Properties

Label 1008.2.cs.p.271.2
Level 1008
Weight 2
Character 1008.271
Analytic conductor 8.049
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.2
Root \(-0.309017 - 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 1008.271
Dual form 1008.2.cs.p.703.2

$q$-expansion

\(f(q)\) \(=\) \(q+(3.35410 + 1.93649i) q^{5} +(2.50000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(3.35410 + 1.93649i) q^{5} +(2.50000 + 0.866025i) q^{7} +(3.35410 - 1.93649i) q^{11} +3.46410i q^{13} +(2.00000 - 3.46410i) q^{19} +(-6.70820 - 3.87298i) q^{23} +(5.00000 + 8.66025i) q^{25} -6.70820 q^{29} +(-0.500000 - 0.866025i) q^{31} +(6.70820 + 7.74597i) q^{35} +(-2.00000 + 3.46410i) q^{37} -7.74597i q^{41} -6.92820i q^{43} +(-6.70820 + 11.6190i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-3.35410 - 5.80948i) q^{53} +15.0000 q^{55} +(-3.35410 - 5.80948i) q^{59} +(9.00000 + 5.19615i) q^{61} +(-6.70820 + 11.6190i) q^{65} +(-6.00000 + 3.46410i) q^{67} -7.74597i q^{71} +(-6.00000 + 3.46410i) q^{73} +(10.0623 - 1.93649i) q^{77} +(10.5000 + 6.06218i) q^{79} +6.70820 q^{83} +(6.70820 + 3.87298i) q^{89} +(-3.00000 + 8.66025i) q^{91} +(13.4164 - 7.74597i) q^{95} -5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{7} + O(q^{10}) \) \( 4q + 10q^{7} + 8q^{19} + 20q^{25} - 2q^{31} - 8q^{37} + 22q^{49} + 60q^{55} + 36q^{61} - 24q^{67} - 24q^{73} + 42q^{79} - 12q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.35410 + 1.93649i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 2.50000 + 0.866025i 0.944911 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.35410 1.93649i 1.01130 0.583874i 0.0997278 0.995015i \(-0.468203\pi\)
0.911572 + 0.411141i \(0.134869\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 2.00000 3.46410i 0.458831 0.794719i −0.540068 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.70820 3.87298i −1.39876 0.807573i −0.404495 0.914540i \(-0.632553\pi\)
−0.994263 + 0.106967i \(0.965886\pi\)
\(24\) 0 0
\(25\) 5.00000 + 8.66025i 1.00000 + 1.73205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.70820 −1.24568 −0.622841 0.782348i \(-0.714022\pi\)
−0.622841 + 0.782348i \(0.714022\pi\)
\(30\) 0 0
\(31\) −0.500000 0.866025i −0.0898027 0.155543i 0.817625 0.575751i \(-0.195290\pi\)
−0.907428 + 0.420208i \(0.861957\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.70820 + 7.74597i 1.13389 + 1.30931i
\(36\) 0 0
\(37\) −2.00000 + 3.46410i −0.328798 + 0.569495i −0.982274 0.187453i \(-0.939977\pi\)
0.653476 + 0.756948i \(0.273310\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.74597i 1.20972i −0.796333 0.604858i \(-0.793230\pi\)
0.796333 0.604858i \(-0.206770\pi\)
\(42\) 0 0
\(43\) 6.92820i 1.05654i −0.849076 0.528271i \(-0.822841\pi\)
0.849076 0.528271i \(-0.177159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.70820 + 11.6190i −0.978492 + 1.69480i −0.310599 + 0.950541i \(0.600530\pi\)
−0.667893 + 0.744257i \(0.732804\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.35410 5.80948i −0.460721 0.797993i 0.538276 0.842769i \(-0.319076\pi\)
−0.998997 + 0.0447760i \(0.985743\pi\)
\(54\) 0 0
\(55\) 15.0000 2.02260
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.35410 5.80948i −0.436667 0.756329i 0.560763 0.827976i \(-0.310508\pi\)
−0.997430 + 0.0716470i \(0.977174\pi\)
\(60\) 0 0
\(61\) 9.00000 + 5.19615i 1.15233 + 0.665299i 0.949454 0.313905i \(-0.101637\pi\)
0.202878 + 0.979204i \(0.434971\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.70820 + 11.6190i −0.832050 + 1.44115i
\(66\) 0 0
\(67\) −6.00000 + 3.46410i −0.733017 + 0.423207i −0.819525 0.573044i \(-0.805762\pi\)
0.0865081 + 0.996251i \(0.472429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.74597i 0.919277i −0.888106 0.459639i \(-0.847979\pi\)
0.888106 0.459639i \(-0.152021\pi\)
\(72\) 0 0
\(73\) −6.00000 + 3.46410i −0.702247 + 0.405442i −0.808184 0.588930i \(-0.799549\pi\)
0.105937 + 0.994373i \(0.466216\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.0623 1.93649i 1.14671 0.220684i
\(78\) 0 0
\(79\) 10.5000 + 6.06218i 1.18134 + 0.682048i 0.956325 0.292306i \(-0.0944227\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.70820 0.736321 0.368161 0.929762i \(-0.379988\pi\)
0.368161 + 0.929762i \(0.379988\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.70820 + 3.87298i 0.711068 + 0.410535i 0.811456 0.584413i \(-0.198675\pi\)
−0.100388 + 0.994948i \(0.532008\pi\)
\(90\) 0 0
\(91\) −3.00000 + 8.66025i −0.314485 + 0.907841i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.4164 7.74597i 1.37649 0.794719i
\(96\) 0 0
\(97\) 5.19615i 0.527589i −0.964579 0.263795i \(-0.915026\pi\)
0.964579 0.263795i \(-0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 4.00000 6.92820i 0.394132 0.682656i −0.598858 0.800855i \(-0.704379\pi\)
0.992990 + 0.118199i \(0.0377120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.0623 5.80948i −0.972760 0.561623i −0.0726833 0.997355i \(-0.523156\pi\)
−0.900076 + 0.435732i \(0.856490\pi\)
\(108\) 0 0
\(109\) 2.00000 + 3.46410i 0.191565 + 0.331801i 0.945769 0.324840i \(-0.105310\pi\)
−0.754204 + 0.656640i \(0.771977\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4164 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(114\) 0 0
\(115\) −15.0000 25.9808i −1.39876 2.42272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.00000 3.46410i 0.181818 0.314918i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 19.3649i 1.73205i
\(126\) 0 0
\(127\) 12.1244i 1.07586i 0.842989 + 0.537931i \(0.180794\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.35410 + 5.80948i −0.293049 + 0.507576i −0.974529 0.224261i \(-0.928003\pi\)
0.681480 + 0.731837i \(0.261337\pi\)
\(132\) 0 0
\(133\) 8.00000 6.92820i 0.693688 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.70820 + 11.6190i 0.560968 + 0.971625i
\(144\) 0 0
\(145\) −22.5000 12.9904i −1.86852 1.07879i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.70820 11.6190i 0.549557 0.951861i −0.448747 0.893659i \(-0.648130\pi\)
0.998305 0.0582028i \(-0.0185370\pi\)
\(150\) 0 0
\(151\) −13.5000 + 7.79423i −1.09861 + 0.634285i −0.935857 0.352381i \(-0.885372\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.87298i 0.311086i
\(156\) 0 0
\(157\) −18.0000 + 10.3923i −1.43656 + 0.829396i −0.997609 0.0691164i \(-0.977982\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −13.4164 15.4919i −1.05736 1.22094i
\(162\) 0 0
\(163\) −9.00000 5.19615i −0.704934 0.406994i 0.104248 0.994551i \(-0.466756\pi\)
−0.809183 + 0.587557i \(0.800090\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.4164 7.74597i −1.02003 0.588915i −0.105918 0.994375i \(-0.533778\pi\)
−0.914113 + 0.405460i \(0.867111\pi\)
\(174\) 0 0
\(175\) 5.00000 + 25.9808i 0.377964 + 1.96396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.1246 + 11.6190i −1.50418 + 0.868441i −0.504196 + 0.863589i \(0.668211\pi\)
−0.999988 + 0.00485178i \(0.998456\pi\)
\(180\) 0 0
\(181\) 24.2487i 1.80239i −0.433411 0.901196i \(-0.642690\pi\)
0.433411 0.901196i \(-0.357310\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.4164 + 7.74597i −0.986394 + 0.569495i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.4164 + 7.74597i 0.970777 + 0.560478i 0.899473 0.436976i \(-0.143951\pi\)
0.0713041 + 0.997455i \(0.477284\pi\)
\(192\) 0 0
\(193\) −3.50000 6.06218i −0.251936 0.436365i 0.712123 0.702055i \(-0.247734\pi\)
−0.964059 + 0.265689i \(0.914400\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.4164 −0.955879 −0.477940 0.878393i \(-0.658616\pi\)
−0.477940 + 0.878393i \(0.658616\pi\)
\(198\) 0 0
\(199\) −4.00000 6.92820i −0.283552 0.491127i 0.688705 0.725042i \(-0.258180\pi\)
−0.972257 + 0.233915i \(0.924846\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −16.7705 5.80948i −1.17706 0.407745i
\(204\) 0 0
\(205\) 15.0000 25.9808i 1.04765 1.81458i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.4919i 1.07160i
\(210\) 0 0
\(211\) 10.3923i 0.715436i −0.933830 0.357718i \(-0.883555\pi\)
0.933830 0.357718i \(-0.116445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.4164 23.2379i 0.914991 1.58481i
\(216\) 0 0
\(217\) −0.500000 2.59808i −0.0339422 0.176369i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 23.0000 1.54019 0.770097 0.637927i \(-0.220208\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.35410 5.80948i −0.222620 0.385588i 0.732983 0.680247i \(-0.238127\pi\)
−0.955603 + 0.294658i \(0.904794\pi\)
\(228\) 0 0
\(229\) −15.0000 8.66025i −0.991228 0.572286i −0.0855868 0.996331i \(-0.527276\pi\)
−0.905641 + 0.424045i \(0.860610\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 0 0
\(235\) −45.0000 + 25.9808i −2.93548 + 1.69480i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.2379i 1.50313i 0.659656 + 0.751567i \(0.270702\pi\)
−0.659656 + 0.751567i \(0.729298\pi\)
\(240\) 0 0
\(241\) 13.5000 7.79423i 0.869611 0.502070i 0.00239235 0.999997i \(-0.499238\pi\)
0.867219 + 0.497927i \(0.165905\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10.0623 + 25.1744i 0.642857 + 1.60833i
\(246\) 0 0
\(247\) 12.0000 + 6.92820i 0.763542 + 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.70820 −0.423418 −0.211709 0.977333i \(-0.567903\pi\)
−0.211709 + 0.977333i \(0.567903\pi\)
\(252\) 0 0
\(253\) −30.0000 −1.88608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.70820 3.87298i −0.418446 0.241590i 0.275966 0.961167i \(-0.411002\pi\)
−0.694412 + 0.719577i \(0.744336\pi\)
\(258\) 0 0
\(259\) −8.00000 + 6.92820i −0.497096 + 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.4164 + 7.74597i −0.827291 + 0.477637i −0.852924 0.522035i \(-0.825173\pi\)
0.0256331 + 0.999671i \(0.491840\pi\)
\(264\) 0 0
\(265\) 25.9808i 1.59599i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.4787 13.5554i 1.43152 0.826490i 0.434285 0.900775i \(-0.357001\pi\)
0.997237 + 0.0742854i \(0.0236676\pi\)
\(270\) 0 0
\(271\) −3.50000 + 6.06218i −0.212610 + 0.368251i −0.952531 0.304443i \(-0.901530\pi\)
0.739921 + 0.672694i \(0.234863\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.5410 + 19.3649i 2.02260 + 1.16775i
\(276\) 0 0
\(277\) 10.0000 + 17.3205i 0.600842 + 1.04069i 0.992694 + 0.120660i \(0.0385012\pi\)
−0.391852 + 0.920028i \(0.628166\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.8328 1.60071 0.800356 0.599525i \(-0.204644\pi\)
0.800356 + 0.599525i \(0.204644\pi\)
\(282\) 0 0
\(283\) −10.0000 17.3205i −0.594438 1.02960i −0.993626 0.112728i \(-0.964041\pi\)
0.399188 0.916869i \(1.63071\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.70820 19.3649i 0.395973 1.14307i
\(288\) 0 0
\(289\) −8.50000 + 14.7224i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.6190i 0.678786i 0.940645 + 0.339393i \(0.110222\pi\)
−0.940645 + 0.339393i \(0.889778\pi\)
\(294\) 0 0
\(295\) 25.9808i 1.51266i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.4164 23.2379i 0.775891 1.34388i
\(300\) 0 0
\(301\) 6.00000 17.3205i 0.345834 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.1246 + 34.8569i 1.15233 + 1.99590i
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.70820 11.6190i −0.380387 0.658850i 0.610730 0.791839i \(-0.290876\pi\)
−0.991118 + 0.132989i \(0.957543\pi\)
\(312\) 0 0
\(313\) 1.50000 + 0.866025i 0.0847850 + 0.0489506i 0.541793 0.840512i \(-0.317746\pi\)
−0.457008 + 0.889463i \(0.651079\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0623 17.4284i 0.565155 0.978878i −0.431880 0.901931i \(-0.642150\pi\)
0.997035 0.0769467i \(-0.0245171\pi\)
\(318\) 0 0
\(319\) −22.5000 + 12.9904i −1.25976 + 0.727322i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −30.0000 + 17.3205i −1.66410 + 0.960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −26.8328 + 23.2379i −1.47934 + 1.28115i
\(330\) 0 0
\(331\) −6.00000 3.46410i −0.329790 0.190404i 0.325958 0.945384i \(-0.394313\pi\)
−0.655748 + 0.754980i \(0.727647\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −26.8328 −1.46603
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.35410 1.93649i −0.181635 0.104867i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.70820 3.87298i 0.360115 0.207913i −0.309016 0.951057i \(-0.600000\pi\)
0.669131 + 0.743144i \(0.266666\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.4164 7.74597i 0.714083 0.412276i −0.0984878 0.995138i \(-0.531401\pi\)
0.812571 + 0.582862i \(0.198067\pi\)
\(354\) 0 0
\(355\) 15.0000 25.9808i 0.796117 1.37892i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 1.50000 + 2.59808i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −26.8328 −1.40449
\(366\) 0 0
\(367\) −5.50000 9.52628i −0.287098 0.497268i 0.686018 0.727585i \(-0.259357\pi\)
−0.973116 + 0.230317i \(0.926024\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.35410 17.4284i −0.174136 0.904839i
\(372\) 0 0
\(373\) 5.00000 8.66025i 0.258890 0.448411i −0.707055 0.707159i \(-0.749977\pi\)
0.965945 + 0.258748i \(0.0833099\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.2379i 1.19681i
\(378\) 0 0
\(379\) 17.3205i 0.889695i 0.895606 + 0.444847i \(0.146742\pi\)
−0.895606 + 0.444847i \(0.853258\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.70820 11.6190i 0.342773 0.593701i −0.642173 0.766559i \(-0.721967\pi\)
0.984947 + 0.172859i \(0.0553004\pi\)
\(384\) 0 0
\(385\) 37.5000 + 12.9904i 1.91118 + 0.662051i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.70820 11.6190i −0.340119 0.589104i 0.644335 0.764743i \(-0.277134\pi\)
−0.984455 + 0.175639i \(0.943801\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 23.4787 + 40.6663i 1.18134 + 2.04614i
\(396\) 0 0
\(397\) 12.0000 + 6.92820i 0.602263 + 0.347717i 0.769931 0.638127i \(-0.220290\pi\)
−0.167668 + 0.985843i \(0.553624\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 3.00000 1.73205i 0.149441 0.0862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.4919i 0.767907i
\(408\) 0 0
\(409\) −7.50000 + 4.33013i −0.370851 + 0.214111i −0.673830 0.738886i \(-0.735352\pi\)
0.302979 + 0.952997i \(0.402019\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.35410 17.4284i −0.165045 0.857597i
\(414\) 0 0
\(415\) 22.5000 + 12.9904i 1.10448 + 0.637673i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.0000 + 20.7846i 0.871081 + 1.00584i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.5410 + 19.3649i −1.61561 + 0.932775i −0.627578 + 0.778554i \(0.715954\pi\)
−0.988036 + 0.154221i \(0.950713\pi\)
\(432\) 0 0
\(433\) 6.92820i 0.332948i 0.986046 + 0.166474i \(0.0532382\pi\)
−0.986046 + 0.166474i \(0.946762\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −26.8328 + 15.4919i −1.28359 + 0.741080i
\(438\) 0 0
\(439\) 18.5000 32.0429i 0.882957 1.52933i 0.0349192 0.999390i \(-0.488883\pi\)
0.848038 0.529936i \(-0.177784\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.35410 1.93649i −0.159358 0.0920055i 0.418200 0.908355i \(-0.362661\pi\)
−0.577558 + 0.816349i \(0.695994\pi\)
\(444\) 0 0
\(445\) 15.0000 + 25.9808i 0.711068 + 1.23161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −13.4164 −0.633159 −0.316580 0.948566i \(-0.602534\pi\)
−0.316580 + 0.948566i \(0.602534\pi\)
\(450\) 0 0
\(451\) −15.0000 25.9808i −0.706322 1.22339i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −26.8328 + 23.2379i −1.25794 + 1.08941i
\(456\) 0 0
\(457\) −14.5000 + 25.1147i −0.678281 + 1.17482i 0.297217 + 0.954810i \(0.403942\pi\)
−0.975498 + 0.220008i \(0.929392\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.4919i 0.721531i 0.932657 + 0.360766i \(0.117485\pi\)
−0.932657 + 0.360766i \(0.882515\pi\)
\(462\) 0 0
\(463\) 24.2487i 1.12693i 0.826139 + 0.563467i \(0.190533\pi\)
−0.826139 + 0.563467i \(0.809467\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) −18.0000 + 3.46410i −0.831163 + 0.159957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13.4164 23.2379i −0.616887 1.06848i
\(474\) 0 0
\(475\) 40.0000 1.83533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13.4164 23.2379i −0.613011 1.06177i −0.990730 0.135846i \(-0.956625\pi\)
0.377719 0.925920i \(1.62329\pi\)
\(480\) 0 0
\(481\) −12.0000 6.92820i −0.547153 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0623 17.4284i 0.456906 0.791384i
\(486\) 0 0
\(487\) 10.5000 6.06218i 0.475800 0.274703i −0.242864 0.970060i \(-0.578087\pi\)
0.718665 + 0.695357i \(0.244754\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.3649i 0.873926i 0.899479 + 0.436963i \(0.143946\pi\)
−0.899479 + 0.436963i \(0.856054\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.70820 19.3649i 0.300904 0.868635i
\(498\) 0 0
\(499\) 15.0000 + 8.66025i 0.671492 + 0.387686i 0.796642 0.604452i \(-0.206608\pi\)
−0.125150 + 0.992138i \(0.539941\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.8328 1.19642 0.598208 0.801341i \(-0.295880\pi\)
0.598208 + 0.801341i \(0.295880\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.0623 + 5.80948i 0.446004 + 0.257500i 0.706141 0.708071i \(-0.250434\pi\)
−0.260137 + 0.965572i \(0.583768\pi\)
\(510\) 0 0
\(511\) −18.0000 + 3.46410i −0.796273 + 0.153243i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 26.8328 15.4919i 1.18240 0.682656i
\(516\) 0 0
\(517\) 51.9615i 2.28527i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13.4164 + 7.74597i −0.587784 + 0.339357i −0.764221 0.644955i \(-0.776876\pi\)
0.176437 + 0.984312i \(0.443543\pi\)
\(522\) 0 0
\(523\) −11.0000 + 19.0526i −0.480996 + 0.833110i −0.999762 0.0218062i \(-0.993058\pi\)
0.518766 + 0.854916i \(0.326392\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.5000 + 32.0429i 0.804348 + 1.39317i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 26.8328 1.16226
\(534\) 0 0
\(535\) −22.5000 38.9711i −0.972760 1.68487i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 26.8328 + 3.87298i 1.15577 + 0.166821i
\(540\) 0 0
\(541\) 16.0000 27.7128i 0.687894 1.19147i −0.284624 0.958639i \(-0.591869\pi\)
0.972518 0.232828i \(-0.0747978\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.4919i 0.663602i
\(546\) 0 0
\(547\) 41.5692i 1.77737i −0.458517 0.888686i \(-0.651619\pi\)
0.458517 0.888686i \(-0.348381\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13.4164 + 23.2379i −0.571558 + 0.989968i
\(552\) 0 0
\(553\) 21.0000 + 24.2487i 0.893011 + 1.03116i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.7705 + 29.0474i 0.710589 + 1.23078i 0.964636 + 0.263585i \(0.0849049\pi\)
−0.254047 + 0.967192i \(0.581762\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.7705 29.0474i −0.706793 1.22420i −0.966041 0.258390i \(-0.916808\pi\)
0.259248 0.965811i \(1.58347\pi\)
\(564\) 0 0
\(565\) 45.0000 + 25.9808i 1.89316 + 1.09302i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.4164 + 23.2379i −0.562445 + 0.974183i 0.434837 + 0.900509i \(0.356806\pi\)
−0.997282 + 0.0736744i \(0.976527\pi\)
\(570\) 0 0
\(571\) 15.0000 8.66025i 0.627730 0.362420i −0.152142 0.988359i \(-0.548617\pi\)
0.779873 + 0.625938i \(0.215284\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 77.4597i 3.23029i
\(576\) 0 0
\(577\) −19.5000 + 11.2583i −0.811796 + 0.468690i −0.847579 0.530669i \(-0.821941\pi\)
0.0357834 + 0.999360i \(0.488607\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 16.7705 + 5.80948i 0.695758 + 0.241018i
\(582\) 0 0
\(583\) −22.5000 12.9904i −0.931855 0.538007i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.70820 0.276877 0.138439 0.990371i \(-0.455792\pi\)
0.138439 + 0.990371i \(0.455792\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.70820 3.87298i −0.275473 0.159044i 0.355899 0.934524i \(-0.384175\pi\)
−0.631372 + 0.775480i \(0.717508\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.8328 15.4919i 1.09636 0.632983i 0.161097 0.986939i \(-0.448497\pi\)
0.935262 + 0.353955i \(0.115164\pi\)
\(600\) 0 0
\(601\) 43.3013i 1.76630i −0.469095 0.883148i \(-0.655420\pi\)
0.469095 0.883148i \(-0.344580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.4164 7.74597i 0.545455 0.314918i
\(606\) 0 0
\(607\) −5.50000 + 9.52628i −0.223238 + 0.386660i −0.955789 0.294052i \(-0.904996\pi\)
0.732551 + 0.680712i \(0.238329\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −40.2492 23.2379i −1.62831 0.940105i
\(612\) 0 0
\(613\) 5.00000 + 8.66025i 0.201948 + 0.349784i 0.949156 0.314806i \(-0.101939\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.2492 1.62037 0.810186 0.586172i \(-0.199366\pi\)
0.810186 + 0.586172i \(0.199366\pi\)
\(618\) 0 0
\(619\) 19.0000 + 32.9090i 0.763674 + 1.32272i 0.940945 + 0.338561i \(0.109940\pi\)
−0.177270 + 0.984162i \(0.556727\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.4164 + 15.4919i 0.537517 + 0.620671i
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 36.3731i 1.44799i −0.689806 0.723994i \(-0.742304\pi\)
0.689806 0.723994i \(-0.257696\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −23.4787 + 40.6663i −0.931724 + 1.61379i
\(636\) 0 0
\(637\) −15.0000 + 19.0526i −0.594322 + 0.754890i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.70820 11.6190i −0.264958 0.458921i 0.702595 0.711590i \(-0.252025\pi\)
−0.967553 + 0.252669i \(0.918691\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.1246 + 34.8569i 0.791180 + 1.37036i 0.925237 + 0.379390i \(0.123866\pi\)
−0.134057 + 0.990974i \(0.542800\pi\)
\(648\) 0 0
\(649\) −22.5000 12.9904i −0.883202 0.509917i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.4787 + 40.6663i −0.918793 + 1.59140i −0.117542 + 0.993068i \(0.537501\pi\)
−0.801251 + 0.598328i \(0.795832\pi\)
\(654\) 0 0
\(655\) −22.5000 + 12.9904i −0.879148 + 0.507576i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.74597i 0.301740i −0.988554 0.150870i \(-0.951793\pi\)
0.988554 0.150870i \(-0.0482075\pi\)
\(660\) 0 0
\(661\) −6.00000 + 3.46410i −0.233373 + 0.134738i −0.612127 0.790759i \(-0.709686\pi\)
0.378754 + 0.925497i \(0.376353\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 40.2492 7.74597i 1.56080 0.300376i
\(666\) 0 0
\(667\) 45.0000 + 25.9808i 1.74241 + 1.00598i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 40.2492 1.55380
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.35410 + 1.93649i 0.128909 + 0.0744254i 0.563068 0.826411i \(-0.309621\pi\)
−0.434159 + 0.900836i \(0.642954\pi\)
\(678\) 0 0
\(679\) 4.50000 12.9904i 0.172694 0.498525i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.35410 1.93649i 0.128341 0.0740978i −0.434455 0.900694i \(-0.643059\pi\)
0.562796 + 0.826596i \(0.309726\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.1246 11.6190i 0.766687 0.442647i
\(690\) 0 0
\(691\) 7.00000 12.1244i 0.266293 0.461232i −0.701609 0.712562i \(-0.747535\pi\)
0.967901 + 0.251330i \(0.0808679\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.70820 3.87298i −0.254457 0.146911i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20.1246 −0.760096 −0.380048 0.924967i \(-0.624093\pi\)
−0.380048 + 0.924967i \(0.624093\pi\)
\(702\) 0 0
\(703\) 8.00000 + 13.8564i 0.301726 + 0.522604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 11.0000 19.0526i 0.413114 0.715534i −0.582115 0.813107i \(-0.697775\pi\)
0.995228 + 0.0975728i \(0.0311079\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.74597i 0.290089i
\(714\) 0 0
\(715\) 51.9615i 1.94325i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.1246 + 34.8569i −0.750521 + 1.29994i 0.197049 + 0.980394i \(0.436864\pi\)
−0.947570 + 0.319547i \(0.896469\pi\)
\(720\) 0 0
\(721\) 16.0000 13.8564i 0.595871 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −33.5410 58.0948i −1.24568 2.15758i
\(726\) 0 0
\(727\) −5.00000 −0.185440 −0.0927199 0.995692i \(-0.529556\pi\)
−0.0927199 + 0.995692i \(0.529556\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 6.00000 + 3.46410i 0.221615 + 0.127950i 0.606698 0.794933i \(-0.292494\pi\)
−0.385083 + 0.922882i \(0.625827\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.4164 + 23.2379i −0.494200 + 0.855979i
\(738\) 0 0
\(739\) −3.00000 + 1.73205i −0.110357 + 0.0637145i −0.554162 0.832409i \(-0.686961\pi\)
0.443806 + 0.896123i \(0.353628\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.4919i 0.568344i 0.958773 + 0.284172i \(0.0917187\pi\)
−0.958773 + 0.284172i \(0.908281\pi\)
\(744\) 0 0
\(745\) 45.0000 25.9808i 1.64867 0.951861i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −20.1246 23.2379i −0.735337 0.849094i
\(750\) 0 0
\(751\) −43.5000 25.1147i −1.58734 0.916450i −0.993744 0.111685i \(-0.964375\pi\)
−0.593594 0.804765i \(1.29771\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −60.3738 −2.19723
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.1246 + 11.6190i 0.729517 + 0.421187i 0.818245 0.574869i \(-0.194947\pi\)
−0.0887287 + 0.996056i \(0.528280\pi\)
\(762\) 0 0
\(763\) 2.00000 + 10.3923i 0.0724049 + 0.376227i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.1246 11.6190i 0.726658 0.419536i
\(768\) 0 0
\(769\) 12.1244i 0.437215i 0.975813 + 0.218608i \(0.0701515\pi\)
−0.975813 + 0.218608i \(0.929848\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −13.4164 + 7.74597i −0.482555 + 0.278603i −0.721480 0.692435i \(-0.756538\pi\)
0.238926 + 0.971038i \(0.423205\pi\)
\(774\) 0 0
\(775\) 5.00000 8.66025i 0.179605 0.311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −26.8328 15.4919i −0.961385 0.555056i
\(780\) 0 0
\(781\) −15.0000 25.9808i −0.536742 0.929665i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −80.4984 −2.87311
\(786\) 0 0
\(787\) 7.00000 + 12.1244i 0.249523 + 0.432187i 0.963394 0.268091i \(-0.0863928\pi\)
−0.713871 + 0.700278i \(0.753059\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 33.5410 + 11.6190i 1.19258 + 0.413122i
\(792\) 0 0
\(793\) −18.0000 + 31.1769i −0.639199 + 1.10712i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 50.3488i 1.78345i −0.452582 0.891723i \(-0.649497\pi\)
0.452582 0.891723i \(-0.350503\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −13.4164 + 23.2379i −0.473455 + 0.820048i
\(804\) 0 0
\(805\) −15.0000 77.9423i −0.528681 2.74710i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13.4164 23.2379i −0.471696 0.817001i 0.527780 0.849381i \(-0.323025\pi\)
−0.999476 + 0.0323801i \(0.989691\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.77