Properties

Label 560.2.g.e.449.3
Level 560
Weight 2
Character 560.449
Analytic conductor 4.472
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(-1.22474 + 1.22474i\)
Character \(\chi\) = 560.449
Dual form 560.2.g.e.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.44949i q^{3} +(-0.224745 + 2.22474i) q^{5} +1.00000i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.44949i q^{3} +(-0.224745 + 2.22474i) q^{5} +1.00000i q^{7} -3.00000 q^{9} -4.89898 q^{11} +0.449490i q^{13} +(-5.44949 - 0.550510i) q^{15} +2.00000i q^{17} +6.44949 q^{19} -2.44949 q^{21} -6.89898i q^{23} +(-4.89898 - 1.00000i) q^{25} +2.89898 q^{29} +0.898979 q^{31} -12.0000i q^{33} +(-2.22474 - 0.224745i) q^{35} +2.00000i q^{37} -1.10102 q^{39} -10.8990 q^{41} +8.89898i q^{43} +(0.674235 - 6.67423i) q^{45} +0.898979i q^{47} -1.00000 q^{49} -4.89898 q^{51} +1.10102i q^{53} +(1.10102 - 10.8990i) q^{55} +15.7980i q^{57} -6.44949 q^{59} +8.44949 q^{61} -3.00000i q^{63} +(-1.00000 - 0.101021i) q^{65} +8.00000i q^{67} +16.8990 q^{69} +10.8990 q^{71} +6.89898i q^{73} +(2.44949 - 12.0000i) q^{75} -4.89898i q^{77} -2.89898 q^{79} -9.00000 q^{81} +2.44949i q^{83} +(-4.44949 - 0.449490i) q^{85} +7.10102i q^{87} +10.0000 q^{89} -0.449490 q^{91} +2.20204i q^{93} +(-1.44949 + 14.3485i) q^{95} -3.79796i q^{97} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} - 12q^{9} + O(q^{10}) \) \( 4q + 4q^{5} - 12q^{9} - 12q^{15} + 16q^{19} - 8q^{29} - 16q^{31} - 4q^{35} - 24q^{39} - 24q^{41} - 12q^{45} - 4q^{49} + 24q^{55} - 16q^{59} + 24q^{61} - 4q^{65} + 48q^{69} + 24q^{71} + 8q^{79} - 36q^{81} - 8q^{85} + 40q^{89} + 8q^{91} + 4q^{95} + O(q^{100}) \)

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
\(3\) 2.44949i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) −0.224745 + 2.22474i −0.100509 + 0.994936i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 0.449490i 0.124666i 0.998055 + 0.0623330i \(0.0198541\pi\)
−0.998055 + 0.0623330i \(0.980146\pi\)
\(14\) 0 0
\(15\) −5.44949 0.550510i −1.40705 0.142141i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 6.44949 1.47961 0.739807 0.672819i \(-0.234917\pi\)
0.739807 + 0.672819i \(0.234917\pi\)
\(20\) 0 0
\(21\) −2.44949 −0.534522
\(22\) 0 0
\(23\) 6.89898i 1.43854i −0.694732 0.719268i \(-0.744477\pi\)
0.694732 0.719268i \(-0.255523\pi\)
\(24\) 0 0
\(25\) −4.89898 1.00000i −0.979796 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.89898 0.538327 0.269163 0.963095i \(-0.413253\pi\)
0.269163 + 0.963095i \(0.413253\pi\)
\(30\) 0 0
\(31\) 0.898979 0.161461 0.0807307 0.996736i \(-0.474275\pi\)
0.0807307 + 0.996736i \(0.474275\pi\)
\(32\) 0 0
\(33\) 12.0000i 2.08893i
\(34\) 0 0
\(35\) −2.22474 0.224745i −0.376051 0.0379888i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −1.10102 −0.176304
\(40\) 0 0
\(41\) −10.8990 −1.70213 −0.851067 0.525057i \(-0.824044\pi\)
−0.851067 + 0.525057i \(0.824044\pi\)
\(42\) 0 0
\(43\) 8.89898i 1.35708i 0.734563 + 0.678541i \(0.237387\pi\)
−0.734563 + 0.678541i \(0.762613\pi\)
\(44\) 0 0
\(45\) 0.674235 6.67423i 0.100509 0.994936i
\(46\) 0 0
\(47\) 0.898979i 0.131130i 0.997848 + 0.0655648i \(0.0208849\pi\)
−0.997848 + 0.0655648i \(0.979115\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −4.89898 −0.685994
\(52\) 0 0
\(53\) 1.10102i 0.151237i 0.997137 + 0.0756184i \(0.0240931\pi\)
−0.997137 + 0.0756184i \(0.975907\pi\)
\(54\) 0 0
\(55\) 1.10102 10.8990i 0.148462 1.46962i
\(56\) 0 0
\(57\) 15.7980i 2.09249i
\(58\) 0 0
\(59\) −6.44949 −0.839652 −0.419826 0.907605i \(-0.637909\pi\)
−0.419826 + 0.907605i \(0.637909\pi\)
\(60\) 0 0
\(61\) 8.44949 1.08185 0.540923 0.841072i \(-0.318075\pi\)
0.540923 + 0.841072i \(0.318075\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) −1.00000 0.101021i −0.124035 0.0125301i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 16.8990 2.03440
\(70\) 0 0
\(71\) 10.8990 1.29347 0.646735 0.762714i \(-0.276134\pi\)
0.646735 + 0.762714i \(0.276134\pi\)
\(72\) 0 0
\(73\) 6.89898i 0.807464i 0.914877 + 0.403732i \(0.132287\pi\)
−0.914877 + 0.403732i \(0.867713\pi\)
\(74\) 0 0
\(75\) 2.44949 12.0000i 0.282843 1.38564i
\(76\) 0 0
\(77\) 4.89898i 0.558291i
\(78\) 0 0
\(79\) −2.89898 −0.326161 −0.163080 0.986613i \(-0.552143\pi\)
−0.163080 + 0.986613i \(0.552143\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 2.44949i 0.268866i 0.990923 + 0.134433i \(0.0429214\pi\)
−0.990923 + 0.134433i \(0.957079\pi\)
\(84\) 0 0
\(85\) −4.44949 0.449490i −0.482615 0.0487540i
\(86\) 0 0
\(87\) 7.10102i 0.761309i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −0.449490 −0.0471193
\(92\) 0 0
\(93\) 2.20204i 0.228341i
\(94\) 0 0
\(95\) −1.44949 + 14.3485i −0.148715 + 1.47212i
\(96\) 0 0
\(97\) 3.79796i 0.385624i −0.981236 0.192812i \(-0.938239\pi\)
0.981236 0.192812i \(-0.0617608\pi\)
\(98\) 0 0
\(99\) 14.6969 1.47710
\(100\) 0 0
\(101\) 8.44949 0.840756 0.420378 0.907349i \(-0.361898\pi\)
0.420378 + 0.907349i \(0.361898\pi\)
\(102\) 0 0
\(103\) 3.10102i 0.305553i 0.988261 + 0.152776i \(0.0488214\pi\)
−0.988261 + 0.152776i \(0.951179\pi\)
\(104\) 0 0
\(105\) 0.550510 5.44949i 0.0537243 0.531816i
\(106\) 0 0
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) 2.89898 0.277672 0.138836 0.990315i \(-0.455664\pi\)
0.138836 + 0.990315i \(0.455664\pi\)
\(110\) 0 0
\(111\) −4.89898 −0.464991
\(112\) 0 0
\(113\) 0.202041i 0.0190064i −0.999955 0.00950321i \(-0.996975\pi\)
0.999955 0.00950321i \(-0.00302501\pi\)
\(114\) 0 0
\(115\) 15.3485 + 1.55051i 1.43125 + 0.144586i
\(116\) 0 0
\(117\) 1.34847i 0.124666i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 26.6969i 2.40718i
\(124\) 0 0
\(125\) 3.32577 10.6742i 0.297465 0.954733i
\(126\) 0 0
\(127\) 5.10102i 0.452642i 0.974053 + 0.226321i \(0.0726699\pi\)
−0.974053 + 0.226321i \(0.927330\pi\)
\(128\) 0 0
\(129\) −21.7980 −1.91920
\(130\) 0 0
\(131\) 1.55051 0.135469 0.0677344 0.997703i \(-0.478423\pi\)
0.0677344 + 0.997703i \(0.478423\pi\)
\(132\) 0 0
\(133\) 6.44949i 0.559242i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.7980i 1.52058i 0.649582 + 0.760291i \(0.274944\pi\)
−0.649582 + 0.760291i \(0.725056\pi\)
\(138\) 0 0
\(139\) 6.44949 0.547039 0.273519 0.961867i \(-0.411812\pi\)
0.273519 + 0.961867i \(0.411812\pi\)
\(140\) 0 0
\(141\) −2.20204 −0.185445
\(142\) 0 0
\(143\) 2.20204i 0.184144i
\(144\) 0 0
\(145\) −0.651531 + 6.44949i −0.0541067 + 0.535601i
\(146\) 0 0
\(147\) 2.44949i 0.202031i
\(148\) 0 0
\(149\) −15.7980 −1.29422 −0.647110 0.762397i \(-0.724022\pi\)
−0.647110 + 0.762397i \(0.724022\pi\)
\(150\) 0 0
\(151\) 19.5959 1.59469 0.797347 0.603522i \(-0.206236\pi\)
0.797347 + 0.603522i \(0.206236\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) −0.202041 + 2.00000i −0.0162283 + 0.160644i
\(156\) 0 0
\(157\) 8.44949i 0.674343i 0.941443 + 0.337171i \(0.109470\pi\)
−0.941443 + 0.337171i \(0.890530\pi\)
\(158\) 0 0
\(159\) −2.69694 −0.213881
\(160\) 0 0
\(161\) 6.89898 0.543716
\(162\) 0 0
\(163\) 16.8990i 1.32363i −0.749667 0.661815i \(-0.769786\pi\)
0.749667 0.661815i \(-0.230214\pi\)
\(164\) 0 0
\(165\) 26.6969 + 2.69694i 2.07835 + 0.209956i
\(166\) 0 0
\(167\) 4.89898i 0.379094i −0.981872 0.189547i \(-0.939298\pi\)
0.981872 0.189547i \(-0.0607020\pi\)
\(168\) 0 0
\(169\) 12.7980 0.984458
\(170\) 0 0
\(171\) −19.3485 −1.47961
\(172\) 0 0
\(173\) 18.2474i 1.38733i −0.720299 0.693664i \(-0.755995\pi\)
0.720299 0.693664i \(-0.244005\pi\)
\(174\) 0 0
\(175\) 1.00000 4.89898i 0.0755929 0.370328i
\(176\) 0 0
\(177\) 15.7980i 1.18745i
\(178\) 0 0
\(179\) −5.79796 −0.433360 −0.216680 0.976243i \(-0.569523\pi\)
−0.216680 + 0.976243i \(0.569523\pi\)
\(180\) 0 0
\(181\) 14.2474 1.05900 0.529502 0.848309i \(-0.322379\pi\)
0.529502 + 0.848309i \(0.322379\pi\)
\(182\) 0 0
\(183\) 20.6969i 1.52996i
\(184\) 0 0
\(185\) −4.44949 0.449490i −0.327133 0.0330471i
\(186\) 0 0
\(187\) 9.79796i 0.716498i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.6969 1.20815 0.604074 0.796928i \(-0.293543\pi\)
0.604074 + 0.796928i \(0.293543\pi\)
\(192\) 0 0
\(193\) 17.5959i 1.26658i −0.773914 0.633291i \(-0.781704\pi\)
0.773914 0.633291i \(-0.218296\pi\)
\(194\) 0 0
\(195\) 0.247449 2.44949i 0.0177202 0.175412i
\(196\) 0 0
\(197\) 9.10102i 0.648421i 0.945985 + 0.324210i \(0.105099\pi\)
−0.945985 + 0.324210i \(0.894901\pi\)
\(198\) 0 0
\(199\) 7.10102 0.503378 0.251689 0.967808i \(-0.419014\pi\)
0.251689 + 0.967808i \(0.419014\pi\)
\(200\) 0 0
\(201\) −19.5959 −1.38219
\(202\) 0 0
\(203\) 2.89898i 0.203468i
\(204\) 0 0
\(205\) 2.44949 24.2474i 0.171080 1.69352i
\(206\) 0 0
\(207\) 20.6969i 1.43854i
\(208\) 0 0
\(209\) −31.5959 −2.18554
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 26.6969i 1.82924i
\(214\) 0 0
\(215\) −19.7980 2.00000i −1.35021 0.136399i
\(216\) 0 0
\(217\) 0.898979i 0.0610267i
\(218\) 0 0
\(219\) −16.8990 −1.14193
\(220\) 0 0
\(221\) −0.898979 −0.0604719
\(222\) 0 0
\(223\) 4.00000i 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 14.6969 + 3.00000i 0.979796 + 0.200000i
\(226\) 0 0
\(227\) 7.34847i 0.487735i 0.969809 + 0.243868i \(0.0784162\pi\)
−0.969809 + 0.243868i \(0.921584\pi\)
\(228\) 0 0
\(229\) −15.1464 −1.00090 −0.500452 0.865764i \(-0.666833\pi\)
−0.500452 + 0.865764i \(0.666833\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 10.2020i 0.668358i −0.942510 0.334179i \(-0.891541\pi\)
0.942510 0.334179i \(-0.108459\pi\)
\(234\) 0 0
\(235\) −2.00000 0.202041i −0.130466 0.0131797i
\(236\) 0 0
\(237\) 7.10102i 0.461261i
\(238\) 0 0
\(239\) −25.7980 −1.66873 −0.834366 0.551211i \(-0.814166\pi\)
−0.834366 + 0.551211i \(0.814166\pi\)
\(240\) 0 0
\(241\) 20.6969 1.33321 0.666604 0.745412i \(-0.267747\pi\)
0.666604 + 0.745412i \(0.267747\pi\)
\(242\) 0 0
\(243\) 22.0454i 1.41421i
\(244\) 0 0
\(245\) 0.224745 2.22474i 0.0143584 0.142134i
\(246\) 0 0
\(247\) 2.89898i 0.184458i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 1.55051 0.0978673 0.0489337 0.998802i \(-0.484418\pi\)
0.0489337 + 0.998802i \(0.484418\pi\)
\(252\) 0 0
\(253\) 33.7980i 2.12486i
\(254\) 0 0
\(255\) 1.10102 10.8990i 0.0689486 0.682521i
\(256\) 0 0
\(257\) 20.6969i 1.29104i 0.763744 + 0.645520i \(0.223359\pi\)
−0.763744 + 0.645520i \(0.776641\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −8.69694 −0.538327
\(262\) 0 0
\(263\) 9.79796i 0.604168i −0.953281 0.302084i \(-0.902318\pi\)
0.953281 0.302084i \(-0.0976823\pi\)
\(264\) 0 0
\(265\) −2.44949 0.247449i −0.150471 0.0152007i
\(266\) 0 0
\(267\) 24.4949i 1.49906i
\(268\) 0 0
\(269\) 15.1464 0.923494 0.461747 0.887012i \(-0.347223\pi\)
0.461747 + 0.887012i \(0.347223\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 1.10102i 0.0666368i
\(274\) 0 0
\(275\) 24.0000 + 4.89898i 1.44725 + 0.295420i
\(276\) 0 0
\(277\) 5.10102i 0.306491i −0.988188 0.153245i \(-0.951028\pi\)
0.988188 0.153245i \(-0.0489725\pi\)
\(278\) 0 0
\(279\) −2.69694 −0.161461
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 28.2474i 1.67914i 0.543254 + 0.839568i \(0.317192\pi\)
−0.543254 + 0.839568i \(0.682808\pi\)
\(284\) 0 0
\(285\) −35.1464 3.55051i −2.08189 0.210314i
\(286\) 0 0
\(287\) 10.8990i 0.643346i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 9.30306 0.545355
\(292\) 0 0
\(293\) 6.24745i 0.364980i 0.983208 + 0.182490i \(0.0584157\pi\)
−0.983208 + 0.182490i \(0.941584\pi\)
\(294\) 0 0
\(295\) 1.44949 14.3485i 0.0843926 0.835400i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.10102 0.179337
\(300\) 0 0
\(301\) −8.89898 −0.512929
\(302\) 0 0
\(303\) 20.6969i 1.18901i
\(304\) 0 0
\(305\) −1.89898 + 18.7980i −0.108735 + 1.07637i
\(306\) 0 0
\(307\) 4.24745i 0.242415i −0.992627 0.121207i \(-0.961323\pi\)
0.992627 0.121207i \(-0.0386766\pi\)
\(308\) 0 0
\(309\) −7.59592 −0.432117
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 17.5959i 0.994580i −0.867584 0.497290i \(-0.834328\pi\)
0.867584 0.497290i \(-0.165672\pi\)
\(314\) 0 0
\(315\) 6.67423 + 0.674235i 0.376051 + 0.0379888i
\(316\) 0 0
\(317\) 26.4949i 1.48810i 0.668123 + 0.744051i \(0.267098\pi\)
−0.668123 + 0.744051i \(0.732902\pi\)
\(318\) 0 0
\(319\) −14.2020 −0.795162
\(320\) 0 0
\(321\) −19.5959 −1.09374
\(322\) 0 0
\(323\) 12.8990i 0.717718i
\(324\) 0 0
\(325\) 0.449490 2.20204i 0.0249332 0.122147i
\(326\) 0 0
\(327\) 7.10102i 0.392687i
\(328\) 0 0
\(329\) −0.898979 −0.0495623
\(330\) 0 0
\(331\) −10.6969 −0.587957 −0.293978 0.955812i \(-0.594979\pi\)
−0.293978 + 0.955812i \(0.594979\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) −17.7980 1.79796i −0.972406 0.0982330i
\(336\) 0 0
\(337\) 29.5959i 1.61219i −0.591785 0.806096i \(-0.701576\pi\)
0.591785 0.806096i \(-0.298424\pi\)
\(338\) 0 0
\(339\) 0.494897 0.0268791
\(340\) 0 0
\(341\) −4.40408 −0.238494
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −3.79796 + 37.5959i −0.204475 + 2.02410i
\(346\) 0 0
\(347\) 19.1010i 1.02540i −0.858569 0.512698i \(-0.828646\pi\)
0.858569 0.512698i \(-0.171354\pi\)
\(348\) 0 0
\(349\) 3.55051 0.190054 0.0950272 0.995475i \(-0.469706\pi\)
0.0950272 + 0.995475i \(0.469706\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.1010i 0.697297i −0.937254 0.348648i \(-0.886641\pi\)
0.937254 0.348648i \(-0.113359\pi\)
\(354\) 0 0
\(355\) −2.44949 + 24.2474i −0.130005 + 1.28692i
\(356\) 0 0
\(357\) 4.89898i 0.259281i
\(358\) 0 0
\(359\) 11.5959 0.612009 0.306005 0.952030i \(-0.401008\pi\)
0.306005 + 0.952030i \(0.401008\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) 0 0
\(363\) 31.8434i 1.67134i
\(364\) 0 0
\(365\) −15.3485 1.55051i −0.803376 0.0811574i
\(366\) 0 0
\(367\) 32.0000i 1.67039i −0.549957 0.835193i \(-0.685356\pi\)
0.549957 0.835193i \(-0.314644\pi\)
\(368\) 0 0
\(369\) 32.6969 1.70213
\(370\) 0 0
\(371\) −1.10102 −0.0571621
\(372\) 0 0
\(373\) 24.6969i 1.27876i −0.768891 0.639380i \(-0.779191\pi\)
0.768891 0.639380i \(-0.220809\pi\)
\(374\) 0 0
\(375\) 26.1464 + 8.14643i 1.35020 + 0.420680i
\(376\) 0 0
\(377\) 1.30306i 0.0671111i
\(378\) 0 0
\(379\) 1.30306 0.0669338 0.0334669 0.999440i \(-0.489345\pi\)
0.0334669 + 0.999440i \(0.489345\pi\)
\(380\) 0 0
\(381\) −12.4949 −0.640133
\(382\) 0 0
\(383\) 16.8990i 0.863498i −0.901994 0.431749i \(-0.857897\pi\)
0.901994 0.431749i \(-0.142103\pi\)
\(384\) 0 0
\(385\) 10.8990 + 1.10102i 0.555463 + 0.0561132i
\(386\) 0 0
\(387\) 26.6969i 1.35708i
\(388\) 0 0
\(389\) −22.8990 −1.16102 −0.580512 0.814252i \(-0.697148\pi\)
−0.580512 + 0.814252i \(0.697148\pi\)
\(390\) 0 0
\(391\) 13.7980 0.697793
\(392\) 0 0
\(393\) 3.79796i 0.191582i
\(394\) 0 0
\(395\) 0.651531 6.44949i 0.0327821 0.324509i
\(396\) 0 0
\(397\) 17.3485i 0.870695i −0.900263 0.435347i \(-0.856626\pi\)
0.900263 0.435347i \(-0.143374\pi\)
\(398\) 0 0
\(399\) −15.7980 −0.790887
\(400\) 0 0
\(401\) 29.3939 1.46786 0.733930 0.679225i \(-0.237684\pi\)
0.733930 + 0.679225i \(0.237684\pi\)
\(402\) 0 0
\(403\) 0.404082i 0.0201288i
\(404\) 0 0
\(405\) 2.02270 20.0227i 0.100509 0.994936i
\(406\) 0 0
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) 14.4949 0.716727 0.358363 0.933582i \(-0.383335\pi\)
0.358363 + 0.933582i \(0.383335\pi\)
\(410\) 0 0
\(411\) −43.5959 −2.15043
\(412\) 0 0
\(413\) 6.44949i 0.317359i
\(414\) 0 0
\(415\) −5.44949 0.550510i −0.267505 0.0270235i
\(416\) 0 0
\(417\) 15.7980i 0.773629i
\(418\) 0 0
\(419\) −6.44949 −0.315078 −0.157539 0.987513i \(-0.550356\pi\)
−0.157539 + 0.987513i \(0.550356\pi\)
\(420\) 0 0
\(421\) −23.7980 −1.15984 −0.579921 0.814673i \(-0.696917\pi\)
−0.579921 + 0.814673i \(0.696917\pi\)
\(422\) 0 0
\(423\) 2.69694i 0.131130i
\(424\) 0 0
\(425\) 2.00000 9.79796i 0.0970143 0.475271i
\(426\) 0 0
\(427\) 8.44949i 0.408899i
\(428\) 0 0
\(429\) 5.39388 0.260419
\(430\) 0 0
\(431\) −17.7980 −0.857298 −0.428649 0.903471i \(-0.641010\pi\)
−0.428649 + 0.903471i \(0.641010\pi\)
\(432\) 0 0
\(433\) 19.7980i 0.951429i 0.879600 + 0.475715i \(0.157810\pi\)
−0.879600 + 0.475715i \(0.842190\pi\)
\(434\) 0 0
\(435\) −15.7980 1.59592i −0.757454 0.0765184i
\(436\) 0 0
\(437\) 44.4949i 2.12848i
\(438\) 0 0
\(439\) 37.3939 1.78471 0.892356 0.451332i \(-0.149051\pi\)
0.892356 + 0.451332i \(0.149051\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 9.79796i 0.465515i −0.972535 0.232758i \(-0.925225\pi\)
0.972535 0.232758i \(-0.0747749\pi\)
\(444\) 0 0
\(445\) −2.24745 + 22.2474i −0.106539 + 1.05463i
\(446\) 0 0
\(447\) 38.6969i 1.83030i
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 53.3939 2.51422
\(452\) 0 0
\(453\) 48.0000i 2.25524i
\(454\) 0 0
\(455\) 0.101021 1.00000i 0.00473591 0.0468807i
\(456\) 0 0
\(457\) 9.59592i 0.448878i −0.974488 0.224439i \(-0.927945\pi\)
0.974488 0.224439i \(-0.0720550\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.65153 0.123494 0.0617470 0.998092i \(-0.480333\pi\)
0.0617470 + 0.998092i \(0.480333\pi\)
\(462\) 0 0
\(463\) 35.5959i 1.65428i −0.561994 0.827141i \(-0.689966\pi\)
0.561994 0.827141i \(-0.310034\pi\)
\(464\) 0 0
\(465\) −4.89898 0.494897i −0.227185 0.0229503i
\(466\) 0 0
\(467\) 5.55051i 0.256847i −0.991719 0.128423i \(-0.959008\pi\)
0.991719 0.128423i \(-0.0409917\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −20.6969 −0.953665
\(472\) 0 0
\(473\) 43.5959i 2.00454i
\(474\) 0 0
\(475\) −31.5959 6.44949i −1.44972 0.295923i
\(476\) 0 0
\(477\) 3.30306i 0.151237i
\(478\) 0 0
\(479\) 38.6969 1.76811 0.884054 0.467385i \(-0.154804\pi\)
0.884054 + 0.467385i \(0.154804\pi\)
\(480\) 0 0
\(481\) −0.898979 −0.0409899
\(482\) 0 0
\(483\) 16.8990i 0.768930i
\(484\) 0 0
\(485\) 8.44949 + 0.853572i 0.383672 + 0.0387587i
\(486\) 0 0
\(487\) 36.6969i 1.66290i 0.555602 + 0.831449i \(0.312488\pi\)
−0.555602 + 0.831449i \(0.687512\pi\)
\(488\) 0 0
\(489\) 41.3939 1.87190
\(490\) 0 0
\(491\) 19.5959 0.884351 0.442176 0.896928i \(-0.354207\pi\)
0.442176 + 0.896928i \(0.354207\pi\)
\(492\) 0 0
\(493\) 5.79796i 0.261127i
\(494\) 0 0
\(495\) −3.30306 + 32.6969i −0.148462 + 1.46962i
\(496\) 0 0
\(497\) 10.8990i 0.488886i
\(498\) 0 0
\(499\) 25.7980 1.15488 0.577438 0.816435i \(-0.304053\pi\)
0.577438 + 0.816435i \(0.304053\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 4.00000i 0.178351i −0.996016 0.0891756i \(-0.971577\pi\)
0.996016 0.0891756i \(-0.0284232\pi\)
\(504\) 0 0
\(505\) −1.89898 + 18.7980i −0.0845035 + 0.836498i
\(506\) 0 0
\(507\) 31.3485i 1.39223i
\(508\) 0 0
\(509\) 36.4495 1.61560 0.807798 0.589460i \(-0.200659\pi\)
0.807798 + 0.589460i \(0.200659\pi\)
\(510\) 0 0
\(511\) −6.89898 −0.305193
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.89898 0.696938i −0.304005 0.0307108i
\(516\) 0 0
\(517\) 4.40408i 0.193691i
\(518\) 0 0
\(519\) 44.6969 1.96198
\(520\) 0 0
\(521\) 3.30306 0.144710 0.0723549 0.997379i \(-0.476949\pi\)
0.0723549 + 0.997379i \(0.476949\pi\)
\(522\) 0 0
\(523\) 1.14643i 0.0501298i 0.999686 + 0.0250649i \(0.00797924\pi\)
−0.999686 + 0.0250649i \(0.992021\pi\)
\(524\) 0 0
\(525\) 12.0000 + 2.44949i 0.523723 + 0.106904i
\(526\) 0 0
\(527\) 1.79796i 0.0783203i
\(528\) 0 0
\(529\) −24.5959 −1.06939
\(530\) 0 0
\(531\) 19.3485 0.839652
\(532\) 0 0
\(533\) 4.89898i 0.212198i
\(534\) 0 0
\(535\) −17.7980 1.79796i −0.769473 0.0777325i
\(536\) 0 0
\(537\) 14.2020i 0.612863i
\(538\) 0 0
\(539\) 4.89898 0.211014
\(540\) 0 0
\(541\) −29.5959 −1.27243 −0.636214 0.771513i \(-0.719500\pi\)
−0.636214 + 0.771513i \(0.719500\pi\)
\(542\) 0 0
\(543\) 34.8990i 1.49766i
\(544\) 0 0
\(545\) −0.651531 + 6.44949i −0.0279085 + 0.276266i
\(546\) 0 0
\(547\) 10.6969i 0.457368i −0.973501 0.228684i \(-0.926558\pi\)
0.973501 0.228684i \(-0.0734423\pi\)
\(548\) 0 0
\(549\) −25.3485 −1.08185
\(550\) 0 0
\(551\) 18.6969 0.796516
\(552\) 0 0
\(553\) 2.89898i 0.123277i
\(554\) 0 0
\(555\) 1.10102 10.8990i 0.0467357 0.462636i
\(556\) 0 0
\(557\) 16.6969i 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 14.0454i 0.591943i 0.955197 + 0.295972i \(0.0956434\pi\)
−0.955197 + 0.295972i \(0.904357\pi\)
\(564\) 0 0
\(565\) 0.449490 + 0.0454077i 0.0189102 + 0.00191032i
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) −14.2020 −0.595381 −0.297690 0.954663i \(-0.596216\pi\)
−0.297690 + 0.954663i \(0.596216\pi\)
\(570\) 0 0
\(571\) 20.8990 0.874595 0.437298 0.899317i \(-0.355936\pi\)
0.437298 + 0.899317i \(0.355936\pi\)
\(572\) 0 0
\(573\) 40.8990i 1.70858i
\(574\) 0 0
\(575\) −6.89898 + 33.7980i −0.287707 + 1.40947i
\(576\) 0 0
\(577\) 46.4949i 1.93561i 0.251705 + 0.967804i \(0.419009\pi\)
−0.251705 + 0.967804i \(0.580991\pi\)
\(578\) 0 0
\(579\) 43.1010 1.79122
\(580\) 0 0
\(581\) −2.44949 −0.101622
\(582\) 0 0
\(583\) 5.39388i 0.223392i
\(584\) 0 0
\(585\) 3.00000 + 0.303062i 0.124035 + 0.0125301i
\(586\) 0 0
\(587\) 33.1464i 1.36810i 0.729435 + 0.684050i \(0.239783\pi\)
−0.729435 + 0.684050i \(0.760217\pi\)
\(588\) 0 0
\(589\) 5.79796 0.238901
\(590\) 0 0
\(591\) −22.2929 −0.917006
\(592\) 0 0
\(593\) 1.10102i 0.0452135i 0.999744 + 0.0226067i \(0.00719656\pi\)
−0.999744 + 0.0226067i \(0.992803\pi\)
\(594\) 0 0
\(595\) 0.449490 4.44949i 0.0184273 0.182411i
\(596\) 0 0
\(597\) 17.3939i 0.711884i
\(598\) 0 0
\(599\) −22.8990 −0.935627 −0.467813 0.883827i \(-0.654958\pi\)
−0.467813 + 0.883827i \(0.654958\pi\)
\(600\) 0 0
\(601\) 19.3939 0.791093 0.395546 0.918446i \(-0.370555\pi\)
0.395546 + 0.918446i \(0.370555\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) −2.92168 + 28.9217i −0.118783 + 1.17583i
\(606\) 0 0
\(607\) 25.3939i 1.03071i 0.856978 + 0.515353i \(0.172339\pi\)
−0.856978 + 0.515353i \(0.827661\pi\)
\(608\) 0 0
\(609\) −7.10102 −0.287748
\(610\) 0 0
\(611\) −0.404082 −0.0163474
\(612\) 0 0
\(613\) 8.20204i 0.331277i 0.986187 + 0.165639i \(0.0529685\pi\)
−0.986187 + 0.165639i \(0.947031\pi\)
\(614\) 0 0
\(615\) 59.3939 + 6.00000i 2.39499 + 0.241943i
\(616\) 0 0
\(617\) 9.59592i 0.386317i −0.981168 0.193159i \(-0.938127\pi\)
0.981168 0.193159i \(-0.0618732\pi\)
\(618\) 0 0
\(619\) −46.4495 −1.86696 −0.933481 0.358626i \(-0.883245\pi\)
−0.933481 + 0.358626i \(0.883245\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 23.0000 + 9.79796i 0.920000 + 0.391918i
\(626\) 0 0
\(627\) 77.3939i 3.09081i
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −6.49490 −0.258558 −0.129279 0.991608i \(-0.541266\pi\)
−0.129279 + 0.991608i \(0.541266\pi\)
\(632\) 0 0
\(633\) 29.3939i 1.16830i
\(634\) 0 0
\(635\) −11.3485 1.14643i −0.450350 0.0454946i
\(636\) 0 0
\(637\) 0.449490i 0.0178094i
\(638\) 0 0
\(639\) −32.6969 −1.29347
\(640\) 0 0
\(641\) 6.20204 0.244966 0.122483 0.992471i \(-0.460914\pi\)
0.122483 + 0.992471i \(0.460914\pi\)
\(642\) 0 0
\(643\) 9.14643i 0.360700i −0.983603 0.180350i \(-0.942277\pi\)
0.983603 0.180350i \(-0.0577230\pi\)
\(644\) 0 0
\(645\) 4.89898 48.4949i 0.192897 1.90948i
\(646\) 0 0
\(647\) 22.2929i 0.876423i −0.898872 0.438211i \(-0.855612\pi\)
0.898872 0.438211i \(-0.144388\pi\)
\(648\) 0 0
\(649\) 31.5959 1.24025
\(650\) 0 0
\(651\) −2.20204 −0.0863048
\(652\) 0 0
\(653\) 39.7980i 1.55741i 0.627387 + 0.778707i \(0.284124\pi\)
−0.627387 + 0.778707i \(0.715876\pi\)
\(654\) 0 0
\(655\) −0.348469 + 3.44949i −0.0136158 + 0.134783i
\(656\) 0 0
\(657\) 20.6969i 0.807464i
\(658\) 0 0
\(659\) 7.10102 0.276616 0.138308 0.990389i \(-0.455834\pi\)
0.138308 + 0.990389i \(0.455834\pi\)
\(660\) 0 0
\(661\) 12.9444 0.503478 0.251739 0.967795i \(-0.418997\pi\)
0.251739 + 0.967795i \(0.418997\pi\)
\(662\) 0 0
\(663\) 2.20204i 0.0855202i
\(664\) 0 0
\(665\) −14.3485 1.44949i −0.556410 0.0562088i
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 0 0
\(669\) 9.79796 0.378811
\(670\) 0 0
\(671\) −41.3939 −1.59799
\(672\) 0 0
\(673\) 1.79796i 0.0693062i −0.999399 0.0346531i \(-0.988967\pi\)
0.999399 0.0346531i \(-0.0110326\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.5505i 1.21258i −0.795242 0.606292i \(-0.792656\pi\)
0.795242 0.606292i \(-0.207344\pi\)
\(678\) 0 0
\(679\) 3.79796 0.145752
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 35.5959i 1.36204i −0.732265 0.681020i \(-0.761537\pi\)
0.732265 0.681020i \(-0.238463\pi\)
\(684\) 0 0
\(685\) −39.5959 4.00000i −1.51288 0.152832i
\(686\) 0 0
\(687\) 37.1010i 1.41549i
\(688\) 0 0
\(689\) −0.494897 −0.0188541
\(690\) 0 0
\(691\) 13.1464 0.500114 0.250057 0.968231i \(-0.419551\pi\)
0.250057 + 0.968231i \(0.419551\pi\)
\(692\) 0 0
\(693\) 14.6969i 0.558291i
\(694\) 0 0
\(695\) −1.44949 + 14.3485i −0.0549823 + 0.544268i
\(696\) 0 0
\(697\) 21.7980i 0.825657i
\(698\) 0 0
\(699\) 24.9898 0.945201
\(700\) 0 0
\(701\) 40.6969 1.53710 0.768551 0.639788i \(-0.220978\pi\)
0.768551 + 0.639788i \(0.220978\pi\)
\(702\) 0 0
\(703\) 12.8990i 0.486494i
\(704\) 0 0
\(705\) 0.494897 4.89898i 0.0186389 0.184506i
\(706\) 0 0
\(707\) 8.44949i 0.317776i
\(708\) 0 0
\(709\) −40.2929 −1.51323 −0.756615 0.653861i \(-0.773148\pi\)
−0.756615 + 0.653861i \(0.773148\pi\)
\(710\) 0 0
\(711\) 8.69694 0.326161
\(712\) 0 0
\(713\) 6.20204i 0.232268i
\(714\) 0 0
\(715\) 4.89898 + 0.494897i 0.183211 + 0.0185081i
\(716\) 0 0
\(717\) 63.1918i 2.35994i
\(718\) 0 0
\(719\) 44.4949 1.65938 0.829690 0.558225i \(-0.188517\pi\)
0.829690 + 0.558225i \(0.188517\pi\)
\(720\) 0 0
\(721\) −3.10102 −0.115488
\(722\) 0 0
\(723\) 50.6969i 1.88544i
\(724\) 0 0
\(725\) −14.2020 2.89898i −0.527451 0.107665i
\(726\) 0 0
\(727\) 6.69694i 0.248376i 0.992259 + 0.124188i \(0.0396325\pi\)
−0.992259 + 0.124188i \(0.960367\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −17.7980 −0.658281
\(732\) 0 0
\(733\) 43.6413i 1.61193i 0.591964 + 0.805965i \(0.298353\pi\)
−0.591964 + 0.805965i \(0.701647\pi\)
\(734\) 0 0
\(735\) 5.44949 + 0.550510i 0.201007 + 0.0203059i
\(736\) 0 0
\(737\) 39.1918i 1.44365i
\(738\) 0 0
\(739\) −44.4949 −1.63677 −0.818386 0.574669i \(-0.805131\pi\)
−0.818386 + 0.574669i \(0.805131\pi\)
\(740\) 0 0
\(741\) −7.10102 −0.260863
\(742\) 0 0
\(743\) 15.3031i 0.561415i −0.959793 0.280707i \(-0.909431\pi\)
0.959793 0.280707i \(-0.0905691\pi\)
\(744\) 0 0
\(745\) 3.55051 35.1464i 0.130081 1.28767i
\(746\) 0 0
\(747\) 7.34847i 0.268866i
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 22.2020 0.810164 0.405082 0.914280i \(-0.367243\pi\)
0.405082 + 0.914280i \(0.367243\pi\)
\(752\) 0 0
\(753\) 3.79796i 0.138405i
\(754\) 0 0
\(755\) −4.40408 + 43.5959i −0.160281 + 1.58662i
\(756\) 0 0
\(757\) 32.2020i 1.17040i −0.810888 0.585202i \(-0.801015\pi\)
0.810888 0.585202i \(-0.198985\pi\)
\(758\) 0 0
\(759\) −82.7878 −3.00501
\(760\) 0 0
\(761\) −30.8990 −1.12009 −0.560044 0.828463i \(-0.689216\pi\)
−0.560044 + 0.828463i \(0.689216\pi\)
\(762\) 0 0
\(763\) 2.89898i 0.104950i
\(764\) 0 0
\(765\) 13.3485 + 1.34847i 0.482615 + 0.0487540i
\(766\) 0 0
\(767\) 2.89898i 0.104676i
\(768\) 0 0
\(769\) 11.3031 0.407599 0.203799 0.979013i \(-0.434671\pi\)
0.203799 + 0.979013i \(0.434671\pi\)
\(770\) 0 0
\(771\) −50.6969 −1.82581
\(772\) 0 0
\(773\) 13.3485i 0.480111i 0.970759 + 0.240056i \(0.0771657\pi\)
−0.970759 + 0.240056i \(0.922834\pi\)
\(774\) 0 0
\(775\) −4.40408 0.898979i −0.158199 0.0322923i
\(776\) 0 0
\(777\) 4.89898i 0.175750i
\(778\) 0 0
\(779\) −70.2929 −2.51850
\(780\) 0 0
\(781\) −53.3939 −1.91058
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −18.7980 1.89898i −0.670928 0.0677775i
\(786\) 0 0
\(787\) 45.5505i 1.62370i −0.583866 0.811850i \(-0.698461\pi\)
0.583866 0.811850i \(-0.301539\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0.202041 0.00718375
\(792\) 0 0
\(793\) 3.79796i 0.134869i
\(794\) 0 0
\(795\) 0.606123 6.00000i 0.0214970 0.212798i
\(796\) 0 0
\(797\) 52.9444i 1.87539i 0.347464 + 0.937693i \(0.387043\pi\)
−0.347464 + 0.937693i \(0.612957\pi\)
\(798\) 0 0
\(799\) −1.79796 −0.0636072
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 33.7980i 1.19270i
\(804\) 0 0
\(805\) −1.55051 + 15.3485i −0.0546483 + 0.540962i
\(806\) 0 0
\(807\) 37.1010i 1.30602i
\(808\) 0 0
\(809\) −8.40408 −0.295472 −0.147736 0.989027i \(-0.547199\pi\)