Properties

Label 3024.2.b.v.1567.8
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 1567.8
Root \(-1.05050 + 1.81952i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.v.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.63904i q^{5} +(1.00000 + 2.44949i) q^{7} +O(q^{10})\) \(q+3.63904i q^{5} +(1.00000 + 2.44949i) q^{7} +1.50734i q^{11} +5.91359i q^{13} -5.14639i q^{17} -5.24264 q^{19} +8.78543i q^{23} -8.24264 q^{25} +8.91380 q^{29} -3.24264 q^{31} +(-8.91380 + 3.63904i) q^{35} +1.00000 q^{37} +6.65373i q^{41} -9.37769i q^{43} -3.69222 q^{47} +(-5.00000 + 4.89898i) q^{49} +12.6060 q^{53} -5.48528 q^{55} +8.91380 q^{59} -3.46410i q^{61} -21.5198 q^{65} -10.8126i q^{67} -3.63904i q^{71} +0.420266i q^{73} +(-3.69222 + 1.50734i) q^{77} -4.47871i q^{79} +3.69222 q^{83} +18.7279 q^{85} -13.9318i q^{89} +(-14.4853 + 5.91359i) q^{91} -19.0782i q^{95} +13.2621i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{19} - 32q^{25} + 8q^{31} + 8q^{37} - 40q^{49} + 24q^{55} + 48q^{85} - 48q^{91} + O(q^{100}) \)

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
\(5\) 3.63904i 1.62743i 0.581264 + 0.813715i \(0.302558\pi\)
−0.581264 + 0.813715i \(0.697442\pi\)
\(6\) 0 0
\(7\) 1.00000 + 2.44949i 0.377964 + 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.50734i 0.454481i 0.973839 + 0.227240i \(0.0729703\pi\)
−0.973839 + 0.227240i \(0.927030\pi\)
\(12\) 0 0
\(13\) 5.91359i 1.64014i 0.572267 + 0.820068i \(0.306064\pi\)
−0.572267 + 0.820068i \(0.693936\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.14639i 1.24818i −0.781352 0.624091i \(-0.785469\pi\)
0.781352 0.624091i \(-0.214531\pi\)
\(18\) 0 0
\(19\) −5.24264 −1.20274 −0.601372 0.798969i \(-0.705379\pi\)
−0.601372 + 0.798969i \(0.705379\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.78543i 1.83189i 0.401305 + 0.915944i \(0.368557\pi\)
−0.401305 + 0.915944i \(0.631443\pi\)
\(24\) 0 0
\(25\) −8.24264 −1.64853
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.91380 1.65525 0.827626 0.561281i \(-0.189691\pi\)
0.827626 + 0.561281i \(0.189691\pi\)
\(30\) 0 0
\(31\) −3.24264 −0.582395 −0.291198 0.956663i \(-0.594054\pi\)
−0.291198 + 0.956663i \(0.594054\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.91380 + 3.63904i −1.50671 + 0.615111i
\(36\) 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
\(41\) 6.65373i 1.03914i 0.854429 + 0.519569i \(0.173907\pi\)
−0.854429 + 0.519569i \(0.826093\pi\)
\(42\) 0 0
\(43\) 9.37769i 1.43008i −0.699081 0.715042i \(-0.746407\pi\)
0.699081 0.715042i \(-0.253593\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.69222 −0.538565 −0.269283 0.963061i \(-0.586787\pi\)
−0.269283 + 0.963061i \(0.586787\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.6060 1.73157 0.865785 0.500416i \(-0.166820\pi\)
0.865785 + 0.500416i \(0.166820\pi\)
\(54\) 0 0
\(55\) −5.48528 −0.739635
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.91380 1.16048 0.580239 0.814446i \(-0.302959\pi\)
0.580239 + 0.814446i \(0.302959\pi\)
\(60\) 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
\(65\) −21.5198 −2.66921
\(66\) 0 0
\(67\) 10.8126i 1.32097i −0.750841 0.660483i \(-0.770352\pi\)
0.750841 0.660483i \(-0.229648\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.63904i 0.431875i −0.976407 0.215938i \(-0.930719\pi\)
0.976407 0.215938i \(-0.0692808\pi\)
\(72\) 0 0
\(73\) 0.420266i 0.0491884i 0.999698 + 0.0245942i \(0.00782937\pi\)
−0.999698 + 0.0245942i \(0.992171\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.69222 + 1.50734i −0.420767 + 0.171777i
\(78\) 0 0
\(79\) 4.47871i 0.503895i −0.967741 0.251947i \(-0.918929\pi\)
0.967741 0.251947i \(-0.0810710\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.69222 0.405273 0.202637 0.979254i \(-0.435049\pi\)
0.202637 + 0.979254i \(0.435049\pi\)
\(84\) 0 0
\(85\) 18.7279 2.03133
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.9318i 1.47677i −0.674380 0.738385i \(-0.735589\pi\)
0.674380 0.738385i \(-0.264411\pi\)
\(90\) 0 0
\(91\) −14.4853 + 5.91359i −1.51847 + 0.619913i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 19.0782i 1.95738i
\(96\) 0 0
\(97\) 13.2621i 1.34656i 0.739388 + 0.673279i \(0.235115\pi\)
−0.739388 + 0.673279i \(0.764885\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.14639i 0.512084i 0.966666 + 0.256042i \(0.0824186\pi\)
−0.966666 + 0.256042i \(0.917581\pi\)
\(102\) 0 0
\(103\) −11.7279 −1.15559 −0.577793 0.816183i \(-0.696086\pi\)
−0.577793 + 0.816183i \(0.696086\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.14639i 0.497520i −0.968565 0.248760i \(-0.919977\pi\)
0.968565 0.248760i \(-0.0800230\pi\)
\(108\) 0 0
\(109\) −11.2426 −1.07685 −0.538425 0.842674i \(-0.680980\pi\)
−0.538425 + 0.842674i \(0.680980\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.22158 −0.491205 −0.245603 0.969371i \(-0.578986\pi\)
−0.245603 + 0.969371i \(0.578986\pi\)
\(114\) 0 0
\(115\) −31.9706 −2.98127
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.6060 5.14639i 1.15559 0.471768i
\(120\) 0 0
\(121\) 8.72792 0.793447
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.8001i 1.05543i
\(126\) 0 0
\(127\) 9.37769i 0.832136i −0.909334 0.416068i \(-0.863408\pi\)
0.909334 0.416068i \(-0.136592\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.91380 0.778802 0.389401 0.921068i \(-0.372682\pi\)
0.389401 + 0.921068i \(0.372682\pi\)
\(132\) 0 0
\(133\) −5.24264 12.8418i −0.454595 1.11352i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.2982 1.39245 0.696226 0.717823i \(-0.254861\pi\)
0.696226 + 0.717823i \(0.254861\pi\)
\(138\) 0 0
\(139\) 1.51472 0.128477 0.0642384 0.997935i \(-0.479538\pi\)
0.0642384 + 0.997935i \(0.479538\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.91380 −0.745409
\(144\) 0 0
\(145\) 32.4377i 2.69381i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.6060 −1.03273 −0.516363 0.856370i \(-0.672714\pi\)
−0.516363 + 0.856370i \(0.672714\pi\)
\(150\) 0 0
\(151\) 18.1610i 1.47792i 0.673747 + 0.738962i \(0.264684\pi\)
−0.673747 + 0.738962i \(0.735316\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.8001i 0.947808i
\(156\) 0 0
\(157\) 1.01461i 0.0809748i −0.999180 0.0404874i \(-0.987109\pi\)
0.999180 0.0404874i \(-0.0128911\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −21.5198 + 8.78543i −1.69600 + 0.692389i
\(162\) 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
\(167\) −17.8276 −1.37954 −0.689771 0.724028i \(-0.742289\pi\)
−0.689771 + 0.724028i \(0.742289\pi\)
\(168\) 0 0
\(169\) −21.9706 −1.69004
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.0488i 0.992085i 0.868298 + 0.496042i \(0.165214\pi\)
−0.868298 + 0.496042i \(0.834786\pi\)
\(174\) 0 0
\(175\) −8.24264 20.1903i −0.623085 1.52624i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.6879i 1.24731i −0.781699 0.623655i \(-0.785647\pi\)
0.781699 0.623655i \(-0.214353\pi\)
\(180\) 0 0
\(181\) 0.594346i 0.0441774i 0.999756 + 0.0220887i \(0.00703162\pi\)
−0.999756 + 0.0220887i \(0.992968\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.63904i 0.267548i
\(186\) 0 0
\(187\) 7.75736 0.567274
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.3269i 1.47080i 0.677631 + 0.735402i \(0.263007\pi\)
−0.677631 + 0.735402i \(0.736993\pi\)
\(192\) 0 0
\(193\) 14.9706 1.07760 0.538802 0.842432i \(-0.318877\pi\)
0.538802 + 0.842432i \(0.318877\pi\)
\(194\) 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 \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.91380 + 21.8343i 0.625626 + 1.53246i
\(204\) 0 0
\(205\) −24.2132 −1.69112
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.90245i 0.546624i
\(210\) 0 0
\(211\) 1.43488i 0.0987811i −0.998780 0.0493905i \(-0.984272\pi\)
0.998780 0.0493905i \(-0.0157279\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 34.1258 2.32736
\(216\) 0 0
\(217\) −3.24264 7.94282i −0.220125 0.539193i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 30.4336 2.04719
\(222\) 0 0
\(223\) −17.9706 −1.20340 −0.601699 0.798723i \(-0.705509\pi\)
−0.601699 + 0.798723i \(0.705509\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.6060 −0.836691 −0.418345 0.908288i \(-0.637390\pi\)
−0.418345 + 0.908288i \(0.637390\pi\)
\(228\) 0 0
\(229\) 12.4215i 0.820838i 0.911897 + 0.410419i \(0.134617\pi\)
−0.911897 + 0.410419i \(0.865383\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.22158 0.342077 0.171039 0.985264i \(-0.445288\pi\)
0.171039 + 0.985264i \(0.445288\pi\)
\(234\) 0 0
\(235\) 13.4361i 0.876477i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.89766i 0.252119i 0.992023 + 0.126059i \(0.0402330\pi\)
−0.992023 + 0.126059i \(0.959767\pi\)
\(240\) 0 0
\(241\) 6.75412i 0.435071i 0.976052 + 0.217536i \(0.0698018\pi\)
−0.976052 + 0.217536i \(0.930198\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17.8276 18.1952i −1.13896 1.16245i
\(246\) 0 0
\(247\) 31.0028i 1.97266i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.38443 −0.466101 −0.233051 0.972465i \(-0.574871\pi\)
−0.233051 + 0.972465i \(0.574871\pi\)
\(252\) 0 0
\(253\) −13.2426 −0.832558
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.0488i 0.813964i −0.913436 0.406982i \(-0.866581\pi\)
0.913436 0.406982i \(-0.133419\pi\)
\(258\) 0 0
\(259\) 1.00000 + 2.44949i 0.0621370 + 0.152204i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.75606i 0.169946i −0.996383 0.0849731i \(-0.972920\pi\)
0.996383 0.0849731i \(-0.0270804\pi\)
\(264\) 0 0
\(265\) 45.8739i 2.81801i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.0929i 1.34703i 0.739175 + 0.673513i \(0.235216\pi\)
−0.739175 + 0.673513i \(0.764784\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.4245i 0.749224i
\(276\) 0 0
\(277\) 10.5147 0.631768 0.315884 0.948798i \(-0.397699\pi\)
0.315884 + 0.948798i \(0.397699\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.1354 0.843246 0.421623 0.906771i \(-0.361461\pi\)
0.421623 + 0.906771i \(0.361461\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −16.2982 + 6.65373i −0.962054 + 0.392757i
\(288\) 0 0
\(289\) −9.48528 −0.557958
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.39511i 0.373606i 0.982397 + 0.186803i \(0.0598126\pi\)
−0.982397 + 0.186803i \(0.940187\pi\)
\(294\) 0 0
\(295\) 32.4377i 1.88860i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −51.9534 −3.00454
\(300\) 0 0
\(301\) 22.9706 9.37769i 1.32400 0.540521i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.6060 0.721818
\(306\) 0 0
\(307\) 15.2426 0.869943 0.434972 0.900444i \(-0.356758\pi\)
0.434972 + 0.900444i \(0.356758\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.91380 −0.505455 −0.252728 0.967537i \(-0.581328\pi\)
−0.252728 + 0.967537i \(0.581328\pi\)
\(312\) 0 0
\(313\) 7.94282i 0.448954i −0.974479 0.224477i \(-0.927933\pi\)
0.974479 0.224477i \(-0.0720674\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.1354 0.793922 0.396961 0.917835i \(-0.370065\pi\)
0.396961 + 0.917835i \(0.370065\pi\)
\(318\) 0 0
\(319\) 13.4361i 0.752279i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 26.9806i 1.50124i
\(324\) 0 0
\(325\) 48.7436i 2.70381i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.69222 9.04405i −0.203559 0.498615i
\(330\) 0 0
\(331\) 23.6544i 1.30016i 0.759865 + 0.650081i \(0.225265\pi\)
−0.759865 + 0.650081i \(0.774735\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 39.3474 2.14978
\(336\) 0 0
\(337\) −2.02944 −0.110550 −0.0552752 0.998471i \(-0.517604\pi\)
−0.0552752 + 0.998471i \(0.517604\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.88777i 0.264687i
\(342\) 0 0
\(343\) −17.0000 7.34847i −0.917914 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.9318i 0.747899i 0.927449 + 0.373949i \(0.121997\pi\)
−0.927449 + 0.373949i \(0.878003\pi\)
\(348\) 0 0
\(349\) 10.9867i 0.588102i 0.955790 + 0.294051i \(0.0950035\pi\)
−0.955790 + 0.294051i \(0.904996\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.6831i 0.675053i −0.941316 0.337526i \(-0.890410\pi\)
0.941316 0.337526i \(-0.109590\pi\)
\(354\) 0 0
\(355\) 13.2426 0.702846
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.9806i 1.42398i 0.702187 + 0.711992i \(0.252207\pi\)
−0.702187 + 0.711992i \(0.747793\pi\)
\(360\) 0 0
\(361\) 8.48528 0.446594
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.52937 −0.0800507
\(366\) 0 0
\(367\) −25.0000 −1.30499 −0.652495 0.757793i \(-0.726278\pi\)
−0.652495 + 0.757793i \(0.726278\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.6060 + 30.8783i 0.654472 + 1.60312i
\(372\) 0 0
\(373\) 20.2132 1.04660 0.523300 0.852149i \(-0.324701\pi\)
0.523300 + 0.852149i \(0.324701\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 52.7126i 2.71484i
\(378\) 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
\(383\) −23.0492 −1.17776 −0.588879 0.808221i \(-0.700431\pi\)
−0.588879 + 0.808221i \(0.700431\pi\)
\(384\) 0 0
\(385\) −5.48528 13.4361i −0.279556 0.684769i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.22158 −0.264745 −0.132372 0.991200i \(-0.542259\pi\)
−0.132372 + 0.991200i \(0.542259\pi\)
\(390\) 0 0
\(391\) 45.2132 2.28653
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.2982 0.820053
\(396\) 0 0
\(397\) 31.4231i 1.57708i −0.614983 0.788540i \(-0.710837\pi\)
0.614983 0.788540i \(-0.289163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.69222 0.184381 0.0921903 0.995741i \(-0.470613\pi\)
0.0921903 + 0.995741i \(0.470613\pi\)
\(402\) 0 0
\(403\) 19.1757i 0.955207i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.50734i 0.0747161i
\(408\) 0 0
\(409\) 7.94282i 0.392747i −0.980529 0.196373i \(-0.937084\pi\)
0.980529 0.196373i \(-0.0629165\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.91380 + 21.8343i 0.438619 + 1.07439i
\(414\) 0 0
\(415\) 13.4361i 0.659554i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.22158 0.255091 0.127546 0.991833i \(-0.459290\pi\)
0.127546 + 0.991833i \(0.459290\pi\)
\(420\) 0 0
\(421\) −9.97056 −0.485935 −0.242968 0.970034i \(-0.578121\pi\)
−0.242968 + 0.970034i \(0.578121\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 42.4198i 2.05766i
\(426\) 0 0
\(427\) 8.48528 3.46410i 0.410632 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0342i 0.483328i 0.970360 + 0.241664i \(0.0776932\pi\)
−0.970360 + 0.241664i \(0.922307\pi\)
\(432\) 0 0
\(433\) 34.4669i 1.65638i 0.560451 + 0.828188i \(0.310628\pi\)
−0.560451 + 0.828188i \(0.689372\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 46.0588i 2.20329i
\(438\) 0 0
\(439\) −1.51472 −0.0722936 −0.0361468 0.999346i \(-0.511508\pi\)
−0.0361468 + 0.999346i \(0.511508\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.8684i 1.51411i −0.653349 0.757057i \(-0.726636\pi\)
0.653349 0.757057i \(-0.273364\pi\)
\(444\) 0 0
\(445\) 50.6985 2.40334
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −10.0294 −0.472268
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −21.5198 52.7126i −1.00886 2.47120i
\(456\) 0 0
\(457\) −15.4853 −0.724371 −0.362185 0.932106i \(-0.617969\pi\)
−0.362185 + 0.932106i \(0.617969\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.3710i 1.36794i −0.729509 0.683971i \(-0.760251\pi\)
0.729509 0.683971i \(-0.239749\pi\)
\(462\) 0 0
\(463\) 17.5667i 0.816394i −0.912894 0.408197i \(-0.866158\pi\)
0.912894 0.408197i \(-0.133842\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.5103 1.92087 0.960433 0.278511i \(-0.0898408\pi\)
0.960433 + 0.278511i \(0.0898408\pi\)
\(468\) 0 0
\(469\) 26.4853 10.8126i 1.22298 0.499278i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14.1354 0.649946
\(474\) 0 0
\(475\) 43.2132 1.98276
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.8276 0.814564 0.407282 0.913302i \(-0.366477\pi\)
0.407282 + 0.913302i \(0.366477\pi\)
\(480\) 0 0
\(481\) 5.91359i 0.269637i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −48.2612 −2.19143
\(486\) 0 0
\(487\) 29.5680i 1.33985i −0.742428 0.669926i \(-0.766326\pi\)
0.742428 0.669926i \(-0.233674\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.65373i 0.300278i −0.988665 0.150139i \(-0.952028\pi\)
0.988665 0.150139i \(-0.0479722\pi\)
\(492\) 0 0
\(493\) 45.8739i 2.06605i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.91380 3.63904i 0.399839 0.163233i
\(498\) 0 0
\(499\) 17.7408i 0.794186i −0.917778 0.397093i \(-0.870019\pi\)
0.917778 0.397093i \(-0.129981\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.2982 −0.726702 −0.363351 0.931652i \(-0.618367\pi\)
−0.363351 + 0.931652i \(0.618367\pi\)
\(504\) 0 0
\(505\) −18.7279 −0.833382
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.14639i 0.228110i −0.993474 0.114055i \(-0.963616\pi\)
0.993474 0.114055i \(-0.0363839\pi\)
\(510\) 0 0
\(511\) −1.02944 + 0.420266i −0.0455396 + 0.0185915i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.6784i 1.88064i
\(516\) 0 0
\(517\) 5.56543i 0.244767i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.3416i 1.02261i −0.859398 0.511307i \(-0.829161\pi\)
0.859398 0.511307i \(-0.170839\pi\)
\(522\) 0 0
\(523\) 7.00000 0.306089 0.153044 0.988219i \(-0.451092\pi\)
0.153044 + 0.988219i \(0.451092\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.6879i 0.726935i
\(528\) 0 0
\(529\) −54.1838 −2.35582
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −39.3474 −1.70433
\(534\) 0 0
\(535\) 18.7279 0.809679
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.38443 7.53671i −0.318070 0.324629i
\(540\) 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
\(545\) 40.9125i 1.75250i
\(546\) 0 0
\(547\) 13.8564i 0.592457i 0.955117 + 0.296229i \(0.0957290\pi\)
−0.955117 + 0.296229i \(0.904271\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −46.7319 −1.99084
\(552\) 0 0
\(553\) 10.9706 4.47871i 0.466516 0.190454i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −43.0396 −1.82365 −0.911824 0.410581i \(-0.865326\pi\)
−0.911824 + 0.410581i \(0.865326\pi\)
\(558\) 0 0
\(559\) 55.4558 2.34553
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.69222 0.155608 0.0778042 0.996969i \(-0.475209\pi\)
0.0778042 + 0.996969i \(0.475209\pi\)
\(564\) 0 0
\(565\) 19.0016i 0.799402i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.2120 1.05694 0.528472 0.848951i \(-0.322765\pi\)
0.528472 + 0.848951i \(0.322765\pi\)
\(570\) 0 0
\(571\) 21.3790i 0.894681i 0.894364 + 0.447341i \(0.147629\pi\)
−0.894364 + 0.447341i \(0.852371\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 72.4151i 3.01992i
\(576\) 0 0
\(577\) 0.840532i 0.0349918i 0.999847 + 0.0174959i \(0.00556940\pi\)
−0.999847 + 0.0174959i \(0.994431\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.69222 + 9.04405i 0.153179 + 0.375210i
\(582\) 0 0
\(583\) 19.0016i 0.786965i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.9905 0.825094 0.412547 0.910936i \(-0.364639\pi\)
0.412547 + 0.910936i \(0.364639\pi\)
\(588\) 0 0
\(589\) 17.0000 0.700473
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.53671i 0.309495i 0.987954 + 0.154748i \(0.0494565\pi\)
−0.987954 + 0.154748i \(0.950544\pi\)
\(594\) 0 0
\(595\) 18.7279 + 45.8739i 0.767770 + 1.88064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.0782i 0.779514i 0.920918 + 0.389757i \(0.127441\pi\)
−0.920918 + 0.389757i \(0.872559\pi\)
\(600\) 0 0
\(601\) 15.4654i 0.630845i 0.948951 + 0.315423i \(0.102146\pi\)
−0.948951 + 0.315423i \(0.897854\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.7613i 1.29128i
\(606\) 0 0
\(607\) 41.9411 1.70234 0.851169 0.524892i \(-0.175894\pi\)
0.851169 + 0.524892i \(0.175894\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.8343i 0.883320i
\(612\) 0 0
\(613\) −40.2132 −1.62420 −0.812098 0.583521i \(-0.801675\pi\)
−0.812098 + 0.583521i \(0.801675\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.1354 0.569069 0.284535 0.958666i \(-0.408161\pi\)
0.284535 + 0.958666i \(0.408161\pi\)
\(618\) 0 0
\(619\) 28.9411 1.16324 0.581621 0.813460i \(-0.302419\pi\)
0.581621 + 0.813460i \(0.302419\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 34.1258 13.9318i 1.36722 0.558166i
\(624\) 0 0
\(625\) 1.72792 0.0691169
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.14639i 0.205200i
\(630\) 0 0
\(631\) 6.75412i 0.268877i −0.990922 0.134439i \(-0.957077\pi\)
0.990922 0.134439i \(-0.0429231\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34.1258 1.35424
\(636\) 0 0
\(637\) −28.9706 29.5680i −1.14786 1.17153i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −37.8181 −1.49372 −0.746862 0.664979i \(-0.768440\pi\)
−0.746862 + 0.664979i \(0.768440\pi\)
\(642\) 0 0
\(643\) −13.7279 −0.541376 −0.270688 0.962667i \(-0.587251\pi\)
−0.270688 + 0.962667i \(0.587251\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 44.5690 1.75219 0.876094 0.482140i \(-0.160140\pi\)
0.876094 + 0.482140i \(0.160140\pi\)
\(648\) 0 0
\(649\) 13.4361i 0.527415i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.52937 −0.0598487 −0.0299244 0.999552i \(-0.509527\pi\)
−0.0299244 + 0.999552i \(0.509527\pi\)
\(654\) 0 0
\(655\) 32.4377i 1.26745i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 34.8831i 1.35885i 0.733744 + 0.679426i \(0.237771\pi\)
−0.733744 + 0.679426i \(0.762229\pi\)
\(660\) 0 0
\(661\) 43.6705i 1.69859i −0.527921 0.849294i \(-0.677028\pi\)
0.527921 0.849294i \(-0.322972\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 46.7319 19.0782i 1.81218 0.739821i
\(666\) 0 0
\(667\) 78.3116i 3.03224i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.22158 0.201577
\(672\) 0 0
\(673\) 18.4853 0.712555 0.356278 0.934380i \(-0.384046\pi\)
0.356278 + 0.934380i \(0.384046\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.9465i 0.651307i −0.945489 0.325653i \(-0.894416\pi\)
0.945489 0.325653i \(-0.105584\pi\)
\(678\) 0 0
\(679\) −32.4853 + 13.2621i −1.24667 + 0.508951i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.0929i 0.845361i 0.906279 + 0.422680i \(0.138911\pi\)
−0.906279 + 0.422680i \(0.861089\pi\)
\(684\) 0 0
\(685\) 59.3100i 2.26612i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 74.5468i 2.84001i
\(690\) 0 0
\(691\) 43.9411 1.67160 0.835800 0.549035i \(-0.185005\pi\)
0.835800 + 0.549035i \(0.185005\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.51213i 0.209087i
\(696\) 0 0
\(697\) 34.2426 1.29703
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.9630 1.20723 0.603613 0.797278i \(-0.293727\pi\)
0.603613 + 0.797278i \(0.293727\pi\)
\(702\) 0 0
\(703\) −5.24264 −0.197730
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.6060 + 5.14639i −0.474098 + 0.193550i
\(708\) 0 0
\(709\) 18.7574 0.704447 0.352224 0.935916i \(-0.385426\pi\)
0.352224 + 0.935916i \(0.385426\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.4880i 1.06688i
\(714\) 0 0
\(715\) 32.4377i 1.21310i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −14.1354 −0.527161 −0.263580 0.964637i \(-0.584903\pi\)
−0.263580 + 0.964637i \(0.584903\pi\)
\(720\) 0 0
\(721\) −11.7279 28.7274i −0.436771 1.06987i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −73.4733 −2.72873
\(726\) 0 0
\(727\) 26.9706 1.00028 0.500141 0.865944i \(-0.333281\pi\)
0.500141 + 0.865944i \(0.333281\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −48.2612 −1.78501
\(732\) 0 0
\(733\) 44.8592i 1.65691i 0.560053 + 0.828457i \(0.310781\pi\)
−0.560053 + 0.828457i \(0.689219\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.2982 0.600353
\(738\) 0 0
\(739\) 12.4215i 0.456933i −0.973552 0.228467i \(-0.926629\pi\)
0.973552 0.228467i \(-0.0733712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.90245i 0.289913i 0.989438 + 0.144956i \(0.0463042\pi\)
−0.989438 + 0.144956i \(0.953696\pi\)
\(744\) 0 0
\(745\) 45.8739i 1.68069i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.6060 5.14639i 0.460614 0.188045i
\(750\) 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
\(755\) −66.0888 −2.40522
\(756\) 0 0
\(757\) 34.4853 1.25339 0.626694 0.779265i \(-0.284407\pi\)
0.626694 + 0.779265i \(0.284407\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.89766i 0.141290i −0.997502 0.0706451i \(-0.977494\pi\)
0.997502 0.0706451i \(-0.0225058\pi\)
\(762\) 0 0
\(763\) −11.2426 27.5387i −0.407011 0.996969i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52.7126i 1.90334i
\(768\) 0 0
\(769\) 37.0905i 1.33752i 0.743479 + 0.668759i \(0.233174\pi\)
−0.743479 + 0.668759i \(0.766826\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.6978i 0.564610i −0.959325 0.282305i \(-0.908901\pi\)
0.959325 0.282305i \(-0.0910990\pi\)
\(774\) 0 0
\(775\) 26.7279 0.960095
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 34.8831i 1.24982i
\(780\) 0 0
\(781\) 5.48528 0.196279
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.69222 0.131781
\(786\) 0 0
\(787\) −21.0294 −0.749618 −0.374809 0.927102i \(-0.622292\pi\)
−0.374809 + 0.927102i \(0.622292\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.22158 12.7902i −0.185658 0.454768i
\(792\) 0 0
\(793\) 20.4853 0.727454
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.0588i 1.63149i 0.578413 + 0.815744i \(0.303672\pi\)
−0.578413 + 0.815744i \(0.696328\pi\)
\(798\) 0 0
\(799\) 19.0016i 0.672227i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.633484 −0.0223552
\(804\) 0 0
\(805\) −31.9706 78.3116i −1.12681 2.76012i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.2982 −0.573015 −0.286508 0.958078i \(-0.592494\pi\)
−0.286508 + 0.958078i \(0.592494\pi\)
\(810\) 0