Properties

Label 177.4.d.a.176.1
Level $177$
Weight $4$
Character 177.176
Analytic conductor $10.443$
Analytic rank $0$
Dimension $2$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-59}) \)
Defining polynomial: \(x^{2} - x + 15\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 176.1
Root \(0.500000 + 3.84057i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.4.d.a.176.2

$q$-expansion

\(f(q)\) \(=\) \(q+(3.50000 - 3.84057i) q^{3} -8.00000 q^{4} +7.68115i q^{5} +29.0000 q^{7} +(-2.50000 - 26.8840i) q^{9} +O(q^{10})\) \(q+(3.50000 - 3.84057i) q^{3} -8.00000 q^{4} +7.68115i q^{5} +29.0000 q^{7} +(-2.50000 - 26.8840i) q^{9} +(-28.0000 + 30.7246i) q^{12} +(29.5000 + 26.8840i) q^{15} +64.0000 q^{16} -61.4492i q^{17} +119.000 q^{19} -61.4492i q^{20} +(101.500 - 111.377i) q^{21} +66.0000 q^{25} +(-112.000 - 84.4926i) q^{27} -232.000 q^{28} -268.840i q^{29} +222.753i q^{35} +(20.0000 + 215.072i) q^{36} +7.68115i q^{41} +(206.500 - 19.2029i) q^{45} +(224.000 - 245.797i) q^{48} +498.000 q^{49} +(-236.000 - 215.072i) q^{51} +698.984i q^{53} +(416.500 - 457.028i) q^{57} +453.188i q^{59} +(-236.000 - 215.072i) q^{60} +(-72.5000 - 779.636i) q^{63} -512.000 q^{64} +491.593i q^{68} +1182.90i q^{71} +(231.000 - 253.478i) q^{75} -952.000 q^{76} -835.000 q^{79} +491.593i q^{80} +(-716.500 + 134.420i) q^{81} +(-812.000 + 891.013i) q^{84} +472.000 q^{85} +(-1032.50 - 940.940i) q^{87} +914.056i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 7q^{3} - 16q^{4} + 58q^{7} - 5q^{9} + O(q^{10}) \) \( 2q + 7q^{3} - 16q^{4} + 58q^{7} - 5q^{9} - 56q^{12} + 59q^{15} + 128q^{16} + 238q^{19} + 203q^{21} + 132q^{25} - 224q^{27} - 464q^{28} + 40q^{36} + 413q^{45} + 448q^{48} + 996q^{49} - 472q^{51} + 833q^{57} - 472q^{60} - 145q^{63} - 1024q^{64} + 462q^{75} - 1904q^{76} - 1670q^{79} - 1433q^{81} - 1624q^{84} + 944q^{85} - 2065q^{87} + O(q^{100}) \)

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 \(0.5\pi\)
\(3\) 3.50000 3.84057i 0.673575 0.739119i
\(4\) −8.00000 −1.00000
\(5\) 7.68115i 0.687023i 0.939149 + 0.343511i \(0.111616\pi\)
−0.939149 + 0.343511i \(0.888384\pi\)
\(6\) 0 0
\(7\) 29.0000 1.56585 0.782926 0.622114i \(-0.213726\pi\)
0.782926 + 0.622114i \(0.213726\pi\)
\(8\) 0 0
\(9\) −2.50000 26.8840i −0.0925926 0.995704i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −28.0000 + 30.7246i −0.673575 + 0.739119i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 29.5000 + 26.8840i 0.507791 + 0.462761i
\(16\) 64.0000 1.00000
\(17\) 61.4492i 0.876683i −0.898808 0.438342i \(-0.855566\pi\)
0.898808 0.438342i \(-0.144434\pi\)
\(18\) 0 0
\(19\) 119.000 1.43687 0.718433 0.695596i \(-0.244859\pi\)
0.718433 + 0.695596i \(0.244859\pi\)
\(20\) 61.4492i 0.687023i
\(21\) 101.500 111.377i 1.05472 1.15735i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 66.0000 0.528000
\(26\) 0 0
\(27\) −112.000 84.4926i −0.798311 0.602245i
\(28\) −232.000 −1.56585
\(29\) 268.840i 1.72146i −0.509061 0.860730i \(-0.670007\pi\)
0.509061 0.860730i \(-0.329993\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 222.753i 1.07578i
\(36\) 20.0000 + 215.072i 0.0925926 + 0.995704i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.68115i 0.0292584i 0.999893 + 0.0146292i \(0.00465678\pi\)
−0.999893 + 0.0146292i \(0.995343\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 206.500 19.2029i 0.684071 0.0636132i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 224.000 245.797i 0.673575 0.739119i
\(49\) 498.000 1.45190
\(50\) 0 0
\(51\) −236.000 215.072i −0.647973 0.590512i
\(52\) 0 0
\(53\) 698.984i 1.81156i 0.423744 + 0.905782i \(0.360715\pi\)
−0.423744 + 0.905782i \(0.639285\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 416.500 457.028i 0.967838 1.06202i
\(58\) 0 0
\(59\) 453.188i 1.00000i
\(60\) −236.000 215.072i −0.507791 0.462761i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −72.5000 779.636i −0.144986 1.55913i
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 491.593i 0.876683i
\(69\) 0 0
\(70\) 0 0
\(71\) 1182.90i 1.97724i 0.150437 + 0.988620i \(0.451932\pi\)
−0.150437 + 0.988620i \(0.548068\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 231.000 253.478i 0.355648 0.390255i
\(76\) −952.000 −1.43687
\(77\) 0 0
\(78\) 0 0
\(79\) −835.000 −1.18918 −0.594588 0.804031i \(-0.702685\pi\)
−0.594588 + 0.804031i \(0.702685\pi\)
\(80\) 491.593i 0.687023i
\(81\) −716.500 + 134.420i −0.982853 + 0.184390i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −812.000 + 891.013i −1.05472 + 1.15735i
\(85\) 472.000 0.602301
\(86\) 0 0
\(87\) −1032.50 940.940i −1.27236 1.15953i
\(88\) 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
\(95\) 914.056i 0.987160i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −528.000 −0.528000
\(101\) 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
\(105\) 855.500 + 779.636i 0.795126 + 0.724616i
\(106\) 0 0
\(107\) 2204.49i 1.99174i −0.0908010 0.995869i \(-0.528943\pi\)
0.0908010 0.995869i \(-0.471057\pi\)
\(108\) 896.000 + 675.941i 0.798311 + 0.602245i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1856.00 1.56585
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2150.72i 1.72146i
\(117\) 0 0
\(118\) 0 0
\(119\) 1782.03i 1.37276i
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 29.5000 + 26.8840i 0.0216254 + 0.0197077i
\(124\) 0 0
\(125\) 1467.10i 1.04977i
\(126\) 0 0
\(127\) 821.000 0.573638 0.286819 0.957985i \(-0.407402\pi\)
0.286819 + 0.957985i \(0.407402\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 3451.00 2.24992
\(134\) 0 0
\(135\) 649.000 860.288i 0.413756 0.548458i
\(136\) 0 0
\(137\) 3172.31i 1.97831i −0.146862 0.989157i \(-0.546917\pi\)
0.146862 0.989157i \(-0.453083\pi\)
\(138\) 0 0
\(139\) −1960.00 −1.19601 −0.598004 0.801493i \(-0.704039\pi\)
−0.598004 + 0.801493i \(0.704039\pi\)
\(140\) 1782.03i 1.07578i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −160.000 1720.58i −0.0925926 0.995704i
\(145\) 2065.00 1.18268
\(146\) 0 0
\(147\) 1743.00 1912.61i 0.977961 1.07312i
\(148\) 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
\(153\) −1652.00 + 153.623i −0.872917 + 0.0811744i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 2684.50 + 2446.44i 1.33896 + 1.22022i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4048.00 −1.94518 −0.972588 0.232533i \(-0.925299\pi\)
−0.972588 + 0.232533i \(0.925299\pi\)
\(164\) 61.4492i 0.0292584i
\(165\) 0 0
\(166\) 0 0
\(167\) 3325.94i 1.54113i 0.637362 + 0.770565i \(0.280026\pi\)
−0.637362 + 0.770565i \(0.719974\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) −297.500 3199.20i −0.133043 1.43069i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 1914.00 0.826770
\(176\) 0 0
\(177\) 1740.50 + 1586.16i 0.739119 + 0.673575i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1652.00 + 153.623i −0.684071 + 0.0636132i
\(181\) −4795.00 −1.96911 −0.984557 0.175066i \(-0.943986\pi\)
−0.984557 + 0.175066i \(0.943986\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3248.00 2450.29i −1.25004 0.943027i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1792.00 + 1966.37i −0.673575 + 0.739119i
\(193\) 1937.00 0.722426 0.361213 0.932483i \(-0.382363\pi\)
0.361213 + 0.932483i \(0.382363\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3984.00 −1.45190
\(197\) 4086.37i 1.47788i 0.673773 + 0.738939i \(0.264673\pi\)
−0.673773 + 0.738939i \(0.735327\pi\)
\(198\) 0 0
\(199\) 5411.00 1.92752 0.963758 0.266779i \(-0.0859592\pi\)
0.963758 + 0.266779i \(0.0859592\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7796.36i 2.69555i
\(204\) 1888.00 + 1720.58i 0.647973 + 0.590512i
\(205\) −59.0000 −0.0201012
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 5591.87i 1.81156i
\(213\) 4543.00 + 4140.14i 1.46141 + 1.33182i
\(214\) 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
\(223\) 5852.00 1.75730 0.878652 0.477462i \(-0.158443\pi\)
0.878652 + 0.477462i \(0.158443\pi\)
\(224\) 0 0
\(225\) −165.000 1774.34i −0.0488889 0.525732i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −3332.00 + 3656.23i −0.967838 + 1.06202i
\(229\) 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
\(236\) 3625.50i 1.00000i
\(237\) −2922.50 + 3206.88i −0.800999 + 0.878941i
\(238\) 0 0
\(239\) 7043.61i 1.90633i −0.302448 0.953166i \(-0.597804\pi\)
0.302448 0.953166i \(-0.402196\pi\)
\(240\) 1888.00 + 1720.58i 0.507791 + 0.462761i
\(241\) −5803.00 −1.55105 −0.775527 0.631314i \(-0.782516\pi\)
−0.775527 + 0.631314i \(0.782516\pi\)
\(242\) 0 0
\(243\) −1991.50 + 3222.24i −0.525740 + 0.850645i
\(244\) 0 0
\(245\) 3825.21i 0.997485i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7750.28i 1.94898i 0.224439 + 0.974488i \(0.427945\pi\)
−0.224439 + 0.974488i \(0.572055\pi\)
\(252\) 580.000 + 6237.09i 0.144986 + 1.55913i
\(253\) 0 0
\(254\) 0 0
\(255\) 1652.00 1812.75i 0.405695 0.445172i
\(256\) 4096.00 1.00000
\(257\) 3325.94i 0.807261i 0.914922 + 0.403631i \(0.132252\pi\)
−0.914922 + 0.403631i \(0.867748\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7227.50 + 672.100i −1.71407 + 0.159394i
\(262\) 0 0
\(263\) 8441.58i 1.97920i 0.143840 + 0.989601i \(0.454055\pi\)
−0.143840 + 0.989601i \(0.545945\pi\)
\(264\) 0 0
\(265\) −5369.00 −1.24459
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −8575.00 −1.92212 −0.961059 0.276342i \(-0.910878\pi\)
−0.961059 + 0.276342i \(0.910878\pi\)
\(272\) 3932.75i 0.876683i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9151.00 −1.98495 −0.992473 0.122460i \(-0.960922\pi\)
−0.992473 + 0.122460i \(0.960922\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 268.840i 0.0570735i −0.999593 0.0285368i \(-0.990915\pi\)
0.999593 0.0285368i \(-0.00908476\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 9463.17i 1.97724i
\(285\) 3510.50 + 3199.20i 0.729628 + 0.664927i
\(286\) 0 0
\(287\) 222.753i 0.0458143i
\(288\) 0 0
\(289\) 1137.00 0.231427
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5384.48i 1.07360i −0.843709 0.536800i \(-0.819633\pi\)
0.843709 0.536800i \(-0.180367\pi\)
\(294\) 0 0
\(295\) −3481.00 −0.687023
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −1848.00 + 2027.82i −0.355648 + 0.390255i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 7616.00 1.43687
\(305\) 0 0
\(306\) 0 0
\(307\) −3661.00 −0.680600 −0.340300 0.940317i \(-0.610529\pi\)
−0.340300 + 0.940317i \(0.610529\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8718.10i 1.58958i 0.606887 + 0.794788i \(0.292418\pi\)
−0.606887 + 0.794788i \(0.707582\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 5988.50 556.883i 1.07115 0.0996089i
\(316\) 6680.00 1.18918
\(317\) 215.072i 0.0381062i 0.999818 + 0.0190531i \(0.00606515\pi\)
−0.999818 + 0.0190531i \(0.993935\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3932.75i 0.687023i
\(321\) −8466.50 7715.71i −1.47213 1.34159i
\(322\) 0 0
\(323\) 7312.45i 1.25968i
\(324\) 5732.00 1075.36i 0.982853 0.184390i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12017.0 1.99551 0.997755 0.0669643i \(-0.0213314\pi\)
0.997755 + 0.0669643i \(0.0213314\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 6496.00 7128.10i 1.05472 1.15735i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −3776.00 −0.602301
\(341\) 0 0
\(342\) 0 0
\(343\) 4495.00 0.707601
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 8260.00 + 7527.52i 1.27236 + 1.15953i
\(349\) 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
\(355\) −9086.00 −1.35841
\(356\) 0 0
\(357\) −6844.00 6237.09i −1.01463 0.924655i
\(358\) 0 0
\(359\) 8011.44i 1.17779i −0.808209 0.588896i \(-0.799563\pi\)
0.808209 0.588896i \(-0.200437\pi\)
\(360\) 0 0
\(361\) 7302.00 1.06459
\(362\) 0 0
\(363\) −4658.50 + 5111.80i −0.673575 + 0.739119i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 206.500 19.2029i 0.0291327 0.00270911i
\(370\) 0 0
\(371\) 20270.5i 2.83664i
\(372\) 0 0
\(373\) 4898.00 0.679916 0.339958 0.940441i \(-0.389587\pi\)
0.339958 + 0.940441i \(0.389587\pi\)
\(374\) 0 0
\(375\) 5634.50 + 5134.85i 0.775905 + 0.707099i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 13025.0 1.76530 0.882651 0.470029i \(-0.155757\pi\)
0.882651 + 0.470029i \(0.155757\pi\)
\(380\) 7312.45i 0.987160i
\(381\) 2873.50 3153.11i 0.386388 0.423986i
\(382\) 0 0
\(383\) 14025.8i 1.87124i −0.353014 0.935618i \(-0.614843\pi\)
0.353014 0.935618i \(-0.385157\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11829.0i 1.54178i 0.636968 + 0.770890i \(0.280188\pi\)
−0.636968 + 0.770890i \(0.719812\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6413.76i 0.816990i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 12078.5 13253.8i 1.51549 1.66296i
\(400\) 4224.00 0.528000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1032.50 5503.54i −0.126680 0.675242i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −12183.5 11103.1i −1.46221 1.33254i
\(412\) 0 0
\(413\) 13142.4i 1.56585i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6860.00 + 7527.52i −0.805601 + 0.883991i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −6844.00 6237.09i −0.795126 0.724616i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4055.64i 0.462889i
\(426\) 0 0
\(427\) 0 0
\(428\) 17635.9i 1.99174i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −7168.00 5407.53i −0.798311 0.602245i
\(433\) −14623.0 −1.62295 −0.811474 0.584388i \(-0.801334\pi\)
−0.811474 + 0.584388i \(0.801334\pi\)
\(434\) 0 0
\(435\) 7227.50 7930.78i 0.796626 0.874142i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −3220.00 −0.350073 −0.175037 0.984562i \(-0.556004\pi\)
−0.175037 + 0.984562i \(0.556004\pi\)
\(440\) 0 0
\(441\) −1245.00 13388.2i −0.134435 1.44566i
\(442\) 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
\(448\) −14848.0 −1.56585
\(449\) 9409.40i 0.988992i 0.869180 + 0.494496i \(0.164647\pi\)
−0.869180 + 0.494496i \(0.835353\pi\)
\(450\) 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
\(459\) −5192.00 + 6882.31i −0.527978 + 0.699866i
\(460\) 0 0
\(461\) 13488.1i 1.36270i 0.731959 + 0.681348i \(0.238606\pi\)
−0.731959 + 0.681348i \(0.761394\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 17205.8i 1.72146i
\(465\) 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
\(475\) 7854.00 0.758666
\(476\) 14256.2i 1.37276i
\(477\) 18791.5 1747.46i 1.80378 0.167737i
\(478\) 0 0
\(479\) 1459.42i 0.139212i 0.997575 + 0.0696059i \(0.0221742\pi\)
−0.997575 + 0.0696059i \(0.977826\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 19721.0 1.83500 0.917499 0.397739i \(-0.130205\pi\)
0.917499 + 0.397739i \(0.130205\pi\)
\(488\) 0 0
\(489\) −14168.0 + 15546.6i −1.31022 + 1.43772i
\(490\) 0 0
\(491\) 9947.08i 0.914268i −0.889398 0.457134i \(-0.848876\pi\)
0.889398 0.457134i \(-0.151124\pi\)
\(492\) −236.000 215.072i −0.0216254 0.0197077i
\(493\) −16520.0 −1.50918
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34304.0i 3.09607i
\(498\) 0 0
\(499\) −9511.00 −0.853248 −0.426624 0.904429i \(-0.640297\pi\)
−0.426624 + 0.904429i \(0.640297\pi\)
\(500\) 11736.8i 1.04977i
\(501\) 12773.5 + 11640.8i 1.13908 + 1.03807i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7689.50 8437.74i 0.673575 0.739119i
\(508\) −6568.00 −0.573638
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −13328.0 10054.6i −1.14707 0.865346i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11552.4i 0.971443i 0.874114 + 0.485721i \(0.161443\pi\)
−0.874114 + 0.485721i \(0.838557\pi\)
\(522\) 0 0
\(523\) 6167.00 0.515610 0.257805 0.966197i \(-0.417001\pi\)
0.257805 + 0.966197i \(0.417001\pi\)
\(524\) 0 0
\(525\) 6699.00 7350.86i 0.556892 0.611081i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 12183.5 1132.97i 0.995704 0.0925926i
\(532\) −27608.0 −2.24992
\(533\) 0 0
\(534\) 0 0
\(535\) 16933.0 1.36837
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −5192.00 + 6882.31i −0.413756 + 0.548458i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −16782.5 + 18415.5i −1.32635 + 1.45541i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12616.0 −0.986145 −0.493072 0.869988i \(-0.664126\pi\)
−0.493072 + 0.869988i \(0.664126\pi\)
\(548\) 25378.5i 1.97831i
\(549\) 0 0
\(550\) 0 0
\(551\) 31992.0i 2.47351i
\(552\) 0 0
\(553\) −24215.0 −1.86207
\(554\) 0 0
\(555\) 0 0
\(556\) 15680.0 1.19601
\(557\) 8011.44i 0.609435i −0.952443 0.304718i \(-0.901438\pi\)
0.952443 0.304718i \(-0.0985621\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 14256.2i 1.07578i
\(561\) 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
\(567\) −20778.5 + 3898.18i −1.53900 + 0.288727i
\(568\) 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
\(576\) 1280.00 + 13764.6i 0.0925926 + 0.995704i
\(577\) 9569.00 0.690403 0.345202 0.938529i \(-0.387811\pi\)
0.345202 + 0.938529i \(0.387811\pi\)
\(578\) 0 0
\(579\) 6779.50 7439.19i 0.486609 0.533959i
\(580\) −16520.0 −1.18268
\(581\) 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 \(0.5\pi\)
\(588\) −13944.0 + 15300.8i −0.977961 + 1.07312i
\(589\) 0 0
\(590\) 0 0
\(591\) 15694.0 + 14302.3i 1.09233 + 0.995461i
\(592\) 0 0
\(593\) 6767.09i 0.468619i −0.972162 0.234309i \(-0.924717\pi\)
0.972162 0.234309i \(-0.0752829\pi\)
\(594\) 0 0
\(595\) 13688.0 0.943115
\(596\) 0 0
\(597\) 18938.5 20781.3i 1.29833 1.42466i
\(598\) 0 0
\(599\) 24464.4i 1.66877i −0.551186 0.834383i \(-0.685824\pi\)
0.551186 0.834383i \(-0.314176\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10223.6i 0.687023i
\(606\) 0 0
\(607\) 25319.0 1.69303 0.846513 0.532368i \(-0.178698\pi\)
0.846513 + 0.532368i \(0.178698\pi\)
\(608\) 0 0
\(609\) −29942.5 27287.3i −1.99233 1.81566i
\(610\) 0 0
\(611\) 0 0
\(612\) 13216.0 1228.98i 0.872917 0.0811744i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −206.500 + 226.594i −0.0135396 + 0.0148571i
\(616\) 0 0
\(617\) 16184.2i 1.05600i 0.849245 + 0.527999i \(0.177057\pi\)
−0.849245 + 0.527999i \(0.822943\pi\)
\(618\) 0 0
\(619\) 30485.0 1.97948 0.989738 0.142894i \(-0.0456409\pi\)
0.989738 + 0.142894i \(0.0456409\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3019.00 −0.193216
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −29860.0 −1.88385 −0.941924 0.335827i \(-0.890984\pi\)
−0.941924 + 0.335827i \(0.890984\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6306.22i 0.394102i
\(636\) −21476.0 19571.6i −1.33896 1.22022i
\(637\) 0 0
\(638\) 0 0
\(639\) 31801.0 2957.24i 1.96875 0.183078i
\(640\) 0 0
\(641\) 17205.8i 1.06020i −0.847936 0.530099i \(-0.822155\pi\)
0.847936 0.530099i \(-0.177845\pi\)
\(642\) 0 0
\(643\) −18907.0 −1.15959 −0.579797 0.814761i \(-0.696868\pi\)
−0.579797 + 0.814761i \(0.696868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32898.3i 1.99902i −0.0312629 0.999511i \(-0.509953\pi\)
0.0312629 0.999511i \(-0.490047\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 32384.0 1.94518
\(653\) 24894.6i 1.49188i 0.666011 + 0.745942i \(0.268001\pi\)
−0.666011 + 0.745942i \(0.731999\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 491.593i 0.0292584i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 31745.0 1.86798 0.933992 0.357294i \(-0.116301\pi\)
0.933992 + 0.357294i \(0.116301\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 26507.6i 1.54575i
\(666\) 0 0
\(667\) 0 0
\(668\) 26607.5i 1.54113i
\(669\) 20482.0 22475.0i 1.18368 1.29886i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −7392.00 5576.51i −0.421508 0.317985i
\(676\) −17576.0 −1.00000
\(677\) 33397.6i 1.89597i 0.318308 + 0.947987i \(0.396885\pi\)
−0.318308 + 0.947987i \(0.603115\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 2380.00 + 25593.6i 0.133043 + 1.43069i
\(685\) 24367.0 1.35915
\(686\) 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
\(695\) 15055.0i 0.821684i
\(696\) 0 0
\(697\) 472.000 0.0256503
\(698\) 0 0
\(699\) 0 0
\(700\) −15312.0 −0.826770
\(701\) 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
\(708\) −13924.0 12689.3i −0.739119 0.673575i
\(709\) −14929.0 −0.790790 −0.395395 0.918511i \(-0.629392\pi\)
−0.395395 + 0.918511i \(0.629392\pi\)
\(710\) 0 0
\(711\) 2087.50 + 22448.1i 0.110109 + 1.18407i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −27051.5 24652.6i −1.40901 1.28406i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 13216.0 1228.98i 0.684071 0.0636132i
\(721\) 0 0
\(722\) 0 0
\(723\) −20310.5 + 22286.8i −1.04475 + 1.14641i
\(724\) 38360.0 1.96911
\(725\) 17743.4i 0.908931i
\(726\) 0 0
\(727\) 39116.0 1.99551 0.997753 0.0670071i \(-0.0213450\pi\)
0.997753 + 0.0670071i \(0.0213450\pi\)
\(728\) 0 0
\(729\) 5405.00 + 18926.3i 0.274602 + 0.961558i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −9898.00 −0.498760 −0.249380 0.968406i \(-0.580227\pi\)
−0.249380 + 0.968406i \(0.580227\pi\)
\(734\) 0 0
\(735\) 14691.0 + 13388.2i 0.737259 + 0.671881i
\(736\) 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
\(743\) 35594.4i 1.75751i −0.477269 0.878757i \(-0.658373\pi\)
0.477269 0.878757i \(-0.341627\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 63930.2i 3.11877i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 29765.5 + 27126.0i 1.44052 + 1.31278i
\(754\) 0 0
\(755\) 0 0
\(756\) 25984.0 + 19602.3i 1.25004 + 0.943027i
\(757\) −19861.0 −0.953580 −0.476790 0.879017i \(-0.658200\pi\)
−0.476790 + 0.879017i \(0.658200\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34419.2i 1.63955i −0.572688 0.819774i \(-0.694099\pi\)
0.572688 0.819774i \(-0.305901\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1180.00 12689.3i −0.0557686 0.599714i
\(766\) 0 0
\(767\) 0 0
\(768\) 14336.0 15731.0i 0.673575 0.739119i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 12773.5 + 11640.8i 0.596662 + 0.543751i
\(772\) −15496.0 −0.722426
\(773\) 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
\(779\) 914.056i 0.0420404i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −22715.0 + 30110.1i −1.03674 + 1.37426i
\(784\) 31872.0 1.45190
\(785\) 0 0
\(786\) 0 0
\(787\) −42784.0 −1.93785 −0.968923 0.247362i \(-0.920436\pi\)
−0.968923 + 0.247362i \(0.920436\pi\)
\(788\) 32691.0i 1.47788i
\(789\) 32420.5 + 29545.5i 1.46286 + 1.33314i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −18791.5 + 20620.0i −0.838322 + 0.919896i
\(796\) −43288.0 −1.92752
\(797\) 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