Properties

Label 177.4.d.b.176.2
Level $177$
Weight $4$
Character 177.176
Analytic conductor $10.443$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-59})\)
Defining polynomial: \(x^{4} - x^{3} - 14 x^{2} - 15 x + 225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 176.2
Root \(3.57603 + 1.48727i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.4.d.b.176.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-5.07603 + 1.11080i) q^{3} -8.00000 q^{4} +22.0271i q^{5} -34.4562 q^{7} +(24.5322 - 11.2769i) q^{9} +O(q^{10})\) \(q+(-5.07603 + 1.11080i) q^{3} -8.00000 q^{4} +22.0271i q^{5} -34.4562 q^{7} +(24.5322 - 11.2769i) q^{9} +(40.6083 - 8.88642i) q^{12} +(-24.4678 - 111.810i) q^{15} +64.0000 q^{16} +61.4492i q^{17} +40.2810 q^{19} -176.217i q^{20} +(174.901 - 38.2740i) q^{21} -360.193 q^{25} +(-112.000 + 84.4926i) q^{27} +275.650 q^{28} -272.118i q^{29} -758.970i q^{35} +(-196.258 + 90.2155i) q^{36} +458.504i q^{41} +(248.398 + 540.374i) q^{45} +(-324.866 + 71.0914i) q^{48} +844.230 q^{49} +(-68.2579 - 311.918i) q^{51} +632.682i q^{53} +(-204.468 + 44.7442i) q^{57} -453.188i q^{59} +(195.742 + 894.483i) q^{60} +(-845.288 + 388.561i) q^{63} -512.000 q^{64} -491.593i q^{68} -1182.90i q^{71} +(1828.35 - 400.104i) q^{75} -322.248 q^{76} -560.354 q^{79} +1409.73i q^{80} +(474.661 - 553.297i) q^{81} +(-1399.21 + 306.192i) q^{84} -1353.55 q^{85} +(302.269 + 1381.28i) q^{87} +887.274i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + O(q^{10}) \) \( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + 56q^{12} - 191q^{15} + 256q^{16} - 238q^{19} + 367q^{21} - 882q^{25} - 448q^{27} + 464q^{28} - 40q^{36} + 49q^{45} - 448q^{48} + 1062q^{49} + 472q^{51} - 911q^{57} + 1528q^{60} - 1931q^{63} - 2048q^{64} + 3402q^{75} + 1904q^{76} + 1670q^{79} + 1433q^{81} - 2936q^{84} - 944q^{85} + 637q^{87} + O(q^{100}) \)

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 \(0.5\pi\)
\(3\) −5.07603 + 1.11080i −0.976883 + 0.213774i
\(4\) −8.00000 −1.00000
\(5\) 22.0271i 1.97016i 0.172085 + 0.985082i \(0.444950\pi\)
−0.172085 + 0.985082i \(0.555050\pi\)
\(6\) 0 0
\(7\) −34.4562 −1.86046 −0.930230 0.366977i \(-0.880393\pi\)
−0.930230 + 0.366977i \(0.880393\pi\)
\(8\) 0 0
\(9\) 24.5322 11.2769i 0.908601 0.417665i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 40.6083 8.88642i 0.976883 0.213774i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −24.4678 111.810i −0.421170 1.92462i
\(16\) 64.0000 1.00000
\(17\) 61.4492i 0.876683i 0.898808 + 0.438342i \(0.144434\pi\)
−0.898808 + 0.438342i \(0.855566\pi\)
\(18\) 0 0
\(19\) 40.2810 0.486374 0.243187 0.969979i \(-0.421807\pi\)
0.243187 + 0.969979i \(0.421807\pi\)
\(20\) 176.217i 1.97016i
\(21\) 174.901 38.2740i 1.81745 0.397718i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −360.193 −2.88155
\(26\) 0 0
\(27\) −112.000 + 84.4926i −0.798311 + 0.602245i
\(28\) 275.650 1.86046
\(29\) 272.118i 1.74245i −0.490884 0.871225i \(-0.663326\pi\)
0.490884 0.871225i \(-0.336674\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 758.970i 3.66541i
\(36\) −196.258 + 90.2155i −0.908601 + 0.417665i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 458.504i 1.74649i 0.487277 + 0.873247i \(0.337990\pi\)
−0.487277 + 0.873247i \(0.662010\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 248.398 + 540.374i 0.822868 + 1.79009i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −324.866 + 71.0914i −0.976883 + 0.213774i
\(49\) 844.230 2.46131
\(50\) 0 0
\(51\) −68.2579 311.918i −0.187412 0.856417i
\(52\) 0 0
\(53\) 632.682i 1.63973i 0.572558 + 0.819864i \(0.305951\pi\)
−0.572558 + 0.819864i \(0.694049\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −204.468 + 44.7442i −0.475130 + 0.103974i
\(58\) 0 0
\(59\) 453.188i 1.00000i
\(60\) 195.742 + 894.483i 0.421170 + 1.92462i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −845.288 + 388.561i −1.69042 + 0.777048i
\(64\) −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 \(-0.548068\pi\)
0.150437 0.988620i \(-0.451932\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1828.35 400.104i 2.81493 0.616000i
\(76\) −322.248 −0.486374
\(77\) 0 0
\(78\) 0 0
\(79\) −560.354 −0.798035 −0.399017 0.916943i \(-0.630649\pi\)
−0.399017 + 0.916943i \(0.630649\pi\)
\(80\) 1409.73i 1.97016i
\(81\) 474.661 553.297i 0.651113 0.758981i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1399.21 + 306.192i −1.81745 + 0.397718i
\(85\) −1353.55 −1.72721
\(86\) 0 0
\(87\) 302.269 + 1381.28i 0.372491 + 1.70217i
\(88\) 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
\(95\) 887.274i 0.958236i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2881.55 2.88155
\(101\) 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
\(105\) 843.066 + 3852.56i 0.783570 + 3.58068i
\(106\) 0 0
\(107\) 928.173i 0.838597i −0.907848 0.419299i \(-0.862276\pi\)
0.907848 0.419299i \(-0.137724\pi\)
\(108\) 896.000 675.941i 0.798311 0.602245i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2205.20 −1.86046
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2176.94i 1.74245i
\(117\) 0 0
\(118\) 0 0
\(119\) 2117.30i 1.63103i
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) −509.307 2327.38i −0.373355 1.70612i
\(124\) 0 0
\(125\) 5180.63i 3.70696i
\(126\) 0 0
\(127\) 1964.29 1.37246 0.686230 0.727385i \(-0.259264\pi\)
0.686230 + 0.727385i \(0.259264\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −1387.93 −0.904879
\(134\) 0 0
\(135\) −1861.13 2467.04i −1.18652 1.57280i
\(136\) 0 0
\(137\) 1178.26i 0.734784i −0.930066 0.367392i \(-0.880251\pi\)
0.930066 0.367392i \(-0.119749\pi\)
\(138\) 0 0
\(139\) −1960.00 −1.19601 −0.598004 0.801493i \(-0.704039\pi\)
−0.598004 + 0.801493i \(0.704039\pi\)
\(140\) 6071.76i 3.66541i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1570.06 721.724i 0.908601 0.417665i
\(145\) 5993.97 3.43291
\(146\) 0 0
\(147\) −4285.34 + 937.773i −2.40441 + 0.526165i
\(148\) 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
\(153\) 692.959 + 1507.49i 0.366159 + 0.796555i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −702.785 3211.52i −0.350531 1.60182i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4048.00 −1.94518 −0.972588 0.232533i \(-0.925299\pi\)
−0.972588 + 0.232533i \(0.925299\pi\)
\(164\) 3668.03i 1.74649i
\(165\) 0 0
\(166\) 0 0
\(167\) 719.468i 0.333378i −0.986010 0.166689i \(-0.946692\pi\)
0.986010 0.166689i \(-0.0533076\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) 988.183 454.247i 0.441920 0.203141i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 12410.9 5.36100
\(176\) 0 0
\(177\) 503.402 + 2300.40i 0.213774 + 0.976883i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1987.19 4322.99i −0.822868 1.79009i
\(181\) 1659.12 0.681334 0.340667 0.940184i \(-0.389347\pi\)
0.340667 + 0.940184i \(0.389347\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3859.09 2911.29i 1.48523 1.12045i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 2598.93 568.731i 0.976883 0.213774i
\(193\) −5299.00 −1.97632 −0.988161 0.153422i \(-0.950971\pi\)
−0.988161 + 0.153422i \(0.950971\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6753.84 −2.46131
\(197\) 4086.37i 1.47788i −0.673773 0.738939i \(-0.735327\pi\)
0.673773 0.738939i \(-0.264673\pi\)
\(198\) 0 0
\(199\) −1408.35 −0.501684 −0.250842 0.968028i \(-0.580707\pi\)
−0.250842 + 0.968028i \(0.580707\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9376.16i 3.24176i
\(204\) 546.063 + 2495.34i 0.187412 + 0.856417i
\(205\) −10099.5 −3.44088
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 5061.46i 1.63973i
\(213\) 1313.96 + 6004.42i 0.422682 + 1.93153i
\(214\) 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
\(225\) −8836.35 + 4061.88i −2.61818 + 1.20352i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 1635.74 357.954i 0.475130 0.103974i
\(229\) 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
\(237\) 2844.38 622.442i 0.779587 0.170599i
\(238\) 0 0
\(239\) 1586.24i 0.429311i −0.976690 0.214655i \(-0.931137\pi\)
0.976690 0.214655i \(-0.0688628\pi\)
\(240\) −1565.94 7155.86i −0.421170 1.92462i
\(241\) −1189.52 −0.317941 −0.158971 0.987283i \(-0.550817\pi\)
−0.158971 + 0.987283i \(0.550817\pi\)
\(242\) 0 0
\(243\) −1794.79 + 3335.81i −0.473811 + 0.880627i
\(244\) 0 0
\(245\) 18595.9i 4.84919i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5420.99i 1.36323i 0.731712 + 0.681614i \(0.238722\pi\)
−0.731712 + 0.681614i \(0.761278\pi\)
\(252\) 6762.30 3108.48i 1.69042 0.777048i
\(253\) 0 0
\(254\) 0 0
\(255\) 6870.65 1503.52i 1.68728 0.369233i
\(256\) 4096.00 1.00000
\(257\) 4866.00i 1.18106i −0.807015 0.590530i \(-0.798919\pi\)
0.807015 0.590530i \(-0.201081\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3068.66 6675.67i −0.727760 1.58319i
\(262\) 0 0
\(263\) 5283.40i 1.23874i 0.785099 + 0.619370i \(0.212612\pi\)
−0.785099 + 0.619370i \(0.787388\pi\)
\(264\) 0 0
\(265\) −13936.2 −3.23053
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 6422.81 1.43970 0.719849 0.694131i \(-0.244211\pi\)
0.719849 + 0.694131i \(0.244211\pi\)
\(272\) 3932.75i 0.876683i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3597.65 0.780367 0.390183 0.920737i \(-0.372412\pi\)
0.390183 + 0.920737i \(0.372412\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8020.94i 1.70281i 0.524510 + 0.851404i \(0.324249\pi\)
−0.524510 + 0.851404i \(0.675751\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 9463.17i 1.97724i
\(285\) −985.586 4503.83i −0.204846 0.936084i
\(286\) 0 0
\(287\) 15798.3i 3.24928i
\(288\) 0 0
\(289\) 1137.00 0.231427
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4636.93i 0.924548i 0.886737 + 0.462274i \(0.152966\pi\)
−0.886737 + 0.462274i \(0.847034\pi\)
\(294\) 0 0
\(295\) 9982.41 1.97016
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −14626.8 + 3200.83i −2.81493 + 0.616000i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 2577.98 0.486374
\(305\) 0 0
\(306\) 0 0
\(307\) −6930.27 −1.28838 −0.644188 0.764867i \(-0.722805\pi\)
−0.644188 + 0.764867i \(0.722805\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1406.08i 0.256371i −0.991750 0.128186i \(-0.959085\pi\)
0.991750 0.128186i \(-0.0409154\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −8558.87 18619.2i −1.53091 3.33040i
\(316\) 4482.83 0.798035
\(317\) 215.072i 0.0381062i −0.999818 0.0190531i \(-0.993935\pi\)
0.999818 0.0190531i \(-0.00606515\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 11277.9i 1.97016i
\(321\) 1031.02 + 4711.44i 0.179270 + 0.819211i
\(322\) 0 0
\(323\) 2475.23i 0.426395i
\(324\) −3797.29 + 4426.38i −0.651113 + 0.758981i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5310.03 −0.881770 −0.440885 0.897564i \(-0.645335\pi\)
−0.440885 + 0.897564i \(0.645335\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 11193.7 2449.54i 1.81745 0.397718i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 10828.4 1.72721
\(341\) 0 0
\(342\) 0 0
\(343\) −17270.5 −2.71871
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −2418.16 11050.2i −0.372491 1.70217i
\(349\) 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
\(355\) 26055.8 3.89549
\(356\) 0 0
\(357\) 2351.91 + 10747.5i 0.348673 + 1.59333i
\(358\) 0 0
\(359\) 13527.7i 1.98875i −0.105895 0.994377i \(-0.533771\pi\)
0.105895 0.994377i \(-0.466229\pi\)
\(360\) 0 0
\(361\) −5236.44 −0.763441
\(362\) 0 0
\(363\) 6756.20 1478.48i 0.976883 0.213774i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 5170.52 + 11248.1i 0.729449 + 1.58687i
\(370\) 0 0
\(371\) 21799.8i 3.05065i
\(372\) 0 0
\(373\) 4898.00 0.679916 0.339958 0.940441i \(-0.389587\pi\)
0.339958 + 0.940441i \(0.389587\pi\)
\(374\) 0 0
\(375\) 5754.66 + 26297.1i 0.792451 + 3.62126i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −12519.3 −1.69677 −0.848383 0.529383i \(-0.822423\pi\)
−0.848383 + 0.529383i \(0.822423\pi\)
\(380\) 7098.19i 0.958236i
\(381\) −9970.79 + 2181.94i −1.34073 + 0.293396i
\(382\) 0 0
\(383\) 14025.8i 1.87124i 0.353014 + 0.935618i \(0.385157\pi\)
−0.353014 + 0.935618i \(0.614843\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11829.0i 1.54178i −0.636968 0.770890i \(-0.719812\pi\)
0.636968 0.770890i \(-0.280188\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12343.0i 1.57226i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 7045.18 1541.72i 0.883961 0.193440i
\(400\) −23052.4 −2.88155
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 12187.5 + 10455.4i 1.49532 + 1.28280i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 1308.81 + 5980.88i 0.157078 + 0.717798i
\(412\) 0 0
\(413\) 15615.1i 1.86046i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 9949.03 2177.17i 1.16836 0.255675i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −6744.53 30820.5i −0.783570 3.58068i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22133.6i 2.52620i
\(426\) 0 0
\(427\) 0 0
\(428\) 7425.39i 0.838597i
\(429\) 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
\(433\) −1808.48 −0.200716 −0.100358 0.994951i \(-0.531999\pi\)
−0.100358 + 0.994951i \(0.531999\pi\)
\(434\) 0 0
\(435\) −30425.6 + 6658.12i −3.35356 + 0.733868i
\(436\) 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
\(441\) 20710.8 9520.33i 2.23635 1.02800i
\(442\) 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
\(448\) 17641.6 1.86046
\(449\) 9618.49i 1.01097i −0.862836 0.505484i \(-0.831314\pi\)
0.862836 0.505484i \(-0.168686\pi\)
\(450\) 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 \(-0.761394\pi\)
0.731959 0.681348i \(-0.238606\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 17415.6i 1.74245i
\(465\) 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
\(475\) −14509.0 −1.40151
\(476\) 16938.4i 1.63103i
\(477\) 7134.72 + 15521.1i 0.684856 + 1.48986i
\(478\) 0 0
\(479\) 1459.42i 0.139212i −0.997575 0.0696059i \(-0.977826\pi\)
0.997575 0.0696059i \(-0.0221742\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −2456.75 −0.228595 −0.114298 0.993447i \(-0.536462\pi\)
−0.114298 + 0.993447i \(0.536462\pi\)
\(488\) 0 0
\(489\) 20547.8 4496.53i 1.90021 0.415828i
\(490\) 0 0
\(491\) 11786.6i 1.08335i 0.840589 + 0.541674i \(0.182209\pi\)
−0.840589 + 0.541674i \(0.817791\pi\)
\(492\) 4074.46 + 18619.1i 0.373355 + 1.70612i
\(493\) 16721.4 1.52758
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 40758.1i 3.67857i
\(498\) 0 0
\(499\) 22217.2 1.99314 0.996571 0.0827472i \(-0.0263694\pi\)
0.996571 + 0.0827472i \(0.0263694\pi\)
\(500\) 41445.0i 3.70696i
\(501\) 799.187 + 3652.04i 0.0712675 + 0.325671i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −11152.0 + 2440.43i −0.976883 + 0.213774i
\(508\) −15714.3 −1.37246
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4511.47 + 3403.45i −0.388278 + 0.292916i
\(514\) 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 \(-0.838557\pi\)
0.874114 0.485721i \(-0.161443\pi\)
\(522\) 0 0
\(523\) −23099.6 −1.93131 −0.965654 0.259833i \(-0.916333\pi\)
−0.965654 + 0.259833i \(0.916333\pi\)
\(524\) 0 0
\(525\) −62998.1 + 13786.1i −5.23707 + 1.14604i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) −5110.57 11117.7i −0.417665 0.908601i
\(532\) 11103.4 0.904879
\(533\) 0 0
\(534\) 0 0
\(535\) 20445.0 1.65217
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 14889.0 + 19736.3i 1.18652 + 1.57280i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −8421.75 + 1842.96i −0.665584 + 0.145652i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12616.0 −0.986145 −0.493072 0.869988i \(-0.664126\pi\)
−0.493072 + 0.869988i \(0.664126\pi\)
\(548\) 9426.07i 0.734784i
\(549\) 0 0
\(550\) 0 0
\(551\) 10961.2i 0.847482i
\(552\) 0 0
\(553\) 19307.7 1.48471
\(554\) 0 0
\(555\) 0 0
\(556\) 15680.0 1.19601
\(557\) 17680.4i 1.34496i 0.740115 + 0.672481i \(0.234771\pi\)
−0.740115 + 0.672481i \(0.765229\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 48574.1i 3.66541i
\(561\) 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
\(567\) −16355.0 + 19064.5i −1.21137 + 1.41205i
\(568\) 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
\(576\) −12560.5 + 5773.79i −0.908601 + 0.417665i
\(577\) −27315.1 −1.97078 −0.985390 0.170311i \(-0.945523\pi\)
−0.985390 + 0.170311i \(0.945523\pi\)
\(578\) 0 0
\(579\) 26897.9 5886.14i 1.93064 0.422486i
\(580\) −47951.8 −3.43291
\(581\) 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 \(0.5\pi\)
\(588\) 34282.7 7502.18i 2.40441 0.526165i
\(589\) 0 0
\(590\) 0 0
\(591\) 4539.15 + 20742.5i 0.315932 + 1.44371i
\(592\) 0 0
\(593\) 27698.9i 1.91814i −0.283163 0.959072i \(-0.591384\pi\)
0.283163 0.959072i \(-0.408616\pi\)
\(594\) 0 0
\(595\) 46638.1 3.21340
\(596\) 0 0
\(597\) 7148.82 1564.40i 0.490086 0.107247i
\(598\) 0 0
\(599\) 26228.1i 1.78906i −0.447004 0.894532i \(-0.647509\pi\)
0.447004 0.894532i \(-0.352491\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 29318.1i 1.97016i
\(606\) 0 0
\(607\) −26449.2 −1.76860 −0.884301 0.466917i \(-0.845365\pi\)
−0.884301 + 0.466917i \(0.845365\pi\)
\(608\) 0 0
\(609\) −10415.1 47593.7i −0.693004 3.16682i
\(610\) 0 0
\(611\) 0 0
\(612\) −5543.67 12059.9i −0.366159 0.796555i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 51265.5 11218.6i 3.36134 0.735571i
\(616\) 0 0
\(617\) 30635.6i 1.99893i 0.0326376 + 0.999467i \(0.489609\pi\)
−0.0326376 + 0.999467i \(0.510391\pi\)
\(618\) 0 0
\(619\) −11430.9 −0.742238 −0.371119 0.928585i \(-0.621026\pi\)
−0.371119 + 0.928585i \(0.621026\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 69090.1 4.42177
\(626\) 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
\(635\) 43267.6i 2.70397i
\(636\) 5622.28 + 25692.1i 0.350531 + 1.60182i
\(637\) 0 0
\(638\) 0 0
\(639\) −13339.5 29019.1i −0.825823 1.79652i
\(640\) 0 0
\(641\) 17205.8i 1.06020i 0.847936 + 0.530099i \(0.177845\pi\)
−0.847936 + 0.530099i \(0.822155\pi\)
\(642\) 0 0
\(643\) −13556.0 −0.831410 −0.415705 0.909500i \(-0.636465\pi\)
−0.415705 + 0.909500i \(0.636465\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15558.0i 0.945362i −0.881234 0.472681i \(-0.843286\pi\)
0.881234 0.472681i \(-0.156714\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 32384.0 1.94518
\(653\) 6801.85i 0.407622i −0.979010 0.203811i \(-0.934667\pi\)
0.979010 0.203811i \(-0.0653328\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 29344.3i 1.74649i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −5355.58 −0.315141 −0.157570 0.987508i \(-0.550366\pi\)
−0.157570 + 0.987508i \(0.550366\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 30572.1i 1.78276i
\(666\) 0 0
\(667\) 0 0
\(668\) 5755.74i 0.333378i
\(669\) −29704.9 + 6500.42i −1.71668 + 0.375666i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 40341.7 30433.7i 2.30037 1.73540i
\(676\) −17576.0 −1.00000
\(677\) 33397.6i 1.89597i −0.318308 0.947987i \(-0.603115\pi\)
0.318308 0.947987i \(-0.396885\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −7905.47 + 3633.97i −0.441920 + 0.203141i
\(685\) 25953.6 1.44765
\(686\) 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
\(695\) 43173.1i 2.35633i
\(696\) 0 0
\(697\) −28174.7 −1.53112
\(698\) 0 0
\(699\) 0 0
\(700\) −99287.2 −5.36100
\(701\) 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
\(708\) −4027.22 18403.2i −0.213774 0.976883i
\(709\) 37498.6 1.98630 0.993152 0.116833i \(-0.0372743\pi\)
0.993152 + 0.116833i \(0.0372743\pi\)
\(710\) 0 0
\(711\) −13746.7 + 6319.08i −0.725095 + 0.333311i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1762.00 + 8051.80i 0.0917755 + 0.419386i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 15897.5 + 34583.9i 0.822868 + 1.79009i
\(721\) 0 0
\(722\) 0 0
\(723\) 6038.05 1321.32i 0.310591 0.0679676i
\(724\) −13273.0 −0.681334
\(725\) 98015.1i 5.02095i
\(726\) 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
\(735\) −20656.4 94393.6i −1.03663 4.73709i
\(736\) 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 \(0.341627\pi\)
−0.477269 + 0.878757i \(0.658373\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 31981.3i 1.56018i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −6021.65 27517.1i −0.291423 1.33171i
\(754\) 0 0
\(755\) 0 0
\(756\) −30872.8 + 23290.4i −1.48523 + 1.12045i
\(757\) 41640.9 1.99929 0.999646 0.0265962i \(-0.00846682\pi\)
0.999646 + 0.0265962i \(0.00846682\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38033.2i 1.81170i −0.423601 0.905849i \(-0.639234\pi\)
0.423601 0.905849i \(-0.360766\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −33205.5 + 15263.9i −1.56934 + 0.721394i
\(766\) 0 0
\(767\) 0 0
\(768\) −20791.4 + 4549.85i −0.976883 + 0.213774i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 5405.16 + 24700.0i 0.252480 + 1.15376i
\(772\) 42392.0 1.97632
\(773\) 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
\(779\) 18469.0i 0.849449i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 22992.0 + 30477.2i 1.04938 + 1.39102i
\(784\) 54030.7 2.46131
\(785\) 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
\(789\) −5868.82 26818.7i −0.264810 1.21010i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 70740.4 15480.3i 3.15585 0.690604i
\(796\) 11266.8 0.501684
\(797\) 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