Properties

Label 1728.1.bn.a.353.2
Level $1728$
Weight $1$
Character 1728.353
Analytic conductor $0.862$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.bn (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

Embedding invariants

Embedding label 353.2
Root \(0.342020 - 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 1728.353
Dual form 1728.1.bn.a.1121.2

$q$-expansion

\(f(q)\) \(=\) \(q+(0.642788 - 0.766044i) q^{3} +(-0.173648 - 0.984808i) q^{9} +O(q^{10})\) \(q+(0.642788 - 0.766044i) q^{3} +(-0.173648 - 0.984808i) q^{9} +(0.524005 + 0.439693i) q^{11} +(1.70574 - 0.984808i) q^{17} +(-1.32683 - 0.766044i) q^{19} +(-0.173648 + 0.984808i) q^{25} +(-0.866025 - 0.500000i) q^{27} +(0.673648 - 0.118782i) q^{33} +(1.26604 - 0.223238i) q^{41} +(0.223238 - 0.266044i) q^{43} +(-0.766044 + 0.642788i) q^{49} +(0.342020 - 1.93969i) q^{51} +(-1.43969 + 0.524005i) q^{57} +(0.984808 - 0.826352i) q^{59} +(-1.85083 + 0.326352i) q^{67} +(-0.173648 + 0.300767i) q^{73} +(0.642788 + 0.766044i) q^{75} +(-0.939693 + 0.342020i) q^{81} +(-0.300767 + 1.70574i) q^{83} +(-1.50000 - 0.866025i) q^{89} +(1.43969 + 1.20805i) q^{97} +(0.342020 - 0.592396i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 6q^{33} + 6q^{41} - 6q^{57} - 18q^{89} + 6q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\)

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
\(3\) 0.642788 0.766044i 0.642788 0.766044i
\(4\) 0 0
\(5\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(6\) 0 0
\(7\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(8\) 0 0
\(9\) −0.173648 0.984808i −0.173648 0.984808i
\(10\) 0 0
\(11\) 0.524005 + 0.439693i 0.524005 + 0.439693i 0.866025 0.500000i \(-0.166667\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(12\) 0 0
\(13\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.70574 0.984808i 1.70574 0.984808i 0.766044 0.642788i \(-0.222222\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(18\) 0 0
\(19\) −1.32683 0.766044i −1.32683 0.766044i −0.342020 0.939693i \(-0.611111\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(24\) 0 0
\(25\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(26\) 0 0
\(27\) −0.866025 0.500000i −0.866025 0.500000i
\(28\) 0 0
\(29\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(30\) 0 0
\(31\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(32\) 0 0
\(33\) 0.673648 0.118782i 0.673648 0.118782i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.26604 0.223238i 1.26604 0.223238i 0.500000 0.866025i \(-0.333333\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(42\) 0 0
\(43\) 0.223238 0.266044i 0.223238 0.266044i −0.642788 0.766044i \(-0.722222\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(48\) 0 0
\(49\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(50\) 0 0
\(51\) 0.342020 1.93969i 0.342020 1.93969i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.43969 + 0.524005i −1.43969 + 0.524005i
\(58\) 0 0
\(59\) 0.984808 0.826352i 0.984808 0.826352i 1.00000i \(-0.5\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(60\) 0 0
\(61\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.85083 + 0.326352i −1.85083 + 0.326352i −0.984808 0.173648i \(-0.944444\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(74\) 0 0
\(75\) 0.642788 + 0.766044i 0.642788 + 0.766044i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(82\) 0 0
\(83\) −0.300767 + 1.70574i −0.300767 + 1.70574i 0.342020 + 0.939693i \(0.388889\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.43969 + 1.20805i 1.43969 + 1.20805i 0.939693 + 0.342020i \(0.111111\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 0 0
\(99\) 0.342020 0.592396i 0.342020 0.592396i
\(100\) 0 0
\(101\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(102\) 0 0
\(103\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.96962 −1.96962 −0.984808 0.173648i \(-0.944444\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.11334 + 1.32683i 1.11334 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.0923963 0.524005i −0.0923963 0.524005i
\(122\) 0 0
\(123\) 0.642788 1.11334i 0.642788 1.11334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(128\) 0 0
\(129\) −0.0603074 0.342020i −0.0603074 0.342020i
\(130\) 0 0
\(131\) 1.62760 + 0.592396i 1.62760 + 0.592396i 0.984808 0.173648i \(-0.0555556\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.673648 0.118782i −0.673648 0.118782i −0.173648 0.984808i \(-0.555556\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) −0.642788 + 1.76604i −0.642788 + 1.76604i 1.00000i \(0.5\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 1.00000i
\(148\) 0 0
\(149\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(150\) 0 0
\(151\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(152\) 0 0
\(153\) −1.26604 1.50881i −1.26604 1.50881i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(168\) 0 0
\(169\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(170\) 0 0
\(171\) −0.524005 + 1.43969i −0.524005 + 1.43969i
\(172\) 0 0
\(173\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.28558i 1.28558i
\(178\) 0 0
\(179\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.32683 + 0.233956i 1.32683 + 0.233956i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(192\) 0 0
\(193\) −1.43969 0.524005i −1.43969 0.524005i −0.500000 0.866025i \(-0.666667\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.358441 0.984808i −0.358441 0.984808i
\(210\) 0 0
\(211\) 0.642788 + 0.766044i 0.642788 + 0.766044i 0.984808 0.173648i \(-0.0555556\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0.118782 + 0.326352i 0.118782 + 0.326352i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(224\) 0 0
\(225\) 1.00000 1.00000
\(226\) 0 0
\(227\) 1.50881 + 1.26604i 1.50881 + 1.26604i 0.866025 + 0.500000i \(0.166667\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(228\) 0 0
\(229\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.11334 0.642788i 1.11334 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(240\) 0 0
\(241\) −0.326352 + 1.85083i −0.326352 + 1.85083i 0.173648 + 0.984808i \(0.444444\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.11334 + 1.32683i 1.11334 + 1.32683i
\(250\) 0 0
\(251\) 0.984808 1.70574i 0.984808 1.70574i 0.342020 0.939693i \(-0.388889\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.673648 + 0.118782i −0.673648 + 0.118782i −0.500000 0.866025i \(-0.666667\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.62760 + 0.592396i −1.62760 + 0.592396i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.524005 + 0.439693i −0.524005 + 0.439693i
\(276\) 0 0
\(277\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.11334 + 1.32683i −1.11334 + 1.32683i −0.173648 + 0.984808i \(0.555556\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(282\) 0 0
\(283\) 0.984808 0.173648i 0.984808 0.173648i 0.342020 0.939693i \(-0.388889\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.43969 2.49362i 1.43969 2.49362i
\(290\) 0 0
\(291\) 1.85083 0.326352i 1.85083 0.326352i
\(292\) 0 0
\(293\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.233956 0.642788i −0.233956 0.642788i
\(298\) 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
\(307\) −1.32683 + 0.766044i −1.32683 + 0.766044i −0.984808 0.173648i \(-0.944444\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(312\) 0 0
\(313\) −1.17365 0.984808i −1.17365 0.984808i −0.173648 0.984808i \(-0.555556\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.26604 + 1.50881i −1.26604 + 1.50881i
\(322\) 0 0
\(323\) −3.01763 −3.01763
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.342020 + 0.939693i 0.342020 + 0.939693i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.0603074 0.342020i −0.0603074 0.342020i 0.939693 0.342020i \(-0.111111\pi\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 1.73205 1.73205
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.20805 0.439693i −1.20805 0.439693i −0.342020 0.939693i \(-0.611111\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.93969 0.342020i −1.93969 0.342020i −0.939693 0.342020i \(-0.888889\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 0.673648 + 1.16679i 0.673648 + 1.16679i
\(362\) 0 0
\(363\) −0.460802 0.266044i −0.460802 0.266044i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(368\) 0 0
\(369\) −0.439693 1.20805i −0.439693 1.20805i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.53209i 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.300767 0.173648i −0.300767 0.173648i
\(388\) 0 0
\(389\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1.50000 0.866025i 1.50000 0.866025i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.439693 + 1.20805i −0.439693 + 1.20805i 0.500000 + 0.866025i \(0.333333\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.326352 0.118782i −0.326352 0.118782i 0.173648 0.984808i \(-0.444444\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) −0.524005 + 0.439693i −0.524005 + 0.439693i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(418\) 0 0
\(419\) −0.300767 1.70574i −0.300767 1.70574i −0.642788 0.766044i \(-0.722222\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(420\) 0 0
\(421\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.673648 + 1.85083i 0.673648 + 1.85083i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(440\) 0 0
\(441\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(442\) 0 0
\(443\) −0.984808 0.826352i −0.984808 0.826352i 1.00000i \(-0.5\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.592396 0.342020i 0.592396 0.342020i −0.173648 0.984808i \(-0.555556\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(450\) 0 0
\(451\) 0.761570 + 0.439693i 0.761570 + 0.439693i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.266044 1.50881i 0.266044 1.50881i −0.500000 0.866025i \(-0.666667\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(458\) 0 0
\(459\) −1.96962 −1.96962
\(460\) 0 0
\(461\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(462\) 0 0
\(463\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.642788 1.11334i 0.642788 1.11334i −0.342020 0.939693i \(-0.611111\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.233956 0.0412527i 0.233956 0.0412527i
\(474\) 0 0
\(475\) 0.984808 1.17365i 0.984808 1.17365i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(480\) 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
\(489\) −0.766044 0.642788i −0.766044 0.642788i
\(490\) 0 0
\(491\) 1.50881 1.26604i 1.50881 1.26604i 0.642788 0.766044i \(-0.277778\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.342020 0.0603074i 0.342020 0.0603074i 1.00000i \(-0.5\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(508\) 0 0
\(509\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.70574 0.984808i −1.70574 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
−0.766044 0.642788i \(-0.777778\pi\)
\(522\) 0 0
\(523\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(530\) 0 0
\(531\) −0.984808 0.826352i −0.984808 0.826352i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.70574 + 0.300767i 1.70574 + 0.300767i
\(538\) 0 0
\(539\) −0.684040 −0.684040
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.118782 0.326352i −0.118782 0.326352i 0.866025 0.500000i \(-0.166667\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.03209 0.866025i 1.03209 0.866025i
\(562\) 0 0
\(563\) 0.642788 + 0.233956i 0.642788 + 0.233956i 0.642788 0.766044i \(-0.277778\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.26604 + 0.223238i 1.26604 + 0.223238i 0.766044 0.642788i \(-0.222222\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 0.118782 0.326352i 0.118782 0.326352i −0.866025 0.500000i \(-0.833333\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(578\) 0 0
\(579\) −1.32683 + 0.766044i −1.32683 + 0.766044i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.642788 + 0.233956i −0.642788 + 0.233956i −0.642788 0.766044i \(-0.722222\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(600\) 0 0
\(601\) 1.76604 0.642788i 1.76604 0.642788i 0.766044 0.642788i \(-0.222222\pi\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 0.642788 + 1.76604i 0.642788 + 1.76604i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.233956 0.642788i 0.233956 0.642788i −0.766044 0.642788i \(-0.777778\pi\)
1.00000 \(0\)
\(618\) 0 0
\(619\) 1.50881 + 0.266044i 1.50881 + 0.266044i 0.866025 0.500000i \(-0.166667\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.939693 0.342020i −0.939693 0.342020i
\(626\) 0 0
\(627\) −0.984808 0.358441i −0.984808 0.358441i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) 1.00000 1.00000
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.439693 1.20805i −0.439693 1.20805i −0.939693 0.342020i \(-0.888889\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(642\) 0 0
\(643\) −1.20805 1.43969i −1.20805 1.43969i −0.866025 0.500000i \(-0.833333\pi\)
−0.342020 0.939693i \(-0.611111\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0.879385 0.879385
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.326352 + 0.118782i 0.326352 + 0.118782i
\(658\) 0 0
\(659\) −1.32683 1.11334i −1.32683 1.11334i −0.984808 0.173648i \(-0.944444\pi\)
−0.342020 0.939693i \(-0.611111\pi\)
\(660\) 0 0
\(661\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.173648 0.984808i 0.173648 0.984808i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(674\) 0 0
\(675\) 0.642788 0.766044i 0.642788 0.766044i
\(676\) 0 0
\(677\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.93969 0.342020i 1.93969 0.342020i
\(682\) 0 0
\(683\) 0.342020 0.592396i 0.342020 0.592396i −0.642788 0.766044i \(-0.722222\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.642788 0.766044i 0.642788 0.766044i −0.342020 0.939693i \(-0.611111\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.93969 1.62760i 1.93969 1.62760i
\(698\) 0 0
\(699\) 0.223238 1.26604i 0.223238 1.26604i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.20805 + 1.43969i 1.20805 + 1.43969i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(728\) 0 0
\(729\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(730\) 0 0
\(731\) 0.118782 0.673648i 0.118782 0.673648i
\(732\) 0 0
\(733\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.11334 0.642788i −1.11334 0.642788i
\(738\) 0 0
\(739\) 1.62760 0.939693i 1.62760 0.939693i 0.642788 0.766044i \(-0.277778\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.73205 1.73205
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(752\) 0 0
\(753\) −0.673648 1.85083i −0.673648 1.85083i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(770\) 0 0
\(771\) −0.342020 + 0.592396i −0.342020 + 0.592396i
\(772\) 0 0
\(773\) 0 0 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
\(779\) −1.85083 0.673648i −1.85083 0.673648i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −0.342020 + 0.939693i −0.342020 + 0.939693i 0.642788 + 0.766044i \(0.277778\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(788\) 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 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.592396 + 1.62760i −0.592396 + 1.62760i
\(802\) 0 0
\(803\) −0.223238 + 0.0812519i −0.223238 + 0.0812519i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.96962i 1.96962i 0.173648 + 0.984808i \(0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(810\) 0 0