Properties

Label 1216.2.h.c.1215.2
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1215.2
Root \(1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.c.1215.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.64575 q^{3} +1.73205i q^{7} +4.00000 q^{9} +O(q^{10})\) \(q-2.64575 q^{3} +1.73205i q^{7} +4.00000 q^{9} +3.46410i q^{11} -4.58258i q^{13} +3.00000 q^{17} +(2.64575 + 3.46410i) q^{19} -4.58258i q^{21} -5.19615i q^{23} -5.00000 q^{25} -2.64575 q^{27} -4.58258i q^{29} -5.29150 q^{31} -9.16515i q^{33} +9.16515i q^{37} +12.1244i q^{39} +9.16515i q^{41} +3.46410i q^{47} +4.00000 q^{49} -7.93725 q^{51} +4.58258i q^{53} +(-7.00000 - 9.16515i) q^{57} -7.93725 q^{59} +8.00000 q^{61} +6.92820i q^{63} -2.64575 q^{67} +13.7477i q^{69} -15.8745 q^{71} -11.0000 q^{73} +13.2288 q^{75} -6.00000 q^{77} +5.29150 q^{79} -5.00000 q^{81} +6.92820i q^{83} +12.1244i q^{87} +18.3303i q^{89} +7.93725 q^{91} +14.0000 q^{93} +9.16515i q^{97} +13.8564i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 16q^{9} + O(q^{10}) \) \( 4q + 16q^{9} + 12q^{17} - 20q^{25} + 16q^{49} - 28q^{57} + 32q^{61} - 44q^{73} - 24q^{77} - 20q^{81} + 56q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −2.64575 −1.52753 −0.763763 0.645497i \(-0.776650\pi\)
−0.763763 + 0.645497i \(0.776650\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.73205i 0.654654i 0.944911 + 0.327327i \(0.106148\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 4.00000 1.33333
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 4.58258i 1.27098i −0.772110 0.635489i \(-0.780799\pi\)
0.772110 0.635489i \(-0.219201\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.64575 + 3.46410i 0.606977 + 0.794719i
\(20\) 0 0
\(21\) 4.58258i 1.00000i
\(22\) 0 0
\(23\) 5.19615i 1.08347i −0.840548 0.541736i \(-0.817767\pi\)
0.840548 0.541736i \(-0.182233\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −2.64575 −0.509175
\(28\) 0 0
\(29\) 4.58258i 0.850963i −0.904967 0.425481i \(-0.860105\pi\)
0.904967 0.425481i \(-0.139895\pi\)
\(30\) 0 0
\(31\) −5.29150 −0.950382 −0.475191 0.879883i \(-0.657621\pi\)
−0.475191 + 0.879883i \(0.657621\pi\)
\(32\) 0 0
\(33\) 9.16515i 1.59545i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.16515i 1.50674i 0.657596 + 0.753371i \(0.271573\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 12.1244i 1.94145i
\(40\) 0 0
\(41\) 9.16515i 1.43136i 0.698430 + 0.715678i \(0.253882\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) −7.93725 −1.11144
\(52\) 0 0
\(53\) 4.58258i 0.629465i 0.949180 + 0.314733i \(0.101915\pi\)
−0.949180 + 0.314733i \(0.898085\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.00000 9.16515i −0.927173 1.21395i
\(58\) 0 0
\(59\) −7.93725 −1.03334 −0.516671 0.856184i \(-0.672829\pi\)
−0.516671 + 0.856184i \(0.672829\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 6.92820i 0.872872i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.64575 −0.323230 −0.161615 0.986854i \(-0.551670\pi\)
−0.161615 + 0.986854i \(0.551670\pi\)
\(68\) 0 0
\(69\) 13.7477i 1.65503i
\(70\) 0 0
\(71\) −15.8745 −1.88396 −0.941979 0.335673i \(-0.891036\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 13.2288 1.52753
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 5.29150 0.595341 0.297670 0.954669i \(-0.403790\pi\)
0.297670 + 0.954669i \(0.403790\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 6.92820i 0.760469i 0.924890 + 0.380235i \(0.124157\pi\)
−0.924890 + 0.380235i \(0.875843\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.1244i 1.29987i
\(88\) 0 0
\(89\) 18.3303i 1.94301i 0.237023 + 0.971504i \(0.423828\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0 0
\(91\) 7.93725 0.832050
\(92\) 0 0
\(93\) 14.0000 1.45173
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.16515i 0.930580i 0.885158 + 0.465290i \(0.154050\pi\)
−0.885158 + 0.465290i \(0.845950\pi\)
\(98\) 0 0
\(99\) 13.8564i 1.39262i
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −10.5830 −1.04277 −0.521387 0.853320i \(-0.674585\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.93725 −0.767323 −0.383662 0.923474i \(-0.625337\pi\)
−0.383662 + 0.923474i \(0.625337\pi\)
\(108\) 0 0
\(109\) 13.7477i 1.31679i 0.752671 + 0.658397i \(0.228765\pi\)
−0.752671 + 0.658397i \(0.771235\pi\)
\(110\) 0 0
\(111\) 24.2487i 2.30159i
\(112\) 0 0
\(113\) 9.16515i 0.862185i −0.902308 0.431092i \(-0.858128\pi\)
0.902308 0.431092i \(-0.141872\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 18.3303i 1.69464i
\(118\) 0 0
\(119\) 5.19615i 0.476331i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 24.2487i 2.18643i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.5830 0.939090 0.469545 0.882909i \(-0.344418\pi\)
0.469545 + 0.882909i \(0.344418\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.7846i 1.81596i 0.419014 + 0.907980i \(0.362376\pi\)
−0.419014 + 0.907980i \(0.637624\pi\)
\(132\) 0 0
\(133\) −6.00000 + 4.58258i −0.520266 + 0.397360i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 17.3205i 1.46911i −0.678551 0.734553i \(-0.737392\pi\)
0.678551 0.734553i \(-0.262608\pi\)
\(140\) 0 0
\(141\) 9.16515i 0.771845i
\(142\) 0 0
\(143\) 15.8745 1.32749
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.5830 −0.872872
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 5.29150 0.430616 0.215308 0.976546i \(-0.430924\pi\)
0.215308 + 0.976546i \(0.430924\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 12.1244i 0.961524i
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) 17.3205i 1.35665i −0.734763 0.678323i \(-0.762707\pi\)
0.734763 0.678323i \(-0.237293\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.8745 1.22841 0.614203 0.789148i \(-0.289478\pi\)
0.614203 + 0.789148i \(0.289478\pi\)
\(168\) 0 0
\(169\) −8.00000 −0.615385
\(170\) 0 0
\(171\) 10.5830 + 13.8564i 0.809303 + 1.05963i
\(172\) 0 0
\(173\) 9.16515i 0.696814i −0.937343 0.348407i \(-0.886723\pi\)
0.937343 0.348407i \(-0.113277\pi\)
\(174\) 0 0
\(175\) 8.66025i 0.654654i
\(176\) 0 0
\(177\) 21.0000 1.57846
\(178\) 0 0
\(179\) −15.8745 −1.18652 −0.593258 0.805012i \(-0.702159\pi\)
−0.593258 + 0.805012i \(0.702159\pi\)
\(180\) 0 0
\(181\) 9.16515i 0.681240i −0.940201 0.340620i \(-0.889363\pi\)
0.940201 0.340620i \(-0.110637\pi\)
\(182\) 0 0
\(183\) −21.1660 −1.56464
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.3923i 0.759961i
\(188\) 0 0
\(189\) 4.58258i 0.333333i
\(190\) 0 0
\(191\) 12.1244i 0.877288i 0.898661 + 0.438644i \(0.144541\pi\)
−0.898661 + 0.438644i \(0.855459\pi\)
\(192\) 0 0
\(193\) 9.16515i 0.659722i −0.944030 0.329861i \(-0.892998\pi\)
0.944030 0.329861i \(-0.107002\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 8.66025i 0.613909i 0.951724 + 0.306955i \(0.0993100\pi\)
−0.951724 + 0.306955i \(0.900690\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) 7.93725 0.557086
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 20.7846i 1.44463i
\(208\) 0 0
\(209\) −12.0000 + 9.16515i −0.830057 + 0.633967i
\(210\) 0 0
\(211\) −2.64575 −0.182141 −0.0910705 0.995844i \(-0.529029\pi\)
−0.0910705 + 0.995844i \(0.529029\pi\)
\(212\) 0 0
\(213\) 42.0000 2.87779
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.16515i 0.622171i
\(218\) 0 0
\(219\) 29.1033 1.96662
\(220\) 0 0
\(221\) 13.7477i 0.924772i
\(222\) 0 0
\(223\) −10.5830 −0.708690 −0.354345 0.935115i \(-0.615296\pi\)
−0.354345 + 0.935115i \(0.615296\pi\)
\(224\) 0 0
\(225\) −20.0000 −1.33333
\(226\) 0 0
\(227\) 7.93725 0.526814 0.263407 0.964685i \(-0.415154\pi\)
0.263407 + 0.964685i \(0.415154\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 15.8745 1.04447
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) 1.73205i 0.112037i −0.998430 0.0560185i \(-0.982159\pi\)
0.998430 0.0560185i \(-0.0178406\pi\)
\(240\) 0 0
\(241\) 9.16515i 0.590379i −0.955439 0.295190i \(-0.904617\pi\)
0.955439 0.295190i \(-0.0953828\pi\)
\(242\) 0 0
\(243\) 21.1660 1.35780
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.8745 12.1244i 1.01007 0.771454i
\(248\) 0 0
\(249\) 18.3303i 1.16164i
\(250\) 0 0
\(251\) 10.3923i 0.655956i 0.944685 + 0.327978i \(0.106367\pi\)
−0.944685 + 0.327978i \(0.893633\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −15.8745 −0.986394
\(260\) 0 0
\(261\) 18.3303i 1.13462i
\(262\) 0 0
\(263\) 3.46410i 0.213606i −0.994280 0.106803i \(-0.965939\pi\)
0.994280 0.106803i \(-0.0340614\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 48.4974i 2.96799i
\(268\) 0 0
\(269\) 27.4955i 1.67643i 0.545342 + 0.838214i \(0.316400\pi\)
−0.545342 + 0.838214i \(0.683600\pi\)
\(270\) 0 0
\(271\) 8.66025i 0.526073i −0.964786 0.263036i \(-0.915276\pi\)
0.964786 0.263036i \(-0.0847240\pi\)
\(272\) 0 0
\(273\) −21.0000 −1.27098
\(274\) 0 0
\(275\) 17.3205i 1.04447i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −21.1660 −1.26718
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 20.7846i 1.23552i −0.786368 0.617758i \(-0.788041\pi\)
0.786368 0.617758i \(-0.211959\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −15.8745 −0.937043
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 24.2487i 1.42148i
\(292\) 0 0
\(293\) 22.9129i 1.33858i 0.742999 + 0.669292i \(0.233403\pi\)
−0.742999 + 0.669292i \(0.766597\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.16515i 0.531816i
\(298\) 0 0
\(299\) −23.8118 −1.37707
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 47.6235 2.73590
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.4575 1.51001 0.755005 0.655719i \(-0.227634\pi\)
0.755005 + 0.655719i \(0.227634\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) 0 0
\(311\) 22.5167i 1.27680i 0.769704 + 0.638401i \(0.220404\pi\)
−0.769704 + 0.638401i \(0.779596\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.7477i 0.772149i 0.922468 + 0.386075i \(0.126169\pi\)
−0.922468 + 0.386075i \(0.873831\pi\)
\(318\) 0 0
\(319\) 15.8745 0.888802
\(320\) 0 0
\(321\) 21.0000 1.17211
\(322\) 0 0
\(323\) 7.93725 + 10.3923i 0.441641 + 0.578243i
\(324\) 0 0
\(325\) 22.9129i 1.27098i
\(326\) 0 0
\(327\) 36.3731i 2.01144i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 2.64575 0.145424 0.0727118 0.997353i \(-0.476835\pi\)
0.0727118 + 0.997353i \(0.476835\pi\)
\(332\) 0 0
\(333\) 36.6606i 2.00899i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.3303i 0.998515i 0.866454 + 0.499258i \(0.166394\pi\)
−0.866454 + 0.499258i \(0.833606\pi\)
\(338\) 0 0
\(339\) 24.2487i 1.31701i
\(340\) 0 0
\(341\) 18.3303i 0.992642i
\(342\) 0 0
\(343\) 19.0526i 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 12.1244i 0.647150i
\(352\) 0 0
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 13.7477i 0.727607i
\(358\) 0 0
\(359\) 5.19615i 0.274242i −0.990554 0.137121i \(-0.956215\pi\)
0.990554 0.137121i \(-0.0437850\pi\)
\(360\) 0 0
\(361\) −5.00000 + 18.3303i −0.263158 + 0.964753i
\(362\) 0 0
\(363\) 2.64575 0.138866
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.46410i 0.180825i 0.995904 + 0.0904123i \(0.0288185\pi\)
−0.995904 + 0.0904123i \(0.971182\pi\)
\(368\) 0 0
\(369\) 36.6606i 1.90847i
\(370\) 0 0
\(371\) −7.93725 −0.412082
\(372\) 0 0
\(373\) 32.0780i 1.66094i −0.557065 0.830469i \(-0.688073\pi\)
0.557065 0.830469i \(-0.311927\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −21.0000 −1.08156
\(378\) 0 0
\(379\) 34.3948 1.76674 0.883370 0.468676i \(-0.155269\pi\)
0.883370 + 0.468676i \(0.155269\pi\)
\(380\) 0 0
\(381\) −28.0000 −1.43448
\(382\) 0 0
\(383\) −31.7490 −1.62230 −0.811149 0.584839i \(-0.801158\pi\)
−0.811149 + 0.584839i \(0.801158\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 15.5885i 0.788342i
\(392\) 0 0
\(393\) 54.9909i 2.77392i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 15.8745 12.1244i 0.794719 0.606977i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 24.2487i 1.20791i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −31.7490 −1.57374
\(408\) 0 0
\(409\) 9.16515i 0.453188i 0.973989 + 0.226594i \(0.0727590\pi\)
−0.973989 + 0.226594i \(0.927241\pi\)
\(410\) 0 0
\(411\) −23.8118 −1.17455
\(412\) 0 0
\(413\) 13.7477i 0.676481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 45.8258i 2.24410i
\(418\) 0 0
\(419\) 6.92820i 0.338465i 0.985576 + 0.169232i \(0.0541289\pi\)
−0.985576 + 0.169232i \(0.945871\pi\)
\(420\) 0 0
\(421\) 22.9129i 1.11671i 0.829604 + 0.558353i \(0.188566\pi\)
−0.829604 + 0.558353i \(0.811434\pi\)
\(422\) 0 0
\(423\) 13.8564i 0.673722i
\(424\) 0 0
\(425\) −15.0000 −0.727607
\(426\) 0 0
\(427\) 13.8564i 0.670559i
\(428\) 0 0
\(429\) −42.0000 −2.02778
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18.0000 13.7477i 0.861057 0.657643i
\(438\) 0 0
\(439\) −10.5830 −0.505099 −0.252550 0.967584i \(-0.581269\pi\)
−0.252550 + 0.967584i \(0.581269\pi\)
\(440\) 0 0
\(441\) 16.0000 0.761905
\(442\) 0 0
\(443\) 17.3205i 0.822922i 0.911427 + 0.411461i \(0.134981\pi\)
−0.911427 + 0.411461i \(0.865019\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 15.8745 0.750838
\(448\) 0 0
\(449\) 9.16515i 0.432530i 0.976335 + 0.216265i \(0.0693876\pi\)
−0.976335 + 0.216265i \(0.930612\pi\)
\(450\) 0 0
\(451\) −31.7490 −1.49500
\(452\) 0 0
\(453\) −14.0000 −0.657777
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) 0 0
\(459\) −7.93725 −0.370479
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 38.1051i 1.77090i −0.464739 0.885448i \(-0.653852\pi\)
0.464739 0.885448i \(-0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.46410i 0.160300i −0.996783 0.0801498i \(-0.974460\pi\)
0.996783 0.0801498i \(-0.0255399\pi\)
\(468\) 0 0
\(469\) 4.58258i 0.211604i
\(470\) 0 0
\(471\) 26.4575 1.21910
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −13.2288 17.3205i −0.606977 0.794719i
\(476\) 0 0
\(477\) 18.3303i 0.839287i
\(478\) 0 0
\(479\) 10.3923i 0.474837i 0.971408 + 0.237418i \(0.0763012\pi\)
−0.971408 + 0.237418i \(0.923699\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) 0 0
\(483\) −23.8118 −1.08347
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 10.5830 0.479562 0.239781 0.970827i \(-0.422924\pi\)
0.239781 + 0.970827i \(0.422924\pi\)
\(488\) 0 0
\(489\) 45.8258i 2.07231i
\(490\) 0 0
\(491\) 13.8564i 0.625331i −0.949863 0.312665i \(-0.898778\pi\)
0.949863 0.312665i \(-0.101222\pi\)
\(492\) 0 0
\(493\) 13.7477i 0.619166i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.4955i 1.23334i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −42.0000 −1.87642
\(502\) 0 0
\(503\) 43.3013i 1.93071i 0.260943 + 0.965354i \(0.415966\pi\)
−0.260943 + 0.965354i \(0.584034\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 21.1660 0.940016
\(508\) 0 0
\(509\) 9.16515i 0.406238i −0.979154 0.203119i \(-0.934892\pi\)
0.979154 0.203119i \(-0.0651079\pi\)
\(510\) 0 0
\(511\) 19.0526i 0.842836i
\(512\) 0 0
\(513\) −7.00000 9.16515i −0.309058 0.404651i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 24.2487i 1.06440i
\(520\) 0 0
\(521\) 27.4955i 1.20460i −0.798271 0.602299i \(-0.794252\pi\)
0.798271 0.602299i \(-0.205748\pi\)
\(522\) 0 0
\(523\) 13.2288 0.578453 0.289227 0.957261i \(-0.406602\pi\)
0.289227 + 0.957261i \(0.406602\pi\)
\(524\) 0 0
\(525\) 22.9129i 1.00000i
\(526\) 0 0
\(527\) −15.8745 −0.691504
\(528\) 0 0
\(529\) −4.00000 −0.173913
\(530\) 0 0
\(531\) −31.7490 −1.37779
\(532\) 0 0
\(533\) 42.0000 1.81922
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 42.0000 1.81243
\(538\) 0 0
\(539\) 13.8564i 0.596838i
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 24.2487i 1.04061i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.4575 1.13124 0.565621 0.824665i \(-0.308637\pi\)
0.565621 + 0.824665i \(0.308637\pi\)
\(548\) 0 0
\(549\) 32.0000 1.36573
\(550\) 0 0
\(551\) 15.8745 12.1244i 0.676277 0.516515i
\(552\) 0 0
\(553\) 9.16515i 0.389742i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 27.4955i 1.16086i
\(562\) 0 0
\(563\) −15.8745 −0.669031 −0.334515 0.942390i \(-0.608573\pi\)
−0.334515 + 0.942390i \(0.608573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 8.66025i 0.363696i
\(568\) 0 0
\(569\) 27.4955i 1.15267i −0.817214 0.576335i \(-0.804482\pi\)
0.817214 0.576335i \(-0.195518\pi\)
\(570\) 0 0
\(571\) 3.46410i 0.144968i 0.997370 + 0.0724841i \(0.0230926\pi\)
−0.997370 + 0.0724841i \(0.976907\pi\)
\(572\) 0 0
\(573\) 32.0780i 1.34008i
\(574\) 0 0
\(575\) 25.9808i 1.08347i
\(576\) 0 0
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 0 0
\(579\) 24.2487i 1.00774i
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −15.8745 −0.657455
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.5692i 1.71575i 0.513862 + 0.857873i \(0.328214\pi\)
−0.513862 + 0.857873i \(0.671786\pi\)
\(588\) 0 0
\(589\) −14.0000 18.3303i −0.576860 0.755287i
\(590\) 0 0
\(591\) −15.8745 −0.652990
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.9129i 0.937762i
\(598\) 0 0
\(599\) −31.7490 −1.29723 −0.648615 0.761117i \(-0.724651\pi\)
−0.648615 + 0.761117i \(0.724651\pi\)
\(600\) 0 0
\(601\) 27.4955i 1.12156i 0.827964 + 0.560781i \(0.189499\pi\)
−0.827964 + 0.560781i \(0.810501\pi\)
\(602\) 0 0
\(603\) −10.5830 −0.430973
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 26.4575 1.07388 0.536939 0.843621i \(-0.319581\pi\)
0.536939 + 0.843621i \(0.319581\pi\)
\(608\) 0 0
\(609\) −21.0000 −0.850963
\(610\) 0 0
\(611\) 15.8745 0.642214
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 13.8564i 0.556936i −0.960446 0.278468i \(-0.910173\pi\)
0.960446 0.278468i \(-0.0898266\pi\)
\(620\) 0 0
\(621\) 13.7477i 0.551677i
\(622\) 0 0
\(623\) −31.7490 −1.27200
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 31.7490 24.2487i 1.26793 0.968400i
\(628\) 0 0
\(629\) 27.4955i 1.09632i
\(630\) 0 0
\(631\) 24.2487i 0.965326i 0.875806 + 0.482663i \(0.160330\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 7.00000 0.278225
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 18.3303i 0.726273i
\(638\) 0 0
\(639\) −63.4980 −2.51194
\(640\) 0 0
\(641\) 36.6606i 1.44801i −0.689796 0.724003i \(-0.742300\pi\)
0.689796 0.724003i \(-0.257700\pi\)
\(642\) 0 0
\(643\) 31.1769i 1.22950i 0.788723 + 0.614749i \(0.210743\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.9090i 1.29378i −0.762581 0.646892i \(-0.776068\pi\)
0.762581 0.646892i \(-0.223932\pi\)
\(648\) 0 0
\(649\) 27.4955i 1.07929i
\(650\) 0 0
\(651\) 24.2487i 0.950382i
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −44.0000 −1.71660
\(658\) 0 0
\(659\) −23.8118 −0.927575 −0.463787 0.885947i \(-0.653510\pi\)
−0.463787 + 0.885947i \(0.653510\pi\)
\(660\) 0 0
\(661\) 4.58258i 0.178242i 0.996021 + 0.0891208i \(0.0284057\pi\)
−0.996021 + 0.0891208i \(0.971594\pi\)
\(662\) 0 0
\(663\) 36.3731i 1.41261i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −23.8118 −0.921995
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 27.7128i 1.06984i
\(672\) 0 0
\(673\) 45.8258i 1.76645i −0.468947 0.883227i \(-0.655366\pi\)
0.468947 0.883227i \(-0.344634\pi\)
\(674\) 0 0
\(675\) 13.2288 0.509175
\(676\) 0 0
\(677\) 13.7477i 0.528368i −0.964472 0.264184i \(-0.914897\pi\)
0.964472 0.264184i \(-0.0851026\pi\)
\(678\) 0 0
\(679\) −15.8745 −0.609208
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) 15.8745 0.607421 0.303711 0.952764i \(-0.401774\pi\)
0.303711 + 0.952764i \(0.401774\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 21.1660 0.807534
\(688\) 0 0
\(689\) 21.0000 0.800036
\(690\) 0 0
\(691\) 20.7846i 0.790684i −0.918534 0.395342i \(-0.870626\pi\)
0.918534 0.395342i \(-0.129374\pi\)
\(692\) 0 0
\(693\) −24.0000 −0.911685
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 27.4955i 1.04146i
\(698\) 0 0
\(699\) −47.6235 −1.80129
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) −31.7490 + 24.2487i −1.19744 + 0.914557i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 31.1769i 1.17253i
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 21.1660 0.793787
\(712\) 0 0
\(713\) 27.4955i 1.02971i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.58258i 0.171139i
\(718\) 0 0
\(719\) 39.8372i 1.48568i 0.669471 + 0.742838i \(0.266521\pi\)
−0.669471 + 0.742838i \(0.733479\pi\)
\(720\) 0 0
\(721\) 18.3303i 0.682656i
\(722\) 0 0
\(723\) 24.2487i 0.901819i
\(724\) 0 0
\(725\) 22.9129i 0.850963i
\(726\) 0 0
\(727\) 25.9808i 0.963573i −0.876289 0.481787i \(-0.839988\pi\)
0.876289 0.481787i \(-0.160012\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −28.0000 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.16515i 0.337603i
\(738\) 0 0
\(739\) 20.7846i 0.764574i −0.924044 0.382287i \(-0.875137\pi\)
0.924044 0.382287i \(-0.124863\pi\)
\(740\) 0 0
\(741\) −42.0000 + 32.0780i −1.54291 + 1.17842i
\(742\) 0 0
\(743\) 15.8745 0.582379 0.291190 0.956665i \(-0.405949\pi\)
0.291190 + 0.956665i \(0.405949\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 27.7128i 1.01396i
\(748\) 0 0
\(749\) 13.7477i 0.502331i
\(750\) 0 0
\(751\) 37.0405 1.35163 0.675814 0.737072i \(-0.263792\pi\)
0.675814 + 0.737072i \(0.263792\pi\)
\(752\) 0 0
\(753\) 27.4955i 1.00199i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) −47.6235 −1.72862
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) −23.8118 −0.862044
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.3731i 1.31336i
\(768\) 0 0
\(769\) 7.00000 0.252426 0.126213 0.992003i \(-0.459718\pi\)
0.126213 + 0.992003i \(0.459718\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.58258i 0.164824i 0.996598 + 0.0824119i \(0.0262623\pi\)
−0.996598 + 0.0824119i \(0.973738\pi\)
\(774\) 0 0
\(775\) 26.4575 0.950382
\(776\) 0 0
\(777\) 42.0000 1.50674
\(778\) 0 0
\(779\) −31.7490 + 24.2487i −1.13753 + 0.868800i
\(780\) 0 0
\(781\) 54.9909i 1.96773i
\(782\) 0 0
\(783\) 12.1244i 0.433289i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −2.64575 −0.0943108 −0.0471554 0.998888i \(-0.515016\pi\)
−0.0471554 + 0.998888i \(0.515016\pi\)
\(788\) 0 0
\(789\) 9.16515i 0.326288i
\(790\) 0 0
\(791\) 15.8745 0.564433
\(792\) 0 0
\(793\) 36.6606i 1.30186i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.0780i 1.13626i 0.822938 + 0.568131i \(0.192333\pi\)
−0.822938 + 0.568131i \(0.807667\pi\)
\(798\) 0 0
\(799\) 10.3923i 0.367653i
\(800\) 0 0
\(801\) 73.3212i 2.59068i
\(802\) 0 0
\(803\) 38.1051i 1.34470i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 72.7461i