# Properties

 Label 4851.2.a.be.1.2 Level $4851$ Weight $2$ Character 4851.1 Self dual yes Analytic conductor $38.735$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.7354300205$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 4851.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +O(q^{10})$$ $$q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +4.82843 q^{10} -1.00000 q^{11} -0.828427 q^{13} +3.00000 q^{16} +4.41421 q^{17} +7.24264 q^{19} +7.65685 q^{20} -2.41421 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -3.24264 q^{29} +5.65685 q^{31} -1.58579 q^{32} +10.6569 q^{34} -9.48528 q^{37} +17.4853 q^{38} +8.82843 q^{40} +1.17157 q^{41} -2.75736 q^{43} -3.82843 q^{44} +16.8995 q^{46} -9.82843 q^{47} -2.41421 q^{50} -3.17157 q^{52} +7.17157 q^{53} -2.00000 q^{55} -7.82843 q^{58} +8.65685 q^{59} -4.00000 q^{61} +13.6569 q^{62} -9.82843 q^{64} -1.65685 q^{65} -3.17157 q^{67} +16.8995 q^{68} +4.17157 q^{71} +0.343146 q^{73} -22.8995 q^{74} +27.7279 q^{76} +13.3137 q^{79} +6.00000 q^{80} +2.82843 q^{82} +2.82843 q^{83} +8.82843 q^{85} -6.65685 q^{86} -4.41421 q^{88} +14.1421 q^{89} +26.7990 q^{92} -23.7279 q^{94} +14.4853 q^{95} -11.4853 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^10 - 2 * q^11 + 4 * q^13 + 6 * q^16 + 6 * q^17 + 6 * q^19 + 4 * q^20 - 2 * q^22 + 14 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^29 - 6 * q^32 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 12 * q^40 + 8 * q^41 - 14 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 2 * q^50 - 12 * q^52 + 20 * q^53 - 4 * q^55 - 10 * q^58 + 6 * q^59 - 8 * q^61 + 16 * q^62 - 14 * q^64 + 8 * q^65 - 12 * q^67 + 14 * q^68 + 14 * q^71 + 12 * q^73 - 26 * q^74 + 30 * q^76 + 4 * q^79 + 12 * q^80 + 12 * q^85 - 2 * q^86 - 6 * q^88 + 14 * q^92 - 22 * q^94 + 12 * q^95 - 6 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41421 1.70711 0.853553 0.521005i $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 0 0
$$4$$ 3.82843 1.91421
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 4.41421 1.56066
$$9$$ 0 0
$$10$$ 4.82843 1.52688
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −0.828427 −0.229764 −0.114882 0.993379i $$-0.536649\pi$$
−0.114882 + 0.993379i $$0.536649\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 4.41421 1.07060 0.535302 0.844661i $$-0.320198\pi$$
0.535302 + 0.844661i $$0.320198\pi$$
$$18$$ 0 0
$$19$$ 7.24264 1.66158 0.830788 0.556589i $$-0.187890\pi$$
0.830788 + 0.556589i $$0.187890\pi$$
$$20$$ 7.65685 1.71212
$$21$$ 0 0
$$22$$ −2.41421 −0.514712
$$23$$ 7.00000 1.45960 0.729800 0.683660i $$-0.239613\pi$$
0.729800 + 0.683660i $$0.239613\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3.24264 −0.602143 −0.301072 0.953602i $$-0.597344\pi$$
−0.301072 + 0.953602i $$0.597344\pi$$
$$30$$ 0 0
$$31$$ 5.65685 1.01600 0.508001 0.861357i $$-0.330385\pi$$
0.508001 + 0.861357i $$0.330385\pi$$
$$32$$ −1.58579 −0.280330
$$33$$ 0 0
$$34$$ 10.6569 1.82764
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.48528 −1.55937 −0.779685 0.626172i $$-0.784621\pi$$
−0.779685 + 0.626172i $$0.784621\pi$$
$$38$$ 17.4853 2.83649
$$39$$ 0 0
$$40$$ 8.82843 1.39590
$$41$$ 1.17157 0.182969 0.0914845 0.995807i $$-0.470839\pi$$
0.0914845 + 0.995807i $$0.470839\pi$$
$$42$$ 0 0
$$43$$ −2.75736 −0.420493 −0.210247 0.977648i $$-0.567427\pi$$
−0.210247 + 0.977648i $$0.567427\pi$$
$$44$$ −3.82843 −0.577157
$$45$$ 0 0
$$46$$ 16.8995 2.49169
$$47$$ −9.82843 −1.43362 −0.716812 0.697267i $$-0.754399\pi$$
−0.716812 + 0.697267i $$0.754399\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −2.41421 −0.341421
$$51$$ 0 0
$$52$$ −3.17157 −0.439818
$$53$$ 7.17157 0.985091 0.492546 0.870287i $$-0.336066\pi$$
0.492546 + 0.870287i $$0.336066\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −7.82843 −1.02792
$$59$$ 8.65685 1.12703 0.563513 0.826107i $$-0.309449\pi$$
0.563513 + 0.826107i $$0.309449\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 13.6569 1.73442
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ −1.65685 −0.205507
$$66$$ 0 0
$$67$$ −3.17157 −0.387469 −0.193735 0.981054i $$-0.562060\pi$$
−0.193735 + 0.981054i $$0.562060\pi$$
$$68$$ 16.8995 2.04936
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.17157 0.495075 0.247537 0.968878i $$-0.420379\pi$$
0.247537 + 0.968878i $$0.420379\pi$$
$$72$$ 0 0
$$73$$ 0.343146 0.0401622 0.0200811 0.999798i $$-0.493608\pi$$
0.0200811 + 0.999798i $$0.493608\pi$$
$$74$$ −22.8995 −2.66201
$$75$$ 0 0
$$76$$ 27.7279 3.18061
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.3137 1.49791 0.748955 0.662621i $$-0.230556\pi$$
0.748955 + 0.662621i $$0.230556\pi$$
$$80$$ 6.00000 0.670820
$$81$$ 0 0
$$82$$ 2.82843 0.312348
$$83$$ 2.82843 0.310460 0.155230 0.987878i $$-0.450388\pi$$
0.155230 + 0.987878i $$0.450388\pi$$
$$84$$ 0 0
$$85$$ 8.82843 0.957577
$$86$$ −6.65685 −0.717827
$$87$$ 0 0
$$88$$ −4.41421 −0.470557
$$89$$ 14.1421 1.49906 0.749532 0.661968i $$-0.230279\pi$$
0.749532 + 0.661968i $$0.230279\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 26.7990 2.79399
$$93$$ 0 0
$$94$$ −23.7279 −2.44735
$$95$$ 14.4853 1.48616
$$96$$ 0 0
$$97$$ −11.4853 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −3.82843 −0.382843
$$101$$ −4.89949 −0.487518 −0.243759 0.969836i $$-0.578381\pi$$
−0.243759 + 0.969836i $$0.578381\pi$$
$$102$$ 0 0
$$103$$ 12.4853 1.23021 0.615106 0.788445i $$-0.289113\pi$$
0.615106 + 0.788445i $$0.289113\pi$$
$$104$$ −3.65685 −0.358584
$$105$$ 0 0
$$106$$ 17.3137 1.68166
$$107$$ −9.65685 −0.933563 −0.466782 0.884373i $$-0.654587\pi$$
−0.466782 + 0.884373i $$0.654587\pi$$
$$108$$ 0 0
$$109$$ −6.82843 −0.654045 −0.327022 0.945017i $$-0.606045\pi$$
−0.327022 + 0.945017i $$0.606045\pi$$
$$110$$ −4.82843 −0.460372
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 7.65685 0.720296 0.360148 0.932895i $$-0.382726\pi$$
0.360148 + 0.932895i $$0.382726\pi$$
$$114$$ 0 0
$$115$$ 14.0000 1.30551
$$116$$ −12.4142 −1.15263
$$117$$ 0 0
$$118$$ 20.8995 1.92395
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −9.65685 −0.874291
$$123$$ 0 0
$$124$$ 21.6569 1.94484
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −16.0711 −1.42608 −0.713038 0.701125i $$-0.752681\pi$$
−0.713038 + 0.701125i $$0.752681\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ 0 0
$$130$$ −4.00000 −0.350823
$$131$$ 5.17157 0.451842 0.225921 0.974146i $$-0.427461\pi$$
0.225921 + 0.974146i $$0.427461\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −7.65685 −0.661451
$$135$$ 0 0
$$136$$ 19.4853 1.67085
$$137$$ −0.485281 −0.0414604 −0.0207302 0.999785i $$-0.506599\pi$$
−0.0207302 + 0.999785i $$0.506599\pi$$
$$138$$ 0 0
$$139$$ 8.41421 0.713684 0.356842 0.934165i $$-0.383853\pi$$
0.356842 + 0.934165i $$0.383853\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 10.0711 0.845145
$$143$$ 0.828427 0.0692766
$$144$$ 0 0
$$145$$ −6.48528 −0.538573
$$146$$ 0.828427 0.0685611
$$147$$ 0 0
$$148$$ −36.3137 −2.98497
$$149$$ 22.2132 1.81978 0.909888 0.414853i $$-0.136167\pi$$
0.909888 + 0.414853i $$0.136167\pi$$
$$150$$ 0 0
$$151$$ −18.8995 −1.53802 −0.769010 0.639237i $$-0.779250\pi$$
−0.769010 + 0.639237i $$0.779250\pi$$
$$152$$ 31.9706 2.59316
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 11.3137 0.908739
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 32.1421 2.55709
$$159$$ 0 0
$$160$$ −3.17157 −0.250735
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −20.9706 −1.64254 −0.821271 0.570539i $$-0.806734\pi$$
−0.821271 + 0.570539i $$0.806734\pi$$
$$164$$ 4.48528 0.350242
$$165$$ 0 0
$$166$$ 6.82843 0.529989
$$167$$ −5.17157 −0.400188 −0.200094 0.979777i $$-0.564125\pi$$
−0.200094 + 0.979777i $$0.564125\pi$$
$$168$$ 0 0
$$169$$ −12.3137 −0.947208
$$170$$ 21.3137 1.63469
$$171$$ 0 0
$$172$$ −10.5563 −0.804914
$$173$$ 14.8284 1.12738 0.563692 0.825985i $$-0.309380\pi$$
0.563692 + 0.825985i $$0.309380\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 34.1421 2.55906
$$179$$ −17.8284 −1.33256 −0.666280 0.745702i $$-0.732114\pi$$
−0.666280 + 0.745702i $$0.732114\pi$$
$$180$$ 0 0
$$181$$ −11.6569 −0.866447 −0.433224 0.901286i $$-0.642624\pi$$
−0.433224 + 0.901286i $$0.642624\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 30.8995 2.27794
$$185$$ −18.9706 −1.39474
$$186$$ 0 0
$$187$$ −4.41421 −0.322799
$$188$$ −37.6274 −2.74426
$$189$$ 0 0
$$190$$ 34.9706 2.53703
$$191$$ 21.6569 1.56703 0.783517 0.621370i $$-0.213423\pi$$
0.783517 + 0.621370i $$0.213423\pi$$
$$192$$ 0 0
$$193$$ −12.1421 −0.874010 −0.437005 0.899459i $$-0.643961\pi$$
−0.437005 + 0.899459i $$0.643961\pi$$
$$194$$ −27.7279 −1.99075
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.4142 1.16946 0.584732 0.811226i $$-0.301200\pi$$
0.584732 + 0.811226i $$0.301200\pi$$
$$198$$ 0 0
$$199$$ −19.7990 −1.40351 −0.701757 0.712417i $$-0.747601\pi$$
−0.701757 + 0.712417i $$0.747601\pi$$
$$200$$ −4.41421 −0.312132
$$201$$ 0 0
$$202$$ −11.8284 −0.832245
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.34315 0.163652
$$206$$ 30.1421 2.10010
$$207$$ 0 0
$$208$$ −2.48528 −0.172323
$$209$$ −7.24264 −0.500984
$$210$$ 0 0
$$211$$ 14.9706 1.03062 0.515308 0.857005i $$-0.327678\pi$$
0.515308 + 0.857005i $$0.327678\pi$$
$$212$$ 27.4558 1.88568
$$213$$ 0 0
$$214$$ −23.3137 −1.59369
$$215$$ −5.51472 −0.376101
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −16.4853 −1.11652
$$219$$ 0 0
$$220$$ −7.65685 −0.516225
$$221$$ −3.65685 −0.245987
$$222$$ 0 0
$$223$$ −22.9706 −1.53822 −0.769111 0.639115i $$-0.779301\pi$$
−0.769111 + 0.639115i $$0.779301\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 18.4853 1.22962
$$227$$ −14.9706 −0.993631 −0.496816 0.867856i $$-0.665497\pi$$
−0.496816 + 0.867856i $$0.665497\pi$$
$$228$$ 0 0
$$229$$ −15.6569 −1.03463 −0.517317 0.855794i $$-0.673069\pi$$
−0.517317 + 0.855794i $$0.673069\pi$$
$$230$$ 33.7990 2.22864
$$231$$ 0 0
$$232$$ −14.3137 −0.939741
$$233$$ −10.4142 −0.682258 −0.341129 0.940017i $$-0.610809\pi$$
−0.341129 + 0.940017i $$0.610809\pi$$
$$234$$ 0 0
$$235$$ −19.6569 −1.28227
$$236$$ 33.1421 2.15737
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.48528 0.419498 0.209749 0.977755i $$-0.432735\pi$$
0.209749 + 0.977755i $$0.432735\pi$$
$$240$$ 0 0
$$241$$ 7.31371 0.471117 0.235559 0.971860i $$-0.424308\pi$$
0.235559 + 0.971860i $$0.424308\pi$$
$$242$$ 2.41421 0.155192
$$243$$ 0 0
$$244$$ −15.3137 −0.980360
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ 24.9706 1.58563
$$249$$ 0 0
$$250$$ −28.9706 −1.83226
$$251$$ 2.51472 0.158728 0.0793638 0.996846i $$-0.474711\pi$$
0.0793638 + 0.996846i $$0.474711\pi$$
$$252$$ 0 0
$$253$$ −7.00000 −0.440086
$$254$$ −38.7990 −2.43447
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ −21.4558 −1.33838 −0.669189 0.743092i $$-0.733359\pi$$
−0.669189 + 0.743092i $$0.733359\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −6.34315 −0.393385
$$261$$ 0 0
$$262$$ 12.4853 0.771343
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 14.3431 0.881092
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.1421 −0.741699
$$269$$ 27.7990 1.69493 0.847467 0.530848i $$-0.178126\pi$$
0.847467 + 0.530848i $$0.178126\pi$$
$$270$$ 0 0
$$271$$ −22.9706 −1.39536 −0.697681 0.716408i $$-0.745785\pi$$
−0.697681 + 0.716408i $$0.745785\pi$$
$$272$$ 13.2426 0.802953
$$273$$ 0 0
$$274$$ −1.17157 −0.0707773
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −25.6569 −1.54157 −0.770785 0.637095i $$-0.780136\pi$$
−0.770785 + 0.637095i $$0.780136\pi$$
$$278$$ 20.3137 1.21834
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −14.4142 −0.859880 −0.429940 0.902857i $$-0.641465\pi$$
−0.429940 + 0.902857i $$0.641465\pi$$
$$282$$ 0 0
$$283$$ 20.3431 1.20927 0.604637 0.796501i $$-0.293318\pi$$
0.604637 + 0.796501i $$0.293318\pi$$
$$284$$ 15.9706 0.947679
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.48528 0.146193
$$290$$ −15.6569 −0.919402
$$291$$ 0 0
$$292$$ 1.31371 0.0768790
$$293$$ −5.72792 −0.334629 −0.167314 0.985904i $$-0.553509\pi$$
−0.167314 + 0.985904i $$0.553509\pi$$
$$294$$ 0 0
$$295$$ 17.3137 1.00804
$$296$$ −41.8701 −2.43365
$$297$$ 0 0
$$298$$ 53.6274 3.10655
$$299$$ −5.79899 −0.335364
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −45.6274 −2.62556
$$303$$ 0 0
$$304$$ 21.7279 1.24618
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ −17.3137 −0.988146 −0.494073 0.869421i $$-0.664492\pi$$
−0.494073 + 0.869421i $$0.664492\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 27.3137 1.55131
$$311$$ −29.6274 −1.68002 −0.840008 0.542573i $$-0.817450\pi$$
−0.840008 + 0.542573i $$0.817450\pi$$
$$312$$ 0 0
$$313$$ 4.79899 0.271255 0.135627 0.990760i $$-0.456695\pi$$
0.135627 + 0.990760i $$0.456695\pi$$
$$314$$ 16.8995 0.953694
$$315$$ 0 0
$$316$$ 50.9706 2.86732
$$317$$ 25.3137 1.42176 0.710880 0.703314i $$-0.248297\pi$$
0.710880 + 0.703314i $$0.248297\pi$$
$$318$$ 0 0
$$319$$ 3.24264 0.181553
$$320$$ −19.6569 −1.09885
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 31.9706 1.77889
$$324$$ 0 0
$$325$$ 0.828427 0.0459529
$$326$$ −50.6274 −2.80399
$$327$$ 0 0
$$328$$ 5.17157 0.285552
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 22.4853 1.23590 0.617951 0.786216i $$-0.287963\pi$$
0.617951 + 0.786216i $$0.287963\pi$$
$$332$$ 10.8284 0.594287
$$333$$ 0 0
$$334$$ −12.4853 −0.683164
$$335$$ −6.34315 −0.346563
$$336$$ 0 0
$$337$$ 3.85786 0.210151 0.105076 0.994464i $$-0.466492\pi$$
0.105076 + 0.994464i $$0.466492\pi$$
$$338$$ −29.7279 −1.61699
$$339$$ 0 0
$$340$$ 33.7990 1.83301
$$341$$ −5.65685 −0.306336
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −12.1716 −0.656247
$$345$$ 0 0
$$346$$ 35.7990 1.92457
$$347$$ −14.9706 −0.803662 −0.401831 0.915714i $$-0.631626\pi$$
−0.401831 + 0.915714i $$0.631626\pi$$
$$348$$ 0 0
$$349$$ 10.9706 0.587241 0.293620 0.955922i $$-0.405140\pi$$
0.293620 + 0.955922i $$0.405140\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.58579 0.0845227
$$353$$ −13.3137 −0.708617 −0.354309 0.935129i $$-0.615284\pi$$
−0.354309 + 0.935129i $$0.615284\pi$$
$$354$$ 0 0
$$355$$ 8.34315 0.442808
$$356$$ 54.1421 2.86953
$$357$$ 0 0
$$358$$ −43.0416 −2.27482
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 33.4558 1.76083
$$362$$ −28.1421 −1.47912
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0.686292 0.0359221
$$366$$ 0 0
$$367$$ −4.14214 −0.216218 −0.108109 0.994139i $$-0.534480\pi$$
−0.108109 + 0.994139i $$0.534480\pi$$
$$368$$ 21.0000 1.09470
$$369$$ 0 0
$$370$$ −45.7990 −2.38098
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.65685 −0.0857887 −0.0428943 0.999080i $$-0.513658\pi$$
−0.0428943 + 0.999080i $$0.513658\pi$$
$$374$$ −10.6569 −0.551053
$$375$$ 0 0
$$376$$ −43.3848 −2.23740
$$377$$ 2.68629 0.138351
$$378$$ 0 0
$$379$$ 18.8284 0.967151 0.483576 0.875303i $$-0.339338\pi$$
0.483576 + 0.875303i $$0.339338\pi$$
$$380$$ 55.4558 2.84482
$$381$$ 0 0
$$382$$ 52.2843 2.67510
$$383$$ −2.31371 −0.118225 −0.0591125 0.998251i $$-0.518827\pi$$
−0.0591125 + 0.998251i $$0.518827\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −29.3137 −1.49203
$$387$$ 0 0
$$388$$ −43.9706 −2.23227
$$389$$ 11.7990 0.598233 0.299116 0.954217i $$-0.403308\pi$$
0.299116 + 0.954217i $$0.403308\pi$$
$$390$$ 0 0
$$391$$ 30.8995 1.56265
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 39.6274 1.99640
$$395$$ 26.6274 1.33977
$$396$$ 0 0
$$397$$ 17.4853 0.877561 0.438781 0.898594i $$-0.355411\pi$$
0.438781 + 0.898594i $$0.355411\pi$$
$$398$$ −47.7990 −2.39595
$$399$$ 0 0
$$400$$ −3.00000 −0.150000
$$401$$ −7.51472 −0.375267 −0.187634 0.982239i $$-0.560082\pi$$
−0.187634 + 0.982239i $$0.560082\pi$$
$$402$$ 0 0
$$403$$ −4.68629 −0.233441
$$404$$ −18.7574 −0.933214
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.48528 0.470168
$$408$$ 0 0
$$409$$ −7.79899 −0.385635 −0.192818 0.981235i $$-0.561763\pi$$
−0.192818 + 0.981235i $$0.561763\pi$$
$$410$$ 5.65685 0.279372
$$411$$ 0 0
$$412$$ 47.7990 2.35489
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 5.65685 0.277684
$$416$$ 1.31371 0.0644099
$$417$$ 0 0
$$418$$ −17.4853 −0.855233
$$419$$ −14.7990 −0.722978 −0.361489 0.932376i $$-0.617731\pi$$
−0.361489 + 0.932376i $$0.617731\pi$$
$$420$$ 0 0
$$421$$ −7.00000 −0.341159 −0.170580 0.985344i $$-0.554564\pi$$
−0.170580 + 0.985344i $$0.554564\pi$$
$$422$$ 36.1421 1.75937
$$423$$ 0 0
$$424$$ 31.6569 1.53739
$$425$$ −4.41421 −0.214121
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −36.9706 −1.78704
$$429$$ 0 0
$$430$$ −13.3137 −0.642044
$$431$$ −6.97056 −0.335760 −0.167880 0.985807i $$-0.553692\pi$$
−0.167880 + 0.985807i $$0.553692\pi$$
$$432$$ 0 0
$$433$$ 0.857864 0.0412263 0.0206132 0.999788i $$-0.493438\pi$$
0.0206132 + 0.999788i $$0.493438\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −26.1421 −1.25198
$$437$$ 50.6985 2.42524
$$438$$ 0 0
$$439$$ 30.8995 1.47475 0.737376 0.675482i $$-0.236065\pi$$
0.737376 + 0.675482i $$0.236065\pi$$
$$440$$ −8.82843 −0.420879
$$441$$ 0 0
$$442$$ −8.82843 −0.419925
$$443$$ −33.6274 −1.59769 −0.798843 0.601539i $$-0.794554\pi$$
−0.798843 + 0.601539i $$0.794554\pi$$
$$444$$ 0 0
$$445$$ 28.2843 1.34080
$$446$$ −55.4558 −2.62591
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 4.00000 0.188772 0.0943858 0.995536i $$-0.469911\pi$$
0.0943858 + 0.995536i $$0.469911\pi$$
$$450$$ 0 0
$$451$$ −1.17157 −0.0551672
$$452$$ 29.3137 1.37880
$$453$$ 0 0
$$454$$ −36.1421 −1.69623
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.8284 0.880757 0.440378 0.897812i $$-0.354844\pi$$
0.440378 + 0.897812i $$0.354844\pi$$
$$458$$ −37.7990 −1.76623
$$459$$ 0 0
$$460$$ 53.5980 2.49902
$$461$$ −6.75736 −0.314722 −0.157361 0.987541i $$-0.550299\pi$$
−0.157361 + 0.987541i $$0.550299\pi$$
$$462$$ 0 0
$$463$$ 18.6274 0.865689 0.432845 0.901468i $$-0.357510\pi$$
0.432845 + 0.901468i $$0.357510\pi$$
$$464$$ −9.72792 −0.451607
$$465$$ 0 0
$$466$$ −25.1421 −1.16469
$$467$$ 8.31371 0.384713 0.192356 0.981325i $$-0.438387\pi$$
0.192356 + 0.981325i $$0.438387\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −47.4558 −2.18897
$$471$$ 0 0
$$472$$ 38.2132 1.75891
$$473$$ 2.75736 0.126784
$$474$$ 0 0
$$475$$ −7.24264 −0.332315
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 15.6569 0.716128
$$479$$ −14.0000 −0.639676 −0.319838 0.947472i $$-0.603629\pi$$
−0.319838 + 0.947472i $$0.603629\pi$$
$$480$$ 0 0
$$481$$ 7.85786 0.358288
$$482$$ 17.6569 0.804248
$$483$$ 0 0
$$484$$ 3.82843 0.174019
$$485$$ −22.9706 −1.04304
$$486$$ 0 0
$$487$$ −32.6274 −1.47849 −0.739245 0.673437i $$-0.764817\pi$$
−0.739245 + 0.673437i $$0.764817\pi$$
$$488$$ −17.6569 −0.799288
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.51472 0.0683583 0.0341791 0.999416i $$-0.489118\pi$$
0.0341791 + 0.999416i $$0.489118\pi$$
$$492$$ 0 0
$$493$$ −14.3137 −0.644657
$$494$$ −14.4853 −0.651724
$$495$$ 0 0
$$496$$ 16.9706 0.762001
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −31.7990 −1.42352 −0.711759 0.702424i $$-0.752101\pi$$
−0.711759 + 0.702424i $$0.752101\pi$$
$$500$$ −45.9411 −2.05455
$$501$$ 0 0
$$502$$ 6.07107 0.270965
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −9.79899 −0.436049
$$506$$ −16.8995 −0.751274
$$507$$ 0 0
$$508$$ −61.5269 −2.72982
$$509$$ 12.0000 0.531891 0.265945 0.963988i $$-0.414316\pi$$
0.265945 + 0.963988i $$0.414316\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −31.2426 −1.38074
$$513$$ 0 0
$$514$$ −51.7990 −2.28476
$$515$$ 24.9706 1.10033
$$516$$ 0 0
$$517$$ 9.82843 0.432254
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −7.31371 −0.320727
$$521$$ −31.1716 −1.36565 −0.682826 0.730581i $$-0.739249\pi$$
−0.682826 + 0.730581i $$0.739249\pi$$
$$522$$ 0 0
$$523$$ −22.2843 −0.974423 −0.487212 0.873284i $$-0.661986\pi$$
−0.487212 + 0.873284i $$0.661986\pi$$
$$524$$ 19.7990 0.864923
$$525$$ 0 0
$$526$$ −43.4558 −1.89476
$$527$$ 24.9706 1.08773
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 34.6274 1.50412
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −0.970563 −0.0420397
$$534$$ 0 0
$$535$$ −19.3137 −0.835004
$$536$$ −14.0000 −0.604708
$$537$$ 0 0
$$538$$ 67.1127 2.89343
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 19.6569 0.845114 0.422557 0.906336i $$-0.361133\pi$$
0.422557 + 0.906336i $$0.361133\pi$$
$$542$$ −55.4558 −2.38203
$$543$$ 0 0
$$544$$ −7.00000 −0.300123
$$545$$ −13.6569 −0.584995
$$546$$ 0 0
$$547$$ −24.2132 −1.03528 −0.517641 0.855598i $$-0.673190\pi$$
−0.517641 + 0.855598i $$0.673190\pi$$
$$548$$ −1.85786 −0.0793640
$$549$$ 0 0
$$550$$ 2.41421 0.102942
$$551$$ −23.4853 −1.00051
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −61.9411 −2.63163
$$555$$ 0 0
$$556$$ 32.2132 1.36614
$$557$$ 8.55635 0.362544 0.181272 0.983433i $$-0.441979\pi$$
0.181272 + 0.983433i $$0.441979\pi$$
$$558$$ 0 0
$$559$$ 2.28427 0.0966144
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −34.7990 −1.46791
$$563$$ −24.8284 −1.04639 −0.523197 0.852212i $$-0.675261\pi$$
−0.523197 + 0.852212i $$0.675261\pi$$
$$564$$ 0 0
$$565$$ 15.3137 0.644253
$$566$$ 49.1127 2.06436
$$567$$ 0 0
$$568$$ 18.4142 0.772643
$$569$$ −1.24264 −0.0520942 −0.0260471 0.999661i $$-0.508292\pi$$
−0.0260471 + 0.999661i $$0.508292\pi$$
$$570$$ 0 0
$$571$$ −3.92893 −0.164421 −0.0822103 0.996615i $$-0.526198\pi$$
−0.0822103 + 0.996615i $$0.526198\pi$$
$$572$$ 3.17157 0.132610
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −7.00000 −0.291920
$$576$$ 0 0
$$577$$ 7.65685 0.318759 0.159380 0.987217i $$-0.449051\pi$$
0.159380 + 0.987217i $$0.449051\pi$$
$$578$$ 6.00000 0.249567
$$579$$ 0 0
$$580$$ −24.8284 −1.03094
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −7.17157 −0.297016
$$584$$ 1.51472 0.0626795
$$585$$ 0 0
$$586$$ −13.8284 −0.571247
$$587$$ −8.68629 −0.358522 −0.179261 0.983802i $$-0.557371\pi$$
−0.179261 + 0.983802i $$0.557371\pi$$
$$588$$ 0 0
$$589$$ 40.9706 1.68816
$$590$$ 41.7990 1.72084
$$591$$ 0 0
$$592$$ −28.4558 −1.16953
$$593$$ 22.2721 0.914605 0.457302 0.889311i $$-0.348816\pi$$
0.457302 + 0.889311i $$0.348816\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 85.0416 3.48344
$$597$$ 0 0
$$598$$ −14.0000 −0.572503
$$599$$ 1.37258 0.0560822 0.0280411 0.999607i $$-0.491073\pi$$
0.0280411 + 0.999607i $$0.491073\pi$$
$$600$$ 0 0
$$601$$ −14.4853 −0.590867 −0.295433 0.955363i $$-0.595464\pi$$
−0.295433 + 0.955363i $$0.595464\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −72.3553 −2.94410
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ −18.6863 −0.758453 −0.379227 0.925304i $$-0.623810\pi$$
−0.379227 + 0.925304i $$0.623810\pi$$
$$608$$ −11.4853 −0.465790
$$609$$ 0 0
$$610$$ −19.3137 −0.781989
$$611$$ 8.14214 0.329396
$$612$$ 0 0
$$613$$ 25.1716 1.01667 0.508335 0.861159i $$-0.330261\pi$$
0.508335 + 0.861159i $$0.330261\pi$$
$$614$$ −41.7990 −1.68687
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −13.1716 −0.530268 −0.265134 0.964212i $$-0.585416\pi$$
−0.265134 + 0.964212i $$0.585416\pi$$
$$618$$ 0 0
$$619$$ 34.6274 1.39179 0.695897 0.718142i $$-0.255007\pi$$
0.695897 + 0.718142i $$0.255007\pi$$
$$620$$ 43.3137 1.73952
$$621$$ 0 0
$$622$$ −71.5269 −2.86797
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 11.5858 0.463061
$$627$$ 0 0
$$628$$ 26.7990 1.06940
$$629$$ −41.8701 −1.66947
$$630$$ 0 0
$$631$$ 26.2843 1.04636 0.523180 0.852222i $$-0.324745\pi$$
0.523180 + 0.852222i $$0.324745\pi$$
$$632$$ 58.7696 2.33773
$$633$$ 0 0
$$634$$ 61.1127 2.42710
$$635$$ −32.1421 −1.27552
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 7.82843 0.309930
$$639$$ 0 0
$$640$$ −41.1127 −1.62512
$$641$$ 12.4853 0.493139 0.246569 0.969125i $$-0.420697\pi$$
0.246569 + 0.969125i $$0.420697\pi$$
$$642$$ 0 0
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 77.1838 3.03675
$$647$$ 19.3137 0.759300 0.379650 0.925130i $$-0.376044\pi$$
0.379650 + 0.925130i $$0.376044\pi$$
$$648$$ 0 0
$$649$$ −8.65685 −0.339811
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −80.2843 −3.14417
$$653$$ −31.1127 −1.21753 −0.608767 0.793349i $$-0.708336\pi$$
−0.608767 + 0.793349i $$0.708336\pi$$
$$654$$ 0 0
$$655$$ 10.3431 0.404140
$$656$$ 3.51472 0.137227
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −34.4853 −1.34336 −0.671678 0.740843i $$-0.734426\pi$$
−0.671678 + 0.740843i $$0.734426\pi$$
$$660$$ 0 0
$$661$$ 11.9706 0.465601 0.232800 0.972525i $$-0.425211\pi$$
0.232800 + 0.972525i $$0.425211\pi$$
$$662$$ 54.2843 2.10982
$$663$$ 0 0
$$664$$ 12.4853 0.484523
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −22.6985 −0.878889
$$668$$ −19.7990 −0.766046
$$669$$ 0 0
$$670$$ −15.3137 −0.591620
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ 27.1716 1.04739 0.523694 0.851907i $$-0.324554\pi$$
0.523694 + 0.851907i $$0.324554\pi$$
$$674$$ 9.31371 0.358751
$$675$$ 0 0
$$676$$ −47.1421 −1.81316
$$677$$ 32.0122 1.23033 0.615164 0.788399i $$-0.289090\pi$$
0.615164 + 0.788399i $$0.289090\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 38.9706 1.49445
$$681$$ 0 0
$$682$$ −13.6569 −0.522948
$$683$$ −3.20101 −0.122483 −0.0612416 0.998123i $$-0.519506\pi$$
−0.0612416 + 0.998123i $$0.519506\pi$$
$$684$$ 0 0
$$685$$ −0.970563 −0.0370833
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −8.27208 −0.315370
$$689$$ −5.94113 −0.226339
$$690$$ 0 0
$$691$$ −11.4558 −0.435801 −0.217900 0.975971i $$-0.569921\pi$$
−0.217900 + 0.975971i $$0.569921\pi$$
$$692$$ 56.7696 2.15805
$$693$$ 0 0
$$694$$ −36.1421 −1.37194
$$695$$ 16.8284 0.638339
$$696$$ 0 0
$$697$$ 5.17157 0.195887
$$698$$ 26.4853 1.00248
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.89949 0.109512 0.0547562 0.998500i $$-0.482562\pi$$
0.0547562 + 0.998500i $$0.482562\pi$$
$$702$$ 0 0
$$703$$ −68.6985 −2.59101
$$704$$ 9.82843 0.370423
$$705$$ 0 0
$$706$$ −32.1421 −1.20969
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 25.7696 0.967796 0.483898 0.875124i $$-0.339221\pi$$
0.483898 + 0.875124i $$0.339221\pi$$
$$710$$ 20.1421 0.755921
$$711$$ 0 0
$$712$$ 62.4264 2.33953
$$713$$ 39.5980 1.48296
$$714$$ 0 0
$$715$$ 1.65685 0.0619628
$$716$$ −68.2548 −2.55080
$$717$$ 0 0
$$718$$ 19.3137 0.720781
$$719$$ −11.2010 −0.417727 −0.208864 0.977945i $$-0.566976\pi$$
−0.208864 + 0.977945i $$0.566976\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 80.7696 3.00593
$$723$$ 0 0
$$724$$ −44.6274 −1.65856
$$725$$ 3.24264 0.120429
$$726$$ 0 0
$$727$$ 46.0833 1.70913 0.854567 0.519342i $$-0.173823\pi$$
0.854567 + 0.519342i $$0.173823\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 1.65685 0.0613229
$$731$$ −12.1716 −0.450182
$$732$$ 0 0
$$733$$ 9.65685 0.356684 0.178342 0.983969i $$-0.442927\pi$$
0.178342 + 0.983969i $$0.442927\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ −11.1005 −0.409170
$$737$$ 3.17157 0.116826
$$738$$ 0 0
$$739$$ −39.6569 −1.45880 −0.729400 0.684087i $$-0.760201\pi$$
−0.729400 + 0.684087i $$0.760201\pi$$
$$740$$ −72.6274 −2.66984
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −33.7990 −1.23996 −0.619982 0.784616i $$-0.712860\pi$$
−0.619982 + 0.784616i $$0.712860\pi$$
$$744$$ 0 0
$$745$$ 44.4264 1.62766
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ −16.8995 −0.617907
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −29.3137 −1.06967 −0.534836 0.844956i $$-0.679627\pi$$
−0.534836 + 0.844956i $$0.679627\pi$$
$$752$$ −29.4853 −1.07522
$$753$$ 0 0
$$754$$ 6.48528 0.236180
$$755$$ −37.7990 −1.37565
$$756$$ 0 0
$$757$$ −7.68629 −0.279363 −0.139682 0.990196i $$-0.544608\pi$$
−0.139682 + 0.990196i $$0.544608\pi$$
$$758$$ 45.4558 1.65103
$$759$$ 0 0
$$760$$ 63.9411 2.31939
$$761$$ 0.201010 0.00728661 0.00364331 0.999993i $$-0.498840\pi$$
0.00364331 + 0.999993i $$0.498840\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 82.9117 2.99964
$$765$$ 0 0
$$766$$ −5.58579 −0.201823
$$767$$ −7.17157 −0.258950
$$768$$ 0 0
$$769$$ 33.7990 1.21882 0.609411 0.792854i $$-0.291406\pi$$
0.609411 + 0.792854i $$0.291406\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −46.4853 −1.67304
$$773$$ 51.6569 1.85797 0.928984 0.370120i $$-0.120683\pi$$
0.928984 + 0.370120i $$0.120683\pi$$
$$774$$ 0 0
$$775$$ −5.65685 −0.203200
$$776$$ −50.6985 −1.81997
$$777$$ 0 0
$$778$$ 28.4853 1.02125
$$779$$ 8.48528 0.304017
$$780$$ 0 0
$$781$$ −4.17157 −0.149271
$$782$$ 74.5980 2.66762
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 14.0000 0.499681
$$786$$ 0 0
$$787$$ 29.5269 1.05252 0.526260 0.850323i $$-0.323594\pi$$
0.526260 + 0.850323i $$0.323594\pi$$
$$788$$ 62.8406 2.23860
$$789$$ 0 0
$$790$$ 64.2843 2.28713
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 3.31371 0.117673
$$794$$ 42.2132 1.49809
$$795$$ 0 0
$$796$$ −75.7990 −2.68662
$$797$$ −11.1716 −0.395717 −0.197859 0.980231i $$-0.563399\pi$$
−0.197859 + 0.980231i $$0.563399\pi$$
$$798$$ 0 0
$$799$$ −43.3848 −1.53484
$$800$$ 1.58579 0.0560660
$$801$$ 0 0
$$802$$ −18.1421 −0.640621
$$803$$ −0.343146 −0.0121094
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −11.3137 −0.398508
$$807$$ 0 0
$$808$$ −21.6274 −0.760850
$$809$$ −10.8284 −0.380707 −0.190354 0.981716i $$-0.560963\pi$$
−0.190354 + 0.981716i $$0.560963\pi$$
$$810$$ 0 0
$$811$$ 14.9706 0.525688 0.262844 0.964838i $$-0.415340\pi$$
0.262844 + 0.964838i $$0.415340\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 22.8995 0.802627
$$815$$ −41.9411 −1.46913
$$816$$ 0 0
$$817$$ −19.9706 −0.698682
$$818$$ −18.8284 −0.658321
$$819$$ 0 0
$$820$$ 8.97056 0.313266
$$821$$ −38.8284 −1.35512 −0.677561 0.735467i $$-0.736963\pi$$
−0.677561 + 0.735467i $$0.736963\pi$$
$$822$$ 0 0
$$823$$ −33.1716 −1.15629 −0.578144 0.815935i $$-0.696223\pi$$
−0.578144 + 0.815935i $$0.696223\pi$$
$$824$$ 55.1127 1.91994
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 37.3137 1.29752 0.648762 0.760991i $$-0.275287\pi$$
0.648762 + 0.760991i $$0.275287\pi$$
$$828$$ 0 0
$$829$$ −6.17157 −0.214348 −0.107174 0.994240i $$-0.534180\pi$$
−0.107174 + 0.994240i $$0.534180\pi$$
$$830$$ 13.6569 0.474036
$$831$$ 0 0
$$832$$ 8.14214 0.282278
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −10.3431 −0.357939
$$836$$ −27.7279 −0.958990
$$837$$ 0 0
$$838$$ −35.7279 −1.23420
$$839$$ −8.97056 −0.309698 −0.154849 0.987938i $$-0.549489\pi$$
−0.154849 + 0.987938i $$0.549489\pi$$
$$840$$ 0 0
$$841$$ −18.4853 −0.637423
$$842$$ −16.8995 −0.582395
$$843$$ 0 0
$$844$$ 57.3137 1.97282
$$845$$ −24.6274 −0.847209
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 21.5147 0.738818
$$849$$ 0 0
$$850$$ −10.6569 −0.365527
$$851$$ −66.3970 −2.27606
$$852$$ 0 0
$$853$$ −12.4853 −0.427488 −0.213744 0.976890i $$-0.568566\pi$$
−0.213744 + 0.976890i $$0.568566\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −42.6274 −1.45698
$$857$$ 44.3553 1.51515 0.757575 0.652748i $$-0.226384\pi$$
0.757575 + 0.652748i $$0.226384\pi$$
$$858$$ 0 0
$$859$$ −24.4853 −0.835427 −0.417714 0.908579i $$-0.637168\pi$$
−0.417714 + 0.908579i $$0.637168\pi$$
$$860$$ −21.1127 −0.719937
$$861$$ 0 0
$$862$$ −16.8284 −0.573179
$$863$$ −2.62742 −0.0894383 −0.0447192 0.999000i $$-0.514239\pi$$
−0.0447192 + 0.999000i $$0.514239\pi$$
$$864$$ 0 0
$$865$$ 29.6569 1.00836
$$866$$ 2.07107 0.0703777
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −13.3137 −0.451637
$$870$$ 0 0
$$871$$ 2.62742 0.0890266
$$872$$ −30.1421 −1.02074
$$873$$ 0 0
$$874$$ 122.397 4.14014
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.48528 0.0839220 0.0419610 0.999119i $$-0.486639\pi$$
0.0419610 + 0.999119i $$0.486639\pi$$
$$878$$ 74.5980 2.51756
$$879$$ 0 0
$$880$$ −6.00000 −0.202260
$$881$$ −16.6863 −0.562175 −0.281088 0.959682i $$-0.590695\pi$$
−0.281088 + 0.959682i $$0.590695\pi$$
$$882$$ 0 0
$$883$$ 39.6569 1.33456 0.667280 0.744807i $$-0.267459\pi$$
0.667280 + 0.744807i $$0.267459\pi$$
$$884$$ −14.0000 −0.470871
$$885$$ 0 0
$$886$$ −81.1838 −2.72742
$$887$$ 13.3137 0.447031 0.223515 0.974700i $$-0.428247\pi$$
0.223515 + 0.974700i $$0.428247\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 68.2843 2.28889
$$891$$ 0 0
$$892$$ −87.9411 −2.94449
$$893$$ −71.1838 −2.38207
$$894$$ 0 0
$$895$$ −35.6569 −1.19188
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 9.65685 0.322253
$$899$$ −18.3431 −0.611778
$$900$$ 0 0
$$901$$ 31.6569 1.05464
$$902$$ −2.82843 −0.0941763
$$903$$ 0 0
$$904$$ 33.7990 1.12414
$$905$$ −23.3137 −0.774974
$$906$$ 0 0
$$907$$ −17.5147 −0.581567 −0.290783 0.956789i $$-0.593916\pi$$
−0.290783 + 0.956789i $$0.593916\pi$$
$$908$$ −57.3137 −1.90202
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 4.51472 0.149579 0.0747897 0.997199i $$-0.476171\pi$$
0.0747897 + 0.997199i $$0.476171\pi$$
$$912$$ 0 0
$$913$$ −2.82843 −0.0936073
$$914$$ 45.4558 1.50355
$$915$$ 0 0
$$916$$ −59.9411 −1.98051
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 56.3553 1.85899 0.929496 0.368833i $$-0.120243\pi$$
0.929496 + 0.368833i $$0.120243\pi$$
$$920$$ 61.7990 2.03745
$$921$$ 0 0
$$922$$ −16.3137 −0.537263
$$923$$ −3.45584 −0.113750
$$924$$ 0 0
$$925$$ 9.48528 0.311874
$$926$$ 44.9706 1.47782
$$927$$ 0 0
$$928$$ 5.14214 0.168799
$$929$$ 22.8284 0.748976 0.374488 0.927232i $$-0.377818\pi$$
0.374488 + 0.927232i $$0.377818\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −39.8701 −1.30599
$$933$$ 0 0
$$934$$ 20.0711 0.656745
$$935$$ −8.82843 −0.288720
$$936$$ 0 0
$$937$$ 5.85786 0.191368 0.0956840 0.995412i $$-0.469496\pi$$
0.0956840 + 0.995412i $$0.469496\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −75.2548 −2.45454
$$941$$ 12.0711 0.393506 0.196753 0.980453i $$-0.436960\pi$$
0.196753 + 0.980453i $$0.436960\pi$$
$$942$$ 0 0
$$943$$ 8.20101 0.267062
$$944$$ 25.9706 0.845270
$$945$$ 0 0
$$946$$ 6.65685 0.216433
$$947$$ −37.4853 −1.21811 −0.609054 0.793129i $$-0.708451\pi$$
−0.609054 + 0.793129i $$0.708451\pi$$
$$948$$ 0 0
$$949$$ −0.284271 −0.00922784
$$950$$ −17.4853 −0.567297
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 14.1421 0.458109 0.229054 0.973414i $$-0.426437\pi$$
0.229054 + 0.973414i $$0.426437\pi$$
$$954$$ 0 0
$$955$$ 43.3137 1.40160
$$956$$ 24.8284 0.803009
$$957$$ 0 0
$$958$$ −33.7990 −1.09200
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 18.9706 0.611635
$$963$$ 0 0
$$964$$ 28.0000 0.901819
$$965$$ −24.2843 −0.781738
$$966$$ 0 0
$$967$$ −21.0416 −0.676653 −0.338327 0.941029i $$-0.609861\pi$$
−0.338327 + 0.941029i $$0.609861\pi$$
$$968$$ 4.41421 0.141878
$$969$$ 0 0
$$970$$ −55.4558 −1.78058
$$971$$ −4.97056 −0.159513 −0.0797565 0.996814i $$-0.525414\pi$$
−0.0797565 + 0.996814i $$0.525414\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −78.7696 −2.52394
$$975$$ 0 0
$$976$$ −12.0000 −0.384111
$$977$$ 12.8284 0.410418 0.205209 0.978718i $$-0.434213\pi$$
0.205209 + 0.978718i $$0.434213\pi$$
$$978$$ 0 0
$$979$$ −14.1421 −0.451985
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 3.65685 0.116695
$$983$$ 2.02944 0.0647290 0.0323645 0.999476i $$-0.489696\pi$$
0.0323645 + 0.999476i $$0.489696\pi$$
$$984$$ 0 0
$$985$$ 32.8284 1.04600
$$986$$ −34.5563 −1.10050
$$987$$ 0 0
$$988$$ −22.9706 −0.730791
$$989$$ −19.3015 −0.613752
$$990$$ 0 0
$$991$$ −24.6863 −0.784186 −0.392093 0.919926i $$-0.628249\pi$$
−0.392093 + 0.919926i $$0.628249\pi$$
$$992$$ −8.97056 −0.284816
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −39.5980 −1.25534
$$996$$ 0 0
$$997$$ 9.85786 0.312202 0.156101 0.987741i $$-0.450108\pi$$
0.156101 + 0.987741i $$0.450108\pi$$
$$998$$ −76.7696 −2.43010
$$999$$ 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 4851.2.a.be.1.2 2
3.2 odd 2 1617.2.a.n.1.1 2
7.2 even 3 693.2.i.f.298.1 4
7.4 even 3 693.2.i.f.100.1 4
7.6 odd 2 4851.2.a.bd.1.2 2
21.2 odd 6 231.2.i.d.67.2 4
21.11 odd 6 231.2.i.d.100.2 yes 4
21.20 even 2 1617.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.i.d.67.2 4 21.2 odd 6
231.2.i.d.100.2 yes 4 21.11 odd 6
693.2.i.f.100.1 4 7.4 even 3
693.2.i.f.298.1 4 7.2 even 3
1617.2.a.m.1.1 2 21.20 even 2
1617.2.a.n.1.1 2 3.2 odd 2
4851.2.a.bd.1.2 2 7.6 odd 2
4851.2.a.be.1.2 2 1.1 even 1 trivial