Properties

Label 4033.2.a.f.1.15
Level 4033
Weight 2
Character 4033.1
Self dual yes
Analytic conductor 32.204
Analytic rank 0
Dimension 85
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4033 = 37 \cdot 109 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4033.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2036671352\)
Analytic rank: \(0\)
Dimension: \(85\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) = 4033.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.96262 q^{2} -1.03474 q^{3} +1.85187 q^{4} +3.48986 q^{5} +2.03080 q^{6} -2.58926 q^{7} +0.290726 q^{8} -1.92931 q^{9} +O(q^{10})\) \(q-1.96262 q^{2} -1.03474 q^{3} +1.85187 q^{4} +3.48986 q^{5} +2.03080 q^{6} -2.58926 q^{7} +0.290726 q^{8} -1.92931 q^{9} -6.84926 q^{10} +5.41222 q^{11} -1.91620 q^{12} +1.24515 q^{13} +5.08173 q^{14} -3.61110 q^{15} -4.27432 q^{16} +1.57417 q^{17} +3.78650 q^{18} +5.32024 q^{19} +6.46276 q^{20} +2.67922 q^{21} -10.6221 q^{22} +0.541267 q^{23} -0.300827 q^{24} +7.17912 q^{25} -2.44376 q^{26} +5.10056 q^{27} -4.79497 q^{28} -3.46260 q^{29} +7.08721 q^{30} +4.16239 q^{31} +7.80740 q^{32} -5.60024 q^{33} -3.08949 q^{34} -9.03616 q^{35} -3.57283 q^{36} +1.00000 q^{37} -10.4416 q^{38} -1.28841 q^{39} +1.01459 q^{40} +3.40187 q^{41} -5.25828 q^{42} +0.894280 q^{43} +10.0227 q^{44} -6.73302 q^{45} -1.06230 q^{46} -5.44804 q^{47} +4.42282 q^{48} -0.295722 q^{49} -14.0899 q^{50} -1.62886 q^{51} +2.30586 q^{52} +9.67344 q^{53} -10.0104 q^{54} +18.8879 q^{55} -0.752767 q^{56} -5.50507 q^{57} +6.79575 q^{58} -4.87176 q^{59} -6.68728 q^{60} -0.171290 q^{61} -8.16918 q^{62} +4.99549 q^{63} -6.77431 q^{64} +4.34541 q^{65} +10.9911 q^{66} -2.32810 q^{67} +2.91516 q^{68} -0.560071 q^{69} +17.7345 q^{70} -12.8303 q^{71} -0.560902 q^{72} +15.9341 q^{73} -1.96262 q^{74} -7.42853 q^{75} +9.85237 q^{76} -14.0136 q^{77} +2.52866 q^{78} +10.9404 q^{79} -14.9168 q^{80} +0.510174 q^{81} -6.67657 q^{82} +12.9999 q^{83} +4.96155 q^{84} +5.49363 q^{85} -1.75513 q^{86} +3.58289 q^{87} +1.57347 q^{88} -7.68383 q^{89} +13.2144 q^{90} -3.22403 q^{91} +1.00235 q^{92} -4.30700 q^{93} +10.6924 q^{94} +18.5669 q^{95} -8.07864 q^{96} +0.658094 q^{97} +0.580389 q^{98} -10.4418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + O(q^{10}) \) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + 9q^{10} + 37q^{11} + 44q^{12} + 14q^{13} + 26q^{14} + 27q^{15} + 85q^{16} + 34q^{17} + 3q^{18} + 15q^{19} + 15q^{20} + 17q^{21} + q^{22} + 72q^{23} + 15q^{24} + 85q^{25} + 33q^{26} + 69q^{27} + 7q^{28} + 19q^{29} - 9q^{30} + 23q^{31} + 51q^{32} + 32q^{33} + 49q^{34} + 40q^{35} + 121q^{36} + 85q^{37} + 84q^{38} + 39q^{39} + 22q^{40} + 55q^{41} - 28q^{42} + 78q^{44} + 28q^{45} + 17q^{46} + 184q^{47} + 97q^{48} + 88q^{49} + 26q^{50} + 27q^{51} + 73q^{52} + 64q^{53} + 31q^{54} + 39q^{55} + 68q^{56} - 33q^{57} + 28q^{58} + 60q^{59} - 22q^{60} + 7q^{61} + 70q^{62} + 28q^{63} + 102q^{64} + 17q^{65} - 15q^{66} + 82q^{67} + 92q^{68} + 22q^{69} - 41q^{70} + 113q^{71} - 19q^{73} + 11q^{74} + 45q^{75} + 34q^{76} + 64q^{77} + 29q^{78} + 23q^{79} + 54q^{80} + 149q^{81} + 4q^{82} + 100q^{83} - 49q^{84} - 5q^{85} - 24q^{86} + 65q^{87} + 14q^{88} + 84q^{89} - 21q^{90} + 32q^{91} + 95q^{92} + 19q^{93} - 47q^{94} + 102q^{95} + 29q^{96} + 7q^{97} + 26q^{98} + 107q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.96262 −1.38778 −0.693890 0.720081i \(-0.744105\pi\)
−0.693890 + 0.720081i \(0.744105\pi\)
\(3\) −1.03474 −0.597408 −0.298704 0.954346i \(-0.596554\pi\)
−0.298704 + 0.954346i \(0.596554\pi\)
\(4\) 1.85187 0.925934
\(5\) 3.48986 1.56071 0.780356 0.625335i \(-0.215038\pi\)
0.780356 + 0.625335i \(0.215038\pi\)
\(6\) 2.03080 0.829071
\(7\) −2.58926 −0.978649 −0.489325 0.872102i \(-0.662757\pi\)
−0.489325 + 0.872102i \(0.662757\pi\)
\(8\) 0.290726 0.102787
\(9\) −1.92931 −0.643104
\(10\) −6.84926 −2.16593
\(11\) 5.41222 1.63184 0.815922 0.578162i \(-0.196230\pi\)
0.815922 + 0.578162i \(0.196230\pi\)
\(12\) −1.91620 −0.553160
\(13\) 1.24515 0.345344 0.172672 0.984979i \(-0.444760\pi\)
0.172672 + 0.984979i \(0.444760\pi\)
\(14\) 5.08173 1.35815
\(15\) −3.61110 −0.932382
\(16\) −4.27432 −1.06858
\(17\) 1.57417 0.381792 0.190896 0.981610i \(-0.438861\pi\)
0.190896 + 0.981610i \(0.438861\pi\)
\(18\) 3.78650 0.892487
\(19\) 5.32024 1.22055 0.610273 0.792191i \(-0.291060\pi\)
0.610273 + 0.792191i \(0.291060\pi\)
\(20\) 6.46276 1.44512
\(21\) 2.67922 0.584653
\(22\) −10.6221 −2.26464
\(23\) 0.541267 0.112862 0.0564310 0.998407i \(-0.482028\pi\)
0.0564310 + 0.998407i \(0.482028\pi\)
\(24\) −0.300827 −0.0614060
\(25\) 7.17912 1.43582
\(26\) −2.44376 −0.479261
\(27\) 5.10056 0.981603
\(28\) −4.79497 −0.906164
\(29\) −3.46260 −0.642988 −0.321494 0.946912i \(-0.604185\pi\)
−0.321494 + 0.946912i \(0.604185\pi\)
\(30\) 7.08721 1.29394
\(31\) 4.16239 0.747588 0.373794 0.927512i \(-0.378057\pi\)
0.373794 + 0.927512i \(0.378057\pi\)
\(32\) 7.80740 1.38017
\(33\) −5.60024 −0.974877
\(34\) −3.08949 −0.529844
\(35\) −9.03616 −1.52739
\(36\) −3.57283 −0.595472
\(37\) 1.00000 0.164399
\(38\) −10.4416 −1.69385
\(39\) −1.28841 −0.206311
\(40\) 1.01459 0.160421
\(41\) 3.40187 0.531283 0.265641 0.964072i \(-0.414416\pi\)
0.265641 + 0.964072i \(0.414416\pi\)
\(42\) −5.25828 −0.811370
\(43\) 0.894280 0.136376 0.0681882 0.997672i \(-0.478278\pi\)
0.0681882 + 0.997672i \(0.478278\pi\)
\(44\) 10.0227 1.51098
\(45\) −6.73302 −1.00370
\(46\) −1.06230 −0.156628
\(47\) −5.44804 −0.794679 −0.397339 0.917672i \(-0.630066\pi\)
−0.397339 + 0.917672i \(0.630066\pi\)
\(48\) 4.42282 0.638378
\(49\) −0.295722 −0.0422460
\(50\) −14.0899 −1.99261
\(51\) −1.62886 −0.228086
\(52\) 2.30586 0.319765
\(53\) 9.67344 1.32875 0.664374 0.747400i \(-0.268698\pi\)
0.664374 + 0.747400i \(0.268698\pi\)
\(54\) −10.0104 −1.36225
\(55\) 18.8879 2.54684
\(56\) −0.752767 −0.100593
\(57\) −5.50507 −0.729164
\(58\) 6.79575 0.892326
\(59\) −4.87176 −0.634250 −0.317125 0.948384i \(-0.602717\pi\)
−0.317125 + 0.948384i \(0.602717\pi\)
\(60\) −6.68728 −0.863324
\(61\) −0.171290 −0.0219315 −0.0109657 0.999940i \(-0.503491\pi\)
−0.0109657 + 0.999940i \(0.503491\pi\)
\(62\) −8.16918 −1.03749
\(63\) 4.99549 0.629373
\(64\) −6.77431 −0.846789
\(65\) 4.34541 0.538982
\(66\) 10.9911 1.35291
\(67\) −2.32810 −0.284422 −0.142211 0.989836i \(-0.545421\pi\)
−0.142211 + 0.989836i \(0.545421\pi\)
\(68\) 2.91516 0.353514
\(69\) −0.560071 −0.0674246
\(70\) 17.7345 2.11968
\(71\) −12.8303 −1.52267 −0.761335 0.648358i \(-0.775456\pi\)
−0.761335 + 0.648358i \(0.775456\pi\)
\(72\) −0.560902 −0.0661029
\(73\) 15.9341 1.86494 0.932472 0.361242i \(-0.117647\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(74\) −1.96262 −0.228150
\(75\) −7.42853 −0.857772
\(76\) 9.85237 1.13014
\(77\) −14.0136 −1.59700
\(78\) 2.52866 0.286314
\(79\) 10.9404 1.23089 0.615446 0.788179i \(-0.288976\pi\)
0.615446 + 0.788179i \(0.288976\pi\)
\(80\) −14.9168 −1.66775
\(81\) 0.510174 0.0566860
\(82\) −6.67657 −0.737304
\(83\) 12.9999 1.42692 0.713460 0.700695i \(-0.247127\pi\)
0.713460 + 0.700695i \(0.247127\pi\)
\(84\) 4.96155 0.541350
\(85\) 5.49363 0.595868
\(86\) −1.75513 −0.189260
\(87\) 3.58289 0.384126
\(88\) 1.57347 0.167733
\(89\) −7.68383 −0.814485 −0.407242 0.913320i \(-0.633510\pi\)
−0.407242 + 0.913320i \(0.633510\pi\)
\(90\) 13.2144 1.39291
\(91\) −3.22403 −0.337970
\(92\) 1.00235 0.104503
\(93\) −4.30700 −0.446615
\(94\) 10.6924 1.10284
\(95\) 18.5669 1.90492
\(96\) −8.07864 −0.824523
\(97\) 0.658094 0.0668193 0.0334096 0.999442i \(-0.489363\pi\)
0.0334096 + 0.999442i \(0.489363\pi\)
\(98\) 0.580389 0.0586281
\(99\) −10.4418 −1.04945
\(100\) 13.2948 1.32948
\(101\) −14.0427 −1.39730 −0.698648 0.715466i \(-0.746215\pi\)
−0.698648 + 0.715466i \(0.746215\pi\)
\(102\) 3.19683 0.316533
\(103\) 11.1308 1.09675 0.548374 0.836233i \(-0.315247\pi\)
0.548374 + 0.836233i \(0.315247\pi\)
\(104\) 0.361999 0.0354970
\(105\) 9.35009 0.912475
\(106\) −18.9853 −1.84401
\(107\) 9.13189 0.882813 0.441407 0.897307i \(-0.354480\pi\)
0.441407 + 0.897307i \(0.354480\pi\)
\(108\) 9.44556 0.908900
\(109\) −1.00000 −0.0957826
\(110\) −37.0697 −3.53445
\(111\) −1.03474 −0.0982133
\(112\) 11.0673 1.04577
\(113\) 3.07986 0.289729 0.144864 0.989452i \(-0.453725\pi\)
0.144864 + 0.989452i \(0.453725\pi\)
\(114\) 10.8043 1.01192
\(115\) 1.88895 0.176145
\(116\) −6.41227 −0.595364
\(117\) −2.40229 −0.222092
\(118\) 9.56141 0.880199
\(119\) −4.07594 −0.373641
\(120\) −1.04984 −0.0958371
\(121\) 18.2921 1.66292
\(122\) 0.336178 0.0304361
\(123\) −3.52005 −0.317393
\(124\) 7.70820 0.692217
\(125\) 7.60481 0.680195
\(126\) −9.80424 −0.873431
\(127\) −9.50813 −0.843710 −0.421855 0.906663i \(-0.638621\pi\)
−0.421855 + 0.906663i \(0.638621\pi\)
\(128\) −2.31943 −0.205011
\(129\) −0.925348 −0.0814724
\(130\) −8.52839 −0.747989
\(131\) −4.89270 −0.427477 −0.213738 0.976891i \(-0.568564\pi\)
−0.213738 + 0.976891i \(0.568564\pi\)
\(132\) −10.3709 −0.902672
\(133\) −13.7755 −1.19449
\(134\) 4.56917 0.394716
\(135\) 17.8002 1.53200
\(136\) 0.457653 0.0392434
\(137\) −7.08113 −0.604982 −0.302491 0.953152i \(-0.597818\pi\)
−0.302491 + 0.953152i \(0.597818\pi\)
\(138\) 1.09921 0.0935706
\(139\) −20.1285 −1.70728 −0.853640 0.520864i \(-0.825610\pi\)
−0.853640 + 0.520864i \(0.825610\pi\)
\(140\) −16.7338 −1.41426
\(141\) 5.63732 0.474748
\(142\) 25.1809 2.11313
\(143\) 6.73904 0.563547
\(144\) 8.24649 0.687208
\(145\) −12.0840 −1.00352
\(146\) −31.2725 −2.58813
\(147\) 0.305995 0.0252381
\(148\) 1.85187 0.152223
\(149\) −6.25724 −0.512613 −0.256307 0.966596i \(-0.582506\pi\)
−0.256307 + 0.966596i \(0.582506\pi\)
\(150\) 14.5794 1.19040
\(151\) −8.91111 −0.725176 −0.362588 0.931950i \(-0.618107\pi\)
−0.362588 + 0.931950i \(0.618107\pi\)
\(152\) 1.54673 0.125457
\(153\) −3.03706 −0.245532
\(154\) 27.5034 2.21629
\(155\) 14.5262 1.16677
\(156\) −2.38597 −0.191030
\(157\) 8.13236 0.649033 0.324517 0.945880i \(-0.394798\pi\)
0.324517 + 0.945880i \(0.394798\pi\)
\(158\) −21.4718 −1.70821
\(159\) −10.0095 −0.793805
\(160\) 27.2467 2.15404
\(161\) −1.40148 −0.110452
\(162\) −1.00128 −0.0786677
\(163\) 0.189580 0.0148491 0.00742454 0.999972i \(-0.497637\pi\)
0.00742454 + 0.999972i \(0.497637\pi\)
\(164\) 6.29981 0.491933
\(165\) −19.5441 −1.52150
\(166\) −25.5138 −1.98025
\(167\) 2.73207 0.211414 0.105707 0.994397i \(-0.466289\pi\)
0.105707 + 0.994397i \(0.466289\pi\)
\(168\) 0.778919 0.0600949
\(169\) −11.4496 −0.880738
\(170\) −10.7819 −0.826934
\(171\) −10.2644 −0.784937
\(172\) 1.65609 0.126276
\(173\) 5.75053 0.437205 0.218602 0.975814i \(-0.429850\pi\)
0.218602 + 0.975814i \(0.429850\pi\)
\(174\) −7.03184 −0.533083
\(175\) −18.5886 −1.40517
\(176\) −23.1335 −1.74376
\(177\) 5.04101 0.378906
\(178\) 15.0804 1.13033
\(179\) −1.15948 −0.0866636 −0.0433318 0.999061i \(-0.513797\pi\)
−0.0433318 + 0.999061i \(0.513797\pi\)
\(180\) −12.4687 −0.929360
\(181\) −9.12591 −0.678323 −0.339162 0.940728i \(-0.610143\pi\)
−0.339162 + 0.940728i \(0.610143\pi\)
\(182\) 6.32754 0.469029
\(183\) 0.177241 0.0131020
\(184\) 0.157361 0.0116008
\(185\) 3.48986 0.256580
\(186\) 8.45299 0.619803
\(187\) 8.51975 0.623026
\(188\) −10.0891 −0.735820
\(189\) −13.2067 −0.960645
\(190\) −36.4397 −2.64361
\(191\) 21.3538 1.54510 0.772552 0.634952i \(-0.218980\pi\)
0.772552 + 0.634952i \(0.218980\pi\)
\(192\) 7.00965 0.505878
\(193\) −18.2224 −1.31168 −0.655838 0.754902i \(-0.727685\pi\)
−0.655838 + 0.754902i \(0.727685\pi\)
\(194\) −1.29159 −0.0927305
\(195\) −4.49638 −0.321992
\(196\) −0.547638 −0.0391170
\(197\) 2.13452 0.152079 0.0760393 0.997105i \(-0.475773\pi\)
0.0760393 + 0.997105i \(0.475773\pi\)
\(198\) 20.4934 1.45640
\(199\) 19.4706 1.38024 0.690118 0.723697i \(-0.257559\pi\)
0.690118 + 0.723697i \(0.257559\pi\)
\(200\) 2.08716 0.147584
\(201\) 2.40898 0.169916
\(202\) 27.5604 1.93914
\(203\) 8.96557 0.629260
\(204\) −3.01643 −0.211192
\(205\) 11.8720 0.829180
\(206\) −21.8455 −1.52204
\(207\) −1.04427 −0.0725819
\(208\) −5.32219 −0.369027
\(209\) 28.7943 1.99174
\(210\) −18.3506 −1.26631
\(211\) 5.50925 0.379272 0.189636 0.981854i \(-0.439269\pi\)
0.189636 + 0.981854i \(0.439269\pi\)
\(212\) 17.9139 1.23033
\(213\) 13.2760 0.909656
\(214\) −17.9224 −1.22515
\(215\) 3.12091 0.212844
\(216\) 1.48287 0.100896
\(217\) −10.7775 −0.731626
\(218\) 1.96262 0.132925
\(219\) −16.4877 −1.11413
\(220\) 34.9778 2.35821
\(221\) 1.96008 0.131850
\(222\) 2.03080 0.136298
\(223\) −23.9134 −1.60136 −0.800679 0.599093i \(-0.795528\pi\)
−0.800679 + 0.599093i \(0.795528\pi\)
\(224\) −20.2154 −1.35070
\(225\) −13.8508 −0.923383
\(226\) −6.04458 −0.402080
\(227\) 16.1320 1.07072 0.535359 0.844625i \(-0.320177\pi\)
0.535359 + 0.844625i \(0.320177\pi\)
\(228\) −10.1947 −0.675158
\(229\) 8.57243 0.566482 0.283241 0.959049i \(-0.408590\pi\)
0.283241 + 0.959049i \(0.408590\pi\)
\(230\) −3.70728 −0.244451
\(231\) 14.5005 0.954062
\(232\) −1.00667 −0.0660910
\(233\) −20.6481 −1.35270 −0.676350 0.736580i \(-0.736439\pi\)
−0.676350 + 0.736580i \(0.736439\pi\)
\(234\) 4.71478 0.308215
\(235\) −19.0129 −1.24027
\(236\) −9.02186 −0.587273
\(237\) −11.3205 −0.735345
\(238\) 7.99951 0.518531
\(239\) 3.53026 0.228353 0.114177 0.993460i \(-0.463577\pi\)
0.114177 + 0.993460i \(0.463577\pi\)
\(240\) 15.4350 0.996325
\(241\) −2.64850 −0.170605 −0.0853025 0.996355i \(-0.527186\pi\)
−0.0853025 + 0.996355i \(0.527186\pi\)
\(242\) −35.9004 −2.30776
\(243\) −15.8296 −1.01547
\(244\) −0.317207 −0.0203071
\(245\) −1.03203 −0.0659338
\(246\) 6.90852 0.440471
\(247\) 6.62452 0.421508
\(248\) 1.21012 0.0768425
\(249\) −13.4515 −0.852454
\(250\) −14.9253 −0.943961
\(251\) 10.5017 0.662862 0.331431 0.943479i \(-0.392469\pi\)
0.331431 + 0.943479i \(0.392469\pi\)
\(252\) 9.25099 0.582758
\(253\) 2.92945 0.184173
\(254\) 18.6608 1.17088
\(255\) −5.68449 −0.355976
\(256\) 18.1008 1.13130
\(257\) −5.32676 −0.332274 −0.166137 0.986103i \(-0.553129\pi\)
−0.166137 + 0.986103i \(0.553129\pi\)
\(258\) 1.81610 0.113066
\(259\) −2.58926 −0.160889
\(260\) 8.04713 0.499062
\(261\) 6.68042 0.413508
\(262\) 9.60249 0.593244
\(263\) 26.0978 1.60926 0.804631 0.593776i \(-0.202363\pi\)
0.804631 + 0.593776i \(0.202363\pi\)
\(264\) −1.62814 −0.100205
\(265\) 33.7589 2.07379
\(266\) 27.0360 1.65768
\(267\) 7.95078 0.486580
\(268\) −4.31133 −0.263356
\(269\) −11.6170 −0.708303 −0.354151 0.935188i \(-0.615230\pi\)
−0.354151 + 0.935188i \(0.615230\pi\)
\(270\) −34.9351 −2.12608
\(271\) −26.0054 −1.57971 −0.789857 0.613291i \(-0.789845\pi\)
−0.789857 + 0.613291i \(0.789845\pi\)
\(272\) −6.72851 −0.407976
\(273\) 3.33604 0.201906
\(274\) 13.8976 0.839582
\(275\) 38.8549 2.34304
\(276\) −1.03718 −0.0624308
\(277\) 12.9258 0.776635 0.388317 0.921526i \(-0.373056\pi\)
0.388317 + 0.921526i \(0.373056\pi\)
\(278\) 39.5046 2.36933
\(279\) −8.03055 −0.480776
\(280\) −2.62705 −0.156996
\(281\) 10.9805 0.655039 0.327519 0.944844i \(-0.393787\pi\)
0.327519 + 0.944844i \(0.393787\pi\)
\(282\) −11.0639 −0.658845
\(283\) 7.02860 0.417807 0.208903 0.977936i \(-0.433011\pi\)
0.208903 + 0.977936i \(0.433011\pi\)
\(284\) −23.7599 −1.40989
\(285\) −19.2119 −1.13802
\(286\) −13.2262 −0.782080
\(287\) −8.80833 −0.519940
\(288\) −15.0629 −0.887591
\(289\) −14.5220 −0.854235
\(290\) 23.7162 1.39266
\(291\) −0.680956 −0.0399184
\(292\) 29.5078 1.72681
\(293\) 0.673916 0.0393706 0.0196853 0.999806i \(-0.493734\pi\)
0.0196853 + 0.999806i \(0.493734\pi\)
\(294\) −0.600552 −0.0350249
\(295\) −17.0018 −0.989881
\(296\) 0.290726 0.0168981
\(297\) 27.6053 1.60182
\(298\) 12.2806 0.711394
\(299\) 0.673961 0.0389762
\(300\) −13.7567 −0.794241
\(301\) −2.31553 −0.133465
\(302\) 17.4891 1.00638
\(303\) 14.5305 0.834756
\(304\) −22.7404 −1.30425
\(305\) −0.597779 −0.0342288
\(306\) 5.96059 0.340745
\(307\) 22.3122 1.27343 0.636713 0.771101i \(-0.280294\pi\)
0.636713 + 0.771101i \(0.280294\pi\)
\(308\) −25.9514 −1.47872
\(309\) −11.5175 −0.655206
\(310\) −28.5093 −1.61922
\(311\) −6.67998 −0.378787 −0.189393 0.981901i \(-0.560652\pi\)
−0.189393 + 0.981901i \(0.560652\pi\)
\(312\) −0.374576 −0.0212062
\(313\) 8.44126 0.477128 0.238564 0.971127i \(-0.423323\pi\)
0.238564 + 0.971127i \(0.423323\pi\)
\(314\) −15.9607 −0.900715
\(315\) 17.4336 0.982270
\(316\) 20.2602 1.13972
\(317\) −29.0662 −1.63252 −0.816261 0.577684i \(-0.803957\pi\)
−0.816261 + 0.577684i \(0.803957\pi\)
\(318\) 19.6448 1.10163
\(319\) −18.7403 −1.04926
\(320\) −23.6414 −1.32159
\(321\) −9.44914 −0.527400
\(322\) 2.75057 0.153283
\(323\) 8.37496 0.465995
\(324\) 0.944774 0.0524875
\(325\) 8.93911 0.495853
\(326\) −0.372074 −0.0206073
\(327\) 1.03474 0.0572213
\(328\) 0.989014 0.0546091
\(329\) 14.1064 0.777712
\(330\) 38.3575 2.11151
\(331\) 17.2168 0.946322 0.473161 0.880976i \(-0.343113\pi\)
0.473161 + 0.880976i \(0.343113\pi\)
\(332\) 24.0740 1.32123
\(333\) −1.92931 −0.105726
\(334\) −5.36201 −0.293396
\(335\) −8.12473 −0.443902
\(336\) −11.4518 −0.624748
\(337\) −32.8879 −1.79152 −0.895759 0.444539i \(-0.853367\pi\)
−0.895759 + 0.444539i \(0.853367\pi\)
\(338\) 22.4712 1.22227
\(339\) −3.18686 −0.173086
\(340\) 10.1735 0.551734
\(341\) 22.5278 1.21995
\(342\) 20.1451 1.08932
\(343\) 18.8905 1.01999
\(344\) 0.259991 0.0140178
\(345\) −1.95457 −0.105230
\(346\) −11.2861 −0.606744
\(347\) 22.1417 1.18863 0.594313 0.804234i \(-0.297424\pi\)
0.594313 + 0.804234i \(0.297424\pi\)
\(348\) 6.63504 0.355675
\(349\) 0.414563 0.0221910 0.0110955 0.999938i \(-0.496468\pi\)
0.0110955 + 0.999938i \(0.496468\pi\)
\(350\) 36.4823 1.95006
\(351\) 6.35099 0.338991
\(352\) 42.2554 2.25222
\(353\) 0.453157 0.0241191 0.0120595 0.999927i \(-0.496161\pi\)
0.0120595 + 0.999927i \(0.496161\pi\)
\(354\) −9.89358 −0.525838
\(355\) −44.7758 −2.37645
\(356\) −14.2294 −0.754159
\(357\) 4.21754 0.223216
\(358\) 2.27562 0.120270
\(359\) 21.1368 1.11556 0.557779 0.829989i \(-0.311654\pi\)
0.557779 + 0.829989i \(0.311654\pi\)
\(360\) −1.95747 −0.103168
\(361\) 9.30491 0.489732
\(362\) 17.9107 0.941364
\(363\) −18.9276 −0.993440
\(364\) −5.97048 −0.312938
\(365\) 55.6077 2.91064
\(366\) −0.347857 −0.0181828
\(367\) 21.9006 1.14320 0.571601 0.820532i \(-0.306323\pi\)
0.571601 + 0.820532i \(0.306323\pi\)
\(368\) −2.31355 −0.120602
\(369\) −6.56327 −0.341670
\(370\) −6.84926 −0.356076
\(371\) −25.0471 −1.30038
\(372\) −7.97599 −0.413536
\(373\) 26.3361 1.36363 0.681815 0.731525i \(-0.261191\pi\)
0.681815 + 0.731525i \(0.261191\pi\)
\(374\) −16.7210 −0.864623
\(375\) −7.86901 −0.406354
\(376\) −1.58389 −0.0816829
\(377\) −4.31147 −0.222052
\(378\) 25.9197 1.33316
\(379\) 29.3425 1.50722 0.753611 0.657320i \(-0.228310\pi\)
0.753611 + 0.657320i \(0.228310\pi\)
\(380\) 34.3834 1.76383
\(381\) 9.83845 0.504039
\(382\) −41.9092 −2.14426
\(383\) 30.5748 1.56230 0.781149 0.624344i \(-0.214634\pi\)
0.781149 + 0.624344i \(0.214634\pi\)
\(384\) 2.40001 0.122475
\(385\) −48.9056 −2.49246
\(386\) 35.7636 1.82032
\(387\) −1.72534 −0.0877042
\(388\) 1.21870 0.0618702
\(389\) 8.58760 0.435409 0.217704 0.976015i \(-0.430143\pi\)
0.217704 + 0.976015i \(0.430143\pi\)
\(390\) 8.82467 0.446855
\(391\) 0.852046 0.0430898
\(392\) −0.0859741 −0.00434235
\(393\) 5.06267 0.255378
\(394\) −4.18926 −0.211052
\(395\) 38.1805 1.92107
\(396\) −19.3369 −0.971717
\(397\) 26.6916 1.33962 0.669808 0.742535i \(-0.266377\pi\)
0.669808 + 0.742535i \(0.266377\pi\)
\(398\) −38.2134 −1.91546
\(399\) 14.2541 0.713596
\(400\) −30.6859 −1.53429
\(401\) −4.96338 −0.247859 −0.123930 0.992291i \(-0.539550\pi\)
−0.123930 + 0.992291i \(0.539550\pi\)
\(402\) −4.72790 −0.235806
\(403\) 5.18282 0.258175
\(404\) −26.0051 −1.29380
\(405\) 1.78043 0.0884705
\(406\) −17.5960 −0.873274
\(407\) 5.41222 0.268274
\(408\) −0.473552 −0.0234443
\(409\) 11.6087 0.574011 0.287006 0.957929i \(-0.407340\pi\)
0.287006 + 0.957929i \(0.407340\pi\)
\(410\) −23.3003 −1.15072
\(411\) 7.32714 0.361421
\(412\) 20.6127 1.01552
\(413\) 12.6143 0.620708
\(414\) 2.04951 0.100728
\(415\) 45.3677 2.22701
\(416\) 9.72142 0.476632
\(417\) 20.8278 1.01994
\(418\) −56.5121 −2.76410
\(419\) 27.8511 1.36062 0.680308 0.732926i \(-0.261846\pi\)
0.680308 + 0.732926i \(0.261846\pi\)
\(420\) 17.3151 0.844892
\(421\) 38.2834 1.86582 0.932910 0.360110i \(-0.117261\pi\)
0.932910 + 0.360110i \(0.117261\pi\)
\(422\) −10.8125 −0.526346
\(423\) 10.5110 0.511061
\(424\) 2.81232 0.136579
\(425\) 11.3012 0.548186
\(426\) −26.0557 −1.26240
\(427\) 0.443516 0.0214632
\(428\) 16.9111 0.817427
\(429\) −6.97317 −0.336668
\(430\) −6.12515 −0.295381
\(431\) 25.1623 1.21203 0.606014 0.795454i \(-0.292768\pi\)
0.606014 + 0.795454i \(0.292768\pi\)
\(432\) −21.8014 −1.04892
\(433\) 4.87160 0.234114 0.117057 0.993125i \(-0.462654\pi\)
0.117057 + 0.993125i \(0.462654\pi\)
\(434\) 21.1522 1.01534
\(435\) 12.5038 0.599511
\(436\) −1.85187 −0.0886884
\(437\) 2.87967 0.137753
\(438\) 32.3590 1.54617
\(439\) −4.55786 −0.217535 −0.108767 0.994067i \(-0.534690\pi\)
−0.108767 + 0.994067i \(0.534690\pi\)
\(440\) 5.49120 0.261783
\(441\) 0.570539 0.0271685
\(442\) −3.84690 −0.182978
\(443\) −4.29707 −0.204160 −0.102080 0.994776i \(-0.532550\pi\)
−0.102080 + 0.994776i \(0.532550\pi\)
\(444\) −1.91620 −0.0909390
\(445\) −26.8155 −1.27118
\(446\) 46.9328 2.22233
\(447\) 6.47462 0.306239
\(448\) 17.5405 0.828709
\(449\) 19.2045 0.906318 0.453159 0.891430i \(-0.350297\pi\)
0.453159 + 0.891430i \(0.350297\pi\)
\(450\) 27.1837 1.28145
\(451\) 18.4117 0.866971
\(452\) 5.70349 0.268270
\(453\) 9.22069 0.433226
\(454\) −31.6609 −1.48592
\(455\) −11.2514 −0.527474
\(456\) −1.60047 −0.0749488
\(457\) −20.9272 −0.978933 −0.489467 0.872022i \(-0.662809\pi\)
−0.489467 + 0.872022i \(0.662809\pi\)
\(458\) −16.8244 −0.786153
\(459\) 8.02915 0.374769
\(460\) 3.49808 0.163099
\(461\) 16.3236 0.760268 0.380134 0.924931i \(-0.375878\pi\)
0.380134 + 0.924931i \(0.375878\pi\)
\(462\) −28.4589 −1.32403
\(463\) 26.2942 1.22200 0.610998 0.791632i \(-0.290768\pi\)
0.610998 + 0.791632i \(0.290768\pi\)
\(464\) 14.8002 0.687084
\(465\) −15.0308 −0.697037
\(466\) 40.5243 1.87725
\(467\) −29.4061 −1.36075 −0.680376 0.732863i \(-0.738184\pi\)
−0.680376 + 0.732863i \(0.738184\pi\)
\(468\) −4.44872 −0.205642
\(469\) 6.02806 0.278350
\(470\) 37.3151 1.72122
\(471\) −8.41489 −0.387738
\(472\) −1.41635 −0.0651928
\(473\) 4.84004 0.222545
\(474\) 22.2178 1.02050
\(475\) 38.1946 1.75249
\(476\) −7.54810 −0.345967
\(477\) −18.6631 −0.854523
\(478\) −6.92855 −0.316904
\(479\) −29.6102 −1.35293 −0.676463 0.736477i \(-0.736488\pi\)
−0.676463 + 0.736477i \(0.736488\pi\)
\(480\) −28.1933 −1.28684
\(481\) 1.24515 0.0567742
\(482\) 5.19800 0.236762
\(483\) 1.45017 0.0659851
\(484\) 33.8745 1.53975
\(485\) 2.29665 0.104286
\(486\) 31.0674 1.40925
\(487\) 10.3597 0.469444 0.234722 0.972062i \(-0.424582\pi\)
0.234722 + 0.972062i \(0.424582\pi\)
\(488\) −0.0497987 −0.00225428
\(489\) −0.196167 −0.00887096
\(490\) 2.02548 0.0915016
\(491\) −27.7722 −1.25334 −0.626671 0.779284i \(-0.715583\pi\)
−0.626671 + 0.779284i \(0.715583\pi\)
\(492\) −6.51868 −0.293885
\(493\) −5.45072 −0.245488
\(494\) −13.0014 −0.584960
\(495\) −36.4406 −1.63788
\(496\) −17.7914 −0.798857
\(497\) 33.2209 1.49016
\(498\) 26.4001 1.18302
\(499\) 33.9045 1.51777 0.758886 0.651223i \(-0.225744\pi\)
0.758886 + 0.651223i \(0.225744\pi\)
\(500\) 14.0831 0.629816
\(501\) −2.82698 −0.126300
\(502\) −20.6108 −0.919907
\(503\) −39.0108 −1.73941 −0.869703 0.493575i \(-0.835690\pi\)
−0.869703 + 0.493575i \(0.835690\pi\)
\(504\) 1.45232 0.0646915
\(505\) −49.0069 −2.18078
\(506\) −5.74940 −0.255592
\(507\) 11.8474 0.526160
\(508\) −17.6078 −0.781220
\(509\) 19.1299 0.847920 0.423960 0.905681i \(-0.360640\pi\)
0.423960 + 0.905681i \(0.360640\pi\)
\(510\) 11.1565 0.494017
\(511\) −41.2575 −1.82513
\(512\) −30.8860 −1.36498
\(513\) 27.1362 1.19809
\(514\) 10.4544 0.461124
\(515\) 38.8448 1.71171
\(516\) −1.71362 −0.0754380
\(517\) −29.4860 −1.29679
\(518\) 5.08173 0.223278
\(519\) −5.95031 −0.261190
\(520\) 1.26333 0.0554005
\(521\) −8.04701 −0.352546 −0.176273 0.984341i \(-0.556404\pi\)
−0.176273 + 0.984341i \(0.556404\pi\)
\(522\) −13.1111 −0.573858
\(523\) 33.8634 1.48074 0.740371 0.672198i \(-0.234650\pi\)
0.740371 + 0.672198i \(0.234650\pi\)
\(524\) −9.06063 −0.395815
\(525\) 19.2344 0.839458
\(526\) −51.2201 −2.23330
\(527\) 6.55231 0.285423
\(528\) 23.9372 1.04173
\(529\) −22.7070 −0.987262
\(530\) −66.2559 −2.87797
\(531\) 9.39915 0.407888
\(532\) −25.5104 −1.10602
\(533\) 4.23585 0.183475
\(534\) −15.6043 −0.675266
\(535\) 31.8690 1.37782
\(536\) −0.676840 −0.0292350
\(537\) 1.19976 0.0517735
\(538\) 22.7998 0.982968
\(539\) −1.60051 −0.0689389
\(540\) 32.9637 1.41853
\(541\) −9.47001 −0.407148 −0.203574 0.979060i \(-0.565256\pi\)
−0.203574 + 0.979060i \(0.565256\pi\)
\(542\) 51.0386 2.19230
\(543\) 9.44295 0.405236
\(544\) 12.2902 0.526937
\(545\) −3.48986 −0.149489
\(546\) −6.54737 −0.280201
\(547\) −26.3719 −1.12758 −0.563791 0.825917i \(-0.690658\pi\)
−0.563791 + 0.825917i \(0.690658\pi\)
\(548\) −13.1133 −0.560174
\(549\) 0.330472 0.0141042
\(550\) −76.2574 −3.25163
\(551\) −18.4218 −0.784796
\(552\) −0.162827 −0.00693040
\(553\) −28.3276 −1.20461
\(554\) −25.3684 −1.07780
\(555\) −3.61110 −0.153283
\(556\) −37.2754 −1.58083
\(557\) 44.1072 1.86888 0.934442 0.356115i \(-0.115899\pi\)
0.934442 + 0.356115i \(0.115899\pi\)
\(558\) 15.7609 0.667212
\(559\) 1.11352 0.0470967
\(560\) 38.6234 1.63214
\(561\) −8.81573 −0.372201
\(562\) −21.5504 −0.909050
\(563\) 0.715150 0.0301400 0.0150700 0.999886i \(-0.495203\pi\)
0.0150700 + 0.999886i \(0.495203\pi\)
\(564\) 10.4396 0.439585
\(565\) 10.7483 0.452183
\(566\) −13.7945 −0.579824
\(567\) −1.32097 −0.0554757
\(568\) −3.73009 −0.156511
\(569\) −11.0535 −0.463388 −0.231694 0.972789i \(-0.574427\pi\)
−0.231694 + 0.972789i \(0.574427\pi\)
\(570\) 37.7056 1.57931
\(571\) 10.1651 0.425396 0.212698 0.977118i \(-0.431775\pi\)
0.212698 + 0.977118i \(0.431775\pi\)
\(572\) 12.4798 0.521808
\(573\) −22.0956 −0.923057
\(574\) 17.2874 0.721562
\(575\) 3.88582 0.162050
\(576\) 13.0697 0.544573
\(577\) −38.0920 −1.58579 −0.792895 0.609358i \(-0.791427\pi\)
−0.792895 + 0.609358i \(0.791427\pi\)
\(578\) 28.5011 1.18549
\(579\) 18.8555 0.783606
\(580\) −22.3779 −0.929193
\(581\) −33.6601 −1.39645
\(582\) 1.33646 0.0553979
\(583\) 52.3547 2.16831
\(584\) 4.63246 0.191693
\(585\) −8.38365 −0.346621
\(586\) −1.32264 −0.0546377
\(587\) 10.2540 0.423226 0.211613 0.977354i \(-0.432128\pi\)
0.211613 + 0.977354i \(0.432128\pi\)
\(588\) 0.566663 0.0233688
\(589\) 22.1449 0.912465
\(590\) 33.3680 1.37374
\(591\) −2.20868 −0.0908530
\(592\) −4.27432 −0.175674
\(593\) −17.3289 −0.711614 −0.355807 0.934559i \(-0.615794\pi\)
−0.355807 + 0.934559i \(0.615794\pi\)
\(594\) −54.1787 −2.22298
\(595\) −14.2245 −0.583146
\(596\) −11.5876 −0.474646
\(597\) −20.1471 −0.824564
\(598\) −1.32273 −0.0540904
\(599\) 12.7294 0.520108 0.260054 0.965594i \(-0.416260\pi\)
0.260054 + 0.965594i \(0.416260\pi\)
\(600\) −2.15967 −0.0881681
\(601\) 6.84116 0.279057 0.139528 0.990218i \(-0.455441\pi\)
0.139528 + 0.990218i \(0.455441\pi\)
\(602\) 4.54449 0.185220
\(603\) 4.49163 0.182913
\(604\) −16.5022 −0.671465
\(605\) 63.8368 2.59533
\(606\) −28.5178 −1.15846
\(607\) 21.3114 0.865004 0.432502 0.901633i \(-0.357631\pi\)
0.432502 + 0.901633i \(0.357631\pi\)
\(608\) 41.5372 1.68456
\(609\) −9.27704 −0.375925
\(610\) 1.17321 0.0475020
\(611\) −6.78366 −0.274437
\(612\) −5.62424 −0.227346
\(613\) −26.1915 −1.05786 −0.528932 0.848664i \(-0.677407\pi\)
−0.528932 + 0.848664i \(0.677407\pi\)
\(614\) −43.7904 −1.76724
\(615\) −12.2845 −0.495359
\(616\) −4.07414 −0.164152
\(617\) 28.6698 1.15420 0.577101 0.816673i \(-0.304184\pi\)
0.577101 + 0.816673i \(0.304184\pi\)
\(618\) 22.6044 0.909282
\(619\) −0.156419 −0.00628701 −0.00314350 0.999995i \(-0.501001\pi\)
−0.00314350 + 0.999995i \(0.501001\pi\)
\(620\) 26.9005 1.08035
\(621\) 2.76076 0.110786
\(622\) 13.1102 0.525673
\(623\) 19.8955 0.797095
\(624\) 5.50709 0.220460
\(625\) −9.35586 −0.374234
\(626\) −16.5670 −0.662149
\(627\) −29.7946 −1.18988
\(628\) 15.0601 0.600962
\(629\) 1.57417 0.0627663
\(630\) −34.2154 −1.36317
\(631\) −11.6621 −0.464262 −0.232131 0.972684i \(-0.574570\pi\)
−0.232131 + 0.972684i \(0.574570\pi\)
\(632\) 3.18067 0.126520
\(633\) −5.70064 −0.226580
\(634\) 57.0459 2.26558
\(635\) −33.1820 −1.31679
\(636\) −18.5363 −0.735011
\(637\) −0.368219 −0.0145894
\(638\) 36.7801 1.45614
\(639\) 24.7535 0.979235
\(640\) −8.09449 −0.319963
\(641\) −26.3124 −1.03928 −0.519638 0.854386i \(-0.673933\pi\)
−0.519638 + 0.854386i \(0.673933\pi\)
\(642\) 18.5450 0.731915
\(643\) −1.48585 −0.0585961 −0.0292981 0.999571i \(-0.509327\pi\)
−0.0292981 + 0.999571i \(0.509327\pi\)
\(644\) −2.59536 −0.102271
\(645\) −3.22933 −0.127155
\(646\) −16.4368 −0.646699
\(647\) −20.2282 −0.795251 −0.397626 0.917548i \(-0.630166\pi\)
−0.397626 + 0.917548i \(0.630166\pi\)
\(648\) 0.148321 0.00582660
\(649\) −26.3670 −1.03500
\(650\) −17.5441 −0.688134
\(651\) 11.1519 0.437079
\(652\) 0.351078 0.0137493
\(653\) 28.5744 1.11820 0.559102 0.829099i \(-0.311146\pi\)
0.559102 + 0.829099i \(0.311146\pi\)
\(654\) −2.03080 −0.0794106
\(655\) −17.0748 −0.667168
\(656\) −14.5407 −0.567718
\(657\) −30.7418 −1.19935
\(658\) −27.6855 −1.07929
\(659\) −1.47999 −0.0576521 −0.0288260 0.999584i \(-0.509177\pi\)
−0.0288260 + 0.999584i \(0.509177\pi\)
\(660\) −36.1930 −1.40881
\(661\) −23.9376 −0.931064 −0.465532 0.885031i \(-0.654137\pi\)
−0.465532 + 0.885031i \(0.654137\pi\)
\(662\) −33.7900 −1.31329
\(663\) −2.02818 −0.0787680
\(664\) 3.77941 0.146669
\(665\) −48.0745 −1.86425
\(666\) 3.78650 0.146724
\(667\) −1.87419 −0.0725689
\(668\) 5.05943 0.195755
\(669\) 24.7442 0.956665
\(670\) 15.9457 0.616038
\(671\) −0.927061 −0.0357888
\(672\) 20.9177 0.806919
\(673\) −32.8577 −1.26657 −0.633285 0.773919i \(-0.718294\pi\)
−0.633285 + 0.773919i \(0.718294\pi\)
\(674\) 64.5464 2.48623
\(675\) 36.6175 1.40941
\(676\) −21.2031 −0.815505
\(677\) 4.93887 0.189816 0.0949082 0.995486i \(-0.469744\pi\)
0.0949082 + 0.995486i \(0.469744\pi\)
\(678\) 6.25458 0.240206
\(679\) −1.70398 −0.0653926
\(680\) 1.59714 0.0612477
\(681\) −16.6924 −0.639655
\(682\) −44.2134 −1.69302
\(683\) 17.9362 0.686311 0.343156 0.939279i \(-0.388504\pi\)
0.343156 + 0.939279i \(0.388504\pi\)
\(684\) −19.0083 −0.726800
\(685\) −24.7122 −0.944203
\(686\) −37.0749 −1.41553
\(687\) −8.87024 −0.338421
\(688\) −3.82244 −0.145729
\(689\) 12.0449 0.458875
\(690\) 3.83607 0.146037
\(691\) −9.34015 −0.355316 −0.177658 0.984092i \(-0.556852\pi\)
−0.177658 + 0.984092i \(0.556852\pi\)
\(692\) 10.6492 0.404823
\(693\) 27.0367 1.02704
\(694\) −43.4556 −1.64955
\(695\) −70.2457 −2.66457
\(696\) 1.04164 0.0394833
\(697\) 5.35512 0.202840
\(698\) −0.813628 −0.0307963
\(699\) 21.3654 0.808114
\(700\) −34.4237 −1.30109
\(701\) 19.9900 0.755011 0.377505 0.926007i \(-0.376782\pi\)
0.377505 + 0.926007i \(0.376782\pi\)
\(702\) −12.4646 −0.470444
\(703\) 5.32024 0.200656
\(704\) −36.6640 −1.38183
\(705\) 19.6734 0.740944
\(706\) −0.889373 −0.0334720
\(707\) 36.3601 1.36746
\(708\) 9.33529 0.350842
\(709\) 18.2840 0.686671 0.343336 0.939213i \(-0.388443\pi\)
0.343336 + 0.939213i \(0.388443\pi\)
\(710\) 87.8777 3.29799
\(711\) −21.1074 −0.791591
\(712\) −2.23389 −0.0837187
\(713\) 2.25296 0.0843742
\(714\) −8.27742 −0.309775
\(715\) 23.5183 0.879535
\(716\) −2.14720 −0.0802448
\(717\) −3.65290 −0.136420
\(718\) −41.4835 −1.54815
\(719\) −4.24470 −0.158301 −0.0791503 0.996863i \(-0.525221\pi\)
−0.0791503 + 0.996863i \(0.525221\pi\)
\(720\) 28.7791 1.07253
\(721\) −28.8205 −1.07333
\(722\) −18.2620 −0.679640
\(723\) 2.74051 0.101921
\(724\) −16.9000 −0.628083
\(725\) −24.8584 −0.923217
\(726\) 37.1476 1.37868
\(727\) 1.79717 0.0666532 0.0333266 0.999445i \(-0.489390\pi\)
0.0333266 + 0.999445i \(0.489390\pi\)
\(728\) −0.937311 −0.0347391
\(729\) 14.8490 0.549963
\(730\) −109.137 −4.03933
\(731\) 1.40775 0.0520675
\(732\) 0.328227 0.0121316
\(733\) 24.0113 0.886877 0.443439 0.896305i \(-0.353758\pi\)
0.443439 + 0.896305i \(0.353758\pi\)
\(734\) −42.9825 −1.58651
\(735\) 1.06788 0.0393894
\(736\) 4.22589 0.155768
\(737\) −12.6002 −0.464133
\(738\) 12.8812 0.474163
\(739\) 0.427427 0.0157231 0.00786157 0.999969i \(-0.497498\pi\)
0.00786157 + 0.999969i \(0.497498\pi\)
\(740\) 6.46276 0.237576
\(741\) −6.85466 −0.251812
\(742\) 49.1578 1.80464
\(743\) 47.4922 1.74232 0.871160 0.490999i \(-0.163368\pi\)
0.871160 + 0.490999i \(0.163368\pi\)
\(744\) −1.25216 −0.0459063
\(745\) −21.8369 −0.800042
\(746\) −51.6876 −1.89242
\(747\) −25.0808 −0.917658
\(748\) 15.7774 0.576881
\(749\) −23.6449 −0.863964
\(750\) 15.4439 0.563930
\(751\) 25.8412 0.942958 0.471479 0.881877i \(-0.343720\pi\)
0.471479 + 0.881877i \(0.343720\pi\)
\(752\) 23.2867 0.849178
\(753\) −10.8665 −0.395999
\(754\) 8.46176 0.308159
\(755\) −31.0985 −1.13179
\(756\) −24.4570 −0.889494
\(757\) −49.8137 −1.81051 −0.905255 0.424868i \(-0.860321\pi\)
−0.905255 + 0.424868i \(0.860321\pi\)
\(758\) −57.5881 −2.09169
\(759\) −3.03123 −0.110027
\(760\) 5.39788 0.195802
\(761\) −6.48467 −0.235069 −0.117534 0.993069i \(-0.537499\pi\)
−0.117534 + 0.993069i \(0.537499\pi\)
\(762\) −19.3091 −0.699496
\(763\) 2.58926 0.0937376
\(764\) 39.5443 1.43066
\(765\) −10.5989 −0.383205
\(766\) −60.0066 −2.16813
\(767\) −6.06610 −0.219034
\(768\) −18.7296 −0.675847
\(769\) −37.4272 −1.34966 −0.674829 0.737974i \(-0.735783\pi\)
−0.674829 + 0.737974i \(0.735783\pi\)
\(770\) 95.9831 3.45899
\(771\) 5.51182 0.198503
\(772\) −33.7455 −1.21453
\(773\) −18.9037 −0.679918 −0.339959 0.940440i \(-0.610413\pi\)
−0.339959 + 0.940440i \(0.610413\pi\)
\(774\) 3.38619 0.121714
\(775\) 29.8823 1.07340
\(776\) 0.191325 0.00686817
\(777\) 2.67922 0.0961163
\(778\) −16.8542 −0.604252
\(779\) 18.0988 0.648455
\(780\) −8.32670 −0.298144
\(781\) −69.4401 −2.48476
\(782\) −1.67224 −0.0597992
\(783\) −17.6612 −0.631159
\(784\) 1.26401 0.0451432
\(785\) 28.3808 1.01295
\(786\) −9.93609 −0.354409
\(787\) −9.43418 −0.336292 −0.168146 0.985762i \(-0.553778\pi\)
−0.168146 + 0.985762i \(0.553778\pi\)
\(788\) 3.95286 0.140815
\(789\) −27.0045 −0.961386
\(790\) −74.9337 −2.66602
\(791\) −7.97456 −0.283543
\(792\) −3.03572 −0.107870
\(793\) −0.213283 −0.00757390
\(794\) −52.3855 −1.85909
\(795\) −34.9318 −1.23890
\(796\) 36.0570 1.27801
\(797\) 32.2257 1.14149 0.570747 0.821126i \(-0.306654\pi\)
0.570747 + 0.821126i \(0.306654\pi\)
\(798\) −27.9753 −0.990314
\(799\) −8.57615 −0.303402
\(800\) 56.0503 1.98168
\(801\) 14.8245 0.523798
\(802\) 9.74121 0.343974
\(803\) 86.2387 3.04330
\(804\) 4.46111 0.157331
\(805\) −4.89097 −0.172384
\(806\) −10.1719 −0.358290
\(807\) 12.0206 0.423146
\(808\) −4.08257 −0.143624
\(809\) −37.1430 −1.30588 −0.652939 0.757410i \(-0.726464\pi\)
−0.652939 + 0.757410i \(0.726464\pi\)
\(810\) −3.49431 −0.122778
\(811\) −22.8875 −0.803687 −0.401844 0.915708i \(-0.631630\pi\)
−0.401844 + 0.915708i \(0.631630\pi\)
\(812\) 16.6031 0.582653
\(813\) 26.9088 0.943734
\(814\) −10.6221 −0.372305
\(815\) 0.661609 0.0231751
\(816\) 6.96226 0.243728
\(817\) 4.75778 0.166454
\(818\) −22.7834 −0.796602
\(819\) 6.22016 0.217350
\(820\) 21.9855 0.767766
\(821\) 7.61464 0.265753 0.132876 0.991133i \(-0.457579\pi\)
0.132876 + 0.991133i \(0.457579\pi\)
\(822\) −14.3804 −0.501573
\(823\) −19.0229 −0.663096 −0.331548 0.943438i \(-0.607571\pi\)
−0.331548 + 0.943438i \(0.607571\pi\)
\(824\) 3.23601 0.112732
\(825\) −40.2048 −1.39975
\(826\) −24.7570 −0.861406
\(827\) 19.0078 0.660968 0.330484 0.943812i \(-0.392788\pi\)
0.330484 + 0.943812i \(0.392788\pi\)
\(828\) −1.93385 −0.0672061
\(829\) −12.3846 −0.430134 −0.215067 0.976599i \(-0.568997\pi\)
−0.215067 + 0.976599i \(0.568997\pi\)
\(830\) −89.0395 −3.09061
\(831\) −13.3748 −0.463968
\(832\) −8.43506 −0.292433
\(833\) −0.465516 −0.0161292
\(834\) −40.8770 −1.41546
\(835\) 9.53453 0.329956
\(836\) 53.3232 1.84422
\(837\) 21.2305 0.733834
\(838\) −54.6611 −1.88824
\(839\) −4.02796 −0.139061 −0.0695303 0.997580i \(-0.522150\pi\)
−0.0695303 + 0.997580i \(0.522150\pi\)
\(840\) 2.71832 0.0937909
\(841\) −17.0104 −0.586566
\(842\) −75.1357 −2.58935
\(843\) −11.3619 −0.391326
\(844\) 10.2024 0.351181
\(845\) −39.9575 −1.37458
\(846\) −20.6290 −0.709240
\(847\) −47.3630 −1.62741
\(848\) −41.3474 −1.41987
\(849\) −7.27278 −0.249601
\(850\) −22.1798 −0.760762
\(851\) 0.541267 0.0185544
\(852\) 24.5854 0.842281
\(853\) 40.9910 1.40350 0.701752 0.712421i \(-0.252401\pi\)
0.701752 + 0.712421i \(0.252401\pi\)
\(854\) −0.870452 −0.0297863
\(855\) −35.8213 −1.22506
\(856\) 2.65488 0.0907420
\(857\) 4.45633 0.152225 0.0761126 0.997099i \(-0.475749\pi\)
0.0761126 + 0.997099i \(0.475749\pi\)
\(858\) 13.6857 0.467221
\(859\) 15.7128 0.536114 0.268057 0.963403i \(-0.413618\pi\)
0.268057 + 0.963403i \(0.413618\pi\)
\(860\) 5.77952 0.197080
\(861\) 9.11434 0.310616
\(862\) −49.3841 −1.68203
\(863\) −27.2952 −0.929140 −0.464570 0.885536i \(-0.653791\pi\)
−0.464570 + 0.885536i \(0.653791\pi\)
\(864\) 39.8221 1.35478
\(865\) 20.0685 0.682351
\(866\) −9.56109 −0.324899
\(867\) 15.0265 0.510327
\(868\) −19.9585 −0.677437
\(869\) 59.2118 2.00862
\(870\) −24.5401 −0.831989
\(871\) −2.89884 −0.0982235
\(872\) −0.290726 −0.00984524
\(873\) −1.26967 −0.0429717
\(874\) −5.65169 −0.191171
\(875\) −19.6909 −0.665672
\(876\) −30.5330 −1.03161
\(877\) −41.6623 −1.40684 −0.703418 0.710777i \(-0.748344\pi\)
−0.703418 + 0.710777i \(0.748344\pi\)
\(878\) 8.94534 0.301891
\(879\) −0.697329 −0.0235203
\(880\) −80.7328 −2.72150
\(881\) 7.59924 0.256025 0.128012 0.991773i \(-0.459140\pi\)
0.128012 + 0.991773i \(0.459140\pi\)
\(882\) −1.11975 −0.0377040
\(883\) 51.3036 1.72650 0.863252 0.504772i \(-0.168424\pi\)
0.863252 + 0.504772i \(0.168424\pi\)
\(884\) 3.62982 0.122084
\(885\) 17.5924 0.591363
\(886\) 8.43350 0.283329
\(887\) 30.8945 1.03733 0.518667 0.854976i \(-0.326428\pi\)
0.518667 + 0.854976i \(0.326428\pi\)
\(888\) −0.300827 −0.0100951
\(889\) 24.6190 0.825696
\(890\) 52.6286 1.76411
\(891\) 2.76117 0.0925027
\(892\) −44.2844 −1.48275
\(893\) −28.9849 −0.969942
\(894\) −12.7072 −0.424993
\(895\) −4.04642 −0.135257
\(896\) 6.00562 0.200634
\(897\) −0.697375 −0.0232847
\(898\) −37.6912 −1.25777
\(899\) −14.4127 −0.480690
\(900\) −25.6498 −0.854992
\(901\) 15.2276 0.507306
\(902\) −36.1350 −1.20317
\(903\) 2.39597 0.0797328
\(904\) 0.895396 0.0297804
\(905\) −31.8481 −1.05867
\(906\) −18.0967 −0.601222
\(907\) −45.0721 −1.49660 −0.748298 0.663363i \(-0.769129\pi\)
−0.748298 + 0.663363i \(0.769129\pi\)
\(908\) 29.8743 0.991413
\(909\) 27.0926 0.898606
\(910\) 22.0822 0.732019
\(911\) 37.0746 1.22834 0.614169 0.789175i \(-0.289491\pi\)
0.614169 + 0.789175i \(0.289491\pi\)
\(912\) 23.5304 0.779170
\(913\) 70.3581 2.32851
\(914\) 41.0721 1.35854
\(915\) 0.618547 0.0204485
\(916\) 15.8750 0.524525
\(917\) 12.6685 0.418350
\(918\) −15.7582 −0.520096
\(919\) −6.55588 −0.216258 −0.108129 0.994137i \(-0.534486\pi\)
−0.108129 + 0.994137i \(0.534486\pi\)
\(920\) 0.549166 0.0181055
\(921\) −23.0874 −0.760755
\(922\) −32.0371 −1.05508
\(923\) −15.9756 −0.525845
\(924\) 26.8530 0.883399
\(925\) 7.17912 0.236048
\(926\) −51.6055 −1.69586
\(927\) −21.4747 −0.705323
\(928\) −27.0339 −0.887431
\(929\) −26.4512 −0.867837 −0.433919 0.900952i \(-0.642869\pi\)
−0.433919 + 0.900952i \(0.642869\pi\)
\(930\) 29.4997 0.967335
\(931\) −1.57331 −0.0515631
\(932\) −38.2375 −1.25251
\(933\) 6.91205 0.226290
\(934\) 57.7129 1.88843
\(935\) 29.7327 0.972364
\(936\) −0.698409 −0.0228282
\(937\) −2.82295 −0.0922219 −0.0461109 0.998936i \(-0.514683\pi\)
−0.0461109 + 0.998936i \(0.514683\pi\)
\(938\) −11.8308 −0.386288
\(939\) −8.73452 −0.285040
\(940\) −35.2094 −1.14840
\(941\) −23.3693 −0.761817 −0.380908 0.924613i \(-0.624389\pi\)
−0.380908 + 0.924613i \(0.624389\pi\)
\(942\) 16.5152 0.538095
\(943\) 1.84132 0.0599616
\(944\) 20.8235 0.677746
\(945\) −46.0895 −1.49929
\(946\) −9.49914 −0.308844
\(947\) 55.2154 1.79426 0.897129 0.441768i \(-0.145649\pi\)
0.897129 + 0.441768i \(0.145649\pi\)
\(948\) −20.9641 −0.680881
\(949\) 19.8404 0.644047
\(950\) −74.9614 −2.43207
\(951\) 30.0760 0.975281
\(952\) −1.18498 −0.0384055
\(953\) −18.7885 −0.608619 −0.304309 0.952573i \(-0.598426\pi\)
−0.304309 + 0.952573i \(0.598426\pi\)
\(954\) 36.6285 1.18589
\(955\) 74.5216 2.41146
\(956\) 6.53757 0.211440
\(957\) 19.3914 0.626834
\(958\) 58.1135 1.87756
\(959\) 18.3349 0.592065
\(960\) 24.4627 0.789531
\(961\) −13.6745 −0.441113
\(962\) −2.44376 −0.0787901
\(963\) −17.6183 −0.567740
\(964\) −4.90468 −0.157969
\(965\) −63.5936 −2.04715
\(966\) −2.84613 −0.0915728
\(967\) −17.0809 −0.549286 −0.274643 0.961546i \(-0.588560\pi\)
−0.274643 + 0.961546i \(0.588560\pi\)
\(968\) 5.31799 0.170927
\(969\) −8.66591 −0.278389
\(970\) −4.50745 −0.144726
\(971\) 17.9005 0.574454 0.287227 0.957863i \(-0.407267\pi\)
0.287227 + 0.957863i \(0.407267\pi\)
\(972\) −29.3143 −0.940256
\(973\) 52.1180 1.67083
\(974\) −20.3322 −0.651486
\(975\) −9.24966 −0.296226
\(976\) 0.732150 0.0234356
\(977\) 30.5331 0.976841 0.488420 0.872608i \(-0.337573\pi\)
0.488420 + 0.872608i \(0.337573\pi\)
\(978\) 0.385000 0.0123109
\(979\) −41.5866 −1.32911
\(980\) −1.91118 −0.0610504
\(981\) 1.92931 0.0615982
\(982\) 54.5063 1.73936
\(983\) 1.79471 0.0572423 0.0286211 0.999590i \(-0.490888\pi\)
0.0286211 + 0.999590i \(0.490888\pi\)
\(984\) −1.02337 −0.0326239
\(985\) 7.44919 0.237351
\(986\) 10.6977 0.340683
\(987\) −14.5965 −0.464611
\(988\) 12.2677 0.390288
\(989\) 0.484044 0.0153917
\(990\) 71.5189 2.27302
\(991\) −13.4650 −0.427731 −0.213865 0.976863i \(-0.568605\pi\)
−0.213865 + 0.976863i \(0.568605\pi\)
\(992\) 32.4975 1.03180
\(993\) −17.8149 −0.565340
\(994\) −65.1999 −2.06802
\(995\) 67.9498 2.15415
\(996\) −24.9104 −0.789316
\(997\) −30.6437 −0.970497 −0.485249 0.874376i \(-0.661271\pi\)
−0.485249 + 0.874376i \(0.661271\pi\)
\(998\) −66.5415 −2.10633
\(999\) 5.10056 0.161375
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 4033.2.a.f.1.15 85
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.f.1.15 85 1.1 even 1 trivial