Properties

Label 5929.2.a.bs.1.3
Level 5929
Weight 2
Character 5929.1
Self dual yes
Analytic conductor 47.343
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 5929.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.3433033584\)
Analytic rank: \(1\)
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 1.3
Root \(1.11447\)
Character \(\chi\) = 5929.1

$q$-expansion

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

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 \(-0.628917\pi\)
−0.394023 + 0.919100i \(0.628917\pi\)
\(3\) 2.85882 1.65054 0.825270 0.564739i \(-0.191023\pi\)
0.825270 + 0.564739i \(0.191023\pi\)
\(4\) −0.757964 −0.378982
\(5\) −3.45608 −1.54561 −0.772803 0.634647i \(-0.781146\pi\)
−0.772803 + 0.634647i \(0.781146\pi\)
\(6\) −3.18606 −1.30070
\(7\) 0 0
\(8\) 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 \(0.592200\pi\)
\(14\) 0 0
\(15\) −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
\(21\) 0 0
\(22\) 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 0
\(29\) −7.97368 −1.48067 −0.740337 0.672235i \(-0.765334\pi\)
−0.740337 + 0.672235i \(0.765334\pi\)
\(30\) 11.0113 2.01037
\(31\) −0.788420 −0.141604 −0.0708022 0.997490i \(-0.522556\pi\)
−0.0708022 + 0.997490i \(0.522556\pi\)
\(32\) −4.01918 −0.710497
\(33\) 0 0
\(34\) −2.15508 −0.369592
\(35\) 0 0
\(36\) −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 \(0.553093\pi\)
\(42\) 0 0
\(43\) −3.08043 −0.469761 −0.234880 0.972024i \(-0.575470\pi\)
−0.234880 + 0.972024i \(0.575470\pi\)
\(44\) 0 0
\(45\) −17.8777 −2.66506
\(46\) 0.899526 0.132628
\(47\) −7.56632 −1.10366 −0.551831 0.833956i \(-0.686070\pi\)
−0.551831 + 0.833956i \(0.686070\pi\)
\(48\) −5.45909 −0.787952
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) 4.63975 0.614550
\(58\) 8.88640 1.16684
\(59\) 3.29664 0.429186 0.214593 0.976704i \(-0.431158\pi\)
0.214593 + 0.976704i \(0.431158\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) −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 \(-0.320916\pi\)
0.533394 + 0.845867i \(0.320916\pi\)
\(80\) 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 \(0.843751\pi\)
\(84\) 0 0
\(85\) −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 \(-0.424416\pi\)
0.235230 + 0.971940i \(0.424416\pi\)
\(90\) 19.9241 2.10019
\(91\) 0 0
\(92\) 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 \(-0.606498\pi\)
−0.328366 + 0.944550i \(0.606498\pi\)
\(98\) 0 0
\(99\) 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 \(-0.644622\pi\)
−0.438872 + 0.898550i \(0.644622\pi\)
\(104\) −6.33067 −0.620773
\(105\) 0 0
\(106\) −12.0613 −1.17149
\(107\) −3.51219 −0.339536 −0.169768 0.985484i \(-0.554302\pi\)
−0.169768 + 0.985484i \(0.554302\pi\)
\(108\) −4.70829 −0.453055
\(109\) −3.87655 −0.371306 −0.185653 0.982615i \(-0.559440\pi\)
−0.185653 + 0.982615i \(0.559440\pi\)
\(110\) 0 0
\(111\) 28.7648 2.73023
\(112\) 0 0
\(113\) 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
\(119\) 0 0
\(120\) −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
\(126\) 0 0
\(127\) −19.4509 −1.72599 −0.862994 0.505214i \(-0.831414\pi\)
−0.862994 + 0.505214i \(0.831414\pi\)
\(128\) −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
\(133\) 0 0
\(134\) −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 \(0.686145\pi\)
\(140\) 0 0
\(141\) −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
\(147\) 0 0
\(148\) −7.62646 −0.626891
\(149\) 3.14650 0.257771 0.128886 0.991659i \(-0.458860\pi\)
0.128886 + 0.991659i \(0.458860\pi\)
\(150\) −22.1255 −1.80654
\(151\) −2.86696 −0.233310 −0.116655 0.993172i \(-0.537217\pi\)
−0.116655 + 0.993172i \(0.537217\pi\)
\(152\) 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 \(-0.172001\pi\)
0.857524 + 0.514443i \(0.172001\pi\)
\(158\) −10.5672 −0.840679
\(159\) 30.9394 2.45365
\(160\) 13.8906 1.09815
\(161\) 0 0
\(162\) −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 \(0.817404\pi\)
\(168\) 0 0
\(169\) −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
\(175\) 0 0
\(176\) 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 \(-0.700033\pi\)
−0.587868 + 0.808957i \(0.700033\pi\)
\(182\) 0 0
\(183\) −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
\(189\) 0 0
\(190\) 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 \(-0.316091\pi\)
0.546154 + 0.837685i \(0.316091\pi\)
\(194\) 7.20845 0.517536
\(195\) 20.3500 1.45729
\(196\) 0 0
\(197\) 20.8082 1.48252 0.741262 0.671216i \(-0.234228\pi\)
0.741262 + 0.671216i \(0.234228\pi\)
\(198\) 0 0
\(199\) −8.44567 −0.598698 −0.299349 0.954144i \(-0.596769\pi\)
−0.299349 + 0.954144i \(0.596769\pi\)
\(200\) 21.3449 1.50932
\(201\) 6.87014 0.484582
\(202\) 17.2128 1.21109
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) 9.85927 0.678740 0.339370 0.940653i \(-0.389786\pi\)
0.339370 + 0.940653i \(0.389786\pi\)
\(212\) −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 0
\(218\) 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 \(-0.697909\pi\)
−0.582457 + 0.812861i \(0.697909\pi\)
\(224\) 0 0
\(225\) 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 \(-0.548285\pi\)
−0.151110 + 0.988517i \(0.548285\pi\)
\(230\) −3.10883 −0.204990
\(231\) 0 0
\(232\) −24.5084 −1.60905
\(233\) −23.8569 −1.56292 −0.781458 0.623957i \(-0.785524\pi\)
−0.781458 + 0.623957i \(0.785524\pi\)
\(234\) 11.8738 0.776215
\(235\) 26.1498 1.70582
\(236\) −2.49873 −0.162654
\(237\) 27.1068 1.76078
\(238\) 0 0
\(239\) −8.83814 −0.571692 −0.285846 0.958276i \(-0.592275\pi\)
−0.285846 + 0.958276i \(0.592275\pi\)
\(240\) 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
\(245\) 0 0
\(246\) 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 \(-0.471160\pi\)
0.0904790 + 0.995898i \(0.471160\pi\)
\(252\) 0 0
\(253\) 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 \(-0.746430\pi\)
−0.699132 + 0.714993i \(0.746430\pi\)
\(258\) 9.81442 0.611019
\(259\) 0 0
\(260\) −5.39543 −0.334611
\(261\) −41.2466 −2.55310
\(262\) −5.69808 −0.352029
\(263\) −0.990706 −0.0610895 −0.0305448 0.999533i \(-0.509724\pi\)
−0.0305448 + 0.999533i \(0.509724\pi\)
\(264\) 0 0
\(265\) −37.4032 −2.29766
\(266\) 0 0
\(267\) 12.6883 0.776511
\(268\) −1.82149 −0.111265
\(269\) 7.19036 0.438404 0.219202 0.975679i \(-0.429655\pi\)
0.219202 + 0.975679i \(0.429655\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −10.1422 −0.612714
\(275\) 0 0
\(276\) 1.74897 0.105275
\(277\) −20.9856 −1.26090 −0.630451 0.776229i \(-0.717130\pi\)
−0.630451 + 0.776229i \(0.717130\pi\)
\(278\) 14.5065 0.870045
\(279\) −4.07837 −0.244166
\(280\) 0 0
\(281\) 28.1580 1.67977 0.839883 0.542768i \(-0.182624\pi\)
0.839883 + 0.542768i \(0.182624\pi\)
\(282\) 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
\(287\) 0 0
\(288\) −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 \(0.541623\pi\)
\(294\) 0 0
\(295\) −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
\(301\) 0 0
\(302\) 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 \(-0.775001\pi\)
−0.760407 + 0.649447i \(0.775001\pi\)
\(312\) −18.0982 −1.02461
\(313\) −3.58869 −0.202845 −0.101422 0.994843i \(-0.532339\pi\)
−0.101422 + 0.994843i \(0.532339\pi\)
\(314\) −23.9493 −1.35154
\(315\) 0 0
\(316\) −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 0
\(323\) 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
\(329\) 0 0
\(330\) 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
\(336\) 0 0
\(337\) 20.4806 1.11565 0.557824 0.829959i \(-0.311636\pi\)
0.557824 + 0.829959i \(0.311636\pi\)
\(338\) 9.76031 0.530891
\(339\) 30.4576 1.65423
\(340\) 5.06556 0.274719
\(341\) 0 0
\(342\) −9.35630 −0.505931
\(343\) 0 0
\(344\) −9.46818 −0.510490
\(345\) 7.97474 0.429346
\(346\) 8.96659 0.482047
\(347\) −27.2699 −1.46392 −0.731961 0.681346i \(-0.761395\pi\)
−0.731961 + 0.681346i \(0.761395\pi\)
\(348\) 17.2780 0.926199
\(349\) 7.98434 0.427392 0.213696 0.976900i \(-0.431450\pi\)
0.213696 + 0.976900i \(0.431450\pi\)
\(350\) 0 0
\(351\) −12.7941 −0.682897
\(352\) 0 0
\(353\) −5.93472 −0.315873 −0.157937 0.987449i \(-0.550484\pi\)
−0.157937 + 0.987449i \(0.550484\pi\)
\(354\) −10.5033 −0.558243
\(355\) 11.0200 0.584883
\(356\) −3.36407 −0.178296
\(357\) 0 0
\(358\) 4.03538 0.213277
\(359\) −28.4203 −1.49996 −0.749982 0.661458i \(-0.769938\pi\)
−0.749982 + 0.661458i \(0.769938\pi\)
\(360\) −54.9501 −2.89612
\(361\) −16.3660 −0.861368
\(362\) 17.6285 0.926535
\(363\) 0 0
\(364\) 0 0
\(365\) −4.23810 −0.221832
\(366\) 3.43020 0.179299
\(367\) 30.4014 1.58694 0.793471 0.608608i \(-0.208272\pi\)
0.793471 + 0.608608i \(0.208272\pi\)
\(368\) 1.54128 0.0803445
\(369\) −10.9982 −0.572541
\(370\) 38.7547 2.01476
\(371\) 0 0
\(372\) 1.70841 0.0885771
\(373\) −14.4226 −0.746772 −0.373386 0.927676i \(-0.621803\pi\)
−0.373386 + 0.927676i \(0.621803\pi\)
\(374\) 0 0
\(375\) −19.2120 −0.992103
\(376\) −23.2563 −1.19935
\(377\) 16.4230 0.845828
\(378\) 0 0
\(379\) −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 \(-0.829815\pi\)
−0.860446 + 0.509541i \(0.829815\pi\)
\(384\) −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
\(392\) 0 0
\(393\) 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 \(-0.452723\pi\)
0.147980 + 0.988990i \(0.452723\pi\)
\(398\) 9.41242 0.471802
\(399\) 0 0
\(400\) −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
\(406\) 0 0
\(407\) 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
\(413\) 0 0
\(414\) 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 \(-0.334980\pi\)
0.495514 + 0.868600i \(0.334980\pi\)
\(420\) 0 0
\(421\) −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
\(427\) 0 0
\(428\) 2.66211 0.128678
\(429\) 0 0
\(430\) −11.8648 −0.572173
\(431\) −7.54197 −0.363284 −0.181642 0.983365i \(-0.558141\pi\)
−0.181642 + 0.983365i \(0.558141\pi\)
\(432\) −11.8617 −0.570698
\(433\) 32.1616 1.54559 0.772794 0.634657i \(-0.218859\pi\)
0.772794 + 0.634657i \(0.218859\pi\)
\(434\) 0 0
\(435\) 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
\(441\) 0 0
\(442\) 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
\(448\) 0 0
\(449\) −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
\(455\) 0 0
\(456\) 14.2610 0.667833
\(457\) 21.6974 1.01496 0.507481 0.861663i \(-0.330577\pi\)
0.507481 + 0.861663i \(0.330577\pi\)
\(458\) 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 \(-0.471096\pi\)
0.0906812 + 0.995880i \(0.471096\pi\)
\(468\) 8.07555 0.373292
\(469\) 0 0
\(470\) −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
\(476\) 0 0
\(477\) 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
\(483\) 0 0
\(484\) 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
\(490\) 0 0
\(491\) −37.2208 −1.67975 −0.839875 0.542780i \(-0.817372\pi\)
−0.839875 + 0.542780i \(0.817372\pi\)
\(492\) 4.60708 0.207703
\(493\) −15.4189 −0.694434
\(494\) 3.72536 0.167612
\(495\) 0 0
\(496\) 1.50554 0.0676006
\(497\) 0 0
\(498\) 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 \(0.640843\pi\)
\(504\) 0 0
\(505\) 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 \(-0.517754\pi\)
−0.0557485 + 0.998445i \(0.517754\pi\)
\(510\) 21.2928 0.942861
\(511\) 0 0
\(512\) 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
\(518\) 0 0
\(519\) −23.0010 −1.00963
\(520\) 21.8793 0.959470
\(521\) −14.8968 −0.652639 −0.326320 0.945260i \(-0.605809\pi\)
−0.326320 + 0.945260i \(0.605809\pi\)
\(522\) 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
\(525\) 0 0
\(526\) 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
\(532\) 0 0
\(533\) 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 \(-0.343132\pi\)
0.473107 + 0.881005i \(0.343132\pi\)
\(542\) 30.2737 1.30037
\(543\) −45.2205 −1.94060
\(544\) −7.77199 −0.333222
\(545\) 13.3977 0.573892
\(546\) 0 0
\(547\) 10.8643 0.464523 0.232261 0.972653i \(-0.425388\pi\)
0.232261 + 0.972653i \(0.425388\pi\)
\(548\) −6.89787 −0.294662
\(549\) −5.56922 −0.237689
\(550\) 0 0
\(551\) −12.9410 −0.551303
\(552\) −7.09233 −0.301869
\(553\) 0 0
\(554\) 23.3878 0.993650
\(555\) −99.4133 −4.21986
\(556\) 9.86610 0.418416
\(557\) −33.6126 −1.42421 −0.712106 0.702072i \(-0.752258\pi\)
−0.712106 + 0.702072i \(0.752258\pi\)
\(558\) 4.54521 0.192414
\(559\) 6.34461 0.268348
\(560\) 0 0
\(561\) 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
\(567\) 0 0
\(568\) −9.80065 −0.411226
\(569\) −3.27416 −0.137260 −0.0686300 0.997642i \(-0.521863\pi\)
−0.0686300 + 0.997642i \(0.521863\pi\)
\(570\) 17.8708 0.748527
\(571\) 43.8897 1.83673 0.918363 0.395738i \(-0.129511\pi\)
0.918363 + 0.395738i \(0.129511\pi\)
\(572\) 0 0
\(573\) 1.22666 0.0512446
\(574\) 0 0
\(575\) −5.60513 −0.233750
\(576\) 42.9261 1.78859
\(577\) 43.8904 1.82718 0.913591 0.406635i \(-0.133298\pi\)
0.913591 + 0.406635i \(0.133298\pi\)
\(578\) 14.7786 0.614709
\(579\) 43.3821 1.80290
\(580\) −20.8877 −0.867315
\(581\) 0 0
\(582\) 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 \(-0.518349\pi\)
−0.0576121 + 0.998339i \(0.518349\pi\)
\(588\) 0 0
\(589\) −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
\(595\) 0 0
\(596\) −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 \(0.349634\pi\)
\(602\) 0 0
\(603\) 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
\(609\) 0 0
\(610\) −4.14683 −0.167900
\(611\) 15.5840 0.630461
\(612\) −7.58182 −0.306477
\(613\) −20.1617 −0.814323 −0.407161 0.913356i \(-0.633481\pi\)
−0.407161 + 0.913356i \(0.633481\pi\)
\(614\) −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 \(-0.285235\pi\)
0.624667 + 0.780891i \(0.285235\pi\)
\(620\) −2.06533 −0.0829457
\(621\) −5.01373 −0.201194
\(622\) 29.8898 1.19847
\(623\) 0 0
\(624\) 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
\(630\) 0 0
\(631\) 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
\(637\) 0 0
\(638\) 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 \(-0.593544\pi\)
−0.289666 + 0.957128i \(0.593544\pi\)
\(644\) 0 0
\(645\) 30.4355 1.19840
\(646\) −3.49760 −0.137611
\(647\) 6.14523 0.241594 0.120797 0.992677i \(-0.461455\pi\)
0.120797 + 0.992677i \(0.461455\pi\)
\(648\) 6.88426 0.270439
\(649\) 0 0
\(650\) 15.9404 0.625236
\(651\) 0 0
\(652\) 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
\(658\) 0 0
\(659\) −18.0090 −0.701531 −0.350765 0.936463i \(-0.614079\pi\)
−0.350765 + 0.936463i \(0.614079\pi\)
\(660\) 0 0
\(661\) 17.1420 0.666745 0.333373 0.942795i \(-0.391813\pi\)
0.333373 + 0.942795i \(0.391813\pi\)
\(662\) 1.38000 0.0536353
\(663\) −11.3861 −0.442201
\(664\) −49.3919 −1.91678
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) 23.1926 0.894008 0.447004 0.894532i \(-0.352491\pi\)
0.447004 + 0.894532i \(0.352491\pi\)
\(674\) −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
\(679\) 0 0
\(680\) −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
\(686\) 0 0
\(687\) −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 \(-0.337653\pi\)
0.488202 + 0.872731i \(0.337653\pi\)
\(692\) 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
\(700\) 0 0
\(701\) 18.4135 0.695467 0.347733 0.937593i \(-0.386951\pi\)
0.347733 + 0.937593i \(0.386951\pi\)
\(702\) 14.2586 0.538154
\(703\) 16.3298 0.615892
\(704\) 0 0
\(705\) 74.7575 2.81553
\(706\) 6.61404 0.248923
\(707\) 0 0
\(708\) −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
\(714\) 0 0
\(715\) 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 \(-0.566498\pi\)
−0.207395 + 0.978257i \(0.566498\pi\)
\(720\) 34.1387 1.27227
\(721\) 0 0
\(722\) 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 \(-0.788756\pi\)
−0.787753 + 0.615991i \(0.788756\pi\)
\(728\) 0 0
\(729\) −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
\(735\) 0 0
\(736\) 3.24402 0.119576
\(737\) 0 0
\(738\) 12.2571 0.451189
\(739\) −29.4481 −1.08327 −0.541633 0.840615i \(-0.682194\pi\)
−0.541633 + 0.840615i \(0.682194\pi\)
\(740\) 26.3576 0.968926
\(741\) −9.55627 −0.351058
\(742\) 0 0
\(743\) −16.9059 −0.620217 −0.310109 0.950701i \(-0.600365\pi\)
−0.310109 + 0.950701i \(0.600365\pi\)
\(744\) −6.92787 −0.253988
\(745\) −10.8745 −0.398412
\(746\) 16.0735 0.588491
\(747\) −83.1244 −3.04136
\(748\) 0 0
\(749\) 0 0
\(750\) 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
\(756\) 0 0
\(757\) 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
\(763\) 0 0
\(764\) −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 \(-0.392566\pi\)
0.331143 + 0.943581i \(0.392566\pi\)
\(774\) 17.7585 0.638317
\(775\) −5.47516 −0.196674
\(776\) −19.8806 −0.713673
\(777\) 0 0
\(778\) −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
\(784\) 0 0
\(785\) −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
\(791\) 0 0
\(792\) 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 \(-0.482376\pi\)
0.0553385 + 0.998468i \(0.482376\pi\)
\(798\) 0 0
\(799\) −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
\(805\) 0 0
\(806\) −1.80975 −0.0637457
\(807\) 20.5559 0.723604
\(808\) −47.4724 −1.67007
\(809\) 32.7257 1.15057 0.575286 0.817952i \(-0.304891\pi\)
0.575286 + 0.817952i \(0.304891\pi\)
\(810\) 8.62685 0.303117
\(811\) 32.6613 1.14689 0.573447 0.819243i \(-0.305606\pi\)
0.573447 + 0.819243i \(0.305606\pi\)
\(812\) 0 0
\(813\) −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
\(819\) 0 0
\(820\) −5.56958 −0.194498
\(821\) 7.43142 0.259358 0.129679 0.991556i \(-0.458605\pi\)
0.129679 + 0.991556i \(0.458605\pi\)
\(822\) −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
\(826\) 0 0
\(827\) −23.8538 −0.829479 −0.414739 0.909940i \(-0.636127\pi\)
−0.414739 + 0.909940i \(0.636127\pi\)
\(828\) 3.16464 0.109979
\(829\) 26.1431 0.907989 0.453994 0.891005i \(-0.349999\pi\)
0.453994 + 0.891005i \(0.349999\pi\)
\(830\) −61.8942 −2.14838
\(831\) −59.9940 −2.08117
\(832\) −17.0917 −0.592549
\(833\) 0 0
\(834\) 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 \(-0.701619\pi\)
−0.591893 + 0.806016i \(0.701619\pi\)
\(840\) 0 0
\(841\) 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 \(0.380696\pi\)
\(854\) 0 0
\(855\) −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 \(-0.326501\pi\)
0.518473 + 0.855094i \(0.326501\pi\)
\(860\) −8.06944 −0.275165
\(861\) 0 0
\(862\) 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 0
\(869\) 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
\(875\) 0 0
\(876\) −2.65719 −0.0897783
\(877\) 9.20488 0.310827 0.155413 0.987850i \(-0.450329\pi\)
0.155413 + 0.987850i \(0.450329\pi\)
\(878\) 5.20149 0.175542
\(879\) −12.7613 −0.430429
\(880\) 0 0
\(881\) 41.9030 1.41175 0.705874 0.708338i \(-0.250555\pi\)
0.705874 + 0.708338i \(0.250555\pi\)
\(882\) 0 0
\(883\) −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
\(889\) 0 0
\(890\) 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
\(896\) 0 0
\(897\) 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
\(903\) 0 0
\(904\) 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
\(910\) 0 0
\(911\) 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
\(917\) 0 0
\(918\) −13.3868 −0.441831
\(919\) 40.7926 1.34563 0.672813 0.739813i \(-0.265086\pi\)
0.672813 + 0.739813i \(0.265086\pi\)
\(920\) 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 \(-0.737599\pi\)
−0.679029 + 0.734112i \(0.737599\pi\)
\(930\) −8.68150 −0.284677
\(931\) 0 0
\(932\) 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 \(0.491699\pi\)
\(938\) 0 0
\(939\) −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
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) 28.6000 0.926446 0.463223 0.886242i \(-0.346693\pi\)
0.463223 + 0.886242i \(0.346693\pi\)
\(954\) −62.3910 −2.01998
\(955\) −1.48294 −0.0479867
\(956\) 6.69900 0.216661
\(957\) 0 0
\(958\) 16.8428 0.544168
\(959\) 0 0
\(960\) −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
\(966\) 0 0
\(967\) 0.213338 0.00686047 0.00343024 0.999994i \(-0.498908\pi\)
0.00343024 + 0.999994i \(0.498908\pi\)
\(968\) 0 0
\(969\) 8.97201 0.288223
\(970\) −24.9129 −0.799907
\(971\) 0.828199 0.0265782 0.0132891 0.999912i \(-0.495770\pi\)
0.0132891 + 0.999912i \(0.495770\pi\)
\(972\) 9.27157 0.297386
\(973\) 0 0
\(974\) −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
\(980\) 0 0
\(981\) −20.0528 −0.640236
\(982\) 41.4813 1.32372
\(983\) −45.1198 −1.43910 −0.719548 0.694442i \(-0.755651\pi\)
−0.719548 + 0.694442i \(0.755651\pi\)
\(984\) −18.6824 −0.595573
\(985\) −71.9148 −2.29140
\(986\) 17.1839 0.547246
\(987\) 0 0
\(988\) 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
\(994\) 0 0
\(995\) 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 5929.2.a.bs.1.3 8
7.6 odd 2 847.2.a.o.1.3 8
11.2 odd 10 539.2.f.e.246.2 16
11.6 odd 10 539.2.f.e.344.2 16
11.10 odd 2 5929.2.a.bt.1.6 8
21.20 even 2 7623.2.a.cw.1.6 8
77.2 odd 30 539.2.q.f.312.2 32
77.6 even 10 77.2.f.b.36.2 yes 16
77.13 even 10 77.2.f.b.15.2 16
77.17 even 30 539.2.q.g.520.2 32
77.20 odd 10 847.2.f.x.323.3 16
77.24 even 30 539.2.q.g.422.3 32
77.27 odd 10 847.2.f.x.729.3 16
77.39 odd 30 539.2.q.f.520.2 32
77.41 even 10 847.2.f.w.372.3 16
77.46 odd 30 539.2.q.f.422.3 32
77.48 odd 10 847.2.f.v.148.2 16
77.61 even 30 539.2.q.g.410.3 32
77.62 even 10 847.2.f.w.148.3 16
77.68 even 30 539.2.q.g.312.2 32
77.69 odd 10 847.2.f.v.372.2 16
77.72 odd 30 539.2.q.f.410.3 32
77.76 even 2 847.2.a.p.1.6 8
231.83 odd 10 693.2.m.i.190.3 16
231.167 odd 10 693.2.m.i.631.3 16
231.230 odd 2 7623.2.a.ct.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.2 16 77.13 even 10
77.2.f.b.36.2 yes 16 77.6 even 10
539.2.f.e.246.2 16 11.2 odd 10
539.2.f.e.344.2 16 11.6 odd 10
539.2.q.f.312.2 32 77.2 odd 30
539.2.q.f.410.3 32 77.72 odd 30
539.2.q.f.422.3 32 77.46 odd 30
539.2.q.f.520.2 32 77.39 odd 30
539.2.q.g.312.2 32 77.68 even 30
539.2.q.g.410.3 32 77.61 even 30
539.2.q.g.422.3 32 77.24 even 30
539.2.q.g.520.2 32 77.17 even 30
693.2.m.i.190.3 16 231.83 odd 10
693.2.m.i.631.3 16 231.167 odd 10
847.2.a.o.1.3 8 7.6 odd 2
847.2.a.p.1.6 8 77.76 even 2
847.2.f.v.148.2 16 77.48 odd 10
847.2.f.v.372.2 16 77.69 odd 10
847.2.f.w.148.3 16 77.62 even 10
847.2.f.w.372.3 16 77.41 even 10
847.2.f.x.323.3 16 77.20 odd 10
847.2.f.x.729.3 16 77.27 odd 10
5929.2.a.bs.1.3 8 1.1 even 1 trivial
5929.2.a.bt.1.6 8 11.10 odd 2
7623.2.a.ct.1.3 8 231.230 odd 2
7623.2.a.cw.1.6 8 21.20 even 2