Properties

Label 1840.2.e.e.369.2
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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 369.2
Root \(1.83051 - 1.83051i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.e.369.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.40815i q^{3} +(1.83051 + 1.28422i) q^{5} -0.706585i q^{7} -2.79917 q^{9} +O(q^{10})\) \(q-2.40815i q^{3} +(1.83051 + 1.28422i) q^{5} -0.706585i q^{7} -2.79917 q^{9} -0.747120 q^{11} -1.29341i q^{13} +(3.09259 - 4.40815i) q^{15} -5.50074i q^{17} +2.44868 q^{19} -1.70156 q^{21} +1.00000i q^{23} +(1.70156 + 4.70156i) q^{25} -0.483617i q^{27} -5.72371 q^{29} -7.52288 q^{31} +1.79917i q^{33} +(0.907411 - 1.29341i) q^{35} -5.07420i q^{37} -3.11473 q^{39} +10.0876 q^{41} -5.34045i q^{43} +(-5.12393 - 3.59475i) q^{45} -7.90888i q^{47} +6.50074 q^{49} -13.2466 q^{51} -5.84621i q^{53} +(-1.36761 - 0.959466i) q^{55} -5.89679i q^{57} +12.4146 q^{59} +1.57346 q^{61} +1.97786i q^{63} +(1.66103 - 2.36761i) q^{65} -5.25938i q^{67} +2.40815 q^{69} -2.68444 q^{71} -10.4589i q^{73} +(11.3221 - 4.09761i) q^{75} +0.527904i q^{77} -6.55005 q^{79} -9.56214 q^{81} +12.7068i q^{83} +(7.06415 - 10.0692i) q^{85} +13.7835i q^{87} -13.6426 q^{89} -0.913908 q^{91} +18.1162i q^{93} +(4.48235 + 3.14464i) q^{95} +1.50074i q^{97} +2.09132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 14q^{9} + O(q^{10}) \) \( 8q - 14q^{9} - 10q^{11} + 16q^{15} - 2q^{19} + 12q^{21} - 12q^{25} - 4q^{29} - 10q^{31} + 16q^{35} + 46q^{41} - 26q^{45} + 18q^{49} - 14q^{51} + 18q^{55} + 32q^{59} + 18q^{61} - 16q^{65} - 6q^{69} - 38q^{71} + 32q^{75} - 12q^{79} + 32q^{81} - 24q^{85} - 60q^{89} + 26q^{91} - 18q^{95} - 54q^{99} + O(q^{100}) \)

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
\(3\) 2.40815i 1.39034i −0.718843 0.695172i \(-0.755328\pi\)
0.718843 0.695172i \(-0.244672\pi\)
\(4\) 0 0
\(5\) 1.83051 + 1.28422i 0.818631 + 0.574320i
\(6\) 0 0
\(7\) 0.706585i 0.267064i −0.991044 0.133532i \(-0.957368\pi\)
0.991044 0.133532i \(-0.0426319\pi\)
\(8\) 0 0
\(9\) −2.79917 −0.933058
\(10\) 0 0
\(11\) −0.747120 −0.225265 −0.112633 0.993637i \(-0.535928\pi\)
−0.112633 + 0.993637i \(0.535928\pi\)
\(12\) 0 0
\(13\) 1.29341i 0.358729i −0.983783 0.179364i \(-0.942596\pi\)
0.983783 0.179364i \(-0.0574041\pi\)
\(14\) 0 0
\(15\) 3.09259 4.40815i 0.798503 1.13818i
\(16\) 0 0
\(17\) 5.50074i 1.33412i −0.745002 0.667062i \(-0.767551\pi\)
0.745002 0.667062i \(-0.232449\pi\)
\(18\) 0 0
\(19\) 2.44868 0.561766 0.280883 0.959742i \(-0.409373\pi\)
0.280883 + 0.959742i \(0.409373\pi\)
\(20\) 0 0
\(21\) −1.70156 −0.371311
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 1.70156 + 4.70156i 0.340312 + 0.940312i
\(26\) 0 0
\(27\) 0.483617i 0.0930721i
\(28\) 0 0
\(29\) −5.72371 −1.06287 −0.531433 0.847100i \(-0.678346\pi\)
−0.531433 + 0.847100i \(0.678346\pi\)
\(30\) 0 0
\(31\) −7.52288 −1.35115 −0.675575 0.737292i \(-0.736104\pi\)
−0.675575 + 0.737292i \(0.736104\pi\)
\(32\) 0 0
\(33\) 1.79917i 0.313196i
\(34\) 0 0
\(35\) 0.907411 1.29341i 0.153380 0.218627i
\(36\) 0 0
\(37\) 5.07420i 0.834193i −0.908862 0.417097i \(-0.863048\pi\)
0.908862 0.417097i \(-0.136952\pi\)
\(38\) 0 0
\(39\) −3.11473 −0.498756
\(40\) 0 0
\(41\) 10.0876 1.57541 0.787707 0.616051i \(-0.211268\pi\)
0.787707 + 0.616051i \(0.211268\pi\)
\(42\) 0 0
\(43\) 5.34045i 0.814410i −0.913337 0.407205i \(-0.866503\pi\)
0.913337 0.407205i \(-0.133497\pi\)
\(44\) 0 0
\(45\) −5.12393 3.59475i −0.763830 0.535874i
\(46\) 0 0
\(47\) 7.90888i 1.15363i −0.816875 0.576815i \(-0.804295\pi\)
0.816875 0.576815i \(-0.195705\pi\)
\(48\) 0 0
\(49\) 6.50074 0.928677
\(50\) 0 0
\(51\) −13.2466 −1.85489
\(52\) 0 0
\(53\) 5.84621i 0.803038i −0.915851 0.401519i \(-0.868482\pi\)
0.915851 0.401519i \(-0.131518\pi\)
\(54\) 0 0
\(55\) −1.36761 0.959466i −0.184409 0.129374i
\(56\) 0 0
\(57\) 5.89679i 0.781049i
\(58\) 0 0
\(59\) 12.4146 1.61625 0.808125 0.589012i \(-0.200483\pi\)
0.808125 + 0.589012i \(0.200483\pi\)
\(60\) 0 0
\(61\) 1.57346 0.201461 0.100731 0.994914i \(-0.467882\pi\)
0.100731 + 0.994914i \(0.467882\pi\)
\(62\) 0 0
\(63\) 1.97786i 0.249186i
\(64\) 0 0
\(65\) 1.66103 2.36761i 0.206025 0.293666i
\(66\) 0 0
\(67\) 5.25938i 0.642535i −0.946988 0.321268i \(-0.895891\pi\)
0.946988 0.321268i \(-0.104109\pi\)
\(68\) 0 0
\(69\) 2.40815 0.289907
\(70\) 0 0
\(71\) −2.68444 −0.318585 −0.159292 0.987231i \(-0.550921\pi\)
−0.159292 + 0.987231i \(0.550921\pi\)
\(72\) 0 0
\(73\) 10.4589i 1.22413i −0.790809 0.612063i \(-0.790340\pi\)
0.790809 0.612063i \(-0.209660\pi\)
\(74\) 0 0
\(75\) 11.3221 4.09761i 1.30736 0.473152i
\(76\) 0 0
\(77\) 0.527904i 0.0601602i
\(78\) 0 0
\(79\) −6.55005 −0.736938 −0.368469 0.929640i \(-0.620118\pi\)
−0.368469 + 0.929640i \(0.620118\pi\)
\(80\) 0 0
\(81\) −9.56214 −1.06246
\(82\) 0 0
\(83\) 12.7068i 1.39475i 0.716706 + 0.697376i \(0.245649\pi\)
−0.716706 + 0.697376i \(0.754351\pi\)
\(84\) 0 0
\(85\) 7.06415 10.0692i 0.766215 1.09216i
\(86\) 0 0
\(87\) 13.7835i 1.47775i
\(88\) 0 0
\(89\) −13.6426 −1.44612 −0.723058 0.690787i \(-0.757264\pi\)
−0.723058 + 0.690787i \(0.757264\pi\)
\(90\) 0 0
\(91\) −0.913908 −0.0958036
\(92\) 0 0
\(93\) 18.1162i 1.87856i
\(94\) 0 0
\(95\) 4.48235 + 3.14464i 0.459879 + 0.322634i
\(96\) 0 0
\(97\) 1.50074i 0.152377i 0.997093 + 0.0761884i \(0.0242750\pi\)
−0.997093 + 0.0761884i \(0.975725\pi\)
\(98\) 0 0
\(99\) 2.09132 0.210185
\(100\) 0 0
\(101\) −16.0472 −1.59676 −0.798380 0.602154i \(-0.794309\pi\)
−0.798380 + 0.602154i \(0.794309\pi\)
\(102\) 0 0
\(103\) 9.49069i 0.935145i 0.883955 + 0.467573i \(0.154871\pi\)
−0.883955 + 0.467573i \(0.845129\pi\)
\(104\) 0 0
\(105\) −3.11473 2.18518i −0.303967 0.213252i
\(106\) 0 0
\(107\) 1.38473i 0.133867i −0.997757 0.0669336i \(-0.978678\pi\)
0.997757 0.0669336i \(-0.0213216\pi\)
\(108\) 0 0
\(109\) 14.0370 1.34450 0.672250 0.740325i \(-0.265328\pi\)
0.672250 + 0.740325i \(0.265328\pi\)
\(110\) 0 0
\(111\) −12.2194 −1.15982
\(112\) 0 0
\(113\) 16.3105i 1.53437i 0.641428 + 0.767183i \(0.278342\pi\)
−0.641428 + 0.767183i \(0.721658\pi\)
\(114\) 0 0
\(115\) −1.28422 + 1.83051i −0.119754 + 0.170696i
\(116\) 0 0
\(117\) 3.62049i 0.334715i
\(118\) 0 0
\(119\) −3.88674 −0.356297
\(120\) 0 0
\(121\) −10.4418 −0.949256
\(122\) 0 0
\(123\) 24.2923i 2.19037i
\(124\) 0 0
\(125\) −2.92310 + 10.7915i −0.261450 + 0.965217i
\(126\) 0 0
\(127\) 2.73523i 0.242712i 0.992609 + 0.121356i \(0.0387243\pi\)
−0.992609 + 0.121356i \(0.961276\pi\)
\(128\) 0 0
\(129\) −12.8606 −1.13231
\(130\) 0 0
\(131\) 8.21942 0.718134 0.359067 0.933312i \(-0.383095\pi\)
0.359067 + 0.933312i \(0.383095\pi\)
\(132\) 0 0
\(133\) 1.73020i 0.150028i
\(134\) 0 0
\(135\) 0.621070 0.885267i 0.0534532 0.0761916i
\(136\) 0 0
\(137\) 22.8449i 1.95177i −0.218275 0.975887i \(-0.570043\pi\)
0.218275 0.975887i \(-0.429957\pi\)
\(138\) 0 0
\(139\) 3.76049 0.318960 0.159480 0.987201i \(-0.449018\pi\)
0.159480 + 0.987201i \(0.449018\pi\)
\(140\) 0 0
\(141\) −19.0458 −1.60394
\(142\) 0 0
\(143\) 0.966336i 0.0808090i
\(144\) 0 0
\(145\) −10.4773 7.35049i −0.870094 0.610425i
\(146\) 0 0
\(147\) 15.6547i 1.29118i
\(148\) 0 0
\(149\) 1.90614 0.156157 0.0780785 0.996947i \(-0.475122\pi\)
0.0780785 + 0.996947i \(0.475122\pi\)
\(150\) 0 0
\(151\) 9.41967 0.766562 0.383281 0.923632i \(-0.374794\pi\)
0.383281 + 0.923632i \(0.374794\pi\)
\(152\) 0 0
\(153\) 15.3975i 1.24482i
\(154\) 0 0
\(155\) −13.7707 9.66103i −1.10609 0.775992i
\(156\) 0 0
\(157\) 4.43304i 0.353795i 0.984229 + 0.176897i \(0.0566061\pi\)
−0.984229 + 0.176897i \(0.943394\pi\)
\(158\) 0 0
\(159\) −14.0785 −1.11650
\(160\) 0 0
\(161\) 0.706585 0.0556867
\(162\) 0 0
\(163\) 9.92453i 0.777349i 0.921375 + 0.388675i \(0.127067\pi\)
−0.921375 + 0.388675i \(0.872933\pi\)
\(164\) 0 0
\(165\) −2.31053 + 3.29341i −0.179875 + 0.256392i
\(166\) 0 0
\(167\) 11.9089i 0.921537i 0.887520 + 0.460769i \(0.152426\pi\)
−0.887520 + 0.460769i \(0.847574\pi\)
\(168\) 0 0
\(169\) 11.3271 0.871314
\(170\) 0 0
\(171\) −6.85429 −0.524161
\(172\) 0 0
\(173\) 2.91391i 0.221540i −0.993846 0.110770i \(-0.964668\pi\)
0.993846 0.110770i \(-0.0353317\pi\)
\(174\) 0 0
\(175\) 3.32206 1.20230i 0.251124 0.0908853i
\(176\) 0 0
\(177\) 29.8963i 2.24714i
\(178\) 0 0
\(179\) 19.3663 1.44751 0.723754 0.690058i \(-0.242415\pi\)
0.723754 + 0.690058i \(0.242415\pi\)
\(180\) 0 0
\(181\) 17.5986 1.30809 0.654045 0.756456i \(-0.273071\pi\)
0.654045 + 0.756456i \(0.273071\pi\)
\(182\) 0 0
\(183\) 3.78913i 0.280100i
\(184\) 0 0
\(185\) 6.51638 9.28839i 0.479094 0.682896i
\(186\) 0 0
\(187\) 4.10971i 0.300532i
\(188\) 0 0
\(189\) −0.341716 −0.0248562
\(190\) 0 0
\(191\) −20.3578 −1.47304 −0.736518 0.676418i \(-0.763531\pi\)
−0.736518 + 0.676418i \(0.763531\pi\)
\(192\) 0 0
\(193\) 1.41317i 0.101722i 0.998706 + 0.0508611i \(0.0161966\pi\)
−0.998706 + 0.0508611i \(0.983803\pi\)
\(194\) 0 0
\(195\) −5.70156 4.00000i −0.408297 0.286446i
\(196\) 0 0
\(197\) 4.92395i 0.350817i 0.984496 + 0.175409i \(0.0561247\pi\)
−0.984496 + 0.175409i \(0.943875\pi\)
\(198\) 0 0
\(199\) −10.3105 −0.730894 −0.365447 0.930832i \(-0.619084\pi\)
−0.365447 + 0.930832i \(0.619084\pi\)
\(200\) 0 0
\(201\) −12.6654 −0.893345
\(202\) 0 0
\(203\) 4.04429i 0.283853i
\(204\) 0 0
\(205\) 18.4654 + 12.9546i 1.28968 + 0.904792i
\(206\) 0 0
\(207\) 2.79917i 0.194556i
\(208\) 0 0
\(209\) −1.82946 −0.126546
\(210\) 0 0
\(211\) −11.4161 −0.785918 −0.392959 0.919556i \(-0.628549\pi\)
−0.392959 + 0.919556i \(0.628549\pi\)
\(212\) 0 0
\(213\) 6.46453i 0.442942i
\(214\) 0 0
\(215\) 6.85830 9.77576i 0.467732 0.666701i
\(216\) 0 0
\(217\) 5.31556i 0.360844i
\(218\) 0 0
\(219\) −25.1867 −1.70196
\(220\) 0 0
\(221\) −7.11473 −0.478589
\(222\) 0 0
\(223\) 18.2094i 1.21939i −0.792636 0.609695i \(-0.791292\pi\)
0.792636 0.609695i \(-0.208708\pi\)
\(224\) 0 0
\(225\) −4.76297 13.1605i −0.317531 0.877366i
\(226\) 0 0
\(227\) 18.9363i 1.25684i 0.777873 + 0.628422i \(0.216299\pi\)
−0.777873 + 0.628422i \(0.783701\pi\)
\(228\) 0 0
\(229\) 8.46433 0.559339 0.279669 0.960096i \(-0.409775\pi\)
0.279669 + 0.960096i \(0.409775\pi\)
\(230\) 0 0
\(231\) 1.27127 0.0836435
\(232\) 0 0
\(233\) 16.6341i 1.08973i −0.838523 0.544867i \(-0.816580\pi\)
0.838523 0.544867i \(-0.183420\pi\)
\(234\) 0 0
\(235\) 10.1567 14.4773i 0.662553 0.944396i
\(236\) 0 0
\(237\) 15.7735i 1.02460i
\(238\) 0 0
\(239\) −16.0840 −1.04039 −0.520194 0.854048i \(-0.674140\pi\)
−0.520194 + 0.854048i \(0.674140\pi\)
\(240\) 0 0
\(241\) −28.1283 −1.81190 −0.905952 0.423381i \(-0.860843\pi\)
−0.905952 + 0.423381i \(0.860843\pi\)
\(242\) 0 0
\(243\) 21.5762i 1.38411i
\(244\) 0 0
\(245\) 11.8997 + 8.34837i 0.760243 + 0.533358i
\(246\) 0 0
\(247\) 3.16716i 0.201522i
\(248\) 0 0
\(249\) 30.5998 1.93919
\(250\) 0 0
\(251\) 20.5770 1.29881 0.649404 0.760444i \(-0.275018\pi\)
0.649404 + 0.760444i \(0.275018\pi\)
\(252\) 0 0
\(253\) 0.747120i 0.0469710i
\(254\) 0 0
\(255\) −24.2481 17.0115i −1.51847 1.06530i
\(256\) 0 0
\(257\) 16.8505i 1.05111i 0.850760 + 0.525554i \(0.176142\pi\)
−0.850760 + 0.525554i \(0.823858\pi\)
\(258\) 0 0
\(259\) −3.58536 −0.222783
\(260\) 0 0
\(261\) 16.0217 0.991715
\(262\) 0 0
\(263\) 24.7630i 1.52695i 0.645837 + 0.763475i \(0.276508\pi\)
−0.645837 + 0.763475i \(0.723492\pi\)
\(264\) 0 0
\(265\) 7.50781 10.7016i 0.461201 0.657392i
\(266\) 0 0
\(267\) 32.8535i 2.01060i
\(268\) 0 0
\(269\) 3.55152 0.216540 0.108270 0.994122i \(-0.465469\pi\)
0.108270 + 0.994122i \(0.465469\pi\)
\(270\) 0 0
\(271\) 27.4761 1.66905 0.834526 0.550969i \(-0.185742\pi\)
0.834526 + 0.550969i \(0.185742\pi\)
\(272\) 0 0
\(273\) 2.20083i 0.133200i
\(274\) 0 0
\(275\) −1.27127 3.51263i −0.0766605 0.211820i
\(276\) 0 0
\(277\) 11.1001i 0.666940i 0.942761 + 0.333470i \(0.108220\pi\)
−0.942761 + 0.333470i \(0.891780\pi\)
\(278\) 0 0
\(279\) 21.0579 1.26070
\(280\) 0 0
\(281\) 13.0458 0.778245 0.389122 0.921186i \(-0.372778\pi\)
0.389122 + 0.921186i \(0.372778\pi\)
\(282\) 0 0
\(283\) 3.35049i 0.199166i −0.995029 0.0995831i \(-0.968249\pi\)
0.995029 0.0995831i \(-0.0317509\pi\)
\(284\) 0 0
\(285\) 7.57277 10.7942i 0.448572 0.639390i
\(286\) 0 0
\(287\) 7.12773i 0.420736i
\(288\) 0 0
\(289\) −13.2581 −0.779889
\(290\) 0 0
\(291\) 3.61400 0.211856
\(292\) 0 0
\(293\) 8.89197i 0.519474i 0.965679 + 0.259737i \(0.0836359\pi\)
−0.965679 + 0.259737i \(0.916364\pi\)
\(294\) 0 0
\(295\) 22.7252 + 15.9431i 1.32311 + 0.928245i
\(296\) 0 0
\(297\) 0.361320i 0.0209659i
\(298\) 0 0
\(299\) 1.29341 0.0748001
\(300\) 0 0
\(301\) −3.77348 −0.217500
\(302\) 0 0
\(303\) 38.6441i 2.22005i
\(304\) 0 0
\(305\) 2.88024 + 2.02067i 0.164922 + 0.115703i
\(306\) 0 0
\(307\) 28.4111i 1.62151i −0.585388 0.810753i \(-0.699058\pi\)
0.585388 0.810753i \(-0.300942\pi\)
\(308\) 0 0
\(309\) 22.8550 1.30017
\(310\) 0 0
\(311\) 28.0105 1.58833 0.794164 0.607704i \(-0.207909\pi\)
0.794164 + 0.607704i \(0.207909\pi\)
\(312\) 0 0
\(313\) 7.42882i 0.419902i 0.977712 + 0.209951i \(0.0673304\pi\)
−0.977712 + 0.209951i \(0.932670\pi\)
\(314\) 0 0
\(315\) −2.54000 + 3.62049i −0.143113 + 0.203992i
\(316\) 0 0
\(317\) 13.1480i 0.738463i 0.929337 + 0.369232i \(0.120379\pi\)
−0.929337 + 0.369232i \(0.879621\pi\)
\(318\) 0 0
\(319\) 4.27629 0.239427
\(320\) 0 0
\(321\) −3.33464 −0.186122
\(322\) 0 0
\(323\) 13.4696i 0.749466i
\(324\) 0 0
\(325\) 6.08107 2.20083i 0.337317 0.122080i
\(326\) 0 0
\(327\) 33.8031i 1.86932i
\(328\) 0 0
\(329\) −5.58830 −0.308093
\(330\) 0 0
\(331\) 15.6225 0.858693 0.429346 0.903140i \(-0.358744\pi\)
0.429346 + 0.903140i \(0.358744\pi\)
\(332\) 0 0
\(333\) 14.2036i 0.778351i
\(334\) 0 0
\(335\) 6.75419 9.62736i 0.369021 0.525999i
\(336\) 0 0
\(337\) 16.6965i 0.909518i −0.890614 0.454759i \(-0.849725\pi\)
0.890614 0.454759i \(-0.150275\pi\)
\(338\) 0 0
\(339\) 39.2782 2.13330
\(340\) 0 0
\(341\) 5.62049 0.304367
\(342\) 0 0
\(343\) 9.53942i 0.515081i
\(344\) 0 0
\(345\) 4.40815 + 3.09259i 0.237327 + 0.166499i
\(346\) 0 0
\(347\) 16.3981i 0.880296i −0.897925 0.440148i \(-0.854926\pi\)
0.897925 0.440148i \(-0.145074\pi\)
\(348\) 0 0
\(349\) 1.86165 0.0996518 0.0498259 0.998758i \(-0.484133\pi\)
0.0498259 + 0.998758i \(0.484133\pi\)
\(350\) 0 0
\(351\) −0.625517 −0.0333876
\(352\) 0 0
\(353\) 25.8408i 1.37537i 0.726010 + 0.687684i \(0.241372\pi\)
−0.726010 + 0.687684i \(0.758628\pi\)
\(354\) 0 0
\(355\) −4.91391 3.44741i −0.260803 0.182970i
\(356\) 0 0
\(357\) 9.35985i 0.495376i
\(358\) 0 0
\(359\) 8.72223 0.460342 0.230171 0.973150i \(-0.426071\pi\)
0.230171 + 0.973150i \(0.426071\pi\)
\(360\) 0 0
\(361\) −13.0040 −0.684419
\(362\) 0 0
\(363\) 25.1454i 1.31979i
\(364\) 0 0
\(365\) 13.4316 19.1452i 0.703040 1.00211i
\(366\) 0 0
\(367\) 25.5958i 1.33609i −0.744121 0.668045i \(-0.767131\pi\)
0.744121 0.668045i \(-0.232869\pi\)
\(368\) 0 0
\(369\) −28.2369 −1.46995
\(370\) 0 0
\(371\) −4.13084 −0.214463
\(372\) 0 0
\(373\) 17.2125i 0.891232i 0.895224 + 0.445616i \(0.147015\pi\)
−0.895224 + 0.445616i \(0.852985\pi\)
\(374\) 0 0
\(375\) 25.9874 + 7.03926i 1.34198 + 0.363506i
\(376\) 0 0
\(377\) 7.40312i 0.381280i
\(378\) 0 0
\(379\) 23.0339 1.18317 0.591585 0.806243i \(-0.298502\pi\)
0.591585 + 0.806243i \(0.298502\pi\)
\(380\) 0 0
\(381\) 6.58683 0.337453
\(382\) 0 0
\(383\) 3.81777i 0.195079i 0.995232 + 0.0975394i \(0.0310972\pi\)
−0.995232 + 0.0975394i \(0.968903\pi\)
\(384\) 0 0
\(385\) −0.677944 + 0.966336i −0.0345513 + 0.0492490i
\(386\) 0 0
\(387\) 14.9488i 0.759892i
\(388\) 0 0
\(389\) 8.25435 0.418512 0.209256 0.977861i \(-0.432896\pi\)
0.209256 + 0.977861i \(0.432896\pi\)
\(390\) 0 0
\(391\) 5.50074 0.278184
\(392\) 0 0
\(393\) 19.7936i 0.998454i
\(394\) 0 0
\(395\) −11.9900 8.41170i −0.603280 0.423238i
\(396\) 0 0
\(397\) 12.4967i 0.627193i 0.949556 + 0.313596i \(0.101534\pi\)
−0.949556 + 0.313596i \(0.898466\pi\)
\(398\) 0 0
\(399\) −4.16658 −0.208590
\(400\) 0 0
\(401\) 34.5660 1.72614 0.863072 0.505082i \(-0.168538\pi\)
0.863072 + 0.505082i \(0.168538\pi\)
\(402\) 0 0
\(403\) 9.73020i 0.484696i
\(404\) 0 0
\(405\) −17.5036 12.2799i −0.869763 0.610193i
\(406\) 0 0
\(407\) 3.79103i 0.187915i
\(408\) 0 0
\(409\) −12.0499 −0.595829 −0.297914 0.954593i \(-0.596291\pi\)
−0.297914 + 0.954593i \(0.596291\pi\)
\(410\) 0 0
\(411\) −55.0140 −2.71364
\(412\) 0 0
\(413\) 8.77201i 0.431642i
\(414\) 0 0
\(415\) −16.3183 + 23.2600i −0.801034 + 1.14179i
\(416\) 0 0
\(417\) 9.05581i 0.443465i
\(418\) 0 0
\(419\) 38.6157 1.88650 0.943250 0.332085i \(-0.107752\pi\)
0.943250 + 0.332085i \(0.107752\pi\)
\(420\) 0 0
\(421\) 32.7384 1.59557 0.797785 0.602941i \(-0.206005\pi\)
0.797785 + 0.602941i \(0.206005\pi\)
\(422\) 0 0
\(423\) 22.1384i 1.07640i
\(424\) 0 0
\(425\) 25.8621 9.35985i 1.25449 0.454019i
\(426\) 0 0
\(427\) 1.11179i 0.0538031i
\(428\) 0 0
\(429\) 2.32708 0.112352
\(430\) 0 0
\(431\) 35.7739 1.72317 0.861584 0.507615i \(-0.169473\pi\)
0.861584 + 0.507615i \(0.169473\pi\)
\(432\) 0 0
\(433\) 10.5682i 0.507877i 0.967220 + 0.253938i \(0.0817261\pi\)
−0.967220 + 0.253938i \(0.918274\pi\)
\(434\) 0 0
\(435\) −17.7011 + 25.2309i −0.848701 + 1.20973i
\(436\) 0 0
\(437\) 2.44868i 0.117136i
\(438\) 0 0
\(439\) −27.6642 −1.32034 −0.660170 0.751117i \(-0.729516\pi\)
−0.660170 + 0.751117i \(0.729516\pi\)
\(440\) 0 0
\(441\) −18.1967 −0.866509
\(442\) 0 0
\(443\) 36.8082i 1.74881i 0.485198 + 0.874404i \(0.338747\pi\)
−0.485198 + 0.874404i \(0.661253\pi\)
\(444\) 0 0
\(445\) −24.9730 17.5201i −1.18384 0.830534i
\(446\) 0 0
\(447\) 4.59027i 0.217112i
\(448\) 0 0
\(449\) −18.2711 −0.862267 −0.431133 0.902288i \(-0.641886\pi\)
−0.431133 + 0.902288i \(0.641886\pi\)
\(450\) 0 0
\(451\) −7.53662 −0.354886
\(452\) 0 0
\(453\) 22.6840i 1.06579i
\(454\) 0 0
\(455\) −1.67292 1.17366i −0.0784278 0.0550219i
\(456\) 0 0
\(457\) 27.6326i 1.29260i −0.763084 0.646299i \(-0.776316\pi\)
0.763084 0.646299i \(-0.223684\pi\)
\(458\) 0 0
\(459\) −2.66025 −0.124170
\(460\) 0 0
\(461\) 35.5515 1.65580 0.827900 0.560876i \(-0.189536\pi\)
0.827900 + 0.560876i \(0.189536\pi\)
\(462\) 0 0
\(463\) 23.9331i 1.11226i −0.831094 0.556132i \(-0.812285\pi\)
0.831094 0.556132i \(-0.187715\pi\)
\(464\) 0 0
\(465\) −23.2652 + 33.1620i −1.07890 + 1.53785i
\(466\) 0 0
\(467\) 19.9035i 0.921024i 0.887654 + 0.460512i \(0.152334\pi\)
−0.887654 + 0.460512i \(0.847666\pi\)
\(468\) 0 0
\(469\) −3.71620 −0.171598
\(470\) 0 0
\(471\) 10.6754 0.491897
\(472\) 0 0
\(473\) 3.98995i 0.183458i
\(474\) 0 0
\(475\) 4.16658 + 11.5126i 0.191176 + 0.528236i
\(476\) 0 0
\(477\) 16.3646i 0.749281i
\(478\) 0 0
\(479\) −3.86206 −0.176462 −0.0882309 0.996100i \(-0.528121\pi\)
−0.0882309 + 0.996100i \(0.528121\pi\)
\(480\) 0 0
\(481\) −6.56304 −0.299249
\(482\) 0 0
\(483\) 1.70156i 0.0774238i
\(484\) 0 0
\(485\) −1.92728 + 2.74712i −0.0875131 + 0.124740i
\(486\) 0 0
\(487\) 15.1499i 0.686506i 0.939243 + 0.343253i \(0.111529\pi\)
−0.939243 + 0.343253i \(0.888471\pi\)
\(488\) 0 0
\(489\) 23.8997 1.08078
\(490\) 0 0
\(491\) 6.86312 0.309728 0.154864 0.987936i \(-0.450506\pi\)
0.154864 + 0.987936i \(0.450506\pi\)
\(492\) 0 0
\(493\) 31.4846i 1.41800i
\(494\) 0 0
\(495\) 3.82819 + 2.68571i 0.172064 + 0.120714i
\(496\) 0 0
\(497\) 1.89679i 0.0850826i
\(498\) 0 0
\(499\) −18.6809 −0.836272 −0.418136 0.908385i \(-0.637316\pi\)
−0.418136 + 0.908385i \(0.637316\pi\)
\(500\) 0 0
\(501\) 28.6784 1.28125
\(502\) 0 0
\(503\) 23.1886i 1.03393i 0.856008 + 0.516963i \(0.172938\pi\)
−0.856008 + 0.516963i \(0.827062\pi\)
\(504\) 0 0
\(505\) −29.3747 20.6082i −1.30716 0.917051i
\(506\) 0 0
\(507\) 27.2773i 1.21143i
\(508\) 0 0
\(509\) −11.5385 −0.511436 −0.255718 0.966751i \(-0.582312\pi\)
−0.255718 + 0.966751i \(0.582312\pi\)
\(510\) 0 0
\(511\) −7.39013 −0.326920
\(512\) 0 0
\(513\) 1.18422i 0.0522847i
\(514\) 0 0
\(515\) −12.1881 + 17.3728i −0.537073 + 0.765539i
\(516\) 0 0
\(517\) 5.90888i 0.259872i
\(518\) 0 0
\(519\) −7.01712 −0.308017
\(520\) 0 0
\(521\) −16.2792 −0.713207 −0.356603 0.934256i \(-0.616065\pi\)
−0.356603 + 0.934256i \(0.616065\pi\)
\(522\) 0 0
\(523\) 30.8347i 1.34831i 0.738591 + 0.674153i \(0.235491\pi\)
−0.738591 + 0.674153i \(0.764509\pi\)
\(524\) 0 0
\(525\) −2.89531 8.00000i −0.126362 0.349149i
\(526\) 0 0
\(527\) 41.3814i 1.80260i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −34.7508 −1.50805
\(532\) 0 0
\(533\) 13.0474i 0.565146i
\(534\) 0 0
\(535\) 1.77830 2.53477i 0.0768827 0.109588i
\(536\) 0 0
\(537\) 46.6370i 2.01254i
\(538\) 0 0
\(539\) −4.85683 −0.209198
\(540\) 0 0
\(541\) −30.5199 −1.31215 −0.656077 0.754694i \(-0.727785\pi\)
−0.656077 + 0.754694i \(0.727785\pi\)
\(542\) 0 0
\(543\) 42.3799i 1.81870i
\(544\) 0 0
\(545\) 25.6949 + 18.0266i 1.10065 + 0.772173i
\(546\) 0 0
\(547\) 28.0416i 1.19897i −0.800385 0.599487i \(-0.795371\pi\)
0.800385 0.599487i \(-0.204629\pi\)
\(548\) 0 0
\(549\) −4.40439 −0.187975
\(550\) 0 0
\(551\) −14.0155 −0.597082
\(552\) 0 0
\(553\) 4.62817i 0.196810i
\(554\) 0 0
\(555\) −22.3678 15.6924i −0.949461 0.666106i
\(556\) 0 0
\(557\) 4.15674i 0.176127i −0.996115 0.0880634i \(-0.971932\pi\)
0.996115 0.0880634i \(-0.0280678\pi\)
\(558\) 0 0
\(559\) −6.90741 −0.292152
\(560\) 0 0
\(561\) 9.89679 0.417843
\(562\) 0 0
\(563\) 21.3480i 0.899709i −0.893102 0.449854i \(-0.851476\pi\)
0.893102 0.449854i \(-0.148524\pi\)
\(564\) 0 0
\(565\) −20.9463 + 29.8567i −0.881218 + 1.25608i
\(566\) 0 0
\(567\) 6.75647i 0.283745i
\(568\) 0 0
\(569\) 25.6430 1.07501 0.537506 0.843260i \(-0.319366\pi\)
0.537506 + 0.843260i \(0.319366\pi\)
\(570\) 0 0
\(571\) 7.10743 0.297437 0.148718 0.988880i \(-0.452485\pi\)
0.148718 + 0.988880i \(0.452485\pi\)
\(572\) 0 0
\(573\) 49.0245i 2.04803i
\(574\) 0 0
\(575\) −4.70156 + 1.70156i −0.196069 + 0.0709600i
\(576\) 0 0
\(577\) 22.7226i 0.945956i −0.881074 0.472978i \(-0.843179\pi\)
0.881074 0.472978i \(-0.156821\pi\)
\(578\) 0 0
\(579\) 3.40312 0.141429
\(580\) 0 0
\(581\) 8.97843 0.372488
\(582\) 0 0
\(583\) 4.36782i 0.180896i
\(584\) 0 0
\(585\) −4.64951 + 6.62736i −0.192233 + 0.274008i
\(586\) 0 0
\(587\) 7.06600i 0.291645i 0.989311 + 0.145822i \(0.0465828\pi\)
−0.989311 + 0.145822i \(0.953417\pi\)
\(588\) 0 0
\(589\) −18.4211 −0.759030
\(590\) 0 0
\(591\) 11.8576 0.487757
\(592\) 0 0
\(593\) 14.9914i 0.615624i 0.951447 + 0.307812i \(0.0995968\pi\)
−0.951447 + 0.307812i \(0.900403\pi\)
\(594\) 0 0
\(595\) −7.11473 4.99143i −0.291676 0.204629i
\(596\) 0 0
\(597\) 24.8293i 1.01620i
\(598\) 0 0
\(599\) −4.03069 −0.164690 −0.0823448 0.996604i \(-0.526241\pi\)
−0.0823448 + 0.996604i \(0.526241\pi\)
\(600\) 0 0
\(601\) 13.9320 0.568300 0.284150 0.958780i \(-0.408289\pi\)
0.284150 + 0.958780i \(0.408289\pi\)
\(602\) 0 0
\(603\) 14.7219i 0.599523i
\(604\) 0 0
\(605\) −19.1139 13.4096i −0.777090 0.545177i
\(606\) 0 0
\(607\) 31.0588i 1.26064i −0.776337 0.630318i \(-0.782925\pi\)
0.776337 0.630318i \(-0.217075\pi\)
\(608\) 0 0
\(609\) 9.73924 0.394654
\(610\) 0 0
\(611\) −10.2295 −0.413840
\(612\) 0 0
\(613\) 19.6722i 0.794554i 0.917699 + 0.397277i \(0.130045\pi\)
−0.917699 + 0.397277i \(0.869955\pi\)
\(614\) 0 0
\(615\) 31.1967 44.4675i 1.25797 1.79310i
\(616\) 0 0
\(617\) 40.9043i 1.64674i 0.567502 + 0.823372i \(0.307910\pi\)
−0.567502 + 0.823372i \(0.692090\pi\)
\(618\) 0 0
\(619\) −3.58556 −0.144116 −0.0720579 0.997400i \(-0.522957\pi\)
−0.0720579 + 0.997400i \(0.522957\pi\)
\(620\) 0 0
\(621\) 0.483617 0.0194069
\(622\) 0 0
\(623\) 9.63969i 0.386206i
\(624\) 0 0
\(625\) −19.2094 + 16.0000i −0.768375 + 0.640000i
\(626\) 0 0
\(627\) 4.40561i 0.175943i
\(628\) 0 0
\(629\) −27.9118 −1.11292
\(630\) 0 0
\(631\) 7.19670 0.286496 0.143248 0.989687i \(-0.454245\pi\)
0.143248 + 0.989687i \(0.454245\pi\)
\(632\) 0 0
\(633\) 27.4917i 1.09270i
\(634\) 0 0
\(635\) −3.51263 + 5.00687i −0.139394 + 0.198692i
\(636\) 0 0
\(637\) 8.40815i 0.333143i
\(638\) 0 0
\(639\) 7.51422 0.297258
\(640\) 0 0
\(641\) −16.3436 −0.645534 −0.322767 0.946478i \(-0.604613\pi\)
−0.322767 + 0.946478i \(0.604613\pi\)
\(642\) 0 0
\(643\) 6.55839i 0.258638i 0.991603 + 0.129319i \(0.0412791\pi\)
−0.991603 + 0.129319i \(0.958721\pi\)
\(644\) 0 0
\(645\) −23.5415 16.5158i −0.926945 0.650309i
\(646\) 0 0
\(647\) 16.5455i 0.650470i 0.945633 + 0.325235i \(0.105443\pi\)
−0.945633 + 0.325235i \(0.894557\pi\)
\(648\) 0 0
\(649\) −9.27523 −0.364085
\(650\) 0 0
\(651\) 12.8006 0.501697
\(652\) 0 0
\(653\) 12.7408i 0.498587i −0.968428 0.249294i \(-0.919802\pi\)
0.968428 0.249294i \(-0.0801984\pi\)
\(654\) 0 0
\(655\) 15.0458 + 10.5555i 0.587887 + 0.412439i
\(656\) 0 0
\(657\) 29.2764i 1.14218i
\(658\) 0 0
\(659\) −23.8094 −0.927484 −0.463742 0.885970i \(-0.653493\pi\)
−0.463742 + 0.885970i \(0.653493\pi\)
\(660\) 0 0
\(661\) 24.9323 0.969754 0.484877 0.874582i \(-0.338864\pi\)
0.484877 + 0.874582i \(0.338864\pi\)
\(662\) 0 0
\(663\) 17.1333i 0.665403i
\(664\) 0 0
\(665\) 2.22196 3.16716i 0.0861639 0.122817i
\(666\) 0 0
\(667\) 5.72371i 0.221623i
\(668\) 0 0
\(669\) −43.8509 −1.69537
\(670\) 0 0
\(671\) −1.17556 −0.0453822
\(672\) 0 0
\(673\) 5.48050i 0.211258i 0.994406 + 0.105629i \(0.0336856\pi\)
−0.994406 + 0.105629i \(0.966314\pi\)
\(674\) 0 0
\(675\) 2.27375 0.822904i 0.0875168 0.0316736i
\(676\) 0 0
\(677\) 1.75255i 0.0673560i −0.999433 0.0336780i \(-0.989278\pi\)
0.999433 0.0336780i \(-0.0107221\pi\)
\(678\) 0 0
\(679\) 1.06040 0.0406944
\(680\) 0 0
\(681\) 45.6013 1.74745
\(682\) 0 0
\(683\) 23.5091i 0.899552i −0.893141 0.449776i \(-0.851504\pi\)
0.893141 0.449776i \(-0.148496\pi\)
\(684\) 0 0
\(685\) 29.3379 41.8180i 1.12094 1.59778i
\(686\) 0 0
\(687\) 20.3834i 0.777673i
\(688\) 0 0
\(689\) −7.56157 −0.288073
\(690\) 0 0
\(691\) 21.4834 0.817269 0.408634 0.912698i \(-0.366005\pi\)
0.408634 + 0.912698i \(0.366005\pi\)
\(692\) 0 0
\(693\) 1.47770i 0.0561330i
\(694\) 0 0
\(695\) 6.88362 + 4.82929i 0.261111 + 0.183185i
\(696\) 0 0
\(697\) 55.4890i 2.10180i
\(698\) 0 0
\(699\) −40.0573 −1.51511
\(700\) 0 0
\(701\) 9.62985 0.363714 0.181857 0.983325i \(-0.441789\pi\)
0.181857 + 0.983325i \(0.441789\pi\)
\(702\) 0 0
\(703\) 12.4251i 0.468621i
\(704\) 0 0
\(705\) −34.8635 24.4589i −1.31304 0.921177i
\(706\) 0 0
\(707\) 11.3387i 0.426437i
\(708\) 0 0
\(709\) −11.8970 −0.446801 −0.223400 0.974727i \(-0.571716\pi\)
−0.223400 + 0.974727i \(0.571716\pi\)
\(710\) 0 0
\(711\) 18.3347 0.687606
\(712\) 0 0
\(713\) 7.52288i 0.281734i
\(714\) 0 0
\(715\) −1.24099 + 1.76889i −0.0464103 + 0.0661528i
\(716\) 0 0
\(717\) 38.7327i 1.44650i
\(718\) 0 0
\(719\) 24.2949 0.906046 0.453023 0.891499i \(-0.350345\pi\)
0.453023 + 0.891499i \(0.350345\pi\)
\(720\) 0 0
\(721\) 6.70598 0.249744
\(722\) 0 0
\(723\) 67.7371i 2.51917i
\(724\) 0 0
\(725\) −9.73924 26.9104i −0.361706 0.999426i
\(726\) 0 0
\(727\) 6.25721i 0.232067i −0.993245 0.116034i \(-0.962982\pi\)
0.993245 0.116034i \(-0.0370180\pi\)
\(728\) 0 0
\(729\) 23.2723 0.861935
\(730\) 0 0
\(731\) −29.3764 −1.08653
\(732\) 0 0
\(733\) 37.3304i 1.37883i 0.724367 + 0.689415i \(0.242132\pi\)
−0.724367 + 0.689415i \(0.757868\pi\)
\(734\) 0 0
\(735\) 20.1041 28.6562i 0.741551 1.05700i
\(736\) 0 0
\(737\) 3.92939i 0.144741i
\(738\) 0 0
\(739\) −26.9889 −0.992802 −0.496401 0.868093i \(-0.665345\pi\)
−0.496401 + 0.868093i \(0.665345\pi\)
\(740\) 0 0
\(741\) −7.62699 −0.280185
\(742\) 0 0
\(743\) 32.1544i 1.17963i −0.807538 0.589815i \(-0.799201\pi\)
0.807538 0.589815i \(-0.200799\pi\)
\(744\) 0 0
\(745\) 3.48922 + 2.44790i 0.127835 + 0.0896842i
\(746\) 0 0
\(747\) 35.5685i 1.30138i
\(748\) 0 0
\(749\) −0.978433 −0.0357512
\(750\) 0 0
\(751\) −39.7865 −1.45183 −0.725915 0.687785i \(-0.758583\pi\)
−0.725915 + 0.687785i \(0.758583\pi\)
\(752\) 0 0
\(753\) 49.5524i 1.80579i
\(754\) 0 0
\(755\) 17.2428 + 12.0969i 0.627531 + 0.440252i
\(756\) 0 0
\(757\) 26.9961i 0.981189i −0.871388 0.490595i \(-0.836780\pi\)
0.871388 0.490595i \(-0.163220\pi\)
\(758\) 0 0
\(759\) −1.79917 −0.0653059
\(760\) 0 0
\(761\) −7.21529 −0.261554 −0.130777 0.991412i \(-0.541747\pi\)
−0.130777 + 0.991412i \(0.541747\pi\)
\(762\) 0 0
\(763\) 9.91833i 0.359068i
\(764\) 0 0
\(765\) −19.7738 + 28.1854i −0.714923 + 1.01904i
\(766\) 0 0
\(767\) 16.0573i 0.579795i
\(768\) 0 0
\(769\) −6.40916 −0.231120 −0.115560 0.993300i \(-0.536866\pi\)
−0.115560 + 0.993300i \(0.536866\pi\)
\(770\) 0 0
\(771\) 40.5786 1.46140
\(772\) 0 0
\(773\) 50.2483i 1.80730i 0.428267 + 0.903652i \(0.359124\pi\)
−0.428267 + 0.903652i \(0.640876\pi\)
\(774\) 0 0
\(775\) −12.8006 35.3693i −0.459813 1.27050i
\(776\) 0 0
\(777\) 8.63406i 0.309745i
\(778\) 0 0
\(779\) 24.7012 0.885014
\(780\) 0 0
\(781\) 2.00560 0.0717660
\(782\) 0 0
\(783\) 2.76808i 0.0989231i
\(784\) 0 0
\(785\) −5.69299 + 8.11473i −0.203192 + 0.289627i
\(786\) 0 0
\(787\) 2.70933i 0.0965772i 0.998833 + 0.0482886i \(0.0153767\pi\)
−0.998833 + 0.0482886i \(0.984623\pi\)
\(788\) 0 0
\(789\) 59.6329 2.12299
\(790\) 0 0
\(791\) 11.5248 0.409774
\(792\) 0 0
\(793\) 2.03514i 0.0722699i
\(794\) 0 0
\(795\) −25.7709 18.0799i −0.914001 0.641229i
\(796\) 0 0
\(797\) 15.1452i 0.536471i 0.963353 + 0.268236i \(0.0864406\pi\)
−0.963353 + 0.268236i \(0.913559\pi\)
\(798\) 0 0
\(799\) −43.5047 −1.53909
\(800\) 0 0
\(801\) 38.1881 1.34931
\(802\) 0 0