Properties

Label 1008.3.cg.i.145.2
Level $1008$
Weight $3$
Character 1008.145
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.cg (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: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1008.145
Dual form 1008.3.cg.i.577.2

$q$-expansion

\(f(q)\) \(=\) \(q+(4.24264 + 2.44949i) q^{5} +(-3.50000 + 6.06218i) q^{7} +O(q^{10})\) \(q+(4.24264 + 2.44949i) q^{5} +(-3.50000 + 6.06218i) q^{7} +(-8.48528 - 14.6969i) q^{11} +1.73205i q^{13} +(-4.24264 + 2.44949i) q^{17} +(-25.5000 - 14.7224i) q^{19} +(-4.24264 + 7.34847i) q^{23} +(-0.500000 - 0.866025i) q^{25} -33.9411 q^{29} +(10.5000 - 6.06218i) q^{31} +(-29.6985 + 17.1464i) q^{35} +(23.5000 - 40.7032i) q^{37} -68.5857i q^{41} -31.0000 q^{43} +(72.1249 + 41.6413i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(38.1838 + 66.1362i) q^{53} -83.1384i q^{55} +(72.1249 - 41.6413i) q^{59} +(-72.0000 - 41.5692i) q^{61} +(-4.24264 + 7.34847i) q^{65} +(-15.5000 - 26.8468i) q^{67} -59.3970 q^{71} +(-70.5000 + 40.7032i) q^{73} +118.794 q^{77} +(20.5000 - 35.5070i) q^{79} -4.89898i q^{83} -24.0000 q^{85} +(50.9117 + 29.3939i) q^{89} +(-10.5000 - 6.06218i) q^{91} +(-72.1249 - 124.924i) q^{95} -41.5692i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{7} + O(q^{10}) \) \( 4 q - 14 q^{7} - 102 q^{19} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 124 q^{43} - 98 q^{49} - 288 q^{61} - 62 q^{67} - 282 q^{73} + 82 q^{79} - 96 q^{85} - 42 q^{91} + O(q^{100}) \)

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
\(5\) 4.24264 + 2.44949i 0.848528 + 0.489898i 0.860154 0.510034i \(-0.170367\pi\)
−0.0116258 + 0.999932i \(0.503701\pi\)
\(6\) 0 0
\(7\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.48528 14.6969i −0.771389 1.33609i −0.936802 0.349861i \(-0.886229\pi\)
0.165412 0.986224i \(-0.447104\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.133235i 0.997779 + 0.0666173i \(0.0212207\pi\)
−0.997779 + 0.0666173i \(0.978779\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.24264 + 2.44949i −0.249567 + 0.144088i −0.619566 0.784945i \(-0.712691\pi\)
0.369999 + 0.929032i \(0.379358\pi\)
\(18\) 0 0
\(19\) −25.5000 14.7224i −1.34211 0.774865i −0.354989 0.934870i \(-0.615515\pi\)
−0.987116 + 0.160006i \(0.948849\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.24264 + 7.34847i −0.184463 + 0.319499i −0.943395 0.331670i \(-0.892388\pi\)
0.758933 + 0.651169i \(0.225721\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.0200000 0.0346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −33.9411 −1.17038 −0.585192 0.810895i \(-0.698981\pi\)
−0.585192 + 0.810895i \(0.698981\pi\)
\(30\) 0 0
\(31\) 10.5000 6.06218i 0.338710 0.195554i −0.320992 0.947082i \(-0.604016\pi\)
0.659701 + 0.751528i \(0.270683\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −29.6985 + 17.1464i −0.848528 + 0.489898i
\(36\) 0 0
\(37\) 23.5000 40.7032i 0.635135 1.10009i −0.351351 0.936244i \(-0.614278\pi\)
0.986486 0.163843i \(-0.0523889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 68.5857i 1.67282i −0.548103 0.836411i \(-0.684650\pi\)
0.548103 0.836411i \(-0.315350\pi\)
\(42\) 0 0
\(43\) −31.0000 −0.720930 −0.360465 0.932773i \(-0.617382\pi\)
−0.360465 + 0.932773i \(0.617382\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 72.1249 + 41.6413i 1.53457 + 0.885986i 0.999142 + 0.0414059i \(0.0131837\pi\)
0.535430 + 0.844580i \(0.320150\pi\)
\(48\) 0 0
\(49\) −24.5000 42.4352i −0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 38.1838 + 66.1362i 0.720448 + 1.24785i 0.960820 + 0.277172i \(0.0893973\pi\)
−0.240372 + 0.970681i \(0.577269\pi\)
\(54\) 0 0
\(55\) 83.1384i 1.51161i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 72.1249 41.6413i 1.22246 0.705785i 0.257015 0.966407i \(-0.417261\pi\)
0.965441 + 0.260622i \(0.0839277\pi\)
\(60\) 0 0
\(61\) −72.0000 41.5692i −1.18033 0.681463i −0.224237 0.974535i \(-0.571989\pi\)
−0.956090 + 0.293072i \(0.905322\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.24264 + 7.34847i −0.0652714 + 0.113053i
\(66\) 0 0
\(67\) −15.5000 26.8468i −0.231343 0.400698i 0.726860 0.686785i \(-0.240979\pi\)
−0.958204 + 0.286087i \(0.907645\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −59.3970 −0.836577 −0.418289 0.908314i \(-0.637370\pi\)
−0.418289 + 0.908314i \(0.637370\pi\)
\(72\) 0 0
\(73\) −70.5000 + 40.7032i −0.965753 + 0.557578i −0.897939 0.440120i \(-0.854936\pi\)
−0.0678144 + 0.997698i \(0.521603\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 118.794 1.54278
\(78\) 0 0
\(79\) 20.5000 35.5070i 0.259494 0.449456i −0.706613 0.707601i \(-0.749778\pi\)
0.966106 + 0.258144i \(0.0831110\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.89898i 0.0590238i −0.999564 0.0295119i \(-0.990605\pi\)
0.999564 0.0295119i \(-0.00939530\pi\)
\(84\) 0 0
\(85\) −24.0000 −0.282353
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 50.9117 + 29.3939i 0.572041 + 0.330268i 0.757964 0.652296i \(-0.226194\pi\)
−0.185923 + 0.982564i \(0.559527\pi\)
\(90\) 0 0
\(91\) −10.5000 6.06218i −0.115385 0.0666173i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −72.1249 124.924i −0.759209 1.31499i
\(96\) 0 0
\(97\) 41.5692i 0.428549i −0.976774 0.214274i \(-0.931261\pi\)
0.976774 0.214274i \(-0.0687387\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −152.735 + 88.1816i −1.51223 + 0.873085i −0.512331 + 0.858788i \(0.671218\pi\)
−0.999898 + 0.0142971i \(0.995449\pi\)
\(102\) 0 0
\(103\) 25.5000 + 14.7224i 0.247573 + 0.142936i 0.618652 0.785665i \(-0.287679\pi\)
−0.371080 + 0.928601i \(0.621012\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −72.1249 + 124.924i −0.674064 + 1.16751i 0.302677 + 0.953093i \(0.402120\pi\)
−0.976741 + 0.214421i \(0.931214\pi\)
\(108\) 0 0
\(109\) −84.5000 146.358i −0.775229 1.34274i −0.934665 0.355528i \(-0.884301\pi\)
0.159436 0.987208i \(-0.449032\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −59.3970 −0.525637 −0.262818 0.964845i \(-0.584652\pi\)
−0.262818 + 0.964845i \(0.584652\pi\)
\(114\) 0 0
\(115\) −36.0000 + 20.7846i −0.313043 + 0.180736i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 34.2929i 0.288175i
\(120\) 0 0
\(121\) −83.5000 + 144.626i −0.690083 + 1.19526i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 127.373i 1.01899i
\(126\) 0 0
\(127\) −209.000 −1.64567 −0.822835 0.568281i \(-0.807609\pi\)
−0.822835 + 0.568281i \(0.807609\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 50.9117 + 29.3939i 0.388639 + 0.224381i 0.681570 0.731753i \(-0.261297\pi\)
−0.292931 + 0.956133i \(0.594631\pi\)
\(132\) 0 0
\(133\) 178.500 103.057i 1.34211 0.774865i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −76.3675 132.272i −0.557427 0.965492i −0.997710 0.0676333i \(-0.978455\pi\)
0.440283 0.897859i \(-0.354878\pi\)
\(138\) 0 0
\(139\) 195.722i 1.40807i −0.710165 0.704035i \(-0.751380\pi\)
0.710165 0.704035i \(-0.248620\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 25.4558 14.6969i 0.178013 0.102776i
\(144\) 0 0
\(145\) −144.000 83.1384i −0.993103 0.573369i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −25.4558 + 44.0908i −0.170845 + 0.295912i −0.938715 0.344693i \(-0.887983\pi\)
0.767871 + 0.640605i \(0.221316\pi\)
\(150\) 0 0
\(151\) 5.00000 + 8.66025i 0.0331126 + 0.0573527i 0.882107 0.471049i \(-0.156125\pi\)
−0.848994 + 0.528402i \(0.822791\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 59.3970 0.383206
\(156\) 0 0
\(157\) 36.0000 20.7846i 0.229299 0.132386i −0.380949 0.924596i \(-0.624403\pi\)
0.610249 + 0.792210i \(0.291069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −29.6985 51.4393i −0.184463 0.319499i
\(162\) 0 0
\(163\) 43.0000 74.4782i 0.263804 0.456921i −0.703446 0.710749i \(-0.748356\pi\)
0.967250 + 0.253828i \(0.0816896\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 181.262i 1.08540i 0.839926 + 0.542701i \(0.182598\pi\)
−0.839926 + 0.542701i \(0.817402\pi\)
\(168\) 0 0
\(169\) 166.000 0.982249
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −38.1838 22.0454i −0.220715 0.127430i 0.385566 0.922680i \(-0.374006\pi\)
−0.606281 + 0.795250i \(0.707340\pi\)
\(174\) 0 0
\(175\) 7.00000 0.0400000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.24264 7.34847i −0.0237019 0.0410529i 0.853931 0.520386i \(-0.174212\pi\)
−0.877633 + 0.479333i \(0.840879\pi\)
\(180\) 0 0
\(181\) 43.3013i 0.239234i −0.992820 0.119617i \(-0.961833\pi\)
0.992820 0.119617i \(-0.0381666\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 199.404 115.126i 1.07786 0.622303i
\(186\) 0 0
\(187\) 72.0000 + 41.5692i 0.385027 + 0.222295i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 38.1838 66.1362i 0.199915 0.346263i −0.748586 0.663038i \(-0.769267\pi\)
0.948501 + 0.316775i \(0.102600\pi\)
\(192\) 0 0
\(193\) 143.500 + 248.549i 0.743523 + 1.28782i 0.950882 + 0.309555i \(0.100180\pi\)
−0.207358 + 0.978265i \(0.566487\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 127.279 0.646087 0.323044 0.946384i \(-0.395294\pi\)
0.323044 + 0.946384i \(0.395294\pi\)
\(198\) 0 0
\(199\) −180.000 + 103.923i −0.904523 + 0.522226i −0.878665 0.477439i \(-0.841565\pi\)
−0.0258579 + 0.999666i \(0.508232\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 118.794 205.757i 0.585192 1.01358i
\(204\) 0 0
\(205\) 168.000 290.985i 0.819512 1.41944i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 499.696i 2.39089i
\(210\) 0 0
\(211\) −82.0000 −0.388626 −0.194313 0.980940i \(-0.562248\pi\)
−0.194313 + 0.980940i \(0.562248\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −131.522 75.9342i −0.611730 0.353182i
\(216\) 0 0
\(217\) 84.8705i 0.391108i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.24264 7.34847i −0.0191975 0.0332510i
\(222\) 0 0
\(223\) 41.5692i 0.186409i −0.995647 0.0932045i \(-0.970289\pi\)
0.995647 0.0932045i \(-0.0297110\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −330.926 + 191.060i −1.45782 + 0.841675i −0.998904 0.0468029i \(-0.985097\pi\)
−0.458920 + 0.888478i \(0.651763\pi\)
\(228\) 0 0
\(229\) −70.5000 40.7032i −0.307860 0.177743i 0.338108 0.941107i \(-0.390213\pi\)
−0.645969 + 0.763364i \(0.723546\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −114.551 + 198.409i −0.491636 + 0.851539i −0.999954 0.00963059i \(-0.996934\pi\)
0.508317 + 0.861170i \(0.330268\pi\)
\(234\) 0 0
\(235\) 204.000 + 353.338i 0.868085 + 1.50357i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 67.8823 0.284026 0.142013 0.989865i \(-0.454642\pi\)
0.142013 + 0.989865i \(0.454642\pi\)
\(240\) 0 0
\(241\) −396.000 + 228.631i −1.64315 + 0.948675i −0.663451 + 0.748220i \(0.730909\pi\)
−0.979703 + 0.200455i \(0.935758\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 240.050i 0.979796i
\(246\) 0 0
\(247\) 25.5000 44.1673i 0.103239 0.178815i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 347.828i 1.38577i −0.721050 0.692884i \(-0.756340\pi\)
0.721050 0.692884i \(-0.243660\pi\)
\(252\) 0 0
\(253\) 144.000 0.569170
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 140.007 + 80.8332i 0.544775 + 0.314526i 0.747012 0.664811i \(-0.231488\pi\)
−0.202237 + 0.979337i \(0.564821\pi\)
\(258\) 0 0
\(259\) 164.500 + 284.922i 0.635135 + 1.10009i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 127.279 + 220.454i 0.483951 + 0.838228i 0.999830 0.0184332i \(-0.00586781\pi\)
−0.515879 + 0.856662i \(0.672534\pi\)
\(264\) 0 0
\(265\) 374.123i 1.41178i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.9706 9.79796i 0.0630876 0.0364236i −0.468124 0.883663i \(-0.655070\pi\)
0.531212 + 0.847239i \(0.321737\pi\)
\(270\) 0 0
\(271\) −36.0000 20.7846i −0.132841 0.0766960i 0.432107 0.901823i \(-0.357770\pi\)
−0.564948 + 0.825127i \(0.691104\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.48528 + 14.6969i −0.0308556 + 0.0534434i
\(276\) 0 0
\(277\) 168.500 + 291.851i 0.608303 + 1.05361i 0.991520 + 0.129954i \(0.0414829\pi\)
−0.383217 + 0.923658i \(0.625184\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 246.073 0.875705 0.437853 0.899047i \(-0.355739\pi\)
0.437853 + 0.899047i \(0.355739\pi\)
\(282\) 0 0
\(283\) 169.500 97.8609i 0.598940 0.345798i −0.169685 0.985498i \(-0.554275\pi\)
0.768624 + 0.639700i \(0.220942\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 415.779 + 240.050i 1.44871 + 0.836411i
\(288\) 0 0
\(289\) −132.500 + 229.497i −0.458478 + 0.794106i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 97.9796i 0.334401i −0.985923 0.167201i \(-0.946527\pi\)
0.985923 0.167201i \(-0.0534728\pi\)
\(294\) 0 0
\(295\) 408.000 1.38305
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.7279 7.34847i −0.0425683 0.0245768i
\(300\) 0 0
\(301\) 108.500 187.928i 0.360465 0.624344i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −203.647 352.727i −0.667694 1.15648i
\(306\) 0 0
\(307\) 71.0141i 0.231316i −0.993289 0.115658i \(-0.963102\pi\)
0.993289 0.115658i \(-0.0368977\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −186.676 + 107.778i −0.600245 + 0.346552i −0.769138 0.639083i \(-0.779314\pi\)
0.168893 + 0.985634i \(0.445981\pi\)
\(312\) 0 0
\(313\) 253.500 + 146.358i 0.809904 + 0.467598i 0.846923 0.531716i \(-0.178453\pi\)
−0.0370184 + 0.999315i \(0.511786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 118.794 205.757i 0.374744 0.649076i −0.615544 0.788102i \(-0.711064\pi\)
0.990289 + 0.139026i \(0.0443972\pi\)
\(318\) 0 0
\(319\) 288.000 + 498.831i 0.902821 + 1.56373i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 144.250 0.446594
\(324\) 0 0
\(325\) 1.50000 0.866025i 0.00461538 0.00266469i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −504.874 + 291.489i −1.53457 + 0.885986i
\(330\) 0 0
\(331\) −92.5000 + 160.215i −0.279456 + 0.484032i −0.971250 0.238063i \(-0.923488\pi\)
0.691794 + 0.722095i \(0.256821\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 151.868i 0.453338i
\(336\) 0 0
\(337\) −359.000 −1.06528 −0.532641 0.846341i \(-0.678800\pi\)
−0.532641 + 0.846341i \(0.678800\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −178.191 102.879i −0.522554 0.301697i
\(342\) 0 0
\(343\) 343.000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 233.345 + 404.166i 0.672465 + 1.16474i 0.977203 + 0.212307i \(0.0680976\pi\)
−0.304738 + 0.952436i \(0.598569\pi\)
\(348\) 0 0
\(349\) 581.969i 1.66753i −0.552117 0.833767i \(-0.686180\pi\)
0.552117 0.833767i \(-0.313820\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 250.316 144.520i 0.709110 0.409405i −0.101621 0.994823i \(-0.532403\pi\)
0.810731 + 0.585418i \(0.199070\pi\)
\(354\) 0 0
\(355\) −252.000 145.492i −0.709859 0.409837i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 169.706 293.939i 0.472718 0.818771i −0.526795 0.849992i \(-0.676606\pi\)
0.999512 + 0.0312215i \(0.00993973\pi\)
\(360\) 0 0
\(361\) 253.000 + 438.209i 0.700831 + 1.21387i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −398.808 −1.09263
\(366\) 0 0
\(367\) −133.500 + 77.0763i −0.363760 + 0.210017i −0.670729 0.741703i \(-0.734019\pi\)
0.306969 + 0.951720i \(0.400685\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −534.573 −1.44090
\(372\) 0 0
\(373\) 144.500 250.281i 0.387399 0.670996i −0.604699 0.796454i \(-0.706707\pi\)
0.992099 + 0.125458i \(0.0400401\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 58.7878i 0.155936i
\(378\) 0 0
\(379\) −7.00000 −0.0184697 −0.00923483 0.999957i \(-0.502940\pi\)
−0.00923483 + 0.999957i \(0.502940\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −428.507 247.398i −1.11882 0.645949i −0.177718 0.984081i \(-0.556871\pi\)
−0.941099 + 0.338132i \(0.890205\pi\)
\(384\) 0 0
\(385\) 504.000 + 290.985i 1.30909 + 0.755804i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 114.551 + 198.409i 0.294476 + 0.510048i 0.974863 0.222805i \(-0.0715214\pi\)
−0.680387 + 0.732853i \(0.738188\pi\)
\(390\) 0 0
\(391\) 41.5692i 0.106315i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 173.948 100.429i 0.440375 0.254251i
\(396\) 0 0
\(397\) 70.5000 + 40.7032i 0.177582 + 0.102527i 0.586156 0.810198i \(-0.300641\pi\)
−0.408574 + 0.912725i \(0.633974\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 46.6690 80.8332i 0.116382 0.201579i −0.801950 0.597392i \(-0.796204\pi\)
0.918331 + 0.395813i \(0.129537\pi\)
\(402\) 0 0
\(403\) 10.5000 + 18.1865i 0.0260546 + 0.0451279i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −797.616 −1.95975
\(408\) 0 0
\(409\) −361.500 + 208.712i −0.883863 + 0.510299i −0.871930 0.489630i \(-0.837132\pi\)
−0.0119329 + 0.999929i \(0.503798\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 582.979i 1.41157i
\(414\) 0 0
\(415\) 12.0000 20.7846i 0.0289157 0.0500834i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 19.5959i 0.0467683i 0.999727 + 0.0233842i \(0.00744408\pi\)
−0.999727 + 0.0233842i \(0.992556\pi\)
\(420\) 0 0
\(421\) 407.000 0.966746 0.483373 0.875415i \(-0.339412\pi\)
0.483373 + 0.875415i \(0.339412\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24264 + 2.44949i 0.00998268 + 0.00576351i
\(426\) 0 0
\(427\) 504.000 290.985i 1.18033 0.681463i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −80.6102 139.621i −0.187031 0.323946i 0.757228 0.653150i \(-0.226553\pi\)
−0.944259 + 0.329204i \(0.893220\pi\)
\(432\) 0 0
\(433\) 168.009i 0.388011i 0.981000 + 0.194006i \(0.0621480\pi\)
−0.981000 + 0.194006i \(0.937852\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 216.375 124.924i 0.495137 0.285867i
\(438\) 0 0
\(439\) 468.000 + 270.200i 1.06606 + 0.615490i 0.927102 0.374809i \(-0.122292\pi\)
0.138957 + 0.990298i \(0.455625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 63.6396 110.227i 0.143656 0.248819i −0.785215 0.619224i \(-0.787447\pi\)
0.928871 + 0.370404i \(0.120781\pi\)
\(444\) 0 0
\(445\) 144.000 + 249.415i 0.323596 + 0.560484i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 110.309 0.245676 0.122838 0.992427i \(-0.460800\pi\)
0.122838 + 0.992427i \(0.460800\pi\)
\(450\) 0 0
\(451\) −1008.00 + 581.969i −2.23503 + 1.29040i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −29.6985 51.4393i −0.0652714 0.113053i
\(456\) 0 0
\(457\) −12.5000 + 21.6506i −0.0273523 + 0.0473756i −0.879378 0.476125i \(-0.842041\pi\)
0.852025 + 0.523501i \(0.175374\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 78.3837i 0.170030i 0.996380 + 0.0850148i \(0.0270938\pi\)
−0.996380 + 0.0850148i \(0.972906\pi\)
\(462\) 0 0
\(463\) −521.000 −1.12527 −0.562635 0.826705i \(-0.690212\pi\)
−0.562635 + 0.826705i \(0.690212\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 190.919 + 110.227i 0.408820 + 0.236032i 0.690283 0.723540i \(-0.257486\pi\)
−0.281463 + 0.959572i \(0.590820\pi\)
\(468\) 0 0
\(469\) 217.000 0.462687
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 263.044 + 455.605i 0.556118 + 0.963224i
\(474\) 0 0
\(475\) 29.4449i 0.0619892i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 759.433 438.459i 1.58545 0.915363i 0.591412 0.806370i \(-0.298571\pi\)
0.994043 0.108993i \(-0.0347626\pi\)
\(480\) 0 0
\(481\) 70.5000 + 40.7032i 0.146570 + 0.0846220i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 101.823 176.363i 0.209945 0.363636i
\(486\) 0 0
\(487\) 63.5000 + 109.985i 0.130390 + 0.225842i 0.923827 0.382810i \(-0.125044\pi\)
−0.793437 + 0.608653i \(0.791710\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 627.911 1.27884 0.639420 0.768857i \(-0.279174\pi\)
0.639420 + 0.768857i \(0.279174\pi\)
\(492\) 0 0
\(493\) 144.000 83.1384i 0.292089 0.168638i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 207.889 360.075i 0.418289 0.724497i
\(498\) 0 0
\(499\) −116.500 + 201.784i −0.233467 + 0.404377i −0.958826 0.283994i \(-0.908340\pi\)
0.725359 + 0.688371i \(0.241674\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 538.888i 1.07135i 0.844425 + 0.535674i \(0.179942\pi\)
−0.844425 + 0.535674i \(0.820058\pi\)
\(504\) 0 0
\(505\) −864.000 −1.71089
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 275.772 + 159.217i 0.541791 + 0.312803i 0.745805 0.666165i \(-0.232065\pi\)
−0.204013 + 0.978968i \(0.565399\pi\)
\(510\) 0 0
\(511\) 569.845i 1.11516i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 72.1249 + 124.924i 0.140048 + 0.242571i
\(516\) 0 0
\(517\) 1413.35i 2.73376i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −492.146 + 284.141i −0.944619 + 0.545376i −0.891405 0.453207i \(-0.850280\pi\)
−0.0532135 + 0.998583i \(0.516946\pi\)
\(522\) 0 0
\(523\) −457.500 264.138i −0.874761 0.505043i −0.00583355 0.999983i \(-0.501857\pi\)
−0.868927 + 0.494939i \(0.835190\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −29.6985 + 51.4393i −0.0563539 + 0.0976078i
\(528\) 0 0
\(529\) 228.500 + 395.774i 0.431947 + 0.748154i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 118.794 0.222878
\(534\) 0 0
\(535\) −612.000 + 353.338i −1.14393 + 0.660446i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −415.779 + 720.150i −0.771389 + 1.33609i
\(540\) 0 0
\(541\) −167.500 + 290.119i −0.309612 + 0.536263i −0.978277 0.207300i \(-0.933532\pi\)
0.668666 + 0.743563i \(0.266866\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 827.928i 1.51913i
\(546\) 0 0
\(547\) 658.000 1.20293 0.601463 0.798901i \(-0.294585\pi\)
0.601463 + 0.798901i \(0.294585\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 865.499 + 499.696i 1.57078 + 0.906889i
\(552\) 0 0
\(553\) 143.500 + 248.549i 0.259494 + 0.449456i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 135.765 + 235.151i 0.243742 + 0.422174i 0.961777 0.273833i \(-0.0882915\pi\)
−0.718035 + 0.696007i \(0.754958\pi\)
\(558\) 0 0
\(559\) 53.6936i 0.0960529i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.7279 7.34847i 0.0226073 0.0130523i −0.488654 0.872478i \(-0.662512\pi\)
0.511261 + 0.859425i \(0.329179\pi\)
\(564\) 0 0
\(565\) −252.000 145.492i −0.446018 0.257508i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 424.264 734.847i 0.745631 1.29147i −0.204268 0.978915i \(-0.565481\pi\)
0.949899 0.312556i \(-0.101185\pi\)
\(570\) 0 0
\(571\) 224.500 + 388.845i 0.393170 + 0.680990i 0.992866 0.119238i \(-0.0380451\pi\)
−0.599696 + 0.800228i \(0.704712\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.48528 0.0147570
\(576\) 0 0
\(577\) −253.500 + 146.358i −0.439341 + 0.253654i −0.703318 0.710875i \(-0.748299\pi\)
0.263977 + 0.964529i \(0.414966\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 29.6985 + 17.1464i 0.0511162 + 0.0295119i
\(582\) 0 0
\(583\) 648.000 1122.37i 1.11149 1.92516i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 529.090i 0.901345i −0.892689 0.450673i \(-0.851184\pi\)
0.892689 0.450673i \(-0.148816\pi\)
\(588\) 0 0
\(589\) −357.000 −0.606112
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −907.925 524.191i −1.53107 0.883964i −0.999313 0.0370681i \(-0.988198\pi\)
−0.531758 0.846896i \(-0.678469\pi\)
\(594\) 0 0
\(595\) 84.0000 145.492i 0.141176 0.244525i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −322.441 558.484i −0.538298 0.932360i −0.998996 0.0448028i \(-0.985734\pi\)
0.460698 0.887557i \(-0.347599\pi\)
\(600\) 0 0
\(601\) 458.993i 0.763716i −0.924221 0.381858i \(-0.875284\pi\)
0.924221 0.381858i \(-0.124716\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −708.521 + 409.065i −1.17111 + 0.676140i
\(606\) 0 0
\(607\) −910.500 525.677i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −72.1249 + 124.924i −0.118044 + 0.204458i
\(612\) 0 0
\(613\) −145.000 251.147i −0.236542 0.409702i 0.723178 0.690662i \(-0.242681\pi\)
−0.959720 + 0.280960i \(0.909347\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −729.734 −1.18271 −0.591357 0.806410i \(-0.701407\pi\)
−0.591357 + 0.806410i \(0.701407\pi\)
\(618\) 0 0
\(619\) 709.500 409.630i 1.14620 0.661761i 0.198244 0.980153i \(-0.436476\pi\)
0.947959 + 0.318392i \(0.103143\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −356.382 + 205.757i −0.572041 + 0.330268i
\(624\) 0 0
\(625\) 299.500 518.749i 0.479200 0.829999i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 230.252i 0.366060i
\(630\) 0 0
\(631\) 58.0000 0.0919176 0.0459588 0.998943i \(-0.485366\pi\)
0.0459588 + 0.998943i \(0.485366\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −886.712 511.943i −1.39640 0.806210i
\(636\) 0 0
\(637\) 73.5000 42.4352i 0.115385 0.0666173i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 479.418 + 830.377i 0.747923 + 1.29544i 0.948817 + 0.315827i \(0.102282\pi\)
−0.200894 + 0.979613i \(0.564385\pi\)
\(642\) 0 0
\(643\) 760.370i 1.18254i −0.806475 0.591268i \(-0.798628\pi\)
0.806475 0.591268i \(-0.201372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −305.470 + 176.363i −0.472133 + 0.272586i −0.717132 0.696937i \(-0.754546\pi\)
0.244999 + 0.969523i \(0.421212\pi\)
\(648\) 0 0
\(649\) −1224.00 706.677i −1.88598 1.08887i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −220.617 + 382.120i −0.337852 + 0.585177i −0.984028 0.178011i \(-0.943034\pi\)
0.646177 + 0.763188i \(0.276367\pi\)
\(654\) 0 0
\(655\) 144.000 + 249.415i 0.219847 + 0.380787i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 161.220 0.244644 0.122322 0.992490i \(-0.460966\pi\)
0.122322 + 0.992490i \(0.460966\pi\)
\(660\) 0 0
\(661\) −721.500 + 416.558i −1.09153 + 0.630194i −0.933983 0.357318i \(-0.883691\pi\)
−0.157545 + 0.987512i \(0.550358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1009.75 1.51842
\(666\) 0 0
\(667\) 144.000 249.415i 0.215892 0.373936i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1410.91i 2.10269i
\(672\) 0 0
\(673\) −263.000 −0.390788 −0.195394 0.980725i \(-0.562598\pi\)
−0.195394 + 0.980725i \(0.562598\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −432.749 249.848i −0.639216 0.369052i 0.145096 0.989418i \(-0.453651\pi\)
−0.784313 + 0.620366i \(0.786984\pi\)
\(678\) 0 0
\(679\) 252.000 + 145.492i 0.371134 + 0.214274i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −479.418 830.377i −0.701930 1.21578i −0.967788 0.251767i \(-0.918988\pi\)
0.265858 0.964012i \(-0.414345\pi\)
\(684\) 0 0
\(685\) 748.246i 1.09233i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −114.551 + 66.1362i −0.166257 + 0.0959887i
\(690\) 0 0
\(691\) −1069.50 617.476i −1.54776 0.893598i −0.998313 0.0580674i \(-0.981506\pi\)
−0.549444 0.835530i \(-0.685161\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 479.418 830.377i 0.689811 1.19479i
\(696\) 0 0
\(697\) 168.000 + 290.985i 0.241033 + 0.417481i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 975.807 1.39202 0.696011 0.718031i \(-0.254956\pi\)
0.696011 + 0.718031i \(0.254956\pi\)
\(702\) 0 0
\(703\) −1198.50 + 691.954i −1.70484 + 0.984288i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1234.54i 1.74617i
\(708\) 0 0
\(709\) 553.000 957.824i 0.779972 1.35095i −0.151986 0.988383i \(-0.548567\pi\)
0.931957 0.362568i \(-0.118100\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 102.879i 0.144290i
\(714\) 0 0
\(715\) 144.000 0.201399
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −593.970 342.929i −0.826105 0.476952i 0.0264120 0.999651i \(-0.491592\pi\)
−0.852517 + 0.522699i \(0.824925\pi\)
\(720\) 0 0
\(721\) −178.500 + 103.057i −0.247573 + 0.142936i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16.9706 + 29.3939i 0.0234077 + 0.0405433i
\(726\) 0 0
\(727\) 427.817i 0.588468i −0.955733 0.294234i \(-0.904935\pi\)
0.955733 0.294234i \(-0.0950646\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 131.522 75.9342i 0.179920 0.103877i
\(732\) 0 0
\(733\) 34.5000 + 19.9186i 0.0470668 + 0.0271741i 0.523349 0.852119i \(-0.324682\pi\)
−0.476282 + 0.879293i \(0.658016\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −263.044 + 455.605i −0.356911 + 0.618189i
\(738\) 0 0
\(739\) −243.500 421.754i −0.329499 0.570710i 0.652913 0.757433i \(-0.273547\pi\)
−0.982413 + 0.186723i \(0.940213\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 509.117 0.685218 0.342609 0.939478i \(-0.388689\pi\)
0.342609 + 0.939478i \(0.388689\pi\)
\(744\) 0 0
\(745\) −216.000 + 124.708i −0.289933 + 0.167393i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −504.874 874.468i −0.674064 1.16751i
\(750\) 0 0
\(751\) −272.500 + 471.984i −0.362850 + 0.628474i −0.988429 0.151687i \(-0.951529\pi\)
0.625579 + 0.780161i \(0.284863\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 48.9898i 0.0648871i
\(756\) 0 0
\(757\) −770.000 −1.01717 −0.508587 0.861011i \(-0.669832\pi\)
−0.508587 + 0.861011i \(0.669832\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 148.492 + 85.7321i 0.195128 + 0.112657i 0.594381 0.804184i \(-0.297397\pi\)
−0.399253 + 0.916841i \(0.630730\pi\)
\(762\) 0 0
\(763\) 1183.00 1.55046
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 72.1249 + 124.924i 0.0940351 + 0.162874i
\(768\) 0 0
\(769\) 704.945i 0.916703i −0.888771 0.458352i \(-0.848440\pi\)
0.888771 0.458352i \(-0.151560\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 797.616 460.504i 1.03185 0.595736i 0.114333 0.993443i \(-0.463527\pi\)
0.917513 + 0.397706i \(0.130194\pi\)
\(774\) 0 0
\(775\) −10.5000 6.06218i −0.0135484 0.00782216i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1009.75 + 1748.94i −1.29621 + 2.24510i
\(780\) 0 0
\(781\) 504.000 + 872.954i 0.645327 + 1.11774i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 203.647 0.259423
\(786\) 0 0
\(787\) −396.000 + 228.631i −0.503177 + 0.290509i −0.730024 0.683421i \(-0.760491\pi\)
0.226848 + 0.973930i \(0.427158\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 207.889 360.075i 0.262818 0.455215i
\(792\) 0 0
\(793\) 72.0000 124.708i 0.0907945 0.157261i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.6969i 0.0184403i 0.999957 + 0.00922016i \(0.00293491\pi\)
−0.999957 + 0.00922016i \(0.997065\pi\)
\(798\) 0 0
\(799\) −408.000 −0.510638
\(800\) 0 0