# Properties

 Label 1216.2.c.i.609.3 Level $1216$ Weight $2$ Character 1216.609 Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2702336256.1 Defining polynomial: $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 609.3 Root $$1.52274 - 1.63746i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.609 Dual form 1216.2.c.i.609.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.31342i q^{5} -4.77753 q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-1.31342i q^{5} -4.77753 q^{7} +3.00000 q^{9} -2.27492i q^{11} +6.09095i q^{13} -4.27492 q^{17} +1.00000i q^{19} -3.46410 q^{23} +3.27492 q^{25} +6.09095i q^{29} +2.62685 q^{31} +6.27492i q^{35} +0.837253i q^{37} -10.5498 q^{41} +10.2749i q^{43} -3.94027i q^{45} +4.77753 q^{47} +15.8248 q^{49} +10.3923i q^{53} -2.98793 q^{55} +8.54983i q^{59} -1.31342i q^{61} -14.3326 q^{63} +8.00000 q^{65} +8.54983i q^{67} -14.8087 q^{71} +4.27492 q^{73} +10.8685i q^{77} -4.30136 q^{79} +9.00000 q^{81} -13.0997i q^{83} +5.61478i q^{85} -10.0000 q^{89} -29.0997i q^{91} +1.31342 q^{95} +1.45017 q^{97} -6.82475i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{9} + O(q^{10})$$ $$8q + 24q^{9} - 4q^{17} - 4q^{25} - 24q^{41} + 36q^{49} + 64q^{65} + 4q^{73} + 72q^{81} - 80q^{89} + 72q^{97} + O(q^{100})$$

## 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
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ − 1.31342i − 0.587381i −0.955901 0.293691i $$-0.905116\pi$$
0.955901 0.293691i $$-0.0948835\pi$$
$$6$$ 0 0
$$7$$ −4.77753 −1.80573 −0.902867 0.429919i $$-0.858542\pi$$
−0.902867 + 0.429919i $$0.858542\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ − 2.27492i − 0.685913i −0.939351 0.342957i $$-0.888572\pi$$
0.939351 0.342957i $$-0.111428\pi$$
$$12$$ 0 0
$$13$$ 6.09095i 1.68933i 0.535299 + 0.844663i $$0.320199\pi$$
−0.535299 + 0.844663i $$0.679801\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.27492 −1.03682 −0.518410 0.855132i $$-0.673476\pi$$
−0.518410 + 0.855132i $$0.673476\pi$$
$$18$$ 0 0
$$19$$ 1.00000i 0.229416i
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 3.27492 0.654983
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.09095i 1.13106i 0.824727 + 0.565530i $$0.191329\pi$$
−0.824727 + 0.565530i $$0.808671\pi$$
$$30$$ 0 0
$$31$$ 2.62685 0.471796 0.235898 0.971778i $$-0.424197\pi$$
0.235898 + 0.971778i $$0.424197\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 6.27492i 1.06065i
$$36$$ 0 0
$$37$$ 0.837253i 0.137644i 0.997629 + 0.0688218i $$0.0219240\pi$$
−0.997629 + 0.0688218i $$0.978076\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.5498 −1.64761 −0.823804 0.566875i $$-0.808152\pi$$
−0.823804 + 0.566875i $$0.808152\pi$$
$$42$$ 0 0
$$43$$ 10.2749i 1.56691i 0.621448 + 0.783455i $$0.286545\pi$$
−0.621448 + 0.783455i $$0.713455\pi$$
$$44$$ 0 0
$$45$$ − 3.94027i − 0.587381i
$$46$$ 0 0
$$47$$ 4.77753 0.696874 0.348437 0.937332i $$-0.386713\pi$$
0.348437 + 0.937332i $$0.386713\pi$$
$$48$$ 0 0
$$49$$ 15.8248 2.26068
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.3923i 1.42749i 0.700404 + 0.713746i $$0.253003\pi$$
−0.700404 + 0.713746i $$0.746997\pi$$
$$54$$ 0 0
$$55$$ −2.98793 −0.402893
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.54983i 1.11309i 0.830816 + 0.556547i $$0.187874\pi$$
−0.830816 + 0.556547i $$0.812126\pi$$
$$60$$ 0 0
$$61$$ − 1.31342i − 0.168167i −0.996459 0.0840834i $$-0.973204\pi$$
0.996459 0.0840834i $$-0.0267962\pi$$
$$62$$ 0 0
$$63$$ −14.3326 −1.80573
$$64$$ 0 0
$$65$$ 8.00000 0.992278
$$66$$ 0 0
$$67$$ 8.54983i 1.04453i 0.852784 + 0.522264i $$0.174913\pi$$
−0.852784 + 0.522264i $$0.825087\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.8087 −1.75748 −0.878738 0.477305i $$-0.841614\pi$$
−0.878738 + 0.477305i $$0.841614\pi$$
$$72$$ 0 0
$$73$$ 4.27492 0.500341 0.250171 0.968202i $$-0.419513\pi$$
0.250171 + 0.968202i $$0.419513\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 10.8685i 1.23858i
$$78$$ 0 0
$$79$$ −4.30136 −0.483940 −0.241970 0.970284i $$-0.577794\pi$$
−0.241970 + 0.970284i $$0.577794\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 13.0997i − 1.43788i −0.695074 0.718938i $$-0.744629\pi$$
0.695074 0.718938i $$-0.255371\pi$$
$$84$$ 0 0
$$85$$ 5.61478i 0.609008i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ − 29.0997i − 3.05047i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.31342 0.134754
$$96$$ 0 0
$$97$$ 1.45017 0.147242 0.0736210 0.997286i $$-0.476544\pi$$
0.0736210 + 0.997286i $$0.476544\pi$$
$$98$$ 0 0
$$99$$ − 6.82475i − 0.685913i
$$100$$ 0 0
$$101$$ − 13.8564i − 1.37876i −0.724398 0.689382i $$-0.757882\pi$$
0.724398 0.689382i $$-0.242118\pi$$
$$102$$ 0 0
$$103$$ 2.62685 0.258831 0.129416 0.991590i $$-0.458690\pi$$
0.129416 + 0.991590i $$0.458690\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.5498i 1.59993i 0.600045 + 0.799966i $$0.295149\pi$$
−0.600045 + 0.799966i $$0.704851\pi$$
$$108$$ 0 0
$$109$$ − 3.46410i − 0.331801i −0.986143 0.165900i $$-0.946947\pi$$
0.986143 0.165900i $$-0.0530530\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.54983 −0.239868 −0.119934 0.992782i $$-0.538268\pi$$
−0.119934 + 0.992782i $$0.538268\pi$$
$$114$$ 0 0
$$115$$ 4.54983i 0.424274i
$$116$$ 0 0
$$117$$ 18.2728i 1.68933i
$$118$$ 0 0
$$119$$ 20.4235 1.87222
$$120$$ 0 0
$$121$$ 5.82475 0.529523
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 10.8685i − 0.972106i
$$126$$ 0 0
$$127$$ −17.4356 −1.54716 −0.773579 0.633699i $$-0.781536\pi$$
−0.773579 + 0.633699i $$0.781536\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 2.27492i − 0.198760i −0.995050 0.0993802i $$-0.968314\pi$$
0.995050 0.0993802i $$-0.0316860\pi$$
$$132$$ 0 0
$$133$$ − 4.77753i − 0.414264i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −20.2749 −1.73220 −0.866102 0.499868i $$-0.833382\pi$$
−0.866102 + 0.499868i $$0.833382\pi$$
$$138$$ 0 0
$$139$$ 10.2749i 0.871507i 0.900066 + 0.435754i $$0.143518\pi$$
−0.900066 + 0.435754i $$0.856482\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 13.8564 1.15873
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 2.98793i − 0.244781i −0.992482 0.122390i $$-0.960944\pi$$
0.992482 0.122390i $$-0.0390560\pi$$
$$150$$ 0 0
$$151$$ 9.55505 0.777579 0.388790 0.921327i $$-0.372893\pi$$
0.388790 + 0.921327i $$0.372893\pi$$
$$152$$ 0 0
$$153$$ −12.8248 −1.03682
$$154$$ 0 0
$$155$$ − 3.45017i − 0.277124i
$$156$$ 0 0
$$157$$ − 5.25370i − 0.419291i −0.977778 0.209645i $$-0.932769\pi$$
0.977778 0.209645i $$-0.0672309\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.5498 1.30431
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.67451 0.129577 0.0647886 0.997899i $$-0.479363\pi$$
0.0647886 + 0.997899i $$0.479363\pi$$
$$168$$ 0 0
$$169$$ −24.0997 −1.85382
$$170$$ 0 0
$$171$$ 3.00000i 0.229416i
$$172$$ 0 0
$$173$$ 8.71780i 0.662802i 0.943490 + 0.331401i $$0.107521\pi$$
−0.943490 + 0.331401i $$0.892479\pi$$
$$174$$ 0 0
$$175$$ −15.6460 −1.18273
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 9.09967i 0.680141i 0.940400 + 0.340071i $$0.110451\pi$$
−0.940400 + 0.340071i $$0.889549\pi$$
$$180$$ 0 0
$$181$$ 3.46410i 0.257485i 0.991678 + 0.128742i $$0.0410940\pi$$
−0.991678 + 0.128742i $$0.958906\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.09967 0.0808493
$$186$$ 0 0
$$187$$ 9.72508i 0.711168i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.0312 0.725834 0.362917 0.931822i $$-0.381781\pi$$
0.362917 + 0.931822i $$0.381781\pi$$
$$192$$ 0 0
$$193$$ −19.0997 −1.37482 −0.687412 0.726268i $$-0.741253\pi$$
−0.687412 + 0.726268i $$0.741253\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 12.1819i − 0.867924i −0.900931 0.433962i $$-0.857115\pi$$
0.900931 0.433962i $$-0.142885\pi$$
$$198$$ 0 0
$$199$$ 18.6339 1.32092 0.660462 0.750859i $$-0.270360\pi$$
0.660462 + 0.750859i $$0.270360\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 29.0997i − 2.04240i
$$204$$ 0 0
$$205$$ 13.8564i 0.967773i
$$206$$ 0 0
$$207$$ −10.3923 −0.722315
$$208$$ 0 0
$$209$$ 2.27492 0.157359
$$210$$ 0 0
$$211$$ − 11.4502i − 0.788262i −0.919054 0.394131i $$-0.871046\pi$$
0.919054 0.394131i $$-0.128954\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 13.4953 0.920373
$$216$$ 0 0
$$217$$ −12.5498 −0.851938
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 26.0383i − 1.75153i
$$222$$ 0 0
$$223$$ 17.4356 1.16757 0.583787 0.811907i $$-0.301570\pi$$
0.583787 + 0.811907i $$0.301570\pi$$
$$224$$ 0 0
$$225$$ 9.82475 0.654983
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ − 22.0980i − 1.46028i −0.683298 0.730140i $$-0.739455\pi$$
0.683298 0.730140i $$-0.260545\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 17.3746 1.13825 0.569123 0.822252i $$-0.307283\pi$$
0.569123 + 0.822252i $$0.307283\pi$$
$$234$$ 0 0
$$235$$ − 6.27492i − 0.409330i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3.82518 0.247431 0.123715 0.992318i $$-0.460519\pi$$
0.123715 + 0.992318i $$0.460519\pi$$
$$240$$ 0 0
$$241$$ 2.54983 0.164249 0.0821246 0.996622i $$-0.473829\pi$$
0.0821246 + 0.996622i $$0.473829\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 20.7846i − 1.32788i
$$246$$ 0 0
$$247$$ −6.09095 −0.387558
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 18.2749i − 1.15350i −0.816920 0.576751i $$-0.804320\pi$$
0.816920 0.576751i $$-0.195680\pi$$
$$252$$ 0 0
$$253$$ 7.88054i 0.495446i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.5498 1.40662 0.703310 0.710883i $$-0.251705\pi$$
0.703310 + 0.710883i $$0.251705\pi$$
$$258$$ 0 0
$$259$$ − 4.00000i − 0.248548i
$$260$$ 0 0
$$261$$ 18.2728i 1.13106i
$$262$$ 0 0
$$263$$ −14.3326 −0.883785 −0.441892 0.897068i $$-0.645693\pi$$
−0.441892 + 0.897068i $$0.645693\pi$$
$$264$$ 0 0
$$265$$ 13.6495 0.838482
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 22.5742i − 1.37637i −0.725534 0.688187i $$-0.758407\pi$$
0.725534 0.688187i $$-0.241593\pi$$
$$270$$ 0 0
$$271$$ −7.04329 −0.427849 −0.213925 0.976850i $$-0.568625\pi$$
−0.213925 + 0.976850i $$0.568625\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 7.45017i − 0.449262i
$$276$$ 0 0
$$277$$ 2.98793i 0.179527i 0.995963 + 0.0897637i $$0.0286112\pi$$
−0.995963 + 0.0897637i $$0.971389\pi$$
$$278$$ 0 0
$$279$$ 7.88054 0.471796
$$280$$ 0 0
$$281$$ −19.0997 −1.13939 −0.569695 0.821856i $$-0.692939\pi$$
−0.569695 + 0.821856i $$0.692939\pi$$
$$282$$ 0 0
$$283$$ 11.3746i 0.676149i 0.941119 + 0.338074i $$0.109776\pi$$
−0.941119 + 0.338074i $$0.890224\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 50.4021 2.97514
$$288$$ 0 0
$$289$$ 1.27492 0.0749951
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 21.6219i − 1.26316i −0.775310 0.631581i $$-0.782406\pi$$
0.775310 0.631581i $$-0.217594\pi$$
$$294$$ 0 0
$$295$$ 11.2296 0.653810
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 21.0997i − 1.22023i
$$300$$ 0 0
$$301$$ − 49.0887i − 2.82942i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1.72508 −0.0987780
$$306$$ 0 0
$$307$$ − 16.5498i − 0.944549i −0.881452 0.472274i $$-0.843433\pi$$
0.881452 0.472274i $$-0.156567\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5.72987 −0.324911 −0.162455 0.986716i $$-0.551941\pi$$
−0.162455 + 0.986716i $$0.551941\pi$$
$$312$$ 0 0
$$313$$ −15.0997 −0.853484 −0.426742 0.904373i $$-0.640339\pi$$
−0.426742 + 0.904373i $$0.640339\pi$$
$$314$$ 0 0
$$315$$ 18.8248i 1.06065i
$$316$$ 0 0
$$317$$ − 8.71780i − 0.489640i −0.969569 0.244820i $$-0.921271\pi$$
0.969569 0.244820i $$-0.0787289\pi$$
$$318$$ 0 0
$$319$$ 13.8564 0.775810
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 4.27492i − 0.237863i
$$324$$ 0 0
$$325$$ 19.9474i 1.10648i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −22.8248 −1.25837
$$330$$ 0 0
$$331$$ 24.5498i 1.34938i 0.738101 + 0.674690i $$0.235723\pi$$
−0.738101 + 0.674690i $$0.764277\pi$$
$$332$$ 0 0
$$333$$ 2.51176i 0.137644i
$$334$$ 0 0
$$335$$ 11.2296 0.613536
$$336$$ 0 0
$$337$$ −11.6495 −0.634589 −0.317294 0.948327i $$-0.602774\pi$$
−0.317294 + 0.948327i $$0.602774\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 5.97586i − 0.323611i
$$342$$ 0 0
$$343$$ −42.1605 −2.27645
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 14.8248i − 0.795834i −0.917421 0.397917i $$-0.869733\pi$$
0.917421 0.397917i $$-0.130267\pi$$
$$348$$ 0 0
$$349$$ 35.2323i 1.88594i 0.332877 + 0.942970i $$0.391981\pi$$
−0.332877 + 0.942970i $$0.608019\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 0 0
$$355$$ 19.4502i 1.03231i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −0.476171 −0.0251313 −0.0125657 0.999921i $$-0.504000\pi$$
−0.0125657 + 0.999921i $$0.504000\pi$$
$$360$$ 0 0
$$361$$ −1.00000 −0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 5.61478i − 0.293891i
$$366$$ 0 0
$$367$$ 1.78959 0.0934161 0.0467080 0.998909i $$-0.485127\pi$$
0.0467080 + 0.998909i $$0.485127\pi$$
$$368$$ 0 0
$$369$$ −31.6495 −1.64761
$$370$$ 0 0
$$371$$ − 49.6495i − 2.57767i
$$372$$ 0 0
$$373$$ 25.2011i 1.30486i 0.757849 + 0.652431i $$0.226251\pi$$
−0.757849 + 0.652431i $$0.773749\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −37.0997 −1.91073
$$378$$ 0 0
$$379$$ − 21.6495i − 1.11206i −0.831162 0.556030i $$-0.812324\pi$$
0.831162 0.556030i $$-0.187676\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5.97586 0.305352 0.152676 0.988276i $$-0.451211\pi$$
0.152676 + 0.988276i $$0.451211\pi$$
$$384$$ 0 0
$$385$$ 14.2749 0.727517
$$386$$ 0 0
$$387$$ 30.8248i 1.56691i
$$388$$ 0 0
$$389$$ 17.7967i 0.902327i 0.892441 + 0.451164i $$0.148991\pi$$
−0.892441 + 0.451164i $$0.851009\pi$$
$$390$$ 0 0
$$391$$ 14.8087 0.748911
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 5.64950i 0.284257i
$$396$$ 0 0
$$397$$ 24.0027i 1.20466i 0.798247 + 0.602331i $$0.205761\pi$$
−0.798247 + 0.602331i $$0.794239\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.90033 0.244711 0.122355 0.992486i $$-0.460955\pi$$
0.122355 + 0.992486i $$0.460955\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ 0 0
$$405$$ − 11.8208i − 0.587381i
$$406$$ 0 0
$$407$$ 1.90468 0.0944116
$$408$$ 0 0
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 40.8471i − 2.00995i
$$414$$ 0 0
$$415$$ −17.2054 −0.844581
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.0000i 0.586238i 0.956076 + 0.293119i $$0.0946933\pi$$
−0.956076 + 0.293119i $$0.905307\pi$$
$$420$$ 0 0
$$421$$ 24.2487i 1.18181i 0.806741 + 0.590905i $$0.201229\pi$$
−0.806741 + 0.590905i $$0.798771\pi$$
$$422$$ 0 0
$$423$$ 14.3326 0.696874
$$424$$ 0 0
$$425$$ −14.0000 −0.679100
$$426$$ 0 0
$$427$$ 6.27492i 0.303665i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −19.1101 −0.920501 −0.460251 0.887789i $$-0.652240\pi$$
−0.460251 + 0.887789i $$0.652240\pi$$
$$432$$ 0 0
$$433$$ 27.6495 1.32875 0.664375 0.747399i $$-0.268698\pi$$
0.664375 + 0.747399i $$0.268698\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3.46410i − 0.165710i
$$438$$ 0 0
$$439$$ −3.57919 −0.170825 −0.0854127 0.996346i $$-0.527221\pi$$
−0.0854127 + 0.996346i $$0.527221\pi$$
$$440$$ 0 0
$$441$$ 47.4743 2.26068
$$442$$ 0 0
$$443$$ 27.3746i 1.30061i 0.759675 + 0.650303i $$0.225358\pi$$
−0.759675 + 0.650303i $$0.774642\pi$$
$$444$$ 0 0
$$445$$ 13.1342i 0.622623i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −5.45017 −0.257209 −0.128605 0.991696i $$-0.541050\pi$$
−0.128605 + 0.991696i $$0.541050\pi$$
$$450$$ 0 0
$$451$$ 24.0000i 1.13012i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −38.2202 −1.79179
$$456$$ 0 0
$$457$$ −7.72508 −0.361364 −0.180682 0.983542i $$-0.557831\pi$$
−0.180682 + 0.983542i $$0.557831\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 35.2323i 1.64093i 0.571696 + 0.820465i $$0.306286\pi$$
−0.571696 + 0.820465i $$0.693714\pi$$
$$462$$ 0 0
$$463$$ 33.4427 1.55421 0.777107 0.629369i $$-0.216687\pi$$
0.777107 + 0.629369i $$0.216687\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 30.8248i − 1.42640i −0.700961 0.713200i $$-0.747245\pi$$
0.700961 0.713200i $$-0.252755\pi$$
$$468$$ 0 0
$$469$$ − 40.8471i − 1.88614i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 23.3746 1.07476
$$474$$ 0 0
$$475$$ 3.27492i 0.150264i
$$476$$ 0 0
$$477$$ 31.1769i 1.42749i
$$478$$ 0 0
$$479$$ 17.3205 0.791394 0.395697 0.918381i $$-0.370503\pi$$
0.395697 + 0.918381i $$0.370503\pi$$
$$480$$ 0 0
$$481$$ −5.09967 −0.232525
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 1.90468i − 0.0864872i
$$486$$ 0 0
$$487$$ −12.9041 −0.584739 −0.292370 0.956305i $$-0.594444\pi$$
−0.292370 + 0.956305i $$0.594444\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ − 26.0383i − 1.17271i
$$494$$ 0 0
$$495$$ −8.96379 −0.402893
$$496$$ 0 0
$$497$$ 70.7492 3.17353
$$498$$ 0 0
$$499$$ − 34.2749i − 1.53436i −0.641434 0.767178i $$-0.721660\pi$$
0.641434 0.767178i $$-0.278340\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6.81312 0.303782 0.151891 0.988397i $$-0.451464\pi$$
0.151891 + 0.988397i $$0.451464\pi$$
$$504$$ 0 0
$$505$$ −18.1993 −0.809860
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 13.0192i 0.577064i 0.957470 + 0.288532i $$0.0931671\pi$$
−0.957470 + 0.288532i $$0.906833\pi$$
$$510$$ 0 0
$$511$$ −20.4235 −0.903484
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 3.45017i − 0.152032i
$$516$$ 0 0
$$517$$ − 10.8685i − 0.477995i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3.09967 0.135799 0.0678995 0.997692i $$-0.478370\pi$$
0.0678995 + 0.997692i $$0.478370\pi$$
$$522$$ 0 0
$$523$$ − 20.5498i − 0.898582i −0.893386 0.449291i $$-0.851677\pi$$
0.893386 0.449291i $$-0.148323\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −11.2296 −0.489167
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 25.6495i 1.11309i
$$532$$ 0 0
$$533$$ − 64.2585i − 2.78335i
$$534$$ 0 0
$$535$$ 21.7370 0.939770
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 36.0000i − 1.55063i
$$540$$ 0 0
$$541$$ 7.28929i 0.313391i 0.987647 + 0.156695i $$0.0500841\pi$$
−0.987647 + 0.156695i $$0.949916\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −4.54983 −0.194893
$$546$$ 0 0
$$547$$ 32.0000i 1.36822i 0.729378 + 0.684111i $$0.239809\pi$$
−0.729378 + 0.684111i $$0.760191\pi$$
$$548$$ 0 0
$$549$$ − 3.94027i − 0.168167i
$$550$$ 0 0
$$551$$ −6.09095 −0.259483
$$552$$ 0 0
$$553$$ 20.5498 0.873868
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 16.8443i 0.713717i 0.934158 + 0.356859i $$0.116152\pi$$
−0.934158 + 0.356859i $$0.883848\pi$$
$$558$$ 0 0
$$559$$ −62.5840 −2.64702
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 0 0
$$565$$ 3.34901i 0.140894i
$$566$$ 0 0
$$567$$ −42.9977 −1.80573
$$568$$ 0 0
$$569$$ −34.5498 −1.44840 −0.724202 0.689588i $$-0.757792\pi$$
−0.724202 + 0.689588i $$0.757792\pi$$
$$570$$ 0 0
$$571$$ 10.9003i 0.456165i 0.973642 + 0.228082i $$0.0732455\pi$$
−0.973642 + 0.228082i $$0.926754\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −11.3446 −0.473104
$$576$$ 0 0
$$577$$ 15.7251 0.654644 0.327322 0.944913i $$-0.393854\pi$$
0.327322 + 0.944913i $$0.393854\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 62.5840i 2.59642i
$$582$$ 0 0
$$583$$ 23.6416 0.979136
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ 0 0
$$587$$ 29.7251i 1.22689i 0.789739 + 0.613443i $$0.210216\pi$$
−0.789739 + 0.613443i $$0.789784\pi$$
$$588$$ 0 0
$$589$$ 2.62685i 0.108237i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −11.0997 −0.455809 −0.227904 0.973684i $$-0.573187\pi$$
−0.227904 + 0.973684i $$0.573187\pi$$
$$594$$ 0 0
$$595$$ − 26.8248i − 1.09971i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 35.5934 1.45431 0.727153 0.686476i $$-0.240843\pi$$
0.727153 + 0.686476i $$0.240843\pi$$
$$600$$ 0 0
$$601$$ −19.0997 −0.779092 −0.389546 0.921007i $$-0.627368\pi$$
−0.389546 + 0.921007i $$0.627368\pi$$
$$602$$ 0 0
$$603$$ 25.6495i 1.04453i
$$604$$ 0 0
$$605$$ − 7.65037i − 0.311032i
$$606$$ 0 0
$$607$$ −46.8229 −1.90048 −0.950242 0.311513i $$-0.899164\pi$$
−0.950242 + 0.311513i $$0.899164\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 29.0997i 1.17725i
$$612$$ 0 0
$$613$$ 16.8443i 0.680336i 0.940365 + 0.340168i $$0.110484\pi$$
−0.940365 + 0.340168i $$0.889516\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 20.2749 0.816237 0.408119 0.912929i $$-0.366185\pi$$
0.408119 + 0.912929i $$0.366185\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 47.7753 1.91408
$$624$$ 0 0
$$625$$ 2.09967 0.0839868
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 3.57919i − 0.142712i
$$630$$ 0 0
$$631$$ −9.07888 −0.361425 −0.180712 0.983536i $$-0.557840\pi$$
−0.180712 + 0.983536i $$0.557840\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 22.9003i 0.908772i
$$636$$ 0 0
$$637$$ 96.3878i 3.81902i
$$638$$ 0 0
$$639$$ −44.4262 −1.75748
$$640$$ 0 0
$$641$$ 2.54983 0.100712 0.0503562 0.998731i $$-0.483964\pi$$
0.0503562 + 0.998731i $$0.483964\pi$$
$$642$$ 0 0
$$643$$ − 7.92442i − 0.312509i −0.987717 0.156254i $$-0.950058\pi$$
0.987717 0.156254i $$-0.0499419\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −3.82518 −0.150384 −0.0751918 0.997169i $$-0.523957\pi$$
−0.0751918 + 0.997169i $$0.523957\pi$$
$$648$$ 0 0
$$649$$ 19.4502 0.763486
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.98793i 0.116927i 0.998290 + 0.0584634i $$0.0186201\pi$$
−0.998290 + 0.0584634i $$0.981380\pi$$
$$654$$ 0 0
$$655$$ −2.98793 −0.116748
$$656$$ 0 0
$$657$$ 12.8248 0.500341
$$658$$ 0 0
$$659$$ 28.5498i 1.11214i 0.831134 + 0.556072i $$0.187692\pi$$
−0.831134 + 0.556072i $$0.812308\pi$$
$$660$$ 0 0
$$661$$ 0.837253i 0.0325654i 0.999867 + 0.0162827i $$0.00518317\pi$$
−0.999867 + 0.0162827i $$0.994817\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −6.27492 −0.243331
$$666$$ 0 0
$$667$$ − 21.0997i − 0.816982i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2.98793 −0.115348
$$672$$ 0 0
$$673$$ −16.9003 −0.651460 −0.325730 0.945463i $$-0.605610\pi$$
−0.325730 + 0.945463i $$0.605610\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 44.3112i 1.70302i 0.524342 + 0.851508i $$0.324312\pi$$
−0.524342 + 0.851508i $$0.675688\pi$$
$$678$$ 0 0
$$679$$ −6.92820 −0.265880
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 38.1993i 1.46166i 0.682561 + 0.730829i $$0.260866\pi$$
−0.682561 + 0.730829i $$0.739134\pi$$
$$684$$ 0 0
$$685$$ 26.6296i 1.01746i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −63.2990 −2.41150
$$690$$ 0 0
$$691$$ 44.4743i 1.69188i 0.533278 + 0.845940i $$0.320960\pi$$
−0.533278 + 0.845940i $$0.679040\pi$$
$$692$$ 0 0
$$693$$ 32.6054i 1.23858i
$$694$$ 0 0
$$695$$ 13.4953 0.511907
$$696$$ 0 0
$$697$$ 45.0997 1.70827
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 48.7276i − 1.84042i −0.391430 0.920208i $$-0.628019\pi$$
0.391430 0.920208i $$-0.371981\pi$$
$$702$$ 0 0
$$703$$ −0.837253 −0.0315776
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 66.1993i 2.48968i
$$708$$ 0 0
$$709$$ − 3.34901i − 0.125775i −0.998021 0.0628874i $$-0.979969\pi$$
0.998021 0.0628874i $$-0.0200309\pi$$
$$710$$ 0 0
$$711$$ −12.9041 −0.483940
$$712$$ 0 0
$$713$$ −9.09967 −0.340785
$$714$$ 0 0
$$715$$ − 18.1993i − 0.680617i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −12.4279 −0.463482 −0.231741 0.972777i $$-0.574442\pi$$
−0.231741 + 0.972777i $$0.574442\pi$$
$$720$$ 0 0
$$721$$ −12.5498 −0.467380
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 19.9474i 0.740826i
$$726$$ 0 0
$$727$$ −19.5863 −0.726415 −0.363207 0.931708i $$-0.618318\pi$$
−0.363207 + 0.931708i $$0.618318\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ − 43.9244i − 1.62460i
$$732$$ 0 0
$$733$$ 1.67451i 0.0618493i 0.999522 + 0.0309247i $$0.00984520\pi$$
−0.999522 + 0.0309247i $$0.990155\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 19.4502 0.716456
$$738$$ 0 0
$$739$$ 14.8248i 0.545337i 0.962108 + 0.272669i $$0.0879063\pi$$
−0.962108 + 0.272669i $$0.912094\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 26.7605 0.981746 0.490873 0.871231i $$-0.336678\pi$$
0.490873 + 0.871231i $$0.336678\pi$$
$$744$$ 0 0
$$745$$ −3.92442 −0.143780
$$746$$ 0 0
$$747$$ − 39.2990i − 1.43788i
$$748$$ 0 0
$$749$$ − 79.0673i − 2.88905i
$$750$$ 0 0
$$751$$ 53.7511 1.96141 0.980703 0.195503i $$-0.0626340\pi$$
0.980703 + 0.195503i $$0.0626340\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 12.5498i − 0.456735i
$$756$$ 0 0
$$757$$ 1.31342i 0.0477372i 0.999715 + 0.0238686i $$0.00759833\pi$$
−0.999715 + 0.0238686i $$0.992402\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −4.27492 −0.154966 −0.0774828 0.996994i $$-0.524688\pi$$
−0.0774828 + 0.996994i $$0.524688\pi$$
$$762$$ 0 0
$$763$$ 16.5498i 0.599144i
$$764$$ 0 0
$$765$$ 16.8443i 0.609008i
$$766$$ 0 0
$$767$$ −52.0766 −1.88038
$$768$$ 0 0
$$769$$ 8.82475 0.318229 0.159114 0.987260i $$-0.449136\pi$$
0.159114 + 0.987260i $$0.449136\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0.115088i 0.00413942i 0.999998 + 0.00206971i $$0.000658809\pi$$
−0.999998 + 0.00206971i $$0.999341\pi$$
$$774$$ 0 0
$$775$$ 8.60271 0.309018
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 10.5498i − 0.377987i
$$780$$ 0 0
$$781$$ 33.6887i 1.20548i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −6.90033 −0.246283
$$786$$ 0 0
$$787$$ − 46.1993i − 1.64683i −0.567441 0.823414i $$-0.692066\pi$$
0.567441 0.823414i $$-0.307934\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.1819 0.433138
$$792$$ 0 0
$$793$$ 8.00000 0.284088
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 24.2487i − 0.858933i −0.903083 0.429467i $$-0.858702\pi$$
0.903083 0.429467i $$-0.141298\pi$$
$$798$$ 0 0
$$799$$ −20.4235 −0.722532
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 0 0
$$803$$ − 9.72508i − 0.343191i
$$804$$ 0 0
$$805$$ − 21.7370i − 0.766127i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 16.8248 0.591527 0.295763 0.955261i $$-0.404426\pi$$
0.295763 + 0.955261i $$0.404426\pi$$
$$810$$ 0 0
$$811$$ 43.2990i 1.52043i 0.649669 + 0.760217i $$0.274907\pi$$
−0.649669 + 0.760217i $$0.725093\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 5.25370 0.184029
$$816$$ 0 0
$$817$$ −10.2749 −0.359474
$$818$$ 0 0
$$819$$ − 87.2990i − 3.05047i
$$820$$ 0 0
$$821$$ 21.3759i 0.746023i 0.927827 + 0.373011i $$0.121675\pi$$
−0.927827 + 0.373011i $$0.878325\pi$$
$$822$$ 0 0
$$823$$ −43.9501 −1.53200 −0.766002 0.642839i $$-0.777757\pi$$
−0.766002 + 0.642839i $$0.777757\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 2.35050i − 0.0817348i −0.999165 0.0408674i $$-0.986988\pi$$
0.999165 0.0408674i $$-0.0130121\pi$$
$$828$$ 0 0
$$829$$ − 35.7084i − 1.24021i −0.784521 0.620103i $$-0.787091\pi$$
0.784521 0.620103i $$-0.212909\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −67.6495 −2.34392
$$834$$ 0 0
$$835$$ − 2.19934i − 0.0761112i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 38.9424 1.34444 0.672220 0.740351i $$-0.265341\pi$$
0.672220 + 0.740351i $$0.265341\pi$$
$$840$$ 0 0
$$841$$ −8.09967 −0.279299
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 31.6531i 1.08890i
$$846$$ 0 0
$$847$$ −27.8279 −0.956178
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 2.90033i − 0.0994221i
$$852$$ 0 0
$$853$$ − 15.5309i − 0.531768i −0.964005 0.265884i $$-0.914336\pi$$
0.964005 0.265884i $$-0.0856639\pi$$
$$854$$ 0 0
$$855$$ 3.94027 0.134754
$$856$$ 0 0
$$857$$ 44.1993 1.50982 0.754910 0.655828i $$-0.227680\pi$$
0.754910 + 0.655828i $$0.227680\pi$$
$$858$$ 0 0
$$859$$ 12.4743i 0.425616i 0.977094 + 0.212808i $$0.0682609\pi$$
−0.977094 + 0.212808i $$0.931739\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −27.9430 −0.951190 −0.475595 0.879664i $$-0.657767\pi$$
−0.475595 + 0.879664i $$0.657767\pi$$
$$864$$ 0 0
$$865$$ 11.4502 0.389317
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 9.78523i 0.331941i
$$870$$ 0 0
$$871$$ −52.0766 −1.76455
$$872$$ 0 0
$$873$$ 4.35050 0.147242
$$874$$ 0 0
$$875$$ 51.9244i 1.75537i
$$876$$ 0 0
$$877$$ − 38.3353i − 1.29449i −0.762282 0.647245i $$-0.775921\pi$$
0.762282 0.647245i $$-0.224079\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −20.8248 −0.701604 −0.350802 0.936450i $$-0.614091\pi$$
−0.350802 + 0.936450i $$0.614091\pi$$
$$882$$ 0 0
$$883$$ − 51.3746i − 1.72889i −0.502725 0.864446i $$-0.667669\pi$$
0.502725 0.864446i $$-0.332331\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −9.55505 −0.320827 −0.160414 0.987050i $$-0.551283\pi$$
−0.160414 + 0.987050i $$0.551283\pi$$
$$888$$ 0 0
$$889$$ 83.2990 2.79376
$$890$$ 0 0
$$891$$ − 20.4743i − 0.685913i
$$892$$ 0 0
$$893$$ 4.77753i 0.159874i
$$894$$ 0 0
$$895$$ 11.9517 0.399502
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 16.0000i 0.533630i
$$900$$ 0 0
$$901$$ − 44.4262i − 1.48005i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4.54983 0.151242
$$906$$ 0 0
$$907$$ 21.6495i 0.718860i 0.933172 + 0.359430i $$0.117029\pi$$
−0.933172 + 0.359430i $$0.882971\pi$$
$$908$$ 0 0
$$909$$ − 41.5692i − 1.37876i
$$910$$ 0 0
$$911$$ 44.4262 1.47191 0.735954 0.677032i $$-0.236734\pi$$
0.735954 + 0.677032i $$0.236734\pi$$
$$912$$ 0 0
$$913$$ −29.8007 −0.986258
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 10.8685i 0.358909i
$$918$$ 0 0
$$919$$ 34.7561 1.14650 0.573249 0.819381i $$-0.305683\pi$$
0.573249 + 0.819381i $$0.305683\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 90.1993i − 2.96895i
$$924$$ 0 0
$$925$$ 2.74194i 0.0901543i
$$926$$ 0 0
$$927$$ 7.88054 0.258831
$$928$$ 0 0
$$929$$ 23.0997 0.757876 0.378938 0.925422i $$-0.376289\pi$$
0.378938 + 0.925422i $$0.376289\pi$$
$$930$$ 0 0
$$931$$ 15.8248i 0.518635i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 12.7732 0.417727
$$936$$ 0 0
$$937$$ 37.9244 1.23894 0.619468 0.785022i $$-0.287348\pi$$
0.619468 + 0.785022i $$0.287348\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 31.1769i 1.01634i 0.861257 + 0.508169i $$0.169678\pi$$
−0.861257 + 0.508169i $$0.830322\pi$$
$$942$$ 0 0
$$943$$ 36.5457 1.19009
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 30.1993i 0.981347i 0.871344 + 0.490673i $$0.163249\pi$$
−0.871344 + 0.490673i $$0.836751\pi$$
$$948$$ 0 0
$$949$$ 26.0383i 0.845239i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 45.2990 1.46738 0.733689 0.679485i $$-0.237797\pi$$
0.733689 + 0.679485i $$0.237797\pi$$
$$954$$ 0 0
$$955$$ − 13.1752i − 0.426341i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 96.8639 3.12790
$$960$$ 0 0
$$961$$ −24.0997 −0.777409
$$962$$ 0 0
$$963$$ 49.6495i 1.59993i
$$964$$ 0 0
$$965$$ 25.0860i 0.807546i
$$966$$ 0 0
$$967$$ −20.6695 −0.664687 −0.332344 0.943158i $$-0.607839\pi$$
−0.332344 + 0.943158i $$0.607839\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 44.0000i 1.41203i 0.708198 + 0.706014i $$0.249508\pi$$
−0.708198 + 0.706014i $$0.750492\pi$$
$$972$$ 0 0
$$973$$ − 49.0887i − 1.57371i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 32.1993 1.03015 0.515074 0.857146i $$-0.327764\pi$$
0.515074 + 0.857146i $$0.327764\pi$$
$$978$$ 0 0
$$979$$ 22.7492i 0.727067i
$$980$$ 0 0
$$981$$ − 10.3923i − 0.331801i
$$982$$ 0 0
$$983$$ −29.6175 −0.944651 −0.472326 0.881424i $$-0.656585\pi$$
−0.472326 + 0.881424i $$0.656585\pi$$
$$984$$ 0 0
$$985$$ −16.0000 −0.509802
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 35.5934i − 1.13180i
$$990$$ 0 0
$$991$$ 40.1249 1.27461 0.637305 0.770612i $$-0.280049\pi$$
0.637305 + 0.770612i $$0.280049\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 24.4743i − 0.775886i
$$996$$ 0 0
$$997$$ 11.5906i 0.367079i 0.983012 + 0.183540i $$0.0587556\pi$$
−0.983012 + 0.183540i $$0.941244\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1216.2.c.i.609.3 8
4.3 odd 2 inner 1216.2.c.i.609.4 yes 8
8.3 odd 2 inner 1216.2.c.i.609.6 yes 8
8.5 even 2 inner 1216.2.c.i.609.5 yes 8
16.3 odd 4 4864.2.a.bj.1.2 4
16.5 even 4 4864.2.a.bj.1.3 4
16.11 odd 4 4864.2.a.bk.1.3 4
16.13 even 4 4864.2.a.bk.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.i.609.3 8 1.1 even 1 trivial
1216.2.c.i.609.4 yes 8 4.3 odd 2 inner
1216.2.c.i.609.5 yes 8 8.5 even 2 inner
1216.2.c.i.609.6 yes 8 8.3 odd 2 inner
4864.2.a.bj.1.2 4 16.3 odd 4
4864.2.a.bj.1.3 4 16.5 even 4
4864.2.a.bk.1.2 4 16.13 even 4
4864.2.a.bk.1.3 4 16.11 odd 4