\(52\!\cdots\!60\)\( T_{2}^{103} + \)\(77\!\cdots\!01\)\( T_{2}^{102} + \)\(10\!\cdots\!28\)\( T_{2}^{101} + \)\(15\!\cdots\!27\)\( T_{2}^{100} + \)\(21\!\cdots\!97\)\( T_{2}^{99} + \)\(30\!\cdots\!87\)\( T_{2}^{98} + \)\(42\!\cdots\!32\)\( T_{2}^{97} + \)\(57\!\cdots\!92\)\( T_{2}^{96} + \)\(76\!\cdots\!03\)\( T_{2}^{95} + \)\(10\!\cdots\!61\)\( T_{2}^{94} + \)\(13\!\cdots\!84\)\( T_{2}^{93} + \)\(16\!\cdots\!36\)\( T_{2}^{92} + \)\(21\!\cdots\!46\)\( T_{2}^{91} + \)\(26\!\cdots\!44\)\( T_{2}^{90} + \)\(32\!\cdots\!23\)\( T_{2}^{89} + \)\(38\!\cdots\!99\)\( T_{2}^{88} + \)\(44\!\cdots\!74\)\( T_{2}^{87} + \)\(51\!\cdots\!26\)\( T_{2}^{86} + \)\(58\!\cdots\!82\)\( T_{2}^{85} + \)\(65\!\cdots\!45\)\( T_{2}^{84} + \)\(72\!\cdots\!00\)\( T_{2}^{83} + \)\(79\!\cdots\!63\)\( T_{2}^{82} + \)\(86\!\cdots\!61\)\( T_{2}^{81} + \)\(93\!\cdots\!29\)\( T_{2}^{80} + \)\(10\!\cdots\!66\)\( T_{2}^{79} + \)\(10\!\cdots\!11\)\( T_{2}^{78} + \)\(11\!\cdots\!07\)\( T_{2}^{77} + \)\(12\!\cdots\!47\)\( T_{2}^{76} + \)\(12\!\cdots\!42\)\( T_{2}^{75} + \)\(13\!\cdots\!34\)\( T_{2}^{74} + \)\(13\!\cdots\!13\)\( T_{2}^{73} + \)\(13\!\cdots\!26\)\( T_{2}^{72} + \)\(13\!\cdots\!42\)\( T_{2}^{71} + \)\(12\!\cdots\!78\)\( T_{2}^{70} + \)\(11\!\cdots\!67\)\( T_{2}^{69} + \)\(10\!\cdots\!85\)\( T_{2}^{68} + \)\(85\!\cdots\!19\)\( T_{2}^{67} + \)\(66\!\cdots\!63\)\( T_{2}^{66} + \)\(47\!\cdots\!83\)\( T_{2}^{65} + \)\(34\!\cdots\!10\)\( T_{2}^{64} + \)\(25\!\cdots\!87\)\( T_{2}^{63} + \)\(22\!\cdots\!17\)\( T_{2}^{62} + \)\(20\!\cdots\!15\)\( T_{2}^{61} + \)\(20\!\cdots\!02\)\( T_{2}^{60} + \)\(18\!\cdots\!40\)\( T_{2}^{59} + \)\(15\!\cdots\!91\)\( T_{2}^{58} + \)\(11\!\cdots\!93\)\( T_{2}^{57} + \)\(87\!\cdots\!52\)\( T_{2}^{56} + \)\(58\!\cdots\!35\)\( T_{2}^{55} + \)\(41\!\cdots\!67\)\( T_{2}^{54} + \)\(26\!\cdots\!33\)\( T_{2}^{53} + \)\(17\!\cdots\!54\)\( T_{2}^{52} + \)\(10\!\cdots\!68\)\( T_{2}^{51} + \)\(68\!\cdots\!95\)\( T_{2}^{50} + \)\(37\!\cdots\!73\)\( T_{2}^{49} + \)\(20\!\cdots\!75\)\( T_{2}^{48} + \)\(97\!\cdots\!47\)\( T_{2}^{47} + \)\(49\!\cdots\!21\)\( T_{2}^{46} + \)\(21\!\cdots\!60\)\( T_{2}^{45} + \)\(11\!\cdots\!69\)\( T_{2}^{44} + \)\(46\!\cdots\!02\)\( T_{2}^{43} + \)\(25\!\cdots\!15\)\( T_{2}^{42} + \)\(89\!\cdots\!87\)\( T_{2}^{41} + \)\(61\!\cdots\!54\)\( T_{2}^{40} + \)\(18\!\cdots\!23\)\( T_{2}^{39} + \)\(13\!\cdots\!54\)\( T_{2}^{38} + \)\(23\!\cdots\!39\)\( T_{2}^{37} + \)\(37\!\cdots\!16\)\( T_{2}^{36} + \)\(49\!\cdots\!13\)\( T_{2}^{35} + \)\(70\!\cdots\!93\)\( T_{2}^{34} + \)\(10\!\cdots\!65\)\( T_{2}^{33} + \)\(12\!\cdots\!30\)\( T_{2}^{32} + \)\(42\!\cdots\!41\)\( T_{2}^{31} + \)\(19\!\cdots\!36\)\( T_{2}^{30} + \)\(55\!\cdots\!09\)\( T_{2}^{29} + \)\(13\!\cdots\!55\)\( T_{2}^{28} + \)\(12\!\cdots\!21\)\( T_{2}^{27} + \)\(46\!\cdots\!61\)\( T_{2}^{26} - \)\(27\!\cdots\!66\)\( T_{2}^{25} + \)\(76\!\cdots\!53\)\( T_{2}^{24} - \)\(30\!\cdots\!23\)\( T_{2}^{23} + \)\(10\!\cdots\!00\)\( T_{2}^{22} - \)\(22\!\cdots\!01\)\( T_{2}^{21} + \)\(83\!\cdots\!14\)\( T_{2}^{20} - \)\(14\!\cdots\!84\)\( T_{2}^{19} + \)\(40\!\cdots\!37\)\( T_{2}^{18} - \)\(63\!\cdots\!00\)\( T_{2}^{17} + \)\(13\!\cdots\!94\)\( T_{2}^{16} - \)\(17\!\cdots\!51\)\( T_{2}^{15} + \)\(30\!\cdots\!89\)\( T_{2}^{14} - \)\(17\!\cdots\!12\)\( T_{2}^{13} + \)\(33\!\cdots\!13\)\( T_{2}^{12} + \)\(17\!\cdots\!29\)\( T_{2}^{11} + \)\(48\!\cdots\!28\)\( T_{2}^{10} + \)\(23\!\cdots\!46\)\( T_{2}^{9} + \)\(26\!\cdots\!74\)\( T_{2}^{8} - \)\(30\!\cdots\!91\)\( T_{2}^{7} + \)\(52\!\cdots\!79\)\( T_{2}^{6} + \)\(61\!\cdots\!98\)\( T_{2}^{5} - \)\(30\!\cdots\!20\)\( T_{2}^{4} + \)\(11\!\cdots\!93\)\( T_{2}^{3} + 119025770219 T_{2}^{2} - 1663167457 T_{2} + 13490929 \)">\(T_{2}^{220} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).