Properties

Label 528.2.b.a.65.2
Level $528$
Weight $2$
Character 528.65
Analytic conductor $4.216$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.2
Root \(0.500000 - 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 528.65
Dual form 528.2.b.a.65.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 + 1.65831i) q^{3} +3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 1.65831i) q^{3} +3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} +3.31662i q^{11} +(-5.50000 - 1.65831i) q^{15} -3.31662i q^{23} -6.00000 q^{25} +(4.00000 - 3.31662i) q^{27} -5.00000 q^{31} +(-5.50000 - 1.65831i) q^{33} -7.00000 q^{37} +(5.50000 - 8.29156i) q^{45} +6.63325i q^{47} +7.00000 q^{49} +13.2665i q^{53} -11.0000 q^{55} -3.31662i q^{59} +13.0000 q^{67} +(5.50000 + 1.65831i) q^{69} +16.5831i q^{71} +(3.00000 - 9.94987i) q^{75} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} +(2.50000 - 8.29156i) q^{93} +17.0000 q^{97} +(5.50000 - 8.29156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{9} + O(q^{10}) \) \( 2 q - q^{3} - 5 q^{9} - 11 q^{15} - 12 q^{25} + 8 q^{27} - 10 q^{31} - 11 q^{33} - 14 q^{37} + 11 q^{45} + 14 q^{49} - 22 q^{55} + 26 q^{67} + 11 q^{69} + 6 q^{75} + 7 q^{81} + 5 q^{93} + 34 q^{97} + 11 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.500000 + 1.65831i −0.288675 + 0.957427i
\(4\) 0 0
\(5\) 3.31662i 1.48324i 0.670820 + 0.741620i \(0.265942\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −2.50000 1.65831i −0.833333 0.552771i
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −5.50000 1.65831i −1.42009 0.428174i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662i 0.691564i −0.938315 0.345782i \(-0.887614\pi\)
0.938315 0.345782i \(-0.112386\pi\)
\(24\) 0 0
\(25\) −6.00000 −1.20000
\(26\) 0 0
\(27\) 4.00000 3.31662i 0.769800 0.638285i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) −5.50000 1.65831i −0.957427 0.288675i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 5.50000 8.29156i 0.819892 1.23603i
\(46\) 0 0
\(47\) 6.63325i 0.967559i 0.875190 + 0.483779i \(0.160736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.2665i 1.82229i 0.412082 + 0.911147i \(0.364802\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −11.0000 −1.48324
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i −0.976417 0.215894i \(-0.930733\pi\)
0.976417 0.215894i \(-0.0692665\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 5.50000 + 1.65831i 0.662122 + 0.199637i
\(70\) 0 0
\(71\) 16.5831i 1.96805i 0.178017 + 0.984027i \(0.443032\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.00000 9.94987i 0.346410 1.14891i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 3.50000 + 8.29156i 0.388889 + 0.921285i
\(82\) 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
\(89\) 16.5831i 1.75781i −0.476999 0.878904i \(-0.658275\pi\)
0.476999 0.878904i \(-0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.50000 8.29156i 0.259238 0.859795i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) 5.50000 8.29156i 0.552771 0.833333i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 3.50000 11.6082i 0.332205 1.10180i
\(112\) 0 0
\(113\) 3.31662i 0.312002i 0.987757 + 0.156001i \(0.0498603\pi\)
−0.987757 + 0.156001i \(0.950140\pi\)
\(114\) 0 0
\(115\) 11.0000 1.02576
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.31662i 0.296648i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 11.0000 + 13.2665i 0.946729 + 1.14180i
\(136\) 0 0
\(137\) 23.2164i 1.98351i 0.128154 + 0.991754i \(0.459095\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −11.0000 3.31662i −0.926367 0.279310i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.50000 + 11.6082i −0.288675 + 0.957427i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5831i 1.33199i
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) −22.0000 6.63325i −1.74471 0.526051i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 5.50000 18.2414i 0.428174 1.42009i
\(166\) 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
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.50000 + 1.65831i 0.413405 + 0.124646i
\(178\) 0 0
\(179\) 16.5831i 1.23948i 0.784807 + 0.619740i \(0.212762\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 23.2164i 1.70690i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.2164i 1.67988i −0.542681 0.839939i \(-0.682591\pi\)
0.542681 0.839939i \(-0.317409\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) −6.50000 + 21.5581i −0.458475 + 1.52059i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.50000 + 8.29156i −0.382276 + 0.576303i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −27.5000 8.29156i −1.88427 0.568128i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) 15.0000 + 9.94987i 1.00000 + 0.663325i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −22.0000 −1.43512
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −15.5000 + 1.65831i −0.994325 + 0.106381i
\(244\) 0 0
\(245\) 23.2164i 1.48324i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.5831i 1.04672i 0.852112 + 0.523359i \(0.175321\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 11.0000 0.691564
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.5330i 1.65508i −0.561405 0.827541i \(-0.689739\pi\)
0.561405 0.827541i \(-0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −44.0000 −2.70290
\(266\) 0 0
\(267\) 27.5000 + 8.29156i 1.68297 + 0.507435i
\(268\) 0 0
\(269\) 13.2665i 0.808873i 0.914566 + 0.404436i \(0.132532\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.8997i 1.20000i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 12.5000 + 8.29156i 0.748355 + 0.496403i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.50000 + 28.1913i −0.498279 + 1.65260i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) 11.0000 + 13.2665i 0.638285 + 0.769800i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −2.00000 + 6.63325i −0.113776 + 0.377352i
\(310\) 0 0
\(311\) 33.1662i 1.88069i −0.340229 0.940343i \(-0.610505\pi\)
0.340229 0.940343i \(-0.389495\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.2164i 1.30396i 0.758236 + 0.651981i \(0.226062\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −35.0000 −1.92377 −0.961887 0.273447i \(-0.911836\pi\)
−0.961887 + 0.273447i \(0.911836\pi\)
\(332\) 0 0
\(333\) 17.5000 + 11.6082i 0.958994 + 0.636125i
\(334\) 0 0
\(335\) 43.1161i 2.35569i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −5.50000 1.65831i −0.298719 0.0900672i
\(340\) 0 0
\(341\) 16.5831i 0.898027i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.50000 + 18.2414i −0.296110 + 0.982086i
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.4829i 1.94179i −0.239511 0.970894i \(-0.576987\pi\)
0.239511 0.970894i \(-0.423013\pi\)
\(354\) 0 0
\(355\) −55.0000 −2.91910
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 5.50000 18.2414i 0.288675 0.957427i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 37.0000 1.93138 0.965692 0.259690i \(-0.0836203\pi\)
0.965692 + 0.259690i \(0.0836203\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 5.50000 + 1.65831i 0.284019 + 0.0856349i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.31662i 0.169472i −0.996403 0.0847358i \(-0.972995\pi\)
0.996403 0.0847358i \(-0.0270046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 36.4829i 1.84976i −0.380265 0.924878i \(-0.624167\pi\)
0.380265 0.924878i \(-0.375833\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.5330i 1.32499i −0.749064 0.662497i \(-0.769497\pi\)
0.749064 0.662497i \(-0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −27.5000 + 11.6082i −1.36649 + 0.576815i
\(406\) 0 0
\(407\) 23.2164i 1.15079i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −38.5000 11.6082i −1.89906 0.572590i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.1662i 1.62028i −0.586238 0.810139i \(-0.699392\pi\)
0.586238 0.810139i \(-0.300608\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 11.0000 16.5831i 0.534838 0.806299i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 29.0000 1.39365 0.696826 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −17.5000 11.6082i −0.833333 0.552771i
\(442\) 0 0
\(443\) 36.4829i 1.73335i 0.498870 + 0.866677i \(0.333748\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 0 0
\(445\) 55.0000 2.60725
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.5831i 0.782606i −0.920262 0.391303i \(-0.872024\pi\)
0.920262 0.391303i \(-0.127976\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 31.0000 1.44069 0.720346 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 0 0
\(465\) 27.5000 + 8.29156i 1.27528 + 0.384512i
\(466\) 0 0
\(467\) 43.1161i 1.99518i −0.0694117 0.997588i \(-0.522112\pi\)
0.0694117 0.997588i \(-0.477888\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −11.5000 + 38.1412i −0.529892 + 1.75745i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 22.0000 33.1662i 1.00731 1.51858i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 56.3826i 2.56020i
\(486\) 0 0
\(487\) 43.0000 1.94852 0.974258 0.225436i \(-0.0723806\pi\)
0.974258 + 0.225436i \(0.0723806\pi\)
\(488\) 0 0
\(489\) −8.00000 + 26.5330i −0.361773 + 1.19986i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 27.5000 + 18.2414i 1.23603 + 0.819892i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 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
\(506\) 0 0
\(507\) −6.50000 + 21.5581i −0.288675 + 0.957427i
\(508\) 0 0
\(509\) 3.31662i 0.147007i 0.997295 + 0.0735034i \(0.0234180\pi\)
−0.997295 + 0.0735034i \(0.976582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.2665i 0.584592i
\(516\) 0 0
\(517\) −22.0000 −0.967559
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 43.1161i 1.88895i 0.328581 + 0.944476i \(0.393430\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12.0000 0.521739
\(530\) 0 0
\(531\) −5.50000 + 8.29156i −0.238680 + 0.359823i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −27.5000 8.29156i −1.18671 0.357807i
\(538\) 0 0
\(539\) 23.2164i 1.00000i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 12.5000 41.4578i 0.536426 1.77912i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 38.5000 + 11.6082i 1.63423 + 0.492740i
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −11.0000 −0.462773
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 38.5000 + 11.6082i 1.60836 + 0.484939i
\(574\) 0 0
\(575\) 19.8997i 0.829877i
\(576\) 0 0
\(577\) 47.0000 1.95664 0.978318 0.207109i \(-0.0664056\pi\)
0.978318 + 0.207109i \(0.0664056\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −44.0000 −1.82229
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.63325i 0.273784i 0.990586 + 0.136892i \(0.0437113\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.0000 33.1662i 0.409273 1.35740i
\(598\) 0 0
\(599\) 33.1662i 1.35514i −0.735460 0.677568i \(-0.763034\pi\)
0.735460 0.677568i \(-0.236966\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −32.5000 21.5581i −1.32350 0.877912i
\(604\) 0 0
\(605\) 36.4829i 1.48324i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.5330i 1.06818i −0.845428 0.534089i \(-0.820655\pi\)
0.845428 0.534089i \(-0.179345\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) −11.0000 13.2665i −0.441415 0.532366i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 27.5000 41.4578i 1.08788 1.64005i
\(640\) 0 0
\(641\) 23.2164i 0.916992i 0.888697 + 0.458496i \(0.151612\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) 0 0
\(643\) −41.0000 −1.61688 −0.808441 0.588577i \(-0.799688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 43.1161i 1.69507i −0.530740 0.847535i \(-0.678086\pi\)
0.530740 0.847535i \(-0.321914\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.31662i 0.129790i 0.997892 + 0.0648948i \(0.0206712\pi\)
−0.997892 + 0.0648948i \(0.979329\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.500000 + 1.65831i −0.0193311 + 0.0641141i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −24.0000 + 19.8997i −0.923760 + 0.765942i
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 46.4327i 1.77670i 0.459167 + 0.888350i \(0.348148\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −77.0000 −2.94202
\(686\) 0 0
\(687\) −2.50000 + 8.29156i −0.0953809 + 0.316343i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −17.0000 −0.646710 −0.323355 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 11.0000 36.4829i 0.414284 1.37402i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.5831i 0.621043i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.5831i 0.618446i 0.950990 + 0.309223i \(0.100069\pi\)
−0.950990 + 0.309223i \(0.899931\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −53.0000 −1.96566 −0.982831 0.184510i \(-0.940930\pi\)
−0.982831 + 0.184510i \(0.940930\pi\)
\(728\) 0 0
\(729\) 5.00000 26.5330i 0.185185 0.982704i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −38.5000 11.6082i −1.42009 0.428174i
\(736\) 0 0
\(737\) 43.1161i 1.58820i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 0 0
\(753\) −27.5000 8.29156i −1.00216 0.302161i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) −5.50000 + 18.2414i −0.199637 + 0.662122i
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 44.0000 + 13.2665i 1.58462 + 0.477781i
\(772\) 0 0
\(773\) 13.2665i 0.477163i 0.971123 + 0.238581i \(0.0766824\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 30.0000 1.07763
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −55.0000 −1.96805
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 76.2824i 2.72263i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 22.0000 72.9657i 0.780260 2.58783i
\(796\) 0 0
\(797\) 56.3826i 1.99717i −0.0531327 0.998587i \(-0.516921\pi\)
0.0531327 0.998587i \(-0.483079\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −27.5000 + 41.4578i −0.971665 + 1.46484i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −22.0000 6.63325i −0.774437 0.233501i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000