Properties

Label 912.3.o.c.721.2
Level $912$
Weight $3$
Character 912.721
Analytic conductor $24.850$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.219615408.1
Defining polynomial: \(x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.2
Root \(0.500000 - 3.19918i\) of defining polynomial
Character \(\chi\) \(=\) 912.721
Dual form 912.3.o.c.721.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} +0.556406 q^{5} +9.52587 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +0.556406 q^{5} +9.52587 q^{7} -3.00000 q^{9} +13.5259 q^{11} +10.5348i q^{13} -0.963723i q^{15} +2.63868 q^{17} +(-0.112811 + 18.9997i) q^{19} -16.4993i q^{21} +22.0212 q^{23} -24.6904 q^{25} +5.19615i q^{27} +48.7824i q^{29} -0.248288i q^{31} -23.4275i q^{33} +5.30025 q^{35} -59.3172i q^{37} +18.2468 q^{39} +69.0705i q^{41} +27.5869 q^{43} -1.66922 q^{45} -44.8855 q^{47} +41.7421 q^{49} -4.57032i q^{51} -3.85489i q^{53} +7.52587 q^{55} +(32.9084 + 0.195394i) q^{57} -59.3539i q^{59} +96.4554 q^{61} -28.5776 q^{63} +5.86162i q^{65} -10.7831i q^{67} -38.1418i q^{69} +52.6373i q^{71} +13.7362 q^{73} +42.7650i q^{75} +128.846 q^{77} -51.3224i q^{79} +9.00000 q^{81} +63.7744 q^{83} +1.46818 q^{85} +84.4936 q^{87} +44.9275i q^{89} +100.353i q^{91} -0.430048 q^{93} +(-0.0627687 + 10.5715i) q^{95} -72.6403i q^{97} -40.5776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + 26 q^{11} - 50 q^{17} + 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 72 q^{39} + 210 q^{43} + 6 q^{45} - 22 q^{47} - 36 q^{49} - 10 q^{55} + 48 q^{57} + 214 q^{61} - 6 q^{63} + 102 q^{73} + 266 q^{77} + 54 q^{81} + 404 q^{83} + 370 q^{85} + 144 q^{87} - 120 q^{93} - 358 q^{95} - 78 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 0.556406 0.111281 0.0556406 0.998451i \(-0.482280\pi\)
0.0556406 + 0.998451i \(0.482280\pi\)
\(6\) 0 0
\(7\) 9.52587 1.36084 0.680419 0.732823i \(-0.261798\pi\)
0.680419 + 0.732823i \(0.261798\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 13.5259 1.22962 0.614812 0.788674i \(-0.289232\pi\)
0.614812 + 0.788674i \(0.289232\pi\)
\(12\) 0 0
\(13\) 10.5348i 0.810370i 0.914235 + 0.405185i \(0.132793\pi\)
−0.914235 + 0.405185i \(0.867207\pi\)
\(14\) 0 0
\(15\) 0.963723i 0.0642482i
\(16\) 0 0
\(17\) 2.63868 0.155216 0.0776082 0.996984i \(-0.475272\pi\)
0.0776082 + 0.996984i \(0.475272\pi\)
\(18\) 0 0
\(19\) −0.112811 + 18.9997i −0.00593742 + 0.999982i
\(20\) 0 0
\(21\) 16.4993i 0.785680i
\(22\) 0 0
\(23\) 22.0212 0.957443 0.478722 0.877967i \(-0.341100\pi\)
0.478722 + 0.877967i \(0.341100\pi\)
\(24\) 0 0
\(25\) −24.6904 −0.987617
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 48.7824i 1.68215i 0.540917 + 0.841076i \(0.318077\pi\)
−0.540917 + 0.841076i \(0.681923\pi\)
\(30\) 0 0
\(31\) 0.248288i 0.00800930i −0.999992 0.00400465i \(-0.998725\pi\)
0.999992 0.00400465i \(-0.00127472\pi\)
\(32\) 0 0
\(33\) 23.4275i 0.709924i
\(34\) 0 0
\(35\) 5.30025 0.151436
\(36\) 0 0
\(37\) 59.3172i 1.60317i −0.597882 0.801584i \(-0.703991\pi\)
0.597882 0.801584i \(-0.296009\pi\)
\(38\) 0 0
\(39\) 18.2468 0.467867
\(40\) 0 0
\(41\) 69.0705i 1.68465i 0.538974 + 0.842323i \(0.318812\pi\)
−0.538974 + 0.842323i \(0.681188\pi\)
\(42\) 0 0
\(43\) 27.5869 0.641557 0.320778 0.947154i \(-0.396056\pi\)
0.320778 + 0.947154i \(0.396056\pi\)
\(44\) 0 0
\(45\) −1.66922 −0.0370937
\(46\) 0 0
\(47\) −44.8855 −0.955011 −0.477505 0.878629i \(-0.658459\pi\)
−0.477505 + 0.878629i \(0.658459\pi\)
\(48\) 0 0
\(49\) 41.7421 0.851881
\(50\) 0 0
\(51\) 4.57032i 0.0896142i
\(52\) 0 0
\(53\) 3.85489i 0.0727338i −0.999339 0.0363669i \(-0.988422\pi\)
0.999339 0.0363669i \(-0.0115785\pi\)
\(54\) 0 0
\(55\) 7.52587 0.136834
\(56\) 0 0
\(57\) 32.9084 + 0.195394i 0.577340 + 0.00342797i
\(58\) 0 0
\(59\) 59.3539i 1.00600i −0.864287 0.503000i \(-0.832230\pi\)
0.864287 0.503000i \(-0.167770\pi\)
\(60\) 0 0
\(61\) 96.4554 1.58124 0.790618 0.612309i \(-0.209759\pi\)
0.790618 + 0.612309i \(0.209759\pi\)
\(62\) 0 0
\(63\) −28.5776 −0.453613
\(64\) 0 0
\(65\) 5.86162i 0.0901788i
\(66\) 0 0
\(67\) 10.7831i 0.160942i −0.996757 0.0804708i \(-0.974358\pi\)
0.996757 0.0804708i \(-0.0256424\pi\)
\(68\) 0 0
\(69\) 38.1418i 0.552780i
\(70\) 0 0
\(71\) 52.6373i 0.741371i 0.928759 + 0.370685i \(0.120877\pi\)
−0.928759 + 0.370685i \(0.879123\pi\)
\(72\) 0 0
\(73\) 13.7362 0.188167 0.0940835 0.995564i \(-0.470008\pi\)
0.0940835 + 0.995564i \(0.470008\pi\)
\(74\) 0 0
\(75\) 42.7650i 0.570201i
\(76\) 0 0
\(77\) 128.846 1.67332
\(78\) 0 0
\(79\) 51.3224i 0.649651i −0.945774 0.324826i \(-0.894694\pi\)
0.945774 0.324826i \(-0.105306\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 63.7744 0.768366 0.384183 0.923257i \(-0.374483\pi\)
0.384183 + 0.923257i \(0.374483\pi\)
\(84\) 0 0
\(85\) 1.46818 0.0172726
\(86\) 0 0
\(87\) 84.4936 0.971191
\(88\) 0 0
\(89\) 44.9275i 0.504804i 0.967622 + 0.252402i \(0.0812205\pi\)
−0.967622 + 0.252402i \(0.918780\pi\)
\(90\) 0 0
\(91\) 100.353i 1.10278i
\(92\) 0 0
\(93\) −0.430048 −0.00462417
\(94\) 0 0
\(95\) −0.0627687 + 10.5715i −0.000660723 + 0.111279i
\(96\) 0 0
\(97\) 72.6403i 0.748870i −0.927253 0.374435i \(-0.877837\pi\)
0.927253 0.374435i \(-0.122163\pi\)
\(98\) 0 0
\(99\) −40.5776 −0.409875
\(100\) 0 0
\(101\) 116.783 1.15627 0.578133 0.815943i \(-0.303782\pi\)
0.578133 + 0.815943i \(0.303782\pi\)
\(102\) 0 0
\(103\) 72.3921i 0.702836i −0.936219 0.351418i \(-0.885700\pi\)
0.936219 0.351418i \(-0.114300\pi\)
\(104\) 0 0
\(105\) 9.18029i 0.0874314i
\(106\) 0 0
\(107\) 154.057i 1.43979i −0.694085 0.719893i \(-0.744191\pi\)
0.694085 0.719893i \(-0.255809\pi\)
\(108\) 0 0
\(109\) 159.670i 1.46487i −0.680839 0.732433i \(-0.738385\pi\)
0.680839 0.732433i \(-0.261615\pi\)
\(110\) 0 0
\(111\) −102.740 −0.925590
\(112\) 0 0
\(113\) 79.6420i 0.704796i 0.935850 + 0.352398i \(0.114634\pi\)
−0.935850 + 0.352398i \(0.885366\pi\)
\(114\) 0 0
\(115\) 12.2527 0.106545
\(116\) 0 0
\(117\) 31.6044i 0.270123i
\(118\) 0 0
\(119\) 25.1357 0.211224
\(120\) 0 0
\(121\) 61.9491 0.511976
\(122\) 0 0
\(123\) 119.634 0.972630
\(124\) 0 0
\(125\) −27.6480 −0.221184
\(126\) 0 0
\(127\) 30.2528i 0.238211i 0.992882 + 0.119106i \(0.0380027\pi\)
−0.992882 + 0.119106i \(0.961997\pi\)
\(128\) 0 0
\(129\) 47.7820i 0.370403i
\(130\) 0 0
\(131\) 142.842 1.09040 0.545199 0.838306i \(-0.316454\pi\)
0.545199 + 0.838306i \(0.316454\pi\)
\(132\) 0 0
\(133\) −1.07462 + 180.988i −0.00807987 + 1.36081i
\(134\) 0 0
\(135\) 2.89117i 0.0214161i
\(136\) 0 0
\(137\) −165.488 −1.20795 −0.603973 0.797005i \(-0.706416\pi\)
−0.603973 + 0.797005i \(0.706416\pi\)
\(138\) 0 0
\(139\) 169.324 1.21816 0.609079 0.793110i \(-0.291539\pi\)
0.609079 + 0.793110i \(0.291539\pi\)
\(140\) 0 0
\(141\) 77.7440i 0.551376i
\(142\) 0 0
\(143\) 142.492i 0.996450i
\(144\) 0 0
\(145\) 27.1428i 0.187192i
\(146\) 0 0
\(147\) 72.2995i 0.491833i
\(148\) 0 0
\(149\) −139.406 −0.935612 −0.467806 0.883831i \(-0.654955\pi\)
−0.467806 + 0.883831i \(0.654955\pi\)
\(150\) 0 0
\(151\) 152.742i 1.01154i −0.862669 0.505769i \(-0.831209\pi\)
0.862669 0.505769i \(-0.168791\pi\)
\(152\) 0 0
\(153\) −7.91603 −0.0517388
\(154\) 0 0
\(155\) 0.138149i 0.000891284i
\(156\) 0 0
\(157\) −295.932 −1.88492 −0.942459 0.334321i \(-0.891493\pi\)
−0.942459 + 0.334321i \(0.891493\pi\)
\(158\) 0 0
\(159\) −6.67687 −0.0419929
\(160\) 0 0
\(161\) 209.771 1.30293
\(162\) 0 0
\(163\) 112.731 0.691602 0.345801 0.938308i \(-0.387607\pi\)
0.345801 + 0.938308i \(0.387607\pi\)
\(164\) 0 0
\(165\) 13.0352i 0.0790011i
\(166\) 0 0
\(167\) 297.205i 1.77967i 0.456284 + 0.889834i \(0.349180\pi\)
−0.456284 + 0.889834i \(0.650820\pi\)
\(168\) 0 0
\(169\) 58.0179 0.343301
\(170\) 0 0
\(171\) 0.338433 56.9990i 0.00197914 0.333327i
\(172\) 0 0
\(173\) 262.849i 1.51936i −0.650299 0.759678i \(-0.725357\pi\)
0.650299 0.759678i \(-0.274643\pi\)
\(174\) 0 0
\(175\) −235.198 −1.34399
\(176\) 0 0
\(177\) −102.804 −0.580814
\(178\) 0 0
\(179\) 73.9185i 0.412953i −0.978452 0.206476i \(-0.933800\pi\)
0.978452 0.206476i \(-0.0661996\pi\)
\(180\) 0 0
\(181\) 34.8893i 0.192759i 0.995345 + 0.0963793i \(0.0307262\pi\)
−0.995345 + 0.0963793i \(0.969274\pi\)
\(182\) 0 0
\(183\) 167.066i 0.912928i
\(184\) 0 0
\(185\) 33.0044i 0.178402i
\(186\) 0 0
\(187\) 35.6904 0.190858
\(188\) 0 0
\(189\) 49.4979i 0.261893i
\(190\) 0 0
\(191\) −111.001 −0.581156 −0.290578 0.956851i \(-0.593848\pi\)
−0.290578 + 0.956851i \(0.593848\pi\)
\(192\) 0 0
\(193\) 83.5616i 0.432962i 0.976287 + 0.216481i \(0.0694579\pi\)
−0.976287 + 0.216481i \(0.930542\pi\)
\(194\) 0 0
\(195\) 10.1526 0.0520648
\(196\) 0 0
\(197\) −365.319 −1.85441 −0.927205 0.374554i \(-0.877796\pi\)
−0.927205 + 0.374554i \(0.877796\pi\)
\(198\) 0 0
\(199\) 10.6303 0.0534184 0.0267092 0.999643i \(-0.491497\pi\)
0.0267092 + 0.999643i \(0.491497\pi\)
\(200\) 0 0
\(201\) −18.6769 −0.0929197
\(202\) 0 0
\(203\) 464.695i 2.28914i
\(204\) 0 0
\(205\) 38.4312i 0.187469i
\(206\) 0 0
\(207\) −66.0636 −0.319148
\(208\) 0 0
\(209\) −1.52587 + 256.987i −0.00730080 + 1.22960i
\(210\) 0 0
\(211\) 230.194i 1.09097i 0.838122 + 0.545483i \(0.183654\pi\)
−0.838122 + 0.545483i \(0.816346\pi\)
\(212\) 0 0
\(213\) 91.1705 0.428031
\(214\) 0 0
\(215\) 15.3495 0.0713932
\(216\) 0 0
\(217\) 2.36516i 0.0108994i
\(218\) 0 0
\(219\) 23.7918i 0.108638i
\(220\) 0 0
\(221\) 27.7980i 0.125783i
\(222\) 0 0
\(223\) 263.814i 1.18302i −0.806297 0.591511i \(-0.798532\pi\)
0.806297 0.591511i \(-0.201468\pi\)
\(224\) 0 0
\(225\) 74.0712 0.329206
\(226\) 0 0
\(227\) 213.928i 0.942414i 0.882023 + 0.471207i \(0.156182\pi\)
−0.882023 + 0.471207i \(0.843818\pi\)
\(228\) 0 0
\(229\) −63.4698 −0.277161 −0.138580 0.990351i \(-0.544254\pi\)
−0.138580 + 0.990351i \(0.544254\pi\)
\(230\) 0 0
\(231\) 223.167i 0.966092i
\(232\) 0 0
\(233\) 288.449 1.23798 0.618989 0.785400i \(-0.287543\pi\)
0.618989 + 0.785400i \(0.287543\pi\)
\(234\) 0 0
\(235\) −24.9745 −0.106275
\(236\) 0 0
\(237\) −88.8931 −0.375076
\(238\) 0 0
\(239\) −279.128 −1.16790 −0.583950 0.811790i \(-0.698493\pi\)
−0.583950 + 0.811790i \(0.698493\pi\)
\(240\) 0 0
\(241\) 242.687i 1.00700i −0.863995 0.503500i \(-0.832045\pi\)
0.863995 0.503500i \(-0.167955\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 23.2256 0.0947982
\(246\) 0 0
\(247\) −200.158 1.18844i −0.810355 0.00481151i
\(248\) 0 0
\(249\) 110.460i 0.443616i
\(250\) 0 0
\(251\) 162.678 0.648118 0.324059 0.946037i \(-0.394952\pi\)
0.324059 + 0.946037i \(0.394952\pi\)
\(252\) 0 0
\(253\) 297.856 1.17730
\(254\) 0 0
\(255\) 2.54295i 0.00997237i
\(256\) 0 0
\(257\) 256.629i 0.998555i 0.866442 + 0.499277i \(0.166401\pi\)
−0.866442 + 0.499277i \(0.833599\pi\)
\(258\) 0 0
\(259\) 565.048i 2.18165i
\(260\) 0 0
\(261\) 146.347i 0.560718i
\(262\) 0 0
\(263\) −71.2027 −0.270733 −0.135366 0.990796i \(-0.543221\pi\)
−0.135366 + 0.990796i \(0.543221\pi\)
\(264\) 0 0
\(265\) 2.14488i 0.00809390i
\(266\) 0 0
\(267\) 77.8168 0.291449
\(268\) 0 0
\(269\) 219.411i 0.815653i 0.913059 + 0.407827i \(0.133713\pi\)
−0.913059 + 0.407827i \(0.866287\pi\)
\(270\) 0 0
\(271\) −323.913 −1.19525 −0.597626 0.801775i \(-0.703889\pi\)
−0.597626 + 0.801775i \(0.703889\pi\)
\(272\) 0 0
\(273\) 173.817 0.636691
\(274\) 0 0
\(275\) −333.959 −1.21440
\(276\) 0 0
\(277\) −37.4953 −0.135362 −0.0676810 0.997707i \(-0.521560\pi\)
−0.0676810 + 0.997707i \(0.521560\pi\)
\(278\) 0 0
\(279\) 0.744865i 0.00266977i
\(280\) 0 0
\(281\) 177.483i 0.631613i −0.948824 0.315806i \(-0.897725\pi\)
0.948824 0.315806i \(-0.102275\pi\)
\(282\) 0 0
\(283\) −72.4063 −0.255853 −0.127926 0.991784i \(-0.540832\pi\)
−0.127926 + 0.991784i \(0.540832\pi\)
\(284\) 0 0
\(285\) 18.3104 + 0.108719i 0.0642470 + 0.000381469i
\(286\) 0 0
\(287\) 657.956i 2.29253i
\(288\) 0 0
\(289\) −282.037 −0.975908
\(290\) 0 0
\(291\) −125.817 −0.432360
\(292\) 0 0
\(293\) 57.9819i 0.197891i 0.995093 + 0.0989453i \(0.0315469\pi\)
−0.995093 + 0.0989453i \(0.968453\pi\)
\(294\) 0 0
\(295\) 33.0249i 0.111949i
\(296\) 0 0
\(297\) 70.2825i 0.236641i
\(298\) 0 0
\(299\) 231.989i 0.775883i
\(300\) 0 0
\(301\) 262.790 0.873055
\(302\) 0 0
\(303\) 202.274i 0.667570i
\(304\) 0 0
\(305\) 53.6683 0.175962
\(306\) 0 0
\(307\) 484.604i 1.57851i −0.614062 0.789257i \(-0.710466\pi\)
0.614062 0.789257i \(-0.289534\pi\)
\(308\) 0 0
\(309\) −125.387 −0.405782
\(310\) 0 0
\(311\) −241.304 −0.775898 −0.387949 0.921681i \(-0.626816\pi\)
−0.387949 + 0.921681i \(0.626816\pi\)
\(312\) 0 0
\(313\) −313.749 −1.00239 −0.501196 0.865334i \(-0.667107\pi\)
−0.501196 + 0.865334i \(0.667107\pi\)
\(314\) 0 0
\(315\) −15.9007 −0.0504785
\(316\) 0 0
\(317\) 184.282i 0.581331i 0.956825 + 0.290665i \(0.0938767\pi\)
−0.956825 + 0.290665i \(0.906123\pi\)
\(318\) 0 0
\(319\) 659.825i 2.06842i
\(320\) 0 0
\(321\) −266.835 −0.831261
\(322\) 0 0
\(323\) −0.297672 + 50.1340i −0.000921585 + 0.155214i
\(324\) 0 0
\(325\) 260.109i 0.800334i
\(326\) 0 0
\(327\) −276.557 −0.845741
\(328\) 0 0
\(329\) −427.573 −1.29961
\(330\) 0 0
\(331\) 460.335i 1.39074i −0.718652 0.695370i \(-0.755241\pi\)
0.718652 0.695370i \(-0.244759\pi\)
\(332\) 0 0
\(333\) 177.952i 0.534389i
\(334\) 0 0
\(335\) 5.99977i 0.0179098i
\(336\) 0 0
\(337\) 493.633i 1.46479i −0.680882 0.732393i \(-0.738403\pi\)
0.680882 0.732393i \(-0.261597\pi\)
\(338\) 0 0
\(339\) 137.944 0.406914
\(340\) 0 0
\(341\) 3.35831i 0.00984843i
\(342\) 0 0
\(343\) −69.1373 −0.201567
\(344\) 0 0
\(345\) 21.2223i 0.0615140i
\(346\) 0 0
\(347\) 415.426 1.19719 0.598596 0.801051i \(-0.295725\pi\)
0.598596 + 0.801051i \(0.295725\pi\)
\(348\) 0 0
\(349\) 34.1332 0.0978030 0.0489015 0.998804i \(-0.484428\pi\)
0.0489015 + 0.998804i \(0.484428\pi\)
\(350\) 0 0
\(351\) −54.7405 −0.155956
\(352\) 0 0
\(353\) −154.833 −0.438620 −0.219310 0.975655i \(-0.570381\pi\)
−0.219310 + 0.975655i \(0.570381\pi\)
\(354\) 0 0
\(355\) 29.2877i 0.0825005i
\(356\) 0 0
\(357\) 43.5363i 0.121950i
\(358\) 0 0
\(359\) −565.544 −1.57533 −0.787665 0.616103i \(-0.788710\pi\)
−0.787665 + 0.616103i \(0.788710\pi\)
\(360\) 0 0
\(361\) −360.975 4.28674i −0.999929 0.0118746i
\(362\) 0 0
\(363\) 107.299i 0.295589i
\(364\) 0 0
\(365\) 7.64289 0.0209394
\(366\) 0 0
\(367\) −197.695 −0.538678 −0.269339 0.963045i \(-0.586805\pi\)
−0.269339 + 0.963045i \(0.586805\pi\)
\(368\) 0 0
\(369\) 207.211i 0.561548i
\(370\) 0 0
\(371\) 36.7212i 0.0989789i
\(372\) 0 0
\(373\) 198.960i 0.533404i 0.963779 + 0.266702i \(0.0859339\pi\)
−0.963779 + 0.266702i \(0.914066\pi\)
\(374\) 0 0
\(375\) 47.8878i 0.127701i
\(376\) 0 0
\(377\) −513.913 −1.36317
\(378\) 0 0
\(379\) 432.419i 1.14095i 0.821316 + 0.570474i \(0.193240\pi\)
−0.821316 + 0.570474i \(0.806760\pi\)
\(380\) 0 0
\(381\) 52.3994 0.137531
\(382\) 0 0
\(383\) 543.758i 1.41973i −0.704336 0.709867i \(-0.748755\pi\)
0.704336 0.709867i \(-0.251245\pi\)
\(384\) 0 0
\(385\) 71.6904 0.186209
\(386\) 0 0
\(387\) −82.7608 −0.213852
\(388\) 0 0
\(389\) 109.783 0.282218 0.141109 0.989994i \(-0.454933\pi\)
0.141109 + 0.989994i \(0.454933\pi\)
\(390\) 0 0
\(391\) 58.1069 0.148611
\(392\) 0 0
\(393\) 247.410i 0.629542i
\(394\) 0 0
\(395\) 28.5561i 0.0722939i
\(396\) 0 0
\(397\) −540.791 −1.36219 −0.681097 0.732193i \(-0.738497\pi\)
−0.681097 + 0.732193i \(0.738497\pi\)
\(398\) 0 0
\(399\) 313.481 + 1.86130i 0.785666 + 0.00466492i
\(400\) 0 0
\(401\) 224.500i 0.559849i 0.960022 + 0.279925i \(0.0903095\pi\)
−0.960022 + 0.279925i \(0.909691\pi\)
\(402\) 0 0
\(403\) 2.61567 0.00649049
\(404\) 0 0
\(405\) 5.00765 0.0123646
\(406\) 0 0
\(407\) 802.317i 1.97129i
\(408\) 0 0
\(409\) 397.737i 0.972463i 0.873830 + 0.486231i \(0.161629\pi\)
−0.873830 + 0.486231i \(0.838371\pi\)
\(410\) 0 0
\(411\) 286.634i 0.697408i
\(412\) 0 0
\(413\) 565.398i 1.36900i
\(414\) 0 0
\(415\) 35.4844 0.0855046
\(416\) 0 0
\(417\) 293.278i 0.703304i
\(418\) 0 0
\(419\) 769.815 1.83727 0.918634 0.395111i \(-0.129294\pi\)
0.918634 + 0.395111i \(0.129294\pi\)
\(420\) 0 0
\(421\) 515.309i 1.22401i 0.790853 + 0.612006i \(0.209637\pi\)
−0.790853 + 0.612006i \(0.790363\pi\)
\(422\) 0 0
\(423\) 134.656 0.318337
\(424\) 0 0
\(425\) −65.1501 −0.153294
\(426\) 0 0
\(427\) 918.822 2.15181
\(428\) 0 0
\(429\) 246.804 0.575301
\(430\) 0 0
\(431\) 484.686i 1.12456i −0.826946 0.562281i \(-0.809924\pi\)
0.826946 0.562281i \(-0.190076\pi\)
\(432\) 0 0
\(433\) 734.040i 1.69524i −0.530602 0.847621i \(-0.678034\pi\)
0.530602 0.847621i \(-0.321966\pi\)
\(434\) 0 0
\(435\) 47.0127 0.108075
\(436\) 0 0
\(437\) −2.48423 + 418.395i −0.00568475 + 0.957426i
\(438\) 0 0
\(439\) 108.108i 0.246260i −0.992391 0.123130i \(-0.960707\pi\)
0.992391 0.123130i \(-0.0392933\pi\)
\(440\) 0 0
\(441\) −125.226 −0.283960
\(442\) 0 0
\(443\) 519.843 1.17346 0.586730 0.809782i \(-0.300415\pi\)
0.586730 + 0.809782i \(0.300415\pi\)
\(444\) 0 0
\(445\) 24.9979i 0.0561751i
\(446\) 0 0
\(447\) 241.459i 0.540176i
\(448\) 0 0
\(449\) 810.462i 1.80504i 0.430651 + 0.902519i \(0.358284\pi\)
−0.430651 + 0.902519i \(0.641716\pi\)
\(450\) 0 0
\(451\) 934.238i 2.07148i
\(452\) 0 0
\(453\) −264.557 −0.584012
\(454\) 0 0
\(455\) 55.8371i 0.122719i
\(456\) 0 0
\(457\) 231.412 0.506373 0.253186 0.967418i \(-0.418521\pi\)
0.253186 + 0.967418i \(0.418521\pi\)
\(458\) 0 0
\(459\) 13.7110i 0.0298714i
\(460\) 0 0
\(461\) 766.366 1.66240 0.831200 0.555974i \(-0.187654\pi\)
0.831200 + 0.555974i \(0.187654\pi\)
\(462\) 0 0
\(463\) −152.091 −0.328490 −0.164245 0.986420i \(-0.552519\pi\)
−0.164245 + 0.986420i \(0.552519\pi\)
\(464\) 0 0
\(465\) −0.239281 −0.000514583
\(466\) 0 0
\(467\) −123.992 −0.265508 −0.132754 0.991149i \(-0.542382\pi\)
−0.132754 + 0.991149i \(0.542382\pi\)
\(468\) 0 0
\(469\) 102.718i 0.219016i
\(470\) 0 0
\(471\) 512.570i 1.08826i
\(472\) 0 0
\(473\) 373.137 0.788874
\(474\) 0 0
\(475\) 2.78535 469.110i 0.00586390 0.987599i
\(476\) 0 0
\(477\) 11.5647i 0.0242446i
\(478\) 0 0
\(479\) 443.502 0.925892 0.462946 0.886387i \(-0.346792\pi\)
0.462946 + 0.886387i \(0.346792\pi\)
\(480\) 0 0
\(481\) 624.895 1.29916
\(482\) 0 0
\(483\) 363.334i 0.752244i
\(484\) 0 0
\(485\) 40.4175i 0.0833350i
\(486\) 0 0
\(487\) 628.314i 1.29017i 0.764110 + 0.645086i \(0.223179\pi\)
−0.764110 + 0.645086i \(0.776821\pi\)
\(488\) 0 0
\(489\) 195.256i 0.399296i
\(490\) 0 0
\(491\) −877.289 −1.78674 −0.893370 0.449322i \(-0.851666\pi\)
−0.893370 + 0.449322i \(0.851666\pi\)
\(492\) 0 0
\(493\) 128.721i 0.261098i
\(494\) 0 0
\(495\) −22.5776 −0.0456113
\(496\) 0 0
\(497\) 501.416i 1.00889i
\(498\) 0 0
\(499\) −206.303 −0.413433 −0.206716 0.978401i \(-0.566278\pi\)
−0.206716 + 0.978401i \(0.566278\pi\)
\(500\) 0 0
\(501\) 514.773 1.02749
\(502\) 0 0
\(503\) 276.617 0.549934 0.274967 0.961454i \(-0.411333\pi\)
0.274967 + 0.961454i \(0.411333\pi\)
\(504\) 0 0
\(505\) 64.9786 0.128671
\(506\) 0 0
\(507\) 100.490i 0.198205i
\(508\) 0 0
\(509\) 728.875i 1.43197i −0.698114 0.715987i \(-0.745977\pi\)
0.698114 0.715987i \(-0.254023\pi\)
\(510\) 0 0
\(511\) 130.849 0.256065
\(512\) 0 0
\(513\) −98.7252 0.586183i −0.192447 0.00114266i
\(514\) 0 0
\(515\) 40.2793i 0.0782123i
\(516\) 0 0
\(517\) −607.115 −1.17430
\(518\) 0 0
\(519\) −455.267 −0.877201
\(520\) 0 0
\(521\) 14.2470i 0.0273455i 0.999907 + 0.0136727i \(0.00435230\pi\)
−0.999907 + 0.0136727i \(0.995648\pi\)
\(522\) 0 0
\(523\) 535.561i 1.02402i 0.858981 + 0.512008i \(0.171098\pi\)
−0.858981 + 0.512008i \(0.828902\pi\)
\(524\) 0 0
\(525\) 407.374i 0.775951i
\(526\) 0 0
\(527\) 0.655153i 0.00124317i
\(528\) 0 0
\(529\) −44.0669 −0.0833023
\(530\) 0 0
\(531\) 178.062i 0.335333i
\(532\) 0 0
\(533\) −727.644 −1.36519
\(534\) 0 0
\(535\) 85.7182i 0.160221i
\(536\) 0 0
\(537\) −128.031 −0.238418
\(538\) 0 0
\(539\) 564.599 1.04749
\(540\) 0 0
\(541\) 1000.18 1.84875 0.924376 0.381482i \(-0.124586\pi\)
0.924376 + 0.381482i \(0.124586\pi\)
\(542\) 0 0
\(543\) 60.4300 0.111289
\(544\) 0 0
\(545\) 88.8415i 0.163012i
\(546\) 0 0
\(547\) 675.400i 1.23473i −0.786675 0.617367i \(-0.788199\pi\)
0.786675 0.617367i \(-0.211801\pi\)
\(548\) 0 0
\(549\) −289.366 −0.527079
\(550\) 0 0
\(551\) −926.850 5.50320i −1.68212 0.00998765i
\(552\) 0 0
\(553\) 488.891i 0.884070i
\(554\) 0 0
\(555\) −57.1654 −0.103001
\(556\) 0 0
\(557\) −366.913 −0.658730 −0.329365 0.944203i \(-0.606835\pi\)
−0.329365 + 0.944203i \(0.606835\pi\)
\(558\) 0 0
\(559\) 290.623i 0.519898i
\(560\) 0 0
\(561\) 61.8176i 0.110192i
\(562\) 0 0
\(563\) 50.3339i 0.0894029i −0.999000 0.0447015i \(-0.985766\pi\)
0.999000 0.0447015i \(-0.0142337\pi\)
\(564\) 0 0
\(565\) 44.3132i 0.0784305i
\(566\) 0 0
\(567\) 85.7328 0.151204
\(568\) 0 0
\(569\) 61.6165i 0.108289i −0.998533 0.0541446i \(-0.982757\pi\)
0.998533 0.0541446i \(-0.0172432\pi\)
\(570\) 0 0
\(571\) −256.504 −0.449218 −0.224609 0.974449i \(-0.572111\pi\)
−0.224609 + 0.974449i \(0.572111\pi\)
\(572\) 0 0
\(573\) 192.259i 0.335530i
\(574\) 0 0
\(575\) −543.712 −0.945587
\(576\) 0 0
\(577\) 429.278 0.743983 0.371991 0.928236i \(-0.378675\pi\)
0.371991 + 0.928236i \(0.378675\pi\)
\(578\) 0 0
\(579\) 144.733 0.249970
\(580\) 0 0
\(581\) 607.506 1.04562
\(582\) 0 0
\(583\) 52.1407i 0.0894352i
\(584\) 0 0
\(585\) 17.5849i 0.0300596i
\(586\) 0 0
\(587\) −491.725 −0.837691 −0.418845 0.908058i \(-0.637565\pi\)
−0.418845 + 0.908058i \(0.637565\pi\)
\(588\) 0 0
\(589\) 4.71739 + 0.0280097i 0.00800916 + 4.75546e-5i
\(590\) 0 0
\(591\) 632.751i 1.07064i
\(592\) 0 0
\(593\) 954.134 1.60899 0.804497 0.593956i \(-0.202435\pi\)
0.804497 + 0.593956i \(0.202435\pi\)
\(594\) 0 0
\(595\) 13.9856 0.0235053
\(596\) 0 0
\(597\) 18.4122i 0.0308411i
\(598\) 0 0
\(599\) 533.310i 0.890334i −0.895448 0.445167i \(-0.853144\pi\)
0.895448 0.445167i \(-0.146856\pi\)
\(600\) 0 0
\(601\) 830.453i 1.38179i 0.722957 + 0.690893i \(0.242782\pi\)
−0.722957 + 0.690893i \(0.757218\pi\)
\(602\) 0 0
\(603\) 32.3493i 0.0536472i
\(604\) 0 0
\(605\) 34.4688 0.0569732
\(606\) 0 0
\(607\) 584.294i 0.962594i −0.876558 0.481297i \(-0.840166\pi\)
0.876558 0.481297i \(-0.159834\pi\)
\(608\) 0 0
\(609\) 804.875 1.32163
\(610\) 0 0
\(611\) 472.860i 0.773912i
\(612\) 0 0
\(613\) −152.216 −0.248314 −0.124157 0.992263i \(-0.539623\pi\)
−0.124157 + 0.992263i \(0.539623\pi\)
\(614\) 0 0
\(615\) 66.5648 0.108235
\(616\) 0 0
\(617\) 647.334 1.04916 0.524582 0.851360i \(-0.324222\pi\)
0.524582 + 0.851360i \(0.324222\pi\)
\(618\) 0 0
\(619\) 104.700 0.169145 0.0845723 0.996417i \(-0.473048\pi\)
0.0845723 + 0.996417i \(0.473048\pi\)
\(620\) 0 0
\(621\) 114.425i 0.184260i
\(622\) 0 0
\(623\) 427.974i 0.686956i
\(624\) 0 0
\(625\) 601.877 0.963003
\(626\) 0 0
\(627\) 445.114 + 2.64288i 0.709911 + 0.00421512i
\(628\) 0 0
\(629\) 156.519i 0.248838i
\(630\) 0 0
\(631\) −628.425 −0.995919 −0.497959 0.867200i \(-0.665917\pi\)
−0.497959 + 0.867200i \(0.665917\pi\)
\(632\) 0 0
\(633\) 398.707 0.629870
\(634\) 0 0
\(635\) 16.8328i 0.0265084i
\(636\) 0 0
\(637\) 439.745i 0.690338i
\(638\) 0 0
\(639\) 157.912i 0.247124i
\(640\) 0 0
\(641\) 462.053i 0.720832i 0.932792 + 0.360416i \(0.117365\pi\)
−0.932792 + 0.360416i \(0.882635\pi\)
\(642\) 0 0
\(643\) 898.660 1.39761 0.698803 0.715314i \(-0.253716\pi\)
0.698803 + 0.715314i \(0.253716\pi\)
\(644\) 0 0
\(645\) 26.5862i 0.0412189i
\(646\) 0 0
\(647\) 773.813 1.19600 0.598001 0.801495i \(-0.295962\pi\)
0.598001 + 0.801495i \(0.295962\pi\)
\(648\) 0 0
\(649\) 802.814i 1.23700i
\(650\) 0 0
\(651\) −4.09658 −0.00629275
\(652\) 0 0
\(653\) −701.376 −1.07408 −0.537041 0.843556i \(-0.680458\pi\)
−0.537041 + 0.843556i \(0.680458\pi\)
\(654\) 0 0
\(655\) 79.4782 0.121341
\(656\) 0 0
\(657\) −41.2086 −0.0627223
\(658\) 0 0
\(659\) 1255.87i 1.90572i 0.303415 + 0.952858i \(0.401873\pi\)
−0.303415 + 0.952858i \(0.598127\pi\)
\(660\) 0 0
\(661\) 399.716i 0.604714i −0.953195 0.302357i \(-0.902227\pi\)
0.953195 0.302357i \(-0.0977735\pi\)
\(662\) 0 0
\(663\) 48.1475 0.0726206
\(664\) 0 0
\(665\) −0.597926 + 100.703i −0.000899137 + 0.151433i
\(666\) 0 0
\(667\) 1074.25i 1.61057i
\(668\) 0 0
\(669\) −456.939 −0.683018
\(670\) 0 0
\(671\) 1304.64 1.94433
\(672\) 0 0
\(673\) 116.954i 0.173779i 0.996218 + 0.0868897i \(0.0276928\pi\)
−0.996218 + 0.0868897i \(0.972307\pi\)
\(674\) 0 0
\(675\) 128.295i 0.190067i
\(676\) 0 0
\(677\) 497.520i 0.734890i 0.930045 + 0.367445i \(0.119767\pi\)
−0.930045 + 0.367445i \(0.880233\pi\)
\(678\) 0 0
\(679\) 691.962i 1.01909i
\(680\) 0 0
\(681\) 370.534 0.544103
\(682\) 0 0
\(683\) 803.013i 1.17572i −0.808964 0.587858i \(-0.799972\pi\)
0.808964 0.587858i \(-0.200028\pi\)
\(684\) 0 0
\(685\) −92.0787 −0.134421
\(686\) 0 0
\(687\) 109.933i 0.160019i
\(688\) 0 0
\(689\) 40.6105 0.0589413
\(690\) 0 0
\(691\) −891.667 −1.29040 −0.645200 0.764014i \(-0.723226\pi\)
−0.645200 + 0.764014i \(0.723226\pi\)
\(692\) 0 0
\(693\) −386.537 −0.557773
\(694\) 0 0
\(695\) 94.2128 0.135558
\(696\) 0 0
\(697\) 182.255i 0.261485i
\(698\) 0 0
\(699\) 499.608i 0.714746i
\(700\) 0 0
\(701\) 1110.82 1.58463 0.792313 0.610114i \(-0.208876\pi\)
0.792313 + 0.610114i \(0.208876\pi\)
\(702\) 0 0
\(703\) 1127.01 + 6.69164i 1.60314 + 0.00951869i
\(704\) 0 0
\(705\) 43.2572i 0.0613577i
\(706\) 0 0
\(707\) 1112.46 1.57349
\(708\) 0 0
\(709\) −9.63354 −0.0135875 −0.00679375 0.999977i \(-0.502163\pi\)
−0.00679375 + 0.999977i \(0.502163\pi\)
\(710\) 0 0
\(711\) 153.967i 0.216550i
\(712\) 0 0
\(713\) 5.46760i 0.00766845i
\(714\) 0 0
\(715\) 79.2835i 0.110886i
\(716\) 0 0
\(717\) 483.464i 0.674287i
\(718\) 0 0
\(719\) −97.4316 −0.135510 −0.0677549 0.997702i \(-0.521584\pi\)
−0.0677549 + 0.997702i \(0.521584\pi\)
\(720\) 0 0
\(721\) 689.597i 0.956445i
\(722\) 0 0
\(723\) −420.346 −0.581392
\(724\) 0 0
\(725\) 1204.46i 1.66132i
\(726\) 0 0
\(727\) 466.358 0.641483 0.320742 0.947167i \(-0.396068\pi\)
0.320742 + 0.947167i \(0.396068\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 72.7931 0.0995801
\(732\) 0 0
\(733\) 452.839 0.617789 0.308895 0.951096i \(-0.400041\pi\)
0.308895 + 0.951096i \(0.400041\pi\)
\(734\) 0 0
\(735\) 40.2279i 0.0547318i
\(736\) 0 0
\(737\) 145.851i 0.197898i
\(738\) 0 0
\(739\) 1346.02 1.82140 0.910701 0.413066i \(-0.135542\pi\)
0.910701 + 0.413066i \(0.135542\pi\)
\(740\) 0 0
\(741\) −2.05844 + 346.683i −0.00277793 + 0.467859i
\(742\) 0 0
\(743\) 787.553i 1.05996i 0.848009 + 0.529982i \(0.177801\pi\)
−0.848009 + 0.529982i \(0.822199\pi\)
\(744\) 0 0
\(745\) −77.5664 −0.104116
\(746\) 0 0
\(747\) −191.323 −0.256122
\(748\) 0 0
\(749\) 1467.53i 1.95932i
\(750\) 0 0
\(751\) 120.158i 0.159997i −0.996795 0.0799986i \(-0.974508\pi\)
0.996795 0.0799986i \(-0.0254916\pi\)
\(752\) 0 0
\(753\) 281.766i 0.374191i
\(754\) 0 0
\(755\) 84.9866i 0.112565i
\(756\) 0 0
\(757\) 263.516 0.348105 0.174053 0.984736i \(-0.444314\pi\)
0.174053 + 0.984736i \(0.444314\pi\)
\(758\) 0 0
\(759\) 515.901i 0.679712i
\(760\) 0 0
\(761\) −60.7691 −0.0798543 −0.0399271 0.999203i \(-0.512713\pi\)
−0.0399271 + 0.999203i \(0.512713\pi\)
\(762\) 0 0
\(763\) 1521.00i 1.99345i
\(764\) 0 0
\(765\) −4.40453 −0.00575755
\(766\) 0 0
\(767\) 625.282 0.815231
\(768\) 0 0
\(769\) −713.480 −0.927802 −0.463901 0.885887i \(-0.653551\pi\)
−0.463901 + 0.885887i \(0.653551\pi\)
\(770\) 0 0
\(771\) 444.494 0.576516
\(772\) 0 0
\(773\) 971.062i 1.25622i −0.778123 0.628112i \(-0.783828\pi\)
0.778123 0.628112i \(-0.216172\pi\)
\(774\) 0 0
\(775\) 6.13034i 0.00791012i
\(776\) 0 0
\(777\) −978.692 −1.25958
\(778\) 0 0
\(779\) −1312.32 7.79191i −1.68462 0.0100025i
\(780\) 0 0
\(781\) 711.965i 0.911607i
\(782\) 0 0
\(783\) −253.481 −0.323730
\(784\) 0 0
\(785\) −164.658 −0.209756
\(786\) 0 0
\(787\) 903.790i 1.14840i −0.818716 0.574199i \(-0.805313\pi\)
0.818716 0.574199i \(-0.194687\pi\)
\(788\) 0 0
\(789\) 123.327i 0.156308i
\(790\) 0 0
\(791\) 758.659i 0.959114i
\(792\) 0 0
\(793\) 1016.14i 1.28139i
\(794\) 0 0
\(795\) −3.71505 −0.00467301
\(796\) 0 0
\(797\) 836.376i 1.04941i −0.851286 0.524703i \(-0.824177\pi\)
0.851286 0.524703i \(-0.175823\pi\)
\(798\) 0 0
\(799\) −118.438 −0.148233
\(800\) 0 0
\(801\) 134.783i 0.168268i
\(802\) 0 0