Properties

Label 147.4.g.a.68.1
Level $147$
Weight $4$
Character 147.68
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 68.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 147.68
Dual form 147.4.g.a.80.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-4.50000 + 2.59808i) q^{3} +(-4.00000 - 6.92820i) q^{4} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\) \(q+(-4.50000 + 2.59808i) q^{3} +(-4.00000 - 6.92820i) q^{4} +(13.5000 - 23.3827i) q^{9} +(36.0000 + 20.7846i) q^{12} +62.3538i q^{13} +(-32.0000 + 55.4256i) q^{16} +(135.000 + 77.9423i) q^{19} +(62.5000 + 108.253i) q^{25} +140.296i q^{27} +(-135.000 + 77.9423i) q^{31} -216.000 q^{36} +(55.0000 - 95.2628i) q^{37} +(-162.000 - 280.592i) q^{39} +520.000 q^{43} -332.554i q^{48} +(432.000 - 249.415i) q^{52} -810.000 q^{57} +(-810.000 - 467.654i) q^{61} +512.000 q^{64} +(440.000 + 762.102i) q^{67} +(-324.000 + 187.061i) q^{73} +(-562.500 - 324.760i) q^{75} -1247.08i q^{76} +(-442.000 + 765.566i) q^{79} +(-364.500 - 631.333i) q^{81} +(405.000 - 701.481i) q^{93} +1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + O(q^{10}) \) \( 2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + 72 q^{12} - 64 q^{16} + 270 q^{19} + 125 q^{25} - 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} + 864 q^{52} - 1620 q^{57} - 1620 q^{61} + 1024 q^{64} + 880 q^{67} - 648 q^{73} - 1125 q^{75} - 884 q^{79} - 729 q^{81} + 810 q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −4.50000 + 2.59808i −0.866025 + 0.500000i
\(4\) −4.00000 6.92820i −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
\(8\) 0 0
\(9\) 13.5000 23.3827i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 36.0000 + 20.7846i 0.866025 + 0.500000i
\(13\) 62.3538i 1.33030i 0.746712 + 0.665148i \(0.231631\pi\)
−0.746712 + 0.665148i \(0.768369\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −32.0000 + 55.4256i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 135.000 + 77.9423i 1.63006 + 0.941115i 0.984073 + 0.177766i \(0.0568871\pi\)
0.645986 + 0.763349i \(0.276446\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 62.5000 + 108.253i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 140.296i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −135.000 + 77.9423i −0.782152 + 0.451576i −0.837192 0.546908i \(-0.815805\pi\)
0.0550403 + 0.998484i \(0.482471\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −216.000 −1.00000
\(37\) 55.0000 95.2628i 0.244377 0.423273i −0.717579 0.696477i \(-0.754750\pi\)
0.961956 + 0.273204i \(0.0880833\pi\)
\(38\) 0 0
\(39\) −162.000 280.592i −0.665148 1.15207i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 520.000 1.84417 0.922084 0.386989i \(-0.126485\pi\)
0.922084 + 0.386989i \(0.126485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 332.554i 1.00000i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 432.000 249.415i 1.15207 0.665148i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −810.000 −1.88223
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −810.000 467.654i −1.70016 0.981589i −0.945584 0.325379i \(-0.894508\pi\)
−0.754578 0.656210i \(-0.772158\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 440.000 + 762.102i 0.802307 + 1.38964i 0.918094 + 0.396362i \(0.129728\pi\)
−0.115787 + 0.993274i \(0.536939\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −324.000 + 187.061i −0.519470 + 0.299916i −0.736718 0.676200i \(-0.763625\pi\)
0.217248 + 0.976117i \(0.430292\pi\)
\(74\) 0 0
\(75\) −562.500 324.760i −0.866025 0.500000i
\(76\) 1247.08i 1.88223i
\(77\) 0 0
\(78\) 0 0
\(79\) −442.000 + 765.566i −0.629480 + 1.09029i 0.358177 + 0.933654i \(0.383399\pi\)
−0.987656 + 0.156637i \(0.949935\pi\)
\(80\) 0 0
\(81\) −364.500 631.333i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 405.000 701.481i 0.451576 0.782152i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1371.78i 1.43591i 0.696088 + 0.717957i \(0.254922\pi\)
−0.696088 + 0.717957i \(0.745078\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 500.000 866.025i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 891.000 + 514.419i 0.852358 + 0.492109i 0.861446 0.507850i \(-0.169560\pi\)
−0.00908799 + 0.999959i \(0.502893\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 972.000 561.184i 0.866025 0.500000i
\(109\) −323.000 559.452i −0.283833 0.491613i 0.688493 0.725243i \(-0.258273\pi\)
−0.972325 + 0.233630i \(0.924939\pi\)
\(110\) 0 0
\(111\) 571.577i 0.488754i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1458.00 + 841.777i 1.15207 + 0.665148i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −665.500 + 1152.68i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 1080.00 + 623.538i 0.782152 + 0.451576i
\(125\) 0 0
\(126\) 0 0
\(127\) 380.000 0.265508 0.132754 0.991149i \(-0.457618\pi\)
0.132754 + 0.991149i \(0.457618\pi\)
\(128\) 0 0
\(129\) −2340.00 + 1351.00i −1.59710 + 0.922084i
\(130\) 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
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 2026.50i 1.23659i 0.785948 + 0.618293i \(0.212175\pi\)
−0.785948 + 0.618293i \(0.787825\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 864.000 + 1496.49i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −880.000 −0.488754
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 874.000 + 1513.81i 0.471027 + 0.815843i 0.999451 0.0331378i \(-0.0105500\pi\)
−0.528424 + 0.848981i \(0.677217\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1296.00 + 2244.74i −0.665148 + 1.15207i
\(157\) −702.000 + 405.300i −0.356852 + 0.206028i −0.667699 0.744432i \(-0.732721\pi\)
0.310847 + 0.950460i \(0.399387\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1700.00 2944.49i 0.816897 1.41491i −0.0910600 0.995845i \(-0.529026\pi\)
0.907957 0.419062i \(-0.137641\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1691.00 −0.769686
\(170\) 0 0
\(171\) 3645.00 2104.44i 1.63006 0.941115i
\(172\) −2080.00 3602.67i −0.922084 1.59710i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 3429.46i 1.40834i −0.710031 0.704171i \(-0.751319\pi\)
0.710031 0.704171i \(-0.248681\pi\)
\(182\) 0 0
\(183\) 4860.00 1.96318
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −2304.00 + 1330.22i −0.866025 + 0.500000i
\(193\) −575.000 995.929i −0.214453 0.371443i 0.738650 0.674089i \(-0.235464\pi\)
−0.953103 + 0.302646i \(0.902130\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1755.00 1013.25i 0.625169 0.360942i −0.153710 0.988116i \(-0.549122\pi\)
0.778879 + 0.627175i \(0.215789\pi\)
\(200\) 0 0
\(201\) −3960.00 2286.31i −1.38964 0.802307i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3456.00 1995.32i −1.15207 0.665148i
\(209\) 0 0
\(210\) 0 0
\(211\) −6032.00 −1.96806 −0.984028 0.178011i \(-0.943034\pi\)
−0.984028 + 0.178011i \(0.943034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 972.000 1683.55i 0.299916 0.519470i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5830.08i 1.75072i −0.483469 0.875362i \(-0.660623\pi\)
0.483469 0.875362i \(-0.339377\pi\)
\(224\) 0 0
\(225\) 3375.00 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 3240.00 + 5611.84i 0.941115 + 1.63006i
\(229\) −4590.00 2650.04i −1.32452 0.764714i −0.340076 0.940398i \(-0.610453\pi\)
−0.984447 + 0.175684i \(0.943786\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4593.40i 1.25896i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1080.00 + 623.538i −0.288668 + 0.166662i −0.637341 0.770582i \(-0.719966\pi\)
0.348673 + 0.937244i \(0.386632\pi\)
\(242\) 0 0
\(243\) 3280.50 + 1894.00i 0.866025 + 0.500000i
\(244\) 7482.46i 1.96318i
\(245\) 0 0
\(246\) 0 0
\(247\) −4860.00 + 8417.77i −1.25196 + 2.16846i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −2048.00 3547.24i −0.500000 0.866025i
\(257\) 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
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 3520.00 6096.82i 0.802307 1.38964i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 7695.00 + 4442.71i 1.72486 + 0.995850i 0.907935 + 0.419111i \(0.137658\pi\)
0.816928 + 0.576739i \(0.195675\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2015.00 + 3490.08i 0.437074 + 0.757035i 0.997462 0.0711951i \(-0.0226813\pi\)
−0.560388 + 0.828230i \(0.689348\pi\)
\(278\) 0 0
\(279\) 4208.88i 0.903151i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 6669.00 3850.35i 1.40082 0.808761i 0.406340 0.913722i \(-0.366805\pi\)
0.994476 + 0.104961i \(0.0334717\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2456.50 4254.78i 0.500000 0.866025i
\(290\) 0 0
\(291\) −3564.00 6173.03i −0.717957 1.24354i
\(292\) 2592.00 + 1496.49i 0.519470 + 0.299916i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 5196.15i 1.00000i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −8640.00 + 4988.31i −1.63006 + 0.941115i
\(305\) 0 0
\(306\) 0 0
\(307\) 1590.02i 0.295594i 0.989018 + 0.147797i \(0.0472182\pi\)
−0.989018 + 0.147797i \(0.952782\pi\)
\(308\) 0 0
\(309\) −5346.00 −0.984218
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 4104.00 + 2369.45i 0.741124 + 0.427888i 0.822478 0.568797i \(-0.192591\pi\)
−0.0813539 + 0.996685i \(0.525924\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 7072.00 1.25896
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −2916.00 + 5050.66i −0.500000 + 0.866025i
\(325\) −6750.00 + 3897.11i −1.15207 + 0.665148i
\(326\) 0 0
\(327\) 2907.00 + 1678.36i 0.491613 + 0.283833i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 496.000 859.097i 0.0823644 0.142659i −0.821901 0.569631i \(-0.807086\pi\)
0.904265 + 0.426971i \(0.140420\pi\)
\(332\) 0 0
\(333\) −1485.00 2572.10i −0.244377 0.423273i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4930.00 0.796897 0.398448 0.917191i \(-0.369549\pi\)
0.398448 + 0.917191i \(0.369549\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 5300.08i 0.812913i 0.913670 + 0.406456i \(0.133236\pi\)
−0.913670 + 0.406456i \(0.866764\pi\)
\(350\) 0 0
\(351\) −8748.00 −1.33030
\(352\) 0 0
\(353\) 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
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 8720.50 + 15104.3i 1.27140 + 2.20212i
\(362\) 0 0
\(363\) 6916.08i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11583.0 6687.45i 1.64749 0.951177i 0.669420 0.742884i \(-0.266543\pi\)
0.978066 0.208293i \(-0.0667908\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −6480.00 −0.903151
\(373\) −6175.00 + 10695.4i −0.857183 + 1.48469i 0.0174213 + 0.999848i \(0.494454\pi\)
−0.874605 + 0.484837i \(0.838879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8584.00 1.16340 0.581702 0.813402i \(-0.302387\pi\)
0.581702 + 0.813402i \(0.302387\pi\)
\(380\) 0 0
\(381\) −1710.00 + 987.269i −0.229937 + 0.132754i
\(382\) 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
\(386\) 0 0
\(387\) 7020.00 12159.0i 0.922084 1.59710i
\(388\) 9504.00 5487.14i 1.24354 0.717957i
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −13662.0 7887.76i −1.72714 0.997167i −0.901182 0.433441i \(-0.857299\pi\)
−0.825962 0.563726i \(-0.809368\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −8000.00 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −4860.00 8417.77i −0.600729 1.04049i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −12420.0 + 7170.69i −1.50154 + 0.866914i −0.501541 + 0.865134i \(0.667233\pi\)
−0.999998 + 0.00177990i \(0.999433\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8230.71i 0.984218i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5265.00 9119.25i −0.618293 1.07091i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 17138.0 1.98398 0.991989 0.126322i \(-0.0403172\pi\)
0.991989 + 0.126322i \(0.0403172\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) −7776.00 4489.48i −0.866025 0.500000i
\(433\) 17833.2i 1.97923i −0.143727 0.989617i \(-0.545909\pi\)
0.143727 0.989617i \(-0.454091\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2584.00 + 4475.62i −0.283833 + 0.491613i
\(437\) 0 0
\(438\) 0 0
\(439\) −9315.00 5378.02i −1.01271 0.584690i −0.100728 0.994914i \(-0.532117\pi\)
−0.911985 + 0.410224i \(0.865450\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 3960.00 2286.31i 0.423273 0.244377i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −7866.00 4541.44i −0.815843 0.471027i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6355.00 11007.2i 0.650491 1.12668i −0.332513 0.943099i \(-0.607897\pi\)
0.983004 0.183585i \(-0.0587702\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −19780.0 −1.98543 −0.992716 0.120482i \(-0.961556\pi\)
−0.992716 + 0.120482i \(0.961556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 13468.4i 1.33030i
\(469\) 0 0
\(470\) 0 0
\(471\) 2106.00 3647.70i 0.206028 0.356852i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 19485.6i 1.88223i
\(476\) 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
\(481\) 5940.00 + 3429.46i 0.563078 + 0.325093i
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 10450.0 + 18099.9i 0.972351 + 1.68416i 0.688415 + 0.725317i \(0.258307\pi\)
0.283936 + 0.958843i \(0.408360\pi\)
\(488\) 0 0
\(489\) 17666.9i 1.63379i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 9976.61i 0.903151i
\(497\) 0 0
\(498\) 0 0
\(499\) −7568.00 + 13108.2i −0.678938 + 1.17596i 0.296363 + 0.955075i \(0.404226\pi\)
−0.975301 + 0.220880i \(0.929107\pi\)
\(500\) 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
\(506\) 0 0
\(507\) 7609.50 4393.35i 0.666568 0.384843i
\(508\) −1520.00 2632.72i −0.132754 0.229937i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −10935.0 + 18940.0i −0.941115 + 1.63006i
\(514\) 0 0
\(515\) 0 0
\(516\) 18720.0 + 10808.0i 1.59710 + 0.922084i
\(517\) 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
\(523\) 17901.0 + 10335.1i 1.49667 + 0.864100i 0.999993 0.00383755i \(-0.00122153\pi\)
0.496673 + 0.867938i \(0.334555\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6083.50 10536.9i −0.500000 0.866025i
\(530\) 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
\(541\) 11339.0 19639.7i 0.901112 1.56077i 0.0750596 0.997179i \(-0.476085\pi\)
0.826053 0.563593i \(-0.190581\pi\)
\(542\) 0 0
\(543\) 8910.00 + 15432.6i 0.704171 + 1.21966i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1640.00 0.128193 0.0640963 0.997944i \(-0.479584\pi\)
0.0640963 + 0.997944i \(0.479584\pi\)
\(548\) 0 0
\(549\) −21870.0 + 12626.7i −1.70016 + 0.981589i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 14040.0 8106.00i 1.07091 0.618293i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 32424.0i 2.45329i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 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
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −11656.0 20188.8i −0.854270 1.47964i −0.877320 0.479905i \(-0.840671\pi\)
0.0230498 0.999734i \(-0.492662\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 6912.00 11971.9i 0.500000 0.866025i
\(577\) −18468.0 + 10662.5i −1.33247 + 0.769300i −0.985677 0.168644i \(-0.946061\pi\)
−0.346789 + 0.937943i \(0.612728\pi\)
\(578\) 0 0
\(579\) 5175.00 + 2987.79i 0.371443 + 0.214453i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −24300.0 −1.69994
\(590\) 0 0
\(591\) 0 0
\(592\) 3520.00 + 6096.82i 0.244377 + 0.423273i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5265.00 + 9119.25i −0.360942 + 0.625169i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 3117.69i 0.211603i 0.994387 + 0.105801i \(0.0337408\pi\)
−0.994387 + 0.105801i \(0.966259\pi\)
\(602\) 0 0
\(603\) 23760.0 1.60461
\(604\) 6992.00 12110.5i 0.471027 0.815843i
\(605\) 0 0
\(606\) 0 0
\(607\) 8073.00 + 4660.95i 0.539824 + 0.311667i 0.745007 0.667056i \(-0.232446\pi\)
−0.205184 + 0.978723i \(0.565779\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8695.00 15060.2i −0.572900 0.992292i −0.996266 0.0863334i \(-0.972485\pi\)
0.423366 0.905959i \(-0.360848\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −13365.0 + 7716.29i −0.867827 + 0.501040i −0.866625 0.498959i \(-0.833716\pi\)
−0.00120126 + 0.999999i \(0.500382\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 20736.0 1.33030
\(625\) −7812.50 + 13531.6i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 5616.00 + 3242.40i 0.356852 + 0.206028i
\(629\) 0 0
\(630\) 0 0
\(631\) 1892.00 0.119365 0.0596825 0.998217i \(-0.480991\pi\)
0.0596825 + 0.998217i \(0.480991\pi\)
\(632\) 0 0
\(633\) 27144.0 15671.6i 1.70439 0.984028i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 29836.3i 1.82991i −0.403561 0.914953i \(-0.632228\pi\)
0.403561 0.914953i \(-0.367772\pi\)
\(644\) 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
\(652\) −27200.0 −1.63379
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10101.3i 0.599833i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 23490.0 13562.0i 1.38223 0.798032i 0.389808 0.920896i \(-0.372541\pi\)
0.992423 + 0.122864i \(0.0392080\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 15147.0 + 26235.4i 0.875362 + 1.51617i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −24050.0 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(674\) 0 0
\(675\) −15187.5 + 8768.51i −0.866025 + 0.500000i
\(676\) 6764.00 + 11715.6i 0.384843 + 0.666568i
\(677\) 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
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) −29160.0 16835.5i −1.63006 0.941115i
\(685\) 0 0
\(686\) 0 0
\(687\) 27540.0 1.52943
\(688\) −16640.0 + 28821.3i −0.922084 + 1.59710i
\(689\) 0 0
\(690\) 0 0
\(691\) −28215.0 16289.9i −1.55333 0.896814i −0.997868 0.0652705i \(-0.979209\pi\)
−0.555460 0.831543i \(-0.687458\pi\)
\(692\) 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
\(703\) 14850.0 8573.65i 0.796698 0.459974i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18073.0 31303.4i 0.957328 1.65814i 0.228381 0.973572i \(-0.426657\pi\)
0.728948 0.684569i \(-0.240010\pi\)
\(710\) 0 0
\(711\) 11934.0 + 20670.3i 0.629480 + 1.09029i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3240.00 5611.84i 0.166662 0.288668i
\(724\) −23760.0 + 13717.8i −1.21966 + 0.704171i
\(725\) 0 0
\(726\) 0 0
\(727\) 37692.9i 1.92290i −0.274971 0.961452i \(-0.588668\pi\)
0.274971 0.961452i \(-0.411332\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −19440.0 33671.1i −0.981589 1.70016i
\(733\) −31806.0 18363.2i −1.60270 0.925321i −0.990944 0.134277i \(-0.957129\pi\)
−0.611759 0.791044i \(-0.709538\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −15688.0 27172.4i −0.780910 1.35258i −0.931412 0.363966i \(-0.881422\pi\)
0.150502 0.988610i \(-0.451911\pi\)
\(740\) 0 0
\(741\) 50506.6i 2.50392i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −11726.0 + 20310.0i −0.569757 + 0.986849i 0.426832 + 0.904331i \(0.359630\pi\)
−0.996590 + 0.0825179i \(0.973704\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41470.0 1.99109 0.995543 0.0943039i \(-0.0300625\pi\)
0.995543 + 0.0943039i \(0.0300625\pi\)
\(758\) 0 0
\(759\) 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
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 18432.0 + 10641.7i 0.866025 + 0.500000i
\(769\) 42400.6i 1.98830i 0.107995 + 0.994151i \(0.465557\pi\)
−0.107995 + 0.994151i \(0.534443\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4600.00 + 7967.43i −0.214453 + 0.371443i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −16875.0 9742.79i −0.782152 0.451576i
\(776\) 0 0
\(777\) 0 0
\(778\) 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
\(787\) 7047.00 4068.59i 0.319185 0.184281i −0.331844 0.943334i \(-0.607671\pi\)
0.651029 + 0.759053i \(0.274338\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 29160.0 50506.6i 1.30580 2.26172i
\(794\) 0 0
\(795\) 0 0
\(796\) −14040.0 8106.00i −0.625169 0.360942i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0