Properties

Label 441.4.e.t.226.2
Level $441$
Weight $4$
Character 441.226
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 226.2
Root \(1.32288 - 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 441.226
Dual form 441.4.e.t.361.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.32288 - 2.29129i) q^{2} +(0.500000 + 0.866025i) q^{4} +23.8118 q^{8} +O(q^{10})\) \(q+(1.32288 - 2.29129i) q^{2} +(0.500000 + 0.866025i) q^{4} +23.8118 q^{8} +(13.2288 + 22.9129i) q^{11} +(27.5000 - 47.6314i) q^{16} +70.0000 q^{22} +(-108.476 + 187.886i) q^{23} +(62.5000 + 108.253i) q^{25} +264.575 q^{29} +(22.4889 + 38.9519i) q^{32} +(225.000 - 389.711i) q^{37} +180.000 q^{43} +(-13.2288 + 22.9129i) q^{44} +(287.000 + 497.099i) q^{46} +330.719 q^{50} +(248.701 + 430.762i) q^{53} +(350.000 - 606.218i) q^{58} +559.000 q^{64} +(370.000 + 640.859i) q^{67} -978.928 q^{71} +(-595.294 - 1031.08i) q^{74} +(692.000 - 1198.58i) q^{79} +(238.118 - 412.432i) q^{86} +(315.000 + 545.596i) q^{88} -216.952 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 110q^{16} + 280q^{22} + 250q^{25} + 900q^{37} + 720q^{43} + 1148q^{46} + 1400q^{58} + 2236q^{64} + 1480q^{67} + 2768q^{79} + 1260q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32288 2.29129i 0.467707 0.810093i −0.531612 0.846988i \(-0.678414\pi\)
0.999319 + 0.0368954i \(0.0117469\pi\)
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.0625000 + 0.108253i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 23.8118 1.05234
\(9\) 0 0
\(10\) 0 0
\(11\) 13.2288 + 22.9129i 0.362602 + 0.628045i 0.988388 0.151950i \(-0.0485552\pi\)
−0.625786 + 0.779994i \(0.715222\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 27.5000 47.6314i 0.429688 0.744241i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 70.0000 0.678366
\(23\) −108.476 + 187.886i −0.983425 + 1.70334i −0.334687 + 0.942329i \(0.608631\pi\)
−0.648737 + 0.761012i \(0.724703\pi\)
\(24\) 0 0
\(25\) 62.5000 + 108.253i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 264.575 1.69415 0.847075 0.531473i \(-0.178361\pi\)
0.847075 + 0.531473i \(0.178361\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 22.4889 + 38.9519i 0.124235 + 0.215181i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 225.000 389.711i 0.999724 1.73157i 0.479500 0.877542i \(-0.340818\pi\)
0.520223 0.854030i \(-0.325849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 180.000 0.638366 0.319183 0.947693i \(-0.396592\pi\)
0.319183 + 0.947693i \(0.396592\pi\)
\(44\) −13.2288 + 22.9129i −0.0453252 + 0.0785056i
\(45\) 0 0
\(46\) 287.000 + 497.099i 0.919910 + 1.59333i
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 330.719 0.935414
\(51\) 0 0
\(52\) 0 0
\(53\) 248.701 + 430.762i 0.644560 + 1.11641i 0.984403 + 0.175928i \(0.0562926\pi\)
−0.339843 + 0.940482i \(0.610374\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 350.000 606.218i 0.792366 1.37242i
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 559.000 1.09180
\(65\) 0 0
\(66\) 0 0
\(67\) 370.000 + 640.859i 0.674667 + 1.16856i 0.976566 + 0.215218i \(0.0690461\pi\)
−0.301899 + 0.953340i \(0.597621\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −978.928 −1.63630 −0.818151 0.575004i \(-0.805000\pi\)
−0.818151 + 0.575004i \(0.805000\pi\)
\(72\) 0 0
\(73\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) −595.294 1031.08i −0.935156 1.61974i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 692.000 1198.58i 0.985520 1.70697i 0.345918 0.938265i \(-0.387568\pi\)
0.639602 0.768706i \(-0.279099\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 238.118 412.432i 0.298568 0.517136i
\(87\) 0 0
\(88\) 315.000 + 545.596i 0.381581 + 0.660917i
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −216.952 −0.245856
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −62.5000 + 108.253i −0.0625000 + 0.108253i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1316.00 1.20586
\(107\) 775.205 1342.69i 0.700392 1.21311i −0.267937 0.963436i \(-0.586342\pi\)
0.968329 0.249678i \(-0.0803246\pi\)
\(108\) 0 0
\(109\) −27.0000 46.7654i −0.0237260 0.0410946i 0.853919 0.520407i \(-0.174220\pi\)
−0.877645 + 0.479312i \(0.840886\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2307.10 −1.92065 −0.960324 0.278886i \(-0.910035\pi\)
−0.960324 + 0.278886i \(0.910035\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 132.288 + 229.129i 0.105884 + 0.183397i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 315.500 546.462i 0.237040 0.410565i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2000.00 −1.39741 −0.698706 0.715409i \(-0.746240\pi\)
−0.698706 + 0.715409i \(0.746240\pi\)
\(128\) 559.576 969.215i 0.386407 0.669276i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1957.86 1.26219
\(135\) 0 0
\(136\) 0 0
\(137\) 391.571 + 678.221i 0.244191 + 0.422951i 0.961904 0.273388i \(-0.0881442\pi\)
−0.717713 + 0.696339i \(0.754811\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1295.00 + 2243.01i −0.765310 + 1.32556i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 450.000 0.249931
\(149\) −1772.65 + 3070.33i −0.974640 + 1.68813i −0.293524 + 0.955952i \(0.594828\pi\)
−0.681117 + 0.732175i \(0.738505\pi\)
\(150\) 0 0
\(151\) −1476.00 2556.51i −0.795465 1.37779i −0.922544 0.385893i \(-0.873893\pi\)
0.127079 0.991893i \(-0.459440\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) −1830.86 3171.14i −0.921870 1.59672i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −890.000 + 1541.53i −0.427670 + 0.740746i −0.996666 0.0815946i \(-0.973999\pi\)
0.568996 + 0.822340i \(0.307332\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −2197.00 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 90.0000 + 155.885i 0.0398979 + 0.0691052i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1455.16 0.623222
\(177\) 0 0
\(178\) 0 0
\(179\) 2156.29 + 3734.80i 0.900383 + 1.55951i 0.826998 + 0.562205i \(0.190047\pi\)
0.0733844 + 0.997304i \(0.476620\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2583.00 + 4473.89i −1.03490 + 1.79250i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1680.05 + 2909.94i −0.636462 + 1.10239i 0.349741 + 0.936846i \(0.386270\pi\)
−0.986203 + 0.165539i \(0.947064\pi\)
\(192\) 0 0
\(193\) −2295.00 3975.06i −0.855947 1.48254i −0.875764 0.482740i \(-0.839642\pi\)
0.0198172 0.999804i \(-0.493692\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5069.26 1.83335 0.916675 0.399634i \(-0.130863\pi\)
0.916675 + 0.399634i \(0.130863\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 1488.24 + 2577.70i 0.526171 + 0.911354i
\(201\) 0 0
\(202\) 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
\(211\) −5868.00 −1.91455 −0.957274 0.289181i \(-0.906617\pi\)
−0.957274 + 0.289181i \(0.906617\pi\)
\(212\) −248.701 + 430.762i −0.0805699 + 0.139551i
\(213\) 0 0
\(214\) −2051.00 3552.44i −0.655156 1.13476i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −142.871 −0.0443872
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −3052.00 + 5286.22i −0.898301 + 1.55590i
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6300.00 1.78282
\(233\) 2656.33 4600.91i 0.746877 1.29363i −0.202436 0.979295i \(-0.564886\pi\)
0.949313 0.314333i \(-0.101781\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 449.778 0.121731 0.0608655 0.998146i \(-0.480614\pi\)
0.0608655 + 0.998146i \(0.480614\pi\)
\(240\) 0 0
\(241\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(242\) −834.735 1445.80i −0.221730 0.384048i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −5740.00 −1.42637
\(254\) −2645.75 + 4582.58i −0.653580 + 1.13203i
\(255\) 0 0
\(256\) 755.500 + 1308.56i 0.184448 + 0.319474i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2013.42 3487.34i −0.472063 0.817637i 0.527426 0.849601i \(-0.323157\pi\)
−0.999489 + 0.0319637i \(0.989824\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −370.000 + 640.859i −0.0843334 + 0.146070i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 2072.00 0.456840
\(275\) −1653.59 + 2864.11i −0.362602 + 0.628045i
\(276\) 0 0
\(277\) −3655.00 6330.65i −0.792807 1.37318i −0.924222 0.381855i \(-0.875285\pi\)
0.131415 0.991327i \(-0.458048\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8360.57 −1.77491 −0.887456 0.460893i \(-0.847529\pi\)
−0.887456 + 0.460893i \(0.847529\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) −489.464 847.777i −0.102269 0.177135i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2456.50 4254.78i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5357.65 9279.72i 1.05205 1.82220i
\(297\) 0 0
\(298\) 4690.00 + 8123.32i 0.911693 + 1.57910i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −7810.26 −1.48818
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1384.00 0.246380
\(317\) 4439.57 7689.56i 0.786597 1.36243i −0.141444 0.989946i \(-0.545175\pi\)
0.928041 0.372479i \(-0.121492\pi\)
\(318\) 0 0
\(319\) 3500.00 + 6062.18i 0.614302 + 1.06400i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 2354.72 + 4078.49i 0.400048 + 0.692904i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5454.00 9446.61i 0.905677 1.56868i 0.0856702 0.996324i \(-0.472697\pi\)
0.820006 0.572354i \(-0.193970\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3330.00 0.538269 0.269135 0.963103i \(-0.413262\pi\)
0.269135 + 0.963103i \(0.413262\pi\)
\(338\) −2906.36 + 5033.96i −0.467707 + 0.810093i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 4286.12 0.671779
\(345\) 0 0
\(346\) 0 0
\(347\) −6130.21 10617.8i −0.948377 1.64264i −0.748845 0.662745i \(-0.769391\pi\)
−0.199532 0.979891i \(-0.563942\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −595.000 + 1030.57i −0.0900955 + 0.156050i
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 11410.0 1.68446
\(359\) 5463.48 9463.02i 0.803207 1.39120i −0.114288 0.993448i \(-0.536459\pi\)
0.917495 0.397747i \(-0.130208\pi\)
\(360\) 0 0
\(361\) 3429.50 + 5940.07i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) 5966.17 + 10333.7i 0.845131 + 1.46381i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6985.00 12098.4i 0.969624 1.67944i 0.272980 0.962020i \(-0.411991\pi\)
0.696643 0.717418i \(-0.254676\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11916.0 −1.61500 −0.807498 0.589870i \(-0.799179\pi\)
−0.807498 + 0.589870i \(0.799179\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4445.00 + 7698.97i 0.595356 + 1.03119i
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12144.0 −1.60133
\(387\) 0 0
\(388\) 0 0
\(389\) −5582.54 9669.23i −0.727624 1.26028i −0.957885 0.287153i \(-0.907291\pi\)
0.230261 0.973129i \(-0.426042\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 6706.00 11615.1i 0.857471 1.48518i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6875.00 0.859375
\(401\) −7990.17 + 13839.4i −0.995037 + 1.72346i −0.411348 + 0.911478i \(0.634942\pi\)
−0.583690 + 0.811977i \(0.698392\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11905.9 1.45001
\(408\) 0 0
\(409\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 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
\(421\) 15262.0 1.76680 0.883402 0.468616i \(-0.155247\pi\)
0.883402 + 0.468616i \(0.155247\pi\)
\(422\) −7762.63 + 13445.3i −0.895448 + 1.55096i
\(423\) 0 0
\(424\) 5922.00 + 10257.2i 0.678297 + 1.17484i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1550.41 0.175098
\(429\) 0 0
\(430\) 0 0
\(431\) 7844.65 + 13587.3i 0.876714 + 1.51851i 0.854926 + 0.518750i \(0.173602\pi\)
0.0217878 + 0.999763i \(0.493064\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 27.0000 46.7654i 0.00296575 0.00513682i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −796.371 + 1379.36i −0.0854102 + 0.147935i −0.905566 0.424205i \(-0.860553\pi\)
0.820156 + 0.572140i \(0.193887\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18837.7 1.97997 0.989987 0.141158i \(-0.0450827\pi\)
0.989987 + 0.141158i \(0.0450827\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1153.55 1998.00i −0.120041 0.207916i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4005.00 6936.86i 0.409947 0.710050i −0.584936 0.811079i \(-0.698880\pi\)
0.994883 + 0.101030i \(0.0322137\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −8440.00 −0.847171 −0.423585 0.905856i \(-0.639229\pi\)
−0.423585 + 0.905856i \(0.639229\pi\)
\(464\) 7275.82 12602.1i 0.727955 1.26086i
\(465\) 0 0
\(466\) −7028.00 12172.9i −0.698639 1.21008i
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2381.18 + 4124.32i 0.231473 + 0.400923i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 595.000 1030.57i 0.0569344 0.0986134i
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 631.000 0.0592600
\(485\) 0 0
\(486\) 0 0
\(487\) 10620.0 + 18394.4i 0.988169 + 1.71156i 0.626908 + 0.779094i \(0.284320\pi\)
0.361261 + 0.932465i \(0.382346\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7646.22 0.702788 0.351394 0.936228i \(-0.385708\pi\)
0.351394 + 0.936228i \(0.385708\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3618.00 + 6266.56i −0.324577 + 0.562184i −0.981427 0.191838i \(-0.938555\pi\)
0.656850 + 0.754022i \(0.271889\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −7593.31 + 13152.0i −0.667122 + 1.15549i
\(507\) 0 0
\(508\) −1000.00 1732.05i −0.0873382 0.151274i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 12951.0 1.11788
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −10654.0 −0.883149
\(527\) 0 0
\(528\) 0 0
\(529\) −17450.5 30225.2i −1.43425 2.48419i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 8810.35 + 15260.0i 0.709980 + 1.22972i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7939.00 + 13750.8i −0.630914 + 1.09277i 0.356452 + 0.934314i \(0.383986\pi\)
−0.987365 + 0.158461i \(0.949347\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12980.0 1.01460 0.507299 0.861770i \(-0.330644\pi\)
0.507299 + 0.861770i \(0.330644\pi\)
\(548\) −391.571 + 678.221i −0.0305239 + 0.0528689i
\(549\) 0 0
\(550\) 4375.00 + 7577.72i 0.339183 + 0.587482i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −19340.4 −1.48321
\(555\) 0 0
\(556\) 0 0
\(557\) −8249.45 14288.5i −0.627541 1.08693i −0.988044 0.154175i \(-0.950728\pi\)
0.360502 0.932758i \(-0.382605\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −11060.0 + 19156.5i −0.830139 + 1.43784i
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −23310.0 −1.72195
\(569\) 1799.11 3116.15i 0.132553 0.229589i −0.792107 0.610382i \(-0.791016\pi\)
0.924660 + 0.380794i \(0.124349\pi\)
\(570\) 0 0
\(571\) 3394.00 + 5878.58i 0.248747 + 0.430842i 0.963178 0.268863i \(-0.0866479\pi\)
−0.714431 + 0.699705i \(0.753315\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −27119.0 −1.96685
\(576\) 0 0
\(577\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(578\) −6499.29 11257.1i −0.467707 0.810093i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6580.00 + 11396.9i −0.467437 + 0.809625i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −12375.0 21434.1i −0.859137 1.48807i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3545.31 −0.243660
\(597\) 0 0
\(598\) 0 0
\(599\) −7871.11 13633.2i −0.536903 0.929943i −0.999069 0.0431495i \(-0.986261\pi\)
0.462166 0.886794i \(-0.347073\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1476.00 2556.51i 0.0994331 0.172223i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −7505.00 12999.0i −0.494493 0.856487i 0.505487 0.862834i \(-0.331313\pi\)
−0.999980 + 0.00634752i \(0.997980\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2497.59 0.162965 0.0814823 0.996675i \(-0.474035\pi\)
0.0814823 + 0.996675i \(0.474035\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7812.50 + 13531.6i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −26192.0 −1.65244 −0.826218 0.563351i \(-0.809512\pi\)
−0.826218 + 0.563351i \(0.809512\pi\)
\(632\) 16477.7 28540.3i 1.03710 1.79632i
\(633\) 0 0
\(634\) −11746.0 20344.7i −0.735794 1.27443i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 18520.3 1.14925
\(639\) 0 0
\(640\) 0 0
\(641\) 15609.9 + 27037.2i 0.961865 + 1.66600i 0.717813 + 0.696236i \(0.245143\pi\)
0.244052 + 0.969762i \(0.421523\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1780.00 −0.106917
\(653\) 9773.41 16928.0i 0.585701 1.01446i −0.409086 0.912496i \(-0.634153\pi\)
0.994788 0.101969i \(-0.0325141\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −33786.2 −1.99716 −0.998578 0.0533186i \(-0.983020\pi\)
−0.998578 + 0.0533186i \(0.983020\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) −14429.9 24993.4i −0.847183 1.46736i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −28700.0 + 49709.9i −1.66607 + 2.88572i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 33570.0 1.92278 0.961388 0.275196i \(-0.0887428\pi\)
0.961388 + 0.275196i \(0.0887428\pi\)
\(674\) 4405.18 7629.99i 0.251752 0.436048i
\(675\) 0 0
\(676\) −1098.50 1902.66i −0.0625000 0.108253i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5347.06 9261.39i −0.299560 0.518854i 0.676475 0.736466i \(-0.263507\pi\)
−0.976035 + 0.217612i \(0.930173\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4950.00 8573.65i 0.274298 0.475098i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −32438.0 −1.77425
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36881.8 −1.98717 −0.993584 0.113093i \(-0.963924\pi\)
−0.993584 + 0.113093i \(0.963924\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 7394.87 + 12808.3i 0.395888 + 0.685697i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6273.00 10865.2i 0.332281 0.575528i −0.650677 0.759354i \(-0.725515\pi\)
0.982959 + 0.183826i \(0.0588483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2156.29 + 3734.80i −0.112548 + 0.194939i
\(717\) 0 0
\(718\) −14455.0 25036.8i −0.751331 1.30134i
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18147.2 0.935414
\(723\) 0 0
\(724\) 0 0
\(725\) 16535.9 + 28641.1i 0.847075 + 1.46718i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −9758.00 −0.488702
\(737\) −9789.28 + 16955.5i −0.489271 + 0.847442i
\(738\) 0 0
\(739\) 12662.0 + 21931.2i 0.630283 + 1.09168i 0.987494 + 0.157658i \(0.0503945\pi\)
−0.357211 + 0.934024i \(0.616272\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31743.7 1.56738 0.783691 0.621151i \(-0.213335\pi\)
0.783691 + 0.621151i \(0.213335\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −18480.6 32009.3i −0.907000 1.57097i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1224.00 + 2120.03i −0.0594732 + 0.103011i −0.894229 0.447610i \(-0.852275\pi\)
0.834756 + 0.550620i \(0.185609\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34830.0 1.67228 0.836141 0.548514i \(-0.184806\pi\)
0.836141 + 0.548514i \(0.184806\pi\)
\(758\) −15763.4 + 27303.0i −0.755346 + 1.30830i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3360.10 −0.159116
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2295.00 3975.06i 0.106993 0.185318i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −29540.0 −1.36126
\(779\) 0 0
\(780\) 0 0
\(781\) −12950.0 22430.1i −0.593326 1.02767i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 2534.63 + 4390.11i 0.114584 + 0.198466i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2811.11 + 4868.99i −0.124235 + 0.215181i
\(801\) 0 0
\(802\) 21140.0 + 36615.6i 0.930772 + 1.61214i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0