Properties

Label 1849.4.a.j.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

\(f(q)\) \(=\) \(q-1.06529 q^{2} -0.463524 q^{3} -6.86516 q^{4} -5.41130 q^{5} +0.493788 q^{6} -4.55619 q^{7} +15.8357 q^{8} -26.7851 q^{9} +O(q^{10})\) \(q-1.06529 q^{2} -0.463524 q^{3} -6.86516 q^{4} -5.41130 q^{5} +0.493788 q^{6} -4.55619 q^{7} +15.8357 q^{8} -26.7851 q^{9} +5.76461 q^{10} +10.2865 q^{11} +3.18216 q^{12} +4.48831 q^{13} +4.85366 q^{14} +2.50827 q^{15} +38.0516 q^{16} -32.2212 q^{17} +28.5340 q^{18} -54.9889 q^{19} +37.1494 q^{20} +2.11190 q^{21} -10.9581 q^{22} +136.229 q^{23} -7.34023 q^{24} -95.7178 q^{25} -4.78136 q^{26} +24.9307 q^{27} +31.2789 q^{28} -87.3472 q^{29} -2.67203 q^{30} +27.1204 q^{31} -167.222 q^{32} -4.76801 q^{33} +34.3250 q^{34} +24.6549 q^{35} +183.884 q^{36} +287.935 q^{37} +58.5791 q^{38} -2.08044 q^{39} -85.6918 q^{40} +285.555 q^{41} -2.24979 q^{42} -70.6181 q^{44} +144.942 q^{45} -145.124 q^{46} -62.1624 q^{47} -17.6378 q^{48} -322.241 q^{49} +101.967 q^{50} +14.9353 q^{51} -30.8129 q^{52} +310.227 q^{53} -26.5584 q^{54} -55.6631 q^{55} -72.1505 q^{56} +25.4886 q^{57} +93.0501 q^{58} +561.966 q^{59} -17.2196 q^{60} +381.217 q^{61} -28.8911 q^{62} +122.038 q^{63} -126.273 q^{64} -24.2876 q^{65} +5.07932 q^{66} -167.929 q^{67} +221.204 q^{68} -63.1455 q^{69} -26.2646 q^{70} +328.805 q^{71} -424.162 q^{72} +1034.34 q^{73} -306.735 q^{74} +44.3675 q^{75} +377.507 q^{76} -46.8670 q^{77} +2.21627 q^{78} -496.972 q^{79} -205.909 q^{80} +711.643 q^{81} -304.199 q^{82} +374.134 q^{83} -14.4985 q^{84} +174.359 q^{85} +40.4875 q^{87} +162.893 q^{88} -270.772 q^{89} -154.406 q^{90} -20.4496 q^{91} -935.236 q^{92} -12.5709 q^{93} +66.2210 q^{94} +297.561 q^{95} +77.5112 q^{96} -1118.35 q^{97} +343.281 q^{98} -275.524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06529 −0.376637 −0.188319 0.982108i \(-0.560304\pi\)
−0.188319 + 0.982108i \(0.560304\pi\)
\(3\) −0.463524 −0.0892052 −0.0446026 0.999005i \(-0.514202\pi\)
−0.0446026 + 0.999005i \(0.514202\pi\)
\(4\) −6.86516 −0.858144
\(5\) −5.41130 −0.484001 −0.242001 0.970276i \(-0.577804\pi\)
−0.242001 + 0.970276i \(0.577804\pi\)
\(6\) 0.493788 0.0335980
\(7\) −4.55619 −0.246011 −0.123005 0.992406i \(-0.539253\pi\)
−0.123005 + 0.992406i \(0.539253\pi\)
\(8\) 15.8357 0.699846
\(9\) −26.7851 −0.992042
\(10\) 5.76461 0.182293
\(11\) 10.2865 0.281953 0.140976 0.990013i \(-0.454976\pi\)
0.140976 + 0.990013i \(0.454976\pi\)
\(12\) 3.18216 0.0765509
\(13\) 4.48831 0.0957564 0.0478782 0.998853i \(-0.484754\pi\)
0.0478782 + 0.998853i \(0.484754\pi\)
\(14\) 4.85366 0.0926569
\(15\) 2.50827 0.0431754
\(16\) 38.0516 0.594556
\(17\) −32.2212 −0.459694 −0.229847 0.973227i \(-0.573823\pi\)
−0.229847 + 0.973227i \(0.573823\pi\)
\(18\) 28.5340 0.373640
\(19\) −54.9889 −0.663964 −0.331982 0.943286i \(-0.607717\pi\)
−0.331982 + 0.943286i \(0.607717\pi\)
\(20\) 37.1494 0.415343
\(21\) 2.11190 0.0219454
\(22\) −10.9581 −0.106194
\(23\) 136.229 1.23503 0.617517 0.786558i \(-0.288139\pi\)
0.617517 + 0.786558i \(0.288139\pi\)
\(24\) −7.34023 −0.0624299
\(25\) −95.7178 −0.765743
\(26\) −4.78136 −0.0360654
\(27\) 24.9307 0.177700
\(28\) 31.2789 0.211113
\(29\) −87.3472 −0.559309 −0.279654 0.960101i \(-0.590220\pi\)
−0.279654 + 0.960101i \(0.590220\pi\)
\(30\) −2.67203 −0.0162615
\(31\) 27.1204 0.157128 0.0785639 0.996909i \(-0.474967\pi\)
0.0785639 + 0.996909i \(0.474967\pi\)
\(32\) −167.222 −0.923778
\(33\) −4.76801 −0.0251517
\(34\) 34.3250 0.173138
\(35\) 24.6549 0.119070
\(36\) 183.884 0.851316
\(37\) 287.935 1.27936 0.639680 0.768642i \(-0.279067\pi\)
0.639680 + 0.768642i \(0.279067\pi\)
\(38\) 58.5791 0.250073
\(39\) −2.08044 −0.00854197
\(40\) −85.6918 −0.338727
\(41\) 285.555 1.08771 0.543856 0.839179i \(-0.316964\pi\)
0.543856 + 0.839179i \(0.316964\pi\)
\(42\) −2.24979 −0.00826547
\(43\) 0 0
\(44\) −70.6181 −0.241956
\(45\) 144.942 0.480150
\(46\) −145.124 −0.465160
\(47\) −62.1624 −0.192922 −0.0964608 0.995337i \(-0.530752\pi\)
−0.0964608 + 0.995337i \(0.530752\pi\)
\(48\) −17.6378 −0.0530375
\(49\) −322.241 −0.939479
\(50\) 101.967 0.288407
\(51\) 14.9353 0.0410071
\(52\) −30.8129 −0.0821728
\(53\) 310.227 0.804019 0.402010 0.915635i \(-0.368312\pi\)
0.402010 + 0.915635i \(0.368312\pi\)
\(54\) −26.5584 −0.0669286
\(55\) −55.6631 −0.136466
\(56\) −72.1505 −0.172170
\(57\) 25.4886 0.0592290
\(58\) 93.0501 0.210657
\(59\) 561.966 1.24003 0.620015 0.784590i \(-0.287126\pi\)
0.620015 + 0.784590i \(0.287126\pi\)
\(60\) −17.2196 −0.0370507
\(61\) 381.217 0.800162 0.400081 0.916480i \(-0.368982\pi\)
0.400081 + 0.916480i \(0.368982\pi\)
\(62\) −28.8911 −0.0591802
\(63\) 122.038 0.244053
\(64\) −126.273 −0.246627
\(65\) −24.2876 −0.0463462
\(66\) 5.07932 0.00947305
\(67\) −167.929 −0.306206 −0.153103 0.988210i \(-0.548927\pi\)
−0.153103 + 0.988210i \(0.548927\pi\)
\(68\) 221.204 0.394484
\(69\) −63.1455 −0.110171
\(70\) −26.2646 −0.0448460
\(71\) 328.805 0.549605 0.274802 0.961501i \(-0.411388\pi\)
0.274802 + 0.961501i \(0.411388\pi\)
\(72\) −424.162 −0.694277
\(73\) 1034.34 1.65836 0.829178 0.558985i \(-0.188809\pi\)
0.829178 + 0.558985i \(0.188809\pi\)
\(74\) −306.735 −0.481854
\(75\) 44.3675 0.0683082
\(76\) 377.507 0.569777
\(77\) −46.8670 −0.0693635
\(78\) 2.21627 0.00321722
\(79\) −496.972 −0.707768 −0.353884 0.935289i \(-0.615139\pi\)
−0.353884 + 0.935289i \(0.615139\pi\)
\(80\) −205.909 −0.287766
\(81\) 711.643 0.976191
\(82\) −304.199 −0.409672
\(83\) 374.134 0.494777 0.247389 0.968916i \(-0.420428\pi\)
0.247389 + 0.968916i \(0.420428\pi\)
\(84\) −14.4985 −0.0188324
\(85\) 174.359 0.222493
\(86\) 0 0
\(87\) 40.4875 0.0498933
\(88\) 162.893 0.197324
\(89\) −270.772 −0.322492 −0.161246 0.986914i \(-0.551551\pi\)
−0.161246 + 0.986914i \(0.551551\pi\)
\(90\) −154.406 −0.180842
\(91\) −20.4496 −0.0235571
\(92\) −935.236 −1.05984
\(93\) −12.5709 −0.0140166
\(94\) 66.2210 0.0726615
\(95\) 297.561 0.321359
\(96\) 77.5112 0.0824058
\(97\) −1118.35 −1.17063 −0.585314 0.810807i \(-0.699029\pi\)
−0.585314 + 0.810807i \(0.699029\pi\)
\(98\) 343.281 0.353843
\(99\) −275.524 −0.279709
\(100\) 657.118 0.657118
\(101\) 4.71204 0.00464223 0.00232112 0.999997i \(-0.499261\pi\)
0.00232112 + 0.999997i \(0.499261\pi\)
\(102\) −15.9105 −0.0154448
\(103\) 1132.30 1.08319 0.541594 0.840640i \(-0.317821\pi\)
0.541594 + 0.840640i \(0.317821\pi\)
\(104\) 71.0756 0.0670148
\(105\) −11.4281 −0.0106216
\(106\) −330.482 −0.302823
\(107\) −1718.40 −1.55256 −0.776279 0.630389i \(-0.782895\pi\)
−0.776279 + 0.630389i \(0.782895\pi\)
\(108\) −171.153 −0.152493
\(109\) −778.298 −0.683921 −0.341961 0.939714i \(-0.611091\pi\)
−0.341961 + 0.939714i \(0.611091\pi\)
\(110\) 59.2974 0.0513980
\(111\) −133.465 −0.114125
\(112\) −173.370 −0.146267
\(113\) 1039.73 0.865568 0.432784 0.901498i \(-0.357531\pi\)
0.432784 + 0.901498i \(0.357531\pi\)
\(114\) −27.1528 −0.0223078
\(115\) −737.178 −0.597758
\(116\) 599.652 0.479968
\(117\) −120.220 −0.0949944
\(118\) −598.658 −0.467042
\(119\) 146.806 0.113090
\(120\) 39.7202 0.0302162
\(121\) −1225.19 −0.920503
\(122\) −406.107 −0.301371
\(123\) −132.361 −0.0970294
\(124\) −186.186 −0.134838
\(125\) 1194.37 0.854622
\(126\) −130.006 −0.0919195
\(127\) −289.845 −0.202516 −0.101258 0.994860i \(-0.532287\pi\)
−0.101258 + 0.994860i \(0.532287\pi\)
\(128\) 1472.29 1.01667
\(129\) 0 0
\(130\) 25.8733 0.0174557
\(131\) 1123.68 0.749441 0.374720 0.927138i \(-0.377739\pi\)
0.374720 + 0.927138i \(0.377739\pi\)
\(132\) 32.7332 0.0215838
\(133\) 250.539 0.163342
\(134\) 178.893 0.115329
\(135\) −134.907 −0.0860073
\(136\) −510.247 −0.321715
\(137\) −2215.34 −1.38153 −0.690764 0.723080i \(-0.742726\pi\)
−0.690764 + 0.723080i \(0.742726\pi\)
\(138\) 67.2684 0.0414947
\(139\) 1955.65 1.19335 0.596676 0.802482i \(-0.296488\pi\)
0.596676 + 0.802482i \(0.296488\pi\)
\(140\) −169.260 −0.102179
\(141\) 28.8137 0.0172096
\(142\) −350.273 −0.207002
\(143\) 46.1688 0.0269988
\(144\) −1019.22 −0.589825
\(145\) 472.662 0.270706
\(146\) −1101.87 −0.624598
\(147\) 149.366 0.0838064
\(148\) −1976.72 −1.09787
\(149\) 1648.94 0.906618 0.453309 0.891353i \(-0.350243\pi\)
0.453309 + 0.891353i \(0.350243\pi\)
\(150\) −47.2643 −0.0257274
\(151\) 2643.96 1.42492 0.712459 0.701713i \(-0.247581\pi\)
0.712459 + 0.701713i \(0.247581\pi\)
\(152\) −870.788 −0.464673
\(153\) 863.051 0.456036
\(154\) 49.9270 0.0261249
\(155\) −146.757 −0.0760501
\(156\) 14.2825 0.00733024
\(157\) −174.029 −0.0884651 −0.0442326 0.999021i \(-0.514084\pi\)
−0.0442326 + 0.999021i \(0.514084\pi\)
\(158\) 529.419 0.266572
\(159\) −143.798 −0.0717227
\(160\) 904.887 0.447110
\(161\) −620.686 −0.303832
\(162\) −758.107 −0.367670
\(163\) −2791.23 −1.34126 −0.670631 0.741791i \(-0.733977\pi\)
−0.670631 + 0.741791i \(0.733977\pi\)
\(164\) −1960.38 −0.933413
\(165\) 25.8011 0.0121734
\(166\) −398.561 −0.186352
\(167\) 1898.85 0.879865 0.439932 0.898031i \(-0.355002\pi\)
0.439932 + 0.898031i \(0.355002\pi\)
\(168\) 33.4434 0.0153584
\(169\) −2176.86 −0.990831
\(170\) −185.743 −0.0837990
\(171\) 1472.88 0.658680
\(172\) 0 0
\(173\) −1658.22 −0.728740 −0.364370 0.931254i \(-0.618716\pi\)
−0.364370 + 0.931254i \(0.618716\pi\)
\(174\) −43.1309 −0.0187917
\(175\) 436.108 0.188381
\(176\) 391.416 0.167637
\(177\) −260.485 −0.110617
\(178\) 288.451 0.121463
\(179\) −3555.98 −1.48484 −0.742420 0.669934i \(-0.766322\pi\)
−0.742420 + 0.669934i \(0.766322\pi\)
\(180\) −995.052 −0.412038
\(181\) −3643.83 −1.49637 −0.748187 0.663488i \(-0.769075\pi\)
−0.748187 + 0.663488i \(0.769075\pi\)
\(182\) 21.7847 0.00887249
\(183\) −176.703 −0.0713786
\(184\) 2157.29 0.864334
\(185\) −1558.10 −0.619212
\(186\) 13.3917 0.00527918
\(187\) −331.442 −0.129612
\(188\) 426.755 0.165555
\(189\) −113.589 −0.0437163
\(190\) −316.989 −0.121036
\(191\) −3297.38 −1.24916 −0.624581 0.780960i \(-0.714730\pi\)
−0.624581 + 0.780960i \(0.714730\pi\)
\(192\) 58.5305 0.0220004
\(193\) −3679.64 −1.37236 −0.686182 0.727430i \(-0.740715\pi\)
−0.686182 + 0.727430i \(0.740715\pi\)
\(194\) 1191.36 0.440902
\(195\) 11.2579 0.00413432
\(196\) 2212.24 0.806208
\(197\) −2520.85 −0.911692 −0.455846 0.890059i \(-0.650663\pi\)
−0.455846 + 0.890059i \(0.650663\pi\)
\(198\) 293.513 0.105349
\(199\) 4762.22 1.69641 0.848203 0.529672i \(-0.177685\pi\)
0.848203 + 0.529672i \(0.177685\pi\)
\(200\) −1515.76 −0.535902
\(201\) 77.8391 0.0273151
\(202\) −5.01969 −0.00174844
\(203\) 397.970 0.137596
\(204\) −102.533 −0.0351900
\(205\) −1545.22 −0.526454
\(206\) −1206.22 −0.407969
\(207\) −3648.92 −1.22521
\(208\) 170.787 0.0569326
\(209\) −565.640 −0.187206
\(210\) 12.1743 0.00400050
\(211\) 581.138 0.189607 0.0948037 0.995496i \(-0.469778\pi\)
0.0948037 + 0.995496i \(0.469778\pi\)
\(212\) −2129.76 −0.689964
\(213\) −152.409 −0.0490276
\(214\) 1830.59 0.584752
\(215\) 0 0
\(216\) 394.795 0.124363
\(217\) −123.565 −0.0386552
\(218\) 829.114 0.257590
\(219\) −479.439 −0.147934
\(220\) 382.136 0.117107
\(221\) −144.619 −0.0440187
\(222\) 142.179 0.0429839
\(223\) −5483.83 −1.64675 −0.823374 0.567500i \(-0.807911\pi\)
−0.823374 + 0.567500i \(0.807911\pi\)
\(224\) 761.893 0.227260
\(225\) 2563.82 0.759649
\(226\) −1107.61 −0.326005
\(227\) 5427.63 1.58698 0.793490 0.608584i \(-0.208262\pi\)
0.793490 + 0.608584i \(0.208262\pi\)
\(228\) −174.983 −0.0508270
\(229\) 513.813 0.148269 0.0741347 0.997248i \(-0.476381\pi\)
0.0741347 + 0.997248i \(0.476381\pi\)
\(230\) 785.309 0.225138
\(231\) 21.7240 0.00618758
\(232\) −1383.20 −0.391430
\(233\) 748.209 0.210373 0.105186 0.994453i \(-0.466456\pi\)
0.105186 + 0.994453i \(0.466456\pi\)
\(234\) 128.069 0.0357784
\(235\) 336.379 0.0933743
\(236\) −3857.99 −1.06412
\(237\) 230.358 0.0631366
\(238\) −156.391 −0.0425938
\(239\) 5232.24 1.41609 0.708045 0.706167i \(-0.249577\pi\)
0.708045 + 0.706167i \(0.249577\pi\)
\(240\) 95.4435 0.0256702
\(241\) −3231.66 −0.863774 −0.431887 0.901928i \(-0.642152\pi\)
−0.431887 + 0.901928i \(0.642152\pi\)
\(242\) 1305.18 0.346696
\(243\) −1002.99 −0.264782
\(244\) −2617.12 −0.686655
\(245\) 1743.74 0.454709
\(246\) 141.003 0.0365449
\(247\) −246.807 −0.0635788
\(248\) 429.471 0.109965
\(249\) −173.420 −0.0441367
\(250\) −1272.35 −0.321882
\(251\) −2668.67 −0.671094 −0.335547 0.942023i \(-0.608921\pi\)
−0.335547 + 0.942023i \(0.608921\pi\)
\(252\) −837.810 −0.209433
\(253\) 1401.32 0.348221
\(254\) 308.769 0.0762752
\(255\) −80.8194 −0.0198475
\(256\) −558.235 −0.136288
\(257\) 4140.36 1.00494 0.502469 0.864595i \(-0.332425\pi\)
0.502469 + 0.864595i \(0.332425\pi\)
\(258\) 0 0
\(259\) −1311.89 −0.314736
\(260\) 166.738 0.0397718
\(261\) 2339.61 0.554858
\(262\) −1197.05 −0.282267
\(263\) 5040.77 1.18185 0.590926 0.806726i \(-0.298763\pi\)
0.590926 + 0.806726i \(0.298763\pi\)
\(264\) −75.5049 −0.0176023
\(265\) −1678.73 −0.389146
\(266\) −266.897 −0.0615208
\(267\) 125.509 0.0287680
\(268\) 1152.86 0.262769
\(269\) −6767.05 −1.53381 −0.766904 0.641762i \(-0.778204\pi\)
−0.766904 + 0.641762i \(0.778204\pi\)
\(270\) 143.716 0.0323935
\(271\) −7973.39 −1.78727 −0.893633 0.448799i \(-0.851852\pi\)
−0.893633 + 0.448799i \(0.851852\pi\)
\(272\) −1226.07 −0.273314
\(273\) 9.47886 0.00210142
\(274\) 2359.98 0.520335
\(275\) −984.597 −0.215903
\(276\) 433.504 0.0945430
\(277\) −1476.35 −0.320236 −0.160118 0.987098i \(-0.551187\pi\)
−0.160118 + 0.987098i \(0.551187\pi\)
\(278\) −2083.34 −0.449461
\(279\) −726.423 −0.155878
\(280\) 390.428 0.0833304
\(281\) −6880.43 −1.46068 −0.730341 0.683082i \(-0.760639\pi\)
−0.730341 + 0.683082i \(0.760639\pi\)
\(282\) −30.6950 −0.00648178
\(283\) −275.458 −0.0578597 −0.0289299 0.999581i \(-0.509210\pi\)
−0.0289299 + 0.999581i \(0.509210\pi\)
\(284\) −2257.30 −0.471640
\(285\) −137.927 −0.0286669
\(286\) −49.1832 −0.0101687
\(287\) −1301.04 −0.267589
\(288\) 4479.06 0.916427
\(289\) −3874.79 −0.788681
\(290\) −503.522 −0.101958
\(291\) 518.380 0.104426
\(292\) −7100.88 −1.42311
\(293\) −4500.88 −0.897420 −0.448710 0.893678i \(-0.648116\pi\)
−0.448710 + 0.893678i \(0.648116\pi\)
\(294\) −159.119 −0.0315646
\(295\) −3040.97 −0.600176
\(296\) 4559.66 0.895355
\(297\) 256.448 0.0501032
\(298\) −1756.60 −0.341466
\(299\) 611.439 0.118262
\(300\) −304.590 −0.0586183
\(301\) 0 0
\(302\) −2816.59 −0.536677
\(303\) −2.18414 −0.000414111 0
\(304\) −2092.41 −0.394764
\(305\) −2062.88 −0.387279
\(306\) −919.400 −0.171760
\(307\) −8303.21 −1.54361 −0.771806 0.635858i \(-0.780646\pi\)
−0.771806 + 0.635858i \(0.780646\pi\)
\(308\) 321.749 0.0595239
\(309\) −524.846 −0.0966260
\(310\) 156.338 0.0286433
\(311\) 9034.89 1.64734 0.823668 0.567073i \(-0.191924\pi\)
0.823668 + 0.567073i \(0.191924\pi\)
\(312\) −32.9452 −0.00597806
\(313\) 4408.24 0.796065 0.398032 0.917371i \(-0.369693\pi\)
0.398032 + 0.917371i \(0.369693\pi\)
\(314\) 185.392 0.0333193
\(315\) −660.385 −0.118122
\(316\) 3411.79 0.607367
\(317\) −1062.85 −0.188314 −0.0941569 0.995557i \(-0.530016\pi\)
−0.0941569 + 0.995557i \(0.530016\pi\)
\(318\) 153.186 0.0270134
\(319\) −898.492 −0.157699
\(320\) 683.301 0.119368
\(321\) 796.518 0.138496
\(322\) 661.211 0.114434
\(323\) 1771.81 0.305220
\(324\) −4885.54 −0.837713
\(325\) −429.611 −0.0733248
\(326\) 2973.47 0.505170
\(327\) 360.760 0.0610093
\(328\) 4521.96 0.761231
\(329\) 283.223 0.0474608
\(330\) −27.4857 −0.00458497
\(331\) 9875.05 1.63982 0.819912 0.572489i \(-0.194022\pi\)
0.819912 + 0.572489i \(0.194022\pi\)
\(332\) −2568.49 −0.424590
\(333\) −7712.39 −1.26918
\(334\) −2022.83 −0.331390
\(335\) 908.714 0.148204
\(336\) 80.3612 0.0130478
\(337\) −4547.10 −0.735004 −0.367502 0.930023i \(-0.619787\pi\)
−0.367502 + 0.930023i \(0.619787\pi\)
\(338\) 2318.98 0.373184
\(339\) −481.937 −0.0772131
\(340\) −1197.00 −0.190931
\(341\) 278.972 0.0443027
\(342\) −1569.05 −0.248084
\(343\) 3030.96 0.477133
\(344\) 0 0
\(345\) 341.699 0.0533231
\(346\) 1766.49 0.274471
\(347\) −10601.8 −1.64016 −0.820081 0.572247i \(-0.806072\pi\)
−0.820081 + 0.572247i \(0.806072\pi\)
\(348\) −277.953 −0.0428156
\(349\) 10990.8 1.68574 0.842870 0.538117i \(-0.180864\pi\)
0.842870 + 0.538117i \(0.180864\pi\)
\(350\) −464.582 −0.0709513
\(351\) 111.897 0.0170160
\(352\) −1720.12 −0.260462
\(353\) −290.232 −0.0437606 −0.0218803 0.999761i \(-0.506965\pi\)
−0.0218803 + 0.999761i \(0.506965\pi\)
\(354\) 277.492 0.0416625
\(355\) −1779.26 −0.266010
\(356\) 1858.89 0.276745
\(357\) −68.0480 −0.0100882
\(358\) 3788.15 0.559246
\(359\) 3441.15 0.505896 0.252948 0.967480i \(-0.418600\pi\)
0.252948 + 0.967480i \(0.418600\pi\)
\(360\) 2295.27 0.336031
\(361\) −3835.22 −0.559152
\(362\) 3881.74 0.563590
\(363\) 567.904 0.0821136
\(364\) 140.389 0.0202154
\(365\) −5597.10 −0.802646
\(366\) 188.240 0.0268838
\(367\) 6785.05 0.965060 0.482530 0.875880i \(-0.339718\pi\)
0.482530 + 0.875880i \(0.339718\pi\)
\(368\) 5183.74 0.734297
\(369\) −7648.62 −1.07906
\(370\) 1659.83 0.233218
\(371\) −1413.45 −0.197797
\(372\) 86.3014 0.0120283
\(373\) −8819.35 −1.22426 −0.612129 0.790758i \(-0.709687\pi\)
−0.612129 + 0.790758i \(0.709687\pi\)
\(374\) 353.082 0.0488167
\(375\) −553.619 −0.0762367
\(376\) −984.386 −0.135016
\(377\) −392.041 −0.0535574
\(378\) 121.005 0.0164652
\(379\) 2684.12 0.363784 0.181892 0.983319i \(-0.441778\pi\)
0.181892 + 0.983319i \(0.441778\pi\)
\(380\) −2042.80 −0.275773
\(381\) 134.350 0.0180655
\(382\) 3512.67 0.470481
\(383\) −4907.07 −0.654673 −0.327336 0.944908i \(-0.606151\pi\)
−0.327336 + 0.944908i \(0.606151\pi\)
\(384\) −682.442 −0.0906920
\(385\) 253.611 0.0335720
\(386\) 3919.89 0.516884
\(387\) 0 0
\(388\) 7677.62 1.00457
\(389\) 1999.95 0.260672 0.130336 0.991470i \(-0.458394\pi\)
0.130336 + 0.991470i \(0.458394\pi\)
\(390\) −11.9929 −0.00155714
\(391\) −4389.48 −0.567738
\(392\) −5102.92 −0.657491
\(393\) −520.854 −0.0668540
\(394\) 2685.44 0.343377
\(395\) 2689.26 0.342561
\(396\) 1891.52 0.240031
\(397\) 9540.63 1.20612 0.603061 0.797695i \(-0.293947\pi\)
0.603061 + 0.797695i \(0.293947\pi\)
\(398\) −5073.15 −0.638929
\(399\) −116.131 −0.0145710
\(400\) −3642.22 −0.455277
\(401\) 6723.57 0.837304 0.418652 0.908147i \(-0.362503\pi\)
0.418652 + 0.908147i \(0.362503\pi\)
\(402\) −82.9213 −0.0102879
\(403\) 121.725 0.0150460
\(404\) −32.3489 −0.00398371
\(405\) −3850.91 −0.472478
\(406\) −423.954 −0.0518238
\(407\) 2961.83 0.360719
\(408\) 236.511 0.0286987
\(409\) 3107.78 0.375720 0.187860 0.982196i \(-0.439845\pi\)
0.187860 + 0.982196i \(0.439845\pi\)
\(410\) 1646.11 0.198282
\(411\) 1026.86 0.123239
\(412\) −7773.38 −0.929532
\(413\) −2560.42 −0.305061
\(414\) 3887.16 0.461458
\(415\) −2024.55 −0.239473
\(416\) −750.543 −0.0884577
\(417\) −906.490 −0.106453
\(418\) 602.571 0.0705089
\(419\) −16291.9 −1.89955 −0.949776 0.312931i \(-0.898689\pi\)
−0.949776 + 0.312931i \(0.898689\pi\)
\(420\) 78.4558 0.00911489
\(421\) −2863.80 −0.331528 −0.165764 0.986165i \(-0.553009\pi\)
−0.165764 + 0.986165i \(0.553009\pi\)
\(422\) −619.081 −0.0714132
\(423\) 1665.03 0.191386
\(424\) 4912.67 0.562690
\(425\) 3084.15 0.352007
\(426\) 162.360 0.0184656
\(427\) −1736.90 −0.196849
\(428\) 11797.1 1.33232
\(429\) −21.4003 −0.00240843
\(430\) 0 0
\(431\) −4365.79 −0.487918 −0.243959 0.969786i \(-0.578446\pi\)
−0.243959 + 0.969786i \(0.578446\pi\)
\(432\) 948.653 0.105653
\(433\) −1671.29 −0.185489 −0.0927447 0.995690i \(-0.529564\pi\)
−0.0927447 + 0.995690i \(0.529564\pi\)
\(434\) 131.633 0.0145590
\(435\) −219.090 −0.0241484
\(436\) 5343.14 0.586903
\(437\) −7491.10 −0.820018
\(438\) 510.742 0.0557174
\(439\) −5823.57 −0.633129 −0.316564 0.948571i \(-0.602529\pi\)
−0.316564 + 0.948571i \(0.602529\pi\)
\(440\) −881.464 −0.0955049
\(441\) 8631.28 0.932003
\(442\) 154.061 0.0165791
\(443\) −1022.42 −0.109654 −0.0548270 0.998496i \(-0.517461\pi\)
−0.0548270 + 0.998496i \(0.517461\pi\)
\(444\) 916.257 0.0979361
\(445\) 1465.23 0.156087
\(446\) 5841.88 0.620226
\(447\) −764.321 −0.0808750
\(448\) 575.323 0.0606729
\(449\) −9630.11 −1.01219 −0.506095 0.862478i \(-0.668911\pi\)
−0.506095 + 0.862478i \(0.668911\pi\)
\(450\) −2731.21 −0.286112
\(451\) 2937.34 0.306683
\(452\) −7137.88 −0.742782
\(453\) −1225.54 −0.127110
\(454\) −5782.00 −0.597715
\(455\) 110.659 0.0114017
\(456\) 403.631 0.0414512
\(457\) −5577.69 −0.570926 −0.285463 0.958390i \(-0.592147\pi\)
−0.285463 + 0.958390i \(0.592147\pi\)
\(458\) −547.360 −0.0558438
\(459\) −803.298 −0.0816879
\(460\) 5060.84 0.512963
\(461\) −4598.29 −0.464563 −0.232282 0.972649i \(-0.574619\pi\)
−0.232282 + 0.972649i \(0.574619\pi\)
\(462\) −23.1423 −0.00233047
\(463\) 463.742 0.0465484 0.0232742 0.999729i \(-0.492591\pi\)
0.0232742 + 0.999729i \(0.492591\pi\)
\(464\) −3323.70 −0.332541
\(465\) 68.0251 0.00678406
\(466\) −797.060 −0.0792341
\(467\) −1238.06 −0.122678 −0.0613388 0.998117i \(-0.519537\pi\)
−0.0613388 + 0.998117i \(0.519537\pi\)
\(468\) 825.329 0.0815189
\(469\) 765.116 0.0753300
\(470\) −358.342 −0.0351683
\(471\) 80.6666 0.00789155
\(472\) 8899.14 0.867831
\(473\) 0 0
\(474\) −245.398 −0.0237796
\(475\) 5263.42 0.508425
\(476\) −1007.85 −0.0970474
\(477\) −8309.49 −0.797621
\(478\) −5573.86 −0.533352
\(479\) 6374.43 0.608048 0.304024 0.952664i \(-0.401670\pi\)
0.304024 + 0.952664i \(0.401670\pi\)
\(480\) −419.437 −0.0398845
\(481\) 1292.34 0.122507
\(482\) 3442.66 0.325330
\(483\) 287.703 0.0271034
\(484\) 8411.11 0.789924
\(485\) 6051.71 0.566586
\(486\) 1068.48 0.0997267
\(487\) −8076.69 −0.751519 −0.375759 0.926717i \(-0.622618\pi\)
−0.375759 + 0.926717i \(0.622618\pi\)
\(488\) 6036.85 0.559990
\(489\) 1293.80 0.119648
\(490\) −1857.59 −0.171260
\(491\) 10183.8 0.936028 0.468014 0.883721i \(-0.344970\pi\)
0.468014 + 0.883721i \(0.344970\pi\)
\(492\) 908.681 0.0832653
\(493\) 2814.43 0.257111
\(494\) 262.921 0.0239461
\(495\) 1490.94 0.135380
\(496\) 1031.97 0.0934214
\(497\) −1498.10 −0.135209
\(498\) 184.743 0.0166235
\(499\) −9449.18 −0.847702 −0.423851 0.905732i \(-0.639322\pi\)
−0.423851 + 0.905732i \(0.639322\pi\)
\(500\) −8199.54 −0.733389
\(501\) −880.162 −0.0784885
\(502\) 2842.91 0.252759
\(503\) −14140.8 −1.25349 −0.626747 0.779223i \(-0.715614\pi\)
−0.626747 + 0.779223i \(0.715614\pi\)
\(504\) 1932.56 0.170800
\(505\) −25.4983 −0.00224685
\(506\) −1492.81 −0.131153
\(507\) 1009.02 0.0883872
\(508\) 1989.83 0.173788
\(509\) −192.153 −0.0167329 −0.00836643 0.999965i \(-0.502663\pi\)
−0.00836643 + 0.999965i \(0.502663\pi\)
\(510\) 86.0962 0.00747530
\(511\) −4712.63 −0.407973
\(512\) −11183.6 −0.965336
\(513\) −1370.91 −0.117987
\(514\) −4410.69 −0.378497
\(515\) −6127.19 −0.524265
\(516\) 0 0
\(517\) −639.430 −0.0543948
\(518\) 1397.54 0.118541
\(519\) 768.624 0.0650074
\(520\) −384.611 −0.0324352
\(521\) 10001.7 0.841038 0.420519 0.907284i \(-0.361848\pi\)
0.420519 + 0.907284i \(0.361848\pi\)
\(522\) −2492.36 −0.208980
\(523\) 1686.23 0.140982 0.0704912 0.997512i \(-0.477543\pi\)
0.0704912 + 0.997512i \(0.477543\pi\)
\(524\) −7714.27 −0.643128
\(525\) −202.146 −0.0168046
\(526\) −5369.88 −0.445129
\(527\) −873.853 −0.0722308
\(528\) −181.431 −0.0149541
\(529\) 6391.44 0.525309
\(530\) 1788.34 0.146567
\(531\) −15052.3 −1.23016
\(532\) −1719.99 −0.140171
\(533\) 1281.66 0.104155
\(534\) −133.704 −0.0108351
\(535\) 9298.77 0.751441
\(536\) −2659.28 −0.214297
\(537\) 1648.28 0.132455
\(538\) 7208.87 0.577689
\(539\) −3314.72 −0.264889
\(540\) 926.160 0.0738067
\(541\) 4269.53 0.339300 0.169650 0.985504i \(-0.445736\pi\)
0.169650 + 0.985504i \(0.445736\pi\)
\(542\) 8493.98 0.673151
\(543\) 1689.00 0.133484
\(544\) 5388.09 0.424656
\(545\) 4211.60 0.331019
\(546\) −10.0977 −0.000791472 0
\(547\) −5089.79 −0.397849 −0.198925 0.980015i \(-0.563745\pi\)
−0.198925 + 0.980015i \(0.563745\pi\)
\(548\) 15208.7 1.18555
\(549\) −10211.0 −0.793795
\(550\) 1048.88 0.0813172
\(551\) 4803.12 0.371361
\(552\) −999.955 −0.0771031
\(553\) 2264.29 0.174119
\(554\) 1572.74 0.120613
\(555\) 722.218 0.0552369
\(556\) −13425.8 −1.02407
\(557\) −211.814 −0.0161128 −0.00805641 0.999968i \(-0.502564\pi\)
−0.00805641 + 0.999968i \(0.502564\pi\)
\(558\) 773.852 0.0587093
\(559\) 0 0
\(560\) 938.158 0.0707936
\(561\) 153.631 0.0115621
\(562\) 7329.66 0.550148
\(563\) 9751.04 0.729942 0.364971 0.931019i \(-0.381079\pi\)
0.364971 + 0.931019i \(0.381079\pi\)
\(564\) −197.811 −0.0147683
\(565\) −5626.27 −0.418936
\(566\) 293.443 0.0217921
\(567\) −3242.38 −0.240154
\(568\) 5206.86 0.384639
\(569\) −13596.2 −1.00173 −0.500865 0.865526i \(-0.666985\pi\)
−0.500865 + 0.865526i \(0.666985\pi\)
\(570\) 146.932 0.0107970
\(571\) −5131.18 −0.376065 −0.188033 0.982163i \(-0.560211\pi\)
−0.188033 + 0.982163i \(0.560211\pi\)
\(572\) −316.956 −0.0231689
\(573\) 1528.41 0.111432
\(574\) 1385.99 0.100784
\(575\) −13039.6 −0.945718
\(576\) 3382.24 0.244664
\(577\) −20274.6 −1.46281 −0.731405 0.681944i \(-0.761135\pi\)
−0.731405 + 0.681944i \(0.761135\pi\)
\(578\) 4127.78 0.297047
\(579\) 1705.60 0.122422
\(580\) −3244.90 −0.232305
\(581\) −1704.62 −0.121721
\(582\) −552.226 −0.0393307
\(583\) 3191.14 0.226695
\(584\) 16379.5 1.16059
\(585\) 650.547 0.0459774
\(586\) 4794.74 0.338002
\(587\) −21105.5 −1.48402 −0.742009 0.670390i \(-0.766127\pi\)
−0.742009 + 0.670390i \(0.766127\pi\)
\(588\) −1025.42 −0.0719180
\(589\) −1491.32 −0.104327
\(590\) 3239.52 0.226049
\(591\) 1168.47 0.0813276
\(592\) 10956.4 0.760651
\(593\) 11899.5 0.824035 0.412017 0.911176i \(-0.364824\pi\)
0.412017 + 0.911176i \(0.364824\pi\)
\(594\) −273.192 −0.0188707
\(595\) −794.411 −0.0547356
\(596\) −11320.2 −0.778009
\(597\) −2207.40 −0.151328
\(598\) −651.361 −0.0445420
\(599\) 25065.9 1.70979 0.854894 0.518802i \(-0.173622\pi\)
0.854894 + 0.518802i \(0.173622\pi\)
\(600\) 702.591 0.0478053
\(601\) −24254.1 −1.64616 −0.823081 0.567924i \(-0.807747\pi\)
−0.823081 + 0.567924i \(0.807747\pi\)
\(602\) 0 0
\(603\) 4498.00 0.303769
\(604\) −18151.2 −1.22279
\(605\) 6629.86 0.445525
\(606\) 2.32675 0.000155970 0
\(607\) −6830.29 −0.456726 −0.228363 0.973576i \(-0.573337\pi\)
−0.228363 + 0.973576i \(0.573337\pi\)
\(608\) 9195.34 0.613355
\(609\) −184.468 −0.0122743
\(610\) 2197.57 0.145864
\(611\) −279.004 −0.0184735
\(612\) −5924.98 −0.391345
\(613\) −9630.02 −0.634507 −0.317253 0.948341i \(-0.602761\pi\)
−0.317253 + 0.948341i \(0.602761\pi\)
\(614\) 8845.33 0.581382
\(615\) 716.247 0.0469624
\(616\) −742.172 −0.0485438
\(617\) 2856.28 0.186369 0.0931844 0.995649i \(-0.470295\pi\)
0.0931844 + 0.995649i \(0.470295\pi\)
\(618\) 559.113 0.0363929
\(619\) 16879.5 1.09603 0.548016 0.836468i \(-0.315383\pi\)
0.548016 + 0.836468i \(0.315383\pi\)
\(620\) 1007.51 0.0652620
\(621\) 3396.29 0.219466
\(622\) −9624.78 −0.620448
\(623\) 1233.69 0.0793366
\(624\) −79.1640 −0.00507868
\(625\) 5501.63 0.352105
\(626\) −4696.06 −0.299828
\(627\) 262.188 0.0166998
\(628\) 1194.74 0.0759159
\(629\) −9277.64 −0.588114
\(630\) 703.502 0.0444892
\(631\) 19957.6 1.25911 0.629555 0.776956i \(-0.283237\pi\)
0.629555 + 0.776956i \(0.283237\pi\)
\(632\) −7869.90 −0.495329
\(633\) −269.371 −0.0169140
\(634\) 1132.24 0.0709260
\(635\) 1568.44 0.0980182
\(636\) 987.194 0.0615484
\(637\) −1446.32 −0.0899611
\(638\) 957.156 0.0593952
\(639\) −8807.08 −0.545231
\(640\) −7967.01 −0.492068
\(641\) 22207.7 1.36841 0.684204 0.729290i \(-0.260150\pi\)
0.684204 + 0.729290i \(0.260150\pi\)
\(642\) −848.524 −0.0521629
\(643\) −19400.5 −1.18986 −0.594932 0.803776i \(-0.702821\pi\)
−0.594932 + 0.803776i \(0.702821\pi\)
\(644\) 4261.11 0.260732
\(645\) 0 0
\(646\) −1887.49 −0.114957
\(647\) 5574.01 0.338697 0.169349 0.985556i \(-0.445834\pi\)
0.169349 + 0.985556i \(0.445834\pi\)
\(648\) 11269.4 0.683183
\(649\) 5780.64 0.349630
\(650\) 457.661 0.0276168
\(651\) 57.2755 0.00344824
\(652\) 19162.2 1.15100
\(653\) 16231.9 0.972747 0.486374 0.873751i \(-0.338319\pi\)
0.486374 + 0.873751i \(0.338319\pi\)
\(654\) −384.314 −0.0229784
\(655\) −6080.59 −0.362730
\(656\) 10865.8 0.646705
\(657\) −27704.9 −1.64516
\(658\) −301.715 −0.0178755
\(659\) −21111.1 −1.24791 −0.623955 0.781461i \(-0.714475\pi\)
−0.623955 + 0.781461i \(0.714475\pi\)
\(660\) −177.129 −0.0104466
\(661\) −12591.6 −0.740935 −0.370467 0.928845i \(-0.620802\pi\)
−0.370467 + 0.928845i \(0.620802\pi\)
\(662\) −10519.8 −0.617619
\(663\) 67.0343 0.00392669
\(664\) 5924.68 0.346268
\(665\) −1355.74 −0.0790579
\(666\) 8215.94 0.478020
\(667\) −11899.2 −0.690766
\(668\) −13035.9 −0.755051
\(669\) 2541.89 0.146898
\(670\) −968.045 −0.0558192
\(671\) 3921.37 0.225608
\(672\) −353.156 −0.0202727
\(673\) 13408.2 0.767979 0.383989 0.923337i \(-0.374550\pi\)
0.383989 + 0.923337i \(0.374550\pi\)
\(674\) 4843.98 0.276830
\(675\) −2386.31 −0.136073
\(676\) 14944.4 0.850276
\(677\) 32507.8 1.84546 0.922731 0.385445i \(-0.125952\pi\)
0.922731 + 0.385445i \(0.125952\pi\)
\(678\) 513.404 0.0290813
\(679\) 5095.40 0.287987
\(680\) 2761.10 0.155711
\(681\) −2515.83 −0.141567
\(682\) −297.187 −0.0166860
\(683\) −14676.1 −0.822203 −0.411101 0.911590i \(-0.634856\pi\)
−0.411101 + 0.911590i \(0.634856\pi\)
\(684\) −10111.6 −0.565243
\(685\) 11987.9 0.668661
\(686\) −3228.86 −0.179706
\(687\) −238.164 −0.0132264
\(688\) 0 0
\(689\) 1392.40 0.0769900
\(690\) −364.009 −0.0200835
\(691\) 2412.12 0.132795 0.0663974 0.997793i \(-0.478849\pi\)
0.0663974 + 0.997793i \(0.478849\pi\)
\(692\) 11383.9 0.625364
\(693\) 1255.34 0.0688115
\(694\) 11294.0 0.617746
\(695\) −10582.6 −0.577584
\(696\) 641.148 0.0349176
\(697\) −9200.93 −0.500014
\(698\) −11708.4 −0.634913
\(699\) −346.813 −0.0187663
\(700\) −2993.95 −0.161658
\(701\) 13310.0 0.717136 0.358568 0.933504i \(-0.383265\pi\)
0.358568 + 0.933504i \(0.383265\pi\)
\(702\) −119.202 −0.00640884
\(703\) −15833.2 −0.849448
\(704\) −1298.90 −0.0695372
\(705\) −155.920 −0.00832947
\(706\) 309.182 0.0164819
\(707\) −21.4689 −0.00114204
\(708\) 1788.27 0.0949254
\(709\) 31617.1 1.67476 0.837381 0.546620i \(-0.184086\pi\)
0.837381 + 0.546620i \(0.184086\pi\)
\(710\) 1895.43 0.100189
\(711\) 13311.5 0.702136
\(712\) −4287.87 −0.225695
\(713\) 3694.59 0.194058
\(714\) 72.4910 0.00379959
\(715\) −249.833 −0.0130674
\(716\) 24412.4 1.27421
\(717\) −2425.27 −0.126323
\(718\) −3665.82 −0.190539
\(719\) 15564.1 0.807294 0.403647 0.914915i \(-0.367742\pi\)
0.403647 + 0.914915i \(0.367742\pi\)
\(720\) 5515.29 0.285476
\(721\) −5158.95 −0.266476
\(722\) 4085.63 0.210597
\(723\) 1497.95 0.0770531
\(724\) 25015.4 1.28410
\(725\) 8360.68 0.428287
\(726\) −604.983 −0.0309270
\(727\) 29265.4 1.49298 0.746488 0.665399i \(-0.231739\pi\)
0.746488 + 0.665399i \(0.231739\pi\)
\(728\) −323.834 −0.0164864
\(729\) −18749.4 −0.952571
\(730\) 5962.54 0.302306
\(731\) 0 0
\(732\) 1213.10 0.0612531
\(733\) −17131.8 −0.863270 −0.431635 0.902048i \(-0.642063\pi\)
−0.431635 + 0.902048i \(0.642063\pi\)
\(734\) −7228.05 −0.363477
\(735\) −808.266 −0.0405624
\(736\) −22780.5 −1.14090
\(737\) −1727.39 −0.0863356
\(738\) 8148.01 0.406412
\(739\) 16971.4 0.844794 0.422397 0.906411i \(-0.361189\pi\)
0.422397 + 0.906411i \(0.361189\pi\)
\(740\) 10696.6 0.531373
\(741\) 114.401 0.00567156
\(742\) 1505.74 0.0744979
\(743\) −18171.5 −0.897238 −0.448619 0.893723i \(-0.648084\pi\)
−0.448619 + 0.893723i \(0.648084\pi\)
\(744\) −199.070 −0.00980948
\(745\) −8922.88 −0.438804
\(746\) 9395.17 0.461101
\(747\) −10021.2 −0.490840
\(748\) 2275.40 0.111226
\(749\) 7829.34 0.381946
\(750\) 589.765 0.0287136
\(751\) 13978.9 0.679223 0.339612 0.940566i \(-0.389704\pi\)
0.339612 + 0.940566i \(0.389704\pi\)
\(752\) −2365.38 −0.114703
\(753\) 1236.99 0.0598651
\(754\) 417.638 0.0201717
\(755\) −14307.3 −0.689663
\(756\) 779.805 0.0375149
\(757\) 19774.5 0.949426 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(758\) −2859.37 −0.137015
\(759\) −649.543 −0.0310631
\(760\) 4712.10 0.224902
\(761\) −34351.4 −1.63632 −0.818158 0.574994i \(-0.805004\pi\)
−0.818158 + 0.574994i \(0.805004\pi\)
\(762\) −143.122 −0.00680415
\(763\) 3546.07 0.168252
\(764\) 22637.0 1.07196
\(765\) −4670.23 −0.220722
\(766\) 5227.46 0.246574
\(767\) 2522.28 0.118741
\(768\) 258.755 0.0121576
\(769\) −19063.9 −0.893969 −0.446984 0.894542i \(-0.647502\pi\)
−0.446984 + 0.894542i \(0.647502\pi\)
\(770\) −270.170 −0.0126445
\(771\) −1919.16 −0.0896456
\(772\) 25261.3 1.17769
\(773\) −36197.4 −1.68426 −0.842129 0.539276i \(-0.818698\pi\)
−0.842129 + 0.539276i \(0.818698\pi\)
\(774\) 0 0
\(775\) −2595.90 −0.120320
\(776\) −17709.8 −0.819260
\(777\) 608.091 0.0280761
\(778\) −2130.53 −0.0981789
\(779\) −15702.3 −0.722201
\(780\) −77.2870 −0.00354785
\(781\) 3382.23 0.154963
\(782\) 4676.07 0.213831
\(783\) −2177.62 −0.0993895
\(784\) −12261.8 −0.558573
\(785\) 941.723 0.0428172
\(786\) 554.861 0.0251797
\(787\) 18293.2 0.828567 0.414284 0.910148i \(-0.364032\pi\)
0.414284 + 0.910148i \(0.364032\pi\)
\(788\) 17306.0 0.782363
\(789\) −2336.51 −0.105427
\(790\) −2864.85 −0.129021
\(791\) −4737.18 −0.212939
\(792\) −4363.12 −0.195753
\(793\) 1711.02 0.0766206
\(794\) −10163.6 −0.454271
\(795\) 778.133 0.0347139
\(796\) −32693.4 −1.45576
\(797\) 31454.9 1.39798 0.698989 0.715133i \(-0.253634\pi\)
0.698989 + 0.715133i \(0.253634\pi\)
\(798\) 123.713 0.00548797
\(799\) 2002.95 0.0886850
\(800\) 16006.1 0.707377
\(801\) 7252.67 0.319926
\(802\) −7162.56 −0.315360
\(803\) 10639.7 0.467578
\(804\) −534.377 −0.0234403
\(805\) 3358.72 0.147055
\(806\) −129.672 −0.00566688
\(807\) 3136.69 0.136824
\(808\) 74.6185 0.00324885
\(809\) 25771.0 1.11998 0.559989 0.828500i \(-0.310805\pi\)
0.559989 + 0.828500i \(0.310805\pi\)
\(810\) 4102.34 0.177953
\(811\) 33599.7 1.45480 0.727402 0.686212i \(-0.240728\pi\)
0.727402 + 0.686212i \(0.240728\pi\)
\(812\) −2732.12 −0.118077
\(813\) 3695.86 0.159433
\(814\) −3155.21 −0.135860
\(815\) 15104.2 0.649173
\(816\) 568.312 0.0243810
\(817\) 0 0
\(818\) −3310.69 −0.141510
\(819\) 547.745 0.0233697
\(820\) 10608.2 0.451773
\(821\) 36051.3 1.53252 0.766260 0.642531i \(-0.222116\pi\)
0.766260 + 0.642531i \(0.222116\pi\)
\(822\) −1093.91 −0.0464166
\(823\) 5404.48 0.228905 0.114452 0.993429i \(-0.463489\pi\)
0.114452 + 0.993429i \(0.463489\pi\)
\(824\) 17930.7 0.758065
\(825\) 456.384 0.0192597
\(826\) 2727.59 0.114897
\(827\) −19906.0 −0.837000 −0.418500 0.908217i \(-0.637444\pi\)
−0.418500 + 0.908217i \(0.637444\pi\)
\(828\) 25050.4 1.05140
\(829\) −25969.3 −1.08800 −0.543999 0.839086i \(-0.683090\pi\)
−0.543999 + 0.839086i \(0.683090\pi\)
\(830\) 2156.74 0.0901944
\(831\) 684.324 0.0285667
\(832\) −566.752 −0.0236161
\(833\) 10383.0 0.431873
\(834\) 965.676 0.0400943
\(835\) −10275.2 −0.425856
\(836\) 3883.21 0.160650
\(837\) 676.130 0.0279217
\(838\) 17355.6 0.715442
\(839\) 26178.3 1.07721 0.538603 0.842560i \(-0.318952\pi\)
0.538603 + 0.842560i \(0.318952\pi\)
\(840\) −180.973 −0.00743351
\(841\) −16759.5 −0.687174
\(842\) 3050.78 0.124866
\(843\) 3189.24 0.130300
\(844\) −3989.60 −0.162711
\(845\) 11779.6 0.479563
\(846\) −1773.74 −0.0720833
\(847\) 5582.19 0.226454
\(848\) 11804.6 0.478034
\(849\) 127.681 0.00516139
\(850\) −3285.52 −0.132579
\(851\) 39225.2 1.58005
\(852\) 1046.31 0.0420728
\(853\) 45066.9 1.80898 0.904490 0.426495i \(-0.140252\pi\)
0.904490 + 0.426495i \(0.140252\pi\)
\(854\) 1850.30 0.0741405
\(855\) −7970.22 −0.318802
\(856\) −27212.1 −1.08655
\(857\) 20163.4 0.803695 0.401848 0.915706i \(-0.368368\pi\)
0.401848 + 0.915706i \(0.368368\pi\)
\(858\) 22.7976 0.000907105 0
\(859\) −38656.8 −1.53545 −0.767726 0.640778i \(-0.778612\pi\)
−0.767726 + 0.640778i \(0.778612\pi\)
\(860\) 0 0
\(861\) 603.063 0.0238703
\(862\) 4650.83 0.183768
\(863\) −11938.0 −0.470886 −0.235443 0.971888i \(-0.575654\pi\)
−0.235443 + 0.971888i \(0.575654\pi\)
\(864\) −4168.95 −0.164156
\(865\) 8973.12 0.352711
\(866\) 1780.41 0.0698622
\(867\) 1796.06 0.0703544
\(868\) 848.296 0.0331717
\(869\) −5112.07 −0.199557
\(870\) 233.394 0.00909519
\(871\) −753.717 −0.0293212
\(872\) −12324.9 −0.478640
\(873\) 29955.1 1.16131
\(874\) 7980.20 0.308849
\(875\) −5441.77 −0.210246
\(876\) 3291.43 0.126949
\(877\) 898.663 0.0346017 0.0173008 0.999850i \(-0.494493\pi\)
0.0173008 + 0.999850i \(0.494493\pi\)
\(878\) 6203.79 0.238460
\(879\) 2086.26 0.0800545
\(880\) −2118.07 −0.0811364
\(881\) −4887.81 −0.186918 −0.0934588 0.995623i \(-0.529792\pi\)
−0.0934588 + 0.995623i \(0.529792\pi\)
\(882\) −9194.82 −0.351027
\(883\) 27593.4 1.05163 0.525816 0.850598i \(-0.323760\pi\)
0.525816 + 0.850598i \(0.323760\pi\)
\(884\) 992.831 0.0377744
\(885\) 1409.56 0.0535388
\(886\) 1089.18 0.0412998
\(887\) −44465.6 −1.68321 −0.841606 0.540092i \(-0.818390\pi\)
−0.841606 + 0.540092i \(0.818390\pi\)
\(888\) −2113.51 −0.0798703
\(889\) 1320.59 0.0498213
\(890\) −1560.90 −0.0587880
\(891\) 7320.28 0.275240
\(892\) 37647.4 1.41315
\(893\) 3418.24 0.128093
\(894\) 814.224 0.0304605
\(895\) 19242.5 0.718665
\(896\) −6708.03 −0.250111
\(897\) −283.417 −0.0105496
\(898\) 10258.9 0.381228
\(899\) −2368.89 −0.0878830
\(900\) −17601.0 −0.651889
\(901\) −9995.91 −0.369603
\(902\) −3129.13 −0.115508
\(903\) 0 0
\(904\) 16464.8 0.605764
\(905\) 19717.8 0.724247
\(906\) 1305.56 0.0478744
\(907\) −13410.8 −0.490958 −0.245479 0.969402i \(-0.578945\pi\)
−0.245479 + 0.969402i \(0.578945\pi\)
\(908\) −37261.5 −1.36186
\(909\) −126.213 −0.00460529
\(910\) −117.884 −0.00429430
\(911\) 24002.1 0.872914 0.436457 0.899725i \(-0.356233\pi\)
0.436457 + 0.899725i \(0.356233\pi\)
\(912\) 969.884 0.0352150
\(913\) 3848.51 0.139504
\(914\) 5941.86 0.215032
\(915\) 956.194 0.0345473
\(916\) −3527.40 −0.127237
\(917\) −5119.71 −0.184371
\(918\) 855.746 0.0307667
\(919\) 28960.1 1.03950 0.519752 0.854317i \(-0.326024\pi\)
0.519752 + 0.854317i \(0.326024\pi\)
\(920\) −11673.7 −0.418339
\(921\) 3848.73 0.137698
\(922\) 4898.52 0.174972
\(923\) 1475.78 0.0526282
\(924\) −149.138 −0.00530984
\(925\) −27560.6 −0.979660
\(926\) −494.020 −0.0175318
\(927\) −30328.7 −1.07457
\(928\) 14606.3 0.516678
\(929\) −30584.7 −1.08014 −0.540071 0.841619i \(-0.681603\pi\)
−0.540071 + 0.841619i \(0.681603\pi\)
\(930\) −72.4665 −0.00255513
\(931\) 17719.7 0.623780
\(932\) −5136.57 −0.180530
\(933\) −4187.88 −0.146951
\(934\) 1318.89 0.0462050
\(935\) 1793.53 0.0627324
\(936\) −1903.77 −0.0664815
\(937\) 45835.7 1.59807 0.799033 0.601287i \(-0.205345\pi\)
0.799033 + 0.601287i \(0.205345\pi\)
\(938\) −815.071 −0.0283721
\(939\) −2043.32 −0.0710131
\(940\) −2309.30 −0.0801287
\(941\) −2785.96 −0.0965140 −0.0482570 0.998835i \(-0.515367\pi\)
−0.0482570 + 0.998835i \(0.515367\pi\)
\(942\) −85.9334 −0.00297225
\(943\) 38900.9 1.34336
\(944\) 21383.7 0.737268
\(945\) 614.663 0.0211587
\(946\) 0 0
\(947\) −12107.3 −0.415454 −0.207727 0.978187i \(-0.566606\pi\)
−0.207727 + 0.978187i \(0.566606\pi\)
\(948\) −1581.44 −0.0541803
\(949\) 4642.42 0.158798
\(950\) −5607.07 −0.191492
\(951\) 492.655 0.0167986
\(952\) 2324.78 0.0791455
\(953\) −17220.7 −0.585343 −0.292671 0.956213i \(-0.594544\pi\)
−0.292671 + 0.956213i \(0.594544\pi\)
\(954\) 8852.02 0.300414
\(955\) 17843.1 0.604597
\(956\) −35920.2 −1.21521
\(957\) 416.472 0.0140675
\(958\) −6790.62 −0.229014
\(959\) 10093.5 0.339871
\(960\) −316.726 −0.0106482
\(961\) −29055.5 −0.975311
\(962\) −1376.72 −0.0461406
\(963\) 46027.5 1.54020
\(964\) 22185.9 0.741243
\(965\) 19911.6 0.664226
\(966\) −306.487 −0.0102081
\(967\) 36590.1 1.21681 0.608407 0.793625i \(-0.291809\pi\)
0.608407 + 0.793625i \(0.291809\pi\)
\(968\) −19401.7 −0.644210
\(969\) −821.276 −0.0272272
\(970\) −6446.83 −0.213397
\(971\) 14970.4 0.494771 0.247386 0.968917i \(-0.420429\pi\)
0.247386 + 0.968917i \(0.420429\pi\)
\(972\) 6885.70 0.227221
\(973\) −8910.30 −0.293578
\(974\) 8604.02 0.283050
\(975\) 199.135 0.00654095
\(976\) 14505.9 0.475741
\(977\) −55910.8 −1.83086 −0.915428 0.402483i \(-0.868147\pi\)
−0.915428 + 0.402483i \(0.868147\pi\)
\(978\) −1378.27 −0.0450637
\(979\) −2785.28 −0.0909276
\(980\) −11971.1 −0.390206
\(981\) 20846.8 0.678479
\(982\) −10848.7 −0.352543
\(983\) −3092.53 −0.100342 −0.0501710 0.998741i \(-0.515977\pi\)
−0.0501710 + 0.998741i \(0.515977\pi\)
\(984\) −2096.04 −0.0679057
\(985\) 13641.1 0.441260
\(986\) −2998.19 −0.0968376
\(987\) −131.281 −0.00423375
\(988\) 1694.37 0.0545598
\(989\) 0 0
\(990\) −1588.29 −0.0509890
\(991\) 12923.0 0.414242 0.207121 0.978315i \(-0.433591\pi\)
0.207121 + 0.978315i \(0.433591\pi\)
\(992\) −4535.12 −0.145151
\(993\) −4577.32 −0.146281
\(994\) 1595.91 0.0509247
\(995\) −25769.8 −0.821062
\(996\) 1190.55 0.0378757
\(997\) 25871.0 0.821807 0.410904 0.911679i \(-0.365213\pi\)
0.410904 + 0.911679i \(0.365213\pi\)
\(998\) 10066.1 0.319276
\(999\) 7178.43 0.227343
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.1.21 yes 50
43.42 odd 2 1849.4.a.i.1.30 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.30 50 43.42 odd 2
1849.4.a.j.1.21 yes 50 1.1 even 1 trivial