Properties

Label 300.2.i.c.257.2
Level $300$
Weight $2$
Character 300.257
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 257.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 300.257
Dual form 300.2.i.c.293.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.22474 - 1.22474i) q^{3} +(3.67423 + 3.67423i) q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 - 1.22474i) q^{3} +(3.67423 + 3.67423i) q^{7} -3.00000i q^{9} +(-1.22474 + 1.22474i) q^{13} -7.00000i q^{19} +9.00000 q^{21} +(-3.67423 - 3.67423i) q^{27} -11.0000 q^{31} +(4.89898 + 4.89898i) q^{37} +3.00000i q^{39} +(-1.22474 + 1.22474i) q^{43} +20.0000i q^{49} +(-8.57321 - 8.57321i) q^{57} -1.00000 q^{61} +(11.0227 - 11.0227i) q^{63} +(-8.57321 - 8.57321i) q^{67} +(-9.79796 + 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} -9.00000 q^{91} +(-13.4722 + 13.4722i) q^{93} +(3.67423 + 3.67423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 36q^{21} - 44q^{31} - 4q^{61} - 36q^{81} - 36q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 1.22474 1.22474i 0.707107 0.707107i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.67423 + 3.67423i 1.38873 + 1.38873i 0.827996 + 0.560734i \(0.189481\pi\)
0.560734 + 0.827996i \(0.310519\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.22474 + 1.22474i −0.339683 + 0.339683i −0.856248 0.516565i \(-0.827210\pi\)
0.516565 + 0.856248i \(0.327210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 7.00000i 1.60591i −0.596040 0.802955i \(-0.703260\pi\)
0.596040 0.802955i \(-0.296740\pi\)
\(20\) 0 0
\(21\) 9.00000 1.96396
\(22\) 0 0
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 3.67423i −0.707107 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −11.0000 −1.97566 −0.987829 0.155543i \(-0.950287\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.89898 + 4.89898i 0.805387 + 0.805387i 0.983932 0.178545i \(-0.0571389\pi\)
−0.178545 + 0.983932i \(0.557139\pi\)
\(38\) 0 0
\(39\) 3.00000i 0.480384i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.22474 + 1.22474i −0.186772 + 0.186772i −0.794299 0.607527i \(-0.792162\pi\)
0.607527 + 0.794299i \(0.292162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 20.0000i 2.85714i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.57321 8.57321i −1.13555 1.13555i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 11.0227 11.0227i 1.38873 1.38873i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.57321 8.57321i −1.04738 1.04738i −0.998820 0.0485648i \(-0.984535\pi\)
−0.0485648 0.998820i \(-0.515465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9.79796 + 9.79796i −1.14676 + 1.14676i −0.159579 + 0.987185i \(0.551014\pi\)
−0.987185 + 0.159579i \(0.948986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) −9.00000 −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
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) 0 0
\(93\) −13.4722 + 13.4722i −1.39700 + 1.39700i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.67423 + 3.67423i 0.373062 + 0.373062i 0.868591 0.495529i \(-0.165026\pi\)
−0.495529 + 0.868591i \(0.665026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.44949 2.44949i 0.241355 0.241355i −0.576055 0.817411i \(-0.695409\pi\)
0.817411 + 0.576055i \(0.195409\pi\)
\(104\) 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
\(109\) 17.0000i 1.62830i −0.580651 0.814152i \(-0.697202\pi\)
0.580651 0.814152i \(-0.302798\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.67423 + 3.67423i 0.339683 + 0.339683i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.34847 7.34847i −0.652071 0.652071i 0.301420 0.953491i \(-0.402539\pi\)
−0.953491 + 0.301420i \(0.902539\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.264135i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 25.7196 25.7196i 2.23018 2.23018i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i −0.734553 0.678551i \(-0.762608\pi\)
0.734553 0.678551i \(-0.237392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 24.4949 + 24.4949i 2.02031 + 2.02031i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.9217 + 15.9217i 1.27069 + 1.27069i 0.945727 + 0.324961i \(0.105351\pi\)
0.324961 + 0.945727i \(0.394649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13.4722 + 13.4722i −1.05522 + 1.05522i −0.0568404 + 0.998383i \(0.518103\pi\)
−0.998383 + 0.0568404i \(0.981897\pi\)
\(164\) 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
\(169\) 10.0000i 0.769231i
\(170\) 0 0
\(171\) −21.0000 −1.60591
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 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
\(181\) 19.0000 1.41226 0.706129 0.708083i \(-0.250440\pi\)
0.706129 + 0.708083i \(0.250440\pi\)
\(182\) 0 0
\(183\) −1.22474 + 1.22474i −0.0905357 + 0.0905357i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 27.0000i 1.96396i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 11.0227 11.0227i 0.793432 0.793432i −0.188619 0.982050i \(-0.560401\pi\)
0.982050 + 0.188619i \(0.0604011\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 17.0000i 1.20510i −0.798082 0.602549i \(-0.794152\pi\)
0.798082 0.602549i \(-0.205848\pi\)
\(200\) 0 0
\(201\) −21.0000 −1.48123
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 29.0000 1.99644 0.998221 0.0596196i \(-0.0189888\pi\)
0.998221 + 0.0596196i \(0.0189888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −40.4166 40.4166i −2.74366 2.74366i
\(218\) 0 0
\(219\) 24.0000i 1.62177i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −13.4722 + 13.4722i −0.902165 + 0.902165i −0.995623 0.0934584i \(-0.970208\pi\)
0.0934584 + 0.995623i \(0.470208\pi\)
\(224\) 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
\(229\) 7.00000i 0.462573i −0.972886 0.231287i \(-0.925707\pi\)
0.972886 0.231287i \(-0.0742935\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.89898 + 4.89898i 0.318223 + 0.318223i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −31.0000 −1.99689 −0.998443 0.0557856i \(-0.982234\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) −11.0227 + 11.0227i −0.707107 + 0.707107i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.57321 + 8.57321i 0.545501 + 0.545501i
\(248\) 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
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 36.0000i 2.23693i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) −11.0227 + 11.0227i −0.667124 + 0.667124i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −8.57321 8.57321i −0.515115 0.515115i 0.400975 0.916089i \(-0.368672\pi\)
−0.916089 + 0.400975i \(0.868672\pi\)
\(278\) 0 0
\(279\) 33.0000i 1.97566i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 23.2702 23.2702i 1.38327 1.38327i 0.544518 0.838749i \(-0.316713\pi\)
0.838749 0.544518i \(-0.183287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 9.00000 0.527589
\(292\) 0 0
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.8207 20.8207i −1.18830 1.18830i −0.977538 0.210760i \(-0.932406\pi\)
−0.210760 0.977538i \(-0.567594\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 23.2702 23.2702i 1.31531 1.31531i 0.397861 0.917446i \(-0.369753\pi\)
0.917446 0.397861i \(-0.130247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.8207 20.8207i −1.15139 1.15139i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) 14.6969 14.6969i 0.805387 0.805387i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 15.9217 + 15.9217i 0.867309 + 0.867309i 0.992174 0.124864i \(-0.0398496\pi\)
−0.124864 + 0.992174i \(0.539850\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −47.7650 + 47.7650i −2.57907 + 2.57907i
\(344\) 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
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 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
\(361\) −30.0000 −1.57895
\(362\) 0 0
\(363\) 13.4722 13.4722i 0.707107 0.707107i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.9217 + 15.9217i 0.831105 + 0.831105i 0.987668 0.156563i \(-0.0500414\pi\)
−0.156563 + 0.987668i \(0.550041\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −25.7196 + 25.7196i −1.33171 + 1.33171i −0.427874 + 0.903838i \(0.640737\pi\)
−0.903838 + 0.427874i \(0.859263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 37.0000i 1.90056i −0.311393 0.950281i \(-0.600796\pi\)
0.311393 0.950281i \(-0.399204\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 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
\(387\) 3.67423 + 3.67423i 0.186772 + 0.186772i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.1691 + 28.1691i 1.41377 + 1.41377i 0.724628 + 0.689140i \(0.242011\pi\)
0.689140 + 0.724628i \(0.257989\pi\)
\(398\) 0 0
\(399\) 63.0000i 3.15394i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 13.4722 13.4722i 0.671098 0.671098i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.00000i 0.346128i −0.984911 0.173064i \(-0.944633\pi\)
0.984911 0.173064i \(-0.0553667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −19.5959 19.5959i −0.959616 0.959616i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.67423 3.67423i −0.177809 0.177809i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −13.4722 + 13.4722i −0.647432 + 0.647432i −0.952372 0.304939i \(-0.901364\pi\)
0.304939 + 0.952372i \(0.401364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 13.0000i 0.620456i 0.950662 + 0.310228i \(0.100405\pi\)
−0.950662 + 0.310228i \(0.899595\pi\)
\(440\) 0 0
\(441\) 60.0000 2.85714
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 23.2702 23.2702i 1.09333 1.09333i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.3939 + 29.3939i 1.37499 + 1.37499i 0.852879 + 0.522108i \(0.174854\pi\)
0.522108 + 0.852879i \(0.325146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 26.9444 26.9444i 1.25221 1.25221i 0.297486 0.954726i \(-0.403852\pi\)
0.954726 0.297486i \(-0.0961480\pi\)
\(464\) 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
\(469\) 63.0000i 2.90907i
\(470\) 0 0
\(471\) 39.0000 1.79703
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 28.1691 + 28.1691i 1.27647 + 1.27647i 0.942632 + 0.333833i \(0.108342\pi\)
0.333833 + 0.942632i \(0.391658\pi\)
\(488\) 0 0
\(489\) 33.0000i 1.49231i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 43.0000i 1.92494i 0.271380 + 0.962472i \(0.412520\pi\)
−0.271380 + 0.962472i \(0.587480\pi\)
\(500\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) 12.2474 + 12.2474i 0.543928 + 0.543928i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −72.0000 −3.18509
\(512\) 0 0
\(513\) −25.7196 + 25.7196i −1.13555 + 1.13555i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 11.0227 11.0227i 0.481989 0.481989i −0.423777 0.905766i \(-0.639296\pi\)
0.905766 + 0.423777i \(0.139296\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 29.0000 1.24681 0.623404 0.781900i \(-0.285749\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 0 0
\(543\) 23.2702 23.2702i 0.998618 0.998618i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17.1464 + 17.1464i 0.733128 + 0.733128i 0.971238 0.238110i \(-0.0765278\pi\)
−0.238110 + 0.971238i \(0.576528\pi\)
\(548\) 0 0
\(549\) 3.00000i 0.128037i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −14.6969 + 14.6969i −0.624977 + 0.624977i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 3.00000i 0.126886i
\(560\) 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
\(565\) 0 0
\(566\) 0 0
\(567\) −33.0681 33.0681i −1.38873 1.38873i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −33.0681 33.0681i −1.37664 1.37664i −0.850227 0.526417i \(-0.823535\pi\)
−0.526417 0.850227i \(-0.676465\pi\)
\(578\) 0 0
\(579\) 27.0000i 1.12208i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 77.0000i 3.17273i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −20.8207 20.8207i −0.852133 0.852133i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 49.0000 1.99875 0.999376 0.0353259i \(-0.0112469\pi\)
0.999376 + 0.0353259i \(0.0112469\pi\)
\(602\) 0 0
\(603\) −25.7196 + 25.7196i −1.04738 + 1.04738i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −31.8434 31.8434i −1.29248 1.29248i −0.933247 0.359235i \(-0.883038\pi\)
−0.359235 0.933247i \(-0.616962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −34.2929 + 34.2929i −1.38508 + 1.38508i −0.549739 + 0.835337i \(0.685273\pi\)
−0.835337 + 0.549739i \(0.814727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 17.0000i 0.683288i −0.939829 0.341644i \(-0.889016\pi\)
0.939829 0.341644i \(-0.110984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) 35.5176 35.5176i 1.41170 1.41170i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −24.4949 24.4949i −0.970523 0.970523i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −22.0454 + 22.0454i −0.869386 + 0.869386i −0.992404 0.123018i \(-0.960743\pi\)
0.123018 + 0.992404i \(0.460743\pi\)
\(644\) 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
\(651\) −99.0000 −3.88012
\(652\) 0 0
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 29.3939 + 29.3939i 1.14676 + 1.14676i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 33.0000i 1.27585i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −9.79796 + 9.79796i −0.377684 + 0.377684i −0.870266 0.492582i \(-0.836053\pi\)
0.492582 + 0.870266i \(0.336053\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 27.0000i 1.03616i
\(680\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) −8.57321 8.57321i −0.327089 0.327089i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 34.2929 34.2929i 1.29338 1.29338i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 53.0000i 1.99046i 0.0975728 + 0.995228i \(0.468892\pi\)
−0.0975728 + 0.995228i \(0.531108\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 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
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) −37.9671 + 37.9671i −1.41201 + 1.41201i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.9217 + 15.9217i 0.590503 + 0.590503i 0.937767 0.347265i \(-0.112889\pi\)
−0.347265 + 0.937767i \(0.612889\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 14.6969 14.6969i 0.542844 0.542844i −0.381518 0.924362i \(-0.624598\pi\)
0.924362 + 0.381518i \(0.124598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 16.0000i 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 0 0
\(741\) 21.0000 0.771454
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −33.0681 33.0681i −1.20188 1.20188i −0.973594 0.228287i \(-0.926688\pi\)
−0.228287 0.973594i \(-0.573312\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 62.4620 62.4620i 2.26128 2.26128i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 47.0000i 1.69486i −0.530904 0.847432i \(-0.678148\pi\)
0.530904 0.847432i \(-0.321852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 44.0908 + 44.0908i 1.58175 + 1.58175i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −33.0681 33.0681i −1.17875 1.17875i −0.980063 0.198688i \(-0.936332\pi\)
−0.198688 0.980063i \(-0.563668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.22474 1.22474i 0.0434920 0.0434920i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 19.0000 0.667180