Properties

Label 1849.4.a.i.1.16
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 1.16
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.04766 q^{2} -1.42990 q^{3} +1.28824 q^{4} +10.9474 q^{5} +4.35784 q^{6} +8.62569 q^{7} +20.4552 q^{8} -24.9554 q^{9} +O(q^{10})\) \(q-3.04766 q^{2} -1.42990 q^{3} +1.28824 q^{4} +10.9474 q^{5} +4.35784 q^{6} +8.62569 q^{7} +20.4552 q^{8} -24.9554 q^{9} -33.3638 q^{10} +42.2441 q^{11} -1.84205 q^{12} +78.4327 q^{13} -26.2882 q^{14} -15.6536 q^{15} -72.6463 q^{16} +75.8963 q^{17} +76.0556 q^{18} -107.684 q^{19} +14.1028 q^{20} -12.3339 q^{21} -128.746 q^{22} -27.7883 q^{23} -29.2488 q^{24} -5.15536 q^{25} -239.036 q^{26} +74.2909 q^{27} +11.1119 q^{28} -12.6364 q^{29} +47.7069 q^{30} -188.821 q^{31} +57.7599 q^{32} -60.4047 q^{33} -231.306 q^{34} +94.4285 q^{35} -32.1484 q^{36} -84.0627 q^{37} +328.185 q^{38} -112.151 q^{39} +223.930 q^{40} -215.325 q^{41} +37.5894 q^{42} +54.4203 q^{44} -273.196 q^{45} +84.6893 q^{46} -598.204 q^{47} +103.877 q^{48} -268.598 q^{49} +15.7118 q^{50} -108.524 q^{51} +101.040 q^{52} +368.465 q^{53} -226.413 q^{54} +462.461 q^{55} +176.440 q^{56} +153.977 q^{57} +38.5114 q^{58} -62.2843 q^{59} -20.1655 q^{60} -186.541 q^{61} +575.462 q^{62} -215.257 q^{63} +405.138 q^{64} +858.631 q^{65} +184.093 q^{66} -717.716 q^{67} +97.7723 q^{68} +39.7344 q^{69} -287.786 q^{70} +474.895 q^{71} -510.467 q^{72} -1229.76 q^{73} +256.195 q^{74} +7.37163 q^{75} -138.723 q^{76} +364.384 q^{77} +341.798 q^{78} -1134.35 q^{79} -795.285 q^{80} +567.567 q^{81} +656.237 q^{82} -632.780 q^{83} -15.8889 q^{84} +830.864 q^{85} +18.0687 q^{87} +864.110 q^{88} +6.66926 q^{89} +832.607 q^{90} +676.536 q^{91} -35.7979 q^{92} +269.994 q^{93} +1823.12 q^{94} -1178.86 q^{95} -82.5908 q^{96} -1024.36 q^{97} +818.594 q^{98} -1054.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.04766 −1.07751 −0.538755 0.842462i \(-0.681105\pi\)
−0.538755 + 0.842462i \(0.681105\pi\)
\(3\) −1.42990 −0.275184 −0.137592 0.990489i \(-0.543936\pi\)
−0.137592 + 0.990489i \(0.543936\pi\)
\(4\) 1.28824 0.161029
\(5\) 10.9474 0.979161 0.489581 0.871958i \(-0.337150\pi\)
0.489581 + 0.871958i \(0.337150\pi\)
\(6\) 4.35784 0.296514
\(7\) 8.62569 0.465743 0.232872 0.972507i \(-0.425188\pi\)
0.232872 + 0.972507i \(0.425188\pi\)
\(8\) 20.4552 0.904000
\(9\) −24.9554 −0.924274
\(10\) −33.3638 −1.05506
\(11\) 42.2441 1.15792 0.578958 0.815358i \(-0.303460\pi\)
0.578958 + 0.815358i \(0.303460\pi\)
\(12\) −1.84205 −0.0443127
\(13\) 78.4327 1.67333 0.836666 0.547713i \(-0.184501\pi\)
0.836666 + 0.547713i \(0.184501\pi\)
\(14\) −26.2882 −0.501843
\(15\) −15.6536 −0.269450
\(16\) −72.6463 −1.13510
\(17\) 75.8963 1.08280 0.541398 0.840766i \(-0.317895\pi\)
0.541398 + 0.840766i \(0.317895\pi\)
\(18\) 76.0556 0.995915
\(19\) −107.684 −1.30023 −0.650117 0.759834i \(-0.725280\pi\)
−0.650117 + 0.759834i \(0.725280\pi\)
\(20\) 14.1028 0.157674
\(21\) −12.3339 −0.128165
\(22\) −128.746 −1.24767
\(23\) −27.7883 −0.251924 −0.125962 0.992035i \(-0.540202\pi\)
−0.125962 + 0.992035i \(0.540202\pi\)
\(24\) −29.2488 −0.248766
\(25\) −5.15536 −0.0412428
\(26\) −239.036 −1.80303
\(27\) 74.2909 0.529529
\(28\) 11.1119 0.0749984
\(29\) −12.6364 −0.0809143 −0.0404572 0.999181i \(-0.512881\pi\)
−0.0404572 + 0.999181i \(0.512881\pi\)
\(30\) 47.7069 0.290335
\(31\) −188.821 −1.09397 −0.546987 0.837141i \(-0.684225\pi\)
−0.546987 + 0.837141i \(0.684225\pi\)
\(32\) 57.7599 0.319082
\(33\) −60.4047 −0.318640
\(34\) −231.306 −1.16673
\(35\) 94.4285 0.456038
\(36\) −32.1484 −0.148835
\(37\) −84.0627 −0.373509 −0.186754 0.982407i \(-0.559797\pi\)
−0.186754 + 0.982407i \(0.559797\pi\)
\(38\) 328.185 1.40102
\(39\) −112.151 −0.460474
\(40\) 223.930 0.885162
\(41\) −215.325 −0.820197 −0.410099 0.912041i \(-0.634506\pi\)
−0.410099 + 0.912041i \(0.634506\pi\)
\(42\) 37.5894 0.138099
\(43\) 0 0
\(44\) 54.4203 0.186458
\(45\) −273.196 −0.905013
\(46\) 84.6893 0.271451
\(47\) −598.204 −1.85653 −0.928266 0.371916i \(-0.878701\pi\)
−0.928266 + 0.371916i \(0.878701\pi\)
\(48\) 103.877 0.312361
\(49\) −268.598 −0.783083
\(50\) 15.7118 0.0444396
\(51\) −108.524 −0.297968
\(52\) 101.040 0.269456
\(53\) 368.465 0.954955 0.477477 0.878644i \(-0.341551\pi\)
0.477477 + 0.878644i \(0.341551\pi\)
\(54\) −226.413 −0.570574
\(55\) 462.461 1.13379
\(56\) 176.440 0.421032
\(57\) 153.977 0.357804
\(58\) 38.5114 0.0871860
\(59\) −62.2843 −0.137436 −0.0687180 0.997636i \(-0.521891\pi\)
−0.0687180 + 0.997636i \(0.521891\pi\)
\(60\) −20.1655 −0.0433893
\(61\) −186.541 −0.391543 −0.195772 0.980649i \(-0.562721\pi\)
−0.195772 + 0.980649i \(0.562721\pi\)
\(62\) 575.462 1.17877
\(63\) −215.257 −0.430474
\(64\) 405.138 0.791285
\(65\) 858.631 1.63846
\(66\) 184.093 0.343338
\(67\) −717.716 −1.30870 −0.654350 0.756192i \(-0.727058\pi\)
−0.654350 + 0.756192i \(0.727058\pi\)
\(68\) 97.7723 0.174362
\(69\) 39.7344 0.0693256
\(70\) −287.786 −0.491386
\(71\) 474.895 0.793798 0.396899 0.917862i \(-0.370086\pi\)
0.396899 + 0.917862i \(0.370086\pi\)
\(72\) −510.467 −0.835543
\(73\) −1229.76 −1.97168 −0.985842 0.167678i \(-0.946373\pi\)
−0.985842 + 0.167678i \(0.946373\pi\)
\(74\) 256.195 0.402460
\(75\) 7.37163 0.0113494
\(76\) −138.723 −0.209376
\(77\) 364.384 0.539291
\(78\) 341.798 0.496166
\(79\) −1134.35 −1.61549 −0.807747 0.589529i \(-0.799313\pi\)
−0.807747 + 0.589529i \(0.799313\pi\)
\(80\) −795.285 −1.11145
\(81\) 567.567 0.778556
\(82\) 656.237 0.883771
\(83\) −632.780 −0.836826 −0.418413 0.908257i \(-0.637414\pi\)
−0.418413 + 0.908257i \(0.637414\pi\)
\(84\) −15.8889 −0.0206384
\(85\) 830.864 1.06023
\(86\) 0 0
\(87\) 18.0687 0.0222663
\(88\) 864.110 1.04676
\(89\) 6.66926 0.00794315 0.00397158 0.999992i \(-0.498736\pi\)
0.00397158 + 0.999992i \(0.498736\pi\)
\(90\) 832.607 0.975162
\(91\) 676.536 0.779344
\(92\) −35.7979 −0.0405672
\(93\) 269.994 0.301044
\(94\) 1823.12 2.00043
\(95\) −1178.86 −1.27314
\(96\) −82.5908 −0.0878061
\(97\) −1024.36 −1.07224 −0.536122 0.844140i \(-0.680111\pi\)
−0.536122 + 0.844140i \(0.680111\pi\)
\(98\) 818.594 0.843781
\(99\) −1054.22 −1.07023
\(100\) −6.64131 −0.00664131
\(101\) 1296.35 1.27714 0.638572 0.769562i \(-0.279526\pi\)
0.638572 + 0.769562i \(0.279526\pi\)
\(102\) 330.744 0.321064
\(103\) 30.1268 0.0288202 0.0144101 0.999896i \(-0.495413\pi\)
0.0144101 + 0.999896i \(0.495413\pi\)
\(104\) 1604.36 1.51269
\(105\) −135.023 −0.125494
\(106\) −1122.96 −1.02897
\(107\) −1362.61 −1.23111 −0.615554 0.788095i \(-0.711068\pi\)
−0.615554 + 0.788095i \(0.711068\pi\)
\(108\) 95.7042 0.0852698
\(109\) −397.621 −0.349406 −0.174703 0.984621i \(-0.555896\pi\)
−0.174703 + 0.984621i \(0.555896\pi\)
\(110\) −1409.42 −1.22167
\(111\) 120.201 0.102784
\(112\) −626.624 −0.528665
\(113\) 348.680 0.290275 0.145137 0.989412i \(-0.453638\pi\)
0.145137 + 0.989412i \(0.453638\pi\)
\(114\) −469.271 −0.385537
\(115\) −304.208 −0.246675
\(116\) −16.2786 −0.0130296
\(117\) −1957.32 −1.54662
\(118\) 189.821 0.148089
\(119\) 654.657 0.504305
\(120\) −320.197 −0.243582
\(121\) 453.562 0.340768
\(122\) 568.514 0.421892
\(123\) 307.893 0.225705
\(124\) −243.246 −0.176162
\(125\) −1424.86 −1.01954
\(126\) 656.031 0.463841
\(127\) 296.445 0.207128 0.103564 0.994623i \(-0.466975\pi\)
0.103564 + 0.994623i \(0.466975\pi\)
\(128\) −1696.80 −1.17170
\(129\) 0 0
\(130\) −2616.82 −1.76546
\(131\) 1527.08 1.01848 0.509241 0.860624i \(-0.329926\pi\)
0.509241 + 0.860624i \(0.329926\pi\)
\(132\) −77.8155 −0.0513104
\(133\) −928.850 −0.605575
\(134\) 2187.35 1.41014
\(135\) 813.289 0.518495
\(136\) 1552.47 0.978848
\(137\) 1851.19 1.15444 0.577220 0.816589i \(-0.304138\pi\)
0.577220 + 0.816589i \(0.304138\pi\)
\(138\) −121.097 −0.0746990
\(139\) −624.022 −0.380783 −0.190392 0.981708i \(-0.560976\pi\)
−0.190392 + 0.981708i \(0.560976\pi\)
\(140\) 121.646 0.0734355
\(141\) 855.371 0.510888
\(142\) −1447.32 −0.855326
\(143\) 3313.32 1.93758
\(144\) 1812.92 1.04914
\(145\) −138.335 −0.0792282
\(146\) 3747.90 2.12451
\(147\) 384.067 0.215492
\(148\) −108.293 −0.0601459
\(149\) 2279.84 1.25350 0.626750 0.779220i \(-0.284385\pi\)
0.626750 + 0.779220i \(0.284385\pi\)
\(150\) −22.4662 −0.0122291
\(151\) 2948.04 1.58880 0.794398 0.607398i \(-0.207787\pi\)
0.794398 + 0.607398i \(0.207787\pi\)
\(152\) −2202.70 −1.17541
\(153\) −1894.02 −1.00080
\(154\) −1110.52 −0.581092
\(155\) −2067.09 −1.07118
\(156\) −144.477 −0.0741499
\(157\) 1507.11 0.766115 0.383058 0.923724i \(-0.374871\pi\)
0.383058 + 0.923724i \(0.374871\pi\)
\(158\) 3457.11 1.74071
\(159\) −526.868 −0.262788
\(160\) 632.319 0.312432
\(161\) −239.693 −0.117332
\(162\) −1729.75 −0.838902
\(163\) −2880.27 −1.38405 −0.692026 0.721873i \(-0.743281\pi\)
−0.692026 + 0.721873i \(0.743281\pi\)
\(164\) −277.389 −0.132076
\(165\) −661.272 −0.312000
\(166\) 1928.50 0.901689
\(167\) 2282.83 1.05779 0.528895 0.848688i \(-0.322607\pi\)
0.528895 + 0.848688i \(0.322607\pi\)
\(168\) −252.291 −0.115861
\(169\) 3954.69 1.80004
\(170\) −2532.19 −1.14241
\(171\) 2687.30 1.20177
\(172\) 0 0
\(173\) −1889.09 −0.830201 −0.415100 0.909776i \(-0.636254\pi\)
−0.415100 + 0.909776i \(0.636254\pi\)
\(174\) −55.0673 −0.0239922
\(175\) −44.4685 −0.0192086
\(176\) −3068.88 −1.31435
\(177\) 89.0602 0.0378202
\(178\) −20.3257 −0.00855883
\(179\) −1064.35 −0.444433 −0.222216 0.974997i \(-0.571329\pi\)
−0.222216 + 0.974997i \(0.571329\pi\)
\(180\) −351.940 −0.145734
\(181\) 1113.89 0.457429 0.228715 0.973494i \(-0.426548\pi\)
0.228715 + 0.973494i \(0.426548\pi\)
\(182\) −2061.85 −0.839751
\(183\) 266.735 0.107746
\(184\) −568.415 −0.227740
\(185\) −920.264 −0.365725
\(186\) −822.851 −0.324378
\(187\) 3206.17 1.25379
\(188\) −770.628 −0.298956
\(189\) 640.810 0.246625
\(190\) 3592.76 1.37182
\(191\) −600.424 −0.227461 −0.113731 0.993512i \(-0.536280\pi\)
−0.113731 + 0.993512i \(0.536280\pi\)
\(192\) −579.306 −0.217749
\(193\) −4418.79 −1.64804 −0.824019 0.566562i \(-0.808273\pi\)
−0.824019 + 0.566562i \(0.808273\pi\)
\(194\) 3121.89 1.15535
\(195\) −1227.76 −0.450879
\(196\) −346.017 −0.126099
\(197\) 2202.16 0.796434 0.398217 0.917291i \(-0.369629\pi\)
0.398217 + 0.917291i \(0.369629\pi\)
\(198\) 3212.90 1.15319
\(199\) −2962.08 −1.05516 −0.527579 0.849506i \(-0.676900\pi\)
−0.527579 + 0.849506i \(0.676900\pi\)
\(200\) −105.454 −0.0372835
\(201\) 1026.26 0.360133
\(202\) −3950.83 −1.37614
\(203\) −108.997 −0.0376853
\(204\) −139.804 −0.0479817
\(205\) −2357.24 −0.803106
\(206\) −91.8163 −0.0310541
\(207\) 693.468 0.232847
\(208\) −5697.85 −1.89940
\(209\) −4549.02 −1.50556
\(210\) 411.505 0.135221
\(211\) 3121.62 1.01849 0.509244 0.860622i \(-0.329925\pi\)
0.509244 + 0.860622i \(0.329925\pi\)
\(212\) 474.670 0.153776
\(213\) −679.051 −0.218440
\(214\) 4152.77 1.32653
\(215\) 0 0
\(216\) 1519.63 0.478694
\(217\) −1628.71 −0.509511
\(218\) 1211.82 0.376488
\(219\) 1758.44 0.542576
\(220\) 595.759 0.182573
\(221\) 5952.75 1.81188
\(222\) −366.332 −0.110750
\(223\) −1921.29 −0.576946 −0.288473 0.957488i \(-0.593148\pi\)
−0.288473 + 0.957488i \(0.593148\pi\)
\(224\) 498.219 0.148610
\(225\) 128.654 0.0381197
\(226\) −1062.66 −0.312774
\(227\) −548.261 −0.160306 −0.0801528 0.996783i \(-0.525541\pi\)
−0.0801528 + 0.996783i \(0.525541\pi\)
\(228\) 198.359 0.0576169
\(229\) 2286.66 0.659854 0.329927 0.944006i \(-0.392976\pi\)
0.329927 + 0.944006i \(0.392976\pi\)
\(230\) 927.124 0.265795
\(231\) −521.032 −0.148404
\(232\) −258.479 −0.0731465
\(233\) 4993.15 1.40392 0.701958 0.712218i \(-0.252309\pi\)
0.701958 + 0.712218i \(0.252309\pi\)
\(234\) 5965.25 1.66650
\(235\) −6548.75 −1.81785
\(236\) −80.2368 −0.0221312
\(237\) 1622.00 0.444558
\(238\) −1995.17 −0.543394
\(239\) −3806.31 −1.03017 −0.515083 0.857140i \(-0.672239\pi\)
−0.515083 + 0.857140i \(0.672239\pi\)
\(240\) 1137.18 0.305852
\(241\) −6268.95 −1.67559 −0.837797 0.545981i \(-0.816157\pi\)
−0.837797 + 0.545981i \(0.816157\pi\)
\(242\) −1382.30 −0.367181
\(243\) −2817.42 −0.743775
\(244\) −240.309 −0.0630500
\(245\) −2940.43 −0.766765
\(246\) −938.352 −0.243200
\(247\) −8445.96 −2.17572
\(248\) −3862.36 −0.988953
\(249\) 904.810 0.230281
\(250\) 4342.48 1.09857
\(251\) 938.494 0.236005 0.118002 0.993013i \(-0.462351\pi\)
0.118002 + 0.993013i \(0.462351\pi\)
\(252\) −277.302 −0.0693190
\(253\) −1173.89 −0.291707
\(254\) −903.464 −0.223183
\(255\) −1188.05 −0.291759
\(256\) 1930.17 0.471234
\(257\) −7223.21 −1.75320 −0.876598 0.481223i \(-0.840193\pi\)
−0.876598 + 0.481223i \(0.840193\pi\)
\(258\) 0 0
\(259\) −725.098 −0.173959
\(260\) 1106.12 0.263841
\(261\) 315.346 0.0747870
\(262\) −4654.01 −1.09743
\(263\) 6840.18 1.60374 0.801870 0.597498i \(-0.203838\pi\)
0.801870 + 0.597498i \(0.203838\pi\)
\(264\) −1235.59 −0.288050
\(265\) 4033.72 0.935055
\(266\) 2830.82 0.652514
\(267\) −9.53637 −0.00218583
\(268\) −924.587 −0.210739
\(269\) −3969.81 −0.899790 −0.449895 0.893081i \(-0.648539\pi\)
−0.449895 + 0.893081i \(0.648539\pi\)
\(270\) −2478.63 −0.558684
\(271\) −7439.71 −1.66764 −0.833820 0.552037i \(-0.813851\pi\)
−0.833820 + 0.552037i \(0.813851\pi\)
\(272\) −5513.59 −1.22908
\(273\) −967.378 −0.214463
\(274\) −5641.81 −1.24392
\(275\) −217.783 −0.0477557
\(276\) 51.1873 0.0111635
\(277\) −627.394 −0.136088 −0.0680442 0.997682i \(-0.521676\pi\)
−0.0680442 + 0.997682i \(0.521676\pi\)
\(278\) 1901.81 0.410298
\(279\) 4712.10 1.01113
\(280\) 1931.55 0.412258
\(281\) 8656.81 1.83780 0.918900 0.394489i \(-0.129079\pi\)
0.918900 + 0.394489i \(0.129079\pi\)
\(282\) −2606.88 −0.550487
\(283\) −5735.35 −1.20470 −0.602352 0.798231i \(-0.705769\pi\)
−0.602352 + 0.798231i \(0.705769\pi\)
\(284\) 611.776 0.127825
\(285\) 1685.65 0.350347
\(286\) −10097.9 −2.08776
\(287\) −1857.32 −0.382001
\(288\) −1441.42 −0.294919
\(289\) 847.243 0.172449
\(290\) 421.598 0.0853692
\(291\) 1464.73 0.295064
\(292\) −1584.22 −0.317499
\(293\) −2132.59 −0.425212 −0.212606 0.977138i \(-0.568195\pi\)
−0.212606 + 0.977138i \(0.568195\pi\)
\(294\) −1170.51 −0.232195
\(295\) −681.848 −0.134572
\(296\) −1719.52 −0.337652
\(297\) 3138.35 0.613150
\(298\) −6948.17 −1.35066
\(299\) −2179.51 −0.421553
\(300\) 9.49640 0.00182758
\(301\) 0 0
\(302\) −8984.63 −1.71194
\(303\) −1853.65 −0.351449
\(304\) 7822.86 1.47589
\(305\) −2042.13 −0.383384
\(306\) 5772.33 1.07837
\(307\) −2974.73 −0.553020 −0.276510 0.961011i \(-0.589178\pi\)
−0.276510 + 0.961011i \(0.589178\pi\)
\(308\) 469.413 0.0868418
\(309\) −43.0783 −0.00793087
\(310\) 6299.78 1.15421
\(311\) −8271.14 −1.50808 −0.754040 0.656828i \(-0.771898\pi\)
−0.754040 + 0.656828i \(0.771898\pi\)
\(312\) −2294.07 −0.416269
\(313\) 4256.50 0.768664 0.384332 0.923195i \(-0.374432\pi\)
0.384332 + 0.923195i \(0.374432\pi\)
\(314\) −4593.15 −0.825498
\(315\) −2356.50 −0.421504
\(316\) −1461.31 −0.260142
\(317\) −5331.95 −0.944708 −0.472354 0.881409i \(-0.656596\pi\)
−0.472354 + 0.881409i \(0.656596\pi\)
\(318\) 1605.71 0.283157
\(319\) −533.812 −0.0936919
\(320\) 4435.19 0.774796
\(321\) 1948.39 0.338781
\(322\) 730.503 0.126427
\(323\) −8172.83 −1.40789
\(324\) 731.160 0.125370
\(325\) −404.349 −0.0690130
\(326\) 8778.09 1.49133
\(327\) 568.558 0.0961509
\(328\) −4404.51 −0.741458
\(329\) −5159.92 −0.864668
\(330\) 2015.33 0.336183
\(331\) 4689.05 0.778650 0.389325 0.921100i \(-0.372708\pi\)
0.389325 + 0.921100i \(0.372708\pi\)
\(332\) −815.169 −0.134754
\(333\) 2097.82 0.345224
\(334\) −6957.29 −1.13978
\(335\) −7857.09 −1.28143
\(336\) 896.009 0.145480
\(337\) 11172.1 1.80588 0.902938 0.429770i \(-0.141405\pi\)
0.902938 + 0.429770i \(0.141405\pi\)
\(338\) −12052.6 −1.93957
\(339\) −498.576 −0.0798789
\(340\) 1070.35 0.170729
\(341\) −7976.56 −1.26673
\(342\) −8189.98 −1.29492
\(343\) −5275.45 −0.830459
\(344\) 0 0
\(345\) 434.987 0.0678809
\(346\) 5757.30 0.894550
\(347\) 7176.25 1.11020 0.555102 0.831782i \(-0.312679\pi\)
0.555102 + 0.831782i \(0.312679\pi\)
\(348\) 23.2768 0.00358553
\(349\) −8305.35 −1.27385 −0.636927 0.770924i \(-0.719795\pi\)
−0.636927 + 0.770924i \(0.719795\pi\)
\(350\) 135.525 0.0206975
\(351\) 5826.84 0.886079
\(352\) 2440.02 0.369469
\(353\) 5746.60 0.866461 0.433231 0.901283i \(-0.357374\pi\)
0.433231 + 0.901283i \(0.357374\pi\)
\(354\) −271.425 −0.0407516
\(355\) 5198.84 0.777256
\(356\) 8.59158 0.00127908
\(357\) −936.093 −0.138777
\(358\) 3243.78 0.478881
\(359\) 1277.28 0.187778 0.0938891 0.995583i \(-0.470070\pi\)
0.0938891 + 0.995583i \(0.470070\pi\)
\(360\) −5588.27 −0.818132
\(361\) 4736.88 0.690608
\(362\) −3394.75 −0.492885
\(363\) −648.548 −0.0937739
\(364\) 871.538 0.125497
\(365\) −13462.7 −1.93060
\(366\) −812.917 −0.116098
\(367\) −3186.62 −0.453243 −0.226622 0.973983i \(-0.572768\pi\)
−0.226622 + 0.973983i \(0.572768\pi\)
\(368\) 2018.72 0.285959
\(369\) 5373.52 0.758087
\(370\) 2804.65 0.394073
\(371\) 3178.27 0.444764
\(372\) 347.816 0.0484770
\(373\) 10717.8 1.48779 0.743894 0.668298i \(-0.232977\pi\)
0.743894 + 0.668298i \(0.232977\pi\)
\(374\) −9771.31 −1.35097
\(375\) 2037.40 0.280562
\(376\) −12236.4 −1.67831
\(377\) −991.105 −0.135397
\(378\) −1952.97 −0.265741
\(379\) 1001.31 0.135709 0.0678544 0.997695i \(-0.478385\pi\)
0.0678544 + 0.997695i \(0.478385\pi\)
\(380\) −1518.65 −0.205013
\(381\) −423.886 −0.0569983
\(382\) 1829.89 0.245092
\(383\) −2126.29 −0.283677 −0.141838 0.989890i \(-0.545301\pi\)
−0.141838 + 0.989890i \(0.545301\pi\)
\(384\) 2426.25 0.322433
\(385\) 3989.04 0.528053
\(386\) 13467.0 1.77578
\(387\) 0 0
\(388\) −1319.61 −0.172663
\(389\) −11830.9 −1.54203 −0.771014 0.636819i \(-0.780250\pi\)
−0.771014 + 0.636819i \(0.780250\pi\)
\(390\) 3741.78 0.485827
\(391\) −2109.03 −0.272783
\(392\) −5494.21 −0.707907
\(393\) −2183.56 −0.280270
\(394\) −6711.44 −0.858166
\(395\) −12418.1 −1.58183
\(396\) −1358.08 −0.172339
\(397\) −4635.30 −0.585992 −0.292996 0.956114i \(-0.594652\pi\)
−0.292996 + 0.956114i \(0.594652\pi\)
\(398\) 9027.43 1.13694
\(399\) 1328.16 0.166645
\(400\) 374.518 0.0468147
\(401\) −12934.8 −1.61081 −0.805406 0.592724i \(-0.798052\pi\)
−0.805406 + 0.592724i \(0.798052\pi\)
\(402\) −3127.69 −0.388048
\(403\) −14809.7 −1.83058
\(404\) 1670.00 0.205658
\(405\) 6213.36 0.762332
\(406\) 332.187 0.0406063
\(407\) −3551.15 −0.432491
\(408\) −2219.88 −0.269363
\(409\) 1367.64 0.165343 0.0826717 0.996577i \(-0.473655\pi\)
0.0826717 + 0.996577i \(0.473655\pi\)
\(410\) 7184.06 0.865355
\(411\) −2647.02 −0.317683
\(412\) 38.8104 0.00464091
\(413\) −537.245 −0.0640099
\(414\) −2113.45 −0.250895
\(415\) −6927.26 −0.819388
\(416\) 4530.27 0.533930
\(417\) 892.288 0.104785
\(418\) 13863.9 1.62226
\(419\) −11334.8 −1.32158 −0.660790 0.750571i \(-0.729779\pi\)
−0.660790 + 0.750571i \(0.729779\pi\)
\(420\) −173.942 −0.0202083
\(421\) 16468.4 1.90646 0.953231 0.302242i \(-0.0977350\pi\)
0.953231 + 0.302242i \(0.0977350\pi\)
\(422\) −9513.63 −1.09743
\(423\) 14928.4 1.71594
\(424\) 7537.03 0.863279
\(425\) −391.272 −0.0446576
\(426\) 2069.52 0.235372
\(427\) −1609.05 −0.182359
\(428\) −1755.36 −0.198245
\(429\) −4737.71 −0.533190
\(430\) 0 0
\(431\) 10893.5 1.21745 0.608726 0.793380i \(-0.291681\pi\)
0.608726 + 0.793380i \(0.291681\pi\)
\(432\) −5396.96 −0.601068
\(433\) −7492.45 −0.831557 −0.415779 0.909466i \(-0.636491\pi\)
−0.415779 + 0.909466i \(0.636491\pi\)
\(434\) 4963.75 0.549004
\(435\) 197.805 0.0218023
\(436\) −512.230 −0.0562646
\(437\) 2992.36 0.327561
\(438\) −5359.12 −0.584631
\(439\) 4521.53 0.491574 0.245787 0.969324i \(-0.420954\pi\)
0.245787 + 0.969324i \(0.420954\pi\)
\(440\) 9459.72 1.02494
\(441\) 6702.96 0.723783
\(442\) −18142.0 −1.95232
\(443\) −8805.39 −0.944372 −0.472186 0.881499i \(-0.656535\pi\)
−0.472186 + 0.881499i \(0.656535\pi\)
\(444\) 154.847 0.0165512
\(445\) 73.0108 0.00777763
\(446\) 5855.43 0.621665
\(447\) −3259.94 −0.344943
\(448\) 3494.59 0.368536
\(449\) 10437.9 1.09709 0.548545 0.836121i \(-0.315182\pi\)
0.548545 + 0.836121i \(0.315182\pi\)
\(450\) −392.094 −0.0410744
\(451\) −9096.20 −0.949719
\(452\) 449.181 0.0467427
\(453\) −4215.40 −0.437211
\(454\) 1670.91 0.172731
\(455\) 7406.28 0.763103
\(456\) 3149.64 0.323454
\(457\) −3984.75 −0.407875 −0.203938 0.978984i \(-0.565374\pi\)
−0.203938 + 0.978984i \(0.565374\pi\)
\(458\) −6968.96 −0.711000
\(459\) 5638.40 0.573373
\(460\) −391.892 −0.0397219
\(461\) −5737.15 −0.579622 −0.289811 0.957084i \(-0.593592\pi\)
−0.289811 + 0.957084i \(0.593592\pi\)
\(462\) 1587.93 0.159907
\(463\) 1479.27 0.148483 0.0742415 0.997240i \(-0.476346\pi\)
0.0742415 + 0.997240i \(0.476346\pi\)
\(464\) 917.986 0.0918457
\(465\) 2955.73 0.294771
\(466\) −15217.4 −1.51273
\(467\) 7229.61 0.716373 0.358187 0.933650i \(-0.383395\pi\)
0.358187 + 0.933650i \(0.383395\pi\)
\(468\) −2521.49 −0.249051
\(469\) −6190.79 −0.609518
\(470\) 19958.4 1.95875
\(471\) −2155.01 −0.210823
\(472\) −1274.04 −0.124242
\(473\) 0 0
\(474\) −4943.31 −0.479016
\(475\) 555.150 0.0536254
\(476\) 843.353 0.0812080
\(477\) −9195.20 −0.882640
\(478\) 11600.3 1.11002
\(479\) 11043.4 1.05342 0.526708 0.850046i \(-0.323426\pi\)
0.526708 + 0.850046i \(0.323426\pi\)
\(480\) −904.151 −0.0859764
\(481\) −6593.27 −0.625004
\(482\) 19105.6 1.80547
\(483\) 342.737 0.0322879
\(484\) 584.295 0.0548737
\(485\) −11214.0 −1.04990
\(486\) 8586.53 0.801426
\(487\) 4969.09 0.462364 0.231182 0.972911i \(-0.425741\pi\)
0.231182 + 0.972911i \(0.425741\pi\)
\(488\) −3815.73 −0.353955
\(489\) 4118.50 0.380869
\(490\) 8961.44 0.826197
\(491\) 7549.09 0.693861 0.346931 0.937891i \(-0.387224\pi\)
0.346931 + 0.937891i \(0.387224\pi\)
\(492\) 396.638 0.0363452
\(493\) −959.053 −0.0876138
\(494\) 25740.4 2.34437
\(495\) −11540.9 −1.04793
\(496\) 13717.1 1.24177
\(497\) 4096.29 0.369706
\(498\) −2757.55 −0.248130
\(499\) 2644.30 0.237225 0.118612 0.992941i \(-0.462155\pi\)
0.118612 + 0.992941i \(0.462155\pi\)
\(500\) −1835.55 −0.164177
\(501\) −3264.22 −0.291087
\(502\) −2860.21 −0.254298
\(503\) 18851.7 1.67109 0.835544 0.549424i \(-0.185153\pi\)
0.835544 + 0.549424i \(0.185153\pi\)
\(504\) −4403.13 −0.389149
\(505\) 14191.6 1.25053
\(506\) 3577.62 0.314318
\(507\) −5654.81 −0.495343
\(508\) 381.891 0.0333537
\(509\) −7646.38 −0.665855 −0.332927 0.942952i \(-0.608036\pi\)
−0.332927 + 0.942952i \(0.608036\pi\)
\(510\) 3620.77 0.314374
\(511\) −10607.6 −0.918298
\(512\) 7691.90 0.663940
\(513\) −7999.96 −0.688512
\(514\) 22013.9 1.88909
\(515\) 329.809 0.0282197
\(516\) 0 0
\(517\) −25270.6 −2.14971
\(518\) 2209.85 0.187443
\(519\) 2701.20 0.228458
\(520\) 17563.5 1.48117
\(521\) −15427.5 −1.29729 −0.648646 0.761090i \(-0.724664\pi\)
−0.648646 + 0.761090i \(0.724664\pi\)
\(522\) −961.066 −0.0805838
\(523\) −14469.9 −1.20980 −0.604899 0.796302i \(-0.706786\pi\)
−0.604899 + 0.796302i \(0.706786\pi\)
\(524\) 1967.23 0.164006
\(525\) 63.5854 0.00528589
\(526\) −20846.6 −1.72805
\(527\) −14330.8 −1.18455
\(528\) 4388.18 0.361688
\(529\) −11394.8 −0.936534
\(530\) −12293.4 −1.00753
\(531\) 1554.33 0.127028
\(532\) −1196.58 −0.0975154
\(533\) −16888.5 −1.37246
\(534\) 29.0636 0.00235525
\(535\) −14917.0 −1.20545
\(536\) −14681.0 −1.18306
\(537\) 1521.92 0.122301
\(538\) 12098.6 0.969534
\(539\) −11346.7 −0.906744
\(540\) 1047.71 0.0834929
\(541\) −2852.57 −0.226694 −0.113347 0.993555i \(-0.536157\pi\)
−0.113347 + 0.993555i \(0.536157\pi\)
\(542\) 22673.7 1.79690
\(543\) −1592.75 −0.125877
\(544\) 4383.76 0.345501
\(545\) −4352.90 −0.342125
\(546\) 2948.24 0.231086
\(547\) 1910.22 0.149315 0.0746574 0.997209i \(-0.476214\pi\)
0.0746574 + 0.997209i \(0.476214\pi\)
\(548\) 2384.77 0.185899
\(549\) 4655.21 0.361893
\(550\) 663.730 0.0514573
\(551\) 1360.74 0.105208
\(552\) 812.775 0.0626703
\(553\) −9784.53 −0.752406
\(554\) 1912.09 0.146637
\(555\) 1315.88 0.100642
\(556\) −803.888 −0.0613173
\(557\) −1503.15 −0.114345 −0.0571726 0.998364i \(-0.518209\pi\)
−0.0571726 + 0.998364i \(0.518209\pi\)
\(558\) −14360.9 −1.08951
\(559\) 0 0
\(560\) −6859.88 −0.517648
\(561\) −4584.49 −0.345022
\(562\) −26383.0 −1.98025
\(563\) −7608.15 −0.569530 −0.284765 0.958597i \(-0.591916\pi\)
−0.284765 + 0.958597i \(0.591916\pi\)
\(564\) 1101.92 0.0822680
\(565\) 3817.12 0.284226
\(566\) 17479.4 1.29808
\(567\) 4895.66 0.362607
\(568\) 9714.06 0.717593
\(569\) 5914.01 0.435726 0.217863 0.975979i \(-0.430091\pi\)
0.217863 + 0.975979i \(0.430091\pi\)
\(570\) −5137.28 −0.377503
\(571\) −7673.35 −0.562381 −0.281191 0.959652i \(-0.590729\pi\)
−0.281191 + 0.959652i \(0.590729\pi\)
\(572\) 4268.34 0.312007
\(573\) 858.545 0.0625938
\(574\) 5660.49 0.411611
\(575\) 143.259 0.0103901
\(576\) −10110.4 −0.731364
\(577\) −14776.8 −1.06615 −0.533074 0.846069i \(-0.678963\pi\)
−0.533074 + 0.846069i \(0.678963\pi\)
\(578\) −2582.11 −0.185816
\(579\) 6318.42 0.453514
\(580\) −178.208 −0.0127581
\(581\) −5458.16 −0.389746
\(582\) −4463.99 −0.317935
\(583\) 15565.5 1.10576
\(584\) −25155.0 −1.78240
\(585\) −21427.5 −1.51439
\(586\) 6499.41 0.458171
\(587\) 7695.08 0.541073 0.270537 0.962710i \(-0.412799\pi\)
0.270537 + 0.962710i \(0.412799\pi\)
\(588\) 494.769 0.0347006
\(589\) 20333.0 1.42242
\(590\) 2078.04 0.145003
\(591\) −3148.87 −0.219166
\(592\) 6106.85 0.423969
\(593\) 20083.7 1.39079 0.695395 0.718628i \(-0.255229\pi\)
0.695395 + 0.718628i \(0.255229\pi\)
\(594\) −9564.63 −0.660676
\(595\) 7166.77 0.493796
\(596\) 2936.97 0.201851
\(597\) 4235.48 0.290363
\(598\) 6642.41 0.454228
\(599\) −15219.6 −1.03815 −0.519077 0.854727i \(-0.673724\pi\)
−0.519077 + 0.854727i \(0.673724\pi\)
\(600\) 150.788 0.0102598
\(601\) −2102.90 −0.142727 −0.0713636 0.997450i \(-0.522735\pi\)
−0.0713636 + 0.997450i \(0.522735\pi\)
\(602\) 0 0
\(603\) 17910.9 1.20960
\(604\) 3797.77 0.255843
\(605\) 4965.31 0.333667
\(606\) 5649.28 0.378691
\(607\) −11150.6 −0.745617 −0.372808 0.927908i \(-0.621605\pi\)
−0.372808 + 0.927908i \(0.621605\pi\)
\(608\) −6219.83 −0.414881
\(609\) 155.855 0.0103704
\(610\) 6223.73 0.413101
\(611\) −46918.8 −3.10660
\(612\) −2439.95 −0.161158
\(613\) 6496.14 0.428021 0.214010 0.976831i \(-0.431347\pi\)
0.214010 + 0.976831i \(0.431347\pi\)
\(614\) 9065.98 0.595885
\(615\) 3370.61 0.221002
\(616\) 7453.54 0.487519
\(617\) −5614.43 −0.366335 −0.183167 0.983082i \(-0.558635\pi\)
−0.183167 + 0.983082i \(0.558635\pi\)
\(618\) 131.288 0.00854559
\(619\) 4482.99 0.291093 0.145546 0.989351i \(-0.453506\pi\)
0.145546 + 0.989351i \(0.453506\pi\)
\(620\) −2662.90 −0.172491
\(621\) −2064.42 −0.133401
\(622\) 25207.6 1.62497
\(623\) 57.5270 0.00369947
\(624\) 8147.35 0.522684
\(625\) −14954.0 −0.957056
\(626\) −12972.4 −0.828243
\(627\) 6504.63 0.414306
\(628\) 1941.51 0.123367
\(629\) −6380.04 −0.404434
\(630\) 7181.81 0.454175
\(631\) 22320.4 1.40818 0.704088 0.710112i \(-0.251356\pi\)
0.704088 + 0.710112i \(0.251356\pi\)
\(632\) −23203.3 −1.46041
\(633\) −4463.59 −0.280272
\(634\) 16250.0 1.01793
\(635\) 3245.29 0.202812
\(636\) −678.730 −0.0423167
\(637\) −21066.8 −1.31036
\(638\) 1626.88 0.100954
\(639\) −11851.2 −0.733686
\(640\) −18575.5 −1.14728
\(641\) −6797.76 −0.418869 −0.209435 0.977823i \(-0.567162\pi\)
−0.209435 + 0.977823i \(0.567162\pi\)
\(642\) −5938.04 −0.365040
\(643\) −6933.22 −0.425225 −0.212612 0.977137i \(-0.568197\pi\)
−0.212612 + 0.977137i \(0.568197\pi\)
\(644\) −308.781 −0.0188939
\(645\) 0 0
\(646\) 24908.0 1.51702
\(647\) 14748.6 0.896180 0.448090 0.893989i \(-0.352104\pi\)
0.448090 + 0.893989i \(0.352104\pi\)
\(648\) 11609.7 0.703814
\(649\) −2631.14 −0.159139
\(650\) 1232.32 0.0743623
\(651\) 2328.89 0.140209
\(652\) −3710.47 −0.222873
\(653\) 10205.7 0.611605 0.305803 0.952095i \(-0.401075\pi\)
0.305803 + 0.952095i \(0.401075\pi\)
\(654\) −1732.77 −0.103604
\(655\) 16717.4 0.997259
\(656\) 15642.6 0.931005
\(657\) 30689.2 1.82238
\(658\) 15725.7 0.931689
\(659\) 25166.3 1.48762 0.743809 0.668392i \(-0.233017\pi\)
0.743809 + 0.668392i \(0.233017\pi\)
\(660\) −851.874 −0.0502412
\(661\) 18594.5 1.09416 0.547080 0.837080i \(-0.315739\pi\)
0.547080 + 0.837080i \(0.315739\pi\)
\(662\) −14290.6 −0.839004
\(663\) −8511.83 −0.498600
\(664\) −12943.6 −0.756491
\(665\) −10168.5 −0.592956
\(666\) −6393.43 −0.371983
\(667\) 351.143 0.0203843
\(668\) 2940.82 0.170335
\(669\) 2747.24 0.158766
\(670\) 23945.7 1.38075
\(671\) −7880.26 −0.453374
\(672\) −712.402 −0.0408951
\(673\) 25367.1 1.45294 0.726471 0.687197i \(-0.241159\pi\)
0.726471 + 0.687197i \(0.241159\pi\)
\(674\) −34048.6 −1.94585
\(675\) −382.996 −0.0218393
\(676\) 5094.58 0.289860
\(677\) −8839.77 −0.501831 −0.250916 0.968009i \(-0.580732\pi\)
−0.250916 + 0.968009i \(0.580732\pi\)
\(678\) 1519.49 0.0860704
\(679\) −8835.78 −0.499391
\(680\) 16995.5 0.958450
\(681\) 783.957 0.0441135
\(682\) 24309.8 1.36492
\(683\) 11252.9 0.630426 0.315213 0.949021i \(-0.397924\pi\)
0.315213 + 0.949021i \(0.397924\pi\)
\(684\) 3461.88 0.193521
\(685\) 20265.7 1.13038
\(686\) 16077.8 0.894828
\(687\) −3269.69 −0.181581
\(688\) 0 0
\(689\) 28899.7 1.59796
\(690\) −1325.69 −0.0731424
\(691\) −19933.8 −1.09742 −0.548711 0.836012i \(-0.684881\pi\)
−0.548711 + 0.836012i \(0.684881\pi\)
\(692\) −2433.59 −0.133687
\(693\) −9093.35 −0.498453
\(694\) −21870.8 −1.19626
\(695\) −6831.40 −0.372848
\(696\) 369.599 0.0201287
\(697\) −16342.4 −0.888107
\(698\) 25311.9 1.37259
\(699\) −7139.70 −0.386335
\(700\) −57.2859 −0.00309315
\(701\) 13753.9 0.741052 0.370526 0.928822i \(-0.379177\pi\)
0.370526 + 0.928822i \(0.379177\pi\)
\(702\) −17758.2 −0.954760
\(703\) 9052.22 0.485649
\(704\) 17114.7 0.916241
\(705\) 9364.05 0.500242
\(706\) −17513.7 −0.933621
\(707\) 11181.9 0.594821
\(708\) 114.730 0.00609016
\(709\) −22572.7 −1.19568 −0.597840 0.801616i \(-0.703974\pi\)
−0.597840 + 0.801616i \(0.703974\pi\)
\(710\) −15844.3 −0.837502
\(711\) 28308.1 1.49316
\(712\) 136.421 0.00718061
\(713\) 5247.01 0.275599
\(714\) 2852.89 0.149533
\(715\) 36272.1 1.89720
\(716\) −1371.14 −0.0715667
\(717\) 5442.63 0.283485
\(718\) −3892.72 −0.202333
\(719\) −12405.4 −0.643456 −0.321728 0.946832i \(-0.604264\pi\)
−0.321728 + 0.946832i \(0.604264\pi\)
\(720\) 19846.7 1.02728
\(721\) 259.865 0.0134228
\(722\) −14436.4 −0.744138
\(723\) 8963.95 0.461097
\(724\) 1434.95 0.0736595
\(725\) 65.1450 0.00333714
\(726\) 1976.55 0.101042
\(727\) −20179.9 −1.02948 −0.514740 0.857346i \(-0.672112\pi\)
−0.514740 + 0.857346i \(0.672112\pi\)
\(728\) 13838.7 0.704526
\(729\) −11295.7 −0.573881
\(730\) 41029.6 2.08024
\(731\) 0 0
\(732\) 343.617 0.0173504
\(733\) 36311.8 1.82975 0.914875 0.403736i \(-0.132289\pi\)
0.914875 + 0.403736i \(0.132289\pi\)
\(734\) 9711.74 0.488374
\(735\) 4204.52 0.211001
\(736\) −1605.05 −0.0803844
\(737\) −30319.2 −1.51536
\(738\) −16376.7 −0.816847
\(739\) 6848.62 0.340907 0.170454 0.985366i \(-0.445477\pi\)
0.170454 + 0.985366i \(0.445477\pi\)
\(740\) −1185.52 −0.0588925
\(741\) 12076.9 0.598724
\(742\) −9686.28 −0.479238
\(743\) 10685.7 0.527616 0.263808 0.964575i \(-0.415022\pi\)
0.263808 + 0.964575i \(0.415022\pi\)
\(744\) 5522.78 0.272144
\(745\) 24958.2 1.22738
\(746\) −32664.1 −1.60311
\(747\) 15791.3 0.773457
\(748\) 4130.30 0.201897
\(749\) −11753.4 −0.573380
\(750\) −6209.31 −0.302309
\(751\) 17077.6 0.829786 0.414893 0.909870i \(-0.363819\pi\)
0.414893 + 0.909870i \(0.363819\pi\)
\(752\) 43457.3 2.10735
\(753\) −1341.95 −0.0649447
\(754\) 3020.55 0.145891
\(755\) 32273.3 1.55569
\(756\) 825.514 0.0397138
\(757\) −13129.5 −0.630381 −0.315190 0.949028i \(-0.602068\pi\)
−0.315190 + 0.949028i \(0.602068\pi\)
\(758\) −3051.64 −0.146228
\(759\) 1678.54 0.0802731
\(760\) −24113.7 −1.15092
\(761\) 3098.58 0.147600 0.0737999 0.997273i \(-0.476487\pi\)
0.0737999 + 0.997273i \(0.476487\pi\)
\(762\) 1291.86 0.0614163
\(763\) −3429.76 −0.162733
\(764\) −773.487 −0.0366280
\(765\) −20734.5 −0.979946
\(766\) 6480.20 0.305665
\(767\) −4885.13 −0.229976
\(768\) −2759.95 −0.129676
\(769\) 38020.8 1.78292 0.891460 0.453099i \(-0.149682\pi\)
0.891460 + 0.453099i \(0.149682\pi\)
\(770\) −12157.3 −0.568983
\(771\) 10328.5 0.482451
\(772\) −5692.44 −0.265383
\(773\) 703.368 0.0327275 0.0163638 0.999866i \(-0.494791\pi\)
0.0163638 + 0.999866i \(0.494791\pi\)
\(774\) 0 0
\(775\) 973.438 0.0451186
\(776\) −20953.4 −0.969309
\(777\) 1036.82 0.0478708
\(778\) 36056.5 1.66155
\(779\) 23187.1 1.06645
\(780\) −1581.64 −0.0726048
\(781\) 20061.5 0.919151
\(782\) 6427.60 0.293927
\(783\) −938.767 −0.0428465
\(784\) 19512.6 0.888877
\(785\) 16498.8 0.750151
\(786\) 6654.76 0.301994
\(787\) 24681.7 1.11792 0.558962 0.829193i \(-0.311200\pi\)
0.558962 + 0.829193i \(0.311200\pi\)
\(788\) 2836.90 0.128249
\(789\) −9780.77 −0.441324
\(790\) 37846.2 1.70444
\(791\) 3007.60 0.135193
\(792\) −21564.2 −0.967488
\(793\) −14630.9 −0.655183
\(794\) 14126.8 0.631413
\(795\) −5767.81 −0.257312
\(796\) −3815.86 −0.169912
\(797\) −23603.5 −1.04903 −0.524516 0.851401i \(-0.675754\pi\)
−0.524516 + 0.851401i \(0.675754\pi\)
\(798\) −4047.78 −0.179561
\(799\) −45401.5 −2.01025
\(800\) −297.773 −0.0131598
\(801\) −166.434 −0.00734165
\(802\) 39421.0 1.73567
\(803\) −51950.2 −2.28304
\(804\) 1322.06 0.0579921
\(805\) −2624.01 −0.114887
\(806\) 45135.0 1.97247
\(807\) 5676.42 0.247608
\(808\) 26517.0 1.15454
\(809\) 22689.9 0.986075 0.493038 0.870008i \(-0.335887\pi\)
0.493038 + 0.870008i \(0.335887\pi\)
\(810\) −18936.2 −0.821421
\(811\) 22849.7 0.989347 0.494674 0.869079i \(-0.335288\pi\)
0.494674 + 0.869079i \(0.335288\pi\)
\(812\) −140.414 −0.00606844
\(813\) 10638.0 0.458908
\(814\) 10822.7 0.466014
\(815\) −31531.4 −1.35521
\(816\) 7883.86 0.338224
\(817\) 0 0
\(818\) −4168.10 −0.178159
\(819\) −16883.2 −0.720327
\(820\) −3036.68 −0.129324
\(821\) −12956.1 −0.550758 −0.275379 0.961336i \(-0.588803\pi\)
−0.275379 + 0.961336i \(0.588803\pi\)
\(822\) 8067.22 0.342307
\(823\) 41910.8 1.77511 0.887557 0.460699i \(-0.152401\pi\)
0.887557 + 0.460699i \(0.152401\pi\)
\(824\) 616.250 0.0260535
\(825\) 311.408 0.0131416
\(826\) 1637.34 0.0689713
\(827\) 11697.3 0.491846 0.245923 0.969289i \(-0.420909\pi\)
0.245923 + 0.969289i \(0.420909\pi\)
\(828\) 893.350 0.0374952
\(829\) 8944.81 0.374748 0.187374 0.982289i \(-0.440002\pi\)
0.187374 + 0.982289i \(0.440002\pi\)
\(830\) 21112.0 0.882899
\(831\) 897.110 0.0374493
\(832\) 31776.1 1.32408
\(833\) −20385.5 −0.847920
\(834\) −2719.39 −0.112907
\(835\) 24991.0 1.03575
\(836\) −5860.21 −0.242440
\(837\) −14027.7 −0.579292
\(838\) 34544.7 1.42402
\(839\) 8980.83 0.369550 0.184775 0.982781i \(-0.440844\pi\)
0.184775 + 0.982781i \(0.440844\pi\)
\(840\) −2761.92 −0.113447
\(841\) −24229.3 −0.993453
\(842\) −50190.1 −2.05423
\(843\) −12378.4 −0.505733
\(844\) 4021.38 0.164007
\(845\) 43293.5 1.76253
\(846\) −45496.7 −1.84895
\(847\) 3912.29 0.158710
\(848\) −26767.7 −1.08397
\(849\) 8200.96 0.331515
\(850\) 1192.47 0.0481191
\(851\) 2335.96 0.0940959
\(852\) −874.778 −0.0351753
\(853\) 28833.8 1.15739 0.578694 0.815545i \(-0.303563\pi\)
0.578694 + 0.815545i \(0.303563\pi\)
\(854\) 4903.83 0.196493
\(855\) 29418.8 1.17673
\(856\) −27872.4 −1.11292
\(857\) −42317.7 −1.68675 −0.843375 0.537325i \(-0.819435\pi\)
−0.843375 + 0.537325i \(0.819435\pi\)
\(858\) 14438.9 0.574518
\(859\) −46572.9 −1.84988 −0.924939 0.380115i \(-0.875884\pi\)
−0.924939 + 0.380115i \(0.875884\pi\)
\(860\) 0 0
\(861\) 2655.78 0.105121
\(862\) −33199.7 −1.31182
\(863\) −10911.3 −0.430388 −0.215194 0.976571i \(-0.569038\pi\)
−0.215194 + 0.976571i \(0.569038\pi\)
\(864\) 4291.04 0.168963
\(865\) −20680.5 −0.812901
\(866\) 22834.4 0.896012
\(867\) −1211.47 −0.0474552
\(868\) −2098.16 −0.0820463
\(869\) −47919.5 −1.87061
\(870\) −602.842 −0.0234922
\(871\) −56292.4 −2.18989
\(872\) −8133.42 −0.315863
\(873\) 25563.2 0.991047
\(874\) −9119.70 −0.352950
\(875\) −12290.4 −0.474846
\(876\) 2265.28 0.0873707
\(877\) −12978.3 −0.499710 −0.249855 0.968283i \(-0.580383\pi\)
−0.249855 + 0.968283i \(0.580383\pi\)
\(878\) −13780.1 −0.529676
\(879\) 3049.39 0.117012
\(880\) −33596.1 −1.28696
\(881\) 16494.3 0.630768 0.315384 0.948964i \(-0.397867\pi\)
0.315384 + 0.948964i \(0.397867\pi\)
\(882\) −20428.3 −0.779884
\(883\) −27803.5 −1.05964 −0.529821 0.848110i \(-0.677741\pi\)
−0.529821 + 0.848110i \(0.677741\pi\)
\(884\) 7668.55 0.291766
\(885\) 974.973 0.0370321
\(886\) 26835.9 1.01757
\(887\) 26424.1 1.00026 0.500132 0.865949i \(-0.333285\pi\)
0.500132 + 0.865949i \(0.333285\pi\)
\(888\) 2458.73 0.0929164
\(889\) 2557.04 0.0964684
\(890\) −222.512 −0.00838048
\(891\) 23976.4 0.901502
\(892\) −2475.07 −0.0929053
\(893\) 64417.1 2.41393
\(894\) 9935.18 0.371680
\(895\) −11651.8 −0.435171
\(896\) −14636.1 −0.545711
\(897\) 3116.48 0.116005
\(898\) −31811.1 −1.18213
\(899\) 2386.01 0.0885182
\(900\) 165.737 0.00613839
\(901\) 27965.1 1.03402
\(902\) 27722.1 1.02333
\(903\) 0 0
\(904\) 7132.30 0.262408
\(905\) 12194.1 0.447897
\(906\) 12847.1 0.471100
\(907\) 12059.4 0.441484 0.220742 0.975332i \(-0.429152\pi\)
0.220742 + 0.975332i \(0.429152\pi\)
\(908\) −706.289 −0.0258139
\(909\) −32350.9 −1.18043
\(910\) −22571.8 −0.822252
\(911\) −11271.8 −0.409937 −0.204969 0.978769i \(-0.565709\pi\)
−0.204969 + 0.978769i \(0.565709\pi\)
\(912\) −11185.9 −0.406142
\(913\) −26731.2 −0.968974
\(914\) 12144.2 0.439490
\(915\) 2920.04 0.105501
\(916\) 2945.75 0.106256
\(917\) 13172.1 0.474351
\(918\) −17183.9 −0.617815
\(919\) 48925.3 1.75614 0.878072 0.478529i \(-0.158830\pi\)
0.878072 + 0.478529i \(0.158830\pi\)
\(920\) −6222.64 −0.222994
\(921\) 4253.57 0.152182
\(922\) 17484.9 0.624549
\(923\) 37247.3 1.32829
\(924\) −671.212 −0.0238975
\(925\) 433.373 0.0154046
\(926\) −4508.32 −0.159992
\(927\) −751.827 −0.0266378
\(928\) −729.876 −0.0258183
\(929\) −13417.4 −0.473854 −0.236927 0.971527i \(-0.576140\pi\)
−0.236927 + 0.971527i \(0.576140\pi\)
\(930\) −9008.05 −0.317619
\(931\) 28923.7 1.01819
\(932\) 6432.36 0.226072
\(933\) 11826.9 0.415000
\(934\) −22033.4 −0.771900
\(935\) 35099.1 1.22766
\(936\) −40037.3 −1.39814
\(937\) −16627.3 −0.579713 −0.289856 0.957070i \(-0.593608\pi\)
−0.289856 + 0.957070i \(0.593608\pi\)
\(938\) 18867.4 0.656763
\(939\) −6086.36 −0.211524
\(940\) −8436.34 −0.292727
\(941\) 2488.17 0.0861976 0.0430988 0.999071i \(-0.486277\pi\)
0.0430988 + 0.999071i \(0.486277\pi\)
\(942\) 6567.73 0.227164
\(943\) 5983.51 0.206628
\(944\) 4524.72 0.156003
\(945\) 7015.18 0.241485
\(946\) 0 0
\(947\) 20531.0 0.704507 0.352254 0.935905i \(-0.385415\pi\)
0.352254 + 0.935905i \(0.385415\pi\)
\(948\) 2089.52 0.0715870
\(949\) −96453.7 −3.29928
\(950\) −1691.91 −0.0577819
\(951\) 7624.15 0.259969
\(952\) 13391.1 0.455892
\(953\) −34038.9 −1.15701 −0.578504 0.815680i \(-0.696363\pi\)
−0.578504 + 0.815680i \(0.696363\pi\)
\(954\) 28023.8 0.951054
\(955\) −6573.05 −0.222722
\(956\) −4903.42 −0.165887
\(957\) 763.296 0.0257825
\(958\) −33656.6 −1.13507
\(959\) 15967.8 0.537672
\(960\) −6341.87 −0.213211
\(961\) 5862.28 0.196780
\(962\) 20094.0 0.673449
\(963\) 34004.5 1.13788
\(964\) −8075.88 −0.269820
\(965\) −48374.1 −1.61370
\(966\) −1044.55 −0.0347906
\(967\) −28484.3 −0.947254 −0.473627 0.880726i \(-0.657055\pi\)
−0.473627 + 0.880726i \(0.657055\pi\)
\(968\) 9277.70 0.308054
\(969\) 11686.3 0.387429
\(970\) 34176.5 1.13128
\(971\) 15621.7 0.516295 0.258148 0.966105i \(-0.416888\pi\)
0.258148 + 0.966105i \(0.416888\pi\)
\(972\) −3629.50 −0.119770
\(973\) −5382.62 −0.177347
\(974\) −15144.1 −0.498202
\(975\) 578.177 0.0189913
\(976\) 13551.5 0.444441
\(977\) −10336.9 −0.338491 −0.169245 0.985574i \(-0.554133\pi\)
−0.169245 + 0.985574i \(0.554133\pi\)
\(978\) −12551.8 −0.410390
\(979\) 281.737 0.00919750
\(980\) −3787.97 −0.123472
\(981\) 9922.80 0.322947
\(982\) −23007.1 −0.747643
\(983\) −19360.5 −0.628182 −0.314091 0.949393i \(-0.601700\pi\)
−0.314091 + 0.949393i \(0.601700\pi\)
\(984\) 6298.00 0.204037
\(985\) 24107.8 0.779838
\(986\) 2922.87 0.0944048
\(987\) 7378.16 0.237943
\(988\) −10880.4 −0.350356
\(989\) 0 0
\(990\) 35172.7 1.12915
\(991\) 32122.8 1.02968 0.514840 0.857286i \(-0.327851\pi\)
0.514840 + 0.857286i \(0.327851\pi\)
\(992\) −10906.3 −0.349067
\(993\) −6704.86 −0.214272
\(994\) −12484.1 −0.398362
\(995\) −32427.0 −1.03317
\(996\) 1165.61 0.0370821
\(997\) −2315.90 −0.0735661 −0.0367831 0.999323i \(-0.511711\pi\)
−0.0367831 + 0.999323i \(0.511711\pi\)
\(998\) −8058.93 −0.255612
\(999\) −6245.09 −0.197784
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.1.16 50
43.42 odd 2 1849.4.a.j.1.35 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.16 50 1.1 even 1 trivial
1849.4.a.j.1.35 yes 50 43.42 odd 2