Properties

Label 3024.2.k.k.1889.7
Level $3024$
Weight $2$
Character 3024.1889
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 24 x^{14} + 230 x^{12} - 1052 x^{10} + 2139 x^{8} - 1244 x^{6} + 1134 x^{4} - 104 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.7
Root \(-0.713245 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.k.1889.8

$q$-expansion

\(f(q)\) \(=\) \(q-0.465643 q^{5} +(-0.656211 - 2.56308i) q^{7} +O(q^{10})\) \(q-0.465643 q^{5} +(-0.656211 - 2.56308i) q^{7} -5.28908i q^{11} -4.50044i q^{13} +3.15854 q^{17} +1.42649i q^{19} -2.27727i q^{23} -4.78318 q^{25} +6.09560i q^{29} -2.76839i q^{31} +(0.305560 + 1.19348i) q^{35} -0.613036 q^{37} -6.93370 q^{41} -3.08379 q^{43} +9.30643 q^{47} +(-6.13877 + 3.36385i) q^{49} +12.1912i q^{53} +2.46283i q^{55} +3.62968 q^{59} -2.27318i q^{61} +2.09560i q^{65} -13.3613 q^{67} -7.38468i q^{71} +10.8175i q^{73} +(-13.5563 + 3.47075i) q^{77} -1.17014 q^{79} -15.0124 q^{83} -1.47075 q^{85} +5.39129 q^{89} +(-11.5350 + 2.95324i) q^{91} -0.664236i q^{95} -10.4528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{7} + O(q^{10}) \) \( 16q + 2q^{7} - 12q^{25} - 8q^{37} - 8q^{43} + 2q^{49} + 28q^{67} + 44q^{79} + 16q^{85} - 18q^{91} + O(q^{100}) \)

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
\(5\) −0.465643 −0.208242 −0.104121 0.994565i \(-0.533203\pi\)
−0.104121 + 0.994565i \(0.533203\pi\)
\(6\) 0 0
\(7\) −0.656211 2.56308i −0.248025 0.968754i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.28908i 1.59472i −0.603505 0.797359i \(-0.706230\pi\)
0.603505 0.797359i \(-0.293770\pi\)
\(12\) 0 0
\(13\) 4.50044i 1.24820i −0.781346 0.624098i \(-0.785466\pi\)
0.781346 0.624098i \(-0.214534\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.15854 0.766059 0.383029 0.923736i \(-0.374881\pi\)
0.383029 + 0.923736i \(0.374881\pi\)
\(18\) 0 0
\(19\) 1.42649i 0.327259i 0.986522 + 0.163630i \(0.0523202\pi\)
−0.986522 + 0.163630i \(0.947680\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.27727i 0.474844i −0.971407 0.237422i \(-0.923698\pi\)
0.971407 0.237422i \(-0.0763024\pi\)
\(24\) 0 0
\(25\) −4.78318 −0.956635
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.09560i 1.13192i 0.824431 + 0.565962i \(0.191495\pi\)
−0.824431 + 0.565962i \(0.808505\pi\)
\(30\) 0 0
\(31\) 2.76839i 0.497217i −0.968604 0.248608i \(-0.920027\pi\)
0.968604 0.248608i \(-0.0799732\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.305560 + 1.19348i 0.0516491 + 0.201735i
\(36\) 0 0
\(37\) −0.613036 −0.100783 −0.0503913 0.998730i \(-0.516047\pi\)
−0.0503913 + 0.998730i \(0.516047\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.93370 −1.08286 −0.541431 0.840745i \(-0.682117\pi\)
−0.541431 + 0.840745i \(0.682117\pi\)
\(42\) 0 0
\(43\) −3.08379 −0.470274 −0.235137 0.971962i \(-0.575554\pi\)
−0.235137 + 0.971962i \(0.575554\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.30643 1.35748 0.678741 0.734377i \(-0.262526\pi\)
0.678741 + 0.734377i \(0.262526\pi\)
\(48\) 0 0
\(49\) −6.13877 + 3.36385i −0.876968 + 0.480549i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.1912i 1.67459i 0.546752 + 0.837295i \(0.315864\pi\)
−0.546752 + 0.837295i \(0.684136\pi\)
\(54\) 0 0
\(55\) 2.46283i 0.332087i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.62968 0.472545 0.236272 0.971687i \(-0.424074\pi\)
0.236272 + 0.971687i \(0.424074\pi\)
\(60\) 0 0
\(61\) 2.27318i 0.291051i −0.989354 0.145526i \(-0.953513\pi\)
0.989354 0.145526i \(-0.0464873\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.09560i 0.259927i
\(66\) 0 0
\(67\) −13.3613 −1.63235 −0.816174 0.577807i \(-0.803909\pi\)
−0.816174 + 0.577807i \(0.803909\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.38468i 0.876400i −0.898877 0.438200i \(-0.855616\pi\)
0.898877 0.438200i \(-0.144384\pi\)
\(72\) 0 0
\(73\) 10.8175i 1.26609i 0.774113 + 0.633047i \(0.218196\pi\)
−0.774113 + 0.633047i \(0.781804\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.5563 + 3.47075i −1.54489 + 0.395529i
\(78\) 0 0
\(79\) −1.17014 −0.131651 −0.0658255 0.997831i \(-0.520968\pi\)
−0.0658255 + 0.997831i \(0.520968\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.0124 −1.64782 −0.823912 0.566717i \(-0.808213\pi\)
−0.823912 + 0.566717i \(0.808213\pi\)
\(84\) 0 0
\(85\) −1.47075 −0.159526
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.39129 0.571476 0.285738 0.958308i \(-0.407761\pi\)
0.285738 + 0.958308i \(0.407761\pi\)
\(90\) 0 0
\(91\) −11.5350 + 2.95324i −1.20920 + 0.309583i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.664236i 0.0681492i
\(96\) 0 0
\(97\) 10.4528i 1.06132i −0.847583 0.530662i \(-0.821943\pi\)
0.847583 0.530662i \(-0.178057\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0867 1.00367 0.501834 0.864964i \(-0.332659\pi\)
0.501834 + 0.864964i \(0.332659\pi\)
\(102\) 0 0
\(103\) 2.35778i 0.232319i −0.993231 0.116159i \(-0.962942\pi\)
0.993231 0.116159i \(-0.0370583\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.7204i 1.42308i −0.702646 0.711539i \(-0.747998\pi\)
0.702646 0.711539i \(-0.252002\pi\)
\(108\) 0 0
\(109\) −16.9862 −1.62698 −0.813491 0.581578i \(-0.802435\pi\)
−0.813491 + 0.581578i \(0.802435\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.94151i 0.653002i 0.945197 + 0.326501i \(0.105870\pi\)
−0.945197 + 0.326501i \(0.894130\pi\)
\(114\) 0 0
\(115\) 1.06040i 0.0988825i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.07267 8.09560i −0.190001 0.742122i
\(120\) 0 0
\(121\) −16.9744 −1.54312
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.55547 0.407454
\(126\) 0 0
\(127\) 3.85772 0.342317 0.171159 0.985243i \(-0.445249\pi\)
0.171159 + 0.985243i \(0.445249\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.7375 1.63711 0.818553 0.574431i \(-0.194776\pi\)
0.818553 + 0.574431i \(0.194776\pi\)
\(132\) 0 0
\(133\) 3.65621 0.936079i 0.317034 0.0811683i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.86953i 0.586903i −0.955974 0.293452i \(-0.905196\pi\)
0.955974 0.293452i \(-0.0948040\pi\)
\(138\) 0 0
\(139\) 16.6007i 1.40806i −0.710172 0.704028i \(-0.751383\pi\)
0.710172 0.704028i \(-0.248617\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −23.8032 −1.99052
\(144\) 0 0
\(145\) 2.83838i 0.235714i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.22607i 0.264290i −0.991230 0.132145i \(-0.957814\pi\)
0.991230 0.132145i \(-0.0421865\pi\)
\(150\) 0 0
\(151\) −2.68758 −0.218712 −0.109356 0.994003i \(-0.534879\pi\)
−0.109356 + 0.994003i \(0.534879\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.28908i 0.103541i
\(156\) 0 0
\(157\) 17.0259i 1.35882i −0.733760 0.679409i \(-0.762236\pi\)
0.733760 0.679409i \(-0.237764\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.83683 + 1.49437i −0.460007 + 0.117773i
\(162\) 0 0
\(163\) −15.9862 −1.25213 −0.626067 0.779769i \(-0.715336\pi\)
−0.626067 + 0.779769i \(0.715336\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.6421 1.44257 0.721284 0.692640i \(-0.243552\pi\)
0.721284 + 0.692640i \(0.243552\pi\)
\(168\) 0 0
\(169\) −7.25393 −0.557995
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.2322 −0.777941 −0.388971 0.921250i \(-0.627169\pi\)
−0.388971 + 0.921250i \(0.627169\pi\)
\(174\) 0 0
\(175\) 3.13877 + 12.2597i 0.237269 + 0.926744i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.6620i 1.02114i 0.859836 + 0.510571i \(0.170566\pi\)
−0.859836 + 0.510571i \(0.829434\pi\)
\(180\) 0 0
\(181\) 11.3033i 0.840165i 0.907486 + 0.420083i \(0.137999\pi\)
−0.907486 + 0.420083i \(0.862001\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.285456 0.0209872
\(186\) 0 0
\(187\) 16.7058i 1.22165i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.66424i 0.626922i 0.949601 + 0.313461i \(0.101489\pi\)
−0.949601 + 0.313461i \(0.898511\pi\)
\(192\) 0 0
\(193\) 16.2775 1.17168 0.585842 0.810425i \(-0.300764\pi\)
0.585842 + 0.810425i \(0.300764\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.9066i 1.70328i −0.524130 0.851639i \(-0.675609\pi\)
0.524130 0.851639i \(-0.324391\pi\)
\(198\) 0 0
\(199\) 25.3806i 1.79919i 0.436729 + 0.899593i \(0.356137\pi\)
−0.436729 + 0.899593i \(0.643863\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 15.6235 4.00000i 1.09656 0.280745i
\(204\) 0 0
\(205\) 3.22863 0.225497
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.54482 0.521886
\(210\) 0 0
\(211\) −16.4451 −1.13213 −0.566065 0.824361i \(-0.691535\pi\)
−0.566065 + 0.824361i \(0.691535\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.43595 0.0979307
\(216\) 0 0
\(217\) −7.09560 + 1.81665i −0.481681 + 0.123322i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.2148i 0.956192i
\(222\) 0 0
\(223\) 22.8464i 1.52991i −0.644084 0.764954i \(-0.722761\pi\)
0.644084 0.764954i \(-0.277239\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.7190 −1.04331 −0.521653 0.853158i \(-0.674684\pi\)
−0.521653 + 0.853158i \(0.674684\pi\)
\(228\) 0 0
\(229\) 1.25145i 0.0826983i −0.999145 0.0413492i \(-0.986834\pi\)
0.999145 0.0413492i \(-0.0131656\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −4.33348 −0.282685
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.9116i 0.964554i 0.876019 + 0.482277i \(0.160190\pi\)
−0.876019 + 0.482277i \(0.839810\pi\)
\(240\) 0 0
\(241\) 0.966217i 0.0622395i 0.999516 + 0.0311198i \(0.00990733\pi\)
−0.999516 + 0.0311198i \(0.990093\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.85848 1.56635i 0.182622 0.100071i
\(246\) 0 0
\(247\) 6.41983 0.408484
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.2532 −1.21525 −0.607626 0.794224i \(-0.707878\pi\)
−0.607626 + 0.794224i \(0.707878\pi\)
\(252\) 0 0
\(253\) −12.0447 −0.757242
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.5437 −1.28148 −0.640739 0.767758i \(-0.721372\pi\)
−0.640739 + 0.767758i \(0.721372\pi\)
\(258\) 0 0
\(259\) 0.402281 + 1.57126i 0.0249965 + 0.0976334i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.89031i 0.486537i 0.969959 + 0.243269i \(0.0782197\pi\)
−0.969959 + 0.243269i \(0.921780\pi\)
\(264\) 0 0
\(265\) 5.67675i 0.348720i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 31.0278 1.89180 0.945900 0.324458i \(-0.105182\pi\)
0.945900 + 0.324458i \(0.105182\pi\)
\(270\) 0 0
\(271\) 0.0605334i 0.00367714i 0.999998 + 0.00183857i \(0.000585236\pi\)
−0.999998 + 0.00183857i \(0.999415\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 25.2986i 1.52556i
\(276\) 0 0
\(277\) −10.6994 −0.642864 −0.321432 0.946933i \(-0.604164\pi\)
−0.321432 + 0.946933i \(0.604164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.92802i 0.472946i 0.971638 + 0.236473i \(0.0759915\pi\)
−0.971638 + 0.236473i \(0.924009\pi\)
\(282\) 0 0
\(283\) 7.60944i 0.452334i −0.974089 0.226167i \(-0.927380\pi\)
0.974089 0.226167i \(-0.0726196\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.54997 + 17.7716i 0.268576 + 1.04903i
\(288\) 0 0
\(289\) −7.02362 −0.413154
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.655636 0.0383026 0.0191513 0.999817i \(-0.493904\pi\)
0.0191513 + 0.999817i \(0.493904\pi\)
\(294\) 0 0
\(295\) −1.69014 −0.0984036
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.2487 −0.592699
\(300\) 0 0
\(301\) 2.02362 + 7.90400i 0.116639 + 0.455579i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.05849i 0.0606091i
\(306\) 0 0
\(307\) 4.45077i 0.254019i −0.991902 0.127009i \(-0.959462\pi\)
0.991902 0.127009i \(-0.0405379\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −23.1628 −1.31344 −0.656722 0.754133i \(-0.728058\pi\)
−0.656722 + 0.754133i \(0.728058\pi\)
\(312\) 0 0
\(313\) 23.5435i 1.33076i −0.746505 0.665380i \(-0.768270\pi\)
0.746505 0.665380i \(-0.231730\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.3217i 0.972882i −0.873713 0.486441i \(-0.838295\pi\)
0.873713 0.486441i \(-0.161705\pi\)
\(318\) 0 0
\(319\) 32.2401 1.80510
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.50563i 0.250700i
\(324\) 0 0
\(325\) 21.5264i 1.19407i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.10699 23.8531i −0.336689 1.31507i
\(330\) 0 0
\(331\) −3.48000 −0.191278 −0.0956391 0.995416i \(-0.530489\pi\)
−0.0956391 + 0.995416i \(0.530489\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.22162 0.339923
\(336\) 0 0
\(337\) 5.94151 0.323655 0.161827 0.986819i \(-0.448261\pi\)
0.161827 + 0.986819i \(0.448261\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.6422 −0.792920
\(342\) 0 0
\(343\) 12.6501 + 13.5268i 0.683043 + 0.730378i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.04440i 0.163432i 0.996656 + 0.0817159i \(0.0260400\pi\)
−0.996656 + 0.0817159i \(0.973960\pi\)
\(348\) 0 0
\(349\) 25.5607i 1.36823i −0.729373 0.684116i \(-0.760188\pi\)
0.729373 0.684116i \(-0.239812\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.0385 1.49234 0.746169 0.665757i \(-0.231891\pi\)
0.746169 + 0.665757i \(0.231891\pi\)
\(354\) 0 0
\(355\) 3.43863i 0.182503i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.33805i 0.334509i −0.985914 0.167255i \(-0.946510\pi\)
0.985914 0.167255i \(-0.0534902\pi\)
\(360\) 0 0
\(361\) 16.9651 0.892901
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.03711i 0.263654i
\(366\) 0 0
\(367\) 2.69255i 0.140550i −0.997528 0.0702750i \(-0.977612\pi\)
0.997528 0.0702750i \(-0.0223877\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 31.2470 8.00000i 1.62227 0.415339i
\(372\) 0 0
\(373\) 15.5247 0.803837 0.401919 0.915675i \(-0.368343\pi\)
0.401919 + 0.915675i \(0.368343\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 27.4329 1.41286
\(378\) 0 0
\(379\) −28.9513 −1.48713 −0.743564 0.668664i \(-0.766866\pi\)
−0.743564 + 0.668664i \(0.766866\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.5816 0.898375 0.449188 0.893437i \(-0.351713\pi\)
0.449188 + 0.893437i \(0.351713\pi\)
\(384\) 0 0
\(385\) 6.31242 1.61613i 0.321711 0.0823658i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.16758i 0.109901i −0.998489 0.0549503i \(-0.982500\pi\)
0.998489 0.0549503i \(-0.0175000\pi\)
\(390\) 0 0
\(391\) 7.19286i 0.363758i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.544868 0.0274153
\(396\) 0 0
\(397\) 28.6843i 1.43962i 0.694169 + 0.719812i \(0.255772\pi\)
−0.694169 + 0.719812i \(0.744228\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.0843i 1.45240i −0.687482 0.726201i \(-0.741284\pi\)
0.687482 0.726201i \(-0.258716\pi\)
\(402\) 0 0
\(403\) −12.4589 −0.620624
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.24240i 0.160720i
\(408\) 0 0
\(409\) 24.6739i 1.22005i 0.792383 + 0.610024i \(0.208840\pi\)
−0.792383 + 0.610024i \(0.791160\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.38184 9.30317i −0.117203 0.457779i
\(414\) 0 0
\(415\) 6.99042 0.343146
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −23.5895 −1.15242 −0.576210 0.817301i \(-0.695469\pi\)
−0.576210 + 0.817301i \(0.695469\pi\)
\(420\) 0 0
\(421\) −5.38696 −0.262545 −0.131272 0.991346i \(-0.541906\pi\)
−0.131272 + 0.991346i \(0.541906\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.1079 −0.732839
\(426\) 0 0
\(427\) −5.82635 + 1.49169i −0.281957 + 0.0721878i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.33576i 0.353351i −0.984269 0.176676i \(-0.943466\pi\)
0.984269 0.176676i \(-0.0565344\pi\)
\(432\) 0 0
\(433\) 19.9270i 0.957633i −0.877915 0.478816i \(-0.841066\pi\)
0.877915 0.478816i \(-0.158934\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.24851 0.155397
\(438\) 0 0
\(439\) 10.6943i 0.510409i 0.966887 + 0.255205i \(0.0821428\pi\)
−0.966887 + 0.255205i \(0.917857\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.6760i 1.12488i 0.826837 + 0.562441i \(0.190138\pi\)
−0.826837 + 0.562441i \(0.809862\pi\)
\(444\) 0 0
\(445\) −2.51042 −0.119005
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.4409i 1.57817i −0.614282 0.789087i \(-0.710554\pi\)
0.614282 0.789087i \(-0.289446\pi\)
\(450\) 0 0
\(451\) 36.6729i 1.72686i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.37119 1.37516i 0.251805 0.0644683i
\(456\) 0 0
\(457\) 11.5385 0.539748 0.269874 0.962896i \(-0.413018\pi\)
0.269874 + 0.962896i \(0.413018\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.87564 0.227081 0.113541 0.993533i \(-0.463781\pi\)
0.113541 + 0.993533i \(0.463781\pi\)
\(462\) 0 0
\(463\) 40.8275 1.89742 0.948708 0.316154i \(-0.102392\pi\)
0.948708 + 0.316154i \(0.102392\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.27002 −0.197593 −0.0987964 0.995108i \(-0.531499\pi\)
−0.0987964 + 0.995108i \(0.531499\pi\)
\(468\) 0 0
\(469\) 8.76786 + 34.2462i 0.404862 + 1.58134i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.3104i 0.749954i
\(474\) 0 0
\(475\) 6.82315i 0.313068i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.6067 −1.39846 −0.699228 0.714899i \(-0.746473\pi\)
−0.699228 + 0.714899i \(0.746473\pi\)
\(480\) 0 0
\(481\) 2.75893i 0.125796i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.86730i 0.221012i
\(486\) 0 0
\(487\) 13.9626 0.632704 0.316352 0.948642i \(-0.397542\pi\)
0.316352 + 0.948642i \(0.397542\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.3729i 0.738897i −0.929251 0.369449i \(-0.879547\pi\)
0.929251 0.369449i \(-0.120453\pi\)
\(492\) 0 0
\(493\) 19.2532i 0.867120i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.9275 + 4.84591i −0.849016 + 0.217369i
\(498\) 0 0
\(499\) −29.2986 −1.31159 −0.655793 0.754941i \(-0.727666\pi\)
−0.655793 + 0.754941i \(0.727666\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.4284 0.821681 0.410840 0.911707i \(-0.365235\pi\)
0.410840 + 0.911707i \(0.365235\pi\)
\(504\) 0 0
\(505\) −4.69683 −0.209006
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −30.6804 −1.35989 −0.679943 0.733265i \(-0.737995\pi\)
−0.679943 + 0.733265i \(0.737995\pi\)
\(510\) 0 0
\(511\) 27.7262 7.09858i 1.22653 0.314023i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.09788i 0.0483785i
\(516\) 0 0
\(517\) 49.2225i 2.16480i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.0125 0.482468 0.241234 0.970467i \(-0.422448\pi\)
0.241234 + 0.970467i \(0.422448\pi\)
\(522\) 0 0
\(523\) 13.4459i 0.587949i 0.955813 + 0.293975i \(0.0949781\pi\)
−0.955813 + 0.293975i \(0.905022\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.74406i 0.380897i
\(528\) 0 0
\(529\) 17.8140 0.774523
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 31.2047i 1.35162i
\(534\) 0 0
\(535\) 6.85448i 0.296345i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.7917 + 32.4685i 0.766341 + 1.39852i
\(540\) 0 0
\(541\) −6.63665 −0.285332 −0.142666 0.989771i \(-0.545567\pi\)
−0.142666 + 0.989771i \(0.545567\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.90950 0.338806
\(546\) 0 0
\(547\) 5.31242 0.227143 0.113571 0.993530i \(-0.463771\pi\)
0.113571 + 0.993530i \(0.463771\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.69531 −0.370433
\(552\) 0 0
\(553\) 0.767859 + 2.99916i 0.0326527 + 0.127537i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.9066i 0.673986i 0.941507 + 0.336993i \(0.109410\pi\)
−0.941507 + 0.336993i \(0.890590\pi\)
\(558\) 0 0
\(559\) 13.8784i 0.586994i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.61560 −0.405249 −0.202625 0.979256i \(-0.564947\pi\)
−0.202625 + 0.979256i \(0.564947\pi\)
\(564\) 0 0
\(565\) 3.23227i 0.135983i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.7154i 0.826514i −0.910614 0.413257i \(-0.864391\pi\)
0.910614 0.413257i \(-0.135609\pi\)
\(570\) 0 0
\(571\) 5.76969 0.241454 0.120727 0.992686i \(-0.461477\pi\)
0.120727 + 0.992686i \(0.461477\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 10.8926i 0.454253i
\(576\) 0 0
\(577\) 32.0587i 1.33462i 0.744780 + 0.667310i \(0.232554\pi\)
−0.744780 + 0.667310i \(0.767446\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.85130 + 38.4780i 0.408701 + 1.59634i
\(582\) 0 0
\(583\) 64.4802 2.67050
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.2298 1.00007 0.500035 0.866005i \(-0.333320\pi\)
0.500035 + 0.866005i \(0.333320\pi\)
\(588\) 0 0
\(589\) 3.94908 0.162719
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −44.7443 −1.83743 −0.918713 0.394925i \(-0.870770\pi\)
−0.918713 + 0.394925i \(0.870770\pi\)
\(594\) 0 0
\(595\) 0.965125 + 3.76966i 0.0395663 + 0.154541i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.60574i 0.310762i 0.987855 + 0.155381i \(0.0496606\pi\)
−0.987855 + 0.155381i \(0.950339\pi\)
\(600\) 0 0
\(601\) 23.3138i 0.950990i −0.879718 0.475495i \(-0.842269\pi\)
0.879718 0.475495i \(-0.157731\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.90400 0.321343
\(606\) 0 0
\(607\) 13.3948i 0.543680i −0.962342 0.271840i \(-0.912368\pi\)
0.962342 0.271840i \(-0.0876322\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 41.8830i 1.69441i
\(612\) 0 0
\(613\) 41.1231 1.66095 0.830474 0.557058i \(-0.188070\pi\)
0.830474 + 0.557058i \(0.188070\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.0607i 0.767356i 0.923467 + 0.383678i \(0.125343\pi\)
−0.923467 + 0.383678i \(0.874657\pi\)
\(618\) 0 0
\(619\) 10.5929i 0.425766i −0.977078 0.212883i \(-0.931715\pi\)
0.977078 0.212883i \(-0.0682854\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.53783 13.8183i −0.141740 0.553620i
\(624\) 0 0
\(625\) 21.7947 0.871786
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.93630 −0.0772053
\(630\) 0 0
\(631\) 26.9277 1.07197 0.535987 0.844226i \(-0.319940\pi\)
0.535987 + 0.844226i \(0.319940\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.79632 −0.0712848
\(636\) 0 0
\(637\) 15.1388 + 27.6272i 0.599820 + 1.09463i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.0821i 0.437717i −0.975757 0.218859i \(-0.929767\pi\)
0.975757 0.218859i \(-0.0702333\pi\)
\(642\) 0 0
\(643\) 34.4756i 1.35958i −0.733405 0.679792i \(-0.762070\pi\)
0.733405 0.679792i \(-0.237930\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −35.1859 −1.38330 −0.691650 0.722233i \(-0.743116\pi\)
−0.691650 + 0.722233i \(0.743116\pi\)
\(648\) 0 0
\(649\) 19.1977i 0.753575i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.4082i 1.26823i 0.773238 + 0.634116i \(0.218636\pi\)
−0.773238 + 0.634116i \(0.781364\pi\)
\(654\) 0 0
\(655\) −8.72501 −0.340914
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.09788i 0.198585i −0.995058 0.0992927i \(-0.968342\pi\)
0.995058 0.0992927i \(-0.0316580\pi\)
\(660\) 0 0
\(661\) 20.4587i 0.795752i −0.917439 0.397876i \(-0.869747\pi\)
0.917439 0.397876i \(-0.130253\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.70249 + 0.435879i −0.0660198 + 0.0169027i
\(666\) 0 0
\(667\) 13.8813 0.537487
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0230 −0.464144
\(672\) 0 0
\(673\) 43.7926 1.68808 0.844041 0.536278i \(-0.180170\pi\)
0.844041 + 0.536278i \(0.180170\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.0624 0.578895 0.289448 0.957194i \(-0.406528\pi\)
0.289448 + 0.957194i \(0.406528\pi\)
\(678\) 0 0
\(679\) −26.7915 + 6.85927i −1.02816 + 0.263235i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.5894i 1.47658i −0.674483 0.738291i \(-0.735633\pi\)
0.674483 0.738291i \(-0.264367\pi\)
\(684\) 0 0
\(685\) 3.19875i 0.122218i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 54.8657 2.09022
\(690\) 0 0
\(691\) 40.1079i 1.52578i −0.646530 0.762889i \(-0.723780\pi\)
0.646530 0.762889i \(-0.276220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.73002i 0.293216i
\(696\) 0 0
\(697\) −21.9004 −0.829536
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.4195i 0.506848i 0.967355 + 0.253424i \(0.0815567\pi\)
−0.967355 + 0.253424i \(0.918443\pi\)
\(702\) 0 0
\(703\) 0.874490i 0.0329820i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.61903 25.8531i −0.248934 0.972308i
\(708\) 0 0
\(709\) 33.2165 1.24747 0.623736 0.781635i \(-0.285614\pi\)
0.623736 + 0.781635i \(0.285614\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.30437 −0.236100
\(714\) 0 0
\(715\) 11.0838 0.414510
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 43.0856 1.60682 0.803411 0.595425i \(-0.203016\pi\)
0.803411 + 0.595425i \(0.203016\pi\)
\(720\) 0 0
\(721\) −6.04318 + 1.54720i −0.225060 + 0.0576207i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.1563i 1.08284i
\(726\) 0 0
\(727\) 22.3861i 0.830256i −0.909763 0.415128i \(-0.863737\pi\)
0.909763 0.415128i \(-0.136263\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9.74028 −0.360257
\(732\) 0 0
\(733\) 35.0569i 1.29486i −0.762127 0.647428i \(-0.775845\pi\)
0.762127 0.647428i \(-0.224155\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 70.6692i 2.60313i
\(738\) 0 0
\(739\) −2.21481 −0.0814733 −0.0407366 0.999170i \(-0.512970\pi\)
−0.0407366 + 0.999170i \(0.512970\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.0516i 1.02911i −0.857456 0.514557i \(-0.827956\pi\)
0.857456 0.514557i \(-0.172044\pi\)
\(744\) 0 0
\(745\) 1.50220i 0.0550363i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −37.7297 + 9.65972i −1.37861 + 0.352958i
\(750\) 0 0
\(751\) −9.62741 −0.351309 −0.175655 0.984452i \(-0.556204\pi\)
−0.175655 + 0.984452i \(0.556204\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.25145 0.0455450
\(756\) 0 0
\(757\) 34.0351 1.23703 0.618513 0.785774i \(-0.287735\pi\)
0.618513 + 0.785774i \(0.287735\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.3692 1.06463 0.532316 0.846546i \(-0.321322\pi\)
0.532316 + 0.846546i \(0.321322\pi\)
\(762\) 0 0
\(763\) 11.1465 + 43.5370i 0.403531 + 1.57614i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.3352i 0.589828i
\(768\) 0 0
\(769\) 41.2653i 1.48807i 0.668143 + 0.744033i \(0.267089\pi\)
−0.668143 + 0.744033i \(0.732911\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.3880 −1.52459 −0.762296 0.647229i \(-0.775928\pi\)
−0.762296 + 0.647229i \(0.775928\pi\)
\(774\) 0 0
\(775\) 13.2417i 0.475655i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.89086i 0.354377i
\(780\) 0 0
\(781\) −39.0582 −1.39761
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.92802i 0.282963i
\(786\) 0 0
\(787\) 33.2592i 1.18556i 0.805364 + 0.592781i \(0.201970\pi\)
−0.805364 + 0.592781i \(0.798030\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 17.7917 4.55510i 0.632598 0.161961i
\(792\) 0 0
\(793\) −10.2303 −0.363289
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.1338 0.890287 0.445143 0.895459i \(-0.353153\pi\)
0.445143 + 0.895459i \(0.353153\pi\)
\(798\) 0 0
\(799\) 29.3948 1.03991
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 57.2147 2.01906
\(804\) 0