Properties

Label 1764.3.z.l.325.4
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 325.4
Root \(-1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.l.901.4

$q$-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 \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

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

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 0.0984097 0.995146i \(-0.468624\pi\)
0.911027 + 0.412348i \(0.135291\pi\)
\(6\) 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 \(0.0444708\pi\)
−0.990257 + 0.139255i \(0.955529\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −8.41362 4.85761i −0.494919 0.285742i 0.231694 0.972789i \(-0.425573\pi\)
−0.726613 + 0.687047i \(0.758907\pi\)
\(18\) 0 0
\(19\) −26.2322 + 15.1451i −1.38064 + 0.797113i −0.992235 0.124376i \(-0.960307\pi\)
−0.388405 + 0.921489i \(0.626974\pi\)
\(20\) 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 \(-0.983673\pi\)
−0.543744 0.839251i \(-0.682994\pi\)
\(32\) 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 \(-0.758937\pi\)
0.726678 0.686978i \(-0.241063\pi\)
\(42\) 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 0.0300061 0.999550i \(-0.490447\pi\)
0.880639 + 0.473789i \(0.157114\pi\)
\(48\) 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 0.222432 0.974948i \(-0.428600\pi\)
−0.733114 + 0.680106i \(0.761934\pi\)
\(60\) 0 0
\(61\) 0.0331519 0.0191403i 0.000543474 0.000313775i −0.499728 0.866182i \(-0.666567\pi\)
0.500272 + 0.865868i \(0.333233\pi\)
\(62\) 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 0.621247 0.783615i \(-0.286626\pi\)
−0.368007 + 0.929823i \(0.619960\pi\)
\(74\) 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 \(-0.935115\pi\)
0.979296 0.202433i \(-0.0648849\pi\)
\(84\) 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 \(-0.326261\pi\)
0.999753 0.0222177i \(-0.00707269\pi\)
\(90\) 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 \(0.0723985\pi\)
−0.974245 + 0.225491i \(0.927601\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −147.005 84.8732i −1.45549 0.840329i −0.456708 0.889617i \(-0.650972\pi\)
−0.998785 + 0.0492878i \(0.984305\pi\)
\(102\) 0 0
\(103\) 51.7204 29.8608i 0.502140 0.289911i −0.227457 0.973788i \(-0.573041\pi\)
0.729597 + 0.683877i \(0.239708\pi\)
\(104\) 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 −0.585143 0.810930i \(-0.698962\pi\)
0.409714 + 0.912214i \(0.365628\pi\)
\(132\) 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 \(-0.902594\pi\)
0.953543 0.301256i \(-0.0974057\pi\)
\(140\) 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 0.578305 0.815821i \(-0.303714\pi\)
−0.417369 + 0.908737i \(0.637048\pi\)
\(158\) 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 \(0.139151\pi\)
−0.905959 + 0.423365i \(0.860849\pi\)
\(168\) 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 0.255621 0.966777i \(-0.417720\pi\)
0.965064 + 0.262015i \(0.0843868\pi\)
\(174\) 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 \(-0.797755\pi\)
0.804852 0.593475i \(-0.202245\pi\)
\(182\) 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 0.331407 0.943488i \(-0.392477\pi\)
−0.651381 + 0.758751i \(0.725810\pi\)
\(200\) 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 \(-0.859097\pi\)
0.903616 0.428344i \(-0.140903\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −373.608 215.702i −1.64585 0.950231i −0.978697 0.205308i \(-0.934180\pi\)
−0.667151 0.744923i \(-0.732486\pi\)
\(228\) 0 0
\(229\) 30.1973 17.4344i 0.131866 0.0761329i −0.432616 0.901578i \(-0.642409\pi\)
0.564482 + 0.825445i \(0.309076\pi\)
\(230\) 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 0.345249 0.938511i \(-0.387794\pi\)
−0.640150 + 0.768250i \(0.721128\pi\)
\(242\) 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 \(0.153579\pi\)
−0.885846 + 0.463979i \(0.846421\pi\)
\(252\) 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 −0.435732 0.900076i \(-0.643511\pi\)
0.561623 + 0.827394i \(0.310177\pi\)
\(258\) 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 0.633856 0.773451i \(-0.281471\pi\)
−0.352901 + 0.935661i \(0.614805\pi\)
\(270\) 0 0
\(271\) −79.6808 + 46.0037i −0.294025 + 0.169756i −0.639756 0.768578i \(-0.720964\pi\)
0.345731 + 0.938334i \(0.387631\pi\)
\(272\) 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 −0.228199 0.973615i \(-0.573284\pi\)
−0.957274 + 0.289181i \(0.906617\pi\)
\(284\) 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 \(0.00808320\pi\)
−0.999678 + 0.0253914i \(0.991917\pi\)
\(294\) 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 \(-0.735710\pi\)
0.674660 0.738128i \(-0.264290\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 376.934 + 217.623i 1.21201 + 0.699752i 0.963196 0.268801i \(-0.0866273\pi\)
0.248810 + 0.968552i \(0.419961\pi\)
\(312\) 0 0
\(313\) 47.5799 27.4703i 0.152012 0.0877644i −0.422064 0.906566i \(-0.638694\pi\)
0.574077 + 0.818801i \(0.305361\pi\)
\(314\) 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 \(0.197851\pi\)
−0.812967 + 0.582309i \(0.802149\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −116.419 67.2146i −0.329799 0.190410i 0.325953 0.945386i \(-0.394315\pi\)
−0.655752 + 0.754976i \(0.727648\pi\)
\(354\) 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 0.905000 0.425413i \(-0.139871\pi\)
0.0840817 + 0.996459i \(0.473204\pi\)
\(368\) 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 −0.178599 0.983922i \(-0.557157\pi\)
0.762802 + 0.646632i \(0.223823\pi\)
\(384\) 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 −0.871634 0.490158i \(-0.836939\pi\)
−0.0113280 + 0.999936i \(0.503606\pi\)
\(398\) 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 0.862239 0.506502i \(-0.169062\pi\)
−0.00752476 + 0.999972i \(0.502395\pi\)
\(410\) 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 \(-0.817447\pi\)
0.840003 0.542583i \(-0.182553\pi\)
\(420\) 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 \(0.0613955\pi\)
−0.981456 + 0.191686i \(0.938604\pi\)
\(434\) 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 −0.528612 0.848864i \(-0.677287\pi\)
0.470832 + 0.882223i \(0.343954\pi\)
\(440\) 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 \(0.0635381\pi\)
−0.980144 + 0.198288i \(0.936462\pi\)
\(462\) 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 −0.854711 0.519104i \(-0.826266\pi\)
0.0222017 + 0.999754i \(0.492932\pi\)
\(468\) 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 0.946926 0.321452i \(-0.104171\pi\)
0.195078 + 0.980788i \(0.437504\pi\)
\(480\) 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 \(-0.846176\pi\)
0.885488 0.464661i \(-0.153824\pi\)
\(504\) 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 −0.445829 0.895118i \(-0.647091\pi\)
0.552281 + 0.833658i \(0.313758\pi\)
\(510\) 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 0.957044 0.289942i \(-0.0936362\pi\)
0.227425 + 0.973796i \(0.426969\pi\)
\(522\) 0 0
\(523\) 242.586 140.057i 0.463835 0.267795i −0.249821 0.968292i \(-0.580372\pi\)
0.713655 + 0.700497i \(0.247038\pi\)
\(524\) 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 0.336877 0.941549i \(-0.390629\pi\)
−0.646967 + 0.762518i \(0.723963\pi\)
\(564\) 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 \(0.0117231\pi\)
0.531549 + 0.847028i \(0.321610\pi\)
\(578\) 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 \(-0.834956\pi\)
0.868563 0.495579i \(-0.165044\pi\)
\(588\) 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 −0.601358 0.798980i \(-0.705373\pi\)
0.391258 + 0.920281i \(0.372040\pi\)
\(594\) 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 \(-0.724624\pi\)
0.648549 0.761173i \(-0.275376\pi\)
\(602\) 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 −0.873324 0.487139i \(-0.838041\pi\)
−0.0147875 + 0.999891i \(0.504707\pi\)
\(608\) 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 0.534034 0.845463i \(-0.320675\pi\)
−0.465175 + 0.885219i \(0.654009\pi\)
\(620\) 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 \(0.0290043\pi\)
−0.995851 + 0.0909935i \(0.970996\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −467.002 269.624i −0.721796 0.416729i 0.0936176 0.995608i \(-0.470157\pi\)
−0.815413 + 0.578879i \(0.803490\pi\)
\(648\) 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 −0.363828 0.931466i \(-0.618530\pi\)
−0.988587 + 0.150649i \(0.951864\pi\)
\(662\) 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 −0.171817 0.985129i \(-0.554964\pi\)
0.767238 + 0.641362i \(0.221630\pi\)
\(678\) 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 −0.759185 0.650875i \(-0.774402\pi\)
0.184081 + 0.982911i \(0.441069\pi\)
\(692\) 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 −0.858021 0.513615i \(-0.828306\pi\)
0.0157934 + 0.999875i \(0.494973\pi\)
\(720\) 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 \(0.0474930\pi\)
−0.988890 + 0.148651i \(0.952507\pi\)
\(728\) 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 −0.00457181 0.999990i \(-0.501455\pi\)
0.863730 + 0.503954i \(0.168122\pi\)
\(734\) 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 −0.940609 0.339491i \(-0.889745\pi\)
−0.176297 + 0.984337i \(0.556412\pi\)
\(762\) 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 \(0.156064\pi\)
−0.882197 + 0.470880i \(0.843936\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1216.68 + 702.449i 1.57397 + 0.908731i 0.995675 + 0.0929015i \(0.0296142\pi\)
0.578293 + 0.815829i \(0.303719\pi\)
\(774\) 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 0.352602 0.935773i \(-0.385297\pi\)
−0.634103 + 0.773249i \(0.718630\pi\)
\(788\) 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 \(-0.903758\pi\)
0.954639 0.297767i \(-0.0962418\pi\)
\(798\) 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