Properties

Label 177.4.d.b.176.4
Level $177$
Weight $4$
Character 177.176
Analytic conductor $10.443$
Analytic rank $0$
Dimension $4$
CM discriminant -59
Inner twists $4$

Related objects

Downloads

Learn more about

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 176.4
Root \(-3.07603 - 2.35330i\) of defining polynomial
Character \(\chi\) \(=\) 177.176
Dual form 177.4.d.b.176.3

$q$-expansion

\(f(q)\) \(=\) \(q+(1.57603 + 4.95138i) q^{3} -8.00000 q^{4} +14.3460i q^{5} +5.45620 q^{7} +(-22.0322 + 15.6071i) q^{9} +O(q^{10})\) \(q+(1.57603 + 4.95138i) q^{3} -8.00000 q^{4} +14.3460i q^{5} +5.45620 q^{7} +(-22.0322 + 15.6071i) q^{9} +(-12.6083 - 39.6110i) q^{12} +(-71.0322 + 22.6097i) q^{15} +64.0000 q^{16} -61.4492i q^{17} -159.281 q^{19} -114.768i q^{20} +(8.59916 + 27.0157i) q^{21} -80.8066 q^{25} +(-112.000 - 84.4926i) q^{27} -43.6496 q^{28} -3.27799i q^{29} +78.2745i q^{35} +(176.258 - 124.857i) q^{36} +450.823i q^{41} +(-223.898 - 316.074i) q^{45} +(100.866 + 316.888i) q^{48} -313.230 q^{49} +(304.258 - 96.8460i) q^{51} -66.3018i q^{53} +(-251.032 - 788.660i) q^{57} +453.188i q^{59} +(568.258 - 180.878i) q^{60} +(-120.212 + 85.1553i) q^{63} -512.000 q^{64} +491.593i q^{68} +1182.90i q^{71} +(-127.354 - 400.104i) q^{75} +1274.25 q^{76} +1395.35 q^{79} +918.141i q^{80} +(241.839 - 687.717i) q^{81} +(-68.7933 - 216.126i) q^{84} +881.547 q^{85} +(16.2306 - 5.16622i) q^{87} -2285.04i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + O(q^{10}) \) \( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + 56q^{12} - 191q^{15} + 256q^{16} - 238q^{19} + 367q^{21} - 882q^{25} - 448q^{27} + 464q^{28} - 40q^{36} + 49q^{45} - 448q^{48} + 1062q^{49} + 472q^{51} - 911q^{57} + 1528q^{60} - 1931q^{63} - 2048q^{64} + 3402q^{75} + 1904q^{76} + 1670q^{79} + 1433q^{81} - 2936q^{84} - 944q^{85} + 637q^{87} + O(q^{100}) \)

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 \(0.5\pi\)
\(3\) 1.57603 + 4.95138i 0.303308 + 0.952893i
\(4\) −8.00000 −1.00000
\(5\) 14.3460i 1.28314i 0.767064 + 0.641571i \(0.221717\pi\)
−0.767064 + 0.641571i \(0.778283\pi\)
\(6\) 0 0
\(7\) 5.45620 0.294607 0.147304 0.989091i \(-0.452941\pi\)
0.147304 + 0.989091i \(0.452941\pi\)
\(8\) 0 0
\(9\) −22.0322 + 15.6071i −0.816009 + 0.578040i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −12.6083 39.6110i −0.303308 0.952893i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −71.0322 + 22.6097i −1.22270 + 0.389187i
\(16\) 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
\(19\) −159.281 −1.92324 −0.961620 0.274384i \(-0.911526\pi\)
−0.961620 + 0.274384i \(0.911526\pi\)
\(20\) 114.768i 1.28314i
\(21\) 8.59916 + 27.0157i 0.0893567 + 0.280729i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −80.8066 −0.646453
\(26\) 0 0
\(27\) −112.000 84.4926i −0.798311 0.602245i
\(28\) −43.6496 −0.294607
\(29\) 3.27799i 0.0209899i −0.999945 0.0104950i \(-0.996659\pi\)
0.999945 0.0104950i \(-0.00334071\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 78.2745i 0.378023i
\(36\) 176.258 124.857i 0.816009 0.578040i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 450.823i 1.71724i 0.512616 + 0.858618i \(0.328677\pi\)
−0.512616 + 0.858618i \(0.671323\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −223.898 316.074i −0.741707 1.04705i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 100.866 + 316.888i 0.303308 + 0.952893i
\(49\) −313.230 −0.913207
\(50\) 0 0
\(51\) 304.258 96.8460i 0.835385 0.265905i
\(52\) 0 0
\(53\) 66.3018i 0.171835i −0.996302 0.0859175i \(-0.972618\pi\)
0.996302 0.0859175i \(-0.0273822\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −251.032 788.660i −0.583334 1.83264i
\(58\) 0 0
\(59\) 453.188i 1.00000i
\(60\) 568.258 180.878i 1.22270 0.389187i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −120.212 + 85.1553i −0.240402 + 0.170295i
\(64\) −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
\(75\) −127.354 400.104i −0.196074 0.616000i
\(76\) 1274.25 1.92324
\(77\) 0 0
\(78\) 0 0
\(79\) 1395.35 1.98721 0.993605 0.112913i \(-0.0360180\pi\)
0.993605 + 0.112913i \(0.0360180\pi\)
\(80\) 918.141i 1.28314i
\(81\) 241.839 687.717i 0.331740 0.943371i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −68.7933 216.126i −0.0893567 0.280729i
\(85\) 881.547 1.12491
\(86\) 0 0
\(87\) 16.2306 5.16622i 0.0200011 0.00636640i
\(88\) 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
\(95\) 2285.04i 2.46779i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 646.453 0.646453
\(101\) 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
\(105\) −387.566 + 123.363i −0.360215 + 0.114657i
\(106\) 0 0
\(107\) 1276.32i 1.15314i 0.817047 + 0.576570i \(0.195609\pi\)
−0.817047 + 0.576570i \(0.804391\pi\)
\(108\) 896.000 + 675.941i 0.798311 + 0.602245i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 349.197 0.294607
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 26.2239i 0.0209899i
\(117\) 0 0
\(118\) 0 0
\(119\) 335.279i 0.258277i
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) −2232.19 + 710.512i −1.63634 + 0.520851i
\(124\) 0 0
\(125\) 633.997i 0.453651i
\(126\) 0 0
\(127\) −2785.29 −1.94610 −0.973049 0.230600i \(-0.925931\pi\)
−0.973049 + 0.230600i \(0.925931\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −869.069 −0.566601
\(134\) 0 0
\(135\) 1212.13 1606.75i 0.772765 1.02435i
\(136\) 0 0
\(137\) 1994.05i 1.24353i 0.783204 + 0.621765i \(0.213584\pi\)
−0.783204 + 0.621765i \(0.786416\pi\)
\(138\) 0 0
\(139\) −1960.00 −1.19601 −0.598004 0.801493i \(-0.704039\pi\)
−0.598004 + 0.801493i \(0.704039\pi\)
\(140\) 626.196i 0.378023i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1410.06 + 998.852i −0.816009 + 0.578040i
\(145\) 47.0259 0.0269330
\(146\) 0 0
\(147\) −493.661 1550.92i −0.276983 0.870188i
\(148\) 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
\(153\) 959.041 + 1353.86i 0.506758 + 0.715381i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 328.285 104.494i 0.163740 0.0521189i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4048.00 −1.94518 −0.972588 0.232533i \(-0.925299\pi\)
−0.972588 + 0.232533i \(0.925299\pi\)
\(164\) 3606.58i 1.71724i
\(165\) 0 0
\(166\) 0 0
\(167\) 4045.40i 1.87451i −0.348648 0.937254i \(-0.613359\pi\)
0.348648 0.937254i \(-0.386641\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) 3509.32 2485.91i 1.56938 1.11171i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −440.897 −0.190450
\(176\) 0 0
\(177\) −2243.90 + 714.239i −0.952893 + 0.303308i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 1791.19 + 2528.59i 0.741707 + 1.04705i
\(181\) 3135.88 1.28778 0.643890 0.765118i \(-0.277320\pi\)
0.643890 + 0.765118i \(0.277320\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −611.095 461.009i −0.235188 0.177426i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −806.929 2535.10i −0.303308 0.952893i
\(193\) 3362.00 1.25390 0.626948 0.779061i \(-0.284304\pi\)
0.626948 + 0.779061i \(0.284304\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2505.84 0.913207
\(197\) 4086.37i 1.47788i 0.673773 + 0.738939i \(0.264673\pi\)
−0.673773 + 0.738939i \(0.735327\pi\)
\(198\) 0 0
\(199\) −4002.65 −1.42583 −0.712916 0.701249i \(-0.752626\pi\)
−0.712916 + 0.701249i \(0.752626\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 17.8854i 0.00618378i
\(204\) −2434.06 + 774.768i −0.835385 + 0.265905i
\(205\) −6467.49 −2.20346
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 530.415i 0.171835i
\(213\) −5856.96 + 1864.28i −1.88410 + 0.599712i
\(214\) 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
\(225\) 1780.35 1261.15i 0.527511 0.373675i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 2008.26 + 6309.28i 0.583334 + 1.83264i
\(229\) 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
\(237\) 2199.12 + 6908.92i 0.602736 + 1.89360i
\(238\) 0 0
\(239\) 5457.37i 1.47702i 0.674242 + 0.738511i \(0.264471\pi\)
−0.674242 + 0.738511i \(0.735529\pi\)
\(240\) −4546.06 + 1447.02i −1.22270 + 0.389187i
\(241\) 6992.52 1.86900 0.934498 0.355969i \(-0.115849\pi\)
0.934498 + 0.355969i \(0.115849\pi\)
\(242\) 0 0
\(243\) 3786.29 + 113.569i 0.999550 + 0.0299814i
\(244\) 0 0
\(245\) 4493.58i 1.17177i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2329.28i 0.585749i −0.956151 0.292875i \(-0.905388\pi\)
0.956151 0.292875i \(-0.0946118\pi\)
\(252\) 961.699 681.243i 0.240402 0.170295i
\(253\) 0 0
\(254\) 0 0
\(255\) 1389.35 + 4364.87i 0.341194 + 1.07192i
\(256\) 4096.00 1.00000
\(257\) 8191.93i 1.98832i −0.107907 0.994161i \(-0.534415\pi\)
0.107907 0.994161i \(-0.465585\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 51.1598 + 72.2214i 0.0121330 + 0.0171279i
\(262\) 0 0
\(263\) 3158.18i 0.740462i −0.928940 0.370231i \(-0.879279\pi\)
0.928940 0.370231i \(-0.120721\pi\)
\(264\) 0 0
\(265\) 951.163 0.220489
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 2152.19 0.482421 0.241210 0.970473i \(-0.422456\pi\)
0.241210 + 0.970473i \(0.422456\pi\)
\(272\) 3932.75i 0.876683i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5553.35 1.20458 0.602290 0.798277i \(-0.294255\pi\)
0.602290 + 0.798277i \(0.294255\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8289.78i 1.75988i 0.475083 + 0.879941i \(0.342418\pi\)
−0.475083 + 0.879941i \(0.657582\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 9463.17i 1.97724i
\(285\) 11314.1 3601.30i 2.35154 0.748500i
\(286\) 0 0
\(287\) 2459.78i 0.505910i
\(288\) 0 0
\(289\) 1137.00 0.231427
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10021.4i 1.99815i 0.0430278 + 0.999074i \(0.486300\pi\)
−0.0430278 + 0.999074i \(0.513700\pi\)
\(294\) 0 0
\(295\) −6501.41 −1.28314
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 1018.83 + 3200.83i 0.196074 + 0.616000i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −10194.0 −1.92324
\(305\) 0 0
\(306\) 0 0
\(307\) 10591.3 1.96898 0.984488 0.175450i \(-0.0561380\pi\)
0.984488 + 0.175450i \(0.0561380\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10124.2i 1.84595i −0.384863 0.922974i \(-0.625751\pi\)
0.384863 0.922974i \(-0.374249\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −1221.63 1724.56i −0.218512 0.308470i
\(316\) −11162.8 −1.98721
\(317\) 215.072i 0.0381062i 0.999818 + 0.0190531i \(0.00606515\pi\)
−0.999818 + 0.0190531i \(0.993935\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7345.13i 1.28314i
\(321\) −6319.52 + 2011.52i −1.09882 + 0.349757i
\(322\) 0 0
\(323\) 9787.69i 1.68607i
\(324\) −1934.71 + 5501.74i −0.331740 + 0.943371i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6706.97 −1.11374 −0.556870 0.830599i \(-0.687998\pi\)
−0.556870 + 0.830599i \(0.687998\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 550.346 + 1729.01i 0.0893567 + 0.280729i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −7052.38 −1.12491
\(341\) 0 0
\(342\) 0 0
\(343\) −3580.52 −0.563644
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −129.844 + 41.3298i −0.0200011 + 0.00636640i
\(349\) 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
\(355\) −16969.8 −2.53708
\(356\) 0 0
\(357\) 1660.09 528.411i 0.246110 0.0783375i
\(358\) 0 0
\(359\) 5516.23i 0.810963i −0.914103 0.405481i \(-0.867104\pi\)
0.914103 0.405481i \(-0.132896\pi\)
\(360\) 0 0
\(361\) 18511.4 2.69885
\(362\) 0 0
\(363\) −2097.70 6590.28i −0.303308 0.952893i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −7036.02 9932.63i −0.992631 1.40128i
\(370\) 0 0
\(371\) 361.756i 0.0506238i
\(372\) 0 0
\(373\) 4898.00 0.679916 0.339958 0.940441i \(-0.389587\pi\)
0.339958 + 0.940441i \(0.389587\pi\)
\(374\) 0 0
\(375\) −3139.16 + 999.201i −0.432281 + 0.137596i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −505.683 −0.0685361 −0.0342681 0.999413i \(-0.510910\pi\)
−0.0342681 + 0.999413i \(0.510910\pi\)
\(380\) 18280.3i 2.46779i
\(381\) −4389.71 13791.0i −0.590267 1.85442i
\(382\) 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
\(395\) 20017.7i 2.54987i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) −1369.68 4303.09i −0.171854 0.539909i
\(400\) −5171.62 −0.646453
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9865.96 + 3469.41i 1.21048 + 0.425670i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −9873.31 + 3142.70i −1.18495 + 0.377172i
\(412\) 0 0
\(413\) 2472.68i 0.294607i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3089.03 9704.70i −0.362758 1.13967i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 3100.53 986.905i 0.360215 0.114657i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4965.50i 0.566734i
\(426\) 0 0
\(427\) 0 0
\(428\) 10210.5i 1.15314i
\(429\) 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
\(433\) 16431.5 1.82366 0.911832 0.410563i \(-0.134668\pi\)
0.911832 + 0.410563i \(0.134668\pi\)
\(434\) 0 0
\(435\) 74.1144 + 232.843i 0.00816900 + 0.0256643i
\(436\) 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
\(441\) 6901.15 4888.60i 0.745185 0.527870i
\(442\) 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
\(448\) −2793.58 −0.294607
\(449\) 19027.9i 1.99996i −0.00634411 0.999980i \(-0.502019\pi\)
0.00634411 0.999980i \(-0.497981\pi\)
\(450\) 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 \(\pi\)
\(464\) 209.791i 0.0209899i
\(465\) 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
\(475\) 12871.0 1.24328
\(476\) 2682.23i 0.258277i
\(477\) 1034.78 + 1460.78i 0.0993274 + 0.140219i
\(478\) 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
\(487\) −17264.3 −1.60640 −0.803201 0.595708i \(-0.796872\pi\)
−0.803201 + 0.595708i \(0.796872\pi\)
\(488\) 0 0
\(489\) −6379.78 20043.2i −0.589987 1.85354i
\(490\) 0 0
\(491\) 21733.7i 1.99762i 0.0488092 + 0.998808i \(0.484457\pi\)
−0.0488092 + 0.998808i \(0.515543\pi\)
\(492\) 17857.5 5684.09i 1.63634 0.520851i
\(493\) −201.430 −0.0184015
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6454.12i 0.582509i
\(498\) 0 0
\(499\) −12706.2 −1.13989 −0.569946 0.821682i \(-0.693036\pi\)
−0.569946 + 0.821682i \(0.693036\pi\)
\(500\) 5071.98i 0.453651i
\(501\) 20030.3 6375.69i 1.78620 0.568553i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3462.55 + 10878.2i 0.303308 + 0.952893i
\(508\) 22282.3 1.94610
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 17839.5 + 13458.1i 1.53534 + 1.15826i
\(514\) 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
\(523\) 16932.6 1.41570 0.707849 0.706364i \(-0.249666\pi\)
0.707849 + 0.706364i \(0.249666\pi\)
\(524\) 0 0
\(525\) −694.869 2183.05i −0.0577649 0.181478i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) −7072.93 9984.74i −0.578040 0.816009i
\(532\) 6952.55 0.566601
\(533\) 0 0
\(534\) 0 0
\(535\) −18310.0 −1.47964
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −9697.02 + 12854.0i −0.772765 + 1.02435i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 4942.25 + 15526.9i 0.390594 + 1.22712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12616.0 −0.986145 −0.493072 0.869988i \(-0.664126\pi\)
−0.493072 + 0.869988i \(0.664126\pi\)
\(548\) 15952.4i 1.24353i
\(549\) 0 0
\(550\) 0 0
\(551\) 522.121i 0.0403686i
\(552\) 0 0
\(553\) 7613.33 0.585446
\(554\) 0 0
\(555\) 0 0
\(556\) 15680.0 1.19601
\(557\) 25691.9i 1.95440i 0.212328 + 0.977198i \(0.431895\pi\)
−0.212328 + 0.977198i \(0.568105\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 5009.57i 0.378023i
\(561\) 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
\(567\) 1319.52 3752.32i 0.0977331 0.277924i
\(568\) 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
\(576\) 11280.5 7990.82i 0.816009 0.578040i
\(577\) 17746.1 1.28038 0.640189 0.768218i \(-0.278856\pi\)
0.640189 + 0.768218i \(0.278856\pi\)
\(578\) 0 0
\(579\) 5298.62 + 16646.5i 0.380316 + 1.19483i
\(580\) −376.207 −0.0269330
\(581\) 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 \(0.5\pi\)
\(588\) 3949.29 + 12407.3i 0.276983 + 0.870188i
\(589\) 0 0
\(590\) 0 0
\(591\) −20233.1 + 6440.26i −1.40826 + 0.448252i
\(592\) 0 0
\(593\) 20931.9i 1.44952i −0.688999 0.724762i \(-0.741950\pi\)
0.688999 0.724762i \(-0.258050\pi\)
\(594\) 0 0
\(595\) 4809.90 0.331406
\(596\) 0 0
\(597\) −6308.32 19818.6i −0.432466 1.35866i
\(598\) 0 0
\(599\) 1763.61i 0.120299i −0.998189 0.0601496i \(-0.980842\pi\)
0.998189 0.0601496i \(-0.0191578\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 19094.5i 1.28314i
\(606\) 0 0
\(607\) 1130.24 0.0755764 0.0377882 0.999286i \(-0.487969\pi\)
0.0377882 + 0.999286i \(0.487969\pi\)
\(608\) 0 0
\(609\) 88.5572 28.1879i 0.00589248 0.00187559i
\(610\) 0 0
\(611\) 0 0
\(612\) −7672.33 10830.9i −0.506758 0.715381i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −10193.0 32022.9i −0.668326 2.09966i
\(616\) 0 0
\(617\) 14451.4i 0.942937i 0.881883 + 0.471469i \(0.156276\pi\)
−0.881883 + 0.471469i \(0.843724\pi\)
\(618\) 0 0
\(619\) −19054.1 −1.23724 −0.618619 0.785691i \(-0.712308\pi\)
−0.618619 + 0.785691i \(0.712308\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19196.1 −1.22855
\(626\) 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
\(635\) 39957.6i 2.49712i
\(636\) −2626.28 + 835.951i −0.163740 + 0.0521189i
\(637\) 0 0
\(638\) 0 0
\(639\) −18461.5 26061.9i −1.14292 1.61344i
\(640\) 0 0
\(641\) 17205.8i 1.06020i −0.847936 0.530099i \(-0.822155\pi\)
0.847936 0.530099i \(-0.177845\pi\)
\(642\) 0 0
\(643\) 32463.0 1.99100 0.995502 0.0947388i \(-0.0302016\pi\)
0.995502 + 0.0947388i \(0.0302016\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17340.3i 1.05366i 0.849971 + 0.526830i \(0.176620\pi\)
−0.849971 + 0.526830i \(0.823380\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 32384.0 1.94518
\(653\) 31696.4i 1.89951i −0.313000 0.949753i \(-0.601334\pi\)
0.313000 0.949753i \(-0.398666\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 28852.7i 1.71724i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −26389.4 −1.55284 −0.776422 0.630214i \(-0.782967\pi\)
−0.776422 + 0.630214i \(0.782967\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12467.6i 0.727029i
\(666\) 0 0
\(667\) 0 0
\(668\) 32363.2i 1.87451i
\(669\) 9222.95 + 28975.4i 0.533004 + 1.67452i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 9050.34 + 6827.56i 0.516071 + 0.389323i
\(676\) −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 \(0.5\pi\)
\(684\) −28074.5 + 19887.3i −1.56938 + 1.11171i
\(685\) −28606.6 −1.59562
\(686\) 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
\(695\) 28118.1i 1.53465i
\(696\) 0 0
\(697\) 27702.7 1.50547
\(698\) 0 0
\(699\) 0 0
\(700\) 3527.18 0.190450
\(701\) 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
\(708\) 17951.2 5713.91i 0.952893 0.303308i
\(709\) −22569.6 −1.19551 −0.597756 0.801678i \(-0.703941\pi\)
−0.597756 + 0.801678i \(0.703941\pi\)
\(710\) 0 0
\(711\) −30742.8 + 21777.4i −1.62158 + 1.14869i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −27021.5 + 8601.00i −1.40744 + 0.447992i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −14329.5 20228.7i −0.741707 1.04705i
\(721\) 0 0
\(722\) 0 0
\(723\) 11020.4 + 34622.6i 0.566881 + 1.78095i
\(724\) −25087.0 −1.28778
\(725\) 264.883i 0.0135690i
\(726\) 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
\(735\) 22249.4 7082.04i 1.11657 0.355408i
\(736\) 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
\(749\) 6963.84i 0.339724i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 11533.2 3671.03i 0.558156 0.177662i
\(754\) 0 0
\(755\) 0 0
\(756\) 4888.76 + 3688.07i 0.235188 + 0.177426i
\(757\) −21779.9 −1.04571 −0.522856 0.852421i \(-0.675133\pi\)
−0.522856 + 0.852421i \(0.675133\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3613.97i 0.172150i −0.996289 0.0860752i \(-0.972567\pi\)
0.996289 0.0860752i \(-0.0274325\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −19422.5 + 13758.4i −0.917935 + 0.650242i
\(766\) 0 0
\(767\) 0 0
\(768\) 6455.43 + 20280.8i 0.303308 + 0.952893i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 40561.3 12910.8i 1.89466 0.603074i
\(772\) −26896.0 −1.25390
\(773\) 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
\(779\) 71807.5i 3.30266i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −276.966 + 367.135i −0.0126411 + 0.0167565i
\(784\) −20046.7 −0.913207
\(785\) 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
\(789\) 15637.3 4977.39i 0.705581 0.224588i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1499.07 + 4709.57i 0.0668759 + 0.210102i
\(796\) 32021.2 1.42583
\(797\) 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