Properties

Label 2520.1.ek.d.1909.1
Level $2520$
Weight $1$
Character 2520.1909
Analytic conductor $1.258$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2520.ek (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.25764383184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.3630312000.4

Embedding invariants

Embedding label 1909.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2520.1909
Dual form 2520.1.ek.d.829.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{7} -1.00000 q^{8} -1.00000 q^{10} +(1.50000 - 0.866025i) q^{11} -1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{20} -1.73205i q^{22} +(-0.500000 + 0.866025i) q^{25} +(-0.500000 + 0.866025i) q^{28} -1.73205i q^{29} +(-1.50000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.500000 + 0.866025i) q^{35} +(0.500000 + 0.866025i) q^{40} +(-1.50000 - 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{53} +(-1.50000 - 0.866025i) q^{55} +(0.500000 + 0.866025i) q^{56} +(-1.50000 - 0.866025i) q^{58} +(-0.500000 - 0.866025i) q^{59} +1.73205i q^{62} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{70} +(1.00000 + 1.73205i) q^{73} +(-1.50000 - 0.866025i) q^{77} +(-0.500000 + 0.866025i) q^{79} +1.00000 q^{80} -1.73205i q^{83} +(-1.50000 + 0.866025i) q^{88} -1.00000 q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - q^{5} - q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - q^{5} - q^{7} - 2q^{8} - 2q^{10} + 3q^{11} - 2q^{14} - q^{16} - q^{20} - q^{25} - q^{28} - 3q^{31} + q^{32} - q^{35} + q^{40} - 3q^{44} - q^{49} + q^{50} + q^{53} - 3q^{55} + q^{56} - 3q^{58} - q^{59} + 2q^{64} + q^{70} + 2q^{73} - 3q^{77} - q^{79} + 2q^{80} - 3q^{88} - 2q^{97} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 0.866025i 0.500000 0.866025i
\(3\) 0 0
\(4\) −0.500000 0.866025i −0.500000 0.866025i
\(5\) −0.500000 0.866025i −0.500000 0.866025i
\(6\) 0 0
\(7\) −0.500000 0.866025i −0.500000 0.866025i
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) −1.00000 −1.00000
\(11\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −1.00000 −1.00000
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(21\) 0 0
\(22\) 1.73205i 1.73205i
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(29\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(30\) 0 0
\(31\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(32\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(33\) 0 0
\(34\) 0 0
\(35\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(36\) 0 0
\(37\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.50000 0.866025i −1.50000 0.866025i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) −1.50000 0.866025i −1.50000 0.866025i
\(56\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(57\) 0 0
\(58\) −1.50000 0.866025i −1.50000 0.866025i
\(59\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 1.73205i 1.73205i
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.50000 0.866025i −1.50000 0.866025i
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 1.00000 1.00000
\(81\) 0 0
\(82\) 0 0
\(83\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.00000 1.00000
\(107\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(108\) 0 0
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(111\) 0 0
\(112\) 1.00000 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(117\) 0 0
\(118\) −1.00000 −1.00000
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.73205i 1.00000 1.73205i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(125\) 1.00000 1.00000
\(126\) 0 0
\(127\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(128\) 0.500000 0.866025i 0.500000 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.00000 1.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(146\) 2.00000 2.00000
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(155\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(159\) 0 0
\(160\) 0.500000 0.866025i 0.500000 0.866025i
\(161\) 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 0
\(166\) −1.50000 0.866025i −1.50000 0.866025i
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 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
\(175\) 1.00000 1.00000
\(176\) 1.73205i 1.73205i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(194\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(198\) 0 0
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 0.500000 0.866025i 0.500000 0.866025i
\(201\) 0 0
\(202\) 2.00000 2.00000
\(203\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(204\) 0 0
\(205\) 0 0
\(206\) −1.00000 1.73205i −1.00000 1.73205i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0.500000 0.866025i 0.500000 0.866025i
\(213\) 0 0
\(214\) −0.500000 0.866025i −0.500000 0.866025i
\(215\) 0 0
\(216\) 0 0
\(217\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(218\) 0 0
\(219\) 0 0
\(220\) 1.73205i 1.73205i
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0.500000 0.866025i 0.500000 0.866025i
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.73205i 1.73205i
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(242\) −1.00000 1.73205i −1.00000 1.73205i
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 1.50000 0.866025i 1.50000 0.866025i
\(249\) 0 0
\(250\) 0.500000 0.866025i 0.500000 0.866025i
\(251\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.50000 0.866025i −1.50000 0.866025i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −0.500000 0.866025i −0.500000 0.866025i
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0.500000 0.866025i 0.500000 0.866025i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.73205i 1.73205i
\(276\) 0 0
\(277\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0.500000 0.866025i 0.500000 0.866025i
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.500000 0.866025i
\(290\) 1.73205i 1.73205i
\(291\) 0 0
\(292\) 1.00000 1.73205i 1.00000 1.73205i
\(293\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(294\) 0 0
\(295\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −1.00000 −1.00000
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 1.73205i 1.73205i
\(309\) 0 0
\(310\) 1.50000 0.866025i 1.50000 0.866025i
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.00000 1.00000
\(317\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −1.50000 2.59808i −1.50000 2.59808i
\(320\) −0.500000 0.866025i −0.500000 0.866025i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 0.500000 0.866025i 0.500000 0.866025i
\(339\) 0 0
\(340\) 0 0
\(341\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0.500000 0.866025i 0.500000 0.866025i
\(351\) 0 0
\(352\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000 1.73205i 1.00000 1.73205i
\(366\) 0 0
\(367\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.500000 0.866025i 0.500000 0.866025i
\(372\) 0 0
\(373\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 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 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 1.73205i 1.73205i
\(386\) 1.73205i 1.73205i
\(387\) 0 0
\(388\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.500000 0.866025i 0.500000 0.866025i
\(393\) 0 0
\(394\) 1.00000 1.73205i 1.00000 1.73205i
\(395\) 1.00000 1.00000
\(396\) 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
\(400\) −0.500000 0.866025i −0.500000 0.866025i
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.00000 1.73205i 1.00000 1.73205i
\(405\) 0 0
\(406\) 1.73205i 1.73205i
\(407\) 0 0
\(408\) 0 0
\(409\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.00000 −2.00000
\(413\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(414\) 0 0
\(415\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.500000 0.866025i −0.500000 0.866025i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.00000 −1.00000
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(434\) 1.50000 0.866025i 1.50000 0.866025i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(441\) 0 0
\(442\) 0 0
\(443\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.500000 0.866025i 0.500000 0.866025i
\(447\) 0 0
\(448\) −0.500000 0.866025i −0.500000 0.866025i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 1.73205i 1.73205i
\(455\) 0 0
\(456\) 0 0
\(457\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.73205i 1.73205i
\(483\) 0 0
\(484\) −2.00000 −2.00000
\(485\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(486\) 0 0
\(487\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.500000 0.866025i 0.500000 0.866025i
\(491\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.73205i 1.73205i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) −0.500000 0.866025i −0.500000 0.866025i
\(501\) 0 0
\(502\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 1.00000 1.73205i 1.00000 1.73205i
\(506\) 0 0
\(507\) 0 0
\(508\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(509\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(510\) 0 0
\(511\) 1.00000 1.73205i 1.00000 1.73205i
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −2.00000 −2.00000
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) −1.00000 −1.00000
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) −0.500000 0.866025i −0.500000 0.866025i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1.00000 −1.00000
\(536\) 0 0
\(537\) 0 0
\(538\) −1.00000 −1.00000
\(539\) 1.73205i 1.73205i
\(540\) 0 0
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 1.50000 0.866025i 1.50000 0.866025i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000 1.00000
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.500000 0.866025i −0.500000 0.866025i
\(561\) 0 0
\(562\) 0 0
\(563\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(578\) −0.500000 0.866025i −0.500000 0.866025i
\(579\) 0 0
\(580\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(581\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(582\) 0 0
\(583\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(584\) −1.00000 1.73205i −1.00000 1.73205i
\(585\) 0 0
\(586\) −1.50000 0.866025i −1.50000 0.866025i
\(587\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(605\) −2.00000 −2.00000
\(606\) 0 0
\(607\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 1.73205i 1.73205i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0.500000 0.866025i 0.500000 0.866025i
\(633\) 0 0
\(634\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(635\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(636\) 0 0
\(637\) 0 0
\(638\) −3.00000 −3.00000
\(639\) 0 0
\(640\) −1.00000 −1.00000
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) −1.50000 0.866025i −1.50000 0.866025i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −1.00000 −1.00000
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.73205i 1.73205i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(674\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(675\) 0 0
\(676\) −0.500000 0.866025i −0.500000 0.866025i
\(677\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(680\) 0 0
\(681\) 0 0
\(682\) 1.50000 + 2.59808i 1.50000 + 2.59808i
\(683\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.500000 0.866025i 0.500000 0.866025i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 2.00000 2.00000
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.500000 0.866025i −0.500000 0.866025i
\(701\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.50000 0.866025i 1.50000 0.866025i
\(705\) 0 0
\(706\) 0 0
\(707\) 1.00000 1.73205i 1.00000 1.73205i
\(708\) 0 0
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) −2.00000 −2.00000
\(722\) 1.00000 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(726\) 0 0
\(727\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.00000 1.73205i −1.00000 1.73205i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 1.00000 1.00000
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.500000 0.866025i −0.500000 0.866025i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.00000 −1.00000
\(750\) 0 0
\(751\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(771\) 0 0
\(772\) −1.50000 0.866025i −1.50000 0.866025i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 1.73205i 1.73205i
\(776\) 1.00000 1.00000
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.500000 0.866025i −0.500000 0.866025i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) −1.00000 1.73205i −1.00000 1.73205i
\(789\) 0 0
\(790\) 0.500000 0.866025i 0.500000 0.866025i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −1.00000
\(801\) 0 0
\(802\) 0 0
\(803\) 3.00000 + 1.73205i 3.00000 + 1.73205i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0