Properties

Label 224.1.v.a.125.1
Level $224$
Weight $1$
Character 224.125
Analytic conductor $0.112$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.111790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.5156108238848.3

Embedding invariants

Embedding label 125.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 224.125
Dual form 224.1.v.a.181.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{2} -1.00000i q^{4} +(0.707107 - 0.707107i) q^{7} +(0.707107 + 0.707107i) q^{8} +(0.707107 + 0.707107i) q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{2} -1.00000i q^{4} +(0.707107 - 0.707107i) q^{7} +(0.707107 + 0.707107i) q^{8} +(0.707107 + 0.707107i) q^{9} +(-0.707107 + 0.292893i) q^{11} +1.00000i q^{14} -1.00000 q^{16} -1.00000 q^{18} +(0.292893 - 0.707107i) q^{22} +(-1.00000 - 1.00000i) q^{23} +(-0.707107 + 0.707107i) q^{25} +(-0.707107 - 0.707107i) q^{28} +(0.707107 + 0.292893i) q^{29} +(0.707107 - 0.707107i) q^{32} +(0.707107 - 0.707107i) q^{36} +(-0.707107 - 1.70711i) q^{37} +(-1.70711 + 0.707107i) q^{43} +(0.292893 + 0.707107i) q^{44} +1.41421 q^{46} -1.00000i q^{49} -1.00000i q^{50} +(-1.70711 + 0.707107i) q^{53} +1.00000 q^{56} +(-0.707107 + 0.292893i) q^{58} +1.00000 q^{63} +1.00000i q^{64} +(1.70711 + 0.707107i) q^{67} +1.00000i q^{72} +(1.70711 + 0.707107i) q^{74} +(-0.292893 + 0.707107i) q^{77} +1.41421i q^{79} +1.00000i q^{81} +(0.707107 - 1.70711i) q^{86} +(-0.707107 - 0.292893i) q^{88} +(-1.00000 + 1.00000i) q^{92} +(0.707107 + 0.707107i) q^{98} +(-0.707107 - 0.292893i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{16} - 4q^{18} + 4q^{22} - 4q^{23} - 4q^{43} + 4q^{44} - 4q^{53} + 4q^{56} + 4q^{63} + 4q^{67} + 4q^{74} - 4q^{77} - 4q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(3\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(4\) 1.00000i 1.00000i
\(5\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(6\) 0 0
\(7\) 0.707107 0.707107i 0.707107 0.707107i
\(8\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(9\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(10\) 0 0
\(11\) −0.707107 + 0.292893i −0.707107 + 0.292893i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(14\) 1.00000i 1.00000i
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −1.00000
\(19\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.292893 0.707107i 0.292893 0.707107i
\(23\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.707107 0.707107i −0.707107 0.707107i
\(29\) 0.707107 + 0.292893i 0.707107 + 0.292893i 0.707107 0.707107i \(-0.250000\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.707107 0.707107i 0.707107 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.707107 0.707107i 0.707107 0.707107i
\(37\) −0.707107 1.70711i −0.707107 1.70711i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(-0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) 0 0
\(43\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(44\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(45\) 0 0
\(46\) 1.41421 1.41421
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) 0 0
\(58\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(59\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(62\) 0 0
\(63\) 1.00000 1.00000
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.70711 + 0.707107i 1.70711 + 0.707107i 1.00000 \(0\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 1.00000i 1.00000i
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(75\) 0 0
\(76\) 0 0
\(77\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(78\) 0 0
\(79\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) 0 0
\(81\) 1.00000i 1.00000i
\(82\) 0 0
\(83\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.707107 1.70711i 0.707107 1.70711i
\(87\) 0 0
\(88\) −0.707107 0.292893i −0.707107 0.292893i
\(89\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(99\) −0.707107 0.292893i −0.707107 0.292893i
\(100\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(101\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.707107 1.70711i 0.707107 1.70711i
\(107\) 1.70711 0.707107i 1.70711 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
1.00000 \(0\)
\(108\) 0 0
\(109\) 0.292893 0.707107i 0.292893 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
1.00000 \(0\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(113\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.292893 0.707107i 0.292893 0.707107i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −0.707107 0.707107i −0.707107 0.707107i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(135\) 0 0
\(136\) 0 0
\(137\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.707107 0.707107i −0.707107 0.707107i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(149\) 1.70711 0.707107i 1.70711 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
1.00000 \(0\)
\(150\) 0 0
\(151\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.292893 0.707107i −0.292893 0.707107i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(158\) −1.00000 1.00000i −1.00000 1.00000i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.41421 −1.41421
\(162\) −0.707107 0.707107i −0.707107 0.707107i
\(163\) 0.707107 + 0.292893i 0.707107 + 0.292893i 0.707107 0.707107i \(-0.250000\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) −0.707107 0.707107i −0.707107 0.707107i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(173\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(174\) 0 0
\(175\) 1.00000i 1.00000i
\(176\) 0.707107 0.292893i 0.707107 0.292893i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.707107 1.70711i 0.707107 1.70711i 1.00000i \(-0.5\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(180\) 0 0
\(181\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.41421i 1.41421i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0 0
\(193\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.00000 −1.00000
\(197\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(198\) 0.707107 0.292893i 0.707107 0.292893i
\(199\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) −1.00000 −1.00000
\(201\) 0 0
\(202\) 0 0
\(203\) 0.707107 0.292893i 0.707107 0.292893i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.41421i 1.41421i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.292893 + 0.707107i −0.292893 + 0.707107i 0.707107 + 0.707107i \(0.250000\pi\)
−1.00000 \(\pi\)
\(212\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(213\) 0 0
\(214\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 1.00000i 1.00000i
\(225\) −1.00000 −1.00000
\(226\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(227\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(233\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(234\) 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 \(0.5\pi\)
\(242\) 0.414214i 0.414214i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(252\) 1.00000i 1.00000i
\(253\) 1.00000 + 0.414214i 1.00000 + 0.414214i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −1.70711 0.707107i −1.70711 0.707107i
\(260\) 0 0
\(261\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(262\) 0 0
\(263\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.707107 1.70711i 0.707107 1.70711i
\(269\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.41421 1.41421
\(275\) 0.292893 0.707107i 0.292893 0.707107i
\(276\) 0 0
\(277\) 0.707107 0.292893i 0.707107 0.292893i 1.00000i \(-0.5\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(282\) 0 0
\(283\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 1.00000
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.707107 1.70711i 0.707107 1.70711i
\(297\) 0 0
\(298\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(299\) 0 0
\(300\) 0 0
\(301\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(302\) −1.41421 −1.41421
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(308\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(312\) 0 0
\(313\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.41421 1.41421
\(317\) −0.707107 0.292893i −0.707107 0.292893i 1.00000i \(-0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) −0.585786 −0.585786
\(320\) 0 0
\(321\) 0 0
\(322\) 1.00000 1.00000i 1.00000 1.00000i
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.707107 0.292893i 0.707107 0.292893i 1.00000i \(-0.5\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(332\) 0 0
\(333\) 0.707107 1.70711i 0.707107 1.70711i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 1.00000 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.707107 0.707107i −0.707107 0.707107i
\(344\) −1.70711 0.707107i −1.70711 0.707107i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(350\) −0.707107 0.707107i −0.707107 0.707107i
\(351\) 0 0
\(352\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(359\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −0.707107 0.707107i −0.707107 0.707107i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(369\) 0 0
\(370\) 0 0
\(371\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(372\) 0 0
\(373\) −0.707107 + 0.292893i −0.707107 + 0.292893i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.707107 1.70711i −0.707107 1.70711i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(-0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.00000 1.00000i 1.00000 1.00000i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(387\) −1.70711 0.707107i −1.70711 0.707107i
\(388\) 0 0
\(389\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.707107 0.707107i 0.707107 0.707107i
\(393\) 0 0
\(394\) −1.70711 0.707107i −1.70711 0.707107i
\(395\) 0 0
\(396\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(397\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.707107 0.707107i 0.707107 0.707107i
\(401\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(407\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(408\) 0 0
\(409\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(420\) 0 0
\(421\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(422\) −0.292893 0.707107i −0.292893 0.707107i
\(423\) 0 0
\(424\) −1.70711 0.707107i −1.70711 0.707107i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.707107 1.70711i −0.707107 1.70711i
\(429\) 0 0
\(430\) 0 0
\(431\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.707107 0.292893i −0.707107 0.292893i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(440\) 0 0
\(441\) 0.707107 0.707107i 0.707107 0.707107i
\(442\) 0 0
\(443\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0.707107 0.707107i 0.707107 0.707107i
\(451\) 0 0
\(452\) −1.41421 −1.41421
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(462\) 0 0
\(463\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(464\) −0.707107 0.292893i −0.707107 0.292893i
\(465\) 0 0
\(466\) −1.41421 −1.41421
\(467\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(468\) 0 0
\(469\) 1.70711 0.707107i 1.70711 0.707107i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.00000 1.00000i 1.00000 1.00000i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.70711 0.707107i −1.70711 0.707107i
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.70711 0.707107i 1.70711 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.707107 + 1.70711i −0.707107 + 1.70711i 1.00000i \(0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(505\) 0 0
\(506\) −1.00000 + 0.414214i −1.00000 + 0.414214i
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.70711 0.707107i 1.70711 0.707107i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(522\) −0.707107 0.292893i −0.707107 0.292893i
\(523\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.00000i 2.00000i
\(527\) 0 0
\(528\) 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
\(536\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(540\) 0 0
\(541\) −1.70711 0.707107i −1.70711 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.707107 + 0.292893i 0.707107 + 0.292893i 0.707107 0.707107i \(-0.250000\pi\)
1.00000i \(0.5\pi\)
\(548\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(549\) 0 0
\(550\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(554\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.292893 0.707107i 0.292893 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
1.00000 \(0\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.41421i 1.41421i
\(563\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(568\) 0 0
\(569\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(570\) 0 0
\(571\) 0.292893 + 0.707107i 0.292893 + 0.707107i 1.00000 \(0\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.41421 1.41421
\(576\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0.707107 0.707107i 0.707107 0.707107i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.00000 1.00000i 1.00000 1.00000i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.707107 1.70711i −0.707107 1.70711i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(600\) 0 0
\(601\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(602\) −0.707107 1.70711i −0.707107 1.70711i
\(603\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(604\) 1.00000 1.00000i 1.00000 1.00000i
\(605\) 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
\(613\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000i 1.00000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(633\) 0 0
\(634\) 0.707107 0.292893i 0.707107 0.292893i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0.414214 0.414214i 0.414214 0.414214i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(644\) 1.41421i 1.41421i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.292893 0.707107i 0.292893 0.707107i
\(653\) −0.292893 + 0.707107i −0.292893 + 0.707107i 0.707107 + 0.707107i \(0.250000\pi\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.292893 0.707107i 0.292893 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
1.00000 \(0\)
\(660\) 0 0
\(661\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(662\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(667\) −0.414214 1.00000i −0.414214 1.00000i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) −1.00000 1.00000i −1.00000 1.00000i
\(675\) 0 0
\(676\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(677\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 1.00000
\(687\) 0 0
\(688\) 1.70711 0.707107i 1.70711 0.707107i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(692\) 0 0
\(693\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(694\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.00000 1.00000
\(701\) −1.70711 0.707107i −1.70711 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.292893 0.707107i −0.292893 0.707107i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.70711 0.707107i −1.70711 0.707107i
\(717\) 0 0
\(718\) 1.41421i 1.41421i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.41421 −1.41421
\(737\) −1.41421 −1.41421
\(738\) 0 0
\(739\) −0.707107 0.292893i −0.707107 0.292893i 1.00000i \(-0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.707107 1.70711i −0.707107 1.70711i
\(743\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.292893 0.707107i 0.292893 0.707107i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.707107 1.70711i 0.707107 1.70711i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(758\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) −0.292893 0.707107i −0.292893 0.707107i
\(764\) 1.41421i 1.41421i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.41421i 1.41421i
\(773\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(774\) 1.70711 0.707107i 1.70711 0.707107i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.70711 0.707107i −1.70711 0.707107i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(788\) 1.70711 0.707107i 1.70711 0.707107i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.00000 1.00000i −1.00000 1.00000i
\(792\) −0.292893 0.707107i −0.292893 0.707107i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 1.00000i
\(801\) 0 0
\(802\) −1.00000 1.00000i −1.00000 1.00000i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\)