Properties

Label 4864.2.a.bi.1.1
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: \(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.792287\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.37228 q^{3} -2.52434 q^{5} -3.31662 q^{7} +8.37228 q^{9} +O(q^{10})\) \(q-3.37228 q^{3} -2.52434 q^{5} -3.31662 q^{7} +8.37228 q^{9} -2.37228 q^{11} +5.84096 q^{13} +8.51278 q^{15} +5.00000 q^{17} +1.00000 q^{19} +11.1846 q^{21} -0.792287 q^{23} +1.37228 q^{25} -18.1168 q^{27} -2.67181 q^{29} +3.46410 q^{31} +8.00000 q^{33} +8.37228 q^{35} -10.0974 q^{37} -19.6974 q^{39} +3.62772 q^{43} -21.1345 q^{45} -0.644810 q^{47} +4.00000 q^{49} -16.8614 q^{51} -6.13592 q^{53} +5.98844 q^{55} -3.37228 q^{57} -1.37228 q^{59} -14.5012 q^{61} -27.7677 q^{63} -14.7446 q^{65} +2.62772 q^{67} +2.67181 q^{69} +6.63325 q^{71} -8.48913 q^{73} -4.62772 q^{75} +7.86797 q^{77} -0.294954 q^{79} +35.9783 q^{81} -8.00000 q^{83} -12.6217 q^{85} +9.01011 q^{87} +15.4891 q^{89} -19.3723 q^{91} -11.6819 q^{93} -2.52434 q^{95} -2.74456 q^{97} -19.8614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 22q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 22q^{9} + 2q^{11} + 20q^{17} + 4q^{19} - 6q^{25} - 38q^{27} + 32q^{33} + 22q^{35} + 26q^{43} + 16q^{49} - 10q^{51} - 2q^{57} + 6q^{59} - 36q^{65} + 22q^{67} + 12q^{73} - 30q^{75} + 52q^{81} - 32q^{83} + 16q^{89} - 66q^{91} + 12q^{97} - 22q^{99} + O(q^{100}) \)

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
\(3\) −3.37228 −1.94699 −0.973494 0.228714i \(-0.926548\pi\)
−0.973494 + 0.228714i \(0.926548\pi\)
\(4\) 0 0
\(5\) −2.52434 −1.12892 −0.564459 0.825461i \(-0.690915\pi\)
−0.564459 + 0.825461i \(0.690915\pi\)
\(6\) 0 0
\(7\) −3.31662 −1.25357 −0.626783 0.779194i \(-0.715629\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(8\) 0 0
\(9\) 8.37228 2.79076
\(10\) 0 0
\(11\) −2.37228 −0.715270 −0.357635 0.933862i \(-0.616417\pi\)
−0.357635 + 0.933862i \(0.616417\pi\)
\(12\) 0 0
\(13\) 5.84096 1.61999 0.809996 0.586436i \(-0.199469\pi\)
0.809996 + 0.586436i \(0.199469\pi\)
\(14\) 0 0
\(15\) 8.51278 2.19799
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 11.1846 2.44068
\(22\) 0 0
\(23\) −0.792287 −0.165203 −0.0826016 0.996583i \(-0.526323\pi\)
−0.0826016 + 0.996583i \(0.526323\pi\)
\(24\) 0 0
\(25\) 1.37228 0.274456
\(26\) 0 0
\(27\) −18.1168 −3.48659
\(28\) 0 0
\(29\) −2.67181 −0.496144 −0.248072 0.968742i \(-0.579797\pi\)
−0.248072 + 0.968742i \(0.579797\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 8.37228 1.41517
\(36\) 0 0
\(37\) −10.0974 −1.65999 −0.829997 0.557768i \(-0.811658\pi\)
−0.829997 + 0.557768i \(0.811658\pi\)
\(38\) 0 0
\(39\) −19.6974 −3.15410
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 3.62772 0.553222 0.276611 0.960982i \(-0.410789\pi\)
0.276611 + 0.960982i \(0.410789\pi\)
\(44\) 0 0
\(45\) −21.1345 −3.15054
\(46\) 0 0
\(47\) −0.644810 −0.0940552 −0.0470276 0.998894i \(-0.514975\pi\)
−0.0470276 + 0.998894i \(0.514975\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) −16.8614 −2.36107
\(52\) 0 0
\(53\) −6.13592 −0.842833 −0.421416 0.906867i \(-0.638467\pi\)
−0.421416 + 0.906867i \(0.638467\pi\)
\(54\) 0 0
\(55\) 5.98844 0.807481
\(56\) 0 0
\(57\) −3.37228 −0.446670
\(58\) 0 0
\(59\) −1.37228 −0.178656 −0.0893279 0.996002i \(-0.528472\pi\)
−0.0893279 + 0.996002i \(0.528472\pi\)
\(60\) 0 0
\(61\) −14.5012 −1.85669 −0.928345 0.371719i \(-0.878768\pi\)
−0.928345 + 0.371719i \(0.878768\pi\)
\(62\) 0 0
\(63\) −27.7677 −3.49840
\(64\) 0 0
\(65\) −14.7446 −1.82884
\(66\) 0 0
\(67\) 2.62772 0.321027 0.160513 0.987034i \(-0.448685\pi\)
0.160513 + 0.987034i \(0.448685\pi\)
\(68\) 0 0
\(69\) 2.67181 0.321649
\(70\) 0 0
\(71\) 6.63325 0.787222 0.393611 0.919277i \(-0.371226\pi\)
0.393611 + 0.919277i \(0.371226\pi\)
\(72\) 0 0
\(73\) −8.48913 −0.993577 −0.496788 0.867872i \(-0.665488\pi\)
−0.496788 + 0.867872i \(0.665488\pi\)
\(74\) 0 0
\(75\) −4.62772 −0.534363
\(76\) 0 0
\(77\) 7.86797 0.896638
\(78\) 0 0
\(79\) −0.294954 −0.0331849 −0.0165924 0.999862i \(-0.505282\pi\)
−0.0165924 + 0.999862i \(0.505282\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) −12.6217 −1.36901
\(86\) 0 0
\(87\) 9.01011 0.965985
\(88\) 0 0
\(89\) 15.4891 1.64184 0.820922 0.571040i \(-0.193460\pi\)
0.820922 + 0.571040i \(0.193460\pi\)
\(90\) 0 0
\(91\) −19.3723 −2.03077
\(92\) 0 0
\(93\) −11.6819 −1.21136
\(94\) 0 0
\(95\) −2.52434 −0.258992
\(96\) 0 0
\(97\) −2.74456 −0.278668 −0.139334 0.990245i \(-0.544496\pi\)
−0.139334 + 0.990245i \(0.544496\pi\)
\(98\) 0 0
\(99\) −19.8614 −1.99615
\(100\) 0 0
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) −11.9769 −1.18012 −0.590058 0.807361i \(-0.700895\pi\)
−0.590058 + 0.807361i \(0.700895\pi\)
\(104\) 0 0
\(105\) −28.2337 −2.75533
\(106\) 0 0
\(107\) 13.3723 1.29275 0.646374 0.763021i \(-0.276285\pi\)
0.646374 + 0.763021i \(0.276285\pi\)
\(108\) 0 0
\(109\) 3.96143 0.379437 0.189718 0.981839i \(-0.439243\pi\)
0.189718 + 0.981839i \(0.439243\pi\)
\(110\) 0 0
\(111\) 34.0511 3.23199
\(112\) 0 0
\(113\) −14.7446 −1.38705 −0.693526 0.720432i \(-0.743944\pi\)
−0.693526 + 0.720432i \(0.743944\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 48.9022 4.52101
\(118\) 0 0
\(119\) −16.5831 −1.52017
\(120\) 0 0
\(121\) −5.37228 −0.488389
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.15759 0.819080
\(126\) 0 0
\(127\) 12.2718 1.08895 0.544475 0.838777i \(-0.316729\pi\)
0.544475 + 0.838777i \(0.316729\pi\)
\(128\) 0 0
\(129\) −12.2337 −1.07712
\(130\) 0 0
\(131\) −4.37228 −0.382008 −0.191004 0.981589i \(-0.561174\pi\)
−0.191004 + 0.981589i \(0.561174\pi\)
\(132\) 0 0
\(133\) −3.31662 −0.287588
\(134\) 0 0
\(135\) 45.7330 3.93607
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) 3.62772 0.307699 0.153850 0.988094i \(-0.450833\pi\)
0.153850 + 0.988094i \(0.450833\pi\)
\(140\) 0 0
\(141\) 2.17448 0.183124
\(142\) 0 0
\(143\) −13.8564 −1.15873
\(144\) 0 0
\(145\) 6.74456 0.560105
\(146\) 0 0
\(147\) −13.4891 −1.11256
\(148\) 0 0
\(149\) −4.69882 −0.384942 −0.192471 0.981303i \(-0.561650\pi\)
−0.192471 + 0.981303i \(0.561650\pi\)
\(150\) 0 0
\(151\) −15.7359 −1.28057 −0.640286 0.768137i \(-0.721184\pi\)
−0.640286 + 0.768137i \(0.721184\pi\)
\(152\) 0 0
\(153\) 41.8614 3.38429
\(154\) 0 0
\(155\) −8.74456 −0.702380
\(156\) 0 0
\(157\) −18.6101 −1.48525 −0.742625 0.669708i \(-0.766419\pi\)
−0.742625 + 0.669708i \(0.766419\pi\)
\(158\) 0 0
\(159\) 20.6920 1.64099
\(160\) 0 0
\(161\) 2.62772 0.207093
\(162\) 0 0
\(163\) 10.7446 0.841579 0.420790 0.907158i \(-0.361753\pi\)
0.420790 + 0.907158i \(0.361753\pi\)
\(164\) 0 0
\(165\) −20.1947 −1.57216
\(166\) 0 0
\(167\) −6.33830 −0.490472 −0.245236 0.969463i \(-0.578865\pi\)
−0.245236 + 0.969463i \(0.578865\pi\)
\(168\) 0 0
\(169\) 21.1168 1.62437
\(170\) 0 0
\(171\) 8.37228 0.640244
\(172\) 0 0
\(173\) 15.1460 1.15153 0.575766 0.817615i \(-0.304704\pi\)
0.575766 + 0.817615i \(0.304704\pi\)
\(174\) 0 0
\(175\) −4.55134 −0.344049
\(176\) 0 0
\(177\) 4.62772 0.347841
\(178\) 0 0
\(179\) 22.9783 1.71748 0.858738 0.512416i \(-0.171249\pi\)
0.858738 + 0.512416i \(0.171249\pi\)
\(180\) 0 0
\(181\) −8.21782 −0.610826 −0.305413 0.952220i \(-0.598795\pi\)
−0.305413 + 0.952220i \(0.598795\pi\)
\(182\) 0 0
\(183\) 48.9022 3.61495
\(184\) 0 0
\(185\) 25.4891 1.87400
\(186\) 0 0
\(187\) −11.8614 −0.867392
\(188\) 0 0
\(189\) 60.0868 4.37067
\(190\) 0 0
\(191\) −7.07568 −0.511978 −0.255989 0.966680i \(-0.582401\pi\)
−0.255989 + 0.966680i \(0.582401\pi\)
\(192\) 0 0
\(193\) 3.25544 0.234332 0.117166 0.993112i \(-0.462619\pi\)
0.117166 + 0.993112i \(0.462619\pi\)
\(194\) 0 0
\(195\) 49.7228 3.56072
\(196\) 0 0
\(197\) 6.33830 0.451585 0.225792 0.974175i \(-0.427503\pi\)
0.225792 + 0.974175i \(0.427503\pi\)
\(198\) 0 0
\(199\) 0.147477 0.0104544 0.00522718 0.999986i \(-0.498336\pi\)
0.00522718 + 0.999986i \(0.498336\pi\)
\(200\) 0 0
\(201\) −8.86141 −0.625035
\(202\) 0 0
\(203\) 8.86141 0.621949
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.63325 −0.461043
\(208\) 0 0
\(209\) −2.37228 −0.164094
\(210\) 0 0
\(211\) −2.62772 −0.180900 −0.0904498 0.995901i \(-0.528830\pi\)
−0.0904498 + 0.995901i \(0.528830\pi\)
\(212\) 0 0
\(213\) −22.3692 −1.53271
\(214\) 0 0
\(215\) −9.15759 −0.624542
\(216\) 0 0
\(217\) −11.4891 −0.779933
\(218\) 0 0
\(219\) 28.6277 1.93448
\(220\) 0 0
\(221\) 29.2048 1.96453
\(222\) 0 0
\(223\) −6.92820 −0.463947 −0.231973 0.972722i \(-0.574518\pi\)
−0.231973 + 0.972722i \(0.574518\pi\)
\(224\) 0 0
\(225\) 11.4891 0.765942
\(226\) 0 0
\(227\) −14.1168 −0.936968 −0.468484 0.883472i \(-0.655200\pi\)
−0.468484 + 0.883472i \(0.655200\pi\)
\(228\) 0 0
\(229\) −12.6217 −0.834065 −0.417032 0.908892i \(-0.636930\pi\)
−0.417032 + 0.908892i \(0.636930\pi\)
\(230\) 0 0
\(231\) −26.5330 −1.74574
\(232\) 0 0
\(233\) 10.8832 0.712979 0.356490 0.934299i \(-0.383974\pi\)
0.356490 + 0.934299i \(0.383974\pi\)
\(234\) 0 0
\(235\) 1.62772 0.106181
\(236\) 0 0
\(237\) 0.994667 0.0646105
\(238\) 0 0
\(239\) −17.4680 −1.12991 −0.564955 0.825122i \(-0.691106\pi\)
−0.564955 + 0.825122i \(0.691106\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) −66.9783 −4.29666
\(244\) 0 0
\(245\) −10.0974 −0.645096
\(246\) 0 0
\(247\) 5.84096 0.371652
\(248\) 0 0
\(249\) 26.9783 1.70968
\(250\) 0 0
\(251\) 17.8614 1.12740 0.563701 0.825979i \(-0.309377\pi\)
0.563701 + 0.825979i \(0.309377\pi\)
\(252\) 0 0
\(253\) 1.87953 0.118165
\(254\) 0 0
\(255\) 42.5639 2.66545
\(256\) 0 0
\(257\) −16.2337 −1.01263 −0.506315 0.862349i \(-0.668993\pi\)
−0.506315 + 0.862349i \(0.668993\pi\)
\(258\) 0 0
\(259\) 33.4891 2.08091
\(260\) 0 0
\(261\) −22.3692 −1.38462
\(262\) 0 0
\(263\) −18.9600 −1.16912 −0.584561 0.811349i \(-0.698733\pi\)
−0.584561 + 0.811349i \(0.698733\pi\)
\(264\) 0 0
\(265\) 15.4891 0.951489
\(266\) 0 0
\(267\) −52.2337 −3.19665
\(268\) 0 0
\(269\) 1.87953 0.114597 0.0572984 0.998357i \(-0.481751\pi\)
0.0572984 + 0.998357i \(0.481751\pi\)
\(270\) 0 0
\(271\) 2.37686 0.144384 0.0721920 0.997391i \(-0.477001\pi\)
0.0721920 + 0.997391i \(0.477001\pi\)
\(272\) 0 0
\(273\) 65.3288 3.95388
\(274\) 0 0
\(275\) −3.25544 −0.196310
\(276\) 0 0
\(277\) 16.3807 0.984224 0.492112 0.870532i \(-0.336225\pi\)
0.492112 + 0.870532i \(0.336225\pi\)
\(278\) 0 0
\(279\) 29.0024 1.73633
\(280\) 0 0
\(281\) −6.23369 −0.371871 −0.185935 0.982562i \(-0.559531\pi\)
−0.185935 + 0.982562i \(0.559531\pi\)
\(282\) 0 0
\(283\) −17.6277 −1.04786 −0.523930 0.851762i \(-0.675534\pi\)
−0.523930 + 0.851762i \(0.675534\pi\)
\(284\) 0 0
\(285\) 8.51278 0.504253
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 9.25544 0.542563
\(292\) 0 0
\(293\) −4.84630 −0.283124 −0.141562 0.989929i \(-0.545212\pi\)
−0.141562 + 0.989929i \(0.545212\pi\)
\(294\) 0 0
\(295\) 3.46410 0.201688
\(296\) 0 0
\(297\) 42.9783 2.49385
\(298\) 0 0
\(299\) −4.62772 −0.267628
\(300\) 0 0
\(301\) −12.0318 −0.693500
\(302\) 0 0
\(303\) −46.7277 −2.68444
\(304\) 0 0
\(305\) 36.6060 2.09605
\(306\) 0 0
\(307\) −15.2554 −0.870674 −0.435337 0.900268i \(-0.643371\pi\)
−0.435337 + 0.900268i \(0.643371\pi\)
\(308\) 0 0
\(309\) 40.3894 2.29767
\(310\) 0 0
\(311\) −23.2164 −1.31648 −0.658240 0.752808i \(-0.728699\pi\)
−0.658240 + 0.752808i \(0.728699\pi\)
\(312\) 0 0
\(313\) −3.88316 −0.219489 −0.109744 0.993960i \(-0.535003\pi\)
−0.109744 + 0.993960i \(0.535003\pi\)
\(314\) 0 0
\(315\) 70.0951 3.94941
\(316\) 0 0
\(317\) 21.5769 1.21188 0.605940 0.795511i \(-0.292797\pi\)
0.605940 + 0.795511i \(0.292797\pi\)
\(318\) 0 0
\(319\) 6.33830 0.354876
\(320\) 0 0
\(321\) −45.0951 −2.51696
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) 8.01544 0.444617
\(326\) 0 0
\(327\) −13.3591 −0.738758
\(328\) 0 0
\(329\) 2.13859 0.117904
\(330\) 0 0
\(331\) 10.8614 0.596997 0.298498 0.954410i \(-0.403514\pi\)
0.298498 + 0.954410i \(0.403514\pi\)
\(332\) 0 0
\(333\) −84.5379 −4.63265
\(334\) 0 0
\(335\) −6.63325 −0.362413
\(336\) 0 0
\(337\) −13.2554 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(338\) 0 0
\(339\) 49.7228 2.70057
\(340\) 0 0
\(341\) −8.21782 −0.445020
\(342\) 0 0
\(343\) 9.94987 0.537243
\(344\) 0 0
\(345\) −6.74456 −0.363115
\(346\) 0 0
\(347\) 37.1168 1.99254 0.996268 0.0863104i \(-0.0275077\pi\)
0.996268 + 0.0863104i \(0.0275077\pi\)
\(348\) 0 0
\(349\) 13.2116 0.707201 0.353600 0.935397i \(-0.384957\pi\)
0.353600 + 0.935397i \(0.384957\pi\)
\(350\) 0 0
\(351\) −105.820 −5.64824
\(352\) 0 0
\(353\) 3.88316 0.206680 0.103340 0.994646i \(-0.467047\pi\)
0.103340 + 0.994646i \(0.467047\pi\)
\(354\) 0 0
\(355\) −16.7446 −0.888709
\(356\) 0 0
\(357\) 55.9230 2.95976
\(358\) 0 0
\(359\) 16.5831 0.875224 0.437612 0.899164i \(-0.355824\pi\)
0.437612 + 0.899164i \(0.355824\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 18.1168 0.950888
\(364\) 0 0
\(365\) 21.4294 1.12167
\(366\) 0 0
\(367\) 4.05401 0.211618 0.105809 0.994386i \(-0.466257\pi\)
0.105809 + 0.994386i \(0.466257\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.3505 1.05655
\(372\) 0 0
\(373\) −14.9436 −0.773753 −0.386876 0.922132i \(-0.626446\pi\)
−0.386876 + 0.922132i \(0.626446\pi\)
\(374\) 0 0
\(375\) −30.8820 −1.59474
\(376\) 0 0
\(377\) −15.6060 −0.803748
\(378\) 0 0
\(379\) 36.3505 1.86720 0.933601 0.358315i \(-0.116649\pi\)
0.933601 + 0.358315i \(0.116649\pi\)
\(380\) 0 0
\(381\) −41.3841 −2.12017
\(382\) 0 0
\(383\) −25.2434 −1.28988 −0.644938 0.764235i \(-0.723117\pi\)
−0.644938 + 0.764235i \(0.723117\pi\)
\(384\) 0 0
\(385\) −19.8614 −1.01223
\(386\) 0 0
\(387\) 30.3723 1.54391
\(388\) 0 0
\(389\) −24.3036 −1.23224 −0.616121 0.787651i \(-0.711297\pi\)
−0.616121 + 0.787651i \(0.711297\pi\)
\(390\) 0 0
\(391\) −3.96143 −0.200338
\(392\) 0 0
\(393\) 14.7446 0.743765
\(394\) 0 0
\(395\) 0.744563 0.0374630
\(396\) 0 0
\(397\) 0.644810 0.0323621 0.0161810 0.999869i \(-0.494849\pi\)
0.0161810 + 0.999869i \(0.494849\pi\)
\(398\) 0 0
\(399\) 11.1846 0.559930
\(400\) 0 0
\(401\) 35.7228 1.78391 0.891956 0.452122i \(-0.149333\pi\)
0.891956 + 0.452122i \(0.149333\pi\)
\(402\) 0 0
\(403\) 20.2337 1.00791
\(404\) 0 0
\(405\) −90.8213 −4.51294
\(406\) 0 0
\(407\) 23.9538 1.18734
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 16.8614 0.831712
\(412\) 0 0
\(413\) 4.55134 0.223957
\(414\) 0 0
\(415\) 20.1947 0.991319
\(416\) 0 0
\(417\) −12.2337 −0.599086
\(418\) 0 0
\(419\) 0.233688 0.0114164 0.00570820 0.999984i \(-0.498183\pi\)
0.00570820 + 0.999984i \(0.498183\pi\)
\(420\) 0 0
\(421\) 20.3971 0.994093 0.497046 0.867724i \(-0.334418\pi\)
0.497046 + 0.867724i \(0.334418\pi\)
\(422\) 0 0
\(423\) −5.39853 −0.262486
\(424\) 0 0
\(425\) 6.86141 0.332827
\(426\) 0 0
\(427\) 48.0951 2.32748
\(428\) 0 0
\(429\) 46.7277 2.25603
\(430\) 0 0
\(431\) −21.7793 −1.04907 −0.524535 0.851389i \(-0.675761\pi\)
−0.524535 + 0.851389i \(0.675761\pi\)
\(432\) 0 0
\(433\) −1.48913 −0.0715628 −0.0357814 0.999360i \(-0.511392\pi\)
−0.0357814 + 0.999360i \(0.511392\pi\)
\(434\) 0 0
\(435\) −22.7446 −1.09052
\(436\) 0 0
\(437\) −0.792287 −0.0379002
\(438\) 0 0
\(439\) 6.92820 0.330665 0.165333 0.986238i \(-0.447130\pi\)
0.165333 + 0.986238i \(0.447130\pi\)
\(440\) 0 0
\(441\) 33.4891 1.59472
\(442\) 0 0
\(443\) 41.3505 1.96462 0.982312 0.187254i \(-0.0599587\pi\)
0.982312 + 0.187254i \(0.0599587\pi\)
\(444\) 0 0
\(445\) −39.0998 −1.85351
\(446\) 0 0
\(447\) 15.8457 0.749478
\(448\) 0 0
\(449\) 37.4891 1.76922 0.884611 0.466330i \(-0.154424\pi\)
0.884611 + 0.466330i \(0.154424\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 53.0660 2.49326
\(454\) 0 0
\(455\) 48.9022 2.29257
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) −90.5842 −4.22811
\(460\) 0 0
\(461\) 7.86797 0.366448 0.183224 0.983071i \(-0.441347\pi\)
0.183224 + 0.983071i \(0.441347\pi\)
\(462\) 0 0
\(463\) 31.5268 1.46517 0.732587 0.680674i \(-0.238313\pi\)
0.732587 + 0.680674i \(0.238313\pi\)
\(464\) 0 0
\(465\) 29.4891 1.36753
\(466\) 0 0
\(467\) 10.8832 0.503612 0.251806 0.967778i \(-0.418975\pi\)
0.251806 + 0.967778i \(0.418975\pi\)
\(468\) 0 0
\(469\) −8.71516 −0.402429
\(470\) 0 0
\(471\) 62.7586 2.89176
\(472\) 0 0
\(473\) −8.60597 −0.395703
\(474\) 0 0
\(475\) 1.37228 0.0629646
\(476\) 0 0
\(477\) −51.3716 −2.35214
\(478\) 0 0
\(479\) 27.4179 1.25275 0.626377 0.779520i \(-0.284537\pi\)
0.626377 + 0.779520i \(0.284537\pi\)
\(480\) 0 0
\(481\) −58.9783 −2.68918
\(482\) 0 0
\(483\) −8.86141 −0.403208
\(484\) 0 0
\(485\) 6.92820 0.314594
\(486\) 0 0
\(487\) −22.6641 −1.02701 −0.513505 0.858087i \(-0.671653\pi\)
−0.513505 + 0.858087i \(0.671653\pi\)
\(488\) 0 0
\(489\) −36.2337 −1.63854
\(490\) 0 0
\(491\) 30.9783 1.39803 0.699014 0.715108i \(-0.253622\pi\)
0.699014 + 0.715108i \(0.253622\pi\)
\(492\) 0 0
\(493\) −13.3591 −0.601662
\(494\) 0 0
\(495\) 50.1369 2.25349
\(496\) 0 0
\(497\) −22.0000 −0.986835
\(498\) 0 0
\(499\) −4.37228 −0.195730 −0.0978651 0.995200i \(-0.531201\pi\)
−0.0978651 + 0.995200i \(0.531201\pi\)
\(500\) 0 0
\(501\) 21.3745 0.954943
\(502\) 0 0
\(503\) 11.4795 0.511848 0.255924 0.966697i \(-0.417620\pi\)
0.255924 + 0.966697i \(0.417620\pi\)
\(504\) 0 0
\(505\) −34.9783 −1.55651
\(506\) 0 0
\(507\) −71.2119 −3.16263
\(508\) 0 0
\(509\) −37.8102 −1.67591 −0.837953 0.545742i \(-0.816248\pi\)
−0.837953 + 0.545742i \(0.816248\pi\)
\(510\) 0 0
\(511\) 28.1552 1.24551
\(512\) 0 0
\(513\) −18.1168 −0.799878
\(514\) 0 0
\(515\) 30.2337 1.33226
\(516\) 0 0
\(517\) 1.52967 0.0672749
\(518\) 0 0
\(519\) −51.0767 −2.24202
\(520\) 0 0
\(521\) 5.76631 0.252627 0.126313 0.991990i \(-0.459686\pi\)
0.126313 + 0.991990i \(0.459686\pi\)
\(522\) 0 0
\(523\) −25.3723 −1.10945 −0.554726 0.832033i \(-0.687177\pi\)
−0.554726 + 0.832033i \(0.687177\pi\)
\(524\) 0 0
\(525\) 15.3484 0.669859
\(526\) 0 0
\(527\) 17.3205 0.754493
\(528\) 0 0
\(529\) −22.3723 −0.972708
\(530\) 0 0
\(531\) −11.4891 −0.498586
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −33.7562 −1.45941
\(536\) 0 0
\(537\) −77.4891 −3.34390
\(538\) 0 0
\(539\) −9.48913 −0.408726
\(540\) 0 0
\(541\) −23.6039 −1.01481 −0.507405 0.861707i \(-0.669395\pi\)
−0.507405 + 0.861707i \(0.669395\pi\)
\(542\) 0 0
\(543\) 27.7128 1.18927
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −2.51087 −0.107357 −0.0536786 0.998558i \(-0.517095\pi\)
−0.0536786 + 0.998558i \(0.517095\pi\)
\(548\) 0 0
\(549\) −121.408 −5.18158
\(550\) 0 0
\(551\) −2.67181 −0.113823
\(552\) 0 0
\(553\) 0.978251 0.0415994
\(554\) 0 0
\(555\) −85.9565 −3.64865
\(556\) 0 0
\(557\) 25.1885 1.06727 0.533635 0.845715i \(-0.320826\pi\)
0.533635 + 0.845715i \(0.320826\pi\)
\(558\) 0 0
\(559\) 21.1894 0.896215
\(560\) 0 0
\(561\) 40.0000 1.68880
\(562\) 0 0
\(563\) −13.4891 −0.568499 −0.284249 0.958750i \(-0.591744\pi\)
−0.284249 + 0.958750i \(0.591744\pi\)
\(564\) 0 0
\(565\) 37.2203 1.56587
\(566\) 0 0
\(567\) −119.326 −5.01124
\(568\) 0 0
\(569\) 26.9783 1.13099 0.565494 0.824753i \(-0.308686\pi\)
0.565494 + 0.824753i \(0.308686\pi\)
\(570\) 0 0
\(571\) 42.9783 1.79858 0.899292 0.437349i \(-0.144083\pi\)
0.899292 + 0.437349i \(0.144083\pi\)
\(572\) 0 0
\(573\) 23.8612 0.996815
\(574\) 0 0
\(575\) −1.08724 −0.0453411
\(576\) 0 0
\(577\) 33.4674 1.39327 0.696633 0.717428i \(-0.254681\pi\)
0.696633 + 0.717428i \(0.254681\pi\)
\(578\) 0 0
\(579\) −10.9783 −0.456241
\(580\) 0 0
\(581\) 26.5330 1.10077
\(582\) 0 0
\(583\) 14.5561 0.602853
\(584\) 0 0
\(585\) −123.446 −5.10385
\(586\) 0 0
\(587\) −31.8614 −1.31506 −0.657530 0.753428i \(-0.728399\pi\)
−0.657530 + 0.753428i \(0.728399\pi\)
\(588\) 0 0
\(589\) 3.46410 0.142736
\(590\) 0 0
\(591\) −21.3745 −0.879230
\(592\) 0 0
\(593\) 4.97825 0.204432 0.102216 0.994762i \(-0.467407\pi\)
0.102216 + 0.994762i \(0.467407\pi\)
\(594\) 0 0
\(595\) 41.8614 1.71615
\(596\) 0 0
\(597\) −0.497333 −0.0203545
\(598\) 0 0
\(599\) 39.6897 1.62168 0.810838 0.585270i \(-0.199012\pi\)
0.810838 + 0.585270i \(0.199012\pi\)
\(600\) 0 0
\(601\) 22.4674 0.916463 0.458232 0.888833i \(-0.348483\pi\)
0.458232 + 0.888833i \(0.348483\pi\)
\(602\) 0 0
\(603\) 22.0000 0.895909
\(604\) 0 0
\(605\) 13.5615 0.551351
\(606\) 0 0
\(607\) −38.8048 −1.57504 −0.787520 0.616289i \(-0.788635\pi\)
−0.787520 + 0.616289i \(0.788635\pi\)
\(608\) 0 0
\(609\) −29.8832 −1.21093
\(610\) 0 0
\(611\) −3.76631 −0.152369
\(612\) 0 0
\(613\) 18.5552 0.749439 0.374719 0.927138i \(-0.377739\pi\)
0.374719 + 0.927138i \(0.377739\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.8397 1.64414 0.822071 0.569384i \(-0.192818\pi\)
0.822071 + 0.569384i \(0.192818\pi\)
\(618\) 0 0
\(619\) −16.2337 −0.652487 −0.326244 0.945286i \(-0.605783\pi\)
−0.326244 + 0.945286i \(0.605783\pi\)
\(620\) 0 0
\(621\) 14.3537 0.575996
\(622\) 0 0
\(623\) −51.3716 −2.05816
\(624\) 0 0
\(625\) −29.9783 −1.19913
\(626\) 0 0
\(627\) 8.00000 0.319489
\(628\) 0 0
\(629\) −50.4868 −2.01304
\(630\) 0 0
\(631\) 37.8651 1.50738 0.753692 0.657227i \(-0.228271\pi\)
0.753692 + 0.657227i \(0.228271\pi\)
\(632\) 0 0
\(633\) 8.86141 0.352209
\(634\) 0 0
\(635\) −30.9783 −1.22933
\(636\) 0 0
\(637\) 23.3639 0.925709
\(638\) 0 0
\(639\) 55.5354 2.19695
\(640\) 0 0
\(641\) −10.9783 −0.433615 −0.216807 0.976214i \(-0.569564\pi\)
−0.216807 + 0.976214i \(0.569564\pi\)
\(642\) 0 0
\(643\) 36.3723 1.43438 0.717191 0.696876i \(-0.245427\pi\)
0.717191 + 0.696876i \(0.245427\pi\)
\(644\) 0 0
\(645\) 30.8820 1.21598
\(646\) 0 0
\(647\) 16.5831 0.651950 0.325975 0.945378i \(-0.394307\pi\)
0.325975 + 0.945378i \(0.394307\pi\)
\(648\) 0 0
\(649\) 3.25544 0.127787
\(650\) 0 0
\(651\) 38.7446 1.51852
\(652\) 0 0
\(653\) −35.3956 −1.38514 −0.692569 0.721352i \(-0.743521\pi\)
−0.692569 + 0.721352i \(0.743521\pi\)
\(654\) 0 0
\(655\) 11.0371 0.431256
\(656\) 0 0
\(657\) −71.0733 −2.77284
\(658\) 0 0
\(659\) −46.8614 −1.82546 −0.912731 0.408562i \(-0.866030\pi\)
−0.912731 + 0.408562i \(0.866030\pi\)
\(660\) 0 0
\(661\) −10.5947 −0.412085 −0.206043 0.978543i \(-0.566059\pi\)
−0.206043 + 0.978543i \(0.566059\pi\)
\(662\) 0 0
\(663\) −98.4868 −3.82491
\(664\) 0 0
\(665\) 8.37228 0.324663
\(666\) 0 0
\(667\) 2.11684 0.0819645
\(668\) 0 0
\(669\) 23.3639 0.903299
\(670\) 0 0
\(671\) 34.4010 1.32803
\(672\) 0 0
\(673\) 16.7446 0.645455 0.322728 0.946492i \(-0.395400\pi\)
0.322728 + 0.946492i \(0.395400\pi\)
\(674\) 0 0
\(675\) −24.8614 −0.956916
\(676\) 0 0
\(677\) 5.84096 0.224486 0.112243 0.993681i \(-0.464196\pi\)
0.112243 + 0.993681i \(0.464196\pi\)
\(678\) 0 0
\(679\) 9.10268 0.349329
\(680\) 0 0
\(681\) 47.6060 1.82426
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 12.6217 0.482250
\(686\) 0 0
\(687\) 42.5639 1.62391
\(688\) 0 0
\(689\) −35.8397 −1.36538
\(690\) 0 0
\(691\) 17.8614 0.679480 0.339740 0.940519i \(-0.389661\pi\)
0.339740 + 0.940519i \(0.389661\pi\)
\(692\) 0 0
\(693\) 65.8728 2.50230
\(694\) 0 0
\(695\) −9.15759 −0.347367
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −36.7011 −1.38816
\(700\) 0 0
\(701\) 27.7128 1.04670 0.523349 0.852118i \(-0.324682\pi\)
0.523349 + 0.852118i \(0.324682\pi\)
\(702\) 0 0
\(703\) −10.0974 −0.380829
\(704\) 0 0
\(705\) −5.48913 −0.206732
\(706\) 0 0
\(707\) −45.9565 −1.72837
\(708\) 0 0
\(709\) 45.7330 1.71754 0.858770 0.512361i \(-0.171229\pi\)
0.858770 + 0.512361i \(0.171229\pi\)
\(710\) 0 0
\(711\) −2.46943 −0.0926110
\(712\) 0 0
\(713\) −2.74456 −0.102785
\(714\) 0 0
\(715\) 34.9783 1.30811
\(716\) 0 0
\(717\) 58.9070 2.19992
\(718\) 0 0
\(719\) −40.5369 −1.51177 −0.755885 0.654704i \(-0.772793\pi\)
−0.755885 + 0.654704i \(0.772793\pi\)
\(720\) 0 0
\(721\) 39.7228 1.47935
\(722\) 0 0
\(723\) −26.9783 −1.00333
\(724\) 0 0
\(725\) −3.66648 −0.136170
\(726\) 0 0
\(727\) 14.0039 0.519375 0.259688 0.965693i \(-0.416380\pi\)
0.259688 + 0.965693i \(0.416380\pi\)
\(728\) 0 0
\(729\) 117.935 4.36795
\(730\) 0 0
\(731\) 18.1386 0.670880
\(732\) 0 0
\(733\) −21.1894 −0.782647 −0.391324 0.920253i \(-0.627983\pi\)
−0.391324 + 0.920253i \(0.627983\pi\)
\(734\) 0 0
\(735\) 34.0511 1.25599
\(736\) 0 0
\(737\) −6.23369 −0.229621
\(738\) 0 0
\(739\) −12.6060 −0.463718 −0.231859 0.972749i \(-0.574481\pi\)
−0.231859 + 0.972749i \(0.574481\pi\)
\(740\) 0 0
\(741\) −19.6974 −0.723601
\(742\) 0 0
\(743\) 6.63325 0.243350 0.121675 0.992570i \(-0.461173\pi\)
0.121675 + 0.992570i \(0.461173\pi\)
\(744\) 0 0
\(745\) 11.8614 0.434568
\(746\) 0 0
\(747\) −66.9783 −2.45061
\(748\) 0 0
\(749\) −44.3508 −1.62054
\(750\) 0 0
\(751\) −9.50744 −0.346932 −0.173466 0.984840i \(-0.555497\pi\)
−0.173466 + 0.984840i \(0.555497\pi\)
\(752\) 0 0
\(753\) −60.2337 −2.19504
\(754\) 0 0
\(755\) 39.7228 1.44566
\(756\) 0 0
\(757\) −18.8502 −0.685121 −0.342561 0.939496i \(-0.611294\pi\)
−0.342561 + 0.939496i \(0.611294\pi\)
\(758\) 0 0
\(759\) −6.33830 −0.230066
\(760\) 0 0
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) 0 0
\(763\) −13.1386 −0.475649
\(764\) 0 0
\(765\) −105.672 −3.82059
\(766\) 0 0
\(767\) −8.01544 −0.289421
\(768\) 0 0
\(769\) 37.4674 1.35111 0.675554 0.737310i \(-0.263904\pi\)
0.675554 + 0.737310i \(0.263904\pi\)
\(770\) 0 0
\(771\) 54.7446 1.97158
\(772\) 0 0
\(773\) 24.1561 0.868836 0.434418 0.900711i \(-0.356954\pi\)
0.434418 + 0.900711i \(0.356954\pi\)
\(774\) 0 0
\(775\) 4.75372 0.170759
\(776\) 0 0
\(777\) −112.935 −4.05151
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −15.7359 −0.563076
\(782\) 0 0
\(783\) 48.4048 1.72985
\(784\) 0 0
\(785\) 46.9783 1.67673
\(786\) 0 0
\(787\) −19.6060 −0.698877 −0.349439 0.936959i \(-0.613628\pi\)
−0.349439 + 0.936959i \(0.613628\pi\)
\(788\) 0 0
\(789\) 63.9384 2.27627
\(790\) 0 0
\(791\) 48.9022 1.73876
\(792\) 0 0
\(793\) −84.7011 −3.00782
\(794\) 0 0
\(795\) −52.2337 −1.85254
\(796\) 0 0
\(797\) −14.6487 −0.518883 −0.259442 0.965759i \(-0.583539\pi\)
−0.259442 + 0.965759i \(0.583539\pi\)
\(798\) 0 0
\(799\) −3.22405 −0.114059
\(800\) 0 0
\(801\) 129.679 4.58199
\(802\) 0 0
\(803\) 20.1386 0.710676
\(804\) 0 0
\(805\) −6.63325 −0.233791
\(806\) 0 0
\(807\) −6.33830 −0.223119
\(808\) 0 0
\(809\) 37.9783 1.33524 0.667622 0.744500i \(-0.267312\pi\)
0.667622 + 0.744500i \(0.267312\pi\)
\(810\) 0 0
\(811\) 28.8614 1.01346 0.506731 0.862105i \(-0.330854\pi\)
0.506731 + 0.862105i \(0.330854\pi\)
\(812\) 0 0
\(813\) −8.01544 −0.281114
\(814\) 0 0
\(815\) −27.1229 −0.950074
\(816\) 0 0
\(817\) 3.62772 0.126918
\(818\) 0 0
\(819\) −162.190 −5.66738
\(820\) 0 0
\(821\) −27.0680 −0.944680 −0.472340 0.881416i \(-0.656591\pi\)
−0.472340 + 0.881416i \(0.656591\pi\)
\(822\) 0 0
\(823\) −7.37063 −0.256924 −0.128462 0.991714i \(-0.541004\pi\)
−0.128462 + 0.991714i \(0.541004\pi\)
\(824\) 0 0
\(825\) 10.9783 0.382214
\(826\) 0 0
\(827\) 40.3505 1.40313 0.701563 0.712608i \(-0.252486\pi\)
0.701563 + 0.712608i \(0.252486\pi\)
\(828\) 0 0
\(829\) 3.26172 0.113284 0.0566421 0.998395i \(-0.481961\pi\)
0.0566421 + 0.998395i \(0.481961\pi\)
\(830\) 0 0
\(831\) −55.2405 −1.91627
\(832\) 0 0
\(833\) 20.0000 0.692959
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) −62.7586 −2.16925
\(838\) 0 0
\(839\) 9.80240 0.338416 0.169208 0.985580i \(-0.445879\pi\)
0.169208 + 0.985580i \(0.445879\pi\)
\(840\) 0 0
\(841\) −21.8614 −0.753842
\(842\) 0 0
\(843\) 21.0217 0.724028
\(844\) 0 0
\(845\) −53.3060 −1.83378
\(846\) 0 0
\(847\) 17.8178 0.612228
\(848\) 0 0
\(849\) 59.4456 2.04017
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 40.3894 1.38291 0.691453 0.722421i \(-0.256971\pi\)
0.691453 + 0.722421i \(0.256971\pi\)
\(854\) 0 0
\(855\) −21.1345 −0.722783
\(856\) 0 0
\(857\) −22.2337 −0.759488 −0.379744 0.925092i \(-0.623988\pi\)
−0.379744 + 0.925092i \(0.623988\pi\)
\(858\) 0 0
\(859\) −8.60597 −0.293632 −0.146816 0.989164i \(-0.546903\pi\)
−0.146816 + 0.989164i \(0.546903\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.76439 0.0941009 0.0470504 0.998893i \(-0.485018\pi\)
0.0470504 + 0.998893i \(0.485018\pi\)
\(864\) 0 0
\(865\) −38.2337 −1.29998
\(866\) 0 0
\(867\) −26.9783 −0.916229
\(868\) 0 0
\(869\) 0.699713 0.0237361
\(870\) 0 0
\(871\) 15.3484 0.520061
\(872\) 0 0
\(873\) −22.9783 −0.777696
\(874\) 0 0
\(875\) −30.3723 −1.02677
\(876\) 0 0
\(877\) −1.38219 −0.0466734 −0.0233367 0.999728i \(-0.507429\pi\)
−0.0233367 + 0.999728i \(0.507429\pi\)
\(878\) 0 0
\(879\) 16.3431 0.551238
\(880\) 0 0
\(881\) −4.37228 −0.147306 −0.0736530 0.997284i \(-0.523466\pi\)
−0.0736530 + 0.997284i \(0.523466\pi\)
\(882\) 0 0
\(883\) −18.8832 −0.635469 −0.317734 0.948180i \(-0.602922\pi\)
−0.317734 + 0.948180i \(0.602922\pi\)
\(884\) 0 0
\(885\) −11.6819 −0.392684
\(886\) 0 0
\(887\) −45.0333 −1.51207 −0.756035 0.654531i \(-0.772866\pi\)
−0.756035 + 0.654531i \(0.772866\pi\)
\(888\) 0 0
\(889\) −40.7011 −1.36507
\(890\) 0 0
\(891\) −85.3505 −2.85935
\(892\) 0 0
\(893\) −0.644810 −0.0215777
\(894\) 0 0
\(895\) −58.0049 −1.93889
\(896\) 0 0
\(897\) 15.6060 0.521068
\(898\) 0 0
\(899\) −9.25544 −0.308686
\(900\) 0 0
\(901\) −30.6796 −1.02209
\(902\) 0 0
\(903\) 40.5746 1.35024
\(904\) 0 0
\(905\) 20.7446 0.689573
\(906\) 0 0
\(907\) −46.8614 −1.55601 −0.778004 0.628260i \(-0.783768\pi\)
−0.778004 + 0.628260i \(0.783768\pi\)
\(908\) 0 0
\(909\) 116.010 3.84780
\(910\) 0 0
\(911\) 47.6126 1.57747 0.788737 0.614730i \(-0.210735\pi\)
0.788737 + 0.614730i \(0.210735\pi\)
\(912\) 0 0
\(913\) 18.9783 0.628088
\(914\) 0 0
\(915\) −123.446 −4.08099
\(916\) 0 0
\(917\) 14.5012 0.478872
\(918\) 0 0
\(919\) 8.71516 0.287486 0.143743 0.989615i \(-0.454086\pi\)
0.143743 + 0.989615i \(0.454086\pi\)
\(920\) 0 0
\(921\) 51.4456 1.69519
\(922\) 0 0
\(923\) 38.7446 1.27529
\(924\) 0 0
\(925\) −13.8564 −0.455596
\(926\) 0 0
\(927\) −100.274 −3.29342
\(928\) 0 0
\(929\) −27.8832 −0.914817 −0.457408 0.889257i \(-0.651222\pi\)
−0.457408 + 0.889257i \(0.651222\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 78.2921 2.56317
\(934\) 0 0
\(935\) 29.9422 0.979215
\(936\) 0 0
\(937\) 21.9783 0.717998 0.358999 0.933338i \(-0.383118\pi\)
0.358999 + 0.933338i \(0.383118\pi\)
\(938\) 0 0
\(939\) 13.0951 0.427342
\(940\) 0 0
\(941\) 10.8896 0.354992 0.177496 0.984122i \(-0.443200\pi\)
0.177496 + 0.984122i \(0.443200\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −151.679 −4.93413
\(946\) 0 0
\(947\) −43.2119 −1.40420 −0.702100 0.712079i \(-0.747754\pi\)
−0.702100 + 0.712079i \(0.747754\pi\)
\(948\) 0 0
\(949\) −49.5847 −1.60959
\(950\) 0 0
\(951\) −72.7634 −2.35951
\(952\) 0 0
\(953\) −26.2337 −0.849793 −0.424896 0.905242i \(-0.639689\pi\)
−0.424896 + 0.905242i \(0.639689\pi\)
\(954\) 0 0
\(955\) 17.8614 0.577982
\(956\) 0 0
\(957\) −21.3745 −0.690940
\(958\) 0 0
\(959\) 16.5831 0.535497
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 111.957 3.60775
\(964\) 0 0
\(965\) −8.21782 −0.264541
\(966\) 0 0
\(967\) 55.1307 1.77288 0.886441 0.462841i \(-0.153170\pi\)
0.886441 + 0.462841i \(0.153170\pi\)
\(968\) 0 0
\(969\) −16.8614 −0.541666
\(970\) 0 0
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) −12.0318 −0.385721
\(974\) 0 0
\(975\) −27.0303 −0.865663
\(976\) 0 0
\(977\) 27.4891 0.879455 0.439728 0.898131i \(-0.355075\pi\)
0.439728 + 0.898131i \(0.355075\pi\)
\(978\) 0 0
\(979\) −36.7446 −1.17436
\(980\) 0 0
\(981\) 33.1662 1.05892
\(982\) 0 0
\(983\) 52.6612 1.67963 0.839816 0.542871i \(-0.182663\pi\)
0.839816 + 0.542871i \(0.182663\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) −7.21194 −0.229559
\(988\) 0 0
\(989\) −2.87419 −0.0913941
\(990\) 0 0
\(991\) 7.92287 0.251678 0.125839 0.992051i \(-0.459838\pi\)
0.125839 + 0.992051i \(0.459838\pi\)
\(992\) 0 0
\(993\) −36.6277 −1.16235
\(994\) 0 0
\(995\) −0.372281 −0.0118021
\(996\) 0 0
\(997\) 32.8164 1.03931 0.519653 0.854378i \(-0.326061\pi\)
0.519653 + 0.854378i \(0.326061\pi\)
\(998\) 0 0
\(999\) 182.932 5.78772
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.bi.1.1 4
4.3 odd 2 4864.2.a.bl.1.3 4
8.3 odd 2 inner 4864.2.a.bi.1.2 4
8.5 even 2 4864.2.a.bl.1.4 4
16.3 odd 4 1216.2.c.h.609.8 yes 8
16.5 even 4 1216.2.c.h.609.7 yes 8
16.11 odd 4 1216.2.c.h.609.1 8
16.13 even 4 1216.2.c.h.609.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.h.609.1 8 16.11 odd 4
1216.2.c.h.609.2 yes 8 16.13 even 4
1216.2.c.h.609.7 yes 8 16.5 even 4
1216.2.c.h.609.8 yes 8 16.3 odd 4
4864.2.a.bi.1.1 4 1.1 even 1 trivial
4864.2.a.bi.1.2 4 8.3 odd 2 inner
4864.2.a.bl.1.3 4 4.3 odd 2
4864.2.a.bl.1.4 4 8.5 even 2