Properties

Label 3381.2.a.bj.1.2
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.06506\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.06506 q^{2} +1.00000 q^{3} +2.26446 q^{4} +0.608853 q^{5} -2.06506 q^{6} -0.546135 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.06506 q^{2} +1.00000 q^{3} +2.26446 q^{4} +0.608853 q^{5} -2.06506 q^{6} -0.546135 q^{8} +1.00000 q^{9} -1.25732 q^{10} -1.10243 q^{11} +2.26446 q^{12} -1.84712 q^{13} +0.608853 q^{15} -3.40113 q^{16} -5.88576 q^{17} -2.06506 q^{18} -1.08814 q^{19} +1.37873 q^{20} +2.27659 q^{22} +1.00000 q^{23} -0.546135 q^{24} -4.62930 q^{25} +3.81441 q^{26} +1.00000 q^{27} -0.804169 q^{29} -1.25732 q^{30} -8.77360 q^{31} +8.11580 q^{32} -1.10243 q^{33} +12.1544 q^{34} +2.26446 q^{36} +9.11528 q^{37} +2.24707 q^{38} -1.84712 q^{39} -0.332516 q^{40} +9.79559 q^{41} +5.23469 q^{43} -2.49642 q^{44} +0.608853 q^{45} -2.06506 q^{46} +5.90878 q^{47} -3.40113 q^{48} +9.55977 q^{50} -5.88576 q^{51} -4.18274 q^{52} +4.51013 q^{53} -2.06506 q^{54} -0.671220 q^{55} -1.08814 q^{57} +1.66066 q^{58} +7.40064 q^{59} +1.37873 q^{60} -5.02768 q^{61} +18.1180 q^{62} -9.95734 q^{64} -1.12462 q^{65} +2.27659 q^{66} +2.06305 q^{67} -13.3281 q^{68} +1.00000 q^{69} +4.30092 q^{71} -0.546135 q^{72} +1.15613 q^{73} -18.8236 q^{74} -4.62930 q^{75} -2.46406 q^{76} +3.81441 q^{78} +15.1716 q^{79} -2.07079 q^{80} +1.00000 q^{81} -20.2285 q^{82} +10.7196 q^{83} -3.58356 q^{85} -10.8099 q^{86} -0.804169 q^{87} +0.602078 q^{88} +7.92167 q^{89} -1.25732 q^{90} +2.26446 q^{92} -8.77360 q^{93} -12.2020 q^{94} -0.662518 q^{95} +8.11580 q^{96} +17.0648 q^{97} -1.10243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} + 10q^{3} + 15q^{4} + 5q^{5} + 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} + 10q^{3} + 15q^{4} + 5q^{5} + 3q^{6} + 9q^{8} + 10q^{9} - 11q^{10} + 8q^{11} + 15q^{12} + 5q^{15} + 37q^{16} + 11q^{17} + 3q^{18} - q^{19} + 15q^{20} + 6q^{22} + 10q^{23} + 9q^{24} + 21q^{25} - q^{26} + 10q^{27} + 22q^{29} - 11q^{30} + 3q^{31} + 11q^{32} + 8q^{33} + 3q^{34} + 15q^{36} - 3q^{37} - 16q^{38} - 39q^{40} + 26q^{41} + 27q^{43} + 16q^{44} + 5q^{45} + 3q^{46} - 11q^{47} + 37q^{48} + 2q^{50} + 11q^{51} - 29q^{52} + 5q^{53} + 3q^{54} + 18q^{55} - q^{57} + 16q^{58} + 10q^{59} + 15q^{60} - 22q^{61} + 32q^{62} + 69q^{64} - 11q^{65} + 6q^{66} - 2q^{67} + 21q^{68} + 10q^{69} + 27q^{71} + 9q^{72} + 8q^{73} + 14q^{74} + 21q^{75} + 22q^{76} - q^{78} + 21q^{79} + 53q^{80} + 10q^{81} - 36q^{82} + 12q^{83} + 23q^{85} + 18q^{86} + 22q^{87} - 10q^{88} - 6q^{89} - 11q^{90} + 15q^{92} + 3q^{93} - 35q^{94} + 44q^{95} + 11q^{96} - 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.06506 −1.46022 −0.730108 0.683331i \(-0.760530\pi\)
−0.730108 + 0.683331i \(0.760530\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.26446 1.13223
\(5\) 0.608853 0.272287 0.136144 0.990689i \(-0.456529\pi\)
0.136144 + 0.990689i \(0.456529\pi\)
\(6\) −2.06506 −0.843056
\(7\) 0 0
\(8\) −0.546135 −0.193088
\(9\) 1.00000 0.333333
\(10\) −1.25732 −0.397599
\(11\) −1.10243 −0.332396 −0.166198 0.986092i \(-0.553149\pi\)
−0.166198 + 0.986092i \(0.553149\pi\)
\(12\) 2.26446 0.653695
\(13\) −1.84712 −0.512299 −0.256149 0.966637i \(-0.582454\pi\)
−0.256149 + 0.966637i \(0.582454\pi\)
\(14\) 0 0
\(15\) 0.608853 0.157205
\(16\) −3.40113 −0.850282
\(17\) −5.88576 −1.42751 −0.713753 0.700397i \(-0.753006\pi\)
−0.713753 + 0.700397i \(0.753006\pi\)
\(18\) −2.06506 −0.486739
\(19\) −1.08814 −0.249637 −0.124818 0.992180i \(-0.539835\pi\)
−0.124818 + 0.992180i \(0.539835\pi\)
\(20\) 1.37873 0.308293
\(21\) 0 0
\(22\) 2.27659 0.485370
\(23\) 1.00000 0.208514
\(24\) −0.546135 −0.111479
\(25\) −4.62930 −0.925860
\(26\) 3.81441 0.748067
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.804169 −0.149330 −0.0746652 0.997209i \(-0.523789\pi\)
−0.0746652 + 0.997209i \(0.523789\pi\)
\(30\) −1.25732 −0.229554
\(31\) −8.77360 −1.57579 −0.787893 0.615813i \(-0.788828\pi\)
−0.787893 + 0.615813i \(0.788828\pi\)
\(32\) 8.11580 1.43468
\(33\) −1.10243 −0.191909
\(34\) 12.1544 2.08447
\(35\) 0 0
\(36\) 2.26446 0.377411
\(37\) 9.11528 1.49854 0.749271 0.662263i \(-0.230404\pi\)
0.749271 + 0.662263i \(0.230404\pi\)
\(38\) 2.24707 0.364524
\(39\) −1.84712 −0.295776
\(40\) −0.332516 −0.0525754
\(41\) 9.79559 1.52981 0.764907 0.644140i \(-0.222785\pi\)
0.764907 + 0.644140i \(0.222785\pi\)
\(42\) 0 0
\(43\) 5.23469 0.798283 0.399142 0.916889i \(-0.369308\pi\)
0.399142 + 0.916889i \(0.369308\pi\)
\(44\) −2.49642 −0.376350
\(45\) 0.608853 0.0907625
\(46\) −2.06506 −0.304476
\(47\) 5.90878 0.861885 0.430942 0.902379i \(-0.358181\pi\)
0.430942 + 0.902379i \(0.358181\pi\)
\(48\) −3.40113 −0.490911
\(49\) 0 0
\(50\) 9.55977 1.35196
\(51\) −5.88576 −0.824171
\(52\) −4.18274 −0.580041
\(53\) 4.51013 0.619515 0.309757 0.950816i \(-0.399752\pi\)
0.309757 + 0.950816i \(0.399752\pi\)
\(54\) −2.06506 −0.281019
\(55\) −0.671220 −0.0905072
\(56\) 0 0
\(57\) −1.08814 −0.144128
\(58\) 1.66066 0.218055
\(59\) 7.40064 0.963481 0.481741 0.876314i \(-0.340005\pi\)
0.481741 + 0.876314i \(0.340005\pi\)
\(60\) 1.37873 0.177993
\(61\) −5.02768 −0.643729 −0.321864 0.946786i \(-0.604310\pi\)
−0.321864 + 0.946786i \(0.604310\pi\)
\(62\) 18.1180 2.30099
\(63\) 0 0
\(64\) −9.95734 −1.24467
\(65\) −1.12462 −0.139492
\(66\) 2.27659 0.280229
\(67\) 2.06305 0.252042 0.126021 0.992028i \(-0.459779\pi\)
0.126021 + 0.992028i \(0.459779\pi\)
\(68\) −13.3281 −1.61627
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.30092 0.510425 0.255213 0.966885i \(-0.417855\pi\)
0.255213 + 0.966885i \(0.417855\pi\)
\(72\) −0.546135 −0.0643627
\(73\) 1.15613 0.135315 0.0676577 0.997709i \(-0.478447\pi\)
0.0676577 + 0.997709i \(0.478447\pi\)
\(74\) −18.8236 −2.18820
\(75\) −4.62930 −0.534545
\(76\) −2.46406 −0.282647
\(77\) 0 0
\(78\) 3.81441 0.431897
\(79\) 15.1716 1.70694 0.853469 0.521144i \(-0.174495\pi\)
0.853469 + 0.521144i \(0.174495\pi\)
\(80\) −2.07079 −0.231521
\(81\) 1.00000 0.111111
\(82\) −20.2285 −2.23386
\(83\) 10.7196 1.17663 0.588315 0.808632i \(-0.299791\pi\)
0.588315 + 0.808632i \(0.299791\pi\)
\(84\) 0 0
\(85\) −3.58356 −0.388692
\(86\) −10.8099 −1.16567
\(87\) −0.804169 −0.0862160
\(88\) 0.602078 0.0641817
\(89\) 7.92167 0.839695 0.419847 0.907595i \(-0.362084\pi\)
0.419847 + 0.907595i \(0.362084\pi\)
\(90\) −1.25732 −0.132533
\(91\) 0 0
\(92\) 2.26446 0.236087
\(93\) −8.77360 −0.909780
\(94\) −12.2020 −1.25854
\(95\) −0.662518 −0.0679729
\(96\) 8.11580 0.828315
\(97\) 17.0648 1.73267 0.866336 0.499461i \(-0.166469\pi\)
0.866336 + 0.499461i \(0.166469\pi\)
\(98\) 0 0
\(99\) −1.10243 −0.110799
\(100\) −10.4829 −1.04829
\(101\) 8.56557 0.852306 0.426153 0.904651i \(-0.359869\pi\)
0.426153 + 0.904651i \(0.359869\pi\)
\(102\) 12.1544 1.20347
\(103\) −10.4373 −1.02842 −0.514208 0.857666i \(-0.671914\pi\)
−0.514208 + 0.857666i \(0.671914\pi\)
\(104\) 1.00878 0.0989187
\(105\) 0 0
\(106\) −9.31369 −0.904625
\(107\) −8.07268 −0.780415 −0.390208 0.920727i \(-0.627597\pi\)
−0.390208 + 0.920727i \(0.627597\pi\)
\(108\) 2.26446 0.217898
\(109\) 3.81163 0.365088 0.182544 0.983198i \(-0.441567\pi\)
0.182544 + 0.983198i \(0.441567\pi\)
\(110\) 1.38611 0.132160
\(111\) 9.11528 0.865184
\(112\) 0 0
\(113\) 8.89428 0.836703 0.418352 0.908285i \(-0.362608\pi\)
0.418352 + 0.908285i \(0.362608\pi\)
\(114\) 2.24707 0.210458
\(115\) 0.608853 0.0567758
\(116\) −1.82101 −0.169077
\(117\) −1.84712 −0.170766
\(118\) −15.2827 −1.40689
\(119\) 0 0
\(120\) −0.332516 −0.0303544
\(121\) −9.78464 −0.889513
\(122\) 10.3825 0.939984
\(123\) 9.79559 0.883239
\(124\) −19.8675 −1.78416
\(125\) −5.86283 −0.524387
\(126\) 0 0
\(127\) 7.91043 0.701937 0.350969 0.936387i \(-0.385852\pi\)
0.350969 + 0.936387i \(0.385852\pi\)
\(128\) 4.33089 0.382800
\(129\) 5.23469 0.460889
\(130\) 2.32241 0.203689
\(131\) −0.475706 −0.0415627 −0.0207813 0.999784i \(-0.506615\pi\)
−0.0207813 + 0.999784i \(0.506615\pi\)
\(132\) −2.49642 −0.217286
\(133\) 0 0
\(134\) −4.26033 −0.368036
\(135\) 0.608853 0.0524017
\(136\) 3.21442 0.275634
\(137\) −8.27302 −0.706812 −0.353406 0.935470i \(-0.614977\pi\)
−0.353406 + 0.935470i \(0.614977\pi\)
\(138\) −2.06506 −0.175789
\(139\) 2.48191 0.210513 0.105257 0.994445i \(-0.466434\pi\)
0.105257 + 0.994445i \(0.466434\pi\)
\(140\) 0 0
\(141\) 5.90878 0.497609
\(142\) −8.88165 −0.745331
\(143\) 2.03633 0.170286
\(144\) −3.40113 −0.283427
\(145\) −0.489621 −0.0406608
\(146\) −2.38749 −0.197590
\(147\) 0 0
\(148\) 20.6412 1.69670
\(149\) 20.5957 1.68726 0.843631 0.536924i \(-0.180414\pi\)
0.843631 + 0.536924i \(0.180414\pi\)
\(150\) 9.55977 0.780552
\(151\) 12.7948 1.04123 0.520613 0.853793i \(-0.325703\pi\)
0.520613 + 0.853793i \(0.325703\pi\)
\(152\) 0.594272 0.0482018
\(153\) −5.88576 −0.475835
\(154\) 0 0
\(155\) −5.34183 −0.429066
\(156\) −4.18274 −0.334887
\(157\) 4.57005 0.364730 0.182365 0.983231i \(-0.441625\pi\)
0.182365 + 0.983231i \(0.441625\pi\)
\(158\) −31.3302 −2.49250
\(159\) 4.51013 0.357677
\(160\) 4.94133 0.390646
\(161\) 0 0
\(162\) −2.06506 −0.162246
\(163\) −21.7544 −1.70393 −0.851967 0.523596i \(-0.824590\pi\)
−0.851967 + 0.523596i \(0.824590\pi\)
\(164\) 22.1818 1.73211
\(165\) −0.671220 −0.0522544
\(166\) −22.1366 −1.71814
\(167\) 3.92341 0.303602 0.151801 0.988411i \(-0.451493\pi\)
0.151801 + 0.988411i \(0.451493\pi\)
\(168\) 0 0
\(169\) −9.58815 −0.737550
\(170\) 7.40026 0.567574
\(171\) −1.08814 −0.0832122
\(172\) 11.8538 0.903842
\(173\) 24.0171 1.82599 0.912993 0.407976i \(-0.133765\pi\)
0.912993 + 0.407976i \(0.133765\pi\)
\(174\) 1.66066 0.125894
\(175\) 0 0
\(176\) 3.74952 0.282630
\(177\) 7.40064 0.556266
\(178\) −16.3587 −1.22614
\(179\) 20.1690 1.50750 0.753752 0.657159i \(-0.228242\pi\)
0.753752 + 0.657159i \(0.228242\pi\)
\(180\) 1.37873 0.102764
\(181\) −16.7087 −1.24195 −0.620973 0.783832i \(-0.713262\pi\)
−0.620973 + 0.783832i \(0.713262\pi\)
\(182\) 0 0
\(183\) −5.02768 −0.371657
\(184\) −0.546135 −0.0402616
\(185\) 5.54986 0.408034
\(186\) 18.1180 1.32848
\(187\) 6.48865 0.474497
\(188\) 13.3802 0.975854
\(189\) 0 0
\(190\) 1.36814 0.0992551
\(191\) 3.50958 0.253944 0.126972 0.991906i \(-0.459474\pi\)
0.126972 + 0.991906i \(0.459474\pi\)
\(192\) −9.95734 −0.718609
\(193\) −13.6422 −0.981987 −0.490993 0.871163i \(-0.663366\pi\)
−0.490993 + 0.871163i \(0.663366\pi\)
\(194\) −35.2399 −2.53008
\(195\) −1.12462 −0.0805360
\(196\) 0 0
\(197\) −13.4651 −0.959349 −0.479675 0.877447i \(-0.659245\pi\)
−0.479675 + 0.877447i \(0.659245\pi\)
\(198\) 2.27659 0.161790
\(199\) −14.2974 −1.01352 −0.506760 0.862087i \(-0.669157\pi\)
−0.506760 + 0.862087i \(0.669157\pi\)
\(200\) 2.52822 0.178772
\(201\) 2.06305 0.145517
\(202\) −17.6884 −1.24455
\(203\) 0 0
\(204\) −13.3281 −0.933153
\(205\) 5.96408 0.416549
\(206\) 21.5536 1.50171
\(207\) 1.00000 0.0695048
\(208\) 6.28229 0.435598
\(209\) 1.19960 0.0829782
\(210\) 0 0
\(211\) 2.83108 0.194900 0.0974499 0.995240i \(-0.468931\pi\)
0.0974499 + 0.995240i \(0.468931\pi\)
\(212\) 10.2130 0.701435
\(213\) 4.30092 0.294694
\(214\) 16.6706 1.13958
\(215\) 3.18716 0.217362
\(216\) −0.546135 −0.0371598
\(217\) 0 0
\(218\) −7.87123 −0.533107
\(219\) 1.15613 0.0781243
\(220\) −1.51995 −0.102475
\(221\) 10.8717 0.731310
\(222\) −18.8236 −1.26336
\(223\) −4.78780 −0.320615 −0.160307 0.987067i \(-0.551249\pi\)
−0.160307 + 0.987067i \(0.551249\pi\)
\(224\) 0 0
\(225\) −4.62930 −0.308620
\(226\) −18.3672 −1.22177
\(227\) −19.2951 −1.28066 −0.640331 0.768099i \(-0.721203\pi\)
−0.640331 + 0.768099i \(0.721203\pi\)
\(228\) −2.46406 −0.163186
\(229\) −8.12954 −0.537215 −0.268607 0.963250i \(-0.586563\pi\)
−0.268607 + 0.963250i \(0.586563\pi\)
\(230\) −1.25732 −0.0829050
\(231\) 0 0
\(232\) 0.439185 0.0288339
\(233\) 7.94940 0.520783 0.260391 0.965503i \(-0.416148\pi\)
0.260391 + 0.965503i \(0.416148\pi\)
\(234\) 3.81441 0.249356
\(235\) 3.59758 0.234680
\(236\) 16.7585 1.09088
\(237\) 15.1716 0.985501
\(238\) 0 0
\(239\) 15.7789 1.02065 0.510326 0.859981i \(-0.329525\pi\)
0.510326 + 0.859981i \(0.329525\pi\)
\(240\) −2.07079 −0.133669
\(241\) 11.2656 0.725680 0.362840 0.931851i \(-0.381807\pi\)
0.362840 + 0.931851i \(0.381807\pi\)
\(242\) 20.2059 1.29888
\(243\) 1.00000 0.0641500
\(244\) −11.3850 −0.728851
\(245\) 0 0
\(246\) −20.2285 −1.28972
\(247\) 2.00993 0.127889
\(248\) 4.79157 0.304265
\(249\) 10.7196 0.679328
\(250\) 12.1071 0.765719
\(251\) 2.27255 0.143442 0.0717209 0.997425i \(-0.477151\pi\)
0.0717209 + 0.997425i \(0.477151\pi\)
\(252\) 0 0
\(253\) −1.10243 −0.0693094
\(254\) −16.3355 −1.02498
\(255\) −3.58356 −0.224411
\(256\) 10.9711 0.685697
\(257\) 0.871285 0.0543492 0.0271746 0.999631i \(-0.491349\pi\)
0.0271746 + 0.999631i \(0.491349\pi\)
\(258\) −10.8099 −0.672998
\(259\) 0 0
\(260\) −2.54667 −0.157938
\(261\) −0.804169 −0.0497768
\(262\) 0.982361 0.0606905
\(263\) −5.33476 −0.328956 −0.164478 0.986381i \(-0.552594\pi\)
−0.164478 + 0.986381i \(0.552594\pi\)
\(264\) 0.602078 0.0370553
\(265\) 2.74601 0.168686
\(266\) 0 0
\(267\) 7.92167 0.484798
\(268\) 4.67171 0.285370
\(269\) −22.7388 −1.38641 −0.693203 0.720742i \(-0.743801\pi\)
−0.693203 + 0.720742i \(0.743801\pi\)
\(270\) −1.25732 −0.0765179
\(271\) 6.42151 0.390079 0.195039 0.980795i \(-0.437517\pi\)
0.195039 + 0.980795i \(0.437517\pi\)
\(272\) 20.0182 1.21378
\(273\) 0 0
\(274\) 17.0843 1.03210
\(275\) 5.10349 0.307752
\(276\) 2.26446 0.136305
\(277\) 14.0353 0.843298 0.421649 0.906759i \(-0.361452\pi\)
0.421649 + 0.906759i \(0.361452\pi\)
\(278\) −5.12530 −0.307395
\(279\) −8.77360 −0.525262
\(280\) 0 0
\(281\) 24.5278 1.46321 0.731603 0.681731i \(-0.238773\pi\)
0.731603 + 0.681731i \(0.238773\pi\)
\(282\) −12.2020 −0.726617
\(283\) 4.39941 0.261518 0.130759 0.991414i \(-0.458259\pi\)
0.130759 + 0.991414i \(0.458259\pi\)
\(284\) 9.73928 0.577920
\(285\) −0.662518 −0.0392442
\(286\) −4.20513 −0.248655
\(287\) 0 0
\(288\) 8.11580 0.478228
\(289\) 17.6422 1.03777
\(290\) 1.01110 0.0593736
\(291\) 17.0648 1.00036
\(292\) 2.61803 0.153208
\(293\) −13.3413 −0.779407 −0.389704 0.920940i \(-0.627423\pi\)
−0.389704 + 0.920940i \(0.627423\pi\)
\(294\) 0 0
\(295\) 4.50590 0.262344
\(296\) −4.97818 −0.289351
\(297\) −1.10243 −0.0639696
\(298\) −42.5312 −2.46377
\(299\) −1.84712 −0.106822
\(300\) −10.4829 −0.605230
\(301\) 0 0
\(302\) −26.4220 −1.52042
\(303\) 8.56557 0.492079
\(304\) 3.70091 0.212262
\(305\) −3.06112 −0.175279
\(306\) 12.1544 0.694823
\(307\) −24.5499 −1.40114 −0.700569 0.713585i \(-0.747070\pi\)
−0.700569 + 0.713585i \(0.747070\pi\)
\(308\) 0 0
\(309\) −10.4373 −0.593756
\(310\) 11.0312 0.626530
\(311\) −26.0773 −1.47871 −0.739355 0.673315i \(-0.764870\pi\)
−0.739355 + 0.673315i \(0.764870\pi\)
\(312\) 1.00878 0.0571108
\(313\) −24.1973 −1.36771 −0.683857 0.729616i \(-0.739699\pi\)
−0.683857 + 0.729616i \(0.739699\pi\)
\(314\) −9.43741 −0.532584
\(315\) 0 0
\(316\) 34.3555 1.93265
\(317\) 3.60976 0.202744 0.101372 0.994849i \(-0.467677\pi\)
0.101372 + 0.994849i \(0.467677\pi\)
\(318\) −9.31369 −0.522286
\(319\) 0.886542 0.0496368
\(320\) −6.06256 −0.338907
\(321\) −8.07268 −0.450573
\(322\) 0 0
\(323\) 6.40453 0.356358
\(324\) 2.26446 0.125804
\(325\) 8.55087 0.474317
\(326\) 44.9240 2.48811
\(327\) 3.81163 0.210784
\(328\) −5.34972 −0.295389
\(329\) 0 0
\(330\) 1.38611 0.0763027
\(331\) 8.23405 0.452584 0.226292 0.974059i \(-0.427340\pi\)
0.226292 + 0.974059i \(0.427340\pi\)
\(332\) 24.2742 1.33222
\(333\) 9.11528 0.499514
\(334\) −8.10206 −0.443325
\(335\) 1.25610 0.0686279
\(336\) 0 0
\(337\) 26.1865 1.42647 0.713235 0.700925i \(-0.247229\pi\)
0.713235 + 0.700925i \(0.247229\pi\)
\(338\) 19.8001 1.07698
\(339\) 8.89428 0.483071
\(340\) −8.11485 −0.440090
\(341\) 9.67231 0.523785
\(342\) 2.24707 0.121508
\(343\) 0 0
\(344\) −2.85885 −0.154139
\(345\) 0.608853 0.0327795
\(346\) −49.5967 −2.66633
\(347\) −5.92735 −0.318197 −0.159098 0.987263i \(-0.550859\pi\)
−0.159098 + 0.987263i \(0.550859\pi\)
\(348\) −1.82101 −0.0976165
\(349\) −29.6319 −1.58616 −0.793080 0.609118i \(-0.791524\pi\)
−0.793080 + 0.609118i \(0.791524\pi\)
\(350\) 0 0
\(351\) −1.84712 −0.0985919
\(352\) −8.94712 −0.476883
\(353\) 14.2388 0.757853 0.378926 0.925427i \(-0.376293\pi\)
0.378926 + 0.925427i \(0.376293\pi\)
\(354\) −15.2827 −0.812269
\(355\) 2.61863 0.138982
\(356\) 17.9383 0.950730
\(357\) 0 0
\(358\) −41.6502 −2.20128
\(359\) 2.78230 0.146844 0.0734220 0.997301i \(-0.476608\pi\)
0.0734220 + 0.997301i \(0.476608\pi\)
\(360\) −0.332516 −0.0175251
\(361\) −17.8159 −0.937682
\(362\) 34.5044 1.81351
\(363\) −9.78464 −0.513561
\(364\) 0 0
\(365\) 0.703916 0.0368447
\(366\) 10.3825 0.542700
\(367\) 6.66447 0.347883 0.173941 0.984756i \(-0.444350\pi\)
0.173941 + 0.984756i \(0.444350\pi\)
\(368\) −3.40113 −0.177296
\(369\) 9.79559 0.509938
\(370\) −11.4608 −0.595818
\(371\) 0 0
\(372\) −19.8675 −1.03008
\(373\) −19.0824 −0.988048 −0.494024 0.869448i \(-0.664475\pi\)
−0.494024 + 0.869448i \(0.664475\pi\)
\(374\) −13.3994 −0.692869
\(375\) −5.86283 −0.302755
\(376\) −3.22700 −0.166420
\(377\) 1.48540 0.0765018
\(378\) 0 0
\(379\) 13.2631 0.681282 0.340641 0.940193i \(-0.389356\pi\)
0.340641 + 0.940193i \(0.389356\pi\)
\(380\) −1.50025 −0.0769611
\(381\) 7.91043 0.405264
\(382\) −7.24748 −0.370813
\(383\) 3.11602 0.159221 0.0796105 0.996826i \(-0.474632\pi\)
0.0796105 + 0.996826i \(0.474632\pi\)
\(384\) 4.33089 0.221010
\(385\) 0 0
\(386\) 28.1719 1.43391
\(387\) 5.23469 0.266094
\(388\) 38.6427 1.96179
\(389\) −9.98409 −0.506214 −0.253107 0.967438i \(-0.581452\pi\)
−0.253107 + 0.967438i \(0.581452\pi\)
\(390\) 2.32241 0.117600
\(391\) −5.88576 −0.297656
\(392\) 0 0
\(393\) −0.475706 −0.0239962
\(394\) 27.8062 1.40086
\(395\) 9.23727 0.464777
\(396\) −2.49642 −0.125450
\(397\) −5.64444 −0.283286 −0.141643 0.989918i \(-0.545239\pi\)
−0.141643 + 0.989918i \(0.545239\pi\)
\(398\) 29.5251 1.47996
\(399\) 0 0
\(400\) 15.7448 0.787242
\(401\) 38.5622 1.92571 0.962853 0.270028i \(-0.0870328\pi\)
0.962853 + 0.270028i \(0.0870328\pi\)
\(402\) −4.26033 −0.212486
\(403\) 16.2059 0.807273
\(404\) 19.3964 0.965008
\(405\) 0.608853 0.0302542
\(406\) 0 0
\(407\) −10.0490 −0.498110
\(408\) 3.21442 0.159138
\(409\) 36.8071 1.81999 0.909997 0.414614i \(-0.136083\pi\)
0.909997 + 0.414614i \(0.136083\pi\)
\(410\) −12.3162 −0.608252
\(411\) −8.27302 −0.408078
\(412\) −23.6349 −1.16441
\(413\) 0 0
\(414\) −2.06506 −0.101492
\(415\) 6.52667 0.320382
\(416\) −14.9908 −0.734987
\(417\) 2.48191 0.121540
\(418\) −2.47725 −0.121166
\(419\) 6.45390 0.315293 0.157647 0.987496i \(-0.449609\pi\)
0.157647 + 0.987496i \(0.449609\pi\)
\(420\) 0 0
\(421\) 29.9287 1.45863 0.729317 0.684176i \(-0.239838\pi\)
0.729317 + 0.684176i \(0.239838\pi\)
\(422\) −5.84635 −0.284596
\(423\) 5.90878 0.287295
\(424\) −2.46314 −0.119621
\(425\) 27.2469 1.32167
\(426\) −8.88165 −0.430317
\(427\) 0 0
\(428\) −18.2803 −0.883612
\(429\) 2.03633 0.0983147
\(430\) −6.58167 −0.317396
\(431\) 7.74839 0.373227 0.186613 0.982433i \(-0.440249\pi\)
0.186613 + 0.982433i \(0.440249\pi\)
\(432\) −3.40113 −0.163637
\(433\) −5.17696 −0.248789 −0.124394 0.992233i \(-0.539699\pi\)
−0.124394 + 0.992233i \(0.539699\pi\)
\(434\) 0 0
\(435\) −0.489621 −0.0234755
\(436\) 8.63130 0.413364
\(437\) −1.08814 −0.0520528
\(438\) −2.38749 −0.114078
\(439\) 30.2921 1.44576 0.722881 0.690973i \(-0.242818\pi\)
0.722881 + 0.690973i \(0.242818\pi\)
\(440\) 0.366577 0.0174759
\(441\) 0 0
\(442\) −22.4507 −1.06787
\(443\) 38.2021 1.81504 0.907519 0.420012i \(-0.137974\pi\)
0.907519 + 0.420012i \(0.137974\pi\)
\(444\) 20.6412 0.979589
\(445\) 4.82313 0.228638
\(446\) 9.88708 0.468167
\(447\) 20.5957 0.974141
\(448\) 0 0
\(449\) 2.95568 0.139487 0.0697435 0.997565i \(-0.477782\pi\)
0.0697435 + 0.997565i \(0.477782\pi\)
\(450\) 9.55977 0.450652
\(451\) −10.7990 −0.508504
\(452\) 20.1408 0.947343
\(453\) 12.7948 0.601152
\(454\) 39.8455 1.87004
\(455\) 0 0
\(456\) 0.594272 0.0278293
\(457\) 17.6954 0.827758 0.413879 0.910332i \(-0.364174\pi\)
0.413879 + 0.910332i \(0.364174\pi\)
\(458\) 16.7880 0.784450
\(459\) −5.88576 −0.274724
\(460\) 1.37873 0.0642834
\(461\) 11.7447 0.547005 0.273503 0.961871i \(-0.411818\pi\)
0.273503 + 0.961871i \(0.411818\pi\)
\(462\) 0 0
\(463\) 27.0872 1.25885 0.629424 0.777062i \(-0.283291\pi\)
0.629424 + 0.777062i \(0.283291\pi\)
\(464\) 2.73508 0.126973
\(465\) −5.34183 −0.247722
\(466\) −16.4160 −0.760455
\(467\) −23.1520 −1.07135 −0.535673 0.844426i \(-0.679942\pi\)
−0.535673 + 0.844426i \(0.679942\pi\)
\(468\) −4.18274 −0.193347
\(469\) 0 0
\(470\) −7.42921 −0.342684
\(471\) 4.57005 0.210577
\(472\) −4.04175 −0.186037
\(473\) −5.77090 −0.265346
\(474\) −31.3302 −1.43904
\(475\) 5.03733 0.231128
\(476\) 0 0
\(477\) 4.51013 0.206505
\(478\) −32.5843 −1.49037
\(479\) −22.3675 −1.02200 −0.510999 0.859581i \(-0.670725\pi\)
−0.510999 + 0.859581i \(0.670725\pi\)
\(480\) 4.94133 0.225540
\(481\) −16.8370 −0.767701
\(482\) −23.2641 −1.05965
\(483\) 0 0
\(484\) −22.1570 −1.00714
\(485\) 10.3900 0.471785
\(486\) −2.06506 −0.0936729
\(487\) 41.8522 1.89650 0.948252 0.317518i \(-0.102849\pi\)
0.948252 + 0.317518i \(0.102849\pi\)
\(488\) 2.74580 0.124296
\(489\) −21.7544 −0.983766
\(490\) 0 0
\(491\) 16.2658 0.734063 0.367032 0.930209i \(-0.380374\pi\)
0.367032 + 0.930209i \(0.380374\pi\)
\(492\) 22.1818 1.00003
\(493\) 4.73315 0.213170
\(494\) −4.15061 −0.186745
\(495\) −0.671220 −0.0301691
\(496\) 29.8401 1.33986
\(497\) 0 0
\(498\) −22.1366 −0.991966
\(499\) −19.8250 −0.887489 −0.443744 0.896153i \(-0.646350\pi\)
−0.443744 + 0.896153i \(0.646350\pi\)
\(500\) −13.2762 −0.593728
\(501\) 3.92341 0.175285
\(502\) −4.69294 −0.209456
\(503\) −9.18201 −0.409406 −0.204703 0.978824i \(-0.565623\pi\)
−0.204703 + 0.978824i \(0.565623\pi\)
\(504\) 0 0
\(505\) 5.21517 0.232072
\(506\) 2.27659 0.101207
\(507\) −9.58815 −0.425825
\(508\) 17.9129 0.794756
\(509\) 32.8014 1.45390 0.726948 0.686692i \(-0.240938\pi\)
0.726948 + 0.686692i \(0.240938\pi\)
\(510\) 7.40026 0.327689
\(511\) 0 0
\(512\) −31.3178 −1.38407
\(513\) −1.08814 −0.0480426
\(514\) −1.79925 −0.0793617
\(515\) −6.35477 −0.280025
\(516\) 11.8538 0.521834
\(517\) −6.51404 −0.286487
\(518\) 0 0
\(519\) 24.0171 1.05423
\(520\) 0.614197 0.0269343
\(521\) −6.86356 −0.300698 −0.150349 0.988633i \(-0.548040\pi\)
−0.150349 + 0.988633i \(0.548040\pi\)
\(522\) 1.66066 0.0726849
\(523\) −9.76622 −0.427047 −0.213524 0.976938i \(-0.568494\pi\)
−0.213524 + 0.976938i \(0.568494\pi\)
\(524\) −1.07722 −0.0470586
\(525\) 0 0
\(526\) 11.0166 0.480347
\(527\) 51.6393 2.24944
\(528\) 3.74952 0.163177
\(529\) 1.00000 0.0434783
\(530\) −5.67067 −0.246318
\(531\) 7.40064 0.321160
\(532\) 0 0
\(533\) −18.0936 −0.783722
\(534\) −16.3587 −0.707910
\(535\) −4.91508 −0.212497
\(536\) −1.12671 −0.0486663
\(537\) 20.1690 0.870357
\(538\) 46.9568 2.02445
\(539\) 0 0
\(540\) 1.37873 0.0593309
\(541\) −4.64207 −0.199578 −0.0997891 0.995009i \(-0.531817\pi\)
−0.0997891 + 0.995009i \(0.531817\pi\)
\(542\) −13.2608 −0.569600
\(543\) −16.7087 −0.717038
\(544\) −47.7676 −2.04802
\(545\) 2.32072 0.0994088
\(546\) 0 0
\(547\) 21.6976 0.927722 0.463861 0.885908i \(-0.346464\pi\)
0.463861 + 0.885908i \(0.346464\pi\)
\(548\) −18.7340 −0.800275
\(549\) −5.02768 −0.214576
\(550\) −10.5390 −0.449385
\(551\) 0.875049 0.0372783
\(552\) −0.546135 −0.0232451
\(553\) 0 0
\(554\) −28.9836 −1.23140
\(555\) 5.54986 0.235579
\(556\) 5.62021 0.238350
\(557\) −32.1596 −1.36265 −0.681323 0.731983i \(-0.738595\pi\)
−0.681323 + 0.731983i \(0.738595\pi\)
\(558\) 18.1180 0.766996
\(559\) −9.66910 −0.408960
\(560\) 0 0
\(561\) 6.48865 0.273951
\(562\) −50.6513 −2.13660
\(563\) 12.3159 0.519055 0.259528 0.965736i \(-0.416433\pi\)
0.259528 + 0.965736i \(0.416433\pi\)
\(564\) 13.3802 0.563409
\(565\) 5.41531 0.227824
\(566\) −9.08503 −0.381872
\(567\) 0 0
\(568\) −2.34888 −0.0985570
\(569\) −26.2350 −1.09983 −0.549914 0.835221i \(-0.685340\pi\)
−0.549914 + 0.835221i \(0.685340\pi\)
\(570\) 1.36814 0.0573050
\(571\) −5.59829 −0.234281 −0.117141 0.993115i \(-0.537373\pi\)
−0.117141 + 0.993115i \(0.537373\pi\)
\(572\) 4.61119 0.192803
\(573\) 3.50958 0.146615
\(574\) 0 0
\(575\) −4.62930 −0.193055
\(576\) −9.95734 −0.414889
\(577\) 10.1348 0.421915 0.210958 0.977495i \(-0.432342\pi\)
0.210958 + 0.977495i \(0.432342\pi\)
\(578\) −36.4321 −1.51538
\(579\) −13.6422 −0.566950
\(580\) −1.10873 −0.0460375
\(581\) 0 0
\(582\) −35.2399 −1.46074
\(583\) −4.97212 −0.205924
\(584\) −0.631406 −0.0261278
\(585\) −1.12462 −0.0464975
\(586\) 27.5506 1.13810
\(587\) −27.2323 −1.12400 −0.561999 0.827138i \(-0.689968\pi\)
−0.561999 + 0.827138i \(0.689968\pi\)
\(588\) 0 0
\(589\) 9.54691 0.393374
\(590\) −9.30495 −0.383079
\(591\) −13.4651 −0.553880
\(592\) −31.0022 −1.27418
\(593\) −38.3547 −1.57504 −0.787519 0.616290i \(-0.788635\pi\)
−0.787519 + 0.616290i \(0.788635\pi\)
\(594\) 2.27659 0.0934095
\(595\) 0 0
\(596\) 46.6381 1.91037
\(597\) −14.2974 −0.585155
\(598\) 3.81441 0.155983
\(599\) −6.88378 −0.281264 −0.140632 0.990062i \(-0.544913\pi\)
−0.140632 + 0.990062i \(0.544913\pi\)
\(600\) 2.52822 0.103214
\(601\) −1.80012 −0.0734283 −0.0367141 0.999326i \(-0.511689\pi\)
−0.0367141 + 0.999326i \(0.511689\pi\)
\(602\) 0 0
\(603\) 2.06305 0.0840141
\(604\) 28.9734 1.17891
\(605\) −5.95741 −0.242203
\(606\) −17.6884 −0.718542
\(607\) −2.93517 −0.119135 −0.0595674 0.998224i \(-0.518972\pi\)
−0.0595674 + 0.998224i \(0.518972\pi\)
\(608\) −8.83113 −0.358150
\(609\) 0 0
\(610\) 6.32139 0.255946
\(611\) −10.9142 −0.441542
\(612\) −13.3281 −0.538756
\(613\) 31.7390 1.28192 0.640962 0.767572i \(-0.278535\pi\)
0.640962 + 0.767572i \(0.278535\pi\)
\(614\) 50.6970 2.04596
\(615\) 5.96408 0.240495
\(616\) 0 0
\(617\) −41.7807 −1.68203 −0.841013 0.541015i \(-0.818040\pi\)
−0.841013 + 0.541015i \(0.818040\pi\)
\(618\) 21.5536 0.867012
\(619\) −27.2170 −1.09394 −0.546972 0.837151i \(-0.684220\pi\)
−0.546972 + 0.837151i \(0.684220\pi\)
\(620\) −12.0964 −0.485803
\(621\) 1.00000 0.0401286
\(622\) 53.8512 2.15924
\(623\) 0 0
\(624\) 6.28229 0.251493
\(625\) 19.5769 0.783076
\(626\) 49.9689 1.99716
\(627\) 1.19960 0.0479075
\(628\) 10.3487 0.412959
\(629\) −53.6503 −2.13918
\(630\) 0 0
\(631\) 25.8728 1.02998 0.514990 0.857196i \(-0.327795\pi\)
0.514990 + 0.857196i \(0.327795\pi\)
\(632\) −8.28574 −0.329589
\(633\) 2.83108 0.112525
\(634\) −7.45437 −0.296051
\(635\) 4.81629 0.191129
\(636\) 10.2130 0.404973
\(637\) 0 0
\(638\) −1.83076 −0.0724805
\(639\) 4.30092 0.170142
\(640\) 2.63687 0.104232
\(641\) −13.3591 −0.527651 −0.263826 0.964570i \(-0.584984\pi\)
−0.263826 + 0.964570i \(0.584984\pi\)
\(642\) 16.6706 0.657934
\(643\) −43.4719 −1.71436 −0.857182 0.515014i \(-0.827787\pi\)
−0.857182 + 0.515014i \(0.827787\pi\)
\(644\) 0 0
\(645\) 3.18716 0.125494
\(646\) −13.2257 −0.520360
\(647\) 46.1515 1.81440 0.907201 0.420698i \(-0.138215\pi\)
0.907201 + 0.420698i \(0.138215\pi\)
\(648\) −0.546135 −0.0214542
\(649\) −8.15871 −0.320257
\(650\) −17.6580 −0.692605
\(651\) 0 0
\(652\) −49.2620 −1.92925
\(653\) −33.5965 −1.31473 −0.657367 0.753570i \(-0.728330\pi\)
−0.657367 + 0.753570i \(0.728330\pi\)
\(654\) −7.87123 −0.307790
\(655\) −0.289635 −0.0113170
\(656\) −33.3161 −1.30077
\(657\) 1.15613 0.0451051
\(658\) 0 0
\(659\) 10.2122 0.397809 0.198905 0.980019i \(-0.436262\pi\)
0.198905 + 0.980019i \(0.436262\pi\)
\(660\) −1.51995 −0.0591641
\(661\) −18.0200 −0.700895 −0.350448 0.936582i \(-0.613971\pi\)
−0.350448 + 0.936582i \(0.613971\pi\)
\(662\) −17.0038 −0.660871
\(663\) 10.8717 0.422222
\(664\) −5.85436 −0.227193
\(665\) 0 0
\(666\) −18.8236 −0.729399
\(667\) −0.804169 −0.0311375
\(668\) 8.88442 0.343748
\(669\) −4.78780 −0.185107
\(670\) −2.59391 −0.100212
\(671\) 5.54268 0.213973
\(672\) 0 0
\(673\) −40.5777 −1.56415 −0.782077 0.623181i \(-0.785840\pi\)
−0.782077 + 0.623181i \(0.785840\pi\)
\(674\) −54.0767 −2.08296
\(675\) −4.62930 −0.178182
\(676\) −21.7120 −0.835078
\(677\) −25.9423 −0.997044 −0.498522 0.866877i \(-0.666124\pi\)
−0.498522 + 0.866877i \(0.666124\pi\)
\(678\) −18.3672 −0.705388
\(679\) 0 0
\(680\) 1.95711 0.0750518
\(681\) −19.2951 −0.739390
\(682\) −19.9739 −0.764839
\(683\) −25.8167 −0.987848 −0.493924 0.869505i \(-0.664438\pi\)
−0.493924 + 0.869505i \(0.664438\pi\)
\(684\) −2.46406 −0.0942156
\(685\) −5.03705 −0.192456
\(686\) 0 0
\(687\) −8.12954 −0.310161
\(688\) −17.8039 −0.678766
\(689\) −8.33076 −0.317377
\(690\) −1.25732 −0.0478652
\(691\) 15.6020 0.593528 0.296764 0.954951i \(-0.404093\pi\)
0.296764 + 0.954951i \(0.404093\pi\)
\(692\) 54.3859 2.06744
\(693\) 0 0
\(694\) 12.2403 0.464636
\(695\) 1.51112 0.0573201
\(696\) 0.439185 0.0166473
\(697\) −57.6545 −2.18382
\(698\) 61.1916 2.31614
\(699\) 7.94940 0.300674
\(700\) 0 0
\(701\) 38.6350 1.45922 0.729612 0.683861i \(-0.239701\pi\)
0.729612 + 0.683861i \(0.239701\pi\)
\(702\) 3.81441 0.143966
\(703\) −9.91871 −0.374091
\(704\) 10.9773 0.413722
\(705\) 3.59758 0.135493
\(706\) −29.4039 −1.10663
\(707\) 0 0
\(708\) 16.7585 0.629823
\(709\) −43.3432 −1.62779 −0.813894 0.581013i \(-0.802656\pi\)
−0.813894 + 0.581013i \(0.802656\pi\)
\(710\) −5.40762 −0.202944
\(711\) 15.1716 0.568979
\(712\) −4.32630 −0.162135
\(713\) −8.77360 −0.328574
\(714\) 0 0
\(715\) 1.23982 0.0463667
\(716\) 45.6720 1.70684
\(717\) 15.7789 0.589274
\(718\) −5.74560 −0.214424
\(719\) 29.4485 1.09824 0.549122 0.835742i \(-0.314962\pi\)
0.549122 + 0.835742i \(0.314962\pi\)
\(720\) −2.07079 −0.0771737
\(721\) 0 0
\(722\) 36.7910 1.36922
\(723\) 11.2656 0.418971
\(724\) −37.8362 −1.40617
\(725\) 3.72274 0.138259
\(726\) 20.2059 0.749910
\(727\) 50.1254 1.85905 0.929524 0.368763i \(-0.120218\pi\)
0.929524 + 0.368763i \(0.120218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.45363 −0.0538012
\(731\) −30.8101 −1.13955
\(732\) −11.3850 −0.420802
\(733\) 20.9760 0.774766 0.387383 0.921919i \(-0.373379\pi\)
0.387383 + 0.921919i \(0.373379\pi\)
\(734\) −13.7625 −0.507984
\(735\) 0 0
\(736\) 8.11580 0.299152
\(737\) −2.27438 −0.0837778
\(738\) −20.2285 −0.744620
\(739\) 11.5852 0.426168 0.213084 0.977034i \(-0.431649\pi\)
0.213084 + 0.977034i \(0.431649\pi\)
\(740\) 12.5675 0.461990
\(741\) 2.00993 0.0738365
\(742\) 0 0
\(743\) −44.7847 −1.64299 −0.821495 0.570216i \(-0.806860\pi\)
−0.821495 + 0.570216i \(0.806860\pi\)
\(744\) 4.79157 0.175668
\(745\) 12.5397 0.459420
\(746\) 39.4062 1.44276
\(747\) 10.7196 0.392210
\(748\) 14.6933 0.537241
\(749\) 0 0
\(750\) 12.1071 0.442088
\(751\) −22.2887 −0.813325 −0.406662 0.913579i \(-0.633307\pi\)
−0.406662 + 0.913579i \(0.633307\pi\)
\(752\) −20.0965 −0.732845
\(753\) 2.27255 0.0828162
\(754\) −3.06743 −0.111709
\(755\) 7.79015 0.283513
\(756\) 0 0
\(757\) −34.3724 −1.24929 −0.624643 0.780910i \(-0.714756\pi\)
−0.624643 + 0.780910i \(0.714756\pi\)
\(758\) −27.3892 −0.994820
\(759\) −1.10243 −0.0400158
\(760\) 0.361824 0.0131248
\(761\) −43.7206 −1.58487 −0.792435 0.609956i \(-0.791187\pi\)
−0.792435 + 0.609956i \(0.791187\pi\)
\(762\) −16.3355 −0.591773
\(763\) 0 0
\(764\) 7.94731 0.287524
\(765\) −3.58356 −0.129564
\(766\) −6.43476 −0.232497
\(767\) −13.6699 −0.493590
\(768\) 10.9711 0.395887
\(769\) 2.96450 0.106903 0.0534513 0.998570i \(-0.482978\pi\)
0.0534513 + 0.998570i \(0.482978\pi\)
\(770\) 0 0
\(771\) 0.871285 0.0313785
\(772\) −30.8923 −1.11184
\(773\) −22.2038 −0.798615 −0.399307 0.916817i \(-0.630749\pi\)
−0.399307 + 0.916817i \(0.630749\pi\)
\(774\) −10.8099 −0.388556
\(775\) 40.6156 1.45896
\(776\) −9.31971 −0.334558
\(777\) 0 0
\(778\) 20.6177 0.739181
\(779\) −10.6590 −0.381898
\(780\) −2.54667 −0.0911855
\(781\) −4.74147 −0.169663
\(782\) 12.1544 0.434642
\(783\) −0.804169 −0.0287387
\(784\) 0 0
\(785\) 2.78249 0.0993112
\(786\) 0.982361 0.0350397
\(787\) 34.6515 1.23519 0.617596 0.786495i \(-0.288107\pi\)
0.617596 + 0.786495i \(0.288107\pi\)
\(788\) −30.4913 −1.08621
\(789\) −5.33476 −0.189923
\(790\) −19.0755 −0.678676
\(791\) 0 0
\(792\) 0.602078 0.0213939
\(793\) 9.28673 0.329781
\(794\) 11.6561 0.413660
\(795\) 2.74601 0.0973909
\(796\) −32.3761 −1.14754
\(797\) 22.4200 0.794158 0.397079 0.917784i \(-0.370024\pi\)
0.397079 + 0.917784i \(0.370024\pi\)
\(798\) 0 0
\(799\) −34.7777 −1.23035
\(800\) −37.5705 −1.32832
\(801\) 7.92167 0.279898
\(802\) −79.6332 −2.81195
\(803\) −1.27456 −0.0449783
\(804\) 4.67171 0.164759
\(805\) 0 0
\(806\) −33.4661 −1.17879
\(807\) −22.7388 −0.800442
\(808\) −4.67796 −0.164570
\(809\) 26.0244 0.914970 0.457485 0.889217i \(-0.348750\pi\)
0.457485 + 0.889217i \(0.348750\pi\)
\(810\) −1.25732 −0.0441776
\(811\) 45.2644 1.58945 0.794724 0.606970i \(-0.207615\pi\)
0.794724 + 0.606970i \(0.207615\pi\)
\(812\) 0 0
\(813\) 6.42151 0.225212
\(814\) 20.7517 0.727348
\(815\) −13.2452 −0.463959
\(816\) 20.0182 0.700778
\(817\) −5.69608 −0.199281
\(818\) −76.0088 −2.65759
\(819\) 0 0
\(820\) 13.5054 0.471630
\(821\) 7.28729 0.254328 0.127164 0.991882i \(-0.459413\pi\)
0.127164 + 0.991882i \(0.459413\pi\)
\(822\) 17.0843 0.595882
\(823\) 3.29869 0.114985 0.0574925 0.998346i \(-0.481689\pi\)
0.0574925 + 0.998346i \(0.481689\pi\)
\(824\) 5.70017 0.198575
\(825\) 5.10349 0.177681
\(826\) 0 0
\(827\) −48.8022 −1.69702 −0.848510 0.529179i \(-0.822500\pi\)
−0.848510 + 0.529179i \(0.822500\pi\)
\(828\) 2.26446 0.0786956
\(829\) −24.7978 −0.861263 −0.430631 0.902528i \(-0.641709\pi\)
−0.430631 + 0.902528i \(0.641709\pi\)
\(830\) −13.4780 −0.467827
\(831\) 14.0353 0.486878
\(832\) 18.3924 0.637642
\(833\) 0 0
\(834\) −5.12530 −0.177475
\(835\) 2.38878 0.0826671
\(836\) 2.71646 0.0939506
\(837\) −8.77360 −0.303260
\(838\) −13.3277 −0.460397
\(839\) 6.27230 0.216544 0.108272 0.994121i \(-0.465468\pi\)
0.108272 + 0.994121i \(0.465468\pi\)
\(840\) 0 0
\(841\) −28.3533 −0.977700
\(842\) −61.8044 −2.12992
\(843\) 24.5278 0.844782
\(844\) 6.41089 0.220672
\(845\) −5.83777 −0.200826
\(846\) −12.2020 −0.419513
\(847\) 0 0
\(848\) −15.3395 −0.526762
\(849\) 4.39941 0.150987
\(850\) −56.2665 −1.92993
\(851\) 9.11528 0.312468
\(852\) 9.73928 0.333662
\(853\) 1.75188 0.0599834 0.0299917 0.999550i \(-0.490452\pi\)
0.0299917 + 0.999550i \(0.490452\pi\)
\(854\) 0 0
\(855\) −0.662518 −0.0226576
\(856\) 4.40878 0.150689
\(857\) 15.0955 0.515653 0.257826 0.966191i \(-0.416994\pi\)
0.257826 + 0.966191i \(0.416994\pi\)
\(858\) −4.20513 −0.143561
\(859\) −9.58801 −0.327139 −0.163569 0.986532i \(-0.552301\pi\)
−0.163569 + 0.986532i \(0.552301\pi\)
\(860\) 7.21721 0.246105
\(861\) 0 0
\(862\) −16.0009 −0.544992
\(863\) 34.3335 1.16872 0.584362 0.811493i \(-0.301345\pi\)
0.584362 + 0.811493i \(0.301345\pi\)
\(864\) 8.11580 0.276105
\(865\) 14.6229 0.497193
\(866\) 10.6907 0.363285
\(867\) 17.6422 0.599159
\(868\) 0 0
\(869\) −16.7257 −0.567379
\(870\) 1.01110 0.0342793
\(871\) −3.81071 −0.129121
\(872\) −2.08166 −0.0704941
\(873\) 17.0648 0.577557
\(874\) 2.24707 0.0760084
\(875\) 0 0
\(876\) 2.61803 0.0884549
\(877\) 17.6058 0.594505 0.297252 0.954799i \(-0.403930\pi\)
0.297252 + 0.954799i \(0.403930\pi\)
\(878\) −62.5549 −2.11112
\(879\) −13.3413 −0.449991
\(880\) 2.28290 0.0769567
\(881\) −39.5494 −1.33245 −0.666227 0.745749i \(-0.732092\pi\)
−0.666227 + 0.745749i \(0.732092\pi\)
\(882\) 0 0
\(883\) 0.340528 0.0114597 0.00572983 0.999984i \(-0.498176\pi\)
0.00572983 + 0.999984i \(0.498176\pi\)
\(884\) 24.6186 0.828013
\(885\) 4.50590 0.151464
\(886\) −78.8896 −2.65035
\(887\) −30.0962 −1.01053 −0.505266 0.862964i \(-0.668606\pi\)
−0.505266 + 0.862964i \(0.668606\pi\)
\(888\) −4.97818 −0.167057
\(889\) 0 0
\(890\) −9.96004 −0.333861
\(891\) −1.10243 −0.0369329
\(892\) −10.8418 −0.363010
\(893\) −6.42959 −0.215158
\(894\) −42.5312 −1.42246
\(895\) 12.2800 0.410474
\(896\) 0 0
\(897\) −1.84712 −0.0616735
\(898\) −6.10364 −0.203681
\(899\) 7.05546 0.235313
\(900\) −10.4829 −0.349429
\(901\) −26.5456 −0.884361
\(902\) 22.3005 0.742526
\(903\) 0 0
\(904\) −4.85748 −0.161557
\(905\) −10.1731 −0.338166
\(906\) −26.4220 −0.877812
\(907\) 31.8853 1.05873 0.529366 0.848393i \(-0.322430\pi\)
0.529366 + 0.848393i \(0.322430\pi\)
\(908\) −43.6931 −1.45001
\(909\) 8.56557 0.284102
\(910\) 0 0
\(911\) 36.7297 1.21691 0.608454 0.793589i \(-0.291790\pi\)
0.608454 + 0.793589i \(0.291790\pi\)
\(912\) 3.70091 0.122549
\(913\) −11.8177 −0.391107
\(914\) −36.5421 −1.20871
\(915\) −3.06112 −0.101198
\(916\) −18.4090 −0.608252
\(917\) 0 0
\(918\) 12.1544 0.401156
\(919\) −29.3315 −0.967556 −0.483778 0.875191i \(-0.660736\pi\)
−0.483778 + 0.875191i \(0.660736\pi\)
\(920\) −0.332516 −0.0109627
\(921\) −24.5499 −0.808947
\(922\) −24.2535 −0.798746
\(923\) −7.94431 −0.261490
\(924\) 0 0
\(925\) −42.1973 −1.38744
\(926\) −55.9366 −1.83819
\(927\) −10.4373 −0.342805
\(928\) −6.52647 −0.214242
\(929\) 1.30704 0.0428826 0.0214413 0.999770i \(-0.493174\pi\)
0.0214413 + 0.999770i \(0.493174\pi\)
\(930\) 11.0312 0.361727
\(931\) 0 0
\(932\) 18.0011 0.589647
\(933\) −26.0773 −0.853734
\(934\) 47.8102 1.56440
\(935\) 3.95064 0.129200
\(936\) 1.00878 0.0329729
\(937\) 4.77786 0.156086 0.0780430 0.996950i \(-0.475133\pi\)
0.0780430 + 0.996950i \(0.475133\pi\)
\(938\) 0 0
\(939\) −24.1973 −0.789650
\(940\) 8.14660 0.265713
\(941\) 33.4004 1.08882 0.544411 0.838819i \(-0.316753\pi\)
0.544411 + 0.838819i \(0.316753\pi\)
\(942\) −9.43741 −0.307488
\(943\) 9.79559 0.318988
\(944\) −25.1705 −0.819231
\(945\) 0 0
\(946\) 11.9172 0.387463
\(947\) 35.8911 1.16630 0.583152 0.812363i \(-0.301819\pi\)
0.583152 + 0.812363i \(0.301819\pi\)
\(948\) 34.3555 1.11582
\(949\) −2.13552 −0.0693219
\(950\) −10.4024 −0.337498
\(951\) 3.60976 0.117055
\(952\) 0 0
\(953\) −42.4876 −1.37631 −0.688154 0.725564i \(-0.741579\pi\)
−0.688154 + 0.725564i \(0.741579\pi\)
\(954\) −9.31369 −0.301542
\(955\) 2.13682 0.0691457
\(956\) 35.7308 1.15562
\(957\) 0.886542 0.0286578
\(958\) 46.1903 1.49234
\(959\) 0 0
\(960\) −6.06256 −0.195668
\(961\) 45.9761 1.48310
\(962\) 34.7694 1.12101
\(963\) −8.07268 −0.260138
\(964\) 25.5105 0.821638
\(965\) −8.30609 −0.267383
\(966\) 0 0
\(967\) −50.9129 −1.63725 −0.818624 0.574330i \(-0.805263\pi\)
−0.818624 + 0.574330i \(0.805263\pi\)
\(968\) 5.34374 0.171754
\(969\) 6.40453 0.205743
\(970\) −21.4559 −0.688908
\(971\) 16.7372 0.537123 0.268561 0.963263i \(-0.413452\pi\)
0.268561 + 0.963263i \(0.413452\pi\)
\(972\) 2.26446 0.0726327
\(973\) 0 0
\(974\) −86.4272 −2.76931
\(975\) 8.55087 0.273847
\(976\) 17.0998 0.547351
\(977\) 47.3729 1.51559 0.757797 0.652491i \(-0.226276\pi\)
0.757797 + 0.652491i \(0.226276\pi\)
\(978\) 44.9240 1.43651
\(979\) −8.73310 −0.279111
\(980\) 0 0
\(981\) 3.81163 0.121696
\(982\) −33.5897 −1.07189
\(983\) −35.2609 −1.12465 −0.562323 0.826918i \(-0.690092\pi\)
−0.562323 + 0.826918i \(0.690092\pi\)
\(984\) −5.34972 −0.170543
\(985\) −8.19827 −0.261219
\(986\) −9.77422 −0.311275
\(987\) 0 0
\(988\) 4.55141 0.144800
\(989\) 5.23469 0.166454
\(990\) 1.38611 0.0440534
\(991\) 16.9164 0.537369 0.268684 0.963228i \(-0.413411\pi\)
0.268684 + 0.963228i \(0.413411\pi\)
\(992\) −71.2048 −2.26075
\(993\) 8.23405 0.261300
\(994\) 0 0
\(995\) −8.70504 −0.275968
\(996\) 24.2742 0.769157
\(997\) 52.3647 1.65841 0.829203 0.558947i \(-0.188795\pi\)
0.829203 + 0.558947i \(0.188795\pi\)
\(998\) 40.9398 1.29593
\(999\) 9.11528 0.288395
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 3381.2.a.bj.1.2 10
7.3 odd 6 483.2.i.h.415.9 yes 20
7.5 odd 6 483.2.i.h.277.9 20
7.6 odd 2 3381.2.a.bi.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.9 20 7.5 odd 6
483.2.i.h.415.9 yes 20 7.3 odd 6
3381.2.a.bi.1.2 10 7.6 odd 2
3381.2.a.bj.1.2 10 1.1 even 1 trivial