Properties

Label 300.2.i.c.257.1
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 257.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 300.257
Dual form 300.2.i.c.293.1

$q$-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 \(-0.810519\pi\)
−0.560734 0.827996i \(-0.689481\pi\)
\(8\) 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 −0.516565 0.856248i \(-0.672790\pi\)
0.856248 + 0.516565i \(0.172790\pi\)
\(14\) 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 0.178545 0.983932i \(-0.442861\pi\)
−0.983932 + 0.178545i \(0.942861\pi\)
\(38\) 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 −0.607527 0.794299i \(-0.707838\pi\)
0.794299 + 0.607527i \(0.207838\pi\)
\(44\) 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 \(0.0154647\pi\)
0.0485648 + 0.998820i \(0.484535\pi\)
\(68\) 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 \(-0.448986\pi\)
0.987185 0.159579i \(-0.0510137\pi\)
\(74\) 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 0.495529 0.868591i \(-0.334974\pi\)
−0.868591 + 0.495529i \(0.834974\pi\)
\(98\) 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 −0.817411 0.576055i \(-0.804591\pi\)
0.576055 + 0.817411i \(0.304591\pi\)
\(104\) 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 0.953491 0.301420i \(-0.0974607\pi\)
−0.301420 + 0.953491i \(0.597461\pi\)
\(128\) 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 \(-0.894649\pi\)
−0.324961 0.945727i \(-0.605351\pi\)
\(158\) 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 \(-0.481897\pi\)
0.998383 0.0568404i \(-0.0181026\pi\)
\(164\) 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 −0.982050 0.188619i \(-0.939599\pi\)
0.188619 + 0.982050i \(0.439599\pi\)
\(194\) 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 −0.0934584 0.995623i \(-0.529792\pi\)
0.995623 + 0.0934584i \(0.0297922\pi\)
\(224\) 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 0.916089 0.400975i \(-0.131328\pi\)
−0.400975 + 0.916089i \(0.631328\pi\)
\(278\) 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 \(0.683287\pi\)
−0.838749 + 0.544518i \(0.816713\pi\)
\(284\) 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 \(0.0675939\pi\)
0.210760 + 0.977538i \(0.432406\pi\)
\(308\) 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 \(0.630247\pi\)
−0.917446 + 0.397861i \(0.869753\pi\)
\(314\) 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 0.124864 0.992174i \(-0.460150\pi\)
−0.992174 + 0.124864i \(0.960150\pi\)
\(338\) 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 0.156563 0.987668i \(-0.449959\pi\)
−0.987668 + 0.156563i \(0.949959\pi\)
\(368\) 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 \(-0.359263\pi\)
0.903838 0.427874i \(-0.140737\pi\)
\(374\) 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 \(-0.757989\pi\)
−0.689140 0.724628i \(-0.742011\pi\)
\(398\) 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 −0.304939 0.952372i \(-0.598636\pi\)
0.952372 + 0.304939i \(0.0986362\pi\)
\(434\) 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 \(-0.825146\pi\)
−0.522108 0.852879i \(-0.674854\pi\)
\(458\) 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 \(0.596148\pi\)
−0.954726 + 0.297486i \(0.903852\pi\)
\(464\) 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 \(-0.891658\pi\)
−0.333833 0.942632i \(-0.608342\pi\)
\(488\) 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 −0.905766 0.423777i \(-0.860704\pi\)
0.423777 + 0.905766i \(0.360704\pi\)
\(524\) 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 0.238110 0.971238i \(-0.423472\pi\)
−0.971238 + 0.238110i \(0.923472\pi\)
\(548\) 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 \(0.176465\pi\)
0.526417 + 0.850227i \(0.323535\pi\)
\(578\) 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 \(0.116962\pi\)
0.359235 + 0.933247i \(0.383038\pi\)
\(608\) 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 \(-0.314727\pi\)
0.835337 0.549739i \(-0.185273\pi\)
\(614\) 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 −0.123018 0.992404i \(-0.539257\pi\)
0.992404 + 0.123018i \(0.0392574\pi\)
\(644\) 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 −0.492582 0.870266i \(-0.663947\pi\)
0.870266 + 0.492582i \(0.163947\pi\)
\(674\) 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 0.347265 0.937767i \(-0.387111\pi\)
−0.937767 + 0.347265i \(0.887111\pi\)
\(728\) 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 −0.924362 0.381518i \(-0.875402\pi\)
0.381518 + 0.924362i \(0.375402\pi\)
\(734\) 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 \(0.0733125\pi\)
0.228287 + 0.973594i \(0.426688\pi\)
\(758\) 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 \(0.0636681\pi\)
0.198688 + 0.980063i \(0.436332\pi\)
\(788\) 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 0.333590