Properties

Label 768.4.c.l.767.4
Level $768$
Weight $4$
Character 768.767
Analytic conductor $45.313$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(767,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.767");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 767.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 768.767
Dual form 768.4.c.l.767.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 + 5.00000i) q^{3} +(-23.0000 + 14.1421i) q^{9} +O(q^{10})\) \(q+(1.41421 + 5.00000i) q^{3} +(-23.0000 + 14.1421i) q^{9} +70.7107 q^{11} -107.480i q^{17} -106.000i q^{19} +125.000 q^{25} +(-103.238 - 95.0000i) q^{27} +(100.000 + 353.553i) q^{33} +56.5685i q^{41} -290.000i q^{43} +343.000 q^{49} +(537.401 - 152.000i) q^{51} +(530.000 - 149.907i) q^{57} +325.269 q^{59} -70.0000i q^{67} +430.000 q^{73} +(176.777 + 625.000i) q^{75} +(329.000 - 650.538i) q^{81} -681.651 q^{83} +1329.36i q^{89} +1910.00 q^{97} +(-1626.35 + 1000.00i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 92 q^{9} + 500 q^{25} + 400 q^{33} + 1372 q^{49} + 2120 q^{57} + 1720 q^{73} + 1316 q^{81} + 7640 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 0
\(3\) 1.41421 + 5.00000i 0.272166 + 0.962250i
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −23.0000 + 14.1421i −0.851852 + 0.523783i
\(10\) 0 0
\(11\) 70.7107 1.93819 0.969094 0.246691i \(-0.0793433\pi\)
0.969094 + 0.246691i \(0.0793433\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 107.480i 1.53340i −0.642006 0.766700i \(-0.721898\pi\)
0.642006 0.766700i \(-0.278102\pi\)
\(18\) 0 0
\(19\) 106.000i 1.27990i −0.768417 0.639949i \(-0.778955\pi\)
0.768417 0.639949i \(-0.221045\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 125.000 1.00000
\(26\) 0 0
\(27\) −103.238 95.0000i −0.735855 0.677139i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 100.000 + 353.553i 0.527508 + 1.86502i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 56.5685i 0.215476i 0.994179 + 0.107738i \(0.0343608\pi\)
−0.994179 + 0.107738i \(0.965639\pi\)
\(42\) 0 0
\(43\) 290.000i 1.02848i −0.857647 0.514239i \(-0.828074\pi\)
0.857647 0.514239i \(-0.171926\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 343.000 1.00000
\(50\) 0 0
\(51\) 537.401 152.000i 1.47551 0.417338i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 530.000 149.907i 1.23158 0.348344i
\(58\) 0 0
\(59\) 325.269 0.717736 0.358868 0.933388i \(-0.383163\pi\)
0.358868 + 0.933388i \(0.383163\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 70.0000i 0.127640i −0.997961 0.0638199i \(-0.979672\pi\)
0.997961 0.0638199i \(-0.0203283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 430.000 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(74\) 0 0
\(75\) 176.777 + 625.000i 0.272166 + 0.962250i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 329.000 650.538i 0.451303 0.892371i
\(82\) 0 0
\(83\) −681.651 −0.901457 −0.450728 0.892661i \(-0.648836\pi\)
−0.450728 + 0.892661i \(0.648836\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1329.36i 1.58328i 0.610988 + 0.791640i \(0.290773\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1910.00 1.99929 0.999645 0.0266459i \(-0.00848265\pi\)
0.999645 + 0.0266459i \(0.00848265\pi\)
\(98\) 0 0
\(99\) −1626.35 + 1000.00i −1.65105 + 1.01519i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1405.73 −1.27006 −0.635032 0.772486i \(-0.719013\pi\)
−0.635032 + 0.772486i \(0.719013\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2387.19i 1.98733i 0.112387 + 0.993665i \(0.464150\pi\)
−0.112387 + 0.993665i \(0.535850\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3669.00 2.75657
\(122\) 0 0
\(123\) −282.843 + 80.0000i −0.207342 + 0.0586452i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 1450.00 410.122i 0.989654 0.279916i
\(130\) 0 0
\(131\) 2729.43 1.82039 0.910197 0.414176i \(-0.135930\pi\)
0.910197 + 0.414176i \(0.135930\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2285.37i 1.42520i −0.701571 0.712599i \(-0.747518\pi\)
0.701571 0.712599i \(-0.252482\pi\)
\(138\) 0 0
\(139\) 1474.00i 0.899446i 0.893168 + 0.449723i \(0.148477\pi\)
−0.893168 + 0.449723i \(0.851523\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 485.075 + 1715.00i 0.272166 + 0.962250i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1520.00 + 2472.05i 0.803168 + 1.30623i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 970.000i 0.466112i −0.972463 0.233056i \(-0.925127\pi\)
0.972463 0.233056i \(-0.0748726\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −2197.00 −1.00000
\(170\) 0 0
\(171\) 1499.07 + 2438.00i 0.670389 + 1.09028i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 460.000 + 1626.35i 0.195343 + 0.690642i
\(178\) 0 0
\(179\) −2870.85 −1.19876 −0.599379 0.800465i \(-0.704586\pi\)
−0.599379 + 0.800465i \(0.704586\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7600.00i 2.97202i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2090.00 0.779490 0.389745 0.920923i \(-0.372563\pi\)
0.389745 + 0.920923i \(0.372563\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 350.000 98.9949i 0.122821 0.0347391i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7495.33i 2.48068i
\(210\) 0 0
\(211\) 6118.00i 1.99612i −0.0622910 0.998058i \(-0.519841\pi\)
0.0622910 0.998058i \(-0.480159\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 608.112 + 2150.00i 0.187636 + 0.663395i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −2875.00 + 1767.77i −0.851852 + 0.523783i
\(226\) 0 0
\(227\) −1903.53 −0.556572 −0.278286 0.960498i \(-0.589766\pi\)
−0.278286 + 0.960498i \(0.589766\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3773.12i 1.06088i 0.847722 + 0.530441i \(0.177974\pi\)
−0.847722 + 0.530441i \(0.822026\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1222.00 −0.326622 −0.163311 0.986575i \(-0.552217\pi\)
−0.163311 + 0.986575i \(0.552217\pi\)
\(242\) 0 0
\(243\) 3717.97 + 725.000i 0.981513 + 0.191394i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −964.000 3408.25i −0.245345 0.867427i
\(250\) 0 0
\(251\) 6689.23 1.68215 0.841077 0.540916i \(-0.181922\pi\)
0.841077 + 0.540916i \(0.181922\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7274.71i 1.76570i 0.469658 + 0.882849i \(0.344377\pi\)
−0.469658 + 0.882849i \(0.655623\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6646.80 + 1880.00i −1.52351 + 0.430914i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8838.83 1.93819
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1216.22i 0.258199i −0.991632 0.129099i \(-0.958791\pi\)
0.991632 0.129099i \(-0.0412086\pi\)
\(282\) 0 0
\(283\) 8030.00i 1.68669i −0.537371 0.843346i \(-0.680582\pi\)
0.537371 0.843346i \(-0.319418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6639.00 −1.35131
\(290\) 0 0
\(291\) 2701.15 + 9550.00i 0.544138 + 1.92382i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7300.00 6717.51i −1.42623 1.31242i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7990.00i 1.48539i −0.669632 0.742693i \(-0.733548\pi\)
0.669632 0.742693i \(-0.266452\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −8390.00 −1.51511 −0.757557 0.652769i \(-0.773607\pi\)
−0.757557 + 0.652769i \(0.773607\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1988.00 7028.64i −0.345668 1.22212i
\(322\) 0 0
\(323\) −11392.9 −1.96259
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8242.00i 1.36864i 0.729180 + 0.684322i \(0.239902\pi\)
−0.729180 + 0.684322i \(0.760098\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11410.0 −1.84434 −0.922170 0.386786i \(-0.873585\pi\)
−0.922170 + 0.386786i \(0.873585\pi\)
\(338\) 0 0
\(339\) −11936.0 + 3376.00i −1.91231 + 0.540882i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −11435.3 −1.76911 −0.884554 0.466437i \(-0.845537\pi\)
−0.884554 + 0.466437i \(0.845537\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12377.2i 1.86621i −0.359605 0.933104i \(-0.617089\pi\)
0.359605 0.933104i \(-0.382911\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −4377.00 −0.638140
\(362\) 0 0
\(363\) 5188.75 + 18345.0i 0.750244 + 2.65251i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −800.000 1301.08i −0.112863 0.183554i
\(370\) 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
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11666.0i 1.58111i −0.612389 0.790557i \(-0.709791\pi\)
0.612389 0.790557i \(-0.290209\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4101.22 + 6670.00i 0.538699 + 0.876112i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 3860.00 + 13647.2i 0.495448 + 1.75167i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14453.3i 1.79990i 0.435989 + 0.899952i \(0.356399\pi\)
−0.435989 + 0.899952i \(0.643601\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −16346.0 −1.97618 −0.988090 0.153877i \(-0.950824\pi\)
−0.988090 + 0.153877i \(0.950824\pi\)
\(410\) 0 0
\(411\) 11426.8 3232.00i 1.37140 0.387890i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7370.00 + 2084.55i −0.865493 + 0.244798i
\(418\) 0 0
\(419\) 3493.11 0.407278 0.203639 0.979046i \(-0.434723\pi\)
0.203639 + 0.979046i \(0.434723\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13435.0i 1.53340i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 5510.00 0.611533 0.305766 0.952107i \(-0.401087\pi\)
0.305766 + 0.952107i \(0.401087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −7889.00 + 4850.75i −0.851852 + 0.523783i
\(442\) 0 0
\(443\) 3736.35 0.400721 0.200361 0.979722i \(-0.435789\pi\)
0.200361 + 0.979722i \(0.435789\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7438.76i 0.781864i −0.920420 0.390932i \(-0.872153\pi\)
0.920420 0.390932i \(-0.127847\pi\)
\(450\) 0 0
\(451\) 4000.00i 0.417633i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18070.0 1.84963 0.924813 0.380422i \(-0.124221\pi\)
0.924813 + 0.380422i \(0.124221\pi\)
\(458\) 0 0
\(459\) −10210.6 + 11096.0i −1.03832 + 1.12836i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13471.8 1.33490 0.667452 0.744653i \(-0.267385\pi\)
0.667452 + 0.744653i \(0.267385\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20506.1i 1.99339i
\(474\) 0 0
\(475\) 13250.0i 1.27990i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 4850.00 1371.79i 0.448517 0.126860i
\(490\) 0 0
\(491\) −18002.9 −1.65471 −0.827354 0.561681i \(-0.810155\pi\)
−0.827354 + 0.561681i \(0.810155\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18214.0i 1.63401i −0.576631 0.817005i \(-0.695633\pi\)
0.576631 0.817005i \(-0.304367\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3107.03 10985.0i −0.272166 0.962250i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −10070.0 + 10943.2i −0.866669 + 0.941819i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21948.6i 1.84565i 0.385215 + 0.922827i \(0.374127\pi\)
−0.385215 + 0.922827i \(0.625873\pi\)
\(522\) 0 0
\(523\) 4750.00i 0.397138i 0.980087 + 0.198569i \(0.0636293\pi\)
−0.980087 + 0.198569i \(0.936371\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) −7481.19 + 4600.00i −0.611405 + 0.375938i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4060.00 14354.3i −0.326261 1.15351i
\(538\) 0 0
\(539\) 24253.8 1.93819
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21850.0i 1.70793i −0.520329 0.853966i \(-0.674191\pi\)
0.520329 0.853966i \(-0.325809\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 38000.0 10748.0i 2.85982 0.808880i
\(562\) 0 0
\(563\) −12391.3 −0.927589 −0.463795 0.885943i \(-0.653512\pi\)
−0.463795 + 0.885943i \(0.653512\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27039.8i 1.99221i 0.0881913 + 0.996104i \(0.471891\pi\)
−0.0881913 + 0.996104i \(0.528109\pi\)
\(570\) 0 0
\(571\) 27038.0i 1.98162i −0.135261 0.990810i \(-0.543187\pi\)
0.135261 0.990810i \(-0.456813\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 19550.0 1.41053 0.705266 0.708943i \(-0.250827\pi\)
0.705266 + 0.708943i \(0.250827\pi\)
\(578\) 0 0
\(579\) 2955.71 + 10450.0i 0.212150 + 0.750064i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26493.9 1.86289 0.931447 0.363876i \(-0.118547\pi\)
0.931447 + 0.363876i \(0.118547\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12173.6i 0.843015i −0.906825 0.421507i \(-0.861501\pi\)
0.906825 0.421507i \(-0.138499\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 14398.0 0.977216 0.488608 0.872503i \(-0.337505\pi\)
0.488608 + 0.872503i \(0.337505\pi\)
\(602\) 0 0
\(603\) 989.949 + 1610.00i 0.0668555 + 0.108730i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 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
\(617\) 11206.2i 0.731192i 0.930774 + 0.365596i \(0.119135\pi\)
−0.930774 + 0.365596i \(0.880865\pi\)
\(618\) 0 0
\(619\) 30706.0i 1.99383i 0.0785136 + 0.996913i \(0.474983\pi\)
−0.0785136 + 0.996913i \(0.525017\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) 37476.7 10600.0i 2.38704 0.675157i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 30590.0 8652.16i 1.92076 0.543274i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31763.2i 1.95721i 0.205745 + 0.978606i \(0.434038\pi\)
−0.205745 + 0.978606i \(0.565962\pi\)
\(642\) 0 0
\(643\) 28550.0i 1.75101i 0.483205 + 0.875507i \(0.339472\pi\)
−0.483205 + 0.875507i \(0.660528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 23000.0 1.39111
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −9890.00 + 6081.12i −0.587284 + 0.361107i
\(658\) 0 0
\(659\) −16107.9 −0.952161 −0.476081 0.879402i \(-0.657943\pi\)
−0.476081 + 0.879402i \(0.657943\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19190.0 1.09914 0.549569 0.835448i \(-0.314792\pi\)
0.549569 + 0.835448i \(0.314792\pi\)
\(674\) 0 0
\(675\) −12904.7 11875.0i −0.735855 0.677139i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2692.00 9517.66i −0.151480 0.535562i
\(682\) 0 0
\(683\) −33632.8 −1.88422 −0.942112 0.335300i \(-0.891162\pi\)
−0.942112 + 0.335300i \(0.891162\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1978.00i 0.108895i −0.998517 0.0544477i \(-0.982660\pi\)
0.998517 0.0544477i \(-0.0173398\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6080.00 0.330411
\(698\) 0 0
\(699\) −18865.6 + 5336.00i −1.02083 + 0.288735i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1728.17 6110.00i −0.0888953 0.314292i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1633.00 + 19615.1i 0.0829650 + 0.996552i
\(730\) 0 0
\(731\) −31169.3 −1.57707
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4949.75i 0.247390i
\(738\) 0 0
\(739\) 36074.0i 1.79567i 0.440327 + 0.897837i \(0.354862\pi\)
−0.440327 + 0.897837i \(0.645138\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 15678.0 9640.00i 0.767908 0.472168i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 9460.00 + 33446.2i 0.457824 + 1.61865i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24381.0i 1.16138i −0.814124 0.580691i \(-0.802782\pi\)
0.814124 0.580691i \(-0.197218\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 40106.0 1.88070 0.940351 0.340207i \(-0.110497\pi\)
0.940351 + 0.340207i \(0.110497\pi\)
\(770\) 0 0
\(771\) −36373.6 + 10288.0i −1.69904 + 0.480562i
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5996.27 0.275788
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 6950.00i 0.314791i 0.987536 + 0.157396i \(0.0503098\pi\)
−0.987536 + 0.157396i \(0.949690\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18800.0 30575.3i −0.829295 1.34872i
\(802\) 0 0
\(803\) 30405.6 1.33623
\(804\)