Properties

Label 1849.4.a.j.1.50
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 1.50
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.46223 q^{2} -9.09504 q^{3} +21.8360 q^{4} +6.15202 q^{5} -49.6792 q^{6} -2.59059 q^{7} +75.5754 q^{8} +55.7197 q^{9} +O(q^{10})\) \(q+5.46223 q^{2} -9.09504 q^{3} +21.8360 q^{4} +6.15202 q^{5} -49.6792 q^{6} -2.59059 q^{7} +75.5754 q^{8} +55.7197 q^{9} +33.6038 q^{10} -20.0476 q^{11} -198.599 q^{12} -34.1627 q^{13} -14.1504 q^{14} -55.9529 q^{15} +238.122 q^{16} +36.1588 q^{17} +304.354 q^{18} -142.784 q^{19} +134.335 q^{20} +23.5615 q^{21} -109.505 q^{22} -132.818 q^{23} -687.361 q^{24} -87.1526 q^{25} -186.604 q^{26} -261.207 q^{27} -56.5681 q^{28} +49.6805 q^{29} -305.628 q^{30} -51.3202 q^{31} +696.077 q^{32} +182.334 q^{33} +197.508 q^{34} -15.9374 q^{35} +1216.69 q^{36} +49.1882 q^{37} -779.922 q^{38} +310.711 q^{39} +464.941 q^{40} +71.4324 q^{41} +128.698 q^{42} -437.760 q^{44} +342.789 q^{45} -725.481 q^{46} -371.774 q^{47} -2165.73 q^{48} -336.289 q^{49} -476.048 q^{50} -328.865 q^{51} -745.976 q^{52} +260.514 q^{53} -1426.77 q^{54} -123.333 q^{55} -195.785 q^{56} +1298.63 q^{57} +271.366 q^{58} +29.4340 q^{59} -1221.79 q^{60} +461.029 q^{61} -280.323 q^{62} -144.347 q^{63} +1897.16 q^{64} -210.169 q^{65} +995.951 q^{66} +891.001 q^{67} +789.563 q^{68} +1207.98 q^{69} -87.0536 q^{70} -475.538 q^{71} +4211.04 q^{72} -1027.68 q^{73} +268.677 q^{74} +792.656 q^{75} -3117.84 q^{76} +51.9352 q^{77} +1697.17 q^{78} -307.975 q^{79} +1464.93 q^{80} +871.251 q^{81} +390.181 q^{82} -1145.51 q^{83} +514.489 q^{84} +222.450 q^{85} -451.846 q^{87} -1515.11 q^{88} +670.531 q^{89} +1872.39 q^{90} +88.5015 q^{91} -2900.20 q^{92} +466.759 q^{93} -2030.72 q^{94} -878.413 q^{95} -6330.85 q^{96} -125.357 q^{97} -1836.89 q^{98} -1117.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.46223 1.93119 0.965595 0.260049i \(-0.0837387\pi\)
0.965595 + 0.260049i \(0.0837387\pi\)
\(3\) −9.09504 −1.75034 −0.875170 0.483815i \(-0.839251\pi\)
−0.875170 + 0.483815i \(0.839251\pi\)
\(4\) 21.8360 2.72950
\(5\) 6.15202 0.550254 0.275127 0.961408i \(-0.411280\pi\)
0.275127 + 0.961408i \(0.411280\pi\)
\(6\) −49.6792 −3.38024
\(7\) −2.59059 −0.139879 −0.0699394 0.997551i \(-0.522281\pi\)
−0.0699394 + 0.997551i \(0.522281\pi\)
\(8\) 75.5754 3.33999
\(9\) 55.7197 2.06369
\(10\) 33.6038 1.06264
\(11\) −20.0476 −0.549508 −0.274754 0.961515i \(-0.588596\pi\)
−0.274754 + 0.961515i \(0.588596\pi\)
\(12\) −198.599 −4.77755
\(13\) −34.1627 −0.728848 −0.364424 0.931233i \(-0.618734\pi\)
−0.364424 + 0.931233i \(0.618734\pi\)
\(14\) −14.1504 −0.270132
\(15\) −55.9529 −0.963131
\(16\) 238.122 3.72066
\(17\) 36.1588 0.515870 0.257935 0.966162i \(-0.416958\pi\)
0.257935 + 0.966162i \(0.416958\pi\)
\(18\) 304.354 3.98538
\(19\) −142.784 −1.72405 −0.862026 0.506864i \(-0.830805\pi\)
−0.862026 + 0.506864i \(0.830805\pi\)
\(20\) 134.335 1.50192
\(21\) 23.5615 0.244835
\(22\) −109.505 −1.06121
\(23\) −132.818 −1.20410 −0.602052 0.798457i \(-0.705650\pi\)
−0.602052 + 0.798457i \(0.705650\pi\)
\(24\) −687.361 −5.84612
\(25\) −87.1526 −0.697221
\(26\) −186.604 −1.40754
\(27\) −261.207 −1.86182
\(28\) −56.5681 −0.381799
\(29\) 49.6805 0.318118 0.159059 0.987269i \(-0.449154\pi\)
0.159059 + 0.987269i \(0.449154\pi\)
\(30\) −305.628 −1.85999
\(31\) −51.3202 −0.297335 −0.148667 0.988887i \(-0.547498\pi\)
−0.148667 + 0.988887i \(0.547498\pi\)
\(32\) 696.077 3.84532
\(33\) 182.334 0.961826
\(34\) 197.508 0.996244
\(35\) −15.9374 −0.0769688
\(36\) 1216.69 5.63284
\(37\) 49.1882 0.218554 0.109277 0.994011i \(-0.465147\pi\)
0.109277 + 0.994011i \(0.465147\pi\)
\(38\) −779.922 −3.32947
\(39\) 310.711 1.27573
\(40\) 464.941 1.83784
\(41\) 71.4324 0.272094 0.136047 0.990702i \(-0.456560\pi\)
0.136047 + 0.990702i \(0.456560\pi\)
\(42\) 128.698 0.472824
\(43\) 0 0
\(44\) −437.760 −1.49988
\(45\) 342.789 1.13555
\(46\) −725.481 −2.32536
\(47\) −371.774 −1.15380 −0.576902 0.816813i \(-0.695739\pi\)
−0.576902 + 0.816813i \(0.695739\pi\)
\(48\) −2165.73 −6.51243
\(49\) −336.289 −0.980434
\(50\) −476.048 −1.34647
\(51\) −328.865 −0.902948
\(52\) −745.976 −1.98939
\(53\) 260.514 0.675177 0.337588 0.941294i \(-0.390389\pi\)
0.337588 + 0.941294i \(0.390389\pi\)
\(54\) −1426.77 −3.59554
\(55\) −123.333 −0.302369
\(56\) −195.785 −0.467194
\(57\) 1298.63 3.01768
\(58\) 271.366 0.614347
\(59\) 29.4340 0.0649489 0.0324745 0.999473i \(-0.489661\pi\)
0.0324745 + 0.999473i \(0.489661\pi\)
\(60\) −1221.79 −2.62886
\(61\) 461.029 0.967683 0.483842 0.875156i \(-0.339241\pi\)
0.483842 + 0.875156i \(0.339241\pi\)
\(62\) −280.323 −0.574210
\(63\) −144.347 −0.288667
\(64\) 1897.16 3.70538
\(65\) −210.169 −0.401051
\(66\) 995.951 1.85747
\(67\) 891.001 1.62467 0.812336 0.583189i \(-0.198195\pi\)
0.812336 + 0.583189i \(0.198195\pi\)
\(68\) 789.563 1.40807
\(69\) 1207.98 2.10759
\(70\) −87.0536 −0.148641
\(71\) −475.538 −0.794873 −0.397436 0.917630i \(-0.630100\pi\)
−0.397436 + 0.917630i \(0.630100\pi\)
\(72\) 4211.04 6.89271
\(73\) −1027.68 −1.64768 −0.823838 0.566826i \(-0.808171\pi\)
−0.823838 + 0.566826i \(0.808171\pi\)
\(74\) 268.677 0.422069
\(75\) 792.656 1.22037
\(76\) −3117.84 −4.70580
\(77\) 51.9352 0.0768645
\(78\) 1697.17 2.46368
\(79\) −307.975 −0.438607 −0.219303 0.975657i \(-0.570378\pi\)
−0.219303 + 0.975657i \(0.570378\pi\)
\(80\) 1464.93 2.04731
\(81\) 871.251 1.19513
\(82\) 390.181 0.525466
\(83\) −1145.51 −1.51490 −0.757448 0.652895i \(-0.773554\pi\)
−0.757448 + 0.652895i \(0.773554\pi\)
\(84\) 514.489 0.668278
\(85\) 222.450 0.283859
\(86\) 0 0
\(87\) −451.846 −0.556815
\(88\) −1515.11 −1.83535
\(89\) 670.531 0.798608 0.399304 0.916819i \(-0.369252\pi\)
0.399304 + 0.916819i \(0.369252\pi\)
\(90\) 1872.39 2.19297
\(91\) 88.5015 0.101950
\(92\) −2900.20 −3.28660
\(93\) 466.759 0.520437
\(94\) −2030.72 −2.22822
\(95\) −878.413 −0.948666
\(96\) −6330.85 −6.73062
\(97\) −125.357 −0.131217 −0.0656085 0.997845i \(-0.520899\pi\)
−0.0656085 + 0.997845i \(0.520899\pi\)
\(98\) −1836.89 −1.89341
\(99\) −1117.05 −1.13402
\(100\) −1903.06 −1.90306
\(101\) −1452.64 −1.43112 −0.715558 0.698554i \(-0.753827\pi\)
−0.715558 + 0.698554i \(0.753827\pi\)
\(102\) −1796.34 −1.74377
\(103\) 840.149 0.803712 0.401856 0.915703i \(-0.368365\pi\)
0.401856 + 0.915703i \(0.368365\pi\)
\(104\) −2581.86 −2.43434
\(105\) 144.951 0.134722
\(106\) 1422.99 1.30390
\(107\) −932.647 −0.842639 −0.421320 0.906912i \(-0.638433\pi\)
−0.421320 + 0.906912i \(0.638433\pi\)
\(108\) −5703.70 −5.08184
\(109\) −74.7800 −0.0657122 −0.0328561 0.999460i \(-0.510460\pi\)
−0.0328561 + 0.999460i \(0.510460\pi\)
\(110\) −673.676 −0.583932
\(111\) −447.368 −0.382543
\(112\) −616.878 −0.520442
\(113\) 2118.16 1.76336 0.881680 0.471849i \(-0.156413\pi\)
0.881680 + 0.471849i \(0.156413\pi\)
\(114\) 7093.42 5.82771
\(115\) −817.097 −0.662563
\(116\) 1084.82 0.868303
\(117\) −1903.53 −1.50412
\(118\) 160.776 0.125429
\(119\) −93.6726 −0.0721593
\(120\) −4228.66 −3.21685
\(121\) −929.092 −0.698041
\(122\) 2518.25 1.86878
\(123\) −649.680 −0.476258
\(124\) −1120.63 −0.811575
\(125\) −1305.17 −0.933902
\(126\) −788.456 −0.557470
\(127\) −2054.14 −1.43524 −0.717619 0.696436i \(-0.754768\pi\)
−0.717619 + 0.696436i \(0.754768\pi\)
\(128\) 4794.09 3.31048
\(129\) 0 0
\(130\) −1147.99 −0.774506
\(131\) −913.318 −0.609137 −0.304569 0.952490i \(-0.598512\pi\)
−0.304569 + 0.952490i \(0.598512\pi\)
\(132\) 3981.44 2.62530
\(133\) 369.896 0.241158
\(134\) 4866.85 3.13755
\(135\) −1606.95 −1.02447
\(136\) 2732.71 1.72300
\(137\) −585.919 −0.365390 −0.182695 0.983170i \(-0.558482\pi\)
−0.182695 + 0.983170i \(0.558482\pi\)
\(138\) 6598.28 4.07016
\(139\) −670.674 −0.409250 −0.204625 0.978840i \(-0.565598\pi\)
−0.204625 + 0.978840i \(0.565598\pi\)
\(140\) −348.008 −0.210086
\(141\) 3381.30 2.01955
\(142\) −2597.50 −1.53505
\(143\) 684.881 0.400508
\(144\) 13268.1 7.67830
\(145\) 305.635 0.175046
\(146\) −5613.40 −3.18198
\(147\) 3058.56 1.71609
\(148\) 1074.07 0.596542
\(149\) 1919.37 1.05531 0.527653 0.849460i \(-0.323072\pi\)
0.527653 + 0.849460i \(0.323072\pi\)
\(150\) 4329.67 2.35678
\(151\) 2246.31 1.21061 0.605304 0.795994i \(-0.293052\pi\)
0.605304 + 0.795994i \(0.293052\pi\)
\(152\) −10791.0 −5.75832
\(153\) 2014.76 1.06460
\(154\) 283.682 0.148440
\(155\) −315.723 −0.163610
\(156\) 6784.67 3.48211
\(157\) −1805.52 −0.917808 −0.458904 0.888486i \(-0.651758\pi\)
−0.458904 + 0.888486i \(0.651758\pi\)
\(158\) −1682.23 −0.847034
\(159\) −2369.39 −1.18179
\(160\) 4282.28 2.11590
\(161\) 344.076 0.168429
\(162\) 4758.98 2.30803
\(163\) −3452.86 −1.65920 −0.829598 0.558361i \(-0.811430\pi\)
−0.829598 + 0.558361i \(0.811430\pi\)
\(164\) 1559.80 0.742681
\(165\) 1121.72 0.529248
\(166\) −6257.06 −2.92555
\(167\) 2197.72 1.01835 0.509175 0.860663i \(-0.329951\pi\)
0.509175 + 0.860663i \(0.329951\pi\)
\(168\) 1780.67 0.817748
\(169\) −1029.91 −0.468781
\(170\) 1215.07 0.548187
\(171\) −7955.90 −3.55791
\(172\) 0 0
\(173\) −109.547 −0.0481427 −0.0240714 0.999710i \(-0.507663\pi\)
−0.0240714 + 0.999710i \(0.507663\pi\)
\(174\) −2468.09 −1.07532
\(175\) 225.777 0.0975264
\(176\) −4773.79 −2.04453
\(177\) −267.704 −0.113683
\(178\) 3662.60 1.54227
\(179\) −2102.20 −0.877799 −0.438899 0.898536i \(-0.644632\pi\)
−0.438899 + 0.898536i \(0.644632\pi\)
\(180\) 7485.13 3.09949
\(181\) −2790.51 −1.14595 −0.572975 0.819573i \(-0.694211\pi\)
−0.572975 + 0.819573i \(0.694211\pi\)
\(182\) 483.416 0.196885
\(183\) −4193.07 −1.69377
\(184\) −10037.7 −4.02170
\(185\) 302.607 0.120260
\(186\) 2549.55 1.00506
\(187\) −724.898 −0.283475
\(188\) −8118.05 −3.14931
\(189\) 676.679 0.260429
\(190\) −4798.10 −1.83206
\(191\) 3833.56 1.45229 0.726144 0.687543i \(-0.241311\pi\)
0.726144 + 0.687543i \(0.241311\pi\)
\(192\) −17254.7 −6.48568
\(193\) 3873.73 1.44475 0.722375 0.691501i \(-0.243050\pi\)
0.722375 + 0.691501i \(0.243050\pi\)
\(194\) −684.728 −0.253405
\(195\) 1911.50 0.701976
\(196\) −7343.20 −2.67609
\(197\) −2456.45 −0.888401 −0.444201 0.895927i \(-0.646512\pi\)
−0.444201 + 0.895927i \(0.646512\pi\)
\(198\) −6101.58 −2.19000
\(199\) −162.945 −0.0580446 −0.0290223 0.999579i \(-0.509239\pi\)
−0.0290223 + 0.999579i \(0.509239\pi\)
\(200\) −6586.59 −2.32871
\(201\) −8103.68 −2.84373
\(202\) −7934.64 −2.76376
\(203\) −128.702 −0.0444980
\(204\) −7181.10 −2.46460
\(205\) 439.454 0.149721
\(206\) 4589.09 1.55212
\(207\) −7400.56 −2.48490
\(208\) −8134.90 −2.71180
\(209\) 2862.49 0.947381
\(210\) 791.756 0.260173
\(211\) 975.959 0.318426 0.159213 0.987244i \(-0.449104\pi\)
0.159213 + 0.987244i \(0.449104\pi\)
\(212\) 5688.58 1.84289
\(213\) 4325.03 1.39130
\(214\) −5094.34 −1.62730
\(215\) 0 0
\(216\) −19740.8 −6.21847
\(217\) 132.950 0.0415908
\(218\) −408.466 −0.126903
\(219\) 9346.74 2.88399
\(220\) −2693.11 −0.825315
\(221\) −1235.28 −0.375991
\(222\) −2443.63 −0.738764
\(223\) −5259.83 −1.57948 −0.789740 0.613441i \(-0.789785\pi\)
−0.789740 + 0.613441i \(0.789785\pi\)
\(224\) −1803.25 −0.537878
\(225\) −4856.12 −1.43885
\(226\) 11569.9 3.40538
\(227\) 2907.49 0.850119 0.425060 0.905165i \(-0.360253\pi\)
0.425060 + 0.905165i \(0.360253\pi\)
\(228\) 28356.9 8.23675
\(229\) 1315.09 0.379492 0.189746 0.981833i \(-0.439234\pi\)
0.189746 + 0.981833i \(0.439234\pi\)
\(230\) −4463.17 −1.27953
\(231\) −472.353 −0.134539
\(232\) 3754.62 1.06251
\(233\) 1070.36 0.300952 0.150476 0.988614i \(-0.451919\pi\)
0.150476 + 0.988614i \(0.451919\pi\)
\(234\) −10397.5 −2.90474
\(235\) −2287.16 −0.634885
\(236\) 642.721 0.177278
\(237\) 2801.05 0.767712
\(238\) −511.661 −0.139353
\(239\) −3598.65 −0.973964 −0.486982 0.873412i \(-0.661902\pi\)
−0.486982 + 0.873412i \(0.661902\pi\)
\(240\) −13323.6 −3.58349
\(241\) 4241.99 1.13382 0.566910 0.823780i \(-0.308139\pi\)
0.566910 + 0.823780i \(0.308139\pi\)
\(242\) −5074.92 −1.34805
\(243\) −871.487 −0.230065
\(244\) 10067.0 2.64129
\(245\) −2068.86 −0.539487
\(246\) −3548.71 −0.919745
\(247\) 4877.90 1.25657
\(248\) −3878.54 −0.993096
\(249\) 10418.5 2.65158
\(250\) −7129.13 −1.80354
\(251\) −2419.46 −0.608425 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(252\) −3151.96 −0.787915
\(253\) 2662.68 0.661665
\(254\) −11220.2 −2.77172
\(255\) −2023.19 −0.496851
\(256\) 11009.2 2.68779
\(257\) −2727.33 −0.661969 −0.330984 0.943636i \(-0.607381\pi\)
−0.330984 + 0.943636i \(0.607381\pi\)
\(258\) 0 0
\(259\) −127.426 −0.0305710
\(260\) −4589.26 −1.09467
\(261\) 2768.18 0.656498
\(262\) −4988.76 −1.17636
\(263\) 1524.72 0.357485 0.178742 0.983896i \(-0.442797\pi\)
0.178742 + 0.983896i \(0.442797\pi\)
\(264\) 13780.0 3.21249
\(265\) 1602.69 0.371518
\(266\) 2020.46 0.465723
\(267\) −6098.50 −1.39784
\(268\) 19455.9 4.43454
\(269\) −1365.12 −0.309415 −0.154707 0.987960i \(-0.549444\pi\)
−0.154707 + 0.987960i \(0.549444\pi\)
\(270\) −8777.53 −1.97846
\(271\) −8496.07 −1.90443 −0.952213 0.305434i \(-0.901198\pi\)
−0.952213 + 0.305434i \(0.901198\pi\)
\(272\) 8610.22 1.91938
\(273\) −804.924 −0.178448
\(274\) −3200.42 −0.705638
\(275\) 1747.20 0.383129
\(276\) 26377.5 5.75267
\(277\) 85.6103 0.0185698 0.00928488 0.999957i \(-0.497044\pi\)
0.00928488 + 0.999957i \(0.497044\pi\)
\(278\) −3663.38 −0.790340
\(279\) −2859.55 −0.613607
\(280\) −1204.47 −0.257075
\(281\) −1787.07 −0.379387 −0.189693 0.981843i \(-0.560749\pi\)
−0.189693 + 0.981843i \(0.560749\pi\)
\(282\) 18469.4 3.90014
\(283\) 4265.75 0.896015 0.448008 0.894030i \(-0.352134\pi\)
0.448008 + 0.894030i \(0.352134\pi\)
\(284\) −10383.8 −2.16960
\(285\) 7989.20 1.66049
\(286\) 3740.98 0.773457
\(287\) −185.052 −0.0380602
\(288\) 38785.2 7.93555
\(289\) −3605.54 −0.733878
\(290\) 1669.45 0.338047
\(291\) 1140.12 0.229674
\(292\) −22440.3 −4.49733
\(293\) 7693.93 1.53408 0.767038 0.641601i \(-0.221730\pi\)
0.767038 + 0.641601i \(0.221730\pi\)
\(294\) 16706.6 3.31410
\(295\) 181.079 0.0357384
\(296\) 3717.41 0.729967
\(297\) 5236.57 1.02309
\(298\) 10484.0 2.03800
\(299\) 4537.41 0.877608
\(300\) 17308.4 3.33101
\(301\) 0 0
\(302\) 12269.8 2.33791
\(303\) 13211.8 2.50494
\(304\) −34000.2 −6.41462
\(305\) 2836.26 0.532471
\(306\) 11005.1 2.05594
\(307\) −2173.11 −0.403993 −0.201996 0.979386i \(-0.564743\pi\)
−0.201996 + 0.979386i \(0.564743\pi\)
\(308\) 1134.06 0.209801
\(309\) −7641.19 −1.40677
\(310\) −1724.55 −0.315961
\(311\) 9690.39 1.76685 0.883427 0.468568i \(-0.155230\pi\)
0.883427 + 0.468568i \(0.155230\pi\)
\(312\) 23482.1 4.26093
\(313\) −2920.33 −0.527369 −0.263685 0.964609i \(-0.584938\pi\)
−0.263685 + 0.964609i \(0.584938\pi\)
\(314\) −9862.15 −1.77246
\(315\) −888.025 −0.158840
\(316\) −6724.95 −1.19718
\(317\) 650.198 0.115201 0.0576006 0.998340i \(-0.481655\pi\)
0.0576006 + 0.998340i \(0.481655\pi\)
\(318\) −12942.1 −2.28226
\(319\) −995.976 −0.174809
\(320\) 11671.3 2.03890
\(321\) 8482.46 1.47491
\(322\) 1879.42 0.325268
\(323\) −5162.91 −0.889387
\(324\) 19024.6 3.26211
\(325\) 2977.37 0.508168
\(326\) −18860.3 −3.20422
\(327\) 680.127 0.115019
\(328\) 5398.53 0.908793
\(329\) 963.114 0.161393
\(330\) 6127.11 1.02208
\(331\) 2920.97 0.485048 0.242524 0.970145i \(-0.422025\pi\)
0.242524 + 0.970145i \(0.422025\pi\)
\(332\) −25013.4 −4.13491
\(333\) 2740.75 0.451027
\(334\) 12004.4 1.96663
\(335\) 5481.46 0.893982
\(336\) 5610.52 0.910950
\(337\) −2962.81 −0.478916 −0.239458 0.970907i \(-0.576970\pi\)
−0.239458 + 0.970907i \(0.576970\pi\)
\(338\) −5625.62 −0.905306
\(339\) −19264.7 −3.08648
\(340\) 4857.41 0.774794
\(341\) 1028.85 0.163388
\(342\) −43457.0 −6.87101
\(343\) 1759.76 0.277021
\(344\) 0 0
\(345\) 7431.53 1.15971
\(346\) −598.370 −0.0929728
\(347\) 7999.88 1.23763 0.618813 0.785538i \(-0.287614\pi\)
0.618813 + 0.785538i \(0.287614\pi\)
\(348\) −9866.49 −1.51983
\(349\) 5314.28 0.815091 0.407546 0.913185i \(-0.366385\pi\)
0.407546 + 0.913185i \(0.366385\pi\)
\(350\) 1233.25 0.188342
\(351\) 8923.51 1.35699
\(352\) −13954.7 −2.11303
\(353\) 2313.24 0.348786 0.174393 0.984676i \(-0.444204\pi\)
0.174393 + 0.984676i \(0.444204\pi\)
\(354\) −1462.26 −0.219543
\(355\) −2925.52 −0.437382
\(356\) 14641.7 2.17980
\(357\) 851.955 0.126303
\(358\) −11482.7 −1.69520
\(359\) 3108.14 0.456940 0.228470 0.973551i \(-0.426628\pi\)
0.228470 + 0.973551i \(0.426628\pi\)
\(360\) 25906.4 3.79274
\(361\) 13528.4 1.97236
\(362\) −15242.4 −2.21305
\(363\) 8450.13 1.22181
\(364\) 1932.52 0.278273
\(365\) −6322.28 −0.906639
\(366\) −22903.5 −3.27100
\(367\) 645.178 0.0917657 0.0458828 0.998947i \(-0.485390\pi\)
0.0458828 + 0.998947i \(0.485390\pi\)
\(368\) −31626.9 −4.48007
\(369\) 3980.19 0.561519
\(370\) 1652.91 0.232245
\(371\) −674.885 −0.0944429
\(372\) 10192.1 1.42053
\(373\) 11994.3 1.66498 0.832492 0.554037i \(-0.186913\pi\)
0.832492 + 0.554037i \(0.186913\pi\)
\(374\) −3959.56 −0.547444
\(375\) 11870.5 1.63465
\(376\) −28097.0 −3.85370
\(377\) −1697.22 −0.231860
\(378\) 3696.18 0.502939
\(379\) −9712.13 −1.31630 −0.658151 0.752886i \(-0.728661\pi\)
−0.658151 + 0.752886i \(0.728661\pi\)
\(380\) −19181.0 −2.58938
\(381\) 18682.5 2.51215
\(382\) 20939.8 2.80464
\(383\) 11940.6 1.59305 0.796525 0.604605i \(-0.206669\pi\)
0.796525 + 0.604605i \(0.206669\pi\)
\(384\) −43602.4 −5.79447
\(385\) 319.506 0.0422950
\(386\) 21159.2 2.79009
\(387\) 0 0
\(388\) −2737.29 −0.358157
\(389\) −2644.94 −0.344740 −0.172370 0.985032i \(-0.555143\pi\)
−0.172370 + 0.985032i \(0.555143\pi\)
\(390\) 10441.1 1.35565
\(391\) −4802.52 −0.621161
\(392\) −25415.2 −3.27464
\(393\) 8306.66 1.06620
\(394\) −13417.7 −1.71567
\(395\) −1894.67 −0.241345
\(396\) −24391.8 −3.09529
\(397\) −8950.96 −1.13158 −0.565788 0.824551i \(-0.691428\pi\)
−0.565788 + 0.824551i \(0.691428\pi\)
\(398\) −890.044 −0.112095
\(399\) −3364.22 −0.422109
\(400\) −20753.0 −2.59412
\(401\) 14096.5 1.75547 0.877736 0.479145i \(-0.159053\pi\)
0.877736 + 0.479145i \(0.159053\pi\)
\(402\) −44264.2 −5.49179
\(403\) 1753.23 0.216712
\(404\) −31719.7 −3.90623
\(405\) 5359.96 0.657626
\(406\) −702.999 −0.0859341
\(407\) −986.106 −0.120097
\(408\) −24854.1 −3.01584
\(409\) −8473.30 −1.02440 −0.512198 0.858868i \(-0.671169\pi\)
−0.512198 + 0.858868i \(0.671169\pi\)
\(410\) 2400.40 0.289140
\(411\) 5328.95 0.639557
\(412\) 18345.5 2.19373
\(413\) −76.2515 −0.00908497
\(414\) −40423.6 −4.79882
\(415\) −7047.22 −0.833577
\(416\) −23779.8 −2.80265
\(417\) 6099.80 0.716327
\(418\) 15635.6 1.82957
\(419\) 500.922 0.0584048 0.0292024 0.999574i \(-0.490703\pi\)
0.0292024 + 0.999574i \(0.490703\pi\)
\(420\) 3165.15 0.367722
\(421\) 13129.3 1.51991 0.759955 0.649976i \(-0.225221\pi\)
0.759955 + 0.649976i \(0.225221\pi\)
\(422\) 5330.92 0.614941
\(423\) −20715.1 −2.38110
\(424\) 19688.5 2.25508
\(425\) −3151.33 −0.359676
\(426\) 23624.3 2.68686
\(427\) −1194.34 −0.135358
\(428\) −20365.3 −2.29998
\(429\) −6229.01 −0.701025
\(430\) 0 0
\(431\) 4610.52 0.515269 0.257635 0.966242i \(-0.417057\pi\)
0.257635 + 0.966242i \(0.417057\pi\)
\(432\) −62199.1 −6.92722
\(433\) −11366.8 −1.26155 −0.630777 0.775964i \(-0.717264\pi\)
−0.630777 + 0.775964i \(0.717264\pi\)
\(434\) 726.202 0.0803198
\(435\) −2779.76 −0.306390
\(436\) −1632.90 −0.179361
\(437\) 18964.3 2.07594
\(438\) 51054.1 5.56954
\(439\) 4689.83 0.509871 0.254935 0.966958i \(-0.417946\pi\)
0.254935 + 0.966958i \(0.417946\pi\)
\(440\) −9320.98 −1.00991
\(441\) −18737.9 −2.02331
\(442\) −6747.39 −0.726110
\(443\) −1822.08 −0.195417 −0.0977086 0.995215i \(-0.531151\pi\)
−0.0977086 + 0.995215i \(0.531151\pi\)
\(444\) −9768.72 −1.04415
\(445\) 4125.12 0.439437
\(446\) −28730.4 −3.05028
\(447\) −17456.7 −1.84715
\(448\) −4914.75 −0.518304
\(449\) 12923.6 1.35836 0.679180 0.733972i \(-0.262336\pi\)
0.679180 + 0.733972i \(0.262336\pi\)
\(450\) −26525.2 −2.77869
\(451\) −1432.05 −0.149518
\(452\) 46252.1 4.81309
\(453\) −20430.2 −2.11898
\(454\) 15881.4 1.64174
\(455\) 544.463 0.0560985
\(456\) 98144.4 10.0790
\(457\) 1087.83 0.111349 0.0556743 0.998449i \(-0.482269\pi\)
0.0556743 + 0.998449i \(0.482269\pi\)
\(458\) 7183.33 0.732871
\(459\) −9444.91 −0.960459
\(460\) −17842.1 −1.80846
\(461\) −1854.32 −0.187341 −0.0936707 0.995603i \(-0.529860\pi\)
−0.0936707 + 0.995603i \(0.529860\pi\)
\(462\) −2580.10 −0.259821
\(463\) −19191.6 −1.92637 −0.963186 0.268834i \(-0.913362\pi\)
−0.963186 + 0.268834i \(0.913362\pi\)
\(464\) 11830.0 1.18361
\(465\) 2871.51 0.286372
\(466\) 5846.58 0.581196
\(467\) −3701.95 −0.366822 −0.183411 0.983036i \(-0.558714\pi\)
−0.183411 + 0.983036i \(0.558714\pi\)
\(468\) −41565.5 −4.10548
\(469\) −2308.22 −0.227257
\(470\) −12493.0 −1.22608
\(471\) 16421.2 1.60648
\(472\) 2224.49 0.216929
\(473\) 0 0
\(474\) 15300.0 1.48260
\(475\) 12444.0 1.20205
\(476\) −2045.43 −0.196959
\(477\) 14515.8 1.39336
\(478\) −19656.7 −1.88091
\(479\) 8304.65 0.792169 0.396085 0.918214i \(-0.370369\pi\)
0.396085 + 0.918214i \(0.370369\pi\)
\(480\) −38947.5 −3.70355
\(481\) −1680.40 −0.159292
\(482\) 23170.7 2.18962
\(483\) −3129.38 −0.294807
\(484\) −20287.6 −1.90530
\(485\) −771.197 −0.0722026
\(486\) −4760.26 −0.444300
\(487\) −15674.8 −1.45851 −0.729254 0.684243i \(-0.760133\pi\)
−0.729254 + 0.684243i \(0.760133\pi\)
\(488\) 34842.4 3.23205
\(489\) 31403.9 2.90416
\(490\) −11300.6 −1.04185
\(491\) 13069.6 1.20127 0.600636 0.799522i \(-0.294914\pi\)
0.600636 + 0.799522i \(0.294914\pi\)
\(492\) −14186.4 −1.29995
\(493\) 1796.38 0.164108
\(494\) 26644.2 2.42668
\(495\) −6872.10 −0.623996
\(496\) −12220.5 −1.10628
\(497\) 1231.92 0.111186
\(498\) 56908.2 5.12072
\(499\) −9756.69 −0.875289 −0.437645 0.899148i \(-0.644187\pi\)
−0.437645 + 0.899148i \(0.644187\pi\)
\(500\) −28499.6 −2.54908
\(501\) −19988.3 −1.78246
\(502\) −13215.6 −1.17498
\(503\) 14365.4 1.27340 0.636700 0.771112i \(-0.280299\pi\)
0.636700 + 0.771112i \(0.280299\pi\)
\(504\) −10909.1 −0.964144
\(505\) −8936.65 −0.787476
\(506\) 14544.2 1.27780
\(507\) 9367.09 0.820527
\(508\) −44854.1 −3.91748
\(509\) −13972.3 −1.21672 −0.608360 0.793662i \(-0.708172\pi\)
−0.608360 + 0.793662i \(0.708172\pi\)
\(510\) −11051.1 −0.959513
\(511\) 2662.29 0.230475
\(512\) 21782.0 1.88015
\(513\) 37296.2 3.20988
\(514\) −14897.3 −1.27839
\(515\) 5168.62 0.442246
\(516\) 0 0
\(517\) 7453.19 0.634025
\(518\) −696.032 −0.0590384
\(519\) 996.332 0.0842662
\(520\) −15883.6 −1.33951
\(521\) −13785.0 −1.15918 −0.579588 0.814910i \(-0.696787\pi\)
−0.579588 + 0.814910i \(0.696787\pi\)
\(522\) 15120.4 1.26782
\(523\) 8431.00 0.704898 0.352449 0.935831i \(-0.385349\pi\)
0.352449 + 0.935831i \(0.385349\pi\)
\(524\) −19943.2 −1.66264
\(525\) −2053.45 −0.170704
\(526\) 8328.40 0.690372
\(527\) −1855.68 −0.153386
\(528\) 43417.8 3.57863
\(529\) 5473.53 0.449867
\(530\) 8754.26 0.717473
\(531\) 1640.06 0.134035
\(532\) 8077.04 0.658241
\(533\) −2440.32 −0.198315
\(534\) −33311.4 −2.69949
\(535\) −5737.67 −0.463665
\(536\) 67337.7 5.42639
\(537\) 19119.6 1.53645
\(538\) −7456.58 −0.597539
\(539\) 6741.80 0.538756
\(540\) −35089.3 −2.79630
\(541\) 375.733 0.0298596 0.0149298 0.999889i \(-0.495248\pi\)
0.0149298 + 0.999889i \(0.495248\pi\)
\(542\) −46407.5 −3.67781
\(543\) 25379.8 2.00580
\(544\) 25169.3 1.98369
\(545\) −460.048 −0.0361584
\(546\) −4396.68 −0.344617
\(547\) 9627.22 0.752523 0.376261 0.926514i \(-0.377209\pi\)
0.376261 + 0.926514i \(0.377209\pi\)
\(548\) −12794.1 −0.997331
\(549\) 25688.4 1.99700
\(550\) 9543.64 0.739894
\(551\) −7093.60 −0.548453
\(552\) 91293.7 7.03934
\(553\) 797.838 0.0613518
\(554\) 467.623 0.0358618
\(555\) −2752.22 −0.210496
\(556\) −14644.8 −1.11705
\(557\) −13673.1 −1.04012 −0.520061 0.854129i \(-0.674091\pi\)
−0.520061 + 0.854129i \(0.674091\pi\)
\(558\) −15619.5 −1.18499
\(559\) 0 0
\(560\) −3795.04 −0.286375
\(561\) 6592.97 0.496177
\(562\) −9761.39 −0.732668
\(563\) −5880.29 −0.440186 −0.220093 0.975479i \(-0.570636\pi\)
−0.220093 + 0.975479i \(0.570636\pi\)
\(564\) 73834.0 5.51236
\(565\) 13031.0 0.970295
\(566\) 23300.5 1.73038
\(567\) −2257.06 −0.167174
\(568\) −35939.0 −2.65487
\(569\) −14311.3 −1.05441 −0.527207 0.849737i \(-0.676761\pi\)
−0.527207 + 0.849737i \(0.676761\pi\)
\(570\) 43638.9 3.20672
\(571\) −875.082 −0.0641349 −0.0320675 0.999486i \(-0.510209\pi\)
−0.0320675 + 0.999486i \(0.510209\pi\)
\(572\) 14955.0 1.09318
\(573\) −34866.4 −2.54200
\(574\) −1010.80 −0.0735015
\(575\) 11575.4 0.839527
\(576\) 105709. 7.64677
\(577\) −1692.97 −0.122147 −0.0610737 0.998133i \(-0.519452\pi\)
−0.0610737 + 0.998133i \(0.519452\pi\)
\(578\) −19694.3 −1.41726
\(579\) −35231.7 −2.52881
\(580\) 6673.85 0.477787
\(581\) 2967.55 0.211902
\(582\) 6227.62 0.443545
\(583\) −5222.69 −0.371015
\(584\) −77667.0 −5.50322
\(585\) −11710.6 −0.827646
\(586\) 42026.1 2.96259
\(587\) −9558.71 −0.672112 −0.336056 0.941842i \(-0.609093\pi\)
−0.336056 + 0.941842i \(0.609093\pi\)
\(588\) 66786.7 4.68407
\(589\) 7327.73 0.512621
\(590\) 989.095 0.0690176
\(591\) 22341.5 1.55501
\(592\) 11712.8 0.813164
\(593\) −380.686 −0.0263624 −0.0131812 0.999913i \(-0.504196\pi\)
−0.0131812 + 0.999913i \(0.504196\pi\)
\(594\) 28603.4 1.97578
\(595\) −576.276 −0.0397059
\(596\) 41911.3 2.88046
\(597\) 1481.99 0.101598
\(598\) 24784.4 1.69483
\(599\) 18267.5 1.24606 0.623031 0.782197i \(-0.285901\pi\)
0.623031 + 0.782197i \(0.285901\pi\)
\(600\) 59905.3 4.07604
\(601\) −3526.45 −0.239346 −0.119673 0.992813i \(-0.538185\pi\)
−0.119673 + 0.992813i \(0.538185\pi\)
\(602\) 0 0
\(603\) 49646.3 3.35282
\(604\) 49050.3 3.30435
\(605\) −5715.80 −0.384099
\(606\) 72165.8 4.83752
\(607\) 1688.55 0.112910 0.0564548 0.998405i \(-0.482020\pi\)
0.0564548 + 0.998405i \(0.482020\pi\)
\(608\) −99389.0 −6.62953
\(609\) 1170.55 0.0778866
\(610\) 15492.3 1.02830
\(611\) 12700.8 0.840948
\(612\) 43994.2 2.90582
\(613\) 19844.4 1.30752 0.653759 0.756703i \(-0.273191\pi\)
0.653759 + 0.756703i \(0.273191\pi\)
\(614\) −11870.0 −0.780187
\(615\) −3996.85 −0.262063
\(616\) 3925.02 0.256727
\(617\) 12892.5 0.841218 0.420609 0.907242i \(-0.361816\pi\)
0.420609 + 0.907242i \(0.361816\pi\)
\(618\) −41738.0 −2.71674
\(619\) −17148.5 −1.11350 −0.556751 0.830679i \(-0.687952\pi\)
−0.556751 + 0.830679i \(0.687952\pi\)
\(620\) −6894.12 −0.446572
\(621\) 34692.8 2.24183
\(622\) 52931.2 3.41213
\(623\) −1737.07 −0.111708
\(624\) 73987.2 4.74657
\(625\) 2864.66 0.183338
\(626\) −15951.5 −1.01845
\(627\) −26034.5 −1.65824
\(628\) −39425.2 −2.50515
\(629\) 1778.58 0.112745
\(630\) −4850.60 −0.306750
\(631\) 978.279 0.0617190 0.0308595 0.999524i \(-0.490176\pi\)
0.0308595 + 0.999524i \(0.490176\pi\)
\(632\) −23275.4 −1.46494
\(633\) −8876.38 −0.557353
\(634\) 3551.54 0.222476
\(635\) −12637.1 −0.789745
\(636\) −51737.9 −3.22569
\(637\) 11488.5 0.714587
\(638\) −5440.25 −0.337589
\(639\) −26496.8 −1.64037
\(640\) 29493.3 1.82160
\(641\) −12867.5 −0.792880 −0.396440 0.918061i \(-0.629754\pi\)
−0.396440 + 0.918061i \(0.629754\pi\)
\(642\) 46333.2 2.84833
\(643\) 13732.1 0.842211 0.421106 0.907012i \(-0.361642\pi\)
0.421106 + 0.907012i \(0.361642\pi\)
\(644\) 7513.24 0.459725
\(645\) 0 0
\(646\) −28201.0 −1.71758
\(647\) −2460.71 −0.149522 −0.0747608 0.997201i \(-0.523819\pi\)
−0.0747608 + 0.997201i \(0.523819\pi\)
\(648\) 65845.2 3.99173
\(649\) −590.083 −0.0356899
\(650\) 16263.1 0.981369
\(651\) −1209.18 −0.0727981
\(652\) −75396.6 −4.52877
\(653\) 26816.4 1.60706 0.803528 0.595267i \(-0.202954\pi\)
0.803528 + 0.595267i \(0.202954\pi\)
\(654\) 3715.01 0.222123
\(655\) −5618.75 −0.335180
\(656\) 17009.7 1.01237
\(657\) −57261.7 −3.40029
\(658\) 5260.75 0.311680
\(659\) −943.089 −0.0557474 −0.0278737 0.999611i \(-0.508874\pi\)
−0.0278737 + 0.999611i \(0.508874\pi\)
\(660\) 24493.9 1.44458
\(661\) 14083.2 0.828706 0.414353 0.910116i \(-0.364008\pi\)
0.414353 + 0.910116i \(0.364008\pi\)
\(662\) 15955.0 0.936721
\(663\) 11234.9 0.658112
\(664\) −86572.6 −5.05974
\(665\) 2275.61 0.132698
\(666\) 14970.6 0.871020
\(667\) −6598.44 −0.383048
\(668\) 47989.3 2.77958
\(669\) 47838.3 2.76463
\(670\) 29941.0 1.72645
\(671\) −9242.53 −0.531750
\(672\) 16400.6 0.941470
\(673\) 4292.74 0.245873 0.122937 0.992415i \(-0.460769\pi\)
0.122937 + 0.992415i \(0.460769\pi\)
\(674\) −16183.6 −0.924878
\(675\) 22764.8 1.29810
\(676\) −22489.2 −1.27954
\(677\) 19977.3 1.13411 0.567054 0.823681i \(-0.308083\pi\)
0.567054 + 0.823681i \(0.308083\pi\)
\(678\) −105228. −5.96058
\(679\) 324.748 0.0183545
\(680\) 16811.7 0.948088
\(681\) −26443.8 −1.48800
\(682\) 5619.81 0.315533
\(683\) 16248.8 0.910314 0.455157 0.890411i \(-0.349583\pi\)
0.455157 + 0.890411i \(0.349583\pi\)
\(684\) −173725. −9.71132
\(685\) −3604.59 −0.201057
\(686\) 9612.21 0.534980
\(687\) −11960.8 −0.664239
\(688\) 0 0
\(689\) −8899.86 −0.492101
\(690\) 40592.7 2.23962
\(691\) 27470.0 1.51231 0.756155 0.654392i \(-0.227075\pi\)
0.756155 + 0.654392i \(0.227075\pi\)
\(692\) −2392.06 −0.131405
\(693\) 2893.81 0.158625
\(694\) 43697.2 2.39009
\(695\) −4126.00 −0.225191
\(696\) −34148.4 −1.85976
\(697\) 2582.91 0.140365
\(698\) 29027.8 1.57410
\(699\) −9735.00 −0.526769
\(700\) 4930.06 0.266198
\(701\) −5373.47 −0.289519 −0.144760 0.989467i \(-0.546241\pi\)
−0.144760 + 0.989467i \(0.546241\pi\)
\(702\) 48742.3 2.62060
\(703\) −7023.30 −0.376798
\(704\) −38033.5 −2.03614
\(705\) 20801.8 1.11126
\(706\) 12635.5 0.673573
\(707\) 3763.18 0.200183
\(708\) −5845.57 −0.310297
\(709\) 22366.6 1.18476 0.592380 0.805658i \(-0.298188\pi\)
0.592380 + 0.805658i \(0.298188\pi\)
\(710\) −15979.9 −0.844667
\(711\) −17160.3 −0.905150
\(712\) 50675.6 2.66735
\(713\) 6816.23 0.358022
\(714\) 4653.58 0.243916
\(715\) 4213.40 0.220381
\(716\) −45903.6 −2.39595
\(717\) 32729.8 1.70477
\(718\) 16977.4 0.882439
\(719\) −19417.5 −1.00716 −0.503582 0.863947i \(-0.667985\pi\)
−0.503582 + 0.863947i \(0.667985\pi\)
\(720\) 81625.7 4.22501
\(721\) −2176.48 −0.112422
\(722\) 73895.2 3.80900
\(723\) −38581.1 −1.98457
\(724\) −60933.5 −3.12787
\(725\) −4329.78 −0.221799
\(726\) 46156.6 2.35955
\(727\) 5295.15 0.270133 0.135066 0.990837i \(-0.456875\pi\)
0.135066 + 0.990837i \(0.456875\pi\)
\(728\) 6688.53 0.340513
\(729\) −15597.6 −0.792439
\(730\) −34533.8 −1.75089
\(731\) 0 0
\(732\) −91559.9 −4.62316
\(733\) 3812.57 0.192115 0.0960576 0.995376i \(-0.469377\pi\)
0.0960576 + 0.995376i \(0.469377\pi\)
\(734\) 3524.11 0.177217
\(735\) 18816.3 0.944286
\(736\) −92451.3 −4.63016
\(737\) −17862.5 −0.892771
\(738\) 21740.7 1.08440
\(739\) −4166.76 −0.207411 −0.103706 0.994608i \(-0.533070\pi\)
−0.103706 + 0.994608i \(0.533070\pi\)
\(740\) 6607.72 0.328249
\(741\) −44364.6 −2.19943
\(742\) −3686.38 −0.182387
\(743\) 25570.4 1.26257 0.631284 0.775552i \(-0.282528\pi\)
0.631284 + 0.775552i \(0.282528\pi\)
\(744\) 35275.5 1.73826
\(745\) 11808.0 0.580686
\(746\) 65515.4 3.21540
\(747\) −63827.6 −3.12628
\(748\) −15828.9 −0.773744
\(749\) 2416.11 0.117867
\(750\) 64839.7 3.15681
\(751\) −8697.56 −0.422608 −0.211304 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(752\) −88527.7 −4.29292
\(753\) 22005.0 1.06495
\(754\) −9270.59 −0.447765
\(755\) 13819.3 0.666141
\(756\) 14776.0 0.710842
\(757\) 6943.97 0.333399 0.166699 0.986008i \(-0.446689\pi\)
0.166699 + 0.986008i \(0.446689\pi\)
\(758\) −53049.9 −2.54203
\(759\) −24217.2 −1.15814
\(760\) −66386.4 −3.16854
\(761\) −14571.8 −0.694121 −0.347060 0.937843i \(-0.612820\pi\)
−0.347060 + 0.937843i \(0.612820\pi\)
\(762\) 102048. 4.85145
\(763\) 193.724 0.00919173
\(764\) 83709.6 3.96402
\(765\) 12394.8 0.585798
\(766\) 65222.6 3.07649
\(767\) −1005.55 −0.0473379
\(768\) −100129. −4.70455
\(769\) −32318.2 −1.51550 −0.757752 0.652543i \(-0.773702\pi\)
−0.757752 + 0.652543i \(0.773702\pi\)
\(770\) 1745.22 0.0816796
\(771\) 24805.1 1.15867
\(772\) 84586.6 3.94344
\(773\) −1785.68 −0.0830874 −0.0415437 0.999137i \(-0.513228\pi\)
−0.0415437 + 0.999137i \(0.513228\pi\)
\(774\) 0 0
\(775\) 4472.69 0.207308
\(776\) −9473.88 −0.438264
\(777\) 1158.95 0.0535097
\(778\) −14447.3 −0.665759
\(779\) −10199.4 −0.469105
\(780\) 41739.5 1.91604
\(781\) 9533.41 0.436789
\(782\) −26232.5 −1.19958
\(783\) −12976.9 −0.592280
\(784\) −80077.9 −3.64786
\(785\) −11107.6 −0.505027
\(786\) 45372.9 2.05903
\(787\) 7591.08 0.343828 0.171914 0.985112i \(-0.445005\pi\)
0.171914 + 0.985112i \(0.445005\pi\)
\(788\) −53639.1 −2.42489
\(789\) −13867.4 −0.625720
\(790\) −10349.1 −0.466083
\(791\) −5487.28 −0.246656
\(792\) −84421.3 −3.78760
\(793\) −15750.0 −0.705293
\(794\) −48892.2 −2.18529
\(795\) −14576.5 −0.650284
\(796\) −3558.07 −0.158433
\(797\) −11200.4 −0.497790 −0.248895 0.968531i \(-0.580067\pi\)
−0.248895 + 0.968531i \(0.580067\pi\)
\(798\) −18376.1 −0.815173
\(799\) −13442.9 −0.595213
\(800\) −60664.9 −2.68104
\(801\) 37361.8 1.64808
\(802\) 76998.2 3.39015
\(803\) 20602.5 0.905411
\(804\) −176952. −7.76196
\(805\) 2116.76 0.0926784
\(806\) 9576.58 0.418512
\(807\) 12415.8 0.541582
\(808\) −109784. −4.77991
\(809\) −23832.0 −1.03571 −0.517854 0.855469i \(-0.673269\pi\)
−0.517854 + 0.855469i \(0.673269\pi\)
\(810\) 29277.3 1.27000
\(811\) 23707.2 1.02648 0.513239 0.858246i \(-0.328445\pi\)
0.513239 + 0.858246i \(0.328445\pi\)
\(812\) −2810.33 −0.121457
\(813\) 77272.1 3.33339
\(814\) −5386.34 −0.231930
\(815\) −21242.1 −0.912978
\(816\) −78310.2 −3.35957
\(817\) 0 0
\(818\) −46283.1 −1.97830
\(819\) 4931.27 0.210394
\(820\) 9595.91 0.408663
\(821\) −27578.1 −1.17233 −0.586165 0.810192i \(-0.699363\pi\)
−0.586165 + 0.810192i \(0.699363\pi\)
\(822\) 29108.0 1.23511
\(823\) 14566.9 0.616975 0.308488 0.951228i \(-0.400177\pi\)
0.308488 + 0.951228i \(0.400177\pi\)
\(824\) 63494.6 2.68439
\(825\) −15890.9 −0.670606
\(826\) −416.504 −0.0175448
\(827\) 46389.9 1.95059 0.975293 0.220917i \(-0.0709049\pi\)
0.975293 + 0.220917i \(0.0709049\pi\)
\(828\) −161598. −6.78253
\(829\) −17471.0 −0.731957 −0.365979 0.930623i \(-0.619266\pi\)
−0.365979 + 0.930623i \(0.619266\pi\)
\(830\) −38493.6 −1.60980
\(831\) −778.629 −0.0325034
\(832\) −64811.9 −2.70066
\(833\) −12159.8 −0.505777
\(834\) 33318.5 1.38336
\(835\) 13520.4 0.560350
\(836\) 62505.3 2.58587
\(837\) 13405.2 0.553585
\(838\) 2736.15 0.112791
\(839\) −31457.8 −1.29445 −0.647225 0.762299i \(-0.724070\pi\)
−0.647225 + 0.762299i \(0.724070\pi\)
\(840\) 10954.7 0.449969
\(841\) −21920.9 −0.898801
\(842\) 71715.2 2.93524
\(843\) 16253.5 0.664056
\(844\) 21311.0 0.869142
\(845\) −6336.04 −0.257949
\(846\) −113151. −4.59835
\(847\) 2406.90 0.0976411
\(848\) 62034.3 2.51211
\(849\) −38797.1 −1.56833
\(850\) −17213.3 −0.694602
\(851\) −6533.06 −0.263161
\(852\) 94441.4 3.79755
\(853\) 12947.7 0.519720 0.259860 0.965646i \(-0.416324\pi\)
0.259860 + 0.965646i \(0.416324\pi\)
\(854\) −6523.74 −0.261403
\(855\) −48944.9 −1.95775
\(856\) −70485.2 −2.81441
\(857\) −22730.2 −0.906007 −0.453004 0.891509i \(-0.649648\pi\)
−0.453004 + 0.891509i \(0.649648\pi\)
\(858\) −34024.3 −1.35381
\(859\) 10019.6 0.397980 0.198990 0.980002i \(-0.436234\pi\)
0.198990 + 0.980002i \(0.436234\pi\)
\(860\) 0 0
\(861\) 1683.06 0.0666183
\(862\) 25183.8 0.995084
\(863\) 33745.3 1.33106 0.665530 0.746371i \(-0.268206\pi\)
0.665530 + 0.746371i \(0.268206\pi\)
\(864\) −181820. −7.15930
\(865\) −673.934 −0.0264907
\(866\) −62088.0 −2.43630
\(867\) 32792.5 1.28454
\(868\) 2903.09 0.113522
\(869\) 6174.18 0.241018
\(870\) −15183.7 −0.591697
\(871\) −30439.0 −1.18414
\(872\) −5651.53 −0.219478
\(873\) −6984.84 −0.270791
\(874\) 103587. 4.00903
\(875\) 3381.15 0.130633
\(876\) 204095. 7.87185
\(877\) −9357.18 −0.360285 −0.180142 0.983641i \(-0.557656\pi\)
−0.180142 + 0.983641i \(0.557656\pi\)
\(878\) 25616.9 0.984658
\(879\) −69976.6 −2.68516
\(880\) −29368.5 −1.12501
\(881\) 18561.7 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(882\) −102351. −3.90740
\(883\) −28177.1 −1.07388 −0.536940 0.843621i \(-0.680420\pi\)
−0.536940 + 0.843621i \(0.680420\pi\)
\(884\) −26973.6 −1.02627
\(885\) −1646.92 −0.0625543
\(886\) −9952.65 −0.377388
\(887\) −13672.0 −0.517543 −0.258771 0.965939i \(-0.583318\pi\)
−0.258771 + 0.965939i \(0.583318\pi\)
\(888\) −33810.0 −1.27769
\(889\) 5321.43 0.200759
\(890\) 22532.4 0.848637
\(891\) −17466.5 −0.656735
\(892\) −114854. −4.31119
\(893\) 53083.5 1.98922
\(894\) −95352.6 −3.56719
\(895\) −12932.8 −0.483012
\(896\) −12419.5 −0.463066
\(897\) −41267.9 −1.53611
\(898\) 70591.9 2.62325
\(899\) −2549.61 −0.0945876
\(900\) −106038. −3.92734
\(901\) 9419.87 0.348304
\(902\) −7822.20 −0.288748
\(903\) 0 0
\(904\) 160081. 5.88960
\(905\) −17167.3 −0.630563
\(906\) −111595. −4.09215
\(907\) −9828.52 −0.359813 −0.179907 0.983684i \(-0.557580\pi\)
−0.179907 + 0.983684i \(0.557580\pi\)
\(908\) 63488.0 2.32040
\(909\) −80940.4 −2.95338
\(910\) 2973.98 0.108337
\(911\) −30471.1 −1.10818 −0.554091 0.832456i \(-0.686934\pi\)
−0.554091 + 0.832456i \(0.686934\pi\)
\(912\) 309233. 11.2278
\(913\) 22964.8 0.832448
\(914\) 5941.95 0.215035
\(915\) −25795.9 −0.932006
\(916\) 28716.3 1.03582
\(917\) 2366.03 0.0852053
\(918\) −51590.3 −1.85483
\(919\) 42727.5 1.53368 0.766839 0.641839i \(-0.221828\pi\)
0.766839 + 0.641839i \(0.221828\pi\)
\(920\) −61752.4 −2.21295
\(921\) 19764.5 0.707125
\(922\) −10128.7 −0.361792
\(923\) 16245.6 0.579341
\(924\) −10314.3 −0.367224
\(925\) −4286.88 −0.152380
\(926\) −104829. −3.72019
\(927\) 46812.9 1.65861
\(928\) 34581.4 1.22327
\(929\) 32717.0 1.15545 0.577723 0.816233i \(-0.303941\pi\)
0.577723 + 0.816233i \(0.303941\pi\)
\(930\) 15684.9 0.553040
\(931\) 48016.8 1.69032
\(932\) 23372.4 0.821449
\(933\) −88134.5 −3.09260
\(934\) −20220.9 −0.708403
\(935\) −4459.59 −0.155983
\(936\) −143860. −5.02374
\(937\) 51528.4 1.79654 0.898271 0.439443i \(-0.144824\pi\)
0.898271 + 0.439443i \(0.144824\pi\)
\(938\) −12608.0 −0.438877
\(939\) 26560.5 0.923076
\(940\) −49942.4 −1.73292
\(941\) 26909.6 0.932229 0.466114 0.884724i \(-0.345653\pi\)
0.466114 + 0.884724i \(0.345653\pi\)
\(942\) 89696.6 3.10241
\(943\) −9487.49 −0.327630
\(944\) 7008.90 0.241653
\(945\) 4162.94 0.143302
\(946\) 0 0
\(947\) 18441.8 0.632818 0.316409 0.948623i \(-0.397523\pi\)
0.316409 + 0.948623i \(0.397523\pi\)
\(948\) 61163.7 2.09547
\(949\) 35108.1 1.20090
\(950\) 67972.2 2.32138
\(951\) −5913.58 −0.201641
\(952\) −7079.34 −0.241011
\(953\) −35819.6 −1.21754 −0.608768 0.793348i \(-0.708336\pi\)
−0.608768 + 0.793348i \(0.708336\pi\)
\(954\) 79288.5 2.69084
\(955\) 23584.2 0.799126
\(956\) −78580.1 −2.65843
\(957\) 9058.43 0.305975
\(958\) 45361.9 1.52983
\(959\) 1517.88 0.0511103
\(960\) −106151. −3.56877
\(961\) −27157.2 −0.911592
\(962\) −9178.73 −0.307624
\(963\) −51966.8 −1.73895
\(964\) 92628.1 3.09476
\(965\) 23831.2 0.794979
\(966\) −17093.4 −0.569329
\(967\) −28557.3 −0.949680 −0.474840 0.880072i \(-0.657494\pi\)
−0.474840 + 0.880072i \(0.657494\pi\)
\(968\) −70216.5 −2.33145
\(969\) 46956.9 1.55673
\(970\) −4212.46 −0.139437
\(971\) −32692.2 −1.08048 −0.540238 0.841513i \(-0.681666\pi\)
−0.540238 + 0.841513i \(0.681666\pi\)
\(972\) −19029.8 −0.627963
\(973\) 1737.44 0.0572454
\(974\) −85619.5 −2.81666
\(975\) −27079.3 −0.889467
\(976\) 109781. 3.60042
\(977\) −2318.74 −0.0759294 −0.0379647 0.999279i \(-0.512087\pi\)
−0.0379647 + 0.999279i \(0.512087\pi\)
\(978\) 171535. 5.60848
\(979\) −13442.6 −0.438842
\(980\) −45175.5 −1.47253
\(981\) −4166.72 −0.135610
\(982\) 71389.5 2.31989
\(983\) −26323.2 −0.854099 −0.427050 0.904228i \(-0.640447\pi\)
−0.427050 + 0.904228i \(0.640447\pi\)
\(984\) −49099.8 −1.59070
\(985\) −15112.2 −0.488846
\(986\) 9812.27 0.316923
\(987\) −8759.56 −0.282492
\(988\) 106514. 3.42981
\(989\) 0 0
\(990\) −37537.0 −1.20506
\(991\) −57890.7 −1.85566 −0.927830 0.373004i \(-0.878328\pi\)
−0.927830 + 0.373004i \(0.878328\pi\)
\(992\) −35722.8 −1.14335
\(993\) −26566.3 −0.849000
\(994\) 6729.05 0.214721
\(995\) −1002.44 −0.0319392
\(996\) 227498. 7.23749
\(997\) −10032.0 −0.318673 −0.159337 0.987224i \(-0.550936\pi\)
−0.159337 + 0.987224i \(0.550936\pi\)
\(998\) −53293.3 −1.69035
\(999\) −12848.3 −0.406908
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.1.50 yes 50
43.42 odd 2 1849.4.a.i.1.1 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.1 50 43.42 odd 2
1849.4.a.j.1.50 yes 50 1.1 even 1 trivial