Properties

Label 600.3.l.b.401.1
Level $600$
Weight $3$
Character 600.401
Analytic conductor $16.349$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 600.401
Dual form 600.3.l.b.401.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 - 2.82843i) q^{3} +6.00000 q^{7} +(-7.00000 + 5.65685i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 2.82843i) q^{3} +6.00000 q^{7} +(-7.00000 + 5.65685i) q^{9} -5.65685i q^{11} -10.0000 q^{13} -22.6274i q^{17} +2.00000 q^{19} +(-6.00000 - 16.9706i) q^{21} -11.3137i q^{23} +(23.0000 + 14.1421i) q^{27} -16.9706i q^{29} -22.0000 q^{31} +(-16.0000 + 5.65685i) q^{33} +6.00000 q^{37} +(10.0000 + 28.2843i) q^{39} -33.9411i q^{41} -82.0000 q^{43} +67.8823i q^{47} -13.0000 q^{49} +(-64.0000 + 22.6274i) q^{51} -62.2254i q^{53} +(-2.00000 - 5.65685i) q^{57} -73.5391i q^{59} -86.0000 q^{61} +(-42.0000 + 33.9411i) q^{63} -2.00000 q^{67} +(-32.0000 + 11.3137i) q^{69} +124.451i q^{71} -82.0000 q^{73} -33.9411i q^{77} +10.0000 q^{79} +(17.0000 - 79.1960i) q^{81} -73.5391i q^{83} +(-48.0000 + 16.9706i) q^{87} +33.9411i q^{89} -60.0000 q^{91} +(22.0000 + 62.2254i) q^{93} +94.0000 q^{97} +(32.0000 + 39.5980i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 12q^{7} - 14q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 12q^{7} - 14q^{9} - 20q^{13} + 4q^{19} - 12q^{21} + 46q^{27} - 44q^{31} - 32q^{33} + 12q^{37} + 20q^{39} - 164q^{43} - 26q^{49} - 128q^{51} - 4q^{57} - 172q^{61} - 84q^{63} - 4q^{67} - 64q^{69} - 164q^{73} + 20q^{79} + 34q^{81} - 96q^{87} - 120q^{91} + 44q^{93} + 188q^{97} + 64q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 (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
\(3\) −1.00000 2.82843i −0.333333 0.942809i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000 0.857143 0.428571 0.903508i \(-0.359017\pi\)
0.428571 + 0.903508i \(0.359017\pi\)
\(8\) 0 0
\(9\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(10\) 0 0
\(11\) 5.65685i 0.514259i −0.966377 0.257130i \(-0.917223\pi\)
0.966377 0.257130i \(-0.0827768\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.769231 −0.384615 0.923077i \(-0.625666\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.6274i 1.33102i −0.746387 0.665512i \(-0.768213\pi\)
0.746387 0.665512i \(-0.231787\pi\)
\(18\) 0 0
\(19\) 2.00000 0.105263 0.0526316 0.998614i \(-0.483239\pi\)
0.0526316 + 0.998614i \(0.483239\pi\)
\(20\) 0 0
\(21\) −6.00000 16.9706i −0.285714 0.808122i
\(22\) 0 0
\(23\) 11.3137i 0.491900i −0.969282 0.245950i \(-0.920900\pi\)
0.969282 0.245950i \(-0.0791000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 23.0000 + 14.1421i 0.851852 + 0.523783i
\(28\) 0 0
\(29\) 16.9706i 0.585192i −0.956236 0.292596i \(-0.905481\pi\)
0.956236 0.292596i \(-0.0945191\pi\)
\(30\) 0 0
\(31\) −22.0000 −0.709677 −0.354839 0.934928i \(-0.615464\pi\)
−0.354839 + 0.934928i \(0.615464\pi\)
\(32\) 0 0
\(33\) −16.0000 + 5.65685i −0.484848 + 0.171420i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.162162 0.0810811 0.996708i \(-0.474163\pi\)
0.0810811 + 0.996708i \(0.474163\pi\)
\(38\) 0 0
\(39\) 10.0000 + 28.2843i 0.256410 + 0.725238i
\(40\) 0 0
\(41\) 33.9411i 0.827832i −0.910315 0.413916i \(-0.864161\pi\)
0.910315 0.413916i \(-0.135839\pi\)
\(42\) 0 0
\(43\) −82.0000 −1.90698 −0.953488 0.301430i \(-0.902536\pi\)
−0.953488 + 0.301430i \(0.902536\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 67.8823i 1.44430i 0.691735 + 0.722152i \(0.256847\pi\)
−0.691735 + 0.722152i \(0.743153\pi\)
\(48\) 0 0
\(49\) −13.0000 −0.265306
\(50\) 0 0
\(51\) −64.0000 + 22.6274i −1.25490 + 0.443675i
\(52\) 0 0
\(53\) 62.2254i 1.17406i −0.809564 0.587032i \(-0.800296\pi\)
0.809564 0.587032i \(-0.199704\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 5.65685i −0.0350877 0.0992431i
\(58\) 0 0
\(59\) 73.5391i 1.24643i −0.782052 0.623213i \(-0.785827\pi\)
0.782052 0.623213i \(-0.214173\pi\)
\(60\) 0 0
\(61\) −86.0000 −1.40984 −0.704918 0.709289i \(-0.749016\pi\)
−0.704918 + 0.709289i \(0.749016\pi\)
\(62\) 0 0
\(63\) −42.0000 + 33.9411i −0.666667 + 0.538748i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.0298507 −0.0149254 0.999889i \(-0.504751\pi\)
−0.0149254 + 0.999889i \(0.504751\pi\)
\(68\) 0 0
\(69\) −32.0000 + 11.3137i −0.463768 + 0.163967i
\(70\) 0 0
\(71\) 124.451i 1.75283i 0.481558 + 0.876414i \(0.340071\pi\)
−0.481558 + 0.876414i \(0.659929\pi\)
\(72\) 0 0
\(73\) −82.0000 −1.12329 −0.561644 0.827379i \(-0.689831\pi\)
−0.561644 + 0.827379i \(0.689831\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 33.9411i 0.440794i
\(78\) 0 0
\(79\) 10.0000 0.126582 0.0632911 0.997995i \(-0.479840\pi\)
0.0632911 + 0.997995i \(0.479840\pi\)
\(80\) 0 0
\(81\) 17.0000 79.1960i 0.209877 0.977728i
\(82\) 0 0
\(83\) 73.5391i 0.886013i −0.896518 0.443007i \(-0.853912\pi\)
0.896518 0.443007i \(-0.146088\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −48.0000 + 16.9706i −0.551724 + 0.195064i
\(88\) 0 0
\(89\) 33.9411i 0.381361i 0.981652 + 0.190680i \(0.0610694\pi\)
−0.981652 + 0.190680i \(0.938931\pi\)
\(90\) 0 0
\(91\) −60.0000 −0.659341
\(92\) 0 0
\(93\) 22.0000 + 62.2254i 0.236559 + 0.669090i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 94.0000 0.969072 0.484536 0.874771i \(-0.338988\pi\)
0.484536 + 0.874771i \(0.338988\pi\)
\(98\) 0 0
\(99\) 32.0000 + 39.5980i 0.323232 + 0.399980i
\(100\) 0 0
\(101\) 50.9117i 0.504076i −0.967717 0.252038i \(-0.918899\pi\)
0.967717 0.252038i \(-0.0811008\pi\)
\(102\) 0 0
\(103\) 134.000 1.30097 0.650485 0.759519i \(-0.274565\pi\)
0.650485 + 0.759519i \(0.274565\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 50.9117i 0.475810i 0.971288 + 0.237905i \(0.0764607\pi\)
−0.971288 + 0.237905i \(0.923539\pi\)
\(108\) 0 0
\(109\) 10.0000 0.0917431 0.0458716 0.998947i \(-0.485394\pi\)
0.0458716 + 0.998947i \(0.485394\pi\)
\(110\) 0 0
\(111\) −6.00000 16.9706i −0.0540541 0.152888i
\(112\) 0 0
\(113\) 67.8823i 0.600728i −0.953825 0.300364i \(-0.902892\pi\)
0.953825 0.300364i \(-0.0971081\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 70.0000 56.5685i 0.598291 0.483492i
\(118\) 0 0
\(119\) 135.765i 1.14088i
\(120\) 0 0
\(121\) 89.0000 0.735537
\(122\) 0 0
\(123\) −96.0000 + 33.9411i −0.780488 + 0.275944i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −106.000 −0.834646 −0.417323 0.908758i \(-0.637032\pi\)
−0.417323 + 0.908758i \(0.637032\pi\)
\(128\) 0 0
\(129\) 82.0000 + 231.931i 0.635659 + 1.79791i
\(130\) 0 0
\(131\) 5.65685i 0.0431821i 0.999767 + 0.0215910i \(0.00687318\pi\)
−0.999767 + 0.0215910i \(0.993127\pi\)
\(132\) 0 0
\(133\) 12.0000 0.0902256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 101.823i 0.743236i −0.928386 0.371618i \(-0.878803\pi\)
0.928386 0.371618i \(-0.121197\pi\)
\(138\) 0 0
\(139\) −78.0000 −0.561151 −0.280576 0.959832i \(-0.590525\pi\)
−0.280576 + 0.959832i \(0.590525\pi\)
\(140\) 0 0
\(141\) 192.000 67.8823i 1.36170 0.481434i
\(142\) 0 0
\(143\) 56.5685i 0.395584i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 13.0000 + 36.7696i 0.0884354 + 0.250133i
\(148\) 0 0
\(149\) 164.049i 1.10100i −0.834836 0.550499i \(-0.814437\pi\)
0.834836 0.550499i \(-0.185563\pi\)
\(150\) 0 0
\(151\) 218.000 1.44371 0.721854 0.692045i \(-0.243290\pi\)
0.721854 + 0.692045i \(0.243290\pi\)
\(152\) 0 0
\(153\) 128.000 + 158.392i 0.836601 + 1.03524i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 86.0000 0.547771 0.273885 0.961762i \(-0.411691\pi\)
0.273885 + 0.961762i \(0.411691\pi\)
\(158\) 0 0
\(159\) −176.000 + 62.2254i −1.10692 + 0.391355i
\(160\) 0 0
\(161\) 67.8823i 0.421629i
\(162\) 0 0
\(163\) 222.000 1.36196 0.680982 0.732301i \(-0.261553\pi\)
0.680982 + 0.732301i \(0.261553\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 169.706i 1.01620i −0.861298 0.508101i \(-0.830348\pi\)
0.861298 0.508101i \(-0.169652\pi\)
\(168\) 0 0
\(169\) −69.0000 −0.408284
\(170\) 0 0
\(171\) −14.0000 + 11.3137i −0.0818713 + 0.0661620i
\(172\) 0 0
\(173\) 186.676i 1.07905i −0.841969 0.539527i \(-0.818603\pi\)
0.841969 0.539527i \(-0.181397\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −208.000 + 73.5391i −1.17514 + 0.415475i
\(178\) 0 0
\(179\) 152.735i 0.853269i −0.904424 0.426634i \(-0.859699\pi\)
0.904424 0.426634i \(-0.140301\pi\)
\(180\) 0 0
\(181\) 90.0000 0.497238 0.248619 0.968601i \(-0.420023\pi\)
0.248619 + 0.968601i \(0.420023\pi\)
\(182\) 0 0
\(183\) 86.0000 + 243.245i 0.469945 + 1.32921i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −128.000 −0.684492
\(188\) 0 0
\(189\) 138.000 + 84.8528i 0.730159 + 0.448957i
\(190\) 0 0
\(191\) 271.529i 1.42162i 0.703385 + 0.710809i \(0.251671\pi\)
−0.703385 + 0.710809i \(0.748329\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.0103627 −0.00518135 0.999987i \(-0.501649\pi\)
−0.00518135 + 0.999987i \(0.501649\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 84.8528i 0.430725i −0.976534 0.215362i \(-0.930907\pi\)
0.976534 0.215362i \(-0.0690933\pi\)
\(198\) 0 0
\(199\) 250.000 1.25628 0.628141 0.778100i \(-0.283816\pi\)
0.628141 + 0.778100i \(0.283816\pi\)
\(200\) 0 0
\(201\) 2.00000 + 5.65685i 0.00995025 + 0.0281436i
\(202\) 0 0
\(203\) 101.823i 0.501593i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 64.0000 + 79.1960i 0.309179 + 0.382589i
\(208\) 0 0
\(209\) 11.3137i 0.0541326i
\(210\) 0 0
\(211\) 34.0000 0.161137 0.0805687 0.996749i \(-0.474326\pi\)
0.0805687 + 0.996749i \(0.474326\pi\)
\(212\) 0 0
\(213\) 352.000 124.451i 1.65258 0.584276i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −132.000 −0.608295
\(218\) 0 0
\(219\) 82.0000 + 231.931i 0.374429 + 1.05905i
\(220\) 0 0
\(221\) 226.274i 1.02387i
\(222\) 0 0
\(223\) 278.000 1.24664 0.623318 0.781968i \(-0.285784\pi\)
0.623318 + 0.781968i \(0.285784\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 220.617i 0.971882i 0.873991 + 0.485941i \(0.161523\pi\)
−0.873991 + 0.485941i \(0.838477\pi\)
\(228\) 0 0
\(229\) 58.0000 0.253275 0.126638 0.991949i \(-0.459581\pi\)
0.126638 + 0.991949i \(0.459581\pi\)
\(230\) 0 0
\(231\) −96.0000 + 33.9411i −0.415584 + 0.146931i
\(232\) 0 0
\(233\) 395.980i 1.69948i 0.527199 + 0.849742i \(0.323242\pi\)
−0.527199 + 0.849742i \(0.676758\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.0000 28.2843i −0.0421941 0.119343i
\(238\) 0 0
\(239\) 22.6274i 0.0946754i 0.998879 + 0.0473377i \(0.0150737\pi\)
−0.998879 + 0.0473377i \(0.984926\pi\)
\(240\) 0 0
\(241\) −30.0000 −0.124481 −0.0622407 0.998061i \(-0.519825\pi\)
−0.0622407 + 0.998061i \(0.519825\pi\)
\(242\) 0 0
\(243\) −241.000 + 31.1127i −0.991770 + 0.128036i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.0000 −0.0809717
\(248\) 0 0
\(249\) −208.000 + 73.5391i −0.835341 + 0.295338i
\(250\) 0 0
\(251\) 107.480i 0.428208i 0.976811 + 0.214104i \(0.0686832\pi\)
−0.976811 + 0.214104i \(0.931317\pi\)
\(252\) 0 0
\(253\) −64.0000 −0.252964
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 181.019i 0.704355i 0.935933 + 0.352178i \(0.114559\pi\)
−0.935933 + 0.352178i \(0.885441\pi\)
\(258\) 0 0
\(259\) 36.0000 0.138996
\(260\) 0 0
\(261\) 96.0000 + 118.794i 0.367816 + 0.455149i
\(262\) 0 0
\(263\) 214.960i 0.817340i −0.912682 0.408670i \(-0.865993\pi\)
0.912682 0.408670i \(-0.134007\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 96.0000 33.9411i 0.359551 0.127120i
\(268\) 0 0
\(269\) 401.637i 1.49307i −0.665344 0.746537i \(-0.731715\pi\)
0.665344 0.746537i \(-0.268285\pi\)
\(270\) 0 0
\(271\) 266.000 0.981550 0.490775 0.871286i \(-0.336714\pi\)
0.490775 + 0.871286i \(0.336714\pi\)
\(272\) 0 0
\(273\) 60.0000 + 169.706i 0.219780 + 0.621632i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −346.000 −1.24910 −0.624549 0.780986i \(-0.714717\pi\)
−0.624549 + 0.780986i \(0.714717\pi\)
\(278\) 0 0
\(279\) 154.000 124.451i 0.551971 0.446060i
\(280\) 0 0
\(281\) 124.451i 0.442885i 0.975173 + 0.221443i \(0.0710766\pi\)
−0.975173 + 0.221443i \(0.928923\pi\)
\(282\) 0 0
\(283\) 46.0000 0.162544 0.0812721 0.996692i \(-0.474102\pi\)
0.0812721 + 0.996692i \(0.474102\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 203.647i 0.709571i
\(288\) 0 0
\(289\) −223.000 −0.771626
\(290\) 0 0
\(291\) −94.0000 265.872i −0.323024 0.913650i
\(292\) 0 0
\(293\) 220.617i 0.752960i −0.926425 0.376480i \(-0.877134\pi\)
0.926425 0.376480i \(-0.122866\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 80.0000 130.108i 0.269360 0.438073i
\(298\) 0 0
\(299\) 113.137i 0.378385i
\(300\) 0 0
\(301\) −492.000 −1.63455
\(302\) 0 0
\(303\) −144.000 + 50.9117i −0.475248 + 0.168025i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 30.0000 0.0977199 0.0488599 0.998806i \(-0.484441\pi\)
0.0488599 + 0.998806i \(0.484441\pi\)
\(308\) 0 0
\(309\) −134.000 379.009i −0.433657 1.22657i
\(310\) 0 0
\(311\) 576.999i 1.85530i −0.373447 0.927651i \(-0.621824\pi\)
0.373447 0.927651i \(-0.378176\pi\)
\(312\) 0 0
\(313\) −210.000 −0.670927 −0.335463 0.942053i \(-0.608893\pi\)
−0.335463 + 0.942053i \(0.608893\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 152.735i 0.481814i 0.970548 + 0.240907i \(0.0774449\pi\)
−0.970548 + 0.240907i \(0.922555\pi\)
\(318\) 0 0
\(319\) −96.0000 −0.300940
\(320\) 0 0
\(321\) 144.000 50.9117i 0.448598 0.158603i
\(322\) 0 0
\(323\) 45.2548i 0.140108i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 28.2843i −0.0305810 0.0864962i
\(328\) 0 0
\(329\) 407.294i 1.23797i
\(330\) 0 0
\(331\) 434.000 1.31118 0.655589 0.755118i \(-0.272420\pi\)
0.655589 + 0.755118i \(0.272420\pi\)
\(332\) 0 0
\(333\) −42.0000 + 33.9411i −0.126126 + 0.101925i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 510.000 1.51335 0.756677 0.653789i \(-0.226822\pi\)
0.756677 + 0.653789i \(0.226822\pi\)
\(338\) 0 0
\(339\) −192.000 + 67.8823i −0.566372 + 0.200243i
\(340\) 0 0
\(341\) 124.451i 0.364958i
\(342\) 0 0
\(343\) −372.000 −1.08455
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 152.735i 0.440159i −0.975482 0.220079i \(-0.929368\pi\)
0.975482 0.220079i \(-0.0706316\pi\)
\(348\) 0 0
\(349\) 426.000 1.22063 0.610315 0.792159i \(-0.291043\pi\)
0.610315 + 0.792159i \(0.291043\pi\)
\(350\) 0 0
\(351\) −230.000 141.421i −0.655271 0.402910i
\(352\) 0 0
\(353\) 45.2548i 0.128201i 0.997943 + 0.0641003i \(0.0204178\pi\)
−0.997943 + 0.0641003i \(0.979582\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −384.000 + 135.765i −1.07563 + 0.380293i
\(358\) 0 0
\(359\) 441.235i 1.22907i 0.788891 + 0.614533i \(0.210655\pi\)
−0.788891 + 0.614533i \(0.789345\pi\)
\(360\) 0 0
\(361\) −357.000 −0.988920
\(362\) 0 0
\(363\) −89.0000 251.730i −0.245179 0.693471i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 566.000 1.54223 0.771117 0.636693i \(-0.219698\pi\)
0.771117 + 0.636693i \(0.219698\pi\)
\(368\) 0 0
\(369\) 192.000 + 237.588i 0.520325 + 0.643870i
\(370\) 0 0
\(371\) 373.352i 1.00634i
\(372\) 0 0
\(373\) −218.000 −0.584450 −0.292225 0.956350i \(-0.594396\pi\)
−0.292225 + 0.956350i \(0.594396\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 169.706i 0.450148i
\(378\) 0 0
\(379\) −142.000 −0.374670 −0.187335 0.982296i \(-0.559985\pi\)
−0.187335 + 0.982296i \(0.559985\pi\)
\(380\) 0 0
\(381\) 106.000 + 299.813i 0.278215 + 0.786911i
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 574.000 463.862i 1.48320 1.19861i
\(388\) 0 0
\(389\) 548.715i 1.41058i −0.708920 0.705289i \(-0.750817\pi\)
0.708920 0.705289i \(-0.249183\pi\)
\(390\) 0 0
\(391\) −256.000 −0.654731
\(392\) 0 0
\(393\) 16.0000 5.65685i 0.0407125 0.0143940i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 310.000 0.780856 0.390428 0.920633i \(-0.372327\pi\)
0.390428 + 0.920633i \(0.372327\pi\)
\(398\) 0 0
\(399\) −12.0000 33.9411i −0.0300752 0.0850655i
\(400\) 0 0
\(401\) 339.411i 0.846412i −0.906033 0.423206i \(-0.860905\pi\)
0.906033 0.423206i \(-0.139095\pi\)
\(402\) 0 0
\(403\) 220.000 0.545906
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.9411i 0.0833934i
\(408\) 0 0
\(409\) −270.000 −0.660147 −0.330073 0.943955i \(-0.607073\pi\)
−0.330073 + 0.943955i \(0.607073\pi\)
\(410\) 0 0
\(411\) −288.000 + 101.823i −0.700730 + 0.247745i
\(412\) 0 0
\(413\) 441.235i 1.06836i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 78.0000 + 220.617i 0.187050 + 0.529058i
\(418\) 0 0
\(419\) 50.9117i 0.121508i 0.998153 + 0.0607538i \(0.0193505\pi\)
−0.998153 + 0.0607538i \(0.980650\pi\)
\(420\) 0 0
\(421\) −454.000 −1.07838 −0.539192 0.842183i \(-0.681270\pi\)
−0.539192 + 0.842183i \(0.681270\pi\)
\(422\) 0 0
\(423\) −384.000 475.176i −0.907801 1.12335i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −516.000 −1.20843
\(428\) 0 0
\(429\) 160.000 56.5685i 0.372960 0.131861i
\(430\) 0 0
\(431\) 248.902i 0.577498i −0.957405 0.288749i \(-0.906761\pi\)
0.957405 0.288749i \(-0.0932393\pi\)
\(432\) 0 0
\(433\) −706.000 −1.63048 −0.815242 0.579120i \(-0.803396\pi\)
−0.815242 + 0.579120i \(0.803396\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.6274i 0.0517790i
\(438\) 0 0
\(439\) −486.000 −1.10706 −0.553531 0.832829i \(-0.686720\pi\)
−0.553531 + 0.832829i \(0.686720\pi\)
\(440\) 0 0
\(441\) 91.0000 73.5391i 0.206349 0.166755i
\(442\) 0 0
\(443\) 707.107i 1.59618i 0.602540 + 0.798089i \(0.294156\pi\)
−0.602540 + 0.798089i \(0.705844\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −464.000 + 164.049i −1.03803 + 0.366999i
\(448\) 0 0
\(449\) 724.077i 1.61264i 0.591477 + 0.806322i \(0.298545\pi\)
−0.591477 + 0.806322i \(0.701455\pi\)
\(450\) 0 0
\(451\) −192.000 −0.425721
\(452\) 0 0
\(453\) −218.000 616.597i −0.481236 1.36114i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −338.000 −0.739606 −0.369803 0.929110i \(-0.620575\pi\)
−0.369803 + 0.929110i \(0.620575\pi\)
\(458\) 0 0
\(459\) 320.000 520.431i 0.697168 1.13384i
\(460\) 0 0
\(461\) 774.989i 1.68110i 0.541731 + 0.840552i \(0.317769\pi\)
−0.541731 + 0.840552i \(0.682231\pi\)
\(462\) 0 0
\(463\) −74.0000 −0.159827 −0.0799136 0.996802i \(-0.525464\pi\)
−0.0799136 + 0.996802i \(0.525464\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 797.616i 1.70796i −0.520307 0.853979i \(-0.674183\pi\)
0.520307 0.853979i \(-0.325817\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.0255864
\(470\) 0 0
\(471\) −86.0000 243.245i −0.182590 0.516443i
\(472\) 0 0
\(473\) 463.862i 0.980681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 352.000 + 435.578i 0.737945 + 0.913161i
\(478\) 0 0
\(479\) 316.784i 0.661344i −0.943746 0.330672i \(-0.892725\pi\)
0.943746 0.330672i \(-0.107275\pi\)
\(480\) 0 0
\(481\) −60.0000 −0.124740
\(482\) 0 0
\(483\) −192.000 + 67.8823i −0.397516 + 0.140543i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 134.000 0.275154 0.137577 0.990491i \(-0.456069\pi\)
0.137577 + 0.990491i \(0.456069\pi\)
\(488\) 0 0
\(489\) −222.000 627.911i −0.453988 1.28407i
\(490\) 0 0
\(491\) 50.9117i 0.103690i −0.998655 0.0518449i \(-0.983490\pi\)
0.998655 0.0518449i \(-0.0165101\pi\)
\(492\) 0 0
\(493\) −384.000 −0.778905
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 746.705i 1.50242i
\(498\) 0 0
\(499\) −30.0000 −0.0601202 −0.0300601 0.999548i \(-0.509570\pi\)
−0.0300601 + 0.999548i \(0.509570\pi\)
\(500\) 0 0
\(501\) −480.000 + 169.706i −0.958084 + 0.338734i
\(502\) 0 0
\(503\) 237.588i 0.472342i −0.971712 0.236171i \(-0.924107\pi\)
0.971712 0.236171i \(-0.0758925\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 69.0000 + 195.161i 0.136095 + 0.384934i
\(508\) 0 0
\(509\) 118.794i 0.233387i 0.993168 + 0.116693i \(0.0372295\pi\)
−0.993168 + 0.116693i \(0.962770\pi\)
\(510\) 0 0
\(511\) −492.000 −0.962818
\(512\) 0 0
\(513\) 46.0000 + 28.2843i 0.0896686 + 0.0551350i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 384.000 0.742747
\(518\) 0 0
\(519\) −528.000 + 186.676i −1.01734 + 0.359684i
\(520\) 0 0
\(521\) 79.1960i 0.152008i −0.997108 0.0760038i \(-0.975784\pi\)
0.997108 0.0760038i \(-0.0242161\pi\)
\(522\) 0 0
\(523\) 494.000 0.944551 0.472275 0.881451i \(-0.343433\pi\)
0.472275 + 0.881451i \(0.343433\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 497.803i 0.944598i
\(528\) 0 0
\(529\) 401.000 0.758034
\(530\) 0 0
\(531\) 416.000 + 514.774i 0.783427 + 0.969442i
\(532\) 0 0
\(533\) 339.411i 0.636794i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −432.000 + 152.735i −0.804469 + 0.284423i
\(538\) 0 0
\(539\) 73.5391i 0.136436i
\(540\) 0 0
\(541\) 234.000 0.432532 0.216266 0.976334i \(-0.430612\pi\)
0.216266 + 0.976334i \(0.430612\pi\)
\(542\) 0 0
\(543\) −90.0000 254.558i −0.165746 0.468800i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −290.000 −0.530165 −0.265082 0.964226i \(-0.585399\pi\)
−0.265082 + 0.964226i \(0.585399\pi\)
\(548\) 0 0
\(549\) 602.000 486.489i 1.09654 0.886137i
\(550\) 0 0
\(551\) 33.9411i 0.0615991i
\(552\) 0 0
\(553\) 60.0000 0.108499
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 175.362i 0.314834i 0.987532 + 0.157417i \(0.0503167\pi\)
−0.987532 + 0.157417i \(0.949683\pi\)
\(558\) 0 0
\(559\) 820.000 1.46691
\(560\) 0 0
\(561\) 128.000 + 362.039i 0.228164 + 0.645345i
\(562\) 0 0
\(563\) 243.245i 0.432051i 0.976388 + 0.216026i \(0.0693094\pi\)
−0.976388 + 0.216026i \(0.930691\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 102.000 475.176i 0.179894 0.838052i
\(568\) 0 0
\(569\) 622.254i 1.09359i 0.837266 + 0.546796i \(0.184153\pi\)
−0.837266 + 0.546796i \(0.815847\pi\)
\(570\) 0 0
\(571\) 402.000 0.704028 0.352014 0.935995i \(-0.385497\pi\)
0.352014 + 0.935995i \(0.385497\pi\)
\(572\) 0 0
\(573\) 768.000 271.529i 1.34031 0.473873i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −98.0000 −0.169844 −0.0849220 0.996388i \(-0.527064\pi\)
−0.0849220 + 0.996388i \(0.527064\pi\)
\(578\) 0 0
\(579\) 2.00000 + 5.65685i 0.00345423 + 0.00977004i
\(580\) 0 0
\(581\) 441.235i 0.759440i
\(582\) 0 0
\(583\) −352.000 −0.603774
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 548.715i 0.934778i 0.884052 + 0.467389i \(0.154805\pi\)
−0.884052 + 0.467389i \(0.845195\pi\)
\(588\) 0 0
\(589\) −44.0000 −0.0747029
\(590\) 0 0
\(591\) −240.000 + 84.8528i −0.406091 + 0.143575i
\(592\) 0 0
\(593\) 701.450i 1.18288i 0.806348 + 0.591442i \(0.201441\pi\)
−0.806348 + 0.591442i \(0.798559\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −250.000 707.107i −0.418760 1.18443i
\(598\) 0 0
\(599\) 644.881i 1.07660i 0.842754 + 0.538298i \(0.180933\pi\)
−0.842754 + 0.538298i \(0.819067\pi\)
\(600\) 0 0
\(601\) −398.000 −0.662230 −0.331115 0.943590i \(-0.607425\pi\)
−0.331115 + 0.943590i \(0.607425\pi\)
\(602\) 0 0
\(603\) 14.0000 11.3137i 0.0232172 0.0187624i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −170.000 −0.280066 −0.140033 0.990147i \(-0.544721\pi\)
−0.140033 + 0.990147i \(0.544721\pi\)
\(608\) 0 0
\(609\) −288.000 + 101.823i −0.472906 + 0.167198i
\(610\) 0 0
\(611\) 678.823i 1.11100i
\(612\) 0 0
\(613\) 1030.00 1.68026 0.840131 0.542384i \(-0.182478\pi\)
0.840131 + 0.542384i \(0.182478\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1052.17i 1.70531i −0.522476 0.852654i \(-0.674992\pi\)
0.522476 0.852654i \(-0.325008\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.0226171 −0.0113086 0.999936i \(-0.503600\pi\)
−0.0113086 + 0.999936i \(0.503600\pi\)
\(620\) 0 0
\(621\) 160.000 260.215i 0.257649 0.419026i
\(622\) 0 0
\(623\) 203.647i 0.326881i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −32.0000 + 11.3137i −0.0510367 + 0.0180442i
\(628\) 0 0
\(629\) 135.765i 0.215842i
\(630\) 0 0
\(631\) 1114.00 1.76545 0.882726 0.469888i \(-0.155706\pi\)
0.882726 + 0.469888i \(0.155706\pi\)
\(632\) 0 0
\(633\) −34.0000 96.1665i −0.0537125 0.151922i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 130.000 0.204082
\(638\) 0 0
\(639\) −704.000 871.156i −1.10172 1.36331i
\(640\) 0 0
\(641\) 452.548i 0.706004i 0.935623 + 0.353002i \(0.114839\pi\)
−0.935623 + 0.353002i \(0.885161\pi\)
\(642\) 0 0
\(643\) 798.000 1.24106 0.620529 0.784184i \(-0.286918\pi\)
0.620529 + 0.784184i \(0.286918\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 147.078i 0.227323i 0.993520 + 0.113662i \(0.0362580\pi\)
−0.993520 + 0.113662i \(0.963742\pi\)
\(648\) 0 0
\(649\) −416.000 −0.640986
\(650\) 0 0
\(651\) 132.000 + 373.352i 0.202765 + 0.573506i
\(652\) 0 0
\(653\) 322.441i 0.493784i −0.969043 0.246892i \(-0.920591\pi\)
0.969043 0.246892i \(-0.0794092\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 574.000 463.862i 0.873668 0.706031i
\(658\) 0 0
\(659\) 797.616i 1.21034i 0.796095 + 0.605172i \(0.206896\pi\)
−0.796095 + 0.605172i \(0.793104\pi\)
\(660\) 0 0
\(661\) 986.000 1.49168 0.745840 0.666126i \(-0.232049\pi\)
0.745840 + 0.666126i \(0.232049\pi\)
\(662\) 0 0
\(663\) 640.000 226.274i 0.965309 0.341288i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −192.000 −0.287856
\(668\) 0 0
\(669\) −278.000 786.303i −0.415546 1.17534i
\(670\) 0 0
\(671\) 486.489i 0.725022i
\(672\) 0 0
\(673\) −34.0000 −0.0505201 −0.0252600 0.999681i \(-0.508041\pi\)
−0.0252600 + 0.999681i \(0.508041\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 401.637i 0.593259i −0.954993 0.296630i \(-0.904137\pi\)
0.954993 0.296630i \(-0.0958627\pi\)
\(678\) 0 0
\(679\) 564.000 0.830633
\(680\) 0 0
\(681\) 624.000 220.617i 0.916300 0.323961i
\(682\) 0 0
\(683\) 130.108i 0.190494i −0.995454 0.0952472i \(-0.969636\pi\)
0.995454 0.0952472i \(-0.0303641\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −58.0000 164.049i −0.0844250 0.238790i
\(688\) 0 0
\(689\) 622.254i 0.903126i
\(690\) 0 0
\(691\) 578.000 0.836469 0.418234 0.908339i \(-0.362649\pi\)
0.418234 + 0.908339i \(0.362649\pi\)
\(692\) 0 0
\(693\) 192.000 + 237.588i 0.277056 + 0.342840i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −768.000 −1.10187
\(698\) 0 0
\(699\) 1120.00 395.980i 1.60229 0.566495i
\(700\) 0 0
\(701\) 1238.85i 1.76726i −0.468183 0.883631i \(-0.655091\pi\)
0.468183 0.883631i \(-0.344909\pi\)
\(702\) 0 0
\(703\) 12.0000 0.0170697
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 305.470i 0.432065i
\(708\) 0 0
\(709\) −1222.00 −1.72355 −0.861777 0.507287i \(-0.830648\pi\)
−0.861777 + 0.507287i \(0.830648\pi\)
\(710\) 0 0
\(711\) −70.0000 + 56.5685i −0.0984529 + 0.0795619i
\(712\) 0 0
\(713\) 248.902i 0.349091i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 64.0000 22.6274i 0.0892608 0.0315585i
\(718\) 0 0
\(719\) 248.902i 0.346177i 0.984906 + 0.173089i \(0.0553747\pi\)
−0.984906 + 0.173089i \(0.944625\pi\)
\(720\) 0 0
\(721\) 804.000 1.11512
\(722\) 0 0
\(723\) 30.0000 + 84.8528i 0.0414938 + 0.117362i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 870.000 1.19670 0.598349 0.801235i \(-0.295823\pi\)
0.598349 + 0.801235i \(0.295823\pi\)
\(728\) 0 0
\(729\) 329.000 + 650.538i 0.451303 + 0.892371i
\(730\) 0 0
\(731\) 1855.45i 2.53823i
\(732\) 0 0
\(733\) 214.000 0.291951 0.145975 0.989288i \(-0.453368\pi\)
0.145975 + 0.989288i \(0.453368\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.3137i 0.0153510i
\(738\) 0 0
\(739\) −958.000 −1.29635 −0.648173 0.761493i \(-0.724467\pi\)
−0.648173 + 0.761493i \(0.724467\pi\)
\(740\) 0 0
\(741\) 20.0000 + 56.5685i 0.0269906 + 0.0763408i
\(742\) 0 0
\(743\) 1006.92i 1.35521i 0.735427 + 0.677604i \(0.236982\pi\)
−0.735427 + 0.677604i \(0.763018\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 416.000 + 514.774i 0.556894 + 0.689121i
\(748\) 0 0
\(749\) 305.470i 0.407837i
\(750\) 0 0
\(751\) −630.000 −0.838881 −0.419441 0.907783i \(-0.637774\pi\)
−0.419441 + 0.907783i \(0.637774\pi\)
\(752\) 0 0
\(753\) 304.000 107.480i 0.403718 0.142736i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −602.000 −0.795244 −0.397622 0.917549i \(-0.630165\pi\)
−0.397622 + 0.917549i \(0.630165\pi\)
\(758\) 0 0
\(759\) 64.0000 + 181.019i 0.0843215 + 0.238497i
\(760\) 0 0
\(761\) 1097.43i 1.44209i −0.692889 0.721044i \(-0.743662\pi\)
0.692889 0.721044i \(-0.256338\pi\)
\(762\) 0 0
\(763\) 60.0000 0.0786370
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 735.391i 0.958789i
\(768\) 0 0
\(769\) 770.000 1.00130 0.500650 0.865650i \(-0.333094\pi\)
0.500650 + 0.865650i \(0.333094\pi\)
\(770\) 0 0
\(771\) 512.000 181.019i 0.664073 0.234785i
\(772\) 0 0
\(773\) 186.676i 0.241496i 0.992683 + 0.120748i \(0.0385293\pi\)
−0.992683 + 0.120748i \(0.961471\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −36.0000 101.823i −0.0463320 0.131047i
\(778\) 0 0
\(779\) 67.8823i 0.0871402i
\(780\) 0 0
\(781\) 704.000 0.901408
\(782\) 0 0
\(783\) 240.000 390.323i 0.306513 0.498497i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −514.000 −0.653113 −0.326557 0.945178i \(-0.605888\pi\)
−0.326557 + 0.945178i \(0.605888\pi\)
\(788\) 0 0
\(789\) −608.000 + 214.960i −0.770596 + 0.272447i
\(790\) 0 0
\(791\) 407.294i 0.514910i
\(792\) 0 0
\(793\) 860.000 1.08449
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 707.107i 0.887211i −0.896222 0.443605i \(-0.853699\pi\)
0.896222 0.443605i \(-0.146301\pi\)
\(798\) 0 0
\(799\) 1536.00 1.92240
\(800\) 0 0
\(801\) −192.000 237.588i