Properties

Label 1849.4.a.i.1.21
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 1.21
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.06456 q^{2} -7.81960 q^{3} -6.86672 q^{4} -6.14081 q^{5} +8.32441 q^{6} +5.24909 q^{7} +15.8265 q^{8} +34.1462 q^{9} +O(q^{10})\) \(q-1.06456 q^{2} -7.81960 q^{3} -6.86672 q^{4} -6.14081 q^{5} +8.32441 q^{6} +5.24909 q^{7} +15.8265 q^{8} +34.1462 q^{9} +6.53724 q^{10} -14.5177 q^{11} +53.6950 q^{12} -49.9353 q^{13} -5.58795 q^{14} +48.0187 q^{15} +38.0856 q^{16} +75.6572 q^{17} -36.3506 q^{18} -128.126 q^{19} +42.1672 q^{20} -41.0458 q^{21} +15.4549 q^{22} -131.825 q^{23} -123.757 q^{24} -87.2904 q^{25} +53.1589 q^{26} -55.8805 q^{27} -36.0440 q^{28} +36.6667 q^{29} -51.1187 q^{30} -251.659 q^{31} -167.156 q^{32} +113.523 q^{33} -80.5414 q^{34} -32.2337 q^{35} -234.472 q^{36} +264.673 q^{37} +136.397 q^{38} +390.474 q^{39} -97.1874 q^{40} -0.0646232 q^{41} +43.6956 q^{42} +99.6890 q^{44} -209.685 q^{45} +140.335 q^{46} +346.105 q^{47} -297.814 q^{48} -315.447 q^{49} +92.9256 q^{50} -591.610 q^{51} +342.892 q^{52} +468.928 q^{53} +59.4880 q^{54} +89.1505 q^{55} +83.0745 q^{56} +1001.89 q^{57} -39.0338 q^{58} +586.702 q^{59} -329.731 q^{60} +592.038 q^{61} +267.906 q^{62} +179.236 q^{63} -126.738 q^{64} +306.643 q^{65} -120.851 q^{66} -472.805 q^{67} -519.517 q^{68} +1030.82 q^{69} +34.3146 q^{70} +548.204 q^{71} +540.414 q^{72} +328.014 q^{73} -281.759 q^{74} +682.576 q^{75} +879.804 q^{76} -76.2047 q^{77} -415.682 q^{78} +801.655 q^{79} -233.876 q^{80} -484.984 q^{81} +0.0687950 q^{82} -1348.75 q^{83} +281.850 q^{84} -464.597 q^{85} -286.719 q^{87} -229.764 q^{88} +1491.49 q^{89} +223.222 q^{90} -262.115 q^{91} +905.205 q^{92} +1967.88 q^{93} -368.448 q^{94} +786.796 q^{95} +1307.09 q^{96} -281.945 q^{97} +335.811 q^{98} -495.724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06456 −0.376378 −0.188189 0.982133i \(-0.560262\pi\)
−0.188189 + 0.982133i \(0.560262\pi\)
\(3\) −7.81960 −1.50488 −0.752442 0.658659i \(-0.771124\pi\)
−0.752442 + 0.658659i \(0.771124\pi\)
\(4\) −6.86672 −0.858340
\(5\) −6.14081 −0.549251 −0.274626 0.961551i \(-0.588554\pi\)
−0.274626 + 0.961551i \(0.588554\pi\)
\(6\) 8.32441 0.566404
\(7\) 5.24909 0.283424 0.141712 0.989908i \(-0.454739\pi\)
0.141712 + 0.989908i \(0.454739\pi\)
\(8\) 15.8265 0.699438
\(9\) 34.1462 1.26467
\(10\) 6.53724 0.206726
\(11\) −14.5177 −0.397932 −0.198966 0.980006i \(-0.563758\pi\)
−0.198966 + 0.980006i \(0.563758\pi\)
\(12\) 53.6950 1.29170
\(13\) −49.9353 −1.06535 −0.532675 0.846320i \(-0.678813\pi\)
−0.532675 + 0.846320i \(0.678813\pi\)
\(14\) −5.58795 −0.106675
\(15\) 48.0187 0.826559
\(16\) 38.0856 0.595087
\(17\) 75.6572 1.07939 0.539693 0.841862i \(-0.318540\pi\)
0.539693 + 0.841862i \(0.318540\pi\)
\(18\) −36.3506 −0.475995
\(19\) −128.126 −1.54706 −0.773528 0.633762i \(-0.781510\pi\)
−0.773528 + 0.633762i \(0.781510\pi\)
\(20\) 42.1672 0.471444
\(21\) −41.0458 −0.426520
\(22\) 15.4549 0.149773
\(23\) −131.825 −1.19510 −0.597552 0.801830i \(-0.703860\pi\)
−0.597552 + 0.801830i \(0.703860\pi\)
\(24\) −123.757 −1.05257
\(25\) −87.2904 −0.698323
\(26\) 53.1589 0.400974
\(27\) −55.8805 −0.398304
\(28\) −36.0440 −0.243274
\(29\) 36.6667 0.234788 0.117394 0.993085i \(-0.462546\pi\)
0.117394 + 0.993085i \(0.462546\pi\)
\(30\) −51.1187 −0.311098
\(31\) −251.659 −1.45804 −0.729022 0.684491i \(-0.760025\pi\)
−0.729022 + 0.684491i \(0.760025\pi\)
\(32\) −167.156 −0.923415
\(33\) 113.523 0.598841
\(34\) −80.5414 −0.406257
\(35\) −32.2337 −0.155671
\(36\) −234.472 −1.08552
\(37\) 264.673 1.17600 0.588000 0.808861i \(-0.299916\pi\)
0.588000 + 0.808861i \(0.299916\pi\)
\(38\) 136.397 0.582277
\(39\) 390.474 1.60323
\(40\) −97.1874 −0.384167
\(41\) −0.0646232 −0.000246157 0 −0.000123079 1.00000i \(-0.500039\pi\)
−0.000123079 1.00000i \(0.500039\pi\)
\(42\) 43.6956 0.160533
\(43\) 0 0
\(44\) 99.6890 0.341561
\(45\) −209.685 −0.694624
\(46\) 140.335 0.449810
\(47\) 346.105 1.07414 0.537070 0.843538i \(-0.319531\pi\)
0.537070 + 0.843538i \(0.319531\pi\)
\(48\) −297.814 −0.895537
\(49\) −315.447 −0.919671
\(50\) 92.9256 0.262833
\(51\) −591.610 −1.62435
\(52\) 342.892 0.914433
\(53\) 468.928 1.21532 0.607662 0.794196i \(-0.292107\pi\)
0.607662 + 0.794196i \(0.292107\pi\)
\(54\) 59.4880 0.149913
\(55\) 89.1505 0.218565
\(56\) 83.0745 0.198237
\(57\) 1001.89 2.32814
\(58\) −39.0338 −0.0883688
\(59\) 586.702 1.29461 0.647306 0.762230i \(-0.275895\pi\)
0.647306 + 0.762230i \(0.275895\pi\)
\(60\) −329.731 −0.709468
\(61\) 592.038 1.24267 0.621333 0.783546i \(-0.286591\pi\)
0.621333 + 0.783546i \(0.286591\pi\)
\(62\) 267.906 0.548775
\(63\) 179.236 0.358439
\(64\) −126.738 −0.247534
\(65\) 306.643 0.585145
\(66\) −120.851 −0.225390
\(67\) −472.805 −0.862125 −0.431062 0.902322i \(-0.641861\pi\)
−0.431062 + 0.902322i \(0.641861\pi\)
\(68\) −519.517 −0.926481
\(69\) 1030.82 1.79849
\(70\) 34.3146 0.0585911
\(71\) 548.204 0.916336 0.458168 0.888866i \(-0.348506\pi\)
0.458168 + 0.888866i \(0.348506\pi\)
\(72\) 540.414 0.884561
\(73\) 328.014 0.525905 0.262953 0.964809i \(-0.415304\pi\)
0.262953 + 0.964809i \(0.415304\pi\)
\(74\) −281.759 −0.442620
\(75\) 682.576 1.05090
\(76\) 879.804 1.32790
\(77\) −76.2047 −0.112784
\(78\) −415.682 −0.603419
\(79\) 801.655 1.14169 0.570843 0.821059i \(-0.306616\pi\)
0.570843 + 0.821059i \(0.306616\pi\)
\(80\) −233.876 −0.326852
\(81\) −484.984 −0.665273
\(82\) 0.0687950 9.26480e−5 0
\(83\) −1348.75 −1.78368 −0.891838 0.452356i \(-0.850584\pi\)
−0.891838 + 0.452356i \(0.850584\pi\)
\(84\) 281.850 0.366099
\(85\) −464.597 −0.592854
\(86\) 0 0
\(87\) −286.719 −0.353328
\(88\) −229.764 −0.278329
\(89\) 1491.49 1.77638 0.888189 0.459478i \(-0.151964\pi\)
0.888189 + 0.459478i \(0.151964\pi\)
\(90\) 223.222 0.261441
\(91\) −262.115 −0.301946
\(92\) 905.205 1.02581
\(93\) 1967.88 2.19419
\(94\) −368.448 −0.404282
\(95\) 786.796 0.849722
\(96\) 1307.09 1.38963
\(97\) −281.945 −0.295125 −0.147563 0.989053i \(-0.547143\pi\)
−0.147563 + 0.989053i \(0.547143\pi\)
\(98\) 335.811 0.346143
\(99\) −495.724 −0.503254
\(100\) 599.399 0.599399
\(101\) 1102.45 1.08612 0.543060 0.839694i \(-0.317266\pi\)
0.543060 + 0.839694i \(0.317266\pi\)
\(102\) 629.802 0.611370
\(103\) −1857.59 −1.77703 −0.888514 0.458849i \(-0.848262\pi\)
−0.888514 + 0.458849i \(0.848262\pi\)
\(104\) −790.299 −0.745146
\(105\) 252.055 0.234267
\(106\) −499.200 −0.457421
\(107\) 187.602 0.169497 0.0847487 0.996402i \(-0.472991\pi\)
0.0847487 + 0.996402i \(0.472991\pi\)
\(108\) 383.716 0.341880
\(109\) 116.470 0.102347 0.0511733 0.998690i \(-0.483704\pi\)
0.0511733 + 0.998690i \(0.483704\pi\)
\(110\) −94.9058 −0.0822628
\(111\) −2069.64 −1.76974
\(112\) 199.915 0.168662
\(113\) −1240.45 −1.03267 −0.516334 0.856387i \(-0.672704\pi\)
−0.516334 + 0.856387i \(0.672704\pi\)
\(114\) −1066.57 −0.876260
\(115\) 809.512 0.656412
\(116\) −251.780 −0.201528
\(117\) −1705.10 −1.34732
\(118\) −624.578 −0.487263
\(119\) 397.132 0.305924
\(120\) 759.967 0.578126
\(121\) −1120.24 −0.841650
\(122\) −630.258 −0.467712
\(123\) 0.505328 0.000370438 0
\(124\) 1728.07 1.25150
\(125\) 1303.64 0.932806
\(126\) −190.807 −0.134909
\(127\) 92.3145 0.0645007 0.0322503 0.999480i \(-0.489733\pi\)
0.0322503 + 0.999480i \(0.489733\pi\)
\(128\) 1472.17 1.01658
\(129\) 0 0
\(130\) −326.439 −0.220235
\(131\) 1721.43 1.14811 0.574055 0.818817i \(-0.305370\pi\)
0.574055 + 0.818817i \(0.305370\pi\)
\(132\) −779.528 −0.514009
\(133\) −672.544 −0.438473
\(134\) 503.328 0.324485
\(135\) 343.152 0.218769
\(136\) 1197.39 0.754964
\(137\) 1677.68 1.04624 0.523118 0.852261i \(-0.324769\pi\)
0.523118 + 0.852261i \(0.324769\pi\)
\(138\) −1097.36 −0.676912
\(139\) 2819.71 1.72061 0.860306 0.509778i \(-0.170272\pi\)
0.860306 + 0.509778i \(0.170272\pi\)
\(140\) 221.340 0.133619
\(141\) −2706.40 −1.61646
\(142\) −583.594 −0.344888
\(143\) 724.946 0.423937
\(144\) 1300.48 0.752592
\(145\) −225.163 −0.128957
\(146\) −349.189 −0.197939
\(147\) 2466.67 1.38400
\(148\) −1817.43 −1.00941
\(149\) 971.666 0.534241 0.267121 0.963663i \(-0.413928\pi\)
0.267121 + 0.963663i \(0.413928\pi\)
\(150\) −726.641 −0.395533
\(151\) −876.678 −0.472470 −0.236235 0.971696i \(-0.575914\pi\)
−0.236235 + 0.971696i \(0.575914\pi\)
\(152\) −2027.78 −1.08207
\(153\) 2583.41 1.36507
\(154\) 81.1242 0.0424492
\(155\) 1545.39 0.800832
\(156\) −2681.28 −1.37611
\(157\) 2191.29 1.11391 0.556955 0.830543i \(-0.311970\pi\)
0.556955 + 0.830543i \(0.311970\pi\)
\(158\) −853.407 −0.429705
\(159\) −3666.83 −1.82892
\(160\) 1026.47 0.507187
\(161\) −691.961 −0.338721
\(162\) 516.293 0.250394
\(163\) 212.798 0.102255 0.0511276 0.998692i \(-0.483718\pi\)
0.0511276 + 0.998692i \(0.483718\pi\)
\(164\) 0.443749 0.000211287 0
\(165\) −697.122 −0.328914
\(166\) 1435.83 0.671335
\(167\) 268.336 0.124338 0.0621691 0.998066i \(-0.480198\pi\)
0.0621691 + 0.998066i \(0.480198\pi\)
\(168\) −649.610 −0.298324
\(169\) 296.533 0.134972
\(170\) 494.590 0.223137
\(171\) −4375.01 −1.95652
\(172\) 0 0
\(173\) −310.456 −0.136437 −0.0682183 0.997670i \(-0.521731\pi\)
−0.0682183 + 0.997670i \(0.521731\pi\)
\(174\) 305.229 0.132985
\(175\) −458.195 −0.197922
\(176\) −552.915 −0.236804
\(177\) −4587.78 −1.94824
\(178\) −1587.78 −0.668589
\(179\) −3308.82 −1.38163 −0.690817 0.723030i \(-0.742749\pi\)
−0.690817 + 0.723030i \(0.742749\pi\)
\(180\) 1439.85 0.596223
\(181\) 1215.01 0.498957 0.249479 0.968380i \(-0.419741\pi\)
0.249479 + 0.968380i \(0.419741\pi\)
\(182\) 279.036 0.113646
\(183\) −4629.50 −1.87007
\(184\) −2086.32 −0.835901
\(185\) −1625.31 −0.645919
\(186\) −2094.92 −0.825842
\(187\) −1098.37 −0.429523
\(188\) −2376.61 −0.921977
\(189\) −293.322 −0.112889
\(190\) −837.589 −0.319816
\(191\) 3262.00 1.23576 0.617880 0.786273i \(-0.287992\pi\)
0.617880 + 0.786273i \(0.287992\pi\)
\(192\) 991.038 0.372511
\(193\) −596.437 −0.222448 −0.111224 0.993795i \(-0.535477\pi\)
−0.111224 + 0.993795i \(0.535477\pi\)
\(194\) 300.146 0.111079
\(195\) −2397.83 −0.880575
\(196\) 2166.09 0.789390
\(197\) 3156.54 1.14160 0.570798 0.821091i \(-0.306634\pi\)
0.570798 + 0.821091i \(0.306634\pi\)
\(198\) 527.727 0.189414
\(199\) 1617.31 0.576120 0.288060 0.957612i \(-0.406990\pi\)
0.288060 + 0.957612i \(0.406990\pi\)
\(200\) −1381.50 −0.488434
\(201\) 3697.15 1.29740
\(202\) −1173.62 −0.408791
\(203\) 192.467 0.0665445
\(204\) 4062.42 1.39425
\(205\) 0.396839 0.000135202 0
\(206\) 1977.51 0.668834
\(207\) −4501.32 −1.51142
\(208\) −1901.81 −0.633977
\(209\) 1860.09 0.615623
\(210\) −268.326 −0.0881728
\(211\) −3160.29 −1.03110 −0.515552 0.856858i \(-0.672413\pi\)
−0.515552 + 0.856858i \(0.672413\pi\)
\(212\) −3220.00 −1.04316
\(213\) −4286.74 −1.37898
\(214\) −199.713 −0.0637950
\(215\) 0 0
\(216\) −884.391 −0.278589
\(217\) −1320.98 −0.413245
\(218\) −123.989 −0.0385209
\(219\) −2564.94 −0.791426
\(220\) −612.171 −0.187603
\(221\) −3777.97 −1.14993
\(222\) 2203.25 0.666091
\(223\) 3030.41 0.910006 0.455003 0.890490i \(-0.349638\pi\)
0.455003 + 0.890490i \(0.349638\pi\)
\(224\) −877.417 −0.261718
\(225\) −2980.64 −0.883152
\(226\) 1320.53 0.388673
\(227\) −4911.55 −1.43608 −0.718041 0.696000i \(-0.754961\pi\)
−0.718041 + 0.696000i \(0.754961\pi\)
\(228\) −6879.72 −1.99833
\(229\) −5122.08 −1.47806 −0.739031 0.673671i \(-0.764716\pi\)
−0.739031 + 0.673671i \(0.764716\pi\)
\(230\) −861.771 −0.247059
\(231\) 595.891 0.169726
\(232\) 580.305 0.164219
\(233\) −1455.82 −0.409332 −0.204666 0.978832i \(-0.565611\pi\)
−0.204666 + 0.978832i \(0.565611\pi\)
\(234\) 1815.18 0.507102
\(235\) −2125.37 −0.589973
\(236\) −4028.72 −1.11122
\(237\) −6268.62 −1.71810
\(238\) −422.769 −0.115143
\(239\) 5646.52 1.52821 0.764107 0.645089i \(-0.223180\pi\)
0.764107 + 0.645089i \(0.223180\pi\)
\(240\) 1828.82 0.491875
\(241\) −2920.77 −0.780677 −0.390339 0.920671i \(-0.627642\pi\)
−0.390339 + 0.920671i \(0.627642\pi\)
\(242\) 1192.56 0.316778
\(243\) 5301.16 1.39946
\(244\) −4065.36 −1.06663
\(245\) 1937.10 0.505130
\(246\) −0.537950 −0.000139425 0
\(247\) 6398.00 1.64816
\(248\) −3982.88 −1.01981
\(249\) 10546.7 2.68422
\(250\) −1387.79 −0.351087
\(251\) −2486.12 −0.625189 −0.312594 0.949887i \(-0.601198\pi\)
−0.312594 + 0.949887i \(0.601198\pi\)
\(252\) −1230.77 −0.307663
\(253\) 1913.79 0.475570
\(254\) −98.2740 −0.0242766
\(255\) 3632.96 0.892177
\(256\) −553.304 −0.135084
\(257\) −7326.49 −1.77826 −0.889132 0.457650i \(-0.848691\pi\)
−0.889132 + 0.457650i \(0.848691\pi\)
\(258\) 0 0
\(259\) 1389.29 0.333307
\(260\) −2105.63 −0.502253
\(261\) 1252.03 0.296930
\(262\) −1832.56 −0.432123
\(263\) −251.220 −0.0589008 −0.0294504 0.999566i \(-0.509376\pi\)
−0.0294504 + 0.999566i \(0.509376\pi\)
\(264\) 1796.66 0.418852
\(265\) −2879.60 −0.667518
\(266\) 715.961 0.165031
\(267\) −11662.9 −2.67324
\(268\) 3246.62 0.739996
\(269\) −4741.17 −1.07463 −0.537313 0.843383i \(-0.680560\pi\)
−0.537313 + 0.843383i \(0.680560\pi\)
\(270\) −365.304 −0.0823397
\(271\) 3799.34 0.851637 0.425819 0.904809i \(-0.359986\pi\)
0.425819 + 0.904809i \(0.359986\pi\)
\(272\) 2881.45 0.642329
\(273\) 2049.63 0.454394
\(274\) −1785.99 −0.393779
\(275\) 1267.26 0.277885
\(276\) −7078.34 −1.54372
\(277\) 4768.63 1.03437 0.517183 0.855875i \(-0.326981\pi\)
0.517183 + 0.855875i \(0.326981\pi\)
\(278\) −3001.75 −0.647600
\(279\) −8593.21 −1.84395
\(280\) −510.145 −0.108882
\(281\) 2410.33 0.511703 0.255851 0.966716i \(-0.417644\pi\)
0.255851 + 0.966716i \(0.417644\pi\)
\(282\) 2881.12 0.608398
\(283\) −8196.95 −1.72176 −0.860880 0.508808i \(-0.830086\pi\)
−0.860880 + 0.508808i \(0.830086\pi\)
\(284\) −3764.36 −0.786528
\(285\) −6152.44 −1.27873
\(286\) −771.746 −0.159560
\(287\) −0.339213 −6.97669e−5 0
\(288\) −5707.74 −1.16782
\(289\) 811.018 0.165076
\(290\) 239.699 0.0485366
\(291\) 2204.70 0.444129
\(292\) −2252.38 −0.451405
\(293\) 4556.22 0.908455 0.454227 0.890886i \(-0.349915\pi\)
0.454227 + 0.890886i \(0.349915\pi\)
\(294\) −2625.91 −0.520906
\(295\) −3602.83 −0.711067
\(296\) 4188.84 0.822538
\(297\) 811.257 0.158498
\(298\) −1034.39 −0.201076
\(299\) 6582.71 1.27320
\(300\) −4687.06 −0.902025
\(301\) 0 0
\(302\) 933.273 0.177827
\(303\) −8620.74 −1.63448
\(304\) −4879.74 −0.920633
\(305\) −3635.59 −0.682536
\(306\) −2750.18 −0.513783
\(307\) 592.948 0.110232 0.0551162 0.998480i \(-0.482447\pi\)
0.0551162 + 0.998480i \(0.482447\pi\)
\(308\) 523.276 0.0968066
\(309\) 14525.6 2.67422
\(310\) −1645.16 −0.301415
\(311\) 462.957 0.0844113 0.0422056 0.999109i \(-0.486562\pi\)
0.0422056 + 0.999109i \(0.486562\pi\)
\(312\) 6179.83 1.12136
\(313\) −2003.19 −0.361748 −0.180874 0.983506i \(-0.557893\pi\)
−0.180874 + 0.983506i \(0.557893\pi\)
\(314\) −2332.75 −0.419251
\(315\) −1100.66 −0.196873
\(316\) −5504.74 −0.979955
\(317\) 7514.15 1.33135 0.665673 0.746244i \(-0.268145\pi\)
0.665673 + 0.746244i \(0.268145\pi\)
\(318\) 3903.55 0.688365
\(319\) −532.317 −0.0934295
\(320\) 778.272 0.135959
\(321\) −1466.98 −0.255074
\(322\) 736.631 0.127487
\(323\) −9693.64 −1.66987
\(324\) 3330.25 0.571031
\(325\) 4358.87 0.743959
\(326\) −226.535 −0.0384866
\(327\) −910.747 −0.154020
\(328\) −1.02276 −0.000172172 0
\(329\) 1816.74 0.304437
\(330\) 742.125 0.123796
\(331\) 45.1750 0.00750164 0.00375082 0.999993i \(-0.498806\pi\)
0.00375082 + 0.999993i \(0.498806\pi\)
\(332\) 9261.52 1.53100
\(333\) 9037.58 1.48726
\(334\) −285.659 −0.0467981
\(335\) 2903.41 0.473523
\(336\) −1563.25 −0.253817
\(337\) −3243.02 −0.524209 −0.262105 0.965039i \(-0.584417\pi\)
−0.262105 + 0.965039i \(0.584417\pi\)
\(338\) −315.676 −0.0508003
\(339\) 9699.82 1.55405
\(340\) 3190.26 0.508870
\(341\) 3653.51 0.580202
\(342\) 4657.45 0.736391
\(343\) −3456.25 −0.544081
\(344\) 0 0
\(345\) −6330.06 −0.987824
\(346\) 330.498 0.0513517
\(347\) 3928.20 0.607715 0.303857 0.952718i \(-0.401725\pi\)
0.303857 + 0.952718i \(0.401725\pi\)
\(348\) 1968.82 0.303275
\(349\) 9860.04 1.51231 0.756154 0.654393i \(-0.227076\pi\)
0.756154 + 0.654393i \(0.227076\pi\)
\(350\) 487.775 0.0744933
\(351\) 2790.41 0.424333
\(352\) 2426.72 0.367456
\(353\) 3705.00 0.558633 0.279316 0.960199i \(-0.409892\pi\)
0.279316 + 0.960199i \(0.409892\pi\)
\(354\) 4883.95 0.733274
\(355\) −3366.42 −0.503299
\(356\) −10241.6 −1.52474
\(357\) −3105.41 −0.460380
\(358\) 3522.42 0.520016
\(359\) 3869.65 0.568892 0.284446 0.958692i \(-0.408190\pi\)
0.284446 + 0.958692i \(0.408190\pi\)
\(360\) −3318.58 −0.485846
\(361\) 9557.21 1.39338
\(362\) −1293.45 −0.187796
\(363\) 8759.80 1.26659
\(364\) 1799.87 0.259172
\(365\) −2014.27 −0.288854
\(366\) 4928.36 0.703852
\(367\) −10876.6 −1.54702 −0.773508 0.633786i \(-0.781500\pi\)
−0.773508 + 0.633786i \(0.781500\pi\)
\(368\) −5020.63 −0.711191
\(369\) −2.20664 −0.000311309 0
\(370\) 1730.23 0.243109
\(371\) 2461.44 0.344452
\(372\) −13512.9 −1.88336
\(373\) 10131.1 1.40634 0.703172 0.711020i \(-0.251766\pi\)
0.703172 + 0.711020i \(0.251766\pi\)
\(374\) 1169.28 0.161663
\(375\) −10193.9 −1.40376
\(376\) 5477.62 0.751294
\(377\) −1830.96 −0.250131
\(378\) 312.258 0.0424889
\(379\) −8606.22 −1.16642 −0.583208 0.812323i \(-0.698203\pi\)
−0.583208 + 0.812323i \(0.698203\pi\)
\(380\) −5402.71 −0.729350
\(381\) −721.862 −0.0970660
\(382\) −3472.58 −0.465112
\(383\) −3345.75 −0.446370 −0.223185 0.974776i \(-0.571645\pi\)
−0.223185 + 0.974776i \(0.571645\pi\)
\(384\) −11511.8 −1.52984
\(385\) 467.959 0.0619465
\(386\) 634.941 0.0837245
\(387\) 0 0
\(388\) 1936.03 0.253318
\(389\) −3603.72 −0.469706 −0.234853 0.972031i \(-0.575461\pi\)
−0.234853 + 0.972031i \(0.575461\pi\)
\(390\) 2552.62 0.331429
\(391\) −9973.51 −1.28998
\(392\) −4992.41 −0.643252
\(393\) −13460.9 −1.72777
\(394\) −3360.32 −0.429671
\(395\) −4922.81 −0.627072
\(396\) 3404.00 0.431963
\(397\) 4178.70 0.528270 0.264135 0.964486i \(-0.414914\pi\)
0.264135 + 0.964486i \(0.414914\pi\)
\(398\) −1721.71 −0.216839
\(399\) 5259.02 0.659851
\(400\) −3324.51 −0.415563
\(401\) −6248.54 −0.778147 −0.389074 0.921207i \(-0.627205\pi\)
−0.389074 + 0.921207i \(0.627205\pi\)
\(402\) −3935.83 −0.488311
\(403\) 12566.7 1.55333
\(404\) −7570.23 −0.932260
\(405\) 2978.20 0.365402
\(406\) −204.892 −0.0250458
\(407\) −3842.44 −0.467968
\(408\) −9363.09 −1.13613
\(409\) −2173.77 −0.262802 −0.131401 0.991329i \(-0.541948\pi\)
−0.131401 + 0.991329i \(0.541948\pi\)
\(410\) −0.422457 −5.08870e−5 0
\(411\) −13118.8 −1.57446
\(412\) 12755.6 1.52529
\(413\) 3079.65 0.366925
\(414\) 4791.91 0.568864
\(415\) 8282.45 0.979685
\(416\) 8346.98 0.983761
\(417\) −22049.1 −2.58932
\(418\) −1980.17 −0.231707
\(419\) −9987.39 −1.16448 −0.582238 0.813018i \(-0.697823\pi\)
−0.582238 + 0.813018i \(0.697823\pi\)
\(420\) −1730.79 −0.201080
\(421\) −6981.99 −0.808269 −0.404134 0.914700i \(-0.632427\pi\)
−0.404134 + 0.914700i \(0.632427\pi\)
\(422\) 3364.30 0.388085
\(423\) 11818.2 1.35844
\(424\) 7421.47 0.850043
\(425\) −6604.15 −0.753761
\(426\) 4563.48 0.519017
\(427\) 3107.66 0.352202
\(428\) −1288.21 −0.145486
\(429\) −5668.79 −0.637976
\(430\) 0 0
\(431\) 222.271 0.0248408 0.0124204 0.999923i \(-0.496046\pi\)
0.0124204 + 0.999923i \(0.496046\pi\)
\(432\) −2128.24 −0.237026
\(433\) 5956.90 0.661132 0.330566 0.943783i \(-0.392760\pi\)
0.330566 + 0.943783i \(0.392760\pi\)
\(434\) 1406.26 0.155536
\(435\) 1760.69 0.194066
\(436\) −799.765 −0.0878481
\(437\) 16890.2 1.84889
\(438\) 2730.52 0.297875
\(439\) 6259.09 0.680478 0.340239 0.940339i \(-0.389492\pi\)
0.340239 + 0.940339i \(0.389492\pi\)
\(440\) 1410.94 0.152872
\(441\) −10771.3 −1.16308
\(442\) 4021.86 0.432806
\(443\) 10832.7 1.16180 0.580902 0.813974i \(-0.302700\pi\)
0.580902 + 0.813974i \(0.302700\pi\)
\(444\) 14211.6 1.51904
\(445\) −9158.96 −0.975677
\(446\) −3226.04 −0.342506
\(447\) −7598.04 −0.803971
\(448\) −665.257 −0.0701573
\(449\) −10794.8 −1.13461 −0.567305 0.823508i \(-0.692014\pi\)
−0.567305 + 0.823508i \(0.692014\pi\)
\(450\) 3173.06 0.332398
\(451\) 0.938180 9.79538e−5 0
\(452\) 8517.81 0.886381
\(453\) 6855.27 0.711013
\(454\) 5228.62 0.540509
\(455\) 1609.60 0.165844
\(456\) 15856.4 1.62839
\(457\) −12577.4 −1.28741 −0.643706 0.765273i \(-0.722604\pi\)
−0.643706 + 0.765273i \(0.722604\pi\)
\(458\) 5452.74 0.556310
\(459\) −4227.76 −0.429924
\(460\) −5558.69 −0.563425
\(461\) −14738.5 −1.48902 −0.744510 0.667611i \(-0.767317\pi\)
−0.744510 + 0.667611i \(0.767317\pi\)
\(462\) −634.359 −0.0638811
\(463\) −877.733 −0.0881030 −0.0440515 0.999029i \(-0.514027\pi\)
−0.0440515 + 0.999029i \(0.514027\pi\)
\(464\) 1396.47 0.139719
\(465\) −12084.4 −1.20516
\(466\) 1549.81 0.154063
\(467\) −3211.66 −0.318239 −0.159120 0.987259i \(-0.550866\pi\)
−0.159120 + 0.987259i \(0.550866\pi\)
\(468\) 11708.4 1.15646
\(469\) −2481.80 −0.244347
\(470\) 2262.57 0.222052
\(471\) −17135.0 −1.67630
\(472\) 9285.43 0.905501
\(473\) 0 0
\(474\) 6673.30 0.646656
\(475\) 11184.2 1.08035
\(476\) −2726.99 −0.262587
\(477\) 16012.1 1.53699
\(478\) −6011.04 −0.575186
\(479\) −6177.44 −0.589258 −0.294629 0.955612i \(-0.595196\pi\)
−0.294629 + 0.955612i \(0.595196\pi\)
\(480\) −8026.62 −0.763257
\(481\) −13216.5 −1.25285
\(482\) 3109.32 0.293829
\(483\) 5410.86 0.509736
\(484\) 7692.35 0.722422
\(485\) 1731.37 0.162098
\(486\) −5643.38 −0.526726
\(487\) 15600.7 1.45161 0.725807 0.687899i \(-0.241467\pi\)
0.725807 + 0.687899i \(0.241467\pi\)
\(488\) 9369.86 0.869168
\(489\) −1663.99 −0.153882
\(490\) −2062.15 −0.190120
\(491\) −108.625 −0.00998406 −0.00499203 0.999988i \(-0.501589\pi\)
−0.00499203 + 0.999988i \(0.501589\pi\)
\(492\) −3.46994 −0.000317962 0
\(493\) 2774.10 0.253427
\(494\) −6811.03 −0.620329
\(495\) 3044.15 0.276413
\(496\) −9584.59 −0.867663
\(497\) 2877.57 0.259712
\(498\) −11227.6 −1.01028
\(499\) 1133.22 0.101663 0.0508317 0.998707i \(-0.483813\pi\)
0.0508317 + 0.998707i \(0.483813\pi\)
\(500\) −8951.70 −0.800664
\(501\) −2098.28 −0.187115
\(502\) 2646.61 0.235307
\(503\) 12385.7 1.09791 0.548956 0.835852i \(-0.315026\pi\)
0.548956 + 0.835852i \(0.315026\pi\)
\(504\) 2836.68 0.250706
\(505\) −6769.95 −0.596552
\(506\) −2037.34 −0.178994
\(507\) −2318.77 −0.203117
\(508\) −633.897 −0.0553635
\(509\) −5798.36 −0.504927 −0.252464 0.967606i \(-0.581241\pi\)
−0.252464 + 0.967606i \(0.581241\pi\)
\(510\) −3867.50 −0.335795
\(511\) 1721.77 0.149054
\(512\) −11188.3 −0.965739
\(513\) 7159.73 0.616199
\(514\) 7799.46 0.669299
\(515\) 11407.1 0.976035
\(516\) 0 0
\(517\) −5024.65 −0.427435
\(518\) −1478.98 −0.125449
\(519\) 2427.64 0.205321
\(520\) 4853.08 0.409272
\(521\) −1523.74 −0.128131 −0.0640653 0.997946i \(-0.520407\pi\)
−0.0640653 + 0.997946i \(0.520407\pi\)
\(522\) −1332.86 −0.111758
\(523\) 7712.81 0.644852 0.322426 0.946595i \(-0.395502\pi\)
0.322426 + 0.946595i \(0.395502\pi\)
\(524\) −11820.6 −0.985468
\(525\) 3582.90 0.297849
\(526\) 267.438 0.0221690
\(527\) −19039.8 −1.57379
\(528\) 4323.58 0.356363
\(529\) 5210.81 0.428274
\(530\) 3065.50 0.251239
\(531\) 20033.7 1.63726
\(532\) 4618.17 0.376359
\(533\) 3.22698 0.000262244 0
\(534\) 12415.8 1.00615
\(535\) −1152.03 −0.0930966
\(536\) −7482.84 −0.603003
\(537\) 25873.6 2.07920
\(538\) 5047.24 0.404465
\(539\) 4579.57 0.365966
\(540\) −2356.33 −0.187778
\(541\) −17909.6 −1.42328 −0.711640 0.702544i \(-0.752047\pi\)
−0.711640 + 0.702544i \(0.752047\pi\)
\(542\) −4044.62 −0.320537
\(543\) −9500.92 −0.750872
\(544\) −12646.6 −0.996722
\(545\) −715.219 −0.0562139
\(546\) −2181.95 −0.171024
\(547\) 6691.92 0.523082 0.261541 0.965192i \(-0.415769\pi\)
0.261541 + 0.965192i \(0.415769\pi\)
\(548\) −11520.2 −0.898025
\(549\) 20215.8 1.57157
\(550\) −1349.07 −0.104590
\(551\) −4697.95 −0.363230
\(552\) 16314.2 1.25793
\(553\) 4207.96 0.323581
\(554\) −5076.48 −0.389312
\(555\) 12709.3 0.972032
\(556\) −19362.2 −1.47687
\(557\) 9117.59 0.693581 0.346790 0.937943i \(-0.387271\pi\)
0.346790 + 0.937943i \(0.387271\pi\)
\(558\) 9147.96 0.694021
\(559\) 0 0
\(560\) −1227.64 −0.0926378
\(561\) 8588.81 0.646381
\(562\) −2565.94 −0.192593
\(563\) 16518.8 1.23656 0.618282 0.785956i \(-0.287829\pi\)
0.618282 + 0.785956i \(0.287829\pi\)
\(564\) 18584.1 1.38747
\(565\) 7617.36 0.567194
\(566\) 8726.12 0.648032
\(567\) −2545.72 −0.188554
\(568\) 8676.13 0.640920
\(569\) 25394.8 1.87101 0.935505 0.353313i \(-0.114945\pi\)
0.935505 + 0.353313i \(0.114945\pi\)
\(570\) 6549.62 0.481286
\(571\) 13697.9 1.00392 0.501960 0.864891i \(-0.332613\pi\)
0.501960 + 0.864891i \(0.332613\pi\)
\(572\) −4978.00 −0.363882
\(573\) −25507.6 −1.85967
\(574\) 0.361111 2.62587e−5 0
\(575\) 11507.1 0.834569
\(576\) −4327.61 −0.313051
\(577\) −763.733 −0.0551033 −0.0275517 0.999620i \(-0.508771\pi\)
−0.0275517 + 0.999620i \(0.508771\pi\)
\(578\) −863.375 −0.0621309
\(579\) 4663.90 0.334759
\(580\) 1546.13 0.110689
\(581\) −7079.73 −0.505537
\(582\) −2347.02 −0.167160
\(583\) −6807.75 −0.483616
\(584\) 5191.29 0.367838
\(585\) 10470.7 0.740018
\(586\) −4850.36 −0.341922
\(587\) −7553.03 −0.531085 −0.265543 0.964099i \(-0.585551\pi\)
−0.265543 + 0.964099i \(0.585551\pi\)
\(588\) −16937.9 −1.18794
\(589\) 32244.0 2.25567
\(590\) 3835.42 0.267630
\(591\) −24682.9 −1.71797
\(592\) 10080.2 0.699822
\(593\) −23780.6 −1.64680 −0.823401 0.567460i \(-0.807926\pi\)
−0.823401 + 0.567460i \(0.807926\pi\)
\(594\) −863.629 −0.0596551
\(595\) −2438.71 −0.168029
\(596\) −6672.16 −0.458561
\(597\) −12646.7 −0.866993
\(598\) −7007.67 −0.479206
\(599\) −5509.20 −0.375792 −0.187896 0.982189i \(-0.560167\pi\)
−0.187896 + 0.982189i \(0.560167\pi\)
\(600\) 10802.8 0.735036
\(601\) −19229.4 −1.30513 −0.652565 0.757733i \(-0.726307\pi\)
−0.652565 + 0.757733i \(0.726307\pi\)
\(602\) 0 0
\(603\) −16144.5 −1.09031
\(604\) 6019.90 0.405540
\(605\) 6879.16 0.462277
\(606\) 9177.26 0.615183
\(607\) −16273.8 −1.08820 −0.544098 0.839022i \(-0.683128\pi\)
−0.544098 + 0.839022i \(0.683128\pi\)
\(608\) 21417.0 1.42858
\(609\) −1505.01 −0.100142
\(610\) 3870.29 0.256891
\(611\) −17282.8 −1.14434
\(612\) −17739.5 −1.17170
\(613\) 22778.0 1.50081 0.750404 0.660979i \(-0.229859\pi\)
0.750404 + 0.660979i \(0.229859\pi\)
\(614\) −631.227 −0.0414890
\(615\) −3.10312 −0.000203463 0
\(616\) −1206.05 −0.0788850
\(617\) −2928.42 −0.191076 −0.0955379 0.995426i \(-0.530457\pi\)
−0.0955379 + 0.995426i \(0.530457\pi\)
\(618\) −15463.4 −1.00652
\(619\) −21693.7 −1.40863 −0.704316 0.709886i \(-0.748746\pi\)
−0.704316 + 0.709886i \(0.748746\pi\)
\(620\) −10611.8 −0.687386
\(621\) 7366.44 0.476015
\(622\) −492.844 −0.0317705
\(623\) 7828.96 0.503468
\(624\) 14871.4 0.954061
\(625\) 2905.92 0.185979
\(626\) 2132.51 0.136154
\(627\) −14545.2 −0.926441
\(628\) −15047.0 −0.956113
\(629\) 20024.4 1.26936
\(630\) 1171.71 0.0740986
\(631\) 6428.64 0.405579 0.202789 0.979222i \(-0.434999\pi\)
0.202789 + 0.979222i \(0.434999\pi\)
\(632\) 12687.4 0.798538
\(633\) 24712.2 1.55169
\(634\) −7999.23 −0.501089
\(635\) −566.886 −0.0354271
\(636\) 25179.1 1.56984
\(637\) 15751.9 0.979772
\(638\) 566.681 0.0351648
\(639\) 18719.1 1.15887
\(640\) −9040.30 −0.558358
\(641\) −16640.2 −1.02535 −0.512675 0.858583i \(-0.671345\pi\)
−0.512675 + 0.858583i \(0.671345\pi\)
\(642\) 1561.68 0.0960041
\(643\) −22067.1 −1.35341 −0.676705 0.736254i \(-0.736593\pi\)
−0.676705 + 0.736254i \(0.736593\pi\)
\(644\) 4751.50 0.290738
\(645\) 0 0
\(646\) 10319.4 0.628502
\(647\) 21490.3 1.30583 0.652914 0.757432i \(-0.273546\pi\)
0.652914 + 0.757432i \(0.273546\pi\)
\(648\) −7675.58 −0.465317
\(649\) −8517.57 −0.515168
\(650\) −4640.27 −0.280010
\(651\) 10329.6 0.621885
\(652\) −1461.22 −0.0877698
\(653\) −6820.47 −0.408738 −0.204369 0.978894i \(-0.565514\pi\)
−0.204369 + 0.978894i \(0.565514\pi\)
\(654\) 969.542 0.0579695
\(655\) −10571.0 −0.630600
\(656\) −2.46121 −0.000146485 0
\(657\) 11200.4 0.665099
\(658\) −1934.02 −0.114583
\(659\) 22489.9 1.32941 0.664705 0.747106i \(-0.268557\pi\)
0.664705 + 0.747106i \(0.268557\pi\)
\(660\) 4786.94 0.282320
\(661\) 4415.60 0.259829 0.129915 0.991525i \(-0.458530\pi\)
0.129915 + 0.991525i \(0.458530\pi\)
\(662\) −48.0913 −0.00282345
\(663\) 29542.2 1.73050
\(664\) −21346.0 −1.24757
\(665\) 4129.96 0.240832
\(666\) −9621.01 −0.559770
\(667\) −4833.59 −0.280596
\(668\) −1842.59 −0.106724
\(669\) −23696.6 −1.36945
\(670\) −3090.84 −0.178223
\(671\) −8595.03 −0.494497
\(672\) 6861.05 0.393855
\(673\) 12380.3 0.709099 0.354550 0.935037i \(-0.384634\pi\)
0.354550 + 0.935037i \(0.384634\pi\)
\(674\) 3452.38 0.197301
\(675\) 4877.83 0.278145
\(676\) −2036.21 −0.115852
\(677\) 13581.0 0.770988 0.385494 0.922710i \(-0.374031\pi\)
0.385494 + 0.922710i \(0.374031\pi\)
\(678\) −10326.0 −0.584908
\(679\) −1479.95 −0.0836456
\(680\) −7352.93 −0.414665
\(681\) 38406.4 2.16114
\(682\) −3889.37 −0.218375
\(683\) −11883.8 −0.665772 −0.332886 0.942967i \(-0.608022\pi\)
−0.332886 + 0.942967i \(0.608022\pi\)
\(684\) 30042.0 1.67936
\(685\) −10302.3 −0.574646
\(686\) 3679.37 0.204780
\(687\) 40052.6 2.22431
\(688\) 0 0
\(689\) −23416.0 −1.29475
\(690\) 6738.71 0.371795
\(691\) −29404.3 −1.61880 −0.809401 0.587256i \(-0.800208\pi\)
−0.809401 + 0.587256i \(0.800208\pi\)
\(692\) 2131.81 0.117109
\(693\) −2602.10 −0.142634
\(694\) −4181.80 −0.228730
\(695\) −17315.3 −0.945048
\(696\) −4537.75 −0.247131
\(697\) −4.88921 −0.000265699 0
\(698\) −10496.6 −0.569199
\(699\) 11384.0 0.615996
\(700\) 3146.30 0.169884
\(701\) −3434.39 −0.185043 −0.0925214 0.995711i \(-0.529493\pi\)
−0.0925214 + 0.995711i \(0.529493\pi\)
\(702\) −2970.55 −0.159710
\(703\) −33911.4 −1.81934
\(704\) 1839.94 0.0985019
\(705\) 16619.5 0.887840
\(706\) −3944.18 −0.210257
\(707\) 5786.87 0.307832
\(708\) 31503.0 1.67225
\(709\) 9505.59 0.503512 0.251756 0.967791i \(-0.418992\pi\)
0.251756 + 0.967791i \(0.418992\pi\)
\(710\) 3583.74 0.189430
\(711\) 27373.5 1.44386
\(712\) 23605.0 1.24247
\(713\) 33175.0 1.74251
\(714\) 3305.89 0.173277
\(715\) −4451.76 −0.232848
\(716\) 22720.7 1.18591
\(717\) −44153.6 −2.29978
\(718\) −4119.46 −0.214118
\(719\) −30161.2 −1.56443 −0.782214 0.623010i \(-0.785909\pi\)
−0.782214 + 0.623010i \(0.785909\pi\)
\(720\) −7985.99 −0.413362
\(721\) −9750.66 −0.503653
\(722\) −10174.2 −0.524438
\(723\) 22839.3 1.17483
\(724\) −8343.16 −0.428275
\(725\) −3200.65 −0.163958
\(726\) −9325.31 −0.476714
\(727\) 13578.6 0.692713 0.346356 0.938103i \(-0.387419\pi\)
0.346356 + 0.938103i \(0.387419\pi\)
\(728\) −4148.35 −0.211192
\(729\) −28358.4 −1.44076
\(730\) 2144.30 0.108718
\(731\) 0 0
\(732\) 31789.5 1.60515
\(733\) −30598.7 −1.54187 −0.770934 0.636915i \(-0.780210\pi\)
−0.770934 + 0.636915i \(0.780210\pi\)
\(734\) 11578.8 0.582263
\(735\) −15147.4 −0.760162
\(736\) 22035.3 1.10358
\(737\) 6864.05 0.343067
\(738\) 2.34909 0.000117170 0
\(739\) −6419.22 −0.319533 −0.159766 0.987155i \(-0.551074\pi\)
−0.159766 + 0.987155i \(0.551074\pi\)
\(740\) 11160.5 0.554418
\(741\) −50029.8 −2.48028
\(742\) −2620.35 −0.129644
\(743\) −30806.9 −1.52113 −0.760563 0.649264i \(-0.775077\pi\)
−0.760563 + 0.649264i \(0.775077\pi\)
\(744\) 31144.5 1.53470
\(745\) −5966.82 −0.293433
\(746\) −10785.1 −0.529317
\(747\) −46054.8 −2.25577
\(748\) 7542.19 0.368676
\(749\) 984.742 0.0480396
\(750\) 10852.0 0.528345
\(751\) −19700.5 −0.957233 −0.478616 0.878024i \(-0.658862\pi\)
−0.478616 + 0.878024i \(0.658862\pi\)
\(752\) 13181.6 0.639207
\(753\) 19440.5 0.940836
\(754\) 1949.16 0.0941437
\(755\) 5383.51 0.259505
\(756\) 2014.16 0.0968971
\(757\) −11597.2 −0.556813 −0.278406 0.960463i \(-0.589806\pi\)
−0.278406 + 0.960463i \(0.589806\pi\)
\(758\) 9161.81 0.439013
\(759\) −14965.1 −0.715678
\(760\) 12452.2 0.594328
\(761\) 35405.7 1.68654 0.843269 0.537492i \(-0.180628\pi\)
0.843269 + 0.537492i \(0.180628\pi\)
\(762\) 768.464 0.0365335
\(763\) 611.360 0.0290075
\(764\) −22399.2 −1.06070
\(765\) −15864.2 −0.749768
\(766\) 3561.74 0.168004
\(767\) −29297.2 −1.37922
\(768\) 4326.62 0.203286
\(769\) −22021.1 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(770\) −498.169 −0.0233153
\(771\) 57290.3 2.67608
\(772\) 4095.57 0.190936
\(773\) 4710.78 0.219191 0.109596 0.993976i \(-0.465044\pi\)
0.109596 + 0.993976i \(0.465044\pi\)
\(774\) 0 0
\(775\) 21967.4 1.01819
\(776\) −4462.19 −0.206422
\(777\) −10863.7 −0.501588
\(778\) 3836.36 0.176787
\(779\) 8.27989 0.000380819 0
\(780\) 16465.2 0.755832
\(781\) −7958.67 −0.364639
\(782\) 10617.4 0.485519
\(783\) −2048.95 −0.0935168
\(784\) −12014.0 −0.547284
\(785\) −13456.3 −0.611816
\(786\) 14329.9 0.650294
\(787\) 23336.9 1.05701 0.528506 0.848929i \(-0.322752\pi\)
0.528506 + 0.848929i \(0.322752\pi\)
\(788\) −21675.1 −0.979877
\(789\) 1964.44 0.0886389
\(790\) 5240.61 0.236016
\(791\) −6511.23 −0.292683
\(792\) −7845.57 −0.351995
\(793\) −29563.6 −1.32388
\(794\) −4448.47 −0.198829
\(795\) 22517.3 1.00454
\(796\) −11105.6 −0.494506
\(797\) −35188.8 −1.56393 −0.781964 0.623324i \(-0.785782\pi\)
−0.781964 + 0.623324i \(0.785782\pi\)
\(798\) −5598.53 −0.248353
\(799\) 26185.3 1.15941
\(800\) 14591.1 0.644842
\(801\) 50928.7 2.24654
\(802\) 6651.92 0.292877
\(803\) −4762.00 −0.209275
\(804\) −25387.3 −1.11361
\(805\) 4249.20 0.186043
\(806\) −13377.9 −0.584638
\(807\) 37074.1 1.61719
\(808\) 17447.9 0.759673
\(809\) 34912.8 1.51727 0.758634 0.651517i \(-0.225867\pi\)
0.758634 + 0.651517i \(0.225867\pi\)
\(810\) −3170.46 −0.137529
\(811\) 12039.2 0.521274 0.260637 0.965437i \(-0.416067\pi\)
0.260637 + 0.965437i \(0.416067\pi\)
\(812\) −1321.62 −0.0571178
\(813\) −29709.4 −1.28161
\(814\) 4090.50 0.176133
\(815\) −1306.75 −0.0561638
\(816\) −22531.8 −0.966631
\(817\) 0 0
\(818\) 2314.10 0.0989128
\(819\) −8950.22 −0.381863
\(820\) −2.72498 −0.000116049 0
\(821\) −15367.9 −0.653282 −0.326641 0.945148i \(-0.605917\pi\)
−0.326641 + 0.945148i \(0.605917\pi\)
\(822\) 13965.7 0.592592
\(823\) −13492.2 −0.571458 −0.285729 0.958311i \(-0.592236\pi\)
−0.285729 + 0.958311i \(0.592236\pi\)
\(824\) −29399.1 −1.24292
\(825\) −9909.44 −0.418185
\(826\) −3278.47 −0.138102
\(827\) −20247.3 −0.851351 −0.425675 0.904876i \(-0.639963\pi\)
−0.425675 + 0.904876i \(0.639963\pi\)
\(828\) 30909.3 1.29731
\(829\) 19359.2 0.811063 0.405531 0.914081i \(-0.367086\pi\)
0.405531 + 0.914081i \(0.367086\pi\)
\(830\) −8817.14 −0.368732
\(831\) −37288.8 −1.55660
\(832\) 6328.68 0.263711
\(833\) −23865.9 −0.992680
\(834\) 23472.5 0.974562
\(835\) −1647.80 −0.0682929
\(836\) −12772.7 −0.528414
\(837\) 14062.8 0.580744
\(838\) 10632.1 0.438283
\(839\) −8186.73 −0.336874 −0.168437 0.985712i \(-0.553872\pi\)
−0.168437 + 0.985712i \(0.553872\pi\)
\(840\) 3989.13 0.163855
\(841\) −23044.6 −0.944875
\(842\) 7432.72 0.304214
\(843\) −18847.9 −0.770053
\(844\) 21700.8 0.885038
\(845\) −1820.95 −0.0741334
\(846\) −12581.1 −0.511286
\(847\) −5880.22 −0.238544
\(848\) 17859.4 0.723224
\(849\) 64096.9 2.59105
\(850\) 7030.49 0.283699
\(851\) −34890.5 −1.40544
\(852\) 29435.8 1.18363
\(853\) −35270.2 −1.41574 −0.707871 0.706342i \(-0.750344\pi\)
−0.707871 + 0.706342i \(0.750344\pi\)
\(854\) −3308.28 −0.132561
\(855\) 26866.1 1.07462
\(856\) 2969.08 0.118553
\(857\) 24977.7 0.995593 0.497797 0.867294i \(-0.334143\pi\)
0.497797 + 0.867294i \(0.334143\pi\)
\(858\) 6034.75 0.240120
\(859\) 31619.8 1.25594 0.627972 0.778236i \(-0.283885\pi\)
0.627972 + 0.778236i \(0.283885\pi\)
\(860\) 0 0
\(861\) 2.65251 0.000104991 0
\(862\) −236.620 −0.00934953
\(863\) 18436.6 0.727220 0.363610 0.931551i \(-0.381544\pi\)
0.363610 + 0.931551i \(0.381544\pi\)
\(864\) 9340.76 0.367800
\(865\) 1906.45 0.0749379
\(866\) −6341.46 −0.248835
\(867\) −6341.84 −0.248420
\(868\) 9070.81 0.354704
\(869\) −11638.2 −0.454313
\(870\) −1874.35 −0.0730420
\(871\) 23609.7 0.918465
\(872\) 1843.30 0.0715850
\(873\) −9627.34 −0.373237
\(874\) −17980.5 −0.695882
\(875\) 6842.90 0.264380
\(876\) 17612.7 0.679313
\(877\) 48236.4 1.85727 0.928635 0.370994i \(-0.120983\pi\)
0.928635 + 0.370994i \(0.120983\pi\)
\(878\) −6663.15 −0.256117
\(879\) −35627.8 −1.36712
\(880\) 3395.35 0.130065
\(881\) 33308.7 1.27378 0.636888 0.770956i \(-0.280221\pi\)
0.636888 + 0.770956i \(0.280221\pi\)
\(882\) 11466.7 0.437759
\(883\) 31091.7 1.18496 0.592479 0.805586i \(-0.298149\pi\)
0.592479 + 0.805586i \(0.298149\pi\)
\(884\) 25942.2 0.987027
\(885\) 28172.7 1.07007
\(886\) −11532.1 −0.437277
\(887\) 6550.91 0.247980 0.123990 0.992283i \(-0.460431\pi\)
0.123990 + 0.992283i \(0.460431\pi\)
\(888\) −32755.0 −1.23782
\(889\) 484.567 0.0182810
\(890\) 9750.23 0.367223
\(891\) 7040.86 0.264734
\(892\) −20809.0 −0.781094
\(893\) −44345.0 −1.66176
\(894\) 8088.55 0.302597
\(895\) 20318.8 0.758864
\(896\) 7727.54 0.288124
\(897\) −51474.2 −1.91602
\(898\) 11491.7 0.427042
\(899\) −9227.52 −0.342330
\(900\) 20467.2 0.758044
\(901\) 35477.8 1.31181
\(902\) −0.998746 −3.68676e−5 0
\(903\) 0 0
\(904\) −19631.9 −0.722287
\(905\) −7461.17 −0.274053
\(906\) −7297.83 −0.267609
\(907\) −46718.5 −1.71032 −0.855160 0.518364i \(-0.826541\pi\)
−0.855160 + 0.518364i \(0.826541\pi\)
\(908\) 33726.2 1.23265
\(909\) 37644.5 1.37359
\(910\) −1713.51 −0.0624200
\(911\) −18054.8 −0.656620 −0.328310 0.944570i \(-0.606479\pi\)
−0.328310 + 0.944570i \(0.606479\pi\)
\(912\) 38157.7 1.38545
\(913\) 19580.8 0.709781
\(914\) 13389.4 0.484553
\(915\) 28428.9 1.02714
\(916\) 35171.9 1.26868
\(917\) 9035.96 0.325402
\(918\) 4500.70 0.161814
\(919\) −42321.3 −1.51910 −0.759548 0.650451i \(-0.774580\pi\)
−0.759548 + 0.650451i \(0.774580\pi\)
\(920\) 12811.7 0.459119
\(921\) −4636.62 −0.165887
\(922\) 15689.9 0.560434
\(923\) −27374.7 −0.976219
\(924\) −4091.81 −0.145683
\(925\) −23103.4 −0.821228
\(926\) 934.396 0.0331600
\(927\) −63429.7 −2.24736
\(928\) −6129.06 −0.216806
\(929\) −32859.5 −1.16048 −0.580240 0.814446i \(-0.697041\pi\)
−0.580240 + 0.814446i \(0.697041\pi\)
\(930\) 12864.5 0.453595
\(931\) 40416.9 1.42278
\(932\) 9996.74 0.351346
\(933\) −3620.14 −0.127029
\(934\) 3418.99 0.119778
\(935\) 6744.88 0.235916
\(936\) −26985.7 −0.942367
\(937\) −14.5879 −0.000508608 0 −0.000254304 1.00000i \(-0.500081\pi\)
−0.000254304 1.00000i \(0.500081\pi\)
\(938\) 2642.01 0.0919668
\(939\) 15664.2 0.544389
\(940\) 14594.3 0.506397
\(941\) 9884.04 0.342413 0.171206 0.985235i \(-0.445233\pi\)
0.171206 + 0.985235i \(0.445233\pi\)
\(942\) 18241.2 0.630924
\(943\) 8.51894 0.000294183 0
\(944\) 22344.9 0.770408
\(945\) 1801.23 0.0620044
\(946\) 0 0
\(947\) 43165.9 1.48121 0.740603 0.671943i \(-0.234540\pi\)
0.740603 + 0.671943i \(0.234540\pi\)
\(948\) 43044.9 1.47472
\(949\) −16379.4 −0.560273
\(950\) −11906.2 −0.406618
\(951\) −58757.6 −2.00352
\(952\) 6285.19 0.213975
\(953\) −11414.8 −0.387998 −0.193999 0.981002i \(-0.562146\pi\)
−0.193999 + 0.981002i \(0.562146\pi\)
\(954\) −17045.8 −0.578488
\(955\) −20031.3 −0.678742
\(956\) −38773.1 −1.31173
\(957\) 4162.50 0.140600
\(958\) 6576.23 0.221783
\(959\) 8806.31 0.296528
\(960\) −6085.78 −0.204602
\(961\) 33541.4 1.12589
\(962\) 14069.7 0.471545
\(963\) 6405.91 0.214359
\(964\) 20056.1 0.670086
\(965\) 3662.61 0.122180
\(966\) −5760.17 −0.191853
\(967\) −21094.6 −0.701508 −0.350754 0.936468i \(-0.614075\pi\)
−0.350754 + 0.936468i \(0.614075\pi\)
\(968\) −17729.4 −0.588682
\(969\) 75800.4 2.51296
\(970\) −1843.14 −0.0610100
\(971\) −17622.1 −0.582409 −0.291205 0.956661i \(-0.594056\pi\)
−0.291205 + 0.956661i \(0.594056\pi\)
\(972\) −36401.6 −1.20121
\(973\) 14800.9 0.487663
\(974\) −16607.8 −0.546355
\(975\) −34084.7 −1.11957
\(976\) 22548.1 0.739495
\(977\) 29292.7 0.959218 0.479609 0.877482i \(-0.340778\pi\)
0.479609 + 0.877482i \(0.340778\pi\)
\(978\) 1771.42 0.0579178
\(979\) −21653.0 −0.706878
\(980\) −13301.5 −0.433573
\(981\) 3977.00 0.129435
\(982\) 115.637 0.00375778
\(983\) −36585.0 −1.18706 −0.593531 0.804811i \(-0.702266\pi\)
−0.593531 + 0.804811i \(0.702266\pi\)
\(984\) 7.99755 0.000259098 0
\(985\) −19383.7 −0.627023
\(986\) −2953.19 −0.0953841
\(987\) −14206.2 −0.458143
\(988\) −43933.2 −1.41468
\(989\) 0 0
\(990\) −3240.67 −0.104036
\(991\) 8841.74 0.283418 0.141709 0.989908i \(-0.454740\pi\)
0.141709 + 0.989908i \(0.454740\pi\)
\(992\) 42066.4 1.34638
\(993\) −353.251 −0.0112891
\(994\) −3063.34 −0.0977497
\(995\) −9931.58 −0.316434
\(996\) −72421.4 −2.30398
\(997\) 6013.86 0.191034 0.0955169 0.995428i \(-0.469550\pi\)
0.0955169 + 0.995428i \(0.469550\pi\)
\(998\) −1206.38 −0.0382638
\(999\) −14790.1 −0.468405
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.1.21 50
43.42 odd 2 1849.4.a.j.1.30 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.21 50 1.1 even 1 trivial
1849.4.a.j.1.30 yes 50 43.42 odd 2