Properties

Label 672.2.j.d.239.8
Level 672
Weight 2
Character 672.239
Analytic conductor 5.366
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.8
Root \(1.37027 - 0.349801i\)
Character \(\chi\) = 672.239
Dual form 672.2.j.d.239.6

$q$-expansion

\(f(q)\) \(=\) \(q+(0.203364 + 1.72007i) q^{3} +3.44014 q^{5} +1.00000i q^{7} +(-2.91729 + 0.699602i) q^{9} +O(q^{10})\) \(q+(0.203364 + 1.72007i) q^{3} +3.44014 q^{5} +1.00000i q^{7} +(-2.91729 + 0.699602i) q^{9} +5.42784i q^{13} +(0.699602 + 5.91729i) q^{15} -3.15559i q^{17} +4.40673 q^{19} +(-1.72007 + 0.203364i) q^{21} -4.55480 q^{23} +6.83457 q^{25} +(-1.79664 - 4.87566i) q^{27} +4.08188 q^{29} +3.44014i q^{35} +3.18654i q^{37} +(-9.33627 + 1.10383i) q^{39} -5.95400i q^{41} -9.83457 q^{43} +(-10.0359 + 2.40673i) q^{45} +10.9622 q^{47} -1.00000 q^{49} +(5.42784 - 0.641735i) q^{51} -6.31119 q^{53} +(0.896171 + 7.57988i) q^{57} +0.641735i q^{59} +10.2413i q^{61} +(-0.699602 - 2.91729i) q^{63} +18.6725i q^{65} +9.02112 q^{67} +(-0.926283 - 7.83457i) q^{69} +3.15559 q^{71} -6.00000 q^{73} +(1.38991 + 11.7559i) q^{75} +1.62691i q^{79} +(8.02112 - 4.08188i) q^{81} -12.5497i q^{83} -10.8557i q^{85} +(0.830108 + 7.02112i) q^{87} -7.80657i q^{89} -5.42784 q^{91} +15.1598 q^{95} -16.0422 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{9} + O(q^{10}) \) \( 12q + 4q^{9} + 48q^{19} + 4q^{25} - 24q^{27} - 40q^{43} - 12q^{49} - 8q^{51} + 40q^{57} + 40q^{67} - 72q^{73} + 24q^{75} + 28q^{81} + 8q^{91} - 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0.203364 + 1.72007i 0.117412 + 0.993083i
\(4\) 0 0
\(5\) 3.44014 1.53848 0.769239 0.638961i \(-0.220636\pi\)
0.769239 + 0.638961i \(0.220636\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.91729 + 0.699602i −0.972429 + 0.233201i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 5.42784i 1.50541i 0.658356 + 0.752706i \(0.271252\pi\)
−0.658356 + 0.752706i \(0.728748\pi\)
\(14\) 0 0
\(15\) 0.699602 + 5.91729i 0.180636 + 1.52784i
\(16\) 0 0
\(17\) 3.15559i 0.765344i −0.923884 0.382672i \(-0.875004\pi\)
0.923884 0.382672i \(-0.124996\pi\)
\(18\) 0 0
\(19\) 4.40673 1.01097 0.505486 0.862835i \(-0.331313\pi\)
0.505486 + 0.862835i \(0.331313\pi\)
\(20\) 0 0
\(21\) −1.72007 + 0.203364i −0.375350 + 0.0443777i
\(22\) 0 0
\(23\) −4.55480 −0.949741 −0.474870 0.880056i \(-0.657505\pi\)
−0.474870 + 0.880056i \(0.657505\pi\)
\(24\) 0 0
\(25\) 6.83457 1.36691
\(26\) 0 0
\(27\) −1.79664 4.87566i −0.345763 0.938322i
\(28\) 0 0
\(29\) 4.08188 0.757985 0.378993 0.925400i \(-0.376271\pi\)
0.378993 + 0.925400i \(0.376271\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44014i 0.581490i
\(36\) 0 0
\(37\) 3.18654i 0.523864i 0.965086 + 0.261932i \(0.0843597\pi\)
−0.965086 + 0.261932i \(0.915640\pi\)
\(38\) 0 0
\(39\) −9.33627 + 1.10383i −1.49500 + 0.176754i
\(40\) 0 0
\(41\) 5.95400i 0.929859i −0.885348 0.464929i \(-0.846080\pi\)
0.885348 0.464929i \(-0.153920\pi\)
\(42\) 0 0
\(43\) −9.83457 −1.49976 −0.749879 0.661575i \(-0.769888\pi\)
−0.749879 + 0.661575i \(0.769888\pi\)
\(44\) 0 0
\(45\) −10.0359 + 2.40673i −1.49606 + 0.358774i
\(46\) 0 0
\(47\) 10.9622 1.59900 0.799498 0.600669i \(-0.205099\pi\)
0.799498 + 0.600669i \(0.205099\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 5.42784 0.641735i 0.760050 0.0898608i
\(52\) 0 0
\(53\) −6.31119 −0.866908 −0.433454 0.901176i \(-0.642705\pi\)
−0.433454 + 0.901176i \(0.642705\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.896171 + 7.57988i 0.118701 + 1.00398i
\(58\) 0 0
\(59\) 0.641735i 0.0835468i 0.999127 + 0.0417734i \(0.0133007\pi\)
−0.999127 + 0.0417734i \(0.986699\pi\)
\(60\) 0 0
\(61\) 10.2413i 1.31126i 0.755081 + 0.655632i \(0.227598\pi\)
−0.755081 + 0.655632i \(0.772402\pi\)
\(62\) 0 0
\(63\) −0.699602 2.91729i −0.0881415 0.367543i
\(64\) 0 0
\(65\) 18.6725i 2.31604i
\(66\) 0 0
\(67\) 9.02112 1.10210 0.551052 0.834471i \(-0.314226\pi\)
0.551052 + 0.834471i \(0.314226\pi\)
\(68\) 0 0
\(69\) −0.926283 7.83457i −0.111511 0.943172i
\(70\) 0 0
\(71\) 3.15559 0.374500 0.187250 0.982312i \(-0.440043\pi\)
0.187250 + 0.982312i \(0.440043\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 1.38991 + 11.7559i 0.160493 + 1.35746i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.62691i 0.183042i 0.995803 + 0.0915210i \(0.0291729\pi\)
−0.995803 + 0.0915210i \(0.970827\pi\)
\(80\) 0 0
\(81\) 8.02112 4.08188i 0.891235 0.453542i
\(82\) 0 0
\(83\) 12.5497i 1.37751i −0.724993 0.688756i \(-0.758157\pi\)
0.724993 0.688756i \(-0.241843\pi\)
\(84\) 0 0
\(85\) 10.8557i 1.17746i
\(86\) 0 0
\(87\) 0.830108 + 7.02112i 0.0889969 + 0.752743i
\(88\) 0 0
\(89\) 7.80657i 0.827494i −0.910392 0.413747i \(-0.864220\pi\)
0.910392 0.413747i \(-0.135780\pi\)
\(90\) 0 0
\(91\) −5.42784 −0.568993
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.1598 1.55536
\(96\) 0 0
\(97\) −16.0422 −1.62884 −0.814421 0.580275i \(-0.802945\pi\)
−0.814421 + 0.580275i \(0.802945\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.75133 0.970293 0.485147 0.874433i \(-0.338766\pi\)
0.485147 + 0.874433i \(0.338766\pi\)
\(102\) 0 0
\(103\) 1.62691i 0.160305i −0.996783 0.0801523i \(-0.974459\pi\)
0.996783 0.0801523i \(-0.0255407\pi\)
\(104\) 0 0
\(105\) −5.91729 + 0.699602i −0.577468 + 0.0682741i
\(106\) 0 0
\(107\) 6.88028i 0.665142i −0.943078 0.332571i \(-0.892084\pi\)
0.943078 0.332571i \(-0.107916\pi\)
\(108\) 0 0
\(109\) 12.8135i 1.22731i −0.789576 0.613653i \(-0.789699\pi\)
0.789576 0.613653i \(-0.210301\pi\)
\(110\) 0 0
\(111\) −5.48108 + 0.648029i −0.520241 + 0.0615082i
\(112\) 0 0
\(113\) 1.39920i 0.131626i −0.997832 0.0658130i \(-0.979036\pi\)
0.997832 0.0658130i \(-0.0209641\pi\)
\(114\) 0 0
\(115\) −15.6691 −1.46116
\(116\) 0 0
\(117\) −3.79733 15.8346i −0.351063 1.46391i
\(118\) 0 0
\(119\) 3.15559 0.289273
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 10.2413 1.21083i 0.923427 0.109177i
\(124\) 0 0
\(125\) 6.31119 0.564490
\(126\) 0 0
\(127\) 10.2077i 0.905783i −0.891566 0.452892i \(-0.850392\pi\)
0.891566 0.452892i \(-0.149608\pi\)
\(128\) 0 0
\(129\) −2.00000 16.9162i −0.176090 1.48938i
\(130\) 0 0
\(131\) 1.21083i 0.105791i −0.998600 0.0528954i \(-0.983155\pi\)
0.998600 0.0528954i \(-0.0168450\pi\)
\(132\) 0 0
\(133\) 4.40673i 0.382112i
\(134\) 0 0
\(135\) −6.18068 16.7730i −0.531948 1.44359i
\(136\) 0 0
\(137\) 7.44938i 0.636443i −0.948016 0.318222i \(-0.896914\pi\)
0.948016 0.318222i \(-0.103086\pi\)
\(138\) 0 0
\(139\) 7.26242 0.615990 0.307995 0.951388i \(-0.400342\pi\)
0.307995 + 0.951388i \(0.400342\pi\)
\(140\) 0 0
\(141\) 2.22931 + 18.8557i 0.187742 + 1.58794i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 14.0422 1.16614
\(146\) 0 0
\(147\) −0.203364 1.72007i −0.0167732 0.141869i
\(148\) 0 0
\(149\) 20.0718 1.64434 0.822171 0.569241i \(-0.192763\pi\)
0.822171 + 0.569241i \(0.192763\pi\)
\(150\) 0 0
\(151\) 13.4615i 1.09548i −0.836649 0.547740i \(-0.815488\pi\)
0.836649 0.547740i \(-0.184512\pi\)
\(152\) 0 0
\(153\) 2.20766 + 9.20577i 0.178479 + 0.744242i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.8009i 0.941817i −0.882182 0.470908i \(-0.843926\pi\)
0.882182 0.470908i \(-0.156074\pi\)
\(158\) 0 0
\(159\) −1.28347 10.8557i −0.101786 0.860912i
\(160\) 0 0
\(161\) 4.55480i 0.358968i
\(162\) 0 0
\(163\) 1.83457 0.143695 0.0718474 0.997416i \(-0.477111\pi\)
0.0718474 + 0.997416i \(0.477111\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.44938 −0.576450 −0.288225 0.957563i \(-0.593065\pi\)
−0.288225 + 0.957563i \(0.593065\pi\)
\(168\) 0 0
\(169\) −16.4615 −1.26627
\(170\) 0 0
\(171\) −12.8557 + 3.08295i −0.983099 + 0.235759i
\(172\) 0 0
\(173\) −15.3481 −1.16690 −0.583449 0.812150i \(-0.698297\pi\)
−0.583449 + 0.812150i \(0.698297\pi\)
\(174\) 0 0
\(175\) 6.83457i 0.516645i
\(176\) 0 0
\(177\) −1.10383 + 0.130506i −0.0829689 + 0.00980942i
\(178\) 0 0
\(179\) 19.5027i 1.45770i 0.684675 + 0.728848i \(0.259944\pi\)
−0.684675 + 0.728848i \(0.740056\pi\)
\(180\) 0 0
\(181\) 10.2413i 0.761230i 0.924734 + 0.380615i \(0.124288\pi\)
−0.924734 + 0.380615i \(0.875712\pi\)
\(182\) 0 0
\(183\) −17.6158 + 2.08271i −1.30219 + 0.153959i
\(184\) 0 0
\(185\) 10.9622i 0.805954i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.87566 1.79664i 0.354652 0.130686i
\(190\) 0 0
\(191\) 13.4034 0.969834 0.484917 0.874560i \(-0.338850\pi\)
0.484917 + 0.874560i \(0.338850\pi\)
\(192\) 0 0
\(193\) 12.2077 0.878726 0.439363 0.898309i \(-0.355204\pi\)
0.439363 + 0.898309i \(0.355204\pi\)
\(194\) 0 0
\(195\) −32.1181 + 3.79733i −2.30003 + 0.271932i
\(196\) 0 0
\(197\) −10.9622 −0.781021 −0.390511 0.920598i \(-0.627702\pi\)
−0.390511 + 0.920598i \(0.627702\pi\)
\(198\) 0 0
\(199\) 7.66914i 0.543651i −0.962347 0.271826i \(-0.912373\pi\)
0.962347 0.271826i \(-0.0876274\pi\)
\(200\) 0 0
\(201\) 1.83457 + 15.5170i 0.129401 + 1.09448i
\(202\) 0 0
\(203\) 4.08188i 0.286492i
\(204\) 0 0
\(205\) 20.4826i 1.43057i
\(206\) 0 0
\(207\) 13.2876 3.18654i 0.923555 0.221480i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.64803 0.182298 0.0911489 0.995837i \(-0.470946\pi\)
0.0911489 + 0.995837i \(0.470946\pi\)
\(212\) 0 0
\(213\) 0.641735 + 5.42784i 0.0439709 + 0.371910i
\(214\) 0 0
\(215\) −33.8323 −2.30734
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.22019 10.3204i −0.0824525 0.697390i
\(220\) 0 0
\(221\) 17.1281 1.15216
\(222\) 0 0
\(223\) 1.95777i 0.131102i 0.997849 + 0.0655510i \(0.0208805\pi\)
−0.997849 + 0.0655510i \(0.979120\pi\)
\(224\) 0 0
\(225\) −19.9384 + 4.78148i −1.32923 + 0.318765i
\(226\) 0 0
\(227\) 1.58758i 0.105371i −0.998611 0.0526856i \(-0.983222\pi\)
0.998611 0.0526856i \(-0.0167781\pi\)
\(228\) 0 0
\(229\) 25.9104i 1.71221i −0.516802 0.856105i \(-0.672878\pi\)
0.516802 0.856105i \(-0.327122\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.31119i 0.413460i −0.978398 0.206730i \(-0.933718\pi\)
0.978398 0.206730i \(-0.0662821\pi\)
\(234\) 0 0
\(235\) 37.7114 2.46002
\(236\) 0 0
\(237\) −2.79841 + 0.330856i −0.181776 + 0.0214914i
\(238\) 0 0
\(239\) −1.04202 −0.0674024 −0.0337012 0.999432i \(-0.510729\pi\)
−0.0337012 + 0.999432i \(0.510729\pi\)
\(240\) 0 0
\(241\) −12.0422 −0.775708 −0.387854 0.921721i \(-0.626784\pi\)
−0.387854 + 0.921721i \(0.626784\pi\)
\(242\) 0 0
\(243\) 8.65232 + 12.9668i 0.555047 + 0.831819i
\(244\) 0 0
\(245\) −3.44014 −0.219783
\(246\) 0 0
\(247\) 23.9190i 1.52193i
\(248\) 0 0
\(249\) 21.5864 2.55217i 1.36798 0.161737i
\(250\) 0 0
\(251\) 9.03695i 0.570407i 0.958467 + 0.285204i \(0.0920613\pi\)
−0.958467 + 0.285204i \(0.907939\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 18.6725 2.20766i 1.16932 0.138249i
\(256\) 0 0
\(257\) 11.3193i 0.706081i −0.935608 0.353041i \(-0.885148\pi\)
0.935608 0.353041i \(-0.114852\pi\)
\(258\) 0 0
\(259\) −3.18654 −0.198002
\(260\) 0 0
\(261\) −11.9080 + 2.85569i −0.737087 + 0.176763i
\(262\) 0 0
\(263\) −9.46678 −0.583747 −0.291873 0.956457i \(-0.594279\pi\)
−0.291873 + 0.956457i \(0.594279\pi\)
\(264\) 0 0
\(265\) −21.7114 −1.33372
\(266\) 0 0
\(267\) 13.4278 1.58758i 0.821771 0.0971581i
\(268\) 0 0
\(269\) −5.29271 −0.322702 −0.161351 0.986897i \(-0.551585\pi\)
−0.161351 + 0.986897i \(0.551585\pi\)
\(270\) 0 0
\(271\) 23.3383i 1.41770i 0.705359 + 0.708850i \(0.250786\pi\)
−0.705359 + 0.708850i \(0.749214\pi\)
\(272\) 0 0
\(273\) −1.10383 9.33627i −0.0668068 0.565057i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.6691i 1.42214i 0.703121 + 0.711071i \(0.251789\pi\)
−0.703121 + 0.711071i \(0.748211\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0718i 1.19738i −0.800981 0.598690i \(-0.795688\pi\)
0.800981 0.598690i \(-0.204312\pi\)
\(282\) 0 0
\(283\) 12.0759 0.717836 0.358918 0.933369i \(-0.383146\pi\)
0.358918 + 0.933369i \(0.383146\pi\)
\(284\) 0 0
\(285\) 3.08295 + 26.0759i 0.182618 + 1.54460i
\(286\) 0 0
\(287\) 5.95400 0.351454
\(288\) 0 0
\(289\) 7.04223 0.414249
\(290\) 0 0
\(291\) −3.26242 27.5938i −0.191246 1.61758i
\(292\) 0 0
\(293\) −15.3481 −0.896648 −0.448324 0.893871i \(-0.647979\pi\)
−0.448324 + 0.893871i \(0.647979\pi\)
\(294\) 0 0
\(295\) 2.20766i 0.128535i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.7227i 1.42975i
\(300\) 0 0
\(301\) 9.83457i 0.566855i
\(302\) 0 0
\(303\) 1.98307 + 16.7730i 0.113924 + 0.963582i
\(304\) 0 0
\(305\) 35.2315i 2.01735i
\(306\) 0 0
\(307\) −24.5585 −1.40163 −0.700813 0.713345i \(-0.747179\pi\)
−0.700813 + 0.713345i \(0.747179\pi\)
\(308\) 0 0
\(309\) 2.79841 0.330856i 0.159196 0.0188217i
\(310\) 0 0
\(311\) −2.79841 −0.158683 −0.0793415 0.996847i \(-0.525282\pi\)
−0.0793415 + 0.996847i \(0.525282\pi\)
\(312\) 0 0
\(313\) 23.2961 1.31677 0.658386 0.752681i \(-0.271240\pi\)
0.658386 + 0.752681i \(0.271240\pi\)
\(314\) 0 0
\(315\) −2.40673 10.0359i −0.135604 0.565458i
\(316\) 0 0
\(317\) −0.714374 −0.0401232 −0.0200616 0.999799i \(-0.506386\pi\)
−0.0200616 + 0.999799i \(0.506386\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 11.8346 1.39920i 0.660541 0.0780959i
\(322\) 0 0
\(323\) 13.9058i 0.773742i
\(324\) 0 0
\(325\) 37.0970i 2.05777i
\(326\) 0 0
\(327\) 22.0401 2.60580i 1.21882 0.144101i
\(328\) 0 0
\(329\) 10.9622i 0.604363i
\(330\) 0 0
\(331\) −11.8768 −0.652808 −0.326404 0.945230i \(-0.605837\pi\)
−0.326404 + 0.945230i \(0.605837\pi\)
\(332\) 0 0
\(333\) −2.22931 9.29606i −0.122165 0.509421i
\(334\) 0 0
\(335\) 31.0339 1.69556
\(336\) 0 0
\(337\) 11.4615 0.624347 0.312173 0.950025i \(-0.398943\pi\)
0.312173 + 0.950025i \(0.398943\pi\)
\(338\) 0 0
\(339\) 2.40673 0.284548i 0.130716 0.0154545i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −3.18654 26.9520i −0.171558 1.45105i
\(346\) 0 0
\(347\) 19.3574i 1.03916i 0.854422 + 0.519579i \(0.173911\pi\)
−0.854422 + 0.519579i \(0.826089\pi\)
\(348\) 0 0
\(349\) 3.86821i 0.207061i 0.994626 + 0.103530i \(0.0330139\pi\)
−0.994626 + 0.103530i \(0.966986\pi\)
\(350\) 0 0
\(351\) 26.4643 9.75186i 1.41256 0.520516i
\(352\) 0 0
\(353\) 7.80657i 0.415502i 0.978182 + 0.207751i \(0.0666143\pi\)
−0.978182 + 0.207751i \(0.933386\pi\)
\(354\) 0 0
\(355\) 10.8557 0.576160
\(356\) 0 0
\(357\) 0.641735 + 5.42784i 0.0339642 + 0.287272i
\(358\) 0 0
\(359\) −17.3695 −0.916728 −0.458364 0.888765i \(-0.651564\pi\)
−0.458364 + 0.888765i \(0.651564\pi\)
\(360\) 0 0
\(361\) 0.419255 0.0220661
\(362\) 0 0
\(363\) 2.23701 + 18.9208i 0.117412 + 0.993083i
\(364\) 0 0
\(365\) −20.6408 −1.08039
\(366\) 0 0
\(367\) 17.2961i 0.902847i 0.892310 + 0.451423i \(0.149084\pi\)
−0.892310 + 0.451423i \(0.850916\pi\)
\(368\) 0 0
\(369\) 4.16543 + 17.3695i 0.216844 + 0.904221i
\(370\) 0 0
\(371\) 6.31119i 0.327660i
\(372\) 0 0
\(373\) 33.2961i 1.72400i 0.506904 + 0.862002i \(0.330790\pi\)
−0.506904 + 0.862002i \(0.669210\pi\)
\(374\) 0 0
\(375\) 1.28347 + 10.8557i 0.0662781 + 0.560585i
\(376\) 0 0
\(377\) 22.1558i 1.14108i
\(378\) 0 0
\(379\) −31.5459 −1.62041 −0.810203 0.586149i \(-0.800643\pi\)
−0.810203 + 0.586149i \(0.800643\pi\)
\(380\) 0 0
\(381\) 17.5579 2.07587i 0.899518 0.106350i
\(382\) 0 0
\(383\) 4.65097 0.237654 0.118827 0.992915i \(-0.462087\pi\)
0.118827 + 0.992915i \(0.462087\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 28.6903 6.88028i 1.45841 0.349744i
\(388\) 0 0
\(389\) 4.08188 0.206959 0.103480 0.994632i \(-0.467002\pi\)
0.103480 + 0.994632i \(0.467002\pi\)
\(390\) 0 0
\(391\) 14.3731i 0.726878i
\(392\) 0 0
\(393\) 2.08271 0.246240i 0.105059 0.0124211i
\(394\) 0 0
\(395\) 5.59681i 0.281606i
\(396\) 0 0
\(397\) 6.98747i 0.350691i 0.984507 + 0.175346i \(0.0561043\pi\)
−0.984507 + 0.175346i \(0.943896\pi\)
\(398\) 0 0
\(399\) −7.57988 + 0.896171i −0.379469 + 0.0448647i
\(400\) 0 0
\(401\) 4.91198i 0.245293i −0.992450 0.122646i \(-0.960862\pi\)
0.992450 0.122646i \(-0.0391381\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 27.5938 14.0422i 1.37115 0.697764i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −30.9652 −1.53113 −0.765565 0.643358i \(-0.777541\pi\)
−0.765565 + 0.643358i \(0.777541\pi\)
\(410\) 0 0
\(411\) 12.8135 1.51494i 0.632041 0.0747263i
\(412\) 0 0
\(413\) −0.641735 −0.0315777
\(414\) 0 0
\(415\) 43.1729i 2.11927i
\(416\) 0 0
\(417\) 1.47692 + 12.4919i 0.0723249 + 0.611729i
\(418\) 0 0
\(419\) 20.7135i 1.01192i 0.862557 + 0.505960i \(0.168862\pi\)
−0.862557 + 0.505960i \(0.831138\pi\)
\(420\) 0 0
\(421\) 14.0422i 0.684376i 0.939631 + 0.342188i \(0.111168\pi\)
−0.939631 + 0.342188i \(0.888832\pi\)
\(422\) 0 0
\(423\) −31.9798 + 7.66914i −1.55491 + 0.372887i
\(424\) 0 0
\(425\) 21.5671i 1.04616i
\(426\) 0 0
\(427\) −10.2413 −0.495611
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −29.9919 −1.44466 −0.722329 0.691549i \(-0.756929\pi\)
−0.722329 + 0.691549i \(0.756929\pi\)
\(432\) 0 0
\(433\) −8.37309 −0.402385 −0.201192 0.979552i \(-0.564482\pi\)
−0.201192 + 0.979552i \(0.564482\pi\)
\(434\) 0 0
\(435\) 2.85569 + 24.1536i 0.136920 + 1.15808i
\(436\) 0 0
\(437\) −20.0718 −0.960162
\(438\) 0 0
\(439\) 12.4153i 0.592551i −0.955103 0.296275i \(-0.904255\pi\)
0.955103 0.296275i \(-0.0957446\pi\)
\(440\) 0 0
\(441\) 2.91729 0.699602i 0.138918 0.0333144i
\(442\) 0 0
\(443\) 36.9684i 1.75642i 0.478276 + 0.878210i \(0.341262\pi\)
−0.478276 + 0.878210i \(0.658738\pi\)
\(444\) 0 0
\(445\) 26.8557i 1.27308i
\(446\) 0 0
\(447\) 4.08188 + 34.5248i 0.193066 + 1.63297i
\(448\) 0 0
\(449\) 21.9243i 1.03467i −0.855782 0.517336i \(-0.826924\pi\)
0.855782 0.517336i \(-0.173076\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 23.1547 2.73758i 1.08790 0.128623i
\(454\) 0 0
\(455\) −18.6725 −0.875383
\(456\) 0 0
\(457\) 19.5459 0.914321 0.457160 0.889384i \(-0.348867\pi\)
0.457160 + 0.889384i \(0.348867\pi\)
\(458\) 0 0
\(459\) −15.3856 + 5.66945i −0.718139 + 0.264627i
\(460\) 0 0
\(461\) −10.3204 −0.480670 −0.240335 0.970690i \(-0.577257\pi\)
−0.240335 + 0.970690i \(0.577257\pi\)
\(462\) 0 0
\(463\) 39.3383i 1.82821i 0.405483 + 0.914103i \(0.367103\pi\)
−0.405483 + 0.914103i \(0.632897\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.0788i 1.20678i −0.797445 0.603392i \(-0.793815\pi\)
0.797445 0.603392i \(-0.206185\pi\)
\(468\) 0 0
\(469\) 9.02112i 0.416556i
\(470\) 0 0
\(471\) 20.2984 2.39989i 0.935302 0.110581i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 30.1181 1.38191
\(476\) 0 0
\(477\) 18.4115 4.41532i 0.843006 0.202163i
\(478\) 0 0
\(479\) −8.16375 −0.373011 −0.186506 0.982454i \(-0.559716\pi\)
−0.186506 + 0.982454i \(0.559716\pi\)
\(480\) 0 0
\(481\) −17.2961 −0.788632
\(482\) 0 0
\(483\) 7.83457 0.926283i 0.356485 0.0421473i
\(484\) 0 0
\(485\) −55.1875 −2.50594
\(486\) 0 0
\(487\) 1.87680i 0.0850460i −0.999095 0.0425230i \(-0.986460\pi\)
0.999095 0.0425230i \(-0.0135396\pi\)
\(488\) 0 0
\(489\) 0.373086 + 3.15559i 0.0168715 + 0.142701i
\(490\) 0 0
\(491\) 26.2377i 1.18409i −0.805905 0.592045i \(-0.798321\pi\)
0.805905 0.592045i \(-0.201679\pi\)
\(492\) 0 0
\(493\) 12.8807i 0.580119i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.15559i 0.141548i
\(498\) 0 0
\(499\) 12.2749 0.549502 0.274751 0.961515i \(-0.411405\pi\)
0.274751 + 0.961515i \(0.411405\pi\)
\(500\) 0 0
\(501\) −1.51494 12.8135i −0.0676824 0.572463i
\(502\) 0 0
\(503\) −33.8323 −1.50851 −0.754254 0.656582i \(-0.772001\pi\)
−0.754254 + 0.656582i \(0.772001\pi\)
\(504\) 0 0
\(505\) 33.5459 1.49278
\(506\) 0 0
\(507\) −3.34768 28.3149i −0.148676 1.25751i
\(508\) 0 0
\(509\) 31.1065 1.37877 0.689387 0.724393i \(-0.257880\pi\)
0.689387 + 0.724393i \(0.257880\pi\)
\(510\) 0 0
\(511\) 6.00000i 0.265424i
\(512\) 0 0
\(513\) −7.91729 21.4857i −0.349557 0.948618i
\(514\) 0 0
\(515\) 5.59681i 0.246625i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −3.12126 26.3999i −0.137008 1.15883i
\(520\) 0 0
\(521\) 36.2735i 1.58917i 0.607151 + 0.794586i \(0.292312\pi\)
−0.607151 + 0.794586i \(0.707688\pi\)
\(522\) 0 0
\(523\) −34.1181 −1.49188 −0.745940 0.666013i \(-0.768000\pi\)
−0.745940 + 0.666013i \(0.768000\pi\)
\(524\) 0 0
\(525\) −11.7559 + 1.38991i −0.513072 + 0.0606605i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −2.25383 −0.0979925
\(530\) 0 0
\(531\) −0.448959 1.87212i −0.0194831 0.0812433i
\(532\) 0 0
\(533\) 32.3174 1.39982
\(534\) 0 0
\(535\) 23.6691i 1.02331i
\(536\) 0 0
\(537\) −33.5459 + 3.96614i −1.44761 + 0.171152i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 36.4153i 1.56562i −0.622263 0.782808i \(-0.713786\pi\)
0.622263 0.782808i \(-0.286214\pi\)
\(542\) 0 0
\(543\) −17.6158 + 2.08271i −0.755965 + 0.0893778i
\(544\) 0 0
\(545\) 44.0801i 1.88818i
\(546\) 0 0
\(547\) 17.8346 0.762551 0.381276 0.924461i \(-0.375485\pi\)
0.381276 + 0.924461i \(0.375485\pi\)
\(548\) 0 0
\(549\) −7.16483 29.8768i −0.305788 1.27511i
\(550\) 0 0
\(551\) 17.9877 0.766303
\(552\) 0 0
\(553\) −1.62691 −0.0691834
\(554\) 0 0
\(555\) −18.8557 + 2.22931i −0.800379 + 0.0946290i
\(556\) 0 0
\(557\) −37.5374 −1.59051 −0.795256 0.606273i \(-0.792664\pi\)
−0.795256 + 0.606273i \(0.792664\pi\)
\(558\) 0 0
\(559\) 53.3805i 2.25776i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.641735i 0.0270459i −0.999909 0.0135229i \(-0.995695\pi\)
0.999909 0.0135229i \(-0.00430462\pi\)
\(564\) 0 0
\(565\) 4.81346i 0.202504i
\(566\) 0 0
\(567\) 4.08188 + 8.02112i 0.171423 + 0.336855i
\(568\) 0 0
\(569\) 0.684829i 0.0287095i −0.999897 0.0143548i \(-0.995431\pi\)
0.999897 0.0143548i \(-0.00456942\pi\)
\(570\) 0 0
\(571\) −6.16543 −0.258015 −0.129008 0.991644i \(-0.541179\pi\)
−0.129008 + 0.991644i \(0.541179\pi\)
\(572\) 0 0
\(573\) 2.72577 + 23.0548i 0.113871 + 0.963126i
\(574\) 0 0
\(575\) −31.1301 −1.29821
\(576\) 0 0
\(577\) 24.9230 1.03756 0.518778 0.854909i \(-0.326387\pi\)
0.518778 + 0.854909i \(0.326387\pi\)
\(578\) 0 0
\(579\) 2.48260 + 20.9980i 0.103173 + 0.872649i
\(580\) 0 0
\(581\) 12.5497 0.520651
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −13.0633 54.4732i −0.540103 2.25219i
\(586\) 0 0
\(587\) 12.1730i 0.502433i −0.967931 0.251217i \(-0.919169\pi\)
0.967931 0.251217i \(-0.0808306\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.22931 18.8557i −0.0917016 0.775619i
\(592\) 0 0
\(593\) 1.30303i 0.0535090i 0.999642 + 0.0267545i \(0.00851723\pi\)
−0.999642 + 0.0267545i \(0.991483\pi\)
\(594\) 0 0
\(595\) 10.8557 0.445040
\(596\) 0 0
\(597\) 13.1915 1.55963i 0.539891 0.0638314i
\(598\) 0 0
\(599\) 37.9338 1.54993 0.774966 0.632003i \(-0.217767\pi\)
0.774966 + 0.632003i \(0.217767\pi\)
\(600\) 0 0
\(601\) −1.58468 −0.0646406 −0.0323203 0.999478i \(-0.510290\pi\)
−0.0323203 + 0.999478i \(0.510290\pi\)
\(602\) 0 0
\(603\) −26.3172 + 6.31119i −1.07172 + 0.257011i
\(604\) 0 0
\(605\) 37.8416 1.53848
\(606\) 0 0
\(607\) 37.7114i 1.53066i −0.643640 0.765329i \(-0.722576\pi\)
0.643640 0.765329i \(-0.277424\pi\)
\(608\) 0 0
\(609\) −7.02112 + 0.830108i −0.284510 + 0.0336377i
\(610\) 0 0
\(611\) 59.5009i 2.40715i
\(612\) 0 0
\(613\) 40.8979i 1.65185i 0.563779 + 0.825926i \(0.309347\pi\)
−0.563779 + 0.825926i \(0.690653\pi\)
\(614\) 0 0
\(615\) 35.2315 4.16543i 1.42067 0.167966i
\(616\) 0 0
\(617\) 38.0299i 1.53103i 0.643420 + 0.765514i \(0.277515\pi\)
−0.643420 + 0.765514i \(0.722485\pi\)
\(618\) 0 0
\(619\) −2.03364 −0.0817390 −0.0408695 0.999164i \(-0.513013\pi\)
−0.0408695 + 0.999164i \(0.513013\pi\)
\(620\) 0 0
\(621\) 8.18331 + 22.2077i 0.328385 + 0.891163i
\(622\) 0 0
\(623\) 7.80657 0.312763
\(624\) 0 0
\(625\) −12.4615 −0.498459
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0554 0.400936
\(630\) 0 0
\(631\) 23.3383i 0.929083i 0.885551 + 0.464541i \(0.153781\pi\)
−0.885551 + 0.464541i \(0.846219\pi\)
\(632\) 0 0
\(633\) 0.538514 + 4.55480i 0.0214040 + 0.181037i
\(634\) 0 0
\(635\) 35.1158i 1.39353i
\(636\) 0 0
\(637\) 5.42784i 0.215059i
\(638\) 0 0
\(639\) −9.20577 + 2.20766i −0.364175 + 0.0873336i
\(640\) 0 0
\(641\) 41.7351i 1.64844i 0.566273 + 0.824218i \(0.308385\pi\)
−0.566273 + 0.824218i \(0.691615\pi\)
\(642\) 0 0
\(643\) −0.0758724 −0.00299212 −0.00149606 0.999999i \(-0.500476\pi\)
−0.00149606 + 0.999999i \(0.500476\pi\)
\(644\) 0 0
\(645\) −6.88028 58.1940i −0.270911 2.29139i
\(646\) 0 0
\(647\) 6.54265 0.257218 0.128609 0.991695i \(-0.458949\pi\)
0.128609 + 0.991695i \(0.458949\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.569096 −0.0222704 −0.0111352 0.999938i \(-0.503545\pi\)
−0.0111352 + 0.999938i \(0.503545\pi\)
\(654\) 0 0
\(655\) 4.16543i 0.162757i
\(656\) 0 0
\(657\) 17.5037 4.19761i 0.682885 0.163764i
\(658\) 0 0
\(659\) 17.0810i 0.665381i 0.943036 + 0.332691i \(0.107956\pi\)
−0.943036 + 0.332691i \(0.892044\pi\)
\(660\) 0 0
\(661\) 6.65662i 0.258912i −0.991585 0.129456i \(-0.958677\pi\)
0.991585 0.129456i \(-0.0413232\pi\)
\(662\) 0 0
\(663\) 3.48324 + 29.4615i 0.135278 + 1.14419i
\(664\) 0 0
\(665\) 15.1598i 0.587871i
\(666\) 0 0
\(667\) −18.5921 −0.719890
\(668\) 0 0
\(669\) −3.36750 + 0.398140i −0.130195 + 0.0153930i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.0422303 −0.00162786 −0.000813928 1.00000i \(-0.500259\pi\)
−0.000813928 1.00000i \(0.500259\pi\)
\(674\) 0 0
\(675\) −12.2792 33.3231i −0.472628 1.28261i
\(676\) 0 0
\(677\) −24.0810 −0.925508 −0.462754 0.886487i \(-0.653139\pi\)
−0.462754 + 0.886487i \(0.653139\pi\)
\(678\) 0 0
\(679\) 16.0422i 0.615644i
\(680\) 0 0
\(681\) 2.73074 0.322856i 0.104642 0.0123719i
\(682\) 0 0
\(683\) 11.3389i 0.433871i −0.976186 0.216936i \(-0.930394\pi\)
0.976186 0.216936i \(-0.0696061\pi\)
\(684\) 0 0
\(685\) 25.6269i 0.979154i
\(686\) 0 0
\(687\) 44.5678 5.26926i 1.70037 0.201035i
\(688\) 0 0
\(689\) 34.2561i 1.30505i
\(690\) 0 0
\(691\) 26.4490 1.00617 0.503083 0.864238i \(-0.332199\pi\)
0.503083 + 0.864238i \(0.332199\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.9837 0.947687
\(696\) 0 0
\(697\) −18.7884 −0.711662
\(698\) 0 0
\(699\) 10.8557 1.28347i 0.410600 0.0485453i
\(700\) 0 0
\(701\) −13.1915 −0.498235 −0.249117 0.968473i \(-0.580141\pi\)
−0.249117 + 0.968473i \(0.580141\pi\)
\(702\) 0 0
\(703\) 14.0422i 0.529613i
\(704\) 0 0
\(705\) 7.66914 + 64.8662i 0.288837 + 2.44300i
\(706\) 0 0
\(707\) 9.75133i 0.366736i
\(708\) 0 0
\(709\) 18.5248i 0.695715i −0.937547 0.347857i \(-0.886909\pi\)
0.937547 0.347857i \(-0.113091\pi\)
\(710\) 0 0
\(711\) −1.13819 4.74617i −0.0426855 0.177995i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.211909 1.79234i −0.00791388 0.0669362i
\(718\) 0 0
\(719\) −10.2478 −0.382178 −0.191089 0.981573i \(-0.561202\pi\)
−0.191089 + 0.981573i \(0.561202\pi\)
\(720\) 0 0
\(721\) 1.62691 0.0605894
\(722\) 0 0
\(723\) −2.44896 20.7135i −0.0910778 0.770343i
\(724\) 0 0
\(725\) 27.8979 1.03610
\(726\) 0 0
\(727\) 41.2961i 1.53159i −0.643087 0.765793i \(-0.722347\pi\)
0.643087 0.765793i \(-0.277653\pi\)
\(728\) 0 0
\(729\) −20.5442 + 17.5196i −0.760896 + 0.648874i
\(730\) 0 0
\(731\) 31.0339i 1.14783i
\(732\) 0 0
\(733\) 13.3606i 0.493484i −0.969081 0.246742i \(-0.920640\pi\)
0.969081 0.246742i \(-0.0793599\pi\)
\(734\) 0 0
\(735\) −0.699602 5.91729i −0.0258052 0.218262i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −49.1729 −1.80885 −0.904426 0.426630i \(-0.859701\pi\)
−0.904426 + 0.426630i \(0.859701\pi\)
\(740\) 0 0
\(741\) −41.1424 + 4.86428i −1.51140 + 0.178694i
\(742\) 0 0
\(743\) 32.2683 1.18381 0.591904 0.806008i \(-0.298376\pi\)
0.591904 + 0.806008i \(0.298376\pi\)
\(744\) 0 0
\(745\) 69.0497 2.52978
\(746\) 0 0
\(747\) 8.77981 + 36.6112i 0.321237 + 1.33953i
\(748\) 0 0
\(749\) 6.88028 0.251400
\(750\) 0 0
\(751\) 49.8768i 1.82003i 0.414575 + 0.910015i \(0.363930\pi\)
−0.414575 + 0.910015i \(0.636070\pi\)
\(752\) 0 0
\(753\) −15.5442 + 1.83779i −0.566462 + 0.0669729i
\(754\) 0 0
\(755\) 46.3094i 1.68537i
\(756\) 0 0
\(757\) 2.52483i 0.0917665i −0.998947 0.0458833i \(-0.985390\pi\)
0.998947 0.0458833i \(-0.0146102\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.1895i 1.23937i −0.784851 0.619684i \(-0.787261\pi\)
0.784851 0.619684i \(-0.212739\pi\)
\(762\) 0 0
\(763\) 12.8135 0.463878
\(764\) 0 0
\(765\) 7.59466 + 31.6691i 0.274585 + 1.14500i
\(766\) 0 0
\(767\) −3.48324 −0.125772
\(768\) 0 0
\(769\) 19.2961 0.695834 0.347917 0.937525i \(-0.386889\pi\)
0.347917 + 0.937525i \(0.386889\pi\)
\(770\) 0 0
\(771\) 19.4701 2.30195i 0.701198 0.0829027i
\(772\) 0 0
\(773\) −9.18223 −0.330262 −0.165131 0.986272i \(-0.552805\pi\)
−0.165131 + 0.986272i \(0.552805\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −0.648029 5.48108i −0.0232479 0.196633i
\(778\) 0 0
\(779\) 26.2377i 0.940062i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −7.33364 19.9019i −0.262083 0.711234i
\(784\) 0 0
\(785\) 40.5969i 1.44896i
\(786\) 0 0
\(787\) −0.737584 −0.0262920 −0.0131460 0.999914i \(-0.504185\pi\)
−0.0131460 + 0.999914i \(0.504185\pi\)
\(788\) 0 0
\(789\) −1.92520 16.2835i −0.0685391 0.579709i
\(790\) 0 0
\(791\) 1.39920 0.0497499
\(792\) 0 0
\(793\) −55.5882 −1.97399
\(794\) 0 0
\(795\) −4.41532 37.3451i −0.156595 1.32449i
\(796\) 0 0
\(797\) −7.32967 −0.259630 −0.129815 0.991538i \(-0.541438\pi\)
−0.129815 + 0.991538i \(0.541438\pi\)
\(798\) 0 0
\(799\) 34.5921i 1.22378i
\(800\) 0