Properties

Label 1280.2.d.l.641.2
Level $1280$
Weight $2$
Character 1280.641
Analytic conductor $10.221$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.2.d.l.641.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{3} +1.00000i q^{5} +2.82843 q^{7} -5.00000 q^{9} +O(q^{10})\) \(q-2.82843i q^{3} +1.00000i q^{5} +2.82843 q^{7} -5.00000 q^{9} -5.65685i q^{11} +2.00000i q^{13} +2.82843 q^{15} +2.00000 q^{17} -8.00000i q^{21} -2.82843 q^{23} -1.00000 q^{25} +5.65685i q^{27} -6.00000i q^{29} +5.65685 q^{31} -16.0000 q^{33} +2.82843i q^{35} -10.0000i q^{37} +5.65685 q^{39} -2.00000 q^{41} -8.48528i q^{43} -5.00000i q^{45} -2.82843 q^{47} +1.00000 q^{49} -5.65685i q^{51} +6.00000i q^{53} +5.65685 q^{55} +11.3137i q^{59} +2.00000i q^{61} -14.1421 q^{63} -2.00000 q^{65} +2.82843i q^{67} +8.00000i q^{69} -5.65685 q^{71} +6.00000 q^{73} +2.82843i q^{75} -16.0000i q^{77} +11.3137 q^{79} +1.00000 q^{81} -2.82843i q^{83} +2.00000i q^{85} -16.9706 q^{87} -10.0000 q^{89} +5.65685i q^{91} -16.0000i q^{93} +2.00000 q^{97} +28.2843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} + O(q^{10}) \) \( 4 q - 20 q^{9} + 8 q^{17} - 4 q^{25} - 64 q^{33} - 8 q^{41} + 4 q^{49} - 8 q^{65} + 24 q^{73} + 4 q^{81} - 40 q^{89} + 8 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) − 5.65685i − 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) − 8.00000i − 1.74574i
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) −16.0000 −2.78524
\(34\) 0 0
\(35\) 2.82843i 0.478091i
\(36\) 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 \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 8.48528i − 1.29399i −0.762493 0.646997i \(-0.776025\pi\)
0.762493 0.646997i \(-0.223975\pi\)
\(44\) 0 0
\(45\) − 5.00000i − 0.745356i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 5.65685i − 0.792118i
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137i 1.47292i 0.676481 + 0.736460i \(0.263504\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) −14.1421 −1.78174
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 2.82843i 0.345547i 0.984962 + 0.172774i \(0.0552729\pi\)
−0.984962 + 0.172774i \(0.944727\pi\)
\(68\) 0 0
\(69\) 8.00000i 0.963087i
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 2.82843i 0.326599i
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 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
\(85\) 2.00000i 0.216930i
\(86\) 0 0
\(87\) −16.9706 −1.81944
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 5.65685i 0.592999i
\(92\) 0 0
\(93\) − 16.0000i − 1.65912i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 28.2843i 2.84268i
\(100\) 0 0
\(101\) − 2.00000i − 0.199007i −0.995037 0.0995037i \(-0.968274\pi\)
0.995037 0.0995037i \(-0.0317255\pi\)
\(102\) 0 0
\(103\) −14.1421 −1.39347 −0.696733 0.717331i \(-0.745364\pi\)
−0.696733 + 0.717331i \(0.745364\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) − 14.1421i − 1.36717i −0.729870 0.683586i \(-0.760419\pi\)
0.729870 0.683586i \(-0.239581\pi\)
\(108\) 0 0
\(109\) 18.0000i 1.72409i 0.506834 + 0.862044i \(0.330816\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) 0 0
\(111\) −28.2843 −2.68462
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) − 2.82843i − 0.263752i
\(116\) 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
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) 0 0
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) − 5.65685i − 0.494242i −0.968985 0.247121i \(-0.920516\pi\)
0.968985 0.247121i \(-0.0794845\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.65685 −0.486864
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 11.3137i 0.959616i 0.877373 + 0.479808i \(0.159294\pi\)
−0.877373 + 0.479808i \(0.840706\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 11.3137 0.946100
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) − 2.82843i − 0.233285i
\(148\) 0 0
\(149\) − 10.0000i − 0.819232i −0.912258 0.409616i \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(150\) 0 0
\(151\) 16.9706 1.38104 0.690522 0.723311i \(-0.257381\pi\)
0.690522 + 0.723311i \(0.257381\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 5.65685i 0.454369i
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 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
\(165\) − 16.0000i − 1.24560i
\(166\) 0 0
\(167\) 14.1421 1.09435 0.547176 0.837018i \(-0.315703\pi\)
0.547176 + 0.837018i \(0.315703\pi\)
\(168\) 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
\(175\) −2.82843 −0.213809
\(176\) 0 0
\(177\) 32.0000 2.40527
\(178\) 0 0
\(179\) − 11.3137i − 0.845626i −0.906217 0.422813i \(-0.861043\pi\)
0.906217 0.422813i \(-0.138957\pi\)
\(180\) 0 0
\(181\) 14.0000i 1.04061i 0.853980 + 0.520306i \(0.174182\pi\)
−0.853980 + 0.520306i \(0.825818\pi\)
\(182\) 0 0
\(183\) 5.65685 0.418167
\(184\) 0 0
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) − 11.3137i − 0.827340i
\(188\) 0 0
\(189\) 16.0000i 1.16383i
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) 5.65685i 0.405096i
\(196\) 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
\(205\) − 2.00000i − 0.139686i
\(206\) 0 0
\(207\) 14.1421 0.982946
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.9706i 1.16830i 0.811645 + 0.584151i \(0.198572\pi\)
−0.811645 + 0.584151i \(0.801428\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) 8.48528 0.578691
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) − 16.9706i − 1.14676i
\(220\) 0 0
\(221\) 4.00000i 0.269069i
\(222\) 0 0
\(223\) −8.48528 −0.568216 −0.284108 0.958792i \(-0.591698\pi\)
−0.284108 + 0.958792i \(0.591698\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) − 19.7990i − 1.31411i −0.753845 0.657053i \(-0.771803\pi\)
0.753845 0.657053i \(-0.228197\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) −45.2548 −2.97755
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) − 2.82843i − 0.184506i
\(236\) 0 0
\(237\) − 32.0000i − 2.07862i
\(238\) 0 0
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) 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
\(245\) 1.00000i 0.0638877i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) − 5.65685i − 0.357057i −0.983935 0.178529i \(-0.942866\pi\)
0.983935 0.178529i \(-0.0571337\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) − 28.2843i − 1.75750i
\(260\) 0 0
\(261\) 30.0000i 1.85695i
\(262\) 0 0
\(263\) 19.7990 1.22086 0.610429 0.792071i \(-0.290997\pi\)
0.610429 + 0.792071i \(0.290997\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 28.2843i 1.73097i
\(268\) 0 0
\(269\) 18.0000i 1.09748i 0.835993 + 0.548740i \(0.184892\pi\)
−0.835993 + 0.548740i \(0.815108\pi\)
\(270\) 0 0
\(271\) 16.9706 1.03089 0.515444 0.856923i \(-0.327627\pi\)
0.515444 + 0.856923i \(0.327627\pi\)
\(272\) 0 0
\(273\) 16.0000 0.968364
\(274\) 0 0
\(275\) 5.65685i 0.341121i
\(276\) 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 \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) − 8.48528i − 0.504398i −0.967675 0.252199i \(-0.918846\pi\)
0.967675 0.252199i \(-0.0811537\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.65685 −0.333914
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 5.65685i − 0.331611i
\(292\) 0 0
\(293\) − 10.0000i − 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 0 0
\(295\) −11.3137 −0.658710
\(296\) 0 0
\(297\) 32.0000 1.85683
\(298\) 0 0
\(299\) − 5.65685i − 0.327144i
\(300\) 0 0
\(301\) − 24.0000i − 1.38334i
\(302\) 0 0
\(303\) −5.65685 −0.324978
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 2.82843i 0.161427i 0.996737 + 0.0807134i \(0.0257199\pi\)
−0.996737 + 0.0807134i \(0.974280\pi\)
\(308\) 0 0
\(309\) 40.0000i 2.27552i
\(310\) 0 0
\(311\) 28.2843 1.60385 0.801927 0.597422i \(-0.203808\pi\)
0.801927 + 0.597422i \(0.203808\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) − 14.1421i − 0.796819i
\(316\) 0 0
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 0 0
\(319\) −33.9411 −1.90034
\(320\) 0 0
\(321\) −40.0000 −2.23258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 2.00000i − 0.110940i
\(326\) 0 0
\(327\) 50.9117 2.81542
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 5.65685i 0.310929i 0.987841 + 0.155464i \(0.0496874\pi\)
−0.987841 + 0.155464i \(0.950313\pi\)
\(332\) 0 0
\(333\) 50.0000i 2.73998i
\(334\) 0 0
\(335\) −2.82843 −0.154533
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) − 5.65685i − 0.307238i
\(340\) 0 0
\(341\) − 32.0000i − 1.73290i
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 8.48528i 0.455514i 0.973718 + 0.227757i \(0.0731391\pi\)
−0.973718 + 0.227757i \(0.926861\pi\)
\(348\) 0 0
\(349\) − 6.00000i − 0.321173i −0.987022 0.160586i \(-0.948662\pi\)
0.987022 0.160586i \(-0.0513385\pi\)
\(350\) 0 0
\(351\) −11.3137 −0.603881
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) − 5.65685i − 0.300235i
\(356\) 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
\(365\) 6.00000i 0.314054i
\(366\) 0 0
\(367\) 8.48528 0.442928 0.221464 0.975169i \(-0.428916\pi\)
0.221464 + 0.975169i \(0.428916\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 16.9706i 0.881068i
\(372\) 0 0
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 0 0
\(375\) −2.82843 −0.146059
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) − 22.6274i − 1.16229i −0.813799 0.581146i \(-0.802604\pi\)
0.813799 0.581146i \(-0.197396\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 36.7696 1.87884 0.939418 0.342773i \(-0.111366\pi\)
0.939418 + 0.342773i \(0.111366\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 42.4264i 2.15666i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) −5.65685 −0.286079
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) 0 0
\(395\) 11.3137i 0.569254i
\(396\) 0 0
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 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
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) −56.5685 −2.80400
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) − 16.9706i − 0.837096i
\(412\) 0 0
\(413\) 32.0000i 1.57462i
\(414\) 0 0
\(415\) 2.82843 0.138842
\(416\) 0 0
\(417\) 32.0000 1.56705
\(418\) 0 0
\(419\) 11.3137i 0.552711i 0.961056 + 0.276355i \(0.0891267\pi\)
−0.961056 + 0.276355i \(0.910873\pi\)
\(420\) 0 0
\(421\) 38.0000i 1.85201i 0.377515 + 0.926003i \(0.376779\pi\)
−0.377515 + 0.926003i \(0.623221\pi\)
\(422\) 0 0
\(423\) 14.1421 0.687614
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 5.65685i 0.273754i
\(428\) 0 0
\(429\) − 32.0000i − 1.54497i
\(430\) 0 0
\(431\) −5.65685 −0.272481 −0.136241 0.990676i \(-0.543502\pi\)
−0.136241 + 0.990676i \(0.543502\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) − 16.9706i − 0.813676i
\(436\) 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 \(0.0214038\pi\)
−0.997740 + 0.0671913i \(0.978596\pi\)
\(444\) 0 0
\(445\) − 10.0000i − 0.474045i
\(446\) 0 0
\(447\) −28.2843 −1.33780
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 11.3137i 0.532742i
\(452\) 0 0
\(453\) − 48.0000i − 2.25524i
\(454\) 0 0
\(455\) −5.65685 −0.265197
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 11.3137i 0.528079i
\(460\) 0 0
\(461\) − 6.00000i − 0.279448i −0.990190 0.139724i \(-0.955378\pi\)
0.990190 0.139724i \(-0.0446215\pi\)
\(462\) 0 0
\(463\) −8.48528 −0.394344 −0.197172 0.980369i \(-0.563176\pi\)
−0.197172 + 0.980369i \(0.563176\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) 14.1421i 0.654420i 0.944952 + 0.327210i \(0.106108\pi\)
−0.944952 + 0.327210i \(0.893892\pi\)
\(468\) 0 0
\(469\) 8.00000i 0.369406i
\(470\) 0 0
\(471\) 50.9117 2.34589
\(472\) 0 0
\(473\) −48.0000 −2.20704
\(474\) 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 \(-0.782454\pi\)
−0.775405 + 0.631464i \(0.782454\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 22.6274i 1.02958i
\(484\) 0 0
\(485\) 2.00000i 0.0908153i
\(486\) 0 0
\(487\) −31.1127 −1.40985 −0.704925 0.709281i \(-0.749020\pi\)
−0.704925 + 0.709281i \(0.749020\pi\)
\(488\) 0 0
\(489\) −40.0000 −1.80886
\(490\) 0 0
\(491\) 39.5980i 1.78703i 0.449032 + 0.893516i \(0.351769\pi\)
−0.449032 + 0.893516i \(0.648231\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) −28.2843 −1.27128
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) − 40.0000i − 1.78707i
\(502\) 0 0
\(503\) 8.48528 0.378340 0.189170 0.981944i \(-0.439420\pi\)
0.189170 + 0.981944i \(0.439420\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) − 25.4558i − 1.13053i
\(508\) 0 0
\(509\) 26.0000i 1.15243i 0.817298 + 0.576215i \(0.195471\pi\)
−0.817298 + 0.576215i \(0.804529\pi\)
\(510\) 0 0
\(511\) 16.9706 0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 14.1421i − 0.623177i
\(516\) 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 \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) − 8.48528i − 0.371035i −0.982641 0.185518i \(-0.940604\pi\)
0.982641 0.185518i \(-0.0593962\pi\)
\(524\) 0 0
\(525\) 8.00000i 0.349149i
\(526\) 0 0
\(527\) 11.3137 0.492833
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) − 56.5685i − 2.45487i
\(532\) 0 0
\(533\) − 4.00000i − 0.173259i
\(534\) 0 0
\(535\) 14.1421 0.611418
\(536\) 0 0
\(537\) −32.0000 −1.38090
\(538\) 0 0
\(539\) − 5.65685i − 0.243658i
\(540\) 0 0
\(541\) 10.0000i 0.429934i 0.976621 + 0.214967i \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) 0 0
\(543\) 39.5980 1.69931
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) − 42.4264i − 1.81402i −0.421107 0.907011i \(-0.638358\pi\)
0.421107 0.907011i \(-0.361642\pi\)
\(548\) 0 0
\(549\) − 10.0000i − 0.426790i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 0 0
\(555\) − 28.2843i − 1.20060i
\(556\) 0 0
\(557\) − 14.0000i − 0.593199i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 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
\(565\) 2.00000i 0.0841406i
\(566\) 0 0
\(567\) 2.82843 0.118783
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 28.2843i 1.18366i 0.806063 + 0.591830i \(0.201594\pi\)
−0.806063 + 0.591830i \(0.798406\pi\)
\(572\) 0 0
\(573\) − 48.0000i − 2.00523i
\(574\) 0 0
\(575\) 2.82843 0.117954
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) − 50.9117i − 2.11582i
\(580\) 0 0
\(581\) − 8.00000i − 0.331896i
\(582\) 0 0
\(583\) 33.9411 1.40570
\(584\) 0 0
\(585\) 10.0000 0.413449
\(586\) 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
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 5.65685i 0.231908i
\(596\) 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 \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) − 14.1421i − 0.575912i
\(604\) 0 0
\(605\) − 21.0000i − 0.853771i
\(606\) 0 0
\(607\) −2.82843 −0.114802 −0.0574012 0.998351i \(-0.518281\pi\)
−0.0574012 + 0.998351i \(0.518281\pi\)
\(608\) 0 0
\(609\) −48.0000 −1.94506
\(610\) 0 0
\(611\) − 5.65685i − 0.228852i
\(612\) 0 0
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 0 0
\(615\) −5.65685 −0.228106
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) − 45.2548i − 1.81895i −0.415764 0.909473i \(-0.636486\pi\)
0.415764 0.909473i \(-0.363514\pi\)
\(620\) 0 0
\(621\) − 16.0000i − 0.642058i
\(622\) 0 0
\(623\) −28.2843 −1.13319
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 20.0000i − 0.797452i
\(630\) 0 0
\(631\) 16.9706 0.675587 0.337794 0.941220i \(-0.390319\pi\)
0.337794 + 0.941220i \(0.390319\pi\)
\(632\) 0 0
\(633\) 48.0000 1.90783
\(634\) 0 0
\(635\) − 2.82843i − 0.112243i
\(636\) 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
\(645\) − 24.0000i − 0.944999i
\(646\) 0 0
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 0 0
\(649\) 64.0000 2.51222
\(650\) 0 0
\(651\) − 45.2548i − 1.77368i
\(652\) 0 0
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) 33.9411i 1.32216i 0.750316 + 0.661079i \(0.229901\pi\)
−0.750316 + 0.661079i \(0.770099\pi\)
\(660\) 0 0
\(661\) 22.0000i 0.855701i 0.903850 + 0.427850i \(0.140729\pi\)
−0.903850 + 0.427850i \(0.859271\pi\)
\(662\) 0 0
\(663\) 11.3137 0.439388
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706i 0.657103i
\(668\) 0 0
\(669\) 24.0000i 0.927894i
\(670\) 0 0
\(671\) 11.3137 0.436761
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) − 5.65685i − 0.217732i
\(676\) 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 \(0.0872111\pi\)
−0.962701 + 0.270567i \(0.912789\pi\)
\(684\) 0 0
\(685\) 6.00000i 0.229248i
\(686\) 0 0
\(687\) 39.5980 1.51076
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 28.2843i 1.07598i 0.842950 + 0.537992i \(0.180817\pi\)
−0.842950 + 0.537992i \(0.819183\pi\)
\(692\) 0 0
\(693\) 80.0000i 3.03895i
\(694\) 0 0
\(695\) −11.3137 −0.429153
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 28.2843i 1.06981i
\(700\) 0 0
\(701\) 2.00000i 0.0755390i 0.999286 + 0.0377695i \(0.0120253\pi\)
−0.999286 + 0.0377695i \(0.987975\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) − 5.65685i − 0.212748i
\(708\) 0 0
\(709\) 14.0000i 0.525781i 0.964826 + 0.262891i \(0.0846758\pi\)
−0.964826 + 0.262891i \(0.915324\pi\)
\(710\) 0 0
\(711\) −56.5685 −2.12149
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 11.3137i 0.423109i
\(716\) 0 0
\(717\) 32.0000i 1.19506i
\(718\) 0 0
\(719\) −11.3137 −0.421930 −0.210965 0.977494i \(-0.567661\pi\)
−0.210965 + 0.977494i \(0.567661\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) − 73.5391i − 2.73495i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 25.4558 0.944105 0.472052 0.881570i \(-0.343513\pi\)
0.472052 + 0.881570i \(0.343513\pi\)
\(728\) 0 0
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) − 16.9706i − 0.627679i
\(732\) 0 0
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) 0 0
\(735\) 2.82843 0.104328
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 11.3137i 0.416181i 0.978110 + 0.208091i \(0.0667249\pi\)
−0.978110 + 0.208091i \(0.933275\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.1421 −0.518825 −0.259412 0.965767i \(-0.583529\pi\)
−0.259412 + 0.965767i \(0.583529\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 14.1421i 0.517434i
\(748\) 0 0
\(749\) − 40.0000i − 1.46157i
\(750\) 0 0
\(751\) −16.9706 −0.619265 −0.309632 0.950856i \(-0.600206\pi\)
−0.309632 + 0.950856i \(0.600206\pi\)
\(752\) 0 0
\(753\) −16.0000 −0.583072
\(754\) 0 0
\(755\) 16.9706i 0.617622i
\(756\) 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 \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 50.9117i 1.84313i
\(764\) 0 0
\(765\) − 10.0000i − 0.361551i
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 39.5980i 1.42609i
\(772\) 0 0
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 0 0
\(775\) −5.65685 −0.203200
\(776\) 0 0
\(777\) −80.0000 −2.86998
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 32.0000i 1.14505i
\(782\) 0 0
\(783\) 33.9411 1.21296
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) − 8.48528i − 0.302468i −0.988498 0.151234i \(-0.951675\pi\)
0.988498 0.151234i \(-0.0483246\pi\)
\(788\) 0 0
\(789\) − 56.0000i − 1.99365i
\(790\) 0 0
\(791\) 5.65685 0.201135
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 16.9706i 0.601884i
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 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
\(805\) − 8.00000i − 0.281963i
\(806\) 0 0
\(807\) 50.9117 1.79218
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) − 5.65685i − 0.198639i −0.995056 0.0993195i \(-0.968333\pi\)
0.995056 0.0993195i \(-0.0316666\pi\)
\(812\) 0 0
\(813\) − 48.0000i − 1.68343i
\(814\) 0 0
\(815\) 14.1421 0.495377
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) − 28.2843i − 0.988332i
\(820\) 0 0
\(821\) − 10.0000i − 0.349002i −0.984657 0.174501i \(-0.944169\pi\)
0.984657 0.174501i \(-0.0558313\pi\)
\(822\) 0 0
\(823\) −25.4558 −0.887335 −0.443667 0.896191i \(-0.646323\pi\)
−0.443667 + 0.896191i \(0.646323\pi\)
\(824\) 0 0
\(825\) 16.0000 0.557048
\(826\) 0 0
\(827\) 31.1127i 1.08189i 0.841057 + 0.540947i \(0.181934\pi\)
−0.841057 + 0.540947i \(0.818066\pi\)
\(828\) 0 0
\(829\) 2.00000i 0.0694629i 0.999397 + 0.0347314i \(0.0110576\pi\)
−0.999397 + 0.0347314i \(0.988942\pi\)
\(830\) 0 0
\(831\) −28.2843 −0.981170
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 14.1421i 0.489409i
\(836\) 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
\(845\) 9.00000i 0.309609i
\(846\) 0 0
\(847\) −59.3970 −2.04090
\(848\) 0 0
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 28.2843i 0.969572i
\(852\) 0 0
\(853\) 22.0000i 0.753266i 0.926363 + 0.376633i \(0.122918\pi\)
−0.926363 + 0.376633i \(0.877082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) 33.9411i 1.15806i 0.815308 + 0.579028i \(0.196568\pi\)
−0.815308 + 0.579028i \(0.803432\pi\)
\(860\) 0 0
\(861\) 16.0000i 0.545279i
\(862\) 0 0
\(863\) −42.4264 −1.44421 −0.722106 0.691783i \(-0.756826\pi\)
−0.722106 + 0.691783i \(0.756826\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 36.7696i 1.24876i
\(868\) 0 0
\(869\) − 64.0000i − 2.17105i
\(870\) 0 0
\(871\) −5.65685 −0.191675
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) − 2.82843i − 0.0956183i
\(876\) 0 0
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 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
\(885\) 32.0000i 1.07567i
\(886\) 0 0
\(887\) 2.82843 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) − 5.65685i − 0.189512i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 11.3137 0.378176
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) − 33.9411i − 1.13200i
\(900\) 0 0
\(901\) 12.0000i 0.399778i
\(902\) 0 0
\(903\) −67.8823 −2.25898
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) − 25.4558i − 0.845247i −0.906305 0.422624i \(-0.861109\pi\)
0.906305 0.422624i \(-0.138891\pi\)
\(908\) 0 0
\(909\) 10.0000i 0.331679i
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 5.65685i 0.187010i
\(916\) 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
\(925\) 10.0000i 0.328798i
\(926\) 0 0
\(927\) 70.7107 2.32244
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 80.0000i − 2.61908i
\(934\) 0 0
\(935\) 11.3137 0.369998
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) − 16.9706i − 0.553813i
\(940\) 0 0
\(941\) − 38.0000i − 1.23876i −0.785090 0.619382i \(-0.787383\pi\)
0.785090 0.619382i \(-0.212617\pi\)
\(942\) 0 0
\(943\) 5.65685 0.184213
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) − 42.4264i − 1.37867i −0.724441 0.689336i \(-0.757902\pi\)
0.724441 0.689336i \(-0.242098\pi\)
\(948\) 0 0
\(949\) 12.0000i 0.389536i
\(950\) 0 0
\(951\) −84.8528 −2.75154
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.9706i 0.549155i
\(956\) 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
\(965\) 18.0000i 0.579441i
\(966\) 0 0
\(967\) −42.4264 −1.36434 −0.682171 0.731193i \(-0.738964\pi\)
−0.682171 + 0.731193i \(0.738964\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28.2843i 0.907685i 0.891082 + 0.453843i \(0.149947\pi\)
−0.891082 + 0.453843i \(0.850053\pi\)
\(972\) 0 0
\(973\) 32.0000i 1.02587i
\(974\) 0 0
\(975\) −5.65685 −0.181164
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 56.5685i 1.80794i
\(980\) 0 0
\(981\) − 90.0000i − 2.87348i
\(982\) 0 0
\(983\) −2.82843 −0.0902128 −0.0451064 0.998982i \(-0.514363\pi\)
−0.0451064 + 0.998982i \(0.514363\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 22.6274i 0.720239i
\(988\) 0 0
\(989\) 24.0000i 0.763156i
\(990\) 0 0
\(991\) −16.9706 −0.539088 −0.269544 0.962988i \(-0.586873\pi\)
−0.269544 + 0.962988i \(0.586873\pi\)
\(992\) 0 0
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) 22.6274i 0.717337i
\(996\) 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 1280.2.d.l.641.2 4
4.3 odd 2 inner 1280.2.d.l.641.4 4
8.3 odd 2 inner 1280.2.d.l.641.1 4
8.5 even 2 inner 1280.2.d.l.641.3 4
16.3 odd 4 160.2.a.c.1.1 2
16.5 even 4 320.2.a.g.1.1 2
16.11 odd 4 320.2.a.g.1.2 2
16.13 even 4 160.2.a.c.1.2 yes 2
48.5 odd 4 2880.2.a.bk.1.1 2
48.11 even 4 2880.2.a.bk.1.2 2
48.29 odd 4 1440.2.a.o.1.1 2
48.35 even 4 1440.2.a.o.1.2 2
80.3 even 4 800.2.c.f.449.2 4
80.13 odd 4 800.2.c.f.449.3 4
80.19 odd 4 800.2.a.m.1.2 2
80.27 even 4 1600.2.c.n.449.1 4
80.29 even 4 800.2.a.m.1.1 2
80.37 odd 4 1600.2.c.n.449.4 4
80.43 even 4 1600.2.c.n.449.3 4
80.53 odd 4 1600.2.c.n.449.2 4
80.59 odd 4 1600.2.a.bc.1.1 2
80.67 even 4 800.2.c.f.449.4 4
80.69 even 4 1600.2.a.bc.1.2 2
80.77 odd 4 800.2.c.f.449.1 4
112.13 odd 4 7840.2.a.bf.1.1 2
112.83 even 4 7840.2.a.bf.1.2 2
240.29 odd 4 7200.2.a.cm.1.2 2
240.77 even 4 7200.2.f.bh.6049.2 4
240.83 odd 4 7200.2.f.bh.6049.1 4
240.173 even 4 7200.2.f.bh.6049.4 4
240.179 even 4 7200.2.a.cm.1.1 2
240.227 odd 4 7200.2.f.bh.6049.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.c.1.1 2 16.3 odd 4
160.2.a.c.1.2 yes 2 16.13 even 4
320.2.a.g.1.1 2 16.5 even 4
320.2.a.g.1.2 2 16.11 odd 4
800.2.a.m.1.1 2 80.29 even 4
800.2.a.m.1.2 2 80.19 odd 4
800.2.c.f.449.1 4 80.77 odd 4
800.2.c.f.449.2 4 80.3 even 4
800.2.c.f.449.3 4 80.13 odd 4
800.2.c.f.449.4 4 80.67 even 4
1280.2.d.l.641.1 4 8.3 odd 2 inner
1280.2.d.l.641.2 4 1.1 even 1 trivial
1280.2.d.l.641.3 4 8.5 even 2 inner
1280.2.d.l.641.4 4 4.3 odd 2 inner
1440.2.a.o.1.1 2 48.29 odd 4
1440.2.a.o.1.2 2 48.35 even 4
1600.2.a.bc.1.1 2 80.59 odd 4
1600.2.a.bc.1.2 2 80.69 even 4
1600.2.c.n.449.1 4 80.27 even 4
1600.2.c.n.449.2 4 80.53 odd 4
1600.2.c.n.449.3 4 80.43 even 4
1600.2.c.n.449.4 4 80.37 odd 4
2880.2.a.bk.1.1 2 48.5 odd 4
2880.2.a.bk.1.2 2 48.11 even 4
7200.2.a.cm.1.1 2 240.179 even 4
7200.2.a.cm.1.2 2 240.29 odd 4
7200.2.f.bh.6049.1 4 240.83 odd 4
7200.2.f.bh.6049.2 4 240.77 even 4
7200.2.f.bh.6049.3 4 240.227 odd 4
7200.2.f.bh.6049.4 4 240.173 even 4
7840.2.a.bf.1.1 2 112.13 odd 4
7840.2.a.bf.1.2 2 112.83 even 4