Properties

Label 800.6.c.h.449.2
Level 800
Weight 6
Character 800.449
Analytic conductor 128.307
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.6.c.h.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-13.4164i q^{3} +138.636i q^{7} +63.0000 q^{9} +O(q^{10})\) \(q-13.4164i q^{3} +138.636i q^{7} +63.0000 q^{9} +259.384 q^{11} -154.000i q^{13} +178.000i q^{17} -965.981 q^{19} +1860.00 q^{21} -2634.09i q^{23} -4105.42i q^{27} -4110.00 q^{29} +3157.33 q^{31} -3480.00i q^{33} +7442.00i q^{37} -2066.13 q^{39} +7270.00 q^{41} +17910.9i q^{43} -7410.33i q^{47} -2413.00 q^{49} +2388.12 q^{51} -32226.0i q^{53} +12960.0i q^{57} -34041.9 q^{59} +26770.0 q^{61} +8734.08i q^{63} +49806.2i q^{67} -35340.0 q^{69} +54103.9 q^{71} +18534.0i q^{73} +35960.0i q^{77} +86741.5 q^{79} -39771.0 q^{81} +78642.5i q^{83} +55141.4i q^{87} +107590. q^{89} +21350.0 q^{91} -42360.0i q^{93} -108838. i q^{97} +16341.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 252q^{9} + O(q^{10}) \) \( 4q + 252q^{9} + 7440q^{21} - 16440q^{29} + 29080q^{41} - 9652q^{49} + 107080q^{61} - 141360q^{69} - 159084q^{81} + 430360q^{89} + O(q^{100}) \)

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
\(3\) − 13.4164i − 0.860663i −0.902671 0.430331i \(-0.858397\pi\)
0.902671 0.430331i \(-0.141603\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 138.636i 1.06938i 0.845049 + 0.534689i \(0.179571\pi\)
−0.845049 + 0.534689i \(0.820429\pi\)
\(8\) 0 0
\(9\) 63.0000 0.259259
\(10\) 0 0
\(11\) 259.384 0.646340 0.323170 0.946341i \(-0.395251\pi\)
0.323170 + 0.946341i \(0.395251\pi\)
\(12\) 0 0
\(13\) − 154.000i − 0.252733i −0.991984 0.126367i \(-0.959668\pi\)
0.991984 0.126367i \(-0.0403316\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 178.000i 0.149382i 0.997207 + 0.0746909i \(0.0237970\pi\)
−0.997207 + 0.0746909i \(0.976203\pi\)
\(18\) 0 0
\(19\) −965.981 −0.613882 −0.306941 0.951729i \(-0.599305\pi\)
−0.306941 + 0.951729i \(0.599305\pi\)
\(20\) 0 0
\(21\) 1860.00 0.920375
\(22\) 0 0
\(23\) − 2634.09i − 1.03827i −0.854692 0.519135i \(-0.826254\pi\)
0.854692 0.519135i \(-0.173746\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4105.42i − 1.08380i
\(28\) 0 0
\(29\) −4110.00 −0.907500 −0.453750 0.891129i \(-0.649914\pi\)
−0.453750 + 0.891129i \(0.649914\pi\)
\(30\) 0 0
\(31\) 3157.33 0.590086 0.295043 0.955484i \(-0.404666\pi\)
0.295043 + 0.955484i \(0.404666\pi\)
\(32\) 0 0
\(33\) − 3480.00i − 0.556281i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7442.00i 0.893687i 0.894612 + 0.446843i \(0.147452\pi\)
−0.894612 + 0.446843i \(0.852548\pi\)
\(38\) 0 0
\(39\) −2066.13 −0.217518
\(40\) 0 0
\(41\) 7270.00 0.675421 0.337711 0.941250i \(-0.390347\pi\)
0.337711 + 0.941250i \(0.390347\pi\)
\(42\) 0 0
\(43\) 17910.9i 1.47722i 0.674131 + 0.738612i \(0.264518\pi\)
−0.674131 + 0.738612i \(0.735482\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 7410.33i − 0.489320i −0.969609 0.244660i \(-0.921324\pi\)
0.969609 0.244660i \(-0.0786763\pi\)
\(48\) 0 0
\(49\) −2413.00 −0.143571
\(50\) 0 0
\(51\) 2388.12 0.128567
\(52\) 0 0
\(53\) − 32226.0i − 1.57586i −0.615767 0.787928i \(-0.711154\pi\)
0.615767 0.787928i \(-0.288846\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12960.0i 0.528345i
\(58\) 0 0
\(59\) −34041.9 −1.27316 −0.636581 0.771210i \(-0.719652\pi\)
−0.636581 + 0.771210i \(0.719652\pi\)
\(60\) 0 0
\(61\) 26770.0 0.921136 0.460568 0.887624i \(-0.347646\pi\)
0.460568 + 0.887624i \(0.347646\pi\)
\(62\) 0 0
\(63\) 8734.08i 0.277246i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 49806.2i 1.35549i 0.735297 + 0.677745i \(0.237043\pi\)
−0.735297 + 0.677745i \(0.762957\pi\)
\(68\) 0 0
\(69\) −35340.0 −0.893601
\(70\) 0 0
\(71\) 54103.9 1.27375 0.636873 0.770969i \(-0.280228\pi\)
0.636873 + 0.770969i \(0.280228\pi\)
\(72\) 0 0
\(73\) 18534.0i 0.407063i 0.979068 + 0.203532i \(0.0652420\pi\)
−0.979068 + 0.203532i \(0.934758\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 35960.0i 0.691183i
\(78\) 0 0
\(79\) 86741.5 1.56372 0.781861 0.623453i \(-0.214271\pi\)
0.781861 + 0.623453i \(0.214271\pi\)
\(80\) 0 0
\(81\) −39771.0 −0.673525
\(82\) 0 0
\(83\) 78642.5i 1.25303i 0.779409 + 0.626516i \(0.215520\pi\)
−0.779409 + 0.626516i \(0.784480\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 55141.4i 0.781052i
\(88\) 0 0
\(89\) 107590. 1.43978 0.719891 0.694087i \(-0.244192\pi\)
0.719891 + 0.694087i \(0.244192\pi\)
\(90\) 0 0
\(91\) 21350.0 0.270268
\(92\) 0 0
\(93\) − 42360.0i − 0.507865i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 108838.i − 1.17450i −0.809407 0.587248i \(-0.800212\pi\)
0.809407 0.587248i \(-0.199788\pi\)
\(98\) 0 0
\(99\) 16341.2 0.167570
\(100\) 0 0
\(101\) 59198.0 0.577436 0.288718 0.957414i \(-0.406771\pi\)
0.288718 + 0.957414i \(0.406771\pi\)
\(102\) 0 0
\(103\) 112908.i 1.04865i 0.851517 + 0.524326i \(0.175683\pi\)
−0.851517 + 0.524326i \(0.824317\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 40039.0i 0.338084i 0.985609 + 0.169042i \(0.0540673\pi\)
−0.985609 + 0.169042i \(0.945933\pi\)
\(108\) 0 0
\(109\) 139614. 1.12554 0.562772 0.826612i \(-0.309735\pi\)
0.562772 + 0.826612i \(0.309735\pi\)
\(110\) 0 0
\(111\) 99844.9 0.769163
\(112\) 0 0
\(113\) 43046.0i 0.317130i 0.987349 + 0.158565i \(0.0506867\pi\)
−0.987349 + 0.158565i \(0.949313\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 9702.00i − 0.0655234i
\(118\) 0 0
\(119\) −24677.2 −0.159746
\(120\) 0 0
\(121\) −93771.0 −0.582244
\(122\) 0 0
\(123\) − 97537.3i − 0.581310i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 157370.i − 0.865790i −0.901444 0.432895i \(-0.857492\pi\)
0.901444 0.432895i \(-0.142508\pi\)
\(128\) 0 0
\(129\) 240300. 1.27139
\(130\) 0 0
\(131\) −267729. −1.36307 −0.681533 0.731787i \(-0.738686\pi\)
−0.681533 + 0.731787i \(0.738686\pi\)
\(132\) 0 0
\(133\) − 133920.i − 0.656472i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 112158.i − 0.510539i −0.966870 0.255269i \(-0.917836\pi\)
0.966870 0.255269i \(-0.0821642\pi\)
\(138\) 0 0
\(139\) −147348. −0.646855 −0.323428 0.946253i \(-0.604835\pi\)
−0.323428 + 0.946253i \(0.604835\pi\)
\(140\) 0 0
\(141\) −99420.0 −0.421139
\(142\) 0 0
\(143\) − 39945.1i − 0.163352i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 32373.8i 0.123566i
\(148\) 0 0
\(149\) 174566. 0.644160 0.322080 0.946712i \(-0.395618\pi\)
0.322080 + 0.946712i \(0.395618\pi\)
\(150\) 0 0
\(151\) 345258. 1.23226 0.616128 0.787646i \(-0.288700\pi\)
0.616128 + 0.787646i \(0.288700\pi\)
\(152\) 0 0
\(153\) 11214.0i 0.0387286i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 26502.0i − 0.0858083i −0.999079 0.0429042i \(-0.986339\pi\)
0.999079 0.0429042i \(-0.0136610\pi\)
\(158\) 0 0
\(159\) −432357. −1.35628
\(160\) 0 0
\(161\) 365180. 1.11030
\(162\) 0 0
\(163\) − 141709.i − 0.417760i −0.977941 0.208880i \(-0.933018\pi\)
0.977941 0.208880i \(-0.0669818\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 411047.i − 1.14051i −0.821466 0.570257i \(-0.806844\pi\)
0.821466 0.570257i \(-0.193156\pi\)
\(168\) 0 0
\(169\) 347577. 0.936126
\(170\) 0 0
\(171\) −60856.8 −0.159155
\(172\) 0 0
\(173\) − 595946.i − 1.51388i −0.653484 0.756940i \(-0.726693\pi\)
0.653484 0.756940i \(-0.273307\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 456720.i 1.09576i
\(178\) 0 0
\(179\) 300939. 0.702014 0.351007 0.936373i \(-0.385839\pi\)
0.351007 + 0.936373i \(0.385839\pi\)
\(180\) 0 0
\(181\) 217022. 0.492388 0.246194 0.969221i \(-0.420820\pi\)
0.246194 + 0.969221i \(0.420820\pi\)
\(182\) 0 0
\(183\) − 359157.i − 0.792788i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 46170.3i 0.0965515i
\(188\) 0 0
\(189\) 569160. 1.15899
\(190\) 0 0
\(191\) −916260. −1.81734 −0.908668 0.417519i \(-0.862900\pi\)
−0.908668 + 0.417519i \(0.862900\pi\)
\(192\) 0 0
\(193\) − 864114.i − 1.66985i −0.550363 0.834926i \(-0.685511\pi\)
0.550363 0.834926i \(-0.314489\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 432522.i 0.794040i 0.917810 + 0.397020i \(0.129956\pi\)
−0.917810 + 0.397020i \(0.870044\pi\)
\(198\) 0 0
\(199\) 795182. 1.42342 0.711711 0.702473i \(-0.247921\pi\)
0.711711 + 0.702473i \(0.247921\pi\)
\(200\) 0 0
\(201\) 668220. 1.16662
\(202\) 0 0
\(203\) − 569795.i − 0.970462i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 165948.i − 0.269181i
\(208\) 0 0
\(209\) −250560. −0.396777
\(210\) 0 0
\(211\) −453126. −0.700669 −0.350334 0.936625i \(-0.613932\pi\)
−0.350334 + 0.936625i \(0.613932\pi\)
\(212\) 0 0
\(213\) − 725880.i − 1.09627i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 437720.i 0.631026i
\(218\) 0 0
\(219\) 248660. 0.350344
\(220\) 0 0
\(221\) 27412.0 0.0377537
\(222\) 0 0
\(223\) 430358.i 0.579519i 0.957099 + 0.289760i \(0.0935754\pi\)
−0.957099 + 0.289760i \(0.906425\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.12403e6i 1.44782i 0.689896 + 0.723909i \(0.257656\pi\)
−0.689896 + 0.723909i \(0.742344\pi\)
\(228\) 0 0
\(229\) 812410. 1.02373 0.511866 0.859065i \(-0.328954\pi\)
0.511866 + 0.859065i \(0.328954\pi\)
\(230\) 0 0
\(231\) 482454. 0.594875
\(232\) 0 0
\(233\) − 846194.i − 1.02113i −0.859840 0.510564i \(-0.829437\pi\)
0.859840 0.510564i \(-0.170563\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.16376e6i − 1.34584i
\(238\) 0 0
\(239\) 56688.8 0.0641952 0.0320976 0.999485i \(-0.489781\pi\)
0.0320976 + 0.999485i \(0.489781\pi\)
\(240\) 0 0
\(241\) 1.27571e6 1.41485 0.707423 0.706790i \(-0.249857\pi\)
0.707423 + 0.706790i \(0.249857\pi\)
\(242\) 0 0
\(243\) − 464033.i − 0.504119i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 148761.i 0.155148i
\(248\) 0 0
\(249\) 1.05510e6 1.07844
\(250\) 0 0
\(251\) −50937.6 −0.0510334 −0.0255167 0.999674i \(-0.508123\pi\)
−0.0255167 + 0.999674i \(0.508123\pi\)
\(252\) 0 0
\(253\) − 683240.i − 0.671076i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.64806e6i 1.55647i 0.627976 + 0.778233i \(0.283884\pi\)
−0.627976 + 0.778233i \(0.716116\pi\)
\(258\) 0 0
\(259\) −1.03173e6 −0.955690
\(260\) 0 0
\(261\) −258930. −0.235278
\(262\) 0 0
\(263\) − 522994.i − 0.466238i −0.972448 0.233119i \(-0.925107\pi\)
0.972448 0.233119i \(-0.0748931\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 1.44347e6i − 1.23917i
\(268\) 0 0
\(269\) 1.93789e6 1.63285 0.816427 0.577448i \(-0.195951\pi\)
0.816427 + 0.577448i \(0.195951\pi\)
\(270\) 0 0
\(271\) 1.00132e6 0.828228 0.414114 0.910225i \(-0.364092\pi\)
0.414114 + 0.910225i \(0.364092\pi\)
\(272\) 0 0
\(273\) − 286440.i − 0.232609i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 445702.i − 0.349016i −0.984656 0.174508i \(-0.944167\pi\)
0.984656 0.174508i \(-0.0558335\pi\)
\(278\) 0 0
\(279\) 198912. 0.152985
\(280\) 0 0
\(281\) 1.24647e6 0.941708 0.470854 0.882211i \(-0.343946\pi\)
0.470854 + 0.882211i \(0.343946\pi\)
\(282\) 0 0
\(283\) 2.27900e6i 1.69153i 0.533557 + 0.845764i \(0.320855\pi\)
−0.533557 + 0.845764i \(0.679145\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00789e6i 0.722281i
\(288\) 0 0
\(289\) 1.38817e6 0.977685
\(290\) 0 0
\(291\) −1.46021e6 −1.01084
\(292\) 0 0
\(293\) − 2.45427e6i − 1.67014i −0.550140 0.835072i \(-0.685426\pi\)
0.550140 0.835072i \(-0.314574\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.06488e6i − 0.700502i
\(298\) 0 0
\(299\) −405650. −0.262406
\(300\) 0 0
\(301\) −2.48310e6 −1.57971
\(302\) 0 0
\(303\) − 794225.i − 0.496977i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 20710.5i − 0.0125413i −0.999980 0.00627067i \(-0.998004\pi\)
0.999980 0.00627067i \(-0.00199603\pi\)
\(308\) 0 0
\(309\) 1.51482e6 0.902537
\(310\) 0 0
\(311\) −47270.5 −0.0277133 −0.0138567 0.999904i \(-0.504411\pi\)
−0.0138567 + 0.999904i \(0.504411\pi\)
\(312\) 0 0
\(313\) 2.79169e6i 1.61067i 0.592822 + 0.805333i \(0.298014\pi\)
−0.592822 + 0.805333i \(0.701986\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 471582.i − 0.263578i −0.991278 0.131789i \(-0.957928\pi\)
0.991278 0.131789i \(-0.0420721\pi\)
\(318\) 0 0
\(319\) −1.06607e6 −0.586554
\(320\) 0 0
\(321\) 537180. 0.290976
\(322\) 0 0
\(323\) − 171945.i − 0.0917028i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 1.87312e6i − 0.968715i
\(328\) 0 0
\(329\) 1.02734e6 0.523268
\(330\) 0 0
\(331\) 3.09092e6 1.55066 0.775331 0.631555i \(-0.217583\pi\)
0.775331 + 0.631555i \(0.217583\pi\)
\(332\) 0 0
\(333\) 468846.i 0.231697i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 665838.i − 0.319370i −0.987168 0.159685i \(-0.948952\pi\)
0.987168 0.159685i \(-0.0510478\pi\)
\(338\) 0 0
\(339\) 577523. 0.272942
\(340\) 0 0
\(341\) 818960. 0.381397
\(342\) 0 0
\(343\) 1.99553e6i 0.915847i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1.80908e6i − 0.806556i −0.915078 0.403278i \(-0.867871\pi\)
0.915078 0.403278i \(-0.132129\pi\)
\(348\) 0 0
\(349\) 2.36181e6 1.03796 0.518981 0.854786i \(-0.326312\pi\)
0.518981 + 0.854786i \(0.326312\pi\)
\(350\) 0 0
\(351\) −632235. −0.273912
\(352\) 0 0
\(353\) − 1.14535e6i − 0.489215i −0.969622 0.244608i \(-0.921341\pi\)
0.969622 0.244608i \(-0.0786591\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 331080.i 0.137487i
\(358\) 0 0
\(359\) 767275. 0.314206 0.157103 0.987582i \(-0.449784\pi\)
0.157103 + 0.987582i \(0.449784\pi\)
\(360\) 0 0
\(361\) −1.54298e6 −0.623149
\(362\) 0 0
\(363\) 1.25807e6i 0.501116i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.99537e6i 1.54843i 0.632923 + 0.774215i \(0.281855\pi\)
−0.632923 + 0.774215i \(0.718145\pi\)
\(368\) 0 0
\(369\) 458010. 0.175109
\(370\) 0 0
\(371\) 4.46769e6 1.68519
\(372\) 0 0
\(373\) − 4.68131e6i − 1.74219i −0.491117 0.871094i \(-0.663411\pi\)
0.491117 0.871094i \(-0.336589\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 632940.i 0.229356i
\(378\) 0 0
\(379\) −337271. −0.120609 −0.0603046 0.998180i \(-0.519207\pi\)
−0.0603046 + 0.998180i \(0.519207\pi\)
\(380\) 0 0
\(381\) −2.11134e6 −0.745153
\(382\) 0 0
\(383\) 598689.i 0.208547i 0.994549 + 0.104274i \(0.0332518\pi\)
−0.994549 + 0.104274i \(0.966748\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.12839e6i 0.382984i
\(388\) 0 0
\(389\) −3.97243e6 −1.33101 −0.665506 0.746393i \(-0.731784\pi\)
−0.665506 + 0.746393i \(0.731784\pi\)
\(390\) 0 0
\(391\) 468868. 0.155099
\(392\) 0 0
\(393\) 3.59196e6i 1.17314i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 179398.i − 0.0571270i −0.999592 0.0285635i \(-0.990907\pi\)
0.999592 0.0285635i \(-0.00909328\pi\)
\(398\) 0 0
\(399\) −1.79673e6 −0.565001
\(400\) 0 0
\(401\) 6.14504e6 1.90838 0.954188 0.299208i \(-0.0967224\pi\)
0.954188 + 0.299208i \(0.0967224\pi\)
\(402\) 0 0
\(403\) − 486229.i − 0.149134i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.93033e6i 0.577626i
\(408\) 0 0
\(409\) 2.64503e6 0.781847 0.390923 0.920423i \(-0.372156\pi\)
0.390923 + 0.920423i \(0.372156\pi\)
\(410\) 0 0
\(411\) −1.50476e6 −0.439402
\(412\) 0 0
\(413\) − 4.71944e6i − 1.36149i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.97688e6i 0.556724i
\(418\) 0 0
\(419\) 2.15984e6 0.601018 0.300509 0.953779i \(-0.402843\pi\)
0.300509 + 0.953779i \(0.402843\pi\)
\(420\) 0 0
\(421\) 1.47209e6 0.404789 0.202395 0.979304i \(-0.435128\pi\)
0.202395 + 0.979304i \(0.435128\pi\)
\(422\) 0 0
\(423\) − 466851.i − 0.126861i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.71129e6i 0.985043i
\(428\) 0 0
\(429\) −535920. −0.140591
\(430\) 0 0
\(431\) −6.63748e6 −1.72112 −0.860558 0.509352i \(-0.829885\pi\)
−0.860558 + 0.509352i \(0.829885\pi\)
\(432\) 0 0
\(433\) 6.27853e6i 1.60931i 0.593746 + 0.804653i \(0.297649\pi\)
−0.593746 + 0.804653i \(0.702351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.54448e6i 0.637376i
\(438\) 0 0
\(439\) 2.44021e6 0.604319 0.302160 0.953257i \(-0.402292\pi\)
0.302160 + 0.953257i \(0.402292\pi\)
\(440\) 0 0
\(441\) −152019. −0.0372221
\(442\) 0 0
\(443\) 2.30646e6i 0.558390i 0.960234 + 0.279195i \(0.0900675\pi\)
−0.960234 + 0.279195i \(0.909932\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 2.34205e6i − 0.554405i
\(448\) 0 0
\(449\) −7.60241e6 −1.77965 −0.889826 0.456300i \(-0.849175\pi\)
−0.889826 + 0.456300i \(0.849175\pi\)
\(450\) 0 0
\(451\) 1.88572e6 0.436552
\(452\) 0 0
\(453\) − 4.63212e6i − 1.06056i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.56938e6i − 0.351510i −0.984434 0.175755i \(-0.943763\pi\)
0.984434 0.175755i \(-0.0562367\pi\)
\(458\) 0 0
\(459\) 730765. 0.161900
\(460\) 0 0
\(461\) −562602. −0.123296 −0.0616480 0.998098i \(-0.519636\pi\)
−0.0616480 + 0.998098i \(0.519636\pi\)
\(462\) 0 0
\(463\) 4.87382e6i 1.05662i 0.849053 + 0.528308i \(0.177173\pi\)
−0.849053 + 0.528308i \(0.822827\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 5.11862e6i − 1.08608i −0.839708 0.543039i \(-0.817274\pi\)
0.839708 0.543039i \(-0.182726\pi\)
\(468\) 0 0
\(469\) −6.90494e6 −1.44953
\(470\) 0 0
\(471\) −355562. −0.0738521
\(472\) 0 0
\(473\) 4.64580e6i 0.954790i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.03024e6i − 0.408555i
\(478\) 0 0
\(479\) −4.00179e6 −0.796922 −0.398461 0.917185i \(-0.630456\pi\)
−0.398461 + 0.917185i \(0.630456\pi\)
\(480\) 0 0
\(481\) 1.14607e6 0.225864
\(482\) 0 0
\(483\) − 4.89940e6i − 0.955598i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.68175e6i 0.321321i 0.987010 + 0.160661i \(0.0513625\pi\)
−0.987010 + 0.160661i \(0.948638\pi\)
\(488\) 0 0
\(489\) −1.90122e6 −0.359551
\(490\) 0 0
\(491\) 115927. 0.0217010 0.0108505 0.999941i \(-0.496546\pi\)
0.0108505 + 0.999941i \(0.496546\pi\)
\(492\) 0 0
\(493\) − 731580.i − 0.135564i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.50076e6i 1.36212i
\(498\) 0 0
\(499\) 7.98867e6 1.43623 0.718113 0.695926i \(-0.245006\pi\)
0.718113 + 0.695926i \(0.245006\pi\)
\(500\) 0 0
\(501\) −5.51478e6 −0.981598
\(502\) 0 0
\(503\) − 8.07650e6i − 1.42332i −0.702523 0.711661i \(-0.747943\pi\)
0.702523 0.711661i \(-0.252057\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 4.66323e6i − 0.805689i
\(508\) 0 0
\(509\) −2.83427e6 −0.484894 −0.242447 0.970165i \(-0.577950\pi\)
−0.242447 + 0.970165i \(0.577950\pi\)
\(510\) 0 0
\(511\) −2.56948e6 −0.435305
\(512\) 0 0
\(513\) 3.96576e6i 0.665324i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.92212e6i − 0.316267i
\(518\) 0 0
\(519\) −7.99545e6 −1.30294
\(520\) 0 0
\(521\) 2.97526e6 0.480209 0.240105 0.970747i \(-0.422818\pi\)
0.240105 + 0.970747i \(0.422818\pi\)
\(522\) 0 0
\(523\) − 7.72888e6i − 1.23556i −0.786352 0.617778i \(-0.788033\pi\)
0.786352 0.617778i \(-0.211967\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 562004.i 0.0881481i
\(528\) 0 0
\(529\) −502077. −0.0780066
\(530\) 0 0
\(531\) −2.14464e6 −0.330079
\(532\) 0 0
\(533\) − 1.11958e6i − 0.170701i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4.03752e6i − 0.604198i
\(538\) 0 0
\(539\) −625893. −0.0927958
\(540\) 0 0
\(541\) −9.83660e6 −1.44495 −0.722474 0.691399i \(-0.756995\pi\)
−0.722474 + 0.691399i \(0.756995\pi\)
\(542\) 0 0
\(543\) − 2.91166e6i − 0.423780i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 3.45608e6i − 0.493873i −0.969032 0.246937i \(-0.920576\pi\)
0.969032 0.246937i \(-0.0794239\pi\)
\(548\) 0 0
\(549\) 1.68651e6 0.238813
\(550\) 0 0
\(551\) 3.97018e6 0.557098
\(552\) 0 0
\(553\) 1.20255e7i 1.67221i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 7.05760e6i − 0.963871i −0.876207 0.481936i \(-0.839934\pi\)
0.876207 0.481936i \(-0.160066\pi\)
\(558\) 0 0
\(559\) 2.75828e6 0.373344
\(560\) 0 0
\(561\) 619440. 0.0830983
\(562\) 0 0
\(563\) − 2.32495e6i − 0.309131i −0.987983 0.154566i \(-0.950602\pi\)
0.987983 0.154566i \(-0.0493978\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 5.51370e6i − 0.720254i
\(568\) 0 0
\(569\) −1.01947e7 −1.32006 −0.660029 0.751240i \(-0.729456\pi\)
−0.660029 + 0.751240i \(0.729456\pi\)
\(570\) 0 0
\(571\) −1.21297e7 −1.55689 −0.778446 0.627712i \(-0.783992\pi\)
−0.778446 + 0.627712i \(0.783992\pi\)
\(572\) 0 0
\(573\) 1.22929e7i 1.56411i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.46358e6i 0.183011i 0.995805 + 0.0915053i \(0.0291678\pi\)
−0.995805 + 0.0915053i \(0.970832\pi\)
\(578\) 0 0
\(579\) −1.15933e7 −1.43718
\(580\) 0 0
\(581\) −1.09027e7 −1.33997
\(582\) 0 0
\(583\) − 8.35891e6i − 1.01854i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.18425e6i 0.620998i 0.950574 + 0.310499i \(0.100496\pi\)
−0.950574 + 0.310499i \(0.899504\pi\)
\(588\) 0 0
\(589\) −3.04992e6 −0.362243
\(590\) 0 0
\(591\) 5.80289e6 0.683401
\(592\) 0 0
\(593\) 1.02722e7i 1.19957i 0.800161 + 0.599785i \(0.204747\pi\)
−0.800161 + 0.599785i \(0.795253\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.06685e7i − 1.22509i
\(598\) 0 0
\(599\) 1.39289e7 1.58617 0.793085 0.609111i \(-0.208474\pi\)
0.793085 + 0.609111i \(0.208474\pi\)
\(600\) 0 0
\(601\) −4.81441e6 −0.543697 −0.271848 0.962340i \(-0.587635\pi\)
−0.271848 + 0.962340i \(0.587635\pi\)
\(602\) 0 0
\(603\) 3.13779e6i 0.351423i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 1.04800e7i − 1.15449i −0.816571 0.577245i \(-0.804128\pi\)
0.816571 0.577245i \(-0.195872\pi\)
\(608\) 0 0
\(609\) −7.64460e6 −0.835240
\(610\) 0 0
\(611\) −1.14119e6 −0.123667
\(612\) 0 0
\(613\) 3.07977e6i 0.331029i 0.986207 + 0.165515i \(0.0529285\pi\)
−0.986207 + 0.165515i \(0.947071\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.15522e6i 0.122166i 0.998133 + 0.0610831i \(0.0194555\pi\)
−0.998133 + 0.0610831i \(0.980545\pi\)
\(618\) 0 0
\(619\) −1.76853e7 −1.85517 −0.927587 0.373606i \(-0.878121\pi\)
−0.927587 + 0.373606i \(0.878121\pi\)
\(620\) 0 0
\(621\) −1.08140e7 −1.12528
\(622\) 0 0
\(623\) 1.49159e7i 1.53967i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.36162e6i 0.341491i
\(628\) 0 0
\(629\) −1.32468e6 −0.133501
\(630\) 0 0
\(631\) −5.42541e6 −0.542449 −0.271225 0.962516i \(-0.587429\pi\)
−0.271225 + 0.962516i \(0.587429\pi\)
\(632\) 0 0
\(633\) 6.07932e6i 0.603039i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 371602.i 0.0362852i
\(638\) 0 0
\(639\) 3.40855e6 0.330230
\(640\) 0 0
\(641\) 1.16005e7 1.11515 0.557573 0.830128i \(-0.311733\pi\)
0.557573 + 0.830128i \(0.311733\pi\)
\(642\) 0 0
\(643\) 9.57873e6i 0.913651i 0.889556 + 0.456826i \(0.151014\pi\)
−0.889556 + 0.456826i \(0.848986\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.28130e7i 1.20334i 0.798744 + 0.601671i \(0.205498\pi\)
−0.798744 + 0.601671i \(0.794502\pi\)
\(648\) 0 0
\(649\) −8.82992e6 −0.822896
\(650\) 0 0
\(651\) 5.87263e6 0.543100
\(652\) 0 0
\(653\) 1.58665e6i 0.145613i 0.997346 + 0.0728064i \(0.0231955\pi\)
−0.997346 + 0.0728064i \(0.976804\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.16764e6i 0.105535i
\(658\) 0 0
\(659\) 1.95559e7 1.75414 0.877071 0.480361i \(-0.159494\pi\)
0.877071 + 0.480361i \(0.159494\pi\)
\(660\) 0 0
\(661\) −7.46471e6 −0.664522 −0.332261 0.943188i \(-0.607811\pi\)
−0.332261 + 0.943188i \(0.607811\pi\)
\(662\) 0 0
\(663\) − 367771.i − 0.0324933i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.08261e7i 0.942231i
\(668\) 0 0
\(669\) 5.77386e6 0.498771
\(670\) 0 0
\(671\) 6.94371e6 0.595367
\(672\) 0 0
\(673\) 1.09694e7i 0.933568i 0.884371 + 0.466784i \(0.154587\pi\)
−0.884371 + 0.466784i \(0.845413\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 6.52708e6i − 0.547327i −0.961825 0.273664i \(-0.911764\pi\)
0.961825 0.273664i \(-0.0882355\pi\)
\(678\) 0 0
\(679\) 1.50889e7 1.25598
\(680\) 0 0
\(681\) 1.50805e7 1.24608
\(682\) 0 0
\(683\) 1.54389e7i 1.26638i 0.773995 + 0.633191i \(0.218255\pi\)
−0.773995 + 0.633191i \(0.781745\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 1.08996e7i − 0.881089i
\(688\) 0 0
\(689\) −4.96280e6 −0.398271
\(690\) 0 0
\(691\) −4.78757e6 −0.381435 −0.190717 0.981645i \(-0.561081\pi\)
−0.190717 + 0.981645i \(0.561081\pi\)
\(692\) 0 0
\(693\) 2.26548e6i 0.179196i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.29406e6i 0.100896i
\(698\) 0 0
\(699\) −1.13529e7 −0.878847
\(700\) 0 0
\(701\) −3.31891e6 −0.255094 −0.127547 0.991833i \(-0.540710\pi\)
−0.127547 + 0.991833i \(0.540710\pi\)
\(702\) 0 0
\(703\) − 7.18883e6i − 0.548618i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.20699e6i 0.617497i
\(708\) 0 0
\(709\) −1.60044e7 −1.19570 −0.597852 0.801607i \(-0.703979\pi\)
−0.597852 + 0.801607i \(0.703979\pi\)
\(710\) 0 0
\(711\) 5.46472e6 0.405409
\(712\) 0 0
\(713\) − 8.31668e6i − 0.612669i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 760560.i − 0.0552504i
\(718\) 0 0
\(719\) −1.56006e7 −1.12543 −0.562715 0.826651i \(-0.690243\pi\)
−0.562715 + 0.826651i \(0.690243\pi\)
\(720\) 0 0
\(721\) −1.56531e7 −1.12141
\(722\) 0 0
\(723\) − 1.71154e7i − 1.21771i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.14545e7i 0.803783i 0.915687 + 0.401891i \(0.131647\pi\)
−0.915687 + 0.401891i \(0.868353\pi\)
\(728\) 0 0
\(729\) −1.58900e7 −1.10740
\(730\) 0 0
\(731\) −3.18814e6 −0.220670
\(732\) 0 0
\(733\) − 1.13267e7i − 0.778650i −0.921101 0.389325i \(-0.872708\pi\)
0.921101 0.389325i \(-0.127292\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.29189e7i 0.876108i
\(738\) 0 0
\(739\) 1.31055e7 0.882756 0.441378 0.897321i \(-0.354490\pi\)
0.441378 + 0.897321i \(0.354490\pi\)
\(740\) 0 0
\(741\) 1.99584e6 0.133530
\(742\) 0 0
\(743\) 4.16787e6i 0.276976i 0.990364 + 0.138488i \(0.0442242\pi\)
−0.990364 + 0.138488i \(0.955776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.95448e6i 0.324860i
\(748\) 0 0
\(749\) −5.55086e6 −0.361539
\(750\) 0 0
\(751\) 2.89308e7 1.87180 0.935902 0.352260i \(-0.114587\pi\)
0.935902 + 0.352260i \(0.114587\pi\)
\(752\) 0 0
\(753\) 683400.i 0.0439225i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.32870e6i 0.528247i 0.964489 + 0.264124i \(0.0850827\pi\)
−0.964489 + 0.264124i \(0.914917\pi\)
\(758\) 0 0
\(759\) −9.16663e6 −0.577571
\(760\) 0 0
\(761\) 4.48292e6 0.280608 0.140304 0.990108i \(-0.455192\pi\)
0.140304 + 0.990108i \(0.455192\pi\)
\(762\) 0 0
\(763\) 1.93556e7i 1.20363i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.24245e6i 0.321770i
\(768\) 0 0
\(769\) −1.32141e6 −0.0805790 −0.0402895 0.999188i \(-0.512828\pi\)
−0.0402895 + 0.999188i \(0.512828\pi\)
\(770\) 0 0
\(771\) 2.21110e7 1.33959
\(772\) 0 0
\(773\) 7.24897e6i 0.436342i 0.975911 + 0.218171i \(0.0700091\pi\)
−0.975911 + 0.218171i \(0.929991\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.38421e7i 0.822527i
\(778\) 0 0
\(779\) −7.02268e6 −0.414629
\(780\) 0 0
\(781\) 1.40337e7 0.823273
\(782\) 0 0
\(783\) 1.68733e7i 0.983547i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.98037e6i 0.286632i 0.989677 + 0.143316i \(0.0457766\pi\)
−0.989677 + 0.143316i \(0.954223\pi\)
\(788\) 0 0
\(789\) −7.01670e6 −0.401273
\(790\) 0 0
\(791\) −5.96773e6 −0.339132
\(792\) 0 0
\(793\) − 4.12258e6i − 0.232802i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.48673e7i 0.829061i 0.910036 + 0.414530i \(0.136054\pi\)
−0.910036 + 0.414530i \(0.863946\pi\)
\(798\) 0 0
\(799\) 1.31904e6 0.0730955
\(800\) 0 0
\(801\) 6.77817e6 0.373277
\(802\) 0 0
\(803\) 4.80742e6i 0.263101i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2.59995e7i − 1.40534i
\(808\) 0 0
\(809\) 684390. 0.0367648 0.0183824 0.999831i \(-0.494148\pi\)
0.0183824 + 0.999831i \(0.494148\pi\)
\(810\) 0 0
\(811\) −3.27890e7 −1.75056 −0.875279 0.483618i \(-0.839323\pi\)
−0.875279 + 0.483618i \(0.839323\pi\)
\(812\) 0 0
\(813\) − 1.34341e7i − 0.712825i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.73016e7i − 0.906841i
\(818\) 0 0
\(819\) 1.34505e6 0.0700694
\(820\) 0 0
\(821\) 7.17265e6 0.371383 0.185691 0.982608i \(-0.440548\pi\)
0.185691 + 0.982608i \(0.440548\pi\)
\(822\) 0 0
\(823\) − 1.32939e6i − 0.0684153i −0.999415 0.0342077i \(-0.989109\pi\)
0.999415 0.0342077i \(-0.0108908\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.92964e7i − 0.981099i −0.871413 0.490550i \(-0.836796\pi\)
0.871413 0.490550i \(-0.163204\pi\)
\(828\) 0 0
\(829\) −7.10811e6 −0.359226 −0.179613 0.983737i \(-0.557485\pi\)
−0.179613 + 0.983737i \(0.557485\pi\)
\(830\) 0 0
\(831\) −5.97972e6 −0.300385
\(832\) 0 0
\(833\) − 429514.i − 0.0214469i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.29622e7i − 0.639534i
\(838\) 0 0
\(839\) 8.24538e6 0.404395 0.202198 0.979345i \(-0.435192\pi\)
0.202198 + 0.979345i \(0.435192\pi\)
\(840\) 0 0
\(841\) −3.61905e6 −0.176443
\(842\) 0 0
\(843\) − 1.67231e7i − 0.810493i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.30001e7i − 0.622640i
\(848\) 0 0
\(849\) 3.05761e7 1.45584
\(850\) 0 0
\(851\) 1.96029e7 0.927889
\(852\) 0 0
\(853\) − 1.86921e7i − 0.879599i −0.898096 0.439800i \(-0.855049\pi\)
0.898096 0.439800i \(-0.144951\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 2.24473e7i − 1.04403i −0.852937 0.522013i \(-0.825181\pi\)
0.852937 0.522013i \(-0.174819\pi\)
\(858\) 0 0
\(859\) −3.24436e6 −0.150019 −0.0750094 0.997183i \(-0.523899\pi\)
−0.0750094 + 0.997183i \(0.523899\pi\)
\(860\) 0 0
\(861\) 1.35222e7 0.621641
\(862\) 0 0
\(863\) 1.01542e6i 0.0464108i 0.999731 + 0.0232054i \(0.00738717\pi\)
−0.999731 + 0.0232054i \(0.992613\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 1.86243e7i − 0.841457i
\(868\) 0 0
\(869\) 2.24994e7 1.01070
\(870\) 0 0
\(871\) 7.67015e6 0.342577
\(872\) 0 0
\(873\) − 6.85679e6i − 0.304499i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.96764e7i 1.74194i 0.491337 + 0.870970i \(0.336508\pi\)
−0.491337 + 0.870970i \(0.663492\pi\)
\(878\) 0 0
\(879\) −3.29275e7 −1.43743
\(880\) 0 0
\(881\) −2.44584e7 −1.06167 −0.530833 0.847477i \(-0.678121\pi\)
−0.530833 + 0.847477i \(0.678121\pi\)
\(882\) 0 0
\(883\) − 3.11179e7i − 1.34310i −0.740960 0.671549i \(-0.765629\pi\)
0.740960 0.671549i \(-0.234371\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.87514e7i 0.800249i 0.916461 + 0.400124i \(0.131033\pi\)
−0.916461 + 0.400124i \(0.868967\pi\)
\(888\) 0 0
\(889\) 2.18172e7 0.925858
\(890\) 0 0
\(891\) −1.03160e7 −0.435327
\(892\) 0 0
\(893\) 7.15824e6i 0.300385i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.44236e6i 0.225843i
\(898\) 0 0
\(899\) −1.29766e7 −0.535503
\(900\) 0 0
\(901\) 5.73623e6 0.235404
\(902\) 0 0
\(903\) 3.33143e7i 1.35960i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.07504e6i 0.204843i 0.994741 + 0.102421i \(0.0326590\pi\)
−0.994741 + 0.102421i \(0.967341\pi\)
\(908\) 0 0
\(909\) 3.72947e6 0.149706
\(910\) 0 0
\(911\) −2.83599e7 −1.13216 −0.566082 0.824349i \(-0.691541\pi\)
−0.566082 + 0.824349i \(0.691541\pi\)
\(912\) 0 0
\(913\) 2.03986e7i 0.809885i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3.71169e7i − 1.45763i
\(918\) 0 0
\(919\) −9.58570e6 −0.374399 −0.187200 0.982322i \(-0.559941\pi\)
−0.187200 + 0.982322i \(0.559941\pi\)
\(920\) 0 0
\(921\) −277860. −0.0107939
\(922\) 0 0
\(923\) − 8.33200e6i − 0.321918i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.11321e6i 0.271873i
\(928\) 0 0
\(929\) 3.82212e7 1.45300 0.726500 0.687167i \(-0.241146\pi\)
0.726500 + 0.687167i \(0.241146\pi\)
\(930\) 0 0
\(931\) 2.33091e6 0.0881357
\(932\) 0 0
\(933\) 634200.i 0.0238519i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 9.37422e6i − 0.348808i −0.984674 0.174404i \(-0.944200\pi\)
0.984674 0.174404i \(-0.0557998\pi\)
\(938\) 0 0
\(939\) 3.74544e7 1.38624
\(940\) 0 0
\(941\) 3.04898e6 0.112249 0.0561243 0.998424i \(-0.482126\pi\)
0.0561243 + 0.998424i \(0.482126\pi\)
\(942\) 0 0
\(943\) − 1.91498e7i − 0.701270i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.95948e7i 1.07236i 0.844104 + 0.536180i \(0.180133\pi\)
−0.844104 + 0.536180i \(0.819867\pi\)
\(948\) 0 0
\(949\) 2.85424e6 0.102878
\(950\) 0 0
\(951\) −6.32694e6 −0.226852
\(952\) 0 0
\(953\) − 1.58338e7i − 0.564744i −0.959305 0.282372i \(-0.908879\pi\)
0.959305 0.282372i \(-0.0911212\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.43028e7i 0.504825i
\(958\) 0 0
\(959\) 1.55492e7 0.545960
\(960\) 0 0
\(961\) −1.86604e7 −0.651798
\(962\) 0 0
\(963\) 2.52246e6i 0.0876513i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.69009e6i 0.298853i 0.988773 + 0.149427i \(0.0477428\pi\)
−0.988773 + 0.149427i \(0.952257\pi\)
\(968\) 0 0
\(969\) −2.30688e6 −0.0789252
\(970\) 0 0
\(971\) 2.81474e7 0.958054 0.479027 0.877800i \(-0.340990\pi\)
0.479027 + 0.877800i \(0.340990\pi\)
\(972\) 0 0
\(973\) − 2.04278e7i − 0.691733i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 4.34090e7i − 1.45493i −0.686143 0.727467i \(-0.740697\pi\)
0.686143 0.727467i \(-0.259303\pi\)
\(978\) 0 0
\(979\) 2.79071e7 0.930590
\(980\) 0 0
\(981\) 8.79568e6 0.291808
\(982\) 0 0
\(983\) − 4.64172e7i − 1.53213i −0.642763 0.766065i \(-0.722212\pi\)
0.642763 0.766065i \(-0.277788\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 1.37832e7i − 0.450358i
\(988\) 0 0
\(989\) 4.71789e7 1.53376
\(990\) 0 0
\(991\) −4.64577e7 −1.50271 −0.751353 0.659901i \(-0.770598\pi\)
−0.751353 + 0.659901i \(0.770598\pi\)
\(992\) 0 0
\(993\) − 4.14690e7i − 1.33460i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.35968e7i 0.433211i 0.976259 + 0.216605i \(0.0694985\pi\)
−0.976259 + 0.216605i \(0.930501\pi\)
\(998\) 0 0
\(999\) 3.05525e7 0.968576
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 800.6.c.h.449.2 4
4.3 odd 2 inner 800.6.c.h.449.3 4
5.2 odd 4 800.6.a.i.1.1 2
5.3 odd 4 160.6.a.b.1.2 yes 2
5.4 even 2 inner 800.6.c.h.449.4 4
20.3 even 4 160.6.a.b.1.1 2
20.7 even 4 800.6.a.i.1.2 2
20.19 odd 2 inner 800.6.c.h.449.1 4
40.3 even 4 320.6.a.t.1.2 2
40.13 odd 4 320.6.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.a.b.1.1 2 20.3 even 4
160.6.a.b.1.2 yes 2 5.3 odd 4
320.6.a.t.1.1 2 40.13 odd 4
320.6.a.t.1.2 2 40.3 even 4
800.6.a.i.1.1 2 5.2 odd 4
800.6.a.i.1.2 2 20.7 even 4
800.6.c.h.449.1 4 20.19 odd 2 inner
800.6.c.h.449.2 4 1.1 even 1 trivial
800.6.c.h.449.3 4 4.3 odd 2 inner
800.6.c.h.449.4 4 5.4 even 2 inner