Properties

Label 1620.2.d.e.649.8
Level $1620$
Weight $2$
Character 1620.649
Analytic conductor $12.936$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.28356903014400.1
Defining polynomial: \(x^{8} - 3 x^{6} + 20 x^{4} - 75 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.8
Root \(-2.07237 - 0.839805i\) of defining polynomial
Character \(\chi\) \(=\) 1620.649
Dual form 1620.2.d.e.649.7

$q$-expansion

\(f(q)\) \(=\) \(q+(2.07237 + 0.839805i) q^{5} -4.93536i q^{7} +O(q^{10})\) \(q+(2.07237 + 0.839805i) q^{5} -4.93536i q^{7} +2.41269 q^{11} +2.90917i q^{13} +6.86869i q^{17} +4.17891 q^{19} -3.35922i q^{23} +(3.58945 + 3.48078i) q^{25} +5.19615 q^{29} -6.17891 q^{31} +(4.14474 - 10.2279i) q^{35} -7.84453i q^{37} +5.87680 q^{41} +4.93536i q^{43} -11.9075i q^{47} -17.3578 q^{49} -8.54830i q^{53} +(5.00000 + 2.02619i) q^{55} -1.05141 q^{59} +9.17891 q^{61} +(-2.44314 + 6.02889i) q^{65} -4.05239i q^{67} +14.1663 q^{71} +2.02619i q^{73} -11.9075i q^{77} -6.00000 q^{79} +5.18908i q^{83} +(-5.76836 + 14.2345i) q^{85} +3.09334 q^{89} +14.3578 q^{91} +(8.66025 + 3.50947i) q^{95} -0.882978i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 12q^{19} + 6q^{25} - 4q^{31} - 48q^{49} + 40q^{55} + 28q^{61} - 48q^{79} + 22q^{85} + 24q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.07237 + 0.839805i 0.926793 + 0.375572i
\(6\) 0 0
\(7\) 4.93536i 1.86539i −0.360663 0.932696i \(-0.617450\pi\)
0.360663 0.932696i \(-0.382550\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.41269 0.727455 0.363727 0.931505i \(-0.381504\pi\)
0.363727 + 0.931505i \(0.381504\pi\)
\(12\) 0 0
\(13\) 2.90917i 0.806859i 0.915011 + 0.403429i \(0.132182\pi\)
−0.915011 + 0.403429i \(0.867818\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.86869i 1.66590i 0.553347 + 0.832951i \(0.313350\pi\)
−0.553347 + 0.832951i \(0.686650\pi\)
\(18\) 0 0
\(19\) 4.17891 0.958707 0.479354 0.877622i \(-0.340871\pi\)
0.479354 + 0.877622i \(0.340871\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.35922i 0.700446i −0.936666 0.350223i \(-0.886106\pi\)
0.936666 0.350223i \(-0.113894\pi\)
\(24\) 0 0
\(25\) 3.58945 + 3.48078i 0.717891 + 0.696156i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.19615 0.964901 0.482451 0.875923i \(-0.339747\pi\)
0.482451 + 0.875923i \(0.339747\pi\)
\(30\) 0 0
\(31\) −6.17891 −1.10976 −0.554882 0.831929i \(-0.687237\pi\)
−0.554882 + 0.831929i \(0.687237\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.14474 10.2279i 0.700590 1.72883i
\(36\) 0 0
\(37\) 7.84453i 1.28963i −0.764337 0.644817i \(-0.776934\pi\)
0.764337 0.644817i \(-0.223066\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.87680 0.917801 0.458901 0.888488i \(-0.348243\pi\)
0.458901 + 0.888488i \(0.348243\pi\)
\(42\) 0 0
\(43\) 4.93536i 0.752636i 0.926491 + 0.376318i \(0.122810\pi\)
−0.926491 + 0.376318i \(0.877190\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.9075i 1.73689i −0.495785 0.868445i \(-0.665120\pi\)
0.495785 0.868445i \(-0.334880\pi\)
\(48\) 0 0
\(49\) −17.3578 −2.47969
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.54830i 1.17420i −0.809515 0.587100i \(-0.800270\pi\)
0.809515 0.587100i \(-0.199730\pi\)
\(54\) 0 0
\(55\) 5.00000 + 2.02619i 0.674200 + 0.273212i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.05141 −0.136882 −0.0684408 0.997655i \(-0.521802\pi\)
−0.0684408 + 0.997655i \(0.521802\pi\)
\(60\) 0 0
\(61\) 9.17891 1.17524 0.587619 0.809137i \(-0.300065\pi\)
0.587619 + 0.809137i \(0.300065\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.44314 + 6.02889i −0.303034 + 0.747791i
\(66\) 0 0
\(67\) 4.05239i 0.495078i −0.968878 0.247539i \(-0.920378\pi\)
0.968878 0.247539i \(-0.0796218\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.1663 1.68123 0.840614 0.541634i \(-0.182194\pi\)
0.840614 + 0.541634i \(0.182194\pi\)
\(72\) 0 0
\(73\) 2.02619i 0.237148i 0.992945 + 0.118574i \(0.0378323\pi\)
−0.992945 + 0.118574i \(0.962168\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.9075i 1.35699i
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.18908i 0.569576i 0.958591 + 0.284788i \(0.0919231\pi\)
−0.958591 + 0.284788i \(0.908077\pi\)
\(84\) 0 0
\(85\) −5.76836 + 14.2345i −0.625667 + 1.54395i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.09334 0.327893 0.163947 0.986469i \(-0.447578\pi\)
0.163947 + 0.986469i \(0.447578\pi\)
\(90\) 0 0
\(91\) 14.3578 1.50511
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.66025 + 3.50947i 0.888523 + 0.360064i
\(96\) 0 0
\(97\) 0.882978i 0.0896528i −0.998995 0.0448264i \(-0.985727\pi\)
0.998995 0.0448264i \(-0.0142735\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.87680 −0.584763 −0.292382 0.956302i \(-0.594448\pi\)
−0.292382 + 0.956302i \(0.594448\pi\)
\(102\) 0 0
\(103\) 9.87073i 0.972592i 0.873794 + 0.486296i \(0.161652\pi\)
−0.873794 + 0.486296i \(0.838348\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.5871i 1.27817i 0.769136 + 0.639085i \(0.220687\pi\)
−0.769136 + 0.639085i \(0.779313\pi\)
\(114\) 0 0
\(115\) 2.82109 6.96156i 0.263068 0.649169i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 33.8995 3.10756
\(120\) 0 0
\(121\) −5.17891 −0.470810
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.51551 + 10.2279i 0.403879 + 0.914812i
\(126\) 0 0
\(127\) 4.93536i 0.437943i −0.975731 0.218971i \(-0.929730\pi\)
0.975731 0.218971i \(-0.0702701\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.1663 1.23771 0.618857 0.785504i \(-0.287596\pi\)
0.618857 + 0.785504i \(0.287596\pi\)
\(132\) 0 0
\(133\) 20.6244i 1.78837i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.03883i 0.430496i 0.976559 + 0.215248i \(0.0690561\pi\)
−0.976559 + 0.215248i \(0.930944\pi\)
\(138\) 0 0
\(139\) −5.82109 −0.493739 −0.246869 0.969049i \(-0.579402\pi\)
−0.246869 + 0.969049i \(0.579402\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.01894i 0.586953i
\(144\) 0 0
\(145\) 10.7684 + 4.36376i 0.894264 + 0.362390i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.1758 −1.07940 −0.539700 0.841857i \(-0.681462\pi\)
−0.539700 + 0.841857i \(0.681462\pi\)
\(150\) 0 0
\(151\) −6.17891 −0.502832 −0.251416 0.967879i \(-0.580896\pi\)
−0.251416 + 0.967879i \(0.580896\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −12.8050 5.18908i −1.02852 0.416797i
\(156\) 0 0
\(157\) 7.84453i 0.626062i 0.949743 + 0.313031i \(0.101344\pi\)
−0.949743 + 0.313031i \(0.898656\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.5790 −1.30661
\(162\) 0 0
\(163\) 10.7537i 0.842295i −0.906992 0.421148i \(-0.861627\pi\)
0.906992 0.421148i \(-0.138373\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.54830i 0.661487i 0.943721 + 0.330744i \(0.107300\pi\)
−0.943721 + 0.330744i \(0.892700\pi\)
\(168\) 0 0
\(169\) 4.53673 0.348979
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.7762i 1.42753i −0.700386 0.713765i \(-0.746989\pi\)
0.700386 0.713765i \(-0.253011\pi\)
\(174\) 0 0
\(175\) 17.1789 17.7153i 1.29860 1.33915i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −7.97961 −0.596424 −0.298212 0.954500i \(-0.596390\pi\)
−0.298212 + 0.954500i \(0.596390\pi\)
\(180\) 0 0
\(181\) 3.82109 0.284020 0.142010 0.989865i \(-0.454644\pi\)
0.142010 + 0.989865i \(0.454644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.58788 16.2568i 0.484351 1.19522i
\(186\) 0 0
\(187\) 16.5720i 1.21187i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.61832 −0.478885 −0.239443 0.970911i \(-0.576965\pi\)
−0.239443 + 0.970911i \(0.576965\pi\)
\(192\) 0 0
\(193\) 21.7676i 1.56687i −0.621474 0.783435i \(-0.713466\pi\)
0.621474 0.783435i \(-0.286534\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.4170i 1.09842i −0.835686 0.549208i \(-0.814930\pi\)
0.835686 0.549208i \(-0.185070\pi\)
\(198\) 0 0
\(199\) −20.3578 −1.44313 −0.721564 0.692348i \(-0.756576\pi\)
−0.721564 + 0.692348i \(0.756576\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 25.6449i 1.79992i
\(204\) 0 0
\(205\) 12.1789 + 4.93536i 0.850612 + 0.344701i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0824 0.697416
\(210\) 0 0
\(211\) 16.1789 1.11380 0.556901 0.830579i \(-0.311990\pi\)
0.556901 + 0.830579i \(0.311990\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.14474 + 10.2279i −0.282669 + 0.697538i
\(216\) 0 0
\(217\) 30.4952i 2.07015i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −19.9822 −1.34415
\(222\) 0 0
\(223\) 25.5598i 1.71161i 0.517298 + 0.855805i \(0.326938\pi\)
−0.517298 + 0.855805i \(0.673062\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.9265i 1.25619i 0.778135 + 0.628097i \(0.216166\pi\)
−0.778135 + 0.628097i \(0.783834\pi\)
\(228\) 0 0
\(229\) −6.82109 −0.450750 −0.225375 0.974272i \(-0.572361\pi\)
−0.225375 + 0.974272i \(0.572361\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.150248i 0.00984309i −0.999988 0.00492154i \(-0.998433\pi\)
0.999988 0.00492154i \(-0.00156658\pi\)
\(234\) 0 0
\(235\) 10.0000 24.6768i 0.652328 1.60974i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.03102 −0.584168 −0.292084 0.956393i \(-0.594349\pi\)
−0.292084 + 0.956393i \(0.594349\pi\)
\(240\) 0 0
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −35.9719 14.5772i −2.29816 0.931302i
\(246\) 0 0
\(247\) 12.1572i 0.773541i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205 1.09326 0.546630 0.837374i \(-0.315910\pi\)
0.546630 + 0.837374i \(0.315910\pi\)
\(252\) 0 0
\(253\) 8.10477i 0.509543i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.1354i 1.38077i 0.723442 + 0.690385i \(0.242559\pi\)
−0.723442 + 0.690385i \(0.757441\pi\)
\(258\) 0 0
\(259\) −38.7156 −2.40567
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.71844i 0.414277i 0.978312 + 0.207138i \(0.0664151\pi\)
−0.978312 + 0.207138i \(0.933585\pi\)
\(264\) 0 0
\(265\) 7.17891 17.7153i 0.440997 1.08824i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −30.0646 −1.83307 −0.916536 0.399952i \(-0.869027\pi\)
−0.916536 + 0.399952i \(0.869027\pi\)
\(270\) 0 0
\(271\) −22.3578 −1.35814 −0.679070 0.734073i \(-0.737617\pi\)
−0.679070 + 0.734073i \(0.737617\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.66025 + 8.39805i 0.522233 + 0.506422i
\(276\) 0 0
\(277\) 10.7537i 0.646128i −0.946377 0.323064i \(-0.895287\pi\)
0.946377 0.323064i \(-0.104713\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.4342 −0.741764 −0.370882 0.928680i \(-0.620945\pi\)
−0.370882 + 0.928680i \(0.620945\pi\)
\(282\) 0 0
\(283\) 1.76596i 0.104975i −0.998622 0.0524876i \(-0.983285\pi\)
0.998622 0.0524876i \(-0.0167150\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 29.0041i 1.71206i
\(288\) 0 0
\(289\) −30.1789 −1.77523
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.6061i 1.20382i 0.798564 + 0.601910i \(0.205593\pi\)
−0.798564 + 0.601910i \(0.794407\pi\)
\(294\) 0 0
\(295\) −2.17891 0.882978i −0.126861 0.0514090i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.77255 0.565161
\(300\) 0 0
\(301\) 24.3578 1.40396
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19.0221 + 7.70850i 1.08920 + 0.441387i
\(306\) 0 0
\(307\) 8.98775i 0.512958i 0.966550 + 0.256479i \(0.0825624\pi\)
−0.966550 + 0.256479i \(0.917438\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.7022 0.606865 0.303433 0.952853i \(-0.401867\pi\)
0.303433 + 0.952853i \(0.401867\pi\)
\(312\) 0 0
\(313\) 6.96156i 0.393490i −0.980455 0.196745i \(-0.936963\pi\)
0.980455 0.196745i \(-0.0630372\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.0430i 1.91204i 0.293297 + 0.956021i \(0.405248\pi\)
−0.293297 + 0.956021i \(0.594752\pi\)
\(318\) 0 0
\(319\) 12.5367 0.701922
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 28.7036i 1.59711i
\(324\) 0 0
\(325\) −10.1262 + 10.4423i −0.561699 + 0.579237i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −58.7680 −3.23998
\(330\) 0 0
\(331\) −32.5367 −1.78838 −0.894190 0.447688i \(-0.852248\pi\)
−0.894190 + 0.447688i \(0.852248\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.40322 8.39805i 0.185938 0.458835i
\(336\) 0 0
\(337\) 9.87073i 0.537693i 0.963183 + 0.268846i \(0.0866424\pi\)
−0.963183 + 0.268846i \(0.913358\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.9078 −0.807303
\(342\) 0 0
\(343\) 51.1196i 2.76020i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.54830i 0.458897i 0.973321 + 0.229448i \(0.0736922\pi\)
−0.973321 + 0.229448i \(0.926308\pi\)
\(348\) 0 0
\(349\) −20.1789 −1.08015 −0.540076 0.841616i \(-0.681605\pi\)
−0.540076 + 0.841616i \(0.681605\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.9852i 1.17015i 0.810978 + 0.585077i \(0.198936\pi\)
−0.810978 + 0.585077i \(0.801064\pi\)
\(354\) 0 0
\(355\) 29.3578 + 11.8969i 1.55815 + 0.631423i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.309878 0.0163548 0.00817738 0.999967i \(-0.497397\pi\)
0.00817738 + 0.999967i \(0.497397\pi\)
\(360\) 0 0
\(361\) −1.53673 −0.0808803
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.70161 + 4.19903i −0.0890662 + 0.219787i
\(366\) 0 0
\(367\) 9.87073i 0.515248i 0.966245 + 0.257624i \(0.0829396\pi\)
−0.966245 + 0.257624i \(0.917060\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −42.1890 −2.19034
\(372\) 0 0
\(373\) 8.98775i 0.465368i 0.972552 + 0.232684i \(0.0747508\pi\)
−0.972552 + 0.232684i \(0.925249\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.1165i 0.778539i
\(378\) 0 0
\(379\) 0.357817 0.0183798 0.00918990 0.999958i \(-0.497075\pi\)
0.00918990 + 0.999958i \(0.497075\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.7374i 0.701947i −0.936385 0.350974i \(-0.885851\pi\)
0.936385 0.350974i \(-0.114149\pi\)
\(384\) 0 0
\(385\) 10.0000 24.6768i 0.509647 1.25765i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.6100 1.29848 0.649239 0.760584i \(-0.275087\pi\)
0.649239 + 0.760584i \(0.275087\pi\)
\(390\) 0 0
\(391\) 23.0735 1.16687
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.4342 5.03883i −0.625634 0.253531i
\(396\) 0 0
\(397\) 11.0139i 0.552774i −0.961046 0.276387i \(-0.910863\pi\)
0.961046 0.276387i \(-0.0891371\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.7237 1.03489 0.517447 0.855715i \(-0.326883\pi\)
0.517447 + 0.855715i \(0.326883\pi\)
\(402\) 0 0
\(403\) 17.9755i 0.895423i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 18.9265i 0.938150i
\(408\) 0 0
\(409\) −6.82109 −0.337281 −0.168641 0.985678i \(-0.553938\pi\)
−0.168641 + 0.985678i \(0.553938\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.18908i 0.255338i
\(414\) 0 0
\(415\) −4.35782 + 10.7537i −0.213917 + 0.527879i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.5790 −0.809936 −0.404968 0.914331i \(-0.632717\pi\)
−0.404968 + 0.914331i \(0.632717\pi\)
\(420\) 0 0
\(421\) 29.7156 1.44825 0.724126 0.689668i \(-0.242244\pi\)
0.724126 + 0.689668i \(0.242244\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −23.9084 + 24.6548i −1.15973 + 1.19594i
\(426\) 0 0
\(427\) 45.3013i 2.19228i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.67116 −0.0804972 −0.0402486 0.999190i \(-0.512815\pi\)
−0.0402486 + 0.999190i \(0.512815\pi\)
\(432\) 0 0
\(433\) 36.5737i 1.75762i −0.477170 0.878811i \(-0.658337\pi\)
0.477170 0.878811i \(-0.341663\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.0379i 0.671523i
\(438\) 0 0
\(439\) −18.5367 −0.884710 −0.442355 0.896840i \(-0.645857\pi\)
−0.442355 + 0.896840i \(0.645857\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.6260i 0.884946i −0.896782 0.442473i \(-0.854101\pi\)
0.896782 0.442473i \(-0.145899\pi\)
\(444\) 0 0
\(445\) 6.41055 + 2.59780i 0.303889 + 0.123148i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −25.1783 −1.18824 −0.594120 0.804377i \(-0.702500\pi\)
−0.594120 + 0.804377i \(0.702500\pi\)
\(450\) 0 0
\(451\) 14.1789 0.667659
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 29.7547 + 12.0578i 1.39492 + 0.565277i
\(456\) 0 0
\(457\) 27.5860i 1.29042i 0.764006 + 0.645209i \(0.223230\pi\)
−0.764006 + 0.645209i \(0.776770\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −10.7022 −0.498450 −0.249225 0.968446i \(-0.580176\pi\)
−0.249225 + 0.968446i \(0.580176\pi\)
\(462\) 0 0
\(463\) 29.6122i 1.37619i −0.725618 0.688097i \(-0.758446\pi\)
0.725618 0.688097i \(-0.241554\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.5672i 0.720366i 0.932882 + 0.360183i \(0.117286\pi\)
−0.932882 + 0.360183i \(0.882714\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.9075i 0.547508i
\(474\) 0 0
\(475\) 15.0000 + 14.5459i 0.688247 + 0.667410i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.3001 1.15599 0.577996 0.816040i \(-0.303835\pi\)
0.577996 + 0.816040i \(0.303835\pi\)
\(480\) 0 0
\(481\) 22.8211 1.04055
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.741529 1.82986i 0.0336711 0.0830896i
\(486\) 0 0
\(487\) 17.9755i 0.814548i −0.913306 0.407274i \(-0.866479\pi\)
0.913306 0.407274i \(-0.133521\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −35.5707 −1.60528 −0.802641 0.596463i \(-0.796572\pi\)
−0.802641 + 0.596463i \(0.796572\pi\)
\(492\) 0 0
\(493\) 35.6908i 1.60743i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 69.9158i 3.13615i
\(498\) 0 0
\(499\) −4.17891 −0.187074 −0.0935368 0.995616i \(-0.529817\pi\)
−0.0935368 + 0.995616i \(0.529817\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.2519i 1.66098i −0.557032 0.830491i \(-0.688060\pi\)
0.557032 0.830491i \(-0.311940\pi\)
\(504\) 0 0
\(505\) −12.1789 4.93536i −0.541954 0.219621i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.54796 0.334557 0.167279 0.985910i \(-0.446502\pi\)
0.167279 + 0.985910i \(0.446502\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.28949 + 20.4558i −0.365279 + 0.901391i
\(516\) 0 0
\(517\) 28.7292i 1.26351i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.03102 −0.395656 −0.197828 0.980237i \(-0.563389\pi\)
−0.197828 + 0.980237i \(0.563389\pi\)
\(522\) 0 0
\(523\) 7.58430i 0.331638i −0.986156 0.165819i \(-0.946973\pi\)
0.986156 0.165819i \(-0.0530268\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.4410i 1.84876i
\(528\) 0 0
\(529\) 11.7156 0.509375
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 17.0966i 0.740536i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −41.8791 −1.80386
\(540\) 0 0
\(541\) 3.35782 0.144364 0.0721819 0.997391i \(-0.477004\pi\)
0.0721819 + 0.997391i \(0.477004\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.5066 5.87864i −0.621395 0.251813i
\(546\) 0 0
\(547\) 19.7415i 0.844084i −0.906576 0.422042i \(-0.861314\pi\)
0.906576 0.422042i \(-0.138686\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21.7142 0.925058
\(552\) 0 0
\(553\) 29.6122i 1.25924i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.2320i 1.66231i 0.556037 + 0.831157i \(0.312321\pi\)
−0.556037 + 0.831157i \(0.687679\pi\)
\(558\) 0 0
\(559\) −14.3578 −0.607271
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.9075i 0.501842i 0.968008 + 0.250921i \(0.0807335\pi\)
−0.968008 + 0.250921i \(0.919267\pi\)
\(564\) 0 0
\(565\) −11.4105 + 28.1576i −0.480045 + 1.18460i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.45462 −0.186748 −0.0933738 0.995631i \(-0.529765\pi\)
−0.0933738 + 0.995631i \(0.529765\pi\)
\(570\) 0 0
\(571\) −18.1789 −0.760764 −0.380382 0.924830i \(-0.624207\pi\)
−0.380382 + 0.924830i \(0.624207\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.6927 12.0578i 0.487619 0.502844i
\(576\) 0 0
\(577\) 7.84453i 0.326572i −0.986579 0.163286i \(-0.947791\pi\)
0.986579 0.163286i \(-0.0522094\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 25.6100 1.06248
\(582\) 0 0
\(583\) 20.6244i 0.854177i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.6070i 0.479073i 0.970887 + 0.239537i \(0.0769955\pi\)
−0.970887 + 0.239537i \(0.923004\pi\)
\(588\) 0 0
\(589\) −25.8211 −1.06394
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.03883i 0.206920i −0.994634 0.103460i \(-0.967009\pi\)
0.994634 0.103460i \(-0.0329914\pi\)
\(594\) 0 0
\(595\) 70.2524 + 28.4690i 2.88007 + 1.16711i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.0106 −0.695035 −0.347518 0.937673i \(-0.612975\pi\)
−0.347518 + 0.937673i \(0.612975\pi\)
\(600\) 0 0
\(601\) −3.35782 −0.136968 −0.0684841 0.997652i \(-0.521816\pi\)
−0.0684841 + 0.997652i \(0.521816\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.7326 4.34927i −0.436343 0.176823i
\(606\) 0 0
\(607\) 4.93536i 0.200320i 0.994971 + 0.100160i \(0.0319355\pi\)
−0.994971 + 0.100160i \(0.968064\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 34.6410 1.40143
\(612\) 0 0
\(613\) 37.7170i 1.52337i 0.647945 + 0.761687i \(0.275628\pi\)
−0.647945 + 0.761687i \(0.724372\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.33933i 0.214953i 0.994208 + 0.107477i \(0.0342771\pi\)
−0.994208 + 0.107477i \(0.965723\pi\)
\(618\) 0 0
\(619\) −11.6422 −0.467939 −0.233969 0.972244i \(-0.575172\pi\)
−0.233969 + 0.972244i \(0.575172\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.2667i 0.611649i
\(624\) 0 0
\(625\) 0.768363 + 24.9882i 0.0307345 + 0.999528i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 53.8817 2.14840
\(630\) 0 0
\(631\) −22.5367 −0.897173 −0.448586 0.893739i \(-0.648072\pi\)
−0.448586 + 0.893739i \(0.648072\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.14474 10.2279i 0.164479 0.405882i
\(636\) 0 0
\(637\) 50.4969i 2.00076i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.3110 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(642\) 0 0
\(643\) 20.6244i 0.813348i 0.913573 + 0.406674i \(0.133312\pi\)
−0.913573 + 0.406674i \(0.866688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 44.2709i 1.74047i 0.492639 + 0.870234i \(0.336032\pi\)
−0.492639 + 0.870234i \(0.663968\pi\)
\(648\) 0 0
\(649\) −2.53673 −0.0995752
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.300496i 0.0117593i −0.999983 0.00587967i \(-0.998128\pi\)
0.999983 0.00587967i \(-0.00187157\pi\)
\(654\) 0 0
\(655\) 29.3578 + 11.8969i 1.14710 + 0.464851i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 41.4474 1.61456 0.807282 0.590166i \(-0.200938\pi\)
0.807282 + 0.590166i \(0.200938\pi\)
\(660\) 0 0
\(661\) −46.4313 −1.80597 −0.902983 0.429675i \(-0.858628\pi\)
−0.902983 + 0.429675i \(0.858628\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 17.3205 42.7415i 0.671660 1.65744i
\(666\) 0 0
\(667\) 17.4550i 0.675861i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 22.1459 0.854933
\(672\) 0 0
\(673\) 14.5459i 0.560701i −0.959898 0.280351i \(-0.909549\pi\)
0.959898 0.280351i \(-0.0904508\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.4937i 1.32570i 0.748752 + 0.662850i \(0.230653\pi\)
−0.748752 + 0.662850i \(0.769347\pi\)
\(678\) 0 0
\(679\) −4.35782 −0.167238
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.5714i 1.70548i −0.522339 0.852738i \(-0.674940\pi\)
0.522339 0.852738i \(-0.325060\pi\)
\(684\) 0 0
\(685\) −4.23164 + 10.4423i −0.161683 + 0.398981i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 24.8685 0.947413
\(690\) 0 0
\(691\) −30.7156 −1.16848 −0.584239 0.811582i \(-0.698607\pi\)
−0.584239 + 0.811582i \(0.698607\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0635 4.88858i −0.457593 0.185435i
\(696\) 0 0
\(697\) 40.3659i 1.52897i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27.3420 −1.03269 −0.516347 0.856379i \(-0.672709\pi\)
−0.516347 + 0.856379i \(0.672709\pi\)
\(702\) 0 0
\(703\) 32.7816i 1.23638i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29.0041i 1.09081i
\(708\) 0 0
\(709\) −15.5367 −0.583494 −0.291747 0.956496i \(-0.594237\pi\)
−0.291747 + 0.956496i \(0.594237\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.7563i 0.777330i
\(714\) 0 0
\(715\) −5.89454 + 14.5459i −0.220443 + 0.543984i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 40.3960 1.50652 0.753259 0.657724i \(-0.228481\pi\)
0.753259 + 0.657724i \(0.228481\pi\)
\(720\) 0 0
\(721\) 48.7156 1.81426
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 18.6514 + 18.0867i 0.692694 + 0.671722i
\(726\) 0 0
\(727\) 47.0672i 1.74563i −0.488055 0.872813i \(-0.662293\pi\)
0.488055 0.872813i \(-0.337707\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −33.8995 −1.25382
\(732\) 0 0
\(733\) 19.7415i 0.729167i −0.931171 0.364584i \(-0.881211\pi\)
0.931171 0.364584i \(-0.118789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.77717i 0.360147i
\(738\) 0 0
\(739\) 22.8945 0.842189 0.421095 0.907017i \(-0.361646\pi\)
0.421095 + 0.907017i \(0.361646\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 49.7604i 1.82553i 0.408481 + 0.912767i \(0.366059\pi\)
−0.408481 + 0.912767i \(0.633941\pi\)
\(744\) 0 0
\(745\) −27.3051 11.0651i −1.00038 0.405393i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −6.35782 −0.232000 −0.116000 0.993249i \(-0.537007\pi\)
−0.116000 + 0.993249i \(0.537007\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.8050 5.18908i −0.466022 0.188850i
\(756\) 0 0
\(757\) 1.76596i 0.0641847i −0.999485 0.0320924i \(-0.989783\pi\)
0.999485 0.0320924i \(-0.0102171\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.73205 −0.0627868 −0.0313934 0.999507i \(-0.509994\pi\)
−0.0313934 + 0.999507i \(0.509994\pi\)
\(762\) 0 0
\(763\) 34.5475i 1.25071i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.05872i 0.110444i
\(768\) 0 0
\(769\) −25.7156 −0.927329 −0.463665 0.886011i \(-0.653466\pi\)
−0.463665 + 0.886011i \(0.653466\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.6837i 1.10362i 0.833971 + 0.551809i \(0.186062\pi\)
−0.833971 + 0.551809i \(0.813938\pi\)
\(774\) 0 0
\(775\) −22.1789 21.5074i −0.796690 0.772569i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.5586 0.879903
\(780\) 0 0
\(781\) 34.1789 1.22302
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.58788 + 16.2568i −0.235132 + 0.580230i
\(786\) 0 0
\(787\) 47.9502i 1.70924i 0.519254 + 0.854620i \(0.326210\pi\)
−0.519254 + 0.854620i \(0.673790\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 67.0574 2.38429
\(792\) 0 0
\(793\) 26.7030i 0.948252i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.7573i 0.416464i −0.978079 0.208232i \(-0.933229\pi\)
0.978079 0.208232i \(-0.0667709\pi\)
\(798\) 0 0
\(799\) 81.7891 2.89349
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.88858i 0.172514i
\(804\) 0 0
\(805\) −34.3578 13.9231i −1.21095 0.490725i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.19615 0.182687