Properties

Label 2352.1.by.a.1007.1
Level $2352$
Weight $1$
Character 2352.1007
Analytic conductor $1.174$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2352.by (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

Embedding invariants

Embedding label 1007.1
Root \(0.222521 - 0.974928i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1007
Dual form 2352.1.by.a.1343.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.222521 + 0.974928i) q^{3} +(0.900969 - 0.433884i) q^{7} +(-0.900969 - 0.433884i) q^{9} +O(q^{10})\) \(q+(-0.222521 + 0.974928i) q^{3} +(0.900969 - 0.433884i) q^{7} +(-0.900969 - 0.433884i) q^{9} +(-0.846011 - 1.75676i) q^{13} +1.80194 q^{19} +(0.222521 + 0.974928i) q^{21} +(0.900969 + 0.433884i) q^{25} +(0.623490 - 0.781831i) q^{27} +1.24698 q^{31} +(-0.777479 - 0.974928i) q^{37} +(1.90097 - 0.433884i) q^{39} +(-0.846011 + 0.193096i) q^{43} +(0.623490 - 0.781831i) q^{49} +(-0.400969 + 1.75676i) q^{57} +(-1.22252 + 0.974928i) q^{61} -1.00000 q^{63} +1.94986i q^{67} +(0.678448 - 1.40881i) q^{73} +(-0.623490 + 0.781831i) q^{75} +1.56366i q^{79} +(0.623490 + 0.781831i) q^{81} +(-1.52446 - 1.21572i) q^{91} +(-0.277479 + 1.21572i) q^{93} +0.867767i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} + q^{7} - q^{9} + O(q^{10}) \) \( 6q - q^{3} + q^{7} - q^{9} + 2q^{19} + q^{21} + q^{25} - q^{27} - 2q^{31} - 5q^{37} + 7q^{39} - q^{49} + 2q^{57} - 7q^{61} - 6q^{63} + q^{75} - q^{81} - 2q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{1}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(4\) 0 0
\(5\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(6\) 0 0
\(7\) 0.900969 0.433884i 0.900969 0.433884i
\(8\) 0 0
\(9\) −0.900969 0.433884i −0.900969 0.433884i
\(10\) 0 0
\(11\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(12\) 0 0
\(13\) −0.846011 1.75676i −0.846011 1.75676i −0.623490 0.781831i \(-0.714286\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(18\) 0 0
\(19\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(20\) 0 0
\(21\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(22\) 0 0
\(23\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(24\) 0 0
\(25\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(26\) 0 0
\(27\) 0.623490 0.781831i 0.623490 0.781831i
\(28\) 0 0
\(29\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(30\) 0 0
\(31\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.777479 0.974928i −0.777479 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.90097 0.433884i 1.90097 0.433884i
\(40\) 0 0
\(41\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(42\) 0 0
\(43\) −0.846011 + 0.193096i −0.846011 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(48\) 0 0
\(49\) 0.623490 0.781831i 0.623490 0.781831i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(58\) 0 0
\(59\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(60\) 0 0
\(61\) −1.22252 + 0.974928i −1.22252 + 0.974928i −0.222521 + 0.974928i \(0.571429\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.00000 −1.00000
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.94986i 1.94986i 0.222521 + 0.974928i \(0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(72\) 0 0
\(73\) 0.678448 1.40881i 0.678448 1.40881i −0.222521 0.974928i \(-0.571429\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(74\) 0 0
\(75\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.56366i 1.56366i 0.623490 + 0.781831i \(0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(80\) 0 0
\(81\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(82\) 0 0
\(83\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(90\) 0 0
\(91\) −1.52446 1.21572i −1.52446 1.21572i
\(92\) 0 0
\(93\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.867767i 0.867767i 0.900969 + 0.433884i \(0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(102\) 0 0
\(103\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(108\) 0 0
\(109\) 0.400969 0.193096i 0.400969 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(110\) 0 0
\(111\) 1.12349 0.541044i 1.12349 0.541044i
\(112\) 0 0
\(113\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.94986i 1.94986i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.678448 0.541044i 0.678448 0.541044i −0.222521 0.974928i \(-0.571429\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(128\) 0 0
\(129\) 0.867767i 0.867767i
\(130\) 0 0
\(131\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(132\) 0 0
\(133\) 1.62349 0.781831i 1.62349 0.781831i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(138\) 0 0
\(139\) 0.0990311 0.433884i 0.0990311 0.433884i −0.900969 0.433884i \(-0.857143\pi\)
1.00000 \(0\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(148\) 0 0
\(149\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(150\) 0 0
\(151\) −0.678448 0.541044i −0.678448 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.90097 0.433884i 1.90097 0.433884i 0.900969 0.433884i \(-0.142857\pi\)
1.00000 \(0\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.90097 + 0.433884i −1.90097 + 0.433884i −0.900969 + 0.433884i \(0.857143\pi\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(168\) 0 0
\(169\) −1.74698 + 2.19064i −1.74698 + 2.19064i
\(170\) 0 0
\(171\) −1.62349 0.781831i −1.62349 0.781831i
\(172\) 0 0
\(173\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(174\) 0 0
\(175\) 1.00000 1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(180\) 0 0
\(181\) 0.846011 1.75676i 0.846011 1.75676i 0.222521 0.974928i \(-0.428571\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(182\) 0 0
\(183\) −0.678448 1.40881i −0.678448 1.40881i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.222521 0.974928i 0.222521 0.974928i
\(190\) 0 0
\(191\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(192\) 0 0
\(193\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i 0.900969 + 0.433884i \(0.142857\pi\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −0.400969 + 1.75676i −0.400969 + 1.75676i 0.222521 + 0.974928i \(0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(200\) 0 0
\(201\) −1.90097 0.433884i −1.90097 0.433884i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.678448 + 1.40881i −0.678448 + 1.40881i 0.222521 + 0.974928i \(0.428571\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.12349 0.541044i 1.12349 0.541044i
\(218\) 0 0
\(219\) 1.22252 + 0.974928i 1.22252 + 0.974928i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.777479 + 0.974928i −0.777479 + 0.974928i 0.222521 + 0.974928i \(0.428571\pi\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −0.623490 0.781831i −0.623490 0.781831i
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −0.846011 + 0.193096i −0.846011 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.52446 0.347948i −1.52446 0.347948i
\(238\) 0 0
\(239\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(240\) 0 0
\(241\) −1.52446 1.21572i −1.52446 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
−0.623490 0.781831i \(-0.714286\pi\)
\(242\) 0 0
\(243\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.52446 3.16557i −1.52446 3.16557i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(258\) 0 0
\(259\) −1.12349 0.541044i −1.12349 0.541044i
\(260\) 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
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(270\) 0 0
\(271\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(272\) 0 0
\(273\) 1.52446 1.21572i 1.52446 1.21572i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.277479 + 0.347948i 0.277479 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(278\) 0 0
\(279\) −1.12349 0.541044i −1.12349 0.541044i
\(280\) 0 0
\(281\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(282\) 0 0
\(283\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.222521 0.974928i 0.222521 0.974928i
\(290\) 0 0
\(291\) −0.846011 0.193096i −0.846011 0.193096i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −0.678448 + 0.541044i −0.678448 + 0.541044i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(308\) 0 0
\(309\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(310\) 0 0
\(311\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(312\) 0 0
\(313\) 1.56366i 1.56366i −0.623490 0.781831i \(-0.714286\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.94986i 1.94986i
\(326\) 0 0
\(327\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.678448 0.541044i 0.678448 0.541044i −0.222521 0.974928i \(-0.571429\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(332\) 0 0
\(333\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.222521 0.974928i 0.222521 0.974928i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(348\) 0 0
\(349\) −0.846011 + 0.193096i −0.846011 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(350\) 0 0
\(351\) −1.90097 0.433884i −1.90097 0.433884i
\(352\) 0 0
\(353\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(360\) 0 0
\(361\) 2.24698 2.24698
\(362\) 0 0
\(363\) −0.623490 0.781831i −0.623490 0.781831i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.376510 + 0.781831i 0.376510 + 0.781831i 1.00000 \(0\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(380\) 0 0
\(381\) 0.376510 + 0.781831i 0.376510 + 0.781831i
\(382\) 0 0
\(383\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.846011 + 0.193096i 0.846011 + 0.193096i
\(388\) 0 0
\(389\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.52446 + 0.347948i 1.52446 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(398\) 0 0
\(399\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(400\) 0 0
\(401\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(402\) 0 0
\(403\) −1.05496 2.19064i −1.05496 2.19064i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.22252 + 0.974928i −1.22252 + 0.974928i −0.222521 + 0.974928i \(0.571429\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(418\) 0 0
\(419\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(420\) 0 0
\(421\) 0.277479 + 0.347948i 0.277479 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.678448 + 1.40881i −0.678448 + 1.40881i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(432\) 0 0
\(433\) −0.846011 0.193096i −0.846011 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(440\) 0 0
\(441\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(442\) 0 0
\(443\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.678448 0.541044i 0.678448 0.541044i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(462\) 0 0
\(463\) 1.52446 + 1.21572i 1.52446 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(468\) 0 0
\(469\) 0.846011 + 1.75676i 0.846011 + 1.75676i
\(470\) 0 0
\(471\) 1.94986i 1.94986i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(480\) 0 0
\(481\) −1.05496 + 2.19064i −1.05496 + 2.19064i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.52446 0.347948i −1.52446 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(488\) 0 0
\(489\) 1.94986i 1.94986i
\(490\) 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
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.678448 1.40881i 0.678448 1.40881i −0.222521 0.974928i \(-0.571429\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.74698 2.19064i −1.74698 2.19064i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 1.56366i 1.56366i
\(512\) 0 0
\(513\) 1.12349 1.40881i 1.12349 1.40881i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(524\) 0 0
\(525\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(530\) 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
\(541\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(542\) 0 0
\(543\) 1.52446 + 1.21572i 1.52446 + 1.21572i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.52446 + 0.347948i 1.52446 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(548\) 0 0
\(549\) 1.52446 0.347948i 1.52446 0.347948i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0.678448 + 1.40881i 0.678448 + 1.40881i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.05496 + 1.32288i 1.05496 + 1.32288i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.22252 + 0.974928i −1.22252 + 0.974928i −0.222521 + 0.974928i \(0.571429\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.846011 + 1.75676i 0.846011 + 1.75676i 0.623490 + 0.781831i \(0.285714\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(578\) 0 0
\(579\) −0.400969 0.193096i −0.400969 0.193096i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 2.24698 2.24698
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.62349 0.781831i −1.62349 0.781831i
\(598\) 0 0
\(599\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(600\) 0 0
\(601\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(602\) 0 0
\(603\) 0.846011 1.75676i 0.846011 1.75676i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.12349 0.541044i −1.12349 0.541044i −0.222521 0.974928i \(-0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(618\) 0 0
\(619\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.52446 0.347948i 1.52446 0.347948i 0.623490 0.781831i \(-0.285714\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(632\) 0 0
\(633\) −1.22252 0.974928i −1.22252 0.974928i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.90097 0.433884i −1.90097 0.433884i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(642\) 0 0
\(643\) −0.0990311 0.433884i −0.0990311 0.433884i 0.900969 0.433884i \(-0.142857\pi\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(652\) 0 0
\(653\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.22252 + 0.974928i −1.22252 + 0.974928i
\(658\) 0 0
\(659\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(660\) 0 0
\(661\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.777479 0.974928i −0.777479 0.974928i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(674\) 0 0
\(675\) 0.900969 0.433884i 0.900969 0.433884i
\(676\) 0 0
\(677\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(678\) 0 0
\(679\) 0.376510 + 0.781831i 0.376510 + 0.781831i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0.867767i 0.867767i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.445042 + 1.94986i −0.445042 + 1.94986i −0.222521 + 0.974928i \(0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(692\) 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 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(702\) 0 0
\(703\) −1.40097 1.75676i −1.40097 1.75676i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.12349 + 1.40881i −1.12349 + 1.40881i −0.222521 + 0.974928i \(0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(710\) 0 0
\(711\) 0.678448 1.40881i 0.678448 1.40881i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(720\) 0 0
\(721\) −0.400969 1.75676i −0.400969 1.75676i
\(722\) 0 0
\(723\) 1.52446 1.21572i 1.52446 1.21572i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −0.445042 + 1.94986i −0.445042 + 1.94986i −0.222521 + 0.974928i \(0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(728\) 0 0
\(729\) −0.222521 0.974928i −0.222521 0.974928i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.376510 + 0.781831i 0.376510 + 0.781831i 1.00000 \(0\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.52446 + 1.21572i 1.52446 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(740\) 0 0
\(741\) 3.42543 0.781831i 3.42543 0.781831i
\(742\) 0 0
\(743\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.846011 + 0.193096i −0.846011 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(762\) 0 0
\(763\) 0.277479 0.347948i 0.277479 0.347948i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(774\) 0 0
\(775\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(776\) 0 0
\(777\) 0.777479 0.974928i 0.777479 0.974928i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.74698 + 1.32288i 2.74698 + 1.32288i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 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 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(810\) 0