Properties

Label 147.4.g.b.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.b.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 \(-0.768369\pi\)
0.746712 0.665148i \(-0.231631\pi\)
\(14\) 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 \(-0.943113\pi\)
−0.645986 0.763349i \(-0.723554\pi\)
\(20\) 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 −0.0550403 0.998484i \(-0.517529\pi\)
0.837192 + 0.546908i \(0.184195\pi\)
\(32\) 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 \(0.105492\pi\)
0.754578 + 0.656210i \(0.227842\pi\)
\(62\) 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 −0.217248 0.976117i \(-0.569708\pi\)
0.736718 + 0.676200i \(0.236375\pi\)
\(74\) 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 \(-0.745078\pi\)
0.696088 0.717957i \(-0.254922\pi\)
\(98\) 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 0.00908799 0.999959i \(-0.497107\pi\)
−0.861446 + 0.507850i \(0.830440\pi\)
\(104\) 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 \(-0.787825\pi\)
0.785948 0.618293i \(-0.212175\pi\)
\(140\) 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 −0.310847 0.950460i \(-0.600613\pi\)
0.667699 + 0.744432i \(0.267279\pi\)
\(158\) 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 \(0.248681\pi\)
−0.710031 + 0.704171i \(0.751319\pi\)
\(182\) 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 −0.778879 0.627175i \(-0.784211\pi\)
0.153710 + 0.988116i \(0.450878\pi\)
\(200\) 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 \(0.339377\pi\)
−0.483469 + 0.875362i \(0.660623\pi\)
\(224\) 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 0.984447 0.175684i \(-0.0562138\pi\)
0.340076 + 0.940398i \(0.389547\pi\)
\(230\) 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 −0.348673 0.937244i \(-0.613368\pi\)
0.637341 + 0.770582i \(0.280034\pi\)
\(242\) 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 \(-0.862342\pi\)
−0.816928 0.576739i \(-0.804325\pi\)
\(272\) 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 −0.994476 0.104961i \(-0.966528\pi\)
−0.406340 + 0.913722i \(0.633195\pi\)
\(284\) 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 \(-0.952782\pi\)
0.989018 0.147797i \(-0.0472182\pi\)
\(308\) 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 0.0813539 0.996685i \(-0.474076\pi\)
−0.822478 + 0.568797i \(0.807409\pi\)
\(314\) 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 \(-0.866764\pi\)
0.913670 0.406456i \(-0.133236\pi\)
\(350\) 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 \(0.733457\pi\)
−0.978066 + 0.208293i \(0.933209\pi\)
\(368\) 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 \(0.142701\pi\)
0.825962 + 0.563726i \(0.190632\pi\)
\(398\) 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 \(-0.332767\pi\)
0.999998 0.00177990i \(-0.000566559\pi\)
\(410\) 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 \(0.454091\pi\)
−0.143727 + 0.989617i \(0.545909\pi\)
\(434\) 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 0.911985 0.410224i \(-0.134550\pi\)
0.100728 + 0.994914i \(0.467883\pi\)
\(440\) 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 −0.496673 0.867938i \(-0.665445\pi\)
−0.999993 + 0.00383755i \(0.998778\pi\)
\(524\) 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 0.346789 0.937943i \(-0.387272\pi\)
0.985677 + 0.168644i \(0.0539387\pi\)
\(578\) 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 \(-0.966259\pi\)
0.994387 0.105801i \(-0.0337408\pi\)
\(602\) 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 0.205184 0.978723i \(-0.434221\pi\)
−0.745007 + 0.667056i \(0.767554\pi\)
\(608\) 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 0.00120126 0.999999i \(-0.499618\pi\)
0.866625 + 0.498959i \(0.166284\pi\)
\(620\) 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 \(0.367772\pi\)
−0.403561 + 0.914953i \(0.632228\pi\)
\(644\) 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 −0.992423 0.122864i \(-0.960792\pi\)
−0.389808 + 0.920896i \(0.627459\pi\)
\(662\) 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 \(0.0207910\pi\)
0.555460 + 0.831543i \(0.312542\pi\)
\(692\) 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 \(0.411332\pi\)
−0.274971 + 0.961452i \(0.588668\pi\)
\(728\) 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 \(0.0428712\pi\)
0.611759 + 0.791044i \(0.290462\pi\)
\(734\) 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 \(-0.534443\pi\)
0.107995 0.994151i \(-0.465557\pi\)
\(770\) 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 −0.651029 0.759053i \(-0.725662\pi\)
0.331844 + 0.943334i \(0.392329\pi\)
\(788\) 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 0
\(803\) 0 0