Properties

Label 80.6.n.a.47.1
Level 80
Weight 6
Character 80.47
Analytic conductor 12.831
Analytic rank 0
Dimension 2
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.8307055850\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 47.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 80.47
Dual form 80.6.n.a.63.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-38.0000 + 41.0000i) q^{5} -243.000i q^{9} +O(q^{10})\) \(q+(-38.0000 + 41.0000i) q^{5} -243.000i q^{9} +(475.000 - 475.000i) q^{13} +(-1525.00 - 1525.00i) q^{17} +(-237.000 - 3116.00i) q^{25} -8564.00i q^{29} +(-475.000 - 475.000i) q^{37} -4952.00 q^{41} +(9963.00 + 9234.00i) q^{45} +16807.0i q^{49} +(-16475.0 + 16475.0i) q^{53} +54948.0 q^{61} +(1425.00 + 37525.0i) q^{65} +(-54475.0 + 54475.0i) q^{73} -59049.0 q^{81} +(120475. - 4575.00i) q^{85} -140464. i q^{89} +(-126475. - 126475. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 76q^{5} + O(q^{10}) \) \( 2q - 76q^{5} + 950q^{13} - 3050q^{17} - 474q^{25} - 950q^{37} - 9904q^{41} + 19926q^{45} - 32950q^{53} + 109896q^{61} + 2850q^{65} - 108950q^{73} - 118098q^{81} + 240950q^{85} - 252950q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) −38.0000 + 41.0000i −0.679765 + 0.733430i
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0 0
\(9\) 243.000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 475.000 475.000i 0.779534 0.779534i −0.200217 0.979752i \(-0.564165\pi\)
0.979752 + 0.200217i \(0.0641648\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1525.00 1525.00i −1.27982 1.27982i −0.940770 0.339046i \(-0.889896\pi\)
−0.339046 0.940770i \(-0.610104\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) −237.000 3116.00i −0.0758400 0.997120i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8564.00i 1.89096i −0.325684 0.945479i \(-0.605595\pi\)
0.325684 0.945479i \(-0.394405\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −475.000 475.000i −0.0570413 0.0570413i 0.678011 0.735052i \(-0.262842\pi\)
−0.735052 + 0.678011i \(0.762842\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4952.00 −0.460067 −0.230033 0.973183i \(-0.573884\pi\)
−0.230033 + 0.973183i \(0.573884\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 9963.00 + 9234.00i 0.733430 + 0.679765i
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 16807.0i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −16475.0 + 16475.0i −0.805630 + 0.805630i −0.983969 0.178339i \(-0.942928\pi\)
0.178339 + 0.983969i \(0.442928\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 54948.0 1.89072 0.945360 0.326028i \(-0.105710\pi\)
0.945360 + 0.326028i \(0.105710\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1425.00 + 37525.0i 0.0418342 + 1.10163i
\(66\) 0 0
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −54475.0 + 54475.0i −1.19644 + 1.19644i −0.221212 + 0.975226i \(0.571001\pi\)
−0.975226 + 0.221212i \(0.928999\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −59049.0 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 120475. 4575.00i 1.80863 0.0686821i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 140464.i 1.87971i −0.341579 0.939853i \(-0.610962\pi\)
0.341579 0.939853i \(-0.389038\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −126475. 126475.i −1.36482 1.36482i −0.867657 0.497162i \(-0.834375\pi\)
−0.497162 0.867657i \(-0.665625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 98002.0 0.955942 0.477971 0.878376i \(-0.341372\pi\)
0.477971 + 0.878376i \(0.341372\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 246486.i 1.98713i −0.113269 0.993564i \(-0.536132\pi\)
0.113269 0.993564i \(-0.463868\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −181425. + 181425.i −1.33660 + 1.33660i −0.437267 + 0.899332i \(0.644053\pi\)
−0.899332 + 0.437267i \(0.855947\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −115425. 115425.i −0.779534 0.779534i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 161051. 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 136762. + 108691.i 0.782871 + 0.622184i
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −178575. 178575.i −0.812867 0.812867i 0.172196 0.985063i \(-0.444914\pi\)
−0.985063 + 0.172196i \(0.944914\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 351124. + 325432.i 1.38689 + 1.28541i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 47614.0i 0.175699i 0.996134 + 0.0878494i \(0.0279994\pi\)
−0.996134 + 0.0878494i \(0.972001\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −370575. + 370575.i −1.27982 + 1.27982i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 432475. + 432475.i 1.40027 + 1.40027i 0.799185 + 0.601086i \(0.205265\pi\)
0.601086 + 0.799185i \(0.294735\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 79957.0i 0.215347i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −556575. + 556575.i −1.41387 + 1.41387i −0.691271 + 0.722596i \(0.742949\pi\)
−0.722596 + 0.691271i \(0.757051\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 439902. 0.998067 0.499033 0.866583i \(-0.333689\pi\)
0.499033 + 0.866583i \(0.333689\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 37525.0 1425.00i 0.0806105 0.00306116i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 702475. 702475.i 1.35749 1.35749i 0.480496 0.876997i \(-0.340457\pi\)
0.876997 0.480496i \(-0.159543\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −700575. 700575.i −1.28614 1.28614i −0.937111 0.349031i \(-0.886511\pi\)
−0.349031 0.937111i \(-0.613489\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 188176. 203032.i 0.312737 0.337427i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.44875e6 −1.99532
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) −757188. + 57591.0i −0.997120 + 0.0758400i
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 976564.i 1.23059i −0.788298 0.615293i \(-0.789038\pi\)
0.788298 0.615293i \(-0.210962\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.07848e6 1.07848e6i 1.30143 1.30143i 0.373999 0.927429i \(-0.377986\pi\)
0.927429 0.373999i \(-0.122014\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.73905e6 1.92872 0.964360 0.264595i \(-0.0852384\pi\)
0.964360 + 0.264595i \(0.0852384\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −689087. 638666.i −0.733430 0.679765i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 682575. + 682575.i 0.644640 + 0.644640i 0.951693 0.307052i \(-0.0993426\pi\)
−0.307052 + 0.951693i \(0.599343\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.08105e6 −1.89096
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) −49425.0 1.30153e6i −0.0432347 1.13851i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.35141e6i 1.98129i 0.136458 + 0.990646i \(0.456428\pi\)
−0.136458 + 0.990646i \(0.543572\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.58447e6 + 1.58447e6i 1.24075 + 1.24075i 0.959688 + 0.281067i \(0.0906882\pi\)
0.281067 + 0.959688i \(0.409312\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −149152. −0.112684 −0.0563421 0.998412i \(-0.517944\pi\)
−0.0563421 + 0.998412i \(0.517944\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.23139e6i 2.27586i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 412575. 412575.i 0.280759 0.280759i −0.552653 0.833412i \(-0.686384\pi\)
0.833412 + 0.552653i \(0.186384\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.08802e6 + 2.25287e6i −1.28524 + 1.38671i
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.40238e6 2.40238e6i 1.38605 1.38605i 0.552620 0.833433i \(-0.313628\pi\)
0.833433 0.552620i \(-0.186372\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.09238e6 + 2.09238e6i 1.16948 + 1.16948i 0.982333 + 0.187144i \(0.0599230\pi\)
0.187144 + 0.982333i \(0.440077\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.59267e6 1.36753e6i −0.836409 0.718169i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −115425. + 115425.i −0.0570413 + 0.0570413i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −602375. 602375.i −0.288930 0.288930i 0.547727 0.836657i \(-0.315493\pi\)
−0.836657 + 0.547727i \(0.815493\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 975636.i 0.428770i 0.976749 + 0.214385i \(0.0687747\pi\)
−0.976749 + 0.214385i \(0.931225\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 517625. 517625.i 0.221095 0.221095i −0.587865 0.808959i \(-0.700031\pi\)
0.808959 + 0.587865i \(0.200031\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −2.47610e6 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −163425. 4.30353e6i −0.0642076 1.69080i
\(366\) 0 0
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 0 0
\(369\) 1.20334e6i 0.460067i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.82762e6 1.82762e6i 0.680166 0.680166i −0.279871 0.960038i \(-0.590292\pi\)
0.960038 + 0.279871i \(0.0902918\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.06790e6 4.06790e6i −1.47407 1.47407i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.28629e6i 1.77124i −0.464414 0.885618i \(-0.653735\pi\)
0.464414 0.885618i \(-0.346265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.26762e6 + 3.26762e6i 1.04053 + 1.04053i 0.999143 + 0.0413901i \(0.0131787\pi\)
0.0413901 + 0.999143i \(0.486821\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.59200e6 0.494405 0.247202 0.968964i \(-0.420489\pi\)
0.247202 + 0.968964i \(0.420489\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.24386e6 2.42101e6i 0.679765 0.733430i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.58449e6i 1.35513i −0.735461 0.677567i \(-0.763034\pi\)
0.735461 0.677567i \(-0.236966\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −4.18485e6 −1.15073 −0.575367 0.817895i \(-0.695141\pi\)
−0.575367 + 0.817895i \(0.695141\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.39048e6 + 5.11332e6i −1.17907 + 1.37319i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −4.15847e6 + 4.15847e6i −1.06589 + 1.06589i −0.0682249 + 0.997670i \(0.521734\pi\)
−0.997670 + 0.0682249i \(0.978266\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 4.08410e6 1.00000
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 5.75902e6 + 5.33763e6i 1.37863 + 1.27776i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.48961e6i 1.98734i 0.112340 + 0.993670i \(0.464165\pi\)
−0.112340 + 0.993670i \(0.535835\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.23763e6 6.23763e6i −1.39710 1.39710i −0.808211 0.588893i \(-0.799564\pi\)
−0.588893 0.808211i \(-0.700436\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.86580e6 1.50466 0.752331 0.658785i \(-0.228929\pi\)
0.752331 + 0.658785i \(0.228929\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00343e6 + 4.00343e6i 0.805630 + 0.805630i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −451250. −0.0889313
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.99152e6 379425.i 1.92876 0.0732439i
\(486\) 0 0
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −1.30601e7 + 1.30601e7i −2.42008 + 2.42008i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) −3.72408e6 + 4.01808e6i −0.649816 + 0.701117i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.12076e6i 0.876073i −0.898957 0.438037i \(-0.855674\pi\)
0.898957 0.438037i \(-0.144326\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.27410e6 1.17405 0.587023 0.809570i \(-0.300300\pi\)
0.587023 + 0.809570i \(0.300300\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.43634e6i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.35220e6 + 2.35220e6i −0.358638 + 0.358638i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.25380e6 −1.21244 −0.606221 0.795297i \(-0.707315\pi\)
−0.606221 + 0.795297i \(0.707315\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.01059e7 + 9.36647e6i 1.45742 + 1.35078i
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) 0 0
\(549\) 1.33524e7i 1.89072i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.07247e6 7.07247e6i −0.965903 0.965903i 0.0335347 0.999438i \(-0.489324\pi\)
−0.999438 + 0.0335347i \(0.989324\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) −544275. 1.43326e7i −0.0717294 1.88887i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.96659e6i 0.513613i −0.966463 0.256807i \(-0.917330\pi\)
0.966463 0.256807i \(-0.0826703\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −9.47762e6 9.47762e6i −1.18511 1.18511i −0.978402 0.206712i \(-0.933724\pi\)
−0.206712 0.978402i \(-0.566276\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 9.11858e6 346275.i 1.10163 0.0418342i
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.39742e6 9.39742e6i 1.09742 1.09742i 0.102706 0.994712i \(-0.467250\pi\)
0.994712 0.102706i \(-0.0327502\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 9.16325e6 1.03482 0.517408 0.855739i \(-0.326897\pi\)
0.517408 + 0.855739i \(0.326897\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.11994e6 + 6.60309e6i −0.679765 + 0.733430i
\(606\) 0 0
\(607\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.30233e7 + 1.30233e7i −1.39982 + 1.39982i −0.599263 + 0.800552i \(0.704540\pi\)
−0.800552 + 0.599263i \(0.795460\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.21693e7 1.21693e7i −1.28693 1.28693i −0.936644 0.350282i \(-0.886086\pi\)
−0.350282 0.936644i \(-0.613914\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.65329e6 + 1.47698e6i −0.988497 + 0.151243i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.44875e6i 0.146005i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 7.98332e6 + 7.98332e6i 0.779534 + 0.779534i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.48270e7 1.42531 0.712655 0.701514i \(-0.247492\pi\)
0.712655 + 0.701514i \(0.247492\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.73532e6 4.73532e6i 0.434577 0.434577i −0.455605 0.890182i \(-0.650577\pi\)
0.890182 + 0.455605i \(0.150577\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.32374e7 + 1.32374e7i 1.19644 + 1.19644i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −2.07531e7 −1.84747 −0.923737 0.383027i \(-0.874881\pi\)
−0.923737 + 0.383027i \(0.874881\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 801325. 801325.i 0.0681979 0.0681979i −0.672185 0.740383i \(-0.734644\pi\)
0.740383 + 0.672185i \(0.234644\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.42763e6 + 9.42763e6i 0.790552 + 0.790552i 0.981584 0.191032i \(-0.0611833\pi\)
−0.191032 + 0.981584i \(0.561183\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 0 0
\(685\) 1.41074e7 535725.i 1.14874 0.0436230i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.56513e7i 1.25603i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.55180e6 + 7.55180e6i 0.588801 + 0.588801i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.51373e7 −1.16346 −0.581731 0.813381i \(-0.697624\pi\)
−0.581731 + 0.813381i \(0.697624\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.64712e7i 1.97769i 0.148942 + 0.988846i \(0.452413\pi\)
−0.148942 + 0.988846i \(0.547587\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.66854e7 + 2.02967e6i −1.88551 + 0.143410i
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 1.43489e7i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 8.30763e6 8.30763e6i 0.571106 0.571106i −0.361331 0.932438i \(-0.617678\pi\)
0.932438 + 0.361331i \(0.117678\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) −1.95217e6 1.80933e6i −0.128863 0.119434i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.41927e7 + 1.41927e7i 0.900170 + 0.900170i 0.995450 0.0952804i \(-0.0303748\pi\)
−0.0952804 + 0.995450i \(0.530375\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.95198e7 1.22184 0.610919 0.791693i \(-0.290800\pi\)
0.610919 + 0.791693i \(0.290800\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.11173e6 2.92754e7i −0.0686821 1.80863i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.57727e7i 1.57161i −0.618477 0.785803i \(-0.712250\pi\)
0.618477 0.785803i \(-0.287750\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.39845e7 1.39845e7i 0.841778 0.841778i −0.147312 0.989090i \(-0.547062\pi\)
0.989090 + 0.147312i \(0.0470622\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.41655e7 + 1.29742e6i −1.97885 + 0.0751464i
\(786\) 0 0
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.61003e7 2.61003e7i 1.47388 1.47388i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.37794e7 2.37794e7i −1.32604 1.32604i −0.908796 0.417241i \(-0.862997\pi\)
−0.417241 0.908796i \(-0.637003\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3.41328e7 −1.87971
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.36197e7i 1.26883i 0.772992 + 0.634415i \(0.218759\pi\)
−0.772992 + 0.634415i \(0.781241\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0