Properties

Label 5586.2.a.br.1.2
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.41421 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.82843 q^{13} +4.41421 q^{15} +1.00000 q^{16} -4.24264 q^{17} +1.00000 q^{18} -1.00000 q^{19} +4.41421 q^{20} -1.00000 q^{22} +0.585786 q^{23} +1.00000 q^{24} +14.4853 q^{25} +2.82843 q^{26} +1.00000 q^{27} +7.82843 q^{29} +4.41421 q^{30} -0.656854 q^{31} +1.00000 q^{32} -1.00000 q^{33} -4.24264 q^{34} +1.00000 q^{36} -8.24264 q^{37} -1.00000 q^{38} +2.82843 q^{39} +4.41421 q^{40} +3.41421 q^{41} -2.24264 q^{43} -1.00000 q^{44} +4.41421 q^{45} +0.585786 q^{46} -8.82843 q^{47} +1.00000 q^{48} +14.4853 q^{50} -4.24264 q^{51} +2.82843 q^{52} -3.82843 q^{53} +1.00000 q^{54} -4.41421 q^{55} -1.00000 q^{57} +7.82843 q^{58} +8.07107 q^{59} +4.41421 q^{60} -0.828427 q^{61} -0.656854 q^{62} +1.00000 q^{64} +12.4853 q^{65} -1.00000 q^{66} -2.00000 q^{67} -4.24264 q^{68} +0.585786 q^{69} +11.8995 q^{71} +1.00000 q^{72} +9.17157 q^{73} -8.24264 q^{74} +14.4853 q^{75} -1.00000 q^{76} +2.82843 q^{78} -5.82843 q^{79} +4.41421 q^{80} +1.00000 q^{81} +3.41421 q^{82} -11.1421 q^{83} -18.7279 q^{85} -2.24264 q^{86} +7.82843 q^{87} -1.00000 q^{88} -12.2426 q^{89} +4.41421 q^{90} +0.585786 q^{92} -0.656854 q^{93} -8.82843 q^{94} -4.41421 q^{95} +1.00000 q^{96} +8.41421 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{15} + 2 q^{16} + 2 q^{18} - 2 q^{19} + 6 q^{20} - 2 q^{22} + 4 q^{23} + 2 q^{24} + 12 q^{25} + 2 q^{27} + 10 q^{29} + 6 q^{30} + 10 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{36} - 8 q^{37} - 2 q^{38} + 6 q^{40} + 4 q^{41} + 4 q^{43} - 2 q^{44} + 6 q^{45} + 4 q^{46} - 12 q^{47} + 2 q^{48} + 12 q^{50} - 2 q^{53} + 2 q^{54} - 6 q^{55} - 2 q^{57} + 10 q^{58} + 2 q^{59} + 6 q^{60} + 4 q^{61} + 10 q^{62} + 2 q^{64} + 8 q^{65} - 2 q^{66} - 4 q^{67} + 4 q^{69} + 4 q^{71} + 2 q^{72} + 24 q^{73} - 8 q^{74} + 12 q^{75} - 2 q^{76} - 6 q^{79} + 6 q^{80} + 2 q^{81} + 4 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} + 10 q^{87} - 2 q^{88} - 16 q^{89} + 6 q^{90} + 4 q^{92} + 10 q^{93} - 12 q^{94} - 6 q^{95} + 2 q^{96} + 14 q^{97} - 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 4.41421 1.97410 0.987048 0.160424i \(-0.0512862\pi\)
0.987048 + 0.160424i \(0.0512862\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.41421 1.39590
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 4.41421 1.13975
\(16\) 1.00000 0.250000
\(17\) −4.24264 −1.02899 −0.514496 0.857493i \(-0.672021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 4.41421 0.987048
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 0.585786 0.122145 0.0610725 0.998133i \(-0.480548\pi\)
0.0610725 + 0.998133i \(0.480548\pi\)
\(24\) 1.00000 0.204124
\(25\) 14.4853 2.89706
\(26\) 2.82843 0.554700
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.82843 1.45370 0.726851 0.686795i \(-0.240983\pi\)
0.726851 + 0.686795i \(0.240983\pi\)
\(30\) 4.41421 0.805921
\(31\) −0.656854 −0.117975 −0.0589873 0.998259i \(-0.518787\pi\)
−0.0589873 + 0.998259i \(0.518787\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.24264 −0.727607
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.24264 −1.35508 −0.677541 0.735485i \(-0.736954\pi\)
−0.677541 + 0.735485i \(0.736954\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.82843 0.452911
\(40\) 4.41421 0.697948
\(41\) 3.41421 0.533211 0.266605 0.963806i \(-0.414098\pi\)
0.266605 + 0.963806i \(0.414098\pi\)
\(42\) 0 0
\(43\) −2.24264 −0.341999 −0.171000 0.985271i \(-0.554700\pi\)
−0.171000 + 0.985271i \(0.554700\pi\)
\(44\) −1.00000 −0.150756
\(45\) 4.41421 0.658032
\(46\) 0.585786 0.0863695
\(47\) −8.82843 −1.28776 −0.643879 0.765127i \(-0.722676\pi\)
−0.643879 + 0.765127i \(0.722676\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 14.4853 2.04853
\(51\) −4.24264 −0.594089
\(52\) 2.82843 0.392232
\(53\) −3.82843 −0.525875 −0.262937 0.964813i \(-0.584691\pi\)
−0.262937 + 0.964813i \(0.584691\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.41421 −0.595212
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 7.82843 1.02792
\(59\) 8.07107 1.05076 0.525382 0.850867i \(-0.323923\pi\)
0.525382 + 0.850867i \(0.323923\pi\)
\(60\) 4.41421 0.569873
\(61\) −0.828427 −0.106069 −0.0530346 0.998593i \(-0.516889\pi\)
−0.0530346 + 0.998593i \(0.516889\pi\)
\(62\) −0.656854 −0.0834206
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.4853 1.54861
\(66\) −1.00000 −0.123091
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −4.24264 −0.514496
\(69\) 0.585786 0.0705204
\(70\) 0 0
\(71\) 11.8995 1.41221 0.706105 0.708107i \(-0.250451\pi\)
0.706105 + 0.708107i \(0.250451\pi\)
\(72\) 1.00000 0.117851
\(73\) 9.17157 1.07345 0.536726 0.843757i \(-0.319661\pi\)
0.536726 + 0.843757i \(0.319661\pi\)
\(74\) −8.24264 −0.958188
\(75\) 14.4853 1.67262
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 2.82843 0.320256
\(79\) −5.82843 −0.655749 −0.327875 0.944721i \(-0.606332\pi\)
−0.327875 + 0.944721i \(0.606332\pi\)
\(80\) 4.41421 0.493524
\(81\) 1.00000 0.111111
\(82\) 3.41421 0.377037
\(83\) −11.1421 −1.22301 −0.611504 0.791241i \(-0.709435\pi\)
−0.611504 + 0.791241i \(0.709435\pi\)
\(84\) 0 0
\(85\) −18.7279 −2.03133
\(86\) −2.24264 −0.241830
\(87\) 7.82843 0.839295
\(88\) −1.00000 −0.106600
\(89\) −12.2426 −1.29772 −0.648859 0.760909i \(-0.724753\pi\)
−0.648859 + 0.760909i \(0.724753\pi\)
\(90\) 4.41421 0.465299
\(91\) 0 0
\(92\) 0.585786 0.0610725
\(93\) −0.656854 −0.0681126
\(94\) −8.82843 −0.910583
\(95\) −4.41421 −0.452889
\(96\) 1.00000 0.102062
\(97\) 8.41421 0.854334 0.427167 0.904173i \(-0.359512\pi\)
0.427167 + 0.904173i \(0.359512\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 14.4853 1.44853
\(101\) 18.9706 1.88764 0.943821 0.330458i \(-0.107203\pi\)
0.943821 + 0.330458i \(0.107203\pi\)
\(102\) −4.24264 −0.420084
\(103\) 1.51472 0.149250 0.0746248 0.997212i \(-0.476224\pi\)
0.0746248 + 0.997212i \(0.476224\pi\)
\(104\) 2.82843 0.277350
\(105\) 0 0
\(106\) −3.82843 −0.371850
\(107\) −13.7279 −1.32713 −0.663564 0.748119i \(-0.730957\pi\)
−0.663564 + 0.748119i \(0.730957\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.2426 0.981067 0.490534 0.871422i \(-0.336802\pi\)
0.490534 + 0.871422i \(0.336802\pi\)
\(110\) −4.41421 −0.420879
\(111\) −8.24264 −0.782357
\(112\) 0 0
\(113\) 1.75736 0.165318 0.0826592 0.996578i \(-0.473659\pi\)
0.0826592 + 0.996578i \(0.473659\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 2.58579 0.241126
\(116\) 7.82843 0.726851
\(117\) 2.82843 0.261488
\(118\) 8.07107 0.743002
\(119\) 0 0
\(120\) 4.41421 0.402961
\(121\) −10.0000 −0.909091
\(122\) −0.828427 −0.0750023
\(123\) 3.41421 0.307849
\(124\) −0.656854 −0.0589873
\(125\) 41.8701 3.74497
\(126\) 0 0
\(127\) −0.171573 −0.0152246 −0.00761232 0.999971i \(-0.502423\pi\)
−0.00761232 + 0.999971i \(0.502423\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.24264 −0.197454
\(130\) 12.4853 1.09503
\(131\) 18.3137 1.60008 0.800038 0.599949i \(-0.204813\pi\)
0.800038 + 0.599949i \(0.204813\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 4.41421 0.379915
\(136\) −4.24264 −0.363803
\(137\) −11.6569 −0.995912 −0.497956 0.867202i \(-0.665916\pi\)
−0.497956 + 0.867202i \(0.665916\pi\)
\(138\) 0.585786 0.0498655
\(139\) 11.8995 1.00930 0.504651 0.863323i \(-0.331621\pi\)
0.504651 + 0.863323i \(0.331621\pi\)
\(140\) 0 0
\(141\) −8.82843 −0.743488
\(142\) 11.8995 0.998583
\(143\) −2.82843 −0.236525
\(144\) 1.00000 0.0833333
\(145\) 34.5563 2.86975
\(146\) 9.17157 0.759045
\(147\) 0 0
\(148\) −8.24264 −0.677541
\(149\) −1.51472 −0.124091 −0.0620453 0.998073i \(-0.519762\pi\)
−0.0620453 + 0.998073i \(0.519762\pi\)
\(150\) 14.4853 1.18272
\(151\) −22.6569 −1.84379 −0.921894 0.387441i \(-0.873359\pi\)
−0.921894 + 0.387441i \(0.873359\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −4.24264 −0.342997
\(154\) 0 0
\(155\) −2.89949 −0.232893
\(156\) 2.82843 0.226455
\(157\) −12.4853 −0.996434 −0.498217 0.867052i \(-0.666012\pi\)
−0.498217 + 0.867052i \(0.666012\pi\)
\(158\) −5.82843 −0.463685
\(159\) −3.82843 −0.303614
\(160\) 4.41421 0.348974
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −5.17157 −0.405069 −0.202534 0.979275i \(-0.564918\pi\)
−0.202534 + 0.979275i \(0.564918\pi\)
\(164\) 3.41421 0.266605
\(165\) −4.41421 −0.343646
\(166\) −11.1421 −0.864797
\(167\) −18.7279 −1.44921 −0.724605 0.689164i \(-0.757978\pi\)
−0.724605 + 0.689164i \(0.757978\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) −18.7279 −1.43637
\(171\) −1.00000 −0.0764719
\(172\) −2.24264 −0.171000
\(173\) −23.7990 −1.80940 −0.904702 0.426045i \(-0.859906\pi\)
−0.904702 + 0.426045i \(0.859906\pi\)
\(174\) 7.82843 0.593472
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 8.07107 0.606659
\(178\) −12.2426 −0.917625
\(179\) −24.4853 −1.83012 −0.915058 0.403322i \(-0.867855\pi\)
−0.915058 + 0.403322i \(0.867855\pi\)
\(180\) 4.41421 0.329016
\(181\) 3.51472 0.261247 0.130623 0.991432i \(-0.458302\pi\)
0.130623 + 0.991432i \(0.458302\pi\)
\(182\) 0 0
\(183\) −0.828427 −0.0612391
\(184\) 0.585786 0.0431847
\(185\) −36.3848 −2.67506
\(186\) −0.656854 −0.0481629
\(187\) 4.24264 0.310253
\(188\) −8.82843 −0.643879
\(189\) 0 0
\(190\) −4.41421 −0.320241
\(191\) 10.1421 0.733859 0.366930 0.930249i \(-0.380409\pi\)
0.366930 + 0.930249i \(0.380409\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.72792 0.124379 0.0621893 0.998064i \(-0.480192\pi\)
0.0621893 + 0.998064i \(0.480192\pi\)
\(194\) 8.41421 0.604105
\(195\) 12.4853 0.894090
\(196\) 0 0
\(197\) 6.82843 0.486505 0.243253 0.969963i \(-0.421786\pi\)
0.243253 + 0.969963i \(0.421786\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 14.4853 1.02426
\(201\) −2.00000 −0.141069
\(202\) 18.9706 1.33476
\(203\) 0 0
\(204\) −4.24264 −0.297044
\(205\) 15.0711 1.05261
\(206\) 1.51472 0.105535
\(207\) 0.585786 0.0407150
\(208\) 2.82843 0.196116
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −0.928932 −0.0639503 −0.0319752 0.999489i \(-0.510180\pi\)
−0.0319752 + 0.999489i \(0.510180\pi\)
\(212\) −3.82843 −0.262937
\(213\) 11.8995 0.815340
\(214\) −13.7279 −0.938421
\(215\) −9.89949 −0.675140
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 10.2426 0.693719
\(219\) 9.17157 0.619757
\(220\) −4.41421 −0.297606
\(221\) −12.0000 −0.807207
\(222\) −8.24264 −0.553210
\(223\) −6.51472 −0.436258 −0.218129 0.975920i \(-0.569995\pi\)
−0.218129 + 0.975920i \(0.569995\pi\)
\(224\) 0 0
\(225\) 14.4853 0.965685
\(226\) 1.75736 0.116898
\(227\) −20.8995 −1.38715 −0.693574 0.720385i \(-0.743965\pi\)
−0.693574 + 0.720385i \(0.743965\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −19.7990 −1.30835 −0.654177 0.756341i \(-0.726985\pi\)
−0.654177 + 0.756341i \(0.726985\pi\)
\(230\) 2.58579 0.170502
\(231\) 0 0
\(232\) 7.82843 0.513961
\(233\) −10.8284 −0.709394 −0.354697 0.934981i \(-0.615416\pi\)
−0.354697 + 0.934981i \(0.615416\pi\)
\(234\) 2.82843 0.184900
\(235\) −38.9706 −2.54216
\(236\) 8.07107 0.525382
\(237\) −5.82843 −0.378597
\(238\) 0 0
\(239\) −28.2843 −1.82956 −0.914779 0.403955i \(-0.867635\pi\)
−0.914779 + 0.403955i \(0.867635\pi\)
\(240\) 4.41421 0.284936
\(241\) 30.5563 1.96831 0.984154 0.177317i \(-0.0567419\pi\)
0.984154 + 0.177317i \(0.0567419\pi\)
\(242\) −10.0000 −0.642824
\(243\) 1.00000 0.0641500
\(244\) −0.828427 −0.0530346
\(245\) 0 0
\(246\) 3.41421 0.217682
\(247\) −2.82843 −0.179969
\(248\) −0.656854 −0.0417103
\(249\) −11.1421 −0.706104
\(250\) 41.8701 2.64809
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −0.585786 −0.0368281
\(254\) −0.171573 −0.0107654
\(255\) −18.7279 −1.17279
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −2.24264 −0.139621
\(259\) 0 0
\(260\) 12.4853 0.774304
\(261\) 7.82843 0.484567
\(262\) 18.3137 1.13142
\(263\) 19.5563 1.20590 0.602948 0.797780i \(-0.293993\pi\)
0.602948 + 0.797780i \(0.293993\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −16.8995 −1.03813
\(266\) 0 0
\(267\) −12.2426 −0.749237
\(268\) −2.00000 −0.122169
\(269\) −1.00000 −0.0609711 −0.0304855 0.999535i \(-0.509705\pi\)
−0.0304855 + 0.999535i \(0.509705\pi\)
\(270\) 4.41421 0.268640
\(271\) 10.8995 0.662097 0.331049 0.943614i \(-0.392598\pi\)
0.331049 + 0.943614i \(0.392598\pi\)
\(272\) −4.24264 −0.257248
\(273\) 0 0
\(274\) −11.6569 −0.704216
\(275\) −14.4853 −0.873495
\(276\) 0.585786 0.0352602
\(277\) −22.2426 −1.33643 −0.668215 0.743968i \(-0.732942\pi\)
−0.668215 + 0.743968i \(0.732942\pi\)
\(278\) 11.8995 0.713684
\(279\) −0.656854 −0.0393248
\(280\) 0 0
\(281\) 21.3137 1.27147 0.635735 0.771908i \(-0.280697\pi\)
0.635735 + 0.771908i \(0.280697\pi\)
\(282\) −8.82843 −0.525725
\(283\) 9.55635 0.568066 0.284033 0.958815i \(-0.408327\pi\)
0.284033 + 0.958815i \(0.408327\pi\)
\(284\) 11.8995 0.706105
\(285\) −4.41421 −0.261475
\(286\) −2.82843 −0.167248
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 34.5563 2.02922
\(291\) 8.41421 0.493250
\(292\) 9.17157 0.536726
\(293\) 23.0000 1.34367 0.671837 0.740699i \(-0.265505\pi\)
0.671837 + 0.740699i \(0.265505\pi\)
\(294\) 0 0
\(295\) 35.6274 2.07431
\(296\) −8.24264 −0.479094
\(297\) −1.00000 −0.0580259
\(298\) −1.51472 −0.0877453
\(299\) 1.65685 0.0958184
\(300\) 14.4853 0.836308
\(301\) 0 0
\(302\) −22.6569 −1.30376
\(303\) 18.9706 1.08983
\(304\) −1.00000 −0.0573539
\(305\) −3.65685 −0.209391
\(306\) −4.24264 −0.242536
\(307\) −4.58579 −0.261725 −0.130862 0.991401i \(-0.541775\pi\)
−0.130862 + 0.991401i \(0.541775\pi\)
\(308\) 0 0
\(309\) 1.51472 0.0861693
\(310\) −2.89949 −0.164680
\(311\) −13.7574 −0.780108 −0.390054 0.920792i \(-0.627544\pi\)
−0.390054 + 0.920792i \(0.627544\pi\)
\(312\) 2.82843 0.160128
\(313\) −1.82843 −0.103349 −0.0516744 0.998664i \(-0.516456\pi\)
−0.0516744 + 0.998664i \(0.516456\pi\)
\(314\) −12.4853 −0.704585
\(315\) 0 0
\(316\) −5.82843 −0.327875
\(317\) 7.48528 0.420415 0.210208 0.977657i \(-0.432586\pi\)
0.210208 + 0.977657i \(0.432586\pi\)
\(318\) −3.82843 −0.214688
\(319\) −7.82843 −0.438308
\(320\) 4.41421 0.246762
\(321\) −13.7279 −0.766218
\(322\) 0 0
\(323\) 4.24264 0.236067
\(324\) 1.00000 0.0555556
\(325\) 40.9706 2.27264
\(326\) −5.17157 −0.286427
\(327\) 10.2426 0.566419
\(328\) 3.41421 0.188518
\(329\) 0 0
\(330\) −4.41421 −0.242994
\(331\) 32.3848 1.78003 0.890014 0.455933i \(-0.150694\pi\)
0.890014 + 0.455933i \(0.150694\pi\)
\(332\) −11.1421 −0.611504
\(333\) −8.24264 −0.451694
\(334\) −18.7279 −1.02475
\(335\) −8.82843 −0.482349
\(336\) 0 0
\(337\) −12.0711 −0.657553 −0.328776 0.944408i \(-0.606636\pi\)
−0.328776 + 0.944408i \(0.606636\pi\)
\(338\) −5.00000 −0.271964
\(339\) 1.75736 0.0954467
\(340\) −18.7279 −1.01566
\(341\) 0.656854 0.0355707
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −2.24264 −0.120915
\(345\) 2.58579 0.139214
\(346\) −23.7990 −1.27944
\(347\) −34.0000 −1.82522 −0.912608 0.408836i \(-0.865935\pi\)
−0.912608 + 0.408836i \(0.865935\pi\)
\(348\) 7.82843 0.419648
\(349\) 16.2426 0.869449 0.434724 0.900564i \(-0.356846\pi\)
0.434724 + 0.900564i \(0.356846\pi\)
\(350\) 0 0
\(351\) 2.82843 0.150970
\(352\) −1.00000 −0.0533002
\(353\) −33.8995 −1.80429 −0.902144 0.431435i \(-0.858007\pi\)
−0.902144 + 0.431435i \(0.858007\pi\)
\(354\) 8.07107 0.428972
\(355\) 52.5269 2.78784
\(356\) −12.2426 −0.648859
\(357\) 0 0
\(358\) −24.4853 −1.29409
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 4.41421 0.232649
\(361\) 1.00000 0.0526316
\(362\) 3.51472 0.184730
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 40.4853 2.11910
\(366\) −0.828427 −0.0433026
\(367\) 3.24264 0.169264 0.0846322 0.996412i \(-0.473028\pi\)
0.0846322 + 0.996412i \(0.473028\pi\)
\(368\) 0.585786 0.0305362
\(369\) 3.41421 0.177737
\(370\) −36.3848 −1.89155
\(371\) 0 0
\(372\) −0.656854 −0.0340563
\(373\) −16.4853 −0.853576 −0.426788 0.904352i \(-0.640355\pi\)
−0.426788 + 0.904352i \(0.640355\pi\)
\(374\) 4.24264 0.219382
\(375\) 41.8701 2.16216
\(376\) −8.82843 −0.455291
\(377\) 22.1421 1.14038
\(378\) 0 0
\(379\) −19.3137 −0.992079 −0.496039 0.868300i \(-0.665213\pi\)
−0.496039 + 0.868300i \(0.665213\pi\)
\(380\) −4.41421 −0.226444
\(381\) −0.171573 −0.00878994
\(382\) 10.1421 0.518917
\(383\) −4.34315 −0.221924 −0.110962 0.993825i \(-0.535393\pi\)
−0.110962 + 0.993825i \(0.535393\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 1.72792 0.0879489
\(387\) −2.24264 −0.114000
\(388\) 8.41421 0.427167
\(389\) 28.8284 1.46166 0.730830 0.682560i \(-0.239133\pi\)
0.730830 + 0.682560i \(0.239133\pi\)
\(390\) 12.4853 0.632217
\(391\) −2.48528 −0.125686
\(392\) 0 0
\(393\) 18.3137 0.923804
\(394\) 6.82843 0.344011
\(395\) −25.7279 −1.29451
\(396\) −1.00000 −0.0502519
\(397\) 5.17157 0.259554 0.129777 0.991543i \(-0.458574\pi\)
0.129777 + 0.991543i \(0.458574\pi\)
\(398\) 8.48528 0.425329
\(399\) 0 0
\(400\) 14.4853 0.724264
\(401\) −14.1421 −0.706225 −0.353112 0.935581i \(-0.614877\pi\)
−0.353112 + 0.935581i \(0.614877\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −1.85786 −0.0925468
\(404\) 18.9706 0.943821
\(405\) 4.41421 0.219344
\(406\) 0 0
\(407\) 8.24264 0.408573
\(408\) −4.24264 −0.210042
\(409\) −11.1005 −0.548885 −0.274442 0.961604i \(-0.588493\pi\)
−0.274442 + 0.961604i \(0.588493\pi\)
\(410\) 15.0711 0.744307
\(411\) −11.6569 −0.574990
\(412\) 1.51472 0.0746248
\(413\) 0 0
\(414\) 0.585786 0.0287898
\(415\) −49.1838 −2.41434
\(416\) 2.82843 0.138675
\(417\) 11.8995 0.582721
\(418\) 1.00000 0.0489116
\(419\) −2.97056 −0.145121 −0.0725607 0.997364i \(-0.523117\pi\)
−0.0725607 + 0.997364i \(0.523117\pi\)
\(420\) 0 0
\(421\) 20.8284 1.01512 0.507558 0.861618i \(-0.330548\pi\)
0.507558 + 0.861618i \(0.330548\pi\)
\(422\) −0.928932 −0.0452197
\(423\) −8.82843 −0.429253
\(424\) −3.82843 −0.185925
\(425\) −61.4558 −2.98105
\(426\) 11.8995 0.576532
\(427\) 0 0
\(428\) −13.7279 −0.663564
\(429\) −2.82843 −0.136558
\(430\) −9.89949 −0.477396
\(431\) −33.8995 −1.63288 −0.816441 0.577429i \(-0.804056\pi\)
−0.816441 + 0.577429i \(0.804056\pi\)
\(432\) 1.00000 0.0481125
\(433\) 25.1716 1.20967 0.604834 0.796351i \(-0.293239\pi\)
0.604834 + 0.796351i \(0.293239\pi\)
\(434\) 0 0
\(435\) 34.5563 1.65685
\(436\) 10.2426 0.490534
\(437\) −0.585786 −0.0280220
\(438\) 9.17157 0.438235
\(439\) 28.4558 1.35812 0.679062 0.734081i \(-0.262387\pi\)
0.679062 + 0.734081i \(0.262387\pi\)
\(440\) −4.41421 −0.210439
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −3.34315 −0.158838 −0.0794188 0.996841i \(-0.525306\pi\)
−0.0794188 + 0.996841i \(0.525306\pi\)
\(444\) −8.24264 −0.391178
\(445\) −54.0416 −2.56182
\(446\) −6.51472 −0.308481
\(447\) −1.51472 −0.0716437
\(448\) 0 0
\(449\) 5.75736 0.271707 0.135853 0.990729i \(-0.456622\pi\)
0.135853 + 0.990729i \(0.456622\pi\)
\(450\) 14.4853 0.682843
\(451\) −3.41421 −0.160769
\(452\) 1.75736 0.0826592
\(453\) −22.6569 −1.06451
\(454\) −20.8995 −0.980862
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −26.1127 −1.22150 −0.610750 0.791823i \(-0.709132\pi\)
−0.610750 + 0.791823i \(0.709132\pi\)
\(458\) −19.7990 −0.925146
\(459\) −4.24264 −0.198030
\(460\) 2.58579 0.120563
\(461\) −5.31371 −0.247484 −0.123742 0.992314i \(-0.539490\pi\)
−0.123742 + 0.992314i \(0.539490\pi\)
\(462\) 0 0
\(463\) −9.65685 −0.448792 −0.224396 0.974498i \(-0.572041\pi\)
−0.224396 + 0.974498i \(0.572041\pi\)
\(464\) 7.82843 0.363426
\(465\) −2.89949 −0.134461
\(466\) −10.8284 −0.501617
\(467\) 20.8284 0.963825 0.481912 0.876219i \(-0.339942\pi\)
0.481912 + 0.876219i \(0.339942\pi\)
\(468\) 2.82843 0.130744
\(469\) 0 0
\(470\) −38.9706 −1.79758
\(471\) −12.4853 −0.575291
\(472\) 8.07107 0.371501
\(473\) 2.24264 0.103117
\(474\) −5.82843 −0.267709
\(475\) −14.4853 −0.664630
\(476\) 0 0
\(477\) −3.82843 −0.175292
\(478\) −28.2843 −1.29369
\(479\) 21.3137 0.973848 0.486924 0.873444i \(-0.338119\pi\)
0.486924 + 0.873444i \(0.338119\pi\)
\(480\) 4.41421 0.201480
\(481\) −23.3137 −1.06301
\(482\) 30.5563 1.39180
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 37.1421 1.68654
\(486\) 1.00000 0.0453609
\(487\) 0.798990 0.0362057 0.0181028 0.999836i \(-0.494237\pi\)
0.0181028 + 0.999836i \(0.494237\pi\)
\(488\) −0.828427 −0.0375011
\(489\) −5.17157 −0.233867
\(490\) 0 0
\(491\) 19.1421 0.863872 0.431936 0.901904i \(-0.357831\pi\)
0.431936 + 0.901904i \(0.357831\pi\)
\(492\) 3.41421 0.153925
\(493\) −33.2132 −1.49585
\(494\) −2.82843 −0.127257
\(495\) −4.41421 −0.198404
\(496\) −0.656854 −0.0294936
\(497\) 0 0
\(498\) −11.1421 −0.499291
\(499\) −13.3137 −0.596003 −0.298002 0.954565i \(-0.596320\pi\)
−0.298002 + 0.954565i \(0.596320\pi\)
\(500\) 41.8701 1.87249
\(501\) −18.7279 −0.836702
\(502\) 21.0000 0.937276
\(503\) −18.2426 −0.813399 −0.406700 0.913562i \(-0.633320\pi\)
−0.406700 + 0.913562i \(0.633320\pi\)
\(504\) 0 0
\(505\) 83.7401 3.72639
\(506\) −0.585786 −0.0260414
\(507\) −5.00000 −0.222058
\(508\) −0.171573 −0.00761232
\(509\) −8.31371 −0.368499 −0.184249 0.982880i \(-0.558985\pi\)
−0.184249 + 0.982880i \(0.558985\pi\)
\(510\) −18.7279 −0.829286
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −6.00000 −0.264649
\(515\) 6.68629 0.294633
\(516\) −2.24264 −0.0987268
\(517\) 8.82843 0.388274
\(518\) 0 0
\(519\) −23.7990 −1.04466
\(520\) 12.4853 0.547516
\(521\) 31.9411 1.39937 0.699683 0.714453i \(-0.253325\pi\)
0.699683 + 0.714453i \(0.253325\pi\)
\(522\) 7.82843 0.342641
\(523\) −23.8995 −1.04505 −0.522526 0.852623i \(-0.675010\pi\)
−0.522526 + 0.852623i \(0.675010\pi\)
\(524\) 18.3137 0.800038
\(525\) 0 0
\(526\) 19.5563 0.852697
\(527\) 2.78680 0.121395
\(528\) −1.00000 −0.0435194
\(529\) −22.6569 −0.985081
\(530\) −16.8995 −0.734067
\(531\) 8.07107 0.350255
\(532\) 0 0
\(533\) 9.65685 0.418285
\(534\) −12.2426 −0.529791
\(535\) −60.5980 −2.61988
\(536\) −2.00000 −0.0863868
\(537\) −24.4853 −1.05662
\(538\) −1.00000 −0.0431131
\(539\) 0 0
\(540\) 4.41421 0.189958
\(541\) 6.38478 0.274503 0.137251 0.990536i \(-0.456173\pi\)
0.137251 + 0.990536i \(0.456173\pi\)
\(542\) 10.8995 0.468173
\(543\) 3.51472 0.150831
\(544\) −4.24264 −0.181902
\(545\) 45.2132 1.93672
\(546\) 0 0
\(547\) −14.7279 −0.629720 −0.314860 0.949138i \(-0.601958\pi\)
−0.314860 + 0.949138i \(0.601958\pi\)
\(548\) −11.6569 −0.497956
\(549\) −0.828427 −0.0353564
\(550\) −14.4853 −0.617654
\(551\) −7.82843 −0.333502
\(552\) 0.585786 0.0249327
\(553\) 0 0
\(554\) −22.2426 −0.944999
\(555\) −36.3848 −1.54445
\(556\) 11.8995 0.504651
\(557\) 21.5858 0.914619 0.457310 0.889308i \(-0.348813\pi\)
0.457310 + 0.889308i \(0.348813\pi\)
\(558\) −0.656854 −0.0278069
\(559\) −6.34315 −0.268286
\(560\) 0 0
\(561\) 4.24264 0.179124
\(562\) 21.3137 0.899065
\(563\) 15.1005 0.636410 0.318205 0.948022i \(-0.396920\pi\)
0.318205 + 0.948022i \(0.396920\pi\)
\(564\) −8.82843 −0.371744
\(565\) 7.75736 0.326355
\(566\) 9.55635 0.401683
\(567\) 0 0
\(568\) 11.8995 0.499292
\(569\) 22.6274 0.948591 0.474295 0.880366i \(-0.342703\pi\)
0.474295 + 0.880366i \(0.342703\pi\)
\(570\) −4.41421 −0.184891
\(571\) −30.5269 −1.27751 −0.638756 0.769410i \(-0.720551\pi\)
−0.638756 + 0.769410i \(0.720551\pi\)
\(572\) −2.82843 −0.118262
\(573\) 10.1421 0.423694
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 1.00000 0.0416667
\(577\) 1.48528 0.0618331 0.0309165 0.999522i \(-0.490157\pi\)
0.0309165 + 0.999522i \(0.490157\pi\)
\(578\) 1.00000 0.0415945
\(579\) 1.72792 0.0718100
\(580\) 34.5563 1.43487
\(581\) 0 0
\(582\) 8.41421 0.348780
\(583\) 3.82843 0.158557
\(584\) 9.17157 0.379522
\(585\) 12.4853 0.516203
\(586\) 23.0000 0.950121
\(587\) 21.6274 0.892659 0.446330 0.894869i \(-0.352731\pi\)
0.446330 + 0.894869i \(0.352731\pi\)
\(588\) 0 0
\(589\) 0.656854 0.0270652
\(590\) 35.6274 1.46676
\(591\) 6.82843 0.280884
\(592\) −8.24264 −0.338770
\(593\) 44.7696 1.83847 0.919233 0.393715i \(-0.128810\pi\)
0.919233 + 0.393715i \(0.128810\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −1.51472 −0.0620453
\(597\) 8.48528 0.347279
\(598\) 1.65685 0.0677538
\(599\) 29.7574 1.21585 0.607926 0.793993i \(-0.292002\pi\)
0.607926 + 0.793993i \(0.292002\pi\)
\(600\) 14.4853 0.591359
\(601\) 32.4142 1.32220 0.661102 0.750296i \(-0.270089\pi\)
0.661102 + 0.750296i \(0.270089\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) −22.6569 −0.921894
\(605\) −44.1421 −1.79463
\(606\) 18.9706 0.770626
\(607\) −22.7990 −0.925382 −0.462691 0.886520i \(-0.653116\pi\)
−0.462691 + 0.886520i \(0.653116\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −3.65685 −0.148062
\(611\) −24.9706 −1.01020
\(612\) −4.24264 −0.171499
\(613\) 45.3553 1.83188 0.915942 0.401310i \(-0.131445\pi\)
0.915942 + 0.401310i \(0.131445\pi\)
\(614\) −4.58579 −0.185067
\(615\) 15.0711 0.607724
\(616\) 0 0
\(617\) −34.5269 −1.39000 −0.695001 0.719009i \(-0.744596\pi\)
−0.695001 + 0.719009i \(0.744596\pi\)
\(618\) 1.51472 0.0609309
\(619\) −33.7990 −1.35850 −0.679248 0.733909i \(-0.737694\pi\)
−0.679248 + 0.733909i \(0.737694\pi\)
\(620\) −2.89949 −0.116447
\(621\) 0.585786 0.0235068
\(622\) −13.7574 −0.551620
\(623\) 0 0
\(624\) 2.82843 0.113228
\(625\) 112.397 4.49588
\(626\) −1.82843 −0.0730786
\(627\) 1.00000 0.0399362
\(628\) −12.4853 −0.498217
\(629\) 34.9706 1.39437
\(630\) 0 0
\(631\) 24.5563 0.977573 0.488786 0.872403i \(-0.337440\pi\)
0.488786 + 0.872403i \(0.337440\pi\)
\(632\) −5.82843 −0.231842
\(633\) −0.928932 −0.0369217
\(634\) 7.48528 0.297279
\(635\) −0.757359 −0.0300549
\(636\) −3.82843 −0.151807
\(637\) 0 0
\(638\) −7.82843 −0.309930
\(639\) 11.8995 0.470737
\(640\) 4.41421 0.174487
\(641\) −21.4142 −0.845811 −0.422905 0.906174i \(-0.638990\pi\)
−0.422905 + 0.906174i \(0.638990\pi\)
\(642\) −13.7279 −0.541798
\(643\) −33.7574 −1.33126 −0.665630 0.746282i \(-0.731837\pi\)
−0.665630 + 0.746282i \(0.731837\pi\)
\(644\) 0 0
\(645\) −9.89949 −0.389792
\(646\) 4.24264 0.166924
\(647\) 27.3553 1.07545 0.537725 0.843120i \(-0.319284\pi\)
0.537725 + 0.843120i \(0.319284\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.07107 −0.316817
\(650\) 40.9706 1.60700
\(651\) 0 0
\(652\) −5.17157 −0.202534
\(653\) 16.2132 0.634472 0.317236 0.948347i \(-0.397245\pi\)
0.317236 + 0.948347i \(0.397245\pi\)
\(654\) 10.2426 0.400519
\(655\) 80.8406 3.15870
\(656\) 3.41421 0.133303
\(657\) 9.17157 0.357817
\(658\) 0 0
\(659\) −30.9706 −1.20644 −0.603221 0.797574i \(-0.706116\pi\)
−0.603221 + 0.797574i \(0.706116\pi\)
\(660\) −4.41421 −0.171823
\(661\) 4.38478 0.170548 0.0852740 0.996358i \(-0.472823\pi\)
0.0852740 + 0.996358i \(0.472823\pi\)
\(662\) 32.3848 1.25867
\(663\) −12.0000 −0.466041
\(664\) −11.1421 −0.432399
\(665\) 0 0
\(666\) −8.24264 −0.319396
\(667\) 4.58579 0.177562
\(668\) −18.7279 −0.724605
\(669\) −6.51472 −0.251874
\(670\) −8.82843 −0.341072
\(671\) 0.828427 0.0319811
\(672\) 0 0
\(673\) −27.7279 −1.06883 −0.534416 0.845221i \(-0.679469\pi\)
−0.534416 + 0.845221i \(0.679469\pi\)
\(674\) −12.0711 −0.464960
\(675\) 14.4853 0.557539
\(676\) −5.00000 −0.192308
\(677\) −0.313708 −0.0120568 −0.00602840 0.999982i \(-0.501919\pi\)
−0.00602840 + 0.999982i \(0.501919\pi\)
\(678\) 1.75736 0.0674910
\(679\) 0 0
\(680\) −18.7279 −0.718183
\(681\) −20.8995 −0.800870
\(682\) 0.656854 0.0251522
\(683\) 22.7574 0.870786 0.435393 0.900240i \(-0.356609\pi\)
0.435393 + 0.900240i \(0.356609\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −51.4558 −1.96603
\(686\) 0 0
\(687\) −19.7990 −0.755379
\(688\) −2.24264 −0.0854999
\(689\) −10.8284 −0.412530
\(690\) 2.58579 0.0984392
\(691\) −51.5563 −1.96130 −0.980648 0.195779i \(-0.937276\pi\)
−0.980648 + 0.195779i \(0.937276\pi\)
\(692\) −23.7990 −0.904702
\(693\) 0 0
\(694\) −34.0000 −1.29062
\(695\) 52.5269 1.99246
\(696\) 7.82843 0.296736
\(697\) −14.4853 −0.548669
\(698\) 16.2426 0.614793
\(699\) −10.8284 −0.409569
\(700\) 0 0
\(701\) −7.04163 −0.265959 −0.132979 0.991119i \(-0.542454\pi\)
−0.132979 + 0.991119i \(0.542454\pi\)
\(702\) 2.82843 0.106752
\(703\) 8.24264 0.310877
\(704\) −1.00000 −0.0376889
\(705\) −38.9706 −1.46772
\(706\) −33.8995 −1.27582
\(707\) 0 0
\(708\) 8.07107 0.303329
\(709\) −13.6152 −0.511330 −0.255665 0.966765i \(-0.582294\pi\)
−0.255665 + 0.966765i \(0.582294\pi\)
\(710\) 52.5269 1.97130
\(711\) −5.82843 −0.218583
\(712\) −12.2426 −0.458812
\(713\) −0.384776 −0.0144100
\(714\) 0 0
\(715\) −12.4853 −0.466923
\(716\) −24.4853 −0.915058
\(717\) −28.2843 −1.05630
\(718\) 10.0000 0.373197
\(719\) −3.17157 −0.118280 −0.0591399 0.998250i \(-0.518836\pi\)
−0.0591399 + 0.998250i \(0.518836\pi\)
\(720\) 4.41421 0.164508
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) 30.5563 1.13640
\(724\) 3.51472 0.130623
\(725\) 113.397 4.21146
\(726\) −10.0000 −0.371135
\(727\) −18.2721 −0.677674 −0.338837 0.940845i \(-0.610033\pi\)
−0.338837 + 0.940845i \(0.610033\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 40.4853 1.49843
\(731\) 9.51472 0.351915
\(732\) −0.828427 −0.0306195
\(733\) 42.1838 1.55809 0.779046 0.626966i \(-0.215704\pi\)
0.779046 + 0.626966i \(0.215704\pi\)
\(734\) 3.24264 0.119688
\(735\) 0 0
\(736\) 0.585786 0.0215924
\(737\) 2.00000 0.0736709
\(738\) 3.41421 0.125679
\(739\) 2.44365 0.0898911 0.0449456 0.998989i \(-0.485689\pi\)
0.0449456 + 0.998989i \(0.485689\pi\)
\(740\) −36.3848 −1.33753
\(741\) −2.82843 −0.103905
\(742\) 0 0
\(743\) 52.9117 1.94114 0.970571 0.240816i \(-0.0774150\pi\)
0.970571 + 0.240816i \(0.0774150\pi\)
\(744\) −0.656854 −0.0240814
\(745\) −6.68629 −0.244967
\(746\) −16.4853 −0.603569
\(747\) −11.1421 −0.407669
\(748\) 4.24264 0.155126
\(749\) 0 0
\(750\) 41.8701 1.52888
\(751\) 46.1127 1.68268 0.841338 0.540509i \(-0.181768\pi\)
0.841338 + 0.540509i \(0.181768\pi\)
\(752\) −8.82843 −0.321940
\(753\) 21.0000 0.765283
\(754\) 22.1421 0.806369
\(755\) −100.012 −3.63982
\(756\) 0 0
\(757\) −41.2132 −1.49792 −0.748960 0.662616i \(-0.769446\pi\)
−0.748960 + 0.662616i \(0.769446\pi\)
\(758\) −19.3137 −0.701505
\(759\) −0.585786 −0.0212627
\(760\) −4.41421 −0.160120
\(761\) 22.1421 0.802652 0.401326 0.915935i \(-0.368549\pi\)
0.401326 + 0.915935i \(0.368549\pi\)
\(762\) −0.171573 −0.00621543
\(763\) 0 0
\(764\) 10.1421 0.366930
\(765\) −18.7279 −0.677109
\(766\) −4.34315 −0.156924
\(767\) 22.8284 0.824287
\(768\) 1.00000 0.0360844
\(769\) −44.6569 −1.61037 −0.805184 0.593026i \(-0.797933\pi\)
−0.805184 + 0.593026i \(0.797933\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 1.72792 0.0621893
\(773\) −10.6274 −0.382242 −0.191121 0.981567i \(-0.561212\pi\)
−0.191121 + 0.981567i \(0.561212\pi\)
\(774\) −2.24264 −0.0806101
\(775\) −9.51472 −0.341779
\(776\) 8.41421 0.302053
\(777\) 0 0
\(778\) 28.8284 1.03355
\(779\) −3.41421 −0.122327
\(780\) 12.4853 0.447045
\(781\) −11.8995 −0.425797
\(782\) −2.48528 −0.0888735
\(783\) 7.82843 0.279765
\(784\) 0 0
\(785\) −55.1127 −1.96706
\(786\) 18.3137 0.653228
\(787\) −41.3137 −1.47267 −0.736337 0.676615i \(-0.763446\pi\)
−0.736337 + 0.676615i \(0.763446\pi\)
\(788\) 6.82843 0.243253
\(789\) 19.5563 0.696224
\(790\) −25.7279 −0.915358
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −2.34315 −0.0832075
\(794\) 5.17157 0.183532
\(795\) −16.8995 −0.599363
\(796\) 8.48528 0.300753
\(797\) 9.00000 0.318796 0.159398 0.987214i \(-0.449045\pi\)
0.159398 + 0.987214i \(0.449045\pi\)
\(798\) 0 0
\(799\) 37.4558 1.32509
\(800\) 14.4853 0.512132
\(801\) −12.2426 −0.432572
\(802\) −14.1421 −0.499376
\(803\) −9.17157 −0.323658
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −1.85786 −0.0654405
\(807\) −1.00000 −0.0352017
\(808\) 18.9706 0.667382
\(809\) 2.87006 0.100906 0.0504529 0.998726i \(-0.483934\pi\)
0.0504529 + 0.998726i \(0.483934\pi\)
\(810\) 4.41421 0.155100
\(811\) 27.5563 0.967634 0.483817 0.875169i \(-0.339250\pi\)
0.483817 + 0.875169i \(0.339250\pi\)
\(812\) 0 0
\(813\) 10.8995 0.382262
\(814\) 8.24264 0.288904
\(815\) −22.8284 −0.799645
\(816\) −4.24264 −0.148522
\(817\) 2.24264 0.0784601
\(818\) −11.1005 −0.388120
\(819\) 0 0
\(820\) 15.0711 0.526305
\(821\) −45.1838 −1.57692 −0.788462 0.615083i \(-0.789122\pi\)
−0.788462 + 0.615083i \(0.789122\pi\)
\(822\) −11.6569 −0.406579
\(823\) −32.2843 −1.12536 −0.562679 0.826675i \(-0.690229\pi\)
−0.562679 + 0.826675i \(0.690229\pi\)
\(824\) 1.51472 0.0527677
\(825\) −14.4853 −0.504313
\(826\) 0 0
\(827\) 2.27208 0.0790079 0.0395039 0.999219i \(-0.487422\pi\)
0.0395039 + 0.999219i \(0.487422\pi\)
\(828\) 0.585786 0.0203575
\(829\) 15.0294 0.521994 0.260997 0.965340i \(-0.415949\pi\)
0.260997 + 0.965340i \(0.415949\pi\)
\(830\) −49.1838 −1.70719
\(831\) −22.2426 −0.771589
\(832\) 2.82843 0.0980581
\(833\) 0 0
\(834\) 11.8995 0.412046
\(835\) −82.6690 −2.86088
\(836\) 1.00000 0.0345857
\(837\) −0.656854 −0.0227042
\(838\) −2.97056 −0.102616
\(839\) −10.7868 −0.372402 −0.186201 0.982512i \(-0.559617\pi\)
−0.186201 + 0.982512i \(0.559617\pi\)
\(840\) 0 0
\(841\) 32.2843 1.11325
\(842\) 20.8284 0.717795
\(843\) 21.3137 0.734083
\(844\) −0.928932 −0.0319752
\(845\) −22.0711 −0.759268
\(846\) −8.82843 −0.303528
\(847\) 0 0
\(848\) −3.82843 −0.131469
\(849\) 9.55635 0.327973
\(850\) −61.4558 −2.10792
\(851\) −4.82843 −0.165516
\(852\) 11.8995 0.407670
\(853\) 41.4975 1.42085 0.710423 0.703775i \(-0.248504\pi\)
0.710423 + 0.703775i \(0.248504\pi\)
\(854\) 0 0
\(855\) −4.41421 −0.150963
\(856\) −13.7279 −0.469211
\(857\) 41.3137 1.41125 0.705625 0.708586i \(-0.250666\pi\)
0.705625 + 0.708586i \(0.250666\pi\)
\(858\) −2.82843 −0.0965609
\(859\) −36.4264 −1.24285 −0.621426 0.783472i \(-0.713447\pi\)
−0.621426 + 0.783472i \(0.713447\pi\)
\(860\) −9.89949 −0.337570
\(861\) 0 0
\(862\) −33.8995 −1.15462
\(863\) 29.7574 1.01295 0.506476 0.862254i \(-0.330948\pi\)
0.506476 + 0.862254i \(0.330948\pi\)
\(864\) 1.00000 0.0340207
\(865\) −105.054 −3.57194
\(866\) 25.1716 0.855365
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 5.82843 0.197716
\(870\) 34.5563 1.17157
\(871\) −5.65685 −0.191675
\(872\) 10.2426 0.346860
\(873\) 8.41421 0.284778
\(874\) −0.585786 −0.0198145
\(875\) 0 0
\(876\) 9.17157 0.309879
\(877\) 15.2721 0.515701 0.257851 0.966185i \(-0.416986\pi\)
0.257851 + 0.966185i \(0.416986\pi\)
\(878\) 28.4558 0.960338
\(879\) 23.0000 0.775771
\(880\) −4.41421 −0.148803
\(881\) −43.9411 −1.48041 −0.740207 0.672379i \(-0.765273\pi\)
−0.740207 + 0.672379i \(0.765273\pi\)
\(882\) 0 0
\(883\) −6.87006 −0.231196 −0.115598 0.993296i \(-0.536878\pi\)
−0.115598 + 0.993296i \(0.536878\pi\)
\(884\) −12.0000 −0.403604
\(885\) 35.6274 1.19760
\(886\) −3.34315 −0.112315
\(887\) 36.9289 1.23995 0.619976 0.784621i \(-0.287142\pi\)
0.619976 + 0.784621i \(0.287142\pi\)
\(888\) −8.24264 −0.276605
\(889\) 0 0
\(890\) −54.0416 −1.81148
\(891\) −1.00000 −0.0335013
\(892\) −6.51472 −0.218129
\(893\) 8.82843 0.295432
\(894\) −1.51472 −0.0506598
\(895\) −108.083 −3.61282
\(896\) 0 0
\(897\) 1.65685 0.0553208
\(898\) 5.75736 0.192126
\(899\) −5.14214 −0.171500
\(900\) 14.4853 0.482843
\(901\) 16.2426 0.541121
\(902\) −3.41421 −0.113681
\(903\) 0 0
\(904\) 1.75736 0.0584489
\(905\) 15.5147 0.515727
\(906\) −22.6569 −0.752724
\(907\) 35.8995 1.19202 0.596012 0.802976i \(-0.296751\pi\)
0.596012 + 0.802976i \(0.296751\pi\)
\(908\) −20.8995 −0.693574
\(909\) 18.9706 0.629214
\(910\) 0 0
\(911\) −23.1716 −0.767708 −0.383854 0.923394i \(-0.625403\pi\)
−0.383854 + 0.923394i \(0.625403\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 11.1421 0.368751
\(914\) −26.1127 −0.863731
\(915\) −3.65685 −0.120892
\(916\) −19.7990 −0.654177
\(917\) 0 0
\(918\) −4.24264 −0.140028
\(919\) −51.6569 −1.70400 −0.852001 0.523540i \(-0.824611\pi\)
−0.852001 + 0.523540i \(0.824611\pi\)
\(920\) 2.58579 0.0852509
\(921\) −4.58579 −0.151107
\(922\) −5.31371 −0.174998
\(923\) 33.6569 1.10783
\(924\) 0 0
\(925\) −119.397 −3.92575
\(926\) −9.65685 −0.317344
\(927\) 1.51472 0.0497499
\(928\) 7.82843 0.256981
\(929\) 17.0711 0.560084 0.280042 0.959988i \(-0.409652\pi\)
0.280042 + 0.959988i \(0.409652\pi\)
\(930\) −2.89949 −0.0950782
\(931\) 0 0
\(932\) −10.8284 −0.354697
\(933\) −13.7574 −0.450396
\(934\) 20.8284 0.681527
\(935\) 18.7279 0.612469
\(936\) 2.82843 0.0924500
\(937\) −29.1421 −0.952032 −0.476016 0.879437i \(-0.657920\pi\)
−0.476016 + 0.879437i \(0.657920\pi\)
\(938\) 0 0
\(939\) −1.82843 −0.0596685
\(940\) −38.9706 −1.27108
\(941\) 3.68629 0.120170 0.0600848 0.998193i \(-0.480863\pi\)
0.0600848 + 0.998193i \(0.480863\pi\)
\(942\) −12.4853 −0.406792
\(943\) 2.00000 0.0651290
\(944\) 8.07107 0.262691
\(945\) 0 0
\(946\) 2.24264 0.0729145
\(947\) −20.3431 −0.661063 −0.330532 0.943795i \(-0.607228\pi\)
−0.330532 + 0.943795i \(0.607228\pi\)
\(948\) −5.82843 −0.189299
\(949\) 25.9411 0.842085
\(950\) −14.4853 −0.469965
\(951\) 7.48528 0.242727
\(952\) 0 0
\(953\) 44.8284 1.45214 0.726068 0.687623i \(-0.241346\pi\)
0.726068 + 0.687623i \(0.241346\pi\)
\(954\) −3.82843 −0.123950
\(955\) 44.7696 1.44871
\(956\) −28.2843 −0.914779
\(957\) −7.82843 −0.253057
\(958\) 21.3137 0.688615
\(959\) 0 0
\(960\) 4.41421 0.142468
\(961\) −30.5685 −0.986082
\(962\) −23.3137 −0.751664
\(963\) −13.7279 −0.442376
\(964\) 30.5563 0.984154
\(965\) 7.62742 0.245535
\(966\) 0 0
\(967\) 0.556349 0.0178910 0.00894549 0.999960i \(-0.497153\pi\)
0.00894549 + 0.999960i \(0.497153\pi\)
\(968\) −10.0000 −0.321412
\(969\) 4.24264 0.136293
\(970\) 37.1421 1.19256
\(971\) 11.0416 0.354343 0.177171 0.984180i \(-0.443305\pi\)
0.177171 + 0.984180i \(0.443305\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 0.798990 0.0256013
\(975\) 40.9706 1.31211
\(976\) −0.828427 −0.0265173
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) −5.17157 −0.165369
\(979\) 12.2426 0.391276
\(980\) 0 0
\(981\) 10.2426 0.327022
\(982\) 19.1421 0.610850
\(983\) −15.0294 −0.479365 −0.239682 0.970851i \(-0.577043\pi\)
−0.239682 + 0.970851i \(0.577043\pi\)
\(984\) 3.41421 0.108841
\(985\) 30.1421 0.960408
\(986\) −33.2132 −1.05772
\(987\) 0 0
\(988\) −2.82843 −0.0899843
\(989\) −1.31371 −0.0417735
\(990\) −4.41421 −0.140293
\(991\) 33.6274 1.06821 0.534105 0.845418i \(-0.320649\pi\)
0.534105 + 0.845418i \(0.320649\pi\)
\(992\) −0.656854 −0.0208551
\(993\) 32.3848 1.02770
\(994\) 0 0
\(995\) 37.4558 1.18743
\(996\) −11.1421 −0.353052
\(997\) −8.82843 −0.279599 −0.139800 0.990180i \(-0.544646\pi\)
−0.139800 + 0.990180i \(0.544646\pi\)
\(998\) −13.3137 −0.421438
\(999\) −8.24264 −0.260786
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 5586.2.a.br.1.2 2
7.3 odd 6 798.2.j.h.457.2 4
7.5 odd 6 798.2.j.h.571.2 yes 4
7.6 odd 2 5586.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.h.457.2 4 7.3 odd 6
798.2.j.h.571.2 yes 4 7.5 odd 6
5586.2.a.bg.1.1 2 7.6 odd 2
5586.2.a.br.1.2 2 1.1 even 1 trivial