# Properties

 Label 4864.2.a.bm.1.6 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.34309996544.1 Defining polynomial: $$x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 2432) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$3.15093$$ of defining polynomial Character $$\chi$$ $$=$$ 4864.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.222191 q^{3} +1.55081 q^{5} -3.06334 q^{7} -2.95063 q^{9} +O(q^{10})$$ $$q+0.222191 q^{3} +1.55081 q^{5} -3.06334 q^{7} -2.95063 q^{9} +0.950631 q^{11} +3.36113 q^{13} +0.344577 q^{15} +1.82843 q^{17} +1.00000 q^{19} -0.680647 q^{21} -3.36113 q^{23} -2.59499 q^{25} -1.32218 q^{27} +4.95873 q^{29} -3.44620 q^{31} +0.211222 q^{33} -4.75066 q^{35} -2.95889 q^{37} +0.746814 q^{39} -4.55687 q^{41} -1.69373 q^{43} -4.57587 q^{45} -3.39941 q^{47} +2.38404 q^{49} +0.406261 q^{51} +4.47142 q^{53} +1.47425 q^{55} +0.222191 q^{57} +0.395014 q^{59} +5.34158 q^{61} +9.03878 q^{63} +5.21247 q^{65} -6.85064 q^{67} -0.746814 q^{69} +5.15207 q^{71} -1.19998 q^{73} -0.576584 q^{75} -2.91210 q^{77} -14.7249 q^{79} +8.55812 q^{81} -3.35564 q^{83} +2.83554 q^{85} +1.10179 q^{87} -9.64561 q^{89} -10.2963 q^{91} -0.765715 q^{93} +1.55081 q^{95} +3.55562 q^{97} -2.80496 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} + 4q^{9} + O(q^{10})$$ $$8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100})$$

## 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
$$3$$ 0.222191 0.128282 0.0641411 0.997941i $$-0.479569\pi$$
0.0641411 + 0.997941i $$0.479569\pi$$
$$4$$ 0 0
$$5$$ 1.55081 0.693543 0.346772 0.937950i $$-0.387278\pi$$
0.346772 + 0.937950i $$0.387278\pi$$
$$6$$ 0 0
$$7$$ −3.06334 −1.15783 −0.578917 0.815387i $$-0.696524\pi$$
−0.578917 + 0.815387i $$0.696524\pi$$
$$8$$ 0 0
$$9$$ −2.95063 −0.983544
$$10$$ 0 0
$$11$$ 0.950631 0.286626 0.143313 0.989677i $$-0.454224\pi$$
0.143313 + 0.989677i $$0.454224\pi$$
$$12$$ 0 0
$$13$$ 3.36113 0.932209 0.466105 0.884730i $$-0.345657\pi$$
0.466105 + 0.884730i $$0.345657\pi$$
$$14$$ 0 0
$$15$$ 0.344577 0.0889693
$$16$$ 0 0
$$17$$ 1.82843 0.443459 0.221729 0.975108i $$-0.428830\pi$$
0.221729 + 0.975108i $$0.428830\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −0.680647 −0.148529
$$22$$ 0 0
$$23$$ −3.36113 −0.700844 −0.350422 0.936592i $$-0.613962\pi$$
−0.350422 + 0.936592i $$0.613962\pi$$
$$24$$ 0 0
$$25$$ −2.59499 −0.518998
$$26$$ 0 0
$$27$$ −1.32218 −0.254453
$$28$$ 0 0
$$29$$ 4.95873 0.920812 0.460406 0.887708i $$-0.347704\pi$$
0.460406 + 0.887708i $$0.347704\pi$$
$$30$$ 0 0
$$31$$ −3.44620 −0.618955 −0.309478 0.950907i $$-0.600154\pi$$
−0.309478 + 0.950907i $$0.600154\pi$$
$$32$$ 0 0
$$33$$ 0.211222 0.0367690
$$34$$ 0 0
$$35$$ −4.75066 −0.803007
$$36$$ 0 0
$$37$$ −2.95889 −0.486439 −0.243219 0.969971i $$-0.578203\pi$$
−0.243219 + 0.969971i $$0.578203\pi$$
$$38$$ 0 0
$$39$$ 0.746814 0.119586
$$40$$ 0 0
$$41$$ −4.55687 −0.711663 −0.355832 0.934550i $$-0.615802\pi$$
−0.355832 + 0.934550i $$0.615802\pi$$
$$42$$ 0 0
$$43$$ −1.69373 −0.258291 −0.129145 0.991626i $$-0.541223\pi$$
−0.129145 + 0.991626i $$0.541223\pi$$
$$44$$ 0 0
$$45$$ −4.57587 −0.682130
$$46$$ 0 0
$$47$$ −3.39941 −0.495855 −0.247927 0.968779i $$-0.579749\pi$$
−0.247927 + 0.968779i $$0.579749\pi$$
$$48$$ 0 0
$$49$$ 2.38404 0.340578
$$50$$ 0 0
$$51$$ 0.406261 0.0568879
$$52$$ 0 0
$$53$$ 4.47142 0.614197 0.307098 0.951678i $$-0.400642\pi$$
0.307098 + 0.951678i $$0.400642\pi$$
$$54$$ 0 0
$$55$$ 1.47425 0.198788
$$56$$ 0 0
$$57$$ 0.222191 0.0294300
$$58$$ 0 0
$$59$$ 0.395014 0.0514264 0.0257132 0.999669i $$-0.491814\pi$$
0.0257132 + 0.999669i $$0.491814\pi$$
$$60$$ 0 0
$$61$$ 5.34158 0.683920 0.341960 0.939715i $$-0.388909\pi$$
0.341960 + 0.939715i $$0.388909\pi$$
$$62$$ 0 0
$$63$$ 9.03878 1.13878
$$64$$ 0 0
$$65$$ 5.21247 0.646528
$$66$$ 0 0
$$67$$ −6.85064 −0.836939 −0.418470 0.908231i $$-0.637433\pi$$
−0.418470 + 0.908231i $$0.637433\pi$$
$$68$$ 0 0
$$69$$ −0.746814 −0.0899058
$$70$$ 0 0
$$71$$ 5.15207 0.611438 0.305719 0.952122i $$-0.401103\pi$$
0.305719 + 0.952122i $$0.401103\pi$$
$$72$$ 0 0
$$73$$ −1.19998 −0.140446 −0.0702232 0.997531i $$-0.522371\pi$$
−0.0702232 + 0.997531i $$0.522371\pi$$
$$74$$ 0 0
$$75$$ −0.576584 −0.0665782
$$76$$ 0 0
$$77$$ −2.91210 −0.331865
$$78$$ 0 0
$$79$$ −14.7249 −1.65669 −0.828343 0.560222i $$-0.810716\pi$$
−0.828343 + 0.560222i $$0.810716\pi$$
$$80$$ 0 0
$$81$$ 8.55812 0.950902
$$82$$ 0 0
$$83$$ −3.35564 −0.368330 −0.184165 0.982895i $$-0.558958\pi$$
−0.184165 + 0.982895i $$0.558958\pi$$
$$84$$ 0 0
$$85$$ 2.83554 0.307558
$$86$$ 0 0
$$87$$ 1.10179 0.118124
$$88$$ 0 0
$$89$$ −9.64561 −1.02243 −0.511216 0.859452i $$-0.670805\pi$$
−0.511216 + 0.859452i $$0.670805\pi$$
$$90$$ 0 0
$$91$$ −10.2963 −1.07934
$$92$$ 0 0
$$93$$ −0.765715 −0.0794010
$$94$$ 0 0
$$95$$ 1.55081 0.159110
$$96$$ 0 0
$$97$$ 3.55562 0.361018 0.180509 0.983573i $$-0.442225\pi$$
0.180509 + 0.983573i $$0.442225\pi$$
$$98$$ 0 0
$$99$$ −2.80496 −0.281909
$$100$$ 0 0
$$101$$ 13.5381 1.34709 0.673545 0.739146i $$-0.264771\pi$$
0.673545 + 0.739146i $$0.264771\pi$$
$$102$$ 0 0
$$103$$ 6.64570 0.654820 0.327410 0.944882i $$-0.393824\pi$$
0.327410 + 0.944882i $$0.393824\pi$$
$$104$$ 0 0
$$105$$ −1.05555 −0.103012
$$106$$ 0 0
$$107$$ −15.8519 −1.53246 −0.766230 0.642566i $$-0.777870\pi$$
−0.766230 + 0.642566i $$0.777870\pi$$
$$108$$ 0 0
$$109$$ −13.4213 −1.28553 −0.642764 0.766064i $$-0.722212\pi$$
−0.642764 + 0.766064i $$0.722212\pi$$
$$110$$ 0 0
$$111$$ −0.657440 −0.0624015
$$112$$ 0 0
$$113$$ 6.65810 0.626342 0.313171 0.949697i $$-0.398609\pi$$
0.313171 + 0.949697i $$0.398609\pi$$
$$114$$ 0 0
$$115$$ −5.21247 −0.486065
$$116$$ 0 0
$$117$$ −9.91745 −0.916869
$$118$$ 0 0
$$119$$ −5.60109 −0.513451
$$120$$ 0 0
$$121$$ −10.0963 −0.917846
$$122$$ 0 0
$$123$$ −1.01250 −0.0912937
$$124$$ 0 0
$$125$$ −11.7784 −1.05349
$$126$$ 0 0
$$127$$ 6.54782 0.581025 0.290512 0.956871i $$-0.406174\pi$$
0.290512 + 0.956871i $$0.406174\pi$$
$$128$$ 0 0
$$129$$ −0.376331 −0.0331341
$$130$$ 0 0
$$131$$ −14.2838 −1.24798 −0.623990 0.781432i $$-0.714489\pi$$
−0.623990 + 0.781432i $$0.714489\pi$$
$$132$$ 0 0
$$133$$ −3.06334 −0.265625
$$134$$ 0 0
$$135$$ −2.05045 −0.176474
$$136$$ 0 0
$$137$$ −11.9138 −1.01786 −0.508931 0.860808i $$-0.669959\pi$$
−0.508931 + 0.860808i $$0.669959\pi$$
$$138$$ 0 0
$$139$$ −11.1950 −0.949551 −0.474775 0.880107i $$-0.657471\pi$$
−0.474775 + 0.880107i $$0.657471\pi$$
$$140$$ 0 0
$$141$$ −0.755320 −0.0636094
$$142$$ 0 0
$$143$$ 3.19519 0.267195
$$144$$ 0 0
$$145$$ 7.69004 0.638623
$$146$$ 0 0
$$147$$ 0.529714 0.0436901
$$148$$ 0 0
$$149$$ −7.67749 −0.628964 −0.314482 0.949263i $$-0.601831\pi$$
−0.314482 + 0.949263i $$0.601831\pi$$
$$150$$ 0 0
$$151$$ 3.10162 0.252406 0.126203 0.992004i $$-0.459721\pi$$
0.126203 + 0.992004i $$0.459721\pi$$
$$152$$ 0 0
$$153$$ −5.39501 −0.436161
$$154$$ 0 0
$$155$$ −5.34440 −0.429272
$$156$$ 0 0
$$157$$ 14.8994 1.18910 0.594550 0.804059i $$-0.297330\pi$$
0.594550 + 0.804059i $$0.297330\pi$$
$$158$$ 0 0
$$159$$ 0.993511 0.0787905
$$160$$ 0 0
$$161$$ 10.2963 0.811460
$$162$$ 0 0
$$163$$ −18.5481 −1.45280 −0.726400 0.687272i $$-0.758808\pi$$
−0.726400 + 0.687272i $$0.758808\pi$$
$$164$$ 0 0
$$165$$ 0.327565 0.0255009
$$166$$ 0 0
$$167$$ −11.4532 −0.886274 −0.443137 0.896454i $$-0.646135\pi$$
−0.443137 + 0.896454i $$0.646135\pi$$
$$168$$ 0 0
$$169$$ −1.70281 −0.130986
$$170$$ 0 0
$$171$$ −2.95063 −0.225640
$$172$$ 0 0
$$173$$ −7.20956 −0.548133 −0.274066 0.961711i $$-0.588369\pi$$
−0.274066 + 0.961711i $$0.588369\pi$$
$$174$$ 0 0
$$175$$ 7.94933 0.600913
$$176$$ 0 0
$$177$$ 0.0877686 0.00659710
$$178$$ 0 0
$$179$$ 4.90251 0.366431 0.183215 0.983073i $$-0.441349\pi$$
0.183215 + 0.983073i $$0.441349\pi$$
$$180$$ 0 0
$$181$$ −19.1775 −1.42545 −0.712725 0.701444i $$-0.752539\pi$$
−0.712725 + 0.701444i $$0.752539\pi$$
$$182$$ 0 0
$$183$$ 1.18685 0.0877347
$$184$$ 0 0
$$185$$ −4.58868 −0.337366
$$186$$ 0 0
$$187$$ 1.73816 0.127107
$$188$$ 0 0
$$189$$ 4.05028 0.294615
$$190$$ 0 0
$$191$$ 22.8812 1.65563 0.827814 0.561003i $$-0.189584\pi$$
0.827814 + 0.561003i $$0.189584\pi$$
$$192$$ 0 0
$$193$$ −2.11248 −0.152060 −0.0760300 0.997106i $$-0.524224\pi$$
−0.0760300 + 0.997106i $$0.524224\pi$$
$$194$$ 0 0
$$195$$ 1.15817 0.0829380
$$196$$ 0 0
$$197$$ 6.05012 0.431053 0.215526 0.976498i $$-0.430853\pi$$
0.215526 + 0.976498i $$0.430853\pi$$
$$198$$ 0 0
$$199$$ 4.03090 0.285743 0.142872 0.989741i $$-0.454366\pi$$
0.142872 + 0.989741i $$0.454366\pi$$
$$200$$ 0 0
$$201$$ −1.52215 −0.107364
$$202$$ 0 0
$$203$$ −15.1903 −1.06615
$$204$$ 0 0
$$205$$ −7.06683 −0.493569
$$206$$ 0 0
$$207$$ 9.91745 0.689310
$$208$$ 0 0
$$209$$ 0.950631 0.0657565
$$210$$ 0 0
$$211$$ −1.66532 −0.114646 −0.0573228 0.998356i $$-0.518256\pi$$
−0.0573228 + 0.998356i $$0.518256\pi$$
$$212$$ 0 0
$$213$$ 1.14475 0.0784366
$$214$$ 0 0
$$215$$ −2.62665 −0.179136
$$216$$ 0 0
$$217$$ 10.5569 0.716647
$$218$$ 0 0
$$219$$ −0.266624 −0.0180168
$$220$$ 0 0
$$221$$ 6.14558 0.413396
$$222$$ 0 0
$$223$$ −28.9352 −1.93764 −0.968821 0.247761i $$-0.920305\pi$$
−0.968821 + 0.247761i $$0.920305\pi$$
$$224$$ 0 0
$$225$$ 7.65685 0.510457
$$226$$ 0 0
$$227$$ −20.0519 −1.33089 −0.665445 0.746447i $$-0.731758\pi$$
−0.665445 + 0.746447i $$0.731758\pi$$
$$228$$ 0 0
$$229$$ −14.9953 −0.990919 −0.495459 0.868631i $$-0.665000\pi$$
−0.495459 + 0.868631i $$0.665000\pi$$
$$230$$ 0 0
$$231$$ −0.647045 −0.0425724
$$232$$ 0 0
$$233$$ 11.2738 0.738570 0.369285 0.929316i $$-0.379603\pi$$
0.369285 + 0.929316i $$0.379603\pi$$
$$234$$ 0 0
$$235$$ −5.27184 −0.343897
$$236$$ 0 0
$$237$$ −3.27176 −0.212523
$$238$$ 0 0
$$239$$ −19.9049 −1.28754 −0.643770 0.765219i $$-0.722631\pi$$
−0.643770 + 0.765219i $$0.722631\pi$$
$$240$$ 0 0
$$241$$ −15.8163 −1.01882 −0.509408 0.860525i $$-0.670135\pi$$
−0.509408 + 0.860525i $$0.670135\pi$$
$$242$$ 0 0
$$243$$ 5.86808 0.376437
$$244$$ 0 0
$$245$$ 3.69720 0.236205
$$246$$ 0 0
$$247$$ 3.36113 0.213863
$$248$$ 0 0
$$249$$ −0.745595 −0.0472501
$$250$$ 0 0
$$251$$ 10.5087 0.663306 0.331653 0.943401i $$-0.392394\pi$$
0.331653 + 0.943401i $$0.392394\pi$$
$$252$$ 0 0
$$253$$ −3.19519 −0.200880
$$254$$ 0 0
$$255$$ 0.630033 0.0394542
$$256$$ 0 0
$$257$$ −1.70129 −0.106123 −0.0530617 0.998591i $$-0.516898\pi$$
−0.0530617 + 0.998591i $$0.516898\pi$$
$$258$$ 0 0
$$259$$ 9.06409 0.563215
$$260$$ 0 0
$$261$$ −14.6314 −0.905659
$$262$$ 0 0
$$263$$ −13.8359 −0.853157 −0.426578 0.904451i $$-0.640281\pi$$
−0.426578 + 0.904451i $$0.640281\pi$$
$$264$$ 0 0
$$265$$ 6.93432 0.425972
$$266$$ 0 0
$$267$$ −2.14317 −0.131160
$$268$$ 0 0
$$269$$ 14.9655 0.912466 0.456233 0.889860i $$-0.349198\pi$$
0.456233 + 0.889860i $$0.349198\pi$$
$$270$$ 0 0
$$271$$ 4.12684 0.250688 0.125344 0.992113i $$-0.459997\pi$$
0.125344 + 0.992113i $$0.459997\pi$$
$$272$$ 0 0
$$273$$ −2.28774 −0.138461
$$274$$ 0 0
$$275$$ −2.46688 −0.148758
$$276$$ 0 0
$$277$$ 25.7721 1.54849 0.774247 0.632884i $$-0.218129\pi$$
0.774247 + 0.632884i $$0.218129\pi$$
$$278$$ 0 0
$$279$$ 10.1685 0.608769
$$280$$ 0 0
$$281$$ −10.2694 −0.612621 −0.306311 0.951932i $$-0.599095\pi$$
−0.306311 + 0.951932i $$0.599095\pi$$
$$282$$ 0 0
$$283$$ 6.99506 0.415813 0.207907 0.978149i $$-0.433335\pi$$
0.207907 + 0.978149i $$0.433335\pi$$
$$284$$ 0 0
$$285$$ 0.344577 0.0204110
$$286$$ 0 0
$$287$$ 13.9592 0.823987
$$288$$ 0 0
$$289$$ −13.6569 −0.803344
$$290$$ 0 0
$$291$$ 0.790027 0.0463122
$$292$$ 0 0
$$293$$ 31.8434 1.86031 0.930157 0.367162i $$-0.119670\pi$$
0.930157 + 0.367162i $$0.119670\pi$$
$$294$$ 0 0
$$295$$ 0.612591 0.0356664
$$296$$ 0 0
$$297$$ −1.25690 −0.0729330
$$298$$ 0 0
$$299$$ −11.2972 −0.653333
$$300$$ 0 0
$$301$$ 5.18846 0.299058
$$302$$ 0 0
$$303$$ 3.00805 0.172808
$$304$$ 0 0
$$305$$ 8.28378 0.474328
$$306$$ 0 0
$$307$$ −6.45813 −0.368585 −0.184292 0.982871i $$-0.558999\pi$$
−0.184292 + 0.982871i $$0.558999\pi$$
$$308$$ 0 0
$$309$$ 1.47662 0.0840018
$$310$$ 0 0
$$311$$ 28.1973 1.59892 0.799462 0.600716i $$-0.205118\pi$$
0.799462 + 0.600716i $$0.205118\pi$$
$$312$$ 0 0
$$313$$ 17.1212 0.967746 0.483873 0.875138i $$-0.339230\pi$$
0.483873 + 0.875138i $$0.339230\pi$$
$$314$$ 0 0
$$315$$ 14.0174 0.789793
$$316$$ 0 0
$$317$$ 29.5841 1.66161 0.830804 0.556564i $$-0.187881\pi$$
0.830804 + 0.556564i $$0.187881\pi$$
$$318$$ 0 0
$$319$$ 4.71392 0.263929
$$320$$ 0 0
$$321$$ −3.52215 −0.196587
$$322$$ 0 0
$$323$$ 1.82843 0.101736
$$324$$ 0 0
$$325$$ −8.72209 −0.483815
$$326$$ 0 0
$$327$$ −2.98210 −0.164910
$$328$$ 0 0
$$329$$ 10.4135 0.574117
$$330$$ 0 0
$$331$$ −20.5519 −1.12964 −0.564818 0.825215i $$-0.691054\pi$$
−0.564818 + 0.825215i $$0.691054\pi$$
$$332$$ 0 0
$$333$$ 8.73060 0.478434
$$334$$ 0 0
$$335$$ −10.6240 −0.580454
$$336$$ 0 0
$$337$$ −19.1832 −1.04497 −0.522487 0.852647i $$-0.674996\pi$$
−0.522487 + 0.852647i $$0.674996\pi$$
$$338$$ 0 0
$$339$$ 1.47937 0.0803485
$$340$$ 0 0
$$341$$ −3.27606 −0.177409
$$342$$ 0 0
$$343$$ 14.1402 0.763501
$$344$$ 0 0
$$345$$ −1.15817 −0.0623536
$$346$$ 0 0
$$347$$ −10.5507 −0.566390 −0.283195 0.959062i $$-0.591394\pi$$
−0.283195 + 0.959062i $$0.591394\pi$$
$$348$$ 0 0
$$349$$ −29.4693 −1.57745 −0.788727 0.614744i $$-0.789259\pi$$
−0.788727 + 0.614744i $$0.789259\pi$$
$$350$$ 0 0
$$351$$ −4.44401 −0.237204
$$352$$ 0 0
$$353$$ −3.98903 −0.212315 −0.106157 0.994349i $$-0.533855\pi$$
−0.106157 + 0.994349i $$0.533855\pi$$
$$354$$ 0 0
$$355$$ 7.98988 0.424059
$$356$$ 0 0
$$357$$ −1.24451 −0.0658667
$$358$$ 0 0
$$359$$ 14.9377 0.788380 0.394190 0.919029i $$-0.371025\pi$$
0.394190 + 0.919029i $$0.371025\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −2.24331 −0.117743
$$364$$ 0 0
$$365$$ −1.86093 −0.0974057
$$366$$ 0 0
$$367$$ 19.1009 0.997059 0.498529 0.866873i $$-0.333874\pi$$
0.498529 + 0.866873i $$0.333874\pi$$
$$368$$ 0 0
$$369$$ 13.4456 0.699952
$$370$$ 0 0
$$371$$ −13.6975 −0.711137
$$372$$ 0 0
$$373$$ −0.461357 −0.0238882 −0.0119441 0.999929i $$-0.503802\pi$$
−0.0119441 + 0.999929i $$0.503802\pi$$
$$374$$ 0 0
$$375$$ −2.61706 −0.135144
$$376$$ 0 0
$$377$$ 16.6669 0.858390
$$378$$ 0 0
$$379$$ 29.9622 1.53906 0.769528 0.638613i $$-0.220492\pi$$
0.769528 + 0.638613i $$0.220492\pi$$
$$380$$ 0 0
$$381$$ 1.45487 0.0745352
$$382$$ 0 0
$$383$$ −16.9077 −0.863944 −0.431972 0.901887i $$-0.642182\pi$$
−0.431972 + 0.901887i $$0.642182\pi$$
$$384$$ 0 0
$$385$$ −4.51612 −0.230163
$$386$$ 0 0
$$387$$ 4.99756 0.254040
$$388$$ 0 0
$$389$$ −7.92418 −0.401772 −0.200886 0.979615i $$-0.564382\pi$$
−0.200886 + 0.979615i $$0.564382\pi$$
$$390$$ 0 0
$$391$$ −6.14558 −0.310795
$$392$$ 0 0
$$393$$ −3.17373 −0.160094
$$394$$ 0 0
$$395$$ −22.8356 −1.14898
$$396$$ 0 0
$$397$$ 13.6718 0.686170 0.343085 0.939304i $$-0.388528\pi$$
0.343085 + 0.939304i $$0.388528\pi$$
$$398$$ 0 0
$$399$$ −0.680647 −0.0340750
$$400$$ 0 0
$$401$$ 12.0107 0.599785 0.299893 0.953973i $$-0.403049\pi$$
0.299893 + 0.953973i $$0.403049\pi$$
$$402$$ 0 0
$$403$$ −11.5831 −0.576996
$$404$$ 0 0
$$405$$ 13.2720 0.659492
$$406$$ 0 0
$$407$$ −2.81281 −0.139426
$$408$$ 0 0
$$409$$ 9.47063 0.468292 0.234146 0.972201i $$-0.424771\pi$$
0.234146 + 0.972201i $$0.424771\pi$$
$$410$$ 0 0
$$411$$ −2.64713 −0.130574
$$412$$ 0 0
$$413$$ −1.21006 −0.0595432
$$414$$ 0 0
$$415$$ −5.20396 −0.255453
$$416$$ 0 0
$$417$$ −2.48744 −0.121811
$$418$$ 0 0
$$419$$ −28.0843 −1.37201 −0.686004 0.727598i $$-0.740637\pi$$
−0.686004 + 0.727598i $$0.740637\pi$$
$$420$$ 0 0
$$421$$ 9.83238 0.479201 0.239601 0.970872i $$-0.422984\pi$$
0.239601 + 0.970872i $$0.422984\pi$$
$$422$$ 0 0
$$423$$ 10.0304 0.487695
$$424$$ 0 0
$$425$$ −4.74475 −0.230154
$$426$$ 0 0
$$427$$ −16.3631 −0.791865
$$428$$ 0 0
$$429$$ 0.709944 0.0342764
$$430$$ 0 0
$$431$$ −8.94284 −0.430761 −0.215381 0.976530i $$-0.569099\pi$$
−0.215381 + 0.976530i $$0.569099\pi$$
$$432$$ 0 0
$$433$$ −29.5274 −1.41900 −0.709499 0.704707i $$-0.751079\pi$$
−0.709499 + 0.704707i $$0.751079\pi$$
$$434$$ 0 0
$$435$$ 1.70866 0.0819240
$$436$$ 0 0
$$437$$ −3.36113 −0.160785
$$438$$ 0 0
$$439$$ −20.5113 −0.978953 −0.489477 0.872016i $$-0.662812\pi$$
−0.489477 + 0.872016i $$0.662812\pi$$
$$440$$ 0 0
$$441$$ −7.03444 −0.334973
$$442$$ 0 0
$$443$$ 13.3188 0.632794 0.316397 0.948627i $$-0.397527\pi$$
0.316397 + 0.948627i $$0.397527\pi$$
$$444$$ 0 0
$$445$$ −14.9585 −0.709101
$$446$$ 0 0
$$447$$ −1.70587 −0.0806850
$$448$$ 0 0
$$449$$ 10.5138 0.496177 0.248089 0.968737i $$-0.420198\pi$$
0.248089 + 0.968737i $$0.420198\pi$$
$$450$$ 0 0
$$451$$ −4.33190 −0.203981
$$452$$ 0 0
$$453$$ 0.689153 0.0323792
$$454$$ 0 0
$$455$$ −15.9676 −0.748571
$$456$$ 0 0
$$457$$ 26.5831 1.24351 0.621753 0.783214i $$-0.286421\pi$$
0.621753 + 0.783214i $$0.286421\pi$$
$$458$$ 0 0
$$459$$ −2.41751 −0.112840
$$460$$ 0 0
$$461$$ 19.2417 0.896175 0.448087 0.893990i $$-0.352105\pi$$
0.448087 + 0.893990i $$0.352105\pi$$
$$462$$ 0 0
$$463$$ 2.24701 0.104427 0.0522137 0.998636i $$-0.483372\pi$$
0.0522137 + 0.998636i $$0.483372\pi$$
$$464$$ 0 0
$$465$$ −1.18748 −0.0550680
$$466$$ 0 0
$$467$$ −27.5682 −1.27570 −0.637852 0.770159i $$-0.720177\pi$$
−0.637852 + 0.770159i $$0.720177\pi$$
$$468$$ 0 0
$$469$$ 20.9858 0.969036
$$470$$ 0 0
$$471$$ 3.31051 0.152540
$$472$$ 0 0
$$473$$ −1.61011 −0.0740329
$$474$$ 0 0
$$475$$ −2.59499 −0.119066
$$476$$ 0 0
$$477$$ −13.1935 −0.604089
$$478$$ 0 0
$$479$$ 10.5447 0.481802 0.240901 0.970550i $$-0.422557\pi$$
0.240901 + 0.970550i $$0.422557\pi$$
$$480$$ 0 0
$$481$$ −9.94521 −0.453463
$$482$$ 0 0
$$483$$ 2.28774 0.104096
$$484$$ 0 0
$$485$$ 5.51409 0.250382
$$486$$ 0 0
$$487$$ −39.7294 −1.80031 −0.900154 0.435571i $$-0.856546\pi$$
−0.900154 + 0.435571i $$0.856546\pi$$
$$488$$ 0 0
$$489$$ −4.12123 −0.186369
$$490$$ 0 0
$$491$$ −31.0150 −1.39969 −0.699844 0.714296i $$-0.746747\pi$$
−0.699844 + 0.714296i $$0.746747\pi$$
$$492$$ 0 0
$$493$$ 9.06667 0.408342
$$494$$ 0 0
$$495$$ −4.34996 −0.195516
$$496$$ 0 0
$$497$$ −15.7825 −0.707943
$$498$$ 0 0
$$499$$ 4.90620 0.219632 0.109816 0.993952i $$-0.464974\pi$$
0.109816 + 0.993952i $$0.464974\pi$$
$$500$$ 0 0
$$501$$ −2.54480 −0.113693
$$502$$ 0 0
$$503$$ −14.6847 −0.654759 −0.327380 0.944893i $$-0.606166\pi$$
−0.327380 + 0.944893i $$0.606166\pi$$
$$504$$ 0 0
$$505$$ 20.9950 0.934265
$$506$$ 0 0
$$507$$ −0.378351 −0.0168031
$$508$$ 0 0
$$509$$ −20.3988 −0.904159 −0.452080 0.891978i $$-0.649318\pi$$
−0.452080 + 0.891978i $$0.649318\pi$$
$$510$$ 0 0
$$511$$ 3.67593 0.162614
$$512$$ 0 0
$$513$$ −1.32218 −0.0583756
$$514$$ 0 0
$$515$$ 10.3062 0.454146
$$516$$ 0 0
$$517$$ −3.23158 −0.142125
$$518$$ 0 0
$$519$$ −1.60190 −0.0703157
$$520$$ 0 0
$$521$$ 22.4356 0.982923 0.491462 0.870899i $$-0.336463\pi$$
0.491462 + 0.870899i $$0.336463\pi$$
$$522$$ 0 0
$$523$$ 9.71531 0.424821 0.212410 0.977181i $$-0.431869\pi$$
0.212410 + 0.977181i $$0.431869\pi$$
$$524$$ 0 0
$$525$$ 1.76627 0.0770865
$$526$$ 0 0
$$527$$ −6.30112 −0.274481
$$528$$ 0 0
$$529$$ −11.7028 −0.508818
$$530$$ 0 0
$$531$$ −1.16554 −0.0505801
$$532$$ 0 0
$$533$$ −15.3162 −0.663419
$$534$$ 0 0
$$535$$ −24.5833 −1.06283
$$536$$ 0 0
$$537$$ 1.08930 0.0470066
$$538$$ 0 0
$$539$$ 2.26635 0.0976185
$$540$$ 0 0
$$541$$ −14.7230 −0.632991 −0.316496 0.948594i $$-0.602506\pi$$
−0.316496 + 0.948594i $$0.602506\pi$$
$$542$$ 0 0
$$543$$ −4.26107 −0.182860
$$544$$ 0 0
$$545$$ −20.8139 −0.891569
$$546$$ 0 0
$$547$$ 10.9906 0.469922 0.234961 0.972005i $$-0.424504\pi$$
0.234961 + 0.972005i $$0.424504\pi$$
$$548$$ 0 0
$$549$$ −15.7610 −0.672665
$$550$$ 0 0
$$551$$ 4.95873 0.211249
$$552$$ 0 0
$$553$$ 45.1075 1.91817
$$554$$ 0 0
$$555$$ −1.01956 −0.0432781
$$556$$ 0 0
$$557$$ −25.9252 −1.09849 −0.549243 0.835663i $$-0.685084\pi$$
−0.549243 + 0.835663i $$0.685084\pi$$
$$558$$ 0 0
$$559$$ −5.69283 −0.240781
$$560$$ 0 0
$$561$$ 0.386204 0.0163055
$$562$$ 0 0
$$563$$ 21.0775 0.888310 0.444155 0.895950i $$-0.353504\pi$$
0.444155 + 0.895950i $$0.353504\pi$$
$$564$$ 0 0
$$565$$ 10.3255 0.434395
$$566$$ 0 0
$$567$$ −26.2164 −1.10099
$$568$$ 0 0
$$569$$ −11.2369 −0.471076 −0.235538 0.971865i $$-0.575685\pi$$
−0.235538 + 0.971865i $$0.575685\pi$$
$$570$$ 0 0
$$571$$ 42.5037 1.77872 0.889362 0.457204i $$-0.151149\pi$$
0.889362 + 0.457204i $$0.151149\pi$$
$$572$$ 0 0
$$573$$ 5.08401 0.212388
$$574$$ 0 0
$$575$$ 8.72209 0.363736
$$576$$ 0 0
$$577$$ −26.6503 −1.10947 −0.554733 0.832029i $$-0.687179\pi$$
−0.554733 + 0.832029i $$0.687179\pi$$
$$578$$ 0 0
$$579$$ −0.469376 −0.0195066
$$580$$ 0 0
$$581$$ 10.2795 0.426464
$$582$$ 0 0
$$583$$ 4.25067 0.176045
$$584$$ 0 0
$$585$$ −15.3801 −0.635888
$$586$$ 0 0
$$587$$ −33.2195 −1.37111 −0.685557 0.728019i $$-0.740441\pi$$
−0.685557 + 0.728019i $$0.740441\pi$$
$$588$$ 0 0
$$589$$ −3.44620 −0.141998
$$590$$ 0 0
$$591$$ 1.34428 0.0552964
$$592$$ 0 0
$$593$$ −25.4149 −1.04367 −0.521833 0.853047i $$-0.674752\pi$$
−0.521833 + 0.853047i $$0.674752\pi$$
$$594$$ 0 0
$$595$$ −8.68623 −0.356101
$$596$$ 0 0
$$597$$ 0.895632 0.0366558
$$598$$ 0 0
$$599$$ 46.6431 1.90578 0.952892 0.303310i $$-0.0980918\pi$$
0.952892 + 0.303310i $$0.0980918\pi$$
$$600$$ 0 0
$$601$$ 47.4644 1.93611 0.968056 0.250734i $$-0.0806718\pi$$
0.968056 + 0.250734i $$0.0806718\pi$$
$$602$$ 0 0
$$603$$ 20.2137 0.823166
$$604$$ 0 0
$$605$$ −15.6574 −0.636566
$$606$$ 0 0
$$607$$ −24.4897 −0.994006 −0.497003 0.867749i $$-0.665566\pi$$
−0.497003 + 0.867749i $$0.665566\pi$$
$$608$$ 0 0
$$609$$ −3.37514 −0.136768
$$610$$ 0 0
$$611$$ −11.4259 −0.462241
$$612$$ 0 0
$$613$$ −18.9950 −0.767201 −0.383600 0.923499i $$-0.625316\pi$$
−0.383600 + 0.923499i $$0.625316\pi$$
$$614$$ 0 0
$$615$$ −1.57019 −0.0633162
$$616$$ 0 0
$$617$$ −0.219907 −0.00885311 −0.00442656 0.999990i $$-0.501409\pi$$
−0.00442656 + 0.999990i $$0.501409\pi$$
$$618$$ 0 0
$$619$$ 15.9136 0.639623 0.319811 0.947481i $$-0.396380\pi$$
0.319811 + 0.947481i $$0.396380\pi$$
$$620$$ 0 0
$$621$$ 4.44401 0.178332
$$622$$ 0 0
$$623$$ 29.5478 1.18381
$$624$$ 0 0
$$625$$ −5.29109 −0.211644
$$626$$ 0 0
$$627$$ 0.211222 0.00843539
$$628$$ 0 0
$$629$$ −5.41012 −0.215716
$$630$$ 0 0
$$631$$ −27.6207 −1.09956 −0.549781 0.835309i $$-0.685289\pi$$
−0.549781 + 0.835309i $$0.685289\pi$$
$$632$$ 0 0
$$633$$ −0.370021 −0.0147070
$$634$$ 0 0
$$635$$ 10.1544 0.402966
$$636$$ 0 0
$$637$$ 8.01308 0.317490
$$638$$ 0 0
$$639$$ −15.2019 −0.601376
$$640$$ 0 0
$$641$$ −4.17066 −0.164731 −0.0823656 0.996602i $$-0.526248\pi$$
−0.0823656 + 0.996602i $$0.526248\pi$$
$$642$$ 0 0
$$643$$ 20.8643 0.822806 0.411403 0.911453i $$-0.365039\pi$$
0.411403 + 0.911453i $$0.365039\pi$$
$$644$$ 0 0
$$645$$ −0.583619 −0.0229800
$$646$$ 0 0
$$647$$ 31.7924 1.24989 0.624943 0.780670i $$-0.285122\pi$$
0.624943 + 0.780670i $$0.285122\pi$$
$$648$$ 0 0
$$649$$ 0.375512 0.0147401
$$650$$ 0 0
$$651$$ 2.34564 0.0919331
$$652$$ 0 0
$$653$$ 27.2099 1.06481 0.532403 0.846491i $$-0.321289\pi$$
0.532403 + 0.846491i $$0.321289\pi$$
$$654$$ 0 0
$$655$$ −22.1514 −0.865528
$$656$$ 0 0
$$657$$ 3.54068 0.138135
$$658$$ 0 0
$$659$$ 32.8090 1.27806 0.639030 0.769182i $$-0.279336\pi$$
0.639030 + 0.769182i $$0.279336\pi$$
$$660$$ 0 0
$$661$$ −12.5824 −0.489398 −0.244699 0.969599i $$-0.578689\pi$$
−0.244699 + 0.969599i $$0.578689\pi$$
$$662$$ 0 0
$$663$$ 1.36549 0.0530314
$$664$$ 0 0
$$665$$ −4.75066 −0.184223
$$666$$ 0 0
$$667$$ −16.6669 −0.645345
$$668$$ 0 0
$$669$$ −6.42915 −0.248565
$$670$$ 0 0
$$671$$ 5.07787 0.196029
$$672$$ 0 0
$$673$$ −2.58049 −0.0994704 −0.0497352 0.998762i $$-0.515838\pi$$
−0.0497352 + 0.998762i $$0.515838\pi$$
$$674$$ 0 0
$$675$$ 3.43104 0.132061
$$676$$ 0 0
$$677$$ −5.86443 −0.225388 −0.112694 0.993630i $$-0.535948\pi$$
−0.112694 + 0.993630i $$0.535948\pi$$
$$678$$ 0 0
$$679$$ −10.8921 −0.417999
$$680$$ 0 0
$$681$$ −4.45535 −0.170729
$$682$$ 0 0
$$683$$ 13.3162 0.509531 0.254765 0.967003i $$-0.418002\pi$$
0.254765 + 0.967003i $$0.418002\pi$$
$$684$$ 0 0
$$685$$ −18.4760 −0.705931
$$686$$ 0 0
$$687$$ −3.33183 −0.127117
$$688$$ 0 0
$$689$$ 15.0290 0.572560
$$690$$ 0 0
$$691$$ 34.9751 1.33051 0.665257 0.746614i $$-0.268322\pi$$
0.665257 + 0.746614i $$0.268322\pi$$
$$692$$ 0 0
$$693$$ 8.59255 0.326404
$$694$$ 0 0
$$695$$ −17.3614 −0.658555
$$696$$ 0 0
$$697$$ −8.33190 −0.315593
$$698$$ 0 0
$$699$$ 2.50494 0.0947454
$$700$$ 0 0
$$701$$ −4.91851 −0.185769 −0.0928847 0.995677i $$-0.529609\pi$$
−0.0928847 + 0.995677i $$0.529609\pi$$
$$702$$ 0 0
$$703$$ −2.95889 −0.111597
$$704$$ 0 0
$$705$$ −1.17136 −0.0441159
$$706$$ 0 0
$$707$$ −41.4717 −1.55971
$$708$$ 0 0
$$709$$ −27.4930 −1.03252 −0.516261 0.856431i $$-0.672676\pi$$
−0.516261 + 0.856431i $$0.672676\pi$$
$$710$$ 0 0
$$711$$ 43.4479 1.62942
$$712$$ 0 0
$$713$$ 11.5831 0.433791
$$714$$ 0 0
$$715$$ 4.95514 0.185312
$$716$$ 0 0
$$717$$ −4.42270 −0.165169
$$718$$ 0 0
$$719$$ 10.8746 0.405553 0.202777 0.979225i $$-0.435004\pi$$
0.202777 + 0.979225i $$0.435004\pi$$
$$720$$ 0 0
$$721$$ −20.3580 −0.758172
$$722$$ 0 0
$$723$$ −3.51424 −0.130696
$$724$$ 0 0
$$725$$ −12.8678 −0.477899
$$726$$ 0 0
$$727$$ −1.13818 −0.0422127 −0.0211063 0.999777i $$-0.506719\pi$$
−0.0211063 + 0.999777i $$0.506719\pi$$
$$728$$ 0 0
$$729$$ −24.3705 −0.902612
$$730$$ 0 0
$$731$$ −3.09686 −0.114541
$$732$$ 0 0
$$733$$ 27.9950 1.03402 0.517010 0.855980i $$-0.327045\pi$$
0.517010 + 0.855980i $$0.327045\pi$$
$$734$$ 0 0
$$735$$ 0.821486 0.0303010
$$736$$ 0 0
$$737$$ −6.51243 −0.239889
$$738$$ 0 0
$$739$$ 44.2375 1.62730 0.813652 0.581352i $$-0.197476\pi$$
0.813652 + 0.581352i $$0.197476\pi$$
$$740$$ 0 0
$$741$$ 0.746814 0.0274349
$$742$$ 0 0
$$743$$ 49.4810 1.81528 0.907640 0.419749i $$-0.137882\pi$$
0.907640 + 0.419749i $$0.137882\pi$$
$$744$$ 0 0
$$745$$ −11.9063 −0.436214
$$746$$ 0 0
$$747$$ 9.90126 0.362268
$$748$$ 0 0
$$749$$ 48.5597 1.77433
$$750$$ 0 0
$$751$$ 25.9833 0.948145 0.474073 0.880486i $$-0.342783\pi$$
0.474073 + 0.880486i $$0.342783\pi$$
$$752$$ 0 0
$$753$$ 2.33495 0.0850904
$$754$$ 0 0
$$755$$ 4.81002 0.175055
$$756$$ 0 0
$$757$$ −43.1605 −1.56869 −0.784347 0.620322i $$-0.787002\pi$$
−0.784347 + 0.620322i $$0.787002\pi$$
$$758$$ 0 0
$$759$$ −0.709944 −0.0257693
$$760$$ 0 0
$$761$$ 4.16498 0.150981 0.0754903 0.997147i $$-0.475948\pi$$
0.0754903 + 0.997147i $$0.475948\pi$$
$$762$$ 0 0
$$763$$ 41.1140 1.48843
$$764$$ 0 0
$$765$$ −8.36664 −0.302497
$$766$$ 0 0
$$767$$ 1.32769 0.0479402
$$768$$ 0 0
$$769$$ 5.24441 0.189118 0.0945591 0.995519i $$-0.469856\pi$$
0.0945591 + 0.995519i $$0.469856\pi$$
$$770$$ 0 0
$$771$$ −0.378011 −0.0136137
$$772$$ 0 0
$$773$$ −3.91186 −0.140700 −0.0703499 0.997522i $$-0.522412\pi$$
−0.0703499 + 0.997522i $$0.522412\pi$$
$$774$$ 0 0
$$775$$ 8.94284 0.321236
$$776$$ 0 0
$$777$$ 2.01396 0.0722505
$$778$$ 0 0
$$779$$ −4.55687 −0.163267
$$780$$ 0 0
$$781$$ 4.89772 0.175254
$$782$$ 0 0
$$783$$ −6.55632 −0.234304
$$784$$ 0 0
$$785$$ 23.1061 0.824692
$$786$$ 0 0
$$787$$ 16.8869 0.601952 0.300976 0.953632i $$-0.402688\pi$$
0.300976 + 0.953632i $$0.402688\pi$$
$$788$$ 0 0
$$789$$ −3.07421 −0.109445
$$790$$ 0 0
$$791$$ −20.3960 −0.725199
$$792$$ 0 0
$$793$$ 17.9537 0.637556
$$794$$ 0 0
$$795$$ 1.54075 0.0546447
$$796$$ 0 0
$$797$$ −25.3722 −0.898729 −0.449365 0.893348i $$-0.648350\pi$$
−0.449365 + 0.893348i $$0.648350\pi$$
$$798$$ 0 0
$$799$$ −6.21557 −0.219891
$$800$$ 0 0
$$801$$ 28.4606 1.00561
$$802$$ 0 0
$$803$$ −1.14073 −0.0402556
$$804$$ 0 0
$$805$$ 15.9676 0.562783
$$806$$ 0 0
$$807$$ 3.32522 0.117053
$$808$$ 0 0
$$809$$ −31.5640 −1.10973 −0.554866 0.831940i $$-0.687230\pi$$
−0.554866 + 0.831940i $$0.687230\pi$$
$$810$$ 0 0
$$811$$ −44.1203 −1.54927 −0.774636 0.632407i $$-0.782067\pi$$
−0.774636 + 0.632407i $$0.782067\pi$$
$$812$$ 0 0
$$813$$ 0.916949 0.0321588
$$814$$ 0 0
$$815$$ −28.7646 −1.00758
$$816$$ 0 0
$$817$$ −1.69373 −0.0592560
$$818$$ 0 0
$$819$$ 30.3805 1.06158
$$820$$ 0 0
$$821$$ −3.81015 −0.132975 −0.0664876 0.997787i $$-0.521179\pi$$
−0.0664876 + 0.997787i $$0.521179\pi$$
$$822$$ 0 0
$$823$$ 34.3301 1.19667 0.598336 0.801245i $$-0.295829\pi$$
0.598336 + 0.801245i $$0.295829\pi$$
$$824$$ 0 0
$$825$$ −0.548119 −0.0190830
$$826$$ 0 0
$$827$$ 30.7484 1.06923 0.534613 0.845097i $$-0.320457\pi$$
0.534613 + 0.845097i $$0.320457\pi$$
$$828$$ 0 0
$$829$$ 10.1917 0.353971 0.176986 0.984213i $$-0.443365\pi$$
0.176986 + 0.984213i $$0.443365\pi$$
$$830$$ 0 0
$$831$$ 5.72633 0.198644
$$832$$ 0 0
$$833$$ 4.35905 0.151032
$$834$$ 0 0
$$835$$ −17.7617 −0.614669
$$836$$ 0 0
$$837$$ 4.55649 0.157495
$$838$$ 0 0
$$839$$ −7.81123 −0.269674 −0.134837 0.990868i $$-0.543051\pi$$
−0.134837 + 0.990868i $$0.543051\pi$$
$$840$$ 0 0
$$841$$ −4.41104 −0.152105
$$842$$ 0 0
$$843$$ −2.28177 −0.0785884
$$844$$ 0 0
$$845$$ −2.64074 −0.0908442
$$846$$ 0 0
$$847$$ 30.9284 1.06271
$$848$$ 0 0
$$849$$ 1.55424 0.0533415
$$850$$ 0 0
$$851$$ 9.94521 0.340918
$$852$$ 0 0
$$853$$ 46.9706 1.60824 0.804122 0.594465i $$-0.202636\pi$$
0.804122 + 0.594465i $$0.202636\pi$$
$$854$$ 0 0
$$855$$ −4.57587 −0.156491
$$856$$ 0 0
$$857$$ 2.24191 0.0765821 0.0382911 0.999267i $$-0.487809\pi$$
0.0382911 + 0.999267i $$0.487809\pi$$
$$858$$ 0 0
$$859$$ 49.9456 1.70412 0.852061 0.523442i $$-0.175352\pi$$
0.852061 + 0.523442i $$0.175352\pi$$
$$860$$ 0 0
$$861$$ 3.10162 0.105703
$$862$$ 0 0
$$863$$ −10.4322 −0.355115 −0.177557 0.984110i $$-0.556820\pi$$
−0.177557 + 0.984110i $$0.556820\pi$$
$$864$$ 0 0
$$865$$ −11.1807 −0.380154
$$866$$ 0 0
$$867$$ −3.03444 −0.103055
$$868$$ 0 0
$$869$$ −13.9980 −0.474849
$$870$$ 0 0
$$871$$ −23.0259 −0.780203
$$872$$ 0 0
$$873$$ −10.4913 −0.355077
$$874$$ 0 0
$$875$$ 36.0812 1.21977
$$876$$ 0 0
$$877$$ −23.6385 −0.798215 −0.399107 0.916904i $$-0.630680\pi$$
−0.399107 + 0.916904i $$0.630680\pi$$
$$878$$ 0 0
$$879$$ 7.07534 0.238645
$$880$$ 0 0
$$881$$ −29.2713 −0.986175 −0.493087 0.869980i $$-0.664132\pi$$
−0.493087 + 0.869980i $$0.664132\pi$$
$$882$$ 0 0
$$883$$ 42.1875 1.41972 0.709862 0.704341i $$-0.248757\pi$$
0.709862 + 0.704341i $$0.248757\pi$$
$$884$$ 0 0
$$885$$ 0.136112 0.00457537
$$886$$ 0 0
$$887$$ 1.32254 0.0444064 0.0222032 0.999753i $$-0.492932\pi$$
0.0222032 + 0.999753i $$0.492932\pi$$
$$888$$ 0 0
$$889$$ −20.0582 −0.672730
$$890$$ 0 0
$$891$$ 8.13561 0.272553
$$892$$ 0 0
$$893$$ −3.39941 −0.113757
$$894$$ 0 0
$$895$$ 7.60286 0.254136
$$896$$ 0 0
$$897$$ −2.51014 −0.0838110
$$898$$ 0 0
$$899$$ −17.0887 −0.569941
$$900$$ 0 0
$$901$$ 8.17567 0.272371
$$902$$ 0 0
$$903$$ 1.15283 0.0383638
$$904$$ 0 0
$$905$$ −29.7406 −0.988611
$$906$$ 0 0
$$907$$ 3.58771 0.119128 0.0595639 0.998224i $$-0.481029\pi$$
0.0595639 + 0.998224i $$0.481029\pi$$
$$908$$ 0 0
$$909$$ −39.9459 −1.32492
$$910$$ 0 0
$$911$$ 14.6828 0.486464 0.243232 0.969968i $$-0.421792\pi$$
0.243232 + 0.969968i $$0.421792\pi$$
$$912$$ 0 0
$$913$$ −3.18998 −0.105573
$$914$$ 0 0
$$915$$ 1.84058 0.0608478
$$916$$ 0 0
$$917$$ 43.7561 1.44495
$$918$$ 0 0
$$919$$ 38.2168 1.26066 0.630328 0.776329i $$-0.282920\pi$$
0.630328 + 0.776329i $$0.282920\pi$$
$$920$$ 0 0
$$921$$ −1.43494 −0.0472829
$$922$$ 0 0
$$923$$ 17.3168 0.569988
$$924$$ 0 0
$$925$$ 7.67829 0.252461
$$926$$ 0 0
$$927$$ −19.6090 −0.644044
$$928$$ 0 0
$$929$$ 36.8919 1.21038 0.605192 0.796080i $$-0.293096\pi$$
0.605192 + 0.796080i $$0.293096\pi$$
$$930$$ 0 0
$$931$$ 2.38404 0.0781339
$$932$$ 0 0
$$933$$ 6.26521 0.205114
$$934$$ 0 0
$$935$$ 2.69555 0.0881541
$$936$$ 0 0
$$937$$ 33.7347 1.10206 0.551032 0.834484i $$-0.314234\pi$$
0.551032 + 0.834484i $$0.314234\pi$$
$$938$$ 0 0
$$939$$ 3.80418 0.124145
$$940$$ 0 0
$$941$$ −3.91029 −0.127472 −0.0637360 0.997967i $$-0.520302\pi$$
−0.0637360 + 0.997967i $$0.520302\pi$$
$$942$$ 0 0
$$943$$ 15.3162 0.498765
$$944$$ 0 0
$$945$$ 6.28122 0.204328
$$946$$ 0 0
$$947$$ 46.8518 1.52248 0.761240 0.648470i $$-0.224591\pi$$
0.761240 + 0.648470i $$0.224591\pi$$
$$948$$ 0 0
$$949$$ −4.03327 −0.130925
$$950$$ 0 0
$$951$$ 6.57333 0.213155
$$952$$ 0 0
$$953$$ 26.5470 0.859941 0.429971 0.902843i $$-0.358524\pi$$
0.429971 + 0.902843i $$0.358524\pi$$
$$954$$ 0 0
$$955$$ 35.4844 1.14825
$$956$$ 0 0
$$957$$ 1.04739 0.0338574
$$958$$ 0 0
$$959$$ 36.4959 1.17851
$$960$$ 0 0
$$961$$ −19.1237 −0.616895
$$962$$ 0 0
$$963$$ 46.7731 1.50724
$$964$$ 0 0
$$965$$ −3.27606 −0.105460
$$966$$ 0 0
$$967$$ −6.80878 −0.218956 −0.109478 0.993989i $$-0.534918\pi$$
−0.109478 + 0.993989i $$0.534918\pi$$
$$968$$ 0 0
$$969$$ 0.406261 0.0130510
$$970$$ 0 0
$$971$$ −48.4549 −1.55499 −0.777496 0.628887i $$-0.783511\pi$$
−0.777496 + 0.628887i $$0.783511\pi$$
$$972$$ 0 0
$$973$$ 34.2942 1.09942
$$974$$ 0 0
$$975$$ −1.93797 −0.0620648
$$976$$ 0 0
$$977$$ 46.0137 1.47211 0.736055 0.676922i $$-0.236686\pi$$
0.736055 + 0.676922i $$0.236686\pi$$
$$978$$ 0 0
$$979$$ −9.16941 −0.293056
$$980$$ 0 0
$$981$$ 39.6013 1.26437
$$982$$ 0 0
$$983$$ −27.8031 −0.886782 −0.443391 0.896328i $$-0.646225\pi$$
−0.443391 + 0.896328i $$0.646225\pi$$
$$984$$ 0 0
$$985$$ 9.38258 0.298954
$$986$$ 0 0
$$987$$ 2.31380 0.0736490
$$988$$ 0 0
$$989$$ 5.69283 0.181022
$$990$$ 0 0
$$991$$ −5.16951 −0.164215 −0.0821074 0.996623i $$-0.526165\pi$$
−0.0821074 + 0.996623i $$0.526165\pi$$
$$992$$ 0 0
$$993$$ −4.56646 −0.144912
$$994$$ 0 0
$$995$$ 6.25116 0.198175
$$996$$ 0 0
$$997$$ −18.7227 −0.592953 −0.296476 0.955040i $$-0.595812\pi$$
−0.296476 + 0.955040i $$0.595812\pi$$
$$998$$ 0 0
$$999$$ 3.91218 0.123776
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 4864.2.a.bm.1.6 8
4.3 odd 2 4864.2.a.br.1.4 8
8.3 odd 2 inner 4864.2.a.bm.1.5 8
8.5 even 2 4864.2.a.br.1.3 8
16.3 odd 4 2432.2.c.i.1217.7 16
16.5 even 4 2432.2.c.i.1217.8 yes 16
16.11 odd 4 2432.2.c.i.1217.10 yes 16
16.13 even 4 2432.2.c.i.1217.9 yes 16

By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.i.1217.7 16 16.3 odd 4
2432.2.c.i.1217.8 yes 16 16.5 even 4
2432.2.c.i.1217.9 yes 16 16.13 even 4
2432.2.c.i.1217.10 yes 16 16.11 odd 4
4864.2.a.bm.1.5 8 8.3 odd 2 inner
4864.2.a.bm.1.6 8 1.1 even 1 trivial
4864.2.a.br.1.3 8 8.5 even 2
4864.2.a.br.1.4 8 4.3 odd 2