Properties

Label 1280.3.e.f.639.4
Level $1280$
Weight $3$
Character 1280.639
Analytic conductor $34.877$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 639.4
Root \(0.273891i\) of defining polynomial
Character \(\chi\) \(=\) 1280.639
Dual form 1280.3.e.f.639.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.547781i q^{3} +(3.30219 - 3.75441i) q^{5} -10.0566 q^{7} +8.69994 q^{9} +O(q^{10})\) \(q+0.547781i q^{3} +(3.30219 - 3.75441i) q^{5} -10.0566 q^{7} +8.69994 q^{9} +17.2087 q^{11} -4.41325 q^{13} +(2.05659 + 1.80888i) q^{15} -27.0176i q^{17} -4.82650 q^{19} -5.50881i q^{21} -15.2653 q^{23} +(-3.19112 - 24.7955i) q^{25} +9.69569i q^{27} +2.38225i q^{29} +38.0352i q^{31} +9.42663i q^{33} +(-33.2087 + 37.7565i) q^{35} +16.5691 q^{37} -2.41749i q^{39} +13.3177 q^{41} -59.7918i q^{43} +(28.7288 - 32.6631i) q^{45} +62.4388 q^{47} +52.1351 q^{49} +14.7997 q^{51} -71.5952 q^{53} +(56.8265 - 64.6086i) q^{55} -2.64386i q^{57} -68.8265 q^{59} -40.9439i q^{61} -87.4917 q^{63} +(-14.5734 + 16.5691i) q^{65} -51.0080i q^{67} -8.36206i q^{69} +40.4527i q^{71} -35.8441i q^{73} +(13.5825 - 1.74804i) q^{75} -173.061 q^{77} -126.800i q^{79} +72.9883 q^{81} -75.1490i q^{83} +(-101.435 - 89.2172i) q^{85} -1.30495 q^{87} +106.523 q^{89} +44.3822 q^{91} -20.8350 q^{93} +(-15.9380 + 18.1206i) q^{95} -85.4351i q^{97} +149.715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{5} - 12 q^{7} - 18 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{5} - 12 q^{7} - 18 q^{9} - 8 q^{11} - 36 q^{15} + 24 q^{19} + 68 q^{23} + 10 q^{25} - 88 q^{35} + 208 q^{37} + 68 q^{41} + 232 q^{45} + 268 q^{47} - 62 q^{49} - 192 q^{51} - 64 q^{53} + 288 q^{55} - 360 q^{59} - 172 q^{63} + 304 q^{75} - 400 q^{77} + 238 q^{81} - 304 q^{85} - 584 q^{87} - 76 q^{89} + 208 q^{91} + 320 q^{93} - 32 q^{95} + 856 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.547781i 0.182594i 0.995824 + 0.0912969i \(0.0291012\pi\)
−0.995824 + 0.0912969i \(0.970899\pi\)
\(4\) 0 0
\(5\) 3.30219 3.75441i 0.660437 0.750881i
\(6\) 0 0
\(7\) −10.0566 −1.43666 −0.718328 0.695705i \(-0.755092\pi\)
−0.718328 + 0.695705i \(0.755092\pi\)
\(8\) 0 0
\(9\) 8.69994 0.966660
\(10\) 0 0
\(11\) 17.2087 1.56443 0.782216 0.623008i \(-0.214089\pi\)
0.782216 + 0.623008i \(0.214089\pi\)
\(12\) 0 0
\(13\) −4.41325 −0.339481 −0.169740 0.985489i \(-0.554293\pi\)
−0.169740 + 0.985489i \(0.554293\pi\)
\(14\) 0 0
\(15\) 2.05659 + 1.80888i 0.137106 + 0.120592i
\(16\) 0 0
\(17\) 27.0176i 1.58927i −0.607086 0.794636i \(-0.707662\pi\)
0.607086 0.794636i \(-0.292338\pi\)
\(18\) 0 0
\(19\) −4.82650 −0.254026 −0.127013 0.991901i \(-0.540539\pi\)
−0.127013 + 0.991901i \(0.540539\pi\)
\(20\) 0 0
\(21\) 5.50881i 0.262324i
\(22\) 0 0
\(23\) −15.2653 −0.663710 −0.331855 0.943330i \(-0.607675\pi\)
−0.331855 + 0.943330i \(0.607675\pi\)
\(24\) 0 0
\(25\) −3.19112 24.7955i −0.127645 0.991820i
\(26\) 0 0
\(27\) 9.69569i 0.359100i
\(28\) 0 0
\(29\) 2.38225i 0.0821465i 0.999156 + 0.0410733i \(0.0130777\pi\)
−0.999156 + 0.0410733i \(0.986922\pi\)
\(30\) 0 0
\(31\) 38.0352i 1.22694i 0.789717 + 0.613472i \(0.210228\pi\)
−0.789717 + 0.613472i \(0.789772\pi\)
\(32\) 0 0
\(33\) 9.42663i 0.285655i
\(34\) 0 0
\(35\) −33.2087 + 37.7565i −0.948821 + 1.07876i
\(36\) 0 0
\(37\) 16.5691 0.447814 0.223907 0.974610i \(-0.428119\pi\)
0.223907 + 0.974610i \(0.428119\pi\)
\(38\) 0 0
\(39\) 2.41749i 0.0619870i
\(40\) 0 0
\(41\) 13.3177 0.324822 0.162411 0.986723i \(-0.448073\pi\)
0.162411 + 0.986723i \(0.448073\pi\)
\(42\) 0 0
\(43\) 59.7918i 1.39051i −0.718765 0.695253i \(-0.755292\pi\)
0.718765 0.695253i \(-0.244708\pi\)
\(44\) 0 0
\(45\) 28.7288 32.6631i 0.638418 0.725846i
\(46\) 0 0
\(47\) 62.4388 1.32849 0.664243 0.747517i \(-0.268754\pi\)
0.664243 + 0.747517i \(0.268754\pi\)
\(48\) 0 0
\(49\) 52.1351 1.06398
\(50\) 0 0
\(51\) 14.7997 0.290191
\(52\) 0 0
\(53\) −71.5952 −1.35085 −0.675427 0.737427i \(-0.736041\pi\)
−0.675427 + 0.737427i \(0.736041\pi\)
\(54\) 0 0
\(55\) 56.8265 64.6086i 1.03321 1.17470i
\(56\) 0 0
\(57\) 2.64386i 0.0463836i
\(58\) 0 0
\(59\) −68.8265 −1.16655 −0.583275 0.812274i \(-0.698229\pi\)
−0.583275 + 0.812274i \(0.698229\pi\)
\(60\) 0 0
\(61\) 40.9439i 0.671212i −0.942002 0.335606i \(-0.891059\pi\)
0.942002 0.335606i \(-0.108941\pi\)
\(62\) 0 0
\(63\) −87.4917 −1.38876
\(64\) 0 0
\(65\) −14.5734 + 16.5691i −0.224206 + 0.254910i
\(66\) 0 0
\(67\) 51.0080i 0.761313i −0.924717 0.380656i \(-0.875698\pi\)
0.924717 0.380656i \(-0.124302\pi\)
\(68\) 0 0
\(69\) 8.36206i 0.121189i
\(70\) 0 0
\(71\) 40.4527i 0.569757i 0.958564 + 0.284878i \(0.0919532\pi\)
−0.958564 + 0.284878i \(0.908047\pi\)
\(72\) 0 0
\(73\) 35.8441i 0.491015i −0.969395 0.245508i \(-0.921045\pi\)
0.969395 0.245508i \(-0.0789547\pi\)
\(74\) 0 0
\(75\) 13.5825 1.74804i 0.181100 0.0233072i
\(76\) 0 0
\(77\) −173.061 −2.24755
\(78\) 0 0
\(79\) 126.800i 1.60506i −0.596612 0.802530i \(-0.703487\pi\)
0.596612 0.802530i \(-0.296513\pi\)
\(80\) 0 0
\(81\) 72.9883 0.901090
\(82\) 0 0
\(83\) 75.1490i 0.905409i −0.891661 0.452705i \(-0.850459\pi\)
0.891661 0.452705i \(-0.149541\pi\)
\(84\) 0 0
\(85\) −101.435 89.2172i −1.19335 1.04961i
\(86\) 0 0
\(87\) −1.30495 −0.0149994
\(88\) 0 0
\(89\) 106.523 1.19689 0.598445 0.801164i \(-0.295785\pi\)
0.598445 + 0.801164i \(0.295785\pi\)
\(90\) 0 0
\(91\) 44.3822 0.487717
\(92\) 0 0
\(93\) −20.8350 −0.224032
\(94\) 0 0
\(95\) −15.9380 + 18.1206i −0.167768 + 0.190744i
\(96\) 0 0
\(97\) 85.4351i 0.880774i −0.897808 0.440387i \(-0.854841\pi\)
0.897808 0.440387i \(-0.145159\pi\)
\(98\) 0 0
\(99\) 149.715 1.51227
\(100\) 0 0
\(101\) 83.2877i 0.824631i −0.911041 0.412315i \(-0.864720\pi\)
0.911041 0.412315i \(-0.135280\pi\)
\(102\) 0 0
\(103\) −47.2653 −0.458887 −0.229443 0.973322i \(-0.573691\pi\)
−0.229443 + 0.973322i \(0.573691\pi\)
\(104\) 0 0
\(105\) −20.6823 18.1911i −0.196974 0.173249i
\(106\) 0 0
\(107\) 189.628i 1.77223i 0.463467 + 0.886114i \(0.346605\pi\)
−0.463467 + 0.886114i \(0.653395\pi\)
\(108\) 0 0
\(109\) 84.5617i 0.775795i −0.921702 0.387898i \(-0.873201\pi\)
0.921702 0.387898i \(-0.126799\pi\)
\(110\) 0 0
\(111\) 9.07626i 0.0817681i
\(112\) 0 0
\(113\) 114.558i 1.01379i −0.862007 0.506896i \(-0.830793\pi\)
0.862007 0.506896i \(-0.169207\pi\)
\(114\) 0 0
\(115\) −50.4090 + 57.3123i −0.438339 + 0.498368i
\(116\) 0 0
\(117\) −38.3950 −0.328162
\(118\) 0 0
\(119\) 271.705i 2.28324i
\(120\) 0 0
\(121\) 175.141 1.44745
\(122\) 0 0
\(123\) 7.29518i 0.0593104i
\(124\) 0 0
\(125\) −103.630 69.8986i −0.829040 0.559189i
\(126\) 0 0
\(127\) 44.4121 0.349701 0.174851 0.984595i \(-0.444056\pi\)
0.174851 + 0.984595i \(0.444056\pi\)
\(128\) 0 0
\(129\) 32.7528 0.253898
\(130\) 0 0
\(131\) −47.1200 −0.359695 −0.179847 0.983695i \(-0.557560\pi\)
−0.179847 + 0.983695i \(0.557560\pi\)
\(132\) 0 0
\(133\) 48.5381 0.364948
\(134\) 0 0
\(135\) 36.4016 + 32.0170i 0.269641 + 0.237163i
\(136\) 0 0
\(137\) 62.8617i 0.458845i 0.973327 + 0.229422i \(0.0736837\pi\)
−0.973327 + 0.229422i \(0.926316\pi\)
\(138\) 0 0
\(139\) −128.320 −0.923167 −0.461584 0.887097i \(-0.652719\pi\)
−0.461584 + 0.887097i \(0.652719\pi\)
\(140\) 0 0
\(141\) 34.2028i 0.242573i
\(142\) 0 0
\(143\) −75.9465 −0.531094
\(144\) 0 0
\(145\) 8.94393 + 7.86663i 0.0616823 + 0.0542526i
\(146\) 0 0
\(147\) 28.5586i 0.194276i
\(148\) 0 0
\(149\) 165.561i 1.11115i −0.831467 0.555574i \(-0.812499\pi\)
0.831467 0.555574i \(-0.187501\pi\)
\(150\) 0 0
\(151\) 218.558i 1.44741i −0.690111 0.723704i \(-0.742438\pi\)
0.690111 0.723704i \(-0.257562\pi\)
\(152\) 0 0
\(153\) 235.052i 1.53628i
\(154\) 0 0
\(155\) 142.800 + 125.599i 0.921289 + 0.810319i
\(156\) 0 0
\(157\) 174.293 1.11014 0.555072 0.831802i \(-0.312691\pi\)
0.555072 + 0.831802i \(0.312691\pi\)
\(158\) 0 0
\(159\) 39.2185i 0.246657i
\(160\) 0 0
\(161\) 153.517 0.953524
\(162\) 0 0
\(163\) 52.6353i 0.322916i 0.986880 + 0.161458i \(0.0516196\pi\)
−0.986880 + 0.161458i \(0.948380\pi\)
\(164\) 0 0
\(165\) 35.3914 + 31.1285i 0.214493 + 0.188657i
\(166\) 0 0
\(167\) 76.6812 0.459169 0.229584 0.973289i \(-0.426263\pi\)
0.229584 + 0.973289i \(0.426263\pi\)
\(168\) 0 0
\(169\) −149.523 −0.884753
\(170\) 0 0
\(171\) −41.9902 −0.245557
\(172\) 0 0
\(173\) −9.72437 −0.0562102 −0.0281051 0.999605i \(-0.508947\pi\)
−0.0281051 + 0.999605i \(0.508947\pi\)
\(174\) 0 0
\(175\) 32.0918 + 249.358i 0.183382 + 1.42490i
\(176\) 0 0
\(177\) 37.7019i 0.213005i
\(178\) 0 0
\(179\) 155.502 0.868728 0.434364 0.900738i \(-0.356973\pi\)
0.434364 + 0.900738i \(0.356973\pi\)
\(180\) 0 0
\(181\) 250.346i 1.38313i 0.722316 + 0.691563i \(0.243077\pi\)
−0.722316 + 0.691563i \(0.756923\pi\)
\(182\) 0 0
\(183\) 22.4283 0.122559
\(184\) 0 0
\(185\) 54.7144 62.2072i 0.295753 0.336255i
\(186\) 0 0
\(187\) 464.939i 2.48631i
\(188\) 0 0
\(189\) 97.5056i 0.515903i
\(190\) 0 0
\(191\) 111.128i 0.581825i −0.956750 0.290912i \(-0.906041\pi\)
0.956750 0.290912i \(-0.0939588\pi\)
\(192\) 0 0
\(193\) 10.2701i 0.0532130i 0.999646 + 0.0266065i \(0.00847011\pi\)
−0.999646 + 0.0266065i \(0.991530\pi\)
\(194\) 0 0
\(195\) −9.07626 7.98302i −0.0465449 0.0409386i
\(196\) 0 0
\(197\) −163.213 −0.828492 −0.414246 0.910165i \(-0.635955\pi\)
−0.414246 + 0.910165i \(0.635955\pi\)
\(198\) 0 0
\(199\) 195.028i 0.980041i −0.871711 0.490021i \(-0.836989\pi\)
0.871711 0.490021i \(-0.163011\pi\)
\(200\) 0 0
\(201\) 27.9412 0.139011
\(202\) 0 0
\(203\) 23.9573i 0.118016i
\(204\) 0 0
\(205\) 43.9775 50.0000i 0.214524 0.243902i
\(206\) 0 0
\(207\) −132.807 −0.641582
\(208\) 0 0
\(209\) −83.0580 −0.397407
\(210\) 0 0
\(211\) −297.038 −1.40776 −0.703881 0.710317i \(-0.748551\pi\)
−0.703881 + 0.710317i \(0.748551\pi\)
\(212\) 0 0
\(213\) −22.1592 −0.104034
\(214\) 0 0
\(215\) −224.483 197.444i −1.04411 0.918342i
\(216\) 0 0
\(217\) 382.505i 1.76270i
\(218\) 0 0
\(219\) 19.6347 0.0896563
\(220\) 0 0
\(221\) 119.236i 0.539527i
\(222\) 0 0
\(223\) 405.335 1.81764 0.908822 0.417184i \(-0.136983\pi\)
0.908822 + 0.417184i \(0.136983\pi\)
\(224\) 0 0
\(225\) −27.7626 215.719i −0.123389 0.958752i
\(226\) 0 0
\(227\) 36.6013i 0.161239i 0.996745 + 0.0806196i \(0.0256899\pi\)
−0.996745 + 0.0806196i \(0.974310\pi\)
\(228\) 0 0
\(229\) 91.1467i 0.398021i −0.979997 0.199010i \(-0.936227\pi\)
0.979997 0.199010i \(-0.0637727\pi\)
\(230\) 0 0
\(231\) 94.7997i 0.410388i
\(232\) 0 0
\(233\) 338.802i 1.45409i 0.686592 + 0.727043i \(0.259106\pi\)
−0.686592 + 0.727043i \(0.740894\pi\)
\(234\) 0 0
\(235\) 206.185 234.421i 0.877382 0.997535i
\(236\) 0 0
\(237\) 69.4585 0.293074
\(238\) 0 0
\(239\) 30.7997i 0.128869i −0.997922 0.0644346i \(-0.979476\pi\)
0.997922 0.0644346i \(-0.0205244\pi\)
\(240\) 0 0
\(241\) 240.282 0.997020 0.498510 0.866884i \(-0.333881\pi\)
0.498510 + 0.866884i \(0.333881\pi\)
\(242\) 0 0
\(243\) 127.243i 0.523633i
\(244\) 0 0
\(245\) 172.160 195.736i 0.702693 0.798923i
\(246\) 0 0
\(247\) 21.3005 0.0862370
\(248\) 0 0
\(249\) 41.1652 0.165322
\(250\) 0 0
\(251\) −86.0268 −0.342736 −0.171368 0.985207i \(-0.554819\pi\)
−0.171368 + 0.985207i \(0.554819\pi\)
\(252\) 0 0
\(253\) −262.697 −1.03833
\(254\) 0 0
\(255\) 48.8715 55.5642i 0.191653 0.217899i
\(256\) 0 0
\(257\) 355.963i 1.38507i 0.721383 + 0.692536i \(0.243507\pi\)
−0.721383 + 0.692536i \(0.756493\pi\)
\(258\) 0 0
\(259\) −166.629 −0.643355
\(260\) 0 0
\(261\) 20.7254i 0.0794077i
\(262\) 0 0
\(263\) 73.1254 0.278043 0.139022 0.990289i \(-0.455604\pi\)
0.139022 + 0.990289i \(0.455604\pi\)
\(264\) 0 0
\(265\) −236.421 + 268.798i −0.892154 + 1.01433i
\(266\) 0 0
\(267\) 58.3514i 0.218545i
\(268\) 0 0
\(269\) 268.002i 0.996290i −0.867094 0.498145i \(-0.834015\pi\)
0.867094 0.498145i \(-0.165985\pi\)
\(270\) 0 0
\(271\) 229.412i 0.846538i 0.906004 + 0.423269i \(0.139117\pi\)
−0.906004 + 0.423269i \(0.860883\pi\)
\(272\) 0 0
\(273\) 24.3118i 0.0890541i
\(274\) 0 0
\(275\) −54.9153 426.699i −0.199692 1.55163i
\(276\) 0 0
\(277\) 126.327 0.456053 0.228026 0.973655i \(-0.426773\pi\)
0.228026 + 0.973655i \(0.426773\pi\)
\(278\) 0 0
\(279\) 330.904i 1.18604i
\(280\) 0 0
\(281\) 458.746 1.63255 0.816275 0.577664i \(-0.196036\pi\)
0.816275 + 0.577664i \(0.196036\pi\)
\(282\) 0 0
\(283\) 465.558i 1.64508i 0.568706 + 0.822541i \(0.307444\pi\)
−0.568706 + 0.822541i \(0.692556\pi\)
\(284\) 0 0
\(285\) −9.92614 8.73053i −0.0348286 0.0306335i
\(286\) 0 0
\(287\) −133.931 −0.466657
\(288\) 0 0
\(289\) −440.952 −1.52579
\(290\) 0 0
\(291\) 46.7997 0.160824
\(292\) 0 0
\(293\) −165.218 −0.563884 −0.281942 0.959431i \(-0.590979\pi\)
−0.281942 + 0.959431i \(0.590979\pi\)
\(294\) 0 0
\(295\) −227.278 + 258.403i −0.770434 + 0.875941i
\(296\) 0 0
\(297\) 166.851i 0.561787i
\(298\) 0 0
\(299\) 67.3697 0.225317
\(300\) 0 0
\(301\) 601.302i 1.99768i
\(302\) 0 0
\(303\) 45.6234 0.150572
\(304\) 0 0
\(305\) −153.720 135.205i −0.504000 0.443293i
\(306\) 0 0
\(307\) 235.339i 0.766577i 0.923629 + 0.383288i \(0.125208\pi\)
−0.923629 + 0.383288i \(0.874792\pi\)
\(308\) 0 0
\(309\) 25.8911i 0.0837898i
\(310\) 0 0
\(311\) 210.665i 0.677381i 0.940898 + 0.338691i \(0.109984\pi\)
−0.940898 + 0.338691i \(0.890016\pi\)
\(312\) 0 0
\(313\) 318.738i 1.01833i 0.860668 + 0.509166i \(0.170046\pi\)
−0.860668 + 0.509166i \(0.829954\pi\)
\(314\) 0 0
\(315\) −288.914 + 328.479i −0.917187 + 1.04279i
\(316\) 0 0
\(317\) −70.3950 −0.222066 −0.111033 0.993817i \(-0.535416\pi\)
−0.111033 + 0.993817i \(0.535416\pi\)
\(318\) 0 0
\(319\) 40.9955i 0.128513i
\(320\) 0 0
\(321\) −103.875 −0.323598
\(322\) 0 0
\(323\) 130.401i 0.403717i
\(324\) 0 0
\(325\) 14.0832 + 109.429i 0.0433330 + 0.336704i
\(326\) 0 0
\(327\) 46.3213 0.141655
\(328\) 0 0
\(329\) −627.922 −1.90858
\(330\) 0 0
\(331\) 280.807 0.848359 0.424180 0.905578i \(-0.360562\pi\)
0.424180 + 0.905578i \(0.360562\pi\)
\(332\) 0 0
\(333\) 144.150 0.432884
\(334\) 0 0
\(335\) −191.505 168.438i −0.571656 0.502800i
\(336\) 0 0
\(337\) 120.969i 0.358959i 0.983762 + 0.179480i \(0.0574414\pi\)
−0.983762 + 0.179480i \(0.942559\pi\)
\(338\) 0 0
\(339\) 62.7530 0.185112
\(340\) 0 0
\(341\) 654.539i 1.91947i
\(342\) 0 0
\(343\) −31.5280 −0.0919182
\(344\) 0 0
\(345\) −31.3946 27.6131i −0.0909988 0.0800380i
\(346\) 0 0
\(347\) 214.333i 0.617675i −0.951115 0.308838i \(-0.900060\pi\)
0.951115 0.308838i \(-0.0999399\pi\)
\(348\) 0 0
\(349\) 418.041i 1.19782i 0.800815 + 0.598912i \(0.204400\pi\)
−0.800815 + 0.598912i \(0.795600\pi\)
\(350\) 0 0
\(351\) 42.7895i 0.121907i
\(352\) 0 0
\(353\) 364.929i 1.03379i 0.856048 + 0.516897i \(0.172913\pi\)
−0.856048 + 0.516897i \(0.827087\pi\)
\(354\) 0 0
\(355\) 151.876 + 133.583i 0.427820 + 0.376289i
\(356\) 0 0
\(357\) −148.835 −0.416905
\(358\) 0 0
\(359\) 242.169i 0.674567i −0.941403 0.337283i \(-0.890492\pi\)
0.941403 0.337283i \(-0.109508\pi\)
\(360\) 0 0
\(361\) −337.705 −0.935471
\(362\) 0 0
\(363\) 95.9389i 0.264295i
\(364\) 0 0
\(365\) −134.573 118.364i −0.368694 0.324285i
\(366\) 0 0
\(367\) −224.564 −0.611891 −0.305945 0.952049i \(-0.598972\pi\)
−0.305945 + 0.952049i \(0.598972\pi\)
\(368\) 0 0
\(369\) 115.863 0.313992
\(370\) 0 0
\(371\) 720.004 1.94071
\(372\) 0 0
\(373\) 572.174 1.53398 0.766989 0.641660i \(-0.221754\pi\)
0.766989 + 0.641660i \(0.221754\pi\)
\(374\) 0 0
\(375\) 38.2891 56.7666i 0.102104 0.151378i
\(376\) 0 0
\(377\) 10.5135i 0.0278872i
\(378\) 0 0
\(379\) −122.896 −0.324263 −0.162132 0.986769i \(-0.551837\pi\)
−0.162132 + 0.986769i \(0.551837\pi\)
\(380\) 0 0
\(381\) 24.3281i 0.0638533i
\(382\) 0 0
\(383\) −289.717 −0.756442 −0.378221 0.925715i \(-0.623464\pi\)
−0.378221 + 0.925715i \(0.623464\pi\)
\(384\) 0 0
\(385\) −571.481 + 649.743i −1.48437 + 1.68764i
\(386\) 0 0
\(387\) 520.185i 1.34415i
\(388\) 0 0
\(389\) 111.590i 0.286863i 0.989660 + 0.143432i \(0.0458137\pi\)
−0.989660 + 0.143432i \(0.954186\pi\)
\(390\) 0 0
\(391\) 412.433i 1.05482i
\(392\) 0 0
\(393\) 25.8114i 0.0656780i
\(394\) 0 0
\(395\) −476.058 418.716i −1.20521 1.06004i
\(396\) 0 0
\(397\) 705.314 1.77661 0.888305 0.459253i \(-0.151883\pi\)
0.888305 + 0.459253i \(0.151883\pi\)
\(398\) 0 0
\(399\) 26.5883i 0.0666373i
\(400\) 0 0
\(401\) 432.052 1.07744 0.538718 0.842486i \(-0.318909\pi\)
0.538718 + 0.842486i \(0.318909\pi\)
\(402\) 0 0
\(403\) 167.859i 0.416524i
\(404\) 0 0
\(405\) 241.021 274.028i 0.595114 0.676612i
\(406\) 0 0
\(407\) 285.134 0.700575
\(408\) 0 0
\(409\) 98.8102 0.241590 0.120795 0.992677i \(-0.461456\pi\)
0.120795 + 0.992677i \(0.461456\pi\)
\(410\) 0 0
\(411\) −34.4345 −0.0837822
\(412\) 0 0
\(413\) 692.160 1.67593
\(414\) 0 0
\(415\) −282.140 248.156i −0.679855 0.597966i
\(416\) 0 0
\(417\) 70.2914i 0.168565i
\(418\) 0 0
\(419\) −204.980 −0.489213 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(420\) 0 0
\(421\) 449.956i 1.06878i 0.845238 + 0.534390i \(0.179459\pi\)
−0.845238 + 0.534390i \(0.820541\pi\)
\(422\) 0 0
\(423\) 543.214 1.28419
\(424\) 0 0
\(425\) −669.915 + 86.2166i −1.57627 + 0.202863i
\(426\) 0 0
\(427\) 411.756i 0.964301i
\(428\) 0 0
\(429\) 41.6021i 0.0969745i
\(430\) 0 0
\(431\) 314.588i 0.729903i −0.931026 0.364952i \(-0.881086\pi\)
0.931026 0.364952i \(-0.118914\pi\)
\(432\) 0 0
\(433\) 330.441i 0.763143i −0.924339 0.381571i \(-0.875383\pi\)
0.924339 0.381571i \(-0.124617\pi\)
\(434\) 0 0
\(435\) −4.30919 + 4.89932i −0.00990619 + 0.0112628i
\(436\) 0 0
\(437\) 73.6781 0.168600
\(438\) 0 0
\(439\) 664.815i 1.51439i −0.653191 0.757193i \(-0.726570\pi\)
0.653191 0.757193i \(-0.273430\pi\)
\(440\) 0 0
\(441\) 453.572 1.02851
\(442\) 0 0
\(443\) 312.860i 0.706229i −0.935580 0.353115i \(-0.885123\pi\)
0.935580 0.353115i \(-0.114877\pi\)
\(444\) 0 0
\(445\) 351.760 399.931i 0.790471 0.898722i
\(446\) 0 0
\(447\) 90.6912 0.202889
\(448\) 0 0
\(449\) −246.330 −0.548620 −0.274310 0.961641i \(-0.588449\pi\)
−0.274310 + 0.961641i \(0.588449\pi\)
\(450\) 0 0
\(451\) 229.181 0.508161
\(452\) 0 0
\(453\) 119.722 0.264287
\(454\) 0 0
\(455\) 146.558 166.629i 0.322107 0.366218i
\(456\) 0 0
\(457\) 707.094i 1.54725i 0.633642 + 0.773626i \(0.281559\pi\)
−0.633642 + 0.773626i \(0.718441\pi\)
\(458\) 0 0
\(459\) 261.955 0.570707
\(460\) 0 0
\(461\) 611.288i 1.32600i 0.748618 + 0.663002i \(0.230718\pi\)
−0.748618 + 0.663002i \(0.769282\pi\)
\(462\) 0 0
\(463\) 334.063 0.721519 0.360760 0.932659i \(-0.382518\pi\)
0.360760 + 0.932659i \(0.382518\pi\)
\(464\) 0 0
\(465\) −68.8010 + 78.2230i −0.147959 + 0.168222i
\(466\) 0 0
\(467\) 128.864i 0.275939i 0.990436 + 0.137970i \(0.0440577\pi\)
−0.990436 + 0.137970i \(0.955942\pi\)
\(468\) 0 0
\(469\) 512.966i 1.09374i
\(470\) 0 0
\(471\) 95.4742i 0.202705i
\(472\) 0 0
\(473\) 1028.94i 2.17535i
\(474\) 0 0
\(475\) 15.4020 + 119.675i 0.0324252 + 0.251948i
\(476\) 0 0
\(477\) −622.874 −1.30582
\(478\) 0 0
\(479\) 510.257i 1.06525i 0.846350 + 0.532627i \(0.178795\pi\)
−0.846350 + 0.532627i \(0.821205\pi\)
\(480\) 0 0
\(481\) −73.1237 −0.152024
\(482\) 0 0
\(483\) 84.0939i 0.174107i
\(484\) 0 0
\(485\) −320.758 282.123i −0.661357 0.581696i
\(486\) 0 0
\(487\) −616.472 −1.26586 −0.632928 0.774211i \(-0.718147\pi\)
−0.632928 + 0.774211i \(0.718147\pi\)
\(488\) 0 0
\(489\) −28.8326 −0.0589624
\(490\) 0 0
\(491\) −603.190 −1.22849 −0.614247 0.789114i \(-0.710540\pi\)
−0.614247 + 0.789114i \(0.710540\pi\)
\(492\) 0 0
\(493\) 64.3627 0.130553
\(494\) 0 0
\(495\) 494.387 562.091i 0.998761 1.13554i
\(496\) 0 0
\(497\) 406.817i 0.818545i
\(498\) 0 0
\(499\) −867.976 −1.73943 −0.869716 0.493553i \(-0.835698\pi\)
−0.869716 + 0.493553i \(0.835698\pi\)
\(500\) 0 0
\(501\) 42.0045i 0.0838413i
\(502\) 0 0
\(503\) −57.8408 −0.114992 −0.0574958 0.998346i \(-0.518312\pi\)
−0.0574958 + 0.998346i \(0.518312\pi\)
\(504\) 0 0
\(505\) −312.696 275.032i −0.619200 0.544617i
\(506\) 0 0
\(507\) 81.9060i 0.161550i
\(508\) 0 0
\(509\) 168.498i 0.331038i 0.986207 + 0.165519i \(0.0529299\pi\)
−0.986207 + 0.165519i \(0.947070\pi\)
\(510\) 0 0
\(511\) 360.470i 0.705420i
\(512\) 0 0
\(513\) 46.7962i 0.0912207i
\(514\) 0 0
\(515\) −156.079 + 177.453i −0.303066 + 0.344569i
\(516\) 0 0
\(517\) 1074.49 2.07833
\(518\) 0 0
\(519\) 5.32682i 0.0102636i
\(520\) 0 0
\(521\) 87.1060 0.167190 0.0835951 0.996500i \(-0.473360\pi\)
0.0835951 + 0.996500i \(0.473360\pi\)
\(522\) 0 0
\(523\) 243.469i 0.465524i −0.972534 0.232762i \(-0.925224\pi\)
0.972534 0.232762i \(-0.0747763\pi\)
\(524\) 0 0
\(525\) −136.594 + 17.5793i −0.260179 + 0.0334844i
\(526\) 0 0
\(527\) 1027.62 1.94995
\(528\) 0 0
\(529\) −295.969 −0.559488
\(530\) 0 0
\(531\) −598.786 −1.12766
\(532\) 0 0
\(533\) −58.7743 −0.110271
\(534\) 0 0
\(535\) 711.942 + 626.189i 1.33073 + 1.17045i
\(536\) 0 0
\(537\) 85.1812i 0.158624i
\(538\) 0 0
\(539\) 897.179 1.66452
\(540\) 0 0
\(541\) 812.616i 1.50206i 0.660266 + 0.751032i \(0.270443\pi\)
−0.660266 + 0.751032i \(0.729557\pi\)
\(542\) 0 0
\(543\) −137.135 −0.252550
\(544\) 0 0
\(545\) −317.479 279.238i −0.582530 0.512364i
\(546\) 0 0
\(547\) 104.713i 0.191431i −0.995409 0.0957155i \(-0.969486\pi\)
0.995409 0.0957155i \(-0.0305139\pi\)
\(548\) 0 0
\(549\) 356.210i 0.648833i
\(550\) 0 0
\(551\) 11.4979i 0.0208674i
\(552\) 0 0
\(553\) 1275.17i 2.30592i
\(554\) 0 0
\(555\) 34.0759 + 29.9715i 0.0613981 + 0.0540027i
\(556\) 0 0
\(557\) −260.018 −0.466818 −0.233409 0.972379i \(-0.574988\pi\)
−0.233409 + 0.972379i \(0.574988\pi\)
\(558\) 0 0
\(559\) 263.876i 0.472050i
\(560\) 0 0
\(561\) 254.685 0.453984
\(562\) 0 0
\(563\) 39.9073i 0.0708833i 0.999372 + 0.0354416i \(0.0112838\pi\)
−0.999372 + 0.0354416i \(0.988716\pi\)
\(564\) 0 0
\(565\) −430.099 378.293i −0.761237 0.669546i
\(566\) 0 0
\(567\) −734.014 −1.29456
\(568\) 0 0
\(569\) 211.588 0.371860 0.185930 0.982563i \(-0.440470\pi\)
0.185930 + 0.982563i \(0.440470\pi\)
\(570\) 0 0
\(571\) 556.938 0.975372 0.487686 0.873019i \(-0.337841\pi\)
0.487686 + 0.873019i \(0.337841\pi\)
\(572\) 0 0
\(573\) 60.8741 0.106237
\(574\) 0 0
\(575\) 48.7136 + 378.512i 0.0847193 + 0.658281i
\(576\) 0 0
\(577\) 516.233i 0.894684i 0.894363 + 0.447342i \(0.147629\pi\)
−0.894363 + 0.447342i \(0.852371\pi\)
\(578\) 0 0
\(579\) −5.62577 −0.00971636
\(580\) 0 0
\(581\) 755.742i 1.30076i
\(582\) 0 0
\(583\) −1232.06 −2.11332
\(584\) 0 0
\(585\) −126.787 + 144.150i −0.216731 + 0.246411i
\(586\) 0 0
\(587\) 934.677i 1.59229i 0.605103 + 0.796147i \(0.293132\pi\)
−0.605103 + 0.796147i \(0.706868\pi\)
\(588\) 0 0
\(589\) 183.577i 0.311676i
\(590\) 0 0
\(591\) 89.4050i 0.151277i
\(592\) 0 0
\(593\) 634.665i 1.07026i 0.844769 + 0.535131i \(0.179738\pi\)
−0.844769 + 0.535131i \(0.820262\pi\)
\(594\) 0 0
\(595\) 1020.09 + 897.221i 1.71444 + 1.50794i
\(596\) 0 0
\(597\) 106.833 0.178949
\(598\) 0 0
\(599\) 914.029i 1.52593i −0.646443 0.762963i \(-0.723744\pi\)
0.646443 0.762963i \(-0.276256\pi\)
\(600\) 0 0
\(601\) 598.569 0.995955 0.497977 0.867190i \(-0.334076\pi\)
0.497977 + 0.867190i \(0.334076\pi\)
\(602\) 0 0
\(603\) 443.766i 0.735930i
\(604\) 0 0
\(605\) 578.348 657.550i 0.955948 1.08686i
\(606\) 0 0
\(607\) −201.830 −0.332504 −0.166252 0.986083i \(-0.553166\pi\)
−0.166252 + 0.986083i \(0.553166\pi\)
\(608\) 0 0
\(609\) 13.1234 0.0215490
\(610\) 0 0
\(611\) −275.558 −0.450995
\(612\) 0 0
\(613\) −71.9205 −0.117325 −0.0586627 0.998278i \(-0.518684\pi\)
−0.0586627 + 0.998278i \(0.518684\pi\)
\(614\) 0 0
\(615\) 27.3891 + 24.0900i 0.0445350 + 0.0391708i
\(616\) 0 0
\(617\) 204.485i 0.331418i 0.986175 + 0.165709i \(0.0529912\pi\)
−0.986175 + 0.165709i \(0.947009\pi\)
\(618\) 0 0
\(619\) 287.512 0.464479 0.232239 0.972659i \(-0.425395\pi\)
0.232239 + 0.972659i \(0.425395\pi\)
\(620\) 0 0
\(621\) 148.008i 0.238338i
\(622\) 0 0
\(623\) −1071.26 −1.71952
\(624\) 0 0
\(625\) −604.633 + 158.251i −0.967414 + 0.253202i
\(626\) 0 0
\(627\) 45.4976i 0.0725640i
\(628\) 0 0
\(629\) 447.658i 0.711699i
\(630\) 0 0
\(631\) 274.203i 0.434554i −0.976110 0.217277i \(-0.930283\pi\)
0.976110 0.217277i \(-0.0697175\pi\)
\(632\) 0 0
\(633\) 162.712i 0.257049i
\(634\) 0 0
\(635\) 146.657 166.741i 0.230956 0.262584i
\(636\) 0 0
\(637\) −230.085 −0.361201
\(638\) 0 0
\(639\) 351.936i 0.550761i
\(640\) 0 0
\(641\) 681.328 1.06291 0.531457 0.847085i \(-0.321645\pi\)
0.531457 + 0.847085i \(0.321645\pi\)
\(642\) 0 0
\(643\) 527.270i 0.820016i −0.912082 0.410008i \(-0.865526\pi\)
0.912082 0.410008i \(-0.134474\pi\)
\(644\) 0 0
\(645\) 108.156 122.967i 0.167684 0.190647i
\(646\) 0 0
\(647\) 551.627 0.852592 0.426296 0.904584i \(-0.359818\pi\)
0.426296 + 0.904584i \(0.359818\pi\)
\(648\) 0 0
\(649\) −1184.42 −1.82499
\(650\) 0 0
\(651\) 209.529 0.321857
\(652\) 0 0
\(653\) −699.422 −1.07109 −0.535545 0.844507i \(-0.679894\pi\)
−0.535545 + 0.844507i \(0.679894\pi\)
\(654\) 0 0
\(655\) −155.599 + 176.908i −0.237556 + 0.270088i
\(656\) 0 0
\(657\) 311.842i 0.474645i
\(658\) 0 0
\(659\) 38.3257 0.0581574 0.0290787 0.999577i \(-0.490743\pi\)
0.0290787 + 0.999577i \(0.490743\pi\)
\(660\) 0 0
\(661\) 392.364i 0.593591i 0.954941 + 0.296796i \(0.0959180\pi\)
−0.954941 + 0.296796i \(0.904082\pi\)
\(662\) 0 0
\(663\) −65.3150 −0.0985143
\(664\) 0 0
\(665\) 160.282 182.232i 0.241026 0.274033i
\(666\) 0 0
\(667\) 36.3658i 0.0545215i
\(668\) 0 0
\(669\) 222.035i 0.331890i
\(670\) 0 0
\(671\) 704.594i 1.05007i
\(672\) 0 0
\(673\) 427.290i 0.634904i 0.948274 + 0.317452i \(0.102827\pi\)
−0.948274 + 0.317452i \(0.897173\pi\)
\(674\) 0 0
\(675\) 240.409 30.9402i 0.356162 0.0458373i
\(676\) 0 0
\(677\) −131.234 −0.193847 −0.0969233 0.995292i \(-0.530900\pi\)
−0.0969233 + 0.995292i \(0.530900\pi\)
\(678\) 0 0
\(679\) 859.186i 1.26537i
\(680\) 0 0
\(681\) −20.0495 −0.0294413
\(682\) 0 0
\(683\) 896.229i 1.31219i −0.754676 0.656097i \(-0.772206\pi\)
0.754676 0.656097i \(-0.227794\pi\)
\(684\) 0 0
\(685\) 236.008 + 207.581i 0.344538 + 0.303038i
\(686\) 0 0
\(687\) 49.9285 0.0726761
\(688\) 0 0
\(689\) 315.968 0.458589
\(690\) 0 0
\(691\) 520.465 0.753206 0.376603 0.926375i \(-0.377092\pi\)
0.376603 + 0.926375i \(0.377092\pi\)
\(692\) 0 0
\(693\) −1505.62 −2.17262
\(694\) 0 0
\(695\) −423.737 + 481.766i −0.609694 + 0.693189i
\(696\) 0 0
\(697\) 359.812i 0.516230i
\(698\) 0 0
\(699\) −185.589 −0.265507
\(700\) 0 0
\(701\) 80.7527i 0.115196i −0.998340 0.0575982i \(-0.981656\pi\)
0.998340 0.0575982i \(-0.0183442\pi\)
\(702\) 0 0
\(703\) −79.9709 −0.113757
\(704\) 0 0
\(705\) 128.411 + 112.944i 0.182144 + 0.160204i
\(706\) 0 0
\(707\) 837.591i 1.18471i
\(708\) 0 0
\(709\) 174.828i 0.246584i −0.992370 0.123292i \(-0.960655\pi\)
0.992370 0.123292i \(-0.0393452\pi\)
\(710\) 0 0
\(711\) 1103.15i 1.55155i
\(712\) 0 0
\(713\) 580.621i 0.814335i
\(714\) 0 0
\(715\) −250.789 + 285.134i −0.350755 + 0.398789i
\(716\) 0 0
\(717\) 16.8715 0.0235307
\(718\) 0 0
\(719\) 889.905i 1.23770i 0.785510 + 0.618849i \(0.212401\pi\)
−0.785510 + 0.618849i \(0.787599\pi\)
\(720\) 0 0
\(721\) 475.328 0.659263
\(722\) 0 0
\(723\) 131.622i 0.182050i
\(724\) 0 0
\(725\) 59.0690 7.60205i 0.0814745 0.0104856i
\(726\) 0 0
\(727\) 408.818 0.562335 0.281168 0.959659i \(-0.409278\pi\)
0.281168 + 0.959659i \(0.409278\pi\)
\(728\) 0 0
\(729\) 587.194 0.805478
\(730\) 0 0
\(731\) −1615.43 −2.20989
\(732\) 0 0
\(733\) −388.369 −0.529835 −0.264917 0.964271i \(-0.585345\pi\)
−0.264917 + 0.964271i \(0.585345\pi\)
\(734\) 0 0
\(735\) 107.221 + 94.3058i 0.145878 + 0.128307i
\(736\) 0 0
\(737\) 877.783i 1.19102i
\(738\) 0 0
\(739\) 109.561 0.148256 0.0741280 0.997249i \(-0.476383\pi\)
0.0741280 + 0.997249i \(0.476383\pi\)
\(740\) 0 0
\(741\) 11.6680i 0.0157463i
\(742\) 0 0
\(743\) 732.717 0.986160 0.493080 0.869984i \(-0.335871\pi\)
0.493080 + 0.869984i \(0.335871\pi\)
\(744\) 0 0
\(745\) −621.583 546.713i −0.834340 0.733844i
\(746\) 0 0
\(747\) 653.791i 0.875222i
\(748\) 0 0
\(749\) 1907.02i 2.54608i
\(750\) 0 0
\(751\) 420.809i 0.560332i −0.959952 0.280166i \(-0.909610\pi\)
0.959952 0.280166i \(-0.0903895\pi\)
\(752\) 0 0
\(753\) 47.1238i 0.0625814i
\(754\) 0 0
\(755\) −820.557 721.721i −1.08683 0.955922i
\(756\) 0 0
\(757\) −1306.99 −1.72654 −0.863269 0.504744i \(-0.831587\pi\)
−0.863269 + 0.504744i \(0.831587\pi\)
\(758\) 0 0
\(759\) 143.901i 0.189592i
\(760\) 0 0
\(761\) 1402.25 1.84264 0.921320 0.388804i \(-0.127112\pi\)
0.921320 + 0.388804i \(0.127112\pi\)
\(762\) 0 0
\(763\) 850.402i 1.11455i
\(764\) 0 0
\(765\) −882.479 776.184i −1.15357 1.01462i
\(766\) 0 0
\(767\) 303.748 0.396021
\(768\) 0 0
\(769\) −359.952 −0.468078 −0.234039 0.972227i \(-0.575194\pi\)
−0.234039 + 0.972227i \(0.575194\pi\)
\(770\) 0 0
\(771\) −194.990 −0.252905
\(772\) 0 0
\(773\) 437.539 0.566027 0.283013 0.959116i \(-0.408666\pi\)
0.283013 + 0.959116i \(0.408666\pi\)
\(774\) 0 0
\(775\) 943.103 121.375i 1.21691 0.156613i
\(776\) 0 0
\(777\) 91.2762i 0.117473i
\(778\) 0 0
\(779\) −64.2778 −0.0825132
\(780\) 0 0
\(781\) 696.141i 0.891346i
\(782\) 0 0
\(783\) −23.0975 −0.0294988
\(784\) 0 0
\(785\) 575.547 654.365i 0.733181 0.833586i
\(786\) 0 0
\(787\) 438.946i 0.557746i −0.960328 0.278873i \(-0.910039\pi\)
0.960328 0.278873i \(-0.0899608\pi\)
\(788\) 0 0
\(789\) 40.0567i 0.0507690i
\(790\) 0 0
\(791\) 1152.07i 1.45647i
\(792\) 0 0
\(793\) 180.696i 0.227864i
\(794\) 0 0
\(795\) −147.242 129.507i −0.185210 0.162902i
\(796\) 0 0
\(797\) 982.414 1.23264 0.616320 0.787496i \(-0.288623\pi\)
0.616320 + 0.787496i \(0.288623\pi\)
\(798\) 0 0
\(799\) 1686.95i 2.11133i
\(800\) 0 0