Properties

Label 1152.3.m.c.415.2
Level $1152$
Weight $3$
Character 1152.415
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 415.2
Root \(1.84258 - 0.777752i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.c.991.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-4.78830 - 4.78830i) q^{5} +10.3302 q^{7} +O(q^{10})\) \(q+(-4.78830 - 4.78830i) q^{5} +10.3302 q^{7} +(0.526169 - 0.526169i) q^{11} +(-17.2840 + 17.2840i) q^{13} -4.71650 q^{17} +(-2.53604 - 2.53604i) q^{19} -12.5864 q^{23} +20.8557i q^{25} +(-2.19683 + 2.19683i) q^{29} +28.0521i q^{31} +(-49.4644 - 49.4644i) q^{35} +(32.1128 + 32.1128i) q^{37} -23.1145i q^{41} +(4.79441 - 4.79441i) q^{43} +39.0095i q^{47} +57.7141 q^{49} +(-27.9768 - 27.9768i) q^{53} -5.03891 q^{55} +(-79.8538 + 79.8538i) q^{59} +(36.7762 - 36.7762i) q^{61} +165.522 q^{65} +(-10.9869 - 10.9869i) q^{67} +52.6605 q^{71} +67.8061i q^{73} +(5.43545 - 5.43545i) q^{77} +56.4602i q^{79} +(58.3697 + 58.3697i) q^{83} +(22.5840 + 22.5840i) q^{85} +131.566i q^{89} +(-178.548 + 178.548i) q^{91} +24.2866i q^{95} +60.9413 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 32q^{11} - 32q^{19} - 128q^{23} + 32q^{29} - 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 160q^{53} + 256q^{55} + 128q^{59} + 32q^{61} + 32q^{65} + 320q^{67} + 512q^{71} + 224q^{77} + 160q^{83} - 160q^{85} - 480q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) −4.78830 4.78830i −0.957661 0.957661i 0.0414785 0.999139i \(-0.486793\pi\)
−0.999139 + 0.0414785i \(0.986793\pi\)
\(6\) 0 0
\(7\) 10.3302 1.47575 0.737875 0.674937i \(-0.235829\pi\)
0.737875 + 0.674937i \(0.235829\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.526169 0.526169i 0.0478335 0.0478335i −0.682785 0.730619i \(-0.739232\pi\)
0.730619 + 0.682785i \(0.239232\pi\)
\(12\) 0 0
\(13\) −17.2840 + 17.2840i −1.32953 + 1.32953i −0.423761 + 0.905774i \(0.639290\pi\)
−0.905774 + 0.423761i \(0.860710\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.71650 −0.277441 −0.138721 0.990332i \(-0.544299\pi\)
−0.138721 + 0.990332i \(0.544299\pi\)
\(18\) 0 0
\(19\) −2.53604 2.53604i −0.133476 0.133476i 0.637213 0.770688i \(-0.280087\pi\)
−0.770688 + 0.637213i \(0.780087\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12.5864 −0.547236 −0.273618 0.961838i \(-0.588220\pi\)
−0.273618 + 0.961838i \(0.588220\pi\)
\(24\) 0 0
\(25\) 20.8557i 0.834229i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.19683 + 2.19683i −0.0757526 + 0.0757526i −0.743968 0.668215i \(-0.767058\pi\)
0.668215 + 0.743968i \(0.267058\pi\)
\(30\) 0 0
\(31\) 28.0521i 0.904908i 0.891788 + 0.452454i \(0.149451\pi\)
−0.891788 + 0.452454i \(0.850549\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −49.4644 49.4644i −1.41327 1.41327i
\(36\) 0 0
\(37\) 32.1128 + 32.1128i 0.867914 + 0.867914i 0.992241 0.124327i \(-0.0396773\pi\)
−0.124327 + 0.992241i \(0.539677\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 23.1145i 0.563768i −0.959449 0.281884i \(-0.909041\pi\)
0.959449 0.281884i \(-0.0909593\pi\)
\(42\) 0 0
\(43\) 4.79441 4.79441i 0.111498 0.111498i −0.649157 0.760655i \(-0.724878\pi\)
0.760655 + 0.649157i \(0.224878\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 39.0095i 0.829989i 0.909824 + 0.414994i \(0.136216\pi\)
−0.909824 + 0.414994i \(0.863784\pi\)
\(48\) 0 0
\(49\) 57.7141 1.17784
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −27.9768 27.9768i −0.527864 0.527864i 0.392071 0.919935i \(-0.371759\pi\)
−0.919935 + 0.392071i \(0.871759\pi\)
\(54\) 0 0
\(55\) −5.03891 −0.0916166
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −79.8538 + 79.8538i −1.35345 + 1.35345i −0.471691 + 0.881764i \(0.656356\pi\)
−0.881764 + 0.471691i \(0.843644\pi\)
\(60\) 0 0
\(61\) 36.7762 36.7762i 0.602888 0.602888i −0.338190 0.941078i \(-0.609815\pi\)
0.941078 + 0.338190i \(0.109815\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 165.522 2.54649
\(66\) 0 0
\(67\) −10.9869 10.9869i −0.163984 0.163984i 0.620345 0.784329i \(-0.286992\pi\)
−0.784329 + 0.620345i \(0.786992\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 52.6605 0.741697 0.370849 0.928693i \(-0.379067\pi\)
0.370849 + 0.928693i \(0.379067\pi\)
\(72\) 0 0
\(73\) 67.8061i 0.928850i 0.885612 + 0.464425i \(0.153739\pi\)
−0.885612 + 0.464425i \(0.846261\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.43545 5.43545i 0.0705903 0.0705903i
\(78\) 0 0
\(79\) 56.4602i 0.714686i 0.933973 + 0.357343i \(0.116317\pi\)
−0.933973 + 0.357343i \(0.883683\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 58.3697 + 58.3697i 0.703249 + 0.703249i 0.965107 0.261857i \(-0.0843349\pi\)
−0.261857 + 0.965107i \(0.584335\pi\)
\(84\) 0 0
\(85\) 22.5840 + 22.5840i 0.265694 + 0.265694i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 131.566i 1.47827i 0.673558 + 0.739135i \(0.264765\pi\)
−0.673558 + 0.739135i \(0.735235\pi\)
\(90\) 0 0
\(91\) −178.548 + 178.548i −1.96206 + 1.96206i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 24.2866i 0.255649i
\(96\) 0 0
\(97\) 60.9413 0.628261 0.314131 0.949380i \(-0.398287\pi\)
0.314131 + 0.949380i \(0.398287\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 109.986 + 109.986i 1.08897 + 1.08897i 0.995635 + 0.0933326i \(0.0297520\pi\)
0.0933326 + 0.995635i \(0.470248\pi\)
\(102\) 0 0
\(103\) −173.295 −1.68248 −0.841239 0.540663i \(-0.818174\pi\)
−0.841239 + 0.540663i \(0.818174\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 25.4747 25.4747i 0.238081 0.238081i −0.577974 0.816055i \(-0.696156\pi\)
0.816055 + 0.577974i \(0.196156\pi\)
\(108\) 0 0
\(109\) −33.0605 + 33.0605i −0.303307 + 0.303307i −0.842306 0.538999i \(-0.818803\pi\)
0.538999 + 0.842306i \(0.318803\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −140.159 −1.24034 −0.620171 0.784466i \(-0.712937\pi\)
−0.620171 + 0.784466i \(0.712937\pi\)
\(114\) 0 0
\(115\) 60.2677 + 60.2677i 0.524067 + 0.524067i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −48.7226 −0.409434
\(120\) 0 0
\(121\) 120.446i 0.995424i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −19.8441 + 19.8441i −0.158752 + 0.158752i
\(126\) 0 0
\(127\) 40.8458i 0.321620i −0.986985 0.160810i \(-0.948589\pi\)
0.986985 0.160810i \(-0.0514107\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 75.0168 + 75.0168i 0.572647 + 0.572647i 0.932867 0.360220i \(-0.117298\pi\)
−0.360220 + 0.932867i \(0.617298\pi\)
\(132\) 0 0
\(133\) −26.1979 26.1979i −0.196977 0.196977i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 134.028i 0.978308i 0.872197 + 0.489154i \(0.162694\pi\)
−0.872197 + 0.489154i \(0.837306\pi\)
\(138\) 0 0
\(139\) 22.8798 22.8798i 0.164603 0.164603i −0.619999 0.784602i \(-0.712867\pi\)
0.784602 + 0.619999i \(0.212867\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.1885i 0.127193i
\(144\) 0 0
\(145\) 21.0381 0.145091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.32124 9.32124i −0.0625587 0.0625587i 0.675135 0.737694i \(-0.264085\pi\)
−0.737694 + 0.675135i \(0.764085\pi\)
\(150\) 0 0
\(151\) 50.5403 0.334704 0.167352 0.985897i \(-0.446478\pi\)
0.167352 + 0.985897i \(0.446478\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 134.322 134.322i 0.866595 0.866595i
\(156\) 0 0
\(157\) 95.8844 95.8844i 0.610729 0.610729i −0.332407 0.943136i \(-0.607861\pi\)
0.943136 + 0.332407i \(0.107861\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −130.021 −0.807584
\(162\) 0 0
\(163\) −140.885 140.885i −0.864324 0.864324i 0.127513 0.991837i \(-0.459301\pi\)
−0.991837 + 0.127513i \(0.959301\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −107.849 −0.645800 −0.322900 0.946433i \(-0.604658\pi\)
−0.322900 + 0.946433i \(0.604658\pi\)
\(168\) 0 0
\(169\) 428.470i 2.53533i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −53.8845 + 53.8845i −0.311471 + 0.311471i −0.845479 0.534008i \(-0.820685\pi\)
0.534008 + 0.845479i \(0.320685\pi\)
\(174\) 0 0
\(175\) 215.445i 1.23111i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −104.178 104.178i −0.582002 0.582002i 0.353451 0.935453i \(-0.385008\pi\)
−0.935453 + 0.353451i \(0.885008\pi\)
\(180\) 0 0
\(181\) 205.498 + 205.498i 1.13535 + 1.13535i 0.989274 + 0.146073i \(0.0466635\pi\)
0.146073 + 0.989274i \(0.453336\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 307.532i 1.66233i
\(186\) 0 0
\(187\) −2.48167 + 2.48167i −0.0132710 + 0.0132710i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 248.255i 1.29977i −0.760034 0.649883i \(-0.774818\pi\)
0.760034 0.649883i \(-0.225182\pi\)
\(192\) 0 0
\(193\) −129.921 −0.673166 −0.336583 0.941654i \(-0.609271\pi\)
−0.336583 + 0.941654i \(0.609271\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 237.001 + 237.001i 1.20305 + 1.20305i 0.973234 + 0.229816i \(0.0738123\pi\)
0.229816 + 0.973234i \(0.426188\pi\)
\(198\) 0 0
\(199\) −246.508 −1.23873 −0.619366 0.785102i \(-0.712610\pi\)
−0.619366 + 0.785102i \(0.712610\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −22.6938 + 22.6938i −0.111792 + 0.111792i
\(204\) 0 0
\(205\) −110.679 + 110.679i −0.539898 + 0.539898i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.66877 −0.0127692
\(210\) 0 0
\(211\) −13.4139 13.4139i −0.0635728 0.0635728i 0.674606 0.738178i \(-0.264314\pi\)
−0.738178 + 0.674606i \(0.764314\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −45.9142 −0.213554
\(216\) 0 0
\(217\) 289.786i 1.33542i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 81.5197 81.5197i 0.368867 0.368867i
\(222\) 0 0
\(223\) 295.580i 1.32547i −0.748854 0.662735i \(-0.769396\pi\)
0.748854 0.662735i \(-0.230604\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 97.0742 + 97.0742i 0.427640 + 0.427640i 0.887824 0.460184i \(-0.152217\pi\)
−0.460184 + 0.887824i \(0.652217\pi\)
\(228\) 0 0
\(229\) −34.2565 34.2565i −0.149592 0.149592i 0.628344 0.777936i \(-0.283733\pi\)
−0.777936 + 0.628344i \(0.783733\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 62.8176i 0.269604i −0.990873 0.134802i \(-0.956960\pi\)
0.990873 0.134802i \(-0.0430398\pi\)
\(234\) 0 0
\(235\) 186.789 186.789i 0.794848 0.794848i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 355.910i 1.48916i 0.667532 + 0.744581i \(0.267351\pi\)
−0.667532 + 0.744581i \(0.732649\pi\)
\(240\) 0 0
\(241\) 66.2545 0.274915 0.137458 0.990508i \(-0.456107\pi\)
0.137458 + 0.990508i \(0.456107\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −276.352 276.352i −1.12797 1.12797i
\(246\) 0 0
\(247\) 87.6655 0.354921
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −325.395 + 325.395i −1.29640 + 1.29640i −0.365638 + 0.930757i \(0.619149\pi\)
−0.930757 + 0.365638i \(0.880851\pi\)
\(252\) 0 0
\(253\) −6.62259 + 6.62259i −0.0261762 + 0.0261762i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 312.011 1.21405 0.607026 0.794682i \(-0.292362\pi\)
0.607026 + 0.794682i \(0.292362\pi\)
\(258\) 0 0
\(259\) 331.733 + 331.733i 1.28082 + 1.28082i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −168.163 −0.639403 −0.319702 0.947518i \(-0.603583\pi\)
−0.319702 + 0.947518i \(0.603583\pi\)
\(264\) 0 0
\(265\) 267.923i 1.01103i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 212.116 212.116i 0.788535 0.788535i −0.192719 0.981254i \(-0.561731\pi\)
0.981254 + 0.192719i \(0.0617306\pi\)
\(270\) 0 0
\(271\) 173.450i 0.640037i 0.947411 + 0.320019i \(0.103689\pi\)
−0.947411 + 0.320019i \(0.896311\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.9736 + 10.9736i 0.0399041 + 0.0399041i
\(276\) 0 0
\(277\) 38.4049 + 38.4049i 0.138646 + 0.138646i 0.773023 0.634377i \(-0.218743\pi\)
−0.634377 + 0.773023i \(0.718743\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 223.573i 0.795632i −0.917465 0.397816i \(-0.869768\pi\)
0.917465 0.397816i \(-0.130232\pi\)
\(282\) 0 0
\(283\) 247.755 247.755i 0.875459 0.875459i −0.117602 0.993061i \(-0.537521\pi\)
0.993061 + 0.117602i \(0.0375206\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 238.778i 0.831980i
\(288\) 0 0
\(289\) −266.755 −0.923026
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −102.262 102.262i −0.349016 0.349016i 0.510727 0.859743i \(-0.329376\pi\)
−0.859743 + 0.510727i \(0.829376\pi\)
\(294\) 0 0
\(295\) 764.729 2.59230
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 217.543 217.543i 0.727570 0.727570i
\(300\) 0 0
\(301\) 49.5275 49.5275i 0.164543 0.164543i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −352.191 −1.15472
\(306\) 0 0
\(307\) 138.292 + 138.292i 0.450463 + 0.450463i 0.895508 0.445045i \(-0.146812\pi\)
−0.445045 + 0.895508i \(0.646812\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 205.789 0.661702 0.330851 0.943683i \(-0.392664\pi\)
0.330851 + 0.943683i \(0.392664\pi\)
\(312\) 0 0
\(313\) 223.861i 0.715209i −0.933873 0.357605i \(-0.883594\pi\)
0.933873 0.357605i \(-0.116406\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −176.488 + 176.488i −0.556744 + 0.556744i −0.928379 0.371635i \(-0.878797\pi\)
0.371635 + 0.928379i \(0.378797\pi\)
\(318\) 0 0
\(319\) 2.31180i 0.00724703i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.9612 + 11.9612i 0.0370316 + 0.0370316i
\(324\) 0 0
\(325\) −360.469 360.469i −1.10914 1.10914i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 402.978i 1.22486i
\(330\) 0 0
\(331\) −183.939 + 183.939i −0.555706 + 0.555706i −0.928082 0.372376i \(-0.878543\pi\)
0.372376 + 0.928082i \(0.378543\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 105.217i 0.314081i
\(336\) 0 0
\(337\) 12.7162 0.0377336 0.0188668 0.999822i \(-0.493994\pi\)
0.0188668 + 0.999822i \(0.493994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.7602 + 14.7602i 0.0432849 + 0.0432849i
\(342\) 0 0
\(343\) 90.0184 0.262444
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 113.546 113.546i 0.327221 0.327221i −0.524308 0.851529i \(-0.675676\pi\)
0.851529 + 0.524308i \(0.175676\pi\)
\(348\) 0 0
\(349\) −90.9653 + 90.9653i −0.260645 + 0.260645i −0.825316 0.564671i \(-0.809003\pi\)
0.564671 + 0.825316i \(0.309003\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −36.2208 −0.102609 −0.0513043 0.998683i \(-0.516338\pi\)
−0.0513043 + 0.998683i \(0.516338\pi\)
\(354\) 0 0
\(355\) −252.155 252.155i −0.710294 0.710294i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 142.121 0.395880 0.197940 0.980214i \(-0.436575\pi\)
0.197940 + 0.980214i \(0.436575\pi\)
\(360\) 0 0
\(361\) 348.137i 0.964369i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 324.676 324.676i 0.889523 0.889523i
\(366\) 0 0
\(367\) 654.218i 1.78261i −0.453404 0.891305i \(-0.649791\pi\)
0.453404 0.891305i \(-0.350209\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −289.007 289.007i −0.778995 0.778995i
\(372\) 0 0
\(373\) −335.277 335.277i −0.898867 0.898867i 0.0964690 0.995336i \(-0.469245\pi\)
−0.995336 + 0.0964690i \(0.969245\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 75.9397i 0.201432i
\(378\) 0 0
\(379\) −98.7497 + 98.7497i −0.260553 + 0.260553i −0.825279 0.564725i \(-0.808982\pi\)
0.564725 + 0.825279i \(0.308982\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 156.144i 0.407687i −0.979003 0.203844i \(-0.934657\pi\)
0.979003 0.203844i \(-0.0653434\pi\)
\(384\) 0 0
\(385\) −52.0532 −0.135203
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −391.047 391.047i −1.00526 1.00526i −0.999986 0.00527486i \(-0.998321\pi\)
−0.00527486 0.999986i \(-0.501679\pi\)
\(390\) 0 0
\(391\) 59.3639 0.151826
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 270.349 270.349i 0.684427 0.684427i
\(396\) 0 0
\(397\) −243.862 + 243.862i −0.614262 + 0.614262i −0.944054 0.329791i \(-0.893022\pi\)
0.329791 + 0.944054i \(0.393022\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 175.261 0.437059 0.218529 0.975830i \(-0.429874\pi\)
0.218529 + 0.975830i \(0.429874\pi\)
\(402\) 0 0
\(403\) −484.852 484.852i −1.20311 1.20311i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.7935 0.0830307
\(408\) 0 0
\(409\) 44.4504i 0.108681i 0.998522 + 0.0543404i \(0.0173056\pi\)
−0.998522 + 0.0543404i \(0.982694\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −824.910 + 824.910i −1.99736 + 1.99736i
\(414\) 0 0
\(415\) 558.984i 1.34695i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.9985 14.9985i −0.0357959 0.0357959i 0.688982 0.724778i \(-0.258058\pi\)
−0.724778 + 0.688982i \(0.758058\pi\)
\(420\) 0 0
\(421\) −312.907 312.907i −0.743247 0.743247i 0.229954 0.973201i \(-0.426142\pi\)
−0.973201 + 0.229954i \(0.926142\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 98.3660i 0.231449i
\(426\) 0 0
\(427\) 379.907 379.907i 0.889712 0.889712i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 532.400i 1.23527i −0.786466 0.617633i \(-0.788092\pi\)
0.786466 0.617633i \(-0.211908\pi\)
\(432\) 0 0
\(433\) 553.451 1.27818 0.639089 0.769133i \(-0.279312\pi\)
0.639089 + 0.769133i \(0.279312\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.9197 + 31.9197i 0.0730427 + 0.0730427i
\(438\) 0 0
\(439\) −645.291 −1.46991 −0.734956 0.678115i \(-0.762797\pi\)
−0.734956 + 0.678115i \(0.762797\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −315.833 + 315.833i −0.712941 + 0.712941i −0.967149 0.254208i \(-0.918185\pi\)
0.254208 + 0.967149i \(0.418185\pi\)
\(444\) 0 0
\(445\) 629.978 629.978i 1.41568 1.41568i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −218.589 −0.486835 −0.243417 0.969922i \(-0.578268\pi\)
−0.243417 + 0.969922i \(0.578268\pi\)
\(450\) 0 0
\(451\) −12.1621 12.1621i −0.0269670 0.0269670i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1709.88 3.75798
\(456\) 0 0
\(457\) 296.561i 0.648930i −0.945898 0.324465i \(-0.894816\pi\)
0.945898 0.324465i \(-0.105184\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 118.061 118.061i 0.256097 0.256097i −0.567368 0.823465i \(-0.692038\pi\)
0.823465 + 0.567368i \(0.192038\pi\)
\(462\) 0 0
\(463\) 409.453i 0.884348i 0.896929 + 0.442174i \(0.145793\pi\)
−0.896929 + 0.442174i \(0.854207\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −494.764 494.764i −1.05945 1.05945i −0.998117 0.0613343i \(-0.980464\pi\)
−0.0613343 0.998117i \(-0.519536\pi\)
\(468\) 0 0
\(469\) −113.497 113.497i −0.241999 0.241999i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.04534i 0.0106667i
\(474\) 0 0
\(475\) 52.8909 52.8909i 0.111349 0.111349i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 558.806i 1.16661i 0.812254 + 0.583305i \(0.198241\pi\)
−0.812254 + 0.583305i \(0.801759\pi\)
\(480\) 0 0
\(481\) −1110.07 −2.30784
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −291.806 291.806i −0.601661 0.601661i
\(486\) 0 0
\(487\) 361.328 0.741946 0.370973 0.928644i \(-0.379024\pi\)
0.370973 + 0.928644i \(0.379024\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 488.975 488.975i 0.995876 0.995876i −0.00411514 0.999992i \(-0.501310\pi\)
0.999992 + 0.00411514i \(0.00130989\pi\)
\(492\) 0 0
\(493\) 10.3613 10.3613i 0.0210169 0.0210169i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 543.996 1.09456
\(498\) 0 0
\(499\) −102.895 102.895i −0.206203 0.206203i 0.596448 0.802652i \(-0.296578\pi\)
−0.802652 + 0.596448i \(0.796578\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −881.975 −1.75343 −0.876715 0.481011i \(-0.840270\pi\)
−0.876715 + 0.481011i \(0.840270\pi\)
\(504\) 0 0
\(505\) 1053.29i 2.08572i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 161.639 161.639i 0.317563 0.317563i −0.530268 0.847830i \(-0.677909\pi\)
0.847830 + 0.530268i \(0.177909\pi\)
\(510\) 0 0
\(511\) 700.454i 1.37075i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 829.791 + 829.791i 1.61124 + 1.61124i
\(516\) 0 0
\(517\) 20.5256 + 20.5256i 0.0397013 + 0.0397013i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 763.931i 1.46628i 0.680078 + 0.733140i \(0.261946\pi\)
−0.680078 + 0.733140i \(0.738054\pi\)
\(522\) 0 0
\(523\) 295.573 295.573i 0.565150 0.565150i −0.365616 0.930766i \(-0.619142\pi\)
0.930766 + 0.365616i \(0.119142\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 132.308i 0.251059i
\(528\) 0 0
\(529\) −370.582 −0.700532
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 399.509 + 399.509i 0.749549 + 0.749549i
\(534\) 0 0
\(535\) −243.961 −0.456002
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 30.3673 30.3673i 0.0563401 0.0563401i
\(540\) 0 0
\(541\) −243.037 + 243.037i −0.449236 + 0.449236i −0.895100 0.445865i \(-0.852896\pi\)
0.445865 + 0.895100i \(0.352896\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 316.607 0.580931
\(546\) 0 0
\(547\) 424.574 + 424.574i 0.776187 + 0.776187i 0.979180 0.202993i \(-0.0650669\pi\)
−0.202993 + 0.979180i \(0.565067\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.1425 0.0202223
\(552\) 0 0
\(553\) 583.248i 1.05470i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −445.773 + 445.773i −0.800311 + 0.800311i −0.983144 0.182833i \(-0.941473\pi\)
0.182833 + 0.983144i \(0.441473\pi\)
\(558\) 0 0
\(559\) 165.733i 0.296481i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 529.295 + 529.295i 0.940133 + 0.940133i 0.998307 0.0581732i \(-0.0185276\pi\)
−0.0581732 + 0.998307i \(0.518528\pi\)
\(564\) 0 0
\(565\) 671.123 + 671.123i 1.18783 + 1.18783i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 346.814i 0.609516i −0.952430 0.304758i \(-0.901424\pi\)
0.952430 0.304758i \(-0.0985755\pi\)
\(570\) 0 0
\(571\) −155.711 + 155.711i −0.272699 + 0.272699i −0.830186 0.557487i \(-0.811766\pi\)
0.557487 + 0.830186i \(0.311766\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 262.499i 0.456520i
\(576\) 0 0
\(577\) 620.510 1.07541 0.537704 0.843134i \(-0.319292\pi\)
0.537704 + 0.843134i \(0.319292\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 602.974 + 602.974i 1.03782 + 1.03782i
\(582\) 0 0
\(583\) −29.4410 −0.0504992
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −561.656 + 561.656i −0.956825 + 0.956825i −0.999106 0.0422810i \(-0.986538\pi\)
0.0422810 + 0.999106i \(0.486538\pi\)
\(588\) 0 0
\(589\) 71.1413 71.1413i 0.120783 0.120783i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −851.739 −1.43632 −0.718161 0.695877i \(-0.755016\pi\)
−0.718161 + 0.695877i \(0.755016\pi\)
\(594\) 0 0
\(595\) 233.299 + 233.299i 0.392099 + 0.392099i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1001.69 −1.67228 −0.836138 0.548519i \(-0.815192\pi\)
−0.836138 + 0.548519i \(0.815192\pi\)
\(600\) 0 0
\(601\) 955.182i 1.58932i 0.607054 + 0.794661i \(0.292351\pi\)
−0.607054 + 0.794661i \(0.707649\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 576.734 576.734i 0.953279 0.953279i
\(606\) 0 0
\(607\) 291.885i 0.480865i 0.970666 + 0.240432i \(0.0772892\pi\)
−0.970666 + 0.240432i \(0.922711\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −674.238 674.238i −1.10350 1.10350i
\(612\) 0 0
\(613\) 332.933 + 332.933i 0.543121 + 0.543121i 0.924442 0.381322i \(-0.124531\pi\)
−0.381322 + 0.924442i \(0.624531\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 970.864i 1.57352i −0.617257 0.786762i \(-0.711756\pi\)
0.617257 0.786762i \(-0.288244\pi\)
\(618\) 0 0
\(619\) −696.761 + 696.761i −1.12562 + 1.12562i −0.134744 + 0.990881i \(0.543021\pi\)
−0.990881 + 0.134744i \(0.956979\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1359.11i 2.18156i
\(624\) 0 0
\(625\) 711.432 1.13829
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −151.460 151.460i −0.240795 0.240795i
\(630\) 0 0
\(631\) 377.591 0.598401 0.299200 0.954190i \(-0.403280\pi\)
0.299200 + 0.954190i \(0.403280\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −195.582 + 195.582i −0.308003 + 0.308003i
\(636\) 0 0
\(637\) −997.527 + 997.527i −1.56598 + 1.56598i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −729.200 −1.13760 −0.568799 0.822477i \(-0.692592\pi\)
−0.568799 + 0.822477i \(0.692592\pi\)
\(642\) 0 0
\(643\) 243.958 + 243.958i 0.379406 + 0.379406i 0.870888 0.491482i \(-0.163545\pi\)
−0.491482 + 0.870888i \(0.663545\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −281.594 −0.435230 −0.217615 0.976035i \(-0.569828\pi\)
−0.217615 + 0.976035i \(0.569828\pi\)
\(648\) 0 0
\(649\) 84.0331i 0.129481i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 323.704 323.704i 0.495718 0.495718i −0.414384 0.910102i \(-0.636003\pi\)
0.910102 + 0.414384i \(0.136003\pi\)
\(654\) 0 0
\(655\) 718.407i 1.09680i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −507.811 507.811i −0.770578 0.770578i 0.207629 0.978208i \(-0.433425\pi\)
−0.978208 + 0.207629i \(0.933425\pi\)
\(660\) 0 0
\(661\) −57.1593 57.1593i −0.0864741 0.0864741i 0.662547 0.749021i \(-0.269476\pi\)
−0.749021 + 0.662547i \(0.769476\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 250.887i 0.377274i
\(666\) 0 0
\(667\) 27.6502 27.6502i 0.0414546 0.0414546i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 38.7009i 0.0576765i
\(672\) 0 0
\(673\) 1110.84 1.65059 0.825293 0.564705i \(-0.191010\pi\)
0.825293 + 0.564705i \(0.191010\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 397.465 + 397.465i 0.587097 + 0.587097i 0.936844 0.349747i \(-0.113732\pi\)
−0.349747 + 0.936844i \(0.613732\pi\)
\(678\) 0 0
\(679\) 629.539 0.927156
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 238.015 238.015i 0.348485 0.348485i −0.511060 0.859545i \(-0.670747\pi\)
0.859545 + 0.511060i \(0.170747\pi\)
\(684\) 0 0
\(685\) 641.768 641.768i 0.936887 0.936887i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 967.099 1.40363
\(690\) 0 0
\(691\) −685.172 685.172i −0.991565 0.991565i 0.00839951 0.999965i \(-0.497326\pi\)
−0.999965 + 0.00839951i \(0.997326\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −219.111 −0.315267
\(696\) 0 0
\(697\) 109.019i 0.156412i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 543.074 543.074i 0.774713 0.774713i −0.204214 0.978926i \(-0.565464\pi\)
0.978926 + 0.204214i \(0.0654637\pi\)
\(702\) 0 0
\(703\) 162.879i 0.231691i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1136.18 + 1136.18i 1.60704 + 1.60704i
\(708\) 0 0
\(709\) 488.019 + 488.019i 0.688320 + 0.688320i 0.961860 0.273541i \(-0.0881948\pi\)
−0.273541 + 0.961860i \(0.588195\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 353.076i 0.495198i
\(714\) 0 0
\(715\) 87.0923 87.0923i 0.121807 0.121807i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 297.369i 0.413587i −0.978385 0.206793i \(-0.933697\pi\)
0.978385 0.206793i \(-0.0663028\pi\)
\(720\) 0 0
\(721\) −1790.18 −2.48292
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −45.8164 45.8164i −0.0631950 0.0631950i
\(726\) 0 0
\(727\) −1158.85 −1.59402 −0.797009 0.603967i \(-0.793586\pi\)
−0.797009 + 0.603967i \(0.793586\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −22.6128 + 22.6128i −0.0309341 + 0.0309341i
\(732\) 0 0
\(733\) 348.835 348.835i 0.475901 0.475901i −0.427917 0.903818i \(-0.640752\pi\)
0.903818 + 0.427917i \(0.140752\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.5619 −0.0156878
\(738\) 0 0
\(739\) 825.489 + 825.489i 1.11703 + 1.11703i 0.992174 + 0.124860i \(0.0398482\pi\)
0.124860 + 0.992174i \(0.460152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −899.725 −1.21094 −0.605468 0.795870i \(-0.707014\pi\)
−0.605468 + 0.795870i \(0.707014\pi\)
\(744\) 0 0
\(745\) 89.2659i 0.119820i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 263.160 263.160i 0.351348 0.351348i
\(750\) 0 0
\(751\) 80.4386i 0.107109i −0.998565 0.0535543i \(-0.982945\pi\)
0.998565 0.0535543i \(-0.0170550\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −242.003 242.003i −0.320533 0.320533i
\(756\) 0 0
\(757\) −233.298 233.298i −0.308187 0.308187i 0.536019 0.844206i \(-0.319928\pi\)
−0.844206 + 0.536019i \(0.819928\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 56.1906i 0.0738378i −0.999318 0.0369189i \(-0.988246\pi\)
0.999318 0.0369189i \(-0.0117543\pi\)
\(762\) 0 0
\(763\) −341.523 + 341.523i −0.447606 + 0.447606i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2760.38i 3.59893i
\(768\) 0 0
\(769\) 517.343 0.672748 0.336374 0.941728i \(-0.390799\pi\)
0.336374 + 0.941728i \(0.390799\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −523.925 523.925i −0.677781 0.677781i 0.281716 0.959498i \(-0.409096\pi\)
−0.959498 + 0.281716i \(0.909096\pi\)
\(774\) 0 0
\(775\) −585.048 −0.754900
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −58.6192 + 58.6192i −0.0752492 + 0.0752492i
\(780\) 0 0
\(781\) 27.7083 27.7083i 0.0354780 0.0354780i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −918.248 −1.16974
\(786\) 0 0
\(787\) −46.6965 46.6965i −0.0593348 0.0593348i 0.676817 0.736152i \(-0.263359\pi\)
−0.736152 + 0.676817i \(0.763359\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1447.87 −1.83044
\(792\) 0 0
\(793\) 1271.27i 1.60312i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 127.126 127.126i 0.159505 0.159505i −0.622842 0.782348i \(-0.714022\pi\)
0.782348 + 0.622842i \(0.214022\pi\)
\(798\) 0 0
\(799\) 183.988i 0.230273i
\(800\) 0