Properties

Label 430.2.a.g.1.2
Level 430
Weight 2
Character 430.1
Self dual yes
Analytic conductor 3.434
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
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 \(1.41421\)
Character \(\chi\) = 430.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.00000 q^{10} +0.585786 q^{11} +1.41421 q^{12} -1.82843 q^{13} +1.00000 q^{14} +1.41421 q^{15} +1.00000 q^{16} +1.41421 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +1.41421 q^{21} +0.585786 q^{22} -5.07107 q^{23} +1.41421 q^{24} +1.00000 q^{25} -1.82843 q^{26} -5.65685 q^{27} +1.00000 q^{28} +1.24264 q^{29} +1.41421 q^{30} -0.414214 q^{31} +1.00000 q^{32} +0.828427 q^{33} +1.41421 q^{34} +1.00000 q^{35} -1.00000 q^{36} +2.24264 q^{37} -1.00000 q^{38} -2.58579 q^{39} +1.00000 q^{40} -1.82843 q^{41} +1.41421 q^{42} +1.00000 q^{43} +0.585786 q^{44} -1.00000 q^{45} -5.07107 q^{46} +7.07107 q^{47} +1.41421 q^{48} -6.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -1.82843 q^{52} -5.65685 q^{53} -5.65685 q^{54} +0.585786 q^{55} +1.00000 q^{56} -1.41421 q^{57} +1.24264 q^{58} -1.17157 q^{59} +1.41421 q^{60} +0.0710678 q^{61} -0.414214 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.82843 q^{65} +0.828427 q^{66} -1.24264 q^{67} +1.41421 q^{68} -7.17157 q^{69} +1.00000 q^{70} +11.8995 q^{71} -1.00000 q^{72} -9.24264 q^{73} +2.24264 q^{74} +1.41421 q^{75} -1.00000 q^{76} +0.585786 q^{77} -2.58579 q^{78} -8.41421 q^{79} +1.00000 q^{80} -5.00000 q^{81} -1.82843 q^{82} +3.65685 q^{83} +1.41421 q^{84} +1.41421 q^{85} +1.00000 q^{86} +1.75736 q^{87} +0.585786 q^{88} -9.07107 q^{89} -1.00000 q^{90} -1.82843 q^{91} -5.07107 q^{92} -0.585786 q^{93} +7.07107 q^{94} -1.00000 q^{95} +1.41421 q^{96} +13.5563 q^{97} -6.00000 q^{98} -0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + 2q^{8} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + 2q^{8} - 2q^{9} + 2q^{10} + 4q^{11} + 2q^{13} + 2q^{14} + 2q^{16} - 2q^{18} - 2q^{19} + 2q^{20} + 4q^{22} + 4q^{23} + 2q^{25} + 2q^{26} + 2q^{28} - 6q^{29} + 2q^{31} + 2q^{32} - 4q^{33} + 2q^{35} - 2q^{36} - 4q^{37} - 2q^{38} - 8q^{39} + 2q^{40} + 2q^{41} + 2q^{43} + 4q^{44} - 2q^{45} + 4q^{46} - 12q^{49} + 2q^{50} + 4q^{51} + 2q^{52} + 4q^{55} + 2q^{56} - 6q^{58} - 8q^{59} - 14q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + 2q^{65} - 4q^{66} + 6q^{67} - 20q^{69} + 2q^{70} + 4q^{71} - 2q^{72} - 10q^{73} - 4q^{74} - 2q^{76} + 4q^{77} - 8q^{78} - 14q^{79} + 2q^{80} - 10q^{81} + 2q^{82} - 4q^{83} + 2q^{86} + 12q^{87} + 4q^{88} - 4q^{89} - 2q^{90} + 2q^{91} + 4q^{92} - 4q^{93} - 2q^{95} - 4q^{97} - 12q^{98} - 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.41421 0.577350
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.00000 −0.333333
\(10\) 1.00000 0.316228
\(11\) 0.585786 0.176621 0.0883106 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) 1.41421 0.408248
\(13\) −1.82843 −0.507114 −0.253557 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.41421 0.365148
\(16\) 1.00000 0.250000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.41421 0.308607
\(22\) 0.585786 0.124890
\(23\) −5.07107 −1.05739 −0.528695 0.848812i \(-0.677319\pi\)
−0.528695 + 0.848812i \(0.677319\pi\)
\(24\) 1.41421 0.288675
\(25\) 1.00000 0.200000
\(26\) −1.82843 −0.358584
\(27\) −5.65685 −1.08866
\(28\) 1.00000 0.188982
\(29\) 1.24264 0.230753 0.115376 0.993322i \(-0.463193\pi\)
0.115376 + 0.993322i \(0.463193\pi\)
\(30\) 1.41421 0.258199
\(31\) −0.414214 −0.0743950 −0.0371975 0.999308i \(-0.511843\pi\)
−0.0371975 + 0.999308i \(0.511843\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.828427 0.144211
\(34\) 1.41421 0.242536
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) 2.24264 0.368688 0.184344 0.982862i \(-0.440984\pi\)
0.184344 + 0.982862i \(0.440984\pi\)
\(38\) −1.00000 −0.162221
\(39\) −2.58579 −0.414057
\(40\) 1.00000 0.158114
\(41\) −1.82843 −0.285552 −0.142776 0.989755i \(-0.545603\pi\)
−0.142776 + 0.989755i \(0.545603\pi\)
\(42\) 1.41421 0.218218
\(43\) 1.00000 0.152499
\(44\) 0.585786 0.0883106
\(45\) −1.00000 −0.149071
\(46\) −5.07107 −0.747688
\(47\) 7.07107 1.03142 0.515711 0.856763i \(-0.327528\pi\)
0.515711 + 0.856763i \(0.327528\pi\)
\(48\) 1.41421 0.204124
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −1.82843 −0.253557
\(53\) −5.65685 −0.777029 −0.388514 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) −5.65685 −0.769800
\(55\) 0.585786 0.0789874
\(56\) 1.00000 0.133631
\(57\) −1.41421 −0.187317
\(58\) 1.24264 0.163167
\(59\) −1.17157 −0.152526 −0.0762629 0.997088i \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) 1.41421 0.182574
\(61\) 0.0710678 0.00909930 0.00454965 0.999990i \(-0.498552\pi\)
0.00454965 + 0.999990i \(0.498552\pi\)
\(62\) −0.414214 −0.0526052
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −1.82843 −0.226788
\(66\) 0.828427 0.101972
\(67\) −1.24264 −0.151813 −0.0759064 0.997115i \(-0.524185\pi\)
−0.0759064 + 0.997115i \(0.524185\pi\)
\(68\) 1.41421 0.171499
\(69\) −7.17157 −0.863356
\(70\) 1.00000 0.119523
\(71\) 11.8995 1.41221 0.706105 0.708107i \(-0.250451\pi\)
0.706105 + 0.708107i \(0.250451\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.24264 −1.08177 −0.540885 0.841097i \(-0.681910\pi\)
−0.540885 + 0.841097i \(0.681910\pi\)
\(74\) 2.24264 0.260702
\(75\) 1.41421 0.163299
\(76\) −1.00000 −0.114708
\(77\) 0.585786 0.0667566
\(78\) −2.58579 −0.292783
\(79\) −8.41421 −0.946673 −0.473336 0.880882i \(-0.656951\pi\)
−0.473336 + 0.880882i \(0.656951\pi\)
\(80\) 1.00000 0.111803
\(81\) −5.00000 −0.555556
\(82\) −1.82843 −0.201916
\(83\) 3.65685 0.401392 0.200696 0.979654i \(-0.435680\pi\)
0.200696 + 0.979654i \(0.435680\pi\)
\(84\) 1.41421 0.154303
\(85\) 1.41421 0.153393
\(86\) 1.00000 0.107833
\(87\) 1.75736 0.188409
\(88\) 0.585786 0.0624450
\(89\) −9.07107 −0.961531 −0.480766 0.876849i \(-0.659641\pi\)
−0.480766 + 0.876849i \(0.659641\pi\)
\(90\) −1.00000 −0.105409
\(91\) −1.82843 −0.191671
\(92\) −5.07107 −0.528695
\(93\) −0.585786 −0.0607432
\(94\) 7.07107 0.729325
\(95\) −1.00000 −0.102598
\(96\) 1.41421 0.144338
\(97\) 13.5563 1.37644 0.688219 0.725503i \(-0.258393\pi\)
0.688219 + 0.725503i \(0.258393\pi\)
\(98\) −6.00000 −0.606092
\(99\) −0.585786 −0.0588738
\(100\) 1.00000 0.100000
\(101\) 1.41421 0.140720 0.0703598 0.997522i \(-0.477585\pi\)
0.0703598 + 0.997522i \(0.477585\pi\)
\(102\) 2.00000 0.198030
\(103\) 0.828427 0.0816274 0.0408137 0.999167i \(-0.487005\pi\)
0.0408137 + 0.999167i \(0.487005\pi\)
\(104\) −1.82843 −0.179292
\(105\) 1.41421 0.138013
\(106\) −5.65685 −0.549442
\(107\) 1.92893 0.186477 0.0932385 0.995644i \(-0.470278\pi\)
0.0932385 + 0.995644i \(0.470278\pi\)
\(108\) −5.65685 −0.544331
\(109\) −10.8284 −1.03718 −0.518588 0.855024i \(-0.673542\pi\)
−0.518588 + 0.855024i \(0.673542\pi\)
\(110\) 0.585786 0.0558525
\(111\) 3.17157 0.301032
\(112\) 1.00000 0.0944911
\(113\) 4.89949 0.460906 0.230453 0.973083i \(-0.425979\pi\)
0.230453 + 0.973083i \(0.425979\pi\)
\(114\) −1.41421 −0.132453
\(115\) −5.07107 −0.472880
\(116\) 1.24264 0.115376
\(117\) 1.82843 0.169038
\(118\) −1.17157 −0.107852
\(119\) 1.41421 0.129641
\(120\) 1.41421 0.129099
\(121\) −10.6569 −0.968805
\(122\) 0.0710678 0.00643418
\(123\) −2.58579 −0.233153
\(124\) −0.414214 −0.0371975
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 3.51472 0.311881 0.155940 0.987766i \(-0.450159\pi\)
0.155940 + 0.987766i \(0.450159\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.41421 0.124515
\(130\) −1.82843 −0.160364
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0.828427 0.0721053
\(133\) −1.00000 −0.0867110
\(134\) −1.24264 −0.107348
\(135\) −5.65685 −0.486864
\(136\) 1.41421 0.121268
\(137\) 3.92893 0.335671 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(138\) −7.17157 −0.610485
\(139\) −6.24264 −0.529494 −0.264747 0.964318i \(-0.585288\pi\)
−0.264747 + 0.964318i \(0.585288\pi\)
\(140\) 1.00000 0.0845154
\(141\) 10.0000 0.842152
\(142\) 11.8995 0.998583
\(143\) −1.07107 −0.0895672
\(144\) −1.00000 −0.0833333
\(145\) 1.24264 0.103196
\(146\) −9.24264 −0.764926
\(147\) −8.48528 −0.699854
\(148\) 2.24264 0.184344
\(149\) −10.5563 −0.864810 −0.432405 0.901680i \(-0.642335\pi\)
−0.432405 + 0.901680i \(0.642335\pi\)
\(150\) 1.41421 0.115470
\(151\) 10.4853 0.853280 0.426640 0.904422i \(-0.359697\pi\)
0.426640 + 0.904422i \(0.359697\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −1.41421 −0.114332
\(154\) 0.585786 0.0472040
\(155\) −0.414214 −0.0332704
\(156\) −2.58579 −0.207029
\(157\) 10.2426 0.817452 0.408726 0.912657i \(-0.365973\pi\)
0.408726 + 0.912657i \(0.365973\pi\)
\(158\) −8.41421 −0.669399
\(159\) −8.00000 −0.634441
\(160\) 1.00000 0.0790569
\(161\) −5.07107 −0.399656
\(162\) −5.00000 −0.392837
\(163\) 13.6569 1.06969 0.534844 0.844951i \(-0.320370\pi\)
0.534844 + 0.844951i \(0.320370\pi\)
\(164\) −1.82843 −0.142776
\(165\) 0.828427 0.0644930
\(166\) 3.65685 0.283827
\(167\) 13.5563 1.04902 0.524511 0.851404i \(-0.324248\pi\)
0.524511 + 0.851404i \(0.324248\pi\)
\(168\) 1.41421 0.109109
\(169\) −9.65685 −0.742835
\(170\) 1.41421 0.108465
\(171\) 1.00000 0.0764719
\(172\) 1.00000 0.0762493
\(173\) 0.514719 0.0391333 0.0195667 0.999809i \(-0.493771\pi\)
0.0195667 + 0.999809i \(0.493771\pi\)
\(174\) 1.75736 0.133225
\(175\) 1.00000 0.0755929
\(176\) 0.585786 0.0441553
\(177\) −1.65685 −0.124537
\(178\) −9.07107 −0.679905
\(179\) 17.4853 1.30691 0.653456 0.756965i \(-0.273319\pi\)
0.653456 + 0.756965i \(0.273319\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −15.0711 −1.12022 −0.560112 0.828417i \(-0.689242\pi\)
−0.560112 + 0.828417i \(0.689242\pi\)
\(182\) −1.82843 −0.135532
\(183\) 0.100505 0.00742955
\(184\) −5.07107 −0.373844
\(185\) 2.24264 0.164882
\(186\) −0.585786 −0.0429519
\(187\) 0.828427 0.0605806
\(188\) 7.07107 0.515711
\(189\) −5.65685 −0.411476
\(190\) −1.00000 −0.0725476
\(191\) 19.8995 1.43988 0.719938 0.694038i \(-0.244170\pi\)
0.719938 + 0.694038i \(0.244170\pi\)
\(192\) 1.41421 0.102062
\(193\) 6.82843 0.491521 0.245760 0.969331i \(-0.420962\pi\)
0.245760 + 0.969331i \(0.420962\pi\)
\(194\) 13.5563 0.973289
\(195\) −2.58579 −0.185172
\(196\) −6.00000 −0.428571
\(197\) 25.1421 1.79130 0.895651 0.444757i \(-0.146710\pi\)
0.895651 + 0.444757i \(0.146710\pi\)
\(198\) −0.585786 −0.0416300
\(199\) −0.100505 −0.00712462 −0.00356231 0.999994i \(-0.501134\pi\)
−0.00356231 + 0.999994i \(0.501134\pi\)
\(200\) 1.00000 0.0707107
\(201\) −1.75736 −0.123955
\(202\) 1.41421 0.0995037
\(203\) 1.24264 0.0872163
\(204\) 2.00000 0.140028
\(205\) −1.82843 −0.127703
\(206\) 0.828427 0.0577193
\(207\) 5.07107 0.352464
\(208\) −1.82843 −0.126779
\(209\) −0.585786 −0.0405197
\(210\) 1.41421 0.0975900
\(211\) −1.65685 −0.114063 −0.0570313 0.998372i \(-0.518163\pi\)
−0.0570313 + 0.998372i \(0.518163\pi\)
\(212\) −5.65685 −0.388514
\(213\) 16.8284 1.15306
\(214\) 1.92893 0.131859
\(215\) 1.00000 0.0681994
\(216\) −5.65685 −0.384900
\(217\) −0.414214 −0.0281186
\(218\) −10.8284 −0.733394
\(219\) −13.0711 −0.883261
\(220\) 0.585786 0.0394937
\(221\) −2.58579 −0.173939
\(222\) 3.17157 0.212862
\(223\) 16.1421 1.08096 0.540479 0.841358i \(-0.318243\pi\)
0.540479 + 0.841358i \(0.318243\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 4.89949 0.325910
\(227\) 9.17157 0.608739 0.304369 0.952554i \(-0.401554\pi\)
0.304369 + 0.952554i \(0.401554\pi\)
\(228\) −1.41421 −0.0936586
\(229\) 0.828427 0.0547440 0.0273720 0.999625i \(-0.491286\pi\)
0.0273720 + 0.999625i \(0.491286\pi\)
\(230\) −5.07107 −0.334376
\(231\) 0.828427 0.0545065
\(232\) 1.24264 0.0815834
\(233\) 8.48528 0.555889 0.277945 0.960597i \(-0.410347\pi\)
0.277945 + 0.960597i \(0.410347\pi\)
\(234\) 1.82843 0.119528
\(235\) 7.07107 0.461266
\(236\) −1.17157 −0.0762629
\(237\) −11.8995 −0.772955
\(238\) 1.41421 0.0916698
\(239\) 11.5858 0.749422 0.374711 0.927142i \(-0.377742\pi\)
0.374711 + 0.927142i \(0.377742\pi\)
\(240\) 1.41421 0.0912871
\(241\) −21.4142 −1.37941 −0.689705 0.724090i \(-0.742260\pi\)
−0.689705 + 0.724090i \(0.742260\pi\)
\(242\) −10.6569 −0.685049
\(243\) 9.89949 0.635053
\(244\) 0.0710678 0.00454965
\(245\) −6.00000 −0.383326
\(246\) −2.58579 −0.164864
\(247\) 1.82843 0.116340
\(248\) −0.414214 −0.0263026
\(249\) 5.17157 0.327735
\(250\) 1.00000 0.0632456
\(251\) −10.9706 −0.692456 −0.346228 0.938150i \(-0.612538\pi\)
−0.346228 + 0.938150i \(0.612538\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.97056 −0.186758
\(254\) 3.51472 0.220533
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 8.89949 0.555135 0.277568 0.960706i \(-0.410472\pi\)
0.277568 + 0.960706i \(0.410472\pi\)
\(258\) 1.41421 0.0880451
\(259\) 2.24264 0.139351
\(260\) −1.82843 −0.113394
\(261\) −1.24264 −0.0769175
\(262\) 6.00000 0.370681
\(263\) 17.4853 1.07819 0.539094 0.842245i \(-0.318767\pi\)
0.539094 + 0.842245i \(0.318767\pi\)
\(264\) 0.828427 0.0509862
\(265\) −5.65685 −0.347498
\(266\) −1.00000 −0.0613139
\(267\) −12.8284 −0.785087
\(268\) −1.24264 −0.0759064
\(269\) 9.17157 0.559201 0.279600 0.960116i \(-0.409798\pi\)
0.279600 + 0.960116i \(0.409798\pi\)
\(270\) −5.65685 −0.344265
\(271\) 8.89949 0.540606 0.270303 0.962775i \(-0.412876\pi\)
0.270303 + 0.962775i \(0.412876\pi\)
\(272\) 1.41421 0.0857493
\(273\) −2.58579 −0.156499
\(274\) 3.92893 0.237355
\(275\) 0.585786 0.0353243
\(276\) −7.17157 −0.431678
\(277\) −0.485281 −0.0291577 −0.0145789 0.999894i \(-0.504641\pi\)
−0.0145789 + 0.999894i \(0.504641\pi\)
\(278\) −6.24264 −0.374409
\(279\) 0.414214 0.0247983
\(280\) 1.00000 0.0597614
\(281\) −1.34315 −0.0801254 −0.0400627 0.999197i \(-0.512756\pi\)
−0.0400627 + 0.999197i \(0.512756\pi\)
\(282\) 10.0000 0.595491
\(283\) 18.7574 1.11501 0.557505 0.830174i \(-0.311759\pi\)
0.557505 + 0.830174i \(0.311759\pi\)
\(284\) 11.8995 0.706105
\(285\) −1.41421 −0.0837708
\(286\) −1.07107 −0.0633336
\(287\) −1.82843 −0.107929
\(288\) −1.00000 −0.0589256
\(289\) −15.0000 −0.882353
\(290\) 1.24264 0.0729704
\(291\) 19.1716 1.12386
\(292\) −9.24264 −0.540885
\(293\) 10.9706 0.640907 0.320454 0.947264i \(-0.396165\pi\)
0.320454 + 0.947264i \(0.396165\pi\)
\(294\) −8.48528 −0.494872
\(295\) −1.17157 −0.0682116
\(296\) 2.24264 0.130351
\(297\) −3.31371 −0.192281
\(298\) −10.5563 −0.611513
\(299\) 9.27208 0.536218
\(300\) 1.41421 0.0816497
\(301\) 1.00000 0.0576390
\(302\) 10.4853 0.603360
\(303\) 2.00000 0.114897
\(304\) −1.00000 −0.0573539
\(305\) 0.0710678 0.00406933
\(306\) −1.41421 −0.0808452
\(307\) −5.92893 −0.338382 −0.169191 0.985583i \(-0.554115\pi\)
−0.169191 + 0.985583i \(0.554115\pi\)
\(308\) 0.585786 0.0333783
\(309\) 1.17157 0.0666485
\(310\) −0.414214 −0.0235257
\(311\) −4.07107 −0.230849 −0.115425 0.993316i \(-0.536823\pi\)
−0.115425 + 0.993316i \(0.536823\pi\)
\(312\) −2.58579 −0.146391
\(313\) −1.31371 −0.0742552 −0.0371276 0.999311i \(-0.511821\pi\)
−0.0371276 + 0.999311i \(0.511821\pi\)
\(314\) 10.2426 0.578026
\(315\) −1.00000 −0.0563436
\(316\) −8.41421 −0.473336
\(317\) 0.857864 0.0481825 0.0240912 0.999710i \(-0.492331\pi\)
0.0240912 + 0.999710i \(0.492331\pi\)
\(318\) −8.00000 −0.448618
\(319\) 0.727922 0.0407558
\(320\) 1.00000 0.0559017
\(321\) 2.72792 0.152258
\(322\) −5.07107 −0.282600
\(323\) −1.41421 −0.0786889
\(324\) −5.00000 −0.277778
\(325\) −1.82843 −0.101423
\(326\) 13.6569 0.756383
\(327\) −15.3137 −0.846850
\(328\) −1.82843 −0.100958
\(329\) 7.07107 0.389841
\(330\) 0.828427 0.0456034
\(331\) −25.3137 −1.39137 −0.695684 0.718348i \(-0.744898\pi\)
−0.695684 + 0.718348i \(0.744898\pi\)
\(332\) 3.65685 0.200696
\(333\) −2.24264 −0.122896
\(334\) 13.5563 0.741770
\(335\) −1.24264 −0.0678927
\(336\) 1.41421 0.0771517
\(337\) −8.34315 −0.454480 −0.227240 0.973839i \(-0.572970\pi\)
−0.227240 + 0.973839i \(0.572970\pi\)
\(338\) −9.65685 −0.525264
\(339\) 6.92893 0.376328
\(340\) 1.41421 0.0766965
\(341\) −0.242641 −0.0131397
\(342\) 1.00000 0.0540738
\(343\) −13.0000 −0.701934
\(344\) 1.00000 0.0539164
\(345\) −7.17157 −0.386105
\(346\) 0.514719 0.0276714
\(347\) −22.2426 −1.19405 −0.597024 0.802224i \(-0.703650\pi\)
−0.597024 + 0.802224i \(0.703650\pi\)
\(348\) 1.75736 0.0942043
\(349\) 23.4558 1.25556 0.627781 0.778390i \(-0.283963\pi\)
0.627781 + 0.778390i \(0.283963\pi\)
\(350\) 1.00000 0.0534522
\(351\) 10.3431 0.552076
\(352\) 0.585786 0.0312225
\(353\) −32.3848 −1.72367 −0.861834 0.507191i \(-0.830684\pi\)
−0.861834 + 0.507191i \(0.830684\pi\)
\(354\) −1.65685 −0.0880608
\(355\) 11.8995 0.631560
\(356\) −9.07107 −0.480766
\(357\) 2.00000 0.105851
\(358\) 17.4853 0.924126
\(359\) 27.5269 1.45281 0.726407 0.687264i \(-0.241189\pi\)
0.726407 + 0.687264i \(0.241189\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) −15.0711 −0.792118
\(363\) −15.0711 −0.791026
\(364\) −1.82843 −0.0958356
\(365\) −9.24264 −0.483782
\(366\) 0.100505 0.00525348
\(367\) −31.3137 −1.63456 −0.817281 0.576239i \(-0.804520\pi\)
−0.817281 + 0.576239i \(0.804520\pi\)
\(368\) −5.07107 −0.264348
\(369\) 1.82843 0.0951841
\(370\) 2.24264 0.116589
\(371\) −5.65685 −0.293689
\(372\) −0.585786 −0.0303716
\(373\) 14.7279 0.762583 0.381291 0.924455i \(-0.375479\pi\)
0.381291 + 0.924455i \(0.375479\pi\)
\(374\) 0.828427 0.0428369
\(375\) 1.41421 0.0730297
\(376\) 7.07107 0.364662
\(377\) −2.27208 −0.117018
\(378\) −5.65685 −0.290957
\(379\) 4.14214 0.212767 0.106384 0.994325i \(-0.466073\pi\)
0.106384 + 0.994325i \(0.466073\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 4.97056 0.254650
\(382\) 19.8995 1.01815
\(383\) −8.31371 −0.424811 −0.212405 0.977182i \(-0.568130\pi\)
−0.212405 + 0.977182i \(0.568130\pi\)
\(384\) 1.41421 0.0721688
\(385\) 0.585786 0.0298544
\(386\) 6.82843 0.347558
\(387\) −1.00000 −0.0508329
\(388\) 13.5563 0.688219
\(389\) 20.8284 1.05604 0.528022 0.849231i \(-0.322934\pi\)
0.528022 + 0.849231i \(0.322934\pi\)
\(390\) −2.58579 −0.130936
\(391\) −7.17157 −0.362682
\(392\) −6.00000 −0.303046
\(393\) 8.48528 0.428026
\(394\) 25.1421 1.26664
\(395\) −8.41421 −0.423365
\(396\) −0.585786 −0.0294369
\(397\) −20.4853 −1.02813 −0.514063 0.857752i \(-0.671860\pi\)
−0.514063 + 0.857752i \(0.671860\pi\)
\(398\) −0.100505 −0.00503786
\(399\) −1.41421 −0.0707992
\(400\) 1.00000 0.0500000
\(401\) −24.1127 −1.20413 −0.602065 0.798447i \(-0.705655\pi\)
−0.602065 + 0.798447i \(0.705655\pi\)
\(402\) −1.75736 −0.0876491
\(403\) 0.757359 0.0377268
\(404\) 1.41421 0.0703598
\(405\) −5.00000 −0.248452
\(406\) 1.24264 0.0616712
\(407\) 1.31371 0.0651181
\(408\) 2.00000 0.0990148
\(409\) −27.2132 −1.34561 −0.672803 0.739822i \(-0.734910\pi\)
−0.672803 + 0.739822i \(0.734910\pi\)
\(410\) −1.82843 −0.0902996
\(411\) 5.55635 0.274074
\(412\) 0.828427 0.0408137
\(413\) −1.17157 −0.0576493
\(414\) 5.07107 0.249229
\(415\) 3.65685 0.179508
\(416\) −1.82843 −0.0896460
\(417\) −8.82843 −0.432330
\(418\) −0.585786 −0.0286518
\(419\) −12.7990 −0.625272 −0.312636 0.949873i \(-0.601212\pi\)
−0.312636 + 0.949873i \(0.601212\pi\)
\(420\) 1.41421 0.0690066
\(421\) 1.24264 0.0605626 0.0302813 0.999541i \(-0.490360\pi\)
0.0302813 + 0.999541i \(0.490360\pi\)
\(422\) −1.65685 −0.0806544
\(423\) −7.07107 −0.343807
\(424\) −5.65685 −0.274721
\(425\) 1.41421 0.0685994
\(426\) 16.8284 0.815340
\(427\) 0.0710678 0.00343921
\(428\) 1.92893 0.0932385
\(429\) −1.51472 −0.0731313
\(430\) 1.00000 0.0482243
\(431\) 1.02944 0.0495862 0.0247931 0.999693i \(-0.492107\pi\)
0.0247931 + 0.999693i \(0.492107\pi\)
\(432\) −5.65685 −0.272166
\(433\) −36.0711 −1.73346 −0.866732 0.498773i \(-0.833784\pi\)
−0.866732 + 0.498773i \(0.833784\pi\)
\(434\) −0.414214 −0.0198829
\(435\) 1.75736 0.0842589
\(436\) −10.8284 −0.518588
\(437\) 5.07107 0.242582
\(438\) −13.0711 −0.624560
\(439\) −15.3137 −0.730883 −0.365442 0.930834i \(-0.619082\pi\)
−0.365442 + 0.930834i \(0.619082\pi\)
\(440\) 0.585786 0.0279263
\(441\) 6.00000 0.285714
\(442\) −2.58579 −0.122993
\(443\) −9.92893 −0.471738 −0.235869 0.971785i \(-0.575794\pi\)
−0.235869 + 0.971785i \(0.575794\pi\)
\(444\) 3.17157 0.150516
\(445\) −9.07107 −0.430010
\(446\) 16.1421 0.764352
\(447\) −14.9289 −0.706114
\(448\) 1.00000 0.0472456
\(449\) 22.2426 1.04970 0.524848 0.851196i \(-0.324122\pi\)
0.524848 + 0.851196i \(0.324122\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −1.07107 −0.0504346
\(452\) 4.89949 0.230453
\(453\) 14.8284 0.696700
\(454\) 9.17157 0.430443
\(455\) −1.82843 −0.0857180
\(456\) −1.41421 −0.0662266
\(457\) −30.4853 −1.42604 −0.713021 0.701143i \(-0.752673\pi\)
−0.713021 + 0.701143i \(0.752673\pi\)
\(458\) 0.828427 0.0387099
\(459\) −8.00000 −0.373408
\(460\) −5.07107 −0.236440
\(461\) −15.0711 −0.701930 −0.350965 0.936389i \(-0.614146\pi\)
−0.350965 + 0.936389i \(0.614146\pi\)
\(462\) 0.828427 0.0385419
\(463\) 30.3137 1.40880 0.704399 0.709804i \(-0.251217\pi\)
0.704399 + 0.709804i \(0.251217\pi\)
\(464\) 1.24264 0.0576881
\(465\) −0.585786 −0.0271652
\(466\) 8.48528 0.393073
\(467\) −20.9289 −0.968475 −0.484238 0.874936i \(-0.660903\pi\)
−0.484238 + 0.874936i \(0.660903\pi\)
\(468\) 1.82843 0.0845191
\(469\) −1.24264 −0.0573798
\(470\) 7.07107 0.326164
\(471\) 14.4853 0.667447
\(472\) −1.17157 −0.0539260
\(473\) 0.585786 0.0269345
\(474\) −11.8995 −0.546562
\(475\) −1.00000 −0.0458831
\(476\) 1.41421 0.0648204
\(477\) 5.65685 0.259010
\(478\) 11.5858 0.529922
\(479\) 9.17157 0.419060 0.209530 0.977802i \(-0.432807\pi\)
0.209530 + 0.977802i \(0.432807\pi\)
\(480\) 1.41421 0.0645497
\(481\) −4.10051 −0.186967
\(482\) −21.4142 −0.975391
\(483\) −7.17157 −0.326318
\(484\) −10.6569 −0.484402
\(485\) 13.5563 0.615562
\(486\) 9.89949 0.449050
\(487\) −33.2132 −1.50503 −0.752517 0.658573i \(-0.771160\pi\)
−0.752517 + 0.658573i \(0.771160\pi\)
\(488\) 0.0710678 0.00321709
\(489\) 19.3137 0.873396
\(490\) −6.00000 −0.271052
\(491\) 16.6274 0.750385 0.375192 0.926947i \(-0.377577\pi\)
0.375192 + 0.926947i \(0.377577\pi\)
\(492\) −2.58579 −0.116576
\(493\) 1.75736 0.0791475
\(494\) 1.82843 0.0822648
\(495\) −0.585786 −0.0263291
\(496\) −0.414214 −0.0185987
\(497\) 11.8995 0.533765
\(498\) 5.17157 0.231744
\(499\) −34.4558 −1.54246 −0.771228 0.636559i \(-0.780357\pi\)
−0.771228 + 0.636559i \(0.780357\pi\)
\(500\) 1.00000 0.0447214
\(501\) 19.1716 0.856523
\(502\) −10.9706 −0.489640
\(503\) −31.6569 −1.41151 −0.705755 0.708456i \(-0.749392\pi\)
−0.705755 + 0.708456i \(0.749392\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 1.41421 0.0629317
\(506\) −2.97056 −0.132058
\(507\) −13.6569 −0.606522
\(508\) 3.51472 0.155940
\(509\) −20.4853 −0.907994 −0.453997 0.891003i \(-0.650002\pi\)
−0.453997 + 0.891003i \(0.650002\pi\)
\(510\) 2.00000 0.0885615
\(511\) −9.24264 −0.408870
\(512\) 1.00000 0.0441942
\(513\) 5.65685 0.249756
\(514\) 8.89949 0.392540
\(515\) 0.828427 0.0365049
\(516\) 1.41421 0.0622573
\(517\) 4.14214 0.182171
\(518\) 2.24264 0.0985360
\(519\) 0.727922 0.0319522
\(520\) −1.82843 −0.0801818
\(521\) −11.0711 −0.485032 −0.242516 0.970147i \(-0.577973\pi\)
−0.242516 + 0.970147i \(0.577973\pi\)
\(522\) −1.24264 −0.0543889
\(523\) 10.7279 0.469099 0.234550 0.972104i \(-0.424638\pi\)
0.234550 + 0.972104i \(0.424638\pi\)
\(524\) 6.00000 0.262111
\(525\) 1.41421 0.0617213
\(526\) 17.4853 0.762394
\(527\) −0.585786 −0.0255173
\(528\) 0.828427 0.0360527
\(529\) 2.71573 0.118075
\(530\) −5.65685 −0.245718
\(531\) 1.17157 0.0508419
\(532\) −1.00000 −0.0433555
\(533\) 3.34315 0.144808
\(534\) −12.8284 −0.555140
\(535\) 1.92893 0.0833950
\(536\) −1.24264 −0.0536739
\(537\) 24.7279 1.06709
\(538\) 9.17157 0.395415
\(539\) −3.51472 −0.151390
\(540\) −5.65685 −0.243432
\(541\) 7.85786 0.337836 0.168918 0.985630i \(-0.445973\pi\)
0.168918 + 0.985630i \(0.445973\pi\)
\(542\) 8.89949 0.382266
\(543\) −21.3137 −0.914659
\(544\) 1.41421 0.0606339
\(545\) −10.8284 −0.463839
\(546\) −2.58579 −0.110661
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 3.92893 0.167836
\(549\) −0.0710678 −0.00303310
\(550\) 0.585786 0.0249780
\(551\) −1.24264 −0.0529383
\(552\) −7.17157 −0.305242
\(553\) −8.41421 −0.357809
\(554\) −0.485281 −0.0206176
\(555\) 3.17157 0.134626
\(556\) −6.24264 −0.264747
\(557\) 41.8284 1.77233 0.886164 0.463372i \(-0.153361\pi\)
0.886164 + 0.463372i \(0.153361\pi\)
\(558\) 0.414214 0.0175351
\(559\) −1.82843 −0.0773342
\(560\) 1.00000 0.0422577
\(561\) 1.17157 0.0494638
\(562\) −1.34315 −0.0566572
\(563\) −1.92893 −0.0812948 −0.0406474 0.999174i \(-0.512942\pi\)
−0.0406474 + 0.999174i \(0.512942\pi\)
\(564\) 10.0000 0.421076
\(565\) 4.89949 0.206123
\(566\) 18.7574 0.788431
\(567\) −5.00000 −0.209980
\(568\) 11.8995 0.499292
\(569\) 22.7990 0.955783 0.477892 0.878419i \(-0.341401\pi\)
0.477892 + 0.878419i \(0.341401\pi\)
\(570\) −1.41421 −0.0592349
\(571\) 33.0000 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(572\) −1.07107 −0.0447836
\(573\) 28.1421 1.17565
\(574\) −1.82843 −0.0763171
\(575\) −5.07107 −0.211478
\(576\) −1.00000 −0.0416667
\(577\) −5.58579 −0.232539 −0.116270 0.993218i \(-0.537094\pi\)
−0.116270 + 0.993218i \(0.537094\pi\)
\(578\) −15.0000 −0.623918
\(579\) 9.65685 0.401325
\(580\) 1.24264 0.0515978
\(581\) 3.65685 0.151712
\(582\) 19.1716 0.794687
\(583\) −3.31371 −0.137240
\(584\) −9.24264 −0.382463
\(585\) 1.82843 0.0755962
\(586\) 10.9706 0.453190
\(587\) −27.5563 −1.13737 −0.568686 0.822555i \(-0.692548\pi\)
−0.568686 + 0.822555i \(0.692548\pi\)
\(588\) −8.48528 −0.349927
\(589\) 0.414214 0.0170674
\(590\) −1.17157 −0.0482329
\(591\) 35.5563 1.46259
\(592\) 2.24264 0.0921720
\(593\) −7.24264 −0.297420 −0.148710 0.988881i \(-0.547512\pi\)
−0.148710 + 0.988881i \(0.547512\pi\)
\(594\) −3.31371 −0.135963
\(595\) 1.41421 0.0579771
\(596\) −10.5563 −0.432405
\(597\) −0.142136 −0.00581722
\(598\) 9.27208 0.379163
\(599\) 9.85786 0.402781 0.201391 0.979511i \(-0.435454\pi\)
0.201391 + 0.979511i \(0.435454\pi\)
\(600\) 1.41421 0.0577350
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 1.00000 0.0407570
\(603\) 1.24264 0.0506042
\(604\) 10.4853 0.426640
\(605\) −10.6569 −0.433263
\(606\) 2.00000 0.0812444
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 1.75736 0.0712118
\(610\) 0.0710678 0.00287745
\(611\) −12.9289 −0.523049
\(612\) −1.41421 −0.0571662
\(613\) 38.1716 1.54174 0.770868 0.636995i \(-0.219823\pi\)
0.770868 + 0.636995i \(0.219823\pi\)
\(614\) −5.92893 −0.239272
\(615\) −2.58579 −0.104269
\(616\) 0.585786 0.0236020
\(617\) −16.2843 −0.655580 −0.327790 0.944751i \(-0.606304\pi\)
−0.327790 + 0.944751i \(0.606304\pi\)
\(618\) 1.17157 0.0471276
\(619\) −0.100505 −0.00403964 −0.00201982 0.999998i \(-0.500643\pi\)
−0.00201982 + 0.999998i \(0.500643\pi\)
\(620\) −0.414214 −0.0166352
\(621\) 28.6863 1.15114
\(622\) −4.07107 −0.163235
\(623\) −9.07107 −0.363425
\(624\) −2.58579 −0.103514
\(625\) 1.00000 0.0400000
\(626\) −1.31371 −0.0525064
\(627\) −0.828427 −0.0330842
\(628\) 10.2426 0.408726
\(629\) 3.17157 0.126459
\(630\) −1.00000 −0.0398410
\(631\) −13.6985 −0.545328 −0.272664 0.962109i \(-0.587905\pi\)
−0.272664 + 0.962109i \(0.587905\pi\)
\(632\) −8.41421 −0.334699
\(633\) −2.34315 −0.0931317
\(634\) 0.857864 0.0340701
\(635\) 3.51472 0.139477
\(636\) −8.00000 −0.317221
\(637\) 10.9706 0.434670
\(638\) 0.727922 0.0288187
\(639\) −11.8995 −0.470737
\(640\) 1.00000 0.0395285
\(641\) 42.7696 1.68930 0.844648 0.535322i \(-0.179810\pi\)
0.844648 + 0.535322i \(0.179810\pi\)
\(642\) 2.72792 0.107662
\(643\) 39.7279 1.56672 0.783358 0.621571i \(-0.213505\pi\)
0.783358 + 0.621571i \(0.213505\pi\)
\(644\) −5.07107 −0.199828
\(645\) 1.41421 0.0556846
\(646\) −1.41421 −0.0556415
\(647\) 5.20101 0.204473 0.102236 0.994760i \(-0.467400\pi\)
0.102236 + 0.994760i \(0.467400\pi\)
\(648\) −5.00000 −0.196419
\(649\) −0.686292 −0.0269393
\(650\) −1.82843 −0.0717168
\(651\) −0.585786 −0.0229588
\(652\) 13.6569 0.534844
\(653\) −6.58579 −0.257722 −0.128861 0.991663i \(-0.541132\pi\)
−0.128861 + 0.991663i \(0.541132\pi\)
\(654\) −15.3137 −0.598813
\(655\) 6.00000 0.234439
\(656\) −1.82843 −0.0713881
\(657\) 9.24264 0.360590
\(658\) 7.07107 0.275659
\(659\) 24.3848 0.949896 0.474948 0.880014i \(-0.342467\pi\)
0.474948 + 0.880014i \(0.342467\pi\)
\(660\) 0.828427 0.0322465
\(661\) 24.8701 0.967333 0.483667 0.875252i \(-0.339305\pi\)
0.483667 + 0.875252i \(0.339305\pi\)
\(662\) −25.3137 −0.983845
\(663\) −3.65685 −0.142020
\(664\) 3.65685 0.141913
\(665\) −1.00000 −0.0387783
\(666\) −2.24264 −0.0869006
\(667\) −6.30152 −0.243996
\(668\) 13.5563 0.524511
\(669\) 22.8284 0.882598
\(670\) −1.24264 −0.0480074
\(671\) 0.0416306 0.00160713
\(672\) 1.41421 0.0545545
\(673\) 20.4142 0.786910 0.393455 0.919344i \(-0.371280\pi\)
0.393455 + 0.919344i \(0.371280\pi\)
\(674\) −8.34315 −0.321366
\(675\) −5.65685 −0.217732
\(676\) −9.65685 −0.371417
\(677\) 43.4558 1.67014 0.835072 0.550141i \(-0.185426\pi\)
0.835072 + 0.550141i \(0.185426\pi\)
\(678\) 6.92893 0.266104
\(679\) 13.5563 0.520245
\(680\) 1.41421 0.0542326
\(681\) 12.9706 0.497033
\(682\) −0.242641 −0.00929119
\(683\) −3.51472 −0.134487 −0.0672435 0.997737i \(-0.521420\pi\)
−0.0672435 + 0.997737i \(0.521420\pi\)
\(684\) 1.00000 0.0382360
\(685\) 3.92893 0.150117
\(686\) −13.0000 −0.496342
\(687\) 1.17157 0.0446983
\(688\) 1.00000 0.0381246
\(689\) 10.3431 0.394042
\(690\) −7.17157 −0.273017
\(691\) 18.4853 0.703213 0.351607 0.936148i \(-0.385635\pi\)
0.351607 + 0.936148i \(0.385635\pi\)
\(692\) 0.514719 0.0195667
\(693\) −0.585786 −0.0222522
\(694\) −22.2426 −0.844319
\(695\) −6.24264 −0.236797
\(696\) 1.75736 0.0666125
\(697\) −2.58579 −0.0979436
\(698\) 23.4558 0.887817
\(699\) 12.0000 0.453882
\(700\) 1.00000 0.0377964
\(701\) 11.2721 0.425741 0.212870 0.977080i \(-0.431719\pi\)
0.212870 + 0.977080i \(0.431719\pi\)
\(702\) 10.3431 0.390377
\(703\) −2.24264 −0.0845828
\(704\) 0.585786 0.0220777
\(705\) 10.0000 0.376622
\(706\) −32.3848 −1.21882
\(707\) 1.41421 0.0531870
\(708\) −1.65685 −0.0622684
\(709\) −7.21320 −0.270898 −0.135449 0.990784i \(-0.543248\pi\)
−0.135449 + 0.990784i \(0.543248\pi\)
\(710\) 11.8995 0.446580
\(711\) 8.41421 0.315558
\(712\) −9.07107 −0.339953
\(713\) 2.10051 0.0786645
\(714\) 2.00000 0.0748481
\(715\) −1.07107 −0.0400557
\(716\) 17.4853 0.653456
\(717\) 16.3848 0.611901
\(718\) 27.5269 1.02730
\(719\) −36.6274 −1.36597 −0.682986 0.730431i \(-0.739319\pi\)
−0.682986 + 0.730431i \(0.739319\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0.828427 0.0308522
\(722\) −18.0000 −0.669891
\(723\) −30.2843 −1.12628
\(724\) −15.0711 −0.560112
\(725\) 1.24264 0.0461505
\(726\) −15.0711 −0.559340
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) −1.82843 −0.0677660
\(729\) 29.0000 1.07407
\(730\) −9.24264 −0.342085
\(731\) 1.41421 0.0523066
\(732\) 0.100505 0.00371477
\(733\) 27.7574 1.02524 0.512621 0.858615i \(-0.328675\pi\)
0.512621 + 0.858615i \(0.328675\pi\)
\(734\) −31.3137 −1.15581
\(735\) −8.48528 −0.312984
\(736\) −5.07107 −0.186922
\(737\) −0.727922 −0.0268134
\(738\) 1.82843 0.0673053
\(739\) 8.02944 0.295368 0.147684 0.989035i \(-0.452818\pi\)
0.147684 + 0.989035i \(0.452818\pi\)
\(740\) 2.24264 0.0824411
\(741\) 2.58579 0.0949912
\(742\) −5.65685 −0.207670
\(743\) 2.31371 0.0848817 0.0424409 0.999099i \(-0.486487\pi\)
0.0424409 + 0.999099i \(0.486487\pi\)
\(744\) −0.585786 −0.0214760
\(745\) −10.5563 −0.386755
\(746\) 14.7279 0.539228
\(747\) −3.65685 −0.133797
\(748\) 0.828427 0.0302903
\(749\) 1.92893 0.0704816
\(750\) 1.41421 0.0516398
\(751\) −8.72792 −0.318486 −0.159243 0.987239i \(-0.550905\pi\)
−0.159243 + 0.987239i \(0.550905\pi\)
\(752\) 7.07107 0.257855
\(753\) −15.5147 −0.565388
\(754\) −2.27208 −0.0827442
\(755\) 10.4853 0.381598
\(756\) −5.65685 −0.205738
\(757\) 36.4264 1.32394 0.661970 0.749530i \(-0.269721\pi\)
0.661970 + 0.749530i \(0.269721\pi\)
\(758\) 4.14214 0.150449
\(759\) −4.20101 −0.152487
\(760\) −1.00000 −0.0362738
\(761\) −44.8284 −1.62503 −0.812515 0.582941i \(-0.801902\pi\)
−0.812515 + 0.582941i \(0.801902\pi\)
\(762\) 4.97056 0.180064
\(763\) −10.8284 −0.392015
\(764\) 19.8995 0.719938
\(765\) −1.41421 −0.0511310
\(766\) −8.31371 −0.300386
\(767\) 2.14214 0.0773480
\(768\) 1.41421 0.0510310
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0.585786 0.0211103
\(771\) 12.5858 0.453266
\(772\) 6.82843 0.245760
\(773\) −25.0711 −0.901744 −0.450872 0.892589i \(-0.648887\pi\)
−0.450872 + 0.892589i \(0.648887\pi\)
\(774\) −1.00000 −0.0359443
\(775\) −0.414214 −0.0148790
\(776\) 13.5563 0.486645
\(777\) 3.17157 0.113780
\(778\) 20.8284 0.746735
\(779\) 1.82843 0.0655102
\(780\) −2.58579 −0.0925860
\(781\) 6.97056 0.249426
\(782\) −7.17157 −0.256455
\(783\) −7.02944 −0.251212
\(784\) −6.00000 −0.214286
\(785\) 10.2426 0.365576
\(786\) 8.48528 0.302660
\(787\) 31.2426 1.11368 0.556840 0.830620i \(-0.312014\pi\)
0.556840 + 0.830620i \(0.312014\pi\)
\(788\) 25.1421 0.895651
\(789\) 24.7279 0.880337
\(790\) −8.41421 −0.299364
\(791\) 4.89949 0.174206
\(792\) −0.585786 −0.0208150
\(793\) −0.129942 −0.00461439
\(794\) −20.4853 −0.726995
\(795\) −8.00000 −0.283731
\(796\) −0.100505 −0.00356231
\(797\) −42.9411 −1.52105 −0.760526 0.649307i \(-0.775059\pi\)
−0.760526 + 0.649307i \(0.775059\pi\)
\(798\) −1.41421 −0.0500626
\(799\) 10.0000 0.353775
\(800\) 1.00000 0.0353553
\(801\) 9.07107 0.320510
\(802\) −24.1127 −0.851449
\(803\) −5.41421 −0.191063
\(804\) −1.75736 −0.0619773
\(805\) −5.07107 −0.178732
\(806\) 0.757359 0.0266768
\(807\) 12.9706 0.456585
\(808\) 1.41421 0.0497519
\(809\) −28.7990 −1.01252 −0.506259 0.862381i \(-0.668972\pi\)
−0.506259 + 0.862381i \(0.668972\pi\)
\(810\) −5.00000 −0.175682
\(811\) −48.1127 −1.68947 −0.844733 0.535188i \(-0.820241\pi\)
−0.844733 + 0.535188i \(0.820241\pi\)
\(812\) 1.24264 0.0436081
\(813\) 12.5858 0.441403
\(814\) 1.31371 0.0460455
\(815\) 13.6569 0.478379
\(816\) 2.00000 0.0700140
\(817\) −1.00000 −0.0349856
\(818\) −27.2132 −0.951487
\(819\) 1.82843 0.0638904
\(820\) −1.82843 −0.0638514
\(821\) −20.1421 −0.702965 −0.351483 0.936194i \(-0.614322\pi\)
−0.351483 + 0.936194i \(0.614322\pi\)
\(822\) 5.55635 0.193800
\(823\) 13.5147 0.471093 0.235547 0.971863i \(-0.424312\pi\)
0.235547 + 0.971863i \(0.424312\pi\)
\(824\) 0.828427 0.0288596
\(825\) 0.828427 0.0288421
\(826\) −1.17157 −0.0407642
\(827\) −43.5269 −1.51358 −0.756790 0.653659i \(-0.773233\pi\)
−0.756790 + 0.653659i \(0.773233\pi\)
\(828\) 5.07107 0.176232
\(829\) 3.24264 0.112622 0.0563108 0.998413i \(-0.482066\pi\)
0.0563108 + 0.998413i \(0.482066\pi\)
\(830\) 3.65685 0.126931
\(831\) −0.686292 −0.0238072
\(832\) −1.82843 −0.0633893
\(833\) −8.48528 −0.293998
\(834\) −8.82843 −0.305703
\(835\) 13.5563 0.469137
\(836\) −0.585786 −0.0202598
\(837\) 2.34315 0.0809910
\(838\) −12.7990 −0.442134
\(839\) −51.0122 −1.76114 −0.880568 0.473919i \(-0.842839\pi\)
−0.880568 + 0.473919i \(0.842839\pi\)
\(840\) 1.41421 0.0487950
\(841\) −27.4558 −0.946753
\(842\) 1.24264 0.0428242
\(843\) −1.89949 −0.0654221
\(844\) −1.65685 −0.0570313
\(845\) −9.65685 −0.332206
\(846\) −7.07107 −0.243108
\(847\) −10.6569 −0.366174
\(848\) −5.65685 −0.194257
\(849\) 26.5269 0.910401
\(850\) 1.41421 0.0485071
\(851\) −11.3726 −0.389847
\(852\) 16.8284 0.576532
\(853\) −35.9411 −1.23060 −0.615300 0.788293i \(-0.710965\pi\)
−0.615300 + 0.788293i \(0.710965\pi\)
\(854\) 0.0710678 0.00243189
\(855\) 1.00000 0.0341993
\(856\) 1.92893 0.0659295
\(857\) 47.0711 1.60792 0.803959 0.594685i \(-0.202723\pi\)
0.803959 + 0.594685i \(0.202723\pi\)
\(858\) −1.51472 −0.0517116
\(859\) −16.5147 −0.563475 −0.281737 0.959492i \(-0.590911\pi\)
−0.281737 + 0.959492i \(0.590911\pi\)
\(860\) 1.00000 0.0340997
\(861\) −2.58579 −0.0881234
\(862\) 1.02944 0.0350628
\(863\) −40.8284 −1.38982 −0.694908 0.719099i \(-0.744555\pi\)
−0.694908 + 0.719099i \(0.744555\pi\)
\(864\) −5.65685 −0.192450
\(865\) 0.514719 0.0175010
\(866\) −36.0711 −1.22574
\(867\) −21.2132 −0.720438
\(868\) −0.414214 −0.0140593
\(869\) −4.92893 −0.167203
\(870\) 1.75736 0.0595801
\(871\) 2.27208 0.0769864
\(872\) −10.8284 −0.366697
\(873\) −13.5563 −0.458813
\(874\) 5.07107 0.171531
\(875\) 1.00000 0.0338062
\(876\) −13.0711 −0.441630
\(877\) −18.6274 −0.629003 −0.314502 0.949257i \(-0.601837\pi\)
−0.314502 + 0.949257i \(0.601837\pi\)
\(878\) −15.3137 −0.516813
\(879\) 15.5147 0.523298
\(880\) 0.585786 0.0197469
\(881\) −1.20101 −0.0404631 −0.0202315 0.999795i \(-0.506440\pi\)
−0.0202315 + 0.999795i \(0.506440\pi\)
\(882\) 6.00000 0.202031
\(883\) 26.2132 0.882145 0.441072 0.897472i \(-0.354598\pi\)
0.441072 + 0.897472i \(0.354598\pi\)
\(884\) −2.58579 −0.0869694
\(885\) −1.65685 −0.0556945
\(886\) −9.92893 −0.333569
\(887\) −56.1127 −1.88408 −0.942040 0.335501i \(-0.891095\pi\)
−0.942040 + 0.335501i \(0.891095\pi\)
\(888\) 3.17157 0.106431
\(889\) 3.51472 0.117880
\(890\) −9.07107 −0.304063
\(891\) −2.92893 −0.0981229
\(892\) 16.1421 0.540479
\(893\) −7.07107 −0.236624
\(894\) −14.9289 −0.499298
\(895\) 17.4853 0.584468
\(896\) 1.00000 0.0334077
\(897\) 13.1127 0.437820
\(898\) 22.2426 0.742247
\(899\) −0.514719 −0.0171668
\(900\) −1.00000 −0.0333333
\(901\) −8.00000 −0.266519
\(902\) −1.07107 −0.0356627
\(903\) 1.41421 0.0470621
\(904\) 4.89949 0.162955
\(905\) −15.0711 −0.500979
\(906\) 14.8284 0.492641
\(907\) 14.8995 0.494730 0.247365 0.968922i \(-0.420435\pi\)
0.247365 + 0.968922i \(0.420435\pi\)
\(908\) 9.17157 0.304369
\(909\) −1.41421 −0.0469065
\(910\) −1.82843 −0.0606118
\(911\) −20.3848 −0.675378 −0.337689 0.941258i \(-0.609645\pi\)
−0.337689 + 0.941258i \(0.609645\pi\)
\(912\) −1.41421 −0.0468293
\(913\) 2.14214 0.0708943
\(914\) −30.4853 −1.00836
\(915\) 0.100505 0.00332259
\(916\) 0.828427 0.0273720
\(917\) 6.00000 0.198137
\(918\) −8.00000 −0.264039
\(919\) −34.8995 −1.15123 −0.575614 0.817722i \(-0.695237\pi\)
−0.575614 + 0.817722i \(0.695237\pi\)
\(920\) −5.07107 −0.167188
\(921\) −8.38478 −0.276288
\(922\) −15.0711 −0.496339
\(923\) −21.7574 −0.716152
\(924\) 0.828427 0.0272533
\(925\) 2.24264 0.0737376
\(926\) 30.3137 0.996170
\(927\) −0.828427 −0.0272091
\(928\) 1.24264 0.0407917
\(929\) −13.7990 −0.452730 −0.226365 0.974043i \(-0.572684\pi\)
−0.226365 + 0.974043i \(0.572684\pi\)
\(930\) −0.585786 −0.0192087
\(931\) 6.00000 0.196642
\(932\) 8.48528 0.277945
\(933\) −5.75736 −0.188487
\(934\) −20.9289 −0.684816
\(935\) 0.828427 0.0270925
\(936\) 1.82843 0.0597640
\(937\) −34.4264 −1.12466 −0.562331 0.826912i \(-0.690095\pi\)
−0.562331 + 0.826912i \(0.690095\pi\)
\(938\) −1.24264 −0.0405737
\(939\) −1.85786 −0.0606291
\(940\) 7.07107 0.230633
\(941\) −53.0711 −1.73007 −0.865034 0.501714i \(-0.832703\pi\)
−0.865034 + 0.501714i \(0.832703\pi\)
\(942\) 14.4853 0.471956
\(943\) 9.27208 0.301940
\(944\) −1.17157 −0.0381314
\(945\) −5.65685 −0.184017
\(946\) 0.585786 0.0190456
\(947\) 59.1838 1.92321 0.961607 0.274430i \(-0.0884893\pi\)
0.961607 + 0.274430i \(0.0884893\pi\)
\(948\) −11.8995 −0.386478
\(949\) 16.8995 0.548581
\(950\) −1.00000 −0.0324443
\(951\) 1.21320 0.0393408
\(952\) 1.41421 0.0458349
\(953\) −9.10051 −0.294794 −0.147397 0.989077i \(-0.547090\pi\)
−0.147397 + 0.989077i \(0.547090\pi\)
\(954\) 5.65685 0.183147
\(955\) 19.8995 0.643933
\(956\) 11.5858 0.374711
\(957\) 1.02944 0.0332770
\(958\) 9.17157 0.296320
\(959\) 3.92893 0.126872
\(960\) 1.41421 0.0456435
\(961\) −30.8284 −0.994465
\(962\) −4.10051 −0.132206
\(963\) −1.92893 −0.0621590
\(964\) −21.4142 −0.689705
\(965\) 6.82843 0.219815
\(966\) −7.17157 −0.230742
\(967\) 36.2426 1.16548 0.582742 0.812657i \(-0.301980\pi\)
0.582742 + 0.812657i \(0.301980\pi\)
\(968\) −10.6569 −0.342524
\(969\) −2.00000 −0.0642493
\(970\) 13.5563 0.435268
\(971\) 21.0711 0.676203 0.338101 0.941110i \(-0.390215\pi\)
0.338101 + 0.941110i \(0.390215\pi\)
\(972\) 9.89949 0.317526
\(973\) −6.24264 −0.200130
\(974\) −33.2132 −1.06422
\(975\) −2.58579 −0.0828114
\(976\) 0.0710678 0.00227483
\(977\) −17.0711 −0.546152 −0.273076 0.961992i \(-0.588041\pi\)
−0.273076 + 0.961992i \(0.588041\pi\)
\(978\) 19.3137 0.617584
\(979\) −5.31371 −0.169827
\(980\) −6.00000 −0.191663
\(981\) 10.8284 0.345725
\(982\) 16.6274 0.530602
\(983\) −27.3431 −0.872111 −0.436055 0.899920i \(-0.643625\pi\)
−0.436055 + 0.899920i \(0.643625\pi\)
\(984\) −2.58579 −0.0824319
\(985\) 25.1421 0.801095
\(986\) 1.75736 0.0559657
\(987\) 10.0000 0.318304
\(988\) 1.82843 0.0581700
\(989\) −5.07107 −0.161251
\(990\) −0.585786 −0.0186175
\(991\) 41.2548 1.31050 0.655251 0.755411i \(-0.272563\pi\)
0.655251 + 0.755411i \(0.272563\pi\)
\(992\) −0.414214 −0.0131513
\(993\) −35.7990 −1.13605
\(994\) 11.8995 0.377429
\(995\) −0.100505 −0.00318622
\(996\) 5.17157 0.163868
\(997\) 37.1716 1.17724 0.588618 0.808411i \(-0.299672\pi\)
0.588618 + 0.808411i \(0.299672\pi\)
\(998\) −34.4558 −1.09068
\(999\) −12.6863 −0.401377
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 430.2.a.g.1.2 2
3.2 odd 2 3870.2.a.bc.1.2 2
4.3 odd 2 3440.2.a.j.1.1 2
5.2 odd 4 2150.2.b.o.1549.3 4
5.3 odd 4 2150.2.b.o.1549.2 4
5.4 even 2 2150.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
430.2.a.g.1.2 2 1.1 even 1 trivial
2150.2.a.v.1.1 2 5.4 even 2
2150.2.b.o.1549.2 4 5.3 odd 4
2150.2.b.o.1549.3 4 5.2 odd 4
3440.2.a.j.1.1 2 4.3 odd 2
3870.2.a.bc.1.2 2 3.2 odd 2