Properties

Label 1764.3.z.m.325.1
Level $1764$
Weight $3$
Character 1764.325
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 325.1
Root \(1.60021 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.m.901.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-5.04718 + 2.91399i) q^{5} +O(q^{10})\) \(q+(-5.04718 + 2.91399i) q^{5} +(-3.43068 + 5.94212i) q^{11} -3.62063i q^{13} +(8.41362 + 4.85761i) q^{17} +(26.2322 - 15.1451i) q^{19} +(-9.07789 - 15.7234i) q^{23} +(4.48269 - 7.76425i) q^{25} +40.4570 q^{29} +(47.8153 + 27.6062i) q^{31} +(-27.4873 - 47.6093i) q^{37} +56.3322i q^{41} -66.0512 q^{43} +(-42.8003 + 24.7108i) q^{47} +(-40.5081 + 70.1621i) q^{53} -39.9879i q^{55} +(30.1302 + 17.3957i) q^{59} +(-0.0331519 + 0.0191403i) q^{61} +(10.5505 + 18.2740i) q^{65} +(32.0449 - 55.5034i) q^{67} -50.2730 q^{71} +(-18.4865 - 10.6732i) q^{73} +(23.7522 + 41.1400i) q^{79} +33.6039i q^{83} -56.6201 q^{85} +(-135.180 + 78.0459i) q^{89} +(-88.2656 + 152.881i) q^{95} -43.7452i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 48q^{17} + 96q^{19} - 8q^{23} - 36q^{25} - 80q^{29} - 48q^{31} - 64q^{37} - 112q^{43} - 264q^{47} - 72q^{53} + 168q^{59} + 144q^{61} + 120q^{65} + 32q^{67} - 224q^{71} - 336q^{73} + 216q^{79} - 96q^{85} + 96q^{89} - 136q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) −5.04718 + 2.91399i −1.00944 + 0.582798i −0.911027 0.412348i \(-0.864709\pi\)
−0.0984097 + 0.995146i \(0.531376\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.43068 + 5.94212i −0.311880 + 0.540193i −0.978769 0.204964i \(-0.934292\pi\)
0.666889 + 0.745157i \(0.267626\pi\)
\(12\) 0 0
\(13\) 3.62063i 0.278510i −0.990257 0.139255i \(-0.955529\pi\)
0.990257 0.139255i \(-0.0444708\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.41362 + 4.85761i 0.494919 + 0.285742i 0.726613 0.687047i \(-0.241093\pi\)
−0.231694 + 0.972789i \(0.574427\pi\)
\(18\) 0 0
\(19\) 26.2322 15.1451i 1.38064 0.797113i 0.388405 0.921489i \(-0.373026\pi\)
0.992235 + 0.124376i \(0.0396930\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.07789 15.7234i −0.394691 0.683625i 0.598371 0.801219i \(-0.295815\pi\)
−0.993062 + 0.117595i \(0.962482\pi\)
\(24\) 0 0
\(25\) 4.48269 7.76425i 0.179308 0.310570i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 40.4570 1.39507 0.697534 0.716552i \(-0.254281\pi\)
0.697534 + 0.716552i \(0.254281\pi\)
\(30\) 0 0
\(31\) 47.8153 + 27.6062i 1.54243 + 0.890522i 0.998685 + 0.0512709i \(0.0163272\pi\)
0.543744 + 0.839251i \(0.317006\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −27.4873 47.6093i −0.742899 1.28674i −0.951170 0.308668i \(-0.900117\pi\)
0.208271 0.978071i \(-0.433216\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 56.3322i 1.37396i 0.726678 + 0.686978i \(0.241063\pi\)
−0.726678 + 0.686978i \(0.758937\pi\)
\(42\) 0 0
\(43\) −66.0512 −1.53608 −0.768038 0.640405i \(-0.778767\pi\)
−0.768038 + 0.640405i \(0.778767\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −42.8003 + 24.7108i −0.910645 + 0.525761i −0.880639 0.473789i \(-0.842886\pi\)
−0.0300061 + 0.999550i \(0.509553\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −40.5081 + 70.1621i −0.764303 + 1.32381i 0.176311 + 0.984335i \(0.443584\pi\)
−0.940614 + 0.339478i \(0.889750\pi\)
\(54\) 0 0
\(55\) 39.9879i 0.727054i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 30.1302 + 17.3957i 0.510682 + 0.294842i 0.733114 0.680106i \(-0.238066\pi\)
−0.222432 + 0.974948i \(0.571400\pi\)
\(60\) 0 0
\(61\) −0.0331519 + 0.0191403i −0.000543474 + 0.000313775i −0.500272 0.865868i \(-0.666767\pi\)
0.499728 + 0.866182i \(0.333433\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.5505 + 18.2740i 0.162315 + 0.281138i
\(66\) 0 0
\(67\) 32.0449 55.5034i 0.478282 0.828409i −0.521408 0.853308i \(-0.674593\pi\)
0.999690 + 0.0248985i \(0.00792626\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −50.2730 −0.708070 −0.354035 0.935232i \(-0.615191\pi\)
−0.354035 + 0.935232i \(0.615191\pi\)
\(72\) 0 0
\(73\) −18.4865 10.6732i −0.253240 0.146208i 0.368007 0.929823i \(-0.380040\pi\)
−0.621247 + 0.783615i \(0.713374\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 23.7522 + 41.1400i 0.300660 + 0.520759i 0.976286 0.216486i \(-0.0694596\pi\)
−0.675625 + 0.737245i \(0.736126\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 33.6039i 0.404866i 0.979296 + 0.202433i \(0.0648849\pi\)
−0.979296 + 0.202433i \(0.935115\pi\)
\(84\) 0 0
\(85\) −56.6201 −0.666119
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −135.180 + 78.0459i −1.51887 + 0.876920i −0.519118 + 0.854703i \(0.673739\pi\)
−0.999753 + 0.0222177i \(0.992927\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −88.2656 + 152.881i −0.929112 + 1.60927i
\(96\) 0 0
\(97\) 43.7452i 0.450981i −0.974245 0.225491i \(-0.927601\pi\)
0.974245 0.225491i \(-0.0723985\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 147.005 + 84.8732i 1.45549 + 0.840329i 0.998785 0.0492878i \(-0.0156951\pi\)
0.456708 + 0.889617i \(0.349028\pi\)
\(102\) 0 0
\(103\) −51.7204 + 29.8608i −0.502140 + 0.289911i −0.729597 0.683877i \(-0.760292\pi\)
0.227457 + 0.973788i \(0.426959\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −28.8839 50.0284i −0.269943 0.467555i 0.698904 0.715216i \(-0.253671\pi\)
−0.968847 + 0.247661i \(0.920338\pi\)
\(108\) 0 0
\(109\) −89.9228 + 155.751i −0.824980 + 1.42891i 0.0769549 + 0.997035i \(0.475480\pi\)
−0.901935 + 0.431872i \(0.857853\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −96.6900 −0.855664 −0.427832 0.903858i \(-0.640722\pi\)
−0.427832 + 0.903858i \(0.640722\pi\)
\(114\) 0 0
\(115\) 91.6355 + 52.9058i 0.796831 + 0.460050i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 36.9608 + 64.0180i 0.305461 + 0.529074i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 93.4495i 0.747596i
\(126\) 0 0
\(127\) −136.454 −1.07444 −0.537221 0.843441i \(-0.680526\pi\)
−0.537221 + 0.843441i \(0.680526\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 22.9812 13.2682i 0.175429 0.101284i −0.409714 0.912214i \(-0.634372\pi\)
0.585143 + 0.810930i \(0.301038\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 105.433 182.615i 0.769583 1.33296i −0.168206 0.985752i \(-0.553797\pi\)
0.937789 0.347206i \(-0.112869\pi\)
\(138\) 0 0
\(139\) 83.7490i 0.602511i 0.953543 + 0.301256i \(0.0974057\pi\)
−0.953543 + 0.301256i \(0.902594\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 21.5142 + 12.4212i 0.150449 + 0.0868619i
\(144\) 0 0
\(145\) −204.194 + 117.891i −1.40823 + 0.813043i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.99740 + 6.92371i 0.0268282 + 0.0464678i 0.879128 0.476586i \(-0.158126\pi\)
−0.852300 + 0.523054i \(0.824793\pi\)
\(150\) 0 0
\(151\) −30.4415 + 52.7263i −0.201600 + 0.349181i −0.949044 0.315144i \(-0.897947\pi\)
0.747444 + 0.664324i \(0.231281\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −321.777 −2.07598
\(156\) 0 0
\(157\) −25.2670 14.5879i −0.160936 0.0929166i 0.417369 0.908737i \(-0.362952\pi\)
−0.578305 + 0.815821i \(0.696286\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 31.2278 + 54.0881i 0.191582 + 0.331829i 0.945775 0.324824i \(-0.105305\pi\)
−0.754193 + 0.656653i \(0.771972\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 141.404i 0.846730i −0.905959 0.423365i \(-0.860849\pi\)
0.905959 0.423365i \(-0.139151\pi\)
\(168\) 0 0
\(169\) 155.891 0.922432
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −211.178 + 121.924i −1.22068 + 0.704763i −0.965064 0.262015i \(-0.915613\pi\)
−0.255621 + 0.966777i \(0.582280\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −143.321 + 248.239i −0.800676 + 1.38681i 0.118496 + 0.992954i \(0.462193\pi\)
−0.919172 + 0.393856i \(0.871141\pi\)
\(180\) 0 0
\(181\) 214.838i 1.18695i 0.804852 + 0.593475i \(0.202245\pi\)
−0.804852 + 0.593475i \(0.797755\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 277.466 + 160.195i 1.49982 + 0.865921i
\(186\) 0 0
\(187\) −57.7290 + 33.3298i −0.308711 + 0.178234i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.6624 39.2525i −0.118652 0.205511i 0.800582 0.599223i \(-0.204524\pi\)
−0.919234 + 0.393713i \(0.871190\pi\)
\(192\) 0 0
\(193\) −157.015 + 271.958i −0.813551 + 1.40911i 0.0968130 + 0.995303i \(0.469135\pi\)
−0.910364 + 0.413809i \(0.864198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −259.231 −1.31590 −0.657948 0.753063i \(-0.728575\pi\)
−0.657948 + 0.753063i \(0.728575\pi\)
\(198\) 0 0
\(199\) 63.6749 + 36.7627i 0.319975 + 0.184737i 0.651381 0.758751i \(-0.274190\pi\)
−0.331407 + 0.943488i \(0.607523\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −164.151 284.319i −0.800739 1.38692i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 207.833i 0.994415i
\(210\) 0 0
\(211\) 263.537 1.24899 0.624496 0.781028i \(-0.285304\pi\)
0.624496 + 0.781028i \(0.285304\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 333.372 192.473i 1.55057 0.895222i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.5876 30.4626i 0.0795819 0.137840i
\(222\) 0 0
\(223\) 191.042i 0.856689i 0.903616 + 0.428344i \(0.140903\pi\)
−0.903616 + 0.428344i \(0.859097\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 373.608 + 215.702i 1.64585 + 0.950231i 0.978697 + 0.205308i \(0.0658196\pi\)
0.667151 + 0.744923i \(0.267514\pi\)
\(228\) 0 0
\(229\) −30.1973 + 17.4344i −0.131866 + 0.0761329i −0.564482 0.825445i \(-0.690924\pi\)
0.432616 + 0.901578i \(0.357591\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −175.386 303.777i −0.752728 1.30376i −0.946496 0.322715i \(-0.895404\pi\)
0.193768 0.981047i \(-0.437929\pi\)
\(234\) 0 0
\(235\) 144.014 249.439i 0.612825 1.06144i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −167.400 −0.700419 −0.350209 0.936671i \(-0.613890\pi\)
−0.350209 + 0.936671i \(0.613890\pi\)
\(240\) 0 0
\(241\) 71.0711 + 41.0329i 0.294901 + 0.170261i 0.640150 0.768250i \(-0.278872\pi\)
−0.345249 + 0.938511i \(0.612206\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −54.8350 94.9770i −0.222004 0.384522i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 232.918i 0.927959i −0.885846 0.463979i \(-0.846421\pi\)
0.885846 0.463979i \(-0.153579\pi\)
\(252\) 0 0
\(253\) 124.574 0.492385
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −32.3538 + 18.6795i −0.125890 + 0.0726827i −0.561623 0.827394i \(-0.689823\pi\)
0.435732 + 0.900076i \(0.356489\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.68631 2.92077i 0.00641181 0.0111056i −0.862802 0.505543i \(-0.831292\pi\)
0.869214 + 0.494437i \(0.164626\pi\)
\(264\) 0 0
\(265\) 472.161i 1.78174i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −75.5768 43.6343i −0.280955 0.162209i 0.352901 0.935661i \(-0.385195\pi\)
−0.633856 + 0.773451i \(0.718529\pi\)
\(270\) 0 0
\(271\) 79.6808 46.0037i 0.294025 0.169756i −0.345731 0.938334i \(-0.612369\pi\)
0.639756 + 0.768578i \(0.279036\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 30.7574 + 53.2734i 0.111845 + 0.193721i
\(276\) 0 0
\(277\) −76.2406 + 132.053i −0.275237 + 0.476724i −0.970195 0.242326i \(-0.922090\pi\)
0.694958 + 0.719050i \(0.255423\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 219.880 0.782491 0.391245 0.920286i \(-0.372044\pi\)
0.391245 + 0.920286i \(0.372044\pi\)
\(282\) 0 0
\(283\) 335.489 + 193.695i 1.18547 + 0.684433i 0.957274 0.289181i \(-0.0933831\pi\)
0.228199 + 0.973615i \(0.426716\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −97.3073 168.541i −0.336704 0.583188i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.8794i 0.0507828i −0.999678 0.0253914i \(-0.991917\pi\)
0.999678 0.0253914i \(-0.00808320\pi\)
\(294\) 0 0
\(295\) −202.764 −0.687335
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −56.9285 + 32.8677i −0.190396 + 0.109925i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.111549 0.193209i 0.000365735 0.000633471i
\(306\) 0 0
\(307\) 453.211i 1.47626i 0.674660 + 0.738128i \(0.264290\pi\)
−0.674660 + 0.738128i \(0.735710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −376.934 217.623i −1.21201 0.699752i −0.248810 0.968552i \(-0.580039\pi\)
−0.963196 + 0.268801i \(0.913373\pi\)
\(312\) 0 0
\(313\) −47.5799 + 27.4703i −0.152012 + 0.0877644i −0.574077 0.818801i \(-0.694639\pi\)
0.422064 + 0.906566i \(0.361306\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 216.846 + 375.587i 0.684055 + 1.18482i 0.973733 + 0.227694i \(0.0731186\pi\)
−0.289677 + 0.957124i \(0.593548\pi\)
\(318\) 0 0
\(319\) −138.795 + 240.400i −0.435095 + 0.753606i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 294.277 0.911073
\(324\) 0 0
\(325\) −28.1115 16.2302i −0.0864968 0.0499390i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 288.274 + 499.305i 0.870919 + 1.50848i 0.861048 + 0.508524i \(0.169809\pi\)
0.00987084 + 0.999951i \(0.496858\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 373.514i 1.11497i
\(336\) 0 0
\(337\) −301.108 −0.893495 −0.446747 0.894660i \(-0.647418\pi\)
−0.446747 + 0.894660i \(0.647418\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −328.078 + 189.416i −0.962107 + 0.555473i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −126.056 + 218.335i −0.363273 + 0.629208i −0.988497 0.151237i \(-0.951674\pi\)
0.625224 + 0.780445i \(0.285008\pi\)
\(348\) 0 0
\(349\) 406.452i 1.16462i −0.812967 0.582309i \(-0.802149\pi\)
0.812967 0.582309i \(-0.197851\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 116.419 + 67.2146i 0.329799 + 0.190410i 0.655752 0.754976i \(-0.272352\pi\)
−0.325953 + 0.945386i \(0.605685\pi\)
\(354\) 0 0
\(355\) 253.737 146.495i 0.714752 0.412662i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −58.3380 101.044i −0.162501 0.281461i 0.773264 0.634084i \(-0.218623\pi\)
−0.935765 + 0.352624i \(0.885290\pi\)
\(360\) 0 0
\(361\) 278.251 481.944i 0.770777 1.33503i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 124.406 0.340839
\(366\) 0 0
\(367\) −362.993 209.574i −0.989081 0.571046i −0.0840817 0.996459i \(-0.526796\pi\)
−0.905000 + 0.425413i \(0.860129\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 63.4962 + 109.979i 0.170231 + 0.294849i 0.938501 0.345278i \(-0.112215\pi\)
−0.768270 + 0.640127i \(0.778882\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 146.480i 0.388541i
\(378\) 0 0
\(379\) −366.675 −0.967479 −0.483740 0.875212i \(-0.660722\pi\)
−0.483740 + 0.875212i \(0.660722\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −223.750 + 129.182i −0.584203 + 0.337290i −0.762802 0.646632i \(-0.776177\pi\)
0.178599 + 0.983922i \(0.442843\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 271.198 469.728i 0.697166 1.20753i −0.272279 0.962218i \(-0.587777\pi\)
0.969445 0.245309i \(-0.0788893\pi\)
\(390\) 0 0
\(391\) 176.387i 0.451118i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −239.763 138.427i −0.606995 0.350449i
\(396\) 0 0
\(397\) 350.536 202.382i 0.882962 0.509778i 0.0113280 0.999936i \(-0.496394\pi\)
0.871634 + 0.490158i \(0.163061\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −236.718 410.008i −0.590319 1.02246i −0.994189 0.107646i \(-0.965669\pi\)
0.403870 0.914816i \(-0.367665\pi\)
\(402\) 0 0
\(403\) 99.9518 173.122i 0.248019 0.429582i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 377.201 0.926783
\(408\) 0 0
\(409\) −349.578 201.829i −0.854714 0.493469i 0.00752476 0.999972i \(-0.497605\pi\)
−0.862239 + 0.506502i \(0.830938\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −97.9215 169.605i −0.235955 0.408687i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 454.684i 1.08517i 0.840003 + 0.542583i \(0.182553\pi\)
−0.840003 + 0.542583i \(0.817447\pi\)
\(420\) 0 0
\(421\) −180.928 −0.429758 −0.214879 0.976641i \(-0.568936\pi\)
−0.214879 + 0.976641i \(0.568936\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 75.4313 43.5503i 0.177485 0.102471i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −403.946 + 699.655i −0.937229 + 1.62333i −0.166619 + 0.986021i \(0.553285\pi\)
−0.770610 + 0.637307i \(0.780048\pi\)
\(432\) 0 0
\(433\) 166.000i 0.383372i −0.981456 0.191686i \(-0.938604\pi\)
0.981456 0.191686i \(-0.0613955\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −476.265 274.972i −1.08985 0.629226i
\(438\) 0 0
\(439\) 25.3654 14.6447i 0.0577800 0.0333593i −0.470832 0.882223i \(-0.656046\pi\)
0.528612 + 0.848864i \(0.322713\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 411.376 + 712.524i 0.928613 + 1.60841i 0.785645 + 0.618678i \(0.212331\pi\)
0.142969 + 0.989727i \(0.454335\pi\)
\(444\) 0 0
\(445\) 454.850 787.824i 1.02214 1.77039i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 428.131 0.953522 0.476761 0.879033i \(-0.341811\pi\)
0.476761 + 0.879033i \(0.341811\pi\)
\(450\) 0 0
\(451\) −334.733 193.258i −0.742201 0.428510i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 93.6555 + 162.216i 0.204935 + 0.354959i 0.950112 0.311908i \(-0.100968\pi\)
−0.745177 + 0.666867i \(0.767635\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 182.821i 0.396576i −0.980144 0.198288i \(-0.936462\pi\)
0.980144 0.198288i \(-0.0635381\pi\)
\(462\) 0 0
\(463\) 232.389 0.501920 0.250960 0.967997i \(-0.419254\pi\)
0.250960 + 0.967997i \(0.419254\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 388.782 224.463i 0.832509 0.480650i −0.0222017 0.999754i \(-0.507068\pi\)
0.854711 + 0.519104i \(0.173734\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 226.601 392.484i 0.479072 0.829777i
\(474\) 0 0
\(475\) 271.564i 0.571713i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −547.020 315.822i −1.14200 0.659336i −0.195078 0.980788i \(-0.562496\pi\)
−0.946926 + 0.321452i \(0.895829\pi\)
\(480\) 0 0
\(481\) −172.376 + 99.5213i −0.358370 + 0.206905i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 127.473 + 220.790i 0.262831 + 0.455237i
\(486\) 0 0
\(487\) −350.767 + 607.547i −0.720262 + 1.24753i 0.240633 + 0.970616i \(0.422645\pi\)
−0.960895 + 0.276914i \(0.910688\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 414.789 0.844785 0.422392 0.906413i \(-0.361190\pi\)
0.422392 + 0.906413i \(0.361190\pi\)
\(492\) 0 0
\(493\) 340.390 + 196.524i 0.690446 + 0.398629i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 306.170 + 530.302i 0.613567 + 1.06273i 0.990634 + 0.136543i \(0.0435992\pi\)
−0.377067 + 0.926186i \(0.623067\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 467.449i 0.929323i 0.885488 + 0.464661i \(0.153824\pi\)
−0.885488 + 0.464661i \(0.846176\pi\)
\(504\) 0 0
\(505\) −989.279 −1.95897
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −54.1843 + 31.2833i −0.106452 + 0.0614603i −0.552281 0.833658i \(-0.686242\pi\)
0.445829 + 0.895118i \(0.352909\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 174.028 301.426i 0.337919 0.585293i
\(516\) 0 0
\(517\) 339.099i 0.655898i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −617.108 356.288i −1.18447 0.683853i −0.227425 0.973796i \(-0.573031\pi\)
−0.957044 + 0.289942i \(0.906364\pi\)
\(522\) 0 0
\(523\) −242.586 + 140.057i −0.463835 + 0.267795i −0.713655 0.700497i \(-0.752962\pi\)
0.249821 + 0.968292i \(0.419628\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 268.200 + 464.536i 0.508918 + 0.881472i
\(528\) 0 0
\(529\) 99.6838 172.657i 0.188438 0.326385i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 203.958 0.382660
\(534\) 0 0
\(535\) 291.564 + 168.335i 0.544980 + 0.314645i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −316.501 548.196i −0.585030 1.01330i −0.994872 0.101145i \(-0.967749\pi\)
0.409842 0.912157i \(-0.365584\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1048.14i 1.92319i
\(546\) 0 0
\(547\) 1047.16 1.91438 0.957189 0.289465i \(-0.0934773\pi\)
0.957189 + 0.289465i \(0.0934773\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1061.27 612.727i 1.92609 1.11203i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 340.866 590.397i 0.611967 1.05996i −0.378941 0.925421i \(-0.623712\pi\)
0.990908 0.134538i \(-0.0429550\pi\)
\(558\) 0 0
\(559\) 239.147i 0.427812i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 174.581 + 100.794i 0.310090 + 0.179031i 0.646967 0.762518i \(-0.276037\pi\)
−0.336877 + 0.941549i \(0.609371\pi\)
\(564\) 0 0
\(565\) 488.012 281.754i 0.863738 0.498679i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −342.400 593.054i −0.601758 1.04227i −0.992555 0.121798i \(-0.961134\pi\)
0.390797 0.920477i \(-0.372199\pi\)
\(570\) 0 0
\(571\) 12.5301 21.7028i 0.0219442 0.0380084i −0.854845 0.518884i \(-0.826348\pi\)
0.876789 + 0.480875i \(0.159681\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −162.773 −0.283084
\(576\) 0 0
\(577\) −883.312 509.981i −1.53087 0.883849i −0.999322 0.0368210i \(-0.988277\pi\)
−0.531549 0.847028i \(-0.678390\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −277.941 481.408i −0.476743 0.825742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 581.810i 0.991158i 0.868563 + 0.495579i \(0.165044\pi\)
−0.868563 + 0.495579i \(0.834956\pi\)
\(588\) 0 0
\(589\) 1672.40 2.83939
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 124.589 71.9314i 0.210099 0.121301i −0.391258 0.920281i \(-0.627960\pi\)
0.601358 + 0.798980i \(0.294627\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 355.368 615.515i 0.593269 1.02757i −0.400520 0.916288i \(-0.631171\pi\)
0.993789 0.111283i \(-0.0354961\pi\)
\(600\) 0 0
\(601\) 914.930i 1.52235i 0.648549 + 0.761173i \(0.275376\pi\)
−0.648549 + 0.761173i \(0.724624\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −373.096 215.407i −0.616687 0.356044i
\(606\) 0 0
\(607\) 539.084 311.240i 0.888112 0.512752i 0.0147875 0.999891i \(-0.495293\pi\)
0.873324 + 0.487139i \(0.161959\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 89.4686 + 154.964i 0.146430 + 0.253624i
\(612\) 0 0
\(613\) 172.510 298.795i 0.281419 0.487431i −0.690316 0.723508i \(-0.742528\pi\)
0.971734 + 0.236077i \(0.0758618\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 854.311 1.38462 0.692310 0.721600i \(-0.256593\pi\)
0.692310 + 0.721600i \(0.256593\pi\)
\(618\) 0 0
\(619\) −42.6240 24.6090i −0.0688595 0.0397561i 0.465175 0.885219i \(-0.345991\pi\)
−0.534034 + 0.845463i \(0.679325\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 384.378 + 665.763i 0.615005 + 1.06522i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 534.089i 0.849109i
\(630\) 0 0
\(631\) 457.699 0.725355 0.362677 0.931915i \(-0.381863\pi\)
0.362677 + 0.931915i \(0.381863\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 688.709 397.626i 1.08458 0.626183i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −161.966 + 280.533i −0.252677 + 0.437649i −0.964262 0.264951i \(-0.914644\pi\)
0.711585 + 0.702600i \(0.247978\pi\)
\(642\) 0 0
\(643\) 117.018i 0.181987i −0.995851 0.0909935i \(-0.970996\pi\)
0.995851 0.0909935i \(-0.0290043\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 467.002 + 269.624i 0.721796 + 0.416729i 0.815413 0.578879i \(-0.196510\pi\)
−0.0936176 + 0.995608i \(0.529843\pi\)
\(648\) 0 0
\(649\) −206.735 + 119.358i −0.318543 + 0.183911i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −398.953 691.007i −0.610954 1.05820i −0.991080 0.133268i \(-0.957453\pi\)
0.380126 0.924935i \(-0.375881\pi\)
\(654\) 0 0
\(655\) −77.3268 + 133.934i −0.118056 + 0.204479i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −295.186 −0.447929 −0.223965 0.974597i \(-0.571900\pi\)
−0.223965 + 0.974597i \(0.571900\pi\)
\(660\) 0 0
\(661\) 893.947 + 516.120i 1.35242 + 0.780817i 0.988587 0.150649i \(-0.0481362\pi\)
0.363828 + 0.931466i \(0.381470\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −367.264 636.120i −0.550621 0.953703i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.262657i 0.000391441i
\(672\) 0 0
\(673\) 418.188 0.621379 0.310689 0.950512i \(-0.399440\pi\)
0.310689 + 0.950512i \(0.399440\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −403.100 + 232.730i −0.595422 + 0.343767i −0.767238 0.641362i \(-0.778370\pi\)
0.171817 + 0.985129i \(0.445036\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 629.705 1090.68i 0.921969 1.59690i 0.125605 0.992080i \(-0.459913\pi\)
0.796364 0.604817i \(-0.206754\pi\)
\(684\) 0 0
\(685\) 1228.92i 1.79405i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 254.031 + 146.665i 0.368695 + 0.212866i
\(690\) 0 0
\(691\) 397.397 229.437i 0.575104 0.332036i −0.184081 0.982911i \(-0.558931\pi\)
0.759185 + 0.650875i \(0.225598\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −244.044 422.697i −0.351142 0.608196i
\(696\) 0 0
\(697\) −273.640 + 473.958i −0.392596 + 0.679997i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −822.907 −1.17390 −0.586952 0.809622i \(-0.699672\pi\)
−0.586952 + 0.809622i \(0.699672\pi\)
\(702\) 0 0
\(703\) −1442.10 832.597i −2.05135 1.18435i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −142.954 247.603i −0.201627 0.349229i 0.747426 0.664346i \(-0.231290\pi\)
−0.949053 + 0.315117i \(0.897956\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1002.42i 1.40592i
\(714\) 0 0
\(715\) −144.782 −0.202492
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 605.561 349.621i 0.842227 0.486260i −0.0157934 0.999875i \(-0.505027\pi\)
0.858021 + 0.513615i \(0.171694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 181.356 314.118i 0.250146 0.433266i
\(726\) 0 0
\(727\) 216.138i 0.297302i −0.988890 0.148651i \(-0.952507\pi\)
0.988890 0.148651i \(-0.0474930\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −555.730 320.851i −0.760233 0.438920i
\(732\) 0 0
\(733\) −629.763 + 363.594i −0.859159 + 0.496035i −0.863730 0.503954i \(-0.831878\pi\)
0.00457181 + 0.999990i \(0.498545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 219.872 + 380.829i 0.298334 + 0.516729i
\(738\) 0 0
\(739\) −198.206 + 343.303i −0.268209 + 0.464551i −0.968399 0.249405i \(-0.919765\pi\)
0.700191 + 0.713956i \(0.253098\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −739.481 −0.995263 −0.497632 0.867388i \(-0.665797\pi\)
−0.497632 + 0.867388i \(0.665797\pi\)
\(744\) 0 0
\(745\) −40.3512 23.2968i −0.0541627 0.0312709i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 374.166 + 648.075i 0.498224 + 0.862949i 0.999998 0.00204963i \(-0.000652418\pi\)
−0.501774 + 0.864999i \(0.667319\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 354.825i 0.469967i
\(756\) 0 0
\(757\) 199.145 0.263071 0.131536 0.991311i \(-0.458009\pi\)
0.131536 + 0.991311i \(0.458009\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 849.966 490.728i 1.11691 0.644846i 0.176297 0.984337i \(-0.443588\pi\)
0.940609 + 0.339491i \(0.110255\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 62.9834 109.090i 0.0821166 0.142230i
\(768\) 0 0
\(769\) 724.214i 0.941760i −0.882197 0.470880i \(-0.843936\pi\)
0.882197 0.470880i \(-0.156064\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1216.68 702.449i −1.57397 0.908731i −0.995675 0.0929015i \(-0.970386\pi\)
−0.578293 0.815829i \(-0.696281\pi\)
\(774\) 0 0
\(775\) 428.682 247.500i 0.553139 0.319355i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 853.159 + 1477.71i 1.09520 + 1.89694i
\(780\) 0 0
\(781\) 172.471 298.728i 0.220833 0.382495i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 170.036 0.216607
\(786\) 0 0
\(787\) 221.541 + 127.907i 0.281501 + 0.162525i 0.634103 0.773249i \(-0.281370\pi\)
−0.352602 + 0.935773i \(0.614703\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.0692998 + 0.120031i 8.73894e−5 + 0.000151363i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 474.641i 0.595534i 0.954639 + 0.297767i \(0.0962418\pi\)
−0.954639 + 0.297767i \(0.903758\pi\)
\(798\) 0 0
\(799\) −480.141 −0.600927
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 126.843 73.2327i 0.157961 0.0911989i
\(804\) 0