Properties

Label 2304.2.k.a.577.2
Level $2304$
Weight $2$
Character 2304.577
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 768)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 577.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.577
Dual form 2304.2.k.a.1729.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.00000 + 2.00000i) q^{5} +4.24264i q^{7} +O(q^{10})\) \(q+(-2.00000 + 2.00000i) q^{5} +4.24264i q^{7} +(2.82843 - 2.82843i) q^{11} +(3.00000 + 3.00000i) q^{13} +6.00000 q^{17} +(1.41421 + 1.41421i) q^{19} +2.82843i q^{23} -3.00000i q^{25} +(-4.00000 - 4.00000i) q^{29} +4.24264 q^{31} +(-8.48528 - 8.48528i) q^{35} +(3.00000 - 3.00000i) q^{37} +10.0000i q^{41} +(4.24264 - 4.24264i) q^{43} +2.82843 q^{47} -11.0000 q^{49} +(-4.00000 + 4.00000i) q^{53} +11.3137i q^{55} +(3.00000 + 3.00000i) q^{61} -12.0000 q^{65} +(-2.82843 - 2.82843i) q^{67} -2.82843i q^{71} +16.0000i q^{73} +(12.0000 + 12.0000i) q^{77} -4.24264 q^{79} +(-11.3137 - 11.3137i) q^{83} +(-12.0000 + 12.0000i) q^{85} +14.0000i q^{89} +(-12.7279 + 12.7279i) q^{91} -5.65685 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + O(q^{10}) \) \( 4q - 8q^{5} + 12q^{13} + 24q^{17} - 16q^{29} + 12q^{37} - 44q^{49} - 16q^{53} + 12q^{61} - 48q^{65} + 48q^{77} - 48q^{85} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) −2.00000 + 2.00000i −0.894427 + 0.894427i −0.994936 0.100509i \(-0.967953\pi\)
0.100509 + 0.994936i \(0.467953\pi\)
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843 2.82843i 0.852803 0.852803i −0.137675 0.990478i \(-0.543963\pi\)
0.990478 + 0.137675i \(0.0439628\pi\)
\(12\) 0 0
\(13\) 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 0.155747i \(-0.0497784\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 1.41421 + 1.41421i 0.324443 + 0.324443i 0.850469 0.526026i \(-0.176318\pi\)
−0.526026 + 0.850469i \(0.676318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 4.00000i −0.742781 0.742781i 0.230331 0.973112i \(-0.426019\pi\)
−0.973112 + 0.230331i \(0.926019\pi\)
\(30\) 0 0
\(31\) 4.24264 0.762001 0.381000 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.48528 8.48528i −1.43427 1.43427i
\(36\) 0 0
\(37\) 3.00000 3.00000i 0.493197 0.493197i −0.416115 0.909312i \(-0.636609\pi\)
0.909312 + 0.416115i \(0.136609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 4.24264 4.24264i 0.646997 0.646997i −0.305269 0.952266i \(-0.598747\pi\)
0.952266 + 0.305269i \(0.0987465\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 + 4.00000i −0.549442 + 0.549442i −0.926279 0.376837i \(-0.877012\pi\)
0.376837 + 0.926279i \(0.377012\pi\)
\(54\) 0 0
\(55\) 11.3137i 1.52554i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) 3.00000 + 3.00000i 0.384111 + 0.384111i 0.872581 0.488470i \(-0.162445\pi\)
−0.488470 + 0.872581i \(0.662445\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −2.82843 2.82843i −0.345547 0.345547i 0.512901 0.858448i \(-0.328571\pi\)
−0.858448 + 0.512901i \(0.828571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.82843i 0.335673i −0.985815 0.167836i \(-0.946322\pi\)
0.985815 0.167836i \(-0.0536780\pi\)
\(72\) 0 0
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 + 12.0000i 1.36753 + 1.36753i
\(78\) 0 0
\(79\) −4.24264 −0.477334 −0.238667 0.971101i \(-0.576710\pi\)
−0.238667 + 0.971101i \(0.576710\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.3137 11.3137i −1.24184 1.24184i −0.959237 0.282604i \(-0.908802\pi\)
−0.282604 0.959237i \(-0.591198\pi\)
\(84\) 0 0
\(85\) −12.0000 + 12.0000i −1.30158 + 1.30158i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 0 0
\(91\) −12.7279 + 12.7279i −1.33425 + 1.33425i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.00000 + 8.00000i −0.796030 + 0.796030i −0.982467 0.186437i \(-0.940306\pi\)
0.186437 + 0.982467i \(0.440306\pi\)
\(102\) 0 0
\(103\) 18.3848i 1.81151i −0.423806 0.905753i \(-0.639306\pi\)
0.423806 0.905753i \(-0.360694\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.82843 2.82843i 0.273434 0.273434i −0.557047 0.830481i \(-0.688066\pi\)
0.830481 + 0.557047i \(0.188066\pi\)
\(108\) 0 0
\(109\) −7.00000 7.00000i −0.670478 0.670478i 0.287348 0.957826i \(-0.407226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −5.65685 5.65685i −0.527504 0.527504i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 25.4558i 2.33353i
\(120\) 0 0
\(121\) 5.00000i 0.454545i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.00000 4.00000i −0.357771 0.357771i
\(126\) 0 0
\(127\) 4.24264 0.376473 0.188237 0.982124i \(-0.439723\pi\)
0.188237 + 0.982124i \(0.439723\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.1421 + 14.1421i 1.23560 + 1.23560i 0.961780 + 0.273824i \(0.0882887\pi\)
0.273824 + 0.961780i \(0.411711\pi\)
\(132\) 0 0
\(133\) −6.00000 + 6.00000i −0.520266 + 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000i 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 0 0
\(139\) 8.48528 8.48528i 0.719712 0.719712i −0.248834 0.968546i \(-0.580047\pi\)
0.968546 + 0.248834i \(0.0800474\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.9706 1.41915
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 + 6.00000i −0.491539 + 0.491539i −0.908791 0.417252i \(-0.862993\pi\)
0.417252 + 0.908791i \(0.362993\pi\)
\(150\) 0 0
\(151\) 1.41421i 0.115087i 0.998343 + 0.0575435i \(0.0183268\pi\)
−0.998343 + 0.0575435i \(0.981673\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.48528 + 8.48528i −0.681554 + 0.681554i
\(156\) 0 0
\(157\) 5.00000 + 5.00000i 0.399043 + 0.399043i 0.877896 0.478852i \(-0.158947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −7.07107 7.07107i −0.553849 0.553849i 0.373701 0.927549i \(-0.378089\pi\)
−0.927549 + 0.373701i \(0.878089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3137i 0.875481i 0.899101 + 0.437741i \(0.144221\pi\)
−0.899101 + 0.437741i \(0.855779\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 + 2.00000i 0.152057 + 0.152057i 0.779036 0.626979i \(-0.215709\pi\)
−0.626979 + 0.779036i \(0.715709\pi\)
\(174\) 0 0
\(175\) 12.7279 0.962140
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) −1.00000 + 1.00000i −0.0743294 + 0.0743294i −0.743294 0.668965i \(-0.766738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000i 0.882258i
\(186\) 0 0
\(187\) 16.9706 16.9706i 1.24101 1.24101i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.6274 1.63726 0.818631 0.574320i \(-0.194733\pi\)
0.818631 + 0.574320i \(0.194733\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.00000 + 4.00000i −0.284988 + 0.284988i −0.835095 0.550106i \(-0.814587\pi\)
0.550106 + 0.835095i \(0.314587\pi\)
\(198\) 0 0
\(199\) 7.07107i 0.501255i 0.968084 + 0.250627i \(0.0806369\pi\)
−0.968084 + 0.250627i \(0.919363\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.9706 16.9706i 1.19110 1.19110i
\(204\) 0 0
\(205\) −20.0000 20.0000i −1.39686 1.39686i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −8.48528 8.48528i −0.584151 0.584151i 0.351890 0.936041i \(-0.385539\pi\)
−0.936041 + 0.351890i \(0.885539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.9706i 1.15738i
\(216\) 0 0
\(217\) 18.0000i 1.22192i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 18.0000 + 18.0000i 1.21081 + 1.21081i
\(222\) 0 0
\(223\) −21.2132 −1.42054 −0.710271 0.703929i \(-0.751427\pi\)
−0.710271 + 0.703929i \(0.751427\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 11.0000 11.0000i 0.726900 0.726900i −0.243101 0.970001i \(-0.578165\pi\)
0.970001 + 0.243101i \(0.0781645\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000i 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) −5.65685 + 5.65685i −0.369012 + 0.369012i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 22.0000 22.0000i 1.40553 1.40553i
\(246\) 0 0
\(247\) 8.48528i 0.539906i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.82843 + 2.82843i −0.178529 + 0.178529i −0.790714 0.612185i \(-0.790291\pi\)
0.612185 + 0.790714i \(0.290291\pi\)
\(252\) 0 0
\(253\) 8.00000 + 8.00000i 0.502956 + 0.502956i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 12.7279 + 12.7279i 0.790875 + 0.790875i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.6274i 1.39527i −0.716455 0.697633i \(-0.754237\pi\)
0.716455 0.697633i \(-0.245763\pi\)
\(264\) 0 0
\(265\) 16.0000i 0.982872i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) −1.41421 −0.0859074 −0.0429537 0.999077i \(-0.513677\pi\)
−0.0429537 + 0.999077i \(0.513677\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.48528 8.48528i −0.511682 0.511682i
\(276\) 0 0
\(277\) 1.00000 1.00000i 0.0600842 0.0600842i −0.676426 0.736510i \(-0.736472\pi\)
0.736510 + 0.676426i \(0.236472\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) 2.82843 2.82843i 0.168133 0.168133i −0.618026 0.786158i \(-0.712067\pi\)
0.786158 + 0.618026i \(0.212067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −42.4264 −2.50435
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.00000 + 4.00000i −0.233682 + 0.233682i −0.814228 0.580545i \(-0.802839\pi\)
0.580545 + 0.814228i \(0.302839\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.48528 + 8.48528i −0.490716 + 0.490716i
\(300\) 0 0
\(301\) 18.0000 + 18.0000i 1.03750 + 1.03750i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −2.82843 2.82843i −0.161427 0.161427i 0.621772 0.783199i \(-0.286413\pi\)
−0.783199 + 0.621772i \(0.786413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2843i 1.60385i −0.597422 0.801927i \(-0.703808\pi\)
0.597422 0.801927i \(-0.296192\pi\)
\(312\) 0 0
\(313\) 26.0000i 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 12.0000i −0.673987 0.673987i 0.284646 0.958633i \(-0.408124\pi\)
−0.958633 + 0.284646i \(0.908124\pi\)
\(318\) 0 0
\(319\) −22.6274 −1.26689
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.48528 + 8.48528i 0.472134 + 0.472134i
\(324\) 0 0
\(325\) 9.00000 9.00000i 0.499230 0.499230i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000i 0.661581i
\(330\) 0 0
\(331\) −2.82843 + 2.82843i −0.155464 + 0.155464i −0.780553 0.625089i \(-0.785063\pi\)
0.625089 + 0.780553i \(0.285063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.0000 12.0000i 0.649836 0.649836i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.9706 + 16.9706i −0.911028 + 0.911028i −0.996353 0.0853256i \(-0.972807\pi\)
0.0853256 + 0.996353i \(0.472807\pi\)
\(348\) 0 0
\(349\) −3.00000 3.00000i −0.160586 0.160586i 0.622240 0.782826i \(-0.286223\pi\)
−0.782826 + 0.622240i \(0.786223\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 5.65685 + 5.65685i 0.300235 + 0.300235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.82843i 0.149279i −0.997211 0.0746393i \(-0.976219\pi\)
0.997211 0.0746393i \(-0.0237806\pi\)
\(360\) 0 0
\(361\) 15.0000i 0.789474i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −32.0000 32.0000i −1.67496 1.67496i
\(366\) 0 0
\(367\) 18.3848 0.959678 0.479839 0.877357i \(-0.340695\pi\)
0.479839 + 0.877357i \(0.340695\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −16.9706 16.9706i −0.881068 0.881068i
\(372\) 0 0
\(373\) 13.0000 13.0000i 0.673114 0.673114i −0.285318 0.958433i \(-0.592099\pi\)
0.958433 + 0.285318i \(0.0920993\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) −26.8701 + 26.8701i −1.38022 + 1.38022i −0.536011 + 0.844211i \(0.680070\pi\)
−0.844211 + 0.536011i \(0.819930\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) −48.0000 −2.44631
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.00000 2.00000i 0.101404 0.101404i −0.654585 0.755989i \(-0.727156\pi\)
0.755989 + 0.654585i \(0.227156\pi\)
\(390\) 0 0
\(391\) 16.9706i 0.858238i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.48528 8.48528i 0.426941 0.426941i
\(396\) 0 0
\(397\) 23.0000 + 23.0000i 1.15434 + 1.15434i 0.985674 + 0.168663i \(0.0539450\pi\)
0.168663 + 0.985674i \(0.446055\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 12.7279 + 12.7279i 0.634023 + 0.634023i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.9706i 0.841200i
\(408\) 0 0
\(409\) 28.0000i 1.38451i 0.721653 + 0.692255i \(0.243383\pi\)
−0.721653 + 0.692255i \(0.756617\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 45.2548 2.22147
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.3137 + 11.3137i 0.552711 + 0.552711i 0.927222 0.374511i \(-0.122190\pi\)
−0.374511 + 0.927222i \(0.622190\pi\)
\(420\) 0 0
\(421\) −25.0000 + 25.0000i −1.21843 + 1.21843i −0.250242 + 0.968183i \(0.580510\pi\)
−0.968183 + 0.250242i \(0.919490\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.0000i 0.873128i
\(426\) 0 0
\(427\) −12.7279 + 12.7279i −0.615947 + 0.615947i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.7696 1.77113 0.885564 0.464518i \(-0.153773\pi\)
0.885564 + 0.464518i \(0.153773\pi\)
\(432\) 0 0
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.00000 + 4.00000i −0.191346 + 0.191346i
\(438\) 0 0
\(439\) 1.41421i 0.0674967i −0.999430 0.0337484i \(-0.989256\pi\)
0.999430 0.0337484i \(-0.0107445\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.9706 16.9706i 0.806296 0.806296i −0.177775 0.984071i \(-0.556890\pi\)
0.984071 + 0.177775i \(0.0568900\pi\)
\(444\) 0 0
\(445\) −28.0000 28.0000i −1.32733 1.32733i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 28.2843 + 28.2843i 1.33185 + 1.33185i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 50.9117i 2.38678i
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 6.00000i −0.279448 0.279448i 0.553441 0.832889i \(-0.313315\pi\)
−0.832889 + 0.553441i \(0.813315\pi\)
\(462\) 0 0
\(463\) 12.7279 0.591517 0.295758 0.955263i \(-0.404428\pi\)
0.295758 + 0.955263i \(0.404428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.9706 + 16.9706i 0.785304 + 0.785304i 0.980720 0.195416i \(-0.0626058\pi\)
−0.195416 + 0.980720i \(0.562606\pi\)
\(468\) 0 0
\(469\) 12.0000 12.0000i 0.554109 0.554109i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) 4.24264 4.24264i 0.194666 0.194666i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −25.4558 −1.16311 −0.581554 0.813508i \(-0.697555\pi\)
−0.581554 + 0.813508i \(0.697555\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000 8.00000i 0.363261 0.363261i
\(486\) 0 0
\(487\) 43.8406i 1.98661i −0.115529 0.993304i \(-0.536856\pi\)
0.115529 0.993304i \(-0.463144\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.65685 5.65685i 0.255290 0.255290i −0.567845 0.823135i \(-0.692223\pi\)
0.823135 + 0.567845i \(0.192223\pi\)
\(492\) 0 0
\(493\) −24.0000 24.0000i −1.08091 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 14.1421 + 14.1421i 0.633089 + 0.633089i 0.948842 0.315753i \(-0.102257\pi\)
−0.315753 + 0.948842i \(0.602257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.1421i 0.630567i −0.948998 0.315283i \(-0.897900\pi\)
0.948998 0.315283i \(-0.102100\pi\)
\(504\) 0 0
\(505\) 32.0000i 1.42398i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16.0000 16.0000i −0.709188 0.709188i 0.257177 0.966364i \(-0.417208\pi\)
−0.966364 + 0.257177i \(0.917208\pi\)
\(510\) 0 0
\(511\) −67.8823 −3.00293
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 36.7696 + 36.7696i 1.62026 + 1.62026i
\(516\) 0 0
\(517\) 8.00000 8.00000i 0.351840 0.351840i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.0000i 0.963837i 0.876216 + 0.481919i \(0.160060\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 15.5563 15.5563i 0.680232 0.680232i −0.279821 0.960052i \(-0.590275\pi\)
0.960052 + 0.279821i \(0.0902750\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.4558 1.10887
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.0000 + 30.0000i −1.29944 + 1.29944i
\(534\) 0 0
\(535\) 11.3137i 0.489134i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −31.1127 + 31.1127i −1.34012 + 1.34012i
\(540\) 0 0
\(541\) 29.0000 + 29.0000i 1.24681 + 1.24681i 0.957122 + 0.289685i \(0.0935507\pi\)
0.289685 + 0.957122i \(0.406449\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 24.0416 + 24.0416i 1.02795 + 1.02795i 0.999598 + 0.0283478i \(0.00902459\pi\)
0.0283478 + 0.999598i \(0.490975\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.3137i 0.481980i
\(552\) 0 0
\(553\) 18.0000i 0.765438i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0000 + 14.0000i 0.593199 + 0.593199i 0.938494 0.345295i \(-0.112221\pi\)
−0.345295 + 0.938494i \(0.612221\pi\)
\(558\) 0 0
\(559\) 25.4558 1.07667
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.82843 + 2.82843i 0.119204 + 0.119204i 0.764192 0.644988i \(-0.223138\pi\)
−0.644988 + 0.764192i \(0.723138\pi\)
\(564\) 0 0
\(565\) 12.0000 12.0000i 0.504844 0.504844i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000i 0.754599i −0.926091 0.377300i \(-0.876853\pi\)
0.926091 0.377300i \(-0.123147\pi\)
\(570\) 0 0
\(571\) −14.1421 + 14.1421i −0.591830 + 0.591830i −0.938125 0.346296i \(-0.887439\pi\)
0.346296 + 0.938125i \(0.387439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 48.0000 48.0000i 1.99138 1.99138i
\(582\) 0 0
\(583\) 22.6274i 0.937132i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.9706 + 16.9706i −0.700450 + 0.700450i −0.964507 0.264057i \(-0.914939\pi\)
0.264057 + 0.964507i \(0.414939\pi\)
\(588\) 0 0
\(589\) 6.00000 + 6.00000i 0.247226 + 0.247226i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −50.9117 50.9117i −2.08718 2.08718i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 48.0833i 1.96463i 0.187239 + 0.982314i \(0.440046\pi\)
−0.187239 + 0.982314i \(0.559954\pi\)
\(600\) 0 0
\(601\) 28.0000i 1.14214i −0.820900 0.571072i \(-0.806528\pi\)
0.820900 0.571072i \(-0.193472\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0000 + 10.0000i 0.406558 + 0.406558i
\(606\) 0 0
\(607\) 4.24264 0.172203 0.0861017 0.996286i \(-0.472559\pi\)
0.0861017 + 0.996286i \(0.472559\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.48528 + 8.48528i 0.343278 + 0.343278i
\(612\) 0 0
\(613\) −13.0000 + 13.0000i −0.525065 + 0.525065i −0.919097 0.394032i \(-0.871080\pi\)
0.394032 + 0.919097i \(0.371080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000i 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 0 0
\(619\) 25.4558 25.4558i 1.02316 1.02316i 0.0234313 0.999725i \(-0.492541\pi\)
0.999725 0.0234313i \(-0.00745910\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −59.3970 −2.37969
\(624\) 0 0
\(625\) 31.0000 1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.0000 18.0000i 0.717707 0.717707i
\(630\) 0 0
\(631\) 35.3553i 1.40747i −0.710461 0.703737i \(-0.751513\pi\)
0.710461 0.703737i \(-0.248487\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.48528 + 8.48528i −0.336728 + 0.336728i
\(636\) 0 0
\(637\) −33.0000 33.0000i −1.30751 1.30751i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −9.89949 9.89949i −0.390398 0.390398i 0.484431 0.874829i \(-0.339027\pi\)
−0.874829 + 0.484431i \(0.839027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.1421i 0.555985i −0.960583 0.277992i \(-0.910331\pi\)
0.960583 0.277992i \(-0.0896690\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 + 6.00000i 0.234798 + 0.234798i 0.814692 0.579894i \(-0.196906\pi\)
−0.579894 + 0.814692i \(0.696906\pi\)
\(654\) 0 0
\(655\) −56.5685 −2.21032
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) −9.00000 + 9.00000i −0.350059 + 0.350059i −0.860132 0.510072i \(-0.829619\pi\)
0.510072 + 0.860132i \(0.329619\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 24.0000i 0.930680i
\(666\) 0 0
\(667\) 11.3137 11.3137i 0.438069 0.438069i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 16.9706 0.655141
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.0000 26.0000i 0.999261 0.999261i −0.000738553 1.00000i \(-0.500235\pi\)
1.00000 0.000738553i \(0.000235089\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.9706 16.9706i 0.649361 0.649361i −0.303478 0.952838i \(-0.598148\pi\)
0.952838 + 0.303478i \(0.0981479\pi\)
\(684\) 0 0
\(685\) 20.0000 + 20.0000i 0.764161 + 0.764161i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −15.5563 15.5563i −0.591791 0.591791i 0.346324 0.938115i \(-0.387430\pi\)
−0.938115 + 0.346324i \(0.887430\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.9411i 1.28746i
\(696\) 0 0
\(697\) 60.0000i 2.27266i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.00000 + 4.00000i 0.151078 + 0.151078i 0.778599 0.627521i \(-0.215931\pi\)
−0.627521 + 0.778599i \(0.715931\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −33.9411 33.9411i −1.27649 1.27649i
\(708\) 0 0
\(709\) 13.0000 13.0000i 0.488225 0.488225i −0.419521 0.907746i \(-0.637802\pi\)
0.907746 + 0.419521i \(0.137802\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) −33.9411 + 33.9411i −1.26933 + 1.26933i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.48528 −0.316448 −0.158224 0.987403i \(-0.550577\pi\)
−0.158224 + 0.987403i \(0.550577\pi\)
\(720\) 0 0
\(721\) 78.0000 2.90487
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.0000 + 12.0000i −0.445669 + 0.445669i
\(726\) 0 0
\(727\) 26.8701i 0.996555i 0.867018 + 0.498278i \(0.166034\pi\)
−0.867018 + 0.498278i \(0.833966\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 25.4558 25.4558i 0.941518 0.941518i
\(732\) 0 0
\(733\) −25.0000 25.0000i −0.923396 0.923396i 0.0738717 0.997268i \(-0.476464\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −25.4558 25.4558i −0.936408 0.936408i 0.0616872 0.998096i \(-0.480352\pi\)
−0.998096 + 0.0616872i \(0.980352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.6274i 0.830119i 0.909794 + 0.415060i \(0.136239\pi\)
−0.909794 + 0.415060i \(0.863761\pi\)
\(744\) 0 0
\(745\) 24.0000i 0.879292i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 + 12.0000i 0.438470 + 0.438470i
\(750\) 0 0
\(751\) −12.7279 −0.464448 −0.232224 0.972662i \(-0.574600\pi\)
−0.232224 + 0.972662i \(0.574600\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.82843 2.82843i −0.102937 0.102937i
\(756\) 0 0
\(757\) 27.0000 27.0000i 0.981332 0.981332i −0.0184972 0.999829i \(-0.505888\pi\)
0.999829 + 0.0184972i \(0.00588818\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000i 1.23250i −0.787551 0.616250i \(-0.788651\pi\)
0.787551 0.616250i \(-0.211349\pi\)
\(762\) 0 0
\(763\) 29.6985 29.6985i 1.07516 1.07516i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.00000 4.00000i 0.143870 0.143870i −0.631503 0.775373i \(-0.717562\pi\)
0.775373 + 0.631503i \(0.217562\pi\)
\(774\) 0 0
\(775\) 12.7279i 0.457200i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.1421 + 14.1421i −0.506695 + 0.506695i
\(780\) 0 0
\(781\) −8.00000 8.00000i −0.286263 0.286263i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 26.8701 + 26.8701i 0.957814 + 0.957814i 0.999145 0.0413314i \(-0.0131599\pi\)
−0.0413314 + 0.999145i \(0.513160\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 25.4558i 0.905106i
\(792\) 0 0
\(793\) 18.0000i 0.639199i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.0000 + 22.0000i 0.779280 + 0.779280i 0.979708 0.200428i \(-0.0642334\pi\)
−0.200428 + 0.979708i \(0.564233\pi\)
\(798\) 0 0
\(799\) 16.9706 0.600375
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 45.2548 + 45.2548i 1.59701 + 1.59701i
\(804\)