Properties

Label 825.2.f.b.626.4
Level $825$
Weight $2$
Character 825.626
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 626.4
Root \(-1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 825.626
Dual form 825.2.f.b.626.3

$q$-expansion

\(f(q)\) \(=\) \(q+(1.65831 + 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +O(q^{10})\) \(q+(1.65831 + 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +3.31662i q^{11} +(-3.31662 - 1.00000i) q^{12} +4.00000 q^{16} +9.00000i q^{23} +(3.31662 + 4.00000i) q^{27} -5.00000 q^{31} +(-1.65831 + 5.50000i) q^{33} +(-5.00000 - 3.31662i) q^{36} +9.94987 q^{37} -6.63325i q^{44} +12.0000i q^{47} +(6.63325 + 2.00000i) q^{48} +7.00000 q^{49} +6.00000i q^{53} -3.31662i q^{59} -8.00000 q^{64} -9.94987 q^{67} +(-4.50000 + 14.9248i) q^{69} -16.5831i q^{71} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} -18.0000i q^{92} +(-8.29156 - 2.50000i) q^{93} -9.94987 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.65831 + 0.500000i 0.957427 + 0.288675i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 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
\(12\) −3.31662 1.00000i −0.957427 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 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
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.31662 + 4.00000i 0.638285 + 0.769800i
\(28\) 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
\(33\) −1.65831 + 5.50000i −0.288675 + 0.957427i
\(34\) 0 0
\(35\) 0 0
\(36\) −5.00000 3.31662i −0.833333 0.552771i
\(37\) 9.94987 1.63575 0.817875 0.575396i \(-0.195152\pi\)
0.817875 + 0.575396i \(0.195152\pi\)
\(38\) 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 \(\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 6.63325 + 2.00000i 0.957427 + 0.288675i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −9.94987 −1.21557 −0.607785 0.794101i \(-0.707942\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) 0 0
\(69\) −4.50000 + 14.9248i −0.541736 + 1.79674i
\(70\) 0 0
\(71\) 16.5831i 1.96805i −0.178017 0.984027i \(-0.556968\pi\)
0.178017 0.984027i \(-0.443032\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(92\) 18.0000i 1.87663i
\(93\) −8.29156 2.50000i −0.859795 0.259238i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.94987 −1.01026 −0.505128 0.863044i \(-0.668555\pi\)
−0.505128 + 0.863044i \(0.668555\pi\)
\(98\) 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
\(103\) 19.8997 1.96078 0.980390 0.197066i \(-0.0631413\pi\)
0.980390 + 0.197066i \(0.0631413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −6.63325 8.00000i −0.638285 0.769800i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 16.5000 + 4.97494i 1.56611 + 0.472200i
\(112\) 0 0
\(113\) 21.0000i 1.97551i −0.156001 0.987757i \(-0.549860\pi\)
0.156001 0.987757i \(-0.450140\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 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 \(0.5\pi\)
\(132\) 3.31662 11.0000i 0.288675 0.957427i
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −6.00000 + 19.8997i −0.505291 + 1.67586i
\(142\) 0 0
\(143\) 0 0
\(144\) 10.0000 + 6.63325i 0.833333 + 0.552771i
\(145\) 0 0
\(146\) 0 0
\(147\) 11.6082 + 3.50000i 0.957427 + 0.288675i
\(148\) −19.8997 −1.63575
\(149\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 9.94987 0.794086 0.397043 0.917800i \(-0.370036\pi\)
0.397043 + 0.917800i \(0.370036\pi\)
\(158\) 0 0
\(159\) −3.00000 + 9.94987i −0.237915 + 0.789076i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −19.8997 −1.55867 −0.779334 0.626608i \(-0.784443\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 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
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 13.2665i 1.00000i
\(177\) 1.65831 5.50000i 0.124646 0.413405i
\(178\) 0 0
\(179\) 16.5831i 1.23948i −0.784807 0.619740i \(-0.787238\pi\)
0.784807 0.619740i \(-0.212762\pi\)
\(180\) 0 0
\(181\) 25.0000 1.85824 0.929118 0.369784i \(-0.120568\pi\)
0.929118 + 0.369784i \(0.120568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 24.0000i 1.75038i
\(189\) 0 0
\(190\) 0 0
\(191\) 23.2164i 1.67988i −0.542681 0.839939i \(-0.682591\pi\)
0.542681 0.839939i \(-0.317409\pi\)
\(192\) −13.2665 4.00000i −0.957427 0.288675i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 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
\(201\) −16.5000 4.97494i −1.16382 0.350905i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −14.9248 + 22.5000i −1.03735 + 1.56386i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 8.29156 27.5000i 0.568128 1.88427i
\(214\) 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
\(223\) 29.8496 1.99888 0.999439 0.0334825i \(-0.0106598\pi\)
0.999439 + 0.0334825i \(0.0106598\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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 \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.63325i 0.431788i
\(237\) 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
\(243\) 1.65831 + 15.5000i 0.106381 + 0.994325i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.5831i 1.04672i −0.852112 0.523359i \(-0.824679\pi\)
0.852112 0.523359i \(-0.175321\pi\)
\(252\) 0 0
\(253\) −29.8496 −1.87663
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 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
\(265\) 0 0
\(266\) 0 0
\(267\) 8.29156 27.5000i 0.507435 1.68297i
\(268\) 19.8997 1.21557
\(269\) 13.2665i 0.808873i −0.914566 0.404436i \(-0.867468\pi\)
0.914566 0.404436i \(-0.132532\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 9.00000 29.8496i 0.541736 1.79674i
\(277\) 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 \(\pi\)
\(284\) 33.1662i 1.96805i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −16.5000 4.97494i −0.967247 0.291636i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −13.2665 + 11.0000i −0.769800 + 0.638285i
\(298\) 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
\(309\) 33.0000 + 9.94987i 1.87730 + 0.566029i
\(310\) 0 0
\(311\) 33.1662i 1.88069i 0.340229 + 0.940343i \(0.389495\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −29.8496 −1.68720 −0.843600 0.536972i \(-0.819568\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.0000i 1.51647i −0.651981 0.758236i \(-0.726062\pi\)
0.651981 0.758236i \(-0.273938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −7.00000 16.5831i −0.388889 0.921285i
\(325\) 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
\(333\) 24.8747 + 16.5000i 1.36312 + 0.904194i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 10.5000 34.8246i 0.570282 1.89141i
\(340\) 0 0
\(341\) 16.5831i 0.898027i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 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
\(353\) 9.00000i 0.479022i −0.970894 0.239511i \(-0.923013\pi\)
0.970894 0.239511i \(-0.0769871\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 33.1662i 1.75781i
\(357\) 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
\(363\) −18.2414 5.50000i −0.957427 0.288675i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.94987 0.519379 0.259690 0.965692i \(-0.416380\pi\)
0.259690 + 0.965692i \(0.416380\pi\)
\(368\) 36.0000i 1.87663i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 16.5831 + 5.00000i 0.859795 + 0.259238i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(383\) 39.0000i 1.99281i 0.0847358 + 0.996403i \(0.472995\pi\)
−0.0847358 + 0.996403i \(0.527005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 19.8997 1.01026
\(389\) 36.4829i 1.84976i 0.380265 + 0.924878i \(0.375833\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 11.0000 16.5831i 0.552771 0.833333i
\(397\) 39.7995 1.99748 0.998740 0.0501886i \(-0.0159822\pi\)
0.998740 + 0.0501886i \(0.0159822\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.5330i 1.32499i 0.749064 + 0.662497i \(0.230503\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.0000i 1.63575i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −1.50000 + 4.97494i −0.0739895 + 0.245396i
\(412\) −39.7995 −1.96078
\(413\) 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 \(0.300608\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −19.8997 + 30.0000i −0.967559 + 1.45865i
\(424\) 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 \(0.5\pi\)
\(432\) 13.2665 + 16.0000i 0.638285 + 0.769800i
\(433\) 29.8496 1.43448 0.717241 0.696826i \(-0.245405\pi\)
0.717241 + 0.696826i \(0.245405\pi\)
\(434\) 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
\(443\) 21.0000i 0.997740i 0.866677 + 0.498870i \(0.166252\pi\)
−0.866677 + 0.498870i \(0.833748\pi\)
\(444\) −33.0000 9.94987i −1.56611 0.472200i
\(445\) 0 0
\(446\) 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
\(452\) 42.0000i 1.97551i
\(453\) 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
\(463\) 29.8496 1.38723 0.693615 0.720346i \(-0.256017\pi\)
0.693615 + 0.720346i \(0.256017\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.00000i 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 16.5000 + 4.97494i 0.760280 + 0.229233i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −9.94987 + 15.0000i −0.455573 + 0.686803i
\(478\) 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
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −9.94987 −0.450872 −0.225436 0.974258i \(-0.572381\pi\)
−0.225436 + 0.974258i \(0.572381\pi\)
\(488\) 0 0
\(489\) −33.0000 9.94987i −1.49231 0.449949i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −20.0000 −0.898027
\(497\) 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
\(507\) 21.5581 + 6.50000i 0.957427 + 0.288675i
\(508\) 0 0
\(509\) 3.31662i 0.147007i −0.997295 0.0735034i \(-0.976582\pi\)
0.997295 0.0735034i \(-0.0234180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −39.7995 −1.75038
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 43.1161i 1.88895i −0.328581 0.944476i \(-0.606570\pi\)
0.328581 0.944476i \(-0.393430\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −6.63325 + 22.0000i −0.288675 + 0.957427i
\(529\) −58.0000 −2.52174
\(530\) 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
\(537\) 8.29156 27.5000i 0.357807 1.18671i
\(538\) 0 0
\(539\) 23.2164i 1.00000i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 41.4578 + 12.5000i 1.77912 + 0.536426i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 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 \(0.5\pi\)
\(564\) 12.0000 39.7995i 0.505291 1.67586i
\(565\) 0 0
\(566\) 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
\(573\) 11.6082 38.5000i 0.484939 1.60836i
\(574\) 0 0
\(575\) 0 0
\(576\) −20.0000 13.2665i −0.833333 0.552771i
\(577\) −9.94987 −0.414219 −0.207109 0.978318i \(-0.566406\pi\)
−0.207109 + 0.978318i \(0.566406\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −19.8997 −0.824163
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 48.0000i 1.98117i −0.136892 0.990586i \(-0.543711\pi\)
0.136892 0.990586i \(-0.456289\pi\)
\(588\) −23.2164 7.00000i −0.957427 0.288675i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 39.7995 1.63575
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −33.1662 10.0000i −1.35740 0.409273i
\(598\) 0 0
\(599\) 33.1662i 1.35514i 0.735460 + 0.677568i \(0.236966\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −24.8747 16.5000i −1.01298 0.671932i
\(604\) 0 0
\(605\) 0 0
\(606\) 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
\(617\) 42.0000i 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 0 0
\(621\) −36.0000 + 29.8496i −1.44463 + 1.19782i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −19.8997 −0.794086
\(629\) 0 0
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 6.00000 19.8997i 0.237915 0.789076i
\(637\) 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 \(-0.848388\pi\)
0.888697 0.458496i \(-0.151612\pi\)
\(642\) 0 0
\(643\) −29.8496 −1.17715 −0.588577 0.808441i \(-0.700312\pi\)
−0.588577 + 0.808441i \(0.700312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.0000i 1.06148i 0.847535 + 0.530740i \(0.178086\pi\)
−0.847535 + 0.530740i \(0.821914\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 39.7995 1.55867
\(653\) 51.0000i 1.99578i −0.0648948 0.997892i \(-0.520671\pi\)
0.0648948 0.997892i \(-0.479329\pi\)
\(654\) 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
\(669\) 49.5000 + 14.9248i 1.91378 + 0.577027i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 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
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.29156 2.50000i −0.316343 0.0953809i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 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
\(704\) 26.5330i 1.00000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −3.31662 + 11.0000i −0.124646 + 0.413405i
\(709\) −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
\(713\) 45.0000i 1.68526i
\(714\) 0 0
\(715\) 0 0
\(716\) 33.1662i 1.23948i
\(717\) 0 0
\(718\) 0 0
\(719\) 16.5831i 0.618446i −0.950990 0.309223i \(-0.899931\pi\)
0.950990 0.309223i \(-0.100069\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −50.0000 −1.85824
\(725\) 0 0
\(726\) 0 0
\(727\) 9.94987 0.369020 0.184510 0.982831i \(-0.440930\pi\)
0.184510 + 0.982831i \(0.440930\pi\)
\(728\) 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
\(735\) 0 0
\(736\) 0 0
\(737\) 33.0000i 1.21557i
\(738\) 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 \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 48.0000i 1.75038i
\(753\) 8.29156 27.5000i 0.302161 1.00216i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −39.7995 −1.44654 −0.723269 0.690567i \(-0.757361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) −49.5000 14.9248i −1.79674 0.541736i
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 46.4327i 1.67988i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 26.5330 + 8.00000i 0.957427 + 0.288675i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −9.00000 + 29.8496i −0.324127 + 1.07501i
\(772\) 0 0
\(773\) 54.0000i 1.94225i −0.238581 0.971123i \(-0.576682\pi\)
0.238581 0.971123i \(-0.423318\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 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
\(795\) 0 0
\(796\) 40.0000 1.41776
\(797\) 3.00000i 0.106265i −0.998587 0.0531327i \(-0.983079\pi\)
0.998587 0.0531327i \(-0.0169206\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 27.5000 41.4578i 0.971665 1.46484i
\(802\) 0 0
\(803\) 0 0
\(804\) 33.0000 + 9.94987i 1.16382 + 0.350905i
\(805\) 0 0
\(806\) 0 0
\(807\) 6.63325 22.0000i 0.233501 0.774437i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)