Properties

 Label 720.6.f.g.289.2 Level $720$ Weight $6$ Character 720.289 Analytic conductor $115.476$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$115.476350265$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 289.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 720.289 Dual form 720.6.f.g.289.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+(55.0000 + 10.0000i) q^{5} -4.00000i q^{7} +O(q^{10})$$ $$q+(55.0000 + 10.0000i) q^{5} -4.00000i q^{7} -500.000 q^{11} -288.000i q^{13} +1516.00i q^{17} -1344.00 q^{19} -4100.00i q^{23} +(2925.00 + 1100.00i) q^{25} -2646.00 q^{29} +5612.00 q^{31} +(40.0000 - 220.000i) q^{35} +7288.00i q^{37} +18986.0 q^{41} +2404.00i q^{43} -8900.00i q^{47} +16791.0 q^{49} -39804.0i q^{53} +(-27500.0 - 5000.00i) q^{55} +28300.0 q^{59} +18290.0 q^{61} +(2880.00 - 15840.0i) q^{65} +65956.0i q^{67} -28800.0 q^{71} -30808.0i q^{73} +2000.00i q^{77} +60228.0 q^{79} -2468.00i q^{83} +(-15160.0 + 83380.0i) q^{85} +22678.0 q^{89} -1152.00 q^{91} +(-73920.0 - 13440.0i) q^{95} +36968.0i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 110 q^{5} + O(q^{10})$$ $$2 q + 110 q^{5} - 1000 q^{11} - 2688 q^{19} + 5850 q^{25} - 5292 q^{29} + 11224 q^{31} + 80 q^{35} + 37972 q^{41} + 33582 q^{49} - 55000 q^{55} + 56600 q^{59} + 36580 q^{61} + 5760 q^{65} - 57600 q^{71} + 120456 q^{79} - 30320 q^{85} + 45356 q^{89} - 2304 q^{91} - 147840 q^{95} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 55.0000 + 10.0000i 0.983870 + 0.178885i
$$6$$ 0 0
$$7$$ 4.00000i 0.0308542i −0.999881 0.0154271i $$-0.995089\pi$$
0.999881 0.0154271i $$-0.00491080\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −500.000 −1.24591 −0.622957 0.782256i $$-0.714069\pi$$
−0.622957 + 0.782256i $$0.714069\pi$$
$$12$$ 0 0
$$13$$ 288.000i 0.472644i −0.971675 0.236322i $$-0.924058\pi$$
0.971675 0.236322i $$-0.0759420\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1516.00i 1.27226i 0.771581 + 0.636132i $$0.219466\pi$$
−0.771581 + 0.636132i $$0.780534\pi$$
$$18$$ 0 0
$$19$$ −1344.00 −0.854113 −0.427056 0.904225i $$-0.640449\pi$$
−0.427056 + 0.904225i $$0.640449\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4100.00i 1.61609i −0.589124 0.808043i $$-0.700527\pi$$
0.589124 0.808043i $$-0.299473\pi$$
$$24$$ 0 0
$$25$$ 2925.00 + 1100.00i 0.936000 + 0.352000i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2646.00 −0.584245 −0.292122 0.956381i $$-0.594361\pi$$
−0.292122 + 0.956381i $$0.594361\pi$$
$$30$$ 0 0
$$31$$ 5612.00 1.04885 0.524425 0.851457i $$-0.324280\pi$$
0.524425 + 0.851457i $$0.324280\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 40.0000 220.000i 0.00551937 0.0303566i
$$36$$ 0 0
$$37$$ 7288.00i 0.875193i 0.899171 + 0.437597i $$0.144170\pi$$
−0.899171 + 0.437597i $$0.855830\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 18986.0 1.76390 0.881950 0.471343i $$-0.156231\pi$$
0.881950 + 0.471343i $$0.156231\pi$$
$$42$$ 0 0
$$43$$ 2404.00i 0.198273i 0.995074 + 0.0991364i $$0.0316080\pi$$
−0.995074 + 0.0991364i $$0.968392\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8900.00i 0.587686i −0.955854 0.293843i $$-0.905066\pi$$
0.955854 0.293843i $$-0.0949343\pi$$
$$48$$ 0 0
$$49$$ 16791.0 0.999048
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 39804.0i 1.94642i −0.229913 0.973211i $$-0.573844\pi$$
0.229913 0.973211i $$-0.426156\pi$$
$$54$$ 0 0
$$55$$ −27500.0 5000.00i −1.22582 0.222876i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 28300.0 1.05842 0.529208 0.848492i $$-0.322489\pi$$
0.529208 + 0.848492i $$0.322489\pi$$
$$60$$ 0 0
$$61$$ 18290.0 0.629345 0.314673 0.949200i $$-0.398105\pi$$
0.314673 + 0.949200i $$0.398105\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2880.00 15840.0i 0.0845491 0.465020i
$$66$$ 0 0
$$67$$ 65956.0i 1.79501i 0.441002 + 0.897506i $$0.354623\pi$$
−0.441002 + 0.897506i $$0.645377\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −28800.0 −0.678026 −0.339013 0.940782i $$-0.610093\pi$$
−0.339013 + 0.940782i $$0.610093\pi$$
$$72$$ 0 0
$$73$$ 30808.0i 0.676638i −0.941031 0.338319i $$-0.890142\pi$$
0.941031 0.338319i $$-0.109858\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2000.00i 0.0384418i
$$78$$ 0 0
$$79$$ 60228.0 1.08575 0.542876 0.839813i $$-0.317335\pi$$
0.542876 + 0.839813i $$0.317335\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 2468.00i 0.0393233i −0.999807 0.0196616i $$-0.993741\pi$$
0.999807 0.0196616i $$-0.00625890\pi$$
$$84$$ 0 0
$$85$$ −15160.0 + 83380.0i −0.227589 + 1.25174i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 22678.0 0.303480 0.151740 0.988420i $$-0.451512\pi$$
0.151740 + 0.988420i $$0.451512\pi$$
$$90$$ 0 0
$$91$$ −1152.00 −0.0145831
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −73920.0 13440.0i −0.840336 0.152788i
$$96$$ 0 0
$$97$$ 36968.0i 0.398930i 0.979905 + 0.199465i $$0.0639204\pi$$
−0.979905 + 0.199465i $$0.936080\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −167918. −1.63792 −0.818962 0.573848i $$-0.805450\pi$$
−0.818962 + 0.573848i $$0.805450\pi$$
$$102$$ 0 0
$$103$$ 154364.i 1.43368i 0.697236 + 0.716841i $$0.254413\pi$$
−0.697236 + 0.716841i $$0.745587\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 136788.i 1.15502i −0.816385 0.577509i $$-0.804025\pi$$
0.816385 0.577509i $$-0.195975\pi$$
$$108$$ 0 0
$$109$$ 53810.0 0.433807 0.216904 0.976193i $$-0.430404\pi$$
0.216904 + 0.976193i $$0.430404\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 82692.0i 0.609211i −0.952479 0.304605i $$-0.901475\pi$$
0.952479 0.304605i $$-0.0985245\pi$$
$$114$$ 0 0
$$115$$ 41000.0 225500.i 0.289094 1.59002i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6064.00 0.0392547
$$120$$ 0 0
$$121$$ 88949.0 0.552303
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 149875. + 89750.0i 0.857935 + 0.513759i
$$126$$ 0 0
$$127$$ 211780.i 1.16513i −0.812783 0.582567i $$-0.802048\pi$$
0.812783 0.582567i $$-0.197952\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 169500. 0.862962 0.431481 0.902122i $$-0.357991\pi$$
0.431481 + 0.902122i $$0.357991\pi$$
$$132$$ 0 0
$$133$$ 5376.00i 0.0263530i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 252036.i 1.14726i 0.819115 + 0.573629i $$0.194465\pi$$
−0.819115 + 0.573629i $$0.805535\pi$$
$$138$$ 0 0
$$139$$ 192016. 0.842947 0.421474 0.906841i $$-0.361513\pi$$
0.421474 + 0.906841i $$0.361513\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 144000.i 0.588874i
$$144$$ 0 0
$$145$$ −145530. 26460.0i −0.574821 0.104513i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 235694. 0.869727 0.434863 0.900496i $$-0.356797\pi$$
0.434863 + 0.900496i $$0.356797\pi$$
$$150$$ 0 0
$$151$$ 371492. 1.32589 0.662944 0.748669i $$-0.269307\pi$$
0.662944 + 0.748669i $$0.269307\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 308660. + 56120.0i 1.03193 + 0.187624i
$$156$$ 0 0
$$157$$ 264952.i 0.857863i −0.903337 0.428932i $$-0.858890\pi$$
0.903337 0.428932i $$-0.141110\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −16400.0 −0.0498631
$$162$$ 0 0
$$163$$ 403124.i 1.18842i 0.804310 + 0.594210i $$0.202535\pi$$
−0.804310 + 0.594210i $$0.797465\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 261900.i 0.726682i 0.931656 + 0.363341i $$0.118364\pi$$
−0.931656 + 0.363341i $$0.881636\pi$$
$$168$$ 0 0
$$169$$ 288349. 0.776608
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 326228.i 0.828716i −0.910114 0.414358i $$-0.864006\pi$$
0.910114 0.414358i $$-0.135994\pi$$
$$174$$ 0 0
$$175$$ 4400.00 11700.0i 0.0108607 0.0288796i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 109516. 0.255473 0.127736 0.991808i $$-0.459229\pi$$
0.127736 + 0.991808i $$0.459229\pi$$
$$180$$ 0 0
$$181$$ −53146.0 −0.120580 −0.0602898 0.998181i $$-0.519202\pi$$
−0.0602898 + 0.998181i $$0.519202\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −72880.0 + 400840.i −0.156559 + 0.861076i
$$186$$ 0 0
$$187$$ 758000.i 1.58513i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 232056. 0.460267 0.230133 0.973159i $$-0.426084\pi$$
0.230133 + 0.973159i $$0.426084\pi$$
$$192$$ 0 0
$$193$$ 1.03067e6i 1.99172i −0.0909274 0.995858i $$-0.528983\pi$$
0.0909274 0.995858i $$-0.471017\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 522796.i 0.959769i −0.877332 0.479884i $$-0.840679\pi$$
0.877332 0.479884i $$-0.159321\pi$$
$$198$$ 0 0
$$199$$ −215292. −0.385385 −0.192693 0.981259i $$-0.561722\pi$$
−0.192693 + 0.981259i $$0.561722\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 10584.0i 0.0180264i
$$204$$ 0 0
$$205$$ 1.04423e6 + 189860.i 1.73545 + 0.315536i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 672000. 1.06415
$$210$$ 0 0
$$211$$ 1.03008e6 1.59281 0.796407 0.604762i $$-0.206732\pi$$
0.796407 + 0.604762i $$0.206732\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −24040.0 + 132220.i −0.0354681 + 0.195075i
$$216$$ 0 0
$$217$$ 22448.0i 0.0323615i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 436608. 0.601327
$$222$$ 0 0
$$223$$ 456020.i 0.614075i −0.951697 0.307038i $$-0.900662\pi$$
0.951697 0.307038i $$-0.0993378\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 434252.i 0.559342i 0.960096 + 0.279671i $$0.0902253\pi$$
−0.960096 + 0.279671i $$0.909775\pi$$
$$228$$ 0 0
$$229$$ 722710. 0.910700 0.455350 0.890313i $$-0.349514\pi$$
0.455350 + 0.890313i $$0.349514\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 565348.i 0.682223i 0.940023 + 0.341111i $$0.110803\pi$$
−0.940023 + 0.341111i $$0.889197\pi$$
$$234$$ 0 0
$$235$$ 89000.0 489500.i 0.105128 0.578207i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −324904. −0.367926 −0.183963 0.982933i $$-0.558893\pi$$
−0.183963 + 0.982933i $$0.558893\pi$$
$$240$$ 0 0
$$241$$ 915262. 1.01509 0.507543 0.861626i $$-0.330554\pi$$
0.507543 + 0.861626i $$0.330554\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 923505. + 167910.i 0.982933 + 0.178715i
$$246$$ 0 0
$$247$$ 387072.i 0.403691i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1.36708e6 1.36965 0.684823 0.728709i $$-0.259879\pi$$
0.684823 + 0.728709i $$0.259879\pi$$
$$252$$ 0 0
$$253$$ 2.05000e6i 2.01350i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 892932.i 0.843307i 0.906757 + 0.421653i $$0.138550\pi$$
−0.906757 + 0.421653i $$0.861450\pi$$
$$258$$ 0 0
$$259$$ 29152.0 0.0270034
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1.86650e6i 1.66394i −0.554818 0.831972i $$-0.687212\pi$$
0.554818 0.831972i $$-0.312788\pi$$
$$264$$ 0 0
$$265$$ 398040. 2.18922e6i 0.348187 1.91503i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −1.37227e6 −1.15627 −0.578133 0.815943i $$-0.696218\pi$$
−0.578133 + 0.815943i $$0.696218\pi$$
$$270$$ 0 0
$$271$$ −458644. −0.379361 −0.189680 0.981846i $$-0.560745\pi$$
−0.189680 + 0.981846i $$0.560745\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.46250e6 550000.i −1.16618 0.438562i
$$276$$ 0 0
$$277$$ 985408.i 0.771643i 0.922573 + 0.385822i $$0.126082\pi$$
−0.922573 + 0.385822i $$0.873918\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −165798. −0.125260 −0.0626302 0.998037i $$-0.519949\pi$$
−0.0626302 + 0.998037i $$0.519949\pi$$
$$282$$ 0 0
$$283$$ 1.66471e6i 1.23558i 0.786342 + 0.617792i $$0.211972\pi$$
−0.786342 + 0.617792i $$0.788028\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 75944.0i 0.0544238i
$$288$$ 0 0
$$289$$ −878399. −0.618653
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.55104e6i 1.73600i −0.496567 0.867998i $$-0.665406\pi$$
0.496567 0.867998i $$-0.334594\pi$$
$$294$$ 0 0
$$295$$ 1.55650e6 + 283000.i 1.04134 + 0.189335i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.18080e6 −0.763833
$$300$$ 0 0
$$301$$ 9616.00 0.00611756
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1.00595e6 + 182900.i 0.619194 + 0.112581i
$$306$$ 0 0
$$307$$ 736020.i 0.445701i 0.974853 + 0.222851i $$0.0715362\pi$$
−0.974853 + 0.222851i $$0.928464\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1.71660e6 1.00639 0.503197 0.864172i $$-0.332157\pi$$
0.503197 + 0.864172i $$0.332157\pi$$
$$312$$ 0 0
$$313$$ 2.83851e6i 1.63768i 0.574020 + 0.818842i $$0.305383\pi$$
−0.574020 + 0.818842i $$0.694617\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.27605e6i 0.713215i −0.934254 0.356607i $$-0.883933\pi$$
0.934254 0.356607i $$-0.116067\pi$$
$$318$$ 0 0
$$319$$ 1.32300e6 0.727919
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 2.03750e6i 1.08666i
$$324$$ 0 0
$$325$$ 316800. 842400.i 0.166371 0.442395i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −35600.0 −0.0181326
$$330$$ 0 0
$$331$$ −443992. −0.222744 −0.111372 0.993779i $$-0.535524\pi$$
−0.111372 + 0.993779i $$0.535524\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −659560. + 3.62758e6i −0.321101 + 1.76606i
$$336$$ 0 0
$$337$$ 2.71326e6i 1.30142i −0.759328 0.650708i $$-0.774472\pi$$
0.759328 0.650708i $$-0.225528\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.80600e6 −1.30678
$$342$$ 0 0
$$343$$ 134392.i 0.0616791i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1.31051e6i 0.584273i 0.956377 + 0.292137i $$0.0943662\pi$$
−0.956377 + 0.292137i $$0.905634\pi$$
$$348$$ 0 0
$$349$$ 298910. 0.131364 0.0656821 0.997841i $$-0.479078\pi$$
0.0656821 + 0.997841i $$0.479078\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 737996.i 0.315223i −0.987501 0.157611i $$-0.949621\pi$$
0.987501 0.157611i $$-0.0503793\pi$$
$$354$$ 0 0
$$355$$ −1.58400e6 288000.i −0.667090 0.121289i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 2.34074e6 0.958557 0.479278 0.877663i $$-0.340898\pi$$
0.479278 + 0.877663i $$0.340898\pi$$
$$360$$ 0 0
$$361$$ −669763. −0.270491
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 308080. 1.69444e6i 0.121041 0.665724i
$$366$$ 0 0
$$367$$ 127292.i 0.0493328i 0.999696 + 0.0246664i $$0.00785236\pi$$
−0.999696 + 0.0246664i $$0.992148\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −159216. −0.0600554
$$372$$ 0 0
$$373$$ 4.03870e6i 1.50303i 0.659713 + 0.751517i $$0.270678\pi$$
−0.659713 + 0.751517i $$0.729322\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 762048.i 0.276140i
$$378$$ 0 0
$$379$$ 1.01214e6 0.361944 0.180972 0.983488i $$-0.442076\pi$$
0.180972 + 0.983488i $$0.442076\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 2.37610e6i 0.827690i −0.910347 0.413845i $$-0.864186\pi$$
0.910347 0.413845i $$-0.135814\pi$$
$$384$$ 0 0
$$385$$ −20000.0 + 110000.i −0.00687667 + 0.0378217i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 1.42497e6 0.477456 0.238728 0.971087i $$-0.423270\pi$$
0.238728 + 0.971087i $$0.423270\pi$$
$$390$$ 0 0
$$391$$ 6.21560e6 2.05609
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3.31254e6 + 602280.i 1.06824 + 0.194225i
$$396$$ 0 0
$$397$$ 1.69345e6i 0.539257i −0.962964 0.269628i $$-0.913099\pi$$
0.962964 0.269628i $$-0.0869009\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.84501e6 0.883532 0.441766 0.897130i $$-0.354352\pi$$
0.441766 + 0.897130i $$0.354352\pi$$
$$402$$ 0 0
$$403$$ 1.61626e6i 0.495733i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.64400e6i 1.09042i
$$408$$ 0 0
$$409$$ 1.89069e6 0.558873 0.279436 0.960164i $$-0.409852\pi$$
0.279436 + 0.960164i $$0.409852\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 113200.i 0.0326566i
$$414$$ 0 0
$$415$$ 24680.0 135740.i 0.00703437 0.0386890i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4.60930e6 −1.28263 −0.641313 0.767280i $$-0.721610\pi$$
−0.641313 + 0.767280i $$0.721610\pi$$
$$420$$ 0 0
$$421$$ −6.04151e6 −1.66127 −0.830635 0.556817i $$-0.812022\pi$$
−0.830635 + 0.556817i $$0.812022\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −1.66760e6 + 4.43430e6i −0.447837 + 1.19084i
$$426$$ 0 0
$$427$$ 73160.0i 0.0194180i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −3800.00 −0.000985350 −0.000492675 1.00000i $$-0.500157\pi$$
−0.000492675 1.00000i $$0.500157\pi$$
$$432$$ 0 0
$$433$$ 250736.i 0.0642683i 0.999484 + 0.0321342i $$0.0102304\pi$$
−0.999484 + 0.0321342i $$0.989770\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5.51040e6i 1.38032i
$$438$$ 0 0
$$439$$ −3.58873e6 −0.888750 −0.444375 0.895841i $$-0.646574\pi$$
−0.444375 + 0.895841i $$0.646574\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1.41479e6i 0.342517i −0.985226 0.171258i $$-0.945217\pi$$
0.985226 0.171258i $$-0.0547833\pi$$
$$444$$ 0 0
$$445$$ 1.24729e6 + 226780.i 0.298585 + 0.0542881i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −829806. −0.194250 −0.0971249 0.995272i $$-0.530965\pi$$
−0.0971249 + 0.995272i $$0.530965\pi$$
$$450$$ 0 0
$$451$$ −9.49300e6 −2.19767
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −63360.0 11520.0i −0.0143478 0.00260870i
$$456$$ 0 0
$$457$$ 4.68198e6i 1.04867i −0.851512 0.524335i $$-0.824314\pi$$
0.851512 0.524335i $$-0.175686\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 141930. 0.0311044 0.0155522 0.999879i $$-0.495049\pi$$
0.0155522 + 0.999879i $$0.495049\pi$$
$$462$$ 0 0
$$463$$ 727476.i 0.157713i −0.996886 0.0788563i $$-0.974873\pi$$
0.996886 0.0788563i $$-0.0251268\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4.47640e6i 0.949809i 0.880037 + 0.474905i $$0.157517\pi$$
−0.880037 + 0.474905i $$0.842483\pi$$
$$468$$ 0 0
$$469$$ 263824. 0.0553837
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.20200e6i 0.247031i
$$474$$ 0 0
$$475$$ −3.93120e6 1.47840e6i −0.799450 0.300648i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 1.32718e6 0.264297 0.132149 0.991230i $$-0.457812\pi$$
0.132149 + 0.991230i $$0.457812\pi$$
$$480$$ 0 0
$$481$$ 2.09894e6 0.413655
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −369680. + 2.03324e6i −0.0713628 + 0.392495i
$$486$$ 0 0
$$487$$ 4.11647e6i 0.786507i −0.919430 0.393253i $$-0.871350\pi$$
0.919430 0.393253i $$-0.128650\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6.12316e6 −1.14623 −0.573115 0.819475i $$-0.694265\pi$$
−0.573115 + 0.819475i $$0.694265\pi$$
$$492$$ 0 0
$$493$$ 4.01134e6i 0.743313i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 115200.i 0.0209200i
$$498$$ 0 0
$$499$$ −7.90490e6 −1.42117 −0.710584 0.703613i $$-0.751569\pi$$
−0.710584 + 0.703613i $$0.751569\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.97628e6i 0.700741i 0.936611 + 0.350370i $$0.113944\pi$$
−0.936611 + 0.350370i $$0.886056\pi$$
$$504$$ 0 0
$$505$$ −9.23549e6 1.67918e6i −1.61150 0.293001i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 781914. 0.133772 0.0668859 0.997761i $$-0.478694\pi$$
0.0668859 + 0.997761i $$0.478694\pi$$
$$510$$ 0 0
$$511$$ −123232. −0.0208772
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1.54364e6 + 8.49002e6i −0.256465 + 1.41056i
$$516$$ 0 0
$$517$$ 4.45000e6i 0.732207i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −5.82694e6 −0.940472 −0.470236 0.882541i $$-0.655831\pi$$
−0.470236 + 0.882541i $$0.655831\pi$$
$$522$$ 0 0
$$523$$ 9.78938e6i 1.56495i 0.622681 + 0.782476i $$0.286043\pi$$
−0.622681 + 0.782476i $$0.713957\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.50779e6i 1.33441i
$$528$$ 0 0
$$529$$ −1.03737e7 −1.61173
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 5.46797e6i 0.833696i
$$534$$ 0 0
$$535$$ 1.36788e6 7.52334e6i 0.206616 1.13639i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −8.39550e6 −1.24473
$$540$$ 0 0
$$541$$ 4.76059e6 0.699307 0.349653 0.936879i $$-0.386299\pi$$
0.349653 + 0.936879i $$0.386299\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 2.95955e6 + 538100.i 0.426810 + 0.0776018i
$$546$$ 0 0
$$547$$ 1.16595e6i 0.166614i −0.996524 0.0833069i $$-0.973452\pi$$
0.996524 0.0833069i $$-0.0265482\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.55622e6 0.499011
$$552$$ 0 0
$$553$$ 240912.i 0.0335001i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.61293e6i 0.220282i −0.993916 0.110141i $$-0.964870\pi$$
0.993916 0.110141i $$-0.0351302\pi$$
$$558$$ 0 0
$$559$$ 692352. 0.0937125
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 3.40603e6i 0.452874i 0.974026 + 0.226437i $$0.0727077\pi$$
−0.974026 + 0.226437i $$0.927292\pi$$
$$564$$ 0 0
$$565$$ 826920. 4.54806e6i 0.108979 0.599384i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1.44009e7 −1.86470 −0.932350 0.361557i $$-0.882245\pi$$
−0.932350 + 0.361557i $$0.882245\pi$$
$$570$$ 0 0
$$571$$ −4.74772e6 −0.609389 −0.304695 0.952450i $$-0.598554\pi$$
−0.304695 + 0.952450i $$0.598554\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.51000e6 1.19925e7i 0.568862 1.51266i
$$576$$ 0 0
$$577$$ 1.09094e7i 1.36415i 0.731283 + 0.682074i $$0.238922\pi$$
−0.731283 + 0.682074i $$0.761078\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −9872.00 −0.00121329
$$582$$ 0 0
$$583$$ 1.99020e7i 2.42508i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.53223e6i 1.02204i −0.859569 0.511019i $$-0.829268\pi$$
0.859569 0.511019i $$-0.170732\pi$$
$$588$$ 0 0
$$589$$ −7.54253e6 −0.895836
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4.63182e6i 0.540897i 0.962734 + 0.270449i $$0.0871721\pi$$
−0.962734 + 0.270449i $$0.912828\pi$$
$$594$$ 0 0
$$595$$ 333520. + 60640.0i 0.0386215 + 0.00702210i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.27598e6 −0.714684 −0.357342 0.933974i $$-0.616317\pi$$
−0.357342 + 0.933974i $$0.616317\pi$$
$$600$$ 0 0
$$601$$ 7.71988e6 0.871815 0.435907 0.899992i $$-0.356428\pi$$
0.435907 + 0.899992i $$0.356428\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 4.89220e6 + 889490.i 0.543395 + 0.0987990i
$$606$$ 0 0
$$607$$ 6.06160e6i 0.667753i −0.942617 0.333876i $$-0.891643\pi$$
0.942617 0.333876i $$-0.108357\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.56320e6 −0.277766
$$612$$ 0 0
$$613$$ 3.66489e6i 0.393921i −0.980411 0.196961i $$-0.936893\pi$$
0.980411 0.196961i $$-0.0631071\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.32522e6i 0.986157i −0.869985 0.493079i $$-0.835871\pi$$
0.869985 0.493079i $$-0.164129\pi$$
$$618$$ 0 0
$$619$$ −7.40162e6 −0.776426 −0.388213 0.921570i $$-0.626907\pi$$
−0.388213 + 0.921570i $$0.626907\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 90712.0i 0.00936364i
$$624$$ 0 0
$$625$$ 7.34562e6 + 6.43500e6i 0.752192 + 0.658944i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −1.10486e7 −1.11348
$$630$$ 0 0
$$631$$ −160052. −0.0160025 −0.00800125 0.999968i $$-0.502547\pi$$
−0.00800125 + 0.999968i $$0.502547\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2.11780e6 1.16479e7i 0.208425 1.14634i
$$636$$ 0 0
$$637$$ 4.83581e6i 0.472194i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.69565e7 1.63002 0.815008 0.579450i $$-0.196732\pi$$
0.815008 + 0.579450i $$0.196732\pi$$
$$642$$ 0 0
$$643$$ 1.10128e7i 1.05044i −0.850967 0.525219i $$-0.823984\pi$$
0.850967 0.525219i $$-0.176016\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 3.33848e6i 0.313537i −0.987635 0.156768i $$-0.949892\pi$$
0.987635 0.156768i $$-0.0501076\pi$$
$$648$$ 0 0
$$649$$ −1.41500e7 −1.31870
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 4.76181e6i 0.437008i 0.975836 + 0.218504i $$0.0701177\pi$$
−0.975836 + 0.218504i $$0.929882\pi$$
$$654$$ 0 0
$$655$$ 9.32250e6 + 1.69500e6i 0.849042 + 0.154371i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −798188. −0.0715965 −0.0357982 0.999359i $$-0.511397\pi$$
−0.0357982 + 0.999359i $$0.511397\pi$$
$$660$$ 0 0
$$661$$ −1.54048e7 −1.37136 −0.685682 0.727901i $$-0.740496\pi$$
−0.685682 + 0.727901i $$0.740496\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −53760.0 + 295680.i −0.00471417 + 0.0259279i
$$666$$ 0 0
$$667$$ 1.08486e7i 0.944189i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9.14500e6 −0.784111
$$672$$ 0 0
$$673$$ 976704.i 0.0831238i −0.999136 0.0415619i $$-0.986767\pi$$
0.999136 0.0415619i $$-0.0132334\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1.93885e7i 1.62582i 0.582388 + 0.812911i $$0.302119\pi$$
−0.582388 + 0.812911i $$0.697881\pi$$
$$678$$ 0 0
$$679$$ 147872. 0.0123087
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 5.25573e6i 0.431103i −0.976492 0.215552i $$-0.930845\pi$$
0.976492 0.215552i $$-0.0691550\pi$$
$$684$$ 0 0
$$685$$ −2.52036e6 + 1.38620e7i −0.205228 + 1.12875i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1.14636e7 −0.919965
$$690$$ 0 0
$$691$$ 5.45034e6 0.434238 0.217119 0.976145i $$-0.430334\pi$$
0.217119 + 0.976145i $$0.430334\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.05609e7 + 1.92016e6i 0.829350 + 0.150791i
$$696$$ 0 0
$$697$$ 2.87828e7i 2.24414i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 4.43961e6 0.341232 0.170616 0.985338i $$-0.445424\pi$$
0.170616 + 0.985338i $$0.445424\pi$$
$$702$$ 0 0
$$703$$ 9.79507e6i 0.747514i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 671672.i 0.0505369i
$$708$$ 0 0
$$709$$ −4.55918e6 −0.340621 −0.170310 0.985390i $$-0.554477\pi$$
−0.170310 + 0.985390i $$0.554477\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 2.30092e7i 1.69503i
$$714$$ 0 0
$$715$$ −1.44000e6 + 7.92000e6i −0.105341 + 0.579375i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.06630e7 1.49063 0.745317 0.666710i $$-0.232298\pi$$
0.745317 + 0.666710i $$0.232298\pi$$
$$720$$ 0 0
$$721$$ 617456. 0.0442352
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.73955e6 2.91060e6i −0.546853 0.205654i
$$726$$ 0 0
$$727$$ 5.48161e6i 0.384656i 0.981331 + 0.192328i $$0.0616037\pi$$
−0.981331 + 0.192328i $$0.938396\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3.64446e6 −0.252255
$$732$$ 0 0
$$733$$ 8.55579e6i 0.588166i −0.955780 0.294083i $$-0.904986\pi$$
0.955780 0.294083i $$-0.0950143\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 3.29780e7i 2.23643i
$$738$$ 0 0
$$739$$ −5.29119e6 −0.356404 −0.178202 0.983994i $$-0.557028\pi$$
−0.178202 + 0.983994i $$0.557028\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2.36432e6i 0.157121i −0.996909 0.0785606i $$-0.974968\pi$$
0.996909 0.0785606i $$-0.0250324\pi$$
$$744$$ 0 0
$$745$$ 1.29632e7 + 2.35694e6i 0.855698 + 0.155581i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −547152. −0.0356372
$$750$$ 0 0
$$751$$ 8.79694e6 0.569157 0.284578 0.958653i $$-0.408146\pi$$
0.284578 + 0.958653i $$0.408146\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 2.04321e7 + 3.71492e6i 1.30450 + 0.237182i
$$756$$ 0 0
$$757$$ 2.95808e7i 1.87616i −0.346421 0.938079i $$-0.612603\pi$$
0.346421 0.938079i $$-0.387397\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.26296e7 0.790549 0.395274 0.918563i $$-0.370649\pi$$
0.395274 + 0.918563i $$0.370649\pi$$
$$762$$ 0 0
$$763$$ 215240.i 0.0133848i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.15040e6i 0.500254i
$$768$$ 0 0
$$769$$ 2.32186e7 1.41586 0.707929 0.706283i $$-0.249630\pi$$
0.707929 + 0.706283i $$0.249630\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1.73201e7i 1.04256i 0.853386 + 0.521280i $$0.174545\pi$$
−0.853386 + 0.521280i $$0.825455\pi$$
$$774$$ 0 0
$$775$$ 1.64151e7 + 6.17320e6i 0.981724 + 0.369195i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.55172e7 −1.50657
$$780$$ 0 0
$$781$$ 1.44000e7 0.844763
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2.64952e6 1.45724e7i 0.153459 0.844026i
$$786$$ 0 0
$$787$$ 556676.i 0.0320380i 0.999872 + 0.0160190i $$0.00509923\pi$$
−0.999872 + 0.0160190i $$0.994901\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −330768. −0.0187967
$$792$$ 0 0
$$793$$ 5.26752e6i 0.297456i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3.00562e6i 0.167606i 0.996482 + 0.0838028i $$0.0267066\pi$$
−0.996482 + 0.0838028i $$0.973293\pi$$
$$798$$ 0 0
$$799$$