Properties

Label 360.2.s.b.233.3
Level $360$
Weight $2$
Character 360.233
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 233.3
Root \(-1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 360.233
Dual form 360.2.s.b.17.3

$q$-expansion

\(f(q)\) \(=\) \(q+(1.58114 + 1.58114i) q^{5} +(-3.23607 + 3.23607i) q^{7} +O(q^{10})\) \(q+(1.58114 + 1.58114i) q^{5} +(-3.23607 + 3.23607i) q^{7} +4.57649i q^{11} +(-4.23607 - 4.23607i) q^{13} +(1.74806 + 1.74806i) q^{17} +2.47214i q^{19} +(2.82843 - 2.82843i) q^{23} +5.00000i q^{25} +5.99070 q^{29} +1.52786 q^{31} -10.2333 q^{35} +(2.23607 - 2.23607i) q^{37} +7.07107i q^{41} +(2.47214 + 2.47214i) q^{43} +(-1.74806 - 1.74806i) q^{47} -13.9443i q^{49} +(-7.23607 + 7.23607i) q^{55} +1.08036 q^{59} +10.4721 q^{61} -13.3956i q^{65} +(-1.52786 + 1.52786i) q^{67} -12.6491i q^{71} +(-0.527864 - 0.527864i) q^{73} +(-14.8098 - 14.8098i) q^{77} -14.4721i q^{79} +(1.08036 - 1.08036i) q^{83} +5.52786i q^{85} +0.746512 q^{89} +27.4164 q^{91} +(-3.90879 + 3.90879i) q^{95} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} - 16q^{13} + 48q^{31} - 16q^{43} - 40q^{55} + 48q^{61} - 48q^{67} - 40q^{73} + 112q^{91} + 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\)

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
\(5\) 1.58114 + 1.58114i 0.707107 + 0.707107i
\(6\) 0 0
\(7\) −3.23607 + 3.23607i −1.22312 + 1.22312i −0.256601 + 0.966517i \(0.582603\pi\)
−0.966517 + 0.256601i \(0.917397\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.57649i 1.37986i 0.723874 + 0.689932i \(0.242360\pi\)
−0.723874 + 0.689932i \(0.757640\pi\)
\(12\) 0 0
\(13\) −4.23607 4.23607i −1.17487 1.17487i −0.981033 0.193841i \(-0.937905\pi\)
−0.193841 0.981033i \(-0.562095\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.74806 + 1.74806i 0.423968 + 0.423968i 0.886567 0.462600i \(-0.153083\pi\)
−0.462600 + 0.886567i \(0.653083\pi\)
\(18\) 0 0
\(19\) 2.47214i 0.567147i 0.958951 + 0.283573i \(0.0915200\pi\)
−0.958951 + 0.283573i \(0.908480\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 2.82843i 0.589768 0.589768i −0.347801 0.937568i \(-0.613071\pi\)
0.937568 + 0.347801i \(0.113071\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.99070 1.11245 0.556223 0.831033i \(-0.312250\pi\)
0.556223 + 0.831033i \(0.312250\pi\)
\(30\) 0 0
\(31\) 1.52786 0.274412 0.137206 0.990543i \(-0.456188\pi\)
0.137206 + 0.990543i \(0.456188\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −10.2333 −1.72975
\(36\) 0 0
\(37\) 2.23607 2.23607i 0.367607 0.367607i −0.498997 0.866604i \(-0.666298\pi\)
0.866604 + 0.498997i \(0.166298\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.07107i 1.10432i 0.833740 + 0.552158i \(0.186195\pi\)
−0.833740 + 0.552158i \(0.813805\pi\)
\(42\) 0 0
\(43\) 2.47214 + 2.47214i 0.376997 + 0.376997i 0.870018 0.493020i \(-0.164107\pi\)
−0.493020 + 0.870018i \(0.664107\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.74806 1.74806i −0.254981 0.254981i 0.568028 0.823009i \(-0.307707\pi\)
−0.823009 + 0.568028i \(0.807707\pi\)
\(48\) 0 0
\(49\) 13.9443i 1.99204i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) −7.23607 + 7.23607i −0.975711 + 0.975711i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.08036 0.140651 0.0703256 0.997524i \(-0.477596\pi\)
0.0703256 + 0.997524i \(0.477596\pi\)
\(60\) 0 0
\(61\) 10.4721 1.34082 0.670410 0.741991i \(-0.266118\pi\)
0.670410 + 0.741991i \(0.266118\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.3956i 1.66152i
\(66\) 0 0
\(67\) −1.52786 + 1.52786i −0.186658 + 0.186658i −0.794250 0.607591i \(-0.792136\pi\)
0.607591 + 0.794250i \(0.292136\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.6491i 1.50117i −0.660772 0.750587i \(-0.729771\pi\)
0.660772 0.750587i \(-0.270229\pi\)
\(72\) 0 0
\(73\) −0.527864 0.527864i −0.0617818 0.0617818i 0.675541 0.737323i \(-0.263910\pi\)
−0.737323 + 0.675541i \(0.763910\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.8098 14.8098i −1.68774 1.68774i
\(78\) 0 0
\(79\) 14.4721i 1.62824i −0.580695 0.814121i \(-0.697219\pi\)
0.580695 0.814121i \(-0.302781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.08036 1.08036i 0.118585 0.118585i −0.645324 0.763909i \(-0.723278\pi\)
0.763909 + 0.645324i \(0.223278\pi\)
\(84\) 0 0
\(85\) 5.52786i 0.599581i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.746512 0.0791302 0.0395651 0.999217i \(-0.487403\pi\)
0.0395651 + 0.999217i \(0.487403\pi\)
\(90\) 0 0
\(91\) 27.4164 2.87402
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.90879 + 3.90879i −0.401033 + 0.401033i
\(96\) 0 0
\(97\) 1.00000 1.00000i 0.101535 0.101535i −0.654515 0.756049i \(-0.727127\pi\)
0.756049 + 0.654515i \(0.227127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.9799i 1.09254i 0.837610 + 0.546268i \(0.183952\pi\)
−0.837610 + 0.546268i \(0.816048\pi\)
\(102\) 0 0
\(103\) −12.1803 12.1803i −1.20016 1.20016i −0.974116 0.226049i \(-0.927419\pi\)
−0.226049 0.974116i \(-0.572581\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.2333 + 10.2333i 0.989295 + 0.989295i 0.999943 0.0106485i \(-0.00338957\pi\)
−0.0106485 + 0.999943i \(0.503390\pi\)
\(108\) 0 0
\(109\) 15.4164i 1.47662i 0.674459 + 0.738312i \(0.264377\pi\)
−0.674459 + 0.738312i \(0.735623\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.16228 3.16228i 0.297482 0.297482i −0.542545 0.840027i \(-0.682539\pi\)
0.840027 + 0.542545i \(0.182539\pi\)
\(114\) 0 0
\(115\) 8.94427 0.834058
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.3137 −1.03713
\(120\) 0 0
\(121\) −9.94427 −0.904025
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.90569 + 7.90569i −0.707107 + 0.707107i
\(126\) 0 0
\(127\) −7.23607 + 7.23607i −0.642097 + 0.642097i −0.951071 0.308973i \(-0.900015\pi\)
0.308973 + 0.951071i \(0.400015\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.41577i 0.211066i 0.994416 + 0.105533i \(0.0336549\pi\)
−0.994416 + 0.105533i \(0.966345\pi\)
\(132\) 0 0
\(133\) −8.00000 8.00000i −0.693688 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.81913 8.81913i −0.753469 0.753469i 0.221656 0.975125i \(-0.428854\pi\)
−0.975125 + 0.221656i \(0.928854\pi\)
\(138\) 0 0
\(139\) 8.94427i 0.758643i −0.925265 0.379322i \(-0.876157\pi\)
0.925265 0.379322i \(-0.123843\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 19.3863 19.3863i 1.62117 1.62117i
\(144\) 0 0
\(145\) 9.47214 + 9.47214i 0.786618 + 0.786618i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.4760 1.18592 0.592959 0.805232i \(-0.297959\pi\)
0.592959 + 0.805232i \(0.297959\pi\)
\(150\) 0 0
\(151\) −4.94427 −0.402359 −0.201180 0.979554i \(-0.564477\pi\)
−0.201180 + 0.979554i \(0.564477\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.41577 + 2.41577i 0.194039 + 0.194039i
\(156\) 0 0
\(157\) −0.708204 + 0.708204i −0.0565208 + 0.0565208i −0.734802 0.678281i \(-0.762725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.3060i 1.44271i
\(162\) 0 0
\(163\) 2.47214 + 2.47214i 0.193633 + 0.193633i 0.797264 0.603631i \(-0.206280\pi\)
−0.603631 + 0.797264i \(0.706280\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.32456 + 6.32456i 0.489409 + 0.489409i 0.908120 0.418711i \(-0.137518\pi\)
−0.418711 + 0.908120i \(0.637518\pi\)
\(168\) 0 0
\(169\) 22.8885i 1.76066i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.99070 + 5.99070i −0.455465 + 0.455465i −0.897163 0.441699i \(-0.854376\pi\)
0.441699 + 0.897163i \(0.354376\pi\)
\(174\) 0 0
\(175\) −16.1803 16.1803i −1.22312 1.22312i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.7295 1.02619 0.513095 0.858332i \(-0.328499\pi\)
0.513095 + 0.858332i \(0.328499\pi\)
\(180\) 0 0
\(181\) −10.4721 −0.778388 −0.389194 0.921156i \(-0.627246\pi\)
−0.389194 + 0.921156i \(0.627246\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.07107 0.519875
\(186\) 0 0
\(187\) −8.00000 + 8.00000i −0.585018 + 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.3060i 1.32457i 0.749251 + 0.662287i \(0.230414\pi\)
−0.749251 + 0.662287i \(0.769586\pi\)
\(192\) 0 0
\(193\) −1.47214 1.47214i −0.105967 0.105967i 0.652136 0.758102i \(-0.273873\pi\)
−0.758102 + 0.652136i \(0.773873\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.82998 + 3.82998i 0.272875 + 0.272875i 0.830256 0.557382i \(-0.188194\pi\)
−0.557382 + 0.830256i \(0.688194\pi\)
\(198\) 0 0
\(199\) 8.00000i 0.567105i −0.958957 0.283552i \(-0.908487\pi\)
0.958957 0.283552i \(-0.0915130\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −19.3863 + 19.3863i −1.36065 + 1.36065i
\(204\) 0 0
\(205\) −11.1803 + 11.1803i −0.780869 + 0.780869i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.81758i 0.533155i
\(216\) 0 0
\(217\) −4.94427 + 4.94427i −0.335639 + 0.335639i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.8098i 0.996217i
\(222\) 0 0
\(223\) 19.2361 + 19.2361i 1.28814 + 1.28814i 0.935914 + 0.352228i \(0.114576\pi\)
0.352228 + 0.935914i \(0.385424\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.4667 + 20.4667i 1.35842 + 1.35842i 0.875864 + 0.482558i \(0.160292\pi\)
0.482558 + 0.875864i \(0.339708\pi\)
\(228\) 0 0
\(229\) 19.8885i 1.31427i −0.753772 0.657136i \(-0.771768\pi\)
0.753772 0.657136i \(-0.228232\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.5579 + 16.5579i −1.08474 + 1.08474i −0.0886844 + 0.996060i \(0.528266\pi\)
−0.996060 + 0.0886844i \(0.971734\pi\)
\(234\) 0 0
\(235\) 5.52786i 0.360598i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.6491 0.818203 0.409101 0.912489i \(-0.365842\pi\)
0.409101 + 0.912489i \(0.365842\pi\)
\(240\) 0 0
\(241\) 2.94427 0.189657 0.0948286 0.995494i \(-0.469770\pi\)
0.0948286 + 0.995494i \(0.469770\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 22.0478 22.0478i 1.40858 1.40858i
\(246\) 0 0
\(247\) 10.4721 10.4721i 0.666326 0.666326i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.7000i 1.93777i −0.247513 0.968885i \(-0.579613\pi\)
0.247513 0.968885i \(-0.420387\pi\)
\(252\) 0 0
\(253\) 12.9443 + 12.9443i 0.813799 + 0.813799i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.16228 + 3.16228i 0.197257 + 0.197257i 0.798823 0.601566i \(-0.205456\pi\)
−0.601566 + 0.798823i \(0.705456\pi\)
\(258\) 0 0
\(259\) 14.4721i 0.899255i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.1421 14.1421i 0.872041 0.872041i −0.120653 0.992695i \(-0.538499\pi\)
0.992695 + 0.120653i \(0.0384989\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −12.3153 −0.750875 −0.375437 0.926848i \(-0.622507\pi\)
−0.375437 + 0.926848i \(0.622507\pi\)
\(270\) 0 0
\(271\) −28.9443 −1.75824 −0.879120 0.476601i \(-0.841869\pi\)
−0.879120 + 0.476601i \(0.841869\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.8825 −1.37986
\(276\) 0 0
\(277\) 15.1803 15.1803i 0.912098 0.912098i −0.0843389 0.996437i \(-0.526878\pi\)
0.996437 + 0.0843389i \(0.0268778\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.5595i 1.04751i −0.851869 0.523755i \(-0.824531\pi\)
0.851869 0.523755i \(-0.175469\pi\)
\(282\) 0 0
\(283\) −8.00000 8.00000i −0.475551 0.475551i 0.428155 0.903705i \(-0.359164\pi\)
−0.903705 + 0.428155i \(0.859164\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −22.8825 22.8825i −1.35071 1.35071i
\(288\) 0 0
\(289\) 10.8885i 0.640503i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.15143 + 8.15143i −0.476212 + 0.476212i −0.903918 0.427706i \(-0.859322\pi\)
0.427706 + 0.903918i \(0.359322\pi\)
\(294\) 0 0
\(295\) 1.70820 + 1.70820i 0.0994555 + 0.0994555i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −23.9628 −1.38581
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.5579 + 16.5579i 0.948103 + 0.948103i
\(306\) 0 0
\(307\) 10.4721 10.4721i 0.597676 0.597676i −0.342017 0.939694i \(-0.611110\pi\)
0.939694 + 0.342017i \(0.111110\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.8098i 0.839789i −0.907573 0.419894i \(-0.862067\pi\)
0.907573 0.419894i \(-0.137933\pi\)
\(312\) 0 0
\(313\) −7.47214 7.47214i −0.422350 0.422350i 0.463662 0.886012i \(-0.346535\pi\)
−0.886012 + 0.463662i \(0.846535\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.81758 + 7.81758i 0.439079 + 0.439079i 0.891702 0.452623i \(-0.149512\pi\)
−0.452623 + 0.891702i \(0.649512\pi\)
\(318\) 0 0
\(319\) 27.4164i 1.53502i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.32145 + 4.32145i −0.240452 + 0.240452i
\(324\) 0 0
\(325\) 21.1803 21.1803i 1.17487 1.17487i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.3137 0.623745
\(330\) 0 0
\(331\) −5.52786 −0.303839 −0.151919 0.988393i \(-0.548545\pi\)
−0.151919 + 0.988393i \(0.548545\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.83153 −0.263975
\(336\) 0 0
\(337\) 14.4164 14.4164i 0.785312 0.785312i −0.195410 0.980722i \(-0.562604\pi\)
0.980722 + 0.195410i \(0.0626037\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.99226i 0.378652i
\(342\) 0 0
\(343\) 22.4721 + 22.4721i 1.21338 + 1.21338i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.99226 6.99226i −0.375364 0.375364i 0.494062 0.869426i \(-0.335511\pi\)
−0.869426 + 0.494062i \(0.835511\pi\)
\(348\) 0 0
\(349\) 5.52786i 0.295900i 0.988995 + 0.147950i \(0.0472674\pi\)
−0.988995 + 0.147950i \(0.952733\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.0540 20.0540i 1.06737 1.06737i 0.0698078 0.997560i \(-0.477761\pi\)
0.997560 0.0698078i \(-0.0222386\pi\)
\(354\) 0 0
\(355\) 20.0000 20.0000i 1.06149 1.06149i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −33.1158 −1.74779 −0.873893 0.486119i \(-0.838412\pi\)
−0.873893 + 0.486119i \(0.838412\pi\)
\(360\) 0 0
\(361\) 12.8885 0.678344
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.66925i 0.0873727i
\(366\) 0 0
\(367\) 2.29180 2.29180i 0.119631 0.119631i −0.644757 0.764388i \(-0.723041\pi\)
0.764388 + 0.644757i \(0.223041\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11.2918 + 11.2918i 0.584667 + 0.584667i 0.936182 0.351515i \(-0.114333\pi\)
−0.351515 + 0.936182i \(0.614333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −25.3770 25.3770i −1.30698 1.30698i
\(378\) 0 0
\(379\) 16.9443i 0.870369i 0.900341 + 0.435184i \(0.143317\pi\)
−0.900341 + 0.435184i \(0.856683\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.90879 3.90879i 0.199730 0.199730i −0.600154 0.799884i \(-0.704894\pi\)
0.799884 + 0.600154i \(0.204894\pi\)
\(384\) 0 0
\(385\) 46.8328i 2.38682i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.7974 −0.953068 −0.476534 0.879156i \(-0.658107\pi\)
−0.476534 + 0.879156i \(0.658107\pi\)
\(390\) 0 0
\(391\) 9.88854 0.500085
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 22.8825 22.8825i 1.15134 1.15134i
\(396\) 0 0
\(397\) −6.23607 + 6.23607i −0.312979 + 0.312979i −0.846063 0.533083i \(-0.821033\pi\)
0.533083 + 0.846063i \(0.321033\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.5378i 1.37517i −0.726104 0.687585i \(-0.758671\pi\)
0.726104 0.687585i \(-0.241329\pi\)
\(402\) 0 0
\(403\) −6.47214 6.47214i −0.322400 0.322400i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.2333 + 10.2333i 0.507248 + 0.507248i
\(408\) 0 0
\(409\) 25.8885i 1.28011i 0.768331 + 0.640053i \(0.221088\pi\)
−0.768331 + 0.640053i \(0.778912\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.49613 + 3.49613i −0.172033 + 0.172033i
\(414\) 0 0
\(415\) 3.41641 0.167705
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.40182 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.74032 + 8.74032i −0.423968 + 0.423968i
\(426\) 0 0
\(427\) −33.8885 + 33.8885i −1.63998 + 1.63998i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685i 0.272481i 0.990676 + 0.136241i \(0.0435020\pi\)
−0.990676 + 0.136241i \(0.956498\pi\)
\(432\) 0 0
\(433\) −16.4164 16.4164i −0.788922 0.788922i 0.192395 0.981318i \(-0.438374\pi\)
−0.981318 + 0.192395i \(0.938374\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.99226 + 6.99226i 0.334485 + 0.334485i
\(438\) 0 0
\(439\) 20.9443i 0.999616i −0.866136 0.499808i \(-0.833404\pi\)
0.866136 0.499808i \(-0.166596\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.4667 + 20.4667i −0.972402 + 0.972402i −0.999629 0.0272274i \(-0.991332\pi\)
0.0272274 + 0.999629i \(0.491332\pi\)
\(444\) 0 0
\(445\) 1.18034 + 1.18034i 0.0559535 + 0.0559535i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.6938 1.82608 0.913038 0.407875i \(-0.133730\pi\)
0.913038 + 0.407875i \(0.133730\pi\)
\(450\) 0 0
\(451\) −32.3607 −1.52380
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 43.3491 + 43.3491i 2.03224 + 2.03224i
\(456\) 0 0
\(457\) 13.0000 13.0000i 0.608114 0.608114i −0.334339 0.942453i \(-0.608513\pi\)
0.942453 + 0.334339i \(0.108513\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.48683i 0.441846i 0.975291 + 0.220923i \(0.0709069\pi\)
−0.975291 + 0.220923i \(0.929093\pi\)
\(462\) 0 0
\(463\) 10.6525 + 10.6525i 0.495063 + 0.495063i 0.909897 0.414834i \(-0.136160\pi\)
−0.414834 + 0.909897i \(0.636160\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.57649 4.57649i −0.211775 0.211775i 0.593246 0.805021i \(-0.297846\pi\)
−0.805021 + 0.593246i \(0.797846\pi\)
\(468\) 0 0
\(469\) 9.88854i 0.456611i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −11.3137 + 11.3137i −0.520205 + 0.520205i
\(474\) 0 0
\(475\) −12.3607 −0.567147
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −39.5980 −1.80928 −0.904639 0.426179i \(-0.859859\pi\)
−0.904639 + 0.426179i \(0.859859\pi\)
\(480\) 0 0
\(481\) −18.9443 −0.863784
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.16228 0.143592
\(486\) 0 0
\(487\) 8.76393 8.76393i 0.397132 0.397132i −0.480088 0.877220i \(-0.659395\pi\)
0.877220 + 0.480088i \(0.159395\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.7000i 1.38547i −0.721191 0.692737i \(-0.756405\pi\)
0.721191 0.692737i \(-0.243595\pi\)
\(492\) 0 0
\(493\) 10.4721 + 10.4721i 0.471641 + 0.471641i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 40.9334 + 40.9334i 1.83611 + 1.83611i
\(498\) 0 0
\(499\) 26.4721i 1.18506i 0.805550 + 0.592528i \(0.201870\pi\)
−0.805550 + 0.592528i \(0.798130\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −29.2070 + 29.2070i −1.30228 + 1.30228i −0.375422 + 0.926854i \(0.622502\pi\)
−0.926854 + 0.375422i \(0.877498\pi\)
\(504\) 0 0
\(505\) −17.3607 + 17.3607i −0.772540 + 0.772540i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.9830 −0.575460 −0.287730 0.957712i \(-0.592901\pi\)
−0.287730 + 0.957712i \(0.592901\pi\)
\(510\) 0 0
\(511\) 3.41641 0.151133
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 38.5176i 1.69729i
\(516\) 0 0
\(517\) 8.00000 8.00000i 0.351840 0.351840i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.24574i 0.273631i 0.990597 + 0.136816i \(0.0436867\pi\)
−0.990597 + 0.136816i \(0.956313\pi\)
\(522\) 0 0
\(523\) −9.88854 9.88854i −0.432396 0.432396i 0.457047 0.889443i \(-0.348907\pi\)
−0.889443 + 0.457047i \(0.848907\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.67080 + 2.67080i 0.116342 + 0.116342i
\(528\) 0 0
\(529\) 7.00000i 0.304348i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 29.9535 29.9535i 1.29743 1.29743i
\(534\) 0 0
\(535\) 32.3607i 1.39907i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 63.8158 2.74874
\(540\) 0 0
\(541\) −25.3050 −1.08794 −0.543972 0.839103i \(-0.683080\pi\)
−0.543972 + 0.839103i \(0.683080\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −24.3755 + 24.3755i −1.04413 + 1.04413i
\(546\) 0 0
\(547\) 12.0000 12.0000i 0.513083 0.513083i −0.402387 0.915470i \(-0.631819\pi\)
0.915470 + 0.402387i \(0.131819\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 14.8098i 0.630920i
\(552\) 0 0
\(553\) 46.8328 + 46.8328i 1.99153 + 1.99153i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.1235 + 26.1235i 1.10689 + 1.10689i 0.993557 + 0.113333i \(0.0361527\pi\)
0.113333 + 0.993557i \(0.463847\pi\)
\(558\) 0 0
\(559\) 20.9443i 0.885848i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.33540 1.33540i 0.0562805 0.0562805i −0.678406 0.734687i \(-0.737329\pi\)
0.734687 + 0.678406i \(0.237329\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.3662 −1.27302 −0.636508 0.771270i \(-0.719622\pi\)
−0.636508 + 0.771270i \(0.719622\pi\)
\(570\) 0 0
\(571\) −7.41641 −0.310367 −0.155184 0.987886i \(-0.549597\pi\)
−0.155184 + 0.987886i \(0.549597\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.1421 + 14.1421i 0.589768 + 0.589768i
\(576\) 0 0
\(577\) −12.0557 + 12.0557i −0.501887 + 0.501887i −0.912024 0.410137i \(-0.865481\pi\)
0.410137 + 0.912024i \(0.365481\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.99226i 0.290088i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.8825 + 22.8825i 0.944460 + 0.944460i 0.998537 0.0540767i \(-0.0172216\pi\)
−0.0540767 + 0.998537i \(0.517222\pi\)
\(588\) 0 0
\(589\) 3.77709i 0.155632i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.1359 + 22.1359i −0.909014 + 0.909014i −0.996193 0.0871785i \(-0.972215\pi\)
0.0871785 + 0.996193i \(0.472215\pi\)
\(594\) 0 0
\(595\) −17.8885 17.8885i −0.733359 0.733359i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.6491 0.516829 0.258414 0.966034i \(-0.416800\pi\)
0.258414 + 0.966034i \(0.416800\pi\)
\(600\) 0 0
\(601\) −30.8328 −1.25770 −0.628848 0.777528i \(-0.716473\pi\)
−0.628848 + 0.777528i \(0.716473\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.7233 15.7233i −0.639242 0.639242i
\(606\) 0 0
\(607\) 30.6525 30.6525i 1.24415 1.24415i 0.285880 0.958265i \(-0.407714\pi\)
0.958265 0.285880i \(-0.0922860\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.8098i 0.599142i
\(612\) 0 0
\(613\) −26.7082 26.7082i −1.07873 1.07873i −0.996623 0.0821110i \(-0.973834\pi\)
−0.0821110 0.996623i \(-0.526166\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −16.5579 16.5579i −0.666596 0.666596i 0.290330 0.956926i \(-0.406235\pi\)
−0.956926 + 0.290330i \(0.906235\pi\)
\(618\) 0 0
\(619\) 5.88854i 0.236681i −0.992973 0.118340i \(-0.962243\pi\)
0.992973 0.118340i \(-0.0377574\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.41577 + 2.41577i −0.0967856 + 0.0967856i
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.81758 0.311707
\(630\) 0 0
\(631\) 8.36068 0.332833 0.166417 0.986056i \(-0.446780\pi\)
0.166417 + 0.986056i \(0.446780\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.8825 −0.908063
\(636\) 0 0
\(637\) −59.0689 + 59.0689i −2.34039 + 2.34039i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.57804i 0.220319i −0.993914 0.110160i \(-0.964864\pi\)
0.993914 0.110160i \(-0.0351362\pi\)
\(642\) 0 0
\(643\) −20.0000 20.0000i −0.788723 0.788723i 0.192562 0.981285i \(-0.438320\pi\)
−0.981285 + 0.192562i \(0.938320\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.3755 24.3755i −0.958299 0.958299i 0.0408656 0.999165i \(-0.486988\pi\)
−0.999165 + 0.0408656i \(0.986988\pi\)
\(648\) 0 0
\(649\) 4.94427i 0.194080i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.3137 11.3137i 0.442740 0.442740i −0.450192 0.892932i \(-0.648644\pi\)
0.892932 + 0.450192i \(0.148644\pi\)
\(654\) 0 0
\(655\) −3.81966 + 3.81966i −0.149246 + 0.149246i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 35.5316 1.38411 0.692057 0.721843i \(-0.256705\pi\)
0.692057 + 0.721843i \(0.256705\pi\)
\(660\) 0 0
\(661\) 41.3050 1.60658 0.803288 0.595591i \(-0.203082\pi\)
0.803288 + 0.595591i \(0.203082\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 25.2982i 0.981023i
\(666\) 0 0
\(667\) 16.9443 16.9443i 0.656085 0.656085i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 47.9256i 1.85015i
\(672\) 0 0
\(673\) 19.4721 + 19.4721i 0.750596 + 0.750596i 0.974590 0.223995i \(-0.0719098\pi\)
−0.223995 + 0.974590i \(0.571910\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.3060 18.3060i −0.703555 0.703555i 0.261617 0.965172i \(-0.415744\pi\)
−0.965172 + 0.261617i \(0.915744\pi\)
\(678\) 0 0
\(679\) 6.47214i 0.248378i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0649 15.0649i 0.576441 0.576441i −0.357480 0.933921i \(-0.616364\pi\)
0.933921 + 0.357480i \(0.116364\pi\)
\(684\) 0 0
\(685\) 27.8885i 1.06557i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −20.3607 −0.774557 −0.387278 0.921963i \(-0.626585\pi\)
−0.387278 + 0.921963i \(0.626585\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.1421 14.1421i 0.536442 0.536442i
\(696\) 0 0
\(697\) −12.3607 + 12.3607i −0.468194 + 0.468194i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.6212i 1.15655i −0.815843 0.578274i \(-0.803727\pi\)
0.815843 0.578274i \(-0.196273\pi\)
\(702\) 0 0
\(703\) 5.52786 + 5.52786i 0.208487 + 0.208487i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −35.5316 35.5316i −1.33630 1.33630i
\(708\) 0 0
\(709\) 15.8885i 0.596707i −0.954455 0.298353i \(-0.903563\pi\)
0.954455 0.298353i \(-0.0964374\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.32145 4.32145i 0.161840 0.161840i
\(714\) 0 0
\(715\) 61.3050 2.29268
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.32145 0.161163 0.0805815 0.996748i \(-0.474322\pi\)
0.0805815 + 0.996748i \(0.474322\pi\)
\(720\) 0 0
\(721\) 78.8328 2.93589
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.9535i 1.11245i
\(726\) 0 0
\(727\) 9.12461 9.12461i 0.338413 0.338413i −0.517357 0.855770i \(-0.673084\pi\)
0.855770 + 0.517357i \(0.173084\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.64290i 0.319669i
\(732\) 0 0
\(733\) 11.1803 + 11.1803i 0.412955 + 0.412955i 0.882767 0.469811i \(-0.155678\pi\)
−0.469811 + 0.882767i \(0.655678\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.99226 6.99226i −0.257563 0.257563i
\(738\) 0 0
\(739\) 12.3607i 0.454695i −0.973814 0.227347i \(-0.926995\pi\)
0.973814 0.227347i \(-0.0730053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.2070 29.2070i 1.07150 1.07150i 0.0742626 0.997239i \(-0.476340\pi\)
0.997239 0.0742626i \(-0.0236603\pi\)
\(744\) 0 0
\(745\) 22.8885 + 22.8885i 0.838571 + 0.838571i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −66.2316 −2.42005
\(750\) 0 0
\(751\) 32.3607 1.18086 0.590429 0.807090i \(-0.298959\pi\)
0.590429 + 0.807090i \(0.298959\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7.81758 7.81758i −0.284511 0.284511i
\(756\) 0 0
\(757\) 10.7082 10.7082i 0.389196 0.389196i −0.485204 0.874401i \(-0.661255\pi\)
0.874401 + 0.485204i \(0.161255\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.0447i 0.944121i −0.881566 0.472060i \(-0.843510\pi\)
0.881566 0.472060i \(-0.156490\pi\)
\(762\) 0 0
\(763\) −49.8885 49.8885i −1.80609 1.80609i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.57649 4.57649i −0.165248 0.165248i
\(768\) 0 0
\(769\) 32.0000i 1.15395i −0.816762 0.576975i \(-0.804233\pi\)
0.816762 0.576975i \(-0.195767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.81758 + 7.81758i −0.281179 + 0.281179i −0.833579 0.552400i \(-0.813712\pi\)
0.552400 + 0.833579i \(0.313712\pi\)
\(774\) 0 0
\(775\) 7.63932i 0.274412i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.4806 −0.626309
\(780\) 0 0
\(781\) 57.8885 2.07141
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.23954 −0.0799325
\(786\) 0 0
\(787\) −30.4721 + 30.4721i −1.08621 + 1.08621i −0.0902997 + 0.995915i \(0.528782\pi\)
−0.995915 + 0.0902997i \(0.971218\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.4667i 0.727712i
\(792\) 0 0
\(793\) −44.3607 44.3607i −1.57529 1.57529i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.66925 1.66925i −0.0591280 0.0591280i 0.676924 0.736052i \(-0.263312\pi\)
−0.736052 + 0.676924i \(0.763312\pi\)
\(798\) 0 0
\(799\) 6.11146i 0.216208i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.41577 2.41577i 0.0852505 0.0852505i
\(804\) 0 0
\(805\) −28.9443 + 28.9443i