Properties

Label 1008.3.dc.a.737.1
Level $1008$
Weight $3$
Character 1008.737
Analytic conductor $27.466$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 737.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1008.737
Dual form 1008.3.dc.a.305.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.22474 - 0.707107i) q^{5} +(-6.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.22474 - 0.707107i) q^{5} +(-6.50000 - 2.59808i) q^{7} +(-6.12372 + 3.53553i) q^{11} +15.0000 q^{13} +(9.79796 - 5.65685i) q^{17} +(-6.50000 + 11.2583i) q^{19} +(-19.5959 - 11.3137i) q^{23} +(-11.5000 - 19.9186i) q^{25} +22.6274i q^{29} +(1.50000 + 2.59808i) q^{31} +(6.12372 + 7.77817i) q^{35} +(-8.50000 + 14.7224i) q^{37} +80.6102i q^{41} +85.0000 q^{43} +(62.4620 + 36.0624i) q^{47} +(35.5000 + 33.7750i) q^{49} +(-29.3939 + 16.9706i) q^{53} +10.0000 q^{55} +(-78.3837 + 45.2548i) q^{59} +(36.0000 - 62.3538i) q^{61} +(-18.3712 - 10.6066i) q^{65} +(21.5000 + 37.2391i) q^{67} +52.3259i q^{71} +(47.5000 + 82.2724i) q^{73} +(48.9898 - 7.07107i) q^{77} +(34.5000 - 59.7558i) q^{79} +60.8112i q^{83} -16.0000 q^{85} +(-117.576 - 67.8823i) q^{89} +(-97.5000 - 38.9711i) q^{91} +(15.9217 - 9.19239i) q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 26 q^{7} + 60 q^{13} - 26 q^{19} - 46 q^{25} + 6 q^{31} - 34 q^{37} + 340 q^{43} + 142 q^{49} + 40 q^{55} + 144 q^{61} + 86 q^{67} + 190 q^{73} + 138 q^{79} - 64 q^{85} - 390 q^{91} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(5\) −1.22474 0.707107i −0.244949 0.141421i 0.372500 0.928032i \(-0.378501\pi\)
−0.617449 + 0.786611i \(0.711834\pi\)
\(6\) 0 0
\(7\) −6.50000 2.59808i −0.928571 0.371154i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.12372 + 3.53553i −0.556702 + 0.321412i −0.751821 0.659367i \(-0.770824\pi\)
0.195119 + 0.980780i \(0.437491\pi\)
\(12\) 0 0
\(13\) 15.0000 1.15385 0.576923 0.816798i \(-0.304253\pi\)
0.576923 + 0.816798i \(0.304253\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.79796 5.65685i 0.576351 0.332756i −0.183331 0.983051i \(-0.558688\pi\)
0.759682 + 0.650295i \(0.225355\pi\)
\(18\) 0 0
\(19\) −6.50000 + 11.2583i −0.342105 + 0.592544i −0.984823 0.173559i \(-0.944473\pi\)
0.642718 + 0.766103i \(0.277807\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −19.5959 11.3137i −0.851996 0.491900i 0.00932753 0.999956i \(-0.497031\pi\)
−0.861324 + 0.508056i \(0.830364\pi\)
\(24\) 0 0
\(25\) −11.5000 19.9186i −0.460000 0.796743i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 22.6274i 0.780256i 0.920761 + 0.390128i \(0.127569\pi\)
−0.920761 + 0.390128i \(0.872431\pi\)
\(30\) 0 0
\(31\) 1.50000 + 2.59808i 0.0483871 + 0.0838089i 0.889205 0.457510i \(-0.151259\pi\)
−0.840817 + 0.541319i \(0.817925\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.12372 + 7.77817i 0.174964 + 0.222234i
\(36\) 0 0
\(37\) −8.50000 + 14.7224i −0.229730 + 0.397904i −0.957728 0.287675i \(-0.907118\pi\)
0.727998 + 0.685579i \(0.240451\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 80.6102i 1.96610i 0.183333 + 0.983051i \(0.441311\pi\)
−0.183333 + 0.983051i \(0.558689\pi\)
\(42\) 0 0
\(43\) 85.0000 1.97674 0.988372 0.152055i \(-0.0485890\pi\)
0.988372 + 0.152055i \(0.0485890\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 62.4620 + 36.0624i 1.32898 + 0.767286i 0.985142 0.171743i \(-0.0549400\pi\)
0.343837 + 0.939029i \(0.388273\pi\)
\(48\) 0 0
\(49\) 35.5000 + 33.7750i 0.724490 + 0.689286i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −29.3939 + 16.9706i −0.554601 + 0.320199i −0.750976 0.660330i \(-0.770417\pi\)
0.196374 + 0.980529i \(0.437083\pi\)
\(54\) 0 0
\(55\) 10.0000 0.181818
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −78.3837 + 45.2548i −1.32854 + 0.767031i −0.985073 0.172135i \(-0.944933\pi\)
−0.343463 + 0.939166i \(0.611600\pi\)
\(60\) 0 0
\(61\) 36.0000 62.3538i 0.590164 1.02219i −0.404046 0.914739i \(-0.632396\pi\)
0.994210 0.107455i \(-0.0342702\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −18.3712 10.6066i −0.282633 0.163178i
\(66\) 0 0
\(67\) 21.5000 + 37.2391i 0.320896 + 0.555807i 0.980673 0.195654i \(-0.0626828\pi\)
−0.659778 + 0.751461i \(0.729349\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 52.3259i 0.736985i 0.929631 + 0.368492i \(0.120126\pi\)
−0.929631 + 0.368492i \(0.879874\pi\)
\(72\) 0 0
\(73\) 47.5000 + 82.2724i 0.650685 + 1.12702i 0.982957 + 0.183836i \(0.0588515\pi\)
−0.332272 + 0.943184i \(0.607815\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 48.9898 7.07107i 0.636231 0.0918320i
\(78\) 0 0
\(79\) 34.5000 59.7558i 0.436709 0.756402i −0.560725 0.828002i \(-0.689477\pi\)
0.997433 + 0.0716005i \(0.0228107\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 60.8112i 0.732665i 0.930484 + 0.366332i \(0.119387\pi\)
−0.930484 + 0.366332i \(0.880613\pi\)
\(84\) 0 0
\(85\) −16.0000 −0.188235
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −117.576 67.8823i −1.32107 0.762722i −0.337173 0.941443i \(-0.609471\pi\)
−0.983900 + 0.178721i \(0.942804\pi\)
\(90\) 0 0
\(91\) −97.5000 38.9711i −1.07143 0.428254i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.9217 9.19239i 0.167597 0.0967620i
\(96\) 0 0
\(97\) 16.0000 0.164948 0.0824742 0.996593i \(-0.473718\pi\)
0.0824742 + 0.996593i \(0.473718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 60.0125 34.6482i 0.594183 0.343052i −0.172567 0.984998i \(-0.555206\pi\)
0.766750 + 0.641946i \(0.221873\pi\)
\(102\) 0 0
\(103\) −30.5000 + 52.8275i −0.296117 + 0.512889i −0.975244 0.221131i \(-0.929025\pi\)
0.679127 + 0.734020i \(0.262358\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 146.969 + 84.8528i 1.37355 + 0.793017i 0.991373 0.131074i \(-0.0418425\pi\)
0.382173 + 0.924091i \(0.375176\pi\)
\(108\) 0 0
\(109\) 32.5000 + 56.2917i 0.298165 + 0.516437i 0.975716 0.219039i \(-0.0702921\pi\)
−0.677551 + 0.735476i \(0.736959\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 137.179i 1.21397i 0.794713 + 0.606985i \(0.207621\pi\)
−0.794713 + 0.606985i \(0.792379\pi\)
\(114\) 0 0
\(115\) 16.0000 + 27.7128i 0.139130 + 0.240981i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −78.3837 + 11.3137i −0.658686 + 0.0950732i
\(120\) 0 0
\(121\) −35.5000 + 61.4878i −0.293388 + 0.508164i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 67.8823i 0.543058i
\(126\) 0 0
\(127\) 171.000 1.34646 0.673228 0.739435i \(-0.264907\pi\)
0.673228 + 0.739435i \(0.264907\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −101.654 58.6899i −0.775983 0.448014i 0.0590215 0.998257i \(-0.481202\pi\)
−0.835005 + 0.550242i \(0.814535\pi\)
\(132\) 0 0
\(133\) 71.5000 56.2917i 0.537594 0.423245i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −186.161 + 107.480i −1.35884 + 0.784527i −0.989468 0.144754i \(-0.953761\pi\)
−0.369373 + 0.929281i \(0.620428\pi\)
\(138\) 0 0
\(139\) 83.0000 0.597122 0.298561 0.954391i \(-0.403493\pi\)
0.298561 + 0.954391i \(0.403493\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −91.8559 + 53.0330i −0.642349 + 0.370860i
\(144\) 0 0
\(145\) 16.0000 27.7128i 0.110345 0.191123i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 88.1816 + 50.9117i 0.591823 + 0.341689i 0.765818 0.643057i \(-0.222334\pi\)
−0.173995 + 0.984747i \(0.555668\pi\)
\(150\) 0 0
\(151\) 20.0000 + 34.6410i 0.132450 + 0.229411i 0.924621 0.380889i \(-0.124382\pi\)
−0.792170 + 0.610300i \(0.791049\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.24264i 0.0273719i
\(156\) 0 0
\(157\) −148.000 256.344i −0.942675 1.63276i −0.760340 0.649525i \(-0.774968\pi\)
−0.182335 0.983237i \(-0.558365\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 97.9796 + 124.451i 0.608569 + 0.772986i
\(162\) 0 0
\(163\) 64.0000 110.851i 0.392638 0.680069i −0.600159 0.799881i \(-0.704896\pi\)
0.992797 + 0.119812i \(0.0382292\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 60.8112i 0.364139i −0.983286 0.182069i \(-0.941720\pi\)
0.983286 0.182069i \(-0.0582796\pi\)
\(168\) 0 0
\(169\) 56.0000 0.331361
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 97.9796 + 56.5685i 0.566356 + 0.326986i 0.755693 0.654926i \(-0.227300\pi\)
−0.189337 + 0.981912i \(0.560634\pi\)
\(174\) 0 0
\(175\) 23.0000 + 159.349i 0.131429 + 0.910564i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −25.7196 + 14.8492i −0.143685 + 0.0829567i −0.570119 0.821562i \(-0.693103\pi\)
0.426434 + 0.904519i \(0.359770\pi\)
\(180\) 0 0
\(181\) 81.0000 0.447514 0.223757 0.974645i \(-0.428168\pi\)
0.223757 + 0.974645i \(0.428168\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.8207 12.0208i 0.112544 0.0649774i
\(186\) 0 0
\(187\) −40.0000 + 69.2820i −0.213904 + 0.370492i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 55.1135 + 31.8198i 0.288552 + 0.166596i 0.637289 0.770625i \(-0.280056\pi\)
−0.348736 + 0.937221i \(0.613389\pi\)
\(192\) 0 0
\(193\) 111.500 + 193.124i 0.577720 + 1.00064i 0.995740 + 0.0922029i \(0.0293909\pi\)
−0.418020 + 0.908438i \(0.637276\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 158.392i 0.804020i −0.915635 0.402010i \(-0.868312\pi\)
0.915635 0.402010i \(-0.131688\pi\)
\(198\) 0 0
\(199\) −68.0000 117.779i −0.341709 0.591857i 0.643042 0.765831i \(-0.277672\pi\)
−0.984750 + 0.173975i \(0.944339\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 58.7878 147.078i 0.289595 0.724523i
\(204\) 0 0
\(205\) 57.0000 98.7269i 0.278049 0.481595i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 91.9239i 0.439827i
\(210\) 0 0
\(211\) −272.000 −1.28910 −0.644550 0.764562i \(-0.722955\pi\)
−0.644550 + 0.764562i \(0.722955\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −104.103 60.1041i −0.484201 0.279554i
\(216\) 0 0
\(217\) −3.00000 20.7846i −0.0138249 0.0957816i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 146.969 84.8528i 0.665020 0.383949i
\(222\) 0 0
\(223\) −248.000 −1.11211 −0.556054 0.831146i \(-0.687685\pi\)
−0.556054 + 0.831146i \(0.687685\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 143.295 82.7315i 0.631256 0.364456i −0.149982 0.988689i \(-0.547922\pi\)
0.781238 + 0.624233i \(0.214588\pi\)
\(228\) 0 0
\(229\) −216.500 + 374.989i −0.945415 + 1.63751i −0.190496 + 0.981688i \(0.561010\pi\)
−0.754918 + 0.655819i \(0.772324\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −197.184 113.844i −0.846283 0.488602i 0.0131120 0.999914i \(-0.495826\pi\)
−0.859395 + 0.511312i \(0.829160\pi\)
\(234\) 0 0
\(235\) −51.0000 88.3346i −0.217021 0.375892i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 343.654i 1.43788i 0.695071 + 0.718941i \(0.255373\pi\)
−0.695071 + 0.718941i \(0.744627\pi\)
\(240\) 0 0
\(241\) −79.0000 136.832i −0.327801 0.567768i 0.654274 0.756257i \(-0.272974\pi\)
−0.982075 + 0.188490i \(0.939641\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −19.5959 66.4680i −0.0799833 0.271298i
\(246\) 0 0
\(247\) −97.5000 + 168.875i −0.394737 + 0.683704i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 237.588i 0.946565i −0.880911 0.473283i \(-0.843069\pi\)
0.880911 0.473283i \(-0.156931\pi\)
\(252\) 0 0
\(253\) 160.000 0.632411
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −275.568 159.099i −1.07225 0.619062i −0.143453 0.989657i \(-0.545821\pi\)
−0.928795 + 0.370595i \(0.879154\pi\)
\(258\) 0 0
\(259\) 93.5000 73.6122i 0.361004 0.284217i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −146.969 + 84.8528i −0.558819 + 0.322634i −0.752671 0.658396i \(-0.771235\pi\)
0.193852 + 0.981031i \(0.437902\pi\)
\(264\) 0 0
\(265\) 48.0000 0.181132
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 420.087 242.538i 1.56166 0.901627i 0.564574 0.825382i \(-0.309040\pi\)
0.997089 0.0762447i \(-0.0242930\pi\)
\(270\) 0 0
\(271\) 108.000 187.061i 0.398524 0.690264i −0.595020 0.803711i \(-0.702856\pi\)
0.993544 + 0.113447i \(0.0361892\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 140.846 + 81.3173i 0.512166 + 0.295699i
\(276\) 0 0
\(277\) −152.500 264.138i −0.550542 0.953566i −0.998236 0.0593790i \(-0.981088\pi\)
0.447694 0.894187i \(-0.352245\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.6274i 0.0805246i −0.999189 0.0402623i \(-0.987181\pi\)
0.999189 0.0402623i \(-0.0128194\pi\)
\(282\) 0 0
\(283\) 78.5000 + 135.966i 0.277385 + 0.480445i 0.970734 0.240157i \(-0.0771989\pi\)
−0.693349 + 0.720602i \(0.743866\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 209.431 523.966i 0.729726 1.82567i
\(288\) 0 0
\(289\) −80.5000 + 139.430i −0.278547 + 0.482457i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 101.823i 0.347520i −0.984788 0.173760i \(-0.944408\pi\)
0.984788 0.173760i \(-0.0555917\pi\)
\(294\) 0 0
\(295\) 128.000 0.433898
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −293.939 169.706i −0.983073 0.567577i
\(300\) 0 0
\(301\) −552.500 220.836i −1.83555 0.733676i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −88.1816 + 50.9117i −0.289120 + 0.166924i
\(306\) 0 0
\(307\) 11.0000 0.0358306 0.0179153 0.999840i \(-0.494297\pi\)
0.0179153 + 0.999840i \(0.494297\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 454.380 262.337i 1.46103 0.843526i 0.461971 0.886895i \(-0.347142\pi\)
0.999059 + 0.0433690i \(0.0138091\pi\)
\(312\) 0 0
\(313\) −276.500 + 478.912i −0.883387 + 1.53007i −0.0358348 + 0.999358i \(0.511409\pi\)
−0.847552 + 0.530713i \(0.821924\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 117.576 + 67.8823i 0.370901 + 0.214140i 0.673852 0.738866i \(-0.264639\pi\)
−0.302951 + 0.953006i \(0.597972\pi\)
\(318\) 0 0
\(319\) −80.0000 138.564i −0.250784 0.434370i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 147.078i 0.455350i
\(324\) 0 0
\(325\) −172.500 298.779i −0.530769 0.919319i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −312.310 396.687i −0.949270 1.20574i
\(330\) 0 0
\(331\) −30.5000 + 52.8275i −0.0921450 + 0.159600i −0.908413 0.418073i \(-0.862706\pi\)
0.816268 + 0.577673i \(0.196039\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 60.8112i 0.181526i
\(336\) 0 0
\(337\) 135.000 0.400593 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.3712 10.6066i −0.0538744 0.0311044i
\(342\) 0 0
\(343\) −143.000 311.769i −0.416910 0.908948i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 88.1816 50.9117i 0.254126 0.146720i −0.367526 0.930013i \(-0.619795\pi\)
0.621652 + 0.783294i \(0.286462\pi\)
\(348\) 0 0
\(349\) −152.000 −0.435530 −0.217765 0.976001i \(-0.569877\pi\)
−0.217765 + 0.976001i \(0.569877\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −334.355 + 193.040i −0.947182 + 0.546856i −0.892205 0.451632i \(-0.850842\pi\)
−0.0549778 + 0.998488i \(0.517509\pi\)
\(354\) 0 0
\(355\) 37.0000 64.0859i 0.104225 0.180524i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −39.1918 22.6274i −0.109169 0.0630290i 0.444421 0.895818i \(-0.353409\pi\)
−0.553591 + 0.832789i \(0.686743\pi\)
\(360\) 0 0
\(361\) 96.0000 + 166.277i 0.265928 + 0.460601i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 134.350i 0.368083i
\(366\) 0 0
\(367\) 50.5000 + 87.4686i 0.137602 + 0.238334i 0.926588 0.376077i \(-0.122727\pi\)
−0.788986 + 0.614411i \(0.789394\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 235.151 33.9411i 0.633830 0.0914855i
\(372\) 0 0
\(373\) 155.500 269.334i 0.416890 0.722075i −0.578735 0.815516i \(-0.696453\pi\)
0.995625 + 0.0934411i \(0.0297867\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 339.411i 0.900295i
\(378\) 0 0
\(379\) −91.0000 −0.240106 −0.120053 0.992768i \(-0.538306\pi\)
−0.120053 + 0.992768i \(0.538306\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −284.141 164.049i −0.741882 0.428326i 0.0808712 0.996725i \(-0.474230\pi\)
−0.822753 + 0.568399i \(0.807563\pi\)
\(384\) 0 0
\(385\) −65.0000 25.9808i −0.168831 0.0674825i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 596.451 344.361i 1.53329 0.885247i 0.534085 0.845431i \(-0.320656\pi\)
0.999207 0.0398163i \(-0.0126773\pi\)
\(390\) 0 0
\(391\) −256.000 −0.654731
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −84.5074 + 48.7904i −0.213943 + 0.123520i
\(396\) 0 0
\(397\) 139.500 241.621i 0.351385 0.608617i −0.635107 0.772424i \(-0.719044\pi\)
0.986492 + 0.163807i \(0.0523774\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 22.5000 + 38.9711i 0.0558313 + 0.0967026i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 120.208i 0.295352i
\(408\) 0 0
\(409\) 111.500 + 193.124i 0.272616 + 0.472185i 0.969531 0.244969i \(-0.0787778\pi\)
−0.696915 + 0.717154i \(0.745444\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 627.069 90.5097i 1.51833 0.219152i
\(414\) 0 0
\(415\) 43.0000 74.4782i 0.103614 0.179466i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 581.242i 1.38721i 0.720355 + 0.693606i \(0.243979\pi\)
−0.720355 + 0.693606i \(0.756021\pi\)
\(420\) 0 0
\(421\) 153.000 0.363420 0.181710 0.983352i \(-0.441837\pi\)
0.181710 + 0.983352i \(0.441837\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −225.353 130.108i −0.530242 0.306136i
\(426\) 0 0
\(427\) −396.000 + 311.769i −0.927400 + 0.730139i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −121.250 + 70.0036i −0.281322 + 0.162421i −0.634022 0.773315i \(-0.718597\pi\)
0.352700 + 0.935737i \(0.385264\pi\)
\(432\) 0 0
\(433\) 137.000 0.316397 0.158199 0.987407i \(-0.449431\pi\)
0.158199 + 0.987407i \(0.449431\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 254.747 147.078i 0.582945 0.336563i
\(438\) 0 0
\(439\) −300.000 + 519.615i −0.683371 + 1.18363i 0.290574 + 0.956852i \(0.406154\pi\)
−0.973946 + 0.226781i \(0.927180\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −548.686 316.784i −1.23857 0.715088i −0.269767 0.962926i \(-0.586947\pi\)
−0.968801 + 0.247838i \(0.920280\pi\)
\(444\) 0 0
\(445\) 96.0000 + 166.277i 0.215730 + 0.373656i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 383.252i 0.853568i 0.904354 + 0.426784i \(0.140353\pi\)
−0.904354 + 0.426784i \(0.859647\pi\)
\(450\) 0 0
\(451\) −285.000 493.634i −0.631929 1.09453i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 91.8559 + 116.673i 0.201881 + 0.256423i
\(456\) 0 0
\(457\) 119.500 206.980i 0.261488 0.452910i −0.705150 0.709059i \(-0.749120\pi\)
0.966638 + 0.256148i \(0.0824535\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 452.548i 0.981667i −0.871253 0.490833i \(-0.836692\pi\)
0.871253 0.490833i \(-0.163308\pi\)
\(462\) 0 0
\(463\) −211.000 −0.455724 −0.227862 0.973693i \(-0.573173\pi\)
−0.227862 + 0.973693i \(0.573173\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −270.669 156.271i −0.579590 0.334627i 0.181380 0.983413i \(-0.441944\pi\)
−0.760971 + 0.648786i \(0.775277\pi\)
\(468\) 0 0
\(469\) −43.0000 297.913i −0.0916844 0.635208i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −520.517 + 300.520i −1.10046 + 0.635350i
\(474\) 0 0
\(475\) 299.000 0.629474
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 607.473 350.725i 1.26821 0.732202i 0.293562 0.955940i \(-0.405159\pi\)
0.974649 + 0.223737i \(0.0718258\pi\)
\(480\) 0 0
\(481\) −127.500 + 220.836i −0.265073 + 0.459119i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −19.5959 11.3137i −0.0404040 0.0233272i
\(486\) 0 0
\(487\) 209.500 + 362.865i 0.430185 + 0.745102i 0.996889 0.0788195i \(-0.0251151\pi\)
−0.566704 + 0.823921i \(0.691782\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 169.706i 0.345633i −0.984954 0.172816i \(-0.944713\pi\)
0.984954 0.172816i \(-0.0552867\pi\)
\(492\) 0 0
\(493\) 128.000 + 221.703i 0.259635 + 0.449701i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 135.947 340.118i 0.273535 0.684343i
\(498\) 0 0
\(499\) 93.5000 161.947i 0.187375 0.324543i −0.756999 0.653416i \(-0.773335\pi\)
0.944374 + 0.328873i \(0.106669\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 173.948i 0.345822i −0.984937 0.172911i \(-0.944683\pi\)
0.984937 0.172911i \(-0.0553172\pi\)
\(504\) 0 0
\(505\) −98.0000 −0.194059
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −224.128 129.401i −0.440331 0.254225i 0.263407 0.964685i \(-0.415154\pi\)
−0.703738 + 0.710460i \(0.748487\pi\)
\(510\) 0 0
\(511\) −95.0000 658.179i −0.185910 1.28802i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 74.7094 43.1335i 0.145067 0.0837544i
\(516\) 0 0
\(517\) −510.000 −0.986460
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 352.727 203.647i 0.677018 0.390877i −0.121712 0.992565i \(-0.538839\pi\)
0.798731 + 0.601689i \(0.205505\pi\)
\(522\) 0 0
\(523\) −405.500 + 702.347i −0.775335 + 1.34292i 0.159272 + 0.987235i \(0.449085\pi\)
−0.934606 + 0.355684i \(0.884248\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.3939 + 16.9706i 0.0557759 + 0.0322022i
\(528\) 0 0
\(529\) −8.50000 14.7224i −0.0160681 0.0278307i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1209.15i 2.26858i
\(534\) 0 0
\(535\) −120.000 207.846i −0.224299 0.388497i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −336.805 81.3173i −0.624870 0.150867i
\(540\) 0 0
\(541\) 172.500 298.779i 0.318854 0.552271i −0.661395 0.750038i \(-0.730035\pi\)
0.980249 + 0.197766i \(0.0633687\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 91.9239i 0.168668i
\(546\) 0 0
\(547\) 864.000 1.57952 0.789762 0.613413i \(-0.210204\pi\)
0.789762 + 0.613413i \(0.210204\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −254.747 147.078i −0.462336 0.266930i
\(552\) 0 0
\(553\) −379.500 + 298.779i −0.686257 + 0.540287i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −821.804 + 474.469i −1.47541 + 0.851829i −0.999616 0.0277272i \(-0.991173\pi\)
−0.475795 + 0.879556i \(0.657840\pi\)
\(558\) 0 0
\(559\) 1275.00 2.28086
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 640.542 369.817i 1.13773 0.656868i 0.191862 0.981422i \(-0.438547\pi\)
0.945867 + 0.324554i \(0.105214\pi\)
\(564\) 0 0
\(565\) 97.0000 168.009i 0.171681 0.297361i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 665.036 + 383.959i 1.16878 + 0.674796i 0.953393 0.301732i \(-0.0975649\pi\)
0.215388 + 0.976528i \(0.430898\pi\)
\(570\) 0 0
\(571\) 238.500 + 413.094i 0.417688 + 0.723457i 0.995706 0.0925665i \(-0.0295071\pi\)
−0.578018 + 0.816024i \(0.696174\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 520.431i 0.905097i
\(576\) 0 0
\(577\) 131.500 + 227.765i 0.227903 + 0.394739i 0.957186 0.289472i \(-0.0934798\pi\)
−0.729283 + 0.684212i \(0.760146\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 157.992 395.273i 0.271931 0.680332i
\(582\) 0 0
\(583\) 120.000 207.846i 0.205832 0.356511i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 531.744i 0.905868i 0.891544 + 0.452934i \(0.149623\pi\)
−0.891544 + 0.452934i \(0.850377\pi\)
\(588\) 0 0
\(589\) −39.0000 −0.0662139
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −694.430 400.930i −1.17105 0.676104i −0.217120 0.976145i \(-0.569666\pi\)
−0.953926 + 0.300041i \(0.903000\pi\)
\(594\) 0 0
\(595\) 104.000 + 41.5692i 0.174790 + 0.0698642i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −676.059 + 390.323i −1.12865 + 0.651624i −0.943594 0.331104i \(-0.892579\pi\)
−0.185052 + 0.982729i \(0.559245\pi\)
\(600\) 0 0
\(601\) 383.000 0.637271 0.318636 0.947877i \(-0.396775\pi\)
0.318636 + 0.947877i \(0.396775\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 86.9569 50.2046i 0.143730 0.0829828i
\(606\) 0 0
\(607\) −382.500 + 662.509i −0.630148 + 1.09145i 0.357373 + 0.933962i \(0.383673\pi\)
−0.987521 + 0.157487i \(0.949661\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 936.930 + 540.937i 1.53344 + 0.885330i
\(612\) 0 0
\(613\) −204.000 353.338i −0.332790 0.576408i 0.650268 0.759705i \(-0.274657\pi\)
−0.983058 + 0.183296i \(0.941323\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 914.996i 1.48298i 0.670966 + 0.741488i \(0.265880\pi\)
−0.670966 + 0.741488i \(0.734120\pi\)
\(618\) 0 0
\(619\) −473.500 820.126i −0.764943 1.32492i −0.940276 0.340412i \(-0.889434\pi\)
0.175333 0.984509i \(-0.443900\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 587.878 + 746.705i 0.943624 + 1.19856i
\(624\) 0 0
\(625\) −239.500 + 414.826i −0.383200 + 0.663722i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 192.333i 0.305776i
\(630\) 0 0
\(631\) 760.000 1.20444 0.602219 0.798331i \(-0.294284\pi\)
0.602219 + 0.798331i \(0.294284\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −209.431 120.915i −0.329813 0.190418i
\(636\) 0 0
\(637\) 532.500 + 506.625i 0.835950 + 0.795329i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −401.716 + 231.931i −0.626703 + 0.361827i −0.779474 0.626435i \(-0.784514\pi\)
0.152771 + 0.988262i \(0.451180\pi\)
\(642\) 0 0
\(643\) 19.0000 0.0295490 0.0147745 0.999891i \(-0.495297\pi\)
0.0147745 + 0.999891i \(0.495297\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1003.07 + 579.120i −1.55033 + 0.895086i −0.552220 + 0.833698i \(0.686219\pi\)
−0.998114 + 0.0613872i \(0.980448\pi\)
\(648\) 0 0
\(649\) 320.000 554.256i 0.493066 0.854016i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 892.839 + 515.481i 1.36729 + 0.789404i 0.990581 0.136928i \(-0.0437230\pi\)
0.376707 + 0.926332i \(0.377056\pi\)
\(654\) 0 0
\(655\) 83.0000 + 143.760i 0.126718 + 0.219481i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 972.979i 1.47645i 0.674556 + 0.738224i \(0.264335\pi\)
−0.674556 + 0.738224i \(0.735665\pi\)
\(660\) 0 0
\(661\) 200.500 + 347.276i 0.303328 + 0.525380i 0.976888 0.213753i \(-0.0685688\pi\)
−0.673559 + 0.739133i \(0.735235\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −127.373 + 18.3848i −0.191539 + 0.0276463i
\(666\) 0 0
\(667\) 256.000 443.405i 0.383808 0.664775i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 509.117i 0.758743i
\(672\) 0 0
\(673\) 665.000 0.988113 0.494056 0.869430i \(-0.335514\pi\)
0.494056 + 0.869430i \(0.335514\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −793.635 458.205i −1.17228 0.676817i −0.218065 0.975934i \(-0.569975\pi\)
−0.954216 + 0.299117i \(0.903308\pi\)
\(678\) 0 0
\(679\) −104.000 41.5692i −0.153166 0.0612212i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 117.576 67.8823i 0.172146 0.0993884i −0.411452 0.911432i \(-0.634978\pi\)
0.583597 + 0.812043i \(0.301645\pi\)
\(684\) 0 0
\(685\) 304.000 0.443796
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −440.908 + 254.558i −0.639925 + 0.369461i
\(690\) 0 0
\(691\) 110.500 191.392i 0.159913 0.276978i −0.774924 0.632054i \(-0.782212\pi\)
0.934837 + 0.355077i \(0.115545\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −101.654 58.6899i −0.146264 0.0844458i
\(696\) 0 0
\(697\) 456.000 + 789.815i 0.654232 + 1.13316i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 192.333i 0.274370i 0.990545 + 0.137185i \(0.0438054\pi\)
−0.990545 + 0.137185i \(0.956195\pi\)
\(702\) 0 0
\(703\) −110.500 191.392i −0.157183 0.272250i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −480.100 + 69.2965i −0.679066 + 0.0980148i
\(708\) 0 0
\(709\) 164.000 284.056i 0.231312 0.400644i −0.726883 0.686762i \(-0.759032\pi\)
0.958194 + 0.286118i \(0.0923649\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 67.8823i 0.0952065i
\(714\) 0 0
\(715\) 150.000 0.209790
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 878.142 + 506.996i 1.22134 + 0.705140i 0.965203 0.261501i \(-0.0842175\pi\)
0.256135 + 0.966641i \(0.417551\pi\)
\(720\) 0 0
\(721\) 335.500 264.138i 0.465326 0.366349i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 450.706 260.215i 0.621664 0.358918i
\(726\) 0 0
\(727\) −1069.00 −1.47043 −0.735213 0.677836i \(-0.762918\pi\)
−0.735213 + 0.677836i \(0.762918\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 832.827 480.833i 1.13930 0.657774i
\(732\) 0 0
\(733\) −247.500 + 428.683i −0.337653 + 0.584833i −0.983991 0.178219i \(-0.942967\pi\)
0.646337 + 0.763052i \(0.276300\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −263.320 152.028i −0.357286 0.206279i
\(738\) 0 0
\(739\) 226.500 + 392.310i 0.306495 + 0.530865i 0.977593 0.210503i \(-0.0675103\pi\)
−0.671098 + 0.741369i \(0.734177\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 538.815i 0.725189i −0.931947 0.362594i \(-0.881891\pi\)
0.931947 0.362594i \(-0.118109\pi\)
\(744\) 0 0
\(745\) −72.0000 124.708i −0.0966443 0.167393i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −734.847 933.381i −0.981104 1.24617i
\(750\) 0 0
\(751\) −169.500 + 293.583i −0.225699 + 0.390922i −0.956529 0.291637i \(-0.905800\pi\)
0.730830 + 0.682560i \(0.239133\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 56.5685i 0.0749252i
\(756\) 0 0
\(757\) 198.000 0.261559 0.130779 0.991411i \(-0.458252\pi\)
0.130779 + 0.991411i \(0.458252\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 313.535 + 181.019i 0.412004 + 0.237870i 0.691650 0.722233i \(-0.256884\pi\)
−0.279647 + 0.960103i \(0.590217\pi\)
\(762\) 0 0
\(763\) −65.0000 450.333i −0.0851900 0.590214i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1175.76 + 678.823i −1.53293 + 0.885036i
\(768\) 0 0
\(769\) −929.000 −1.20806 −0.604031 0.796961i \(-0.706440\pi\)
−0.604031 + 0.796961i \(0.706440\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 148.194 85.5599i 0.191713 0.110686i −0.401071 0.916047i \(-0.631362\pi\)
0.592784 + 0.805361i \(0.298029\pi\)
\(774\) 0 0
\(775\) 34.5000 59.7558i 0.0445161 0.0771042i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −907.536 523.966i −1.16500 0.672614i
\(780\) 0 0
\(781\) −185.000 320.429i −0.236876 0.410281i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 418.607i 0.533258i
\(786\) 0 0
\(787\) 45.0000 + 77.9423i 0.0571792 + 0.0990372i 0.893198 0.449663i \(-0.148456\pi\)
−0.836019 + 0.548701i \(0.815123\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 356.401 891.662i 0.450570 1.12726i
\(792\) 0 0
\(793\) 540.000 935.307i 0.680958 1.17945i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1154.00i 1.44793i 0.689838 + 0.723964i \(0.257682\pi\)
−0.689838 + 0.723964i \(0.742318\pi\)
\(798\)