Properties

Label 6762.2.a.cp.1.4
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.42048.1
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.77137\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.82843 q^{10} +4.14794 q^{11} +1.00000 q^{12} -2.18558 q^{13} +1.82843 q^{15} +1.00000 q^{16} -3.35716 q^{17} +1.00000 q^{18} -7.33352 q^{19} +1.82843 q^{20} +4.14794 q^{22} +1.00000 q^{23} +1.00000 q^{24} -1.65685 q^{25} -2.18558 q^{26} +1.00000 q^{27} +5.18558 q^{29} +1.82843 q^{30} +10.5191 q^{31} +1.00000 q^{32} +4.14794 q^{33} -3.35716 q^{34} +1.00000 q^{36} -1.03764 q^{37} -7.33352 q^{38} -2.18558 q^{39} +1.82843 q^{40} +7.33352 q^{41} +5.36098 q^{43} +4.14794 q^{44} +1.82843 q^{45} +1.00000 q^{46} +7.22323 q^{47} +1.00000 q^{48} -1.65685 q^{50} -3.35716 q^{51} -2.18558 q^{52} +4.54274 q^{53} +1.00000 q^{54} +7.58420 q^{55} -7.33352 q^{57} +5.18558 q^{58} -0.471271 q^{59} +1.82843 q^{60} -14.7423 q^{61} +10.5191 q^{62} +1.00000 q^{64} -3.99618 q^{65} +4.14794 q^{66} +13.4050 q^{67} -3.35716 q^{68} +1.00000 q^{69} +14.1856 q^{71} +1.00000 q^{72} +11.5191 q^{73} -1.03764 q^{74} -1.65685 q^{75} -7.33352 q^{76} -2.18558 q^{78} -5.18558 q^{79} +1.82843 q^{80} +1.00000 q^{81} +7.33352 q^{82} +5.80479 q^{83} -6.13831 q^{85} +5.36098 q^{86} +5.18558 q^{87} +4.14794 q^{88} +6.69450 q^{89} +1.82843 q^{90} +1.00000 q^{92} +10.5191 q^{93} +7.22323 q^{94} -13.4088 q^{95} +1.00000 q^{96} -4.47127 q^{97} +4.14794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{10} + 6 q^{11} + 4 q^{12} + 6 q^{13} - 4 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 4 q^{19} - 4 q^{20} + 6 q^{22} + 4 q^{23} + 4 q^{24} + 16 q^{25} + 6 q^{26} + 4 q^{27} + 6 q^{29} - 4 q^{30} + 2 q^{31} + 4 q^{32} + 6 q^{33} - 10 q^{34} + 4 q^{36} - 4 q^{38} + 6 q^{39} - 4 q^{40} + 4 q^{41} + 20 q^{43} + 6 q^{44} - 4 q^{45} + 4 q^{46} + 10 q^{47} + 4 q^{48} + 16 q^{50} - 10 q^{51} + 6 q^{52} + 4 q^{54} + 10 q^{55} - 4 q^{57} + 6 q^{58} + 6 q^{59} - 4 q^{60} + 2 q^{62} + 4 q^{64} - 14 q^{65} + 6 q^{66} + 18 q^{67} - 10 q^{68} + 4 q^{69} + 42 q^{71} + 4 q^{72} + 6 q^{73} + 16 q^{75} - 4 q^{76} + 6 q^{78} - 6 q^{79} - 4 q^{80} + 4 q^{81} + 4 q^{82} - 10 q^{83} + 34 q^{85} + 20 q^{86} + 6 q^{87} + 6 q^{88} - 4 q^{90} + 4 q^{92} + 2 q^{93} + 10 q^{94} - 20 q^{95} + 4 q^{96} - 10 q^{97} + 6 q^{99} + O(q^{100}) \)

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
\(5\) 1.82843 0.817697 0.408849 0.912602i \(-0.365930\pi\)
0.408849 + 0.912602i \(0.365930\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.82843 0.578199
\(11\) 4.14794 1.25065 0.625325 0.780364i \(-0.284966\pi\)
0.625325 + 0.780364i \(0.284966\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.18558 −0.606172 −0.303086 0.952963i \(-0.598017\pi\)
−0.303086 + 0.952963i \(0.598017\pi\)
\(14\) 0 0
\(15\) 1.82843 0.472098
\(16\) 1.00000 0.250000
\(17\) −3.35716 −0.814230 −0.407115 0.913377i \(-0.633465\pi\)
−0.407115 + 0.913377i \(0.633465\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.33352 −1.68243 −0.841213 0.540704i \(-0.818158\pi\)
−0.841213 + 0.540704i \(0.818158\pi\)
\(20\) 1.82843 0.408849
\(21\) 0 0
\(22\) 4.14794 0.884344
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −1.65685 −0.331371
\(26\) −2.18558 −0.428628
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.18558 0.962939 0.481469 0.876463i \(-0.340103\pi\)
0.481469 + 0.876463i \(0.340103\pi\)
\(30\) 1.82843 0.333824
\(31\) 10.5191 1.88929 0.944643 0.328099i \(-0.106408\pi\)
0.944643 + 0.328099i \(0.106408\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.14794 0.722063
\(34\) −3.35716 −0.575747
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.03764 −0.170588 −0.0852938 0.996356i \(-0.527183\pi\)
−0.0852938 + 0.996356i \(0.527183\pi\)
\(38\) −7.33352 −1.18965
\(39\) −2.18558 −0.349973
\(40\) 1.82843 0.289100
\(41\) 7.33352 1.14530 0.572652 0.819799i \(-0.305915\pi\)
0.572652 + 0.819799i \(0.305915\pi\)
\(42\) 0 0
\(43\) 5.36098 0.817541 0.408771 0.912637i \(-0.365958\pi\)
0.408771 + 0.912637i \(0.365958\pi\)
\(44\) 4.14794 0.625325
\(45\) 1.82843 0.272566
\(46\) 1.00000 0.147442
\(47\) 7.22323 1.05362 0.526808 0.849984i \(-0.323389\pi\)
0.526808 + 0.849984i \(0.323389\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.65685 −0.234315
\(51\) −3.35716 −0.470096
\(52\) −2.18558 −0.303086
\(53\) 4.54274 0.623993 0.311997 0.950083i \(-0.399002\pi\)
0.311997 + 0.950083i \(0.399002\pi\)
\(54\) 1.00000 0.136083
\(55\) 7.58420 1.02265
\(56\) 0 0
\(57\) −7.33352 −0.971349
\(58\) 5.18558 0.680900
\(59\) −0.471271 −0.0613543 −0.0306771 0.999529i \(-0.509766\pi\)
−0.0306771 + 0.999529i \(0.509766\pi\)
\(60\) 1.82843 0.236049
\(61\) −14.7423 −1.88756 −0.943781 0.330571i \(-0.892759\pi\)
−0.943781 + 0.330571i \(0.892759\pi\)
\(62\) 10.5191 1.33593
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.99618 −0.495665
\(66\) 4.14794 0.510576
\(67\) 13.4050 1.63768 0.818840 0.574022i \(-0.194618\pi\)
0.818840 + 0.574022i \(0.194618\pi\)
\(68\) −3.35716 −0.407115
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 14.1856 1.68352 0.841759 0.539853i \(-0.181520\pi\)
0.841759 + 0.539853i \(0.181520\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.5191 1.34821 0.674105 0.738636i \(-0.264530\pi\)
0.674105 + 0.738636i \(0.264530\pi\)
\(74\) −1.03764 −0.120624
\(75\) −1.65685 −0.191317
\(76\) −7.33352 −0.841213
\(77\) 0 0
\(78\) −2.18558 −0.247469
\(79\) −5.18558 −0.583424 −0.291712 0.956506i \(-0.594225\pi\)
−0.291712 + 0.956506i \(0.594225\pi\)
\(80\) 1.82843 0.204424
\(81\) 1.00000 0.111111
\(82\) 7.33352 0.809852
\(83\) 5.80479 0.637159 0.318579 0.947896i \(-0.396794\pi\)
0.318579 + 0.947896i \(0.396794\pi\)
\(84\) 0 0
\(85\) −6.13831 −0.665794
\(86\) 5.36098 0.578089
\(87\) 5.18558 0.555953
\(88\) 4.14794 0.442172
\(89\) 6.69450 0.709615 0.354808 0.934939i \(-0.384546\pi\)
0.354808 + 0.934939i \(0.384546\pi\)
\(90\) 1.82843 0.192733
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 10.5191 1.09078
\(94\) 7.22323 0.745019
\(95\) −13.4088 −1.37571
\(96\) 1.00000 0.102062
\(97\) −4.47127 −0.453989 −0.226994 0.973896i \(-0.572890\pi\)
−0.226994 + 0.973896i \(0.572890\pi\)
\(98\) 0 0
\(99\) 4.14794 0.416884
\(100\) −1.65685 −0.165685
\(101\) 15.3061 1.52301 0.761505 0.648159i \(-0.224461\pi\)
0.761505 + 0.648159i \(0.224461\pi\)
\(102\) −3.35716 −0.332408
\(103\) 7.40499 0.729635 0.364818 0.931079i \(-0.381131\pi\)
0.364818 + 0.931079i \(0.381131\pi\)
\(104\) −2.18558 −0.214314
\(105\) 0 0
\(106\) 4.54274 0.441230
\(107\) −5.50127 −0.531828 −0.265914 0.963997i \(-0.585674\pi\)
−0.265914 + 0.963997i \(0.585674\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.6951 −1.79066 −0.895331 0.445401i \(-0.853061\pi\)
−0.895331 + 0.445401i \(0.853061\pi\)
\(110\) 7.58420 0.723125
\(111\) −1.03764 −0.0984888
\(112\) 0 0
\(113\) 5.28187 0.496876 0.248438 0.968648i \(-0.420083\pi\)
0.248438 + 0.968648i \(0.420083\pi\)
\(114\) −7.33352 −0.686847
\(115\) 1.82843 0.170502
\(116\) 5.18558 0.481469
\(117\) −2.18558 −0.202057
\(118\) −0.471271 −0.0433840
\(119\) 0 0
\(120\) 1.82843 0.166912
\(121\) 6.20540 0.564127
\(122\) −14.7423 −1.33471
\(123\) 7.33352 0.661241
\(124\) 10.5191 0.944643
\(125\) −12.1716 −1.08866
\(126\) 0 0
\(127\) 6.49108 0.575991 0.287995 0.957632i \(-0.407011\pi\)
0.287995 + 0.957632i \(0.407011\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.36098 0.472008
\(130\) −3.99618 −0.350488
\(131\) −4.40881 −0.385200 −0.192600 0.981277i \(-0.561692\pi\)
−0.192600 + 0.981277i \(0.561692\pi\)
\(132\) 4.14794 0.361032
\(133\) 0 0
\(134\) 13.4050 1.15801
\(135\) 1.82843 0.157366
\(136\) −3.35716 −0.287874
\(137\) 1.68049 0.143574 0.0717869 0.997420i \(-0.477130\pi\)
0.0717869 + 0.997420i \(0.477130\pi\)
\(138\) 1.00000 0.0851257
\(139\) −13.3137 −1.12925 −0.564627 0.825346i \(-0.690980\pi\)
−0.564627 + 0.825346i \(0.690980\pi\)
\(140\) 0 0
\(141\) 7.22323 0.608305
\(142\) 14.1856 1.19043
\(143\) −9.06566 −0.758109
\(144\) 1.00000 0.0833333
\(145\) 9.48146 0.787392
\(146\) 11.5191 0.953328
\(147\) 0 0
\(148\) −1.03764 −0.0852938
\(149\) 19.3577 1.58585 0.792923 0.609322i \(-0.208558\pi\)
0.792923 + 0.609322i \(0.208558\pi\)
\(150\) −1.65685 −0.135282
\(151\) 5.58420 0.454436 0.227218 0.973844i \(-0.427037\pi\)
0.227218 + 0.973844i \(0.427037\pi\)
\(152\) −7.33352 −0.594827
\(153\) −3.35716 −0.271410
\(154\) 0 0
\(155\) 19.2334 1.54487
\(156\) −2.18558 −0.174987
\(157\) −22.7902 −1.81885 −0.909427 0.415864i \(-0.863479\pi\)
−0.909427 + 0.415864i \(0.863479\pi\)
\(158\) −5.18558 −0.412543
\(159\) 4.54274 0.360263
\(160\) 1.82843 0.144550
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −25.4368 −1.99237 −0.996183 0.0872876i \(-0.972180\pi\)
−0.996183 + 0.0872876i \(0.972180\pi\)
\(164\) 7.33352 0.572652
\(165\) 7.58420 0.590429
\(166\) 5.80479 0.450539
\(167\) −22.1185 −1.71158 −0.855791 0.517323i \(-0.826929\pi\)
−0.855791 + 0.517323i \(0.826929\pi\)
\(168\) 0 0
\(169\) −8.22323 −0.632556
\(170\) −6.13831 −0.470787
\(171\) −7.33352 −0.560808
\(172\) 5.36098 0.408771
\(173\) −13.6849 −1.04044 −0.520221 0.854032i \(-0.674150\pi\)
−0.520221 + 0.854032i \(0.674150\pi\)
\(174\) 5.18558 0.393118
\(175\) 0 0
\(176\) 4.14794 0.312663
\(177\) −0.471271 −0.0354229
\(178\) 6.69450 0.501774
\(179\) −2.77677 −0.207546 −0.103773 0.994601i \(-0.533092\pi\)
−0.103773 + 0.994601i \(0.533092\pi\)
\(180\) 1.82843 0.136283
\(181\) −20.9629 −1.55816 −0.779081 0.626923i \(-0.784314\pi\)
−0.779081 + 0.626923i \(0.784314\pi\)
\(182\) 0 0
\(183\) −14.7423 −1.08978
\(184\) 1.00000 0.0737210
\(185\) −1.89726 −0.139489
\(186\) 10.5191 0.771298
\(187\) −13.9253 −1.01832
\(188\) 7.22323 0.526808
\(189\) 0 0
\(190\) −13.4088 −0.972777
\(191\) −8.96236 −0.648493 −0.324247 0.945973i \(-0.605111\pi\)
−0.324247 + 0.945973i \(0.605111\pi\)
\(192\) 1.00000 0.0721688
\(193\) 11.6390 0.837795 0.418898 0.908033i \(-0.362417\pi\)
0.418898 + 0.908033i \(0.362417\pi\)
\(194\) −4.47127 −0.321019
\(195\) −3.99618 −0.286172
\(196\) 0 0
\(197\) 9.58157 0.682658 0.341329 0.939944i \(-0.389123\pi\)
0.341329 + 0.939944i \(0.389123\pi\)
\(198\) 4.14794 0.294781
\(199\) −9.95273 −0.705530 −0.352765 0.935712i \(-0.614759\pi\)
−0.352765 + 0.935712i \(0.614759\pi\)
\(200\) −1.65685 −0.117157
\(201\) 13.4050 0.945515
\(202\) 15.3061 1.07693
\(203\) 0 0
\(204\) −3.35716 −0.235048
\(205\) 13.4088 0.936512
\(206\) 7.40499 0.515930
\(207\) 1.00000 0.0695048
\(208\) −2.18558 −0.151543
\(209\) −30.4190 −2.10413
\(210\) 0 0
\(211\) 0.303519 0.0208951 0.0104476 0.999945i \(-0.496674\pi\)
0.0104476 + 0.999945i \(0.496674\pi\)
\(212\) 4.54274 0.311997
\(213\) 14.1856 0.971980
\(214\) −5.50127 −0.376059
\(215\) 9.80216 0.668501
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −18.6951 −1.26619
\(219\) 11.5191 0.778389
\(220\) 7.58420 0.511327
\(221\) 7.33734 0.493563
\(222\) −1.03764 −0.0696421
\(223\) −16.9375 −1.13422 −0.567111 0.823642i \(-0.691939\pi\)
−0.567111 + 0.823642i \(0.691939\pi\)
\(224\) 0 0
\(225\) −1.65685 −0.110457
\(226\) 5.28187 0.351345
\(227\) −21.5369 −1.42946 −0.714728 0.699402i \(-0.753450\pi\)
−0.714728 + 0.699402i \(0.753450\pi\)
\(228\) −7.33352 −0.485674
\(229\) −0.666478 −0.0440421 −0.0220211 0.999758i \(-0.507010\pi\)
−0.0220211 + 0.999758i \(0.507010\pi\)
\(230\) 1.82843 0.120563
\(231\) 0 0
\(232\) 5.18558 0.340450
\(233\) −1.67667 −0.109842 −0.0549211 0.998491i \(-0.517491\pi\)
−0.0549211 + 0.998491i \(0.517491\pi\)
\(234\) −2.18558 −0.142876
\(235\) 13.2071 0.861539
\(236\) −0.471271 −0.0306771
\(237\) −5.18558 −0.336840
\(238\) 0 0
\(239\) 10.7220 0.693546 0.346773 0.937949i \(-0.387278\pi\)
0.346773 + 0.937949i \(0.387278\pi\)
\(240\) 1.82843 0.118024
\(241\) 18.2034 1.17258 0.586292 0.810099i \(-0.300587\pi\)
0.586292 + 0.810099i \(0.300587\pi\)
\(242\) 6.20540 0.398898
\(243\) 1.00000 0.0641500
\(244\) −14.7423 −0.943781
\(245\) 0 0
\(246\) 7.33352 0.467568
\(247\) 16.0280 1.01984
\(248\) 10.5191 0.667964
\(249\) 5.80479 0.367864
\(250\) −12.1716 −0.769798
\(251\) −0.794604 −0.0501549 −0.0250775 0.999686i \(-0.507983\pi\)
−0.0250775 + 0.999686i \(0.507983\pi\)
\(252\) 0 0
\(253\) 4.14794 0.260779
\(254\) 6.49108 0.407287
\(255\) −6.13831 −0.384396
\(256\) 1.00000 0.0625000
\(257\) −0.591190 −0.0368774 −0.0184387 0.999830i \(-0.505870\pi\)
−0.0184387 + 0.999830i \(0.505870\pi\)
\(258\) 5.36098 0.333760
\(259\) 0 0
\(260\) −3.99618 −0.247833
\(261\) 5.18558 0.320980
\(262\) −4.40881 −0.272377
\(263\) 2.86725 0.176802 0.0884012 0.996085i \(-0.471824\pi\)
0.0884012 + 0.996085i \(0.471824\pi\)
\(264\) 4.14794 0.255288
\(265\) 8.30607 0.510238
\(266\) 0 0
\(267\) 6.69450 0.409697
\(268\) 13.4050 0.818840
\(269\) −7.17794 −0.437647 −0.218823 0.975764i \(-0.570222\pi\)
−0.218823 + 0.975764i \(0.570222\pi\)
\(270\) 1.82843 0.111275
\(271\) −10.5191 −0.638990 −0.319495 0.947588i \(-0.603513\pi\)
−0.319495 + 0.947588i \(0.603513\pi\)
\(272\) −3.35716 −0.203557
\(273\) 0 0
\(274\) 1.68049 0.101522
\(275\) −6.87253 −0.414429
\(276\) 1.00000 0.0601929
\(277\) −30.9285 −1.85831 −0.929156 0.369688i \(-0.879465\pi\)
−0.929156 + 0.369688i \(0.879465\pi\)
\(278\) −13.3137 −0.798503
\(279\) 10.5191 0.629762
\(280\) 0 0
\(281\) −2.09949 −0.125245 −0.0626225 0.998037i \(-0.519946\pi\)
−0.0626225 + 0.998037i \(0.519946\pi\)
\(282\) 7.22323 0.430137
\(283\) 20.3475 1.20954 0.604768 0.796402i \(-0.293266\pi\)
0.604768 + 0.796402i \(0.293266\pi\)
\(284\) 14.1856 0.841759
\(285\) −13.4088 −0.794269
\(286\) −9.06566 −0.536064
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −5.72950 −0.337030
\(290\) 9.48146 0.556771
\(291\) −4.47127 −0.262111
\(292\) 11.5191 0.674105
\(293\) −31.4853 −1.83939 −0.919695 0.392634i \(-0.871564\pi\)
−0.919695 + 0.392634i \(0.871564\pi\)
\(294\) 0 0
\(295\) −0.861685 −0.0501693
\(296\) −1.03764 −0.0603118
\(297\) 4.14794 0.240688
\(298\) 19.3577 1.12136
\(299\) −2.18558 −0.126396
\(300\) −1.65685 −0.0956585
\(301\) 0 0
\(302\) 5.58420 0.321335
\(303\) 15.3061 0.879311
\(304\) −7.33352 −0.420606
\(305\) −26.9553 −1.54345
\(306\) −3.35716 −0.191916
\(307\) 1.64865 0.0940933 0.0470466 0.998893i \(-0.485019\pi\)
0.0470466 + 0.998893i \(0.485019\pi\)
\(308\) 0 0
\(309\) 7.40499 0.421255
\(310\) 19.2334 1.09238
\(311\) 12.1779 0.690548 0.345274 0.938502i \(-0.387786\pi\)
0.345274 + 0.938502i \(0.387786\pi\)
\(312\) −2.18558 −0.123734
\(313\) 12.5669 0.710325 0.355163 0.934805i \(-0.384425\pi\)
0.355163 + 0.934805i \(0.384425\pi\)
\(314\) −22.7902 −1.28612
\(315\) 0 0
\(316\) −5.18558 −0.291712
\(317\) −9.17794 −0.515485 −0.257742 0.966214i \(-0.582979\pi\)
−0.257742 + 0.966214i \(0.582979\pi\)
\(318\) 4.54274 0.254744
\(319\) 21.5095 1.20430
\(320\) 1.82843 0.102212
\(321\) −5.50127 −0.307051
\(322\) 0 0
\(323\) 24.6198 1.36988
\(324\) 1.00000 0.0555556
\(325\) 3.62119 0.200868
\(326\) −25.4368 −1.40882
\(327\) −18.6951 −1.03384
\(328\) 7.33352 0.404926
\(329\) 0 0
\(330\) 7.58420 0.417497
\(331\) −29.3896 −1.61540 −0.807698 0.589596i \(-0.799287\pi\)
−0.807698 + 0.589596i \(0.799287\pi\)
\(332\) 5.80479 0.318579
\(333\) −1.03764 −0.0568626
\(334\) −22.1185 −1.21027
\(335\) 24.5100 1.33913
\(336\) 0 0
\(337\) 8.12048 0.442351 0.221175 0.975234i \(-0.429011\pi\)
0.221175 + 0.975234i \(0.429011\pi\)
\(338\) −8.22323 −0.447285
\(339\) 5.28187 0.286872
\(340\) −6.13831 −0.332897
\(341\) 43.6326 2.36284
\(342\) −7.33352 −0.396551
\(343\) 0 0
\(344\) 5.36098 0.289044
\(345\) 1.82843 0.0984392
\(346\) −13.6849 −0.735703
\(347\) 28.2614 1.51715 0.758577 0.651584i \(-0.225895\pi\)
0.758577 + 0.651584i \(0.225895\pi\)
\(348\) 5.18558 0.277976
\(349\) 24.1307 1.29169 0.645843 0.763471i \(-0.276506\pi\)
0.645843 + 0.763471i \(0.276506\pi\)
\(350\) 0 0
\(351\) −2.18558 −0.116658
\(352\) 4.14794 0.221086
\(353\) 13.4088 0.713679 0.356839 0.934166i \(-0.383854\pi\)
0.356839 + 0.934166i \(0.383854\pi\)
\(354\) −0.471271 −0.0250478
\(355\) 25.9373 1.37661
\(356\) 6.69450 0.354808
\(357\) 0 0
\(358\) −2.77677 −0.146757
\(359\) −3.88509 −0.205047 −0.102523 0.994731i \(-0.532692\pi\)
−0.102523 + 0.994731i \(0.532692\pi\)
\(360\) 1.82843 0.0963666
\(361\) 34.7805 1.83055
\(362\) −20.9629 −1.10179
\(363\) 6.20540 0.325699
\(364\) 0 0
\(365\) 21.0618 1.10243
\(366\) −14.7423 −0.770594
\(367\) −1.43745 −0.0750342 −0.0375171 0.999296i \(-0.511945\pi\)
−0.0375171 + 0.999296i \(0.511945\pi\)
\(368\) 1.00000 0.0521286
\(369\) 7.33352 0.381768
\(370\) −1.89726 −0.0986337
\(371\) 0 0
\(372\) 10.5191 0.545390
\(373\) 18.2959 0.947325 0.473662 0.880707i \(-0.342932\pi\)
0.473662 + 0.880707i \(0.342932\pi\)
\(374\) −13.9253 −0.720059
\(375\) −12.1716 −0.628537
\(376\) 7.22323 0.372509
\(377\) −11.3335 −0.583706
\(378\) 0 0
\(379\) −5.97637 −0.306985 −0.153493 0.988150i \(-0.549052\pi\)
−0.153493 + 0.988150i \(0.549052\pi\)
\(380\) −13.4088 −0.687857
\(381\) 6.49108 0.332548
\(382\) −8.96236 −0.458554
\(383\) −5.89726 −0.301336 −0.150668 0.988584i \(-0.548142\pi\)
−0.150668 + 0.988584i \(0.548142\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 11.6390 0.592411
\(387\) 5.36098 0.272514
\(388\) −4.47127 −0.226994
\(389\) −17.1135 −0.867689 −0.433844 0.900988i \(-0.642843\pi\)
−0.433844 + 0.900988i \(0.642843\pi\)
\(390\) −3.99618 −0.202354
\(391\) −3.35716 −0.169779
\(392\) 0 0
\(393\) −4.40881 −0.222395
\(394\) 9.58157 0.482712
\(395\) −9.48146 −0.477064
\(396\) 4.14794 0.208442
\(397\) −12.8450 −0.644671 −0.322336 0.946625i \(-0.604468\pi\)
−0.322336 + 0.946625i \(0.604468\pi\)
\(398\) −9.95273 −0.498885
\(399\) 0 0
\(400\) −1.65685 −0.0828427
\(401\) 17.4330 0.870563 0.435281 0.900294i \(-0.356649\pi\)
0.435281 + 0.900294i \(0.356649\pi\)
\(402\) 13.4050 0.668580
\(403\) −22.9904 −1.14523
\(404\) 15.3061 0.761505
\(405\) 1.82843 0.0908553
\(406\) 0 0
\(407\) −4.30408 −0.213346
\(408\) −3.35716 −0.166204
\(409\) −9.06765 −0.448366 −0.224183 0.974547i \(-0.571971\pi\)
−0.224183 + 0.974547i \(0.571971\pi\)
\(410\) 13.4088 0.662214
\(411\) 1.68049 0.0828924
\(412\) 7.40499 0.364818
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 10.6136 0.521003
\(416\) −2.18558 −0.107157
\(417\) −13.3137 −0.651975
\(418\) −30.4190 −1.48784
\(419\) −6.63902 −0.324338 −0.162169 0.986763i \(-0.551849\pi\)
−0.162169 + 0.986763i \(0.551849\pi\)
\(420\) 0 0
\(421\) −0.942543 −0.0459367 −0.0229684 0.999736i \(-0.507312\pi\)
−0.0229684 + 0.999736i \(0.507312\pi\)
\(422\) 0.303519 0.0147751
\(423\) 7.22323 0.351205
\(424\) 4.54274 0.220615
\(425\) 5.56232 0.269812
\(426\) 14.1856 0.687294
\(427\) 0 0
\(428\) −5.50127 −0.265914
\(429\) −9.06566 −0.437694
\(430\) 9.80216 0.472702
\(431\) −35.3697 −1.70370 −0.851850 0.523785i \(-0.824519\pi\)
−0.851850 + 0.523785i \(0.824519\pi\)
\(432\) 1.00000 0.0481125
\(433\) 32.7627 1.57448 0.787238 0.616650i \(-0.211510\pi\)
0.787238 + 0.616650i \(0.211510\pi\)
\(434\) 0 0
\(435\) 9.48146 0.454601
\(436\) −18.6951 −0.895331
\(437\) −7.33352 −0.350810
\(438\) 11.5191 0.550404
\(439\) −19.3163 −0.921919 −0.460959 0.887421i \(-0.652495\pi\)
−0.460959 + 0.887421i \(0.652495\pi\)
\(440\) 7.58420 0.361563
\(441\) 0 0
\(442\) 7.33734 0.349002
\(443\) 14.4088 0.684583 0.342292 0.939594i \(-0.388797\pi\)
0.342292 + 0.939594i \(0.388797\pi\)
\(444\) −1.03764 −0.0492444
\(445\) 12.2404 0.580251
\(446\) −16.9375 −0.802016
\(447\) 19.3577 0.915589
\(448\) 0 0
\(449\) −11.0580 −0.521860 −0.260930 0.965358i \(-0.584029\pi\)
−0.260930 + 0.965358i \(0.584029\pi\)
\(450\) −1.65685 −0.0781049
\(451\) 30.4190 1.43237
\(452\) 5.28187 0.248438
\(453\) 5.58420 0.262369
\(454\) −21.5369 −1.01078
\(455\) 0 0
\(456\) −7.33352 −0.343424
\(457\) −25.8450 −1.20898 −0.604489 0.796614i \(-0.706623\pi\)
−0.604489 + 0.796614i \(0.706623\pi\)
\(458\) −0.666478 −0.0311425
\(459\) −3.35716 −0.156699
\(460\) 1.82843 0.0852509
\(461\) −39.3341 −1.83197 −0.915986 0.401211i \(-0.868589\pi\)
−0.915986 + 0.401211i \(0.868589\pi\)
\(462\) 0 0
\(463\) 25.0778 1.16547 0.582733 0.812664i \(-0.301983\pi\)
0.582733 + 0.812664i \(0.301983\pi\)
\(464\) 5.18558 0.240735
\(465\) 19.2334 0.891928
\(466\) −1.67667 −0.0776701
\(467\) −22.9629 −1.06260 −0.531299 0.847185i \(-0.678296\pi\)
−0.531299 + 0.847185i \(0.678296\pi\)
\(468\) −2.18558 −0.101029
\(469\) 0 0
\(470\) 13.2071 0.609200
\(471\) −22.7902 −1.05012
\(472\) −0.471271 −0.0216920
\(473\) 22.2370 1.02246
\(474\) −5.18558 −0.238182
\(475\) 12.1506 0.557507
\(476\) 0 0
\(477\) 4.54274 0.207998
\(478\) 10.7220 0.490411
\(479\) −16.6116 −0.759002 −0.379501 0.925191i \(-0.623904\pi\)
−0.379501 + 0.925191i \(0.623904\pi\)
\(480\) 1.82843 0.0834559
\(481\) 2.26786 0.103405
\(482\) 18.2034 0.829143
\(483\) 0 0
\(484\) 6.20540 0.282063
\(485\) −8.17539 −0.371226
\(486\) 1.00000 0.0453609
\(487\) −14.8226 −0.671677 −0.335839 0.941920i \(-0.609020\pi\)
−0.335839 + 0.941920i \(0.609020\pi\)
\(488\) −14.7423 −0.667354
\(489\) −25.4368 −1.15029
\(490\) 0 0
\(491\) 22.2466 1.00398 0.501988 0.864875i \(-0.332602\pi\)
0.501988 + 0.864875i \(0.332602\pi\)
\(492\) 7.33352 0.330621
\(493\) −17.4088 −0.784053
\(494\) 16.0280 0.721135
\(495\) 7.58420 0.340885
\(496\) 10.5191 0.472322
\(497\) 0 0
\(498\) 5.80479 0.260119
\(499\) −11.2857 −0.505217 −0.252608 0.967569i \(-0.581288\pi\)
−0.252608 + 0.967569i \(0.581288\pi\)
\(500\) −12.1716 −0.544329
\(501\) −22.1185 −0.988182
\(502\) −0.794604 −0.0354649
\(503\) 27.1339 1.20984 0.604920 0.796286i \(-0.293205\pi\)
0.604920 + 0.796286i \(0.293205\pi\)
\(504\) 0 0
\(505\) 27.9860 1.24536
\(506\) 4.14794 0.184398
\(507\) −8.22323 −0.365206
\(508\) 6.49108 0.287995
\(509\) −38.7952 −1.71957 −0.859783 0.510660i \(-0.829401\pi\)
−0.859783 + 0.510660i \(0.829401\pi\)
\(510\) −6.13831 −0.271809
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −7.33352 −0.323783
\(514\) −0.591190 −0.0260763
\(515\) 13.5395 0.596621
\(516\) 5.36098 0.236004
\(517\) 29.9615 1.31771
\(518\) 0 0
\(519\) −13.6849 −0.600699
\(520\) −3.99618 −0.175244
\(521\) 1.52991 0.0670266 0.0335133 0.999438i \(-0.489330\pi\)
0.0335133 + 0.999438i \(0.489330\pi\)
\(522\) 5.18558 0.226967
\(523\) 8.42282 0.368304 0.184152 0.982898i \(-0.441046\pi\)
0.184152 + 0.982898i \(0.441046\pi\)
\(524\) −4.40881 −0.192600
\(525\) 0 0
\(526\) 2.86725 0.125018
\(527\) −35.3143 −1.53831
\(528\) 4.14794 0.180516
\(529\) 1.00000 0.0434783
\(530\) 8.30607 0.360792
\(531\) −0.471271 −0.0204514
\(532\) 0 0
\(533\) −16.0280 −0.694251
\(534\) 6.69450 0.289699
\(535\) −10.0587 −0.434875
\(536\) 13.4050 0.579007
\(537\) −2.77677 −0.119827
\(538\) −7.17794 −0.309463
\(539\) 0 0
\(540\) 1.82843 0.0786830
\(541\) −5.65383 −0.243077 −0.121539 0.992587i \(-0.538783\pi\)
−0.121539 + 0.992587i \(0.538783\pi\)
\(542\) −10.5191 −0.451834
\(543\) −20.9629 −0.899605
\(544\) −3.35716 −0.143937
\(545\) −34.1826 −1.46422
\(546\) 0 0
\(547\) −26.2480 −1.12229 −0.561143 0.827719i \(-0.689638\pi\)
−0.561143 + 0.827719i \(0.689638\pi\)
\(548\) 1.68049 0.0717869
\(549\) −14.7423 −0.629187
\(550\) −6.87253 −0.293046
\(551\) −38.0286 −1.62007
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −30.9285 −1.31403
\(555\) −1.89726 −0.0805341
\(556\) −13.3137 −0.564627
\(557\) 37.2422 1.57800 0.789001 0.614392i \(-0.210599\pi\)
0.789001 + 0.614392i \(0.210599\pi\)
\(558\) 10.5191 0.445309
\(559\) −11.7169 −0.495570
\(560\) 0 0
\(561\) −13.9253 −0.587926
\(562\) −2.09949 −0.0885615
\(563\) 13.4897 0.568522 0.284261 0.958747i \(-0.408252\pi\)
0.284261 + 0.958747i \(0.408252\pi\)
\(564\) 7.22323 0.304153
\(565\) 9.65751 0.406294
\(566\) 20.3475 0.855271
\(567\) 0 0
\(568\) 14.1856 0.595214
\(569\) −16.7379 −0.701691 −0.350846 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(570\) −13.4088 −0.561633
\(571\) −26.1671 −1.09506 −0.547530 0.836786i \(-0.684432\pi\)
−0.547530 + 0.836786i \(0.684432\pi\)
\(572\) −9.06566 −0.379054
\(573\) −8.96236 −0.374408
\(574\) 0 0
\(575\) −1.65685 −0.0690956
\(576\) 1.00000 0.0416667
\(577\) 19.7972 0.824166 0.412083 0.911146i \(-0.364801\pi\)
0.412083 + 0.911146i \(0.364801\pi\)
\(578\) −5.72950 −0.238316
\(579\) 11.6390 0.483701
\(580\) 9.48146 0.393696
\(581\) 0 0
\(582\) −4.47127 −0.185340
\(583\) 18.8430 0.780397
\(584\) 11.5191 0.476664
\(585\) −3.99618 −0.165222
\(586\) −31.4853 −1.30064
\(587\) −3.20078 −0.132110 −0.0660551 0.997816i \(-0.521041\pi\)
−0.0660551 + 0.997816i \(0.521041\pi\)
\(588\) 0 0
\(589\) −77.1421 −3.17858
\(590\) −0.861685 −0.0354750
\(591\) 9.58157 0.394133
\(592\) −1.03764 −0.0426469
\(593\) −32.1033 −1.31832 −0.659162 0.752001i \(-0.729089\pi\)
−0.659162 + 0.752001i \(0.729089\pi\)
\(594\) 4.14794 0.170192
\(595\) 0 0
\(596\) 19.3577 0.792923
\(597\) −9.95273 −0.407338
\(598\) −2.18558 −0.0893751
\(599\) 45.1842 1.84617 0.923087 0.384590i \(-0.125657\pi\)
0.923087 + 0.384590i \(0.125657\pi\)
\(600\) −1.65685 −0.0676408
\(601\) 15.5166 0.632934 0.316467 0.948604i \(-0.397503\pi\)
0.316467 + 0.948604i \(0.397503\pi\)
\(602\) 0 0
\(603\) 13.4050 0.545893
\(604\) 5.58420 0.227218
\(605\) 11.3461 0.461285
\(606\) 15.3061 0.621766
\(607\) 4.86225 0.197353 0.0986763 0.995120i \(-0.468539\pi\)
0.0986763 + 0.995120i \(0.468539\pi\)
\(608\) −7.33352 −0.297414
\(609\) 0 0
\(610\) −26.9553 −1.09139
\(611\) −15.7870 −0.638672
\(612\) −3.35716 −0.135705
\(613\) 6.80998 0.275052 0.137526 0.990498i \(-0.456085\pi\)
0.137526 + 0.990498i \(0.456085\pi\)
\(614\) 1.64865 0.0665340
\(615\) 13.4088 0.540695
\(616\) 0 0
\(617\) 7.70087 0.310025 0.155013 0.987912i \(-0.450458\pi\)
0.155013 + 0.987912i \(0.450458\pi\)
\(618\) 7.40499 0.297872
\(619\) 3.13657 0.126069 0.0630346 0.998011i \(-0.479922\pi\)
0.0630346 + 0.998011i \(0.479922\pi\)
\(620\) 19.2334 0.772433
\(621\) 1.00000 0.0401286
\(622\) 12.1779 0.488291
\(623\) 0 0
\(624\) −2.18558 −0.0874933
\(625\) −13.9706 −0.558823
\(626\) 12.5669 0.502276
\(627\) −30.4190 −1.21482
\(628\) −22.7902 −0.909427
\(629\) 3.48353 0.138898
\(630\) 0 0
\(631\) −22.1045 −0.879966 −0.439983 0.898006i \(-0.645016\pi\)
−0.439983 + 0.898006i \(0.645016\pi\)
\(632\) −5.18558 −0.206271
\(633\) 0.303519 0.0120638
\(634\) −9.17794 −0.364503
\(635\) 11.8685 0.470986
\(636\) 4.54274 0.180131
\(637\) 0 0
\(638\) 21.5095 0.851568
\(639\) 14.1856 0.561173
\(640\) 1.82843 0.0722749
\(641\) −18.5483 −0.732614 −0.366307 0.930494i \(-0.619378\pi\)
−0.366307 + 0.930494i \(0.619378\pi\)
\(642\) −5.50127 −0.217118
\(643\) −12.4114 −0.489456 −0.244728 0.969592i \(-0.578699\pi\)
−0.244728 + 0.969592i \(0.578699\pi\)
\(644\) 0 0
\(645\) 9.80216 0.385959
\(646\) 24.6198 0.968652
\(647\) 37.8202 1.48686 0.743432 0.668811i \(-0.233197\pi\)
0.743432 + 0.668811i \(0.233197\pi\)
\(648\) 1.00000 0.0392837
\(649\) −1.95480 −0.0767328
\(650\) 3.62119 0.142035
\(651\) 0 0
\(652\) −25.4368 −0.996183
\(653\) 19.5095 0.763465 0.381732 0.924273i \(-0.375328\pi\)
0.381732 + 0.924273i \(0.375328\pi\)
\(654\) −18.6951 −0.731035
\(655\) −8.06119 −0.314977
\(656\) 7.33352 0.286326
\(657\) 11.5191 0.449403
\(658\) 0 0
\(659\) −33.6772 −1.31188 −0.655939 0.754814i \(-0.727727\pi\)
−0.655939 + 0.754814i \(0.727727\pi\)
\(660\) 7.58420 0.295215
\(661\) 27.3966 1.06561 0.532803 0.846239i \(-0.321139\pi\)
0.532803 + 0.846239i \(0.321139\pi\)
\(662\) −29.3896 −1.14226
\(663\) 7.33734 0.284959
\(664\) 5.80479 0.225270
\(665\) 0 0
\(666\) −1.03764 −0.0402079
\(667\) 5.18558 0.200787
\(668\) −22.1185 −0.855791
\(669\) −16.9375 −0.654843
\(670\) 24.5100 0.946906
\(671\) −61.1503 −2.36068
\(672\) 0 0
\(673\) 7.79715 0.300558 0.150279 0.988644i \(-0.451983\pi\)
0.150279 + 0.988644i \(0.451983\pi\)
\(674\) 8.12048 0.312789
\(675\) −1.65685 −0.0637723
\(676\) −8.22323 −0.316278
\(677\) −0.0566580 −0.00217755 −0.00108877 0.999999i \(-0.500347\pi\)
−0.00108877 + 0.999999i \(0.500347\pi\)
\(678\) 5.28187 0.202849
\(679\) 0 0
\(680\) −6.13831 −0.235394
\(681\) −21.5369 −0.825297
\(682\) 43.6326 1.67078
\(683\) −35.9635 −1.37610 −0.688052 0.725661i \(-0.741534\pi\)
−0.688052 + 0.725661i \(0.741534\pi\)
\(684\) −7.33352 −0.280404
\(685\) 3.07265 0.117400
\(686\) 0 0
\(687\) −0.666478 −0.0254277
\(688\) 5.36098 0.204385
\(689\) −9.92853 −0.378247
\(690\) 1.82843 0.0696070
\(691\) −22.0362 −0.838298 −0.419149 0.907918i \(-0.637671\pi\)
−0.419149 + 0.907918i \(0.637671\pi\)
\(692\) −13.6849 −0.520221
\(693\) 0 0
\(694\) 28.2614 1.07279
\(695\) −24.3431 −0.923388
\(696\) 5.18558 0.196559
\(697\) −24.6198 −0.932540
\(698\) 24.1307 0.913359
\(699\) −1.67667 −0.0634174
\(700\) 0 0
\(701\) 37.5337 1.41763 0.708814 0.705396i \(-0.249231\pi\)
0.708814 + 0.705396i \(0.249231\pi\)
\(702\) −2.18558 −0.0824895
\(703\) 7.60959 0.287001
\(704\) 4.14794 0.156331
\(705\) 13.2071 0.497410
\(706\) 13.4088 0.504647
\(707\) 0 0
\(708\) −0.471271 −0.0177115
\(709\) 40.9788 1.53899 0.769495 0.638653i \(-0.220508\pi\)
0.769495 + 0.638653i \(0.220508\pi\)
\(710\) 25.9373 0.973410
\(711\) −5.18558 −0.194475
\(712\) 6.69450 0.250887
\(713\) 10.5191 0.393944
\(714\) 0 0
\(715\) −16.5759 −0.619904
\(716\) −2.77677 −0.103773
\(717\) 10.7220 0.400419
\(718\) −3.88509 −0.144990
\(719\) 17.7397 0.661579 0.330789 0.943705i \(-0.392685\pi\)
0.330789 + 0.943705i \(0.392685\pi\)
\(720\) 1.82843 0.0681415
\(721\) 0 0
\(722\) 34.7805 1.29440
\(723\) 18.2034 0.676992
\(724\) −20.9629 −0.779081
\(725\) −8.59176 −0.319090
\(726\) 6.20540 0.230304
\(727\) 14.4240 0.534957 0.267478 0.963564i \(-0.413810\pi\)
0.267478 + 0.963564i \(0.413810\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 21.0618 0.779534
\(731\) −17.9976 −0.665667
\(732\) −14.7423 −0.544892
\(733\) −31.6854 −1.17033 −0.585164 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(734\) −1.43745 −0.0530572
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 55.6031 2.04817
\(738\) 7.33352 0.269951
\(739\) −4.20021 −0.154507 −0.0772536 0.997011i \(-0.524615\pi\)
−0.0772536 + 0.997011i \(0.524615\pi\)
\(740\) −1.89726 −0.0697445
\(741\) 16.0280 0.588804
\(742\) 0 0
\(743\) −42.3168 −1.55245 −0.776227 0.630454i \(-0.782869\pi\)
−0.776227 + 0.630454i \(0.782869\pi\)
\(744\) 10.5191 0.385649
\(745\) 35.3942 1.29674
\(746\) 18.2959 0.669860
\(747\) 5.80479 0.212386
\(748\) −13.9253 −0.509159
\(749\) 0 0
\(750\) −12.1716 −0.444443
\(751\) 16.4444 0.600064 0.300032 0.953929i \(-0.403003\pi\)
0.300032 + 0.953929i \(0.403003\pi\)
\(752\) 7.22323 0.263404
\(753\) −0.794604 −0.0289570
\(754\) −11.3335 −0.412743
\(755\) 10.2103 0.371591
\(756\) 0 0
\(757\) 45.4215 1.65087 0.825437 0.564494i \(-0.190929\pi\)
0.825437 + 0.564494i \(0.190929\pi\)
\(758\) −5.97637 −0.217071
\(759\) 4.14794 0.150561
\(760\) −13.4088 −0.486389
\(761\) 15.6421 0.567027 0.283513 0.958968i \(-0.408500\pi\)
0.283513 + 0.958968i \(0.408500\pi\)
\(762\) 6.49108 0.235147
\(763\) 0 0
\(764\) −8.96236 −0.324247
\(765\) −6.13831 −0.221931
\(766\) −5.89726 −0.213077
\(767\) 1.03000 0.0371912
\(768\) 1.00000 0.0360844
\(769\) −14.1205 −0.509198 −0.254599 0.967047i \(-0.581943\pi\)
−0.254599 + 0.967047i \(0.581943\pi\)
\(770\) 0 0
\(771\) −0.591190 −0.0212912
\(772\) 11.6390 0.418898
\(773\) 17.6052 0.633215 0.316608 0.948557i \(-0.397456\pi\)
0.316608 + 0.948557i \(0.397456\pi\)
\(774\) 5.36098 0.192696
\(775\) −17.4286 −0.626055
\(776\) −4.47127 −0.160509
\(777\) 0 0
\(778\) −17.1135 −0.613549
\(779\) −53.7805 −1.92689
\(780\) −3.99618 −0.143086
\(781\) 58.8409 2.10549
\(782\) −3.35716 −0.120052
\(783\) 5.18558 0.185318
\(784\) 0 0
\(785\) −41.6702 −1.48727
\(786\) −4.40881 −0.157257
\(787\) −8.75832 −0.312201 −0.156100 0.987741i \(-0.549892\pi\)
−0.156100 + 0.987741i \(0.549892\pi\)
\(788\) 9.58157 0.341329
\(789\) 2.86725 0.102077
\(790\) −9.48146 −0.337335
\(791\) 0 0
\(792\) 4.14794 0.147391
\(793\) 32.2206 1.14419
\(794\) −12.8450 −0.455851
\(795\) 8.30607 0.294586
\(796\) −9.95273 −0.352765
\(797\) 16.5708 0.586966 0.293483 0.955964i \(-0.405186\pi\)
0.293483 + 0.955964i \(0.405186\pi\)
\(798\) 0 0
\(799\) −24.2495 −0.857886
\(800\) −1.65685 −0.0585786
\(801\) 6.69450 0.236538
\(802\) 17.4330 0.615581
\(803\) 47.7805 1.68614
\(804\) 13.4050 0.472758
\(805\) 0 0
\(806\) −22.9904 −0.809801
\(807\) −7.17794 −0.252676
\(808\) 15.3061 0.538466
\(809\) 27.8553 0.979339 0.489669 0.871908i \(-0.337117\pi\)
0.489669 + 0.871908i \(0.337117\pi\)
\(810\) 1.82843 0.0642444
\(811\) 31.5623 1.10830 0.554151 0.832416i \(-0.313043\pi\)
0.554151 + 0.832416i \(0.313043\pi\)
\(812\) 0 0
\(813\) −10.5191 −0.368921
\(814\) −4.30408 −0.150858
\(815\) −46.5094 −1.62915
\(816\) −3.35716 −0.117524
\(817\) −39.3148 −1.37545
\(818\) −9.06765 −0.317043
\(819\) 0 0
\(820\) 13.4088 0.468256
\(821\) −24.9930 −0.872262 −0.436131 0.899883i \(-0.643652\pi\)
−0.436131 + 0.899883i \(0.643652\pi\)
\(822\) 1.68049 0.0586138
\(823\) −8.08803 −0.281931 −0.140965 0.990015i \(-0.545021\pi\)
−0.140965 + 0.990015i \(0.545021\pi\)
\(824\) 7.40499 0.257965
\(825\) −6.87253 −0.239271
\(826\) 0 0
\(827\) −19.3864 −0.674130 −0.337065 0.941481i \(-0.609434\pi\)
−0.337065 + 0.941481i \(0.609434\pi\)
\(828\) 1.00000 0.0347524
\(829\) 26.8252 0.931677 0.465838 0.884870i \(-0.345753\pi\)
0.465838 + 0.884870i \(0.345753\pi\)
\(830\) 10.6136 0.368405
\(831\) −30.9285 −1.07290
\(832\) −2.18558 −0.0757715
\(833\) 0 0
\(834\) −13.3137 −0.461016
\(835\) −40.4421 −1.39956
\(836\) −30.4190 −1.05206
\(837\) 10.5191 0.363593
\(838\) −6.63902 −0.229341
\(839\) −4.82979 −0.166743 −0.0833715 0.996519i \(-0.526569\pi\)
−0.0833715 + 0.996519i \(0.526569\pi\)
\(840\) 0 0
\(841\) −2.10973 −0.0727493
\(842\) −0.942543 −0.0324822
\(843\) −2.09949 −0.0723102
\(844\) 0.303519 0.0104476
\(845\) −15.0356 −0.517239
\(846\) 7.22323 0.248340
\(847\) 0 0
\(848\) 4.54274 0.155998
\(849\) 20.3475 0.698325
\(850\) 5.56232 0.190786
\(851\) −1.03764 −0.0355700
\(852\) 14.1856 0.485990
\(853\) 15.3330 0.524990 0.262495 0.964933i \(-0.415455\pi\)
0.262495 + 0.964933i \(0.415455\pi\)
\(854\) 0 0
\(855\) −13.4088 −0.458572
\(856\) −5.50127 −0.188030
\(857\) −37.6025 −1.28448 −0.642239 0.766505i \(-0.721994\pi\)
−0.642239 + 0.766505i \(0.721994\pi\)
\(858\) −9.06566 −0.309497
\(859\) 37.1257 1.26671 0.633356 0.773861i \(-0.281677\pi\)
0.633356 + 0.773861i \(0.281677\pi\)
\(860\) 9.80216 0.334251
\(861\) 0 0
\(862\) −35.3697 −1.20470
\(863\) 33.1461 1.12831 0.564154 0.825670i \(-0.309202\pi\)
0.564154 + 0.825670i \(0.309202\pi\)
\(864\) 1.00000 0.0340207
\(865\) −25.0218 −0.850767
\(866\) 32.7627 1.11332
\(867\) −5.72950 −0.194584
\(868\) 0 0
\(869\) −21.5095 −0.729659
\(870\) 9.48146 0.321452
\(871\) −29.2977 −0.992715
\(872\) −18.6951 −0.633095
\(873\) −4.47127 −0.151330
\(874\) −7.33352 −0.248060
\(875\) 0 0
\(876\) 11.5191 0.389194
\(877\) 32.8603 1.10961 0.554806 0.831980i \(-0.312792\pi\)
0.554806 + 0.831980i \(0.312792\pi\)
\(878\) −19.3163 −0.651895
\(879\) −31.4853 −1.06197
\(880\) 7.58420 0.255663
\(881\) 10.6907 0.360178 0.180089 0.983650i \(-0.442361\pi\)
0.180089 + 0.983650i \(0.442361\pi\)
\(882\) 0 0
\(883\) 5.83725 0.196439 0.0982196 0.995165i \(-0.468685\pi\)
0.0982196 + 0.995165i \(0.468685\pi\)
\(884\) 7.33734 0.246782
\(885\) −0.861685 −0.0289652
\(886\) 14.4088 0.484073
\(887\) 54.0917 1.81622 0.908111 0.418730i \(-0.137525\pi\)
0.908111 + 0.418730i \(0.137525\pi\)
\(888\) −1.03764 −0.0348211
\(889\) 0 0
\(890\) 12.2404 0.410299
\(891\) 4.14794 0.138961
\(892\) −16.9375 −0.567111
\(893\) −52.9717 −1.77263
\(894\) 19.3577 0.647419
\(895\) −5.07713 −0.169710
\(896\) 0 0
\(897\) −2.18558 −0.0729745
\(898\) −11.0580 −0.369011
\(899\) 54.5477 1.81927
\(900\) −1.65685 −0.0552285
\(901\) −15.2507 −0.508074
\(902\) 30.4190 1.01284
\(903\) 0 0
\(904\) 5.28187 0.175672
\(905\) −38.3292 −1.27410
\(906\) 5.58420 0.185523
\(907\) −13.1289 −0.435939 −0.217969 0.975956i \(-0.569943\pi\)
−0.217969 + 0.975956i \(0.569943\pi\)
\(908\) −21.5369 −0.714728
\(909\) 15.3061 0.507670
\(910\) 0 0
\(911\) 2.77432 0.0919172 0.0459586 0.998943i \(-0.485366\pi\)
0.0459586 + 0.998943i \(0.485366\pi\)
\(912\) −7.33352 −0.242837
\(913\) 24.0779 0.796863
\(914\) −25.8450 −0.854876
\(915\) −26.9553 −0.891114
\(916\) −0.666478 −0.0220211
\(917\) 0 0
\(918\) −3.35716 −0.110803
\(919\) −21.1868 −0.698887 −0.349443 0.936957i \(-0.613629\pi\)
−0.349443 + 0.936957i \(0.613629\pi\)
\(920\) 1.82843 0.0602815
\(921\) 1.64865 0.0543248
\(922\) −39.3341 −1.29540
\(923\) −31.0038 −1.02050
\(924\) 0 0
\(925\) 1.71923 0.0565278
\(926\) 25.0778 0.824109
\(927\) 7.40499 0.243212
\(928\) 5.18558 0.170225
\(929\) −36.0478 −1.18269 −0.591346 0.806418i \(-0.701403\pi\)
−0.591346 + 0.806418i \(0.701403\pi\)
\(930\) 19.2334 0.630689
\(931\) 0 0
\(932\) −1.67667 −0.0549211
\(933\) 12.1779 0.398688
\(934\) −22.9629 −0.751370
\(935\) −25.4614 −0.832675
\(936\) −2.18558 −0.0714380
\(937\) −14.4789 −0.473005 −0.236503 0.971631i \(-0.576001\pi\)
−0.236503 + 0.971631i \(0.576001\pi\)
\(938\) 0 0
\(939\) 12.5669 0.410107
\(940\) 13.2071 0.430770
\(941\) −40.2996 −1.31373 −0.656865 0.754008i \(-0.728118\pi\)
−0.656865 + 0.754008i \(0.728118\pi\)
\(942\) −22.7902 −0.742544
\(943\) 7.33352 0.238812
\(944\) −0.471271 −0.0153386
\(945\) 0 0
\(946\) 22.2370 0.722987
\(947\) −55.4602 −1.80222 −0.901108 0.433595i \(-0.857245\pi\)
−0.901108 + 0.433595i \(0.857245\pi\)
\(948\) −5.18558 −0.168420
\(949\) −25.1760 −0.817246
\(950\) 12.1506 0.394217
\(951\) −9.17794 −0.297615
\(952\) 0 0
\(953\) −42.0164 −1.36105 −0.680523 0.732727i \(-0.738247\pi\)
−0.680523 + 0.732727i \(0.738247\pi\)
\(954\) 4.54274 0.147077
\(955\) −16.3870 −0.530271
\(956\) 10.7220 0.346773
\(957\) 21.5095 0.695303
\(958\) −16.6116 −0.536695
\(959\) 0 0
\(960\) 1.82843 0.0590122
\(961\) 79.6516 2.56941
\(962\) 2.26786 0.0731187
\(963\) −5.50127 −0.177276
\(964\) 18.2034 0.586292
\(965\) 21.2811 0.685063
\(966\) 0 0
\(967\) 6.71167 0.215833 0.107916 0.994160i \(-0.465582\pi\)
0.107916 + 0.994160i \(0.465582\pi\)
\(968\) 6.20540 0.199449
\(969\) 24.6198 0.790901
\(970\) −8.17539 −0.262496
\(971\) 37.4616 1.20220 0.601101 0.799173i \(-0.294729\pi\)
0.601101 + 0.799173i \(0.294729\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −14.8226 −0.474948
\(975\) 3.62119 0.115971
\(976\) −14.7423 −0.471891
\(977\) 53.2632 1.70404 0.852020 0.523509i \(-0.175377\pi\)
0.852020 + 0.523509i \(0.175377\pi\)
\(978\) −25.4368 −0.813380
\(979\) 27.7684 0.887481
\(980\) 0 0
\(981\) −18.6951 −0.596887
\(982\) 22.2466 0.709918
\(983\) −15.9049 −0.507287 −0.253644 0.967298i \(-0.581629\pi\)
−0.253644 + 0.967298i \(0.581629\pi\)
\(984\) 7.33352 0.233784
\(985\) 17.5192 0.558208
\(986\) −17.4088 −0.554409
\(987\) 0 0
\(988\) 16.0280 0.509919
\(989\) 5.36098 0.170469
\(990\) 7.58420 0.241042
\(991\) 15.5969 0.495453 0.247727 0.968830i \(-0.420317\pi\)
0.247727 + 0.968830i \(0.420317\pi\)
\(992\) 10.5191 0.333982
\(993\) −29.3896 −0.932650
\(994\) 0 0
\(995\) −18.1978 −0.576910
\(996\) 5.80479 0.183932
\(997\) −8.75450 −0.277258 −0.138629 0.990344i \(-0.544270\pi\)
−0.138629 + 0.990344i \(0.544270\pi\)
\(998\) −11.2857 −0.357242
\(999\) −1.03764 −0.0328296
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 6762.2.a.cp.1.4 4
7.2 even 3 966.2.i.l.277.1 8
7.4 even 3 966.2.i.l.415.1 yes 8
7.6 odd 2 6762.2.a.cm.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.l.277.1 8 7.2 even 3
966.2.i.l.415.1 yes 8 7.4 even 3
6762.2.a.cm.1.2 4 7.6 odd 2
6762.2.a.cp.1.4 4 1.1 even 1 trivial