Properties

Label 1849.4.a.i.1.43
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 1.43
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.15048 q^{2} -4.65912 q^{3} +9.22650 q^{4} -19.9023 q^{5} -19.3376 q^{6} +27.8552 q^{7} +5.09055 q^{8} -5.29261 q^{9} +O(q^{10})\) \(q+4.15048 q^{2} -4.65912 q^{3} +9.22650 q^{4} -19.9023 q^{5} -19.3376 q^{6} +27.8552 q^{7} +5.09055 q^{8} -5.29261 q^{9} -82.6043 q^{10} -0.733230 q^{11} -42.9873 q^{12} +67.6888 q^{13} +115.613 q^{14} +92.7274 q^{15} -52.6837 q^{16} -45.0640 q^{17} -21.9669 q^{18} +96.4948 q^{19} -183.629 q^{20} -129.781 q^{21} -3.04326 q^{22} -76.2235 q^{23} -23.7175 q^{24} +271.103 q^{25} +280.941 q^{26} +150.455 q^{27} +257.006 q^{28} -182.776 q^{29} +384.863 q^{30} -37.9792 q^{31} -259.387 q^{32} +3.41621 q^{33} -187.037 q^{34} -554.384 q^{35} -48.8323 q^{36} +95.8006 q^{37} +400.500 q^{38} -315.370 q^{39} -101.314 q^{40} +33.5766 q^{41} -538.653 q^{42} -6.76515 q^{44} +105.335 q^{45} -316.364 q^{46} +533.353 q^{47} +245.460 q^{48} +432.913 q^{49} +1125.21 q^{50} +209.958 q^{51} +624.530 q^{52} -0.134055 q^{53} +624.461 q^{54} +14.5930 q^{55} +141.798 q^{56} -449.581 q^{57} -758.607 q^{58} -110.795 q^{59} +855.549 q^{60} +533.253 q^{61} -157.632 q^{62} -147.427 q^{63} -655.112 q^{64} -1347.16 q^{65} +14.1789 q^{66} -411.431 q^{67} -415.783 q^{68} +355.134 q^{69} -2300.96 q^{70} -216.076 q^{71} -26.9423 q^{72} -136.316 q^{73} +397.619 q^{74} -1263.10 q^{75} +890.309 q^{76} -20.4243 q^{77} -1308.94 q^{78} +81.5918 q^{79} +1048.53 q^{80} -558.088 q^{81} +139.359 q^{82} -926.163 q^{83} -1197.42 q^{84} +896.879 q^{85} +851.573 q^{87} -3.73254 q^{88} -1125.82 q^{89} +437.193 q^{90} +1885.49 q^{91} -703.276 q^{92} +176.950 q^{93} +2213.67 q^{94} -1920.47 q^{95} +1208.52 q^{96} -1512.02 q^{97} +1796.80 q^{98} +3.88071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.15048 1.46742 0.733708 0.679465i \(-0.237788\pi\)
0.733708 + 0.679465i \(0.237788\pi\)
\(3\) −4.65912 −0.896648 −0.448324 0.893871i \(-0.647979\pi\)
−0.448324 + 0.893871i \(0.647979\pi\)
\(4\) 9.22650 1.15331
\(5\) −19.9023 −1.78012 −0.890060 0.455844i \(-0.849338\pi\)
−0.890060 + 0.455844i \(0.849338\pi\)
\(6\) −19.3376 −1.31576
\(7\) 27.8552 1.50404 0.752020 0.659140i \(-0.229079\pi\)
0.752020 + 0.659140i \(0.229079\pi\)
\(8\) 5.09055 0.224973
\(9\) −5.29261 −0.196023
\(10\) −82.6043 −2.61218
\(11\) −0.733230 −0.0200979 −0.0100490 0.999950i \(-0.503199\pi\)
−0.0100490 + 0.999950i \(0.503199\pi\)
\(12\) −42.9873 −1.03411
\(13\) 67.6888 1.44411 0.722057 0.691834i \(-0.243197\pi\)
0.722057 + 0.691834i \(0.243197\pi\)
\(14\) 115.613 2.20705
\(15\) 92.7274 1.59614
\(16\) −52.6837 −0.823183
\(17\) −45.0640 −0.642919 −0.321460 0.946923i \(-0.604173\pi\)
−0.321460 + 0.946923i \(0.604173\pi\)
\(18\) −21.9669 −0.287647
\(19\) 96.4948 1.16513 0.582564 0.812785i \(-0.302050\pi\)
0.582564 + 0.812785i \(0.302050\pi\)
\(20\) −183.629 −2.05303
\(21\) −129.781 −1.34859
\(22\) −3.04326 −0.0294920
\(23\) −76.2235 −0.691031 −0.345515 0.938413i \(-0.612296\pi\)
−0.345515 + 0.938413i \(0.612296\pi\)
\(24\) −23.7175 −0.201721
\(25\) 271.103 2.16883
\(26\) 280.941 2.11912
\(27\) 150.455 1.07241
\(28\) 257.006 1.73463
\(29\) −182.776 −1.17036 −0.585182 0.810902i \(-0.698977\pi\)
−0.585182 + 0.810902i \(0.698977\pi\)
\(30\) 384.863 2.34220
\(31\) −37.9792 −0.220041 −0.110020 0.993929i \(-0.535092\pi\)
−0.110020 + 0.993929i \(0.535092\pi\)
\(32\) −259.387 −1.43293
\(33\) 3.41621 0.0180208
\(34\) −187.037 −0.943430
\(35\) −554.384 −2.67737
\(36\) −48.8323 −0.226075
\(37\) 95.8006 0.425663 0.212831 0.977089i \(-0.431731\pi\)
0.212831 + 0.977089i \(0.431731\pi\)
\(38\) 400.500 1.70973
\(39\) −315.370 −1.29486
\(40\) −101.314 −0.400478
\(41\) 33.5766 0.127897 0.0639485 0.997953i \(-0.479631\pi\)
0.0639485 + 0.997953i \(0.479631\pi\)
\(42\) −538.653 −1.97895
\(43\) 0 0
\(44\) −6.76515 −0.0231792
\(45\) 105.335 0.348944
\(46\) −316.364 −1.01403
\(47\) 533.353 1.65527 0.827634 0.561268i \(-0.189686\pi\)
0.827634 + 0.561268i \(0.189686\pi\)
\(48\) 245.460 0.738106
\(49\) 432.913 1.26214
\(50\) 1125.21 3.18257
\(51\) 209.958 0.576472
\(52\) 624.530 1.66551
\(53\) −0.134055 −0.000347430 0 −0.000173715 1.00000i \(-0.500055\pi\)
−0.000173715 1.00000i \(0.500055\pi\)
\(54\) 624.461 1.57367
\(55\) 14.5930 0.0357767
\(56\) 141.798 0.338368
\(57\) −449.581 −1.04471
\(58\) −758.607 −1.71741
\(59\) −110.795 −0.244480 −0.122240 0.992501i \(-0.539008\pi\)
−0.122240 + 0.992501i \(0.539008\pi\)
\(60\) 855.549 1.84085
\(61\) 533.253 1.11928 0.559640 0.828736i \(-0.310939\pi\)
0.559640 + 0.828736i \(0.310939\pi\)
\(62\) −157.632 −0.322891
\(63\) −147.427 −0.294826
\(64\) −655.112 −1.27952
\(65\) −1347.16 −2.57070
\(66\) 14.1789 0.0264440
\(67\) −411.431 −0.750214 −0.375107 0.926982i \(-0.622394\pi\)
−0.375107 + 0.926982i \(0.622394\pi\)
\(68\) −415.783 −0.741486
\(69\) 355.134 0.619611
\(70\) −2300.96 −3.92882
\(71\) −216.076 −0.361176 −0.180588 0.983559i \(-0.557800\pi\)
−0.180588 + 0.983559i \(0.557800\pi\)
\(72\) −26.9423 −0.0440997
\(73\) −136.316 −0.218557 −0.109278 0.994011i \(-0.534854\pi\)
−0.109278 + 0.994011i \(0.534854\pi\)
\(74\) 397.619 0.624625
\(75\) −1263.10 −1.94467
\(76\) 890.309 1.34376
\(77\) −20.4243 −0.0302281
\(78\) −1308.94 −1.90010
\(79\) 81.5918 0.116200 0.0581000 0.998311i \(-0.481496\pi\)
0.0581000 + 0.998311i \(0.481496\pi\)
\(80\) 1048.53 1.46536
\(81\) −558.088 −0.765552
\(82\) 139.359 0.187678
\(83\) −926.163 −1.22481 −0.612407 0.790542i \(-0.709799\pi\)
−0.612407 + 0.790542i \(0.709799\pi\)
\(84\) −1197.42 −1.55535
\(85\) 896.879 1.14447
\(86\) 0 0
\(87\) 851.573 1.04941
\(88\) −3.73254 −0.00452148
\(89\) −1125.82 −1.34086 −0.670429 0.741974i \(-0.733890\pi\)
−0.670429 + 0.741974i \(0.733890\pi\)
\(90\) 437.193 0.512046
\(91\) 1885.49 2.17201
\(92\) −703.276 −0.796974
\(93\) 176.950 0.197299
\(94\) 2213.67 2.42897
\(95\) −1920.47 −2.07407
\(96\) 1208.52 1.28483
\(97\) −1512.02 −1.58270 −0.791352 0.611361i \(-0.790622\pi\)
−0.791352 + 0.611361i \(0.790622\pi\)
\(98\) 1796.80 1.85208
\(99\) 3.88071 0.00393965
\(100\) 2501.33 2.50133
\(101\) −1324.79 −1.30517 −0.652583 0.757717i \(-0.726315\pi\)
−0.652583 + 0.757717i \(0.726315\pi\)
\(102\) 871.429 0.845925
\(103\) −1115.34 −1.06696 −0.533482 0.845811i \(-0.679117\pi\)
−0.533482 + 0.845811i \(0.679117\pi\)
\(104\) 344.573 0.324886
\(105\) 2582.94 2.40066
\(106\) −0.556391 −0.000509825 0
\(107\) −647.960 −0.585427 −0.292713 0.956200i \(-0.594558\pi\)
−0.292713 + 0.956200i \(0.594558\pi\)
\(108\) 1388.17 1.23682
\(109\) −2058.34 −1.80875 −0.904374 0.426740i \(-0.859662\pi\)
−0.904374 + 0.426740i \(0.859662\pi\)
\(110\) 60.5680 0.0524994
\(111\) −446.346 −0.381670
\(112\) −1467.52 −1.23810
\(113\) 911.539 0.758853 0.379426 0.925222i \(-0.376121\pi\)
0.379426 + 0.925222i \(0.376121\pi\)
\(114\) −1865.98 −1.53302
\(115\) 1517.03 1.23012
\(116\) −1686.38 −1.34980
\(117\) −358.250 −0.283079
\(118\) −459.854 −0.358754
\(119\) −1255.27 −0.966976
\(120\) 472.033 0.359088
\(121\) −1330.46 −0.999596
\(122\) 2213.26 1.64245
\(123\) −156.437 −0.114679
\(124\) −350.415 −0.253776
\(125\) −2907.79 −2.08065
\(126\) −611.893 −0.432633
\(127\) −375.272 −0.262205 −0.131102 0.991369i \(-0.541852\pi\)
−0.131102 + 0.991369i \(0.541852\pi\)
\(128\) −643.932 −0.444657
\(129\) 0 0
\(130\) −5591.38 −3.77228
\(131\) −2701.28 −1.80161 −0.900807 0.434219i \(-0.857024\pi\)
−0.900807 + 0.434219i \(0.857024\pi\)
\(132\) 31.5196 0.0207836
\(133\) 2687.88 1.75240
\(134\) −1707.64 −1.10088
\(135\) −2994.41 −1.90902
\(136\) −229.400 −0.144639
\(137\) −2022.29 −1.26114 −0.630568 0.776134i \(-0.717178\pi\)
−0.630568 + 0.776134i \(0.717178\pi\)
\(138\) 1473.98 0.909228
\(139\) −564.430 −0.344419 −0.172210 0.985060i \(-0.555091\pi\)
−0.172210 + 0.985060i \(0.555091\pi\)
\(140\) −5115.02 −3.08785
\(141\) −2484.96 −1.48419
\(142\) −896.820 −0.529996
\(143\) −49.6314 −0.0290237
\(144\) 278.835 0.161363
\(145\) 3637.66 2.08339
\(146\) −565.779 −0.320714
\(147\) −2017.00 −1.13169
\(148\) 883.904 0.490922
\(149\) −754.273 −0.414714 −0.207357 0.978265i \(-0.566486\pi\)
−0.207357 + 0.978265i \(0.566486\pi\)
\(150\) −5242.48 −2.85364
\(151\) 2555.27 1.37712 0.688560 0.725179i \(-0.258243\pi\)
0.688560 + 0.725179i \(0.258243\pi\)
\(152\) 491.211 0.262122
\(153\) 238.506 0.126027
\(154\) −84.7707 −0.0443572
\(155\) 755.875 0.391699
\(156\) −2909.76 −1.49338
\(157\) −1811.53 −0.920864 −0.460432 0.887695i \(-0.652305\pi\)
−0.460432 + 0.887695i \(0.652305\pi\)
\(158\) 338.645 0.170514
\(159\) 0.624576 0.000311523 0
\(160\) 5162.41 2.55078
\(161\) −2123.22 −1.03934
\(162\) −2316.33 −1.12338
\(163\) 2865.26 1.37684 0.688419 0.725313i \(-0.258305\pi\)
0.688419 + 0.725313i \(0.258305\pi\)
\(164\) 309.794 0.147505
\(165\) −67.9905 −0.0320791
\(166\) −3844.02 −1.79731
\(167\) 14.5741 0.00675318 0.00337659 0.999994i \(-0.498925\pi\)
0.00337659 + 0.999994i \(0.498925\pi\)
\(168\) −660.655 −0.303397
\(169\) 2384.77 1.08547
\(170\) 3722.48 1.67942
\(171\) −510.710 −0.228392
\(172\) 0 0
\(173\) 1036.88 0.455678 0.227839 0.973699i \(-0.426834\pi\)
0.227839 + 0.973699i \(0.426834\pi\)
\(174\) 3534.44 1.53991
\(175\) 7551.64 3.26200
\(176\) 38.6293 0.0165443
\(177\) 516.209 0.219213
\(178\) −4672.68 −1.96760
\(179\) 1266.66 0.528909 0.264455 0.964398i \(-0.414808\pi\)
0.264455 + 0.964398i \(0.414808\pi\)
\(180\) 971.877 0.402441
\(181\) −60.5440 −0.0248630 −0.0124315 0.999923i \(-0.503957\pi\)
−0.0124315 + 0.999923i \(0.503957\pi\)
\(182\) 7825.67 3.18724
\(183\) −2484.49 −1.00360
\(184\) −388.019 −0.155463
\(185\) −1906.66 −0.757731
\(186\) 734.426 0.289520
\(187\) 33.0423 0.0129213
\(188\) 4920.98 1.90904
\(189\) 4190.96 1.61295
\(190\) −7970.89 −3.04352
\(191\) 2237.34 0.847581 0.423791 0.905760i \(-0.360699\pi\)
0.423791 + 0.905760i \(0.360699\pi\)
\(192\) 3052.24 1.14728
\(193\) −1150.33 −0.429029 −0.214515 0.976721i \(-0.568817\pi\)
−0.214515 + 0.976721i \(0.568817\pi\)
\(194\) −6275.60 −2.32249
\(195\) 6276.60 2.30501
\(196\) 3994.27 1.45564
\(197\) 3565.84 1.28962 0.644811 0.764342i \(-0.276936\pi\)
0.644811 + 0.764342i \(0.276936\pi\)
\(198\) 16.1068 0.00578111
\(199\) −2313.80 −0.824226 −0.412113 0.911133i \(-0.635209\pi\)
−0.412113 + 0.911133i \(0.635209\pi\)
\(200\) 1380.06 0.487926
\(201\) 1916.91 0.672678
\(202\) −5498.52 −1.91522
\(203\) −5091.26 −1.76028
\(204\) 1937.18 0.664852
\(205\) −668.252 −0.227672
\(206\) −4629.18 −1.56568
\(207\) 403.422 0.135458
\(208\) −3566.10 −1.18877
\(209\) −70.7529 −0.0234167
\(210\) 10720.5 3.52277
\(211\) 1691.58 0.551912 0.275956 0.961170i \(-0.411006\pi\)
0.275956 + 0.961170i \(0.411006\pi\)
\(212\) −1.23685 −0.000400695 0
\(213\) 1006.72 0.323848
\(214\) −2689.34 −0.859065
\(215\) 0 0
\(216\) 765.899 0.241263
\(217\) −1057.92 −0.330950
\(218\) −8543.12 −2.65419
\(219\) 635.115 0.195968
\(220\) 134.642 0.0412617
\(221\) −3050.33 −0.928448
\(222\) −1852.55 −0.560068
\(223\) −406.066 −0.121938 −0.0609691 0.998140i \(-0.519419\pi\)
−0.0609691 + 0.998140i \(0.519419\pi\)
\(224\) −7225.29 −2.15518
\(225\) −1434.84 −0.425139
\(226\) 3783.32 1.11355
\(227\) 4463.18 1.30499 0.652493 0.757795i \(-0.273723\pi\)
0.652493 + 0.757795i \(0.273723\pi\)
\(228\) −4148.06 −1.20488
\(229\) −5509.43 −1.58984 −0.794920 0.606714i \(-0.792487\pi\)
−0.794920 + 0.606714i \(0.792487\pi\)
\(230\) 6296.39 1.80509
\(231\) 95.1592 0.0271040
\(232\) −930.428 −0.263300
\(233\) 4222.96 1.18736 0.593680 0.804701i \(-0.297674\pi\)
0.593680 + 0.804701i \(0.297674\pi\)
\(234\) −1486.91 −0.415395
\(235\) −10615.0 −2.94657
\(236\) −1022.25 −0.281962
\(237\) −380.146 −0.104190
\(238\) −5209.96 −1.41896
\(239\) −876.821 −0.237309 −0.118654 0.992936i \(-0.537858\pi\)
−0.118654 + 0.992936i \(0.537858\pi\)
\(240\) −4885.22 −1.31392
\(241\) 3169.86 0.847257 0.423628 0.905836i \(-0.360756\pi\)
0.423628 + 0.905836i \(0.360756\pi\)
\(242\) −5522.06 −1.46682
\(243\) −1462.09 −0.385980
\(244\) 4920.06 1.29088
\(245\) −8615.99 −2.24676
\(246\) −649.290 −0.168281
\(247\) 6531.61 1.68258
\(248\) −193.335 −0.0495031
\(249\) 4315.10 1.09823
\(250\) −12068.7 −3.05318
\(251\) 3990.59 1.00352 0.501761 0.865006i \(-0.332686\pi\)
0.501761 + 0.865006i \(0.332686\pi\)
\(252\) −1360.23 −0.340027
\(253\) 55.8894 0.0138883
\(254\) −1557.56 −0.384763
\(255\) −4178.67 −1.02619
\(256\) 2568.27 0.627018
\(257\) −4425.70 −1.07419 −0.537096 0.843521i \(-0.680479\pi\)
−0.537096 + 0.843521i \(0.680479\pi\)
\(258\) 0 0
\(259\) 2668.55 0.640214
\(260\) −12429.6 −2.96481
\(261\) 967.361 0.229418
\(262\) −11211.6 −2.64372
\(263\) −72.8494 −0.0170802 −0.00854009 0.999964i \(-0.502718\pi\)
−0.00854009 + 0.999964i \(0.502718\pi\)
\(264\) 17.3904 0.00405418
\(265\) 2.66800 0.000618467 0
\(266\) 11156.0 2.57150
\(267\) 5245.32 1.20228
\(268\) −3796.07 −0.865231
\(269\) 3884.39 0.880430 0.440215 0.897892i \(-0.354902\pi\)
0.440215 + 0.897892i \(0.354902\pi\)
\(270\) −12428.2 −2.80133
\(271\) 6716.63 1.50556 0.752779 0.658273i \(-0.228713\pi\)
0.752779 + 0.658273i \(0.228713\pi\)
\(272\) 2374.14 0.529240
\(273\) −8784.70 −1.94752
\(274\) −8393.46 −1.85061
\(275\) −198.781 −0.0435889
\(276\) 3276.65 0.714605
\(277\) −2043.39 −0.443233 −0.221616 0.975134i \(-0.571133\pi\)
−0.221616 + 0.975134i \(0.571133\pi\)
\(278\) −2342.66 −0.505407
\(279\) 201.009 0.0431330
\(280\) −2822.12 −0.602335
\(281\) −2091.71 −0.444060 −0.222030 0.975040i \(-0.571268\pi\)
−0.222030 + 0.975040i \(0.571268\pi\)
\(282\) −10313.8 −2.17793
\(283\) −4108.63 −0.863013 −0.431507 0.902110i \(-0.642018\pi\)
−0.431507 + 0.902110i \(0.642018\pi\)
\(284\) −1993.63 −0.416549
\(285\) 8947.71 1.85971
\(286\) −205.994 −0.0425899
\(287\) 935.283 0.192362
\(288\) 1372.84 0.280886
\(289\) −2882.24 −0.586655
\(290\) 15098.1 3.05720
\(291\) 7044.67 1.41913
\(292\) −1257.72 −0.252064
\(293\) −771.427 −0.153813 −0.0769066 0.997038i \(-0.524504\pi\)
−0.0769066 + 0.997038i \(0.524504\pi\)
\(294\) −8371.50 −1.66067
\(295\) 2205.09 0.435204
\(296\) 487.678 0.0957624
\(297\) −110.318 −0.0215532
\(298\) −3130.60 −0.608559
\(299\) −5159.48 −0.997927
\(300\) −11654.0 −2.24281
\(301\) 0 0
\(302\) 10605.6 2.02081
\(303\) 6172.36 1.17027
\(304\) −5083.71 −0.959114
\(305\) −10613.0 −1.99245
\(306\) 989.916 0.184934
\(307\) −6628.00 −1.23218 −0.616092 0.787675i \(-0.711285\pi\)
−0.616092 + 0.787675i \(0.711285\pi\)
\(308\) −188.445 −0.0348624
\(309\) 5196.49 0.956692
\(310\) 3137.24 0.574785
\(311\) 4804.84 0.876069 0.438035 0.898958i \(-0.355675\pi\)
0.438035 + 0.898958i \(0.355675\pi\)
\(312\) −1605.41 −0.291308
\(313\) −969.070 −0.175000 −0.0875001 0.996165i \(-0.527888\pi\)
−0.0875001 + 0.996165i \(0.527888\pi\)
\(314\) −7518.71 −1.35129
\(315\) 2934.14 0.524826
\(316\) 752.806 0.134015
\(317\) −3279.49 −0.581055 −0.290528 0.956867i \(-0.593831\pi\)
−0.290528 + 0.956867i \(0.593831\pi\)
\(318\) 2.59229 0.000457133 0
\(319\) 134.017 0.0235219
\(320\) 13038.3 2.27769
\(321\) 3018.92 0.524921
\(322\) −8812.40 −1.52514
\(323\) −4348.44 −0.749083
\(324\) −5149.19 −0.882921
\(325\) 18350.6 3.13203
\(326\) 11892.2 2.02040
\(327\) 9590.07 1.62181
\(328\) 170.923 0.0287733
\(329\) 14856.7 2.48959
\(330\) −282.193 −0.0470734
\(331\) 6633.82 1.10159 0.550797 0.834639i \(-0.314324\pi\)
0.550797 + 0.834639i \(0.314324\pi\)
\(332\) −8545.24 −1.41259
\(333\) −507.036 −0.0834396
\(334\) 60.4897 0.00990973
\(335\) 8188.45 1.33547
\(336\) 6837.34 1.11014
\(337\) −4061.37 −0.656489 −0.328244 0.944593i \(-0.606457\pi\)
−0.328244 + 0.944593i \(0.606457\pi\)
\(338\) 9897.93 1.59283
\(339\) −4246.97 −0.680424
\(340\) 8275.05 1.31993
\(341\) 27.8475 0.00442236
\(342\) −2119.69 −0.335146
\(343\) 2504.56 0.394267
\(344\) 0 0
\(345\) −7068.01 −1.10298
\(346\) 4303.54 0.668669
\(347\) 7916.21 1.22468 0.612341 0.790594i \(-0.290228\pi\)
0.612341 + 0.790594i \(0.290228\pi\)
\(348\) 7857.04 1.21029
\(349\) −12863.0 −1.97290 −0.986448 0.164077i \(-0.947535\pi\)
−0.986448 + 0.164077i \(0.947535\pi\)
\(350\) 31342.9 4.78672
\(351\) 10184.1 1.54868
\(352\) 190.191 0.0287988
\(353\) −7949.47 −1.19861 −0.599303 0.800522i \(-0.704555\pi\)
−0.599303 + 0.800522i \(0.704555\pi\)
\(354\) 2142.52 0.321676
\(355\) 4300.42 0.642937
\(356\) −10387.3 −1.54643
\(357\) 5848.44 0.867037
\(358\) 5257.26 0.776130
\(359\) −3114.11 −0.457817 −0.228909 0.973448i \(-0.573516\pi\)
−0.228909 + 0.973448i \(0.573516\pi\)
\(360\) 536.215 0.0785028
\(361\) 2452.25 0.357523
\(362\) −251.287 −0.0364843
\(363\) 6198.78 0.896286
\(364\) 17396.4 2.50500
\(365\) 2713.02 0.389057
\(366\) −10311.8 −1.47270
\(367\) −4292.76 −0.610573 −0.305287 0.952260i \(-0.598752\pi\)
−0.305287 + 0.952260i \(0.598752\pi\)
\(368\) 4015.74 0.568845
\(369\) −177.708 −0.0250707
\(370\) −7913.54 −1.11191
\(371\) −3.73412 −0.000522549 0
\(372\) 1632.62 0.227547
\(373\) −2868.12 −0.398138 −0.199069 0.979985i \(-0.563792\pi\)
−0.199069 + 0.979985i \(0.563792\pi\)
\(374\) 137.141 0.0189610
\(375\) 13547.8 1.86561
\(376\) 2715.06 0.372390
\(377\) −12371.9 −1.69014
\(378\) 17394.5 2.36687
\(379\) 5154.61 0.698613 0.349307 0.937009i \(-0.386417\pi\)
0.349307 + 0.937009i \(0.386417\pi\)
\(380\) −17719.2 −2.39205
\(381\) 1748.44 0.235105
\(382\) 9286.03 1.24376
\(383\) 4471.13 0.596512 0.298256 0.954486i \(-0.403595\pi\)
0.298256 + 0.954486i \(0.403595\pi\)
\(384\) 3000.16 0.398701
\(385\) 406.491 0.0538096
\(386\) −4774.43 −0.629565
\(387\) 0 0
\(388\) −13950.6 −1.82535
\(389\) −8980.63 −1.17053 −0.585265 0.810842i \(-0.699010\pi\)
−0.585265 + 0.810842i \(0.699010\pi\)
\(390\) 26050.9 3.38241
\(391\) 3434.94 0.444277
\(392\) 2203.77 0.283946
\(393\) 12585.6 1.61541
\(394\) 14800.0 1.89241
\(395\) −1623.87 −0.206850
\(396\) 35.8053 0.00454365
\(397\) 6091.78 0.770120 0.385060 0.922892i \(-0.374181\pi\)
0.385060 + 0.922892i \(0.374181\pi\)
\(398\) −9603.39 −1.20948
\(399\) −12523.2 −1.57129
\(400\) −14282.7 −1.78534
\(401\) −6404.00 −0.797507 −0.398753 0.917058i \(-0.630557\pi\)
−0.398753 + 0.917058i \(0.630557\pi\)
\(402\) 7956.09 0.987099
\(403\) −2570.76 −0.317764
\(404\) −12223.2 −1.50526
\(405\) 11107.3 1.36277
\(406\) −21131.2 −2.58306
\(407\) −70.2439 −0.00855494
\(408\) 1068.80 0.129690
\(409\) 12843.8 1.55277 0.776386 0.630258i \(-0.217051\pi\)
0.776386 + 0.630258i \(0.217051\pi\)
\(410\) −2773.57 −0.334090
\(411\) 9422.07 1.13079
\(412\) −10290.6 −1.23054
\(413\) −3086.23 −0.367708
\(414\) 1674.39 0.198773
\(415\) 18432.8 2.18032
\(416\) −17557.6 −2.06931
\(417\) 2629.75 0.308823
\(418\) −293.659 −0.0343620
\(419\) −8367.04 −0.975552 −0.487776 0.872969i \(-0.662192\pi\)
−0.487776 + 0.872969i \(0.662192\pi\)
\(420\) 23831.5 2.76871
\(421\) −2176.45 −0.251956 −0.125978 0.992033i \(-0.540207\pi\)
−0.125978 + 0.992033i \(0.540207\pi\)
\(422\) 7020.88 0.809884
\(423\) −2822.83 −0.324470
\(424\) −0.682411 −7.81623e−5 0
\(425\) −12217.0 −1.39438
\(426\) 4178.39 0.475220
\(427\) 14853.9 1.68344
\(428\) −5978.40 −0.675179
\(429\) 231.239 0.0260240
\(430\) 0 0
\(431\) 2491.13 0.278408 0.139204 0.990264i \(-0.455546\pi\)
0.139204 + 0.990264i \(0.455546\pi\)
\(432\) −7926.54 −0.882791
\(433\) −15117.3 −1.67781 −0.838906 0.544277i \(-0.816804\pi\)
−0.838906 + 0.544277i \(0.816804\pi\)
\(434\) −4390.87 −0.485642
\(435\) −16948.3 −1.86807
\(436\) −18991.3 −2.08605
\(437\) −7355.18 −0.805139
\(438\) 2636.03 0.287567
\(439\) 2947.73 0.320473 0.160237 0.987079i \(-0.448774\pi\)
0.160237 + 0.987079i \(0.448774\pi\)
\(440\) 74.2864 0.00804878
\(441\) −2291.24 −0.247408
\(442\) −12660.3 −1.36242
\(443\) −6749.23 −0.723850 −0.361925 0.932207i \(-0.617880\pi\)
−0.361925 + 0.932207i \(0.617880\pi\)
\(444\) −4118.21 −0.440184
\(445\) 22406.4 2.38689
\(446\) −1685.37 −0.178934
\(447\) 3514.25 0.371853
\(448\) −18248.3 −1.92444
\(449\) −5120.14 −0.538162 −0.269081 0.963118i \(-0.586720\pi\)
−0.269081 + 0.963118i \(0.586720\pi\)
\(450\) −5955.29 −0.623856
\(451\) −24.6194 −0.00257047
\(452\) 8410.31 0.875194
\(453\) −11905.3 −1.23479
\(454\) 18524.4 1.91496
\(455\) −37525.6 −3.86643
\(456\) −2288.61 −0.235031
\(457\) 11669.3 1.19446 0.597230 0.802070i \(-0.296268\pi\)
0.597230 + 0.802070i \(0.296268\pi\)
\(458\) −22866.8 −2.33296
\(459\) −6780.11 −0.689474
\(460\) 13996.8 1.41871
\(461\) −12772.3 −1.29038 −0.645188 0.764024i \(-0.723221\pi\)
−0.645188 + 0.764024i \(0.723221\pi\)
\(462\) 394.957 0.0397728
\(463\) 6707.16 0.673235 0.336618 0.941641i \(-0.390717\pi\)
0.336618 + 0.941641i \(0.390717\pi\)
\(464\) 9629.30 0.963425
\(465\) −3521.71 −0.351216
\(466\) 17527.3 1.74235
\(467\) 13188.3 1.30681 0.653407 0.757007i \(-0.273339\pi\)
0.653407 + 0.757007i \(0.273339\pi\)
\(468\) −3305.40 −0.326479
\(469\) −11460.5 −1.12835
\(470\) −44057.3 −4.32385
\(471\) 8440.12 0.825691
\(472\) −564.009 −0.0550013
\(473\) 0 0
\(474\) −1577.79 −0.152891
\(475\) 26160.0 2.52696
\(476\) −11581.7 −1.11523
\(477\) 0.709499 6.81042e−5 0
\(478\) −3639.23 −0.348231
\(479\) 2983.64 0.284606 0.142303 0.989823i \(-0.454549\pi\)
0.142303 + 0.989823i \(0.454549\pi\)
\(480\) −24052.3 −2.28715
\(481\) 6484.62 0.614706
\(482\) 13156.5 1.24328
\(483\) 9892.35 0.931920
\(484\) −12275.5 −1.15285
\(485\) 30092.7 2.81740
\(486\) −6068.38 −0.566394
\(487\) 903.861 0.0841024 0.0420512 0.999115i \(-0.486611\pi\)
0.0420512 + 0.999115i \(0.486611\pi\)
\(488\) 2714.55 0.251807
\(489\) −13349.6 −1.23454
\(490\) −35760.5 −3.29693
\(491\) 14745.6 1.35531 0.677656 0.735379i \(-0.262996\pi\)
0.677656 + 0.735379i \(0.262996\pi\)
\(492\) −1443.37 −0.132260
\(493\) 8236.60 0.752450
\(494\) 27109.3 2.46904
\(495\) −77.2351 −0.00701305
\(496\) 2000.89 0.181134
\(497\) −6018.85 −0.543224
\(498\) 17909.8 1.61156
\(499\) 846.074 0.0759028 0.0379514 0.999280i \(-0.487917\pi\)
0.0379514 + 0.999280i \(0.487917\pi\)
\(500\) −26828.8 −2.39964
\(501\) −67.9026 −0.00605522
\(502\) 16562.9 1.47258
\(503\) 239.890 0.0212647 0.0106324 0.999943i \(-0.496616\pi\)
0.0106324 + 0.999943i \(0.496616\pi\)
\(504\) −750.484 −0.0663278
\(505\) 26366.5 2.32335
\(506\) 231.968 0.0203799
\(507\) −11110.9 −0.973280
\(508\) −3462.44 −0.302404
\(509\) 5624.56 0.489792 0.244896 0.969549i \(-0.421246\pi\)
0.244896 + 0.969549i \(0.421246\pi\)
\(510\) −17343.5 −1.50585
\(511\) −3797.13 −0.328718
\(512\) 15811.0 1.36475
\(513\) 14518.1 1.24950
\(514\) −18368.8 −1.57629
\(515\) 22197.8 1.89932
\(516\) 0 0
\(517\) −391.071 −0.0332675
\(518\) 11075.8 0.939461
\(519\) −4830.93 −0.408582
\(520\) −6857.80 −0.578336
\(521\) −12416.2 −1.04408 −0.522038 0.852922i \(-0.674828\pi\)
−0.522038 + 0.852922i \(0.674828\pi\)
\(522\) 4015.01 0.336652
\(523\) −8470.79 −0.708225 −0.354112 0.935203i \(-0.615217\pi\)
−0.354112 + 0.935203i \(0.615217\pi\)
\(524\) −24923.3 −2.07782
\(525\) −35184.0 −2.92487
\(526\) −302.360 −0.0250637
\(527\) 1711.49 0.141468
\(528\) −179.979 −0.0148344
\(529\) −6356.97 −0.522477
\(530\) 11.0735 0.000907549 0
\(531\) 586.397 0.0479237
\(532\) 24799.8 2.02106
\(533\) 2272.76 0.184698
\(534\) 21770.6 1.76424
\(535\) 12895.9 1.04213
\(536\) −2094.41 −0.168778
\(537\) −5901.53 −0.474245
\(538\) 16122.1 1.29196
\(539\) −317.425 −0.0253664
\(540\) −27627.9 −2.20170
\(541\) 1109.59 0.0881792 0.0440896 0.999028i \(-0.485961\pi\)
0.0440896 + 0.999028i \(0.485961\pi\)
\(542\) 27877.3 2.20928
\(543\) 282.082 0.0222933
\(544\) 11689.0 0.921255
\(545\) 40965.9 3.21979
\(546\) −36460.7 −2.85783
\(547\) −13123.6 −1.02582 −0.512912 0.858441i \(-0.671433\pi\)
−0.512912 + 0.858441i \(0.671433\pi\)
\(548\) −18658.6 −1.45448
\(549\) −2822.30 −0.219404
\(550\) −825.037 −0.0639631
\(551\) −17636.9 −1.36362
\(552\) 1807.83 0.139395
\(553\) 2272.76 0.174769
\(554\) −8481.06 −0.650407
\(555\) 8883.34 0.679417
\(556\) −5207.71 −0.397223
\(557\) 16074.4 1.22279 0.611395 0.791325i \(-0.290608\pi\)
0.611395 + 0.791325i \(0.290608\pi\)
\(558\) 834.285 0.0632941
\(559\) 0 0
\(560\) 29207.0 2.20397
\(561\) −153.948 −0.0115859
\(562\) −8681.60 −0.651621
\(563\) −13329.8 −0.997839 −0.498919 0.866648i \(-0.666270\pi\)
−0.498919 + 0.866648i \(0.666270\pi\)
\(564\) −22927.4 −1.71174
\(565\) −18141.8 −1.35085
\(566\) −17052.8 −1.26640
\(567\) −15545.7 −1.15142
\(568\) −1099.95 −0.0812548
\(569\) −17313.5 −1.27561 −0.637803 0.770199i \(-0.720157\pi\)
−0.637803 + 0.770199i \(0.720157\pi\)
\(570\) 37137.3 2.72897
\(571\) 8514.71 0.624045 0.312022 0.950075i \(-0.398994\pi\)
0.312022 + 0.950075i \(0.398994\pi\)
\(572\) −457.924 −0.0334734
\(573\) −10424.0 −0.759982
\(574\) 3881.87 0.282276
\(575\) −20664.4 −1.49872
\(576\) 3467.26 0.250814
\(577\) −23224.4 −1.67564 −0.837820 0.545947i \(-0.816170\pi\)
−0.837820 + 0.545947i \(0.816170\pi\)
\(578\) −11962.7 −0.860868
\(579\) 5359.53 0.384688
\(580\) 33562.9 2.40280
\(581\) −25798.5 −1.84217
\(582\) 29238.8 2.08245
\(583\) 0.0982928 6.98263e−6 0
\(584\) −693.926 −0.0491693
\(585\) 7130.02 0.503915
\(586\) −3201.79 −0.225708
\(587\) −15872.8 −1.11608 −0.558042 0.829813i \(-0.688447\pi\)
−0.558042 + 0.829813i \(0.688447\pi\)
\(588\) −18609.8 −1.30520
\(589\) −3664.79 −0.256376
\(590\) 9152.18 0.638626
\(591\) −16613.7 −1.15634
\(592\) −5047.13 −0.350399
\(593\) −21818.4 −1.51092 −0.755461 0.655194i \(-0.772587\pi\)
−0.755461 + 0.655194i \(0.772587\pi\)
\(594\) −457.874 −0.0316276
\(595\) 24982.8 1.72133
\(596\) −6959.30 −0.478295
\(597\) 10780.3 0.739041
\(598\) −21414.3 −1.46437
\(599\) −10835.2 −0.739087 −0.369544 0.929213i \(-0.620486\pi\)
−0.369544 + 0.929213i \(0.620486\pi\)
\(600\) −6429.88 −0.437498
\(601\) 7125.29 0.483605 0.241803 0.970325i \(-0.422261\pi\)
0.241803 + 0.970325i \(0.422261\pi\)
\(602\) 0 0
\(603\) 2177.55 0.147059
\(604\) 23576.2 1.58825
\(605\) 26479.3 1.77940
\(606\) 25618.3 1.71728
\(607\) 10703.4 0.715710 0.357855 0.933777i \(-0.383508\pi\)
0.357855 + 0.933777i \(0.383508\pi\)
\(608\) −25029.5 −1.66954
\(609\) 23720.8 1.57835
\(610\) −44049.0 −2.92376
\(611\) 36102.0 2.39040
\(612\) 2200.58 0.145348
\(613\) −3774.44 −0.248692 −0.124346 0.992239i \(-0.539683\pi\)
−0.124346 + 0.992239i \(0.539683\pi\)
\(614\) −27509.4 −1.80813
\(615\) 3113.47 0.204142
\(616\) −103.971 −0.00680049
\(617\) −19188.0 −1.25199 −0.625997 0.779826i \(-0.715308\pi\)
−0.625997 + 0.779826i \(0.715308\pi\)
\(618\) 21567.9 1.40387
\(619\) −19479.3 −1.26485 −0.632424 0.774622i \(-0.717940\pi\)
−0.632424 + 0.774622i \(0.717940\pi\)
\(620\) 6974.07 0.451751
\(621\) −11468.2 −0.741069
\(622\) 19942.4 1.28556
\(623\) −31359.9 −2.01671
\(624\) 16614.9 1.06591
\(625\) 23984.0 1.53498
\(626\) −4022.11 −0.256798
\(627\) 329.646 0.0209965
\(628\) −16714.0 −1.06204
\(629\) −4317.16 −0.273667
\(630\) 12178.1 0.770138
\(631\) −7063.66 −0.445642 −0.222821 0.974859i \(-0.571527\pi\)
−0.222821 + 0.974859i \(0.571527\pi\)
\(632\) 415.347 0.0261418
\(633\) −7881.28 −0.494870
\(634\) −13611.5 −0.852650
\(635\) 7468.79 0.466756
\(636\) 5.76265 0.000359283 0
\(637\) 29303.4 1.82267
\(638\) 556.234 0.0345165
\(639\) 1143.61 0.0707988
\(640\) 12815.8 0.791543
\(641\) 8274.88 0.509888 0.254944 0.966956i \(-0.417943\pi\)
0.254944 + 0.966956i \(0.417943\pi\)
\(642\) 12530.0 0.770278
\(643\) 18880.9 1.15799 0.578996 0.815331i \(-0.303445\pi\)
0.578996 + 0.815331i \(0.303445\pi\)
\(644\) −19589.9 −1.19868
\(645\) 0 0
\(646\) −18048.1 −1.09922
\(647\) −13808.8 −0.839075 −0.419537 0.907738i \(-0.637808\pi\)
−0.419537 + 0.907738i \(0.637808\pi\)
\(648\) −2840.97 −0.172228
\(649\) 81.2386 0.00491355
\(650\) 76164.0 4.59599
\(651\) 4928.97 0.296746
\(652\) 26436.3 1.58792
\(653\) 30854.7 1.84906 0.924530 0.381109i \(-0.124458\pi\)
0.924530 + 0.381109i \(0.124458\pi\)
\(654\) 39803.4 2.37987
\(655\) 53761.7 3.20709
\(656\) −1768.94 −0.105283
\(657\) 721.471 0.0428421
\(658\) 61662.4 3.65327
\(659\) −11484.2 −0.678848 −0.339424 0.940633i \(-0.610232\pi\)
−0.339424 + 0.940633i \(0.610232\pi\)
\(660\) −627.314 −0.0369972
\(661\) 6233.04 0.366773 0.183387 0.983041i \(-0.441294\pi\)
0.183387 + 0.983041i \(0.441294\pi\)
\(662\) 27533.5 1.61650
\(663\) 14211.8 0.832491
\(664\) −4714.68 −0.275550
\(665\) −53495.2 −3.11948
\(666\) −2104.44 −0.122441
\(667\) 13931.8 0.808758
\(668\) 134.468 0.00778852
\(669\) 1891.91 0.109336
\(670\) 33986.0 1.95969
\(671\) −390.997 −0.0224952
\(672\) 33663.5 1.93244
\(673\) −27119.3 −1.55330 −0.776651 0.629931i \(-0.783083\pi\)
−0.776651 + 0.629931i \(0.783083\pi\)
\(674\) −16856.6 −0.963343
\(675\) 40788.9 2.32587
\(676\) 22003.0 1.25188
\(677\) 22288.2 1.26530 0.632649 0.774439i \(-0.281968\pi\)
0.632649 + 0.774439i \(0.281968\pi\)
\(678\) −17627.0 −0.998465
\(679\) −42117.6 −2.38045
\(680\) 4565.60 0.257475
\(681\) −20794.5 −1.17011
\(682\) 115.580 0.00648945
\(683\) 2868.96 0.160729 0.0803644 0.996766i \(-0.474392\pi\)
0.0803644 + 0.996766i \(0.474392\pi\)
\(684\) −4712.06 −0.263407
\(685\) 40248.2 2.24497
\(686\) 10395.1 0.578554
\(687\) 25669.1 1.42553
\(688\) 0 0
\(689\) −9.07398 −0.000501729 0
\(690\) −29335.6 −1.61853
\(691\) 32355.3 1.78126 0.890631 0.454726i \(-0.150263\pi\)
0.890631 + 0.454726i \(0.150263\pi\)
\(692\) 9566.74 0.525539
\(693\) 108.098 0.00592540
\(694\) 32856.1 1.79712
\(695\) 11233.5 0.613108
\(696\) 4334.97 0.236087
\(697\) −1513.09 −0.0822274
\(698\) −53387.6 −2.89506
\(699\) −19675.2 −1.06464
\(700\) 69675.2 3.76211
\(701\) −5159.16 −0.277973 −0.138986 0.990294i \(-0.544384\pi\)
−0.138986 + 0.990294i \(0.544384\pi\)
\(702\) 42269.0 2.27256
\(703\) 9244.26 0.495952
\(704\) 480.348 0.0257156
\(705\) 49456.5 2.64204
\(706\) −32994.1 −1.75885
\(707\) −36902.4 −1.96302
\(708\) 4762.80 0.252821
\(709\) 20468.6 1.08422 0.542112 0.840306i \(-0.317625\pi\)
0.542112 + 0.840306i \(0.317625\pi\)
\(710\) 17848.8 0.943457
\(711\) −431.834 −0.0227778
\(712\) −5731.03 −0.301656
\(713\) 2894.91 0.152055
\(714\) 24273.8 1.27230
\(715\) 987.782 0.0516657
\(716\) 11686.9 0.609997
\(717\) 4085.21 0.212782
\(718\) −12925.0 −0.671809
\(719\) −16111.8 −0.835702 −0.417851 0.908515i \(-0.637217\pi\)
−0.417851 + 0.908515i \(0.637217\pi\)
\(720\) −5549.46 −0.287245
\(721\) −31068.0 −1.60476
\(722\) 10178.0 0.524635
\(723\) −14768.8 −0.759691
\(724\) −558.609 −0.0286748
\(725\) −49551.1 −2.53832
\(726\) 25727.9 1.31522
\(727\) −9681.53 −0.493904 −0.246952 0.969028i \(-0.579429\pi\)
−0.246952 + 0.969028i \(0.579429\pi\)
\(728\) 9598.15 0.488642
\(729\) 21880.4 1.11164
\(730\) 11260.3 0.570909
\(731\) 0 0
\(732\) −22923.1 −1.15746
\(733\) −18580.8 −0.936288 −0.468144 0.883652i \(-0.655077\pi\)
−0.468144 + 0.883652i \(0.655077\pi\)
\(734\) −17817.0 −0.895966
\(735\) 40142.9 2.01455
\(736\) 19771.4 0.990196
\(737\) 301.674 0.0150778
\(738\) −737.573 −0.0367892
\(739\) 27864.0 1.38700 0.693502 0.720455i \(-0.256067\pi\)
0.693502 + 0.720455i \(0.256067\pi\)
\(740\) −17591.8 −0.873900
\(741\) −30431.6 −1.50868
\(742\) −15.4984 −0.000766798 0
\(743\) −9787.89 −0.483288 −0.241644 0.970365i \(-0.577687\pi\)
−0.241644 + 0.970365i \(0.577687\pi\)
\(744\) 900.770 0.0443869
\(745\) 15011.8 0.738241
\(746\) −11904.1 −0.584235
\(747\) 4901.82 0.240091
\(748\) 304.864 0.0149023
\(749\) −18049.1 −0.880505
\(750\) 56229.7 2.73763
\(751\) −19515.4 −0.948239 −0.474119 0.880461i \(-0.657234\pi\)
−0.474119 + 0.880461i \(0.657234\pi\)
\(752\) −28099.1 −1.36259
\(753\) −18592.6 −0.899805
\(754\) −51349.2 −2.48014
\(755\) −50855.9 −2.45144
\(756\) 38667.9 1.86023
\(757\) −23470.3 −1.12687 −0.563436 0.826160i \(-0.690521\pi\)
−0.563436 + 0.826160i \(0.690521\pi\)
\(758\) 21394.1 1.02516
\(759\) −260.395 −0.0124529
\(760\) −9776.26 −0.466608
\(761\) 24891.6 1.18570 0.592851 0.805312i \(-0.298002\pi\)
0.592851 + 0.805312i \(0.298002\pi\)
\(762\) 7256.85 0.344997
\(763\) −57335.6 −2.72043
\(764\) 20642.8 0.977526
\(765\) −4746.83 −0.224343
\(766\) 18557.4 0.875332
\(767\) −7499.60 −0.353057
\(768\) −11965.9 −0.562215
\(769\) −32481.7 −1.52317 −0.761586 0.648064i \(-0.775579\pi\)
−0.761586 + 0.648064i \(0.775579\pi\)
\(770\) 1687.13 0.0789612
\(771\) 20619.8 0.963172
\(772\) −10613.5 −0.494805
\(773\) −1476.22 −0.0686879 −0.0343440 0.999410i \(-0.510934\pi\)
−0.0343440 + 0.999410i \(0.510934\pi\)
\(774\) 0 0
\(775\) −10296.3 −0.477230
\(776\) −7697.00 −0.356065
\(777\) −12433.1 −0.574047
\(778\) −37274.0 −1.71766
\(779\) 3239.96 0.149016
\(780\) 57911.0 2.65839
\(781\) 158.434 0.00725890
\(782\) 14256.6 0.651939
\(783\) −27499.5 −1.25511
\(784\) −22807.5 −1.03897
\(785\) 36053.6 1.63925
\(786\) 52236.1 2.37049
\(787\) 7837.58 0.354993 0.177496 0.984121i \(-0.443200\pi\)
0.177496 + 0.984121i \(0.443200\pi\)
\(788\) 32900.2 1.48734
\(789\) 339.414 0.0153149
\(790\) −6739.83 −0.303535
\(791\) 25391.1 1.14135
\(792\) 19.7549 0.000886313 0
\(793\) 36095.2 1.61637
\(794\) 25283.8 1.13009
\(795\) −12.4305 −0.000554547 0
\(796\) −21348.3 −0.950590
\(797\) 10321.0 0.458706 0.229353 0.973343i \(-0.426339\pi\)
0.229353 + 0.973343i \(0.426339\pi\)
\(798\) −51977.2 −2.30573
\(799\) −24035.0 −1.06420
\(800\) −70320.7 −3.10777
\(801\) 5958.52 0.262839
\(802\) −26579.7 −1.17028
\(803\) 99.9514 0.00439254
\(804\) 17686.3 0.775807
\(805\) 42257.1 1.85015
\(806\) −10669.9 −0.466292
\(807\) −18097.8 −0.789435
\(808\) −6743.92 −0.293626
\(809\) 30215.4 1.31312 0.656561 0.754273i \(-0.272010\pi\)
0.656561 + 0.754273i \(0.272010\pi\)
\(810\) 46100.4 1.99976
\(811\) 43448.1 1.88122 0.940611 0.339487i \(-0.110253\pi\)
0.940611 + 0.339487i \(0.110253\pi\)
\(812\) −46974.5 −2.03015
\(813\) −31293.6 −1.34996
\(814\) −291.546 −0.0125537
\(815\) −57025.4 −2.45094
\(816\) −11061.4 −0.474542
\(817\) 0 0
\(818\) 53307.8 2.27856
\(819\) −9979.15 −0.425763
\(820\) −6165.63 −0.262577
\(821\) 28087.5 1.19398 0.596991 0.802248i \(-0.296363\pi\)
0.596991 + 0.802248i \(0.296363\pi\)
\(822\) 39106.1 1.65935
\(823\) −44076.2 −1.86683 −0.933413 0.358803i \(-0.883185\pi\)
−0.933413 + 0.358803i \(0.883185\pi\)
\(824\) −5677.67 −0.240038
\(825\) 926.144 0.0390839
\(826\) −12809.3 −0.539581
\(827\) 6095.24 0.256291 0.128145 0.991755i \(-0.459098\pi\)
0.128145 + 0.991755i \(0.459098\pi\)
\(828\) 3722.17 0.156225
\(829\) −8358.62 −0.350189 −0.175095 0.984552i \(-0.556023\pi\)
−0.175095 + 0.984552i \(0.556023\pi\)
\(830\) 76505.0 3.19943
\(831\) 9520.40 0.397424
\(832\) −44343.7 −1.84777
\(833\) −19508.8 −0.811453
\(834\) 10914.7 0.453172
\(835\) −290.059 −0.0120215
\(836\) −652.802 −0.0270067
\(837\) −5714.16 −0.235974
\(838\) −34727.2 −1.43154
\(839\) −12358.3 −0.508530 −0.254265 0.967135i \(-0.581834\pi\)
−0.254265 + 0.967135i \(0.581834\pi\)
\(840\) 13148.6 0.540082
\(841\) 9017.93 0.369754
\(842\) −9033.31 −0.369725
\(843\) 9745.52 0.398165
\(844\) 15607.4 0.636526
\(845\) −47462.4 −1.93226
\(846\) −11716.1 −0.476133
\(847\) −37060.3 −1.50343
\(848\) 7.06249 0.000285999 0
\(849\) 19142.6 0.773819
\(850\) −50706.4 −2.04614
\(851\) −7302.26 −0.294146
\(852\) 9288.54 0.373498
\(853\) −17084.9 −0.685786 −0.342893 0.939374i \(-0.611407\pi\)
−0.342893 + 0.939374i \(0.611407\pi\)
\(854\) 61650.8 2.47031
\(855\) 10164.3 0.406564
\(856\) −3298.47 −0.131705
\(857\) −16684.6 −0.665037 −0.332519 0.943097i \(-0.607898\pi\)
−0.332519 + 0.943097i \(0.607898\pi\)
\(858\) 959.752 0.0381881
\(859\) 48813.4 1.93887 0.969436 0.245343i \(-0.0789005\pi\)
0.969436 + 0.245343i \(0.0789005\pi\)
\(860\) 0 0
\(861\) −4357.59 −0.172481
\(862\) 10339.4 0.408540
\(863\) −38260.4 −1.50915 −0.754577 0.656212i \(-0.772158\pi\)
−0.754577 + 0.656212i \(0.772158\pi\)
\(864\) −39026.1 −1.53669
\(865\) −20636.3 −0.811161
\(866\) −62744.2 −2.46205
\(867\) 13428.7 0.526023
\(868\) −9760.88 −0.381689
\(869\) −59.8256 −0.00233538
\(870\) −70343.6 −2.74123
\(871\) −27849.3 −1.08339
\(872\) −10478.1 −0.406919
\(873\) 8002.53 0.310246
\(874\) −30527.5 −1.18147
\(875\) −80997.3 −3.12938
\(876\) 5859.88 0.226013
\(877\) 29665.4 1.14222 0.571112 0.820872i \(-0.306512\pi\)
0.571112 + 0.820872i \(0.306512\pi\)
\(878\) 12234.5 0.470268
\(879\) 3594.17 0.137916
\(880\) −768.814 −0.0294508
\(881\) 3377.77 0.129171 0.0645857 0.997912i \(-0.479427\pi\)
0.0645857 + 0.997912i \(0.479427\pi\)
\(882\) −9509.77 −0.363050
\(883\) 32125.9 1.22437 0.612186 0.790713i \(-0.290290\pi\)
0.612186 + 0.790713i \(0.290290\pi\)
\(884\) −28143.8 −1.07079
\(885\) −10273.8 −0.390225
\(886\) −28012.6 −1.06219
\(887\) −25106.0 −0.950368 −0.475184 0.879886i \(-0.657618\pi\)
−0.475184 + 0.879886i \(0.657618\pi\)
\(888\) −2272.15 −0.0858652
\(889\) −10453.3 −0.394366
\(890\) 92997.3 3.50256
\(891\) 409.207 0.0153860
\(892\) −3746.57 −0.140633
\(893\) 51465.8 1.92860
\(894\) 14585.8 0.545663
\(895\) −25209.5 −0.941522
\(896\) −17936.9 −0.668782
\(897\) 24038.6 0.894789
\(898\) −21251.1 −0.789707
\(899\) 6941.67 0.257528
\(900\) −13238.6 −0.490318
\(901\) 6.04103 0.000223370 0
\(902\) −102.182 −0.00377194
\(903\) 0 0
\(904\) 4640.23 0.170721
\(905\) 1204.97 0.0442591
\(906\) −49412.8 −1.81195
\(907\) 17495.5 0.640496 0.320248 0.947334i \(-0.396234\pi\)
0.320248 + 0.947334i \(0.396234\pi\)
\(908\) 41179.5 1.50506
\(909\) 7011.61 0.255842
\(910\) −155749. −5.67366
\(911\) −44170.3 −1.60639 −0.803197 0.595713i \(-0.796870\pi\)
−0.803197 + 0.595713i \(0.796870\pi\)
\(912\) 23685.6 0.859987
\(913\) 679.091 0.0246162
\(914\) 48433.3 1.75277
\(915\) 49447.2 1.78653
\(916\) −50832.7 −1.83358
\(917\) −75244.6 −2.70970
\(918\) −28140.7 −1.01174
\(919\) 2965.31 0.106438 0.0532191 0.998583i \(-0.483052\pi\)
0.0532191 + 0.998583i \(0.483052\pi\)
\(920\) 7722.50 0.276743
\(921\) 30880.7 1.10483
\(922\) −53011.0 −1.89352
\(923\) −14625.9 −0.521580
\(924\) 877.986 0.0312593
\(925\) 25971.8 0.923188
\(926\) 27837.9 0.987917
\(927\) 5903.05 0.209149
\(928\) 47409.7 1.67705
\(929\) −34759.3 −1.22757 −0.613786 0.789472i \(-0.710354\pi\)
−0.613786 + 0.789472i \(0.710354\pi\)
\(930\) −14616.8 −0.515380
\(931\) 41773.9 1.47055
\(932\) 38963.1 1.36940
\(933\) −22386.3 −0.785526
\(934\) 54737.8 1.91764
\(935\) −657.619 −0.0230015
\(936\) −1823.69 −0.0636850
\(937\) 27998.7 0.976178 0.488089 0.872794i \(-0.337694\pi\)
0.488089 + 0.872794i \(0.337694\pi\)
\(938\) −47566.6 −1.65576
\(939\) 4515.01 0.156914
\(940\) −97939.1 −3.39832
\(941\) −23821.3 −0.825242 −0.412621 0.910903i \(-0.635387\pi\)
−0.412621 + 0.910903i \(0.635387\pi\)
\(942\) 35030.6 1.21163
\(943\) −2559.32 −0.0883808
\(944\) 5837.12 0.201252
\(945\) −83409.9 −2.87124
\(946\) 0 0
\(947\) −15304.0 −0.525145 −0.262573 0.964912i \(-0.584571\pi\)
−0.262573 + 0.964912i \(0.584571\pi\)
\(948\) −3507.41 −0.120164
\(949\) −9227.09 −0.315621
\(950\) 108577. 3.70810
\(951\) 15279.5 0.521002
\(952\) −6390.00 −0.217543
\(953\) 528.850 0.0179760 0.00898800 0.999960i \(-0.497139\pi\)
0.00898800 + 0.999960i \(0.497139\pi\)
\(954\) 2.94476 9.99373e−5 0
\(955\) −44528.2 −1.50880
\(956\) −8089.98 −0.273691
\(957\) −624.399 −0.0210909
\(958\) 12383.6 0.417635
\(959\) −56331.2 −1.89680
\(960\) −60746.8 −2.04229
\(961\) −28348.6 −0.951582
\(962\) 26914.3 0.902029
\(963\) 3429.40 0.114757
\(964\) 29246.7 0.977151
\(965\) 22894.3 0.763724
\(966\) 41058.0 1.36752
\(967\) 22602.1 0.751639 0.375819 0.926693i \(-0.377361\pi\)
0.375819 + 0.926693i \(0.377361\pi\)
\(968\) −6772.78 −0.224882
\(969\) 20259.9 0.671663
\(970\) 124899. 4.13430
\(971\) 23322.2 0.770797 0.385398 0.922750i \(-0.374064\pi\)
0.385398 + 0.922750i \(0.374064\pi\)
\(972\) −13490.0 −0.445156
\(973\) −15722.3 −0.518021
\(974\) 3751.46 0.123413
\(975\) −85497.8 −2.80833
\(976\) −28093.8 −0.921373
\(977\) 27151.1 0.889090 0.444545 0.895757i \(-0.353365\pi\)
0.444545 + 0.895757i \(0.353365\pi\)
\(978\) −55407.2 −1.81158
\(979\) 825.483 0.0269485
\(980\) −79495.4 −2.59121
\(981\) 10894.0 0.354556
\(982\) 61201.2 1.98881
\(983\) −16794.4 −0.544922 −0.272461 0.962167i \(-0.587837\pi\)
−0.272461 + 0.962167i \(0.587837\pi\)
\(984\) −796.351 −0.0257995
\(985\) −70968.6 −2.29568
\(986\) 34185.9 1.10416
\(987\) −69219.0 −2.23229
\(988\) 60263.9 1.94054
\(989\) 0 0
\(990\) −320.563 −0.0102911
\(991\) 6415.98 0.205661 0.102831 0.994699i \(-0.467210\pi\)
0.102831 + 0.994699i \(0.467210\pi\)
\(992\) 9851.32 0.315302
\(993\) −30907.7 −0.987742
\(994\) −24981.1 −0.797136
\(995\) 46050.1 1.46722
\(996\) 39813.3 1.26660
\(997\) −38417.5 −1.22036 −0.610178 0.792264i \(-0.708902\pi\)
−0.610178 + 0.792264i \(0.708902\pi\)
\(998\) 3511.62 0.111381
\(999\) 14413.7 0.456486
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.i.1.43 50
43.42 odd 2 1849.4.a.j.1.8 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.43 50 1.1 even 1 trivial
1849.4.a.j.1.8 yes 50 43.42 odd 2