Properties

Label 864.3.g.b.703.4
Level $864$
Weight $3$
Character 864.703
Analytic conductor $23.542$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5422948407\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.4
Root \(0.500000 + 2.19293i\) of defining polynomial
Character \(\chi\) \(=\) 864.703
Dual form 864.3.g.b.703.3

$q$-expansion

\(f(q)\) \(=\) \(q-5.13244 q^{5} +4.02516i q^{7} +O(q^{10})\) \(q-5.13244 q^{5} +4.02516i q^{7} -4.30344i q^{11} -18.4862 q^{13} +23.5751 q^{17} +21.7259i q^{19} -30.7461i q^{23} +1.34199 q^{25} +12.6840 q^{29} -24.5298i q^{31} -20.6589i q^{35} +18.2314 q^{37} +38.0990 q^{41} -34.9724i q^{43} +29.6295i q^{47} +32.7981 q^{49} +39.3043 q^{53} +22.0872i q^{55} -65.3429i q^{59} -29.8541 q^{61} +94.8794 q^{65} -11.8634i q^{67} -140.651i q^{71} +119.285 q^{73} +17.3220 q^{77} +9.18859i q^{79} -113.180i q^{83} -120.998 q^{85} +7.88346 q^{89} -74.4099i q^{91} -111.507i q^{95} +55.5080 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} + 8 q^{13} - 24 q^{17} + 24 q^{25} + 128 q^{29} + 24 q^{37} - 160 q^{41} - 144 q^{49} + 48 q^{53} - 136 q^{61} + 280 q^{65} + 72 q^{73} - 520 q^{77} + 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −5.13244 −1.02649 −0.513244 0.858242i \(-0.671557\pi\)
−0.513244 + 0.858242i \(0.671557\pi\)
\(6\) 0 0
\(7\) 4.02516i 0.575023i 0.957777 + 0.287511i \(0.0928279\pi\)
−0.957777 + 0.287511i \(0.907172\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.30344i − 0.391222i −0.980682 0.195611i \(-0.937331\pi\)
0.980682 0.195611i \(-0.0626689\pi\)
\(12\) 0 0
\(13\) −18.4862 −1.42202 −0.711008 0.703184i \(-0.751761\pi\)
−0.711008 + 0.703184i \(0.751761\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 23.5751 1.38677 0.693384 0.720568i \(-0.256119\pi\)
0.693384 + 0.720568i \(0.256119\pi\)
\(18\) 0 0
\(19\) 21.7259i 1.14347i 0.820438 + 0.571735i \(0.193729\pi\)
−0.820438 + 0.571735i \(0.806271\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 30.7461i − 1.33679i −0.743809 0.668393i \(-0.766983\pi\)
0.743809 0.668393i \(-0.233017\pi\)
\(24\) 0 0
\(25\) 1.34199 0.0536794
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 12.6840 0.437378 0.218689 0.975795i \(-0.429822\pi\)
0.218689 + 0.975795i \(0.429822\pi\)
\(30\) 0 0
\(31\) − 24.5298i − 0.791283i −0.918405 0.395642i \(-0.870522\pi\)
0.918405 0.395642i \(-0.129478\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 20.6589i − 0.590254i
\(36\) 0 0
\(37\) 18.2314 0.492740 0.246370 0.969176i \(-0.420762\pi\)
0.246370 + 0.969176i \(0.420762\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 38.0990 0.929244 0.464622 0.885509i \(-0.346190\pi\)
0.464622 + 0.885509i \(0.346190\pi\)
\(42\) 0 0
\(43\) − 34.9724i − 0.813312i −0.913581 0.406656i \(-0.866695\pi\)
0.913581 0.406656i \(-0.133305\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 29.6295i 0.630415i 0.949023 + 0.315208i \(0.102074\pi\)
−0.949023 + 0.315208i \(0.897926\pi\)
\(48\) 0 0
\(49\) 32.7981 0.669349
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 39.3043 0.741591 0.370796 0.928715i \(-0.379085\pi\)
0.370796 + 0.928715i \(0.379085\pi\)
\(54\) 0 0
\(55\) 22.0872i 0.401585i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 65.3429i − 1.10751i −0.832681 0.553753i \(-0.813195\pi\)
0.832681 0.553753i \(-0.186805\pi\)
\(60\) 0 0
\(61\) −29.8541 −0.489412 −0.244706 0.969597i \(-0.578691\pi\)
−0.244706 + 0.969597i \(0.578691\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 94.8794 1.45968
\(66\) 0 0
\(67\) − 11.8634i − 0.177066i −0.996073 0.0885329i \(-0.971782\pi\)
0.996073 0.0885329i \(-0.0282178\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 140.651i − 1.98100i −0.137529 0.990498i \(-0.543916\pi\)
0.137529 0.990498i \(-0.456084\pi\)
\(72\) 0 0
\(73\) 119.285 1.63404 0.817022 0.576607i \(-0.195624\pi\)
0.817022 + 0.576607i \(0.195624\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.3220 0.224961
\(78\) 0 0
\(79\) 9.18859i 0.116311i 0.998308 + 0.0581556i \(0.0185220\pi\)
−0.998308 + 0.0581556i \(0.981478\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 113.180i − 1.36362i −0.731529 0.681810i \(-0.761193\pi\)
0.731529 0.681810i \(-0.238807\pi\)
\(84\) 0 0
\(85\) −120.998 −1.42350
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.88346 0.0885782 0.0442891 0.999019i \(-0.485898\pi\)
0.0442891 + 0.999019i \(0.485898\pi\)
\(90\) 0 0
\(91\) − 74.4099i − 0.817691i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 111.507i − 1.17376i
\(96\) 0 0
\(97\) 55.5080 0.572248 0.286124 0.958193i \(-0.407633\pi\)
0.286124 + 0.958193i \(0.407633\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 195.186 1.93253 0.966265 0.257550i \(-0.0829152\pi\)
0.966265 + 0.257550i \(0.0829152\pi\)
\(102\) 0 0
\(103\) − 29.3579i − 0.285028i −0.989793 0.142514i \(-0.954481\pi\)
0.989793 0.142514i \(-0.0455186\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 133.256i 1.24538i 0.782468 + 0.622691i \(0.213961\pi\)
−0.782468 + 0.622691i \(0.786039\pi\)
\(108\) 0 0
\(109\) −198.485 −1.82096 −0.910481 0.413552i \(-0.864288\pi\)
−0.910481 + 0.413552i \(0.864288\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −178.745 −1.58182 −0.790908 0.611934i \(-0.790392\pi\)
−0.790908 + 0.611934i \(0.790392\pi\)
\(114\) 0 0
\(115\) 157.802i 1.37220i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 94.8934i 0.797424i
\(120\) 0 0
\(121\) 102.480 0.846946
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 121.423 0.971388
\(126\) 0 0
\(127\) 172.213i 1.35601i 0.735058 + 0.678004i \(0.237155\pi\)
−0.735058 + 0.678004i \(0.762845\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 206.300i − 1.57481i −0.616435 0.787406i \(-0.711424\pi\)
0.616435 0.787406i \(-0.288576\pi\)
\(132\) 0 0
\(133\) −87.4503 −0.657521
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 85.1089 0.621233 0.310616 0.950535i \(-0.399465\pi\)
0.310616 + 0.950535i \(0.399465\pi\)
\(138\) 0 0
\(139\) 150.326i 1.08148i 0.841189 + 0.540742i \(0.181856\pi\)
−0.841189 + 0.540742i \(0.818144\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 79.5542i 0.556323i
\(144\) 0 0
\(145\) −65.0998 −0.448964
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −111.133 −0.745861 −0.372931 0.927859i \(-0.621647\pi\)
−0.372931 + 0.927859i \(0.621647\pi\)
\(150\) 0 0
\(151\) − 166.295i − 1.10129i −0.834739 0.550646i \(-0.814381\pi\)
0.834739 0.550646i \(-0.185619\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 125.898i 0.812243i
\(156\) 0 0
\(157\) 50.1560 0.319465 0.159732 0.987160i \(-0.448937\pi\)
0.159732 + 0.987160i \(0.448937\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 123.758 0.768682
\(162\) 0 0
\(163\) − 265.994i − 1.63187i −0.578145 0.815934i \(-0.696223\pi\)
0.578145 0.815934i \(-0.303777\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 223.651i − 1.33923i −0.742708 0.669615i \(-0.766459\pi\)
0.742708 0.669615i \(-0.233541\pi\)
\(168\) 0 0
\(169\) 172.740 1.02213
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 55.4735 0.320656 0.160328 0.987064i \(-0.448745\pi\)
0.160328 + 0.987064i \(0.448745\pi\)
\(174\) 0 0
\(175\) 5.40171i 0.0308669i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 10.7562i − 0.0600907i −0.999549 0.0300453i \(-0.990435\pi\)
0.999549 0.0300453i \(-0.00956517\pi\)
\(180\) 0 0
\(181\) −36.3638 −0.200905 −0.100453 0.994942i \(-0.532029\pi\)
−0.100453 + 0.994942i \(0.532029\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −93.5715 −0.505792
\(186\) 0 0
\(187\) − 101.454i − 0.542534i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 214.445i 1.12275i 0.827563 + 0.561373i \(0.189727\pi\)
−0.827563 + 0.561373i \(0.810273\pi\)
\(192\) 0 0
\(193\) 280.031 1.45094 0.725470 0.688254i \(-0.241622\pi\)
0.725470 + 0.688254i \(0.241622\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −254.078 −1.28974 −0.644869 0.764293i \(-0.723088\pi\)
−0.644869 + 0.764293i \(0.723088\pi\)
\(198\) 0 0
\(199\) − 150.197i − 0.754757i −0.926059 0.377379i \(-0.876826\pi\)
0.926059 0.377379i \(-0.123174\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 51.0550i 0.251503i
\(204\) 0 0
\(205\) −195.541 −0.953858
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 93.4962 0.447350
\(210\) 0 0
\(211\) 294.457i 1.39553i 0.716326 + 0.697765i \(0.245822\pi\)
−0.716326 + 0.697765i \(0.754178\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 179.494i 0.834855i
\(216\) 0 0
\(217\) 98.7363 0.455006
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −435.813 −1.97201
\(222\) 0 0
\(223\) 90.5784i 0.406181i 0.979160 + 0.203091i \(0.0650986\pi\)
−0.979160 + 0.203091i \(0.934901\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 401.033i 1.76667i 0.468746 + 0.883333i \(0.344706\pi\)
−0.468746 + 0.883333i \(0.655294\pi\)
\(228\) 0 0
\(229\) −155.539 −0.679209 −0.339605 0.940568i \(-0.610293\pi\)
−0.339605 + 0.940568i \(0.610293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −238.595 −1.02401 −0.512007 0.858981i \(-0.671098\pi\)
−0.512007 + 0.858981i \(0.671098\pi\)
\(234\) 0 0
\(235\) − 152.072i − 0.647114i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 61.3314i 0.256617i 0.991734 + 0.128308i \(0.0409547\pi\)
−0.991734 + 0.128308i \(0.959045\pi\)
\(240\) 0 0
\(241\) 270.776 1.12355 0.561776 0.827290i \(-0.310118\pi\)
0.561776 + 0.827290i \(0.310118\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −168.334 −0.687079
\(246\) 0 0
\(247\) − 401.630i − 1.62603i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 229.191i − 0.913110i −0.889695 0.456555i \(-0.849083\pi\)
0.889695 0.456555i \(-0.150917\pi\)
\(252\) 0 0
\(253\) −132.314 −0.522979
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −366.617 −1.42653 −0.713263 0.700897i \(-0.752783\pi\)
−0.713263 + 0.700897i \(0.752783\pi\)
\(258\) 0 0
\(259\) 73.3842i 0.283337i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 159.531i 0.606581i 0.952898 + 0.303290i \(0.0980852\pi\)
−0.952898 + 0.303290i \(0.901915\pi\)
\(264\) 0 0
\(265\) −201.727 −0.761235
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 91.2480 0.339212 0.169606 0.985512i \(-0.445751\pi\)
0.169606 + 0.985512i \(0.445751\pi\)
\(270\) 0 0
\(271\) 506.145i 1.86769i 0.357675 + 0.933846i \(0.383570\pi\)
−0.357675 + 0.933846i \(0.616430\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 5.77515i − 0.0210006i
\(276\) 0 0
\(277\) −28.7143 −0.103662 −0.0518308 0.998656i \(-0.516506\pi\)
−0.0518308 + 0.998656i \(0.516506\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 194.297 0.691447 0.345724 0.938336i \(-0.387633\pi\)
0.345724 + 0.938336i \(0.387633\pi\)
\(282\) 0 0
\(283\) − 212.089i − 0.749432i −0.927140 0.374716i \(-0.877740\pi\)
0.927140 0.374716i \(-0.122260\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 153.355i 0.534336i
\(288\) 0 0
\(289\) 266.784 0.923127
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −118.777 −0.405382 −0.202691 0.979243i \(-0.564969\pi\)
−0.202691 + 0.979243i \(0.564969\pi\)
\(294\) 0 0
\(295\) 335.369i 1.13684i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 568.378i 1.90093i
\(300\) 0 0
\(301\) 140.769 0.467673
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 153.225 0.502376
\(306\) 0 0
\(307\) 118.198i 0.385011i 0.981296 + 0.192506i \(0.0616614\pi\)
−0.981296 + 0.192506i \(0.938339\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 57.5386i − 0.185012i −0.995712 0.0925059i \(-0.970512\pi\)
0.995712 0.0925059i \(-0.0294877\pi\)
\(312\) 0 0
\(313\) −401.012 −1.28119 −0.640594 0.767879i \(-0.721312\pi\)
−0.640594 + 0.767879i \(0.721312\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.34020 0.0263098 0.0131549 0.999913i \(-0.495813\pi\)
0.0131549 + 0.999913i \(0.495813\pi\)
\(318\) 0 0
\(319\) − 54.5847i − 0.171112i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 512.190i 1.58573i
\(324\) 0 0
\(325\) −24.8082 −0.0763330
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −119.264 −0.362503
\(330\) 0 0
\(331\) − 253.990i − 0.767340i −0.923470 0.383670i \(-0.874660\pi\)
0.923470 0.383670i \(-0.125340\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 60.8883i 0.181756i
\(336\) 0 0
\(337\) −620.332 −1.84075 −0.920374 0.391040i \(-0.872115\pi\)
−0.920374 + 0.391040i \(0.872115\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −105.562 −0.309567
\(342\) 0 0
\(343\) 329.250i 0.959914i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 496.172i − 1.42989i −0.699181 0.714945i \(-0.746452\pi\)
0.699181 0.714945i \(-0.253548\pi\)
\(348\) 0 0
\(349\) 63.4267 0.181738 0.0908692 0.995863i \(-0.471035\pi\)
0.0908692 + 0.995863i \(0.471035\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −439.982 −1.24641 −0.623204 0.782059i \(-0.714170\pi\)
−0.623204 + 0.782059i \(0.714170\pi\)
\(354\) 0 0
\(355\) 721.882i 2.03347i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 489.550i 1.36365i 0.731516 + 0.681824i \(0.238813\pi\)
−0.731516 + 0.681824i \(0.761187\pi\)
\(360\) 0 0
\(361\) −111.016 −0.307524
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −612.224 −1.67733
\(366\) 0 0
\(367\) − 393.063i − 1.07102i −0.844530 0.535509i \(-0.820120\pi\)
0.844530 0.535509i \(-0.179880\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 158.206i 0.426432i
\(372\) 0 0
\(373\) 11.4757 0.0307658 0.0153829 0.999882i \(-0.495103\pi\)
0.0153829 + 0.999882i \(0.495103\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −234.478 −0.621959
\(378\) 0 0
\(379\) − 661.804i − 1.74618i −0.487556 0.873092i \(-0.662112\pi\)
0.487556 0.873092i \(-0.337888\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 95.8558i − 0.250276i −0.992139 0.125138i \(-0.960063\pi\)
0.992139 0.125138i \(-0.0399374\pi\)
\(384\) 0 0
\(385\) −88.9043 −0.230920
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 471.609 1.21236 0.606182 0.795326i \(-0.292700\pi\)
0.606182 + 0.795326i \(0.292700\pi\)
\(390\) 0 0
\(391\) − 724.840i − 1.85381i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 47.1599i − 0.119392i
\(396\) 0 0
\(397\) 718.221 1.80912 0.904560 0.426346i \(-0.140199\pi\)
0.904560 + 0.426346i \(0.140199\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −225.675 −0.562780 −0.281390 0.959593i \(-0.590796\pi\)
−0.281390 + 0.959593i \(0.590796\pi\)
\(402\) 0 0
\(403\) 453.462i 1.12522i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 78.4576i − 0.192770i
\(408\) 0 0
\(409\) 588.917 1.43990 0.719948 0.694028i \(-0.244166\pi\)
0.719948 + 0.694028i \(0.244166\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 263.015 0.636841
\(414\) 0 0
\(415\) 580.892i 1.39974i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 121.906i − 0.290944i −0.989362 0.145472i \(-0.953530\pi\)
0.989362 0.145472i \(-0.0464701\pi\)
\(420\) 0 0
\(421\) 289.293 0.687157 0.343578 0.939124i \(-0.388361\pi\)
0.343578 + 0.939124i \(0.388361\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 31.6374 0.0744410
\(426\) 0 0
\(427\) − 120.168i − 0.281423i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 337.402i − 0.782835i −0.920213 0.391418i \(-0.871985\pi\)
0.920213 0.391418i \(-0.128015\pi\)
\(432\) 0 0
\(433\) −7.18171 −0.0165859 −0.00829297 0.999966i \(-0.502640\pi\)
−0.00829297 + 0.999966i \(0.502640\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 667.987 1.52857
\(438\) 0 0
\(439\) − 780.145i − 1.77710i −0.458783 0.888548i \(-0.651715\pi\)
0.458783 0.888548i \(-0.348285\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 478.844i − 1.08091i −0.841372 0.540456i \(-0.818252\pi\)
0.841372 0.540456i \(-0.181748\pi\)
\(444\) 0 0
\(445\) −40.4614 −0.0909245
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 55.5835 0.123794 0.0618970 0.998083i \(-0.480285\pi\)
0.0618970 + 0.998083i \(0.480285\pi\)
\(450\) 0 0
\(451\) − 163.957i − 0.363540i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 381.905i 0.839351i
\(456\) 0 0
\(457\) 31.6281 0.0692080 0.0346040 0.999401i \(-0.488983\pi\)
0.0346040 + 0.999401i \(0.488983\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 498.508 1.08136 0.540681 0.841228i \(-0.318167\pi\)
0.540681 + 0.841228i \(0.318167\pi\)
\(462\) 0 0
\(463\) 63.5797i 0.137321i 0.997640 + 0.0686606i \(0.0218726\pi\)
−0.997640 + 0.0686606i \(0.978127\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 550.457i − 1.17871i −0.807875 0.589354i \(-0.799382\pi\)
0.807875 0.589354i \(-0.200618\pi\)
\(468\) 0 0
\(469\) 47.7521 0.101817
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −150.502 −0.318185
\(474\) 0 0
\(475\) 29.1559i 0.0613808i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 760.357i − 1.58738i −0.608320 0.793692i \(-0.708156\pi\)
0.608320 0.793692i \(-0.291844\pi\)
\(480\) 0 0
\(481\) −337.029 −0.700683
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −284.892 −0.587406
\(486\) 0 0
\(487\) 340.198i 0.698558i 0.937019 + 0.349279i \(0.113573\pi\)
−0.937019 + 0.349279i \(0.886427\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 488.166i 0.994229i 0.867685 + 0.497114i \(0.165607\pi\)
−0.867685 + 0.497114i \(0.834393\pi\)
\(492\) 0 0
\(493\) 299.026 0.606543
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 566.141 1.13912
\(498\) 0 0
\(499\) 195.282i 0.391347i 0.980669 + 0.195674i \(0.0626893\pi\)
−0.980669 + 0.195674i \(0.937311\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 167.239i − 0.332483i −0.986085 0.166241i \(-0.946837\pi\)
0.986085 0.166241i \(-0.0531631\pi\)
\(504\) 0 0
\(505\) −1001.78 −1.98372
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 374.972 0.736683 0.368342 0.929691i \(-0.379926\pi\)
0.368342 + 0.929691i \(0.379926\pi\)
\(510\) 0 0
\(511\) 480.142i 0.939612i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 150.678i 0.292578i
\(516\) 0 0
\(517\) 127.509 0.246632
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −49.7644 −0.0955170 −0.0477585 0.998859i \(-0.515208\pi\)
−0.0477585 + 0.998859i \(0.515208\pi\)
\(522\) 0 0
\(523\) − 878.700i − 1.68012i −0.542497 0.840058i \(-0.682521\pi\)
0.542497 0.840058i \(-0.317479\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 578.291i − 1.09733i
\(528\) 0 0
\(529\) −416.320 −0.786995
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −704.306 −1.32140
\(534\) 0 0
\(535\) − 683.928i − 1.27837i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 141.145i − 0.261864i
\(540\) 0 0
\(541\) 183.963 0.340043 0.170022 0.985440i \(-0.445616\pi\)
0.170022 + 0.985440i \(0.445616\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1018.71 1.86920
\(546\) 0 0
\(547\) 642.639i 1.17484i 0.809281 + 0.587421i \(0.199857\pi\)
−0.809281 + 0.587421i \(0.800143\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 275.571i 0.500129i
\(552\) 0 0
\(553\) −36.9855 −0.0668816
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 231.980 0.416480 0.208240 0.978078i \(-0.433226\pi\)
0.208240 + 0.978078i \(0.433226\pi\)
\(558\) 0 0
\(559\) 646.507i 1.15654i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 356.868i 0.633869i 0.948447 + 0.316935i \(0.102654\pi\)
−0.948447 + 0.316935i \(0.897346\pi\)
\(564\) 0 0
\(565\) 917.400 1.62372
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 440.786 0.774669 0.387334 0.921939i \(-0.373396\pi\)
0.387334 + 0.921939i \(0.373396\pi\)
\(570\) 0 0
\(571\) 712.649i 1.24807i 0.781396 + 0.624036i \(0.214508\pi\)
−0.781396 + 0.624036i \(0.785492\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 41.2608i − 0.0717579i
\(576\) 0 0
\(577\) −253.401 −0.439170 −0.219585 0.975593i \(-0.570470\pi\)
−0.219585 + 0.975593i \(0.570470\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 455.569 0.784112
\(582\) 0 0
\(583\) − 169.144i − 0.290126i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 800.800i − 1.36422i −0.731248 0.682112i \(-0.761062\pi\)
0.731248 0.682112i \(-0.238938\pi\)
\(588\) 0 0
\(589\) 532.932 0.904809
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 618.665 1.04328 0.521640 0.853166i \(-0.325321\pi\)
0.521640 + 0.853166i \(0.325321\pi\)
\(594\) 0 0
\(595\) − 487.035i − 0.818546i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 249.660i − 0.416795i −0.978044 0.208398i \(-0.933175\pi\)
0.978044 0.208398i \(-0.0668248\pi\)
\(600\) 0 0
\(601\) −1137.02 −1.89188 −0.945941 0.324339i \(-0.894858\pi\)
−0.945941 + 0.324339i \(0.894858\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −525.975 −0.869380
\(606\) 0 0
\(607\) 520.853i 0.858077i 0.903286 + 0.429038i \(0.141148\pi\)
−0.903286 + 0.429038i \(0.858852\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 547.737i − 0.896460i
\(612\) 0 0
\(613\) 148.439 0.242152 0.121076 0.992643i \(-0.461365\pi\)
0.121076 + 0.992643i \(0.461365\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1081.44 1.75273 0.876366 0.481646i \(-0.159961\pi\)
0.876366 + 0.481646i \(0.159961\pi\)
\(618\) 0 0
\(619\) − 306.359i − 0.494925i −0.968897 0.247463i \(-0.920403\pi\)
0.968897 0.247463i \(-0.0795967\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 31.7322i 0.0509345i
\(624\) 0 0
\(625\) −656.749 −1.05080
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 429.806 0.683316
\(630\) 0 0
\(631\) − 807.507i − 1.27973i −0.768489 0.639863i \(-0.778991\pi\)
0.768489 0.639863i \(-0.221009\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 883.874i − 1.39193i
\(636\) 0 0
\(637\) −606.312 −0.951824
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −387.743 −0.604904 −0.302452 0.953165i \(-0.597805\pi\)
−0.302452 + 0.953165i \(0.597805\pi\)
\(642\) 0 0
\(643\) 978.913i 1.52242i 0.648508 + 0.761208i \(0.275393\pi\)
−0.648508 + 0.761208i \(0.724607\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 786.067i − 1.21494i −0.794342 0.607470i \(-0.792184\pi\)
0.794342 0.607470i \(-0.207816\pi\)
\(648\) 0 0
\(649\) −281.199 −0.433280
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 251.026 0.384419 0.192209 0.981354i \(-0.438435\pi\)
0.192209 + 0.981354i \(0.438435\pi\)
\(654\) 0 0
\(655\) 1058.82i 1.61653i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 933.805i − 1.41700i −0.705709 0.708501i \(-0.749372\pi\)
0.705709 0.708501i \(-0.250628\pi\)
\(660\) 0 0
\(661\) 473.319 0.716065 0.358033 0.933709i \(-0.383448\pi\)
0.358033 + 0.933709i \(0.383448\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 448.834 0.674938
\(666\) 0 0
\(667\) − 389.982i − 0.584681i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 128.475i 0.191469i
\(672\) 0 0
\(673\) 254.846 0.378671 0.189336 0.981912i \(-0.439367\pi\)
0.189336 + 0.981912i \(0.439367\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 104.575 0.154468 0.0772338 0.997013i \(-0.475391\pi\)
0.0772338 + 0.997013i \(0.475391\pi\)
\(678\) 0 0
\(679\) 223.429i 0.329055i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 623.576i 0.912995i 0.889725 + 0.456497i \(0.150896\pi\)
−0.889725 + 0.456497i \(0.849104\pi\)
\(684\) 0 0
\(685\) −436.817 −0.637689
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −726.588 −1.05455
\(690\) 0 0
\(691\) − 544.623i − 0.788167i −0.919075 0.394083i \(-0.871062\pi\)
0.919075 0.394083i \(-0.128938\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 771.541i − 1.11013i
\(696\) 0 0
\(697\) 898.186 1.28865
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 411.842 0.587507 0.293753 0.955881i \(-0.405096\pi\)
0.293753 + 0.955881i \(0.405096\pi\)
\(702\) 0 0
\(703\) 396.093i 0.563433i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 785.653i 1.11125i
\(708\) 0 0
\(709\) 679.236 0.958020 0.479010 0.877809i \(-0.340996\pi\)
0.479010 + 0.877809i \(0.340996\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −754.194 −1.05778
\(714\) 0 0
\(715\) − 408.308i − 0.571060i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 619.766i − 0.861983i −0.902356 0.430991i \(-0.858164\pi\)
0.902356 0.430991i \(-0.141836\pi\)
\(720\) 0 0
\(721\) 118.170 0.163898
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 17.0217 0.0234782
\(726\) 0 0
\(727\) − 1224.73i − 1.68463i −0.538982 0.842317i \(-0.681191\pi\)
0.538982 0.842317i \(-0.318809\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 824.477i − 1.12788i
\(732\) 0 0
\(733\) 797.758 1.08835 0.544173 0.838973i \(-0.316844\pi\)
0.544173 + 0.838973i \(0.316844\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −51.0534 −0.0692719
\(738\) 0 0
\(739\) 422.067i 0.571133i 0.958359 + 0.285567i \(0.0921818\pi\)
−0.958359 + 0.285567i \(0.907818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 805.671i 1.08435i 0.840266 + 0.542174i \(0.182399\pi\)
−0.840266 + 0.542174i \(0.817601\pi\)
\(744\) 0 0
\(745\) 570.385 0.765618
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −536.376 −0.716123
\(750\) 0 0
\(751\) − 1014.96i − 1.35148i −0.737138 0.675742i \(-0.763823\pi\)
0.737138 0.675742i \(-0.236177\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 853.501i 1.13046i
\(756\) 0 0
\(757\) −14.8738 −0.0196483 −0.00982416 0.999952i \(-0.503127\pi\)
−0.00982416 + 0.999952i \(0.503127\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 884.373 1.16212 0.581060 0.813861i \(-0.302638\pi\)
0.581060 + 0.813861i \(0.302638\pi\)
\(762\) 0 0
\(763\) − 798.933i − 1.04709i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1207.94i 1.57489i
\(768\) 0 0
\(769\) −775.760 −1.00879 −0.504395 0.863473i \(-0.668285\pi\)
−0.504395 + 0.863473i \(0.668285\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −794.115 −1.02732 −0.513658 0.857995i \(-0.671710\pi\)
−0.513658 + 0.857995i \(0.671710\pi\)
\(774\) 0 0
\(775\) − 32.9186i − 0.0424756i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 827.736i 1.06256i
\(780\) 0 0
\(781\) −605.281 −0.775008
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −257.423 −0.327927
\(786\) 0 0
\(787\) − 344.287i − 0.437468i −0.975785 0.218734i \(-0.929807\pi\)
0.975785 0.218734i \(-0.0701927\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 719.478i − 0.909581i
\(792\) 0 0
\(793\) 551.890 0.695952
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 986.969 1.23836 0.619178 0.785251i \(-0.287466\pi\)
0.619178 + 0.785251i \(0.287466\pi\)
\(798\) 0 0
\(799\) 698.518i 0.874240i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 513.336i − 0.639273i
\(804\) 0 0
\(805\) −635.180 −0.789043
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −33.7831 −0.0417591 −0.0208795 0.999782i \(-0.506647\pi\)
−0.0208795 + 0.999782i \(0.506647\pi\)
\(810\) 0 0
\(811\) − 174.359i − 0.214993i −0.994205 0.107496i \(-0.965717\pi\)
0.994205 0.107496i \(-0.0342834\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1365.20i 1.67509i
\(816\) 0 0
\(817\) 759.808 0.929997
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 826.976 1.00728 0.503640 0.863914i \(-0.331994\pi\)
0.503640 + 0.863914i \(0.331994\pi\)
\(822\) 0 0
\(823\) − 789.862i − 0.959735i −0.877341 0.479868i \(-0.840685\pi\)
0.877341 0.479868i \(-0.159315\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 94.0797i − 0.113760i −0.998381 0.0568801i \(-0.981885\pi\)
0.998381 0.0568801i \(-0.0181153\pi\)
\(828\) 0 0
\(829\) 383.996 0.463204 0.231602 0.972811i \(-0.425603\pi\)
0.231602 + 0.972811i \(0.425603\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 773.217 0.928232
\(834\) 0 0
\(835\) 1147.88i 1.37470i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1148.75i − 1.36919i −0.728924 0.684594i \(-0.759979\pi\)
0.728924 0.684594i \(-0.240021\pi\)
\(840\) 0 0
\(841\) −680.117 −0.808700
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −886.576 −1.04920
\(846\) 0 0
\(847\) 412.500i 0.487013i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 560.543i − 0.658687i
\(852\) 0 0
\(853\) 165.565 0.194097 0.0970485 0.995280i \(-0.469060\pi\)
0.0970485 + 0.995280i \(0.469060\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1294.69 −1.51073 −0.755363 0.655307i \(-0.772539\pi\)
−0.755363 + 0.655307i \(0.772539\pi\)
\(858\) 0 0
\(859\) 14.0574i 0.0163648i 0.999967 + 0.00818241i \(0.00260457\pi\)
−0.999967 + 0.00818241i \(0.997395\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 409.653i 0.474685i 0.971426 + 0.237343i \(0.0762764\pi\)
−0.971426 + 0.237343i \(0.923724\pi\)
\(864\) 0 0
\(865\) −284.715 −0.329150
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 39.5425 0.0455035
\(870\) 0 0
\(871\) 219.309i 0.251790i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 488.749i 0.558570i
\(876\) 0 0
\(877\) −10.0270 −0.0114333 −0.00571666 0.999984i \(-0.501820\pi\)
−0.00571666 + 0.999984i \(0.501820\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1388.38 −1.57591 −0.787956 0.615731i \(-0.788861\pi\)
−0.787956 + 0.615731i \(0.788861\pi\)
\(882\) 0 0
\(883\) 1445.91i 1.63750i 0.574153 + 0.818748i \(0.305332\pi\)
−0.574153 + 0.818748i \(0.694668\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 661.924i 0.746250i 0.927781 + 0.373125i \(0.121714\pi\)
−0.927781 + 0.373125i \(0.878286\pi\)
\(888\) 0 0
\(889\) −693.185 −0.779736
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −643.729 −0.720861
\(894\) 0 0
\(895\) 55.2058i 0.0616824i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 311.135i − 0.346090i
\(900\) 0 0
\(901\) 926.602 1.02842
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 186.635 0.206227
\(906\) 0 0
\(907\) 314.099i 0.346306i 0.984895 + 0.173153i \(0.0553955\pi\)
−0.984895 + 0.173153i \(0.944605\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 620.050i − 0.680626i −0.940312 0.340313i \(-0.889467\pi\)
0.940312 0.340313i \(-0.110533\pi\)
\(912\) 0 0
\(913\) −487.065 −0.533478
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 830.391 0.905552
\(918\) 0 0
\(919\) 259.390i 0.282253i 0.989992 + 0.141126i \(0.0450724\pi\)
−0.989992 + 0.141126i \(0.954928\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2600.10i 2.81701i
\(924\) 0 0
\(925\) 24.4662 0.0264500
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −511.871 −0.550991 −0.275496 0.961302i \(-0.588842\pi\)
−0.275496 + 0.961302i \(0.588842\pi\)
\(930\) 0 0
\(931\) 712.569i 0.765380i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 520.706i 0.556905i
\(936\) 0 0
\(937\) 531.153 0.566866 0.283433 0.958992i \(-0.408527\pi\)
0.283433 + 0.958992i \(0.408527\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1401.22 1.48908 0.744540 0.667578i \(-0.232669\pi\)
0.744540 + 0.667578i \(0.232669\pi\)
\(942\) 0 0
\(943\) − 1171.39i − 1.24220i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 246.077i − 0.259849i −0.991524 0.129925i \(-0.958526\pi\)
0.991524 0.129925i \(-0.0414736\pi\)
\(948\) 0 0
\(949\) −2205.13 −2.32363
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1711.93 −1.79636 −0.898178 0.439631i \(-0.855109\pi\)
−0.898178 + 0.439631i \(0.855109\pi\)
\(954\) 0 0
\(955\) − 1100.62i − 1.15249i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 342.577i 0.357223i
\(960\) 0 0
\(961\) 359.290 0.373871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1437.25 −1.48937
\(966\) 0 0
\(967\) 801.048i 0.828385i 0.910189 + 0.414192i \(0.135936\pi\)
−0.910189 + 0.414192i \(0.864064\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 526.798i 0.542532i 0.962504 + 0.271266i \(0.0874423\pi\)
−0.962504 + 0.271266i \(0.912558\pi\)
\(972\) 0 0
\(973\) −605.087 −0.621877
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −630.886 −0.645738 −0.322869 0.946444i \(-0.604647\pi\)
−0.322869 + 0.946444i \(0.604647\pi\)
\(978\) 0 0
\(979\) − 33.9260i − 0.0346537i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1553.23i 1.58009i 0.613051 + 0.790043i \(0.289942\pi\)
−0.613051 + 0.790043i \(0.710058\pi\)
\(984\) 0 0
\(985\) 1304.04 1.32390
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1075.26 −1.08722
\(990\) 0 0
\(991\) 487.053i 0.491476i 0.969336 + 0.245738i \(0.0790303\pi\)
−0.969336 + 0.245738i \(0.920970\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 770.876i 0.774750i
\(996\) 0 0
\(997\) −1665.23 −1.67024 −0.835122 0.550065i \(-0.814603\pi\)
−0.835122 + 0.550065i \(0.814603\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 864.3.g.b.703.4 yes 8
3.2 odd 2 864.3.g.d.703.6 yes 8
4.3 odd 2 inner 864.3.g.b.703.3 8
8.3 odd 2 1728.3.g.m.703.5 8
8.5 even 2 1728.3.g.m.703.6 8
12.11 even 2 864.3.g.d.703.5 yes 8
24.5 odd 2 1728.3.g.j.703.4 8
24.11 even 2 1728.3.g.j.703.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.b.703.3 8 4.3 odd 2 inner
864.3.g.b.703.4 yes 8 1.1 even 1 trivial
864.3.g.d.703.5 yes 8 12.11 even 2
864.3.g.d.703.6 yes 8 3.2 odd 2
1728.3.g.j.703.3 8 24.11 even 2
1728.3.g.j.703.4 8 24.5 odd 2
1728.3.g.m.703.5 8 8.3 odd 2
1728.3.g.m.703.6 8 8.5 even 2