Properties

Label 1575.1.y.a.76.1
Level $1575$
Weight $1$
Character 1575.76
Analytic conductor $0.786$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -35
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.2835.1
Artin image $C_{12}\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 76.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1575.76
Dual form 1575.1.y.a.601.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(0.500000 - 0.866025i) q^{4} +(0.866025 - 0.500000i) q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(0.500000 - 0.866025i) q^{4} +(0.866025 - 0.500000i) q^{7} -1.00000 q^{9} +(0.500000 + 0.866025i) q^{11} +(-0.866025 - 0.500000i) q^{12} +(-0.866025 - 0.500000i) q^{13} +(-0.500000 - 0.866025i) q^{16} -1.00000i q^{17} +(-0.500000 - 0.866025i) q^{21} +1.00000i q^{27} -1.00000i q^{28} +(1.00000 + 1.73205i) q^{29} +(0.866025 - 0.500000i) q^{33} +(-0.500000 + 0.866025i) q^{36} +(-0.500000 + 0.866025i) q^{39} +1.00000 q^{44} +(-0.866025 + 0.500000i) q^{47} +(-0.866025 + 0.500000i) q^{48} +(0.500000 - 0.866025i) q^{49} -1.00000 q^{51} +(-0.866025 + 0.500000i) q^{52} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(-0.866025 - 0.500000i) q^{68} -1.00000 q^{71} +1.00000i q^{73} +(0.866025 + 0.500000i) q^{77} +(-0.500000 - 0.866025i) q^{79} +1.00000 q^{81} +(0.866025 - 0.500000i) q^{83} -1.00000 q^{84} +(1.73205 - 1.00000i) q^{87} -1.00000 q^{91} +(-0.866025 + 0.500000i) q^{97} +(-0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 4q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 4q^{9} + 2q^{11} - 2q^{16} - 2q^{21} + 4q^{29} - 2q^{36} - 2q^{39} + 4q^{44} + 2q^{49} - 4q^{51} - 4q^{64} - 4q^{71} - 2q^{79} + 4q^{81} - 4q^{84} - 4q^{91} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 1.00000i 1.00000i
\(4\) 0.500000 0.866025i 0.500000 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) 0.866025 0.500000i 0.866025 0.500000i
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −0.866025 0.500000i −0.866025 0.500000i
\(13\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −0.500000 0.866025i −0.500000 0.866025i
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 1.00000i 1.00000i
\(29\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0.866025 0.500000i 0.866025 0.500000i
\(34\) 0 0
\(35\) 0 0
\(36\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(40\) 0 0
\(41\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 1.00000 1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.500000\pi\)
\(48\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(49\) 0.500000 0.866025i 0.500000 0.866025i
\(50\) 0 0
\(51\) −1.00000 −1.00000
\(52\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) −0.866025 0.500000i −0.866025 0.500000i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(78\) 0 0
\(79\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(-1.00000\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 1.73205 1.00000i 1.73205 1.00000i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.500000\pi\)
\(98\) 0 0
\(99\) −0.500000 0.866025i −0.500000 0.866025i
\(100\) 0 0
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(109\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.866025 0.500000i −0.866025 0.500000i
\(113\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 2.00000
\(117\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(118\) 0 0
\(119\) −0.500000 0.866025i −0.500000 0.866025i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 1.00000i 1.00000i
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(142\) 0 0
\(143\) 1.00000i 1.00000i
\(144\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.866025 0.500000i −0.866025 0.500000i
\(148\) 0 0
\(149\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(1.00000\pi\)
\(150\) 0 0
\(151\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0 0
\(153\) 1.00000i 1.00000i
\(154\) 0 0
\(155\) 0 0
\(156\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(157\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.500000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.500000\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.500000 0.866025i 0.500000 0.866025i
\(177\) 0 0
\(178\) 0 0
\(179\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.866025 0.500000i 0.866025 0.500000i
\(188\) 1.00000i 1.00000i
\(189\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(190\) 0 0
\(191\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(192\) 1.00000i 1.00000i
\(193\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.500000 0.866025i −0.500000 0.866025i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(204\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000i 1.00000i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(212\) 0 0
\(213\) 1.00000i 1.00000i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.00000 1.00000
\(220\) 0 0
\(221\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(222\) 0 0
\(223\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.500000\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0.500000 0.866025i 0.500000 0.866025i
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(238\) 0 0
\(239\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(240\) 0 0
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 0 0
\(243\) 1.00000i 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −0.500000 0.866025i −0.500000 0.866025i
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.00000i 1.00000i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.500000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.00000 1.73205i −1.00000 1.73205i
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(273\) 1.00000i 1.00000i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(292\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(293\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(298\) 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
\(307\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(308\) 0.866025 0.500000i 0.866025 0.500000i
\(309\) 1.00000 1.73205i 1.00000 1.73205i
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.00000 −1.00000
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.500000 0.866025i 0.500000 0.866025i
\(325\) 0 0
\(326\) 0 0
\(327\) 1.00000i 1.00000i
\(328\) 0 0
\(329\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(330\) 0 0
\(331\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 1.00000i 1.00000i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(337\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(348\) 2.00000i 2.00000i
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) 0.500000 0.866025i 0.500000 0.866025i
\(352\) 0 0
\(353\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(358\) 0 0
\(359\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.500000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000i 2.00000i
\(378\) 0 0
\(379\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 1.00000i 1.00000i
\(389\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(-1.00000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.00000 −1.00000
\(397\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.73205 1.00000i 1.73205 1.00000i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 0 0
\(423\) 0.866025 0.500000i 0.866025 0.500000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.00000 −1.00000
\(430\) 0 0
\(431\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0.866025 0.500000i 0.866025 0.500000i
\(433\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.500000 0.866025i 0.500000 0.866025i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(442\) 0 0
\(443\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(448\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(449\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.866025 0.500000i 0.866025 0.500000i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(458\) 0 0
\(459\) 1.00000 1.00000
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(464\) 1.00000 1.73205i 1.00000 1.73205i
\(465\) 0 0
\(466\) 0 0
\(467\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(468\) 0.866025 0.500000i 0.866025 0.500000i
\(469\) 0 0
\(470\) 0 0
\(471\) 0.500000 0.866025i 0.500000 0.866025i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 −1.00000
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 1.73205 1.00000i 1.73205 1.00000i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(498\) 0 0
\(499\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(1.00000\pi\)
\(500\) 0 0
\(501\) 0.500000 0.866025i 0.500000 0.866025i
\(502\) 0 0
\(503\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.866025 0.500000i −0.866025 0.500000i
\(518\) 0 0
\(519\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.866025 0.500000i −0.866025 0.500000i
\(529\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.00000i 1.00000i
\(538\) 0 0
\(539\) 1.00000 1.00000
\(540\) 0 0
\(541\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.866025 0.500000i −0.866025 0.500000i
\(554\) 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
\(561\) −0.500000 0.866025i −0.500000 0.866025i
\(562\) 0 0
\(563\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 1.00000 1.00000
\(565\) 0 0
\(566\) 0 0
\(567\) 0.866025 0.500000i 0.866025 0.500000i
\(568\) 0 0
\(569\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(-1.00000\pi\)
\(570\) 0 0
\(571\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(572\) −0.866025 0.500000i −0.866025 0.500000i
\(573\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 1.00000
\(577\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.500000 0.866025i 0.500000 0.866025i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(588\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(597\) 0 0
\(598\) 0 0
\(599\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(1.00000\pi\)
\(600\) 0 0
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.00000 1.00000
\(605\) 0 0
\(606\) 0 0
\(607\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(608\) 0 0
\(609\) 1.00000 1.73205i 1.00000 1.73205i
\(610\) 0 0
\(611\) 1.00000 1.00000
\(612\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(618\) 0 0
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.00000 1.00000
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0.866025 0.500000i 0.866025 0.500000i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0 0
\(633\) −0.866025 0.500000i −0.866025 0.500000i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(638\) 0 0
\(639\) 1.00000 1.00000
\(640\) 0 0
\(641\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(642\) 0 0
\(643\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.00000i 1.00000i
\(658\) 0 0
\(659\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(-1.00000\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0.866025 0.500000i 0.866025 0.500000i
\(669\) −0.500000 0.866025i −0.500000 0.866025i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.500000\pi\)
\(678\) 0 0
\(679\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(680\) 0 0
\(681\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 2.00000i 2.00000i
\(693\) −0.866025 0.500000i −0.866025 0.500000i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.500000 0.866025i −0.500000 0.866025i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.500000 0.866025i 0.500000 0.866025i
\(717\) −1.73205 1.00000i −1.73205 1.00000i
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 2.00000 2.00000
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.500000\pi\)
\(728\) 0 0
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(748\) 1.00000i 1.00000i
\(749\) 0 0
\(750\) 0 0
\(751\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 1.00000 1.00000
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0.866025 0.500000i 0.866025 0.500000i
\(764\) −2.00000 −2.00000
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(769\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) 0.500000 0.866025i 0.500000 0.866025i
\(772\) 0 0
\(773\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −0.500000 0.866025i −0.500000 0.866025i
\(782\) 0 0
\(783\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(784\) −1.00000 −1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.500000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.500000\pi\)
\(798\) 0 0
\(799\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1.73205 1.00000i 1.73205 1.00000i
\(813\) 0 0
\(814\) 0