Properties

Label 177.2.d.b.176.1
Level $177$
Weight $2$
Character 177.176
Analytic conductor $1.413$
Analytic rank $0$
Dimension $6$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.149721291.1
Defining polynomial: \(x^{6} - 7 x^{3} + 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 176.1
Root \(-1.24353 - 1.20567i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.2.d.b.176.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.24353 - 1.20567i) q^{3} -2.00000 q^{4} +3.58579i q^{5} +5.15952 q^{7} +(0.0927106 + 2.99857i) q^{9} +O(q^{10})\) \(q+(-1.24353 - 1.20567i) q^{3} -2.00000 q^{4} +3.58579i q^{5} +5.15952 q^{7} +(0.0927106 + 2.99857i) q^{9} +(2.48705 + 2.41134i) q^{12} +(4.32329 - 4.45902i) q^{15} +4.00000 q^{16} +7.68115i q^{17} -2.30163 q^{19} -7.17158i q^{20} +(-6.41600 - 6.22069i) q^{21} -7.85789 q^{25} +(3.50000 - 3.84057i) q^{27} -10.3190 q^{28} -3.64824i q^{29} +18.5010i q^{35} +(-0.185421 - 5.99713i) q^{36} +0.0624503i q^{41} +(-10.7522 + 0.332441i) q^{45} +(-4.97410 - 4.82269i) q^{48} +19.6207 q^{49} +(9.26094 - 9.55170i) q^{51} -14.4056i q^{53} +(2.86213 + 2.77501i) q^{57} -7.68115i q^{59} +(-8.64657 + 8.91804i) q^{60} +(0.478343 + 15.4712i) q^{63} -8.00000 q^{64} -15.3623i q^{68} +7.68115i q^{71} +(9.77149 + 9.47404i) q^{75} +4.60326 q^{76} +13.1769 q^{79} +14.3432i q^{80} +(-8.98281 + 0.555998i) q^{81} +(12.8320 + 12.4414i) q^{84} -27.5430 q^{85} +(-4.39858 + 4.53668i) q^{87} -8.25316i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 12q^{4} + O(q^{10}) \) \( 6q - 12q^{4} + 3q^{15} + 24q^{16} - 15q^{21} - 30q^{25} + 21q^{27} - 33q^{45} + 42q^{49} + 39q^{57} - 6q^{60} + 12q^{63} - 48q^{64} - 24q^{75} + 30q^{84} - 51q^{87} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.24353 1.20567i −0.717950 0.696095i
\(4\) −2.00000 −1.00000
\(5\) 3.58579i 1.60361i 0.597583 + 0.801807i \(0.296128\pi\)
−0.597583 + 0.801807i \(0.703872\pi\)
\(6\) 0 0
\(7\) 5.15952 1.95012 0.975058 0.221950i \(-0.0712420\pi\)
0.975058 + 0.221950i \(0.0712420\pi\)
\(8\) 0 0
\(9\) 0.0927106 + 2.99857i 0.0309035 + 0.999522i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.48705 + 2.41134i 0.717950 + 0.696095i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 4.32329 4.45902i 1.11627 1.15131i
\(16\) 4.00000 1.00000
\(17\) 7.68115i 1.86295i 0.363803 + 0.931476i \(0.381478\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.30163 −0.528030 −0.264015 0.964519i \(-0.585047\pi\)
−0.264015 + 0.964519i \(0.585047\pi\)
\(20\) 7.17158i 1.60361i
\(21\) −6.41600 6.22069i −1.40009 1.35747i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −7.85789 −1.57158
\(26\) 0 0
\(27\) 3.50000 3.84057i 0.673575 0.739119i
\(28\) −10.3190 −1.95012
\(29\) 3.64824i 0.677461i −0.940883 0.338731i \(-0.890002\pi\)
0.940883 0.338731i \(-0.109998\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 18.5010i 3.12723i
\(36\) −0.185421 5.99713i −0.0309035 0.999522i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.0624503i 0.00975310i 0.999988 + 0.00487655i \(0.00155226\pi\)
−0.999988 + 0.00487655i \(0.998448\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −10.7522 + 0.332441i −1.60285 + 0.0495574i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −4.97410 4.82269i −0.717950 0.696095i
\(49\) 19.6207 2.80295
\(50\) 0 0
\(51\) 9.26094 9.55170i 1.29679 1.33751i
\(52\) 0 0
\(53\) 14.4056i 1.97876i −0.145342 0.989382i \(-0.546428\pi\)
0.145342 0.989382i \(-0.453572\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.86213 + 2.77501i 0.379099 + 0.367559i
\(58\) 0 0
\(59\) 7.68115i 1.00000i
\(60\) −8.64657 + 8.91804i −1.11627 + 1.15131i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0.478343 + 15.4712i 0.0602655 + 1.94918i
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 15.3623i 1.86295i
\(69\) 0 0
\(70\) 0 0
\(71\) 7.68115i 0.911584i 0.890086 + 0.455792i \(0.150644\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 9.77149 + 9.47404i 1.12831 + 1.09397i
\(76\) 4.60326 0.528030
\(77\) 0 0
\(78\) 0 0
\(79\) 13.1769 1.48252 0.741261 0.671217i \(-0.234228\pi\)
0.741261 + 0.671217i \(0.234228\pi\)
\(80\) 14.3432i 1.60361i
\(81\) −8.98281 + 0.555998i −0.998090 + 0.0617776i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 12.8320 + 12.4414i 1.40009 + 1.35747i
\(85\) −27.5430 −2.98746
\(86\) 0 0
\(87\) −4.39858 + 4.53668i −0.471577 + 0.486383i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.25316i 0.846756i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 15.7158 1.57158
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 22.3061 23.0064i 2.17685 2.24520i
\(106\) 0 0
\(107\) 10.8823i 1.05203i −0.850476 0.526014i \(-0.823686\pi\)
0.850476 0.526014i \(-0.176314\pi\)
\(108\) −7.00000 + 7.68115i −0.673575 + 0.739119i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 20.6381 1.95012
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.29648i 0.677461i
\(117\) 0 0
\(118\) 0 0
\(119\) 39.6310i 3.63297i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0.0752946 0.0776586i 0.00678908 0.00700224i
\(124\) 0 0
\(125\) 10.2478i 0.916592i
\(126\) 0 0
\(127\) −18.3365 −1.62710 −0.813549 0.581496i \(-0.802467\pi\)
−0.813549 + 0.581496i \(0.802467\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −11.8753 −1.02972
\(134\) 0 0
\(135\) 13.7715 + 12.5503i 1.18526 + 1.08015i
\(136\) 0 0
\(137\) 10.6949i 0.913729i −0.889536 0.456864i \(-0.848972\pi\)
0.889536 0.456864i \(-0.151028\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 37.0019i 3.12723i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.370843 + 11.9943i 0.0309035 + 0.999522i
\(145\) 13.0818 1.08639
\(146\) 0 0
\(147\) −24.3988 23.6561i −2.01238 1.95112i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −23.0324 + 0.712124i −1.86206 + 0.0575718i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −17.3684 + 17.9137i −1.37741 + 1.42065i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0.124901i 0.00975310i
\(165\) 0 0
\(166\) 0 0
\(167\) 25.1630i 1.94717i −0.228325 0.973585i \(-0.573325\pi\)
0.228325 0.973585i \(-0.426675\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −0.213386 6.90159i −0.0163180 0.527778i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −40.5430 −3.06476
\(176\) 0 0
\(177\) −9.26094 + 9.55170i −0.696095 + 0.717950i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 21.5045 0.664882i 1.60285 0.0495574i
\(181\) 12.0644 0.896741 0.448370 0.893848i \(-0.352005\pi\)
0.448370 + 0.893848i \(0.352005\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 18.0583 19.8155i 1.31355 1.44137i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 9.94820 + 9.64538i 0.717950 + 0.696095i
\(193\) −3.41416 −0.245756 −0.122878 0.992422i \(-0.539212\pi\)
−0.122878 + 0.992422i \(0.539212\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −39.2413 −2.80295
\(197\) 7.68115i 0.547259i 0.961835 + 0.273629i \(0.0882242\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) 28.0992 1.99190 0.995951 0.0898948i \(-0.0286531\pi\)
0.995951 + 0.0898948i \(0.0286531\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18.8232i 1.32113i
\(204\) −18.5219 + 19.1034i −1.29679 + 1.33751i
\(205\) −0.223934 −0.0156402
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 28.8112i 1.97876i
\(213\) 9.26094 9.55170i 0.634549 0.654472i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) −0.728510 23.5624i −0.0485674 1.57083i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −5.72427 5.55002i −0.379099 0.367559i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 15.3623i 1.00000i
\(237\) −16.3859 15.8871i −1.06438 1.03198i
\(238\) 0 0
\(239\) 18.1163i 1.17185i −0.810367 0.585923i \(-0.800732\pi\)
0.810367 0.585923i \(-0.199268\pi\)
\(240\) 17.2931 17.8361i 1.11627 1.15131i
\(241\) 21.1944 1.36525 0.682624 0.730770i \(-0.260839\pi\)
0.682624 + 0.730770i \(0.260839\pi\)
\(242\) 0 0
\(243\) 11.8407 + 10.1389i 0.759581 + 0.650412i
\(244\) 0 0
\(245\) 70.3556i 4.49486i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.8665i 1.12772i 0.825869 + 0.563862i \(0.190685\pi\)
−0.825869 + 0.563862i \(0.809315\pi\)
\(252\) −0.956685 30.9424i −0.0602655 1.94918i
\(253\) 0 0
\(254\) 0 0
\(255\) 34.2504 + 33.2078i 2.14484 + 2.07955i
\(256\) 16.0000 1.00000
\(257\) 25.2879i 1.57741i 0.614769 + 0.788707i \(0.289249\pi\)
−0.614769 + 0.788707i \(0.710751\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 10.9395 0.338231i 0.677138 0.0209360i
\(262\) 0 0
\(263\) 32.3970i 1.99769i −0.0480951 0.998843i \(-0.515315\pi\)
0.0480951 0.998843i \(-0.484685\pi\)
\(264\) 0 0
\(265\) 51.6555 3.17317
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 13.7332 0.834233 0.417116 0.908853i \(-0.363041\pi\)
0.417116 + 0.908853i \(0.363041\pi\)
\(272\) 30.7246i 1.86295i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −33.2588 −1.99833 −0.999163 0.0409112i \(-0.986974\pi\)
−0.999163 + 0.0409112i \(0.986974\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.8737i 1.72246i −0.508216 0.861230i \(-0.669695\pi\)
0.508216 0.861230i \(-0.330305\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 15.3623i 0.911584i
\(285\) −9.95060 + 10.2630i −0.589423 + 0.607929i
\(286\) 0 0
\(287\) 0.322214i 0.0190197i
\(288\) 0 0
\(289\) −42.0000 −2.47059
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 32.3346i 1.88901i 0.328504 + 0.944503i \(0.393456\pi\)
−0.328504 + 0.944503i \(0.606544\pi\)
\(294\) 0 0
\(295\) 27.5430 1.60361
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −19.5430 18.9481i −1.12831 1.09397i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −9.20652 −0.528030
\(305\) 0 0
\(306\) 0 0
\(307\) 4.04699 0.230974 0.115487 0.993309i \(-0.463157\pi\)
0.115487 + 0.993309i \(0.463157\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.6325i 0.602912i 0.953480 + 0.301456i \(0.0974727\pi\)
−0.953480 + 0.301456i \(0.902527\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −55.4764 + 1.71524i −3.12574 + 0.0966426i
\(316\) −26.3539 −1.48252
\(317\) 30.7246i 1.72566i 0.505490 + 0.862832i \(0.331312\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 28.6863i 1.60361i
\(321\) −13.1204 + 13.5324i −0.732312 + 0.755304i
\(322\) 0 0
\(323\) 17.6792i 0.983694i
\(324\) 17.9656 1.11200i 0.998090 0.0617776i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −18.8927 −1.03844 −0.519219 0.854641i \(-0.673777\pi\)
−0.519219 + 0.854641i \(0.673777\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −25.6640 24.8828i −1.40009 1.35747i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 55.0860 2.98746
\(341\) 0 0
\(342\) 0 0
\(343\) 65.1167 3.51597
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 8.79716 9.07336i 0.471577 0.486383i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −27.5430 −1.46183
\(356\) 0 0
\(357\) 47.7820 49.2822i 2.52889 2.60829i
\(358\) 0 0
\(359\) 36.0453i 1.90240i 0.308581 + 0.951198i \(0.400146\pi\)
−0.308581 + 0.951198i \(0.599854\pi\)
\(360\) 0 0
\(361\) −13.7025 −0.721184
\(362\) 0 0
\(363\) 13.6788 + 13.2624i 0.717950 + 0.696095i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −0.187261 + 0.00578981i −0.00974844 + 0.000301405i
\(370\) 0 0
\(371\) 74.3261i 3.85882i
\(372\) 0 0
\(373\) −31.0000 −1.60512 −0.802560 0.596572i \(-0.796529\pi\)
−0.802560 + 0.596572i \(0.796529\pi\)
\(374\) 0 0
\(375\) −12.3555 + 12.7434i −0.638035 + 0.658067i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −24.6851 −1.26799 −0.633994 0.773338i \(-0.718586\pi\)
−0.633994 + 0.773338i \(0.718586\pi\)
\(380\) 16.5063i 0.846756i
\(381\) 22.8019 + 22.1078i 1.16817 + 1.13261i
\(382\) 0 0
\(383\) 15.3623i 0.784976i −0.919757 0.392488i \(-0.871614\pi\)
0.919757 0.392488i \(-0.128386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 38.4057i 1.94725i −0.228159 0.973624i \(-0.573271\pi\)
0.228159 0.973624i \(-0.426729\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 47.2497i 2.37739i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 14.7672 + 14.3177i 0.739287 + 0.716783i
\(400\) −31.4316 −1.57158
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.99369 32.2105i −0.0990674 1.60055i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −12.8946 + 13.2994i −0.636042 + 0.656011i
\(412\) 0 0
\(413\) 39.6310i 1.95012i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.21763 6.02836i −0.304478 0.295210i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −44.6122 + 46.0128i −2.17685 + 2.24520i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 60.3576i 2.92777i
\(426\) 0 0
\(427\) 0 0
\(428\) 21.7645i 1.05203i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 14.0000 15.3623i 0.673575 0.739119i
\(433\) −40.7199 −1.95687 −0.978437 0.206545i \(-0.933778\pi\)
−0.978437 + 0.206545i \(0.933778\pi\)
\(434\) 0 0
\(435\) −16.2676 15.7724i −0.779971 0.756228i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 1.81905 + 58.8339i 0.0866212 + 2.80161i
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −41.2762 −1.95012
\(449\) 32.5219i 1.53480i 0.641166 + 0.767402i \(0.278451\pi\)
−0.641166 + 0.767402i \(0.721549\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 29.5000 + 26.8840i 1.37694 + 1.25484i
\(460\) 0 0
\(461\) 30.7246i 1.43099i 0.698620 + 0.715493i \(0.253798\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 14.5930i 0.677461i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 18.0860 0.829841
\(476\) 79.2621i 3.63297i
\(477\) 43.1962 1.33555i 1.97782 0.0611508i
\(478\) 0 0
\(479\) 38.4057i 1.75480i −0.479757 0.877401i \(-0.659275\pi\)
0.479757 0.877401i \(-0.340725\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −16.6677 −0.755284 −0.377642 0.925952i \(-0.623265\pi\)
−0.377642 + 0.925952i \(0.623265\pi\)
\(488\) 0 0
\(489\) −13.6788 13.2624i −0.618576 0.599746i
\(490\) 0 0
\(491\) 6.98423i 0.315194i −0.987504 0.157597i \(-0.949625\pi\)
0.987504 0.157597i \(-0.0503747\pi\)
\(492\) −0.150589 + 0.155317i −0.00678908 + 0.00700224i
\(493\) 28.0227 1.26208
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 39.6310i 1.77770i
\(498\) 0 0
\(499\) 35.0041 1.56700 0.783500 0.621392i \(-0.213432\pi\)
0.783500 + 0.621392i \(0.213432\pi\)
\(500\) 20.4956i 0.916592i
\(501\) −30.3383 + 31.2908i −1.35542 + 1.39797i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −16.1658 15.6737i −0.717950 0.696095i
\(508\) 36.6729 1.62710
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −8.05570 + 8.83958i −0.355668 + 0.390277i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.68115i 0.336517i 0.985743 + 0.168259i \(0.0538144\pi\)
−0.985743 + 0.168259i \(0.946186\pi\)
\(522\) 0 0
\(523\) −3.97042 −0.173614 −0.0868072 0.996225i \(-0.527666\pi\)
−0.0868072 + 0.996225i \(0.527666\pi\)
\(524\) 0 0
\(525\) 50.4162 + 48.8815i 2.20034 + 2.13336i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 23.0324 0.712124i 0.999522 0.0309035i
\(532\) 23.7506 1.02972
\(533\) 0 0
\(534\) 0 0
\(535\) 39.0215 1.68705
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −27.5430 25.1005i −1.18526 1.08015i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −15.0024 14.5457i −0.643815 0.624217i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 21.3898i 0.913729i
\(549\) 0 0
\(550\) 0 0
\(551\) 8.39690i 0.357720i
\(552\) 0 0
\(553\) 67.9867 2.89109
\(554\) 0 0
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) 43.0919i 1.82586i 0.408112 + 0.912932i \(0.366187\pi\)
−0.408112 + 0.912932i \(0.633813\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 74.0039i 3.12723i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −46.3470 + 2.86869i −1.94639 + 0.120473i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.741685 23.9885i −0.0309035 0.999522i
\(577\) 44.1341 1.83733 0.918663 0.395043i \(-0.129270\pi\)
0.918663 + 0.395043i \(0.129270\pi\)
\(578\) 0 0
\(579\) 4.24559 + 4.11635i 0.176441 + 0.171070i
\(580\) −26.1637 −1.08639
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 48.7976 + 47.3122i 2.01238 + 1.95112i
\(589\) 0 0
\(590\) 0 0
\(591\) 9.26094 9.55170i 0.380944 0.392904i
\(592\) 0 0
\(593\) 3.83559i 0.157509i −0.996894 0.0787544i \(-0.974906\pi\)
0.996894 0.0787544i \(-0.0250943\pi\)
\(594\) 0 0
\(595\) −142.109 −5.82589
\(596\) 0 0
\(597\) −34.9421 33.8785i −1.43009 1.38655i
\(598\) 0 0
\(599\) 32.2097i 1.31605i −0.752996 0.658026i \(-0.771392\pi\)
0.752996 0.658026i \(-0.228608\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 39.4437i 1.60361i
\(606\) 0 0
\(607\) −32.1462 −1.30478 −0.652388 0.757885i \(-0.726233\pi\)
−0.652388 + 0.757885i \(0.726233\pi\)
\(608\) 0 0
\(609\) −22.6946 + 23.4071i −0.919631 + 0.948504i
\(610\) 0 0
\(611\) 0 0
\(612\) 46.0649 1.42425i 1.86206 0.0575718i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0.278467 + 0.269991i 0.0112289 + 0.0108871i
\(616\) 0 0
\(617\) 46.8651i 1.88672i −0.331775 0.943358i \(-0.607648\pi\)
0.331775 0.943358i \(-0.392352\pi\)
\(618\) 0 0
\(619\) −26.9101 −1.08161 −0.540805 0.841148i \(-0.681880\pi\)
−0.540805 + 0.841148i \(0.681880\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −2.54298 −0.101719
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 65.7507i 2.60924i
\(636\) 34.7369 35.8275i 1.37741 1.42065i
\(637\) 0 0
\(638\) 0 0
\(639\) −23.0324 + 0.712124i −0.911149 + 0.0281712i
\(640\) 0 0
\(641\) 38.4057i 1.51694i −0.651711 0.758468i \(-0.725948\pi\)
0.651711 0.758468i \(-0.274052\pi\)
\(642\) 0 0
\(643\) −48.1811 −1.90008 −0.950038 0.312134i \(-0.898956\pi\)
−0.950038 + 0.312134i \(0.898956\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.9756i 0.981893i −0.871190 0.490947i \(-0.836651\pi\)
0.871190 0.490947i \(-0.163349\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) 14.1558i 0.553960i 0.960876 + 0.276980i \(0.0893335\pi\)
−0.960876 + 0.276980i \(0.910666\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.249801i 0.00975310i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 51.0390 1.98519 0.992593 0.121489i \(-0.0387669\pi\)
0.992593 + 0.121489i \(0.0387669\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 42.5824i 1.65127i
\(666\) 0 0
\(667\) 0 0
\(668\) 50.3260i 1.94717i
\(669\) 23.6270 + 22.9078i 0.913472 + 0.885665i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −27.5026 + 30.1788i −1.05858 + 1.16158i
\(676\) −26.0000 −1.00000
\(677\) 30.7246i 1.18084i 0.807096 + 0.590421i \(0.201038\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0.426771 + 13.8032i 0.0163180 + 0.527778i
\(685\) 38.3497 1.46527
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.9290i 0.680084i
\(696\) 0 0
\(697\) −0.479690 −0.0181696
\(698\) 0 0
\(699\) 0 0
\(700\) 81.0860 3.06476
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 18.5219 19.1034i 0.696095 0.717950i
\(709\) −49.2936 −1.85126 −0.925630 0.378430i \(-0.876464\pi\)
−0.925630 + 0.378430i \(0.876464\pi\)
\(710\) 0 0
\(711\) 1.22164 + 39.5119i 0.0458152 + 1.48181i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −21.8423 + 22.5281i −0.815716 + 0.841327i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −43.0089 + 1.32976i −1.60285 + 0.0495574i
\(721\) 0 0
\(722\) 0 0
\(723\) −26.3557 25.5534i −0.980180 0.950343i
\(724\) −24.1288 −0.896741
\(725\) 28.6675i 1.06468i
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) −2.50000 26.8840i −0.0925926 0.995704i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −49.0000 −1.80986 −0.904928 0.425564i \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) 0 0
\(735\) 84.8258 87.4890i 3.12885 3.22708i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 53.7680i 1.97256i 0.165089 + 0.986279i \(0.447209\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 56.1473i 2.05158i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 21.5411 22.2174i 0.785003 0.809649i
\(754\) 0 0
\(755\) 0 0
\(756\) −36.1167 + 39.6310i −1.31355 + 1.44137i
\(757\) 42.4653 1.54343 0.771713 0.635970i \(-0.219400\pi\)
0.771713 + 0.635970i \(0.219400\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 54.0367i 1.95883i 0.201863 + 0.979414i \(0.435300\pi\)
−0.201863 + 0.979414i \(0.564700\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.55353 82.5895i −0.0923230 2.98603i
\(766\) 0 0
\(767\) 0 0
\(768\) −19.8964 19.2908i −0.717950 0.696095i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 30.4889 31.4461i 1.09803 1.13250i
\(772\) 6.82831 0.245756
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.143737i 0.00514993i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −14.0113 12.7688i −0.500724 0.456321i
\(784\) 78.4827 2.80295
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 15.3623i 0.547259i
\(789\) −39.0602 + 40.2865i −1.39058 + 1.43424i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −64.2349 62.2796i −2.27818 2.20883i
\(796\) −56.1985 −1.99190
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0