Properties

Label 3920.2.a.ca.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.585786 q^{3} -1.00000 q^{5} -2.65685 q^{9} +O(q^{10})\) \(q+0.585786 q^{3} -1.00000 q^{5} -2.65685 q^{9} -4.82843 q^{11} -0.828427 q^{13} -0.585786 q^{15} -5.41421 q^{17} +3.41421 q^{19} +6.82843 q^{23} +1.00000 q^{25} -3.31371 q^{27} +0.828427 q^{29} +2.82843 q^{31} -2.82843 q^{33} +3.65685 q^{37} -0.485281 q^{39} -11.0711 q^{41} +3.17157 q^{43} +2.65685 q^{45} +10.8284 q^{47} -3.17157 q^{51} -10.4853 q^{53} +4.82843 q^{55} +2.00000 q^{57} +11.4142 q^{59} +13.3137 q^{61} +0.828427 q^{65} +9.65685 q^{67} +4.00000 q^{69} -12.4853 q^{71} +6.58579 q^{73} +0.585786 q^{75} +1.17157 q^{79} +6.02944 q^{81} +6.24264 q^{83} +5.41421 q^{85} +0.485281 q^{87} +12.7279 q^{89} +1.65685 q^{93} -3.41421 q^{95} +16.2426 q^{97} +12.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 2q^{5} + 6q^{9} - 4q^{11} + 4q^{13} - 4q^{15} - 8q^{17} + 4q^{19} + 8q^{23} + 2q^{25} + 16q^{27} - 4q^{29} - 4q^{37} + 16q^{39} - 8q^{41} + 12q^{43} - 6q^{45} + 16q^{47} - 12q^{51} - 4q^{53} + 4q^{55} + 4q^{57} + 20q^{59} + 4q^{61} - 4q^{65} + 8q^{67} + 8q^{69} - 8q^{71} + 16q^{73} + 4q^{75} + 8q^{79} + 46q^{81} + 4q^{83} + 8q^{85} - 16q^{87} - 8q^{93} - 4q^{95} + 24q^{97} + 20q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.585786 0.338204 0.169102 0.985599i \(-0.445913\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.65685 −0.885618
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) −0.828427 −0.229764 −0.114882 0.993379i \(-0.536649\pi\)
−0.114882 + 0.993379i \(0.536649\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −5.41421 −1.31314 −0.656570 0.754265i \(-0.727993\pi\)
−0.656570 + 0.754265i \(0.727993\pi\)
\(18\) 0 0
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.82843 1.42383 0.711913 0.702268i \(-0.247829\pi\)
0.711913 + 0.702268i \(0.247829\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.31371 −0.637723
\(28\) 0 0
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 0 0
\(39\) −0.485281 −0.0777072
\(40\) 0 0
\(41\) −11.0711 −1.72901 −0.864505 0.502624i \(-0.832368\pi\)
−0.864505 + 0.502624i \(0.832368\pi\)
\(42\) 0 0
\(43\) 3.17157 0.483660 0.241830 0.970319i \(-0.422252\pi\)
0.241830 + 0.970319i \(0.422252\pi\)
\(44\) 0 0
\(45\) 2.65685 0.396060
\(46\) 0 0
\(47\) 10.8284 1.57949 0.789744 0.613436i \(-0.210213\pi\)
0.789744 + 0.613436i \(0.210213\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.17157 −0.444109
\(52\) 0 0
\(53\) −10.4853 −1.44026 −0.720132 0.693837i \(-0.755919\pi\)
−0.720132 + 0.693837i \(0.755919\pi\)
\(54\) 0 0
\(55\) 4.82843 0.651065
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 11.4142 1.48600 0.743002 0.669289i \(-0.233401\pi\)
0.743002 + 0.669289i \(0.233401\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.828427 0.102754
\(66\) 0 0
\(67\) 9.65685 1.17977 0.589886 0.807486i \(-0.299173\pi\)
0.589886 + 0.807486i \(0.299173\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.4853 −1.48173 −0.740865 0.671654i \(-0.765584\pi\)
−0.740865 + 0.671654i \(0.765584\pi\)
\(72\) 0 0
\(73\) 6.58579 0.770808 0.385404 0.922748i \(-0.374062\pi\)
0.385404 + 0.922748i \(0.374062\pi\)
\(74\) 0 0
\(75\) 0.585786 0.0676408
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.17157 0.131812 0.0659061 0.997826i \(-0.479006\pi\)
0.0659061 + 0.997826i \(0.479006\pi\)
\(80\) 0 0
\(81\) 6.02944 0.669937
\(82\) 0 0
\(83\) 6.24264 0.685219 0.342609 0.939478i \(-0.388689\pi\)
0.342609 + 0.939478i \(0.388689\pi\)
\(84\) 0 0
\(85\) 5.41421 0.587254
\(86\) 0 0
\(87\) 0.485281 0.0520276
\(88\) 0 0
\(89\) 12.7279 1.34916 0.674579 0.738203i \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.65685 0.171808
\(94\) 0 0
\(95\) −3.41421 −0.350291
\(96\) 0 0
\(97\) 16.2426 1.64919 0.824595 0.565723i \(-0.191403\pi\)
0.824595 + 0.565723i \(0.191403\pi\)
\(98\) 0 0
\(99\) 12.8284 1.28931
\(100\) 0 0
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) 0 0
\(103\) −9.17157 −0.903702 −0.451851 0.892093i \(-0.649236\pi\)
−0.451851 + 0.892093i \(0.649236\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.65685 0.160174 0.0800871 0.996788i \(-0.474480\pi\)
0.0800871 + 0.996788i \(0.474480\pi\)
\(108\) 0 0
\(109\) −14.4853 −1.38744 −0.693719 0.720246i \(-0.744029\pi\)
−0.693719 + 0.720246i \(0.744029\pi\)
\(110\) 0 0
\(111\) 2.14214 0.203323
\(112\) 0 0
\(113\) 7.31371 0.688016 0.344008 0.938967i \(-0.388215\pi\)
0.344008 + 0.938967i \(0.388215\pi\)
\(114\) 0 0
\(115\) −6.82843 −0.636754
\(116\) 0 0
\(117\) 2.20101 0.203483
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) −6.48528 −0.584758
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.82843 0.250982 0.125491 0.992095i \(-0.459949\pi\)
0.125491 + 0.992095i \(0.459949\pi\)
\(128\) 0 0
\(129\) 1.85786 0.163576
\(130\) 0 0
\(131\) 2.24264 0.195940 0.0979702 0.995189i \(-0.468765\pi\)
0.0979702 + 0.995189i \(0.468765\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.31371 0.285199
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) 0.100505 0.00852473 0.00426236 0.999991i \(-0.498643\pi\)
0.00426236 + 0.999991i \(0.498643\pi\)
\(140\) 0 0
\(141\) 6.34315 0.534189
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −0.828427 −0.0687971
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −11.3137 −0.920697 −0.460348 0.887738i \(-0.652275\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(152\) 0 0
\(153\) 14.3848 1.16294
\(154\) 0 0
\(155\) −2.82843 −0.227185
\(156\) 0 0
\(157\) 10.4853 0.836817 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(158\) 0 0
\(159\) −6.14214 −0.487103
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.14214 −0.637741 −0.318871 0.947798i \(-0.603304\pi\)
−0.318871 + 0.947798i \(0.603304\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) −23.7990 −1.84162 −0.920811 0.390010i \(-0.872471\pi\)
−0.920811 + 0.390010i \(0.872471\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) 0 0
\(171\) −9.07107 −0.693682
\(172\) 0 0
\(173\) 3.17157 0.241130 0.120565 0.992705i \(-0.461529\pi\)
0.120565 + 0.992705i \(0.461529\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.68629 0.502572
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −14.4853 −1.07668 −0.538341 0.842727i \(-0.680949\pi\)
−0.538341 + 0.842727i \(0.680949\pi\)
\(182\) 0 0
\(183\) 7.79899 0.576518
\(184\) 0 0
\(185\) −3.65685 −0.268857
\(186\) 0 0
\(187\) 26.1421 1.91170
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.1421 1.31272 0.656359 0.754448i \(-0.272096\pi\)
0.656359 + 0.754448i \(0.272096\pi\)
\(192\) 0 0
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) 0 0
\(195\) 0.485281 0.0347517
\(196\) 0 0
\(197\) 13.7990 0.983137 0.491569 0.870839i \(-0.336424\pi\)
0.491569 + 0.870839i \(0.336424\pi\)
\(198\) 0 0
\(199\) −0.485281 −0.0344007 −0.0172003 0.999852i \(-0.505475\pi\)
−0.0172003 + 0.999852i \(0.505475\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 11.0711 0.773237
\(206\) 0 0
\(207\) −18.1421 −1.26097
\(208\) 0 0
\(209\) −16.4853 −1.14031
\(210\) 0 0
\(211\) 26.6274 1.83311 0.916553 0.399912i \(-0.130959\pi\)
0.916553 + 0.399912i \(0.130959\pi\)
\(212\) 0 0
\(213\) −7.31371 −0.501127
\(214\) 0 0
\(215\) −3.17157 −0.216299
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.85786 0.260690
\(220\) 0 0
\(221\) 4.48528 0.301713
\(222\) 0 0
\(223\) 15.3137 1.02548 0.512741 0.858543i \(-0.328630\pi\)
0.512741 + 0.858543i \(0.328630\pi\)
\(224\) 0 0
\(225\) −2.65685 −0.177124
\(226\) 0 0
\(227\) 9.75736 0.647619 0.323809 0.946122i \(-0.395036\pi\)
0.323809 + 0.946122i \(0.395036\pi\)
\(228\) 0 0
\(229\) −12.1421 −0.802375 −0.401187 0.915996i \(-0.631402\pi\)
−0.401187 + 0.915996i \(0.631402\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.686292 0.0449605 0.0224802 0.999747i \(-0.492844\pi\)
0.0224802 + 0.999747i \(0.492844\pi\)
\(234\) 0 0
\(235\) −10.8284 −0.706369
\(236\) 0 0
\(237\) 0.686292 0.0445794
\(238\) 0 0
\(239\) 9.65685 0.624650 0.312325 0.949975i \(-0.398892\pi\)
0.312325 + 0.949975i \(0.398892\pi\)
\(240\) 0 0
\(241\) 10.5858 0.681890 0.340945 0.940083i \(-0.389253\pi\)
0.340945 + 0.940083i \(0.389253\pi\)
\(242\) 0 0
\(243\) 13.4731 0.864299
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.82843 −0.179969
\(248\) 0 0
\(249\) 3.65685 0.231744
\(250\) 0 0
\(251\) 3.41421 0.215503 0.107752 0.994178i \(-0.465635\pi\)
0.107752 + 0.994178i \(0.465635\pi\)
\(252\) 0 0
\(253\) −32.9706 −2.07284
\(254\) 0 0
\(255\) 3.17157 0.198612
\(256\) 0 0
\(257\) −9.89949 −0.617514 −0.308757 0.951141i \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.20101 −0.136239
\(262\) 0 0
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 10.4853 0.644106
\(266\) 0 0
\(267\) 7.45584 0.456290
\(268\) 0 0
\(269\) −1.51472 −0.0923540 −0.0461770 0.998933i \(-0.514704\pi\)
−0.0461770 + 0.998933i \(0.514704\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.82843 −0.291165
\(276\) 0 0
\(277\) 20.1421 1.21022 0.605112 0.796140i \(-0.293128\pi\)
0.605112 + 0.796140i \(0.293128\pi\)
\(278\) 0 0
\(279\) −7.51472 −0.449894
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 6.24264 0.371086 0.185543 0.982636i \(-0.440596\pi\)
0.185543 + 0.982636i \(0.440596\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.3137 0.724336
\(290\) 0 0
\(291\) 9.51472 0.557763
\(292\) 0 0
\(293\) 19.6569 1.14837 0.574183 0.818727i \(-0.305320\pi\)
0.574183 + 0.818727i \(0.305320\pi\)
\(294\) 0 0
\(295\) −11.4142 −0.664561
\(296\) 0 0
\(297\) 16.0000 0.928414
\(298\) 0 0
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 5.45584 0.313430
\(304\) 0 0
\(305\) −13.3137 −0.762341
\(306\) 0 0
\(307\) 29.0711 1.65917 0.829587 0.558378i \(-0.188576\pi\)
0.829587 + 0.558378i \(0.188576\pi\)
\(308\) 0 0
\(309\) −5.37258 −0.305636
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 22.3848 1.26526 0.632631 0.774453i \(-0.281975\pi\)
0.632631 + 0.774453i \(0.281975\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.48528 −0.364250 −0.182125 0.983275i \(-0.558298\pi\)
−0.182125 + 0.983275i \(0.558298\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 0.970563 0.0541715
\(322\) 0 0
\(323\) −18.4853 −1.02855
\(324\) 0 0
\(325\) −0.828427 −0.0459529
\(326\) 0 0
\(327\) −8.48528 −0.469237
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.79899 0.318741 0.159371 0.987219i \(-0.449054\pi\)
0.159371 + 0.987219i \(0.449054\pi\)
\(332\) 0 0
\(333\) −9.71573 −0.532419
\(334\) 0 0
\(335\) −9.65685 −0.527610
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) 4.28427 0.232690
\(340\) 0 0
\(341\) −13.6569 −0.739560
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 8.82843 0.473935 0.236967 0.971518i \(-0.423847\pi\)
0.236967 + 0.971518i \(0.423847\pi\)
\(348\) 0 0
\(349\) 14.4853 0.775379 0.387690 0.921790i \(-0.373273\pi\)
0.387690 + 0.921790i \(0.373273\pi\)
\(350\) 0 0
\(351\) 2.74517 0.146526
\(352\) 0 0
\(353\) −34.3848 −1.83012 −0.915058 0.403321i \(-0.867856\pi\)
−0.915058 + 0.403321i \(0.867856\pi\)
\(354\) 0 0
\(355\) 12.4853 0.662650
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −7.34315 −0.386481
\(362\) 0 0
\(363\) 7.21320 0.378595
\(364\) 0 0
\(365\) −6.58579 −0.344716
\(366\) 0 0
\(367\) −8.97056 −0.468260 −0.234130 0.972205i \(-0.575224\pi\)
−0.234130 + 0.972205i \(0.575224\pi\)
\(368\) 0 0
\(369\) 29.4142 1.53124
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.5147 −0.699766 −0.349883 0.936793i \(-0.613779\pi\)
−0.349883 + 0.936793i \(0.613779\pi\)
\(374\) 0 0
\(375\) −0.585786 −0.0302499
\(376\) 0 0
\(377\) −0.686292 −0.0353458
\(378\) 0 0
\(379\) −17.5147 −0.899671 −0.449835 0.893112i \(-0.648517\pi\)
−0.449835 + 0.893112i \(0.648517\pi\)
\(380\) 0 0
\(381\) 1.65685 0.0848832
\(382\) 0 0
\(383\) −15.5147 −0.792765 −0.396383 0.918085i \(-0.629735\pi\)
−0.396383 + 0.918085i \(0.629735\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.42641 −0.428338
\(388\) 0 0
\(389\) −0.142136 −0.00720656 −0.00360328 0.999994i \(-0.501147\pi\)
−0.00360328 + 0.999994i \(0.501147\pi\)
\(390\) 0 0
\(391\) −36.9706 −1.86968
\(392\) 0 0
\(393\) 1.31371 0.0662678
\(394\) 0 0
\(395\) −1.17157 −0.0589482
\(396\) 0 0
\(397\) 5.79899 0.291043 0.145521 0.989355i \(-0.453514\pi\)
0.145521 + 0.989355i \(0.453514\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −2.34315 −0.116720
\(404\) 0 0
\(405\) −6.02944 −0.299605
\(406\) 0 0
\(407\) −17.6569 −0.875218
\(408\) 0 0
\(409\) 13.4142 0.663290 0.331645 0.943404i \(-0.392396\pi\)
0.331645 + 0.943404i \(0.392396\pi\)
\(410\) 0 0
\(411\) −9.37258 −0.462315
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.24264 −0.306439
\(416\) 0 0
\(417\) 0.0588745 0.00288310
\(418\) 0 0
\(419\) 32.8701 1.60581 0.802904 0.596109i \(-0.203287\pi\)
0.802904 + 0.596109i \(0.203287\pi\)
\(420\) 0 0
\(421\) −5.31371 −0.258974 −0.129487 0.991581i \(-0.541333\pi\)
−0.129487 + 0.991581i \(0.541333\pi\)
\(422\) 0 0
\(423\) −28.7696 −1.39882
\(424\) 0 0
\(425\) −5.41421 −0.262628
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.34315 0.113128
\(430\) 0 0
\(431\) 33.6569 1.62119 0.810597 0.585605i \(-0.199143\pi\)
0.810597 + 0.585605i \(0.199143\pi\)
\(432\) 0 0
\(433\) −13.4142 −0.644646 −0.322323 0.946630i \(-0.604464\pi\)
−0.322323 + 0.946630i \(0.604464\pi\)
\(434\) 0 0
\(435\) −0.485281 −0.0232675
\(436\) 0 0
\(437\) 23.3137 1.11525
\(438\) 0 0
\(439\) 8.97056 0.428142 0.214071 0.976818i \(-0.431328\pi\)
0.214071 + 0.976818i \(0.431328\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.9706 −1.75652 −0.878262 0.478179i \(-0.841297\pi\)
−0.878262 + 0.478179i \(0.841297\pi\)
\(444\) 0 0
\(445\) −12.7279 −0.603361
\(446\) 0 0
\(447\) −3.51472 −0.166240
\(448\) 0 0
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 0 0
\(451\) 53.4558 2.51714
\(452\) 0 0
\(453\) −6.62742 −0.311383
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.3431 0.483832 0.241916 0.970297i \(-0.422224\pi\)
0.241916 + 0.970297i \(0.422224\pi\)
\(458\) 0 0
\(459\) 17.9411 0.837420
\(460\) 0 0
\(461\) 7.17157 0.334013 0.167007 0.985956i \(-0.446590\pi\)
0.167007 + 0.985956i \(0.446590\pi\)
\(462\) 0 0
\(463\) −16.9706 −0.788689 −0.394344 0.918963i \(-0.629028\pi\)
−0.394344 + 0.918963i \(0.629028\pi\)
\(464\) 0 0
\(465\) −1.65685 −0.0768348
\(466\) 0 0
\(467\) 3.89949 0.180447 0.0902236 0.995922i \(-0.471242\pi\)
0.0902236 + 0.995922i \(0.471242\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.14214 0.283015
\(472\) 0 0
\(473\) −15.3137 −0.704125
\(474\) 0 0
\(475\) 3.41421 0.156655
\(476\) 0 0
\(477\) 27.8579 1.27552
\(478\) 0 0
\(479\) −22.8284 −1.04306 −0.521529 0.853234i \(-0.674638\pi\)
−0.521529 + 0.853234i \(0.674638\pi\)
\(480\) 0 0
\(481\) −3.02944 −0.138130
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.2426 −0.737540
\(486\) 0 0
\(487\) 7.79899 0.353406 0.176703 0.984264i \(-0.443457\pi\)
0.176703 + 0.984264i \(0.443457\pi\)
\(488\) 0 0
\(489\) −4.76955 −0.215687
\(490\) 0 0
\(491\) 24.2843 1.09593 0.547967 0.836500i \(-0.315402\pi\)
0.547967 + 0.836500i \(0.315402\pi\)
\(492\) 0 0
\(493\) −4.48528 −0.202007
\(494\) 0 0
\(495\) −12.8284 −0.576595
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −41.6569 −1.86482 −0.932408 0.361406i \(-0.882297\pi\)
−0.932408 + 0.361406i \(0.882297\pi\)
\(500\) 0 0
\(501\) −13.9411 −0.622844
\(502\) 0 0
\(503\) 6.34315 0.282827 0.141413 0.989951i \(-0.454835\pi\)
0.141413 + 0.989951i \(0.454835\pi\)
\(504\) 0 0
\(505\) −9.31371 −0.414455
\(506\) 0 0
\(507\) −7.21320 −0.320350
\(508\) 0 0
\(509\) 33.7990 1.49811 0.749057 0.662506i \(-0.230507\pi\)
0.749057 + 0.662506i \(0.230507\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −11.3137 −0.499512
\(514\) 0 0
\(515\) 9.17157 0.404148
\(516\) 0 0
\(517\) −52.2843 −2.29946
\(518\) 0 0
\(519\) 1.85786 0.0815512
\(520\) 0 0
\(521\) −4.92893 −0.215940 −0.107970 0.994154i \(-0.534435\pi\)
−0.107970 + 0.994154i \(0.534435\pi\)
\(522\) 0 0
\(523\) 4.10051 0.179303 0.0896513 0.995973i \(-0.471425\pi\)
0.0896513 + 0.995973i \(0.471425\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.3137 −0.667076
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) −30.3259 −1.31603
\(532\) 0 0
\(533\) 9.17157 0.397265
\(534\) 0 0
\(535\) −1.65685 −0.0716321
\(536\) 0 0
\(537\) −2.34315 −0.101114
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.9706 0.815608 0.407804 0.913069i \(-0.366295\pi\)
0.407804 + 0.913069i \(0.366295\pi\)
\(542\) 0 0
\(543\) −8.48528 −0.364138
\(544\) 0 0
\(545\) 14.4853 0.620481
\(546\) 0 0
\(547\) −6.48528 −0.277291 −0.138645 0.990342i \(-0.544275\pi\)
−0.138645 + 0.990342i \(0.544275\pi\)
\(548\) 0 0
\(549\) −35.3726 −1.50967
\(550\) 0 0
\(551\) 2.82843 0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.14214 −0.0909286
\(556\) 0 0
\(557\) 20.8284 0.882529 0.441264 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(558\) 0 0
\(559\) −2.62742 −0.111128
\(560\) 0 0
\(561\) 15.3137 0.646545
\(562\) 0 0
\(563\) −39.4142 −1.66111 −0.830556 0.556936i \(-0.811977\pi\)
−0.830556 + 0.556936i \(0.811977\pi\)
\(564\) 0 0
\(565\) −7.31371 −0.307690
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.68629 −0.280304 −0.140152 0.990130i \(-0.544759\pi\)
−0.140152 + 0.990130i \(0.544759\pi\)
\(570\) 0 0
\(571\) 41.7990 1.74923 0.874617 0.484815i \(-0.161113\pi\)
0.874617 + 0.484815i \(0.161113\pi\)
\(572\) 0 0
\(573\) 10.6274 0.443967
\(574\) 0 0
\(575\) 6.82843 0.284765
\(576\) 0 0
\(577\) −25.8995 −1.07821 −0.539105 0.842239i \(-0.681237\pi\)
−0.539105 + 0.842239i \(0.681237\pi\)
\(578\) 0 0
\(579\) −3.31371 −0.137713
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 50.6274 2.09677
\(584\) 0 0
\(585\) −2.20101 −0.0910006
\(586\) 0 0
\(587\) 2.92893 0.120890 0.0604450 0.998172i \(-0.480748\pi\)
0.0604450 + 0.998172i \(0.480748\pi\)
\(588\) 0 0
\(589\) 9.65685 0.397904
\(590\) 0 0
\(591\) 8.08326 0.332501
\(592\) 0 0
\(593\) −28.7279 −1.17971 −0.589857 0.807508i \(-0.700816\pi\)
−0.589857 + 0.807508i \(0.700816\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.284271 −0.0116344
\(598\) 0 0
\(599\) 5.17157 0.211305 0.105652 0.994403i \(-0.466307\pi\)
0.105652 + 0.994403i \(0.466307\pi\)
\(600\) 0 0
\(601\) −9.41421 −0.384014 −0.192007 0.981394i \(-0.561500\pi\)
−0.192007 + 0.981394i \(0.561500\pi\)
\(602\) 0 0
\(603\) −25.6569 −1.04483
\(604\) 0 0
\(605\) −12.3137 −0.500623
\(606\) 0 0
\(607\) −40.2843 −1.63509 −0.817544 0.575866i \(-0.804665\pi\)
−0.817544 + 0.575866i \(0.804665\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.97056 −0.362910
\(612\) 0 0
\(613\) −23.6569 −0.955491 −0.477746 0.878498i \(-0.658546\pi\)
−0.477746 + 0.878498i \(0.658546\pi\)
\(614\) 0 0
\(615\) 6.48528 0.261512
\(616\) 0 0
\(617\) −10.6863 −0.430214 −0.215107 0.976590i \(-0.569010\pi\)
−0.215107 + 0.976590i \(0.569010\pi\)
\(618\) 0 0
\(619\) 14.9289 0.600044 0.300022 0.953932i \(-0.403006\pi\)
0.300022 + 0.953932i \(0.403006\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −9.65685 −0.385658
\(628\) 0 0
\(629\) −19.7990 −0.789437
\(630\) 0 0
\(631\) −4.48528 −0.178556 −0.0892781 0.996007i \(-0.528456\pi\)
−0.0892781 + 0.996007i \(0.528456\pi\)
\(632\) 0 0
\(633\) 15.5980 0.619964
\(634\) 0 0
\(635\) −2.82843 −0.112243
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 33.1716 1.31225
\(640\) 0 0
\(641\) 20.6274 0.814734 0.407367 0.913265i \(-0.366447\pi\)
0.407367 + 0.913265i \(0.366447\pi\)
\(642\) 0 0
\(643\) 47.2132 1.86191 0.930953 0.365138i \(-0.118978\pi\)
0.930953 + 0.365138i \(0.118978\pi\)
\(644\) 0 0
\(645\) −1.85786 −0.0731533
\(646\) 0 0
\(647\) 39.1127 1.53768 0.768839 0.639442i \(-0.220835\pi\)
0.768839 + 0.639442i \(0.220835\pi\)
\(648\) 0 0
\(649\) −55.1127 −2.16336
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.6569 0.612700 0.306350 0.951919i \(-0.400892\pi\)
0.306350 + 0.951919i \(0.400892\pi\)
\(654\) 0 0
\(655\) −2.24264 −0.0876272
\(656\) 0 0
\(657\) −17.4975 −0.682642
\(658\) 0 0
\(659\) 32.8284 1.27881 0.639407 0.768868i \(-0.279180\pi\)
0.639407 + 0.768868i \(0.279180\pi\)
\(660\) 0 0
\(661\) −18.2843 −0.711176 −0.355588 0.934643i \(-0.615719\pi\)
−0.355588 + 0.934643i \(0.615719\pi\)
\(662\) 0 0
\(663\) 2.62742 0.102040
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.65685 0.219034
\(668\) 0 0
\(669\) 8.97056 0.346822
\(670\) 0 0
\(671\) −64.2843 −2.48167
\(672\) 0 0
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) 0 0
\(675\) −3.31371 −0.127545
\(676\) 0 0
\(677\) 11.4558 0.440284 0.220142 0.975468i \(-0.429348\pi\)
0.220142 + 0.975468i \(0.429348\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 5.71573 0.219027
\(682\) 0 0
\(683\) −22.3431 −0.854937 −0.427468 0.904030i \(-0.640594\pi\)
−0.427468 + 0.904030i \(0.640594\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) −7.11270 −0.271366
\(688\) 0 0
\(689\) 8.68629 0.330921
\(690\) 0 0
\(691\) 10.2426 0.389648 0.194824 0.980838i \(-0.437586\pi\)
0.194824 + 0.980838i \(0.437586\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.100505 −0.00381237
\(696\) 0 0
\(697\) 59.9411 2.27043
\(698\) 0 0
\(699\) 0.402020 0.0152058
\(700\) 0 0
\(701\) −14.4853 −0.547102 −0.273551 0.961858i \(-0.588198\pi\)
−0.273551 + 0.961858i \(0.588198\pi\)
\(702\) 0 0
\(703\) 12.4853 0.470891
\(704\) 0 0
\(705\) −6.34315 −0.238897
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.1127 −0.642681 −0.321340 0.946964i \(-0.604133\pi\)
−0.321340 + 0.946964i \(0.604133\pi\)
\(710\) 0 0
\(711\) −3.11270 −0.116735
\(712\) 0 0
\(713\) 19.3137 0.723304
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 5.65685 0.211259
\(718\) 0 0
\(719\) −9.45584 −0.352643 −0.176322 0.984333i \(-0.556420\pi\)
−0.176322 + 0.984333i \(0.556420\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 6.20101 0.230618
\(724\) 0 0
\(725\) 0.828427 0.0307670
\(726\) 0 0
\(727\) −20.4853 −0.759757 −0.379879 0.925036i \(-0.624034\pi\)
−0.379879 + 0.925036i \(0.624034\pi\)
\(728\) 0 0
\(729\) −10.1960 −0.377628
\(730\) 0 0
\(731\) −17.1716 −0.635114
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −46.6274 −1.71754
\(738\) 0 0
\(739\) −8.82843 −0.324759 −0.162379 0.986728i \(-0.551917\pi\)
−0.162379 + 0.986728i \(0.551917\pi\)
\(740\) 0 0
\(741\) −1.65685 −0.0608661
\(742\) 0 0
\(743\) −12.2010 −0.447612 −0.223806 0.974634i \(-0.571848\pi\)
−0.223806 + 0.974634i \(0.571848\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −16.5858 −0.606842
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.6863 0.608891 0.304446 0.952530i \(-0.401529\pi\)
0.304446 + 0.952530i \(0.401529\pi\)
\(752\) 0 0
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) 11.3137 0.411748
\(756\) 0 0
\(757\) −7.65685 −0.278293 −0.139147 0.990272i \(-0.544436\pi\)
−0.139147 + 0.990272i \(0.544436\pi\)
\(758\) 0 0
\(759\) −19.3137 −0.701043
\(760\) 0 0
\(761\) 14.3848 0.521448 0.260724 0.965413i \(-0.416039\pi\)
0.260724 + 0.965413i \(0.416039\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −14.3848 −0.520083
\(766\) 0 0
\(767\) −9.45584 −0.341431
\(768\) 0 0
\(769\) 11.5563 0.416733 0.208366 0.978051i \(-0.433185\pi\)
0.208366 + 0.978051i \(0.433185\pi\)
\(770\) 0 0
\(771\) −5.79899 −0.208846
\(772\) 0 0
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) 2.82843 0.101600
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −37.7990 −1.35429
\(780\) 0 0
\(781\) 60.2843 2.15714
\(782\) 0 0
\(783\) −2.74517 −0.0981042
\(784\) 0 0
\(785\) −10.4853 −0.374236
\(786\) 0 0
\(787\) −26.7279 −0.952748 −0.476374 0.879243i \(-0.658049\pi\)
−0.476374 + 0.879243i \(0.658049\pi\)
\(788\) 0 0
\(789\) −16.4020 −0.583927
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −11.0294 −0.391667
\(794\) 0 0
\(795\) 6.14214 0.217839
\(796\) 0 0
\(797\) 2.20101 0.0779638 0.0389819 0.999240i \(-0.487589\pi\)
0.0389819 + 0.999240i \(0.487589\pi\)
\(798\) 0 0
\(799\) −58.6274 −2.07409
\(800\) 0 0
\(801\) −33.8162 −1.19484
\(802\) 0 0
\(803\) −31.7990 −1.12216
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −0.887302 −0.0312345
\(808\) 0 0
\(809\) 36.9706 1.29982 0.649908 0.760013i \(-0.274807\pi\)
0.649908 + 0.760013i \(0.274807\pi\)
\(810\) 0 0
\(811\) 35.4142 1.24356 0.621781 0.783191i \(-0.286410\pi\)
0.621781 + 0.783191i \(0.286410\pi\)
\(812\) 0 0
\(813\) 7.02944 0.246533
\(814\) 0 0
\(815\) 8.14214 0.285207
\(816\) 0 0
\(817\) 10.8284 0.378839
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.31371 −0.185450 −0.0927249 0.995692i \(-0.529558\pi\)
−0.0927249 + 0.995692i \(0.529558\pi\)
\(822\) 0 0
\(823\) 36.2843 1.26479 0.632395 0.774646i \(-0.282072\pi\)
0.632395 + 0.774646i \(0.282072\pi\)
\(824\) 0 0
\(825\) −2.82843 −0.0984732
\(826\) 0 0
\(827\) 50.6274 1.76049 0.880244 0.474522i \(-0.157379\pi\)
0.880244 + 0.474522i \(0.157379\pi\)
\(828\) 0 0
\(829\) 38.9706 1.35350 0.676752 0.736211i \(-0.263387\pi\)
0.676752 + 0.736211i \(0.263387\pi\)
\(830\) 0 0
\(831\) 11.7990 0.409302
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 23.7990 0.823598
\(836\) 0 0
\(837\) −9.37258 −0.323964
\(838\) 0 0
\(839\) −13.8579 −0.478427 −0.239213 0.970967i \(-0.576890\pi\)
−0.239213 + 0.970967i \(0.576890\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 0 0
\(843\) 4.68629 0.161404
\(844\) 0 0
\(845\) 12.3137 0.423604
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 3.65685 0.125503
\(850\) 0 0
\(851\) 24.9706 0.855980
\(852\) 0 0
\(853\) 48.8284 1.67185 0.835927 0.548841i \(-0.184931\pi\)
0.835927 + 0.548841i \(0.184931\pi\)
\(854\) 0 0
\(855\) 9.07107 0.310224
\(856\) 0 0
\(857\) 19.0711 0.651455 0.325728 0.945464i \(-0.394391\pi\)
0.325728 + 0.945464i \(0.394391\pi\)
\(858\) 0 0
\(859\) 35.2132 1.20146 0.600729 0.799452i \(-0.294877\pi\)
0.600729 + 0.799452i \(0.294877\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28.9706 −0.986169 −0.493085 0.869981i \(-0.664131\pi\)
−0.493085 + 0.869981i \(0.664131\pi\)
\(864\) 0 0
\(865\) −3.17157 −0.107837
\(866\) 0 0
\(867\) 7.21320 0.244973
\(868\) 0 0
\(869\) −5.65685 −0.191896
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −43.1543 −1.46055
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.2843 0.887557 0.443778 0.896137i \(-0.353638\pi\)
0.443778 + 0.896137i \(0.353638\pi\)
\(878\) 0 0
\(879\) 11.5147 0.388382
\(880\) 0 0
\(881\) −34.3848 −1.15845 −0.579226 0.815167i \(-0.696645\pi\)
−0.579226 + 0.815167i \(0.696645\pi\)
\(882\) 0 0
\(883\) 30.3431 1.02113 0.510564 0.859840i \(-0.329437\pi\)
0.510564 + 0.859840i \(0.329437\pi\)
\(884\) 0 0
\(885\) −6.68629 −0.224757
\(886\) 0 0
\(887\) 7.11270 0.238821 0.119411 0.992845i \(-0.461900\pi\)
0.119411 + 0.992845i \(0.461900\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −29.1127 −0.975312
\(892\) 0 0
\(893\) 36.9706 1.23717
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −3.31371 −0.110642
\(898\) 0 0
\(899\) 2.34315 0.0781483
\(900\) 0 0
\(901\) 56.7696 1.89127
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.4853 0.481507
\(906\) 0 0
\(907\) 56.2843 1.86889 0.934444 0.356109i \(-0.115897\pi\)
0.934444 + 0.356109i \(0.115897\pi\)
\(908\) 0 0
\(909\) −24.7452 −0.820745
\(910\) 0 0
\(911\) 20.2843 0.672048 0.336024 0.941853i \(-0.390918\pi\)
0.336024 + 0.941853i \(0.390918\pi\)
\(912\) 0 0
\(913\) −30.1421 −0.997559
\(914\) 0 0
\(915\) −7.79899 −0.257827
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32.4853 −1.07159 −0.535795 0.844348i \(-0.679988\pi\)
−0.535795 + 0.844348i \(0.679988\pi\)
\(920\) 0 0
\(921\) 17.0294 0.561139
\(922\) 0 0
\(923\) 10.3431 0.340449
\(924\) 0 0
\(925\) 3.65685 0.120237
\(926\) 0 0
\(927\) 24.3675 0.800335
\(928\) 0 0
\(929\) −25.2132 −0.827218 −0.413609 0.910455i \(-0.635732\pi\)
−0.413609 + 0.910455i \(0.635732\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.34315 0.0767111
\(934\) 0 0
\(935\) −26.1421 −0.854939
\(936\) 0 0
\(937\) −11.7574 −0.384096 −0.192048 0.981386i \(-0.561513\pi\)
−0.192048 + 0.981386i \(0.561513\pi\)
\(938\) 0 0
\(939\) 13.1127 0.427917
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) −75.5980 −2.46181
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.828427 −0.0269203 −0.0134601 0.999909i \(-0.504285\pi\)
−0.0134601 + 0.999909i \(0.504285\pi\)
\(948\) 0 0
\(949\) −5.45584 −0.177104
\(950\) 0 0
\(951\) −3.79899 −0.123191
\(952\) 0 0
\(953\) 11.6569 0.377603 0.188801 0.982015i \(-0.439540\pi\)
0.188801 + 0.982015i \(0.439540\pi\)
\(954\) 0 0
\(955\) −18.1421 −0.587066
\(956\) 0 0
\(957\) −2.34315 −0.0757431
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) −4.40202 −0.141853
\(964\) 0 0
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) −13.4558 −0.432711 −0.216355 0.976315i \(-0.569417\pi\)
−0.216355 + 0.976315i \(0.569417\pi\)
\(968\) 0 0
\(969\) −10.8284 −0.347859
\(970\) 0 0
\(971\) 37.3553 1.19879 0.599395 0.800453i \(-0.295408\pi\)
0.599395 + 0.800453i \(0.295408\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −0.485281 −0.0155414
\(976\) 0 0
\(977\) −35.3137 −1.12979 −0.564893 0.825164i \(-0.691082\pi\)
−0.564893 + 0.825164i \(0.691082\pi\)
\(978\) 0 0
\(979\) −61.4558 −1.96414
\(980\) 0 0
\(981\) 38.4853 1.22874
\(982\) 0 0
\(983\) −51.7990 −1.65213 −0.826066 0.563574i \(-0.809426\pi\)
−0.826066 + 0.563574i \(0.809426\pi\)
\(984\) 0 0
\(985\) −13.7990 −0.439672
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.6569 0.688648
\(990\) 0 0
\(991\) 28.7696 0.913895 0.456947 0.889494i \(-0.348943\pi\)
0.456947 + 0.889494i \(0.348943\pi\)
\(992\) 0 0
\(993\) 3.39697 0.107800
\(994\) 0 0
\(995\) 0.485281 0.0153845
\(996\) 0 0
\(997\) −38.2843 −1.21248 −0.606238 0.795284i \(-0.707322\pi\)
−0.606238 + 0.795284i \(0.707322\pi\)
\(998\) 0 0
\(999\) −12.1177 −0.383389
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 3920.2.a.ca.1.1 2
4.3 odd 2 490.2.a.l.1.2 2
7.6 odd 2 3920.2.a.bm.1.2 2
12.11 even 2 4410.2.a.by.1.1 2
20.3 even 4 2450.2.c.w.99.4 4
20.7 even 4 2450.2.c.w.99.1 4
20.19 odd 2 2450.2.a.bs.1.1 2
28.3 even 6 490.2.e.i.471.2 4
28.11 odd 6 490.2.e.j.471.1 4
28.19 even 6 490.2.e.i.361.2 4
28.23 odd 6 490.2.e.j.361.1 4
28.27 even 2 490.2.a.m.1.1 yes 2
84.83 odd 2 4410.2.a.bt.1.1 2
140.27 odd 4 2450.2.c.t.99.2 4
140.83 odd 4 2450.2.c.t.99.3 4
140.139 even 2 2450.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.2 2 4.3 odd 2
490.2.a.m.1.1 yes 2 28.27 even 2
490.2.e.i.361.2 4 28.19 even 6
490.2.e.i.471.2 4 28.3 even 6
490.2.e.j.361.1 4 28.23 odd 6
490.2.e.j.471.1 4 28.11 odd 6
2450.2.a.bn.1.2 2 140.139 even 2
2450.2.a.bs.1.1 2 20.19 odd 2
2450.2.c.t.99.2 4 140.27 odd 4
2450.2.c.t.99.3 4 140.83 odd 4
2450.2.c.w.99.1 4 20.7 even 4
2450.2.c.w.99.4 4 20.3 even 4
3920.2.a.bm.1.2 2 7.6 odd 2
3920.2.a.ca.1.1 2 1.1 even 1 trivial
4410.2.a.bt.1.1 2 84.83 odd 2
4410.2.a.by.1.1 2 12.11 even 2