Properties

Label 177.6.d.a.176.6
Level $177$
Weight $6$
Character 177.176
Analytic conductor $28.388$
Analytic rank $0$
Dimension $6$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.3879361069\)
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 176.6
Root \(-1.24353 - 1.20567i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.6.d.a.176.5

$q$-expansion

\(f(q)\) \(=\) \(q+(11.8407 + 10.1389i) q^{3} -32.0000 q^{4} -111.597i q^{5} +113.183 q^{7} +(37.4045 + 240.104i) q^{9} +O(q^{10})\) \(q+(11.8407 + 10.1389i) q^{3} -32.0000 q^{4} -111.597i q^{5} +113.183 q^{7} +(37.4045 + 240.104i) q^{9} +(-378.902 - 324.446i) q^{12} +(1131.47 - 1321.38i) q^{15} +1024.00 q^{16} -683.622i q^{17} -3060.71 q^{19} +3571.09i q^{20} +(1340.17 + 1147.55i) q^{21} -9328.81 q^{25} +(-1991.50 + 3222.24i) q^{27} -3621.85 q^{28} -8946.38i q^{29} -12630.8i q^{35} +(-1196.94 - 7683.33i) q^{36} +524.845i q^{41} +(26794.8 - 4174.22i) q^{45} +(12124.9 + 10382.3i) q^{48} -3996.62 q^{49} +(6931.19 - 8094.56i) q^{51} -30496.7i q^{53} +(-36241.0 - 31032.3i) q^{57} -26738.1i q^{59} +(-36207.0 + 42284.3i) q^{60} +(4233.55 + 27175.7i) q^{63} -32768.0 q^{64} +21875.9i q^{68} +59459.7i q^{71} +(-110460. - 94584.1i) q^{75} +97942.7 q^{76} -95289.8 q^{79} -114275. i q^{80} +(-56250.8 + 17961.9i) q^{81} +(-42885.3 - 36721.7i) q^{84} -76289.9 q^{85} +(90706.7 - 105931. i) q^{87} +341565. i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 192q^{4} + O(q^{10}) \) \( 6q - 192q^{4} + 3153q^{15} + 6144q^{16} - 3075q^{21} - 18750q^{25} - 11949q^{27} + 81147q^{45} + 100842q^{49} - 131001q^{57} - 100896q^{60} + 180732q^{63} - 196608q^{64} - 197304q^{75} + 98400q^{84} + 221799q^{87} + O(q^{100}) \)

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 \(0.5\pi\)
\(3\) 11.8407 + 10.1389i 0.759581 + 0.650412i
\(4\) −32.0000 −1.00000
\(5\) 111.597i 1.99630i −0.0607856 0.998151i \(-0.519361\pi\)
0.0607856 0.998151i \(-0.480639\pi\)
\(6\) 0 0
\(7\) 113.183 0.873044 0.436522 0.899694i \(-0.356210\pi\)
0.436522 + 0.899694i \(0.356210\pi\)
\(8\) 0 0
\(9\) 37.4045 + 240.104i 0.153928 + 0.988082i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −378.902 324.446i −0.759581 0.650412i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1131.47 1321.38i 1.29842 1.51635i
\(16\) 1024.00 1.00000
\(17\) 683.622i 0.573712i −0.957974 0.286856i \(-0.907390\pi\)
0.957974 0.286856i \(-0.0926101\pi\)
\(18\) 0 0
\(19\) −3060.71 −1.94508 −0.972541 0.232730i \(-0.925234\pi\)
−0.972541 + 0.232730i \(0.925234\pi\)
\(20\) 3571.09i 1.99630i
\(21\) 1340.17 + 1147.55i 0.663148 + 0.567838i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −9328.81 −2.98522
\(26\) 0 0
\(27\) −1991.50 + 3222.24i −0.525740 + 0.850645i
\(28\) −3621.85 −0.873044
\(29\) 8946.38i 1.97539i −0.156396 0.987694i \(-0.549988\pi\)
0.156396 0.987694i \(-0.450012\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12630.8i 1.74286i
\(36\) −1196.94 7683.33i −0.153928 0.988082i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 524.845i 0.0487609i 0.999703 + 0.0243804i \(0.00776130\pi\)
−0.999703 + 0.0243804i \(0.992239\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 26794.8 4174.22i 1.97251 0.307287i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 12124.9 + 10382.3i 0.759581 + 0.650412i
\(49\) −3996.62 −0.237795
\(50\) 0 0
\(51\) 6931.19 8094.56i 0.373149 0.435781i
\(52\) 0 0
\(53\) 30496.7i 1.49129i −0.666342 0.745646i \(-0.732141\pi\)
0.666342 0.745646i \(-0.267859\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −36241.0 31032.3i −1.47745 1.26511i
\(58\) 0 0
\(59\) 26738.1i 1.00000i
\(60\) −36207.0 + 42284.3i −1.29842 + 1.51635i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 4233.55 + 27175.7i 0.134386 + 0.862639i
\(64\) −32768.0 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 21875.9i 0.573712i
\(69\) 0 0
\(70\) 0 0
\(71\) 59459.7i 1.39984i 0.714223 + 0.699918i \(0.246780\pi\)
−0.714223 + 0.699918i \(0.753220\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −110460. 94584.1i −2.26752 1.94162i
\(76\) 97942.7 1.94508
\(77\) 0 0
\(78\) 0 0
\(79\) −95289.8 −1.71782 −0.858912 0.512124i \(-0.828859\pi\)
−0.858912 + 0.512124i \(0.828859\pi\)
\(80\) 114275.i 1.99630i
\(81\) −56250.8 + 17961.9i −0.952612 + 0.304187i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −42885.3 36721.7i −0.663148 0.567838i
\(85\) −76289.9 −1.14530
\(86\) 0 0
\(87\) 90706.7 105931.i 1.28482 1.50047i
\(88\) 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
\(95\) 341565.i 3.88297i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 298522. 2.98522
\(101\) 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
\(105\) 128063. 149558.i 1.13358 1.32384i
\(106\) 0 0
\(107\) 86105.4i 0.727061i −0.931582 0.363531i \(-0.881571\pi\)
0.931582 0.363531i \(-0.118429\pi\)
\(108\) 63728.0 103112.i 0.525740 0.850645i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 115899. 0.873044
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 286284.i 1.97539i
\(117\) 0 0
\(118\) 0 0
\(119\) 77374.4i 0.500875i
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) −5321.36 + 6214.53i −0.0317147 + 0.0370379i
\(124\) 0 0
\(125\) 692325.i 3.96310i
\(126\) 0 0
\(127\) 363256. 1.99850 0.999249 0.0387483i \(-0.0123371\pi\)
0.999249 + 0.0387483i \(0.0123371\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −346420. −1.69814
\(134\) 0 0
\(135\) 359591. + 222245.i 1.69814 + 1.04954i
\(136\) 0 0
\(137\) 305627.i 1.39120i −0.718427 0.695602i \(-0.755138\pi\)
0.718427 0.695602i \(-0.244862\pi\)
\(138\) 0 0
\(139\) 399275. 1.75281 0.876406 0.481574i \(-0.159935\pi\)
0.876406 + 0.481574i \(0.159935\pi\)
\(140\) 404187.i 1.74286i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 38302.2 + 245866.i 0.153928 + 0.988082i
\(145\) −998387. −3.94347
\(146\) 0 0
\(147\) −47322.8 40521.4i −0.180625 0.154665i
\(148\) 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
\(153\) 164140. 25570.5i 0.566874 0.0883102i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 309203. 361102.i 0.969955 1.13276i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 537581. 1.58480 0.792401 0.610001i \(-0.208831\pi\)
0.792401 + 0.610001i \(0.208831\pi\)
\(164\) 16795.0i 0.0487609i
\(165\) 0 0
\(166\) 0 0
\(167\) 293269.i 0.813720i −0.913491 0.406860i \(-0.866624\pi\)
0.913491 0.406860i \(-0.133376\pi\)
\(168\) 0 0
\(169\) 371293. 1.00000
\(170\) 0 0
\(171\) −114484. 734888.i −0.299403 1.92190i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −1.05586e6 −2.60623
\(176\) 0 0
\(177\) 271095. 316598.i 0.650412 0.759581i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −857434. + 133575.i −1.97251 + 0.307287i
\(181\) 642636. 1.45804 0.729018 0.684494i \(-0.239977\pi\)
0.729018 + 0.684494i \(0.239977\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −225404. + 364703.i −0.458994 + 0.742651i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −387996. 332232.i −0.759581 0.650412i
\(193\) −597929. −1.15546 −0.577732 0.816226i \(-0.696062\pi\)
−0.577732 + 0.816226i \(0.696062\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 127892. 0.237795
\(197\) 1.07084e6i 1.96588i 0.183921 + 0.982941i \(0.441121\pi\)
−0.183921 + 0.982941i \(0.558879\pi\)
\(198\) 0 0
\(199\) 1.00601e6 1.80082 0.900412 0.435039i \(-0.143265\pi\)
0.900412 + 0.435039i \(0.143265\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.01258e6i 1.72460i
\(204\) −221798. + 259026.i −0.373149 + 0.435781i
\(205\) 58570.9 0.0973414
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 975894.i 1.49129i
\(213\) −602858. + 704045.i −0.910471 + 1.06329i
\(214\) 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
\(223\) 447431. 0.602510 0.301255 0.953544i \(-0.402595\pi\)
0.301255 + 0.953544i \(0.402595\pi\)
\(224\) 0 0
\(225\) −348939. 2.23989e6i −0.459509 2.94964i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 1.15971e6 + 993034.i 1.47745 + 1.26511i
\(229\) 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
\(236\) 855618.i 1.00000i
\(237\) −1.12830e6 966136.i −1.30483 1.11729i
\(238\) 0 0
\(239\) 20308.4i 0.0229975i −0.999934 0.0114987i \(-0.996340\pi\)
0.999934 0.0114987i \(-0.00366024\pi\)
\(240\) 1.15863e6 1.35310e6i 1.29842 1.51635i
\(241\) −1.04065e6 −1.15415 −0.577077 0.816689i \(-0.695807\pi\)
−0.577077 + 0.816689i \(0.695807\pi\)
\(242\) 0 0
\(243\) −848164. 357641.i −0.921434 0.388536i
\(244\) 0 0
\(245\) 446009.i 0.474710i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 291051.i 0.291598i 0.989314 + 0.145799i \(0.0465753\pi\)
−0.989314 + 0.145799i \(0.953425\pi\)
\(252\) −135474. 869622.i −0.134386 0.862639i
\(253\) 0 0
\(254\) 0 0
\(255\) −903326. 773498.i −0.869950 0.744918i
\(256\) 1.04858e6 1.00000
\(257\) 2.08757e6i 1.97155i −0.168069 0.985775i \(-0.553753\pi\)
0.168069 0.985775i \(-0.446247\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.14806e6 334635.i 1.95185 0.304067i
\(262\) 0 0
\(263\) 2.17886e6i 1.94241i −0.238255 0.971203i \(-0.576575\pi\)
0.238255 0.971203i \(-0.423425\pi\)
\(264\) 0 0
\(265\) −3.40333e6 −2.97707
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 2.02182e6 1.67232 0.836159 0.548488i \(-0.184796\pi\)
0.836159 + 0.548488i \(0.184796\pi\)
\(272\) 700029.i 0.573712i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.50077e6 −1.95828 −0.979140 0.203189i \(-0.934870\pi\)
−0.979140 + 0.203189i \(0.934870\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.35294e6i 1.77764i 0.458255 + 0.888821i \(0.348475\pi\)
−0.458255 + 0.888821i \(0.651525\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.90271e6i 1.39984i
\(285\) −3.46310e6 + 4.04437e6i −2.52553 + 2.94943i
\(286\) 0 0
\(287\) 59403.5i 0.0425704i
\(288\) 0 0
\(289\) 952518. 0.670855
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 301590.i 0.205234i −0.994721 0.102617i \(-0.967278\pi\)
0.994721 0.102617i \(-0.0327215\pi\)
\(294\) 0 0
\(295\) −2.98388e6 −1.99630
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 3.53471e6 + 3.02669e6i 2.26752 + 1.94162i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −3.13417e6 −1.94508
\(305\) 0 0
\(306\) 0 0
\(307\) 1.80647e6 1.09392 0.546959 0.837160i \(-0.315786\pi\)
0.546959 + 0.837160i \(0.315786\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.40870e6i 1.99842i 0.0396890 + 0.999212i \(0.487363\pi\)
−0.0396890 + 0.999212i \(0.512637\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 3.03271e6 472450.i 1.72209 0.268275i
\(316\) 3.04927e6 1.71782
\(317\) 3.15415e6i 1.76293i −0.472250 0.881465i \(-0.656558\pi\)
0.472250 0.881465i \(-0.343442\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3.65680e6i 1.99630i
\(321\) 873016. 1.01955e6i 0.472890 0.552262i
\(322\) 0 0
\(323\) 2.09237e6i 1.11592i
\(324\) 1.80003e6 574782.i 0.952612 0.304187i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.59606e6 −0.800717 −0.400358 0.916359i \(-0.631114\pi\)
−0.400358 + 0.916359i \(0.631114\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 1.37233e6 + 1.17509e6i 0.663148 + 0.567838i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2.44128e6 1.14530
\(341\) 0 0
\(342\) 0 0
\(343\) −2.35462e6 −1.08065
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −2.90261e6 + 3.38981e6i −1.28482 + 1.50047i
\(349\) 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
\(355\) 6.63551e6 2.79450
\(356\) 0 0
\(357\) 784493. 916167.i 0.325775 0.380456i
\(358\) 0 0
\(359\) 11181.8i 0.00457905i 0.999997 + 0.00228952i \(0.000728778\pi\)
−0.999997 + 0.00228952i \(0.999271\pi\)
\(360\) 0 0
\(361\) 6.89184e6 2.78335
\(362\) 0 0
\(363\) −1.90696e6 1.63288e6i −0.759581 0.650412i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −126017. + 19631.6i −0.0481797 + 0.00750566i
\(370\) 0 0
\(371\) 3.45170e6i 1.30196i
\(372\) 0 0
\(373\) 5.36607e6 1.99703 0.998514 0.0544950i \(-0.0173549\pi\)
0.998514 + 0.0544950i \(0.0173549\pi\)
\(374\) 0 0
\(375\) −7.01943e6 + 8.19761e6i −2.57765 + 3.01030i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.60967e6 0.575624 0.287812 0.957687i \(-0.407072\pi\)
0.287812 + 0.957687i \(0.407072\pi\)
\(380\) 1.09301e7i 3.88297i
\(381\) 4.30121e6 + 3.68303e6i 1.51802 + 1.29985i
\(382\) 0 0
\(383\) 5.18018e6i 1.80446i −0.431252 0.902231i \(-0.641928\pi\)
0.431252 0.902231i \(-0.358072\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.43319e6i 0.815273i −0.913144 0.407636i \(-0.866353\pi\)
0.913144 0.407636i \(-0.133647\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.06340e7i 3.42929i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) −4.10186e6 3.51233e6i −1.28988 1.10449i
\(400\) −9.55271e6 −2.98522
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.00449e6 + 6.27740e6i 0.607249 + 1.90170i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 3.09873e6 3.61884e6i 0.904856 1.05673i
\(412\) 0 0
\(413\) 3.02629e6i 0.873044i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.72770e6 + 4.04822e6i 1.33140 + 1.14005i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −4.09802e6 + 4.78586e6i −1.13358 + 1.32384i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.37738e6i 1.71266i
\(426\) 0 0
\(427\) 0 0
\(428\) 2.75537e6i 0.727061i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −2.03930e6 + 3.29957e6i −0.525740 + 0.850645i
\(433\) −3.94851e6 −1.01208 −0.506039 0.862511i \(-0.668891\pi\)
−0.506039 + 0.862511i \(0.668891\pi\)
\(434\) 0 0
\(435\) −1.18216e7 1.01226e7i −2.99539 2.56488i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −7.86257e6 −1.94717 −0.973584 0.228328i \(-0.926674\pi\)
−0.973584 + 0.228328i \(0.926674\pi\)
\(440\) 0 0
\(441\) −149491. 959604.i −0.0366033 0.234961i
\(442\) 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
\(448\) −3.70878e6 −0.873044
\(449\) 8.05890e6i 1.88651i −0.332065 0.943256i \(-0.607745\pi\)
0.332065 0.943256i \(-0.392255\pi\)
\(450\) 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
\(459\) 2.20279e6 + 1.36143e6i 0.488025 + 0.301623i
\(460\) 0 0
\(461\) 6.82636e6i 1.49602i −0.663689 0.748009i \(-0.731010\pi\)
0.663689 0.748009i \(-0.268990\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 9.16110e6i 1.97539i
\(465\) 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
\(475\) 2.85528e7 5.80650
\(476\) 2.47598e6i 0.500875i
\(477\) 7.32237e6 1.14071e6i 1.47352 0.229552i
\(478\) 0 0
\(479\) 8.05733e6i 1.60455i 0.596957 + 0.802273i \(0.296376\pi\)
−0.596957 + 0.802273i \(0.703624\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −9.77648e6 −1.86793 −0.933964 0.357366i \(-0.883675\pi\)
−0.933964 + 0.357366i \(0.883675\pi\)
\(488\) 0 0
\(489\) 6.36534e6 + 5.45049e6i 1.20379 + 1.03077i
\(490\) 0 0
\(491\) 7.59906e6i 1.42251i −0.702933 0.711256i \(-0.748127\pi\)
0.702933 0.711256i \(-0.251873\pi\)
\(492\) 170284. 198865.i 0.0317147 0.0370379i
\(493\) −6.11595e6 −1.13330
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.72983e6i 1.22212i
\(498\) 0 0
\(499\) −1.08778e7 −1.95565 −0.977823 0.209434i \(-0.932838\pi\)
−0.977823 + 0.209434i \(0.932838\pi\)
\(500\) 2.21544e7i 3.96310i
\(501\) 2.97343e6 3.47251e6i 0.529253 0.618087i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.39637e6 + 3.76451e6i 0.759581 + 0.650412i
\(508\) −1.16242e7 −1.99850
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.09540e6 9.86234e6i 1.02261 1.65458i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.27107e6i 1.49636i 0.663497 + 0.748179i \(0.269072\pi\)
−0.663497 + 0.748179i \(0.730928\pi\)
\(522\) 0 0
\(523\) −5.26744e6 −0.842064 −0.421032 0.907046i \(-0.638332\pi\)
−0.421032 + 0.907046i \(0.638332\pi\)
\(524\) 0 0
\(525\) −1.25022e7 1.07053e7i −1.97964 1.69512i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) 6.41992e6 1.00012e6i 0.988082 0.153928i
\(532\) 1.10854e7 1.69814
\(533\) 0 0
\(534\) 0 0
\(535\) −9.60908e6 −1.45143
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −1.15069e7 7.11183e6i −1.69814 1.04954i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 7.60926e6 + 6.51563e6i 1.10750 + 0.948325i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.06008e7 1.51485 0.757427 0.652920i \(-0.226456\pi\)
0.757427 + 0.652920i \(0.226456\pi\)
\(548\) 9.78008e6i 1.39120i
\(549\) 0 0
\(550\) 0 0
\(551\) 2.73823e7i 3.84229i
\(552\) 0 0
\(553\) −1.07852e7 −1.49973
\(554\) 0 0
\(555\) 0 0
\(556\) −1.27768e7 −1.75281
\(557\) 7.41742e6i 1.01301i −0.862236 0.506506i \(-0.830937\pi\)
0.862236 0.506506i \(-0.169063\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.29340e7i 1.74286i
\(561\) 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
\(567\) −6.36663e6 + 2.03298e6i −0.831672 + 0.265568i
\(568\) 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
\(576\) −1.22567e6 7.86773e6i −0.153928 0.988082i
\(577\) −7.09748e6 −0.887493 −0.443746 0.896152i \(-0.646351\pi\)
−0.443746 + 0.896152i \(0.646351\pi\)
\(578\) 0 0
\(579\) −7.07991e6 6.06236e6i −0.877670 0.751528i
\(580\) 3.19484e7 3.94347
\(581\) 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 \(0.5\pi\)
\(588\) 1.51433e6 + 1.29668e6i 0.180625 + 0.154665i
\(589\) 0 0
\(590\) 0 0
\(591\) −1.08571e7 + 1.26795e7i −1.27863 + 1.49325i
\(592\) 0 0
\(593\) 6.57743e6i 0.768103i −0.923312 0.384052i \(-0.874528\pi\)
0.923312 0.384052i \(-0.125472\pi\)
\(594\) 0 0
\(595\) −8.63472e6 −0.999898
\(596\) 0 0
\(597\) 1.19119e7 + 1.01999e7i 1.36787 + 1.17128i
\(598\) 0 0
\(599\) 7.62940e6i 0.868807i 0.900718 + 0.434403i \(0.143041\pi\)
−0.900718 + 0.434403i \(0.856959\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.79728e7i 1.99630i
\(606\) 0 0
\(607\) 7.27115e6 0.800998 0.400499 0.916297i \(-0.368837\pi\)
0.400499 + 0.916297i \(0.368837\pi\)
\(608\) 0 0
\(609\) 1.02665e7 1.19896e7i 1.12170 1.30997i
\(610\) 0 0
\(611\) 0 0
\(612\) −5.25249e6 + 818257.i −0.566874 + 0.0883102i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 693521. + 593846.i 0.0739387 + 0.0633120i
\(616\) 0 0
\(617\) 2.26641e6i 0.239677i 0.992793 + 0.119839i \(0.0382377\pi\)
−0.992793 + 0.119839i \(0.961762\pi\)
\(618\) 0 0
\(619\) −5.35363e6 −0.561593 −0.280796 0.959767i \(-0.590599\pi\)
−0.280796 + 0.959767i \(0.590599\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.81086e7 4.92632
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.77761e7 1.77731 0.888655 0.458577i \(-0.151641\pi\)
0.888655 + 0.458577i \(0.151641\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.05382e7i 3.98960i
\(636\) −9.89451e6 + 1.15553e7i −0.969955 + 1.13276i
\(637\) 0 0
\(638\) 0 0
\(639\) −1.42765e7 + 2.22406e6i −1.38315 + 0.215474i
\(640\) 0 0
\(641\) 1.91009e7i 1.83615i 0.396404 + 0.918076i \(0.370258\pi\)
−0.396404 + 0.918076i \(0.629742\pi\)
\(642\) 0 0
\(643\) 343814. 0.0327941 0.0163971 0.999866i \(-0.494780\pi\)
0.0163971 + 0.999866i \(0.494780\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.15940e7i 1.08886i −0.838805 0.544432i \(-0.816745\pi\)
0.838805 0.544432i \(-0.183255\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.72026e7 −1.58480
\(653\) 2.14876e7i 1.97199i 0.166766 + 0.985997i \(0.446668\pi\)
−0.166766 + 0.985997i \(0.553332\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 537441.i 0.0487609i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.84281e7 1.64050 0.820250 0.572005i \(-0.193834\pi\)
0.820250 + 0.572005i \(0.193834\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.86593e7i 3.39000i
\(666\) 0 0
\(667\) 0 0
\(668\) 9.38461e6i 0.813720i
\(669\) 5.29790e6 + 4.53647e6i 0.457655 + 0.391880i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.85783e7 3.00597e7i 1.56945 2.53936i
\(676\) −1.18814e7 −1.00000
\(677\) 388942.i 0.0326147i −0.999867 0.0163074i \(-0.994809\pi\)
0.999867 0.0163074i \(-0.00519102\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 3.66350e6 + 2.35164e7i 0.299403 + 1.92190i
\(685\) −3.41070e7 −2.77726
\(686\) 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
\(695\) 4.45578e7i 3.49914i
\(696\) 0 0
\(697\) 358796. 0.0279747
\(698\) 0 0
\(699\) 0 0
\(700\) 3.37876e7 2.60623
\(701\) 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
\(708\) −8.67505e6 + 1.01311e7i −0.650412 + 0.759581i
\(709\) 9.67292e6 0.722673 0.361336 0.932435i \(-0.382321\pi\)
0.361336 + 0.932435i \(0.382321\pi\)
\(710\) 0 0
\(711\) −3.56426e6 2.28794e7i −0.264421 1.69735i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 205905. 240465.i 0.0149578 0.0174685i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 2.74379e7 4.27440e6i 1.97251 0.307287i
\(721\) 0 0
\(722\) 0 0
\(723\) −1.23221e7 1.05511e7i −0.876675 0.750676i
\(724\) −2.05643e7 −1.45804
\(725\) 8.34592e7i 5.89697i
\(726\) 0 0
\(727\) −1.14089e7 −0.800586 −0.400293 0.916387i \(-0.631092\pi\)
−0.400293 + 0.916387i \(0.631092\pi\)
\(728\) 0 0
\(729\) −6.41676e6 1.28342e7i −0.447195 0.894436i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.70725e7 1.17365 0.586824 0.809714i \(-0.300378\pi\)
0.586824 + 0.809714i \(0.300378\pi\)
\(734\) 0 0
\(735\) −4.52205e6 + 5.28106e6i −0.308757 + 0.360581i
\(736\) 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
\(743\) 2.03276e7i 1.35087i 0.737419 + 0.675436i \(0.236044\pi\)
−0.737419 + 0.675436i \(0.763956\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.74567e6i 0.634756i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −2.95094e6 + 3.44625e6i −0.189659 + 0.221492i
\(754\) 0 0
\(755\) 0 0
\(756\) 7.21292e6 1.16705e7i 0.458994 0.742651i
\(757\) −3.00805e7 −1.90785 −0.953927 0.300039i \(-0.903000\pi\)
−0.953927 + 0.300039i \(0.903000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.68229e7i 1.05303i 0.850166 + 0.526514i \(0.176501\pi\)
−0.850166 + 0.526514i \(0.823499\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.85359e6 1.83175e7i −0.176294 1.13165i
\(766\) 0 0
\(767\) 0 0
\(768\) 1.24159e7 + 1.06314e7i 0.759581 + 0.650412i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 2.11657e7 2.47183e7i 1.28232 1.49755i
\(772\) 1.91337e7 1.15546
\(773\) 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
\(779\) 1.60640e6i 0.0948439i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.88274e7 + 1.78167e7i 1.68036 + 1.03854i
\(784\) −4.09254e6 −0.237795
\(785\) 0 0
\(786\) 0 0
\(787\) 3.71139e6 0.213599 0.106800 0.994281i \(-0.465940\pi\)
0.106800 + 0.994281i \(0.465940\pi\)
\(788\) 3.42668e7i 1.96588i
\(789\) 2.20913e7 2.57992e7i 1.26336 1.47542i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −4.02978e7 3.45061e7i −2.26133 1.93632i
\(796\) −3.21924e7 −1.80082
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0