Properties

Label 252.2.b.c.55.2
Level $252$
Weight $2$
Character 252.55
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 55.2
Root \(-1.32288 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 252.55
Dual form 252.2.b.c.55.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} +O(q^{10})\) \(q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} -4.00000i q^{11} +(1.32288 + 3.50000i) q^{14} +(0.500000 - 3.96863i) q^{16} +(2.00000 + 5.29150i) q^{22} -8.00000i q^{23} +5.00000 q^{25} +(-3.50000 - 3.96863i) q^{28} +10.5830 q^{29} +(1.32288 + 5.50000i) q^{32} -6.00000 q^{37} +5.29150i q^{43} +(-5.29150 - 6.00000i) q^{44} +(4.00000 + 10.5830i) q^{46} -7.00000 q^{49} +(-6.61438 + 2.50000i) q^{50} -10.5830 q^{53} +(6.61438 + 3.50000i) q^{56} +(-14.0000 + 5.29150i) q^{58} +(-4.50000 - 6.61438i) q^{64} +15.8745i q^{67} +16.0000i q^{71} +(7.93725 - 3.00000i) q^{74} -10.5830 q^{77} -15.8745i q^{79} +(-2.64575 - 7.00000i) q^{86} +(10.0000 + 5.29150i) q^{88} +(-10.5830 - 12.0000i) q^{92} +(9.26013 - 3.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} + 2 q^{16} + 8 q^{22} + 20 q^{25} - 14 q^{28} - 24 q^{37} + 16 q^{46} - 28 q^{49} - 56 q^{58} - 18 q^{64} + 40 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32288 + 0.500000i −0.935414 + 0.353553i
\(3\) 0 0
\(4\) 1.50000 1.32288i 0.750000 0.661438i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 2.64575i 1.00000i
\(8\) −1.32288 + 2.50000i −0.467707 + 0.883883i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000i 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.32288 + 3.50000i 0.353553 + 0.935414i
\(15\) 0 0
\(16\) 0.500000 3.96863i 0.125000 0.992157i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 + 5.29150i 0.426401 + 1.12815i
\(23\) 8.00000i 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −3.50000 3.96863i −0.661438 0.750000i
\(29\) 10.5830 1.96521 0.982607 0.185695i \(-0.0594537\pi\)
0.982607 + 0.185695i \(0.0594537\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.32288 + 5.50000i 0.233854 + 0.972272i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 5.29150i 0.806947i 0.914991 + 0.403473i \(0.132197\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −5.29150 6.00000i −0.797724 0.904534i
\(45\) 0 0
\(46\) 4.00000 + 10.5830i 0.589768 + 1.56038i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −6.61438 + 2.50000i −0.935414 + 0.353553i
\(51\) 0 0
\(52\) 0 0
\(53\) −10.5830 −1.45369 −0.726844 0.686803i \(-0.759014\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.61438 + 3.50000i 0.883883 + 0.467707i
\(57\) 0 0
\(58\) −14.0000 + 5.29150i −1.83829 + 0.694808i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.50000 6.61438i −0.562500 0.826797i
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8745i 1.93938i 0.244339 + 0.969690i \(0.421429\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000i 1.89885i 0.313993 + 0.949425i \(0.398333\pi\)
−0.313993 + 0.949425i \(0.601667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 7.93725 3.00000i 0.922687 0.348743i
\(75\) 0 0
\(76\) 0 0
\(77\) −10.5830 −1.20605
\(78\) 0 0
\(79\) 15.8745i 1.78602i −0.450035 0.893011i \(-0.648589\pi\)
0.450035 0.893011i \(-0.351411\pi\)
\(80\) 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
\(86\) −2.64575 7.00000i −0.285299 0.754829i
\(87\) 0 0
\(88\) 10.0000 + 5.29150i 1.06600 + 0.564076i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −10.5830 12.0000i −1.10335 1.25109i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9.26013 3.50000i 0.935414 0.353553i
\(99\) 0 0
\(100\) 7.50000 6.61438i 0.750000 0.661438i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 14.0000 5.29150i 1.35980 0.513956i
\(107\) 20.0000i 1.93347i 0.255774 + 0.966736i \(0.417670\pi\)
−0.255774 + 0.966736i \(0.582330\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −10.5000 1.32288i −0.992157 0.125000i
\(113\) 21.1660 1.99113 0.995565 0.0940721i \(-0.0299884\pi\)
0.995565 + 0.0940721i \(0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 15.8745 14.0000i 1.47391 1.29987i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.8745i 1.40863i −0.709885 0.704317i \(-0.751253\pi\)
0.709885 0.704317i \(-0.248747\pi\)
\(128\) 9.26013 + 6.50000i 0.818488 + 0.574524i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.93725 21.0000i −0.685674 1.81412i
\(135\) 0 0
\(136\) 0 0
\(137\) −21.1660 −1.80833 −0.904167 0.427179i \(-0.859507\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 21.1660i −0.671345 1.77621i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −9.00000 + 7.93725i −0.739795 + 0.652438i
\(149\) 10.5830 0.866994 0.433497 0.901155i \(-0.357280\pi\)
0.433497 + 0.901155i \(0.357280\pi\)
\(150\) 0 0
\(151\) 5.29150i 0.430616i −0.976546 0.215308i \(-0.930924\pi\)
0.976546 0.215308i \(-0.0690756\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 14.0000 5.29150i 1.12815 0.426401i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 7.93725 + 21.0000i 0.631454 + 1.67067i
\(159\) 0 0
\(160\) 0 0
\(161\) −21.1660 −1.66812
\(162\) 0 0
\(163\) 15.8745i 1.24339i 0.783260 + 0.621694i \(0.213555\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 7.00000 + 7.93725i 0.533745 + 0.605210i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 13.2288i 1.00000i
\(176\) −15.8745 2.00000i −1.19659 0.150756i
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000i 0.298974i 0.988764 + 0.149487i \(0.0477622\pi\)
−0.988764 + 0.149487i \(0.952238\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 20.0000 + 10.5830i 1.47442 + 0.780189i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000i 0.578860i 0.957199 + 0.289430i \(0.0934657\pi\)
−0.957199 + 0.289430i \(0.906534\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −10.5000 + 9.26013i −0.750000 + 0.661438i
\(197\) 10.5830 0.754008 0.377004 0.926212i \(-0.376954\pi\)
0.377004 + 0.926212i \(0.376954\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −6.61438 + 12.5000i −0.467707 + 0.883883i
\(201\) 0 0
\(202\) 0 0
\(203\) 28.0000i 1.96521i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 26.4575i 1.82141i 0.413057 + 0.910705i \(0.364461\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −15.8745 + 14.0000i −1.09027 + 0.961524i
\(213\) 0 0
\(214\) −10.0000 26.4575i −0.683586 1.80860i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −23.8118 + 9.00000i −1.61274 + 0.609557i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 14.5516 3.50000i 0.972272 0.233854i
\(225\) 0 0
\(226\) −28.0000 + 10.5830i −1.86253 + 0.703971i
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −14.0000 + 26.4575i −0.919145 + 1.73702i
\(233\) 21.1660 1.38663 0.693316 0.720634i \(-0.256149\pi\)
0.693316 + 0.720634i \(0.256149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000i 1.03495i −0.855697 0.517477i \(-0.826871\pi\)
0.855697 0.517477i \(-0.173129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 6.61438 2.50000i 0.425188 0.160706i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 7.93725 + 21.0000i 0.498028 + 1.31766i
\(255\) 0 0
\(256\) −15.5000 3.96863i −0.968750 0.248039i
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 15.8745i 0.986394i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 32.0000i 1.97320i −0.163144 0.986602i \(-0.552164\pi\)
0.163144 0.986602i \(-0.447836\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 21.0000 + 23.8118i 1.28278 + 1.45453i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 28.0000 10.5830i 1.69154 0.639343i
\(275\) 20.0000i 1.20605i
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 21.1660 1.26266 0.631329 0.775515i \(-0.282510\pi\)
0.631329 + 0.775515i \(0.282510\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 21.1660 + 24.0000i 1.25597 + 1.42414i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.93725 15.0000i 0.461344 0.871857i
\(297\) 0 0
\(298\) −14.0000 + 5.29150i −0.810998 + 0.306529i
\(299\) 0 0
\(300\) 0 0
\(301\) 14.0000 0.806947
\(302\) 2.64575 + 7.00000i 0.152246 + 0.402805i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −15.8745 + 14.0000i −0.904534 + 0.797724i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −21.0000 23.8118i −1.18134 1.33952i
\(317\) −10.5830 −0.594401 −0.297200 0.954815i \(-0.596053\pi\)
−0.297200 + 0.954815i \(0.596053\pi\)
\(318\) 0 0
\(319\) 42.3320i 2.37014i
\(320\) 0 0
\(321\) 0 0
\(322\) 28.0000 10.5830i 1.56038 0.589768i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −7.93725 21.0000i −0.439604 1.16308i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.29150i 0.290847i 0.989369 + 0.145424i \(0.0464545\pi\)
−0.989369 + 0.145424i \(0.953545\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) −17.1974 + 6.50000i −0.935414 + 0.353553i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.5203i 1.00000i
\(344\) −13.2288 7.00000i −0.713247 0.377415i
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 6.61438 + 17.5000i 0.353553 + 0.935414i
\(351\) 0 0
\(352\) 22.0000 5.29150i 1.17260 0.282038i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −2.00000 5.29150i −0.105703 0.279665i
\(359\) 8.00000i 0.422224i 0.977462 + 0.211112i \(0.0677085\pi\)
−0.977462 + 0.211112i \(0.932292\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −31.7490 4.00000i −1.65503 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 28.0000i 1.45369i
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 37.0405i 1.90264i −0.308199 0.951322i \(-0.599726\pi\)
0.308199 0.951322i \(-0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.00000 10.5830i −0.204658 0.541474i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.8118 9.00000i 1.21199 0.458088i
\(387\) 0 0
\(388\) 0 0
\(389\) −10.5830 −0.536580 −0.268290 0.963338i \(-0.586458\pi\)
−0.268290 + 0.963338i \(0.586458\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.26013 17.5000i 0.467707 0.883883i
\(393\) 0 0
\(394\) −14.0000 + 5.29150i −0.705310 + 0.266582i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.50000 19.8431i 0.125000 0.992157i
\(401\) −21.1660 −1.05698 −0.528490 0.848939i \(-0.677242\pi\)
−0.528490 + 0.848939i \(0.677242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 14.0000 + 37.0405i 0.694808 + 1.83829i
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 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
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −13.2288 35.0000i −0.643966 1.70377i
\(423\) 0 0
\(424\) 14.0000 26.4575i 0.679900 1.28489i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 26.4575 + 30.0000i 1.27887 + 1.45010i
\(429\) 0 0
\(430\) 0 0
\(431\) 32.0000i 1.54139i 0.637207 + 0.770693i \(0.280090\pi\)
−0.637207 + 0.770693i \(0.719910\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 27.0000 23.8118i 1.29307 1.14038i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000i 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −17.5000 + 11.9059i −0.826797 + 0.562500i
\(449\) −42.3320 −1.99777 −0.998886 0.0471929i \(-0.984972\pi\)
−0.998886 + 0.0471929i \(0.984972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 31.7490 28.0000i 1.49335 1.31701i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 15.8745i 0.737751i −0.929479 0.368875i \(-0.879743\pi\)
0.929479 0.368875i \(-0.120257\pi\)
\(464\) 5.29150 42.0000i 0.245652 1.94980i
\(465\) 0 0
\(466\) −28.0000 + 10.5830i −1.29707 + 0.490248i
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 42.0000 1.93938
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.1660 0.973214
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 8.00000 + 21.1660i 0.365911 + 0.968111i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −7.50000 + 6.61438i −0.340909 + 0.300654i
\(485\) 0 0
\(486\) 0 0
\(487\) 37.0405i 1.67847i 0.543772 + 0.839233i \(0.316996\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 44.0000i 1.98569i 0.119401 + 0.992846i \(0.461903\pi\)
−0.119401 + 0.992846i \(0.538097\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.3320 1.89885
\(498\) 0 0
\(499\) 26.4575i 1.18440i 0.805791 + 0.592200i \(0.201741\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 42.3320 16.0000i 1.88189 0.711287i
\(507\) 0 0
\(508\) −21.0000 23.8118i −0.931724 1.05648i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.4889 2.50000i 0.993878 0.110485i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −7.93725 21.0000i −0.348743 0.922687i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 16.0000 + 42.3320i 0.697633 + 1.84576i
\(527\) 0 0
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −39.6863 21.0000i −1.71419 0.907062i
\(537\) 0 0
\(538\) 0 0
\(539\) 28.0000i 1.20605i
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.8745i 0.678745i 0.940652 + 0.339372i \(0.110215\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −31.7490 + 28.0000i −1.35625 + 1.19610i
\(549\) 0 0
\(550\) 10.0000 + 26.4575i 0.426401 + 1.12815i
\(551\) 0 0
\(552\) 0 0
\(553\) −42.0000 −1.78602
\(554\) 13.2288 5.00000i 0.562036 0.212430i
\(555\) 0 0
\(556\) 0 0
\(557\) 10.5830 0.448416 0.224208 0.974541i \(-0.428020\pi\)
0.224208 + 0.974541i \(0.428020\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −28.0000 + 10.5830i −1.18111 + 0.446417i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −40.0000 21.1660i −1.67836 0.888106i
\(569\) −42.3320 −1.77465 −0.887325 0.461144i \(-0.847439\pi\)
−0.887325 + 0.461144i \(0.847439\pi\)
\(570\) 0 0
\(571\) 47.6235i 1.99298i −0.0836974 0.996491i \(-0.526673\pi\)
0.0836974 0.996491i \(-0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 40.0000i 1.66812i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −22.4889 + 8.50000i −0.935414 + 0.353553i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 42.3320i 1.75321i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.00000 + 23.8118i −0.123299 + 0.978657i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.8745 14.0000i 0.650245 0.573462i
\(597\) 0 0
\(598\) 0 0
\(599\) 32.0000i 1.30748i 0.756717 + 0.653742i \(0.226802\pi\)
−0.756717 + 0.653742i \(0.773198\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −18.5203 + 7.00000i −0.754829 + 0.285299i
\(603\) 0 0
\(604\) −7.00000 7.93725i −0.284826 0.322962i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 14.0000 26.4575i 0.564076 1.06600i
\(617\) −42.3320 −1.70422 −0.852111 0.523360i \(-0.824678\pi\)
−0.852111 + 0.523360i \(0.824678\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 47.6235i 1.89586i 0.318475 + 0.947931i \(0.396829\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 39.6863 + 21.0000i 1.57864 + 0.835335i
\(633\) 0 0
\(634\) 14.0000 5.29150i 0.556011 0.210152i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 21.1660 + 56.0000i 0.837970 + 2.21706i
\(639\) 0 0
\(640\) 0 0
\(641\) 21.1660 0.836007 0.418004 0.908445i \(-0.362730\pi\)
0.418004 + 0.908445i \(0.362730\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) −31.7490 + 28.0000i −1.25109 + 1.10335i
\(645\) 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
\(652\) 21.0000 + 23.8118i 0.822423 + 0.932541i
\(653\) 10.5830 0.414145 0.207072 0.978326i \(-0.433606\pi\)
0.207072 + 0.978326i \(0.433606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 44.0000i 1.71400i −0.515319 0.856998i \(-0.672327\pi\)
0.515319 0.856998i \(-0.327673\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −2.64575 7.00000i −0.102830 0.272063i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 84.6640i 3.27820i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) −39.6863 + 15.0000i −1.52866 + 0.577778i
\(675\) 0 0
\(676\) 19.5000 17.1974i 0.750000 0.661438i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 52.0000i 1.98972i −0.101237 0.994862i \(-0.532280\pi\)
0.101237 0.994862i \(-0.467720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9.26013 24.5000i −0.353553 0.935414i
\(687\) 0 0
\(688\) 21.0000 + 2.64575i 0.800617 + 0.100868i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −2.00000 5.29150i −0.0759190 0.200863i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −17.5000 19.8431i −0.661438 0.750000i
\(701\) −52.9150 −1.99857 −0.999286 0.0377695i \(-0.987975\pi\)
−0.999286 + 0.0377695i \(0.987975\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −26.4575 + 18.0000i −0.997155 + 0.678401i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 5.29150 + 6.00000i 0.197753 + 0.224231i
\(717\) 0 0
\(718\) −4.00000 10.5830i −0.149279 0.394954i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 25.1346 9.50000i 0.935414 0.353553i
\(723\) 0 0
\(724\) 0 0
\(725\) 52.9150 1.96521
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 44.0000 10.5830i 1.62186 0.390095i
\(737\) 63.4980 2.33898
\(738\) 0 0
\(739\) 15.8745i 0.583953i 0.956425 + 0.291977i \(0.0943129\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −14.0000 37.0405i −0.513956 1.35980i
\(743\) 40.0000i 1.46746i −0.679442 0.733729i \(-0.737778\pi\)
0.679442 0.733729i \(-0.262222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −29.1033 + 11.0000i −1.06555 + 0.402739i
\(747\) 0 0
\(748\) 0 0
\(749\) 52.9150 1.93347
\(750\) 0 0
\(751\) 26.4575i 0.965448i −0.875772 0.482724i \(-0.839647\pi\)
0.875772 0.482724i \(-0.160353\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 18.5203 + 49.0000i 0.672686 + 1.77976i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 47.6235i 1.72409i
\(764\) 10.5830 + 12.0000i 0.382880 + 0.434145i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −27.0000 + 23.8118i −0.971751 + 0.857004i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 14.0000 5.29150i 0.501924 0.189710i
\(779\) 0 0
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) 0 0
\(783\) 0 0
\(784\) −3.50000 + 27.7804i −0.125000 + 0.992157i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 15.8745 14.0000i 0.565506 0.498729i
\(789\) 0 0
\(790\) 0 0
\(791\) 56.0000i 1.99113i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 6.61438 + 27.5000i 0.233854 + 0.972272i
\(801\) 0 0
\(802\) 28.0000 10.5830i 0.988714 0.373699i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0