# Properties

 Label 588.3.d.c.97.4 Level $588$ Weight $3$ Character 588.97 Analytic conductor $16.022$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 588.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0218395444$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 97.4 Root $$1.60021 + 0.923880i$$ of defining polynomial Character $$\chi$$ $$=$$ 588.97 Dual form 588.3.d.c.97.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +5.37964i q^{5} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +5.37964i q^{5} -3.00000 q^{9} -8.59159 q^{11} -21.0158i q^{13} +9.31781 q^{15} -5.48622i q^{17} -7.24098i q^{19} +28.0556 q^{23} -3.94054 q^{25} +5.19615i q^{27} +40.3447 q^{29} -40.5101i q^{31} +14.8811i q^{33} +66.6370 q^{37} -36.4005 q^{39} +33.6357i q^{41} +0.932907 q^{43} -16.1389i q^{45} -85.6544i q^{47} -9.50241 q^{51} -44.5954 q^{53} -46.2197i q^{55} -12.5418 q^{57} -63.6950i q^{59} +32.0084i q^{61} +113.058 q^{65} -47.7341 q^{67} -48.5938i q^{69} +14.9676 q^{71} -140.298i q^{73} +6.82521i q^{75} -122.307 q^{79} +9.00000 q^{81} +33.1852i q^{83} +29.5139 q^{85} -69.8791i q^{87} +36.1246i q^{89} -70.1655 q^{93} +38.9539 q^{95} +16.2175i q^{97} +25.7748 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{9} + O(q^{10})$$ $$8 q - 24 q^{9} - 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} - 112 q^{43} - 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} - 64 q^{67} + 224 q^{71} - 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} - 272 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.73205i − 0.577350i
$$4$$ 0 0
$$5$$ 5.37964i 1.07593i 0.842968 + 0.537964i $$0.180806\pi$$
−0.842968 + 0.537964i $$0.819194\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −3.00000 −0.333333
$$10$$ 0 0
$$11$$ −8.59159 −0.781054 −0.390527 0.920592i $$-0.627707\pi$$
−0.390527 + 0.920592i $$0.627707\pi$$
$$12$$ 0 0
$$13$$ − 21.0158i − 1.61660i −0.588769 0.808301i $$-0.700387\pi$$
0.588769 0.808301i $$-0.299613\pi$$
$$14$$ 0 0
$$15$$ 9.31781 0.621187
$$16$$ 0 0
$$17$$ − 5.48622i − 0.322719i −0.986896 0.161359i $$-0.948412\pi$$
0.986896 0.161359i $$-0.0515878\pi$$
$$18$$ 0 0
$$19$$ − 7.24098i − 0.381104i −0.981677 0.190552i $$-0.938972\pi$$
0.981677 0.190552i $$-0.0610278\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 28.0556 1.21981 0.609905 0.792474i $$-0.291208\pi$$
0.609905 + 0.792474i $$0.291208\pi$$
$$24$$ 0 0
$$25$$ −3.94054 −0.157621
$$26$$ 0 0
$$27$$ 5.19615i 0.192450i
$$28$$ 0 0
$$29$$ 40.3447 1.39120 0.695599 0.718431i $$-0.255139\pi$$
0.695599 + 0.718431i $$0.255139\pi$$
$$30$$ 0 0
$$31$$ − 40.5101i − 1.30678i −0.757023 0.653388i $$-0.773347\pi$$
0.757023 0.653388i $$-0.226653\pi$$
$$32$$ 0 0
$$33$$ 14.8811i 0.450941i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 66.6370 1.80100 0.900500 0.434856i $$-0.143201\pi$$
0.900500 + 0.434856i $$0.143201\pi$$
$$38$$ 0 0
$$39$$ −36.4005 −0.933346
$$40$$ 0 0
$$41$$ 33.6357i 0.820383i 0.911999 + 0.410191i $$0.134538\pi$$
−0.911999 + 0.410191i $$0.865462\pi$$
$$42$$ 0 0
$$43$$ 0.932907 0.0216955 0.0108478 0.999941i $$-0.496547\pi$$
0.0108478 + 0.999941i $$0.496547\pi$$
$$44$$ 0 0
$$45$$ − 16.1389i − 0.358643i
$$46$$ 0 0
$$47$$ − 85.6544i − 1.82243i −0.411927 0.911217i $$-0.635144\pi$$
0.411927 0.911217i $$-0.364856\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −9.50241 −0.186322
$$52$$ 0 0
$$53$$ −44.5954 −0.841422 −0.420711 0.907195i $$-0.638219\pi$$
−0.420711 + 0.907195i $$0.638219\pi$$
$$54$$ 0 0
$$55$$ − 46.2197i − 0.840358i
$$56$$ 0 0
$$57$$ −12.5418 −0.220031
$$58$$ 0 0
$$59$$ − 63.6950i − 1.07958i −0.841801 0.539788i $$-0.818504\pi$$
0.841801 0.539788i $$-0.181496\pi$$
$$60$$ 0 0
$$61$$ 32.0084i 0.524727i 0.964969 + 0.262364i $$0.0845020\pi$$
−0.964969 + 0.262364i $$0.915498\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 113.058 1.73935
$$66$$ 0 0
$$67$$ −47.7341 −0.712450 −0.356225 0.934400i $$-0.615936\pi$$
−0.356225 + 0.934400i $$0.615936\pi$$
$$68$$ 0 0
$$69$$ − 48.5938i − 0.704258i
$$70$$ 0 0
$$71$$ 14.9676 0.210811 0.105405 0.994429i $$-0.466386\pi$$
0.105405 + 0.994429i $$0.466386\pi$$
$$72$$ 0 0
$$73$$ − 140.298i − 1.92188i −0.276750 0.960942i $$-0.589258\pi$$
0.276750 0.960942i $$-0.410742\pi$$
$$74$$ 0 0
$$75$$ 6.82521i 0.0910028i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −122.307 −1.54820 −0.774098 0.633066i $$-0.781796\pi$$
−0.774098 + 0.633066i $$0.781796\pi$$
$$80$$ 0 0
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ 33.1852i 0.399822i 0.979814 + 0.199911i $$0.0640652\pi$$
−0.979814 + 0.199911i $$0.935935\pi$$
$$84$$ 0 0
$$85$$ 29.5139 0.347222
$$86$$ 0 0
$$87$$ − 69.8791i − 0.803208i
$$88$$ 0 0
$$89$$ 36.1246i 0.405894i 0.979190 + 0.202947i $$0.0650519\pi$$
−0.979190 + 0.202947i $$0.934948\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −70.1655 −0.754468
$$94$$ 0 0
$$95$$ 38.9539 0.410041
$$96$$ 0 0
$$97$$ 16.2175i 0.167191i 0.996500 + 0.0835956i $$0.0266404\pi$$
−0.996500 + 0.0835956i $$0.973360\pi$$
$$98$$ 0 0
$$99$$ 25.7748 0.260351
$$100$$ 0 0
$$101$$ 119.464i 1.18281i 0.806373 + 0.591407i $$0.201427\pi$$
−0.806373 + 0.591407i $$0.798573\pi$$
$$102$$ 0 0
$$103$$ − 8.93188i − 0.0867173i −0.999060 0.0433586i $$-0.986194\pi$$
0.999060 0.0433586i $$-0.0138058\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 173.856 1.62482 0.812410 0.583086i $$-0.198155\pi$$
0.812410 + 0.583086i $$0.198155\pi$$
$$108$$ 0 0
$$109$$ −160.315 −1.47078 −0.735388 0.677646i $$-0.763000\pi$$
−0.735388 + 0.677646i $$0.763000\pi$$
$$110$$ 0 0
$$111$$ − 115.419i − 1.03981i
$$112$$ 0 0
$$113$$ −81.4420 −0.720725 −0.360363 0.932812i $$-0.617347\pi$$
−0.360363 + 0.932812i $$0.617347\pi$$
$$114$$ 0 0
$$115$$ 150.929i 1.31243i
$$116$$ 0 0
$$117$$ 63.0475i 0.538867i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −47.1846 −0.389955
$$122$$ 0 0
$$123$$ 58.2587 0.473648
$$124$$ 0 0
$$125$$ 113.292i 0.906339i
$$126$$ 0 0
$$127$$ 117.172 0.922613 0.461307 0.887241i $$-0.347381\pi$$
0.461307 + 0.887241i $$0.347381\pi$$
$$128$$ 0 0
$$129$$ − 1.61584i − 0.0125259i
$$130$$ 0 0
$$131$$ − 244.150i − 1.86374i −0.362795 0.931869i $$-0.618177\pi$$
0.362795 0.931869i $$-0.381823\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −27.9534 −0.207062
$$136$$ 0 0
$$137$$ 245.412 1.79133 0.895663 0.444733i $$-0.146701\pi$$
0.895663 + 0.444733i $$0.146701\pi$$
$$138$$ 0 0
$$139$$ − 17.1371i − 0.123288i −0.998098 0.0616441i $$-0.980366\pi$$
0.998098 0.0616441i $$-0.0196344\pi$$
$$140$$ 0 0
$$141$$ −148.358 −1.05218
$$142$$ 0 0
$$143$$ 180.559i 1.26265i
$$144$$ 0 0
$$145$$ 217.040i 1.49683i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 134.242 0.900952 0.450476 0.892789i $$-0.351254\pi$$
0.450476 + 0.892789i $$0.351254\pi$$
$$150$$ 0 0
$$151$$ 199.009 1.31794 0.658971 0.752168i $$-0.270992\pi$$
0.658971 + 0.752168i $$0.270992\pi$$
$$152$$ 0 0
$$153$$ 16.4587i 0.107573i
$$154$$ 0 0
$$155$$ 217.930 1.40600
$$156$$ 0 0
$$157$$ 77.0189i 0.490566i 0.969451 + 0.245283i $$0.0788810\pi$$
−0.969451 + 0.245283i $$0.921119\pi$$
$$158$$ 0 0
$$159$$ 77.2415i 0.485795i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −230.379 −1.41337 −0.706684 0.707529i $$-0.749810\pi$$
−0.706684 + 0.707529i $$0.749810\pi$$
$$164$$ 0 0
$$165$$ −80.0548 −0.485181
$$166$$ 0 0
$$167$$ 229.231i 1.37264i 0.727298 + 0.686321i $$0.240776\pi$$
−0.727298 + 0.686321i $$0.759224\pi$$
$$168$$ 0 0
$$169$$ −272.665 −1.61340
$$170$$ 0 0
$$171$$ 21.7230i 0.127035i
$$172$$ 0 0
$$173$$ − 112.473i − 0.650133i −0.945691 0.325066i $$-0.894613\pi$$
0.945691 0.325066i $$-0.105387\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −110.323 −0.623293
$$178$$ 0 0
$$179$$ 105.494 0.589352 0.294676 0.955597i $$-0.404788\pi$$
0.294676 + 0.955597i $$0.404788\pi$$
$$180$$ 0 0
$$181$$ 15.2683i 0.0843553i 0.999110 + 0.0421776i $$0.0134295\pi$$
−0.999110 + 0.0421776i $$0.986570\pi$$
$$182$$ 0 0
$$183$$ 55.4401 0.302951
$$184$$ 0 0
$$185$$ 358.483i 1.93775i
$$186$$ 0 0
$$187$$ 47.1353i 0.252061i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −306.186 −1.60307 −0.801534 0.597950i $$-0.795982\pi$$
−0.801534 + 0.597950i $$0.795982\pi$$
$$192$$ 0 0
$$193$$ −1.84100 −0.00953885 −0.00476943 0.999989i $$-0.501518\pi$$
−0.00476943 + 0.999989i $$0.501518\pi$$
$$194$$ 0 0
$$195$$ − 195.822i − 1.00421i
$$196$$ 0 0
$$197$$ −255.334 −1.29611 −0.648056 0.761593i $$-0.724418\pi$$
−0.648056 + 0.761593i $$0.724418\pi$$
$$198$$ 0 0
$$199$$ − 36.6483i − 0.184162i −0.995752 0.0920812i $$-0.970648\pi$$
0.995752 0.0920812i $$-0.0293519\pi$$
$$200$$ 0 0
$$201$$ 82.6779i 0.411333i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −180.948 −0.882673
$$206$$ 0 0
$$207$$ −84.1669 −0.406603
$$208$$ 0 0
$$209$$ 62.2116i 0.297663i
$$210$$ 0 0
$$211$$ −126.571 −0.599862 −0.299931 0.953961i $$-0.596964\pi$$
−0.299931 + 0.953961i $$0.596964\pi$$
$$212$$ 0 0
$$213$$ − 25.9246i − 0.121712i
$$214$$ 0 0
$$215$$ 5.01871i 0.0233428i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −243.002 −1.10960
$$220$$ 0 0
$$221$$ −115.297 −0.521708
$$222$$ 0 0
$$223$$ − 212.193i − 0.951536i −0.879571 0.475768i $$-0.842170\pi$$
0.879571 0.475768i $$-0.157830\pi$$
$$224$$ 0 0
$$225$$ 11.8216 0.0525405
$$226$$ 0 0
$$227$$ − 105.489i − 0.464708i −0.972631 0.232354i $$-0.925357\pi$$
0.972631 0.232354i $$-0.0746428\pi$$
$$228$$ 0 0
$$229$$ 6.99086i 0.0305278i 0.999884 + 0.0152639i $$0.00485883\pi$$
−0.999884 + 0.0152639i $$0.995141\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 336.046 1.44226 0.721128 0.692801i $$-0.243624\pi$$
0.721128 + 0.692801i $$0.243624\pi$$
$$234$$ 0 0
$$235$$ 460.790 1.96081
$$236$$ 0 0
$$237$$ 211.843i 0.893851i
$$238$$ 0 0
$$239$$ −232.382 −0.972309 −0.486155 0.873873i $$-0.661601\pi$$
−0.486155 + 0.873873i $$0.661601\pi$$
$$240$$ 0 0
$$241$$ − 63.7346i − 0.264459i −0.991219 0.132229i $$-0.957786\pi$$
0.991219 0.132229i $$-0.0422136\pi$$
$$242$$ 0 0
$$243$$ − 15.5885i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −152.175 −0.616094
$$248$$ 0 0
$$249$$ 57.4784 0.230837
$$250$$ 0 0
$$251$$ 415.450i 1.65518i 0.561334 + 0.827589i $$0.310288\pi$$
−0.561334 + 0.827589i $$0.689712\pi$$
$$252$$ 0 0
$$253$$ −241.043 −0.952737
$$254$$ 0 0
$$255$$ − 51.1196i − 0.200469i
$$256$$ 0 0
$$257$$ 265.598i 1.03345i 0.856150 + 0.516727i $$0.172850\pi$$
−0.856150 + 0.516727i $$0.827150\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −121.034 −0.463732
$$262$$ 0 0
$$263$$ 156.373 0.594575 0.297288 0.954788i $$-0.403918\pi$$
0.297288 + 0.954788i $$0.403918\pi$$
$$264$$ 0 0
$$265$$ − 239.907i − 0.905310i
$$266$$ 0 0
$$267$$ 62.5696 0.234343
$$268$$ 0 0
$$269$$ 332.861i 1.23740i 0.785626 + 0.618701i $$0.212341\pi$$
−0.785626 + 0.618701i $$0.787659\pi$$
$$270$$ 0 0
$$271$$ − 209.089i − 0.771546i −0.922594 0.385773i $$-0.873935\pi$$
0.922594 0.385773i $$-0.126065\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 33.8555 0.123111
$$276$$ 0 0
$$277$$ −263.231 −0.950293 −0.475146 0.879907i $$-0.657605\pi$$
−0.475146 + 0.879907i $$0.657605\pi$$
$$278$$ 0 0
$$279$$ 121.530i 0.435592i
$$280$$ 0 0
$$281$$ 391.519 1.39331 0.696653 0.717408i $$-0.254672\pi$$
0.696653 + 0.717408i $$0.254672\pi$$
$$282$$ 0 0
$$283$$ 109.394i 0.386550i 0.981145 + 0.193275i $$0.0619109\pi$$
−0.981145 + 0.193275i $$0.938089\pi$$
$$284$$ 0 0
$$285$$ − 67.4701i − 0.236737i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 258.901 0.895853
$$290$$ 0 0
$$291$$ 28.0896 0.0965278
$$292$$ 0 0
$$293$$ 35.3685i 0.120712i 0.998177 + 0.0603558i $$0.0192235\pi$$
−0.998177 + 0.0603558i $$0.980776\pi$$
$$294$$ 0 0
$$295$$ 342.656 1.16155
$$296$$ 0 0
$$297$$ − 44.6432i − 0.150314i
$$298$$ 0 0
$$299$$ − 589.613i − 1.97195i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 206.918 0.682898
$$304$$ 0 0
$$305$$ −172.194 −0.564569
$$306$$ 0 0
$$307$$ 125.621i 0.409189i 0.978847 + 0.204594i $$0.0655875\pi$$
−0.978847 + 0.204594i $$0.934412\pi$$
$$308$$ 0 0
$$309$$ −15.4705 −0.0500662
$$310$$ 0 0
$$311$$ − 315.735i − 1.01523i −0.861585 0.507613i $$-0.830528\pi$$
0.861585 0.507613i $$-0.169472\pi$$
$$312$$ 0 0
$$313$$ − 51.6046i − 0.164871i −0.996596 0.0824355i $$-0.973730\pi$$
0.996596 0.0824355i $$-0.0262698\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −4.99391 −0.0157537 −0.00787683 0.999969i $$-0.502507\pi$$
−0.00787683 + 0.999969i $$0.502507\pi$$
$$318$$ 0 0
$$319$$ −346.625 −1.08660
$$320$$ 0 0
$$321$$ − 301.127i − 0.938091i
$$322$$ 0 0
$$323$$ −39.7256 −0.122990
$$324$$ 0 0
$$325$$ 82.8137i 0.254811i
$$326$$ 0 0
$$327$$ 277.673i 0.849153i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 454.780 1.37396 0.686980 0.726677i $$-0.258936\pi$$
0.686980 + 0.726677i $$0.258936\pi$$
$$332$$ 0 0
$$333$$ −199.911 −0.600333
$$334$$ 0 0
$$335$$ − 256.792i − 0.766545i
$$336$$ 0 0
$$337$$ −183.824 −0.545471 −0.272736 0.962089i $$-0.587928\pi$$
−0.272736 + 0.962089i $$0.587928\pi$$
$$338$$ 0 0
$$339$$ 141.062i 0.416111i
$$340$$ 0 0
$$341$$ 348.046i 1.02066i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 261.417 0.757731
$$346$$ 0 0
$$347$$ 144.776 0.417223 0.208611 0.977999i $$-0.433106\pi$$
0.208611 + 0.977999i $$0.433106\pi$$
$$348$$ 0 0
$$349$$ 187.069i 0.536015i 0.963417 + 0.268007i $$0.0863651\pi$$
−0.963417 + 0.268007i $$0.913635\pi$$
$$350$$ 0 0
$$351$$ 109.201 0.311115
$$352$$ 0 0
$$353$$ − 222.601i − 0.630597i −0.948993 0.315299i $$-0.897895\pi$$
0.948993 0.315299i $$-0.102105\pi$$
$$354$$ 0 0
$$355$$ 80.5201i 0.226817i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 62.9856 0.175447 0.0877237 0.996145i $$-0.472041\pi$$
0.0877237 + 0.996145i $$0.472041\pi$$
$$360$$ 0 0
$$361$$ 308.568 0.854759
$$362$$ 0 0
$$363$$ 81.7261i 0.225141i
$$364$$ 0 0
$$365$$ 754.750 2.06781
$$366$$ 0 0
$$367$$ − 547.100i − 1.49074i −0.666653 0.745368i $$-0.732274\pi$$
0.666653 0.745368i $$-0.267726\pi$$
$$368$$ 0 0
$$369$$ − 100.907i − 0.273461i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −32.4417 −0.0869750 −0.0434875 0.999054i $$-0.513847\pi$$
−0.0434875 + 0.999054i $$0.513847\pi$$
$$374$$ 0 0
$$375$$ 196.228 0.523275
$$376$$ 0 0
$$377$$ − 847.878i − 2.24901i
$$378$$ 0 0
$$379$$ 508.859 1.34263 0.671317 0.741170i $$-0.265729\pi$$
0.671317 + 0.741170i $$0.265729\pi$$
$$380$$ 0 0
$$381$$ − 202.948i − 0.532671i
$$382$$ 0 0
$$383$$ 26.2839i 0.0686263i 0.999411 + 0.0343131i $$0.0109244\pi$$
−0.999411 + 0.0343131i $$0.989076\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2.79872 −0.00723184
$$388$$ 0 0
$$389$$ −360.128 −0.925779 −0.462890 0.886416i $$-0.653187\pi$$
−0.462890 + 0.886416i $$0.653187\pi$$
$$390$$ 0 0
$$391$$ − 153.919i − 0.393656i
$$392$$ 0 0
$$393$$ −422.880 −1.07603
$$394$$ 0 0
$$395$$ − 657.970i − 1.66575i
$$396$$ 0 0
$$397$$ 570.992i 1.43827i 0.694872 + 0.719134i $$0.255461\pi$$
−0.694872 + 0.719134i $$0.744539\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 68.4014 0.170577 0.0852885 0.996356i $$-0.472819\pi$$
0.0852885 + 0.996356i $$0.472819\pi$$
$$402$$ 0 0
$$403$$ −851.353 −2.11254
$$404$$ 0 0
$$405$$ 48.4168i 0.119548i
$$406$$ 0 0
$$407$$ −572.518 −1.40668
$$408$$ 0 0
$$409$$ − 15.3649i − 0.0375671i −0.999824 0.0187835i $$-0.994021\pi$$
0.999824 0.0187835i $$-0.00597934\pi$$
$$410$$ 0 0
$$411$$ − 425.066i − 1.03422i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −178.524 −0.430179
$$416$$ 0 0
$$417$$ −29.6823 −0.0711805
$$418$$ 0 0
$$419$$ − 366.079i − 0.873696i −0.899535 0.436848i $$-0.856095\pi$$
0.899535 0.436848i $$-0.143905\pi$$
$$420$$ 0 0
$$421$$ 607.135 1.44213 0.721063 0.692870i $$-0.243654\pi$$
0.721063 + 0.692870i $$0.243654\pi$$
$$422$$ 0 0
$$423$$ 256.963i 0.607478i
$$424$$ 0 0
$$425$$ 21.6186i 0.0508674i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 312.738 0.728993
$$430$$ 0 0
$$431$$ −586.353 −1.36045 −0.680224 0.733004i $$-0.738118\pi$$
−0.680224 + 0.733004i $$0.738118\pi$$
$$432$$ 0 0
$$433$$ 518.769i 1.19808i 0.800719 + 0.599040i $$0.204451\pi$$
−0.800719 + 0.599040i $$0.795549\pi$$
$$434$$ 0 0
$$435$$ 375.924 0.864194
$$436$$ 0 0
$$437$$ − 203.150i − 0.464875i
$$438$$ 0 0
$$439$$ − 221.524i − 0.504610i −0.967648 0.252305i $$-0.918811\pi$$
0.967648 0.252305i $$-0.0811886\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −226.415 −0.511095 −0.255548 0.966796i $$-0.582256\pi$$
−0.255548 + 0.966796i $$0.582256\pi$$
$$444$$ 0 0
$$445$$ −194.337 −0.436713
$$446$$ 0 0
$$447$$ − 232.514i − 0.520165i
$$448$$ 0 0
$$449$$ −378.422 −0.842810 −0.421405 0.906873i $$-0.638463\pi$$
−0.421405 + 0.906873i $$0.638463\pi$$
$$450$$ 0 0
$$451$$ − 288.984i − 0.640763i
$$452$$ 0 0
$$453$$ − 344.694i − 0.760914i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 438.153 0.958760 0.479380 0.877607i $$-0.340862\pi$$
0.479380 + 0.877607i $$0.340862\pi$$
$$458$$ 0 0
$$459$$ 28.5072 0.0621073
$$460$$ 0 0
$$461$$ 46.3981i 0.100647i 0.998733 + 0.0503234i $$0.0160252\pi$$
−0.998733 + 0.0503234i $$0.983975\pi$$
$$462$$ 0 0
$$463$$ 367.455 0.793639 0.396820 0.917897i $$-0.370114\pi$$
0.396820 + 0.917897i $$0.370114\pi$$
$$464$$ 0 0
$$465$$ − 377.465i − 0.811753i
$$466$$ 0 0
$$467$$ 529.465i 1.13376i 0.823801 + 0.566879i $$0.191849\pi$$
−0.823801 + 0.566879i $$0.808151\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 133.401 0.283229
$$472$$ 0 0
$$473$$ −8.01516 −0.0169454
$$474$$ 0 0
$$475$$ 28.5334i 0.0600702i
$$476$$ 0 0
$$477$$ 133.786 0.280474
$$478$$ 0 0
$$479$$ − 289.761i − 0.604928i −0.953161 0.302464i $$-0.902191\pi$$
0.953161 0.302464i $$-0.0978093\pi$$
$$480$$ 0 0
$$481$$ − 1400.43i − 2.91150i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −87.2445 −0.179886
$$486$$ 0 0
$$487$$ 439.636 0.902744 0.451372 0.892336i $$-0.350935\pi$$
0.451372 + 0.892336i $$0.350935\pi$$
$$488$$ 0 0
$$489$$ 399.028i 0.816009i
$$490$$ 0 0
$$491$$ −320.561 −0.652874 −0.326437 0.945219i $$-0.605848\pi$$
−0.326437 + 0.945219i $$0.605848\pi$$
$$492$$ 0 0
$$493$$ − 221.340i − 0.448965i
$$494$$ 0 0
$$495$$ 138.659i 0.280119i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 771.973 1.54704 0.773520 0.633772i $$-0.218494\pi$$
0.773520 + 0.633772i $$0.218494\pi$$
$$500$$ 0 0
$$501$$ 397.040 0.792496
$$502$$ 0 0
$$503$$ 101.632i 0.202052i 0.994884 + 0.101026i $$0.0322126\pi$$
−0.994884 + 0.101026i $$0.967787\pi$$
$$504$$ 0 0
$$505$$ −642.675 −1.27262
$$506$$ 0 0
$$507$$ 472.270i 0.931499i
$$508$$ 0 0
$$509$$ 605.407i 1.18941i 0.803946 + 0.594703i $$0.202730\pi$$
−0.803946 + 0.594703i $$0.797270\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 37.6253 0.0733436
$$514$$ 0 0
$$515$$ 48.0503 0.0933016
$$516$$ 0 0
$$517$$ 735.907i 1.42342i
$$518$$ 0 0
$$519$$ −194.809 −0.375354
$$520$$ 0 0
$$521$$ − 383.293i − 0.735688i −0.929888 0.367844i $$-0.880096\pi$$
0.929888 0.367844i $$-0.119904\pi$$
$$522$$ 0 0
$$523$$ 448.095i 0.856777i 0.903595 + 0.428389i $$0.140919\pi$$
−0.903595 + 0.428389i $$0.859081\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −222.247 −0.421721
$$528$$ 0 0
$$529$$ 258.119 0.487937
$$530$$ 0 0
$$531$$ 191.085i 0.359859i
$$532$$ 0 0
$$533$$ 706.882 1.32623
$$534$$ 0 0
$$535$$ 935.282i 1.74819i
$$536$$ 0 0
$$537$$ − 182.721i − 0.340262i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −271.258 −0.501400 −0.250700 0.968065i $$-0.580661\pi$$
−0.250700 + 0.968065i $$0.580661\pi$$
$$542$$ 0 0
$$543$$ 26.4455 0.0487025
$$544$$ 0 0
$$545$$ − 862.435i − 1.58245i
$$546$$ 0 0
$$547$$ −590.544 −1.07961 −0.539803 0.841791i $$-0.681501\pi$$
−0.539803 + 0.841791i $$0.681501\pi$$
$$548$$ 0 0
$$549$$ − 96.0251i − 0.174909i
$$550$$ 0 0
$$551$$ − 292.135i − 0.530191i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 620.911 1.11876
$$556$$ 0 0
$$557$$ 188.295 0.338051 0.169026 0.985612i $$-0.445938\pi$$
0.169026 + 0.985612i $$0.445938\pi$$
$$558$$ 0 0
$$559$$ − 19.6058i − 0.0350730i
$$560$$ 0 0
$$561$$ 81.6408 0.145527
$$562$$ 0 0
$$563$$ 204.157i 0.362623i 0.983426 + 0.181312i $$0.0580342\pi$$
−0.983426 + 0.181312i $$0.941966\pi$$
$$564$$ 0 0
$$565$$ − 438.129i − 0.775449i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −660.913 −1.16153 −0.580767 0.814070i $$-0.697247\pi$$
−0.580767 + 0.814070i $$0.697247\pi$$
$$570$$ 0 0
$$571$$ −533.978 −0.935162 −0.467581 0.883950i $$-0.654874\pi$$
−0.467581 + 0.883950i $$0.654874\pi$$
$$572$$ 0 0
$$573$$ 530.329i 0.925531i
$$574$$ 0 0
$$575$$ −110.554 −0.192268
$$576$$ 0 0
$$577$$ − 53.4337i − 0.0926062i −0.998927 0.0463031i $$-0.985256\pi$$
0.998927 0.0463031i $$-0.0147440\pi$$
$$578$$ 0 0
$$579$$ 3.18870i 0.00550726i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 383.145 0.657196
$$584$$ 0 0
$$585$$ −339.173 −0.579783
$$586$$ 0 0
$$587$$ 413.063i 0.703684i 0.936059 + 0.351842i $$0.114445\pi$$
−0.936059 + 0.351842i $$0.885555\pi$$
$$588$$ 0 0
$$589$$ −293.333 −0.498018
$$590$$ 0 0
$$591$$ 442.252i 0.748311i
$$592$$ 0 0
$$593$$ − 122.452i − 0.206496i −0.994656 0.103248i $$-0.967076\pi$$
0.994656 0.103248i $$-0.0329235\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −63.4768 −0.106326
$$598$$ 0 0
$$599$$ −645.136 −1.07702 −0.538511 0.842619i $$-0.681013\pi$$
−0.538511 + 0.842619i $$0.681013\pi$$
$$600$$ 0 0
$$601$$ − 683.488i − 1.13725i −0.822596 0.568626i $$-0.807475\pi$$
0.822596 0.568626i $$-0.192525\pi$$
$$602$$ 0 0
$$603$$ 143.202 0.237483
$$604$$ 0 0
$$605$$ − 253.836i − 0.419564i
$$606$$ 0 0
$$607$$ 257.253i 0.423811i 0.977290 + 0.211905i $$0.0679669\pi$$
−0.977290 + 0.211905i $$0.932033\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1800.10 −2.94615
$$612$$ 0 0
$$613$$ 885.713 1.44488 0.722441 0.691433i $$-0.243020\pi$$
0.722441 + 0.691433i $$0.243020\pi$$
$$614$$ 0 0
$$615$$ 313.411i 0.509612i
$$616$$ 0 0
$$617$$ 46.0724 0.0746717 0.0373358 0.999303i $$-0.488113\pi$$
0.0373358 + 0.999303i $$0.488113\pi$$
$$618$$ 0 0
$$619$$ 1096.33i 1.77112i 0.464522 + 0.885562i $$0.346226\pi$$
−0.464522 + 0.885562i $$0.653774\pi$$
$$620$$ 0 0
$$621$$ 145.781i 0.234753i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −707.986 −1.13278
$$626$$ 0 0
$$627$$ 107.754 0.171856
$$628$$ 0 0
$$629$$ − 365.585i − 0.581217i
$$630$$ 0 0
$$631$$ 606.319 0.960886 0.480443 0.877026i $$-0.340476\pi$$
0.480443 + 0.877026i $$0.340476\pi$$
$$632$$ 0 0
$$633$$ 219.227i 0.346330i
$$634$$ 0 0
$$635$$ 630.343i 0.992666i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −44.9027 −0.0702702
$$640$$ 0 0
$$641$$ 939.184 1.46519 0.732593 0.680667i $$-0.238310\pi$$
0.732593 + 0.680667i $$0.238310\pi$$
$$642$$ 0 0
$$643$$ − 992.960i − 1.54426i −0.635464 0.772131i $$-0.719191\pi$$
0.635464 0.772131i $$-0.280809\pi$$
$$644$$ 0 0
$$645$$ 8.69266 0.0134770
$$646$$ 0 0
$$647$$ 265.184i 0.409867i 0.978776 + 0.204933i $$0.0656978\pi$$
−0.978776 + 0.204933i $$0.934302\pi$$
$$648$$ 0 0
$$649$$ 547.241i 0.843206i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1134.14 −1.73682 −0.868409 0.495849i $$-0.834857\pi$$
−0.868409 + 0.495849i $$0.834857\pi$$
$$654$$ 0 0
$$655$$ 1313.44 2.00525
$$656$$ 0 0
$$657$$ 420.893i 0.640628i
$$658$$ 0 0
$$659$$ −924.147 −1.40235 −0.701174 0.712990i $$-0.747340\pi$$
−0.701174 + 0.712990i $$0.747340\pi$$
$$660$$ 0 0
$$661$$ 234.367i 0.354564i 0.984160 + 0.177282i $$0.0567305\pi$$
−0.984160 + 0.177282i $$0.943269\pi$$
$$662$$ 0 0
$$663$$ 199.701i 0.301208i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1131.90 1.69700
$$668$$ 0 0
$$669$$ −367.528 −0.549370
$$670$$ 0 0
$$671$$ − 275.003i − 0.409840i
$$672$$ 0 0
$$673$$ −465.127 −0.691125 −0.345563 0.938396i $$-0.612312\pi$$
−0.345563 + 0.938396i $$0.612312\pi$$
$$674$$ 0 0
$$675$$ − 20.4756i − 0.0303343i
$$676$$ 0 0
$$677$$ 1295.93i 1.91422i 0.289724 + 0.957110i $$0.406437\pi$$
−0.289724 + 0.957110i $$0.593563\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −182.712 −0.268300
$$682$$ 0 0
$$683$$ 16.2612 0.0238085 0.0119042 0.999929i $$-0.496211\pi$$
0.0119042 + 0.999929i $$0.496211\pi$$
$$684$$ 0 0
$$685$$ 1320.23i 1.92734i
$$686$$ 0 0
$$687$$ 12.1085 0.0176252
$$688$$ 0 0
$$689$$ 937.209i 1.36025i
$$690$$ 0 0
$$691$$ − 68.5364i − 0.0991843i −0.998770 0.0495922i $$-0.984208\pi$$
0.998770 0.0495922i $$-0.0157922\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 92.1912 0.132649
$$696$$ 0 0
$$697$$ 184.533 0.264753
$$698$$ 0 0
$$699$$ − 582.048i − 0.832687i
$$700$$ 0 0
$$701$$ 923.360 1.31720 0.658602 0.752491i $$-0.271148\pi$$
0.658602 + 0.752491i $$0.271148\pi$$
$$702$$ 0 0
$$703$$ − 482.518i − 0.686369i
$$704$$ 0 0
$$705$$ − 798.111i − 1.13207i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −912.863 −1.28754 −0.643768 0.765221i $$-0.722630\pi$$
−0.643768 + 0.765221i $$0.722630\pi$$
$$710$$ 0 0
$$711$$ 366.922 0.516065
$$712$$ 0 0
$$713$$ − 1136.54i − 1.59402i
$$714$$ 0 0
$$715$$ −971.345 −1.35852
$$716$$ 0 0
$$717$$ 402.497i 0.561363i
$$718$$ 0 0
$$719$$ 545.020i 0.758025i 0.925391 + 0.379013i $$0.123736\pi$$
−0.925391 + 0.379013i $$0.876264\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −110.392 −0.152685
$$724$$ 0 0
$$725$$ −158.980 −0.219283
$$726$$ 0 0
$$727$$ 750.292i 1.03204i 0.856577 + 0.516019i $$0.172587\pi$$
−0.856577 + 0.516019i $$0.827413\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −0.0370370
$$730$$ 0 0
$$731$$ − 5.11813i − 0.00700155i
$$732$$ 0 0
$$733$$ − 926.599i − 1.26412i −0.774920 0.632059i $$-0.782210\pi$$
0.774920 0.632059i $$-0.217790\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 410.112 0.556461
$$738$$ 0 0
$$739$$ 488.147 0.660551 0.330275 0.943885i $$-0.392858\pi$$
0.330275 + 0.943885i $$0.392858\pi$$
$$740$$ 0 0
$$741$$ 263.575i 0.355702i
$$742$$ 0 0
$$743$$ −1091.72 −1.46935 −0.734674 0.678421i $$-0.762665\pi$$
−0.734674 + 0.678421i $$0.762665\pi$$
$$744$$ 0 0
$$745$$ 722.173i 0.969360i
$$746$$ 0 0
$$747$$ − 99.5556i − 0.133274i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −480.443 −0.639738 −0.319869 0.947462i $$-0.603639\pi$$
−0.319869 + 0.947462i $$0.603639\pi$$
$$752$$ 0 0
$$753$$ 719.580 0.955618
$$754$$ 0 0
$$755$$ 1070.60i 1.41801i
$$756$$ 0 0
$$757$$ 941.400 1.24359 0.621796 0.783179i $$-0.286403\pi$$
0.621796 + 0.783179i $$0.286403\pi$$
$$758$$ 0 0
$$759$$ 417.498i 0.550063i
$$760$$ 0 0
$$761$$ − 1192.34i − 1.56681i −0.621509 0.783407i $$-0.713480\pi$$
0.621509 0.783407i $$-0.286520\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −88.5417 −0.115741
$$766$$ 0 0
$$767$$ −1338.60 −1.74524
$$768$$ 0 0
$$769$$ − 908.294i − 1.18114i −0.806988 0.590568i $$-0.798904\pi$$
0.806988 0.590568i $$-0.201096\pi$$
$$770$$ 0 0
$$771$$ 460.029 0.596665
$$772$$ 0 0
$$773$$ 1189.38i 1.53866i 0.638854 + 0.769328i $$0.279409\pi$$
−0.638854 + 0.769328i $$0.720591\pi$$
$$774$$ 0 0
$$775$$ 159.632i 0.205976i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 243.556 0.312652
$$780$$ 0 0
$$781$$ −128.595 −0.164654
$$782$$ 0 0
$$783$$ 209.637i 0.267736i
$$784$$ 0 0
$$785$$ −414.334 −0.527814
$$786$$ 0 0
$$787$$ 1057.12i 1.34323i 0.740902 + 0.671613i $$0.234398\pi$$
−0.740902 + 0.671613i $$0.765602\pi$$
$$788$$ 0 0
$$789$$ − 270.846i − 0.343278i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 672.682 0.848275
$$794$$ 0 0
$$795$$ −415.531 −0.522681
$$796$$ 0 0
$$797$$ 1037.94i 1.30231i 0.758945 + 0.651154i $$0.225715\pi$$
−0.758945 + 0.651154i $$0.774285\pi$$
$$798$$ 0 0
$$799$$ −469.919 −0.588133
$$800$$ 0 0
$$801$$ − 108.374i − 0.135298i
$$802$$ 0 0
$$803$$ 1205.38i 1.50109i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 576.533 0.714415
$$808$$ 0 0
$$809$$ 1087.36 1.34408 0.672041 0.740514i $$-0.265418\pi$$
0.672041 + 0.740514i $$0.265418\pi$$
$$810$$ 0 0
$$811$$ 926.995i 1.14303i 0.820593 + 0.571514i $$0.193644\pi$$
−0.820593 + 0.571514i $$0.806356\pi$$
$$812$$ 0 0
$$813$$ −362.153 −0.445452
$$814$$ 0 0
$$815$$ − 1239.36i − 1.52068i
$$816$$ 0 0
$$817$$ − 6.75517i − 0.00826826i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −430.300 −0.524117 −0.262058 0.965052i $$-0.584401\pi$$
−0.262058 + 0.965052i $$0.584401\pi$$
$$822$$ 0 0
$$823$$ −1044.74 −1.26942 −0.634712 0.772749i $$-0.718881\pi$$
−0.634712 + 0.772749i $$0.718881\pi$$
$$824$$ 0 0
$$825$$ − 58.6394i − 0.0710781i
$$826$$ 0 0
$$827$$ −610.313 −0.737984 −0.368992 0.929433i $$-0.620297\pi$$
−0.368992 + 0.929433i $$0.620297\pi$$
$$828$$ 0 0
$$829$$ − 1413.90i − 1.70555i −0.522279 0.852775i $$-0.674918\pi$$
0.522279 0.852775i $$-0.325082\pi$$
$$830$$ 0 0
$$831$$ 455.930i 0.548652i
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −1233.18 −1.47687
$$836$$ 0 0
$$837$$ 210.497 0.251489
$$838$$ 0 0
$$839$$ 46.3303i 0.0552209i 0.999619 + 0.0276105i $$0.00878980\pi$$
−0.999619 + 0.0276105i $$0.991210\pi$$
$$840$$ 0 0
$$841$$ 786.696 0.935430
$$842$$ 0 0
$$843$$ − 678.131i − 0.804426i
$$844$$ 0 0
$$845$$ − 1466.84i − 1.73591i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 189.475 0.223175
$$850$$ 0 0
$$851$$ 1869.54 2.19688
$$852$$ 0 0
$$853$$ 773.111i 0.906343i 0.891423 + 0.453172i $$0.149708\pi$$
−0.891423 + 0.453172i $$0.850292\pi$$
$$854$$ 0 0
$$855$$ −116.862 −0.136680
$$856$$ 0 0
$$857$$ 438.159i 0.511271i 0.966773 + 0.255636i $$0.0822847\pi$$
−0.966773 + 0.255636i $$0.917715\pi$$
$$858$$ 0 0
$$859$$ 72.3419i 0.0842164i 0.999113 + 0.0421082i $$0.0134074\pi$$
−0.999113 + 0.0421082i $$0.986593\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 426.495 0.494200 0.247100 0.968990i $$-0.420522\pi$$
0.247100 + 0.968990i $$0.420522\pi$$
$$864$$ 0 0
$$865$$ 605.064 0.699496
$$866$$ 0 0
$$867$$ − 448.430i − 0.517221i
$$868$$ 0 0
$$869$$ 1050.82 1.20922
$$870$$ 0 0
$$871$$ 1003.17i 1.15175i
$$872$$ 0 0
$$873$$ − 48.6526i − 0.0557304i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 416.485 0.474898 0.237449 0.971400i $$-0.423689\pi$$
0.237449 + 0.971400i $$0.423689\pi$$
$$878$$ 0 0
$$879$$ 61.2600 0.0696929
$$880$$ 0 0
$$881$$ − 1249.01i − 1.41771i −0.705353 0.708857i $$-0.749211\pi$$
0.705353 0.708857i $$-0.250789\pi$$
$$882$$ 0 0
$$883$$ 81.5906 0.0924016 0.0462008 0.998932i $$-0.485289\pi$$
0.0462008 + 0.998932i $$0.485289\pi$$
$$884$$ 0 0
$$885$$ − 593.498i − 0.670619i
$$886$$ 0 0
$$887$$ − 917.730i − 1.03464i −0.855791 0.517322i $$-0.826929\pi$$
0.855791 0.517322i $$-0.173071\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −77.3243 −0.0867837
$$892$$ 0 0
$$893$$ −620.222 −0.694537
$$894$$ 0 0
$$895$$ 567.520i 0.634100i
$$896$$ 0 0
$$897$$ −1021.24 −1.13850
$$898$$ 0 0
$$899$$ − 1634.37i − 1.81798i
$$900$$ 0 0
$$901$$ 244.660i 0.271543i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −82.1380 −0.0907602
$$906$$ 0 0
$$907$$ −1451.40 −1.60022 −0.800110 0.599853i $$-0.795226\pi$$
−0.800110 + 0.599853i $$0.795226\pi$$
$$908$$ 0 0
$$909$$ − 358.393i − 0.394272i
$$910$$ 0 0
$$911$$ −562.064 −0.616975 −0.308487 0.951228i $$-0.599823\pi$$
−0.308487 + 0.951228i $$0.599823\pi$$
$$912$$ 0 0
$$913$$ − 285.114i − 0.312282i
$$914$$ 0 0
$$915$$ 298.248i 0.325954i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 538.158 0.585591 0.292795 0.956175i $$-0.405414\pi$$
0.292795 + 0.956175i $$0.405414\pi$$
$$920$$ 0 0
$$921$$ 217.582 0.236245
$$922$$ 0 0
$$923$$ − 314.556i − 0.340797i
$$924$$ 0 0
$$925$$ −262.586 −0.283876
$$926$$ 0 0
$$927$$ 26.7956i 0.0289058i
$$928$$ 0 0
$$929$$ 827.080i 0.890291i 0.895458 + 0.445145i $$0.146848\pi$$
−0.895458 + 0.445145i $$0.853152\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −546.869 −0.586141
$$934$$ 0 0
$$935$$ −253.571 −0.271199
$$936$$ 0 0
$$937$$ − 1501.10i − 1.60203i −0.598644 0.801016i $$-0.704293\pi$$
0.598644 0.801016i $$-0.295707\pi$$
$$938$$ 0 0
$$939$$ −89.3818 −0.0951883
$$940$$ 0 0
$$941$$ − 766.672i − 0.814742i −0.913263 0.407371i $$-0.866446\pi$$
0.913263 0.407371i $$-0.133554\pi$$
$$942$$ 0 0
$$943$$ 943.671i 1.00071i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1362.23 1.43847 0.719237 0.694765i $$-0.244492\pi$$
0.719237 + 0.694765i $$0.244492\pi$$
$$948$$ 0 0
$$949$$ −2948.47 −3.10692
$$950$$ 0 0
$$951$$ 8.64971i 0.00909538i
$$952$$ 0 0
$$953$$ 1431.37 1.50196 0.750980 0.660325i $$-0.229581\pi$$
0.750980 + 0.660325i $$0.229581\pi$$
$$954$$ 0 0
$$955$$ − 1647.17i − 1.72479i
$$956$$ 0 0
$$957$$ 600.373i 0.627348i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −680.067 −0.707666
$$962$$ 0 0
$$963$$ −521.567 −0.541607
$$964$$ 0 0
$$965$$ − 9.90391i − 0.0102631i
$$966$$ 0 0
$$967$$ −426.276 −0.440823 −0.220411 0.975407i $$-0.570740\pi$$
−0.220411 + 0.975407i $$0.570740\pi$$
$$968$$ 0 0
$$969$$ 68.8068i 0.0710080i
$$970$$ 0 0
$$971$$ − 1728.41i − 1.78003i −0.455934 0.890013i $$-0.650695\pi$$
0.455934 0.890013i $$-0.349305\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 143.437 0.147115
$$976$$ 0 0
$$977$$ −712.320 −0.729089 −0.364545 0.931186i $$-0.618775\pi$$
−0.364545 + 0.931186i $$0.618775\pi$$
$$978$$ 0 0
$$979$$ − 310.368i − 0.317025i
$$980$$ 0 0
$$981$$ 480.944 0.490259
$$982$$ 0 0
$$983$$ 1042.56i 1.06059i 0.847812 + 0.530297i $$0.177920\pi$$
−0.847812 + 0.530297i $$0.822080\pi$$
$$984$$ 0 0
$$985$$ − 1373.61i − 1.39452i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 26.1733 0.0264644
$$990$$ 0 0
$$991$$ 1397.41 1.41010 0.705052 0.709156i $$-0.250924\pi$$
0.705052 + 0.709156i $$0.250924\pi$$
$$992$$ 0 0
$$993$$ − 787.703i − 0.793256i
$$994$$ 0 0
$$995$$ 197.155 0.198146
$$996$$ 0 0
$$997$$ − 717.962i − 0.720122i −0.932929 0.360061i $$-0.882756\pi$$
0.932929 0.360061i $$-0.117244\pi$$
$$998$$ 0 0
$$999$$ 346.256i 0.346603i
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 588.3.d.c.97.4 8
3.2 odd 2 1764.3.d.h.685.2 8
4.3 odd 2 2352.3.f.j.97.8 8
7.2 even 3 588.3.m.f.325.4 8
7.3 odd 6 588.3.m.f.313.4 8
7.4 even 3 588.3.m.e.313.1 8
7.5 odd 6 588.3.m.e.325.1 8
7.6 odd 2 inner 588.3.d.c.97.5 yes 8
21.2 odd 6 1764.3.z.l.325.1 8
21.5 even 6 1764.3.z.m.325.4 8
21.11 odd 6 1764.3.z.m.901.4 8
21.17 even 6 1764.3.z.l.901.1 8
21.20 even 2 1764.3.d.h.685.7 8
28.27 even 2 2352.3.f.j.97.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.d.c.97.4 8 1.1 even 1 trivial
588.3.d.c.97.5 yes 8 7.6 odd 2 inner
588.3.m.e.313.1 8 7.4 even 3
588.3.m.e.325.1 8 7.5 odd 6
588.3.m.f.313.4 8 7.3 odd 6
588.3.m.f.325.4 8 7.2 even 3
1764.3.d.h.685.2 8 3.2 odd 2
1764.3.d.h.685.7 8 21.20 even 2
1764.3.z.l.325.1 8 21.2 odd 6
1764.3.z.l.901.1 8 21.17 even 6
1764.3.z.m.325.4 8 21.5 even 6
1764.3.z.m.901.4 8 21.11 odd 6
2352.3.f.j.97.1 8 28.27 even 2
2352.3.f.j.97.8 8 4.3 odd 2