Properties

Label 847.2.a.p.1.6
Level 847
Weight 2
Character 847.1
Self dual yes
Analytic conductor 6.763
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.11447\)
Character \(\chi\) = 847.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.11447 q^{2} -2.85882 q^{3} -0.757964 q^{4} +3.45608 q^{5} -3.18606 q^{6} +1.00000 q^{7} -3.07366 q^{8} +5.17284 q^{9} +O(q^{10})\) \(q+1.11447 q^{2} -2.85882 q^{3} -0.757964 q^{4} +3.45608 q^{5} -3.18606 q^{6} +1.00000 q^{7} -3.07366 q^{8} +5.17284 q^{9} +3.85168 q^{10} +2.16688 q^{12} -2.05965 q^{13} +1.11447 q^{14} -9.88030 q^{15} -1.90956 q^{16} +1.93373 q^{17} +5.76496 q^{18} +1.62296 q^{19} -2.61958 q^{20} -2.85882 q^{21} -0.807136 q^{23} +8.78703 q^{24} +6.94447 q^{25} -2.29541 q^{26} -6.21176 q^{27} -0.757964 q^{28} +7.97368 q^{29} -11.0113 q^{30} +0.788420 q^{31} +4.01918 q^{32} +2.15508 q^{34} +3.45608 q^{35} -3.92083 q^{36} +10.0618 q^{37} +1.80873 q^{38} +5.88817 q^{39} -10.6228 q^{40} -2.12613 q^{41} -3.18606 q^{42} +3.08043 q^{43} +17.8777 q^{45} -0.899526 q^{46} +7.56632 q^{47} +5.45909 q^{48} +1.00000 q^{49} +7.73938 q^{50} -5.52818 q^{51} +1.56114 q^{52} +10.8224 q^{53} -6.92280 q^{54} -3.07366 q^{56} -4.63975 q^{57} +8.88640 q^{58} -3.29664 q^{59} +7.48891 q^{60} -1.07663 q^{61} +0.878667 q^{62} +5.17284 q^{63} +8.29836 q^{64} -7.11832 q^{65} +2.40314 q^{67} -1.46570 q^{68} +2.30745 q^{69} +3.85168 q^{70} -3.18859 q^{71} -15.8995 q^{72} +1.22628 q^{73} +11.2135 q^{74} -19.8530 q^{75} -1.23015 q^{76} +6.56217 q^{78} -9.48182 q^{79} -6.59959 q^{80} +2.23976 q^{81} -2.36950 q^{82} -16.0694 q^{83} +2.16688 q^{84} +6.68312 q^{85} +3.43303 q^{86} -22.7953 q^{87} -4.43830 q^{89} +19.9241 q^{90} -2.05965 q^{91} +0.611780 q^{92} -2.25395 q^{93} +8.43241 q^{94} +5.60908 q^{95} -11.4901 q^{96} +6.46807 q^{97} +1.11447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{2} + 4q^{3} + 7q^{4} + 10q^{5} - q^{6} + 8q^{7} + 14q^{9} + O(q^{10}) \) \( 8q + q^{2} + 4q^{3} + 7q^{4} + 10q^{5} - q^{6} + 8q^{7} + 14q^{9} + 6q^{10} + 9q^{12} - 6q^{13} + q^{14} + 11q^{15} + q^{16} - 5q^{17} + 8q^{18} - 13q^{19} + 23q^{20} + 4q^{21} + 16q^{23} - 10q^{24} + 16q^{25} - 6q^{26} + 10q^{27} + 7q^{28} + 9q^{29} - 36q^{30} + 9q^{31} + 16q^{32} - 12q^{34} + 10q^{35} - 14q^{36} + 7q^{37} - 10q^{38} + 13q^{39} + 5q^{40} - 10q^{41} - q^{42} - 4q^{43} + 35q^{45} + 4q^{46} + 16q^{47} - 3q^{48} + 8q^{49} + 6q^{50} + 13q^{51} - 41q^{52} + 37q^{53} + 30q^{54} + 2q^{57} - 15q^{58} + q^{59} + 5q^{60} + 19q^{61} - 18q^{62} + 14q^{63} - 4q^{64} - 4q^{65} - 19q^{67} + 9q^{68} + 20q^{69} + 6q^{70} + 13q^{71} - 35q^{72} - 25q^{73} + 33q^{74} - 13q^{75} + 26q^{76} - 29q^{78} + 4q^{80} + 8q^{81} - 13q^{82} - 25q^{83} + 9q^{84} + 3q^{85} + 4q^{86} - 36q^{87} + 37q^{89} - 2q^{90} - 6q^{91} + 35q^{92} + 21q^{93} - 42q^{94} + 21q^{95} - 6q^{96} + 15q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.11447 0.788047 0.394023 0.919100i \(-0.371083\pi\)
0.394023 + 0.919100i \(0.371083\pi\)
\(3\) −2.85882 −1.65054 −0.825270 0.564739i \(-0.808977\pi\)
−0.825270 + 0.564739i \(0.808977\pi\)
\(4\) −0.757964 −0.378982
\(5\) 3.45608 1.54561 0.772803 0.634647i \(-0.218854\pi\)
0.772803 + 0.634647i \(0.218854\pi\)
\(6\) −3.18606 −1.30070
\(7\) 1.00000 0.377964
\(8\) −3.07366 −1.08670
\(9\) 5.17284 1.72428
\(10\) 3.85168 1.21801
\(11\) 0 0
\(12\) 2.16688 0.625525
\(13\) −2.05965 −0.571245 −0.285622 0.958342i \(-0.592200\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(14\) 1.11447 0.297854
\(15\) −9.88030 −2.55108
\(16\) −1.90956 −0.477390
\(17\) 1.93373 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(18\) 5.76496 1.35881
\(19\) 1.62296 0.372333 0.186166 0.982518i \(-0.440394\pi\)
0.186166 + 0.982518i \(0.440394\pi\)
\(20\) −2.61958 −0.585757
\(21\) −2.85882 −0.623845
\(22\) 0 0
\(23\) −0.807136 −0.168299 −0.0841497 0.996453i \(-0.526817\pi\)
−0.0841497 + 0.996453i \(0.526817\pi\)
\(24\) 8.78703 1.79365
\(25\) 6.94447 1.38889
\(26\) −2.29541 −0.450168
\(27\) −6.21176 −1.19545
\(28\) −0.757964 −0.143242
\(29\) 7.97368 1.48067 0.740337 0.672235i \(-0.234666\pi\)
0.740337 + 0.672235i \(0.234666\pi\)
\(30\) −11.0113 −2.01037
\(31\) 0.788420 0.141604 0.0708022 0.997490i \(-0.477444\pi\)
0.0708022 + 0.997490i \(0.477444\pi\)
\(32\) 4.01918 0.710497
\(33\) 0 0
\(34\) 2.15508 0.369592
\(35\) 3.45608 0.584184
\(36\) −3.92083 −0.653471
\(37\) 10.0618 1.65414 0.827072 0.562096i \(-0.190005\pi\)
0.827072 + 0.562096i \(0.190005\pi\)
\(38\) 1.80873 0.293415
\(39\) 5.88817 0.942862
\(40\) −10.6228 −1.67961
\(41\) −2.12613 −0.332046 −0.166023 0.986122i \(-0.553093\pi\)
−0.166023 + 0.986122i \(0.553093\pi\)
\(42\) −3.18606 −0.491619
\(43\) 3.08043 0.469761 0.234880 0.972024i \(-0.424530\pi\)
0.234880 + 0.972024i \(0.424530\pi\)
\(44\) 0 0
\(45\) 17.8777 2.66506
\(46\) −0.899526 −0.132628
\(47\) 7.56632 1.10366 0.551831 0.833956i \(-0.313930\pi\)
0.551831 + 0.833956i \(0.313930\pi\)
\(48\) 5.45909 0.787952
\(49\) 1.00000 0.142857
\(50\) 7.73938 1.09451
\(51\) −5.52818 −0.774100
\(52\) 1.56114 0.216492
\(53\) 10.8224 1.48658 0.743289 0.668971i \(-0.233265\pi\)
0.743289 + 0.668971i \(0.233265\pi\)
\(54\) −6.92280 −0.942073
\(55\) 0 0
\(56\) −3.07366 −0.410735
\(57\) −4.63975 −0.614550
\(58\) 8.88640 1.16684
\(59\) −3.29664 −0.429186 −0.214593 0.976704i \(-0.568842\pi\)
−0.214593 + 0.976704i \(0.568842\pi\)
\(60\) 7.48891 0.966814
\(61\) −1.07663 −0.137848 −0.0689240 0.997622i \(-0.521957\pi\)
−0.0689240 + 0.997622i \(0.521957\pi\)
\(62\) 0.878667 0.111591
\(63\) 5.17284 0.651717
\(64\) 8.29836 1.03729
\(65\) −7.11832 −0.882919
\(66\) 0 0
\(67\) 2.40314 0.293590 0.146795 0.989167i \(-0.453104\pi\)
0.146795 + 0.989167i \(0.453104\pi\)
\(68\) −1.46570 −0.177742
\(69\) 2.30745 0.277785
\(70\) 3.85168 0.460364
\(71\) −3.18859 −0.378417 −0.189208 0.981937i \(-0.560592\pi\)
−0.189208 + 0.981937i \(0.560592\pi\)
\(72\) −15.8995 −1.87378
\(73\) 1.22628 0.143525 0.0717624 0.997422i \(-0.477138\pi\)
0.0717624 + 0.997422i \(0.477138\pi\)
\(74\) 11.2135 1.30354
\(75\) −19.8530 −2.29243
\(76\) −1.23015 −0.141107
\(77\) 0 0
\(78\) 6.56217 0.743020
\(79\) −9.48182 −1.06679 −0.533394 0.845867i \(-0.679084\pi\)
−0.533394 + 0.845867i \(0.679084\pi\)
\(80\) −6.59959 −0.737857
\(81\) 2.23976 0.248862
\(82\) −2.36950 −0.261668
\(83\) −16.0694 −1.76385 −0.881923 0.471394i \(-0.843751\pi\)
−0.881923 + 0.471394i \(0.843751\pi\)
\(84\) 2.16688 0.236426
\(85\) 6.68312 0.724886
\(86\) 3.43303 0.370193
\(87\) −22.7953 −2.44391
\(88\) 0 0
\(89\) −4.43830 −0.470459 −0.235230 0.971940i \(-0.575584\pi\)
−0.235230 + 0.971940i \(0.575584\pi\)
\(90\) 19.9241 2.10019
\(91\) −2.05965 −0.215910
\(92\) 0.611780 0.0637825
\(93\) −2.25395 −0.233724
\(94\) 8.43241 0.869737
\(95\) 5.60908 0.575479
\(96\) −11.4901 −1.17270
\(97\) 6.46807 0.656733 0.328366 0.944550i \(-0.393502\pi\)
0.328366 + 0.944550i \(0.393502\pi\)
\(98\) 1.11447 0.112578
\(99\) 0 0
\(100\) −5.26366 −0.526366
\(101\) −15.4449 −1.53682 −0.768412 0.639955i \(-0.778953\pi\)
−0.768412 + 0.639955i \(0.778953\pi\)
\(102\) −6.16097 −0.610027
\(103\) 8.90812 0.877743 0.438872 0.898550i \(-0.355378\pi\)
0.438872 + 0.898550i \(0.355378\pi\)
\(104\) 6.33067 0.620773
\(105\) −9.88030 −0.964218
\(106\) 12.0613 1.17149
\(107\) 3.51219 0.339536 0.169768 0.985484i \(-0.445698\pi\)
0.169768 + 0.985484i \(0.445698\pi\)
\(108\) 4.70829 0.453055
\(109\) 3.87655 0.371306 0.185653 0.982615i \(-0.440560\pi\)
0.185653 + 0.982615i \(0.440560\pi\)
\(110\) 0 0
\(111\) −28.7648 −2.73023
\(112\) −1.90956 −0.180437
\(113\) 10.6539 1.00224 0.501118 0.865379i \(-0.332922\pi\)
0.501118 + 0.865379i \(0.332922\pi\)
\(114\) −5.17084 −0.484294
\(115\) −2.78952 −0.260124
\(116\) −6.04376 −0.561149
\(117\) −10.6543 −0.984986
\(118\) −3.67399 −0.338218
\(119\) 1.93373 0.177265
\(120\) 30.3687 2.77227
\(121\) 0 0
\(122\) −1.19987 −0.108631
\(123\) 6.07823 0.548056
\(124\) −0.597594 −0.0536655
\(125\) 6.72025 0.601078
\(126\) 5.76496 0.513583
\(127\) 19.4509 1.72599 0.862994 0.505214i \(-0.168586\pi\)
0.862994 + 0.505214i \(0.168586\pi\)
\(128\) 1.20989 0.106941
\(129\) −8.80638 −0.775359
\(130\) −7.93313 −0.695782
\(131\) 5.11284 0.446711 0.223355 0.974737i \(-0.428299\pi\)
0.223355 + 0.974737i \(0.428299\pi\)
\(132\) 0 0
\(133\) 1.62296 0.140728
\(134\) 2.67822 0.231363
\(135\) −21.4683 −1.84770
\(136\) −5.94362 −0.509661
\(137\) 9.10052 0.777510 0.388755 0.921341i \(-0.372905\pi\)
0.388755 + 0.921341i \(0.372905\pi\)
\(138\) 2.57158 0.218907
\(139\) −13.0166 −1.10405 −0.552026 0.833827i \(-0.686145\pi\)
−0.552026 + 0.833827i \(0.686145\pi\)
\(140\) −2.61958 −0.221395
\(141\) −21.6307 −1.82164
\(142\) −3.55358 −0.298210
\(143\) 0 0
\(144\) −9.87786 −0.823155
\(145\) 27.5577 2.28854
\(146\) 1.36664 0.113104
\(147\) −2.85882 −0.235791
\(148\) −7.62646 −0.626891
\(149\) −3.14650 −0.257771 −0.128886 0.991659i \(-0.541140\pi\)
−0.128886 + 0.991659i \(0.541140\pi\)
\(150\) −22.1255 −1.80654
\(151\) 2.86696 0.233310 0.116655 0.993172i \(-0.462783\pi\)
0.116655 + 0.993172i \(0.462783\pi\)
\(152\) −4.98843 −0.404615
\(153\) 10.0029 0.808684
\(154\) 0 0
\(155\) 2.72484 0.218864
\(156\) −4.46302 −0.357328
\(157\) −21.4895 −1.71505 −0.857524 0.514443i \(-0.827999\pi\)
−0.857524 + 0.514443i \(0.827999\pi\)
\(158\) −10.5672 −0.840679
\(159\) −30.9394 −2.45365
\(160\) 13.8906 1.09815
\(161\) −0.807136 −0.0636112
\(162\) 2.49614 0.196115
\(163\) −8.22245 −0.644032 −0.322016 0.946734i \(-0.604361\pi\)
−0.322016 + 0.946734i \(0.604361\pi\)
\(164\) 1.61153 0.125840
\(165\) 0 0
\(166\) −17.9088 −1.38999
\(167\) −21.7086 −1.67986 −0.839930 0.542695i \(-0.817404\pi\)
−0.839930 + 0.542695i \(0.817404\pi\)
\(168\) 8.78703 0.677934
\(169\) −8.75783 −0.673679
\(170\) 7.44811 0.571244
\(171\) 8.39531 0.642006
\(172\) −2.33485 −0.178031
\(173\) −8.04563 −0.611698 −0.305849 0.952080i \(-0.598940\pi\)
−0.305849 + 0.952080i \(0.598940\pi\)
\(174\) −25.4046 −1.92592
\(175\) 6.94447 0.524953
\(176\) 0 0
\(177\) 9.42449 0.708388
\(178\) −4.94634 −0.370744
\(179\) −3.62091 −0.270640 −0.135320 0.990802i \(-0.543206\pi\)
−0.135320 + 0.990802i \(0.543206\pi\)
\(180\) −13.5507 −1.01001
\(181\) 15.8179 1.17574 0.587868 0.808957i \(-0.299967\pi\)
0.587868 + 0.808957i \(0.299967\pi\)
\(182\) −2.29541 −0.170147
\(183\) 3.07788 0.227524
\(184\) 2.48086 0.182891
\(185\) 34.7743 2.55665
\(186\) −2.51195 −0.184185
\(187\) 0 0
\(188\) −5.73500 −0.418268
\(189\) −6.21176 −0.451839
\(190\) 6.25113 0.453504
\(191\) 0.429081 0.0310472 0.0155236 0.999880i \(-0.495058\pi\)
0.0155236 + 0.999880i \(0.495058\pi\)
\(192\) −23.7235 −1.71210
\(193\) −15.1748 −1.09231 −0.546154 0.837685i \(-0.683909\pi\)
−0.546154 + 0.837685i \(0.683909\pi\)
\(194\) 7.20845 0.517536
\(195\) 20.3500 1.45729
\(196\) −0.757964 −0.0541403
\(197\) −20.8082 −1.48252 −0.741262 0.671216i \(-0.765772\pi\)
−0.741262 + 0.671216i \(0.765772\pi\)
\(198\) 0 0
\(199\) 8.44567 0.598698 0.299349 0.954144i \(-0.403231\pi\)
0.299349 + 0.954144i \(0.403231\pi\)
\(200\) −21.3449 −1.50932
\(201\) −6.87014 −0.484582
\(202\) −17.2128 −1.21109
\(203\) 7.97368 0.559642
\(204\) 4.19016 0.293370
\(205\) −7.34808 −0.513212
\(206\) 9.92781 0.691703
\(207\) −4.17518 −0.290195
\(208\) 3.93303 0.272707
\(209\) 0 0
\(210\) −11.0113 −0.759849
\(211\) −9.85927 −0.678740 −0.339370 0.940653i \(-0.610214\pi\)
−0.339370 + 0.940653i \(0.610214\pi\)
\(212\) −8.20303 −0.563386
\(213\) 9.11561 0.624592
\(214\) 3.91422 0.267570
\(215\) 10.6462 0.726065
\(216\) 19.0928 1.29910
\(217\) 0.788420 0.0535214
\(218\) 4.32028 0.292606
\(219\) −3.50570 −0.236893
\(220\) 0 0
\(221\) −3.98281 −0.267913
\(222\) −32.0574 −2.15155
\(223\) 17.3959 1.16491 0.582457 0.812861i \(-0.302091\pi\)
0.582457 + 0.812861i \(0.302091\pi\)
\(224\) 4.01918 0.268542
\(225\) 35.9227 2.39484
\(226\) 11.8734 0.789809
\(227\) −12.6315 −0.838381 −0.419190 0.907898i \(-0.637686\pi\)
−0.419190 + 0.907898i \(0.637686\pi\)
\(228\) 3.51676 0.232903
\(229\) 4.57341 0.302220 0.151110 0.988517i \(-0.451715\pi\)
0.151110 + 0.988517i \(0.451715\pi\)
\(230\) −3.10883 −0.204990
\(231\) 0 0
\(232\) −24.5084 −1.60905
\(233\) 23.8569 1.56292 0.781458 0.623957i \(-0.214476\pi\)
0.781458 + 0.623957i \(0.214476\pi\)
\(234\) −11.8738 −0.776215
\(235\) 26.1498 1.70582
\(236\) 2.49873 0.162654
\(237\) 27.1068 1.76078
\(238\) 2.15508 0.139693
\(239\) 8.83814 0.571692 0.285846 0.958276i \(-0.407725\pi\)
0.285846 + 0.958276i \(0.407725\pi\)
\(240\) 18.8670 1.21786
\(241\) 18.9464 1.22045 0.610224 0.792229i \(-0.291079\pi\)
0.610224 + 0.792229i \(0.291079\pi\)
\(242\) 0 0
\(243\) 12.2322 0.784696
\(244\) 0.816045 0.0522419
\(245\) 3.45608 0.220801
\(246\) 6.77398 0.431893
\(247\) −3.34273 −0.212693
\(248\) −2.42333 −0.153882
\(249\) 45.9395 2.91130
\(250\) 7.48950 0.473678
\(251\) −2.86691 −0.180958 −0.0904790 0.995898i \(-0.528840\pi\)
−0.0904790 + 0.995898i \(0.528840\pi\)
\(252\) −3.92083 −0.246989
\(253\) 0 0
\(254\) 21.6774 1.36016
\(255\) −19.1058 −1.19645
\(256\) −15.2483 −0.953021
\(257\) 22.4159 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(258\) −9.81442 −0.611019
\(259\) 10.0618 0.625208
\(260\) 5.39543 0.334611
\(261\) 41.2466 2.55310
\(262\) 5.69808 0.352029
\(263\) 0.990706 0.0610895 0.0305448 0.999533i \(-0.490276\pi\)
0.0305448 + 0.999533i \(0.490276\pi\)
\(264\) 0 0
\(265\) 37.4032 2.29766
\(266\) 1.80873 0.110901
\(267\) 12.6883 0.776511
\(268\) −1.82149 −0.111265
\(269\) −7.19036 −0.438404 −0.219202 0.975679i \(-0.570345\pi\)
−0.219202 + 0.975679i \(0.570345\pi\)
\(270\) −23.9257 −1.45607
\(271\) −27.1643 −1.65011 −0.825056 0.565050i \(-0.808857\pi\)
−0.825056 + 0.565050i \(0.808857\pi\)
\(272\) −3.69257 −0.223895
\(273\) 5.88817 0.356368
\(274\) 10.1422 0.612714
\(275\) 0 0
\(276\) −1.74897 −0.105275
\(277\) 20.9856 1.26090 0.630451 0.776229i \(-0.282870\pi\)
0.630451 + 0.776229i \(0.282870\pi\)
\(278\) −14.5065 −0.870045
\(279\) 4.07837 0.244166
\(280\) −10.6228 −0.634834
\(281\) −28.1580 −1.67977 −0.839883 0.542768i \(-0.817376\pi\)
−0.839883 + 0.542768i \(0.817376\pi\)
\(282\) −24.1067 −1.43553
\(283\) −26.1917 −1.55694 −0.778469 0.627684i \(-0.784003\pi\)
−0.778469 + 0.627684i \(0.784003\pi\)
\(284\) 2.41684 0.143413
\(285\) −16.0353 −0.949851
\(286\) 0 0
\(287\) −2.12613 −0.125502
\(288\) 20.7906 1.22510
\(289\) −13.2607 −0.780041
\(290\) 30.7121 1.80348
\(291\) −18.4910 −1.08396
\(292\) −0.929473 −0.0543933
\(293\) −4.46385 −0.260781 −0.130391 0.991463i \(-0.541623\pi\)
−0.130391 + 0.991463i \(0.541623\pi\)
\(294\) −3.18606 −0.185815
\(295\) −11.3934 −0.663351
\(296\) −30.9264 −1.79756
\(297\) 0 0
\(298\) −3.50667 −0.203136
\(299\) 1.66242 0.0961402
\(300\) 15.0479 0.868788
\(301\) 3.08043 0.177553
\(302\) 3.19513 0.183859
\(303\) 44.1542 2.53659
\(304\) −3.09914 −0.177748
\(305\) −3.72091 −0.213059
\(306\) 11.1479 0.637281
\(307\) 12.8841 0.735334 0.367667 0.929957i \(-0.380157\pi\)
0.367667 + 0.929957i \(0.380157\pi\)
\(308\) 0 0
\(309\) −25.4667 −1.44875
\(310\) 3.03674 0.172475
\(311\) 26.8199 1.52081 0.760407 0.649447i \(-0.224999\pi\)
0.760407 + 0.649447i \(0.224999\pi\)
\(312\) −18.0982 −1.02461
\(313\) 3.58869 0.202845 0.101422 0.994843i \(-0.467661\pi\)
0.101422 + 0.994843i \(0.467661\pi\)
\(314\) −23.9493 −1.35154
\(315\) 17.8777 1.00730
\(316\) 7.18688 0.404294
\(317\) −16.9181 −0.950213 −0.475106 0.879928i \(-0.657590\pi\)
−0.475106 + 0.879928i \(0.657590\pi\)
\(318\) −34.4809 −1.93359
\(319\) 0 0
\(320\) 28.6798 1.60325
\(321\) −10.0407 −0.560418
\(322\) −0.899526 −0.0501286
\(323\) 3.13836 0.174623
\(324\) −1.69766 −0.0943144
\(325\) −14.3032 −0.793399
\(326\) −9.16365 −0.507528
\(327\) −11.0823 −0.612855
\(328\) 6.53501 0.360836
\(329\) 7.56632 0.417145
\(330\) 0 0
\(331\) −1.23826 −0.0680610 −0.0340305 0.999421i \(-0.510834\pi\)
−0.0340305 + 0.999421i \(0.510834\pi\)
\(332\) 12.1800 0.668466
\(333\) 52.0479 2.85221
\(334\) −24.1935 −1.32381
\(335\) 8.30544 0.453774
\(336\) 5.45909 0.297818
\(337\) −20.4806 −1.11565 −0.557824 0.829959i \(-0.688364\pi\)
−0.557824 + 0.829959i \(0.688364\pi\)
\(338\) −9.76031 −0.530891
\(339\) −30.4576 −1.65423
\(340\) −5.06556 −0.274719
\(341\) 0 0
\(342\) 9.35630 0.505931
\(343\) 1.00000 0.0539949
\(344\) −9.46818 −0.510490
\(345\) 7.97474 0.429346
\(346\) −8.96659 −0.482047
\(347\) 27.2699 1.46392 0.731961 0.681346i \(-0.238605\pi\)
0.731961 + 0.681346i \(0.238605\pi\)
\(348\) 17.2780 0.926199
\(349\) 7.98434 0.427392 0.213696 0.976900i \(-0.431450\pi\)
0.213696 + 0.976900i \(0.431450\pi\)
\(350\) 7.73938 0.413688
\(351\) 12.7941 0.682897
\(352\) 0 0
\(353\) 5.93472 0.315873 0.157937 0.987449i \(-0.449516\pi\)
0.157937 + 0.987449i \(0.449516\pi\)
\(354\) 10.5033 0.558243
\(355\) −11.0200 −0.584883
\(356\) 3.36407 0.178296
\(357\) −5.52818 −0.292582
\(358\) −4.03538 −0.213277
\(359\) 28.4203 1.49996 0.749982 0.661458i \(-0.230062\pi\)
0.749982 + 0.661458i \(0.230062\pi\)
\(360\) −54.9501 −2.89612
\(361\) −16.3660 −0.861368
\(362\) 17.6285 0.926535
\(363\) 0 0
\(364\) 1.56114 0.0818261
\(365\) 4.23810 0.221832
\(366\) 3.43020 0.179299
\(367\) −30.4014 −1.58694 −0.793471 0.608608i \(-0.791728\pi\)
−0.793471 + 0.608608i \(0.791728\pi\)
\(368\) 1.54128 0.0803445
\(369\) −10.9982 −0.572541
\(370\) 38.7547 2.01476
\(371\) 10.8224 0.561873
\(372\) 1.70841 0.0885771
\(373\) 14.4226 0.746772 0.373386 0.927676i \(-0.378197\pi\)
0.373386 + 0.927676i \(0.378197\pi\)
\(374\) 0 0
\(375\) −19.2120 −0.992103
\(376\) −23.2563 −1.19935
\(377\) −16.4230 −0.845828
\(378\) −6.92280 −0.356070
\(379\) −22.4072 −1.15098 −0.575490 0.817809i \(-0.695189\pi\)
−0.575490 + 0.817809i \(0.695189\pi\)
\(380\) −4.25148 −0.218096
\(381\) −55.6066 −2.84881
\(382\) 0.478196 0.0244667
\(383\) 33.6785 1.72089 0.860446 0.509541i \(-0.170185\pi\)
0.860446 + 0.509541i \(0.170185\pi\)
\(384\) −3.45887 −0.176510
\(385\) 0 0
\(386\) −16.9118 −0.860790
\(387\) 15.9346 0.809999
\(388\) −4.90256 −0.248890
\(389\) 2.42783 0.123096 0.0615479 0.998104i \(-0.480396\pi\)
0.0615479 + 0.998104i \(0.480396\pi\)
\(390\) 22.6794 1.14841
\(391\) −1.56078 −0.0789321
\(392\) −3.07366 −0.155243
\(393\) −14.6167 −0.737313
\(394\) −23.1900 −1.16830
\(395\) −32.7699 −1.64883
\(396\) 0 0
\(397\) −5.89696 −0.295960 −0.147980 0.988990i \(-0.547277\pi\)
−0.147980 + 0.988990i \(0.547277\pi\)
\(398\) 9.41242 0.471802
\(399\) −4.63975 −0.232278
\(400\) −13.2609 −0.663045
\(401\) 11.2396 0.561278 0.280639 0.959813i \(-0.409454\pi\)
0.280639 + 0.959813i \(0.409454\pi\)
\(402\) −7.65654 −0.381873
\(403\) −1.62387 −0.0808908
\(404\) 11.7067 0.582429
\(405\) 7.74079 0.384643
\(406\) 8.88640 0.441025
\(407\) 0 0
\(408\) 16.9917 0.841216
\(409\) 29.3344 1.45049 0.725246 0.688490i \(-0.241726\pi\)
0.725246 + 0.688490i \(0.241726\pi\)
\(410\) −8.18919 −0.404435
\(411\) −26.0167 −1.28331
\(412\) −6.75204 −0.332649
\(413\) −3.29664 −0.162217
\(414\) −4.65310 −0.228688
\(415\) −55.5371 −2.72621
\(416\) −8.27811 −0.405868
\(417\) 37.2120 1.82228
\(418\) 0 0
\(419\) −20.2858 −0.991027 −0.495514 0.868600i \(-0.665020\pi\)
−0.495514 + 0.868600i \(0.665020\pi\)
\(420\) 7.48891 0.365422
\(421\) −3.06003 −0.149137 −0.0745683 0.997216i \(-0.523758\pi\)
−0.0745683 + 0.997216i \(0.523758\pi\)
\(422\) −10.9878 −0.534879
\(423\) 39.1394 1.90302
\(424\) −33.2645 −1.61547
\(425\) 13.4287 0.651389
\(426\) 10.1590 0.492207
\(427\) −1.07663 −0.0521017
\(428\) −2.66211 −0.128678
\(429\) 0 0
\(430\) 11.8648 0.572173
\(431\) 7.54197 0.363284 0.181642 0.983365i \(-0.441859\pi\)
0.181642 + 0.983365i \(0.441859\pi\)
\(432\) 11.8617 0.570698
\(433\) −32.1616 −1.54559 −0.772794 0.634657i \(-0.781141\pi\)
−0.772794 + 0.634657i \(0.781141\pi\)
\(434\) 0.878667 0.0421774
\(435\) −78.7823 −3.77732
\(436\) −2.93828 −0.140718
\(437\) −1.30995 −0.0626633
\(438\) −3.90698 −0.186683
\(439\) −4.66725 −0.222756 −0.111378 0.993778i \(-0.535526\pi\)
−0.111378 + 0.993778i \(0.535526\pi\)
\(440\) 0 0
\(441\) 5.17284 0.246326
\(442\) −4.43871 −0.211128
\(443\) 17.2772 0.820862 0.410431 0.911892i \(-0.365378\pi\)
0.410431 + 0.911892i \(0.365378\pi\)
\(444\) 21.8027 1.03471
\(445\) −15.3391 −0.727144
\(446\) 19.3871 0.918007
\(447\) 8.99527 0.425461
\(448\) 8.29836 0.392061
\(449\) −16.7401 −0.790013 −0.395007 0.918678i \(-0.629258\pi\)
−0.395007 + 0.918678i \(0.629258\pi\)
\(450\) 40.0346 1.88725
\(451\) 0 0
\(452\) −8.07529 −0.379829
\(453\) −8.19612 −0.385088
\(454\) −14.0774 −0.660683
\(455\) −7.11832 −0.333712
\(456\) 14.2610 0.667833
\(457\) −21.6974 −1.01496 −0.507481 0.861663i \(-0.669423\pi\)
−0.507481 + 0.861663i \(0.669423\pi\)
\(458\) 5.09692 0.238163
\(459\) −12.0119 −0.560665
\(460\) 2.11436 0.0985825
\(461\) −6.07778 −0.283070 −0.141535 0.989933i \(-0.545204\pi\)
−0.141535 + 0.989933i \(0.545204\pi\)
\(462\) 0 0
\(463\) −5.14719 −0.239210 −0.119605 0.992822i \(-0.538163\pi\)
−0.119605 + 0.992822i \(0.538163\pi\)
\(464\) −15.2262 −0.706860
\(465\) −7.78982 −0.361244
\(466\) 26.5877 1.23165
\(467\) −3.91927 −0.181362 −0.0906812 0.995880i \(-0.528904\pi\)
−0.0906812 + 0.995880i \(0.528904\pi\)
\(468\) 8.07555 0.373292
\(469\) 2.40314 0.110967
\(470\) 29.1431 1.34427
\(471\) 61.4346 2.83076
\(472\) 10.1327 0.466397
\(473\) 0 0
\(474\) 30.2096 1.38757
\(475\) 11.2706 0.517131
\(476\) −1.46570 −0.0671801
\(477\) 55.9828 2.56328
\(478\) 9.84982 0.450520
\(479\) −15.1129 −0.690527 −0.345263 0.938506i \(-0.612210\pi\)
−0.345263 + 0.938506i \(0.612210\pi\)
\(480\) −39.7106 −1.81253
\(481\) −20.7237 −0.944922
\(482\) 21.1152 0.961770
\(483\) 2.30745 0.104993
\(484\) 0 0
\(485\) 22.3541 1.01505
\(486\) 13.6324 0.618377
\(487\) 25.0768 1.13634 0.568169 0.822912i \(-0.307652\pi\)
0.568169 + 0.822912i \(0.307652\pi\)
\(488\) 3.30919 0.149800
\(489\) 23.5065 1.06300
\(490\) 3.85168 0.174001
\(491\) 37.2208 1.67975 0.839875 0.542780i \(-0.182628\pi\)
0.839875 + 0.542780i \(0.182628\pi\)
\(492\) −4.60708 −0.207703
\(493\) 15.4189 0.694434
\(494\) −3.72536 −0.167612
\(495\) 0 0
\(496\) −1.50554 −0.0676006
\(497\) −3.18859 −0.143028
\(498\) 51.1980 2.29424
\(499\) −31.8134 −1.42416 −0.712081 0.702098i \(-0.752247\pi\)
−0.712081 + 0.702098i \(0.752247\pi\)
\(500\) −5.09371 −0.227798
\(501\) 62.0609 2.77267
\(502\) −3.19508 −0.142603
\(503\) −19.2058 −0.856346 −0.428173 0.903697i \(-0.640843\pi\)
−0.428173 + 0.903697i \(0.640843\pi\)
\(504\) −15.8995 −0.708222
\(505\) −53.3788 −2.37532
\(506\) 0 0
\(507\) 25.0370 1.11193
\(508\) −14.7431 −0.654119
\(509\) 2.51549 0.111497 0.0557485 0.998445i \(-0.482246\pi\)
0.0557485 + 0.998445i \(0.482246\pi\)
\(510\) −21.2928 −0.942861
\(511\) 1.22628 0.0542472
\(512\) −19.4135 −0.857966
\(513\) −10.0814 −0.445106
\(514\) 24.9817 1.10190
\(515\) 30.7872 1.35664
\(516\) 6.67492 0.293847
\(517\) 0 0
\(518\) 11.2135 0.492693
\(519\) 23.0010 1.00963
\(520\) 21.8793 0.959470
\(521\) 14.8968 0.652639 0.326320 0.945260i \(-0.394191\pi\)
0.326320 + 0.945260i \(0.394191\pi\)
\(522\) 45.9679 2.01196
\(523\) −9.90502 −0.433116 −0.216558 0.976270i \(-0.569483\pi\)
−0.216558 + 0.976270i \(0.569483\pi\)
\(524\) −3.87535 −0.169295
\(525\) −19.8530 −0.866455
\(526\) 1.10411 0.0481414
\(527\) 1.52459 0.0664122
\(528\) 0 0
\(529\) −22.3485 −0.971675
\(530\) 41.6846 1.81066
\(531\) −17.0530 −0.740036
\(532\) −1.23015 −0.0533336
\(533\) 4.37910 0.189680
\(534\) 14.1407 0.611927
\(535\) 12.1384 0.524789
\(536\) −7.38643 −0.319045
\(537\) 10.3515 0.446701
\(538\) −8.01342 −0.345483
\(539\) 0 0
\(540\) 16.2722 0.700245
\(541\) −22.0084 −0.946214 −0.473107 0.881005i \(-0.656868\pi\)
−0.473107 + 0.881005i \(0.656868\pi\)
\(542\) −30.2737 −1.30037
\(543\) −45.2205 −1.94060
\(544\) 7.77199 0.333222
\(545\) 13.3977 0.573892
\(546\) 6.56217 0.280835
\(547\) −10.8643 −0.464523 −0.232261 0.972653i \(-0.574612\pi\)
−0.232261 + 0.972653i \(0.574612\pi\)
\(548\) −6.89787 −0.294662
\(549\) −5.56922 −0.237689
\(550\) 0 0
\(551\) 12.9410 0.551303
\(552\) −7.09233 −0.301869
\(553\) −9.48182 −0.403208
\(554\) 23.3878 0.993650
\(555\) −99.4133 −4.21986
\(556\) 9.86610 0.418416
\(557\) 33.6126 1.42421 0.712106 0.702072i \(-0.247742\pi\)
0.712106 + 0.702072i \(0.247742\pi\)
\(558\) 4.54521 0.192414
\(559\) −6.34461 −0.268348
\(560\) −6.59959 −0.278884
\(561\) 0 0
\(562\) −31.3811 −1.32373
\(563\) −2.05348 −0.0865440 −0.0432720 0.999063i \(-0.513778\pi\)
−0.0432720 + 0.999063i \(0.513778\pi\)
\(564\) 16.3953 0.690367
\(565\) 36.8208 1.54906
\(566\) −29.1898 −1.22694
\(567\) 2.23976 0.0940611
\(568\) 9.80065 0.411226
\(569\) 3.27416 0.137260 0.0686300 0.997642i \(-0.478137\pi\)
0.0686300 + 0.997642i \(0.478137\pi\)
\(570\) −17.8708 −0.748527
\(571\) −43.8897 −1.83673 −0.918363 0.395738i \(-0.870489\pi\)
−0.918363 + 0.395738i \(0.870489\pi\)
\(572\) 0 0
\(573\) −1.22666 −0.0512446
\(574\) −2.36950 −0.0989012
\(575\) −5.60513 −0.233750
\(576\) 42.9261 1.78859
\(577\) −43.8904 −1.82718 −0.913591 0.406635i \(-0.866702\pi\)
−0.913591 + 0.406635i \(0.866702\pi\)
\(578\) −14.7786 −0.614709
\(579\) 43.3821 1.80290
\(580\) −20.8877 −0.867315
\(581\) −16.0694 −0.666671
\(582\) −20.6076 −0.854214
\(583\) 0 0
\(584\) −3.76915 −0.155969
\(585\) −36.8219 −1.52240
\(586\) −4.97481 −0.205508
\(587\) 2.79166 0.115224 0.0576121 0.998339i \(-0.481651\pi\)
0.0576121 + 0.998339i \(0.481651\pi\)
\(588\) 2.16688 0.0893607
\(589\) 1.27957 0.0527239
\(590\) −12.6976 −0.522752
\(591\) 59.4869 2.44696
\(592\) −19.2136 −0.789673
\(593\) 23.2526 0.954871 0.477435 0.878667i \(-0.341566\pi\)
0.477435 + 0.878667i \(0.341566\pi\)
\(594\) 0 0
\(595\) 6.68312 0.273981
\(596\) 2.38493 0.0976907
\(597\) −24.1446 −0.988175
\(598\) 1.85271 0.0757630
\(599\) 10.4595 0.427362 0.213681 0.976904i \(-0.431455\pi\)
0.213681 + 0.976904i \(0.431455\pi\)
\(600\) 61.0213 2.49118
\(601\) 22.3096 0.910029 0.455015 0.890484i \(-0.349634\pi\)
0.455015 + 0.890484i \(0.349634\pi\)
\(602\) 3.43303 0.139920
\(603\) 12.4311 0.506232
\(604\) −2.17306 −0.0884204
\(605\) 0 0
\(606\) 49.2083 1.99895
\(607\) 18.9884 0.770716 0.385358 0.922767i \(-0.374078\pi\)
0.385358 + 0.922767i \(0.374078\pi\)
\(608\) 6.52296 0.264541
\(609\) −22.7953 −0.923712
\(610\) −4.14683 −0.167900
\(611\) −15.5840 −0.630461
\(612\) −7.58182 −0.306477
\(613\) 20.1617 0.814323 0.407161 0.913356i \(-0.366519\pi\)
0.407161 + 0.913356i \(0.366519\pi\)
\(614\) 14.3589 0.579478
\(615\) 21.0068 0.847077
\(616\) 0 0
\(617\) 7.03919 0.283387 0.141694 0.989911i \(-0.454745\pi\)
0.141694 + 0.989911i \(0.454745\pi\)
\(618\) −28.3818 −1.14168
\(619\) −31.0831 −1.24933 −0.624667 0.780891i \(-0.714765\pi\)
−0.624667 + 0.780891i \(0.714765\pi\)
\(620\) −2.06533 −0.0829457
\(621\) 5.01373 0.201194
\(622\) 29.8898 1.19847
\(623\) −4.43830 −0.177817
\(624\) −11.2438 −0.450113
\(625\) −11.4966 −0.459866
\(626\) 3.99947 0.159851
\(627\) 0 0
\(628\) 16.2883 0.649973
\(629\) 19.4567 0.775791
\(630\) 19.9241 0.793797
\(631\) 21.9720 0.874690 0.437345 0.899294i \(-0.355919\pi\)
0.437345 + 0.899294i \(0.355919\pi\)
\(632\) 29.1439 1.15928
\(633\) 28.1859 1.12029
\(634\) −18.8546 −0.748812
\(635\) 67.2238 2.66770
\(636\) 23.4510 0.929891
\(637\) −2.05965 −0.0816064
\(638\) 0 0
\(639\) −16.4941 −0.652496
\(640\) 4.18149 0.165288
\(641\) −13.2909 −0.524961 −0.262480 0.964937i \(-0.584540\pi\)
−0.262480 + 0.964937i \(0.584540\pi\)
\(642\) −11.1900 −0.441636
\(643\) 14.6904 0.579333 0.289666 0.957128i \(-0.406456\pi\)
0.289666 + 0.957128i \(0.406456\pi\)
\(644\) 0.611780 0.0241075
\(645\) −30.4355 −1.19840
\(646\) 3.49760 0.137611
\(647\) −6.14523 −0.241594 −0.120797 0.992677i \(-0.538545\pi\)
−0.120797 + 0.992677i \(0.538545\pi\)
\(648\) −6.88426 −0.270439
\(649\) 0 0
\(650\) −15.9404 −0.625236
\(651\) −2.25395 −0.0883392
\(652\) 6.23232 0.244077
\(653\) −40.6691 −1.59151 −0.795753 0.605622i \(-0.792924\pi\)
−0.795753 + 0.605622i \(0.792924\pi\)
\(654\) −12.3509 −0.482959
\(655\) 17.6704 0.690438
\(656\) 4.05998 0.158516
\(657\) 6.34333 0.247477
\(658\) 8.43241 0.328730
\(659\) 18.0090 0.701531 0.350765 0.936463i \(-0.385921\pi\)
0.350765 + 0.936463i \(0.385921\pi\)
\(660\) 0 0
\(661\) −17.1420 −0.666745 −0.333373 0.942795i \(-0.608187\pi\)
−0.333373 + 0.942795i \(0.608187\pi\)
\(662\) −1.38000 −0.0536353
\(663\) 11.3861 0.442201
\(664\) 49.3919 1.91678
\(665\) 5.60908 0.217511
\(666\) 58.0057 2.24767
\(667\) −6.43584 −0.249197
\(668\) 16.4543 0.636637
\(669\) −49.7317 −1.92274
\(670\) 9.25613 0.357596
\(671\) 0 0
\(672\) −11.4901 −0.443240
\(673\) −23.1926 −0.894008 −0.447004 0.894532i \(-0.647509\pi\)
−0.447004 + 0.894532i \(0.647509\pi\)
\(674\) −22.8249 −0.879183
\(675\) −43.1374 −1.66036
\(676\) 6.63812 0.255312
\(677\) −27.5705 −1.05962 −0.529811 0.848116i \(-0.677737\pi\)
−0.529811 + 0.848116i \(0.677737\pi\)
\(678\) −33.9440 −1.30361
\(679\) 6.46807 0.248222
\(680\) −20.5416 −0.787735
\(681\) 36.1111 1.38378
\(682\) 0 0
\(683\) 21.9351 0.839322 0.419661 0.907681i \(-0.362149\pi\)
0.419661 + 0.907681i \(0.362149\pi\)
\(684\) −6.36335 −0.243309
\(685\) 31.4521 1.20172
\(686\) 1.11447 0.0425505
\(687\) −13.0746 −0.498826
\(688\) −5.88227 −0.224259
\(689\) −22.2905 −0.849200
\(690\) 8.88758 0.338344
\(691\) −25.6666 −0.976404 −0.488202 0.872731i \(-0.662347\pi\)
−0.488202 + 0.872731i \(0.662347\pi\)
\(692\) 6.09830 0.231823
\(693\) 0 0
\(694\) 30.3913 1.15364
\(695\) −44.9863 −1.70643
\(696\) 70.0650 2.65581
\(697\) −4.11137 −0.155729
\(698\) 8.89828 0.336805
\(699\) −68.2025 −2.57966
\(700\) −5.26366 −0.198948
\(701\) −18.4135 −0.695467 −0.347733 0.937593i \(-0.613049\pi\)
−0.347733 + 0.937593i \(0.613049\pi\)
\(702\) 14.2586 0.538154
\(703\) 16.3298 0.615892
\(704\) 0 0
\(705\) −74.7575 −2.81553
\(706\) 6.61404 0.248923
\(707\) −15.4449 −0.580865
\(708\) −7.14342 −0.268466
\(709\) 23.6621 0.888647 0.444323 0.895866i \(-0.353444\pi\)
0.444323 + 0.895866i \(0.353444\pi\)
\(710\) −12.2815 −0.460915
\(711\) −49.0480 −1.83944
\(712\) 13.6418 0.511249
\(713\) −0.636362 −0.0238319
\(714\) −6.16097 −0.230569
\(715\) 0 0
\(716\) 2.74452 0.102568
\(717\) −25.2666 −0.943600
\(718\) 31.6734 1.18204
\(719\) 11.1222 0.414790 0.207395 0.978257i \(-0.433502\pi\)
0.207395 + 0.978257i \(0.433502\pi\)
\(720\) −34.1387 −1.27227
\(721\) 8.90812 0.331756
\(722\) −18.2394 −0.678799
\(723\) −54.1644 −2.01440
\(724\) −11.9894 −0.445583
\(725\) 55.3730 2.05650
\(726\) 0 0
\(727\) 42.4803 1.57551 0.787753 0.615991i \(-0.211244\pi\)
0.787753 + 0.615991i \(0.211244\pi\)
\(728\) 6.33067 0.234630
\(729\) −41.6889 −1.54403
\(730\) 4.72323 0.174814
\(731\) 5.95671 0.220317
\(732\) −2.33292 −0.0862274
\(733\) 22.1884 0.819548 0.409774 0.912187i \(-0.365607\pi\)
0.409774 + 0.912187i \(0.365607\pi\)
\(734\) −33.8814 −1.25058
\(735\) −9.88030 −0.364440
\(736\) −3.24402 −0.119576
\(737\) 0 0
\(738\) −12.2571 −0.451189
\(739\) 29.4481 1.08327 0.541633 0.840615i \(-0.317806\pi\)
0.541633 + 0.840615i \(0.317806\pi\)
\(740\) −26.3576 −0.968926
\(741\) 9.55627 0.351058
\(742\) 12.0613 0.442783
\(743\) 16.9059 0.620217 0.310109 0.950701i \(-0.399635\pi\)
0.310109 + 0.950701i \(0.399635\pi\)
\(744\) 6.92787 0.253988
\(745\) −10.8745 −0.398412
\(746\) 16.0735 0.588491
\(747\) −83.1244 −3.04136
\(748\) 0 0
\(749\) 3.51219 0.128333
\(750\) −21.4111 −0.781823
\(751\) −1.55147 −0.0566138 −0.0283069 0.999599i \(-0.509012\pi\)
−0.0283069 + 0.999599i \(0.509012\pi\)
\(752\) −14.4484 −0.526877
\(753\) 8.19598 0.298678
\(754\) −18.3029 −0.666552
\(755\) 9.90845 0.360605
\(756\) 4.70829 0.171239
\(757\) 12.5467 0.456016 0.228008 0.973659i \(-0.426779\pi\)
0.228008 + 0.973659i \(0.426779\pi\)
\(758\) −24.9721 −0.907027
\(759\) 0 0
\(760\) −17.2404 −0.625375
\(761\) −9.03436 −0.327495 −0.163748 0.986502i \(-0.552358\pi\)
−0.163748 + 0.986502i \(0.552358\pi\)
\(762\) −61.9717 −2.24500
\(763\) 3.87655 0.140340
\(764\) −0.325228 −0.0117663
\(765\) 34.5707 1.24991
\(766\) 37.5336 1.35614
\(767\) 6.78993 0.245170
\(768\) 43.5922 1.57300
\(769\) −16.1383 −0.581963 −0.290981 0.956729i \(-0.593982\pi\)
−0.290981 + 0.956729i \(0.593982\pi\)
\(770\) 0 0
\(771\) −64.0829 −2.30789
\(772\) 11.5020 0.413965
\(773\) −18.4134 −0.662285 −0.331143 0.943581i \(-0.607434\pi\)
−0.331143 + 0.943581i \(0.607434\pi\)
\(774\) 17.7585 0.638317
\(775\) 5.47516 0.196674
\(776\) −19.8806 −0.713673
\(777\) −28.7648 −1.03193
\(778\) 2.70573 0.0970053
\(779\) −3.45063 −0.123632
\(780\) −15.4246 −0.552288
\(781\) 0 0
\(782\) −1.73944 −0.0622022
\(783\) −49.5306 −1.77008
\(784\) −1.90956 −0.0681986
\(785\) −74.2694 −2.65079
\(786\) −16.2898 −0.581037
\(787\) 46.9870 1.67491 0.837453 0.546509i \(-0.184043\pi\)
0.837453 + 0.546509i \(0.184043\pi\)
\(788\) 15.7719 0.561850
\(789\) −2.83225 −0.100831
\(790\) −36.5210 −1.29936
\(791\) 10.6539 0.378810
\(792\) 0 0
\(793\) 2.21748 0.0787450
\(794\) −6.57197 −0.233230
\(795\) −106.929 −3.79238
\(796\) −6.40152 −0.226896
\(797\) −3.12454 −0.110677 −0.0553385 0.998468i \(-0.517624\pi\)
−0.0553385 + 0.998468i \(0.517624\pi\)
\(798\) −5.17084 −0.183046
\(799\) 14.6312 0.517615
\(800\) 27.9111 0.986805
\(801\) −22.9586 −0.811203
\(802\) 12.5261 0.442314
\(803\) 0 0
\(804\) 5.20732 0.183648
\(805\) −2.78952 −0.0983178
\(806\) −1.80975 −0.0637457
\(807\) 20.5559 0.723604
\(808\) 47.4724 1.67007
\(809\) −32.7257 −1.15057 −0.575286 0.817952i \(-0.695109\pi\)
−0.575286 + 0.817952i \(0.695109\pi\)
\(810\) 8.62685 0.303117
\(811\) 32.6613 1.14689 0.573447 0.819243i \(-0.305606\pi\)
0.573447 + 0.819243i \(0.305606\pi\)
\(812\) −6.04376 −0.212095
\(813\) 77.6578 2.72358
\(814\) 0 0
\(815\) −28.4174 −0.995419
\(816\) 10.5564 0.369548
\(817\) 4.99941 0.174907
\(818\) 32.6922 1.14306
\(819\) −10.6543 −0.372290
\(820\) 5.56958 0.194498
\(821\) −7.43142 −0.259358 −0.129679 0.991556i \(-0.541395\pi\)
−0.129679 + 0.991556i \(0.541395\pi\)
\(822\) −28.9948 −1.01131
\(823\) 6.55866 0.228621 0.114310 0.993445i \(-0.463534\pi\)
0.114310 + 0.993445i \(0.463534\pi\)
\(824\) −27.3805 −0.953846
\(825\) 0 0
\(826\) −3.67399 −0.127835
\(827\) 23.8538 0.829479 0.414739 0.909940i \(-0.363873\pi\)
0.414739 + 0.909940i \(0.363873\pi\)
\(828\) 3.16464 0.109979
\(829\) −26.1431 −0.907989 −0.453994 0.891005i \(-0.650001\pi\)
−0.453994 + 0.891005i \(0.650001\pi\)
\(830\) −61.8942 −2.14838
\(831\) −59.9940 −2.08117
\(832\) −17.0917 −0.592549
\(833\) 1.93373 0.0669997
\(834\) 41.4716 1.43604
\(835\) −75.0265 −2.59640
\(836\) 0 0
\(837\) −4.89747 −0.169281
\(838\) −22.6079 −0.780976
\(839\) 34.2890 1.18379 0.591893 0.806016i \(-0.298381\pi\)
0.591893 + 0.806016i \(0.298381\pi\)
\(840\) 30.3687 1.04782
\(841\) 34.5795 1.19240
\(842\) −3.41030 −0.117527
\(843\) 80.4986 2.77252
\(844\) 7.47297 0.257230
\(845\) −30.2677 −1.04124
\(846\) 43.6195 1.49967
\(847\) 0 0
\(848\) −20.6661 −0.709678
\(849\) 74.8774 2.56979
\(850\) 14.9659 0.513325
\(851\) −8.12121 −0.278392
\(852\) −6.90931 −0.236709
\(853\) 21.3842 0.732181 0.366090 0.930579i \(-0.380696\pi\)
0.366090 + 0.930579i \(0.380696\pi\)
\(854\) −1.19987 −0.0410585
\(855\) 29.0149 0.992287
\(856\) −10.7953 −0.368975
\(857\) 42.8697 1.46440 0.732200 0.681090i \(-0.238494\pi\)
0.732200 + 0.681090i \(0.238494\pi\)
\(858\) 0 0
\(859\) −30.3915 −1.03695 −0.518473 0.855094i \(-0.673499\pi\)
−0.518473 + 0.855094i \(0.673499\pi\)
\(860\) −8.06944 −0.275165
\(861\) 6.07823 0.207146
\(862\) 8.40527 0.286285
\(863\) 11.8184 0.402304 0.201152 0.979560i \(-0.435532\pi\)
0.201152 + 0.979560i \(0.435532\pi\)
\(864\) −24.9661 −0.849365
\(865\) −27.8063 −0.945444
\(866\) −35.8430 −1.21800
\(867\) 37.9099 1.28749
\(868\) −0.597594 −0.0202837
\(869\) 0 0
\(870\) −87.8003 −2.97671
\(871\) −4.94963 −0.167712
\(872\) −11.9152 −0.403499
\(873\) 33.4583 1.13239
\(874\) −1.45989 −0.0493816
\(875\) 6.72025 0.227186
\(876\) 2.65719 0.0897783
\(877\) −9.20488 −0.310827 −0.155413 0.987850i \(-0.549671\pi\)
−0.155413 + 0.987850i \(0.549671\pi\)
\(878\) −5.20149 −0.175542
\(879\) 12.7613 0.430429
\(880\) 0 0
\(881\) −41.9030 −1.41175 −0.705874 0.708338i \(-0.749445\pi\)
−0.705874 + 0.708338i \(0.749445\pi\)
\(882\) 5.76496 0.194116
\(883\) −16.0478 −0.540053 −0.270026 0.962853i \(-0.587032\pi\)
−0.270026 + 0.962853i \(0.587032\pi\)
\(884\) 3.01883 0.101534
\(885\) 32.5718 1.09489
\(886\) 19.2548 0.646878
\(887\) −33.8483 −1.13652 −0.568258 0.822850i \(-0.692382\pi\)
−0.568258 + 0.822850i \(0.692382\pi\)
\(888\) 88.4131 2.96695
\(889\) 19.4509 0.652362
\(890\) −17.0949 −0.573024
\(891\) 0 0
\(892\) −13.1855 −0.441482
\(893\) 12.2798 0.410929
\(894\) 10.0249 0.335284
\(895\) −12.5141 −0.418302
\(896\) 1.20989 0.0404197
\(897\) −4.75255 −0.158683
\(898\) −18.6563 −0.622568
\(899\) 6.28660 0.209670
\(900\) −27.2281 −0.907603
\(901\) 20.9277 0.697202
\(902\) 0 0
\(903\) −8.80638 −0.293058
\(904\) −32.7465 −1.08913
\(905\) 54.6679 1.81722
\(906\) −9.13431 −0.303467
\(907\) −44.1013 −1.46436 −0.732181 0.681111i \(-0.761497\pi\)
−0.732181 + 0.681111i \(0.761497\pi\)
\(908\) 9.57421 0.317731
\(909\) −79.8940 −2.64992
\(910\) −7.93313 −0.262981
\(911\) 49.5756 1.64251 0.821257 0.570558i \(-0.193273\pi\)
0.821257 + 0.570558i \(0.193273\pi\)
\(912\) 8.85988 0.293380
\(913\) 0 0
\(914\) −24.1810 −0.799837
\(915\) 10.6374 0.351662
\(916\) −3.46648 −0.114536
\(917\) 5.11284 0.168841
\(918\) −13.3868 −0.441831
\(919\) −40.7926 −1.34563 −0.672813 0.739813i \(-0.734914\pi\)
−0.672813 + 0.739813i \(0.734914\pi\)
\(920\) 8.57404 0.282678
\(921\) −36.8333 −1.21370
\(922\) −6.77348 −0.223073
\(923\) 6.56740 0.216169
\(924\) 0 0
\(925\) 69.8737 2.29743
\(926\) −5.73637 −0.188509
\(927\) 46.0803 1.51348
\(928\) 32.0476 1.05201
\(929\) 41.3929 1.35806 0.679029 0.734112i \(-0.262401\pi\)
0.679029 + 0.734112i \(0.262401\pi\)
\(930\) −8.68150 −0.284677
\(931\) 1.62296 0.0531904
\(932\) −18.0827 −0.592318
\(933\) −76.6731 −2.51016
\(934\) −4.36790 −0.142922
\(935\) 0 0
\(936\) 32.7476 1.07039
\(937\) 1.59644 0.0521534 0.0260767 0.999660i \(-0.491699\pi\)
0.0260767 + 0.999660i \(0.491699\pi\)
\(938\) 2.67822 0.0874469
\(939\) −10.2594 −0.334803
\(940\) −19.8206 −0.646477
\(941\) 7.10787 0.231710 0.115855 0.993266i \(-0.463039\pi\)
0.115855 + 0.993266i \(0.463039\pi\)
\(942\) 68.4668 2.23077
\(943\) 1.71608 0.0558832
\(944\) 6.29513 0.204889
\(945\) −21.4683 −0.698364
\(946\) 0 0
\(947\) −2.45986 −0.0799347 −0.0399674 0.999201i \(-0.512725\pi\)
−0.0399674 + 0.999201i \(0.512725\pi\)
\(948\) −20.5460 −0.667303
\(949\) −2.52570 −0.0819878
\(950\) 12.5607 0.407523
\(951\) 48.3656 1.56836
\(952\) −5.94362 −0.192634
\(953\) −28.6000 −0.926446 −0.463223 0.886242i \(-0.653307\pi\)
−0.463223 + 0.886242i \(0.653307\pi\)
\(954\) 62.3910 2.01998
\(955\) 1.48294 0.0479867
\(956\) −6.69900 −0.216661
\(957\) 0 0
\(958\) −16.8428 −0.544168
\(959\) 9.10052 0.293871
\(960\) −81.9903 −2.64622
\(961\) −30.3784 −0.979948
\(962\) −23.0959 −0.744643
\(963\) 18.1680 0.585456
\(964\) −14.3607 −0.462528
\(965\) −52.4454 −1.68828
\(966\) 2.57158 0.0827392
\(967\) −0.213338 −0.00686047 −0.00343024 0.999994i \(-0.501092\pi\)
−0.00343024 + 0.999994i \(0.501092\pi\)
\(968\) 0 0
\(969\) −8.97201 −0.288223
\(970\) 24.9129 0.799907
\(971\) −0.828199 −0.0265782 −0.0132891 0.999912i \(-0.504230\pi\)
−0.0132891 + 0.999912i \(0.504230\pi\)
\(972\) −9.27157 −0.297386
\(973\) −13.0166 −0.417292
\(974\) 27.9473 0.895488
\(975\) 40.8903 1.30954
\(976\) 2.05589 0.0658073
\(977\) 9.75714 0.312159 0.156079 0.987745i \(-0.450114\pi\)
0.156079 + 0.987745i \(0.450114\pi\)
\(978\) 26.1972 0.837694
\(979\) 0 0
\(980\) −2.61958 −0.0836795
\(981\) 20.0528 0.640236
\(982\) 41.4813 1.32372
\(983\) 45.1198 1.43910 0.719548 0.694442i \(-0.244349\pi\)
0.719548 + 0.694442i \(0.244349\pi\)
\(984\) −18.6824 −0.595573
\(985\) −71.9148 −2.29140
\(986\) 17.1839 0.547246
\(987\) −21.6307 −0.688514
\(988\) 2.53367 0.0806069
\(989\) −2.48632 −0.0790604
\(990\) 0 0
\(991\) 53.5405 1.70077 0.850384 0.526162i \(-0.176369\pi\)
0.850384 + 0.526162i \(0.176369\pi\)
\(992\) 3.16880 0.100609
\(993\) 3.53997 0.112337
\(994\) −3.55358 −0.112713
\(995\) 29.1889 0.925351
\(996\) −34.8205 −1.10333
\(997\) −31.1607 −0.986869 −0.493435 0.869783i \(-0.664259\pi\)
−0.493435 + 0.869783i \(0.664259\pi\)
\(998\) −35.4549 −1.12231
\(999\) −62.5013 −1.97745
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 847.2.a.p.1.6 8
3.2 odd 2 7623.2.a.ct.1.3 8
7.6 odd 2 5929.2.a.bt.1.6 8
11.2 odd 10 847.2.f.x.323.3 16
11.3 even 5 847.2.f.w.372.3 16
11.4 even 5 847.2.f.w.148.3 16
11.5 even 5 77.2.f.b.36.2 yes 16
11.6 odd 10 847.2.f.x.729.3 16
11.7 odd 10 847.2.f.v.148.2 16
11.8 odd 10 847.2.f.v.372.2 16
11.9 even 5 77.2.f.b.15.2 16
11.10 odd 2 847.2.a.o.1.3 8
33.5 odd 10 693.2.m.i.190.3 16
33.20 odd 10 693.2.m.i.631.3 16
33.32 even 2 7623.2.a.cw.1.6 8
77.5 odd 30 539.2.q.f.410.3 32
77.9 even 15 539.2.q.g.312.2 32
77.16 even 15 539.2.q.g.410.3 32
77.20 odd 10 539.2.f.e.246.2 16
77.27 odd 10 539.2.f.e.344.2 16
77.31 odd 30 539.2.q.f.422.3 32
77.38 odd 30 539.2.q.f.520.2 32
77.53 even 15 539.2.q.g.422.3 32
77.60 even 15 539.2.q.g.520.2 32
77.75 odd 30 539.2.q.f.312.2 32
77.76 even 2 5929.2.a.bs.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.2 16 11.9 even 5
77.2.f.b.36.2 yes 16 11.5 even 5
539.2.f.e.246.2 16 77.20 odd 10
539.2.f.e.344.2 16 77.27 odd 10
539.2.q.f.312.2 32 77.75 odd 30
539.2.q.f.410.3 32 77.5 odd 30
539.2.q.f.422.3 32 77.31 odd 30
539.2.q.f.520.2 32 77.38 odd 30
539.2.q.g.312.2 32 77.9 even 15
539.2.q.g.410.3 32 77.16 even 15
539.2.q.g.422.3 32 77.53 even 15
539.2.q.g.520.2 32 77.60 even 15
693.2.m.i.190.3 16 33.5 odd 10
693.2.m.i.631.3 16 33.20 odd 10
847.2.a.o.1.3 8 11.10 odd 2
847.2.a.p.1.6 8 1.1 even 1 trivial
847.2.f.v.148.2 16 11.7 odd 10
847.2.f.v.372.2 16 11.8 odd 10
847.2.f.w.148.3 16 11.4 even 5
847.2.f.w.372.3 16 11.3 even 5
847.2.f.x.323.3 16 11.2 odd 10
847.2.f.x.729.3 16 11.6 odd 10
5929.2.a.bs.1.3 8 77.76 even 2
5929.2.a.bt.1.6 8 7.6 odd 2
7623.2.a.ct.1.3 8 3.2 odd 2
7623.2.a.cw.1.6 8 33.32 even 2