Properties

Label 966.2.a.n.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.70156 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.70156 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.70156 q^{10} -4.00000 q^{11} +1.00000 q^{12} -0.701562 q^{13} -1.00000 q^{14} -2.70156 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +7.40312 q^{19} -2.70156 q^{20} +1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +2.29844 q^{25} +0.701562 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.70156 q^{29} +2.70156 q^{30} -2.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -4.00000 q^{34} -2.70156 q^{35} +1.00000 q^{36} +10.7016 q^{37} -7.40312 q^{38} -0.701562 q^{39} +2.70156 q^{40} -6.70156 q^{41} -1.00000 q^{42} +4.70156 q^{43} -4.00000 q^{44} -2.70156 q^{45} -1.00000 q^{46} +8.10469 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.29844 q^{50} +4.00000 q^{51} -0.701562 q^{52} +3.40312 q^{53} -1.00000 q^{54} +10.8062 q^{55} -1.00000 q^{56} +7.40312 q^{57} -6.70156 q^{58} +5.40312 q^{59} -2.70156 q^{60} -2.00000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.89531 q^{65} +4.00000 q^{66} -10.8062 q^{67} +4.00000 q^{68} +1.00000 q^{69} +2.70156 q^{70} +5.40312 q^{71} -1.00000 q^{72} -11.4031 q^{73} -10.7016 q^{74} +2.29844 q^{75} +7.40312 q^{76} -4.00000 q^{77} +0.701562 q^{78} +8.00000 q^{79} -2.70156 q^{80} +1.00000 q^{81} +6.70156 q^{82} +0.596876 q^{83} +1.00000 q^{84} -10.8062 q^{85} -4.70156 q^{86} +6.70156 q^{87} +4.00000 q^{88} +1.40312 q^{89} +2.70156 q^{90} -0.701562 q^{91} +1.00000 q^{92} -2.00000 q^{93} -8.10469 q^{94} -20.0000 q^{95} -1.00000 q^{96} +12.7016 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - q^{10} - 8q^{11} + 2q^{12} + 5q^{13} - 2q^{14} + q^{15} + 2q^{16} + 8q^{17} - 2q^{18} + 2q^{19} + q^{20} + 2q^{21} + 8q^{22} + 2q^{23} - 2q^{24} + 11q^{25} - 5q^{26} + 2q^{27} + 2q^{28} + 7q^{29} - q^{30} - 4q^{31} - 2q^{32} - 8q^{33} - 8q^{34} + q^{35} + 2q^{36} + 15q^{37} - 2q^{38} + 5q^{39} - q^{40} - 7q^{41} - 2q^{42} + 3q^{43} - 8q^{44} + q^{45} - 2q^{46} - 3q^{47} + 2q^{48} + 2q^{49} - 11q^{50} + 8q^{51} + 5q^{52} - 6q^{53} - 2q^{54} - 4q^{55} - 2q^{56} + 2q^{57} - 7q^{58} - 2q^{59} + q^{60} - 4q^{61} + 4q^{62} + 2q^{63} + 2q^{64} + 23q^{65} + 8q^{66} + 4q^{67} + 8q^{68} + 2q^{69} - q^{70} - 2q^{71} - 2q^{72} - 10q^{73} - 15q^{74} + 11q^{75} + 2q^{76} - 8q^{77} - 5q^{78} + 16q^{79} + q^{80} + 2q^{81} + 7q^{82} + 14q^{83} + 2q^{84} + 4q^{85} - 3q^{86} + 7q^{87} + 8q^{88} - 10q^{89} - q^{90} + 5q^{91} + 2q^{92} - 4q^{93} + 3q^{94} - 40q^{95} - 2q^{96} + 19q^{97} - 2q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) −2.70156 −1.20818 −0.604088 0.796918i \(-0.706462\pi\)
−0.604088 + 0.796918i \(0.706462\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.70156 0.854309
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −0.701562 −0.194578 −0.0972892 0.995256i \(-0.531017\pi\)
−0.0972892 + 0.995256i \(0.531017\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.70156 −0.697540
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.40312 1.69839 0.849197 0.528077i \(-0.177087\pi\)
0.849197 + 0.528077i \(0.177087\pi\)
\(20\) −2.70156 −0.604088
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 2.29844 0.459688
\(26\) 0.701562 0.137588
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 6.70156 1.24445 0.622224 0.782839i \(-0.286229\pi\)
0.622224 + 0.782839i \(0.286229\pi\)
\(30\) 2.70156 0.493236
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −4.00000 −0.685994
\(35\) −2.70156 −0.456647
\(36\) 1.00000 0.166667
\(37\) 10.7016 1.75933 0.879663 0.475598i \(-0.157768\pi\)
0.879663 + 0.475598i \(0.157768\pi\)
\(38\) −7.40312 −1.20095
\(39\) −0.701562 −0.112340
\(40\) 2.70156 0.427154
\(41\) −6.70156 −1.04661 −0.523304 0.852146i \(-0.675301\pi\)
−0.523304 + 0.852146i \(0.675301\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.70156 0.716982 0.358491 0.933533i \(-0.383291\pi\)
0.358491 + 0.933533i \(0.383291\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.70156 −0.402725
\(46\) −1.00000 −0.147442
\(47\) 8.10469 1.18219 0.591095 0.806602i \(-0.298696\pi\)
0.591095 + 0.806602i \(0.298696\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −2.29844 −0.325048
\(51\) 4.00000 0.560112
\(52\) −0.701562 −0.0972892
\(53\) 3.40312 0.467455 0.233728 0.972302i \(-0.424908\pi\)
0.233728 + 0.972302i \(0.424908\pi\)
\(54\) −1.00000 −0.136083
\(55\) 10.8062 1.45711
\(56\) −1.00000 −0.133631
\(57\) 7.40312 0.980568
\(58\) −6.70156 −0.879958
\(59\) 5.40312 0.703427 0.351713 0.936108i \(-0.385599\pi\)
0.351713 + 0.936108i \(0.385599\pi\)
\(60\) −2.70156 −0.348770
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 1.89531 0.235085
\(66\) 4.00000 0.492366
\(67\) −10.8062 −1.32019 −0.660097 0.751181i \(-0.729485\pi\)
−0.660097 + 0.751181i \(0.729485\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 2.70156 0.322898
\(71\) 5.40312 0.641233 0.320616 0.947209i \(-0.396110\pi\)
0.320616 + 0.947209i \(0.396110\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.4031 −1.33463 −0.667317 0.744773i \(-0.732558\pi\)
−0.667317 + 0.744773i \(0.732558\pi\)
\(74\) −10.7016 −1.24403
\(75\) 2.29844 0.265401
\(76\) 7.40312 0.849197
\(77\) −4.00000 −0.455842
\(78\) 0.701562 0.0794363
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.70156 −0.302044
\(81\) 1.00000 0.111111
\(82\) 6.70156 0.740064
\(83\) 0.596876 0.0655156 0.0327578 0.999463i \(-0.489571\pi\)
0.0327578 + 0.999463i \(0.489571\pi\)
\(84\) 1.00000 0.109109
\(85\) −10.8062 −1.17210
\(86\) −4.70156 −0.506982
\(87\) 6.70156 0.718483
\(88\) 4.00000 0.426401
\(89\) 1.40312 0.148731 0.0743654 0.997231i \(-0.476307\pi\)
0.0743654 + 0.997231i \(0.476307\pi\)
\(90\) 2.70156 0.284770
\(91\) −0.701562 −0.0735437
\(92\) 1.00000 0.104257
\(93\) −2.00000 −0.207390
\(94\) −8.10469 −0.835935
\(95\) −20.0000 −2.05196
\(96\) −1.00000 −0.102062
\(97\) 12.7016 1.28965 0.644824 0.764331i \(-0.276931\pi\)
0.644824 + 0.764331i \(0.276931\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
\(100\) 2.29844 0.229844
\(101\) 9.40312 0.935646 0.467823 0.883822i \(-0.345039\pi\)
0.467823 + 0.883822i \(0.345039\pi\)
\(102\) −4.00000 −0.396059
\(103\) 8.70156 0.857390 0.428695 0.903449i \(-0.358973\pi\)
0.428695 + 0.903449i \(0.358973\pi\)
\(104\) 0.701562 0.0687938
\(105\) −2.70156 −0.263645
\(106\) −3.40312 −0.330541
\(107\) −6.80625 −0.657985 −0.328992 0.944333i \(-0.606709\pi\)
−0.328992 + 0.944333i \(0.606709\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.29844 −0.124368 −0.0621839 0.998065i \(-0.519807\pi\)
−0.0621839 + 0.998065i \(0.519807\pi\)
\(110\) −10.8062 −1.03034
\(111\) 10.7016 1.01575
\(112\) 1.00000 0.0944911
\(113\) −12.1047 −1.13871 −0.569357 0.822091i \(-0.692808\pi\)
−0.569357 + 0.822091i \(0.692808\pi\)
\(114\) −7.40312 −0.693366
\(115\) −2.70156 −0.251922
\(116\) 6.70156 0.622224
\(117\) −0.701562 −0.0648594
\(118\) −5.40312 −0.497398
\(119\) 4.00000 0.366679
\(120\) 2.70156 0.246618
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) −6.70156 −0.604260
\(124\) −2.00000 −0.179605
\(125\) 7.29844 0.652792
\(126\) −1.00000 −0.0890871
\(127\) 0.701562 0.0622536 0.0311268 0.999515i \(-0.490090\pi\)
0.0311268 + 0.999515i \(0.490090\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.70156 0.413949
\(130\) −1.89531 −0.166230
\(131\) 5.40312 0.472073 0.236037 0.971744i \(-0.424151\pi\)
0.236037 + 0.971744i \(0.424151\pi\)
\(132\) −4.00000 −0.348155
\(133\) 7.40312 0.641932
\(134\) 10.8062 0.933518
\(135\) −2.70156 −0.232513
\(136\) −4.00000 −0.342997
\(137\) −13.2984 −1.13616 −0.568081 0.822973i \(-0.692314\pi\)
−0.568081 + 0.822973i \(0.692314\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 11.2984 0.958321 0.479160 0.877727i \(-0.340941\pi\)
0.479160 + 0.877727i \(0.340941\pi\)
\(140\) −2.70156 −0.228324
\(141\) 8.10469 0.682538
\(142\) −5.40312 −0.453420
\(143\) 2.80625 0.234670
\(144\) 1.00000 0.0833333
\(145\) −18.1047 −1.50351
\(146\) 11.4031 0.943729
\(147\) 1.00000 0.0824786
\(148\) 10.7016 0.879663
\(149\) 18.2094 1.49177 0.745885 0.666075i \(-0.232027\pi\)
0.745885 + 0.666075i \(0.232027\pi\)
\(150\) −2.29844 −0.187667
\(151\) −20.9109 −1.70171 −0.850854 0.525402i \(-0.823915\pi\)
−0.850854 + 0.525402i \(0.823915\pi\)
\(152\) −7.40312 −0.600473
\(153\) 4.00000 0.323381
\(154\) 4.00000 0.322329
\(155\) 5.40312 0.433989
\(156\) −0.701562 −0.0561699
\(157\) 11.4031 0.910068 0.455034 0.890474i \(-0.349627\pi\)
0.455034 + 0.890474i \(0.349627\pi\)
\(158\) −8.00000 −0.636446
\(159\) 3.40312 0.269885
\(160\) 2.70156 0.213577
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 14.8062 1.15971 0.579857 0.814718i \(-0.303108\pi\)
0.579857 + 0.814718i \(0.303108\pi\)
\(164\) −6.70156 −0.523304
\(165\) 10.8062 0.841265
\(166\) −0.596876 −0.0463265
\(167\) −0.596876 −0.0461876 −0.0230938 0.999733i \(-0.507352\pi\)
−0.0230938 + 0.999733i \(0.507352\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −12.5078 −0.962139
\(170\) 10.8062 0.828801
\(171\) 7.40312 0.566131
\(172\) 4.70156 0.358491
\(173\) 6.59688 0.501551 0.250776 0.968045i \(-0.419314\pi\)
0.250776 + 0.968045i \(0.419314\pi\)
\(174\) −6.70156 −0.508044
\(175\) 2.29844 0.173746
\(176\) −4.00000 −0.301511
\(177\) 5.40312 0.406124
\(178\) −1.40312 −0.105169
\(179\) −5.89531 −0.440636 −0.220318 0.975428i \(-0.570710\pi\)
−0.220318 + 0.975428i \(0.570710\pi\)
\(180\) −2.70156 −0.201363
\(181\) 8.80625 0.654563 0.327282 0.944927i \(-0.393867\pi\)
0.327282 + 0.944927i \(0.393867\pi\)
\(182\) 0.701562 0.0520032
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) −28.9109 −2.12557
\(186\) 2.00000 0.146647
\(187\) −16.0000 −1.17004
\(188\) 8.10469 0.591095
\(189\) 1.00000 0.0727393
\(190\) 20.0000 1.45095
\(191\) −2.80625 −0.203053 −0.101527 0.994833i \(-0.532373\pi\)
−0.101527 + 0.994833i \(0.532373\pi\)
\(192\) 1.00000 0.0721688
\(193\) −3.89531 −0.280391 −0.140195 0.990124i \(-0.544773\pi\)
−0.140195 + 0.990124i \(0.544773\pi\)
\(194\) −12.7016 −0.911919
\(195\) 1.89531 0.135726
\(196\) 1.00000 0.0714286
\(197\) −9.50781 −0.677403 −0.338702 0.940894i \(-0.609988\pi\)
−0.338702 + 0.940894i \(0.609988\pi\)
\(198\) 4.00000 0.284268
\(199\) −8.70156 −0.616837 −0.308419 0.951251i \(-0.599800\pi\)
−0.308419 + 0.951251i \(0.599800\pi\)
\(200\) −2.29844 −0.162524
\(201\) −10.8062 −0.762214
\(202\) −9.40312 −0.661602
\(203\) 6.70156 0.470357
\(204\) 4.00000 0.280056
\(205\) 18.1047 1.26449
\(206\) −8.70156 −0.606267
\(207\) 1.00000 0.0695048
\(208\) −0.701562 −0.0486446
\(209\) −29.6125 −2.04834
\(210\) 2.70156 0.186425
\(211\) −14.8062 −1.01930 −0.509652 0.860381i \(-0.670226\pi\)
−0.509652 + 0.860381i \(0.670226\pi\)
\(212\) 3.40312 0.233728
\(213\) 5.40312 0.370216
\(214\) 6.80625 0.465266
\(215\) −12.7016 −0.866239
\(216\) −1.00000 −0.0680414
\(217\) −2.00000 −0.135769
\(218\) 1.29844 0.0879413
\(219\) −11.4031 −0.770552
\(220\) 10.8062 0.728557
\(221\) −2.80625 −0.188769
\(222\) −10.7016 −0.718242
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.29844 0.153229
\(226\) 12.1047 0.805192
\(227\) −3.89531 −0.258541 −0.129271 0.991609i \(-0.541264\pi\)
−0.129271 + 0.991609i \(0.541264\pi\)
\(228\) 7.40312 0.490284
\(229\) −27.4031 −1.81085 −0.905425 0.424507i \(-0.860447\pi\)
−0.905425 + 0.424507i \(0.860447\pi\)
\(230\) 2.70156 0.178136
\(231\) −4.00000 −0.263181
\(232\) −6.70156 −0.439979
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0.701562 0.0458626
\(235\) −21.8953 −1.42829
\(236\) 5.40312 0.351713
\(237\) 8.00000 0.519656
\(238\) −4.00000 −0.259281
\(239\) −14.8062 −0.957737 −0.478868 0.877887i \(-0.658953\pi\)
−0.478868 + 0.877887i \(0.658953\pi\)
\(240\) −2.70156 −0.174385
\(241\) 28.9109 1.86232 0.931159 0.364615i \(-0.118799\pi\)
0.931159 + 0.364615i \(0.118799\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −2.70156 −0.172596
\(246\) 6.70156 0.427276
\(247\) −5.19375 −0.330470
\(248\) 2.00000 0.127000
\(249\) 0.596876 0.0378255
\(250\) −7.29844 −0.461594
\(251\) 13.2984 0.839390 0.419695 0.907665i \(-0.362137\pi\)
0.419695 + 0.907665i \(0.362137\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) −0.701562 −0.0440199
\(255\) −10.8062 −0.676714
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −4.70156 −0.292706
\(259\) 10.7016 0.664963
\(260\) 1.89531 0.117542
\(261\) 6.70156 0.414816
\(262\) −5.40312 −0.333806
\(263\) 3.29844 0.203390 0.101695 0.994816i \(-0.467573\pi\)
0.101695 + 0.994816i \(0.467573\pi\)
\(264\) 4.00000 0.246183
\(265\) −9.19375 −0.564768
\(266\) −7.40312 −0.453915
\(267\) 1.40312 0.0858698
\(268\) −10.8062 −0.660097
\(269\) 28.2094 1.71996 0.859978 0.510331i \(-0.170477\pi\)
0.859978 + 0.510331i \(0.170477\pi\)
\(270\) 2.70156 0.164412
\(271\) −11.4031 −0.692690 −0.346345 0.938107i \(-0.612577\pi\)
−0.346345 + 0.938107i \(0.612577\pi\)
\(272\) 4.00000 0.242536
\(273\) −0.701562 −0.0424605
\(274\) 13.2984 0.803388
\(275\) −9.19375 −0.554404
\(276\) 1.00000 0.0601929
\(277\) −8.59688 −0.516536 −0.258268 0.966073i \(-0.583152\pi\)
−0.258268 + 0.966073i \(0.583152\pi\)
\(278\) −11.2984 −0.677635
\(279\) −2.00000 −0.119737
\(280\) 2.70156 0.161449
\(281\) −17.5078 −1.04443 −0.522214 0.852814i \(-0.674894\pi\)
−0.522214 + 0.852814i \(0.674894\pi\)
\(282\) −8.10469 −0.482627
\(283\) −20.8062 −1.23680 −0.618402 0.785862i \(-0.712219\pi\)
−0.618402 + 0.785862i \(0.712219\pi\)
\(284\) 5.40312 0.320616
\(285\) −20.0000 −1.18470
\(286\) −2.80625 −0.165937
\(287\) −6.70156 −0.395581
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 18.1047 1.06314
\(291\) 12.7016 0.744579
\(292\) −11.4031 −0.667317
\(293\) 4.80625 0.280784 0.140392 0.990096i \(-0.455164\pi\)
0.140392 + 0.990096i \(0.455164\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −14.5969 −0.849863
\(296\) −10.7016 −0.622016
\(297\) −4.00000 −0.232104
\(298\) −18.2094 −1.05484
\(299\) −0.701562 −0.0405724
\(300\) 2.29844 0.132700
\(301\) 4.70156 0.270994
\(302\) 20.9109 1.20329
\(303\) 9.40312 0.540195
\(304\) 7.40312 0.424598
\(305\) 5.40312 0.309382
\(306\) −4.00000 −0.228665
\(307\) −32.9109 −1.87833 −0.939163 0.343471i \(-0.888397\pi\)
−0.939163 + 0.343471i \(0.888397\pi\)
\(308\) −4.00000 −0.227921
\(309\) 8.70156 0.495015
\(310\) −5.40312 −0.306877
\(311\) 0.596876 0.0338457 0.0169229 0.999857i \(-0.494613\pi\)
0.0169229 + 0.999857i \(0.494613\pi\)
\(312\) 0.701562 0.0397181
\(313\) 32.2094 1.82058 0.910291 0.413970i \(-0.135858\pi\)
0.910291 + 0.413970i \(0.135858\pi\)
\(314\) −11.4031 −0.643516
\(315\) −2.70156 −0.152216
\(316\) 8.00000 0.450035
\(317\) −14.7016 −0.825722 −0.412861 0.910794i \(-0.635470\pi\)
−0.412861 + 0.910794i \(0.635470\pi\)
\(318\) −3.40312 −0.190838
\(319\) −26.8062 −1.50086
\(320\) −2.70156 −0.151022
\(321\) −6.80625 −0.379888
\(322\) −1.00000 −0.0557278
\(323\) 29.6125 1.64768
\(324\) 1.00000 0.0555556
\(325\) −1.61250 −0.0894452
\(326\) −14.8062 −0.820042
\(327\) −1.29844 −0.0718038
\(328\) 6.70156 0.370032
\(329\) 8.10469 0.446826
\(330\) −10.8062 −0.594864
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0.596876 0.0327578
\(333\) 10.7016 0.586442
\(334\) 0.596876 0.0326596
\(335\) 29.1938 1.59503
\(336\) 1.00000 0.0545545
\(337\) 34.2094 1.86350 0.931752 0.363096i \(-0.118280\pi\)
0.931752 + 0.363096i \(0.118280\pi\)
\(338\) 12.5078 0.680335
\(339\) −12.1047 −0.657436
\(340\) −10.8062 −0.586051
\(341\) 8.00000 0.433224
\(342\) −7.40312 −0.400315
\(343\) 1.00000 0.0539949
\(344\) −4.70156 −0.253491
\(345\) −2.70156 −0.145447
\(346\) −6.59688 −0.354650
\(347\) −16.7016 −0.896587 −0.448293 0.893886i \(-0.647968\pi\)
−0.448293 + 0.893886i \(0.647968\pi\)
\(348\) 6.70156 0.359241
\(349\) 25.4031 1.35980 0.679899 0.733306i \(-0.262024\pi\)
0.679899 + 0.733306i \(0.262024\pi\)
\(350\) −2.29844 −0.122857
\(351\) −0.701562 −0.0374466
\(352\) 4.00000 0.213201
\(353\) 10.7016 0.569587 0.284793 0.958589i \(-0.408075\pi\)
0.284793 + 0.958589i \(0.408075\pi\)
\(354\) −5.40312 −0.287173
\(355\) −14.5969 −0.774722
\(356\) 1.40312 0.0743654
\(357\) 4.00000 0.211702
\(358\) 5.89531 0.311577
\(359\) −22.1047 −1.16664 −0.583320 0.812242i \(-0.698247\pi\)
−0.583320 + 0.812242i \(0.698247\pi\)
\(360\) 2.70156 0.142385
\(361\) 35.8062 1.88454
\(362\) −8.80625 −0.462846
\(363\) 5.00000 0.262432
\(364\) −0.701562 −0.0367718
\(365\) 30.8062 1.61247
\(366\) 2.00000 0.104542
\(367\) 2.10469 0.109864 0.0549319 0.998490i \(-0.482506\pi\)
0.0549319 + 0.998490i \(0.482506\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.70156 −0.348869
\(370\) 28.9109 1.50301
\(371\) 3.40312 0.176681
\(372\) −2.00000 −0.103695
\(373\) 24.8062 1.28442 0.642209 0.766529i \(-0.278018\pi\)
0.642209 + 0.766529i \(0.278018\pi\)
\(374\) 16.0000 0.827340
\(375\) 7.29844 0.376890
\(376\) −8.10469 −0.417967
\(377\) −4.70156 −0.242143
\(378\) −1.00000 −0.0514344
\(379\) 7.29844 0.374896 0.187448 0.982275i \(-0.439978\pi\)
0.187448 + 0.982275i \(0.439978\pi\)
\(380\) −20.0000 −1.02598
\(381\) 0.701562 0.0359421
\(382\) 2.80625 0.143580
\(383\) −21.6125 −1.10435 −0.552174 0.833729i \(-0.686201\pi\)
−0.552174 + 0.833729i \(0.686201\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 10.8062 0.550737
\(386\) 3.89531 0.198266
\(387\) 4.70156 0.238994
\(388\) 12.7016 0.644824
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −1.89531 −0.0959729
\(391\) 4.00000 0.202289
\(392\) −1.00000 −0.0505076
\(393\) 5.40312 0.272552
\(394\) 9.50781 0.478997
\(395\) −21.6125 −1.08744
\(396\) −4.00000 −0.201008
\(397\) 21.4031 1.07419 0.537096 0.843521i \(-0.319521\pi\)
0.537096 + 0.843521i \(0.319521\pi\)
\(398\) 8.70156 0.436170
\(399\) 7.40312 0.370620
\(400\) 2.29844 0.114922
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 10.8062 0.538967
\(403\) 1.40312 0.0698946
\(404\) 9.40312 0.467823
\(405\) −2.70156 −0.134242
\(406\) −6.70156 −0.332593
\(407\) −42.8062 −2.12183
\(408\) −4.00000 −0.198030
\(409\) −7.40312 −0.366061 −0.183030 0.983107i \(-0.558591\pi\)
−0.183030 + 0.983107i \(0.558591\pi\)
\(410\) −18.1047 −0.894127
\(411\) −13.2984 −0.655964
\(412\) 8.70156 0.428695
\(413\) 5.40312 0.265870
\(414\) −1.00000 −0.0491473
\(415\) −1.61250 −0.0791544
\(416\) 0.701562 0.0343969
\(417\) 11.2984 0.553287
\(418\) 29.6125 1.44839
\(419\) 38.2094 1.86665 0.933325 0.359033i \(-0.116893\pi\)
0.933325 + 0.359033i \(0.116893\pi\)
\(420\) −2.70156 −0.131823
\(421\) −9.29844 −0.453178 −0.226589 0.973990i \(-0.572757\pi\)
−0.226589 + 0.973990i \(0.572757\pi\)
\(422\) 14.8062 0.720757
\(423\) 8.10469 0.394063
\(424\) −3.40312 −0.165270
\(425\) 9.19375 0.445962
\(426\) −5.40312 −0.261782
\(427\) −2.00000 −0.0967868
\(428\) −6.80625 −0.328992
\(429\) 2.80625 0.135487
\(430\) 12.7016 0.612524
\(431\) 40.9109 1.97061 0.985305 0.170803i \(-0.0546362\pi\)
0.985305 + 0.170803i \(0.0546362\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.3141 −1.26457 −0.632286 0.774735i \(-0.717883\pi\)
−0.632286 + 0.774735i \(0.717883\pi\)
\(434\) 2.00000 0.0960031
\(435\) −18.1047 −0.868053
\(436\) −1.29844 −0.0621839
\(437\) 7.40312 0.354139
\(438\) 11.4031 0.544862
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) −10.8062 −0.515168
\(441\) 1.00000 0.0476190
\(442\) 2.80625 0.133480
\(443\) −7.29844 −0.346759 −0.173380 0.984855i \(-0.555469\pi\)
−0.173380 + 0.984855i \(0.555469\pi\)
\(444\) 10.7016 0.507874
\(445\) −3.79063 −0.179693
\(446\) 22.0000 1.04173
\(447\) 18.2094 0.861274
\(448\) 1.00000 0.0472456
\(449\) 38.4187 1.81309 0.906546 0.422106i \(-0.138709\pi\)
0.906546 + 0.422106i \(0.138709\pi\)
\(450\) −2.29844 −0.108349
\(451\) 26.8062 1.26226
\(452\) −12.1047 −0.569357
\(453\) −20.9109 −0.982481
\(454\) 3.89531 0.182816
\(455\) 1.89531 0.0888537
\(456\) −7.40312 −0.346683
\(457\) −10.2094 −0.477574 −0.238787 0.971072i \(-0.576750\pi\)
−0.238787 + 0.971072i \(0.576750\pi\)
\(458\) 27.4031 1.28046
\(459\) 4.00000 0.186704
\(460\) −2.70156 −0.125961
\(461\) 36.2094 1.68644 0.843219 0.537570i \(-0.180658\pi\)
0.843219 + 0.537570i \(0.180658\pi\)
\(462\) 4.00000 0.186097
\(463\) −28.7016 −1.33387 −0.666937 0.745114i \(-0.732395\pi\)
−0.666937 + 0.745114i \(0.732395\pi\)
\(464\) 6.70156 0.311112
\(465\) 5.40312 0.250564
\(466\) 26.0000 1.20443
\(467\) 2.70156 0.125013 0.0625067 0.998045i \(-0.480091\pi\)
0.0625067 + 0.998045i \(0.480091\pi\)
\(468\) −0.701562 −0.0324297
\(469\) −10.8062 −0.498986
\(470\) 21.8953 1.00996
\(471\) 11.4031 0.525428
\(472\) −5.40312 −0.248699
\(473\) −18.8062 −0.864712
\(474\) −8.00000 −0.367452
\(475\) 17.0156 0.780730
\(476\) 4.00000 0.183340
\(477\) 3.40312 0.155818
\(478\) 14.8062 0.677222
\(479\) 29.4031 1.34346 0.671732 0.740795i \(-0.265551\pi\)
0.671732 + 0.740795i \(0.265551\pi\)
\(480\) 2.70156 0.123309
\(481\) −7.50781 −0.342327
\(482\) −28.9109 −1.31686
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) −34.3141 −1.55812
\(486\) −1.00000 −0.0453609
\(487\) −12.7016 −0.575563 −0.287781 0.957696i \(-0.592918\pi\)
−0.287781 + 0.957696i \(0.592918\pi\)
\(488\) 2.00000 0.0905357
\(489\) 14.8062 0.669562
\(490\) 2.70156 0.122044
\(491\) 41.6125 1.87795 0.938973 0.343991i \(-0.111779\pi\)
0.938973 + 0.343991i \(0.111779\pi\)
\(492\) −6.70156 −0.302130
\(493\) 26.8062 1.20729
\(494\) 5.19375 0.233678
\(495\) 10.8062 0.485705
\(496\) −2.00000 −0.0898027
\(497\) 5.40312 0.242363
\(498\) −0.596876 −0.0267466
\(499\) −10.5969 −0.474381 −0.237191 0.971463i \(-0.576227\pi\)
−0.237191 + 0.971463i \(0.576227\pi\)
\(500\) 7.29844 0.326396
\(501\) −0.596876 −0.0266664
\(502\) −13.2984 −0.593538
\(503\) 17.4031 0.775967 0.387983 0.921666i \(-0.373172\pi\)
0.387983 + 0.921666i \(0.373172\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −25.4031 −1.13042
\(506\) 4.00000 0.177822
\(507\) −12.5078 −0.555491
\(508\) 0.701562 0.0311268
\(509\) −13.6125 −0.603363 −0.301682 0.953409i \(-0.597548\pi\)
−0.301682 + 0.953409i \(0.597548\pi\)
\(510\) 10.8062 0.478509
\(511\) −11.4031 −0.504445
\(512\) −1.00000 −0.0441942
\(513\) 7.40312 0.326856
\(514\) −2.00000 −0.0882162
\(515\) −23.5078 −1.03588
\(516\) 4.70156 0.206975
\(517\) −32.4187 −1.42577
\(518\) −10.7016 −0.470200
\(519\) 6.59688 0.289571
\(520\) −1.89531 −0.0831150
\(521\) −33.4031 −1.46342 −0.731709 0.681617i \(-0.761277\pi\)
−0.731709 + 0.681617i \(0.761277\pi\)
\(522\) −6.70156 −0.293319
\(523\) −4.80625 −0.210163 −0.105081 0.994464i \(-0.533510\pi\)
−0.105081 + 0.994464i \(0.533510\pi\)
\(524\) 5.40312 0.236037
\(525\) 2.29844 0.100312
\(526\) −3.29844 −0.143819
\(527\) −8.00000 −0.348485
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 9.19375 0.399351
\(531\) 5.40312 0.234476
\(532\) 7.40312 0.320966
\(533\) 4.70156 0.203647
\(534\) −1.40312 −0.0607191
\(535\) 18.3875 0.794961
\(536\) 10.8062 0.466759
\(537\) −5.89531 −0.254402
\(538\) −28.2094 −1.21619
\(539\) −4.00000 −0.172292
\(540\) −2.70156 −0.116257
\(541\) 20.5969 0.885529 0.442764 0.896638i \(-0.353998\pi\)
0.442764 + 0.896638i \(0.353998\pi\)
\(542\) 11.4031 0.489806
\(543\) 8.80625 0.377912
\(544\) −4.00000 −0.171499
\(545\) 3.50781 0.150258
\(546\) 0.701562 0.0300241
\(547\) −35.0156 −1.49716 −0.748580 0.663045i \(-0.769264\pi\)
−0.748580 + 0.663045i \(0.769264\pi\)
\(548\) −13.2984 −0.568081
\(549\) −2.00000 −0.0853579
\(550\) 9.19375 0.392023
\(551\) 49.6125 2.11356
\(552\) −1.00000 −0.0425628
\(553\) 8.00000 0.340195
\(554\) 8.59688 0.365246
\(555\) −28.9109 −1.22720
\(556\) 11.2984 0.479160
\(557\) −24.5969 −1.04220 −0.521102 0.853495i \(-0.674479\pi\)
−0.521102 + 0.853495i \(0.674479\pi\)
\(558\) 2.00000 0.0846668
\(559\) −3.29844 −0.139509
\(560\) −2.70156 −0.114162
\(561\) −16.0000 −0.675521
\(562\) 17.5078 0.738522
\(563\) 16.1047 0.678732 0.339366 0.940654i \(-0.389788\pi\)
0.339366 + 0.940654i \(0.389788\pi\)
\(564\) 8.10469 0.341269
\(565\) 32.7016 1.37577
\(566\) 20.8062 0.874552
\(567\) 1.00000 0.0419961
\(568\) −5.40312 −0.226710
\(569\) 25.2984 1.06057 0.530283 0.847821i \(-0.322086\pi\)
0.530283 + 0.847821i \(0.322086\pi\)
\(570\) 20.0000 0.837708
\(571\) 10.8062 0.452227 0.226114 0.974101i \(-0.427398\pi\)
0.226114 + 0.974101i \(0.427398\pi\)
\(572\) 2.80625 0.117335
\(573\) −2.80625 −0.117233
\(574\) 6.70156 0.279718
\(575\) 2.29844 0.0958515
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 1.00000 0.0415945
\(579\) −3.89531 −0.161884
\(580\) −18.1047 −0.751756
\(581\) 0.596876 0.0247626
\(582\) −12.7016 −0.526497
\(583\) −13.6125 −0.563772
\(584\) 11.4031 0.471865
\(585\) 1.89531 0.0783616
\(586\) −4.80625 −0.198544
\(587\) −10.8062 −0.446022 −0.223011 0.974816i \(-0.571589\pi\)
−0.223011 + 0.974816i \(0.571589\pi\)
\(588\) 1.00000 0.0412393
\(589\) −14.8062 −0.610081
\(590\) 14.5969 0.600944
\(591\) −9.50781 −0.391099
\(592\) 10.7016 0.439831
\(593\) 37.2984 1.53166 0.765832 0.643041i \(-0.222328\pi\)
0.765832 + 0.643041i \(0.222328\pi\)
\(594\) 4.00000 0.164122
\(595\) −10.8062 −0.443013
\(596\) 18.2094 0.745885
\(597\) −8.70156 −0.356131
\(598\) 0.701562 0.0286890
\(599\) −37.6125 −1.53680 −0.768402 0.639967i \(-0.778948\pi\)
−0.768402 + 0.639967i \(0.778948\pi\)
\(600\) −2.29844 −0.0938333
\(601\) −0.596876 −0.0243471 −0.0121735 0.999926i \(-0.503875\pi\)
−0.0121735 + 0.999926i \(0.503875\pi\)
\(602\) −4.70156 −0.191621
\(603\) −10.8062 −0.440064
\(604\) −20.9109 −0.850854
\(605\) −13.5078 −0.549171
\(606\) −9.40312 −0.381976
\(607\) 11.6125 0.471337 0.235668 0.971834i \(-0.424272\pi\)
0.235668 + 0.971834i \(0.424272\pi\)
\(608\) −7.40312 −0.300236
\(609\) 6.70156 0.271561
\(610\) −5.40312 −0.218766
\(611\) −5.68594 −0.230029
\(612\) 4.00000 0.161690
\(613\) −2.70156 −0.109115 −0.0545575 0.998511i \(-0.517375\pi\)
−0.0545575 + 0.998511i \(0.517375\pi\)
\(614\) 32.9109 1.32818
\(615\) 18.1047 0.730051
\(616\) 4.00000 0.161165
\(617\) −44.8062 −1.80383 −0.901916 0.431912i \(-0.857839\pi\)
−0.901916 + 0.431912i \(0.857839\pi\)
\(618\) −8.70156 −0.350028
\(619\) −32.8062 −1.31859 −0.659297 0.751882i \(-0.729146\pi\)
−0.659297 + 0.751882i \(0.729146\pi\)
\(620\) 5.40312 0.216995
\(621\) 1.00000 0.0401286
\(622\) −0.596876 −0.0239325
\(623\) 1.40312 0.0562150
\(624\) −0.701562 −0.0280850
\(625\) −31.2094 −1.24837
\(626\) −32.2094 −1.28735
\(627\) −29.6125 −1.18261
\(628\) 11.4031 0.455034
\(629\) 42.8062 1.70680
\(630\) 2.70156 0.107633
\(631\) 21.6125 0.860380 0.430190 0.902738i \(-0.358447\pi\)
0.430190 + 0.902738i \(0.358447\pi\)
\(632\) −8.00000 −0.318223
\(633\) −14.8062 −0.588496
\(634\) 14.7016 0.583874
\(635\) −1.89531 −0.0752132
\(636\) 3.40312 0.134943
\(637\) −0.701562 −0.0277969
\(638\) 26.8062 1.06127
\(639\) 5.40312 0.213744
\(640\) 2.70156 0.106789
\(641\) 49.7172 1.96371 0.981855 0.189631i \(-0.0607293\pi\)
0.981855 + 0.189631i \(0.0607293\pi\)
\(642\) 6.80625 0.268621
\(643\) −24.8062 −0.978263 −0.489131 0.872210i \(-0.662686\pi\)
−0.489131 + 0.872210i \(0.662686\pi\)
\(644\) 1.00000 0.0394055
\(645\) −12.7016 −0.500124
\(646\) −29.6125 −1.16509
\(647\) −19.4031 −0.762816 −0.381408 0.924407i \(-0.624561\pi\)
−0.381408 + 0.924407i \(0.624561\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −21.6125 −0.848365
\(650\) 1.61250 0.0632473
\(651\) −2.00000 −0.0783862
\(652\) 14.8062 0.579857
\(653\) −12.1047 −0.473693 −0.236846 0.971547i \(-0.576114\pi\)
−0.236846 + 0.971547i \(0.576114\pi\)
\(654\) 1.29844 0.0507729
\(655\) −14.5969 −0.570347
\(656\) −6.70156 −0.261652
\(657\) −11.4031 −0.444878
\(658\) −8.10469 −0.315954
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 10.8062 0.420633
\(661\) −34.4187 −1.33873 −0.669367 0.742932i \(-0.733435\pi\)
−0.669367 + 0.742932i \(0.733435\pi\)
\(662\) 20.0000 0.777322
\(663\) −2.80625 −0.108986
\(664\) −0.596876 −0.0231633
\(665\) −20.0000 −0.775567
\(666\) −10.7016 −0.414677
\(667\) 6.70156 0.259486
\(668\) −0.596876 −0.0230938
\(669\) −22.0000 −0.850569
\(670\) −29.1938 −1.12785
\(671\) 8.00000 0.308837
\(672\) −1.00000 −0.0385758
\(673\) 1.29844 0.0500511 0.0250256 0.999687i \(-0.492033\pi\)
0.0250256 + 0.999687i \(0.492033\pi\)
\(674\) −34.2094 −1.31770
\(675\) 2.29844 0.0884669
\(676\) −12.5078 −0.481070
\(677\) 26.4187 1.01535 0.507677 0.861547i \(-0.330504\pi\)
0.507677 + 0.861547i \(0.330504\pi\)
\(678\) 12.1047 0.464878
\(679\) 12.7016 0.487441
\(680\) 10.8062 0.414401
\(681\) −3.89531 −0.149269
\(682\) −8.00000 −0.306336
\(683\) −17.6125 −0.673923 −0.336962 0.941518i \(-0.609399\pi\)
−0.336962 + 0.941518i \(0.609399\pi\)
\(684\) 7.40312 0.283066
\(685\) 35.9266 1.37268
\(686\) −1.00000 −0.0381802
\(687\) −27.4031 −1.04549
\(688\) 4.70156 0.179245
\(689\) −2.38750 −0.0909566
\(690\) 2.70156 0.102847
\(691\) 31.5078 1.19861 0.599307 0.800519i \(-0.295443\pi\)
0.599307 + 0.800519i \(0.295443\pi\)
\(692\) 6.59688 0.250776
\(693\) −4.00000 −0.151947
\(694\) 16.7016 0.633983
\(695\) −30.5234 −1.15782
\(696\) −6.70156 −0.254022
\(697\) −26.8062 −1.01536
\(698\) −25.4031 −0.961522
\(699\) −26.0000 −0.983410
\(700\) 2.29844 0.0868728
\(701\) −17.0156 −0.642671 −0.321336 0.946965i \(-0.604132\pi\)
−0.321336 + 0.946965i \(0.604132\pi\)
\(702\) 0.701562 0.0264788
\(703\) 79.2250 2.98803
\(704\) −4.00000 −0.150756
\(705\) −21.8953 −0.824625
\(706\) −10.7016 −0.402759
\(707\) 9.40312 0.353641
\(708\) 5.40312 0.203062
\(709\) 48.8062 1.83296 0.916479 0.400084i \(-0.131019\pi\)
0.916479 + 0.400084i \(0.131019\pi\)
\(710\) 14.5969 0.547811
\(711\) 8.00000 0.300023
\(712\) −1.40312 −0.0525843
\(713\) −2.00000 −0.0749006
\(714\) −4.00000 −0.149696
\(715\) −7.58125 −0.283523
\(716\) −5.89531 −0.220318
\(717\) −14.8062 −0.552949
\(718\) 22.1047 0.824940
\(719\) 38.9109 1.45113 0.725567 0.688152i \(-0.241578\pi\)
0.725567 + 0.688152i \(0.241578\pi\)
\(720\) −2.70156 −0.100681
\(721\) 8.70156 0.324063
\(722\) −35.8062 −1.33257
\(723\) 28.9109 1.07521
\(724\) 8.80625 0.327282
\(725\) 15.4031 0.572058
\(726\) −5.00000 −0.185567
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0.701562 0.0260016
\(729\) 1.00000 0.0370370
\(730\) −30.8062 −1.14019
\(731\) 18.8062 0.695574
\(732\) −2.00000 −0.0739221
\(733\) 13.7906 0.509368 0.254684 0.967024i \(-0.418028\pi\)
0.254684 + 0.967024i \(0.418028\pi\)
\(734\) −2.10469 −0.0776854
\(735\) −2.70156 −0.0996486
\(736\) −1.00000 −0.0368605
\(737\) 43.2250 1.59221
\(738\) 6.70156 0.246688
\(739\) 35.0156 1.28807 0.644035 0.764996i \(-0.277259\pi\)
0.644035 + 0.764996i \(0.277259\pi\)
\(740\) −28.9109 −1.06279
\(741\) −5.19375 −0.190797
\(742\) −3.40312 −0.124933
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 2.00000 0.0733236
\(745\) −49.1938 −1.80232
\(746\) −24.8062 −0.908221
\(747\) 0.596876 0.0218385
\(748\) −16.0000 −0.585018
\(749\) −6.80625 −0.248695
\(750\) −7.29844 −0.266501
\(751\) −21.6125 −0.788651 −0.394326 0.918971i \(-0.629022\pi\)
−0.394326 + 0.918971i \(0.629022\pi\)
\(752\) 8.10469 0.295548
\(753\) 13.2984 0.484622
\(754\) 4.70156 0.171221
\(755\) 56.4922 2.05596
\(756\) 1.00000 0.0363696
\(757\) 50.4187 1.83250 0.916250 0.400606i \(-0.131201\pi\)
0.916250 + 0.400606i \(0.131201\pi\)
\(758\) −7.29844 −0.265091
\(759\) −4.00000 −0.145191
\(760\) 20.0000 0.725476
\(761\) 15.6125 0.565953 0.282976 0.959127i \(-0.408678\pi\)
0.282976 + 0.959127i \(0.408678\pi\)
\(762\) −0.701562 −0.0254149
\(763\) −1.29844 −0.0470066
\(764\) −2.80625 −0.101527
\(765\) −10.8062 −0.390701
\(766\) 21.6125 0.780891
\(767\) −3.79063 −0.136872
\(768\) 1.00000 0.0360844
\(769\) −28.9109 −1.04255 −0.521277 0.853387i \(-0.674544\pi\)
−0.521277 + 0.853387i \(0.674544\pi\)
\(770\) −10.8062 −0.389430
\(771\) 2.00000 0.0720282
\(772\) −3.89531 −0.140195
\(773\) 12.1047 0.435375 0.217688 0.976018i \(-0.430149\pi\)
0.217688 + 0.976018i \(0.430149\pi\)
\(774\) −4.70156 −0.168994
\(775\) −4.59688 −0.165125
\(776\) −12.7016 −0.455960
\(777\) 10.7016 0.383916
\(778\) 6.00000 0.215110
\(779\) −49.6125 −1.77755
\(780\) 1.89531 0.0678631
\(781\) −21.6125 −0.773356
\(782\) −4.00000 −0.143040
\(783\) 6.70156 0.239494
\(784\) 1.00000 0.0357143
\(785\) −30.8062 −1.09952
\(786\) −5.40312 −0.192723
\(787\) −31.4031 −1.11940 −0.559700 0.828695i \(-0.689084\pi\)
−0.559700 + 0.828695i \(0.689084\pi\)
\(788\) −9.50781 −0.338702
\(789\) 3.29844 0.117427
\(790\) 21.6125 0.768938
\(791\) −12.1047 −0.430393
\(792\) 4.00000 0.142134
\(793\) 1.40312 0.0498264
\(794\) −21.4031 −0.759568
\(795\) −9.19375 −0.326069
\(796\) −8.70156 −0.308419
\(797\) 9.29844 0.329368 0.164684 0.986346i \(-0.447340\pi\)
0.164684 + 0.986346i \(0.447340\pi\)
\(798\) −7.40312 −0.262068
\(799\) 32.4187 1.14689
\(800\) −2.29844 −0.0812621
\(801\) 1.40312 0.0495770
\(802\) −30.0000 −1.05934
\(803\) 45.6125 1.60963
\(804\) −10.8062 −0.381107
\(805\) −2.70156 −0.0952176
\(806\) −1.40312 −0.0494229
\(807\) 28.2094 0.993017
\(808\) −9.40312 −0.330801
\(809\) −33.0156 −1.16077 −0.580384 0.814343i \(-0.697097\pi\)
−0.580384 + 0.814343i \(0.697097\pi\)
\(810\) 2.70156 0.0949232
\(811\) 20.4922 0.719578 0.359789 0.933034i \(-0.382849\pi\)
0.359789 + 0.933034i \(0.382849\pi\)
\(812\) 6.70156 0.235179
\(813\) −11.4031 −0.399925
\(814\) 42.8062 1.50036
\(815\) −40.0000 −1.40114
\(816\) 4.00000 0.140028
\(817\) 34.8062 1.21772
\(818\) 7.40312 0.258844
\(819\) −0.701562 −0.0245146
\(820\) 18.1047 0.632243
\(821\) −23.6125 −0.824082 −0.412041 0.911165i \(-0.635184\pi\)
−0.412041 + 0.911165i \(0.635184\pi\)
\(822\) 13.2984 0.463836
\(823\) −7.29844 −0.254408 −0.127204 0.991877i \(-0.540600\pi\)
−0.127204 + 0.991877i \(0.540600\pi\)
\(824\) −8.70156 −0.303133
\(825\) −9.19375 −0.320085
\(826\) −5.40312 −0.187999
\(827\) −34.5969 −1.20305 −0.601526 0.798854i \(-0.705440\pi\)
−0.601526 + 0.798854i \(0.705440\pi\)
\(828\) 1.00000 0.0347524
\(829\) 42.5969 1.47945 0.739725 0.672909i \(-0.234955\pi\)
0.739725 + 0.672909i \(0.234955\pi\)
\(830\) 1.61250 0.0559706
\(831\) −8.59688 −0.298222
\(832\) −0.701562 −0.0243223
\(833\) 4.00000 0.138592
\(834\) −11.2984 −0.391233
\(835\) 1.61250 0.0558028
\(836\) −29.6125 −1.02417
\(837\) −2.00000 −0.0691301
\(838\) −38.2094 −1.31992
\(839\) 3.79063 0.130867 0.0654335 0.997857i \(-0.479157\pi\)
0.0654335 + 0.997857i \(0.479157\pi\)
\(840\) 2.70156 0.0932127
\(841\) 15.9109 0.548653
\(842\) 9.29844 0.320445
\(843\) −17.5078 −0.603001
\(844\) −14.8062 −0.509652
\(845\) 33.7906 1.16243
\(846\) −8.10469 −0.278645
\(847\) 5.00000 0.171802
\(848\) 3.40312 0.116864
\(849\) −20.8062 −0.714069
\(850\) −9.19375 −0.315343
\(851\) 10.7016 0.366845
\(852\) 5.40312 0.185108
\(853\) −10.1047 −0.345978 −0.172989 0.984924i \(-0.555342\pi\)
−0.172989 + 0.984924i \(0.555342\pi\)
\(854\) 2.00000 0.0684386
\(855\) −20.0000 −0.683986
\(856\) 6.80625 0.232633
\(857\) −6.70156 −0.228921 −0.114461 0.993428i \(-0.536514\pi\)
−0.114461 + 0.993428i \(0.536514\pi\)
\(858\) −2.80625 −0.0958037
\(859\) −47.5078 −1.62095 −0.810473 0.585776i \(-0.800790\pi\)
−0.810473 + 0.585776i \(0.800790\pi\)
\(860\) −12.7016 −0.433120
\(861\) −6.70156 −0.228389
\(862\) −40.9109 −1.39343
\(863\) −57.6125 −1.96115 −0.980576 0.196139i \(-0.937160\pi\)
−0.980576 + 0.196139i \(0.937160\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −17.8219 −0.605962
\(866\) 26.3141 0.894188
\(867\) −1.00000 −0.0339618
\(868\) −2.00000 −0.0678844
\(869\) −32.0000 −1.08553
\(870\) 18.1047 0.613806
\(871\) 7.58125 0.256881
\(872\) 1.29844 0.0439707
\(873\) 12.7016 0.429883
\(874\) −7.40312 −0.250414
\(875\) 7.29844 0.246732
\(876\) −11.4031 −0.385276
\(877\) −4.59688 −0.155225 −0.0776127 0.996984i \(-0.524730\pi\)
−0.0776127 + 0.996984i \(0.524730\pi\)
\(878\) 22.0000 0.742464
\(879\) 4.80625 0.162111
\(880\) 10.8062 0.364279
\(881\) −32.2094 −1.08516 −0.542581 0.840004i \(-0.682553\pi\)
−0.542581 + 0.840004i \(0.682553\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −37.4031 −1.25872 −0.629358 0.777116i \(-0.716682\pi\)
−0.629358 + 0.777116i \(0.716682\pi\)
\(884\) −2.80625 −0.0943844
\(885\) −14.5969 −0.490669
\(886\) 7.29844 0.245196
\(887\) 10.2094 0.342797 0.171399 0.985202i \(-0.445171\pi\)
0.171399 + 0.985202i \(0.445171\pi\)
\(888\) −10.7016 −0.359121
\(889\) 0.701562 0.0235296
\(890\) 3.79063 0.127062
\(891\) −4.00000 −0.134005
\(892\) −22.0000 −0.736614
\(893\) 60.0000 2.00782
\(894\) −18.2094 −0.609013
\(895\) 15.9266 0.532366
\(896\) −1.00000 −0.0334077
\(897\) −0.701562 −0.0234245
\(898\) −38.4187 −1.28205
\(899\) −13.4031 −0.447019
\(900\) 2.29844 0.0766146
\(901\) 13.6125 0.453498
\(902\) −26.8062 −0.892550
\(903\) 4.70156 0.156458
\(904\) 12.1047 0.402596
\(905\) −23.7906 −0.790827
\(906\) 20.9109 0.694719
\(907\) −11.5078 −0.382111 −0.191055 0.981579i \(-0.561191\pi\)
−0.191055 + 0.981579i \(0.561191\pi\)
\(908\) −3.89531 −0.129271
\(909\) 9.40312 0.311882
\(910\) −1.89531 −0.0628290
\(911\) 33.8953 1.12300 0.561501 0.827476i \(-0.310224\pi\)
0.561501 + 0.827476i \(0.310224\pi\)
\(912\) 7.40312 0.245142
\(913\) −2.38750 −0.0790148
\(914\) 10.2094 0.337696
\(915\) 5.40312 0.178622
\(916\) −27.4031 −0.905425
\(917\) 5.40312 0.178427
\(918\) −4.00000 −0.132020
\(919\) 24.4187 0.805500 0.402750 0.915310i \(-0.368054\pi\)
0.402750 + 0.915310i \(0.368054\pi\)
\(920\) 2.70156 0.0890679
\(921\) −32.9109 −1.08445
\(922\) −36.2094 −1.19249
\(923\) −3.79063 −0.124770
\(924\) −4.00000 −0.131590
\(925\) 24.5969 0.808740
\(926\) 28.7016 0.943192
\(927\) 8.70156 0.285797
\(928\) −6.70156 −0.219990
\(929\) −1.08907 −0.0357311 −0.0178655 0.999840i \(-0.505687\pi\)
−0.0178655 + 0.999840i \(0.505687\pi\)
\(930\) −5.40312 −0.177175
\(931\) 7.40312 0.242628
\(932\) −26.0000 −0.851658
\(933\) 0.596876 0.0195408
\(934\) −2.70156 −0.0883978
\(935\) 43.2250 1.41361
\(936\) 0.701562 0.0229313
\(937\) 5.89531 0.192592 0.0962958 0.995353i \(-0.469301\pi\)
0.0962958 + 0.995353i \(0.469301\pi\)
\(938\) 10.8062 0.352837
\(939\) 32.2094 1.05111
\(940\) −21.8953 −0.714146
\(941\) −32.3141 −1.05341 −0.526704 0.850049i \(-0.676572\pi\)
−0.526704 + 0.850049i \(0.676572\pi\)
\(942\) −11.4031 −0.371534
\(943\) −6.70156 −0.218233
\(944\) 5.40312 0.175857
\(945\) −2.70156 −0.0878818
\(946\) 18.8062 0.611444
\(947\) −28.9109 −0.939479 −0.469740 0.882805i \(-0.655652\pi\)
−0.469740 + 0.882805i \(0.655652\pi\)
\(948\) 8.00000 0.259828
\(949\) 8.00000 0.259691
\(950\) −17.0156 −0.552060
\(951\) −14.7016 −0.476731
\(952\) −4.00000 −0.129641
\(953\) −41.2250 −1.33541 −0.667704 0.744427i \(-0.732723\pi\)
−0.667704 + 0.744427i \(0.732723\pi\)
\(954\) −3.40312 −0.110180
\(955\) 7.58125 0.245324
\(956\) −14.8062 −0.478868
\(957\) −26.8062 −0.866523
\(958\) −29.4031 −0.949972
\(959\) −13.2984 −0.429429
\(960\) −2.70156 −0.0871925
\(961\) −27.0000 −0.870968
\(962\) 7.50781 0.242062
\(963\) −6.80625 −0.219328
\(964\) 28.9109 0.931159
\(965\) 10.5234 0.338761
\(966\) −1.00000 −0.0321745
\(967\) 10.8062 0.347506 0.173753 0.984789i \(-0.444411\pi\)
0.173753 + 0.984789i \(0.444411\pi\)
\(968\) −5.00000 −0.160706
\(969\) 29.6125 0.951290
\(970\) 34.3141 1.10176
\(971\) −39.8219 −1.27794 −0.638972 0.769230i \(-0.720640\pi\)
−0.638972 + 0.769230i \(0.720640\pi\)
\(972\) 1.00000 0.0320750
\(973\) 11.2984 0.362211
\(974\) 12.7016 0.406984
\(975\) −1.61250 −0.0516412
\(976\) −2.00000 −0.0640184
\(977\) −53.7172 −1.71856 −0.859282 0.511501i \(-0.829090\pi\)
−0.859282 + 0.511501i \(0.829090\pi\)
\(978\) −14.8062 −0.473452
\(979\) −5.61250 −0.179376
\(980\) −2.70156 −0.0862982
\(981\) −1.29844 −0.0414559
\(982\) −41.6125 −1.32791
\(983\) 3.79063 0.120902 0.0604511 0.998171i \(-0.480746\pi\)
0.0604511 + 0.998171i \(0.480746\pi\)
\(984\) 6.70156 0.213638
\(985\) 25.6859 0.818422
\(986\) −26.8062 −0.853685
\(987\) 8.10469 0.257975
\(988\) −5.19375 −0.165235
\(989\) 4.70156 0.149501
\(990\) −10.8062 −0.343445
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 2.00000 0.0635001
\(993\) −20.0000 −0.634681
\(994\) −5.40312 −0.171377
\(995\) 23.5078 0.745248
\(996\) 0.596876 0.0189127
\(997\) −10.5969 −0.335606 −0.167803 0.985821i \(-0.553667\pi\)
−0.167803 + 0.985821i \(0.553667\pi\)
\(998\) 10.5969 0.335438
\(999\) 10.7016 0.338582
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 966.2.a.n.1.1 2
3.2 odd 2 2898.2.a.bb.1.2 2
4.3 odd 2 7728.2.a.bc.1.1 2
7.6 odd 2 6762.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.n.1.1 2 1.1 even 1 trivial
2898.2.a.bb.1.2 2 3.2 odd 2
6762.2.a.bo.1.2 2 7.6 odd 2
7728.2.a.bc.1.1 2 4.3 odd 2