# Properties

 Label 1216.3.e.g.1025.2 Level $1216$ Weight $3$ Character 1216.1025 Analytic conductor $33.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1336001462$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-13})$$ Defining polynomial: $$x^{2} + 13$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1025.2 Root $$3.60555i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1025 Dual form 1216.3.e.g.1025.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +O(q^{10})$$ $$q+3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +10.0000 q^{11} -3.60555i q^{13} -14.4222i q^{15} +15.0000 q^{17} +(6.00000 - 18.0278i) q^{19} -18.0278i q^{21} +35.0000 q^{23} -9.00000 q^{25} +18.0278i q^{27} -18.0278i q^{29} -36.0555i q^{31} +36.0555i q^{33} +20.0000 q^{35} +21.6333i q^{37} +13.0000 q^{39} +36.0555i q^{41} +20.0000 q^{43} +16.0000 q^{45} +10.0000 q^{47} -24.0000 q^{49} +54.0833i q^{51} +75.7166i q^{53} -40.0000 q^{55} +(65.0000 + 21.6333i) q^{57} -18.0278i q^{59} +40.0000 q^{61} +20.0000 q^{63} +14.4222i q^{65} -39.6611i q^{67} +126.194i q^{69} +108.167i q^{71} +105.000 q^{73} -32.4500i q^{75} -50.0000 q^{77} -36.0555i q^{79} -101.000 q^{81} +40.0000 q^{83} -60.0000 q^{85} +65.0000 q^{87} +18.0278i q^{91} +130.000 q^{93} +(-24.0000 + 72.1110i) q^{95} +122.589i q^{97} -40.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} - 10q^{7} - 8q^{9} + O(q^{10})$$ $$2q - 8q^{5} - 10q^{7} - 8q^{9} + 20q^{11} + 30q^{17} + 12q^{19} + 70q^{23} - 18q^{25} + 40q^{35} + 26q^{39} + 40q^{43} + 32q^{45} + 20q^{47} - 48q^{49} - 80q^{55} + 130q^{57} + 80q^{61} + 40q^{63} + 210q^{73} - 100q^{77} - 202q^{81} + 80q^{83} - 120q^{85} + 130q^{87} + 260q^{93} - 48q^{95} - 80q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 3.60555i 1.20185i 0.799305 + 0.600925i $$0.205201\pi$$
−0.799305 + 0.600925i $$0.794799\pi$$
$$4$$ 0 0
$$5$$ −4.00000 −0.800000 −0.400000 0.916515i $$-0.630990\pi$$
−0.400000 + 0.916515i $$0.630990\pi$$
$$6$$ 0 0
$$7$$ −5.00000 −0.714286 −0.357143 0.934050i $$-0.616249\pi$$
−0.357143 + 0.934050i $$0.616249\pi$$
$$8$$ 0 0
$$9$$ −4.00000 −0.444444
$$10$$ 0 0
$$11$$ 10.0000 0.909091 0.454545 0.890724i $$-0.349802\pi$$
0.454545 + 0.890724i $$0.349802\pi$$
$$12$$ 0 0
$$13$$ 3.60555i 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 0 0
$$15$$ 14.4222i 0.961480i
$$16$$ 0 0
$$17$$ 15.0000 0.882353 0.441176 0.897420i $$-0.354561\pi$$
0.441176 + 0.897420i $$0.354561\pi$$
$$18$$ 0 0
$$19$$ 6.00000 18.0278i 0.315789 0.948829i
$$20$$ 0 0
$$21$$ 18.0278i 0.858465i
$$22$$ 0 0
$$23$$ 35.0000 1.52174 0.760870 0.648905i $$-0.224773\pi$$
0.760870 + 0.648905i $$0.224773\pi$$
$$24$$ 0 0
$$25$$ −9.00000 −0.360000
$$26$$ 0 0
$$27$$ 18.0278i 0.667695i
$$28$$ 0 0
$$29$$ 18.0278i 0.621647i −0.950468 0.310823i $$-0.899395\pi$$
0.950468 0.310823i $$-0.100605\pi$$
$$30$$ 0 0
$$31$$ 36.0555i 1.16308i −0.813517 0.581541i $$-0.802450\pi$$
0.813517 0.581541i $$-0.197550\pi$$
$$32$$ 0 0
$$33$$ 36.0555i 1.09259i
$$34$$ 0 0
$$35$$ 20.0000 0.571429
$$36$$ 0 0
$$37$$ 21.6333i 0.584684i 0.956314 + 0.292342i $$0.0944346\pi$$
−0.956314 + 0.292342i $$0.905565\pi$$
$$38$$ 0 0
$$39$$ 13.0000 0.333333
$$40$$ 0 0
$$41$$ 36.0555i 0.879403i 0.898144 + 0.439701i $$0.144916\pi$$
−0.898144 + 0.439701i $$0.855084\pi$$
$$42$$ 0 0
$$43$$ 20.0000 0.465116 0.232558 0.972582i $$-0.425290\pi$$
0.232558 + 0.972582i $$0.425290\pi$$
$$44$$ 0 0
$$45$$ 16.0000 0.355556
$$46$$ 0 0
$$47$$ 10.0000 0.212766 0.106383 0.994325i $$-0.466073\pi$$
0.106383 + 0.994325i $$0.466073\pi$$
$$48$$ 0 0
$$49$$ −24.0000 −0.489796
$$50$$ 0 0
$$51$$ 54.0833i 1.06046i
$$52$$ 0 0
$$53$$ 75.7166i 1.42861i 0.699832 + 0.714307i $$0.253258\pi$$
−0.699832 + 0.714307i $$0.746742\pi$$
$$54$$ 0 0
$$55$$ −40.0000 −0.727273
$$56$$ 0 0
$$57$$ 65.0000 + 21.6333i 1.14035 + 0.379532i
$$58$$ 0 0
$$59$$ 18.0278i 0.305555i −0.988261 0.152778i $$-0.951178\pi$$
0.988261 0.152778i $$-0.0488218\pi$$
$$60$$ 0 0
$$61$$ 40.0000 0.655738 0.327869 0.944723i $$-0.393670\pi$$
0.327869 + 0.944723i $$0.393670\pi$$
$$62$$ 0 0
$$63$$ 20.0000 0.317460
$$64$$ 0 0
$$65$$ 14.4222i 0.221880i
$$66$$ 0 0
$$67$$ 39.6611i 0.591956i −0.955195 0.295978i $$-0.904354\pi$$
0.955195 0.295978i $$-0.0956455\pi$$
$$68$$ 0 0
$$69$$ 126.194i 1.82890i
$$70$$ 0 0
$$71$$ 108.167i 1.52347i 0.647887 + 0.761736i $$0.275653\pi$$
−0.647887 + 0.761736i $$0.724347\pi$$
$$72$$ 0 0
$$73$$ 105.000 1.43836 0.719178 0.694826i $$-0.244519\pi$$
0.719178 + 0.694826i $$0.244519\pi$$
$$74$$ 0 0
$$75$$ 32.4500i 0.432666i
$$76$$ 0 0
$$77$$ −50.0000 −0.649351
$$78$$ 0 0
$$79$$ 36.0555i 0.456399i −0.973614 0.228199i $$-0.926716\pi$$
0.973614 0.228199i $$-0.0732838\pi$$
$$80$$ 0 0
$$81$$ −101.000 −1.24691
$$82$$ 0 0
$$83$$ 40.0000 0.481928 0.240964 0.970534i $$-0.422536\pi$$
0.240964 + 0.970534i $$0.422536\pi$$
$$84$$ 0 0
$$85$$ −60.0000 −0.705882
$$86$$ 0 0
$$87$$ 65.0000 0.747126
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 18.0278i 0.198107i
$$92$$ 0 0
$$93$$ 130.000 1.39785
$$94$$ 0 0
$$95$$ −24.0000 + 72.1110i −0.252632 + 0.759063i
$$96$$ 0 0
$$97$$ 122.589i 1.26380i 0.775049 + 0.631901i $$0.217725\pi$$
−0.775049 + 0.631901i $$0.782275\pi$$
$$98$$ 0 0
$$99$$ −40.0000 −0.404040
$$100$$ 0 0
$$101$$ 50.0000 0.495050 0.247525 0.968882i $$-0.420383\pi$$
0.247525 + 0.968882i $$0.420383\pi$$
$$102$$ 0 0
$$103$$ 57.6888i 0.560086i 0.959988 + 0.280043i $$0.0903487\pi$$
−0.959988 + 0.280043i $$0.909651\pi$$
$$104$$ 0 0
$$105$$ 72.1110i 0.686772i
$$106$$ 0 0
$$107$$ 75.7166i 0.707632i −0.935315 0.353816i $$-0.884884\pi$$
0.935315 0.353816i $$-0.115116\pi$$
$$108$$ 0 0
$$109$$ 198.305i 1.81931i 0.415359 + 0.909657i $$0.363656\pi$$
−0.415359 + 0.909657i $$0.636344\pi$$
$$110$$ 0 0
$$111$$ −78.0000 −0.702703
$$112$$ 0 0
$$113$$ 122.589i 1.08486i −0.840102 0.542428i $$-0.817505\pi$$
0.840102 0.542428i $$-0.182495\pi$$
$$114$$ 0 0
$$115$$ −140.000 −1.21739
$$116$$ 0 0
$$117$$ 14.4222i 0.123267i
$$118$$ 0 0
$$119$$ −75.0000 −0.630252
$$120$$ 0 0
$$121$$ −21.0000 −0.173554
$$122$$ 0 0
$$123$$ −130.000 −1.05691
$$124$$ 0 0
$$125$$ 136.000 1.08800
$$126$$ 0 0
$$127$$ 129.800i 1.02205i −0.859567 0.511023i $$-0.829267\pi$$
0.859567 0.511023i $$-0.170733\pi$$
$$128$$ 0 0
$$129$$ 72.1110i 0.559000i
$$130$$ 0 0
$$131$$ −112.000 −0.854962 −0.427481 0.904024i $$-0.640599\pi$$
−0.427481 + 0.904024i $$0.640599\pi$$
$$132$$ 0 0
$$133$$ −30.0000 + 90.1388i −0.225564 + 0.677735i
$$134$$ 0 0
$$135$$ 72.1110i 0.534156i
$$136$$ 0 0
$$137$$ 125.000 0.912409 0.456204 0.889875i $$-0.349209\pi$$
0.456204 + 0.889875i $$0.349209\pi$$
$$138$$ 0 0
$$139$$ −50.0000 −0.359712 −0.179856 0.983693i $$-0.557563\pi$$
−0.179856 + 0.983693i $$0.557563\pi$$
$$140$$ 0 0
$$141$$ 36.0555i 0.255713i
$$142$$ 0 0
$$143$$ 36.0555i 0.252136i
$$144$$ 0 0
$$145$$ 72.1110i 0.497317i
$$146$$ 0 0
$$147$$ 86.5332i 0.588661i
$$148$$ 0 0
$$149$$ −70.0000 −0.469799 −0.234899 0.972020i $$-0.575476\pi$$
−0.234899 + 0.972020i $$0.575476\pi$$
$$150$$ 0 0
$$151$$ 36.0555i 0.238778i 0.992848 + 0.119389i $$0.0380936\pi$$
−0.992848 + 0.119389i $$0.961906\pi$$
$$152$$ 0 0
$$153$$ −60.0000 −0.392157
$$154$$ 0 0
$$155$$ 144.222i 0.930465i
$$156$$ 0 0
$$157$$ −10.0000 −0.0636943 −0.0318471 0.999493i $$-0.510139\pi$$
−0.0318471 + 0.999493i $$0.510139\pi$$
$$158$$ 0 0
$$159$$ −273.000 −1.71698
$$160$$ 0 0
$$161$$ −175.000 −1.08696
$$162$$ 0 0
$$163$$ 270.000 1.65644 0.828221 0.560402i $$-0.189353\pi$$
0.828221 + 0.560402i $$0.189353\pi$$
$$164$$ 0 0
$$165$$ 144.222i 0.874073i
$$166$$ 0 0
$$167$$ 122.589i 0.734064i 0.930208 + 0.367032i $$0.119626\pi$$
−0.930208 + 0.367032i $$0.880374\pi$$
$$168$$ 0 0
$$169$$ 156.000 0.923077
$$170$$ 0 0
$$171$$ −24.0000 + 72.1110i −0.140351 + 0.421702i
$$172$$ 0 0
$$173$$ 122.589i 0.708605i 0.935131 + 0.354303i $$0.115282\pi$$
−0.935131 + 0.354303i $$0.884718\pi$$
$$174$$ 0 0
$$175$$ 45.0000 0.257143
$$176$$ 0 0
$$177$$ 65.0000 0.367232
$$178$$ 0 0
$$179$$ 36.0555i 0.201427i −0.994915 0.100714i $$-0.967887\pi$$
0.994915 0.100714i $$-0.0321126\pi$$
$$180$$ 0 0
$$181$$ 108.167i 0.597605i −0.954315 0.298803i $$-0.903413\pi$$
0.954315 0.298803i $$-0.0965872\pi$$
$$182$$ 0 0
$$183$$ 144.222i 0.788099i
$$184$$ 0 0
$$185$$ 86.5332i 0.467747i
$$186$$ 0 0
$$187$$ 150.000 0.802139
$$188$$ 0 0
$$189$$ 90.1388i 0.476925i
$$190$$ 0 0
$$191$$ 193.000 1.01047 0.505236 0.862981i $$-0.331406\pi$$
0.505236 + 0.862981i $$0.331406\pi$$
$$192$$ 0 0
$$193$$ 266.811i 1.38244i 0.722645 + 0.691220i $$0.242926\pi$$
−0.722645 + 0.691220i $$0.757074\pi$$
$$194$$ 0 0
$$195$$ −52.0000 −0.266667
$$196$$ 0 0
$$197$$ −90.0000 −0.456853 −0.228426 0.973561i $$-0.573358\pi$$
−0.228426 + 0.973561i $$0.573358\pi$$
$$198$$ 0 0
$$199$$ 123.000 0.618090 0.309045 0.951047i $$-0.399991\pi$$
0.309045 + 0.951047i $$0.399991\pi$$
$$200$$ 0 0
$$201$$ 143.000 0.711443
$$202$$ 0 0
$$203$$ 90.1388i 0.444033i
$$204$$ 0 0
$$205$$ 144.222i 0.703522i
$$206$$ 0 0
$$207$$ −140.000 −0.676329
$$208$$ 0 0
$$209$$ 60.0000 180.278i 0.287081 0.862572i
$$210$$ 0 0
$$211$$ 234.361i 1.11071i 0.831612 + 0.555357i $$0.187419\pi$$
−0.831612 + 0.555357i $$0.812581\pi$$
$$212$$ 0 0
$$213$$ −390.000 −1.83099
$$214$$ 0 0
$$215$$ −80.0000 −0.372093
$$216$$ 0 0
$$217$$ 180.278i 0.830772i
$$218$$ 0 0
$$219$$ 378.583i 1.72869i
$$220$$ 0 0
$$221$$ 54.0833i 0.244721i
$$222$$ 0 0
$$223$$ 201.911i 0.905430i −0.891655 0.452715i $$-0.850456\pi$$
0.891655 0.452715i $$-0.149544\pi$$
$$224$$ 0 0
$$225$$ 36.0000 0.160000
$$226$$ 0 0
$$227$$ 255.994i 1.12773i −0.825868 0.563864i $$-0.809314\pi$$
0.825868 0.563864i $$-0.190686\pi$$
$$228$$ 0 0
$$229$$ 160.000 0.698690 0.349345 0.936994i $$-0.386404\pi$$
0.349345 + 0.936994i $$0.386404\pi$$
$$230$$ 0 0
$$231$$ 180.278i 0.780422i
$$232$$ 0 0
$$233$$ −270.000 −1.15880 −0.579399 0.815044i $$-0.696713\pi$$
−0.579399 + 0.815044i $$0.696713\pi$$
$$234$$ 0 0
$$235$$ −40.0000 −0.170213
$$236$$ 0 0
$$237$$ 130.000 0.548523
$$238$$ 0 0
$$239$$ 197.000 0.824268 0.412134 0.911123i $$-0.364784\pi$$
0.412134 + 0.911123i $$0.364784\pi$$
$$240$$ 0 0
$$241$$ 396.611i 1.64569i 0.568268 + 0.822844i $$0.307614\pi$$
−0.568268 + 0.822844i $$0.692386\pi$$
$$242$$ 0 0
$$243$$ 201.911i 0.830909i
$$244$$ 0 0
$$245$$ 96.0000 0.391837
$$246$$ 0 0
$$247$$ −65.0000 21.6333i −0.263158 0.0875842i
$$248$$ 0 0
$$249$$ 144.222i 0.579205i
$$250$$ 0 0
$$251$$ 402.000 1.60159 0.800797 0.598936i $$-0.204410\pi$$
0.800797 + 0.598936i $$0.204410\pi$$
$$252$$ 0 0
$$253$$ 350.000 1.38340
$$254$$ 0 0
$$255$$ 216.333i 0.848365i
$$256$$ 0 0
$$257$$ 418.244i 1.62741i −0.581279 0.813704i $$-0.697448\pi$$
0.581279 0.813704i $$-0.302552\pi$$
$$258$$ 0 0
$$259$$ 108.167i 0.417631i
$$260$$ 0 0
$$261$$ 72.1110i 0.276287i
$$262$$ 0 0
$$263$$ 310.000 1.17871 0.589354 0.807875i $$-0.299382\pi$$
0.589354 + 0.807875i $$0.299382\pi$$
$$264$$ 0 0
$$265$$ 302.866i 1.14289i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 108.167i 0.402106i −0.979580 0.201053i $$-0.935564\pi$$
0.979580 0.201053i $$-0.0644364\pi$$
$$270$$ 0 0
$$271$$ 105.000 0.387454 0.193727 0.981055i $$-0.437942\pi$$
0.193727 + 0.981055i $$0.437942\pi$$
$$272$$ 0 0
$$273$$ −65.0000 −0.238095
$$274$$ 0 0
$$275$$ −90.0000 −0.327273
$$276$$ 0 0
$$277$$ 50.0000 0.180505 0.0902527 0.995919i $$-0.471233\pi$$
0.0902527 + 0.995919i $$0.471233\pi$$
$$278$$ 0 0
$$279$$ 144.222i 0.516925i
$$280$$ 0 0
$$281$$ 288.444i 1.02649i −0.858242 0.513246i $$-0.828443\pi$$
0.858242 0.513246i $$-0.171557\pi$$
$$282$$ 0 0
$$283$$ 320.000 1.13074 0.565371 0.824837i $$-0.308733\pi$$
0.565371 + 0.824837i $$0.308733\pi$$
$$284$$ 0 0
$$285$$ −260.000 86.5332i −0.912281 0.303625i
$$286$$ 0 0
$$287$$ 180.278i 0.628145i
$$288$$ 0 0
$$289$$ −64.0000 −0.221453
$$290$$ 0 0
$$291$$ −442.000 −1.51890
$$292$$ 0 0
$$293$$ 219.939i 0.750644i 0.926895 + 0.375322i $$0.122468\pi$$
−0.926895 + 0.375322i $$0.877532\pi$$
$$294$$ 0 0
$$295$$ 72.1110i 0.244444i
$$296$$ 0 0
$$297$$ 180.278i 0.606995i
$$298$$ 0 0
$$299$$ 126.194i 0.422054i
$$300$$ 0 0
$$301$$ −100.000 −0.332226
$$302$$ 0 0
$$303$$ 180.278i 0.594975i
$$304$$ 0 0
$$305$$ −160.000 −0.524590
$$306$$ 0 0
$$307$$ 237.966i 0.775135i −0.921841 0.387567i $$-0.873315\pi$$
0.921841 0.387567i $$-0.126685\pi$$
$$308$$ 0 0
$$309$$ −208.000 −0.673139
$$310$$ 0 0
$$311$$ 395.000 1.27010 0.635048 0.772472i $$-0.280980\pi$$
0.635048 + 0.772472i $$0.280980\pi$$
$$312$$ 0 0
$$313$$ 125.000 0.399361 0.199681 0.979861i $$-0.436010\pi$$
0.199681 + 0.979861i $$0.436010\pi$$
$$314$$ 0 0
$$315$$ −80.0000 −0.253968
$$316$$ 0 0
$$317$$ 3.60555i 0.0113740i −0.999984 0.00568699i $$-0.998190\pi$$
0.999984 0.00568699i $$-0.00181023\pi$$
$$318$$ 0 0
$$319$$ 180.278i 0.565133i
$$320$$ 0 0
$$321$$ 273.000 0.850467
$$322$$ 0 0
$$323$$ 90.0000 270.416i 0.278638 0.837202i
$$324$$ 0 0
$$325$$ 32.4500i 0.0998460i
$$326$$ 0 0
$$327$$ −715.000 −2.18654
$$328$$ 0 0
$$329$$ −50.0000 −0.151976
$$330$$ 0 0
$$331$$ 198.305i 0.599110i 0.954079 + 0.299555i $$0.0968382\pi$$
−0.954079 + 0.299555i $$0.903162\pi$$
$$332$$ 0 0
$$333$$ 86.5332i 0.259860i
$$334$$ 0 0
$$335$$ 158.644i 0.473565i
$$336$$ 0 0
$$337$$ 57.6888i 0.171183i 0.996330 + 0.0855917i $$0.0272781\pi$$
−0.996330 + 0.0855917i $$0.972722\pi$$
$$338$$ 0 0
$$339$$ 442.000 1.30383
$$340$$ 0 0
$$341$$ 360.555i 1.05735i
$$342$$ 0 0
$$343$$ 365.000 1.06414
$$344$$ 0 0
$$345$$ 504.777i 1.46312i
$$346$$ 0 0
$$347$$ 40.0000 0.115274 0.0576369 0.998338i $$-0.481643\pi$$
0.0576369 + 0.998338i $$0.481643\pi$$
$$348$$ 0 0
$$349$$ −98.0000 −0.280802 −0.140401 0.990095i $$-0.544839\pi$$
−0.140401 + 0.990095i $$0.544839\pi$$
$$350$$ 0 0
$$351$$ 65.0000 0.185185
$$352$$ 0 0
$$353$$ −185.000 −0.524079 −0.262040 0.965057i $$-0.584395\pi$$
−0.262040 + 0.965057i $$0.584395\pi$$
$$354$$ 0 0
$$355$$ 432.666i 1.21878i
$$356$$ 0 0
$$357$$ 270.416i 0.757469i
$$358$$ 0 0
$$359$$ −225.000 −0.626741 −0.313370 0.949631i $$-0.601458\pi$$
−0.313370 + 0.949631i $$0.601458\pi$$
$$360$$ 0 0
$$361$$ −289.000 216.333i −0.800554 0.599261i
$$362$$ 0 0
$$363$$ 75.7166i 0.208586i
$$364$$ 0 0
$$365$$ −420.000 −1.15068
$$366$$ 0 0
$$367$$ 50.0000 0.136240 0.0681199 0.997677i $$-0.478300\pi$$
0.0681199 + 0.997677i $$0.478300\pi$$
$$368$$ 0 0
$$369$$ 144.222i 0.390846i
$$370$$ 0 0
$$371$$ 378.583i 1.02044i
$$372$$ 0 0
$$373$$ 436.272i 1.16963i 0.811167 + 0.584815i $$0.198833\pi$$
−0.811167 + 0.584815i $$0.801167\pi$$
$$374$$ 0 0
$$375$$ 490.355i 1.30761i
$$376$$ 0 0
$$377$$ −65.0000 −0.172414
$$378$$ 0 0
$$379$$ 486.749i 1.28430i 0.766579 + 0.642150i $$0.221957\pi$$
−0.766579 + 0.642150i $$0.778043\pi$$
$$380$$ 0 0
$$381$$ 468.000 1.22835
$$382$$ 0 0
$$383$$ 201.911i 0.527182i 0.964634 + 0.263591i $$0.0849070\pi$$
−0.964634 + 0.263591i $$0.915093\pi$$
$$384$$ 0 0
$$385$$ 200.000 0.519481
$$386$$ 0 0
$$387$$ −80.0000 −0.206718
$$388$$ 0 0
$$389$$ 478.000 1.22879 0.614396 0.788998i $$-0.289400\pi$$
0.614396 + 0.788998i $$0.289400\pi$$
$$390$$ 0 0
$$391$$ 525.000 1.34271
$$392$$ 0 0
$$393$$ 403.822i 1.02754i
$$394$$ 0 0
$$395$$ 144.222i 0.365119i
$$396$$ 0 0
$$397$$ −750.000 −1.88917 −0.944584 0.328269i $$-0.893535\pi$$
−0.944584 + 0.328269i $$0.893535\pi$$
$$398$$ 0 0
$$399$$ −325.000 108.167i −0.814536 0.271094i
$$400$$ 0 0
$$401$$ 288.444i 0.719312i −0.933085 0.359656i $$-0.882894\pi$$
0.933085 0.359656i $$-0.117106\pi$$
$$402$$ 0 0
$$403$$ −130.000 −0.322581
$$404$$ 0 0
$$405$$ 404.000 0.997531
$$406$$ 0 0
$$407$$ 216.333i 0.531531i
$$408$$ 0 0
$$409$$ 36.0555i 0.0881553i −0.999028 0.0440776i $$-0.985965\pi$$
0.999028 0.0440776i $$-0.0140349\pi$$
$$410$$ 0 0
$$411$$ 450.694i 1.09658i
$$412$$ 0 0
$$413$$ 90.1388i 0.218254i
$$414$$ 0 0
$$415$$ −160.000 −0.385542
$$416$$ 0 0
$$417$$ 180.278i 0.432320i
$$418$$ 0 0
$$419$$ −112.000 −0.267303 −0.133652 0.991028i $$-0.542670\pi$$
−0.133652 + 0.991028i $$0.542670\pi$$
$$420$$ 0 0
$$421$$ 630.971i 1.49874i −0.662149 0.749372i $$-0.730355\pi$$
0.662149 0.749372i $$-0.269645\pi$$
$$422$$ 0 0
$$423$$ −40.0000 −0.0945626
$$424$$ 0 0
$$425$$ −135.000 −0.317647
$$426$$ 0 0
$$427$$ −200.000 −0.468384
$$428$$ 0 0
$$429$$ 130.000 0.303030
$$430$$ 0 0
$$431$$ 432.666i 1.00387i −0.864907 0.501933i $$-0.832622\pi$$
0.864907 0.501933i $$-0.167378\pi$$
$$432$$ 0 0
$$433$$ 735.532i 1.69869i −0.527839 0.849345i $$-0.676997\pi$$
0.527839 0.849345i $$-0.323003\pi$$
$$434$$ 0 0
$$435$$ −260.000 −0.597701
$$436$$ 0 0
$$437$$ 210.000 630.971i 0.480549 1.44387i
$$438$$ 0 0
$$439$$ 793.221i 1.80688i 0.428712 + 0.903441i $$0.358967\pi$$
−0.428712 + 0.903441i $$0.641033\pi$$
$$440$$ 0 0
$$441$$ 96.0000 0.217687
$$442$$ 0 0
$$443$$ −670.000 −1.51242 −0.756208 0.654332i $$-0.772950\pi$$
−0.756208 + 0.654332i $$0.772950\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 252.389i 0.564628i
$$448$$ 0 0
$$449$$ 36.0555i 0.0803018i −0.999194 0.0401509i $$-0.987216\pi$$
0.999194 0.0401509i $$-0.0127839\pi$$
$$450$$ 0 0
$$451$$ 360.555i 0.799457i
$$452$$ 0 0
$$453$$ −130.000 −0.286976
$$454$$ 0 0
$$455$$ 72.1110i 0.158486i
$$456$$ 0 0
$$457$$ −755.000 −1.65208 −0.826039 0.563612i $$-0.809411\pi$$
−0.826039 + 0.563612i $$0.809411\pi$$
$$458$$ 0 0
$$459$$ 270.416i 0.589142i
$$460$$ 0 0
$$461$$ −772.000 −1.67462 −0.837310 0.546728i $$-0.815873\pi$$
−0.837310 + 0.546728i $$0.815873\pi$$
$$462$$ 0 0
$$463$$ −350.000 −0.755940 −0.377970 0.925818i $$-0.623378\pi$$
−0.377970 + 0.925818i $$0.623378\pi$$
$$464$$ 0 0
$$465$$ −520.000 −1.11828
$$466$$ 0 0
$$467$$ −70.0000 −0.149893 −0.0749465 0.997188i $$-0.523879\pi$$
−0.0749465 + 0.997188i $$0.523879\pi$$
$$468$$ 0 0
$$469$$ 198.305i 0.422826i
$$470$$ 0 0
$$471$$ 36.0555i 0.0765510i
$$472$$ 0 0
$$473$$ 200.000 0.422833
$$474$$ 0 0
$$475$$ −54.0000 + 162.250i −0.113684 + 0.341579i
$$476$$ 0 0
$$477$$ 302.866i 0.634940i
$$478$$ 0 0
$$479$$ −370.000 −0.772443 −0.386221 0.922406i $$-0.626220\pi$$
−0.386221 + 0.922406i $$0.626220\pi$$
$$480$$ 0 0
$$481$$ 78.0000 0.162162
$$482$$ 0 0
$$483$$ 630.971i 1.30636i
$$484$$ 0 0
$$485$$ 490.355i 1.01104i
$$486$$ 0 0
$$487$$ 519.199i 1.06612i −0.846078 0.533059i $$-0.821042\pi$$
0.846078 0.533059i $$-0.178958\pi$$
$$488$$ 0 0
$$489$$ 973.499i 1.99080i
$$490$$ 0 0
$$491$$ 632.000 1.28717 0.643585 0.765375i $$-0.277446\pi$$
0.643585 + 0.765375i $$0.277446\pi$$
$$492$$ 0 0
$$493$$ 270.416i 0.548512i
$$494$$ 0 0
$$495$$ 160.000 0.323232
$$496$$ 0 0
$$497$$ 540.833i 1.08819i
$$498$$ 0 0
$$499$$ −380.000 −0.761523 −0.380762 0.924673i $$-0.624338\pi$$
−0.380762 + 0.924673i $$0.624338\pi$$
$$500$$ 0 0
$$501$$ −442.000 −0.882236
$$502$$ 0 0
$$503$$ −45.0000 −0.0894632 −0.0447316 0.998999i $$-0.514243\pi$$
−0.0447316 + 0.998999i $$0.514243\pi$$
$$504$$ 0 0
$$505$$ −200.000 −0.396040
$$506$$ 0 0
$$507$$ 562.466i 1.10940i
$$508$$ 0 0
$$509$$ 829.277i 1.62923i −0.580004 0.814614i $$-0.696949\pi$$
0.580004 0.814614i $$-0.303051\pi$$
$$510$$ 0 0
$$511$$ −525.000 −1.02740
$$512$$ 0 0
$$513$$ 325.000 + 108.167i 0.633528 + 0.210851i
$$514$$ 0 0
$$515$$ 230.755i 0.448069i
$$516$$ 0 0
$$517$$ 100.000 0.193424
$$518$$ 0 0
$$519$$ −442.000 −0.851638
$$520$$ 0 0
$$521$$ 612.944i 1.17648i 0.808688 + 0.588238i $$0.200178\pi$$
−0.808688 + 0.588238i $$0.799822\pi$$
$$522$$ 0 0
$$523$$ 465.116i 0.889323i −0.895699 0.444662i $$-0.853324\pi$$
0.895699 0.444662i $$-0.146676\pi$$
$$524$$ 0 0
$$525$$ 162.250i 0.309047i
$$526$$ 0 0
$$527$$ 540.833i 1.02625i
$$528$$ 0 0
$$529$$ 696.000 1.31569
$$530$$ 0 0
$$531$$ 72.1110i 0.135802i
$$532$$ 0 0
$$533$$ 130.000 0.243902
$$534$$ 0 0
$$535$$ 302.866i 0.566105i
$$536$$ 0 0
$$537$$ 130.000 0.242086
$$538$$ 0 0
$$539$$ −240.000 −0.445269
$$540$$ 0 0
$$541$$ 600.000 1.10906 0.554529 0.832165i $$-0.312899\pi$$
0.554529 + 0.832165i $$0.312899\pi$$
$$542$$ 0 0
$$543$$ 390.000 0.718232
$$544$$ 0 0
$$545$$ 793.221i 1.45545i
$$546$$ 0 0
$$547$$ 598.522i 1.09419i 0.837071 + 0.547095i $$0.184266\pi$$
−0.837071 + 0.547095i $$0.815734\pi$$
$$548$$ 0 0
$$549$$ −160.000 −0.291439
$$550$$ 0 0
$$551$$ −325.000 108.167i −0.589837 0.196310i
$$552$$ 0 0
$$553$$ 180.278i 0.325999i
$$554$$ 0 0
$$555$$ 312.000 0.562162
$$556$$ 0 0
$$557$$ −380.000 −0.682226 −0.341113 0.940022i $$-0.610804\pi$$
−0.341113 + 0.940022i $$0.610804\pi$$
$$558$$ 0 0
$$559$$ 72.1110i 0.129000i
$$560$$ 0 0
$$561$$ 540.833i 0.964051i
$$562$$ 0 0
$$563$$ 122.589i 0.217742i −0.994056 0.108871i $$-0.965276\pi$$
0.994056 0.108871i $$-0.0347235\pi$$
$$564$$ 0 0
$$565$$ 490.355i 0.867885i
$$566$$ 0 0
$$567$$ 505.000 0.890653
$$568$$ 0 0
$$569$$ 36.0555i 0.0633665i 0.999498 + 0.0316832i $$0.0100868\pi$$
−0.999498 + 0.0316832i $$0.989913\pi$$
$$570$$ 0 0
$$571$$ 790.000 1.38354 0.691769 0.722119i $$-0.256832\pi$$
0.691769 + 0.722119i $$0.256832\pi$$
$$572$$ 0 0
$$573$$ 695.871i 1.21444i
$$574$$ 0 0
$$575$$ −315.000 −0.547826
$$576$$ 0 0
$$577$$ 675.000 1.16984 0.584922 0.811090i $$-0.301125\pi$$
0.584922 + 0.811090i $$0.301125\pi$$
$$578$$ 0 0
$$579$$ −962.000 −1.66149
$$580$$ 0 0
$$581$$ −200.000 −0.344234
$$582$$ 0 0
$$583$$ 757.166i 1.29874i
$$584$$ 0 0
$$585$$ 57.6888i 0.0986134i
$$586$$ 0 0
$$587$$ 280.000 0.477002 0.238501 0.971142i $$-0.423344\pi$$
0.238501 + 0.971142i $$0.423344\pi$$
$$588$$ 0 0
$$589$$ −650.000 216.333i −1.10357 0.367289i
$$590$$ 0 0
$$591$$ 324.500i 0.549069i
$$592$$ 0 0
$$593$$ 750.000 1.26476 0.632378 0.774660i $$-0.282079\pi$$
0.632378 + 0.774660i $$0.282079\pi$$
$$594$$ 0 0
$$595$$ 300.000 0.504202
$$596$$ 0 0
$$597$$ 443.483i 0.742852i
$$598$$ 0 0
$$599$$ 504.777i 0.842700i 0.906898 + 0.421350i $$0.138444\pi$$
−0.906898 + 0.421350i $$0.861556\pi$$
$$600$$ 0 0
$$601$$ 612.944i 1.01987i 0.860212 + 0.509937i $$0.170331\pi$$
−0.860212 + 0.509937i $$0.829669\pi$$
$$602$$ 0 0
$$603$$ 158.644i 0.263092i
$$604$$ 0 0
$$605$$ 84.0000 0.138843
$$606$$ 0 0
$$607$$ 987.921i 1.62755i −0.581182 0.813774i $$-0.697410\pi$$
0.581182 0.813774i $$-0.302590\pi$$
$$608$$ 0 0
$$609$$ −325.000 −0.533662
$$610$$ 0 0
$$611$$ 36.0555i 0.0590107i
$$612$$ 0 0
$$613$$ 1200.00 1.95759 0.978793 0.204853i $$-0.0656715\pi$$
0.978793 + 0.204853i $$0.0656715\pi$$
$$614$$ 0 0
$$615$$ 520.000 0.845528
$$616$$ 0 0
$$617$$ −350.000 −0.567261 −0.283630 0.958934i $$-0.591539\pi$$
−0.283630 + 0.958934i $$0.591539\pi$$
$$618$$ 0 0
$$619$$ −560.000 −0.904685 −0.452342 0.891844i $$-0.649412\pi$$
−0.452342 + 0.891844i $$0.649412\pi$$
$$620$$ 0 0
$$621$$ 630.971i 1.01606i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −319.000 −0.510400
$$626$$ 0 0
$$627$$ 650.000 + 216.333i 1.03668 + 0.345029i
$$628$$ 0 0
$$629$$ 324.500i 0.515898i
$$630$$ 0 0
$$631$$ −1050.00 −1.66403 −0.832013 0.554757i $$-0.812811\pi$$
−0.832013 + 0.554757i $$0.812811\pi$$
$$632$$ 0 0
$$633$$ −845.000 −1.33491
$$634$$ 0 0
$$635$$ 519.199i 0.817637i
$$636$$ 0 0
$$637$$ 86.5332i 0.135845i
$$638$$ 0 0
$$639$$ 432.666i 0.677099i
$$640$$ 0 0
$$641$$ 1225.89i 1.91246i 0.292615 + 0.956230i $$0.405475\pi$$
−0.292615 + 0.956230i $$0.594525\pi$$
$$642$$ 0 0
$$643$$ 1030.00 1.60187 0.800933 0.598754i $$-0.204337\pi$$
0.800933 + 0.598754i $$0.204337\pi$$
$$644$$ 0 0
$$645$$ 288.444i 0.447200i
$$646$$ 0 0
$$647$$ 555.000 0.857805 0.428903 0.903351i $$-0.358900\pi$$
0.428903 + 0.903351i $$0.358900\pi$$
$$648$$ 0 0
$$649$$ 180.278i 0.277777i
$$650$$ 0 0
$$651$$ −650.000 −0.998464
$$652$$ 0 0
$$653$$ −50.0000 −0.0765697 −0.0382848 0.999267i $$-0.512189\pi$$
−0.0382848 + 0.999267i $$0.512189\pi$$
$$654$$ 0 0
$$655$$ 448.000 0.683969
$$656$$ 0 0
$$657$$ −420.000 −0.639269
$$658$$ 0 0
$$659$$ 198.305i 0.300919i −0.988616 0.150459i $$-0.951925\pi$$
0.988616 0.150459i $$-0.0480752\pi$$
$$660$$ 0 0
$$661$$ 198.305i 0.300008i −0.988685 0.150004i $$-0.952071\pi$$
0.988685 0.150004i $$-0.0479287\pi$$
$$662$$ 0 0
$$663$$ 195.000 0.294118
$$664$$ 0 0
$$665$$ 120.000 360.555i 0.180451 0.542188i
$$666$$ 0 0
$$667$$ 630.971i 0.945984i
$$668$$ 0 0
$$669$$ 728.000 1.08819
$$670$$ 0 0
$$671$$ 400.000 0.596125
$$672$$ 0 0
$$673$$ 598.522i 0.889334i 0.895696 + 0.444667i $$0.146678\pi$$
−0.895696 + 0.444667i $$0.853322\pi$$
$$674$$ 0 0
$$675$$ 162.250i 0.240370i
$$676$$ 0 0
$$677$$ 68.5055i 0.101190i −0.998719 0.0505949i $$-0.983888\pi$$
0.998719 0.0505949i $$-0.0161117\pi$$
$$678$$ 0 0
$$679$$ 612.944i 0.902715i
$$680$$ 0 0
$$681$$ 923.000 1.35536
$$682$$ 0 0
$$683$$ 237.966i 0.348413i 0.984709 + 0.174207i $$0.0557361\pi$$
−0.984709 + 0.174207i $$0.944264\pi$$
$$684$$ 0 0
$$685$$ −500.000 −0.729927
$$686$$ 0 0
$$687$$ 576.888i 0.839721i
$$688$$ 0 0
$$689$$ 273.000 0.396226
$$690$$ 0 0
$$691$$ 820.000 1.18669 0.593343 0.804950i $$-0.297808\pi$$
0.593343 + 0.804950i $$0.297808\pi$$
$$692$$ 0 0
$$693$$ 200.000 0.288600
$$694$$ 0 0
$$695$$ 200.000 0.287770
$$696$$ 0 0
$$697$$ 540.833i 0.775944i
$$698$$ 0 0
$$699$$ 973.499i 1.39270i
$$700$$ 0 0
$$701$$ 540.000 0.770328 0.385164 0.922848i $$-0.374145\pi$$
0.385164 + 0.922848i $$0.374145\pi$$
$$702$$ 0 0
$$703$$ 390.000 + 129.800i 0.554765 + 0.184637i
$$704$$ 0 0
$$705$$ 144.222i 0.204570i
$$706$$ 0 0
$$707$$ −250.000 −0.353607
$$708$$ 0 0
$$709$$ −268.000 −0.377997 −0.188999 0.981977i $$-0.560524\pi$$
−0.188999 + 0.981977i $$0.560524\pi$$
$$710$$ 0 0
$$711$$ 144.222i 0.202844i
$$712$$ 0 0
$$713$$ 1261.94i 1.76991i
$$714$$ 0 0
$$715$$ 144.222i 0.201709i
$$716$$ 0 0
$$717$$ 710.294i 0.990647i
$$718$$ 0 0
$$719$$ 105.000 0.146036 0.0730181 0.997331i $$-0.476737\pi$$
0.0730181 + 0.997331i $$0.476737\pi$$
$$720$$ 0 0
$$721$$ 288.444i 0.400061i
$$722$$ 0 0
$$723$$ −1430.00 −1.97787
$$724$$ 0 0
$$725$$ 162.250i 0.223793i
$$726$$ 0 0
$$727$$ 695.000 0.955983 0.477992 0.878364i $$-0.341365\pi$$
0.477992 + 0.878364i $$0.341365\pi$$
$$728$$ 0 0
$$729$$ −181.000 −0.248285
$$730$$ 0 0
$$731$$ 300.000 0.410397
$$732$$ 0 0
$$733$$ −160.000 −0.218281 −0.109141 0.994026i $$-0.534810\pi$$
−0.109141 + 0.994026i $$0.534810\pi$$
$$734$$ 0 0
$$735$$ 346.133i 0.470929i
$$736$$ 0 0
$$737$$ 396.611i 0.538142i
$$738$$ 0 0
$$739$$ −1028.00 −1.39107 −0.695535 0.718493i $$-0.744832\pi$$
−0.695535 + 0.718493i $$0.744832\pi$$
$$740$$ 0 0
$$741$$ 78.0000 234.361i 0.105263 0.316276i
$$742$$ 0 0
$$743$$ 526.410i 0.708493i −0.935152 0.354247i $$-0.884737\pi$$
0.935152 0.354247i $$-0.115263\pi$$
$$744$$ 0 0
$$745$$ 280.000 0.375839
$$746$$ 0 0
$$747$$ −160.000 −0.214190
$$748$$ 0 0
$$749$$ 378.583i 0.505451i
$$750$$ 0 0
$$751$$ 36.0555i 0.0480100i 0.999712 + 0.0240050i $$0.00764176\pi$$
−0.999712 + 0.0240050i $$0.992358\pi$$
$$752$$ 0 0
$$753$$ 1449.43i 1.92488i
$$754$$ 0 0
$$755$$ 144.222i 0.191023i
$$756$$ 0 0
$$757$$ −60.0000 −0.0792602 −0.0396301 0.999214i $$-0.512618\pi$$
−0.0396301 + 0.999214i $$0.512618\pi$$
$$758$$ 0 0
$$759$$ 1261.94i 1.66264i
$$760$$ 0 0
$$761$$ −655.000 −0.860710 −0.430355 0.902660i $$-0.641612\pi$$
−0.430355 + 0.902660i $$0.641612\pi$$
$$762$$ 0 0
$$763$$ 991.527i 1.29951i
$$764$$ 0 0
$$765$$ 240.000 0.313725
$$766$$ 0 0
$$767$$ −65.0000 −0.0847458
$$768$$ 0 0
$$769$$ −185.000 −0.240572 −0.120286 0.992739i $$-0.538381\pi$$
−0.120286 + 0.992739i $$0.538381\pi$$
$$770$$ 0 0
$$771$$ 1508.00 1.95590
$$772$$ 0 0
$$773$$ 320.894i 0.415128i 0.978221 + 0.207564i $$0.0665536\pi$$
−0.978221 + 0.207564i $$0.933446\pi$$
$$774$$ 0 0
$$775$$ 324.500i 0.418709i
$$776$$ 0 0
$$777$$ 390.000 0.501931
$$778$$ 0 0
$$779$$ 650.000 + 216.333i 0.834403 + 0.277706i
$$780$$ 0 0
$$781$$ 1081.67i 1.38497i
$$782$$ 0 0
$$783$$ 325.000 0.415070
$$784$$ 0 0
$$785$$ 40.0000 0.0509554
$$786$$ 0 0
$$787$$ 68.5055i 0.0870463i −0.999052 0.0435232i $$-0.986142\pi$$
0.999052 0.0435232i $$-0.0138582\pi$$
$$788$$ 0 0
$$789$$ 1117.72i 1.41663i
$$790$$ 0 0
$$791$$ 612.944i 0.774897i
$$792$$ 0 0
$$793$$ 144.222i 0.181869i
$$794$$ 0 0
$$795$$ 1092.00 1.37358
$$796$$ 0 0
$$797$$ 1258.34i 1.57884i −0.613852 0.789421i $$-0.710381\pi$$
0.613852 0.789421i $$-0.289619\pi$$
$$798$$ 0 0
$$799$$ 150.000 0.187735
$$800$$ 0 0
$$801$$ 0