# Properties

 Label 1216.2.h.d.1215.7 Level $1216$ Weight $2$ Character 1216.1215 Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.14453810176.1 Defining polynomial: $$x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1215.7 Root $$0.331077 + 1.37491i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1215 Dual form 1216.2.h.d.1215.8

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.35829 q^{3} +2.56155 q^{5} -4.15286i q^{7} +2.56155 q^{9} +O(q^{10})$$ $$q+2.35829 q^{3} +2.56155 q^{5} -4.15286i q^{7} +2.56155 q^{9} -2.33205i q^{11} +4.29400i q^{13} +6.04090 q^{15} -1.00000 q^{17} +(3.68260 + 2.33205i) q^{19} -9.79366i q^{21} +1.82081i q^{23} +1.56155 q^{25} -1.03399 q^{27} +1.20565i q^{29} +1.32431 q^{31} -5.49966i q^{33} -10.6378i q^{35} -5.49966i q^{37} +10.1265i q^{39} -5.49966i q^{41} +1.30957i q^{43} +6.56155 q^{45} -6.99614i q^{47} -10.2462 q^{49} -2.35829 q^{51} +9.79366i q^{53} -5.97366i q^{55} +(8.68466 + 5.49966i) q^{57} +6.33122 q^{59} -11.6847 q^{61} -10.6378i q^{63} +10.9993i q^{65} -0.290319 q^{67} +4.29400i q^{69} -2.06798 q^{71} -0.123106 q^{73} +3.68260 q^{75} -9.68466 q^{77} +13.4061 q^{79} -10.1231 q^{81} +11.9473i q^{83} -2.56155 q^{85} +2.84329i q^{87} +14.0877i q^{89} +17.8324 q^{91} +3.12311 q^{93} +(9.43318 + 5.97366i) q^{95} +2.41131i q^{97} -5.97366i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{5} + 4q^{9} + O(q^{10})$$ $$8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100})$$

## 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
$$3$$ 2.35829 1.36156 0.680781 0.732487i $$-0.261641\pi$$
0.680781 + 0.732487i $$0.261641\pi$$
$$4$$ 0 0
$$5$$ 2.56155 1.14556 0.572781 0.819709i $$-0.305865\pi$$
0.572781 + 0.819709i $$0.305865\pi$$
$$6$$ 0 0
$$7$$ 4.15286i 1.56963i −0.619729 0.784816i $$-0.712757\pi$$
0.619729 0.784816i $$-0.287243\pi$$
$$8$$ 0 0
$$9$$ 2.56155 0.853851
$$10$$ 0 0
$$11$$ 2.33205i 0.703139i −0.936162 0.351569i $$-0.885648\pi$$
0.936162 0.351569i $$-0.114352\pi$$
$$12$$ 0 0
$$13$$ 4.29400i 1.19094i 0.803377 + 0.595471i $$0.203034\pi$$
−0.803377 + 0.595471i $$0.796966\pi$$
$$14$$ 0 0
$$15$$ 6.04090 1.55975
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ 3.68260 + 2.33205i 0.844847 + 0.535008i
$$20$$ 0 0
$$21$$ 9.79366i 2.13715i
$$22$$ 0 0
$$23$$ 1.82081i 0.379665i 0.981817 + 0.189832i $$0.0607944\pi$$
−0.981817 + 0.189832i $$0.939206\pi$$
$$24$$ 0 0
$$25$$ 1.56155 0.312311
$$26$$ 0 0
$$27$$ −1.03399 −0.198991
$$28$$ 0 0
$$29$$ 1.20565i 0.223884i 0.993715 + 0.111942i $$0.0357071\pi$$
−0.993715 + 0.111942i $$0.964293\pi$$
$$30$$ 0 0
$$31$$ 1.32431 0.237853 0.118926 0.992903i $$-0.462055\pi$$
0.118926 + 0.992903i $$0.462055\pi$$
$$32$$ 0 0
$$33$$ 5.49966i 0.957367i
$$34$$ 0 0
$$35$$ 10.6378i 1.79811i
$$36$$ 0 0
$$37$$ 5.49966i 0.904138i −0.891983 0.452069i $$-0.850686\pi$$
0.891983 0.452069i $$-0.149314\pi$$
$$38$$ 0 0
$$39$$ 10.1265i 1.62154i
$$40$$ 0 0
$$41$$ 5.49966i 0.858902i −0.903090 0.429451i $$-0.858707\pi$$
0.903090 0.429451i $$-0.141293\pi$$
$$42$$ 0 0
$$43$$ 1.30957i 0.199707i 0.995002 + 0.0998536i $$0.0318375\pi$$
−0.995002 + 0.0998536i $$0.968163\pi$$
$$44$$ 0 0
$$45$$ 6.56155 0.978139
$$46$$ 0 0
$$47$$ 6.99614i 1.02049i −0.860028 0.510246i $$-0.829554\pi$$
0.860028 0.510246i $$-0.170446\pi$$
$$48$$ 0 0
$$49$$ −10.2462 −1.46374
$$50$$ 0 0
$$51$$ −2.35829 −0.330227
$$52$$ 0 0
$$53$$ 9.79366i 1.34526i 0.739978 + 0.672631i $$0.234836\pi$$
−0.739978 + 0.672631i $$0.765164\pi$$
$$54$$ 0 0
$$55$$ 5.97366i 0.805489i
$$56$$ 0 0
$$57$$ 8.68466 + 5.49966i 1.15031 + 0.728447i
$$58$$ 0 0
$$59$$ 6.33122 0.824254 0.412127 0.911126i $$-0.364786\pi$$
0.412127 + 0.911126i $$0.364786\pi$$
$$60$$ 0 0
$$61$$ −11.6847 −1.49607 −0.748034 0.663661i $$-0.769002\pi$$
−0.748034 + 0.663661i $$0.769002\pi$$
$$62$$ 0 0
$$63$$ 10.6378i 1.34023i
$$64$$ 0 0
$$65$$ 10.9993i 1.36430i
$$66$$ 0 0
$$67$$ −0.290319 −0.0354681 −0.0177341 0.999843i $$-0.505645\pi$$
−0.0177341 + 0.999843i $$0.505645\pi$$
$$68$$ 0 0
$$69$$ 4.29400i 0.516937i
$$70$$ 0 0
$$71$$ −2.06798 −0.245423 −0.122712 0.992442i $$-0.539159\pi$$
−0.122712 + 0.992442i $$0.539159\pi$$
$$72$$ 0 0
$$73$$ −0.123106 −0.0144084 −0.00720421 0.999974i $$-0.502293\pi$$
−0.00720421 + 0.999974i $$0.502293\pi$$
$$74$$ 0 0
$$75$$ 3.68260 0.425230
$$76$$ 0 0
$$77$$ −9.68466 −1.10367
$$78$$ 0 0
$$79$$ 13.4061 1.50830 0.754152 0.656700i $$-0.228048\pi$$
0.754152 + 0.656700i $$0.228048\pi$$
$$80$$ 0 0
$$81$$ −10.1231 −1.12479
$$82$$ 0 0
$$83$$ 11.9473i 1.31139i 0.755026 + 0.655695i $$0.227624\pi$$
−0.755026 + 0.655695i $$0.772376\pi$$
$$84$$ 0 0
$$85$$ −2.56155 −0.277839
$$86$$ 0 0
$$87$$ 2.84329i 0.304832i
$$88$$ 0 0
$$89$$ 14.0877i 1.49329i 0.665223 + 0.746644i $$0.268336\pi$$
−0.665223 + 0.746644i $$0.731664\pi$$
$$90$$ 0 0
$$91$$ 17.8324 1.86934
$$92$$ 0 0
$$93$$ 3.12311 0.323851
$$94$$ 0 0
$$95$$ 9.43318 + 5.97366i 0.967824 + 0.612885i
$$96$$ 0 0
$$97$$ 2.41131i 0.244831i 0.992479 + 0.122416i $$0.0390641\pi$$
−0.992479 + 0.122416i $$0.960936\pi$$
$$98$$ 0 0
$$99$$ 5.97366i 0.600376i
$$100$$ 0 0
$$101$$ 8.24621 0.820529 0.410264 0.911967i $$-0.365436\pi$$
0.410264 + 0.911967i $$0.365436\pi$$
$$102$$ 0 0
$$103$$ −12.0818 −1.19045 −0.595227 0.803557i $$-0.702938\pi$$
−0.595227 + 0.803557i $$0.702938\pi$$
$$104$$ 0 0
$$105$$ 25.0870i 2.44824i
$$106$$ 0 0
$$107$$ −5.75058 −0.555929 −0.277965 0.960591i $$-0.589660\pi$$
−0.277965 + 0.960591i $$0.589660\pi$$
$$108$$ 0 0
$$109$$ 12.2050i 1.16902i 0.811385 + 0.584512i $$0.198714\pi$$
−0.811385 + 0.584512i $$0.801286\pi$$
$$110$$ 0 0
$$111$$ 12.9698i 1.23104i
$$112$$ 0 0
$$113$$ 5.49966i 0.517364i 0.965963 + 0.258682i $$0.0832882\pi$$
−0.965963 + 0.258682i $$0.916712\pi$$
$$114$$ 0 0
$$115$$ 4.66410i 0.434929i
$$116$$ 0 0
$$117$$ 10.9993i 1.01689i
$$118$$ 0 0
$$119$$ 4.15286i 0.380692i
$$120$$ 0 0
$$121$$ 5.56155 0.505596
$$122$$ 0 0
$$123$$ 12.9698i 1.16945i
$$124$$ 0 0
$$125$$ −8.80776 −0.787790
$$126$$ 0 0
$$127$$ −6.04090 −0.536043 −0.268021 0.963413i $$-0.586370\pi$$
−0.268021 + 0.963413i $$0.586370\pi$$
$$128$$ 0 0
$$129$$ 3.08835i 0.271914i
$$130$$ 0 0
$$131$$ 5.97366i 0.521921i −0.965349 0.260961i $$-0.915961\pi$$
0.965349 0.260961i $$-0.0840393\pi$$
$$132$$ 0 0
$$133$$ 9.68466 15.2933i 0.839766 1.32610i
$$134$$ 0 0
$$135$$ −2.64861 −0.227956
$$136$$ 0 0
$$137$$ −12.1231 −1.03575 −0.517873 0.855457i $$-0.673276\pi$$
−0.517873 + 0.855457i $$0.673276\pi$$
$$138$$ 0 0
$$139$$ 18.9435i 1.60676i −0.595464 0.803382i $$-0.703032\pi$$
0.595464 0.803382i $$-0.296968\pi$$
$$140$$ 0 0
$$141$$ 16.4990i 1.38946i
$$142$$ 0 0
$$143$$ 10.0138 0.837397
$$144$$ 0 0
$$145$$ 3.08835i 0.256473i
$$146$$ 0 0
$$147$$ −24.1636 −1.99298
$$148$$ 0 0
$$149$$ −2.80776 −0.230021 −0.115010 0.993364i $$-0.536690\pi$$
−0.115010 + 0.993364i $$0.536690\pi$$
$$150$$ 0 0
$$151$$ 8.85254 0.720409 0.360205 0.932873i $$-0.382707\pi$$
0.360205 + 0.932873i $$0.382707\pi$$
$$152$$ 0 0
$$153$$ −2.56155 −0.207089
$$154$$ 0 0
$$155$$ 3.39228 0.272475
$$156$$ 0 0
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ 0 0
$$159$$ 23.0963i 1.83166i
$$160$$ 0 0
$$161$$ 7.56155 0.595934
$$162$$ 0 0
$$163$$ 17.6339i 1.38119i 0.723240 + 0.690597i $$0.242652\pi$$
−0.723240 + 0.690597i $$0.757348\pi$$
$$164$$ 0 0
$$165$$ 14.0877i 1.09672i
$$166$$ 0 0
$$167$$ 10.7575 0.832439 0.416220 0.909264i $$-0.363355\pi$$
0.416220 + 0.909264i $$0.363355\pi$$
$$168$$ 0 0
$$169$$ −5.43845 −0.418342
$$170$$ 0 0
$$171$$ 9.43318 + 5.97366i 0.721373 + 0.456817i
$$172$$ 0 0
$$173$$ 2.41131i 0.183328i −0.995790 0.0916642i $$-0.970781\pi$$
0.995790 0.0916642i $$-0.0292186\pi$$
$$174$$ 0 0
$$175$$ 6.48490i 0.490213i
$$176$$ 0 0
$$177$$ 14.9309 1.12227
$$178$$ 0 0
$$179$$ −8.10887 −0.606085 −0.303043 0.952977i $$-0.598002\pi$$
−0.303043 + 0.952977i $$0.598002\pi$$
$$180$$ 0 0
$$181$$ 19.5873i 1.45591i 0.685623 + 0.727957i $$0.259530\pi$$
−0.685623 + 0.727957i $$0.740470\pi$$
$$182$$ 0 0
$$183$$ −27.5559 −2.03699
$$184$$ 0 0
$$185$$ 14.0877i 1.03575i
$$186$$ 0 0
$$187$$ 2.33205i 0.170536i
$$188$$ 0 0
$$189$$ 4.29400i 0.312343i
$$190$$ 0 0
$$191$$ 7.79447i 0.563988i −0.959416 0.281994i $$-0.909004\pi$$
0.959416 0.281994i $$-0.0909959\pi$$
$$192$$ 0 0
$$193$$ 5.49966i 0.395874i −0.980215 0.197937i $$-0.936576\pi$$
0.980215 0.197937i $$-0.0634241\pi$$
$$194$$ 0 0
$$195$$ 25.9396i 1.85757i
$$196$$ 0 0
$$197$$ 22.4924 1.60252 0.801259 0.598317i $$-0.204164\pi$$
0.801259 + 0.598317i $$0.204164\pi$$
$$198$$ 0 0
$$199$$ 5.17534i 0.366870i 0.983032 + 0.183435i $$0.0587217\pi$$
−0.983032 + 0.183435i $$0.941278\pi$$
$$200$$ 0 0
$$201$$ −0.684658 −0.0482921
$$202$$ 0 0
$$203$$ 5.00691 0.351416
$$204$$ 0 0
$$205$$ 14.0877i 0.983925i
$$206$$ 0 0
$$207$$ 4.66410i 0.324177i
$$208$$ 0 0
$$209$$ 5.43845 8.58800i 0.376185 0.594045i
$$210$$ 0 0
$$211$$ −25.1976 −1.73467 −0.867336 0.497723i $$-0.834170\pi$$
−0.867336 + 0.497723i $$0.834170\pi$$
$$212$$ 0 0
$$213$$ −4.87689 −0.334159
$$214$$ 0 0
$$215$$ 3.35453i 0.228777i
$$216$$ 0 0
$$217$$ 5.49966i 0.373341i
$$218$$ 0 0
$$219$$ −0.290319 −0.0196180
$$220$$ 0 0
$$221$$ 4.29400i 0.288846i
$$222$$ 0 0
$$223$$ −22.2586 −1.49055 −0.745274 0.666758i $$-0.767681\pi$$
−0.745274 + 0.666758i $$0.767681\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ 24.6169 1.63388 0.816942 0.576720i $$-0.195668\pi$$
0.816942 + 0.576720i $$0.195668\pi$$
$$228$$ 0 0
$$229$$ 6.56155 0.433600 0.216800 0.976216i $$-0.430438\pi$$
0.216800 + 0.976216i $$0.430438\pi$$
$$230$$ 0 0
$$231$$ −22.8393 −1.50271
$$232$$ 0 0
$$233$$ 10.8078 0.708040 0.354020 0.935238i $$-0.384814\pi$$
0.354020 + 0.935238i $$0.384814\pi$$
$$234$$ 0 0
$$235$$ 17.9210i 1.16904i
$$236$$ 0 0
$$237$$ 31.6155 2.05365
$$238$$ 0 0
$$239$$ 9.83943i 0.636460i −0.948014 0.318230i $$-0.896912\pi$$
0.948014 0.318230i $$-0.103088\pi$$
$$240$$ 0 0
$$241$$ 22.6757i 1.46067i −0.683090 0.730334i $$-0.739365\pi$$
0.683090 0.730334i $$-0.260635\pi$$
$$242$$ 0 0
$$243$$ −20.7713 −1.33248
$$244$$ 0 0
$$245$$ −26.2462 −1.67681
$$246$$ 0 0
$$247$$ −10.0138 + 15.8131i −0.637164 + 1.00616i
$$248$$ 0 0
$$249$$ 28.1753i 1.78554i
$$250$$ 0 0
$$251$$ 21.5626i 1.36102i 0.732739 + 0.680510i $$0.238242\pi$$
−0.732739 + 0.680510i $$0.761758\pi$$
$$252$$ 0 0
$$253$$ 4.24621 0.266957
$$254$$ 0 0
$$255$$ −6.04090 −0.378296
$$256$$ 0 0
$$257$$ 17.1760i 1.07141i 0.844405 + 0.535705i $$0.179954\pi$$
−0.844405 + 0.535705i $$0.820046\pi$$
$$258$$ 0 0
$$259$$ −22.8393 −1.41916
$$260$$ 0 0
$$261$$ 3.08835i 0.191164i
$$262$$ 0 0
$$263$$ 14.2794i 0.880504i −0.897874 0.440252i $$-0.854889\pi$$
0.897874 0.440252i $$-0.145111\pi$$
$$264$$ 0 0
$$265$$ 25.0870i 1.54108i
$$266$$ 0 0
$$267$$ 33.2228i 2.03321i
$$268$$ 0 0
$$269$$ 5.49966i 0.335320i 0.985845 + 0.167660i $$0.0536211\pi$$
−0.985845 + 0.167660i $$0.946379\pi$$
$$270$$ 0 0
$$271$$ 22.0738i 1.34089i 0.741959 + 0.670445i $$0.233897\pi$$
−0.741959 + 0.670445i $$0.766103\pi$$
$$272$$ 0 0
$$273$$ 42.0540 2.54522
$$274$$ 0 0
$$275$$ 3.64162i 0.219598i
$$276$$ 0 0
$$277$$ −13.0540 −0.784337 −0.392169 0.919893i $$-0.628275\pi$$
−0.392169 + 0.919893i $$0.628275\pi$$
$$278$$ 0 0
$$279$$ 3.39228 0.203091
$$280$$ 0 0
$$281$$ 28.1753i 1.68080i −0.541968 0.840399i $$-0.682321\pi$$
0.541968 0.840399i $$-0.317679\pi$$
$$282$$ 0 0
$$283$$ 5.97366i 0.355097i −0.984112 0.177549i $$-0.943183\pi$$
0.984112 0.177549i $$-0.0568167\pi$$
$$284$$ 0 0
$$285$$ 22.2462 + 14.0877i 1.31775 + 0.834481i
$$286$$ 0 0
$$287$$ −22.8393 −1.34816
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 5.68658i 0.333353i
$$292$$ 0 0
$$293$$ 4.29400i 0.250858i −0.992103 0.125429i $$-0.959969\pi$$
0.992103 0.125429i $$-0.0400308\pi$$
$$294$$ 0 0
$$295$$ 16.2177 0.944233
$$296$$ 0 0
$$297$$ 2.41131i 0.139918i
$$298$$ 0 0
$$299$$ −7.81855 −0.452159
$$300$$ 0 0
$$301$$ 5.43845 0.313467
$$302$$ 0 0
$$303$$ 19.4470 1.11720
$$304$$ 0 0
$$305$$ −29.9309 −1.71384
$$306$$ 0 0
$$307$$ −13.4061 −0.765126 −0.382563 0.923929i $$-0.624959\pi$$
−0.382563 + 0.923929i $$0.624959\pi$$
$$308$$ 0 0
$$309$$ −28.4924 −1.62088
$$310$$ 0 0
$$311$$ 4.15286i 0.235487i −0.993044 0.117743i $$-0.962434\pi$$
0.993044 0.117743i $$-0.0375660\pi$$
$$312$$ 0 0
$$313$$ −0.438447 −0.0247825 −0.0123913 0.999923i $$-0.503944\pi$$
−0.0123913 + 0.999923i $$0.503944\pi$$
$$314$$ 0 0
$$315$$ 27.2492i 1.53532i
$$316$$ 0 0
$$317$$ 4.29400i 0.241175i 0.992703 + 0.120588i $$0.0384779\pi$$
−0.992703 + 0.120588i $$0.961522\pi$$
$$318$$ 0 0
$$319$$ 2.81164 0.157422
$$320$$ 0 0
$$321$$ −13.5616 −0.756932
$$322$$ 0 0
$$323$$ −3.68260 2.33205i −0.204905 0.129759i
$$324$$ 0 0
$$325$$ 6.70531i 0.371944i
$$326$$ 0 0
$$327$$ 28.7829i 1.59170i
$$328$$ 0 0
$$329$$ −29.0540 −1.60180
$$330$$ 0 0
$$331$$ −23.8733 −1.31219 −0.656097 0.754677i $$-0.727794\pi$$
−0.656097 + 0.754677i $$0.727794\pi$$
$$332$$ 0 0
$$333$$ 14.0877i 0.771999i
$$334$$ 0 0
$$335$$ −0.743668 −0.0406309
$$336$$ 0 0
$$337$$ 6.17669i 0.336466i 0.985747 + 0.168233i $$0.0538061\pi$$
−0.985747 + 0.168233i $$0.946194\pi$$
$$338$$ 0 0
$$339$$ 12.9698i 0.704423i
$$340$$ 0 0
$$341$$ 3.08835i 0.167243i
$$342$$ 0 0
$$343$$ 13.4810i 0.727908i
$$344$$ 0 0
$$345$$ 10.9993i 0.592183i
$$346$$ 0 0
$$347$$ 29.8683i 1.60342i −0.597716 0.801708i $$-0.703925\pi$$
0.597716 0.801708i $$-0.296075\pi$$
$$348$$ 0 0
$$349$$ −9.05398 −0.484648 −0.242324 0.970195i $$-0.577910\pi$$
−0.242324 + 0.970195i $$0.577910\pi$$
$$350$$ 0 0
$$351$$ 4.43994i 0.236987i
$$352$$ 0 0
$$353$$ 18.6847 0.994484 0.497242 0.867612i $$-0.334346\pi$$
0.497242 + 0.867612i $$0.334346\pi$$
$$354$$ 0 0
$$355$$ −5.29723 −0.281148
$$356$$ 0 0
$$357$$ 9.79366i 0.518335i
$$358$$ 0 0
$$359$$ 6.19782i 0.327108i −0.986534 0.163554i $$-0.947704\pi$$
0.986534 0.163554i $$-0.0522958\pi$$
$$360$$ 0 0
$$361$$ 8.12311 + 17.1760i 0.427532 + 0.904000i
$$362$$ 0 0
$$363$$ 13.1158 0.688400
$$364$$ 0 0
$$365$$ −0.315342 −0.0165057
$$366$$ 0 0
$$367$$ 1.02248i 0.0533730i −0.999644 0.0266865i $$-0.991504\pi$$
0.999644 0.0266865i $$-0.00849559\pi$$
$$368$$ 0 0
$$369$$ 14.0877i 0.733374i
$$370$$ 0 0
$$371$$ 40.6716 2.11157
$$372$$ 0 0
$$373$$ 6.70531i 0.347188i 0.984817 + 0.173594i $$0.0555380\pi$$
−0.984817 + 0.173594i $$0.944462\pi$$
$$374$$ 0 0
$$375$$ −20.7713 −1.07263
$$376$$ 0 0
$$377$$ −5.17708 −0.266633
$$378$$ 0 0
$$379$$ 15.7644 0.809762 0.404881 0.914369i $$-0.367313\pi$$
0.404881 + 0.914369i $$0.367313\pi$$
$$380$$ 0 0
$$381$$ −14.2462 −0.729856
$$382$$ 0 0
$$383$$ 22.2586 1.13736 0.568682 0.822558i $$-0.307454\pi$$
0.568682 + 0.822558i $$0.307454\pi$$
$$384$$ 0 0
$$385$$ −24.8078 −1.26432
$$386$$ 0 0
$$387$$ 3.35453i 0.170520i
$$388$$ 0 0
$$389$$ −14.8078 −0.750783 −0.375392 0.926866i $$-0.622492\pi$$
−0.375392 + 0.926866i $$0.622492\pi$$
$$390$$ 0 0
$$391$$ 1.82081i 0.0920822i
$$392$$ 0 0
$$393$$ 14.0877i 0.710628i
$$394$$ 0 0
$$395$$ 34.3404 1.72785
$$396$$ 0 0
$$397$$ 3.93087 0.197285 0.0986423 0.995123i $$-0.468550\pi$$
0.0986423 + 0.995123i $$0.468550\pi$$
$$398$$ 0 0
$$399$$ 22.8393 36.0661i 1.14339 1.80557i
$$400$$ 0 0
$$401$$ 7.91096i 0.395055i −0.980297 0.197527i $$-0.936709\pi$$
0.980297 0.197527i $$-0.0632911\pi$$
$$402$$ 0 0
$$403$$ 5.68658i 0.283269i
$$404$$ 0 0
$$405$$ −25.9309 −1.28852
$$406$$ 0 0
$$407$$ −12.8255 −0.635734
$$408$$ 0 0
$$409$$ 27.4983i 1.35970i 0.733350 + 0.679851i $$0.237956\pi$$
−0.733350 + 0.679851i $$0.762044\pi$$
$$410$$ 0 0
$$411$$ −28.5899 −1.41023
$$412$$ 0 0
$$413$$ 26.2926i 1.29378i
$$414$$ 0 0
$$415$$ 30.6037i 1.50228i
$$416$$ 0 0
$$417$$ 44.6743i 2.18771i
$$418$$ 0 0
$$419$$ 39.9319i 1.95080i −0.220440 0.975401i $$-0.570749\pi$$
0.220440 0.975401i $$-0.429251\pi$$
$$420$$ 0 0
$$421$$ 23.2043i 1.13091i −0.824780 0.565454i $$-0.808701\pi$$
0.824780 0.565454i $$-0.191299\pi$$
$$422$$ 0 0
$$423$$ 17.9210i 0.871348i
$$424$$ 0 0
$$425$$ −1.56155 −0.0757464
$$426$$ 0 0
$$427$$ 48.5247i 2.34827i
$$428$$ 0 0
$$429$$ 23.6155 1.14017
$$430$$ 0 0
$$431$$ 5.29723 0.255158 0.127579 0.991828i $$-0.459279\pi$$
0.127579 + 0.991828i $$0.459279\pi$$
$$432$$ 0 0
$$433$$ 18.9103i 0.908770i 0.890805 + 0.454385i $$0.150141\pi$$
−0.890805 + 0.454385i $$0.849859\pi$$
$$434$$ 0 0
$$435$$ 7.28323i 0.349204i
$$436$$ 0 0
$$437$$ −4.24621 + 6.70531i −0.203124 + 0.320758i
$$438$$ 0 0
$$439$$ −19.6100 −0.935935 −0.467968 0.883746i $$-0.655014\pi$$
−0.467968 + 0.883746i $$0.655014\pi$$
$$440$$ 0 0
$$441$$ −26.2462 −1.24982
$$442$$ 0 0
$$443$$ 12.2344i 0.581275i 0.956833 + 0.290637i $$0.0938673\pi$$
−0.956833 + 0.290637i $$0.906133\pi$$
$$444$$ 0 0
$$445$$ 36.0863i 1.71065i
$$446$$ 0 0
$$447$$ −6.62153 −0.313188
$$448$$ 0 0
$$449$$ 14.7647i 0.696789i −0.937348 0.348395i $$-0.886727\pi$$
0.937348 0.348395i $$-0.113273\pi$$
$$450$$ 0 0
$$451$$ −12.8255 −0.603927
$$452$$ 0 0
$$453$$ 20.8769 0.980882
$$454$$ 0 0
$$455$$ 45.6786 2.14144
$$456$$ 0 0
$$457$$ 3.87689 0.181353 0.0906767 0.995880i $$-0.471097\pi$$
0.0906767 + 0.995880i $$0.471097\pi$$
$$458$$ 0 0
$$459$$ 1.03399 0.0482624
$$460$$ 0 0
$$461$$ −31.6847 −1.47570 −0.737851 0.674964i $$-0.764159\pi$$
−0.737851 + 0.674964i $$0.764159\pi$$
$$462$$ 0 0
$$463$$ 16.3243i 0.758656i −0.925262 0.379328i $$-0.876155\pi$$
0.925262 0.379328i $$-0.123845\pi$$
$$464$$ 0 0
$$465$$ 8.00000 0.370991
$$466$$ 0 0
$$467$$ 2.33205i 0.107914i −0.998543 0.0539572i $$-0.982817\pi$$
0.998543 0.0539572i $$-0.0171834\pi$$
$$468$$ 0 0
$$469$$ 1.20565i 0.0556719i
$$470$$ 0 0
$$471$$ −14.1498 −0.651987
$$472$$ 0 0
$$473$$ 3.05398 0.140422
$$474$$ 0 0
$$475$$ 5.75058 + 3.64162i 0.263855 + 0.167089i
$$476$$ 0 0
$$477$$ 25.0870i 1.14865i
$$478$$ 0 0
$$479$$ 41.5286i 1.89749i −0.316045 0.948744i $$-0.602355\pi$$
0.316045 0.948744i $$-0.397645\pi$$
$$480$$ 0 0
$$481$$ 23.6155 1.07678
$$482$$ 0 0
$$483$$ 17.8324 0.811401
$$484$$ 0 0
$$485$$ 6.17669i 0.280469i
$$486$$ 0 0
$$487$$ −18.8664 −0.854916 −0.427458 0.904035i $$-0.640591\pi$$
−0.427458 + 0.904035i $$0.640591\pi$$
$$488$$ 0 0
$$489$$ 41.5859i 1.88058i
$$490$$ 0 0
$$491$$ 21.2755i 0.960151i −0.877227 0.480075i $$-0.840609\pi$$
0.877227 0.480075i $$-0.159391\pi$$
$$492$$ 0 0
$$493$$ 1.20565i 0.0542999i
$$494$$ 0 0
$$495$$ 15.3019i 0.687767i
$$496$$ 0 0
$$497$$ 8.58800i 0.385225i
$$498$$ 0 0
$$499$$ 1.30957i 0.0586243i 0.999570 + 0.0293122i $$0.00933169\pi$$
−0.999570 + 0.0293122i $$0.990668\pi$$
$$500$$ 0 0
$$501$$ 25.3693 1.13342
$$502$$ 0 0
$$503$$ 16.3873i 0.730672i 0.930876 + 0.365336i $$0.119046\pi$$
−0.930876 + 0.365336i $$0.880954\pi$$
$$504$$ 0 0
$$505$$ 21.1231 0.939966
$$506$$ 0 0
$$507$$ −12.8255 −0.569599
$$508$$ 0 0
$$509$$ 8.58800i 0.380657i −0.981721 0.190328i $$-0.939045\pi$$
0.981721 0.190328i $$-0.0609552\pi$$
$$510$$ 0 0
$$511$$ 0.511240i 0.0226159i
$$512$$ 0 0
$$513$$ −3.80776 2.41131i −0.168117 0.106462i
$$514$$ 0 0
$$515$$ −30.9481 −1.36374
$$516$$ 0 0
$$517$$ −16.3153 −0.717548
$$518$$ 0 0
$$519$$ 5.68658i 0.249613i
$$520$$ 0 0
$$521$$ 22.6757i 0.993439i 0.867911 + 0.496719i $$0.165462\pi$$
−0.867911 + 0.496719i $$0.834538\pi$$
$$522$$ 0 0
$$523$$ 2.19526 0.0959922 0.0479961 0.998848i $$-0.484716\pi$$
0.0479961 + 0.998848i $$0.484716\pi$$
$$524$$ 0 0
$$525$$ 15.2933i 0.667455i
$$526$$ 0 0
$$527$$ −1.32431 −0.0576877
$$528$$ 0 0
$$529$$ 19.6847 0.855855
$$530$$ 0 0
$$531$$ 16.2177 0.703790
$$532$$ 0 0
$$533$$ 23.6155 1.02290
$$534$$ 0 0
$$535$$ −14.7304 −0.636851
$$536$$ 0 0
$$537$$ −19.1231 −0.825223
$$538$$ 0 0
$$539$$ 23.8947i 1.02922i
$$540$$ 0 0
$$541$$ 32.8078 1.41052 0.705258 0.708951i $$-0.250831\pi$$
0.705258 + 0.708951i $$0.250831\pi$$
$$542$$ 0 0
$$543$$ 46.1927i 1.98232i
$$544$$ 0 0
$$545$$ 31.2637i 1.33919i
$$546$$ 0 0
$$547$$ 0.580639 0.0248263 0.0124132 0.999923i $$-0.496049\pi$$
0.0124132 + 0.999923i $$0.496049\pi$$
$$548$$ 0 0
$$549$$ −29.9309 −1.27742
$$550$$ 0 0
$$551$$ −2.81164 + 4.43994i −0.119780 + 0.189148i
$$552$$ 0 0
$$553$$ 55.6736i 2.36748i
$$554$$ 0 0
$$555$$ 33.2228i 1.41023i
$$556$$ 0 0
$$557$$ −9.93087 −0.420784 −0.210392 0.977617i $$-0.567474\pi$$
−0.210392 + 0.977617i $$0.567474\pi$$
$$558$$ 0 0
$$559$$ −5.62329 −0.237840
$$560$$ 0 0
$$561$$ 5.49966i 0.232196i
$$562$$ 0 0
$$563$$ 21.3519 0.899877 0.449938 0.893060i $$-0.351446\pi$$
0.449938 + 0.893060i $$0.351446\pi$$
$$564$$ 0 0
$$565$$ 14.0877i 0.592672i
$$566$$ 0 0
$$567$$ 42.0398i 1.76551i
$$568$$ 0 0
$$569$$ 41.5859i 1.74337i −0.490064 0.871687i $$-0.663027\pi$$
0.490064 0.871687i $$-0.336973\pi$$
$$570$$ 0 0
$$571$$ 15.5889i 0.652377i −0.945305 0.326188i $$-0.894236\pi$$
0.945305 0.326188i $$-0.105764\pi$$
$$572$$ 0 0
$$573$$ 18.3817i 0.767905i
$$574$$ 0 0
$$575$$ 2.84329i 0.118573i
$$576$$ 0 0
$$577$$ 27.0000 1.12402 0.562012 0.827129i $$-0.310027\pi$$
0.562012 + 0.827129i $$0.310027\pi$$
$$578$$ 0 0
$$579$$ 12.9698i 0.539007i
$$580$$ 0 0
$$581$$ 49.6155 2.05840
$$582$$ 0 0
$$583$$ 22.8393 0.945906
$$584$$ 0 0
$$585$$ 28.1753i 1.16491i
$$586$$ 0 0
$$587$$ 5.97366i 0.246559i −0.992372 0.123280i $$-0.960659\pi$$
0.992372 0.123280i $$-0.0393412\pi$$
$$588$$ 0 0
$$589$$ 4.87689 + 3.08835i 0.200949 + 0.127253i
$$590$$ 0 0
$$591$$ 53.0438 2.18193
$$592$$ 0 0
$$593$$ 12.7386 0.523113 0.261556 0.965188i $$-0.415764\pi$$
0.261556 + 0.965188i $$0.415764\pi$$
$$594$$ 0 0
$$595$$ 10.6378i 0.436106i
$$596$$ 0 0
$$597$$ 12.2050i 0.499516i
$$598$$ 0 0
$$599$$ 24.9073 1.01768 0.508841 0.860860i $$-0.330074\pi$$
0.508841 + 0.860860i $$0.330074\pi$$
$$600$$ 0 0
$$601$$ 35.4092i 1.44437i −0.691698 0.722187i $$-0.743137\pi$$
0.691698 0.722187i $$-0.256863\pi$$
$$602$$ 0 0
$$603$$ −0.743668 −0.0302845
$$604$$ 0 0
$$605$$ 14.2462 0.579191
$$606$$ 0 0
$$607$$ −31.5288 −1.27971 −0.639857 0.768494i $$-0.721006\pi$$
−0.639857 + 0.768494i $$0.721006\pi$$
$$608$$ 0 0
$$609$$ 11.8078 0.478475
$$610$$ 0 0
$$611$$ 30.0414 1.21535
$$612$$ 0 0
$$613$$ 29.3002 1.18342 0.591712 0.806150i $$-0.298452\pi$$
0.591712 + 0.806150i $$0.298452\pi$$
$$614$$ 0 0
$$615$$ 33.2228i 1.33967i
$$616$$ 0 0
$$617$$ 41.5464 1.67259 0.836297 0.548276i $$-0.184716\pi$$
0.836297 + 0.548276i $$0.184716\pi$$
$$618$$ 0 0
$$619$$ 6.70906i 0.269660i −0.990869 0.134830i $$-0.956951\pi$$
0.990869 0.134830i $$-0.0430488\pi$$
$$620$$ 0 0
$$621$$ 1.88269i 0.0755499i
$$622$$ 0 0
$$623$$ 58.5040 2.34391
$$624$$ 0 0
$$625$$ −30.3693 −1.21477
$$626$$ 0 0
$$627$$ 12.8255 20.2530i 0.512200 0.808828i
$$628$$ 0 0
$$629$$ 5.49966i 0.219286i
$$630$$ 0 0
$$631$$ 9.61528i 0.382778i 0.981514 + 0.191389i $$0.0612992\pi$$
−0.981514 + 0.191389i $$0.938701\pi$$
$$632$$ 0 0
$$633$$ −59.4233 −2.36186
$$634$$ 0 0
$$635$$ −15.4741 −0.614070
$$636$$ 0 0
$$637$$ 43.9972i 1.74323i
$$638$$ 0 0
$$639$$ −5.29723 −0.209555
$$640$$ 0 0
$$641$$ 26.8212i 1.05938i 0.848193 + 0.529688i $$0.177691\pi$$
−0.848193 + 0.529688i $$0.822309\pi$$
$$642$$ 0 0
$$643$$ 14.2794i 0.563124i 0.959543 + 0.281562i $$0.0908524\pi$$
−0.959543 + 0.281562i $$0.909148\pi$$
$$644$$ 0 0
$$645$$ 7.91096i 0.311494i
$$646$$ 0 0
$$647$$ 32.1374i 1.26345i −0.775191 0.631726i $$-0.782347\pi$$
0.775191 0.631726i $$-0.217653\pi$$
$$648$$ 0 0
$$649$$ 14.7647i 0.579565i
$$650$$ 0 0
$$651$$ 12.9698i 0.508327i
$$652$$ 0 0
$$653$$ 25.1922 0.985848 0.492924 0.870072i $$-0.335928\pi$$
0.492924 + 0.870072i $$0.335928\pi$$
$$654$$ 0 0
$$655$$ 15.3019i 0.597893i
$$656$$ 0 0
$$657$$ −0.315342 −0.0123026
$$658$$ 0 0
$$659$$ 41.9960 1.63593 0.817965 0.575268i $$-0.195102\pi$$
0.817965 + 0.575268i $$0.195102\pi$$
$$660$$ 0 0
$$661$$ 37.2919i 1.45049i −0.688492 0.725244i $$-0.741727\pi$$
0.688492 0.725244i $$-0.258273\pi$$
$$662$$ 0 0
$$663$$ 10.1265i 0.393281i
$$664$$ 0 0
$$665$$ 24.8078 39.1746i 0.962004 1.51913i
$$666$$ 0 0
$$667$$ −2.19526 −0.0850010
$$668$$ 0 0
$$669$$ −52.4924 −2.02947
$$670$$ 0 0
$$671$$ 27.2492i 1.05194i
$$672$$ 0 0
$$673$$ 24.4099i 0.940934i −0.882418 0.470467i $$-0.844086\pi$$
0.882418 0.470467i $$-0.155914\pi$$
$$674$$ 0 0
$$675$$ −1.61463 −0.0621470
$$676$$ 0 0
$$677$$ 40.3803i 1.55194i −0.630770 0.775970i $$-0.717261\pi$$
0.630770 0.775970i $$-0.282739\pi$$
$$678$$ 0 0
$$679$$ 10.0138 0.384295
$$680$$ 0 0
$$681$$ 58.0540 2.22463
$$682$$ 0 0
$$683$$ −31.5288 −1.20642 −0.603208 0.797584i $$-0.706111\pi$$
−0.603208 + 0.797584i $$0.706111\pi$$
$$684$$ 0 0
$$685$$ −31.0540 −1.18651
$$686$$ 0 0
$$687$$ 15.4741 0.590373
$$688$$ 0 0
$$689$$ −42.0540 −1.60213
$$690$$ 0 0
$$691$$ 8.01862i 0.305043i −0.988300 0.152521i $$-0.951261\pi$$
0.988300 0.152521i $$-0.0487393\pi$$
$$692$$ 0 0
$$693$$ −24.8078 −0.942369
$$694$$ 0 0
$$695$$ 48.5247i 1.84065i
$$696$$ 0 0
$$697$$ 5.49966i 0.208314i
$$698$$ 0 0
$$699$$ 25.4879 0.964041
$$700$$ 0 0
$$701$$ −4.63068 −0.174898 −0.0874492 0.996169i $$-0.527872\pi$$
−0.0874492 + 0.996169i $$0.527872\pi$$
$$702$$ 0 0
$$703$$ 12.8255 20.2530i 0.483721 0.763858i
$$704$$ 0 0
$$705$$ 42.2630i 1.59172i
$$706$$ 0 0
$$707$$ 34.2453i 1.28793i
$$708$$ 0 0
$$709$$ −12.6307 −0.474355 −0.237178 0.971466i $$-0.576222\pi$$
−0.237178 + 0.971466i $$0.576222\pi$$
$$710$$ 0 0
$$711$$ 34.3404 1.28787
$$712$$ 0 0
$$713$$ 2.41131i 0.0903042i
$$714$$ 0 0
$$715$$ 25.6509 0.959290
$$716$$ 0 0
$$717$$ 23.2043i 0.866580i
$$718$$ 0 0
$$719$$ 26.4509i 0.986450i −0.869902 0.493225i $$-0.835818\pi$$
0.869902 0.493225i $$-0.164182\pi$$
$$720$$ 0 0
$$721$$ 50.1739i 1.86858i
$$722$$ 0 0
$$723$$ 53.4759i 1.98879i
$$724$$ 0 0
$$725$$ 1.88269i 0.0699215i
$$726$$ 0 0
$$727$$ 44.6589i 1.65631i −0.560500 0.828154i $$-0.689391\pi$$
0.560500 0.828154i $$-0.310609\pi$$
$$728$$ 0 0
$$729$$ −18.6155 −0.689464
$$730$$ 0 0
$$731$$ 1.30957i 0.0484361i
$$732$$ 0 0
$$733$$ 5.12311 0.189226 0.0946131 0.995514i $$-0.469839\pi$$
0.0946131 + 0.995514i $$0.469839\pi$$
$$734$$ 0 0
$$735$$ −61.8963 −2.28308
$$736$$ 0 0
$$737$$ 0.677039i 0.0249390i
$$738$$ 0 0
$$739$$ 25.2042i 0.927152i 0.886057 + 0.463576i $$0.153434\pi$$
−0.886057 + 0.463576i $$0.846566\pi$$
$$740$$ 0 0
$$741$$ −23.6155 + 37.2919i −0.867538 + 1.36995i
$$742$$ 0 0
$$743$$ 11.9188 0.437257 0.218628 0.975808i $$-0.429842\pi$$
0.218628 + 0.975808i $$0.429842\pi$$
$$744$$ 0 0
$$745$$ −7.19224 −0.263503
$$746$$ 0 0
$$747$$ 30.6037i 1.11973i
$$748$$ 0 0
$$749$$ 23.8813i 0.872604i
$$750$$ 0 0
$$751$$ −35.6647 −1.30142 −0.650712 0.759324i $$-0.725530\pi$$
−0.650712 + 0.759324i $$0.725530\pi$$
$$752$$ 0 0
$$753$$ 50.8510i 1.85311i
$$754$$ 0 0
$$755$$ 22.6762 0.825273
$$756$$ 0 0
$$757$$ 44.8078 1.62857 0.814283 0.580468i $$-0.197130\pi$$
0.814283 + 0.580468i $$0.197130\pi$$
$$758$$ 0 0
$$759$$ 10.0138 0.363479
$$760$$ 0 0
$$761$$ 30.6155 1.10981 0.554906 0.831913i $$-0.312754\pi$$
0.554906 + 0.831913i $$0.312754\pi$$
$$762$$ 0 0
$$763$$ 50.6855 1.83494
$$764$$ 0 0
$$765$$ −6.56155 −0.237233
$$766$$ 0 0
$$767$$ 27.1862i 0.981638i
$$768$$ 0 0
$$769$$ −28.2311 −1.01804 −0.509019 0.860755i $$-0.669992\pi$$
−0.509019 + 0.860755i $$0.669992\pi$$
$$770$$ 0 0
$$771$$ 40.5061i 1.45879i
$$772$$ 0 0
$$773$$ 41.0573i 1.47673i 0.674402 + 0.738365i $$0.264402\pi$$
−0.674402 + 0.738365i $$0.735598\pi$$
$$774$$ 0 0
$$775$$ 2.06798 0.0742839
$$776$$ 0 0
$$777$$ −53.8617 −1.93228
$$778$$ 0 0
$$779$$ 12.8255 20.2530i 0.459520 0.725640i
$$780$$ 0 0
$$781$$ 4.82262i 0.172567i
$$782$$ 0 0
$$783$$ 1.24663i 0.0445510i
$$784$$ 0 0
$$785$$ −15.3693 −0.548554
$$786$$ 0 0
$$787$$ −26.3588 −0.939591 −0.469796 0.882775i $$-0.655672\pi$$
−0.469796 + 0.882775i $$0.655672\pi$$
$$788$$ 0 0
$$789$$ 33.6750i 1.19886i
$$790$$ 0 0
$$791$$ 22.8393 0.812071
$$792$$ 0 0
$$793$$ 50.1739i 1.78173i
$$794$$ 0 0
$$795$$ 59.1625i 2.09828i
$$796$$ 0 0
$$797$$ 23.2043i 0.821938i 0.911649 + 0.410969i $$0.134809\pi$$
−0.911649 + 0.410969i $$0.865191\pi$$
$$798$$ 0 0
$$799$$ 6.99614i 0.247506i
$$800$$ 0 0
$$801$$ 36.0863i 1.27505i
$$802$$ 0 0
$$803$$ 0.287088i 0.0101311i
$$804$$ 0 0
$$805$$ 19.3693 0.682679
$$806$$ 0