Properties

Label 4864.2.a.bj.1.3
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
Defining polynomial: \(x^{4} - 11 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.31342\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.31342 q^{5} +4.77753 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.31342 q^{5} +4.77753 q^{7} -3.00000 q^{9} +2.27492 q^{11} +6.09095 q^{13} -4.27492 q^{17} +1.00000 q^{19} +3.46410 q^{23} -3.27492 q^{25} +6.09095 q^{29} +2.62685 q^{31} +6.27492 q^{35} -0.837253 q^{37} +10.5498 q^{41} -10.2749 q^{43} -3.94027 q^{45} +4.77753 q^{47} +15.8248 q^{49} -10.3923 q^{53} +2.98793 q^{55} -8.54983 q^{59} -1.31342 q^{61} -14.3326 q^{63} +8.00000 q^{65} +8.54983 q^{67} +14.8087 q^{71} -4.27492 q^{73} +10.8685 q^{77} -4.30136 q^{79} +9.00000 q^{81} -13.0997 q^{83} -5.61478 q^{85} +10.0000 q^{89} +29.0997 q^{91} +1.31342 q^{95} +1.45017 q^{97} -6.82475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 6q^{11} - 2q^{17} + 4q^{19} + 2q^{25} + 10q^{35} + 12q^{41} - 26q^{43} + 18q^{49} - 4q^{59} + 32q^{65} + 4q^{67} - 2q^{73} + 36q^{81} + 8q^{83} + 40q^{89} + 56q^{91} + 36q^{97} + 18q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 1.31342 0.587381 0.293691 0.955901i \(-0.405116\pi\)
0.293691 + 0.955901i \(0.405116\pi\)
\(6\) 0 0
\(7\) 4.77753 1.80573 0.902867 0.429919i \(-0.141458\pi\)
0.902867 + 0.429919i \(0.141458\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 2.27492 0.685913 0.342957 0.939351i \(-0.388572\pi\)
0.342957 + 0.939351i \(0.388572\pi\)
\(12\) 0 0
\(13\) 6.09095 1.68933 0.844663 0.535299i \(-0.179801\pi\)
0.844663 + 0.535299i \(0.179801\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.27492 −1.03682 −0.518410 0.855132i \(-0.673476\pi\)
−0.518410 + 0.855132i \(0.673476\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) −3.27492 −0.654983
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.09095 1.13106 0.565530 0.824727i \(-0.308671\pi\)
0.565530 + 0.824727i \(0.308671\pi\)
\(30\) 0 0
\(31\) 2.62685 0.471796 0.235898 0.971778i \(-0.424197\pi\)
0.235898 + 0.971778i \(0.424197\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.27492 1.06065
\(36\) 0 0
\(37\) −0.837253 −0.137644 −0.0688218 0.997629i \(-0.521924\pi\)
−0.0688218 + 0.997629i \(0.521924\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.5498 1.64761 0.823804 0.566875i \(-0.191848\pi\)
0.823804 + 0.566875i \(0.191848\pi\)
\(42\) 0 0
\(43\) −10.2749 −1.56691 −0.783455 0.621448i \(-0.786545\pi\)
−0.783455 + 0.621448i \(0.786545\pi\)
\(44\) 0 0
\(45\) −3.94027 −0.587381
\(46\) 0 0
\(47\) 4.77753 0.696874 0.348437 0.937332i \(-0.386713\pi\)
0.348437 + 0.937332i \(0.386713\pi\)
\(48\) 0 0
\(49\) 15.8248 2.26068
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 0 0
\(55\) 2.98793 0.402893
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.54983 −1.11309 −0.556547 0.830816i \(-0.687874\pi\)
−0.556547 + 0.830816i \(0.687874\pi\)
\(60\) 0 0
\(61\) −1.31342 −0.168167 −0.0840834 0.996459i \(-0.526796\pi\)
−0.0840834 + 0.996459i \(0.526796\pi\)
\(62\) 0 0
\(63\) −14.3326 −1.80573
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 8.54983 1.04453 0.522264 0.852784i \(-0.325087\pi\)
0.522264 + 0.852784i \(0.325087\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.8087 1.75748 0.878738 0.477305i \(-0.158386\pi\)
0.878738 + 0.477305i \(0.158386\pi\)
\(72\) 0 0
\(73\) −4.27492 −0.500341 −0.250171 0.968202i \(-0.580487\pi\)
−0.250171 + 0.968202i \(0.580487\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.8685 1.23858
\(78\) 0 0
\(79\) −4.30136 −0.483940 −0.241970 0.970284i \(-0.577794\pi\)
−0.241970 + 0.970284i \(0.577794\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −13.0997 −1.43788 −0.718938 0.695074i \(-0.755371\pi\)
−0.718938 + 0.695074i \(0.755371\pi\)
\(84\) 0 0
\(85\) −5.61478 −0.609008
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 29.0997 3.05047
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.31342 0.134754
\(96\) 0 0
\(97\) 1.45017 0.147242 0.0736210 0.997286i \(-0.476544\pi\)
0.0736210 + 0.997286i \(0.476544\pi\)
\(98\) 0 0
\(99\) −6.82475 −0.685913
\(100\) 0 0
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) −2.62685 −0.258831 −0.129416 0.991590i \(-0.541310\pi\)
−0.129416 + 0.991590i \(0.541310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.5498 −1.59993 −0.799966 0.600045i \(-0.795149\pi\)
−0.799966 + 0.600045i \(0.795149\pi\)
\(108\) 0 0
\(109\) −3.46410 −0.331801 −0.165900 0.986143i \(-0.553053\pi\)
−0.165900 + 0.986143i \(0.553053\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.54983 −0.239868 −0.119934 0.992782i \(-0.538268\pi\)
−0.119934 + 0.992782i \(0.538268\pi\)
\(114\) 0 0
\(115\) 4.54983 0.424274
\(116\) 0 0
\(117\) −18.2728 −1.68933
\(118\) 0 0
\(119\) −20.4235 −1.87222
\(120\) 0 0
\(121\) −5.82475 −0.529523
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.8685 −0.972106
\(126\) 0 0
\(127\) −17.4356 −1.54716 −0.773579 0.633699i \(-0.781536\pi\)
−0.773579 + 0.633699i \(0.781536\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.27492 −0.198760 −0.0993802 0.995050i \(-0.531686\pi\)
−0.0993802 + 0.995050i \(0.531686\pi\)
\(132\) 0 0
\(133\) 4.77753 0.414264
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.2749 1.73220 0.866102 0.499868i \(-0.166618\pi\)
0.866102 + 0.499868i \(0.166618\pi\)
\(138\) 0 0
\(139\) −10.2749 −0.871507 −0.435754 0.900066i \(-0.643518\pi\)
−0.435754 + 0.900066i \(0.643518\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.98793 0.244781 0.122390 0.992482i \(-0.460944\pi\)
0.122390 + 0.992482i \(0.460944\pi\)
\(150\) 0 0
\(151\) −9.55505 −0.777579 −0.388790 0.921327i \(-0.627107\pi\)
−0.388790 + 0.921327i \(0.627107\pi\)
\(152\) 0 0
\(153\) 12.8248 1.03682
\(154\) 0 0
\(155\) 3.45017 0.277124
\(156\) 0 0
\(157\) −5.25370 −0.419291 −0.209645 0.977778i \(-0.567231\pi\)
−0.209645 + 0.977778i \(0.567231\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.5498 1.30431
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.67451 −0.129577 −0.0647886 0.997899i \(-0.520637\pi\)
−0.0647886 + 0.997899i \(0.520637\pi\)
\(168\) 0 0
\(169\) 24.0997 1.85382
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 0 0
\(173\) 8.71780 0.662802 0.331401 0.943490i \(-0.392479\pi\)
0.331401 + 0.943490i \(0.392479\pi\)
\(174\) 0 0
\(175\) −15.6460 −1.18273
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.09967 0.680141 0.340071 0.940400i \(-0.389549\pi\)
0.340071 + 0.940400i \(0.389549\pi\)
\(180\) 0 0
\(181\) −3.46410 −0.257485 −0.128742 0.991678i \(-0.541094\pi\)
−0.128742 + 0.991678i \(0.541094\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.09967 −0.0808493
\(186\) 0 0
\(187\) −9.72508 −0.711168
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0312 0.725834 0.362917 0.931822i \(-0.381781\pi\)
0.362917 + 0.931822i \(0.381781\pi\)
\(192\) 0 0
\(193\) −19.0997 −1.37482 −0.687412 0.726268i \(-0.741253\pi\)
−0.687412 + 0.726268i \(0.741253\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.1819 0.867924 0.433962 0.900931i \(-0.357115\pi\)
0.433962 + 0.900931i \(0.357115\pi\)
\(198\) 0 0
\(199\) −18.6339 −1.32092 −0.660462 0.750859i \(-0.729640\pi\)
−0.660462 + 0.750859i \(0.729640\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 29.0997 2.04240
\(204\) 0 0
\(205\) 13.8564 0.967773
\(206\) 0 0
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) 2.27492 0.157359
\(210\) 0 0
\(211\) −11.4502 −0.788262 −0.394131 0.919054i \(-0.628954\pi\)
−0.394131 + 0.919054i \(0.628954\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.4953 −0.920373
\(216\) 0 0
\(217\) 12.5498 0.851938
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −26.0383 −1.75153
\(222\) 0 0
\(223\) 17.4356 1.16757 0.583787 0.811907i \(-0.301570\pi\)
0.583787 + 0.811907i \(0.301570\pi\)
\(224\) 0 0
\(225\) 9.82475 0.654983
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 22.0980 1.46028 0.730140 0.683298i \(-0.239455\pi\)
0.730140 + 0.683298i \(0.239455\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.3746 −1.13825 −0.569123 0.822252i \(-0.692717\pi\)
−0.569123 + 0.822252i \(0.692717\pi\)
\(234\) 0 0
\(235\) 6.27492 0.409330
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.82518 0.247431 0.123715 0.992318i \(-0.460519\pi\)
0.123715 + 0.992318i \(0.460519\pi\)
\(240\) 0 0
\(241\) 2.54983 0.164249 0.0821246 0.996622i \(-0.473829\pi\)
0.0821246 + 0.996622i \(0.473829\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.7846 1.32788
\(246\) 0 0
\(247\) 6.09095 0.387558
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.2749 1.15350 0.576751 0.816920i \(-0.304320\pi\)
0.576751 + 0.816920i \(0.304320\pi\)
\(252\) 0 0
\(253\) 7.88054 0.495446
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.5498 1.40662 0.703310 0.710883i \(-0.251705\pi\)
0.703310 + 0.710883i \(0.251705\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −18.2728 −1.13106
\(262\) 0 0
\(263\) 14.3326 0.883785 0.441892 0.897068i \(-0.354307\pi\)
0.441892 + 0.897068i \(0.354307\pi\)
\(264\) 0 0
\(265\) −13.6495 −0.838482
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −22.5742 −1.37637 −0.688187 0.725534i \(-0.741593\pi\)
−0.688187 + 0.725534i \(0.741593\pi\)
\(270\) 0 0
\(271\) −7.04329 −0.427849 −0.213925 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.45017 −0.449262
\(276\) 0 0
\(277\) −2.98793 −0.179527 −0.0897637 0.995963i \(-0.528611\pi\)
−0.0897637 + 0.995963i \(0.528611\pi\)
\(278\) 0 0
\(279\) −7.88054 −0.471796
\(280\) 0 0
\(281\) 19.0997 1.13939 0.569695 0.821856i \(-0.307061\pi\)
0.569695 + 0.821856i \(0.307061\pi\)
\(282\) 0 0
\(283\) −11.3746 −0.676149 −0.338074 0.941119i \(-0.609776\pi\)
−0.338074 + 0.941119i \(0.609776\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 50.4021 2.97514
\(288\) 0 0
\(289\) 1.27492 0.0749951
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.6219 1.26316 0.631581 0.775310i \(-0.282406\pi\)
0.631581 + 0.775310i \(0.282406\pi\)
\(294\) 0 0
\(295\) −11.2296 −0.653810
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 21.0997 1.22023
\(300\) 0 0
\(301\) −49.0887 −2.82942
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.72508 −0.0987780
\(306\) 0 0
\(307\) −16.5498 −0.944549 −0.472274 0.881452i \(-0.656567\pi\)
−0.472274 + 0.881452i \(0.656567\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.72987 0.324911 0.162455 0.986716i \(-0.448059\pi\)
0.162455 + 0.986716i \(0.448059\pi\)
\(312\) 0 0
\(313\) 15.0997 0.853484 0.426742 0.904373i \(-0.359661\pi\)
0.426742 + 0.904373i \(0.359661\pi\)
\(314\) 0 0
\(315\) −18.8248 −1.06065
\(316\) 0 0
\(317\) −8.71780 −0.489640 −0.244820 0.969569i \(-0.578729\pi\)
−0.244820 + 0.969569i \(0.578729\pi\)
\(318\) 0 0
\(319\) 13.8564 0.775810
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.27492 −0.237863
\(324\) 0 0
\(325\) −19.9474 −1.10648
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 22.8248 1.25837
\(330\) 0 0
\(331\) −24.5498 −1.34938 −0.674690 0.738101i \(-0.735723\pi\)
−0.674690 + 0.738101i \(0.735723\pi\)
\(332\) 0 0
\(333\) 2.51176 0.137644
\(334\) 0 0
\(335\) 11.2296 0.613536
\(336\) 0 0
\(337\) −11.6495 −0.634589 −0.317294 0.948327i \(-0.602774\pi\)
−0.317294 + 0.948327i \(0.602774\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.97586 0.323611
\(342\) 0 0
\(343\) 42.1605 2.27645
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.8248 0.795834 0.397917 0.917421i \(-0.369733\pi\)
0.397917 + 0.917421i \(0.369733\pi\)
\(348\) 0 0
\(349\) 35.2323 1.88594 0.942970 0.332877i \(-0.108019\pi\)
0.942970 + 0.332877i \(0.108019\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 19.4502 1.03231
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.476171 0.0251313 0.0125657 0.999921i \(-0.496000\pi\)
0.0125657 + 0.999921i \(0.496000\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.61478 −0.293891
\(366\) 0 0
\(367\) 1.78959 0.0934161 0.0467080 0.998909i \(-0.485127\pi\)
0.0467080 + 0.998909i \(0.485127\pi\)
\(368\) 0 0
\(369\) −31.6495 −1.64761
\(370\) 0 0
\(371\) −49.6495 −2.57767
\(372\) 0 0
\(373\) −25.2011 −1.30486 −0.652431 0.757849i \(-0.726251\pi\)
−0.652431 + 0.757849i \(0.726251\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 37.0997 1.91073
\(378\) 0 0
\(379\) 21.6495 1.11206 0.556030 0.831162i \(-0.312324\pi\)
0.556030 + 0.831162i \(0.312324\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.97586 0.305352 0.152676 0.988276i \(-0.451211\pi\)
0.152676 + 0.988276i \(0.451211\pi\)
\(384\) 0 0
\(385\) 14.2749 0.727517
\(386\) 0 0
\(387\) 30.8248 1.56691
\(388\) 0 0
\(389\) −17.7967 −0.902327 −0.451164 0.892441i \(-0.648991\pi\)
−0.451164 + 0.892441i \(0.648991\pi\)
\(390\) 0 0
\(391\) −14.8087 −0.748911
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.64950 −0.284257
\(396\) 0 0
\(397\) 24.0027 1.20466 0.602331 0.798247i \(-0.294239\pi\)
0.602331 + 0.798247i \(0.294239\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.90033 0.244711 0.122355 0.992486i \(-0.460955\pi\)
0.122355 + 0.992486i \(0.460955\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 11.8208 0.587381
\(406\) 0 0
\(407\) −1.90468 −0.0944116
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −40.8471 −2.00995
\(414\) 0 0
\(415\) −17.2054 −0.844581
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −24.2487 −1.18181 −0.590905 0.806741i \(-0.701229\pi\)
−0.590905 + 0.806741i \(0.701229\pi\)
\(422\) 0 0
\(423\) −14.3326 −0.696874
\(424\) 0 0
\(425\) 14.0000 0.679100
\(426\) 0 0
\(427\) −6.27492 −0.303665
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.1101 −0.920501 −0.460251 0.887789i \(-0.652240\pi\)
−0.460251 + 0.887789i \(0.652240\pi\)
\(432\) 0 0
\(433\) 27.6495 1.32875 0.664375 0.747399i \(-0.268698\pi\)
0.664375 + 0.747399i \(0.268698\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410 0.165710
\(438\) 0 0
\(439\) 3.57919 0.170825 0.0854127 0.996346i \(-0.472779\pi\)
0.0854127 + 0.996346i \(0.472779\pi\)
\(440\) 0 0
\(441\) −47.4743 −2.26068
\(442\) 0 0
\(443\) −27.3746 −1.30061 −0.650303 0.759675i \(-0.725358\pi\)
−0.650303 + 0.759675i \(0.725358\pi\)
\(444\) 0 0
\(445\) 13.1342 0.622623
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.45017 −0.257209 −0.128605 0.991696i \(-0.541050\pi\)
−0.128605 + 0.991696i \(0.541050\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 38.2202 1.79179
\(456\) 0 0
\(457\) 7.72508 0.361364 0.180682 0.983542i \(-0.442169\pi\)
0.180682 + 0.983542i \(0.442169\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 35.2323 1.64093 0.820465 0.571696i \(-0.193714\pi\)
0.820465 + 0.571696i \(0.193714\pi\)
\(462\) 0 0
\(463\) 33.4427 1.55421 0.777107 0.629369i \(-0.216687\pi\)
0.777107 + 0.629369i \(0.216687\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.8248 −1.42640 −0.713200 0.700961i \(-0.752755\pi\)
−0.713200 + 0.700961i \(0.752755\pi\)
\(468\) 0 0
\(469\) 40.8471 1.88614
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −23.3746 −1.07476
\(474\) 0 0
\(475\) −3.27492 −0.150264
\(476\) 0 0
\(477\) 31.1769 1.42749
\(478\) 0 0
\(479\) 17.3205 0.791394 0.395697 0.918381i \(-0.370503\pi\)
0.395697 + 0.918381i \(0.370503\pi\)
\(480\) 0 0
\(481\) −5.09967 −0.232525
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.90468 0.0864872
\(486\) 0 0
\(487\) 12.9041 0.584739 0.292370 0.956305i \(-0.405556\pi\)
0.292370 + 0.956305i \(0.405556\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −26.0383 −1.17271
\(494\) 0 0
\(495\) −8.96379 −0.402893
\(496\) 0 0
\(497\) 70.7492 3.17353
\(498\) 0 0
\(499\) −34.2749 −1.53436 −0.767178 0.641434i \(-0.778340\pi\)
−0.767178 + 0.641434i \(0.778340\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.81312 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(504\) 0 0
\(505\) 18.1993 0.809860
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.0192 0.577064 0.288532 0.957470i \(-0.406833\pi\)
0.288532 + 0.957470i \(0.406833\pi\)
\(510\) 0 0
\(511\) −20.4235 −0.903484
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.45017 −0.152032
\(516\) 0 0
\(517\) 10.8685 0.477995
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.09967 −0.135799 −0.0678995 0.997692i \(-0.521630\pi\)
−0.0678995 + 0.997692i \(0.521630\pi\)
\(522\) 0 0
\(523\) 20.5498 0.898582 0.449291 0.893386i \(-0.351677\pi\)
0.449291 + 0.893386i \(0.351677\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.2296 −0.489167
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 25.6495 1.11309
\(532\) 0 0
\(533\) 64.2585 2.78335
\(534\) 0 0
\(535\) −21.7370 −0.939770
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 7.28929 0.313391 0.156695 0.987647i \(-0.449916\pi\)
0.156695 + 0.987647i \(0.449916\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.54983 −0.194893
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 0 0
\(549\) 3.94027 0.168167
\(550\) 0 0
\(551\) 6.09095 0.259483
\(552\) 0 0
\(553\) −20.5498 −0.873868
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.8443 0.713717 0.356859 0.934158i \(-0.383848\pi\)
0.356859 + 0.934158i \(0.383848\pi\)
\(558\) 0 0
\(559\) −62.5840 −2.64702
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −3.34901 −0.140894
\(566\) 0 0
\(567\) 42.9977 1.80573
\(568\) 0 0
\(569\) 34.5498 1.44840 0.724202 0.689588i \(-0.242208\pi\)
0.724202 + 0.689588i \(0.242208\pi\)
\(570\) 0 0
\(571\) −10.9003 −0.456165 −0.228082 0.973642i \(-0.573246\pi\)
−0.228082 + 0.973642i \(0.573246\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −11.3446 −0.473104
\(576\) 0 0
\(577\) 15.7251 0.654644 0.327322 0.944913i \(-0.393854\pi\)
0.327322 + 0.944913i \(0.393854\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −62.5840 −2.59642
\(582\) 0 0
\(583\) −23.6416 −0.979136
\(584\) 0 0
\(585\) −24.0000 −0.992278
\(586\) 0 0
\(587\) −29.7251 −1.22689 −0.613443 0.789739i \(-0.710216\pi\)
−0.613443 + 0.789739i \(0.710216\pi\)
\(588\) 0 0
\(589\) 2.62685 0.108237
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11.0997 −0.455809 −0.227904 0.973684i \(-0.573187\pi\)
−0.227904 + 0.973684i \(0.573187\pi\)
\(594\) 0 0
\(595\) −26.8248 −1.09971
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −35.5934 −1.45431 −0.727153 0.686476i \(-0.759157\pi\)
−0.727153 + 0.686476i \(0.759157\pi\)
\(600\) 0 0
\(601\) 19.0997 0.779092 0.389546 0.921007i \(-0.372632\pi\)
0.389546 + 0.921007i \(0.372632\pi\)
\(602\) 0 0
\(603\) −25.6495 −1.04453
\(604\) 0 0
\(605\) −7.65037 −0.311032
\(606\) 0 0
\(607\) −46.8229 −1.90048 −0.950242 0.311513i \(-0.899164\pi\)
−0.950242 + 0.311513i \(0.899164\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29.0997 1.17725
\(612\) 0 0
\(613\) −16.8443 −0.680336 −0.340168 0.940365i \(-0.610484\pi\)
−0.340168 + 0.940365i \(0.610484\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.2749 −0.816237 −0.408119 0.912929i \(-0.633815\pi\)
−0.408119 + 0.912929i \(0.633815\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 47.7753 1.91408
\(624\) 0 0
\(625\) 2.09967 0.0839868
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.57919 0.142712
\(630\) 0 0
\(631\) 9.07888 0.361425 0.180712 0.983536i \(-0.442160\pi\)
0.180712 + 0.983536i \(0.442160\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.9003 −0.908772
\(636\) 0 0
\(637\) 96.3878 3.81902
\(638\) 0 0
\(639\) −44.4262 −1.75748
\(640\) 0 0
\(641\) 2.54983 0.100712 0.0503562 0.998731i \(-0.483964\pi\)
0.0503562 + 0.998731i \(0.483964\pi\)
\(642\) 0 0
\(643\) −7.92442 −0.312509 −0.156254 0.987717i \(-0.549942\pi\)
−0.156254 + 0.987717i \(0.549942\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.82518 0.150384 0.0751918 0.997169i \(-0.476043\pi\)
0.0751918 + 0.997169i \(0.476043\pi\)
\(648\) 0 0
\(649\) −19.4502 −0.763486
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.98793 0.116927 0.0584634 0.998290i \(-0.481380\pi\)
0.0584634 + 0.998290i \(0.481380\pi\)
\(654\) 0 0
\(655\) −2.98793 −0.116748
\(656\) 0 0
\(657\) 12.8248 0.500341
\(658\) 0 0
\(659\) 28.5498 1.11214 0.556072 0.831134i \(-0.312308\pi\)
0.556072 + 0.831134i \(0.312308\pi\)
\(660\) 0 0
\(661\) −0.837253 −0.0325654 −0.0162827 0.999867i \(-0.505183\pi\)
−0.0162827 + 0.999867i \(0.505183\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.27492 0.243331
\(666\) 0 0
\(667\) 21.0997 0.816982
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.98793 −0.115348
\(672\) 0 0
\(673\) −16.9003 −0.651460 −0.325730 0.945463i \(-0.605610\pi\)
−0.325730 + 0.945463i \(0.605610\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −44.3112 −1.70302 −0.851508 0.524342i \(-0.824312\pi\)
−0.851508 + 0.524342i \(0.824312\pi\)
\(678\) 0 0
\(679\) 6.92820 0.265880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −38.1993 −1.46166 −0.730829 0.682561i \(-0.760866\pi\)
−0.730829 + 0.682561i \(0.760866\pi\)
\(684\) 0 0
\(685\) 26.6296 1.01746
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −63.2990 −2.41150
\(690\) 0 0
\(691\) 44.4743 1.69188 0.845940 0.533278i \(-0.179040\pi\)
0.845940 + 0.533278i \(0.179040\pi\)
\(692\) 0 0
\(693\) −32.6054 −1.23858
\(694\) 0 0
\(695\) −13.4953 −0.511907
\(696\) 0 0
\(697\) −45.0997 −1.70827
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −48.7276 −1.84042 −0.920208 0.391430i \(-0.871981\pi\)
−0.920208 + 0.391430i \(0.871981\pi\)
\(702\) 0 0
\(703\) −0.837253 −0.0315776
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 66.1993 2.48968
\(708\) 0 0
\(709\) 3.34901 0.125775 0.0628874 0.998021i \(-0.479969\pi\)
0.0628874 + 0.998021i \(0.479969\pi\)
\(710\) 0 0
\(711\) 12.9041 0.483940
\(712\) 0 0
\(713\) 9.09967 0.340785
\(714\) 0 0
\(715\) 18.1993 0.680617
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.4279 −0.463482 −0.231741 0.972777i \(-0.574442\pi\)
−0.231741 + 0.972777i \(0.574442\pi\)
\(720\) 0 0
\(721\) −12.5498 −0.467380
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −19.9474 −0.740826
\(726\) 0 0
\(727\) 19.5863 0.726415 0.363207 0.931708i \(-0.381682\pi\)
0.363207 + 0.931708i \(0.381682\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 43.9244 1.62460
\(732\) 0 0
\(733\) 1.67451 0.0618493 0.0309247 0.999522i \(-0.490155\pi\)
0.0309247 + 0.999522i \(0.490155\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.4502 0.716456
\(738\) 0 0
\(739\) 14.8248 0.545337 0.272669 0.962108i \(-0.412094\pi\)
0.272669 + 0.962108i \(0.412094\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −26.7605 −0.981746 −0.490873 0.871231i \(-0.663322\pi\)
−0.490873 + 0.871231i \(0.663322\pi\)
\(744\) 0 0
\(745\) 3.92442 0.143780
\(746\) 0 0
\(747\) 39.2990 1.43788
\(748\) 0 0
\(749\) −79.0673 −2.88905
\(750\) 0 0
\(751\) 53.7511 1.96141 0.980703 0.195503i \(-0.0626340\pi\)
0.980703 + 0.195503i \(0.0626340\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.5498 −0.456735
\(756\) 0 0
\(757\) −1.31342 −0.0477372 −0.0238686 0.999715i \(-0.507598\pi\)
−0.0238686 + 0.999715i \(0.507598\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.27492 0.154966 0.0774828 0.996994i \(-0.475312\pi\)
0.0774828 + 0.996994i \(0.475312\pi\)
\(762\) 0 0
\(763\) −16.5498 −0.599144
\(764\) 0 0
\(765\) 16.8443 0.609008
\(766\) 0 0
\(767\) −52.0766 −1.88038
\(768\) 0 0
\(769\) 8.82475 0.318229 0.159114 0.987260i \(-0.449136\pi\)
0.159114 + 0.987260i \(0.449136\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.115088 −0.00413942 −0.00206971 0.999998i \(-0.500659\pi\)
−0.00206971 + 0.999998i \(0.500659\pi\)
\(774\) 0 0
\(775\) −8.60271 −0.309018
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.5498 0.377987
\(780\) 0 0
\(781\) 33.6887 1.20548
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.90033 −0.246283
\(786\) 0 0
\(787\) −46.1993 −1.64683 −0.823414 0.567441i \(-0.807934\pi\)
−0.823414 + 0.567441i \(0.807934\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.1819 −0.433138
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24.2487 −0.858933 −0.429467 0.903083i \(-0.641298\pi\)
−0.429467 + 0.903083i \(0.641298\pi\)
\(798\) 0 0
\(799\) −20.4235 −0.722532
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) −9.72508 −0.343191
\(804\) 0 0
\(805\) 21.7370 0.766127
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.8248 −0.591527 −0.295763 0.955261i \(-0.595574\pi\)
−0.295763 + 0.955261i \(0.595574\pi\)
\(810\) 0 0
\(811\) −43.2990 −1.52043 −0.760217 0.649669i \(-0.774907\pi\)
−0.760217 + 0.649669i \(0.774907\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.25370 0.184029
\(816\) 0 0
\(817\) −10.2749 −0.359474
\(818\) 0 0
\(819\) −87.2990 −3.05047
\(820\) 0 0
\(821\) −21.3759 −0.746023 −0.373011 0.927827i \(-0.621675\pi\)
−0.373011 + 0.927827i \(0.621675\pi\)
\(822\) 0 0
\(823\) 43.9501 1.53200 0.766002 0.642839i \(-0.222243\pi\)
0.766002 + 0.642839i \(0.222243\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.35050 0.0817348 0.0408674 0.999165i \(-0.486988\pi\)
0.0408674 + 0.999165i \(0.486988\pi\)
\(828\) 0 0
\(829\) −35.7084 −1.24021 −0.620103 0.784521i \(-0.712909\pi\)
−0.620103 + 0.784521i \(0.712909\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −67.6495 −2.34392
\(834\) 0 0
\(835\) −2.19934 −0.0761112
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −38.9424 −1.34444 −0.672220 0.740351i \(-0.734659\pi\)
−0.672220 + 0.740351i \(0.734659\pi\)
\(840\) 0 0
\(841\) 8.09967 0.279299
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 31.6531 1.08890
\(846\) 0 0
\(847\) −27.8279 −0.956178
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.90033 −0.0994221
\(852\) 0 0
\(853\) 15.5309 0.531768 0.265884 0.964005i \(-0.414336\pi\)
0.265884 + 0.964005i \(0.414336\pi\)
\(854\) 0 0
\(855\) −3.94027 −0.134754
\(856\) 0 0
\(857\) −44.1993 −1.50982 −0.754910 0.655828i \(-0.772320\pi\)
−0.754910 + 0.655828i \(0.772320\pi\)
\(858\) 0 0
\(859\) −12.4743 −0.425616 −0.212808 0.977094i \(-0.568261\pi\)
−0.212808 + 0.977094i \(0.568261\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −27.9430 −0.951190 −0.475595 0.879664i \(-0.657767\pi\)
−0.475595 + 0.879664i \(0.657767\pi\)
\(864\) 0 0
\(865\) 11.4502 0.389317
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.78523 −0.331941
\(870\) 0 0
\(871\) 52.0766 1.76455
\(872\) 0 0
\(873\) −4.35050 −0.147242
\(874\) 0 0
\(875\) −51.9244 −1.75537
\(876\) 0 0
\(877\) −38.3353 −1.29449 −0.647245 0.762282i \(-0.724079\pi\)
−0.647245 + 0.762282i \(0.724079\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −20.8248 −0.701604 −0.350802 0.936450i \(-0.614091\pi\)
−0.350802 + 0.936450i \(0.614091\pi\)
\(882\) 0 0
\(883\) −51.3746 −1.72889 −0.864446 0.502725i \(-0.832331\pi\)
−0.864446 + 0.502725i \(0.832331\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.55505 0.320827 0.160414 0.987050i \(-0.448717\pi\)
0.160414 + 0.987050i \(0.448717\pi\)
\(888\) 0 0
\(889\) −83.2990 −2.79376
\(890\) 0 0
\(891\) 20.4743 0.685913
\(892\) 0 0
\(893\) 4.77753 0.159874
\(894\) 0 0
\(895\) 11.9517 0.399502
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) 44.4262 1.48005
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.54983 −0.151242
\(906\) 0 0
\(907\) −21.6495 −0.718860 −0.359430 0.933172i \(-0.617029\pi\)
−0.359430 + 0.933172i \(0.617029\pi\)
\(908\) 0 0
\(909\) −41.5692 −1.37876
\(910\) 0 0
\(911\) 44.4262 1.47191 0.735954 0.677032i \(-0.236734\pi\)
0.735954 + 0.677032i \(0.236734\pi\)
\(912\) 0 0
\(913\) −29.8007 −0.986258
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.8685 −0.358909
\(918\) 0 0
\(919\) −34.7561 −1.14650 −0.573249 0.819381i \(-0.694317\pi\)
−0.573249 + 0.819381i \(0.694317\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 90.1993 2.96895
\(924\) 0 0
\(925\) 2.74194 0.0901543
\(926\) 0 0
\(927\) 7.88054 0.258831
\(928\) 0 0
\(929\) 23.0997 0.757876 0.378938 0.925422i \(-0.376289\pi\)
0.378938 + 0.925422i \(0.376289\pi\)
\(930\) 0 0
\(931\) 15.8248 0.518635
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.7732 −0.417727
\(936\) 0 0
\(937\) −37.9244 −1.23894 −0.619468 0.785022i \(-0.712652\pi\)
−0.619468 + 0.785022i \(0.712652\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.1769 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(942\) 0 0
\(943\) 36.5457 1.19009
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.1993 0.981347 0.490673 0.871344i \(-0.336751\pi\)
0.490673 + 0.871344i \(0.336751\pi\)
\(948\) 0 0
\(949\) −26.0383 −0.845239
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −45.2990 −1.46738 −0.733689 0.679485i \(-0.762203\pi\)
−0.733689 + 0.679485i \(0.762203\pi\)
\(954\) 0 0
\(955\) 13.1752 0.426341
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 96.8639 3.12790
\(960\) 0 0
\(961\) −24.0997 −0.777409
\(962\) 0 0
\(963\) 49.6495 1.59993
\(964\) 0 0
\(965\) −25.0860 −0.807546
\(966\) 0 0
\(967\) 20.6695 0.664687 0.332344 0.943158i \(-0.392161\pi\)
0.332344 + 0.943158i \(0.392161\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) −49.0887 −1.57371
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.1993 1.03015 0.515074 0.857146i \(-0.327764\pi\)
0.515074 + 0.857146i \(0.327764\pi\)
\(978\) 0 0
\(979\) 22.7492 0.727067
\(980\) 0 0
\(981\) 10.3923 0.331801
\(982\) 0 0
\(983\) 29.6175 0.944651 0.472326 0.881424i \(-0.343415\pi\)
0.472326 + 0.881424i \(0.343415\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −35.5934 −1.13180
\(990\) 0 0
\(991\) 40.1249 1.27461 0.637305 0.770612i \(-0.280049\pi\)
0.637305 + 0.770612i \(0.280049\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −24.4743 −0.775886
\(996\) 0 0
\(997\) −11.5906 −0.367079 −0.183540 0.983012i \(-0.558756\pi\)
−0.183540 + 0.983012i \(0.558756\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4864.2.a.bj.1.3 4
4.3 odd 2 4864.2.a.bk.1.3 4
8.3 odd 2 inner 4864.2.a.bj.1.2 4
8.5 even 2 4864.2.a.bk.1.2 4
16.3 odd 4 1216.2.c.i.609.4 yes 8
16.5 even 4 1216.2.c.i.609.5 yes 8
16.11 odd 4 1216.2.c.i.609.6 yes 8
16.13 even 4 1216.2.c.i.609.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.i.609.3 8 16.13 even 4
1216.2.c.i.609.4 yes 8 16.3 odd 4
1216.2.c.i.609.5 yes 8 16.5 even 4
1216.2.c.i.609.6 yes 8 16.11 odd 4
4864.2.a.bj.1.2 4 8.3 odd 2 inner
4864.2.a.bj.1.3 4 1.1 even 1 trivial
4864.2.a.bk.1.2 4 8.5 even 2
4864.2.a.bk.1.3 4 4.3 odd 2