Properties

Label 1560.1.cs.b.1403.2
Level $1560$
Weight $1$
Character 1560.1403
Analytic conductor $0.779$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -104
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1560.cs (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1521000.2

Embedding invariants

Embedding label 1403.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1560.1403
Dual form 1560.1.cs.b.467.2

$q$-expansion

\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{2} +1.00000 q^{3} +1.00000i q^{4} +(-0.707107 + 0.707107i) q^{5} +(0.707107 + 0.707107i) q^{6} +(-0.707107 + 0.707107i) q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+(0.707107 + 0.707107i) q^{2} +1.00000 q^{3} +1.00000i q^{4} +(-0.707107 + 0.707107i) q^{5} +(0.707107 + 0.707107i) q^{6} +(-0.707107 + 0.707107i) q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{12} +(0.707107 + 0.707107i) q^{13} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} +(-1.00000 - 1.00000i) q^{17} +(0.707107 + 0.707107i) q^{18} +(-0.707107 - 0.707107i) q^{20} +(-0.707107 + 0.707107i) q^{24} -1.00000i q^{25} +1.00000i q^{26} +1.00000 q^{27} -1.00000 q^{30} -1.41421 q^{31} +(-0.707107 - 0.707107i) q^{32} -1.41421i q^{34} +1.00000i q^{36} +(1.41421 - 1.41421i) q^{37} +(0.707107 + 0.707107i) q^{39} -1.00000i q^{40} +(1.00000 + 1.00000i) q^{43} +(-0.707107 + 0.707107i) q^{45} -1.00000 q^{48} +1.00000i q^{49} +(0.707107 - 0.707107i) q^{50} +(-1.00000 - 1.00000i) q^{51} +(-0.707107 + 0.707107i) q^{52} +(0.707107 + 0.707107i) q^{54} +(-0.707107 - 0.707107i) q^{60} +(-1.00000 - 1.00000i) q^{62} -1.00000i q^{64} -1.00000 q^{65} +(1.00000 - 1.00000i) q^{68} -1.41421i q^{71} +(-0.707107 + 0.707107i) q^{72} +2.00000 q^{74} -1.00000i q^{75} +1.00000i q^{78} +(0.707107 - 0.707107i) q^{80} +1.00000 q^{81} +1.41421 q^{85} +1.41421i q^{86} -1.00000 q^{90} -1.41421 q^{93} +(-0.707107 - 0.707107i) q^{96} +(-0.707107 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 4 q^{10} - 4 q^{16} - 4 q^{17} + 4 q^{27} - 4 q^{30} + 4 q^{43} - 4 q^{48} - 4 q^{51} - 4 q^{62} - 4 q^{65} + 4 q^{68} + 8 q^{74} + 4 q^{81} - 4 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(3\) 1.00000 1.00000
\(4\) 1.00000i 1.00000i
\(5\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(6\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(9\) 1.00000 1.00000
\(10\) −1.00000 −1.00000
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 1.00000i
\(13\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(14\) 0 0
\(15\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(16\) −1.00000 −1.00000
\(17\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(18\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −0.707107 0.707107i −0.707107 0.707107i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(25\) 1.00000i 1.00000i
\(26\) 1.00000i 1.00000i
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −1.00000 −1.00000
\(31\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) −0.707107 0.707107i −0.707107 0.707107i
\(33\) 0 0
\(34\) 1.41421i 1.41421i
\(35\) 0 0
\(36\) 1.00000i 1.00000i
\(37\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(38\) 0 0
\(39\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(40\) 1.00000i 1.00000i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −1.00000 −1.00000
\(49\) 1.00000i 1.00000i
\(50\) 0.707107 0.707107i 0.707107 0.707107i
\(51\) −1.00000 1.00000i −1.00000 1.00000i
\(52\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −0.707107 0.707107i −0.707107 0.707107i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −1.00000 1.00000i −1.00000 1.00000i
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) −1.00000 −1.00000
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 1.00000 1.00000i 1.00000 1.00000i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(72\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 2.00000 2.00000
\(75\) 1.00000i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 1.00000i 1.00000i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0.707107 0.707107i 0.707107 0.707107i
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 1.41421 1.41421
\(86\) 1.41421i 1.41421i
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −1.00000 −1.00000
\(91\) 0 0
\(92\) 0 0
\(93\) −1.41421 −1.41421
\(94\) 0 0
\(95\) 0 0
\(96\) −0.707107 0.707107i −0.707107 0.707107i
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 1.41421i 1.41421i
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) −1.00000 −1.00000
\(105\) 0 0
\(106\) 0 0
\(107\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(108\) 1.00000i 1.00000i
\(109\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(110\) 0 0
\(111\) 1.41421 1.41421i 1.41421 1.41421i
\(112\) 0 0
\(113\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000i 1.00000i
\(121\) −1.00000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.41421i 1.41421i
\(125\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0.707107 0.707107i 0.707107 0.707107i
\(129\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(130\) −0.707107 0.707107i −0.707107 0.707107i
\(131\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(136\) 1.41421 1.41421
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.00000 1.00000i 1.00000 1.00000i
\(143\) 0 0
\(144\) −1.00000 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 1.00000i
\(148\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(149\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0.707107 0.707107i 0.707107 0.707107i
\(151\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(152\) 0 0
\(153\) −1.00000 1.00000i −1.00000 1.00000i
\(154\) 0 0
\(155\) 1.00000 1.00000i 1.00000 1.00000i
\(156\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.00000 1.00000
\(161\) 0 0
\(162\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(171\) 0 0
\(172\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(180\) −0.707107 0.707107i −0.707107 0.707107i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.00000i 2.00000i
\(186\) −1.00000 1.00000i −1.00000 1.00000i
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 1.00000i
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) −1.00000 −1.00000
\(196\) −1.00000 −1.00000
\(197\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 1.00000 1.00000i 1.00000 1.00000i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −0.707107 0.707107i −0.707107 0.707107i
\(209\) 0 0
\(210\) 0 0
\(211\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 1.41421i 1.41421i
\(214\) 1.41421i 1.41421i
\(215\) −1.41421 −1.41421
\(216\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(217\) 0 0
\(218\) 1.00000 1.00000i 1.00000 1.00000i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.41421i 1.41421i
\(222\) 2.00000 2.00000
\(223\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(224\) 0 0
\(225\) 1.00000i 1.00000i
\(226\) 1.41421 1.41421
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(234\) 1.00000i 1.00000i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0.707107 0.707107i 0.707107 0.707107i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −0.707107 0.707107i −0.707107 0.707107i
\(243\) 1.00000 1.00000
\(244\) 0 0
\(245\) −0.707107 0.707107i −0.707107 0.707107i
\(246\) 0 0
\(247\) 0 0
\(248\) 1.00000 1.00000i 1.00000 1.00000i
\(249\) 0 0
\(250\) 1.00000i 1.00000i
\(251\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 1.41421 1.41421
\(256\) 1.00000 1.00000
\(257\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(258\) 1.41421i 1.41421i
\(259\) 0 0
\(260\) 1.00000i 1.00000i
\(261\) 0 0
\(262\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −1.00000 −1.00000
\(271\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(272\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 1.41421 1.41421i 1.41421 1.41421i
\(279\) −1.41421 −1.41421
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(284\) 1.41421 1.41421
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.707107 0.707107i −0.707107 0.707107i
\(289\) 1.00000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(295\) 0 0
\(296\) 2.00000i 2.00000i
\(297\) 0 0
\(298\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(299\) 0 0
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) −1.00000 1.00000i −1.00000 1.00000i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 1.41421i 1.41421i
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.41421 1.41421
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −1.00000 −1.00000
\(313\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(321\) −1.00000 1.00000i −1.00000 1.00000i
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000i 1.00000i
\(325\) 0.707107 0.707107i 0.707107 0.707107i
\(326\) 0 0
\(327\) 1.41421i 1.41421i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 1.41421 1.41421i 1.41421 1.41421i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(338\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(339\) 1.00000 1.00000i 1.00000 1.00000i
\(340\) 1.41421i 1.41421i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.41421 −1.41421
\(345\) 0 0
\(346\) 0 0
\(347\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(350\) 0 0
\(351\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(352\) 0 0
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(356\) 0 0
\(357\) 0 0
\(358\) −1.41421 1.41421i −1.41421 1.41421i
\(359\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 1.00000i 1.00000i
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) −1.00000 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(371\) 0 0
\(372\) 1.41421i 1.41421i
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(384\) 0.707107 0.707107i 0.707107 0.707107i
\(385\) 0 0
\(386\) 0 0
\(387\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −0.707107 0.707107i −0.707107 0.707107i
\(391\) 0 0
\(392\) −0.707107 0.707107i −0.707107 0.707107i
\(393\) 2.00000i 2.00000i
\(394\) 2.00000i 2.00000i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000i 1.00000i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −1.00000 1.00000i −1.00000 1.00000i
\(404\) 0 0
\(405\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(406\) 0 0
\(407\) 0 0
\(408\) 1.41421 1.41421
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000i 1.00000i
\(417\) 2.00000i 2.00000i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) −1.41421 1.41421i −1.41421 1.41421i
\(423\) 0 0
\(424\) 0 0
\(425\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(426\) 1.00000 1.00000i 1.00000 1.00000i
\(427\) 0 0
\(428\) 1.00000 1.00000i 1.00000 1.00000i
\(429\) 0 0
\(430\) −1.00000 1.00000i −1.00000 1.00000i
\(431\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) −1.00000 −1.00000
\(433\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.41421 1.41421
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 1.00000i 1.00000i
\(442\) 1.00000 1.00000i 1.00000 1.00000i
\(443\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(444\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(445\) 0 0
\(446\) 2.00000i 2.00000i
\(447\) 1.41421 1.41421
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0.707107 0.707107i 0.707107 0.707107i
\(451\) 0 0
\(452\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(453\) −1.41421 −1.41421
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(459\) −1.00000 1.00000i −1.00000 1.00000i
\(460\) 0 0
\(461\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 1.00000 1.00000i 1.00000 1.00000i
\(466\) −1.41421 −1.41421
\(467\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(468\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(469\) 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
\(478\) −1.00000 1.00000i −1.00000 1.00000i
\(479\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(480\) 1.00000 1.00000
\(481\) 2.00000 2.00000
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000i 1.00000i
\(485\) 0 0
\(486\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.00000i 1.00000i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.41421 1.41421
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(501\) 0 0
\(502\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 1.00000i
\(508\) 0 0
\(509\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(511\) 0 0
\(512\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(513\) 0 0
\(514\) 1.41421i 1.41421i
\(515\) 0 0
\(516\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0.707107 0.707107i 0.707107 0.707107i
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(524\) −2.00000 −2.00000
\(525\) 0 0
\(526\) 0 0
\(527\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.41421 1.41421
\(536\) 0 0
\(537\) −2.00000 −2.00000
\(538\) 0 0
\(539\) 0 0
\(540\) −0.707107 0.707107i −0.707107 0.707107i
\(541\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(542\) −1.00000 1.00000i −1.00000 1.00000i
\(543\) 0 0
\(544\) 1.41421i 1.41421i
\(545\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(546\) 0 0
\(547\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.00000i 2.00000i
\(556\) 2.00000 2.00000
\(557\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(558\) −1.00000 1.00000i −1.00000 1.00000i
\(559\) 1.41421i 1.41421i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.41421i 1.41421i
\(566\) 1.41421i 1.41421i
\(567\) 0 0
\(568\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000i 1.00000i
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.00000 −1.00000
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) −1.00000 −1.00000
\(589\) 0 0
\(590\) 0 0
\(591\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(592\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.41421i 1.41421i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.41421i 1.41421i
\(605\) 0.707107 0.707107i 0.707107 0.707107i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.00000 1.00000i 1.00000 1.00000i
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −0.707107 0.707107i −0.707107 0.707107i
\(625\) −1.00000 −1.00000
\(626\) 1.41421i 1.41421i
\(627\) 0 0
\(628\) 0 0
\(629\) −2.82843 −2.82843
\(630\) 0 0
\(631\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(632\) 0 0
\(633\) −2.00000 −2.00000
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(638\) 0 0
\(639\) 1.41421i 1.41421i
\(640\) 1.00000i 1.00000i
\(641\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(642\) 1.41421i 1.41421i
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) −1.41421 −1.41421
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(649\) 0 0
\(650\) 1.00000 1.00000
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 1.00000 1.00000i 1.00000 1.00000i
\(655\) −1.41421 1.41421i −1.41421 1.41421i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 0 0
\(663\) 1.41421i 1.41421i
\(664\) 0 0
\(665\) 0 0
\(666\) 2.00000 2.00000
\(667\) 0 0
\(668\) 0 0
\(669\) −1.41421 1.41421i −1.41421 1.41421i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(674\) 1.41421 1.41421
\(675\) 1.00000i 1.00000i
\(676\) −1.00000 −1.00000
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 1.41421 1.41421
\(679\) 0 0
\(680\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.41421i 1.41421i
\(688\) −1.00000 1.00000i −1.00000 1.00000i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.41421i 1.41421i
\(695\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(696\) 0 0
\(697\) 0 0
\(698\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(699\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 1.00000i 1.00000i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(710\) 1.41421i 1.41421i
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000i 2.00000i
\(717\) −1.41421 −1.41421
\(718\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0.707107 0.707107i 0.707107 0.707107i
\(721\) 0 0
\(722\) −0.707107 0.707107i −0.707107 0.707107i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −0.707107 0.707107i −0.707107 0.707107i
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 2.00000i 2.00000i
\(732\) 0 0
\(733\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) −0.707107 0.707107i −0.707107 0.707107i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −2.00000 −2.00000
\(741\) 0 0
\(742\) 0 0
\(743\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 1.00000 1.00000i 1.00000 1.00000i
\(745\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 1.00000i 1.00000i
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 2.00000i 2.00000i
\(754\) 0 0
\(755\) 1.00000 1.00000i 1.00000 1.00000i
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.41421 1.41421
\(766\) 2.00000 2.00000
\(767\) 0 0
\(768\) 1.00000 1.00000
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(772\) 0 0
\(773\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(774\) 1.41421i 1.41421i
\(775\) 1.41421i 1.41421i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000i 1.00000i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) 0 0
\(786\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0