Properties

Label 1200.1.bd.a.443.1
Level $1200$
Weight $1$
Character 1200.443
Analytic conductor $0.599$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1200.bd (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.153600.1

Embedding invariants

Embedding label 443.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.443
Dual form 1200.1.bd.a.707.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{12} +1.00000 q^{16} +(1.00000 + 1.00000i) q^{17} +1.00000i q^{18} +(1.00000 - 1.00000i) q^{19} +(-1.00000 + 1.00000i) q^{23} +1.00000i q^{24} -1.00000 q^{27} +2.00000i q^{31} +1.00000i q^{32} +(-1.00000 + 1.00000i) q^{34} -1.00000 q^{36} +(1.00000 + 1.00000i) q^{38} +(-1.00000 - 1.00000i) q^{46} +(1.00000 - 1.00000i) q^{47} -1.00000 q^{48} +1.00000i q^{49} +(-1.00000 - 1.00000i) q^{51} +2.00000 q^{53} -1.00000i q^{54} +(-1.00000 + 1.00000i) q^{57} +(-1.00000 + 1.00000i) q^{61} -2.00000 q^{62} -1.00000 q^{64} +(-1.00000 - 1.00000i) q^{68} +(1.00000 - 1.00000i) q^{69} -1.00000i q^{72} +(-1.00000 + 1.00000i) q^{76} +1.00000 q^{81} +(1.00000 - 1.00000i) q^{92} -2.00000i q^{93} +(1.00000 + 1.00000i) q^{94} -1.00000i q^{96} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{12} + 2q^{16} + 2q^{17} + 2q^{19} - 2q^{23} - 2q^{27} - 2q^{34} - 2q^{36} + 2q^{38} - 2q^{46} + 2q^{47} - 2q^{48} - 2q^{51} + 4q^{53} - 2q^{57} - 2q^{61} - 4q^{62} - 2q^{64} - 2q^{68} + 2q^{69} - 2q^{76} + 2q^{81} + 2q^{92} + 2q^{94} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i
\(3\) −1.00000 −1.00000
\(4\) −1.00000 −1.00000
\(5\) 0 0
\(6\) 1.00000i 1.00000i
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 1.00000 1.00000
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(18\) 1.00000i 1.00000i
\(19\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(24\) 1.00000i 1.00000i
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) 0 0
\(31\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 1.00000i −1.00000 1.00000i
\(47\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(48\) −1.00000 −1.00000
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) −1.00000 1.00000i −1.00000 1.00000i
\(52\) 0 0
\(53\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(54\) 1.00000i 1.00000i
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(62\) −2.00000 −2.00000
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −1.00000 1.00000i −1.00000 1.00000i
\(69\) 1.00000 1.00000i 1.00000 1.00000i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.00000i 1.00000i
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 1.00000i 1.00000 1.00000i
\(93\) 2.00000i 2.00000i
\(94\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(95\) 0 0
\(96\) 1.00000i 1.00000i
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) −1.00000 −1.00000
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(102\) 1.00000 1.00000i 1.00000 1.00000i
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 2.00000i 2.00000i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 1.00000
\(109\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(114\) −1.00000 1.00000i −1.00000 1.00000i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000i 1.00000i
\(122\) −1.00000 1.00000i −1.00000 1.00000i
\(123\) 0 0
\(124\) 2.00000i 2.00000i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 1.00000 1.00000i 1.00000 1.00000i
\(137\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(138\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(139\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 1.00000i
\(148\) 0 0
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(152\) −1.00000 1.00000i −1.00000 1.00000i
\(153\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −2.00000 −2.00000
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 1.00000i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) 1.00000 1.00000i 1.00000 1.00000i
\(172\) 0 0
\(173\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 1.00000 1.00000i 1.00000 1.00000i
\(184\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(185\) 0 0
\(186\) 2.00000 2.00000
\(187\) 0 0
\(188\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 1.00000
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.00000i 1.00000i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(212\) −2.00000 −2.00000
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) 1.00000 1.00000i 1.00000 1.00000i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 1.00000 1.00000i 1.00000 1.00000i
\(229\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 1.00000 1.00000
\(243\) −1.00000 −1.00000
\(244\) 1.00000 1.00000i 1.00000 1.00000i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 2.00000 2.00000
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(273\) 0 0
\(274\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(275\) 0 0
\(276\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(279\) 2.00000i 2.00000i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −1.00000 1.00000i −1.00000 1.00000i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000i 1.00000i
\(289\) 1.00000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(294\) 1.00000 1.00000
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 2.00000i 2.00000i
\(303\) 0 0
\(304\) 1.00000 1.00000i 1.00000 1.00000i
\(305\) 0 0
\(306\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 2.00000i 2.00000i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00000 2.00000
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.00000 1.00000i 1.00000 1.00000i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) 1.00000i 1.00000i
\(339\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −2.00000 −2.00000
\(347\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(348\) 0 0
\(349\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000i 1.00000i
\(362\) 1.00000 1.00000i 1.00000 1.00000i
\(363\) 1.00000i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2.00000i 2.00000i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.00000 1.00000i −1.00000 1.00000i
\(377\) 0 0
\(378\) 0 0
\(379\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(384\) 1.00000i 1.00000i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0 0
\(391\) −2.00000 −2.00000
\(392\) 1.00000 1.00000
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 1.00000i −1.00000 1.00000i
\(415\) 0 0
\(416\) 0 0
\(417\) −1.00000 1.00000i −1.00000 1.00000i
\(418\) 0 0
\(419\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(420\) 0 0
\(421\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(422\) 1.00000 1.00000i 1.00000 1.00000i
\(423\) 1.00000 1.00000i 1.00000 1.00000i
\(424\) 2.00000i 2.00000i
\(425\) 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 \(0.5\pi\)
\(432\) −1.00000 −1.00000
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(437\) 2.00000i 2.00000i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.00000i 1.00000i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(453\) −2.00000 −2.00000
\(454\) 0 0
\(455\) 0 0
\(456\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) −1.00000 1.00000i −1.00000 1.00000i
\(459\) −1.00000 1.00000i −1.00000 1.00000i
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.00000 1.00000i 1.00000 1.00000i
\(467\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 2.00000
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000i 1.00000i
\(485\) 0 0
\(486\) 1.00000i 1.00000i
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 2.00000i 2.00000i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(500\) 0 0
\(501\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(502\) 0 0
\(503\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 1.00000
\(508\) 0 0
\(509\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(514\) 1.00000 1.00000i 1.00000 1.00000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.00000i 2.00000i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1.00000 1.00000i −1.00000 1.00000i
\(527\) −2.00000 + 2.00000i −2.00000 + 2.00000i
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(542\) 0 0
\(543\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(544\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(549\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(550\) 0 0
\(551\) 0 0
\(552\) −1.00000 1.00000i −1.00000 1.00000i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.00000 1.00000i −1.00000 1.00000i
\(557\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(558\) −2.00000 −2.00000
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(564\) 1.00000 1.00000i 1.00000 1.00000i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −1.00000 −1.00000
\(577\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) −1.00000 −1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 2.00000i 2.00000i
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.00000i 1.00000i
\(589\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.00000 −2.00000
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 1.00000i −1.00000 1.00000i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(618\) 0 0
\(619\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 1.00000 1.00000i 1.00000 1.00000i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(634\) 0 0
\(635\) 0 0
\(636\) 2.00000 2.00000
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.00000i 2.00000i
\(647\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(648\) 1.00000i 1.00000i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(662\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(678\) −1.00000 1.00000i −1.00000 1.00000i
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(684\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(685\) 0 0
\(686\) 0 0
\(687\) 1.00000 1.00000i 1.00000 1.00000i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(692\) 2.00000i 2.00000i
\(693\) 0 0
\(694\) 2.00000i 2.00000i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 1.00000 1.00000i 1.00000 1.00000i
\(699\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(700\) 0 0
\(701\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −1.00000 1.00000i −1.00000 1.00000i
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.00000 2.00000i −2.00000 2.00000i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 1.00000
\(723\) 0 0
\(724\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(725\) 0 0
\(726\) −1.00000 −1.00000
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.00000 1.00000i −1.00000 1.00000i
\(737\) 0 0
\(738\) 0 0
\(739\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(744\) −2.00000 −2.00000
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(752\) 1.00000 1.00000i 1.00000 1.00000i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 1.00000 1.00000i 1.00000 1.00000i
\(759\) 0 0
\(760\) 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
\(766\) 1.00000 1.00000i 1.00000 1.00000i
\(767\) 0 0
\(768\) −1.00000 −1.00000
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 2.00000i 2.00000i
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 1.00000 1.00000i 1.00000 1.00000i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 2.00000 2.00000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0