Properties

Label 684.3.y.c.145.1
Level $684$
Weight $3$
Character 684.145
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 145.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.3.y.c.217.1

$q$-expansion

\(f(q)\) \(=\) \(q-11.0000 q^{7} +O(q^{10})\) \(q-11.0000 q^{7} +(22.5000 + 12.9904i) q^{13} +(-13.0000 - 13.8564i) q^{19} +(12.5000 - 21.6506i) q^{25} -60.6218i q^{31} -57.1577i q^{37} +(-41.5000 - 71.8801i) q^{43} +72.0000 q^{49} +(-60.5000 + 104.789i) q^{61} +(115.500 + 66.6840i) q^{67} +(-71.5000 - 123.842i) q^{73} +(76.5000 - 44.1673i) q^{79} +(-247.500 - 142.894i) q^{91} +(168.000 - 96.9948i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 22 q^{7} + O(q^{10}) \) \( 2 q - 22 q^{7} + 45 q^{13} - 26 q^{19} + 25 q^{25} - 83 q^{43} + 144 q^{49} - 121 q^{61} + 231 q^{67} - 143 q^{73} + 153 q^{79} - 495 q^{91} + 336 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −11.0000 −1.57143 −0.785714 0.618590i \(-0.787704\pi\)
−0.785714 + 0.618590i \(0.787704\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 22.5000 + 12.9904i 1.73077 + 0.999260i 0.884615 + 0.466321i \(0.154421\pi\)
0.846154 + 0.532939i \(0.178912\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −13.0000 13.8564i −0.684211 0.729285i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 12.5000 21.6506i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 60.6218i 1.95554i −0.209677 0.977771i \(-0.567241\pi\)
0.209677 0.977771i \(-0.432759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 57.1577i 1.54480i −0.635135 0.772401i \(-0.719056\pi\)
0.635135 0.772401i \(-0.280944\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −41.5000 71.8801i −0.965116 1.67163i −0.709302 0.704904i \(-0.750990\pi\)
−0.255814 0.966726i \(-0.582343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 72.0000 1.46939
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −60.5000 + 104.789i −0.991803 + 1.71785i −0.385246 + 0.922814i \(0.625883\pi\)
−0.606557 + 0.795040i \(0.707450\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 115.500 + 66.6840i 1.72388 + 0.995283i 0.910448 + 0.413624i \(0.135737\pi\)
0.813433 + 0.581659i \(0.197596\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) −71.5000 123.842i −0.979452 1.69646i −0.664384 0.747392i \(-0.731306\pi\)
−0.315068 0.949069i \(-0.602027\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 76.5000 44.1673i 0.968354 0.559080i 0.0696203 0.997574i \(-0.477821\pi\)
0.898734 + 0.438494i \(0.144488\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) −247.500 142.894i −2.71978 1.57027i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 168.000 96.9948i 1.73196 0.999947i 0.860825 0.508902i \(-0.169948\pi\)
0.871134 0.491045i \(-0.163385\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 133.368i 1.29483i −0.762136 0.647417i \(-0.775849\pi\)
0.762136 0.647417i \(-0.224151\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 36.0000 20.7846i 0.330275 0.190684i −0.325688 0.945477i \(-0.605596\pi\)
0.655963 + 0.754793i \(0.272263\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −121.000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 180.000 + 103.923i 1.41732 + 0.818292i 0.996063 0.0886483i \(-0.0282547\pi\)
0.421260 + 0.906940i \(0.361588\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 143.000 + 152.420i 1.07519 + 1.14602i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0 0
\(139\) −114.500 + 198.320i −0.823741 + 1.42676i 0.0791367 + 0.996864i \(0.474784\pi\)
−0.902878 + 0.429898i \(0.858550\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 96.9948i 0.642350i −0.947020 0.321175i \(-0.895922\pi\)
0.947020 0.321175i \(-0.104078\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −96.5000 167.143i −0.614650 1.06460i −0.990446 0.137902i \(-0.955964\pi\)
0.375796 0.926702i \(-0.377369\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −299.000 −1.83436 −0.917178 0.398478i \(-0.869539\pi\)
−0.917178 + 0.398478i \(0.869539\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) 253.000 + 438.209i 1.49704 + 2.59295i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) −137.500 + 238.157i −0.785714 + 1.36090i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 156.000 + 90.0666i 0.861878 + 0.497606i 0.864641 0.502390i \(-0.167546\pi\)
−0.00276243 + 0.999996i \(0.500879\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −262.500 + 151.554i −1.36010 + 0.785256i −0.989637 0.143590i \(-0.954135\pi\)
−0.370466 + 0.928846i \(0.620802\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 54.5000 94.3968i 0.273869 0.474356i −0.695980 0.718061i \(-0.745030\pi\)
0.969849 + 0.243706i \(0.0783631\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 43.5000 25.1147i 0.206161 0.119027i −0.393365 0.919382i \(-0.628689\pi\)
0.599526 + 0.800355i \(0.295356\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 666.840i 3.07299i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −127.500 + 73.6122i −0.571749 + 0.330099i −0.757848 0.652432i \(-0.773749\pi\)
0.186099 + 0.982531i \(0.440416\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 409.000 1.78603 0.893013 0.450031i \(-0.148587\pi\)
0.893013 + 0.450031i \(0.148587\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −46.5000 26.8468i −0.192946 0.111397i 0.400415 0.916334i \(-0.368866\pi\)
−0.593361 + 0.804936i \(0.702199\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −112.500 480.644i −0.455466 1.94593i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 628.734i 2.42755i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −121.000 209.578i −0.446494 0.773351i 0.551661 0.834069i \(-0.313994\pi\)
−0.998155 + 0.0607176i \(0.980661\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −122.000 −0.440433 −0.220217 0.975451i \(-0.570676\pi\)
−0.220217 + 0.975451i \(0.570676\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) −229.000 396.640i −0.809187 1.40155i −0.913428 0.407001i \(-0.866574\pi\)
0.104240 0.994552i \(-0.466759\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 144.500 250.281i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 456.500 + 790.681i 1.51661 + 2.62685i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −432.000 + 249.415i −1.40717 + 0.812428i −0.995114 0.0987325i \(-0.968521\pi\)
−0.412052 + 0.911160i \(0.635188\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −71.0000 + 122.976i −0.226837 + 0.392893i −0.956869 0.290520i \(-0.906172\pi\)
0.730032 + 0.683413i \(0.239505\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 562.500 324.760i 1.73077 0.999260i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 36.3731i 0.109888i 0.998489 + 0.0549442i \(0.0174981\pi\)
−0.998489 + 0.0549442i \(0.982502\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −565.500 + 326.492i −1.67804 + 0.968818i −0.715134 + 0.698988i \(0.753634\pi\)
−0.962908 + 0.269830i \(0.913033\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −253.000 −0.737609
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 671.000 1.92264 0.961318 0.275441i \(-0.0888238\pi\)
0.961318 + 0.275441i \(0.0888238\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −23.0000 + 360.267i −0.0637119 + 0.997968i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −245.500 + 425.218i −0.668937 + 1.15863i 0.309264 + 0.950976i \(0.399917\pi\)
−0.978202 + 0.207657i \(0.933416\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 263.272i 0.705822i −0.935657 0.352911i \(-0.885192\pi\)
0.935657 0.352911i \(-0.114808\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 448.601i 1.18364i −0.806069 0.591822i \(-0.798409\pi\)
0.806069 0.591822i \(-0.201591\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −215.500 373.257i −0.542821 0.940194i −0.998741 0.0501728i \(-0.984023\pi\)
0.455919 0.890021i \(-0.349311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 787.500 1363.99i 1.95409 3.38459i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −456.000 263.272i −1.11491 0.643696i −0.174817 0.984601i \(-0.555933\pi\)
−0.940098 + 0.340905i \(0.889267\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 660.000 381.051i 1.56770 0.905110i 0.571259 0.820770i \(-0.306455\pi\)
0.996437 0.0843398i \(-0.0268781\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 665.500 1152.68i 1.55855 2.69948i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) −610.500 352.472i −1.40993 0.814024i −0.414550 0.910027i \(-0.636061\pi\)
−0.995381 + 0.0960028i \(0.969394\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −448.500 + 258.942i −1.02164 + 0.589844i −0.914579 0.404408i \(-0.867478\pi\)
−0.107062 + 0.994252i \(0.534144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 767.000 1.67834 0.839168 0.543872i \(-0.183042\pi\)
0.839168 + 0.543872i \(0.183042\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) −397.000 −0.857451 −0.428726 0.903435i \(-0.641037\pi\)
−0.428726 + 0.903435i \(0.641037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1270.50 733.524i −2.70896 1.56402i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −462.500 + 108.253i −0.973684 + 0.227901i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 742.500 1286.05i 1.54366 2.67370i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 152.420i 0.312978i 0.987680 + 0.156489i \(0.0500176\pi\)
−0.987680 + 0.156489i \(0.949982\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 425.500 + 736.988i 0.852705 + 1.47693i 0.878758 + 0.477269i \(0.158373\pi\)
−0.0260521 + 0.999661i \(0.508294\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 786.500 + 1362.26i 1.53914 + 2.66587i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 580.500 + 335.152i 1.10994 + 0.640826i 0.938815 0.344423i \(-0.111925\pi\)
0.171128 + 0.985249i \(0.445259\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 264.500 + 458.127i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 120.500 208.712i 0.222736 0.385790i −0.732902 0.680334i \(-0.761835\pi\)
0.955638 + 0.294545i \(0.0951680\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 40.5000 + 23.3827i 0.0740402 + 0.0427471i 0.536563 0.843860i \(-0.319722\pi\)
−0.462523 + 0.886607i \(0.653056\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −841.500 + 485.840i −1.52170 + 0.878554i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) 2156.40i 3.85761i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −181.000 −0.316988 −0.158494 0.987360i \(-0.550664\pi\)
−0.158494 + 0.987360i \(0.550664\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −962.000 −1.66724 −0.833622 0.552335i \(-0.813737\pi\)
−0.833622 + 0.552335i \(0.813737\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) 0 0
\(589\) −840.000 + 788.083i −1.42615 + 1.33800i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 84.8705i 0.141215i −0.997504 0.0706077i \(-0.977506\pi\)
0.997504 0.0706077i \(-0.0224939\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1155.28i 1.90326i −0.307249 0.951629i \(-0.599408\pi\)
0.307249 0.951629i \(-0.400592\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −563.000 975.145i −0.918434 1.59077i −0.801794 0.597600i \(-0.796121\pi\)
−0.116639 0.993174i \(-0.537212\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(618\) 0 0
\(619\) 949.000 1.53312 0.766559 0.642174i \(-0.221967\pi\)
0.766559 + 0.642174i \(0.221967\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −312.500 541.266i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 293.500 508.357i 0.465135 0.805637i −0.534073 0.845438i \(-0.679339\pi\)
0.999208 + 0.0398015i \(0.0126726\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1620.00 + 935.307i 2.54317 + 1.46830i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 618.500 + 1071.27i 0.961897 + 1.66606i 0.717729 + 0.696322i \(0.245181\pi\)
0.244168 + 0.969733i \(0.421485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) 1140.00 + 658.179i 1.72466 + 0.995733i 0.908472 + 0.417946i \(0.137250\pi\)
0.816188 + 0.577787i \(0.196084\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 652.983i 0.970257i −0.874443 0.485129i \(-0.838773\pi\)
0.874443 0.485129i \(-0.161227\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1848.00 + 1066.94i −2.72165 + 1.57135i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1318.00 −1.90738 −0.953690 0.300790i \(-0.902750\pi\)
−0.953690 + 0.300790i \(0.902750\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(702\) 0 0
\(703\) −792.000 + 743.050i −1.12660 + 1.05697i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 228.500 395.774i 0.322285 0.558214i −0.658674 0.752428i \(-0.728882\pi\)
0.980959 + 0.194214i \(0.0622158\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 1467.05i 2.03474i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 714.500 + 1237.55i 0.982806 + 1.70227i 0.651307 + 0.758815i \(0.274221\pi\)
0.331499 + 0.943455i \(0.392446\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1034.00 1.41064 0.705321 0.708888i \(-0.250803\pi\)
0.705321 + 0.708888i \(0.250803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −54.5000 94.3968i −0.0737483 0.127736i 0.826793 0.562506i \(-0.190163\pi\)
−0.900541 + 0.434771i \(0.856829\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1291.50 + 745.648i 1.71971 + 0.992873i 0.919441 + 0.393229i \(0.128642\pi\)
0.800266 + 0.599645i \(0.204691\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −336.500 582.835i −0.444518 0.769927i 0.553501 0.832849i \(-0.313292\pi\)
−0.998018 + 0.0629213i \(0.979958\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −396.000 + 228.631i −0.519004 + 0.299647i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 431.500 747.380i 0.561118 0.971885i −0.436281 0.899811i \(-0.643705\pi\)
0.997399 0.0720749i \(-0.0229621\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) −1312.50 757.772i −1.69355 0.977771i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1449.73i 1.84209i 0.389454 + 0.921046i \(0.372664\pi\)
−0.389454 + 0.921046i \(0.627336\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2722.50 + 1571.84i −3.43317 + 1.98214i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0