Properties

Label 2151.4.a.h.1.12
Level $2151$
Weight $4$
Character 2151.1
Self dual yes
Analytic conductor $126.913$
Analytic rank $0$
Dimension $59$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(126.913108422\)
Analytic rank: \(0\)
Dimension: \(59\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.78499 q^{2} +6.32612 q^{4} -5.39134 q^{5} +9.96233 q^{7} +6.33562 q^{8} +O(q^{10})\) \(q-3.78499 q^{2} +6.32612 q^{4} -5.39134 q^{5} +9.96233 q^{7} +6.33562 q^{8} +20.4061 q^{10} +23.9107 q^{11} -68.8672 q^{13} -37.7073 q^{14} -74.5892 q^{16} -101.415 q^{17} -4.75784 q^{19} -34.1062 q^{20} -90.5017 q^{22} -138.385 q^{23} -95.9335 q^{25} +260.661 q^{26} +63.0229 q^{28} -147.063 q^{29} -284.339 q^{31} +231.634 q^{32} +383.854 q^{34} -53.7103 q^{35} +299.816 q^{37} +18.0083 q^{38} -34.1575 q^{40} +128.369 q^{41} -253.017 q^{43} +151.262 q^{44} +523.784 q^{46} +113.991 q^{47} -243.752 q^{49} +363.107 q^{50} -435.662 q^{52} +472.814 q^{53} -128.911 q^{55} +63.1175 q^{56} +556.632 q^{58} +756.581 q^{59} -476.984 q^{61} +1076.22 q^{62} -280.018 q^{64} +371.286 q^{65} -50.4211 q^{67} -641.563 q^{68} +203.293 q^{70} -722.838 q^{71} +145.341 q^{73} -1134.80 q^{74} -30.0986 q^{76} +238.206 q^{77} -632.943 q^{79} +402.135 q^{80} -485.877 q^{82} -401.146 q^{83} +546.762 q^{85} +957.667 q^{86} +151.489 q^{88} +1337.22 q^{89} -686.078 q^{91} -875.438 q^{92} -431.455 q^{94} +25.6511 q^{95} -367.601 q^{97} +922.598 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 59q + 8q^{2} + 238q^{4} + 80q^{5} - 10q^{7} + 96q^{8} + O(q^{10}) \) \( 59q + 8q^{2} + 238q^{4} + 80q^{5} - 10q^{7} + 96q^{8} - 36q^{10} + 132q^{11} + 104q^{13} + 280q^{14} + 822q^{16} + 408q^{17} + 20q^{19} + 800q^{20} - 2q^{22} + 276q^{23} + 1477q^{25} + 780q^{26} + 224q^{28} + 696q^{29} - 380q^{31} + 896q^{32} - 72q^{34} + 700q^{35} + 224q^{37} + 988q^{38} - 258q^{40} + 2706q^{41} - 156q^{43} + 1584q^{44} + 428q^{46} + 1316q^{47} + 2135q^{49} + 1400q^{50} + 1092q^{52} + 1484q^{53} - 992q^{55} + 3360q^{56} - 120q^{58} + 3186q^{59} - 254q^{61} + 1240q^{62} + 3054q^{64} + 5120q^{65} + 288q^{67} + 9420q^{68} + 1108q^{70} + 4468q^{71} - 1770q^{73} + 6214q^{74} + 720q^{76} + 6352q^{77} - 746q^{79} + 7040q^{80} + 276q^{82} + 5484q^{83} + 588q^{85} + 10152q^{86} + 1186q^{88} + 11570q^{89} + 1768q^{91} + 15366q^{92} - 2142q^{94} + 5736q^{95} + 2390q^{97} + 6912q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.78499 −1.33819 −0.669097 0.743175i \(-0.733319\pi\)
−0.669097 + 0.743175i \(0.733319\pi\)
\(3\) 0 0
\(4\) 6.32612 0.790765
\(5\) −5.39134 −0.482216 −0.241108 0.970498i \(-0.577511\pi\)
−0.241108 + 0.970498i \(0.577511\pi\)
\(6\) 0 0
\(7\) 9.96233 0.537915 0.268958 0.963152i \(-0.413321\pi\)
0.268958 + 0.963152i \(0.413321\pi\)
\(8\) 6.33562 0.279997
\(9\) 0 0
\(10\) 20.4061 0.645299
\(11\) 23.9107 0.655395 0.327698 0.944783i \(-0.393727\pi\)
0.327698 + 0.944783i \(0.393727\pi\)
\(12\) 0 0
\(13\) −68.8672 −1.46926 −0.734628 0.678470i \(-0.762643\pi\)
−0.734628 + 0.678470i \(0.762643\pi\)
\(14\) −37.7073 −0.719835
\(15\) 0 0
\(16\) −74.5892 −1.16546
\(17\) −101.415 −1.44687 −0.723433 0.690395i \(-0.757437\pi\)
−0.723433 + 0.690395i \(0.757437\pi\)
\(18\) 0 0
\(19\) −4.75784 −0.0574486 −0.0287243 0.999587i \(-0.509144\pi\)
−0.0287243 + 0.999587i \(0.509144\pi\)
\(20\) −34.1062 −0.381319
\(21\) 0 0
\(22\) −90.5017 −0.877046
\(23\) −138.385 −1.25457 −0.627287 0.778788i \(-0.715835\pi\)
−0.627287 + 0.778788i \(0.715835\pi\)
\(24\) 0 0
\(25\) −95.9335 −0.767468
\(26\) 260.661 1.96615
\(27\) 0 0
\(28\) 63.0229 0.425365
\(29\) −147.063 −0.941687 −0.470843 0.882217i \(-0.656050\pi\)
−0.470843 + 0.882217i \(0.656050\pi\)
\(30\) 0 0
\(31\) −284.339 −1.64738 −0.823689 0.567042i \(-0.808088\pi\)
−0.823689 + 0.567042i \(0.808088\pi\)
\(32\) 231.634 1.27961
\(33\) 0 0
\(34\) 383.854 1.93619
\(35\) −53.7103 −0.259391
\(36\) 0 0
\(37\) 299.816 1.33215 0.666073 0.745887i \(-0.267974\pi\)
0.666073 + 0.745887i \(0.267974\pi\)
\(38\) 18.0083 0.0768774
\(39\) 0 0
\(40\) −34.1575 −0.135019
\(41\) 128.369 0.488974 0.244487 0.969653i \(-0.421380\pi\)
0.244487 + 0.969653i \(0.421380\pi\)
\(42\) 0 0
\(43\) −253.017 −0.897321 −0.448660 0.893702i \(-0.648099\pi\)
−0.448660 + 0.893702i \(0.648099\pi\)
\(44\) 151.262 0.518263
\(45\) 0 0
\(46\) 523.784 1.67886
\(47\) 113.991 0.353773 0.176886 0.984231i \(-0.443397\pi\)
0.176886 + 0.984231i \(0.443397\pi\)
\(48\) 0 0
\(49\) −243.752 −0.710647
\(50\) 363.107 1.02702
\(51\) 0 0
\(52\) −435.662 −1.16184
\(53\) 472.814 1.22540 0.612698 0.790317i \(-0.290084\pi\)
0.612698 + 0.790317i \(0.290084\pi\)
\(54\) 0 0
\(55\) −128.911 −0.316042
\(56\) 63.1175 0.150615
\(57\) 0 0
\(58\) 556.632 1.26016
\(59\) 756.581 1.66947 0.834733 0.550655i \(-0.185622\pi\)
0.834733 + 0.550655i \(0.185622\pi\)
\(60\) 0 0
\(61\) −476.984 −1.00117 −0.500587 0.865686i \(-0.666882\pi\)
−0.500587 + 0.865686i \(0.666882\pi\)
\(62\) 1076.22 2.20451
\(63\) 0 0
\(64\) −280.018 −0.546910
\(65\) 371.286 0.708498
\(66\) 0 0
\(67\) −50.4211 −0.0919390 −0.0459695 0.998943i \(-0.514638\pi\)
−0.0459695 + 0.998943i \(0.514638\pi\)
\(68\) −641.563 −1.14413
\(69\) 0 0
\(70\) 203.293 0.347116
\(71\) −722.838 −1.20824 −0.604121 0.796893i \(-0.706475\pi\)
−0.604121 + 0.796893i \(0.706475\pi\)
\(72\) 0 0
\(73\) 145.341 0.233026 0.116513 0.993189i \(-0.462828\pi\)
0.116513 + 0.993189i \(0.462828\pi\)
\(74\) −1134.80 −1.78267
\(75\) 0 0
\(76\) −30.0986 −0.0454283
\(77\) 238.206 0.352547
\(78\) 0 0
\(79\) −632.943 −0.901413 −0.450707 0.892672i \(-0.648828\pi\)
−0.450707 + 0.892672i \(0.648828\pi\)
\(80\) 402.135 0.562001
\(81\) 0 0
\(82\) −485.877 −0.654343
\(83\) −401.146 −0.530500 −0.265250 0.964180i \(-0.585454\pi\)
−0.265250 + 0.964180i \(0.585454\pi\)
\(84\) 0 0
\(85\) 546.762 0.697702
\(86\) 957.667 1.20079
\(87\) 0 0
\(88\) 151.489 0.183509
\(89\) 1337.22 1.59264 0.796320 0.604876i \(-0.206777\pi\)
0.796320 + 0.604876i \(0.206777\pi\)
\(90\) 0 0
\(91\) −686.078 −0.790335
\(92\) −875.438 −0.992073
\(93\) 0 0
\(94\) −431.455 −0.473417
\(95\) 25.6511 0.0277026
\(96\) 0 0
\(97\) −367.601 −0.384786 −0.192393 0.981318i \(-0.561625\pi\)
−0.192393 + 0.981318i \(0.561625\pi\)
\(98\) 922.598 0.950984
\(99\) 0 0
\(100\) −606.887 −0.606887
\(101\) −934.506 −0.920662 −0.460331 0.887747i \(-0.652269\pi\)
−0.460331 + 0.887747i \(0.652269\pi\)
\(102\) 0 0
\(103\) −1470.29 −1.40653 −0.703264 0.710929i \(-0.748275\pi\)
−0.703264 + 0.710929i \(0.748275\pi\)
\(104\) −436.316 −0.411388
\(105\) 0 0
\(106\) −1789.59 −1.63982
\(107\) −1090.90 −0.985618 −0.492809 0.870138i \(-0.664030\pi\)
−0.492809 + 0.870138i \(0.664030\pi\)
\(108\) 0 0
\(109\) −169.075 −0.148573 −0.0742864 0.997237i \(-0.523668\pi\)
−0.0742864 + 0.997237i \(0.523668\pi\)
\(110\) 487.925 0.422926
\(111\) 0 0
\(112\) −743.082 −0.626917
\(113\) 1473.23 1.22646 0.613230 0.789904i \(-0.289870\pi\)
0.613230 + 0.789904i \(0.289870\pi\)
\(114\) 0 0
\(115\) 746.078 0.604975
\(116\) −930.338 −0.744653
\(117\) 0 0
\(118\) −2863.65 −2.23407
\(119\) −1010.33 −0.778292
\(120\) 0 0
\(121\) −759.278 −0.570457
\(122\) 1805.38 1.33977
\(123\) 0 0
\(124\) −1798.76 −1.30269
\(125\) 1191.13 0.852301
\(126\) 0 0
\(127\) 146.269 0.102199 0.0510995 0.998694i \(-0.483727\pi\)
0.0510995 + 0.998694i \(0.483727\pi\)
\(128\) −793.208 −0.547737
\(129\) 0 0
\(130\) −1405.31 −0.948108
\(131\) 2900.64 1.93458 0.967290 0.253675i \(-0.0816393\pi\)
0.967290 + 0.253675i \(0.0816393\pi\)
\(132\) 0 0
\(133\) −47.3992 −0.0309025
\(134\) 190.843 0.123032
\(135\) 0 0
\(136\) −642.526 −0.405119
\(137\) −924.185 −0.576339 −0.288170 0.957579i \(-0.593047\pi\)
−0.288170 + 0.957579i \(0.593047\pi\)
\(138\) 0 0
\(139\) 2798.11 1.70743 0.853714 0.520743i \(-0.174345\pi\)
0.853714 + 0.520743i \(0.174345\pi\)
\(140\) −339.778 −0.205117
\(141\) 0 0
\(142\) 2735.93 1.61686
\(143\) −1646.66 −0.962943
\(144\) 0 0
\(145\) 792.866 0.454096
\(146\) −550.113 −0.311833
\(147\) 0 0
\(148\) 1896.67 1.05341
\(149\) −2584.90 −1.42123 −0.710615 0.703581i \(-0.751583\pi\)
−0.710615 + 0.703581i \(0.751583\pi\)
\(150\) 0 0
\(151\) 101.170 0.0545240 0.0272620 0.999628i \(-0.491321\pi\)
0.0272620 + 0.999628i \(0.491321\pi\)
\(152\) −30.1438 −0.0160855
\(153\) 0 0
\(154\) −901.608 −0.471777
\(155\) 1532.96 0.794391
\(156\) 0 0
\(157\) −2156.84 −1.09640 −0.548200 0.836348i \(-0.684687\pi\)
−0.548200 + 0.836348i \(0.684687\pi\)
\(158\) 2395.68 1.20627
\(159\) 0 0
\(160\) −1248.82 −0.617048
\(161\) −1378.63 −0.674854
\(162\) 0 0
\(163\) 2839.46 1.36444 0.682219 0.731148i \(-0.261015\pi\)
0.682219 + 0.731148i \(0.261015\pi\)
\(164\) 812.080 0.386663
\(165\) 0 0
\(166\) 1518.33 0.709912
\(167\) 1018.41 0.471900 0.235950 0.971765i \(-0.424180\pi\)
0.235950 + 0.971765i \(0.424180\pi\)
\(168\) 0 0
\(169\) 2545.69 1.15871
\(170\) −2069.49 −0.933661
\(171\) 0 0
\(172\) −1600.62 −0.709569
\(173\) 110.834 0.0487085 0.0243542 0.999703i \(-0.492247\pi\)
0.0243542 + 0.999703i \(0.492247\pi\)
\(174\) 0 0
\(175\) −955.721 −0.412833
\(176\) −1783.48 −0.763834
\(177\) 0 0
\(178\) −5061.35 −2.13126
\(179\) 1730.63 0.722647 0.361323 0.932441i \(-0.382325\pi\)
0.361323 + 0.932441i \(0.382325\pi\)
\(180\) 0 0
\(181\) −2118.36 −0.869926 −0.434963 0.900448i \(-0.643239\pi\)
−0.434963 + 0.900448i \(0.643239\pi\)
\(182\) 2596.79 1.05762
\(183\) 0 0
\(184\) −876.753 −0.351277
\(185\) −1616.41 −0.642381
\(186\) 0 0
\(187\) −2424.90 −0.948269
\(188\) 721.122 0.279751
\(189\) 0 0
\(190\) −97.0891 −0.0370715
\(191\) 259.696 0.0983821 0.0491910 0.998789i \(-0.484336\pi\)
0.0491910 + 0.998789i \(0.484336\pi\)
\(192\) 0 0
\(193\) −1595.13 −0.594923 −0.297461 0.954734i \(-0.596140\pi\)
−0.297461 + 0.954734i \(0.596140\pi\)
\(194\) 1391.37 0.514919
\(195\) 0 0
\(196\) −1542.00 −0.561955
\(197\) 415.809 0.150381 0.0751907 0.997169i \(-0.476043\pi\)
0.0751907 + 0.997169i \(0.476043\pi\)
\(198\) 0 0
\(199\) 5337.78 1.90143 0.950717 0.310061i \(-0.100349\pi\)
0.950717 + 0.310061i \(0.100349\pi\)
\(200\) −607.798 −0.214889
\(201\) 0 0
\(202\) 3537.09 1.23202
\(203\) −1465.09 −0.506548
\(204\) 0 0
\(205\) −692.083 −0.235791
\(206\) 5565.04 1.88221
\(207\) 0 0
\(208\) 5136.75 1.71235
\(209\) −113.763 −0.0376515
\(210\) 0 0
\(211\) 2425.65 0.791414 0.395707 0.918377i \(-0.370500\pi\)
0.395707 + 0.918377i \(0.370500\pi\)
\(212\) 2991.08 0.969000
\(213\) 0 0
\(214\) 4129.03 1.31895
\(215\) 1364.10 0.432702
\(216\) 0 0
\(217\) −2832.67 −0.886150
\(218\) 639.946 0.198819
\(219\) 0 0
\(220\) −815.504 −0.249915
\(221\) 6984.16 2.12582
\(222\) 0 0
\(223\) −2394.31 −0.718990 −0.359495 0.933147i \(-0.617051\pi\)
−0.359495 + 0.933147i \(0.617051\pi\)
\(224\) 2307.61 0.688321
\(225\) 0 0
\(226\) −5576.16 −1.64124
\(227\) −3203.94 −0.936796 −0.468398 0.883518i \(-0.655169\pi\)
−0.468398 + 0.883518i \(0.655169\pi\)
\(228\) 0 0
\(229\) −1598.36 −0.461235 −0.230618 0.973044i \(-0.574075\pi\)
−0.230618 + 0.973044i \(0.574075\pi\)
\(230\) −2823.90 −0.809575
\(231\) 0 0
\(232\) −931.735 −0.263670
\(233\) −1658.01 −0.466178 −0.233089 0.972455i \(-0.574883\pi\)
−0.233089 + 0.972455i \(0.574883\pi\)
\(234\) 0 0
\(235\) −614.565 −0.170595
\(236\) 4786.22 1.32015
\(237\) 0 0
\(238\) 3824.08 1.04151
\(239\) −239.000 −0.0646846
\(240\) 0 0
\(241\) −1185.09 −0.316756 −0.158378 0.987379i \(-0.550626\pi\)
−0.158378 + 0.987379i \(0.550626\pi\)
\(242\) 2873.86 0.763383
\(243\) 0 0
\(244\) −3017.46 −0.791693
\(245\) 1314.15 0.342685
\(246\) 0 0
\(247\) 327.659 0.0844066
\(248\) −1801.46 −0.461262
\(249\) 0 0
\(250\) −4508.40 −1.14054
\(251\) 3398.24 0.854563 0.427281 0.904119i \(-0.359471\pi\)
0.427281 + 0.904119i \(0.359471\pi\)
\(252\) 0 0
\(253\) −3308.87 −0.822242
\(254\) −553.626 −0.136762
\(255\) 0 0
\(256\) 5242.42 1.27989
\(257\) −554.979 −0.134703 −0.0673514 0.997729i \(-0.521455\pi\)
−0.0673514 + 0.997729i \(0.521455\pi\)
\(258\) 0 0
\(259\) 2986.86 0.716581
\(260\) 2348.80 0.560255
\(261\) 0 0
\(262\) −10978.9 −2.58884
\(263\) 4424.11 1.03727 0.518635 0.854996i \(-0.326440\pi\)
0.518635 + 0.854996i \(0.326440\pi\)
\(264\) 0 0
\(265\) −2549.10 −0.590905
\(266\) 179.405 0.0413535
\(267\) 0 0
\(268\) −318.970 −0.0727021
\(269\) −475.646 −0.107809 −0.0539045 0.998546i \(-0.517167\pi\)
−0.0539045 + 0.998546i \(0.517167\pi\)
\(270\) 0 0
\(271\) −3360.51 −0.753270 −0.376635 0.926362i \(-0.622919\pi\)
−0.376635 + 0.926362i \(0.622919\pi\)
\(272\) 7564.45 1.68626
\(273\) 0 0
\(274\) 3498.03 0.771254
\(275\) −2293.84 −0.502995
\(276\) 0 0
\(277\) −5384.17 −1.16788 −0.583941 0.811796i \(-0.698490\pi\)
−0.583941 + 0.811796i \(0.698490\pi\)
\(278\) −10590.8 −2.28487
\(279\) 0 0
\(280\) −340.288 −0.0726289
\(281\) −5548.30 −1.17788 −0.588939 0.808178i \(-0.700454\pi\)
−0.588939 + 0.808178i \(0.700454\pi\)
\(282\) 0 0
\(283\) 960.377 0.201726 0.100863 0.994900i \(-0.467840\pi\)
0.100863 + 0.994900i \(0.467840\pi\)
\(284\) −4572.76 −0.955434
\(285\) 0 0
\(286\) 6232.59 1.28861
\(287\) 1278.86 0.263027
\(288\) 0 0
\(289\) 5371.98 1.09342
\(290\) −3000.99 −0.607669
\(291\) 0 0
\(292\) 919.444 0.184268
\(293\) −1616.62 −0.322334 −0.161167 0.986927i \(-0.551526\pi\)
−0.161167 + 0.986927i \(0.551526\pi\)
\(294\) 0 0
\(295\) −4078.98 −0.805043
\(296\) 1899.52 0.372997
\(297\) 0 0
\(298\) 9783.81 1.90188
\(299\) 9530.16 1.84329
\(300\) 0 0
\(301\) −2520.64 −0.482683
\(302\) −382.928 −0.0729637
\(303\) 0 0
\(304\) 354.883 0.0669538
\(305\) 2571.58 0.482782
\(306\) 0 0
\(307\) −3044.79 −0.566043 −0.283021 0.959114i \(-0.591337\pi\)
−0.283021 + 0.959114i \(0.591337\pi\)
\(308\) 1506.92 0.278782
\(309\) 0 0
\(310\) −5802.25 −1.06305
\(311\) −3266.24 −0.595535 −0.297767 0.954638i \(-0.596242\pi\)
−0.297767 + 0.954638i \(0.596242\pi\)
\(312\) 0 0
\(313\) 9926.21 1.79253 0.896266 0.443517i \(-0.146269\pi\)
0.896266 + 0.443517i \(0.146269\pi\)
\(314\) 8163.61 1.46720
\(315\) 0 0
\(316\) −4004.07 −0.712806
\(317\) 4692.00 0.831322 0.415661 0.909520i \(-0.363550\pi\)
0.415661 + 0.909520i \(0.363550\pi\)
\(318\) 0 0
\(319\) −3516.38 −0.617177
\(320\) 1509.67 0.263729
\(321\) 0 0
\(322\) 5218.11 0.903087
\(323\) 482.516 0.0831204
\(324\) 0 0
\(325\) 6606.67 1.12761
\(326\) −10747.3 −1.82588
\(327\) 0 0
\(328\) 813.300 0.136912
\(329\) 1135.62 0.190300
\(330\) 0 0
\(331\) 287.433 0.0477304 0.0238652 0.999715i \(-0.492403\pi\)
0.0238652 + 0.999715i \(0.492403\pi\)
\(332\) −2537.70 −0.419500
\(333\) 0 0
\(334\) −3854.68 −0.631494
\(335\) 271.837 0.0443345
\(336\) 0 0
\(337\) 818.172 0.132251 0.0661256 0.997811i \(-0.478936\pi\)
0.0661256 + 0.997811i \(0.478936\pi\)
\(338\) −9635.39 −1.55058
\(339\) 0 0
\(340\) 3458.88 0.551718
\(341\) −6798.73 −1.07968
\(342\) 0 0
\(343\) −5845.42 −0.920183
\(344\) −1603.02 −0.251247
\(345\) 0 0
\(346\) −419.506 −0.0651814
\(347\) 6501.16 1.00577 0.502883 0.864355i \(-0.332273\pi\)
0.502883 + 0.864355i \(0.332273\pi\)
\(348\) 0 0
\(349\) 301.109 0.0461833 0.0230917 0.999733i \(-0.492649\pi\)
0.0230917 + 0.999733i \(0.492649\pi\)
\(350\) 3617.39 0.552451
\(351\) 0 0
\(352\) 5538.53 0.838650
\(353\) −8116.43 −1.22378 −0.611890 0.790943i \(-0.709590\pi\)
−0.611890 + 0.790943i \(0.709590\pi\)
\(354\) 0 0
\(355\) 3897.06 0.582633
\(356\) 8459.40 1.25940
\(357\) 0 0
\(358\) −6550.43 −0.967042
\(359\) −3735.04 −0.549103 −0.274551 0.961572i \(-0.588529\pi\)
−0.274551 + 0.961572i \(0.588529\pi\)
\(360\) 0 0
\(361\) −6836.36 −0.996700
\(362\) 8017.97 1.16413
\(363\) 0 0
\(364\) −4340.21 −0.624969
\(365\) −783.582 −0.112369
\(366\) 0 0
\(367\) −6786.94 −0.965328 −0.482664 0.875806i \(-0.660331\pi\)
−0.482664 + 0.875806i \(0.660331\pi\)
\(368\) 10322.0 1.46215
\(369\) 0 0
\(370\) 6118.08 0.859631
\(371\) 4710.33 0.659159
\(372\) 0 0
\(373\) −8021.78 −1.11354 −0.556772 0.830665i \(-0.687960\pi\)
−0.556772 + 0.830665i \(0.687960\pi\)
\(374\) 9178.22 1.26897
\(375\) 0 0
\(376\) 722.205 0.0990555
\(377\) 10127.8 1.38358
\(378\) 0 0
\(379\) 5810.90 0.787561 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(380\) 162.272 0.0219062
\(381\) 0 0
\(382\) −982.947 −0.131654
\(383\) 7403.68 0.987755 0.493878 0.869531i \(-0.335579\pi\)
0.493878 + 0.869531i \(0.335579\pi\)
\(384\) 0 0
\(385\) −1284.25 −0.170004
\(386\) 6037.55 0.796122
\(387\) 0 0
\(388\) −2325.49 −0.304275
\(389\) 457.630 0.0596472 0.0298236 0.999555i \(-0.490505\pi\)
0.0298236 + 0.999555i \(0.490505\pi\)
\(390\) 0 0
\(391\) 14034.3 1.81520
\(392\) −1544.32 −0.198979
\(393\) 0 0
\(394\) −1573.83 −0.201240
\(395\) 3412.41 0.434676
\(396\) 0 0
\(397\) −291.335 −0.0368305 −0.0184152 0.999830i \(-0.505862\pi\)
−0.0184152 + 0.999830i \(0.505862\pi\)
\(398\) −20203.4 −2.54449
\(399\) 0 0
\(400\) 7155.60 0.894450
\(401\) 3567.33 0.444249 0.222125 0.975018i \(-0.428701\pi\)
0.222125 + 0.975018i \(0.428701\pi\)
\(402\) 0 0
\(403\) 19581.6 2.42042
\(404\) −5911.80 −0.728027
\(405\) 0 0
\(406\) 5545.35 0.677860
\(407\) 7168.80 0.873082
\(408\) 0 0
\(409\) 3528.46 0.426580 0.213290 0.976989i \(-0.431582\pi\)
0.213290 + 0.976989i \(0.431582\pi\)
\(410\) 2619.52 0.315534
\(411\) 0 0
\(412\) −9301.25 −1.11223
\(413\) 7537.31 0.898031
\(414\) 0 0
\(415\) 2162.71 0.255815
\(416\) −15952.0 −1.88007
\(417\) 0 0
\(418\) 430.592 0.0503851
\(419\) 3459.28 0.403334 0.201667 0.979454i \(-0.435364\pi\)
0.201667 + 0.979454i \(0.435364\pi\)
\(420\) 0 0
\(421\) 15493.6 1.79361 0.896805 0.442426i \(-0.145882\pi\)
0.896805 + 0.442426i \(0.145882\pi\)
\(422\) −9181.04 −1.05907
\(423\) 0 0
\(424\) 2995.57 0.343108
\(425\) 9729.08 1.11042
\(426\) 0 0
\(427\) −4751.88 −0.538547
\(428\) −6901.15 −0.779392
\(429\) 0 0
\(430\) −5163.11 −0.579040
\(431\) 8338.54 0.931910 0.465955 0.884808i \(-0.345711\pi\)
0.465955 + 0.884808i \(0.345711\pi\)
\(432\) 0 0
\(433\) 16692.7 1.85266 0.926330 0.376713i \(-0.122946\pi\)
0.926330 + 0.376713i \(0.122946\pi\)
\(434\) 10721.6 1.18584
\(435\) 0 0
\(436\) −1069.59 −0.117486
\(437\) 658.412 0.0720735
\(438\) 0 0
\(439\) 12115.7 1.31720 0.658601 0.752493i \(-0.271149\pi\)
0.658601 + 0.752493i \(0.271149\pi\)
\(440\) −816.729 −0.0884909
\(441\) 0 0
\(442\) −26434.9 −2.84476
\(443\) 4961.23 0.532088 0.266044 0.963961i \(-0.414283\pi\)
0.266044 + 0.963961i \(0.414283\pi\)
\(444\) 0 0
\(445\) −7209.40 −0.767996
\(446\) 9062.42 0.962149
\(447\) 0 0
\(448\) −2789.63 −0.294191
\(449\) −12845.1 −1.35011 −0.675056 0.737767i \(-0.735880\pi\)
−0.675056 + 0.737767i \(0.735880\pi\)
\(450\) 0 0
\(451\) 3069.40 0.320471
\(452\) 9319.84 0.969841
\(453\) 0 0
\(454\) 12126.9 1.25362
\(455\) 3698.88 0.381112
\(456\) 0 0
\(457\) −17334.8 −1.77437 −0.887186 0.461413i \(-0.847343\pi\)
−0.887186 + 0.461413i \(0.847343\pi\)
\(458\) 6049.79 0.617223
\(459\) 0 0
\(460\) 4719.78 0.478393
\(461\) 11143.0 1.12577 0.562886 0.826535i \(-0.309691\pi\)
0.562886 + 0.826535i \(0.309691\pi\)
\(462\) 0 0
\(463\) 11700.8 1.17447 0.587235 0.809416i \(-0.300216\pi\)
0.587235 + 0.809416i \(0.300216\pi\)
\(464\) 10969.3 1.09749
\(465\) 0 0
\(466\) 6275.53 0.623837
\(467\) 10130.2 1.00379 0.501894 0.864929i \(-0.332637\pi\)
0.501894 + 0.864929i \(0.332637\pi\)
\(468\) 0 0
\(469\) −502.312 −0.0494554
\(470\) 2326.12 0.228289
\(471\) 0 0
\(472\) 4793.41 0.467446
\(473\) −6049.82 −0.588100
\(474\) 0 0
\(475\) 456.436 0.0440899
\(476\) −6391.46 −0.615446
\(477\) 0 0
\(478\) 904.612 0.0865606
\(479\) −4302.76 −0.410434 −0.205217 0.978716i \(-0.565790\pi\)
−0.205217 + 0.978716i \(0.565790\pi\)
\(480\) 0 0
\(481\) −20647.5 −1.95726
\(482\) 4485.54 0.423881
\(483\) 0 0
\(484\) −4803.28 −0.451097
\(485\) 1981.86 0.185550
\(486\) 0 0
\(487\) 12589.1 1.17139 0.585696 0.810531i \(-0.300821\pi\)
0.585696 + 0.810531i \(0.300821\pi\)
\(488\) −3021.99 −0.280326
\(489\) 0 0
\(490\) −4974.03 −0.458580
\(491\) 4301.47 0.395362 0.197681 0.980266i \(-0.436659\pi\)
0.197681 + 0.980266i \(0.436659\pi\)
\(492\) 0 0
\(493\) 14914.4 1.36249
\(494\) −1240.18 −0.112952
\(495\) 0 0
\(496\) 21208.6 1.91995
\(497\) −7201.15 −0.649931
\(498\) 0 0
\(499\) 10328.7 0.926603 0.463302 0.886201i \(-0.346665\pi\)
0.463302 + 0.886201i \(0.346665\pi\)
\(500\) 7535.21 0.673970
\(501\) 0 0
\(502\) −12862.3 −1.14357
\(503\) 6664.46 0.590763 0.295381 0.955379i \(-0.404553\pi\)
0.295381 + 0.955379i \(0.404553\pi\)
\(504\) 0 0
\(505\) 5038.24 0.443958
\(506\) 12524.0 1.10032
\(507\) 0 0
\(508\) 925.314 0.0808153
\(509\) −9576.76 −0.833953 −0.416977 0.908917i \(-0.636910\pi\)
−0.416977 + 0.908917i \(0.636910\pi\)
\(510\) 0 0
\(511\) 1447.93 0.125348
\(512\) −13496.8 −1.16500
\(513\) 0 0
\(514\) 2100.59 0.180259
\(515\) 7926.85 0.678250
\(516\) 0 0
\(517\) 2725.61 0.231861
\(518\) −11305.2 −0.958925
\(519\) 0 0
\(520\) 2352.33 0.198378
\(521\) −20897.5 −1.75727 −0.878636 0.477493i \(-0.841546\pi\)
−0.878636 + 0.477493i \(0.841546\pi\)
\(522\) 0 0
\(523\) 1660.23 0.138808 0.0694041 0.997589i \(-0.477890\pi\)
0.0694041 + 0.997589i \(0.477890\pi\)
\(524\) 18349.8 1.52980
\(525\) 0 0
\(526\) −16745.2 −1.38807
\(527\) 28836.2 2.38353
\(528\) 0 0
\(529\) 6983.31 0.573955
\(530\) 9648.30 0.790746
\(531\) 0 0
\(532\) −299.853 −0.0244366
\(533\) −8840.44 −0.718428
\(534\) 0 0
\(535\) 5881.40 0.475280
\(536\) −319.449 −0.0257427
\(537\) 0 0
\(538\) 1800.31 0.144270
\(539\) −5828.28 −0.465755
\(540\) 0 0
\(541\) 23034.4 1.83055 0.915274 0.402831i \(-0.131974\pi\)
0.915274 + 0.402831i \(0.131974\pi\)
\(542\) 12719.5 1.00802
\(543\) 0 0
\(544\) −23491.1 −1.85142
\(545\) 911.539 0.0716441
\(546\) 0 0
\(547\) −15457.2 −1.20823 −0.604117 0.796896i \(-0.706474\pi\)
−0.604117 + 0.796896i \(0.706474\pi\)
\(548\) −5846.51 −0.455749
\(549\) 0 0
\(550\) 8682.14 0.673105
\(551\) 699.702 0.0540986
\(552\) 0 0
\(553\) −6305.59 −0.484884
\(554\) 20379.0 1.56285
\(555\) 0 0
\(556\) 17701.2 1.35017
\(557\) −13685.6 −1.04107 −0.520535 0.853840i \(-0.674267\pi\)
−0.520535 + 0.853840i \(0.674267\pi\)
\(558\) 0 0
\(559\) 17424.6 1.31839
\(560\) 4006.21 0.302309
\(561\) 0 0
\(562\) 21000.2 1.57623
\(563\) 180.930 0.0135440 0.00677202 0.999977i \(-0.497844\pi\)
0.00677202 + 0.999977i \(0.497844\pi\)
\(564\) 0 0
\(565\) −7942.69 −0.591418
\(566\) −3635.01 −0.269949
\(567\) 0 0
\(568\) −4579.63 −0.338304
\(569\) −990.907 −0.0730070 −0.0365035 0.999334i \(-0.511622\pi\)
−0.0365035 + 0.999334i \(0.511622\pi\)
\(570\) 0 0
\(571\) 14592.9 1.06952 0.534759 0.845004i \(-0.320402\pi\)
0.534759 + 0.845004i \(0.320402\pi\)
\(572\) −10417.0 −0.761461
\(573\) 0 0
\(574\) −4840.46 −0.351981
\(575\) 13275.7 0.962845
\(576\) 0 0
\(577\) −71.1062 −0.00513031 −0.00256516 0.999997i \(-0.500817\pi\)
−0.00256516 + 0.999997i \(0.500817\pi\)
\(578\) −20332.9 −1.46321
\(579\) 0 0
\(580\) 5015.77 0.359083
\(581\) −3996.35 −0.285364
\(582\) 0 0
\(583\) 11305.3 0.803118
\(584\) 920.824 0.0652466
\(585\) 0 0
\(586\) 6118.87 0.431345
\(587\) −9855.47 −0.692979 −0.346490 0.938054i \(-0.612626\pi\)
−0.346490 + 0.938054i \(0.612626\pi\)
\(588\) 0 0
\(589\) 1352.84 0.0946395
\(590\) 15438.9 1.07730
\(591\) 0 0
\(592\) −22363.0 −1.55256
\(593\) 3528.47 0.244345 0.122173 0.992509i \(-0.461014\pi\)
0.122173 + 0.992509i \(0.461014\pi\)
\(594\) 0 0
\(595\) 5447.02 0.375304
\(596\) −16352.4 −1.12386
\(597\) 0 0
\(598\) −36071.5 −2.46668
\(599\) −18538.1 −1.26452 −0.632259 0.774757i \(-0.717872\pi\)
−0.632259 + 0.774757i \(0.717872\pi\)
\(600\) 0 0
\(601\) −4577.09 −0.310654 −0.155327 0.987863i \(-0.549643\pi\)
−0.155327 + 0.987863i \(0.549643\pi\)
\(602\) 9540.60 0.645923
\(603\) 0 0
\(604\) 640.015 0.0431156
\(605\) 4093.52 0.275083
\(606\) 0 0
\(607\) −6748.64 −0.451267 −0.225633 0.974212i \(-0.572445\pi\)
−0.225633 + 0.974212i \(0.572445\pi\)
\(608\) −1102.08 −0.0735117
\(609\) 0 0
\(610\) −9733.41 −0.646056
\(611\) −7850.25 −0.519782
\(612\) 0 0
\(613\) 3105.57 0.204621 0.102311 0.994753i \(-0.467376\pi\)
0.102311 + 0.994753i \(0.467376\pi\)
\(614\) 11524.5 0.757475
\(615\) 0 0
\(616\) 1509.18 0.0987123
\(617\) −17559.7 −1.14575 −0.572873 0.819644i \(-0.694171\pi\)
−0.572873 + 0.819644i \(0.694171\pi\)
\(618\) 0 0
\(619\) 6603.88 0.428808 0.214404 0.976745i \(-0.431219\pi\)
0.214404 + 0.976745i \(0.431219\pi\)
\(620\) 9697.71 0.628177
\(621\) 0 0
\(622\) 12362.7 0.796942
\(623\) 13321.8 0.856705
\(624\) 0 0
\(625\) 5569.92 0.356475
\(626\) −37570.6 −2.39876
\(627\) 0 0
\(628\) −13644.4 −0.866994
\(629\) −30405.8 −1.92744
\(630\) 0 0
\(631\) −20581.4 −1.29847 −0.649234 0.760589i \(-0.724910\pi\)
−0.649234 + 0.760589i \(0.724910\pi\)
\(632\) −4010.09 −0.252393
\(633\) 0 0
\(634\) −17759.2 −1.11247
\(635\) −788.585 −0.0492819
\(636\) 0 0
\(637\) 16786.5 1.04412
\(638\) 13309.5 0.825903
\(639\) 0 0
\(640\) 4276.45 0.264127
\(641\) 17570.3 1.08266 0.541332 0.840809i \(-0.317920\pi\)
0.541332 + 0.840809i \(0.317920\pi\)
\(642\) 0 0
\(643\) 13570.7 0.832311 0.416156 0.909293i \(-0.363377\pi\)
0.416156 + 0.909293i \(0.363377\pi\)
\(644\) −8721.40 −0.533651
\(645\) 0 0
\(646\) −1826.31 −0.111231
\(647\) −8755.42 −0.532011 −0.266006 0.963971i \(-0.585704\pi\)
−0.266006 + 0.963971i \(0.585704\pi\)
\(648\) 0 0
\(649\) 18090.4 1.09416
\(650\) −25006.1 −1.50896
\(651\) 0 0
\(652\) 17962.7 1.07895
\(653\) 3949.95 0.236713 0.118356 0.992971i \(-0.462237\pi\)
0.118356 + 0.992971i \(0.462237\pi\)
\(654\) 0 0
\(655\) −15638.3 −0.932885
\(656\) −9574.97 −0.569878
\(657\) 0 0
\(658\) −4298.30 −0.254658
\(659\) −9579.76 −0.566274 −0.283137 0.959080i \(-0.591375\pi\)
−0.283137 + 0.959080i \(0.591375\pi\)
\(660\) 0 0
\(661\) 10579.5 0.622533 0.311267 0.950323i \(-0.399247\pi\)
0.311267 + 0.950323i \(0.399247\pi\)
\(662\) −1087.93 −0.0638726
\(663\) 0 0
\(664\) −2541.51 −0.148539
\(665\) 255.545 0.0149017
\(666\) 0 0
\(667\) 20351.3 1.18142
\(668\) 6442.61 0.373162
\(669\) 0 0
\(670\) −1028.90 −0.0593281
\(671\) −11405.0 −0.656165
\(672\) 0 0
\(673\) −4256.15 −0.243778 −0.121889 0.992544i \(-0.538895\pi\)
−0.121889 + 0.992544i \(0.538895\pi\)
\(674\) −3096.77 −0.176978
\(675\) 0 0
\(676\) 16104.3 0.916268
\(677\) 18996.7 1.07844 0.539219 0.842165i \(-0.318719\pi\)
0.539219 + 0.842165i \(0.318719\pi\)
\(678\) 0 0
\(679\) −3662.17 −0.206982
\(680\) 3464.07 0.195355
\(681\) 0 0
\(682\) 25733.1 1.44483
\(683\) −28376.8 −1.58977 −0.794883 0.606763i \(-0.792468\pi\)
−0.794883 + 0.606763i \(0.792468\pi\)
\(684\) 0 0
\(685\) 4982.59 0.277920
\(686\) 22124.8 1.23138
\(687\) 0 0
\(688\) 18872.4 1.04579
\(689\) −32561.3 −1.80042
\(690\) 0 0
\(691\) −23570.2 −1.29761 −0.648807 0.760953i \(-0.724732\pi\)
−0.648807 + 0.760953i \(0.724732\pi\)
\(692\) 701.150 0.0385170
\(693\) 0 0
\(694\) −24606.8 −1.34591
\(695\) −15085.5 −0.823348
\(696\) 0 0
\(697\) −13018.6 −0.707480
\(698\) −1139.69 −0.0618023
\(699\) 0 0
\(700\) −6046.01 −0.326454
\(701\) 25244.6 1.36016 0.680082 0.733136i \(-0.261944\pi\)
0.680082 + 0.733136i \(0.261944\pi\)
\(702\) 0 0
\(703\) −1426.47 −0.0765298
\(704\) −6695.43 −0.358442
\(705\) 0 0
\(706\) 30720.6 1.63765
\(707\) −9309.86 −0.495238
\(708\) 0 0
\(709\) 14943.8 0.791576 0.395788 0.918342i \(-0.370472\pi\)
0.395788 + 0.918342i \(0.370472\pi\)
\(710\) −14750.3 −0.779676
\(711\) 0 0
\(712\) 8472.11 0.445935
\(713\) 39348.1 2.06676
\(714\) 0 0
\(715\) 8877.71 0.464346
\(716\) 10948.2 0.571443
\(717\) 0 0
\(718\) 14137.1 0.734806
\(719\) 25609.1 1.32832 0.664158 0.747592i \(-0.268790\pi\)
0.664158 + 0.747592i \(0.268790\pi\)
\(720\) 0 0
\(721\) −14647.6 −0.756593
\(722\) 25875.5 1.33378
\(723\) 0 0
\(724\) −13401.0 −0.687907
\(725\) 14108.3 0.722715
\(726\) 0 0
\(727\) −2306.17 −0.117649 −0.0588247 0.998268i \(-0.518735\pi\)
−0.0588247 + 0.998268i \(0.518735\pi\)
\(728\) −4346.73 −0.221292
\(729\) 0 0
\(730\) 2965.85 0.150371
\(731\) 25659.7 1.29830
\(732\) 0 0
\(733\) −14863.6 −0.748977 −0.374488 0.927232i \(-0.622182\pi\)
−0.374488 + 0.927232i \(0.622182\pi\)
\(734\) 25688.5 1.29180
\(735\) 0 0
\(736\) −32054.6 −1.60536
\(737\) −1205.60 −0.0602564
\(738\) 0 0
\(739\) 35857.8 1.78491 0.892457 0.451133i \(-0.148980\pi\)
0.892457 + 0.451133i \(0.148980\pi\)
\(740\) −10225.6 −0.507973
\(741\) 0 0
\(742\) −17828.5 −0.882083
\(743\) 4247.42 0.209721 0.104861 0.994487i \(-0.466560\pi\)
0.104861 + 0.994487i \(0.466560\pi\)
\(744\) 0 0
\(745\) 13936.1 0.685340
\(746\) 30362.3 1.49014
\(747\) 0 0
\(748\) −15340.2 −0.749858
\(749\) −10867.9 −0.530179
\(750\) 0 0
\(751\) 22971.0 1.11614 0.558072 0.829792i \(-0.311541\pi\)
0.558072 + 0.829792i \(0.311541\pi\)
\(752\) −8502.51 −0.412307
\(753\) 0 0
\(754\) −38333.6 −1.85150
\(755\) −545.443 −0.0262923
\(756\) 0 0
\(757\) 12971.9 0.622818 0.311409 0.950276i \(-0.399199\pi\)
0.311409 + 0.950276i \(0.399199\pi\)
\(758\) −21994.2 −1.05391
\(759\) 0 0
\(760\) 162.516 0.00775666
\(761\) −25835.3 −1.23066 −0.615328 0.788271i \(-0.710976\pi\)
−0.615328 + 0.788271i \(0.710976\pi\)
\(762\) 0 0
\(763\) −1684.38 −0.0799196
\(764\) 1642.87 0.0777971
\(765\) 0 0
\(766\) −28022.8 −1.32181
\(767\) −52103.6 −2.45287
\(768\) 0 0
\(769\) −39553.5 −1.85479 −0.927395 0.374082i \(-0.877958\pi\)
−0.927395 + 0.374082i \(0.877958\pi\)
\(770\) 4860.87 0.227498
\(771\) 0 0
\(772\) −10091.0 −0.470444
\(773\) 28781.1 1.33918 0.669588 0.742732i \(-0.266471\pi\)
0.669588 + 0.742732i \(0.266471\pi\)
\(774\) 0 0
\(775\) 27277.6 1.26431
\(776\) −2328.98 −0.107739
\(777\) 0 0
\(778\) −1732.12 −0.0798195
\(779\) −610.761 −0.0280909
\(780\) 0 0
\(781\) −17283.6 −0.791876
\(782\) −53119.5 −2.42909
\(783\) 0 0
\(784\) 18181.3 0.828228
\(785\) 11628.3 0.528701
\(786\) 0 0
\(787\) −17177.2 −0.778020 −0.389010 0.921234i \(-0.627183\pi\)
−0.389010 + 0.921234i \(0.627183\pi\)
\(788\) 2630.46 0.118916
\(789\) 0 0
\(790\) −12915.9 −0.581681
\(791\) 14676.8 0.659732
\(792\) 0 0
\(793\) 32848.6 1.47098
\(794\) 1102.70 0.0492863
\(795\) 0 0
\(796\) 33767.4 1.50359
\(797\) −28084.7 −1.24820 −0.624098 0.781346i \(-0.714533\pi\)
−0.624098 + 0.781346i \(0.714533\pi\)
\(798\) 0 0
\(799\) −11560.4 −0.511862
\(800\) −22221.5 −0.982059
\(801\) 0 0
\(802\) −13502.3 −0.594492
\(803\) 3475.20 0.152724
\(804\) 0 0
\(805\) 7432.68 0.325425
\(806\) −74116.0 −3.23899
\(807\) 0 0
\(808\) −5920.68 −0.257783
\(809\) 15309.4 0.665329 0.332664 0.943045i \(-0.392052\pi\)
0.332664 + 0.943045i \(0.392052\pi\)
\(810\) 0 0
\(811\) 27565.6 1.19354 0.596768 0.802414i \(-0.296451\pi\)
0.596768 + 0.802414i \(0.296451\pi\)
\(812\) −9268.34 −0.400560
\(813\) 0 0
\(814\) −27133.8 −1.16835
\(815\) −15308.5 −0.657954
\(816\) 0 0
\(817\) 1203.82 0.0515498
\(818\) −13355.2 −0.570847
\(819\) 0 0
\(820\) −4378.20 −0.186455
\(821\) 15159.1 0.644404 0.322202 0.946671i \(-0.395577\pi\)
0.322202 + 0.946671i \(0.395577\pi\)
\(822\) 0 0
\(823\) −37930.0 −1.60651 −0.803253 0.595637i \(-0.796900\pi\)
−0.803253 + 0.595637i \(0.796900\pi\)
\(824\) −9315.22 −0.393824
\(825\) 0 0
\(826\) −28528.6 −1.20174
\(827\) 30447.1 1.28023 0.640115 0.768279i \(-0.278887\pi\)
0.640115 + 0.768279i \(0.278887\pi\)
\(828\) 0 0
\(829\) −6923.16 −0.290050 −0.145025 0.989428i \(-0.546326\pi\)
−0.145025 + 0.989428i \(0.546326\pi\)
\(830\) −8185.83 −0.342331
\(831\) 0 0
\(832\) 19284.1 0.803551
\(833\) 24720.1 1.02821
\(834\) 0 0
\(835\) −5490.62 −0.227558
\(836\) −719.680 −0.0297735
\(837\) 0 0
\(838\) −13093.3 −0.539739
\(839\) −45271.4 −1.86286 −0.931432 0.363916i \(-0.881440\pi\)
−0.931432 + 0.363916i \(0.881440\pi\)
\(840\) 0 0
\(841\) −2761.46 −0.113226
\(842\) −58642.9 −2.40020
\(843\) 0 0
\(844\) 15344.9 0.625822
\(845\) −13724.7 −0.558749
\(846\) 0 0
\(847\) −7564.18 −0.306858
\(848\) −35266.8 −1.42814
\(849\) 0 0
\(850\) −36824.4 −1.48596
\(851\) −41489.9 −1.67127
\(852\) 0 0
\(853\) 6399.49 0.256875 0.128438 0.991718i \(-0.459004\pi\)
0.128438 + 0.991718i \(0.459004\pi\)
\(854\) 17985.8 0.720680
\(855\) 0 0
\(856\) −6911.52 −0.275971
\(857\) 39912.0 1.59086 0.795430 0.606045i \(-0.207245\pi\)
0.795430 + 0.606045i \(0.207245\pi\)
\(858\) 0 0
\(859\) 2195.35 0.0871994 0.0435997 0.999049i \(-0.486117\pi\)
0.0435997 + 0.999049i \(0.486117\pi\)
\(860\) 8629.47 0.342166
\(861\) 0 0
\(862\) −31561.2 −1.24708
\(863\) −29140.2 −1.14942 −0.574708 0.818359i \(-0.694884\pi\)
−0.574708 + 0.818359i \(0.694884\pi\)
\(864\) 0 0
\(865\) −597.544 −0.0234880
\(866\) −63181.8 −2.47922
\(867\) 0 0
\(868\) −17919.8 −0.700736
\(869\) −15134.1 −0.590782
\(870\) 0 0
\(871\) 3472.36 0.135082
\(872\) −1071.19 −0.0416000
\(873\) 0 0
\(874\) −2492.08 −0.0964483
\(875\) 11866.4 0.458466
\(876\) 0 0
\(877\) −13975.7 −0.538114 −0.269057 0.963124i \(-0.586712\pi\)
−0.269057 + 0.963124i \(0.586712\pi\)
\(878\) −45857.8 −1.76267
\(879\) 0 0
\(880\) 9615.34 0.368333
\(881\) 4596.29 0.175770 0.0878848 0.996131i \(-0.471989\pi\)
0.0878848 + 0.996131i \(0.471989\pi\)
\(882\) 0 0
\(883\) 9476.47 0.361165 0.180582 0.983560i \(-0.442202\pi\)
0.180582 + 0.983560i \(0.442202\pi\)
\(884\) 44182.6 1.68102
\(885\) 0 0
\(886\) −18778.2 −0.712037
\(887\) −35132.1 −1.32990 −0.664949 0.746889i \(-0.731547\pi\)
−0.664949 + 0.746889i \(0.731547\pi\)
\(888\) 0 0
\(889\) 1457.18 0.0549744
\(890\) 27287.5 1.02773
\(891\) 0 0
\(892\) −15146.7 −0.568552
\(893\) −542.351 −0.0203237
\(894\) 0 0
\(895\) −9330.44 −0.348472
\(896\) −7902.20 −0.294636
\(897\) 0 0
\(898\) 48618.7 1.80671
\(899\) 41815.7 1.55131
\(900\) 0 0
\(901\) −47950.4 −1.77298
\(902\) −11617.6 −0.428853
\(903\) 0 0
\(904\) 9333.84 0.343406
\(905\) 11420.8 0.419492
\(906\) 0 0
\(907\) −17329.8 −0.634430 −0.317215 0.948354i \(-0.602748\pi\)
−0.317215 + 0.948354i \(0.602748\pi\)
\(908\) −20268.5 −0.740785
\(909\) 0 0
\(910\) −14000.2 −0.510002
\(911\) 42270.8 1.53731 0.768657 0.639661i \(-0.220925\pi\)
0.768657 + 0.639661i \(0.220925\pi\)
\(912\) 0 0
\(913\) −9591.68 −0.347687
\(914\) 65612.0 2.37445
\(915\) 0 0
\(916\) −10111.4 −0.364729
\(917\) 28897.1 1.04064
\(918\) 0 0
\(919\) −18991.4 −0.681686 −0.340843 0.940120i \(-0.610713\pi\)
−0.340843 + 0.940120i \(0.610713\pi\)
\(920\) 4726.87 0.169392
\(921\) 0 0
\(922\) −42176.1 −1.50650
\(923\) 49779.8 1.77521
\(924\) 0 0
\(925\) −28762.4 −1.02238
\(926\) −44287.2 −1.57167
\(927\) 0 0
\(928\) −34064.8 −1.20499
\(929\) 19692.9 0.695483 0.347741 0.937590i \(-0.386949\pi\)
0.347741 + 0.937590i \(0.386949\pi\)
\(930\) 0 0
\(931\) 1159.73 0.0408257
\(932\) −10488.7 −0.368637
\(933\) 0 0
\(934\) −38342.6 −1.34326
\(935\) 13073.5 0.457270
\(936\) 0 0
\(937\) 41739.0 1.45523 0.727616 0.685984i \(-0.240628\pi\)
0.727616 + 0.685984i \(0.240628\pi\)
\(938\) 1901.24 0.0661810
\(939\) 0 0
\(940\) −3887.81 −0.134900
\(941\) −8040.14 −0.278535 −0.139267 0.990255i \(-0.544475\pi\)
−0.139267 + 0.990255i \(0.544475\pi\)
\(942\) 0 0
\(943\) −17764.4 −0.613454
\(944\) −56432.8 −1.94569
\(945\) 0 0
\(946\) 22898.5 0.786992
\(947\) 14958.3 0.513284 0.256642 0.966506i \(-0.417384\pi\)
0.256642 + 0.966506i \(0.417384\pi\)
\(948\) 0 0
\(949\) −10009.2 −0.342374
\(950\) −1727.60 −0.0590009
\(951\) 0 0
\(952\) −6401.06 −0.217920
\(953\) 21420.2 0.728089 0.364044 0.931382i \(-0.381396\pi\)
0.364044 + 0.931382i \(0.381396\pi\)
\(954\) 0 0
\(955\) −1400.11 −0.0474414
\(956\) −1511.94 −0.0511503
\(957\) 0 0
\(958\) 16285.9 0.549241
\(959\) −9207.04 −0.310022
\(960\) 0 0
\(961\) 51057.4 1.71385
\(962\) 78150.3 2.61920
\(963\) 0 0
\(964\) −7497.00 −0.250479
\(965\) 8599.89 0.286881
\(966\) 0 0
\(967\) −31297.0 −1.04079 −0.520395 0.853925i \(-0.674215\pi\)
−0.520395 + 0.853925i \(0.674215\pi\)
\(968\) −4810.50 −0.159727
\(969\) 0 0
\(970\) −7501.32 −0.248302
\(971\) −47837.1 −1.58101 −0.790507 0.612452i \(-0.790183\pi\)
−0.790507 + 0.612452i \(0.790183\pi\)
\(972\) 0 0
\(973\) 27875.7 0.918451
\(974\) −47649.7 −1.56755
\(975\) 0 0
\(976\) 35577.9 1.16682
\(977\) 37480.5 1.22734 0.613668 0.789564i \(-0.289693\pi\)
0.613668 + 0.789564i \(0.289693\pi\)
\(978\) 0 0
\(979\) 31973.8 1.04381
\(980\) 8313.46 0.270983
\(981\) 0 0
\(982\) −16281.0 −0.529071
\(983\) −22314.5 −0.724030 −0.362015 0.932172i \(-0.617911\pi\)
−0.362015 + 0.932172i \(0.617911\pi\)
\(984\) 0 0
\(985\) −2241.77 −0.0725163
\(986\) −56450.7 −1.82328
\(987\) 0 0
\(988\) 2072.81 0.0667458
\(989\) 35013.7 1.12575
\(990\) 0 0
\(991\) −18810.4 −0.602957 −0.301479 0.953473i \(-0.597480\pi\)
−0.301479 + 0.953473i \(0.597480\pi\)
\(992\) −65862.5 −2.10800
\(993\) 0 0
\(994\) 27256.3 0.869735
\(995\) −28777.8 −0.916901
\(996\) 0 0
\(997\) 33209.9 1.05493 0.527466 0.849576i \(-0.323142\pi\)
0.527466 + 0.849576i \(0.323142\pi\)
\(998\) −39093.9 −1.23998
\(999\) 0 0
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 2151.4.a.h.1.12 yes 59
3.2 odd 2 2151.4.a.g.1.48 59
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.4.a.g.1.48 59 3.2 odd 2
2151.4.a.h.1.12 yes 59 1.1 even 1 trivial