Properties

Label 490.2.a.l.1.2
Level $490$
Weight $2$
Character 490.1
Self dual yes
Analytic conductor $3.913$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.91266969904\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 490.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.585786 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.585786 q^{6} -1.00000 q^{8} -2.65685 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.585786 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.585786 q^{6} -1.00000 q^{8} -2.65685 q^{9} +1.00000 q^{10} +4.82843 q^{11} -0.585786 q^{12} -0.828427 q^{13} +0.585786 q^{15} +1.00000 q^{16} -5.41421 q^{17} +2.65685 q^{18} -3.41421 q^{19} -1.00000 q^{20} -4.82843 q^{22} -6.82843 q^{23} +0.585786 q^{24} +1.00000 q^{25} +0.828427 q^{26} +3.31371 q^{27} +0.828427 q^{29} -0.585786 q^{30} -2.82843 q^{31} -1.00000 q^{32} -2.82843 q^{33} +5.41421 q^{34} -2.65685 q^{36} +3.65685 q^{37} +3.41421 q^{38} +0.485281 q^{39} +1.00000 q^{40} -11.0711 q^{41} -3.17157 q^{43} +4.82843 q^{44} +2.65685 q^{45} +6.82843 q^{46} -10.8284 q^{47} -0.585786 q^{48} -1.00000 q^{50} +3.17157 q^{51} -0.828427 q^{52} -10.4853 q^{53} -3.31371 q^{54} -4.82843 q^{55} +2.00000 q^{57} -0.828427 q^{58} -11.4142 q^{59} +0.585786 q^{60} +13.3137 q^{61} +2.82843 q^{62} +1.00000 q^{64} +0.828427 q^{65} +2.82843 q^{66} -9.65685 q^{67} -5.41421 q^{68} +4.00000 q^{69} +12.4853 q^{71} +2.65685 q^{72} +6.58579 q^{73} -3.65685 q^{74} -0.585786 q^{75} -3.41421 q^{76} -0.485281 q^{78} -1.17157 q^{79} -1.00000 q^{80} +6.02944 q^{81} +11.0711 q^{82} -6.24264 q^{83} +5.41421 q^{85} +3.17157 q^{86} -0.485281 q^{87} -4.82843 q^{88} +12.7279 q^{89} -2.65685 q^{90} -6.82843 q^{92} +1.65685 q^{93} +10.8284 q^{94} +3.41421 q^{95} +0.585786 q^{96} +16.2426 q^{97} -12.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} + 4q^{6} - 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} + 4q^{6} - 2q^{8} + 6q^{9} + 2q^{10} + 4q^{11} - 4q^{12} + 4q^{13} + 4q^{15} + 2q^{16} - 8q^{17} - 6q^{18} - 4q^{19} - 2q^{20} - 4q^{22} - 8q^{23} + 4q^{24} + 2q^{25} - 4q^{26} - 16q^{27} - 4q^{29} - 4q^{30} - 2q^{32} + 8q^{34} + 6q^{36} - 4q^{37} + 4q^{38} - 16q^{39} + 2q^{40} - 8q^{41} - 12q^{43} + 4q^{44} - 6q^{45} + 8q^{46} - 16q^{47} - 4q^{48} - 2q^{50} + 12q^{51} + 4q^{52} - 4q^{53} + 16q^{54} - 4q^{55} + 4q^{57} + 4q^{58} - 20q^{59} + 4q^{60} + 4q^{61} + 2q^{64} - 4q^{65} - 8q^{67} - 8q^{68} + 8q^{69} + 8q^{71} - 6q^{72} + 16q^{73} + 4q^{74} - 4q^{75} - 4q^{76} + 16q^{78} - 8q^{79} - 2q^{80} + 46q^{81} + 8q^{82} - 4q^{83} + 8q^{85} + 12q^{86} + 16q^{87} - 4q^{88} + 6q^{90} - 8q^{92} - 8q^{93} + 16q^{94} + 4q^{95} + 4q^{96} + 24q^{97} - 20q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.585786 −0.338204 −0.169102 0.985599i \(-0.554087\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.585786 0.239146
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −2.65685 −0.885618
\(10\) 1.00000 0.316228
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) −0.585786 −0.169102
\(13\) −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
\(16\) 1.00000 0.250000
\(17\) −5.41421 −1.31314 −0.656570 0.754265i \(-0.727993\pi\)
−0.656570 + 0.754265i \(0.727993\pi\)
\(18\) 2.65685 0.626227
\(19\) −3.41421 −0.783274 −0.391637 0.920120i \(-0.628091\pi\)
−0.391637 + 0.920120i \(0.628091\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −4.82843 −1.02942
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0.585786 0.119573
\(25\) 1.00000 0.200000
\(26\) 0.828427 0.162468
\(27\) 3.31371 0.637723
\(28\) 0 0
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) −0.585786 −0.106949
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.82843 −0.492366
\(34\) 5.41421 0.928530
\(35\) 0 0
\(36\) −2.65685 −0.442809
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 3.41421 0.553859
\(39\) 0.485281 0.0777072
\(40\) 1.00000 0.158114
\(41\) −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 \(-0.577748\pi\)
−0.241830 + 0.970319i \(0.577748\pi\)
\(44\) 4.82843 0.727913
\(45\) 2.65685 0.396060
\(46\) 6.82843 1.00680
\(47\) −10.8284 −1.57949 −0.789744 0.613436i \(-0.789787\pi\)
−0.789744 + 0.613436i \(0.789787\pi\)
\(48\) −0.585786 −0.0845510
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 3.17157 0.444109
\(52\) −0.828427 −0.114882
\(53\) −10.4853 −1.44026 −0.720132 0.693837i \(-0.755919\pi\)
−0.720132 + 0.693837i \(0.755919\pi\)
\(54\) −3.31371 −0.450939
\(55\) −4.82843 −0.651065
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −0.828427 −0.108778
\(59\) −11.4142 −1.48600 −0.743002 0.669289i \(-0.766599\pi\)
−0.743002 + 0.669289i \(0.766599\pi\)
\(60\) 0.585786 0.0756247
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 2.82843 0.359211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.828427 0.102754
\(66\) 2.82843 0.348155
\(67\) −9.65685 −1.17977 −0.589886 0.807486i \(-0.700827\pi\)
−0.589886 + 0.807486i \(0.700827\pi\)
\(68\) −5.41421 −0.656570
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 12.4853 1.48173 0.740865 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) 2.65685 0.313113
\(73\) 6.58579 0.770808 0.385404 0.922748i \(-0.374062\pi\)
0.385404 + 0.922748i \(0.374062\pi\)
\(74\) −3.65685 −0.425101
\(75\) −0.585786 −0.0676408
\(76\) −3.41421 −0.391637
\(77\) 0 0
\(78\) −0.485281 −0.0549473
\(79\) −1.17157 −0.131812 −0.0659061 0.997826i \(-0.520994\pi\)
−0.0659061 + 0.997826i \(0.520994\pi\)
\(80\) −1.00000 −0.111803
\(81\) 6.02944 0.669937
\(82\) 11.0711 1.22259
\(83\) −6.24264 −0.685219 −0.342609 0.939478i \(-0.611311\pi\)
−0.342609 + 0.939478i \(0.611311\pi\)
\(84\) 0 0
\(85\) 5.41421 0.587254
\(86\) 3.17157 0.341999
\(87\) −0.485281 −0.0520276
\(88\) −4.82843 −0.514712
\(89\) 12.7279 1.34916 0.674579 0.738203i \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(90\) −2.65685 −0.280057
\(91\) 0 0
\(92\) −6.82843 −0.711913
\(93\) 1.65685 0.171808
\(94\) 10.8284 1.11687
\(95\) 3.41421 0.350291
\(96\) 0.585786 0.0597866
\(97\) 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
\(100\) 1.00000 0.100000
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) −3.17157 −0.314033
\(103\) 9.17157 0.903702 0.451851 0.892093i \(-0.350764\pi\)
0.451851 + 0.892093i \(0.350764\pi\)
\(104\) 0.828427 0.0812340
\(105\) 0 0
\(106\) 10.4853 1.01842
\(107\) −1.65685 −0.160174 −0.0800871 0.996788i \(-0.525520\pi\)
−0.0800871 + 0.996788i \(0.525520\pi\)
\(108\) 3.31371 0.318862
\(109\) −14.4853 −1.38744 −0.693719 0.720246i \(-0.744029\pi\)
−0.693719 + 0.720246i \(0.744029\pi\)
\(110\) 4.82843 0.460372
\(111\) −2.14214 −0.203323
\(112\) 0 0
\(113\) 7.31371 0.688016 0.344008 0.938967i \(-0.388215\pi\)
0.344008 + 0.938967i \(0.388215\pi\)
\(114\) −2.00000 −0.187317
\(115\) 6.82843 0.636754
\(116\) 0.828427 0.0769175
\(117\) 2.20101 0.203483
\(118\) 11.4142 1.05076
\(119\) 0 0
\(120\) −0.585786 −0.0534747
\(121\) 12.3137 1.11943
\(122\) −13.3137 −1.20537
\(123\) 6.48528 0.584758
\(124\) −2.82843 −0.254000
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.85786 0.163576
\(130\) −0.828427 −0.0726579
\(131\) −2.24264 −0.195940 −0.0979702 0.995189i \(-0.531235\pi\)
−0.0979702 + 0.995189i \(0.531235\pi\)
\(132\) −2.82843 −0.246183
\(133\) 0 0
\(134\) 9.65685 0.834225
\(135\) −3.31371 −0.285199
\(136\) 5.41421 0.464265
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −4.00000 −0.340503
\(139\) −0.100505 −0.00852473 −0.00426236 0.999991i \(-0.501357\pi\)
−0.00426236 + 0.999991i \(0.501357\pi\)
\(140\) 0 0
\(141\) 6.34315 0.534189
\(142\) −12.4853 −1.04774
\(143\) −4.00000 −0.334497
\(144\) −2.65685 −0.221405
\(145\) −0.828427 −0.0687971
\(146\) −6.58579 −0.545044
\(147\) 0 0
\(148\) 3.65685 0.300592
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0.585786 0.0478293
\(151\) 11.3137 0.920697 0.460348 0.887738i \(-0.347725\pi\)
0.460348 + 0.887738i \(0.347725\pi\)
\(152\) 3.41421 0.276929
\(153\) 14.3848 1.16294
\(154\) 0 0
\(155\) 2.82843 0.227185
\(156\) 0.485281 0.0388536
\(157\) 10.4853 0.836817 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(158\) 1.17157 0.0932053
\(159\) 6.14214 0.487103
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −6.02944 −0.473717
\(163\) 8.14214 0.637741 0.318871 0.947798i \(-0.396696\pi\)
0.318871 + 0.947798i \(0.396696\pi\)
\(164\) −11.0711 −0.864505
\(165\) 2.82843 0.220193
\(166\) 6.24264 0.484523
\(167\) 23.7990 1.84162 0.920811 0.390010i \(-0.127529\pi\)
0.920811 + 0.390010i \(0.127529\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) −5.41421 −0.415251
\(171\) 9.07107 0.693682
\(172\) −3.17157 −0.241830
\(173\) 3.17157 0.241130 0.120565 0.992705i \(-0.461529\pi\)
0.120565 + 0.992705i \(0.461529\pi\)
\(174\) 0.485281 0.0367891
\(175\) 0 0
\(176\) 4.82843 0.363956
\(177\) 6.68629 0.502572
\(178\) −12.7279 −0.953998
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 2.65685 0.198030
\(181\) −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
\(184\) 6.82843 0.503398
\(185\) −3.65685 −0.268857
\(186\) −1.65685 −0.121486
\(187\) −26.1421 −1.91170
\(188\) −10.8284 −0.789744
\(189\) 0 0
\(190\) −3.41421 −0.247693
\(191\) −18.1421 −1.31272 −0.656359 0.754448i \(-0.727904\pi\)
−0.656359 + 0.754448i \(0.727904\pi\)
\(192\) −0.585786 −0.0422755
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) −16.2426 −1.16615
\(195\) −0.485281 −0.0347517
\(196\) 0 0
\(197\) 13.7990 0.983137 0.491569 0.870839i \(-0.336424\pi\)
0.491569 + 0.870839i \(0.336424\pi\)
\(198\) 12.8284 0.911677
\(199\) 0.485281 0.0344007 0.0172003 0.999852i \(-0.494525\pi\)
0.0172003 + 0.999852i \(0.494525\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 5.65685 0.399004
\(202\) −9.31371 −0.655310
\(203\) 0 0
\(204\) 3.17157 0.222055
\(205\) 11.0711 0.773237
\(206\) −9.17157 −0.639014
\(207\) 18.1421 1.26097
\(208\) −0.828427 −0.0574411
\(209\) −16.4853 −1.14031
\(210\) 0 0
\(211\) −26.6274 −1.83311 −0.916553 0.399912i \(-0.869041\pi\)
−0.916553 + 0.399912i \(0.869041\pi\)
\(212\) −10.4853 −0.720132
\(213\) −7.31371 −0.501127
\(214\) 1.65685 0.113260
\(215\) 3.17157 0.216299
\(216\) −3.31371 −0.225469
\(217\) 0 0
\(218\) 14.4853 0.981067
\(219\) −3.85786 −0.260690
\(220\) −4.82843 −0.325532
\(221\) 4.48528 0.301713
\(222\) 2.14214 0.143771
\(223\) −15.3137 −1.02548 −0.512741 0.858543i \(-0.671370\pi\)
−0.512741 + 0.858543i \(0.671370\pi\)
\(224\) 0 0
\(225\) −2.65685 −0.177124
\(226\) −7.31371 −0.486501
\(227\) −9.75736 −0.647619 −0.323809 0.946122i \(-0.604964\pi\)
−0.323809 + 0.946122i \(0.604964\pi\)
\(228\) 2.00000 0.132453
\(229\) −12.1421 −0.802375 −0.401187 0.915996i \(-0.631402\pi\)
−0.401187 + 0.915996i \(0.631402\pi\)
\(230\) −6.82843 −0.450253
\(231\) 0 0
\(232\) −0.828427 −0.0543889
\(233\) 0.686292 0.0449605 0.0224802 0.999747i \(-0.492844\pi\)
0.0224802 + 0.999747i \(0.492844\pi\)
\(234\) −2.20101 −0.143885
\(235\) 10.8284 0.706369
\(236\) −11.4142 −0.743002
\(237\) 0.686292 0.0445794
\(238\) 0 0
\(239\) −9.65685 −0.624650 −0.312325 0.949975i \(-0.601108\pi\)
−0.312325 + 0.949975i \(0.601108\pi\)
\(240\) 0.585786 0.0378124
\(241\) 10.5858 0.681890 0.340945 0.940083i \(-0.389253\pi\)
0.340945 + 0.940083i \(0.389253\pi\)
\(242\) −12.3137 −0.791555
\(243\) −13.4731 −0.864299
\(244\) 13.3137 0.852323
\(245\) 0 0
\(246\) −6.48528 −0.413486
\(247\) 2.82843 0.179969
\(248\) 2.82843 0.179605
\(249\) 3.65685 0.231744
\(250\) 1.00000 0.0632456
\(251\) −3.41421 −0.215503 −0.107752 0.994178i \(-0.534365\pi\)
−0.107752 + 0.994178i \(0.534365\pi\)
\(252\) 0 0
\(253\) −32.9706 −2.07284
\(254\) 2.82843 0.177471
\(255\) −3.17157 −0.198612
\(256\) 1.00000 0.0625000
\(257\) −9.89949 −0.617514 −0.308757 0.951141i \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) −1.85786 −0.115666
\(259\) 0 0
\(260\) 0.828427 0.0513769
\(261\) −2.20101 −0.136239
\(262\) 2.24264 0.138551
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) 2.82843 0.174078
\(265\) 10.4853 0.644106
\(266\) 0 0
\(267\) −7.45584 −0.456290
\(268\) −9.65685 −0.589886
\(269\) −1.51472 −0.0923540 −0.0461770 0.998933i \(-0.514704\pi\)
−0.0461770 + 0.998933i \(0.514704\pi\)
\(270\) 3.31371 0.201666
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −5.41421 −0.328285
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) 4.82843 0.291165
\(276\) 4.00000 0.240772
\(277\) 20.1421 1.21022 0.605112 0.796140i \(-0.293128\pi\)
0.605112 + 0.796140i \(0.293128\pi\)
\(278\) 0.100505 0.00602789
\(279\) 7.51472 0.449894
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) −6.34315 −0.377729
\(283\) −6.24264 −0.371086 −0.185543 0.982636i \(-0.559404\pi\)
−0.185543 + 0.982636i \(0.559404\pi\)
\(284\) 12.4853 0.740865
\(285\) −2.00000 −0.118470
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 2.65685 0.156557
\(289\) 12.3137 0.724336
\(290\) 0.828427 0.0486469
\(291\) −9.51472 −0.557763
\(292\) 6.58579 0.385404
\(293\) 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
\(296\) −3.65685 −0.212550
\(297\) 16.0000 0.928414
\(298\) 6.00000 0.347571
\(299\) 5.65685 0.327144
\(300\) −0.585786 −0.0338204
\(301\) 0 0
\(302\) −11.3137 −0.651031
\(303\) −5.45584 −0.313430
\(304\) −3.41421 −0.195819
\(305\) −13.3137 −0.762341
\(306\) −14.3848 −0.822323
\(307\) −29.0711 −1.65917 −0.829587 0.558378i \(-0.811424\pi\)
−0.829587 + 0.558378i \(0.811424\pi\)
\(308\) 0 0
\(309\) −5.37258 −0.305636
\(310\) −2.82843 −0.160644
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) −0.485281 −0.0274736
\(313\) 22.3848 1.26526 0.632631 0.774453i \(-0.281975\pi\)
0.632631 + 0.774453i \(0.281975\pi\)
\(314\) −10.4853 −0.591719
\(315\) 0 0
\(316\) −1.17157 −0.0659061
\(317\) −6.48528 −0.364250 −0.182125 0.983275i \(-0.558298\pi\)
−0.182125 + 0.983275i \(0.558298\pi\)
\(318\) −6.14214 −0.344434
\(319\) 4.00000 0.223957
\(320\) −1.00000 −0.0559017
\(321\) 0.970563 0.0541715
\(322\) 0 0
\(323\) 18.4853 1.02855
\(324\) 6.02944 0.334969
\(325\) −0.828427 −0.0459529
\(326\) −8.14214 −0.450951
\(327\) 8.48528 0.469237
\(328\) 11.0711 0.611297
\(329\) 0 0
\(330\) −2.82843 −0.155700
\(331\) −5.79899 −0.318741 −0.159371 0.987219i \(-0.550946\pi\)
−0.159371 + 0.987219i \(0.550946\pi\)
\(332\) −6.24264 −0.342609
\(333\) −9.71573 −0.532419
\(334\) −23.7990 −1.30222
\(335\) 9.65685 0.527610
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 12.3137 0.669777
\(339\) −4.28427 −0.232690
\(340\) 5.41421 0.293627
\(341\) −13.6569 −0.739560
\(342\) −9.07107 −0.490507
\(343\) 0 0
\(344\) 3.17157 0.171000
\(345\) −4.00000 −0.215353
\(346\) −3.17157 −0.170505
\(347\) −8.82843 −0.473935 −0.236967 0.971518i \(-0.576153\pi\)
−0.236967 + 0.971518i \(0.576153\pi\)
\(348\) −0.485281 −0.0260138
\(349\) 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
\(352\) −4.82843 −0.257356
\(353\) −34.3848 −1.83012 −0.915058 0.403321i \(-0.867856\pi\)
−0.915058 + 0.403321i \(0.867856\pi\)
\(354\) −6.68629 −0.355372
\(355\) −12.4853 −0.662650
\(356\) 12.7279 0.674579
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) −2.65685 −0.140029
\(361\) −7.34315 −0.386481
\(362\) 14.4853 0.761329
\(363\) −7.21320 −0.378595
\(364\) 0 0
\(365\) −6.58579 −0.344716
\(366\) 7.79899 0.407660
\(367\) 8.97056 0.468260 0.234130 0.972205i \(-0.424776\pi\)
0.234130 + 0.972205i \(0.424776\pi\)
\(368\) −6.82843 −0.355956
\(369\) 29.4142 1.53124
\(370\) 3.65685 0.190111
\(371\) 0 0
\(372\) 1.65685 0.0859039
\(373\) −13.5147 −0.699766 −0.349883 0.936793i \(-0.613779\pi\)
−0.349883 + 0.936793i \(0.613779\pi\)
\(374\) 26.1421 1.35178
\(375\) 0.585786 0.0302499
\(376\) 10.8284 0.558433
\(377\) −0.686292 −0.0353458
\(378\) 0 0
\(379\) 17.5147 0.899671 0.449835 0.893112i \(-0.351483\pi\)
0.449835 + 0.893112i \(0.351483\pi\)
\(380\) 3.41421 0.175145
\(381\) 1.65685 0.0848832
\(382\) 18.1421 0.928232
\(383\) 15.5147 0.792765 0.396383 0.918085i \(-0.370265\pi\)
0.396383 + 0.918085i \(0.370265\pi\)
\(384\) 0.585786 0.0298933
\(385\) 0 0
\(386\) 5.65685 0.287926
\(387\) 8.42641 0.428338
\(388\) 16.2426 0.824595
\(389\) −0.142136 −0.00720656 −0.00360328 0.999994i \(-0.501147\pi\)
−0.00360328 + 0.999994i \(0.501147\pi\)
\(390\) 0.485281 0.0245732
\(391\) 36.9706 1.86968
\(392\) 0 0
\(393\) 1.31371 0.0662678
\(394\) −13.7990 −0.695183
\(395\) 1.17157 0.0589482
\(396\) −12.8284 −0.644653
\(397\) 5.79899 0.291043 0.145521 0.989355i \(-0.453514\pi\)
0.145521 + 0.989355i \(0.453514\pi\)
\(398\) −0.485281 −0.0243250
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −5.65685 −0.282138
\(403\) 2.34315 0.116720
\(404\) 9.31371 0.463374
\(405\) −6.02944 −0.299605
\(406\) 0 0
\(407\) 17.6569 0.875218
\(408\) −3.17157 −0.157016
\(409\) 13.4142 0.663290 0.331645 0.943404i \(-0.392396\pi\)
0.331645 + 0.943404i \(0.392396\pi\)
\(410\) −11.0711 −0.546761
\(411\) 9.37258 0.462315
\(412\) 9.17157 0.451851
\(413\) 0 0
\(414\) −18.1421 −0.891637
\(415\) 6.24264 0.306439
\(416\) 0.828427 0.0406170
\(417\) 0.0588745 0.00288310
\(418\) 16.4853 0.806321
\(419\) −32.8701 −1.60581 −0.802904 0.596109i \(-0.796713\pi\)
−0.802904 + 0.596109i \(0.796713\pi\)
\(420\) 0 0
\(421\) −5.31371 −0.258974 −0.129487 0.991581i \(-0.541333\pi\)
−0.129487 + 0.991581i \(0.541333\pi\)
\(422\) 26.6274 1.29620
\(423\) 28.7696 1.39882
\(424\) 10.4853 0.509210
\(425\) −5.41421 −0.262628
\(426\) 7.31371 0.354350
\(427\) 0 0
\(428\) −1.65685 −0.0800871
\(429\) 2.34315 0.113128
\(430\) −3.17157 −0.152947
\(431\) −33.6569 −1.62119 −0.810597 0.585605i \(-0.800857\pi\)
−0.810597 + 0.585605i \(0.800857\pi\)
\(432\) 3.31371 0.159431
\(433\) −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
\(436\) −14.4853 −0.693719
\(437\) 23.3137 1.11525
\(438\) 3.85786 0.184336
\(439\) −8.97056 −0.428142 −0.214071 0.976818i \(-0.568672\pi\)
−0.214071 + 0.976818i \(0.568672\pi\)
\(440\) 4.82843 0.230186
\(441\) 0 0
\(442\) −4.48528 −0.213343
\(443\) 36.9706 1.75652 0.878262 0.478179i \(-0.158703\pi\)
0.878262 + 0.478179i \(0.158703\pi\)
\(444\) −2.14214 −0.101661
\(445\) −12.7279 −0.603361
\(446\) 15.3137 0.725125
\(447\) 3.51472 0.166240
\(448\) 0 0
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 2.65685 0.125245
\(451\) −53.4558 −2.51714
\(452\) 7.31371 0.344008
\(453\) −6.62742 −0.311383
\(454\) 9.75736 0.457936
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 10.3431 0.483832 0.241916 0.970297i \(-0.422224\pi\)
0.241916 + 0.970297i \(0.422224\pi\)
\(458\) 12.1421 0.567365
\(459\) −17.9411 −0.837420
\(460\) 6.82843 0.318377
\(461\) 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 \(-0.370972\pi\)
0.394344 + 0.918963i \(0.370972\pi\)
\(464\) 0.828427 0.0384588
\(465\) −1.65685 −0.0768348
\(466\) −0.686292 −0.0317918
\(467\) −3.89949 −0.180447 −0.0902236 0.995922i \(-0.528758\pi\)
−0.0902236 + 0.995922i \(0.528758\pi\)
\(468\) 2.20101 0.101742
\(469\) 0 0
\(470\) −10.8284 −0.499478
\(471\) −6.14214 −0.283015
\(472\) 11.4142 0.525382
\(473\) −15.3137 −0.704125
\(474\) −0.686292 −0.0315224
\(475\) −3.41421 −0.156655
\(476\) 0 0
\(477\) 27.8579 1.27552
\(478\) 9.65685 0.441694
\(479\) 22.8284 1.04306 0.521529 0.853234i \(-0.325362\pi\)
0.521529 + 0.853234i \(0.325362\pi\)
\(480\) −0.585786 −0.0267374
\(481\) −3.02944 −0.138130
\(482\) −10.5858 −0.482169
\(483\) 0 0
\(484\) 12.3137 0.559714
\(485\) −16.2426 −0.737540
\(486\) 13.4731 0.611152
\(487\) −7.79899 −0.353406 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(488\) −13.3137 −0.602683
\(489\) −4.76955 −0.215687
\(490\) 0 0
\(491\) −24.2843 −1.09593 −0.547967 0.836500i \(-0.684598\pi\)
−0.547967 + 0.836500i \(0.684598\pi\)
\(492\) 6.48528 0.292379
\(493\) −4.48528 −0.202007
\(494\) −2.82843 −0.127257
\(495\) 12.8284 0.576595
\(496\) −2.82843 −0.127000
\(497\) 0 0
\(498\) −3.65685 −0.163868
\(499\) 41.6569 1.86482 0.932408 0.361406i \(-0.117703\pi\)
0.932408 + 0.361406i \(0.117703\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −13.9411 −0.622844
\(502\) 3.41421 0.152384
\(503\) −6.34315 −0.282827 −0.141413 0.989951i \(-0.545165\pi\)
−0.141413 + 0.989951i \(0.545165\pi\)
\(504\) 0 0
\(505\) −9.31371 −0.414455
\(506\) 32.9706 1.46572
\(507\) 7.21320 0.320350
\(508\) −2.82843 −0.125491
\(509\) 33.7990 1.49811 0.749057 0.662506i \(-0.230507\pi\)
0.749057 + 0.662506i \(0.230507\pi\)
\(510\) 3.17157 0.140440
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −11.3137 −0.499512
\(514\) 9.89949 0.436648
\(515\) −9.17157 −0.404148
\(516\) 1.85786 0.0817879
\(517\) −52.2843 −2.29946
\(518\) 0 0
\(519\) −1.85786 −0.0815512
\(520\) −0.828427 −0.0363289
\(521\) −4.92893 −0.215940 −0.107970 0.994154i \(-0.534435\pi\)
−0.107970 + 0.994154i \(0.534435\pi\)
\(522\) 2.20101 0.0963356
\(523\) −4.10051 −0.179303 −0.0896513 0.995973i \(-0.528575\pi\)
−0.0896513 + 0.995973i \(0.528575\pi\)
\(524\) −2.24264 −0.0979702
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 15.3137 0.667076
\(528\) −2.82843 −0.123091
\(529\) 23.6274 1.02728
\(530\) −10.4853 −0.455452
\(531\) 30.3259 1.31603
\(532\) 0 0
\(533\) 9.17157 0.397265
\(534\) 7.45584 0.322646
\(535\) 1.65685 0.0716321
\(536\) 9.65685 0.417113
\(537\) −2.34315 −0.101114
\(538\) 1.51472 0.0653042
\(539\) 0 0
\(540\) −3.31371 −0.142599
\(541\) 18.9706 0.815608 0.407804 0.913069i \(-0.366295\pi\)
0.407804 + 0.913069i \(0.366295\pi\)
\(542\) 12.0000 0.515444
\(543\) 8.48528 0.364138
\(544\) 5.41421 0.232132
\(545\) 14.4853 0.620481
\(546\) 0 0
\(547\) 6.48528 0.277291 0.138645 0.990342i \(-0.455725\pi\)
0.138645 + 0.990342i \(0.455725\pi\)
\(548\) −16.0000 −0.683486
\(549\) −35.3726 −1.50967
\(550\) −4.82843 −0.205885
\(551\) −2.82843 −0.120495
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −20.1421 −0.855757
\(555\) 2.14214 0.0909286
\(556\) −0.100505 −0.00426236
\(557\) 20.8284 0.882529 0.441264 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(558\) −7.51472 −0.318123
\(559\) 2.62742 0.111128
\(560\) 0 0
\(561\) 15.3137 0.646545
\(562\) −8.00000 −0.337460
\(563\) 39.4142 1.66111 0.830556 0.556936i \(-0.188023\pi\)
0.830556 + 0.556936i \(0.188023\pi\)
\(564\) 6.34315 0.267095
\(565\) −7.31371 −0.307690
\(566\) 6.24264 0.262398
\(567\) 0 0
\(568\) −12.4853 −0.523871
\(569\) −6.68629 −0.280304 −0.140152 0.990130i \(-0.544759\pi\)
−0.140152 + 0.990130i \(0.544759\pi\)
\(570\) 2.00000 0.0837708
\(571\) −41.7990 −1.74923 −0.874617 0.484815i \(-0.838887\pi\)
−0.874617 + 0.484815i \(0.838887\pi\)
\(572\) −4.00000 −0.167248
\(573\) 10.6274 0.443967
\(574\) 0 0
\(575\) −6.82843 −0.284765
\(576\) −2.65685 −0.110702
\(577\) −25.8995 −1.07821 −0.539105 0.842239i \(-0.681237\pi\)
−0.539105 + 0.842239i \(0.681237\pi\)
\(578\) −12.3137 −0.512183
\(579\) 3.31371 0.137713
\(580\) −0.828427 −0.0343986
\(581\) 0 0
\(582\) 9.51472 0.394398
\(583\) −50.6274 −2.09677
\(584\) −6.58579 −0.272522
\(585\) −2.20101 −0.0910006
\(586\) −19.6569 −0.812017
\(587\) −2.92893 −0.120890 −0.0604450 0.998172i \(-0.519252\pi\)
−0.0604450 + 0.998172i \(0.519252\pi\)
\(588\) 0 0
\(589\) 9.65685 0.397904
\(590\) −11.4142 −0.469916
\(591\) −8.08326 −0.332501
\(592\) 3.65685 0.150296
\(593\) −28.7279 −1.17971 −0.589857 0.807508i \(-0.700816\pi\)
−0.589857 + 0.807508i \(0.700816\pi\)
\(594\) −16.0000 −0.656488
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −0.284271 −0.0116344
\(598\) −5.65685 −0.231326
\(599\) −5.17157 −0.211305 −0.105652 0.994403i \(-0.533693\pi\)
−0.105652 + 0.994403i \(0.533693\pi\)
\(600\) 0.585786 0.0239146
\(601\) −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
\(604\) 11.3137 0.460348
\(605\) −12.3137 −0.500623
\(606\) 5.45584 0.221629
\(607\) 40.2843 1.63509 0.817544 0.575866i \(-0.195335\pi\)
0.817544 + 0.575866i \(0.195335\pi\)
\(608\) 3.41421 0.138465
\(609\) 0 0
\(610\) 13.3137 0.539056
\(611\) 8.97056 0.362910
\(612\) 14.3848 0.581470
\(613\) −23.6569 −0.955491 −0.477746 0.878498i \(-0.658546\pi\)
−0.477746 + 0.878498i \(0.658546\pi\)
\(614\) 29.0711 1.17321
\(615\) −6.48528 −0.261512
\(616\) 0 0
\(617\) −10.6863 −0.430214 −0.215107 0.976590i \(-0.569010\pi\)
−0.215107 + 0.976590i \(0.569010\pi\)
\(618\) 5.37258 0.216117
\(619\) −14.9289 −0.600044 −0.300022 0.953932i \(-0.596994\pi\)
−0.300022 + 0.953932i \(0.596994\pi\)
\(620\) 2.82843 0.113592
\(621\) −22.6274 −0.908007
\(622\) 4.00000 0.160385
\(623\) 0 0
\(624\) 0.485281 0.0194268
\(625\) 1.00000 0.0400000
\(626\) −22.3848 −0.894676
\(627\) 9.65685 0.385658
\(628\) 10.4853 0.418408
\(629\) −19.7990 −0.789437
\(630\) 0 0
\(631\) 4.48528 0.178556 0.0892781 0.996007i \(-0.471544\pi\)
0.0892781 + 0.996007i \(0.471544\pi\)
\(632\) 1.17157 0.0466027
\(633\) 15.5980 0.619964
\(634\) 6.48528 0.257563
\(635\) 2.82843 0.112243
\(636\) 6.14214 0.243552
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) −33.1716 −1.31225
\(640\) 1.00000 0.0395285
\(641\) 20.6274 0.814734 0.407367 0.913265i \(-0.366447\pi\)
0.407367 + 0.913265i \(0.366447\pi\)
\(642\) −0.970563 −0.0383051
\(643\) −47.2132 −1.86191 −0.930953 0.365138i \(-0.881022\pi\)
−0.930953 + 0.365138i \(0.881022\pi\)
\(644\) 0 0
\(645\) −1.85786 −0.0731533
\(646\) −18.4853 −0.727294
\(647\) −39.1127 −1.53768 −0.768839 0.639442i \(-0.779165\pi\)
−0.768839 + 0.639442i \(0.779165\pi\)
\(648\) −6.02944 −0.236859
\(649\) −55.1127 −2.16336
\(650\) 0.828427 0.0324936
\(651\) 0 0
\(652\) 8.14214 0.318871
\(653\) 15.6569 0.612700 0.306350 0.951919i \(-0.400892\pi\)
0.306350 + 0.951919i \(0.400892\pi\)
\(654\) −8.48528 −0.331801
\(655\) 2.24264 0.0876272
\(656\) −11.0711 −0.432253
\(657\) −17.4975 −0.682642
\(658\) 0 0
\(659\) −32.8284 −1.27881 −0.639407 0.768868i \(-0.720820\pi\)
−0.639407 + 0.768868i \(0.720820\pi\)
\(660\) 2.82843 0.110096
\(661\) −18.2843 −0.711176 −0.355588 0.934643i \(-0.615719\pi\)
−0.355588 + 0.934643i \(0.615719\pi\)
\(662\) 5.79899 0.225384
\(663\) −2.62742 −0.102040
\(664\) 6.24264 0.242261
\(665\) 0 0
\(666\) 9.71573 0.376477
\(667\) −5.65685 −0.219034
\(668\) 23.7990 0.920811
\(669\) 8.97056 0.346822
\(670\) −9.65685 −0.373077
\(671\) 64.2843 2.48167
\(672\) 0 0
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) −6.00000 −0.231111
\(675\) 3.31371 0.127545
\(676\) −12.3137 −0.473604
\(677\) 11.4558 0.440284 0.220142 0.975468i \(-0.429348\pi\)
0.220142 + 0.975468i \(0.429348\pi\)
\(678\) 4.28427 0.164536
\(679\) 0 0
\(680\) −5.41421 −0.207626
\(681\) 5.71573 0.219027
\(682\) 13.6569 0.522948
\(683\) 22.3431 0.854937 0.427468 0.904030i \(-0.359406\pi\)
0.427468 + 0.904030i \(0.359406\pi\)
\(684\) 9.07107 0.346841
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 7.11270 0.271366
\(688\) −3.17157 −0.120915
\(689\) 8.68629 0.330921
\(690\) 4.00000 0.152277
\(691\) −10.2426 −0.389648 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(692\) 3.17157 0.120565
\(693\) 0 0
\(694\) 8.82843 0.335123
\(695\) 0.100505 0.00381237
\(696\) 0.485281 0.0183945
\(697\) 59.9411 2.27043
\(698\) −14.4853 −0.548276
\(699\) −0.402020 −0.0152058
\(700\) 0 0
\(701\) −14.4853 −0.547102 −0.273551 0.961858i \(-0.588198\pi\)
−0.273551 + 0.961858i \(0.588198\pi\)
\(702\) 2.74517 0.103610
\(703\) −12.4853 −0.470891
\(704\) 4.82843 0.181978
\(705\) −6.34315 −0.238897
\(706\) 34.3848 1.29409
\(707\) 0 0
\(708\) 6.68629 0.251286
\(709\) −17.1127 −0.642681 −0.321340 0.946964i \(-0.604133\pi\)
−0.321340 + 0.946964i \(0.604133\pi\)
\(710\) 12.4853 0.468564
\(711\) 3.11270 0.116735
\(712\) −12.7279 −0.476999
\(713\) 19.3137 0.723304
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 4.00000 0.149487
\(717\) 5.65685 0.211259
\(718\) 28.2843 1.05556
\(719\) 9.45584 0.352643 0.176322 0.984333i \(-0.443580\pi\)
0.176322 + 0.984333i \(0.443580\pi\)
\(720\) 2.65685 0.0990151
\(721\) 0 0
\(722\) 7.34315 0.273284
\(723\) −6.20101 −0.230618
\(724\) −14.4853 −0.538341
\(725\) 0.828427 0.0307670
\(726\) 7.21320 0.267707
\(727\) 20.4853 0.759757 0.379879 0.925036i \(-0.375966\pi\)
0.379879 + 0.925036i \(0.375966\pi\)
\(728\) 0 0
\(729\) −10.1960 −0.377628
\(730\) 6.58579 0.243751
\(731\) 17.1716 0.635114
\(732\) −7.79899 −0.288259
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −8.97056 −0.331110
\(735\) 0 0
\(736\) 6.82843 0.251699
\(737\) −46.6274 −1.71754
\(738\) −29.4142 −1.08275
\(739\) 8.82843 0.324759 0.162379 0.986728i \(-0.448083\pi\)
0.162379 + 0.986728i \(0.448083\pi\)
\(740\) −3.65685 −0.134429
\(741\) −1.65685 −0.0608661
\(742\) 0 0
\(743\) 12.2010 0.447612 0.223806 0.974634i \(-0.428152\pi\)
0.223806 + 0.974634i \(0.428152\pi\)
\(744\) −1.65685 −0.0607432
\(745\) 6.00000 0.219823
\(746\) 13.5147 0.494809
\(747\) 16.5858 0.606842
\(748\) −26.1421 −0.955851
\(749\) 0 0
\(750\) −0.585786 −0.0213899
\(751\) −16.6863 −0.608891 −0.304446 0.952530i \(-0.598471\pi\)
−0.304446 + 0.952530i \(0.598471\pi\)
\(752\) −10.8284 −0.394872
\(753\) 2.00000 0.0728841
\(754\) 0.686292 0.0249933
\(755\) −11.3137 −0.411748
\(756\) 0 0
\(757\) −7.65685 −0.278293 −0.139147 0.990272i \(-0.544436\pi\)
−0.139147 + 0.990272i \(0.544436\pi\)
\(758\) −17.5147 −0.636163
\(759\) 19.3137 0.701043
\(760\) −3.41421 −0.123847
\(761\) 14.3848 0.521448 0.260724 0.965413i \(-0.416039\pi\)
0.260724 + 0.965413i \(0.416039\pi\)
\(762\) −1.65685 −0.0600215
\(763\) 0 0
\(764\) −18.1421 −0.656359
\(765\) −14.3848 −0.520083
\(766\) −15.5147 −0.560570
\(767\) 9.45584 0.341431
\(768\) −0.585786 −0.0211377
\(769\) 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
\(772\) −5.65685 −0.203595
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) −8.42641 −0.302881
\(775\) −2.82843 −0.101600
\(776\) −16.2426 −0.583077
\(777\) 0 0
\(778\) 0.142136 0.00509581
\(779\) 37.7990 1.35429
\(780\) −0.485281 −0.0173759
\(781\) 60.2843 2.15714
\(782\) −36.9706 −1.32206
\(783\) 2.74517 0.0981042
\(784\) 0 0
\(785\) −10.4853 −0.374236
\(786\) −1.31371 −0.0468584
\(787\) 26.7279 0.952748 0.476374 0.879243i \(-0.341951\pi\)
0.476374 + 0.879243i \(0.341951\pi\)
\(788\) 13.7990 0.491569
\(789\) −16.4020 −0.583927
\(790\) −1.17157 −0.0416827
\(791\) 0 0
\(792\) 12.8284 0.455838
\(793\) −11.0294 −0.391667
\(794\) −5.79899 −0.205798
\(795\) −6.14214 −0.217839
\(796\) 0.485281 0.0172003
\(797\) 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
\(800\) −1.00000 −0.0353553
\(801\) −33.8162 −1.19484
\(802\) 6.00000 0.211867
\(803\) 31.7990 1.12216
\(804\) 5.65685 0.199502
\(805\) 0 0
\(806\) −2.34315 −0.0825338
\(807\) 0.887302 0.0312345
\(808\) −9.31371 −0.327655
\(809\) 36.9706 1.29982 0.649908 0.760013i \(-0.274807\pi\)
0.649908 + 0.760013i \(0.274807\pi\)
\(810\) 6.02944 0.211853
\(811\) −35.4142 −1.24356 −0.621781 0.783191i \(-0.713590\pi\)
−0.621781 + 0.783191i \(0.713590\pi\)
\(812\) 0 0
\(813\) 7.02944 0.246533
\(814\) −17.6569 −0.618872
\(815\) −8.14214 −0.285207
\(816\) 3.17157 0.111027
\(817\) 10.8284 0.378839
\(818\) −13.4142 −0.469017
\(819\) 0 0
\(820\) 11.0711 0.386618
\(821\) −5.31371 −0.185450 −0.0927249 0.995692i \(-0.529558\pi\)
−0.0927249 + 0.995692i \(0.529558\pi\)
\(822\) −9.37258 −0.326906
\(823\) −36.2843 −1.26479 −0.632395 0.774646i \(-0.717928\pi\)
−0.632395 + 0.774646i \(0.717928\pi\)
\(824\) −9.17157 −0.319507
\(825\) −2.82843 −0.0984732
\(826\) 0 0
\(827\) −50.6274 −1.76049 −0.880244 0.474522i \(-0.842621\pi\)
−0.880244 + 0.474522i \(0.842621\pi\)
\(828\) 18.1421 0.630483
\(829\) 38.9706 1.35350 0.676752 0.736211i \(-0.263387\pi\)
0.676752 + 0.736211i \(0.263387\pi\)
\(830\) −6.24264 −0.216685
\(831\) −11.7990 −0.409302
\(832\) −0.828427 −0.0287205
\(833\) 0 0
\(834\) −0.0588745 −0.00203866
\(835\) −23.7990 −0.823598
\(836\) −16.4853 −0.570155
\(837\) −9.37258 −0.323964
\(838\) 32.8701 1.13548
\(839\) 13.8579 0.478427 0.239213 0.970967i \(-0.423110\pi\)
0.239213 + 0.970967i \(0.423110\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 5.31371 0.183122
\(843\) −4.68629 −0.161404
\(844\) −26.6274 −0.916553
\(845\) 12.3137 0.423604
\(846\) −28.7696 −0.989118
\(847\) 0 0
\(848\) −10.4853 −0.360066
\(849\) 3.65685 0.125503
\(850\) 5.41421 0.185706
\(851\) −24.9706 −0.855980
\(852\) −7.31371 −0.250564
\(853\) 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
\(856\) 1.65685 0.0566301
\(857\) 19.0711 0.651455 0.325728 0.945464i \(-0.394391\pi\)
0.325728 + 0.945464i \(0.394391\pi\)
\(858\) −2.34315 −0.0799937
\(859\) −35.2132 −1.20146 −0.600729 0.799452i \(-0.705123\pi\)
−0.600729 + 0.799452i \(0.705123\pi\)
\(860\) 3.17157 0.108150
\(861\) 0 0
\(862\) 33.6569 1.14636
\(863\) 28.9706 0.986169 0.493085 0.869981i \(-0.335869\pi\)
0.493085 + 0.869981i \(0.335869\pi\)
\(864\) −3.31371 −0.112735
\(865\) −3.17157 −0.107837
\(866\) 13.4142 0.455834
\(867\) −7.21320 −0.244973
\(868\) 0 0
\(869\) −5.65685 −0.191896
\(870\) −0.485281 −0.0164526
\(871\) 8.00000 0.271070
\(872\) 14.4853 0.490534
\(873\) −43.1543 −1.46055
\(874\) −23.3137 −0.788598
\(875\) 0 0
\(876\) −3.85786 −0.130345
\(877\) 26.2843 0.887557 0.443778 0.896137i \(-0.353638\pi\)
0.443778 + 0.896137i \(0.353638\pi\)
\(878\) 8.97056 0.302742
\(879\) −11.5147 −0.388382
\(880\) −4.82843 −0.162766
\(881\) −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 \(-0.670563\pi\)
−0.510564 + 0.859840i \(0.670563\pi\)
\(884\) 4.48528 0.150856
\(885\) −6.68629 −0.224757
\(886\) −36.9706 −1.24205
\(887\) −7.11270 −0.238821 −0.119411 0.992845i \(-0.538100\pi\)
−0.119411 + 0.992845i \(0.538100\pi\)
\(888\) 2.14214 0.0718854
\(889\) 0 0
\(890\) 12.7279 0.426641
\(891\) 29.1127 0.975312
\(892\) −15.3137 −0.512741
\(893\) 36.9706 1.23717
\(894\) −3.51472 −0.117550
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) −3.31371 −0.110642
\(898\) −28.6274 −0.955309
\(899\) −2.34315 −0.0781483
\(900\) −2.65685 −0.0885618
\(901\) 56.7696 1.89127
\(902\) 53.4558 1.77988
\(903\) 0 0
\(904\) −7.31371 −0.243250
\(905\) 14.4853 0.481507
\(906\) 6.62742 0.220181
\(907\) −56.2843 −1.86889 −0.934444 0.356109i \(-0.884103\pi\)
−0.934444 + 0.356109i \(0.884103\pi\)
\(908\) −9.75736 −0.323809
\(909\) −24.7452 −0.820745
\(910\) 0 0
\(911\) −20.2843 −0.672048 −0.336024 0.941853i \(-0.609082\pi\)
−0.336024 + 0.941853i \(0.609082\pi\)
\(912\) 2.00000 0.0662266
\(913\) −30.1421 −0.997559
\(914\) −10.3431 −0.342121
\(915\) 7.79899 0.257827
\(916\) −12.1421 −0.401187
\(917\) 0 0
\(918\) 17.9411 0.592145
\(919\) 32.4853 1.07159 0.535795 0.844348i \(-0.320012\pi\)
0.535795 + 0.844348i \(0.320012\pi\)
\(920\) −6.82843 −0.225127
\(921\) 17.0294 0.561139
\(922\) −7.17157 −0.236183
\(923\) −10.3431 −0.340449
\(924\) 0 0
\(925\) 3.65685 0.120237
\(926\) −16.9706 −0.557687
\(927\) −24.3675 −0.800335
\(928\) −0.828427 −0.0271945
\(929\) −25.2132 −0.827218 −0.413609 0.910455i \(-0.635732\pi\)
−0.413609 + 0.910455i \(0.635732\pi\)
\(930\) 1.65685 0.0543304
\(931\) 0 0
\(932\) 0.686292 0.0224802
\(933\) 2.34315 0.0767111
\(934\) 3.89949 0.127595
\(935\) 26.1421 0.854939
\(936\) −2.20101 −0.0719423
\(937\) −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
\(940\) 10.8284 0.353184
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 6.14214 0.200122
\(943\) 75.5980 2.46181
\(944\) −11.4142 −0.371501
\(945\) 0 0
\(946\) 15.3137 0.497892
\(947\) 0.828427 0.0269203 0.0134601 0.999909i \(-0.495715\pi\)
0.0134601 + 0.999909i \(0.495715\pi\)
\(948\) 0.686292 0.0222897
\(949\) −5.45584 −0.177104
\(950\) 3.41421 0.110772
\(951\) 3.79899 0.123191
\(952\) 0 0
\(953\) 11.6569 0.377603 0.188801 0.982015i \(-0.439540\pi\)
0.188801 + 0.982015i \(0.439540\pi\)
\(954\) −27.8579 −0.901932
\(955\) 18.1421 0.587066
\(956\) −9.65685 −0.312325
\(957\) −2.34315 −0.0757431
\(958\) −22.8284 −0.737553
\(959\) 0 0
\(960\) 0.585786 0.0189062
\(961\) −23.0000 −0.741935
\(962\) 3.02944 0.0976730
\(963\) 4.40202 0.141853
\(964\) 10.5858 0.340945
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) 13.4558 0.432711 0.216355 0.976315i \(-0.430583\pi\)
0.216355 + 0.976315i \(0.430583\pi\)
\(968\) −12.3137 −0.395778
\(969\) −10.8284 −0.347859
\(970\) 16.2426 0.521520
\(971\) −37.3553 −1.19879 −0.599395 0.800453i \(-0.704592\pi\)
−0.599395 + 0.800453i \(0.704592\pi\)
\(972\) −13.4731 −0.432150
\(973\) 0 0
\(974\) 7.79899 0.249896
\(975\) 0.485281 0.0155414
\(976\) 13.3137 0.426161
\(977\) −35.3137 −1.12979 −0.564893 0.825164i \(-0.691082\pi\)
−0.564893 + 0.825164i \(0.691082\pi\)
\(978\) 4.76955 0.152513
\(979\) 61.4558 1.96414
\(980\) 0 0
\(981\) 38.4853 1.22874
\(982\) 24.2843 0.774942
\(983\) 51.7990 1.65213 0.826066 0.563574i \(-0.190574\pi\)
0.826066 + 0.563574i \(0.190574\pi\)
\(984\) −6.48528 −0.206743
\(985\) −13.7990 −0.439672
\(986\) 4.48528 0.142840
\(987\) 0 0
\(988\) 2.82843 0.0899843
\(989\) 21.6569 0.688648
\(990\) −12.8284 −0.407714
\(991\) −28.7696 −0.913895 −0.456947 0.889494i \(-0.651057\pi\)
−0.456947 + 0.889494i \(0.651057\pi\)
\(992\) 2.82843 0.0898027
\(993\) 3.39697 0.107800
\(994\) 0 0
\(995\) −0.485281 −0.0153845
\(996\) 3.65685 0.115872
\(997\) −38.2843 −1.21248 −0.606238 0.795284i \(-0.707322\pi\)
−0.606238 + 0.795284i \(0.707322\pi\)
\(998\) −41.6569 −1.31862
\(999\) 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 490.2.a.l.1.2 2
3.2 odd 2 4410.2.a.by.1.1 2
4.3 odd 2 3920.2.a.ca.1.1 2
5.2 odd 4 2450.2.c.w.99.1 4
5.3 odd 4 2450.2.c.w.99.4 4
5.4 even 2 2450.2.a.bs.1.1 2
7.2 even 3 490.2.e.j.361.1 4
7.3 odd 6 490.2.e.i.471.2 4
7.4 even 3 490.2.e.j.471.1 4
7.5 odd 6 490.2.e.i.361.2 4
7.6 odd 2 490.2.a.m.1.1 yes 2
21.20 even 2 4410.2.a.bt.1.1 2
28.27 even 2 3920.2.a.bm.1.2 2
35.13 even 4 2450.2.c.t.99.3 4
35.27 even 4 2450.2.c.t.99.2 4
35.34 odd 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 1.1 even 1 trivial
490.2.a.m.1.1 yes 2 7.6 odd 2
490.2.e.i.361.2 4 7.5 odd 6
490.2.e.i.471.2 4 7.3 odd 6
490.2.e.j.361.1 4 7.2 even 3
490.2.e.j.471.1 4 7.4 even 3
2450.2.a.bn.1.2 2 35.34 odd 2
2450.2.a.bs.1.1 2 5.4 even 2
2450.2.c.t.99.2 4 35.27 even 4
2450.2.c.t.99.3 4 35.13 even 4
2450.2.c.w.99.1 4 5.2 odd 4
2450.2.c.w.99.4 4 5.3 odd 4
3920.2.a.bm.1.2 2 28.27 even 2
3920.2.a.ca.1.1 2 4.3 odd 2
4410.2.a.bt.1.1 2 21.20 even 2
4410.2.a.by.1.1 2 3.2 odd 2