# Properties

 Label 4600.2.e.v.4049.10 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.10 Root $$-0.144312i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.v.4049.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.08344i q^{3} +0.555022i q^{7} -6.50759 q^{9} +O(q^{10})$$ $$q+3.08344i q^{3} +0.555022i q^{7} -6.50759 q^{9} +4.65190 q^{11} +5.02256i q^{13} -1.32867i q^{17} +0.196402 q^{19} -1.71138 q^{21} +1.00000i q^{23} -10.8154i q^{27} +0.812298 q^{29} -2.11145 q^{31} +14.3439i q^{33} +5.64564i q^{37} -15.4868 q^{39} -4.89714 q^{41} +1.66507i q^{43} +9.89310i q^{47} +6.69195 q^{49} +4.09688 q^{51} +2.23261i q^{53} +0.605593i q^{57} -2.43488 q^{59} -5.71138 q^{61} -3.61185i q^{63} -6.91830i q^{67} -3.08344 q^{69} +0.0120411 q^{71} +15.2989i q^{73} +2.58191i q^{77} -10.6351 q^{79} +13.8260 q^{81} -5.64020i q^{83} +2.50467i q^{87} -9.06693 q^{89} -2.78763 q^{91} -6.51051i q^{93} +14.0720i q^{97} -30.2727 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 8 q^{9} + O(q^{10})$$ $$10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.08344i 1.78022i 0.455742 + 0.890112i $$0.349374\pi$$
−0.455742 + 0.890112i $$0.650626\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.555022i 0.209779i 0.994484 + 0.104889i $$0.0334488\pi$$
−0.994484 + 0.104889i $$0.966551\pi$$
$$8$$ 0 0
$$9$$ −6.50759 −2.16920
$$10$$ 0 0
$$11$$ 4.65190 1.40260 0.701301 0.712866i $$-0.252603\pi$$
0.701301 + 0.712866i $$0.252603\pi$$
$$12$$ 0 0
$$13$$ 5.02256i 1.39301i 0.717553 + 0.696504i $$0.245262\pi$$
−0.717553 + 0.696504i $$0.754738\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 1.32867i − 0.322251i −0.986934 0.161125i $$-0.948488\pi$$
0.986934 0.161125i $$-0.0515123\pi$$
$$18$$ 0 0
$$19$$ 0.196402 0.0450577 0.0225288 0.999746i $$-0.492828\pi$$
0.0225288 + 0.999746i $$0.492828\pi$$
$$20$$ 0 0
$$21$$ −1.71138 −0.373453
$$22$$ 0 0
$$23$$ 1.00000i 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 10.8154i − 2.08143i
$$28$$ 0 0
$$29$$ 0.812298 0.150840 0.0754200 0.997152i $$-0.475970\pi$$
0.0754200 + 0.997152i $$0.475970\pi$$
$$30$$ 0 0
$$31$$ −2.11145 −0.379227 −0.189613 0.981859i $$-0.560723\pi$$
−0.189613 + 0.981859i $$0.560723\pi$$
$$32$$ 0 0
$$33$$ 14.3439i 2.49694i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.64564i 0.928138i 0.885799 + 0.464069i $$0.153611\pi$$
−0.885799 + 0.464069i $$0.846389\pi$$
$$38$$ 0 0
$$39$$ −15.4868 −2.47987
$$40$$ 0 0
$$41$$ −4.89714 −0.764804 −0.382402 0.923996i $$-0.624903\pi$$
−0.382402 + 0.923996i $$0.624903\pi$$
$$42$$ 0 0
$$43$$ 1.66507i 0.253920i 0.991908 + 0.126960i $$0.0405220\pi$$
−0.991908 + 0.126960i $$0.959478\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.89310i 1.44306i 0.692385 + 0.721528i $$0.256560\pi$$
−0.692385 + 0.721528i $$0.743440\pi$$
$$48$$ 0 0
$$49$$ 6.69195 0.955993
$$50$$ 0 0
$$51$$ 4.09688 0.573678
$$52$$ 0 0
$$53$$ 2.23261i 0.306673i 0.988174 + 0.153336i $$0.0490018\pi$$
−0.988174 + 0.153336i $$0.950998\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.605593i 0.0802127i
$$58$$ 0 0
$$59$$ −2.43488 −0.316994 −0.158497 0.987359i $$-0.550665\pi$$
−0.158497 + 0.987359i $$0.550665\pi$$
$$60$$ 0 0
$$61$$ −5.71138 −0.731267 −0.365633 0.930759i $$-0.619148\pi$$
−0.365633 + 0.930759i $$0.619148\pi$$
$$62$$ 0 0
$$63$$ − 3.61185i − 0.455051i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 6.91830i − 0.845205i −0.906315 0.422602i $$-0.861117\pi$$
0.906315 0.422602i $$-0.138883\pi$$
$$68$$ 0 0
$$69$$ −3.08344 −0.371202
$$70$$ 0 0
$$71$$ 0.0120411 0.00142902 0.000714510 1.00000i $$-0.499773\pi$$
0.000714510 1.00000i $$0.499773\pi$$
$$72$$ 0 0
$$73$$ 15.2989i 1.79061i 0.445458 + 0.895303i $$0.353041\pi$$
−0.445458 + 0.895303i $$0.646959\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.58191i 0.294236i
$$78$$ 0 0
$$79$$ −10.6351 −1.19654 −0.598272 0.801293i $$-0.704146\pi$$
−0.598272 + 0.801293i $$0.704146\pi$$
$$80$$ 0 0
$$81$$ 13.8260 1.53622
$$82$$ 0 0
$$83$$ − 5.64020i − 0.619092i −0.950884 0.309546i $$-0.899823\pi$$
0.950884 0.309546i $$-0.100177\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.50467i 0.268529i
$$88$$ 0 0
$$89$$ −9.06693 −0.961093 −0.480547 0.876969i $$-0.659562\pi$$
−0.480547 + 0.876969i $$0.659562\pi$$
$$90$$ 0 0
$$91$$ −2.78763 −0.292223
$$92$$ 0 0
$$93$$ − 6.51051i − 0.675108i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0720i 1.42880i 0.699739 + 0.714398i $$0.253299\pi$$
−0.699739 + 0.714398i $$0.746701\pi$$
$$98$$ 0 0
$$99$$ −30.2727 −3.04252
$$100$$ 0 0
$$101$$ 7.53015 0.749278 0.374639 0.927171i $$-0.377767\pi$$
0.374639 + 0.927171i $$0.377767\pi$$
$$102$$ 0 0
$$103$$ − 14.4476i − 1.42357i −0.702399 0.711784i $$-0.747888\pi$$
0.702399 0.711784i $$-0.252112\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 2.35862i − 0.228016i −0.993480 0.114008i $$-0.963631\pi$$
0.993480 0.114008i $$-0.0363690\pi$$
$$108$$ 0 0
$$109$$ 10.3578 0.992101 0.496050 0.868294i $$-0.334783\pi$$
0.496050 + 0.868294i $$0.334783\pi$$
$$110$$ 0 0
$$111$$ −17.4080 −1.65229
$$112$$ 0 0
$$113$$ − 19.2723i − 1.81298i −0.422226 0.906491i $$-0.638751\pi$$
0.422226 0.906491i $$-0.361249\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 32.6848i − 3.02171i
$$118$$ 0 0
$$119$$ 0.737442 0.0676012
$$120$$ 0 0
$$121$$ 10.6402 0.967291
$$122$$ 0 0
$$123$$ − 15.1000i − 1.36152i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 18.4934i 1.64102i 0.571632 + 0.820510i $$0.306311\pi$$
−0.571632 + 0.820510i $$0.693689\pi$$
$$128$$ 0 0
$$129$$ −5.13413 −0.452035
$$130$$ 0 0
$$131$$ −14.7727 −1.29070 −0.645351 0.763887i $$-0.723289\pi$$
−0.645351 + 0.763887i $$0.723289\pi$$
$$132$$ 0 0
$$133$$ 0.109007i 0.00945213i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 11.6173i − 0.992533i −0.868170 0.496266i $$-0.834704\pi$$
0.868170 0.496266i $$-0.165296\pi$$
$$138$$ 0 0
$$139$$ −11.1389 −0.944787 −0.472393 0.881388i $$-0.656610\pi$$
−0.472393 + 0.881388i $$0.656610\pi$$
$$140$$ 0 0
$$141$$ −30.5047 −2.56896
$$142$$ 0 0
$$143$$ 23.3645i 1.95384i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 20.6342i 1.70188i
$$148$$ 0 0
$$149$$ 12.4796 1.02237 0.511184 0.859472i $$-0.329207\pi$$
0.511184 + 0.859472i $$0.329207\pi$$
$$150$$ 0 0
$$151$$ 1.32673 0.107968 0.0539840 0.998542i $$-0.482808\pi$$
0.0539840 + 0.998542i $$0.482808\pi$$
$$152$$ 0 0
$$153$$ 8.64646i 0.699025i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 23.1754i − 1.84960i −0.380458 0.924798i $$-0.624234\pi$$
0.380458 0.924798i $$-0.375766\pi$$
$$158$$ 0 0
$$159$$ −6.88412 −0.545946
$$160$$ 0 0
$$161$$ −0.555022 −0.0437418
$$162$$ 0 0
$$163$$ 25.0682i 1.96349i 0.190196 + 0.981746i $$0.439088\pi$$
−0.190196 + 0.981746i $$0.560912\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4.39603i 0.340175i 0.985429 + 0.170087i $$0.0544050\pi$$
−0.985429 + 0.170087i $$0.945595\pi$$
$$168$$ 0 0
$$169$$ −12.2261 −0.940473
$$170$$ 0 0
$$171$$ −1.27810 −0.0977389
$$172$$ 0 0
$$173$$ 19.4211i 1.47656i 0.674493 + 0.738281i $$0.264362\pi$$
−0.674493 + 0.738281i $$0.735638\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 7.50779i − 0.564320i
$$178$$ 0 0
$$179$$ −10.4169 −0.778592 −0.389296 0.921113i $$-0.627282\pi$$
−0.389296 + 0.921113i $$0.627282\pi$$
$$180$$ 0 0
$$181$$ −9.13693 −0.679143 −0.339571 0.940580i $$-0.610282\pi$$
−0.339571 + 0.940580i $$0.610282\pi$$
$$182$$ 0 0
$$183$$ − 17.6107i − 1.30182i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.18086i − 0.451989i
$$188$$ 0 0
$$189$$ 6.00280 0.436640
$$190$$ 0 0
$$191$$ −1.49819 −0.108405 −0.0542026 0.998530i $$-0.517262\pi$$
−0.0542026 + 0.998530i $$0.517262\pi$$
$$192$$ 0 0
$$193$$ 14.4113i 1.03735i 0.854972 + 0.518675i $$0.173575\pi$$
−0.854972 + 0.518675i $$0.826425\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 24.4686i − 1.74331i −0.490116 0.871657i $$-0.663046\pi$$
0.490116 0.871657i $$-0.336954\pi$$
$$198$$ 0 0
$$199$$ 1.61185 0.114261 0.0571307 0.998367i $$-0.481805\pi$$
0.0571307 + 0.998367i $$0.481805\pi$$
$$200$$ 0 0
$$201$$ 21.3321 1.50465
$$202$$ 0 0
$$203$$ 0.450843i 0.0316430i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.50759i − 0.452309i
$$208$$ 0 0
$$209$$ 0.913642 0.0631979
$$210$$ 0 0
$$211$$ 6.25081 0.430324 0.215162 0.976578i $$-0.430972\pi$$
0.215162 + 0.976578i $$0.430972\pi$$
$$212$$ 0 0
$$213$$ 0.0371281i 0.00254398i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 1.17190i − 0.0795536i
$$218$$ 0 0
$$219$$ −47.1733 −3.18768
$$220$$ 0 0
$$221$$ 6.67334 0.448898
$$222$$ 0 0
$$223$$ 21.2953i 1.42604i 0.701144 + 0.713019i $$0.252673\pi$$
−0.701144 + 0.713019i $$0.747327\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 10.5174i − 0.698061i −0.937111 0.349031i $$-0.886511\pi$$
0.937111 0.349031i $$-0.113489\pi$$
$$228$$ 0 0
$$229$$ 24.0063 1.58638 0.793190 0.608975i $$-0.208419\pi$$
0.793190 + 0.608975i $$0.208419\pi$$
$$230$$ 0 0
$$231$$ −7.96115 −0.523805
$$232$$ 0 0
$$233$$ − 11.3112i − 0.741021i −0.928828 0.370510i $$-0.879183\pi$$
0.928828 0.370510i $$-0.120817\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 32.7927i − 2.13012i
$$238$$ 0 0
$$239$$ 25.5592 1.65329 0.826644 0.562726i $$-0.190247\pi$$
0.826644 + 0.562726i $$0.190247\pi$$
$$240$$ 0 0
$$241$$ −16.1617 −1.04106 −0.520532 0.853842i $$-0.674266\pi$$
−0.520532 + 0.853842i $$0.674266\pi$$
$$242$$ 0 0
$$243$$ 10.1852i 0.653380i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.986440i 0.0627657i
$$248$$ 0 0
$$249$$ 17.3912 1.10212
$$250$$ 0 0
$$251$$ −13.3497 −0.842627 −0.421313 0.906915i $$-0.638431\pi$$
−0.421313 + 0.906915i $$0.638431\pi$$
$$252$$ 0 0
$$253$$ 4.65190i 0.292463i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 10.1834i − 0.635222i −0.948221 0.317611i $$-0.897119\pi$$
0.948221 0.317611i $$-0.102881\pi$$
$$258$$ 0 0
$$259$$ −3.13345 −0.194703
$$260$$ 0 0
$$261$$ −5.28610 −0.327201
$$262$$ 0 0
$$263$$ − 23.3073i − 1.43719i −0.695430 0.718594i $$-0.744786\pi$$
0.695430 0.718594i $$-0.255214\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 27.9573i − 1.71096i
$$268$$ 0 0
$$269$$ −9.80817 −0.598015 −0.299007 0.954251i $$-0.596656\pi$$
−0.299007 + 0.954251i $$0.596656\pi$$
$$270$$ 0 0
$$271$$ −6.08332 −0.369535 −0.184768 0.982782i $$-0.559153\pi$$
−0.184768 + 0.982782i $$0.559153\pi$$
$$272$$ 0 0
$$273$$ − 8.59549i − 0.520223i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 0.0445722i − 0.00267809i −0.999999 0.00133904i $$-0.999574\pi$$
0.999999 0.00133904i $$-0.000426231\pi$$
$$278$$ 0 0
$$279$$ 13.7404 0.822617
$$280$$ 0 0
$$281$$ 27.9996 1.67032 0.835158 0.550010i $$-0.185376\pi$$
0.835158 + 0.550010i $$0.185376\pi$$
$$282$$ 0 0
$$283$$ − 12.1798i − 0.724013i −0.932176 0.362006i $$-0.882092\pi$$
0.932176 0.362006i $$-0.117908\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 2.71802i − 0.160440i
$$288$$ 0 0
$$289$$ 15.2346 0.896155
$$290$$ 0 0
$$291$$ −43.3902 −2.54358
$$292$$ 0 0
$$293$$ 2.71056i 0.158352i 0.996861 + 0.0791762i $$0.0252290\pi$$
−0.996861 + 0.0791762i $$0.974771\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 50.3124i − 2.91942i
$$298$$ 0 0
$$299$$ −5.02256 −0.290462
$$300$$ 0 0
$$301$$ −0.924148 −0.0532670
$$302$$ 0 0
$$303$$ 23.2188i 1.33388i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 32.1631i − 1.83564i −0.396994 0.917821i $$-0.629947\pi$$
0.396994 0.917821i $$-0.370053\pi$$
$$308$$ 0 0
$$309$$ 44.5484 2.53427
$$310$$ 0 0
$$311$$ −4.07951 −0.231328 −0.115664 0.993288i $$-0.536900\pi$$
−0.115664 + 0.993288i $$0.536900\pi$$
$$312$$ 0 0
$$313$$ − 13.7806i − 0.778925i −0.921042 0.389462i $$-0.872661\pi$$
0.921042 0.389462i $$-0.127339\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10.4630i 0.587658i 0.955858 + 0.293829i $$0.0949297\pi$$
−0.955858 + 0.293829i $$0.905070\pi$$
$$318$$ 0 0
$$319$$ 3.77873 0.211568
$$320$$ 0 0
$$321$$ 7.27266 0.405920
$$322$$ 0 0
$$323$$ − 0.260954i − 0.0145199i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 31.9377i 1.76616i
$$328$$ 0 0
$$329$$ −5.49088 −0.302722
$$330$$ 0 0
$$331$$ −12.6451 −0.695038 −0.347519 0.937673i $$-0.612976\pi$$
−0.347519 + 0.937673i $$0.612976\pi$$
$$332$$ 0 0
$$333$$ − 36.7395i − 2.01331i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 24.5733i 1.33859i 0.742997 + 0.669295i $$0.233404\pi$$
−0.742997 + 0.669295i $$0.766596\pi$$
$$338$$ 0 0
$$339$$ 59.4248 3.22751
$$340$$ 0 0
$$341$$ −9.82224 −0.531904
$$342$$ 0 0
$$343$$ 7.59933i 0.410325i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 3.50569i − 0.188195i −0.995563 0.0940977i $$-0.970003\pi$$
0.995563 0.0940977i $$-0.0299966\pi$$
$$348$$ 0 0
$$349$$ 18.1085 0.969327 0.484664 0.874701i $$-0.338942\pi$$
0.484664 + 0.874701i $$0.338942\pi$$
$$350$$ 0 0
$$351$$ 54.3212 2.89945
$$352$$ 0 0
$$353$$ 20.1915i 1.07469i 0.843364 + 0.537343i $$0.180572\pi$$
−0.843364 + 0.537343i $$0.819428\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 2.27386i 0.120345i
$$358$$ 0 0
$$359$$ 9.43409 0.497912 0.248956 0.968515i $$-0.419912\pi$$
0.248956 + 0.968515i $$0.419912\pi$$
$$360$$ 0 0
$$361$$ −18.9614 −0.997970
$$362$$ 0 0
$$363$$ 32.8084i 1.72199i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 30.3412i − 1.58380i −0.610652 0.791899i $$-0.709092\pi$$
0.610652 0.791899i $$-0.290908\pi$$
$$368$$ 0 0
$$369$$ 31.8686 1.65901
$$370$$ 0 0
$$371$$ −1.23915 −0.0643333
$$372$$ 0 0
$$373$$ − 14.7893i − 0.765758i −0.923798 0.382879i $$-0.874933\pi$$
0.923798 0.382879i $$-0.125067\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.07982i 0.210121i
$$378$$ 0 0
$$379$$ 11.2405 0.577385 0.288693 0.957422i $$-0.406779\pi$$
0.288693 + 0.957422i $$0.406779\pi$$
$$380$$ 0 0
$$381$$ −57.0231 −2.92138
$$382$$ 0 0
$$383$$ 30.4602i 1.55644i 0.627991 + 0.778221i $$0.283878\pi$$
−0.627991 + 0.778221i $$0.716122\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 10.8356i − 0.550803i
$$388$$ 0 0
$$389$$ −1.94011 −0.0983673 −0.0491836 0.998790i $$-0.515662\pi$$
−0.0491836 + 0.998790i $$0.515662\pi$$
$$390$$ 0 0
$$391$$ 1.32867 0.0671939
$$392$$ 0 0
$$393$$ − 45.5509i − 2.29774i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 18.3206i 0.919487i 0.888052 + 0.459743i $$0.152059\pi$$
−0.888052 + 0.459743i $$0.847941\pi$$
$$398$$ 0 0
$$399$$ −0.336117 −0.0168269
$$400$$ 0 0
$$401$$ −1.32199 −0.0660170 −0.0330085 0.999455i $$-0.510509\pi$$
−0.0330085 + 0.999455i $$0.510509\pi$$
$$402$$ 0 0
$$403$$ − 10.6049i − 0.528266i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 26.2630i 1.30181i
$$408$$ 0 0
$$409$$ −21.9963 −1.08765 −0.543823 0.839200i $$-0.683024\pi$$
−0.543823 + 0.839200i $$0.683024\pi$$
$$410$$ 0 0
$$411$$ 35.8212 1.76693
$$412$$ 0 0
$$413$$ − 1.35141i − 0.0664985i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 34.3460i − 1.68193i
$$418$$ 0 0
$$419$$ −28.3194 −1.38349 −0.691746 0.722141i $$-0.743158\pi$$
−0.691746 + 0.722141i $$0.743158\pi$$
$$420$$ 0 0
$$421$$ 14.7650 0.719602 0.359801 0.933029i $$-0.382845\pi$$
0.359801 + 0.933029i $$0.382845\pi$$
$$422$$ 0 0
$$423$$ − 64.3802i − 3.13027i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 3.16994i − 0.153404i
$$428$$ 0 0
$$429$$ −72.0429 −3.47826
$$430$$ 0 0
$$431$$ −27.0514 −1.30302 −0.651510 0.758640i $$-0.725864\pi$$
−0.651510 + 0.758640i $$0.725864\pi$$
$$432$$ 0 0
$$433$$ − 38.2700i − 1.83914i −0.392926 0.919570i $$-0.628537\pi$$
0.392926 0.919570i $$-0.371463\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.196402i 0.00939517i
$$438$$ 0 0
$$439$$ 2.39959 0.114526 0.0572630 0.998359i $$-0.481763\pi$$
0.0572630 + 0.998359i $$0.481763\pi$$
$$440$$ 0 0
$$441$$ −43.5485 −2.07374
$$442$$ 0 0
$$443$$ 10.6542i 0.506198i 0.967440 + 0.253099i $$0.0814499\pi$$
−0.967440 + 0.253099i $$0.918550\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 38.4800i 1.82004i
$$448$$ 0 0
$$449$$ 28.9533 1.36639 0.683195 0.730236i $$-0.260590\pi$$
0.683195 + 0.730236i $$0.260590\pi$$
$$450$$ 0 0
$$451$$ −22.7810 −1.07272
$$452$$ 0 0
$$453$$ 4.09090i 0.192207i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.5855i 1.80496i 0.430736 + 0.902478i $$0.358254\pi$$
−0.430736 + 0.902478i $$0.641746\pi$$
$$458$$ 0 0
$$459$$ −14.3702 −0.670743
$$460$$ 0 0
$$461$$ −8.07659 −0.376164 −0.188082 0.982153i $$-0.560227\pi$$
−0.188082 + 0.982153i $$0.560227\pi$$
$$462$$ 0 0
$$463$$ 6.07771i 0.282455i 0.989977 + 0.141228i $$0.0451050\pi$$
−0.989977 + 0.141228i $$0.954895\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 10.2961i 0.476446i 0.971210 + 0.238223i $$0.0765649\pi$$
−0.971210 + 0.238223i $$0.923435\pi$$
$$468$$ 0 0
$$469$$ 3.83981 0.177306
$$470$$ 0 0
$$471$$ 71.4598 3.29270
$$472$$ 0 0
$$473$$ 7.74572i 0.356149i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 14.5289i − 0.665233i
$$478$$ 0 0
$$479$$ 33.9760 1.55240 0.776201 0.630486i $$-0.217144\pi$$
0.776201 + 0.630486i $$0.217144\pi$$
$$480$$ 0 0
$$481$$ −28.3556 −1.29290
$$482$$ 0 0
$$483$$ − 1.71138i − 0.0778703i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 16.4951i − 0.747464i −0.927537 0.373732i $$-0.878078\pi$$
0.927537 0.373732i $$-0.121922\pi$$
$$488$$ 0 0
$$489$$ −77.2962 −3.49546
$$490$$ 0 0
$$491$$ 27.4876 1.24050 0.620249 0.784405i $$-0.287032\pi$$
0.620249 + 0.784405i $$0.287032\pi$$
$$492$$ 0 0
$$493$$ − 1.07928i − 0.0486082i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0.00668310i 0 0.000299778i
$$498$$ 0 0
$$499$$ −18.2961 −0.819045 −0.409523 0.912300i $$-0.634305\pi$$
−0.409523 + 0.912300i $$0.634305\pi$$
$$500$$ 0 0
$$501$$ −13.5549 −0.605587
$$502$$ 0 0
$$503$$ − 1.97389i − 0.0880115i −0.999031 0.0440058i $$-0.985988\pi$$
0.999031 0.0440058i $$-0.0140120\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 37.6986i − 1.67425i
$$508$$ 0 0
$$509$$ 38.3441 1.69957 0.849787 0.527126i $$-0.176731\pi$$
0.849787 + 0.527126i $$0.176731\pi$$
$$510$$ 0 0
$$511$$ −8.49125 −0.375631
$$512$$ 0 0
$$513$$ − 2.12417i − 0.0937844i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 46.0217i 2.02403i
$$518$$ 0 0
$$519$$ −59.8839 −2.62861
$$520$$ 0 0
$$521$$ −23.1814 −1.01560 −0.507799 0.861476i $$-0.669541\pi$$
−0.507799 + 0.861476i $$0.669541\pi$$
$$522$$ 0 0
$$523$$ − 43.5123i − 1.90266i −0.308172 0.951331i $$-0.599717\pi$$
0.308172 0.951331i $$-0.400283\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.80542i 0.122206i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 15.8452 0.687622
$$532$$ 0 0
$$533$$ − 24.5962i − 1.06538i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 32.1197i − 1.38607i
$$538$$ 0 0
$$539$$ 31.1303 1.34088
$$540$$ 0 0
$$541$$ 18.6112 0.800158 0.400079 0.916481i $$-0.368983\pi$$
0.400079 + 0.916481i $$0.368983\pi$$
$$542$$ 0 0
$$543$$ − 28.1732i − 1.20903i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.78956i − 0.290300i −0.989410 0.145150i $$-0.953633\pi$$
0.989410 0.145150i $$-0.0463665\pi$$
$$548$$ 0 0
$$549$$ 37.1673 1.58626
$$550$$ 0 0
$$551$$ 0.159537 0.00679649
$$552$$ 0 0
$$553$$ − 5.90272i − 0.251009i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 24.9869i 1.05873i 0.848395 + 0.529364i $$0.177570\pi$$
−0.848395 + 0.529364i $$0.822430\pi$$
$$558$$ 0 0
$$559$$ −8.36290 −0.353713
$$560$$ 0 0
$$561$$ 19.0583 0.804642
$$562$$ 0 0
$$563$$ 44.7551i 1.88620i 0.332508 + 0.943100i $$0.392105\pi$$
−0.332508 + 0.943100i $$0.607895\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 7.67371i 0.322265i
$$568$$ 0 0
$$569$$ 27.4895 1.15242 0.576211 0.817301i $$-0.304531\pi$$
0.576211 + 0.817301i $$0.304531\pi$$
$$570$$ 0 0
$$571$$ −1.40495 −0.0587952 −0.0293976 0.999568i $$-0.509359\pi$$
−0.0293976 + 0.999568i $$0.509359\pi$$
$$572$$ 0 0
$$573$$ − 4.61957i − 0.192985i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 19.1987i − 0.799253i −0.916678 0.399627i $$-0.869140\pi$$
0.916678 0.399627i $$-0.130860\pi$$
$$578$$ 0 0
$$579$$ −44.4364 −1.84671
$$580$$ 0 0
$$581$$ 3.13043 0.129872
$$582$$ 0 0
$$583$$ 10.3859i 0.430139i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 22.8565i − 0.943390i −0.881762 0.471695i $$-0.843642\pi$$
0.881762 0.471695i $$-0.156358\pi$$
$$588$$ 0 0
$$589$$ −0.414692 −0.0170871
$$590$$ 0 0
$$591$$ 75.4473 3.10349
$$592$$ 0 0
$$593$$ 26.5387i 1.08981i 0.838497 + 0.544906i $$0.183435\pi$$
−0.838497 + 0.544906i $$0.816565\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.97005i 0.203411i
$$598$$ 0 0
$$599$$ 8.82768 0.360689 0.180345 0.983603i $$-0.442279\pi$$
0.180345 + 0.983603i $$0.442279\pi$$
$$600$$ 0 0
$$601$$ −30.3072 −1.23626 −0.618129 0.786077i $$-0.712109\pi$$
−0.618129 + 0.786077i $$0.712109\pi$$
$$602$$ 0 0
$$603$$ 45.0215i 1.83342i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 38.0102i 1.54278i 0.636360 + 0.771392i $$0.280439\pi$$
−0.636360 + 0.771392i $$0.719561\pi$$
$$608$$ 0 0
$$609$$ −1.39015 −0.0563316
$$610$$ 0 0
$$611$$ −49.6887 −2.01019
$$612$$ 0 0
$$613$$ 31.7417i 1.28204i 0.767526 + 0.641018i $$0.221488\pi$$
−0.767526 + 0.641018i $$0.778512\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 46.6699i 1.87886i 0.342741 + 0.939430i $$0.388645\pi$$
−0.342741 + 0.939430i $$0.611355\pi$$
$$618$$ 0 0
$$619$$ −23.2223 −0.933383 −0.466692 0.884420i $$-0.654554\pi$$
−0.466692 + 0.884420i $$0.654554\pi$$
$$620$$ 0 0
$$621$$ 10.8154 0.434009
$$622$$ 0 0
$$623$$ − 5.03235i − 0.201617i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 2.81716i 0.112506i
$$628$$ 0 0
$$629$$ 7.50121 0.299093
$$630$$ 0 0
$$631$$ −13.0906 −0.521129 −0.260565 0.965456i $$-0.583909\pi$$
−0.260565 + 0.965456i $$0.583909\pi$$
$$632$$ 0 0
$$633$$ 19.2740i 0.766072i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 33.6107i 1.33171i
$$638$$ 0 0
$$639$$ −0.0783588 −0.00309983
$$640$$ 0 0
$$641$$ 22.8746 0.903494 0.451747 0.892146i $$-0.350801\pi$$
0.451747 + 0.892146i $$0.350801\pi$$
$$642$$ 0 0
$$643$$ 30.6333i 1.20806i 0.796962 + 0.604030i $$0.206439\pi$$
−0.796962 + 0.604030i $$0.793561\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 8.74454i − 0.343783i −0.985116 0.171892i $$-0.945012\pi$$
0.985116 0.171892i $$-0.0549879\pi$$
$$648$$ 0 0
$$649$$ −11.3268 −0.444616
$$650$$ 0 0
$$651$$ 3.61348 0.141623
$$652$$ 0 0
$$653$$ 24.7443i 0.968320i 0.874979 + 0.484160i $$0.160875\pi$$
−0.874979 + 0.484160i $$0.839125\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 99.5593i − 3.88418i
$$658$$ 0 0
$$659$$ −8.30726 −0.323605 −0.161803 0.986823i $$-0.551731\pi$$
−0.161803 + 0.986823i $$0.551731\pi$$
$$660$$ 0 0
$$661$$ −14.1755 −0.551364 −0.275682 0.961249i $$-0.588904\pi$$
−0.275682 + 0.961249i $$0.588904\pi$$
$$662$$ 0 0
$$663$$ 20.5768i 0.799138i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0.812298i 0.0314523i
$$668$$ 0 0
$$669$$ −65.6627 −2.53867
$$670$$ 0 0
$$671$$ −26.5688 −1.02568
$$672$$ 0 0
$$673$$ − 7.30229i − 0.281482i −0.990046 0.140741i $$-0.955051\pi$$
0.990046 0.140741i $$-0.0449486\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 39.7090i 1.52614i 0.646315 + 0.763071i $$0.276309\pi$$
−0.646315 + 0.763071i $$0.723691\pi$$
$$678$$ 0 0
$$679$$ −7.81027 −0.299731
$$680$$ 0 0
$$681$$ 32.4296 1.24271
$$682$$ 0 0
$$683$$ 38.5993i 1.47696i 0.674274 + 0.738481i $$0.264457\pi$$
−0.674274 + 0.738481i $$0.735543\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 74.0219i 2.82411i
$$688$$ 0 0
$$689$$ −11.2134 −0.427198
$$690$$ 0 0
$$691$$ −21.4163 −0.814713 −0.407357 0.913269i $$-0.633549\pi$$
−0.407357 + 0.913269i $$0.633549\pi$$
$$692$$ 0 0
$$693$$ − 16.8020i − 0.638255i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.50669i 0.246459i
$$698$$ 0 0
$$699$$ 34.8773 1.31918
$$700$$ 0 0
$$701$$ −5.40147 −0.204011 −0.102005 0.994784i $$-0.532526\pi$$
−0.102005 + 0.994784i $$0.532526\pi$$
$$702$$ 0 0
$$703$$ 1.10881i 0.0418197i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 4.17940i 0.157182i
$$708$$ 0 0
$$709$$ 15.3161 0.575208 0.287604 0.957749i $$-0.407141\pi$$
0.287604 + 0.957749i $$0.407141\pi$$
$$710$$ 0 0
$$711$$ 69.2090 2.59554
$$712$$ 0 0
$$713$$ − 2.11145i − 0.0790742i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 78.8102i 2.94322i
$$718$$ 0 0
$$719$$ −11.6505 −0.434491 −0.217245 0.976117i $$-0.569707\pi$$
−0.217245 + 0.976117i $$0.569707\pi$$
$$720$$ 0 0
$$721$$ 8.01875 0.298634
$$722$$ 0 0
$$723$$ − 49.8334i − 1.85333i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 28.8188i − 1.06883i −0.845223 0.534415i $$-0.820532\pi$$
0.845223 0.534415i $$-0.179468\pi$$
$$728$$ 0 0
$$729$$ 10.0725 0.373056
$$730$$ 0 0
$$731$$ 2.21233 0.0818259
$$732$$ 0 0
$$733$$ − 25.2304i − 0.931906i −0.884810 0.465953i $$-0.845712\pi$$
0.884810 0.465953i $$-0.154288\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 32.1833i − 1.18549i
$$738$$ 0 0
$$739$$ 1.22800 0.0451728 0.0225864 0.999745i $$-0.492810\pi$$
0.0225864 + 0.999745i $$0.492810\pi$$
$$740$$ 0 0
$$741$$ −3.04163 −0.111737
$$742$$ 0 0
$$743$$ 35.1117i 1.28812i 0.764974 + 0.644061i $$0.222752\pi$$
−0.764974 + 0.644061i $$0.777248\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 36.7041i 1.34293i
$$748$$ 0 0
$$749$$ 1.30909 0.0478329
$$750$$ 0 0
$$751$$ 31.7367 1.15809 0.579044 0.815296i $$-0.303426\pi$$
0.579044 + 0.815296i $$0.303426\pi$$
$$752$$ 0 0
$$753$$ − 41.1630i − 1.50006i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.22287i 0.117137i 0.998283 + 0.0585685i $$0.0186536\pi$$
−0.998283 + 0.0585685i $$0.981346\pi$$
$$758$$ 0 0
$$759$$ −14.3439 −0.520649
$$760$$ 0 0
$$761$$ −9.60077 −0.348027 −0.174014 0.984743i $$-0.555674\pi$$
−0.174014 + 0.984743i $$0.555674\pi$$
$$762$$ 0 0
$$763$$ 5.74882i 0.208121i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 12.2293i − 0.441575i
$$768$$ 0 0
$$769$$ −14.9778 −0.540113 −0.270056 0.962844i $$-0.587042\pi$$
−0.270056 + 0.962844i $$0.587042\pi$$
$$770$$ 0 0
$$771$$ 31.3998 1.13084
$$772$$ 0 0
$$773$$ 28.7430i 1.03381i 0.856042 + 0.516906i $$0.172916\pi$$
−0.856042 + 0.516906i $$0.827084\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 9.66181i − 0.346616i
$$778$$ 0 0
$$779$$ −0.961806 −0.0344603
$$780$$ 0 0
$$781$$ 0.0560142 0.00200435
$$782$$ 0 0
$$783$$ − 8.78536i − 0.313963i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 34.1965i 1.21897i 0.792797 + 0.609486i $$0.208624\pi$$
−0.792797 + 0.609486i $$0.791376\pi$$
$$788$$ 0 0
$$789$$ 71.8666 2.55852
$$790$$ 0 0
$$791$$ 10.6965 0.380324
$$792$$ 0 0
$$793$$ − 28.6857i − 1.01866i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0.303391i 0.0107467i 0.999986 + 0.00537334i $$0.00171039\pi$$
−0.999986 + 0.00537334i $$0.998290\pi$$
$$798$$ 0 0
$$799$$ 13.1447 0.465026
$$800$$ 0 0
$$801$$ 59.0039 2.08480
$$802$$ 0 0
$$803$$ 71.1692i 2.51151i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 30.2429i − 1.06460i
$$808$$ 0 0
$$809$$ 13.1865 0.463611 0.231806 0.972762i $$-0.425537\pi$$
0.231806 + 0.972762i $$0.425537\pi$$
$$810$$ 0 0
$$811$$ 14.7284 0.517185 0.258593 0.965986i $$-0.416741\pi$$
0.258593 + 0.965986i $$0.416741\pi$$
$$812$$ 0 0
$$813$$ − 18.7575i − 0.657856i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0.327022i 0.0114410i
$$818$$ 0 0
$$819$$ 18.1408 0.633890
$$820$$ 0 0
$$821$$ 36.6177 1.27797 0.638984 0.769220i $$-0.279355\pi$$
0.638984 + 0.769220i $$0.279355\pi$$
$$822$$ 0 0
$$823$$ 0.420867i 0.0146705i 0.999973 + 0.00733525i $$0.00233490\pi$$
−0.999973 + 0.00733525i $$0.997665\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 35.1118i − 1.22096i −0.792032 0.610479i $$-0.790977\pi$$
0.792032 0.610479i $$-0.209023\pi$$
$$828$$ 0 0
$$829$$ 35.6234 1.23725 0.618626 0.785685i $$-0.287689\pi$$
0.618626 + 0.785685i $$0.287689\pi$$
$$830$$ 0 0
$$831$$ 0.137436 0.00476759
$$832$$ 0 0
$$833$$ − 8.89141i − 0.308069i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 22.8362i 0.789335i
$$838$$ 0 0
$$839$$ 19.9998 0.690469 0.345235 0.938516i $$-0.387799\pi$$
0.345235 + 0.938516i $$0.387799\pi$$
$$840$$ 0 0
$$841$$ −28.3402 −0.977247
$$842$$ 0 0
$$843$$ 86.3350i 2.97354i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.90554i 0.202917i
$$848$$ 0 0
$$849$$ 37.5556 1.28890
$$850$$ 0 0
$$851$$ −5.64564 −0.193530
$$852$$ 0 0
$$853$$ 42.4994i 1.45515i 0.686026 + 0.727577i $$0.259353\pi$$
−0.686026 + 0.727577i $$0.740647\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0100i 0.615208i 0.951514 + 0.307604i $$0.0995273\pi$$
−0.951514 + 0.307604i $$0.900473\pi$$
$$858$$ 0 0
$$859$$ −22.3869 −0.763832 −0.381916 0.924197i $$-0.624736\pi$$
−0.381916 + 0.924197i $$0.624736\pi$$
$$860$$ 0 0
$$861$$ 8.38084 0.285618
$$862$$ 0 0
$$863$$ 18.7613i 0.638642i 0.947647 + 0.319321i $$0.103455\pi$$
−0.947647 + 0.319321i $$0.896545\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 46.9750i 1.59536i
$$868$$ 0 0
$$869$$ −49.4735 −1.67827
$$870$$ 0 0
$$871$$ 34.7476 1.17738
$$872$$ 0 0
$$873$$ − 91.5749i − 3.09934i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31.6345i 1.06822i 0.845415 + 0.534111i $$0.179353\pi$$
−0.845415 + 0.534111i $$0.820647\pi$$
$$878$$ 0 0
$$879$$ −8.35783 −0.281903
$$880$$ 0 0
$$881$$ 39.6850 1.33702 0.668512 0.743702i $$-0.266932\pi$$
0.668512 + 0.743702i $$0.266932\pi$$
$$882$$ 0 0
$$883$$ − 10.3778i − 0.349242i −0.984636 0.174621i $$-0.944130\pi$$
0.984636 0.174621i $$-0.0558701\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 24.0199i − 0.806508i −0.915088 0.403254i $$-0.867879\pi$$
0.915088 0.403254i $$-0.132121\pi$$
$$888$$ 0 0
$$889$$ −10.2642 −0.344251
$$890$$ 0 0
$$891$$ 64.3170 2.15470
$$892$$ 0 0
$$893$$ 1.94302i 0.0650207i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 15.4868i − 0.517088i
$$898$$ 0 0
$$899$$ −1.71512 −0.0572025
$$900$$ 0 0
$$901$$ 2.96641 0.0988254
$$902$$ 0 0
$$903$$ − 2.84955i − 0.0948271i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 18.5181i 0.614885i 0.951567 + 0.307442i $$0.0994731\pi$$
−0.951567 + 0.307442i $$0.900527\pi$$
$$908$$ 0 0
$$909$$ −49.0032 −1.62533
$$910$$ 0 0
$$911$$ −3.34956 −0.110976 −0.0554879 0.998459i $$-0.517671\pi$$
−0.0554879 + 0.998459i $$0.517671\pi$$
$$912$$ 0 0
$$913$$ − 26.2376i − 0.868339i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 8.19920i − 0.270761i
$$918$$ 0 0
$$919$$ 32.9754 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$920$$ 0 0
$$921$$ 99.1728 3.26785
$$922$$ 0 0
$$923$$ 0.0604774i 0.00199064i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 94.0193i 3.08800i
$$928$$ 0 0
$$929$$ 36.5008 1.19755 0.598777 0.800916i $$-0.295654\pi$$
0.598777 + 0.800916i $$0.295654\pi$$
$$930$$ 0 0
$$931$$ 1.31431 0.0430748
$$932$$ 0 0
$$933$$ − 12.5789i − 0.411816i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12.4598i 0.407043i 0.979071 + 0.203521i $$0.0652386\pi$$
−0.979071 + 0.203521i $$0.934761\pi$$
$$938$$ 0 0
$$939$$ 42.4916 1.38666
$$940$$ 0 0
$$941$$ −45.2938 −1.47653 −0.738267 0.674508i $$-0.764356\pi$$
−0.738267 + 0.674508i $$0.764356\pi$$
$$942$$ 0 0
$$943$$ − 4.89714i − 0.159473i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 41.4016i 1.34537i 0.739929 + 0.672685i $$0.234859\pi$$
−0.739929 + 0.672685i $$0.765141\pi$$
$$948$$ 0 0
$$949$$ −76.8399 −2.49433
$$950$$ 0 0
$$951$$ −32.2619 −1.04616
$$952$$ 0 0
$$953$$ 15.3738i 0.498006i 0.968503 + 0.249003i $$0.0801029\pi$$
−0.968503 + 0.249003i $$0.919897\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 11.6515i 0.376639i
$$958$$ 0 0
$$959$$ 6.44785 0.208212
$$960$$ 0 0
$$961$$ −26.5418 −0.856187
$$962$$ 0 0
$$963$$ 15.3489i 0.494612i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 53.4925i 1.72020i 0.510124 + 0.860101i $$0.329600\pi$$
−0.510124 + 0.860101i $$0.670400\pi$$
$$968$$ 0 0
$$969$$ 0.804635 0.0258486
$$970$$ 0 0
$$971$$ 19.8881 0.638241 0.319120 0.947714i $$-0.396613\pi$$
0.319120 + 0.947714i $$0.396613\pi$$
$$972$$ 0 0
$$973$$ − 6.18231i − 0.198196i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 14.8758i − 0.475919i −0.971275 0.237960i $$-0.923521\pi$$
0.971275 0.237960i $$-0.0764786\pi$$
$$978$$ 0 0
$$979$$ −42.1785 −1.34803
$$980$$ 0 0
$$981$$ −67.4045 −2.15206
$$982$$ 0 0
$$983$$ 1.16403i 0.0371269i 0.999828 + 0.0185634i $$0.00590926\pi$$
−0.999828 + 0.0185634i $$0.994091\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 16.9308i − 0.538913i
$$988$$ 0 0
$$989$$ −1.66507 −0.0529460
$$990$$ 0 0
$$991$$ −47.8693 −1.52062 −0.760310 0.649561i $$-0.774953\pi$$
−0.760310 + 0.649561i $$0.774953\pi$$
$$992$$ 0 0
$$993$$ − 38.9904i − 1.23732i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 11.4353i − 0.362159i −0.983468 0.181079i $$-0.942041\pi$$
0.983468 0.181079i $$-0.0579591\pi$$
$$998$$ 0 0
$$999$$ 61.0601 1.93186
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 4600.2.e.v.4049.10 10
5.2 odd 4 4600.2.a.bg.1.5 yes 5
5.3 odd 4 4600.2.a.bc.1.1 5
5.4 even 2 inner 4600.2.e.v.4049.1 10
20.3 even 4 9200.2.a.cw.1.5 5
20.7 even 4 9200.2.a.cs.1.1 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.1 5 5.3 odd 4
4600.2.a.bg.1.5 yes 5 5.2 odd 4
4600.2.e.v.4049.1 10 5.4 even 2 inner
4600.2.e.v.4049.10 10 1.1 even 1 trivial
9200.2.a.cs.1.1 5 20.7 even 4
9200.2.a.cw.1.5 5 20.3 even 4