Properties

Label 441.2.e.h.361.1
Level $441$
Weight $2$
Character 441.361
Analytic conductor $3.521$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
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 361.1
Root \(-1.32288 - 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 441.361
Dual form 441.2.e.h.226.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.32288 - 2.29129i) q^{2} +(-2.50000 + 4.33013i) q^{4} +7.93725 q^{8} +O(q^{10})\) \(q+(-1.32288 - 2.29129i) q^{2} +(-2.50000 + 4.33013i) q^{4} +7.93725 q^{8} +(2.64575 - 4.58258i) q^{11} +(-5.50000 - 9.52628i) q^{16} -14.0000 q^{22} +(-2.64575 - 4.58258i) q^{23} +(2.50000 - 4.33013i) q^{25} -10.5830 q^{29} +(-6.61438 + 11.4564i) q^{32} +(-3.00000 - 5.19615i) q^{37} +12.0000 q^{43} +(13.2288 + 22.9129i) q^{44} +(-7.00000 + 12.1244i) q^{46} -13.2288 q^{50} +(5.29150 - 9.16515i) q^{53} +(14.0000 + 24.2487i) q^{58} +13.0000 q^{64} +(-2.00000 + 3.46410i) q^{67} -5.29150 q^{71} +(-7.93725 + 13.7477i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(-15.8745 - 27.4955i) q^{86} +(21.0000 - 36.3731i) q^{88} +26.4575 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{4} + O(q^{10}) \) \( 4q - 10q^{4} - 22q^{16} - 56q^{22} + 10q^{25} - 12q^{37} + 48q^{43} - 28q^{46} + 56q^{58} + 52q^{64} - 8q^{67} - 16q^{79} + 84q^{88} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 −0.935414 1.62019i −0.773893 0.633316i \(-0.781693\pi\)
−0.161521 0.986869i \(-0.551640\pi\)
\(3\) 0 0
\(4\) −2.50000 + 4.33013i −1.25000 + 2.16506i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 7.93725 2.80624
\(9\) 0 0
\(10\) 0 0
\(11\) 2.64575 4.58258i 0.797724 1.38170i −0.123371 0.992361i \(-0.539370\pi\)
0.921095 0.389338i \(-0.127296\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −5.50000 9.52628i −1.37500 2.38157i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −14.0000 −2.98481
\(23\) −2.64575 4.58258i −0.551677 0.955533i −0.998154 0.0607377i \(-0.980655\pi\)
0.446476 0.894795i \(-0.352679\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.5830 −1.96521 −0.982607 0.185695i \(-0.940546\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) −6.61438 + 11.4564i −1.16927 + 2.02523i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 5.19615i −0.493197 0.854242i 0.506772 0.862080i \(-0.330838\pi\)
−0.999969 + 0.00783774i \(0.997505\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 13.2288 + 22.9129i 1.99431 + 3.45425i
\(45\) 0 0
\(46\) −7.00000 + 12.1244i −1.03209 + 1.78764i
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −13.2288 −1.87083
\(51\) 0 0
\(52\) 0 0
\(53\) 5.29150 9.16515i 0.726844 1.25893i −0.231367 0.972867i \(-0.574320\pi\)
0.958211 0.286064i \(-0.0923469\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 14.0000 + 24.2487i 1.83829 + 3.18401i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 + 3.46410i −0.244339 + 0.423207i −0.961946 0.273241i \(-0.911904\pi\)
0.717607 + 0.696449i \(0.245238\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.29150 −0.627986 −0.313993 0.949425i \(-0.601667\pi\)
−0.313993 + 0.949425i \(0.601667\pi\)
\(72\) 0 0
\(73\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) −7.93725 + 13.7477i −0.922687 + 1.59814i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 6.92820i −0.450035 0.779484i 0.548352 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 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
\(86\) −15.8745 27.4955i −1.71179 2.96491i
\(87\) 0 0
\(88\) 21.0000 36.3731i 2.23861 3.87738i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 26.4575 2.75839
\(93\) 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
\(100\) 12.5000 + 21.6506i 1.25000 + 2.16506i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −28.0000 −2.71960
\(107\) 2.64575 + 4.58258i 0.255774 + 0.443014i 0.965106 0.261861i \(-0.0843362\pi\)
−0.709331 + 0.704875i \(0.751003\pi\)
\(108\) 0 0
\(109\) 9.00000 15.5885i 0.862044 1.49310i −0.00790932 0.999969i \(-0.502518\pi\)
0.869953 0.493135i \(-0.164149\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 21.1660 1.99113 0.995565 0.0940721i \(-0.0299884\pi\)
0.995565 + 0.0940721i \(0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 26.4575 45.8258i 2.45652 4.25481i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.50000 14.7224i −0.772727 1.33840i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −3.96863 6.87386i −0.350780 0.607569i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 10.5830 0.914232
\(135\) 0 0
\(136\) 0 0
\(137\) −10.5830 + 18.3303i −0.904167 + 1.56606i −0.0821359 + 0.996621i \(0.526174\pi\)
−0.822031 + 0.569442i \(0.807159\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.00000 + 12.1244i 0.587427 + 1.01745i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 30.0000 2.46598
\(149\) −5.29150 9.16515i −0.433497 0.750838i 0.563675 0.825997i \(-0.309387\pi\)
−0.997172 + 0.0751583i \(0.976054\pi\)
\(150\) 0 0
\(151\) −12.0000 + 20.7846i −0.976546 + 1.69143i −0.301811 + 0.953368i \(0.597591\pi\)
−0.674735 + 0.738060i \(0.735742\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) −10.5830 + 18.3303i −0.841939 + 1.45828i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 + 17.3205i 0.783260 + 1.35665i 0.930033 + 0.367477i \(0.119778\pi\)
−0.146772 + 0.989170i \(0.546888\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −30.0000 + 51.9615i −2.28748 + 3.96203i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −58.2065 −4.38748
\(177\) 0 0
\(178\) 0 0
\(179\) −13.2288 + 22.9129i −0.988764 + 1.71259i −0.364922 + 0.931038i \(0.618904\pi\)
−0.623841 + 0.781551i \(0.714429\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −21.0000 36.3731i −1.54814 2.68146i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.2288 + 22.9129i 0.957199 + 1.65792i 0.729253 + 0.684244i \(0.239868\pi\)
0.227946 + 0.973674i \(0.426799\pi\)
\(192\) 0 0
\(193\) 9.00000 15.5885i 0.647834 1.12208i −0.335805 0.941932i \(-0.609008\pi\)
0.983639 0.180150i \(-0.0576584\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.5830 0.754008 0.377004 0.926212i \(-0.376954\pi\)
0.377004 + 0.926212i \(0.376954\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 19.8431 34.3693i 1.40312 2.43028i
\(201\) 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
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 26.4575 + 45.8258i 1.81711 + 3.14733i
\(213\) 0 0
\(214\) 7.00000 12.1244i 0.478510 0.828804i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −47.6235 −3.22547
\(219\) 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
\(226\) −28.0000 48.4974i −1.86253 3.22600i
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −84.0000 −5.51487
\(233\) 10.5830 + 18.3303i 0.693316 + 1.20086i 0.970745 + 0.240112i \(0.0771842\pi\)
−0.277429 + 0.960746i \(0.589482\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.4575 1.71139 0.855697 0.517477i \(-0.173129\pi\)
0.855697 + 0.517477i \(0.173129\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) −22.4889 + 38.9519i −1.44564 + 2.50392i
\(243\) 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
\(253\) −28.0000 −1.76034
\(254\) −21.1660 36.6606i −1.32807 2.30029i
\(255\) 0 0
\(256\) 2.50000 4.33013i 0.156250 0.270633i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.64575 + 4.58258i −0.163144 + 0.282574i −0.935995 0.352014i \(-0.885497\pi\)
0.772851 + 0.634588i \(0.218830\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −10.0000 17.3205i −0.610847 1.05802i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 56.0000 3.38308
\(275\) −13.2288 22.9129i −0.797724 1.38170i
\(276\) 0 0
\(277\) 5.00000 8.66025i 0.300421 0.520344i −0.675810 0.737075i \(-0.736206\pi\)
0.976231 + 0.216731i \(0.0695395\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.1660 −1.26266 −0.631329 0.775515i \(-0.717490\pi\)
−0.631329 + 0.775515i \(0.717490\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 13.2288 22.9129i 0.784982 1.35963i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 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
\(296\) −23.8118 41.2432i −1.38403 2.39721i
\(297\) 0 0
\(298\) −14.0000 + 24.2487i −0.810998 + 1.40469i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 63.4980 3.65390
\(303\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 40.0000 2.25018
\(317\) 5.29150 + 9.16515i 0.297200 + 0.514766i 0.975494 0.220024i \(-0.0706137\pi\)
−0.678294 + 0.734791i \(0.737280\pi\)
\(318\) 0 0
\(319\) −28.0000 + 48.4974i −1.56770 + 2.71533i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 26.4575 45.8258i 1.46535 2.53805i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 18.0000 + 31.1769i 0.989369 + 1.71364i 0.620625 + 0.784107i \(0.286879\pi\)
0.368744 + 0.929531i \(0.379788\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 17.1974 + 29.7867i 0.935414 + 1.62019i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 95.2470 5.13538
\(345\) 0 0
\(346\) 0 0
\(347\) 18.5203 32.0780i 0.994220 1.72204i 0.404128 0.914702i \(-0.367575\pi\)
0.590091 0.807337i \(-0.299092\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 35.0000 + 60.6218i 1.86551 + 3.23115i
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 70.0000 3.69961
\(359\) −18.5203 32.0780i −0.977462 1.69301i −0.671559 0.740951i \(-0.734375\pi\)
−0.305903 0.952063i \(-0.598958\pi\)
\(360\) 0 0
\(361\) 9.50000 16.4545i 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) −29.1033 + 50.4083i −1.51711 + 2.62772i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 19.0526i −0.569558 0.986504i −0.996610 0.0822766i \(-0.973781\pi\)
0.427051 0.904227i \(-0.359552\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 35.0000 60.6218i 1.79076 3.10168i
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −47.6235 −2.42397
\(387\) 0 0
\(388\) 0 0
\(389\) −5.29150 + 9.16515i −0.268290 + 0.464692i −0.968420 0.249323i \(-0.919792\pi\)
0.700130 + 0.714015i \(0.253125\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −14.0000 24.2487i −0.705310 1.22163i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −55.0000 −2.75000
\(401\) −10.5830 18.3303i −0.528490 0.915372i −0.999448 0.0332161i \(-0.989425\pi\)
0.470958 0.882156i \(-0.343908\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −31.7490 −1.57374
\(408\) 0 0
\(409\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 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
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −15.8745 27.4955i −0.772759 1.33846i
\(423\) 0 0
\(424\) 42.0000 72.7461i 2.03970 3.53286i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −26.4575 −1.27887
\(429\) 0 0
\(430\) 0 0
\(431\) 13.2288 22.9129i 0.637207 1.10367i −0.348836 0.937184i \(-0.613423\pi\)
0.986043 0.166491i \(-0.0532436\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 45.0000 + 77.9423i 2.15511 + 3.73276i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.5203 + 32.0780i 0.879924 + 1.52407i 0.851423 + 0.524479i \(0.175740\pi\)
0.0285009 + 0.999594i \(0.490927\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −42.3320 −1.99777 −0.998886 0.0471929i \(-0.984972\pi\)
−0.998886 + 0.0471929i \(0.984972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −52.9150 + 91.6515i −2.48891 + 4.31092i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.00000 5.19615i −0.140334 0.243066i 0.787288 0.616585i \(-0.211484\pi\)
−0.927622 + 0.373519i \(0.878151\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 58.2065 + 100.817i 2.70217 + 4.68030i
\(465\) 0 0
\(466\) 28.0000 48.4974i 1.29707 2.24660i
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 31.7490 54.9909i 1.45982 2.52848i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −35.0000 60.6218i −1.60086 2.77278i
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 85.0000 3.86364
\(485\) 0 0
\(486\) 0 0
\(487\) −12.0000 + 20.7846i −0.543772 + 0.941841i 0.454911 + 0.890537i \(0.349671\pi\)
−0.998683 + 0.0513038i \(0.983662\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.29150 0.238802 0.119401 0.992846i \(-0.461903\pi\)
0.119401 + 0.992846i \(0.461903\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18.0000 + 31.1769i 0.805791 + 1.39567i 0.915756 + 0.401735i \(0.131593\pi\)
−0.109965 + 0.993935i \(0.535074\pi\)
\(500\) 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
\(506\) 37.0405 + 64.1561i 1.64665 + 2.85208i
\(507\) 0 0
\(508\) −40.0000 + 69.2820i −1.77471 + 3.07389i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −29.1033 −1.28619
\(513\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) 0 0
\(528\) 0 0
\(529\) −2.50000 + 4.33013i −0.108696 + 0.188266i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −15.8745 + 27.4955i −0.685674 + 1.18762i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 + 29.4449i 0.730887 + 1.26593i 0.956504 + 0.291718i \(0.0942267\pi\)
−0.225617 + 0.974216i \(0.572440\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −52.9150 91.6515i −2.26042 3.91516i
\(549\) 0 0
\(550\) −35.0000 + 60.6218i −1.49241 + 2.58492i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −26.4575 −1.12407
\(555\) 0 0
\(556\) 0 0
\(557\) −5.29150 + 9.16515i −0.224208 + 0.388340i −0.956082 0.293101i \(-0.905313\pi\)
0.731873 + 0.681441i \(0.238646\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 28.0000 + 48.4974i 1.18111 + 2.04574i
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −42.0000 −1.76228
\(569\) −21.1660 36.6606i −0.887325 1.53689i −0.843025 0.537874i \(-0.819228\pi\)
−0.0443003 0.999018i \(-0.514106\pi\)
\(570\) 0 0
\(571\) −2.00000 + 3.46410i −0.0836974 + 0.144968i −0.904835 0.425762i \(-0.860006\pi\)
0.821138 + 0.570730i \(0.193340\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −26.4575 −1.10335
\(576\) 0 0
\(577\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(578\) 22.4889 38.9519i 0.935414 1.62019i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −28.0000 48.4974i −1.15964 2.00856i
\(584\) 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
\(592\) −33.0000 + 57.1577i −1.35629 + 2.34917i
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 52.9150 2.16748
\(597\) 0 0
\(598\) 0 0
\(599\) −18.5203 + 32.0780i −0.756717 + 1.31067i 0.187799 + 0.982208i \(0.439865\pi\)
−0.944516 + 0.328465i \(0.893469\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −60.0000 103.923i −2.44137 4.22857i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 19.0000 32.9090i 0.767403 1.32918i −0.171564 0.985173i \(-0.554882\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.3320 1.70422 0.852111 0.523360i \(-0.175322\pi\)
0.852111 + 0.523360i \(0.175322\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −31.7490 54.9909i −1.26291 2.18742i
\(633\) 0 0
\(634\) 14.0000 24.2487i 0.556011 0.963039i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 148.162 5.86579
\(639\) 0 0
\(640\) 0 0
\(641\) 10.5830 18.3303i 0.418004 0.724003i −0.577735 0.816224i \(-0.696063\pi\)
0.995739 + 0.0922210i \(0.0293966\pi\)
\(642\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −100.000 −3.91630
\(653\) 5.29150 + 9.16515i 0.207072 + 0.358660i 0.950791 0.309833i \(-0.100273\pi\)
−0.743719 + 0.668493i \(0.766940\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −26.4575 −1.03064 −0.515319 0.856998i \(-0.672327\pi\)
−0.515319 + 0.856998i \(0.672327\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 47.6235 82.4864i 1.85094 3.20592i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.0000 + 48.4974i 1.08416 + 1.87783i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 39.6863 + 68.7386i 1.52866 + 2.64771i
\(675\) 0 0
\(676\) 32.5000 56.2917i 1.25000 2.16506i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.64575 + 4.58258i −0.101237 + 0.175347i −0.912194 0.409757i \(-0.865613\pi\)
0.810958 + 0.585105i \(0.198947\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −66.0000 114.315i −2.51623 4.35823i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −98.0000 −3.72003
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 52.9150 1.99857 0.999286 0.0377695i \(-0.0120253\pi\)
0.999286 + 0.0377695i \(0.0120253\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 34.3948 59.5735i 1.29630 2.24526i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −66.1438 114.564i −2.47191 4.28147i
\(717\) 0 0
\(718\) −49.0000 + 84.8705i −1.82866 + 3.16734i
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −50.2693 −1.87083
\(723\) 0 0
\(724\) 0 0
\(725\) −26.4575 + 45.8258i −0.982607 + 1.70193i
\(726\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 70.0000 2.58023
\(737\) 10.5830 + 18.3303i 0.389830 + 0.675205i
\(738\) 0 0
\(739\) 26.0000 45.0333i 0.956425 1.65658i 0.225354 0.974277i \(-0.427646\pi\)
0.731072 0.682300i \(-0.239020\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.0405 1.35888 0.679442 0.733729i \(-0.262222\pi\)
0.679442 + 0.733729i \(0.262222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −29.1033 + 50.4083i −1.06555 + 1.84558i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 41.5692i −0.875772 1.51688i −0.855938 0.517079i \(-0.827019\pi\)
−0.0198348 0.999803i \(-0.506314\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) −15.8745 27.4955i −0.576588 0.998680i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −132.288 −4.78600
\(765\) 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
\(772\) 45.0000 + 77.9423i 1.61959 + 2.80520i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 28.0000 1.00385
\(779\) 0 0
\(780\) 0 0
\(781\) −14.0000 + 24.2487i −0.500959 + 0.867687i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) −26.4575 + 45.8258i −0.942510 + 1.63247i
\(789\) 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
\(800\) 33.0719 + 57.2822i 1.16927 + 2.02523i
\(801\) 0 0
\(802\) −28.0000 + 48.4974i −0.988714 + 1.71250i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0