Properties

Label 800.4.f.a.49.4
Level $800$
Weight $4$
Character 800.49
Analytic conductor $47.202$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(-1.32288 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 800.49
Dual form 800.4.f.a.49.3

$q$-expansion

\(f(q)\) \(=\) \(q+5.29150 q^{3} +8.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+5.29150 q^{3} +8.00000i q^{7} +1.00000 q^{9} -15.8745i q^{11} -52.9150 q^{13} -14.0000i q^{17} +37.0405i q^{19} +42.3320i q^{21} -152.000i q^{23} -137.579 q^{27} -158.745i q^{29} -224.000 q^{31} -84.0000i q^{33} +243.409 q^{37} -280.000 q^{39} -70.0000 q^{41} -439.195 q^{43} -336.000i q^{47} +279.000 q^{49} -74.0810i q^{51} -31.7490 q^{53} +196.000i q^{57} -534.442i q^{59} -95.2470i q^{61} +8.00000i q^{63} -174.620 q^{67} -804.308i q^{69} +72.0000 q^{71} +294.000i q^{73} +126.996 q^{77} -464.000 q^{79} -755.000 q^{81} -545.025 q^{83} -840.000i q^{87} -266.000 q^{89} -423.320i q^{91} -1185.30 q^{93} +994.000i q^{97} -15.8745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 896 q^{31} - 1120 q^{39} - 280 q^{41} + 1116 q^{49} + 288 q^{71} - 1856 q^{79} - 3020 q^{81} - 1064 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 0
\(3\) 5.29150 1.01835 0.509175 0.860663i \(-0.329951\pi\)
0.509175 + 0.860663i \(0.329951\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.00000i 0.431959i 0.976398 + 0.215980i \(0.0692945\pi\)
−0.976398 + 0.215980i \(0.930705\pi\)
\(8\) 0 0
\(9\) 1.00000 0.0370370
\(10\) 0 0
\(11\) − 15.8745i − 0.435122i −0.976047 0.217561i \(-0.930190\pi\)
0.976047 0.217561i \(-0.0698101\pi\)
\(12\) 0 0
\(13\) −52.9150 −1.12892 −0.564461 0.825460i \(-0.690916\pi\)
−0.564461 + 0.825460i \(0.690916\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 14.0000i − 0.199735i −0.995001 0.0998676i \(-0.968158\pi\)
0.995001 0.0998676i \(-0.0318419\pi\)
\(18\) 0 0
\(19\) 37.0405i 0.447246i 0.974676 + 0.223623i \(0.0717885\pi\)
−0.974676 + 0.223623i \(0.928212\pi\)
\(20\) 0 0
\(21\) 42.3320i 0.439886i
\(22\) 0 0
\(23\) − 152.000i − 1.37801i −0.724757 0.689004i \(-0.758048\pi\)
0.724757 0.689004i \(-0.241952\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −137.579 −0.980633
\(28\) 0 0
\(29\) − 158.745i − 1.01649i −0.861212 0.508245i \(-0.830294\pi\)
0.861212 0.508245i \(-0.169706\pi\)
\(30\) 0 0
\(31\) −224.000 −1.29779 −0.648897 0.760877i \(-0.724769\pi\)
−0.648897 + 0.760877i \(0.724769\pi\)
\(32\) 0 0
\(33\) − 84.0000i − 0.443107i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 243.409 1.08152 0.540760 0.841177i \(-0.318137\pi\)
0.540760 + 0.841177i \(0.318137\pi\)
\(38\) 0 0
\(39\) −280.000 −1.14964
\(40\) 0 0
\(41\) −70.0000 −0.266638 −0.133319 0.991073i \(-0.542564\pi\)
−0.133319 + 0.991073i \(0.542564\pi\)
\(42\) 0 0
\(43\) −439.195 −1.55759 −0.778797 0.627276i \(-0.784170\pi\)
−0.778797 + 0.627276i \(0.784170\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 336.000i − 1.04278i −0.853319 0.521390i \(-0.825414\pi\)
0.853319 0.521390i \(-0.174586\pi\)
\(48\) 0 0
\(49\) 279.000 0.813411
\(50\) 0 0
\(51\) − 74.0810i − 0.203400i
\(52\) 0 0
\(53\) −31.7490 −0.0822842 −0.0411421 0.999153i \(-0.513100\pi\)
−0.0411421 + 0.999153i \(0.513100\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 196.000i 0.455453i
\(58\) 0 0
\(59\) − 534.442i − 1.17929i −0.807661 0.589647i \(-0.799267\pi\)
0.807661 0.589647i \(-0.200733\pi\)
\(60\) 0 0
\(61\) − 95.2470i − 0.199920i −0.994991 0.0999601i \(-0.968128\pi\)
0.994991 0.0999601i \(-0.0318715\pi\)
\(62\) 0 0
\(63\) 8.00000i 0.0159985i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −174.620 −0.318406 −0.159203 0.987246i \(-0.550892\pi\)
−0.159203 + 0.987246i \(0.550892\pi\)
\(68\) 0 0
\(69\) − 804.308i − 1.40329i
\(70\) 0 0
\(71\) 72.0000 0.120350 0.0601748 0.998188i \(-0.480834\pi\)
0.0601748 + 0.998188i \(0.480834\pi\)
\(72\) 0 0
\(73\) 294.000i 0.471371i 0.971829 + 0.235686i \(0.0757336\pi\)
−0.971829 + 0.235686i \(0.924266\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 126.996 0.187955
\(78\) 0 0
\(79\) −464.000 −0.660811 −0.330406 0.943839i \(-0.607186\pi\)
−0.330406 + 0.943839i \(0.607186\pi\)
\(80\) 0 0
\(81\) −755.000 −1.03567
\(82\) 0 0
\(83\) −545.025 −0.720774 −0.360387 0.932803i \(-0.617355\pi\)
−0.360387 + 0.932803i \(0.617355\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 840.000i − 1.03514i
\(88\) 0 0
\(89\) −266.000 −0.316808 −0.158404 0.987374i \(-0.550635\pi\)
−0.158404 + 0.987374i \(0.550635\pi\)
\(90\) 0 0
\(91\) − 423.320i − 0.487649i
\(92\) 0 0
\(93\) −1185.30 −1.32161
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 994.000i 1.04047i 0.854024 + 0.520234i \(0.174155\pi\)
−0.854024 + 0.520234i \(0.825845\pi\)
\(98\) 0 0
\(99\) − 15.8745i − 0.0161156i
\(100\) 0 0
\(101\) − 751.393i − 0.740262i −0.928980 0.370131i \(-0.879313\pi\)
0.928980 0.370131i \(-0.120687\pi\)
\(102\) 0 0
\(103\) 1176.00i 1.12500i 0.826798 + 0.562499i \(0.190160\pi\)
−0.826798 + 0.562499i \(0.809840\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 269.867 0.243822 0.121911 0.992541i \(-0.461098\pi\)
0.121911 + 0.992541i \(0.461098\pi\)
\(108\) 0 0
\(109\) − 1894.36i − 1.66465i −0.554290 0.832324i \(-0.687010\pi\)
0.554290 0.832324i \(-0.312990\pi\)
\(110\) 0 0
\(111\) 1288.00 1.10137
\(112\) 0 0
\(113\) 1710.00i 1.42357i 0.702398 + 0.711784i \(0.252113\pi\)
−0.702398 + 0.711784i \(0.747887\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −52.9150 −0.0418119
\(118\) 0 0
\(119\) 112.000 0.0862775
\(120\) 0 0
\(121\) 1079.00 0.810669
\(122\) 0 0
\(123\) −370.405 −0.271531
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1664.00i 1.16265i 0.813673 + 0.581323i \(0.197465\pi\)
−0.813673 + 0.581323i \(0.802535\pi\)
\(128\) 0 0
\(129\) −2324.00 −1.58618
\(130\) 0 0
\(131\) − 672.021i − 0.448204i −0.974566 0.224102i \(-0.928055\pi\)
0.974566 0.224102i \(-0.0719449\pi\)
\(132\) 0 0
\(133\) −296.324 −0.193192
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1062.00i − 0.662283i −0.943581 0.331142i \(-0.892566\pi\)
0.943581 0.331142i \(-0.107434\pi\)
\(138\) 0 0
\(139\) − 2693.37i − 1.64352i −0.569835 0.821759i \(-0.692993\pi\)
0.569835 0.821759i \(-0.307007\pi\)
\(140\) 0 0
\(141\) − 1777.94i − 1.06191i
\(142\) 0 0
\(143\) 840.000i 0.491219i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1476.33 0.828337
\(148\) 0 0
\(149\) 793.725i 0.436406i 0.975903 + 0.218203i \(0.0700195\pi\)
−0.975903 + 0.218203i \(0.929980\pi\)
\(150\) 0 0
\(151\) −744.000 −0.400966 −0.200483 0.979697i \(-0.564251\pi\)
−0.200483 + 0.979697i \(0.564251\pi\)
\(152\) 0 0
\(153\) − 14.0000i − 0.00739760i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 179.911 0.0914552 0.0457276 0.998954i \(-0.485439\pi\)
0.0457276 + 0.998954i \(0.485439\pi\)
\(158\) 0 0
\(159\) −168.000 −0.0837941
\(160\) 0 0
\(161\) 1216.00 0.595244
\(162\) 0 0
\(163\) −1772.65 −0.851809 −0.425905 0.904768i \(-0.640044\pi\)
−0.425905 + 0.904768i \(0.640044\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1960.00i 0.908200i 0.890951 + 0.454100i \(0.150039\pi\)
−0.890951 + 0.454100i \(0.849961\pi\)
\(168\) 0 0
\(169\) 603.000 0.274465
\(170\) 0 0
\(171\) 37.0405i 0.0165647i
\(172\) 0 0
\(173\) −2000.19 −0.879026 −0.439513 0.898236i \(-0.644849\pi\)
−0.439513 + 0.898236i \(0.644849\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2828.00i − 1.20094i
\(178\) 0 0
\(179\) − 3264.86i − 1.36328i −0.731688 0.681639i \(-0.761267\pi\)
0.731688 0.681639i \(-0.238733\pi\)
\(180\) 0 0
\(181\) 2338.84i 0.960469i 0.877140 + 0.480235i \(0.159448\pi\)
−0.877140 + 0.480235i \(0.840552\pi\)
\(182\) 0 0
\(183\) − 504.000i − 0.203589i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −222.243 −0.0869092
\(188\) 0 0
\(189\) − 1100.63i − 0.423594i
\(190\) 0 0
\(191\) −3904.00 −1.47897 −0.739486 0.673172i \(-0.764931\pi\)
−0.739486 + 0.673172i \(0.764931\pi\)
\(192\) 0 0
\(193\) − 3330.00i − 1.24196i −0.783826 0.620981i \(-0.786734\pi\)
0.783826 0.620981i \(-0.213266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1195.88 −0.432502 −0.216251 0.976338i \(-0.569383\pi\)
−0.216251 + 0.976338i \(0.569383\pi\)
\(198\) 0 0
\(199\) −1736.00 −0.618401 −0.309200 0.950997i \(-0.600061\pi\)
−0.309200 + 0.950997i \(0.600061\pi\)
\(200\) 0 0
\(201\) −924.000 −0.324248
\(202\) 0 0
\(203\) 1269.96 0.439083
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 152.000i − 0.0510373i
\(208\) 0 0
\(209\) 588.000 0.194607
\(210\) 0 0
\(211\) − 2915.62i − 0.951277i −0.879641 0.475638i \(-0.842217\pi\)
0.879641 0.475638i \(-0.157783\pi\)
\(212\) 0 0
\(213\) 380.988 0.122558
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1792.00i − 0.560594i
\(218\) 0 0
\(219\) 1555.70i 0.480021i
\(220\) 0 0
\(221\) 740.810i 0.225486i
\(222\) 0 0
\(223\) 1568.00i 0.470857i 0.971892 + 0.235428i \(0.0756493\pi\)
−0.971892 + 0.235428i \(0.924351\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1264.67 0.369775 0.184888 0.982760i \(-0.440808\pi\)
0.184888 + 0.982760i \(0.440808\pi\)
\(228\) 0 0
\(229\) 5153.92i 1.48725i 0.668595 + 0.743626i \(0.266896\pi\)
−0.668595 + 0.743626i \(0.733104\pi\)
\(230\) 0 0
\(231\) 672.000 0.191404
\(232\) 0 0
\(233\) 838.000i 0.235619i 0.993036 + 0.117809i \(0.0375872\pi\)
−0.993036 + 0.117809i \(0.962413\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2455.26 −0.672937
\(238\) 0 0
\(239\) 6288.00 1.70183 0.850914 0.525305i \(-0.176049\pi\)
0.850914 + 0.525305i \(0.176049\pi\)
\(240\) 0 0
\(241\) −2926.00 −0.782076 −0.391038 0.920375i \(-0.627884\pi\)
−0.391038 + 0.920375i \(0.627884\pi\)
\(242\) 0 0
\(243\) −280.450 −0.0740364
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1960.00i − 0.504906i
\(248\) 0 0
\(249\) −2884.00 −0.734000
\(250\) 0 0
\(251\) 5444.96i 1.36925i 0.728894 + 0.684627i \(0.240035\pi\)
−0.728894 + 0.684627i \(0.759965\pi\)
\(252\) 0 0
\(253\) −2412.93 −0.599602
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2562.00i 0.621841i 0.950436 + 0.310921i \(0.100637\pi\)
−0.950436 + 0.310921i \(0.899363\pi\)
\(258\) 0 0
\(259\) 1947.27i 0.467172i
\(260\) 0 0
\(261\) − 158.745i − 0.0376478i
\(262\) 0 0
\(263\) − 5896.00i − 1.38237i −0.722679 0.691184i \(-0.757089\pi\)
0.722679 0.691184i \(-0.242911\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1407.54 −0.322622
\(268\) 0 0
\(269\) − 5365.58i − 1.21615i −0.793878 0.608077i \(-0.791941\pi\)
0.793878 0.608077i \(-0.208059\pi\)
\(270\) 0 0
\(271\) 1680.00 0.376578 0.188289 0.982114i \(-0.439706\pi\)
0.188289 + 0.982114i \(0.439706\pi\)
\(272\) 0 0
\(273\) − 2240.00i − 0.496597i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1576.87 −0.342039 −0.171019 0.985268i \(-0.554706\pi\)
−0.171019 + 0.985268i \(0.554706\pi\)
\(278\) 0 0
\(279\) −224.000 −0.0480664
\(280\) 0 0
\(281\) −2742.00 −0.582114 −0.291057 0.956706i \(-0.594007\pi\)
−0.291057 + 0.956706i \(0.594007\pi\)
\(282\) 0 0
\(283\) 2989.70 0.627983 0.313991 0.949426i \(-0.398334\pi\)
0.313991 + 0.949426i \(0.398334\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 560.000i − 0.115177i
\(288\) 0 0
\(289\) 4717.00 0.960106
\(290\) 0 0
\(291\) 5259.75i 1.05956i
\(292\) 0 0
\(293\) 9238.96 1.84214 0.921068 0.389401i \(-0.127318\pi\)
0.921068 + 0.389401i \(0.127318\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2184.00i 0.426695i
\(298\) 0 0
\(299\) 8043.08i 1.55566i
\(300\) 0 0
\(301\) − 3513.56i − 0.672818i
\(302\) 0 0
\(303\) − 3976.00i − 0.753846i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2587.54 −0.481039 −0.240520 0.970644i \(-0.577318\pi\)
−0.240520 + 0.970644i \(0.577318\pi\)
\(308\) 0 0
\(309\) 6222.81i 1.14564i
\(310\) 0 0
\(311\) 2744.00 0.500315 0.250157 0.968205i \(-0.419518\pi\)
0.250157 + 0.968205i \(0.419518\pi\)
\(312\) 0 0
\(313\) − 2282.00i − 0.412097i −0.978542 0.206048i \(-0.933940\pi\)
0.978542 0.206048i \(-0.0660604\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9577.62 1.69695 0.848474 0.529237i \(-0.177522\pi\)
0.848474 + 0.529237i \(0.177522\pi\)
\(318\) 0 0
\(319\) −2520.00 −0.442298
\(320\) 0 0
\(321\) 1428.00 0.248297
\(322\) 0 0
\(323\) 518.567 0.0893308
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 10024.0i − 1.69519i
\(328\) 0 0
\(329\) 2688.00 0.450438
\(330\) 0 0
\(331\) − 4249.08i − 0.705590i −0.935701 0.352795i \(-0.885231\pi\)
0.935701 0.352795i \(-0.114769\pi\)
\(332\) 0 0
\(333\) 243.409 0.0400563
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6130.00i 0.990868i 0.868646 + 0.495434i \(0.164991\pi\)
−0.868646 + 0.495434i \(0.835009\pi\)
\(338\) 0 0
\(339\) 9048.47i 1.44969i
\(340\) 0 0
\(341\) 3555.89i 0.564699i
\(342\) 0 0
\(343\) 4976.00i 0.783320i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2481.71 −0.383935 −0.191967 0.981401i \(-0.561487\pi\)
−0.191967 + 0.981401i \(0.561487\pi\)
\(348\) 0 0
\(349\) − 328.073i − 0.0503191i −0.999683 0.0251595i \(-0.991991\pi\)
0.999683 0.0251595i \(-0.00800937\pi\)
\(350\) 0 0
\(351\) 7280.00 1.10706
\(352\) 0 0
\(353\) 10206.0i 1.53884i 0.638743 + 0.769420i \(0.279455\pi\)
−0.638743 + 0.769420i \(0.720545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 592.648 0.0878607
\(358\) 0 0
\(359\) −3176.00 −0.466916 −0.233458 0.972367i \(-0.575004\pi\)
−0.233458 + 0.972367i \(0.575004\pi\)
\(360\) 0 0
\(361\) 5487.00 0.799971
\(362\) 0 0
\(363\) 5709.53 0.825545
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11760.0i 1.67266i 0.548225 + 0.836331i \(0.315304\pi\)
−0.548225 + 0.836331i \(0.684696\pi\)
\(368\) 0 0
\(369\) −70.0000 −0.00987549
\(370\) 0 0
\(371\) − 253.992i − 0.0355434i
\(372\) 0 0
\(373\) 10974.6 1.52344 0.761719 0.647908i \(-0.224356\pi\)
0.761719 + 0.647908i \(0.224356\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8400.00i 1.14754i
\(378\) 0 0
\(379\) − 3074.36i − 0.416674i −0.978057 0.208337i \(-0.933195\pi\)
0.978057 0.208337i \(-0.0668051\pi\)
\(380\) 0 0
\(381\) 8805.06i 1.18398i
\(382\) 0 0
\(383\) 2688.00i 0.358617i 0.983793 + 0.179309i \(0.0573861\pi\)
−0.983793 + 0.179309i \(0.942614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −439.195 −0.0576887
\(388\) 0 0
\(389\) 10487.8i 1.36697i 0.729966 + 0.683484i \(0.239536\pi\)
−0.729966 + 0.683484i \(0.760464\pi\)
\(390\) 0 0
\(391\) −2128.00 −0.275237
\(392\) 0 0
\(393\) − 3556.00i − 0.456429i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5704.24 −0.721127 −0.360564 0.932735i \(-0.617416\pi\)
−0.360564 + 0.932735i \(0.617416\pi\)
\(398\) 0 0
\(399\) −1568.00 −0.196737
\(400\) 0 0
\(401\) 12402.0 1.54445 0.772227 0.635346i \(-0.219143\pi\)
0.772227 + 0.635346i \(0.219143\pi\)
\(402\) 0 0
\(403\) 11853.0 1.46511
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3864.00i − 0.470593i
\(408\) 0 0
\(409\) 12278.0 1.48437 0.742186 0.670194i \(-0.233789\pi\)
0.742186 + 0.670194i \(0.233789\pi\)
\(410\) 0 0
\(411\) − 5619.58i − 0.674436i
\(412\) 0 0
\(413\) 4275.53 0.509407
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 14252.0i − 1.67368i
\(418\) 0 0
\(419\) 8207.12i 0.956907i 0.878113 + 0.478454i \(0.158802\pi\)
−0.878113 + 0.478454i \(0.841198\pi\)
\(420\) 0 0
\(421\) 1449.87i 0.167844i 0.996472 + 0.0839221i \(0.0267447\pi\)
−0.996472 + 0.0839221i \(0.973255\pi\)
\(422\) 0 0
\(423\) − 336.000i − 0.0386215i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 761.976 0.0863574
\(428\) 0 0
\(429\) 4444.86i 0.500233i
\(430\) 0 0
\(431\) −7632.00 −0.852948 −0.426474 0.904500i \(-0.640244\pi\)
−0.426474 + 0.904500i \(0.640244\pi\)
\(432\) 0 0
\(433\) − 3794.00i − 0.421081i −0.977585 0.210540i \(-0.932478\pi\)
0.977585 0.210540i \(-0.0675224\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5630.16 0.616309
\(438\) 0 0
\(439\) −1848.00 −0.200912 −0.100456 0.994942i \(-0.532030\pi\)
−0.100456 + 0.994942i \(0.532030\pi\)
\(440\) 0 0
\(441\) 279.000 0.0301263
\(442\) 0 0
\(443\) −12334.5 −1.32287 −0.661433 0.750004i \(-0.730051\pi\)
−0.661433 + 0.750004i \(0.730051\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4200.00i 0.444414i
\(448\) 0 0
\(449\) 3582.00 0.376492 0.188246 0.982122i \(-0.439720\pi\)
0.188246 + 0.982122i \(0.439720\pi\)
\(450\) 0 0
\(451\) 1111.22i 0.116020i
\(452\) 0 0
\(453\) −3936.88 −0.408324
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2714.00i 0.277802i 0.990306 + 0.138901i \(0.0443570\pi\)
−0.990306 + 0.138901i \(0.955643\pi\)
\(458\) 0 0
\(459\) 1926.11i 0.195867i
\(460\) 0 0
\(461\) − 8349.99i − 0.843596i −0.906690 0.421798i \(-0.861399\pi\)
0.906690 0.421798i \(-0.138601\pi\)
\(462\) 0 0
\(463\) 2224.00i 0.223236i 0.993751 + 0.111618i \(0.0356032\pi\)
−0.993751 + 0.111618i \(0.964397\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10292.0 −1.01982 −0.509910 0.860228i \(-0.670321\pi\)
−0.509910 + 0.860228i \(0.670321\pi\)
\(468\) 0 0
\(469\) − 1396.96i − 0.137538i
\(470\) 0 0
\(471\) 952.000 0.0931334
\(472\) 0 0
\(473\) 6972.00i 0.677744i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −31.7490 −0.00304756
\(478\) 0 0
\(479\) 17696.0 1.68800 0.843999 0.536345i \(-0.180195\pi\)
0.843999 + 0.536345i \(0.180195\pi\)
\(480\) 0 0
\(481\) −12880.0 −1.22095
\(482\) 0 0
\(483\) 6434.47 0.606166
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 1304.00i − 0.121334i −0.998158 0.0606672i \(-0.980677\pi\)
0.998158 0.0606672i \(-0.0193228\pi\)
\(488\) 0 0
\(489\) −9380.00 −0.867440
\(490\) 0 0
\(491\) 16662.9i 1.53154i 0.643112 + 0.765772i \(0.277643\pi\)
−0.643112 + 0.765772i \(0.722357\pi\)
\(492\) 0 0
\(493\) −2222.43 −0.203029
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 576.000i 0.0519862i
\(498\) 0 0
\(499\) − 3095.53i − 0.277705i −0.990313 0.138853i \(-0.955659\pi\)
0.990313 0.138853i \(-0.0443414\pi\)
\(500\) 0 0
\(501\) 10371.3i 0.924865i
\(502\) 0 0
\(503\) − 19320.0i − 1.71260i −0.516481 0.856298i \(-0.672758\pi\)
0.516481 0.856298i \(-0.327242\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3190.78 0.279502
\(508\) 0 0
\(509\) − 4476.61i − 0.389828i −0.980820 0.194914i \(-0.937557\pi\)
0.980820 0.194914i \(-0.0624427\pi\)
\(510\) 0 0
\(511\) −2352.00 −0.203613
\(512\) 0 0
\(513\) − 5096.00i − 0.438585i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5333.83 −0.453737
\(518\) 0 0
\(519\) −10584.0 −0.895156
\(520\) 0 0
\(521\) −2982.00 −0.250756 −0.125378 0.992109i \(-0.540014\pi\)
−0.125378 + 0.992109i \(0.540014\pi\)
\(522\) 0 0
\(523\) 2016.06 0.168559 0.0842794 0.996442i \(-0.473141\pi\)
0.0842794 + 0.996442i \(0.473141\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3136.00i 0.259215i
\(528\) 0 0
\(529\) −10937.0 −0.898907
\(530\) 0 0
\(531\) − 534.442i − 0.0436776i
\(532\) 0 0
\(533\) 3704.05 0.301014
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 17276.0i − 1.38830i
\(538\) 0 0
\(539\) − 4428.99i − 0.353933i
\(540\) 0 0
\(541\) − 15419.4i − 1.22539i −0.790321 0.612693i \(-0.790086\pi\)
0.790321 0.612693i \(-0.209914\pi\)
\(542\) 0 0
\(543\) 12376.0i 0.978094i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12609.7 0.985649 0.492824 0.870129i \(-0.335965\pi\)
0.492824 + 0.870129i \(0.335965\pi\)
\(548\) 0 0
\(549\) − 95.2470i − 0.00740445i
\(550\) 0 0
\(551\) 5880.00 0.454621
\(552\) 0 0
\(553\) − 3712.00i − 0.285444i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7143.53 −0.543413 −0.271706 0.962380i \(-0.587588\pi\)
−0.271706 + 0.962380i \(0.587588\pi\)
\(558\) 0 0
\(559\) 23240.0 1.75840
\(560\) 0 0
\(561\) −1176.00 −0.0885040
\(562\) 0 0
\(563\) −7572.14 −0.566834 −0.283417 0.958997i \(-0.591468\pi\)
−0.283417 + 0.958997i \(0.591468\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 6040.00i − 0.447365i
\(568\) 0 0
\(569\) −15594.0 −1.14892 −0.574459 0.818533i \(-0.694788\pi\)
−0.574459 + 0.818533i \(0.694788\pi\)
\(570\) 0 0
\(571\) − 16737.0i − 1.22666i −0.789827 0.613330i \(-0.789830\pi\)
0.789827 0.613330i \(-0.210170\pi\)
\(572\) 0 0
\(573\) −20658.0 −1.50611
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6594.00i 0.475757i 0.971295 + 0.237879i \(0.0764520\pi\)
−0.971295 + 0.237879i \(0.923548\pi\)
\(578\) 0 0
\(579\) − 17620.7i − 1.26475i
\(580\) 0 0
\(581\) − 4360.20i − 0.311345i
\(582\) 0 0
\(583\) 504.000i 0.0358037i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23213.8 1.63226 0.816130 0.577868i \(-0.196115\pi\)
0.816130 + 0.577868i \(0.196115\pi\)
\(588\) 0 0
\(589\) − 8297.08i − 0.580433i
\(590\) 0 0
\(591\) −6328.00 −0.440438
\(592\) 0 0
\(593\) − 14322.0i − 0.991794i −0.868381 0.495897i \(-0.834839\pi\)
0.868381 0.495897i \(-0.165161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9186.05 −0.629749
\(598\) 0 0
\(599\) −16088.0 −1.09739 −0.548696 0.836022i \(-0.684876\pi\)
−0.548696 + 0.836022i \(0.684876\pi\)
\(600\) 0 0
\(601\) −21238.0 −1.44146 −0.720729 0.693217i \(-0.756193\pi\)
−0.720729 + 0.693217i \(0.756193\pi\)
\(602\) 0 0
\(603\) −174.620 −0.0117928
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13664.0i 0.913681i 0.889549 + 0.456841i \(0.151019\pi\)
−0.889549 + 0.456841i \(0.848981\pi\)
\(608\) 0 0
\(609\) 6720.00 0.447140
\(610\) 0 0
\(611\) 17779.4i 1.17722i
\(612\) 0 0
\(613\) −20393.5 −1.34369 −0.671846 0.740690i \(-0.734499\pi\)
−0.671846 + 0.740690i \(0.734499\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 3782.00i − 0.246771i −0.992359 0.123385i \(-0.960625\pi\)
0.992359 0.123385i \(-0.0393751\pi\)
\(618\) 0 0
\(619\) − 5825.94i − 0.378295i −0.981949 0.189147i \(-0.939428\pi\)
0.981949 0.189147i \(-0.0605724\pi\)
\(620\) 0 0
\(621\) 20912.0i 1.35132i
\(622\) 0 0
\(623\) − 2128.00i − 0.136848i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3111.40 0.198178
\(628\) 0 0
\(629\) − 3407.73i − 0.216017i
\(630\) 0 0
\(631\) −2056.00 −0.129712 −0.0648558 0.997895i \(-0.520659\pi\)
−0.0648558 + 0.997895i \(0.520659\pi\)
\(632\) 0 0
\(633\) − 15428.0i − 0.968733i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −14763.3 −0.918278
\(638\) 0 0
\(639\) 72.0000 0.00445740
\(640\) 0 0
\(641\) 11842.0 0.729689 0.364845 0.931068i \(-0.381122\pi\)
0.364845 + 0.931068i \(0.381122\pi\)
\(642\) 0 0
\(643\) −16250.2 −0.996649 −0.498325 0.866991i \(-0.666051\pi\)
−0.498325 + 0.866991i \(0.666051\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 19320.0i − 1.17395i −0.809604 0.586976i \(-0.800318\pi\)
0.809604 0.586976i \(-0.199682\pi\)
\(648\) 0 0
\(649\) −8484.00 −0.513137
\(650\) 0 0
\(651\) − 9482.37i − 0.570881i
\(652\) 0 0
\(653\) 2317.68 0.138894 0.0694470 0.997586i \(-0.477877\pi\)
0.0694470 + 0.997586i \(0.477877\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 294.000i 0.0174582i
\(658\) 0 0
\(659\) − 27732.8i − 1.63932i −0.572847 0.819662i \(-0.694161\pi\)
0.572847 0.819662i \(-0.305839\pi\)
\(660\) 0 0
\(661\) − 22467.7i − 1.32208i −0.750352 0.661039i \(-0.770116\pi\)
0.750352 0.661039i \(-0.229884\pi\)
\(662\) 0 0
\(663\) 3920.00i 0.229623i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24129.3 −1.40073
\(668\) 0 0
\(669\) 8297.08i 0.479497i
\(670\) 0 0
\(671\) −1512.00 −0.0869897
\(672\) 0 0
\(673\) 10078.0i 0.577234i 0.957445 + 0.288617i \(0.0931954\pi\)
−0.957445 + 0.288617i \(0.906805\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16160.2 0.917413 0.458707 0.888588i \(-0.348313\pi\)
0.458707 + 0.888588i \(0.348313\pi\)
\(678\) 0 0
\(679\) −7952.00 −0.449440
\(680\) 0 0
\(681\) 6692.00 0.376561
\(682\) 0 0
\(683\) −16356.0 −0.916320 −0.458160 0.888870i \(-0.651491\pi\)
−0.458160 + 0.888870i \(0.651491\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 27272.0i 1.51454i
\(688\) 0 0
\(689\) 1680.00 0.0928925
\(690\) 0 0
\(691\) − 29246.1i − 1.61009i −0.593211 0.805047i \(-0.702140\pi\)
0.593211 0.805047i \(-0.297860\pi\)
\(692\) 0 0
\(693\) 126.996 0.00696130
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 980.000i 0.0532570i
\(698\) 0 0
\(699\) 4434.28i 0.239943i
\(700\) 0 0
\(701\) − 2465.84i − 0.132858i −0.997791 0.0664290i \(-0.978839\pi\)
0.997791 0.0664290i \(-0.0211606\pi\)
\(702\) 0 0
\(703\) 9016.00i 0.483705i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6011.15 0.319763
\(708\) 0 0
\(709\) − 31674.9i − 1.67782i −0.544267 0.838912i \(-0.683192\pi\)
0.544267 0.838912i \(-0.316808\pi\)
\(710\) 0 0
\(711\) −464.000 −0.0244745
\(712\) 0 0
\(713\) 34048.0i 1.78837i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 33273.0 1.73306
\(718\) 0 0
\(719\) −9296.00 −0.482173 −0.241086 0.970504i \(-0.577504\pi\)
−0.241086 + 0.970504i \(0.577504\pi\)
\(720\) 0 0
\(721\) −9408.00 −0.485953
\(722\) 0 0
\(723\) −15482.9 −0.796427
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 21672.0i − 1.10560i −0.833315 0.552799i \(-0.813560\pi\)
0.833315 0.552799i \(-0.186440\pi\)
\(728\) 0 0
\(729\) 18901.0 0.960270
\(730\) 0 0
\(731\) 6148.73i 0.311106i
\(732\) 0 0
\(733\) 9471.79 0.477283 0.238642 0.971108i \(-0.423298\pi\)
0.238642 + 0.971108i \(0.423298\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2772.00i 0.138545i
\(738\) 0 0
\(739\) − 6863.08i − 0.341627i −0.985303 0.170814i \(-0.945360\pi\)
0.985303 0.170814i \(-0.0546396\pi\)
\(740\) 0 0
\(741\) − 10371.3i − 0.514171i
\(742\) 0 0
\(743\) 17432.0i 0.860724i 0.902656 + 0.430362i \(0.141614\pi\)
−0.902656 + 0.430362i \(0.858386\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −545.025 −0.0266953
\(748\) 0 0
\(749\) 2158.93i 0.105321i
\(750\) 0 0
\(751\) 11632.0 0.565190 0.282595 0.959239i \(-0.408805\pi\)
0.282595 + 0.959239i \(0.408805\pi\)
\(752\) 0 0
\(753\) 28812.0i 1.39438i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16731.7 −0.803336 −0.401668 0.915785i \(-0.631569\pi\)
−0.401668 + 0.915785i \(0.631569\pi\)
\(758\) 0 0
\(759\) −12768.0 −0.610605
\(760\) 0 0
\(761\) 39466.0 1.87995 0.939975 0.341244i \(-0.110848\pi\)
0.939975 + 0.341244i \(0.110848\pi\)
\(762\) 0 0
\(763\) 15154.9 0.719060
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 28280.0i 1.33133i
\(768\) 0 0
\(769\) −35266.0 −1.65374 −0.826869 0.562395i \(-0.809880\pi\)
−0.826869 + 0.562395i \(0.809880\pi\)
\(770\) 0 0
\(771\) 13556.8i 0.633252i
\(772\) 0 0
\(773\) −16244.9 −0.755872 −0.377936 0.925832i \(-0.623366\pi\)
−0.377936 + 0.925832i \(0.623366\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10304.0i 0.475745i
\(778\) 0 0
\(779\) − 2592.84i − 0.119253i
\(780\) 0 0
\(781\) − 1142.96i − 0.0523668i
\(782\) 0 0
\(783\) 21840.0i 0.996805i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −34844.5 −1.57824 −0.789119 0.614240i \(-0.789463\pi\)
−0.789119 + 0.614240i \(0.789463\pi\)
\(788\) 0 0
\(789\) − 31198.7i − 1.40774i
\(790\) 0 0
\(791\) −13680.0 −0.614924
\(792\) 0 0
\(793\) 5040.00i 0.225694i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2550.50 0.113354 0.0566772 0.998393i \(-0.481949\pi\)
0.0566772 + 0.998393i \(0.481949\pi\)
\(798\) 0 0
\(799\) −4704.00 −0.208280
\(800\) 0 0
\(801\) −266.000 −0.0117336
\(802\) 0 0
\(803\) 4667.11 0.205104
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 28392.0i − 1.23847i
\(808\) 0 0
\(809\) 24390.0 1.05996 0.529979 0.848010i \(-0.322200\pi\)
0.529979 + 0.848010i \(0.322200\pi\)
\(810\) 0 0
\(811\) − 9582.91i − 0.414922i −0.978243 0.207461i \(-0.933480\pi\)
0.978243 0.207461i \(-0.0665200\pi\)
\(812\) 0 0
\(813\) 8889.72 0.383489
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16268.0i − 0.696628i
\(818\) 0 0
\(819\) − 423.320i − 0.0180611i
\(820\) 0 0
\(821\) 8773.31i 0.372948i 0.982460 + 0.186474i \(0.0597061\pi\)
−0.982460 + 0.186474i \(0.940294\pi\)
\(822\) 0 0
\(823\) − 21688.0i − 0.918586i −0.888285 0.459293i \(-0.848103\pi\)
0.888285 0.459293i \(-0.151897\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19446.3 0.817670 0.408835 0.912608i \(-0.365935\pi\)
0.408835 + 0.912608i \(0.365935\pi\)
\(828\) 0 0
\(829\) − 19546.8i − 0.818925i −0.912327 0.409462i \(-0.865716\pi\)
0.912327 0.409462i \(-0.134284\pi\)
\(830\) 0 0
\(831\) −8344.00 −0.348315
\(832\) 0 0
\(833\) − 3906.00i − 0.162467i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 30817.7 1.27266
\(838\) 0 0
\(839\) −18760.0 −0.771951 −0.385976 0.922509i \(-0.626135\pi\)
−0.385976 + 0.922509i \(0.626135\pi\)
\(840\) 0 0
\(841\) −811.000 −0.0332527
\(842\) 0 0
\(843\) −14509.3 −0.592796
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8632.00i 0.350176i
\(848\) 0 0
\(849\) 15820.0 0.639506
\(850\) 0 0
\(851\) − 36998.2i − 1.49034i
\(852\) 0 0
\(853\) −28732.9 −1.15333 −0.576667 0.816979i \(-0.695647\pi\)
−0.576667 + 0.816979i \(0.695647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8778.00i 0.349884i 0.984579 + 0.174942i \(0.0559738\pi\)
−0.984579 + 0.174942i \(0.944026\pi\)
\(858\) 0 0
\(859\) 5646.03i 0.224261i 0.993693 + 0.112130i \(0.0357675\pi\)
−0.993693 + 0.112130i \(0.964233\pi\)
\(860\) 0 0
\(861\) − 2963.24i − 0.117290i
\(862\) 0 0
\(863\) − 9312.00i − 0.367305i −0.982991 0.183652i \(-0.941208\pi\)
0.982991 0.183652i \(-0.0587921\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 24960.0 0.977724
\(868\) 0 0
\(869\) 7365.77i 0.287534i
\(870\) 0 0
\(871\) 9240.00 0.359455
\(872\) 0 0
\(873\) 994.000i 0.0385359i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 137.579 0.00529728 0.00264864 0.999996i \(-0.499157\pi\)
0.00264864 + 0.999996i \(0.499157\pi\)
\(878\) 0 0
\(879\) 48888.0 1.87594
\(880\) 0 0
\(881\) −31150.0 −1.19123 −0.595613 0.803272i \(-0.703091\pi\)
−0.595613 + 0.803272i \(0.703091\pi\)
\(882\) 0 0
\(883\) 12577.9 0.479366 0.239683 0.970851i \(-0.422957\pi\)
0.239683 + 0.970851i \(0.422957\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 37128.0i − 1.40545i −0.711460 0.702726i \(-0.751966\pi\)
0.711460 0.702726i \(-0.248034\pi\)
\(888\) 0 0
\(889\) −13312.0 −0.502216
\(890\) 0 0
\(891\) 11985.3i 0.450641i
\(892\) 0 0
\(893\) 12445.6 0.466379
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 42560.0i 1.58421i
\(898\) 0 0
\(899\) 35558.9i 1.31919i
\(900\) 0 0
\(901\) 444.486i 0.0164351i
\(902\) 0 0
\(903\) − 18592.0i − 0.685164i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −35204.4 −1.28880 −0.644400 0.764688i \(-0.722893\pi\)
−0.644400 + 0.764688i \(0.722893\pi\)
\(908\) 0 0
\(909\) − 751.393i − 0.0274171i
\(910\) 0 0
\(911\) 10512.0 0.382303 0.191152 0.981561i \(-0.438778\pi\)
0.191152 + 0.981561i \(0.438778\pi\)
\(912\) 0 0
\(913\) 8652.00i 0.313625i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5376.17 0.193606
\(918\) 0 0
\(919\) −46104.0 −1.65488 −0.827438 0.561557i \(-0.810202\pi\)
−0.827438 + 0.561557i \(0.810202\pi\)
\(920\) 0 0
\(921\) −13692.0 −0.489866
\(922\) 0 0
\(923\) −3809.88 −0.135865
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1176.00i 0.0416666i
\(928\) 0 0
\(929\) 5726.00 0.202222 0.101111 0.994875i \(-0.467760\pi\)
0.101111 + 0.994875i \(0.467760\pi\)
\(930\) 0 0
\(931\) 10334.3i 0.363795i
\(932\) 0 0
\(933\) 14519.9 0.509496
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1274.00i 0.0444181i 0.999753 + 0.0222091i \(0.00706994\pi\)
−0.999753 + 0.0222091i \(0.992930\pi\)
\(938\) 0 0
\(939\) − 12075.2i − 0.419659i
\(940\) 0 0
\(941\) 26446.9i 0.916201i 0.888900 + 0.458101i \(0.151470\pi\)
−0.888900 + 0.458101i \(0.848530\pi\)
\(942\) 0 0
\(943\) 10640.0i 0.367430i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23922.9 −0.820897 −0.410448 0.911884i \(-0.634628\pi\)
−0.410448 + 0.911884i \(0.634628\pi\)
\(948\) 0 0
\(949\) − 15557.0i − 0.532141i
\(950\) 0 0
\(951\) 50680.0 1.72809
\(952\) 0 0
\(953\) − 38250.0i − 1.30015i −0.759872 0.650073i \(-0.774738\pi\)
0.759872 0.650073i \(-0.225262\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −13334.6 −0.450414
\(958\) 0 0
\(959\) 8496.00 0.286079
\(960\) 0 0
\(961\) 20385.0 0.684267
\(962\) 0 0
\(963\) 269.867 0.00903046
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 4664.00i − 0.155103i −0.996988 0.0775513i \(-0.975290\pi\)
0.996988 0.0775513i \(-0.0247101\pi\)
\(968\) 0 0
\(969\) 2744.00 0.0909701
\(970\) 0 0
\(971\) 30971.2i 1.02360i 0.859106 + 0.511798i \(0.171020\pi\)
−0.859106 + 0.511798i \(0.828980\pi\)
\(972\) 0 0
\(973\) 21547.0 0.709933
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 4814.00i − 0.157639i −0.996889 0.0788196i \(-0.974885\pi\)
0.996889 0.0788196i \(-0.0251151\pi\)
\(978\) 0 0
\(979\) 4222.62i 0.137850i
\(980\) 0 0
\(981\) − 1894.36i − 0.0616536i
\(982\) 0 0
\(983\) − 12376.0i − 0.401560i −0.979636 0.200780i \(-0.935652\pi\)
0.979636 0.200780i \(-0.0643476\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 14223.6 0.458704
\(988\) 0 0
\(989\) 66757.6i 2.14638i
\(990\) 0 0
\(991\) −45344.0 −1.45348 −0.726740 0.686912i \(-0.758966\pi\)
−0.726740 + 0.686912i \(0.758966\pi\)
\(992\) 0 0
\(993\) − 22484.0i − 0.718538i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −26002.4 −0.825984 −0.412992 0.910735i \(-0.635516\pi\)
−0.412992 + 0.910735i \(0.635516\pi\)
\(998\) 0 0
\(999\) −33488.0 −1.06057
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 800.4.f.a.49.4 4
4.3 odd 2 200.4.f.a.149.1 4
5.2 odd 4 800.4.d.a.401.1 2
5.3 odd 4 32.4.b.a.17.2 2
5.4 even 2 inner 800.4.f.a.49.1 4
8.3 odd 2 200.4.f.a.149.3 4
8.5 even 2 inner 800.4.f.a.49.2 4
15.8 even 4 288.4.d.a.145.1 2
20.3 even 4 8.4.b.a.5.2 yes 2
20.7 even 4 200.4.d.a.101.1 2
20.19 odd 2 200.4.f.a.149.4 4
40.3 even 4 8.4.b.a.5.1 2
40.13 odd 4 32.4.b.a.17.1 2
40.19 odd 2 200.4.f.a.149.2 4
40.27 even 4 200.4.d.a.101.2 2
40.29 even 2 inner 800.4.f.a.49.3 4
40.37 odd 4 800.4.d.a.401.2 2
60.23 odd 4 72.4.d.b.37.1 2
80.3 even 4 256.4.a.l.1.2 2
80.13 odd 4 256.4.a.j.1.1 2
80.43 even 4 256.4.a.l.1.1 2
80.53 odd 4 256.4.a.j.1.2 2
120.53 even 4 288.4.d.a.145.2 2
120.83 odd 4 72.4.d.b.37.2 2
240.53 even 4 2304.4.a.v.1.2 2
240.83 odd 4 2304.4.a.bn.1.1 2
240.173 even 4 2304.4.a.v.1.1 2
240.203 odd 4 2304.4.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.b.a.5.1 2 40.3 even 4
8.4.b.a.5.2 yes 2 20.3 even 4
32.4.b.a.17.1 2 40.13 odd 4
32.4.b.a.17.2 2 5.3 odd 4
72.4.d.b.37.1 2 60.23 odd 4
72.4.d.b.37.2 2 120.83 odd 4
200.4.d.a.101.1 2 20.7 even 4
200.4.d.a.101.2 2 40.27 even 4
200.4.f.a.149.1 4 4.3 odd 2
200.4.f.a.149.2 4 40.19 odd 2
200.4.f.a.149.3 4 8.3 odd 2
200.4.f.a.149.4 4 20.19 odd 2
256.4.a.j.1.1 2 80.13 odd 4
256.4.a.j.1.2 2 80.53 odd 4
256.4.a.l.1.1 2 80.43 even 4
256.4.a.l.1.2 2 80.3 even 4
288.4.d.a.145.1 2 15.8 even 4
288.4.d.a.145.2 2 120.53 even 4
800.4.d.a.401.1 2 5.2 odd 4
800.4.d.a.401.2 2 40.37 odd 4
800.4.f.a.49.1 4 5.4 even 2 inner
800.4.f.a.49.2 4 8.5 even 2 inner
800.4.f.a.49.3 4 40.29 even 2 inner
800.4.f.a.49.4 4 1.1 even 1 trivial
2304.4.a.v.1.1 2 240.173 even 4
2304.4.a.v.1.2 2 240.53 even 4
2304.4.a.bn.1.1 2 240.83 odd 4
2304.4.a.bn.1.2 2 240.203 odd 4