Properties

Label 1764.3.bk.g.1745.6
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 1745.6
Root \(1.33156 + 3.03759i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1745
Dual form 1764.3.bk.g.557.6

$q$-expansion

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

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 0.775375 0.631501i \(-0.217561\pi\)
−0.159209 + 0.987245i \(0.550894\pi\)
\(6\) 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 \(-0.447825\pi\)
0.163178 + 0.986597i \(0.447825\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.11532 1.22128i 0.124431 0.0718400i −0.436493 0.899708i \(-0.643780\pi\)
0.560923 + 0.827868i \(0.310446\pi\)
\(18\) 0 0
\(19\) 10.3749 17.9698i 0.546047 0.945781i −0.452493 0.891768i \(-0.649465\pi\)
0.998540 0.0540134i \(-0.0172014\pi\)
\(20\) 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 0.991104 0.133089i \(-0.0424895\pi\)
−0.610810 + 0.791777i \(0.709156\pi\)
\(32\) 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 \(-0.783567\pi\)
0.777608 0.628749i \(-0.216433\pi\)
\(42\) 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 0.347621 0.937635i \(-0.386989\pi\)
−0.638205 + 0.769866i \(0.720323\pi\)
\(48\) 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 −0.379774 0.925079i \(-0.623998\pi\)
0.611256 + 0.791433i \(0.290665\pi\)
\(60\) 0 0
\(61\) −4.01093 + 6.94714i −0.0657530 + 0.113888i −0.897028 0.441974i \(-0.854278\pi\)
0.831275 + 0.555862i \(0.187612\pi\)
\(62\) 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 0.959461 0.281843i \(-0.0909456\pi\)
−0.723813 + 0.689996i \(0.757612\pi\)
\(74\) 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 \(-0.756084\pi\)
0.720492 0.693463i \(-0.243916\pi\)
\(84\) 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 −0.0343770 0.999409i \(-0.510945\pi\)
−0.882702 + 0.469933i \(0.844278\pi\)
\(90\) 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 \(-0.207624\pi\)
0.794707 + 0.606993i \(0.207624\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 165.724 95.6808i 1.64083 0.947335i 0.660293 0.751008i \(-0.270432\pi\)
0.980539 0.196326i \(-0.0629012\pi\)
\(102\) 0 0
\(103\) 40.5368 70.2118i 0.393561 0.681668i −0.599355 0.800483i \(-0.704576\pi\)
0.992916 + 0.118815i \(0.0379096\pi\)
\(104\) 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 0.147904 0.989002i \(-0.452747\pi\)
−0.782549 + 0.622590i \(0.786081\pi\)
\(132\) 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 \(-0.382971\pi\)
0.359431 + 0.933172i \(0.382971\pi\)
\(140\) 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 0.983439 0.181239i \(-0.0580108\pi\)
−0.648677 + 0.761064i \(0.724677\pi\)
\(158\) 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 \(-0.616608\pi\)
0.358195 0.933647i \(-0.383392\pi\)
\(168\) 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 0.329316 0.944220i \(-0.393182\pi\)
−0.653061 + 0.757306i \(0.726515\pi\)
\(174\) 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 \(-0.281319\pi\)
0.634226 + 0.773147i \(0.281319\pi\)
\(182\) 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 0.936147 0.351608i \(-0.114365\pi\)
−0.772575 + 0.634923i \(0.781032\pi\)
\(200\) 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 \(-0.606308\pi\)
−0.327801 + 0.944747i \(0.606308\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.3815 12.3446i 0.0941918 0.0543816i −0.452164 0.891935i \(-0.649348\pi\)
0.546356 + 0.837553i \(0.316015\pi\)
\(228\) 0 0
\(229\) 104.919 181.726i 0.458164 0.793563i −0.540700 0.841215i \(-0.681841\pi\)
0.998864 + 0.0476526i \(0.0151741\pi\)
\(230\) 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 \(0.123946\pi\)
−0.133810 + 0.991007i \(0.542721\pi\)
\(242\) 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 \(0.0459809\pi\)
−0.989585 + 0.143951i \(0.954019\pi\)
\(252\) 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 0.825903 0.563813i \(-0.190666\pi\)
−0.0753246 + 0.997159i \(0.523999\pi\)
\(258\) 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 0.0849131 0.996388i \(-0.472939\pi\)
0.905354 + 0.424657i \(0.139605\pi\)
\(270\) 0 0
\(271\) 181.055 313.597i 0.668101 1.15718i −0.310334 0.950628i \(-0.600441\pi\)
0.978435 0.206557i \(-0.0662258\pi\)
\(272\) 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 \(-0.987691\pi\)
0.466145 0.884708i \(-0.345643\pi\)
\(284\) 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 \(-0.868147\pi\)
0.915428 0.402483i \(-0.131853\pi\)
\(294\) 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 \(-0.464215\pi\)
0.112184 + 0.993687i \(0.464215\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −432.931 + 249.953i −1.39206 + 0.803707i −0.993543 0.113453i \(-0.963809\pi\)
−0.398518 + 0.917160i \(0.630475\pi\)
\(312\) 0 0
\(313\) −144.006 + 249.426i −0.460083 + 0.796888i −0.998965 0.0454940i \(-0.985514\pi\)
0.538881 + 0.842382i \(0.318847\pi\)
\(314\) 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 \(-0.650709\pi\)
−0.455974 + 0.889993i \(0.650709\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 146.502 84.5829i 0.415019 0.239611i −0.277925 0.960603i \(-0.589647\pi\)
0.692944 + 0.720991i \(0.256313\pi\)
\(354\) 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 0.805519 0.592571i \(-0.201887\pi\)
−0.915940 + 0.401314i \(0.868554\pi\)
\(368\) 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 −0.121864 0.992547i \(-0.538887\pi\)
−0.920503 + 0.390736i \(0.872220\pi\)
\(384\) 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 −0.991494 0.130152i \(-0.958454\pi\)
0.608462 + 0.793583i \(0.291787\pi\)
\(398\) 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 0.874815 0.484457i \(-0.160983\pi\)
−0.856960 + 0.515383i \(0.827650\pi\)
\(410\) 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 \(0.264996\pi\)
−0.673021 + 0.739623i \(0.735004\pi\)
\(420\) 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 \(-0.717829\pi\)
−0.632153 + 0.774844i \(0.717829\pi\)
\(434\) 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 \(0.366979\pi\)
−0.994419 + 0.105503i \(0.966355\pi\)
\(440\) 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 \(0.0729731\pi\)
−0.973837 + 0.227249i \(0.927027\pi\)
\(462\) 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 0.699600 0.714535i \(-0.253362\pi\)
−0.269006 + 0.963139i \(0.586695\pi\)
\(468\) 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 \(0.712359\pi\)
−0.989715 + 0.143054i \(0.954308\pi\)
\(480\) 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 \(0.200616\pi\)
−0.807879 + 0.589349i \(0.799384\pi\)
\(504\) 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 0.984297 0.176518i \(-0.0564833\pi\)
0.339280 + 0.940685i \(0.389817\pi\)
\(510\) 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 −0.0947375 0.995502i \(-0.530201\pi\)
0.814762 + 0.579796i \(0.196868\pi\)
\(522\) 0 0
\(523\) 138.617 240.092i 0.265042 0.459066i −0.702533 0.711651i \(-0.747948\pi\)
0.967575 + 0.252585i \(0.0812809\pi\)
\(524\) 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 −0.950097 0.311955i \(-0.899016\pi\)
−0.204888 + 0.978786i \(0.565683\pi\)
\(564\) 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 0.928498 0.371337i \(-0.121100\pi\)
−0.785836 + 0.618435i \(0.787767\pi\)
\(578\) 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 \(0.346874\pi\)
−0.462719 + 0.886505i \(0.653126\pi\)
\(588\) 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 0.882685 0.469964i \(-0.155733\pi\)
0.0343417 + 0.999410i \(0.489067\pi\)
\(594\) 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 \(-0.637169\pi\)
−0.417715 + 0.908578i \(0.637169\pi\)
\(602\) 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 \(-0.525411\pi\)
0.903140 0.429346i \(-0.141256\pi\)
\(608\) 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 0.753418 0.657542i \(-0.228404\pi\)
−0.946157 + 0.323708i \(0.895070\pi\)
\(620\) 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 \(-0.696657\pi\)
−0.579256 + 0.815146i \(0.696657\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −772.015 + 445.723i −1.19322 + 0.688908i −0.959036 0.283284i \(-0.908576\pi\)
−0.234187 + 0.972192i \(0.575243\pi\)
\(648\) 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 0.995182 0.0980435i \(-0.0312584\pi\)
−0.582499 + 0.812831i \(0.697925\pi\)
\(662\) 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 0.628237 0.778022i \(-0.283777\pi\)
−0.359668 + 0.933080i \(0.617110\pi\)
\(678\) 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 \(0.574668\pi\)
−0.726092 + 0.687597i \(0.758665\pi\)
\(692\) 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 0.244710 0.969596i \(-0.421307\pi\)
−0.717340 + 0.696723i \(0.754641\pi\)
\(720\) 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 \(-0.873671\pi\)
−0.922274 + 0.386537i \(0.873671\pi\)
\(728\) 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 −0.818117 0.575052i \(-0.804982\pi\)
0.907068 + 0.420983i \(0.138315\pi\)
\(734\) 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 0.831751 0.555149i \(-0.187339\pi\)
−0.0648975 + 0.997892i \(0.520672\pi\)
\(762\) 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 \(-0.360417\pi\)
0.424592 + 0.905385i \(0.360417\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −399.499 + 230.651i −0.516816 + 0.298384i −0.735631 0.677383i \(-0.763114\pi\)
0.218815 + 0.975766i \(0.429781\pi\)
\(774\) 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 \(-0.998568\pi\)
0.496098 0.868267i \(-0.334766\pi\)
\(788\) 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 \(0.0881066\pi\)
−0.961936 + 0.273274i \(0.911893\pi\)
\(798\) 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
\(805\) 0