Properties

Label 1840.1.bq.a.719.2
Level $1840$
Weight $1$
Character 1840.719
Analytic conductor $0.918$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1840.bq (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
Defining polynomial: \(x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

Embedding invariants

Embedding label 719.2
Root \(0.909632 + 0.415415i\) of defining polynomial
Character \(\chi\) \(=\) 1840.719
Dual form 1840.1.bq.a.1039.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.74557 + 0.512546i) q^{3} +(-0.841254 + 0.540641i) q^{5} +(0.281733 - 1.95949i) q^{7} +(1.94306 + 1.24873i) q^{9} +O(q^{10})\) \(q+(1.74557 + 0.512546i) q^{3} +(-0.841254 + 0.540641i) q^{5} +(0.281733 - 1.95949i) q^{7} +(1.94306 + 1.24873i) q^{9} +(-1.74557 + 0.512546i) q^{15} +(1.49611 - 3.27603i) q^{21} +(0.909632 + 0.415415i) q^{23} +(0.415415 - 0.909632i) q^{25} +(1.56036 + 1.80075i) q^{27} +(-1.10181 + 1.27155i) q^{29} +(0.822373 + 1.80075i) q^{35} +(0.239446 - 0.153882i) q^{41} +(-1.45027 - 0.425839i) q^{43} -2.30972 q^{45} +1.08128 q^{47} +(-2.80075 - 0.822373i) q^{49} +(-1.25667 + 0.368991i) q^{61} +(2.99430 - 3.45561i) q^{63} +(-0.234072 + 0.512546i) q^{67} +(1.37491 + 1.19136i) q^{69} +(1.19136 - 1.37491i) q^{75} +(0.841254 + 1.84209i) q^{81} +(0.474017 + 0.304632i) q^{83} +(-2.57501 + 1.65486i) q^{87} +(-0.797176 - 0.234072i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{5} + 2 q^{9} + O(q^{10}) \) \( 20 q + 2 q^{5} + 2 q^{9} - 2 q^{25} - 4 q^{29} + 4 q^{41} - 24 q^{45} - 20 q^{49} + 4 q^{61} - 2 q^{81} - 4 q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{9}{11}\right)\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.74557 + 0.512546i 1.74557 + 0.512546i 0.989821 0.142315i \(-0.0454545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(4\) 0 0
\(5\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(6\) 0 0
\(7\) 0.281733 1.95949i 0.281733 1.95949i 1.00000i \(-0.5\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(8\) 0 0
\(9\) 1.94306 + 1.24873i 1.94306 + 1.24873i
\(10\) 0 0
\(11\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(12\) 0 0
\(13\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(14\) 0 0
\(15\) −1.74557 + 0.512546i −1.74557 + 0.512546i
\(16\) 0 0
\(17\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(18\) 0 0
\(19\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(20\) 0 0
\(21\) 1.49611 3.27603i 1.49611 3.27603i
\(22\) 0 0
\(23\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(24\) 0 0
\(25\) 0.415415 0.909632i 0.415415 0.909632i
\(26\) 0 0
\(27\) 1.56036 + 1.80075i 1.56036 + 1.80075i
\(28\) 0 0
\(29\) −1.10181 + 1.27155i −1.10181 + 1.27155i −0.142315 + 0.989821i \(0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(30\) 0 0
\(31\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.822373 + 1.80075i 0.822373 + 1.80075i
\(36\) 0 0
\(37\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.239446 0.153882i 0.239446 0.153882i −0.415415 0.909632i \(-0.636364\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(42\) 0 0
\(43\) −1.45027 0.425839i −1.45027 0.425839i −0.540641 0.841254i \(-0.681818\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(44\) 0 0
\(45\) −2.30972 −2.30972
\(46\) 0 0
\(47\) 1.08128 1.08128 0.540641 0.841254i \(-0.318182\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(48\) 0 0
\(49\) −2.80075 0.822373i −2.80075 0.822373i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(60\) 0 0
\(61\) −1.25667 + 0.368991i −1.25667 + 0.368991i −0.841254 0.540641i \(-0.818182\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(62\) 0 0
\(63\) 2.99430 3.45561i 2.99430 3.45561i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.234072 + 0.512546i −0.234072 + 0.512546i −0.989821 0.142315i \(-0.954545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(68\) 0 0
\(69\) 1.37491 + 1.19136i 1.37491 + 1.19136i
\(70\) 0 0
\(71\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(72\) 0 0
\(73\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(74\) 0 0
\(75\) 1.19136 1.37491i 1.19136 1.37491i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(80\) 0 0
\(81\) 0.841254 + 1.84209i 0.841254 + 1.84209i
\(82\) 0 0
\(83\) 0.474017 + 0.304632i 0.474017 + 0.304632i 0.755750 0.654861i \(-0.227273\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.57501 + 1.65486i −2.57501 + 1.65486i
\(88\) 0 0
\(89\) −0.797176 0.234072i −0.797176 0.234072i −0.142315 0.989821i \(-0.545455\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.61435 1.03748i −1.61435 1.03748i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(102\) 0 0
\(103\) 0.449181 + 0.983568i 0.449181 + 0.983568i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(104\) 0 0
\(105\) 0.512546 + 3.56484i 0.512546 + 3.56484i
\(106\) 0 0
\(107\) −1.03748 + 0.304632i −1.03748 + 0.304632i −0.755750 0.654861i \(-0.772727\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(108\) 0 0
\(109\) 0.544078 0.627899i 0.544078 0.627899i −0.415415 0.909632i \(-0.636364\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(114\) 0 0
\(115\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(122\) 0 0
\(123\) 0.496841 0.145886i 0.496841 0.145886i
\(124\) 0 0
\(125\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(126\) 0 0
\(127\) 0.234072 + 0.512546i 0.234072 + 0.512546i 0.989821 0.142315i \(-0.0454545\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(128\) 0 0
\(129\) −2.31329 1.48666i −2.31329 1.48666i
\(130\) 0 0
\(131\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.28621 0.671292i −2.28621 0.671292i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 1.88745 + 0.554206i 1.88745 + 0.554206i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.239446 1.66538i 0.239446 1.66538i
\(146\) 0 0
\(147\) −4.46740 2.87102i −4.46740 2.87102i
\(148\) 0 0
\(149\) −0.698939 1.53046i −0.698939 1.53046i −0.841254 0.540641i \(-0.818182\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(150\) 0 0
\(151\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.07028 1.66538i 1.07028 1.66538i
\(162\) 0 0
\(163\) 0.627899 1.37491i 0.627899 1.37491i −0.281733 0.959493i \(-0.590909\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.989821 + 1.14231i −0.989821 + 1.14231i 1.00000i \(0.5\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(168\) 0 0
\(169\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(174\) 0 0
\(175\) −1.66538 1.07028i −1.66538 1.07028i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(180\) 0 0
\(181\) 1.84125 + 0.540641i 1.84125 + 0.540641i 1.00000 \(0\)
0.841254 + 0.540641i \(0.181818\pi\)
\(182\) 0 0
\(183\) −2.38273 −2.38273
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.96815 2.55018i 3.96815 2.55018i
\(190\) 0 0
\(191\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(192\) 0 0
\(193\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(198\) 0 0
\(199\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(200\) 0 0
\(201\) −0.671292 + 0.774713i −0.671292 + 0.774713i
\(202\) 0 0
\(203\) 2.18119 + 2.51722i 2.18119 + 2.51722i
\(204\) 0 0
\(205\) −0.118239 + 0.258908i −0.118239 + 0.258908i
\(206\) 0 0
\(207\) 1.24873 + 1.94306i 1.24873 + 1.94306i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.45027 0.425839i 1.45027 0.425839i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.215109 1.49611i 0.215109 1.49611i −0.540641 0.841254i \(-0.681818\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(224\) 0 0
\(225\) 1.94306 1.24873i 1.94306 1.24873i
\(226\) 0 0
\(227\) 1.03748 + 0.304632i 1.03748 + 0.304632i 0.755750 0.654861i \(-0.227273\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(228\) 0 0
\(229\) 0.830830 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(234\) 0 0
\(235\) −0.909632 + 0.584585i −0.909632 + 0.584585i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(240\) 0 0
\(241\) 0.118239 + 0.258908i 0.118239 + 0.258908i 0.959493 0.281733i \(-0.0909091\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(242\) 0 0
\(243\) 0.185215 + 1.28820i 0.185215 + 1.28820i
\(244\) 0 0
\(245\) 2.80075 0.822373i 2.80075 0.822373i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.671292 + 0.774713i 0.671292 + 0.774713i
\(250\) 0 0
\(251\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.72871 + 1.09485i −3.72871 + 1.09485i
\(262\) 0 0
\(263\) −0.215109 1.49611i −0.215109 1.49611i −0.755750 0.654861i \(-0.772727\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.27155 0.817178i −1.27155 0.817178i
\(268\) 0 0
\(269\) −0.273100 + 1.89945i −0.273100 + 1.89945i 0.142315 + 0.989821i \(0.454545\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(270\) 0 0
\(271\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.10181 0.708089i 1.10181 0.708089i 0.142315 0.989821i \(-0.454545\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(282\) 0 0
\(283\) 0.258908 1.80075i 0.258908 1.80075i −0.281733 0.959493i \(-0.590909\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.234072 0.512546i −0.234072 0.512546i
\(288\) 0 0
\(289\) −0.142315 0.989821i −0.142315 0.989821i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.24302 + 2.72183i −1.24302 + 2.72183i
\(302\) 0 0
\(303\) −2.28621 2.63843i −2.28621 2.63843i
\(304\) 0 0
\(305\) 0.857685 0.989821i 0.857685 0.989821i
\(306\) 0 0
\(307\) 1.89945 0.557730i 1.89945 0.557730i 0.909632 0.415415i \(-0.136364\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(308\) 0 0
\(309\) 0.279953 + 1.94711i 0.279953 + 1.94711i
\(310\) 0 0
\(311\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(312\) 0 0
\(313\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(314\) 0 0
\(315\) −0.650724 + 4.52588i −0.650724 + 4.52588i
\(316\) 0 0
\(317\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.96714 −1.96714
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.27155 0.817178i 1.27155 0.817178i
\(328\) 0 0
\(329\) 0.304632 2.11876i 0.304632 2.11876i
\(330\) 0 0
\(331\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.0801894 0.557730i −0.0801894 0.557730i
\(336\) 0 0
\(337\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.57812 + 3.45561i −1.57812 + 3.45561i
\(344\) 0 0
\(345\) −1.80075 0.258908i −1.80075 0.258908i
\(346\) 0 0
\(347\) 0.234072 0.512546i 0.234072 0.512546i −0.755750 0.654861i \(-0.772727\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(348\) 0 0
\(349\) 0.186393 + 0.215109i 0.186393 + 0.215109i 0.841254 0.540641i \(-0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(360\) 0 0
\(361\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(362\) 0 0
\(363\) −1.53046 + 0.983568i −1.53046 + 0.983568i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.563465 0.563465 0.281733 0.959493i \(-0.409091\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(368\) 0 0
\(369\) 0.657415 0.657415
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(374\) 0 0
\(375\) −0.258908 + 1.80075i −0.258908 + 1.80075i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(380\) 0 0
\(381\) 0.145886 + 1.01466i 0.145886 + 1.01466i
\(382\) 0 0
\(383\) −1.89945 + 0.557730i −1.89945 + 0.557730i −0.909632 + 0.415415i \(0.863636\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.28621 2.63843i −2.28621 2.63843i
\(388\) 0 0
\(389\) −0.345139 + 0.755750i −0.345139 + 0.755750i 0.654861 + 0.755750i \(0.272727\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.273100 + 1.89945i 0.273100 + 1.89945i 0.415415 + 0.909632i \(0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.70362 1.09485i −1.70362 1.09485i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.41542 + 0.909632i −1.41542 + 0.909632i −0.415415 + 0.909632i \(0.636364\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −0.563465 −0.563465
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(420\) 0 0
\(421\) −0.0405070 + 0.281733i −0.0405070 + 0.281733i 0.959493 + 0.281733i \(0.0909091\pi\)
−1.00000 \(1.00000\pi\)
\(422\) 0 0
\(423\) 2.10100 + 1.35023i 2.10100 + 1.35023i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.368991 + 2.56639i 0.368991 + 2.56639i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(432\) 0 0
\(433\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(434\) 0 0
\(435\) 1.27155 2.78431i 1.27155 2.78431i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(440\) 0 0
\(441\) −4.41510 5.09530i −4.41510 5.09530i
\(442\) 0 0
\(443\) 0.989821 1.14231i 0.989821 1.14231i 1.00000i \(-0.5\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(444\) 0 0
\(445\) 0.797176 0.234072i 0.797176 0.234072i
\(446\) 0 0
\(447\) −0.435615 3.02977i −0.435615 3.02977i
\(448\) 0 0
\(449\) −0.345139 0.755750i −0.345139 0.755750i 0.654861 0.755750i \(-0.272727\pi\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.830830 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(462\) 0 0
\(463\) −1.89945 0.557730i −1.89945 0.557730i −0.989821 0.142315i \(-0.954545\pi\)
−0.909632 0.415415i \(-0.863636\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(468\) 0 0
\(469\) 0.938384 + 0.603063i 0.938384 + 0.603063i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 2.72183 2.35848i 2.72183 2.35848i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.29639 + 1.49611i 1.29639 + 1.49611i 0.755750 + 0.654861i \(0.227273\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(488\) 0 0
\(489\) 1.80075 2.07817i 1.80075 2.07817i
\(490\) 0 0
\(491\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(500\) 0 0
\(501\) −2.31329 + 1.48666i −2.31329 + 1.48666i
\(502\) 0 0
\(503\) 1.74557 + 0.512546i 1.74557 + 0.512546i 0.989821 0.142315i \(-0.0454545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(504\) 0 0
\(505\) 1.91899 1.91899
\(506\) 0 0
\(507\) −1.81926 −1.81926
\(508\) 0 0
\(509\) 1.25667 + 0.368991i 1.25667 + 0.368991i 0.841254 0.540641i \(-0.181818\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.909632 0.584585i −0.909632 0.584585i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.61435 + 0.474017i −1.61435 + 0.474017i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(522\) 0 0
\(523\) −0.368991 + 0.425839i −0.368991 + 0.425839i −0.909632 0.415415i \(-0.863636\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(524\) 0 0
\(525\) −2.35848 2.72183i −2.35848 2.72183i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0.708089 0.817178i 0.708089 0.817178i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.797176 1.74557i −0.797176 1.74557i −0.654861 0.755750i \(-0.727273\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(542\) 0 0
\(543\) 2.93694 + 1.88745i 2.93694 + 1.88745i
\(544\) 0 0
\(545\) −0.118239 + 0.822373i −0.118239 + 0.822373i
\(546\) 0 0
\(547\) 0.909632 0.584585i 0.909632 0.584585i 1.00000i \(-0.5\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(548\) 0 0
\(549\) −2.90255 0.852267i −2.90255 0.852267i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.822373 + 1.80075i 0.822373 + 1.80075i 0.540641 + 0.841254i \(0.318182\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.84657 1.12945i 3.84657 1.12945i
\(568\) 0 0
\(569\) −1.25667 + 1.45027i −1.25667 + 1.45027i −0.415415 + 0.909632i \(0.636364\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(570\) 0 0
\(571\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.755750 0.654861i 0.755750 0.654861i
\(576\) 0 0
\(577\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.730471 0.843008i 0.730471 0.843008i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.627899 + 1.37491i 0.627899 + 1.37491i 0.909632 + 0.415415i \(0.136364\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0.797176 + 0.234072i 0.797176 + 0.234072i 0.654861 0.755750i \(-0.272727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(602\) 0 0
\(603\) −1.09485 + 0.703616i −1.09485 + 0.703616i
\(604\) 0 0
\(605\) 0.142315 0.989821i 0.142315 0.989821i
\(606\) 0 0
\(607\) 1.27155 + 0.817178i 1.27155 + 0.817178i 0.989821 0.142315i \(-0.0454545\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(608\) 0 0
\(609\) 2.51722 + 5.51195i 2.51722 + 5.51195i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(614\) 0 0
\(615\) −0.339098 + 0.391340i −0.339098 + 0.391340i
\(616\) 0 0
\(617\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(618\) 0 0
\(619\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(620\) 0 0
\(621\) 0.671292 + 2.28621i 0.671292 + 2.28621i
\(622\) 0 0
\(623\) −0.683252 + 1.49611i −0.683252 + 1.49611i
\(624\) 0 0
\(625\) −0.654861 0.755750i −0.654861 0.755750i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.474017 0.304632i −0.474017 0.304632i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.61435 + 0.474017i 1.61435 + 0.474017i 0.959493 0.281733i \(-0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(642\) 0 0
\(643\) −1.51150 −1.51150 −0.755750 0.654861i \(-0.772727\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(644\) 0 0
\(645\) 2.74982 2.74982
\(646\) 0 0
\(647\) 0.540641 + 0.158746i 0.540641 + 0.158746i 0.540641 0.841254i \(-0.318182\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(660\) 0 0
\(661\) 1.10181 1.27155i 1.10181 1.27155i 0.142315 0.989821i \(-0.454545\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.53046 + 0.698939i −1.53046 + 0.698939i
\(668\) 0 0
\(669\) 1.14231 2.50132i 1.14231 2.50132i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(674\) 0 0
\(675\) 2.28621 0.671292i 2.28621 0.671292i
\(676\) 0 0
\(677\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.65486 + 1.06351i 1.65486 + 1.06351i
\(682\) 0 0
\(683\) 0.281733 1.95949i 0.281733 1.95949i 1.00000i \(-0.5\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.45027 + 0.425839i 1.45027 + 0.425839i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0.544078 + 1.19136i 0.544078 + 1.19136i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −1.88745 + 0.554206i −1.88745 + 0.554206i
\(706\) 0 0
\(707\) −2.48775 + 2.87102i −2.48775 + 2.87102i
\(708\) 0 0
\(709\) −1.30972 1.51150i −1.30972 1.51150i −0.654861 0.755750i \(-0.727273\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(710\) 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 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(720\) 0 0
\(721\) 2.05384 0.603063i 2.05384 0.603063i
\(722\) 0 0
\(723\) 0.0736930 + 0.512546i 0.0736930 + 0.512546i
\(724\) 0 0
\(725\) 0.698939 + 1.53046i 0.698939 + 1.53046i
\(726\) 0 0
\(727\) −0.909632 0.584585i −0.909632 0.584585i 1.00000i \(-0.5\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(728\) 0 0
\(729\) −0.0487549 + 0.339098i −0.0487549 + 0.339098i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(734\) 0 0
\(735\) 5.31041 5.31041
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.153882 1.07028i 0.153882 1.07028i −0.755750 0.654861i \(-0.772727\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(744\) 0 0
\(745\) 1.41542 + 0.909632i 1.41542 + 0.909632i
\(746\) 0 0
\(747\) 0.540641 + 1.18384i 0.540641 + 1.18384i
\(748\) 0 0
\(749\) 0.304632 + 2.11876i 0.304632 + 2.11876i
\(750\) 0 0
\(751\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.118239 + 0.258908i −0.118239 + 0.258908i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(762\) 0 0
\(763\) −1.07708 1.24302i −1.07708 1.24302i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.186393 + 1.29639i 0.186393 + 1.29639i 0.841254 + 0.540641i \(0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(774\) 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
\(783\) −4.00896 −4.00896
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.53046 0.983568i 1.53046 0.983568i 0.540641 0.841254i \(-0.318182\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(788\) 0 0
\(789\) 0.391340 2.72183i 0.391340 2.72183i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.25667 1.45027i −1.25667 1.45027i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.97964i 1.97964i
\(806\) 0 0
\(807\) −1.45027 + 3.17565i −1.45027 + 3.17565i
\(808\) 0 0