Properties

Label 1045.1.br.a.18.1
Level $1045$
Weight $1$
Character 1045.18
Analytic conductor $0.522$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -19
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.br (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.521522938201\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

Embedding invariants

Embedding label 18.1
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1045.18
Dual form 1045.1.br.a.987.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.951057 + 0.309017i) q^{4} +(-0.809017 - 0.587785i) q^{5} +(-0.412215 + 0.809017i) q^{7} +(0.587785 - 0.809017i) q^{9} +O(q^{10})\) \(q+(-0.951057 + 0.309017i) q^{4} +(-0.809017 - 0.587785i) q^{5} +(-0.412215 + 0.809017i) q^{7} +(0.587785 - 0.809017i) q^{9} +(0.951057 + 0.309017i) q^{11} +(0.809017 - 0.587785i) q^{16} +(0.309017 + 0.0489435i) q^{17} +(0.309017 - 0.951057i) q^{19} +(0.951057 + 0.309017i) q^{20} +(1.39680 + 1.39680i) q^{23} +(0.309017 + 0.951057i) q^{25} +(0.142040 - 0.896802i) q^{28} +(0.809017 - 0.412215i) q^{35} +(-0.309017 + 0.951057i) q^{36} +(1.39680 - 1.39680i) q^{43} -1.00000 q^{44} +(-0.951057 + 0.309017i) q^{45} +(0.142040 + 0.278768i) q^{47} +(0.103198 + 0.142040i) q^{49} +(-0.587785 - 0.809017i) q^{55} +(-0.951057 - 1.30902i) q^{61} +(0.412215 + 0.809017i) q^{63} +(-0.587785 + 0.809017i) q^{64} +(-0.309017 + 0.0489435i) q^{68} +(-1.26007 - 0.642040i) q^{73} +1.00000i q^{76} +(-0.642040 + 0.642040i) q^{77} -1.00000 q^{80} +(-0.309017 - 0.951057i) q^{81} +(-0.278768 + 1.76007i) q^{83} +(-0.221232 - 0.221232i) q^{85} +(-1.76007 - 0.896802i) q^{92} +(-0.809017 + 0.587785i) q^{95} +(0.809017 - 0.587785i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 8 q^{7} + 2 q^{16} - 2 q^{17} - 2 q^{19} + 2 q^{23} - 2 q^{25} - 2 q^{28} + 2 q^{35} + 2 q^{36} + 2 q^{43} - 8 q^{44} - 2 q^{47} + 10 q^{49} + 8 q^{63} + 2 q^{68} + 2 q^{73} - 2 q^{77} - 8 q^{80} + 2 q^{81} - 2 q^{83} - 2 q^{85} - 2 q^{92} - 2 q^{95} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(3\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(4\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(5\) −0.809017 0.587785i −0.809017 0.587785i
\(6\) 0 0
\(7\) −0.412215 + 0.809017i −0.412215 + 0.809017i 0.587785 + 0.809017i \(0.300000\pi\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0.587785 0.809017i 0.587785 0.809017i
\(10\) 0 0
\(11\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(12\) 0 0
\(13\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.809017 0.587785i 0.809017 0.587785i
\(17\) 0.309017 + 0.0489435i 0.309017 + 0.0489435i 0.309017 0.951057i \(-0.400000\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0.309017 0.951057i 0.309017 0.951057i
\(20\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(21\) 0 0
\(22\) 0 0
\(23\) 1.39680 + 1.39680i 1.39680 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(24\) 0 0
\(25\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.142040 0.896802i 0.142040 0.896802i
\(29\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.809017 0.412215i 0.809017 0.412215i
\(36\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(37\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(42\) 0 0
\(43\) 1.39680 1.39680i 1.39680 1.39680i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(44\) −1.00000 −1.00000
\(45\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(46\) 0 0
\(47\) 0.142040 + 0.278768i 0.142040 + 0.278768i 0.951057 0.309017i \(-0.100000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) 0 0
\(49\) 0.103198 + 0.142040i 0.103198 + 0.142040i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(54\) 0 0
\(55\) −0.587785 0.809017i −0.587785 0.809017i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(60\) 0 0
\(61\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(62\) 0 0
\(63\) 0.412215 + 0.809017i 0.412215 + 0.809017i
\(64\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) 0 0
\(73\) −1.26007 0.642040i −1.26007 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.00000i 1.00000i
\(77\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(78\) 0 0
\(79\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) −1.00000 −1.00000
\(81\) −0.309017 0.951057i −0.309017 0.951057i
\(82\) 0 0
\(83\) −0.278768 + 1.76007i −0.278768 + 1.76007i 0.309017 + 0.951057i \(0.400000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) 0 0
\(85\) −0.221232 0.221232i −0.221232 0.221232i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.76007 0.896802i −1.76007 0.896802i
\(93\) 0 0
\(94\) 0 0
\(95\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(96\) 0 0
\(97\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(98\) 0 0
\(99\) 0.809017 0.587785i 0.809017 0.587785i
\(100\) −0.587785 0.809017i −0.587785 0.809017i
\(101\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(102\) 0 0
\(103\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.142040 + 0.896802i 0.142040 + 0.896802i
\(113\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(114\) 0 0
\(115\) −0.309017 1.95106i −0.309017 1.95106i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.166977 + 0.229825i −0.166977 + 0.229825i
\(120\) 0 0
\(121\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.309017 0.951057i 0.309017 0.951057i
\(126\) 0 0
\(127\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(132\) 0 0
\(133\) 0.642040 + 0.642040i 0.642040 + 0.642040i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.278768 + 1.76007i −0.278768 + 1.76007i 0.309017 + 0.951057i \(0.400000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(138\) 0 0
\(139\) 0.363271 + 1.11803i 0.363271 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(140\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000i 1.00000i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(152\) 0 0
\(153\) 0.221232 0.221232i 0.221232 0.221232i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.809017 + 1.58779i 0.809017 + 1.58779i 0.809017 + 0.587785i \(0.200000\pi\)
1.00000i \(0.500000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.70582 + 0.554254i −1.70582 + 0.554254i
\(162\) 0 0
\(163\) −0.142040 0.896802i −0.142040 0.896802i −0.951057 0.309017i \(-0.900000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(168\) 0 0
\(169\) 0.951057 0.309017i 0.951057 0.309017i
\(170\) 0 0
\(171\) −0.587785 0.809017i −0.587785 0.809017i
\(172\) −0.896802 + 1.76007i −0.896802 + 1.76007i
\(173\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(174\) 0 0
\(175\) −0.896802 0.142040i −0.896802 0.142040i
\(176\) 0.951057 0.309017i 0.951057 0.309017i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(180\) 0.809017 0.587785i 0.809017 0.587785i
\(181\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.278768 + 0.142040i 0.278768 + 0.142040i
\(188\) −0.221232 0.221232i −0.221232 0.221232i
\(189\) 0 0
\(190\) 0 0
\(191\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.142040 0.103198i −0.142040 0.103198i
\(197\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.95106 0.309017i 1.95106 0.309017i
\(208\) 0 0
\(209\) 0.587785 0.809017i 0.587785 0.809017i
\(210\) 0 0
\(211\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.95106 + 0.309017i −1.95106 + 0.309017i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(224\) 0 0
\(225\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(226\) 0 0
\(227\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(228\) 0 0
\(229\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.76007 + 0.278768i −1.76007 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(234\) 0 0
\(235\) 0.0489435 0.309017i 0.0489435 0.309017i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(245\) 0.175571i 0.175571i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(252\) −0.642040 0.642040i −0.642040 0.642040i
\(253\) 0.896802 + 1.76007i 0.896802 + 1.76007i
\(254\) 0 0
\(255\) 0 0
\(256\) 0.309017 0.951057i 0.309017 0.951057i
\(257\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(270\) 0 0
\(271\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(272\) 0.278768 0.142040i 0.278768 0.142040i
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000i 1.00000i
\(276\) 0 0
\(277\) −0.221232 1.39680i −0.221232 1.39680i −0.809017 0.587785i \(-0.800000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(282\) 0 0
\(283\) −0.809017 1.58779i −0.809017 1.58779i −0.809017 0.587785i \(-0.800000\pi\)
1.00000i \(-0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.857960 0.278768i −0.857960 0.278768i
\(290\) 0 0
\(291\) 0 0
\(292\) 1.39680 + 0.221232i 1.39680 + 0.221232i
\(293\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.554254 + 1.70582i 0.554254 + 1.70582i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.309017 0.951057i −0.309017 0.951057i
\(305\) 1.61803i 1.61803i
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0.412215 0.809017i 0.412215 0.809017i
\(309\) 0 0
\(310\) 0 0
\(311\) −0.363271 + 1.11803i −0.363271 + 1.11803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(312\) 0 0
\(313\) −0.309017 0.0489435i −0.309017 0.0489435i 1.00000i \(-0.5\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(314\) 0 0
\(315\) 0.142040 0.896802i 0.142040 0.896802i
\(316\) 0 0
\(317\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.951057 0.309017i 0.951057 0.309017i
\(321\) 0 0
\(322\) 0 0
\(323\) 0.142040 0.278768i 0.142040 0.278768i
\(324\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.284079 −0.284079
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −0.278768 1.76007i −0.278768 1.76007i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0.278768 + 0.142040i 0.278768 + 0.142040i
\(341\) 0 0
\(342\) 0 0
\(343\) −1.05425 + 0.166977i −1.05425 + 0.166977i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.95106 + 0.309017i 1.95106 + 0.309017i 1.00000 \(0\)
0.951057 + 0.309017i \(0.100000\pi\)
\(348\) 0 0
\(349\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.642040 0.642040i −0.642040 0.642040i 0.309017 0.951057i \(-0.400000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.363271 1.11803i −0.363271 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(360\) 0 0
\(361\) −0.809017 0.587785i −0.809017 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.642040 + 1.26007i 0.642040 + 1.26007i
\(366\) 0 0
\(367\) −1.58779 0.809017i −1.58779 0.809017i −0.587785 0.809017i \(-0.700000\pi\)
−1.00000 \(\pi\)
\(368\) 1.95106 + 0.309017i 1.95106 + 0.309017i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(380\) 0.587785 0.809017i 0.587785 0.809017i
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(384\) 0 0
\(385\) 0.896802 0.142040i 0.896802 0.142040i
\(386\) 0 0
\(387\) −0.309017 1.95106i −0.309017 1.95106i
\(388\) 0 0
\(389\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(390\) 0 0
\(391\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(397\) 0.221232 0.221232i 0.221232 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(401\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.190983 0.587785i 0.190983 0.587785i
\(405\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.26007 1.26007i 1.26007 1.26007i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(420\) 0 0
\(421\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(422\) 0 0
\(423\) 0.309017 + 0.0489435i 0.309017 + 0.0489435i
\(424\) 0 0
\(425\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i
\(426\) 0 0
\(427\) 1.45106 0.229825i 1.45106 0.229825i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(432\) 0 0
\(433\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.76007 0.896802i 1.76007 0.896802i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0.175571 0.175571
\(442\) 0 0
\(443\) 0.809017 0.412215i 0.809017 0.412215i 1.00000i \(-0.5\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.412215 0.809017i −0.412215 0.809017i
\(449\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.76007 0.278768i −1.76007 0.278768i −0.809017 0.587785i \(-0.800000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0.896802 + 1.76007i 0.896802 + 1.76007i
\(461\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(462\) 0 0
\(463\) 1.26007 + 1.26007i 1.26007 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.142040 + 0.896802i −0.142040 + 0.896802i 0.809017 + 0.587785i \(0.200000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.76007 0.896802i 1.76007 0.896802i
\(474\) 0 0
\(475\) 1.00000 1.00000
\(476\) 0.0877853 0.270175i 0.0877853 0.270175i
\(477\) 0 0
\(478\) 0 0
\(479\) −0.951057 + 0.690983i −0.951057 + 0.690983i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.951057 0.309017i −0.951057 0.309017i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.00000 −1.00000
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.587785 + 0.190983i −0.587785 + 0.190983i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(0.5\pi\)
\(500\) 1.00000i 1.00000i
\(501\) 0 0
\(502\) 0 0
\(503\) 0.642040 + 1.26007i 0.642040 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(504\) 0 0
\(505\) 0.587785 0.190983i 0.587785 0.190983i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(510\) 0 0
\(511\) 1.03884 0.754763i 1.03884 0.754763i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(522\) 0 0
\(523\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(524\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 2.90211i 2.90211i
\(530\) 0 0
\(531\) 0 0
\(532\) −0.809017 0.412215i −0.809017 0.412215i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.0542543 + 0.166977i 0.0542543 + 0.166977i
\(540\) 0 0
\(541\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(548\) −0.278768 1.76007i −0.278768 1.76007i
\(549\) −1.61803 −1.61803
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.690983 0.951057i −0.690983 0.951057i
\(557\) 0.809017 1.58779i 0.809017 1.58779i 1.00000i \(-0.5\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.412215 0.809017i 0.412215 0.809017i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.896802 + 0.142040i 0.896802 + 0.142040i
\(568\) 0 0
\(569\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(570\) 0 0
\(571\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.896802 + 1.76007i −0.896802 + 1.76007i
\(576\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(577\) 0.221232 1.39680i 0.221232 1.39680i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.30902 0.951057i −1.30902 0.951057i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.26007 0.642040i −1.26007 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.39680 + 1.39680i −1.39680 + 1.39680i −0.587785 + 0.809017i \(0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(594\) 0 0
\(595\) 0.270175 0.0877853i 0.270175 0.0877853i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(600\) 0 0
\(601\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.309017 0.951057i −0.309017 0.951057i
\(606\) 0 0
\(607\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −0.142040 + 0.278768i −0.142040 + 0.278768i
\(613\) −0.896802 1.76007i −0.896802 1.76007i −0.587785 0.809017i \(-0.700000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(618\) 0 0
\(619\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.26007 1.26007i −1.26007 1.26007i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(642\) 0 0
\(643\) −1.95106 0.309017i −1.95106 0.309017i −0.951057 0.309017i \(-0.900000\pi\)
−1.00000 \(\pi\)
\(644\) 1.45106 1.05425i 1.45106 1.05425i
\(645\) 0 0
\(646\) 0 0
\(647\) 1.39680 0.221232i 1.39680 0.221232i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.412215 + 0.809017i 0.412215 + 0.809017i
\(653\) −0.896802 + 1.76007i −0.896802 + 1.76007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(654\) 0 0
\(655\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(656\) 0 0
\(657\) −1.26007 + 0.642040i −1.26007 + 0.642040i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.142040 0.896802i −0.142040 0.896802i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.500000 1.53884i −0.500000 1.53884i
\(672\) 0 0
\(673\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(677\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(685\) 1.26007 1.26007i 1.26007 1.26007i
\(686\) 0 0
\(687\) 0 0
\(688\) 0.309017 1.95106i 0.309017 1.95106i
\(689\) 0 0
\(690\) 0 0
\(691\) −1.53884 1.11803i −1.53884 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(692\) 0 0
\(693\) 0.142040 + 0.896802i 0.142040 + 0.896802i
\(694\) 0 0
\(695\) 0.363271 1.11803i 0.363271 1.11803i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.896802 0.142040i 0.896802 0.142040i
\(701\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(705\) 0 0
\(706\) 0 0
\(707\) −0.254763 0.500000i −0.254763 0.500000i
\(708\) 0 0
\(709\) −1.11803 1.53884i −1.11803 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(710\) 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
\(719\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(720\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.39680 + 1.39680i −1.39680 + 1.39680i −0.587785 + 0.809017i \(0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(728\) 0 0
\(729\) −0.951057 0.309017i −0.951057 0.309017i
\(730\) 0 0
\(731\) 0.500000 0.363271i 0.500000 0.363271i
\(732\) 0 0
\(733\) −1.58779 0.809017i −1.58779 0.809017i −0.587785 0.809017i \(-0.700000\pi\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.26007 + 1.26007i 1.26007 + 1.26007i
\(748\) −0.309017 0.0489435i −0.309017 0.0489435i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(752\) 0.278768 + 0.142040i 0.278768 + 0.142040i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.39680 + 0.221232i −1.39680 + 0.221232i −0.809017 0.587785i \(-0.800000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.17557 + 1.61803i −1.17557 + 1.61803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(765\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.90211 1.90211 0.951057 0.309017i \(-0.100000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.166977 + 0.0542543i 0.166977 + 0.0542543i
\(785\) 0.278768 1.76007i 0.278768 1.76007i
\(786\) 0 0
\(787\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(788\) 1.26007 + 0.642040i 1.26007 + 0.642040i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(797\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(798\) 0 0
\(799\) 0.0302487 + 0.0930960i 0.0302487 + 0.0930960i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\)