Properties

Label 560.2.g.f.449.1
Level $560$
Weight $2$
Character 560.449
Analytic conductor $4.472$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(1.32001 + 1.32001i\) of defining polynomial
Character \(\chi\) \(=\) 560.449
Dual form 560.2.g.f.449.6

$q$-expansion

\(f(q)\) \(=\) \(q-3.12489i q^{3} +(1.32001 + 1.80487i) q^{5} -1.00000i q^{7} -6.76491 q^{9} +O(q^{10})\) \(q-3.12489i q^{3} +(1.32001 + 1.80487i) q^{5} -1.00000i q^{7} -6.76491 q^{9} -2.48486 q^{11} -4.15516i q^{13} +(5.64002 - 4.12489i) q^{15} -5.76491i q^{17} -1.60975 q^{19} -3.12489 q^{21} -7.28005i q^{23} +(-1.51514 + 4.76491i) q^{25} +11.7649i q^{27} -1.45459 q^{29} +2.24977 q^{31} +7.76491i q^{33} +(1.80487 - 1.32001i) q^{35} +6.00000i q^{37} -12.9844 q^{39} +11.2800 q^{41} +5.28005i q^{43} +(-8.92976 - 12.2098i) q^{45} +3.45459i q^{47} -1.00000 q^{49} -18.0147 q^{51} -9.21949i q^{53} +(-3.28005 - 4.48486i) q^{55} +5.03028i q^{57} -5.92007 q^{59} +5.35998 q^{61} +6.76491i q^{63} +(7.49954 - 5.48486i) q^{65} -7.52982i q^{67} -22.7493 q^{69} +4.24977 q^{71} +7.28005i q^{73} +(14.8898 + 4.73463i) q^{75} +2.48486i q^{77} +16.9844 q^{79} +16.4693 q^{81} +10.1093i q^{83} +(10.4049 - 7.60975i) q^{85} +4.54541i q^{87} +11.4693 q^{89} -4.15516 q^{91} -7.03028i q^{93} +(-2.12489 - 2.90539i) q^{95} +2.73463i q^{97} +16.8099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 8q^{9} + O(q^{10}) \) \( 6q - 8q^{9} - 14q^{11} + 18q^{15} + 8q^{19} - 2q^{21} - 10q^{25} - 6q^{29} - 20q^{31} + 2q^{35} - 10q^{39} + 36q^{41} - 28q^{45} - 6q^{49} - 42q^{51} + 12q^{55} + 12q^{59} + 48q^{61} - 22q^{65} - 36q^{69} - 8q^{71} + 40q^{75} + 34q^{79} + 30q^{81} + 14q^{85} - 10q^{91} + 4q^{95} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(3\) 3.12489i 1.80415i −0.431576 0.902077i \(-0.642042\pi\)
0.431576 0.902077i \(-0.357958\pi\)
\(4\) 0 0
\(5\) 1.32001 + 1.80487i 0.590327 + 0.807164i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −6.76491 −2.25497
\(10\) 0 0
\(11\) −2.48486 −0.749214 −0.374607 0.927184i \(-0.622222\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(12\) 0 0
\(13\) 4.15516i 1.15243i −0.817297 0.576217i \(-0.804528\pi\)
0.817297 0.576217i \(-0.195472\pi\)
\(14\) 0 0
\(15\) 5.64002 4.12489i 1.45625 1.06504i
\(16\) 0 0
\(17\) 5.76491i 1.39820i −0.715026 0.699098i \(-0.753585\pi\)
0.715026 0.699098i \(-0.246415\pi\)
\(18\) 0 0
\(19\) −1.60975 −0.369301 −0.184651 0.982804i \(-0.559115\pi\)
−0.184651 + 0.982804i \(0.559115\pi\)
\(20\) 0 0
\(21\) −3.12489 −0.681906
\(22\) 0 0
\(23\) 7.28005i 1.51799i −0.651094 0.758997i \(-0.725690\pi\)
0.651094 0.758997i \(-0.274310\pi\)
\(24\) 0 0
\(25\) −1.51514 + 4.76491i −0.303028 + 0.952982i
\(26\) 0 0
\(27\) 11.7649i 2.26416i
\(28\) 0 0
\(29\) −1.45459 −0.270110 −0.135055 0.990838i \(-0.543121\pi\)
−0.135055 + 0.990838i \(0.543121\pi\)
\(30\) 0 0
\(31\) 2.24977 0.404071 0.202035 0.979378i \(-0.435244\pi\)
0.202035 + 0.979378i \(0.435244\pi\)
\(32\) 0 0
\(33\) 7.76491i 1.35170i
\(34\) 0 0
\(35\) 1.80487 1.32001i 0.305079 0.223123i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −12.9844 −2.07917
\(40\) 0 0
\(41\) 11.2800 1.76165 0.880824 0.473444i \(-0.156990\pi\)
0.880824 + 0.473444i \(0.156990\pi\)
\(42\) 0 0
\(43\) 5.28005i 0.805200i 0.915376 + 0.402600i \(0.131893\pi\)
−0.915376 + 0.402600i \(0.868107\pi\)
\(44\) 0 0
\(45\) −8.92976 12.2098i −1.33117 1.82013i
\(46\) 0 0
\(47\) 3.45459i 0.503903i 0.967740 + 0.251952i \(0.0810724\pi\)
−0.967740 + 0.251952i \(0.918928\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −18.0147 −2.52256
\(52\) 0 0
\(53\) 9.21949i 1.26639i −0.773990 0.633197i \(-0.781742\pi\)
0.773990 0.633197i \(-0.218258\pi\)
\(54\) 0 0
\(55\) −3.28005 4.48486i −0.442281 0.604739i
\(56\) 0 0
\(57\) 5.03028i 0.666276i
\(58\) 0 0
\(59\) −5.92007 −0.770728 −0.385364 0.922765i \(-0.625924\pi\)
−0.385364 + 0.922765i \(0.625924\pi\)
\(60\) 0 0
\(61\) 5.35998 0.686275 0.343137 0.939285i \(-0.388510\pi\)
0.343137 + 0.939285i \(0.388510\pi\)
\(62\) 0 0
\(63\) 6.76491i 0.852298i
\(64\) 0 0
\(65\) 7.49954 5.48486i 0.930204 0.680313i
\(66\) 0 0
\(67\) 7.52982i 0.919914i −0.887941 0.459957i \(-0.847865\pi\)
0.887941 0.459957i \(-0.152135\pi\)
\(68\) 0 0
\(69\) −22.7493 −2.73870
\(70\) 0 0
\(71\) 4.24977 0.504355 0.252178 0.967681i \(-0.418853\pi\)
0.252178 + 0.967681i \(0.418853\pi\)
\(72\) 0 0
\(73\) 7.28005i 0.852065i 0.904708 + 0.426033i \(0.140089\pi\)
−0.904708 + 0.426033i \(0.859911\pi\)
\(74\) 0 0
\(75\) 14.8898 + 4.73463i 1.71933 + 0.546708i
\(76\) 0 0
\(77\) 2.48486i 0.283176i
\(78\) 0 0
\(79\) 16.9844 1.91089 0.955447 0.295162i \(-0.0953735\pi\)
0.955447 + 0.295162i \(0.0953735\pi\)
\(80\) 0 0
\(81\) 16.4693 1.82992
\(82\) 0 0
\(83\) 10.1093i 1.10964i 0.831971 + 0.554819i \(0.187213\pi\)
−0.831971 + 0.554819i \(0.812787\pi\)
\(84\) 0 0
\(85\) 10.4049 7.60975i 1.12857 0.825393i
\(86\) 0 0
\(87\) 4.54541i 0.487320i
\(88\) 0 0
\(89\) 11.4693 1.21574 0.607870 0.794037i \(-0.292024\pi\)
0.607870 + 0.794037i \(0.292024\pi\)
\(90\) 0 0
\(91\) −4.15516 −0.435579
\(92\) 0 0
\(93\) 7.03028i 0.729006i
\(94\) 0 0
\(95\) −2.12489 2.90539i −0.218009 0.298087i
\(96\) 0 0
\(97\) 2.73463i 0.277660i 0.990316 + 0.138830i \(0.0443341\pi\)
−0.990316 + 0.138830i \(0.955666\pi\)
\(98\) 0 0
\(99\) 16.8099 1.68945
\(100\) 0 0
\(101\) −4.57947 −0.455674 −0.227837 0.973699i \(-0.573165\pi\)
−0.227837 + 0.973699i \(0.573165\pi\)
\(102\) 0 0
\(103\) 2.48486i 0.244841i 0.992478 + 0.122420i \(0.0390656\pi\)
−0.992478 + 0.122420i \(0.960934\pi\)
\(104\) 0 0
\(105\) −4.12489 5.64002i −0.402548 0.550410i
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 9.95413 0.953433 0.476716 0.879057i \(-0.341827\pi\)
0.476716 + 0.879057i \(0.341827\pi\)
\(110\) 0 0
\(111\) 18.7493 1.77961
\(112\) 0 0
\(113\) 18.4995i 1.74029i −0.492795 0.870145i \(-0.664025\pi\)
0.492795 0.870145i \(-0.335975\pi\)
\(114\) 0 0
\(115\) 13.1396 9.60975i 1.22527 0.896114i
\(116\) 0 0
\(117\) 28.1093i 2.59870i
\(118\) 0 0
\(119\) −5.76491 −0.528468
\(120\) 0 0
\(121\) −4.82546 −0.438678
\(122\) 0 0
\(123\) 35.2489i 3.17828i
\(124\) 0 0
\(125\) −10.6001 + 3.55510i −0.948098 + 0.317978i
\(126\) 0 0
\(127\) 7.15894i 0.635253i −0.948216 0.317627i \(-0.897114\pi\)
0.948216 0.317627i \(-0.102886\pi\)
\(128\) 0 0
\(129\) 16.4995 1.45270
\(130\) 0 0
\(131\) −7.85952 −0.686689 −0.343345 0.939209i \(-0.611560\pi\)
−0.343345 + 0.939209i \(0.611560\pi\)
\(132\) 0 0
\(133\) 1.60975i 0.139583i
\(134\) 0 0
\(135\) −21.2342 + 15.5298i −1.82755 + 1.33659i
\(136\) 0 0
\(137\) 10.5601i 0.902210i 0.892471 + 0.451105i \(0.148970\pi\)
−0.892471 + 0.451105i \(0.851030\pi\)
\(138\) 0 0
\(139\) 9.79897 0.831137 0.415569 0.909562i \(-0.363583\pi\)
0.415569 + 0.909562i \(0.363583\pi\)
\(140\) 0 0
\(141\) 10.7952 0.909119
\(142\) 0 0
\(143\) 10.3250i 0.863420i
\(144\) 0 0
\(145\) −1.92007 2.62534i −0.159453 0.218023i
\(146\) 0 0
\(147\) 3.12489i 0.257736i
\(148\) 0 0
\(149\) −2.49954 −0.204770 −0.102385 0.994745i \(-0.532647\pi\)
−0.102385 + 0.994745i \(0.532647\pi\)
\(150\) 0 0
\(151\) 3.76491 0.306384 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(152\) 0 0
\(153\) 38.9991i 3.15289i
\(154\) 0 0
\(155\) 2.96972 + 4.06055i 0.238534 + 0.326151i
\(156\) 0 0
\(157\) 5.67030i 0.452539i 0.974065 + 0.226270i \(0.0726530\pi\)
−0.974065 + 0.226270i \(0.927347\pi\)
\(158\) 0 0
\(159\) −28.8099 −2.28477
\(160\) 0 0
\(161\) −7.28005 −0.573748
\(162\) 0 0
\(163\) 10.2498i 0.802824i 0.915898 + 0.401412i \(0.131480\pi\)
−0.915898 + 0.401412i \(0.868520\pi\)
\(164\) 0 0
\(165\) −14.0147 + 10.2498i −1.09104 + 0.797944i
\(166\) 0 0
\(167\) 2.95504i 0.228668i 0.993442 + 0.114334i \(0.0364734\pi\)
−0.993442 + 0.114334i \(0.963527\pi\)
\(168\) 0 0
\(169\) −4.26537 −0.328105
\(170\) 0 0
\(171\) 10.8898 0.832763
\(172\) 0 0
\(173\) 22.4049i 1.70342i −0.524017 0.851708i \(-0.675567\pi\)
0.524017 0.851708i \(-0.324433\pi\)
\(174\) 0 0
\(175\) 4.76491 + 1.51514i 0.360193 + 0.114534i
\(176\) 0 0
\(177\) 18.4995i 1.39051i
\(178\) 0 0
\(179\) 14.5601 1.08827 0.544136 0.838997i \(-0.316857\pi\)
0.544136 + 0.838997i \(0.316857\pi\)
\(180\) 0 0
\(181\) −9.67030 −0.718788 −0.359394 0.933186i \(-0.617017\pi\)
−0.359394 + 0.933186i \(0.617017\pi\)
\(182\) 0 0
\(183\) 16.7493i 1.23814i
\(184\) 0 0
\(185\) −10.8292 + 7.92007i −0.796182 + 0.582295i
\(186\) 0 0
\(187\) 14.3250i 1.04755i
\(188\) 0 0
\(189\) 11.7649 0.855771
\(190\) 0 0
\(191\) −8.01468 −0.579922 −0.289961 0.957038i \(-0.593642\pi\)
−0.289961 + 0.957038i \(0.593642\pi\)
\(192\) 0 0
\(193\) 3.46927i 0.249723i −0.992174 0.124862i \(-0.960151\pi\)
0.992174 0.124862i \(-0.0398487\pi\)
\(194\) 0 0
\(195\) −17.1396 23.4352i −1.22739 1.67823i
\(196\) 0 0
\(197\) 16.2791i 1.15984i −0.814673 0.579920i \(-0.803084\pi\)
0.814673 0.579920i \(-0.196916\pi\)
\(198\) 0 0
\(199\) 26.8704 1.90479 0.952397 0.304862i \(-0.0986102\pi\)
0.952397 + 0.304862i \(0.0986102\pi\)
\(200\) 0 0
\(201\) −23.5298 −1.65967
\(202\) 0 0
\(203\) 1.45459i 0.102092i
\(204\) 0 0
\(205\) 14.8898 + 20.3591i 1.03995 + 1.42194i
\(206\) 0 0
\(207\) 49.2489i 3.42303i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −17.2342 −1.18645 −0.593225 0.805037i \(-0.702145\pi\)
−0.593225 + 0.805037i \(0.702145\pi\)
\(212\) 0 0
\(213\) 13.2800i 0.909934i
\(214\) 0 0
\(215\) −9.52982 + 6.96972i −0.649928 + 0.475331i
\(216\) 0 0
\(217\) 2.24977i 0.152724i
\(218\) 0 0
\(219\) 22.7493 1.53726
\(220\) 0 0
\(221\) −23.9541 −1.61133
\(222\) 0 0
\(223\) 0.235091i 0.0157429i −0.999969 0.00787143i \(-0.997494\pi\)
0.999969 0.00787143i \(-0.00250558\pi\)
\(224\) 0 0
\(225\) 10.2498 32.2342i 0.683318 2.14894i
\(226\) 0 0
\(227\) 0.564792i 0.0374865i 0.999824 + 0.0187433i \(0.00596652\pi\)
−0.999824 + 0.0187433i \(0.994033\pi\)
\(228\) 0 0
\(229\) −7.11021 −0.469856 −0.234928 0.972013i \(-0.575485\pi\)
−0.234928 + 0.972013i \(0.575485\pi\)
\(230\) 0 0
\(231\) 7.76491 0.510893
\(232\) 0 0
\(233\) 8.49954i 0.556823i −0.960462 0.278412i \(-0.910192\pi\)
0.960462 0.278412i \(-0.0898080\pi\)
\(234\) 0 0
\(235\) −6.23509 + 4.56009i −0.406732 + 0.297468i
\(236\) 0 0
\(237\) 53.0743i 3.44755i
\(238\) 0 0
\(239\) −7.26537 −0.469958 −0.234979 0.972001i \(-0.575502\pi\)
−0.234979 + 0.972001i \(0.575502\pi\)
\(240\) 0 0
\(241\) 7.28005 0.468949 0.234475 0.972122i \(-0.424663\pi\)
0.234475 + 0.972122i \(0.424663\pi\)
\(242\) 0 0
\(243\) 16.1698i 1.03730i
\(244\) 0 0
\(245\) −1.32001 1.80487i −0.0843325 0.115309i
\(246\) 0 0
\(247\) 6.68876i 0.425596i
\(248\) 0 0
\(249\) 31.5904 2.00196
\(250\) 0 0
\(251\) −22.9192 −1.44664 −0.723322 0.690511i \(-0.757386\pi\)
−0.723322 + 0.690511i \(0.757386\pi\)
\(252\) 0 0
\(253\) 18.0899i 1.13730i
\(254\) 0 0
\(255\) −23.7796 32.5142i −1.48914 2.03612i
\(256\) 0 0
\(257\) 0.719953i 0.0449094i 0.999748 + 0.0224547i \(0.00714816\pi\)
−0.999748 + 0.0224547i \(0.992852\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 9.84014 0.609089
\(262\) 0 0
\(263\) 18.5601i 1.14446i −0.820092 0.572232i \(-0.806078\pi\)
0.820092 0.572232i \(-0.193922\pi\)
\(264\) 0 0
\(265\) 16.6400 12.1698i 1.02219 0.747587i
\(266\) 0 0
\(267\) 35.8401i 2.19338i
\(268\) 0 0
\(269\) 22.1698 1.35172 0.675860 0.737030i \(-0.263773\pi\)
0.675860 + 0.737030i \(0.263773\pi\)
\(270\) 0 0
\(271\) 7.87890 0.478609 0.239304 0.970945i \(-0.423081\pi\)
0.239304 + 0.970945i \(0.423081\pi\)
\(272\) 0 0
\(273\) 12.9844i 0.785852i
\(274\) 0 0
\(275\) 3.76491 11.8401i 0.227033 0.713987i
\(276\) 0 0
\(277\) 2.78051i 0.167064i −0.996505 0.0835322i \(-0.973380\pi\)
0.996505 0.0835322i \(-0.0266201\pi\)
\(278\) 0 0
\(279\) −15.2195 −0.911167
\(280\) 0 0
\(281\) −23.7044 −1.41408 −0.707042 0.707172i \(-0.749971\pi\)
−0.707042 + 0.707172i \(0.749971\pi\)
\(282\) 0 0
\(283\) 26.8439i 1.59571i 0.602852 + 0.797853i \(0.294031\pi\)
−0.602852 + 0.797853i \(0.705969\pi\)
\(284\) 0 0
\(285\) −9.07901 + 6.64002i −0.537794 + 0.393321i
\(286\) 0 0
\(287\) 11.2800i 0.665840i
\(288\) 0 0
\(289\) −16.2342 −0.954951
\(290\) 0 0
\(291\) 8.54541 0.500941
\(292\) 0 0
\(293\) 9.93475i 0.580394i 0.956967 + 0.290197i \(0.0937209\pi\)
−0.956967 + 0.290197i \(0.906279\pi\)
\(294\) 0 0
\(295\) −7.81456 10.6850i −0.454981 0.622104i
\(296\) 0 0
\(297\) 29.2342i 1.69634i
\(298\) 0 0
\(299\) −30.2498 −1.74939
\(300\) 0 0
\(301\) 5.28005 0.304337
\(302\) 0 0
\(303\) 14.3103i 0.822107i
\(304\) 0 0
\(305\) 7.07523 + 9.67408i 0.405127 + 0.553936i
\(306\) 0 0
\(307\) 32.0946i 1.83174i 0.401479 + 0.915868i \(0.368496\pi\)
−0.401479 + 0.915868i \(0.631504\pi\)
\(308\) 0 0
\(309\) 7.76491 0.441730
\(310\) 0 0
\(311\) 10.0606 0.570482 0.285241 0.958456i \(-0.407926\pi\)
0.285241 + 0.958456i \(0.407926\pi\)
\(312\) 0 0
\(313\) 20.8245i 1.17707i −0.808471 0.588536i \(-0.799704\pi\)
0.808471 0.588536i \(-0.200296\pi\)
\(314\) 0 0
\(315\) −12.2098 + 8.92976i −0.687945 + 0.503135i
\(316\) 0 0
\(317\) 15.7502i 0.884621i 0.896862 + 0.442311i \(0.145841\pi\)
−0.896862 + 0.442311i \(0.854159\pi\)
\(318\) 0 0
\(319\) 3.61445 0.202370
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.28005i 0.516356i
\(324\) 0 0
\(325\) 19.7990 + 6.29564i 1.09825 + 0.349219i
\(326\) 0 0
\(327\) 31.1055i 1.72014i
\(328\) 0 0
\(329\) 3.45459 0.190457
\(330\) 0 0
\(331\) 25.7796 1.41697 0.708487 0.705724i \(-0.249378\pi\)
0.708487 + 0.705724i \(0.249378\pi\)
\(332\) 0 0
\(333\) 40.5895i 2.22429i
\(334\) 0 0
\(335\) 13.5904 9.93945i 0.742521 0.543050i
\(336\) 0 0
\(337\) 0.0605522i 0.00329849i 0.999999 + 0.00164924i \(0.000524971\pi\)
−0.999999 + 0.00164924i \(0.999475\pi\)
\(338\) 0 0
\(339\) −57.8089 −3.13975
\(340\) 0 0
\(341\) −5.59037 −0.302736
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −30.0294 41.0596i −1.61673 2.21058i
\(346\) 0 0
\(347\) 20.9310i 1.12363i 0.827262 + 0.561817i \(0.189897\pi\)
−0.827262 + 0.561817i \(0.810103\pi\)
\(348\) 0 0
\(349\) −5.48108 −0.293396 −0.146698 0.989181i \(-0.546864\pi\)
−0.146698 + 0.989181i \(0.546864\pi\)
\(350\) 0 0
\(351\) 48.8851 2.60929
\(352\) 0 0
\(353\) 26.9239i 1.43301i 0.697581 + 0.716506i \(0.254260\pi\)
−0.697581 + 0.716506i \(0.745740\pi\)
\(354\) 0 0
\(355\) 5.60975 + 7.67030i 0.297734 + 0.407097i
\(356\) 0 0
\(357\) 18.0147i 0.953438i
\(358\) 0 0
\(359\) 6.06055 0.319864 0.159932 0.987128i \(-0.448873\pi\)
0.159932 + 0.987128i \(0.448873\pi\)
\(360\) 0 0
\(361\) −16.4087 −0.863616
\(362\) 0 0
\(363\) 15.0790i 0.791443i
\(364\) 0 0
\(365\) −13.1396 + 9.60975i −0.687756 + 0.502997i
\(366\) 0 0
\(367\) 30.7034i 1.60271i −0.598191 0.801353i \(-0.704114\pi\)
0.598191 0.801353i \(-0.295886\pi\)
\(368\) 0 0
\(369\) −76.3085 −3.97246
\(370\) 0 0
\(371\) −9.21949 −0.478652
\(372\) 0 0
\(373\) 10.8099i 0.559714i 0.960042 + 0.279857i \(0.0902870\pi\)
−0.960042 + 0.279857i \(0.909713\pi\)
\(374\) 0 0
\(375\) 11.1093 + 33.1240i 0.573681 + 1.71051i
\(376\) 0 0
\(377\) 6.04404i 0.311284i
\(378\) 0 0
\(379\) −9.28005 −0.476684 −0.238342 0.971181i \(-0.576604\pi\)
−0.238342 + 0.971181i \(0.576604\pi\)
\(380\) 0 0
\(381\) −22.3709 −1.14609
\(382\) 0 0
\(383\) 5.28005i 0.269798i −0.990859 0.134899i \(-0.956929\pi\)
0.990859 0.134899i \(-0.0430710\pi\)
\(384\) 0 0
\(385\) −4.48486 + 3.28005i −0.228570 + 0.167167i
\(386\) 0 0
\(387\) 35.7190i 1.81570i
\(388\) 0 0
\(389\) −5.48395 −0.278047 −0.139024 0.990289i \(-0.544396\pi\)
−0.139024 + 0.990289i \(0.544396\pi\)
\(390\) 0 0
\(391\) −41.9688 −2.12245
\(392\) 0 0
\(393\) 24.5601i 1.23889i
\(394\) 0 0
\(395\) 22.4196 + 30.6547i 1.12805 + 1.54241i
\(396\) 0 0
\(397\) 6.40493i 0.321454i 0.986999 + 0.160727i \(0.0513839\pi\)
−0.986999 + 0.160727i \(0.948616\pi\)
\(398\) 0 0
\(399\) 5.03028 0.251829
\(400\) 0 0
\(401\) −1.20482 −0.0601656 −0.0300828 0.999547i \(-0.509577\pi\)
−0.0300828 + 0.999547i \(0.509577\pi\)
\(402\) 0 0
\(403\) 9.34816i 0.465665i
\(404\) 0 0
\(405\) 21.7396 + 29.7249i 1.08025 + 1.47704i
\(406\) 0 0
\(407\) 14.9092i 0.739020i
\(408\) 0 0
\(409\) 2.31032 0.114238 0.0571191 0.998367i \(-0.481809\pi\)
0.0571191 + 0.998367i \(0.481809\pi\)
\(410\) 0 0
\(411\) 32.9991 1.62772
\(412\) 0 0
\(413\) 5.92007i 0.291308i
\(414\) 0 0
\(415\) −18.2460 + 13.3444i −0.895660 + 0.655050i
\(416\) 0 0
\(417\) 30.6206i 1.49950i
\(418\) 0 0
\(419\) −19.7384 −0.964285 −0.482142 0.876093i \(-0.660141\pi\)
−0.482142 + 0.876093i \(0.660141\pi\)
\(420\) 0 0
\(421\) 30.1433 1.46910 0.734548 0.678556i \(-0.237394\pi\)
0.734548 + 0.678556i \(0.237394\pi\)
\(422\) 0 0
\(423\) 23.3700i 1.13629i
\(424\) 0 0
\(425\) 27.4693 + 8.73463i 1.33246 + 0.423692i
\(426\) 0 0
\(427\) 5.35998i 0.259387i
\(428\) 0 0
\(429\) 32.2645 1.55774
\(430\) 0 0
\(431\) 4.61353 0.222226 0.111113 0.993808i \(-0.464558\pi\)
0.111113 + 0.993808i \(0.464558\pi\)
\(432\) 0 0
\(433\) 11.4399i 0.549767i 0.961478 + 0.274883i \(0.0886393\pi\)
−0.961478 + 0.274883i \(0.911361\pi\)
\(434\) 0 0
\(435\) −8.20390 + 6.00000i −0.393347 + 0.287678i
\(436\) 0 0
\(437\) 11.7190i 0.560598i
\(438\) 0 0
\(439\) 12.0294 0.574130 0.287065 0.957911i \(-0.407320\pi\)
0.287065 + 0.957911i \(0.407320\pi\)
\(440\) 0 0
\(441\) 6.76491 0.322139
\(442\) 0 0
\(443\) 14.4390i 0.686017i −0.939332 0.343009i \(-0.888554\pi\)
0.939332 0.343009i \(-0.111446\pi\)
\(444\) 0 0
\(445\) 15.1396 + 20.7006i 0.717684 + 0.981301i
\(446\) 0 0
\(447\) 7.81078i 0.369437i
\(448\) 0 0
\(449\) −19.2342 −0.907717 −0.453858 0.891074i \(-0.649953\pi\)
−0.453858 + 0.891074i \(0.649953\pi\)
\(450\) 0 0
\(451\) −28.0294 −1.31985
\(452\) 0 0
\(453\) 11.7649i 0.552764i
\(454\) 0 0
\(455\) −5.48486 7.49954i −0.257134 0.351584i
\(456\) 0 0
\(457\) 34.8780i 1.63152i 0.578388 + 0.815762i \(0.303682\pi\)
−0.578388 + 0.815762i \(0.696318\pi\)
\(458\) 0 0
\(459\) 67.8236 3.16574
\(460\) 0 0
\(461\) −15.2607 −0.710760 −0.355380 0.934722i \(-0.615649\pi\)
−0.355380 + 0.934722i \(0.615649\pi\)
\(462\) 0 0
\(463\) 24.9991i 1.16181i 0.813973 + 0.580903i \(0.197300\pi\)
−0.813973 + 0.580903i \(0.802700\pi\)
\(464\) 0 0
\(465\) 12.6888 9.28005i 0.588427 0.430352i
\(466\) 0 0
\(467\) 5.06433i 0.234349i −0.993111 0.117175i \(-0.962616\pi\)
0.993111 0.117175i \(-0.0373837\pi\)
\(468\) 0 0
\(469\) −7.52982 −0.347695
\(470\) 0 0
\(471\) 17.7190 0.816450
\(472\) 0 0
\(473\) 13.1202i 0.603267i
\(474\) 0 0
\(475\) 2.43899 7.67030i 0.111909 0.351937i
\(476\) 0 0
\(477\) 62.3690i 2.85568i
\(478\) 0 0
\(479\) 41.8089 1.91030 0.955150 0.296123i \(-0.0956938\pi\)
0.955150 + 0.296123i \(0.0956938\pi\)
\(480\) 0 0
\(481\) 24.9310 1.13675
\(482\) 0 0
\(483\) 22.7493i 1.03513i
\(484\) 0 0
\(485\) −4.93567 + 3.60975i −0.224117 + 0.163910i
\(486\) 0 0
\(487\) 21.3406i 0.967035i −0.875335 0.483517i \(-0.839359\pi\)
0.875335 0.483517i \(-0.160641\pi\)
\(488\) 0 0
\(489\) 32.0294 1.44842
\(490\) 0 0
\(491\) −8.35620 −0.377110 −0.188555 0.982063i \(-0.560380\pi\)
−0.188555 + 0.982063i \(0.560380\pi\)
\(492\) 0 0
\(493\) 8.38555i 0.377666i
\(494\) 0 0
\(495\) 22.1892 + 30.3397i 0.997331 + 1.36367i
\(496\) 0 0
\(497\) 4.24977i 0.190628i
\(498\) 0 0
\(499\) −38.8539 −1.73934 −0.869670 0.493634i \(-0.835668\pi\)
−0.869670 + 0.493634i \(0.835668\pi\)
\(500\) 0 0
\(501\) 9.23417 0.412552
\(502\) 0 0
\(503\) 17.7044i 0.789398i −0.918810 0.394699i \(-0.870849\pi\)
0.918810 0.394699i \(-0.129151\pi\)
\(504\) 0 0
\(505\) −6.04496 8.26537i −0.268997 0.367804i
\(506\) 0 0
\(507\) 13.3288i 0.591952i
\(508\) 0 0
\(509\) −7.11021 −0.315154 −0.157577 0.987507i \(-0.550368\pi\)
−0.157577 + 0.987507i \(0.550368\pi\)
\(510\) 0 0
\(511\) 7.28005 0.322050
\(512\) 0 0
\(513\) 18.9385i 0.836157i
\(514\) 0 0
\(515\) −4.48486 + 3.28005i −0.197627 + 0.144536i
\(516\) 0 0
\(517\) 8.58417i 0.377531i
\(518\) 0 0
\(519\) −70.0128 −3.07322
\(520\) 0 0
\(521\) 18.1892 0.796884 0.398442 0.917194i \(-0.369551\pi\)
0.398442 + 0.917194i \(0.369551\pi\)
\(522\) 0 0
\(523\) 13.9882i 0.611661i −0.952086 0.305830i \(-0.901066\pi\)
0.952086 0.305830i \(-0.0989340\pi\)
\(524\) 0 0
\(525\) 4.73463 14.8898i 0.206636 0.649844i
\(526\) 0 0
\(527\) 12.9697i 0.564970i
\(528\) 0 0
\(529\) −29.9991 −1.30431
\(530\) 0 0
\(531\) 40.0487 1.73797
\(532\) 0 0
\(533\) 46.8704i 2.03018i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 45.4986i 1.96341i
\(538\) 0 0
\(539\) 2.48486 0.107031
\(540\) 0 0
\(541\) −20.8245 −0.895317 −0.447659 0.894205i \(-0.647742\pi\)
−0.447659 + 0.894205i \(0.647742\pi\)
\(542\) 0 0
\(543\) 30.2186i 1.29680i
\(544\) 0 0
\(545\) 13.1396 + 17.9659i 0.562837 + 0.769576i
\(546\) 0 0
\(547\) 3.09839i 0.132478i 0.997804 + 0.0662388i \(0.0210999\pi\)
−0.997804 + 0.0662388i \(0.978900\pi\)
\(548\) 0 0
\(549\) −36.2598 −1.54753
\(550\) 0 0
\(551\) 2.34152 0.0997519
\(552\) 0 0
\(553\) 16.9844i 0.722250i
\(554\) 0 0
\(555\) 24.7493 + 33.8401i 1.05055 + 1.43643i
\(556\) 0 0
\(557\) 18.8099i 0.797000i 0.917168 + 0.398500i \(0.130469\pi\)
−0.917168 + 0.398500i \(0.869531\pi\)
\(558\) 0 0
\(559\) 21.9394 0.927940
\(560\) 0 0
\(561\) 44.7640 1.88994
\(562\) 0 0
\(563\) 28.9503i 1.22011i −0.792358 0.610056i \(-0.791147\pi\)
0.792358 0.610056i \(-0.208853\pi\)
\(564\) 0 0
\(565\) 33.3893 24.4196i 1.40470 1.02734i
\(566\) 0 0
\(567\) 16.4693i 0.691644i
\(568\) 0 0
\(569\) 29.9688 1.25636 0.628179 0.778069i \(-0.283801\pi\)
0.628179 + 0.778069i \(0.283801\pi\)
\(570\) 0 0
\(571\) −0.280964 −0.0117580 −0.00587898 0.999983i \(-0.501871\pi\)
−0.00587898 + 0.999983i \(0.501871\pi\)
\(572\) 0 0
\(573\) 25.0450i 1.04627i
\(574\) 0 0
\(575\) 34.6888 + 11.0303i 1.44662 + 0.459994i
\(576\) 0 0
\(577\) 25.9541i 1.08048i −0.841510 0.540242i \(-0.818333\pi\)
0.841510 0.540242i \(-0.181667\pi\)
\(578\) 0 0
\(579\) −10.8411 −0.450539
\(580\) 0 0
\(581\) 10.1093 0.419404
\(582\) 0 0
\(583\) 22.9092i 0.948801i
\(584\) 0 0
\(585\) −50.7337 + 37.1046i −2.09758 + 1.53409i
\(586\) 0 0
\(587\) 22.2304i 0.917547i −0.888553 0.458773i \(-0.848289\pi\)
0.888553 0.458773i \(-0.151711\pi\)
\(588\) 0 0
\(589\) −3.62156 −0.149224
\(590\) 0 0
\(591\) −50.8704 −2.09253
\(592\) 0 0
\(593\) 35.0743i 1.44033i 0.693803 + 0.720165i \(0.255934\pi\)
−0.693803 + 0.720165i \(0.744066\pi\)
\(594\) 0 0
\(595\) −7.60975 10.4049i −0.311969 0.426561i
\(596\) 0 0
\(597\) 83.9670i 3.43654i
\(598\) 0 0
\(599\) −21.7262 −0.887707 −0.443853 0.896099i \(-0.646389\pi\)
−0.443853 + 0.896099i \(0.646389\pi\)
\(600\) 0 0
\(601\) −23.9688 −0.977708 −0.488854 0.872366i \(-0.662585\pi\)
−0.488854 + 0.872366i \(0.662585\pi\)
\(602\) 0 0
\(603\) 50.9385i 2.07438i
\(604\) 0 0
\(605\) −6.36967 8.70935i −0.258964 0.354085i
\(606\) 0 0
\(607\) 43.3241i 1.75847i 0.476388 + 0.879235i \(0.341946\pi\)
−0.476388 + 0.879235i \(0.658054\pi\)
\(608\) 0 0
\(609\) 4.54541 0.184189
\(610\) 0 0
\(611\) 14.3544 0.580715
\(612\) 0 0
\(613\) 8.90917i 0.359838i −0.983681 0.179919i \(-0.942416\pi\)
0.983681 0.179919i \(-0.0575836\pi\)
\(614\) 0 0
\(615\) 63.6197 46.5289i 2.56540 1.87623i
\(616\) 0 0
\(617\) 27.2413i 1.09669i −0.836251 0.548347i \(-0.815257\pi\)
0.836251 0.548347i \(-0.184743\pi\)
\(618\) 0 0
\(619\) 24.1405 0.970288 0.485144 0.874434i \(-0.338767\pi\)
0.485144 + 0.874434i \(0.338767\pi\)
\(620\) 0 0
\(621\) 85.6491 3.43698
\(622\) 0 0
\(623\) 11.4693i 0.459506i
\(624\) 0 0
\(625\) −20.4087 14.4390i −0.816349 0.577560i
\(626\) 0 0
\(627\) 12.4995i 0.499184i
\(628\) 0 0
\(629\) 34.5895 1.37917
\(630\) 0 0
\(631\) −8.01468 −0.319059 −0.159530 0.987193i \(-0.550998\pi\)
−0.159530 + 0.987193i \(0.550998\pi\)
\(632\) 0 0
\(633\) 53.8548i 2.14054i
\(634\) 0 0
\(635\) 12.9210 9.44989i 0.512754 0.375007i
\(636\) 0 0
\(637\) 4.15516i 0.164633i
\(638\) 0 0
\(639\) −28.7493 −1.13731
\(640\) 0 0
\(641\) −49.1495 −1.94129 −0.970645 0.240516i \(-0.922683\pi\)
−0.970645 + 0.240516i \(0.922683\pi\)
\(642\) 0 0
\(643\) 7.24599i 0.285754i −0.989740 0.142877i \(-0.954365\pi\)
0.989740 0.142877i \(-0.0456353\pi\)
\(644\) 0 0
\(645\) 21.7796 + 29.7796i 0.857570 + 1.17257i
\(646\) 0 0
\(647\) 30.2791i 1.19040i 0.803579 + 0.595198i \(0.202926\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(648\) 0 0
\(649\) 14.7106 0.577440
\(650\) 0 0
\(651\) −7.03028 −0.275538
\(652\) 0 0
\(653\) 7.96881i 0.311844i 0.987769 + 0.155922i \(0.0498348\pi\)
−0.987769 + 0.155922i \(0.950165\pi\)
\(654\) 0 0
\(655\) −10.3747 14.1854i −0.405371 0.554271i
\(656\) 0 0
\(657\) 49.2489i 1.92138i
\(658\) 0 0
\(659\) 20.9239 0.815078 0.407539 0.913188i \(-0.366387\pi\)
0.407539 + 0.913188i \(0.366387\pi\)
\(660\) 0 0
\(661\) 46.1992 1.79694 0.898470 0.439034i \(-0.144679\pi\)
0.898470 + 0.439034i \(0.144679\pi\)
\(662\) 0 0
\(663\) 74.8539i 2.90708i
\(664\) 0 0
\(665\) −2.90539 + 2.12489i −0.112666 + 0.0823995i
\(666\) 0 0
\(667\) 10.5895i 0.410025i
\(668\) 0 0
\(669\) −0.734633 −0.0284025
\(670\) 0 0
\(671\) −13.3188 −0.514167
\(672\) 0 0
\(673\) 40.3784i 1.55647i 0.627970 + 0.778237i \(0.283886\pi\)
−0.627970 + 0.778237i \(0.716114\pi\)
\(674\) 0 0
\(675\) −56.0587 17.8255i −2.15770 0.686102i
\(676\) 0 0
\(677\) 44.3737i 1.70542i 0.522383 + 0.852711i \(0.325043\pi\)
−0.522383 + 0.852711i \(0.674957\pi\)
\(678\) 0 0
\(679\) 2.73463 0.104946
\(680\) 0 0
\(681\) 1.76491 0.0676315
\(682\) 0 0
\(683\) 17.9394i 0.686434i −0.939256 0.343217i \(-0.888483\pi\)
0.939256 0.343217i \(-0.111517\pi\)
\(684\) 0 0
\(685\) −19.0596 + 13.9394i −0.728231 + 0.532599i
\(686\) 0 0
\(687\) 22.2186i 0.847692i
\(688\) 0 0
\(689\) −38.3085 −1.45944
\(690\) 0 0
\(691\) 12.7905 0.486573 0.243287 0.969954i \(-0.421774\pi\)
0.243287 + 0.969954i \(0.421774\pi\)
\(692\) 0 0
\(693\) 16.8099i 0.638554i
\(694\) 0 0
\(695\) 12.9348 + 17.6859i 0.490643 + 0.670864i
\(696\) 0 0
\(697\) 65.0284i 2.46313i
\(698\) 0 0
\(699\) −26.5601 −1.00460
\(700\) 0 0
\(701\) −19.4234 −0.733611 −0.366806 0.930298i \(-0.619549\pi\)
−0.366806 + 0.930298i \(0.619549\pi\)
\(702\) 0 0
\(703\) 9.65848i 0.364277i
\(704\) 0 0
\(705\) 14.2498 + 19.4839i 0.536677 + 0.733808i
\(706\) 0 0
\(707\) 4.57947i 0.172229i
\(708\) 0 0
\(709\) 13.4839 0.506400 0.253200 0.967414i \(-0.418517\pi\)
0.253200 + 0.967414i \(0.418517\pi\)
\(710\) 0 0
\(711\) −114.898 −4.30901
\(712\) 0 0
\(713\) 16.3784i 0.613377i
\(714\) 0 0
\(715\) −18.6353 + 13.6291i −0.696922 + 0.509700i
\(716\) 0 0
\(717\) 22.7034i 0.847876i
\(718\) 0 0
\(719\) 15.3700 0.573203 0.286601 0.958050i \(-0.407474\pi\)
0.286601 + 0.958050i \(0.407474\pi\)
\(720\) 0 0
\(721\) 2.48486 0.0925411
\(722\) 0 0
\(723\) 22.7493i 0.846056i
\(724\) 0 0
\(725\) 2.20390 6.93097i 0.0818507 0.257410i
\(726\) 0 0
\(727\) 13.1589i 0.488038i −0.969770 0.244019i \(-0.921534\pi\)
0.969770 0.244019i \(-0.0784660\pi\)
\(728\) 0 0
\(729\) −1.12110 −0.0415224
\(730\) 0 0
\(731\) 30.4390 1.12583
\(732\) 0 0
\(733\) 10.0265i 0.370337i 0.982707 + 0.185169i \(0.0592831\pi\)
−0.982707 + 0.185169i \(0.940717\pi\)
\(734\) 0 0
\(735\) −5.64002 + 4.12489i −0.208035 + 0.152149i
\(736\) 0 0
\(737\) 18.7106i 0.689212i
\(738\) 0 0
\(739\) 19.2266 0.707262 0.353631 0.935385i \(-0.384947\pi\)
0.353631 + 0.935385i \(0.384947\pi\)
\(740\) 0 0
\(741\) 20.9016 0.767840
\(742\) 0 0
\(743\) 5.21949i 0.191485i −0.995406 0.0957423i \(-0.969478\pi\)
0.995406 0.0957423i \(-0.0305225\pi\)
\(744\) 0 0
\(745\) −3.29942 4.51136i −0.120882 0.165283i
\(746\) 0 0
\(747\) 68.3884i 2.50220i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29.8548 −1.08942 −0.544709 0.838625i \(-0.683360\pi\)
−0.544709 + 0.838625i \(0.683360\pi\)
\(752\) 0 0
\(753\) 71.6197i 2.60997i
\(754\) 0 0
\(755\) 4.96972 + 6.79518i 0.180867 + 0.247302i
\(756\) 0 0
\(757\) 18.0294i 0.655288i −0.944801 0.327644i \(-0.893745\pi\)
0.944801 0.327644i \(-0.106255\pi\)
\(758\) 0 0
\(759\) 56.5289 2.05187
\(760\) 0 0
\(761\) 8.22041 0.297990 0.148995 0.988838i \(-0.452396\pi\)
0.148995 + 0.988838i \(0.452396\pi\)
\(762\) 0 0
\(763\) 9.95413i 0.360364i
\(764\) 0 0
\(765\) −70.3884 + 51.4792i −2.54490 + 1.86124i
\(766\) 0 0
\(767\) 24.5988i 0.888213i
\(768\) 0 0
\(769\) −50.6888 −1.82788 −0.913942 0.405845i \(-0.866977\pi\)
−0.913942 + 0.405845i \(0.866977\pi\)
\(770\) 0 0
\(771\) 2.24977 0.0810235
\(772\) 0 0
\(773\) 6.99622i 0.251637i −0.992053 0.125818i \(-0.959844\pi\)
0.992053 0.125818i \(-0.0401556\pi\)
\(774\) 0 0
\(775\) −3.40871 + 10.7200i −0.122445 + 0.385072i
\(776\) 0 0
\(777\) 18.7493i 0.672628i
\(778\) 0 0
\(779\) −18.1580 −0.650579
\(780\) 0 0
\(781\) −10.5601 −0.377870
\(782\) 0 0
\(783\) 17.1131i 0.611571i
\(784\) 0 0
\(785\) −10.2342 + 7.48486i −0.365273 + 0.267146i
\(786\) 0 0
\(787\) 14.3444i 0.511322i 0.966767 + 0.255661i \(0.0822931\pi\)
−0.966767 + 0.255661i \(0.917707\pi\)
\(788\) 0 0
\(789\) −57.9982 −2.06479
\(790\) 0 0
\(791\) −18.4995 −0.657768
\(792\) 0 0
\(793\) 22.2716i 0.790887i
\(794\) 0 0
\(795\) −38.0294 51.9982i −1.34876 1.84418i
\(796\) 0 0
\(797\) 13.9348i 0.493594i −0.969067 0.246797i \(-0.920622\pi\)
0.969067 0.246797i \(-0.0793781\pi\)
\(798\) 0 0
\(799\) 19.9154 0.704555
\(800\) 0 0
\(801\) −77.5885 −2.74146
\(802\) 0 0