Properties

Label 1849.4.a.j.1.37
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 1.37
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.22072 q^{2} +8.71909 q^{3} +2.37305 q^{4} -13.5152 q^{5} +28.0818 q^{6} +24.0687 q^{7} -18.1228 q^{8} +49.0226 q^{9} +O(q^{10})\) \(q+3.22072 q^{2} +8.71909 q^{3} +2.37305 q^{4} -13.5152 q^{5} +28.0818 q^{6} +24.0687 q^{7} -18.1228 q^{8} +49.0226 q^{9} -43.5288 q^{10} -71.6649 q^{11} +20.6909 q^{12} +26.7691 q^{13} +77.5186 q^{14} -117.841 q^{15} -77.3530 q^{16} +35.6223 q^{17} +157.888 q^{18} -88.4072 q^{19} -32.0724 q^{20} +209.857 q^{21} -230.813 q^{22} -64.6600 q^{23} -158.015 q^{24} +57.6615 q^{25} +86.2157 q^{26} +192.017 q^{27} +57.1164 q^{28} -106.675 q^{29} -379.532 q^{30} -7.27119 q^{31} -104.150 q^{32} -624.853 q^{33} +114.730 q^{34} -325.294 q^{35} +116.333 q^{36} +195.473 q^{37} -284.735 q^{38} +233.402 q^{39} +244.934 q^{40} -404.629 q^{41} +675.892 q^{42} -170.065 q^{44} -662.552 q^{45} -208.252 q^{46} +31.1933 q^{47} -674.449 q^{48} +236.303 q^{49} +185.712 q^{50} +310.594 q^{51} +63.5244 q^{52} +34.4808 q^{53} +618.434 q^{54} +968.567 q^{55} -436.193 q^{56} -770.831 q^{57} -343.572 q^{58} -486.120 q^{59} -279.642 q^{60} -491.320 q^{61} -23.4185 q^{62} +1179.91 q^{63} +283.386 q^{64} -361.790 q^{65} -2012.48 q^{66} +102.537 q^{67} +84.5337 q^{68} -563.777 q^{69} -1047.68 q^{70} +119.002 q^{71} -888.429 q^{72} -123.974 q^{73} +629.565 q^{74} +502.756 q^{75} -209.795 q^{76} -1724.88 q^{77} +751.723 q^{78} +34.0155 q^{79} +1045.44 q^{80} +350.606 q^{81} -1303.20 q^{82} -681.077 q^{83} +498.003 q^{84} -481.444 q^{85} -930.114 q^{87} +1298.77 q^{88} -1563.77 q^{89} -2133.90 q^{90} +644.297 q^{91} -153.442 q^{92} -63.3982 q^{93} +100.465 q^{94} +1194.84 q^{95} -908.094 q^{96} +93.1966 q^{97} +761.066 q^{98} -3513.20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.22072 1.13870 0.569349 0.822096i \(-0.307195\pi\)
0.569349 + 0.822096i \(0.307195\pi\)
\(3\) 8.71909 1.67799 0.838995 0.544139i \(-0.183143\pi\)
0.838995 + 0.544139i \(0.183143\pi\)
\(4\) 2.37305 0.296632
\(5\) −13.5152 −1.20884 −0.604420 0.796666i \(-0.706595\pi\)
−0.604420 + 0.796666i \(0.706595\pi\)
\(6\) 28.0818 1.91072
\(7\) 24.0687 1.29959 0.649794 0.760110i \(-0.274855\pi\)
0.649794 + 0.760110i \(0.274855\pi\)
\(8\) −18.1228 −0.800924
\(9\) 49.0226 1.81565
\(10\) −43.5288 −1.37650
\(11\) −71.6649 −1.96434 −0.982171 0.187987i \(-0.939804\pi\)
−0.982171 + 0.187987i \(0.939804\pi\)
\(12\) 20.6909 0.497745
\(13\) 26.7691 0.571108 0.285554 0.958363i \(-0.407822\pi\)
0.285554 + 0.958363i \(0.407822\pi\)
\(14\) 77.5186 1.47984
\(15\) −117.841 −2.02842
\(16\) −77.3530 −1.20864
\(17\) 35.6223 0.508217 0.254108 0.967176i \(-0.418218\pi\)
0.254108 + 0.967176i \(0.418218\pi\)
\(18\) 157.888 2.06748
\(19\) −88.4072 −1.06747 −0.533737 0.845650i \(-0.679213\pi\)
−0.533737 + 0.845650i \(0.679213\pi\)
\(20\) −32.0724 −0.358580
\(21\) 209.857 2.18070
\(22\) −230.813 −2.23679
\(23\) −64.6600 −0.586197 −0.293099 0.956082i \(-0.594686\pi\)
−0.293099 + 0.956082i \(0.594686\pi\)
\(24\) −158.015 −1.34394
\(25\) 57.6615 0.461292
\(26\) 86.2157 0.650319
\(27\) 192.017 1.36866
\(28\) 57.1164 0.385499
\(29\) −106.675 −0.683074 −0.341537 0.939868i \(-0.610947\pi\)
−0.341537 + 0.939868i \(0.610947\pi\)
\(30\) −379.532 −2.30976
\(31\) −7.27119 −0.0421272 −0.0210636 0.999778i \(-0.506705\pi\)
−0.0210636 + 0.999778i \(0.506705\pi\)
\(32\) −104.150 −0.575353
\(33\) −624.853 −3.29615
\(34\) 114.730 0.578705
\(35\) −325.294 −1.57099
\(36\) 116.333 0.538580
\(37\) 195.473 0.868529 0.434265 0.900785i \(-0.357008\pi\)
0.434265 + 0.900785i \(0.357008\pi\)
\(38\) −284.735 −1.21553
\(39\) 233.402 0.958314
\(40\) 244.934 0.968188
\(41\) −404.629 −1.54128 −0.770639 0.637272i \(-0.780063\pi\)
−0.770639 + 0.637272i \(0.780063\pi\)
\(42\) 675.892 2.48315
\(43\) 0 0
\(44\) −170.065 −0.582687
\(45\) −662.552 −2.19483
\(46\) −208.252 −0.667502
\(47\) 31.1933 0.0968087 0.0484043 0.998828i \(-0.484586\pi\)
0.0484043 + 0.998828i \(0.484586\pi\)
\(48\) −674.449 −2.02809
\(49\) 236.303 0.688930
\(50\) 185.712 0.525272
\(51\) 310.594 0.852783
\(52\) 63.5244 0.169409
\(53\) 34.4808 0.0893641 0.0446820 0.999001i \(-0.485773\pi\)
0.0446820 + 0.999001i \(0.485773\pi\)
\(54\) 618.434 1.55849
\(55\) 968.567 2.37457
\(56\) −436.193 −1.04087
\(57\) −770.831 −1.79121
\(58\) −343.572 −0.777814
\(59\) −486.120 −1.07267 −0.536335 0.844005i \(-0.680191\pi\)
−0.536335 + 0.844005i \(0.680191\pi\)
\(60\) −279.642 −0.601694
\(61\) −491.320 −1.03126 −0.515632 0.856810i \(-0.672443\pi\)
−0.515632 + 0.856810i \(0.672443\pi\)
\(62\) −23.4185 −0.0479701
\(63\) 1179.91 2.35960
\(64\) 283.386 0.553488
\(65\) −361.790 −0.690378
\(66\) −2012.48 −3.75332
\(67\) 102.537 0.186969 0.0934843 0.995621i \(-0.470200\pi\)
0.0934843 + 0.995621i \(0.470200\pi\)
\(68\) 84.5337 0.150753
\(69\) −563.777 −0.983634
\(70\) −1047.68 −1.78889
\(71\) 119.002 0.198915 0.0994573 0.995042i \(-0.468289\pi\)
0.0994573 + 0.995042i \(0.468289\pi\)
\(72\) −888.429 −1.45420
\(73\) −123.974 −0.198768 −0.0993840 0.995049i \(-0.531687\pi\)
−0.0993840 + 0.995049i \(0.531687\pi\)
\(74\) 629.565 0.988992
\(75\) 502.756 0.774044
\(76\) −209.795 −0.316647
\(77\) −1724.88 −2.55284
\(78\) 751.723 1.09123
\(79\) 34.0155 0.0484436 0.0242218 0.999707i \(-0.492289\pi\)
0.0242218 + 0.999707i \(0.492289\pi\)
\(80\) 1045.44 1.46105
\(81\) 350.606 0.480941
\(82\) −1303.20 −1.75505
\(83\) −681.077 −0.900698 −0.450349 0.892853i \(-0.648700\pi\)
−0.450349 + 0.892853i \(0.648700\pi\)
\(84\) 498.003 0.646864
\(85\) −481.444 −0.614352
\(86\) 0 0
\(87\) −930.114 −1.14619
\(88\) 1298.77 1.57329
\(89\) −1563.77 −1.86247 −0.931234 0.364422i \(-0.881267\pi\)
−0.931234 + 0.364422i \(0.881267\pi\)
\(90\) −2133.90 −2.49925
\(91\) 644.297 0.742205
\(92\) −153.442 −0.173885
\(93\) −63.3982 −0.0706891
\(94\) 100.465 0.110236
\(95\) 1194.84 1.29040
\(96\) −908.094 −0.965437
\(97\) 93.1966 0.0975535 0.0487767 0.998810i \(-0.484468\pi\)
0.0487767 + 0.998810i \(0.484468\pi\)
\(98\) 761.066 0.784483
\(99\) −3513.20 −3.56656
\(100\) 136.834 0.136834
\(101\) −358.186 −0.352880 −0.176440 0.984311i \(-0.556458\pi\)
−0.176440 + 0.984311i \(0.556458\pi\)
\(102\) 1000.34 0.971062
\(103\) −280.395 −0.268234 −0.134117 0.990965i \(-0.542820\pi\)
−0.134117 + 0.990965i \(0.542820\pi\)
\(104\) −485.131 −0.457414
\(105\) −2836.27 −2.63611
\(106\) 111.053 0.101759
\(107\) 2033.32 1.83709 0.918543 0.395322i \(-0.129367\pi\)
0.918543 + 0.395322i \(0.129367\pi\)
\(108\) 455.667 0.405987
\(109\) 1575.03 1.38404 0.692022 0.721876i \(-0.256720\pi\)
0.692022 + 0.721876i \(0.256720\pi\)
\(110\) 3119.49 2.70392
\(111\) 1704.35 1.45738
\(112\) −1861.79 −1.57074
\(113\) −939.934 −0.782492 −0.391246 0.920286i \(-0.627956\pi\)
−0.391246 + 0.920286i \(0.627956\pi\)
\(114\) −2482.63 −2.03965
\(115\) 873.895 0.708618
\(116\) −253.147 −0.202621
\(117\) 1312.29 1.03693
\(118\) −1565.66 −1.22145
\(119\) 857.384 0.660473
\(120\) 2135.61 1.62461
\(121\) 3804.85 2.85864
\(122\) −1582.41 −1.17430
\(123\) −3528.00 −2.58625
\(124\) −17.2549 −0.0124963
\(125\) 910.095 0.651211
\(126\) 3800.17 2.68687
\(127\) 622.014 0.434605 0.217302 0.976104i \(-0.430274\pi\)
0.217302 + 0.976104i \(0.430274\pi\)
\(128\) 1745.91 1.20561
\(129\) 0 0
\(130\) −1165.23 −0.786131
\(131\) −2763.95 −1.84342 −0.921708 0.387884i \(-0.873206\pi\)
−0.921708 + 0.387884i \(0.873206\pi\)
\(132\) −1482.81 −0.977743
\(133\) −2127.85 −1.38728
\(134\) 330.243 0.212901
\(135\) −2595.16 −1.65449
\(136\) −645.577 −0.407043
\(137\) 685.313 0.427374 0.213687 0.976902i \(-0.431453\pi\)
0.213687 + 0.976902i \(0.431453\pi\)
\(138\) −1815.77 −1.12006
\(139\) −57.4962 −0.0350846 −0.0175423 0.999846i \(-0.505584\pi\)
−0.0175423 + 0.999846i \(0.505584\pi\)
\(140\) −771.941 −0.466007
\(141\) 271.977 0.162444
\(142\) 383.272 0.226503
\(143\) −1918.40 −1.12185
\(144\) −3792.05 −2.19447
\(145\) 1441.74 0.825726
\(146\) −399.286 −0.226337
\(147\) 2060.35 1.15602
\(148\) 463.868 0.257633
\(149\) 648.505 0.356561 0.178281 0.983980i \(-0.442947\pi\)
0.178281 + 0.983980i \(0.442947\pi\)
\(150\) 1619.24 0.881402
\(151\) −2521.32 −1.35882 −0.679410 0.733759i \(-0.737764\pi\)
−0.679410 + 0.733759i \(0.737764\pi\)
\(152\) 1602.19 0.854965
\(153\) 1746.30 0.922745
\(154\) −5555.36 −2.90691
\(155\) 98.2718 0.0509250
\(156\) 553.876 0.284266
\(157\) 892.524 0.453702 0.226851 0.973930i \(-0.427157\pi\)
0.226851 + 0.973930i \(0.427157\pi\)
\(158\) 109.554 0.0551626
\(159\) 300.641 0.149952
\(160\) 1407.61 0.695509
\(161\) −1556.28 −0.761815
\(162\) 1129.21 0.547647
\(163\) 3013.59 1.44811 0.724057 0.689740i \(-0.242275\pi\)
0.724057 + 0.689740i \(0.242275\pi\)
\(164\) −960.206 −0.457192
\(165\) 8445.03 3.98451
\(166\) −2193.56 −1.02562
\(167\) 3783.09 1.75296 0.876480 0.481439i \(-0.159886\pi\)
0.876480 + 0.481439i \(0.159886\pi\)
\(168\) −3803.21 −1.74657
\(169\) −1480.42 −0.673836
\(170\) −1550.60 −0.699561
\(171\) −4333.95 −1.93816
\(172\) 0 0
\(173\) 1859.79 0.817326 0.408663 0.912685i \(-0.365995\pi\)
0.408663 + 0.912685i \(0.365995\pi\)
\(174\) −2995.64 −1.30517
\(175\) 1387.84 0.599490
\(176\) 5543.50 2.37419
\(177\) −4238.53 −1.79993
\(178\) −5036.48 −2.12079
\(179\) 949.068 0.396294 0.198147 0.980172i \(-0.436508\pi\)
0.198147 + 0.980172i \(0.436508\pi\)
\(180\) −1572.27 −0.651057
\(181\) −3778.60 −1.55172 −0.775859 0.630906i \(-0.782683\pi\)
−0.775859 + 0.630906i \(0.782683\pi\)
\(182\) 2075.10 0.845147
\(183\) −4283.87 −1.73045
\(184\) 1171.82 0.469499
\(185\) −2641.86 −1.04991
\(186\) −204.188 −0.0804934
\(187\) −2552.87 −0.998312
\(188\) 74.0233 0.0287165
\(189\) 4621.61 1.77869
\(190\) 3848.26 1.46938
\(191\) −2236.60 −0.847300 −0.423650 0.905826i \(-0.639251\pi\)
−0.423650 + 0.905826i \(0.639251\pi\)
\(192\) 2470.87 0.928748
\(193\) −1426.80 −0.532142 −0.266071 0.963953i \(-0.585726\pi\)
−0.266071 + 0.963953i \(0.585726\pi\)
\(194\) 300.161 0.111084
\(195\) −3154.48 −1.15845
\(196\) 560.760 0.204358
\(197\) 3733.48 1.35025 0.675125 0.737703i \(-0.264090\pi\)
0.675125 + 0.737703i \(0.264090\pi\)
\(198\) −11315.0 −4.06124
\(199\) −2509.13 −0.893808 −0.446904 0.894582i \(-0.647473\pi\)
−0.446904 + 0.894582i \(0.647473\pi\)
\(200\) −1044.99 −0.369460
\(201\) 894.030 0.313731
\(202\) −1153.62 −0.401823
\(203\) −2567.54 −0.887715
\(204\) 737.057 0.252962
\(205\) 5468.65 1.86316
\(206\) −903.074 −0.305438
\(207\) −3169.80 −1.06433
\(208\) −2070.67 −0.690265
\(209\) 6335.69 2.09689
\(210\) −9134.84 −3.00173
\(211\) 1516.64 0.494833 0.247416 0.968909i \(-0.420418\pi\)
0.247416 + 0.968909i \(0.420418\pi\)
\(212\) 81.8247 0.0265082
\(213\) 1037.59 0.333777
\(214\) 6548.75 2.09188
\(215\) 0 0
\(216\) −3479.90 −1.09619
\(217\) −175.008 −0.0547480
\(218\) 5072.74 1.57601
\(219\) −1080.94 −0.333531
\(220\) 2298.46 0.704374
\(221\) 953.576 0.290247
\(222\) 5489.23 1.65952
\(223\) −2914.69 −0.875256 −0.437628 0.899156i \(-0.644181\pi\)
−0.437628 + 0.899156i \(0.644181\pi\)
\(224\) −2506.76 −0.747722
\(225\) 2826.72 0.837546
\(226\) −3027.27 −0.891021
\(227\) −3001.30 −0.877549 −0.438774 0.898597i \(-0.644587\pi\)
−0.438774 + 0.898597i \(0.644587\pi\)
\(228\) −1829.22 −0.531330
\(229\) −576.669 −0.166408 −0.0832038 0.996533i \(-0.526515\pi\)
−0.0832038 + 0.996533i \(0.526515\pi\)
\(230\) 2814.57 0.806902
\(231\) −15039.4 −4.28364
\(232\) 1933.26 0.547090
\(233\) −5444.61 −1.53085 −0.765425 0.643525i \(-0.777471\pi\)
−0.765425 + 0.643525i \(0.777471\pi\)
\(234\) 4226.52 1.18075
\(235\) −421.584 −0.117026
\(236\) −1153.59 −0.318188
\(237\) 296.584 0.0812879
\(238\) 2761.39 0.752078
\(239\) −1814.68 −0.491138 −0.245569 0.969379i \(-0.578975\pi\)
−0.245569 + 0.969379i \(0.578975\pi\)
\(240\) 9115.33 2.45163
\(241\) 3802.66 1.01639 0.508196 0.861241i \(-0.330312\pi\)
0.508196 + 0.861241i \(0.330312\pi\)
\(242\) 12254.4 3.25513
\(243\) −2127.50 −0.561642
\(244\) −1165.93 −0.305906
\(245\) −3193.69 −0.832805
\(246\) −11362.7 −2.94496
\(247\) −2366.58 −0.609643
\(248\) 131.774 0.0337407
\(249\) −5938.38 −1.51136
\(250\) 2931.16 0.741532
\(251\) 589.071 0.148135 0.0740673 0.997253i \(-0.476402\pi\)
0.0740673 + 0.997253i \(0.476402\pi\)
\(252\) 2799.99 0.699933
\(253\) 4633.85 1.15149
\(254\) 2003.33 0.494883
\(255\) −4197.76 −1.03088
\(256\) 3356.00 0.819335
\(257\) 1674.00 0.406308 0.203154 0.979147i \(-0.434881\pi\)
0.203154 + 0.979147i \(0.434881\pi\)
\(258\) 0 0
\(259\) 4704.79 1.12873
\(260\) −858.548 −0.204788
\(261\) −5229.51 −1.24022
\(262\) −8901.92 −2.09909
\(263\) 5843.39 1.37003 0.685017 0.728527i \(-0.259795\pi\)
0.685017 + 0.728527i \(0.259795\pi\)
\(264\) 11324.1 2.63996
\(265\) −466.016 −0.108027
\(266\) −6853.21 −1.57969
\(267\) −13634.7 −3.12520
\(268\) 243.326 0.0554608
\(269\) −1405.83 −0.318643 −0.159322 0.987227i \(-0.550931\pi\)
−0.159322 + 0.987227i \(0.550931\pi\)
\(270\) −8358.28 −1.88396
\(271\) 3263.18 0.731455 0.365727 0.930722i \(-0.380820\pi\)
0.365727 + 0.930722i \(0.380820\pi\)
\(272\) −2755.50 −0.614252
\(273\) 5617.69 1.24541
\(274\) 2207.20 0.486650
\(275\) −4132.31 −0.906136
\(276\) −1337.87 −0.291777
\(277\) 706.424 0.153231 0.0766153 0.997061i \(-0.475589\pi\)
0.0766153 + 0.997061i \(0.475589\pi\)
\(278\) −185.179 −0.0399507
\(279\) −356.453 −0.0764884
\(280\) 5895.25 1.25825
\(281\) −7794.16 −1.65466 −0.827332 0.561714i \(-0.810142\pi\)
−0.827332 + 0.561714i \(0.810142\pi\)
\(282\) 875.963 0.184975
\(283\) −4923.94 −1.03427 −0.517134 0.855905i \(-0.673001\pi\)
−0.517134 + 0.855905i \(0.673001\pi\)
\(284\) 282.398 0.0590044
\(285\) 10418.0 2.16529
\(286\) −6178.64 −1.27745
\(287\) −9738.89 −2.00303
\(288\) −5105.71 −1.04464
\(289\) −3644.05 −0.741716
\(290\) 4643.46 0.940252
\(291\) 812.590 0.163694
\(292\) −294.197 −0.0589609
\(293\) 5279.68 1.05270 0.526352 0.850267i \(-0.323560\pi\)
0.526352 + 0.850267i \(0.323560\pi\)
\(294\) 6635.81 1.31635
\(295\) 6570.03 1.29668
\(296\) −3542.53 −0.695626
\(297\) −13760.9 −2.68851
\(298\) 2088.66 0.406015
\(299\) −1730.89 −0.334782
\(300\) 1193.07 0.229606
\(301\) 0 0
\(302\) −8120.46 −1.54729
\(303\) −3123.06 −0.592129
\(304\) 6838.57 1.29019
\(305\) 6640.31 1.24663
\(306\) 5624.35 1.05073
\(307\) 5804.15 1.07902 0.539512 0.841978i \(-0.318609\pi\)
0.539512 + 0.841978i \(0.318609\pi\)
\(308\) −4093.24 −0.757253
\(309\) −2444.79 −0.450095
\(310\) 316.506 0.0579882
\(311\) −8304.16 −1.51410 −0.757051 0.653356i \(-0.773361\pi\)
−0.757051 + 0.653356i \(0.773361\pi\)
\(312\) −4229.91 −0.767536
\(313\) −2.78390 −0.000502733 0 −0.000251367 1.00000i \(-0.500080\pi\)
−0.000251367 1.00000i \(0.500080\pi\)
\(314\) 2874.57 0.516629
\(315\) −15946.8 −2.85238
\(316\) 80.7206 0.0143699
\(317\) 3284.80 0.581997 0.290998 0.956724i \(-0.406013\pi\)
0.290998 + 0.956724i \(0.406013\pi\)
\(318\) 968.281 0.170750
\(319\) 7644.89 1.34179
\(320\) −3830.03 −0.669078
\(321\) 17728.7 3.08261
\(322\) −5012.36 −0.867477
\(323\) −3149.27 −0.542508
\(324\) 832.007 0.142662
\(325\) 1543.54 0.263448
\(326\) 9705.94 1.64896
\(327\) 13732.9 2.32241
\(328\) 7333.02 1.23445
\(329\) 750.782 0.125811
\(330\) 27199.1 4.53716
\(331\) 7993.89 1.32744 0.663722 0.747979i \(-0.268976\pi\)
0.663722 + 0.747979i \(0.268976\pi\)
\(332\) −1616.23 −0.267176
\(333\) 9582.60 1.57695
\(334\) 12184.3 1.99609
\(335\) −1385.81 −0.226015
\(336\) −16233.1 −2.63568
\(337\) 7345.18 1.18729 0.593646 0.804726i \(-0.297688\pi\)
0.593646 + 0.804726i \(0.297688\pi\)
\(338\) −4768.01 −0.767295
\(339\) −8195.37 −1.31301
\(340\) −1142.49 −0.182236
\(341\) 521.089 0.0827523
\(342\) −13958.5 −2.20698
\(343\) −2568.06 −0.404263
\(344\) 0 0
\(345\) 7619.57 1.18906
\(346\) 5989.88 0.930687
\(347\) 8573.21 1.32632 0.663162 0.748476i \(-0.269214\pi\)
0.663162 + 0.748476i \(0.269214\pi\)
\(348\) −2207.21 −0.339997
\(349\) 4612.19 0.707407 0.353704 0.935358i \(-0.384922\pi\)
0.353704 + 0.935358i \(0.384922\pi\)
\(350\) 4469.84 0.682638
\(351\) 5140.12 0.781651
\(352\) 7463.90 1.13019
\(353\) 7509.11 1.13221 0.566104 0.824334i \(-0.308450\pi\)
0.566104 + 0.824334i \(0.308450\pi\)
\(354\) −13651.1 −2.04957
\(355\) −1608.34 −0.240456
\(356\) −3710.92 −0.552467
\(357\) 7475.61 1.10827
\(358\) 3056.69 0.451259
\(359\) −10602.9 −1.55877 −0.779384 0.626547i \(-0.784468\pi\)
−0.779384 + 0.626547i \(0.784468\pi\)
\(360\) 12007.3 1.75789
\(361\) 956.838 0.139501
\(362\) −12169.8 −1.76694
\(363\) 33174.9 4.79678
\(364\) 1528.95 0.220162
\(365\) 1675.54 0.240278
\(366\) −13797.2 −1.97046
\(367\) 10146.4 1.44315 0.721576 0.692335i \(-0.243418\pi\)
0.721576 + 0.692335i \(0.243418\pi\)
\(368\) 5001.65 0.708503
\(369\) −19836.0 −2.79842
\(370\) −8508.71 −1.19553
\(371\) 829.908 0.116137
\(372\) −150.447 −0.0209686
\(373\) −5897.78 −0.818702 −0.409351 0.912377i \(-0.634245\pi\)
−0.409351 + 0.912377i \(0.634245\pi\)
\(374\) −8222.09 −1.13678
\(375\) 7935.21 1.09273
\(376\) −565.311 −0.0775363
\(377\) −2855.60 −0.390109
\(378\) 14884.9 2.02539
\(379\) −1623.92 −0.220093 −0.110047 0.993926i \(-0.535100\pi\)
−0.110047 + 0.993926i \(0.535100\pi\)
\(380\) 2835.43 0.382775
\(381\) 5423.40 0.729263
\(382\) −7203.45 −0.964819
\(383\) 8375.64 1.11743 0.558715 0.829360i \(-0.311295\pi\)
0.558715 + 0.829360i \(0.311295\pi\)
\(384\) 15222.7 2.02300
\(385\) 23312.2 3.08597
\(386\) −4595.33 −0.605949
\(387\) 0 0
\(388\) 221.161 0.0289375
\(389\) 13745.0 1.79151 0.895757 0.444544i \(-0.146634\pi\)
0.895757 + 0.444544i \(0.146634\pi\)
\(390\) −10159.7 −1.31912
\(391\) −2303.34 −0.297915
\(392\) −4282.48 −0.551780
\(393\) −24099.2 −3.09323
\(394\) 12024.5 1.53753
\(395\) −459.727 −0.0585605
\(396\) −8337.01 −1.05796
\(397\) 6849.80 0.865949 0.432975 0.901406i \(-0.357464\pi\)
0.432975 + 0.901406i \(0.357464\pi\)
\(398\) −8081.22 −1.01778
\(399\) −18552.9 −2.32784
\(400\) −4460.29 −0.557537
\(401\) 1819.07 0.226533 0.113267 0.993565i \(-0.463869\pi\)
0.113267 + 0.993565i \(0.463869\pi\)
\(402\) 2879.42 0.357245
\(403\) −194.643 −0.0240592
\(404\) −849.995 −0.104675
\(405\) −4738.52 −0.581381
\(406\) −8269.34 −1.01084
\(407\) −14008.6 −1.70609
\(408\) −5628.85 −0.683014
\(409\) −6571.50 −0.794474 −0.397237 0.917716i \(-0.630031\pi\)
−0.397237 + 0.917716i \(0.630031\pi\)
\(410\) 17613.0 2.12157
\(411\) 5975.31 0.717130
\(412\) −665.392 −0.0795668
\(413\) −11700.3 −1.39403
\(414\) −10209.1 −1.21195
\(415\) 9204.92 1.08880
\(416\) −2788.00 −0.328589
\(417\) −501.314 −0.0588716
\(418\) 20405.5 2.38772
\(419\) −12326.7 −1.43723 −0.718613 0.695410i \(-0.755223\pi\)
−0.718613 + 0.695410i \(0.755223\pi\)
\(420\) −6730.63 −0.781955
\(421\) 5434.97 0.629179 0.314590 0.949228i \(-0.398133\pi\)
0.314590 + 0.949228i \(0.398133\pi\)
\(422\) 4884.67 0.563465
\(423\) 1529.18 0.175771
\(424\) −624.889 −0.0715738
\(425\) 2054.04 0.234436
\(426\) 3341.79 0.380071
\(427\) −11825.4 −1.34022
\(428\) 4825.17 0.544938
\(429\) −16726.7 −1.88246
\(430\) 0 0
\(431\) −5331.48 −0.595843 −0.297922 0.954590i \(-0.596293\pi\)
−0.297922 + 0.954590i \(0.596293\pi\)
\(432\) −14853.1 −1.65422
\(433\) −3987.83 −0.442593 −0.221297 0.975207i \(-0.571029\pi\)
−0.221297 + 0.975207i \(0.571029\pi\)
\(434\) −563.652 −0.0623414
\(435\) 12570.7 1.38556
\(436\) 3737.64 0.410551
\(437\) 5716.41 0.625751
\(438\) −3481.41 −0.379791
\(439\) 1810.69 0.196856 0.0984278 0.995144i \(-0.468619\pi\)
0.0984278 + 0.995144i \(0.468619\pi\)
\(440\) −17553.2 −1.90185
\(441\) 11584.2 1.25086
\(442\) 3071.21 0.330503
\(443\) −10313.9 −1.10616 −0.553078 0.833129i \(-0.686547\pi\)
−0.553078 + 0.833129i \(0.686547\pi\)
\(444\) 4044.51 0.432306
\(445\) 21134.8 2.25142
\(446\) −9387.41 −0.996651
\(447\) 5654.38 0.598307
\(448\) 6820.74 0.719307
\(449\) 2382.45 0.250411 0.125206 0.992131i \(-0.460041\pi\)
0.125206 + 0.992131i \(0.460041\pi\)
\(450\) 9104.08 0.953712
\(451\) 28997.7 3.02760
\(452\) −2230.51 −0.232112
\(453\) −21983.6 −2.28009
\(454\) −9666.37 −0.999262
\(455\) −8707.82 −0.897207
\(456\) 13969.6 1.43462
\(457\) 968.005 0.0990840 0.0495420 0.998772i \(-0.484224\pi\)
0.0495420 + 0.998772i \(0.484224\pi\)
\(458\) −1857.29 −0.189488
\(459\) 6840.10 0.695574
\(460\) 2073.80 0.210199
\(461\) 5755.46 0.581472 0.290736 0.956803i \(-0.406100\pi\)
0.290736 + 0.956803i \(0.406100\pi\)
\(462\) −48437.7 −4.87777
\(463\) 14706.8 1.47621 0.738104 0.674687i \(-0.235721\pi\)
0.738104 + 0.674687i \(0.235721\pi\)
\(464\) 8251.67 0.825591
\(465\) 856.841 0.0854517
\(466\) −17535.6 −1.74317
\(467\) −4670.26 −0.462770 −0.231385 0.972862i \(-0.574326\pi\)
−0.231385 + 0.972862i \(0.574326\pi\)
\(468\) 3114.13 0.307587
\(469\) 2467.93 0.242982
\(470\) −1357.81 −0.133257
\(471\) 7782.00 0.761307
\(472\) 8809.88 0.859126
\(473\) 0 0
\(474\) 955.216 0.0925623
\(475\) −5097.70 −0.492417
\(476\) 2034.62 0.195917
\(477\) 1690.34 0.162254
\(478\) −5844.59 −0.559258
\(479\) 1645.32 0.156944 0.0784722 0.996916i \(-0.474996\pi\)
0.0784722 + 0.996916i \(0.474996\pi\)
\(480\) 12273.1 1.16706
\(481\) 5232.63 0.496024
\(482\) 12247.3 1.15736
\(483\) −13569.4 −1.27832
\(484\) 9029.13 0.847964
\(485\) −1259.57 −0.117926
\(486\) −6852.08 −0.639541
\(487\) 15977.1 1.48664 0.743318 0.668938i \(-0.233251\pi\)
0.743318 + 0.668938i \(0.233251\pi\)
\(488\) 8904.11 0.825964
\(489\) 26275.8 2.42992
\(490\) −10286.0 −0.948313
\(491\) 6890.84 0.633359 0.316679 0.948533i \(-0.397432\pi\)
0.316679 + 0.948533i \(0.397432\pi\)
\(492\) −8372.13 −0.767164
\(493\) −3800.03 −0.347150
\(494\) −7622.09 −0.694199
\(495\) 47481.7 4.31140
\(496\) 562.448 0.0509167
\(497\) 2864.22 0.258507
\(498\) −19125.9 −1.72098
\(499\) −5724.01 −0.513511 −0.256756 0.966476i \(-0.582654\pi\)
−0.256756 + 0.966476i \(0.582654\pi\)
\(500\) 2159.71 0.193170
\(501\) 32985.1 2.94145
\(502\) 1897.23 0.168681
\(503\) −19916.3 −1.76546 −0.882728 0.469885i \(-0.844295\pi\)
−0.882728 + 0.469885i \(0.844295\pi\)
\(504\) −21383.3 −1.88986
\(505\) 4840.97 0.426575
\(506\) 14924.3 1.31120
\(507\) −12907.9 −1.13069
\(508\) 1476.07 0.128918
\(509\) −12749.9 −1.11028 −0.555138 0.831758i \(-0.687335\pi\)
−0.555138 + 0.831758i \(0.687335\pi\)
\(510\) −13519.8 −1.17386
\(511\) −2983.89 −0.258317
\(512\) −3158.53 −0.272634
\(513\) −16975.7 −1.46101
\(514\) 5391.49 0.462662
\(515\) 3789.60 0.324252
\(516\) 0 0
\(517\) −2235.46 −0.190165
\(518\) 15152.8 1.28528
\(519\) 16215.7 1.37147
\(520\) 6556.66 0.552940
\(521\) 3723.80 0.313134 0.156567 0.987667i \(-0.449957\pi\)
0.156567 + 0.987667i \(0.449957\pi\)
\(522\) −16842.8 −1.41224
\(523\) 13767.0 1.15103 0.575517 0.817790i \(-0.304801\pi\)
0.575517 + 0.817790i \(0.304801\pi\)
\(524\) −6559.01 −0.546816
\(525\) 12100.7 1.00594
\(526\) 18819.9 1.56005
\(527\) −259.017 −0.0214098
\(528\) 48334.3 3.98386
\(529\) −7986.08 −0.656373
\(530\) −1500.91 −0.123010
\(531\) −23830.9 −1.94759
\(532\) −5049.50 −0.411510
\(533\) −10831.5 −0.880236
\(534\) −43913.5 −3.55866
\(535\) −27480.7 −2.22074
\(536\) −1858.26 −0.149748
\(537\) 8275.02 0.664978
\(538\) −4527.79 −0.362838
\(539\) −16934.6 −1.35329
\(540\) −6158.45 −0.490773
\(541\) −14720.0 −1.16980 −0.584899 0.811106i \(-0.698866\pi\)
−0.584899 + 0.811106i \(0.698866\pi\)
\(542\) 10509.8 0.832906
\(543\) −32946.0 −2.60377
\(544\) −3710.07 −0.292404
\(545\) −21286.9 −1.67309
\(546\) 18093.0 1.41815
\(547\) −19914.4 −1.55663 −0.778315 0.627874i \(-0.783925\pi\)
−0.778315 + 0.627874i \(0.783925\pi\)
\(548\) 1626.29 0.126773
\(549\) −24085.8 −1.87242
\(550\) −13309.0 −1.03181
\(551\) 9430.88 0.729164
\(552\) 10217.2 0.787816
\(553\) 818.709 0.0629567
\(554\) 2275.20 0.174483
\(555\) −23034.7 −1.76174
\(556\) −136.441 −0.0104072
\(557\) 2514.93 0.191313 0.0956563 0.995414i \(-0.469505\pi\)
0.0956563 + 0.995414i \(0.469505\pi\)
\(558\) −1148.03 −0.0870971
\(559\) 0 0
\(560\) 25162.5 1.89877
\(561\) −22258.7 −1.67516
\(562\) −25102.8 −1.88416
\(563\) −10579.1 −0.791932 −0.395966 0.918265i \(-0.629590\pi\)
−0.395966 + 0.918265i \(0.629590\pi\)
\(564\) 645.417 0.0481861
\(565\) 12703.4 0.945907
\(566\) −15858.6 −1.17772
\(567\) 8438.64 0.625026
\(568\) −2156.65 −0.159315
\(569\) −8023.71 −0.591162 −0.295581 0.955318i \(-0.595513\pi\)
−0.295581 + 0.955318i \(0.595513\pi\)
\(570\) 33553.4 2.46561
\(571\) −14485.2 −1.06162 −0.530812 0.847490i \(-0.678113\pi\)
−0.530812 + 0.847490i \(0.678113\pi\)
\(572\) −4552.47 −0.332777
\(573\) −19501.1 −1.42176
\(574\) −31366.3 −2.28084
\(575\) −3728.39 −0.270408
\(576\) 13892.3 1.00494
\(577\) −10236.0 −0.738529 −0.369264 0.929324i \(-0.620390\pi\)
−0.369264 + 0.929324i \(0.620390\pi\)
\(578\) −11736.5 −0.844590
\(579\) −12440.4 −0.892930
\(580\) 3421.34 0.244937
\(581\) −16392.7 −1.17054
\(582\) 2617.13 0.186398
\(583\) −2471.06 −0.175542
\(584\) 2246.76 0.159198
\(585\) −17735.9 −1.25349
\(586\) 17004.4 1.19871
\(587\) 16260.3 1.14333 0.571665 0.820487i \(-0.306298\pi\)
0.571665 + 0.820487i \(0.306298\pi\)
\(588\) 4889.32 0.342912
\(589\) 642.825 0.0449697
\(590\) 21160.2 1.47653
\(591\) 32552.5 2.26571
\(592\) −15120.4 −1.04974
\(593\) −7324.06 −0.507189 −0.253594 0.967311i \(-0.581613\pi\)
−0.253594 + 0.967311i \(0.581613\pi\)
\(594\) −44320.0 −3.06140
\(595\) −11587.7 −0.798405
\(596\) 1538.94 0.105767
\(597\) −21877.4 −1.49980
\(598\) −5574.71 −0.381215
\(599\) −12995.7 −0.886462 −0.443231 0.896407i \(-0.646168\pi\)
−0.443231 + 0.896407i \(0.646168\pi\)
\(600\) −9111.37 −0.619950
\(601\) 18095.7 1.22819 0.614093 0.789233i \(-0.289522\pi\)
0.614093 + 0.789233i \(0.289522\pi\)
\(602\) 0 0
\(603\) 5026.63 0.339470
\(604\) −5983.22 −0.403069
\(605\) −51423.5 −3.45564
\(606\) −10058.5 −0.674255
\(607\) −12834.4 −0.858207 −0.429103 0.903255i \(-0.641170\pi\)
−0.429103 + 0.903255i \(0.641170\pi\)
\(608\) 9207.62 0.614175
\(609\) −22386.6 −1.48958
\(610\) 21386.6 1.41954
\(611\) 835.015 0.0552882
\(612\) 4144.06 0.273715
\(613\) −11358.5 −0.748391 −0.374196 0.927350i \(-0.622081\pi\)
−0.374196 + 0.927350i \(0.622081\pi\)
\(614\) 18693.6 1.22868
\(615\) 47681.7 3.12636
\(616\) 31259.7 2.04463
\(617\) 9745.84 0.635904 0.317952 0.948107i \(-0.397005\pi\)
0.317952 + 0.948107i \(0.397005\pi\)
\(618\) −7873.99 −0.512521
\(619\) −19796.2 −1.28543 −0.642713 0.766107i \(-0.722191\pi\)
−0.642713 + 0.766107i \(0.722191\pi\)
\(620\) 233.204 0.0151060
\(621\) −12415.8 −0.802303
\(622\) −26745.4 −1.72410
\(623\) −37638.0 −2.42044
\(624\) −18054.4 −1.15826
\(625\) −19507.8 −1.24850
\(626\) −8.96618 −0.000572461 0
\(627\) 55241.5 3.51855
\(628\) 2118.01 0.134582
\(629\) 6963.21 0.441401
\(630\) −51360.1 −3.24800
\(631\) −15835.2 −0.999031 −0.499516 0.866305i \(-0.666489\pi\)
−0.499516 + 0.866305i \(0.666489\pi\)
\(632\) −616.457 −0.0387996
\(633\) 13223.7 0.830325
\(634\) 10579.4 0.662718
\(635\) −8406.66 −0.525367
\(636\) 713.437 0.0444806
\(637\) 6325.61 0.393453
\(638\) 24622.1 1.52789
\(639\) 5833.79 0.361160
\(640\) −23596.4 −1.45739
\(641\) 7595.24 0.468009 0.234005 0.972236i \(-0.424817\pi\)
0.234005 + 0.972236i \(0.424817\pi\)
\(642\) 57099.2 3.51016
\(643\) 1315.59 0.0806873 0.0403437 0.999186i \(-0.487155\pi\)
0.0403437 + 0.999186i \(0.487155\pi\)
\(644\) −3693.14 −0.225979
\(645\) 0 0
\(646\) −10142.9 −0.617753
\(647\) 15048.3 0.914386 0.457193 0.889367i \(-0.348855\pi\)
0.457193 + 0.889367i \(0.348855\pi\)
\(648\) −6353.98 −0.385197
\(649\) 34837.8 2.10709
\(650\) 4971.33 0.299987
\(651\) −1525.91 −0.0918667
\(652\) 7151.41 0.429557
\(653\) 15506.6 0.929282 0.464641 0.885499i \(-0.346183\pi\)
0.464641 + 0.885499i \(0.346183\pi\)
\(654\) 44229.7 2.64453
\(655\) 37355.4 2.22839
\(656\) 31299.3 1.86285
\(657\) −6077.53 −0.360894
\(658\) 2418.06 0.143261
\(659\) −1105.73 −0.0653611 −0.0326806 0.999466i \(-0.510404\pi\)
−0.0326806 + 0.999466i \(0.510404\pi\)
\(660\) 20040.5 1.18193
\(661\) 25902.8 1.52421 0.762104 0.647455i \(-0.224167\pi\)
0.762104 + 0.647455i \(0.224167\pi\)
\(662\) 25746.1 1.51156
\(663\) 8314.32 0.487031
\(664\) 12343.0 0.721390
\(665\) 28758.4 1.67699
\(666\) 30862.9 1.79567
\(667\) 6897.64 0.400416
\(668\) 8977.47 0.519983
\(669\) −25413.5 −1.46867
\(670\) −4463.32 −0.257363
\(671\) 35210.4 2.02576
\(672\) −21856.7 −1.25467
\(673\) −27050.7 −1.54937 −0.774685 0.632347i \(-0.782092\pi\)
−0.774685 + 0.632347i \(0.782092\pi\)
\(674\) 23656.8 1.35197
\(675\) 11072.0 0.631351
\(676\) −3513.11 −0.199881
\(677\) −22070.9 −1.25296 −0.626479 0.779438i \(-0.715505\pi\)
−0.626479 + 0.779438i \(0.715505\pi\)
\(678\) −26395.0 −1.49513
\(679\) 2243.12 0.126779
\(680\) 8725.13 0.492049
\(681\) −26168.7 −1.47252
\(682\) 1678.28 0.0942298
\(683\) −6638.47 −0.371909 −0.185955 0.982558i \(-0.559538\pi\)
−0.185955 + 0.982558i \(0.559538\pi\)
\(684\) −10284.7 −0.574920
\(685\) −9262.17 −0.516627
\(686\) −8271.01 −0.460333
\(687\) −5028.03 −0.279230
\(688\) 0 0
\(689\) 923.018 0.0510365
\(690\) 24540.5 1.35397
\(691\) 29895.5 1.64584 0.822922 0.568154i \(-0.192342\pi\)
0.822922 + 0.568154i \(0.192342\pi\)
\(692\) 4413.39 0.242445
\(693\) −84558.2 −4.63507
\(694\) 27611.9 1.51028
\(695\) 777.074 0.0424116
\(696\) 16856.3 0.918012
\(697\) −14413.8 −0.783303
\(698\) 14854.6 0.805523
\(699\) −47472.0 −2.56875
\(700\) 3293.42 0.177828
\(701\) −28435.1 −1.53207 −0.766034 0.642800i \(-0.777773\pi\)
−0.766034 + 0.642800i \(0.777773\pi\)
\(702\) 16554.9 0.890064
\(703\) −17281.2 −0.927133
\(704\) −20308.8 −1.08724
\(705\) −3675.83 −0.196369
\(706\) 24184.7 1.28924
\(707\) −8621.08 −0.458598
\(708\) −10058.3 −0.533916
\(709\) 2985.59 0.158147 0.0790735 0.996869i \(-0.474804\pi\)
0.0790735 + 0.996869i \(0.474804\pi\)
\(710\) −5180.01 −0.273806
\(711\) 1667.53 0.0879567
\(712\) 28340.0 1.49169
\(713\) 470.155 0.0246949
\(714\) 24076.9 1.26198
\(715\) 25927.6 1.35614
\(716\) 2252.19 0.117554
\(717\) −15822.4 −0.824125
\(718\) −34148.9 −1.77496
\(719\) 13.7437 0.000712871 0 0.000356436 1.00000i \(-0.499887\pi\)
0.000356436 1.00000i \(0.499887\pi\)
\(720\) 51250.4 2.65276
\(721\) −6748.74 −0.348594
\(722\) 3081.71 0.158850
\(723\) 33155.7 1.70550
\(724\) −8966.82 −0.460289
\(725\) −6151.07 −0.315097
\(726\) 106847. 5.46208
\(727\) 25528.1 1.30232 0.651158 0.758942i \(-0.274283\pi\)
0.651158 + 0.758942i \(0.274283\pi\)
\(728\) −11676.5 −0.594450
\(729\) −28016.2 −1.42337
\(730\) 5396.44 0.273604
\(731\) 0 0
\(732\) −10165.9 −0.513307
\(733\) −20204.9 −1.01812 −0.509062 0.860730i \(-0.670008\pi\)
−0.509062 + 0.860730i \(0.670008\pi\)
\(734\) 32678.7 1.64331
\(735\) −27846.1 −1.39744
\(736\) 6734.34 0.337271
\(737\) −7348.30 −0.367270
\(738\) −63886.1 −3.18656
\(739\) −20377.5 −1.01434 −0.507172 0.861845i \(-0.669309\pi\)
−0.507172 + 0.861845i \(0.669309\pi\)
\(740\) −6269.29 −0.311437
\(741\) −20634.4 −1.02298
\(742\) 2672.90 0.132244
\(743\) 31394.6 1.55014 0.775071 0.631874i \(-0.217714\pi\)
0.775071 + 0.631874i \(0.217714\pi\)
\(744\) 1148.95 0.0566165
\(745\) −8764.70 −0.431025
\(746\) −18995.1 −0.932253
\(747\) −33388.2 −1.63535
\(748\) −6058.10 −0.296131
\(749\) 48939.3 2.38745
\(750\) 25557.1 1.24428
\(751\) 765.609 0.0372004 0.0186002 0.999827i \(-0.494079\pi\)
0.0186002 + 0.999827i \(0.494079\pi\)
\(752\) −2412.90 −0.117007
\(753\) 5136.16 0.248569
\(754\) −9197.10 −0.444216
\(755\) 34076.2 1.64260
\(756\) 10967.3 0.527616
\(757\) 20525.6 0.985491 0.492745 0.870174i \(-0.335993\pi\)
0.492745 + 0.870174i \(0.335993\pi\)
\(758\) −5230.20 −0.250619
\(759\) 40403.0 1.93219
\(760\) −21654.0 −1.03352
\(761\) 16665.0 0.793829 0.396915 0.917856i \(-0.370081\pi\)
0.396915 + 0.917856i \(0.370081\pi\)
\(762\) 17467.3 0.830410
\(763\) 37909.0 1.79869
\(764\) −5307.56 −0.251336
\(765\) −23601.6 −1.11545
\(766\) 26975.6 1.27241
\(767\) −13013.0 −0.612610
\(768\) 29261.3 1.37484
\(769\) 16234.8 0.761303 0.380652 0.924719i \(-0.375700\pi\)
0.380652 + 0.924719i \(0.375700\pi\)
\(770\) 75082.0 3.51399
\(771\) 14595.8 0.681781
\(772\) −3385.88 −0.157850
\(773\) 10674.0 0.496658 0.248329 0.968676i \(-0.420119\pi\)
0.248329 + 0.968676i \(0.420119\pi\)
\(774\) 0 0
\(775\) −419.268 −0.0194330
\(776\) −1688.99 −0.0781329
\(777\) 41021.5 1.89400
\(778\) 44268.8 2.03999
\(779\) 35772.1 1.64527
\(780\) −7485.76 −0.343632
\(781\) −8528.26 −0.390736
\(782\) −7418.42 −0.339235
\(783\) −20483.5 −0.934894
\(784\) −18278.8 −0.832669
\(785\) −12062.7 −0.548452
\(786\) −77616.7 −3.52226
\(787\) −5413.70 −0.245206 −0.122603 0.992456i \(-0.539124\pi\)
−0.122603 + 0.992456i \(0.539124\pi\)
\(788\) 8859.74 0.400527
\(789\) 50949.1 2.29890
\(790\) −1480.65 −0.0666827
\(791\) −22623.0 −1.01692
\(792\) 63669.1 2.85655
\(793\) −13152.2 −0.588963
\(794\) 22061.3 0.986054
\(795\) −4063.23 −0.181268
\(796\) −5954.31 −0.265132
\(797\) 25337.3 1.12609 0.563044 0.826427i \(-0.309630\pi\)
0.563044 + 0.826427i \(0.309630\pi\)
\(798\) −59753.8 −2.65070
\(799\) 1111.18 0.0491998
\(800\) −6005.45 −0.265406
\(801\) −76660.2 −3.38159
\(802\) 5858.70 0.257953
\(803\) 8884.58 0.390448
\(804\) 2121.58 0.0930627
\(805\) 21033.5 0.920912
\(806\) −626.891 −0.0273961
\(807\) −12257.6 −0.534680
\(808\) 6491.34 0.282630
\(809\) −12894.1 −0.560360 −0.280180 0.959948i \(-0.590394\pi\)
−0.280180 + 0.959948i \(0.590394\pi\)
\(810\) −15261.5 −0.662017
\(811\) −8554.95 −0.370413 −0.185207 0.982700i \(-0.559295\pi\)
−0.185207 + 0.982700i \(0.559295\pi\)
\(812\) −6092.92 −0.263324
\(813\) 28452.0 1.22737
\(814\) −45117.7 −1.94272
\(815\) −40729.4 −1.75054
\(816\) −24025.4 −1.03071
\(817\) 0 0
\(818\) −21165.0 −0.904665
\(819\) 31585.1 1.34759
\(820\) 12977.4 0.552672
\(821\) −30480.2 −1.29569 −0.647847 0.761770i \(-0.724331\pi\)
−0.647847 + 0.761770i \(0.724331\pi\)
\(822\) 19244.8 0.816594
\(823\) −13207.1 −0.559380 −0.279690 0.960090i \(-0.590232\pi\)
−0.279690 + 0.960090i \(0.590232\pi\)
\(824\) 5081.55 0.214835
\(825\) −36030.0 −1.52049
\(826\) −37683.4 −1.58738
\(827\) 33164.4 1.39449 0.697243 0.716835i \(-0.254410\pi\)
0.697243 + 0.716835i \(0.254410\pi\)
\(828\) −7522.11 −0.315714
\(829\) 39344.0 1.64834 0.824169 0.566343i \(-0.191642\pi\)
0.824169 + 0.566343i \(0.191642\pi\)
\(830\) 29646.5 1.23981
\(831\) 6159.38 0.257120
\(832\) 7585.98 0.316101
\(833\) 8417.66 0.350126
\(834\) −1614.59 −0.0670370
\(835\) −51129.3 −2.11905
\(836\) 15034.9 0.622003
\(837\) −1396.19 −0.0576577
\(838\) −39700.8 −1.63657
\(839\) 11863.9 0.488186 0.244093 0.969752i \(-0.421510\pi\)
0.244093 + 0.969752i \(0.421510\pi\)
\(840\) 51401.3 2.11132
\(841\) −13009.3 −0.533410
\(842\) 17504.5 0.716445
\(843\) −67958.0 −2.77651
\(844\) 3599.07 0.146783
\(845\) 20008.2 0.814559
\(846\) 4925.05 0.200150
\(847\) 91577.9 3.71506
\(848\) −2667.19 −0.108009
\(849\) −42932.3 −1.73549
\(850\) 6615.49 0.266952
\(851\) −12639.3 −0.509130
\(852\) 2462.26 0.0990088
\(853\) −15340.4 −0.615760 −0.307880 0.951425i \(-0.599620\pi\)
−0.307880 + 0.951425i \(0.599620\pi\)
\(854\) −38086.5 −1.52610
\(855\) 58574.4 2.34293
\(856\) −36849.4 −1.47136
\(857\) 2347.71 0.0935778 0.0467889 0.998905i \(-0.485101\pi\)
0.0467889 + 0.998905i \(0.485101\pi\)
\(858\) −53872.1 −2.14355
\(859\) −38761.4 −1.53961 −0.769803 0.638281i \(-0.779646\pi\)
−0.769803 + 0.638281i \(0.779646\pi\)
\(860\) 0 0
\(861\) −84914.3 −3.36106
\(862\) −17171.2 −0.678485
\(863\) 33659.3 1.32767 0.663834 0.747880i \(-0.268928\pi\)
0.663834 + 0.747880i \(0.268928\pi\)
\(864\) −19998.6 −0.787461
\(865\) −25135.5 −0.988016
\(866\) −12843.7 −0.503980
\(867\) −31772.8 −1.24459
\(868\) −415.304 −0.0162400
\(869\) −2437.72 −0.0951598
\(870\) 40486.7 1.57773
\(871\) 2744.82 0.106779
\(872\) −28544.1 −1.10851
\(873\) 4568.74 0.177123
\(874\) 18411.0 0.712541
\(875\) 21904.8 0.846306
\(876\) −2565.13 −0.0989358
\(877\) 26492.9 1.02007 0.510034 0.860154i \(-0.329633\pi\)
0.510034 + 0.860154i \(0.329633\pi\)
\(878\) 5831.74 0.224159
\(879\) 46034.0 1.76643
\(880\) −74921.6 −2.87001
\(881\) −34532.1 −1.32056 −0.660281 0.751018i \(-0.729563\pi\)
−0.660281 + 0.751018i \(0.729563\pi\)
\(882\) 37309.5 1.42435
\(883\) 8248.77 0.314375 0.157187 0.987569i \(-0.449757\pi\)
0.157187 + 0.987569i \(0.449757\pi\)
\(884\) 2262.89 0.0860963
\(885\) 57284.7 2.17582
\(886\) −33218.2 −1.25958
\(887\) −33247.6 −1.25856 −0.629282 0.777177i \(-0.716651\pi\)
−0.629282 + 0.777177i \(0.716651\pi\)
\(888\) −30887.6 −1.16725
\(889\) 14971.1 0.564807
\(890\) 68069.2 2.56369
\(891\) −25126.1 −0.944734
\(892\) −6916.72 −0.259629
\(893\) −2757.71 −0.103341
\(894\) 18211.2 0.681290
\(895\) −12826.9 −0.479056
\(896\) 42021.8 1.56680
\(897\) −15091.8 −0.561761
\(898\) 7673.20 0.285142
\(899\) 775.657 0.0287760
\(900\) 6707.96 0.248443
\(901\) 1228.29 0.0454163
\(902\) 93393.4 3.44752
\(903\) 0 0
\(904\) 17034.3 0.626716
\(905\) 51068.7 1.87578
\(906\) −70803.1 −2.59633
\(907\) 11992.7 0.439042 0.219521 0.975608i \(-0.429551\pi\)
0.219521 + 0.975608i \(0.429551\pi\)
\(908\) −7122.26 −0.260309
\(909\) −17559.2 −0.640707
\(910\) −28045.5 −1.02165
\(911\) −33624.5 −1.22287 −0.611433 0.791296i \(-0.709407\pi\)
−0.611433 + 0.791296i \(0.709407\pi\)
\(912\) 59626.1 2.16493
\(913\) 48809.3 1.76928
\(914\) 3117.68 0.112827
\(915\) 57897.5 2.09184
\(916\) −1368.47 −0.0493618
\(917\) −66524.7 −2.39568
\(918\) 22030.1 0.792049
\(919\) −21470.7 −0.770678 −0.385339 0.922775i \(-0.625915\pi\)
−0.385339 + 0.922775i \(0.625915\pi\)
\(920\) −15837.5 −0.567549
\(921\) 50606.9 1.81059
\(922\) 18536.7 0.662120
\(923\) 3185.57 0.113602
\(924\) −35689.3 −1.27066
\(925\) 11271.3 0.400646
\(926\) 47366.6 1.68095
\(927\) −13745.7 −0.487020
\(928\) 11110.3 0.393009
\(929\) −7941.69 −0.280472 −0.140236 0.990118i \(-0.544786\pi\)
−0.140236 + 0.990118i \(0.544786\pi\)
\(930\) 2759.65 0.0973036
\(931\) −20890.9 −0.735415
\(932\) −12920.3 −0.454099
\(933\) −72404.8 −2.54065
\(934\) −15041.6 −0.526955
\(935\) 34502.6 1.20680
\(936\) −23782.4 −0.830504
\(937\) 43111.2 1.50307 0.751537 0.659691i \(-0.229313\pi\)
0.751537 + 0.659691i \(0.229313\pi\)
\(938\) 7948.53 0.276683
\(939\) −24.2731 −0.000843581 0
\(940\) −1000.44 −0.0347137
\(941\) −16162.2 −0.559906 −0.279953 0.960014i \(-0.590319\pi\)
−0.279953 + 0.960014i \(0.590319\pi\)
\(942\) 25063.7 0.866898
\(943\) 26163.3 0.903493
\(944\) 37602.9 1.29647
\(945\) −62462.1 −2.15015
\(946\) 0 0
\(947\) 7629.78 0.261811 0.130905 0.991395i \(-0.458212\pi\)
0.130905 + 0.991395i \(0.458212\pi\)
\(948\) 703.811 0.0241126
\(949\) −3318.67 −0.113518
\(950\) −16418.3 −0.560715
\(951\) 28640.5 0.976585
\(952\) −15538.2 −0.528988
\(953\) 12245.8 0.416243 0.208122 0.978103i \(-0.433265\pi\)
0.208122 + 0.978103i \(0.433265\pi\)
\(954\) 5444.11 0.184758
\(955\) 30228.1 1.02425
\(956\) −4306.34 −0.145687
\(957\) 66656.5 2.25151
\(958\) 5299.10 0.178712
\(959\) 16494.6 0.555410
\(960\) −33394.4 −1.12271
\(961\) −29738.1 −0.998225
\(962\) 16852.9 0.564821
\(963\) 99678.5 3.33551
\(964\) 9023.91 0.301494
\(965\) 19283.6 0.643274
\(966\) −43703.2 −1.45562
\(967\) −9078.20 −0.301898 −0.150949 0.988542i \(-0.548233\pi\)
−0.150949 + 0.988542i \(0.548233\pi\)
\(968\) −68954.7 −2.28955
\(969\) −27458.8 −0.910324
\(970\) −4056.74 −0.134283
\(971\) 5977.14 0.197544 0.0987722 0.995110i \(-0.468508\pi\)
0.0987722 + 0.995110i \(0.468508\pi\)
\(972\) −5048.67 −0.166601
\(973\) −1383.86 −0.0455955
\(974\) 51457.8 1.69283
\(975\) 13458.3 0.442063
\(976\) 38005.1 1.24643
\(977\) −45996.0 −1.50619 −0.753093 0.657914i \(-0.771439\pi\)
−0.753093 + 0.657914i \(0.771439\pi\)
\(978\) 84627.0 2.76695
\(979\) 112068. 3.65853
\(980\) −7578.80 −0.247037
\(981\) 77212.2 2.51294
\(982\) 22193.5 0.721204
\(983\) 8525.77 0.276633 0.138316 0.990388i \(-0.455831\pi\)
0.138316 + 0.990388i \(0.455831\pi\)
\(984\) 63937.3 2.07139
\(985\) −50458.8 −1.63224
\(986\) −12238.8 −0.395298
\(987\) 6546.14 0.211110
\(988\) −5616.02 −0.180839
\(989\) 0 0
\(990\) 152925. 4.90938
\(991\) 12323.8 0.395033 0.197516 0.980300i \(-0.436712\pi\)
0.197516 + 0.980300i \(0.436712\pi\)
\(992\) 757.294 0.0242380
\(993\) 69699.5 2.22744
\(994\) 9224.87 0.294361
\(995\) 33911.5 1.08047
\(996\) −14092.1 −0.448318
\(997\) −3674.25 −0.116715 −0.0583574 0.998296i \(-0.518586\pi\)
−0.0583574 + 0.998296i \(0.518586\pi\)
\(998\) −18435.5 −0.584734
\(999\) 37534.2 1.18872
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.1.37 yes 50
43.42 odd 2 1849.4.a.i.1.14 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.14 50 43.42 odd 2
1849.4.a.j.1.37 yes 50 1.1 even 1 trivial