Properties

Label 1849.4.a.j.1.41
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 1.41
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.81936 q^{2} -8.34786 q^{3} +6.58755 q^{4} -2.56409 q^{5} -31.8835 q^{6} +3.42361 q^{7} -5.39467 q^{8} +42.6867 q^{9} +O(q^{10})\) \(q+3.81936 q^{2} -8.34786 q^{3} +6.58755 q^{4} -2.56409 q^{5} -31.8835 q^{6} +3.42361 q^{7} -5.39467 q^{8} +42.6867 q^{9} -9.79321 q^{10} -24.5225 q^{11} -54.9919 q^{12} -46.2969 q^{13} +13.0760 q^{14} +21.4047 q^{15} -73.3046 q^{16} +97.5824 q^{17} +163.036 q^{18} +131.921 q^{19} -16.8911 q^{20} -28.5798 q^{21} -93.6603 q^{22} +175.382 q^{23} +45.0339 q^{24} -118.425 q^{25} -176.825 q^{26} -130.950 q^{27} +22.5532 q^{28} +83.1501 q^{29} +81.7523 q^{30} +128.550 q^{31} -236.820 q^{32} +204.710 q^{33} +372.703 q^{34} -8.77845 q^{35} +281.201 q^{36} -301.363 q^{37} +503.853 q^{38} +386.480 q^{39} +13.8324 q^{40} -201.066 q^{41} -109.157 q^{42} -161.543 q^{44} -109.453 q^{45} +669.846 q^{46} -335.227 q^{47} +611.936 q^{48} -331.279 q^{49} -452.310 q^{50} -814.604 q^{51} -304.983 q^{52} +373.842 q^{53} -500.147 q^{54} +62.8779 q^{55} -18.4692 q^{56} -1101.25 q^{57} +317.581 q^{58} +578.431 q^{59} +141.004 q^{60} +892.641 q^{61} +490.980 q^{62} +146.143 q^{63} -318.064 q^{64} +118.710 q^{65} +781.862 q^{66} +2.66116 q^{67} +642.829 q^{68} -1464.06 q^{69} -33.5281 q^{70} -467.961 q^{71} -230.281 q^{72} +481.550 q^{73} -1151.01 q^{74} +988.598 q^{75} +869.034 q^{76} -83.9554 q^{77} +1476.11 q^{78} -296.213 q^{79} +187.960 q^{80} -59.3866 q^{81} -767.945 q^{82} +122.365 q^{83} -188.271 q^{84} -250.210 q^{85} -694.125 q^{87} +132.291 q^{88} -746.209 q^{89} -418.040 q^{90} -158.503 q^{91} +1155.33 q^{92} -1073.12 q^{93} -1280.36 q^{94} -338.257 q^{95} +1976.94 q^{96} -1723.34 q^{97} -1265.27 q^{98} -1046.78 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.81936 1.35035 0.675175 0.737658i \(-0.264068\pi\)
0.675175 + 0.737658i \(0.264068\pi\)
\(3\) −8.34786 −1.60655 −0.803273 0.595611i \(-0.796910\pi\)
−0.803273 + 0.595611i \(0.796910\pi\)
\(4\) 6.58755 0.823443
\(5\) −2.56409 −0.229339 −0.114670 0.993404i \(-0.536581\pi\)
−0.114670 + 0.993404i \(0.536581\pi\)
\(6\) −31.8835 −2.16940
\(7\) 3.42361 0.184858 0.0924288 0.995719i \(-0.470537\pi\)
0.0924288 + 0.995719i \(0.470537\pi\)
\(8\) −5.39467 −0.238413
\(9\) 42.6867 1.58099
\(10\) −9.79321 −0.309688
\(11\) −24.5225 −0.672164 −0.336082 0.941833i \(-0.609102\pi\)
−0.336082 + 0.941833i \(0.609102\pi\)
\(12\) −54.9919 −1.32290
\(13\) −46.2969 −0.987727 −0.493864 0.869539i \(-0.664416\pi\)
−0.493864 + 0.869539i \(0.664416\pi\)
\(14\) 13.0760 0.249622
\(15\) 21.4047 0.368444
\(16\) −73.3046 −1.14538
\(17\) 97.5824 1.39219 0.696095 0.717950i \(-0.254919\pi\)
0.696095 + 0.717950i \(0.254919\pi\)
\(18\) 163.036 2.13489
\(19\) 131.921 1.59288 0.796439 0.604719i \(-0.206715\pi\)
0.796439 + 0.604719i \(0.206715\pi\)
\(20\) −16.8911 −0.188848
\(21\) −28.5798 −0.296982
\(22\) −93.6603 −0.907656
\(23\) 175.382 1.58998 0.794991 0.606621i \(-0.207475\pi\)
0.794991 + 0.606621i \(0.207475\pi\)
\(24\) 45.0339 0.383021
\(25\) −118.425 −0.947403
\(26\) −176.825 −1.33378
\(27\) −130.950 −0.933385
\(28\) 22.5532 0.152220
\(29\) 83.1501 0.532434 0.266217 0.963913i \(-0.414226\pi\)
0.266217 + 0.963913i \(0.414226\pi\)
\(30\) 81.7523 0.497529
\(31\) 128.550 0.744783 0.372392 0.928076i \(-0.378538\pi\)
0.372392 + 0.928076i \(0.378538\pi\)
\(32\) −236.820 −1.30826
\(33\) 204.710 1.07986
\(34\) 372.703 1.87994
\(35\) −8.77845 −0.0423951
\(36\) 281.201 1.30186
\(37\) −301.363 −1.33902 −0.669510 0.742803i \(-0.733496\pi\)
−0.669510 + 0.742803i \(0.733496\pi\)
\(38\) 503.853 2.15094
\(39\) 386.480 1.58683
\(40\) 13.8324 0.0546775
\(41\) −201.066 −0.765884 −0.382942 0.923772i \(-0.625089\pi\)
−0.382942 + 0.923772i \(0.625089\pi\)
\(42\) −109.157 −0.401030
\(43\) 0 0
\(44\) −161.543 −0.553489
\(45\) −109.453 −0.362583
\(46\) 669.846 2.14703
\(47\) −335.227 −1.04038 −0.520191 0.854050i \(-0.674139\pi\)
−0.520191 + 0.854050i \(0.674139\pi\)
\(48\) 611.936 1.84011
\(49\) −331.279 −0.965828
\(50\) −452.310 −1.27933
\(51\) −814.604 −2.23662
\(52\) −304.983 −0.813338
\(53\) 373.842 0.968891 0.484445 0.874822i \(-0.339021\pi\)
0.484445 + 0.874822i \(0.339021\pi\)
\(54\) −500.147 −1.26040
\(55\) 62.8779 0.154154
\(56\) −18.4692 −0.0440724
\(57\) −1101.25 −2.55903
\(58\) 317.581 0.718972
\(59\) 578.431 1.27636 0.638180 0.769887i \(-0.279688\pi\)
0.638180 + 0.769887i \(0.279688\pi\)
\(60\) 141.004 0.303393
\(61\) 892.641 1.87362 0.936811 0.349835i \(-0.113762\pi\)
0.936811 + 0.349835i \(0.113762\pi\)
\(62\) 490.980 1.00572
\(63\) 146.143 0.292258
\(64\) −318.064 −0.621218
\(65\) 118.710 0.226525
\(66\) 781.862 1.45819
\(67\) 2.66116 0.00485242 0.00242621 0.999997i \(-0.499228\pi\)
0.00242621 + 0.999997i \(0.499228\pi\)
\(68\) 642.829 1.14639
\(69\) −1464.06 −2.55438
\(70\) −33.5281 −0.0572482
\(71\) −467.961 −0.782208 −0.391104 0.920346i \(-0.627907\pi\)
−0.391104 + 0.920346i \(0.627907\pi\)
\(72\) −230.281 −0.376928
\(73\) 481.550 0.772070 0.386035 0.922484i \(-0.373844\pi\)
0.386035 + 0.922484i \(0.373844\pi\)
\(74\) −1151.01 −1.80814
\(75\) 988.598 1.52205
\(76\) 869.034 1.31164
\(77\) −83.9554 −0.124255
\(78\) 1476.11 2.14277
\(79\) −296.213 −0.421856 −0.210928 0.977502i \(-0.567649\pi\)
−0.210928 + 0.977502i \(0.567649\pi\)
\(80\) 187.960 0.262682
\(81\) −59.3866 −0.0814631
\(82\) −767.945 −1.03421
\(83\) 122.365 0.161823 0.0809116 0.996721i \(-0.474217\pi\)
0.0809116 + 0.996721i \(0.474217\pi\)
\(84\) −188.271 −0.244548
\(85\) −250.210 −0.319284
\(86\) 0 0
\(87\) −694.125 −0.855379
\(88\) 132.291 0.160253
\(89\) −746.209 −0.888742 −0.444371 0.895843i \(-0.646573\pi\)
−0.444371 + 0.895843i \(0.646573\pi\)
\(90\) −418.040 −0.489614
\(91\) −158.503 −0.182589
\(92\) 1155.33 1.30926
\(93\) −1073.12 −1.19653
\(94\) −1280.36 −1.40488
\(95\) −338.257 −0.365310
\(96\) 1976.94 2.10177
\(97\) −1723.34 −1.80390 −0.901949 0.431842i \(-0.857864\pi\)
−0.901949 + 0.431842i \(0.857864\pi\)
\(98\) −1265.27 −1.30420
\(99\) −1046.78 −1.06268
\(100\) −780.133 −0.780133
\(101\) −384.032 −0.378343 −0.189171 0.981944i \(-0.560580\pi\)
−0.189171 + 0.981944i \(0.560580\pi\)
\(102\) −3111.27 −3.02021
\(103\) 187.491 0.179360 0.0896798 0.995971i \(-0.471416\pi\)
0.0896798 + 0.995971i \(0.471416\pi\)
\(104\) 249.757 0.235487
\(105\) 73.2813 0.0681097
\(106\) 1427.84 1.30834
\(107\) −1145.03 −1.03453 −0.517263 0.855827i \(-0.673049\pi\)
−0.517263 + 0.855827i \(0.673049\pi\)
\(108\) −862.641 −0.768590
\(109\) −1108.95 −0.974482 −0.487241 0.873268i \(-0.661997\pi\)
−0.487241 + 0.873268i \(0.661997\pi\)
\(110\) 240.154 0.208161
\(111\) 2515.73 2.15120
\(112\) −250.966 −0.211733
\(113\) 1450.93 1.20789 0.603947 0.797025i \(-0.293594\pi\)
0.603947 + 0.797025i \(0.293594\pi\)
\(114\) −4206.09 −3.45558
\(115\) −449.695 −0.364646
\(116\) 547.755 0.438429
\(117\) −1976.26 −1.56159
\(118\) 2209.24 1.72353
\(119\) 334.084 0.257357
\(120\) −115.471 −0.0878419
\(121\) −729.648 −0.548196
\(122\) 3409.32 2.53004
\(123\) 1678.47 1.23043
\(124\) 846.830 0.613287
\(125\) 624.165 0.446616
\(126\) 558.172 0.394650
\(127\) −2025.38 −1.41515 −0.707574 0.706640i \(-0.750210\pi\)
−0.707574 + 0.706640i \(0.750210\pi\)
\(128\) 679.755 0.469394
\(129\) 0 0
\(130\) 453.395 0.305888
\(131\) −2553.08 −1.70278 −0.851390 0.524534i \(-0.824240\pi\)
−0.851390 + 0.524534i \(0.824240\pi\)
\(132\) 1348.54 0.889205
\(133\) 451.645 0.294455
\(134\) 10.1639 0.00655246
\(135\) 335.769 0.214062
\(136\) −526.425 −0.331916
\(137\) 1338.26 0.834564 0.417282 0.908777i \(-0.362983\pi\)
0.417282 + 0.908777i \(0.362983\pi\)
\(138\) −5591.78 −3.44930
\(139\) 687.366 0.419436 0.209718 0.977762i \(-0.432745\pi\)
0.209718 + 0.977762i \(0.432745\pi\)
\(140\) −57.8285 −0.0349100
\(141\) 2798.43 1.67142
\(142\) −1787.31 −1.05625
\(143\) 1135.31 0.663915
\(144\) −3129.13 −1.81084
\(145\) −213.205 −0.122108
\(146\) 1839.21 1.04256
\(147\) 2765.47 1.55165
\(148\) −1985.24 −1.10261
\(149\) 467.843 0.257230 0.128615 0.991695i \(-0.458947\pi\)
0.128615 + 0.991695i \(0.458947\pi\)
\(150\) 3775.82 2.05530
\(151\) 1726.18 0.930293 0.465147 0.885234i \(-0.346002\pi\)
0.465147 + 0.885234i \(0.346002\pi\)
\(152\) −711.668 −0.379763
\(153\) 4165.47 2.20104
\(154\) −320.656 −0.167787
\(155\) −329.614 −0.170808
\(156\) 2545.96 1.30666
\(157\) −488.504 −0.248324 −0.124162 0.992262i \(-0.539624\pi\)
−0.124162 + 0.992262i \(0.539624\pi\)
\(158\) −1131.35 −0.569653
\(159\) −3120.78 −1.55657
\(160\) 607.228 0.300035
\(161\) 600.438 0.293920
\(162\) −226.819 −0.110004
\(163\) −1013.08 −0.486814 −0.243407 0.969924i \(-0.578265\pi\)
−0.243407 + 0.969924i \(0.578265\pi\)
\(164\) −1324.53 −0.630662
\(165\) −524.896 −0.247655
\(166\) 467.357 0.218518
\(167\) −916.697 −0.424767 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(168\) 154.179 0.0708044
\(169\) −53.5953 −0.0243948
\(170\) −955.645 −0.431145
\(171\) 5631.26 2.51832
\(172\) 0 0
\(173\) −4040.24 −1.77557 −0.887786 0.460257i \(-0.847757\pi\)
−0.887786 + 0.460257i \(0.847757\pi\)
\(174\) −2651.12 −1.15506
\(175\) −405.442 −0.175135
\(176\) 1797.61 0.769886
\(177\) −4828.65 −2.05053
\(178\) −2850.04 −1.20011
\(179\) 818.629 0.341828 0.170914 0.985286i \(-0.445328\pi\)
0.170914 + 0.985286i \(0.445328\pi\)
\(180\) −721.025 −0.298567
\(181\) −1761.43 −0.723350 −0.361675 0.932304i \(-0.617795\pi\)
−0.361675 + 0.932304i \(0.617795\pi\)
\(182\) −605.379 −0.246559
\(183\) −7451.64 −3.01006
\(184\) −946.126 −0.379072
\(185\) 772.722 0.307090
\(186\) −4098.63 −1.61573
\(187\) −2392.96 −0.935779
\(188\) −2208.33 −0.856696
\(189\) −448.323 −0.172543
\(190\) −1291.93 −0.493296
\(191\) −1206.69 −0.457137 −0.228568 0.973528i \(-0.573404\pi\)
−0.228568 + 0.973528i \(0.573404\pi\)
\(192\) 2655.15 0.998016
\(193\) −1227.19 −0.457694 −0.228847 0.973462i \(-0.573496\pi\)
−0.228847 + 0.973462i \(0.573496\pi\)
\(194\) −6582.05 −2.43589
\(195\) −990.971 −0.363922
\(196\) −2182.32 −0.795305
\(197\) 2906.38 1.05112 0.525560 0.850756i \(-0.323856\pi\)
0.525560 + 0.850756i \(0.323856\pi\)
\(198\) −3998.05 −1.43499
\(199\) −1960.29 −0.698299 −0.349149 0.937067i \(-0.613529\pi\)
−0.349149 + 0.937067i \(0.613529\pi\)
\(200\) 638.866 0.225873
\(201\) −22.2149 −0.00779563
\(202\) −1466.76 −0.510895
\(203\) 284.673 0.0984244
\(204\) −5366.24 −1.84173
\(205\) 515.552 0.175647
\(206\) 716.097 0.242198
\(207\) 7486.46 2.51374
\(208\) 3393.78 1.13133
\(209\) −3235.02 −1.07067
\(210\) 279.888 0.0919719
\(211\) −5227.84 −1.70568 −0.852841 0.522170i \(-0.825123\pi\)
−0.852841 + 0.522170i \(0.825123\pi\)
\(212\) 2462.70 0.797827
\(213\) 3906.47 1.25665
\(214\) −4373.29 −1.39697
\(215\) 0 0
\(216\) 706.434 0.222531
\(217\) 440.105 0.137679
\(218\) −4235.50 −1.31589
\(219\) −4019.91 −1.24037
\(220\) 414.211 0.126937
\(221\) −4517.77 −1.37510
\(222\) 9608.50 2.90487
\(223\) −1121.01 −0.336631 −0.168315 0.985733i \(-0.553833\pi\)
−0.168315 + 0.985733i \(0.553833\pi\)
\(224\) −810.778 −0.241841
\(225\) −5055.19 −1.49783
\(226\) 5541.63 1.63108
\(227\) −499.397 −0.146018 −0.0730092 0.997331i \(-0.523260\pi\)
−0.0730092 + 0.997331i \(0.523260\pi\)
\(228\) −7254.57 −2.10722
\(229\) 1384.89 0.399633 0.199817 0.979833i \(-0.435965\pi\)
0.199817 + 0.979833i \(0.435965\pi\)
\(230\) −1717.55 −0.492399
\(231\) 700.847 0.199621
\(232\) −448.567 −0.126939
\(233\) 259.379 0.0729291 0.0364646 0.999335i \(-0.488390\pi\)
0.0364646 + 0.999335i \(0.488390\pi\)
\(234\) −7548.07 −2.10869
\(235\) 859.554 0.238601
\(236\) 3810.44 1.05101
\(237\) 2472.75 0.677730
\(238\) 1275.99 0.347521
\(239\) 4228.11 1.14432 0.572162 0.820140i \(-0.306105\pi\)
0.572162 + 0.820140i \(0.306105\pi\)
\(240\) −1569.06 −0.422010
\(241\) −7388.11 −1.97473 −0.987365 0.158463i \(-0.949346\pi\)
−0.987365 + 0.158463i \(0.949346\pi\)
\(242\) −2786.79 −0.740256
\(243\) 4031.41 1.06426
\(244\) 5880.32 1.54282
\(245\) 849.430 0.221502
\(246\) 6410.69 1.66151
\(247\) −6107.52 −1.57333
\(248\) −693.485 −0.177566
\(249\) −1021.49 −0.259976
\(250\) 2383.92 0.603088
\(251\) −2006.43 −0.504561 −0.252280 0.967654i \(-0.581180\pi\)
−0.252280 + 0.967654i \(0.581180\pi\)
\(252\) 962.721 0.240658
\(253\) −4300.79 −1.06873
\(254\) −7735.68 −1.91094
\(255\) 2088.72 0.512944
\(256\) 5140.74 1.25506
\(257\) 356.050 0.0864195 0.0432098 0.999066i \(-0.486242\pi\)
0.0432098 + 0.999066i \(0.486242\pi\)
\(258\) 0 0
\(259\) −1031.75 −0.247528
\(260\) 782.005 0.186530
\(261\) 3549.40 0.841772
\(262\) −9751.16 −2.29935
\(263\) 2558.23 0.599800 0.299900 0.953971i \(-0.403047\pi\)
0.299900 + 0.953971i \(0.403047\pi\)
\(264\) −1104.34 −0.257453
\(265\) −958.567 −0.222205
\(266\) 1725.00 0.397618
\(267\) 6229.25 1.42780
\(268\) 17.5305 0.00399569
\(269\) 7645.48 1.73291 0.866455 0.499255i \(-0.166393\pi\)
0.866455 + 0.499255i \(0.166393\pi\)
\(270\) 1282.42 0.289059
\(271\) −2563.79 −0.574683 −0.287342 0.957828i \(-0.592772\pi\)
−0.287342 + 0.957828i \(0.592772\pi\)
\(272\) −7153.24 −1.59459
\(273\) 1323.16 0.293337
\(274\) 5111.31 1.12695
\(275\) 2904.08 0.636810
\(276\) −9644.57 −2.10339
\(277\) 5655.56 1.22675 0.613375 0.789792i \(-0.289811\pi\)
0.613375 + 0.789792i \(0.289811\pi\)
\(278\) 2625.30 0.566386
\(279\) 5487.38 1.17749
\(280\) 47.3568 0.0101075
\(281\) −6410.59 −1.36094 −0.680469 0.732777i \(-0.738224\pi\)
−0.680469 + 0.732777i \(0.738224\pi\)
\(282\) 10688.2 2.25700
\(283\) −8030.69 −1.68684 −0.843419 0.537256i \(-0.819461\pi\)
−0.843419 + 0.537256i \(0.819461\pi\)
\(284\) −3082.72 −0.644104
\(285\) 2823.72 0.586887
\(286\) 4336.18 0.896517
\(287\) −688.372 −0.141579
\(288\) −10109.0 −2.06834
\(289\) 4609.33 0.938191
\(290\) −814.306 −0.164889
\(291\) 14386.2 2.89805
\(292\) 3172.23 0.635756
\(293\) −3612.22 −0.720232 −0.360116 0.932908i \(-0.617263\pi\)
−0.360116 + 0.932908i \(0.617263\pi\)
\(294\) 10562.3 2.09526
\(295\) −1483.15 −0.292720
\(296\) 1625.75 0.319240
\(297\) 3211.23 0.627388
\(298\) 1786.86 0.347350
\(299\) −8119.63 −1.57047
\(300\) 6512.44 1.25332
\(301\) 0 0
\(302\) 6592.90 1.25622
\(303\) 3205.84 0.607825
\(304\) −9670.39 −1.82446
\(305\) −2288.81 −0.429696
\(306\) 15909.5 2.97217
\(307\) −1282.10 −0.238349 −0.119174 0.992873i \(-0.538025\pi\)
−0.119174 + 0.992873i \(0.538025\pi\)
\(308\) −553.060 −0.102317
\(309\) −1565.15 −0.288149
\(310\) −1258.92 −0.230651
\(311\) −9145.70 −1.66754 −0.833770 0.552111i \(-0.813822\pi\)
−0.833770 + 0.552111i \(0.813822\pi\)
\(312\) −2084.93 −0.378321
\(313\) 3256.64 0.588103 0.294051 0.955790i \(-0.404996\pi\)
0.294051 + 0.955790i \(0.404996\pi\)
\(314\) −1865.78 −0.335324
\(315\) −374.723 −0.0670262
\(316\) −1951.32 −0.347374
\(317\) 6460.36 1.14464 0.572319 0.820031i \(-0.306044\pi\)
0.572319 + 0.820031i \(0.306044\pi\)
\(318\) −11919.4 −2.10191
\(319\) −2039.05 −0.357883
\(320\) 815.545 0.142470
\(321\) 9558.54 1.66201
\(322\) 2293.29 0.396895
\(323\) 12873.1 2.21759
\(324\) −391.212 −0.0670802
\(325\) 5482.73 0.935776
\(326\) −3869.33 −0.657369
\(327\) 9257.39 1.56555
\(328\) 1084.69 0.182597
\(329\) −1147.69 −0.192322
\(330\) −2004.77 −0.334421
\(331\) 6472.81 1.07486 0.537429 0.843309i \(-0.319396\pi\)
0.537429 + 0.843309i \(0.319396\pi\)
\(332\) 806.086 0.133252
\(333\) −12864.2 −2.11697
\(334\) −3501.20 −0.573584
\(335\) −6.82345 −0.00111285
\(336\) 2095.03 0.340159
\(337\) 1072.36 0.173339 0.0866695 0.996237i \(-0.472378\pi\)
0.0866695 + 0.996237i \(0.472378\pi\)
\(338\) −204.700 −0.0329415
\(339\) −12112.2 −1.94054
\(340\) −1648.27 −0.262912
\(341\) −3152.37 −0.500616
\(342\) 21507.8 3.40061
\(343\) −2308.47 −0.363398
\(344\) 0 0
\(345\) 3753.99 0.585820
\(346\) −15431.2 −2.39764
\(347\) −1353.35 −0.209371 −0.104686 0.994505i \(-0.533384\pi\)
−0.104686 + 0.994505i \(0.533384\pi\)
\(348\) −4572.58 −0.704357
\(349\) −8713.86 −1.33651 −0.668256 0.743932i \(-0.732959\pi\)
−0.668256 + 0.743932i \(0.732959\pi\)
\(350\) −1548.53 −0.236493
\(351\) 6062.60 0.921930
\(352\) 5807.40 0.879363
\(353\) 6064.05 0.914325 0.457163 0.889383i \(-0.348866\pi\)
0.457163 + 0.889383i \(0.348866\pi\)
\(354\) −18442.4 −2.76893
\(355\) 1199.90 0.179391
\(356\) −4915.69 −0.731829
\(357\) −2788.89 −0.413455
\(358\) 3126.64 0.461587
\(359\) 2672.80 0.392939 0.196469 0.980510i \(-0.437052\pi\)
0.196469 + 0.980510i \(0.437052\pi\)
\(360\) 590.461 0.0864445
\(361\) 10544.1 1.53726
\(362\) −6727.56 −0.976775
\(363\) 6091.00 0.880701
\(364\) −1044.14 −0.150352
\(365\) −1234.74 −0.177066
\(366\) −28460.5 −4.06463
\(367\) 3206.83 0.456117 0.228059 0.973647i \(-0.426762\pi\)
0.228059 + 0.973647i \(0.426762\pi\)
\(368\) −12856.3 −1.82114
\(369\) −8582.85 −1.21085
\(370\) 2951.31 0.414679
\(371\) 1279.89 0.179107
\(372\) −7069.22 −0.985273
\(373\) 2341.81 0.325079 0.162539 0.986702i \(-0.448032\pi\)
0.162539 + 0.986702i \(0.448032\pi\)
\(374\) −9139.60 −1.26363
\(375\) −5210.44 −0.717510
\(376\) 1808.44 0.248041
\(377\) −3849.59 −0.525900
\(378\) −1712.31 −0.232994
\(379\) −12165.1 −1.64876 −0.824382 0.566034i \(-0.808477\pi\)
−0.824382 + 0.566034i \(0.808477\pi\)
\(380\) −2228.28 −0.300812
\(381\) 16907.6 2.27350
\(382\) −4608.80 −0.617295
\(383\) −10219.7 −1.36345 −0.681727 0.731606i \(-0.738771\pi\)
−0.681727 + 0.731606i \(0.738771\pi\)
\(384\) −5674.50 −0.754103
\(385\) 215.269 0.0284965
\(386\) −4687.08 −0.618047
\(387\) 0 0
\(388\) −11352.6 −1.48541
\(389\) 9610.28 1.25260 0.626299 0.779583i \(-0.284569\pi\)
0.626299 + 0.779583i \(0.284569\pi\)
\(390\) −3784.88 −0.491422
\(391\) 17114.2 2.21356
\(392\) 1787.14 0.230266
\(393\) 21312.8 2.73559
\(394\) 11100.5 1.41938
\(395\) 759.518 0.0967481
\(396\) −6895.74 −0.875060
\(397\) 8283.95 1.04725 0.523627 0.851948i \(-0.324579\pi\)
0.523627 + 0.851948i \(0.324579\pi\)
\(398\) −7487.07 −0.942947
\(399\) −3770.26 −0.473056
\(400\) 8681.13 1.08514
\(401\) −12685.4 −1.57975 −0.789874 0.613269i \(-0.789854\pi\)
−0.789874 + 0.613269i \(0.789854\pi\)
\(402\) −84.8470 −0.0105268
\(403\) −5951.47 −0.735643
\(404\) −2529.83 −0.311544
\(405\) 152.273 0.0186827
\(406\) 1087.27 0.132907
\(407\) 7390.16 0.900040
\(408\) 4394.52 0.533238
\(409\) −981.857 −0.118703 −0.0593517 0.998237i \(-0.518903\pi\)
−0.0593517 + 0.998237i \(0.518903\pi\)
\(410\) 1969.08 0.237185
\(411\) −11171.6 −1.34077
\(412\) 1235.11 0.147693
\(413\) 1980.32 0.235945
\(414\) 28593.5 3.39443
\(415\) −313.756 −0.0371124
\(416\) 10964.0 1.29220
\(417\) −5738.03 −0.673844
\(418\) −12355.7 −1.44578
\(419\) 10930.0 1.27439 0.637193 0.770704i \(-0.280095\pi\)
0.637193 + 0.770704i \(0.280095\pi\)
\(420\) 482.744 0.0560845
\(421\) 12352.3 1.42996 0.714980 0.699145i \(-0.246436\pi\)
0.714980 + 0.699145i \(0.246436\pi\)
\(422\) −19967.0 −2.30327
\(423\) −14309.8 −1.64483
\(424\) −2016.76 −0.230996
\(425\) −11556.2 −1.31896
\(426\) 14920.2 1.69692
\(427\) 3056.05 0.346353
\(428\) −7542.94 −0.851873
\(429\) −9477.44 −1.06661
\(430\) 0 0
\(431\) −6534.96 −0.730343 −0.365172 0.930940i \(-0.618990\pi\)
−0.365172 + 0.930940i \(0.618990\pi\)
\(432\) 9599.26 1.06908
\(433\) 12272.7 1.36210 0.681048 0.732239i \(-0.261524\pi\)
0.681048 + 0.732239i \(0.261524\pi\)
\(434\) 1680.92 0.185914
\(435\) 1779.80 0.196172
\(436\) −7305.29 −0.802431
\(437\) 23136.5 2.53265
\(438\) −15353.5 −1.67493
\(439\) −2806.41 −0.305108 −0.152554 0.988295i \(-0.548750\pi\)
−0.152554 + 0.988295i \(0.548750\pi\)
\(440\) −339.205 −0.0367522
\(441\) −14141.2 −1.52696
\(442\) −17255.0 −1.85687
\(443\) −9336.79 −1.00136 −0.500682 0.865631i \(-0.666917\pi\)
−0.500682 + 0.865631i \(0.666917\pi\)
\(444\) 16572.5 1.77139
\(445\) 1913.35 0.203824
\(446\) −4281.56 −0.454569
\(447\) −3905.49 −0.413251
\(448\) −1088.93 −0.114837
\(449\) 7212.30 0.758062 0.379031 0.925384i \(-0.376257\pi\)
0.379031 + 0.925384i \(0.376257\pi\)
\(450\) −19307.6 −2.02260
\(451\) 4930.64 0.514800
\(452\) 9558.07 0.994632
\(453\) −14409.9 −1.49456
\(454\) −1907.38 −0.197176
\(455\) 406.415 0.0418748
\(456\) 5940.90 0.610106
\(457\) −1378.40 −0.141091 −0.0705455 0.997509i \(-0.522474\pi\)
−0.0705455 + 0.997509i \(0.522474\pi\)
\(458\) 5289.40 0.539645
\(459\) −12778.4 −1.29945
\(460\) −2962.39 −0.300265
\(461\) −11245.2 −1.13610 −0.568048 0.822996i \(-0.692301\pi\)
−0.568048 + 0.822996i \(0.692301\pi\)
\(462\) 2676.79 0.269558
\(463\) 2077.54 0.208535 0.104267 0.994549i \(-0.466750\pi\)
0.104267 + 0.994549i \(0.466750\pi\)
\(464\) −6095.28 −0.609842
\(465\) 2751.57 0.274411
\(466\) 990.663 0.0984798
\(467\) −2016.77 −0.199839 −0.0999197 0.994996i \(-0.531859\pi\)
−0.0999197 + 0.994996i \(0.531859\pi\)
\(468\) −13018.7 −1.28588
\(469\) 9.11076 0.000897006 0
\(470\) 3282.95 0.322194
\(471\) 4077.96 0.398944
\(472\) −3120.44 −0.304301
\(473\) 0 0
\(474\) 9444.32 0.915173
\(475\) −15622.8 −1.50910
\(476\) 2200.80 0.211919
\(477\) 15958.1 1.53181
\(478\) 16148.7 1.54524
\(479\) 10427.0 0.994620 0.497310 0.867573i \(-0.334321\pi\)
0.497310 + 0.867573i \(0.334321\pi\)
\(480\) −5069.05 −0.482019
\(481\) 13952.2 1.32259
\(482\) −28217.9 −2.66658
\(483\) −5012.37 −0.472196
\(484\) −4806.59 −0.451408
\(485\) 4418.79 0.413705
\(486\) 15397.4 1.43712
\(487\) −4165.93 −0.387631 −0.193815 0.981038i \(-0.562086\pi\)
−0.193815 + 0.981038i \(0.562086\pi\)
\(488\) −4815.50 −0.446696
\(489\) 8457.06 0.782089
\(490\) 3244.28 0.299106
\(491\) −11303.6 −1.03895 −0.519475 0.854486i \(-0.673873\pi\)
−0.519475 + 0.854486i \(0.673873\pi\)
\(492\) 11057.0 1.01319
\(493\) 8113.99 0.741249
\(494\) −23326.8 −2.12454
\(495\) 2684.05 0.243715
\(496\) −9423.31 −0.853063
\(497\) −1602.12 −0.144597
\(498\) −3901.43 −0.351059
\(499\) 13070.3 1.17256 0.586281 0.810108i \(-0.300592\pi\)
0.586281 + 0.810108i \(0.300592\pi\)
\(500\) 4111.72 0.367763
\(501\) 7652.45 0.682408
\(502\) −7663.29 −0.681333
\(503\) 9968.27 0.883625 0.441812 0.897107i \(-0.354336\pi\)
0.441812 + 0.897107i \(0.354336\pi\)
\(504\) −788.391 −0.0696780
\(505\) 984.694 0.0867689
\(506\) −16426.3 −1.44316
\(507\) 447.406 0.0391913
\(508\) −13342.3 −1.16529
\(509\) 11910.6 1.03719 0.518594 0.855021i \(-0.326456\pi\)
0.518594 + 0.855021i \(0.326456\pi\)
\(510\) 7977.59 0.692654
\(511\) 1648.64 0.142723
\(512\) 14196.3 1.22538
\(513\) −17275.0 −1.48677
\(514\) 1359.89 0.116697
\(515\) −480.744 −0.0411342
\(516\) 0 0
\(517\) 8220.61 0.699307
\(518\) −3940.62 −0.334249
\(519\) 33727.4 2.85254
\(520\) −640.399 −0.0540065
\(521\) −7264.31 −0.610854 −0.305427 0.952215i \(-0.598799\pi\)
−0.305427 + 0.952215i \(0.598799\pi\)
\(522\) 13556.5 1.13669
\(523\) −11647.5 −0.973823 −0.486912 0.873451i \(-0.661877\pi\)
−0.486912 + 0.873451i \(0.661877\pi\)
\(524\) −16818.6 −1.40214
\(525\) 3384.57 0.281362
\(526\) 9770.83 0.809940
\(527\) 12544.2 1.03688
\(528\) −15006.2 −1.23686
\(529\) 18591.7 1.52804
\(530\) −3661.12 −0.300054
\(531\) 24691.3 2.01791
\(532\) 2975.23 0.242467
\(533\) 9308.74 0.756485
\(534\) 23791.8 1.92803
\(535\) 2935.96 0.237257
\(536\) −14.3561 −0.00115688
\(537\) −6833.80 −0.549162
\(538\) 29200.9 2.34003
\(539\) 8123.78 0.649195
\(540\) 2211.89 0.176268
\(541\) 3578.45 0.284380 0.142190 0.989839i \(-0.454586\pi\)
0.142190 + 0.989839i \(0.454586\pi\)
\(542\) −9792.05 −0.776023
\(543\) 14704.2 1.16209
\(544\) −23109.4 −1.82134
\(545\) 2843.46 0.223487
\(546\) 5053.62 0.396108
\(547\) −15440.8 −1.20695 −0.603475 0.797382i \(-0.706218\pi\)
−0.603475 + 0.797382i \(0.706218\pi\)
\(548\) 8815.86 0.687217
\(549\) 38103.9 2.96218
\(550\) 11091.8 0.859917
\(551\) 10969.2 0.848102
\(552\) 7898.12 0.608997
\(553\) −1014.12 −0.0779832
\(554\) 21600.7 1.65654
\(555\) −6450.57 −0.493354
\(556\) 4528.06 0.345382
\(557\) −2684.18 −0.204187 −0.102094 0.994775i \(-0.532554\pi\)
−0.102094 + 0.994775i \(0.532554\pi\)
\(558\) 20958.3 1.59003
\(559\) 0 0
\(560\) 643.501 0.0485587
\(561\) 19976.1 1.50337
\(562\) −24484.4 −1.83774
\(563\) −10345.1 −0.774412 −0.387206 0.921993i \(-0.626560\pi\)
−0.387206 + 0.921993i \(0.626560\pi\)
\(564\) 18434.8 1.37632
\(565\) −3720.32 −0.277018
\(566\) −30672.1 −2.27782
\(567\) −203.316 −0.0150591
\(568\) 2524.50 0.186489
\(569\) 60.6533 0.00446875 0.00223438 0.999998i \(-0.499289\pi\)
0.00223438 + 0.999998i \(0.499289\pi\)
\(570\) 10784.8 0.792502
\(571\) −2225.56 −0.163111 −0.0815557 0.996669i \(-0.525989\pi\)
−0.0815557 + 0.996669i \(0.525989\pi\)
\(572\) 7478.94 0.546696
\(573\) 10073.3 0.734411
\(574\) −2629.14 −0.191182
\(575\) −20769.6 −1.50635
\(576\) −13577.1 −0.982139
\(577\) −16249.2 −1.17238 −0.586190 0.810173i \(-0.699373\pi\)
−0.586190 + 0.810173i \(0.699373\pi\)
\(578\) 17604.7 1.26689
\(579\) 10244.4 0.735306
\(580\) −1404.50 −0.100549
\(581\) 418.930 0.0299142
\(582\) 54946.0 3.91337
\(583\) −9167.54 −0.651253
\(584\) −2597.80 −0.184072
\(585\) 5067.32 0.358133
\(586\) −13796.4 −0.972564
\(587\) 10104.6 0.710496 0.355248 0.934772i \(-0.384396\pi\)
0.355248 + 0.934772i \(0.384396\pi\)
\(588\) 18217.7 1.27769
\(589\) 16958.4 1.18635
\(590\) −5664.69 −0.395274
\(591\) −24262.0 −1.68867
\(592\) 22091.3 1.53369
\(593\) −10930.9 −0.756962 −0.378481 0.925609i \(-0.623553\pi\)
−0.378481 + 0.925609i \(0.623553\pi\)
\(594\) 12264.8 0.847193
\(595\) −856.623 −0.0590220
\(596\) 3081.94 0.211814
\(597\) 16364.2 1.12185
\(598\) −31011.8 −2.12068
\(599\) −21584.2 −1.47230 −0.736149 0.676819i \(-0.763358\pi\)
−0.736149 + 0.676819i \(0.763358\pi\)
\(600\) −5333.16 −0.362876
\(601\) −18226.1 −1.23704 −0.618519 0.785770i \(-0.712267\pi\)
−0.618519 + 0.785770i \(0.712267\pi\)
\(602\) 0 0
\(603\) 113.596 0.00767162
\(604\) 11371.3 0.766044
\(605\) 1870.89 0.125723
\(606\) 12244.3 0.820776
\(607\) −24757.1 −1.65546 −0.827728 0.561130i \(-0.810367\pi\)
−0.827728 + 0.561130i \(0.810367\pi\)
\(608\) −31241.4 −2.08389
\(609\) −2376.41 −0.158123
\(610\) −8741.82 −0.580239
\(611\) 15520.0 1.02761
\(612\) 27440.2 1.81243
\(613\) −19983.4 −1.31668 −0.658339 0.752722i \(-0.728741\pi\)
−0.658339 + 0.752722i \(0.728741\pi\)
\(614\) −4896.79 −0.321854
\(615\) −4303.76 −0.282186
\(616\) 452.911 0.0296239
\(617\) 26758.7 1.74597 0.872985 0.487746i \(-0.162181\pi\)
0.872985 + 0.487746i \(0.162181\pi\)
\(618\) −5977.87 −0.389102
\(619\) 3557.65 0.231008 0.115504 0.993307i \(-0.463152\pi\)
0.115504 + 0.993307i \(0.463152\pi\)
\(620\) −2171.35 −0.140651
\(621\) −22966.3 −1.48407
\(622\) −34930.8 −2.25176
\(623\) −2554.73 −0.164291
\(624\) −28330.8 −1.81753
\(625\) 13202.8 0.844977
\(626\) 12438.3 0.794144
\(627\) 27005.5 1.72009
\(628\) −3218.05 −0.204481
\(629\) −29407.7 −1.86417
\(630\) −1431.20 −0.0905088
\(631\) 22683.1 1.43106 0.715532 0.698580i \(-0.246184\pi\)
0.715532 + 0.698580i \(0.246184\pi\)
\(632\) 1597.97 0.100576
\(633\) 43641.2 2.74026
\(634\) 24674.5 1.54566
\(635\) 5193.27 0.324549
\(636\) −20558.3 −1.28174
\(637\) 15337.2 0.953974
\(638\) −7787.86 −0.483267
\(639\) −19975.7 −1.23666
\(640\) −1742.96 −0.107651
\(641\) 3214.41 0.198068 0.0990341 0.995084i \(-0.468425\pi\)
0.0990341 + 0.995084i \(0.468425\pi\)
\(642\) 36507.6 2.24430
\(643\) 5096.07 0.312550 0.156275 0.987714i \(-0.450051\pi\)
0.156275 + 0.987714i \(0.450051\pi\)
\(644\) 3955.42 0.242027
\(645\) 0 0
\(646\) 49167.2 2.99452
\(647\) 2031.59 0.123447 0.0617234 0.998093i \(-0.480340\pi\)
0.0617234 + 0.998093i \(0.480340\pi\)
\(648\) 320.371 0.0194218
\(649\) −14184.5 −0.857923
\(650\) 20940.6 1.26362
\(651\) −3673.94 −0.221187
\(652\) −6673.72 −0.400864
\(653\) 6868.13 0.411594 0.205797 0.978595i \(-0.434021\pi\)
0.205797 + 0.978595i \(0.434021\pi\)
\(654\) 35357.3 2.11404
\(655\) 6546.35 0.390514
\(656\) 14739.1 0.877232
\(657\) 20555.8 1.22063
\(658\) −4383.44 −0.259702
\(659\) −11046.0 −0.652947 −0.326474 0.945206i \(-0.605860\pi\)
−0.326474 + 0.945206i \(0.605860\pi\)
\(660\) −3457.78 −0.203930
\(661\) 19990.5 1.17631 0.588156 0.808748i \(-0.299854\pi\)
0.588156 + 0.808748i \(0.299854\pi\)
\(662\) 24722.0 1.45143
\(663\) 37713.7 2.20917
\(664\) −660.120 −0.0385807
\(665\) −1158.06 −0.0675302
\(666\) −49133.0 −2.85865
\(667\) 14583.0 0.846561
\(668\) −6038.78 −0.349772
\(669\) 9358.06 0.540812
\(670\) −26.0612 −0.00150274
\(671\) −21889.8 −1.25938
\(672\) 6768.26 0.388529
\(673\) 17975.7 1.02958 0.514792 0.857315i \(-0.327869\pi\)
0.514792 + 0.857315i \(0.327869\pi\)
\(674\) 4095.74 0.234068
\(675\) 15507.8 0.884292
\(676\) −353.062 −0.0200877
\(677\) 742.490 0.0421510 0.0210755 0.999778i \(-0.493291\pi\)
0.0210755 + 0.999778i \(0.493291\pi\)
\(678\) −46260.7 −2.62040
\(679\) −5900.03 −0.333464
\(680\) 1349.80 0.0761214
\(681\) 4168.90 0.234585
\(682\) −12040.0 −0.676007
\(683\) 29791.7 1.66903 0.834514 0.550987i \(-0.185748\pi\)
0.834514 + 0.550987i \(0.185748\pi\)
\(684\) 37096.2 2.07370
\(685\) −3431.42 −0.191399
\(686\) −8816.88 −0.490714
\(687\) −11560.9 −0.642029
\(688\) 0 0
\(689\) −17307.7 −0.957000
\(690\) 14337.8 0.791062
\(691\) 3361.09 0.185039 0.0925196 0.995711i \(-0.470508\pi\)
0.0925196 + 0.995711i \(0.470508\pi\)
\(692\) −26615.3 −1.46208
\(693\) −3583.78 −0.196445
\(694\) −5168.95 −0.282724
\(695\) −1762.47 −0.0961933
\(696\) 3744.58 0.203934
\(697\) −19620.5 −1.06626
\(698\) −33281.4 −1.80476
\(699\) −2165.26 −0.117164
\(700\) −2670.87 −0.144213
\(701\) 23288.1 1.25475 0.627373 0.778719i \(-0.284130\pi\)
0.627373 + 0.778719i \(0.284130\pi\)
\(702\) 23155.3 1.24493
\(703\) −39755.9 −2.13289
\(704\) 7799.71 0.417561
\(705\) −7175.44 −0.383323
\(706\) 23160.8 1.23466
\(707\) −1314.78 −0.0699395
\(708\) −31809.0 −1.68850
\(709\) 17503.7 0.927171 0.463585 0.886052i \(-0.346563\pi\)
0.463585 + 0.886052i \(0.346563\pi\)
\(710\) 4582.84 0.242241
\(711\) −12644.4 −0.666949
\(712\) 4025.55 0.211888
\(713\) 22545.3 1.18419
\(714\) −10651.8 −0.558309
\(715\) −2911.05 −0.152262
\(716\) 5392.76 0.281476
\(717\) −35295.6 −1.83841
\(718\) 10208.4 0.530605
\(719\) −10970.4 −0.569024 −0.284512 0.958673i \(-0.591832\pi\)
−0.284512 + 0.958673i \(0.591832\pi\)
\(720\) 8023.38 0.415297
\(721\) 641.896 0.0331560
\(722\) 40271.6 2.07584
\(723\) 61674.9 3.17249
\(724\) −11603.5 −0.595638
\(725\) −9847.09 −0.504430
\(726\) 23263.8 1.18925
\(727\) −747.962 −0.0381573 −0.0190787 0.999818i \(-0.506073\pi\)
−0.0190787 + 0.999818i \(0.506073\pi\)
\(728\) 855.069 0.0435315
\(729\) −32050.2 −1.62832
\(730\) −4715.91 −0.239101
\(731\) 0 0
\(732\) −49088.0 −2.47861
\(733\) −21811.3 −1.09907 −0.549534 0.835471i \(-0.685195\pi\)
−0.549534 + 0.835471i \(0.685195\pi\)
\(734\) 12248.0 0.615918
\(735\) −7090.92 −0.355854
\(736\) −41533.8 −2.08010
\(737\) −65.2581 −0.00326162
\(738\) −32781.0 −1.63508
\(739\) −13215.5 −0.657834 −0.328917 0.944359i \(-0.606684\pi\)
−0.328917 + 0.944359i \(0.606684\pi\)
\(740\) 5090.34 0.252871
\(741\) 50984.7 2.52762
\(742\) 4888.37 0.241857
\(743\) −25963.2 −1.28196 −0.640980 0.767558i \(-0.721472\pi\)
−0.640980 + 0.767558i \(0.721472\pi\)
\(744\) 5789.12 0.285268
\(745\) −1199.59 −0.0589929
\(746\) 8944.24 0.438970
\(747\) 5223.36 0.255841
\(748\) −15763.8 −0.770561
\(749\) −3920.13 −0.191240
\(750\) −19900.6 −0.968889
\(751\) 3717.93 0.180651 0.0903256 0.995912i \(-0.471209\pi\)
0.0903256 + 0.995912i \(0.471209\pi\)
\(752\) 24573.7 1.19164
\(753\) 16749.4 0.810600
\(754\) −14703.0 −0.710148
\(755\) −4426.08 −0.213353
\(756\) −2953.35 −0.142080
\(757\) −120.818 −0.00580079 −0.00290039 0.999996i \(-0.500923\pi\)
−0.00290039 + 0.999996i \(0.500923\pi\)
\(758\) −46463.1 −2.22641
\(759\) 35902.4 1.71696
\(760\) 1824.78 0.0870945
\(761\) −23248.7 −1.10744 −0.553722 0.832702i \(-0.686793\pi\)
−0.553722 + 0.832702i \(0.686793\pi\)
\(762\) 64576.3 3.07002
\(763\) −3796.62 −0.180140
\(764\) −7949.14 −0.376426
\(765\) −10680.7 −0.504784
\(766\) −39032.8 −1.84114
\(767\) −26779.6 −1.26070
\(768\) −42914.2 −2.01632
\(769\) 12713.4 0.596174 0.298087 0.954539i \(-0.403651\pi\)
0.298087 + 0.954539i \(0.403651\pi\)
\(770\) 822.192 0.0384802
\(771\) −2972.26 −0.138837
\(772\) −8084.16 −0.376885
\(773\) 6472.11 0.301146 0.150573 0.988599i \(-0.451888\pi\)
0.150573 + 0.988599i \(0.451888\pi\)
\(774\) 0 0
\(775\) −15223.6 −0.705610
\(776\) 9296.82 0.430073
\(777\) 8612.88 0.397665
\(778\) 36705.2 1.69144
\(779\) −26524.8 −1.21996
\(780\) −6528.07 −0.299670
\(781\) 11475.6 0.525772
\(782\) 65365.2 2.98907
\(783\) −10888.5 −0.496966
\(784\) 24284.3 1.10624
\(785\) 1252.57 0.0569505
\(786\) 81401.3 3.69401
\(787\) −25596.9 −1.15938 −0.579689 0.814838i \(-0.696826\pi\)
−0.579689 + 0.814838i \(0.696826\pi\)
\(788\) 19145.9 0.865538
\(789\) −21355.8 −0.963606
\(790\) 2900.88 0.130644
\(791\) 4967.42 0.223288
\(792\) 5647.05 0.253358
\(793\) −41326.5 −1.85063
\(794\) 31639.4 1.41416
\(795\) 8001.98 0.356982
\(796\) −12913.5 −0.575010
\(797\) −26835.0 −1.19265 −0.596327 0.802742i \(-0.703374\pi\)
−0.596327 + 0.802742i \(0.703374\pi\)
\(798\) −14400.0 −0.638791
\(799\) −32712.3 −1.44841
\(800\) 28045.5 1.23945
\(801\) −31853.2 −1.40509
\(802\) −48450.2 −2.13321
\(803\) −11808.8 −0.518958
\(804\) −146.342 −0.00641926
\(805\) −1539.58 −0.0674075
\(806\) −22730.9 −0.993375
\(807\) −63823.3 −2.78400
\(808\) 2071.73 0.0902018
\(809\) −22386.4 −0.972887 −0.486443 0.873712i \(-0.661706\pi\)
−0.486443 + 0.873712i \(0.661706\pi\)
\(810\) 581.585 0.0252282
\(811\) 12050.9 0.521781 0.260891 0.965368i \(-0.415984\pi\)
0.260891 + 0.965368i \(0.415984\pi\)
\(812\) 1875.30 0.0810470
\(813\) 21402.2 0.923255
\(814\) 28225.7 1.21537
\(815\) 2597.64 0.111646
\(816\) 59714.2 2.56178
\(817\) 0 0
\(818\) −3750.07 −0.160291
\(819\) −6765.95 −0.288671
\(820\) 3396.23 0.144636
\(821\) 2118.86 0.0900717 0.0450358 0.998985i \(-0.485660\pi\)
0.0450358 + 0.998985i \(0.485660\pi\)
\(822\) −42668.4 −1.81050
\(823\) −15733.7 −0.666396 −0.333198 0.942857i \(-0.608128\pi\)
−0.333198 + 0.942857i \(0.608128\pi\)
\(824\) −1011.45 −0.0427617
\(825\) −24242.9 −1.02306
\(826\) 7563.57 0.318608
\(827\) −19983.4 −0.840256 −0.420128 0.907465i \(-0.638015\pi\)
−0.420128 + 0.907465i \(0.638015\pi\)
\(828\) 49317.4 2.06993
\(829\) 34475.6 1.44437 0.722187 0.691698i \(-0.243137\pi\)
0.722187 + 0.691698i \(0.243137\pi\)
\(830\) −1198.35 −0.0501147
\(831\) −47211.8 −1.97083
\(832\) 14725.4 0.613594
\(833\) −32327.0 −1.34461
\(834\) −21915.6 −0.909924
\(835\) 2350.50 0.0974159
\(836\) −21310.8 −0.881640
\(837\) −16833.7 −0.695170
\(838\) 41745.8 1.72087
\(839\) −9582.61 −0.394313 −0.197156 0.980372i \(-0.563171\pi\)
−0.197156 + 0.980372i \(0.563171\pi\)
\(840\) −395.328 −0.0162382
\(841\) −17475.1 −0.716514
\(842\) 47177.9 1.93095
\(843\) 53514.7 2.18641
\(844\) −34438.6 −1.40453
\(845\) 137.423 0.00559468
\(846\) −54654.2 −2.22110
\(847\) −2498.03 −0.101338
\(848\) −27404.4 −1.10975
\(849\) 67039.1 2.70998
\(850\) −44137.5 −1.78106
\(851\) −52853.5 −2.12902
\(852\) 25734.1 1.03478
\(853\) −31343.1 −1.25811 −0.629055 0.777361i \(-0.716558\pi\)
−0.629055 + 0.777361i \(0.716558\pi\)
\(854\) 11672.2 0.467698
\(855\) −14439.1 −0.577550
\(856\) 6177.06 0.246644
\(857\) −3018.06 −0.120298 −0.0601488 0.998189i \(-0.519158\pi\)
−0.0601488 + 0.998189i \(0.519158\pi\)
\(858\) −36197.8 −1.44030
\(859\) −20259.2 −0.804697 −0.402349 0.915487i \(-0.631806\pi\)
−0.402349 + 0.915487i \(0.631806\pi\)
\(860\) 0 0
\(861\) 5746.43 0.227454
\(862\) −24959.4 −0.986219
\(863\) −40225.7 −1.58667 −0.793336 0.608785i \(-0.791657\pi\)
−0.793336 + 0.608785i \(0.791657\pi\)
\(864\) 31011.6 1.22111
\(865\) 10359.6 0.407209
\(866\) 46873.9 1.83931
\(867\) −38478.0 −1.50725
\(868\) 2899.22 0.113371
\(869\) 7263.88 0.283556
\(870\) 6797.71 0.264901
\(871\) −123.203 −0.00479286
\(872\) 5982.44 0.232329
\(873\) −73563.5 −2.85194
\(874\) 88366.6 3.41996
\(875\) 2136.90 0.0825604
\(876\) −26481.3 −1.02137
\(877\) 9078.63 0.349559 0.174780 0.984608i \(-0.444079\pi\)
0.174780 + 0.984608i \(0.444079\pi\)
\(878\) −10718.7 −0.412003
\(879\) 30154.3 1.15708
\(880\) −4609.24 −0.176565
\(881\) −1661.65 −0.0635440 −0.0317720 0.999495i \(-0.510115\pi\)
−0.0317720 + 0.999495i \(0.510115\pi\)
\(882\) −54010.4 −2.06193
\(883\) 10862.1 0.413974 0.206987 0.978344i \(-0.433634\pi\)
0.206987 + 0.978344i \(0.433634\pi\)
\(884\) −29761.0 −1.13232
\(885\) 12381.1 0.470268
\(886\) −35660.6 −1.35219
\(887\) −12633.9 −0.478248 −0.239124 0.970989i \(-0.576860\pi\)
−0.239124 + 0.970989i \(0.576860\pi\)
\(888\) −13571.5 −0.512873
\(889\) −6934.12 −0.261601
\(890\) 7307.78 0.275233
\(891\) 1456.31 0.0547565
\(892\) −7384.73 −0.277196
\(893\) −44223.4 −1.65720
\(894\) −14916.5 −0.558034
\(895\) −2099.04 −0.0783946
\(896\) 2327.22 0.0867710
\(897\) 67781.5 2.52303
\(898\) 27546.4 1.02365
\(899\) 10689.0 0.396548
\(900\) −33301.3 −1.23338
\(901\) 36480.4 1.34888
\(902\) 18831.9 0.695160
\(903\) 0 0
\(904\) −7827.29 −0.287977
\(905\) 4516.48 0.165893
\(906\) −55036.6 −2.01818
\(907\) 43157.9 1.57997 0.789987 0.613124i \(-0.210087\pi\)
0.789987 + 0.613124i \(0.210087\pi\)
\(908\) −3289.80 −0.120238
\(909\) −16393.1 −0.598156
\(910\) 1552.25 0.0565456
\(911\) 8726.36 0.317363 0.158681 0.987330i \(-0.449276\pi\)
0.158681 + 0.987330i \(0.449276\pi\)
\(912\) 80727.0 2.93107
\(913\) −3000.70 −0.108772
\(914\) −5264.59 −0.190522
\(915\) 19106.7 0.690325
\(916\) 9123.02 0.329076
\(917\) −8740.76 −0.314772
\(918\) −48805.6 −1.75471
\(919\) −25907.4 −0.929932 −0.464966 0.885328i \(-0.653934\pi\)
−0.464966 + 0.885328i \(0.653934\pi\)
\(920\) 2425.95 0.0869363
\(921\) 10702.8 0.382918
\(922\) −42949.4 −1.53413
\(923\) 21665.2 0.772609
\(924\) 4616.86 0.164376
\(925\) 35689.0 1.26859
\(926\) 7934.89 0.281595
\(927\) 8003.37 0.283566
\(928\) −19691.6 −0.696560
\(929\) −32728.1 −1.15584 −0.577919 0.816094i \(-0.696135\pi\)
−0.577919 + 0.816094i \(0.696135\pi\)
\(930\) 10509.3 0.370551
\(931\) −43702.5 −1.53845
\(932\) 1708.67 0.0600530
\(933\) 76347.0 2.67898
\(934\) −7702.78 −0.269853
\(935\) 6135.78 0.214611
\(936\) 10661.3 0.372302
\(937\) −5170.31 −0.180263 −0.0901316 0.995930i \(-0.528729\pi\)
−0.0901316 + 0.995930i \(0.528729\pi\)
\(938\) 34.7973 0.00121127
\(939\) −27186.0 −0.944814
\(940\) 5662.36 0.196474
\(941\) −23338.4 −0.808511 −0.404255 0.914646i \(-0.632469\pi\)
−0.404255 + 0.914646i \(0.632469\pi\)
\(942\) 15575.2 0.538714
\(943\) −35263.3 −1.21774
\(944\) −42401.6 −1.46192
\(945\) 1149.54 0.0395710
\(946\) 0 0
\(947\) 6979.11 0.239483 0.119742 0.992805i \(-0.461793\pi\)
0.119742 + 0.992805i \(0.461793\pi\)
\(948\) 16289.3 0.558073
\(949\) −22294.3 −0.762595
\(950\) −59669.0 −2.03781
\(951\) −53930.2 −1.83891
\(952\) −1802.27 −0.0613572
\(953\) −15961.8 −0.542552 −0.271276 0.962502i \(-0.587446\pi\)
−0.271276 + 0.962502i \(0.587446\pi\)
\(954\) 60949.8 2.06847
\(955\) 3094.07 0.104840
\(956\) 27852.9 0.942287
\(957\) 17021.7 0.574955
\(958\) 39824.6 1.34308
\(959\) 4581.68 0.154276
\(960\) −6808.05 −0.228884
\(961\) −13265.9 −0.445298
\(962\) 53288.4 1.78595
\(963\) −48877.5 −1.63557
\(964\) −48669.5 −1.62608
\(965\) 3146.62 0.104967
\(966\) −19144.1 −0.637630
\(967\) −15404.6 −0.512283 −0.256142 0.966639i \(-0.582451\pi\)
−0.256142 + 0.966639i \(0.582451\pi\)
\(968\) 3936.21 0.130697
\(969\) −107463. −3.56265
\(970\) 16877.0 0.558646
\(971\) −814.407 −0.0269161 −0.0134581 0.999909i \(-0.504284\pi\)
−0.0134581 + 0.999909i \(0.504284\pi\)
\(972\) 26557.1 0.876357
\(973\) 2353.27 0.0775360
\(974\) −15911.2 −0.523437
\(975\) −45769.1 −1.50337
\(976\) −65434.7 −2.14602
\(977\) 3952.69 0.129435 0.0647174 0.997904i \(-0.479385\pi\)
0.0647174 + 0.997904i \(0.479385\pi\)
\(978\) 32300.6 1.05609
\(979\) 18298.9 0.597380
\(980\) 5595.66 0.182395
\(981\) −47337.6 −1.54064
\(982\) −43172.6 −1.40295
\(983\) −2736.55 −0.0887918 −0.0443959 0.999014i \(-0.514136\pi\)
−0.0443959 + 0.999014i \(0.514136\pi\)
\(984\) −9054.80 −0.293350
\(985\) −7452.22 −0.241063
\(986\) 30990.3 1.00094
\(987\) 9580.73 0.308975
\(988\) −40233.6 −1.29555
\(989\) 0 0
\(990\) 10251.4 0.329101
\(991\) −24139.6 −0.773783 −0.386892 0.922125i \(-0.626451\pi\)
−0.386892 + 0.922125i \(0.626451\pi\)
\(992\) −30443.2 −0.974367
\(993\) −54034.1 −1.72681
\(994\) −6119.07 −0.195257
\(995\) 5026.37 0.160147
\(996\) −6729.09 −0.214076
\(997\) 27188.4 0.863656 0.431828 0.901956i \(-0.357869\pi\)
0.431828 + 0.901956i \(0.357869\pi\)
\(998\) 49920.3 1.58337
\(999\) 39463.5 1.24982
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.1.41 yes 50
43.42 odd 2 1849.4.a.i.1.10 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.10 50 43.42 odd 2
1849.4.a.j.1.41 yes 50 1.1 even 1 trivial