Properties

Label 6762.2.a.bo.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $2$
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: \(1\)
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: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

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

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 \(-0.293538\pi\)
0.604088 + 0.796918i \(0.293538\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −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 \(-0.468983\pi\)
0.0972892 + 0.995256i \(0.468983\pi\)
\(14\) 0 0
\(15\) −2.70156 −0.697540
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.40312 −1.69839 −0.849197 0.528077i \(-0.822913\pi\)
−0.849197 + 0.528077i \(0.822913\pi\)
\(20\) 2.70156 0.604088
\(21\) 0 0
\(22\) 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
\(28\) 0 0
\(29\) 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 \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 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 \(-0.324699\pi\)
0.523304 + 0.852146i \(0.324699\pi\)
\(42\) 0 0
\(43\) 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 \(-0.701304\pi\)
−0.591095 + 0.806602i \(0.701304\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) 7.40312 0.980568
\(58\) −6.70156 −0.879958
\(59\) −5.40312 −0.703427 −0.351713 0.936108i \(-0.614401\pi\)
−0.351713 + 0.936108i \(0.614401\pi\)
\(60\) −2.70156 −0.348770
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) 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 \(-0.267442\pi\)
0.667317 + 0.744773i \(0.267442\pi\)
\(74\) −10.7016 −1.24403
\(75\) −2.29844 −0.265401
\(76\) −7.40312 −0.849197
\(77\) 0 0
\(78\) 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 \(-0.510429\pi\)
−0.0327578 + 0.999463i \(0.510429\pi\)
\(84\) 0 0
\(85\) −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 \(-0.523693\pi\)
−0.0743654 + 0.997231i \(0.523693\pi\)
\(90\) −2.70156 −0.284770
\(91\) 0 0
\(92\) 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 \(-0.723069\pi\)
−0.644824 + 0.764331i \(0.723069\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 2.29844 0.229844
\(101\) −9.40312 −0.935646 −0.467823 0.883822i \(-0.654961\pi\)
−0.467823 + 0.883822i \(0.654961\pi\)
\(102\) −4.00000 −0.396059
\(103\) −8.70156 −0.857390 −0.428695 0.903449i \(-0.641027\pi\)
−0.428695 + 0.903449i \(0.641027\pi\)
\(104\) −0.701562 −0.0687938
\(105\) 0 0
\(106\) −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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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
\(126\) 0 0
\(127\) 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 \(-0.575849\pi\)
−0.236037 + 0.971744i \(0.575849\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 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 \(-0.659059\pi\)
−0.479160 + 0.877727i \(0.659059\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) 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
\(154\) 0 0
\(155\) 5.40312 0.433989
\(156\) −0.701562 −0.0561699
\(157\) −11.4031 −0.910068 −0.455034 0.890474i \(-0.650373\pi\)
−0.455034 + 0.890474i \(0.650373\pi\)
\(158\) −8.00000 −0.636446
\(159\) −3.40312 −0.269885
\(160\) −2.70156 −0.213577
\(161\) 0 0
\(162\) −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 \(-0.492648\pi\)
0.0230938 + 0.999733i \(0.492648\pi\)
\(168\) 0 0
\(169\) −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 \(-0.580686\pi\)
−0.250776 + 0.968045i \(0.580686\pi\)
\(174\) 6.70156 0.508044
\(175\) 0 0
\(176\) −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 \(-0.606133\pi\)
−0.327282 + 0.944927i \(0.606133\pi\)
\(182\) 0 0
\(183\) −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
\(189\) 0 0
\(190\) 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
\(196\) 0 0
\(197\) −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 \(-0.400200\pi\)
0.308419 + 0.951251i \(0.400200\pi\)
\(200\) −2.29844 −0.162524
\(201\) 10.8062 0.762214
\(202\) 9.40312 0.661602
\(203\) 0 0
\(204\) 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
\(210\) 0 0
\(211\) −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
\(217\) 0 0
\(218\) 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 \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) 2.29844 0.153229
\(226\) 12.1047 0.805192
\(227\) 3.89531 0.258541 0.129271 0.991609i \(-0.458736\pi\)
0.129271 + 0.991609i \(0.458736\pi\)
\(228\) 7.40312 0.490284
\(229\) 27.4031 1.81085 0.905425 0.424507i \(-0.139553\pi\)
0.905425 + 0.424507i \(0.139553\pi\)
\(230\) −2.70156 −0.178136
\(231\) 0 0
\(232\) −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
\(238\) 0 0
\(239\) −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 \(-0.881201\pi\)
−0.931159 + 0.364615i \(0.881201\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 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 \(-0.637863\pi\)
−0.419695 + 0.907665i \(0.637863\pi\)
\(252\) 0 0
\(253\) −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 \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 4.70156 0.292706
\(259\) 0 0
\(260\) 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
\(266\) 0 0
\(267\) 1.40312 0.0858698
\(268\) −10.8062 −0.660097
\(269\) −28.2094 −1.71996 −0.859978 0.510331i \(-0.829523\pi\)
−0.859978 + 0.510331i \(0.829523\pi\)
\(270\) 2.70156 0.164412
\(271\) 11.4031 0.692690 0.346345 0.938107i \(-0.387423\pi\)
0.346345 + 0.938107i \(0.387423\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 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
\(280\) 0 0
\(281\) −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 \(-0.287781\pi\)
0.618402 + 0.785862i \(0.287781\pi\)
\(284\) 5.40312 0.320616
\(285\) 20.0000 1.18470
\(286\) 2.80625 0.165937
\(287\) 0 0
\(288\) −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 \(-0.544836\pi\)
−0.140392 + 0.990096i \(0.544836\pi\)
\(294\) 0 0
\(295\) −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
\(301\) 0 0
\(302\) 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 \(-0.111603\pi\)
0.939163 + 0.343471i \(0.111603\pi\)
\(308\) 0 0
\(309\) 8.70156 0.495015
\(310\) −5.40312 −0.306877
\(311\) −0.596876 −0.0338457 −0.0169229 0.999857i \(-0.505387\pi\)
−0.0169229 + 0.999857i \(0.505387\pi\)
\(312\) 0.701562 0.0397181
\(313\) −32.2094 −1.82058 −0.910291 0.413970i \(-0.864142\pi\)
−0.910291 + 0.413970i \(0.864142\pi\)
\(314\) 11.4031 0.643516
\(315\) 0 0
\(316\) 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
\(322\) 0 0
\(323\) 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
\(329\) 0 0
\(330\) −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
\(336\) 0 0
\(337\) 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
\(343\) 0 0
\(344\) −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 \(-0.737976\pi\)
−0.679899 + 0.733306i \(0.737976\pi\)
\(350\) 0 0
\(351\) −0.701562 −0.0374466
\(352\) 4.00000 0.213201
\(353\) −10.7016 −0.569587 −0.284793 0.958589i \(-0.591925\pi\)
−0.284793 + 0.958589i \(0.591925\pi\)
\(354\) −5.40312 −0.287173
\(355\) 14.5969 0.774722
\(356\) −1.40312 −0.0743654
\(357\) 0 0
\(358\) 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 0
\(365\) 30.8062 1.61247
\(366\) 2.00000 0.104542
\(367\) −2.10469 −0.109864 −0.0549319 0.998490i \(-0.517494\pi\)
−0.0549319 + 0.998490i \(0.517494\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.70156 0.348869
\(370\) −28.9109 −1.50301
\(371\) 0 0
\(372\) −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
\(378\) 0 0
\(379\) 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 \(-0.313799\pi\)
0.552174 + 0.833729i \(0.313799\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 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
\(392\) 0 0
\(393\) 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 \(-0.680479\pi\)
−0.537096 + 0.843521i \(0.680479\pi\)
\(398\) −8.70156 −0.436170
\(399\) 0 0
\(400\) 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
\(406\) 0 0
\(407\) −42.8062 −2.12183
\(408\) −4.00000 −0.198030
\(409\) 7.40312 0.366061 0.183030 0.983107i \(-0.441409\pi\)
0.183030 + 0.983107i \(0.441409\pi\)
\(410\) −18.1047 −0.894127
\(411\) 13.2984 0.655964
\(412\) −8.70156 −0.428695
\(413\) 0 0
\(414\) −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 \(-0.883107\pi\)
−0.933325 + 0.359033i \(0.883107\pi\)
\(420\) 0 0
\(421\) −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
\(427\) 0 0
\(428\) −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 \(-0.282117\pi\)
0.632286 + 0.774735i \(0.282117\pi\)
\(434\) 0 0
\(435\) −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 \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 10.8062 0.515168
\(441\) 0 0
\(442\) 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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) −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 \(-0.819342\pi\)
−0.843219 + 0.537570i \(0.819342\pi\)
\(462\) 0 0
\(463\) −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 \(-0.519909\pi\)
−0.0625067 + 0.998045i \(0.519909\pi\)
\(468\) 0.701562 0.0324297
\(469\) 0 0
\(470\) 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
\(476\) 0 0
\(477\) 3.40312 0.155818
\(478\) 14.8062 0.677222
\(479\) −29.4031 −1.34346 −0.671732 0.740795i \(-0.734449\pi\)
−0.671732 + 0.740795i \(0.734449\pi\)
\(480\) 2.70156 0.123309
\(481\) 7.50781 0.342327
\(482\) 28.9109 1.31686
\(483\) 0 0
\(484\) 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
\(490\) 0 0
\(491\) 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
\(497\) 0 0
\(498\) −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 \(-0.626828\pi\)
−0.387983 + 0.921666i \(0.626828\pi\)
\(504\) 0 0
\(505\) −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 \(-0.402452\pi\)
0.301682 + 0.953409i \(0.402452\pi\)
\(510\) −10.8062 −0.478509
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) 6.59688 0.289571
\(520\) −1.89531 −0.0831150
\(521\) 33.4031 1.46342 0.731709 0.681617i \(-0.238723\pi\)
0.731709 + 0.681617i \(0.238723\pi\)
\(522\) −6.70156 −0.293319
\(523\) 4.80625 0.210163 0.105081 0.994464i \(-0.466490\pi\)
0.105081 + 0.994464i \(0.466490\pi\)
\(524\) −5.40312 −0.236037
\(525\) 0 0
\(526\) −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
\(532\) 0 0
\(533\) 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
\(539\) 0 0
\(540\) −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 0
\(547\) −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
\(553\) 0 0
\(554\) 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
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 17.5078 0.738522
\(563\) −16.1047 −0.678732 −0.339366 0.940654i \(-0.610212\pi\)
−0.339366 + 0.940654i \(0.610212\pi\)
\(564\) 8.10469 0.341269
\(565\) −32.7016 −1.37577
\(566\) −20.8062 −0.874552
\(567\) 0 0
\(568\) −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
\(574\) 0 0
\(575\) 2.29844 0.0958515
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 3.89531 0.161884
\(580\) 18.1047 0.751756
\(581\) 0 0
\(582\) −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 \(-0.428411\pi\)
0.223011 + 0.974816i \(0.428411\pi\)
\(588\) 0 0
\(589\) −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 \(-0.777672\pi\)
−0.765832 + 0.643041i \(0.777672\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 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 \(-0.496125\pi\)
0.0121735 + 0.999926i \(0.496125\pi\)
\(602\) 0 0
\(603\) −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 \(-0.575728\pi\)
−0.235668 + 0.971834i \(0.575728\pi\)
\(608\) 7.40312 0.300236
\(609\) 0 0
\(610\) −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
\(616\) 0 0
\(617\) −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 \(-0.270854\pi\)
0.659297 + 0.751882i \(0.270854\pi\)
\(620\) 5.40312 0.216995
\(621\) −1.00000 −0.0401286
\(622\) 0.596876 0.0239325
\(623\) 0 0
\(624\) −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
\(630\) 0 0
\(631\) 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 0
\(638\) 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 \(-0.337314\pi\)
0.489131 + 0.872210i \(0.337314\pi\)
\(644\) 0 0
\(645\) −12.7016 −0.500124
\(646\) −29.6125 −1.16509
\(647\) 19.4031 0.762816 0.381408 0.924407i \(-0.375439\pi\)
0.381408 + 0.924407i \(0.375439\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 21.6125 0.848365
\(650\) −1.61250 −0.0632473
\(651\) 0 0
\(652\) 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
\(658\) 0 0
\(659\) 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 \(-0.266565\pi\)
0.669367 + 0.742932i \(0.266565\pi\)
\(662\) 20.0000 0.777322
\(663\) 2.80625 0.108986
\(664\) 0.596876 0.0231633
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) 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 \(-0.669496\pi\)
−0.507677 + 0.861547i \(0.669496\pi\)
\(678\) −12.1047 −0.464878
\(679\) 0 0
\(680\) 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
\(686\) 0 0
\(687\) −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 \(-0.704557\pi\)
−0.599307 + 0.800519i \(0.704557\pi\)
\(692\) −6.59688 −0.250776
\(693\) 0 0
\(694\) 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
\(700\) 0 0
\(701\) −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
\(707\) 0 0
\(708\) 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
\(714\) 0 0
\(715\) −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 \(-0.758422\pi\)
−0.725567 + 0.688152i \(0.758422\pi\)
\(720\) 2.70156 0.100681
\(721\) 0 0
\(722\) −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 \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 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 \(-0.581972\pi\)
−0.254684 + 0.967024i \(0.581972\pi\)
\(734\) 2.10469 0.0776854
\(735\) 0 0
\(736\) −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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) −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
\(756\) 0 0
\(757\) 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 \(-0.591322\pi\)
−0.282976 + 0.959127i \(0.591322\pi\)
\(762\) 0.701562 0.0254149
\(763\) 0 0
\(764\) −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 \(-0.325456\pi\)
0.521277 + 0.853387i \(0.325456\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) −3.89531 −0.140195
\(773\) −12.1047 −0.435375 −0.217688 0.976018i \(-0.569851\pi\)
−0.217688 + 0.976018i \(0.569851\pi\)
\(774\) −4.70156 −0.168994
\(775\) 4.59688 0.165125
\(776\) 12.7016 0.455960
\(777\) 0 0
\(778\) 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
\(784\) 0 0
\(785\) −30.8062 −1.09952
\(786\) −5.40312 −0.192723
\(787\) 31.4031 1.11940 0.559700 0.828695i \(-0.310916\pi\)
0.559700 + 0.828695i \(0.310916\pi\)
\(788\) −9.50781 −0.338702
\(789\) −3.29844 −0.117427
\(790\) −21.6125 −0.768938
\(791\) 0 0
\(792\) 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 \(-0.552660\pi\)
−0.164684 + 0.986346i \(0.552660\pi\)
\(798\) 0 0
\(799\) 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
\(805\) 0 0
\(806\) −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 \(-0.617151\pi\)
−0.359789 + 0.933034i \(0.617151\pi\)
\(812\) 0 0
\(813\) −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 0
\(820\) 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
\(826\) 0 0
\(827\) −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 \(-0.765045\pi\)
−0.739725 + 0.672909i \(0.765045\pi\)
\(830\) 1.61250 0.0559706
\(831\) 8.59688 0.298222
\(832\) 0.701562 0.0243223
\(833\) 0 0
\(834\) −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 \(-0.520843\pi\)
−0.0654335 + 0.997857i \(0.520843\pi\)
\(840\) 0 0
\(841\) 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
\(847\) 0 0
\(848\) 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 \(-0.444658\pi\)
0.172989 + 0.984924i \(0.444658\pi\)
\(854\) 0 0
\(855\) −20.0000 −0.683986
\(856\) 6.80625 0.232633
\(857\) 6.70156 0.228921 0.114461 0.993428i \(-0.463486\pi\)
0.114461 + 0.993428i \(0.463486\pi\)
\(858\) −2.80625 −0.0958037
\(859\) 47.5078 1.62095 0.810473 0.585776i \(-0.199210\pi\)
0.810473 + 0.585776i \(0.199210\pi\)
\(860\) 12.7016 0.433120
\(861\) 0 0
\(862\) −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
\(868\) 0 0
\(869\) −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
\(875\) 0 0
\(876\) −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 \(-0.317447\pi\)
0.542581 + 0.840004i \(0.317447\pi\)
\(882\) 0 0
\(883\) −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 \(-0.554829\pi\)
−0.171399 + 0.985202i \(0.554829\pi\)
\(888\) 10.7016 0.359121
\(889\) 0 0
\(890\) 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
\(896\) 0 0
\(897\) −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
\(903\) 0 0
\(904\) 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
\(910\) 0 0
\(911\) 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
\(917\) 0 0
\(918\) −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
\(924\) 0 0
\(925\) 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 \(-0.494313\pi\)
0.0178655 + 0.999840i \(0.494313\pi\)
\(930\) 5.40312 0.177175
\(931\) 0 0
\(932\) −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 \(-0.530699\pi\)
−0.0962958 + 0.995353i \(0.530699\pi\)
\(938\) 0 0
\(939\) 32.2094 1.05111
\(940\) −21.8953 −0.714146
\(941\) 32.3141 1.05341 0.526704 0.850049i \(-0.323428\pi\)
0.526704 + 0.850049i \(0.323428\pi\)
\(942\) −11.4031 −0.371534
\(943\) 6.70156 0.218233
\(944\) −5.40312 −0.175857
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) −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
\(959\) 0 0
\(960\) −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
\(966\) 0 0
\(967\) 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 \(-0.279360\pi\)
0.638972 + 0.769230i \(0.279360\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 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
\(980\) 0 0
\(981\) −1.29844 −0.0414559
\(982\) −41.6125 −1.32791
\(983\) −3.79063 −0.120902 −0.0604511 0.998171i \(-0.519254\pi\)
−0.0604511 + 0.998171i \(0.519254\pi\)
\(984\) 6.70156 0.213638
\(985\) −25.6859 −0.818422
\(986\) 26.8062 0.853685
\(987\) 0 0
\(988\) −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
\(994\) 0 0
\(995\) 23.5078 0.745248
\(996\) 0.596876 0.0189127
\(997\) 10.5969 0.335606 0.167803 0.985821i \(-0.446333\pi\)
0.167803 + 0.985821i \(0.446333\pi\)
\(998\) 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 6762.2.a.bo.1.2 2
7.6 odd 2 966.2.a.n.1.1 2
21.20 even 2 2898.2.a.bb.1.2 2
28.27 even 2 7728.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.n.1.1 2 7.6 odd 2
2898.2.a.bb.1.2 2 21.20 even 2
6762.2.a.bo.1.2 2 1.1 even 1 trivial
7728.2.a.bc.1.1 2 28.27 even 2