Properties

Label 1840.2.e.e.369.8
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11574317056.3
Defining polynomial: \(x^{8} + 45 x^{4} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 369.8
Root \(0.386289 - 0.386289i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.e.369.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.25886i q^{3} +(0.386289 - 2.20245i) q^{5} -1.44270i q^{7} -7.62018 q^{9} +O(q^{10})\) \(q+3.25886i q^{3} +(0.386289 - 2.20245i) q^{5} -1.44270i q^{7} -7.62018 q^{9} +2.03144 q^{11} -0.557299i q^{13} +(7.17748 + 1.25886i) q^{15} -3.91861i q^{17} -6.73300 q^{19} +4.70156 q^{21} +1.00000i q^{23} +(-4.70156 - 1.70156i) q^{25} -15.0565i q^{27} +9.69520 q^{29} +3.07502 q^{31} +6.62018i q^{33} +(-3.17748 - 0.557299i) q^{35} -3.65798i q^{37} +1.81616 q^{39} +7.03321 q^{41} -5.06465i q^{43} +(-2.94359 + 16.7830i) q^{45} -0.659753i q^{47} +4.91861 q^{49} +12.7702 q^{51} -11.1275i q^{53} +(0.784722 - 4.47414i) q^{55} -21.9419i q^{57} +10.7226 q^{59} +1.73937 q^{61} +10.9936i q^{63} +(-1.22742 - 0.215278i) q^{65} -12.0129i q^{67} -3.25886 q^{69} -12.4363 q^{71} +9.26464i q^{73} +(5.54515 - 15.3217i) q^{75} -2.93076i q^{77} +5.92439 q^{79} +26.2066 q^{81} -11.3775i q^{83} +(-8.63055 - 1.51372i) q^{85} +31.5953i q^{87} -5.25308 q^{89} -0.804016 q^{91} +10.0211i q^{93} +(-2.60088 + 14.8291i) q^{95} -0.0813861i q^{97} -15.4799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 14q^{9} + O(q^{10}) \) \( 8q - 14q^{9} - 10q^{11} + 16q^{15} - 2q^{19} + 12q^{21} - 12q^{25} - 4q^{29} - 10q^{31} + 16q^{35} + 46q^{41} - 26q^{45} + 18q^{49} - 14q^{51} + 18q^{55} + 32q^{59} + 18q^{61} - 16q^{65} - 6q^{69} - 38q^{71} + 32q^{75} - 12q^{79} + 32q^{81} - 24q^{85} - 60q^{89} + 26q^{91} - 18q^{95} - 54q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 3.25886i 1.88150i 0.339095 + 0.940752i \(0.389879\pi\)
−0.339095 + 0.940752i \(0.610121\pi\)
\(4\) 0 0
\(5\) 0.386289 2.20245i 0.172754 0.984965i
\(6\) 0 0
\(7\) 1.44270i 0.545290i −0.962115 0.272645i \(-0.912102\pi\)
0.962115 0.272645i \(-0.0878984\pi\)
\(8\) 0 0
\(9\) −7.62018 −2.54006
\(10\) 0 0
\(11\) 2.03144 0.612502 0.306251 0.951951i \(-0.400925\pi\)
0.306251 + 0.951951i \(0.400925\pi\)
\(12\) 0 0
\(13\) 0.557299i 0.154567i −0.997009 0.0772835i \(-0.975375\pi\)
0.997009 0.0772835i \(-0.0246246\pi\)
\(14\) 0 0
\(15\) 7.17748 + 1.25886i 1.85322 + 0.325037i
\(16\) 0 0
\(17\) 3.91861i 0.950403i −0.879877 0.475202i \(-0.842375\pi\)
0.879877 0.475202i \(-0.157625\pi\)
\(18\) 0 0
\(19\) −6.73300 −1.54466 −0.772328 0.635224i \(-0.780908\pi\)
−0.772328 + 0.635224i \(0.780908\pi\)
\(20\) 0 0
\(21\) 4.70156 1.02596
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) −4.70156 1.70156i −0.940312 0.340312i
\(26\) 0 0
\(27\) 15.0565i 2.89763i
\(28\) 0 0
\(29\) 9.69520 1.80035 0.900176 0.435525i \(-0.143437\pi\)
0.900176 + 0.435525i \(0.143437\pi\)
\(30\) 0 0
\(31\) 3.07502 0.552290 0.276145 0.961116i \(-0.410943\pi\)
0.276145 + 0.961116i \(0.410943\pi\)
\(32\) 0 0
\(33\) 6.62018i 1.15242i
\(34\) 0 0
\(35\) −3.17748 0.557299i −0.537091 0.0942007i
\(36\) 0 0
\(37\) 3.65798i 0.601368i −0.953724 0.300684i \(-0.902785\pi\)
0.953724 0.300684i \(-0.0972150\pi\)
\(38\) 0 0
\(39\) 1.81616 0.290818
\(40\) 0 0
\(41\) 7.03321 1.09840 0.549202 0.835690i \(-0.314932\pi\)
0.549202 + 0.835690i \(0.314932\pi\)
\(42\) 0 0
\(43\) 5.06465i 0.772352i −0.922425 0.386176i \(-0.873796\pi\)
0.922425 0.386176i \(-0.126204\pi\)
\(44\) 0 0
\(45\) −2.94359 + 16.7830i −0.438804 + 2.50187i
\(46\) 0 0
\(47\) 0.659753i 0.0962348i −0.998842 0.0481174i \(-0.984678\pi\)
0.998842 0.0481174i \(-0.0153222\pi\)
\(48\) 0 0
\(49\) 4.91861 0.702659
\(50\) 0 0
\(51\) 12.7702 1.78819
\(52\) 0 0
\(53\) 11.1275i 1.52848i −0.644930 0.764242i \(-0.723113\pi\)
0.644930 0.764242i \(-0.276887\pi\)
\(54\) 0 0
\(55\) 0.784722 4.47414i 0.105812 0.603293i
\(56\) 0 0
\(57\) 21.9419i 2.90628i
\(58\) 0 0
\(59\) 10.7226 1.39597 0.697984 0.716114i \(-0.254081\pi\)
0.697984 + 0.716114i \(0.254081\pi\)
\(60\) 0 0
\(61\) 1.73937 0.222703 0.111351 0.993781i \(-0.464482\pi\)
0.111351 + 0.993781i \(0.464482\pi\)
\(62\) 0 0
\(63\) 10.9936i 1.38507i
\(64\) 0 0
\(65\) −1.22742 0.215278i −0.152243 0.0267020i
\(66\) 0 0
\(67\) 12.0129i 1.46761i −0.679359 0.733806i \(-0.737742\pi\)
0.679359 0.733806i \(-0.262258\pi\)
\(68\) 0 0
\(69\) −3.25886 −0.392321
\(70\) 0 0
\(71\) −12.4363 −1.47592 −0.737961 0.674844i \(-0.764211\pi\)
−0.737961 + 0.674844i \(0.764211\pi\)
\(72\) 0 0
\(73\) 9.26464i 1.08434i 0.840267 + 0.542172i \(0.182398\pi\)
−0.840267 + 0.542172i \(0.817602\pi\)
\(74\) 0 0
\(75\) 5.54515 15.3217i 0.640299 1.76920i
\(76\) 0 0
\(77\) 2.93076i 0.333991i
\(78\) 0 0
\(79\) 5.92439 0.666546 0.333273 0.942830i \(-0.391847\pi\)
0.333273 + 0.942830i \(0.391847\pi\)
\(80\) 0 0
\(81\) 26.2066 2.91184
\(82\) 0 0
\(83\) 11.3775i 1.24884i −0.781089 0.624420i \(-0.785336\pi\)
0.781089 0.624420i \(-0.214664\pi\)
\(84\) 0 0
\(85\) −8.63055 1.51372i −0.936114 0.164186i
\(86\) 0 0
\(87\) 31.5953i 3.38737i
\(88\) 0 0
\(89\) −5.25308 −0.556826 −0.278413 0.960462i \(-0.589808\pi\)
−0.278413 + 0.960462i \(0.589808\pi\)
\(90\) 0 0
\(91\) −0.804016 −0.0842838
\(92\) 0 0
\(93\) 10.0211i 1.03914i
\(94\) 0 0
\(95\) −2.60088 + 14.8291i −0.266845 + 1.52143i
\(96\) 0 0
\(97\) 0.0813861i 0.00826350i −0.999991 0.00413175i \(-0.998685\pi\)
0.999991 0.00413175i \(-0.00131518\pi\)
\(98\) 0 0
\(99\) −15.4799 −1.55579
\(100\) 0 0
\(101\) 8.31281 0.827156 0.413578 0.910469i \(-0.364279\pi\)
0.413578 + 0.910469i \(0.364279\pi\)
\(102\) 0 0
\(103\) 6.36991i 0.627646i −0.949481 0.313823i \(-0.898390\pi\)
0.949481 0.313823i \(-0.101610\pi\)
\(104\) 0 0
\(105\) 1.81616 10.3550i 0.177239 1.01054i
\(106\) 0 0
\(107\) 16.9226i 1.63597i 0.575239 + 0.817986i \(0.304909\pi\)
−0.575239 + 0.817986i \(0.695091\pi\)
\(108\) 0 0
\(109\) 0.218825 0.0209597 0.0104798 0.999945i \(-0.496664\pi\)
0.0104798 + 0.999945i \(0.496664\pi\)
\(110\) 0 0
\(111\) 11.9208 1.13148
\(112\) 0 0
\(113\) 0.580599i 0.0546182i −0.999627 0.0273091i \(-0.991306\pi\)
0.999627 0.0273091i \(-0.00869384\pi\)
\(114\) 0 0
\(115\) 2.20245 + 0.386289i 0.205379 + 0.0360216i
\(116\) 0 0
\(117\) 4.24672i 0.392609i
\(118\) 0 0
\(119\) −5.65339 −0.518245
\(120\) 0 0
\(121\) −6.87326 −0.624842
\(122\) 0 0
\(123\) 22.9203i 2.06665i
\(124\) 0 0
\(125\) −5.56376 + 9.69766i −0.497638 + 0.867385i
\(126\) 0 0
\(127\) 1.56944i 0.139266i −0.997573 0.0696328i \(-0.977817\pi\)
0.997573 0.0696328i \(-0.0221828\pi\)
\(128\) 0 0
\(129\) 16.5050 1.45318
\(130\) 0 0
\(131\) −15.9208 −1.39101 −0.695505 0.718521i \(-0.744819\pi\)
−0.695505 + 0.718521i \(0.744819\pi\)
\(132\) 0 0
\(133\) 9.71371i 0.842285i
\(134\) 0 0
\(135\) −33.1612 5.81616i −2.85406 0.500575i
\(136\) 0 0
\(137\) 6.47013i 0.552781i −0.961045 0.276390i \(-0.910862\pi\)
0.961045 0.276390i \(-0.0891383\pi\)
\(138\) 0 0
\(139\) −0.656206 −0.0556586 −0.0278293 0.999613i \(-0.508859\pi\)
−0.0278293 + 0.999613i \(0.508859\pi\)
\(140\) 0 0
\(141\) 2.15004 0.181066
\(142\) 0 0
\(143\) 1.13212i 0.0946725i
\(144\) 0 0
\(145\) 3.74514 21.3532i 0.311017 1.77328i
\(146\) 0 0
\(147\) 16.0291i 1.32206i
\(148\) 0 0
\(149\) −23.8349 −1.95263 −0.976314 0.216357i \(-0.930582\pi\)
−0.976314 + 0.216357i \(0.930582\pi\)
\(150\) 0 0
\(151\) 14.8669 1.20985 0.604925 0.796282i \(-0.293203\pi\)
0.604925 + 0.796282i \(0.293203\pi\)
\(152\) 0 0
\(153\) 29.8605i 2.41408i
\(154\) 0 0
\(155\) 1.18785 6.77258i 0.0954101 0.543987i
\(156\) 0 0
\(157\) 8.24213i 0.657793i 0.944366 + 0.328897i \(0.106677\pi\)
−0.944366 + 0.328897i \(0.893323\pi\)
\(158\) 0 0
\(159\) 36.2631 2.87585
\(160\) 0 0
\(161\) 1.44270 0.113701
\(162\) 0 0
\(163\) 10.3154i 0.807962i −0.914767 0.403981i \(-0.867626\pi\)
0.914767 0.403981i \(-0.132374\pi\)
\(164\) 0 0
\(165\) 14.5806 + 2.55730i 1.13510 + 0.199085i
\(166\) 0 0
\(167\) 4.65975i 0.360582i 0.983613 + 0.180291i \(0.0577040\pi\)
−0.983613 + 0.180291i \(0.942296\pi\)
\(168\) 0 0
\(169\) 12.6894 0.976109
\(170\) 0 0
\(171\) 51.3067 3.92352
\(172\) 0 0
\(173\) 2.80402i 0.213185i −0.994303 0.106593i \(-0.966006\pi\)
0.994303 0.106593i \(-0.0339941\pi\)
\(174\) 0 0
\(175\) −2.45485 + 6.78295i −0.185569 + 0.512743i
\(176\) 0 0
\(177\) 34.9436i 2.62652i
\(178\) 0 0
\(179\) −4.44212 −0.332019 −0.166010 0.986124i \(-0.553088\pi\)
−0.166010 + 0.986124i \(0.553088\pi\)
\(180\) 0 0
\(181\) 2.42019 0.179891 0.0899455 0.995947i \(-0.471331\pi\)
0.0899455 + 0.995947i \(0.471331\pi\)
\(182\) 0 0
\(183\) 5.66835i 0.419016i
\(184\) 0 0
\(185\) −8.05651 1.41304i −0.592327 0.103888i
\(186\) 0 0
\(187\) 7.96042i 0.582124i
\(188\) 0 0
\(189\) −21.7220 −1.58005
\(190\) 0 0
\(191\) 20.8934 1.51179 0.755897 0.654690i \(-0.227201\pi\)
0.755897 + 0.654690i \(0.227201\pi\)
\(192\) 0 0
\(193\) 2.88540i 0.207696i 0.994593 + 0.103848i \(0.0331155\pi\)
−0.994593 + 0.103848i \(0.966884\pi\)
\(194\) 0 0
\(195\) 0.701562 4.00000i 0.0502399 0.286446i
\(196\) 0 0
\(197\) 19.0925i 1.36029i 0.733079 + 0.680144i \(0.238083\pi\)
−0.733079 + 0.680144i \(0.761917\pi\)
\(198\) 0 0
\(199\) 6.58060 0.466486 0.233243 0.972418i \(-0.425066\pi\)
0.233243 + 0.972418i \(0.425066\pi\)
\(200\) 0 0
\(201\) 39.1485 2.76132
\(202\) 0 0
\(203\) 13.9873i 0.981714i
\(204\) 0 0
\(205\) 2.71685 15.4903i 0.189753 1.08189i
\(206\) 0 0
\(207\) 7.62018i 0.529639i
\(208\) 0 0
\(209\) −13.6777 −0.946105
\(210\) 0 0
\(211\) −6.55986 −0.451599 −0.225800 0.974174i \(-0.572499\pi\)
−0.225800 + 0.974174i \(0.572499\pi\)
\(212\) 0 0
\(213\) 40.5283i 2.77695i
\(214\) 0 0
\(215\) −11.1546 1.95642i −0.760740 0.133427i
\(216\) 0 0
\(217\) 4.43634i 0.301158i
\(218\) 0 0
\(219\) −30.1922 −2.04020
\(220\) 0 0
\(221\) −2.18384 −0.146901
\(222\) 0 0
\(223\) 20.2094i 1.35332i 0.736296 + 0.676660i \(0.236573\pi\)
−0.736296 + 0.676660i \(0.763427\pi\)
\(224\) 0 0
\(225\) 35.8267 + 12.9662i 2.38845 + 0.864414i
\(226\) 0 0
\(227\) 15.0098i 0.996234i −0.867110 0.498117i \(-0.834025\pi\)
0.867110 0.498117i \(-0.165975\pi\)
\(228\) 0 0
\(229\) −13.7081 −0.905859 −0.452929 0.891546i \(-0.649621\pi\)
−0.452929 + 0.891546i \(0.649621\pi\)
\(230\) 0 0
\(231\) 9.55093 0.628405
\(232\) 0 0
\(233\) 9.19822i 0.602595i 0.953530 + 0.301298i \(0.0974198\pi\)
−0.953530 + 0.301298i \(0.902580\pi\)
\(234\) 0 0
\(235\) −1.45307 0.254855i −0.0947880 0.0166249i
\(236\) 0 0
\(237\) 19.3068i 1.25411i
\(238\) 0 0
\(239\) −2.72618 −0.176342 −0.0881709 0.996105i \(-0.528102\pi\)
−0.0881709 + 0.996105i \(0.528102\pi\)
\(240\) 0 0
\(241\) 3.26109 0.210066 0.105033 0.994469i \(-0.466505\pi\)
0.105033 + 0.994469i \(0.466505\pi\)
\(242\) 0 0
\(243\) 40.2340i 2.58101i
\(244\) 0 0
\(245\) 1.90000 10.8330i 0.121387 0.692095i
\(246\) 0 0
\(247\) 3.75229i 0.238753i
\(248\) 0 0
\(249\) 37.0776 2.34970
\(250\) 0 0
\(251\) −19.9941 −1.26202 −0.631008 0.775776i \(-0.717359\pi\)
−0.631008 + 0.775776i \(0.717359\pi\)
\(252\) 0 0
\(253\) 2.03144i 0.127715i
\(254\) 0 0
\(255\) 4.93299 28.1258i 0.308916 1.76130i
\(256\) 0 0
\(257\) 26.7935i 1.67133i −0.549237 0.835667i \(-0.685081\pi\)
0.549237 0.835667i \(-0.314919\pi\)
\(258\) 0 0
\(259\) −5.27737 −0.327920
\(260\) 0 0
\(261\) −73.8791 −4.57300
\(262\) 0 0
\(263\) 15.8267i 0.975918i −0.872867 0.487959i \(-0.837742\pi\)
0.872867 0.487959i \(-0.162258\pi\)
\(264\) 0 0
\(265\) −24.5078 4.29844i −1.50550 0.264051i
\(266\) 0 0
\(267\) 17.1191i 1.04767i
\(268\) 0 0
\(269\) −12.0872 −0.736967 −0.368484 0.929634i \(-0.620123\pi\)
−0.368484 + 0.929634i \(0.620123\pi\)
\(270\) 0 0
\(271\) −8.40254 −0.510418 −0.255209 0.966886i \(-0.582144\pi\)
−0.255209 + 0.966886i \(0.582144\pi\)
\(272\) 0 0
\(273\) 2.62018i 0.158580i
\(274\) 0 0
\(275\) −9.55093 3.45662i −0.575943 0.208442i
\(276\) 0 0
\(277\) 13.8488i 0.832093i −0.909343 0.416046i \(-0.863415\pi\)
0.909343 0.416046i \(-0.136585\pi\)
\(278\) 0 0
\(279\) −23.4322 −1.40285
\(280\) 0 0
\(281\) −8.15004 −0.486191 −0.243095 0.970002i \(-0.578163\pi\)
−0.243095 + 0.970002i \(0.578163\pi\)
\(282\) 0 0
\(283\) 17.3532i 1.03154i −0.856727 0.515770i \(-0.827506\pi\)
0.856727 0.515770i \(-0.172494\pi\)
\(284\) 0 0
\(285\) −48.3259 8.47591i −2.86258 0.502070i
\(286\) 0 0
\(287\) 10.1468i 0.598948i
\(288\) 0 0
\(289\) 1.64446 0.0967332
\(290\) 0 0
\(291\) 0.265226 0.0155478
\(292\) 0 0
\(293\) 7.02252i 0.410260i −0.978735 0.205130i \(-0.934238\pi\)
0.978735 0.205130i \(-0.0657617\pi\)
\(294\) 0 0
\(295\) 4.14203 23.6160i 0.241158 1.37498i
\(296\) 0 0
\(297\) 30.5864i 1.77480i
\(298\) 0 0
\(299\) 0.557299 0.0322294
\(300\) 0 0
\(301\) −7.30678 −0.421156
\(302\) 0 0
\(303\) 27.0903i 1.55630i
\(304\) 0 0
\(305\) 0.671897 3.83086i 0.0384727 0.219355i
\(306\) 0 0
\(307\) 16.4156i 0.936887i −0.883493 0.468444i \(-0.844815\pi\)
0.883493 0.468444i \(-0.155185\pi\)
\(308\) 0 0
\(309\) 20.7587 1.18092
\(310\) 0 0
\(311\) −7.35181 −0.416883 −0.208441 0.978035i \(-0.566839\pi\)
−0.208441 + 0.978035i \(0.566839\pi\)
\(312\) 0 0
\(313\) 4.08972i 0.231165i −0.993298 0.115582i \(-0.963127\pi\)
0.993298 0.115582i \(-0.0368734\pi\)
\(314\) 0 0
\(315\) 24.2129 + 4.24672i 1.36424 + 0.239275i
\(316\) 0 0
\(317\) 8.91580i 0.500761i 0.968148 + 0.250381i \(0.0805557\pi\)
−0.968148 + 0.250381i \(0.919444\pi\)
\(318\) 0 0
\(319\) 19.6952 1.10272
\(320\) 0 0
\(321\) −55.1485 −3.07809
\(322\) 0 0
\(323\) 26.3840i 1.46805i
\(324\) 0 0
\(325\) −0.948279 + 2.62018i −0.0526011 + 0.145341i
\(326\) 0 0
\(327\) 0.713121i 0.0394357i
\(328\) 0 0
\(329\) −0.951826 −0.0524759
\(330\) 0 0
\(331\) −21.3240 −1.17207 −0.586036 0.810285i \(-0.699312\pi\)
−0.586036 + 0.810285i \(0.699312\pi\)
\(332\) 0 0
\(333\) 27.8744i 1.52751i
\(334\) 0 0
\(335\) −26.4579 4.64046i −1.44555 0.253535i
\(336\) 0 0
\(337\) 3.15417i 0.171819i −0.996303 0.0859094i \(-0.972620\pi\)
0.996303 0.0859094i \(-0.0273796\pi\)
\(338\) 0 0
\(339\) 1.89209 0.102764
\(340\) 0 0
\(341\) 6.24672 0.338279
\(342\) 0 0
\(343\) 17.1950i 0.928443i
\(344\) 0 0
\(345\) −1.25886 + 7.17748i −0.0677748 + 0.386422i
\(346\) 0 0
\(347\) 3.54739i 0.190434i 0.995457 + 0.0952169i \(0.0303545\pi\)
−0.995457 + 0.0952169i \(0.969646\pi\)
\(348\) 0 0
\(349\) 18.9726 1.01558 0.507789 0.861481i \(-0.330463\pi\)
0.507789 + 0.861481i \(0.330463\pi\)
\(350\) 0 0
\(351\) −8.39098 −0.447877
\(352\) 0 0
\(353\) 33.5710i 1.78680i 0.449257 + 0.893402i \(0.351689\pi\)
−0.449257 + 0.893402i \(0.648311\pi\)
\(354\) 0 0
\(355\) −4.80402 + 27.3904i −0.254971 + 1.45373i
\(356\) 0 0
\(357\) 18.4236i 0.975081i
\(358\) 0 0
\(359\) −3.53243 −0.186434 −0.0932171 0.995646i \(-0.529715\pi\)
−0.0932171 + 0.995646i \(0.529715\pi\)
\(360\) 0 0
\(361\) 26.3333 1.38596
\(362\) 0 0
\(363\) 22.3990i 1.17564i
\(364\) 0 0
\(365\) 20.4049 + 3.57883i 1.06804 + 0.187324i
\(366\) 0 0
\(367\) 8.07444i 0.421482i 0.977542 + 0.210741i \(0.0675877\pi\)
−0.977542 + 0.210741i \(0.932412\pi\)
\(368\) 0 0
\(369\) −53.5943 −2.79001
\(370\) 0 0
\(371\) −16.0537 −0.833466
\(372\) 0 0
\(373\) 1.31459i 0.0680668i −0.999421 0.0340334i \(-0.989165\pi\)
0.999421 0.0340334i \(-0.0108353\pi\)
\(374\) 0 0
\(375\) −31.6033 18.1315i −1.63199 0.936308i
\(376\) 0 0
\(377\) 5.40312i 0.278275i
\(378\) 0 0
\(379\) 0.311952 0.0160239 0.00801196 0.999968i \(-0.497450\pi\)
0.00801196 + 0.999968i \(0.497450\pi\)
\(380\) 0 0
\(381\) 5.11460 0.262029
\(382\) 0 0
\(383\) 10.6805i 0.545748i −0.962050 0.272874i \(-0.912026\pi\)
0.962050 0.272874i \(-0.0879742\pi\)
\(384\) 0 0
\(385\) −6.45485 1.13212i −0.328969 0.0576981i
\(386\) 0 0
\(387\) 38.5935i 1.96182i
\(388\) 0 0
\(389\) 7.86867 0.398957 0.199479 0.979902i \(-0.436075\pi\)
0.199479 + 0.979902i \(0.436075\pi\)
\(390\) 0 0
\(391\) 3.91861 0.198173
\(392\) 0 0
\(393\) 51.8838i 2.61719i
\(394\) 0 0
\(395\) 2.28853 13.0482i 0.115148 0.656525i
\(396\) 0 0
\(397\) 29.2334i 1.46718i −0.679591 0.733591i \(-0.737843\pi\)
0.679591 0.733591i \(-0.262157\pi\)
\(398\) 0 0
\(399\) −31.6556 −1.58476
\(400\) 0 0
\(401\) 23.7130 1.18417 0.592086 0.805874i \(-0.298304\pi\)
0.592086 + 0.805874i \(0.298304\pi\)
\(402\) 0 0
\(403\) 1.71371i 0.0853658i
\(404\) 0 0
\(405\) 10.1233 57.7186i 0.503031 2.86806i
\(406\) 0 0
\(407\) 7.43096i 0.368339i
\(408\) 0 0
\(409\) 33.2509 1.64415 0.822076 0.569378i \(-0.192816\pi\)
0.822076 + 0.569378i \(0.192816\pi\)
\(410\) 0 0
\(411\) 21.0853 1.04006
\(412\) 0 0
\(413\) 15.4695i 0.761207i
\(414\) 0 0
\(415\) −25.0583 4.39499i −1.23006 0.215741i
\(416\) 0 0
\(417\) 2.13848i 0.104722i
\(418\) 0 0
\(419\) 7.28229 0.355763 0.177882 0.984052i \(-0.443076\pi\)
0.177882 + 0.984052i \(0.443076\pi\)
\(420\) 0 0
\(421\) −2.71515 −0.132329 −0.0661643 0.997809i \(-0.521076\pi\)
−0.0661643 + 0.997809i \(0.521076\pi\)
\(422\) 0 0
\(423\) 5.02743i 0.244442i
\(424\) 0 0
\(425\) −6.66777 + 18.4236i −0.323434 + 0.893676i
\(426\) 0 0
\(427\) 2.50938i 0.121438i
\(428\) 0 0
\(429\) 3.68942 0.178127
\(430\) 0 0
\(431\) −10.3336 −0.497750 −0.248875 0.968536i \(-0.580061\pi\)
−0.248875 + 0.968536i \(0.580061\pi\)
\(432\) 0 0
\(433\) 28.4153i 1.36555i 0.730628 + 0.682775i \(0.239227\pi\)
−0.730628 + 0.682775i \(0.760773\pi\)
\(434\) 0 0
\(435\) 69.5870 + 12.2049i 3.33644 + 0.585181i
\(436\) 0 0
\(437\) 6.73300i 0.322083i
\(438\) 0 0
\(439\) 6.37313 0.304173 0.152087 0.988367i \(-0.451401\pi\)
0.152087 + 0.988367i \(0.451401\pi\)
\(440\) 0 0
\(441\) −37.4807 −1.78480
\(442\) 0 0
\(443\) 9.43114i 0.448087i 0.974579 + 0.224044i \(0.0719258\pi\)
−0.974579 + 0.224044i \(0.928074\pi\)
\(444\) 0 0
\(445\) −2.02921 + 11.5696i −0.0961936 + 0.548454i
\(446\) 0 0
\(447\) 77.6745i 3.67388i
\(448\) 0 0
\(449\) −11.3185 −0.534154 −0.267077 0.963675i \(-0.586058\pi\)
−0.267077 + 0.963675i \(0.586058\pi\)
\(450\) 0 0
\(451\) 14.2875 0.672774
\(452\) 0 0
\(453\) 48.4491i 2.27634i
\(454\) 0 0
\(455\) −0.310582 + 1.77080i −0.0145603 + 0.0830166i
\(456\) 0 0
\(457\) 4.96456i 0.232232i −0.993236 0.116116i \(-0.962956\pi\)
0.993236 0.116116i \(-0.0370445\pi\)
\(458\) 0 0
\(459\) −59.0007 −2.75391
\(460\) 0 0
\(461\) 19.9128 0.927433 0.463717 0.885984i \(-0.346516\pi\)
0.463717 + 0.885984i \(0.346516\pi\)
\(462\) 0 0
\(463\) 29.9046i 1.38978i 0.719114 + 0.694892i \(0.244548\pi\)
−0.719114 + 0.694892i \(0.755452\pi\)
\(464\) 0 0
\(465\) 22.0709 + 3.87102i 1.02351 + 0.179514i
\(466\) 0 0
\(467\) 15.1032i 0.698895i 0.936956 + 0.349447i \(0.113631\pi\)
−0.936956 + 0.349447i \(0.886369\pi\)
\(468\) 0 0
\(469\) −17.3311 −0.800274
\(470\) 0 0
\(471\) −26.8599 −1.23764
\(472\) 0 0
\(473\) 10.2885i 0.473067i
\(474\) 0 0
\(475\) 31.6556 + 11.4566i 1.45246 + 0.525666i
\(476\) 0 0
\(477\) 84.7937i 3.88244i
\(478\) 0 0
\(479\) 28.6678 1.30986 0.654932 0.755688i \(-0.272697\pi\)
0.654932 + 0.755688i \(0.272697\pi\)
\(480\) 0 0
\(481\) −2.03859 −0.0929516
\(482\) 0 0
\(483\) 4.70156i 0.213928i
\(484\) 0 0
\(485\) −0.179249 0.0314385i −0.00813926 0.00142755i
\(486\) 0 0
\(487\) 9.15319i 0.414770i 0.978259 + 0.207385i \(0.0664954\pi\)
−0.978259 + 0.207385i \(0.933505\pi\)
\(488\) 0 0
\(489\) 33.6164 1.52018
\(490\) 0 0
\(491\) 20.8098 0.939133 0.469566 0.882897i \(-0.344410\pi\)
0.469566 + 0.882897i \(0.344410\pi\)
\(492\) 0 0
\(493\) 37.9917i 1.71106i
\(494\) 0 0
\(495\) −5.97972 + 34.0937i −0.268768 + 1.53240i
\(496\) 0 0
\(497\) 17.9419i 0.804805i
\(498\) 0 0
\(499\) −18.1293 −0.811579 −0.405789 0.913967i \(-0.633003\pi\)
−0.405789 + 0.913967i \(0.633003\pi\)
\(500\) 0 0
\(501\) −15.1855 −0.678438
\(502\) 0 0
\(503\) 26.4296i 1.17844i 0.807974 + 0.589218i \(0.200564\pi\)
−0.807974 + 0.589218i \(0.799436\pi\)
\(504\) 0 0
\(505\) 3.21115 18.3085i 0.142894 0.814720i
\(506\) 0 0
\(507\) 41.3531i 1.83655i
\(508\) 0 0
\(509\) 12.0501 0.534113 0.267057 0.963681i \(-0.413949\pi\)
0.267057 + 0.963681i \(0.413949\pi\)
\(510\) 0 0
\(511\) 13.3661 0.591282
\(512\) 0 0
\(513\) 101.376i 4.47584i
\(514\) 0 0
\(515\) −14.0294 2.46062i −0.618209 0.108428i
\(516\) 0 0
\(517\) 1.34025i 0.0589440i
\(518\) 0 0
\(519\) 9.13790 0.401109
\(520\) 0 0
\(521\) −25.3697 −1.11146 −0.555732 0.831361i \(-0.687562\pi\)
−0.555732 + 0.831361i \(0.687562\pi\)
\(522\) 0 0
\(523\) 25.0018i 1.09325i 0.837377 + 0.546626i \(0.184088\pi\)
−0.837377 + 0.546626i \(0.815912\pi\)
\(524\) 0 0
\(525\) −22.1047 8.00000i −0.964728 0.349149i
\(526\) 0 0
\(527\) 12.0498i 0.524898i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −81.7083 −3.54584
\(532\) 0 0
\(533\) 3.91960i 0.169777i
\(534\) 0 0
\(535\) 37.2712 + 6.53702i 1.61137 + 0.282620i
\(536\) 0 0
\(537\) 14.4762i 0.624696i
\(538\) 0 0
\(539\) 9.99186 0.430380
\(540\) 0 0
\(541\) 24.7900 1.06580 0.532902 0.846177i \(-0.321101\pi\)
0.532902 + 0.846177i \(0.321101\pi\)
\(542\) 0 0
\(543\) 7.88705i 0.338466i
\(544\) 0 0
\(545\) 0.0845298 0.481952i 0.00362086 0.0206445i
\(546\) 0 0
\(547\) 30.9508i 1.32336i −0.749785 0.661681i \(-0.769843\pi\)
0.749785 0.661681i \(-0.230157\pi\)
\(548\) 0 0
\(549\) −13.2543 −0.565678
\(550\) 0 0
\(551\) −65.2778 −2.78093
\(552\) 0 0
\(553\) 8.54713i 0.363461i
\(554\) 0 0
\(555\) 4.60489 26.2551i 0.195467 1.11447i
\(556\) 0 0
\(557\) 7.45307i 0.315797i 0.987455 + 0.157898i \(0.0504718\pi\)
−0.987455 + 0.157898i \(0.949528\pi\)
\(558\) 0 0
\(559\) −2.82252 −0.119380
\(560\) 0 0
\(561\) 25.9419 1.09527
\(562\) 0 0
\(563\) 7.96161i 0.335542i 0.985826 + 0.167771i \(0.0536569\pi\)
−0.985826 + 0.167771i \(0.946343\pi\)
\(564\) 0 0
\(565\) −1.27874 0.224279i −0.0537970 0.00943549i
\(566\) 0 0
\(567\) 37.8082i 1.58780i
\(568\) 0 0
\(569\) −32.3873 −1.35774 −0.678872 0.734257i \(-0.737531\pi\)
−0.678872 + 0.734257i \(0.737531\pi\)
\(570\) 0 0
\(571\) 6.38994 0.267410 0.133705 0.991021i \(-0.457312\pi\)
0.133705 + 0.991021i \(0.457312\pi\)
\(572\) 0 0
\(573\) 68.0887i 2.84445i
\(574\) 0 0
\(575\) 1.70156 4.70156i 0.0709600 0.196069i
\(576\) 0 0
\(577\) 39.1728i 1.63078i 0.578910 + 0.815392i \(0.303478\pi\)
−0.578910 + 0.815392i \(0.696522\pi\)
\(578\) 0 0
\(579\) −9.40312 −0.390781
\(580\) 0 0
\(581\) −16.4143 −0.680979
\(582\) 0 0
\(583\) 22.6049i 0.936199i
\(584\) 0 0
\(585\) 9.35318 + 1.64046i 0.386706 + 0.0678246i
\(586\) 0 0
\(587\) 21.3811i 0.882491i −0.897386 0.441246i \(-0.854537\pi\)
0.897386 0.441246i \(-0.145463\pi\)
\(588\) 0 0
\(589\) −20.7041 −0.853098
\(590\) 0 0
\(591\) −62.2199 −2.55939
\(592\) 0 0
\(593\) 2.45130i 0.100663i −0.998733 0.0503314i \(-0.983972\pi\)
0.998733 0.0503314i \(-0.0160277\pi\)
\(594\) 0 0
\(595\) −2.18384 + 12.4513i −0.0895287 + 0.510453i
\(596\) 0 0
\(597\) 21.4453i 0.877696i
\(598\) 0 0
\(599\) 38.5828 1.57645 0.788226 0.615386i \(-0.211000\pi\)
0.788226 + 0.615386i \(0.211000\pi\)
\(600\) 0 0
\(601\) −35.3416 −1.44162 −0.720808 0.693135i \(-0.756229\pi\)
−0.720808 + 0.693135i \(0.756229\pi\)
\(602\) 0 0
\(603\) 91.5406i 3.72782i
\(604\) 0 0
\(605\) −2.65506 + 15.1380i −0.107944 + 0.615447i
\(606\) 0 0
\(607\) 17.8129i 0.723005i −0.932371 0.361502i \(-0.882264\pi\)
0.932371 0.361502i \(-0.117736\pi\)
\(608\) 0 0
\(609\) 45.5826 1.84710
\(610\) 0 0
\(611\) −0.367680 −0.0148747
\(612\) 0 0
\(613\) 37.7547i 1.52490i −0.647048 0.762450i \(-0.723997\pi\)
0.647048 0.762450i \(-0.276003\pi\)
\(614\) 0 0
\(615\) 50.4807 + 8.85384i 2.03558 + 0.357021i
\(616\) 0 0
\(617\) 23.1248i 0.930971i −0.885055 0.465486i \(-0.845880\pi\)
0.885055 0.465486i \(-0.154120\pi\)
\(618\) 0 0
\(619\) 19.5428 0.785491 0.392746 0.919647i \(-0.371525\pi\)
0.392746 + 0.919647i \(0.371525\pi\)
\(620\) 0 0
\(621\) 15.0565 0.604197
\(622\) 0 0
\(623\) 7.57863i 0.303631i
\(624\) 0 0
\(625\) 19.2094 + 16.0000i 0.768375 + 0.640000i
\(626\) 0 0
\(627\) 44.5736i 1.78010i
\(628\) 0 0
\(629\) −14.3342 −0.571542
\(630\) 0 0
\(631\) 26.4807 1.05418 0.527090 0.849809i \(-0.323283\pi\)
0.527090 + 0.849809i \(0.323283\pi\)
\(632\) 0 0
\(633\) 21.3777i 0.849686i
\(634\) 0 0
\(635\) −3.45662 0.606258i −0.137172 0.0240586i
\(636\) 0 0
\(637\) 2.74114i 0.108608i
\(638\) 0 0
\(639\) 94.7671 3.74893
\(640\) 0 0
\(641\) −35.9594 −1.42031 −0.710156 0.704044i \(-0.751376\pi\)
−0.710156 + 0.704044i \(0.751376\pi\)
\(642\) 0 0
\(643\) 14.6934i 0.579452i −0.957110 0.289726i \(-0.906436\pi\)
0.957110 0.289726i \(-0.0935642\pi\)
\(644\) 0 0
\(645\) 6.37569 36.3514i 0.251043 1.43134i
\(646\) 0 0
\(647\) 26.7763i 1.05269i 0.850272 + 0.526343i \(0.176437\pi\)
−0.850272 + 0.526343i \(0.823563\pi\)
\(648\) 0 0
\(649\) 21.7824 0.855033
\(650\) 0 0
\(651\) 14.4574 0.566630
\(652\) 0 0
\(653\) 18.8331i 0.736996i 0.929629 + 0.368498i \(0.120128\pi\)
−0.929629 + 0.368498i \(0.879872\pi\)
\(654\) 0 0
\(655\) −6.15004 + 35.0649i −0.240302 + 1.37010i
\(656\) 0 0
\(657\) 70.5982i 2.75430i
\(658\) 0 0
\(659\) −18.0885 −0.704629 −0.352315 0.935882i \(-0.614605\pi\)
−0.352315 + 0.935882i \(0.614605\pi\)
\(660\) 0 0
\(661\) 30.3235 1.17945 0.589724 0.807605i \(-0.299237\pi\)
0.589724 + 0.807605i \(0.299237\pi\)
\(662\) 0 0
\(663\) 7.11683i 0.276395i
\(664\) 0 0
\(665\) 21.3939 + 3.75229i 0.829621 + 0.145508i
\(666\) 0 0
\(667\) 9.69520i 0.375400i
\(668\) 0 0
\(669\) −65.8595 −2.54628
\(670\) 0 0
\(671\) 3.53341 0.136406
\(672\) 0 0
\(673\) 11.1496i 0.429787i 0.976637 + 0.214894i \(0.0689404\pi\)
−0.976637 + 0.214894i \(0.931060\pi\)
\(674\) 0 0
\(675\) −25.6196 + 70.7891i −0.986099 + 2.72467i
\(676\) 0 0
\(677\) 43.5275i 1.67290i 0.548045 + 0.836449i \(0.315372\pi\)
−0.548045 + 0.836449i \(0.684628\pi\)
\(678\) 0 0
\(679\) −0.117416 −0.00450600
\(680\) 0 0
\(681\) 48.9148 1.87442
\(682\) 0 0
\(683\) 24.1375i 0.923596i −0.886985 0.461798i \(-0.847205\pi\)
0.886985 0.461798i \(-0.152795\pi\)
\(684\) 0 0
\(685\) −14.2501 2.49934i −0.544470 0.0954948i
\(686\) 0 0
\(687\) 44.6729i 1.70438i
\(688\) 0 0
\(689\) −6.20136 −0.236253
\(690\) 0 0
\(691\) 20.8241 0.792186 0.396093 0.918210i \(-0.370366\pi\)
0.396093 + 0.918210i \(0.370366\pi\)
\(692\) 0 0
\(693\) 22.3329i 0.848356i
\(694\) 0 0
\(695\) −0.253485 + 1.44526i −0.00961523 + 0.0548218i
\(696\) 0 0
\(697\) 27.5604i 1.04393i
\(698\) 0 0
\(699\) −29.9757 −1.13379
\(700\) 0 0
\(701\) −31.5301 −1.19087 −0.595437 0.803402i \(-0.703021\pi\)
−0.595437 + 0.803402i \(0.703021\pi\)
\(702\) 0 0
\(703\) 24.6292i 0.928907i
\(704\) 0 0
\(705\) 0.830537 4.73536i 0.0312798 0.178344i
\(706\) 0 0
\(707\) 11.9929i 0.451040i
\(708\) 0 0
\(709\) −3.12175 −0.117240 −0.0586199 0.998280i \(-0.518670\pi\)
−0.0586199 + 0.998280i \(0.518670\pi\)
\(710\) 0 0
\(711\) −45.1449 −1.69307
\(712\) 0 0
\(713\) 3.07502i 0.115160i
\(714\) 0 0
\(715\) −2.49343 0.437325i −0.0932491 0.0163550i
\(716\) 0 0
\(717\) 8.88423i 0.331788i
\(718\) 0 0
\(719\) 20.3945 0.760588 0.380294 0.924866i \(-0.375823\pi\)
0.380294 + 0.924866i \(0.375823\pi\)
\(720\) 0 0
\(721\) −9.18988 −0.342249
\(722\) 0 0
\(723\) 10.6275i 0.395239i
\(724\) 0 0
\(725\) −45.5826 16.4970i −1.69289 0.612682i
\(726\) 0 0
\(727\) 39.8896i 1.47942i 0.672924 + 0.739712i \(0.265038\pi\)
−0.672924 + 0.739712i \(0.734962\pi\)
\(728\) 0 0
\(729\) −52.4973 −1.94434
\(730\) 0 0
\(731\) −19.8464 −0.734046
\(732\) 0 0
\(733\) 22.7761i 0.841255i 0.907233 + 0.420628i \(0.138190\pi\)
−0.907233 + 0.420628i \(0.861810\pi\)
\(734\) 0 0
\(735\) 35.3032 + 6.19185i 1.30218 + 0.228390i
\(736\) 0 0
\(737\) 24.4035i 0.898915i
\(738\) 0 0
\(739\) 33.7661 1.24211 0.621053 0.783769i \(-0.286705\pi\)
0.621053 + 0.783769i \(0.286705\pi\)
\(740\) 0 0
\(741\) −12.2282 −0.449214
\(742\) 0 0
\(743\) 47.5880i 1.74584i 0.487868 + 0.872918i \(0.337775\pi\)
−0.487868 + 0.872918i \(0.662225\pi\)
\(744\) 0 0
\(745\) −9.20714 + 52.4951i −0.337324 + 1.92327i
\(746\) 0 0
\(747\) 86.6983i 3.17212i
\(748\) 0 0
\(749\) 24.4143 0.892078
\(750\) 0 0
\(751\) −51.2698 −1.87086 −0.935430 0.353512i \(-0.884987\pi\)
−0.935430 + 0.353512i \(0.884987\pi\)
\(752\) 0 0
\(753\) 65.1580i 2.37449i
\(754\) 0 0
\(755\) 5.74291 32.7436i 0.209006 1.19166i
\(756\) 0 0
\(757\) 26.2807i 0.955189i −0.878580 0.477594i \(-0.841509\pi\)
0.878580 0.477594i \(-0.158491\pi\)
\(758\) 0 0
\(759\) −6.62018 −0.240297
\(760\) 0 0
\(761\) −7.18003 −0.260276 −0.130138 0.991496i \(-0.541542\pi\)
−0.130138 + 0.991496i \(0.541542\pi\)
\(762\) 0 0
\(763\) 0.315700i 0.0114291i
\(764\) 0 0
\(765\) 65.7663 + 11.5348i 2.37779 + 0.417041i
\(766\) 0 0
\(767\) 5.97571i 0.215770i
\(768\) 0 0
\(769\) 32.2666 1.16356 0.581782 0.813345i \(-0.302356\pi\)
0.581782 + 0.813345i \(0.302356\pi\)
\(770\) 0 0
\(771\) 87.3164 3.14462
\(772\) 0 0
\(773\) 3.75316i 0.134992i −0.997720 0.0674958i \(-0.978499\pi\)
0.997720 0.0674958i \(-0.0215009\pi\)
\(774\) 0 0
\(775\) −14.4574 5.23234i −0.519325 0.187951i
\(776\) 0 0
\(777\) 17.1982i 0.616983i
\(778\) 0 0
\(779\) −47.3546 −1.69666
\(780\) 0 0
\(781\) −25.2637 −0.904005
\(782\) 0 0
\(783\) 145.976i 5.21675i
\(784\) 0 0
\(785\) 18.1529 + 3.18384i 0.647903 + 0.113636i
\(786\) 0 0
\(787\) 21.9373i 0.781981i 0.920395 + 0.390991i \(0.127867\pi\)
−0.920395 + 0.390991i \(0.872133\pi\)
\(788\) 0 0
\(789\) 51.5771 1.83619
\(790\) 0 0
\(791\) −0.837631 −0.0297827
\(792\) 0 0
\(793\) 0.969347i 0.0344225i
\(794\) 0 0
\(795\) 14.0080 79.8676i 0.496813 2.83261i
\(796\) 0 0
\(797\) 7.57883i 0.268456i −0.990950 0.134228i \(-0.957145\pi\)
0.990950 0.134228i \(-0.0428554\pi\)
\(798\) 0 0
\(799\) −2.58532 −0.0914619
\(800\) 0 0
\(801\) 40.0294 1.41437
\(802\)