Properties

Label 1764.3.bk.g.1745.4
Level $1764$
Weight $3$
Character 1764.1745
Analytic conductor $48.066$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.bk (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.329365073333488765586374656.1
Defining polynomial: \(x^{16} - 199 x^{12} + 24960 x^{8} - 2913559 x^{4} + 214358881\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1745.4
Root \(3.29641 - 0.365632i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1745
Dual form 1764.3.bk.g.557.4

$q$-expansion

\(f(q)\) \(=\) \(q+(-3.08083 - 1.77872i) q^{5} +O(q^{10})\) \(q+(-3.08083 - 1.77872i) q^{5} +(8.03119 - 4.63681i) q^{11} -4.24264 q^{13} +(-2.11532 + 1.22128i) q^{17} +(-10.3749 + 17.9698i) q^{19} +(0.682720 + 0.394169i) q^{23} +(-6.17232 - 10.6908i) q^{25} +7.69694i q^{29} +(-11.7891 - 20.4193i) q^{31} +(11.6723 - 20.2170i) q^{37} +51.5574i q^{41} -49.3446 q^{43} +(13.6574 + 7.88512i) q^{47} +(65.1929 - 37.6391i) q^{53} -32.9903 q^{55} +(-13.6574 + 7.88512i) q^{59} +(4.01093 - 6.94714i) q^{61} +(13.0709 + 7.54647i) q^{65} +(43.6723 + 75.6427i) q^{67} +40.0614i q^{71} +(-17.2023 - 29.7952i) q^{73} +(-4.34463 + 7.52512i) q^{79} +115.115i q^{83} +8.68926 q^{85} +(81.6200 + 47.1234i) q^{89} +(63.9266 - 36.9080i) q^{95} -154.173 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 216q^{25} - 128q^{37} - 160q^{43} + 384q^{67} + 560q^{79} - 1120q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −3.08083 1.77872i −0.616166 0.355744i 0.159209 0.987245i \(-0.449106\pi\)
−0.775375 + 0.631501i \(0.782439\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8.03119 4.63681i 0.730108 0.421528i −0.0883536 0.996089i \(-0.528161\pi\)
0.818462 + 0.574561i \(0.194827\pi\)
\(12\) 0 0
\(13\) −4.24264 −0.326357 −0.163178 0.986597i \(-0.552175\pi\)
−0.163178 + 0.986597i \(0.552175\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.11532 + 1.22128i −0.124431 + 0.0718400i −0.560923 0.827868i \(-0.689554\pi\)
0.436493 + 0.899708i \(0.356220\pi\)
\(18\) 0 0
\(19\) −10.3749 + 17.9698i −0.546047 + 0.945781i 0.452493 + 0.891768i \(0.350535\pi\)
−0.998540 + 0.0540134i \(0.982799\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.682720 + 0.394169i 0.0296835 + 0.0171378i 0.514768 0.857329i \(-0.327878\pi\)
−0.485085 + 0.874467i \(0.661211\pi\)
\(24\) 0 0
\(25\) −6.17232 10.6908i −0.246893 0.427631i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.69694i 0.265412i 0.991155 + 0.132706i \(0.0423666\pi\)
−0.991155 + 0.132706i \(0.957633\pi\)
\(30\) 0 0
\(31\) −11.7891 20.4193i −0.380294 0.658688i 0.610810 0.791777i \(-0.290844\pi\)
−0.991104 + 0.133089i \(0.957510\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.6723 20.2170i 0.315468 0.546407i −0.664069 0.747671i \(-0.731172\pi\)
0.979537 + 0.201265i \(0.0645052\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 51.5574i 1.25750i 0.777608 + 0.628749i \(0.216433\pi\)
−0.777608 + 0.628749i \(0.783567\pi\)
\(42\) 0 0
\(43\) −49.3446 −1.14755 −0.573775 0.819013i \(-0.694522\pi\)
−0.573775 + 0.819013i \(0.694522\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.6574 + 7.88512i 0.290584 + 0.167769i 0.638205 0.769866i \(-0.279677\pi\)
−0.347621 + 0.937635i \(0.613011\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 65.1929 37.6391i 1.23005 0.710172i 0.263013 0.964792i \(-0.415284\pi\)
0.967041 + 0.254620i \(0.0819505\pi\)
\(54\) 0 0
\(55\) −32.9903 −0.599824
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.6574 + 7.88512i −0.231482 + 0.133646i −0.611256 0.791433i \(-0.709335\pi\)
0.379774 + 0.925079i \(0.376002\pi\)
\(60\) 0 0
\(61\) 4.01093 6.94714i 0.0657530 0.113888i −0.831275 0.555862i \(-0.812388\pi\)
0.897028 + 0.441974i \(0.145722\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.0709 + 7.54647i 0.201090 + 0.116099i
\(66\) 0 0
\(67\) 43.6723 + 75.6427i 0.651826 + 1.12900i 0.982680 + 0.185313i \(0.0593299\pi\)
−0.330854 + 0.943682i \(0.607337\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 40.0614i 0.564245i 0.959378 + 0.282123i \(0.0910385\pi\)
−0.959378 + 0.282123i \(0.908962\pi\)
\(72\) 0 0
\(73\) −17.2023 29.7952i −0.235648 0.408154i 0.723813 0.689996i \(-0.242388\pi\)
−0.959461 + 0.281843i \(0.909054\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.34463 + 7.52512i −0.0549953 + 0.0952547i −0.892212 0.451616i \(-0.850848\pi\)
0.837217 + 0.546871i \(0.184181\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 115.115i 1.38693i 0.720492 + 0.693463i \(0.243916\pi\)
−0.720492 + 0.693463i \(0.756084\pi\)
\(84\) 0 0
\(85\) 8.68926 0.102227
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 81.6200 + 47.1234i 0.917079 + 0.529476i 0.882702 0.469933i \(-0.155722\pi\)
0.0343770 + 0.999409i \(0.489055\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 63.9266 36.9080i 0.672912 0.388506i
\(96\) 0 0
\(97\) −154.173 −1.58941 −0.794707 0.606993i \(-0.792376\pi\)
−0.794707 + 0.606993i \(0.792376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −165.724 + 95.6808i −1.64083 + 0.947335i −0.660293 + 0.751008i \(0.729568\pi\)
−0.980539 + 0.196326i \(0.937099\pi\)
\(102\) 0 0
\(103\) −40.5368 + 70.2118i −0.393561 + 0.681668i −0.992916 0.118815i \(-0.962090\pi\)
0.599355 + 0.800483i \(0.295424\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −143.879 83.0684i −1.34466 0.776340i −0.357173 0.934038i \(-0.616259\pi\)
−0.987487 + 0.157698i \(0.949593\pi\)
\(108\) 0 0
\(109\) 64.3446 + 111.448i 0.590318 + 1.02246i 0.994189 + 0.107645i \(0.0343309\pi\)
−0.403872 + 0.914816i \(0.632336\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 188.138i 1.66494i 0.554069 + 0.832471i \(0.313074\pi\)
−0.554069 + 0.832471i \(0.686926\pi\)
\(114\) 0 0
\(115\) −1.40223 2.42873i −0.0121933 0.0211194i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −17.5000 + 30.3109i −0.144628 + 0.250503i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 132.851i 1.06281i
\(126\) 0 0
\(127\) −40.6554 −0.320121 −0.160061 0.987107i \(-0.551169\pi\)
−0.160061 + 0.987107i \(0.551169\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 83.1384 + 48.0000i 0.634645 + 0.366412i 0.782549 0.622590i \(-0.213919\pi\)
−0.147904 + 0.989002i \(0.547253\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 100.830 58.2145i 0.735988 0.424923i −0.0846205 0.996413i \(-0.526968\pi\)
0.820609 + 0.571490i \(0.193634\pi\)
\(138\) 0 0
\(139\) −99.9218 −0.718862 −0.359431 0.933172i \(-0.617029\pi\)
−0.359431 + 0.933172i \(0.617029\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −34.0735 + 19.6723i −0.238276 + 0.137569i
\(144\) 0 0
\(145\) 13.6907 23.7130i 0.0944186 0.163538i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 94.0655 + 54.3087i 0.631312 + 0.364488i 0.781260 0.624206i \(-0.214577\pi\)
−0.149948 + 0.988694i \(0.547911\pi\)
\(150\) 0 0
\(151\) 112.017 + 194.019i 0.741834 + 1.28489i 0.951659 + 0.307155i \(0.0993771\pi\)
−0.209825 + 0.977739i \(0.567290\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 83.8780i 0.541149i
\(156\) 0 0
\(157\) −52.5576 91.0325i −0.334762 0.579825i 0.648677 0.761064i \(-0.275323\pi\)
−0.983439 + 0.181239i \(0.941989\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.9831 19.0232i 0.0673807 0.116707i −0.830367 0.557217i \(-0.811869\pi\)
0.897748 + 0.440510i \(0.145202\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 311.838i 1.86729i 0.358195 + 0.933647i \(0.383392\pi\)
−0.358195 + 0.933647i \(0.616608\pi\)
\(168\) 0 0
\(169\) −151.000 −0.893491
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 56.0079 + 32.3362i 0.323745 + 0.186914i 0.653061 0.757306i \(-0.273485\pi\)
−0.329316 + 0.944220i \(0.606818\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 289.483 167.133i 1.61722 0.933703i 0.629587 0.776930i \(-0.283224\pi\)
0.987635 0.156773i \(-0.0501091\pi\)
\(180\) 0 0
\(181\) −229.590 −1.26845 −0.634226 0.773147i \(-0.718681\pi\)
−0.634226 + 0.773147i \(0.718681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −71.9209 + 41.5235i −0.388762 + 0.224452i
\(186\) 0 0
\(187\) −11.3257 + 19.6167i −0.0605652 + 0.104902i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −65.4535 37.7896i −0.342688 0.197851i 0.318772 0.947832i \(-0.396730\pi\)
−0.661460 + 0.749980i \(0.730063\pi\)
\(192\) 0 0
\(193\) −4.68926 8.12204i −0.0242967 0.0420831i 0.853621 0.520894i \(-0.174401\pi\)
−0.877918 + 0.478811i \(0.841068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 38.5566i 0.195719i −0.995200 0.0978594i \(-0.968800\pi\)
0.995200 0.0978594i \(-0.0311996\pi\)
\(198\) 0 0
\(199\) −32.5509 56.3798i −0.163572 0.283315i 0.772575 0.634923i \(-0.218968\pi\)
−0.936147 + 0.351608i \(0.885635\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 91.7062 158.840i 0.447347 0.774828i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 192.426i 0.920697i
\(210\) 0 0
\(211\) −176.068 −0.834444 −0.417222 0.908804i \(-0.636996\pi\)
−0.417222 + 0.908804i \(0.636996\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 152.023 + 87.7702i 0.707082 + 0.408234i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.97454 5.18146i 0.0406088 0.0234455i
\(222\) 0 0
\(223\) 146.199 0.655602 0.327801 0.944747i \(-0.393692\pi\)
0.327801 + 0.944747i \(0.393692\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.3815 + 12.3446i −0.0941918 + 0.0543816i −0.546356 0.837553i \(-0.683985\pi\)
0.452164 + 0.891935i \(0.350652\pi\)
\(228\) 0 0
\(229\) −104.919 + 181.726i −0.458164 + 0.793563i −0.998864 0.0476526i \(-0.984826\pi\)
0.540700 + 0.841215i \(0.318159\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 34.6942 + 20.0307i 0.148902 + 0.0859687i 0.572600 0.819835i \(-0.305935\pi\)
−0.423698 + 0.905804i \(0.639268\pi\)
\(234\) 0 0
\(235\) −28.0508 48.5855i −0.119365 0.206747i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 229.662i 0.960928i 0.877014 + 0.480464i \(0.159532\pi\)
−0.877014 + 0.480464i \(0.840468\pi\)
\(240\) 0 0
\(241\) −190.711 330.321i −0.791332 1.37063i −0.925142 0.379621i \(-0.876054\pi\)
0.133810 0.991007i \(-0.457279\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 44.0169 76.2396i 0.178206 0.308662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 72.2636i 0.287903i −0.989585 0.143951i \(-0.954019\pi\)
0.989585 0.143951i \(-0.0459809\pi\)
\(252\) 0 0
\(253\) 7.31074 0.0288962
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −192.899 111.370i −0.750578 0.433347i 0.0753246 0.997159i \(-0.476001\pi\)
−0.825903 + 0.563813i \(0.809334\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 278.882 161.013i 1.06039 0.612215i 0.134848 0.990866i \(-0.456945\pi\)
0.925540 + 0.378651i \(0.123612\pi\)
\(264\) 0 0
\(265\) −267.798 −1.01056
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −266.382 + 153.796i −0.990267 + 0.571731i −0.905354 0.424657i \(-0.860395\pi\)
−0.0849131 + 0.996388i \(0.527061\pi\)
\(270\) 0 0
\(271\) −181.055 + 313.597i −0.668101 + 1.15718i 0.310334 + 0.950628i \(0.399559\pi\)
−0.978435 + 0.206557i \(0.933774\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −99.1421 57.2397i −0.360517 0.208144i
\(276\) 0 0
\(277\) −14.3446 24.8456i −0.0517857 0.0896954i 0.838971 0.544177i \(-0.183158\pi\)
−0.890756 + 0.454481i \(0.849825\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 89.7693i 0.319464i −0.987160 0.159732i \(-0.948937\pi\)
0.987160 0.159732i \(-0.0510629\pi\)
\(282\) 0 0
\(283\) 150.869 + 261.313i 0.533107 + 0.923369i 0.999252 + 0.0386609i \(0.0123092\pi\)
−0.466145 + 0.884708i \(0.654357\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −141.517 + 245.115i −0.489678 + 0.848147i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 235.855i 0.804966i 0.915428 + 0.402483i \(0.131853\pi\)
−0.915428 + 0.402483i \(0.868147\pi\)
\(294\) 0 0
\(295\) 56.1017 0.190175
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.89654 1.67232i −0.00968741 0.00559303i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.7140 + 14.2686i −0.0810296 + 0.0467824i
\(306\) 0 0
\(307\) −68.8810 −0.224368 −0.112184 0.993687i \(-0.535785\pi\)
−0.112184 + 0.993687i \(0.535785\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 432.931 249.953i 1.39206 0.803707i 0.398518 0.917160i \(-0.369525\pi\)
0.993543 + 0.113453i \(0.0361912\pi\)
\(312\) 0 0
\(313\) 144.006 249.426i 0.460083 0.796888i −0.538881 0.842382i \(-0.681153\pi\)
0.998965 + 0.0454940i \(0.0144862\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 142.972 + 82.5451i 0.451017 + 0.260395i 0.708259 0.705952i \(-0.249481\pi\)
−0.257243 + 0.966347i \(0.582814\pi\)
\(318\) 0 0
\(319\) 35.6893 + 61.8156i 0.111879 + 0.193779i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 50.6826i 0.156912i
\(324\) 0 0
\(325\) 26.1869 + 45.3571i 0.0805751 + 0.139560i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 132.655 229.766i 0.400772 0.694157i −0.593048 0.805167i \(-0.702075\pi\)
0.993819 + 0.111011i \(0.0354088\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 310.723i 0.927532i
\(336\) 0 0
\(337\) 220.655 0.654764 0.327382 0.944892i \(-0.393834\pi\)
0.327382 + 0.944892i \(0.393834\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −189.361 109.328i −0.555311 0.320609i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −162.151 + 93.6177i −0.467293 + 0.269792i −0.715106 0.699016i \(-0.753622\pi\)
0.247813 + 0.968808i \(0.420288\pi\)
\(348\) 0 0
\(349\) 318.270 0.911948 0.455974 0.889993i \(-0.349291\pi\)
0.455974 + 0.889993i \(0.349291\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −146.502 + 84.5829i −0.415019 + 0.239611i −0.692944 0.720991i \(-0.743687\pi\)
0.277925 + 0.960603i \(0.410353\pi\)
\(354\) 0 0
\(355\) 71.2580 123.422i 0.200727 0.347669i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −96.2128 55.5485i −0.268002 0.154731i 0.359977 0.932961i \(-0.382784\pi\)
−0.627979 + 0.778230i \(0.716118\pi\)
\(360\) 0 0
\(361\) −34.7768 60.2353i −0.0963347 0.166857i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 122.392i 0.335321i
\(366\) 0 0
\(367\) 40.5248 + 70.1910i 0.110422 + 0.191256i 0.915940 0.401314i \(-0.131446\pi\)
−0.805519 + 0.592571i \(0.798113\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −219.723 + 380.572i −0.589070 + 1.02030i 0.405284 + 0.914191i \(0.367172\pi\)
−0.994355 + 0.106109i \(0.966161\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.6554i 0.0866190i
\(378\) 0 0
\(379\) −81.3785 −0.214719 −0.107360 0.994220i \(-0.534240\pi\)
−0.107360 + 0.994220i \(0.534240\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 399.226 + 230.493i 1.04237 + 0.601810i 0.920503 0.390736i \(-0.127780\pi\)
0.121864 + 0.992547i \(0.461113\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −480.866 + 277.628i −1.23616 + 0.713697i −0.968307 0.249764i \(-0.919647\pi\)
−0.267852 + 0.963460i \(0.586314\pi\)
\(390\) 0 0
\(391\) −1.92556 −0.00492471
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 26.7702 15.4558i 0.0677726 0.0391285i
\(396\) 0 0
\(397\) 152.064 263.382i 0.383033 0.663432i −0.608462 0.793583i \(-0.708213\pi\)
0.991494 + 0.130152i \(0.0415464\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 213.465 + 123.244i 0.532333 + 0.307342i 0.741966 0.670438i \(-0.233894\pi\)
−0.209633 + 0.977780i \(0.567227\pi\)
\(402\) 0 0
\(403\) 50.0169 + 86.6319i 0.124112 + 0.214967i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 216.489i 0.531915i
\(408\) 0 0
\(409\) −7.30278 12.6488i −0.0178552 0.0309261i 0.856960 0.515383i \(-0.172350\pi\)
−0.874815 + 0.484457i \(0.839017\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 204.757 354.650i 0.493390 0.854577i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 619.804i 1.47925i −0.673021 0.739623i \(-0.735004\pi\)
0.673021 0.739623i \(-0.264996\pi\)
\(420\) 0 0
\(421\) −307.379 −0.730115 −0.365058 0.930985i \(-0.618951\pi\)
−0.365058 + 0.930985i \(0.618951\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 26.1128 + 15.0763i 0.0614420 + 0.0354736i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −452.837 + 261.446i −1.05067 + 0.606603i −0.922836 0.385193i \(-0.874135\pi\)
−0.127831 + 0.991796i \(0.540801\pi\)
\(432\) 0 0
\(433\) 547.444 1.26431 0.632153 0.774844i \(-0.282171\pi\)
0.632153 + 0.774844i \(0.282171\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −14.1663 + 8.17891i −0.0324171 + 0.0187161i
\(438\) 0 0
\(439\) 258.386 447.537i 0.588578 1.01945i −0.405841 0.913944i \(-0.633021\pi\)
0.994419 0.105503i \(-0.0336452\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −474.560 273.987i −1.07124 0.618481i −0.142721 0.989763i \(-0.545585\pi\)
−0.928520 + 0.371282i \(0.878918\pi\)
\(444\) 0 0
\(445\) −167.638 290.358i −0.376716 0.652490i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 575.681i 1.28214i −0.767482 0.641070i \(-0.778491\pi\)
0.767482 0.641070i \(-0.221509\pi\)
\(450\) 0 0
\(451\) 239.062 + 414.068i 0.530071 + 0.918110i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −203.412 + 352.321i −0.445104 + 0.770942i −0.998059 0.0622683i \(-0.980167\pi\)
0.552956 + 0.833211i \(0.313500\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 209.524i 0.454498i −0.973837 0.227249i \(-0.927027\pi\)
0.973837 0.227249i \(-0.0729731\pi\)
\(462\) 0 0
\(463\) 870.169 1.87942 0.939708 0.341978i \(-0.111097\pi\)
0.939708 + 0.341978i \(0.111097\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −201.088 116.098i −0.430594 0.248604i 0.269006 0.963139i \(-0.413305\pi\)
−0.699600 + 0.714535i \(0.746638\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −396.296 + 228.802i −0.837835 + 0.483724i
\(474\) 0 0
\(475\) 256.148 0.539260
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 770.453 444.821i 1.60846 0.928645i 0.618746 0.785591i \(-0.287641\pi\)
0.989715 0.143054i \(-0.0456923\pi\)
\(480\) 0 0
\(481\) −49.5214 + 85.7737i −0.102955 + 0.178324i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 474.982 + 274.231i 0.979344 + 0.565425i
\(486\) 0 0
\(487\) 32.7062 + 56.6488i 0.0671585 + 0.116322i 0.897649 0.440710i \(-0.145273\pi\)
−0.830491 + 0.557032i \(0.811940\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 82.4878i 0.168000i −0.996466 0.0839998i \(-0.973230\pi\)
0.996466 0.0839998i \(-0.0267695\pi\)
\(492\) 0 0
\(493\) −9.40013 16.2815i −0.0190672 0.0330254i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 33.3277 57.7252i 0.0667889 0.115682i −0.830697 0.556724i \(-0.812058\pi\)
0.897486 + 0.441043i \(0.145391\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 592.885i 1.17870i −0.807879 0.589349i \(-0.799384\pi\)
0.807879 0.589349i \(-0.200616\pi\)
\(504\) 0 0
\(505\) 680.757 1.34803
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −673.701 388.961i −1.32358 0.764168i −0.339280 0.940685i \(-0.610183\pi\)
−0.984297 + 0.176518i \(0.943517\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 249.774 144.207i 0.484998 0.280014i
\(516\) 0 0
\(517\) 146.247 0.282877
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −375.133 + 216.583i −0.720024 + 0.415706i −0.814762 0.579796i \(-0.803132\pi\)
0.0947375 + 0.995502i \(0.469799\pi\)
\(522\) 0 0
\(523\) −138.617 + 240.092i −0.265042 + 0.459066i −0.967575 0.252585i \(-0.918719\pi\)
0.702533 + 0.711651i \(0.252052\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 49.8755 + 28.7956i 0.0946404 + 0.0546406i
\(528\) 0 0
\(529\) −264.189 457.589i −0.499413 0.865008i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 218.740i 0.410393i
\(534\) 0 0
\(535\) 295.511 + 511.840i 0.552356 + 0.956709i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −360.379 + 624.194i −0.666134 + 1.15378i 0.312842 + 0.949805i \(0.398719\pi\)
−0.978977 + 0.203973i \(0.934615\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 457.804i 0.840008i
\(546\) 0 0
\(547\) −744.825 −1.36165 −0.680827 0.732444i \(-0.738379\pi\)
−0.680827 + 0.732444i \(0.738379\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −138.313 79.8550i −0.251022 0.144927i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −620.101 + 358.016i −1.11329 + 0.642757i −0.939679 0.342057i \(-0.888876\pi\)
−0.173609 + 0.984815i \(0.555543\pi\)
\(558\) 0 0
\(559\) 209.352 0.374511
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 650.256 375.426i 1.15498 0.666831i 0.204888 0.978786i \(-0.434317\pi\)
0.950097 + 0.311955i \(0.100984\pi\)
\(564\) 0 0
\(565\) 334.645 579.623i 0.592293 1.02588i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 147.006 + 84.8741i 0.258359 + 0.149164i 0.623586 0.781755i \(-0.285675\pi\)
−0.365227 + 0.930919i \(0.619009\pi\)
\(570\) 0 0
\(571\) −313.757 543.443i −0.549487 0.951739i −0.998310 0.0581185i \(-0.981490\pi\)
0.448823 0.893621i \(-0.351843\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.73173i 0.0169247i
\(576\) 0 0
\(577\) −82.3160 142.576i −0.142662 0.247098i 0.785836 0.618435i \(-0.212233\pi\)
−0.928498 + 0.371337i \(0.878900\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 349.051 604.574i 0.598715 1.03700i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1040.76i 1.77301i −0.462719 0.886505i \(-0.653126\pi\)
0.462719 0.886505i \(-0.346874\pi\)
\(588\) 0 0
\(589\) 489.243 0.830633
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −543.797 313.961i −0.917027 0.529446i −0.0343417 0.999410i \(-0.510933\pi\)
−0.882685 + 0.469964i \(0.844267\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −228.485 + 131.916i −0.381445 + 0.220227i −0.678447 0.734650i \(-0.737347\pi\)
0.297002 + 0.954877i \(0.404013\pi\)
\(600\) 0 0
\(601\) 502.094 0.835431 0.417715 0.908578i \(-0.362831\pi\)
0.417715 + 0.908578i \(0.362831\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 107.829 62.2552i 0.178230 0.102901i
\(606\) 0 0
\(607\) −499.801 + 865.680i −0.823395 + 1.42616i 0.0797451 + 0.996815i \(0.474589\pi\)
−0.903140 + 0.429346i \(0.858744\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −57.9436 33.4537i −0.0948340 0.0547524i
\(612\) 0 0
\(613\) −460.085 796.890i −0.750546 1.29998i −0.947558 0.319583i \(-0.896457\pi\)
0.197012 0.980401i \(-0.436876\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 520.713i 0.843943i −0.906609 0.421972i \(-0.861338\pi\)
0.906609 0.421972i \(-0.138662\pi\)
\(618\) 0 0
\(619\) 119.305 + 206.643i 0.192739 + 0.333833i 0.946157 0.323708i \(-0.104930\pi\)
−0.753418 + 0.657542i \(0.771596\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 81.9972 142.023i 0.131195 0.227237i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 57.0207i 0.0906529i
\(630\) 0 0
\(631\) 534.034 0.846329 0.423165 0.906053i \(-0.360919\pi\)
0.423165 + 0.906053i \(0.360919\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 125.252 + 72.3145i 0.197248 + 0.113881i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −868.646 + 501.513i −1.35514 + 0.782392i −0.988964 0.148153i \(-0.952667\pi\)
−0.366178 + 0.930545i \(0.619334\pi\)
\(642\) 0 0
\(643\) 744.923 1.15851 0.579256 0.815146i \(-0.303343\pi\)
0.579256 + 0.815146i \(0.303343\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 772.015 445.723i 1.19322 0.688908i 0.234187 0.972192i \(-0.424757\pi\)
0.959036 + 0.283284i \(0.0914239\pi\)
\(648\) 0 0
\(649\) −73.1236 + 126.654i −0.112671 + 0.195152i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 70.5924 + 40.7565i 0.108105 + 0.0624143i 0.553078 0.833130i \(-0.313453\pi\)
−0.444973 + 0.895544i \(0.646787\pi\)
\(654\) 0 0
\(655\) −170.757 295.760i −0.260698 0.451542i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 185.430i 0.281380i −0.990054 0.140690i \(-0.955068\pi\)
0.990054 0.140690i \(-0.0449322\pi\)
\(660\) 0 0
\(661\) −272.783 472.475i −0.412683 0.714788i 0.582499 0.812831i \(-0.302075\pi\)
−0.995182 + 0.0980435i \(0.968742\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.03389 + 5.25486i −0.00454857 + 0.00787835i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 74.3917i 0.110867i
\(672\) 0 0
\(673\) −994.136 −1.47717 −0.738585 0.674160i \(-0.764506\pi\)
−0.738585 + 0.674160i \(0.764506\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −181.821 104.975i −0.268569 0.155058i 0.359668 0.933080i \(-0.382890\pi\)
−0.628237 + 0.778022i \(0.716223\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 355.619 205.317i 0.520672 0.300610i −0.216538 0.976274i \(-0.569476\pi\)
0.737209 + 0.675664i \(0.236143\pi\)
\(684\) 0 0
\(685\) −414.189 −0.604655
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −276.590 + 159.689i −0.401437 + 0.231770i
\(690\) 0 0
\(691\) 662.339 1147.21i 0.958523 1.66021i 0.232431 0.972613i \(-0.425332\pi\)
0.726092 0.687597i \(-0.241335\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 307.842 + 177.733i 0.442938 + 0.255731i
\(696\) 0 0
\(697\) −62.9661 109.060i −0.0903387 0.156471i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 580.339i 0.827873i −0.910306 0.413936i \(-0.864154\pi\)
0.910306 0.413936i \(-0.135846\pi\)
\(702\) 0 0
\(703\) 242.198 + 419.499i 0.344521 + 0.596727i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 163.689 283.518i 0.230873 0.399885i −0.727192 0.686434i \(-0.759175\pi\)
0.958065 + 0.286550i \(0.0925083\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.5876i 0.0260695i
\(714\) 0 0
\(715\) 139.966 0.195757
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 339.821 + 196.196i 0.472630 + 0.272873i 0.717340 0.696723i \(-0.245359\pi\)
−0.244710 + 0.969596i \(0.578693\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 82.2862 47.5080i 0.113498 0.0655282i
\(726\) 0 0
\(727\) 1340.99 1.84455 0.922274 0.386537i \(-0.126329\pi\)
0.922274 + 0.386537i \(0.126329\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 104.380 60.2636i 0.142790 0.0824400i
\(732\) 0 0
\(733\) −65.2016 + 112.933i −0.0889518 + 0.154069i −0.907068 0.420983i \(-0.861685\pi\)
0.818117 + 0.575052i \(0.195018\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 701.481 + 405.000i 0.951806 + 0.549526i
\(738\) 0 0
\(739\) 101.017 + 174.966i 0.136694 + 0.236761i 0.926243 0.376926i \(-0.123019\pi\)
−0.789549 + 0.613687i \(0.789686\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 821.453i 1.10559i −0.833317 0.552795i \(-0.813561\pi\)
0.833317 0.552795i \(-0.186439\pi\)
\(744\) 0 0
\(745\) −193.200 334.632i −0.259329 0.449171i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −402.068 + 696.402i −0.535377 + 0.927299i 0.463768 + 0.885956i \(0.346497\pi\)
−0.999145 + 0.0413429i \(0.986836\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 796.987i 1.05561i
\(756\) 0 0
\(757\) 518.000 0.684280 0.342140 0.939649i \(-0.388848\pi\)
0.342140 + 0.939649i \(0.388848\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −583.576 336.927i −0.766854 0.442743i 0.0648975 0.997892i \(-0.479328\pi\)
−0.831751 + 0.555149i \(0.812661\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 57.9436 33.4537i 0.0755457 0.0436164i
\(768\) 0 0
\(769\) −653.023 −0.849185 −0.424592 0.905385i \(-0.639583\pi\)
−0.424592 + 0.905385i \(0.639583\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 399.499 230.651i 0.516816 0.298384i −0.218815 0.975766i \(-0.570219\pi\)
0.735631 + 0.677383i \(0.236886\pi\)
\(774\) 0 0
\(775\) −145.532 + 252.069i −0.187783 + 0.325250i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −926.479 534.903i −1.18932 0.686653i
\(780\) 0 0
\(781\) 185.757 + 321.741i 0.237845 + 0.411960i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 373.941i 0.476358i
\(786\) 0 0
\(787\) 396.563 + 686.867i 0.503892 + 0.872767i 0.999990 + 0.00450000i \(0.00143240\pi\)
−0.496098 + 0.868267i \(0.665234\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −17.0169 + 29.4742i −0.0214589 + 0.0371680i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 435.599i 0.546548i −0.961936 0.273274i \(-0.911893\pi\)
0.961936 0.273274i \(-0.0881066\pi\)
\(798\) 0 0
\(799\) −38.5198 −0.0482100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −276.309 159.527i −0.344096 0.198664i
\(804\) 0 0
\(805\)