Properties

Label 3744.1.ci.a.3535.1
Level $3744$
Weight $1$
Character 3744.3535
Analytic conductor $1.868$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -104
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.ci (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 3535.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 3744.3535
Dual form 3744.1.ci.a.1039.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.766044 + 0.642788i) q^{3} +(-0.766044 - 1.32683i) q^{5} +(0.173648 - 0.300767i) q^{7} +(0.173648 - 0.984808i) q^{9} +O(q^{10})\) \(q+(-0.766044 + 0.642788i) q^{3} +(-0.766044 - 1.32683i) q^{5} +(0.173648 - 0.300767i) q^{7} +(0.173648 - 0.984808i) q^{9} +(-0.500000 - 0.866025i) q^{13} +(1.43969 + 0.524005i) q^{15} -1.87939 q^{17} +(0.0603074 + 0.342020i) q^{21} +(-0.673648 + 1.16679i) q^{25} +(0.500000 + 0.866025i) q^{27} +(-0.500000 - 0.866025i) q^{31} -0.532089 q^{35} +1.53209 q^{37} +(0.939693 + 0.342020i) q^{39} +(-0.939693 + 1.62760i) q^{43} +(-1.43969 + 0.524005i) q^{45} +(-0.939693 + 1.62760i) q^{47} +(0.439693 + 0.761570i) q^{49} +(1.43969 - 1.20805i) q^{51} +(-0.266044 - 0.223238i) q^{63} +(-0.766044 + 1.32683i) q^{65} -0.347296 q^{71} +(-0.233956 - 1.32683i) q^{75} +(-0.939693 - 0.342020i) q^{81} +(1.43969 + 2.49362i) q^{85} -0.347296 q^{91} +(0.939693 + 0.342020i) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 3q^{13} + 3q^{15} + 6q^{21} - 3q^{25} + 3q^{27} - 3q^{31} + 6q^{35} - 3q^{45} - 3q^{49} + 3q^{51} + 3q^{63} - 6q^{75} + 3q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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
\(3\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(4\) 0 0
\(5\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(6\) 0 0
\(7\) 0.173648 0.300767i 0.173648 0.300767i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(8\) 0 0
\(9\) 0.173648 0.984808i 0.173648 0.984808i
\(10\) 0 0
\(11\) 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 0
\(15\) 1.43969 + 0.524005i 1.43969 + 0.524005i
\(16\) 0 0
\(17\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0.0603074 + 0.342020i 0.0603074 + 0.342020i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −0.673648 + 1.16679i −0.673648 + 1.16679i
\(26\) 0 0
\(27\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(28\) 0 0
\(29\) 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 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.532089 −0.532089
\(36\) 0 0
\(37\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(38\) 0 0
\(39\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −0.939693 + 1.62760i −0.939693 + 1.62760i −0.173648 + 0.984808i \(0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(44\) 0 0
\(45\) −1.43969 + 0.524005i −1.43969 + 0.524005i
\(46\) 0 0
\(47\) −0.939693 + 1.62760i −0.939693 + 1.62760i −0.173648 + 0.984808i \(0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(48\) 0 0
\(49\) 0.439693 + 0.761570i 0.439693 + 0.761570i
\(50\) 0 0
\(51\) 1.43969 1.20805i 1.43969 1.20805i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 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 \(0.666667\pi\)
\(62\) 0 0
\(63\) −0.266044 0.223238i −0.266044 0.223238i
\(64\) 0 0
\(65\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −0.233956 1.32683i −0.233956 1.32683i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.939693 0.342020i −0.939693 0.342020i
\(82\) 0 0
\(83\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 1.43969 + 2.49362i 1.43969 + 2.49362i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −0.347296 −0.347296
\(92\) 0 0
\(93\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 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 0
\(105\) 0.407604 0.342020i 0.407604 0.342020i
\(106\) 0 0
\(107\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(110\) 0 0
\(111\) −1.17365 + 0.984808i −1.17365 + 0.984808i
\(112\) 0 0
\(113\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(118\) 0 0
\(119\) −0.326352 + 0.565258i −0.326352 + 0.565258i
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.532089 0.532089
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −0.326352 1.85083i −0.326352 1.85083i
\(130\) 0 0
\(131\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.766044 1.32683i 0.766044 1.32683i
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(140\) 0 0
\(141\) −0.326352 1.85083i −0.326352 1.85083i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.826352 0.300767i −0.826352 0.300767i
\(148\) 0 0
\(149\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(152\) 0 0
\(153\) −0.326352 + 1.85083i −0.326352 + 1.85083i
\(154\) 0 0
\(155\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(156\) 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 0
\(161\) 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 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0.233956 + 0.405223i 0.233956 + 0.405223i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.17365 2.03282i −1.17365 2.03282i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.347296 0.347296
\(190\) 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
\(195\) −0.266044 1.50881i −0.266044 1.50881i
\(196\) 0 0
\(197\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(212\) 0 0
\(213\) 0.266044 0.223238i 0.266044 0.223238i
\(214\) 0 0
\(215\) 2.87939 2.87939
\(216\) 0 0
\(217\) −0.347296 −0.347296
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(222\) 0 0
\(223\) −0.939693 + 1.62760i −0.939693 + 1.62760i −0.173648 + 0.984808i \(0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(224\) 0 0
\(225\) 1.03209 + 0.866025i 1.03209 + 0.866025i
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(234\) 0 0
\(235\) 2.87939 2.87939
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(240\) 0 0
\(241\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) 0.939693 0.342020i 0.939693 0.342020i
\(244\) 0 0
\(245\) 0.673648 1.16679i 0.673648 1.16679i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 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
\(255\) −2.70574 0.984808i −2.70574 0.984808i
\(256\) 0 0
\(257\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(258\) 0 0
\(259\) 0.266044 0.460802i 0.266044 0.460802i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 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
\(271\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(272\) 0 0
\(273\) 0.266044 0.223238i 0.266044 0.223238i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(280\) 0 0
\(281\) 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 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.53209 2.53209
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.173648 0.300767i −0.173648 0.300767i 0.766044 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.326352 + 0.565258i 0.326352 + 0.565258i
\(302\) 0 0
\(303\) 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 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(314\) 0 0
\(315\) −0.0923963 + 0.524005i −0.0923963 + 0.524005i
\(316\) 0 0
\(317\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 1.53209 1.28558i 1.53209 1.28558i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.34730 1.34730
\(326\) 0 0
\(327\) 1.43969 1.20805i 1.43969 1.20805i
\(328\) 0 0
\(329\) 0.326352 + 0.565258i 0.326352 + 0.565258i
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0.266044 1.50881i 0.266044 1.50881i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.173648 0.300767i −0.173648 0.300767i 0.766044 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(338\) 0 0
\(339\) −0.939693 0.342020i −0.939693 0.342020i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.652704 0.652704
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(348\) 0 0
\(349\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(350\) 0 0
\(351\) 0.500000 0.866025i 0.500000 0.866025i
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0.266044 + 0.460802i 0.266044 + 0.460802i
\(356\) 0 0
\(357\) −0.113341 0.642788i −0.113341 0.642788i
\(358\) 0 0
\(359\) 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
\(363\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(364\) 0 0
\(365\) 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
\(370\) 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
\(375\) −0.407604 + 0.342020i −0.407604 + 0.342020i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.43969 + 1.20805i 1.43969 + 1.20805i
\(388\) 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 0
\(393\) 1.76604 + 0.642788i 1.76604 + 0.642788i
\(394\) 0 0
\(395\) 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
\(400\) 0 0
\(401\) 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
\(405\) 0.266044 + 1.50881i 0.266044 + 1.50881i
\(406\) 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 0
\(417\) −0.326352 0.118782i −0.326352 0.118782i
\(418\) 0 0
\(419\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(420\) 0 0
\(421\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(422\) 0 0
\(423\) 1.43969 + 1.20805i 1.43969 + 1.20805i
\(424\) 0 0
\(425\) 1.26604 2.19285i 1.26604 2.19285i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(432\) 0 0
\(433\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 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
\(441\) 0.826352 0.300767i 0.826352 0.300767i
\(442\) 0 0
\(443\) 0.173648 0.300767i 0.173648 0.300767i −0.766044 0.642788i \(-0.777778\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.939693 0.342020i −0.939693 0.342020i
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.266044 + 1.50881i 0.266044 + 1.50881i
\(454\) 0 0
\(455\) 0.266044 + 0.460802i 0.266044 + 0.460802i
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) −0.939693 1.62760i −0.939693 1.62760i
\(460\) 0 0
\(461\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(462\) 0 0
\(463\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −0.266044 1.50881i −0.266044 1.50881i
\(466\) 0 0
\(467\) 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 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(480\) 0 0
\(481\) −0.766044 1.32683i −0.766044 1.32683i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.0603074 + 0.104455i −0.0603074 + 0.104455i
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 0 0
\(501\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.173648 0.984808i −0.173648 0.984808i
\(508\) 0 0
\(509\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(522\) 0 0
\(523\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) 0 0
\(525\) −0.439693 0.160035i −0.439693 0.160035i
\(526\) 0 0
\(527\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(528\) 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
\(535\) 1.53209 + 2.65366i 1.53209 + 2.65366i
\(536\) 0 0
\(537\) 0.266044 0.223238i 0.266044 0.223238i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.43969 + 2.49362i 1.43969 + 2.49362i
\(546\) 0 0
\(547\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.20574 + 0.802823i 2.20574 + 0.802823i
\(556\) 0 0
\(557\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(558\) 0 0
\(559\) 1.87939 1.87939
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(564\) 0 0
\(565\) 0.766044 1.32683i 0.766044 1.32683i
\(566\) 0 0
\(567\) −0.266044 + 0.223238i −0.266044 + 0.223238i
\(568\) 0 0
\(569\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(570\) 0 0
\(571\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(572\) 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 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.17365 + 0.984808i 1.17365 + 0.984808i
\(586\) 0 0
\(587\) 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
\(591\) −0.266044 + 0.223238i −0.266044 + 0.223238i
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 1.00000 1.00000
\(596\) 0 0
\(597\) 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
\(601\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(606\) 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
\(611\) 1.87939 1.87939
\(612\) 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 \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.266044 + 0.460802i 0.266044 + 0.460802i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.87939 −2.87939
\(630\) 0 0
\(631\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(632\) 0 0
\(633\) −1.43969 0.524005i −1.43969 0.524005i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.439693 0.761570i 0.439693 0.761570i
\(638\) 0 0
\(639\) −0.0603074 + 0.342020i −0.0603074 + 0.342020i
\(640\) 0 0
\(641\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(642\) 0 0
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) −2.20574 + 1.85083i −2.20574 + 1.85083i
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.266044 0.223238i 0.266044 0.223238i
\(652\) 0 0
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) −1.43969 + 2.49362i −1.43969 + 2.49362i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −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
\(663\) −1.76604 0.642788i −1.76604 0.642788i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.326352 1.85083i −0.326352 1.85083i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(674\) 0 0
\(675\) −1.34730 −1.34730
\(676\) 0 0
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.43969 + 0.524005i 1.43969 + 0.524005i
\(688\) 0 0
\(689\) 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
\(694\) 0 0
\(695\) 0.266044 0.460802i 0.266044 0.460802i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −0.266044 + 0.223238i −0.266044 + 0.223238i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −2.20574 + 1.85083i −2.20574 + 1.85083i
\(706\) 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 \(0\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.76604 + 0.642788i 1.76604 + 0.642788i
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 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 \(0.666667\pi\)
\(728\) 0 0
\(729\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(730\) 0 0
\(731\) 1.76604 3.05888i 1.76604 3.05888i
\(732\) 0 0
\(733\) −0.173648 0.300767i −0.173648 0.300767i 0.766044 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(734\) 0 0
\(735\) 0.233956 + 1.32683i 0.233956 + 1.32683i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(744\) 0 0
\(745\) 0.766044 1.32683i 0.766044 1.32683i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.347296 + 0.601535i −0.347296 + 0.601535i
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0 0
\(753\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(754\) 0 0
\(755\) −2.34730 −2.34730
\(756\) 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
\(763\) −0.326352 + 0.565258i −0.326352 + 0.565258i
\(764\) 0 0
\(765\) 2.70574 0.984808i 2.70574 0.984808i
\(766\) 0 0
\(767\) 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
\(771\) 1.43969 + 0.524005i 1.43969 + 0.524005i
\(772\) 0 0
\(773\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(774\) 0 0
\(775\) 1.34730 1.34730
\(776\) 0 0
\(777\) 0.0923963 + 0.524005i 0.0923963 + 0.524005i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.347296 0.347296
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 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
\(799\) 1.76604 3.05888i 1.76604 3.05888i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)