Properties

Label 1849.4.a.j.1.2
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 1.2
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.20801 q^{2} -5.00842 q^{3} +19.1234 q^{4} +13.8519 q^{5} +26.0839 q^{6} +26.6225 q^{7} -57.9307 q^{8} -1.91570 q^{9} +O(q^{10})\) \(q-5.20801 q^{2} -5.00842 q^{3} +19.1234 q^{4} +13.8519 q^{5} +26.0839 q^{6} +26.6225 q^{7} -57.9307 q^{8} -1.91570 q^{9} -72.1409 q^{10} +47.0765 q^{11} -95.7780 q^{12} -60.3088 q^{13} -138.650 q^{14} -69.3762 q^{15} +148.717 q^{16} +69.2105 q^{17} +9.97699 q^{18} +9.96884 q^{19} +264.895 q^{20} -133.337 q^{21} -245.175 q^{22} -26.9731 q^{23} +290.142 q^{24} +66.8755 q^{25} +314.089 q^{26} +144.822 q^{27} +509.112 q^{28} +158.022 q^{29} +361.312 q^{30} -129.716 q^{31} -311.073 q^{32} -235.779 q^{33} -360.449 q^{34} +368.773 q^{35} -36.6347 q^{36} +59.9333 q^{37} -51.9178 q^{38} +302.052 q^{39} -802.451 q^{40} -415.611 q^{41} +694.419 q^{42} +900.263 q^{44} -26.5361 q^{45} +140.476 q^{46} -398.299 q^{47} -744.836 q^{48} +365.758 q^{49} -348.289 q^{50} -346.636 q^{51} -1153.31 q^{52} -280.790 q^{53} -754.235 q^{54} +652.100 q^{55} -1542.26 q^{56} -49.9282 q^{57} -822.981 q^{58} +143.280 q^{59} -1326.71 q^{60} -791.993 q^{61} +675.562 q^{62} -51.0008 q^{63} +430.337 q^{64} -835.392 q^{65} +1227.94 q^{66} -635.977 q^{67} +1323.54 q^{68} +135.093 q^{69} -1920.57 q^{70} -707.876 q^{71} +110.978 q^{72} +168.798 q^{73} -312.133 q^{74} -334.941 q^{75} +190.638 q^{76} +1253.30 q^{77} -1573.09 q^{78} -631.664 q^{79} +2060.01 q^{80} -673.606 q^{81} +2164.50 q^{82} +516.463 q^{83} -2549.85 q^{84} +958.698 q^{85} -791.441 q^{87} -2727.18 q^{88} -1272.00 q^{89} +138.200 q^{90} -1605.57 q^{91} -515.817 q^{92} +649.672 q^{93} +2074.34 q^{94} +138.088 q^{95} +1557.98 q^{96} -298.811 q^{97} -1904.87 q^{98} -90.1846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.20801 −1.84131 −0.920655 0.390377i \(-0.872345\pi\)
−0.920655 + 0.390377i \(0.872345\pi\)
\(3\) −5.00842 −0.963871 −0.481936 0.876207i \(-0.660066\pi\)
−0.481936 + 0.876207i \(0.660066\pi\)
\(4\) 19.1234 2.39042
\(5\) 13.8519 1.23895 0.619476 0.785015i \(-0.287345\pi\)
0.619476 + 0.785015i \(0.287345\pi\)
\(6\) 26.0839 1.77479
\(7\) 26.6225 1.43748 0.718740 0.695279i \(-0.244719\pi\)
0.718740 + 0.695279i \(0.244719\pi\)
\(8\) −57.9307 −2.56020
\(9\) −1.91570 −0.0709519
\(10\) −72.1409 −2.28130
\(11\) 47.0765 1.29037 0.645187 0.764025i \(-0.276780\pi\)
0.645187 + 0.764025i \(0.276780\pi\)
\(12\) −95.7780 −2.30406
\(13\) −60.3088 −1.28667 −0.643333 0.765587i \(-0.722449\pi\)
−0.643333 + 0.765587i \(0.722449\pi\)
\(14\) −138.650 −2.64685
\(15\) −69.3762 −1.19419
\(16\) 148.717 2.32370
\(17\) 69.2105 0.987413 0.493706 0.869629i \(-0.335642\pi\)
0.493706 + 0.869629i \(0.335642\pi\)
\(18\) 9.97699 0.130644
\(19\) 9.96884 0.120369 0.0601845 0.998187i \(-0.480831\pi\)
0.0601845 + 0.998187i \(0.480831\pi\)
\(20\) 264.895 2.96162
\(21\) −133.337 −1.38555
\(22\) −245.175 −2.37598
\(23\) −26.9731 −0.244534 −0.122267 0.992497i \(-0.539016\pi\)
−0.122267 + 0.992497i \(0.539016\pi\)
\(24\) 290.142 2.46770
\(25\) 66.8755 0.535004
\(26\) 314.089 2.36915
\(27\) 144.822 1.03226
\(28\) 509.112 3.43619
\(29\) 158.022 1.01186 0.505930 0.862574i \(-0.331149\pi\)
0.505930 + 0.862574i \(0.331149\pi\)
\(30\) 361.312 2.19888
\(31\) −129.716 −0.751537 −0.375769 0.926713i \(-0.622621\pi\)
−0.375769 + 0.926713i \(0.622621\pi\)
\(32\) −311.073 −1.71845
\(33\) −235.779 −1.24375
\(34\) −360.449 −1.81813
\(35\) 368.773 1.78097
\(36\) −36.6347 −0.169605
\(37\) 59.9333 0.266296 0.133148 0.991096i \(-0.457491\pi\)
0.133148 + 0.991096i \(0.457491\pi\)
\(38\) −51.9178 −0.221636
\(39\) 302.052 1.24018
\(40\) −802.451 −3.17197
\(41\) −415.611 −1.58311 −0.791554 0.611099i \(-0.790728\pi\)
−0.791554 + 0.611099i \(0.790728\pi\)
\(42\) 694.419 2.55122
\(43\) 0 0
\(44\) 900.263 3.08454
\(45\) −26.5361 −0.0879061
\(46\) 140.476 0.450263
\(47\) −398.299 −1.23612 −0.618062 0.786129i \(-0.712082\pi\)
−0.618062 + 0.786129i \(0.712082\pi\)
\(48\) −744.836 −2.23975
\(49\) 365.758 1.06635
\(50\) −348.289 −0.985109
\(51\) −346.636 −0.951739
\(52\) −1153.31 −3.07567
\(53\) −280.790 −0.727725 −0.363863 0.931453i \(-0.618542\pi\)
−0.363863 + 0.931453i \(0.618542\pi\)
\(54\) −754.235 −1.90071
\(55\) 652.100 1.59871
\(56\) −1542.26 −3.68024
\(57\) −49.9282 −0.116020
\(58\) −822.981 −1.86315
\(59\) 143.280 0.316159 0.158080 0.987426i \(-0.449470\pi\)
0.158080 + 0.987426i \(0.449470\pi\)
\(60\) −1326.71 −2.85462
\(61\) −791.993 −1.66237 −0.831183 0.555999i \(-0.812336\pi\)
−0.831183 + 0.555999i \(0.812336\pi\)
\(62\) 675.562 1.38381
\(63\) −51.0008 −0.101992
\(64\) 430.337 0.840502
\(65\) −835.392 −1.59412
\(66\) 1227.94 2.29014
\(67\) −635.977 −1.15966 −0.579828 0.814739i \(-0.696880\pi\)
−0.579828 + 0.814739i \(0.696880\pi\)
\(68\) 1323.54 2.36033
\(69\) 135.093 0.235699
\(70\) −1920.57 −3.27932
\(71\) −707.876 −1.18323 −0.591615 0.806220i \(-0.701510\pi\)
−0.591615 + 0.806220i \(0.701510\pi\)
\(72\) 110.978 0.181651
\(73\) 168.798 0.270634 0.135317 0.990802i \(-0.456795\pi\)
0.135317 + 0.990802i \(0.456795\pi\)
\(74\) −312.133 −0.490334
\(75\) −334.941 −0.515675
\(76\) 190.638 0.287733
\(77\) 1253.30 1.85489
\(78\) −1573.09 −2.28356
\(79\) −631.664 −0.899591 −0.449796 0.893131i \(-0.648503\pi\)
−0.449796 + 0.893131i \(0.648503\pi\)
\(80\) 2060.01 2.87895
\(81\) −673.606 −0.924014
\(82\) 2164.50 2.91499
\(83\) 516.463 0.683002 0.341501 0.939881i \(-0.389065\pi\)
0.341501 + 0.939881i \(0.389065\pi\)
\(84\) −2549.85 −3.31204
\(85\) 958.698 1.22336
\(86\) 0 0
\(87\) −791.441 −0.975303
\(88\) −2727.18 −3.30361
\(89\) −1272.00 −1.51496 −0.757481 0.652857i \(-0.773570\pi\)
−0.757481 + 0.652857i \(0.773570\pi\)
\(90\) 138.200 0.161862
\(91\) −1605.57 −1.84956
\(92\) −515.817 −0.584540
\(93\) 649.672 0.724385
\(94\) 2074.34 2.27609
\(95\) 138.088 0.149131
\(96\) 1557.98 1.65637
\(97\) −298.811 −0.312780 −0.156390 0.987695i \(-0.549986\pi\)
−0.156390 + 0.987695i \(0.549986\pi\)
\(98\) −1904.87 −1.96348
\(99\) −90.1846 −0.0915545
\(100\) 1278.89 1.27889
\(101\) −842.187 −0.829711 −0.414855 0.909887i \(-0.636168\pi\)
−0.414855 + 0.909887i \(0.636168\pi\)
\(102\) 1805.28 1.75245
\(103\) 330.701 0.316359 0.158179 0.987410i \(-0.449438\pi\)
0.158179 + 0.987410i \(0.449438\pi\)
\(104\) 3493.73 3.29412
\(105\) −1846.97 −1.71663
\(106\) 1462.36 1.33997
\(107\) −691.923 −0.625147 −0.312573 0.949894i \(-0.601191\pi\)
−0.312573 + 0.949894i \(0.601191\pi\)
\(108\) 2769.49 2.46754
\(109\) −1114.65 −0.979483 −0.489742 0.871868i \(-0.662909\pi\)
−0.489742 + 0.871868i \(0.662909\pi\)
\(110\) −3396.15 −2.94373
\(111\) −300.171 −0.256675
\(112\) 3959.21 3.34027
\(113\) −1732.15 −1.44201 −0.721005 0.692929i \(-0.756320\pi\)
−0.721005 + 0.692929i \(0.756320\pi\)
\(114\) 260.027 0.213629
\(115\) −373.629 −0.302966
\(116\) 3021.92 2.41877
\(117\) 115.534 0.0912913
\(118\) −746.201 −0.582148
\(119\) 1842.56 1.41939
\(120\) 4019.02 3.05737
\(121\) 885.201 0.665064
\(122\) 4124.71 3.06093
\(123\) 2081.55 1.52591
\(124\) −2480.61 −1.79649
\(125\) −805.135 −0.576108
\(126\) 265.613 0.187799
\(127\) 2426.06 1.69510 0.847550 0.530715i \(-0.178077\pi\)
0.847550 + 0.530715i \(0.178077\pi\)
\(128\) 247.383 0.170826
\(129\) 0 0
\(130\) 4350.73 2.93527
\(131\) −260.318 −0.173619 −0.0868094 0.996225i \(-0.527667\pi\)
−0.0868094 + 0.996225i \(0.527667\pi\)
\(132\) −4508.90 −2.97310
\(133\) 265.396 0.173028
\(134\) 3312.18 2.13529
\(135\) 2006.06 1.27892
\(136\) −4009.42 −2.52797
\(137\) −209.330 −0.130542 −0.0652711 0.997868i \(-0.520791\pi\)
−0.0652711 + 0.997868i \(0.520791\pi\)
\(138\) −703.565 −0.433996
\(139\) 253.533 0.154708 0.0773540 0.997004i \(-0.475353\pi\)
0.0773540 + 0.997004i \(0.475353\pi\)
\(140\) 7052.18 4.25727
\(141\) 1994.85 1.19146
\(142\) 3686.62 2.17869
\(143\) −2839.13 −1.66028
\(144\) −284.897 −0.164871
\(145\) 2188.91 1.25365
\(146\) −879.100 −0.498321
\(147\) −1831.87 −1.02782
\(148\) 1146.13 0.636561
\(149\) −2506.92 −1.37836 −0.689178 0.724592i \(-0.742028\pi\)
−0.689178 + 0.724592i \(0.742028\pi\)
\(150\) 1744.38 0.949518
\(151\) 1804.46 0.972482 0.486241 0.873825i \(-0.338368\pi\)
0.486241 + 0.873825i \(0.338368\pi\)
\(152\) −577.502 −0.308168
\(153\) −132.587 −0.0700588
\(154\) −6527.18 −3.41542
\(155\) −1796.81 −0.931120
\(156\) 5776.25 2.96455
\(157\) 676.565 0.343922 0.171961 0.985104i \(-0.444990\pi\)
0.171961 + 0.985104i \(0.444990\pi\)
\(158\) 3289.71 1.65643
\(159\) 1406.31 0.701433
\(160\) −4308.95 −2.12908
\(161\) −718.092 −0.351513
\(162\) 3508.15 1.70140
\(163\) −509.421 −0.244791 −0.122396 0.992481i \(-0.539058\pi\)
−0.122396 + 0.992481i \(0.539058\pi\)
\(164\) −7947.88 −3.78430
\(165\) −3265.99 −1.54095
\(166\) −2689.74 −1.25762
\(167\) −1635.97 −0.758057 −0.379028 0.925385i \(-0.623742\pi\)
−0.379028 + 0.925385i \(0.623742\pi\)
\(168\) 7724.29 3.54727
\(169\) 1440.15 0.655507
\(170\) −4992.91 −2.25258
\(171\) −19.0973 −0.00854040
\(172\) 0 0
\(173\) −336.002 −0.147663 −0.0738317 0.997271i \(-0.523523\pi\)
−0.0738317 + 0.997271i \(0.523523\pi\)
\(174\) 4121.83 1.79584
\(175\) 1780.39 0.769058
\(176\) 7001.07 2.99844
\(177\) −717.604 −0.304737
\(178\) 6624.59 2.78952
\(179\) −885.817 −0.369883 −0.184941 0.982750i \(-0.559210\pi\)
−0.184941 + 0.982750i \(0.559210\pi\)
\(180\) −507.461 −0.210133
\(181\) −3866.02 −1.58762 −0.793808 0.608168i \(-0.791905\pi\)
−0.793808 + 0.608168i \(0.791905\pi\)
\(182\) 8361.83 3.40561
\(183\) 3966.64 1.60231
\(184\) 1562.57 0.626056
\(185\) 830.190 0.329929
\(186\) −3383.50 −1.33382
\(187\) 3258.19 1.27413
\(188\) −7616.82 −2.95486
\(189\) 3855.53 1.48385
\(190\) −719.162 −0.274597
\(191\) 3601.46 1.36436 0.682180 0.731184i \(-0.261032\pi\)
0.682180 + 0.731184i \(0.261032\pi\)
\(192\) −2155.31 −0.810136
\(193\) −907.399 −0.338425 −0.169213 0.985580i \(-0.554122\pi\)
−0.169213 + 0.985580i \(0.554122\pi\)
\(194\) 1556.21 0.575926
\(195\) 4184.00 1.53652
\(196\) 6994.52 2.54902
\(197\) 3133.94 1.13342 0.566711 0.823917i \(-0.308216\pi\)
0.566711 + 0.823917i \(0.308216\pi\)
\(198\) 469.682 0.168580
\(199\) 702.456 0.250230 0.125115 0.992142i \(-0.460070\pi\)
0.125115 + 0.992142i \(0.460070\pi\)
\(200\) −3874.15 −1.36972
\(201\) 3185.24 1.11776
\(202\) 4386.12 1.52775
\(203\) 4206.94 1.45453
\(204\) −6628.85 −2.27506
\(205\) −5757.00 −1.96140
\(206\) −1722.30 −0.582515
\(207\) 51.6724 0.0173502
\(208\) −8968.93 −2.98982
\(209\) 469.299 0.155321
\(210\) 9619.04 3.16084
\(211\) −2634.32 −0.859497 −0.429749 0.902949i \(-0.641398\pi\)
−0.429749 + 0.902949i \(0.641398\pi\)
\(212\) −5369.65 −1.73957
\(213\) 3545.34 1.14048
\(214\) 3603.54 1.15109
\(215\) 0 0
\(216\) −8389.65 −2.64279
\(217\) −3453.36 −1.08032
\(218\) 5805.09 1.80353
\(219\) −845.410 −0.260856
\(220\) 12470.4 3.82160
\(221\) −4174.00 −1.27047
\(222\) 1563.29 0.472619
\(223\) 5852.83 1.75755 0.878776 0.477234i \(-0.158361\pi\)
0.878776 + 0.477234i \(0.158361\pi\)
\(224\) −8281.54 −2.47024
\(225\) −128.114 −0.0379596
\(226\) 9021.07 2.65519
\(227\) −2690.65 −0.786718 −0.393359 0.919385i \(-0.628687\pi\)
−0.393359 + 0.919385i \(0.628687\pi\)
\(228\) −954.796 −0.277337
\(229\) 909.642 0.262493 0.131246 0.991350i \(-0.458102\pi\)
0.131246 + 0.991350i \(0.458102\pi\)
\(230\) 1945.87 0.557855
\(231\) −6277.03 −1.78787
\(232\) −9154.33 −2.59057
\(233\) −2711.93 −0.762509 −0.381255 0.924470i \(-0.624508\pi\)
−0.381255 + 0.924470i \(0.624508\pi\)
\(234\) −601.700 −0.168096
\(235\) −5517.20 −1.53150
\(236\) 2739.99 0.755755
\(237\) 3163.64 0.867090
\(238\) −9596.06 −2.61353
\(239\) 2199.65 0.595329 0.297665 0.954671i \(-0.403792\pi\)
0.297665 + 0.954671i \(0.403792\pi\)
\(240\) −10317.4 −2.77494
\(241\) 2700.40 0.721776 0.360888 0.932609i \(-0.382474\pi\)
0.360888 + 0.932609i \(0.382474\pi\)
\(242\) −4610.14 −1.22459
\(243\) −536.491 −0.141629
\(244\) −15145.6 −3.97376
\(245\) 5066.44 1.32116
\(246\) −10840.8 −2.80968
\(247\) −601.209 −0.154874
\(248\) 7514.53 1.92409
\(249\) −2586.66 −0.658326
\(250\) 4193.15 1.06079
\(251\) −5537.81 −1.39260 −0.696302 0.717749i \(-0.745173\pi\)
−0.696302 + 0.717749i \(0.745173\pi\)
\(252\) −975.307 −0.243804
\(253\) −1269.80 −0.315540
\(254\) −12634.9 −3.12121
\(255\) −4801.57 −1.17916
\(256\) −4731.07 −1.15505
\(257\) −468.256 −0.113654 −0.0568268 0.998384i \(-0.518098\pi\)
−0.0568268 + 0.998384i \(0.518098\pi\)
\(258\) 0 0
\(259\) 1595.57 0.382796
\(260\) −15975.5 −3.81062
\(261\) −302.723 −0.0717934
\(262\) 1355.74 0.319686
\(263\) −6435.62 −1.50889 −0.754443 0.656365i \(-0.772093\pi\)
−0.754443 + 0.656365i \(0.772093\pi\)
\(264\) 13658.9 3.18426
\(265\) −3889.47 −0.901617
\(266\) −1382.18 −0.318598
\(267\) 6370.71 1.46023
\(268\) −12162.0 −2.77207
\(269\) −882.896 −0.200116 −0.100058 0.994982i \(-0.531903\pi\)
−0.100058 + 0.994982i \(0.531903\pi\)
\(270\) −10447.6 −2.35489
\(271\) 6424.80 1.44014 0.720071 0.693900i \(-0.244109\pi\)
0.720071 + 0.693900i \(0.244109\pi\)
\(272\) 10292.8 2.29445
\(273\) 8041.38 1.78273
\(274\) 1090.19 0.240369
\(275\) 3148.27 0.690356
\(276\) 2583.43 0.563421
\(277\) 1236.99 0.268315 0.134158 0.990960i \(-0.457167\pi\)
0.134158 + 0.990960i \(0.457167\pi\)
\(278\) −1320.41 −0.284866
\(279\) 248.497 0.0533230
\(280\) −21363.3 −4.55964
\(281\) 1461.25 0.310217 0.155109 0.987897i \(-0.450427\pi\)
0.155109 + 0.987897i \(0.450427\pi\)
\(282\) −10389.2 −2.19386
\(283\) 1698.07 0.356678 0.178339 0.983969i \(-0.442928\pi\)
0.178339 + 0.983969i \(0.442928\pi\)
\(284\) −13537.0 −2.82842
\(285\) −691.601 −0.143743
\(286\) 14786.2 3.05709
\(287\) −11064.6 −2.27569
\(288\) 595.923 0.121927
\(289\) −122.902 −0.0250158
\(290\) −11399.9 −2.30835
\(291\) 1496.57 0.301480
\(292\) 3227.98 0.646929
\(293\) 3559.87 0.709795 0.354897 0.934905i \(-0.384516\pi\)
0.354897 + 0.934905i \(0.384516\pi\)
\(294\) 9540.39 1.89254
\(295\) 1984.70 0.391707
\(296\) −3471.98 −0.681772
\(297\) 6817.72 1.33200
\(298\) 13056.1 2.53798
\(299\) 1626.72 0.314634
\(300\) −6405.21 −1.23268
\(301\) 0 0
\(302\) −9397.64 −1.79064
\(303\) 4218.03 0.799734
\(304\) 1482.53 0.279701
\(305\) −10970.6 −2.05959
\(306\) 690.513 0.129000
\(307\) −5590.48 −1.03930 −0.519651 0.854379i \(-0.673938\pi\)
−0.519651 + 0.854379i \(0.673938\pi\)
\(308\) 23967.2 4.43396
\(309\) −1656.29 −0.304929
\(310\) 9357.83 1.71448
\(311\) 2708.95 0.493924 0.246962 0.969025i \(-0.420568\pi\)
0.246962 + 0.969025i \(0.420568\pi\)
\(312\) −17498.1 −3.17511
\(313\) 6730.92 1.21551 0.607754 0.794125i \(-0.292071\pi\)
0.607754 + 0.794125i \(0.292071\pi\)
\(314\) −3523.56 −0.633267
\(315\) −706.458 −0.126363
\(316\) −12079.5 −2.15040
\(317\) 5984.74 1.06037 0.530184 0.847883i \(-0.322123\pi\)
0.530184 + 0.847883i \(0.322123\pi\)
\(318\) −7324.10 −1.29156
\(319\) 7439.13 1.30568
\(320\) 5960.99 1.04134
\(321\) 3465.44 0.602561
\(322\) 3739.83 0.647244
\(323\) 689.949 0.118854
\(324\) −12881.6 −2.20878
\(325\) −4033.18 −0.688372
\(326\) 2653.07 0.450737
\(327\) 5582.62 0.944096
\(328\) 24076.6 4.05308
\(329\) −10603.7 −1.77690
\(330\) 17009.3 2.83737
\(331\) −2114.28 −0.351092 −0.175546 0.984471i \(-0.556169\pi\)
−0.175546 + 0.984471i \(0.556169\pi\)
\(332\) 9876.51 1.63266
\(333\) −114.814 −0.0188942
\(334\) 8520.17 1.39582
\(335\) −8809.51 −1.43676
\(336\) −19829.4 −3.21959
\(337\) 10476.9 1.69351 0.846756 0.531982i \(-0.178552\pi\)
0.846756 + 0.531982i \(0.178552\pi\)
\(338\) −7500.32 −1.20699
\(339\) 8675.35 1.38991
\(340\) 18333.6 2.92434
\(341\) −6106.58 −0.969764
\(342\) 99.4591 0.0157255
\(343\) 605.864 0.0953749
\(344\) 0 0
\(345\) 1871.29 0.292021
\(346\) 1749.90 0.271894
\(347\) −11253.4 −1.74096 −0.870482 0.492199i \(-0.836193\pi\)
−0.870482 + 0.492199i \(0.836193\pi\)
\(348\) −15135.0 −2.33139
\(349\) 1170.22 0.179486 0.0897428 0.995965i \(-0.471396\pi\)
0.0897428 + 0.995965i \(0.471396\pi\)
\(350\) −9272.31 −1.41607
\(351\) −8734.04 −1.32817
\(352\) −14644.2 −2.21744
\(353\) −804.464 −0.121295 −0.0606477 0.998159i \(-0.519317\pi\)
−0.0606477 + 0.998159i \(0.519317\pi\)
\(354\) 3737.29 0.561115
\(355\) −9805.43 −1.46597
\(356\) −24324.9 −3.62140
\(357\) −9228.31 −1.36811
\(358\) 4613.34 0.681069
\(359\) 9746.71 1.43290 0.716451 0.697638i \(-0.245765\pi\)
0.716451 + 0.697638i \(0.245765\pi\)
\(360\) 1537.26 0.225057
\(361\) −6759.62 −0.985511
\(362\) 20134.3 2.92329
\(363\) −4433.46 −0.641037
\(364\) −30703.9 −4.42122
\(365\) 2338.17 0.335303
\(366\) −20658.3 −2.95034
\(367\) −3663.53 −0.521075 −0.260538 0.965464i \(-0.583900\pi\)
−0.260538 + 0.965464i \(0.583900\pi\)
\(368\) −4011.36 −0.568224
\(369\) 796.186 0.112325
\(370\) −4323.64 −0.607501
\(371\) −7475.32 −1.04609
\(372\) 12423.9 1.73159
\(373\) 13468.9 1.86969 0.934846 0.355053i \(-0.115537\pi\)
0.934846 + 0.355053i \(0.115537\pi\)
\(374\) −16968.7 −2.34607
\(375\) 4032.46 0.555294
\(376\) 23073.7 3.16472
\(377\) −9530.12 −1.30193
\(378\) −20079.6 −2.73223
\(379\) 2326.69 0.315341 0.157670 0.987492i \(-0.449602\pi\)
0.157670 + 0.987492i \(0.449602\pi\)
\(380\) 2640.70 0.356487
\(381\) −12150.7 −1.63386
\(382\) −18756.5 −2.51221
\(383\) −1329.44 −0.177366 −0.0886831 0.996060i \(-0.528266\pi\)
−0.0886831 + 0.996060i \(0.528266\pi\)
\(384\) −1239.00 −0.164655
\(385\) 17360.5 2.29812
\(386\) 4725.75 0.623145
\(387\) 0 0
\(388\) −5714.28 −0.747677
\(389\) −10163.9 −1.32475 −0.662375 0.749172i \(-0.730451\pi\)
−0.662375 + 0.749172i \(0.730451\pi\)
\(390\) −21790.3 −2.82922
\(391\) −1866.82 −0.241456
\(392\) −21188.6 −2.73007
\(393\) 1303.78 0.167346
\(394\) −16321.6 −2.08698
\(395\) −8749.75 −1.11455
\(396\) −1724.63 −0.218854
\(397\) −13306.5 −1.68221 −0.841103 0.540875i \(-0.818093\pi\)
−0.841103 + 0.540875i \(0.818093\pi\)
\(398\) −3658.40 −0.460751
\(399\) −1329.21 −0.166777
\(400\) 9945.51 1.24319
\(401\) 15275.3 1.90228 0.951140 0.308759i \(-0.0999137\pi\)
0.951140 + 0.308759i \(0.0999137\pi\)
\(402\) −16588.8 −2.05814
\(403\) 7823.01 0.966977
\(404\) −16105.5 −1.98336
\(405\) −9330.74 −1.14481
\(406\) −21909.8 −2.67824
\(407\) 2821.45 0.343622
\(408\) 20080.8 2.43664
\(409\) 16043.3 1.93958 0.969792 0.243931i \(-0.0784372\pi\)
0.969792 + 0.243931i \(0.0784372\pi\)
\(410\) 29982.5 3.61154
\(411\) 1048.41 0.125826
\(412\) 6324.13 0.756232
\(413\) 3814.46 0.454473
\(414\) −269.111 −0.0319470
\(415\) 7154.00 0.846207
\(416\) 18760.4 2.21107
\(417\) −1269.80 −0.149119
\(418\) −2444.11 −0.285994
\(419\) 8887.02 1.03618 0.518090 0.855326i \(-0.326643\pi\)
0.518090 + 0.855326i \(0.326643\pi\)
\(420\) −35320.3 −4.10346
\(421\) 600.862 0.0695587 0.0347794 0.999395i \(-0.488927\pi\)
0.0347794 + 0.999395i \(0.488927\pi\)
\(422\) 13719.6 1.58260
\(423\) 763.021 0.0877053
\(424\) 16266.3 1.86312
\(425\) 4628.49 0.528270
\(426\) −18464.2 −2.09998
\(427\) −21084.8 −2.38962
\(428\) −13231.9 −1.49436
\(429\) 14219.6 1.60030
\(430\) 0 0
\(431\) −16834.7 −1.88143 −0.940717 0.339193i \(-0.889846\pi\)
−0.940717 + 0.339193i \(0.889846\pi\)
\(432\) 21537.5 2.39866
\(433\) −1169.19 −0.129764 −0.0648820 0.997893i \(-0.520667\pi\)
−0.0648820 + 0.997893i \(0.520667\pi\)
\(434\) 17985.1 1.98920
\(435\) −10963.0 −1.20836
\(436\) −21315.8 −2.34138
\(437\) −268.891 −0.0294343
\(438\) 4402.91 0.480317
\(439\) 404.634 0.0439912 0.0219956 0.999758i \(-0.492998\pi\)
0.0219956 + 0.999758i \(0.492998\pi\)
\(440\) −37776.6 −4.09302
\(441\) −700.682 −0.0756595
\(442\) 21738.3 2.33933
\(443\) 8522.87 0.914071 0.457036 0.889448i \(-0.348911\pi\)
0.457036 + 0.889448i \(0.348911\pi\)
\(444\) −5740.29 −0.613563
\(445\) −17619.6 −1.87697
\(446\) −30481.6 −3.23620
\(447\) 12555.7 1.32856
\(448\) 11456.6 1.20820
\(449\) 2477.35 0.260386 0.130193 0.991489i \(-0.458440\pi\)
0.130193 + 0.991489i \(0.458440\pi\)
\(450\) 667.217 0.0698954
\(451\) −19565.5 −2.04280
\(452\) −33124.6 −3.44702
\(453\) −9037.49 −0.937347
\(454\) 14013.0 1.44859
\(455\) −22240.2 −2.29151
\(456\) 2892.37 0.297035
\(457\) 4605.88 0.471453 0.235727 0.971819i \(-0.424253\pi\)
0.235727 + 0.971819i \(0.424253\pi\)
\(458\) −4737.42 −0.483330
\(459\) 10023.2 1.01927
\(460\) −7145.06 −0.724218
\(461\) −11285.7 −1.14019 −0.570093 0.821580i \(-0.693093\pi\)
−0.570093 + 0.821580i \(0.693093\pi\)
\(462\) 32690.9 3.29203
\(463\) 508.870 0.0510782 0.0255391 0.999674i \(-0.491870\pi\)
0.0255391 + 0.999674i \(0.491870\pi\)
\(464\) 23500.5 2.35126
\(465\) 8999.20 0.897480
\(466\) 14123.8 1.40402
\(467\) −568.788 −0.0563606 −0.0281803 0.999603i \(-0.508971\pi\)
−0.0281803 + 0.999603i \(0.508971\pi\)
\(468\) 2209.39 0.218225
\(469\) −16931.3 −1.66698
\(470\) 28733.6 2.81997
\(471\) −3388.52 −0.331497
\(472\) −8300.29 −0.809431
\(473\) 0 0
\(474\) −16476.3 −1.59658
\(475\) 666.672 0.0643979
\(476\) 35235.9 3.39293
\(477\) 537.909 0.0516335
\(478\) −11455.8 −1.09619
\(479\) −3278.52 −0.312734 −0.156367 0.987699i \(-0.549978\pi\)
−0.156367 + 0.987699i \(0.549978\pi\)
\(480\) 21581.1 2.05216
\(481\) −3614.50 −0.342634
\(482\) −14063.7 −1.32901
\(483\) 3596.51 0.338813
\(484\) 16928.0 1.58979
\(485\) −4139.11 −0.387520
\(486\) 2794.05 0.260783
\(487\) 3439.30 0.320020 0.160010 0.987115i \(-0.448847\pi\)
0.160010 + 0.987115i \(0.448847\pi\)
\(488\) 45880.7 4.25599
\(489\) 2551.40 0.235947
\(490\) −26386.1 −2.43266
\(491\) 3116.14 0.286414 0.143207 0.989693i \(-0.454259\pi\)
0.143207 + 0.989693i \(0.454259\pi\)
\(492\) 39806.3 3.64758
\(493\) 10936.8 0.999124
\(494\) 3131.10 0.285172
\(495\) −1249.23 −0.113432
\(496\) −19290.9 −1.74635
\(497\) −18845.4 −1.70087
\(498\) 13471.4 1.21218
\(499\) 16962.6 1.52175 0.760874 0.648899i \(-0.224770\pi\)
0.760874 + 0.648899i \(0.224770\pi\)
\(500\) −15396.9 −1.37714
\(501\) 8193.65 0.730669
\(502\) 28841.0 2.56422
\(503\) 17725.1 1.57122 0.785610 0.618722i \(-0.212349\pi\)
0.785610 + 0.618722i \(0.212349\pi\)
\(504\) 2954.51 0.261120
\(505\) −11665.9 −1.02797
\(506\) 6613.14 0.581008
\(507\) −7212.88 −0.631825
\(508\) 46394.4 4.05201
\(509\) 3044.24 0.265095 0.132548 0.991177i \(-0.457684\pi\)
0.132548 + 0.991177i \(0.457684\pi\)
\(510\) 25006.6 2.17120
\(511\) 4493.82 0.389031
\(512\) 22660.4 1.95597
\(513\) 1443.71 0.124252
\(514\) 2438.68 0.209272
\(515\) 4580.85 0.391954
\(516\) 0 0
\(517\) −18750.5 −1.59506
\(518\) −8309.76 −0.704846
\(519\) 1682.84 0.142329
\(520\) 48394.9 4.08126
\(521\) 12593.9 1.05902 0.529509 0.848304i \(-0.322376\pi\)
0.529509 + 0.848304i \(0.322376\pi\)
\(522\) 1576.59 0.132194
\(523\) 15100.1 1.26249 0.631243 0.775585i \(-0.282545\pi\)
0.631243 + 0.775585i \(0.282545\pi\)
\(524\) −4978.16 −0.415023
\(525\) −8916.97 −0.741273
\(526\) 33516.8 2.77833
\(527\) −8977.71 −0.742078
\(528\) −35064.3 −2.89011
\(529\) −11439.5 −0.940203
\(530\) 20256.4 1.66016
\(531\) −274.481 −0.0224321
\(532\) 5075.26 0.413610
\(533\) 25065.0 2.03693
\(534\) −33178.7 −2.68873
\(535\) −9584.45 −0.774527
\(536\) 36842.6 2.96895
\(537\) 4436.54 0.356520
\(538\) 4598.13 0.368475
\(539\) 17218.6 1.37599
\(540\) 38362.7 3.05716
\(541\) −11318.5 −0.899485 −0.449743 0.893158i \(-0.648484\pi\)
−0.449743 + 0.893158i \(0.648484\pi\)
\(542\) −33460.4 −2.65175
\(543\) 19362.6 1.53026
\(544\) −21529.5 −1.69682
\(545\) −15440.0 −1.21353
\(546\) −41879.6 −3.28257
\(547\) −4299.81 −0.336100 −0.168050 0.985778i \(-0.553747\pi\)
−0.168050 + 0.985778i \(0.553747\pi\)
\(548\) −4003.10 −0.312051
\(549\) 1517.22 0.117948
\(550\) −16396.2 −1.27116
\(551\) 1575.30 0.121797
\(552\) −7826.02 −0.603438
\(553\) −16816.5 −1.29314
\(554\) −6442.24 −0.494052
\(555\) −4157.94 −0.318009
\(556\) 4848.42 0.369818
\(557\) −8356.80 −0.635707 −0.317854 0.948140i \(-0.602962\pi\)
−0.317854 + 0.948140i \(0.602962\pi\)
\(558\) −1294.17 −0.0981842
\(559\) 0 0
\(560\) 54842.7 4.13844
\(561\) −16318.4 −1.22810
\(562\) −7610.22 −0.571206
\(563\) −17758.8 −1.32938 −0.664692 0.747118i \(-0.731437\pi\)
−0.664692 + 0.747118i \(0.731437\pi\)
\(564\) 38148.2 2.84810
\(565\) −23993.6 −1.78658
\(566\) −8843.57 −0.656754
\(567\) −17933.1 −1.32825
\(568\) 41007.7 3.02931
\(569\) −6954.03 −0.512352 −0.256176 0.966630i \(-0.582463\pi\)
−0.256176 + 0.966630i \(0.582463\pi\)
\(570\) 3601.87 0.264676
\(571\) 20289.4 1.48701 0.743505 0.668730i \(-0.233162\pi\)
0.743505 + 0.668730i \(0.233162\pi\)
\(572\) −54293.8 −3.96877
\(573\) −18037.6 −1.31507
\(574\) 57624.5 4.19025
\(575\) −1803.84 −0.130827
\(576\) −824.397 −0.0596352
\(577\) −5613.49 −0.405013 −0.202507 0.979281i \(-0.564909\pi\)
−0.202507 + 0.979281i \(0.564909\pi\)
\(578\) 640.078 0.0460618
\(579\) 4544.64 0.326198
\(580\) 41859.3 2.99675
\(581\) 13749.5 0.981801
\(582\) −7794.17 −0.555118
\(583\) −13218.6 −0.939037
\(584\) −9778.57 −0.692877
\(585\) 1600.36 0.113106
\(586\) −18539.8 −1.30695
\(587\) −8010.42 −0.563246 −0.281623 0.959525i \(-0.590873\pi\)
−0.281623 + 0.959525i \(0.590873\pi\)
\(588\) −35031.5 −2.45693
\(589\) −1293.12 −0.0904617
\(590\) −10336.3 −0.721253
\(591\) −15696.1 −1.09247
\(592\) 8913.08 0.618793
\(593\) −6472.49 −0.448218 −0.224109 0.974564i \(-0.571947\pi\)
−0.224109 + 0.974564i \(0.571947\pi\)
\(594\) −35506.8 −2.45263
\(595\) 25522.9 1.75855
\(596\) −47940.8 −3.29485
\(597\) −3518.20 −0.241190
\(598\) −8471.96 −0.579338
\(599\) −25455.3 −1.73635 −0.868176 0.496257i \(-0.834708\pi\)
−0.868176 + 0.496257i \(0.834708\pi\)
\(600\) 19403.4 1.32023
\(601\) −13600.1 −0.923064 −0.461532 0.887124i \(-0.652700\pi\)
−0.461532 + 0.887124i \(0.652700\pi\)
\(602\) 0 0
\(603\) 1218.34 0.0822799
\(604\) 34507.4 2.32464
\(605\) 12261.7 0.823983
\(606\) −21967.5 −1.47256
\(607\) −8144.66 −0.544616 −0.272308 0.962210i \(-0.587787\pi\)
−0.272308 + 0.962210i \(0.587787\pi\)
\(608\) −3101.04 −0.206848
\(609\) −21070.1 −1.40198
\(610\) 57135.1 3.79235
\(611\) 24020.9 1.59048
\(612\) −2535.51 −0.167470
\(613\) 21750.5 1.43311 0.716554 0.697531i \(-0.245718\pi\)
0.716554 + 0.697531i \(0.245718\pi\)
\(614\) 29115.3 1.91368
\(615\) 28833.5 1.89053
\(616\) −72604.3 −4.74888
\(617\) 11659.4 0.760758 0.380379 0.924831i \(-0.375793\pi\)
0.380379 + 0.924831i \(0.375793\pi\)
\(618\) 8625.99 0.561469
\(619\) 17054.3 1.10738 0.553692 0.832722i \(-0.313219\pi\)
0.553692 + 0.832722i \(0.313219\pi\)
\(620\) −34361.2 −2.22577
\(621\) −3906.30 −0.252423
\(622\) −14108.2 −0.909467
\(623\) −33863.8 −2.17773
\(624\) 44920.2 2.88180
\(625\) −19512.1 −1.24877
\(626\) −35054.7 −2.23813
\(627\) −2350.45 −0.149709
\(628\) 12938.2 0.822119
\(629\) 4148.01 0.262944
\(630\) 3679.24 0.232674
\(631\) −15049.7 −0.949475 −0.474737 0.880128i \(-0.657457\pi\)
−0.474737 + 0.880128i \(0.657457\pi\)
\(632\) 36592.7 2.30313
\(633\) 13193.8 0.828445
\(634\) −31168.6 −1.95247
\(635\) 33605.5 2.10015
\(636\) 26893.5 1.67672
\(637\) −22058.4 −1.37203
\(638\) −38743.1 −2.40416
\(639\) 1356.08 0.0839525
\(640\) 3426.73 0.211646
\(641\) 22558.3 1.39002 0.695008 0.719002i \(-0.255401\pi\)
0.695008 + 0.719002i \(0.255401\pi\)
\(642\) −18048.1 −1.10950
\(643\) 16170.3 0.991748 0.495874 0.868395i \(-0.334848\pi\)
0.495874 + 0.868395i \(0.334848\pi\)
\(644\) −13732.3 −0.840265
\(645\) 0 0
\(646\) −3593.26 −0.218847
\(647\) −16223.2 −0.985778 −0.492889 0.870092i \(-0.664059\pi\)
−0.492889 + 0.870092i \(0.664059\pi\)
\(648\) 39022.5 2.36566
\(649\) 6745.10 0.407964
\(650\) 21004.9 1.26751
\(651\) 17295.9 1.04129
\(652\) −9741.86 −0.585155
\(653\) 6142.50 0.368108 0.184054 0.982916i \(-0.441078\pi\)
0.184054 + 0.982916i \(0.441078\pi\)
\(654\) −29074.3 −1.73837
\(655\) −3605.90 −0.215106
\(656\) −61808.3 −3.67867
\(657\) −323.366 −0.0192020
\(658\) 55224.2 3.27183
\(659\) −14009.5 −0.828123 −0.414061 0.910249i \(-0.635890\pi\)
−0.414061 + 0.910249i \(0.635890\pi\)
\(660\) −62456.9 −3.68353
\(661\) −26667.6 −1.56921 −0.784605 0.619996i \(-0.787134\pi\)
−0.784605 + 0.619996i \(0.787134\pi\)
\(662\) 11011.2 0.646470
\(663\) 20905.2 1.22457
\(664\) −29919.0 −1.74862
\(665\) 3676.24 0.214373
\(666\) 597.954 0.0347901
\(667\) −4262.35 −0.247434
\(668\) −31285.4 −1.81208
\(669\) −29313.4 −1.69405
\(670\) 45880.0 2.64552
\(671\) −37284.3 −2.14507
\(672\) 41477.4 2.38099
\(673\) 26390.1 1.51153 0.755767 0.654841i \(-0.227264\pi\)
0.755767 + 0.654841i \(0.227264\pi\)
\(674\) −54563.8 −3.11828
\(675\) 9685.05 0.552264
\(676\) 27540.5 1.56694
\(677\) 10340.1 0.587006 0.293503 0.955958i \(-0.405179\pi\)
0.293503 + 0.955958i \(0.405179\pi\)
\(678\) −45181.3 −2.55926
\(679\) −7955.10 −0.449615
\(680\) −55538.1 −3.13204
\(681\) 13475.9 0.758295
\(682\) 31803.1 1.78564
\(683\) 3929.56 0.220147 0.110074 0.993923i \(-0.464891\pi\)
0.110074 + 0.993923i \(0.464891\pi\)
\(684\) −365.205 −0.0204152
\(685\) −2899.62 −0.161736
\(686\) −3155.35 −0.175615
\(687\) −4555.87 −0.253009
\(688\) 0 0
\(689\) 16934.1 0.936339
\(690\) −9745.72 −0.537700
\(691\) −2660.45 −0.146466 −0.0732332 0.997315i \(-0.523332\pi\)
−0.0732332 + 0.997315i \(0.523332\pi\)
\(692\) −6425.50 −0.352978
\(693\) −2400.94 −0.131608
\(694\) 58607.9 3.20566
\(695\) 3511.92 0.191676
\(696\) 45848.8 2.49697
\(697\) −28764.6 −1.56318
\(698\) −6094.52 −0.330489
\(699\) 13582.5 0.734961
\(700\) 34047.2 1.83837
\(701\) 27310.9 1.47150 0.735749 0.677255i \(-0.236830\pi\)
0.735749 + 0.677255i \(0.236830\pi\)
\(702\) 45487.0 2.44558
\(703\) 597.465 0.0320538
\(704\) 20258.8 1.08456
\(705\) 27632.5 1.47617
\(706\) 4189.66 0.223342
\(707\) −22421.1 −1.19269
\(708\) −13723.0 −0.728450
\(709\) 2281.77 0.120866 0.0604328 0.998172i \(-0.480752\pi\)
0.0604328 + 0.998172i \(0.480752\pi\)
\(710\) 51066.8 2.69930
\(711\) 1210.08 0.0638277
\(712\) 73687.8 3.87861
\(713\) 3498.84 0.183777
\(714\) 48061.1 2.51911
\(715\) −39327.4 −2.05701
\(716\) −16939.8 −0.884177
\(717\) −11016.8 −0.573821
\(718\) −50761.0 −2.63842
\(719\) −7427.41 −0.385251 −0.192626 0.981272i \(-0.561700\pi\)
−0.192626 + 0.981272i \(0.561700\pi\)
\(720\) −3946.37 −0.204267
\(721\) 8804.10 0.454760
\(722\) 35204.2 1.81463
\(723\) −13524.7 −0.695699
\(724\) −73931.3 −3.79508
\(725\) 10567.8 0.541350
\(726\) 23089.5 1.18035
\(727\) 31960.7 1.63048 0.815240 0.579124i \(-0.196605\pi\)
0.815240 + 0.579124i \(0.196605\pi\)
\(728\) 93011.9 4.73523
\(729\) 20874.3 1.06053
\(730\) −12177.2 −0.617396
\(731\) 0 0
\(732\) 75855.5 3.83019
\(733\) 3379.50 0.170293 0.0851463 0.996368i \(-0.472864\pi\)
0.0851463 + 0.996368i \(0.472864\pi\)
\(734\) 19079.7 0.959462
\(735\) −25374.9 −1.27342
\(736\) 8390.61 0.420220
\(737\) −29939.6 −1.49639
\(738\) −4146.54 −0.206824
\(739\) −32194.7 −1.60257 −0.801285 0.598283i \(-0.795850\pi\)
−0.801285 + 0.598283i \(0.795850\pi\)
\(740\) 15876.0 0.788669
\(741\) 3011.11 0.149279
\(742\) 38931.6 1.92618
\(743\) −8237.10 −0.406716 −0.203358 0.979104i \(-0.565185\pi\)
−0.203358 + 0.979104i \(0.565185\pi\)
\(744\) −37636.0 −1.85457
\(745\) −34725.6 −1.70772
\(746\) −70146.4 −3.44268
\(747\) −989.388 −0.0484603
\(748\) 62307.7 3.04571
\(749\) −18420.7 −0.898636
\(750\) −21001.1 −1.02247
\(751\) −33321.1 −1.61904 −0.809522 0.587089i \(-0.800274\pi\)
−0.809522 + 0.587089i \(0.800274\pi\)
\(752\) −59233.7 −2.87238
\(753\) 27735.7 1.34229
\(754\) 49633.0 2.39725
\(755\) 24995.2 1.20486
\(756\) 73730.7 3.54704
\(757\) 40005.7 1.92078 0.960391 0.278656i \(-0.0898890\pi\)
0.960391 + 0.278656i \(0.0898890\pi\)
\(758\) −12117.4 −0.580640
\(759\) 6359.70 0.304140
\(760\) −7999.51 −0.381806
\(761\) −1344.71 −0.0640549 −0.0320274 0.999487i \(-0.510196\pi\)
−0.0320274 + 0.999487i \(0.510196\pi\)
\(762\) 63281.1 3.00844
\(763\) −29674.6 −1.40799
\(764\) 68872.1 3.26140
\(765\) −1836.58 −0.0867996
\(766\) 6923.74 0.326586
\(767\) −8641.02 −0.406791
\(768\) 23695.2 1.11332
\(769\) −41407.9 −1.94175 −0.970875 0.239586i \(-0.922988\pi\)
−0.970875 + 0.239586i \(0.922988\pi\)
\(770\) −90413.9 −4.23155
\(771\) 2345.22 0.109547
\(772\) −17352.5 −0.808979
\(773\) 10328.8 0.480598 0.240299 0.970699i \(-0.422755\pi\)
0.240299 + 0.970699i \(0.422755\pi\)
\(774\) 0 0
\(775\) −8674.82 −0.402076
\(776\) 17310.4 0.800780
\(777\) −7991.30 −0.368966
\(778\) 52933.5 2.43928
\(779\) −4143.16 −0.190557
\(780\) 80012.2 3.67294
\(781\) −33324.3 −1.52681
\(782\) 9722.44 0.444596
\(783\) 22885.1 1.04450
\(784\) 54394.3 2.47787
\(785\) 9371.72 0.426103
\(786\) −6790.11 −0.308136
\(787\) −35833.2 −1.62302 −0.811509 0.584341i \(-0.801353\pi\)
−0.811509 + 0.584341i \(0.801353\pi\)
\(788\) 59931.5 2.70936
\(789\) 32232.3 1.45437
\(790\) 45568.8 2.05223
\(791\) −46114.2 −2.07286
\(792\) 5224.46 0.234398
\(793\) 47764.2 2.13891
\(794\) 69300.6 3.09746
\(795\) 19480.1 0.869043
\(796\) 13433.3 0.598156
\(797\) 9312.94 0.413904 0.206952 0.978351i \(-0.433646\pi\)
0.206952 + 0.978351i \(0.433646\pi\)
\(798\) 6922.56 0.307088
\(799\) −27566.5 −1.22056
\(800\) −20803.2 −0.919379
\(801\) 2436.77 0.107489
\(802\) −79554.2 −3.50269
\(803\) 7946.41 0.349219
\(804\) 60912.6 2.67192
\(805\) −9946.95 −0.435508
\(806\) −40742.3 −1.78050
\(807\) 4421.92 0.192886
\(808\) 48788.5 2.12422
\(809\) 29114.8 1.26529 0.632646 0.774441i \(-0.281969\pi\)
0.632646 + 0.774441i \(0.281969\pi\)
\(810\) 48594.6 2.10795
\(811\) −41592.3 −1.80087 −0.900434 0.434993i \(-0.856751\pi\)
−0.900434 + 0.434993i \(0.856751\pi\)
\(812\) 80451.0 3.47694
\(813\) −32178.1 −1.38811
\(814\) −14694.1 −0.632714
\(815\) −7056.46 −0.303285
\(816\) −51550.5 −2.21156
\(817\) 0 0
\(818\) −83553.7 −3.57138
\(819\) 3075.79 0.131229
\(820\) −110093. −4.68857
\(821\) −39650.4 −1.68551 −0.842757 0.538294i \(-0.819069\pi\)
−0.842757 + 0.538294i \(0.819069\pi\)
\(822\) −5460.15 −0.231684
\(823\) −18879.4 −0.799629 −0.399815 0.916596i \(-0.630925\pi\)
−0.399815 + 0.916596i \(0.630925\pi\)
\(824\) −19157.8 −0.809942
\(825\) −15767.9 −0.665414
\(826\) −19865.8 −0.836825
\(827\) 19740.1 0.830024 0.415012 0.909816i \(-0.363777\pi\)
0.415012 + 0.909816i \(0.363777\pi\)
\(828\) 988.152 0.0414742
\(829\) −26241.2 −1.09939 −0.549695 0.835365i \(-0.685256\pi\)
−0.549695 + 0.835365i \(0.685256\pi\)
\(830\) −37258.1 −1.55813
\(831\) −6195.35 −0.258621
\(832\) −25953.1 −1.08144
\(833\) 25314.3 1.05293
\(834\) 6613.15 0.274574
\(835\) −22661.4 −0.939197
\(836\) 8974.58 0.371283
\(837\) −18785.7 −0.775782
\(838\) −46283.7 −1.90793
\(839\) −9598.19 −0.394954 −0.197477 0.980308i \(-0.563275\pi\)
−0.197477 + 0.980308i \(0.563275\pi\)
\(840\) 106996. 4.39491
\(841\) 581.966 0.0238618
\(842\) −3129.30 −0.128079
\(843\) −7318.57 −0.299009
\(844\) −50377.1 −2.05456
\(845\) 19948.8 0.812143
\(846\) −3973.82 −0.161493
\(847\) 23566.3 0.956017
\(848\) −41758.1 −1.69101
\(849\) −8504.65 −0.343791
\(850\) −24105.2 −0.972709
\(851\) −1616.59 −0.0651186
\(852\) 67798.9 2.72623
\(853\) −13278.1 −0.532981 −0.266490 0.963838i \(-0.585864\pi\)
−0.266490 + 0.963838i \(0.585864\pi\)
\(854\) 109810. 4.40003
\(855\) −264.534 −0.0105812
\(856\) 40083.6 1.60050
\(857\) −18835.6 −0.750773 −0.375386 0.926868i \(-0.622490\pi\)
−0.375386 + 0.926868i \(0.622490\pi\)
\(858\) −74055.6 −2.94664
\(859\) −8470.49 −0.336449 −0.168224 0.985749i \(-0.553803\pi\)
−0.168224 + 0.985749i \(0.553803\pi\)
\(860\) 0 0
\(861\) 55416.2 2.19347
\(862\) 87675.2 3.46430
\(863\) −14654.0 −0.578017 −0.289008 0.957327i \(-0.593326\pi\)
−0.289008 + 0.957327i \(0.593326\pi\)
\(864\) −45050.2 −1.77389
\(865\) −4654.27 −0.182948
\(866\) 6089.17 0.238936
\(867\) 615.548 0.0241120
\(868\) −66040.0 −2.58242
\(869\) −29736.5 −1.16081
\(870\) 57095.3 2.22496
\(871\) 38355.0 1.49209
\(872\) 64572.2 2.50767
\(873\) 572.433 0.0221924
\(874\) 1400.39 0.0541977
\(875\) −21434.7 −0.828143
\(876\) −16167.1 −0.623557
\(877\) 31034.9 1.19495 0.597477 0.801886i \(-0.296170\pi\)
0.597477 + 0.801886i \(0.296170\pi\)
\(878\) −2107.34 −0.0810015
\(879\) −17829.3 −0.684151
\(880\) 96978.2 3.71493
\(881\) −2331.10 −0.0891449 −0.0445725 0.999006i \(-0.514193\pi\)
−0.0445725 + 0.999006i \(0.514193\pi\)
\(882\) 3649.16 0.139313
\(883\) 38070.6 1.45094 0.725469 0.688255i \(-0.241623\pi\)
0.725469 + 0.688255i \(0.241623\pi\)
\(884\) −79821.1 −3.03696
\(885\) −9940.20 −0.377555
\(886\) −44387.2 −1.68309
\(887\) 19683.4 0.745102 0.372551 0.928012i \(-0.378483\pi\)
0.372551 + 0.928012i \(0.378483\pi\)
\(888\) 17389.1 0.657141
\(889\) 64587.7 2.43667
\(890\) 91763.2 3.45608
\(891\) −31711.0 −1.19232
\(892\) 111926. 4.20129
\(893\) −3970.58 −0.148791
\(894\) −65390.3 −2.44629
\(895\) −12270.3 −0.458268
\(896\) 6585.95 0.245559
\(897\) −8147.28 −0.303266
\(898\) −12902.1 −0.479451
\(899\) −20498.0 −0.760451
\(900\) −2449.97 −0.0907394
\(901\) −19433.6 −0.718565
\(902\) 101897. 3.76143
\(903\) 0 0
\(904\) 100345. 3.69184
\(905\) −53551.7 −1.96698
\(906\) 47067.4 1.72595
\(907\) −3322.49 −0.121633 −0.0608167 0.998149i \(-0.519371\pi\)
−0.0608167 + 0.998149i \(0.519371\pi\)
\(908\) −51454.4 −1.88059
\(909\) 1613.38 0.0588695
\(910\) 115827. 4.21938
\(911\) 36251.9 1.31842 0.659209 0.751959i \(-0.270891\pi\)
0.659209 + 0.751959i \(0.270891\pi\)
\(912\) −7425.16 −0.269596
\(913\) 24313.3 0.881327
\(914\) −23987.5 −0.868092
\(915\) 54945.5 1.98518
\(916\) 17395.4 0.627468
\(917\) −6930.31 −0.249574
\(918\) −52201.0 −1.87679
\(919\) 20127.9 0.722477 0.361239 0.932473i \(-0.382354\pi\)
0.361239 + 0.932473i \(0.382354\pi\)
\(920\) 21644.6 0.775654
\(921\) 27999.5 1.00175
\(922\) 58775.9 2.09944
\(923\) 42691.1 1.52242
\(924\) −120038. −4.27377
\(925\) 4008.07 0.142470
\(926\) −2650.20 −0.0940508
\(927\) −633.525 −0.0224463
\(928\) −49156.4 −1.73883
\(929\) −39220.1 −1.38511 −0.692557 0.721364i \(-0.743516\pi\)
−0.692557 + 0.721364i \(0.743516\pi\)
\(930\) −46867.9 −1.65254
\(931\) 3646.18 0.128355
\(932\) −51861.4 −1.82272
\(933\) −13567.5 −0.476079
\(934\) 2962.26 0.103777
\(935\) 45132.2 1.57859
\(936\) −6692.94 −0.233724
\(937\) 12798.4 0.446218 0.223109 0.974794i \(-0.428379\pi\)
0.223109 + 0.974794i \(0.428379\pi\)
\(938\) 88178.5 3.06943
\(939\) −33711.3 −1.17159
\(940\) −105507. −3.66093
\(941\) −45827.7 −1.58761 −0.793805 0.608172i \(-0.791903\pi\)
−0.793805 + 0.608172i \(0.791903\pi\)
\(942\) 17647.5 0.610388
\(943\) 11210.3 0.387124
\(944\) 21308.1 0.734659
\(945\) 53406.4 1.83842
\(946\) 0 0
\(947\) 8956.69 0.307342 0.153671 0.988122i \(-0.450890\pi\)
0.153671 + 0.988122i \(0.450890\pi\)
\(948\) 60499.5 2.07271
\(949\) −10180.0 −0.348215
\(950\) −3472.03 −0.118576
\(951\) −29974.1 −1.02206
\(952\) −106741. −3.63391
\(953\) −32038.7 −1.08902 −0.544510 0.838754i \(-0.683285\pi\)
−0.544510 + 0.838754i \(0.683285\pi\)
\(954\) −2801.44 −0.0950732
\(955\) 49887.2 1.69038
\(956\) 42064.8 1.42309
\(957\) −37258.3 −1.25851
\(958\) 17074.6 0.575840
\(959\) −5572.89 −0.187652
\(960\) −29855.2 −1.00372
\(961\) −12964.8 −0.435191
\(962\) 18824.4 0.630896
\(963\) 1325.52 0.0443553
\(964\) 51640.8 1.72535
\(965\) −12569.2 −0.419293
\(966\) −18730.7 −0.623860
\(967\) −33896.0 −1.12722 −0.563610 0.826041i \(-0.690588\pi\)
−0.563610 + 0.826041i \(0.690588\pi\)
\(968\) −51280.3 −1.70270
\(969\) −3455.56 −0.114560
\(970\) 21556.5 0.713545
\(971\) −37446.8 −1.23762 −0.618808 0.785542i \(-0.712384\pi\)
−0.618808 + 0.785542i \(0.712384\pi\)
\(972\) −10259.5 −0.338554
\(973\) 6749.69 0.222390
\(974\) −17911.9 −0.589256
\(975\) 20199.9 0.663502
\(976\) −117783. −3.86284
\(977\) 23276.6 0.762214 0.381107 0.924531i \(-0.375543\pi\)
0.381107 + 0.924531i \(0.375543\pi\)
\(978\) −13287.7 −0.434452
\(979\) −59881.3 −1.95487
\(980\) 96887.5 3.15812
\(981\) 2135.33 0.0694962
\(982\) −16228.9 −0.527377
\(983\) 20647.7 0.669947 0.334974 0.942228i \(-0.391273\pi\)
0.334974 + 0.942228i \(0.391273\pi\)
\(984\) −120586. −3.90664
\(985\) 43411.1 1.40426
\(986\) −56958.9 −1.83970
\(987\) 53107.8 1.71271
\(988\) −11497.1 −0.370216
\(989\) 0 0
\(990\) 6506.00 0.208863
\(991\) −20706.8 −0.663745 −0.331873 0.943324i \(-0.607680\pi\)
−0.331873 + 0.943324i \(0.607680\pi\)
\(992\) 40351.1 1.29148
\(993\) 10589.2 0.338408
\(994\) 98147.2 3.13183
\(995\) 9730.36 0.310023
\(996\) −49465.7 −1.57368
\(997\) 55958.1 1.77754 0.888771 0.458351i \(-0.151560\pi\)
0.888771 + 0.458351i \(0.151560\pi\)
\(998\) −88341.7 −2.80201
\(999\) 8679.66 0.274887
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.1.2 yes 50
43.42 odd 2 1849.4.a.i.1.49 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.49 50 43.42 odd 2
1849.4.a.j.1.2 yes 50 1.1 even 1 trivial