# Properties

 Label 320.2.d.b.161.4 Level $320$ Weight $2$ Character 320.161 Analytic conductor $2.555$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 320.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.55521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 320.161 Dual form 320.2.d.b.161.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.73205i q^{3} +1.00000i q^{5} +4.73205 q^{7} -4.46410 q^{9} +O(q^{10})$$ $$q+2.73205i q^{3} +1.00000i q^{5} +4.73205 q^{7} -4.46410 q^{9} +3.46410i q^{11} -3.46410i q^{13} -2.73205 q^{15} -3.46410 q^{17} -2.00000i q^{19} +12.9282i q^{21} -2.19615 q^{23} -1.00000 q^{25} -4.00000i q^{27} -2.53590 q^{31} -9.46410 q^{33} +4.73205i q^{35} -6.00000i q^{37} +9.46410 q^{39} +9.46410 q^{41} -0.196152i q^{43} -4.46410i q^{45} +2.19615 q^{47} +15.3923 q^{49} -9.46410i q^{51} -10.3923i q^{53} -3.46410 q^{55} +5.46410 q^{57} +6.00000i q^{59} +0.928203i q^{61} -21.1244 q^{63} +3.46410 q^{65} -0.196152i q^{67} -6.00000i q^{69} +16.3923 q^{71} +6.39230 q^{73} -2.73205i q^{75} +16.3923i q^{77} -12.0000 q^{79} -2.46410 q^{81} +1.26795i q^{83} -3.46410i q^{85} -12.9282 q^{89} -16.3923i q^{91} -6.92820i q^{93} +2.00000 q^{95} +14.3923 q^{97} -15.4641i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q + 12 q^{7} - 4 q^{9} - 4 q^{15} + 12 q^{23} - 4 q^{25} - 24 q^{31} - 24 q^{33} + 24 q^{39} + 24 q^{41} - 12 q^{47} + 20 q^{49} + 8 q^{57} - 36 q^{63} + 24 q^{71} - 16 q^{73} - 48 q^{79} + 4 q^{81} - 24 q^{89} + 8 q^{95} + 16 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.73205i 1.57735i 0.614810 + 0.788675i $$0.289233\pi$$
−0.614810 + 0.788675i $$0.710767\pi$$
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ 4.73205 1.78855 0.894274 0.447521i $$-0.147693\pi$$
0.894274 + 0.447521i $$0.147693\pi$$
$$8$$ 0 0
$$9$$ −4.46410 −1.48803
$$10$$ 0 0
$$11$$ 3.46410i 1.04447i 0.852803 + 0.522233i $$0.174901\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ − 3.46410i − 0.960769i −0.877058 0.480384i $$-0.840497\pi$$
0.877058 0.480384i $$-0.159503\pi$$
$$14$$ 0 0
$$15$$ −2.73205 −0.705412
$$16$$ 0 0
$$17$$ −3.46410 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$18$$ 0 0
$$19$$ − 2.00000i − 0.458831i −0.973329 0.229416i $$-0.926318\pi$$
0.973329 0.229416i $$-0.0736815\pi$$
$$20$$ 0 0
$$21$$ 12.9282i 2.82117i
$$22$$ 0 0
$$23$$ −2.19615 −0.457929 −0.228965 0.973435i $$-0.573534\pi$$
−0.228965 + 0.973435i $$0.573534\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −2.53590 −0.455461 −0.227730 0.973724i $$-0.573130\pi$$
−0.227730 + 0.973724i $$0.573130\pi$$
$$32$$ 0 0
$$33$$ −9.46410 −1.64749
$$34$$ 0 0
$$35$$ 4.73205i 0.799863i
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 9.46410 1.51547
$$40$$ 0 0
$$41$$ 9.46410 1.47804 0.739022 0.673681i $$-0.235288\pi$$
0.739022 + 0.673681i $$0.235288\pi$$
$$42$$ 0 0
$$43$$ − 0.196152i − 0.0299130i −0.999888 0.0149565i $$-0.995239\pi$$
0.999888 0.0149565i $$-0.00476097\pi$$
$$44$$ 0 0
$$45$$ − 4.46410i − 0.665469i
$$46$$ 0 0
$$47$$ 2.19615 0.320342 0.160171 0.987089i $$-0.448795\pi$$
0.160171 + 0.987089i $$0.448795\pi$$
$$48$$ 0 0
$$49$$ 15.3923 2.19890
$$50$$ 0 0
$$51$$ − 9.46410i − 1.32524i
$$52$$ 0 0
$$53$$ − 10.3923i − 1.42749i −0.700404 0.713746i $$-0.746997\pi$$
0.700404 0.713746i $$-0.253003\pi$$
$$54$$ 0 0
$$55$$ −3.46410 −0.467099
$$56$$ 0 0
$$57$$ 5.46410 0.723738
$$58$$ 0 0
$$59$$ 6.00000i 0.781133i 0.920575 + 0.390567i $$0.127721\pi$$
−0.920575 + 0.390567i $$0.872279\pi$$
$$60$$ 0 0
$$61$$ 0.928203i 0.118844i 0.998233 + 0.0594221i $$0.0189258\pi$$
−0.998233 + 0.0594221i $$0.981074\pi$$
$$62$$ 0 0
$$63$$ −21.1244 −2.66142
$$64$$ 0 0
$$65$$ 3.46410 0.429669
$$66$$ 0 0
$$67$$ − 0.196152i − 0.0239638i −0.999928 0.0119819i $$-0.996186\pi$$
0.999928 0.0119819i $$-0.00381405\pi$$
$$68$$ 0 0
$$69$$ − 6.00000i − 0.722315i
$$70$$ 0 0
$$71$$ 16.3923 1.94541 0.972704 0.232048i $$-0.0745426\pi$$
0.972704 + 0.232048i $$0.0745426\pi$$
$$72$$ 0 0
$$73$$ 6.39230 0.748163 0.374081 0.927396i $$-0.377958\pi$$
0.374081 + 0.927396i $$0.377958\pi$$
$$74$$ 0 0
$$75$$ − 2.73205i − 0.315470i
$$76$$ 0 0
$$77$$ 16.3923i 1.86808i
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ 1.26795i 0.139176i 0.997576 + 0.0695878i $$0.0221684\pi$$
−0.997576 + 0.0695878i $$0.977832\pi$$
$$84$$ 0 0
$$85$$ − 3.46410i − 0.375735i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −12.9282 −1.37039 −0.685193 0.728361i $$-0.740282\pi$$
−0.685193 + 0.728361i $$0.740282\pi$$
$$90$$ 0 0
$$91$$ − 16.3923i − 1.71838i
$$92$$ 0 0
$$93$$ − 6.92820i − 0.718421i
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 14.3923 1.46132 0.730659 0.682743i $$-0.239213\pi$$
0.730659 + 0.682743i $$0.239213\pi$$
$$98$$ 0 0
$$99$$ − 15.4641i − 1.55420i
$$100$$ 0 0
$$101$$ − 12.0000i − 1.19404i −0.802225 0.597022i $$-0.796350\pi$$
0.802225 0.597022i $$-0.203650\pi$$
$$102$$ 0 0
$$103$$ −2.19615 −0.216393 −0.108197 0.994130i $$-0.534508\pi$$
−0.108197 + 0.994130i $$0.534508\pi$$
$$104$$ 0 0
$$105$$ −12.9282 −1.26166
$$106$$ 0 0
$$107$$ − 13.2679i − 1.28266i −0.767265 0.641331i $$-0.778383\pi$$
0.767265 0.641331i $$-0.221617\pi$$
$$108$$ 0 0
$$109$$ 12.9282i 1.23830i 0.785274 + 0.619149i $$0.212522\pi$$
−0.785274 + 0.619149i $$0.787478\pi$$
$$110$$ 0 0
$$111$$ 16.3923 1.55589
$$112$$ 0 0
$$113$$ −12.9282 −1.21618 −0.608092 0.793867i $$-0.708065\pi$$
−0.608092 + 0.793867i $$0.708065\pi$$
$$114$$ 0 0
$$115$$ − 2.19615i − 0.204792i
$$116$$ 0 0
$$117$$ 15.4641i 1.42966i
$$118$$ 0 0
$$119$$ −16.3923 −1.50268
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ 25.8564i 2.33139i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −14.1962 −1.25970 −0.629852 0.776715i $$-0.716885\pi$$
−0.629852 + 0.776715i $$0.716885\pi$$
$$128$$ 0 0
$$129$$ 0.535898 0.0471832
$$130$$ 0 0
$$131$$ 10.3923i 0.907980i 0.891007 + 0.453990i $$0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$132$$ 0 0
$$133$$ − 9.46410i − 0.820642i
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ −0.928203 −0.0793018 −0.0396509 0.999214i $$-0.512625\pi$$
−0.0396509 + 0.999214i $$0.512625\pi$$
$$138$$ 0 0
$$139$$ 10.0000i 0.848189i 0.905618 + 0.424094i $$0.139408\pi$$
−0.905618 + 0.424094i $$0.860592\pi$$
$$140$$ 0 0
$$141$$ 6.00000i 0.505291i
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 42.0526i 3.46844i
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ 9.46410 0.770178 0.385089 0.922880i $$-0.374171\pi$$
0.385089 + 0.922880i $$0.374171\pi$$
$$152$$ 0 0
$$153$$ 15.4641 1.25020
$$154$$ 0 0
$$155$$ − 2.53590i − 0.203688i
$$156$$ 0 0
$$157$$ − 12.9282i − 1.03178i −0.856654 0.515891i $$-0.827461\pi$$
0.856654 0.515891i $$-0.172539\pi$$
$$158$$ 0 0
$$159$$ 28.3923 2.25166
$$160$$ 0 0
$$161$$ −10.3923 −0.819028
$$162$$ 0 0
$$163$$ − 16.1962i − 1.26858i −0.773095 0.634290i $$-0.781292\pi$$
0.773095 0.634290i $$-0.218708\pi$$
$$164$$ 0 0
$$165$$ − 9.46410i − 0.736779i
$$166$$ 0 0
$$167$$ 2.19615 0.169943 0.0849717 0.996383i $$-0.472920\pi$$
0.0849717 + 0.996383i $$0.472920\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 8.92820i 0.682757i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ −4.73205 −0.357709
$$176$$ 0 0
$$177$$ −16.3923 −1.23212
$$178$$ 0 0
$$179$$ 7.85641i 0.587215i 0.955926 + 0.293608i $$0.0948559\pi$$
−0.955926 + 0.293608i $$0.905144\pi$$
$$180$$ 0 0
$$181$$ − 6.92820i − 0.514969i −0.966282 0.257485i $$-0.917106\pi$$
0.966282 0.257485i $$-0.0828937\pi$$
$$182$$ 0 0
$$183$$ −2.53590 −0.187459
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ − 12.0000i − 0.877527i
$$188$$ 0 0
$$189$$ − 18.9282i − 1.37682i
$$190$$ 0 0
$$191$$ −4.39230 −0.317816 −0.158908 0.987293i $$-0.550797\pi$$
−0.158908 + 0.987293i $$0.550797\pi$$
$$192$$ 0 0
$$193$$ −14.3923 −1.03598 −0.517990 0.855386i $$-0.673320\pi$$
−0.517990 + 0.855386i $$0.673320\pi$$
$$194$$ 0 0
$$195$$ 9.46410i 0.677738i
$$196$$ 0 0
$$197$$ 10.3923i 0.740421i 0.928948 + 0.370211i $$0.120714\pi$$
−0.928948 + 0.370211i $$0.879286\pi$$
$$198$$ 0 0
$$199$$ 6.92820 0.491127 0.245564 0.969380i $$-0.421027\pi$$
0.245564 + 0.969380i $$0.421027\pi$$
$$200$$ 0 0
$$201$$ 0.535898 0.0377994
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9.46410i 0.661002i
$$206$$ 0 0
$$207$$ 9.80385 0.681415
$$208$$ 0 0
$$209$$ 6.92820 0.479234
$$210$$ 0 0
$$211$$ − 14.3923i − 0.990807i −0.868663 0.495404i $$-0.835020\pi$$
0.868663 0.495404i $$-0.164980\pi$$
$$212$$ 0 0
$$213$$ 44.7846i 3.06859i
$$214$$ 0 0
$$215$$ 0.196152 0.0133775
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 0 0
$$219$$ 17.4641i 1.18011i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 9.12436 0.611012 0.305506 0.952190i $$-0.401174\pi$$
0.305506 + 0.952190i $$0.401174\pi$$
$$224$$ 0 0
$$225$$ 4.46410 0.297607
$$226$$ 0 0
$$227$$ − 12.5885i − 0.835525i −0.908556 0.417763i $$-0.862814\pi$$
0.908556 0.417763i $$-0.137186\pi$$
$$228$$ 0 0
$$229$$ − 5.07180i − 0.335154i −0.985859 0.167577i $$-0.946406\pi$$
0.985859 0.167577i $$-0.0535942\pi$$
$$230$$ 0 0
$$231$$ −44.7846 −2.94661
$$232$$ 0 0
$$233$$ −22.3923 −1.46697 −0.733484 0.679706i $$-0.762107\pi$$
−0.733484 + 0.679706i $$0.762107\pi$$
$$234$$ 0 0
$$235$$ 2.19615i 0.143261i
$$236$$ 0 0
$$237$$ − 32.7846i − 2.12959i
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 0 0
$$241$$ 0.392305 0.0252706 0.0126353 0.999920i $$-0.495978\pi$$
0.0126353 + 0.999920i $$0.495978\pi$$
$$242$$ 0 0
$$243$$ − 18.7321i − 1.20166i
$$244$$ 0 0
$$245$$ 15.3923i 0.983378i
$$246$$ 0 0
$$247$$ −6.92820 −0.440831
$$248$$ 0 0
$$249$$ −3.46410 −0.219529
$$250$$ 0 0
$$251$$ 8.53590i 0.538781i 0.963031 + 0.269391i $$0.0868223\pi$$
−0.963031 + 0.269391i $$0.913178\pi$$
$$252$$ 0 0
$$253$$ − 7.60770i − 0.478292i
$$254$$ 0 0
$$255$$ 9.46410 0.592665
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ − 28.3923i − 1.76421i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −9.80385 −0.604531 −0.302266 0.953224i $$-0.597743\pi$$
−0.302266 + 0.953224i $$0.597743\pi$$
$$264$$ 0 0
$$265$$ 10.3923 0.638394
$$266$$ 0 0
$$267$$ − 35.3205i − 2.16158i
$$268$$ 0 0
$$269$$ − 26.7846i − 1.63309i −0.577284 0.816543i $$-0.695888\pi$$
0.577284 0.816543i $$-0.304112\pi$$
$$270$$ 0 0
$$271$$ −16.3923 −0.995762 −0.497881 0.867245i $$-0.665888\pi$$
−0.497881 + 0.867245i $$0.665888\pi$$
$$272$$ 0 0
$$273$$ 44.7846 2.71049
$$274$$ 0 0
$$275$$ − 3.46410i − 0.208893i
$$276$$ 0 0
$$277$$ − 7.85641i − 0.472046i −0.971748 0.236023i $$-0.924156\pi$$
0.971748 0.236023i $$-0.0758440\pi$$
$$278$$ 0 0
$$279$$ 11.3205 0.677741
$$280$$ 0 0
$$281$$ −7.60770 −0.453837 −0.226919 0.973914i $$-0.572865\pi$$
−0.226919 + 0.973914i $$0.572865\pi$$
$$282$$ 0 0
$$283$$ − 20.5885i − 1.22386i −0.790913 0.611928i $$-0.790394\pi$$
0.790913 0.611928i $$-0.209606\pi$$
$$284$$ 0 0
$$285$$ 5.46410i 0.323665i
$$286$$ 0 0
$$287$$ 44.7846 2.64355
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 39.3205i 2.30501i
$$292$$ 0 0
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ 13.8564 0.804030
$$298$$ 0 0
$$299$$ 7.60770i 0.439964i
$$300$$ 0 0
$$301$$ − 0.928203i − 0.0535007i
$$302$$ 0 0
$$303$$ 32.7846 1.88343
$$304$$ 0 0
$$305$$ −0.928203 −0.0531488
$$306$$ 0 0
$$307$$ 23.8038i 1.35856i 0.733880 + 0.679279i $$0.237707\pi$$
−0.733880 + 0.679279i $$0.762293\pi$$
$$308$$ 0 0
$$309$$ − 6.00000i − 0.341328i
$$310$$ 0 0
$$311$$ −28.3923 −1.60998 −0.804990 0.593288i $$-0.797829\pi$$
−0.804990 + 0.593288i $$0.797829\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ − 21.1244i − 1.19022i
$$316$$ 0 0
$$317$$ − 1.60770i − 0.0902972i −0.998980 0.0451486i $$-0.985624\pi$$
0.998980 0.0451486i $$-0.0143761\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 36.2487 2.02321
$$322$$ 0 0
$$323$$ 6.92820i 0.385496i
$$324$$ 0 0
$$325$$ 3.46410i 0.192154i
$$326$$ 0 0
$$327$$ −35.3205 −1.95323
$$328$$ 0 0
$$329$$ 10.3923 0.572946
$$330$$ 0 0
$$331$$ 26.3923i 1.45065i 0.688405 + 0.725326i $$0.258311\pi$$
−0.688405 + 0.725326i $$0.741689\pi$$
$$332$$ 0 0
$$333$$ 26.7846i 1.46779i
$$334$$ 0 0
$$335$$ 0.196152 0.0107170
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ − 35.3205i − 1.91835i
$$340$$ 0 0
$$341$$ − 8.78461i − 0.475713i
$$342$$ 0 0
$$343$$ 39.7128 2.14429
$$344$$ 0 0
$$345$$ 6.00000 0.323029
$$346$$ 0 0
$$347$$ 10.0526i 0.539650i 0.962909 + 0.269825i $$0.0869658\pi$$
−0.962909 + 0.269825i $$0.913034\pi$$
$$348$$ 0 0
$$349$$ 32.7846i 1.75492i 0.479650 + 0.877460i $$0.340764\pi$$
−0.479650 + 0.877460i $$0.659236\pi$$
$$350$$ 0 0
$$351$$ −13.8564 −0.739600
$$352$$ 0 0
$$353$$ 26.7846 1.42560 0.712800 0.701367i $$-0.247427\pi$$
0.712800 + 0.701367i $$0.247427\pi$$
$$354$$ 0 0
$$355$$ 16.3923i 0.870013i
$$356$$ 0 0
$$357$$ − 44.7846i − 2.37025i
$$358$$ 0 0
$$359$$ −32.7846 −1.73031 −0.865153 0.501508i $$-0.832779\pi$$
−0.865153 + 0.501508i $$0.832779\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ − 2.73205i − 0.143395i
$$364$$ 0 0
$$365$$ 6.39230i 0.334589i
$$366$$ 0 0
$$367$$ −28.0526 −1.46433 −0.732166 0.681126i $$-0.761490\pi$$
−0.732166 + 0.681126i $$0.761490\pi$$
$$368$$ 0 0
$$369$$ −42.2487 −2.19938
$$370$$ 0 0
$$371$$ − 49.1769i − 2.55314i
$$372$$ 0 0
$$373$$ 19.8564i 1.02813i 0.857753 + 0.514063i $$0.171860\pi$$
−0.857753 + 0.514063i $$0.828140\pi$$
$$374$$ 0 0
$$375$$ 2.73205 0.141082
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 2.00000i 0.102733i 0.998680 + 0.0513665i $$0.0163577\pi$$
−0.998680 + 0.0513665i $$0.983642\pi$$
$$380$$ 0 0
$$381$$ − 38.7846i − 1.98700i
$$382$$ 0 0
$$383$$ 14.1962 0.725390 0.362695 0.931908i $$-0.381857\pi$$
0.362695 + 0.931908i $$0.381857\pi$$
$$384$$ 0 0
$$385$$ −16.3923 −0.835429
$$386$$ 0 0
$$387$$ 0.875644i 0.0445115i
$$388$$ 0 0
$$389$$ − 14.7846i − 0.749609i −0.927104 0.374805i $$-0.877710\pi$$
0.927104 0.374805i $$-0.122290\pi$$
$$390$$ 0 0
$$391$$ 7.60770 0.384738
$$392$$ 0 0
$$393$$ −28.3923 −1.43220
$$394$$ 0 0
$$395$$ − 12.0000i − 0.603786i
$$396$$ 0 0
$$397$$ 20.5359i 1.03067i 0.856990 + 0.515334i $$0.172332\pi$$
−0.856990 + 0.515334i $$0.827668\pi$$
$$398$$ 0 0
$$399$$ 25.8564 1.29444
$$400$$ 0 0
$$401$$ −31.8564 −1.59083 −0.795417 0.606063i $$-0.792748\pi$$
−0.795417 + 0.606063i $$0.792748\pi$$
$$402$$ 0 0
$$403$$ 8.78461i 0.437593i
$$404$$ 0 0
$$405$$ − 2.46410i − 0.122442i
$$406$$ 0 0
$$407$$ 20.7846 1.03025
$$408$$ 0 0
$$409$$ 24.3923 1.20612 0.603061 0.797695i $$-0.293948\pi$$
0.603061 + 0.797695i $$0.293948\pi$$
$$410$$ 0 0
$$411$$ − 2.53590i − 0.125087i
$$412$$ 0 0
$$413$$ 28.3923i 1.39709i
$$414$$ 0 0
$$415$$ −1.26795 −0.0622412
$$416$$ 0 0
$$417$$ −27.3205 −1.33789
$$418$$ 0 0
$$419$$ 12.9282i 0.631584i 0.948828 + 0.315792i $$0.102270\pi$$
−0.948828 + 0.315792i $$0.897730\pi$$
$$420$$ 0 0
$$421$$ − 6.00000i − 0.292422i −0.989253 0.146211i $$-0.953292\pi$$
0.989253 0.146211i $$-0.0467079\pi$$
$$422$$ 0 0
$$423$$ −9.80385 −0.476679
$$424$$ 0 0
$$425$$ 3.46410 0.168034
$$426$$ 0 0
$$427$$ 4.39230i 0.212559i
$$428$$ 0 0
$$429$$ 32.7846i 1.58286i
$$430$$ 0 0
$$431$$ 7.60770 0.366450 0.183225 0.983071i $$-0.441346\pi$$
0.183225 + 0.983071i $$0.441346\pi$$
$$432$$ 0 0
$$433$$ 5.60770 0.269489 0.134744 0.990880i $$-0.456979\pi$$
0.134744 + 0.990880i $$0.456979\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.39230i 0.210112i
$$438$$ 0 0
$$439$$ 5.07180 0.242064 0.121032 0.992649i $$-0.461380\pi$$
0.121032 + 0.992649i $$0.461380\pi$$
$$440$$ 0 0
$$441$$ −68.7128 −3.27204
$$442$$ 0 0
$$443$$ − 17.6603i − 0.839064i −0.907741 0.419532i $$-0.862194\pi$$
0.907741 0.419532i $$-0.137806\pi$$
$$444$$ 0 0
$$445$$ − 12.9282i − 0.612856i
$$446$$ 0 0
$$447$$ −49.1769 −2.32599
$$448$$ 0 0
$$449$$ −9.46410 −0.446639 −0.223319 0.974745i $$-0.571689\pi$$
−0.223319 + 0.974745i $$0.571689\pi$$
$$450$$ 0 0
$$451$$ 32.7846i 1.54377i
$$452$$ 0 0
$$453$$ 25.8564i 1.21484i
$$454$$ 0 0
$$455$$ 16.3923 0.768483
$$456$$ 0 0
$$457$$ −18.7846 −0.878707 −0.439353 0.898314i $$-0.644792\pi$$
−0.439353 + 0.898314i $$0.644792\pi$$
$$458$$ 0 0
$$459$$ 13.8564i 0.646762i
$$460$$ 0 0
$$461$$ − 12.0000i − 0.558896i −0.960161 0.279448i $$-0.909849\pi$$
0.960161 0.279448i $$-0.0901514\pi$$
$$462$$ 0 0
$$463$$ −26.1962 −1.21744 −0.608719 0.793386i $$-0.708316\pi$$
−0.608719 + 0.793386i $$0.708316\pi$$
$$464$$ 0 0
$$465$$ 6.92820 0.321288
$$466$$ 0 0
$$467$$ 28.9808i 1.34107i 0.741878 + 0.670535i $$0.233935\pi$$
−0.741878 + 0.670535i $$0.766065\pi$$
$$468$$ 0 0
$$469$$ − 0.928203i − 0.0428604i
$$470$$ 0 0
$$471$$ 35.3205 1.62748
$$472$$ 0 0
$$473$$ 0.679492 0.0312431
$$474$$ 0 0
$$475$$ 2.00000i 0.0917663i
$$476$$ 0 0
$$477$$ 46.3923i 2.12416i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −20.7846 −0.947697
$$482$$ 0 0
$$483$$ − 28.3923i − 1.29189i
$$484$$ 0 0
$$485$$ 14.3923i 0.653521i
$$486$$ 0 0
$$487$$ 14.8756 0.674080 0.337040 0.941490i $$-0.390574\pi$$
0.337040 + 0.941490i $$0.390574\pi$$
$$488$$ 0 0
$$489$$ 44.2487 2.00100
$$490$$ 0 0
$$491$$ 1.60770i 0.0725543i 0.999342 + 0.0362771i $$0.0115499\pi$$
−0.999342 + 0.0362771i $$0.988450\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 15.4641 0.695060
$$496$$ 0 0
$$497$$ 77.5692 3.47946
$$498$$ 0 0
$$499$$ 43.5692i 1.95043i 0.221268 + 0.975213i $$0.428980\pi$$
−0.221268 + 0.975213i $$0.571020\pi$$
$$500$$ 0 0
$$501$$ 6.00000i 0.268060i
$$502$$ 0 0
$$503$$ −2.19615 −0.0979216 −0.0489608 0.998801i $$-0.515591\pi$$
−0.0489608 + 0.998801i $$0.515591\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 2.73205i 0.121335i
$$508$$ 0 0
$$509$$ − 32.7846i − 1.45315i −0.687086 0.726576i $$-0.741110\pi$$
0.687086 0.726576i $$-0.258890\pi$$
$$510$$ 0 0
$$511$$ 30.2487 1.33812
$$512$$ 0 0
$$513$$ −8.00000 −0.353209
$$514$$ 0 0
$$515$$ − 2.19615i − 0.0967740i
$$516$$ 0 0
$$517$$ 7.60770i 0.334586i
$$518$$ 0 0
$$519$$ −16.3923 −0.719542
$$520$$ 0 0
$$521$$ 4.14359 0.181534 0.0907671 0.995872i $$-0.471068\pi$$
0.0907671 + 0.995872i $$0.471068\pi$$
$$522$$ 0 0
$$523$$ 36.9808i 1.61706i 0.588458 + 0.808528i $$0.299735\pi$$
−0.588458 + 0.808528i $$0.700265\pi$$
$$524$$ 0 0
$$525$$ − 12.9282i − 0.564233i
$$526$$ 0 0
$$527$$ 8.78461 0.382664
$$528$$ 0 0
$$529$$ −18.1769 −0.790301
$$530$$ 0 0
$$531$$ − 26.7846i − 1.16235i
$$532$$ 0 0
$$533$$ − 32.7846i − 1.42006i
$$534$$ 0 0
$$535$$ 13.2679 0.573623
$$536$$ 0 0
$$537$$ −21.4641 −0.926244
$$538$$ 0 0
$$539$$ 53.3205i 2.29668i
$$540$$ 0 0
$$541$$ − 39.7128i − 1.70739i −0.520776 0.853694i $$-0.674357\pi$$
0.520776 0.853694i $$-0.325643\pi$$
$$542$$ 0 0
$$543$$ 18.9282 0.812287
$$544$$ 0 0
$$545$$ −12.9282 −0.553783
$$546$$ 0 0
$$547$$ 12.1962i 0.521470i 0.965410 + 0.260735i $$0.0839649\pi$$
−0.965410 + 0.260735i $$0.916035\pi$$
$$548$$ 0 0
$$549$$ − 4.14359i − 0.176844i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −56.7846 −2.41473
$$554$$ 0 0
$$555$$ 16.3923i 0.695815i
$$556$$ 0 0
$$557$$ 26.7846i 1.13490i 0.823408 + 0.567450i $$0.192070\pi$$
−0.823408 + 0.567450i $$0.807930\pi$$
$$558$$ 0 0
$$559$$ −0.679492 −0.0287394
$$560$$ 0 0
$$561$$ 32.7846 1.38417
$$562$$ 0 0
$$563$$ 15.8038i 0.666053i 0.942918 + 0.333026i $$0.108070\pi$$
−0.942918 + 0.333026i $$0.891930\pi$$
$$564$$ 0 0
$$565$$ − 12.9282i − 0.543894i
$$566$$ 0 0
$$567$$ −11.6603 −0.489685
$$568$$ 0 0
$$569$$ 6.24871 0.261960 0.130980 0.991385i $$-0.458188\pi$$
0.130980 + 0.991385i $$0.458188\pi$$
$$570$$ 0 0
$$571$$ − 9.60770i − 0.402070i −0.979584 0.201035i $$-0.935570\pi$$
0.979584 0.201035i $$-0.0644304\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ 0 0
$$575$$ 2.19615 0.0915859
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ − 39.3205i − 1.63410i
$$580$$ 0 0
$$581$$ 6.00000i 0.248922i
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ −15.4641 −0.639362
$$586$$ 0 0
$$587$$ − 31.5167i − 1.30083i −0.759578 0.650416i $$-0.774595\pi$$
0.759578 0.650416i $$-0.225405\pi$$
$$588$$ 0 0
$$589$$ 5.07180i 0.208980i
$$590$$ 0 0
$$591$$ −28.3923 −1.16790
$$592$$ 0 0
$$593$$ 12.9282 0.530898 0.265449 0.964125i $$-0.414480\pi$$
0.265449 + 0.964125i $$0.414480\pi$$
$$594$$ 0 0
$$595$$ − 16.3923i − 0.672019i
$$596$$ 0 0
$$597$$ 18.9282i 0.774680i
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 0.392305 0.0160024 0.00800122 0.999968i $$-0.497453\pi$$
0.00800122 + 0.999968i $$0.497453\pi$$
$$602$$ 0 0
$$603$$ 0.875644i 0.0356590i
$$604$$ 0 0
$$605$$ − 1.00000i − 0.0406558i
$$606$$ 0 0
$$607$$ 2.19615 0.0891391 0.0445695 0.999006i $$-0.485808\pi$$
0.0445695 + 0.999006i $$0.485808\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 7.60770i − 0.307774i
$$612$$ 0 0
$$613$$ 34.3923i 1.38909i 0.719448 + 0.694546i $$0.244395\pi$$
−0.719448 + 0.694546i $$0.755605\pi$$
$$614$$ 0 0
$$615$$ −25.8564 −1.04263
$$616$$ 0 0
$$617$$ 34.3923 1.38458 0.692291 0.721618i $$-0.256601\pi$$
0.692291 + 0.721618i $$0.256601\pi$$
$$618$$ 0 0
$$619$$ − 34.7846i − 1.39811i −0.715067 0.699056i $$-0.753604\pi$$
0.715067 0.699056i $$-0.246396\pi$$
$$620$$ 0 0
$$621$$ 8.78461i 0.352514i
$$622$$ 0 0
$$623$$ −61.1769 −2.45100
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 18.9282i 0.755920i
$$628$$ 0 0
$$629$$ 20.7846i 0.828737i
$$630$$ 0 0
$$631$$ 14.5359 0.578665 0.289332 0.957229i $$-0.406567\pi$$
0.289332 + 0.957229i $$0.406567\pi$$
$$632$$ 0 0
$$633$$ 39.3205 1.56285
$$634$$ 0 0
$$635$$ − 14.1962i − 0.563357i
$$636$$ 0 0
$$637$$ − 53.3205i − 2.11264i
$$638$$ 0 0
$$639$$ −73.1769 −2.89483
$$640$$ 0 0
$$641$$ 16.3923 0.647457 0.323729 0.946150i $$-0.395064\pi$$
0.323729 + 0.946150i $$0.395064\pi$$
$$642$$ 0 0
$$643$$ − 20.5885i − 0.811929i −0.913889 0.405965i $$-0.866936\pi$$
0.913889 0.405965i $$-0.133064\pi$$
$$644$$ 0 0
$$645$$ 0.535898i 0.0211010i
$$646$$ 0 0
$$647$$ −5.41154 −0.212750 −0.106375 0.994326i $$-0.533924\pi$$
−0.106375 + 0.994326i $$0.533924\pi$$
$$648$$ 0 0
$$649$$ −20.7846 −0.815867
$$650$$ 0 0
$$651$$ − 32.7846i − 1.28493i
$$652$$ 0 0
$$653$$ − 43.1769i − 1.68964i −0.535048 0.844822i $$-0.679707\pi$$
0.535048 0.844822i $$-0.320293\pi$$
$$654$$ 0 0
$$655$$ −10.3923 −0.406061
$$656$$ 0 0
$$657$$ −28.5359 −1.11329
$$658$$ 0 0
$$659$$ − 28.6410i − 1.11570i −0.829943 0.557848i $$-0.811627\pi$$
0.829943 0.557848i $$-0.188373\pi$$
$$660$$ 0 0
$$661$$ 47.5692i 1.85023i 0.379689 + 0.925114i $$0.376031\pi$$
−0.379689 + 0.925114i $$0.623969\pi$$
$$662$$ 0 0
$$663$$ −32.7846 −1.27325
$$664$$ 0 0
$$665$$ 9.46410 0.367002
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 24.9282i 0.963780i
$$670$$ 0 0
$$671$$ −3.21539 −0.124129
$$672$$ 0 0
$$673$$ 23.1769 0.893404 0.446702 0.894683i $$-0.352598\pi$$
0.446702 + 0.894683i $$0.352598\pi$$
$$674$$ 0 0
$$675$$ 4.00000i 0.153960i
$$676$$ 0 0
$$677$$ 19.1769i 0.737029i 0.929622 + 0.368514i $$0.120133\pi$$
−0.929622 + 0.368514i $$0.879867\pi$$
$$678$$ 0 0
$$679$$ 68.1051 2.61363
$$680$$ 0 0
$$681$$ 34.3923 1.31792
$$682$$ 0 0
$$683$$ 5.66025i 0.216584i 0.994119 + 0.108292i $$0.0345381\pi$$
−0.994119 + 0.108292i $$0.965462\pi$$
$$684$$ 0 0
$$685$$ − 0.928203i − 0.0354648i
$$686$$ 0 0
$$687$$ 13.8564 0.528655
$$688$$ 0 0
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ 26.3923i 1.00401i 0.864865 + 0.502005i $$0.167404\pi$$
−0.864865 + 0.502005i $$0.832596\pi$$
$$692$$ 0 0
$$693$$ − 73.1769i − 2.77976i
$$694$$ 0 0
$$695$$ −10.0000 −0.379322
$$696$$ 0 0
$$697$$ −32.7846 −1.24181
$$698$$ 0 0
$$699$$ − 61.1769i − 2.31392i
$$700$$ 0 0
$$701$$ 26.7846i 1.01164i 0.862639 + 0.505820i $$0.168810\pi$$
−0.862639 + 0.505820i $$0.831190\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 0 0
$$705$$ −6.00000 −0.225973
$$706$$ 0 0
$$707$$ − 56.7846i − 2.13561i
$$708$$ 0 0
$$709$$ 37.8564i 1.42173i 0.703330 + 0.710864i $$0.251696\pi$$
−0.703330 + 0.710864i $$0.748304\pi$$
$$710$$ 0 0
$$711$$ 53.5692 2.00900
$$712$$ 0 0
$$713$$ 5.56922 0.208569
$$714$$ 0 0
$$715$$ 12.0000i 0.448775i
$$716$$ 0 0
$$717$$ − 56.7846i − 2.12066i
$$718$$ 0 0
$$719$$ 3.21539 0.119914 0.0599569 0.998201i $$-0.480904\pi$$
0.0599569 + 0.998201i $$0.480904\pi$$
$$720$$ 0 0
$$721$$ −10.3923 −0.387030
$$722$$ 0 0
$$723$$ 1.07180i 0.0398606i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −4.05256 −0.150301 −0.0751505 0.997172i $$-0.523944\pi$$
−0.0751505 + 0.997172i $$0.523944\pi$$
$$728$$ 0 0
$$729$$ 43.7846 1.62165
$$730$$ 0 0
$$731$$ 0.679492i 0.0251319i
$$732$$ 0 0
$$733$$ − 2.78461i − 0.102852i −0.998677 0.0514260i $$-0.983623\pi$$
0.998677 0.0514260i $$-0.0163766\pi$$
$$734$$ 0 0
$$735$$ −42.0526 −1.55113
$$736$$ 0 0
$$737$$ 0.679492 0.0250294
$$738$$ 0 0
$$739$$ 2.00000i 0.0735712i 0.999323 + 0.0367856i $$0.0117119\pi$$
−0.999323 + 0.0367856i $$0.988288\pi$$
$$740$$ 0 0
$$741$$ − 18.9282i − 0.695345i
$$742$$ 0 0
$$743$$ 5.41154 0.198530 0.0992651 0.995061i $$-0.468351\pi$$
0.0992651 + 0.995061i $$0.468351\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 0 0
$$747$$ − 5.66025i − 0.207098i
$$748$$ 0 0
$$749$$ − 62.7846i − 2.29410i
$$750$$ 0 0
$$751$$ −19.6077 −0.715495 −0.357747 0.933818i $$-0.616455\pi$$
−0.357747 + 0.933818i $$0.616455\pi$$
$$752$$ 0 0
$$753$$ −23.3205 −0.849847
$$754$$ 0 0
$$755$$ 9.46410i 0.344434i
$$756$$ 0 0
$$757$$ − 36.9282i − 1.34218i −0.741377 0.671089i $$-0.765827\pi$$
0.741377 0.671089i $$-0.234173\pi$$
$$758$$ 0 0
$$759$$ 20.7846 0.754434
$$760$$ 0 0
$$761$$ 33.7128 1.22209 0.611044 0.791596i $$-0.290750\pi$$
0.611044 + 0.791596i $$0.290750\pi$$
$$762$$ 0 0
$$763$$ 61.1769i 2.21475i
$$764$$ 0 0
$$765$$ 15.4641i 0.559106i
$$766$$ 0 0
$$767$$ 20.7846 0.750489
$$768$$ 0 0
$$769$$ 34.7846 1.25437 0.627183 0.778872i $$-0.284208\pi$$
0.627183 + 0.778872i $$0.284208\pi$$
$$770$$ 0 0
$$771$$ 16.3923i 0.590354i
$$772$$ 0 0
$$773$$ 25.6077i 0.921045i 0.887648 + 0.460522i $$0.152338\pi$$
−0.887648 + 0.460522i $$0.847662\pi$$
$$774$$ 0 0
$$775$$ 2.53590 0.0910922
$$776$$ 0 0
$$777$$ 77.5692 2.78278
$$778$$ 0 0
$$779$$ − 18.9282i − 0.678173i
$$780$$ 0 0
$$781$$ 56.7846i 2.03191i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 12.9282 0.461427
$$786$$ 0 0
$$787$$ − 16.1962i − 0.577330i −0.957430 0.288665i $$-0.906789\pi$$
0.957430 0.288665i $$-0.0932115\pi$$
$$788$$ 0 0
$$789$$ − 26.7846i − 0.953557i
$$790$$ 0 0
$$791$$ −61.1769 −2.17520
$$792$$ 0 0
$$793$$ 3.21539 0.114182
$$794$$ 0 0
$$795$$ 28.3923i 1.00697i
$$796$$ 0 0
$$797$$ 22.3923i 0.793176i 0.917997 + 0.396588i $$0.129806\pi$$
−0.917997 + 0.396588i $$0.870194\pi$$
$$798$$ 0 0
$$799$$ −7.60770 −0.269141
$$800$$ 0 0
$$801$$ 57.7128 2.03918
$$802$$ 0 0
$$803$$ 22.1436i 0.781430i
$$804$$ 0 0
$$805$$ − 10.3923i − 0.366281i
$$806$$ 0 0
$$807$$ 73.1769 2.57595
$$808$$ 0 0
$$809$$ −47.5692 −1.67244 −0.836222 0.548391i $$-0.815241\pi$$
−0.836222 + 0.548391i $$0.815241\pi$$
$$810$$ 0 0
$$811$$ − 17.6077i − 0.618290i −0.951015 0.309145i $$-0.899957\pi$$
0.951015 0.309145i $$-0.100043\pi$$
$$812$$ 0 0
$$813$$ − 44.7846i − 1.57066i
$$814$$ 0 0
$$815$$ 16.1962 0.567326
$$816$$ 0 0
$$817$$ −0.392305 −0.0137250
$$818$$ 0 0
$$819$$ 73.1769i 2.55701i
$$820$$ 0 0
$$821$$ − 9.21539i − 0.321619i −0.986985 0.160810i $$-0.948589\pi$$
0.986985 0.160810i $$-0.0514105\pi$$
$$822$$ 0 0
$$823$$ −48.8372 −1.70236 −0.851178 0.524877i $$-0.824111\pi$$
−0.851178 + 0.524877i $$0.824111\pi$$
$$824$$ 0 0
$$825$$ 9.46410 0.329498
$$826$$ 0 0
$$827$$ 1.26795i 0.0440909i 0.999757 + 0.0220455i $$0.00701786\pi$$
−0.999757 + 0.0220455i $$0.992982\pi$$
$$828$$ 0 0
$$829$$ − 9.21539i − 0.320064i −0.987112 0.160032i $$-0.948840\pi$$
0.987112 0.160032i $$-0.0511597\pi$$
$$830$$ 0 0
$$831$$ 21.4641 0.744581
$$832$$ 0 0
$$833$$ −53.3205 −1.84745
$$834$$ 0 0
$$835$$ 2.19615i 0.0760010i
$$836$$ 0 0
$$837$$ 10.1436i 0.350614i
$$838$$ 0 0
$$839$$ −32.7846 −1.13185 −0.565925 0.824457i $$-0.691481\pi$$
−0.565925 + 0.824457i $$0.691481\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ − 20.7846i − 0.715860i
$$844$$ 0 0
$$845$$ 1.00000i 0.0344010i
$$846$$ 0 0
$$847$$ −4.73205 −0.162595
$$848$$ 0 0
$$849$$ 56.2487 1.93045
$$850$$ 0 0
$$851$$ 13.1769i 0.451699i
$$852$$ 0 0
$$853$$ − 3.46410i − 0.118609i −0.998240 0.0593043i $$-0.981112\pi$$
0.998240 0.0593043i $$-0.0188882\pi$$
$$854$$ 0 0
$$855$$ −8.92820 −0.305338
$$856$$ 0 0
$$857$$ −43.8564 −1.49811 −0.749053 0.662510i $$-0.769491\pi$$
−0.749053 + 0.662510i $$0.769491\pi$$
$$858$$ 0 0
$$859$$ 31.5692i 1.07713i 0.842585 + 0.538564i $$0.181033\pi$$
−0.842585 + 0.538564i $$0.818967\pi$$
$$860$$ 0 0
$$861$$ 122.354i 4.16981i
$$862$$ 0 0
$$863$$ −10.9808 −0.373789 −0.186895 0.982380i $$-0.559842\pi$$
−0.186895 + 0.982380i $$0.559842\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ − 13.6603i − 0.463927i
$$868$$ 0 0
$$869$$ − 41.5692i − 1.41014i
$$870$$ 0 0
$$871$$ −0.679492 −0.0230237
$$872$$ 0 0
$$873$$ −64.2487 −2.17449
$$874$$ 0 0
$$875$$ − 4.73205i − 0.159973i
$$876$$ 0 0
$$877$$ 19.8564i 0.670503i 0.942129 + 0.335252i $$0.108821\pi$$
−0.942129 + 0.335252i $$0.891179\pi$$
$$878$$ 0 0
$$879$$ 81.9615 2.76449
$$880$$ 0 0
$$881$$ −28.3923 −0.956561 −0.478281 0.878207i $$-0.658740\pi$$
−0.478281 + 0.878207i $$0.658740\pi$$
$$882$$ 0 0
$$883$$ − 16.1962i − 0.545044i −0.962150 0.272522i $$-0.912142\pi$$
0.962150 0.272522i $$-0.0878577\pi$$
$$884$$ 0 0
$$885$$ − 16.3923i − 0.551021i
$$886$$ 0 0
$$887$$ 51.3731 1.72494 0.862469 0.506109i $$-0.168917\pi$$
0.862469 + 0.506109i $$0.168917\pi$$
$$888$$ 0 0
$$889$$ −67.1769 −2.25304
$$890$$ 0 0
$$891$$ − 8.53590i − 0.285963i
$$892$$ 0 0
$$893$$ − 4.39230i − 0.146983i
$$894$$ 0 0
$$895$$ −7.85641 −0.262611
$$896$$ 0 0
$$897$$ −20.7846 −0.693978
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 36.0000i 1.19933i
$$902$$ 0 0
$$903$$ 2.53590 0.0843894
$$904$$ 0 0
$$905$$ 6.92820 0.230301
$$906$$ 0 0
$$907$$ 32.5885i 1.08208i 0.840996 + 0.541041i $$0.181970\pi$$
−0.840996 + 0.541041i $$0.818030\pi$$
$$908$$ 0 0
$$909$$ 53.5692i 1.77678i
$$910$$ 0 0
$$911$$ 25.1769 0.834148 0.417074 0.908872i $$-0.363056\pi$$
0.417074 + 0.908872i $$0.363056\pi$$
$$912$$ 0 0
$$913$$ −4.39230 −0.145364
$$914$$ 0 0
$$915$$ − 2.53590i − 0.0838342i
$$916$$ 0 0
$$917$$ 49.1769i 1.62396i
$$918$$ 0 0
$$919$$ −15.7128 −0.518318 −0.259159 0.965835i $$-0.583445\pi$$
−0.259159 + 0.965835i $$0.583445\pi$$
$$920$$ 0 0
$$921$$ −65.0333 −2.14292
$$922$$ 0 0
$$923$$ − 56.7846i − 1.86909i
$$924$$ 0 0
$$925$$ 6.00000i 0.197279i
$$926$$ 0 0
$$927$$ 9.80385 0.322001
$$928$$ 0 0
$$929$$ 28.3923 0.931521 0.465761 0.884911i $$-0.345781\pi$$
0.465761 + 0.884911i $$0.345781\pi$$
$$930$$ 0 0
$$931$$ − 30.7846i − 1.00892i
$$932$$ 0 0
$$933$$ − 77.5692i − 2.53950i
$$934$$ 0 0
$$935$$ 12.0000 0.392442
$$936$$ 0 0
$$937$$ −30.3923 −0.992873 −0.496437 0.868073i $$-0.665359\pi$$
−0.496437 + 0.868073i $$0.665359\pi$$
$$938$$ 0 0
$$939$$ 60.1051i 1.96146i
$$940$$ 0 0
$$941$$ − 20.7846i − 0.677559i −0.940866 0.338779i $$-0.889986\pi$$
0.940866 0.338779i $$-0.110014\pi$$
$$942$$ 0 0
$$943$$ −20.7846 −0.676840
$$944$$ 0 0
$$945$$ 18.9282 0.615734
$$946$$ 0 0
$$947$$ − 56.1962i − 1.82613i −0.407815 0.913065i $$-0.633709\pi$$
0.407815 0.913065i $$-0.366291\pi$$
$$948$$ 0 0
$$949$$ − 22.1436i − 0.718811i
$$950$$ 0 0
$$951$$ 4.39230 0.142430
$$952$$ 0 0
$$953$$ 11.0718 0.358651 0.179325 0.983790i $$-0.442609\pi$$
0.179325 + 0.983790i $$0.442609\pi$$
$$954$$ 0 0
$$955$$ − 4.39230i − 0.142132i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −4.39230 −0.141835
$$960$$ 0 0
$$961$$ −24.5692 −0.792555
$$962$$ 0 0
$$963$$ 59.2295i 1.90864i
$$964$$ 0 0
$$965$$ − 14.3923i − 0.463305i
$$966$$ 0 0
$$967$$ 2.19615 0.0706235 0.0353118 0.999376i $$-0.488758\pi$$
0.0353118 + 0.999376i $$0.488758\pi$$
$$968$$ 0 0
$$969$$ −18.9282 −0.608061
$$970$$ 0 0
$$971$$ 57.0333i 1.83029i 0.403129 + 0.915143i $$0.367923\pi$$
−0.403129 + 0.915143i $$0.632077\pi$$
$$972$$ 0 0
$$973$$ 47.3205i 1.51703i
$$974$$ 0 0
$$975$$ −9.46410 −0.303094
$$976$$ 0 0
$$977$$ −29.3205 −0.938046 −0.469023 0.883186i $$-0.655394\pi$$
−0.469023 + 0.883186i $$0.655394\pi$$
$$978$$ 0 0
$$979$$ − 44.7846i − 1.43132i
$$980$$ 0 0
$$981$$ − 57.7128i − 1.84263i
$$982$$ 0 0
$$983$$ −42.5885 −1.35836 −0.679180 0.733971i $$-0.737665\pi$$
−0.679180 + 0.733971i $$0.737665\pi$$
$$984$$ 0 0
$$985$$ −10.3923 −0.331126
$$986$$ 0 0
$$987$$ 28.3923i 0.903737i
$$988$$ 0 0
$$989$$ 0.430781i 0.0136980i
$$990$$ 0 0
$$991$$ −32.1051 −1.01985 −0.509926 0.860218i $$-0.670327\pi$$
−0.509926 + 0.860218i $$0.670327\pi$$
$$992$$ 0 0
$$993$$ −72.1051 −2.28819
$$994$$ 0 0
$$995$$ 6.92820i 0.219639i
$$996$$ 0 0
$$997$$ 46.3923i 1.46926i 0.678468 + 0.734630i $$0.262644\pi$$
−0.678468 + 0.734630i $$0.737356\pi$$
$$998$$ 0 0
$$999$$ −24.0000 −0.759326
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 320.2.d.b.161.4 yes 4
3.2 odd 2 2880.2.k.l.1441.2 4
4.3 odd 2 320.2.d.a.161.1 4
5.2 odd 4 1600.2.f.h.1249.4 4
5.3 odd 4 1600.2.f.d.1249.1 4
5.4 even 2 1600.2.d.b.801.1 4
8.3 odd 2 320.2.d.a.161.4 yes 4
8.5 even 2 inner 320.2.d.b.161.1 yes 4
12.11 even 2 2880.2.k.e.1441.1 4
16.3 odd 4 1280.2.a.p.1.2 2
16.5 even 4 1280.2.a.m.1.2 2
16.11 odd 4 1280.2.a.b.1.1 2
16.13 even 4 1280.2.a.c.1.1 2
20.3 even 4 1600.2.f.i.1249.4 4
20.7 even 4 1600.2.f.e.1249.1 4
20.19 odd 2 1600.2.d.h.801.4 4
24.5 odd 2 2880.2.k.l.1441.4 4
24.11 even 2 2880.2.k.e.1441.3 4
40.3 even 4 1600.2.f.e.1249.2 4
40.13 odd 4 1600.2.f.h.1249.3 4
40.19 odd 2 1600.2.d.h.801.1 4
40.27 even 4 1600.2.f.i.1249.3 4
40.29 even 2 1600.2.d.b.801.4 4
40.37 odd 4 1600.2.f.d.1249.2 4
80.19 odd 4 6400.2.a.y.1.1 2
80.29 even 4 6400.2.a.ck.1.2 2
80.59 odd 4 6400.2.a.cd.1.2 2
80.69 even 4 6400.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.1 4 4.3 odd 2
320.2.d.a.161.4 yes 4 8.3 odd 2
320.2.d.b.161.1 yes 4 8.5 even 2 inner
320.2.d.b.161.4 yes 4 1.1 even 1 trivial
1280.2.a.b.1.1 2 16.11 odd 4
1280.2.a.c.1.1 2 16.13 even 4
1280.2.a.m.1.2 2 16.5 even 4
1280.2.a.p.1.2 2 16.3 odd 4
1600.2.d.b.801.1 4 5.4 even 2
1600.2.d.b.801.4 4 40.29 even 2
1600.2.d.h.801.1 4 40.19 odd 2
1600.2.d.h.801.4 4 20.19 odd 2
1600.2.f.d.1249.1 4 5.3 odd 4
1600.2.f.d.1249.2 4 40.37 odd 4
1600.2.f.e.1249.1 4 20.7 even 4
1600.2.f.e.1249.2 4 40.3 even 4
1600.2.f.h.1249.3 4 40.13 odd 4
1600.2.f.h.1249.4 4 5.2 odd 4
1600.2.f.i.1249.3 4 40.27 even 4
1600.2.f.i.1249.4 4 20.3 even 4
2880.2.k.e.1441.1 4 12.11 even 2
2880.2.k.e.1441.3 4 24.11 even 2
2880.2.k.l.1441.2 4 3.2 odd 2
2880.2.k.l.1441.4 4 24.5 odd 2
6400.2.a.y.1.1 2 80.19 odd 4
6400.2.a.bf.1.1 2 80.69 even 4
6400.2.a.cd.1.2 2 80.59 odd 4
6400.2.a.ck.1.2 2 80.29 even 4