Properties

Label 6864.2.a.ca.1.4
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70164.1
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} + 10 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.20666\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.20666 q^{5} +1.53005 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.20666 q^{5} +1.53005 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.20666 q^{15} +2.73671 q^{17} +7.78961 q^{19} +1.53005 q^{21} +6.88326 q^{23} +5.28265 q^{25} +1.00000 q^{27} +1.22974 q^{29} +8.11301 q^{31} -1.00000 q^{33} +4.90635 q^{35} -6.41331 q^{37} -1.00000 q^{39} +3.53005 q^{41} -3.98234 q^{43} +3.20666 q^{45} +2.93990 q^{47} -4.65894 q^{49} +2.73671 q^{51} -10.2260 q^{53} -3.20666 q^{55} +7.78961 q^{57} -12.6729 q^{59} -6.30573 q^{61} +1.53005 q^{63} -3.20666 q^{65} +5.80727 q^{67} +6.88326 q^{69} -10.2260 q^{71} -15.4486 q^{73} +5.28265 q^{75} -1.53005 q^{77} +1.98411 q^{79} +1.00000 q^{81} -5.16591 q^{83} +8.77568 q^{85} +1.22974 q^{87} -4.11301 q^{89} -1.53005 q^{91} +8.11301 q^{93} +24.9786 q^{95} +15.9185 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} - 4q^{11} - 4q^{13} + q^{15} - 6q^{17} + q^{21} + 9q^{23} + q^{25} + 4q^{27} - q^{29} + 8q^{31} - 4q^{33} + 7q^{35} - 2q^{37} - 4q^{39} + 9q^{41} + 5q^{43} + q^{45} + 22q^{47} + 9q^{49} - 6q^{51} + 8q^{53} - q^{55} - q^{59} - 11q^{61} + q^{63} - q^{65} + 13q^{67} + 9q^{69} + 8q^{71} - 3q^{73} + q^{75} - q^{77} + 6q^{79} + 4q^{81} + 18q^{83} + 26q^{85} - q^{87} + 8q^{89} - q^{91} + 8q^{93} + 36q^{95} + 10q^{97} - 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.20666 1.43406 0.717030 0.697042i \(-0.245501\pi\)
0.717030 + 0.697042i \(0.245501\pi\)
\(6\) 0 0
\(7\) 1.53005 0.578305 0.289152 0.957283i \(-0.406627\pi\)
0.289152 + 0.957283i \(0.406627\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.20666 0.827955
\(16\) 0 0
\(17\) 2.73671 0.663749 0.331875 0.943324i \(-0.392319\pi\)
0.331875 + 0.943324i \(0.392319\pi\)
\(18\) 0 0
\(19\) 7.78961 1.78706 0.893530 0.449004i \(-0.148221\pi\)
0.893530 + 0.449004i \(0.148221\pi\)
\(20\) 0 0
\(21\) 1.53005 0.333885
\(22\) 0 0
\(23\) 6.88326 1.43526 0.717630 0.696425i \(-0.245227\pi\)
0.717630 + 0.696425i \(0.245227\pi\)
\(24\) 0 0
\(25\) 5.28265 1.05653
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.22974 0.228358 0.114179 0.993460i \(-0.463576\pi\)
0.114179 + 0.993460i \(0.463576\pi\)
\(30\) 0 0
\(31\) 8.11301 1.45714 0.728569 0.684972i \(-0.240186\pi\)
0.728569 + 0.684972i \(0.240186\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 4.90635 0.829324
\(36\) 0 0
\(37\) −6.41331 −1.05434 −0.527171 0.849759i \(-0.676747\pi\)
−0.527171 + 0.849759i \(0.676747\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 3.53005 0.551301 0.275651 0.961258i \(-0.411107\pi\)
0.275651 + 0.961258i \(0.411107\pi\)
\(42\) 0 0
\(43\) −3.98234 −0.607301 −0.303650 0.952784i \(-0.598205\pi\)
−0.303650 + 0.952784i \(0.598205\pi\)
\(44\) 0 0
\(45\) 3.20666 0.478020
\(46\) 0 0
\(47\) 2.93990 0.428828 0.214414 0.976743i \(-0.431216\pi\)
0.214414 + 0.976743i \(0.431216\pi\)
\(48\) 0 0
\(49\) −4.65894 −0.665563
\(50\) 0 0
\(51\) 2.73671 0.383216
\(52\) 0 0
\(53\) −10.2260 −1.40465 −0.702325 0.711856i \(-0.747855\pi\)
−0.702325 + 0.711856i \(0.747855\pi\)
\(54\) 0 0
\(55\) −3.20666 −0.432385
\(56\) 0 0
\(57\) 7.78961 1.03176
\(58\) 0 0
\(59\) −12.6729 −1.64987 −0.824934 0.565229i \(-0.808788\pi\)
−0.824934 + 0.565229i \(0.808788\pi\)
\(60\) 0 0
\(61\) −6.30573 −0.807366 −0.403683 0.914899i \(-0.632270\pi\)
−0.403683 + 0.914899i \(0.632270\pi\)
\(62\) 0 0
\(63\) 1.53005 0.192768
\(64\) 0 0
\(65\) −3.20666 −0.397737
\(66\) 0 0
\(67\) 5.80727 0.709471 0.354736 0.934967i \(-0.384571\pi\)
0.354736 + 0.934967i \(0.384571\pi\)
\(68\) 0 0
\(69\) 6.88326 0.828647
\(70\) 0 0
\(71\) −10.2260 −1.21360 −0.606802 0.794853i \(-0.707548\pi\)
−0.606802 + 0.794853i \(0.707548\pi\)
\(72\) 0 0
\(73\) −15.4486 −1.80812 −0.904058 0.427409i \(-0.859426\pi\)
−0.904058 + 0.427409i \(0.859426\pi\)
\(74\) 0 0
\(75\) 5.28265 0.609987
\(76\) 0 0
\(77\) −1.53005 −0.174366
\(78\) 0 0
\(79\) 1.98411 0.223230 0.111615 0.993752i \(-0.464398\pi\)
0.111615 + 0.993752i \(0.464398\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.16591 −0.567032 −0.283516 0.958967i \(-0.591501\pi\)
−0.283516 + 0.958967i \(0.591501\pi\)
\(84\) 0 0
\(85\) 8.77568 0.951856
\(86\) 0 0
\(87\) 1.22974 0.131842
\(88\) 0 0
\(89\) −4.11301 −0.435978 −0.217989 0.975951i \(-0.569950\pi\)
−0.217989 + 0.975951i \(0.569950\pi\)
\(90\) 0 0
\(91\) −1.53005 −0.160393
\(92\) 0 0
\(93\) 8.11301 0.841279
\(94\) 0 0
\(95\) 24.9786 2.56275
\(96\) 0 0
\(97\) 15.9185 1.61628 0.808140 0.588991i \(-0.200475\pi\)
0.808140 + 0.588991i \(0.200475\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 9.82858 0.977981 0.488990 0.872289i \(-0.337365\pi\)
0.488990 + 0.872289i \(0.337365\pi\)
\(102\) 0 0
\(103\) 10.3057 1.01545 0.507727 0.861518i \(-0.330486\pi\)
0.507727 + 0.861518i \(0.330486\pi\)
\(104\) 0 0
\(105\) 4.90635 0.478811
\(106\) 0 0
\(107\) −2.86018 −0.276504 −0.138252 0.990397i \(-0.544148\pi\)
−0.138252 + 0.990397i \(0.544148\pi\)
\(108\) 0 0
\(109\) −10.4364 −0.999626 −0.499813 0.866133i \(-0.666598\pi\)
−0.499813 + 0.866133i \(0.666598\pi\)
\(110\) 0 0
\(111\) −6.41331 −0.608725
\(112\) 0 0
\(113\) 0.906349 0.0852621 0.0426311 0.999091i \(-0.486426\pi\)
0.0426311 + 0.999091i \(0.486426\pi\)
\(114\) 0 0
\(115\) 22.0723 2.05825
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 4.18730 0.383849
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 3.53005 0.318294
\(124\) 0 0
\(125\) 0.906349 0.0810663
\(126\) 0 0
\(127\) −11.9026 −1.05619 −0.528093 0.849186i \(-0.677093\pi\)
−0.528093 + 0.849186i \(0.677093\pi\)
\(128\) 0 0
\(129\) −3.98234 −0.350625
\(130\) 0 0
\(131\) 19.1324 1.67160 0.835801 0.549033i \(-0.185004\pi\)
0.835801 + 0.549033i \(0.185004\pi\)
\(132\) 0 0
\(133\) 11.9185 1.03347
\(134\) 0 0
\(135\) 3.20666 0.275985
\(136\) 0 0
\(137\) 5.02113 0.428984 0.214492 0.976726i \(-0.431190\pi\)
0.214492 + 0.976726i \(0.431190\pi\)
\(138\) 0 0
\(139\) −14.1361 −1.19901 −0.599504 0.800372i \(-0.704635\pi\)
−0.599504 + 0.800372i \(0.704635\pi\)
\(140\) 0 0
\(141\) 2.93990 0.247584
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 3.94336 0.327479
\(146\) 0 0
\(147\) −4.65894 −0.384263
\(148\) 0 0
\(149\) 7.03702 0.576495 0.288247 0.957556i \(-0.406927\pi\)
0.288247 + 0.957556i \(0.406927\pi\)
\(150\) 0 0
\(151\) −10.8497 −0.882937 −0.441469 0.897277i \(-0.645542\pi\)
−0.441469 + 0.897277i \(0.645542\pi\)
\(152\) 0 0
\(153\) 2.73671 0.221250
\(154\) 0 0
\(155\) 26.0156 2.08962
\(156\) 0 0
\(157\) −16.6624 −1.32981 −0.664903 0.746930i \(-0.731527\pi\)
−0.664903 + 0.746930i \(0.731527\pi\)
\(158\) 0 0
\(159\) −10.2260 −0.810975
\(160\) 0 0
\(161\) 10.5317 0.830018
\(162\) 0 0
\(163\) 9.95925 0.780069 0.390034 0.920800i \(-0.372463\pi\)
0.390034 + 0.920800i \(0.372463\pi\)
\(164\) 0 0
\(165\) −3.20666 −0.249638
\(166\) 0 0
\(167\) 6.68034 0.516940 0.258470 0.966019i \(-0.416782\pi\)
0.258470 + 0.966019i \(0.416782\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 7.78961 0.595686
\(172\) 0 0
\(173\) −0.169641 −0.0128976 −0.00644878 0.999979i \(-0.502053\pi\)
−0.00644878 + 0.999979i \(0.502053\pi\)
\(174\) 0 0
\(175\) 8.08272 0.610996
\(176\) 0 0
\(177\) −12.6729 −0.952551
\(178\) 0 0
\(179\) 7.66941 0.573238 0.286619 0.958045i \(-0.407469\pi\)
0.286619 + 0.958045i \(0.407469\pi\)
\(180\) 0 0
\(181\) 1.03702 0.0770808 0.0385404 0.999257i \(-0.487729\pi\)
0.0385404 + 0.999257i \(0.487729\pi\)
\(182\) 0 0
\(183\) −6.30573 −0.466133
\(184\) 0 0
\(185\) −20.5653 −1.51199
\(186\) 0 0
\(187\) −2.73671 −0.200128
\(188\) 0 0
\(189\) 1.53005 0.111295
\(190\) 0 0
\(191\) 0.199927 0.0144662 0.00723311 0.999974i \(-0.497698\pi\)
0.00723311 + 0.999974i \(0.497698\pi\)
\(192\) 0 0
\(193\) 20.5069 1.47612 0.738059 0.674736i \(-0.235743\pi\)
0.738059 + 0.674736i \(0.235743\pi\)
\(194\) 0 0
\(195\) −3.20666 −0.229633
\(196\) 0 0
\(197\) −10.4826 −0.746852 −0.373426 0.927660i \(-0.621817\pi\)
−0.373426 + 0.927660i \(0.621817\pi\)
\(198\) 0 0
\(199\) 20.1184 1.42616 0.713079 0.701084i \(-0.247300\pi\)
0.713079 + 0.701084i \(0.247300\pi\)
\(200\) 0 0
\(201\) 5.80727 0.409613
\(202\) 0 0
\(203\) 1.88157 0.132060
\(204\) 0 0
\(205\) 11.3197 0.790599
\(206\) 0 0
\(207\) 6.88326 0.478420
\(208\) 0 0
\(209\) −7.78961 −0.538819
\(210\) 0 0
\(211\) 16.0546 1.10524 0.552622 0.833432i \(-0.313627\pi\)
0.552622 + 0.833432i \(0.313627\pi\)
\(212\) 0 0
\(213\) −10.2260 −0.700675
\(214\) 0 0
\(215\) −12.7700 −0.870906
\(216\) 0 0
\(217\) 12.4133 0.842671
\(218\) 0 0
\(219\) −15.4486 −1.04392
\(220\) 0 0
\(221\) −2.73671 −0.184091
\(222\) 0 0
\(223\) 15.5124 1.03879 0.519393 0.854535i \(-0.326158\pi\)
0.519393 + 0.854535i \(0.326158\pi\)
\(224\) 0 0
\(225\) 5.28265 0.352176
\(226\) 0 0
\(227\) −2.02309 −0.134277 −0.0671385 0.997744i \(-0.521387\pi\)
−0.0671385 + 0.997744i \(0.521387\pi\)
\(228\) 0 0
\(229\) 7.19946 0.475754 0.237877 0.971295i \(-0.423549\pi\)
0.237877 + 0.971295i \(0.423549\pi\)
\(230\) 0 0
\(231\) −1.53005 −0.100670
\(232\) 0 0
\(233\) −16.7367 −1.09646 −0.548229 0.836328i \(-0.684698\pi\)
−0.548229 + 0.836328i \(0.684698\pi\)
\(234\) 0 0
\(235\) 9.42724 0.614965
\(236\) 0 0
\(237\) 1.98411 0.128882
\(238\) 0 0
\(239\) −8.23647 −0.532773 −0.266387 0.963866i \(-0.585830\pi\)
−0.266387 + 0.963866i \(0.585830\pi\)
\(240\) 0 0
\(241\) 4.72082 0.304095 0.152047 0.988373i \(-0.451413\pi\)
0.152047 + 0.988373i \(0.451413\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −14.9396 −0.954458
\(246\) 0 0
\(247\) −7.78961 −0.495641
\(248\) 0 0
\(249\) −5.16591 −0.327376
\(250\) 0 0
\(251\) 11.6607 0.736018 0.368009 0.929822i \(-0.380040\pi\)
0.368009 + 0.929822i \(0.380040\pi\)
\(252\) 0 0
\(253\) −6.88326 −0.432747
\(254\) 0 0
\(255\) 8.77568 0.549554
\(256\) 0 0
\(257\) −9.55314 −0.595908 −0.297954 0.954580i \(-0.596304\pi\)
−0.297954 + 0.954580i \(0.596304\pi\)
\(258\) 0 0
\(259\) −9.81270 −0.609731
\(260\) 0 0
\(261\) 1.22974 0.0761192
\(262\) 0 0
\(263\) 8.60062 0.530337 0.265168 0.964202i \(-0.414572\pi\)
0.265168 + 0.964202i \(0.414572\pi\)
\(264\) 0 0
\(265\) −32.7913 −2.01435
\(266\) 0 0
\(267\) −4.11301 −0.251712
\(268\) 0 0
\(269\) 22.7913 1.38961 0.694805 0.719198i \(-0.255491\pi\)
0.694805 + 0.719198i \(0.255491\pi\)
\(270\) 0 0
\(271\) −10.2491 −0.622588 −0.311294 0.950314i \(-0.600762\pi\)
−0.311294 + 0.950314i \(0.600762\pi\)
\(272\) 0 0
\(273\) −1.53005 −0.0926029
\(274\) 0 0
\(275\) −5.28265 −0.318556
\(276\) 0 0
\(277\) −14.1184 −0.848294 −0.424147 0.905593i \(-0.639426\pi\)
−0.424147 + 0.905593i \(0.639426\pi\)
\(278\) 0 0
\(279\) 8.11301 0.485713
\(280\) 0 0
\(281\) −22.4625 −1.34000 −0.670000 0.742361i \(-0.733706\pi\)
−0.670000 + 0.742361i \(0.733706\pi\)
\(282\) 0 0
\(283\) −27.1165 −1.61191 −0.805953 0.591979i \(-0.798347\pi\)
−0.805953 + 0.591979i \(0.798347\pi\)
\(284\) 0 0
\(285\) 24.9786 1.47960
\(286\) 0 0
\(287\) 5.40116 0.318820
\(288\) 0 0
\(289\) −9.51043 −0.559437
\(290\) 0 0
\(291\) 15.9185 0.933159
\(292\) 0 0
\(293\) 2.74391 0.160301 0.0801504 0.996783i \(-0.474460\pi\)
0.0801504 + 0.996783i \(0.474460\pi\)
\(294\) 0 0
\(295\) −40.6375 −2.36601
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −6.88326 −0.398069
\(300\) 0 0
\(301\) −6.09318 −0.351205
\(302\) 0 0
\(303\) 9.82858 0.564637
\(304\) 0 0
\(305\) −20.2203 −1.15781
\(306\) 0 0
\(307\) −15.1110 −0.862433 −0.431216 0.902248i \(-0.641915\pi\)
−0.431216 + 0.902248i \(0.641915\pi\)
\(308\) 0 0
\(309\) 10.3057 0.586273
\(310\) 0 0
\(311\) −7.94159 −0.450326 −0.225163 0.974321i \(-0.572291\pi\)
−0.225163 + 0.974321i \(0.572291\pi\)
\(312\) 0 0
\(313\) 15.2613 0.862617 0.431308 0.902205i \(-0.358052\pi\)
0.431308 + 0.902205i \(0.358052\pi\)
\(314\) 0 0
\(315\) 4.90635 0.276441
\(316\) 0 0
\(317\) 1.83905 0.103291 0.0516456 0.998665i \(-0.483553\pi\)
0.0516456 + 0.998665i \(0.483553\pi\)
\(318\) 0 0
\(319\) −1.22974 −0.0688524
\(320\) 0 0
\(321\) −2.86018 −0.159639
\(322\) 0 0
\(323\) 21.3179 1.18616
\(324\) 0 0
\(325\) −5.28265 −0.293028
\(326\) 0 0
\(327\) −10.4364 −0.577134
\(328\) 0 0
\(329\) 4.49819 0.247993
\(330\) 0 0
\(331\) 30.6726 1.68592 0.842959 0.537977i \(-0.180811\pi\)
0.842959 + 0.537977i \(0.180811\pi\)
\(332\) 0 0
\(333\) −6.41331 −0.351447
\(334\) 0 0
\(335\) 18.6219 1.01742
\(336\) 0 0
\(337\) 7.27918 0.396522 0.198261 0.980149i \(-0.436471\pi\)
0.198261 + 0.980149i \(0.436471\pi\)
\(338\) 0 0
\(339\) 0.906349 0.0492261
\(340\) 0 0
\(341\) −8.11301 −0.439344
\(342\) 0 0
\(343\) −17.8388 −0.963204
\(344\) 0 0
\(345\) 22.0723 1.18833
\(346\) 0 0
\(347\) −19.1197 −1.02640 −0.513201 0.858269i \(-0.671540\pi\)
−0.513201 + 0.858269i \(0.671540\pi\)
\(348\) 0 0
\(349\) −12.0387 −0.644417 −0.322209 0.946669i \(-0.604425\pi\)
−0.322209 + 0.946669i \(0.604425\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −25.7066 −1.36823 −0.684113 0.729376i \(-0.739810\pi\)
−0.684113 + 0.729376i \(0.739810\pi\)
\(354\) 0 0
\(355\) −32.7913 −1.74038
\(356\) 0 0
\(357\) 4.18730 0.221616
\(358\) 0 0
\(359\) 21.7417 1.14748 0.573741 0.819037i \(-0.305492\pi\)
0.573741 + 0.819037i \(0.305492\pi\)
\(360\) 0 0
\(361\) 41.6780 2.19358
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −49.5382 −2.59295
\(366\) 0 0
\(367\) 27.5474 1.43797 0.718983 0.695028i \(-0.244608\pi\)
0.718983 + 0.695028i \(0.244608\pi\)
\(368\) 0 0
\(369\) 3.53005 0.183767
\(370\) 0 0
\(371\) −15.6463 −0.812316
\(372\) 0 0
\(373\) 6.98038 0.361430 0.180715 0.983535i \(-0.442159\pi\)
0.180715 + 0.983535i \(0.442159\pi\)
\(374\) 0 0
\(375\) 0.906349 0.0468036
\(376\) 0 0
\(377\) −1.22974 −0.0633350
\(378\) 0 0
\(379\) 26.2575 1.34876 0.674379 0.738385i \(-0.264411\pi\)
0.674379 + 0.738385i \(0.264411\pi\)
\(380\) 0 0
\(381\) −11.9026 −0.609789
\(382\) 0 0
\(383\) 11.9647 0.611366 0.305683 0.952133i \(-0.401115\pi\)
0.305683 + 0.952133i \(0.401115\pi\)
\(384\) 0 0
\(385\) −4.90635 −0.250051
\(386\) 0 0
\(387\) −3.98234 −0.202434
\(388\) 0 0
\(389\) 28.9324 1.46693 0.733466 0.679726i \(-0.237901\pi\)
0.733466 + 0.679726i \(0.237901\pi\)
\(390\) 0 0
\(391\) 18.8375 0.952652
\(392\) 0 0
\(393\) 19.1324 0.965100
\(394\) 0 0
\(395\) 6.36237 0.320126
\(396\) 0 0
\(397\) 1.13982 0.0572062 0.0286031 0.999591i \(-0.490894\pi\)
0.0286031 + 0.999591i \(0.490894\pi\)
\(398\) 0 0
\(399\) 11.9185 0.596671
\(400\) 0 0
\(401\) −26.8581 −1.34123 −0.670616 0.741805i \(-0.733970\pi\)
−0.670616 + 0.741805i \(0.733970\pi\)
\(402\) 0 0
\(403\) −8.11301 −0.404138
\(404\) 0 0
\(405\) 3.20666 0.159340
\(406\) 0 0
\(407\) 6.41331 0.317896
\(408\) 0 0
\(409\) −19.2891 −0.953785 −0.476893 0.878962i \(-0.658237\pi\)
−0.476893 + 0.878962i \(0.658237\pi\)
\(410\) 0 0
\(411\) 5.02113 0.247674
\(412\) 0 0
\(413\) −19.3901 −0.954127
\(414\) 0 0
\(415\) −16.5653 −0.813158
\(416\) 0 0
\(417\) −14.1361 −0.692247
\(418\) 0 0
\(419\) −7.34097 −0.358630 −0.179315 0.983792i \(-0.557388\pi\)
−0.179315 + 0.983792i \(0.557388\pi\)
\(420\) 0 0
\(421\) −22.2665 −1.08520 −0.542601 0.839990i \(-0.682561\pi\)
−0.542601 + 0.839990i \(0.682561\pi\)
\(422\) 0 0
\(423\) 2.93990 0.142943
\(424\) 0 0
\(425\) 14.4571 0.701270
\(426\) 0 0
\(427\) −9.64809 −0.466904
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 21.2474 1.02345 0.511726 0.859149i \(-0.329006\pi\)
0.511726 + 0.859149i \(0.329006\pi\)
\(432\) 0 0
\(433\) −39.5774 −1.90197 −0.950985 0.309236i \(-0.899927\pi\)
−0.950985 + 0.309236i \(0.899927\pi\)
\(434\) 0 0
\(435\) 3.94336 0.189070
\(436\) 0 0
\(437\) 53.6179 2.56489
\(438\) 0 0
\(439\) −3.16442 −0.151030 −0.0755148 0.997145i \(-0.524060\pi\)
−0.0755148 + 0.997145i \(0.524060\pi\)
\(440\) 0 0
\(441\) −4.65894 −0.221854
\(442\) 0 0
\(443\) 16.8266 0.799457 0.399729 0.916634i \(-0.369104\pi\)
0.399729 + 0.916634i \(0.369104\pi\)
\(444\) 0 0
\(445\) −13.1890 −0.625218
\(446\) 0 0
\(447\) 7.03702 0.332839
\(448\) 0 0
\(449\) 31.3251 1.47832 0.739161 0.673529i \(-0.235222\pi\)
0.739161 + 0.673529i \(0.235222\pi\)
\(450\) 0 0
\(451\) −3.53005 −0.166224
\(452\) 0 0
\(453\) −10.8497 −0.509764
\(454\) 0 0
\(455\) −4.90635 −0.230013
\(456\) 0 0
\(457\) 12.5209 0.585703 0.292851 0.956158i \(-0.405396\pi\)
0.292851 + 0.956158i \(0.405396\pi\)
\(458\) 0 0
\(459\) 2.73671 0.127739
\(460\) 0 0
\(461\) −6.35845 −0.296143 −0.148071 0.988977i \(-0.547307\pi\)
−0.148071 + 0.988977i \(0.547307\pi\)
\(462\) 0 0
\(463\) −19.1974 −0.892180 −0.446090 0.894988i \(-0.647184\pi\)
−0.446090 + 0.894988i \(0.647184\pi\)
\(464\) 0 0
\(465\) 26.0156 1.20645
\(466\) 0 0
\(467\) 11.6376 0.538525 0.269263 0.963067i \(-0.413220\pi\)
0.269263 + 0.963067i \(0.413220\pi\)
\(468\) 0 0
\(469\) 8.88542 0.410291
\(470\) 0 0
\(471\) −16.6624 −0.767763
\(472\) 0 0
\(473\) 3.98234 0.183108
\(474\) 0 0
\(475\) 41.1498 1.88808
\(476\) 0 0
\(477\) −10.2260 −0.468217
\(478\) 0 0
\(479\) −30.3358 −1.38608 −0.693038 0.720901i \(-0.743728\pi\)
−0.693038 + 0.720901i \(0.743728\pi\)
\(480\) 0 0
\(481\) 6.41331 0.292422
\(482\) 0 0
\(483\) 10.5317 0.479211
\(484\) 0 0
\(485\) 51.0452 2.31784
\(486\) 0 0
\(487\) −34.7136 −1.57302 −0.786512 0.617575i \(-0.788115\pi\)
−0.786512 + 0.617575i \(0.788115\pi\)
\(488\) 0 0
\(489\) 9.95925 0.450373
\(490\) 0 0
\(491\) −17.4926 −0.789428 −0.394714 0.918804i \(-0.629156\pi\)
−0.394714 + 0.918804i \(0.629156\pi\)
\(492\) 0 0
\(493\) 3.36545 0.151572
\(494\) 0 0
\(495\) −3.20666 −0.144128
\(496\) 0 0
\(497\) −15.6463 −0.701833
\(498\) 0 0
\(499\) −6.07946 −0.272154 −0.136077 0.990698i \(-0.543449\pi\)
−0.136077 + 0.990698i \(0.543449\pi\)
\(500\) 0 0
\(501\) 6.68034 0.298456
\(502\) 0 0
\(503\) 19.8514 0.885130 0.442565 0.896736i \(-0.354069\pi\)
0.442565 + 0.896736i \(0.354069\pi\)
\(504\) 0 0
\(505\) 31.5169 1.40248
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 8.48406 0.376049 0.188025 0.982164i \(-0.439791\pi\)
0.188025 + 0.982164i \(0.439791\pi\)
\(510\) 0 0
\(511\) −23.6371 −1.04564
\(512\) 0 0
\(513\) 7.78961 0.343920
\(514\) 0 0
\(515\) 33.0469 1.45622
\(516\) 0 0
\(517\) −2.93990 −0.129297
\(518\) 0 0
\(519\) −0.169641 −0.00744641
\(520\) 0 0
\(521\) 24.4041 1.06916 0.534581 0.845117i \(-0.320469\pi\)
0.534581 + 0.845117i \(0.320469\pi\)
\(522\) 0 0
\(523\) −27.4078 −1.19846 −0.599230 0.800577i \(-0.704526\pi\)
−0.599230 + 0.800577i \(0.704526\pi\)
\(524\) 0 0
\(525\) 8.08272 0.352759
\(526\) 0 0
\(527\) 22.2029 0.967175
\(528\) 0 0
\(529\) 24.3793 1.05997
\(530\) 0 0
\(531\) −12.6729 −0.549956
\(532\) 0 0
\(533\) −3.53005 −0.152903
\(534\) 0 0
\(535\) −9.17160 −0.396523
\(536\) 0 0
\(537\) 7.66941 0.330959
\(538\) 0 0
\(539\) 4.65894 0.200675
\(540\) 0 0
\(541\) 43.5826 1.87376 0.936881 0.349648i \(-0.113699\pi\)
0.936881 + 0.349648i \(0.113699\pi\)
\(542\) 0 0
\(543\) 1.03702 0.0445026
\(544\) 0 0
\(545\) −33.4659 −1.43352
\(546\) 0 0
\(547\) −33.7767 −1.44419 −0.722094 0.691795i \(-0.756820\pi\)
−0.722094 + 0.691795i \(0.756820\pi\)
\(548\) 0 0
\(549\) −6.30573 −0.269122
\(550\) 0 0
\(551\) 9.57922 0.408089
\(552\) 0 0
\(553\) 3.03579 0.129095
\(554\) 0 0
\(555\) −20.5653 −0.872948
\(556\) 0 0
\(557\) 16.5458 0.701066 0.350533 0.936550i \(-0.386000\pi\)
0.350533 + 0.936550i \(0.386000\pi\)
\(558\) 0 0
\(559\) 3.98234 0.168435
\(560\) 0 0
\(561\) −2.73671 −0.115544
\(562\) 0 0
\(563\) 24.6324 1.03813 0.519066 0.854734i \(-0.326280\pi\)
0.519066 + 0.854734i \(0.326280\pi\)
\(564\) 0 0
\(565\) 2.90635 0.122271
\(566\) 0 0
\(567\) 1.53005 0.0642561
\(568\) 0 0
\(569\) −7.34817 −0.308051 −0.154026 0.988067i \(-0.549224\pi\)
−0.154026 + 0.988067i \(0.549224\pi\)
\(570\) 0 0
\(571\) −39.9044 −1.66995 −0.834973 0.550290i \(-0.814517\pi\)
−0.834973 + 0.550290i \(0.814517\pi\)
\(572\) 0 0
\(573\) 0.199927 0.00835208
\(574\) 0 0
\(575\) 36.3618 1.51639
\(576\) 0 0
\(577\) −11.4626 −0.477193 −0.238596 0.971119i \(-0.576687\pi\)
−0.238596 + 0.971119i \(0.576687\pi\)
\(578\) 0 0
\(579\) 20.5069 0.852237
\(580\) 0 0
\(581\) −7.90410 −0.327917
\(582\) 0 0
\(583\) 10.2260 0.423518
\(584\) 0 0
\(585\) −3.20666 −0.132579
\(586\) 0 0
\(587\) 29.4995 1.21757 0.608787 0.793333i \(-0.291656\pi\)
0.608787 + 0.793333i \(0.291656\pi\)
\(588\) 0 0
\(589\) 63.1971 2.60399
\(590\) 0 0
\(591\) −10.4826 −0.431195
\(592\) 0 0
\(593\) −11.3763 −0.467169 −0.233584 0.972337i \(-0.575045\pi\)
−0.233584 + 0.972337i \(0.575045\pi\)
\(594\) 0 0
\(595\) 13.4272 0.550463
\(596\) 0 0
\(597\) 20.1184 0.823393
\(598\) 0 0
\(599\) 26.2242 1.07149 0.535747 0.844379i \(-0.320030\pi\)
0.535747 + 0.844379i \(0.320030\pi\)
\(600\) 0 0
\(601\) −6.51912 −0.265920 −0.132960 0.991121i \(-0.542448\pi\)
−0.132960 + 0.991121i \(0.542448\pi\)
\(602\) 0 0
\(603\) 5.80727 0.236490
\(604\) 0 0
\(605\) 3.20666 0.130369
\(606\) 0 0
\(607\) 3.34817 0.135898 0.0679491 0.997689i \(-0.478354\pi\)
0.0679491 + 0.997689i \(0.478354\pi\)
\(608\) 0 0
\(609\) 1.88157 0.0762451
\(610\) 0 0
\(611\) −2.93990 −0.118936
\(612\) 0 0
\(613\) −0.948585 −0.0383130 −0.0191565 0.999816i \(-0.506098\pi\)
−0.0191565 + 0.999816i \(0.506098\pi\)
\(614\) 0 0
\(615\) 11.3197 0.456453
\(616\) 0 0
\(617\) −2.02066 −0.0813486 −0.0406743 0.999172i \(-0.512951\pi\)
−0.0406743 + 0.999172i \(0.512951\pi\)
\(618\) 0 0
\(619\) −29.2345 −1.17503 −0.587517 0.809212i \(-0.699895\pi\)
−0.587517 + 0.809212i \(0.699895\pi\)
\(620\) 0 0
\(621\) 6.88326 0.276216
\(622\) 0 0
\(623\) −6.29311 −0.252128
\(624\) 0 0
\(625\) −23.5069 −0.940275
\(626\) 0 0
\(627\) −7.78961 −0.311087
\(628\) 0 0
\(629\) −17.5514 −0.699819
\(630\) 0 0
\(631\) −45.2253 −1.80039 −0.900195 0.435487i \(-0.856576\pi\)
−0.900195 + 0.435487i \(0.856576\pi\)
\(632\) 0 0
\(633\) 16.0546 0.638113
\(634\) 0 0
\(635\) −38.1676 −1.51463
\(636\) 0 0
\(637\) 4.65894 0.184594
\(638\) 0 0
\(639\) −10.2260 −0.404535
\(640\) 0 0
\(641\) −46.2629 −1.82728 −0.913638 0.406528i \(-0.866739\pi\)
−0.913638 + 0.406528i \(0.866739\pi\)
\(642\) 0 0
\(643\) −38.2506 −1.50846 −0.754228 0.656613i \(-0.771989\pi\)
−0.754228 + 0.656613i \(0.771989\pi\)
\(644\) 0 0
\(645\) −12.7700 −0.502818
\(646\) 0 0
\(647\) −35.4306 −1.39292 −0.696461 0.717595i \(-0.745243\pi\)
−0.696461 + 0.717595i \(0.745243\pi\)
\(648\) 0 0
\(649\) 12.6729 0.497454
\(650\) 0 0
\(651\) 12.4133 0.486516
\(652\) 0 0
\(653\) 3.06704 0.120022 0.0600112 0.998198i \(-0.480886\pi\)
0.0600112 + 0.998198i \(0.480886\pi\)
\(654\) 0 0
\(655\) 61.3509 2.39718
\(656\) 0 0
\(657\) −15.4486 −0.602705
\(658\) 0 0
\(659\) −44.3109 −1.72611 −0.863054 0.505112i \(-0.831451\pi\)
−0.863054 + 0.505112i \(0.831451\pi\)
\(660\) 0 0
\(661\) 16.6746 0.648569 0.324284 0.945960i \(-0.394877\pi\)
0.324284 + 0.945960i \(0.394877\pi\)
\(662\) 0 0
\(663\) −2.73671 −0.106285
\(664\) 0 0
\(665\) 38.2185 1.48205
\(666\) 0 0
\(667\) 8.46464 0.327752
\(668\) 0 0
\(669\) 15.5124 0.599744
\(670\) 0 0
\(671\) 6.30573 0.243430
\(672\) 0 0
\(673\) −6.19815 −0.238921 −0.119461 0.992839i \(-0.538117\pi\)
−0.119461 + 0.992839i \(0.538117\pi\)
\(674\) 0 0
\(675\) 5.28265 0.203329
\(676\) 0 0
\(677\) 27.1465 1.04332 0.521662 0.853152i \(-0.325312\pi\)
0.521662 + 0.853152i \(0.325312\pi\)
\(678\) 0 0
\(679\) 24.3561 0.934702
\(680\) 0 0
\(681\) −2.02309 −0.0775249
\(682\) 0 0
\(683\) 8.55267 0.327259 0.163629 0.986522i \(-0.447680\pi\)
0.163629 + 0.986522i \(0.447680\pi\)
\(684\) 0 0
\(685\) 16.1010 0.615189
\(686\) 0 0
\(687\) 7.19946 0.274676
\(688\) 0 0
\(689\) 10.2260 0.389580
\(690\) 0 0
\(691\) 5.53331 0.210497 0.105249 0.994446i \(-0.466436\pi\)
0.105249 + 0.994446i \(0.466436\pi\)
\(692\) 0 0
\(693\) −1.53005 −0.0581218
\(694\) 0 0
\(695\) −45.3296 −1.71945
\(696\) 0 0
\(697\) 9.66072 0.365926
\(698\) 0 0
\(699\) −16.7367 −0.633040
\(700\) 0 0
\(701\) 46.6639 1.76247 0.881236 0.472677i \(-0.156712\pi\)
0.881236 + 0.472677i \(0.156712\pi\)
\(702\) 0 0
\(703\) −49.9572 −1.88417
\(704\) 0 0
\(705\) 9.42724 0.355050
\(706\) 0 0
\(707\) 15.0382 0.565571
\(708\) 0 0
\(709\) 14.1001 0.529541 0.264770 0.964311i \(-0.414704\pi\)
0.264770 + 0.964311i \(0.414704\pi\)
\(710\) 0 0
\(711\) 1.98411 0.0744100
\(712\) 0 0
\(713\) 55.8439 2.09137
\(714\) 0 0
\(715\) 3.20666 0.119922
\(716\) 0 0
\(717\) −8.23647 −0.307597
\(718\) 0 0
\(719\) −16.3127 −0.608360 −0.304180 0.952615i \(-0.598382\pi\)
−0.304180 + 0.952615i \(0.598382\pi\)
\(720\) 0 0
\(721\) 15.7683 0.587242
\(722\) 0 0
\(723\) 4.72082 0.175569
\(724\) 0 0
\(725\) 6.49630 0.241266
\(726\) 0 0
\(727\) 43.9533 1.63014 0.815069 0.579364i \(-0.196699\pi\)
0.815069 + 0.579364i \(0.196699\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.8985 −0.403095
\(732\) 0 0
\(733\) −40.2029 −1.48493 −0.742464 0.669885i \(-0.766343\pi\)
−0.742464 + 0.669885i \(0.766343\pi\)
\(734\) 0 0
\(735\) −14.9396 −0.551057
\(736\) 0 0
\(737\) −5.80727 −0.213914
\(738\) 0 0
\(739\) −46.1467 −1.69753 −0.848766 0.528768i \(-0.822654\pi\)
−0.848766 + 0.528768i \(0.822654\pi\)
\(740\) 0 0
\(741\) −7.78961 −0.286158
\(742\) 0 0
\(743\) 40.8174 1.49744 0.748722 0.662884i \(-0.230668\pi\)
0.748722 + 0.662884i \(0.230668\pi\)
\(744\) 0 0
\(745\) 22.5653 0.826728
\(746\) 0 0
\(747\) −5.16591 −0.189011
\(748\) 0 0
\(749\) −4.37621 −0.159903
\(750\) 0 0
\(751\) −19.9272 −0.727154 −0.363577 0.931564i \(-0.618445\pi\)
−0.363577 + 0.931564i \(0.618445\pi\)
\(752\) 0 0
\(753\) 11.6607 0.424940
\(754\) 0 0
\(755\) −34.7913 −1.26619
\(756\) 0 0
\(757\) −1.62323 −0.0589974 −0.0294987 0.999565i \(-0.509391\pi\)
−0.0294987 + 0.999565i \(0.509391\pi\)
\(758\) 0 0
\(759\) −6.88326 −0.249847
\(760\) 0 0
\(761\) 16.5017 0.598187 0.299093 0.954224i \(-0.403316\pi\)
0.299093 + 0.954224i \(0.403316\pi\)
\(762\) 0 0
\(763\) −15.9682 −0.578089
\(764\) 0 0
\(765\) 8.77568 0.317285
\(766\) 0 0
\(767\) 12.6729 0.457591
\(768\) 0 0
\(769\) 39.2699 1.41611 0.708055 0.706157i \(-0.249573\pi\)
0.708055 + 0.706157i \(0.249573\pi\)
\(770\) 0 0
\(771\) −9.55314 −0.344048
\(772\) 0 0
\(773\) 30.2224 1.08702 0.543511 0.839402i \(-0.317094\pi\)
0.543511 + 0.839402i \(0.317094\pi\)
\(774\) 0 0
\(775\) 42.8581 1.53951
\(776\) 0 0
\(777\) −9.81270 −0.352029
\(778\) 0 0
\(779\) 27.4977 0.985208
\(780\) 0 0
\(781\) 10.2260 0.365915
\(782\) 0 0
\(783\) 1.22974 0.0439474
\(784\) 0 0
\(785\) −53.4306 −1.90702
\(786\) 0 0
\(787\) −38.6405 −1.37739 −0.688693 0.725053i \(-0.741815\pi\)
−0.688693 + 0.725053i \(0.741815\pi\)
\(788\) 0 0
\(789\) 8.60062 0.306190
\(790\) 0 0
\(791\) 1.38676 0.0493075
\(792\) 0 0
\(793\) 6.30573 0.223923
\(794\) 0 0
\(795\) −32.7913 −1.16299
\(796\) 0 0
\(797\) 4.00693 0.141933 0.0709664 0.997479i \(-0.477392\pi\)
0.0709664 + 0.997479i \(0.477392\pi\)
\(798\) 0 0
\(799\) 8.04564 0.284634
\(800\) 0 0
\(801\) −4.11301 −0.145326
\(802\) 0 0
\(803\) 15.4486 0.545168
\(804\) 0 0
\(805\) 33.7717 1.19030
\(806\) 0 0
\(807\) 22.7913 0.802292
\(808\) 0 0
\(809\) 52.9409 1.86130 0.930651 0.365909i \(-0.119242\pi\)
0.930651 + 0.365909i \(0.119242\pi\)
\(810\) 0 0
\(811\) −21.6947 −0.761802 −0.380901 0.924616i \(-0.624386\pi\)
−0.380901 + 0.924616i \(0.624386\pi\)
\(812\) 0 0
\(813\) −10.2491 −0.359452
\(814\) 0 0
\(815\) 31.9359 1.11867
\(816\) 0 0
\(817\) −31.0209 −1.08528
\(818\) 0 0
\(819\) −1.53005 −0.0534643
\(820\) 0 0
\(821\) −10.5497 −0.368186 −0.184093 0.982909i \(-0.558935\pi\)
−0.184093 + 0.982909i \(0.558935\pi\)
\(822\) 0 0
\(823\) 10.3723 0.361556 0.180778 0.983524i \(-0.442139\pi\)
0.180778 + 0.983524i \(0.442139\pi\)
\(824\) 0 0
\(825\) −5.28265 −0.183918
\(826\) 0 0
\(827\) −41.3296 −1.43717 −0.718585 0.695439i \(-0.755210\pi\)
−0.718585 + 0.695439i \(0.755210\pi\)
\(828\) 0 0
\(829\) −40.4799 −1.40592 −0.702962 0.711227i \(-0.748140\pi\)
−0.702962 + 0.711227i \(0.748140\pi\)
\(830\) 0 0
\(831\) −14.1184 −0.489763
\(832\) 0 0
\(833\) −12.7502 −0.441767
\(834\) 0 0
\(835\) 21.4216 0.741323
\(836\) 0 0
\(837\) 8.11301 0.280426
\(838\) 0 0
\(839\) −40.6427 −1.40314 −0.701571 0.712600i \(-0.747518\pi\)
−0.701571 + 0.712600i \(0.747518\pi\)
\(840\) 0 0
\(841\) −27.4877 −0.947853
\(842\) 0 0
\(843\) −22.4625 −0.773649
\(844\) 0 0
\(845\) 3.20666 0.110312
\(846\) 0 0
\(847\) 1.53005 0.0525732
\(848\) 0 0
\(849\) −27.1165 −0.930635
\(850\) 0 0
\(851\) −44.1445 −1.51325
\(852\) 0 0
\(853\) −32.3815 −1.10872 −0.554361 0.832276i \(-0.687037\pi\)
−0.554361 + 0.832276i \(0.687037\pi\)
\(854\) 0 0
\(855\) 24.9786 0.854250
\(856\) 0 0
\(857\) −36.8956 −1.26033 −0.630165 0.776461i \(-0.717013\pi\)
−0.630165 + 0.776461i \(0.717013\pi\)
\(858\) 0 0
\(859\) −23.5013 −0.801853 −0.400927 0.916110i \(-0.631312\pi\)
−0.400927 + 0.916110i \(0.631312\pi\)
\(860\) 0 0
\(861\) 5.40116 0.184071
\(862\) 0 0
\(863\) −9.31042 −0.316930 −0.158465 0.987365i \(-0.550655\pi\)
−0.158465 + 0.987365i \(0.550655\pi\)
\(864\) 0 0
\(865\) −0.543980 −0.0184959
\(866\) 0 0
\(867\) −9.51043 −0.322991
\(868\) 0 0
\(869\) −1.98411 −0.0673064
\(870\) 0 0
\(871\) −5.80727 −0.196772
\(872\) 0 0
\(873\) 15.9185 0.538760
\(874\) 0 0
\(875\) 1.38676 0.0468810
\(876\) 0 0
\(877\) −15.3601 −0.518675 −0.259338 0.965787i \(-0.583504\pi\)
−0.259338 + 0.965787i \(0.583504\pi\)
\(878\) 0 0
\(879\) 2.74391 0.0925497
\(880\) 0 0
\(881\) 44.9232 1.51350 0.756750 0.653704i \(-0.226786\pi\)
0.756750 + 0.653704i \(0.226786\pi\)
\(882\) 0 0
\(883\) −26.2892 −0.884702 −0.442351 0.896842i \(-0.645855\pi\)
−0.442351 + 0.896842i \(0.645855\pi\)
\(884\) 0 0
\(885\) −40.6375 −1.36602
\(886\) 0 0
\(887\) 31.5400 1.05901 0.529504 0.848307i \(-0.322378\pi\)
0.529504 + 0.848307i \(0.322378\pi\)
\(888\) 0 0
\(889\) −18.2116 −0.610798
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 22.9007 0.766341
\(894\) 0 0
\(895\) 24.5931 0.822058
\(896\) 0 0
\(897\) −6.88326 −0.229825
\(898\) 0 0
\(899\) 9.97691 0.332749
\(900\) 0 0
\(901\) −27.9856 −0.932336
\(902\) 0 0
\(903\) −6.09318 −0.202768
\(904\) 0 0
\(905\) 3.32535 0.110538
\(906\) 0 0
\(907\) 42.6502 1.41618 0.708088 0.706124i \(-0.249558\pi\)
0.708088 + 0.706124i \(0.249558\pi\)
\(908\) 0 0
\(909\) 9.82858 0.325994
\(910\) 0 0
\(911\) −27.8052 −0.921228 −0.460614 0.887600i \(-0.652371\pi\)
−0.460614 + 0.887600i \(0.652371\pi\)
\(912\) 0 0
\(913\) 5.16591 0.170967
\(914\) 0 0
\(915\) −20.2203 −0.668463
\(916\) 0 0
\(917\) 29.2735 0.966696
\(918\) 0 0
\(919\) 21.6547 0.714324 0.357162 0.934042i \(-0.383744\pi\)
0.357162 + 0.934042i \(0.383744\pi\)
\(920\) 0 0
\(921\) −15.1110 −0.497926
\(922\) 0 0
\(923\) 10.2260 0.336593
\(924\) 0 0
\(925\) −33.8793 −1.11394
\(926\) 0 0
\(927\) 10.3057 0.338485
\(928\) 0 0
\(929\) −48.8228 −1.60182 −0.800912 0.598782i \(-0.795652\pi\)
−0.800912 + 0.598782i \(0.795652\pi\)
\(930\) 0 0
\(931\) −36.2914 −1.18940
\(932\) 0 0
\(933\) −7.94159 −0.259996
\(934\) 0 0
\(935\) −8.77568 −0.286995
\(936\) 0 0
\(937\) −53.2181 −1.73856 −0.869279 0.494321i \(-0.835417\pi\)
−0.869279 + 0.494321i \(0.835417\pi\)
\(938\) 0 0
\(939\) 15.2613 0.498032
\(940\) 0 0
\(941\) −27.8636 −0.908329 −0.454164 0.890918i \(-0.650062\pi\)
−0.454164 + 0.890918i \(0.650062\pi\)
\(942\) 0 0
\(943\) 24.2983 0.791260
\(944\) 0 0
\(945\) 4.90635 0.159604
\(946\) 0 0
\(947\) 60.2359 1.95740 0.978702 0.205285i \(-0.0658122\pi\)
0.978702 + 0.205285i \(0.0658122\pi\)
\(948\) 0 0
\(949\) 15.4486 0.501481
\(950\) 0 0
\(951\) 1.83905 0.0596352
\(952\) 0 0
\(953\) 45.0928 1.46070 0.730350 0.683073i \(-0.239357\pi\)
0.730350 + 0.683073i \(0.239357\pi\)
\(954\) 0 0
\(955\) 0.641098 0.0207454
\(956\) 0 0
\(957\) −1.22974 −0.0397520
\(958\) 0 0
\(959\) 7.68258 0.248084
\(960\) 0 0
\(961\) 34.8209 1.12325
\(962\) 0 0
\(963\) −2.86018 −0.0921679
\(964\) 0 0
\(965\) 65.7585 2.11684
\(966\) 0 0
\(967\) −23.8936 −0.768368 −0.384184 0.923257i \(-0.625517\pi\)
−0.384184 + 0.923257i \(0.625517\pi\)
\(968\) 0 0
\(969\) 21.3179 0.684829
\(970\) 0 0
\(971\) −45.4503 −1.45857 −0.729285 0.684210i \(-0.760147\pi\)
−0.729285 + 0.684210i \(0.760147\pi\)
\(972\) 0 0
\(973\) −21.6289 −0.693392
\(974\) 0 0
\(975\) −5.28265 −0.169180
\(976\) 0 0
\(977\) 19.6461 0.628533 0.314266 0.949335i \(-0.398241\pi\)
0.314266 + 0.949335i \(0.398241\pi\)
\(978\) 0 0
\(979\) 4.11301 0.131452
\(980\) 0 0
\(981\) −10.4364 −0.333209
\(982\) 0 0
\(983\) 29.6742 0.946459 0.473230 0.880939i \(-0.343088\pi\)
0.473230 + 0.880939i \(0.343088\pi\)
\(984\) 0 0
\(985\) −33.6140 −1.07103
\(986\) 0 0
\(987\) 4.49819 0.143179
\(988\) 0 0
\(989\) −27.4115 −0.871634
\(990\) 0 0
\(991\) −2.55267 −0.0810882 −0.0405441 0.999178i \(-0.512909\pi\)
−0.0405441 + 0.999178i \(0.512909\pi\)
\(992\) 0 0
\(993\) 30.6726 0.973366
\(994\) 0 0
\(995\) 64.5129 2.04520
\(996\) 0 0
\(997\) −32.7260 −1.03644 −0.518221 0.855247i \(-0.673406\pi\)
−0.518221 + 0.855247i \(0.673406\pi\)
\(998\) 0 0
\(999\) −6.41331 −0.202908
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 6864.2.a.ca.1.4 4
4.3 odd 2 3432.2.a.r.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.r.1.4 4 4.3 odd 2
6864.2.a.ca.1.4 4 1.1 even 1 trivial