Properties

Label 1200.2.h.n.1151.3
Level $1200$
Weight $2$
Character 1200.1151
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.3
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1151
Dual form 1200.2.h.n.1151.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 + 1.41421i) q^{3} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(1.00000 + 1.41421i) q^{3} +(-1.00000 + 2.82843i) q^{9} -4.89898 q^{11} -4.89898 q^{13} -3.46410i q^{17} +3.46410i q^{19} -6.00000 q^{23} +(-5.00000 + 1.41421i) q^{27} -2.82843i q^{29} +3.46410i q^{31} +(-4.89898 - 6.92820i) q^{33} -4.89898 q^{37} +(-4.89898 - 6.92820i) q^{39} +5.65685i q^{41} +8.48528i q^{43} +6.00000 q^{47} +7.00000 q^{49} +(4.89898 - 3.46410i) q^{51} -10.3923i q^{53} +(-4.89898 + 3.46410i) q^{57} +4.89898 q^{59} -2.00000 q^{61} +8.48528i q^{67} +(-6.00000 - 8.48528i) q^{69} +9.79796 q^{71} -9.79796 q^{73} +10.3923i q^{79} +(-7.00000 - 5.65685i) q^{81} -6.00000 q^{83} +(4.00000 - 2.82843i) q^{87} +5.65685i q^{89} +(-4.89898 + 3.46410i) q^{93} +(4.89898 - 13.8564i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{9} - 24q^{23} - 20q^{27} + 24q^{47} + 28q^{49} - 8q^{61} - 24q^{69} - 28q^{81} - 24q^{83} + 16q^{87} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000 + 1.41421i 0.577350 + 0.816497i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) −4.89898 −1.35873 −0.679366 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) 0 0
\(29\) 2.82843i 0.525226i −0.964901 0.262613i \(-0.915416\pi\)
0.964901 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) −4.89898 6.92820i −0.852803 1.20605i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.89898 −0.805387 −0.402694 0.915335i \(-0.631926\pi\)
−0.402694 + 0.915335i \(0.631926\pi\)
\(38\) 0 0
\(39\) −4.89898 6.92820i −0.784465 1.10940i
\(40\) 0 0
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.89898 3.46410i 0.685994 0.485071i
\(52\) 0 0
\(53\) 10.3923i 1.42749i −0.700404 0.713746i \(-0.746997\pi\)
0.700404 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.89898 + 3.46410i −0.648886 + 0.458831i
\(58\) 0 0
\(59\) 4.89898 0.637793 0.318896 0.947790i \(-0.396688\pi\)
0.318896 + 0.947790i \(0.396688\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528i 1.03664i 0.855186 + 0.518321i \(0.173443\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) −6.00000 8.48528i −0.722315 1.02151i
\(70\) 0 0
\(71\) 9.79796 1.16280 0.581402 0.813617i \(-0.302504\pi\)
0.581402 + 0.813617i \(0.302504\pi\)
\(72\) 0 0
\(73\) −9.79796 −1.14676 −0.573382 0.819288i \(-0.694369\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923i 1.16923i 0.811312 + 0.584613i \(0.198754\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000 2.82843i 0.428845 0.303239i
\(88\) 0 0
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.89898 + 3.46410i −0.508001 + 0.359211i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 4.89898 13.8564i 0.492366 1.39262i
\(100\) 0 0
\(101\) 14.1421i 1.40720i −0.710599 0.703598i \(-0.751576\pi\)
0.710599 0.703598i \(-0.248424\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.89898 6.92820i −0.464991 0.657596i
\(112\) 0 0
\(113\) 3.46410i 0.325875i 0.986636 + 0.162938i \(0.0520969\pi\)
−0.986636 + 0.162938i \(0.947903\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.89898 13.8564i 0.452911 1.28103i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) −8.00000 + 5.65685i −0.721336 + 0.510061i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.9706i 1.50589i −0.658081 0.752947i \(-0.728632\pi\)
0.658081 0.752947i \(-0.271368\pi\)
\(128\) 0 0
\(129\) −12.0000 + 8.48528i −1.05654 + 0.747087i
\(130\) 0 0
\(131\) 4.89898 0.428026 0.214013 0.976831i \(-0.431347\pi\)
0.214013 + 0.976831i \(0.431347\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.46410i 0.295958i −0.988990 0.147979i \(-0.952723\pi\)
0.988990 0.147979i \(-0.0472768\pi\)
\(138\) 0 0
\(139\) 17.3205i 1.46911i 0.678551 + 0.734553i \(0.262608\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 6.00000 + 8.48528i 0.505291 + 0.714590i
\(142\) 0 0
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.00000 + 9.89949i 0.577350 + 0.816497i
\(148\) 0 0
\(149\) 2.82843i 0.231714i 0.993266 + 0.115857i \(0.0369614\pi\)
−0.993266 + 0.115857i \(0.963039\pi\)
\(150\) 0 0
\(151\) 3.46410i 0.281905i 0.990016 + 0.140952i \(0.0450164\pi\)
−0.990016 + 0.140952i \(0.954984\pi\)
\(152\) 0 0
\(153\) 9.79796 + 3.46410i 0.792118 + 0.280056i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.89898 0.390981 0.195491 0.980706i \(-0.437370\pi\)
0.195491 + 0.980706i \(0.437370\pi\)
\(158\) 0 0
\(159\) 14.6969 10.3923i 1.16554 0.824163i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.48528i 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) 0 0
\(171\) −9.79796 3.46410i −0.749269 0.264906i
\(172\) 0 0
\(173\) 3.46410i 0.263371i 0.991292 + 0.131685i \(0.0420389\pi\)
−0.991292 + 0.131685i \(0.957961\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.89898 + 6.92820i 0.368230 + 0.520756i
\(178\) 0 0
\(179\) −24.4949 −1.83083 −0.915417 0.402506i \(-0.868139\pi\)
−0.915417 + 0.402506i \(0.868139\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −2.00000 2.82843i −0.147844 0.209083i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.9706i 1.24101i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.5959 −1.41791 −0.708955 0.705253i \(-0.750833\pi\)
−0.708955 + 0.705253i \(0.750833\pi\)
\(192\) 0 0
\(193\) 19.5959 1.41055 0.705273 0.708936i \(-0.250825\pi\)
0.705273 + 0.708936i \(0.250825\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.2487i 1.72765i 0.503793 + 0.863825i \(0.331938\pi\)
−0.503793 + 0.863825i \(0.668062\pi\)
\(198\) 0 0
\(199\) 10.3923i 0.736691i 0.929689 + 0.368345i \(0.120076\pi\)
−0.929689 + 0.368345i \(0.879924\pi\)
\(200\) 0 0
\(201\) −12.0000 + 8.48528i −0.846415 + 0.598506i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000 16.9706i 0.417029 1.17954i
\(208\) 0 0
\(209\) 16.9706i 1.17388i
\(210\) 0 0
\(211\) 24.2487i 1.66935i 0.550743 + 0.834675i \(0.314345\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 9.79796 + 13.8564i 0.671345 + 0.949425i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −9.79796 13.8564i −0.662085 0.936329i
\(220\) 0 0
\(221\) 16.9706i 1.14156i
\(222\) 0 0
\(223\) 16.9706i 1.13643i −0.822879 0.568216i \(-0.807634\pi\)
0.822879 0.568216i \(-0.192366\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.46410i 0.226941i 0.993541 + 0.113470i \(0.0361967\pi\)
−0.993541 + 0.113470i \(0.963803\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.6969 + 10.3923i −0.954669 + 0.675053i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 1.00000 15.5563i 0.0641500 0.997940i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.9706i 1.07981i
\(248\) 0 0
\(249\) −6.00000 8.48528i −0.380235 0.537733i
\(250\) 0 0
\(251\) 14.6969 0.927663 0.463831 0.885924i \(-0.346474\pi\)
0.463831 + 0.885924i \(0.346474\pi\)
\(252\) 0 0
\(253\) 29.3939 1.84798
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.46410i 0.216085i −0.994146 0.108042i \(-0.965542\pi\)
0.994146 0.108042i \(-0.0344582\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 8.00000 + 2.82843i 0.495188 + 0.175075i
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −8.00000 + 5.65685i −0.489592 + 0.346194i
\(268\) 0 0
\(269\) 19.7990i 1.20717i 0.797300 + 0.603583i \(0.206261\pi\)
−0.797300 + 0.603583i \(0.793739\pi\)
\(270\) 0 0
\(271\) 10.3923i 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.6969 0.883053 0.441527 0.897248i \(-0.354437\pi\)
0.441527 + 0.897248i \(0.354437\pi\)
\(278\) 0 0
\(279\) −9.79796 3.46410i −0.586588 0.207390i
\(280\) 0 0
\(281\) 5.65685i 0.337460i 0.985662 + 0.168730i \(0.0539665\pi\)
−0.985662 + 0.168730i \(0.946033\pi\)
\(282\) 0 0
\(283\) 8.48528i 0.504398i 0.967675 + 0.252199i \(0.0811537\pi\)
−0.967675 + 0.252199i \(0.918846\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.3205i 1.01187i 0.862570 + 0.505937i \(0.168853\pi\)
−0.862570 + 0.505937i \(0.831147\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 24.4949 6.92820i 1.42134 0.402015i
\(298\) 0 0
\(299\) 29.3939 1.69989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 20.0000 14.1421i 1.14897 0.812444i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.48528i 0.484281i 0.970241 + 0.242140i \(0.0778494\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.79796 −0.555591 −0.277796 0.960640i \(-0.589604\pi\)
−0.277796 + 0.960640i \(0.589604\pi\)
\(312\) 0 0
\(313\) 9.79796 0.553813 0.276907 0.960897i \(-0.410691\pi\)
0.276907 + 0.960897i \(0.410691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.3205i 0.972817i −0.873732 0.486408i \(-0.838307\pi\)
0.873732 0.486408i \(-0.161693\pi\)
\(318\) 0 0
\(319\) 13.8564i 0.775810i
\(320\) 0 0
\(321\) −18.0000 25.4558i −1.00466 1.42081i
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000 + 2.82843i 0.110600 + 0.156412i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.3205i 0.952021i −0.879440 0.476011i \(-0.842082\pi\)
0.879440 0.476011i \(-0.157918\pi\)
\(332\) 0 0
\(333\) 4.89898 13.8564i 0.268462 0.759326i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) −4.89898 + 3.46410i −0.266076 + 0.188144i
\(340\) 0 0
\(341\) 16.9706i 0.919007i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 24.4949 6.92820i 1.30744 0.369800i
\(352\) 0 0
\(353\) 31.1769i 1.65938i 0.558225 + 0.829690i \(0.311483\pi\)
−0.558225 + 0.829690i \(0.688517\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.79796 0.517116 0.258558 0.965996i \(-0.416753\pi\)
0.258558 + 0.965996i \(0.416753\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 13.0000 + 18.3848i 0.682323 + 0.964951i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.9706i 0.885856i −0.896557 0.442928i \(-0.853940\pi\)
0.896557 0.442928i \(-0.146060\pi\)
\(368\) 0 0
\(369\) −16.0000 5.65685i −0.832927 0.294484i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14.6969 −0.760979 −0.380489 0.924785i \(-0.624244\pi\)
−0.380489 + 0.924785i \(0.624244\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.8564i 0.713641i
\(378\) 0 0
\(379\) 10.3923i 0.533817i −0.963722 0.266908i \(-0.913998\pi\)
0.963722 0.266908i \(-0.0860021\pi\)
\(380\) 0 0
\(381\) 24.0000 16.9706i 1.22956 0.869428i
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −24.0000 8.48528i −1.21999 0.431331i
\(388\) 0 0
\(389\) 2.82843i 0.143407i 0.997426 + 0.0717035i \(0.0228435\pi\)
−0.997426 + 0.0717035i \(0.977156\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) 0 0
\(393\) 4.89898 + 6.92820i 0.247121 + 0.349482i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.89898 0.245873 0.122936 0.992415i \(-0.460769\pi\)
0.122936 + 0.992415i \(0.460769\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6274i 1.12996i −0.825105 0.564980i \(-0.808884\pi\)
0.825105 0.564980i \(-0.191116\pi\)
\(402\) 0 0
\(403\) 16.9706i 0.845364i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 4.89898 3.46410i 0.241649 0.170872i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −24.4949 + 17.3205i −1.19952 + 0.848189i
\(418\) 0 0
\(419\) 14.6969 0.717992 0.358996 0.933339i \(-0.383119\pi\)
0.358996 + 0.933339i \(0.383119\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) −6.00000 + 16.9706i −0.291730 + 0.825137i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 24.0000 + 33.9411i 1.15873 + 1.63869i
\(430\) 0 0
\(431\) 19.5959 0.943902 0.471951 0.881625i \(-0.343550\pi\)
0.471951 + 0.881625i \(0.343550\pi\)
\(432\) 0 0
\(433\) −19.5959 −0.941720 −0.470860 0.882208i \(-0.656056\pi\)
−0.470860 + 0.882208i \(0.656056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7846i 0.994263i
\(438\) 0 0
\(439\) 17.3205i 0.826663i −0.910581 0.413331i \(-0.864365\pi\)
0.910581 0.413331i \(-0.135635\pi\)
\(440\) 0 0
\(441\) −7.00000 + 19.7990i −0.333333 + 0.942809i
\(442\) 0 0
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.00000 + 2.82843i −0.189194 + 0.133780i
\(448\) 0 0
\(449\) 22.6274i 1.06785i −0.845531 0.533927i \(-0.820716\pi\)
0.845531 0.533927i \(-0.179284\pi\)
\(450\) 0 0
\(451\) 27.7128i 1.30495i
\(452\) 0 0
\(453\) −4.89898 + 3.46410i −0.230174 + 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.79796 −0.458329 −0.229165 0.973388i \(-0.573599\pi\)
−0.229165 + 0.973388i \(0.573599\pi\)
\(458\) 0 0
\(459\) 4.89898 + 17.3205i 0.228665 + 0.808452i
\(460\) 0 0
\(461\) 36.7696i 1.71253i 0.516538 + 0.856264i \(0.327221\pi\)
−0.516538 + 0.856264i \(0.672779\pi\)
\(462\) 0 0
\(463\) 16.9706i 0.788689i −0.918963 0.394344i \(-0.870972\pi\)
0.918963 0.394344i \(-0.129028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.89898 + 6.92820i 0.225733 + 0.319235i
\(472\) 0 0
\(473\) 41.5692i 1.91135i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 29.3939 + 10.3923i 1.34585 + 0.475831i
\(478\) 0 0
\(479\) −19.5959 −0.895360 −0.447680 0.894194i \(-0.647750\pi\)
−0.447680 + 0.894194i \(0.647750\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 33.9411i 1.53802i 0.639237 + 0.769010i \(0.279250\pi\)
−0.639237 + 0.769010i \(0.720750\pi\)
\(488\) 0 0
\(489\) 12.0000 8.48528i 0.542659 0.383718i
\(490\) 0 0
\(491\) −24.4949 −1.10544 −0.552720 0.833367i \(-0.686410\pi\)
−0.552720 + 0.833367i \(0.686410\pi\)
\(492\) 0 0
\(493\) −9.79796 −0.441278
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.46410i 0.155074i 0.996989 + 0.0775372i \(0.0247057\pi\)
−0.996989 + 0.0775372i \(0.975294\pi\)
\(500\) 0 0
\(501\) 6.00000 + 8.48528i 0.268060 + 0.379094i
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 11.0000 + 15.5563i 0.488527 + 0.690882i
\(508\) 0 0
\(509\) 36.7696i 1.62978i −0.579614 0.814891i \(-0.696797\pi\)
0.579614 0.814891i \(-0.303203\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.89898 17.3205i −0.216295 0.764719i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −29.3939 −1.29274
\(518\) 0 0
\(519\) −4.89898 + 3.46410i −0.215041 + 0.152057i
\(520\) 0 0
\(521\) 5.65685i 0.247831i −0.992293 0.123916i \(-0.960455\pi\)
0.992293 0.123916i \(-0.0395452\pi\)
\(522\) 0 0
\(523\) 42.4264i 1.85518i 0.373603 + 0.927589i \(0.378122\pi\)
−0.373603 + 0.927589i \(0.621878\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −4.89898 + 13.8564i −0.212598 + 0.601317i
\(532\) 0 0
\(533\) 27.7128i 1.20038i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −24.4949 34.6410i −1.05703 1.49487i
\(538\) 0 0
\(539\) −34.2929 −1.47710
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) −10.0000 14.1421i −0.429141 0.606897i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 25.4558i 1.08841i −0.838951 0.544207i \(-0.816831\pi\)
0.838951 0.544207i \(-0.183169\pi\)
\(548\) 0 0
\(549\) 2.00000 5.65685i 0.0853579 0.241429i
\(550\) 0 0
\(551\) 9.79796 0.417407
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.46410i 0.146779i −0.997303 0.0733893i \(-0.976618\pi\)
0.997303 0.0733893i \(-0.0233816\pi\)
\(558\) 0 0
\(559\) 41.5692i 1.75819i
\(560\) 0 0
\(561\) −24.0000 + 16.9706i −1.01328 + 0.716498i
\(562\) 0 0
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.2843i 1.18574i 0.805299 + 0.592869i \(0.202005\pi\)
−0.805299 + 0.592869i \(0.797995\pi\)
\(570\) 0 0
\(571\) 38.1051i 1.59465i 0.603550 + 0.797325i \(0.293752\pi\)
−0.603550 + 0.797325i \(0.706248\pi\)
\(572\) 0 0
\(573\) −19.5959 27.7128i −0.818631 1.15772i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 19.5959 0.815789 0.407894 0.913029i \(-0.366263\pi\)
0.407894 + 0.913029i \(0.366263\pi\)
\(578\) 0 0
\(579\) 19.5959 + 27.7128i 0.814379 + 1.15171i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 50.9117i 2.10855i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −34.2929 + 24.2487i −1.41062 + 0.997459i
\(592\) 0 0
\(593\) 24.2487i 0.995775i −0.867242 0.497888i \(-0.834109\pi\)
0.867242 0.497888i \(-0.165891\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −14.6969 + 10.3923i −0.601506 + 0.425329i
\(598\) 0 0
\(599\) −9.79796 −0.400334 −0.200167 0.979762i \(-0.564148\pi\)
−0.200167 + 0.979762i \(0.564148\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −24.0000 8.48528i −0.977356 0.345547i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.9706i 0.688814i 0.938820 + 0.344407i \(0.111920\pi\)
−0.938820 + 0.344407i \(0.888080\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −29.3939 −1.18915
\(612\) 0 0
\(613\) −14.6969 −0.593604 −0.296802 0.954939i \(-0.595920\pi\)
−0.296802 + 0.954939i \(0.595920\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.2487i 0.976216i 0.872783 + 0.488108i \(0.162313\pi\)
−0.872783 + 0.488108i \(0.837687\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i −0.977947 0.208851i \(-0.933028\pi\)
0.977947 0.208851i \(-0.0669724\pi\)
\(620\) 0 0
\(621\) 30.0000 8.48528i 1.20386 0.340503i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 24.0000 16.9706i 0.958468 0.677739i
\(628\) 0 0
\(629\) 16.9706i 0.676661i
\(630\) 0 0
\(631\) 38.1051i 1.51694i −0.651707 0.758470i \(-0.725947\pi\)
0.651707 0.758470i \(-0.274053\pi\)
\(632\) 0 0
\(633\) −34.2929 + 24.2487i −1.36302 + 0.963800i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −34.2929 −1.35873
\(638\) 0 0
\(639\) −9.79796 + 27.7128i −0.387601 + 1.09630i
\(640\) 0 0
\(641\) 22.6274i 0.893729i 0.894602 + 0.446865i \(0.147459\pi\)
−0.894602 + 0.446865i \(0.852541\pi\)
\(642\) 0 0
\(643\) 8.48528i 0.334627i −0.985904 0.167313i \(-0.946491\pi\)
0.985904 0.167313i \(-0.0535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 31.1769i 1.22005i 0.792383 + 0.610023i \(0.208840\pi\)
−0.792383 + 0.610023i \(0.791160\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.79796 27.7128i 0.382255 1.08118i
\(658\) 0 0
\(659\) 14.6969 0.572511 0.286256 0.958153i \(-0.407589\pi\)
0.286256 + 0.958153i \(0.407589\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) −24.0000 + 16.9706i −0.932083 + 0.659082i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706i 0.657103i
\(668\) 0 0
\(669\) 24.0000 16.9706i 0.927894 0.656120i
\(670\) 0 0
\(671\) 9.79796 0.378246
\(672\) 0 0
\(673\) −39.1918 −1.51073 −0.755367 0.655302i \(-0.772541\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.3923i 0.399409i 0.979856 + 0.199704i \(0.0639982\pi\)
−0.979856 + 0.199704i \(0.936002\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6.00000 + 8.48528i 0.229920 + 0.325157i
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000 + 14.1421i 0.381524 + 0.539556i
\(688\) 0 0
\(689\) 50.9117i 1.93958i
\(690\) 0 0
\(691\) 31.1769i 1.18603i −0.805193 0.593013i \(-0.797938\pi\)
0.805193 0.593013i \(-0.202062\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 19.5959 0.742248
\(698\) 0 0
\(699\) −4.89898 + 3.46410i −0.185296 + 0.131024i
\(700\) 0 0
\(701\) 19.7990i 0.747798i −0.927470 0.373899i \(-0.878021\pi\)
0.927470 0.373899i \(-0.121979\pi\)
\(702\) 0 0
\(703\) 16.9706i 0.640057i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −29.3939 10.3923i −1.10236 0.389742i
\(712\) 0 0
\(713\) 20.7846i 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 19.5959 0.730804 0.365402 0.930850i \(-0.380931\pi\)
0.365402 + 0.930850i \(0.380931\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 10.0000 + 14.1421i 0.371904 + 0.525952i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 29.3939 1.08717
\(732\) 0 0
\(733\) −44.0908 −1.62853 −0.814266 0.580492i \(-0.802860\pi\)
−0.814266 + 0.580492i \(0.802860\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 41.5692i 1.53122i
\(738\) 0 0
\(739\) 24.2487i 0.892003i −0.895032 0.446002i \(-0.852848\pi\)
0.895032 0.446002i \(-0.147152\pi\)
\(740\) 0 0
\(741\) 24.0000 16.9706i 0.881662 0.623429i
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000 16.9706i 0.219529 0.620920i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 31.1769i 1.13766i 0.822455 + 0.568831i \(0.192604\pi\)
−0.822455 + 0.568831i \(0.807396\pi\)
\(752\) 0 0
\(753\) 14.6969 + 20.7846i 0.535586 + 0.757433i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.6969 0.534169 0.267085 0.963673i \(-0.413940\pi\)
0.267085 + 0.963673i \(0.413940\pi\)
\(758\) 0 0
\(759\) 29.3939 + 41.5692i 1.06693 + 1.50887i
\(760\) 0 0
\(761\) 39.5980i 1.43543i −0.696339 0.717713i \(-0.745189\pi\)
0.696339 0.717713i \(-0.254811\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 4.89898 3.46410i 0.176432 0.124757i
\(772\) 0 0
\(773\) 24.2487i 0.872166i −0.899907 0.436083i \(-0.856365\pi\)
0.899907 0.436083i \(-0.143635\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −19.5959 −0.702097
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 4.00000 + 14.1421i 0.142948 + 0.505399i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 25.4558i 0.907403i −0.891154 0.453701i \(-0.850103\pi\)
0.891154 0.453701i \(-0.149897\pi\)
\(788\) 0 0
\(789\) −6.00000 8.48528i −0.213606 0.302084i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 9.79796 0.347936
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.3205i 0.613524i −0.951786 0.306762i \(-0.900754\pi\)
0.951786 0.306762i \(-0.0992455\pi\)
\(798\) 0 0
\(799\) 20.7846i 0.735307i
\(800\) 0 0
\(801\) −16.0000 5.65685i −0.565332 0.199875i
\(802\) 0 0
\(803\) 48.0000 1.69388
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −28.0000