Properties

Label 1849.4.a.j.1.27
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 1.27
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.902811 q^{2} +9.58399 q^{3} -7.18493 q^{4} +2.03345 q^{5} +8.65253 q^{6} -26.8278 q^{7} -13.7091 q^{8} +64.8528 q^{9} +O(q^{10})\) \(q+0.902811 q^{2} +9.58399 q^{3} -7.18493 q^{4} +2.03345 q^{5} +8.65253 q^{6} -26.8278 q^{7} -13.7091 q^{8} +64.8528 q^{9} +1.83582 q^{10} +38.6799 q^{11} -68.8603 q^{12} -37.4866 q^{13} -24.2205 q^{14} +19.4886 q^{15} +45.1027 q^{16} -12.0881 q^{17} +58.5498 q^{18} +64.0712 q^{19} -14.6102 q^{20} -257.118 q^{21} +34.9206 q^{22} -85.2739 q^{23} -131.388 q^{24} -120.865 q^{25} -33.8433 q^{26} +362.781 q^{27} +192.756 q^{28} -129.469 q^{29} +17.5945 q^{30} +261.836 q^{31} +150.392 q^{32} +370.707 q^{33} -10.9133 q^{34} -54.5531 q^{35} -465.963 q^{36} -310.147 q^{37} +57.8442 q^{38} -359.271 q^{39} -27.8768 q^{40} -333.686 q^{41} -232.129 q^{42} -277.912 q^{44} +131.875 q^{45} -76.9862 q^{46} -475.663 q^{47} +432.264 q^{48} +376.733 q^{49} -109.118 q^{50} -115.852 q^{51} +269.339 q^{52} +14.1574 q^{53} +327.523 q^{54} +78.6536 q^{55} +367.786 q^{56} +614.057 q^{57} -116.886 q^{58} -652.202 q^{59} -140.024 q^{60} +233.602 q^{61} +236.388 q^{62} -1739.86 q^{63} -225.046 q^{64} -76.2271 q^{65} +334.679 q^{66} +963.472 q^{67} +86.8523 q^{68} -817.264 q^{69} -49.2511 q^{70} -887.185 q^{71} -889.075 q^{72} +345.150 q^{73} -280.004 q^{74} -1158.37 q^{75} -460.347 q^{76} -1037.70 q^{77} -324.354 q^{78} -396.269 q^{79} +91.7141 q^{80} +1725.86 q^{81} -301.255 q^{82} +539.877 q^{83} +1847.37 q^{84} -24.5806 q^{85} -1240.83 q^{87} -530.267 q^{88} -295.412 q^{89} +119.058 q^{90} +1005.68 q^{91} +612.687 q^{92} +2509.43 q^{93} -429.433 q^{94} +130.286 q^{95} +1441.36 q^{96} +191.963 q^{97} +340.119 q^{98} +2508.50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.902811 0.319192 0.159596 0.987182i \(-0.448981\pi\)
0.159596 + 0.987182i \(0.448981\pi\)
\(3\) 9.58399 1.84444 0.922220 0.386667i \(-0.126374\pi\)
0.922220 + 0.386667i \(0.126374\pi\)
\(4\) −7.18493 −0.898117
\(5\) 2.03345 0.181877 0.0909387 0.995856i \(-0.471013\pi\)
0.0909387 + 0.995856i \(0.471013\pi\)
\(6\) 8.65253 0.588730
\(7\) −26.8278 −1.44857 −0.724284 0.689502i \(-0.757829\pi\)
−0.724284 + 0.689502i \(0.757829\pi\)
\(8\) −13.7091 −0.605863
\(9\) 64.8528 2.40196
\(10\) 1.83582 0.0580538
\(11\) 38.6799 1.06022 0.530110 0.847929i \(-0.322151\pi\)
0.530110 + 0.847929i \(0.322151\pi\)
\(12\) −68.8603 −1.65652
\(13\) −37.4866 −0.799762 −0.399881 0.916567i \(-0.630949\pi\)
−0.399881 + 0.916567i \(0.630949\pi\)
\(14\) −24.2205 −0.462371
\(15\) 19.4886 0.335462
\(16\) 45.1027 0.704730
\(17\) −12.0881 −0.172459 −0.0862294 0.996275i \(-0.527482\pi\)
−0.0862294 + 0.996275i \(0.527482\pi\)
\(18\) 58.5498 0.766685
\(19\) 64.0712 0.773628 0.386814 0.922158i \(-0.373576\pi\)
0.386814 + 0.922158i \(0.373576\pi\)
\(20\) −14.6102 −0.163347
\(21\) −257.118 −2.67179
\(22\) 34.9206 0.338414
\(23\) −85.2739 −0.773080 −0.386540 0.922273i \(-0.626330\pi\)
−0.386540 + 0.922273i \(0.626330\pi\)
\(24\) −131.388 −1.11748
\(25\) −120.865 −0.966921
\(26\) −33.8433 −0.255278
\(27\) 362.781 2.58582
\(28\) 192.756 1.30098
\(29\) −129.469 −0.829029 −0.414514 0.910043i \(-0.636049\pi\)
−0.414514 + 0.910043i \(0.636049\pi\)
\(30\) 17.5945 0.107077
\(31\) 261.836 1.51700 0.758501 0.651671i \(-0.225932\pi\)
0.758501 + 0.651671i \(0.225932\pi\)
\(32\) 150.392 0.830808
\(33\) 370.707 1.95551
\(34\) −10.9133 −0.0550474
\(35\) −54.5531 −0.263462
\(36\) −465.963 −2.15724
\(37\) −310.147 −1.37805 −0.689024 0.724739i \(-0.741961\pi\)
−0.689024 + 0.724739i \(0.741961\pi\)
\(38\) 57.8442 0.246936
\(39\) −359.271 −1.47511
\(40\) −27.8768 −0.110193
\(41\) −333.686 −1.27105 −0.635524 0.772081i \(-0.719216\pi\)
−0.635524 + 0.772081i \(0.719216\pi\)
\(42\) −232.129 −0.852815
\(43\) 0 0
\(44\) −277.912 −0.952201
\(45\) 131.875 0.436861
\(46\) −76.9862 −0.246761
\(47\) −475.663 −1.47622 −0.738112 0.674678i \(-0.764282\pi\)
−0.738112 + 0.674678i \(0.764282\pi\)
\(48\) 432.264 1.29983
\(49\) 376.733 1.09835
\(50\) −109.118 −0.308633
\(51\) −115.852 −0.318090
\(52\) 269.339 0.718280
\(53\) 14.1574 0.0366919 0.0183460 0.999832i \(-0.494160\pi\)
0.0183460 + 0.999832i \(0.494160\pi\)
\(54\) 327.523 0.825374
\(55\) 78.6536 0.192830
\(56\) 367.786 0.877634
\(57\) 614.057 1.42691
\(58\) −116.886 −0.264619
\(59\) −652.202 −1.43914 −0.719571 0.694419i \(-0.755662\pi\)
−0.719571 + 0.694419i \(0.755662\pi\)
\(60\) −140.024 −0.301284
\(61\) 233.602 0.490323 0.245162 0.969482i \(-0.421159\pi\)
0.245162 + 0.969482i \(0.421159\pi\)
\(62\) 236.388 0.484215
\(63\) −1739.86 −3.47939
\(64\) −225.046 −0.439543
\(65\) −76.2271 −0.145459
\(66\) 334.679 0.624183
\(67\) 963.472 1.75682 0.878409 0.477909i \(-0.158605\pi\)
0.878409 + 0.477909i \(0.158605\pi\)
\(68\) 86.8523 0.154888
\(69\) −817.264 −1.42590
\(70\) −49.2511 −0.0840948
\(71\) −887.185 −1.48295 −0.741475 0.670980i \(-0.765874\pi\)
−0.741475 + 0.670980i \(0.765874\pi\)
\(72\) −889.075 −1.45526
\(73\) 345.150 0.553380 0.276690 0.960959i \(-0.410762\pi\)
0.276690 + 0.960959i \(0.410762\pi\)
\(74\) −280.004 −0.439862
\(75\) −1158.37 −1.78343
\(76\) −460.347 −0.694808
\(77\) −1037.70 −1.53580
\(78\) −324.354 −0.470844
\(79\) −396.269 −0.564351 −0.282176 0.959363i \(-0.591056\pi\)
−0.282176 + 0.959363i \(0.591056\pi\)
\(80\) 91.7141 0.128174
\(81\) 1725.86 2.36744
\(82\) −301.255 −0.405708
\(83\) 539.877 0.713966 0.356983 0.934111i \(-0.383805\pi\)
0.356983 + 0.934111i \(0.383805\pi\)
\(84\) 1847.37 2.39958
\(85\) −24.5806 −0.0313663
\(86\) 0 0
\(87\) −1240.83 −1.52909
\(88\) −530.267 −0.642349
\(89\) −295.412 −0.351838 −0.175919 0.984405i \(-0.556290\pi\)
−0.175919 + 0.984405i \(0.556290\pi\)
\(90\) 119.058 0.139443
\(91\) 1005.68 1.15851
\(92\) 612.687 0.694316
\(93\) 2509.43 2.79802
\(94\) −429.433 −0.471199
\(95\) 130.286 0.140705
\(96\) 1441.36 1.53237
\(97\) 191.963 0.200937 0.100469 0.994940i \(-0.467966\pi\)
0.100469 + 0.994940i \(0.467966\pi\)
\(98\) 340.119 0.350584
\(99\) 2508.50 2.54660
\(100\) 868.407 0.868407
\(101\) −1717.20 −1.69176 −0.845878 0.533377i \(-0.820923\pi\)
−0.845878 + 0.533377i \(0.820923\pi\)
\(102\) −104.593 −0.101532
\(103\) −31.9437 −0.0305583 −0.0152792 0.999883i \(-0.504864\pi\)
−0.0152792 + 0.999883i \(0.504864\pi\)
\(104\) 513.908 0.484547
\(105\) −522.836 −0.485939
\(106\) 12.7815 0.0117118
\(107\) 1275.01 1.15196 0.575980 0.817463i \(-0.304620\pi\)
0.575980 + 0.817463i \(0.304620\pi\)
\(108\) −2606.56 −2.32237
\(109\) −1181.87 −1.03856 −0.519278 0.854606i \(-0.673799\pi\)
−0.519278 + 0.854606i \(0.673799\pi\)
\(110\) 71.0094 0.0615498
\(111\) −2972.44 −2.54173
\(112\) −1210.01 −1.02085
\(113\) −1007.21 −0.838495 −0.419248 0.907872i \(-0.637706\pi\)
−0.419248 + 0.907872i \(0.637706\pi\)
\(114\) 554.378 0.455458
\(115\) −173.400 −0.140606
\(116\) 930.228 0.744564
\(117\) −2431.11 −1.92099
\(118\) −588.815 −0.459363
\(119\) 324.298 0.249818
\(120\) −267.171 −0.203244
\(121\) 165.132 0.124066
\(122\) 210.899 0.156507
\(123\) −3198.04 −2.34437
\(124\) −1881.27 −1.36245
\(125\) −499.955 −0.357738
\(126\) −1570.77 −1.11059
\(127\) −694.035 −0.484926 −0.242463 0.970161i \(-0.577955\pi\)
−0.242463 + 0.970161i \(0.577955\pi\)
\(128\) −1406.31 −0.971106
\(129\) 0 0
\(130\) −68.8187 −0.0464292
\(131\) 1014.00 0.676286 0.338143 0.941095i \(-0.390201\pi\)
0.338143 + 0.941095i \(0.390201\pi\)
\(132\) −2663.51 −1.75628
\(133\) −1718.89 −1.12065
\(134\) 869.833 0.560762
\(135\) 737.697 0.470303
\(136\) 165.717 0.104486
\(137\) −633.186 −0.394867 −0.197433 0.980316i \(-0.563261\pi\)
−0.197433 + 0.980316i \(0.563261\pi\)
\(138\) −737.835 −0.455135
\(139\) −437.269 −0.266825 −0.133412 0.991061i \(-0.542594\pi\)
−0.133412 + 0.991061i \(0.542594\pi\)
\(140\) 391.960 0.236619
\(141\) −4558.74 −2.72280
\(142\) −800.961 −0.473346
\(143\) −1449.98 −0.847924
\(144\) 2925.04 1.69273
\(145\) −263.269 −0.150782
\(146\) 311.605 0.176635
\(147\) 3610.60 2.02583
\(148\) 2228.38 1.23765
\(149\) −2052.33 −1.12841 −0.564205 0.825635i \(-0.690817\pi\)
−0.564205 + 0.825635i \(0.690817\pi\)
\(150\) −1045.79 −0.569255
\(151\) −1021.62 −0.550583 −0.275292 0.961361i \(-0.588774\pi\)
−0.275292 + 0.961361i \(0.588774\pi\)
\(152\) −878.360 −0.468713
\(153\) −783.948 −0.414238
\(154\) −936.845 −0.490215
\(155\) 532.430 0.275909
\(156\) 2581.34 1.32482
\(157\) −183.529 −0.0932942 −0.0466471 0.998911i \(-0.514854\pi\)
−0.0466471 + 0.998911i \(0.514854\pi\)
\(158\) −357.756 −0.180136
\(159\) 135.685 0.0676760
\(160\) 305.815 0.151105
\(161\) 2287.71 1.11986
\(162\) 1558.13 0.755666
\(163\) −832.248 −0.399918 −0.199959 0.979804i \(-0.564081\pi\)
−0.199959 + 0.979804i \(0.564081\pi\)
\(164\) 2397.51 1.14155
\(165\) 753.815 0.355663
\(166\) 487.407 0.227892
\(167\) −319.899 −0.148231 −0.0741153 0.997250i \(-0.523613\pi\)
−0.0741153 + 0.997250i \(0.523613\pi\)
\(168\) 3524.86 1.61874
\(169\) −791.756 −0.360380
\(170\) −22.1916 −0.0100119
\(171\) 4155.20 1.85822
\(172\) 0 0
\(173\) −3014.11 −1.32462 −0.662309 0.749231i \(-0.730423\pi\)
−0.662309 + 0.749231i \(0.730423\pi\)
\(174\) −1120.24 −0.488074
\(175\) 3242.55 1.40065
\(176\) 1744.57 0.747169
\(177\) −6250.69 −2.65441
\(178\) −266.701 −0.112304
\(179\) −3477.58 −1.45211 −0.726053 0.687639i \(-0.758647\pi\)
−0.726053 + 0.687639i \(0.758647\pi\)
\(180\) −947.513 −0.392352
\(181\) 7.73635 0.00317701 0.00158850 0.999999i \(-0.499494\pi\)
0.00158850 + 0.999999i \(0.499494\pi\)
\(182\) 907.943 0.369787
\(183\) 2238.84 0.904372
\(184\) 1169.03 0.468381
\(185\) −630.668 −0.250636
\(186\) 2265.54 0.893105
\(187\) −467.567 −0.182844
\(188\) 3417.60 1.32582
\(189\) −9732.63 −3.74574
\(190\) 117.623 0.0449121
\(191\) 1301.75 0.493147 0.246574 0.969124i \(-0.420695\pi\)
0.246574 + 0.969124i \(0.420695\pi\)
\(192\) −2156.84 −0.810710
\(193\) −760.523 −0.283646 −0.141823 0.989892i \(-0.545296\pi\)
−0.141823 + 0.989892i \(0.545296\pi\)
\(194\) 173.306 0.0641375
\(195\) −730.560 −0.268290
\(196\) −2706.80 −0.986444
\(197\) −200.537 −0.0725263 −0.0362632 0.999342i \(-0.511545\pi\)
−0.0362632 + 0.999342i \(0.511545\pi\)
\(198\) 2264.70 0.812855
\(199\) 1878.65 0.669217 0.334608 0.942357i \(-0.391396\pi\)
0.334608 + 0.942357i \(0.391396\pi\)
\(200\) 1656.95 0.585822
\(201\) 9233.90 3.24034
\(202\) −1550.30 −0.539995
\(203\) 3473.38 1.20090
\(204\) 832.391 0.285682
\(205\) −678.534 −0.231175
\(206\) −28.8391 −0.00975397
\(207\) −5530.25 −1.85690
\(208\) −1690.75 −0.563616
\(209\) 2478.26 0.820216
\(210\) −472.022 −0.155108
\(211\) 907.815 0.296192 0.148096 0.988973i \(-0.452685\pi\)
0.148096 + 0.988973i \(0.452685\pi\)
\(212\) −101.720 −0.0329536
\(213\) −8502.77 −2.73521
\(214\) 1151.09 0.367697
\(215\) 0 0
\(216\) −4973.41 −1.56666
\(217\) −7024.49 −2.19748
\(218\) −1067.00 −0.331498
\(219\) 3307.91 1.02068
\(220\) −565.121 −0.173184
\(221\) 453.142 0.137926
\(222\) −2683.55 −0.811298
\(223\) 2724.67 0.818193 0.409097 0.912491i \(-0.365844\pi\)
0.409097 + 0.912491i \(0.365844\pi\)
\(224\) −4034.70 −1.20348
\(225\) −7838.44 −2.32250
\(226\) −909.317 −0.267641
\(227\) −2297.13 −0.671655 −0.335828 0.941923i \(-0.609016\pi\)
−0.335828 + 0.941923i \(0.609016\pi\)
\(228\) −4411.96 −1.28153
\(229\) 3034.87 0.875763 0.437882 0.899033i \(-0.355729\pi\)
0.437882 + 0.899033i \(0.355729\pi\)
\(230\) −156.548 −0.0448802
\(231\) −9945.28 −2.83269
\(232\) 1774.91 0.502278
\(233\) −4994.74 −1.40436 −0.702180 0.711999i \(-0.747790\pi\)
−0.702180 + 0.711999i \(0.747790\pi\)
\(234\) −2194.83 −0.613166
\(235\) −967.237 −0.268492
\(236\) 4686.02 1.29252
\(237\) −3797.84 −1.04091
\(238\) 292.780 0.0797399
\(239\) −2939.14 −0.795470 −0.397735 0.917500i \(-0.630204\pi\)
−0.397735 + 0.917500i \(0.630204\pi\)
\(240\) 878.987 0.236410
\(241\) 4981.47 1.33147 0.665736 0.746187i \(-0.268118\pi\)
0.665736 + 0.746187i \(0.268118\pi\)
\(242\) 149.083 0.0396010
\(243\) 6745.54 1.78077
\(244\) −1678.42 −0.440368
\(245\) 766.068 0.199765
\(246\) −2887.23 −0.748304
\(247\) −2401.81 −0.618719
\(248\) −3589.54 −0.919097
\(249\) 5174.17 1.31687
\(250\) −451.365 −0.114187
\(251\) 5028.93 1.26463 0.632317 0.774709i \(-0.282104\pi\)
0.632317 + 0.774709i \(0.282104\pi\)
\(252\) 12500.8 3.12490
\(253\) −3298.38 −0.819634
\(254\) −626.583 −0.154785
\(255\) −235.580 −0.0578533
\(256\) 530.733 0.129573
\(257\) 846.260 0.205402 0.102701 0.994712i \(-0.467252\pi\)
0.102701 + 0.994712i \(0.467252\pi\)
\(258\) 0 0
\(259\) 8320.56 1.99620
\(260\) 547.687 0.130639
\(261\) −8396.44 −1.99129
\(262\) 915.449 0.215865
\(263\) 7725.74 1.81137 0.905684 0.423954i \(-0.139358\pi\)
0.905684 + 0.423954i \(0.139358\pi\)
\(264\) −5082.07 −1.18477
\(265\) 28.7884 0.00667343
\(266\) −1551.83 −0.357703
\(267\) −2831.23 −0.648944
\(268\) −6922.48 −1.57783
\(269\) −1281.93 −0.290559 −0.145279 0.989391i \(-0.546408\pi\)
−0.145279 + 0.989391i \(0.546408\pi\)
\(270\) 666.001 0.150117
\(271\) 1156.22 0.259171 0.129585 0.991568i \(-0.458635\pi\)
0.129585 + 0.991568i \(0.458635\pi\)
\(272\) −545.207 −0.121537
\(273\) 9638.46 2.13680
\(274\) −571.648 −0.126038
\(275\) −4675.05 −1.02515
\(276\) 5871.98 1.28062
\(277\) −2086.27 −0.452533 −0.226266 0.974065i \(-0.572652\pi\)
−0.226266 + 0.974065i \(0.572652\pi\)
\(278\) −394.771 −0.0851684
\(279\) 16980.8 3.64377
\(280\) 747.875 0.159622
\(281\) 1154.08 0.245006 0.122503 0.992468i \(-0.460908\pi\)
0.122503 + 0.992468i \(0.460908\pi\)
\(282\) −4115.68 −0.869097
\(283\) 5288.51 1.11085 0.555423 0.831568i \(-0.312556\pi\)
0.555423 + 0.831568i \(0.312556\pi\)
\(284\) 6374.37 1.33186
\(285\) 1248.66 0.259523
\(286\) −1309.05 −0.270650
\(287\) 8952.07 1.84120
\(288\) 9753.36 1.99556
\(289\) −4766.88 −0.970258
\(290\) −237.683 −0.0481283
\(291\) 1839.77 0.370616
\(292\) −2479.88 −0.497000
\(293\) −3079.73 −0.614060 −0.307030 0.951700i \(-0.599335\pi\)
−0.307030 + 0.951700i \(0.599335\pi\)
\(294\) 3259.69 0.646630
\(295\) −1326.22 −0.261747
\(296\) 4251.84 0.834909
\(297\) 14032.3 2.74154
\(298\) −1852.86 −0.360179
\(299\) 3196.63 0.618280
\(300\) 8322.80 1.60172
\(301\) 0 0
\(302\) −922.328 −0.175742
\(303\) −16457.6 −3.12034
\(304\) 2889.78 0.545199
\(305\) 475.019 0.0891787
\(306\) −707.757 −0.132222
\(307\) −639.484 −0.118884 −0.0594419 0.998232i \(-0.518932\pi\)
−0.0594419 + 0.998232i \(0.518932\pi\)
\(308\) 7455.79 1.37933
\(309\) −306.148 −0.0563630
\(310\) 480.684 0.0880678
\(311\) 7068.39 1.28878 0.644391 0.764696i \(-0.277111\pi\)
0.644391 + 0.764696i \(0.277111\pi\)
\(312\) 4925.29 0.893717
\(313\) −5479.55 −0.989528 −0.494764 0.869027i \(-0.664746\pi\)
−0.494764 + 0.869027i \(0.664746\pi\)
\(314\) −165.692 −0.0297787
\(315\) −3537.92 −0.632823
\(316\) 2847.17 0.506853
\(317\) 6514.68 1.15426 0.577131 0.816652i \(-0.304172\pi\)
0.577131 + 0.816652i \(0.304172\pi\)
\(318\) 122.498 0.0216016
\(319\) −5007.85 −0.878953
\(320\) −457.620 −0.0799429
\(321\) 12219.7 2.12472
\(322\) 2065.37 0.357450
\(323\) −774.500 −0.133419
\(324\) −12400.2 −2.12623
\(325\) 4530.82 0.773307
\(326\) −751.363 −0.127651
\(327\) −11327.0 −1.91555
\(328\) 4574.54 0.770081
\(329\) 12761.0 2.13841
\(330\) 680.553 0.113525
\(331\) 3774.59 0.626799 0.313399 0.949621i \(-0.398532\pi\)
0.313399 + 0.949621i \(0.398532\pi\)
\(332\) −3878.98 −0.641225
\(333\) −20113.9 −3.31001
\(334\) −288.808 −0.0473140
\(335\) 1959.17 0.319526
\(336\) −11596.7 −1.88289
\(337\) 11690.7 1.88972 0.944859 0.327478i \(-0.106199\pi\)
0.944859 + 0.327478i \(0.106199\pi\)
\(338\) −714.806 −0.115031
\(339\) −9653.05 −1.54655
\(340\) 176.610 0.0281706
\(341\) 10127.8 1.60836
\(342\) 3751.36 0.593129
\(343\) −904.985 −0.142462
\(344\) 0 0
\(345\) −1661.87 −0.259339
\(346\) −2721.18 −0.422807
\(347\) 11140.5 1.72349 0.861746 0.507340i \(-0.169371\pi\)
0.861746 + 0.507340i \(0.169371\pi\)
\(348\) 8915.29 1.37330
\(349\) −1226.84 −0.188170 −0.0940852 0.995564i \(-0.529993\pi\)
−0.0940852 + 0.995564i \(0.529993\pi\)
\(350\) 2927.41 0.447076
\(351\) −13599.4 −2.06804
\(352\) 5817.15 0.880839
\(353\) −1582.18 −0.238558 −0.119279 0.992861i \(-0.538058\pi\)
−0.119279 + 0.992861i \(0.538058\pi\)
\(354\) −5643.19 −0.847267
\(355\) −1804.05 −0.269715
\(356\) 2122.52 0.315992
\(357\) 3108.07 0.460774
\(358\) −3139.60 −0.463500
\(359\) −613.266 −0.0901586 −0.0450793 0.998983i \(-0.514354\pi\)
−0.0450793 + 0.998983i \(0.514354\pi\)
\(360\) −1807.89 −0.264678
\(361\) −2753.88 −0.401499
\(362\) 6.98446 0.00101407
\(363\) 1582.63 0.228833
\(364\) −7225.77 −1.04048
\(365\) 701.846 0.100647
\(366\) 2021.25 0.288668
\(367\) 2202.45 0.313261 0.156631 0.987657i \(-0.449937\pi\)
0.156631 + 0.987657i \(0.449937\pi\)
\(368\) −3846.08 −0.544812
\(369\) −21640.5 −3.05300
\(370\) −569.374 −0.0800009
\(371\) −379.813 −0.0531507
\(372\) −18030.1 −2.51295
\(373\) −6268.78 −0.870201 −0.435101 0.900382i \(-0.643287\pi\)
−0.435101 + 0.900382i \(0.643287\pi\)
\(374\) −422.124 −0.0583624
\(375\) −4791.56 −0.659827
\(376\) 6520.92 0.894390
\(377\) 4853.36 0.663026
\(378\) −8786.72 −1.19561
\(379\) 3247.19 0.440097 0.220049 0.975489i \(-0.429378\pi\)
0.220049 + 0.975489i \(0.429378\pi\)
\(380\) −936.093 −0.126370
\(381\) −6651.62 −0.894417
\(382\) 1175.23 0.157409
\(383\) 8938.80 1.19256 0.596281 0.802776i \(-0.296644\pi\)
0.596281 + 0.802776i \(0.296644\pi\)
\(384\) −13478.1 −1.79115
\(385\) −2110.11 −0.279327
\(386\) −686.609 −0.0905375
\(387\) 0 0
\(388\) −1379.24 −0.180465
\(389\) 4626.84 0.603060 0.301530 0.953457i \(-0.402503\pi\)
0.301530 + 0.953457i \(0.402503\pi\)
\(390\) −659.558 −0.0856359
\(391\) 1030.80 0.133324
\(392\) −5164.68 −0.665448
\(393\) 9718.15 1.24737
\(394\) −181.047 −0.0231498
\(395\) −805.794 −0.102643
\(396\) −18023.4 −2.28714
\(397\) 10840.9 1.37051 0.685253 0.728305i \(-0.259692\pi\)
0.685253 + 0.728305i \(0.259692\pi\)
\(398\) 1696.07 0.213609
\(399\) −16473.8 −2.06698
\(400\) −5451.34 −0.681418
\(401\) −12391.8 −1.54319 −0.771594 0.636116i \(-0.780540\pi\)
−0.771594 + 0.636116i \(0.780540\pi\)
\(402\) 8336.47 1.03429
\(403\) −9815.33 −1.21324
\(404\) 12337.9 1.51939
\(405\) 3509.45 0.430583
\(406\) 3135.81 0.383319
\(407\) −11996.4 −1.46103
\(408\) 1588.23 0.192719
\(409\) −7910.71 −0.956380 −0.478190 0.878256i \(-0.658707\pi\)
−0.478190 + 0.878256i \(0.658707\pi\)
\(410\) −612.588 −0.0737891
\(411\) −6068.45 −0.728308
\(412\) 229.513 0.0274449
\(413\) 17497.2 2.08469
\(414\) −4992.77 −0.592708
\(415\) 1097.81 0.129854
\(416\) −5637.69 −0.664448
\(417\) −4190.78 −0.492142
\(418\) 2237.41 0.261806
\(419\) 12901.8 1.50428 0.752142 0.659001i \(-0.229021\pi\)
0.752142 + 0.659001i \(0.229021\pi\)
\(420\) 3756.54 0.436430
\(421\) 16075.1 1.86093 0.930464 0.366382i \(-0.119404\pi\)
0.930464 + 0.366382i \(0.119404\pi\)
\(422\) 819.586 0.0945422
\(423\) −30848.1 −3.54582
\(424\) −194.086 −0.0222303
\(425\) 1461.03 0.166754
\(426\) −7676.40 −0.873058
\(427\) −6267.05 −0.710266
\(428\) −9160.86 −1.03460
\(429\) −13896.6 −1.56394
\(430\) 0 0
\(431\) −9277.47 −1.03684 −0.518422 0.855125i \(-0.673480\pi\)
−0.518422 + 0.855125i \(0.673480\pi\)
\(432\) 16362.4 1.82231
\(433\) 182.499 0.0202548 0.0101274 0.999949i \(-0.496776\pi\)
0.0101274 + 0.999949i \(0.496776\pi\)
\(434\) −6341.79 −0.701418
\(435\) −2523.17 −0.278107
\(436\) 8491.65 0.932744
\(437\) −5463.60 −0.598076
\(438\) 2986.42 0.325792
\(439\) −5806.09 −0.631229 −0.315614 0.948888i \(-0.602211\pi\)
−0.315614 + 0.948888i \(0.602211\pi\)
\(440\) −1078.27 −0.116829
\(441\) 24432.2 2.63818
\(442\) 409.102 0.0440249
\(443\) 13300.1 1.42643 0.713214 0.700946i \(-0.247239\pi\)
0.713214 + 0.700946i \(0.247239\pi\)
\(444\) 21356.8 2.28277
\(445\) −600.706 −0.0639914
\(446\) 2459.86 0.261161
\(447\) −19669.5 −2.08128
\(448\) 6037.49 0.636707
\(449\) −7189.18 −0.755631 −0.377815 0.925881i \(-0.623325\pi\)
−0.377815 + 0.925881i \(0.623325\pi\)
\(450\) −7076.63 −0.741323
\(451\) −12906.9 −1.34759
\(452\) 7236.71 0.753066
\(453\) −9791.17 −1.01552
\(454\) −2073.87 −0.214387
\(455\) 2045.01 0.210707
\(456\) −8418.19 −0.864513
\(457\) 4199.15 0.429820 0.214910 0.976634i \(-0.431054\pi\)
0.214910 + 0.976634i \(0.431054\pi\)
\(458\) 2739.91 0.279537
\(459\) −4385.34 −0.445948
\(460\) 1245.87 0.126280
\(461\) −1185.60 −0.119781 −0.0598905 0.998205i \(-0.519075\pi\)
−0.0598905 + 0.998205i \(0.519075\pi\)
\(462\) −8978.71 −0.904172
\(463\) 2522.44 0.253192 0.126596 0.991954i \(-0.459595\pi\)
0.126596 + 0.991954i \(0.459595\pi\)
\(464\) −5839.41 −0.584241
\(465\) 5102.80 0.508897
\(466\) −4509.30 −0.448261
\(467\) −9629.56 −0.954182 −0.477091 0.878854i \(-0.658309\pi\)
−0.477091 + 0.878854i \(0.658309\pi\)
\(468\) 17467.4 1.72528
\(469\) −25847.9 −2.54487
\(470\) −873.232 −0.0857004
\(471\) −1758.94 −0.172075
\(472\) 8941.11 0.871924
\(473\) 0 0
\(474\) −3428.73 −0.332251
\(475\) −7743.97 −0.748037
\(476\) −2330.06 −0.224366
\(477\) 918.149 0.0881324
\(478\) −2653.49 −0.253908
\(479\) 12358.5 1.17886 0.589431 0.807819i \(-0.299352\pi\)
0.589431 + 0.807819i \(0.299352\pi\)
\(480\) 2930.93 0.278704
\(481\) 11626.3 1.10211
\(482\) 4497.33 0.424995
\(483\) 21925.4 2.06551
\(484\) −1186.47 −0.111426
\(485\) 390.348 0.0365459
\(486\) 6089.95 0.568407
\(487\) 2723.31 0.253398 0.126699 0.991941i \(-0.459562\pi\)
0.126699 + 0.991941i \(0.459562\pi\)
\(488\) −3202.49 −0.297069
\(489\) −7976.25 −0.737625
\(490\) 691.615 0.0637632
\(491\) 4199.46 0.385986 0.192993 0.981200i \(-0.438181\pi\)
0.192993 + 0.981200i \(0.438181\pi\)
\(492\) 22977.7 2.10552
\(493\) 1565.04 0.142973
\(494\) −2168.38 −0.197490
\(495\) 5100.91 0.463169
\(496\) 11809.5 1.06908
\(497\) 23801.3 2.14815
\(498\) 4671.30 0.420333
\(499\) −19985.3 −1.79292 −0.896458 0.443129i \(-0.853868\pi\)
−0.896458 + 0.443129i \(0.853868\pi\)
\(500\) 3592.14 0.321291
\(501\) −3065.90 −0.273402
\(502\) 4540.18 0.403661
\(503\) −520.717 −0.0461583 −0.0230792 0.999734i \(-0.507347\pi\)
−0.0230792 + 0.999734i \(0.507347\pi\)
\(504\) 23852.0 2.10804
\(505\) −3491.83 −0.307692
\(506\) −2977.82 −0.261621
\(507\) −7588.18 −0.664700
\(508\) 4986.60 0.435520
\(509\) 1746.69 0.152103 0.0760517 0.997104i \(-0.475769\pi\)
0.0760517 + 0.997104i \(0.475769\pi\)
\(510\) −212.684 −0.0184663
\(511\) −9259.63 −0.801609
\(512\) 11729.6 1.01246
\(513\) 23243.8 2.00046
\(514\) 764.013 0.0655626
\(515\) −64.9560 −0.00555787
\(516\) 0 0
\(517\) −18398.6 −1.56512
\(518\) 7511.90 0.637169
\(519\) −28887.2 −2.44318
\(520\) 1045.01 0.0881281
\(521\) 12223.8 1.02790 0.513948 0.857821i \(-0.328182\pi\)
0.513948 + 0.857821i \(0.328182\pi\)
\(522\) −7580.40 −0.635604
\(523\) 12923.8 1.08053 0.540265 0.841495i \(-0.318324\pi\)
0.540265 + 0.841495i \(0.318324\pi\)
\(524\) −7285.51 −0.607383
\(525\) 31076.5 2.58341
\(526\) 6974.88 0.578174
\(527\) −3165.10 −0.261620
\(528\) 16719.9 1.37811
\(529\) −4895.37 −0.402348
\(530\) 25.9905 0.00213011
\(531\) −42297.1 −3.45676
\(532\) 12350.1 1.00648
\(533\) 12508.7 1.01654
\(534\) −2556.06 −0.207138
\(535\) 2592.67 0.209516
\(536\) −13208.4 −1.06439
\(537\) −33329.1 −2.67832
\(538\) −1157.34 −0.0927441
\(539\) 14572.0 1.16449
\(540\) −5300.30 −0.422387
\(541\) −10061.3 −0.799574 −0.399787 0.916608i \(-0.630916\pi\)
−0.399787 + 0.916608i \(0.630916\pi\)
\(542\) 1043.85 0.0827252
\(543\) 74.1451 0.00585980
\(544\) −1817.96 −0.143280
\(545\) −2403.27 −0.188890
\(546\) 8701.71 0.682049
\(547\) −23951.3 −1.87218 −0.936090 0.351761i \(-0.885583\pi\)
−0.936090 + 0.351761i \(0.885583\pi\)
\(548\) 4549.40 0.354636
\(549\) 15149.8 1.17774
\(550\) −4220.68 −0.327219
\(551\) −8295.25 −0.641360
\(552\) 11204.0 0.863900
\(553\) 10631.0 0.817501
\(554\) −1883.50 −0.144445
\(555\) −6044.31 −0.462282
\(556\) 3141.75 0.239640
\(557\) 8803.19 0.669665 0.334832 0.942278i \(-0.391320\pi\)
0.334832 + 0.942278i \(0.391320\pi\)
\(558\) 15330.4 1.16306
\(559\) 0 0
\(560\) −2460.49 −0.185669
\(561\) −4481.15 −0.337245
\(562\) 1041.92 0.0782038
\(563\) −21876.5 −1.63762 −0.818812 0.574061i \(-0.805367\pi\)
−0.818812 + 0.574061i \(0.805367\pi\)
\(564\) 32754.3 2.44540
\(565\) −2048.10 −0.152503
\(566\) 4774.53 0.354573
\(567\) −46301.1 −3.42939
\(568\) 12162.5 0.898466
\(569\) −22198.4 −1.63551 −0.817756 0.575565i \(-0.804782\pi\)
−0.817756 + 0.575565i \(0.804782\pi\)
\(570\) 1127.30 0.0828376
\(571\) 13270.8 0.972619 0.486310 0.873787i \(-0.338343\pi\)
0.486310 + 0.873787i \(0.338343\pi\)
\(572\) 10418.0 0.761534
\(573\) 12475.9 0.909580
\(574\) 8082.03 0.587696
\(575\) 10306.6 0.747507
\(576\) −14594.9 −1.05576
\(577\) 15373.6 1.10921 0.554603 0.832115i \(-0.312870\pi\)
0.554603 + 0.832115i \(0.312870\pi\)
\(578\) −4303.59 −0.309699
\(579\) −7288.85 −0.523168
\(580\) 1891.57 0.135419
\(581\) −14483.7 −1.03423
\(582\) 1660.97 0.118298
\(583\) 547.607 0.0389015
\(584\) −4731.71 −0.335273
\(585\) −4943.54 −0.349385
\(586\) −2780.41 −0.196003
\(587\) −863.918 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(588\) −25942.0 −1.81944
\(589\) 16776.1 1.17360
\(590\) −1197.33 −0.0835477
\(591\) −1921.95 −0.133770
\(592\) −13988.4 −0.971151
\(593\) 1387.83 0.0961071 0.0480536 0.998845i \(-0.484698\pi\)
0.0480536 + 0.998845i \(0.484698\pi\)
\(594\) 12668.5 0.875078
\(595\) 659.444 0.0454363
\(596\) 14745.8 1.01344
\(597\) 18005.0 1.23433
\(598\) 2885.95 0.197350
\(599\) −23119.8 −1.57704 −0.788521 0.615008i \(-0.789153\pi\)
−0.788521 + 0.615008i \(0.789153\pi\)
\(600\) 15880.2 1.08051
\(601\) −29375.9 −1.99379 −0.996897 0.0787233i \(-0.974916\pi\)
−0.996897 + 0.0787233i \(0.974916\pi\)
\(602\) 0 0
\(603\) 62483.9 4.21980
\(604\) 7340.26 0.494488
\(605\) 335.789 0.0225649
\(606\) −14858.1 −0.995987
\(607\) 10354.5 0.692383 0.346191 0.938164i \(-0.387475\pi\)
0.346191 + 0.938164i \(0.387475\pi\)
\(608\) 9635.81 0.642736
\(609\) 33288.8 2.21499
\(610\) 428.853 0.0284651
\(611\) 17831.0 1.18063
\(612\) 5632.61 0.372034
\(613\) −9960.96 −0.656312 −0.328156 0.944624i \(-0.606427\pi\)
−0.328156 + 0.944624i \(0.606427\pi\)
\(614\) −577.334 −0.0379467
\(615\) −6503.06 −0.426388
\(616\) 14225.9 0.930485
\(617\) −13796.7 −0.900218 −0.450109 0.892974i \(-0.648615\pi\)
−0.450109 + 0.892974i \(0.648615\pi\)
\(618\) −276.394 −0.0179906
\(619\) 25631.5 1.66432 0.832162 0.554533i \(-0.187103\pi\)
0.832162 + 0.554533i \(0.187103\pi\)
\(620\) −3825.47 −0.247798
\(621\) −30935.7 −1.99905
\(622\) 6381.42 0.411369
\(623\) 7925.27 0.509662
\(624\) −16204.1 −1.03956
\(625\) 14091.5 0.901856
\(626\) −4947.00 −0.315849
\(627\) 23751.7 1.51284
\(628\) 1318.64 0.0837890
\(629\) 3749.09 0.237656
\(630\) −3194.08 −0.201992
\(631\) 17655.1 1.11385 0.556925 0.830563i \(-0.311981\pi\)
0.556925 + 0.830563i \(0.311981\pi\)
\(632\) 5432.50 0.341920
\(633\) 8700.49 0.546309
\(634\) 5881.53 0.368431
\(635\) −1411.29 −0.0881972
\(636\) −974.884 −0.0607809
\(637\) −14122.4 −0.878417
\(638\) −4521.15 −0.280555
\(639\) −57536.5 −3.56198
\(640\) −2859.67 −0.176622
\(641\) −374.249 −0.0230608 −0.0115304 0.999934i \(-0.503670\pi\)
−0.0115304 + 0.999934i \(0.503670\pi\)
\(642\) 11032.1 0.678194
\(643\) −18264.0 −1.12016 −0.560080 0.828438i \(-0.689230\pi\)
−0.560080 + 0.828438i \(0.689230\pi\)
\(644\) −16437.1 −1.00576
\(645\) 0 0
\(646\) −699.227 −0.0425863
\(647\) 1121.95 0.0681735 0.0340868 0.999419i \(-0.489148\pi\)
0.0340868 + 0.999419i \(0.489148\pi\)
\(648\) −23660.0 −1.43434
\(649\) −25227.1 −1.52581
\(650\) 4090.47 0.246833
\(651\) −67322.6 −4.05312
\(652\) 5979.65 0.359173
\(653\) 14019.2 0.840142 0.420071 0.907491i \(-0.362005\pi\)
0.420071 + 0.907491i \(0.362005\pi\)
\(654\) −10226.2 −0.611429
\(655\) 2061.92 0.123001
\(656\) −15050.1 −0.895745
\(657\) 22384.0 1.32920
\(658\) 11520.8 0.682563
\(659\) −26897.3 −1.58994 −0.794969 0.606650i \(-0.792513\pi\)
−0.794969 + 0.606650i \(0.792513\pi\)
\(660\) −5416.11 −0.319427
\(661\) 4456.60 0.262241 0.131121 0.991366i \(-0.458142\pi\)
0.131121 + 0.991366i \(0.458142\pi\)
\(662\) 3407.74 0.200069
\(663\) 4342.91 0.254396
\(664\) −7401.24 −0.432566
\(665\) −3495.28 −0.203821
\(666\) −18159.0 −1.05653
\(667\) 11040.3 0.640905
\(668\) 2298.45 0.133128
\(669\) 26113.2 1.50911
\(670\) 1768.76 0.101990
\(671\) 9035.71 0.519851
\(672\) −38668.5 −2.21975
\(673\) −12270.1 −0.702787 −0.351394 0.936228i \(-0.614292\pi\)
−0.351394 + 0.936228i \(0.614292\pi\)
\(674\) 10554.5 0.603183
\(675\) −43847.5 −2.50028
\(676\) 5688.71 0.323664
\(677\) −5587.24 −0.317186 −0.158593 0.987344i \(-0.550696\pi\)
−0.158593 + 0.987344i \(0.550696\pi\)
\(678\) −8714.88 −0.493647
\(679\) −5149.96 −0.291071
\(680\) 336.978 0.0190037
\(681\) −22015.7 −1.23883
\(682\) 9143.47 0.513375
\(683\) −16419.2 −0.919860 −0.459930 0.887955i \(-0.652126\pi\)
−0.459930 + 0.887955i \(0.652126\pi\)
\(684\) −29854.8 −1.66890
\(685\) −1287.55 −0.0718173
\(686\) −817.031 −0.0454729
\(687\) 29086.1 1.61529
\(688\) 0 0
\(689\) −530.714 −0.0293448
\(690\) −1500.35 −0.0827788
\(691\) 21390.2 1.17760 0.588801 0.808278i \(-0.299600\pi\)
0.588801 + 0.808278i \(0.299600\pi\)
\(692\) 21656.2 1.18966
\(693\) −67297.6 −3.68892
\(694\) 10057.7 0.550125
\(695\) −889.165 −0.0485294
\(696\) 17010.7 0.926422
\(697\) 4033.63 0.219203
\(698\) −1107.61 −0.0600625
\(699\) −47869.5 −2.59026
\(700\) −23297.5 −1.25795
\(701\) 2029.82 0.109365 0.0546827 0.998504i \(-0.482585\pi\)
0.0546827 + 0.998504i \(0.482585\pi\)
\(702\) −12277.7 −0.660103
\(703\) −19871.5 −1.06610
\(704\) −8704.74 −0.466012
\(705\) −9269.98 −0.495217
\(706\) −1428.41 −0.0761459
\(707\) 46068.6 2.45062
\(708\) 44910.8 2.38397
\(709\) 17269.2 0.914751 0.457376 0.889274i \(-0.348789\pi\)
0.457376 + 0.889274i \(0.348789\pi\)
\(710\) −1628.71 −0.0860909
\(711\) −25699.2 −1.35555
\(712\) 4049.84 0.213166
\(713\) −22327.7 −1.17276
\(714\) 2806.00 0.147075
\(715\) −2948.46 −0.154218
\(716\) 24986.2 1.30416
\(717\) −28168.7 −1.46720
\(718\) −553.663 −0.0287779
\(719\) 13317.0 0.690739 0.345369 0.938467i \(-0.387754\pi\)
0.345369 + 0.938467i \(0.387754\pi\)
\(720\) 5947.92 0.307869
\(721\) 856.981 0.0442658
\(722\) −2486.24 −0.128155
\(723\) 47742.4 2.45582
\(724\) −55.5851 −0.00285332
\(725\) 15648.3 0.801605
\(726\) 1428.81 0.0730416
\(727\) 22136.1 1.12927 0.564636 0.825340i \(-0.309017\pi\)
0.564636 + 0.825340i \(0.309017\pi\)
\(728\) −13787.1 −0.701899
\(729\) 18051.0 0.917084
\(730\) 633.634 0.0321258
\(731\) 0 0
\(732\) −16085.9 −0.812231
\(733\) 17545.9 0.884139 0.442069 0.896981i \(-0.354244\pi\)
0.442069 + 0.896981i \(0.354244\pi\)
\(734\) 1988.39 0.0999904
\(735\) 7341.99 0.368453
\(736\) −12824.5 −0.642280
\(737\) 37267.0 1.86261
\(738\) −19537.2 −0.974493
\(739\) 13702.8 0.682094 0.341047 0.940046i \(-0.389219\pi\)
0.341047 + 0.940046i \(0.389219\pi\)
\(740\) 4531.31 0.225100
\(741\) −23018.9 −1.14119
\(742\) −342.900 −0.0169653
\(743\) 12371.8 0.610871 0.305435 0.952213i \(-0.401198\pi\)
0.305435 + 0.952213i \(0.401198\pi\)
\(744\) −34402.1 −1.69522
\(745\) −4173.30 −0.205232
\(746\) −5659.52 −0.277761
\(747\) 35012.5 1.71492
\(748\) 3359.44 0.164215
\(749\) −34205.7 −1.66869
\(750\) −4325.87 −0.210611
\(751\) −9938.30 −0.482895 −0.241447 0.970414i \(-0.577622\pi\)
−0.241447 + 0.970414i \(0.577622\pi\)
\(752\) −21453.7 −1.04034
\(753\) 48197.2 2.33254
\(754\) 4381.67 0.211633
\(755\) −2077.41 −0.100139
\(756\) 69928.3 3.36411
\(757\) 10311.6 0.495088 0.247544 0.968877i \(-0.420377\pi\)
0.247544 + 0.968877i \(0.420377\pi\)
\(758\) 2931.60 0.140475
\(759\) −31611.7 −1.51177
\(760\) −1786.10 −0.0852483
\(761\) −19234.7 −0.916238 −0.458119 0.888891i \(-0.651477\pi\)
−0.458119 + 0.888891i \(0.651477\pi\)
\(762\) −6005.16 −0.285491
\(763\) 31707.0 1.50442
\(764\) −9352.97 −0.442904
\(765\) −1594.12 −0.0753406
\(766\) 8070.05 0.380656
\(767\) 24448.8 1.15097
\(768\) 5086.54 0.238990
\(769\) 13634.6 0.639370 0.319685 0.947524i \(-0.396423\pi\)
0.319685 + 0.947524i \(0.396423\pi\)
\(770\) −1905.03 −0.0891590
\(771\) 8110.55 0.378851
\(772\) 5464.31 0.254747
\(773\) −11204.3 −0.521335 −0.260667 0.965429i \(-0.583943\pi\)
−0.260667 + 0.965429i \(0.583943\pi\)
\(774\) 0 0
\(775\) −31646.8 −1.46682
\(776\) −2631.65 −0.121740
\(777\) 79744.2 3.68186
\(778\) 4177.17 0.192492
\(779\) −21379.6 −0.983318
\(780\) 5249.02 0.240955
\(781\) −34316.2 −1.57225
\(782\) 930.618 0.0425561
\(783\) −46969.0 −2.14372
\(784\) 16991.7 0.774038
\(785\) −373.197 −0.0169681
\(786\) 8773.65 0.398150
\(787\) −40995.2 −1.85683 −0.928413 0.371551i \(-0.878826\pi\)
−0.928413 + 0.371551i \(0.878826\pi\)
\(788\) 1440.85 0.0651371
\(789\) 74043.4 3.34096
\(790\) −727.480 −0.0327627
\(791\) 27021.2 1.21462
\(792\) −34389.3 −1.54289
\(793\) −8756.96 −0.392142
\(794\) 9787.31 0.437454
\(795\) 275.908 0.0123087
\(796\) −13498.0 −0.601034
\(797\) −1380.09 −0.0613368 −0.0306684 0.999530i \(-0.509764\pi\)
−0.0306684 + 0.999530i \(0.509764\pi\)
\(798\) −14872.8 −0.659762
\(799\) 5749.86 0.254588
\(800\) −18177.2 −0.803325
\(801\) −19158.3 −0.845100
\(802\) −11187.5 −0.492573
\(803\) 13350.4 0.586705
\(804\) −66345.0 −2.91021
\(805\) 4651.95 0.203677
\(806\) −8861.39 −0.387257
\(807\) −12286.0 −0.535918
\(808\) 23541.2 1.02497
\(809\) −18158.2 −0.789131 −0.394566 0.918868i \(-0.629105\pi\)
−0.394566 + 0.918868i \(0.629105\pi\)
\(810\) 3168.37 0.137439
\(811\) 478.154 0.0207031 0.0103516 0.999946i \(-0.496705\pi\)
0.0103516 + 0.999946i \(0.496705\pi\)
\(812\) −24956.0 −1.07855
\(813\) 11081.2 0.478024
\(814\) −10830.5 −0.466350
\(815\) −1692.34 −0.0727361
\(816\) −5225.25 −0.224167
\(817\) 0 0
\(818\) −7141.88 −0.305269
\(819\) 65221.4 2.78269
\(820\) 4875.22 0.207622
\(821\) 7722.38 0.328274 0.164137 0.986438i \(-0.447516\pi\)
0.164137 + 0.986438i \(0.447516\pi\)
\(822\) −5478.66 −0.232470
\(823\) −1765.67 −0.0747842 −0.0373921 0.999301i \(-0.511905\pi\)
−0.0373921 + 0.999301i \(0.511905\pi\)
\(824\) 437.920 0.0185142
\(825\) −44805.6 −1.89082
\(826\) 15796.6 0.665418
\(827\) −19472.6 −0.818776 −0.409388 0.912360i \(-0.634258\pi\)
−0.409388 + 0.912360i \(0.634258\pi\)
\(828\) 39734.5 1.66772
\(829\) −26318.0 −1.10261 −0.551304 0.834304i \(-0.685870\pi\)
−0.551304 + 0.834304i \(0.685870\pi\)
\(830\) 991.118 0.0414485
\(831\) −19994.7 −0.834669
\(832\) 8436.20 0.351530
\(833\) −4553.99 −0.189420
\(834\) −3783.48 −0.157088
\(835\) −650.498 −0.0269598
\(836\) −17806.2 −0.736650
\(837\) 94989.0 3.92270
\(838\) 11647.9 0.480156
\(839\) −6566.60 −0.270208 −0.135104 0.990831i \(-0.543137\pi\)
−0.135104 + 0.990831i \(0.543137\pi\)
\(840\) 7167.63 0.294413
\(841\) −7626.71 −0.312711
\(842\) 14512.8 0.593994
\(843\) 11060.7 0.451898
\(844\) −6522.59 −0.266015
\(845\) −1610.00 −0.0655450
\(846\) −27850.0 −1.13180
\(847\) −4430.15 −0.179719
\(848\) 638.538 0.0258579
\(849\) 50685.0 2.04889
\(850\) 1319.04 0.0532265
\(851\) 26447.4 1.06534
\(852\) 61091.8 2.45654
\(853\) 39036.9 1.56694 0.783468 0.621432i \(-0.213449\pi\)
0.783468 + 0.621432i \(0.213449\pi\)
\(854\) −5657.96 −0.226711
\(855\) 8449.39 0.337968
\(856\) −17479.3 −0.697931
\(857\) −6060.17 −0.241554 −0.120777 0.992680i \(-0.538539\pi\)
−0.120777 + 0.992680i \(0.538539\pi\)
\(858\) −12546.0 −0.499198
\(859\) −13664.8 −0.542767 −0.271383 0.962471i \(-0.587481\pi\)
−0.271383 + 0.962471i \(0.587481\pi\)
\(860\) 0 0
\(861\) 85796.5 3.39598
\(862\) −8375.80 −0.330952
\(863\) 8805.11 0.347311 0.173655 0.984806i \(-0.444442\pi\)
0.173655 + 0.984806i \(0.444442\pi\)
\(864\) 54559.4 2.14832
\(865\) −6129.05 −0.240918
\(866\) 164.762 0.00646517
\(867\) −45685.7 −1.78958
\(868\) 50470.5 1.97359
\(869\) −15327.6 −0.598337
\(870\) −2277.95 −0.0887697
\(871\) −36117.3 −1.40504
\(872\) 16202.4 0.629223
\(873\) 12449.3 0.482642
\(874\) −4932.60 −0.190901
\(875\) 13412.7 0.518208
\(876\) −23767.1 −0.916686
\(877\) 24442.4 0.941120 0.470560 0.882368i \(-0.344052\pi\)
0.470560 + 0.882368i \(0.344052\pi\)
\(878\) −5241.80 −0.201483
\(879\) −29516.1 −1.13260
\(880\) 3547.49 0.135893
\(881\) −21839.8 −0.835190 −0.417595 0.908633i \(-0.637127\pi\)
−0.417595 + 0.908633i \(0.637127\pi\)
\(882\) 22057.7 0.842086
\(883\) 17757.9 0.676785 0.338393 0.941005i \(-0.390117\pi\)
0.338393 + 0.941005i \(0.390117\pi\)
\(884\) −3255.80 −0.123874
\(885\) −12710.5 −0.482777
\(886\) 12007.5 0.455304
\(887\) 42106.9 1.59392 0.796962 0.604030i \(-0.206439\pi\)
0.796962 + 0.604030i \(0.206439\pi\)
\(888\) 40749.6 1.53994
\(889\) 18619.5 0.702449
\(890\) −542.324 −0.0204256
\(891\) 66756.1 2.51000
\(892\) −19576.5 −0.734833
\(893\) −30476.3 −1.14205
\(894\) −17757.8 −0.664329
\(895\) −7071.50 −0.264105
\(896\) 37728.3 1.40671
\(897\) 30636.4 1.14038
\(898\) −6490.47 −0.241191
\(899\) −33899.7 −1.25764
\(900\) 56318.7 2.08588
\(901\) −171.137 −0.00632784
\(902\) −11652.5 −0.430140
\(903\) 0 0
\(904\) 13807.9 0.508014
\(905\) 15.7315 0.000577826 0
\(906\) −8839.58 −0.324145
\(907\) 1377.33 0.0504227 0.0252114 0.999682i \(-0.491974\pi\)
0.0252114 + 0.999682i \(0.491974\pi\)
\(908\) 16504.7 0.603225
\(909\) −111365. −4.06352
\(910\) 1846.26 0.0672559
\(911\) 39864.1 1.44979 0.724895 0.688860i \(-0.241888\pi\)
0.724895 + 0.688860i \(0.241888\pi\)
\(912\) 27695.6 1.00559
\(913\) 20882.4 0.756961
\(914\) 3791.04 0.137195
\(915\) 4552.58 0.164485
\(916\) −21805.3 −0.786537
\(917\) −27203.4 −0.979646
\(918\) −3959.13 −0.142343
\(919\) 5830.88 0.209296 0.104648 0.994509i \(-0.466628\pi\)
0.104648 + 0.994509i \(0.466628\pi\)
\(920\) 2377.17 0.0851879
\(921\) −6128.81 −0.219274
\(922\) −1070.38 −0.0382331
\(923\) 33257.5 1.18601
\(924\) 71456.2 2.54409
\(925\) 37485.9 1.33246
\(926\) 2277.29 0.0808169
\(927\) −2071.64 −0.0733998
\(928\) −19471.2 −0.688763
\(929\) 4121.82 0.145568 0.0727840 0.997348i \(-0.476812\pi\)
0.0727840 + 0.997348i \(0.476812\pi\)
\(930\) 4606.87 0.162436
\(931\) 24137.7 0.849712
\(932\) 35886.8 1.26128
\(933\) 67743.3 2.37708
\(934\) −8693.67 −0.304567
\(935\) −950.774 −0.0332552
\(936\) 33328.4 1.16386
\(937\) −33608.5 −1.17176 −0.585881 0.810397i \(-0.699251\pi\)
−0.585881 + 0.810397i \(0.699251\pi\)
\(938\) −23335.7 −0.812302
\(939\) −52515.9 −1.82512
\(940\) 6949.53 0.241137
\(941\) −30075.8 −1.04192 −0.520958 0.853582i \(-0.674425\pi\)
−0.520958 + 0.853582i \(0.674425\pi\)
\(942\) −1587.99 −0.0549251
\(943\) 28454.7 0.982621
\(944\) −29416.1 −1.01421
\(945\) −19790.8 −0.681265
\(946\) 0 0
\(947\) 8945.08 0.306944 0.153472 0.988153i \(-0.450955\pi\)
0.153472 + 0.988153i \(0.450955\pi\)
\(948\) 27287.2 0.934860
\(949\) −12938.5 −0.442573
\(950\) −6991.34 −0.238767
\(951\) 62436.6 2.12897
\(952\) −4445.84 −0.151356
\(953\) 51373.5 1.74622 0.873112 0.487519i \(-0.162098\pi\)
0.873112 + 0.487519i \(0.162098\pi\)
\(954\) 828.915 0.0281311
\(955\) 2647.04 0.0896924
\(956\) 21117.5 0.714425
\(957\) −47995.2 −1.62118
\(958\) 11157.4 0.376283
\(959\) 16987.0 0.571991
\(960\) −4385.82 −0.147450
\(961\) 38767.0 1.30130
\(962\) 10496.4 0.351785
\(963\) 82687.9 2.76696
\(964\) −35791.5 −1.19582
\(965\) −1546.49 −0.0515888
\(966\) 19794.5 0.659294
\(967\) 22479.1 0.747549 0.373775 0.927520i \(-0.378063\pi\)
0.373775 + 0.927520i \(0.378063\pi\)
\(968\) −2263.82 −0.0751673
\(969\) −7422.80 −0.246083
\(970\) 352.410 0.0116652
\(971\) 45003.4 1.48736 0.743680 0.668536i \(-0.233079\pi\)
0.743680 + 0.668536i \(0.233079\pi\)
\(972\) −48466.3 −1.59934
\(973\) 11731.0 0.386514
\(974\) 2458.63 0.0808826
\(975\) 43423.3 1.42632
\(976\) 10536.1 0.345545
\(977\) −21946.4 −0.718657 −0.359329 0.933211i \(-0.616994\pi\)
−0.359329 + 0.933211i \(0.616994\pi\)
\(978\) −7201.05 −0.235444
\(979\) −11426.5 −0.373026
\(980\) −5504.15 −0.179412
\(981\) −76647.5 −2.49456
\(982\) 3791.32 0.123204
\(983\) −950.865 −0.0308524 −0.0154262 0.999881i \(-0.504911\pi\)
−0.0154262 + 0.999881i \(0.504911\pi\)
\(984\) 43842.3 1.42037
\(985\) −407.783 −0.0131909
\(986\) 1412.93 0.0456359
\(987\) 122301. 3.94417
\(988\) 17256.8 0.555681
\(989\) 0 0
\(990\) 4605.16 0.147840
\(991\) −51696.8 −1.65712 −0.828559 0.559901i \(-0.810839\pi\)
−0.828559 + 0.559901i \(0.810839\pi\)
\(992\) 39378.1 1.26034
\(993\) 36175.6 1.15609
\(994\) 21488.0 0.685674
\(995\) 3820.15 0.121715
\(996\) −37176.1 −1.18270
\(997\) −44474.6 −1.41276 −0.706381 0.707832i \(-0.749674\pi\)
−0.706381 + 0.707832i \(0.749674\pi\)
\(998\) −18042.9 −0.572284
\(999\) −112515. −3.56339
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.j.1.27 yes 50
43.42 odd 2 1849.4.a.i.1.24 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.24 50 43.42 odd 2
1849.4.a.j.1.27 yes 50 1.1 even 1 trivial