# Properties

 Label 768.2.d.e.385.1 Level $768$ Weight $2$ Character 768.385 Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 385.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.385 Dual form 768.2.d.e.385.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} -4.00000 q^{33} -6.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} -2.00000i q^{45} -7.00000 q^{49} -2.00000i q^{51} +2.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -4.00000i q^{59} -2.00000i q^{61} +4.00000 q^{65} -4.00000i q^{67} -8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +4.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} -8.00000i q^{93} +8.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{15} + 4q^{17} + 16q^{23} + 2q^{25} + 16q^{31} - 8q^{33} - 4q^{39} + 12q^{41} - 14q^{49} + 16q^{55} - 8q^{57} + 8q^{65} - 16q^{71} - 20q^{73} - 16q^{79} + 2q^{81} + 12q^{87} + 12q^{89} + 16q^{95} + 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 2.00000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ − 2.00000i − 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ − 2.00000i − 0.280056i
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ − 2.00000i − 0.256074i −0.991769 0.128037i $$-0.959132\pi$$
0.991769 0.128037i $$-0.0408676\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ − 8.00000i − 0.963087i
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ − 1.00000i − 0.115470i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 4.00000i 0.433861i
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 18.0000i 1.79107i 0.444994 + 0.895533i $$0.353206\pi$$
−0.444994 + 0.895533i $$0.646794\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ 16.0000i 1.49201i
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 6.00000i − 0.541002i
$$124$$ 0 0
$$125$$ 12.0000i 1.07331i
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ − 4.00000i − 0.349482i −0.984614 0.174741i $$-0.944091\pi$$
0.984614 0.174741i $$-0.0559088\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 12.0000i 1.01783i 0.860818 + 0.508913i $$0.169953\pi$$
−0.860818 + 0.508913i $$0.830047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ 7.00000i 0.577350i
$$148$$ 0 0
$$149$$ − 14.0000i − 1.14692i −0.819232 0.573462i $$-0.805600\pi$$
0.819232 0.573462i $$-0.194400\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 16.0000i 1.28515i
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 0 0
$$165$$ − 8.00000i − 0.622799i
$$166$$ 0 0
$$167$$ −24.0000 −1.85718 −0.928588 0.371113i $$-0.878976\pi$$
−0.928588 + 0.371113i $$0.878976\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000i 0.305888i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 12.0000i 0.896922i 0.893802 + 0.448461i $$0.148028\pi$$
−0.893802 + 0.448461i $$0.851972\pi$$
$$180$$ 0 0
$$181$$ − 6.00000i − 0.445976i −0.974821 0.222988i $$-0.928419\pi$$
0.974821 0.222988i $$-0.0715812\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ − 4.00000i − 0.286446i
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 12.0000i 0.838116i
$$206$$ 0 0
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ − 20.0000i − 1.37686i −0.725304 0.688428i $$-0.758301\pi$$
0.725304 0.688428i $$-0.241699\pi$$
$$212$$ 0 0
$$213$$ 8.00000i 0.548151i
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 10.0000i 0.675737i
$$220$$ 0 0
$$221$$ − 4.00000i − 0.269069i
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ − 22.0000i − 1.45380i −0.686743 0.726900i $$-0.740960\pi$$
0.686743 0.726900i $$-0.259040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ − 14.0000i − 0.894427i
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ − 20.0000i − 1.26239i −0.775625 0.631194i $$-0.782565\pi$$
0.775625 0.631194i $$-0.217435\pi$$
$$252$$ 0 0
$$253$$ − 32.0000i − 2.01182i
$$254$$ 0 0
$$255$$ 4.00000 0.250490
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ − 6.00000i − 0.371391i
$$262$$ 0 0
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ −4.00000 −0.245718
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 0 0
$$269$$ − 10.0000i − 0.609711i −0.952399 0.304855i $$-0.901392\pi$$
0.952399 0.304855i $$-0.0986081\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 0 0
$$285$$ − 8.00000i − 0.473879i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 2.00000i − 0.117242i
$$292$$ 0 0
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ − 16.0000i − 0.925304i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 18.0000 1.03407
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 16.0000i 0.910208i
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ − 2.00000i − 0.110940i
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 20.0000i − 1.09930i −0.835395 0.549650i $$-0.814761\pi$$
0.835395 0.549650i $$-0.185239\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 0 0
$$339$$ − 18.0000i − 0.977626i
$$340$$ 0 0
$$341$$ − 32.0000i − 1.73290i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 0 0
$$349$$ 30.0000i 1.60586i 0.596071 + 0.802932i $$0.296728\pi$$
−0.596071 + 0.802932i $$0.703272\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 0 0
$$355$$ − 16.0000i − 0.849192i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ − 20.0000i − 1.04685i
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ − 20.0000i − 1.02733i −0.857991 0.513665i $$-0.828287\pi$$
0.857991 0.513665i $$-0.171713\pi$$
$$380$$ 0 0
$$381$$ 8.00000i 0.409852i
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ 2.00000i 0.101404i 0.998714 + 0.0507020i $$0.0161459\pi$$
−0.998714 + 0.0507020i $$0.983854\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ − 16.0000i − 0.805047i
$$396$$ 0 0
$$397$$ 14.0000i 0.702640i 0.936255 + 0.351320i $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.797017i
$$404$$ 0 0
$$405$$ 2.00000i 0.0993808i
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ − 6.00000i − 0.295958i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ 12.0000i 0.586238i 0.956076 + 0.293119i $$0.0946933\pi$$
−0.956076 + 0.293119i $$0.905307\pi$$
$$420$$ 0 0
$$421$$ 10.0000i 0.487370i 0.969854 + 0.243685i $$0.0783563\pi$$
−0.969854 + 0.243685i $$0.921644\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 8.00000i 0.386244i
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 12.0000i 0.575356i
$$436$$ 0 0
$$437$$ − 32.0000i − 1.53077i
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 0 0
$$445$$ 12.0000i 0.568855i
$$446$$ 0 0
$$447$$ −14.0000 −0.662177
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ − 24.0000i − 1.13012i
$$452$$ 0 0
$$453$$ − 16.0000i − 0.751746i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ 2.00000i 0.0933520i
$$460$$ 0 0
$$461$$ − 26.0000i − 1.21094i −0.795868 0.605470i $$-0.792985\pi$$
0.795868 0.605470i $$-0.207015\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ 16.0000 0.741982
$$466$$ 0 0
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.00000i 0.181631i
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 12.0000i 0.537194i 0.963253 + 0.268597i $$0.0865599\pi$$
−0.963253 + 0.268597i $$0.913440\pi$$
$$500$$ 0 0
$$501$$ 24.0000i 1.07224i
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ −36.0000 −1.60198
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ 6.00000i 0.265945i 0.991120 + 0.132973i $$0.0424523\pi$$
−0.991120 + 0.132973i $$0.957548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ − 32.0000i − 1.41009i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 0 0
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 4.00000i 0.173585i
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 28.0000i 1.20605i
$$540$$ 0 0
$$541$$ − 18.0000i − 0.773880i −0.922105 0.386940i $$-0.873532\pi$$
0.922105 0.386940i $$-0.126468\pi$$
$$542$$ 0 0
$$543$$ −6.00000 −0.257485
$$544$$ 0 0
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ 44.0000i 1.88130i 0.339372 + 0.940652i $$0.389785\pi$$
−0.339372 + 0.940652i $$0.610215\pi$$
$$548$$ 0 0
$$549$$ 2.00000i 0.0853579i
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ − 12.0000i − 0.509372i
$$556$$ 0 0
$$557$$ − 26.0000i − 1.10166i −0.834619 0.550828i $$-0.814312\pi$$
0.834619 0.550828i $$-0.185688\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ 28.0000i 1.18006i 0.807382 + 0.590030i $$0.200884\pi$$
−0.807382 + 0.590030i $$0.799116\pi$$
$$564$$ 0 0
$$565$$ 36.0000i 1.51453i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ − 36.0000i − 1.50655i −0.657704 0.753277i $$-0.728472\pi$$
0.657704 0.753277i $$-0.271528\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ − 2.00000i − 0.0831172i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ −4.00000 −0.165380
$$586$$ 0 0
$$587$$ 44.0000i 1.81607i 0.418890 + 0.908037i $$0.362419\pi$$
−0.418890 + 0.908037i $$0.637581\pi$$
$$588$$ 0 0
$$589$$ − 32.0000i − 1.31854i
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000i 0.654836i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ − 10.0000i − 0.406558i
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 38.0000i − 1.53481i −0.641165 0.767403i $$-0.721549\pi$$
0.641165 0.767403i $$-0.278451\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 44.0000i 1.76851i 0.467005 + 0.884255i $$0.345333\pi$$
−0.467005 + 0.884255i $$0.654667\pi$$
$$620$$ 0 0
$$621$$ 8.00000i 0.321029i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 16.0000i 0.638978i
$$628$$ 0 0
$$629$$ − 12.0000i − 0.478471i
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ −20.0000 −0.794929
$$634$$ 0 0
$$635$$ − 16.0000i − 0.634941i
$$636$$ 0 0
$$637$$ 14.0000i 0.554700i
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −14.0000 −0.552967 −0.276483 0.961019i $$-0.589169\pi$$
−0.276483 + 0.961019i $$0.589169\pi$$
$$642$$ 0 0
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ − 8.00000i − 0.315000i
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 0 0
$$655$$ 8.00000 0.312586
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 12.0000i 0.467454i 0.972302 + 0.233727i $$0.0750921\pi$$
−0.972302 + 0.233727i $$0.924908\pi$$
$$660$$ 0 0
$$661$$ 10.0000i 0.388955i 0.980907 + 0.194477i $$0.0623011\pi$$
−0.980907 + 0.194477i $$0.937699\pi$$
$$662$$ 0 0
$$663$$ −4.00000 −0.155347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 48.0000i 1.85857i
$$668$$ 0 0
$$669$$ 8.00000i 0.309298i
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ 1.00000i 0.0384900i
$$676$$ 0 0
$$677$$ 2.00000i 0.0768662i 0.999261 + 0.0384331i $$0.0122367\pi$$
−0.999261 + 0.0384331i $$0.987763\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 12.0000i 0.458496i
$$686$$ 0 0
$$687$$ −22.0000 −0.839352
$$688$$ 0 0
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ − 4.00000i − 0.152167i −0.997101 0.0760836i $$-0.975758\pi$$
0.997101 0.0760836i $$-0.0242416\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −24.0000 −0.910372
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 10.0000i 0.378235i
$$700$$ 0 0
$$701$$ 6.00000i 0.226617i 0.993560 + 0.113308i $$0.0361448\pi$$
−0.993560 + 0.113308i $$0.963855\pi$$
$$702$$ 0 0
$$703$$ −24.0000 −0.905177
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000i 0.375558i 0.982211 + 0.187779i $$0.0601289\pi$$
−0.982211 + 0.187779i $$0.939871\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 64.0000 2.39682
$$714$$ 0 0
$$715$$ − 16.0000i − 0.598366i
$$716$$ 0 0
$$717$$ 16.0000i 0.597531i
$$718$$ 0 0
$$719$$ −32.0000 −1.19340 −0.596699 0.802465i $$-0.703521\pi$$
−0.596699 + 0.802465i $$0.703521\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ − 18.0000i − 0.669427i
$$724$$ 0 0
$$725$$ 6.00000i 0.222834i
$$726$$ 0 0
$$727$$ −48.0000 −1.78022 −0.890111 0.455744i $$-0.849373\pi$$
−0.890111 + 0.455744i $$0.849373\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 8.00000i − 0.295891i
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 0 0
$$735$$ −14.0000 −0.516398
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ − 4.00000i − 0.147142i −0.997290 0.0735712i $$-0.976560\pi$$
0.997290 0.0735712i $$-0.0234396\pi$$
$$740$$ 0 0
$$741$$ 8.00000i 0.293887i
$$742$$ 0 0
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ 28.0000 1.02584
$$746$$ 0 0
$$747$$ 4.00000i 0.146352i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ −20.0000 −0.728841
$$754$$ 0 0
$$755$$ 32.0000i 1.16460i
$$756$$ 0 0
$$757$$ − 38.0000i − 1.38113i −0.723269 0.690567i $$-0.757361\pi$$
0.723269 0.690567i $$-0.242639\pi$$
$$758$$ 0 0
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ − 4.00000i − 0.144620i
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ − 2.00000i − 0.0720282i
$$772$$ 0 0
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 24.0000i − 0.859889i
$$780$$ 0 0
$$781$$ 32.0000i 1.14505i
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 0 0
$$785$$ 4.00000 0.142766
$$786$$ 0 0
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ 0 0
$$789$$ − 8.00000i − 0.284808i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 4.00000i 0.141865i
$$796$$ 0 0
$$797$$ 22.0000i 0.779280i 0.920967 + 0.389640i $$0.127401\pi$$
−0.920967 + 0.389640i $$0.872599\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ − 4.00000i − 0.140459i −0.997531 0.0702295i $$-0.977627\pi$$
0.997531 0.0702295i $$-0.0223732\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 30.0000i − 1.04701i −0.852023 0.523504i $$-0.824625\pi$$
0.852023 0.523504i $$-0.175375\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 0 0
$$829$$ − 50.0000i − 1.73657i −0.496064 0.868286i $$-0.665222\pi$$
0.496064 0.868286i $$-0.334778\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ − 48.0000i − 1.66111i
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 26.0000i 0.895488i
$$844$$ 0 0
$$845$$ 18.0000i 0.619219i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ − 48.0000i − 1.64542i
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 12.0000i 0.409435i 0.978821 + 0.204717i $$0.0656275\pi$$
−0.978821 + 0.204717i $$0.934372\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 32.0000i 1.08553i
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 18.0000i − 0.607817i −0.952701 0.303908i $$-0.901708\pi$$
0.952701 0.303908i $$-0.0982917\pi$$
$$878$$ 0 0
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 0 0
$$883$$ − 4.00000i − 0.134611i −0.997732 0.0673054i $$-0.978560\pi$$
0.997732 0.0673054i $$-0.0214402\pi$$
$$884$$ 0 0
$$885$$ − 8.00000i − 0.268917i
$$886$$ 0 0
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ 0 0
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ 48.0000i 1.60089i
$$900$$ 0 0
$$901$$ 4.00000i 0.133259i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.0000 0.398893
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 0 0
$$909$$ − 18.0000i − 0.597022i
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ − 4.00000i − 0.132236i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ − 6.00000i − 0.197279i
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ 28.0000i 0.917663i
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ − 6.00000i − 0.195803i
$$940$$ 0 0
$$941$$ 6.00000i 0.195594i 0.995206 + 0.0977972i $$0.0311797\pi$$
−0.995206 + 0.0977972i $$0.968820\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ 20.0000i 0.649227i
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 24.0000i − 0.775810i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 4.00000i 0.128765i
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ − 36.0000i − 1.15529i −0.816286 0.577647i $$-0.803971\pi$$
0.816286 0.577647i $$-0.196029\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ − 24.0000i − 0.767043i
$$980$$ 0 0
$$981$$ 2.00000i 0.0638551i
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ −36.0000 −1.14706
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 32.0000i − 1.01754i
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ − 32.0000i − 1.01447i
$$996$$ 0 0
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 0 0
$$999$$ 6.00000 0.189832
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 768.2.d.e.385.1 2
3.2 odd 2 2304.2.d.i.1153.1 2
4.3 odd 2 768.2.d.d.385.2 2
8.3 odd 2 768.2.d.d.385.1 2
8.5 even 2 inner 768.2.d.e.385.2 2
12.11 even 2 2304.2.d.k.1153.1 2
16.3 odd 4 192.2.a.b.1.1 1
16.5 even 4 24.2.a.a.1.1 1
16.11 odd 4 48.2.a.a.1.1 1
16.13 even 4 192.2.a.d.1.1 1
24.5 odd 2 2304.2.d.i.1153.2 2
24.11 even 2 2304.2.d.k.1153.2 2
48.5 odd 4 72.2.a.a.1.1 1
48.11 even 4 144.2.a.b.1.1 1
48.29 odd 4 576.2.a.d.1.1 1
48.35 even 4 576.2.a.b.1.1 1
80.3 even 4 4800.2.f.bg.3649.1 2
80.13 odd 4 4800.2.f.d.3649.2 2
80.19 odd 4 4800.2.a.cc.1.1 1
80.27 even 4 1200.2.f.b.49.1 2
80.29 even 4 4800.2.a.q.1.1 1
80.37 odd 4 600.2.f.e.49.2 2
80.43 even 4 1200.2.f.b.49.2 2
80.53 odd 4 600.2.f.e.49.1 2
80.59 odd 4 1200.2.a.d.1.1 1
80.67 even 4 4800.2.f.bg.3649.2 2
80.69 even 4 600.2.a.h.1.1 1
80.77 odd 4 4800.2.f.d.3649.1 2
112.5 odd 12 1176.2.q.a.361.1 2
112.11 odd 12 2352.2.q.l.961.1 2
112.13 odd 4 9408.2.a.h.1.1 1
112.27 even 4 2352.2.a.i.1.1 1
112.37 even 12 1176.2.q.i.361.1 2
112.53 even 12 1176.2.q.i.961.1 2
112.59 even 12 2352.2.q.r.961.1 2
112.69 odd 4 1176.2.a.i.1.1 1
112.75 even 12 2352.2.q.r.1537.1 2
112.83 even 4 9408.2.a.cc.1.1 1
112.101 odd 12 1176.2.q.a.961.1 2
112.107 odd 12 2352.2.q.l.1537.1 2
144.5 odd 12 648.2.i.b.217.1 2
144.11 even 12 1296.2.i.e.433.1 2
144.43 odd 12 1296.2.i.m.433.1 2
144.59 even 12 1296.2.i.e.865.1 2
144.85 even 12 648.2.i.g.217.1 2
144.101 odd 12 648.2.i.b.433.1 2
144.133 even 12 648.2.i.g.433.1 2
144.139 odd 12 1296.2.i.m.865.1 2
176.21 odd 4 2904.2.a.c.1.1 1
176.43 even 4 5808.2.a.s.1.1 1
208.5 odd 4 4056.2.c.e.337.2 2
208.21 odd 4 4056.2.c.e.337.1 2
208.155 odd 4 8112.2.a.be.1.1 1
208.181 even 4 4056.2.a.i.1.1 1
240.53 even 4 1800.2.f.c.649.1 2
240.59 even 4 3600.2.a.v.1.1 1
240.107 odd 4 3600.2.f.r.2449.2 2
240.149 odd 4 1800.2.a.m.1.1 1
240.197 even 4 1800.2.f.c.649.2 2
240.203 odd 4 3600.2.f.r.2449.1 2
272.101 even 4 6936.2.a.p.1.1 1
304.37 odd 4 8664.2.a.j.1.1 1
336.5 even 12 3528.2.s.y.361.1 2
336.53 odd 12 3528.2.s.j.3313.1 2
336.101 even 12 3528.2.s.y.3313.1 2
336.149 odd 12 3528.2.s.j.361.1 2
336.251 odd 4 7056.2.a.q.1.1 1
336.293 even 4 3528.2.a.d.1.1 1
528.197 even 4 8712.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 1 16.5 even 4
48.2.a.a.1.1 1 16.11 odd 4
72.2.a.a.1.1 1 48.5 odd 4
144.2.a.b.1.1 1 48.11 even 4
192.2.a.b.1.1 1 16.3 odd 4
192.2.a.d.1.1 1 16.13 even 4
576.2.a.b.1.1 1 48.35 even 4
576.2.a.d.1.1 1 48.29 odd 4
600.2.a.h.1.1 1 80.69 even 4
600.2.f.e.49.1 2 80.53 odd 4
600.2.f.e.49.2 2 80.37 odd 4
648.2.i.b.217.1 2 144.5 odd 12
648.2.i.b.433.1 2 144.101 odd 12
648.2.i.g.217.1 2 144.85 even 12
648.2.i.g.433.1 2 144.133 even 12
768.2.d.d.385.1 2 8.3 odd 2
768.2.d.d.385.2 2 4.3 odd 2
768.2.d.e.385.1 2 1.1 even 1 trivial
768.2.d.e.385.2 2 8.5 even 2 inner
1176.2.a.i.1.1 1 112.69 odd 4
1176.2.q.a.361.1 2 112.5 odd 12
1176.2.q.a.961.1 2 112.101 odd 12
1176.2.q.i.361.1 2 112.37 even 12
1176.2.q.i.961.1 2 112.53 even 12
1200.2.a.d.1.1 1 80.59 odd 4
1200.2.f.b.49.1 2 80.27 even 4
1200.2.f.b.49.2 2 80.43 even 4
1296.2.i.e.433.1 2 144.11 even 12
1296.2.i.e.865.1 2 144.59 even 12
1296.2.i.m.433.1 2 144.43 odd 12
1296.2.i.m.865.1 2 144.139 odd 12
1800.2.a.m.1.1 1 240.149 odd 4
1800.2.f.c.649.1 2 240.53 even 4
1800.2.f.c.649.2 2 240.197 even 4
2304.2.d.i.1153.1 2 3.2 odd 2
2304.2.d.i.1153.2 2 24.5 odd 2
2304.2.d.k.1153.1 2 12.11 even 2
2304.2.d.k.1153.2 2 24.11 even 2
2352.2.a.i.1.1 1 112.27 even 4
2352.2.q.l.961.1 2 112.11 odd 12
2352.2.q.l.1537.1 2 112.107 odd 12
2352.2.q.r.961.1 2 112.59 even 12
2352.2.q.r.1537.1 2 112.75 even 12
2904.2.a.c.1.1 1 176.21 odd 4
3528.2.a.d.1.1 1 336.293 even 4
3528.2.s.j.361.1 2 336.149 odd 12
3528.2.s.j.3313.1 2 336.53 odd 12
3528.2.s.y.361.1 2 336.5 even 12
3528.2.s.y.3313.1 2 336.101 even 12
3600.2.a.v.1.1 1 240.59 even 4
3600.2.f.r.2449.1 2 240.203 odd 4
3600.2.f.r.2449.2 2 240.107 odd 4
4056.2.a.i.1.1 1 208.181 even 4
4056.2.c.e.337.1 2 208.21 odd 4
4056.2.c.e.337.2 2 208.5 odd 4
4800.2.a.q.1.1 1 80.29 even 4
4800.2.a.cc.1.1 1 80.19 odd 4
4800.2.f.d.3649.1 2 80.77 odd 4
4800.2.f.d.3649.2 2 80.13 odd 4
4800.2.f.bg.3649.1 2 80.3 even 4
4800.2.f.bg.3649.2 2 80.67 even 4
5808.2.a.s.1.1 1 176.43 even 4
6936.2.a.p.1.1 1 272.101 even 4
7056.2.a.q.1.1 1 336.251 odd 4
8112.2.a.be.1.1 1 208.155 odd 4
8664.2.a.j.1.1 1 304.37 odd 4
8712.2.a.u.1.1 1 528.197 even 4
9408.2.a.h.1.1 1 112.13 odd 4
9408.2.a.cc.1.1 1 112.83 even 4