Properties

Label 900.3.c.i.451.1
Level $900$
Weight $3$
Character 900.451
Analytic conductor $24.523$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-15}) \)
Defining polynomial: \(x^{2} - x + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 451.1
Root \(0.500000 + 1.93649i\) of defining polynomial
Character \(\chi\) \(=\) 900.451
Dual form 900.3.c.i.451.2

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 - 1.93649i) q^{2} +(-3.50000 - 1.93649i) q^{4} +(-5.50000 + 5.80948i) q^{8} +O(q^{10})\) \(q+(0.500000 - 1.93649i) q^{2} +(-3.50000 - 1.93649i) q^{4} +(-5.50000 + 5.80948i) q^{8} +(8.50000 + 13.5554i) q^{16} +14.0000 q^{17} +30.9839i q^{19} +30.9839i q^{23} -61.9677i q^{31} +(30.5000 - 9.68246i) q^{32} +(7.00000 - 27.1109i) q^{34} +(60.0000 + 15.4919i) q^{38} +(60.0000 + 15.4919i) q^{46} +92.9516i q^{47} +49.0000 q^{49} +86.0000 q^{53} +118.000 q^{61} +(-120.000 - 30.9839i) q^{62} +(-3.50000 - 63.9042i) q^{64} +(-49.0000 - 27.1109i) q^{68} +(60.0000 - 108.444i) q^{76} -123.935i q^{79} +61.9677i q^{83} +(60.0000 - 108.444i) q^{92} +(180.000 + 46.4758i) q^{94} +(24.5000 - 94.8881i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 7q^{4} - 11q^{8} + O(q^{10}) \) \( 2q + q^{2} - 7q^{4} - 11q^{8} + 17q^{16} + 28q^{17} + 61q^{32} + 14q^{34} + 120q^{38} + 120q^{46} + 98q^{49} + 172q^{53} + 236q^{61} - 240q^{62} - 7q^{64} - 98q^{68} + 120q^{76} + 120q^{92} + 360q^{94} + 49q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 1.93649i 0.250000 0.968246i
\(3\) 0 0
\(4\) −3.50000 1.93649i −0.875000 0.484123i
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −5.50000 + 5.80948i −0.687500 + 0.726184i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 8.50000 + 13.5554i 0.531250 + 0.847215i
\(17\) 14.0000 0.823529 0.411765 0.911290i \(-0.364913\pi\)
0.411765 + 0.911290i \(0.364913\pi\)
\(18\) 0 0
\(19\) 30.9839i 1.63073i 0.578947 + 0.815365i \(0.303464\pi\)
−0.578947 + 0.815365i \(0.696536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 30.9839i 1.34712i 0.739130 + 0.673562i \(0.235237\pi\)
−0.739130 + 0.673562i \(0.764763\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 61.9677i 1.99896i −0.0322581 0.999480i \(-0.510270\pi\)
0.0322581 0.999480i \(-0.489730\pi\)
\(32\) 30.5000 9.68246i 0.953125 0.302577i
\(33\) 0 0
\(34\) 7.00000 27.1109i 0.205882 0.797379i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 60.0000 + 15.4919i 1.57895 + 0.407682i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 60.0000 + 15.4919i 1.30435 + 0.336781i
\(47\) 92.9516i 1.97769i 0.148936 + 0.988847i \(0.452415\pi\)
−0.148936 + 0.988847i \(0.547585\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 86.0000 1.62264 0.811321 0.584601i \(-0.198749\pi\)
0.811321 + 0.584601i \(0.198749\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 118.000 1.93443 0.967213 0.253966i \(-0.0817352\pi\)
0.967213 + 0.253966i \(0.0817352\pi\)
\(62\) −120.000 30.9839i −1.93548 0.499740i
\(63\) 0 0
\(64\) −3.50000 63.9042i −0.0546875 0.998504i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −49.0000 27.1109i −0.720588 0.398689i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 60.0000 108.444i 0.789474 1.42689i
\(77\) 0 0
\(78\) 0 0
\(79\) 123.935i 1.56880i −0.620253 0.784402i \(-0.712970\pi\)
0.620253 0.784402i \(-0.287030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 61.9677i 0.746599i 0.927711 + 0.373300i \(0.121774\pi\)
−0.927711 + 0.373300i \(0.878226\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 60.0000 108.444i 0.652174 1.17873i
\(93\) 0 0
\(94\) 180.000 + 46.4758i 1.91489 + 0.494423i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 24.5000 94.8881i 0.250000 0.968246i
\(99\) 0 0
\(100\) 0 0
\(101\) 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
\(105\) 0 0
\(106\) 43.0000 166.538i 0.405660 1.57112i
\(107\) 185.903i 1.73741i 0.495327 + 0.868707i \(0.335048\pi\)
−0.495327 + 0.868707i \(0.664952\pi\)
\(108\) 0 0
\(109\) 22.0000 0.201835 0.100917 0.994895i \(-0.467822\pi\)
0.100917 + 0.994895i \(0.467822\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −206.000 −1.82301 −0.911504 0.411290i \(-0.865078\pi\)
−0.911504 + 0.411290i \(0.865078\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 59.0000 228.506i 0.483607 1.87300i
\(123\) 0 0
\(124\) −120.000 + 216.887i −0.967742 + 1.74909i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −125.500 25.1744i −0.980469 0.196675i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −77.0000 + 81.3327i −0.566176 + 0.598034i
\(137\) 226.000 1.64964 0.824818 0.565399i \(-0.191278\pi\)
0.824818 + 0.565399i \(0.191278\pi\)
\(138\) 0 0
\(139\) 92.9516i 0.668717i −0.942446 0.334358i \(-0.891480\pi\)
0.942446 0.334358i \(-0.108520\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 185.903i 1.23115i 0.788079 + 0.615574i \(0.211076\pi\)
−0.788079 + 0.615574i \(0.788924\pi\)
\(152\) −180.000 170.411i −1.18421 1.12113i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −240.000 61.9677i −1.51899 0.392201i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 120.000 + 30.9839i 0.722892 + 0.186650i
\(167\) 216.887i 1.29872i 0.760479 + 0.649362i \(0.224964\pi\)
−0.760479 + 0.649362i \(0.775036\pi\)
\(168\) 0 0
\(169\) −169.000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 154.000 0.890173 0.445087 0.895487i \(-0.353173\pi\)
0.445087 + 0.895487i \(0.353173\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −122.000 −0.674033 −0.337017 0.941499i \(-0.609418\pi\)
−0.337017 + 0.941499i \(0.609418\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −180.000 170.411i −0.978261 0.926148i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 180.000 325.331i 0.957447 1.73048i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −171.500 94.8881i −0.875000 0.484123i
\(197\) −374.000 −1.89848 −0.949239 0.314557i \(-0.898144\pi\)
−0.949239 + 0.314557i \(0.898144\pi\)
\(198\) 0 0
\(199\) 371.806i 1.86837i 0.356784 + 0.934187i \(0.383873\pi\)
−0.356784 + 0.934187i \(0.616127\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 216.887i 1.02790i −0.857820 0.513950i \(-0.828182\pi\)
0.857820 0.513950i \(-0.171818\pi\)
\(212\) −301.000 166.538i −1.41981 0.785558i
\(213\) 0 0
\(214\) 360.000 + 92.9516i 1.68224 + 0.434353i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 11.0000 42.6028i 0.0504587 0.195426i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −103.000 + 398.917i −0.455752 + 1.76512i
\(227\) 433.774i 1.91090i 0.295154 + 0.955450i \(0.404629\pi\)
−0.295154 + 0.955450i \(0.595371\pi\)
\(228\) 0 0
\(229\) 218.000 0.951965 0.475983 0.879455i \(-0.342093\pi\)
0.475983 + 0.879455i \(0.342093\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −34.0000 −0.145923 −0.0729614 0.997335i \(-0.523245\pi\)
−0.0729614 + 0.997335i \(0.523245\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −478.000 −1.98340 −0.991701 0.128564i \(-0.958963\pi\)
−0.991701 + 0.128564i \(0.958963\pi\)
\(242\) 60.5000 234.315i 0.250000 0.968246i
\(243\) 0 0
\(244\) −413.000 228.506i −1.69262 0.936500i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 360.000 + 340.823i 1.45161 + 1.37428i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −111.500 + 230.443i −0.435547 + 0.900166i
\(257\) 466.000 1.81323 0.906615 0.421959i \(-0.138657\pi\)
0.906615 + 0.421959i \(0.138657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 278.855i 1.06028i 0.847909 + 0.530142i \(0.177861\pi\)
−0.847909 + 0.530142i \(0.822139\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 247.871i 0.914653i −0.889299 0.457326i \(-0.848807\pi\)
0.889299 0.457326i \(-0.151193\pi\)
\(272\) 119.000 + 189.776i 0.437500 + 0.697707i
\(273\) 0 0
\(274\) 113.000 437.647i 0.412409 1.59725i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −180.000 46.4758i −0.647482 0.167179i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −93.0000 −0.321799
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 394.000 1.34471 0.672355 0.740229i \(-0.265283\pi\)
0.672355 + 0.740229i \(0.265283\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 360.000 + 92.9516i 1.19205 + 0.307787i
\(303\) 0 0
\(304\) −420.000 + 263.363i −1.38158 + 0.866325i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −240.000 + 433.774i −0.759494 + 1.37270i
\(317\) 134.000 0.422713 0.211356 0.977409i \(-0.432212\pi\)
0.211356 + 0.977409i \(0.432212\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 433.774i 1.34295i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 650.661i 1.96574i 0.184290 + 0.982872i \(0.441001\pi\)
−0.184290 + 0.982872i \(0.558999\pi\)
\(332\) 120.000 216.887i 0.361446 0.653274i
\(333\) 0 0
\(334\) 420.000 + 108.444i 1.25749 + 0.324681i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) −84.5000 + 327.267i −0.250000 + 0.968246i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 77.0000 298.220i 0.222543 0.861907i
\(347\) 371.806i 1.07149i −0.844380 0.535744i \(-0.820031\pi\)
0.844380 0.535744i \(-0.179969\pi\)
\(348\) 0 0
\(349\) −458.000 −1.31232 −0.656160 0.754621i \(-0.727821\pi\)
−0.656160 + 0.754621i \(0.727821\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −274.000 −0.776204 −0.388102 0.921616i \(-0.626869\pi\)
−0.388102 + 0.921616i \(0.626869\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −599.000 −1.65928
\(362\) −61.0000 + 236.252i −0.168508 + 0.652630i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −420.000 + 263.363i −1.14130 + 0.715660i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −540.000 511.234i −1.43617 1.35966i
\(377\) 0 0
\(378\) 0 0
\(379\) 154.919i 0.408758i 0.978892 + 0.204379i \(0.0655175\pi\)
−0.978892 + 0.204379i \(0.934482\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 340.823i 0.889876i −0.895561 0.444938i \(-0.853226\pi\)
0.895561 0.444938i \(-0.146774\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 433.774i 1.10940i
\(392\) −269.500 + 284.664i −0.687500 + 0.726184i
\(393\) 0 0
\(394\) −187.000 + 724.248i −0.474619 + 1.83819i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 720.000 + 185.903i 1.80905 + 0.467093i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 142.000 0.347188 0.173594 0.984817i \(-0.444462\pi\)
0.173594 + 0.984817i \(0.444462\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 602.000 1.42993 0.714964 0.699161i \(-0.246443\pi\)
0.714964 + 0.699161i \(0.246443\pi\)
\(422\) −420.000 108.444i −0.995261 0.256975i
\(423\) 0 0
\(424\) −473.000 + 499.615i −1.11557 + 1.17834i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 360.000 650.661i 0.841121 1.52024i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −77.0000 42.6028i −0.176606 0.0977129i
\(437\) −960.000 −2.19680
\(438\) 0 0
\(439\) 619.677i 1.41157i −0.708428 0.705783i \(-0.750595\pi\)
0.708428 0.705783i \(-0.249405\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 681.645i 1.53870i −0.638826 0.769351i \(-0.720580\pi\)
0.638826 0.769351i \(-0.279420\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 721.000 + 398.917i 1.59513 + 0.882560i
\(453\) 0 0
\(454\) 840.000 + 216.887i 1.85022 + 0.477725i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 109.000 422.155i 0.237991 0.921736i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −17.0000 + 65.8407i −0.0364807 + 0.141289i
\(467\) 867.548i 1.85771i −0.370450 0.928853i \(-0.620796\pi\)
0.370450 0.928853i \(-0.379204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −239.000 + 925.643i −0.495851 + 1.92042i
\(483\) 0 0
\(484\) −423.500 234.315i −0.875000 0.484123i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −649.000 + 685.518i −1.32992 + 1.40475i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 840.000 526.726i 1.69355 1.06195i
\(497\) 0 0
\(498\) 0 0
\(499\) 340.823i 0.683011i −0.939880 0.341506i \(-0.889063\pi\)
0.939880 0.341506i \(-0.110937\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 154.919i 0.307991i −0.988072 0.153995i \(-0.950786\pi\)
0.988072 0.153995i \(-0.0492141\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 390.500 + 331.140i 0.762695 + 0.646758i
\(513\) 0 0
\(514\) 233.000 902.405i 0.453307 1.75565i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 540.000 + 139.427i 1.02662 + 0.265071i
\(527\) 867.548i 1.64620i
\(528\) 0 0
\(529\) −431.000 −0.814745
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1078.00 1.99261 0.996303 0.0859072i \(-0.0273789\pi\)
0.996303 + 0.0859072i \(0.0273789\pi\)
\(542\) −480.000 123.935i −0.885609 0.228663i
\(543\) 0 0
\(544\) 427.000 135.554i 0.784926 0.249181i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −791.000 437.647i −1.44343 0.798626i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −180.000 + 325.331i −0.323741 + 0.585127i
\(557\) 614.000 1.10233 0.551167 0.834395i \(-0.314183\pi\)
0.551167 + 0.834395i \(0.314183\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1115.42i 1.98121i −0.136767 0.990603i \(-0.543671\pi\)
0.136767 0.990603i \(-0.456329\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 1084.44i 1.89919i −0.313485 0.949593i \(-0.601497\pi\)
0.313485 0.949593i \(-0.398503\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −46.5000 + 180.094i −0.0804498 + 0.311581i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 197.000 762.978i 0.336177 1.30201i
\(587\) 805.581i 1.37237i −0.727428 0.686184i \(-0.759284\pi\)
0.727428 0.686184i \(-0.240716\pi\)
\(588\) 0 0
\(589\) 1920.00 3.25976
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1166.00 −1.96627 −0.983137 0.182873i \(-0.941460\pi\)
−0.983137 + 0.182873i \(0.941460\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −242.000 −0.402662 −0.201331 0.979523i \(-0.564527\pi\)
−0.201331 + 0.979523i \(0.564527\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 360.000 650.661i 0.596026 1.07725i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 300.000 + 945.008i 0.493421 + 1.55429i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1186.00 −1.92220 −0.961102 0.276193i \(-0.910927\pi\)
−0.961102 + 0.276193i \(0.910927\pi\)
\(618\) 0 0
\(619\) 1022.47i 1.65181i 0.563813 + 0.825903i \(0.309334\pi\)
−0.563813 + 0.825903i \(0.690666\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1239.35i 1.96411i −0.188590 0.982056i \(-0.560392\pi\)
0.188590 0.982056i \(-0.439608\pi\)
\(632\) 720.000 + 681.645i 1.13924 + 1.07855i
\(633\) 0 0
\(634\) 67.0000 259.490i 0.105678 0.409290i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 840.000 + 216.887i 1.30031 + 0.335738i
\(647\) 1084.44i 1.67610i −0.545595 0.838049i \(-0.683696\pi\)
0.545595 0.838049i \(-0.316304\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1114.00 −1.70597 −0.852986 0.521933i \(-0.825211\pi\)
−0.852986 + 0.521933i \(0.825211\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 838.000 1.26778 0.633888 0.773425i \(-0.281458\pi\)
0.633888 + 0.773425i \(0.281458\pi\)
\(662\) 1260.00 + 325.331i 1.90332 + 0.491436i
\(663\) 0 0
\(664\) −360.000 340.823i −0.542169 0.513287i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 420.000 759.105i 0.628743 1.13638i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 591.500 + 327.267i 0.875000 + 0.484123i
\(677\) 374.000 0.552437 0.276219 0.961095i \(-0.410919\pi\)
0.276219 + 0.961095i \(0.410919\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1363.29i 1.99603i 0.0629575 + 0.998016i \(0.479947\pi\)
−0.0629575 + 0.998016i \(0.520053\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 402.790i 0.582909i −0.956585 0.291455i \(-0.905861\pi\)
0.956585 0.291455i \(-0.0941392\pi\)
\(692\) −539.000 298.220i −0.778902 0.430953i
\(693\) 0 0
\(694\) −720.000 185.903i −1.03746 0.267872i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −229.000 + 886.913i −0.328080 + 1.27065i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −137.000 + 530.599i −0.194051 + 0.751556i
\(707\) 0 0
\(708\) 0 0
\(709\) −742.000 −1.04654 −0.523272 0.852166i \(-0.675289\pi\)
−0.523272 + 0.852166i \(0.675289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1920.00 2.69285
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −299.500 + 1159.96i −0.414820 + 1.60659i
\(723\) 0 0
\(724\) 427.000 + 236.252i 0.589779 + 0.326315i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 300.000 + 945.008i 0.407609 + 1.28398i
\(737\) 0 0
\(738\) 0 0
\(739\) 216.887i 0.293487i −0.989175 0.146744i \(-0.953121\pi\)
0.989175 0.146744i \(-0.0468792\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1394.27i 1.87655i −0.345895 0.938273i \(-0.612425\pi\)
0.345895 0.938273i \(-0.387575\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 433.774i 0.577595i 0.957390 + 0.288798i \(0.0932555\pi\)
−0.957390 + 0.288798i \(0.906745\pi\)
\(752\) −1260.00 + 790.089i −1.67553 + 1.05065i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 300.000 + 77.4597i 0.395778 + 0.102190i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −660.000 170.411i −0.861619 0.222469i
\(767\) 0 0
\(768\) 0 0
\(769\) −578.000 −0.751625 −0.375813 0.926696i \(-0.622636\pi\)
−0.375813 + 0.926696i \(0.622636\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1526.00 1.97413 0.987063 0.160330i \(-0.0512560\pi\)
0.987063 + 0.160330i \(0.0512560\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 840.000 + 216.887i 1.07417 + 0.277349i
\(783\) 0 0
\(784\) 416.500 + 664.217i 0.531250 + 0.847215i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1309.00 + 724.248i 1.66117 + 0.919096i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 720.000 1301.32i 0.904523 1.63483i
\(797\) 826.000 1.03639 0.518193 0.855264i \(-0.326605\pi\)
0.518193 + 0.855264i \(0.326605\pi\)
\(798\) 0 0
\(799\) 1301.32i 1.62869i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 </