# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$.