Properties

Label 2304.2.k.a.1729.1
Level $2304$
Weight $2$
Character 2304.1729
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 1729.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1729
Dual form 2304.2.k.a.577.2

$q$-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 \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

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

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 0.100509 0.994936i \(-0.467953\pi\)
−0.994936 + 0.100509i \(0.967953\pi\)
\(6\) 0 0
\(7\) 4.24264i 1.60357i −0.597614 0.801784i \(-0.703885\pi\)
0.597614 0.801784i \(-0.296115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843 + 2.82843i 0.852803 + 0.852803i 0.990478 0.137675i \(-0.0439628\pi\)
−0.137675 + 0.990478i \(0.543963\pi\)
\(12\) 0 0
\(13\) 3.00000 3.00000i 0.832050 0.832050i −0.155747 0.987797i \(-0.549778\pi\)
0.987797 + 0.155747i \(0.0497784\pi\)
\(14\) 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 −0.526026 0.850469i \(-0.676318\pi\)
0.850469 + 0.526026i \(0.176318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843i 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 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 −0.973112 0.230331i \(-0.926019\pi\)
0.230331 + 0.973112i \(0.426019\pi\)
\(30\) 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 0.909312 0.416115i \(-0.136609\pi\)
−0.416115 + 0.909312i \(0.636609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000i 1.56174i −0.624695 0.780869i \(-0.714777\pi\)
0.624695 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) 4.24264 + 4.24264i 0.646997 + 0.646997i 0.952266 0.305269i \(-0.0987465\pi\)
−0.305269 + 0.952266i \(0.598747\pi\)
\(44\) 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 0.376837 0.926279i \(-0.377012\pi\)
−0.926279 + 0.376837i \(0.877012\pi\)
\(54\) 0 0
\(55\) 11.3137i 1.52554i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 3.00000 3.00000i 0.384111 0.384111i −0.488470 0.872581i \(-0.662445\pi\)
0.872581 + 0.488470i \(0.162445\pi\)
\(62\) 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 −0.858448 0.512901i \(-0.828571\pi\)
0.512901 + 0.858448i \(0.328571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.82843i 0.335673i 0.985815 + 0.167836i \(0.0536780\pi\)
−0.985815 + 0.167836i \(0.946322\pi\)
\(72\) 0 0
\(73\) 16.0000i 1.87266i −0.351123 0.936329i \(-0.614200\pi\)
0.351123 0.936329i \(-0.385800\pi\)
\(74\) 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 −0.282604 + 0.959237i \(0.591198\pi\)
−0.959237 + 0.282604i \(0.908802\pi\)
\(84\) 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 \(-0.733878\pi\)
0.670402 0.741999i \(-0.266122\pi\)
\(90\) 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 0.186437 0.982467i \(-0.440306\pi\)
−0.982467 + 0.186437i \(0.940306\pi\)
\(102\) 0 0
\(103\) 18.3848i 1.81151i 0.423806 + 0.905753i \(0.360694\pi\)
−0.423806 + 0.905753i \(0.639306\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.82843 + 2.82843i 0.273434 + 0.273434i 0.830481 0.557047i \(-0.188066\pi\)
−0.557047 + 0.830481i \(0.688066\pi\)
\(108\) 0 0
\(109\) −7.00000 + 7.00000i −0.670478 + 0.670478i −0.957826 0.287348i \(-0.907226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 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 0.273824 0.961780i \(-0.411711\pi\)
0.961780 0.273824i \(-0.0882887\pi\)
\(132\) 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 \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 0 0
\(139\) 8.48528 + 8.48528i 0.719712 + 0.719712i 0.968546 0.248834i \(-0.0800474\pi\)
−0.248834 + 0.968546i \(0.580047\pi\)
\(140\) 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 0.417252 0.908791i \(-0.362993\pi\)
−0.908791 + 0.417252i \(0.862993\pi\)
\(150\) 0 0
\(151\) 1.41421i 0.115087i −0.998343 0.0575435i \(-0.981673\pi\)
0.998343 0.0575435i \(-0.0183268\pi\)
\(152\) 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 −0.478852 0.877896i \(-0.658947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 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 −0.927549 0.373701i \(-0.878089\pi\)
0.373701 + 0.927549i \(0.378089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3137i 0.875481i −0.899101 0.437741i \(-0.855779\pi\)
0.899101 0.437741i \(-0.144221\pi\)
\(168\) 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 −0.626979 0.779036i \(-0.715709\pi\)
0.779036 + 0.626979i \(0.215709\pi\)
\(174\) 0 0
\(175\) 12.7279 0.962140
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(180\) 0 0
\(181\) −1.00000 1.00000i −0.0743294 0.0743294i 0.668965 0.743294i \(-0.266738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 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 0.550106 0.835095i \(-0.314587\pi\)
−0.835095 + 0.550106i \(0.814587\pi\)
\(198\) 0 0
\(199\) 7.07107i 0.501255i −0.968084 0.250627i \(-0.919363\pi\)
0.968084 0.250627i \(-0.0806369\pi\)
\(200\) 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 −0.936041 0.351890i \(-0.885539\pi\)
0.351890 + 0.936041i \(0.385539\pi\)
\(212\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 11.0000 + 11.0000i 0.726900 + 0.726900i 0.970001 0.243101i \(-0.0781645\pi\)
−0.243101 + 0.970001i \(0.578165\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 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 0.612185 0.790714i \(-0.290291\pi\)
−0.790714 + 0.612185i \(0.790291\pi\)
\(252\) 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 \(0.245763\pi\)
−0.716455 + 0.697633i \(0.754237\pi\)
\(264\) 0 0
\(265\) 16.0000i 0.982872i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(270\) 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 0.736510 0.676426i \(-0.236472\pi\)
−0.676426 + 0.736510i \(0.736472\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000i 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 0 0
\(283\) 2.82843 + 2.82843i 0.168133 + 0.168133i 0.786158 0.618026i \(-0.212067\pi\)
−0.618026 + 0.786158i \(0.712067\pi\)
\(284\) 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 0.580545 0.814228i \(-0.302839\pi\)
−0.814228 + 0.580545i \(0.802839\pi\)
\(294\) 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 −0.783199 0.621772i \(-0.786413\pi\)
0.621772 + 0.783199i \(0.286413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2843i 1.60385i 0.597422 + 0.801927i \(0.296192\pi\)
−0.597422 + 0.801927i \(0.703808\pi\)
\(312\) 0 0
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 + 12.0000i −0.673987 + 0.673987i −0.958633 0.284646i \(-0.908124\pi\)
0.284646 + 0.958633i \(0.408124\pi\)
\(318\) 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 0.625089 0.780553i \(-0.285063\pi\)
−0.780553 + 0.625089i \(0.785063\pi\)
\(332\) 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 0.0853256 0.996353i \(-0.472807\pi\)
−0.996353 + 0.0853256i \(0.972807\pi\)
\(348\) 0 0
\(349\) −3.00000 + 3.00000i −0.160586 + 0.160586i −0.782826 0.622240i \(-0.786223\pi\)
0.622240 + 0.782826i \(0.286223\pi\)
\(350\) 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 \(0.0237806\pi\)
−0.997211 + 0.0746393i \(0.976219\pi\)
\(360\) 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 0.958433 0.285318i \(-0.0920993\pi\)
−0.285318 + 0.958433i \(0.592099\pi\)
\(374\) 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 −0.844211 0.536011i \(-0.819930\pi\)
−0.536011 0.844211i \(-0.680070\pi\)
\(380\) 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 0.755989 0.654585i \(-0.227156\pi\)
−0.654585 + 0.755989i \(0.727156\pi\)
\(390\) 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 0.168663 0.985674i \(-0.446055\pi\)
0.985674 0.168663i \(-0.0539450\pi\)
\(398\) 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 \(-0.756617\pi\)
0.721653 0.692255i \(-0.243383\pi\)
\(410\) 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 −0.374511 0.927222i \(-0.622190\pi\)
0.927222 + 0.374511i \(0.122190\pi\)
\(420\) 0 0
\(421\) −25.0000 25.0000i −1.21843 1.21843i −0.968183 0.250242i \(-0.919490\pi\)
−0.250242 0.968183i \(-0.580510\pi\)
\(422\) 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 \(0.0107445\pi\)
−0.999430 + 0.0337484i \(0.989256\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.9706 + 16.9706i 0.806296 + 0.806296i 0.984071 0.177775i \(-0.0568900\pi\)
−0.177775 + 0.984071i \(0.556890\pi\)
\(444\) 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 \(-0.909444\pi\)
0.959805 0.280668i \(-0.0905560\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 + 6.00000i −0.279448 + 0.279448i −0.832889 0.553441i \(-0.813315\pi\)
0.553441 + 0.832889i \(0.313315\pi\)
\(462\) 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 −0.195416 0.980720i \(-0.562606\pi\)
0.980720 + 0.195416i \(0.0626058\pi\)
\(468\) 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 \(0.463144\pi\)
−0.115529 + 0.993304i \(0.536856\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.65685 + 5.65685i 0.255290 + 0.255290i 0.823135 0.567845i \(-0.192223\pi\)
−0.567845 + 0.823135i \(0.692223\pi\)
\(492\) 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 −0.315753 0.948842i \(-0.602257\pi\)
0.948842 + 0.315753i \(0.102257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.1421i 0.630567i 0.948998 + 0.315283i \(0.102100\pi\)
−0.948998 + 0.315283i \(0.897900\pi\)
\(504\) 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 −0.966364 0.257177i \(-0.917208\pi\)
0.257177 + 0.966364i \(0.417208\pi\)
\(510\) 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 \(-0.839940\pi\)
0.876216 0.481919i \(-0.160060\pi\)
\(522\) 0 0
\(523\) 15.5563 + 15.5563i 0.680232 + 0.680232i 0.960052 0.279821i \(-0.0902750\pi\)
−0.279821 + 0.960052i \(0.590275\pi\)
\(524\) 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 0.289685 0.957122i \(-0.406449\pi\)
0.957122 0.289685i \(-0.0935507\pi\)
\(542\) 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 0.0283478 0.999598i \(-0.490975\pi\)
0.999598 0.0283478i \(-0.00902459\pi\)
\(548\) 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 −0.345295 0.938494i \(-0.612221\pi\)
0.938494 + 0.345295i \(0.112221\pi\)
\(558\) 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 −0.644988 0.764192i \(-0.723138\pi\)
0.764192 + 0.644988i \(0.223138\pi\)
\(564\) 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 \(0.123147\pi\)
−0.926091 + 0.377300i \(0.876853\pi\)
\(570\) 0 0
\(571\) −14.1421 14.1421i −0.591830 0.591830i 0.346296 0.938125i \(-0.387439\pi\)
−0.938125 + 0.346296i \(0.887439\pi\)
\(572\) 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 0.264057 0.964507i \(-0.414939\pi\)
−0.964507 + 0.264057i \(0.914939\pi\)
\(588\) 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 \(-0.559954\pi\)
0.187239 0.982314i \(-0.440046\pi\)
\(600\) 0 0
\(601\) 28.0000i 1.14214i 0.820900 + 0.571072i \(0.193472\pi\)
−0.820900 + 0.571072i \(0.806528\pi\)
\(602\) 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 0.394032 0.919097i \(-0.371080\pi\)
−0.919097 + 0.394032i \(0.871080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000i 0.885687i 0.896599 + 0.442843i \(0.146030\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(618\) 0 0
\(619\) 25.4558 + 25.4558i 1.02316 + 1.02316i 0.999725 + 0.0234313i \(0.00745910\pi\)
0.0234313 + 0.999725i \(0.492541\pi\)
\(620\) 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 \(0.248487\pi\)
−0.710461 + 0.703737i \(0.751513\pi\)
\(632\) 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 −0.874829 0.484431i \(-0.839027\pi\)
0.484431 + 0.874829i \(0.339027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.1421i 0.555985i 0.960583 + 0.277992i \(0.0896690\pi\)
−0.960583 + 0.277992i \(0.910331\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 6.00000i 0.234798 0.234798i −0.579894 0.814692i \(-0.696906\pi\)
0.814692 + 0.579894i \(0.196906\pi\)
\(654\) 0 0
\(655\) −56.5685 −2.21032
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0 0
\(661\) −9.00000 9.00000i −0.350059 0.350059i 0.510072 0.860132i \(-0.329619\pi\)
−0.860132 + 0.510072i \(0.829619\pi\)
\(662\) 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 1.00000 0.000738553i \(-0.000235089\pi\)
−0.000738553 1.00000i \(0.500235\pi\)
\(678\) 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 0.952838 0.303478i \(-0.0981479\pi\)
−0.303478 + 0.952838i \(0.598148\pi\)
\(684\) 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 −0.938115 0.346324i \(-0.887430\pi\)
0.346324 + 0.938115i \(0.387430\pi\)
\(692\) 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 −0.627521 0.778599i \(-0.715931\pi\)
0.778599 + 0.627521i \(0.215931\pi\)
\(702\) 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 0.907746 0.419521i \(-0.137802\pi\)
−0.419521 + 0.907746i \(0.637802\pi\)
\(710\) 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 \(-0.833966\pi\)
0.867018 0.498278i \(-0.166034\pi\)
\(728\) 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 −0.997268 0.0738717i \(-0.976464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 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 −0.998096 0.0616872i \(-0.980352\pi\)
0.0616872 + 0.998096i \(0.480352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.6274i 0.830119i −0.909794 0.415060i \(-0.863761\pi\)
0.909794 0.415060i \(-0.136239\pi\)
\(744\) 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 0.999829 0.0184972i \(-0.00588818\pi\)
−0.0184972 + 0.999829i \(0.505888\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000i 1.23250i 0.787551 + 0.616250i \(0.211349\pi\)
−0.787551 + 0.616250i \(0.788651\pi\)
\(762\) 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 0.775373 0.631503i \(-0.217562\pi\)
−0.631503 + 0.775373i \(0.717562\pi\)
\(774\) 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 −0.0413314 0.999145i \(-0.513160\pi\)
0.999145 + 0.0413314i \(0.0131599\pi\)
\(788\) 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 −0.200428 0.979708i \(-0.564233\pi\)
0.979708 + 0.200428i \(0.0642334\pi\)
\(798\) 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
\(804\)