Properties

Label 2808.1.bs.b.883.1
Level $2808$
Weight $1$
Character 2808.883
Analytic conductor $1.401$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -104
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2808.bs (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.40137455547\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 936)
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.62171080298496.1

Embedding invariants

Embedding label 883.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 2808.883
Dual form 2808.1.bs.b.1819.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.939693 - 1.62760i) q^{5} +(-0.766044 + 1.32683i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.939693 - 1.62760i) q^{5} +(-0.766044 + 1.32683i) q^{7} -1.00000 q^{8} -1.87939 q^{10} +(-0.500000 - 0.866025i) q^{13} +(0.766044 + 1.32683i) q^{14} +(-0.500000 + 0.866025i) q^{16} -0.347296 q^{17} +(-0.939693 + 1.62760i) q^{20} +(-1.26604 + 2.19285i) q^{25} -1.00000 q^{26} +1.53209 q^{28} +(0.500000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.173648 + 0.300767i) q^{34} +2.87939 q^{35} -1.87939 q^{37} +(0.939693 + 1.62760i) q^{40} +(-0.173648 + 0.300767i) q^{43} +(0.173648 - 0.300767i) q^{47} +(-0.673648 - 1.16679i) q^{49} +(1.26604 + 2.19285i) q^{50} +(-0.500000 + 0.866025i) q^{52} +(0.766044 - 1.32683i) q^{56} +1.00000 q^{62} +1.00000 q^{64} +(-0.939693 + 1.62760i) q^{65} +(0.173648 + 0.300767i) q^{68} +(1.43969 - 2.49362i) q^{70} -1.53209 q^{71} +(-0.939693 + 1.62760i) q^{74} +1.87939 q^{80} +(0.326352 + 0.565258i) q^{85} +(0.173648 + 0.300767i) q^{86} +1.53209 q^{91} +(-0.173648 - 0.300767i) q^{94} -1.34730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} - 3q^{4} - 6q^{8} + O(q^{10}) \) \( 6q + 3q^{2} - 3q^{4} - 6q^{8} - 3q^{13} - 3q^{16} - 3q^{25} - 6q^{26} + 3q^{31} + 3q^{32} + 6q^{35} - 3q^{49} + 3q^{50} - 3q^{52} + 6q^{62} + 6q^{64} + 3q^{70} + 3q^{85} - 6q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(1081\) \(1405\) \(2081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 0.866025i 0.500000 0.866025i
\(3\) 0 0
\(4\) −0.500000 0.866025i −0.500000 0.866025i
\(5\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(6\) 0 0
\(7\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) −1.87939 −1.87939
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) −0.500000 0.866025i −0.500000 0.866025i
\(14\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −1.26604 + 2.19285i −1.26604 + 2.19285i
\(26\) −1.00000 −1.00000
\(27\) 0 0
\(28\) 1.53209 1.53209
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(33\) 0 0
\(34\) −0.173648 + 0.300767i −0.173648 + 0.300767i
\(35\) 2.87939 2.87939
\(36\) 0 0
\(37\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.173648 0.300767i 0.173648 0.300767i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(48\) 0 0
\(49\) −0.673648 1.16679i −0.673648 1.16679i
\(50\) 1.26604 + 2.19285i 1.26604 + 2.19285i
\(51\) 0 0
\(52\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.766044 1.32683i 0.766044 1.32683i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 1.00000 1.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(69\) 0 0
\(70\) 1.43969 2.49362i 1.43969 2.49362i
\(71\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 1.87939 1.87939
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0.326352 + 0.565258i 0.326352 + 0.565258i
\(86\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.53209 1.53209
\(92\) 0 0
\(93\) 0 0
\(94\) −0.173648 0.300767i −0.173648 0.300767i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −1.34730 −1.34730
\(99\) 0 0
\(100\) 2.53209 2.53209
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(105\) 0 0
\(106\) 0 0
\(107\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.766044 1.32683i −0.766044 1.32683i
\(113\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.266044 0.460802i 0.266044 0.460802i
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0.500000 0.866025i 0.500000 0.866025i
\(125\) 2.87939 2.87939
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0.500000 0.866025i 0.500000 0.866025i
\(129\) 0 0
\(130\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(131\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.347296 0.347296
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(140\) −1.43969 2.49362i −1.43969 2.49362i
\(141\) 0 0
\(142\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(149\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.939693 1.62760i 0.939693 1.62760i
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.939693 1.62760i 0.939693 1.62760i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(170\) 0.652704 0.652704
\(171\) 0 0
\(172\) 0.347296 0.347296
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) −1.93969 3.35965i −1.93969 3.35965i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0.766044 1.32683i 0.766044 1.32683i
\(183\) 0 0
\(184\) 0 0
\(185\) 1.76604 + 3.05888i 1.76604 + 3.05888i
\(186\) 0 0
\(187\) 0 0
\(188\) −0.347296 −0.347296
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.673648 + 1.16679i −0.673648 + 1.16679i
\(197\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 1.26604 2.19285i 1.26604 2.19285i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000 1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 0.939693 + 1.62760i 0.939693 + 1.62760i 0.766044 + 0.642788i \(0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(215\) 0.652704 0.652704
\(216\) 0 0
\(217\) −1.53209 −1.53209
\(218\) 0.173648 0.300767i 0.173648 0.300767i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(222\) 0 0
\(223\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(224\) −1.53209 −1.53209
\(225\) 0 0
\(226\) −1.00000 −1.00000
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) 0.939693 + 1.62760i 0.939693 + 1.62760i 0.766044 + 0.642788i \(0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(234\) 0 0
\(235\) −0.652704 −0.652704
\(236\) 0 0
\(237\) 0 0
\(238\) −0.266044 0.460802i −0.266044 0.460802i
\(239\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(240\) 0 0
\(241\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) −1.00000 −1.00000
\(243\) 0 0
\(244\) 0 0
\(245\) −1.26604 + 2.19285i −1.26604 + 2.19285i
\(246\) 0 0
\(247\) 0 0
\(248\) −0.500000 0.866025i −0.500000 0.866025i
\(249\) 0 0
\(250\) 1.43969 2.49362i 1.43969 2.49362i
\(251\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(258\) 0 0
\(259\) 1.43969 2.49362i 1.43969 2.49362i
\(260\) 1.87939 1.87939
\(261\) 0 0
\(262\) 0.347296 0.347296
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 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
\(271\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(272\) 0.173648 0.300767i 0.173648 0.300767i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) −1.53209 −1.53209
\(279\) 0 0
\(280\) −2.87939 −2.87939
\(281\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.879385 −0.879385
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.87939 1.87939
\(297\) 0 0
\(298\) −1.00000 −1.00000
\(299\) 0 0
\(300\) 0 0
\(301\) −0.266044 0.460802i −0.266044 0.460802i
\(302\) −0.939693 1.62760i −0.939693 1.62760i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.939693 1.62760i −0.939693 1.62760i
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.939693 1.62760i −0.939693 1.62760i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.53209 2.53209
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.266044 + 0.460802i 0.266044 + 0.460802i
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −1.00000 −1.00000
\(335\) 0 0
\(336\) 0 0
\(337\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(338\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(339\) 0 0
\(340\) 0.326352 0.565258i 0.326352 0.565258i
\(341\) 0 0
\(342\) 0 0
\(343\) 0.532089 0.532089
\(344\) 0.173648 0.300767i 0.173648 0.300767i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(348\) 0 0
\(349\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(350\) −3.87939 −3.87939
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 1.43969 + 2.49362i 1.43969 + 2.49362i
\(356\) 0 0
\(357\) 0 0
\(358\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(359\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) −0.766044 1.32683i −0.766044 1.32683i
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 3.53209 3.53209
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −0.173648 + 0.300767i −0.173648 + 0.300767i
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.673648 + 1.16679i 0.673648 + 1.16679i
\(393\) 0 0
\(394\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.26604 2.19285i −1.26604 2.19285i
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 0.500000 0.866025i 0.500000 0.866025i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.500000 0.866025i 0.500000 0.866025i
\(417\) 0 0
\(418\) 0 0
\(419\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(420\) 0 0
\(421\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(422\) 1.87939 1.87939
\(423\) 0 0
\(424\) 0 0
\(425\) 0.439693 0.761570i 0.439693 0.761570i
\(426\) 0 0
\(427\) 0 0
\(428\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(429\) 0 0
\(430\) 0.326352 0.565258i 0.326352 0.565258i
\(431\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(432\) 0 0
\(433\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(434\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(435\) 0 0
\(436\) −0.173648 0.300767i −0.173648 0.300767i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.347296 0.347296
\(443\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(447\) 0 0
\(448\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(453\) 0 0
\(454\) 0 0
\(455\) −1.43969 2.49362i −1.43969 2.49362i
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 1.87939 1.87939
\(459\) 0 0
\(460\) 0 0
\(461\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(462\) 0 0
\(463\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(467\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.326352 + 0.565258i −0.326352 + 0.565258i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −0.532089 −0.532089
\(477\) 0 0
\(478\) 0.347296 0.347296
\(479\) −0.939693 + 1.62760i −0.939693 + 1.62760i −0.173648 + 0.984808i \(0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(480\) 0 0
\(481\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.26604 + 2.19285i 1.26604 + 2.19285i
\(491\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 1.17365 2.03282i 1.17365 2.03282i
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) −1.43969 2.49362i −1.43969 2.49362i
\(501\) 0 0
\(502\) 0.500000 0.866025i 0.500000 0.866025i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) −1.87939 −1.87939
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −1.43969 2.49362i −1.43969 2.49362i
\(519\) 0 0
\(520\) 0.939693 1.62760i 0.939693 1.62760i
\(521\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(522\) 0 0
\(523\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0.173648 0.300767i 0.173648 0.300767i
\(525\) 0 0
\(526\) 0 0
\(527\) −0.173648 0.300767i −0.173648 0.300767i
\(528\) 0 0
\(529\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.87939 + 3.25519i 1.87939 + 3.25519i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(542\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(543\) 0 0
\(544\) −0.173648 0.300767i −0.173648 0.300767i
\(545\) −0.326352 0.565258i −0.326352 0.565258i
\(546\) 0 0
\(547\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(557\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(558\) 0 0
\(559\) 0.347296 0.347296
\(560\) −1.43969 + 2.49362i −1.43969 + 2.49362i
\(561\) 0 0
\(562\) 0 0
\(563\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(564\) 0 0
\(565\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(566\) 1.00000 1.00000
\(567\) 0 0
\(568\) 1.53209 1.53209
\(569\) −0.939693 + 1.62760i −0.939693 + 1.62760i −0.173648 + 0.984808i \(0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(570\) 0 0
\(571\) −0.173648 0.300767i −0.173648 0.300767i 0.766044 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −0.439693 + 0.761570i −0.439693 + 0.761570i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 1.53209 1.53209
\(587\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.939693 1.62760i 0.939693 1.62760i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) −1.00000 −1.00000
\(596\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(602\) −0.532089 −0.532089
\(603\) 0 0
\(604\) −1.87939 −1.87939
\(605\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.347296 −0.347296
\(612\) 0 0
\(613\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(618\) 0 0
\(619\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) −1.87939 −1.87939
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.43969 2.49362i −1.43969 2.49362i
\(626\) −0.939693 1.62760i −0.939693 1.62760i
\(627\) 0 0
\(628\) 0 0
\(629\) 0.652704 0.652704
\(630\) 0 0
\(631\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(635\) 0 0
\(636\) 0 0
\(637\) −0.673648 + 1.16679i −0.673648 + 1.16679i
\(638\) 0 0
\(639\) 0 0
\(640\) −1.87939 −1.87939
\(641\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.26604 2.19285i 1.26604 2.19285i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0.326352 0.565258i 0.326352 0.565258i
\(656\) 0 0
\(657\) 0 0
\(658\) 0.532089 0.532089
\(659\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(674\) −1.53209 −1.53209
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.326352 0.565258i −0.326352 0.565258i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.266044 0.460802i 0.266044 0.460802i
\(687\) 0 0
\(688\) −0.173648 0.300767i −0.173648 0.300767i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.87939 −1.87939
\(695\) −1.43969 + 2.49362i −1.43969 + 2.49362i
\(696\) 0 0
\(697\) 0 0
\(698\) −0.939693 1.62760i −0.939693 1.62760i
\(699\) 0 0
\(700\) −1.93969 + 3.35965i −1.93969 + 3.35965i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(710\) 2.87939 2.87939
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(717\) 0 0
\(718\) 0.500000 0.866025i 0.500000 0.866025i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.500000 0.866025i 0.500000 0.866025i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) −1.53209 −1.53209
\(729\) 0 0
\(730\) 0 0
\(731\) 0.0603074 0.104455i 0.0603074 0.104455i
\(732\) 0 0
\(733\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1.76604 3.05888i 1.76604 3.05888i
\(741\) 0 0
\(742\) 0 0
\(743\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(744\) 0 0
\(745\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.53209 2.65366i 1.53209 2.65366i
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(753\) 0 0
\(754\) 0 0
\(755\) −3.53209 −3.53209
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −0.266044 + 0.460802i −0.266044 + 0.460802i
\(764\) 0 0
\(765\) 0 0
\(766\) 1.53209 1.53209
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(774\) 0 0
\(775\) −2.53209 −2.53209
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.34730 1.34730
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(789\) 0 0
\(790\) 0 0
\(791\) 1.53209 1.53209
\(792\) 0 0
\(793\) 0 0
\(794\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) −0.0603074 + 0.104455i −0.0603074 + 0.104455i
\(800\) −2.53209 −2.53209
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −0.500000 0.866025i −0.500000 0.866025i