Properties

Label 3024.2.b.u.1567.4
Level $3024$
Weight $2$
Character 3024.1567
Analytic conductor $24.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4161798144.13
Defining polynomial: \(x^{8} + 6 x^{6} + 29 x^{4} + 42 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.4
Root \(0.629640 + 1.09057i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.u.1567.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.18114i q^{5} +(-1.00000 + 2.44949i) q^{7} +O(q^{10})\) \(q-2.18114i q^{5} +(-1.00000 + 2.44949i) q^{7} -5.26573i q^{11} +1.01461i q^{13} -3.08459i q^{17} -3.24264 q^{19} -0.903457i q^{23} +0.242641 q^{25} +5.34267 q^{29} -5.24264 q^{31} +(5.34267 + 2.18114i) q^{35} +1.00000 q^{37} +8.35032i q^{41} +4.47871i q^{43} -12.8984 q^{47} +(-5.00000 - 4.89898i) q^{49} -7.55568 q^{53} -11.4853 q^{55} -5.34267 q^{59} -3.46410i q^{61} +2.21301 q^{65} -3.88437i q^{67} -2.18114i q^{71} -14.2767i q^{73} +(12.8984 + 5.26573i) q^{77} +9.37769i q^{79} +12.8984 q^{83} -6.72792 q^{85} -3.98805i q^{89} +(-2.48528 - 1.01461i) q^{91} +7.07264i q^{95} -6.33386i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} + 8q^{19} - 32q^{25} - 8q^{31} + 8q^{37} - 40q^{49} - 24q^{55} + 48q^{85} + 48q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.18114i 0.975434i −0.873002 0.487717i \(-0.837830\pi\)
0.873002 0.487717i \(-0.162170\pi\)
\(6\) 0 0
\(7\) −1.00000 + 2.44949i −0.377964 + 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.26573i 1.58768i −0.608128 0.793839i \(-0.708079\pi\)
0.608128 0.793839i \(-0.291921\pi\)
\(12\) 0 0
\(13\) 1.01461i 0.281403i 0.990052 + 0.140701i \(0.0449357\pi\)
−0.990052 + 0.140701i \(0.955064\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.08459i 0.748124i −0.927404 0.374062i \(-0.877965\pi\)
0.927404 0.374062i \(-0.122035\pi\)
\(18\) 0 0
\(19\) −3.24264 −0.743913 −0.371956 0.928250i \(-0.621313\pi\)
−0.371956 + 0.928250i \(0.621313\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.903457i 0.188384i −0.995554 0.0941919i \(-0.969973\pi\)
0.995554 0.0941919i \(-0.0300267\pi\)
\(24\) 0 0
\(25\) 0.242641 0.0485281
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.34267 0.992109 0.496055 0.868291i \(-0.334782\pi\)
0.496055 + 0.868291i \(0.334782\pi\)
\(30\) 0 0
\(31\) −5.24264 −0.941606 −0.470803 0.882238i \(-0.656036\pi\)
−0.470803 + 0.882238i \(0.656036\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.34267 + 2.18114i 0.903077 + 0.368679i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.35032i 1.30410i 0.758175 + 0.652051i \(0.226091\pi\)
−0.758175 + 0.652051i \(0.773909\pi\)
\(42\) 0 0
\(43\) 4.47871i 0.682997i 0.939882 + 0.341499i \(0.110934\pi\)
−0.939882 + 0.341499i \(0.889066\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.8984 −1.88142 −0.940709 0.339214i \(-0.889839\pi\)
−0.940709 + 0.339214i \(0.889839\pi\)
\(48\) 0 0
\(49\) −5.00000 4.89898i −0.714286 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.55568 −1.03785 −0.518926 0.854819i \(-0.673668\pi\)
−0.518926 + 0.854819i \(0.673668\pi\)
\(54\) 0 0
\(55\) −11.4853 −1.54868
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.34267 −0.695557 −0.347778 0.937577i \(-0.613064\pi\)
−0.347778 + 0.937577i \(0.613064\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i −0.975100 0.221766i \(-0.928818\pi\)
0.975100 0.221766i \(-0.0711822\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.21301 0.274490
\(66\) 0 0
\(67\) 3.88437i 0.474551i −0.971442 0.237276i \(-0.923746\pi\)
0.971442 0.237276i \(-0.0762544\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.18114i 0.258853i −0.991589 0.129427i \(-0.958686\pi\)
0.991589 0.129427i \(-0.0413137\pi\)
\(72\) 0 0
\(73\) 14.2767i 1.67096i −0.549522 0.835479i \(-0.685190\pi\)
0.549522 0.835479i \(-0.314810\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.8984 + 5.26573i 1.46990 + 0.600086i
\(78\) 0 0
\(79\) 9.37769i 1.05507i 0.849532 + 0.527536i \(0.176884\pi\)
−0.849532 + 0.527536i \(0.823116\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.8984 1.41578 0.707889 0.706324i \(-0.249648\pi\)
0.707889 + 0.706324i \(0.249648\pi\)
\(84\) 0 0
\(85\) −6.72792 −0.729746
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.98805i 0.422732i −0.977407 0.211366i \(-0.932209\pi\)
0.977407 0.211366i \(-0.0677913\pi\)
\(90\) 0 0
\(91\) −2.48528 1.01461i −0.260528 0.106360i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.07264i 0.725638i
\(96\) 0 0
\(97\) 6.33386i 0.643106i −0.946892 0.321553i \(-0.895795\pi\)
0.946892 0.321553i \(-0.104205\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.08459i 0.306929i 0.988154 + 0.153464i \(0.0490430\pi\)
−0.988154 + 0.153464i \(0.950957\pi\)
\(102\) 0 0
\(103\) −13.7279 −1.35265 −0.676326 0.736602i \(-0.736429\pi\)
−0.676326 + 0.736602i \(0.736429\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.08459i 0.298199i 0.988822 + 0.149099i \(0.0476375\pi\)
−0.988822 + 0.149099i \(0.952363\pi\)
\(108\) 0 0
\(109\) −2.75736 −0.264107 −0.132054 0.991243i \(-0.542157\pi\)
−0.132054 + 0.991243i \(0.542157\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.2410 −1.71597 −0.857986 0.513674i \(-0.828284\pi\)
−0.857986 + 0.513674i \(0.828284\pi\)
\(114\) 0 0
\(115\) −1.97056 −0.183756
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.55568 + 3.08459i 0.692628 + 0.282764i
\(120\) 0 0
\(121\) −16.7279 −1.52072
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.4349i 1.02277i
\(126\) 0 0
\(127\) 4.47871i 0.397422i 0.980058 + 0.198711i \(0.0636754\pi\)
−0.980058 + 0.198711i \(0.936325\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.34267 −0.466792 −0.233396 0.972382i \(-0.574984\pi\)
−0.233396 + 0.972382i \(0.574984\pi\)
\(132\) 0 0
\(133\) 3.24264 7.94282i 0.281173 0.688729i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.4540 −1.74751 −0.873753 0.486370i \(-0.838321\pi\)
−0.873753 + 0.486370i \(0.838321\pi\)
\(138\) 0 0
\(139\) −18.4853 −1.56790 −0.783951 0.620823i \(-0.786798\pi\)
−0.783951 + 0.620823i \(0.786798\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.34267 0.446777
\(144\) 0 0
\(145\) 11.6531i 0.967738i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.55568 0.618985 0.309493 0.950902i \(-0.399841\pi\)
0.309493 + 0.950902i \(0.399841\pi\)
\(150\) 0 0
\(151\) 11.2328i 0.914115i 0.889437 + 0.457058i \(0.151097\pi\)
−0.889437 + 0.457058i \(0.848903\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.4349i 0.918475i
\(156\) 0 0
\(157\) 5.91359i 0.471956i −0.971758 0.235978i \(-0.924171\pi\)
0.971758 0.235978i \(-0.0758293\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.21301 + 0.903457i 0.174409 + 0.0712024i
\(162\) 0 0
\(163\) 2.44949i 0.191859i 0.995388 + 0.0959294i \(0.0305823\pi\)
−0.995388 + 0.0959294i \(0.969418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.6853 0.826857 0.413428 0.910537i \(-0.364331\pi\)
0.413428 + 0.910537i \(0.364331\pi\)
\(168\) 0 0
\(169\) 11.9706 0.920813
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.9903i 1.06366i −0.846851 0.531831i \(-0.821504\pi\)
0.846851 0.531831i \(-0.178496\pi\)
\(174\) 0 0
\(175\) −0.242641 + 0.594346i −0.0183419 + 0.0449283i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.1714i 1.20871i −0.796716 0.604354i \(-0.793431\pi\)
0.796716 0.604354i \(-0.206569\pi\)
\(180\) 0 0
\(181\) 20.1903i 1.50073i 0.661023 + 0.750365i \(0.270122\pi\)
−0.661023 + 0.750365i \(0.729878\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.18114i 0.160360i
\(186\) 0 0
\(187\) −16.2426 −1.18778
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.3526i 1.32794i 0.747757 + 0.663972i \(0.231131\pi\)
−0.747757 + 0.663972i \(0.768869\pi\)
\(192\) 0 0
\(193\) −18.9706 −1.36553 −0.682765 0.730638i \(-0.739223\pi\)
−0.682765 + 0.730638i \(0.739223\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.34267 + 13.0868i −0.374982 + 0.918515i
\(204\) 0 0
\(205\) 18.2132 1.27207
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 17.0749i 1.18109i
\(210\) 0 0
\(211\) 8.36308i 0.575738i −0.957670 0.287869i \(-0.907053\pi\)
0.957670 0.287869i \(-0.0929468\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.76869 0.666219
\(216\) 0 0
\(217\) 5.24264 12.8418i 0.355894 0.871758i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.12967 0.210524
\(222\) 0 0
\(223\) −15.9706 −1.06947 −0.534734 0.845020i \(-0.679588\pi\)
−0.534734 + 0.845020i \(0.679588\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.55568 −0.501488 −0.250744 0.968053i \(-0.580675\pi\)
−0.250744 + 0.968053i \(0.580675\pi\)
\(228\) 0 0
\(229\) 22.2195i 1.46831i 0.678985 + 0.734153i \(0.262420\pi\)
−0.678985 + 0.734153i \(0.737580\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.2410 1.19501 0.597505 0.801865i \(-0.296159\pi\)
0.597505 + 0.801865i \(0.296159\pi\)
\(234\) 0 0
\(235\) 28.1331i 1.83520i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 28.5098i 1.84415i −0.387017 0.922073i \(-0.626495\pi\)
0.387017 0.922073i \(-0.373505\pi\)
\(240\) 0 0
\(241\) 27.5387i 1.77393i −0.461841 0.886963i \(-0.652811\pi\)
0.461841 0.886963i \(-0.347189\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.6853 + 10.9057i −0.682662 + 0.696739i
\(246\) 0 0
\(247\) 3.29002i 0.209339i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −25.7967 −1.62827 −0.814137 0.580673i \(-0.802789\pi\)
−0.814137 + 0.580673i \(0.802789\pi\)
\(252\) 0 0
\(253\) −4.75736 −0.299093
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.9903i 0.872690i 0.899780 + 0.436345i \(0.143727\pi\)
−0.899780 + 0.436345i \(0.856273\pi\)
\(258\) 0 0
\(259\) −1.00000 + 2.44949i −0.0621370 + 0.152204i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.1595i 1.24309i −0.783380 0.621543i \(-0.786506\pi\)
0.783380 0.621543i \(-0.213494\pi\)
\(264\) 0 0
\(265\) 16.4800i 1.01236i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.6041i 1.07334i 0.843792 + 0.536671i \(0.180318\pi\)
−0.843792 + 0.536671i \(0.819682\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.27768i 0.0770470i
\(276\) 0 0
\(277\) 27.4853 1.65143 0.825715 0.564087i \(-0.190772\pi\)
0.825715 + 0.564087i \(0.190772\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.5837 1.40689 0.703443 0.710752i \(-0.251645\pi\)
0.703443 + 0.710752i \(0.251645\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.4540 8.35032i −1.20736 0.492904i
\(288\) 0 0
\(289\) 7.48528 0.440311
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.3406i 1.30515i −0.757723 0.652576i \(-0.773688\pi\)
0.757723 0.652576i \(-0.226312\pi\)
\(294\) 0 0
\(295\) 11.6531i 0.678470i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.916658 0.0530117
\(300\) 0 0
\(301\) −10.9706 4.47871i −0.632333 0.258149i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.55568 −0.432637
\(306\) 0 0
\(307\) −6.75736 −0.385663 −0.192831 0.981232i \(-0.561767\pi\)
−0.192831 + 0.981232i \(0.561767\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.34267 0.302955 0.151478 0.988461i \(-0.451597\pi\)
0.151478 + 0.988461i \(0.451597\pi\)
\(312\) 0 0
\(313\) 12.8418i 0.725861i −0.931816 0.362931i \(-0.881776\pi\)
0.931816 0.362931i \(-0.118224\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.5837 1.32459 0.662296 0.749242i \(-0.269582\pi\)
0.662296 + 0.749242i \(0.269582\pi\)
\(318\) 0 0
\(319\) 28.1331i 1.57515i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.0022i 0.556539i
\(324\) 0 0
\(325\) 0.246186i 0.0136559i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.8984 31.5944i 0.711109 1.74185i
\(330\) 0 0
\(331\) 4.05845i 0.223072i −0.993760 0.111536i \(-0.964423\pi\)
0.993760 0.111536i \(-0.0355771\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.47234 −0.462893
\(336\) 0 0
\(337\) −35.9706 −1.95944 −0.979721 0.200368i \(-0.935786\pi\)
−0.979721 + 0.200368i \(0.935786\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 27.6063i 1.49497i
\(342\) 0 0
\(343\) 17.0000 7.34847i 0.917914 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.98805i 0.214090i −0.994254 0.107045i \(-0.965861\pi\)
0.994254 0.107045i \(-0.0341389\pi\)
\(348\) 0 0
\(349\) 30.5826i 1.63705i 0.574473 + 0.818524i \(0.305207\pi\)
−0.574473 + 0.818524i \(0.694793\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.4132i 1.56551i −0.622330 0.782755i \(-0.713814\pi\)
0.622330 0.782755i \(-0.286186\pi\)
\(354\) 0 0
\(355\) −4.75736 −0.252494
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0022i 0.527897i 0.964537 + 0.263949i \(0.0850250\pi\)
−0.964537 + 0.263949i \(0.914975\pi\)
\(360\) 0 0
\(361\) −8.48528 −0.446594
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −31.1394 −1.62991
\(366\) 0 0
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.55568 18.5076i 0.392271 0.960865i
\(372\) 0 0
\(373\) −22.2132 −1.15016 −0.575078 0.818099i \(-0.695028\pi\)
−0.575078 + 0.818099i \(0.695028\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.42074i 0.279182i
\(378\) 0 0
\(379\) 17.1464i 0.880753i 0.897813 + 0.440376i \(0.145155\pi\)
−0.897813 + 0.440376i \(0.854845\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 28.9264 1.47807 0.739034 0.673668i \(-0.235282\pi\)
0.739034 + 0.673668i \(0.235282\pi\)
\(384\) 0 0
\(385\) 11.4853 28.1331i 0.585344 1.43379i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.2410 −0.924857 −0.462428 0.886657i \(-0.653022\pi\)
−0.462428 + 0.886657i \(0.653022\pi\)
\(390\) 0 0
\(391\) −2.78680 −0.140934
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 20.4540 1.02915
\(396\) 0 0
\(397\) 17.5667i 0.881647i 0.897594 + 0.440824i \(0.145314\pi\)
−0.897594 + 0.440824i \(0.854686\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.8984 −0.644113 −0.322057 0.946720i \(-0.604374\pi\)
−0.322057 + 0.946720i \(0.604374\pi\)
\(402\) 0 0
\(403\) 5.31925i 0.264970i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.26573i 0.261013i
\(408\) 0 0
\(409\) 12.8418i 0.634986i −0.948261 0.317493i \(-0.897159\pi\)
0.948261 0.317493i \(-0.102841\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.34267 13.0868i 0.262896 0.643960i
\(414\) 0 0
\(415\) 28.1331i 1.38100i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.2410 −0.891132 −0.445566 0.895249i \(-0.646998\pi\)
−0.445566 + 0.895249i \(0.646998\pi\)
\(420\) 0 0
\(421\) 23.9706 1.16825 0.584127 0.811662i \(-0.301437\pi\)
0.584127 + 0.811662i \(0.301437\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.748448i 0.0363051i
\(426\) 0 0
\(427\) 8.48528 + 3.46410i 0.410632 + 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.5217i 1.18117i 0.806975 + 0.590585i \(0.201103\pi\)
−0.806975 + 0.590585i \(0.798897\pi\)
\(432\) 0 0
\(433\) 0.174080i 0.00836574i 0.999991 + 0.00418287i \(0.00133145\pi\)
−0.999991 + 0.00418287i \(0.998669\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.92959i 0.140141i
\(438\) 0 0
\(439\) 18.4853 0.882254 0.441127 0.897445i \(-0.354579\pi\)
0.441127 + 0.897445i \(0.354579\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 37.6086i 1.78684i −0.449225 0.893418i \(-0.648300\pi\)
0.449225 0.893418i \(-0.351700\pi\)
\(444\) 0 0
\(445\) −8.69848 −0.412348
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 43.9706 2.07049
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.21301 + 5.42074i −0.103747 + 0.254128i
\(456\) 0 0
\(457\) 1.48528 0.0694785 0.0347393 0.999396i \(-0.488940\pi\)
0.0347393 + 0.999396i \(0.488940\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13.2418i 0.616734i −0.951268 0.308367i \(-0.900218\pi\)
0.951268 0.308367i \(-0.0997824\pi\)
\(462\) 0 0
\(463\) 31.4231i 1.46036i −0.683257 0.730178i \(-0.739437\pi\)
0.683257 0.730178i \(-0.260563\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 35.5654 1.64577 0.822885 0.568208i \(-0.192363\pi\)
0.822885 + 0.568208i \(0.192363\pi\)
\(468\) 0 0
\(469\) 9.51472 + 3.88437i 0.439349 + 0.179363i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.5837 1.08438
\(474\) 0 0
\(475\) −0.786797 −0.0361007
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.6853 −0.488226 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(480\) 0 0
\(481\) 1.01461i 0.0462623i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13.8150 −0.627307
\(486\) 0 0
\(487\) 5.07306i 0.229882i 0.993372 + 0.114941i \(0.0366679\pi\)
−0.993372 + 0.114941i \(0.963332\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.35032i 0.376845i 0.982088 + 0.188422i \(0.0603374\pi\)
−0.982088 + 0.188422i \(0.939663\pi\)
\(492\) 0 0
\(493\) 16.4800i 0.742221i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.34267 + 2.18114i 0.239652 + 0.0978374i
\(498\) 0 0
\(499\) 3.04384i 0.136261i 0.997676 + 0.0681304i \(0.0217034\pi\)
−0.997676 + 0.0681304i \(0.978297\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20.4540 −0.912000 −0.456000 0.889980i \(-0.650718\pi\)
−0.456000 + 0.889980i \(0.650718\pi\)
\(504\) 0 0
\(505\) 6.72792 0.299389
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.08459i 0.136722i −0.997661 0.0683611i \(-0.978223\pi\)
0.997661 0.0683611i \(-0.0217770\pi\)
\(510\) 0 0
\(511\) 34.9706 + 14.2767i 1.54701 + 0.631563i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 29.9425i 1.31942i
\(516\) 0 0
\(517\) 67.9193i 2.98709i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.82109i 0.342648i 0.985215 + 0.171324i \(0.0548045\pi\)
−0.985215 + 0.171324i \(0.945195\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.1714i 0.704438i
\(528\) 0 0
\(529\) 22.1838 0.964512
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.47234 −0.366978
\(534\) 0 0
\(535\) 6.72792 0.290873
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −25.7967 + 26.3287i −1.11114 + 1.13406i
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.01418i 0.257619i
\(546\) 0 0
\(547\) 13.8564i 0.592457i −0.955117 0.296229i \(-0.904271\pi\)
0.955117 0.296229i \(-0.0957290\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −17.3244 −0.738043
\(552\) 0 0
\(553\) −22.9706 9.37769i −0.976808 0.398780i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.42602 0.187536 0.0937681 0.995594i \(-0.470109\pi\)
0.0937681 + 0.995594i \(0.470109\pi\)
\(558\) 0 0
\(559\) −4.54416 −0.192197
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.8984 0.543601 0.271800 0.962354i \(-0.412381\pi\)
0.271800 + 0.962354i \(0.412381\pi\)
\(564\) 0 0
\(565\) 39.7862i 1.67382i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.1114 −0.633501 −0.316751 0.948509i \(-0.602592\pi\)
−0.316751 + 0.948509i \(0.602592\pi\)
\(570\) 0 0
\(571\) 40.9749i 1.71475i −0.514696 0.857373i \(-0.672095\pi\)
0.514696 0.857373i \(-0.327905\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.219215i 0.00914191i
\(576\) 0 0
\(577\) 28.5533i 1.18869i −0.804210 0.594346i \(-0.797411\pi\)
0.804210 0.594346i \(-0.202589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.8984 + 31.5944i −0.535114 + 1.31076i
\(582\) 0 0
\(583\) 39.7862i 1.64778i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.3524 1.37660 0.688300 0.725426i \(-0.258357\pi\)
0.688300 + 0.725426i \(0.258357\pi\)
\(588\) 0 0
\(589\) 17.0000 0.700473
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26.3287i 1.08119i 0.841284 + 0.540594i \(0.181801\pi\)
−0.841284 + 0.540594i \(0.818199\pi\)
\(594\) 0 0
\(595\) 6.72792 16.4800i 0.275818 0.675613i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.07264i 0.288980i −0.989506 0.144490i \(-0.953846\pi\)
0.989506 0.144490i \(-0.0461542\pi\)
\(600\) 0 0
\(601\) 39.9603i 1.63001i 0.579452 + 0.815007i \(0.303267\pi\)
−0.579452 + 0.815007i \(0.696733\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 36.4859i 1.48336i
\(606\) 0 0
\(607\) 25.9411 1.05292 0.526459 0.850201i \(-0.323519\pi\)
0.526459 + 0.850201i \(0.323519\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.0868i 0.529436i
\(612\) 0 0
\(613\) 2.21320 0.0893904 0.0446952 0.999001i \(-0.485768\pi\)
0.0446952 + 0.999001i \(0.485768\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.5837 0.949444 0.474722 0.880136i \(-0.342549\pi\)
0.474722 + 0.880136i \(0.342549\pi\)
\(618\) 0 0
\(619\) 38.9411 1.56518 0.782588 0.622540i \(-0.213899\pi\)
0.782588 + 0.622540i \(0.213899\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.76869 + 3.98805i 0.391374 + 0.159778i
\(624\) 0 0
\(625\) −23.7279 −0.949117
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.08459i 0.122991i
\(630\) 0 0
\(631\) 27.5387i 1.09630i −0.836380 0.548150i \(-0.815332\pi\)
0.836380 0.548150i \(-0.184668\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.76869 0.387659
\(636\) 0 0
\(637\) 4.97056 5.07306i 0.196941 0.201002i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 22.6670 0.895294 0.447647 0.894210i \(-0.352262\pi\)
0.447647 + 0.894210i \(0.352262\pi\)
\(642\) 0 0
\(643\) −11.7279 −0.462504 −0.231252 0.972894i \(-0.574282\pi\)
−0.231252 + 0.972894i \(0.574282\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −26.7134 −1.05021 −0.525105 0.851037i \(-0.675974\pi\)
−0.525105 + 0.851037i \(0.675974\pi\)
\(648\) 0 0
\(649\) 28.1331i 1.10432i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.1394 −1.21858 −0.609289 0.792948i \(-0.708545\pi\)
−0.609289 + 0.792948i \(0.708545\pi\)
\(654\) 0 0
\(655\) 11.6531i 0.455324i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.0771i 1.05477i 0.849625 + 0.527387i \(0.176828\pi\)
−0.849625 + 0.527387i \(0.823172\pi\)
\(660\) 0 0
\(661\) 29.8141i 1.15964i 0.814746 + 0.579818i \(0.196876\pi\)
−0.814746 + 0.579818i \(0.803124\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −17.3244 7.07264i −0.671810 0.274265i
\(666\) 0 0
\(667\) 4.82687i 0.186897i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −18.2410 −0.704187
\(672\) 0 0
\(673\) 1.51472 0.0583881 0.0291941 0.999574i \(-0.490706\pi\)
0.0291941 + 0.999574i \(0.490706\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.5195i 0.558030i −0.960287 0.279015i \(-0.909992\pi\)
0.960287 0.279015i \(-0.0900080\pi\)
\(678\) 0 0
\(679\) 15.5147 + 6.33386i 0.595400 + 0.243071i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.6041i 0.673602i −0.941576 0.336801i \(-0.890655\pi\)
0.941576 0.336801i \(-0.109345\pi\)
\(684\) 0 0
\(685\) 44.6131i 1.70458i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.66608i 0.292055i
\(690\) 0 0
\(691\) 23.9411 0.910763 0.455382 0.890296i \(-0.349503\pi\)
0.455382 + 0.890296i \(0.349503\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 40.3189i 1.52938i
\(696\) 0 0
\(697\) 25.7574 0.975630
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 34.2690 1.29432 0.647162 0.762353i \(-0.275956\pi\)
0.647162 + 0.762353i \(0.275956\pi\)
\(702\) 0 0
\(703\) −3.24264 −0.122299
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.55568 3.08459i −0.284161 0.116008i
\(708\) 0 0
\(709\) 27.2426 1.02312 0.511559 0.859248i \(-0.329068\pi\)
0.511559 + 0.859248i \(0.329068\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.73650i 0.177383i
\(714\) 0 0
\(715\) 11.6531i 0.435801i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.5837 0.879524 0.439762 0.898114i \(-0.355063\pi\)
0.439762 + 0.898114i \(0.355063\pi\)
\(720\) 0 0
\(721\) 13.7279 33.6264i 0.511255 1.25231i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.29635 0.0481452
\(726\) 0 0
\(727\) 6.97056 0.258524 0.129262 0.991610i \(-0.458739\pi\)
0.129262 + 0.991610i \(0.458739\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 13.8150 0.510967
\(732\) 0 0
\(733\) 10.5664i 0.390278i 0.980776 + 0.195139i \(0.0625159\pi\)
−0.980776 + 0.195139i \(0.937484\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −20.4540 −0.753434
\(738\) 0 0
\(739\) 22.2195i 0.817357i 0.912678 + 0.408679i \(0.134010\pi\)
−0.912678 + 0.408679i \(0.865990\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.0749i 0.626416i 0.949684 + 0.313208i \(0.101404\pi\)
−0.949684 + 0.313208i \(0.898596\pi\)
\(744\) 0 0
\(745\) 16.4800i 0.603780i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7.55568 3.08459i −0.276079 0.112709i
\(750\) 0 0
\(751\) 2.44949i 0.0893832i 0.999001 + 0.0446916i \(0.0142305\pi\)
−0.999001 + 0.0446916i \(0.985769\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.5004 0.891659
\(756\) 0 0
\(757\) 17.5147 0.636583 0.318292 0.947993i \(-0.396891\pi\)
0.318292 + 0.947993i \(0.396891\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.5098i 1.03348i −0.856143 0.516740i \(-0.827146\pi\)
0.856143 0.516740i \(-0.172854\pi\)
\(762\) 0 0
\(763\) 2.75736 6.75412i 0.0998231 0.244516i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.42074i 0.195732i
\(768\) 0 0
\(769\) 32.1915i 1.16086i 0.814312 + 0.580428i \(0.197115\pi\)
−0.814312 + 0.580428i \(0.802885\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.9447i 1.43671i −0.695676 0.718356i \(-0.744895\pi\)
0.695676 0.718356i \(-0.255105\pi\)
\(774\) 0 0
\(775\) −1.27208 −0.0456944
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.0771i 0.970138i
\(780\) 0 0
\(781\) −11.4853 −0.410976
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.8984 −0.460362
\(786\) 0 0
\(787\) 54.9706 1.95949 0.979744 0.200252i \(-0.0641760\pi\)
0.979744 + 0.200252i \(0.0641760\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.2410 44.6812i 0.648576 1.58868i
\(792\) 0 0
\(793\) 3.51472 0.124811
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.92959i 0.103771i −0.998653 0.0518856i \(-0.983477\pi\)
0.998653 0.0518856i \(-0.0165231\pi\)
\(798\) 0 0
\(799\) 39.7862i 1.40753i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −75.1771 −2.65294
\(804\) 0 0
\(805\) 1.97056 4.82687i 0.0694532 0.170125i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.4540