Properties

Label 5292.2.x.a.4409.2
Level $5292$
Weight $2$
Character 5292.4409
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 4409.2
Root \(-0.811340 + 1.53027i\) of defining polynomial
Character \(\chi\) \(=\) 5292.4409
Dual form 5292.2.x.a.881.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.37166 - 2.37578i) q^{5} +O(q^{10})\) \(q+(-1.37166 - 2.37578i) q^{5} +(-0.362306 - 0.209178i) q^{11} +(1.32512 - 0.765056i) q^{13} -3.90581 q^{17} -5.91199i q^{19} +(7.72884 - 4.46225i) q^{23} +(-1.26290 + 2.18740i) q^{25} +(-6.00378 - 3.46629i) q^{29} +(-3.05626 + 1.76453i) q^{31} +9.09722 q^{37} +(-1.06236 - 1.84006i) q^{41} +(-5.77846 + 10.0086i) q^{43} +(-0.885373 + 1.53351i) q^{47} -3.92050i q^{53} +1.14768i q^{55} +(-2.02728 - 3.51135i) q^{59} +(-1.61459 - 0.932184i) q^{61} +(-3.63521 - 2.09879i) q^{65} +(6.38441 + 11.0581i) q^{67} -8.51021i q^{71} -1.90594i q^{73} +(0.433633 - 0.751074i) q^{79} +(-3.45880 + 5.99082i) q^{83} +(5.35744 + 9.27936i) q^{85} -9.77729 q^{89} +(-14.0456 + 8.10924i) q^{95} +(0.200411 + 0.115707i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 6 q^{11} - 3 q^{13} + 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} + 6 q^{31} - 2 q^{37} + 6 q^{41} - 2 q^{43} - 18 q^{47} - 15 q^{59} + 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} + 42 q^{89} + 6 q^{95} - 3 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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
\(5\) −1.37166 2.37578i −0.613425 1.06248i −0.990659 0.136365i \(-0.956458\pi\)
0.377234 0.926118i \(-0.376875\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.362306 0.209178i −0.109240 0.0630695i 0.444385 0.895836i \(-0.353422\pi\)
−0.553624 + 0.832767i \(0.686756\pi\)
\(12\) 0 0
\(13\) 1.32512 0.765056i 0.367521 0.212188i −0.304854 0.952399i \(-0.598608\pi\)
0.672375 + 0.740211i \(0.265274\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.90581 −0.947298 −0.473649 0.880714i \(-0.657064\pi\)
−0.473649 + 0.880714i \(0.657064\pi\)
\(18\) 0 0
\(19\) 5.91199i 1.35630i −0.734922 0.678152i \(-0.762781\pi\)
0.734922 0.678152i \(-0.237219\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.72884 4.46225i 1.61157 0.930443i 0.622569 0.782565i \(-0.286089\pi\)
0.989006 0.147878i \(-0.0472444\pi\)
\(24\) 0 0
\(25\) −1.26290 + 2.18740i −0.252579 + 0.437480i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00378 3.46629i −1.11487 0.643673i −0.174787 0.984606i \(-0.555924\pi\)
−0.940088 + 0.340933i \(0.889257\pi\)
\(30\) 0 0
\(31\) −3.05626 + 1.76453i −0.548921 + 0.316920i −0.748687 0.662924i \(-0.769315\pi\)
0.199766 + 0.979844i \(0.435982\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.09722 1.49557 0.747787 0.663939i \(-0.231116\pi\)
0.747787 + 0.663939i \(0.231116\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.06236 1.84006i −0.165913 0.287370i 0.771066 0.636755i \(-0.219724\pi\)
−0.936979 + 0.349385i \(0.886390\pi\)
\(42\) 0 0
\(43\) −5.77846 + 10.0086i −0.881208 + 1.52630i −0.0312079 + 0.999513i \(0.509935\pi\)
−0.850000 + 0.526783i \(0.823398\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.885373 + 1.53351i −0.129145 + 0.223686i −0.923346 0.383970i \(-0.874557\pi\)
0.794201 + 0.607656i \(0.207890\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.92050i 0.538523i −0.963067 0.269261i \(-0.913220\pi\)
0.963067 0.269261i \(-0.0867795\pi\)
\(54\) 0 0
\(55\) 1.14768i 0.154753i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.02728 3.51135i −0.263929 0.457139i 0.703353 0.710840i \(-0.251685\pi\)
−0.967283 + 0.253702i \(0.918352\pi\)
\(60\) 0 0
\(61\) −1.61459 0.932184i −0.206727 0.119354i 0.393062 0.919512i \(-0.371416\pi\)
−0.599789 + 0.800158i \(0.704749\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.63521 2.09879i −0.450893 0.260323i
\(66\) 0 0
\(67\) 6.38441 + 11.0581i 0.779979 + 1.35096i 0.931953 + 0.362579i \(0.118104\pi\)
−0.151974 + 0.988385i \(0.548563\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.51021i 1.00998i −0.863126 0.504988i \(-0.831497\pi\)
0.863126 0.504988i \(-0.168503\pi\)
\(72\) 0 0
\(73\) 1.90594i 0.223074i −0.993760 0.111537i \(-0.964423\pi\)
0.993760 0.111537i \(-0.0355773\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.433633 0.751074i 0.0487875 0.0845024i −0.840600 0.541656i \(-0.817798\pi\)
0.889388 + 0.457153i \(0.151131\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.45880 + 5.99082i −0.379653 + 0.657578i −0.991012 0.133775i \(-0.957290\pi\)
0.611359 + 0.791354i \(0.290623\pi\)
\(84\) 0 0
\(85\) 5.35744 + 9.27936i 0.581096 + 1.00649i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.77729 −1.03639 −0.518195 0.855262i \(-0.673396\pi\)
−0.518195 + 0.855262i \(0.673396\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −14.0456 + 8.10924i −1.44105 + 0.831990i
\(96\) 0 0
\(97\) 0.200411 + 0.115707i 0.0203486 + 0.0117483i 0.510140 0.860091i \(-0.329594\pi\)
−0.489791 + 0.871840i \(0.662927\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.14031 12.3674i 0.710487 1.23060i −0.254187 0.967155i \(-0.581808\pi\)
0.964674 0.263445i \(-0.0848587\pi\)
\(102\) 0 0
\(103\) 9.30617 5.37292i 0.916964 0.529410i 0.0342991 0.999412i \(-0.489080\pi\)
0.882665 + 0.470002i \(0.155747\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.35702i 0.614556i 0.951620 + 0.307278i \(0.0994181\pi\)
−0.951620 + 0.307278i \(0.900582\pi\)
\(108\) 0 0
\(109\) −5.16072 −0.494308 −0.247154 0.968976i \(-0.579495\pi\)
−0.247154 + 0.968976i \(0.579495\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.19186 + 5.30692i −0.864697 + 0.499233i −0.865582 0.500766i \(-0.833052\pi\)
0.000885276 1.00000i \(0.499718\pi\)
\(114\) 0 0
\(115\) −21.2027 12.2414i −1.97716 1.14151i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.41249 9.37471i −0.492044 0.852246i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.78753 −0.607096
\(126\) 0 0
\(127\) 10.2909 0.913169 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.83048 17.0269i −0.858893 1.48765i −0.872986 0.487746i \(-0.837819\pi\)
0.0140928 0.999901i \(-0.495514\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.66411 2.69282i −0.398481 0.230063i 0.287347 0.957827i \(-0.407227\pi\)
−0.685829 + 0.727763i \(0.740560\pi\)
\(138\) 0 0
\(139\) 14.7839 8.53549i 1.25395 0.723971i 0.282062 0.959396i \(-0.408982\pi\)
0.971892 + 0.235425i \(0.0756484\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.640131 −0.0535304
\(144\) 0 0
\(145\) 19.0182i 1.57938i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.31162 5.37607i 0.762838 0.440425i −0.0674758 0.997721i \(-0.521495\pi\)
0.830314 + 0.557296i \(0.188161\pi\)
\(150\) 0 0
\(151\) −3.78223 + 6.55102i −0.307794 + 0.533115i −0.977879 0.209169i \(-0.932924\pi\)
0.670086 + 0.742284i \(0.266257\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.38430 + 4.84068i 0.673443 + 0.388812i
\(156\) 0 0
\(157\) −10.6317 + 6.13820i −0.848500 + 0.489882i −0.860144 0.510051i \(-0.829627\pi\)
0.0116445 + 0.999932i \(0.496293\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.8349 −0.926981 −0.463490 0.886102i \(-0.653403\pi\)
−0.463490 + 0.886102i \(0.653403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.78854 11.7581i −0.525313 0.909869i −0.999565 0.0294798i \(-0.990615\pi\)
0.474252 0.880389i \(-0.342718\pi\)
\(168\) 0 0
\(169\) −5.32938 + 9.23075i −0.409952 + 0.710058i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.31085 + 14.3948i −0.631862 + 1.09442i 0.355308 + 0.934749i \(0.384376\pi\)
−0.987171 + 0.159668i \(0.948957\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.0988i 1.27803i 0.769195 + 0.639014i \(0.220657\pi\)
−0.769195 + 0.639014i \(0.779343\pi\)
\(180\) 0 0
\(181\) 18.2171i 1.35407i 0.735952 + 0.677034i \(0.236735\pi\)
−0.735952 + 0.677034i \(0.763265\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.4783 21.6130i −0.917422 1.58902i
\(186\) 0 0
\(187\) 1.41510 + 0.817009i 0.103482 + 0.0597456i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.1860 10.4997i −1.31589 0.759730i −0.332826 0.942988i \(-0.608002\pi\)
−0.983065 + 0.183258i \(0.941336\pi\)
\(192\) 0 0
\(193\) 3.48741 + 6.04038i 0.251030 + 0.434796i 0.963810 0.266592i \(-0.0858975\pi\)
−0.712780 + 0.701388i \(0.752564\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.0756i 1.14534i 0.819786 + 0.572670i \(0.194092\pi\)
−0.819786 + 0.572670i \(0.805908\pi\)
\(198\) 0 0
\(199\) 6.29261i 0.446071i 0.974810 + 0.223036i \(0.0715966\pi\)
−0.974810 + 0.223036i \(0.928403\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.91440 + 5.04788i −0.203550 + 0.352559i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.23666 + 2.14195i −0.0855414 + 0.148162i
\(210\) 0 0
\(211\) −1.29814 2.24844i −0.0893674 0.154789i 0.817876 0.575394i \(-0.195151\pi\)
−0.907244 + 0.420605i \(0.861818\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 31.7043 2.16222
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.17565 + 2.98816i −0.348152 + 0.201006i
\(222\) 0 0
\(223\) −20.7215 11.9636i −1.38762 0.801141i −0.394571 0.918866i \(-0.629107\pi\)
−0.993046 + 0.117725i \(0.962440\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.86609 3.23216i 0.123857 0.214526i −0.797429 0.603413i \(-0.793807\pi\)
0.921285 + 0.388887i \(0.127140\pi\)
\(228\) 0 0
\(229\) −18.2455 + 10.5341i −1.20570 + 0.696111i −0.961817 0.273694i \(-0.911754\pi\)
−0.243882 + 0.969805i \(0.578421\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.7643i 0.836217i 0.908397 + 0.418109i \(0.137307\pi\)
−0.908397 + 0.418109i \(0.862693\pi\)
\(234\) 0 0
\(235\) 4.85772 0.316883
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.0521 + 6.38091i −0.714899 + 0.412747i −0.812872 0.582442i \(-0.802097\pi\)
0.0979736 + 0.995189i \(0.468764\pi\)
\(240\) 0 0
\(241\) 2.63438 + 1.52096i 0.169695 + 0.0979737i 0.582442 0.812872i \(-0.302097\pi\)
−0.412747 + 0.910846i \(0.635431\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.52301 7.83408i −0.287792 0.498470i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.32067 0.398957 0.199478 0.979902i \(-0.436075\pi\)
0.199478 + 0.979902i \(0.436075\pi\)
\(252\) 0 0
\(253\) −3.73361 −0.234730
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.2538 + 21.2242i 0.764372 + 1.32393i 0.940578 + 0.339577i \(0.110284\pi\)
−0.176206 + 0.984353i \(0.556383\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −21.1163 12.1915i −1.30208 0.751759i −0.321323 0.946970i \(-0.604128\pi\)
−0.980761 + 0.195211i \(0.937461\pi\)
\(264\) 0 0
\(265\) −9.31427 + 5.37760i −0.572171 + 0.330343i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.89049 0.603034 0.301517 0.953461i \(-0.402507\pi\)
0.301517 + 0.953461i \(0.402507\pi\)
\(270\) 0 0
\(271\) 5.89481i 0.358084i −0.983841 0.179042i \(-0.942700\pi\)
0.983841 0.179042i \(-0.0572998\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.915111 0.528340i 0.0551833 0.0318601i
\(276\) 0 0
\(277\) −11.6469 + 20.1731i −0.699796 + 1.21208i 0.268741 + 0.963213i \(0.413392\pi\)
−0.968537 + 0.248870i \(0.919941\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.7962 12.5840i −1.30025 0.750700i −0.319803 0.947484i \(-0.603617\pi\)
−0.980447 + 0.196784i \(0.936950\pi\)
\(282\) 0 0
\(283\) −8.62942 + 4.98220i −0.512966 + 0.296161i −0.734052 0.679093i \(-0.762373\pi\)
0.221086 + 0.975254i \(0.429040\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.74463 −0.102626
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.79065 + 11.7618i 0.396714 + 0.687129i 0.993318 0.115406i \(-0.0368171\pi\)
−0.596604 + 0.802536i \(0.703484\pi\)
\(294\) 0 0
\(295\) −5.56147 + 9.63275i −0.323801 + 0.560841i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.82774 11.8260i 0.394858 0.683915i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.11455i 0.292858i
\(306\) 0 0
\(307\) 16.9849i 0.969381i −0.874686 0.484691i \(-0.838932\pi\)
0.874686 0.484691i \(-0.161068\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.00148940 + 0.00257972i 8.44563e−5 + 0.000146283i 0.866068 0.499927i \(-0.166640\pi\)
−0.865983 + 0.500073i \(0.833306\pi\)
\(312\) 0 0
\(313\) −10.6154 6.12878i −0.600015 0.346419i 0.169032 0.985611i \(-0.445936\pi\)
−0.769048 + 0.639191i \(0.779269\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.0008 + 11.5475i 1.12336 + 0.648571i 0.942256 0.334894i \(-0.108700\pi\)
0.181102 + 0.983464i \(0.442034\pi\)
\(318\) 0 0
\(319\) 1.45014 + 2.51172i 0.0811922 + 0.140629i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 23.0911i 1.28482i
\(324\) 0 0
\(325\) 3.86475i 0.214378i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.73106 2.99829i 0.0951479 0.164801i −0.814522 0.580132i \(-0.803001\pi\)
0.909670 + 0.415331i \(0.136334\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.5145 30.3359i 0.956917 1.65743i
\(336\) 0 0
\(337\) −9.13018 15.8139i −0.497352 0.861440i 0.502643 0.864494i \(-0.332361\pi\)
−0.999995 + 0.00305455i \(0.999028\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.47640 0.0799518
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.62386 + 2.66959i −0.248222 + 0.143311i −0.618950 0.785431i \(-0.712442\pi\)
0.370728 + 0.928741i \(0.379108\pi\)
\(348\) 0 0
\(349\) −0.0136817 0.00789914i −0.000732365 0.000422831i 0.499634 0.866237i \(-0.333468\pi\)
−0.500366 + 0.865814i \(0.666801\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −17.1543 + 29.7121i −0.913029 + 1.58141i −0.103268 + 0.994654i \(0.532930\pi\)
−0.809761 + 0.586760i \(0.800403\pi\)
\(354\) 0 0
\(355\) −20.2184 + 11.6731i −1.07308 + 0.619544i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.26718i 0.330769i 0.986229 + 0.165385i \(0.0528865\pi\)
−0.986229 + 0.165385i \(0.947113\pi\)
\(360\) 0 0
\(361\) −15.9517 −0.839561
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.52811 + 2.61430i −0.237012 + 0.136839i
\(366\) 0 0
\(367\) −16.4888 9.51984i −0.860711 0.496931i 0.00353959 0.999994i \(-0.498873\pi\)
−0.864250 + 0.503062i \(0.832207\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.41901 9.38600i −0.280586 0.485989i 0.690943 0.722909i \(-0.257195\pi\)
−0.971529 + 0.236920i \(0.923862\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.6076 −0.546320
\(378\) 0 0
\(379\) 0.700312 0.0359726 0.0179863 0.999838i \(-0.494274\pi\)
0.0179863 + 0.999838i \(0.494274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.0235 32.9497i −0.972056 1.68365i −0.689327 0.724451i \(-0.742094\pi\)
−0.282729 0.959200i \(-0.591240\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.6958 + 9.63934i 0.846512 + 0.488734i 0.859473 0.511182i \(-0.170792\pi\)
−0.0129603 + 0.999916i \(0.504125\pi\)
\(390\) 0 0
\(391\) −30.1874 + 17.4287i −1.52664 + 0.881407i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.37919 −0.119710
\(396\) 0 0
\(397\) 20.0468i 1.00612i −0.864252 0.503059i \(-0.832208\pi\)
0.864252 0.503059i \(-0.167792\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.4232 15.2554i 1.31951 0.761820i 0.335861 0.941912i \(-0.390973\pi\)
0.983650 + 0.180092i \(0.0576395\pi\)
\(402\) 0 0
\(403\) −2.69993 + 4.67642i −0.134493 + 0.232949i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.29598 1.90294i −0.163376 0.0943250i
\(408\) 0 0
\(409\) −0.150631 + 0.0869667i −0.00744821 + 0.00430023i −0.503719 0.863867i \(-0.668035\pi\)
0.496271 + 0.868168i \(0.334702\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 18.9772 0.931554
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0690 + 24.3682i 0.687316 + 1.19047i 0.972703 + 0.232054i \(0.0745445\pi\)
−0.285387 + 0.958412i \(0.592122\pi\)
\(420\) 0 0
\(421\) −1.56130 + 2.70424i −0.0760929 + 0.131797i −0.901561 0.432652i \(-0.857578\pi\)
0.825468 + 0.564449i \(0.190911\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.93264 8.54358i 0.239268 0.414424i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.91744i 0.477706i 0.971056 + 0.238853i \(0.0767715\pi\)
−0.971056 + 0.238853i \(0.923229\pi\)
\(432\) 0 0
\(433\) 17.1274i 0.823092i −0.911389 0.411546i \(-0.864989\pi\)
0.911389 0.411546i \(-0.135011\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −26.3808 45.6929i −1.26196 2.18579i
\(438\) 0 0
\(439\) −18.5795 10.7269i −0.886750 0.511965i −0.0138721 0.999904i \(-0.504416\pi\)
−0.872878 + 0.487938i \(0.837749\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.84340 3.37369i −0.277628 0.160289i 0.354721 0.934972i \(-0.384576\pi\)
−0.632349 + 0.774683i \(0.717909\pi\)
\(444\) 0 0
\(445\) 13.4111 + 23.2287i 0.635747 + 1.10115i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.81624i 0.274485i 0.990537 + 0.137243i \(0.0438240\pi\)
−0.990537 + 0.137243i \(0.956176\pi\)
\(450\) 0 0
\(451\) 0.888889i 0.0418562i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.6949 28.9164i 0.780954 1.35265i −0.150432 0.988620i \(-0.548066\pi\)
0.931386 0.364032i \(-0.118600\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.5154 32.0696i 0.862347 1.49363i −0.00730959 0.999973i \(-0.502327\pi\)
0.869657 0.493656i \(-0.164340\pi\)
\(462\) 0 0
\(463\) 10.5618 + 18.2935i 0.490848 + 0.850173i 0.999944 0.0105362i \(-0.00335383\pi\)
−0.509097 + 0.860709i \(0.670020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.6094 0.861141 0.430570 0.902557i \(-0.358312\pi\)
0.430570 + 0.902557i \(0.358312\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.18715 2.41745i 0.192525 0.111155i
\(474\) 0 0
\(475\) 12.9319 + 7.46624i 0.593356 + 0.342574i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7.16703 + 12.4137i −0.327470 + 0.567194i −0.982009 0.188834i \(-0.939529\pi\)
0.654539 + 0.756028i \(0.272863\pi\)
\(480\) 0 0
\(481\) 12.0549 6.95988i 0.549655 0.317343i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.634844i 0.0288268i
\(486\) 0 0
\(487\) 11.2966 0.511897 0.255949 0.966690i \(-0.417612\pi\)
0.255949 + 0.966690i \(0.417612\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.84097 5.10434i 0.398988 0.230356i −0.287059 0.957913i \(-0.592678\pi\)
0.686047 + 0.727557i \(0.259344\pi\)
\(492\) 0 0
\(493\) 23.4496 + 13.5387i 1.05612 + 0.609751i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9.56672 + 16.5701i 0.428265 + 0.741777i 0.996719 0.0809379i \(-0.0257915\pi\)
−0.568454 + 0.822715i \(0.692458\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −0.268917 −0.0119904 −0.00599520 0.999982i \(-0.501908\pi\)
−0.00599520 + 0.999982i \(0.501908\pi\)
\(504\) 0 0
\(505\) −39.1763 −1.74332
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.9439 18.9553i −0.485079 0.840181i 0.514774 0.857326i \(-0.327876\pi\)
−0.999853 + 0.0171449i \(0.994542\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25.5298 14.7396i −1.12498 0.649506i
\(516\) 0 0
\(517\) 0.641553 0.370401i 0.0282155 0.0162902i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.71215 0.0750105 0.0375053 0.999296i \(-0.488059\pi\)
0.0375053 + 0.999296i \(0.488059\pi\)
\(522\) 0 0
\(523\) 8.27136i 0.361681i 0.983512 + 0.180841i \(0.0578818\pi\)
−0.983512 + 0.180841i \(0.942118\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.9372 6.89193i 0.519992 0.300217i
\(528\) 0 0
\(529\) 28.3233 49.0574i 1.23145 2.13293i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.81550 1.62553i −0.121953 0.0704096i
\(534\) 0 0
\(535\) 15.1029 8.71966i 0.652955 0.376984i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.3993 0.877036 0.438518 0.898722i \(-0.355503\pi\)
0.438518 + 0.898722i \(0.355503\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.07875 + 12.2608i 0.303220 + 0.525193i
\(546\) 0 0
\(547\) 18.9630 32.8449i 0.810801 1.40435i −0.101503 0.994835i \(-0.532365\pi\)
0.912304 0.409513i \(-0.134301\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.4927 + 35.4943i −0.873017 + 1.51211i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.8493i 0.713926i 0.934119 + 0.356963i \(0.116188\pi\)
−0.934119 + 0.356963i \(0.883812\pi\)
\(558\) 0 0
\(559\) 17.6834i 0.747928i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.28035 14.3420i −0.348975 0.604443i 0.637093 0.770787i \(-0.280137\pi\)
−0.986068 + 0.166345i \(0.946804\pi\)
\(564\) 0 0
\(565\) 25.2162 + 14.5586i 1.06085 + 0.612484i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.49856 3.17460i −0.230512 0.133086i 0.380296 0.924865i \(-0.375822\pi\)
−0.610808 + 0.791779i \(0.709155\pi\)
\(570\) 0 0
\(571\) −22.8703 39.6125i −0.957092 1.65773i −0.729507 0.683973i \(-0.760250\pi\)
−0.227585 0.973758i \(-0.573083\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.5414i 0.940043i
\(576\) 0 0
\(577\) 17.7499i 0.738939i −0.929243 0.369470i \(-0.879539\pi\)
0.929243 0.369470i \(-0.120461\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.820082 + 1.42042i −0.0339643 + 0.0588280i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.41148 + 7.64091i −0.182081 + 0.315374i −0.942589 0.333955i \(-0.891617\pi\)
0.760508 + 0.649329i \(0.224950\pi\)
\(588\) 0 0
\(589\) 10.4319 + 18.0686i 0.429839 + 0.744503i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.49698 0.348929 0.174465 0.984663i \(-0.444181\pi\)
0.174465 + 0.984663i \(0.444181\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.21158 + 1.85421i −0.131222 + 0.0757609i −0.564174 0.825656i \(-0.690805\pi\)
0.432952 + 0.901417i \(0.357472\pi\)
\(600\) 0 0
\(601\) 6.14043 + 3.54518i 0.250473 + 0.144611i 0.619981 0.784617i \(-0.287140\pi\)
−0.369508 + 0.929228i \(0.620474\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −14.8482 + 25.7178i −0.603664 + 1.04558i
\(606\) 0 0
\(607\) 29.4396 16.9970i 1.19492 0.689886i 0.235500 0.971874i \(-0.424327\pi\)
0.959418 + 0.281988i \(0.0909939\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.70944i 0.109612i
\(612\) 0 0
\(613\) 23.3523 0.943190 0.471595 0.881815i \(-0.343678\pi\)
0.471595 + 0.881815i \(0.343678\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.0817 22.5638i 1.57337 0.908386i 0.577618 0.816307i \(-0.303982\pi\)
0.995752 0.0920787i \(-0.0293511\pi\)
\(618\) 0 0
\(619\) 7.97914 + 4.60676i 0.320709 + 0.185161i 0.651708 0.758470i \(-0.274053\pi\)
−0.331000 + 0.943631i \(0.607386\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15.6247 + 27.0627i 0.624987 + 1.08251i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −35.5320 −1.41675
\(630\) 0 0
\(631\) 17.6136 0.701188 0.350594 0.936528i \(-0.385980\pi\)
0.350594 + 0.936528i \(0.385980\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −14.1156 24.4489i −0.560160 0.970226i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5759 + 9.57009i 0.654708 + 0.377996i 0.790258 0.612775i \(-0.209947\pi\)
−0.135550 + 0.990771i \(0.543280\pi\)
\(642\) 0 0
\(643\) −2.01129 + 1.16122i −0.0793177 + 0.0457941i −0.539134 0.842220i \(-0.681248\pi\)
0.459817 + 0.888014i \(0.347915\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.8620 1.01674 0.508370 0.861139i \(-0.330248\pi\)
0.508370 + 0.861139i \(0.330248\pi\)
\(648\) 0 0
\(649\) 1.69625i 0.0665835i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.1140 + 11.6128i −0.787123 + 0.454446i −0.838949 0.544211i \(-0.816829\pi\)
0.0518258 + 0.998656i \(0.483496\pi\)
\(654\) 0 0
\(655\) −26.9681 + 46.7102i −1.05373 + 1.82512i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13.7002 7.90981i −0.533684 0.308122i 0.208832 0.977952i \(-0.433034\pi\)
−0.742515 + 0.669829i \(0.766367\pi\)
\(660\) 0 0
\(661\) −15.8006 + 9.12248i −0.614572 + 0.354823i −0.774753 0.632264i \(-0.782126\pi\)
0.160181 + 0.987088i \(0.448792\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −61.8697 −2.39560
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.389984 + 0.675472i 0.0150552 + 0.0260763i
\(672\) 0 0
\(673\) 14.4184 24.9733i 0.555787 0.962651i −0.442055 0.896988i \(-0.645750\pi\)
0.997842 0.0656633i \(-0.0209163\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16.7668 + 29.0409i −0.644400 + 1.11613i 0.340040 + 0.940411i \(0.389559\pi\)
−0.984440 + 0.175722i \(0.943774\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.0481i 0.843649i −0.906678 0.421824i \(-0.861390\pi\)
0.906678 0.421824i \(-0.138610\pi\)
\(684\) 0 0
\(685\) 14.7745i 0.564506i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.99941 5.19512i −0.114268 0.197918i
\(690\) 0 0
\(691\) −22.8662 13.2018i −0.869869 0.502219i −0.00256453 0.999997i \(-0.500816\pi\)
−0.867305 + 0.497777i \(0.834150\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −40.5569 23.4156i −1.53841 0.888203i
\(696\) 0 0
\(697\) 4.14938 + 7.18694i 0.157169 + 0.272225i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 20.5140i 0.774804i −0.921911 0.387402i \(-0.873373\pi\)
0.921911 0.387402i \(-0.126627\pi\)
\(702\) 0 0
\(703\) 53.7827i 2.02845i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.13054 5.42226i 0.117570 0.203637i −0.801234 0.598351i \(-0.795823\pi\)
0.918804 + 0.394714i \(0.129156\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.7476 + 27.2756i −0.589751 + 1.02148i
\(714\) 0 0
\(715\) 0.878041 + 1.52081i 0.0328369 + 0.0568751i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23.2223 −0.866045 −0.433023 0.901383i \(-0.642553\pi\)
−0.433023 + 0.901383i \(0.642553\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.1643 8.75512i 0.563189 0.325157i
\(726\) 0 0
\(727\) 2.50999 + 1.44914i 0.0930903 + 0.0537457i 0.545822 0.837901i \(-0.316217\pi\)
−0.452732 + 0.891647i \(0.649551\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 22.5696 39.0917i 0.834767 1.44586i
\(732\) 0 0
\(733\) −10.2963 + 5.94457i −0.380302 + 0.219568i −0.677950 0.735108i \(-0.737131\pi\)
0.297647 + 0.954676i \(0.403798\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.34190i 0.196772i
\(738\) 0 0
\(739\) −34.4509 −1.26730 −0.633648 0.773621i \(-0.718443\pi\)
−0.633648 + 0.773621i \(0.718443\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.44069 1.40913i 0.0895401 0.0516960i −0.454561 0.890715i \(-0.650204\pi\)
0.544101 + 0.839019i \(0.316871\pi\)
\(744\) 0 0
\(745\) −25.5447 14.7483i −0.935887 0.540335i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.86045 6.68649i −0.140870 0.243993i 0.786955 0.617011i \(-0.211657\pi\)
−0.927824 + 0.373017i \(0.878323\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.7517 0.755233
\(756\) 0 0
\(757\) 1.17924 0.0428603 0.0214302 0.999770i \(-0.493178\pi\)
0.0214302 + 0.999770i \(0.493178\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.56644 + 2.71316i 0.0567835 + 0.0983520i 0.893020 0.450017i \(-0.148582\pi\)
−0.836236 + 0.548369i \(0.815249\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.37276 3.10196i −0.193999 0.112005i
\(768\) 0 0
\(769\) 5.53497 3.19562i 0.199596 0.115237i −0.396871 0.917874i \(-0.629904\pi\)
0.596467 + 0.802637i \(0.296571\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 47.9558 1.72485 0.862425 0.506185i \(-0.168945\pi\)
0.862425 + 0.506185i \(0.168945\pi\)
\(774\) 0 0
\(775\) 8.91369i 0.320189i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.8784 + 6.28067i −0.389761 + 0.225028i
\(780\) 0 0
\(781\) −1.78015 + 3.08331i −0.0636987 + 0.110329i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 29.1661 + 16.8390i 1.04098 + 0.601011i
\(786\) 0 0
\(787\) 5.23136 3.02033i 0.186478 0.107663i −0.403855 0.914823i \(-0.632330\pi\)
0.590333 + 0.807160i \(0.298997\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.85269 −0.101302
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.782501 1.35533i −0.0277176 0.0480083i 0.851834 0.523812i \(-0.175491\pi\)
−0.879551 + 0.475804i \(0.842157\pi\)
\(798\) 0 0
\(799\) 3.45810 5.98961i 0.122339 0.211897i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.398681 + 0.690535i −0.0140691 + 0.0243685i
\(804\) 0 0
\(805\) 0 0