Properties

Label 1444.1.j.b.1029.1
Level $1444$
Weight $1$
Character 1444.1029
Analytic conductor $0.721$
Analytic rank $0$
Dimension $12$
Projective image $S_{4}$
CM/RM no
Inner twists $12$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.j (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
Defining polynomial: \(x^{12} - 8 x^{6} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.27436.1

Embedding invariants

Embedding label 1029.1
Root \(0.483690 - 1.32893i\) of defining polynomial
Character \(\chi\) \(=\) 1444.1029
Dual form 1444.1.j.b.849.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.483690 - 1.32893i) q^{3} +(0.173648 - 0.984808i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.766044 + 0.642788i) q^{9} +O(q^{10})\) \(q+(-0.483690 - 1.32893i) q^{3} +(0.173648 - 0.984808i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.766044 + 0.642788i) q^{9} +(0.500000 - 0.866025i) q^{11} +(-1.39273 + 0.245576i) q^{15} +(0.766044 + 0.642788i) q^{17} +(-0.909039 + 1.08335i) q^{21} +(-0.909039 - 1.08335i) q^{29} +(-1.39273 - 0.245576i) q^{33} +(-0.939693 + 0.342020i) q^{35} +1.41421i q^{37} +(0.483690 + 1.32893i) q^{41} +(-0.173648 + 0.984808i) q^{43} +(0.500000 + 0.866025i) q^{45} +(-0.766044 + 0.642788i) q^{47} +(0.483690 - 1.32893i) q^{51} +(-0.766044 - 0.642788i) q^{55} +(0.909039 - 1.08335i) q^{59} +(-0.173648 - 0.984808i) q^{61} +(0.939693 + 0.342020i) q^{63} +(1.39273 + 0.245576i) q^{71} +(-0.939693 + 0.342020i) q^{73} -1.00000 q^{77} +(-0.173648 + 0.984808i) q^{81} +(0.766044 - 0.642788i) q^{85} +(-1.00000 + 1.73205i) q^{87} +(0.173648 + 0.984808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{7} + O(q^{10}) \) \( 12q - 6q^{7} + 6q^{11} + 6q^{45} - 12q^{77} - 12q^{87} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(1\) \(e\left(\frac{13}{18}\right)\)

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
\(3\) −0.483690 1.32893i −0.483690 1.32893i −0.906308 0.422618i \(-0.861111\pi\)
0.422618 0.906308i \(-0.361111\pi\)
\(4\) 0 0
\(5\) 0.173648 0.984808i 0.173648 0.984808i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(6\) 0 0
\(7\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(12\) 0 0
\(13\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(14\) 0 0
\(15\) −1.39273 + 0.245576i −1.39273 + 0.245576i
\(16\) 0 0
\(17\) 0.766044 + 0.642788i 0.766044 + 0.642788i 0.939693 0.342020i \(-0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −0.909039 + 1.08335i −0.909039 + 1.08335i
\(22\) 0 0
\(23\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.909039 1.08335i −0.909039 1.08335i −0.996195 0.0871557i \(-0.972222\pi\)
0.0871557 0.996195i \(-0.472222\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) −1.39273 0.245576i −1.39273 0.245576i
\(34\) 0 0
\(35\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(36\) 0 0
\(37\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.483690 + 1.32893i 0.483690 + 1.32893i 0.906308 + 0.422618i \(0.138889\pi\)
−0.422618 + 0.906308i \(0.638889\pi\)
\(42\) 0 0
\(43\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(46\) 0 0
\(47\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.483690 1.32893i 0.483690 1.32893i
\(52\) 0 0
\(53\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(54\) 0 0
\(55\) −0.766044 0.642788i −0.766044 0.642788i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.909039 1.08335i 0.909039 1.08335i −0.0871557 0.996195i \(-0.527778\pi\)
0.996195 0.0871557i \(-0.0277778\pi\)
\(60\) 0 0
\(61\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(62\) 0 0
\(63\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.39273 + 0.245576i 1.39273 + 0.245576i 0.819152 0.573576i \(-0.194444\pi\)
0.573576 + 0.819152i \(0.305556\pi\)
\(72\) 0 0
\(73\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −1.00000
\(78\) 0 0
\(79\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(80\) 0 0
\(81\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0.766044 0.642788i 0.766044 0.642788i
\(86\) 0 0
\(87\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(88\) 0 0
\(89\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(98\) 0 0
\(99\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(100\) 0 0
\(101\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(102\) 0 0
\(103\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(104\) 0 0
\(105\) 0.909039 + 1.08335i 0.909039 + 1.08335i
\(106\) 0 0
\(107\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(108\) 0 0
\(109\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(110\) 0 0
\(111\) 1.87939 0.684040i 1.87939 0.684040i
\(112\) 0 0
\(113\) 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 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.173648 0.984808i 0.173648 0.984808i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 1.53209 1.28558i 1.53209 1.28558i
\(124\) 0 0
\(125\) 0.500000 0.866025i 0.500000 0.866025i
\(126\) 0 0
\(127\) −0.483690 + 1.32893i −0.483690 + 1.32893i 0.422618 + 0.906308i \(0.361111\pi\)
−0.906308 + 0.422618i \(0.861111\pi\)
\(128\) 0 0
\(129\) 1.39273 0.245576i 1.39273 0.245576i
\(130\) 0 0
\(131\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.173648 + 0.984808i 0.173648 + 0.984808i 0.939693 + 0.342020i \(0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(138\) 0 0
\(139\) −0.939693 0.342020i −0.939693 0.342020i −0.173648 0.984808i \(-0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(140\) 0 0
\(141\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(150\) 0 0
\(151\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(152\) 0 0
\(153\) −1.00000 −1.00000
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(164\) 0 0
\(165\) −0.483690 + 1.32893i −0.483690 + 1.32893i
\(166\) 0 0
\(167\) 1.39273 0.245576i 1.39273 0.245576i 0.573576 0.819152i \(-0.305556\pi\)
0.819152 + 0.573576i \(0.194444\pi\)
\(168\) 0 0
\(169\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.909039 1.08335i 0.909039 1.08335i −0.0871557 0.996195i \(-0.527778\pi\)
0.996195 0.0871557i \(-0.0277778\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.87939 0.684040i −1.87939 0.684040i
\(178\) 0 0
\(179\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(180\) 0 0
\(181\) 0.909039 + 1.08335i 0.909039 + 1.08335i 0.996195 + 0.0871557i \(0.0277778\pi\)
−0.0871557 + 0.996195i \(0.527778\pi\)
\(182\) 0 0
\(183\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(184\) 0 0
\(185\) 1.39273 + 0.245576i 1.39273 + 0.245576i
\(186\) 0 0
\(187\) 0.939693 0.342020i 0.939693 0.342020i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0 0
\(199\) 0.766044 0.642788i 0.766044 0.642788i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.483690 + 1.32893i −0.483690 + 1.32893i
\(204\) 0 0
\(205\) 1.39273 0.245576i 1.39273 0.245576i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.909039 + 1.08335i −0.909039 + 1.08335i 0.0871557 + 0.996195i \(0.472222\pi\)
−0.996195 + 0.0871557i \(0.972222\pi\)
\(212\) 0 0
\(213\) −0.347296 1.96962i −0.347296 1.96962i
\(214\) 0 0
\(215\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0.909039 + 1.08335i 0.909039 + 1.08335i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.39273 0.245576i −1.39273 0.245576i −0.573576 0.819152i \(-0.694444\pi\)
−0.819152 + 0.573576i \(0.805556\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0.483690 + 1.32893i 0.483690 + 1.32893i
\(232\) 0 0
\(233\) 0.173648 0.984808i 0.173648 0.984808i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(234\) 0 0
\(235\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0.483690 1.32893i 0.483690 1.32893i −0.422618 0.906308i \(-0.638889\pi\)
0.906308 0.422618i \(-0.138889\pi\)
\(242\) 0 0
\(243\) 1.39273 0.245576i 1.39273 0.245576i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.173648 + 0.984808i 0.173648 + 0.984808i 0.939693 + 0.342020i \(0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.22474 0.707107i −1.22474 0.707107i
\(256\) 0 0
\(257\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(258\) 0 0
\(259\) 1.22474 0.707107i 1.22474 0.707107i
\(260\) 0 0
\(261\) 1.39273 + 0.245576i 1.39273 + 0.245576i
\(262\) 0 0
\(263\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(270\) 0 0
\(271\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.39273 + 0.245576i −1.39273 + 0.245576i −0.819152 0.573576i \(-0.805556\pi\)
−0.573576 + 0.819152i \(0.694444\pi\)
\(282\) 0 0
\(283\) 0.766044 + 0.642788i 0.766044 + 0.642788i 0.939693 0.342020i \(-0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.909039 1.08335i 0.909039 1.08335i
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) −0.909039 1.08335i −0.909039 1.08335i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.939693 0.342020i 0.939693 0.342020i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.00000 −1.00000
\(306\) 0 0
\(307\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(308\) 0 0
\(309\) −0.347296 + 1.96962i −0.347296 + 1.96962i
\(310\) 0 0
\(311\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(314\) 0 0
\(315\) 0.500000 0.866025i 0.500000 0.866025i
\(316\) 0 0
\(317\) 0.483690 1.32893i 0.483690 1.32893i −0.422618 0.906308i \(-0.638889\pi\)
0.906308 0.422618i \(-0.138889\pi\)
\(318\) 0 0
\(319\) −1.39273 + 0.245576i −1.39273 + 0.245576i
\(320\) 0 0
\(321\) −1.53209 1.28558i −1.53209 1.28558i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) −0.909039 1.08335i −0.909039 1.08335i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(338\) 0 0
\(339\) −1.87939 + 0.684040i −1.87939 + 0.684040i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(348\) 0 0
\(349\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0.483690 1.32893i 0.483690 1.32893i
\(356\) 0 0
\(357\) −1.39273 + 0.245576i −1.39273 + 0.245576i
\(358\) 0 0
\(359\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(366\) 0 0
\(367\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(368\) 0 0
\(369\) −1.22474 0.707107i −1.22474 0.707107i
\(370\) 0 0
\(371\) 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
\(375\) −1.39273 0.245576i −1.39273 0.245576i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0 0
\(381\) 2.00000 2.00000
\(382\) 0 0
\(383\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(384\) 0 0
\(385\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(386\) 0 0
\(387\) −0.500000 0.866025i −0.500000 0.866025i
\(388\) 0 0
\(389\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −0.483690 + 1.32893i −0.483690 + 1.32893i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(406\) 0 0
\(407\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(408\) 0 0
\(409\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(410\) 0 0
\(411\) 1.22474 0.707107i 1.22474 0.707107i
\(412\) 0 0
\(413\) −1.39273 0.245576i −1.39273 0.245576i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.41421i 1.41421i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(422\) 0 0
\(423\) 0.173648 0.984808i 0.173648 0.984808i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(432\) 0 0
\(433\) −1.39273 + 0.245576i −1.39273 + 0.245576i −0.819152 0.573576i \(-0.805556\pi\)
−0.573576 + 0.819152i \(0.694444\pi\)
\(434\) 0 0
\(435\) 1.53209 + 1.28558i 1.53209 + 1.28558i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.909039 + 1.08335i −0.909039 + 1.08335i 0.0871557 + 0.996195i \(0.472222\pi\)
−0.996195 + 0.0871557i \(0.972222\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.909039 1.08335i −0.909039 1.08335i
\(448\) 0 0
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 1.39273 + 0.245576i 1.39273 + 0.245576i
\(452\) 0 0
\(453\) −1.87939 + 0.684040i −1.87939 + 0.684040i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.173648 0.984808i 0.173648 0.984808i −0.766044 0.642788i \(-0.777778\pi\)
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 0
\(467\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.347296 1.96962i −0.347296 1.96962i −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(492\) 0 0
\(493\) 1.41421i 1.41421i
\(494\) 0 0
\(495\) 1.00000 1.00000
\(496\) 0 0
\(497\) −0.483690 1.32893i −0.483690 1.32893i
\(498\) 0 0
\(499\) 0.173648 0.984808i 0.173648 0.984808i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(500\) 0 0
\(501\) −1.00000 1.73205i −1.00000 1.73205i
\(502\) 0 0
\(503\) −1.53209 + 1.28558i −1.53209 + 1.28558i −0.766044 + 0.642788i \(0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.483690 1.32893i 0.483690 1.32893i
\(508\) 0 0
\(509\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(510\) 0 0
\(511\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.909039 + 1.08335i −0.909039 + 1.08335i
\(516\) 0 0
\(517\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(518\) 0 0
\(519\) −1.87939 0.684040i −1.87939 0.684040i
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 0.909039 + 1.08335i 0.909039 + 1.08335i 0.996195 + 0.0871557i \(0.0277778\pi\)
−0.0871557 + 0.996195i \(0.527778\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.939693 0.342020i 0.939693 0.342020i
\(530\) 0 0
\(531\) 1.41421i 1.41421i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.483690 1.32893i −0.483690 1.32893i
\(536\) 0 0
\(537\) 0.347296 1.96962i 0.347296 1.96962i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(542\) 0 0
\(543\) 1.00000 1.73205i 1.00000 1.73205i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(548\) 0 0
\(549\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.347296 1.96962i −0.347296 1.96962i
\(556\) 0 0
\(557\) −0.939693 0.342020i −0.939693 0.342020i −0.173648 0.984808i \(-0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.909039 1.08335i −0.909039 1.08335i
\(562\) 0 0
\(563\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(564\) 0 0
\(565\) −1.39273 0.245576i −1.39273 0.245576i
\(566\) 0 0
\(567\) 0.939693 0.342020i 0.939693 0.342020i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) −0.483690 1.32893i −0.483690 1.32893i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.766044 + 0.642788i 0.766044 + 0.642788i 0.939693 0.342020i \(-0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(594\) 0 0
\(595\) −0.939693 0.342020i −0.939693 0.342020i
\(596\) 0 0
\(597\) −1.22474 0.707107i −1.22474 0.707107i
\(598\) 0 0
\(599\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(600\) 0 0
\(601\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 2.00000 2.00000
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.173648 0.984808i 0.173648 0.984808i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(614\) 0 0
\(615\) −1.00000 1.73205i −1.00000 1.73205i
\(616\) 0 0
\(617\) 0.766044 0.642788i 0.766044 0.642788i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.766044 0.642788i −0.766044 0.642788i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.909039 + 1.08335i −0.909039 + 1.08335i
\(630\) 0 0
\(631\) 0.173648 + 0.984808i 0.173648 + 0.984808i 0.939693 + 0.342020i \(0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(632\) 0 0
\(633\) 1.87939 + 0.684040i 1.87939 + 0.684040i
\(634\) 0 0
\(635\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(640\) 0 0
\(641\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(642\) 0 0
\(643\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(644\) 0 0
\(645\) 1.41421i 1.41421i
\(646\) 0 0
\(647\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0 0
\(649\) −0.483690 1.32893i −0.483690 1.32893i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(656\) 0 0
\(657\) 0.500000 0.866025i 0.500000 0.866025i
\(658\) 0 0
\(659\) −0.483690 + 1.32893i −0.483690 + 1.32893i 0.422618 + 0.906308i \(0.361111\pi\)
−0.906308 + 0.422618i \(0.861111\pi\)
\(660\) 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
\(669\) 0.347296 + 1.96962i 0.347296 + 1.96962i
\(670\) 0 0
\(671\) −0.939693 0.342020i −0.939693 0.342020i
\(672\) 0 0
\(673\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 1.00000 1.00000
\(686\) 0 0
\(687\) 0.483690 + 1.32893i 0.483690 + 1.32893i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0.766044 0.642788i 0.766044 0.642788i
\(694\) 0 0
\(695\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(696\) 0 0
\(697\) −0.483690 + 1.32893i −0.483690 + 1.32893i
\(698\) 0 0
\(699\) −1.39273 + 0.245576i −1.39273 + 0.245576i
\(700\) 0 0
\(701\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0.909039 1.08335i 0.909039 1.08335i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.87939 + 0.684040i 1.87939 + 0.684040i 0.939693 + 0.342020i \(0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.39273 + 0.245576i 1.39273 + 0.245576i
\(718\) 0 0
\(719\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(720\) 0 0
\(721\) 1.41421i 1.41421i
\(722\) 0 0
\(723\) −2.00000 −2.00000
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(728\) 0 0
\(729\) −0.500000 0.866025i −0.500000 0.866025i
\(730\) 0 0
\(731\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(744\) 0 0
\(745\) −0.173648 0.984808i −0.173648 0.984808i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.22474 0.707107i −1.22474 0.707107i
\(750\) 0 0
\(751\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(752\) 0 0
\(753\) 1.22474 0.707107i 1.22474 0.707107i
\(754\) 0 0
\(755\) −1.39273 0.245576i −1.39273 0.245576i
\(756\) 0 0
\(757\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.766044 0.642788i 0.766044 0.642788i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.483690 1.32893i 0.483690 1.32893i −0.422618 0.906308i \(-0.638889\pi\)
0.906308 0.422618i \(-0.138889\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.53209 1.28558i −1.53209 1.28558i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0.909039 1.08335i 0.909039 1.08335i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) 0 0
\(789\) 0.909039 + 1.08335i 0.909039 + 1.08335i
\(790\) 0 0
\(791\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) −1.00000 −1.00000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0