Properties

Label 21.4.c.a.20.2
Level $21$
Weight $4$
Character 21.20
Analytic conductor $1.239$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 20.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.20
Dual form 21.4.c.a.20.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.19615i q^{3} +8.00000 q^{4} +(-10.0000 - 15.5885i) q^{7} -27.0000 q^{9} +O(q^{10})\) \(q+5.19615i q^{3} +8.00000 q^{4} +(-10.0000 - 15.5885i) q^{7} -27.0000 q^{9} +41.5692i q^{12} -62.3538i q^{13} +64.0000 q^{16} +155.885i q^{19} +(81.0000 - 51.9615i) q^{21} -125.000 q^{25} -140.296i q^{27} +(-80.0000 - 124.708i) q^{28} +155.885i q^{31} -216.000 q^{36} -110.000 q^{37} +324.000 q^{39} +520.000 q^{43} +332.554i q^{48} +(-143.000 + 311.769i) q^{49} -498.831i q^{52} -810.000 q^{57} -935.307i q^{61} +(270.000 + 420.888i) q^{63} +512.000 q^{64} -880.000 q^{67} +374.123i q^{73} -649.519i q^{75} +1247.08i q^{76} +884.000 q^{79} +729.000 q^{81} +(648.000 - 415.692i) q^{84} +(-972.000 + 623.538i) q^{91} -810.000 q^{93} -1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{4} - 20 q^{7} - 54 q^{9} + O(q^{10}) \) \( 2 q + 16 q^{4} - 20 q^{7} - 54 q^{9} + 128 q^{16} + 162 q^{21} - 250 q^{25} - 160 q^{28} - 432 q^{36} - 220 q^{37} + 648 q^{39} + 1040 q^{43} - 286 q^{49} - 1620 q^{57} + 540 q^{63} + 1024 q^{64} - 1760 q^{67} + 1768 q^{79} + 1458 q^{81} + 1296 q^{84} - 1944 q^{91} - 1620 q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 5.19615i 1.00000i
\(4\) 8.00000 1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −10.0000 15.5885i −0.539949 0.841698i
\(8\) 0 0
\(9\) −27.0000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 41.5692i 1.00000i
\(13\) 62.3538i 1.33030i −0.746712 0.665148i \(-0.768369\pi\)
0.746712 0.665148i \(-0.231631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 155.885i 1.88223i 0.338086 + 0.941115i \(0.390220\pi\)
−0.338086 + 0.941115i \(0.609780\pi\)
\(20\) 0 0
\(21\) 81.0000 51.9615i 0.841698 0.539949i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 140.296i 1.00000i
\(28\) −80.0000 124.708i −0.539949 0.841698i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 155.885i 0.903151i 0.892233 + 0.451576i \(0.149138\pi\)
−0.892233 + 0.451576i \(0.850862\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −216.000 −1.00000
\(37\) −110.000 −0.488754 −0.244377 0.969680i \(-0.578583\pi\)
−0.244377 + 0.969680i \(0.578583\pi\)
\(38\) 0 0
\(39\) 324.000 1.33030
\(40\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 332.554i 1.00000i
\(49\) −143.000 + 311.769i −0.416910 + 0.908948i
\(50\) 0 0
\(51\) 0 0
\(52\) 498.831i 1.33030i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −810.000 −1.88223
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 935.307i 1.96318i −0.191006 0.981589i \(-0.561175\pi\)
0.191006 0.981589i \(-0.438825\pi\)
\(62\) 0 0
\(63\) 270.000 + 420.888i 0.539949 + 0.841698i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −880.000 −1.60461 −0.802307 0.596912i \(-0.796394\pi\)
−0.802307 + 0.596912i \(0.796394\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 374.123i 0.599833i 0.953966 + 0.299916i \(0.0969588\pi\)
−0.953966 + 0.299916i \(0.903041\pi\)
\(74\) 0 0
\(75\) 649.519i 1.00000i
\(76\) 1247.08i 1.88223i
\(77\) 0 0
\(78\) 0 0
\(79\) 884.000 1.25896 0.629480 0.777017i \(-0.283268\pi\)
0.629480 + 0.777017i \(0.283268\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 648.000 415.692i 0.841698 0.539949i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −972.000 + 623.538i −1.11971 + 0.718292i
\(92\) 0 0
\(93\) −810.000 −0.903151
\(94\) 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
\(100\) −1000.00 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1028.84i 0.984218i 0.870534 + 0.492109i \(0.163774\pi\)
−0.870534 + 0.492109i \(0.836226\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1122.37i 1.00000i
\(109\) 646.000 0.567666 0.283833 0.958874i \(-0.408394\pi\)
0.283833 + 0.958874i \(0.408394\pi\)
\(110\) 0 0
\(111\) 571.577i 0.488754i
\(112\) −640.000 997.661i −0.539949 0.841698i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1683.55i 1.33030i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1331.00 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1247.08i 0.903151i
\(125\) 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
\(129\) 2702.00i 1.84417i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2430.00 1558.85i 1.58427 1.01631i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 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
\(144\) −1728.00 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −1620.00 743.050i −0.908948 0.416910i
\(148\) −880.000 −0.488754
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1748.00 −0.942054 −0.471027 0.882119i \(-0.656117\pi\)
−0.471027 + 0.882119i \(0.656117\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 2592.00 1.33030
\(157\) 810.600i 0.412057i 0.978546 + 0.206028i \(0.0660539\pi\)
−0.978546 + 0.206028i \(0.933946\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3400.00 −1.63379 −0.816897 0.576783i \(-0.804308\pi\)
−0.816897 + 0.576783i \(0.804308\pi\)
\(164\) 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
\(171\) 4208.88i 1.88223i
\(172\) 4160.00 1.84417
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 1250.00 + 1948.56i 0.539949 + 0.841698i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 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
\(189\) −2187.00 + 1402.96i −0.841698 + 0.539949i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2660.43i 1.00000i
\(193\) 1150.00 0.428906 0.214453 0.976734i \(-0.431203\pi\)
0.214453 + 0.976734i \(0.431203\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1144.00 + 2494.15i −0.416910 + 0.908948i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 2026.50i 0.721883i −0.932588 0.360942i \(-0.882455\pi\)
0.932588 0.360942i \(-0.117545\pi\)
\(200\) 0 0
\(201\) 4572.61i 1.60461i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 3990.65i 1.33030i
\(209\) 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
\(217\) 2430.00 1558.85i 0.760180 0.487656i
\(218\) 0 0
\(219\) −1944.00 −0.599833
\(220\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −6480.00 −1.88223
\(229\) 5300.08i 1.52943i −0.644370 0.764714i \(-0.722880\pi\)
0.644370 0.764714i \(-0.277120\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 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
\(241\) 1247.08i 0.333325i 0.986014 + 0.166662i \(0.0532990\pi\)
−0.986014 + 0.166662i \(0.946701\pi\)
\(242\) 0 0
\(243\) 3788.00i 1.00000i
\(244\) 7482.46i 1.96318i
\(245\) 0 0
\(246\) 0 0
\(247\) 9720.00 2.50392
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 2160.00 + 3367.11i 0.539949 + 0.841698i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 1100.00 + 1714.73i 0.263902 + 0.411383i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −7040.00 −1.60461
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8885.42i 1.99170i 0.0910064 + 0.995850i \(0.470992\pi\)
−0.0910064 + 0.995850i \(0.529008\pi\)
\(272\) 0 0
\(273\) −3240.00 5050.66i −0.718292 1.11971i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4030.00 −0.874149 −0.437074 0.899425i \(-0.643985\pi\)
−0.437074 + 0.899425i \(0.643985\pi\)
\(278\) 0 0
\(279\) 4208.88i 0.903151i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 7700.70i 1.61752i −0.588137 0.808761i \(-0.700138\pi\)
0.588137 0.808761i \(-0.299862\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) 7128.00 1.43591
\(292\) 2992.98i 0.599833i
\(293\) 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
\(301\) −5200.00 8106.00i −0.995758 1.55223i
\(302\) 0 0
\(303\) 0 0
\(304\) 9976.61i 1.88223i
\(305\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 4738.89i 0.855776i 0.903832 + 0.427888i \(0.140742\pi\)
−0.903832 + 0.427888i \(0.859258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 7072.00 1.25896
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5832.00 1.00000
\(325\) 7794.23i 1.33030i
\(326\) 0 0
\(327\) 3356.71i 0.567666i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −992.000 −0.164729 −0.0823644 0.996602i \(-0.526247\pi\)
−0.0823644 + 0.996602i \(0.526247\pi\)
\(332\) 0 0
\(333\) 2970.00 0.488754
\(334\) 0 0
\(335\) 0 0
\(336\) 5184.00 3325.54i 0.841698 0.539949i
\(337\) 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
\(343\) 6290.00 888.542i 0.990169 0.139874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −17441.0 −2.54279
\(362\) 0 0
\(363\) 6916.08i 1.00000i
\(364\) −7776.00 + 4988.31i −1.11971 + 0.718292i
\(365\) 0 0
\(366\) 0 0
\(367\) 13374.9i 1.90235i −0.308646 0.951177i \(-0.599876\pi\)
0.308646 0.951177i \(-0.400124\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −6480.00 −0.903151
\(373\) 12350.0 1.71437 0.857183 0.515011i \(-0.172212\pi\)
0.857183 + 0.515011i \(0.172212\pi\)
\(374\) 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
\(381\) 1974.54i 0.265508i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −14040.0 −1.84417
\(388\) 10974.3i 1.43591i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 15775.5i 1.99433i −0.0752196 0.997167i \(-0.523966\pi\)
0.0752196 0.997167i \(-0.476034\pi\)
\(398\) 0 0
\(399\) 8100.00 + 12626.7i 1.01631 + 1.58427i
\(400\) −8000.00 −1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 9720.00 1.20146
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14341.4i 1.73383i 0.498458 + 0.866914i \(0.333900\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8230.71i 0.984218i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10530.0 1.23659
\(418\) 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
\(427\) −14580.0 + 9353.07i −1.65240 + 1.06002i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 8978.95i 1.00000i
\(433\) 17833.2i 1.97923i 0.143727 + 0.989617i \(0.454091\pi\)
−0.143727 + 0.989617i \(0.545909\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5168.00 0.567666
\(437\) 0 0
\(438\) 0 0
\(439\) 10756.0i 1.16938i −0.811257 0.584690i \(-0.801216\pi\)
0.811257 0.584690i \(-0.198784\pi\)
\(440\) 0 0
\(441\) 3861.00 8417.77i 0.416910 0.908948i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 4572.61i 0.488754i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −5120.00 7981.29i −0.539949 0.841698i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 9082.87i 0.942054i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12710.0 −1.30098 −0.650491 0.759514i \(-0.725437\pi\)
−0.650491 + 0.759514i \(0.725437\pi\)
\(458\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 13468.4i 1.33030i
\(469\) 8800.00 + 13717.8i 0.866410 + 1.35060i
\(470\) 0 0
\(471\) −4212.00 −0.412057
\(472\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 6858.92i 0.650187i
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −20900.0 −1.94470 −0.972351 0.233526i \(-0.924974\pi\)
−0.972351 + 0.233526i \(0.924974\pi\)
\(488\) 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
\(499\) 15136.0 1.35788 0.678938 0.734195i \(-0.262440\pi\)
0.678938 + 0.734195i \(0.262440\pi\)
\(500\) 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
\(507\) 8786.69i 0.769686i
\(508\) 3040.00 0.265508
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 5832.00 3741.23i 0.504878 0.323879i
\(512\) 0 0
\(513\) 21870.0 1.88223
\(514\) 0 0
\(515\) 0 0
\(516\) 21616.0i 1.84417i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 20670.3i 1.72820i 0.503320 + 0.864100i \(0.332112\pi\)
−0.503320 + 0.864100i \(0.667888\pi\)
\(524\) 0 0
\(525\) −10125.0 + 6495.19i −0.841698 + 0.539949i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12167.0 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 19440.0 12470.8i 1.58427 1.01631i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22678.0 −1.80222 −0.901112 0.433586i \(-0.857248\pi\)
−0.901112 + 0.433586i \(0.857248\pi\)
\(542\) 0 0
\(543\) −17820.0 −1.40834
\(544\) 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
\(549\) 25253.3i 1.96318i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8840.00 13780.2i −0.679774 1.05966i
\(554\) 0 0
\(555\) 0 0
\(556\) 16212.0i 1.23659i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 32424.0i 2.45329i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7290.00 11364.0i −0.539949 0.841698i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 23312.0 1.70854 0.854270 0.519829i \(-0.174004\pi\)
0.854270 + 0.519829i \(0.174004\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −13824.0 −1.00000
\(577\) 21325.0i 1.53860i 0.638888 + 0.769300i \(0.279395\pi\)
−0.638888 + 0.769300i \(0.720605\pi\)
\(578\) 0 0
\(579\) 5975.58i 0.428906i
\(580\) 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 \(0.5\pi\)
\(588\) −12960.0 5944.40i −0.908948 0.416910i
\(589\) −24300.0 −1.69994
\(590\) 0 0
\(591\) 0 0
\(592\) −7040.00 −0.488754
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10530.0 0.721883
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 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
\(604\) −13984.0 −0.942054
\(605\) 0 0
\(606\) 0 0
\(607\) 9321.90i 0.623335i 0.950191 + 0.311667i \(0.100887\pi\)
−0.950191 + 0.311667i \(0.899113\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17390.0 1.14580 0.572900 0.819625i \(-0.305818\pi\)
0.572900 + 0.819625i \(0.305818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 15432.6i 1.00208i 0.865424 + 0.501040i \(0.167049\pi\)
−0.865424 + 0.501040i \(0.832951\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 20736.0 1.33030
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 6484.80i 0.412057i
\(629\) 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
\(633\) 31343.2i 1.96806i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 19440.0 + 8916.60i 1.20917 + 0.554613i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8100.00 + 12626.7i 0.487656 + 0.760180i
\(652\) −27200.0 −1.63379
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 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
\(661\) 27123.9i 1.59606i −0.602615 0.798032i \(-0.705875\pi\)
0.602615 0.798032i \(-0.294125\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −30294.0 −1.75072
\(670\) 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
\(675\) 17537.0i 1.00000i
\(676\) −13528.0 −0.769686
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −21384.0 + 13717.8i −1.20860 + 0.775320i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 33671.1i 1.88223i
\(685\) 0 0
\(686\) 0 0
\(687\) 27540.0 1.52943
\(688\) 33280.0 1.84417
\(689\) 0 0
\(690\) 0 0
\(691\) 32579.9i 1.79363i −0.442408 0.896814i \(-0.645876\pi\)
0.442408 0.896814i \(-0.354124\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 10000.0 + 15588.5i 0.539949 + 0.841698i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 17147.3i 0.919947i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36146.0 −1.91466 −0.957328 0.289003i \(-0.906676\pi\)
−0.957328 + 0.289003i \(0.906676\pi\)
\(710\) 0 0
\(711\) −23868.0 −1.25896
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16038.0 10288.4i 0.828414 0.531428i
\(722\) 0 0
\(723\) −6480.00 −0.333325
\(724\) 27435.7i 1.40834i
\(725\) 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
\(732\) 38880.0 1.96318
\(733\) 36726.4i 1.85064i −0.379184 0.925321i \(-0.623795\pi\)
0.379184 0.925321i \(-0.376205\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 31376.0 1.56182 0.780910 0.624644i \(-0.214756\pi\)
0.780910 + 0.624644i \(0.214756\pi\)
\(740\) 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
\(751\) 23452.0 1.13951 0.569757 0.821813i \(-0.307037\pi\)
0.569757 + 0.821813i \(0.307037\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −17496.0 + 11223.7i −0.841698 + 0.539949i
\(757\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −6460.00 10070.1i −0.306511 0.477803i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 21283.4i 1.00000i
\(769\) 42400.6i 1.98830i −0.107995 0.994151i \(-0.534443\pi\)
0.107995 0.994151i \(-0.465557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9200.00 0.428906
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 19485.6i 0.903151i
\(776\) 0 0
\(777\) −8910.00 + 5715.77i −0.411383 + 0.263902i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −9152.00 + 19953.2i −0.416910 + 0.908948i
\(785\) 0 0
\(786\) 0 0
\(787\) 8137.17i 0.368563i −0.982874 0.184281i \(-0.941004\pi\)
0.982874 0.184281i \(-0.0589958\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −58320.0 −2.61161
\(794\) 0 0
\(795\) 0 0
\(796\) 16212.0i 0.721883i
\(797\) 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
\(804\) 36580.9i 1.60461i
\(805\) 0 0
\(806\) 0 0
\(807\)