Properties

Label 1620.2.d.e.649.6
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 649.6
Root \(-1.20635 - 1.88274i\) of defining polynomial
Character \(\chi\) \(=\) 1620.649
Dual form 1620.2.d.e.649.5

$q$-expansion

\(f(q)\) \(=\) \(q+(1.20635 + 1.88274i) q^{5} -1.28148i q^{7} +O(q^{10})\) \(q+(1.20635 + 1.88274i) q^{5} -1.28148i q^{7} +4.14474 q^{11} -6.52202i q^{13} -5.98507i q^{17} -7.17891 q^{19} -7.53098i q^{23} +(-2.08945 + 4.54249i) q^{25} -5.19615 q^{29} +5.17891 q^{31} +(2.41269 - 1.54591i) q^{35} +5.24054i q^{37} +0.680643 q^{41} +1.28148i q^{43} -5.31139i q^{47} +5.35782 q^{49} +2.21958i q^{53} +(5.00000 + 7.80350i) q^{55} +7.60885 q^{59} -2.17891 q^{61} +(12.2793 - 7.86782i) q^{65} -15.6070i q^{67} +5.50603 q^{71} +7.80350i q^{73} -5.31139i q^{77} -6.00000 q^{79} -9.75056i q^{83} +(11.2684 - 7.22007i) q^{85} +10.0215 q^{89} -8.35782 q^{91} +(-8.66025 - 13.5161i) q^{95} +14.3255i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 12q^{19} + 6q^{25} - 4q^{31} - 48q^{49} + 40q^{55} + 28q^{61} - 48q^{79} + 22q^{85} + 24q^{91} + O(q^{100}) \)

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
\(5\) 1.20635 + 1.88274i 0.539495 + 0.841989i
\(6\) 0 0
\(7\) 1.28148i 0.484353i −0.970232 0.242176i \(-0.922139\pi\)
0.970232 0.242176i \(-0.0778613\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.14474 1.24969 0.624844 0.780750i \(-0.285163\pi\)
0.624844 + 0.780750i \(0.285163\pi\)
\(12\) 0 0
\(13\) 6.52202i 1.80888i −0.426598 0.904441i \(-0.640288\pi\)
0.426598 0.904441i \(-0.359712\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.98507i 1.45159i −0.687909 0.725797i \(-0.741471\pi\)
0.687909 0.725797i \(-0.258529\pi\)
\(18\) 0 0
\(19\) −7.17891 −1.64695 −0.823477 0.567349i \(-0.807969\pi\)
−0.823477 + 0.567349i \(0.807969\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.53098i 1.57032i −0.619295 0.785159i \(-0.712581\pi\)
0.619295 0.785159i \(-0.287419\pi\)
\(24\) 0 0
\(25\) −2.08945 + 4.54249i −0.417891 + 0.908497i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.19615 −0.964901 −0.482451 0.875923i \(-0.660253\pi\)
−0.482451 + 0.875923i \(0.660253\pi\)
\(30\) 0 0
\(31\) 5.17891 0.930159 0.465080 0.885269i \(-0.346026\pi\)
0.465080 + 0.885269i \(0.346026\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.41269 1.54591i 0.407820 0.261306i
\(36\) 0 0
\(37\) 5.24054i 0.861540i 0.902462 + 0.430770i \(0.141758\pi\)
−0.902462 + 0.430770i \(0.858242\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.680643 0.106299 0.0531493 0.998587i \(-0.483074\pi\)
0.0531493 + 0.998587i \(0.483074\pi\)
\(42\) 0 0
\(43\) 1.28148i 0.195423i 0.995215 + 0.0977117i \(0.0311523\pi\)
−0.995215 + 0.0977117i \(0.968848\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.31139i 0.774747i −0.921923 0.387373i \(-0.873382\pi\)
0.921923 0.387373i \(-0.126618\pi\)
\(48\) 0 0
\(49\) 5.35782 0.765402
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.21958i 0.304883i 0.988312 + 0.152442i \(0.0487136\pi\)
−0.988312 + 0.152442i \(0.951286\pi\)
\(54\) 0 0
\(55\) 5.00000 + 7.80350i 0.674200 + 1.05222i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.60885 0.990587 0.495294 0.868726i \(-0.335060\pi\)
0.495294 + 0.868726i \(0.335060\pi\)
\(60\) 0 0
\(61\) −2.17891 −0.278981 −0.139490 0.990223i \(-0.544546\pi\)
−0.139490 + 0.990223i \(0.544546\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.2793 7.86782i 1.52306 0.975883i
\(66\) 0 0
\(67\) 15.6070i 1.90670i −0.301871 0.953349i \(-0.597611\pi\)
0.301871 0.953349i \(-0.402389\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.50603 0.653446 0.326723 0.945120i \(-0.394056\pi\)
0.326723 + 0.945120i \(0.394056\pi\)
\(72\) 0 0
\(73\) 7.80350i 0.913330i 0.889639 + 0.456665i \(0.150956\pi\)
−0.889639 + 0.456665i \(0.849044\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.31139i 0.605290i
\(78\) 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
\(83\) 9.75056i 1.07026i −0.844769 0.535132i \(-0.820262\pi\)
0.844769 0.535132i \(-0.179738\pi\)
\(84\) 0 0
\(85\) 11.2684 7.22007i 1.22223 0.783127i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0215 1.06228 0.531141 0.847284i \(-0.321764\pi\)
0.531141 + 0.847284i \(0.321764\pi\)
\(90\) 0 0
\(91\) −8.35782 −0.876137
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.66025 13.5161i −0.888523 1.38672i
\(96\) 0 0
\(97\) 14.3255i 1.45454i 0.686354 + 0.727268i \(0.259210\pi\)
−0.686354 + 0.727268i \(0.740790\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.680643 −0.0677265 −0.0338633 0.999426i \(-0.510781\pi\)
−0.0338633 + 0.999426i \(0.510781\pi\)
\(102\) 0 0
\(103\) 2.56295i 0.252535i 0.991996 + 0.126268i \(0.0402998\pi\)
−0.991996 + 0.126268i \(0.959700\pi\)
\(104\) 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
\(113\) 9.07688i 0.853881i 0.904280 + 0.426941i \(0.140409\pi\)
−0.904280 + 0.426941i \(0.859591\pi\)
\(114\) 0 0
\(115\) 14.1789 9.08497i 1.32219 0.847178i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.66973 −0.703083
\(120\) 0 0
\(121\) 6.17891 0.561719
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.0729 + 1.54591i −0.990395 + 0.138270i
\(126\) 0 0
\(127\) 1.28148i 0.113713i −0.998382 0.0568563i \(-0.981892\pi\)
0.998382 0.0568563i \(-0.0181077\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.50603 0.481064 0.240532 0.970641i \(-0.422678\pi\)
0.240532 + 0.970641i \(0.422678\pi\)
\(132\) 0 0
\(133\) 9.19961i 0.797707i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.2965i 0.965122i 0.875862 + 0.482561i \(0.160293\pi\)
−0.875862 + 0.482561i \(0.839707\pi\)
\(138\) 0 0
\(139\) −17.1789 −1.45710 −0.728548 0.684995i \(-0.759804\pi\)
−0.728548 + 0.684995i \(0.759804\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 27.0321i 2.26054i
\(144\) 0 0
\(145\) −6.26836 9.78303i −0.520559 0.812436i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.7332 1.61661 0.808303 0.588766i \(-0.200386\pi\)
0.808303 + 0.588766i \(0.200386\pi\)
\(150\) 0 0
\(151\) 5.17891 0.421454 0.210727 0.977545i \(-0.432417\pi\)
0.210727 + 0.977545i \(0.432417\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.24756 + 9.75056i 0.501816 + 0.783184i
\(156\) 0 0
\(157\) 5.24054i 0.418241i −0.977890 0.209120i \(-0.932940\pi\)
0.977890 0.209120i \(-0.0670600\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.65078 −0.760588
\(162\) 0 0
\(163\) 11.7626i 0.921315i 0.887578 + 0.460657i \(0.152386\pi\)
−0.887578 + 0.460657i \(0.847614\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.21958i 0.171757i −0.996306 0.0858783i \(-0.972630\pi\)
0.996306 0.0858783i \(-0.0273696\pi\)
\(168\) 0 0
\(169\) −29.5367 −2.27206
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.673677i 0.0512187i 0.999672 + 0.0256094i \(0.00815261\pi\)
−0.999672 + 0.0256094i \(0.991847\pi\)
\(174\) 0 0
\(175\) 5.82109 + 2.67759i 0.440033 + 0.202407i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.5370 1.08655 0.543275 0.839555i \(-0.317184\pi\)
0.543275 + 0.839555i \(0.317184\pi\)
\(180\) 0 0
\(181\) 15.1789 1.12824 0.564120 0.825693i \(-0.309216\pi\)
0.564120 + 0.825693i \(0.309216\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.86660 + 6.32191i −0.725407 + 0.464796i
\(186\) 0 0
\(187\) 24.8066i 1.81404i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.2906 1.90232 0.951162 0.308692i \(-0.0998913\pi\)
0.951162 + 0.308692i \(0.0998913\pi\)
\(192\) 0 0
\(193\) 12.9294i 0.930679i −0.885132 0.465339i \(-0.845932\pi\)
0.885132 0.465339i \(-0.154068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.20466i 0.584558i 0.956333 + 0.292279i \(0.0944135\pi\)
−0.956333 + 0.292279i \(0.905586\pi\)
\(198\) 0 0
\(199\) 2.35782 0.167141 0.0835706 0.996502i \(-0.473368\pi\)
0.0835706 + 0.996502i \(0.473368\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.65875i 0.467353i
\(204\) 0 0
\(205\) 0.821092 + 1.28148i 0.0573475 + 0.0895022i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −29.7547 −2.05818
\(210\) 0 0
\(211\) 4.82109 0.331898 0.165949 0.986134i \(-0.446931\pi\)
0.165949 + 0.986134i \(0.446931\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.41269 + 1.54591i −0.164544 + 0.105430i
\(216\) 0 0
\(217\) 6.63665i 0.450525i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −39.0348 −2.62576
\(222\) 0 0
\(223\) 7.91813i 0.530237i −0.964216 0.265119i \(-0.914589\pi\)
0.964216 0.265119i \(-0.0854111\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.7207i 1.44165i −0.693115 0.720827i \(-0.743762\pi\)
0.693115 0.720827i \(-0.256238\pi\)
\(228\) 0 0
\(229\) −18.1789 −1.20130 −0.600648 0.799514i \(-0.705091\pi\)
−0.600648 + 0.799514i \(0.705091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0470i 1.37884i 0.724363 + 0.689418i \(0.242134\pi\)
−0.724363 + 0.689418i \(0.757866\pi\)
\(234\) 0 0
\(235\) 10.0000 6.40739i 0.652328 0.417972i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.1459 1.43250 0.716249 0.697844i \(-0.245857\pi\)
0.716249 + 0.697844i \(0.245857\pi\)
\(240\) 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
\(245\) 6.46339 + 10.0874i 0.412931 + 0.644460i
\(246\) 0 0
\(247\) 46.8210i 2.97915i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.3205 −1.09326 −0.546630 0.837374i \(-0.684090\pi\)
−0.546630 + 0.837374i \(0.684090\pi\)
\(252\) 0 0
\(253\) 31.2140i 1.96241i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.85730i 0.427747i 0.976861 + 0.213873i \(0.0686080\pi\)
−0.976861 + 0.213873i \(0.931392\pi\)
\(258\) 0 0
\(259\) 6.71563 0.417289
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.0620i 0.928760i 0.885636 + 0.464380i \(0.153723\pi\)
−0.885636 + 0.464380i \(0.846277\pi\)
\(264\) 0 0
\(265\) −4.17891 + 2.67759i −0.256708 + 0.164483i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.28001 −0.565812 −0.282906 0.959148i \(-0.591298\pi\)
−0.282906 + 0.959148i \(0.591298\pi\)
\(270\) 0 0
\(271\) 0.357817 0.0217358 0.0108679 0.999941i \(-0.496541\pi\)
0.0108679 + 0.999941i \(0.496541\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.66025 + 18.8274i −0.522233 + 1.13534i
\(276\) 0 0
\(277\) 11.7626i 0.706744i 0.935483 + 0.353372i \(0.114965\pi\)
−0.935483 + 0.353372i \(0.885035\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.23808 −0.431788 −0.215894 0.976417i \(-0.569267\pi\)
−0.215894 + 0.976417i \(0.569267\pi\)
\(282\) 0 0
\(283\) 28.6510i 1.70313i 0.524252 + 0.851563i \(0.324345\pi\)
−0.524252 + 0.851563i \(0.675655\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.872228i 0.0514860i
\(288\) 0 0
\(289\) −18.8211 −1.10712
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.9552i 1.04895i −0.851424 0.524477i \(-0.824261\pi\)
0.851424 0.524477i \(-0.175739\pi\)
\(294\) 0 0
\(295\) 9.17891 + 14.3255i 0.534417 + 0.834064i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −49.1172 −2.84052
\(300\) 0 0
\(301\) 1.64218 0.0946539
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.62852 4.10233i −0.150509 0.234899i
\(306\) 0 0
\(307\) 16.8885i 0.963876i 0.876205 + 0.481938i \(0.160067\pi\)
−0.876205 + 0.481938i \(0.839933\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.97013 0.508650 0.254325 0.967119i \(-0.418147\pi\)
0.254325 + 0.967119i \(0.418147\pi\)
\(312\) 0 0
\(313\) 9.08497i 0.513513i −0.966476 0.256757i \(-0.917346\pi\)
0.966476 0.256757i \(-0.0826538\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.1687i 0.683462i 0.939798 + 0.341731i \(0.111013\pi\)
−0.939798 + 0.341731i \(0.888987\pi\)
\(318\) 0 0
\(319\) −21.5367 −1.20583
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 42.9663i 2.39071i
\(324\) 0 0
\(325\) 29.6262 + 13.6275i 1.64336 + 0.755915i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.80643 −0.375251
\(330\) 0 0
\(331\) 1.53673 0.0844660 0.0422330 0.999108i \(-0.486553\pi\)
0.0422330 + 0.999108i \(0.486553\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 29.3840 18.8274i 1.60542 1.02865i
\(336\) 0 0
\(337\) 2.56295i 0.139613i 0.997561 + 0.0698065i \(0.0222382\pi\)
−0.997561 + 0.0698065i \(0.977762\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.4653 1.16241
\(342\) 0 0
\(343\) 15.8363i 0.855078i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.21958i 0.119153i −0.998224 0.0595767i \(-0.981025\pi\)
0.998224 0.0595767i \(-0.0189751\pi\)
\(348\) 0 0
\(349\) −8.82109 −0.472182 −0.236091 0.971731i \(-0.575866\pi\)
−0.236091 + 0.971731i \(0.575866\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.9043i 1.48520i 0.669737 + 0.742599i \(0.266407\pi\)
−0.669737 + 0.742599i \(0.733593\pi\)
\(354\) 0 0
\(355\) 6.64218 + 10.3664i 0.352530 + 0.550194i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.3624 1.02191 0.510955 0.859607i \(-0.329292\pi\)
0.510955 + 0.859607i \(0.329292\pi\)
\(360\) 0 0
\(361\) 32.5367 1.71246
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.6920 + 9.41372i −0.769014 + 0.492737i
\(366\) 0 0
\(367\) 2.56295i 0.133785i 0.997760 + 0.0668926i \(0.0213085\pi\)
−0.997760 + 0.0668926i \(0.978692\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.84434 0.147671
\(372\) 0 0
\(373\) 16.8885i 0.874452i 0.899352 + 0.437226i \(0.144039\pi\)
−0.899352 + 0.437226i \(0.855961\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.8894i 1.74539i
\(378\) 0 0
\(379\) −22.3578 −1.14844 −0.574222 0.818700i \(-0.694695\pi\)
−0.574222 + 0.818700i \(0.694695\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.9701i 0.611646i 0.952088 + 0.305823i \(0.0989316\pi\)
−0.952088 + 0.305823i \(0.901068\pi\)
\(384\) 0 0
\(385\) 10.0000 6.40739i 0.509647 0.326551i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.4951 −0.633528 −0.316764 0.948504i \(-0.602596\pi\)
−0.316764 + 0.948504i \(0.602596\pi\)
\(390\) 0 0
\(391\) −45.0735 −2.27946
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.23808 11.2965i −0.364187 0.568387i
\(396\) 0 0
\(397\) 24.6920i 1.23925i −0.784896 0.619627i \(-0.787284\pi\)
0.784896 0.619627i \(-0.212716\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0635 0.602421 0.301210 0.953558i \(-0.402609\pi\)
0.301210 + 0.953558i \(0.402609\pi\)
\(402\) 0 0
\(403\) 33.7769i 1.68255i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.7207i 1.07666i
\(408\) 0 0
\(409\) −18.1789 −0.898889 −0.449445 0.893308i \(-0.648378\pi\)
−0.449445 + 0.893308i \(0.648378\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.75056i 0.479794i
\(414\) 0 0
\(415\) 18.3578 11.7626i 0.901150 0.577401i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.65078 −0.471471 −0.235736 0.971817i \(-0.575750\pi\)
−0.235736 + 0.971817i \(0.575750\pi\)
\(420\) 0 0
\(421\) −15.7156 −0.765933 −0.382967 0.923762i \(-0.625098\pi\)
−0.382967 + 0.923762i \(0.625098\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 27.1871 + 12.5055i 1.31877 + 0.606608i
\(426\) 0 0
\(427\) 2.79222i 0.135125i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.1160 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(432\) 0 0
\(433\) 16.7738i 0.806099i −0.915178 0.403050i \(-0.867950\pi\)
0.915178 0.403050i \(-0.132050\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 54.0642i 2.58624i
\(438\) 0 0
\(439\) 15.5367 0.741527 0.370764 0.928727i \(-0.379096\pi\)
0.370764 + 0.928727i \(0.379096\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.3734i 0.967967i −0.875077 0.483984i \(-0.839189\pi\)
0.875077 0.483984i \(-0.160811\pi\)
\(444\) 0 0
\(445\) 12.0895 + 18.8680i 0.573095 + 0.894429i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.8386 −1.59694 −0.798471 0.602033i \(-0.794358\pi\)
−0.798471 + 0.602033i \(0.794358\pi\)
\(450\) 0 0
\(451\) 2.82109 0.132840
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.0824 15.7356i −0.472671 0.737698i
\(456\) 0 0
\(457\) 0.114633i 0.00536233i −0.999996 0.00268116i \(-0.999147\pi\)
0.999996 0.00268116i \(-0.000853442\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.97013 −0.417781 −0.208890 0.977939i \(-0.566985\pi\)
−0.208890 + 0.977939i \(0.566985\pi\)
\(462\) 0 0
\(463\) 7.68886i 0.357332i −0.983910 0.178666i \(-0.942822\pi\)
0.983910 0.178666i \(-0.0571781\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.2517i 1.35361i −0.736164 0.676803i \(-0.763365\pi\)
0.736164 0.676803i \(-0.236635\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.31139i 0.244218i
\(474\) 0 0
\(475\) 15.0000 32.6101i 0.688247 1.49625i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −31.8576 −1.45561 −0.727804 0.685785i \(-0.759459\pi\)
−0.727804 + 0.685785i \(0.759459\pi\)
\(480\) 0 0
\(481\) 34.1789 1.55842
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26.9713 + 17.2815i −1.22470 + 0.784714i
\(486\) 0 0
\(487\) 33.7769i 1.53058i −0.643686 0.765290i \(-0.722596\pi\)
0.643686 0.765290i \(-0.277404\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −23.4463 −1.05812 −0.529058 0.848586i \(-0.677455\pi\)
−0.529058 + 0.848586i \(0.677455\pi\)
\(492\) 0 0
\(493\) 31.0993i 1.40064i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.05585i 0.316498i
\(498\) 0 0
\(499\) 7.17891 0.321372 0.160686 0.987006i \(-0.448629\pi\)
0.160686 + 0.987006i \(0.448629\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.7467i 1.81681i −0.418096 0.908403i \(-0.637302\pi\)
0.418096 0.908403i \(-0.362698\pi\)
\(504\) 0 0
\(505\) −0.821092 1.28148i −0.0365381 0.0570250i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.7967 1.40936 0.704681 0.709524i \(-0.251090\pi\)
0.704681 + 0.709524i \(0.251090\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.82539 + 3.09181i −0.212632 + 0.136242i
\(516\) 0 0
\(517\) 22.0144i 0.968191i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.1459 0.970229 0.485115 0.874451i \(-0.338778\pi\)
0.485115 + 0.874451i \(0.338778\pi\)
\(522\) 0 0
\(523\) 41.6951i 1.82320i 0.411081 + 0.911599i \(0.365151\pi\)
−0.411081 + 0.911599i \(0.634849\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 30.9961i 1.35021i
\(528\) 0 0
\(529\) −33.7156 −1.46590
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.43917i 0.192282i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22.2068 0.956514
\(540\) 0 0
\(541\) −19.3578 −0.832258 −0.416129 0.909306i \(-0.636613\pi\)
−0.416129 + 0.909306i \(0.636613\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.44443 13.1792i −0.361720 0.564535i
\(546\) 0 0
\(547\) 5.12591i 0.219168i −0.993978 0.109584i \(-0.965048\pi\)
0.993978 0.109584i \(-0.0349519\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 37.3027 1.58915
\(552\) 0 0
\(553\) 7.68886i 0.326964i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.41813i 0.102460i 0.998687 + 0.0512298i \(0.0163141\pi\)
−0.998687 + 0.0512298i \(0.983686\pi\)
\(558\) 0 0
\(559\) 8.35782 0.353498
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.31139i 0.223849i 0.993717 + 0.111924i \(0.0357014\pi\)
−0.993717 + 0.111924i \(0.964299\pi\)
\(564\) 0 0
\(565\) −17.0895 + 10.9499i −0.718959 + 0.460665i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −21.7751 −0.912861 −0.456430 0.889759i \(-0.650872\pi\)
−0.456430 + 0.889759i \(0.650872\pi\)
\(570\) 0 0
\(571\) −6.82109 −0.285454 −0.142727 0.989762i \(-0.545587\pi\)
−0.142727 + 0.989762i \(0.545587\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 34.2094 + 15.7356i 1.42663 + 0.656221i
\(576\) 0 0
\(577\) 5.24054i 0.218167i 0.994033 + 0.109083i \(0.0347915\pi\)
−0.994033 + 0.109083i \(0.965208\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.4951 −0.518385
\(582\) 0 0
\(583\) 9.19961i 0.381009i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 47.4055i 1.95663i 0.207117 + 0.978316i \(0.433592\pi\)
−0.207117 + 0.978316i \(0.566408\pi\)
\(588\) 0 0
\(589\) −37.1789 −1.53193
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 11.2965i 0.463890i −0.972729 0.231945i \(-0.925491\pi\)
0.972729 0.231945i \(-0.0745090\pi\)
\(594\) 0 0
\(595\) −9.25236 14.4401i −0.379310 0.591988i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.6829 1.49882 0.749412 0.662104i \(-0.230336\pi\)
0.749412 + 0.662104i \(0.230336\pi\)
\(600\) 0 0
\(601\) 19.3578 0.789622 0.394811 0.918762i \(-0.370810\pi\)
0.394811 + 0.918762i \(0.370810\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.45391 + 11.6333i 0.303044 + 0.472961i
\(606\) 0 0
\(607\) 1.28148i 0.0520135i 0.999662 + 0.0260068i \(0.00827915\pi\)
−0.999662 + 0.0260068i \(0.991721\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −34.6410 −1.40143
\(612\) 0 0
\(613\) 38.9028i 1.57127i 0.618690 + 0.785636i \(0.287664\pi\)
−0.618690 + 0.785636i \(0.712336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.7976i 1.23986i −0.784655 0.619932i \(-0.787160\pi\)
0.784655 0.619932i \(-0.212840\pi\)
\(618\) 0 0
\(619\) −34.3578 −1.38096 −0.690479 0.723353i \(-0.742600\pi\)
−0.690479 + 0.723353i \(0.742600\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.8424i 0.514519i
\(624\) 0 0
\(625\) −16.2684 18.9826i −0.650735 0.759305i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 31.3650 1.25061
\(630\) 0 0
\(631\) 11.5367 0.459270 0.229635 0.973277i \(-0.426247\pi\)
0.229635 + 0.973277i \(0.426247\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.41269 1.54591i 0.0957448 0.0613474i
\(636\) 0 0
\(637\) 34.9438i 1.38452i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.91872 0.312771 0.156385 0.987696i \(-0.450016\pi\)
0.156385 + 0.987696i \(0.450016\pi\)
\(642\) 0 0
\(643\) 9.19961i 0.362797i −0.983410 0.181399i \(-0.941938\pi\)
0.983410 0.181399i \(-0.0580624\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.7146i 0.539177i 0.962976 + 0.269588i \(0.0868876\pi\)
−0.962976 + 0.269588i \(0.913112\pi\)
\(648\) 0 0
\(649\) 31.5367 1.23792
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0941i 1.64727i 0.567122 + 0.823634i \(0.308057\pi\)
−0.567122 + 0.823634i \(0.691943\pi\)
\(654\) 0 0
\(655\) 6.64218 + 10.3664i 0.259532 + 0.405051i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.1269 0.939852 0.469926 0.882706i \(-0.344281\pi\)
0.469926 + 0.882706i \(0.344281\pi\)
\(660\) 0 0
\(661\) 44.4313 1.72818 0.864088 0.503341i \(-0.167896\pi\)
0.864088 + 0.503341i \(0.167896\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −17.3205 + 11.0979i −0.671660 + 0.430359i
\(666\) 0 0
\(667\) 39.1321i 1.51520i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.03102 −0.348639
\(672\) 0 0
\(673\) 32.6101i 1.25703i 0.777799 + 0.628513i \(0.216336\pi\)
−0.777799 + 0.628513i \(0.783664\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 50.9724i 1.95903i −0.201376 0.979514i \(-0.564541\pi\)
0.201376 0.979514i \(-0.435459\pi\)
\(678\) 0 0
\(679\) 18.3578 0.704508
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28.3795i 1.08591i 0.839762 + 0.542955i \(0.182695\pi\)
−0.839762 + 0.542955i \(0.817305\pi\)
\(684\) 0 0
\(685\) −21.2684 + 13.6275i −0.812622 + 0.520678i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.4762 0.551498
\(690\) 0 0
\(691\) 14.7156 0.559809 0.279905 0.960028i \(-0.409697\pi\)
0.279905 + 0.960028i \(0.409697\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.7237 32.3435i −0.786096 1.22686i
\(696\) 0 0
\(697\) 4.07370i 0.154302i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.2272 0.537353 0.268676 0.963230i \(-0.413414\pi\)
0.268676 + 0.963230i \(0.413414\pi\)
\(702\) 0 0
\(703\) 37.6214i 1.41892i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.872228i 0.0328035i
\(708\) 0 0
\(709\) 18.5367 0.696161 0.348081 0.937465i \(-0.386834\pi\)
0.348081 + 0.937465i \(0.386834\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 39.0022i 1.46065i
\(714\) 0 0
\(715\) 50.8945 32.6101i 1.90335 1.21955i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.7358 1.18355 0.591773 0.806105i \(-0.298428\pi\)
0.591773 + 0.806105i \(0.298428\pi\)
\(720\) 0 0
\(721\) 3.28437 0.122316
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.8571 23.6035i 0.403223 0.876610i
\(726\) 0 0
\(727\) 31.4432i 1.16617i 0.812413 + 0.583083i \(0.198154\pi\)
−0.812413 + 0.583083i \(0.801846\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.66973 0.283675
\(732\) 0 0
\(733\) 5.12591i 0.189330i −0.995509 0.0946649i \(-0.969822\pi\)
0.995509 0.0946649i \(-0.0301780\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 64.6870i 2.38278i
\(738\) 0 0
\(739\) −33.8945 −1.24683 −0.623415 0.781891i \(-0.714255\pi\)
−0.623415 + 0.781891i \(0.714255\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.1300i 1.39885i −0.714704 0.699427i \(-0.753438\pi\)
0.714704 0.699427i \(-0.246562\pi\)
\(744\) 0 0
\(745\) 23.8051 + 37.1526i 0.872151 + 1.36117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.3578 0.596905 0.298453 0.954424i \(-0.403530\pi\)
0.298453 + 0.954424i \(0.403530\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.24756 + 9.75056i 0.227372 + 0.354859i
\(756\) 0 0
\(757\) 28.6510i 1.04134i 0.853758 + 0.520670i \(0.174318\pi\)
−0.853758 + 0.520670i \(0.825682\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.73205 0.0627868 0.0313934 0.999507i \(-0.490006\pi\)
0.0313934 + 0.999507i \(0.490006\pi\)
\(762\) 0 0
\(763\) 8.97034i 0.324748i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 49.6250i 1.79186i
\(768\) 0 0
\(769\) 19.7156 0.710964 0.355482 0.934683i \(-0.384317\pi\)
0.355482 + 0.934683i \(0.384317\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.63772i 0.166807i 0.996516 + 0.0834036i \(0.0265791\pi\)
−0.996516 + 0.0834036i \(0.973421\pi\)
\(774\) 0 0
\(775\) −10.8211 + 23.5251i −0.388705 + 0.845047i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.88627 −0.175069
\(780\) 0 0
\(781\) 22.8211 0.816603
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.86660 6.32191i 0.352154 0.225639i
\(786\) 0 0
\(787\) 45.7688i 1.63148i −0.578419 0.815740i \(-0.696330\pi\)
0.578419 0.815740i \(-0.303670\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.6318 0.413580
\(792\) 0 0
\(793\) 14.2109i 0.504643i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.3584i 0.933663i −0.884346 0.466832i \(-0.845395\pi\)
0.884346 0.466832i \(-0.154605\pi\)
\(798\) 0 0
\(799\) −31.7891 −1.12462
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 32.3435i 1.14138i
\(804\) 0 0
\(805\) −11.6422 18.1699i −0.410333 0.640406i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.19615 −0.182687 −0.0913435 0.99581