Properties

Label 8016.2.a.w.1.4
Level 8016
Weight 2
Character 8016.1
Self dual yes
Analytic conductor 64.008
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.05123\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.597616 q^{5} -2.60994 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.597616 q^{5} -2.60994 q^{7} +1.00000 q^{9} +4.70008 q^{11} -4.30432 q^{13} -0.597616 q^{15} +2.94088 q^{17} +4.08825 q^{19} -2.60994 q^{21} -2.33094 q^{23} -4.64285 q^{25} +1.00000 q^{27} -2.00408 q^{29} -5.83579 q^{31} +4.70008 q^{33} +1.55974 q^{35} -1.82343 q^{37} -4.30432 q^{39} +8.73365 q^{41} -11.8623 q^{43} -0.597616 q^{45} +2.91468 q^{47} -0.188202 q^{49} +2.94088 q^{51} +9.63168 q^{53} -2.80885 q^{55} +4.08825 q^{57} -6.28216 q^{59} -8.58971 q^{61} -2.60994 q^{63} +2.57233 q^{65} +5.46847 q^{67} -2.33094 q^{69} -9.16011 q^{71} -2.56015 q^{73} -4.64285 q^{75} -12.2669 q^{77} -4.32690 q^{79} +1.00000 q^{81} +8.40318 q^{83} -1.75752 q^{85} -2.00408 q^{87} +8.59052 q^{89} +11.2340 q^{91} -5.83579 q^{93} -2.44321 q^{95} +6.10370 q^{97} +4.70008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} + O(q^{10}) \) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} - q^{11} - 2q^{13} - 3q^{15} + 11q^{17} - 2q^{19} - 8q^{21} - 17q^{23} + 4q^{25} + 7q^{27} - 7q^{29} - 10q^{31} - q^{33} - 10q^{35} - 21q^{37} - 2q^{39} + 8q^{41} + 12q^{43} - 3q^{45} - 25q^{47} - 7q^{49} + 11q^{51} - 7q^{53} - 15q^{55} - 2q^{57} - 3q^{59} - 14q^{61} - 8q^{63} + 4q^{65} - 4q^{67} - 17q^{69} - 27q^{71} - 12q^{73} + 4q^{75} + 16q^{77} - 8q^{79} + 7q^{81} - 15q^{83} - 3q^{85} - 7q^{87} + 14q^{89} + 3q^{91} - 10q^{93} - 37q^{95} + 3q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) −0.597616 −0.267262 −0.133631 0.991031i \(-0.542664\pi\)
−0.133631 + 0.991031i \(0.542664\pi\)
\(6\) 0 0
\(7\) −2.60994 −0.986465 −0.493233 0.869897i \(-0.664185\pi\)
−0.493233 + 0.869897i \(0.664185\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.70008 1.41713 0.708564 0.705646i \(-0.249343\pi\)
0.708564 + 0.705646i \(0.249343\pi\)
\(12\) 0 0
\(13\) −4.30432 −1.19380 −0.596902 0.802314i \(-0.703602\pi\)
−0.596902 + 0.802314i \(0.703602\pi\)
\(14\) 0 0
\(15\) −0.597616 −0.154304
\(16\) 0 0
\(17\) 2.94088 0.713269 0.356635 0.934244i \(-0.383924\pi\)
0.356635 + 0.934244i \(0.383924\pi\)
\(18\) 0 0
\(19\) 4.08825 0.937910 0.468955 0.883222i \(-0.344631\pi\)
0.468955 + 0.883222i \(0.344631\pi\)
\(20\) 0 0
\(21\) −2.60994 −0.569536
\(22\) 0 0
\(23\) −2.33094 −0.486035 −0.243018 0.970022i \(-0.578137\pi\)
−0.243018 + 0.970022i \(0.578137\pi\)
\(24\) 0 0
\(25\) −4.64285 −0.928571
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00408 −0.372148 −0.186074 0.982536i \(-0.559576\pi\)
−0.186074 + 0.982536i \(0.559576\pi\)
\(30\) 0 0
\(31\) −5.83579 −1.04814 −0.524070 0.851675i \(-0.675587\pi\)
−0.524070 + 0.851675i \(0.675587\pi\)
\(32\) 0 0
\(33\) 4.70008 0.818179
\(34\) 0 0
\(35\) 1.55974 0.263645
\(36\) 0 0
\(37\) −1.82343 −0.299770 −0.149885 0.988703i \(-0.547890\pi\)
−0.149885 + 0.988703i \(0.547890\pi\)
\(38\) 0 0
\(39\) −4.30432 −0.689243
\(40\) 0 0
\(41\) 8.73365 1.36397 0.681984 0.731367i \(-0.261118\pi\)
0.681984 + 0.731367i \(0.261118\pi\)
\(42\) 0 0
\(43\) −11.8623 −1.80898 −0.904489 0.426496i \(-0.859748\pi\)
−0.904489 + 0.426496i \(0.859748\pi\)
\(44\) 0 0
\(45\) −0.597616 −0.0890874
\(46\) 0 0
\(47\) 2.91468 0.425150 0.212575 0.977145i \(-0.431815\pi\)
0.212575 + 0.977145i \(0.431815\pi\)
\(48\) 0 0
\(49\) −0.188202 −0.0268860
\(50\) 0 0
\(51\) 2.94088 0.411806
\(52\) 0 0
\(53\) 9.63168 1.32301 0.661507 0.749939i \(-0.269917\pi\)
0.661507 + 0.749939i \(0.269917\pi\)
\(54\) 0 0
\(55\) −2.80885 −0.378745
\(56\) 0 0
\(57\) 4.08825 0.541502
\(58\) 0 0
\(59\) −6.28216 −0.817867 −0.408934 0.912564i \(-0.634099\pi\)
−0.408934 + 0.912564i \(0.634099\pi\)
\(60\) 0 0
\(61\) −8.58971 −1.09980 −0.549900 0.835231i \(-0.685334\pi\)
−0.549900 + 0.835231i \(0.685334\pi\)
\(62\) 0 0
\(63\) −2.60994 −0.328822
\(64\) 0 0
\(65\) 2.57233 0.319059
\(66\) 0 0
\(67\) 5.46847 0.668080 0.334040 0.942559i \(-0.391588\pi\)
0.334040 + 0.942559i \(0.391588\pi\)
\(68\) 0 0
\(69\) −2.33094 −0.280612
\(70\) 0 0
\(71\) −9.16011 −1.08711 −0.543553 0.839375i \(-0.682921\pi\)
−0.543553 + 0.839375i \(0.682921\pi\)
\(72\) 0 0
\(73\) −2.56015 −0.299643 −0.149822 0.988713i \(-0.547870\pi\)
−0.149822 + 0.988713i \(0.547870\pi\)
\(74\) 0 0
\(75\) −4.64285 −0.536111
\(76\) 0 0
\(77\) −12.2669 −1.39795
\(78\) 0 0
\(79\) −4.32690 −0.486815 −0.243407 0.969924i \(-0.578265\pi\)
−0.243407 + 0.969924i \(0.578265\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.40318 0.922369 0.461185 0.887304i \(-0.347425\pi\)
0.461185 + 0.887304i \(0.347425\pi\)
\(84\) 0 0
\(85\) −1.75752 −0.190630
\(86\) 0 0
\(87\) −2.00408 −0.214860
\(88\) 0 0
\(89\) 8.59052 0.910593 0.455297 0.890340i \(-0.349533\pi\)
0.455297 + 0.890340i \(0.349533\pi\)
\(90\) 0 0
\(91\) 11.2340 1.17765
\(92\) 0 0
\(93\) −5.83579 −0.605144
\(94\) 0 0
\(95\) −2.44321 −0.250668
\(96\) 0 0
\(97\) 6.10370 0.619737 0.309868 0.950779i \(-0.399715\pi\)
0.309868 + 0.950779i \(0.399715\pi\)
\(98\) 0 0
\(99\) 4.70008 0.472376
\(100\) 0 0
\(101\) 3.97986 0.396011 0.198005 0.980201i \(-0.436554\pi\)
0.198005 + 0.980201i \(0.436554\pi\)
\(102\) 0 0
\(103\) 0.527144 0.0519410 0.0259705 0.999663i \(-0.491732\pi\)
0.0259705 + 0.999663i \(0.491732\pi\)
\(104\) 0 0
\(105\) 1.55974 0.152215
\(106\) 0 0
\(107\) −17.5907 −1.70056 −0.850279 0.526332i \(-0.823567\pi\)
−0.850279 + 0.526332i \(0.823567\pi\)
\(108\) 0 0
\(109\) −3.62048 −0.346779 −0.173389 0.984853i \(-0.555472\pi\)
−0.173389 + 0.984853i \(0.555472\pi\)
\(110\) 0 0
\(111\) −1.82343 −0.173072
\(112\) 0 0
\(113\) −13.2414 −1.24564 −0.622822 0.782364i \(-0.714014\pi\)
−0.622822 + 0.782364i \(0.714014\pi\)
\(114\) 0 0
\(115\) 1.39301 0.129899
\(116\) 0 0
\(117\) −4.30432 −0.397935
\(118\) 0 0
\(119\) −7.67554 −0.703615
\(120\) 0 0
\(121\) 11.0908 1.00825
\(122\) 0 0
\(123\) 8.73365 0.787487
\(124\) 0 0
\(125\) 5.76273 0.515434
\(126\) 0 0
\(127\) 11.7823 1.04551 0.522756 0.852483i \(-0.324904\pi\)
0.522756 + 0.852483i \(0.324904\pi\)
\(128\) 0 0
\(129\) −11.8623 −1.04441
\(130\) 0 0
\(131\) −0.718799 −0.0628018 −0.0314009 0.999507i \(-0.509997\pi\)
−0.0314009 + 0.999507i \(0.509997\pi\)
\(132\) 0 0
\(133\) −10.6701 −0.925215
\(134\) 0 0
\(135\) −0.597616 −0.0514346
\(136\) 0 0
\(137\) 17.6668 1.50937 0.754687 0.656086i \(-0.227789\pi\)
0.754687 + 0.656086i \(0.227789\pi\)
\(138\) 0 0
\(139\) −21.9527 −1.86200 −0.931002 0.365013i \(-0.881064\pi\)
−0.931002 + 0.365013i \(0.881064\pi\)
\(140\) 0 0
\(141\) 2.91468 0.245460
\(142\) 0 0
\(143\) −20.2307 −1.69177
\(144\) 0 0
\(145\) 1.19767 0.0994611
\(146\) 0 0
\(147\) −0.188202 −0.0155227
\(148\) 0 0
\(149\) −5.43757 −0.445463 −0.222731 0.974880i \(-0.571497\pi\)
−0.222731 + 0.974880i \(0.571497\pi\)
\(150\) 0 0
\(151\) −0.821226 −0.0668304 −0.0334152 0.999442i \(-0.510638\pi\)
−0.0334152 + 0.999442i \(0.510638\pi\)
\(152\) 0 0
\(153\) 2.94088 0.237756
\(154\) 0 0
\(155\) 3.48757 0.280128
\(156\) 0 0
\(157\) −9.50629 −0.758685 −0.379342 0.925256i \(-0.623850\pi\)
−0.379342 + 0.925256i \(0.623850\pi\)
\(158\) 0 0
\(159\) 9.63168 0.763842
\(160\) 0 0
\(161\) 6.08362 0.479457
\(162\) 0 0
\(163\) 12.2885 0.962506 0.481253 0.876582i \(-0.340182\pi\)
0.481253 + 0.876582i \(0.340182\pi\)
\(164\) 0 0
\(165\) −2.80885 −0.218668
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 5.52719 0.425169
\(170\) 0 0
\(171\) 4.08825 0.312637
\(172\) 0 0
\(173\) 21.7951 1.65705 0.828525 0.559952i \(-0.189180\pi\)
0.828525 + 0.559952i \(0.189180\pi\)
\(174\) 0 0
\(175\) 12.1176 0.916003
\(176\) 0 0
\(177\) −6.28216 −0.472196
\(178\) 0 0
\(179\) −16.7053 −1.24861 −0.624307 0.781179i \(-0.714619\pi\)
−0.624307 + 0.781179i \(0.714619\pi\)
\(180\) 0 0
\(181\) 3.62717 0.269606 0.134803 0.990872i \(-0.456960\pi\)
0.134803 + 0.990872i \(0.456960\pi\)
\(182\) 0 0
\(183\) −8.58971 −0.634969
\(184\) 0 0
\(185\) 1.08971 0.0801171
\(186\) 0 0
\(187\) 13.8224 1.01079
\(188\) 0 0
\(189\) −2.60994 −0.189845
\(190\) 0 0
\(191\) −19.6156 −1.41933 −0.709667 0.704537i \(-0.751154\pi\)
−0.709667 + 0.704537i \(0.751154\pi\)
\(192\) 0 0
\(193\) −4.10177 −0.295252 −0.147626 0.989043i \(-0.547163\pi\)
−0.147626 + 0.989043i \(0.547163\pi\)
\(194\) 0 0
\(195\) 2.57233 0.184209
\(196\) 0 0
\(197\) −14.5624 −1.03753 −0.518763 0.854918i \(-0.673607\pi\)
−0.518763 + 0.854918i \(0.673607\pi\)
\(198\) 0 0
\(199\) −18.9202 −1.34122 −0.670608 0.741812i \(-0.733967\pi\)
−0.670608 + 0.741812i \(0.733967\pi\)
\(200\) 0 0
\(201\) 5.46847 0.385716
\(202\) 0 0
\(203\) 5.23053 0.367111
\(204\) 0 0
\(205\) −5.21937 −0.364537
\(206\) 0 0
\(207\) −2.33094 −0.162012
\(208\) 0 0
\(209\) 19.2151 1.32914
\(210\) 0 0
\(211\) −14.7857 −1.01789 −0.508944 0.860800i \(-0.669964\pi\)
−0.508944 + 0.860800i \(0.669964\pi\)
\(212\) 0 0
\(213\) −9.16011 −0.627641
\(214\) 0 0
\(215\) 7.08908 0.483472
\(216\) 0 0
\(217\) 15.2311 1.03395
\(218\) 0 0
\(219\) −2.56015 −0.172999
\(220\) 0 0
\(221\) −12.6585 −0.851504
\(222\) 0 0
\(223\) −13.1986 −0.883844 −0.441922 0.897054i \(-0.645703\pi\)
−0.441922 + 0.897054i \(0.645703\pi\)
\(224\) 0 0
\(225\) −4.64285 −0.309524
\(226\) 0 0
\(227\) −0.0436562 −0.00289756 −0.00144878 0.999999i \(-0.500461\pi\)
−0.00144878 + 0.999999i \(0.500461\pi\)
\(228\) 0 0
\(229\) −17.2711 −1.14131 −0.570654 0.821190i \(-0.693310\pi\)
−0.570654 + 0.821190i \(0.693310\pi\)
\(230\) 0 0
\(231\) −12.2669 −0.807106
\(232\) 0 0
\(233\) −4.50687 −0.295255 −0.147627 0.989043i \(-0.547164\pi\)
−0.147627 + 0.989043i \(0.547164\pi\)
\(234\) 0 0
\(235\) −1.74186 −0.113626
\(236\) 0 0
\(237\) −4.32690 −0.281063
\(238\) 0 0
\(239\) −5.31017 −0.343486 −0.171743 0.985142i \(-0.554940\pi\)
−0.171743 + 0.985142i \(0.554940\pi\)
\(240\) 0 0
\(241\) −10.0415 −0.646830 −0.323415 0.946257i \(-0.604831\pi\)
−0.323415 + 0.946257i \(0.604831\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.112473 0.00718562
\(246\) 0 0
\(247\) −17.5972 −1.11968
\(248\) 0 0
\(249\) 8.40318 0.532530
\(250\) 0 0
\(251\) −13.4734 −0.850436 −0.425218 0.905091i \(-0.639802\pi\)
−0.425218 + 0.905091i \(0.639802\pi\)
\(252\) 0 0
\(253\) −10.9556 −0.688774
\(254\) 0 0
\(255\) −1.75752 −0.110060
\(256\) 0 0
\(257\) −8.27420 −0.516130 −0.258065 0.966127i \(-0.583085\pi\)
−0.258065 + 0.966127i \(0.583085\pi\)
\(258\) 0 0
\(259\) 4.75904 0.295712
\(260\) 0 0
\(261\) −2.00408 −0.124049
\(262\) 0 0
\(263\) 2.94503 0.181598 0.0907990 0.995869i \(-0.471058\pi\)
0.0907990 + 0.995869i \(0.471058\pi\)
\(264\) 0 0
\(265\) −5.75605 −0.353591
\(266\) 0 0
\(267\) 8.59052 0.525731
\(268\) 0 0
\(269\) 2.30723 0.140675 0.0703373 0.997523i \(-0.477592\pi\)
0.0703373 + 0.997523i \(0.477592\pi\)
\(270\) 0 0
\(271\) −17.9041 −1.08759 −0.543797 0.839217i \(-0.683014\pi\)
−0.543797 + 0.839217i \(0.683014\pi\)
\(272\) 0 0
\(273\) 11.2340 0.679915
\(274\) 0 0
\(275\) −21.8218 −1.31590
\(276\) 0 0
\(277\) 18.7020 1.12369 0.561846 0.827242i \(-0.310091\pi\)
0.561846 + 0.827242i \(0.310091\pi\)
\(278\) 0 0
\(279\) −5.83579 −0.349380
\(280\) 0 0
\(281\) −12.7088 −0.758144 −0.379072 0.925367i \(-0.623757\pi\)
−0.379072 + 0.925367i \(0.623757\pi\)
\(282\) 0 0
\(283\) −31.7962 −1.89009 −0.945044 0.326944i \(-0.893981\pi\)
−0.945044 + 0.326944i \(0.893981\pi\)
\(284\) 0 0
\(285\) −2.44321 −0.144723
\(286\) 0 0
\(287\) −22.7943 −1.34551
\(288\) 0 0
\(289\) −8.35120 −0.491247
\(290\) 0 0
\(291\) 6.10370 0.357805
\(292\) 0 0
\(293\) 10.5156 0.614327 0.307164 0.951657i \(-0.400620\pi\)
0.307164 + 0.951657i \(0.400620\pi\)
\(294\) 0 0
\(295\) 3.75432 0.218585
\(296\) 0 0
\(297\) 4.70008 0.272726
\(298\) 0 0
\(299\) 10.0331 0.580231
\(300\) 0 0
\(301\) 30.9598 1.78449
\(302\) 0 0
\(303\) 3.97986 0.228637
\(304\) 0 0
\(305\) 5.13335 0.293935
\(306\) 0 0
\(307\) −28.2239 −1.61082 −0.805412 0.592716i \(-0.798056\pi\)
−0.805412 + 0.592716i \(0.798056\pi\)
\(308\) 0 0
\(309\) 0.527144 0.0299882
\(310\) 0 0
\(311\) 5.06229 0.287056 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(312\) 0 0
\(313\) 4.34882 0.245810 0.122905 0.992418i \(-0.460779\pi\)
0.122905 + 0.992418i \(0.460779\pi\)
\(314\) 0 0
\(315\) 1.55974 0.0878816
\(316\) 0 0
\(317\) 18.2380 1.02435 0.512175 0.858881i \(-0.328840\pi\)
0.512175 + 0.858881i \(0.328840\pi\)
\(318\) 0 0
\(319\) −9.41934 −0.527382
\(320\) 0 0
\(321\) −17.5907 −0.981818
\(322\) 0 0
\(323\) 12.0231 0.668982
\(324\) 0 0
\(325\) 19.9843 1.10853
\(326\) 0 0
\(327\) −3.62048 −0.200213
\(328\) 0 0
\(329\) −7.60715 −0.419396
\(330\) 0 0
\(331\) −6.31296 −0.346992 −0.173496 0.984835i \(-0.555506\pi\)
−0.173496 + 0.984835i \(0.555506\pi\)
\(332\) 0 0
\(333\) −1.82343 −0.0999232
\(334\) 0 0
\(335\) −3.26805 −0.178553
\(336\) 0 0
\(337\) 10.2585 0.558815 0.279407 0.960173i \(-0.409862\pi\)
0.279407 + 0.960173i \(0.409862\pi\)
\(338\) 0 0
\(339\) −13.2414 −0.719173
\(340\) 0 0
\(341\) −27.4287 −1.48535
\(342\) 0 0
\(343\) 18.7608 1.01299
\(344\) 0 0
\(345\) 1.39301 0.0749971
\(346\) 0 0
\(347\) −25.2848 −1.35736 −0.678680 0.734434i \(-0.737448\pi\)
−0.678680 + 0.734434i \(0.737448\pi\)
\(348\) 0 0
\(349\) 17.7879 0.952165 0.476083 0.879401i \(-0.342056\pi\)
0.476083 + 0.879401i \(0.342056\pi\)
\(350\) 0 0
\(351\) −4.30432 −0.229748
\(352\) 0 0
\(353\) −21.9434 −1.16793 −0.583964 0.811780i \(-0.698499\pi\)
−0.583964 + 0.811780i \(0.698499\pi\)
\(354\) 0 0
\(355\) 5.47423 0.290542
\(356\) 0 0
\(357\) −7.67554 −0.406233
\(358\) 0 0
\(359\) 13.6414 0.719967 0.359983 0.932959i \(-0.382782\pi\)
0.359983 + 0.932959i \(0.382782\pi\)
\(360\) 0 0
\(361\) −2.28618 −0.120325
\(362\) 0 0
\(363\) 11.0908 0.582115
\(364\) 0 0
\(365\) 1.52999 0.0800833
\(366\) 0 0
\(367\) −6.32286 −0.330051 −0.165025 0.986289i \(-0.552771\pi\)
−0.165025 + 0.986289i \(0.552771\pi\)
\(368\) 0 0
\(369\) 8.73365 0.454656
\(370\) 0 0
\(371\) −25.1381 −1.30511
\(372\) 0 0
\(373\) 1.51797 0.0785974 0.0392987 0.999228i \(-0.487488\pi\)
0.0392987 + 0.999228i \(0.487488\pi\)
\(374\) 0 0
\(375\) 5.76273 0.297586
\(376\) 0 0
\(377\) 8.62620 0.444272
\(378\) 0 0
\(379\) 1.06679 0.0547973 0.0273986 0.999625i \(-0.491278\pi\)
0.0273986 + 0.999625i \(0.491278\pi\)
\(380\) 0 0
\(381\) 11.7823 0.603626
\(382\) 0 0
\(383\) −11.8956 −0.607837 −0.303919 0.952698i \(-0.598295\pi\)
−0.303919 + 0.952698i \(0.598295\pi\)
\(384\) 0 0
\(385\) 7.33093 0.373619
\(386\) 0 0
\(387\) −11.8623 −0.602993
\(388\) 0 0
\(389\) −21.6928 −1.09987 −0.549935 0.835207i \(-0.685348\pi\)
−0.549935 + 0.835207i \(0.685348\pi\)
\(390\) 0 0
\(391\) −6.85503 −0.346674
\(392\) 0 0
\(393\) −0.718799 −0.0362586
\(394\) 0 0
\(395\) 2.58583 0.130107
\(396\) 0 0
\(397\) −0.338020 −0.0169648 −0.00848238 0.999964i \(-0.502700\pi\)
−0.00848238 + 0.999964i \(0.502700\pi\)
\(398\) 0 0
\(399\) −10.6701 −0.534173
\(400\) 0 0
\(401\) 23.5413 1.17560 0.587798 0.809008i \(-0.299995\pi\)
0.587798 + 0.809008i \(0.299995\pi\)
\(402\) 0 0
\(403\) 25.1191 1.25127
\(404\) 0 0
\(405\) −0.597616 −0.0296958
\(406\) 0 0
\(407\) −8.57026 −0.424812
\(408\) 0 0
\(409\) −9.86567 −0.487826 −0.243913 0.969797i \(-0.578431\pi\)
−0.243913 + 0.969797i \(0.578431\pi\)
\(410\) 0 0
\(411\) 17.6668 0.871437
\(412\) 0 0
\(413\) 16.3961 0.806798
\(414\) 0 0
\(415\) −5.02188 −0.246514
\(416\) 0 0
\(417\) −21.9527 −1.07503
\(418\) 0 0
\(419\) −4.01568 −0.196179 −0.0980895 0.995178i \(-0.531273\pi\)
−0.0980895 + 0.995178i \(0.531273\pi\)
\(420\) 0 0
\(421\) 13.6265 0.664116 0.332058 0.943259i \(-0.392257\pi\)
0.332058 + 0.943259i \(0.392257\pi\)
\(422\) 0 0
\(423\) 2.91468 0.141717
\(424\) 0 0
\(425\) −13.6541 −0.662321
\(426\) 0 0
\(427\) 22.4186 1.08491
\(428\) 0 0
\(429\) −20.2307 −0.976746
\(430\) 0 0
\(431\) −34.0798 −1.64157 −0.820783 0.571241i \(-0.806462\pi\)
−0.820783 + 0.571241i \(0.806462\pi\)
\(432\) 0 0
\(433\) −12.6902 −0.609853 −0.304926 0.952376i \(-0.598632\pi\)
−0.304926 + 0.952376i \(0.598632\pi\)
\(434\) 0 0
\(435\) 1.19767 0.0574239
\(436\) 0 0
\(437\) −9.52948 −0.455857
\(438\) 0 0
\(439\) 16.2168 0.773985 0.386993 0.922083i \(-0.373514\pi\)
0.386993 + 0.922083i \(0.373514\pi\)
\(440\) 0 0
\(441\) −0.188202 −0.00896202
\(442\) 0 0
\(443\) 8.51185 0.404410 0.202205 0.979343i \(-0.435189\pi\)
0.202205 + 0.979343i \(0.435189\pi\)
\(444\) 0 0
\(445\) −5.13384 −0.243367
\(446\) 0 0
\(447\) −5.43757 −0.257188
\(448\) 0 0
\(449\) −17.0909 −0.806568 −0.403284 0.915075i \(-0.632131\pi\)
−0.403284 + 0.915075i \(0.632131\pi\)
\(450\) 0 0
\(451\) 41.0489 1.93292
\(452\) 0 0
\(453\) −0.821226 −0.0385846
\(454\) 0 0
\(455\) −6.71364 −0.314740
\(456\) 0 0
\(457\) −21.6512 −1.01280 −0.506400 0.862299i \(-0.669024\pi\)
−0.506400 + 0.862299i \(0.669024\pi\)
\(458\) 0 0
\(459\) 2.94088 0.137269
\(460\) 0 0
\(461\) −0.243769 −0.0113534 −0.00567672 0.999984i \(-0.501807\pi\)
−0.00567672 + 0.999984i \(0.501807\pi\)
\(462\) 0 0
\(463\) 29.8493 1.38721 0.693606 0.720354i \(-0.256021\pi\)
0.693606 + 0.720354i \(0.256021\pi\)
\(464\) 0 0
\(465\) 3.48757 0.161732
\(466\) 0 0
\(467\) −21.9645 −1.01640 −0.508198 0.861240i \(-0.669688\pi\)
−0.508198 + 0.861240i \(0.669688\pi\)
\(468\) 0 0
\(469\) −14.2724 −0.659038
\(470\) 0 0
\(471\) −9.50629 −0.438027
\(472\) 0 0
\(473\) −55.7536 −2.56355
\(474\) 0 0
\(475\) −18.9812 −0.870916
\(476\) 0 0
\(477\) 9.63168 0.441004
\(478\) 0 0
\(479\) 25.4694 1.16372 0.581862 0.813287i \(-0.302324\pi\)
0.581862 + 0.813287i \(0.302324\pi\)
\(480\) 0 0
\(481\) 7.84862 0.357866
\(482\) 0 0
\(483\) 6.08362 0.276814
\(484\) 0 0
\(485\) −3.64767 −0.165632
\(486\) 0 0
\(487\) −10.6138 −0.480959 −0.240480 0.970654i \(-0.577305\pi\)
−0.240480 + 0.970654i \(0.577305\pi\)
\(488\) 0 0
\(489\) 12.2885 0.555703
\(490\) 0 0
\(491\) −21.3302 −0.962617 −0.481308 0.876551i \(-0.659838\pi\)
−0.481308 + 0.876551i \(0.659838\pi\)
\(492\) 0 0
\(493\) −5.89376 −0.265442
\(494\) 0 0
\(495\) −2.80885 −0.126248
\(496\) 0 0
\(497\) 23.9074 1.07239
\(498\) 0 0
\(499\) 1.09710 0.0491129 0.0245565 0.999698i \(-0.492183\pi\)
0.0245565 + 0.999698i \(0.492183\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 32.3037 1.44035 0.720175 0.693793i \(-0.244062\pi\)
0.720175 + 0.693793i \(0.244062\pi\)
\(504\) 0 0
\(505\) −2.37843 −0.105839
\(506\) 0 0
\(507\) 5.52719 0.245471
\(508\) 0 0
\(509\) −30.6050 −1.35654 −0.678271 0.734812i \(-0.737270\pi\)
−0.678271 + 0.734812i \(0.737270\pi\)
\(510\) 0 0
\(511\) 6.68185 0.295588
\(512\) 0 0
\(513\) 4.08825 0.180501
\(514\) 0 0
\(515\) −0.315030 −0.0138819
\(516\) 0 0
\(517\) 13.6992 0.602492
\(518\) 0 0
\(519\) 21.7951 0.956699
\(520\) 0 0
\(521\) 36.2872 1.58977 0.794885 0.606759i \(-0.207531\pi\)
0.794885 + 0.606759i \(0.207531\pi\)
\(522\) 0 0
\(523\) 32.5672 1.42406 0.712031 0.702148i \(-0.247775\pi\)
0.712031 + 0.702148i \(0.247775\pi\)
\(524\) 0 0
\(525\) 12.1176 0.528855
\(526\) 0 0
\(527\) −17.1624 −0.747605
\(528\) 0 0
\(529\) −17.5667 −0.763770
\(530\) 0 0
\(531\) −6.28216 −0.272622
\(532\) 0 0
\(533\) −37.5925 −1.62831
\(534\) 0 0
\(535\) 10.5125 0.454495
\(536\) 0 0
\(537\) −16.7053 −0.720887
\(538\) 0 0
\(539\) −0.884567 −0.0381010
\(540\) 0 0
\(541\) −10.6189 −0.456541 −0.228270 0.973598i \(-0.573307\pi\)
−0.228270 + 0.973598i \(0.573307\pi\)
\(542\) 0 0
\(543\) 3.62717 0.155657
\(544\) 0 0
\(545\) 2.16366 0.0926808
\(546\) 0 0
\(547\) 3.13950 0.134235 0.0671177 0.997745i \(-0.478620\pi\)
0.0671177 + 0.997745i \(0.478620\pi\)
\(548\) 0 0
\(549\) −8.58971 −0.366600
\(550\) 0 0
\(551\) −8.19318 −0.349041
\(552\) 0 0
\(553\) 11.2930 0.480226
\(554\) 0 0
\(555\) 1.08971 0.0462556
\(556\) 0 0
\(557\) 27.4739 1.16411 0.582053 0.813151i \(-0.302250\pi\)
0.582053 + 0.813151i \(0.302250\pi\)
\(558\) 0 0
\(559\) 51.0590 2.15957
\(560\) 0 0
\(561\) 13.8224 0.583582
\(562\) 0 0
\(563\) 16.0913 0.678169 0.339084 0.940756i \(-0.389883\pi\)
0.339084 + 0.940756i \(0.389883\pi\)
\(564\) 0 0
\(565\) 7.91326 0.332913
\(566\) 0 0
\(567\) −2.60994 −0.109607
\(568\) 0 0
\(569\) −9.40907 −0.394449 −0.197224 0.980358i \(-0.563193\pi\)
−0.197224 + 0.980358i \(0.563193\pi\)
\(570\) 0 0
\(571\) 38.6950 1.61934 0.809668 0.586888i \(-0.199647\pi\)
0.809668 + 0.586888i \(0.199647\pi\)
\(572\) 0 0
\(573\) −19.6156 −0.819453
\(574\) 0 0
\(575\) 10.8222 0.451318
\(576\) 0 0
\(577\) 26.8472 1.11767 0.558833 0.829280i \(-0.311249\pi\)
0.558833 + 0.829280i \(0.311249\pi\)
\(578\) 0 0
\(579\) −4.10177 −0.170464
\(580\) 0 0
\(581\) −21.9318 −0.909885
\(582\) 0 0
\(583\) 45.2697 1.87488
\(584\) 0 0
\(585\) 2.57233 0.106353
\(586\) 0 0
\(587\) 37.0850 1.53066 0.765332 0.643636i \(-0.222575\pi\)
0.765332 + 0.643636i \(0.222575\pi\)
\(588\) 0 0
\(589\) −23.8582 −0.983060
\(590\) 0 0
\(591\) −14.5624 −0.599016
\(592\) 0 0
\(593\) −24.0629 −0.988145 −0.494072 0.869421i \(-0.664492\pi\)
−0.494072 + 0.869421i \(0.664492\pi\)
\(594\) 0 0
\(595\) 4.58703 0.188050
\(596\) 0 0
\(597\) −18.9202 −0.774352
\(598\) 0 0
\(599\) −37.7712 −1.54329 −0.771644 0.636054i \(-0.780565\pi\)
−0.771644 + 0.636054i \(0.780565\pi\)
\(600\) 0 0
\(601\) −17.0915 −0.697175 −0.348588 0.937276i \(-0.613339\pi\)
−0.348588 + 0.937276i \(0.613339\pi\)
\(602\) 0 0
\(603\) 5.46847 0.222693
\(604\) 0 0
\(605\) −6.62803 −0.269468
\(606\) 0 0
\(607\) 14.5907 0.592217 0.296109 0.955154i \(-0.404311\pi\)
0.296109 + 0.955154i \(0.404311\pi\)
\(608\) 0 0
\(609\) 5.23053 0.211952
\(610\) 0 0
\(611\) −12.5457 −0.507546
\(612\) 0 0
\(613\) 36.1169 1.45875 0.729375 0.684115i \(-0.239811\pi\)
0.729375 + 0.684115i \(0.239811\pi\)
\(614\) 0 0
\(615\) −5.21937 −0.210465
\(616\) 0 0
\(617\) 29.4728 1.18653 0.593265 0.805007i \(-0.297838\pi\)
0.593265 + 0.805007i \(0.297838\pi\)
\(618\) 0 0
\(619\) −20.6128 −0.828497 −0.414249 0.910164i \(-0.635956\pi\)
−0.414249 + 0.910164i \(0.635956\pi\)
\(620\) 0 0
\(621\) −2.33094 −0.0935375
\(622\) 0 0
\(623\) −22.4208 −0.898269
\(624\) 0 0
\(625\) 19.7704 0.790815
\(626\) 0 0
\(627\) 19.2151 0.767378
\(628\) 0 0
\(629\) −5.36249 −0.213816
\(630\) 0 0
\(631\) 21.1872 0.843450 0.421725 0.906724i \(-0.361425\pi\)
0.421725 + 0.906724i \(0.361425\pi\)
\(632\) 0 0
\(633\) −14.7857 −0.587678
\(634\) 0 0
\(635\) −7.04130 −0.279426
\(636\) 0 0
\(637\) 0.810084 0.0320967
\(638\) 0 0
\(639\) −9.16011 −0.362368
\(640\) 0 0
\(641\) −11.0393 −0.436025 −0.218012 0.975946i \(-0.569957\pi\)
−0.218012 + 0.975946i \(0.569957\pi\)
\(642\) 0 0
\(643\) −29.4130 −1.15994 −0.579968 0.814639i \(-0.696935\pi\)
−0.579968 + 0.814639i \(0.696935\pi\)
\(644\) 0 0
\(645\) 7.08908 0.279132
\(646\) 0 0
\(647\) −33.6968 −1.32476 −0.662378 0.749169i \(-0.730453\pi\)
−0.662378 + 0.749169i \(0.730453\pi\)
\(648\) 0 0
\(649\) −29.5267 −1.15902
\(650\) 0 0
\(651\) 15.2311 0.596953
\(652\) 0 0
\(653\) −3.89310 −0.152349 −0.0761745 0.997095i \(-0.524271\pi\)
−0.0761745 + 0.997095i \(0.524271\pi\)
\(654\) 0 0
\(655\) 0.429566 0.0167845
\(656\) 0 0
\(657\) −2.56015 −0.0998810
\(658\) 0 0
\(659\) 48.8140 1.90152 0.950761 0.309926i \(-0.100304\pi\)
0.950761 + 0.309926i \(0.100304\pi\)
\(660\) 0 0
\(661\) −6.28000 −0.244264 −0.122132 0.992514i \(-0.538973\pi\)
−0.122132 + 0.992514i \(0.538973\pi\)
\(662\) 0 0
\(663\) −12.6585 −0.491616
\(664\) 0 0
\(665\) 6.37663 0.247275
\(666\) 0 0
\(667\) 4.67139 0.180877
\(668\) 0 0
\(669\) −13.1986 −0.510287
\(670\) 0 0
\(671\) −40.3723 −1.55856
\(672\) 0 0
\(673\) −25.2666 −0.973954 −0.486977 0.873415i \(-0.661901\pi\)
−0.486977 + 0.873415i \(0.661901\pi\)
\(674\) 0 0
\(675\) −4.64285 −0.178704
\(676\) 0 0
\(677\) 8.38840 0.322392 0.161196 0.986922i \(-0.448465\pi\)
0.161196 + 0.986922i \(0.448465\pi\)
\(678\) 0 0
\(679\) −15.9303 −0.611349
\(680\) 0 0
\(681\) −0.0436562 −0.00167291
\(682\) 0 0
\(683\) −40.1864 −1.53769 −0.768844 0.639436i \(-0.779168\pi\)
−0.768844 + 0.639436i \(0.779168\pi\)
\(684\) 0 0
\(685\) −10.5579 −0.403398
\(686\) 0 0
\(687\) −17.2711 −0.658935
\(688\) 0 0
\(689\) −41.4579 −1.57942
\(690\) 0 0
\(691\) 37.9709 1.44448 0.722240 0.691642i \(-0.243112\pi\)
0.722240 + 0.691642i \(0.243112\pi\)
\(692\) 0 0
\(693\) −12.2669 −0.465983
\(694\) 0 0
\(695\) 13.1193 0.497643
\(696\) 0 0
\(697\) 25.6847 0.972876
\(698\) 0 0
\(699\) −4.50687 −0.170465
\(700\) 0 0
\(701\) −10.1397 −0.382972 −0.191486 0.981495i \(-0.561331\pi\)
−0.191486 + 0.981495i \(0.561331\pi\)
\(702\) 0 0
\(703\) −7.45463 −0.281157
\(704\) 0 0
\(705\) −1.74186 −0.0656023
\(706\) 0 0
\(707\) −10.3872 −0.390651
\(708\) 0 0
\(709\) 13.7876 0.517804 0.258902 0.965904i \(-0.416639\pi\)
0.258902 + 0.965904i \(0.416639\pi\)
\(710\) 0 0
\(711\) −4.32690 −0.162272
\(712\) 0 0
\(713\) 13.6029 0.509432
\(714\) 0 0
\(715\) 12.0902 0.452147
\(716\) 0 0
\(717\) −5.31017 −0.198312
\(718\) 0 0
\(719\) 19.1050 0.712496 0.356248 0.934391i \(-0.384056\pi\)
0.356248 + 0.934391i \(0.384056\pi\)
\(720\) 0 0
\(721\) −1.37581 −0.0512380
\(722\) 0 0
\(723\) −10.0415 −0.373447
\(724\) 0 0
\(725\) 9.30465 0.345566
\(726\) 0 0
\(727\) 52.7736 1.95726 0.978632 0.205618i \(-0.0659203\pi\)
0.978632 + 0.205618i \(0.0659203\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −34.8856 −1.29029
\(732\) 0 0
\(733\) 51.1372 1.88880 0.944399 0.328803i \(-0.106645\pi\)
0.944399 + 0.328803i \(0.106645\pi\)
\(734\) 0 0
\(735\) 0.112473 0.00414862
\(736\) 0 0
\(737\) 25.7023 0.946756
\(738\) 0 0
\(739\) −8.74479 −0.321682 −0.160841 0.986980i \(-0.551421\pi\)
−0.160841 + 0.986980i \(0.551421\pi\)
\(740\) 0 0
\(741\) −17.5972 −0.646448
\(742\) 0 0
\(743\) −28.4343 −1.04315 −0.521576 0.853205i \(-0.674656\pi\)
−0.521576 + 0.853205i \(0.674656\pi\)
\(744\) 0 0
\(745\) 3.24958 0.119055
\(746\) 0 0
\(747\) 8.40318 0.307456
\(748\) 0 0
\(749\) 45.9107 1.67754
\(750\) 0 0
\(751\) −29.7346 −1.08503 −0.542515 0.840046i \(-0.682528\pi\)
−0.542515 + 0.840046i \(0.682528\pi\)
\(752\) 0 0
\(753\) −13.4734 −0.490999
\(754\) 0 0
\(755\) 0.490778 0.0178612
\(756\) 0 0
\(757\) 12.8251 0.466135 0.233067 0.972461i \(-0.425124\pi\)
0.233067 + 0.972461i \(0.425124\pi\)
\(758\) 0 0
\(759\) −10.9556 −0.397664
\(760\) 0 0
\(761\) 50.6844 1.83731 0.918653 0.395065i \(-0.129278\pi\)
0.918653 + 0.395065i \(0.129278\pi\)
\(762\) 0 0
\(763\) 9.44923 0.342085
\(764\) 0 0
\(765\) −1.75752 −0.0635433
\(766\) 0 0
\(767\) 27.0404 0.976374
\(768\) 0 0
\(769\) −8.47689 −0.305684 −0.152842 0.988251i \(-0.548843\pi\)
−0.152842 + 0.988251i \(0.548843\pi\)
\(770\) 0 0
\(771\) −8.27420 −0.297988
\(772\) 0 0
\(773\) 31.4837 1.13239 0.566195 0.824272i \(-0.308415\pi\)
0.566195 + 0.824272i \(0.308415\pi\)
\(774\) 0 0
\(775\) 27.0947 0.973272
\(776\) 0 0
\(777\) 4.75904 0.170730
\(778\) 0 0
\(779\) 35.7054 1.27928
\(780\) 0 0
\(781\) −43.0533 −1.54057
\(782\) 0 0
\(783\) −2.00408 −0.0716199
\(784\) 0 0
\(785\) 5.68111 0.202768
\(786\) 0 0
\(787\) 25.4579 0.907475 0.453738 0.891135i \(-0.350090\pi\)
0.453738 + 0.891135i \(0.350090\pi\)
\(788\) 0 0
\(789\) 2.94503 0.104846
\(790\) 0 0
\(791\) 34.5592 1.22878
\(792\) 0 0
\(793\) 36.9729 1.31294
\(794\) 0 0
\(795\) −5.75605 −0.204146
\(796\) 0 0
\(797\) 17.5432 0.621413 0.310707 0.950506i \(-0.399434\pi\)
0.310707 + 0.950506i \(0.399434\pi\)
\(798\) 0 0
\(799\) 8.57174 0.303246
\(800\) 0 0
\(801\) 8.59052 0.303531
\(802\) 0 0
\(803\) −12.0329 −0.424633
\(804\) 0 0
\(805\) −3.63567 −0.128141
\(806\) 0 0
\(807\) 2.30723 0.0812185
\(808\) 0 0
\(809\) 29.8265 1.04865 0.524323 0.851520i \(-0.324319\pi\)
0.524323 + 0.851520i \(0.324319\pi\)
\(810\) 0 0
\(811\) 9.08561 0.319039 0.159519 0.987195i \(-0.449006\pi\)
0.159519 + 0.987195i \(0.449006\pi\)
\(812\) 0 0
\(813\) −17.9041 −0.627923
\(814\) 0 0
\(815\) −7.34378 −0.257241
\(816\) 0 0
\(817\) −48.4960 −1.69666
\(818\) 0 0
\(819\) 11.2340 0.392549
\(820\) 0 0
\(821\) 21.2210 0.740618 0.370309 0.928909i \(-0.379252\pi\)
0.370309 + 0.928909i \(0.379252\pi\)
\(822\) 0 0
\(823\) 2.12448 0.0740547 0.0370273 0.999314i \(-0.488211\pi\)
0.0370273 + 0.999314i \(0.488211\pi\)
\(824\) 0 0
\(825\) −21.8218 −0.759738
\(826\) 0 0
\(827\) 16.6543 0.579127 0.289564 0.957159i \(-0.406490\pi\)
0.289564 + 0.957159i \(0.406490\pi\)
\(828\) 0 0
\(829\) −44.1703 −1.53410 −0.767049 0.641588i \(-0.778276\pi\)
−0.767049 + 0.641588i \(0.778276\pi\)
\(830\) 0 0
\(831\) 18.7020 0.648764
\(832\) 0 0
\(833\) −0.553481 −0.0191770
\(834\) 0 0
\(835\) −0.597616 −0.0206814
\(836\) 0 0
\(837\) −5.83579 −0.201715
\(838\) 0 0
\(839\) 13.2467 0.457328 0.228664 0.973505i \(-0.426564\pi\)
0.228664 + 0.973505i \(0.426564\pi\)
\(840\) 0 0
\(841\) −24.9837 −0.861506
\(842\) 0 0
\(843\) −12.7088 −0.437714
\(844\) 0 0
\(845\) −3.30314 −0.113631
\(846\) 0 0
\(847\) −28.9463 −0.994607
\(848\) 0 0
\(849\) −31.7962 −1.09124
\(850\) 0 0
\(851\) 4.25030 0.145699
\(852\) 0 0
\(853\) −0.557679 −0.0190946 −0.00954728 0.999954i \(-0.503039\pi\)
−0.00954728 + 0.999954i \(0.503039\pi\)
\(854\) 0 0
\(855\) −2.44321 −0.0835559
\(856\) 0 0
\(857\) −20.2809 −0.692784 −0.346392 0.938090i \(-0.612593\pi\)
−0.346392 + 0.938090i \(0.612593\pi\)
\(858\) 0 0
\(859\) −53.5678 −1.82771 −0.913855 0.406040i \(-0.866909\pi\)
−0.913855 + 0.406040i \(0.866909\pi\)
\(860\) 0 0
\(861\) −22.7943 −0.776829
\(862\) 0 0
\(863\) 48.0933 1.63711 0.818557 0.574425i \(-0.194774\pi\)
0.818557 + 0.574425i \(0.194774\pi\)
\(864\) 0 0
\(865\) −13.0251 −0.442867
\(866\) 0 0
\(867\) −8.35120 −0.283622
\(868\) 0 0
\(869\) −20.3368 −0.689879
\(870\) 0 0
\(871\) −23.5381 −0.797557
\(872\) 0 0
\(873\) 6.10370 0.206579
\(874\) 0 0
\(875\) −15.0404 −0.508458
\(876\) 0 0
\(877\) 56.7094 1.91494 0.957470 0.288532i \(-0.0931670\pi\)
0.957470 + 0.288532i \(0.0931670\pi\)
\(878\) 0 0
\(879\) 10.5156 0.354682
\(880\) 0 0
\(881\) 1.31381 0.0442635 0.0221318 0.999755i \(-0.492955\pi\)
0.0221318 + 0.999755i \(0.492955\pi\)
\(882\) 0 0
\(883\) 20.3942 0.686321 0.343160 0.939277i \(-0.388503\pi\)
0.343160 + 0.939277i \(0.388503\pi\)
\(884\) 0 0
\(885\) 3.75432 0.126200
\(886\) 0 0
\(887\) 16.2392 0.545260 0.272630 0.962119i \(-0.412106\pi\)
0.272630 + 0.962119i \(0.412106\pi\)
\(888\) 0 0
\(889\) −30.7512 −1.03136
\(890\) 0 0
\(891\) 4.70008 0.157459
\(892\) 0 0
\(893\) 11.9160 0.398752
\(894\) 0 0
\(895\) 9.98337 0.333707
\(896\) 0 0
\(897\) 10.0331 0.334996
\(898\) 0 0
\(899\) 11.6954 0.390063
\(900\) 0 0
\(901\) 28.3257 0.943664
\(902\) 0 0
\(903\) 30.9598 1.03028
\(904\) 0 0
\(905\) −2.16766 −0.0720553
\(906\) 0 0
\(907\) 37.6423 1.24989 0.624946 0.780668i \(-0.285121\pi\)
0.624946 + 0.780668i \(0.285121\pi\)
\(908\) 0 0
\(909\) 3.97986 0.132004
\(910\) 0 0
\(911\) −10.1491 −0.336254 −0.168127 0.985765i \(-0.553772\pi\)
−0.168127 + 0.985765i \(0.553772\pi\)
\(912\) 0 0
\(913\) 39.4957 1.30712
\(914\) 0 0
\(915\) 5.13335 0.169703
\(916\) 0 0
\(917\) 1.87602 0.0619518
\(918\) 0 0
\(919\) −17.3876 −0.573563 −0.286781 0.957996i \(-0.592585\pi\)
−0.286781 + 0.957996i \(0.592585\pi\)
\(920\) 0 0
\(921\) −28.2239 −0.930009
\(922\) 0 0
\(923\) 39.4281 1.29779
\(924\) 0 0
\(925\) 8.46591 0.278357
\(926\) 0 0
\(927\) 0.527144 0.0173137
\(928\) 0 0
\(929\) −27.3989 −0.898930 −0.449465 0.893298i \(-0.648385\pi\)
−0.449465 + 0.893298i \(0.648385\pi\)
\(930\) 0 0
\(931\) −0.769419 −0.0252167
\(932\) 0 0
\(933\) 5.06229 0.165732
\(934\) 0 0
\(935\) −8.26049 −0.270147
\(936\) 0 0
\(937\) 10.8491 0.354424 0.177212 0.984173i \(-0.443292\pi\)
0.177212 + 0.984173i \(0.443292\pi\)
\(938\) 0 0
\(939\) 4.34882 0.141918
\(940\) 0 0
\(941\) −18.6872 −0.609184 −0.304592 0.952483i \(-0.598520\pi\)
−0.304592 + 0.952483i \(0.598520\pi\)
\(942\) 0 0
\(943\) −20.3576 −0.662936
\(944\) 0 0
\(945\) 1.55974 0.0507385
\(946\) 0 0
\(947\) 3.23655 0.105174 0.0525869 0.998616i \(-0.483253\pi\)
0.0525869 + 0.998616i \(0.483253\pi\)
\(948\) 0 0
\(949\) 11.0197 0.357715
\(950\) 0 0
\(951\) 18.2380 0.591409
\(952\) 0 0
\(953\) −19.3359 −0.626350 −0.313175 0.949695i \(-0.601393\pi\)
−0.313175 + 0.949695i \(0.601393\pi\)
\(954\) 0 0
\(955\) 11.7226 0.379334
\(956\) 0 0
\(957\) −9.41934 −0.304484
\(958\) 0 0
\(959\) −46.1092 −1.48894
\(960\) 0 0
\(961\) 3.05648 0.0985960
\(962\) 0 0
\(963\) −17.5907 −0.566853
\(964\) 0 0
\(965\) 2.45129 0.0789097
\(966\) 0 0
\(967\) 23.8874 0.768166 0.384083 0.923299i \(-0.374518\pi\)
0.384083 + 0.923299i \(0.374518\pi\)
\(968\) 0 0
\(969\) 12.0231 0.386237
\(970\) 0 0
\(971\) −11.7505 −0.377092 −0.188546 0.982064i \(-0.560377\pi\)
−0.188546 + 0.982064i \(0.560377\pi\)
\(972\) 0 0
\(973\) 57.2953 1.83680
\(974\) 0 0
\(975\) 19.9843 0.640011
\(976\) 0 0
\(977\) −10.6651 −0.341207 −0.170604 0.985340i \(-0.554572\pi\)
−0.170604 + 0.985340i \(0.554572\pi\)
\(978\) 0 0
\(979\) 40.3762 1.29043
\(980\) 0 0
\(981\) −3.62048 −0.115593
\(982\) 0 0
\(983\) 19.1379 0.610403 0.305201 0.952288i \(-0.401276\pi\)
0.305201 + 0.952288i \(0.401276\pi\)
\(984\) 0 0
\(985\) 8.70272 0.277292
\(986\) 0 0
\(987\) −7.60715 −0.242138
\(988\) 0 0
\(989\) 27.6503 0.879227
\(990\) 0 0
\(991\) 24.4631 0.777097 0.388549 0.921428i \(-0.372977\pi\)
0.388549 + 0.921428i \(0.372977\pi\)
\(992\) 0 0
\(993\) −6.31296 −0.200336
\(994\) 0 0
\(995\) 11.3070 0.358456
\(996\) 0 0
\(997\) 26.5133 0.839685 0.419842 0.907597i \(-0.362085\pi\)
0.419842 + 0.907597i \(0.362085\pi\)
\(998\) 0 0
\(999\) −1.82343 −0.0576907
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 8016.2.a.w.1.4 7
4.3 odd 2 4008.2.a.g.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.4 7 4.3 odd 2
8016.2.a.w.1.4 7 1.1 even 1 trivial