# Properties

 Label 4056.2.c.e.337.1 Level $4056$ Weight $2$ Character 4056.337 Analytic conductor $32.387$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.3873230598$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4056.337 Dual form 4056.2.c.e.337.2

## $q$-expansion

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$$.

 $$n$$ $$1015$$ $$2029$$ $$2705$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ − 2.00000i − 0.894427i −0.894427 0.447214i $$-0.852416\pi$$
0.894427 0.447214i $$-0.147584\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 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
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 2.00000i 0.516398i
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 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
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 8.00000i 1.43684i 0.695608 + 0.718421i $$0.255135\pi$$
−0.695608 + 0.718421i $$0.744865\pi$$
$$32$$ 0 0
$$33$$ 4.00000i 0.696311i
$$34$$ 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
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ − 2.00000i − 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 0 0
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 8.00000i 0.949425i 0.880141 + 0.474713i $$0.157448\pi$$
−0.880141 + 0.474713i $$0.842552\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 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
$$89$$ 6.00000i 0.635999i 0.948091 + 0.317999i $$0.103011\pi$$
−0.948091 + 0.317999i $$0.896989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 0 0
$$99$$ − 4.00000i − 0.402015i
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 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
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ 6.00000i 0.569495i
$$112$$ 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
$$117$$ 0 0
$$118$$ 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 $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 2.00000i 0.172133i
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 12.0000i − 0.996546i
$$146$$ 0 0
$$147$$ −7.00000 −0.577350
$$148$$ 0 0
$$149$$ 14.0000i 1.14692i 0.819232 + 0.573462i $$0.194400\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ 16.0000i 1.30206i 0.759051 + 0.651031i $$0.225663\pi$$
−0.759051 + 0.651031i $$0.774337\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 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 $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 4.00000i − 0.305888i
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000i 0.300658i
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 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
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 4.00000i 0.282138i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ − 8.00000i − 0.548151i
$$214$$ 0 0
$$215$$ 8.00000i 0.545595i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 10.0000i 0.675737i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 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
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ − 16.0000i − 1.03495i −0.855697 0.517477i $$-0.826871\pi$$
0.855697 0.517477i $$-0.173129\pi$$
$$240$$ 0 0
$$241$$ − 18.0000i − 1.15948i −0.814801 0.579741i $$-0.803154\pi$$
0.814801 0.579741i $$-0.196846\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ − 14.0000i − 0.894427i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.00000i 0.253490i
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ − 32.0000i − 2.01182i
$$254$$ 0 0
$$255$$ − 4.00000i − 0.250490i
$$256$$ 0 0
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ 4.00000i 0.245718i
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ − 8.00000i − 0.485965i −0.970031 0.242983i $$-0.921874\pi$$
0.970031 0.242983i $$-0.0781258\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 0 0
$$279$$ 8.00000i 0.478947i
$$280$$ 0 0
$$281$$ − 26.0000i − 1.55103i −0.631329 0.775515i $$-0.717490\pi$$
0.631329 0.775515i $$-0.282510\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 0 0
$$285$$ 8.00000 0.473879
$$286$$ 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
$$297$$ 4.00000i 0.232104i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −18.0000 −1.03407
$$304$$ 0 0
$$305$$ 4.00000i 0.229039i
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 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 $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 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
$$319$$ − 24.0000i − 1.34374i
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.00000i 0.110600i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000i 1.09930i 0.835395 + 0.549650i $$0.185239\pi$$
−0.835395 + 0.549650i $$0.814761\pi$$
$$332$$ 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 $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 16.0000i 0.861411i
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ − 30.0000i − 1.60586i −0.596071 0.802932i $$-0.703272\pi$$
0.596071 0.802932i $$-0.296728\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000i 1.26667i 0.773877 + 0.633336i $$0.218315\pi$$
−0.773877 + 0.633336i $$0.781685\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ −20.0000 −1.04685
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ − 6.00000i − 0.312348i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ 12.0000i 0.619677i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 20.0000i 1.02733i 0.857991 + 0.513665i $$0.171713\pi$$
−0.857991 + 0.513665i $$0.828287\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 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 $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000i 1.49813i 0.662497 + 0.749064i $$0.269497\pi$$
−0.662497 + 0.749064i $$0.730503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 2.00000i − 0.0993808i
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ − 6.00000i − 0.296681i −0.988936 0.148340i $$-0.952607\pi$$
0.988936 0.148340i $$-0.0473931\pi$$
$$410$$ 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
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ − 10.0000i − 0.487370i −0.969854 0.243685i $$-0.921644\pi$$
0.969854 0.243685i $$-0.0783563\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 32.0000i 1.54139i 0.637207 + 0.770693i $$0.280090\pi$$
−0.637207 + 0.770693i $$0.719910\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 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
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ 0 0
$$447$$ − 14.0000i − 0.662177i
$$448$$ 0 0
$$449$$ 14.0000i 0.660701i 0.943858 + 0.330350i $$0.107167\pi$$
−0.943858 + 0.330350i $$0.892833\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ − 16.0000i − 0.751746i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ − 26.0000i − 1.21094i −0.795868 0.605470i $$-0.792985\pi$$
0.795868 0.605470i $$-0.207015\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 0 0
$$465$$ −16.0000 −0.741982
$$466$$ 0 0
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 0 0
$$479$$ 16.0000i 0.731059i 0.930800 + 0.365529i $$0.119112\pi$$
−0.930800 + 0.365529i $$0.880888\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.00000 0.181631
$$486$$ 0 0
$$487$$ − 32.0000i − 1.45006i −0.688718 0.725029i $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 0 0
$$489$$ 12.0000i 0.542659i
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 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 $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ − 36.0000i − 1.60198i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 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
$$513$$ 4.00000i 0.176604i
$$514$$ 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 $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ − 4.00000i − 0.173585i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 24.0000i 1.03761i
$$536$$ 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 $$0.126468\pi$$
−0.922105 + 0.386940i $$0.873532\pi$$
$$542$$ 0 0
$$543$$ 6.00000 0.257485
$$544$$ 0 0
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ − 24.0000i − 1.02243i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 12.0000 0.509372
$$556$$ 0 0
$$557$$ 26.0000i 1.10166i 0.834619 + 0.550828i $$0.185688\pi$$
−0.834619 + 0.550828i $$0.814312\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ − 8.00000i − 0.337760i
$$562$$ 0 0
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 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
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 0 0
$$579$$ 2.00000i 0.0831172i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 8.00000i 0.331326i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 44.0000i − 1.81607i −0.418890 0.908037i $$-0.637581\pi$$
0.418890 0.908037i $$-0.362419\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 18.0000i 0.740421i
$$592$$ 0 0
$$593$$ 14.0000i 0.574911i 0.957794 + 0.287456i $$0.0928094\pi$$
−0.957794 + 0.287456i $$0.907191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 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 $$0.278451\pi$$
−0.641165 + 0.767403i $$0.721549\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 0 0
$$619$$ 44.0000i 1.76851i 0.467005 + 0.884255i $$0.345333\pi$$
−0.467005 + 0.884255i $$0.654667\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 16.0000 0.638978
$$628$$ 0 0
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ − 16.0000i − 0.636950i −0.947931 0.318475i $$-0.896829\pi$$
0.947931 0.318475i $$-0.103171\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ − 16.0000i − 0.634941i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 8.00000i 0.316475i
$$640$$ 0 0
$$641$$ 14.0000 0.552967 0.276483 0.961019i $$-0.410831\pi$$
0.276483 + 0.961019i $$0.410831\pi$$
$$642$$ 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
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 8.00000i 0.312586i
$$656$$ 0 0
$$657$$ − 10.0000i − 0.390137i
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 10.0000i 0.388955i 0.980907 + 0.194477i $$0.0623011\pi$$
−0.980907 + 0.194477i $$0.937699\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 48.0000 1.85857
$$668$$ 0 0
$$669$$ 8.00000i 0.309298i
$$670$$ 0 0
$$671$$ 8.00000i 0.308837i
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ − 12.0000i − 0.459841i
$$682$$ 0 0
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 22.0000i 0.839352i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 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
$$695$$ 24.0000i 0.910372i
$$696$$ 0 0
$$697$$ 12.0000i 0.454532i
$$698$$ 0 0
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 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
$$713$$ 64.0000i 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 16.0000i 0.597531i
$$718$$ 0 0
$$719$$ 32.0000 1.19340 0.596699 0.802465i $$-0.296479\pi$$
0.596699 + 0.802465i $$0.296479\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 18.0000i 0.669427i
$$724$$ 0 0
$$725$$ 6.00000 0.222834
$$726$$ 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
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 0 0
$$735$$ 14.0000i 0.516398i
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 4.00000i 0.147142i 0.997290 + 0.0735712i $$0.0234396\pi$$
−0.997290 + 0.0735712i $$0.976560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 8.00000i − 0.293492i −0.989174 0.146746i $$-0.953120\pi$$
0.989174 0.146746i $$-0.0468799\pi$$
$$744$$ 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 $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ 0 0
$$753$$ 20.0000 0.728841
$$754$$ 0 0
$$755$$ 32.0000 1.16460
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 0 0
$$759$$ 32.0000i 1.16153i
$$760$$ 0 0
$$761$$ 22.0000i 0.797499i 0.917060 + 0.398750i $$0.130556\pi$$
−0.917060 + 0.398750i $$0.869444\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 4.00000i 0.144620i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 2.00000i 0.0721218i 0.999350 + 0.0360609i $$0.0114810\pi$$
−0.999350 + 0.0360609i $$0.988519\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ 0 0
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ 0 0
$$775$$ 8.00000i 0.287368i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 0 0
$$785$$ 4.00000i 0.142766i
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 0 0
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 4.00000i − 0.141865i
$$796$$ 0 0
$$797$$ −22.0000 −0.779280 −0.389640 0.920967i $$-0.627401\pi$$
−0.389640 + 0.920967i $$0.627401\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000i 0.212000i
$$802$$ 0 0
$$803$$ −40.0000 −1.41157
$$804$$ 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 $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ 4.00000i 0.140459i 0.997531 + 0.0702295i $$0.0223732\pi$$
−0.997531 + 0.0702295i $$0.977627\pi$$
$$812$$ 0 0
$$813$$ 8.00000i 0.280572i
$$814$$ 0 0
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ 16.0000i 0.559769i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 30.0000i 1.04701i 0.852023 + 0.523504i $$0.175375\pi$$
−0.852023 + 0.523504i $$0.824625\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ 4.00000i 0.139262i
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 0 0
$$829$$ 50.0000 1.73657 0.868286 0.496064i $$-0.165222\pi$$
0.868286 + 0.496064i $$0.165222\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ −48.0000 −1.66111
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 0 0
$$839$$ 24.0000i 0.828572i 0.910147 + 0.414286i $$0.135969\pi$$
−0.910147 + 0.414286i $$0.864031\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 26.0000i 0.895488i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 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
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ 0 0
$$865$$ 12.0000i 0.408012i
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 32.0000i 1.08553i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 2.00000i 0.0676897i
$$874$$ 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
$$879$$ − 18.0000i − 0.607125i
$$880$$ 0 0
$$881$$ −50.0000 −1.68454 −0.842271 0.539054i $$-0.818782\pi$$
−0.842271 + 0.539054i $$0.818782\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ 8.00000 0.268917
$$886$$ 0 0
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000i 0.802232i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 48.0000i 1.60089i
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.0000i 0.398893i
$$906$$ 0 0
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ 0 0
$$909$$ 18.0000 0.597022
$$910$$ 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 $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 12.0000i 0.395413i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 6.00000i − 0.197279i
$$926$$ 0 0
$$927$$ −16.0000 −0.525509
$$928$$ 0 0
$$929$$ 50.0000i 1.64045i 0.572043 + 0.820223i $$0.306151\pi$$
−0.572043 + 0.820223i $$0.693849\pi$$
$$930$$ 0 0
$$931$$ − 28.0000i − 0.917663i
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 6.00000i 0.195594i 0.995206 + 0.0977972i $$0.0311797\pi$$
−0.995206 + 0.0977972i $$0.968820\pi$$
$$942$$ 0 0
$$943$$ − 48.0000i − 1.56310i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 6.00000i − 0.194563i
$$952$$ 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
$$963$$ −12.0000 −0.386695
$$964$$ 0 0
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ − 16.0000i − 0.514525i −0.966342 0.257263i $$-0.917179\pi$$
0.966342 0.257263i $$-0.0828206\pi$$
$$968$$ 0 0
$$969$$ − 8.00000i − 0.256997i
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ − 2.00000i − 0.0638551i
$$982$$ 0 0
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 0 0
$$985$$ −36.0000 −1.14706
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 32.0000i 1.01447i
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 0 0
$$999$$ 6.00000i 0.189832i
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 4056.2.c.e.337.1 2
13.5 odd 4 24.2.a.a.1.1 1
13.8 odd 4 4056.2.a.i.1.1 1
13.12 even 2 inner 4056.2.c.e.337.2 2
39.5 even 4 72.2.a.a.1.1 1
52.31 even 4 48.2.a.a.1.1 1
52.47 even 4 8112.2.a.be.1.1 1
65.18 even 4 600.2.f.e.49.1 2
65.44 odd 4 600.2.a.h.1.1 1
65.57 even 4 600.2.f.e.49.2 2
91.5 even 12 1176.2.q.a.361.1 2
91.18 odd 12 1176.2.q.i.961.1 2
91.31 even 12 1176.2.q.a.961.1 2
91.44 odd 12 1176.2.q.i.361.1 2
91.83 even 4 1176.2.a.i.1.1 1
104.5 odd 4 192.2.a.d.1.1 1
104.83 even 4 192.2.a.b.1.1 1
117.5 even 12 648.2.i.b.217.1 2
117.31 odd 12 648.2.i.g.217.1 2
117.70 odd 12 648.2.i.g.433.1 2
117.83 even 12 648.2.i.b.433.1 2
143.109 even 4 2904.2.a.c.1.1 1
156.83 odd 4 144.2.a.b.1.1 1
195.44 even 4 1800.2.a.m.1.1 1
195.83 odd 4 1800.2.f.c.649.1 2
195.122 odd 4 1800.2.f.c.649.2 2
208.5 odd 4 768.2.d.e.385.2 2
208.83 even 4 768.2.d.d.385.2 2
208.109 odd 4 768.2.d.e.385.1 2
208.187 even 4 768.2.d.d.385.1 2
221.135 odd 4 6936.2.a.p.1.1 1
247.18 even 4 8664.2.a.j.1.1 1
260.83 odd 4 1200.2.f.b.49.2 2
260.187 odd 4 1200.2.f.b.49.1 2
260.239 even 4 1200.2.a.d.1.1 1
273.5 odd 12 3528.2.s.y.361.1 2
273.44 even 12 3528.2.s.j.361.1 2
273.83 odd 4 3528.2.a.d.1.1 1
273.122 odd 12 3528.2.s.y.3313.1 2
273.200 even 12 3528.2.s.j.3313.1 2
312.5 even 4 576.2.a.d.1.1 1
312.83 odd 4 576.2.a.b.1.1 1
364.31 odd 12 2352.2.q.r.961.1 2
364.83 odd 4 2352.2.a.i.1.1 1
364.135 even 12 2352.2.q.l.1537.1 2
364.187 odd 12 2352.2.q.r.1537.1 2
364.291 even 12 2352.2.q.l.961.1 2
429.395 odd 4 8712.2.a.u.1.1 1
468.31 even 12 1296.2.i.m.865.1 2
468.83 odd 12 1296.2.i.e.433.1 2
468.187 even 12 1296.2.i.m.433.1 2
468.239 odd 12 1296.2.i.e.865.1 2
520.83 odd 4 4800.2.f.bg.3649.1 2
520.109 odd 4 4800.2.a.q.1.1 1
520.187 odd 4 4800.2.f.bg.3649.2 2
520.213 even 4 4800.2.f.d.3649.2 2
520.317 even 4 4800.2.f.d.3649.1 2
520.499 even 4 4800.2.a.cc.1.1 1
572.395 odd 4 5808.2.a.s.1.1 1
624.5 even 4 2304.2.d.i.1153.2 2
624.83 odd 4 2304.2.d.k.1153.1 2
624.317 even 4 2304.2.d.i.1153.1 2
624.395 odd 4 2304.2.d.k.1153.2 2
728.83 odd 4 9408.2.a.cc.1.1 1
728.629 even 4 9408.2.a.h.1.1 1
780.83 even 4 3600.2.f.r.2449.1 2
780.239 odd 4 3600.2.a.v.1.1 1
780.707 even 4 3600.2.f.r.2449.2 2
1092.83 even 4 7056.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 1 13.5 odd 4
48.2.a.a.1.1 1 52.31 even 4
72.2.a.a.1.1 1 39.5 even 4
144.2.a.b.1.1 1 156.83 odd 4
192.2.a.b.1.1 1 104.83 even 4
192.2.a.d.1.1 1 104.5 odd 4
576.2.a.b.1.1 1 312.83 odd 4
576.2.a.d.1.1 1 312.5 even 4
600.2.a.h.1.1 1 65.44 odd 4
600.2.f.e.49.1 2 65.18 even 4
600.2.f.e.49.2 2 65.57 even 4
648.2.i.b.217.1 2 117.5 even 12
648.2.i.b.433.1 2 117.83 even 12
648.2.i.g.217.1 2 117.31 odd 12
648.2.i.g.433.1 2 117.70 odd 12
768.2.d.d.385.1 2 208.187 even 4
768.2.d.d.385.2 2 208.83 even 4
768.2.d.e.385.1 2 208.109 odd 4
768.2.d.e.385.2 2 208.5 odd 4
1176.2.a.i.1.1 1 91.83 even 4
1176.2.q.a.361.1 2 91.5 even 12
1176.2.q.a.961.1 2 91.31 even 12
1176.2.q.i.361.1 2 91.44 odd 12
1176.2.q.i.961.1 2 91.18 odd 12
1200.2.a.d.1.1 1 260.239 even 4
1200.2.f.b.49.1 2 260.187 odd 4
1200.2.f.b.49.2 2 260.83 odd 4
1296.2.i.e.433.1 2 468.83 odd 12
1296.2.i.e.865.1 2 468.239 odd 12
1296.2.i.m.433.1 2 468.187 even 12
1296.2.i.m.865.1 2 468.31 even 12
1800.2.a.m.1.1 1 195.44 even 4
1800.2.f.c.649.1 2 195.83 odd 4
1800.2.f.c.649.2 2 195.122 odd 4
2304.2.d.i.1153.1 2 624.317 even 4
2304.2.d.i.1153.2 2 624.5 even 4
2304.2.d.k.1153.1 2 624.83 odd 4
2304.2.d.k.1153.2 2 624.395 odd 4
2352.2.a.i.1.1 1 364.83 odd 4
2352.2.q.l.961.1 2 364.291 even 12
2352.2.q.l.1537.1 2 364.135 even 12
2352.2.q.r.961.1 2 364.31 odd 12
2352.2.q.r.1537.1 2 364.187 odd 12
2904.2.a.c.1.1 1 143.109 even 4
3528.2.a.d.1.1 1 273.83 odd 4
3528.2.s.j.361.1 2 273.44 even 12
3528.2.s.j.3313.1 2 273.200 even 12
3528.2.s.y.361.1 2 273.5 odd 12
3528.2.s.y.3313.1 2 273.122 odd 12
3600.2.a.v.1.1 1 780.239 odd 4
3600.2.f.r.2449.1 2 780.83 even 4
3600.2.f.r.2449.2 2 780.707 even 4
4056.2.a.i.1.1 1 13.8 odd 4
4056.2.c.e.337.1 2 1.1 even 1 trivial
4056.2.c.e.337.2 2 13.12 even 2 inner
4800.2.a.q.1.1 1 520.109 odd 4
4800.2.a.cc.1.1 1 520.499 even 4
4800.2.f.d.3649.1 2 520.317 even 4
4800.2.f.d.3649.2 2 520.213 even 4
4800.2.f.bg.3649.1 2 520.83 odd 4
4800.2.f.bg.3649.2 2 520.187 odd 4
5808.2.a.s.1.1 1 572.395 odd 4
6936.2.a.p.1.1 1 221.135 odd 4
7056.2.a.q.1.1 1 1092.83 even 4
8112.2.a.be.1.1 1 52.47 even 4
8664.2.a.j.1.1 1 247.18 even 4
8712.2.a.u.1.1 1 429.395 odd 4
9408.2.a.h.1.1 1 728.629 even 4
9408.2.a.cc.1.1 1 728.83 odd 4