Properties

Label 1764.3.z.l.901.4
Level $1764$
Weight $3$
Character 1764.901
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 901.4
Root \(-1.60021 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1764.901
Dual form 1764.3.z.l.325.4

$q$-expansion

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

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 \(-0.135291\pi\)
0.0984097 + 0.995146i \(0.468624\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.43068 5.94212i −0.311880 0.540193i 0.666889 0.745157i \(-0.267626\pi\)
−0.978769 + 0.204964i \(0.934292\pi\)
\(12\) 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 \(-0.758907\pi\)
0.231694 + 0.972789i \(0.425573\pi\)
\(18\) 0 0
\(19\) −26.2322 15.1451i −1.38064 0.797113i −0.388405 0.921489i \(-0.626974\pi\)
−0.992235 + 0.124376i \(0.960307\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.07789 + 15.7234i −0.394691 + 0.683625i −0.993062 0.117595i \(-0.962482\pi\)
0.598371 + 0.801219i \(0.295815\pi\)
\(24\) 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 −0.543744 + 0.839251i \(0.682994\pi\)
−0.998685 + 0.0512709i \(0.983673\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −27.4873 + 47.6093i −0.742899 + 1.28674i 0.208271 + 0.978071i \(0.433216\pi\)
−0.951170 + 0.308668i \(0.900117\pi\)
\(38\) 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 \(-0.157114\pi\)
0.0300061 + 0.999550i \(0.490447\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −40.5081 70.1621i −0.764303 1.32381i −0.940614 0.339478i \(-0.889750\pi\)
0.176311 0.984335i \(-0.443584\pi\)
\(54\) 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 \(-0.761934\pi\)
0.222432 + 0.974948i \(0.428600\pi\)
\(60\) 0 0
\(61\) 0.0331519 + 0.0191403i 0.000543474 + 0.000313775i 0.500272 0.865868i \(-0.333233\pi\)
−0.499728 + 0.866182i \(0.666567\pi\)
\(62\) 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 0.999690 0.0248985i \(-0.00792626\pi\)
−0.521408 + 0.853308i \(0.674593\pi\)
\(68\) 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 \(-0.619960\pi\)
0.621247 + 0.783615i \(0.286626\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 23.7522 41.1400i 0.300660 0.520759i −0.675625 0.737245i \(-0.736126\pi\)
0.976286 + 0.216486i \(0.0694596\pi\)
\(80\) 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 0.999753 + 0.0222177i \(0.00707269\pi\)
0.519118 + 0.854703i \(0.326261\pi\)
\(90\) 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 \(-0.984305\pi\)
−0.456708 + 0.889617i \(0.650972\pi\)
\(102\) 0 0
\(103\) 51.7204 + 29.8608i 0.502140 + 0.289911i 0.729597 0.683877i \(-0.239708\pi\)
−0.227457 + 0.973788i \(0.573041\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −28.8839 + 50.0284i −0.269943 + 0.467555i −0.968847 0.247661i \(-0.920338\pi\)
0.698904 + 0.715216i \(0.253671\pi\)
\(108\) 0 0
\(109\) −89.9228 155.751i −0.824980 1.42891i −0.901935 0.431872i \(-0.857853\pi\)
0.0769549 0.997035i \(-0.475480\pi\)
\(110\) 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 \(-0.365628\pi\)
−0.585143 + 0.810930i \(0.698962\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 105.433 + 182.615i 0.769583 + 1.33296i 0.937789 + 0.347206i \(0.112869\pi\)
−0.168206 + 0.985752i \(0.553797\pi\)
\(138\) 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 −0.852300 0.523054i \(-0.824793\pi\)
0.879128 + 0.476586i \(0.158126\pi\)
\(150\) 0 0
\(151\) −30.4415 52.7263i −0.201600 0.349181i 0.747444 0.664324i \(-0.231281\pi\)
−0.949044 + 0.315144i \(0.897947\pi\)
\(152\) 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 \(-0.637048\pi\)
0.578305 + 0.815821i \(0.303714\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 31.2278 54.0881i 0.191582 0.331829i −0.754193 0.656653i \(-0.771972\pi\)
0.945775 + 0.324824i \(0.105305\pi\)
\(164\) 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 \(-0.0843868\pi\)
0.255621 + 0.966777i \(0.417720\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −143.321 248.239i −0.800676 1.38681i −0.919172 0.393856i \(-0.871141\pi\)
0.118496 0.992954i \(-0.462193\pi\)
\(180\) 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 −0.919234 0.393713i \(-0.871190\pi\)
0.800582 + 0.599223i \(0.204524\pi\)
\(192\) 0 0
\(193\) −157.015 271.958i −0.813551 1.40911i −0.910364 0.413809i \(-0.864198\pi\)
0.0968130 0.995303i \(-0.469135\pi\)
\(194\) 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 \(-0.725810\pi\)
0.331407 + 0.943488i \(0.392477\pi\)
\(200\) 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 −0.667151 + 0.744923i \(0.732486\pi\)
−0.978697 + 0.205308i \(0.934180\pi\)
\(228\) 0 0
\(229\) 30.1973 + 17.4344i 0.131866 + 0.0761329i 0.564482 0.825445i \(-0.309076\pi\)
−0.432616 + 0.901578i \(0.642409\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −175.386 + 303.777i −0.752728 + 1.30376i 0.193768 + 0.981047i \(0.437929\pi\)
−0.946496 + 0.322715i \(0.895404\pi\)
\(234\) 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 \(-0.721128\pi\)
0.345249 + 0.938511i \(0.387794\pi\)
\(242\) 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 \(-0.310177\pi\)
−0.435732 + 0.900076i \(0.643511\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.68631 + 2.92077i 0.00641181 + 0.0111056i 0.869214 0.494437i \(-0.164626\pi\)
−0.862802 + 0.505543i \(0.831292\pi\)
\(264\) 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 \(-0.614805\pi\)
0.633856 + 0.773451i \(0.281471\pi\)
\(270\) 0 0
\(271\) −79.6808 46.0037i −0.294025 0.169756i 0.345731 0.938334i \(-0.387631\pi\)
−0.639756 + 0.768578i \(0.720964\pi\)
\(272\) 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 0.694958 0.719050i \(-0.255423\pi\)
−0.970195 + 0.242326i \(0.922090\pi\)
\(278\) 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 \(-0.906617\pi\)
−0.228199 + 0.973615i \(0.573284\pi\)
\(284\) 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 \(-0.419961\pi\)
0.963196 + 0.268801i \(0.0866273\pi\)
\(312\) 0 0
\(313\) 47.5799 + 27.4703i 0.152012 + 0.0877644i 0.574077 0.818801i \(-0.305361\pi\)
−0.422064 + 0.906566i \(0.638694\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 216.846 375.587i 0.684055 1.18482i −0.289677 0.957124i \(-0.593548\pi\)
0.973733 0.227694i \(-0.0731186\pi\)
\(318\) 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 0.00987084 0.999951i \(-0.496858\pi\)
0.861048 0.508524i \(-0.169809\pi\)
\(332\) 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 0.625224 0.780445i \(-0.285008\pi\)
−0.988497 + 0.151237i \(0.951674\pi\)
\(348\) 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 \(-0.727648\pi\)
0.325953 + 0.945386i \(0.394315\pi\)
\(354\) 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 −0.935765 0.352624i \(-0.885290\pi\)
0.773264 + 0.634084i \(0.218623\pi\)
\(360\) 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 \(-0.473204\pi\)
0.905000 + 0.425413i \(0.139871\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 63.4962 109.979i 0.170231 0.294849i −0.768270 0.640127i \(-0.778882\pi\)
0.938501 + 0.345278i \(0.112215\pi\)
\(374\) 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 \(-0.223823\pi\)
−0.178599 + 0.983922i \(0.557157\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 271.198 + 469.728i 0.697166 + 1.20753i 0.969445 + 0.245309i \(0.0788893\pi\)
−0.272279 + 0.962218i \(0.587777\pi\)
\(390\) 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 \(-0.503606\pi\)
−0.871634 + 0.490158i \(0.836939\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −236.718 + 410.008i −0.590319 + 1.02246i 0.403870 + 0.914816i \(0.367665\pi\)
−0.994189 + 0.107646i \(0.965669\pi\)
\(402\) 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 \(-0.502395\pi\)
0.862239 + 0.506502i \(0.169062\pi\)
\(410\) 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 −0.770610 0.637307i \(-0.780048\pi\)
−0.166619 0.986021i \(-0.553285\pi\)
\(432\) 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 \(-0.343954\pi\)
−0.528612 + 0.848864i \(0.677287\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 411.376 712.524i 0.928613 1.60841i 0.142969 0.989727i \(-0.454335\pi\)
0.785645 0.618678i \(-0.212331\pi\)
\(444\) 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 −0.745177 0.666867i \(-0.767635\pi\)
0.950112 + 0.311908i \(0.100968\pi\)
\(458\) 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 \(-0.492932\pi\)
−0.854711 + 0.519104i \(0.826266\pi\)
\(468\) 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 \(-0.437504\pi\)
0.946926 + 0.321452i \(0.104171\pi\)
\(480\) 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 −0.960895 0.276914i \(-0.910688\pi\)
0.240633 0.970616i \(-0.422645\pi\)
\(488\) 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 −0.377067 0.926186i \(-0.623067\pi\)
0.990634 0.136543i \(-0.0435992\pi\)
\(500\) 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 \(-0.313758\pi\)
−0.445829 + 0.895118i \(0.647091\pi\)
\(510\) 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 \(-0.426969\pi\)
0.957044 + 0.289942i \(0.0936362\pi\)
\(522\) 0 0
\(523\) 242.586 + 140.057i 0.463835 + 0.267795i 0.713655 0.700497i \(-0.247038\pi\)
−0.249821 + 0.968292i \(0.580372\pi\)
\(524\) 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 0.409842 + 0.912157i \(0.365584\pi\)
−0.994872 + 0.101145i \(0.967749\pi\)
\(542\) 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 0.990908 + 0.134538i \(0.0429550\pi\)
−0.378941 + 0.925421i \(0.623712\pi\)
\(558\) 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 \(-0.723963\pi\)
0.336877 + 0.941549i \(0.390629\pi\)
\(564\) 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 0.390797 + 0.920477i \(0.372199\pi\)
−0.992555 + 0.121798i \(0.961134\pi\)
\(570\) 0 0
\(571\) 12.5301 + 21.7028i 0.0219442 + 0.0380084i 0.876789 0.480875i \(-0.159681\pi\)
−0.854845 + 0.518884i \(0.826348\pi\)
\(572\) 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 0.531549 0.847028i \(-0.321610\pi\)
0.999322 0.0368210i \(-0.0117231\pi\)
\(578\) 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 \(-0.372040\pi\)
−0.601358 + 0.798980i \(0.705373\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 355.368 + 615.515i 0.593269 + 1.02757i 0.993789 + 0.111283i \(0.0354961\pi\)
−0.400520 + 0.916288i \(0.631171\pi\)
\(600\) 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 \(-0.504707\pi\)
−0.873324 + 0.487139i \(0.838041\pi\)
\(608\) 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 0.971734 0.236077i \(-0.0758618\pi\)
−0.690316 + 0.723508i \(0.742528\pi\)
\(614\) 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 \(-0.654009\pi\)
0.534034 + 0.845463i \(0.320675\pi\)
\(620\) 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 0.711585 0.702600i \(-0.247978\pi\)
−0.964262 + 0.264951i \(0.914644\pi\)
\(642\) 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 \(-0.803490\pi\)
0.0936176 + 0.995608i \(0.470157\pi\)
\(648\) 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 0.380126 + 0.924935i \(0.375881\pi\)
−0.991080 + 0.133268i \(0.957453\pi\)
\(654\) 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 \(-0.951864\pi\)
−0.363828 + 0.931466i \(0.618530\pi\)
\(662\) 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 \(-0.221630\pi\)
−0.171817 + 0.985129i \(0.554964\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 629.705 + 1090.68i 0.921969 + 1.59690i 0.796364 + 0.604817i \(0.206754\pi\)
0.125605 + 0.992080i \(0.459913\pi\)
\(684\) 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 \(-0.441069\pi\)
−0.759185 + 0.650875i \(0.774402\pi\)
\(692\) 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 −0.949053 0.315117i \(-0.897956\pi\)
0.747426 + 0.664346i \(0.231290\pi\)
\(710\) 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 \(-0.494973\pi\)
−0.858021 + 0.513615i \(0.828306\pi\)
\(720\) 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 \(-0.168122\pi\)
−0.00457181 + 0.999990i \(0.501455\pi\)
\(734\) 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 0.700191 0.713956i \(-0.253098\pi\)
−0.968399 + 0.249405i \(0.919765\pi\)
\(740\) 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 −0.501774 0.864999i \(-0.667319\pi\)
0.999998 + 0.00204963i \(0.000652418\pi\)
\(752\) 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 \(-0.556412\pi\)
−0.940609 + 0.339491i \(0.889745\pi\)
\(762\) 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 0.578293 0.815829i \(-0.303719\pi\)
0.995675 0.0929015i \(-0.0296142\pi\)
\(774\) 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 \(-0.718630\pi\)
0.352602 + 0.935773i \(0.385297\pi\)
\(788\) 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