Properties

Label 840.1.bp.a.293.1
Level $840$
Weight $1$
Character 840.293
Analytic conductor $0.419$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 840.bp (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.419214610612\)
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 4.0.441000.2

Embedding invariants

Embedding label 293.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 840.293
Dual form 840.1.bp.a.797.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000i q^{4} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{6} -1.00000 q^{7} +(0.707107 + 0.707107i) q^{8} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000i q^{4} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{6} -1.00000 q^{7} +(0.707107 + 0.707107i) q^{8} -1.00000i q^{9} -1.00000 q^{10} +1.41421 q^{11} +(-0.707107 - 0.707107i) q^{12} +(0.707107 - 0.707107i) q^{14} +1.00000 q^{15} -1.00000 q^{16} +(0.707107 + 0.707107i) q^{18} +(0.707107 - 0.707107i) q^{20} +(-0.707107 + 0.707107i) q^{21} +(-1.00000 + 1.00000i) q^{22} +1.00000 q^{24} +1.00000i q^{25} +(-0.707107 - 0.707107i) q^{27} +1.00000i q^{28} +1.41421i q^{29} +(-0.707107 + 0.707107i) q^{30} -2.00000i q^{31} +(0.707107 - 0.707107i) q^{32} +(1.00000 - 1.00000i) q^{33} +(-0.707107 - 0.707107i) q^{35} -1.00000 q^{36} +1.00000i q^{40} -1.00000i q^{42} -1.41421i q^{44} +(0.707107 - 0.707107i) q^{45} +(-0.707107 + 0.707107i) q^{48} +1.00000 q^{49} +(-0.707107 - 0.707107i) q^{50} +1.00000 q^{54} +(1.00000 + 1.00000i) q^{55} +(-0.707107 - 0.707107i) q^{56} +(-1.00000 - 1.00000i) q^{58} -1.41421 q^{59} -1.00000i q^{60} +(1.41421 + 1.41421i) q^{62} +1.00000i q^{63} +1.00000i q^{64} +1.41421i q^{66} +1.00000 q^{70} +(0.707107 - 0.707107i) q^{72} +(-1.00000 + 1.00000i) q^{73} +(0.707107 + 0.707107i) q^{75} -1.41421 q^{77} +(-0.707107 - 0.707107i) q^{80} -1.00000 q^{81} +(0.707107 + 0.707107i) q^{84} +(1.00000 + 1.00000i) q^{87} +(1.00000 + 1.00000i) q^{88} +1.00000i q^{90} +(-1.41421 - 1.41421i) q^{93} -1.00000i q^{96} +(-1.00000 - 1.00000i) q^{97} +(-0.707107 + 0.707107i) q^{98} -1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 4 q^{10} + 4 q^{15} - 4 q^{16} - 4 q^{22} + 4 q^{24} + 4 q^{33} - 4 q^{36} + 4 q^{49} + 4 q^{54} + 4 q^{55} - 4 q^{58} + 4 q^{70} - 4 q^{73} - 4 q^{81} + 4 q^{87} + 4 q^{88} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(3\) 0.707107 0.707107i 0.707107 0.707107i
\(4\) 1.00000i 1.00000i
\(5\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(6\) 1.00000i 1.00000i
\(7\) −1.00000 −1.00000
\(8\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(9\) 1.00000i 1.00000i
\(10\) −1.00000 −1.00000
\(11\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) −0.707107 0.707107i −0.707107 0.707107i
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0.707107 0.707107i 0.707107 0.707107i
\(15\) 1.00000 1.00000
\(16\) −1.00000 −1.00000
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0.707107 0.707107i 0.707107 0.707107i
\(21\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(22\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 1.00000 1.00000
\(25\) 1.00000i 1.00000i
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.707107 0.707107i
\(28\) 1.00000i 1.00000i
\(29\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(31\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(32\) 0.707107 0.707107i 0.707107 0.707107i
\(33\) 1.00000 1.00000i 1.00000 1.00000i
\(34\) 0 0
\(35\) −0.707107 0.707107i −0.707107 0.707107i
\(36\) −1.00000 −1.00000
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000i 1.00000i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000i 1.00000i
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 1.41421i 1.41421i
\(45\) 0.707107 0.707107i 0.707107 0.707107i
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(49\) 1.00000 1.00000
\(50\) −0.707107 0.707107i −0.707107 0.707107i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 1.00000 1.00000
\(55\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(56\) −0.707107 0.707107i −0.707107 0.707107i
\(57\) 0 0
\(58\) −1.00000 1.00000i −1.00000 1.00000i
\(59\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 1.00000i 1.00000i
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(63\) 1.00000i 1.00000i
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 1.41421i 1.41421i
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.00000 1.00000
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.707107 0.707107i 0.707107 0.707107i
\(73\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(76\) 0 0
\(77\) −1.41421 −1.41421
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −0.707107 0.707107i −0.707107 0.707107i
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(88\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 1.00000i 1.00000i
\(91\) 0 0
\(92\) 0 0
\(93\) −1.41421 1.41421i −1.41421 1.41421i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000i 1.00000i
\(97\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(98\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(99\) 1.41421i 1.41421i
\(100\) 1.00000 1.00000
\(101\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(102\) 0 0
\(103\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −1.00000 −1.00000
\(106\) 0 0
\(107\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) −1.41421 −1.41421
\(111\) 0 0
\(112\) 1.00000 1.00000
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.41421 1.41421
\(117\) 0 0
\(118\) 1.00000 1.00000i 1.00000 1.00000i
\(119\) 0 0
\(120\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −2.00000 −2.00000
\(125\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(126\) −0.707107 0.707107i −0.707107 0.707107i
\(127\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(128\) −0.707107 0.707107i −0.707107 0.707107i
\(129\) 0 0
\(130\) 0 0
\(131\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(132\) −1.00000 1.00000i −1.00000 1.00000i
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000i 1.00000i
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000i 1.00000i
\(145\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(146\) 1.41421i 1.41421i
\(147\) 0.707107 0.707107i 0.707107 0.707107i
\(148\) 0 0
\(149\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(150\) −1.00000 −1.00000
\(151\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 1.00000 1.00000i 1.00000 1.00000i
\(155\) 1.41421 1.41421i 1.41421 1.41421i
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 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
\(165\) 1.41421 1.41421
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) −1.00000 −1.00000
\(169\) 1.00000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) −1.41421 −1.41421
\(175\) 1.00000i 1.00000i
\(176\) −1.41421 −1.41421
\(177\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(178\) 0 0
\(179\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) −0.707107 0.707107i −0.707107 0.707107i
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 2.00000 2.00000
\(187\) 0 0
\(188\) 0 0
\(189\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(193\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(194\) 1.41421 1.41421
\(195\) 0 0
\(196\) 1.00000i 1.00000i
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(201\) 0 0
\(202\) −1.00000 1.00000i −1.00000 1.00000i
\(203\) 1.41421i 1.41421i
\(204\) 0 0
\(205\) 0 0
\(206\) 1.41421i 1.41421i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0.707107 0.707107i 0.707107 0.707107i
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 2.00000i 2.00000i
\(215\) 0 0
\(216\) 1.00000i 1.00000i
\(217\) 2.00000i 2.00000i
\(218\) 0 0
\(219\) 1.41421i 1.41421i
\(220\) 1.00000 1.00000i 1.00000 1.00000i
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(224\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(225\) 1.00000 1.00000
\(226\) 0 0
\(227\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(232\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.41421i 1.41421i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −1.00000
\(241\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(243\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(244\) 0 0
\(245\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(246\) 0 0
\(247\) 0 0
\(248\) 1.41421 1.41421i 1.41421 1.41421i
\(249\) 0 0
\(250\) 1.00000i 1.00000i
\(251\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 1.00000 1.00000
\(253\) 0 0
\(254\) 1.41421i 1.41421i
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.41421 1.41421
\(262\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 1.41421 1.41421
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.41421i 1.41421i
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 0 0
\(279\) −2.00000 −2.00000
\(280\) 1.00000i 1.00000i
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.707107 0.707107i −0.707107 0.707107i
\(289\) 1.00000i 1.00000i
\(290\) 1.41421i 1.41421i
\(291\) −1.41421 −1.41421
\(292\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(293\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(294\) 1.00000i 1.00000i
\(295\) −1.00000 1.00000i −1.00000 1.00000i
\(296\) 0 0
\(297\) −1.00000 1.00000i −1.00000 1.00000i
\(298\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(299\) 0 0
\(300\) 0.707107 0.707107i 0.707107 0.707107i
\(301\) 0 0
\(302\) 1.41421 1.41421i 1.41421 1.41421i
\(303\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 1.41421i 1.41421i
\(309\) 1.41421i 1.41421i
\(310\) 2.00000i 2.00000i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(314\) 0 0
\(315\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(316\) 0 0
\(317\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(318\) 0 0
\(319\) 2.00000i 2.00000i
\(320\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(321\) 2.00000i 2.00000i
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0.707107 0.707107i 0.707107 0.707107i
\(337\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) 2.82843i 2.82843i
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 1.00000 1.00000i 1.00000 1.00000i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(351\) 0 0
\(352\) 1.00000 1.00000i 1.00000 1.00000i
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 1.41421i 1.41421i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.00000 1.00000i −1.00000 1.00000i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 1.00000 1.00000
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0.707107 0.707107i 0.707107 0.707107i
\(364\) 0 0
\(365\) −1.41421 −1.41421
\(366\) 0 0
\(367\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 1.00000i 1.00000i
\(376\) 0 0
\(377\) 0 0
\(378\) −1.00000 −1.00000
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 1.41421i 1.41421i
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) −1.00000 −1.00000
\(385\) −1.00000 1.00000i −1.00000 1.00000i
\(386\) 1.41421 1.41421
\(387\) 0 0
\(388\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(389\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(393\) −1.00000 1.00000i −1.00000 1.00000i
\(394\) 0 0
\(395\) 0 0
\(396\) −1.41421 −1.41421
\(397\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.41421 1.41421
\(405\) −0.707107 0.707107i −0.707107 0.707107i
\(406\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(407\) 0 0
\(408\) 0 0
\(409\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(413\) 1.41421 1.41421
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) 1.00000i 1.00000i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(433\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(434\) −1.41421 1.41421i −1.41421 1.41421i
\(435\) 1.41421i 1.41421i
\(436\) 0 0
\(437\) 0 0
\(438\) −1.00000 1.00000i −1.00000 1.00000i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 1.41421i 1.41421i
\(441\) 1.00000i 1.00000i
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.41421i 1.41421i
\(447\) −1.00000 1.00000i −1.00000 1.00000i
\(448\) 1.00000i 1.00000i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(451\) 0 0
\(452\) 0 0
\(453\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(454\) −2.00000 −2.00000
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 1.41421i 1.41421i
\(463\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(464\) 1.41421i 1.41421i
\(465\) 2.00000i 2.00000i
\(466\) 0 0
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.00000 1.00000i −1.00000 1.00000i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0.707107 0.707107i 0.707107 0.707107i
\(481\) 0 0
\(482\) −1.41421 1.41421i −1.41421 1.41421i
\(483\) 0 0
\(484\) 1.00000i 1.00000i
\(485\) 1.41421i 1.41421i
\(486\) 1.00000i 1.00000i
\(487\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(488\) 0 0
\(489\) 0 0
\(490\) −1.00000 −1.00000
\(491\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.00000 1.00000i 1.00000 1.00000i
\(496\) 2.00000i 2.00000i
\(497\) 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
\(502\) −1.00000 1.00000i −1.00000 1.00000i
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(505\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(506\) 0 0
\(507\) −0.707107 0.707107i −0.707107 0.707107i
\(508\) −1.00000 1.00000i −1.00000 1.00000i
\(509\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 1.00000 1.00000i 1.00000 1.00000i
\(512\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(513\) 0 0
\(514\) 0 0
\(515\) −1.41421 −1.41421
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) −1.41421 −1.41421
\(525\) −0.707107 0.707107i −0.707107 0.707107i
\(526\) 0 0
\(527\) 0 0
\(528\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 1.41421i 1.41421i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −2.00000 −2.00000
\(536\) 0 0
\(537\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(538\) 1.00000 1.00000i 1.00000 1.00000i
\(539\) 1.41421 1.41421
\(540\) −1.00000 −1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −1.00000 1.00000i −1.00000 1.00000i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 1.41421 1.41421i 1.41421 1.41421i
\(559\) 0 0
\(560\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(561\) 0 0
\(562\) 0 0
\(563\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 1.00000
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 1.00000
\(577\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(578\) −0.707107 0.707107i −0.707107 0.707107i
\(579\) −1.41421 −1.41421
\(580\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(581\) 0 0
\(582\) 1.00000 1.00000i 1.00000 1.00000i
\(583\) 0 0
\(584\) −1.41421 −1.41421
\(585\) 0 0
\(586\) 2.00000i 2.00000i
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) −0.707107 0.707107i −0.707107 0.707107i
\(589\) 0 0
\(590\) 1.41421 1.41421
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 1.41421 1.41421
\(595\) 0 0
\(596\) −1.41421 −1.41421
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000i 1.00000i
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.00000i 2.00000i
\(605\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(606\) −1.41421 −1.41421
\(607\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −1.00000 1.00000i −1.00000 1.00000i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −1.00000 1.00000i −1.00000 1.00000i
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) −1.00000 1.00000i −1.00000 1.00000i
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −1.41421 1.41421i −1.41421 1.41421i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 1.41421i 1.41421i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 1.00000i 1.00000i
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000i 2.00000i
\(635\) 1.41421 1.41421
\(636\) 0 0
\(637\) 0 0
\(638\) −1.41421 1.41421i −1.41421 1.41421i
\(639\) 0 0
\(640\) 1.00000i 1.00000i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −1.41421 1.41421i −1.41421 1.41421i
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 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
\(649\) −2.00000 −2.00000
\(650\) 0 0
\(651\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(652\) 0 0
\(653\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 1.00000 1.00000i 1.00000 1.00000i
\(656\) 0 0
\(657\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(658\) 0 0
\(659\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(660\) 1.41421i 1.41421i
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.41421i 1.41421i
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000i 1.00000i
\(673\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(674\) 1.41421i 1.41421i
\(675\) 0.707107 0.707107i 0.707107 0.707107i
\(676\) −1.00000 −1.00000
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(680\) 0 0
\(681\) 2.00000 2.00000
\(682\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(683\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.707107 0.707107i 0.707107 0.707107i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 1.41421i 1.41421i
\(694\) 0 0
\(695\) 0 0
\(696\) 1.41421i 1.41421i
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.00000 −1.00000
\(701\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.41421i 1.41421i
\(705\) 0 0
\(706\) 0 0
\(707\) 1.41421i 1.41421i
\(708\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.41421 1.41421
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(721\) 1.00000 1.00000i 1.00000 1.00000i
\(722\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(723\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(724\) 0 0
\(725\) −1.41421 −1.41421
\(726\) 1.00000i 1.00000i
\(727\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 1.00000 1.00000i 1.00000 1.00000i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) −1.41421 −1.41421
\(735\) 1.00000 1.00000
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 2.00000i 2.00000i
\(745\) 1.00000 1.00000i 1.00000 1.00000i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.41421 1.41421i 1.41421 1.41421i
\(750\) −0.707107 0.707107i −0.707107 0.707107i
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(754\) 0 0
\(755\) −1.41421 1.41421i −1.41421 1.41421i
\(756\) 0.707107 0.707107i 0.707107 0.707107i
\(757\) 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 \(0.5\pi\)
\(762\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.707107 0.707107i 0.707107 0.707107i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 1.41421 1.41421
\(771\) 0 0
\(772\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 2.00000 2.00000
\(776\) 1.41421i 1.41421i
\(777\) 0 0
\(778\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.00000 1.00000i 1.00000 1.00000i
\(784\) −1.00000 −1.00000
\(785\) 0 0
\(786\) 1.41421 1.41421
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 1.00000 1.00000i 1.00000 1.00000i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(801\) 0 0
\(802\) 0