# Properties

 Label 512.2.a.c.1.1 Level $512$ Weight $2$ Character 512.1 Self dual yes Analytic conductor $4.088$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.08834058349$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 512.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421 q^{3} +2.82843 q^{5} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.41421 q^{3} +2.82843 q^{5} -4.00000 q^{7} -1.00000 q^{9} -1.41421 q^{11} -2.82843 q^{13} -4.00000 q^{15} -4.00000 q^{17} +7.07107 q^{19} +5.65685 q^{21} -4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} -8.48528 q^{29} -8.00000 q^{31} +2.00000 q^{33} -11.3137 q^{35} -2.82843 q^{37} +4.00000 q^{39} +2.00000 q^{41} -4.24264 q^{43} -2.82843 q^{45} +9.00000 q^{49} +5.65685 q^{51} -2.82843 q^{53} -4.00000 q^{55} -10.0000 q^{57} -4.24264 q^{59} +8.48528 q^{61} +4.00000 q^{63} -8.00000 q^{65} -4.24264 q^{67} +5.65685 q^{69} +4.00000 q^{71} -4.00000 q^{73} -4.24264 q^{75} +5.65685 q^{77} +8.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} -11.3137 q^{85} +12.0000 q^{87} +12.0000 q^{89} +11.3137 q^{91} +11.3137 q^{93} +20.0000 q^{95} -4.00000 q^{97} +1.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 2 q^{9} + O(q^{10})$$ $$2 q - 8 q^{7} - 2 q^{9} - 8 q^{15} - 8 q^{17} - 8 q^{23} + 6 q^{25} - 16 q^{31} + 4 q^{33} + 8 q^{39} + 4 q^{41} + 18 q^{49} - 8 q^{55} - 20 q^{57} + 8 q^{63} - 16 q^{65} + 8 q^{71} - 8 q^{73} + 16 q^{79} - 10 q^{81} + 24 q^{87} + 24 q^{89} + 40 q^{95} - 8 q^{97} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.41421 −0.816497 −0.408248 0.912871i $$-0.633860\pi$$
−0.408248 + 0.912871i $$0.633860\pi$$
$$4$$ 0 0
$$5$$ 2.82843 1.26491 0.632456 0.774597i $$-0.282047\pi$$
0.632456 + 0.774597i $$0.282047\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.41421 −0.426401 −0.213201 0.977008i $$-0.568389\pi$$
−0.213201 + 0.977008i $$0.568389\pi$$
$$12$$ 0 0
$$13$$ −2.82843 −0.784465 −0.392232 0.919866i $$-0.628297\pi$$
−0.392232 + 0.919866i $$0.628297\pi$$
$$14$$ 0 0
$$15$$ −4.00000 −1.03280
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 7.07107 1.62221 0.811107 0.584898i $$-0.198865\pi$$
0.811107 + 0.584898i $$0.198865\pi$$
$$20$$ 0 0
$$21$$ 5.65685 1.23443
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 5.65685 1.08866
$$28$$ 0 0
$$29$$ −8.48528 −1.57568 −0.787839 0.615882i $$-0.788800\pi$$
−0.787839 + 0.615882i $$0.788800\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ −11.3137 −1.91237
$$36$$ 0 0
$$37$$ −2.82843 −0.464991 −0.232495 0.972598i $$-0.574689\pi$$
−0.232495 + 0.972598i $$0.574689\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −4.24264 −0.646997 −0.323498 0.946229i $$-0.604859\pi$$
−0.323498 + 0.946229i $$0.604859\pi$$
$$44$$ 0 0
$$45$$ −2.82843 −0.421637
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 5.65685 0.792118
$$52$$ 0 0
$$53$$ −2.82843 −0.388514 −0.194257 0.980951i $$-0.562230\pi$$
−0.194257 + 0.980951i $$0.562230\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ −10.0000 −1.32453
$$58$$ 0 0
$$59$$ −4.24264 −0.552345 −0.276172 0.961108i $$-0.589066\pi$$
−0.276172 + 0.961108i $$0.589066\pi$$
$$60$$ 0 0
$$61$$ 8.48528 1.08643 0.543214 0.839594i $$-0.317207\pi$$
0.543214 + 0.839594i $$0.317207\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ −4.24264 −0.518321 −0.259161 0.965834i $$-0.583446\pi$$
−0.259161 + 0.965834i $$0.583446\pi$$
$$68$$ 0 0
$$69$$ 5.65685 0.681005
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ −4.24264 −0.489898
$$76$$ 0 0
$$77$$ 5.65685 0.644658
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 0 0
$$83$$ 9.89949 1.08661 0.543305 0.839535i $$-0.317173\pi$$
0.543305 + 0.839535i $$0.317173\pi$$
$$84$$ 0 0
$$85$$ −11.3137 −1.22714
$$86$$ 0 0
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 11.3137 1.18600
$$92$$ 0 0
$$93$$ 11.3137 1.17318
$$94$$ 0 0
$$95$$ 20.0000 2.05196
$$96$$ 0 0
$$97$$ −4.00000 −0.406138 −0.203069 0.979164i $$-0.565092\pi$$
−0.203069 + 0.979164i $$0.565092\pi$$
$$98$$ 0 0
$$99$$ 1.41421 0.142134
$$100$$ 0 0
$$101$$ −2.82843 −0.281439 −0.140720 0.990050i $$-0.544942\pi$$
−0.140720 + 0.990050i $$0.544942\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 16.0000 1.56144
$$106$$ 0 0
$$107$$ 9.89949 0.957020 0.478510 0.878082i $$-0.341177\pi$$
0.478510 + 0.878082i $$0.341177\pi$$
$$108$$ 0 0
$$109$$ 14.1421 1.35457 0.677285 0.735720i $$-0.263156\pi$$
0.677285 + 0.735720i $$0.263156\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ −11.3137 −1.05501
$$116$$ 0 0
$$117$$ 2.82843 0.261488
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 0 0
$$123$$ −2.82843 −0.255031
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 7.07107 0.617802 0.308901 0.951094i $$-0.400039\pi$$
0.308901 + 0.951094i $$0.400039\pi$$
$$132$$ 0 0
$$133$$ −28.2843 −2.45256
$$134$$ 0 0
$$135$$ 16.0000 1.37706
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −4.24264 −0.359856 −0.179928 0.983680i $$-0.557586\pi$$
−0.179928 + 0.983680i $$0.557586\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ −24.0000 −1.99309
$$146$$ 0 0
$$147$$ −12.7279 −1.04978
$$148$$ 0 0
$$149$$ 2.82843 0.231714 0.115857 0.993266i $$-0.463039\pi$$
0.115857 + 0.993266i $$0.463039\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ −22.6274 −1.81748
$$156$$ 0 0
$$157$$ 8.48528 0.677199 0.338600 0.940931i $$-0.390047\pi$$
0.338600 + 0.940931i $$0.390047\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ 9.89949 0.775388 0.387694 0.921788i $$-0.373272\pi$$
0.387694 + 0.921788i $$0.373272\pi$$
$$164$$ 0 0
$$165$$ 5.65685 0.440386
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −5.00000 −0.384615
$$170$$ 0 0
$$171$$ −7.07107 −0.540738
$$172$$ 0 0
$$173$$ −14.1421 −1.07521 −0.537603 0.843198i $$-0.680670\pi$$
−0.537603 + 0.843198i $$0.680670\pi$$
$$174$$ 0 0
$$175$$ −12.0000 −0.907115
$$176$$ 0 0
$$177$$ 6.00000 0.450988
$$178$$ 0 0
$$179$$ −1.41421 −0.105703 −0.0528516 0.998602i $$-0.516831\pi$$
−0.0528516 + 0.998602i $$0.516831\pi$$
$$180$$ 0 0
$$181$$ −8.48528 −0.630706 −0.315353 0.948974i $$-0.602123\pi$$
−0.315353 + 0.948974i $$0.602123\pi$$
$$182$$ 0 0
$$183$$ −12.0000 −0.887066
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 5.65685 0.413670
$$188$$ 0 0
$$189$$ −22.6274 −1.64590
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 11.3137 0.810191
$$196$$ 0 0
$$197$$ 14.1421 1.00759 0.503793 0.863825i $$-0.331938\pi$$
0.503793 + 0.863825i $$0.331938\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 6.00000 0.423207
$$202$$ 0 0
$$203$$ 33.9411 2.38220
$$204$$ 0 0
$$205$$ 5.65685 0.395092
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ −10.0000 −0.691714
$$210$$ 0 0
$$211$$ −24.0416 −1.65509 −0.827547 0.561396i $$-0.810264\pi$$
−0.827547 + 0.561396i $$0.810264\pi$$
$$212$$ 0 0
$$213$$ −5.65685 −0.387601
$$214$$ 0 0
$$215$$ −12.0000 −0.818393
$$216$$ 0 0
$$217$$ 32.0000 2.17230
$$218$$ 0 0
$$219$$ 5.65685 0.382255
$$220$$ 0 0
$$221$$ 11.3137 0.761042
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ 21.2132 1.40797 0.703985 0.710215i $$-0.251402\pi$$
0.703985 + 0.710215i $$0.251402\pi$$
$$228$$ 0 0
$$229$$ 25.4558 1.68217 0.841085 0.540903i $$-0.181918\pi$$
0.841085 + 0.540903i $$0.181918\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −11.3137 −0.734904
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ 0 0
$$243$$ −9.89949 −0.635053
$$244$$ 0 0
$$245$$ 25.4558 1.62631
$$246$$ 0 0
$$247$$ −20.0000 −1.27257
$$248$$ 0 0
$$249$$ −14.0000 −0.887214
$$250$$ 0 0
$$251$$ −4.24264 −0.267793 −0.133897 0.990995i $$-0.542749\pi$$
−0.133897 + 0.990995i $$0.542749\pi$$
$$252$$ 0 0
$$253$$ 5.65685 0.355643
$$254$$ 0 0
$$255$$ 16.0000 1.00196
$$256$$ 0 0
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 11.3137 0.703000
$$260$$ 0 0
$$261$$ 8.48528 0.525226
$$262$$ 0 0
$$263$$ −28.0000 −1.72655 −0.863277 0.504730i $$-0.831592\pi$$
−0.863277 + 0.504730i $$0.831592\pi$$
$$264$$ 0 0
$$265$$ −8.00000 −0.491436
$$266$$ 0 0
$$267$$ −16.9706 −1.03858
$$268$$ 0 0
$$269$$ −19.7990 −1.20717 −0.603583 0.797300i $$-0.706261\pi$$
−0.603583 + 0.797300i $$0.706261\pi$$
$$270$$ 0 0
$$271$$ −32.0000 −1.94386 −0.971931 0.235267i $$-0.924404\pi$$
−0.971931 + 0.235267i $$0.924404\pi$$
$$272$$ 0 0
$$273$$ −16.0000 −0.968364
$$274$$ 0 0
$$275$$ −4.24264 −0.255841
$$276$$ 0 0
$$277$$ 8.48528 0.509831 0.254916 0.966963i $$-0.417952\pi$$
0.254916 + 0.966963i $$0.417952\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ 0 0
$$283$$ −24.0416 −1.42913 −0.714563 0.699571i $$-0.753375\pi$$
−0.714563 + 0.699571i $$0.753375\pi$$
$$284$$ 0 0
$$285$$ −28.2843 −1.67542
$$286$$ 0 0
$$287$$ −8.00000 −0.472225
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 5.65685 0.331611
$$292$$ 0 0
$$293$$ −8.48528 −0.495715 −0.247858 0.968796i $$-0.579727\pi$$
−0.247858 + 0.968796i $$0.579727\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ −8.00000 −0.464207
$$298$$ 0 0
$$299$$ 11.3137 0.654289
$$300$$ 0 0
$$301$$ 16.9706 0.978167
$$302$$ 0 0
$$303$$ 4.00000 0.229794
$$304$$ 0 0
$$305$$ 24.0000 1.37424
$$306$$ 0 0
$$307$$ 7.07107 0.403567 0.201784 0.979430i $$-0.435326\pi$$
0.201784 + 0.979430i $$0.435326\pi$$
$$308$$ 0 0
$$309$$ 16.9706 0.965422
$$310$$ 0 0
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ 11.3137 0.637455
$$316$$ 0 0
$$317$$ 2.82843 0.158860 0.0794301 0.996840i $$-0.474690\pi$$
0.0794301 + 0.996840i $$0.474690\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ −14.0000 −0.781404
$$322$$ 0 0
$$323$$ −28.2843 −1.57378
$$324$$ 0 0
$$325$$ −8.48528 −0.470679
$$326$$ 0 0
$$327$$ −20.0000 −1.10600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.7279 −0.699590 −0.349795 0.936826i $$-0.613749\pi$$
−0.349795 + 0.936826i $$0.613749\pi$$
$$332$$ 0 0
$$333$$ 2.82843 0.154997
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ 30.0000 1.63420 0.817102 0.576493i $$-0.195579\pi$$
0.817102 + 0.576493i $$0.195579\pi$$
$$338$$ 0 0
$$339$$ −19.7990 −1.07533
$$340$$ 0 0
$$341$$ 11.3137 0.612672
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ −15.5563 −0.835109 −0.417554 0.908652i $$-0.637113\pi$$
−0.417554 + 0.908652i $$0.637113\pi$$
$$348$$ 0 0
$$349$$ −19.7990 −1.05982 −0.529908 0.848055i $$-0.677773\pi$$
−0.529908 + 0.848055i $$0.677773\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 11.3137 0.600469
$$356$$ 0 0
$$357$$ −22.6274 −1.19757
$$358$$ 0 0
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ 31.0000 1.63158
$$362$$ 0 0
$$363$$ 12.7279 0.668043
$$364$$ 0 0
$$365$$ −11.3137 −0.592187
$$366$$ 0 0
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 11.3137 0.587378
$$372$$ 0 0
$$373$$ −36.7696 −1.90386 −0.951928 0.306323i $$-0.900901\pi$$
−0.951928 + 0.306323i $$0.900901\pi$$
$$374$$ 0 0
$$375$$ 8.00000 0.413118
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 29.6985 1.52551 0.762754 0.646688i $$-0.223847\pi$$
0.762754 + 0.646688i $$0.223847\pi$$
$$380$$ 0 0
$$381$$ 11.3137 0.579619
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ 4.24264 0.215666
$$388$$ 0 0
$$389$$ 19.7990 1.00385 0.501924 0.864912i $$-0.332626\pi$$
0.501924 + 0.864912i $$0.332626\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ −10.0000 −0.504433
$$394$$ 0 0
$$395$$ 22.6274 1.13851
$$396$$ 0 0
$$397$$ 2.82843 0.141955 0.0709773 0.997478i $$-0.477388\pi$$
0.0709773 + 0.997478i $$0.477388\pi$$
$$398$$ 0 0
$$399$$ 40.0000 2.00250
$$400$$ 0 0
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ 0 0
$$403$$ 22.6274 1.12715
$$404$$ 0 0
$$405$$ −14.1421 −0.702728
$$406$$ 0 0
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 25.4558 1.25564
$$412$$ 0 0
$$413$$ 16.9706 0.835067
$$414$$ 0 0
$$415$$ 28.0000 1.37447
$$416$$ 0 0
$$417$$ 6.00000 0.293821
$$418$$ 0 0
$$419$$ −26.8701 −1.31269 −0.656344 0.754462i $$-0.727898\pi$$
−0.656344 + 0.754462i $$0.727898\pi$$
$$420$$ 0 0
$$421$$ −8.48528 −0.413547 −0.206774 0.978389i $$-0.566296\pi$$
−0.206774 + 0.978389i $$0.566296\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ −33.9411 −1.64253
$$428$$ 0 0
$$429$$ −5.65685 −0.273115
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ −20.0000 −0.961139 −0.480569 0.876957i $$-0.659570\pi$$
−0.480569 + 0.876957i $$0.659570\pi$$
$$434$$ 0 0
$$435$$ 33.9411 1.62735
$$436$$ 0 0
$$437$$ −28.2843 −1.35302
$$438$$ 0 0
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ −35.3553 −1.67978 −0.839891 0.542754i $$-0.817381\pi$$
−0.839891 + 0.542754i $$0.817381\pi$$
$$444$$ 0 0
$$445$$ 33.9411 1.60896
$$446$$ 0 0
$$447$$ −4.00000 −0.189194
$$448$$ 0 0
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ −2.82843 −0.133185
$$452$$ 0 0
$$453$$ 5.65685 0.265782
$$454$$ 0 0
$$455$$ 32.0000 1.50018
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 0 0
$$459$$ −22.6274 −1.05616
$$460$$ 0 0
$$461$$ 14.1421 0.658665 0.329332 0.944214i $$-0.393176\pi$$
0.329332 + 0.944214i $$0.393176\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 32.0000 1.48396
$$466$$ 0 0
$$467$$ 18.3848 0.850746 0.425373 0.905018i $$-0.360143\pi$$
0.425373 + 0.905018i $$0.360143\pi$$
$$468$$ 0 0
$$469$$ 16.9706 0.783628
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 21.2132 0.973329
$$476$$ 0 0
$$477$$ 2.82843 0.129505
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ 0 0
$$483$$ −22.6274 −1.02958
$$484$$ 0 0
$$485$$ −11.3137 −0.513729
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ −15.5563 −0.702048 −0.351024 0.936366i $$-0.614166\pi$$
−0.351024 + 0.936366i $$0.614166\pi$$
$$492$$ 0 0
$$493$$ 33.9411 1.52863
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ −16.0000 −0.717698
$$498$$ 0 0
$$499$$ −12.7279 −0.569780 −0.284890 0.958560i $$-0.591957\pi$$
−0.284890 + 0.958560i $$0.591957\pi$$
$$500$$ 0 0
$$501$$ −16.9706 −0.758189
$$502$$ 0 0
$$503$$ −20.0000 −0.891756 −0.445878 0.895094i $$-0.647108\pi$$
−0.445878 + 0.895094i $$0.647108\pi$$
$$504$$ 0 0
$$505$$ −8.00000 −0.355995
$$506$$ 0 0
$$507$$ 7.07107 0.314037
$$508$$ 0 0
$$509$$ −2.82843 −0.125368 −0.0626839 0.998033i $$-0.519966\pi$$
−0.0626839 + 0.998033i $$0.519966\pi$$
$$510$$ 0 0
$$511$$ 16.0000 0.707798
$$512$$ 0 0
$$513$$ 40.0000 1.76604
$$514$$ 0 0
$$515$$ −33.9411 −1.49562
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 20.0000 0.877903
$$520$$ 0 0
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 0 0
$$523$$ 9.89949 0.432875 0.216437 0.976297i $$-0.430556\pi$$
0.216437 + 0.976297i $$0.430556\pi$$
$$524$$ 0 0
$$525$$ 16.9706 0.740656
$$526$$ 0 0
$$527$$ 32.0000 1.39394
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 4.24264 0.184115
$$532$$ 0 0
$$533$$ −5.65685 −0.245026
$$534$$ 0 0
$$535$$ 28.0000 1.21055
$$536$$ 0 0
$$537$$ 2.00000 0.0863064
$$538$$ 0 0
$$539$$ −12.7279 −0.548230
$$540$$ 0 0
$$541$$ 8.48528 0.364811 0.182405 0.983223i $$-0.441612\pi$$
0.182405 + 0.983223i $$0.441612\pi$$
$$542$$ 0 0
$$543$$ 12.0000 0.514969
$$544$$ 0 0
$$545$$ 40.0000 1.71341
$$546$$ 0 0
$$547$$ −26.8701 −1.14888 −0.574440 0.818546i $$-0.694780\pi$$
−0.574440 + 0.818546i $$0.694780\pi$$
$$548$$ 0 0
$$549$$ −8.48528 −0.362143
$$550$$ 0 0
$$551$$ −60.0000 −2.55609
$$552$$ 0 0
$$553$$ −32.0000 −1.36078
$$554$$ 0 0
$$555$$ 11.3137 0.480240
$$556$$ 0 0
$$557$$ 31.1127 1.31829 0.659144 0.752017i $$-0.270919\pi$$
0.659144 + 0.752017i $$0.270919\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ 7.07107 0.298010 0.149005 0.988836i $$-0.452393\pi$$
0.149005 + 0.988836i $$0.452393\pi$$
$$564$$ 0 0
$$565$$ 39.5980 1.66590
$$566$$ 0 0
$$567$$ 20.0000 0.839921
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 32.5269 1.36121 0.680604 0.732651i $$-0.261717\pi$$
0.680604 + 0.732651i $$0.261717\pi$$
$$572$$ 0 0
$$573$$ 11.3137 0.472637
$$574$$ 0 0
$$575$$ −12.0000 −0.500435
$$576$$ 0 0
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ 0 0
$$579$$ 5.65685 0.235091
$$580$$ 0 0
$$581$$ −39.5980 −1.64280
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ 0 0
$$585$$ 8.00000 0.330759
$$586$$ 0 0
$$587$$ −1.41421 −0.0583708 −0.0291854 0.999574i $$-0.509291\pi$$
−0.0291854 + 0.999574i $$0.509291\pi$$
$$588$$ 0 0
$$589$$ −56.5685 −2.33087
$$590$$ 0 0
$$591$$ −20.0000 −0.822690
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 45.2548 1.85527
$$596$$ 0 0
$$597$$ 5.65685 0.231520
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 0 0
$$603$$ 4.24264 0.172774
$$604$$ 0 0
$$605$$ −25.4558 −1.03493
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ −48.0000 −1.94506
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −8.48528 −0.342717 −0.171359 0.985209i $$-0.554816\pi$$
−0.171359 + 0.985209i $$0.554816\pi$$
$$614$$ 0 0
$$615$$ −8.00000 −0.322591
$$616$$ 0 0
$$617$$ 28.0000 1.12724 0.563619 0.826035i $$-0.309409\pi$$
0.563619 + 0.826035i $$0.309409\pi$$
$$618$$ 0 0
$$619$$ −1.41421 −0.0568420 −0.0284210 0.999596i $$-0.509048\pi$$
−0.0284210 + 0.999596i $$0.509048\pi$$
$$620$$ 0 0
$$621$$ −22.6274 −0.908007
$$622$$ 0 0
$$623$$ −48.0000 −1.92308
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 14.1421 0.564782
$$628$$ 0 0
$$629$$ 11.3137 0.451107
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 0 0
$$633$$ 34.0000 1.35138
$$634$$ 0 0
$$635$$ −22.6274 −0.897942
$$636$$ 0 0
$$637$$ −25.4558 −1.00860
$$638$$ 0 0
$$639$$ −4.00000 −0.158238
$$640$$ 0 0
$$641$$ −4.00000 −0.157991 −0.0789953 0.996875i $$-0.525171\pi$$
−0.0789953 + 0.996875i $$0.525171\pi$$
$$642$$ 0 0
$$643$$ −4.24264 −0.167313 −0.0836567 0.996495i $$-0.526660\pi$$
−0.0836567 + 0.996495i $$0.526660\pi$$
$$644$$ 0 0
$$645$$ 16.9706 0.668215
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ 6.00000 0.235521
$$650$$ 0 0
$$651$$ −45.2548 −1.77368
$$652$$ 0 0
$$653$$ 48.0833 1.88164 0.940822 0.338902i $$-0.110055\pi$$
0.940822 + 0.338902i $$0.110055\pi$$
$$654$$ 0 0
$$655$$ 20.0000 0.781465
$$656$$ 0 0
$$657$$ 4.00000 0.156055
$$658$$ 0 0
$$659$$ 32.5269 1.26707 0.633534 0.773715i $$-0.281604\pi$$
0.633534 + 0.773715i $$0.281604\pi$$
$$660$$ 0 0
$$661$$ −31.1127 −1.21014 −0.605072 0.796171i $$-0.706856\pi$$
−0.605072 + 0.796171i $$0.706856\pi$$
$$662$$ 0 0
$$663$$ −16.0000 −0.621389
$$664$$ 0 0
$$665$$ −80.0000 −3.10227
$$666$$ 0 0
$$667$$ 33.9411 1.31421
$$668$$ 0 0
$$669$$ 33.9411 1.31224
$$670$$ 0 0
$$671$$ −12.0000 −0.463255
$$672$$ 0 0
$$673$$ −20.0000 −0.770943 −0.385472 0.922720i $$-0.625961\pi$$
−0.385472 + 0.922720i $$0.625961\pi$$
$$674$$ 0 0
$$675$$ 16.9706 0.653197
$$676$$ 0 0
$$677$$ 42.4264 1.63058 0.815290 0.579053i $$-0.196578\pi$$
0.815290 + 0.579053i $$0.196578\pi$$
$$678$$ 0 0
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ −30.0000 −1.14960
$$682$$ 0 0
$$683$$ 29.6985 1.13638 0.568190 0.822897i $$-0.307644\pi$$
0.568190 + 0.822897i $$0.307644\pi$$
$$684$$ 0 0
$$685$$ −50.9117 −1.94524
$$686$$ 0 0
$$687$$ −36.0000 −1.37349
$$688$$ 0 0
$$689$$ 8.00000 0.304776
$$690$$ 0 0
$$691$$ −46.6690 −1.77537 −0.887687 0.460447i $$-0.847689\pi$$
−0.887687 + 0.460447i $$0.847689\pi$$
$$692$$ 0 0
$$693$$ −5.65685 −0.214886
$$694$$ 0 0
$$695$$ −12.0000 −0.455186
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ 0 0
$$699$$ −16.9706 −0.641886
$$700$$ 0 0
$$701$$ −48.0833 −1.81608 −0.908040 0.418884i $$-0.862421\pi$$
−0.908040 + 0.418884i $$0.862421\pi$$
$$702$$ 0 0
$$703$$ −20.0000 −0.754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 11.3137 0.425496
$$708$$ 0 0
$$709$$ 8.48528 0.318671 0.159336 0.987224i $$-0.449065\pi$$
0.159336 + 0.987224i $$0.449065\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 11.3137 0.423109
$$716$$ 0 0
$$717$$ −11.3137 −0.422518
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 48.0000 1.78761
$$722$$ 0 0
$$723$$ −16.9706 −0.631142
$$724$$ 0 0
$$725$$ −25.4558 −0.945406
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 0 0
$$731$$ 16.9706 0.627679
$$732$$ 0 0
$$733$$ −19.7990 −0.731292 −0.365646 0.930754i $$-0.619152\pi$$
−0.365646 + 0.930754i $$0.619152\pi$$
$$734$$ 0 0
$$735$$ −36.0000 −1.32788
$$736$$ 0 0
$$737$$ 6.00000 0.221013
$$738$$ 0 0
$$739$$ −24.0416 −0.884386 −0.442193 0.896920i $$-0.645799\pi$$
−0.442193 + 0.896920i $$0.645799\pi$$
$$740$$ 0 0
$$741$$ 28.2843 1.03905
$$742$$ 0 0
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 0 0
$$747$$ −9.89949 −0.362204
$$748$$ 0 0
$$749$$ −39.5980 −1.44688
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 6.00000 0.218652
$$754$$ 0 0
$$755$$ −11.3137 −0.411748
$$756$$ 0 0
$$757$$ 25.4558 0.925208 0.462604 0.886565i $$-0.346915\pi$$
0.462604 + 0.886565i $$0.346915\pi$$
$$758$$ 0 0
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ −56.5685 −2.04792
$$764$$ 0 0
$$765$$ 11.3137 0.409048
$$766$$ 0 0
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ 44.0000 1.58668 0.793340 0.608778i $$-0.208340\pi$$
0.793340 + 0.608778i $$0.208340\pi$$
$$770$$ 0 0
$$771$$ 2.82843 0.101863
$$772$$ 0 0
$$773$$ 8.48528 0.305194 0.152597 0.988288i $$-0.451236\pi$$
0.152597 + 0.988288i $$0.451236\pi$$
$$774$$ 0 0
$$775$$ −24.0000 −0.862105
$$776$$ 0 0
$$777$$ −16.0000 −0.573997
$$778$$ 0 0
$$779$$ 14.1421 0.506695
$$780$$ 0 0
$$781$$ −5.65685 −0.202418
$$782$$ 0 0
$$783$$ −48.0000 −1.71538
$$784$$ 0 0
$$785$$ 24.0000 0.856597
$$786$$ 0 0
$$787$$ −46.6690 −1.66357 −0.831786 0.555097i $$-0.812681\pi$$
−0.831786 + 0.555097i $$0.812681\pi$$
$$788$$ 0 0
$$789$$ 39.5980 1.40973
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ 0 0
$$793$$ −24.0000 −0.852265
$$794$$ 0 0
$$795$$ 11.3137 0.401256
$$796$$ 0 0
$$797$$ 19.7990 0.701316 0.350658 0.936504i $$-0.385958\pi$$
0.350658 + 0.936504i $$0.385958\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ 0 0
$$803$$ 5.65685 0.199626
$$804$$ 0 0
$$805$$ 45.2548 1.59502
$$806$$ 0 0
$$807$$ 28.0000 0.985647
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −24.0416 −0.844216 −0.422108 0.906546i $$-0.638710\pi$$
−0.422108 + 0.906546i $$0.638710\pi$$
$$812$$ 0 0
$$813$$ 45.2548 1.58716
$$814$$ 0 0
$$815$$ 28.0000 0.980797
$$816$$ 0 0
$$817$$ −30.0000 −1.04957
$$818$$ 0 0
$$819$$ −11.3137 −0.395333
$$820$$ 0 0
$$821$$ 8.48528 0.296138 0.148069 0.988977i $$-0.452694\pi$$
0.148069 + 0.988977i $$0.452694\pi$$
$$822$$ 0 0
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 0 0
$$825$$ 6.00000 0.208893
$$826$$ 0 0
$$827$$ 21.2132 0.737655 0.368828 0.929498i $$-0.379759\pi$$
0.368828 + 0.929498i $$0.379759\pi$$
$$828$$ 0 0
$$829$$ −25.4558 −0.884118 −0.442059 0.896986i $$-0.645752\pi$$
−0.442059 + 0.896986i $$0.645752\pi$$
$$830$$ 0 0
$$831$$ −12.0000 −0.416275
$$832$$ 0 0
$$833$$ −36.0000 −1.24733
$$834$$ 0 0
$$835$$ 33.9411 1.17458
$$836$$ 0 0
$$837$$ −45.2548 −1.56424
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 43.0000 1.48276
$$842$$ 0 0
$$843$$ 28.2843 0.974162
$$844$$ 0 0
$$845$$ −14.1421 −0.486504
$$846$$ 0 0
$$847$$ 36.0000 1.23697
$$848$$ 0 0
$$849$$ 34.0000 1.16688
$$850$$ 0 0
$$851$$ 11.3137 0.387829
$$852$$ 0 0
$$853$$ 42.4264 1.45265 0.726326 0.687350i $$-0.241226\pi$$
0.726326 + 0.687350i $$0.241226\pi$$
$$854$$ 0 0
$$855$$ −20.0000 −0.683986
$$856$$ 0 0
$$857$$ −34.0000 −1.16142 −0.580709 0.814111i $$-0.697225\pi$$
−0.580709 + 0.814111i $$0.697225\pi$$
$$858$$ 0 0
$$859$$ −38.1838 −1.30281 −0.651407 0.758729i $$-0.725821\pi$$
−0.651407 + 0.758729i $$0.725821\pi$$
$$860$$ 0 0
$$861$$ 11.3137 0.385570
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −40.0000 −1.36004
$$866$$ 0 0
$$867$$ 1.41421 0.0480292
$$868$$ 0 0
$$869$$ −11.3137 −0.383791
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ 0 0
$$873$$ 4.00000 0.135379
$$874$$ 0 0
$$875$$ 22.6274 0.764946
$$876$$ 0 0
$$877$$ −8.48528 −0.286528 −0.143264 0.989685i $$-0.545760\pi$$
−0.143264 + 0.989685i $$0.545760\pi$$
$$878$$ 0 0
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ −34.0000 −1.14549 −0.572745 0.819734i $$-0.694121\pi$$
−0.572745 + 0.819734i $$0.694121\pi$$
$$882$$ 0 0
$$883$$ 41.0122 1.38017 0.690085 0.723728i $$-0.257573\pi$$
0.690085 + 0.723728i $$0.257573\pi$$
$$884$$ 0 0
$$885$$ 16.9706 0.570459
$$886$$ 0 0
$$887$$ −28.0000 −0.940148 −0.470074 0.882627i $$-0.655773\pi$$
−0.470074 + 0.882627i $$0.655773\pi$$
$$888$$ 0 0
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 7.07107 0.236890
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ 67.8823 2.26400
$$900$$ 0 0
$$901$$ 11.3137 0.376914
$$902$$ 0 0
$$903$$ −24.0000 −0.798670
$$904$$ 0 0
$$905$$ −24.0000 −0.797787
$$906$$ 0 0
$$907$$ 7.07107 0.234791 0.117395 0.993085i $$-0.462545\pi$$
0.117395 + 0.993085i $$0.462545\pi$$
$$908$$ 0 0
$$909$$ 2.82843 0.0938130
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ −14.0000 −0.463332
$$914$$ 0 0
$$915$$ −33.9411 −1.12206
$$916$$ 0 0
$$917$$ −28.2843 −0.934029
$$918$$ 0 0
$$919$$ 44.0000 1.45143 0.725713 0.687998i $$-0.241510\pi$$
0.725713 + 0.687998i $$0.241510\pi$$
$$920$$ 0 0
$$921$$ −10.0000 −0.329511
$$922$$ 0 0
$$923$$ −11.3137 −0.372395
$$924$$ 0 0
$$925$$ −8.48528 −0.278994
$$926$$ 0 0
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ 60.0000 1.96854 0.984268 0.176682i $$-0.0565363\pi$$
0.984268 + 0.176682i $$0.0565363\pi$$
$$930$$ 0 0
$$931$$ 63.6396 2.08570
$$932$$ 0 0
$$933$$ −28.2843 −0.925985
$$934$$ 0 0
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ −20.0000 −0.653372 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$938$$ 0 0
$$939$$ 19.7990 0.646116
$$940$$ 0 0
$$941$$ 36.7696 1.19865 0.599327 0.800505i $$-0.295435\pi$$
0.599327 + 0.800505i $$0.295435\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ −64.0000 −2.08192
$$946$$ 0 0
$$947$$ −60.8112 −1.97610 −0.988049 0.154140i $$-0.950739\pi$$
−0.988049 + 0.154140i $$0.950739\pi$$
$$948$$ 0 0
$$949$$ 11.3137 0.367259
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ 0 0
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 0 0
$$955$$ −22.6274 −0.732206
$$956$$ 0 0
$$957$$ −16.9706 −0.548580
$$958$$ 0 0
$$959$$ 72.0000 2.32500
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −9.89949 −0.319007
$$964$$ 0 0
$$965$$ −11.3137 −0.364201
$$966$$ 0 0
$$967$$ −20.0000 −0.643157 −0.321578 0.946883i $$-0.604213\pi$$
−0.321578 + 0.946883i $$0.604213\pi$$
$$968$$ 0 0
$$969$$ 40.0000 1.28499
$$970$$ 0 0
$$971$$ −15.5563 −0.499227 −0.249614 0.968346i $$-0.580304\pi$$
−0.249614 + 0.968346i $$0.580304\pi$$
$$972$$ 0 0
$$973$$ 16.9706 0.544051
$$974$$ 0 0
$$975$$ 12.0000 0.384308
$$976$$ 0 0
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 0 0
$$979$$ −16.9706 −0.542382
$$980$$ 0 0
$$981$$ −14.1421 −0.451524
$$982$$ 0 0
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 40.0000 1.27451
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.9706 0.539633
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 18.0000 0.571213
$$994$$ 0 0
$$995$$ −11.3137 −0.358669
$$996$$ 0 0
$$997$$ −14.1421 −0.447886 −0.223943 0.974602i $$-0.571893\pi$$
−0.223943 + 0.974602i $$0.571893\pi$$
$$998$$ 0 0
$$999$$ −16.0000 −0.506218
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 512.2.a.c.1.1 2
3.2 odd 2 4608.2.a.f.1.1 2
4.3 odd 2 512.2.a.d.1.2 yes 2
8.3 odd 2 512.2.a.d.1.1 yes 2
8.5 even 2 inner 512.2.a.c.1.2 yes 2
12.11 even 2 4608.2.a.m.1.1 2
16.3 odd 4 512.2.b.a.257.2 2
16.5 even 4 512.2.b.b.257.2 2
16.11 odd 4 512.2.b.a.257.1 2
16.13 even 4 512.2.b.b.257.1 2
24.5 odd 2 4608.2.a.f.1.2 2
24.11 even 2 4608.2.a.m.1.2 2
32.3 odd 8 1024.2.e.d.257.1 2
32.5 even 8 1024.2.e.e.769.1 2
32.11 odd 8 1024.2.e.d.769.1 2
32.13 even 8 1024.2.e.e.257.1 2
32.19 odd 8 1024.2.e.c.257.1 2
32.21 even 8 1024.2.e.b.769.1 2
32.27 odd 8 1024.2.e.c.769.1 2
32.29 even 8 1024.2.e.b.257.1 2
48.5 odd 4 4608.2.d.b.2305.1 2
48.11 even 4 4608.2.d.a.2305.1 2
48.29 odd 4 4608.2.d.b.2305.2 2
48.35 even 4 4608.2.d.a.2305.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.c.1.1 2 1.1 even 1 trivial
512.2.a.c.1.2 yes 2 8.5 even 2 inner
512.2.a.d.1.1 yes 2 8.3 odd 2
512.2.a.d.1.2 yes 2 4.3 odd 2
512.2.b.a.257.1 2 16.11 odd 4
512.2.b.a.257.2 2 16.3 odd 4
512.2.b.b.257.1 2 16.13 even 4
512.2.b.b.257.2 2 16.5 even 4
1024.2.e.b.257.1 2 32.29 even 8
1024.2.e.b.769.1 2 32.21 even 8
1024.2.e.c.257.1 2 32.19 odd 8
1024.2.e.c.769.1 2 32.27 odd 8
1024.2.e.d.257.1 2 32.3 odd 8
1024.2.e.d.769.1 2 32.11 odd 8
1024.2.e.e.257.1 2 32.13 even 8
1024.2.e.e.769.1 2 32.5 even 8
4608.2.a.f.1.1 2 3.2 odd 2
4608.2.a.f.1.2 2 24.5 odd 2
4608.2.a.m.1.1 2 12.11 even 2
4608.2.a.m.1.2 2 24.11 even 2
4608.2.d.a.2305.1 2 48.11 even 4
4608.2.d.a.2305.2 2 48.35 even 4
4608.2.d.b.2305.1 2 48.5 odd 4
4608.2.d.b.2305.2 2 48.29 odd 4