Properties

Label 4730.2.a.bf.1.11
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-2.24942\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.24942 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24942 q^{6} -1.43432 q^{7} +1.00000 q^{8} +2.05989 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.24942 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24942 q^{6} -1.43432 q^{7} +1.00000 q^{8} +2.05989 q^{9} -1.00000 q^{10} +1.00000 q^{11} +2.24942 q^{12} +4.69328 q^{13} -1.43432 q^{14} -2.24942 q^{15} +1.00000 q^{16} +2.69172 q^{17} +2.05989 q^{18} +5.09763 q^{19} -1.00000 q^{20} -3.22639 q^{21} +1.00000 q^{22} +2.31042 q^{23} +2.24942 q^{24} +1.00000 q^{25} +4.69328 q^{26} -2.11469 q^{27} -1.43432 q^{28} -5.94499 q^{29} -2.24942 q^{30} -6.78435 q^{31} +1.00000 q^{32} +2.24942 q^{33} +2.69172 q^{34} +1.43432 q^{35} +2.05989 q^{36} +3.88177 q^{37} +5.09763 q^{38} +10.5572 q^{39} -1.00000 q^{40} +7.01704 q^{41} -3.22639 q^{42} +1.00000 q^{43} +1.00000 q^{44} -2.05989 q^{45} +2.31042 q^{46} +0.722561 q^{47} +2.24942 q^{48} -4.94272 q^{49} +1.00000 q^{50} +6.05480 q^{51} +4.69328 q^{52} +9.28737 q^{53} -2.11469 q^{54} -1.00000 q^{55} -1.43432 q^{56} +11.4667 q^{57} -5.94499 q^{58} -14.7355 q^{59} -2.24942 q^{60} +11.9068 q^{61} -6.78435 q^{62} -2.95455 q^{63} +1.00000 q^{64} -4.69328 q^{65} +2.24942 q^{66} +9.71230 q^{67} +2.69172 q^{68} +5.19710 q^{69} +1.43432 q^{70} +12.2764 q^{71} +2.05989 q^{72} -6.20325 q^{73} +3.88177 q^{74} +2.24942 q^{75} +5.09763 q^{76} -1.43432 q^{77} +10.5572 q^{78} -10.4505 q^{79} -1.00000 q^{80} -10.9365 q^{81} +7.01704 q^{82} +4.45987 q^{83} -3.22639 q^{84} -2.69172 q^{85} +1.00000 q^{86} -13.3728 q^{87} +1.00000 q^{88} -6.38594 q^{89} -2.05989 q^{90} -6.73167 q^{91} +2.31042 q^{92} -15.2609 q^{93} +0.722561 q^{94} -5.09763 q^{95} +2.24942 q^{96} +0.328464 q^{97} -4.94272 q^{98} +2.05989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

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
\(3\) 2.24942 1.29870 0.649352 0.760488i \(-0.275040\pi\)
0.649352 + 0.760488i \(0.275040\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.24942 0.918322
\(7\) −1.43432 −0.542123 −0.271061 0.962562i \(-0.587375\pi\)
−0.271061 + 0.962562i \(0.587375\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.05989 0.686631
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 2.24942 0.649352
\(13\) 4.69328 1.30168 0.650840 0.759215i \(-0.274417\pi\)
0.650840 + 0.759215i \(0.274417\pi\)
\(14\) −1.43432 −0.383339
\(15\) −2.24942 −0.580798
\(16\) 1.00000 0.250000
\(17\) 2.69172 0.652837 0.326418 0.945225i \(-0.394158\pi\)
0.326418 + 0.945225i \(0.394158\pi\)
\(18\) 2.05989 0.485522
\(19\) 5.09763 1.16948 0.584738 0.811222i \(-0.301197\pi\)
0.584738 + 0.811222i \(0.301197\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.22639 −0.704057
\(22\) 1.00000 0.213201
\(23\) 2.31042 0.481755 0.240877 0.970556i \(-0.422565\pi\)
0.240877 + 0.970556i \(0.422565\pi\)
\(24\) 2.24942 0.459161
\(25\) 1.00000 0.200000
\(26\) 4.69328 0.920427
\(27\) −2.11469 −0.406973
\(28\) −1.43432 −0.271061
\(29\) −5.94499 −1.10396 −0.551979 0.833858i \(-0.686127\pi\)
−0.551979 + 0.833858i \(0.686127\pi\)
\(30\) −2.24942 −0.410686
\(31\) −6.78435 −1.21851 −0.609253 0.792976i \(-0.708531\pi\)
−0.609253 + 0.792976i \(0.708531\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.24942 0.391574
\(34\) 2.69172 0.461625
\(35\) 1.43432 0.242445
\(36\) 2.05989 0.343316
\(37\) 3.88177 0.638159 0.319080 0.947728i \(-0.396626\pi\)
0.319080 + 0.947728i \(0.396626\pi\)
\(38\) 5.09763 0.826944
\(39\) 10.5572 1.69050
\(40\) −1.00000 −0.158114
\(41\) 7.01704 1.09588 0.547939 0.836518i \(-0.315413\pi\)
0.547939 + 0.836518i \(0.315413\pi\)
\(42\) −3.22639 −0.497843
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) −2.05989 −0.307071
\(46\) 2.31042 0.340652
\(47\) 0.722561 0.105396 0.0526982 0.998610i \(-0.483218\pi\)
0.0526982 + 0.998610i \(0.483218\pi\)
\(48\) 2.24942 0.324676
\(49\) −4.94272 −0.706103
\(50\) 1.00000 0.141421
\(51\) 6.05480 0.847842
\(52\) 4.69328 0.650840
\(53\) 9.28737 1.27572 0.637859 0.770153i \(-0.279820\pi\)
0.637859 + 0.770153i \(0.279820\pi\)
\(54\) −2.11469 −0.287773
\(55\) −1.00000 −0.134840
\(56\) −1.43432 −0.191669
\(57\) 11.4667 1.51880
\(58\) −5.94499 −0.780616
\(59\) −14.7355 −1.91840 −0.959200 0.282728i \(-0.908760\pi\)
−0.959200 + 0.282728i \(0.908760\pi\)
\(60\) −2.24942 −0.290399
\(61\) 11.9068 1.52451 0.762255 0.647277i \(-0.224092\pi\)
0.762255 + 0.647277i \(0.224092\pi\)
\(62\) −6.78435 −0.861614
\(63\) −2.95455 −0.372238
\(64\) 1.00000 0.125000
\(65\) −4.69328 −0.582129
\(66\) 2.24942 0.276885
\(67\) 9.71230 1.18655 0.593273 0.805001i \(-0.297835\pi\)
0.593273 + 0.805001i \(0.297835\pi\)
\(68\) 2.69172 0.326418
\(69\) 5.19710 0.625657
\(70\) 1.43432 0.171434
\(71\) 12.2764 1.45695 0.728473 0.685075i \(-0.240231\pi\)
0.728473 + 0.685075i \(0.240231\pi\)
\(72\) 2.05989 0.242761
\(73\) −6.20325 −0.726035 −0.363018 0.931782i \(-0.618254\pi\)
−0.363018 + 0.931782i \(0.618254\pi\)
\(74\) 3.88177 0.451247
\(75\) 2.24942 0.259741
\(76\) 5.09763 0.584738
\(77\) −1.43432 −0.163456
\(78\) 10.5572 1.19536
\(79\) −10.4505 −1.17577 −0.587885 0.808944i \(-0.700039\pi\)
−0.587885 + 0.808944i \(0.700039\pi\)
\(80\) −1.00000 −0.111803
\(81\) −10.9365 −1.21517
\(82\) 7.01704 0.774903
\(83\) 4.45987 0.489534 0.244767 0.969582i \(-0.421288\pi\)
0.244767 + 0.969582i \(0.421288\pi\)
\(84\) −3.22639 −0.352028
\(85\) −2.69172 −0.291958
\(86\) 1.00000 0.107833
\(87\) −13.3728 −1.43371
\(88\) 1.00000 0.106600
\(89\) −6.38594 −0.676908 −0.338454 0.940983i \(-0.609904\pi\)
−0.338454 + 0.940983i \(0.609904\pi\)
\(90\) −2.05989 −0.217132
\(91\) −6.73167 −0.705671
\(92\) 2.31042 0.240877
\(93\) −15.2609 −1.58248
\(94\) 0.722561 0.0745265
\(95\) −5.09763 −0.523006
\(96\) 2.24942 0.229581
\(97\) 0.328464 0.0333505 0.0166753 0.999861i \(-0.494692\pi\)
0.0166753 + 0.999861i \(0.494692\pi\)
\(98\) −4.94272 −0.499290
\(99\) 2.05989 0.207027
\(100\) 1.00000 0.100000
\(101\) 13.3472 1.32809 0.664047 0.747691i \(-0.268837\pi\)
0.664047 + 0.747691i \(0.268837\pi\)
\(102\) 6.05480 0.599515
\(103\) 15.0788 1.48576 0.742879 0.669426i \(-0.233460\pi\)
0.742879 + 0.669426i \(0.233460\pi\)
\(104\) 4.69328 0.460214
\(105\) 3.22639 0.314864
\(106\) 9.28737 0.902069
\(107\) −1.72911 −0.167160 −0.0835798 0.996501i \(-0.526635\pi\)
−0.0835798 + 0.996501i \(0.526635\pi\)
\(108\) −2.11469 −0.203487
\(109\) 9.13341 0.874822 0.437411 0.899262i \(-0.355895\pi\)
0.437411 + 0.899262i \(0.355895\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 8.73174 0.828780
\(112\) −1.43432 −0.135531
\(113\) 4.62839 0.435403 0.217701 0.976015i \(-0.430144\pi\)
0.217701 + 0.976015i \(0.430144\pi\)
\(114\) 11.4667 1.07396
\(115\) −2.31042 −0.215447
\(116\) −5.94499 −0.551979
\(117\) 9.66765 0.893775
\(118\) −14.7355 −1.35651
\(119\) −3.86079 −0.353918
\(120\) −2.24942 −0.205343
\(121\) 1.00000 0.0909091
\(122\) 11.9068 1.07799
\(123\) 15.7843 1.42322
\(124\) −6.78435 −0.609253
\(125\) −1.00000 −0.0894427
\(126\) −2.95455 −0.263212
\(127\) −7.81461 −0.693435 −0.346717 0.937970i \(-0.612704\pi\)
−0.346717 + 0.937970i \(0.612704\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.24942 0.198050
\(130\) −4.69328 −0.411628
\(131\) −6.71338 −0.586551 −0.293275 0.956028i \(-0.594745\pi\)
−0.293275 + 0.956028i \(0.594745\pi\)
\(132\) 2.24942 0.195787
\(133\) −7.31164 −0.633999
\(134\) 9.71230 0.839015
\(135\) 2.11469 0.182004
\(136\) 2.69172 0.230813
\(137\) −14.4584 −1.23526 −0.617632 0.786467i \(-0.711908\pi\)
−0.617632 + 0.786467i \(0.711908\pi\)
\(138\) 5.19710 0.442406
\(139\) 1.30594 0.110768 0.0553840 0.998465i \(-0.482362\pi\)
0.0553840 + 0.998465i \(0.482362\pi\)
\(140\) 1.43432 0.121222
\(141\) 1.62534 0.136879
\(142\) 12.2764 1.03022
\(143\) 4.69328 0.392472
\(144\) 2.05989 0.171658
\(145\) 5.94499 0.493705
\(146\) −6.20325 −0.513385
\(147\) −11.1183 −0.917019
\(148\) 3.88177 0.319080
\(149\) 9.28161 0.760379 0.380190 0.924909i \(-0.375859\pi\)
0.380190 + 0.924909i \(0.375859\pi\)
\(150\) 2.24942 0.183664
\(151\) −5.35962 −0.436160 −0.218080 0.975931i \(-0.569979\pi\)
−0.218080 + 0.975931i \(0.569979\pi\)
\(152\) 5.09763 0.413472
\(153\) 5.54465 0.448258
\(154\) −1.43432 −0.115581
\(155\) 6.78435 0.544932
\(156\) 10.5572 0.845249
\(157\) 18.6665 1.48975 0.744876 0.667203i \(-0.232509\pi\)
0.744876 + 0.667203i \(0.232509\pi\)
\(158\) −10.4505 −0.831395
\(159\) 20.8912 1.65678
\(160\) −1.00000 −0.0790569
\(161\) −3.31388 −0.261170
\(162\) −10.9365 −0.859254
\(163\) 23.7568 1.86078 0.930388 0.366577i \(-0.119470\pi\)
0.930388 + 0.366577i \(0.119470\pi\)
\(164\) 7.01704 0.547939
\(165\) −2.24942 −0.175117
\(166\) 4.45987 0.346153
\(167\) −4.94648 −0.382770 −0.191385 0.981515i \(-0.561298\pi\)
−0.191385 + 0.981515i \(0.561298\pi\)
\(168\) −3.22639 −0.248922
\(169\) 9.02685 0.694373
\(170\) −2.69172 −0.206445
\(171\) 10.5006 0.802999
\(172\) 1.00000 0.0762493
\(173\) 24.4374 1.85794 0.928971 0.370154i \(-0.120695\pi\)
0.928971 + 0.370154i \(0.120695\pi\)
\(174\) −13.3728 −1.01379
\(175\) −1.43432 −0.108425
\(176\) 1.00000 0.0753778
\(177\) −33.1464 −2.49143
\(178\) −6.38594 −0.478646
\(179\) −16.6363 −1.24346 −0.621729 0.783233i \(-0.713569\pi\)
−0.621729 + 0.783233i \(0.713569\pi\)
\(180\) −2.05989 −0.153535
\(181\) −14.4304 −1.07260 −0.536302 0.844026i \(-0.680179\pi\)
−0.536302 + 0.844026i \(0.680179\pi\)
\(182\) −6.73167 −0.498985
\(183\) 26.7834 1.97989
\(184\) 2.31042 0.170326
\(185\) −3.88177 −0.285394
\(186\) −15.2609 −1.11898
\(187\) 2.69172 0.196838
\(188\) 0.722561 0.0526982
\(189\) 3.03315 0.220629
\(190\) −5.09763 −0.369821
\(191\) 6.94965 0.502859 0.251430 0.967876i \(-0.419099\pi\)
0.251430 + 0.967876i \(0.419099\pi\)
\(192\) 2.24942 0.162338
\(193\) 3.04767 0.219376 0.109688 0.993966i \(-0.465015\pi\)
0.109688 + 0.993966i \(0.465015\pi\)
\(194\) 0.328464 0.0235824
\(195\) −10.5572 −0.756014
\(196\) −4.94272 −0.353051
\(197\) −12.9605 −0.923398 −0.461699 0.887037i \(-0.652760\pi\)
−0.461699 + 0.887037i \(0.652760\pi\)
\(198\) 2.05989 0.146390
\(199\) 5.53937 0.392675 0.196338 0.980536i \(-0.437095\pi\)
0.196338 + 0.980536i \(0.437095\pi\)
\(200\) 1.00000 0.0707107
\(201\) 21.8471 1.54097
\(202\) 13.3472 0.939104
\(203\) 8.52703 0.598480
\(204\) 6.05480 0.423921
\(205\) −7.01704 −0.490092
\(206\) 15.0788 1.05059
\(207\) 4.75921 0.330788
\(208\) 4.69328 0.325420
\(209\) 5.09763 0.352610
\(210\) 3.22639 0.222642
\(211\) 1.49551 0.102955 0.0514775 0.998674i \(-0.483607\pi\)
0.0514775 + 0.998674i \(0.483607\pi\)
\(212\) 9.28737 0.637859
\(213\) 27.6149 1.89214
\(214\) −1.72911 −0.118200
\(215\) −1.00000 −0.0681994
\(216\) −2.11469 −0.143887
\(217\) 9.73095 0.660580
\(218\) 9.13341 0.618593
\(219\) −13.9537 −0.942905
\(220\) −1.00000 −0.0674200
\(221\) 12.6330 0.849785
\(222\) 8.73174 0.586036
\(223\) −20.6795 −1.38480 −0.692400 0.721514i \(-0.743447\pi\)
−0.692400 + 0.721514i \(0.743447\pi\)
\(224\) −1.43432 −0.0958347
\(225\) 2.05989 0.137326
\(226\) 4.62839 0.307876
\(227\) −21.9344 −1.45583 −0.727917 0.685665i \(-0.759512\pi\)
−0.727917 + 0.685665i \(0.759512\pi\)
\(228\) 11.4667 0.759401
\(229\) −21.3370 −1.40999 −0.704995 0.709212i \(-0.749051\pi\)
−0.704995 + 0.709212i \(0.749051\pi\)
\(230\) −2.31042 −0.152344
\(231\) −3.22639 −0.212281
\(232\) −5.94499 −0.390308
\(233\) −18.4304 −1.20741 −0.603706 0.797207i \(-0.706310\pi\)
−0.603706 + 0.797207i \(0.706310\pi\)
\(234\) 9.66765 0.631994
\(235\) −0.722561 −0.0471347
\(236\) −14.7355 −0.959200
\(237\) −23.5075 −1.52698
\(238\) −3.86079 −0.250258
\(239\) 4.81697 0.311584 0.155792 0.987790i \(-0.450207\pi\)
0.155792 + 0.987790i \(0.450207\pi\)
\(240\) −2.24942 −0.145199
\(241\) 5.77599 0.372064 0.186032 0.982544i \(-0.440437\pi\)
0.186032 + 0.982544i \(0.440437\pi\)
\(242\) 1.00000 0.0642824
\(243\) −18.2568 −1.17117
\(244\) 11.9068 0.762255
\(245\) 4.94272 0.315779
\(246\) 15.7843 1.00637
\(247\) 23.9246 1.52228
\(248\) −6.78435 −0.430807
\(249\) 10.0321 0.635760
\(250\) −1.00000 −0.0632456
\(251\) −27.9375 −1.76340 −0.881701 0.471809i \(-0.843601\pi\)
−0.881701 + 0.471809i \(0.843601\pi\)
\(252\) −2.95455 −0.186119
\(253\) 2.31042 0.145255
\(254\) −7.81461 −0.490332
\(255\) −6.05480 −0.379166
\(256\) 1.00000 0.0625000
\(257\) −9.69635 −0.604841 −0.302421 0.953175i \(-0.597795\pi\)
−0.302421 + 0.953175i \(0.597795\pi\)
\(258\) 2.24942 0.140043
\(259\) −5.56771 −0.345961
\(260\) −4.69328 −0.291065
\(261\) −12.2461 −0.758012
\(262\) −6.71338 −0.414754
\(263\) −20.4163 −1.25892 −0.629461 0.777032i \(-0.716724\pi\)
−0.629461 + 0.777032i \(0.716724\pi\)
\(264\) 2.24942 0.138442
\(265\) −9.28737 −0.570518
\(266\) −7.31164 −0.448305
\(267\) −14.3647 −0.879103
\(268\) 9.71230 0.593273
\(269\) 13.9619 0.851273 0.425637 0.904894i \(-0.360050\pi\)
0.425637 + 0.904894i \(0.360050\pi\)
\(270\) 2.11469 0.128696
\(271\) −18.0976 −1.09935 −0.549675 0.835379i \(-0.685248\pi\)
−0.549675 + 0.835379i \(0.685248\pi\)
\(272\) 2.69172 0.163209
\(273\) −15.1424 −0.916457
\(274\) −14.4584 −0.873463
\(275\) 1.00000 0.0603023
\(276\) 5.19710 0.312828
\(277\) −8.13107 −0.488549 −0.244274 0.969706i \(-0.578550\pi\)
−0.244274 + 0.969706i \(0.578550\pi\)
\(278\) 1.30594 0.0783248
\(279\) −13.9751 −0.836664
\(280\) 1.43432 0.0857171
\(281\) −5.99636 −0.357713 −0.178856 0.983875i \(-0.557240\pi\)
−0.178856 + 0.983875i \(0.557240\pi\)
\(282\) 1.62534 0.0967878
\(283\) −21.0031 −1.24850 −0.624252 0.781223i \(-0.714596\pi\)
−0.624252 + 0.781223i \(0.714596\pi\)
\(284\) 12.2764 0.728473
\(285\) −11.4667 −0.679229
\(286\) 4.69328 0.277519
\(287\) −10.0647 −0.594100
\(288\) 2.05989 0.121380
\(289\) −9.75467 −0.573804
\(290\) 5.94499 0.349102
\(291\) 0.738855 0.0433124
\(292\) −6.20325 −0.363018
\(293\) −29.8493 −1.74382 −0.871908 0.489671i \(-0.837117\pi\)
−0.871908 + 0.489671i \(0.837117\pi\)
\(294\) −11.1183 −0.648430
\(295\) 14.7355 0.857935
\(296\) 3.88177 0.225623
\(297\) −2.11469 −0.122707
\(298\) 9.28161 0.537669
\(299\) 10.8434 0.627091
\(300\) 2.24942 0.129870
\(301\) −1.43432 −0.0826729
\(302\) −5.35962 −0.308412
\(303\) 30.0234 1.72480
\(304\) 5.09763 0.292369
\(305\) −11.9068 −0.681782
\(306\) 5.54465 0.316966
\(307\) 3.81085 0.217497 0.108748 0.994069i \(-0.465316\pi\)
0.108748 + 0.994069i \(0.465316\pi\)
\(308\) −1.43432 −0.0817281
\(309\) 33.9185 1.92956
\(310\) 6.78435 0.385325
\(311\) 17.9922 1.02025 0.510123 0.860101i \(-0.329600\pi\)
0.510123 + 0.860101i \(0.329600\pi\)
\(312\) 10.5572 0.597681
\(313\) 4.23321 0.239275 0.119638 0.992818i \(-0.461827\pi\)
0.119638 + 0.992818i \(0.461827\pi\)
\(314\) 18.6665 1.05341
\(315\) 2.95455 0.166470
\(316\) −10.4505 −0.587885
\(317\) 22.6847 1.27410 0.637050 0.770822i \(-0.280154\pi\)
0.637050 + 0.770822i \(0.280154\pi\)
\(318\) 20.8912 1.17152
\(319\) −5.94499 −0.332856
\(320\) −1.00000 −0.0559017
\(321\) −3.88950 −0.217091
\(322\) −3.31388 −0.184675
\(323\) 13.7214 0.763477
\(324\) −10.9365 −0.607584
\(325\) 4.69328 0.260336
\(326\) 23.7568 1.31577
\(327\) 20.5449 1.13613
\(328\) 7.01704 0.387451
\(329\) −1.03639 −0.0571378
\(330\) −2.24942 −0.123827
\(331\) 20.9408 1.15101 0.575505 0.817798i \(-0.304805\pi\)
0.575505 + 0.817798i \(0.304805\pi\)
\(332\) 4.45987 0.244767
\(333\) 7.99604 0.438180
\(334\) −4.94648 −0.270660
\(335\) −9.71230 −0.530640
\(336\) −3.22639 −0.176014
\(337\) 12.2906 0.669512 0.334756 0.942305i \(-0.391346\pi\)
0.334756 + 0.942305i \(0.391346\pi\)
\(338\) 9.02685 0.490996
\(339\) 10.4112 0.565459
\(340\) −2.69172 −0.145979
\(341\) −6.78435 −0.367393
\(342\) 10.5006 0.567806
\(343\) 17.1297 0.924917
\(344\) 1.00000 0.0539164
\(345\) −5.19710 −0.279802
\(346\) 24.4374 1.31376
\(347\) −31.6551 −1.69933 −0.849667 0.527319i \(-0.823197\pi\)
−0.849667 + 0.527319i \(0.823197\pi\)
\(348\) −13.3728 −0.716857
\(349\) 21.3475 1.14270 0.571352 0.820705i \(-0.306419\pi\)
0.571352 + 0.820705i \(0.306419\pi\)
\(350\) −1.43432 −0.0766677
\(351\) −9.92484 −0.529749
\(352\) 1.00000 0.0533002
\(353\) −16.1131 −0.857611 −0.428806 0.903397i \(-0.641065\pi\)
−0.428806 + 0.903397i \(0.641065\pi\)
\(354\) −33.1464 −1.76171
\(355\) −12.2764 −0.651566
\(356\) −6.38594 −0.338454
\(357\) −8.68453 −0.459634
\(358\) −16.6363 −0.879257
\(359\) −21.6717 −1.14379 −0.571894 0.820328i \(-0.693791\pi\)
−0.571894 + 0.820328i \(0.693791\pi\)
\(360\) −2.05989 −0.108566
\(361\) 6.98581 0.367674
\(362\) −14.4304 −0.758446
\(363\) 2.24942 0.118064
\(364\) −6.73167 −0.352835
\(365\) 6.20325 0.324693
\(366\) 26.7834 1.39999
\(367\) −15.0819 −0.787269 −0.393635 0.919267i \(-0.628782\pi\)
−0.393635 + 0.919267i \(0.628782\pi\)
\(368\) 2.31042 0.120439
\(369\) 14.4544 0.752464
\(370\) −3.88177 −0.201804
\(371\) −13.3211 −0.691596
\(372\) −15.2609 −0.791239
\(373\) −23.7371 −1.22906 −0.614529 0.788894i \(-0.710654\pi\)
−0.614529 + 0.788894i \(0.710654\pi\)
\(374\) 2.69172 0.139185
\(375\) −2.24942 −0.116160
\(376\) 0.722561 0.0372632
\(377\) −27.9015 −1.43700
\(378\) 3.03315 0.156008
\(379\) 11.5566 0.593622 0.296811 0.954936i \(-0.404077\pi\)
0.296811 + 0.954936i \(0.404077\pi\)
\(380\) −5.09763 −0.261503
\(381\) −17.5784 −0.900566
\(382\) 6.94965 0.355575
\(383\) 24.9628 1.27554 0.637769 0.770228i \(-0.279857\pi\)
0.637769 + 0.770228i \(0.279857\pi\)
\(384\) 2.24942 0.114790
\(385\) 1.43432 0.0730998
\(386\) 3.04767 0.155122
\(387\) 2.05989 0.104710
\(388\) 0.328464 0.0166753
\(389\) −26.8036 −1.35899 −0.679497 0.733678i \(-0.737802\pi\)
−0.679497 + 0.733678i \(0.737802\pi\)
\(390\) −10.5572 −0.534582
\(391\) 6.21898 0.314507
\(392\) −4.94272 −0.249645
\(393\) −15.1012 −0.761756
\(394\) −12.9605 −0.652941
\(395\) 10.4505 0.525820
\(396\) 2.05989 0.103514
\(397\) −6.69149 −0.335836 −0.167918 0.985801i \(-0.553704\pi\)
−0.167918 + 0.985801i \(0.553704\pi\)
\(398\) 5.53937 0.277663
\(399\) −16.4470 −0.823377
\(400\) 1.00000 0.0500000
\(401\) −7.04199 −0.351660 −0.175830 0.984421i \(-0.556261\pi\)
−0.175830 + 0.984421i \(0.556261\pi\)
\(402\) 21.8471 1.08963
\(403\) −31.8409 −1.58611
\(404\) 13.3472 0.664047
\(405\) 10.9365 0.543440
\(406\) 8.52703 0.423190
\(407\) 3.88177 0.192412
\(408\) 6.05480 0.299757
\(409\) 0.530901 0.0262514 0.0131257 0.999914i \(-0.495822\pi\)
0.0131257 + 0.999914i \(0.495822\pi\)
\(410\) −7.01704 −0.346547
\(411\) −32.5230 −1.60424
\(412\) 15.0788 0.742879
\(413\) 21.1355 1.04001
\(414\) 4.75921 0.233902
\(415\) −4.45987 −0.218926
\(416\) 4.69328 0.230107
\(417\) 2.93760 0.143855
\(418\) 5.09763 0.249333
\(419\) −21.8429 −1.06710 −0.533548 0.845770i \(-0.679142\pi\)
−0.533548 + 0.845770i \(0.679142\pi\)
\(420\) 3.22639 0.157432
\(421\) −29.9008 −1.45727 −0.728637 0.684900i \(-0.759846\pi\)
−0.728637 + 0.684900i \(0.759846\pi\)
\(422\) 1.49551 0.0728002
\(423\) 1.48840 0.0723684
\(424\) 9.28737 0.451034
\(425\) 2.69172 0.130567
\(426\) 27.6149 1.33795
\(427\) −17.0782 −0.826472
\(428\) −1.72911 −0.0835798
\(429\) 10.5572 0.509704
\(430\) −1.00000 −0.0482243
\(431\) −3.74480 −0.180381 −0.0901903 0.995925i \(-0.528748\pi\)
−0.0901903 + 0.995925i \(0.528748\pi\)
\(432\) −2.11469 −0.101743
\(433\) 15.2090 0.730898 0.365449 0.930831i \(-0.380915\pi\)
0.365449 + 0.930831i \(0.380915\pi\)
\(434\) 9.73095 0.467100
\(435\) 13.3728 0.641176
\(436\) 9.13341 0.437411
\(437\) 11.7776 0.563401
\(438\) −13.9537 −0.666734
\(439\) −35.0097 −1.67092 −0.835462 0.549548i \(-0.814800\pi\)
−0.835462 + 0.549548i \(0.814800\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −10.1815 −0.484832
\(442\) 12.6330 0.600889
\(443\) −31.3131 −1.48773 −0.743864 0.668330i \(-0.767009\pi\)
−0.743864 + 0.668330i \(0.767009\pi\)
\(444\) 8.73174 0.414390
\(445\) 6.38594 0.302723
\(446\) −20.6795 −0.979201
\(447\) 20.8782 0.987507
\(448\) −1.43432 −0.0677653
\(449\) 1.28870 0.0608177 0.0304088 0.999538i \(-0.490319\pi\)
0.0304088 + 0.999538i \(0.490319\pi\)
\(450\) 2.05989 0.0971043
\(451\) 7.01704 0.330420
\(452\) 4.62839 0.217701
\(453\) −12.0560 −0.566443
\(454\) −21.9344 −1.02943
\(455\) 6.73167 0.315586
\(456\) 11.4667 0.536978
\(457\) −35.6533 −1.66779 −0.833895 0.551922i \(-0.813894\pi\)
−0.833895 + 0.551922i \(0.813894\pi\)
\(458\) −21.3370 −0.997014
\(459\) −5.69215 −0.265687
\(460\) −2.31042 −0.107724
\(461\) −10.0290 −0.467096 −0.233548 0.972345i \(-0.575034\pi\)
−0.233548 + 0.972345i \(0.575034\pi\)
\(462\) −3.22639 −0.150105
\(463\) −26.9399 −1.25200 −0.626001 0.779822i \(-0.715309\pi\)
−0.626001 + 0.779822i \(0.715309\pi\)
\(464\) −5.94499 −0.275989
\(465\) 15.2609 0.707706
\(466\) −18.4304 −0.853770
\(467\) −17.3201 −0.801479 −0.400740 0.916192i \(-0.631247\pi\)
−0.400740 + 0.916192i \(0.631247\pi\)
\(468\) 9.66765 0.446887
\(469\) −13.9306 −0.643254
\(470\) −0.722561 −0.0333293
\(471\) 41.9889 1.93475
\(472\) −14.7355 −0.678257
\(473\) 1.00000 0.0459800
\(474\) −23.5075 −1.07974
\(475\) 5.09763 0.233895
\(476\) −3.86079 −0.176959
\(477\) 19.1310 0.875948
\(478\) 4.81697 0.220323
\(479\) −29.6771 −1.35598 −0.677990 0.735071i \(-0.737149\pi\)
−0.677990 + 0.735071i \(0.737149\pi\)
\(480\) −2.24942 −0.102672
\(481\) 18.2182 0.830680
\(482\) 5.77599 0.263089
\(483\) −7.45431 −0.339183
\(484\) 1.00000 0.0454545
\(485\) −0.328464 −0.0149148
\(486\) −18.2568 −0.828143
\(487\) 20.7138 0.938631 0.469316 0.883030i \(-0.344501\pi\)
0.469316 + 0.883030i \(0.344501\pi\)
\(488\) 11.9068 0.538996
\(489\) 53.4390 2.41660
\(490\) 4.94272 0.223289
\(491\) 0.702895 0.0317212 0.0158606 0.999874i \(-0.494951\pi\)
0.0158606 + 0.999874i \(0.494951\pi\)
\(492\) 15.7843 0.711611
\(493\) −16.0022 −0.720704
\(494\) 23.9246 1.07642
\(495\) −2.05989 −0.0925854
\(496\) −6.78435 −0.304627
\(497\) −17.6084 −0.789843
\(498\) 10.0321 0.449550
\(499\) 7.52040 0.336659 0.168330 0.985731i \(-0.446163\pi\)
0.168330 + 0.985731i \(0.446163\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −11.1267 −0.497105
\(502\) −27.9375 −1.24691
\(503\) 20.6329 0.919975 0.459988 0.887925i \(-0.347854\pi\)
0.459988 + 0.887925i \(0.347854\pi\)
\(504\) −2.95455 −0.131606
\(505\) −13.3472 −0.593942
\(506\) 2.31042 0.102710
\(507\) 20.3052 0.901785
\(508\) −7.81461 −0.346717
\(509\) 41.8117 1.85327 0.926636 0.375959i \(-0.122687\pi\)
0.926636 + 0.375959i \(0.122687\pi\)
\(510\) −6.05480 −0.268111
\(511\) 8.89746 0.393600
\(512\) 1.00000 0.0441942
\(513\) −10.7799 −0.475945
\(514\) −9.69635 −0.427687
\(515\) −15.0788 −0.664451
\(516\) 2.24942 0.0990252
\(517\) 0.722561 0.0317782
\(518\) −5.56771 −0.244631
\(519\) 54.9700 2.41292
\(520\) −4.69328 −0.205814
\(521\) −23.7301 −1.03964 −0.519818 0.854277i \(-0.674000\pi\)
−0.519818 + 0.854277i \(0.674000\pi\)
\(522\) −12.2461 −0.535995
\(523\) 24.1764 1.05716 0.528581 0.848883i \(-0.322724\pi\)
0.528581 + 0.848883i \(0.322724\pi\)
\(524\) −6.71338 −0.293275
\(525\) −3.22639 −0.140811
\(526\) −20.4163 −0.890193
\(527\) −18.2616 −0.795486
\(528\) 2.24942 0.0978935
\(529\) −17.6620 −0.767912
\(530\) −9.28737 −0.403417
\(531\) −30.3536 −1.31723
\(532\) −7.31164 −0.317000
\(533\) 32.9329 1.42648
\(534\) −14.3647 −0.621620
\(535\) 1.72911 0.0747561
\(536\) 9.71230 0.419508
\(537\) −37.4221 −1.61488
\(538\) 13.9619 0.601941
\(539\) −4.94272 −0.212898
\(540\) 2.11469 0.0910019
\(541\) −10.3446 −0.444750 −0.222375 0.974961i \(-0.571381\pi\)
−0.222375 + 0.974961i \(0.571381\pi\)
\(542\) −18.0976 −0.777357
\(543\) −32.4601 −1.39300
\(544\) 2.69172 0.115406
\(545\) −9.13341 −0.391232
\(546\) −15.1424 −0.648033
\(547\) 11.4357 0.488955 0.244477 0.969655i \(-0.421384\pi\)
0.244477 + 0.969655i \(0.421384\pi\)
\(548\) −14.4584 −0.617632
\(549\) 24.5268 1.04678
\(550\) 1.00000 0.0426401
\(551\) −30.3054 −1.29105
\(552\) 5.19710 0.221203
\(553\) 14.9893 0.637412
\(554\) −8.13107 −0.345456
\(555\) −8.73174 −0.370642
\(556\) 1.30594 0.0553840
\(557\) 42.4216 1.79746 0.898731 0.438500i \(-0.144490\pi\)
0.898731 + 0.438500i \(0.144490\pi\)
\(558\) −13.9751 −0.591611
\(559\) 4.69328 0.198504
\(560\) 1.43432 0.0606112
\(561\) 6.05480 0.255634
\(562\) −5.99636 −0.252941
\(563\) 27.0026 1.13802 0.569012 0.822330i \(-0.307326\pi\)
0.569012 + 0.822330i \(0.307326\pi\)
\(564\) 1.62534 0.0684393
\(565\) −4.62839 −0.194718
\(566\) −21.0031 −0.882826
\(567\) 15.6865 0.658771
\(568\) 12.2764 0.515108
\(569\) −35.6919 −1.49628 −0.748141 0.663540i \(-0.769053\pi\)
−0.748141 + 0.663540i \(0.769053\pi\)
\(570\) −11.4667 −0.480288
\(571\) −4.76591 −0.199447 −0.0997235 0.995015i \(-0.531796\pi\)
−0.0997235 + 0.995015i \(0.531796\pi\)
\(572\) 4.69328 0.196236
\(573\) 15.6327 0.653065
\(574\) −10.0647 −0.420092
\(575\) 2.31042 0.0963510
\(576\) 2.05989 0.0858289
\(577\) −17.3230 −0.721167 −0.360583 0.932727i \(-0.617422\pi\)
−0.360583 + 0.932727i \(0.617422\pi\)
\(578\) −9.75467 −0.405741
\(579\) 6.85550 0.284905
\(580\) 5.94499 0.246852
\(581\) −6.39689 −0.265388
\(582\) 0.738855 0.0306265
\(583\) 9.28737 0.384643
\(584\) −6.20325 −0.256692
\(585\) −9.66765 −0.399708
\(586\) −29.8493 −1.23306
\(587\) 34.5220 1.42488 0.712438 0.701735i \(-0.247591\pi\)
0.712438 + 0.701735i \(0.247591\pi\)
\(588\) −11.1183 −0.458509
\(589\) −34.5841 −1.42501
\(590\) 14.7355 0.606651
\(591\) −29.1536 −1.19922
\(592\) 3.88177 0.159540
\(593\) 6.36844 0.261521 0.130760 0.991414i \(-0.458258\pi\)
0.130760 + 0.991414i \(0.458258\pi\)
\(594\) −2.11469 −0.0867669
\(595\) 3.86079 0.158277
\(596\) 9.28161 0.380190
\(597\) 12.4604 0.509969
\(598\) 10.8434 0.443420
\(599\) 41.4623 1.69410 0.847051 0.531511i \(-0.178376\pi\)
0.847051 + 0.531511i \(0.178376\pi\)
\(600\) 2.24942 0.0918322
\(601\) −20.1939 −0.823725 −0.411863 0.911246i \(-0.635122\pi\)
−0.411863 + 0.911246i \(0.635122\pi\)
\(602\) −1.43432 −0.0584586
\(603\) 20.0063 0.814720
\(604\) −5.35962 −0.218080
\(605\) −1.00000 −0.0406558
\(606\) 30.0234 1.21962
\(607\) 20.7568 0.842491 0.421246 0.906947i \(-0.361593\pi\)
0.421246 + 0.906947i \(0.361593\pi\)
\(608\) 5.09763 0.206736
\(609\) 19.1809 0.777249
\(610\) −11.9068 −0.482093
\(611\) 3.39118 0.137192
\(612\) 5.54465 0.224129
\(613\) −18.5042 −0.747376 −0.373688 0.927554i \(-0.621907\pi\)
−0.373688 + 0.927554i \(0.621907\pi\)
\(614\) 3.81085 0.153793
\(615\) −15.7843 −0.636484
\(616\) −1.43432 −0.0577905
\(617\) −9.52812 −0.383587 −0.191794 0.981435i \(-0.561430\pi\)
−0.191794 + 0.981435i \(0.561430\pi\)
\(618\) 33.9185 1.36440
\(619\) −8.29009 −0.333207 −0.166604 0.986024i \(-0.553280\pi\)
−0.166604 + 0.986024i \(0.553280\pi\)
\(620\) 6.78435 0.272466
\(621\) −4.88582 −0.196061
\(622\) 17.9922 0.721423
\(623\) 9.15949 0.366967
\(624\) 10.5572 0.422624
\(625\) 1.00000 0.0400000
\(626\) 4.23321 0.169193
\(627\) 11.4667 0.457936
\(628\) 18.6665 0.744876
\(629\) 10.4486 0.416614
\(630\) 2.95455 0.117712
\(631\) 39.4944 1.57225 0.786123 0.618070i \(-0.212085\pi\)
0.786123 + 0.618070i \(0.212085\pi\)
\(632\) −10.4505 −0.415698
\(633\) 3.36403 0.133708
\(634\) 22.6847 0.900925
\(635\) 7.81461 0.310113
\(636\) 20.8912 0.828390
\(637\) −23.1976 −0.919121
\(638\) −5.94499 −0.235365
\(639\) 25.2882 1.00038
\(640\) −1.00000 −0.0395285
\(641\) 15.3879 0.607787 0.303893 0.952706i \(-0.401713\pi\)
0.303893 + 0.952706i \(0.401713\pi\)
\(642\) −3.88950 −0.153506
\(643\) 6.81634 0.268810 0.134405 0.990926i \(-0.457088\pi\)
0.134405 + 0.990926i \(0.457088\pi\)
\(644\) −3.31388 −0.130585
\(645\) −2.24942 −0.0885709
\(646\) 13.7214 0.539860
\(647\) −28.1040 −1.10488 −0.552442 0.833551i \(-0.686304\pi\)
−0.552442 + 0.833551i \(0.686304\pi\)
\(648\) −10.9365 −0.429627
\(649\) −14.7355 −0.578419
\(650\) 4.69328 0.184085
\(651\) 21.8890 0.857897
\(652\) 23.7568 0.930388
\(653\) −13.1673 −0.515277 −0.257639 0.966241i \(-0.582944\pi\)
−0.257639 + 0.966241i \(0.582944\pi\)
\(654\) 20.5449 0.803369
\(655\) 6.71338 0.262313
\(656\) 7.01704 0.273970
\(657\) −12.7780 −0.498519
\(658\) −1.03639 −0.0404025
\(659\) 34.7212 1.35254 0.676272 0.736652i \(-0.263594\pi\)
0.676272 + 0.736652i \(0.263594\pi\)
\(660\) −2.24942 −0.0875586
\(661\) 30.9514 1.20387 0.601935 0.798545i \(-0.294396\pi\)
0.601935 + 0.798545i \(0.294396\pi\)
\(662\) 20.9408 0.813887
\(663\) 28.4169 1.10362
\(664\) 4.45987 0.173077
\(665\) 7.31164 0.283533
\(666\) 7.99604 0.309840
\(667\) −13.7354 −0.531837
\(668\) −4.94648 −0.191385
\(669\) −46.5168 −1.79844
\(670\) −9.71230 −0.375219
\(671\) 11.9068 0.459657
\(672\) −3.22639 −0.124461
\(673\) −26.2301 −1.01110 −0.505549 0.862798i \(-0.668710\pi\)
−0.505549 + 0.862798i \(0.668710\pi\)
\(674\) 12.2906 0.473416
\(675\) −2.11469 −0.0813946
\(676\) 9.02685 0.347187
\(677\) −9.28209 −0.356740 −0.178370 0.983963i \(-0.557082\pi\)
−0.178370 + 0.983963i \(0.557082\pi\)
\(678\) 10.4112 0.399840
\(679\) −0.471124 −0.0180801
\(680\) −2.69172 −0.103223
\(681\) −49.3396 −1.89070
\(682\) −6.78435 −0.259786
\(683\) 19.0813 0.730124 0.365062 0.930983i \(-0.381048\pi\)
0.365062 + 0.930983i \(0.381048\pi\)
\(684\) 10.5006 0.401499
\(685\) 14.4584 0.552427
\(686\) 17.1297 0.654015
\(687\) −47.9960 −1.83116
\(688\) 1.00000 0.0381246
\(689\) 43.5882 1.66058
\(690\) −5.19710 −0.197850
\(691\) −22.5886 −0.859311 −0.429656 0.902993i \(-0.641365\pi\)
−0.429656 + 0.902993i \(0.641365\pi\)
\(692\) 24.4374 0.928971
\(693\) −2.95455 −0.112234
\(694\) −31.6551 −1.20161
\(695\) −1.30594 −0.0495370
\(696\) −13.3728 −0.506894
\(697\) 18.8879 0.715430
\(698\) 21.3475 0.808014
\(699\) −41.4576 −1.56807
\(700\) −1.43432 −0.0542123
\(701\) −40.5973 −1.53334 −0.766669 0.642042i \(-0.778087\pi\)
−0.766669 + 0.642042i \(0.778087\pi\)
\(702\) −9.92484 −0.374589
\(703\) 19.7878 0.746312
\(704\) 1.00000 0.0376889
\(705\) −1.62534 −0.0612140
\(706\) −16.1131 −0.606423
\(707\) −19.1442 −0.719990
\(708\) −33.1464 −1.24572
\(709\) 39.4834 1.48283 0.741414 0.671048i \(-0.234155\pi\)
0.741414 + 0.671048i \(0.234155\pi\)
\(710\) −12.2764 −0.460727
\(711\) −21.5269 −0.807321
\(712\) −6.38594 −0.239323
\(713\) −15.6747 −0.587021
\(714\) −8.68453 −0.325010
\(715\) −4.69328 −0.175519
\(716\) −16.6363 −0.621729
\(717\) 10.8354 0.404655
\(718\) −21.6717 −0.808780
\(719\) 17.8020 0.663902 0.331951 0.943297i \(-0.392293\pi\)
0.331951 + 0.943297i \(0.392293\pi\)
\(720\) −2.05989 −0.0767677
\(721\) −21.6278 −0.805463
\(722\) 6.98581 0.259985
\(723\) 12.9926 0.483201
\(724\) −14.4304 −0.536302
\(725\) −5.94499 −0.220791
\(726\) 2.24942 0.0834838
\(727\) 39.2561 1.45593 0.727964 0.685615i \(-0.240467\pi\)
0.727964 + 0.685615i \(0.240467\pi\)
\(728\) −6.73167 −0.249492
\(729\) −8.25756 −0.305836
\(730\) 6.20325 0.229593
\(731\) 2.69172 0.0995567
\(732\) 26.7834 0.989944
\(733\) 38.5716 1.42467 0.712337 0.701838i \(-0.247637\pi\)
0.712337 + 0.701838i \(0.247637\pi\)
\(734\) −15.0819 −0.556683
\(735\) 11.1183 0.410103
\(736\) 2.31042 0.0851630
\(737\) 9.71230 0.357757
\(738\) 14.4544 0.532073
\(739\) 23.2463 0.855128 0.427564 0.903985i \(-0.359372\pi\)
0.427564 + 0.903985i \(0.359372\pi\)
\(740\) −3.88177 −0.142697
\(741\) 53.8164 1.97700
\(742\) −13.3211 −0.489032
\(743\) 18.0655 0.662760 0.331380 0.943497i \(-0.392486\pi\)
0.331380 + 0.943497i \(0.392486\pi\)
\(744\) −15.2609 −0.559491
\(745\) −9.28161 −0.340052
\(746\) −23.7371 −0.869076
\(747\) 9.18686 0.336130
\(748\) 2.69172 0.0984189
\(749\) 2.48011 0.0906211
\(750\) −2.24942 −0.0821372
\(751\) −37.8733 −1.38202 −0.691008 0.722847i \(-0.742833\pi\)
−0.691008 + 0.722847i \(0.742833\pi\)
\(752\) 0.722561 0.0263491
\(753\) −62.8433 −2.29014
\(754\) −27.9015 −1.01611
\(755\) 5.35962 0.195057
\(756\) 3.03315 0.110315
\(757\) −0.0136388 −0.000495711 0 −0.000247856 1.00000i \(-0.500079\pi\)
−0.000247856 1.00000i \(0.500079\pi\)
\(758\) 11.5566 0.419754
\(759\) 5.19710 0.188643
\(760\) −5.09763 −0.184910
\(761\) 11.8236 0.428606 0.214303 0.976767i \(-0.431252\pi\)
0.214303 + 0.976767i \(0.431252\pi\)
\(762\) −17.5784 −0.636796
\(763\) −13.1003 −0.474261
\(764\) 6.94965 0.251430
\(765\) −5.54465 −0.200467
\(766\) 24.9628 0.901942
\(767\) −69.1578 −2.49714
\(768\) 2.24942 0.0811690
\(769\) −35.4685 −1.27903 −0.639514 0.768779i \(-0.720864\pi\)
−0.639514 + 0.768779i \(0.720864\pi\)
\(770\) 1.43432 0.0516894
\(771\) −21.8112 −0.785510
\(772\) 3.04767 0.109688
\(773\) 2.29853 0.0826723 0.0413361 0.999145i \(-0.486839\pi\)
0.0413361 + 0.999145i \(0.486839\pi\)
\(774\) 2.05989 0.0740414
\(775\) −6.78435 −0.243701
\(776\) 0.328464 0.0117912
\(777\) −12.5241 −0.449300
\(778\) −26.8036 −0.960954
\(779\) 35.7703 1.28160
\(780\) −10.5572 −0.378007
\(781\) 12.2764 0.439286
\(782\) 6.21898 0.222390
\(783\) 12.5718 0.449281
\(784\) −4.94272 −0.176526
\(785\) −18.6665 −0.666237
\(786\) −15.1012 −0.538643
\(787\) −6.00131 −0.213924 −0.106962 0.994263i \(-0.534112\pi\)
−0.106962 + 0.994263i \(0.534112\pi\)
\(788\) −12.9605 −0.461699
\(789\) −45.9248 −1.63497
\(790\) 10.4505 0.371811
\(791\) −6.63861 −0.236042
\(792\) 2.05989 0.0731951
\(793\) 55.8819 1.98443
\(794\) −6.69149 −0.237472
\(795\) −20.8912 −0.740934
\(796\) 5.53937 0.196338
\(797\) 48.1581 1.70585 0.852924 0.522036i \(-0.174827\pi\)
0.852924 + 0.522036i \(0.174827\pi\)
\(798\) −16.4470 −0.582216
\(799\) 1.94493 0.0688066
\(800\) 1.00000 0.0353553
\(801\) −13.1544 −0.464786
\(802\) −7.04199 −0.248661
\(803\) −6.20325 −0.218908
\(804\) 21.8471 0.770486
\(805\) 3.31388 0.116799
\(806\) −31.8409 −1.12155
\(807\) 31.4062 1.10555
\(808\) 13.3472 0.469552
\(809\) −30.9838 −1.08933 −0.544666 0.838653i \(-0.683343\pi\)
−0.544666 + 0.838653i \(0.683343\pi\)
\(810\) 10.9365 0.384270
\(811\) −4.33544 −0.152238 −0.0761190 0.997099i \(-0.524253\pi\)
−0.0761190 + 0.997099i \(0.524253\pi\)
\(812\) 8.52703 0.299240
\(813\) −40.7091 −1.42773
\(814\) 3.88177 0.136056
\(815\) −23.7568 −0.832164
\(816\) 6.05480 0.211960
\(817\) 5.09763 0.178343
\(818\) 0.530901 0.0185625
\(819\) −13.8665 −0.484536
\(820\) −7.01704 −0.245046
\(821\) 12.9425 0.451695 0.225847 0.974163i \(-0.427485\pi\)
0.225847 + 0.974163i \(0.427485\pi\)
\(822\) −32.5230 −1.13437
\(823\) 26.9475 0.939331 0.469665 0.882844i \(-0.344375\pi\)
0.469665 + 0.882844i \(0.344375\pi\)
\(824\) 15.0788 0.525295
\(825\) 2.24942 0.0783148
\(826\) 21.1355 0.735397
\(827\) −12.4487 −0.432884 −0.216442 0.976295i \(-0.569445\pi\)
−0.216442 + 0.976295i \(0.569445\pi\)
\(828\) 4.75921 0.165394
\(829\) −1.36229 −0.0473144 −0.0236572 0.999720i \(-0.507531\pi\)
−0.0236572 + 0.999720i \(0.507531\pi\)
\(830\) −4.45987 −0.154804
\(831\) −18.2902 −0.634480
\(832\) 4.69328 0.162710
\(833\) −13.3044 −0.460970
\(834\) 2.93760 0.101721
\(835\) 4.94648 0.171180
\(836\) 5.09763 0.176305
\(837\) 14.3468 0.495899
\(838\) −21.8429 −0.754551
\(839\) 26.2011 0.904564 0.452282 0.891875i \(-0.350610\pi\)
0.452282 + 0.891875i \(0.350610\pi\)
\(840\) 3.22639 0.111321
\(841\) 6.34294 0.218722
\(842\) −29.9008 −1.03045
\(843\) −13.4883 −0.464563
\(844\) 1.49551 0.0514775
\(845\) −9.02685 −0.310533
\(846\) 1.48840 0.0511722
\(847\) −1.43432 −0.0492839
\(848\) 9.28737 0.318929
\(849\) −47.2448 −1.62144
\(850\) 2.69172 0.0923251
\(851\) 8.96851 0.307436
\(852\) 27.6149 0.946070
\(853\) 25.9974 0.890135 0.445068 0.895497i \(-0.353180\pi\)
0.445068 + 0.895497i \(0.353180\pi\)
\(854\) −17.0782 −0.584404
\(855\) −10.5006 −0.359112
\(856\) −1.72911 −0.0590999
\(857\) 4.12590 0.140938 0.0704690 0.997514i \(-0.477550\pi\)
0.0704690 + 0.997514i \(0.477550\pi\)
\(858\) 10.5572 0.360415
\(859\) 49.0374 1.67313 0.836567 0.547864i \(-0.184559\pi\)
0.836567 + 0.547864i \(0.184559\pi\)
\(860\) −1.00000 −0.0340997
\(861\) −22.6397 −0.771560
\(862\) −3.74480 −0.127548
\(863\) −12.7928 −0.435472 −0.217736 0.976008i \(-0.569867\pi\)
−0.217736 + 0.976008i \(0.569867\pi\)
\(864\) −2.11469 −0.0719433
\(865\) −24.4374 −0.830897
\(866\) 15.2090 0.516823
\(867\) −21.9424 −0.745201
\(868\) 9.73095 0.330290
\(869\) −10.4505 −0.354508
\(870\) 13.3728 0.453380
\(871\) 45.5825 1.54451
\(872\) 9.13341 0.309296
\(873\) 0.676602 0.0228995
\(874\) 11.7776 0.398385
\(875\) 1.43432 0.0484889
\(876\) −13.9537 −0.471452
\(877\) 36.9386 1.24733 0.623664 0.781693i \(-0.285643\pi\)
0.623664 + 0.781693i \(0.285643\pi\)
\(878\) −35.0097 −1.18152
\(879\) −67.1437 −2.26470
\(880\) −1.00000 −0.0337100
\(881\) 50.0472 1.68613 0.843066 0.537811i \(-0.180748\pi\)
0.843066 + 0.537811i \(0.180748\pi\)
\(882\) −10.1815 −0.342828
\(883\) 44.2846 1.49030 0.745148 0.666899i \(-0.232379\pi\)
0.745148 + 0.666899i \(0.232379\pi\)
\(884\) 12.6330 0.424893
\(885\) 33.1464 1.11420
\(886\) −31.3131 −1.05198
\(887\) −37.3745 −1.25491 −0.627456 0.778652i \(-0.715904\pi\)
−0.627456 + 0.778652i \(0.715904\pi\)
\(888\) 8.73174 0.293018
\(889\) 11.2087 0.375927
\(890\) 6.38594 0.214057
\(891\) −10.9365 −0.366387
\(892\) −20.6795 −0.692400
\(893\) 3.68335 0.123259
\(894\) 20.8782 0.698273
\(895\) 16.6363 0.556091
\(896\) −1.43432 −0.0479173
\(897\) 24.3914 0.814406
\(898\) 1.28870 0.0430046
\(899\) 40.3329 1.34518
\(900\) 2.05989 0.0686631
\(901\) 24.9989 0.832836
\(902\) 7.01704 0.233642
\(903\) −3.22639 −0.107368
\(904\) 4.62839 0.153938
\(905\) 14.4304 0.479683
\(906\) −12.0560 −0.400535
\(907\) −13.2860 −0.441154 −0.220577 0.975370i \(-0.570794\pi\)
−0.220577 + 0.975370i \(0.570794\pi\)
\(908\) −21.9344 −0.727917
\(909\) 27.4938 0.911911
\(910\) 6.73167 0.223153
\(911\) −37.1564 −1.23105 −0.615523 0.788119i \(-0.711055\pi\)
−0.615523 + 0.788119i \(0.711055\pi\)
\(912\) 11.4667 0.379701
\(913\) 4.45987 0.147600
\(914\) −35.6533 −1.17931
\(915\) −26.7834 −0.885433
\(916\) −21.3370 −0.704995
\(917\) 9.62915 0.317982
\(918\) −5.69215 −0.187869
\(919\) 12.2813 0.405122 0.202561 0.979270i \(-0.435074\pi\)
0.202561 + 0.979270i \(0.435074\pi\)
\(920\) −2.31042 −0.0761721
\(921\) 8.57221 0.282464
\(922\) −10.0290 −0.330286
\(923\) 57.6167 1.89648
\(924\) −3.22639 −0.106141
\(925\) 3.88177 0.127632
\(926\) −26.9399 −0.885299
\(927\) 31.0607 1.02017
\(928\) −5.94499 −0.195154
\(929\) −16.4206 −0.538741 −0.269371 0.963037i \(-0.586816\pi\)
−0.269371 + 0.963037i \(0.586816\pi\)
\(930\) 15.2609 0.500424
\(931\) −25.1962 −0.825770
\(932\) −18.4304 −0.603706
\(933\) 40.4721 1.32500
\(934\) −17.3201 −0.566731
\(935\) −2.69172 −0.0880285
\(936\) 9.66765 0.315997
\(937\) 37.7828 1.23431 0.617156 0.786841i \(-0.288285\pi\)
0.617156 + 0.786841i \(0.288285\pi\)
\(938\) −13.9306 −0.454849
\(939\) 9.52228 0.310748
\(940\) −0.722561 −0.0235673
\(941\) −17.0274 −0.555077 −0.277539 0.960715i \(-0.589519\pi\)
−0.277539 + 0.960715i \(0.589519\pi\)
\(942\) 41.9889 1.36807
\(943\) 16.2123 0.527945
\(944\) −14.7355 −0.479600
\(945\) −3.03315 −0.0986684
\(946\) 1.00000 0.0325128
\(947\) −2.03265 −0.0660521 −0.0330260 0.999454i \(-0.510514\pi\)
−0.0330260 + 0.999454i \(0.510514\pi\)
\(948\) −23.5075 −0.763489
\(949\) −29.1136 −0.945066
\(950\) 5.09763 0.165389
\(951\) 51.0274 1.65468
\(952\) −3.86079 −0.125129
\(953\) 1.53425 0.0496994 0.0248497 0.999691i \(-0.492089\pi\)
0.0248497 + 0.999691i \(0.492089\pi\)
\(954\) 19.1310 0.619389
\(955\) −6.94965 −0.224885
\(956\) 4.81697 0.155792
\(957\) −13.3728 −0.432281
\(958\) −29.6771 −0.958823
\(959\) 20.7380 0.669664
\(960\) −2.24942 −0.0725997
\(961\) 15.0275 0.484757
\(962\) 18.2182 0.587379
\(963\) −3.56179 −0.114777
\(964\) 5.77599 0.186032
\(965\) −3.04767 −0.0981081
\(966\) −7.45431 −0.239838
\(967\) −12.6438 −0.406597 −0.203298 0.979117i \(-0.565166\pi\)
−0.203298 + 0.979117i \(0.565166\pi\)
\(968\) 1.00000 0.0321412
\(969\) 30.8651 0.991531
\(970\) −0.328464 −0.0105464
\(971\) 22.1070 0.709446 0.354723 0.934971i \(-0.384575\pi\)
0.354723 + 0.934971i \(0.384575\pi\)
\(972\) −18.2568 −0.585586
\(973\) −1.87313 −0.0600499
\(974\) 20.7138 0.663713
\(975\) 10.5572 0.338100
\(976\) 11.9068 0.381128
\(977\) 6.61636 0.211676 0.105838 0.994383i \(-0.466248\pi\)
0.105838 + 0.994383i \(0.466248\pi\)
\(978\) 53.4390 1.70879
\(979\) −6.38594 −0.204096
\(980\) 4.94272 0.157889
\(981\) 18.8139 0.600680
\(982\) 0.702895 0.0224303
\(983\) −45.9786 −1.46649 −0.733245 0.679964i \(-0.761995\pi\)
−0.733245 + 0.679964i \(0.761995\pi\)
\(984\) 15.7843 0.503185
\(985\) 12.9605 0.412956
\(986\) −16.0022 −0.509615
\(987\) −2.33127 −0.0742050
\(988\) 23.9246 0.761142
\(989\) 2.31042 0.0734669
\(990\) −2.05989 −0.0654677
\(991\) 17.3138 0.549991 0.274995 0.961446i \(-0.411324\pi\)
0.274995 + 0.961446i \(0.411324\pi\)
\(992\) −6.78435 −0.215403
\(993\) 47.1047 1.49482
\(994\) −17.6084 −0.558503
\(995\) −5.53937 −0.175610
\(996\) 10.0321 0.317880
\(997\) −42.1164 −1.33384 −0.666921 0.745129i \(-0.732388\pi\)
−0.666921 + 0.745129i \(0.732388\pi\)
\(998\) 7.52040 0.238054
\(999\) −8.20876 −0.259714
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 4730.2.a.bf.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.11 13 1.1 even 1 trivial