Properties

Label 800.2.c.f.449.1
Level $800$
Weight $2$
Character 800.449
Analytic conductor $6.388$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.2.c.f.449.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{3} -2.82843i q^{7} -5.00000 q^{9} +O(q^{10})\) \(q-2.82843i q^{3} -2.82843i q^{7} -5.00000 q^{9} -5.65685 q^{11} +2.00000i q^{13} +2.00000i q^{17} -8.00000 q^{21} -2.82843i q^{23} +5.65685i q^{27} -6.00000 q^{29} +5.65685 q^{31} +16.0000i q^{33} -10.0000i q^{37} +5.65685 q^{39} +2.00000 q^{41} +8.48528i q^{43} -2.82843i q^{47} -1.00000 q^{49} +5.65685 q^{51} -6.00000i q^{53} -11.3137 q^{59} -2.00000 q^{61} +14.1421i q^{63} -2.82843i q^{67} -8.00000 q^{69} +5.65685 q^{71} +6.00000i q^{73} +16.0000i q^{77} -11.3137 q^{79} +1.00000 q^{81} -2.82843i q^{83} +16.9706i q^{87} -10.0000 q^{89} +5.65685 q^{91} -16.0000i q^{93} +2.00000i q^{97} +28.2843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} + O(q^{10}) \) \( 4 q - 20 q^{9} - 32 q^{21} - 24 q^{29} + 8 q^{41} - 4 q^{49} - 8 q^{61} - 32 q^{69} + 4 q^{81} - 40 q^{89} + O(q^{100}) \)

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
\(3\) − 2.82843i − 1.63299i −0.577350 0.816497i \(-0.695913\pi\)
0.577350 0.816497i \(-0.304087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) − 2.82843i − 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 16.0000i 2.78524i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.82843i − 0.412568i −0.978492 0.206284i \(-0.933863\pi\)
0.978492 0.206284i \(-0.0661372\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 5.65685 0.792118
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 14.1421i 1.78174i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.82843i − 0.345547i −0.984962 0.172774i \(-0.944727\pi\)
0.984962 0.172774i \(-0.0552729\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000i 1.82337i
\(78\) 0 0
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 2.82843i − 0.310460i −0.987878 0.155230i \(-0.950388\pi\)
0.987878 0.155230i \(-0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 16.9706i 1.81944i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 5.65685 0.592999
\(92\) 0 0
\(93\) − 16.0000i − 1.65912i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 28.2843 2.84268
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) − 14.1421i − 1.39347i −0.717331 0.696733i \(-0.754636\pi\)
0.717331 0.696733i \(-0.245364\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 14.1421i − 1.36717i −0.729870 0.683586i \(-0.760419\pi\)
0.729870 0.683586i \(-0.239581\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −28.2843 −2.68462
\(112\) 0 0
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 10.0000i − 0.924500i
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) − 5.65685i − 0.510061i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.82843i − 0.250982i −0.992095 0.125491i \(-0.959949\pi\)
0.992095 0.125491i \(-0.0400507\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) −11.3137 −0.959616 −0.479808 0.877373i \(-0.659294\pi\)
−0.479808 + 0.877373i \(0.659294\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) − 11.3137i − 0.946100i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.82843i 0.233285i
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.9706 −1.38104 −0.690522 0.723311i \(-0.742619\pi\)
−0.690522 + 0.723311i \(0.742619\pi\)
\(152\) 0 0
\(153\) − 10.0000i − 0.808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) −16.9706 −1.34585
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) − 14.1421i − 1.10770i −0.832617 0.553849i \(-0.813159\pi\)
0.832617 0.553849i \(-0.186841\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 14.1421i − 1.09435i −0.837018 0.547176i \(-0.815703\pi\)
0.837018 0.547176i \(-0.184297\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 32.0000i 2.40527i
\(178\) 0 0
\(179\) −11.3137 −0.845626 −0.422813 0.906217i \(-0.638957\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 5.65685i 0.418167i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 11.3137i − 0.827340i
\(188\) 0 0
\(189\) 16.0000 1.16383
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) − 18.0000i − 1.29567i −0.761781 0.647834i \(-0.775675\pi\)
0.761781 0.647834i \(-0.224325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 22.6274 1.60402 0.802008 0.597314i \(-0.203765\pi\)
0.802008 + 0.597314i \(0.203765\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 16.9706i 1.19110i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 14.1421i 0.982946i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.9706 −1.16830 −0.584151 0.811645i \(-0.698572\pi\)
−0.584151 + 0.811645i \(0.698572\pi\)
\(212\) 0 0
\(213\) − 16.0000i − 1.09630i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 16.0000i − 1.08615i
\(218\) 0 0
\(219\) 16.9706 1.14676
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 8.48528i 0.568216i 0.958792 + 0.284108i \(0.0916975\pi\)
−0.958792 + 0.284108i \(0.908302\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.7990i 1.31411i 0.753845 + 0.657053i \(0.228197\pi\)
−0.753845 + 0.657053i \(0.771803\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 45.2548 2.97755
\(232\) 0 0
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 32.0000i 2.07862i
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 14.1421i 0.907218i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −5.65685 −0.357057 −0.178529 0.983935i \(-0.557134\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 14.0000i − 0.873296i −0.899632 0.436648i \(-0.856166\pi\)
0.899632 0.436648i \(-0.143834\pi\)
\(258\) 0 0
\(259\) −28.2843 −1.75750
\(260\) 0 0
\(261\) 30.0000 1.85695
\(262\) 0 0
\(263\) 19.7990i 1.22086i 0.792071 + 0.610429i \(0.209003\pi\)
−0.792071 + 0.610429i \(0.790997\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 28.2843i 1.73097i
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.9706 1.03089 0.515444 0.856923i \(-0.327627\pi\)
0.515444 + 0.856923i \(0.327627\pi\)
\(272\) 0 0
\(273\) − 16.0000i − 0.968364i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) −28.2843 −1.69334
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 8.48528i 0.504398i 0.967675 + 0.252199i \(0.0811537\pi\)
−0.967675 + 0.252199i \(0.918846\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 5.65685i − 0.333914i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 5.65685 0.331611
\(292\) 0 0
\(293\) 10.0000i 0.584206i 0.956387 + 0.292103i \(0.0943550\pi\)
−0.956387 + 0.292103i \(0.905645\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 32.0000i − 1.85683i
\(298\) 0 0
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 5.65685i 0.324978i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.82843i − 0.161427i −0.996737 0.0807134i \(-0.974280\pi\)
0.996737 0.0807134i \(-0.0257199\pi\)
\(308\) 0 0
\(309\) −40.0000 −2.27552
\(310\) 0 0
\(311\) −28.2843 −1.60385 −0.801927 0.597422i \(-0.796192\pi\)
−0.801927 + 0.597422i \(0.796192\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 0 0
\(319\) 33.9411 1.90034
\(320\) 0 0
\(321\) −40.0000 −2.23258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 50.9117i − 2.81542i
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 5.65685 0.310929 0.155464 0.987841i \(-0.450313\pi\)
0.155464 + 0.987841i \(0.450313\pi\)
\(332\) 0 0
\(333\) 50.0000i 2.73998i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) −5.65685 −0.307238
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.48528i 0.455514i 0.973718 + 0.227757i \(0.0731391\pi\)
−0.973718 + 0.227757i \(0.926861\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −11.3137 −0.603881
\(352\) 0 0
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 16.0000i − 0.846810i
\(358\) 0 0
\(359\) −22.6274 −1.19423 −0.597115 0.802156i \(-0.703686\pi\)
−0.597115 + 0.802156i \(0.703686\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 59.3970i − 3.11753i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −16.9706 −0.881068
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 22.6274 1.16229 0.581146 0.813799i \(-0.302604\pi\)
0.581146 + 0.813799i \(0.302604\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) − 36.7696i − 1.87884i −0.342773 0.939418i \(-0.611366\pi\)
0.342773 0.939418i \(-0.388634\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 42.4264i − 2.15666i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) − 16.0000i − 0.807093i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 34.0000i − 1.70641i −0.521575 0.853206i \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 11.3137i 0.563576i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 56.5685i 2.80400i
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −16.9706 −0.837096
\(412\) 0 0
\(413\) 32.0000i 1.57462i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 32.0000i 1.56705i
\(418\) 0 0
\(419\) 11.3137 0.552711 0.276355 0.961056i \(-0.410873\pi\)
0.276355 + 0.961056i \(0.410873\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) 14.1421i 0.687614i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.65685i 0.273754i
\(428\) 0 0
\(429\) −32.0000 −1.54497
\(430\) 0 0
\(431\) −5.65685 −0.272481 −0.136241 0.990676i \(-0.543502\pi\)
−0.136241 + 0.990676i \(0.543502\pi\)
\(432\) 0 0
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) − 2.82843i − 0.134383i −0.997740 0.0671913i \(-0.978596\pi\)
0.997740 0.0671913i \(-0.0214038\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 28.2843i − 1.33780i
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −11.3137 −0.532742
\(452\) 0 0
\(453\) 48.0000i 2.25524i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 0 0
\(459\) −11.3137 −0.528079
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 8.48528i 0.394344i 0.980369 + 0.197172i \(0.0631758\pi\)
−0.980369 + 0.197172i \(0.936824\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 14.1421i − 0.654420i −0.944952 0.327210i \(-0.893892\pi\)
0.944952 0.327210i \(-0.106108\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −50.9117 −2.34589
\(472\) 0 0
\(473\) − 48.0000i − 2.20704i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 30.0000i 1.37361i
\(478\) 0 0
\(479\) 33.9411 1.55081 0.775405 0.631464i \(-0.217546\pi\)
0.775405 + 0.631464i \(0.217546\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 22.6274i 1.02958i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 31.1127i 1.40985i 0.709281 + 0.704925i \(0.249020\pi\)
−0.709281 + 0.704925i \(0.750980\pi\)
\(488\) 0 0
\(489\) −40.0000 −1.80886
\(490\) 0 0
\(491\) 39.5980 1.78703 0.893516 0.449032i \(-0.148231\pi\)
0.893516 + 0.449032i \(0.148231\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 16.0000i − 0.717698i
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −40.0000 −1.78707
\(502\) 0 0
\(503\) 8.48528i 0.378340i 0.981944 + 0.189170i \(0.0605797\pi\)
−0.981944 + 0.189170i \(0.939420\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 25.4558i − 1.13053i
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) 16.9706 0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 5.65685 0.248308
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 8.48528i 0.371035i 0.982641 + 0.185518i \(0.0593962\pi\)
−0.982641 + 0.185518i \(0.940604\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.3137i 0.492833i
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 56.5685 2.45487
\(532\) 0 0
\(533\) 4.00000i 0.173259i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 32.0000i 1.38090i
\(538\) 0 0
\(539\) 5.65685 0.243658
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) − 39.5980i − 1.69931i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 42.4264i 1.81402i 0.421107 + 0.907011i \(0.361642\pi\)
−0.421107 + 0.907011i \(0.638358\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 0 0
\(559\) −16.9706 −0.717778
\(560\) 0 0
\(561\) −32.0000 −1.35104
\(562\) 0 0
\(563\) 19.7990i 0.834428i 0.908808 + 0.417214i \(0.136993\pi\)
−0.908808 + 0.417214i \(0.863007\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.82843i − 0.118783i
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 28.2843 1.18366 0.591830 0.806063i \(-0.298406\pi\)
0.591830 + 0.806063i \(0.298406\pi\)
\(572\) 0 0
\(573\) − 48.0000i − 2.00523i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 0 0
\(579\) −50.9117 −2.11582
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 33.9411i 1.40570i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 25.4558i − 1.05068i −0.850894 0.525338i \(-0.823939\pi\)
0.850894 0.525338i \(-0.176061\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 16.9706 0.698076
\(592\) 0 0
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 64.0000i − 2.61935i
\(598\) 0 0
\(599\) −11.3137 −0.462266 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 14.1421i 0.575912i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 2.82843i − 0.114802i −0.998351 0.0574012i \(-0.981719\pi\)
0.998351 0.0574012i \(-0.0182814\pi\)
\(608\) 0 0
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) 5.65685 0.228852
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.0000i 0.402585i 0.979531 + 0.201292i \(0.0645141\pi\)
−0.979531 + 0.201292i \(0.935486\pi\)
\(618\) 0 0
\(619\) 45.2548 1.81895 0.909473 0.415764i \(-0.136486\pi\)
0.909473 + 0.415764i \(0.136486\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) 28.2843i 1.13319i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −16.9706 −0.675587 −0.337794 0.941220i \(-0.609681\pi\)
−0.337794 + 0.941220i \(0.609681\pi\)
\(632\) 0 0
\(633\) 48.0000i 1.90783i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) −28.2843 −1.11891
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 31.1127i 1.22697i 0.789708 + 0.613483i \(0.210232\pi\)
−0.789708 + 0.613483i \(0.789768\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 14.1421i − 0.555985i −0.960583 0.277992i \(-0.910331\pi\)
0.960583 0.277992i \(-0.0896690\pi\)
\(648\) 0 0
\(649\) 64.0000 2.51222
\(650\) 0 0
\(651\) −45.2548 −1.77368
\(652\) 0 0
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 30.0000i − 1.17041i
\(658\) 0 0
\(659\) 33.9411 1.32216 0.661079 0.750316i \(-0.270099\pi\)
0.661079 + 0.750316i \(0.270099\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 11.3137i 0.439388i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706i 0.657103i
\(668\) 0 0
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 11.3137 0.436761
\(672\) 0 0
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 38.0000i 1.46046i 0.683202 + 0.730229i \(0.260587\pi\)
−0.683202 + 0.730229i \(0.739413\pi\)
\(678\) 0 0
\(679\) 5.65685 0.217090
\(680\) 0 0
\(681\) 56.0000 2.14592
\(682\) 0 0
\(683\) − 14.1421i − 0.541134i −0.962701 0.270567i \(-0.912789\pi\)
0.962701 0.270567i \(-0.0872111\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 39.5980i 1.51076i
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −28.2843 −1.07598 −0.537992 0.842950i \(-0.680817\pi\)
−0.537992 + 0.842950i \(0.680817\pi\)
\(692\) 0 0
\(693\) − 80.0000i − 3.03895i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 0 0
\(699\) −28.2843 −1.06981
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.65685i 0.212748i
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 56.5685 2.12149
\(712\) 0 0
\(713\) − 16.0000i − 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 32.0000i − 1.19506i
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) − 73.5391i − 2.73495i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 25.4558i − 0.944105i −0.881570 0.472052i \(-0.843513\pi\)
0.881570 0.472052i \(-0.156487\pi\)
\(728\) 0 0
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) −16.9706 −0.627679
\(732\) 0 0
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) 11.3137 0.416181 0.208091 0.978110i \(-0.433275\pi\)
0.208091 + 0.978110i \(0.433275\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 14.1421i − 0.518825i −0.965767 0.259412i \(-0.916471\pi\)
0.965767 0.259412i \(-0.0835289\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 14.1421i 0.517434i
\(748\) 0 0
\(749\) −40.0000 −1.46157
\(750\) 0 0
\(751\) −16.9706 −0.619265 −0.309632 0.950856i \(-0.600206\pi\)
−0.309632 + 0.950856i \(0.600206\pi\)
\(752\) 0 0
\(753\) 16.0000i 0.583072i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 0 0
\(759\) 45.2548 1.64265
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) − 50.9117i − 1.84313i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 22.6274i − 0.817029i
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) −39.5980 −1.42609
\(772\) 0 0
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 80.0000i 2.86998i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) − 33.9411i − 1.21296i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 8.48528i 0.302468i 0.988498 + 0.151234i \(0.0483246\pi\)
−0.988498 + 0.151234i \(0.951675\pi\)
\(788\) 0 0
\(789\) 56.0000 1.99365
\(790\) 0 0
\(791\) −5.65685 −0.201135
\(792\) 0 0
\(793\) − 4.00000i − 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) 0 0
\(799\) 5.65685 0.200125
\(800\) 0 0
\(801\) 50.0000 1.76666
\(802\) 0 0
\(803\) − 33.9411i − 1.19776i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 50.9117i − 1.79218i
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −5.65685 −0.198639 −0.0993195 0.995056i \(-0.531667\pi\)
−0.0993195 + 0.995056i \(0.531667\pi\)
\(812\) 0 0
\(813\) − 48.0000i − 1.68343i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −28.2843 −0.988332
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) − 25.4558i − 0.887335i −0.896191 0.443667i \(-0.853677\pi\)
0.896191 0.443667i \(-0.146323\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.1127i 1.08189i 0.841057 + 0.540947i \(0.181934\pi\)
−0.841057 + 0.540947i \(0.818066\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −28.2843 −0.981170
\(832\) 0 0
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 32.0000i 1.10608i
\(838\) 0 0
\(839\) −11.3137 −0.390593 −0.195296 0.980744i \(-0.562567\pi\)
−0.195296 + 0.980744i \(0.562567\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 84.8528i 2.92249i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 59.3970i − 2.04090i
\(848\) 0 0
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) −28.2843 −0.969572
\(852\) 0 0
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 54.0000i − 1.84460i −0.386469 0.922302i \(-0.626305\pi\)
0.386469 0.922302i \(-0.373695\pi\)
\(858\) 0 0
\(859\) −33.9411 −1.15806 −0.579028 0.815308i \(-0.696568\pi\)
−0.579028 + 0.815308i \(0.696568\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) 42.4264i 1.44421i 0.691783 + 0.722106i \(0.256826\pi\)
−0.691783 + 0.722106i \(0.743174\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 36.7696i − 1.24876i
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 5.65685 0.191675
\(872\) 0 0
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 0 0
\(879\) 28.2843 0.954005
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) − 2.82843i − 0.0951842i −0.998867 0.0475921i \(-0.984845\pi\)
0.998867 0.0475921i \(-0.0151548\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.82843i − 0.0949693i −0.998872 0.0474846i \(-0.984879\pi\)
0.998872 0.0474846i \(-0.0151205\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −5.65685 −0.189512
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 16.0000i − 0.534224i
\(898\) 0 0
\(899\) −33.9411 −1.13200
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) − 67.8823i − 2.25898i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 25.4558i − 0.845247i −0.906305 0.422624i \(-0.861109\pi\)
0.906305 0.422624i \(-0.138891\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) 16.0000i 0.529523i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 16.0000i − 0.528367i
\(918\) 0 0
\(919\) −33.9411 −1.11961 −0.559807 0.828623i \(-0.689125\pi\)
−0.559807 + 0.828623i \(0.689125\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 0 0
\(923\) 11.3137i 0.372395i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 70.7107i 2.32244i
\(928\) 0 0
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 80.0000i 2.61908i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 10.0000i 0.326686i 0.986569 + 0.163343i \(0.0522277\pi\)
−0.986569 + 0.163343i \(0.947772\pi\)
\(938\) 0 0
\(939\) 16.9706 0.553813
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) − 5.65685i − 0.184213i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.4264i 1.37867i 0.724441 + 0.689336i \(0.242098\pi\)
−0.724441 + 0.689336i \(0.757902\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 84.8528 2.75154
\(952\) 0 0
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 96.0000i − 3.10324i
\(958\) 0 0
\(959\) −16.9706 −0.548008
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 70.7107i 2.27862i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42.4264i 1.36434i 0.731193 + 0.682171i \(0.238964\pi\)
−0.731193 + 0.682171i \(0.761036\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28.2843 0.907685 0.453843 0.891082i \(-0.350053\pi\)
0.453843 + 0.891082i \(0.350053\pi\)
\(972\) 0 0
\(973\) 32.0000i 1.02587i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) 56.5685 1.80794
\(980\) 0 0
\(981\) −90.0000 −2.87348
\(982\) 0 0
\(983\) − 2.82843i − 0.0902128i −0.998982 0.0451064i \(-0.985637\pi\)
0.998982 0.0451064i \(-0.0143627\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 22.6274i 0.720239i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −16.9706 −0.539088 −0.269544 0.962988i \(-0.586873\pi\)
−0.269544 + 0.962988i \(0.586873\pi\)
\(992\) 0 0
\(993\) − 16.0000i − 0.507745i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 0 0
\(999\) 56.5685 1.78975
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 800.2.c.f.449.1 4
3.2 odd 2 7200.2.f.bh.6049.2 4
4.3 odd 2 inner 800.2.c.f.449.4 4
5.2 odd 4 800.2.a.m.1.1 2
5.3 odd 4 160.2.a.c.1.2 yes 2
5.4 even 2 inner 800.2.c.f.449.3 4
8.3 odd 2 1600.2.c.n.449.1 4
8.5 even 2 1600.2.c.n.449.4 4
12.11 even 2 7200.2.f.bh.6049.3 4
15.2 even 4 7200.2.a.cm.1.2 2
15.8 even 4 1440.2.a.o.1.1 2
15.14 odd 2 7200.2.f.bh.6049.4 4
20.3 even 4 160.2.a.c.1.1 2
20.7 even 4 800.2.a.m.1.2 2
20.19 odd 2 inner 800.2.c.f.449.2 4
35.13 even 4 7840.2.a.bf.1.1 2
40.3 even 4 320.2.a.g.1.2 2
40.13 odd 4 320.2.a.g.1.1 2
40.19 odd 2 1600.2.c.n.449.3 4
40.27 even 4 1600.2.a.bc.1.1 2
40.29 even 2 1600.2.c.n.449.2 4
40.37 odd 4 1600.2.a.bc.1.2 2
60.23 odd 4 1440.2.a.o.1.2 2
60.47 odd 4 7200.2.a.cm.1.1 2
60.59 even 2 7200.2.f.bh.6049.1 4
80.3 even 4 1280.2.d.l.641.1 4
80.13 odd 4 1280.2.d.l.641.3 4
80.43 even 4 1280.2.d.l.641.4 4
80.53 odd 4 1280.2.d.l.641.2 4
120.53 even 4 2880.2.a.bk.1.1 2
120.83 odd 4 2880.2.a.bk.1.2 2
140.83 odd 4 7840.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.c.1.1 2 20.3 even 4
160.2.a.c.1.2 yes 2 5.3 odd 4
320.2.a.g.1.1 2 40.13 odd 4
320.2.a.g.1.2 2 40.3 even 4
800.2.a.m.1.1 2 5.2 odd 4
800.2.a.m.1.2 2 20.7 even 4
800.2.c.f.449.1 4 1.1 even 1 trivial
800.2.c.f.449.2 4 20.19 odd 2 inner
800.2.c.f.449.3 4 5.4 even 2 inner
800.2.c.f.449.4 4 4.3 odd 2 inner
1280.2.d.l.641.1 4 80.3 even 4
1280.2.d.l.641.2 4 80.53 odd 4
1280.2.d.l.641.3 4 80.13 odd 4
1280.2.d.l.641.4 4 80.43 even 4
1440.2.a.o.1.1 2 15.8 even 4
1440.2.a.o.1.2 2 60.23 odd 4
1600.2.a.bc.1.1 2 40.27 even 4
1600.2.a.bc.1.2 2 40.37 odd 4
1600.2.c.n.449.1 4 8.3 odd 2
1600.2.c.n.449.2 4 40.29 even 2
1600.2.c.n.449.3 4 40.19 odd 2
1600.2.c.n.449.4 4 8.5 even 2
2880.2.a.bk.1.1 2 120.53 even 4
2880.2.a.bk.1.2 2 120.83 odd 4
7200.2.a.cm.1.1 2 60.47 odd 4
7200.2.a.cm.1.2 2 15.2 even 4
7200.2.f.bh.6049.1 4 60.59 even 2
7200.2.f.bh.6049.2 4 3.2 odd 2
7200.2.f.bh.6049.3 4 12.11 even 2
7200.2.f.bh.6049.4 4 15.14 odd 2
7840.2.a.bf.1.1 2 35.13 even 4
7840.2.a.bf.1.2 2 140.83 odd 4