Properties

Label 1849.4.a.i.1.13
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.43471 q^{2} -1.13642 q^{3} +3.79726 q^{4} -2.82850 q^{5} +3.90328 q^{6} -33.5669 q^{7} +14.4352 q^{8} -25.7086 q^{9} +O(q^{10})\) \(q-3.43471 q^{2} -1.13642 q^{3} +3.79726 q^{4} -2.82850 q^{5} +3.90328 q^{6} -33.5669 q^{7} +14.4352 q^{8} -25.7086 q^{9} +9.71508 q^{10} -8.65202 q^{11} -4.31528 q^{12} +54.3373 q^{13} +115.293 q^{14} +3.21436 q^{15} -79.9589 q^{16} -76.6586 q^{17} +88.3015 q^{18} +95.8312 q^{19} -10.7405 q^{20} +38.1461 q^{21} +29.7172 q^{22} -162.323 q^{23} -16.4045 q^{24} -117.000 q^{25} -186.633 q^{26} +59.8990 q^{27} -127.462 q^{28} +17.8240 q^{29} -11.0404 q^{30} +135.293 q^{31} +159.154 q^{32} +9.83232 q^{33} +263.300 q^{34} +94.9439 q^{35} -97.6220 q^{36} +204.476 q^{37} -329.153 q^{38} -61.7500 q^{39} -40.8300 q^{40} -302.873 q^{41} -131.021 q^{42} -32.8540 q^{44} +72.7166 q^{45} +557.532 q^{46} -79.5496 q^{47} +90.8669 q^{48} +783.738 q^{49} +401.860 q^{50} +87.1163 q^{51} +206.333 q^{52} +445.730 q^{53} -205.736 q^{54} +24.4722 q^{55} -484.546 q^{56} -108.904 q^{57} -61.2204 q^{58} -219.828 q^{59} +12.2058 q^{60} +116.310 q^{61} -464.693 q^{62} +862.957 q^{63} +93.0220 q^{64} -153.693 q^{65} -33.7712 q^{66} +859.592 q^{67} -291.092 q^{68} +184.467 q^{69} -326.105 q^{70} +1028.88 q^{71} -371.108 q^{72} +136.880 q^{73} -702.318 q^{74} +132.961 q^{75} +363.896 q^{76} +290.422 q^{77} +212.094 q^{78} -584.277 q^{79} +226.164 q^{80} +626.060 q^{81} +1040.28 q^{82} +27.3094 q^{83} +144.851 q^{84} +216.829 q^{85} -20.2556 q^{87} -124.894 q^{88} -1120.94 q^{89} -249.761 q^{90} -1823.94 q^{91} -616.381 q^{92} -153.750 q^{93} +273.230 q^{94} -271.058 q^{95} -180.866 q^{96} +643.857 q^{97} -2691.92 q^{98} +222.431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.43471 −1.21435 −0.607177 0.794566i \(-0.707698\pi\)
−0.607177 + 0.794566i \(0.707698\pi\)
\(3\) −1.13642 −0.218704 −0.109352 0.994003i \(-0.534878\pi\)
−0.109352 + 0.994003i \(0.534878\pi\)
\(4\) 3.79726 0.474657
\(5\) −2.82850 −0.252989 −0.126494 0.991967i \(-0.540373\pi\)
−0.126494 + 0.991967i \(0.540373\pi\)
\(6\) 3.90328 0.265584
\(7\) −33.5669 −1.81244 −0.906222 0.422803i \(-0.861046\pi\)
−0.906222 + 0.422803i \(0.861046\pi\)
\(8\) 14.4352 0.637952
\(9\) −25.7086 −0.952169
\(10\) 9.71508 0.307218
\(11\) −8.65202 −0.237153 −0.118576 0.992945i \(-0.537833\pi\)
−0.118576 + 0.992945i \(0.537833\pi\)
\(12\) −4.31528 −0.103809
\(13\) 54.3373 1.15927 0.579633 0.814878i \(-0.303196\pi\)
0.579633 + 0.814878i \(0.303196\pi\)
\(14\) 115.293 2.20095
\(15\) 3.21436 0.0553296
\(16\) −79.9589 −1.24936
\(17\) −76.6586 −1.09367 −0.546836 0.837240i \(-0.684168\pi\)
−0.546836 + 0.837240i \(0.684168\pi\)
\(18\) 88.3015 1.15627
\(19\) 95.8312 1.15711 0.578557 0.815642i \(-0.303616\pi\)
0.578557 + 0.815642i \(0.303616\pi\)
\(20\) −10.7405 −0.120083
\(21\) 38.1461 0.396389
\(22\) 29.7172 0.287988
\(23\) −162.323 −1.47159 −0.735796 0.677203i \(-0.763192\pi\)
−0.735796 + 0.677203i \(0.763192\pi\)
\(24\) −16.4045 −0.139523
\(25\) −117.000 −0.935997
\(26\) −186.633 −1.40776
\(27\) 59.8990 0.426947
\(28\) −127.462 −0.860289
\(29\) 17.8240 0.114132 0.0570661 0.998370i \(-0.481825\pi\)
0.0570661 + 0.998370i \(0.481825\pi\)
\(30\) −11.0404 −0.0671898
\(31\) 135.293 0.783851 0.391925 0.919997i \(-0.371809\pi\)
0.391925 + 0.919997i \(0.371809\pi\)
\(32\) 159.154 0.879211
\(33\) 9.83232 0.0518663
\(34\) 263.300 1.32811
\(35\) 94.9439 0.458527
\(36\) −97.6220 −0.451954
\(37\) 204.476 0.908533 0.454267 0.890866i \(-0.349901\pi\)
0.454267 + 0.890866i \(0.349901\pi\)
\(38\) −329.153 −1.40515
\(39\) −61.7500 −0.253536
\(40\) −40.8300 −0.161395
\(41\) −302.873 −1.15368 −0.576840 0.816857i \(-0.695714\pi\)
−0.576840 + 0.816857i \(0.695714\pi\)
\(42\) −131.021 −0.481356
\(43\) 0 0
\(44\) −32.8540 −0.112566
\(45\) 72.7166 0.240888
\(46\) 557.532 1.78704
\(47\) −79.5496 −0.246883 −0.123442 0.992352i \(-0.539393\pi\)
−0.123442 + 0.992352i \(0.539393\pi\)
\(48\) 90.8669 0.273240
\(49\) 783.738 2.28495
\(50\) 401.860 1.13663
\(51\) 87.1163 0.239191
\(52\) 206.333 0.550254
\(53\) 445.730 1.15520 0.577601 0.816319i \(-0.303989\pi\)
0.577601 + 0.816319i \(0.303989\pi\)
\(54\) −205.736 −0.518465
\(55\) 24.4722 0.0599970
\(56\) −484.546 −1.15625
\(57\) −108.904 −0.253066
\(58\) −61.2204 −0.138597
\(59\) −219.828 −0.485070 −0.242535 0.970143i \(-0.577979\pi\)
−0.242535 + 0.970143i \(0.577979\pi\)
\(60\) 12.2058 0.0262626
\(61\) 116.310 0.244130 0.122065 0.992522i \(-0.461048\pi\)
0.122065 + 0.992522i \(0.461048\pi\)
\(62\) −464.693 −0.951873
\(63\) 862.957 1.72575
\(64\) 93.0220 0.181684
\(65\) −153.693 −0.293281
\(66\) −33.7712 −0.0629841
\(67\) 859.592 1.56740 0.783701 0.621139i \(-0.213330\pi\)
0.783701 + 0.621139i \(0.213330\pi\)
\(68\) −291.092 −0.519120
\(69\) 184.467 0.321843
\(70\) −326.105 −0.556815
\(71\) 1028.88 1.71979 0.859897 0.510468i \(-0.170528\pi\)
0.859897 + 0.510468i \(0.170528\pi\)
\(72\) −371.108 −0.607438
\(73\) 136.880 0.219461 0.109730 0.993961i \(-0.465001\pi\)
0.109730 + 0.993961i \(0.465001\pi\)
\(74\) −702.318 −1.10328
\(75\) 132.961 0.204706
\(76\) 363.896 0.549233
\(77\) 290.422 0.429826
\(78\) 212.094 0.307883
\(79\) −584.277 −0.832105 −0.416053 0.909341i \(-0.636587\pi\)
−0.416053 + 0.909341i \(0.636587\pi\)
\(80\) 226.164 0.316073
\(81\) 626.060 0.858793
\(82\) 1040.28 1.40098
\(83\) 27.3094 0.0361156 0.0180578 0.999837i \(-0.494252\pi\)
0.0180578 + 0.999837i \(0.494252\pi\)
\(84\) 144.851 0.188149
\(85\) 216.829 0.276687
\(86\) 0 0
\(87\) −20.2556 −0.0249612
\(88\) −124.894 −0.151292
\(89\) −1120.94 −1.33506 −0.667528 0.744585i \(-0.732647\pi\)
−0.667528 + 0.744585i \(0.732647\pi\)
\(90\) −249.761 −0.292523
\(91\) −1823.94 −2.10110
\(92\) −616.381 −0.698502
\(93\) −153.750 −0.171431
\(94\) 273.230 0.299804
\(95\) −271.058 −0.292737
\(96\) −180.866 −0.192287
\(97\) 643.857 0.673956 0.336978 0.941513i \(-0.390595\pi\)
0.336978 + 0.941513i \(0.390595\pi\)
\(98\) −2691.92 −2.77474
\(99\) 222.431 0.225810
\(100\) −444.278 −0.444278
\(101\) 1533.90 1.51118 0.755590 0.655045i \(-0.227350\pi\)
0.755590 + 0.655045i \(0.227350\pi\)
\(102\) −299.220 −0.290462
\(103\) −1838.68 −1.75893 −0.879467 0.475961i \(-0.842101\pi\)
−0.879467 + 0.475961i \(0.842101\pi\)
\(104\) 784.371 0.739556
\(105\) −107.896 −0.100282
\(106\) −1530.96 −1.40283
\(107\) 596.356 0.538803 0.269402 0.963028i \(-0.413174\pi\)
0.269402 + 0.963028i \(0.413174\pi\)
\(108\) 227.452 0.202654
\(109\) −1348.07 −1.18460 −0.592300 0.805718i \(-0.701780\pi\)
−0.592300 + 0.805718i \(0.701780\pi\)
\(110\) −84.0550 −0.0728576
\(111\) −232.371 −0.198700
\(112\) 2683.97 2.26439
\(113\) 1033.37 0.860275 0.430137 0.902763i \(-0.358465\pi\)
0.430137 + 0.902763i \(0.358465\pi\)
\(114\) 374.056 0.307312
\(115\) 459.129 0.372296
\(116\) 67.6824 0.0541737
\(117\) −1396.93 −1.10382
\(118\) 755.045 0.589047
\(119\) 2573.19 1.98222
\(120\) 46.4000 0.0352977
\(121\) −1256.14 −0.943759
\(122\) −399.490 −0.296460
\(123\) 344.191 0.252314
\(124\) 513.743 0.372060
\(125\) 684.495 0.489785
\(126\) −2964.01 −2.09567
\(127\) 1189.24 0.830928 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(128\) −1592.74 −1.09984
\(129\) 0 0
\(130\) 527.891 0.356147
\(131\) 1423.04 0.949094 0.474547 0.880230i \(-0.342612\pi\)
0.474547 + 0.880230i \(0.342612\pi\)
\(132\) 37.3359 0.0246187
\(133\) −3216.76 −2.09720
\(134\) −2952.45 −1.90338
\(135\) −169.424 −0.108013
\(136\) −1106.58 −0.697711
\(137\) −565.231 −0.352488 −0.176244 0.984346i \(-0.556395\pi\)
−0.176244 + 0.984346i \(0.556395\pi\)
\(138\) −633.591 −0.390832
\(139\) 26.2087 0.0159928 0.00799638 0.999968i \(-0.497455\pi\)
0.00799638 + 0.999968i \(0.497455\pi\)
\(140\) 360.527 0.217643
\(141\) 90.4018 0.0539943
\(142\) −3533.90 −2.08844
\(143\) −470.127 −0.274923
\(144\) 2055.63 1.18960
\(145\) −50.4152 −0.0288742
\(146\) −470.144 −0.266503
\(147\) −890.655 −0.499728
\(148\) 776.450 0.431242
\(149\) 1937.30 1.06517 0.532583 0.846378i \(-0.321222\pi\)
0.532583 + 0.846378i \(0.321222\pi\)
\(150\) −456.682 −0.248586
\(151\) −290.926 −0.156790 −0.0783948 0.996922i \(-0.524979\pi\)
−0.0783948 + 0.996922i \(0.524979\pi\)
\(152\) 1383.34 0.738184
\(153\) 1970.78 1.04136
\(154\) −997.515 −0.521961
\(155\) −382.676 −0.198305
\(156\) −234.481 −0.120343
\(157\) 3041.68 1.54619 0.773097 0.634288i \(-0.218706\pi\)
0.773097 + 0.634288i \(0.218706\pi\)
\(158\) 2006.82 1.01047
\(159\) −506.537 −0.252648
\(160\) −450.167 −0.222430
\(161\) 5448.67 2.66718
\(162\) −2150.34 −1.04288
\(163\) 1241.97 0.596801 0.298401 0.954441i \(-0.403547\pi\)
0.298401 + 0.954441i \(0.403547\pi\)
\(164\) −1150.09 −0.547603
\(165\) −27.8107 −0.0131216
\(166\) −93.7998 −0.0438571
\(167\) 1158.03 0.536593 0.268296 0.963336i \(-0.413539\pi\)
0.268296 + 0.963336i \(0.413539\pi\)
\(168\) 550.647 0.252877
\(169\) 755.542 0.343897
\(170\) −744.744 −0.335996
\(171\) −2463.68 −1.10177
\(172\) 0 0
\(173\) 773.746 0.340039 0.170020 0.985441i \(-0.445617\pi\)
0.170020 + 0.985441i \(0.445617\pi\)
\(174\) 69.5721 0.0303117
\(175\) 3927.32 1.69644
\(176\) 691.806 0.296289
\(177\) 249.816 0.106087
\(178\) 3850.12 1.62123
\(179\) 2989.47 1.24829 0.624143 0.781310i \(-0.285448\pi\)
0.624143 + 0.781310i \(0.285448\pi\)
\(180\) 276.124 0.114339
\(181\) 3175.06 1.30387 0.651934 0.758276i \(-0.273958\pi\)
0.651934 + 0.758276i \(0.273958\pi\)
\(182\) 6264.70 2.55148
\(183\) −132.176 −0.0533922
\(184\) −2343.16 −0.938806
\(185\) −578.361 −0.229848
\(186\) 528.087 0.208178
\(187\) 663.251 0.259368
\(188\) −302.071 −0.117185
\(189\) −2010.63 −0.773818
\(190\) 931.008 0.355486
\(191\) −3155.65 −1.19547 −0.597735 0.801693i \(-0.703933\pi\)
−0.597735 + 0.801693i \(0.703933\pi\)
\(192\) −105.712 −0.0397350
\(193\) −295.876 −0.110350 −0.0551752 0.998477i \(-0.517572\pi\)
−0.0551752 + 0.998477i \(0.517572\pi\)
\(194\) −2211.46 −0.818422
\(195\) 174.660 0.0641417
\(196\) 2976.06 1.08457
\(197\) −4324.76 −1.56409 −0.782047 0.623219i \(-0.785825\pi\)
−0.782047 + 0.623219i \(0.785825\pi\)
\(198\) −763.986 −0.274213
\(199\) −5148.55 −1.83402 −0.917012 0.398860i \(-0.869406\pi\)
−0.917012 + 0.398860i \(0.869406\pi\)
\(200\) −1688.91 −0.597121
\(201\) −976.857 −0.342797
\(202\) −5268.52 −1.83511
\(203\) −598.297 −0.206858
\(204\) 330.803 0.113534
\(205\) 856.677 0.291868
\(206\) 6315.33 2.13597
\(207\) 4173.08 1.40120
\(208\) −4344.75 −1.44834
\(209\) −829.133 −0.274413
\(210\) 370.592 0.121778
\(211\) 3094.70 1.00971 0.504853 0.863206i \(-0.331547\pi\)
0.504853 + 0.863206i \(0.331547\pi\)
\(212\) 1692.55 0.548325
\(213\) −1169.24 −0.376126
\(214\) −2048.31 −0.654298
\(215\) 0 0
\(216\) 864.655 0.272372
\(217\) −4541.37 −1.42068
\(218\) 4630.22 1.43852
\(219\) −155.553 −0.0479969
\(220\) 92.9273 0.0284780
\(221\) −4165.42 −1.26786
\(222\) 798.128 0.241292
\(223\) 509.504 0.153000 0.0764998 0.997070i \(-0.475626\pi\)
0.0764998 + 0.997070i \(0.475626\pi\)
\(224\) −5342.32 −1.59352
\(225\) 3007.89 0.891227
\(226\) −3549.32 −1.04468
\(227\) −3714.74 −1.08615 −0.543075 0.839684i \(-0.682740\pi\)
−0.543075 + 0.839684i \(0.682740\pi\)
\(228\) −413.538 −0.120120
\(229\) 2358.34 0.680538 0.340269 0.940328i \(-0.389482\pi\)
0.340269 + 0.940328i \(0.389482\pi\)
\(230\) −1576.98 −0.452099
\(231\) −330.041 −0.0940047
\(232\) 257.293 0.0728110
\(233\) −1098.23 −0.308787 −0.154393 0.988009i \(-0.549342\pi\)
−0.154393 + 0.988009i \(0.549342\pi\)
\(234\) 4798.07 1.34042
\(235\) 225.006 0.0624586
\(236\) −834.742 −0.230242
\(237\) 663.984 0.181985
\(238\) −8838.18 −2.40712
\(239\) 1499.69 0.405886 0.202943 0.979191i \(-0.434949\pi\)
0.202943 + 0.979191i \(0.434949\pi\)
\(240\) −257.017 −0.0691265
\(241\) −2578.81 −0.689278 −0.344639 0.938735i \(-0.611999\pi\)
−0.344639 + 0.938735i \(0.611999\pi\)
\(242\) 4314.49 1.14606
\(243\) −2328.74 −0.614769
\(244\) 441.657 0.115878
\(245\) −2216.80 −0.578066
\(246\) −1182.20 −0.306399
\(247\) 5207.21 1.34140
\(248\) 1952.99 0.500059
\(249\) −31.0349 −0.00789862
\(250\) −2351.05 −0.594773
\(251\) −3641.51 −0.915738 −0.457869 0.889020i \(-0.651387\pi\)
−0.457869 + 0.889020i \(0.651387\pi\)
\(252\) 3276.87 0.819140
\(253\) 1404.42 0.348992
\(254\) −4084.69 −1.00904
\(255\) −246.408 −0.0605125
\(256\) 4726.42 1.15391
\(257\) 149.473 0.0362796 0.0181398 0.999835i \(-0.494226\pi\)
0.0181398 + 0.999835i \(0.494226\pi\)
\(258\) 0 0
\(259\) −6863.64 −1.64666
\(260\) −583.612 −0.139208
\(261\) −458.230 −0.108673
\(262\) −4887.72 −1.15254
\(263\) −502.011 −0.117701 −0.0588504 0.998267i \(-0.518743\pi\)
−0.0588504 + 0.998267i \(0.518743\pi\)
\(264\) 141.932 0.0330882
\(265\) −1260.75 −0.292253
\(266\) 11048.6 2.54675
\(267\) 1273.86 0.291982
\(268\) 3264.09 0.743978
\(269\) −6223.49 −1.41061 −0.705303 0.708906i \(-0.749189\pi\)
−0.705303 + 0.708906i \(0.749189\pi\)
\(270\) 581.924 0.131166
\(271\) 2572.19 0.576565 0.288283 0.957545i \(-0.406916\pi\)
0.288283 + 0.957545i \(0.406916\pi\)
\(272\) 6129.53 1.36639
\(273\) 2072.76 0.459520
\(274\) 1941.41 0.428046
\(275\) 1012.28 0.221974
\(276\) 700.468 0.152765
\(277\) 6947.90 1.50707 0.753536 0.657407i \(-0.228347\pi\)
0.753536 + 0.657407i \(0.228347\pi\)
\(278\) −90.0194 −0.0194209
\(279\) −3478.19 −0.746358
\(280\) 1370.54 0.292519
\(281\) −2013.18 −0.427388 −0.213694 0.976901i \(-0.568550\pi\)
−0.213694 + 0.976901i \(0.568550\pi\)
\(282\) −310.504 −0.0655683
\(283\) 4743.92 0.996455 0.498227 0.867046i \(-0.333984\pi\)
0.498227 + 0.867046i \(0.333984\pi\)
\(284\) 3906.92 0.816313
\(285\) 308.036 0.0640227
\(286\) 1614.75 0.333854
\(287\) 10166.5 2.09098
\(288\) −4091.62 −0.837157
\(289\) 963.534 0.196119
\(290\) 173.162 0.0350635
\(291\) −731.691 −0.147397
\(292\) 519.770 0.104169
\(293\) −7007.19 −1.39715 −0.698575 0.715537i \(-0.746182\pi\)
−0.698575 + 0.715537i \(0.746182\pi\)
\(294\) 3059.15 0.606847
\(295\) 621.782 0.122717
\(296\) 2951.66 0.579601
\(297\) −518.248 −0.101252
\(298\) −6654.06 −1.29349
\(299\) −8820.18 −1.70597
\(300\) 504.886 0.0971654
\(301\) 0 0
\(302\) 999.248 0.190398
\(303\) −1743.16 −0.330501
\(304\) −7662.56 −1.44565
\(305\) −328.981 −0.0617620
\(306\) −6769.07 −1.26458
\(307\) −1306.12 −0.242815 −0.121407 0.992603i \(-0.538741\pi\)
−0.121407 + 0.992603i \(0.538741\pi\)
\(308\) 1102.81 0.204020
\(309\) 2089.51 0.384686
\(310\) 1314.38 0.240813
\(311\) −4743.40 −0.864866 −0.432433 0.901666i \(-0.642345\pi\)
−0.432433 + 0.901666i \(0.642345\pi\)
\(312\) −891.374 −0.161744
\(313\) −5460.04 −0.986005 −0.493002 0.870028i \(-0.664101\pi\)
−0.493002 + 0.870028i \(0.664101\pi\)
\(314\) −10447.3 −1.87763
\(315\) −2440.87 −0.436595
\(316\) −2218.65 −0.394965
\(317\) 7010.45 1.24210 0.621051 0.783770i \(-0.286706\pi\)
0.621051 + 0.783770i \(0.286706\pi\)
\(318\) 1739.81 0.306804
\(319\) −154.214 −0.0270668
\(320\) −263.113 −0.0459639
\(321\) −677.711 −0.117838
\(322\) −18714.6 −3.23890
\(323\) −7346.28 −1.26550
\(324\) 2377.31 0.407633
\(325\) −6357.44 −1.08507
\(326\) −4265.81 −0.724728
\(327\) 1531.97 0.259077
\(328\) −4372.04 −0.735993
\(329\) 2670.24 0.447462
\(330\) 95.5218 0.0159343
\(331\) 2338.01 0.388244 0.194122 0.980977i \(-0.437814\pi\)
0.194122 + 0.980977i \(0.437814\pi\)
\(332\) 103.701 0.0171425
\(333\) −5256.79 −0.865077
\(334\) −3977.50 −0.651614
\(335\) −2431.35 −0.396535
\(336\) −3050.12 −0.495231
\(337\) 2441.37 0.394628 0.197314 0.980340i \(-0.436778\pi\)
0.197314 + 0.980340i \(0.436778\pi\)
\(338\) −2595.07 −0.417613
\(339\) −1174.34 −0.188146
\(340\) 823.354 0.131331
\(341\) −1170.56 −0.185892
\(342\) 8462.04 1.33794
\(343\) −14794.2 −2.32890
\(344\) 0 0
\(345\) −521.764 −0.0814227
\(346\) −2657.59 −0.412928
\(347\) −2549.11 −0.394361 −0.197180 0.980367i \(-0.563178\pi\)
−0.197180 + 0.980367i \(0.563178\pi\)
\(348\) −76.9156 −0.0118480
\(349\) 4610.07 0.707081 0.353541 0.935419i \(-0.384978\pi\)
0.353541 + 0.935419i \(0.384978\pi\)
\(350\) −13489.2 −2.06008
\(351\) 3254.75 0.494945
\(352\) −1377.01 −0.208507
\(353\) −6868.97 −1.03569 −0.517845 0.855474i \(-0.673266\pi\)
−0.517845 + 0.855474i \(0.673266\pi\)
\(354\) −858.048 −0.128827
\(355\) −2910.18 −0.435088
\(356\) −4256.52 −0.633694
\(357\) −2924.23 −0.433519
\(358\) −10268.0 −1.51586
\(359\) −11827.0 −1.73873 −0.869366 0.494168i \(-0.835473\pi\)
−0.869366 + 0.494168i \(0.835473\pi\)
\(360\) 1049.68 0.153675
\(361\) 2324.62 0.338915
\(362\) −10905.4 −1.58336
\(363\) 1427.51 0.206404
\(364\) −6925.96 −0.997304
\(365\) −387.165 −0.0555210
\(366\) 453.988 0.0648370
\(367\) 10023.2 1.42563 0.712816 0.701351i \(-0.247419\pi\)
0.712816 + 0.701351i \(0.247419\pi\)
\(368\) 12979.1 1.83855
\(369\) 7786.44 1.09850
\(370\) 1986.51 0.279118
\(371\) −14961.8 −2.09374
\(372\) −583.828 −0.0813711
\(373\) −12954.9 −1.79833 −0.899166 0.437608i \(-0.855826\pi\)
−0.899166 + 0.437608i \(0.855826\pi\)
\(374\) −2278.08 −0.314964
\(375\) −777.874 −0.107118
\(376\) −1148.32 −0.157500
\(377\) 968.509 0.132310
\(378\) 6905.92 0.939689
\(379\) −13013.5 −1.76374 −0.881872 0.471488i \(-0.843717\pi\)
−0.881872 + 0.471488i \(0.843717\pi\)
\(380\) −1029.28 −0.138950
\(381\) −1351.47 −0.181727
\(382\) 10838.8 1.45173
\(383\) −5448.90 −0.726960 −0.363480 0.931602i \(-0.618412\pi\)
−0.363480 + 0.931602i \(0.618412\pi\)
\(384\) 1810.02 0.240539
\(385\) −821.457 −0.108741
\(386\) 1016.25 0.134005
\(387\) 0 0
\(388\) 2444.89 0.319898
\(389\) 4972.81 0.648153 0.324077 0.946031i \(-0.394946\pi\)
0.324077 + 0.946031i \(0.394946\pi\)
\(390\) −599.906 −0.0778908
\(391\) 12443.4 1.60944
\(392\) 11313.4 1.45769
\(393\) −1617.17 −0.207571
\(394\) 14854.3 1.89937
\(395\) 1652.63 0.210513
\(396\) 844.627 0.107182
\(397\) 10080.8 1.27441 0.637204 0.770695i \(-0.280091\pi\)
0.637204 + 0.770695i \(0.280091\pi\)
\(398\) 17683.8 2.22716
\(399\) 3655.59 0.458667
\(400\) 9355.16 1.16939
\(401\) −431.785 −0.0537713 −0.0268857 0.999639i \(-0.508559\pi\)
−0.0268857 + 0.999639i \(0.508559\pi\)
\(402\) 3355.23 0.416277
\(403\) 7351.47 0.908691
\(404\) 5824.63 0.717293
\(405\) −1770.81 −0.217265
\(406\) 2054.98 0.251199
\(407\) −1769.13 −0.215461
\(408\) 1257.54 0.152592
\(409\) −15428.6 −1.86526 −0.932632 0.360828i \(-0.882494\pi\)
−0.932632 + 0.360828i \(0.882494\pi\)
\(410\) −2942.44 −0.354431
\(411\) 642.339 0.0770906
\(412\) −6981.93 −0.834891
\(413\) 7378.93 0.879161
\(414\) −14333.3 −1.70156
\(415\) −77.2444 −0.00913682
\(416\) 8648.01 1.01924
\(417\) −29.7841 −0.00349768
\(418\) 2847.84 0.333235
\(419\) −8702.42 −1.01466 −0.507328 0.861753i \(-0.669367\pi\)
−0.507328 + 0.861753i \(0.669367\pi\)
\(420\) −409.710 −0.0475995
\(421\) −4123.21 −0.477323 −0.238661 0.971103i \(-0.576709\pi\)
−0.238661 + 0.971103i \(0.576709\pi\)
\(422\) −10629.4 −1.22614
\(423\) 2045.11 0.235074
\(424\) 6434.21 0.736964
\(425\) 8969.02 1.02367
\(426\) 4015.99 0.456750
\(427\) −3904.15 −0.442471
\(428\) 2264.52 0.255747
\(429\) 534.262 0.0601268
\(430\) 0 0
\(431\) −2310.06 −0.258171 −0.129086 0.991633i \(-0.541204\pi\)
−0.129086 + 0.991633i \(0.541204\pi\)
\(432\) −4789.46 −0.533410
\(433\) −9811.69 −1.08896 −0.544480 0.838774i \(-0.683273\pi\)
−0.544480 + 0.838774i \(0.683273\pi\)
\(434\) 15598.3 1.72522
\(435\) 57.2928 0.00631490
\(436\) −5118.96 −0.562279
\(437\) −15555.6 −1.70280
\(438\) 534.281 0.0582853
\(439\) −8625.01 −0.937698 −0.468849 0.883278i \(-0.655331\pi\)
−0.468849 + 0.883278i \(0.655331\pi\)
\(440\) 353.262 0.0382752
\(441\) −20148.8 −2.17566
\(442\) 14307.0 1.53963
\(443\) −10509.3 −1.12712 −0.563558 0.826076i \(-0.690568\pi\)
−0.563558 + 0.826076i \(0.690568\pi\)
\(444\) −882.373 −0.0943144
\(445\) 3170.59 0.337754
\(446\) −1750.00 −0.185796
\(447\) −2201.58 −0.232956
\(448\) −3122.46 −0.329291
\(449\) 12991.4 1.36549 0.682743 0.730658i \(-0.260787\pi\)
0.682743 + 0.730658i \(0.260787\pi\)
\(450\) −10331.2 −1.08227
\(451\) 2620.47 0.273599
\(452\) 3923.96 0.408336
\(453\) 330.614 0.0342905
\(454\) 12759.1 1.31897
\(455\) 5159.00 0.531555
\(456\) −1572.06 −0.161444
\(457\) 9830.48 1.00624 0.503119 0.864217i \(-0.332186\pi\)
0.503119 + 0.864217i \(0.332186\pi\)
\(458\) −8100.21 −0.826414
\(459\) −4591.77 −0.466940
\(460\) 1743.43 0.176713
\(461\) 13731.6 1.38730 0.693651 0.720311i \(-0.256001\pi\)
0.693651 + 0.720311i \(0.256001\pi\)
\(462\) 1133.60 0.114155
\(463\) 2930.80 0.294181 0.147090 0.989123i \(-0.453009\pi\)
0.147090 + 0.989123i \(0.453009\pi\)
\(464\) −1425.19 −0.142592
\(465\) 434.881 0.0433702
\(466\) 3772.10 0.374977
\(467\) −15928.6 −1.57835 −0.789174 0.614170i \(-0.789491\pi\)
−0.789174 + 0.614170i \(0.789491\pi\)
\(468\) −5304.52 −0.523935
\(469\) −28853.9 −2.84083
\(470\) −772.831 −0.0758469
\(471\) −3456.63 −0.338159
\(472\) −3173.26 −0.309451
\(473\) 0 0
\(474\) −2280.60 −0.220994
\(475\) −11212.2 −1.08306
\(476\) 9771.07 0.940875
\(477\) −11459.1 −1.09995
\(478\) −5150.99 −0.492889
\(479\) 642.583 0.0612952 0.0306476 0.999530i \(-0.490243\pi\)
0.0306476 + 0.999530i \(0.490243\pi\)
\(480\) 511.579 0.0486464
\(481\) 11110.7 1.05323
\(482\) 8857.49 0.837028
\(483\) −6191.98 −0.583323
\(484\) −4769.90 −0.447962
\(485\) −1821.15 −0.170503
\(486\) 7998.56 0.746547
\(487\) 18634.0 1.73386 0.866928 0.498434i \(-0.166091\pi\)
0.866928 + 0.498434i \(0.166091\pi\)
\(488\) 1678.95 0.155743
\(489\) −1411.40 −0.130523
\(490\) 7614.08 0.701977
\(491\) 18317.8 1.68365 0.841823 0.539754i \(-0.181483\pi\)
0.841823 + 0.539754i \(0.181483\pi\)
\(492\) 1306.98 0.119763
\(493\) −1366.36 −0.124823
\(494\) −17885.3 −1.62894
\(495\) −629.145 −0.0571272
\(496\) −10817.9 −0.979310
\(497\) −34536.3 −3.11703
\(498\) 106.596 0.00959173
\(499\) −17631.0 −1.58171 −0.790854 0.612005i \(-0.790363\pi\)
−0.790854 + 0.612005i \(0.790363\pi\)
\(500\) 2599.21 0.232480
\(501\) −1316.01 −0.117355
\(502\) 12507.6 1.11203
\(503\) −8292.71 −0.735096 −0.367548 0.930004i \(-0.619803\pi\)
−0.367548 + 0.930004i \(0.619803\pi\)
\(504\) 12457.0 1.10095
\(505\) −4338.65 −0.382311
\(506\) −4823.78 −0.423801
\(507\) −858.613 −0.0752118
\(508\) 4515.85 0.394406
\(509\) 2838.44 0.247174 0.123587 0.992334i \(-0.460560\pi\)
0.123587 + 0.992334i \(0.460560\pi\)
\(510\) 846.342 0.0734836
\(511\) −4594.65 −0.397760
\(512\) −3492.00 −0.301418
\(513\) 5740.20 0.494027
\(514\) −513.396 −0.0440563
\(515\) 5200.69 0.444990
\(516\) 0 0
\(517\) 688.265 0.0585490
\(518\) 23574.7 1.99963
\(519\) −879.300 −0.0743680
\(520\) −2218.59 −0.187099
\(521\) 10587.5 0.890299 0.445150 0.895456i \(-0.353150\pi\)
0.445150 + 0.895456i \(0.353150\pi\)
\(522\) 1573.89 0.131968
\(523\) 17462.6 1.46001 0.730007 0.683439i \(-0.239517\pi\)
0.730007 + 0.683439i \(0.239517\pi\)
\(524\) 5403.64 0.450494
\(525\) −4463.08 −0.371019
\(526\) 1724.26 0.142931
\(527\) −10371.4 −0.857276
\(528\) −786.182 −0.0647996
\(529\) 14181.7 1.16559
\(530\) 4330.30 0.354899
\(531\) 5651.45 0.461868
\(532\) −12214.9 −0.995454
\(533\) −16457.3 −1.33742
\(534\) −4375.36 −0.354570
\(535\) −1686.79 −0.136311
\(536\) 12408.4 0.999927
\(537\) −3397.29 −0.273005
\(538\) 21375.9 1.71298
\(539\) −6780.91 −0.541882
\(540\) −643.348 −0.0512690
\(541\) 1614.09 0.128272 0.0641360 0.997941i \(-0.479571\pi\)
0.0641360 + 0.997941i \(0.479571\pi\)
\(542\) −8834.72 −0.700154
\(543\) −3608.20 −0.285161
\(544\) −12200.5 −0.961569
\(545\) 3813.00 0.299690
\(546\) −7119.33 −0.558020
\(547\) −22672.4 −1.77222 −0.886109 0.463477i \(-0.846602\pi\)
−0.886109 + 0.463477i \(0.846602\pi\)
\(548\) −2146.33 −0.167311
\(549\) −2990.15 −0.232453
\(550\) −3476.90 −0.269556
\(551\) 1708.10 0.132064
\(552\) 2662.82 0.205321
\(553\) 19612.4 1.50814
\(554\) −23864.0 −1.83012
\(555\) 657.261 0.0502688
\(556\) 99.5212 0.00759108
\(557\) −15228.6 −1.15845 −0.579226 0.815167i \(-0.696645\pi\)
−0.579226 + 0.815167i \(0.696645\pi\)
\(558\) 11946.6 0.906343
\(559\) 0 0
\(560\) −7591.61 −0.572865
\(561\) −753.732 −0.0567247
\(562\) 6914.69 0.519001
\(563\) −4374.53 −0.327468 −0.163734 0.986505i \(-0.552354\pi\)
−0.163734 + 0.986505i \(0.552354\pi\)
\(564\) 343.279 0.0256288
\(565\) −2922.88 −0.217640
\(566\) −16294.0 −1.21005
\(567\) −21014.9 −1.55651
\(568\) 14852.1 1.09715
\(569\) 818.728 0.0603214 0.0301607 0.999545i \(-0.490398\pi\)
0.0301607 + 0.999545i \(0.490398\pi\)
\(570\) −1058.02 −0.0777463
\(571\) −1233.69 −0.0904173 −0.0452087 0.998978i \(-0.514395\pi\)
−0.0452087 + 0.998978i \(0.514395\pi\)
\(572\) −1785.20 −0.130494
\(573\) 3586.14 0.261454
\(574\) −34919.1 −2.53919
\(575\) 18991.7 1.37741
\(576\) −2391.46 −0.172993
\(577\) −7097.43 −0.512080 −0.256040 0.966666i \(-0.582418\pi\)
−0.256040 + 0.966666i \(0.582418\pi\)
\(578\) −3309.46 −0.238158
\(579\) 336.239 0.0241341
\(580\) −191.439 −0.0137053
\(581\) −916.691 −0.0654574
\(582\) 2513.15 0.178992
\(583\) −3856.47 −0.273960
\(584\) 1975.90 0.140005
\(585\) 3951.22 0.279253
\(586\) 24067.7 1.69663
\(587\) −12554.1 −0.882730 −0.441365 0.897328i \(-0.645506\pi\)
−0.441365 + 0.897328i \(0.645506\pi\)
\(588\) −3382.05 −0.237199
\(589\) 12965.3 0.907005
\(590\) −2135.64 −0.149022
\(591\) 4914.75 0.342074
\(592\) −16349.7 −1.13508
\(593\) 6924.59 0.479526 0.239763 0.970831i \(-0.422930\pi\)
0.239763 + 0.970831i \(0.422930\pi\)
\(594\) 1780.03 0.122956
\(595\) −7278.27 −0.501479
\(596\) 7356.42 0.505589
\(597\) 5850.91 0.401108
\(598\) 30294.8 2.07165
\(599\) 16074.6 1.09648 0.548240 0.836321i \(-0.315298\pi\)
0.548240 + 0.836321i \(0.315298\pi\)
\(600\) 1919.32 0.130593
\(601\) 126.013 0.00855269 0.00427635 0.999991i \(-0.498639\pi\)
0.00427635 + 0.999991i \(0.498639\pi\)
\(602\) 0 0
\(603\) −22098.9 −1.49243
\(604\) −1104.72 −0.0744214
\(605\) 3553.00 0.238760
\(606\) 5987.25 0.401346
\(607\) 174.746 0.0116849 0.00584245 0.999983i \(-0.498140\pi\)
0.00584245 + 0.999983i \(0.498140\pi\)
\(608\) 15251.9 1.01735
\(609\) 679.917 0.0452407
\(610\) 1129.96 0.0750010
\(611\) −4322.51 −0.286203
\(612\) 7483.56 0.494289
\(613\) 2579.36 0.169950 0.0849750 0.996383i \(-0.472919\pi\)
0.0849750 + 0.996383i \(0.472919\pi\)
\(614\) 4486.14 0.294863
\(615\) −973.544 −0.0638327
\(616\) 4192.30 0.274209
\(617\) 3453.87 0.225361 0.112680 0.993631i \(-0.464056\pi\)
0.112680 + 0.993631i \(0.464056\pi\)
\(618\) −7176.86 −0.467145
\(619\) 23838.1 1.54787 0.773936 0.633264i \(-0.218285\pi\)
0.773936 + 0.633264i \(0.218285\pi\)
\(620\) −1453.12 −0.0941270
\(621\) −9722.98 −0.628292
\(622\) 16292.2 1.05025
\(623\) 37626.7 2.41971
\(624\) 4937.46 0.316757
\(625\) 12688.9 0.812087
\(626\) 18753.7 1.19736
\(627\) 942.243 0.0600153
\(628\) 11550.0 0.733913
\(629\) −15674.9 −0.993638
\(630\) 8383.69 0.530181
\(631\) −11630.5 −0.733762 −0.366881 0.930268i \(-0.619574\pi\)
−0.366881 + 0.930268i \(0.619574\pi\)
\(632\) −8434.17 −0.530844
\(633\) −3516.88 −0.220827
\(634\) −24078.9 −1.50835
\(635\) −3363.76 −0.210215
\(636\) −1923.45 −0.119921
\(637\) 42586.2 2.64886
\(638\) 529.680 0.0328687
\(639\) −26451.0 −1.63753
\(640\) 4505.05 0.278247
\(641\) 16501.9 1.01683 0.508414 0.861113i \(-0.330232\pi\)
0.508414 + 0.861113i \(0.330232\pi\)
\(642\) 2327.74 0.143098
\(643\) −7764.80 −0.476227 −0.238113 0.971237i \(-0.576529\pi\)
−0.238113 + 0.971237i \(0.576529\pi\)
\(644\) 20690.0 1.26600
\(645\) 0 0
\(646\) 25232.4 1.53677
\(647\) −23206.2 −1.41009 −0.705046 0.709161i \(-0.749074\pi\)
−0.705046 + 0.709161i \(0.749074\pi\)
\(648\) 9037.32 0.547869
\(649\) 1901.95 0.115036
\(650\) 21836.0 1.31766
\(651\) 5160.91 0.310710
\(652\) 4716.08 0.283276
\(653\) −9085.23 −0.544460 −0.272230 0.962232i \(-0.587761\pi\)
−0.272230 + 0.962232i \(0.587761\pi\)
\(654\) −5261.88 −0.314611
\(655\) −4025.06 −0.240110
\(656\) 24217.4 1.44136
\(657\) −3518.99 −0.208963
\(658\) −9171.50 −0.543377
\(659\) −21346.5 −1.26183 −0.630913 0.775854i \(-0.717320\pi\)
−0.630913 + 0.775854i \(0.717320\pi\)
\(660\) −105.604 −0.00622825
\(661\) −998.527 −0.0587567 −0.0293784 0.999568i \(-0.509353\pi\)
−0.0293784 + 0.999568i \(0.509353\pi\)
\(662\) −8030.41 −0.471466
\(663\) 4733.66 0.277286
\(664\) 394.216 0.0230400
\(665\) 9098.59 0.530569
\(666\) 18055.6 1.05051
\(667\) −2893.24 −0.167956
\(668\) 4397.33 0.254698
\(669\) −579.010 −0.0334616
\(670\) 8351.00 0.481534
\(671\) −1006.31 −0.0578961
\(672\) 6071.11 0.348509
\(673\) 24703.1 1.41491 0.707455 0.706758i \(-0.249843\pi\)
0.707455 + 0.706758i \(0.249843\pi\)
\(674\) −8385.40 −0.479219
\(675\) −7008.16 −0.399621
\(676\) 2868.99 0.163233
\(677\) 20119.2 1.14216 0.571082 0.820893i \(-0.306524\pi\)
0.571082 + 0.820893i \(0.306524\pi\)
\(678\) 4033.52 0.228475
\(679\) −21612.3 −1.22151
\(680\) 3129.97 0.176513
\(681\) 4221.51 0.237545
\(682\) 4020.54 0.225739
\(683\) −9869.00 −0.552894 −0.276447 0.961029i \(-0.589157\pi\)
−0.276447 + 0.961029i \(0.589157\pi\)
\(684\) −9355.23 −0.522962
\(685\) 1598.75 0.0891755
\(686\) 50813.9 2.82811
\(687\) −2680.06 −0.148836
\(688\) 0 0
\(689\) 24219.8 1.33919
\(690\) 1792.11 0.0988760
\(691\) −410.966 −0.0226250 −0.0113125 0.999936i \(-0.503601\pi\)
−0.0113125 + 0.999936i \(0.503601\pi\)
\(692\) 2938.11 0.161402
\(693\) −7466.32 −0.409267
\(694\) 8755.45 0.478894
\(695\) −74.1312 −0.00404598
\(696\) −292.393 −0.0159241
\(697\) 23217.8 1.26175
\(698\) −15834.3 −0.858647
\(699\) 1248.05 0.0675329
\(700\) 14913.0 0.805228
\(701\) 9413.69 0.507204 0.253602 0.967309i \(-0.418385\pi\)
0.253602 + 0.967309i \(0.418385\pi\)
\(702\) −11179.1 −0.601039
\(703\) 19595.2 1.05128
\(704\) −804.828 −0.0430868
\(705\) −255.701 −0.0136599
\(706\) 23593.0 1.25769
\(707\) −51488.5 −2.73893
\(708\) 948.617 0.0503548
\(709\) 7139.41 0.378175 0.189087 0.981960i \(-0.439447\pi\)
0.189087 + 0.981960i \(0.439447\pi\)
\(710\) 9995.63 0.528351
\(711\) 15020.9 0.792304
\(712\) −16181.1 −0.851702
\(713\) −21961.2 −1.15351
\(714\) 10043.9 0.526446
\(715\) 1329.75 0.0695524
\(716\) 11351.8 0.592508
\(717\) −1704.27 −0.0887688
\(718\) 40622.4 2.11144
\(719\) 1775.83 0.0921100 0.0460550 0.998939i \(-0.485335\pi\)
0.0460550 + 0.998939i \(0.485335\pi\)
\(720\) −5814.34 −0.300955
\(721\) 61718.7 3.18797
\(722\) −7984.39 −0.411563
\(723\) 2930.61 0.150748
\(724\) 12056.5 0.618891
\(725\) −2085.40 −0.106827
\(726\) −4903.07 −0.250647
\(727\) 12260.4 0.625463 0.312732 0.949842i \(-0.398756\pi\)
0.312732 + 0.949842i \(0.398756\pi\)
\(728\) −26328.9 −1.34040
\(729\) −14257.2 −0.724341
\(730\) 1329.80 0.0674222
\(731\) 0 0
\(732\) −501.908 −0.0253430
\(733\) 7176.95 0.361646 0.180823 0.983516i \(-0.442124\pi\)
0.180823 + 0.983516i \(0.442124\pi\)
\(734\) −34426.8 −1.73122
\(735\) 2519.22 0.126425
\(736\) −25834.3 −1.29384
\(737\) −7437.21 −0.371714
\(738\) −26744.2 −1.33397
\(739\) −14928.3 −0.743092 −0.371546 0.928415i \(-0.621172\pi\)
−0.371546 + 0.928415i \(0.621172\pi\)
\(740\) −2196.19 −0.109099
\(741\) −5917.57 −0.293370
\(742\) 51389.5 2.54254
\(743\) 10031.8 0.495333 0.247666 0.968845i \(-0.420336\pi\)
0.247666 + 0.968845i \(0.420336\pi\)
\(744\) −2219.41 −0.109365
\(745\) −5479.64 −0.269475
\(746\) 44496.3 2.18381
\(747\) −702.084 −0.0343881
\(748\) 2518.54 0.123111
\(749\) −20017.8 −0.976550
\(750\) 2671.77 0.130079
\(751\) −7723.92 −0.375300 −0.187650 0.982236i \(-0.560087\pi\)
−0.187650 + 0.982236i \(0.560087\pi\)
\(752\) 6360.70 0.308445
\(753\) 4138.29 0.200276
\(754\) −3326.55 −0.160671
\(755\) 822.884 0.0396660
\(756\) −7634.87 −0.367298
\(757\) −2723.98 −0.130786 −0.0653928 0.997860i \(-0.520830\pi\)
−0.0653928 + 0.997860i \(0.520830\pi\)
\(758\) 44697.7 2.14181
\(759\) −1596.01 −0.0763261
\(760\) −3912.78 −0.186752
\(761\) 35688.3 1.70000 0.849999 0.526784i \(-0.176602\pi\)
0.849999 + 0.526784i \(0.176602\pi\)
\(762\) 4641.93 0.220681
\(763\) 45250.4 2.14702
\(764\) −11982.8 −0.567439
\(765\) −5574.35 −0.263452
\(766\) 18715.4 0.882788
\(767\) −11944.8 −0.562325
\(768\) −5371.20 −0.252365
\(769\) 8833.25 0.414220 0.207110 0.978318i \(-0.433594\pi\)
0.207110 + 0.978318i \(0.433594\pi\)
\(770\) 2821.47 0.132050
\(771\) −169.864 −0.00793449
\(772\) −1123.52 −0.0523786
\(773\) 15932.9 0.741356 0.370678 0.928761i \(-0.379125\pi\)
0.370678 + 0.928761i \(0.379125\pi\)
\(774\) 0 0
\(775\) −15829.2 −0.733682
\(776\) 9294.21 0.429952
\(777\) 7799.98 0.360132
\(778\) −17080.2 −0.787088
\(779\) −29024.7 −1.33494
\(780\) 663.228 0.0304453
\(781\) −8901.87 −0.407854
\(782\) −42739.6 −1.95443
\(783\) 1067.64 0.0487285
\(784\) −62666.8 −2.85472
\(785\) −8603.39 −0.391169
\(786\) 5554.51 0.252064
\(787\) 28093.0 1.27244 0.636218 0.771510i \(-0.280498\pi\)
0.636218 + 0.771510i \(0.280498\pi\)
\(788\) −16422.2 −0.742409
\(789\) 570.495 0.0257416
\(790\) −5676.30 −0.255638
\(791\) −34687.0 −1.55920
\(792\) 3210.84 0.144056
\(793\) 6319.95 0.283011
\(794\) −34624.6 −1.54758
\(795\) 1432.74 0.0639169
\(796\) −19550.4 −0.870533
\(797\) 31595.5 1.40423 0.702115 0.712064i \(-0.252239\pi\)
0.702115 + 0.712064i \(0.252239\pi\)
\(798\) −12555.9 −0.556985
\(799\) 6098.16 0.270009
\(800\) −18621.0 −0.822939
\(801\) 28817.9 1.27120
\(802\) 1483.06 0.0652975
\(803\) −1184.29 −0.0520457
\(804\) −3709.38 −0.162711
\(805\) −15411.6 −0.674765
\(806\) −25250.2 −1.10347
\(807\) 7072.50 0.308505
\(808\) 22142.2 0.964061
\(809\) −8768.27 −0.381058 −0.190529 0.981682i \(-0.561020\pi\)
−0.190529 + 0.981682i \(0.561020\pi\)
\(810\) 6082.23 0.263837
\(811\) −8266.76 −0.357935 −0.178968 0.983855i \(-0.557276\pi\)
−0.178968 + 0.983855i \(0.557276\pi\)
\(812\) −2271.89 −0.0981868
\(813\) −2923.08 −0.126097
\(814\) 6076.47 0.261646
\(815\) −3512.91 −0.150984
\(816\) −6965.72 −0.298835
\(817\) 0 0
\(818\) 52992.7 2.26509
\(819\) 46890.7 2.00060
\(820\) 3253.02 0.138537
\(821\) −43950.7 −1.86832 −0.934160 0.356855i \(-0.883849\pi\)
−0.934160 + 0.356855i \(0.883849\pi\)
\(822\) −2206.25 −0.0936154
\(823\) 46213.0 1.95733 0.978666 0.205457i \(-0.0658680\pi\)
0.978666 + 0.205457i \(0.0658680\pi\)
\(824\) −26541.7 −1.12212
\(825\) −1150.38 −0.0485467
\(826\) −25344.5 −1.06761
\(827\) 802.687 0.0337511 0.0168755 0.999858i \(-0.494628\pi\)
0.0168755 + 0.999858i \(0.494628\pi\)
\(828\) 15846.3 0.665092
\(829\) 7150.10 0.299558 0.149779 0.988720i \(-0.452144\pi\)
0.149779 + 0.988720i \(0.452144\pi\)
\(830\) 265.313 0.0110953
\(831\) −7895.73 −0.329603
\(832\) 5054.57 0.210620
\(833\) −60080.2 −2.49899
\(834\) 102.300 0.00424742
\(835\) −3275.48 −0.135752
\(836\) −3148.43 −0.130252
\(837\) 8103.93 0.334663
\(838\) 29890.3 1.23215
\(839\) 588.326 0.0242089 0.0121044 0.999927i \(-0.496147\pi\)
0.0121044 + 0.999927i \(0.496147\pi\)
\(840\) −1557.50 −0.0639750
\(841\) −24071.3 −0.986974
\(842\) 14162.0 0.579639
\(843\) 2287.81 0.0934715
\(844\) 11751.4 0.479264
\(845\) −2137.05 −0.0870021
\(846\) −7024.35 −0.285464
\(847\) 42164.8 1.71051
\(848\) −35640.1 −1.44326
\(849\) −5391.08 −0.217929
\(850\) −30806.0 −1.24310
\(851\) −33191.2 −1.33699
\(852\) −4439.90 −0.178531
\(853\) −41492.4 −1.66550 −0.832751 0.553648i \(-0.813235\pi\)
−0.832751 + 0.553648i \(0.813235\pi\)
\(854\) 13409.6 0.537317
\(855\) 6968.52 0.278735
\(856\) 8608.53 0.343731
\(857\) 11068.4 0.441176 0.220588 0.975367i \(-0.429202\pi\)
0.220588 + 0.975367i \(0.429202\pi\)
\(858\) −1835.04 −0.0730153
\(859\) 8924.67 0.354489 0.177244 0.984167i \(-0.443282\pi\)
0.177244 + 0.984167i \(0.443282\pi\)
\(860\) 0 0
\(861\) −11553.4 −0.457306
\(862\) 7934.40 0.313511
\(863\) 5135.14 0.202552 0.101276 0.994858i \(-0.467708\pi\)
0.101276 + 0.994858i \(0.467708\pi\)
\(864\) 9533.18 0.375377
\(865\) −2188.54 −0.0860260
\(866\) 33700.3 1.32238
\(867\) −1094.98 −0.0428921
\(868\) −17244.8 −0.674338
\(869\) 5055.18 0.197336
\(870\) −196.784 −0.00766852
\(871\) 46707.9 1.81703
\(872\) −19459.6 −0.755718
\(873\) −16552.6 −0.641720
\(874\) 53429.0 2.06781
\(875\) −22976.4 −0.887707
\(876\) −590.676 −0.0227821
\(877\) 6291.03 0.242227 0.121114 0.992639i \(-0.461353\pi\)
0.121114 + 0.992639i \(0.461353\pi\)
\(878\) 29624.4 1.13870
\(879\) 7963.11 0.305562
\(880\) −1956.77 −0.0749577
\(881\) 48382.5 1.85022 0.925112 0.379695i \(-0.123971\pi\)
0.925112 + 0.379695i \(0.123971\pi\)
\(882\) 69205.2 2.64202
\(883\) −21310.2 −0.812169 −0.406084 0.913836i \(-0.633106\pi\)
−0.406084 + 0.913836i \(0.633106\pi\)
\(884\) −15817.2 −0.601798
\(885\) −706.605 −0.0268387
\(886\) 36096.5 1.36872
\(887\) −6283.54 −0.237859 −0.118929 0.992903i \(-0.537946\pi\)
−0.118929 + 0.992903i \(0.537946\pi\)
\(888\) −3354.33 −0.126761
\(889\) −39919.1 −1.50601
\(890\) −10890.1 −0.410153
\(891\) −5416.69 −0.203665
\(892\) 1934.72 0.0726224
\(893\) −7623.34 −0.285672
\(894\) 7561.81 0.282891
\(895\) −8455.70 −0.315802
\(896\) 53463.3 1.99340
\(897\) 10023.4 0.373102
\(898\) −44621.8 −1.65818
\(899\) 2411.47 0.0894627
\(900\) 11421.7 0.423027
\(901\) −34169.0 −1.26341
\(902\) −9000.55 −0.332246
\(903\) 0 0
\(904\) 14916.9 0.548814
\(905\) −8980.64 −0.329864
\(906\) −1135.57 −0.0416409
\(907\) 9911.20 0.362840 0.181420 0.983406i \(-0.441931\pi\)
0.181420 + 0.983406i \(0.441931\pi\)
\(908\) −14105.8 −0.515549
\(909\) −39434.5 −1.43890
\(910\) −17719.7 −0.645496
\(911\) −44778.4 −1.62851 −0.814256 0.580505i \(-0.802855\pi\)
−0.814256 + 0.580505i \(0.802855\pi\)
\(912\) 8707.88 0.316170
\(913\) −236.281 −0.00856491
\(914\) −33764.9 −1.22193
\(915\) 373.861 0.0135076
\(916\) 8955.21 0.323022
\(917\) −47767.0 −1.72018
\(918\) 15771.4 0.567031
\(919\) −9209.38 −0.330565 −0.165283 0.986246i \(-0.552854\pi\)
−0.165283 + 0.986246i \(0.552854\pi\)
\(920\) 6627.63 0.237507
\(921\) 1484.30 0.0531046
\(922\) −47164.3 −1.68468
\(923\) 55906.4 1.99370
\(924\) −1253.25 −0.0446200
\(925\) −23923.7 −0.850384
\(926\) −10066.5 −0.357240
\(927\) 47269.7 1.67480
\(928\) 2836.77 0.100346
\(929\) −37773.3 −1.33402 −0.667009 0.745049i \(-0.732426\pi\)
−0.667009 + 0.745049i \(0.732426\pi\)
\(930\) −1493.69 −0.0526668
\(931\) 75106.5 2.64395
\(932\) −4170.25 −0.146568
\(933\) 5390.49 0.189150
\(934\) 54710.2 1.91667
\(935\) −1876.00 −0.0656170
\(936\) −20165.0 −0.704182
\(937\) −7402.89 −0.258102 −0.129051 0.991638i \(-0.541193\pi\)
−0.129051 + 0.991638i \(0.541193\pi\)
\(938\) 99104.7 3.44977
\(939\) 6204.89 0.215643
\(940\) 854.406 0.0296464
\(941\) −29080.2 −1.00742 −0.503712 0.863872i \(-0.668033\pi\)
−0.503712 + 0.863872i \(0.668033\pi\)
\(942\) 11872.5 0.410645
\(943\) 49163.2 1.69775
\(944\) 17577.2 0.606026
\(945\) 5687.05 0.195767
\(946\) 0 0
\(947\) −26077.7 −0.894837 −0.447419 0.894325i \(-0.647657\pi\)
−0.447419 + 0.894325i \(0.647657\pi\)
\(948\) 2521.32 0.0863804
\(949\) 7437.70 0.254413
\(950\) 38510.7 1.31521
\(951\) −7966.81 −0.271653
\(952\) 37144.6 1.26456
\(953\) 31004.8 1.05388 0.526939 0.849903i \(-0.323340\pi\)
0.526939 + 0.849903i \(0.323340\pi\)
\(954\) 39358.7 1.33573
\(955\) 8925.75 0.302440
\(956\) 5694.70 0.192657
\(957\) 175.251 0.00591962
\(958\) −2207.09 −0.0744341
\(959\) 18973.0 0.638865
\(960\) 299.006 0.0100525
\(961\) −11486.8 −0.385578
\(962\) −38162.1 −1.27900
\(963\) −15331.5 −0.513032
\(964\) −9792.42 −0.327171
\(965\) 836.885 0.0279174
\(966\) 21267.7 0.708361
\(967\) 8359.12 0.277985 0.138992 0.990293i \(-0.455614\pi\)
0.138992 + 0.990293i \(0.455614\pi\)
\(968\) −18132.7 −0.602073
\(969\) 8348.46 0.276771
\(970\) 6255.12 0.207051
\(971\) 23620.0 0.780641 0.390321 0.920679i \(-0.372364\pi\)
0.390321 + 0.920679i \(0.372364\pi\)
\(972\) −8842.83 −0.291805
\(973\) −879.745 −0.0289860
\(974\) −64002.5 −2.10552
\(975\) 7224.72 0.237309
\(976\) −9299.98 −0.305005
\(977\) 20068.3 0.657156 0.328578 0.944477i \(-0.393431\pi\)
0.328578 + 0.944477i \(0.393431\pi\)
\(978\) 4847.75 0.158501
\(979\) 9698.43 0.316612
\(980\) −8417.76 −0.274383
\(981\) 34656.8 1.12794
\(982\) −62916.3 −2.04454
\(983\) 26290.9 0.853052 0.426526 0.904475i \(-0.359737\pi\)
0.426526 + 0.904475i \(0.359737\pi\)
\(984\) 4968.47 0.160965
\(985\) 12232.6 0.395698
\(986\) 4693.07 0.151580
\(987\) −3034.51 −0.0978617
\(988\) 19773.1 0.636707
\(989\) 0 0
\(990\) 2160.93 0.0693727
\(991\) −21747.2 −0.697097 −0.348549 0.937291i \(-0.613325\pi\)
−0.348549 + 0.937291i \(0.613325\pi\)
\(992\) 21532.5 0.689170
\(993\) −2656.96 −0.0849106
\(994\) 118622. 3.78518
\(995\) 14562.6 0.463987
\(996\) −117.847 −0.00374914
\(997\) 39252.8 1.24689 0.623445 0.781868i \(-0.285733\pi\)
0.623445 + 0.781868i \(0.285733\pi\)
\(998\) 60557.5 1.92075
\(999\) 12247.9 0.387896
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 1849.4.a.i.1.13 50
43.42 odd 2 1849.4.a.j.1.38 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.13 50 1.1 even 1 trivial
1849.4.a.j.1.38 yes 50 43.42 odd 2