Properties

Label 1849.4.a.j.1.42
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 1.42
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.90499 q^{2} -3.52020 q^{3} +7.24897 q^{4} +10.7056 q^{5} -13.7464 q^{6} -28.1424 q^{7} -2.93275 q^{8} -14.6082 q^{9} +O(q^{10})\) \(q+3.90499 q^{2} -3.52020 q^{3} +7.24897 q^{4} +10.7056 q^{5} -13.7464 q^{6} -28.1424 q^{7} -2.93275 q^{8} -14.6082 q^{9} +41.8054 q^{10} +63.0835 q^{11} -25.5179 q^{12} -23.3564 q^{13} -109.896 q^{14} -37.6860 q^{15} -69.4442 q^{16} +49.0078 q^{17} -57.0448 q^{18} +57.3067 q^{19} +77.6048 q^{20} +99.0669 q^{21} +246.341 q^{22} +137.960 q^{23} +10.3239 q^{24} -10.3896 q^{25} -91.2066 q^{26} +146.469 q^{27} -204.003 q^{28} +129.150 q^{29} -147.164 q^{30} -69.2254 q^{31} -247.717 q^{32} -222.067 q^{33} +191.375 q^{34} -301.282 q^{35} -105.894 q^{36} -51.1938 q^{37} +223.782 q^{38} +82.2194 q^{39} -31.3969 q^{40} -179.964 q^{41} +386.856 q^{42} +457.291 q^{44} -156.389 q^{45} +538.732 q^{46} -432.101 q^{47} +244.458 q^{48} +448.993 q^{49} -40.5715 q^{50} -172.517 q^{51} -169.310 q^{52} -753.298 q^{53} +571.961 q^{54} +675.348 q^{55} +82.5346 q^{56} -201.731 q^{57} +504.330 q^{58} -845.732 q^{59} -273.185 q^{60} -156.589 q^{61} -270.325 q^{62} +411.108 q^{63} -411.780 q^{64} -250.045 q^{65} -867.170 q^{66} +618.378 q^{67} +355.256 q^{68} -485.647 q^{69} -1176.50 q^{70} -723.463 q^{71} +42.8421 q^{72} -315.462 q^{73} -199.912 q^{74} +36.5737 q^{75} +415.415 q^{76} -1775.32 q^{77} +321.066 q^{78} -1029.04 q^{79} -743.443 q^{80} -121.181 q^{81} -702.757 q^{82} -587.747 q^{83} +718.134 q^{84} +524.659 q^{85} -454.635 q^{87} -185.008 q^{88} -1076.10 q^{89} -610.700 q^{90} +657.305 q^{91} +1000.07 q^{92} +243.688 q^{93} -1687.35 q^{94} +613.504 q^{95} +872.015 q^{96} -402.331 q^{97} +1753.32 q^{98} -921.534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.90499 1.38062 0.690312 0.723512i \(-0.257473\pi\)
0.690312 + 0.723512i \(0.257473\pi\)
\(3\) −3.52020 −0.677464 −0.338732 0.940883i \(-0.609998\pi\)
−0.338732 + 0.940883i \(0.609998\pi\)
\(4\) 7.24897 0.906122
\(5\) 10.7056 0.957540 0.478770 0.877940i \(-0.341083\pi\)
0.478770 + 0.877940i \(0.341083\pi\)
\(6\) −13.7464 −0.935322
\(7\) −28.1424 −1.51955 −0.759773 0.650189i \(-0.774690\pi\)
−0.759773 + 0.650189i \(0.774690\pi\)
\(8\) −2.93275 −0.129611
\(9\) −14.6082 −0.541043
\(10\) 41.8054 1.32200
\(11\) 63.0835 1.72913 0.864563 0.502524i \(-0.167595\pi\)
0.864563 + 0.502524i \(0.167595\pi\)
\(12\) −25.5179 −0.613865
\(13\) −23.3564 −0.498300 −0.249150 0.968465i \(-0.580151\pi\)
−0.249150 + 0.968465i \(0.580151\pi\)
\(14\) −109.896 −2.09792
\(15\) −37.6860 −0.648699
\(16\) −69.4442 −1.08507
\(17\) 49.0078 0.699184 0.349592 0.936902i \(-0.386320\pi\)
0.349592 + 0.936902i \(0.386320\pi\)
\(18\) −57.0448 −0.746977
\(19\) 57.3067 0.691950 0.345975 0.938244i \(-0.387548\pi\)
0.345975 + 0.938244i \(0.387548\pi\)
\(20\) 77.6048 0.867648
\(21\) 99.0669 1.02944
\(22\) 246.341 2.38727
\(23\) 137.960 1.25072 0.625361 0.780336i \(-0.284952\pi\)
0.625361 + 0.780336i \(0.284952\pi\)
\(24\) 10.3239 0.0878064
\(25\) −10.3896 −0.0831171
\(26\) −91.2066 −0.687965
\(27\) 146.469 1.04400
\(28\) −204.003 −1.37689
\(29\) 129.150 0.826985 0.413493 0.910508i \(-0.364309\pi\)
0.413493 + 0.910508i \(0.364309\pi\)
\(30\) −147.164 −0.895609
\(31\) −69.2254 −0.401072 −0.200536 0.979686i \(-0.564268\pi\)
−0.200536 + 0.979686i \(0.564268\pi\)
\(32\) −247.717 −1.36846
\(33\) −222.067 −1.17142
\(34\) 191.375 0.965310
\(35\) −301.282 −1.45503
\(36\) −105.894 −0.490251
\(37\) −51.1938 −0.227465 −0.113733 0.993511i \(-0.536281\pi\)
−0.113733 + 0.993511i \(0.536281\pi\)
\(38\) 223.782 0.955323
\(39\) 82.2194 0.337580
\(40\) −31.3969 −0.124107
\(41\) −179.964 −0.685503 −0.342751 0.939426i \(-0.611359\pi\)
−0.342751 + 0.939426i \(0.611359\pi\)
\(42\) 386.856 1.42127
\(43\) 0 0
\(44\) 457.291 1.56680
\(45\) −156.389 −0.518070
\(46\) 538.732 1.72678
\(47\) −432.101 −1.34103 −0.670515 0.741896i \(-0.733927\pi\)
−0.670515 + 0.741896i \(0.733927\pi\)
\(48\) 244.458 0.735092
\(49\) 448.993 1.30902
\(50\) −40.5715 −0.114753
\(51\) −172.517 −0.473672
\(52\) −169.310 −0.451521
\(53\) −753.298 −1.95233 −0.976164 0.217033i \(-0.930362\pi\)
−0.976164 + 0.217033i \(0.930362\pi\)
\(54\) 571.961 1.44137
\(55\) 675.348 1.65571
\(56\) 82.5346 0.196949
\(57\) −201.731 −0.468771
\(58\) 504.330 1.14176
\(59\) −845.732 −1.86619 −0.933093 0.359635i \(-0.882901\pi\)
−0.933093 + 0.359635i \(0.882901\pi\)
\(60\) −273.185 −0.587800
\(61\) −156.589 −0.328676 −0.164338 0.986404i \(-0.552549\pi\)
−0.164338 + 0.986404i \(0.552549\pi\)
\(62\) −270.325 −0.553730
\(63\) 411.108 0.822139
\(64\) −411.780 −0.804258
\(65\) −250.045 −0.477142
\(66\) −867.170 −1.61729
\(67\) 618.378 1.12757 0.563783 0.825923i \(-0.309345\pi\)
0.563783 + 0.825923i \(0.309345\pi\)
\(68\) 355.256 0.633546
\(69\) −485.647 −0.847318
\(70\) −1176.50 −2.00884
\(71\) −723.463 −1.20928 −0.604642 0.796497i \(-0.706684\pi\)
−0.604642 + 0.796497i \(0.706684\pi\)
\(72\) 42.8421 0.0701249
\(73\) −315.462 −0.505781 −0.252890 0.967495i \(-0.581381\pi\)
−0.252890 + 0.967495i \(0.581381\pi\)
\(74\) −199.912 −0.314044
\(75\) 36.5737 0.0563088
\(76\) 415.415 0.626991
\(77\) −1775.32 −2.62749
\(78\) 321.066 0.466071
\(79\) −1029.04 −1.46552 −0.732760 0.680487i \(-0.761768\pi\)
−0.732760 + 0.680487i \(0.761768\pi\)
\(80\) −743.443 −1.03899
\(81\) −121.181 −0.166230
\(82\) −702.757 −0.946421
\(83\) −587.747 −0.777272 −0.388636 0.921391i \(-0.627054\pi\)
−0.388636 + 0.921391i \(0.627054\pi\)
\(84\) 718.134 0.932795
\(85\) 524.659 0.669497
\(86\) 0 0
\(87\) −454.635 −0.560252
\(88\) −185.008 −0.224113
\(89\) −1076.10 −1.28165 −0.640823 0.767689i \(-0.721407\pi\)
−0.640823 + 0.767689i \(0.721407\pi\)
\(90\) −610.700 −0.715260
\(91\) 657.305 0.757190
\(92\) 1000.07 1.13331
\(93\) 243.688 0.271712
\(94\) −1687.35 −1.85146
\(95\) 613.504 0.662570
\(96\) 872.015 0.927079
\(97\) −402.331 −0.421139 −0.210570 0.977579i \(-0.567532\pi\)
−0.210570 + 0.977579i \(0.567532\pi\)
\(98\) 1753.32 1.80726
\(99\) −921.534 −0.935532
\(100\) −75.3142 −0.0753142
\(101\) −67.5492 −0.0665485 −0.0332743 0.999446i \(-0.510593\pi\)
−0.0332743 + 0.999446i \(0.510593\pi\)
\(102\) −673.679 −0.653962
\(103\) −352.283 −0.337005 −0.168502 0.985701i \(-0.553893\pi\)
−0.168502 + 0.985701i \(0.553893\pi\)
\(104\) 68.4986 0.0645850
\(105\) 1060.57 0.985727
\(106\) −2941.62 −2.69543
\(107\) 1370.03 1.23781 0.618907 0.785464i \(-0.287576\pi\)
0.618907 + 0.785464i \(0.287576\pi\)
\(108\) 1061.75 0.945992
\(109\) −430.759 −0.378525 −0.189262 0.981927i \(-0.560610\pi\)
−0.189262 + 0.981927i \(0.560610\pi\)
\(110\) 2637.23 2.28591
\(111\) 180.213 0.154100
\(112\) 1954.32 1.64881
\(113\) 670.846 0.558477 0.279238 0.960222i \(-0.409918\pi\)
0.279238 + 0.960222i \(0.409918\pi\)
\(114\) −787.759 −0.647197
\(115\) 1476.94 1.19762
\(116\) 936.206 0.749349
\(117\) 341.194 0.269602
\(118\) −3302.58 −2.57650
\(119\) −1379.19 −1.06244
\(120\) 110.524 0.0840782
\(121\) 2648.53 1.98988
\(122\) −611.480 −0.453777
\(123\) 633.509 0.464403
\(124\) −501.813 −0.363420
\(125\) −1449.43 −1.03713
\(126\) 1605.38 1.13507
\(127\) −2571.14 −1.79647 −0.898237 0.439512i \(-0.855151\pi\)
−0.898237 + 0.439512i \(0.855151\pi\)
\(128\) 373.738 0.258079
\(129\) 0 0
\(130\) −976.424 −0.658754
\(131\) 1438.60 0.959476 0.479738 0.877412i \(-0.340732\pi\)
0.479738 + 0.877412i \(0.340732\pi\)
\(132\) −1609.76 −1.06145
\(133\) −1612.75 −1.05145
\(134\) 2414.76 1.55674
\(135\) 1568.04 0.999672
\(136\) −143.728 −0.0906216
\(137\) 1790.29 1.11646 0.558230 0.829686i \(-0.311481\pi\)
0.558230 + 0.829686i \(0.311481\pi\)
\(138\) −1896.45 −1.16983
\(139\) 2724.94 1.66278 0.831390 0.555689i \(-0.187546\pi\)
0.831390 + 0.555689i \(0.187546\pi\)
\(140\) −2183.98 −1.31843
\(141\) 1521.08 0.908499
\(142\) −2825.12 −1.66957
\(143\) −1473.40 −0.861624
\(144\) 1014.45 0.587067
\(145\) 1382.63 0.791871
\(146\) −1231.88 −0.698293
\(147\) −1580.55 −0.886813
\(148\) −371.103 −0.206111
\(149\) 1234.42 0.678711 0.339355 0.940658i \(-0.389791\pi\)
0.339355 + 0.940658i \(0.389791\pi\)
\(150\) 142.820 0.0777413
\(151\) −863.120 −0.465164 −0.232582 0.972577i \(-0.574717\pi\)
−0.232582 + 0.972577i \(0.574717\pi\)
\(152\) −168.066 −0.0896840
\(153\) −715.913 −0.378288
\(154\) −6932.61 −3.62757
\(155\) −741.101 −0.384043
\(156\) 596.006 0.305889
\(157\) 33.1386 0.0168455 0.00842276 0.999965i \(-0.497319\pi\)
0.00842276 + 0.999965i \(0.497319\pi\)
\(158\) −4018.40 −2.02333
\(159\) 2651.76 1.32263
\(160\) −2651.96 −1.31035
\(161\) −3882.52 −1.90053
\(162\) −473.213 −0.229501
\(163\) −3113.81 −1.49627 −0.748136 0.663545i \(-0.769051\pi\)
−0.748136 + 0.663545i \(0.769051\pi\)
\(164\) −1304.55 −0.621149
\(165\) −2377.36 −1.12168
\(166\) −2295.15 −1.07312
\(167\) −602.453 −0.279157 −0.139578 0.990211i \(-0.544575\pi\)
−0.139578 + 0.990211i \(0.544575\pi\)
\(168\) −290.539 −0.133426
\(169\) −1651.48 −0.751697
\(170\) 2048.79 0.924323
\(171\) −837.145 −0.374375
\(172\) 0 0
\(173\) 2194.51 0.964423 0.482211 0.876055i \(-0.339834\pi\)
0.482211 + 0.876055i \(0.339834\pi\)
\(174\) −1775.35 −0.773498
\(175\) 292.389 0.126300
\(176\) −4380.78 −1.87621
\(177\) 2977.15 1.26427
\(178\) −4202.17 −1.76947
\(179\) 2282.67 0.953153 0.476577 0.879133i \(-0.341877\pi\)
0.476577 + 0.879133i \(0.341877\pi\)
\(180\) −1133.66 −0.469435
\(181\) −3042.27 −1.24934 −0.624669 0.780890i \(-0.714766\pi\)
−0.624669 + 0.780890i \(0.714766\pi\)
\(182\) 2566.77 1.04539
\(183\) 551.227 0.222666
\(184\) −404.602 −0.162107
\(185\) −548.062 −0.217807
\(186\) 951.598 0.375132
\(187\) 3091.58 1.20898
\(188\) −3132.29 −1.21514
\(189\) −4121.99 −1.58641
\(190\) 2395.73 0.914760
\(191\) −1473.13 −0.558074 −0.279037 0.960280i \(-0.590015\pi\)
−0.279037 + 0.960280i \(0.590015\pi\)
\(192\) 1449.55 0.544855
\(193\) 535.690 0.199792 0.0998959 0.994998i \(-0.468149\pi\)
0.0998959 + 0.994998i \(0.468149\pi\)
\(194\) −1571.10 −0.581435
\(195\) 880.209 0.323247
\(196\) 3254.74 1.18613
\(197\) 1305.23 0.472051 0.236025 0.971747i \(-0.424155\pi\)
0.236025 + 0.971747i \(0.424155\pi\)
\(198\) −3598.58 −1.29162
\(199\) −3474.39 −1.23765 −0.618827 0.785527i \(-0.712392\pi\)
−0.618827 + 0.785527i \(0.712392\pi\)
\(200\) 30.4702 0.0107729
\(201\) −2176.82 −0.763885
\(202\) −263.779 −0.0918785
\(203\) −3634.59 −1.25664
\(204\) −1250.57 −0.429204
\(205\) −1926.62 −0.656396
\(206\) −1375.66 −0.465277
\(207\) −2015.34 −0.676694
\(208\) 1621.97 0.540688
\(209\) 3615.11 1.19647
\(210\) 4141.53 1.36092
\(211\) 597.942 0.195090 0.0975450 0.995231i \(-0.468901\pi\)
0.0975450 + 0.995231i \(0.468901\pi\)
\(212\) −5460.64 −1.76905
\(213\) 2546.74 0.819247
\(214\) 5349.97 1.70896
\(215\) 0 0
\(216\) −429.558 −0.135313
\(217\) 1948.17 0.609448
\(218\) −1682.11 −0.522600
\(219\) 1110.49 0.342648
\(220\) 4895.58 1.50027
\(221\) −1144.65 −0.348403
\(222\) 703.730 0.212753
\(223\) 3216.08 0.965761 0.482881 0.875686i \(-0.339591\pi\)
0.482881 + 0.875686i \(0.339591\pi\)
\(224\) 6971.35 2.07943
\(225\) 151.773 0.0449699
\(226\) 2619.65 0.771046
\(227\) 4588.34 1.34158 0.670790 0.741647i \(-0.265955\pi\)
0.670790 + 0.741647i \(0.265955\pi\)
\(228\) −1462.34 −0.424764
\(229\) 4872.42 1.40602 0.703009 0.711181i \(-0.251839\pi\)
0.703009 + 0.711181i \(0.251839\pi\)
\(230\) 5767.46 1.65346
\(231\) 6249.49 1.78003
\(232\) −378.765 −0.107186
\(233\) 1175.26 0.330447 0.165224 0.986256i \(-0.447165\pi\)
0.165224 + 0.986256i \(0.447165\pi\)
\(234\) 1332.36 0.372219
\(235\) −4625.91 −1.28409
\(236\) −6130.69 −1.69099
\(237\) 3622.43 0.992837
\(238\) −5385.75 −1.46683
\(239\) −3114.29 −0.842874 −0.421437 0.906858i \(-0.638474\pi\)
−0.421437 + 0.906858i \(0.638474\pi\)
\(240\) 2617.07 0.703880
\(241\) −3808.63 −1.01799 −0.508994 0.860770i \(-0.669983\pi\)
−0.508994 + 0.860770i \(0.669983\pi\)
\(242\) 10342.5 2.74727
\(243\) −3528.09 −0.931386
\(244\) −1135.11 −0.297820
\(245\) 4806.75 1.25344
\(246\) 2473.85 0.641166
\(247\) −1338.48 −0.344799
\(248\) 203.021 0.0519832
\(249\) 2068.99 0.526574
\(250\) −5660.02 −1.43188
\(251\) 670.351 0.168574 0.0842872 0.996442i \(-0.473139\pi\)
0.0842872 + 0.996442i \(0.473139\pi\)
\(252\) 2980.11 0.744958
\(253\) 8702.98 2.16266
\(254\) −10040.3 −2.48025
\(255\) −1846.91 −0.453560
\(256\) 4753.68 1.16057
\(257\) 5924.21 1.43791 0.718953 0.695058i \(-0.244621\pi\)
0.718953 + 0.695058i \(0.244621\pi\)
\(258\) 0 0
\(259\) 1440.72 0.345644
\(260\) −1812.57 −0.432349
\(261\) −1886.64 −0.447434
\(262\) 5617.73 1.32467
\(263\) −4838.94 −1.13453 −0.567266 0.823535i \(-0.691999\pi\)
−0.567266 + 0.823535i \(0.691999\pi\)
\(264\) 651.267 0.151828
\(265\) −8064.52 −1.86943
\(266\) −6297.76 −1.45166
\(267\) 3788.10 0.868269
\(268\) 4482.61 1.02171
\(269\) −3925.10 −0.889657 −0.444828 0.895616i \(-0.646735\pi\)
−0.444828 + 0.895616i \(0.646735\pi\)
\(270\) 6123.20 1.38017
\(271\) −3220.97 −0.721992 −0.360996 0.932567i \(-0.617563\pi\)
−0.360996 + 0.932567i \(0.617563\pi\)
\(272\) −3403.30 −0.758660
\(273\) −2313.85 −0.512969
\(274\) 6991.08 1.54141
\(275\) −655.415 −0.143720
\(276\) −3520.44 −0.767774
\(277\) −500.239 −0.108507 −0.0542535 0.998527i \(-0.517278\pi\)
−0.0542535 + 0.998527i \(0.517278\pi\)
\(278\) 10640.9 2.29567
\(279\) 1011.26 0.216997
\(280\) 883.584 0.188587
\(281\) 2081.73 0.441942 0.220971 0.975280i \(-0.429077\pi\)
0.220971 + 0.975280i \(0.429077\pi\)
\(282\) 5939.82 1.25430
\(283\) 4247.16 0.892111 0.446055 0.895005i \(-0.352828\pi\)
0.446055 + 0.895005i \(0.352828\pi\)
\(284\) −5244.36 −1.09576
\(285\) −2159.66 −0.448867
\(286\) −5753.63 −1.18958
\(287\) 5064.61 1.04165
\(288\) 3618.69 0.740393
\(289\) −2511.24 −0.511142
\(290\) 5399.17 1.09328
\(291\) 1416.29 0.285306
\(292\) −2286.77 −0.458299
\(293\) −3078.85 −0.613885 −0.306943 0.951728i \(-0.599306\pi\)
−0.306943 + 0.951728i \(0.599306\pi\)
\(294\) −6172.03 −1.22435
\(295\) −9054.09 −1.78695
\(296\) 150.139 0.0294819
\(297\) 9239.79 1.80521
\(298\) 4820.42 0.937044
\(299\) −3222.24 −0.623235
\(300\) 265.121 0.0510227
\(301\) 0 0
\(302\) −3370.48 −0.642216
\(303\) 237.787 0.0450842
\(304\) −3979.61 −0.750811
\(305\) −1676.39 −0.314720
\(306\) −2795.64 −0.522274
\(307\) −2386.54 −0.443671 −0.221835 0.975084i \(-0.571205\pi\)
−0.221835 + 0.975084i \(0.571205\pi\)
\(308\) −12869.2 −2.38082
\(309\) 1240.11 0.228308
\(310\) −2893.99 −0.530219
\(311\) 5088.76 0.927836 0.463918 0.885878i \(-0.346443\pi\)
0.463918 + 0.885878i \(0.346443\pi\)
\(312\) −241.129 −0.0437540
\(313\) −2442.28 −0.441041 −0.220520 0.975382i \(-0.570776\pi\)
−0.220520 + 0.975382i \(0.570776\pi\)
\(314\) 129.406 0.0232573
\(315\) 4401.17 0.787231
\(316\) −7459.49 −1.32794
\(317\) −7346.87 −1.30171 −0.650854 0.759203i \(-0.725589\pi\)
−0.650854 + 0.759203i \(0.725589\pi\)
\(318\) 10355.1 1.82606
\(319\) 8147.24 1.42996
\(320\) −4408.36 −0.770109
\(321\) −4822.80 −0.838575
\(322\) −15161.2 −2.62391
\(323\) 2808.47 0.483800
\(324\) −878.441 −0.150624
\(325\) 242.665 0.0414173
\(326\) −12159.4 −2.06579
\(327\) 1516.36 0.256437
\(328\) 527.789 0.0888484
\(329\) 12160.4 2.03776
\(330\) −9283.59 −1.54862
\(331\) −6905.46 −1.14670 −0.573351 0.819310i \(-0.694357\pi\)
−0.573351 + 0.819310i \(0.694357\pi\)
\(332\) −4260.56 −0.704303
\(333\) 747.848 0.123068
\(334\) −2352.57 −0.385411
\(335\) 6620.12 1.07969
\(336\) −6879.62 −1.11701
\(337\) −1575.18 −0.254616 −0.127308 0.991863i \(-0.540634\pi\)
−0.127308 + 0.991863i \(0.540634\pi\)
\(338\) −6449.01 −1.03781
\(339\) −2361.51 −0.378348
\(340\) 3803.24 0.606645
\(341\) −4366.98 −0.693505
\(342\) −3269.05 −0.516871
\(343\) −2982.91 −0.469568
\(344\) 0 0
\(345\) −5199.15 −0.811341
\(346\) 8569.53 1.33151
\(347\) 9101.68 1.40808 0.704040 0.710160i \(-0.251377\pi\)
0.704040 + 0.710160i \(0.251377\pi\)
\(348\) −3295.64 −0.507657
\(349\) −1577.99 −0.242028 −0.121014 0.992651i \(-0.538615\pi\)
−0.121014 + 0.992651i \(0.538615\pi\)
\(350\) 1141.78 0.174373
\(351\) −3421.00 −0.520226
\(352\) −15626.9 −2.36623
\(353\) −9222.14 −1.39050 −0.695248 0.718770i \(-0.744705\pi\)
−0.695248 + 0.718770i \(0.744705\pi\)
\(354\) 11625.8 1.74549
\(355\) −7745.12 −1.15794
\(356\) −7800.63 −1.16133
\(357\) 4855.05 0.719766
\(358\) 8913.80 1.31595
\(359\) −3662.12 −0.538382 −0.269191 0.963087i \(-0.586756\pi\)
−0.269191 + 0.963087i \(0.586756\pi\)
\(360\) 458.651 0.0671474
\(361\) −3574.94 −0.521205
\(362\) −11880.0 −1.72486
\(363\) −9323.36 −1.34807
\(364\) 4764.79 0.686106
\(365\) −3377.21 −0.484305
\(366\) 2152.54 0.307418
\(367\) −8159.43 −1.16054 −0.580271 0.814423i \(-0.697053\pi\)
−0.580271 + 0.814423i \(0.697053\pi\)
\(368\) −9580.50 −1.35711
\(369\) 2628.94 0.370886
\(370\) −2140.18 −0.300710
\(371\) 21199.6 2.96665
\(372\) 1766.48 0.246204
\(373\) −5635.31 −0.782267 −0.391133 0.920334i \(-0.627917\pi\)
−0.391133 + 0.920334i \(0.627917\pi\)
\(374\) 12072.6 1.66914
\(375\) 5102.29 0.702617
\(376\) 1267.24 0.173812
\(377\) −3016.48 −0.412087
\(378\) −16096.4 −2.19023
\(379\) −9744.43 −1.32068 −0.660340 0.750967i \(-0.729588\pi\)
−0.660340 + 0.750967i \(0.729588\pi\)
\(380\) 4447.27 0.600369
\(381\) 9050.95 1.21705
\(382\) −5752.57 −0.770490
\(383\) 10689.7 1.42616 0.713078 0.701084i \(-0.247300\pi\)
0.713078 + 0.701084i \(0.247300\pi\)
\(384\) −1315.63 −0.174839
\(385\) −19005.9 −2.51592
\(386\) 2091.87 0.275837
\(387\) 0 0
\(388\) −2916.48 −0.381603
\(389\) 8888.68 1.15854 0.579272 0.815134i \(-0.303337\pi\)
0.579272 + 0.815134i \(0.303337\pi\)
\(390\) 3437.21 0.446282
\(391\) 6761.10 0.874484
\(392\) −1316.79 −0.169663
\(393\) −5064.18 −0.650010
\(394\) 5096.93 0.651724
\(395\) −11016.5 −1.40329
\(396\) −6680.17 −0.847705
\(397\) −14090.4 −1.78130 −0.890649 0.454692i \(-0.849749\pi\)
−0.890649 + 0.454692i \(0.849749\pi\)
\(398\) −13567.5 −1.70873
\(399\) 5677.20 0.712319
\(400\) 721.500 0.0901875
\(401\) −1953.93 −0.243328 −0.121664 0.992571i \(-0.538823\pi\)
−0.121664 + 0.992571i \(0.538823\pi\)
\(402\) −8500.46 −1.05464
\(403\) 1616.86 0.199854
\(404\) −489.663 −0.0603011
\(405\) −1297.32 −0.159172
\(406\) −14193.1 −1.73495
\(407\) −3229.49 −0.393316
\(408\) 505.950 0.0613928
\(409\) 8069.26 0.975548 0.487774 0.872970i \(-0.337809\pi\)
0.487774 + 0.872970i \(0.337809\pi\)
\(410\) −7523.45 −0.906236
\(411\) −6302.20 −0.756361
\(412\) −2553.69 −0.305367
\(413\) 23800.9 2.83575
\(414\) −7869.88 −0.934260
\(415\) −6292.20 −0.744269
\(416\) 5785.78 0.681902
\(417\) −9592.35 −1.12647
\(418\) 14117.0 1.65187
\(419\) −8731.19 −1.01801 −0.509006 0.860763i \(-0.669987\pi\)
−0.509006 + 0.860763i \(0.669987\pi\)
\(420\) 7688.07 0.893189
\(421\) 3707.32 0.429177 0.214589 0.976705i \(-0.431159\pi\)
0.214589 + 0.976705i \(0.431159\pi\)
\(422\) 2334.96 0.269346
\(423\) 6312.20 0.725555
\(424\) 2209.24 0.253042
\(425\) −509.173 −0.0581141
\(426\) 9944.99 1.13107
\(427\) 4406.80 0.499438
\(428\) 9931.34 1.12161
\(429\) 5186.68 0.583719
\(430\) 0 0
\(431\) −4090.48 −0.457149 −0.228575 0.973526i \(-0.573406\pi\)
−0.228575 + 0.973526i \(0.573406\pi\)
\(432\) −10171.4 −1.13281
\(433\) 10181.0 1.12995 0.564973 0.825110i \(-0.308887\pi\)
0.564973 + 0.825110i \(0.308887\pi\)
\(434\) 7607.58 0.841418
\(435\) −4867.15 −0.536464
\(436\) −3122.56 −0.342990
\(437\) 7906.01 0.865437
\(438\) 4336.45 0.473068
\(439\) 10981.6 1.19390 0.596952 0.802277i \(-0.296378\pi\)
0.596952 + 0.802277i \(0.296378\pi\)
\(440\) −1980.63 −0.214597
\(441\) −6558.97 −0.708235
\(442\) −4469.83 −0.481014
\(443\) −2325.24 −0.249380 −0.124690 0.992196i \(-0.539794\pi\)
−0.124690 + 0.992196i \(0.539794\pi\)
\(444\) 1306.36 0.139633
\(445\) −11520.3 −1.22723
\(446\) 12558.8 1.33335
\(447\) −4345.42 −0.459802
\(448\) 11588.5 1.22211
\(449\) −3298.40 −0.346684 −0.173342 0.984862i \(-0.555457\pi\)
−0.173342 + 0.984862i \(0.555457\pi\)
\(450\) 592.674 0.0620865
\(451\) −11352.7 −1.18532
\(452\) 4862.94 0.506048
\(453\) 3038.36 0.315132
\(454\) 17917.4 1.85222
\(455\) 7036.86 0.725040
\(456\) 591.628 0.0607577
\(457\) 10638.0 1.08890 0.544448 0.838795i \(-0.316739\pi\)
0.544448 + 0.838795i \(0.316739\pi\)
\(458\) 19026.8 1.94118
\(459\) 7178.13 0.729948
\(460\) 10706.3 1.08519
\(461\) 8689.61 0.877908 0.438954 0.898510i \(-0.355349\pi\)
0.438954 + 0.898510i \(0.355349\pi\)
\(462\) 24404.2 2.45755
\(463\) −10497.9 −1.05373 −0.526866 0.849948i \(-0.676633\pi\)
−0.526866 + 0.849948i \(0.676633\pi\)
\(464\) −8968.72 −0.897333
\(465\) 2608.83 0.260175
\(466\) 4589.40 0.456223
\(467\) −8248.45 −0.817330 −0.408665 0.912685i \(-0.634005\pi\)
−0.408665 + 0.912685i \(0.634005\pi\)
\(468\) 2473.31 0.244292
\(469\) −17402.6 −1.71339
\(470\) −18064.2 −1.77284
\(471\) −116.655 −0.0114122
\(472\) 2480.32 0.241877
\(473\) 0 0
\(474\) 14145.6 1.37073
\(475\) −595.396 −0.0575129
\(476\) −9997.75 −0.962702
\(477\) 11004.3 1.05629
\(478\) −12161.3 −1.16369
\(479\) −14009.0 −1.33630 −0.668150 0.744027i \(-0.732914\pi\)
−0.668150 + 0.744027i \(0.732914\pi\)
\(480\) 9335.46 0.887716
\(481\) 1195.70 0.113346
\(482\) −14872.7 −1.40546
\(483\) 13667.2 1.28754
\(484\) 19199.1 1.80307
\(485\) −4307.20 −0.403258
\(486\) −13777.2 −1.28589
\(487\) −3912.41 −0.364041 −0.182021 0.983295i \(-0.558264\pi\)
−0.182021 + 0.983295i \(0.558264\pi\)
\(488\) 459.238 0.0425998
\(489\) 10961.2 1.01367
\(490\) 18770.3 1.73053
\(491\) 7916.93 0.727670 0.363835 0.931463i \(-0.381467\pi\)
0.363835 + 0.931463i \(0.381467\pi\)
\(492\) 4592.29 0.420806
\(493\) 6329.36 0.578215
\(494\) −5226.75 −0.476038
\(495\) −9865.59 −0.895809
\(496\) 4807.30 0.435190
\(497\) 20360.0 1.83756
\(498\) 8079.39 0.727000
\(499\) −4207.97 −0.377504 −0.188752 0.982025i \(-0.560444\pi\)
−0.188752 + 0.982025i \(0.560444\pi\)
\(500\) −10506.9 −0.939764
\(501\) 2120.76 0.189119
\(502\) 2617.72 0.232738
\(503\) 5613.91 0.497638 0.248819 0.968550i \(-0.419958\pi\)
0.248819 + 0.968550i \(0.419958\pi\)
\(504\) −1205.68 −0.106558
\(505\) −723.157 −0.0637229
\(506\) 33985.1 2.98581
\(507\) 5813.54 0.509247
\(508\) −18638.2 −1.62782
\(509\) −1936.88 −0.168665 −0.0843325 0.996438i \(-0.526876\pi\)
−0.0843325 + 0.996438i \(0.526876\pi\)
\(510\) −7212.15 −0.626195
\(511\) 8877.84 0.768557
\(512\) 15573.2 1.34423
\(513\) 8393.67 0.722396
\(514\) 23134.0 1.98521
\(515\) −3771.41 −0.322695
\(516\) 0 0
\(517\) −27258.4 −2.31881
\(518\) 5625.99 0.477204
\(519\) −7725.11 −0.653362
\(520\) 733.320 0.0618427
\(521\) 3402.75 0.286137 0.143068 0.989713i \(-0.454303\pi\)
0.143068 + 0.989713i \(0.454303\pi\)
\(522\) −7367.34 −0.617739
\(523\) −13408.0 −1.12101 −0.560506 0.828150i \(-0.689393\pi\)
−0.560506 + 0.828150i \(0.689393\pi\)
\(524\) 10428.4 0.869402
\(525\) −1029.27 −0.0855638
\(526\) −18896.0 −1.56636
\(527\) −3392.58 −0.280423
\(528\) 15421.2 1.27107
\(529\) 6865.89 0.564304
\(530\) −31491.9 −2.58098
\(531\) 12354.6 1.00969
\(532\) −11690.8 −0.952742
\(533\) 4203.31 0.341586
\(534\) 14792.5 1.19875
\(535\) 14667.1 1.18526
\(536\) −1813.55 −0.146144
\(537\) −8035.45 −0.645727
\(538\) −15327.5 −1.22828
\(539\) 28324.1 2.26346
\(540\) 11366.7 0.905825
\(541\) 11248.0 0.893881 0.446940 0.894564i \(-0.352514\pi\)
0.446940 + 0.894564i \(0.352514\pi\)
\(542\) −12577.9 −0.996800
\(543\) 10709.4 0.846381
\(544\) −12140.1 −0.956802
\(545\) −4611.54 −0.362453
\(546\) −9035.56 −0.708217
\(547\) 21075.0 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(548\) 12977.8 1.01165
\(549\) 2287.48 0.177828
\(550\) −2559.39 −0.198423
\(551\) 7401.16 0.572232
\(552\) 1424.28 0.109821
\(553\) 28959.6 2.22692
\(554\) −1953.43 −0.149807
\(555\) 1929.29 0.147556
\(556\) 19753.0 1.50668
\(557\) −6250.35 −0.475468 −0.237734 0.971330i \(-0.576405\pi\)
−0.237734 + 0.971330i \(0.576405\pi\)
\(558\) 3948.95 0.299592
\(559\) 0 0
\(560\) 20922.3 1.57880
\(561\) −10883.0 −0.819038
\(562\) 8129.16 0.610156
\(563\) −7955.02 −0.595496 −0.297748 0.954644i \(-0.596236\pi\)
−0.297748 + 0.954644i \(0.596236\pi\)
\(564\) 11026.3 0.823211
\(565\) 7181.82 0.534764
\(566\) 16585.1 1.23167
\(567\) 3410.34 0.252594
\(568\) 2121.74 0.156736
\(569\) −13925.5 −1.02599 −0.512995 0.858391i \(-0.671464\pi\)
−0.512995 + 0.858391i \(0.671464\pi\)
\(570\) −8433.45 −0.619717
\(571\) −11857.0 −0.869005 −0.434502 0.900671i \(-0.643076\pi\)
−0.434502 + 0.900671i \(0.643076\pi\)
\(572\) −10680.7 −0.780736
\(573\) 5185.73 0.378075
\(574\) 19777.3 1.43813
\(575\) −1433.35 −0.103956
\(576\) 6015.35 0.435138
\(577\) 2884.39 0.208109 0.104054 0.994572i \(-0.466818\pi\)
0.104054 + 0.994572i \(0.466818\pi\)
\(578\) −9806.38 −0.705695
\(579\) −1885.74 −0.135352
\(580\) 10022.7 0.717532
\(581\) 16540.6 1.18110
\(582\) 5530.59 0.393901
\(583\) −47520.7 −3.37582
\(584\) 925.171 0.0655545
\(585\) 3652.70 0.258154
\(586\) −12022.9 −0.847545
\(587\) −12049.5 −0.847254 −0.423627 0.905837i \(-0.639243\pi\)
−0.423627 + 0.905837i \(0.639243\pi\)
\(588\) −11457.4 −0.803560
\(589\) −3967.08 −0.277522
\(590\) −35356.2 −2.46710
\(591\) −4594.69 −0.319797
\(592\) 3555.11 0.246815
\(593\) 20939.8 1.45008 0.725038 0.688709i \(-0.241822\pi\)
0.725038 + 0.688709i \(0.241822\pi\)
\(594\) 36081.3 2.49231
\(595\) −14765.1 −1.01733
\(596\) 8948.30 0.614995
\(597\) 12230.6 0.838466
\(598\) −12582.8 −0.860453
\(599\) 18862.1 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(600\) −107.261 −0.00729822
\(601\) 6381.91 0.433151 0.216575 0.976266i \(-0.430511\pi\)
0.216575 + 0.976266i \(0.430511\pi\)
\(602\) 0 0
\(603\) −9033.36 −0.610061
\(604\) −6256.74 −0.421495
\(605\) 28354.1 1.90539
\(606\) 928.557 0.0622443
\(607\) −3055.87 −0.204339 −0.102170 0.994767i \(-0.532578\pi\)
−0.102170 + 0.994767i \(0.532578\pi\)
\(608\) −14195.8 −0.946903
\(609\) 12794.5 0.851329
\(610\) −6546.28 −0.434510
\(611\) 10092.3 0.668236
\(612\) −5189.63 −0.342775
\(613\) 339.125 0.0223444 0.0111722 0.999938i \(-0.496444\pi\)
0.0111722 + 0.999938i \(0.496444\pi\)
\(614\) −9319.42 −0.612543
\(615\) 6782.11 0.444685
\(616\) 5206.57 0.340550
\(617\) 5525.71 0.360546 0.180273 0.983617i \(-0.442302\pi\)
0.180273 + 0.983617i \(0.442302\pi\)
\(618\) 4842.62 0.315208
\(619\) −9145.73 −0.593858 −0.296929 0.954900i \(-0.595962\pi\)
−0.296929 + 0.954900i \(0.595962\pi\)
\(620\) −5372.22 −0.347990
\(621\) 20206.9 1.30575
\(622\) 19871.6 1.28099
\(623\) 30284.0 1.94752
\(624\) −5709.65 −0.366297
\(625\) −14218.4 −0.909974
\(626\) −9537.08 −0.608911
\(627\) −12725.9 −0.810565
\(628\) 240.221 0.0152641
\(629\) −2508.90 −0.159040
\(630\) 17186.5 1.08687
\(631\) 13263.0 0.836753 0.418376 0.908274i \(-0.362599\pi\)
0.418376 + 0.908274i \(0.362599\pi\)
\(632\) 3017.92 0.189947
\(633\) −2104.88 −0.132166
\(634\) −28689.5 −1.79717
\(635\) −27525.7 −1.72020
\(636\) 19222.6 1.19847
\(637\) −10486.9 −0.652284
\(638\) 31814.9 1.97424
\(639\) 10568.5 0.654275
\(640\) 4001.10 0.247121
\(641\) 19417.2 1.19647 0.598233 0.801322i \(-0.295870\pi\)
0.598233 + 0.801322i \(0.295870\pi\)
\(642\) −18833.0 −1.15776
\(643\) 265.029 0.0162546 0.00812732 0.999967i \(-0.497413\pi\)
0.00812732 + 0.999967i \(0.497413\pi\)
\(644\) −28144.3 −1.72211
\(645\) 0 0
\(646\) 10967.1 0.667946
\(647\) 16950.6 1.02998 0.514989 0.857197i \(-0.327796\pi\)
0.514989 + 0.857197i \(0.327796\pi\)
\(648\) 355.395 0.0215451
\(649\) −53351.8 −3.22687
\(650\) 947.604 0.0571817
\(651\) −6857.95 −0.412879
\(652\) −22571.9 −1.35580
\(653\) 14811.0 0.887596 0.443798 0.896127i \(-0.353631\pi\)
0.443798 + 0.896127i \(0.353631\pi\)
\(654\) 5921.37 0.354043
\(655\) 15401.1 0.918736
\(656\) 12497.4 0.743815
\(657\) 4608.31 0.273649
\(658\) 47486.1 2.81337
\(659\) 12313.7 0.727882 0.363941 0.931422i \(-0.381431\pi\)
0.363941 + 0.931422i \(0.381431\pi\)
\(660\) −17233.4 −1.01638
\(661\) −12928.7 −0.760769 −0.380385 0.924828i \(-0.624208\pi\)
−0.380385 + 0.924828i \(0.624208\pi\)
\(662\) −26965.8 −1.58316
\(663\) 4029.39 0.236031
\(664\) 1723.72 0.100743
\(665\) −17265.5 −1.00681
\(666\) 2920.34 0.169911
\(667\) 17817.5 1.03433
\(668\) −4367.17 −0.252950
\(669\) −11321.3 −0.654268
\(670\) 25851.5 1.49064
\(671\) −9878.21 −0.568322
\(672\) −24540.6 −1.40874
\(673\) −1829.04 −0.104761 −0.0523806 0.998627i \(-0.516681\pi\)
−0.0523806 + 0.998627i \(0.516681\pi\)
\(674\) −6151.07 −0.351529
\(675\) −1521.76 −0.0867743
\(676\) −11971.5 −0.681129
\(677\) −31239.7 −1.77347 −0.886734 0.462281i \(-0.847031\pi\)
−0.886734 + 0.462281i \(0.847031\pi\)
\(678\) −9221.70 −0.522356
\(679\) 11322.5 0.639940
\(680\) −1538.69 −0.0867738
\(681\) −16151.9 −0.908872
\(682\) −17053.0 −0.957469
\(683\) −13225.0 −0.740908 −0.370454 0.928851i \(-0.620798\pi\)
−0.370454 + 0.928851i \(0.620798\pi\)
\(684\) −6068.44 −0.339229
\(685\) 19166.2 1.06906
\(686\) −11648.2 −0.648297
\(687\) −17151.9 −0.952527
\(688\) 0 0
\(689\) 17594.3 0.972846
\(690\) −20302.6 −1.12016
\(691\) −32827.9 −1.80728 −0.903641 0.428290i \(-0.859116\pi\)
−0.903641 + 0.428290i \(0.859116\pi\)
\(692\) 15907.9 0.873885
\(693\) 25934.2 1.42158
\(694\) 35542.0 1.94403
\(695\) 29172.2 1.59218
\(696\) 1333.33 0.0726146
\(697\) −8819.62 −0.479292
\(698\) −6162.03 −0.334150
\(699\) −4137.17 −0.223866
\(700\) 2119.52 0.114443
\(701\) −2531.25 −0.136382 −0.0681911 0.997672i \(-0.521723\pi\)
−0.0681911 + 0.997672i \(0.521723\pi\)
\(702\) −13359.0 −0.718236
\(703\) −2933.75 −0.157395
\(704\) −25976.5 −1.39066
\(705\) 16284.2 0.869924
\(706\) −36012.4 −1.91975
\(707\) 1901.00 0.101124
\(708\) 21581.3 1.14559
\(709\) 22917.0 1.21392 0.606958 0.794734i \(-0.292389\pi\)
0.606958 + 0.794734i \(0.292389\pi\)
\(710\) −30244.6 −1.59868
\(711\) 15032.4 0.792909
\(712\) 3155.94 0.166115
\(713\) −9550.32 −0.501630
\(714\) 18958.9 0.993726
\(715\) −15773.7 −0.825040
\(716\) 16547.0 0.863673
\(717\) 10963.0 0.571017
\(718\) −14300.5 −0.743303
\(719\) −16579.6 −0.859967 −0.429983 0.902837i \(-0.641481\pi\)
−0.429983 + 0.902837i \(0.641481\pi\)
\(720\) 10860.3 0.562140
\(721\) 9914.08 0.512094
\(722\) −13960.1 −0.719588
\(723\) 13407.1 0.689650
\(724\) −22053.3 −1.13205
\(725\) −1341.82 −0.0687366
\(726\) −36407.7 −1.86118
\(727\) −13124.1 −0.669528 −0.334764 0.942302i \(-0.608657\pi\)
−0.334764 + 0.942302i \(0.608657\pi\)
\(728\) −1927.71 −0.0981398
\(729\) 15691.5 0.797210
\(730\) −13188.0 −0.668643
\(731\) 0 0
\(732\) 3995.83 0.201762
\(733\) 19053.7 0.960115 0.480057 0.877237i \(-0.340616\pi\)
0.480057 + 0.877237i \(0.340616\pi\)
\(734\) −31862.5 −1.60227
\(735\) −16920.8 −0.849159
\(736\) −34175.0 −1.71156
\(737\) 39009.5 1.94970
\(738\) 10266.0 0.512054
\(739\) 14039.8 0.698867 0.349434 0.936961i \(-0.386374\pi\)
0.349434 + 0.936961i \(0.386374\pi\)
\(740\) −3972.89 −0.197360
\(741\) 4711.72 0.233589
\(742\) 82784.3 4.09583
\(743\) −19227.3 −0.949368 −0.474684 0.880156i \(-0.657438\pi\)
−0.474684 + 0.880156i \(0.657438\pi\)
\(744\) −714.675 −0.0352167
\(745\) 13215.3 0.649893
\(746\) −22005.9 −1.08002
\(747\) 8585.90 0.420538
\(748\) 22410.8 1.09548
\(749\) −38556.0 −1.88092
\(750\) 19924.4 0.970049
\(751\) −12740.9 −0.619071 −0.309536 0.950888i \(-0.600174\pi\)
−0.309536 + 0.950888i \(0.600174\pi\)
\(752\) 30006.9 1.45510
\(753\) −2359.77 −0.114203
\(754\) −11779.3 −0.568937
\(755\) −9240.24 −0.445413
\(756\) −29880.2 −1.43748
\(757\) 8644.05 0.415024 0.207512 0.978232i \(-0.433463\pi\)
0.207512 + 0.978232i \(0.433463\pi\)
\(758\) −38051.9 −1.82336
\(759\) −30636.3 −1.46512
\(760\) −1799.25 −0.0858761
\(761\) 26953.2 1.28391 0.641954 0.766743i \(-0.278124\pi\)
0.641954 + 0.766743i \(0.278124\pi\)
\(762\) 35343.9 1.68028
\(763\) 12122.6 0.575186
\(764\) −10678.7 −0.505683
\(765\) −7664.29 −0.362226
\(766\) 41743.2 1.96899
\(767\) 19753.3 0.929921
\(768\) −16733.9 −0.786242
\(769\) 6330.39 0.296853 0.148426 0.988923i \(-0.452579\pi\)
0.148426 + 0.988923i \(0.452579\pi\)
\(770\) −74217.9 −3.47354
\(771\) −20854.4 −0.974130
\(772\) 3883.20 0.181036
\(773\) 1515.26 0.0705047 0.0352523 0.999378i \(-0.488777\pi\)
0.0352523 + 0.999378i \(0.488777\pi\)
\(774\) 0 0
\(775\) 719.227 0.0333360
\(776\) 1179.94 0.0545841
\(777\) −5071.62 −0.234161
\(778\) 34710.2 1.59951
\(779\) −10313.1 −0.474334
\(780\) 6380.62 0.292901
\(781\) −45638.6 −2.09101
\(782\) 26402.0 1.20733
\(783\) 18916.5 0.863373
\(784\) −31180.0 −1.42037
\(785\) 354.769 0.0161303
\(786\) −19775.6 −0.897419
\(787\) 6916.26 0.313263 0.156632 0.987657i \(-0.449936\pi\)
0.156632 + 0.987657i \(0.449936\pi\)
\(788\) 9461.60 0.427735
\(789\) 17034.1 0.768604
\(790\) −43019.4 −1.93742
\(791\) −18879.2 −0.848631
\(792\) 2702.63 0.121255
\(793\) 3657.37 0.163779
\(794\) −55022.8 −2.45930
\(795\) 28388.8 1.26647
\(796\) −25185.8 −1.12147
\(797\) 17973.3 0.798804 0.399402 0.916776i \(-0.369218\pi\)
0.399402 + 0.916776i \(0.369218\pi\)
\(798\) 22169.4 0.983445
\(799\) −21176.3 −0.937627
\(800\) 2573.69 0.113742
\(801\) 15719.9 0.693425
\(802\) −7630.08 −0.335944
\(803\) −19900.4 −0.874559
\(804\) −15779.7 −0.692173
\(805\) −41564.7 −1.81983
\(806\) 6313.81 0.275924
\(807\) 13817.2 0.602710
\(808\) 198.105 0.00862539
\(809\) 35822.1 1.55678 0.778391 0.627779i \(-0.216036\pi\)
0.778391 + 0.627779i \(0.216036\pi\)
\(810\) −5066.04 −0.219756
\(811\) −30499.7 −1.32058 −0.660288 0.751012i \(-0.729566\pi\)
−0.660288 + 0.751012i \(0.729566\pi\)
\(812\) −26347.1 −1.13867
\(813\) 11338.5 0.489124
\(814\) −12611.1 −0.543022
\(815\) −33335.3 −1.43274
\(816\) 11980.3 0.513965
\(817\) 0 0
\(818\) 31510.4 1.34687
\(819\) −9602.01 −0.409672
\(820\) −13966.0 −0.594775
\(821\) 30273.0 1.28689 0.643445 0.765493i \(-0.277505\pi\)
0.643445 + 0.765493i \(0.277505\pi\)
\(822\) −24610.0 −1.04425
\(823\) 42243.7 1.78922 0.894608 0.446852i \(-0.147455\pi\)
0.894608 + 0.446852i \(0.147455\pi\)
\(824\) 1033.16 0.0436794
\(825\) 2307.19 0.0973651
\(826\) 92942.4 3.91511
\(827\) 22125.2 0.930313 0.465156 0.885228i \(-0.345998\pi\)
0.465156 + 0.885228i \(0.345998\pi\)
\(828\) −14609.1 −0.613167
\(829\) 17028.7 0.713429 0.356715 0.934213i \(-0.383897\pi\)
0.356715 + 0.934213i \(0.383897\pi\)
\(830\) −24571.0 −1.02756
\(831\) 1760.94 0.0735096
\(832\) 9617.70 0.400762
\(833\) 22004.2 0.915245
\(834\) −37458.1 −1.55524
\(835\) −6449.63 −0.267304
\(836\) 26205.8 1.08415
\(837\) −10139.4 −0.418720
\(838\) −34095.2 −1.40549
\(839\) 4109.20 0.169089 0.0845443 0.996420i \(-0.473057\pi\)
0.0845443 + 0.996420i \(0.473057\pi\)
\(840\) −3110.40 −0.127761
\(841\) −7709.26 −0.316096
\(842\) 14477.0 0.592532
\(843\) −7328.13 −0.299400
\(844\) 4334.46 0.176775
\(845\) −17680.1 −0.719780
\(846\) 24649.1 1.00172
\(847\) −74535.9 −3.02371
\(848\) 52312.1 2.11840
\(849\) −14950.9 −0.604373
\(850\) −1988.32 −0.0802338
\(851\) −7062.69 −0.284496
\(852\) 18461.2 0.742337
\(853\) −9742.74 −0.391073 −0.195536 0.980696i \(-0.562645\pi\)
−0.195536 + 0.980696i \(0.562645\pi\)
\(854\) 17208.5 0.689535
\(855\) −8962.16 −0.358479
\(856\) −4017.97 −0.160434
\(857\) 28781.0 1.14719 0.573593 0.819140i \(-0.305549\pi\)
0.573593 + 0.819140i \(0.305549\pi\)
\(858\) 20254.0 0.805896
\(859\) 28477.1 1.13111 0.565556 0.824710i \(-0.308661\pi\)
0.565556 + 0.824710i \(0.308661\pi\)
\(860\) 0 0
\(861\) −17828.5 −0.705682
\(862\) −15973.3 −0.631151
\(863\) 29789.0 1.17501 0.587503 0.809222i \(-0.300111\pi\)
0.587503 + 0.809222i \(0.300111\pi\)
\(864\) −36282.9 −1.42867
\(865\) 23493.5 0.923474
\(866\) 39756.6 1.56003
\(867\) 8840.08 0.346280
\(868\) 14122.2 0.552234
\(869\) −64915.5 −2.53407
\(870\) −19006.2 −0.740655
\(871\) −14443.1 −0.561866
\(872\) 1263.31 0.0490608
\(873\) 5877.31 0.227854
\(874\) 30872.9 1.19484
\(875\) 40790.4 1.57596
\(876\) 8049.91 0.310481
\(877\) 8110.77 0.312293 0.156147 0.987734i \(-0.450093\pi\)
0.156147 + 0.987734i \(0.450093\pi\)
\(878\) 42883.1 1.64833
\(879\) 10838.2 0.415885
\(880\) −46899.0 −1.79655
\(881\) 31359.8 1.19925 0.599624 0.800282i \(-0.295317\pi\)
0.599624 + 0.800282i \(0.295317\pi\)
\(882\) −25612.7 −0.977806
\(883\) −11827.4 −0.450764 −0.225382 0.974270i \(-0.572363\pi\)
−0.225382 + 0.974270i \(0.572363\pi\)
\(884\) −8297.50 −0.315696
\(885\) 31872.3 1.21059
\(886\) −9080.04 −0.344300
\(887\) −15000.4 −0.567829 −0.283915 0.958850i \(-0.591633\pi\)
−0.283915 + 0.958850i \(0.591633\pi\)
\(888\) −528.520 −0.0199729
\(889\) 72358.1 2.72982
\(890\) −44986.8 −1.69434
\(891\) −7644.55 −0.287432
\(892\) 23313.3 0.875097
\(893\) −24762.3 −0.927926
\(894\) −16968.9 −0.634814
\(895\) 24437.4 0.912683
\(896\) −10517.9 −0.392163
\(897\) 11343.0 0.422219
\(898\) −12880.2 −0.478640
\(899\) −8940.46 −0.331681
\(900\) 1100.20 0.0407482
\(901\) −36917.4 −1.36504
\(902\) −44332.4 −1.63648
\(903\) 0 0
\(904\) −1967.42 −0.0723845
\(905\) −32569.4 −1.19629
\(906\) 11864.8 0.435078
\(907\) 7030.05 0.257364 0.128682 0.991686i \(-0.458925\pi\)
0.128682 + 0.991686i \(0.458925\pi\)
\(908\) 33260.8 1.21564
\(909\) 986.770 0.0360056
\(910\) 27478.9 1.00101
\(911\) 10526.7 0.382838 0.191419 0.981508i \(-0.438691\pi\)
0.191419 + 0.981508i \(0.438691\pi\)
\(912\) 14009.1 0.508647
\(913\) −37077.1 −1.34400
\(914\) 41541.4 1.50336
\(915\) 5901.22 0.213211
\(916\) 35320.0 1.27402
\(917\) −40485.7 −1.45797
\(918\) 28030.5 1.00778
\(919\) −23229.7 −0.833816 −0.416908 0.908949i \(-0.636886\pi\)
−0.416908 + 0.908949i \(0.636886\pi\)
\(920\) −4331.51 −0.155224
\(921\) 8401.11 0.300571
\(922\) 33932.9 1.21206
\(923\) 16897.5 0.602587
\(924\) 45302.4 1.61292
\(925\) 531.886 0.0189063
\(926\) −40994.1 −1.45481
\(927\) 5146.21 0.182334
\(928\) −31992.7 −1.13169
\(929\) 22527.4 0.795586 0.397793 0.917475i \(-0.369776\pi\)
0.397793 + 0.917475i \(0.369776\pi\)
\(930\) 10187.5 0.359204
\(931\) 25730.3 0.905776
\(932\) 8519.46 0.299425
\(933\) −17913.5 −0.628575
\(934\) −32210.2 −1.12842
\(935\) 33097.3 1.15764
\(936\) −1000.64 −0.0349432
\(937\) 17122.8 0.596987 0.298494 0.954412i \(-0.403516\pi\)
0.298494 + 0.954412i \(0.403516\pi\)
\(938\) −67957.1 −2.36554
\(939\) 8597.32 0.298789
\(940\) −33533.1 −1.16354
\(941\) −21511.1 −0.745208 −0.372604 0.927990i \(-0.621535\pi\)
−0.372604 + 0.927990i \(0.621535\pi\)
\(942\) −455.536 −0.0157560
\(943\) −24827.7 −0.857373
\(944\) 58731.2 2.02493
\(945\) −44128.5 −1.51905
\(946\) 0 0
\(947\) −18487.6 −0.634387 −0.317194 0.948361i \(-0.602741\pi\)
−0.317194 + 0.948361i \(0.602741\pi\)
\(948\) 26258.9 0.899631
\(949\) 7368.05 0.252031
\(950\) −2325.02 −0.0794037
\(951\) 25862.5 0.881859
\(952\) 4044.84 0.137704
\(953\) 27815.6 0.945471 0.472736 0.881204i \(-0.343267\pi\)
0.472736 + 0.881204i \(0.343267\pi\)
\(954\) 42971.7 1.45834
\(955\) −15770.8 −0.534378
\(956\) −22575.4 −0.763747
\(957\) −28680.0 −0.968747
\(958\) −54705.1 −1.84493
\(959\) −50383.1 −1.69651
\(960\) 15518.3 0.521721
\(961\) −24998.8 −0.839141
\(962\) 4669.22 0.156488
\(963\) −20013.7 −0.669711
\(964\) −27608.6 −0.922421
\(965\) 5734.89 0.191309
\(966\) 53370.5 1.77761
\(967\) −1372.88 −0.0456556 −0.0228278 0.999739i \(-0.507267\pi\)
−0.0228278 + 0.999739i \(0.507267\pi\)
\(968\) −7767.48 −0.257909
\(969\) −9886.39 −0.327757
\(970\) −16819.6 −0.556747
\(971\) −30280.5 −1.00077 −0.500385 0.865803i \(-0.666808\pi\)
−0.500385 + 0.865803i \(0.666808\pi\)
\(972\) −25575.0 −0.843949
\(973\) −76686.3 −2.52667
\(974\) −15277.9 −0.502604
\(975\) −854.229 −0.0280587
\(976\) 10874.2 0.356634
\(977\) −31533.9 −1.03261 −0.516305 0.856405i \(-0.672693\pi\)
−0.516305 + 0.856405i \(0.672693\pi\)
\(978\) 42803.6 1.39950
\(979\) −67884.2 −2.21613
\(980\) 34844.0 1.13577
\(981\) 6292.59 0.204798
\(982\) 30915.6 1.00464
\(983\) 22289.4 0.723217 0.361609 0.932330i \(-0.382228\pi\)
0.361609 + 0.932330i \(0.382228\pi\)
\(984\) −1857.93 −0.0601916
\(985\) 13973.3 0.452007
\(986\) 24716.1 0.798297
\(987\) −42806.9 −1.38051
\(988\) −9702.59 −0.312430
\(989\) 0 0
\(990\) −38525.1 −1.23677
\(991\) −2037.39 −0.0653074 −0.0326537 0.999467i \(-0.510396\pi\)
−0.0326537 + 0.999467i \(0.510396\pi\)
\(992\) 17148.3 0.548850
\(993\) 24308.6 0.776849
\(994\) 79505.5 2.53698
\(995\) −37195.5 −1.18510
\(996\) 14998.0 0.477140
\(997\) −26780.4 −0.850695 −0.425347 0.905030i \(-0.639848\pi\)
−0.425347 + 0.905030i \(0.639848\pi\)
\(998\) −16432.1 −0.521191
\(999\) −7498.32 −0.237474
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.1.42 yes 50
43.42 odd 2 1849.4.a.i.1.9 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.9 50 43.42 odd 2
1849.4.a.j.1.42 yes 50 1.1 even 1 trivial