Properties

Label 216.2.l.a.179.2
Level $216$
Weight $2$
Character 216.179
Analytic conductor $1.725$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 179.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 216.179
Dual form 216.2.l.a.35.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{8} +O(q^{10})\) \(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{8} +(3.27526 - 1.89097i) q^{11} +(-2.00000 - 3.46410i) q^{16} +8.02458i q^{17} -8.34847 q^{19} +(2.67423 - 4.63191i) q^{22} +(2.50000 + 4.33013i) q^{25} +(-4.89898 - 2.82843i) q^{32} +(5.67423 + 9.82806i) q^{34} +(-10.2247 + 5.90326i) q^{38} +(0.398979 + 0.230351i) q^{41} +(1.17423 + 2.03383i) q^{43} -7.56388i q^{44} +(-3.50000 + 6.06218i) q^{49} +(6.12372 + 3.53553i) q^{50} +(-10.6237 - 6.13361i) q^{59} -8.00000 q^{64} +(7.17423 - 12.4261i) q^{67} +(13.8990 + 8.02458i) q^{68} +13.6969 q^{73} +(-8.34847 + 14.4600i) q^{76} +0.651531 q^{82} +(2.44949 - 1.41421i) q^{83} +(2.87628 + 1.66062i) q^{86} +(-5.34847 - 9.26382i) q^{88} -5.65685i q^{89} +(-9.84847 - 17.0580i) q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} + 18q^{11} - 8q^{16} - 4q^{19} - 4q^{22} + 10q^{25} + 8q^{34} - 36q^{38} - 18q^{41} - 10q^{43} - 14q^{49} - 18q^{59} - 32q^{64} + 14q^{67} + 36q^{68} - 4q^{73} - 4q^{76} + 32q^{82} + 36q^{86} + 8q^{88} - 10q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22474 0.707107i 0.866025 0.500000i
\(3\) 0 0
\(4\) 1.00000 1.73205i 0.500000 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.27526 1.89097i 0.987527 0.570149i 0.0829925 0.996550i \(-0.473552\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 3.46410i −0.500000 0.866025i
\(17\) 8.02458i 1.94625i 0.230285 + 0.973123i \(0.426034\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) −8.34847 −1.91527 −0.957635 0.287984i \(-0.907015\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.67423 4.63191i 0.570149 0.987527i
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −4.89898 2.82843i −0.866025 0.500000i
\(33\) 0 0
\(34\) 5.67423 + 9.82806i 0.973123 + 1.68550i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −10.2247 + 5.90326i −1.65867 + 0.957635i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.398979 + 0.230351i 0.0623101 + 0.0359748i 0.530831 0.847477i \(-0.321880\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 1.17423 + 2.03383i 0.179069 + 0.310157i 0.941562 0.336840i \(-0.109358\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 7.56388i 1.14030i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(50\) 6.12372 + 3.53553i 0.866025 + 0.500000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.6237 6.13361i −1.38309 0.798528i −0.390567 0.920575i \(-0.627721\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.17423 12.4261i 0.876472 1.51809i 0.0212861 0.999773i \(-0.493224\pi\)
0.855186 0.518321i \(-0.173443\pi\)
\(68\) 13.8990 + 8.02458i 1.68550 + 0.973123i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.6969 1.60311 0.801553 0.597924i \(-0.204008\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −8.34847 + 14.4600i −0.957635 + 1.65867i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.651531 0.0719495
\(83\) 2.44949 1.41421i 0.268866 0.155230i −0.359506 0.933143i \(-0.617055\pi\)
0.628372 + 0.777913i \(0.283721\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.87628 + 1.66062i 0.310157 + 0.179069i
\(87\) 0 0
\(88\) −5.34847 9.26382i −0.570149 0.987527i
\(89\) 5.65685i 0.599625i −0.953998 0.299813i \(-0.903076\pi\)
0.953998 0.299813i \(-0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.84847 17.0580i −0.999961 1.73198i −0.507673 0.861550i \(-0.669494\pi\)
−0.492287 0.870433i \(-0.663839\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 0 0
\(100\) 10.0000 1.00000
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.70334i 0.454689i −0.973814 0.227345i \(-0.926996\pi\)
0.973814 0.227345i \(-0.0730044\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.79796 5.65685i −0.921714 0.532152i −0.0375328 0.999295i \(-0.511950\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −17.3485 −1.59706
\(119\) 0 0
\(120\) 0 0
\(121\) 1.65153 2.86054i 0.150139 0.260049i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −9.79796 + 5.65685i −0.866025 + 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.2474 + 7.07107i 1.07006 + 0.617802i 0.928199 0.372084i \(-0.121357\pi\)
0.141865 + 0.989886i \(0.454690\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 20.2918i 1.75294i
\(135\) 0 0
\(136\) 22.6969 1.94625
\(137\) 14.2980 8.25493i 1.22156 0.705266i 0.256307 0.966595i \(-0.417494\pi\)
0.965250 + 0.261329i \(0.0841608\pi\)
\(138\) 0 0
\(139\) −1.82577 + 3.16232i −0.154859 + 0.268224i −0.933008 0.359856i \(-0.882826\pi\)
0.778148 + 0.628080i \(0.216159\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 16.7753 9.68520i 1.38833 0.801553i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 23.6130i 1.91527i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0.797959 0.460702i 0.0623101 0.0359748i
\(165\) 0 0
\(166\) 2.00000 3.46410i 0.155230 0.268866i
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) −6.50000 11.2583i −0.500000 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 4.69694 0.358138
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.1010 7.56388i −0.987527 0.570149i
\(177\) 0 0
\(178\) −4.00000 6.92820i −0.299813 0.519291i
\(179\) 19.7990i 1.47985i 0.672692 + 0.739923i \(0.265138\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.1742 + 26.2825i 1.10965 + 1.92197i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) −12.8485 + 22.2542i −0.924853 + 1.60189i −0.133056 + 0.991109i \(0.542479\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −24.1237 13.9278i −1.73198 0.999961i
\(195\) 0 0
\(196\) 7.00000 + 12.1244i 0.500000 + 0.866025i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 12.2474 7.07107i 0.866025 0.500000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −27.3434 + 15.7867i −1.89138 + 1.09199i
\(210\) 0 0
\(211\) −7.00000 + 12.1244i −0.481900 + 0.834675i −0.999784 0.0207756i \(-0.993386\pi\)
0.517884 + 0.855451i \(0.326720\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −3.32577 5.76039i −0.227345 0.393772i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) −23.7247 + 13.6975i −1.57467 + 0.909134i −0.579082 + 0.815270i \(0.696589\pi\)
−0.995585 + 0.0938647i \(0.970078\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.1523i 1.51676i −0.651813 0.758380i \(-0.725991\pi\)
0.651813 0.758380i \(-0.274009\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −21.2474 + 12.2672i −1.38309 + 0.798528i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) −0.848469 1.46959i −0.0546547 0.0946647i 0.837404 0.546585i \(-0.184072\pi\)
−0.892058 + 0.451920i \(0.850739\pi\)
\(242\) 4.67123i 0.300278i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.7525i 1.30989i 0.755678 + 0.654943i \(0.227307\pi\)
−0.755678 + 0.654943i \(0.772693\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 27.3990 + 15.8188i 1.70910 + 0.986750i 0.935674 + 0.352865i \(0.114792\pi\)
0.773427 + 0.633885i \(0.218541\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −14.3485 24.8523i −0.876472 1.51809i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 27.7980 16.0492i 1.68550 0.973123i
\(273\) 0 0
\(274\) 11.6742 20.2204i 0.705266 1.22156i
\(275\) 16.3763 + 9.45485i 0.987527 + 0.570149i
\(276\) 0 0
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 5.16404i 0.309719i
\(279\) 0 0
\(280\) 0 0
\(281\) 24.4949 14.1421i 1.46124 0.843649i 0.462174 0.886789i \(-0.347070\pi\)
0.999069 + 0.0431402i \(0.0137362\pi\)
\(282\) 0 0
\(283\) 11.0000 19.0526i 0.653882 1.13256i −0.328291 0.944577i \(-0.606473\pi\)
0.982173 0.187980i \(-0.0601941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −47.3939 −2.78788
\(290\) 0 0
\(291\) 0 0
\(292\) 13.6969 23.7238i 0.801553 1.38833i
\(293\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 16.6969 + 28.9199i 0.957635 + 1.65867i
\(305\) 0 0
\(306\) 0 0
\(307\) 9.65153 0.550842 0.275421 0.961324i \(-0.411183\pi\)
0.275421 + 0.961324i \(0.411183\pi\)
\(308\) 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
\(313\) 12.1969 + 21.1257i 0.689412 + 1.19410i 0.972028 + 0.234863i \(0.0754642\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 66.9930i 3.72759i
\(324\) 0 0
\(325\) 0 0
\(326\) 2.44949 1.41421i 0.135665 0.0783260i
\(327\) 0 0
\(328\) 0.651531 1.12848i 0.0359748 0.0623101i
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 22.5167i −0.714545 1.23763i −0.963135 0.269019i \(-0.913301\pi\)
0.248590 0.968609i \(-0.420033\pi\)
\(332\) 5.65685i 0.310460i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.1969 31.5180i 0.991250 1.71690i 0.381314 0.924445i \(-0.375472\pi\)
0.609936 0.792451i \(-0.291195\pi\)
\(338\) −15.9217 9.19239i −0.866025 0.500000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 5.75255 3.32124i 0.310157 0.179069i
\(345\) 0 0
\(346\) 0 0
\(347\) 11.4217 + 6.59431i 0.613148 + 0.354001i 0.774197 0.632945i \(-0.218154\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −21.3939 −1.14030
\(353\) −12.7020 + 7.33353i −0.676061 + 0.390324i −0.798369 0.602168i \(-0.794304\pi\)
0.122308 + 0.992492i \(0.460970\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.79796 5.65685i −0.519291 0.299813i
\(357\) 0 0
\(358\) 14.0000 + 24.2487i 0.739923 + 1.28158i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 50.6969 2.66826
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 37.1691 + 21.4596i 1.92197 + 1.10965i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −26.3485 −1.35343 −0.676715 0.736245i \(-0.736597\pi\)
−0.676715 + 0.736245i \(0.736597\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 36.3410i 1.84971i
\(387\) 0 0
\(388\) −39.3939 −1.99992
\(389\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 17.1464 + 9.89949i 0.866025 + 0.500000i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 10.0000 17.3205i 0.500000 0.866025i
\(401\) −21.6464 12.4976i −1.08097 0.624099i −0.149813 0.988714i \(-0.547867\pi\)
−0.931158 + 0.364615i \(0.881200\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 9.19694 15.9296i 0.454759 0.787666i −0.543915 0.839140i \(-0.683059\pi\)
0.998674 + 0.0514740i \(0.0163919\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −22.3258 + 38.6694i −1.09199 + 1.89138i
\(419\) −31.8434 18.3848i −1.55565 0.898155i −0.997665 0.0683046i \(-0.978241\pi\)
−0.557986 0.829851i \(-0.688426\pi\)
\(420\) 0 0
\(421\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 19.7990i 0.963800i
\(423\) 0 0
\(424\) 0 0
\(425\) −34.7474 + 20.0614i −1.68550 + 0.973123i
\(426\) 0 0
\(427\) 0 0
\(428\) −8.14643 4.70334i −0.393772 0.227345i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −4.30306 −0.206792 −0.103396 0.994640i \(-0.532971\pi\)
−0.103396 + 0.994640i \(0.532971\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.2753 17.4794i 1.43842 0.830473i 0.440681 0.897664i \(-0.354737\pi\)
0.997740 + 0.0671913i \(0.0214038\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.2015i 1.85003i 0.379927 + 0.925016i \(0.375949\pi\)
−0.379927 + 0.925016i \(0.624051\pi\)
\(450\) 0 0
\(451\) 1.74235 0.0820439
\(452\) −19.5959 + 11.3137i −0.921714 + 0.532152i
\(453\) 0 0
\(454\) −19.3712 + 33.5519i −0.909134 + 1.57467i
\(455\) 0 0
\(456\) 0 0
\(457\) 21.1969 + 36.7142i 0.991551 + 1.71742i 0.608114 + 0.793849i \(0.291926\pi\)
0.383437 + 0.923567i \(0.374740\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −16.3712 28.3557i −0.758380 1.31355i
\(467\) 10.4244i 0.482384i −0.970477 0.241192i \(-0.922462\pi\)
0.970477 0.241192i \(-0.0775384\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −17.3485 + 30.0484i −0.798528 + 1.38309i
\(473\) 7.69184 + 4.44088i 0.353671 + 0.204192i
\(474\) 0 0
\(475\) −20.8712 36.1499i −0.957635 1.65867i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.07832 1.19992i −0.0946647 0.0546547i
\(483\) 0 0
\(484\) −3.30306 5.72107i −0.150139 0.260049i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −37.6237 21.7221i −1.69793 0.980303i −0.947717 0.319113i \(-0.896615\pi\)
−0.750218 0.661190i \(-0.770052\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.8712 + 25.7576i −0.665725 + 1.15307i 0.313363 + 0.949633i \(0.398544\pi\)
−0.979088 + 0.203436i \(0.934789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14.6742 + 25.4165i 0.654943 + 1.13439i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 44.7423 1.97350
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.4313i 0.763678i −0.924229 0.381839i \(-0.875291\pi\)
0.924229 0.381839i \(-0.124709\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 24.4949 14.1421i 1.07006 0.617802i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −35.1464 20.2918i −1.51809 0.876472i
\(537\) 0 0
\(538\) 0 0
\(539\) 26.4736i 1.14030i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 22.6969 39.3123i 0.973123 1.68550i
\(545\) 0 0
\(546\) 0 0
\(547\) −7.82577 13.5546i −0.334606 0.579554i 0.648803 0.760956i \(-0.275270\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 33.0197i 1.41053i
\(549\) 0 0
\(550\) 26.7423 1.14030
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 3.65153 + 6.32464i 0.154859 + 0.268224i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 20.0000 34.6410i 0.843649 1.46124i
\(563\) 38.4217 + 22.1828i 1.61928 + 0.934892i 0.987106 + 0.160066i \(0.0511708\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 31.1127i 1.30776i
\(567\) 0 0
\(568\) 0 0
\(569\) 41.2980 23.8434i 1.73130 0.999567i 0.850935 0.525271i \(-0.176036\pi\)
0.880366 0.474295i \(-0.157297\pi\)
\(570\) 0 0
\(571\) −23.8712 + 41.3461i −0.998978 + 1.73028i −0.460336 + 0.887745i \(0.652271\pi\)
−0.538642 + 0.842535i \(0.681062\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −12.3939 −0.515964 −0.257982 0.966150i \(-0.583058\pi\)
−0.257982 + 0.966150i \(0.583058\pi\)
\(578\) −58.0454 + 33.5125i −2.41437 + 1.39394i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 38.7408i 1.60311i
\(585\) 0 0
\(586\) 0 0
\(587\) 25.3207 14.6189i 1.04510 0.603386i 0.123823 0.992304i \(-0.460484\pi\)
0.921272 + 0.388918i \(0.127151\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 45.2548i 1.85839i 0.369586 + 0.929197i \(0.379500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) −18.8485 32.6465i −0.768845 1.33168i −0.938190 0.346122i \(-0.887498\pi\)
0.169344 0.985557i \(-0.445835\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 40.8990 + 23.6130i 1.65867 + 0.957635i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 11.8207 6.82466i 0.477043 0.275421i
\(615\) 0 0
\(616\) 0 0
\(617\) 5.35357 + 3.09089i 0.215527 + 0.124434i 0.603877 0.797077i \(-0.293622\pi\)
−0.388351 + 0.921512i \(0.626955\pi\)
\(618\) 0 0
\(619\) −11.8712 20.5615i −0.477143 0.826435i 0.522514 0.852631i \(-0.324994\pi\)
−0.999657 + 0.0261952i \(0.991661\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 29.8763 + 17.2491i 1.19410 + 0.689412i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19.2526 11.1155i 0.760430 0.439034i −0.0690201 0.997615i \(-0.521987\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 16.1742 28.0146i 0.637850 1.10479i −0.348054 0.937474i \(-0.613157\pi\)
0.985904 0.167313i \(-0.0535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −47.3712 82.0493i −1.86379 3.22819i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −46.3939 −1.82112
\(650\) 0 0
\(651\) 0 0
\(652\) 2.00000 3.46410i 0.0783260 0.135665i
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.84281i 0.0719495i
\(657\) 0 0
\(658\) 0 0
\(659\) −41.6413 + 24.0416i −1.62212 + 0.936529i −0.635763 + 0.771885i \(0.719314\pi\)
−0.986353 + 0.164644i \(0.947352\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) −31.8434 18.3848i −1.23763 0.714545i
\(663\) 0 0
\(664\) −4.00000 6.92820i −0.155230 0.268866i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.00000 + 8.66025i 0.192736 + 0.333828i 0.946156 0.323711i \(-0.104931\pi\)
−0.753420 + 0.657539i \(0.771597\pi\)
\(674\) 51.4687i 1.98250i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 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
\(683\) 51.9294i 1.98702i 0.113728 + 0.993512i \(0.463721\pi\)
−0.113728 + 0.993512i \(0.536279\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.69694 8.13534i 0.179069 0.310157i
\(689\) 0 0
\(690\) 0 0
\(691\) 23.0000 + 39.8372i 0.874961 + 1.51548i 0.856804 + 0.515642i \(0.172447\pi\)
0.0181572 + 0.999835i \(0.494220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 18.6515 0.708002
\(695\) 0 0
\(696\) 0 0
\(697\) −1.84847 + 3.20164i −0.0700158 + 0.121271i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −26.2020 + 15.1278i −0.987527 + 0.570149i
\(705\) 0 0
\(706\) −10.3712 + 17.9634i −0.390324 + 0.676061i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −16.0000 −0.599625
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 34.2929 + 19.7990i 1.28158 + 0.739923i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 62.0908 35.8481i 2.31078 1.33413i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.3207 + 9.42274i −0.603642 + 0.348513i
\(732\) 0 0
\(733\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 54.2650i 1.99888i
\(738\) 0 0
\(739\) 53.7423 1.97694 0.988472 0.151403i \(-0.0483792\pi\)
0.988472 + 0.151403i \(0.0483792\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 60.6969 2.21930
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −32.2702 + 18.6312i −1.17210 + 0.676715i
\(759\) 0 0
\(760\) 0 0
\(761\) −9.79796 5.65685i −0.355176 0.205061i 0.311787 0.950152i \(-0.399073\pi\)
−0.666962 + 0.745091i \(0.732406\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 11.0000 19.0526i 0.396670 0.687053i −0.596643 0.802507i \(-0.703499\pi\)
0.993313 + 0.115454i \(0.0368323\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 25.6969 + 44.5084i 0.924853 + 1.60189i
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −48.2474 + 27.8557i −1.73198 + 0.999961i
\(777\) 0 0
\(778\) 0 0
\(779\) −3.33087 1.92308i −0.119341 0.0689014i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −25.0000 + 43.3013i −0.891154 + 1.54352i −0.0526599 + 0.998613i \(0.516770\pi\)
−0.838494 + 0.544911i \(0.816563\pi\)
\(788\) 0 0
\(789\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.2843i 1.00000i
\(801\) 0 0
\(802\) −35.3485 −1.24820
\(803\) 44.8610 25.9005i 1.58311 0.914009i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0