Properties

Label 1849.4.a.j.1.22
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 1.22
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.05440 q^{2} +4.53621 q^{3} -6.88824 q^{4} +14.4725 q^{5} -4.78298 q^{6} +15.4664 q^{7} +15.6982 q^{8} -6.42280 q^{9} +O(q^{10})\) \(q-1.05440 q^{2} +4.53621 q^{3} -6.88824 q^{4} +14.4725 q^{5} -4.78298 q^{6} +15.4664 q^{7} +15.6982 q^{8} -6.42280 q^{9} -15.2598 q^{10} -62.4686 q^{11} -31.2465 q^{12} -21.6145 q^{13} -16.3078 q^{14} +65.6505 q^{15} +38.5538 q^{16} +25.1556 q^{17} +6.77220 q^{18} -45.2306 q^{19} -99.6903 q^{20} +70.1588 q^{21} +65.8669 q^{22} +195.169 q^{23} +71.2102 q^{24} +84.4542 q^{25} +22.7903 q^{26} -151.613 q^{27} -106.536 q^{28} +39.6769 q^{29} -69.2218 q^{30} -254.845 q^{31} -166.236 q^{32} -283.371 q^{33} -26.5241 q^{34} +223.838 q^{35} +44.2418 q^{36} +110.141 q^{37} +47.6911 q^{38} -98.0478 q^{39} +227.192 q^{40} -62.1285 q^{41} -73.9754 q^{42} +430.299 q^{44} -92.9542 q^{45} -205.786 q^{46} +586.112 q^{47} +174.888 q^{48} -103.791 q^{49} -89.0485 q^{50} +114.111 q^{51} +148.886 q^{52} -515.058 q^{53} +159.861 q^{54} -904.079 q^{55} +242.794 q^{56} -205.175 q^{57} -41.8353 q^{58} +250.178 q^{59} -452.216 q^{60} -812.162 q^{61} +268.709 q^{62} -99.3374 q^{63} -133.151 q^{64} -312.816 q^{65} +298.786 q^{66} -558.014 q^{67} -173.278 q^{68} +885.326 q^{69} -236.014 q^{70} -227.436 q^{71} -100.826 q^{72} +568.115 q^{73} -116.133 q^{74} +383.102 q^{75} +311.559 q^{76} -966.164 q^{77} +103.382 q^{78} -1336.40 q^{79} +557.971 q^{80} -514.332 q^{81} +65.5083 q^{82} -16.2755 q^{83} -483.270 q^{84} +364.066 q^{85} +179.983 q^{87} -980.643 q^{88} -858.571 q^{89} +98.0109 q^{90} -334.298 q^{91} -1344.37 q^{92} -1156.03 q^{93} -617.997 q^{94} -654.601 q^{95} -754.083 q^{96} -312.788 q^{97} +109.437 q^{98} +401.223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05440 −0.372787 −0.186393 0.982475i \(-0.559680\pi\)
−0.186393 + 0.982475i \(0.559680\pi\)
\(3\) 4.53621 0.872994 0.436497 0.899706i \(-0.356219\pi\)
0.436497 + 0.899706i \(0.356219\pi\)
\(4\) −6.88824 −0.861030
\(5\) 14.4725 1.29446 0.647231 0.762294i \(-0.275927\pi\)
0.647231 + 0.762294i \(0.275927\pi\)
\(6\) −4.78298 −0.325441
\(7\) 15.4664 0.835106 0.417553 0.908653i \(-0.362888\pi\)
0.417553 + 0.908653i \(0.362888\pi\)
\(8\) 15.6982 0.693767
\(9\) −6.42280 −0.237881
\(10\) −15.2598 −0.482559
\(11\) −62.4686 −1.71227 −0.856136 0.516750i \(-0.827142\pi\)
−0.856136 + 0.516750i \(0.827142\pi\)
\(12\) −31.2465 −0.751674
\(13\) −21.6145 −0.461137 −0.230568 0.973056i \(-0.574059\pi\)
−0.230568 + 0.973056i \(0.574059\pi\)
\(14\) −16.3078 −0.311316
\(15\) 65.6505 1.13006
\(16\) 38.5538 0.602403
\(17\) 25.1556 0.358891 0.179445 0.983768i \(-0.442570\pi\)
0.179445 + 0.983768i \(0.442570\pi\)
\(18\) 6.77220 0.0886790
\(19\) −45.2306 −0.546137 −0.273069 0.961995i \(-0.588039\pi\)
−0.273069 + 0.961995i \(0.588039\pi\)
\(20\) −99.6903 −1.11457
\(21\) 70.1588 0.729043
\(22\) 65.8669 0.638313
\(23\) 195.169 1.76937 0.884684 0.466191i \(-0.154374\pi\)
0.884684 + 0.466191i \(0.154374\pi\)
\(24\) 71.2102 0.605655
\(25\) 84.4542 0.675634
\(26\) 22.7903 0.171906
\(27\) −151.613 −1.08066
\(28\) −106.536 −0.719051
\(29\) 39.6769 0.254063 0.127031 0.991899i \(-0.459455\pi\)
0.127031 + 0.991899i \(0.459455\pi\)
\(30\) −69.2218 −0.421271
\(31\) −254.845 −1.47650 −0.738250 0.674527i \(-0.764348\pi\)
−0.738250 + 0.674527i \(0.764348\pi\)
\(32\) −166.236 −0.918335
\(33\) −283.371 −1.49480
\(34\) −26.5241 −0.133790
\(35\) 223.838 1.08101
\(36\) 44.2418 0.204823
\(37\) 110.141 0.489380 0.244690 0.969601i \(-0.421314\pi\)
0.244690 + 0.969601i \(0.421314\pi\)
\(38\) 47.6911 0.203593
\(39\) −98.0478 −0.402570
\(40\) 227.192 0.898056
\(41\) −62.1285 −0.236655 −0.118327 0.992975i \(-0.537753\pi\)
−0.118327 + 0.992975i \(0.537753\pi\)
\(42\) −73.9754 −0.271777
\(43\) 0 0
\(44\) 430.299 1.47432
\(45\) −92.9542 −0.307929
\(46\) −205.786 −0.659597
\(47\) 586.112 1.81900 0.909502 0.415698i \(-0.136463\pi\)
0.909502 + 0.415698i \(0.136463\pi\)
\(48\) 174.888 0.525894
\(49\) −103.791 −0.302598
\(50\) −89.0485 −0.251867
\(51\) 114.111 0.313309
\(52\) 148.886 0.397053
\(53\) −515.058 −1.33488 −0.667441 0.744663i \(-0.732610\pi\)
−0.667441 + 0.744663i \(0.732610\pi\)
\(54\) 159.861 0.402857
\(55\) −904.079 −2.21647
\(56\) 242.794 0.579369
\(57\) −205.175 −0.476775
\(58\) −41.8353 −0.0947112
\(59\) 250.178 0.552041 0.276020 0.961152i \(-0.410984\pi\)
0.276020 + 0.961152i \(0.410984\pi\)
\(60\) −452.216 −0.973014
\(61\) −812.162 −1.70470 −0.852350 0.522971i \(-0.824823\pi\)
−0.852350 + 0.522971i \(0.824823\pi\)
\(62\) 268.709 0.550420
\(63\) −99.3374 −0.198656
\(64\) −133.151 −0.260060
\(65\) −312.816 −0.596924
\(66\) 298.786 0.557243
\(67\) −558.014 −1.01750 −0.508748 0.860915i \(-0.669892\pi\)
−0.508748 + 0.860915i \(0.669892\pi\)
\(68\) −173.278 −0.309016
\(69\) 885.326 1.54465
\(70\) −236.014 −0.402988
\(71\) −227.436 −0.380165 −0.190083 0.981768i \(-0.560876\pi\)
−0.190083 + 0.981768i \(0.560876\pi\)
\(72\) −100.826 −0.165034
\(73\) 568.115 0.910860 0.455430 0.890272i \(-0.349485\pi\)
0.455430 + 0.890272i \(0.349485\pi\)
\(74\) −116.133 −0.182434
\(75\) 383.102 0.589824
\(76\) 311.559 0.470241
\(77\) −966.164 −1.42993
\(78\) 103.382 0.150073
\(79\) −1336.40 −1.90325 −0.951624 0.307264i \(-0.900586\pi\)
−0.951624 + 0.307264i \(0.900586\pi\)
\(80\) 557.971 0.779788
\(81\) −514.332 −0.705531
\(82\) 65.5083 0.0882217
\(83\) −16.2755 −0.0215237 −0.0107619 0.999942i \(-0.503426\pi\)
−0.0107619 + 0.999942i \(0.503426\pi\)
\(84\) −483.270 −0.627728
\(85\) 364.066 0.464570
\(86\) 0 0
\(87\) 179.983 0.221795
\(88\) −980.643 −1.18792
\(89\) −858.571 −1.02257 −0.511283 0.859412i \(-0.670830\pi\)
−0.511283 + 0.859412i \(0.670830\pi\)
\(90\) 98.0109 0.114792
\(91\) −334.298 −0.385098
\(92\) −1344.37 −1.52348
\(93\) −1156.03 −1.28898
\(94\) −617.997 −0.678101
\(95\) −654.601 −0.706954
\(96\) −754.083 −0.801701
\(97\) −312.788 −0.327410 −0.163705 0.986509i \(-0.552345\pi\)
−0.163705 + 0.986509i \(0.552345\pi\)
\(98\) 109.437 0.112804
\(99\) 401.223 0.407318
\(100\) −581.741 −0.581741
\(101\) −524.823 −0.517048 −0.258524 0.966005i \(-0.583236\pi\)
−0.258524 + 0.966005i \(0.583236\pi\)
\(102\) −120.319 −0.116798
\(103\) 517.358 0.494920 0.247460 0.968898i \(-0.420404\pi\)
0.247460 + 0.968898i \(0.420404\pi\)
\(104\) −339.308 −0.319922
\(105\) 1015.37 0.943718
\(106\) 543.078 0.497626
\(107\) 1251.80 1.13099 0.565494 0.824752i \(-0.308685\pi\)
0.565494 + 0.824752i \(0.308685\pi\)
\(108\) 1044.35 0.930483
\(109\) −775.264 −0.681256 −0.340628 0.940198i \(-0.610640\pi\)
−0.340628 + 0.940198i \(0.610640\pi\)
\(110\) 953.261 0.826272
\(111\) 499.622 0.427226
\(112\) 596.287 0.503070
\(113\) 64.5912 0.0537719 0.0268859 0.999639i \(-0.491441\pi\)
0.0268859 + 0.999639i \(0.491441\pi\)
\(114\) 216.337 0.177735
\(115\) 2824.58 2.29038
\(116\) −273.304 −0.218756
\(117\) 138.825 0.109696
\(118\) −263.788 −0.205793
\(119\) 389.067 0.299712
\(120\) 1030.59 0.783997
\(121\) 2571.33 1.93188
\(122\) 856.344 0.635490
\(123\) −281.828 −0.206598
\(124\) 1755.43 1.27131
\(125\) −586.800 −0.419880
\(126\) 104.741 0.0740564
\(127\) −231.994 −0.162095 −0.0810477 0.996710i \(-0.525827\pi\)
−0.0810477 + 0.996710i \(0.525827\pi\)
\(128\) 1470.29 1.01528
\(129\) 0 0
\(130\) 329.833 0.222525
\(131\) −378.667 −0.252552 −0.126276 0.991995i \(-0.540302\pi\)
−0.126276 + 0.991995i \(0.540302\pi\)
\(132\) 1951.93 1.28707
\(133\) −699.553 −0.456082
\(134\) 588.370 0.379309
\(135\) −2194.22 −1.39888
\(136\) 394.897 0.248987
\(137\) 189.827 0.118380 0.0591899 0.998247i \(-0.481148\pi\)
0.0591899 + 0.998247i \(0.481148\pi\)
\(138\) −933.488 −0.575824
\(139\) 1933.32 1.17973 0.589865 0.807502i \(-0.299181\pi\)
0.589865 + 0.807502i \(0.299181\pi\)
\(140\) −1541.85 −0.930785
\(141\) 2658.73 1.58798
\(142\) 239.809 0.141721
\(143\) 1350.23 0.789592
\(144\) −247.623 −0.143300
\(145\) 574.225 0.328875
\(146\) −599.020 −0.339557
\(147\) −470.818 −0.264166
\(148\) −758.677 −0.421371
\(149\) −3149.21 −1.73150 −0.865749 0.500479i \(-0.833157\pi\)
−0.865749 + 0.500479i \(0.833157\pi\)
\(150\) −403.943 −0.219879
\(151\) −2085.86 −1.12414 −0.562068 0.827091i \(-0.689994\pi\)
−0.562068 + 0.827091i \(0.689994\pi\)
\(152\) −710.037 −0.378892
\(153\) −161.570 −0.0853734
\(154\) 1018.72 0.533059
\(155\) −3688.25 −1.91128
\(156\) 675.377 0.346624
\(157\) 3511.99 1.78527 0.892634 0.450782i \(-0.148855\pi\)
0.892634 + 0.450782i \(0.148855\pi\)
\(158\) 1409.10 0.709506
\(159\) −2336.41 −1.16534
\(160\) −2405.86 −1.18875
\(161\) 3018.55 1.47761
\(162\) 542.312 0.263013
\(163\) −2706.27 −1.30044 −0.650219 0.759747i \(-0.725323\pi\)
−0.650219 + 0.759747i \(0.725323\pi\)
\(164\) 427.956 0.203767
\(165\) −4101.09 −1.93497
\(166\) 17.1609 0.00802376
\(167\) 1012.98 0.469380 0.234690 0.972070i \(-0.424593\pi\)
0.234690 + 0.972070i \(0.424593\pi\)
\(168\) 1101.36 0.505786
\(169\) −1729.81 −0.787353
\(170\) −383.871 −0.173186
\(171\) 290.507 0.129916
\(172\) 0 0
\(173\) 456.153 0.200466 0.100233 0.994964i \(-0.468041\pi\)
0.100233 + 0.994964i \(0.468041\pi\)
\(174\) −189.774 −0.0826823
\(175\) 1306.20 0.564226
\(176\) −2408.40 −1.03148
\(177\) 1134.86 0.481928
\(178\) 905.278 0.381199
\(179\) −2095.61 −0.875045 −0.437523 0.899207i \(-0.644144\pi\)
−0.437523 + 0.899207i \(0.644144\pi\)
\(180\) 640.291 0.265136
\(181\) 3117.69 1.28031 0.640155 0.768246i \(-0.278870\pi\)
0.640155 + 0.768246i \(0.278870\pi\)
\(182\) 352.484 0.143559
\(183\) −3684.14 −1.48819
\(184\) 3063.79 1.22753
\(185\) 1594.02 0.633484
\(186\) 1218.92 0.480513
\(187\) −1571.44 −0.614519
\(188\) −4037.28 −1.56622
\(189\) −2344.90 −0.902468
\(190\) 690.211 0.263543
\(191\) 3068.03 1.16228 0.581138 0.813805i \(-0.302608\pi\)
0.581138 + 0.813805i \(0.302608\pi\)
\(192\) −603.999 −0.227031
\(193\) −2193.07 −0.817933 −0.408966 0.912549i \(-0.634111\pi\)
−0.408966 + 0.912549i \(0.634111\pi\)
\(194\) 329.804 0.122054
\(195\) −1419.00 −0.521111
\(196\) 714.938 0.260546
\(197\) 302.542 0.109417 0.0547087 0.998502i \(-0.482577\pi\)
0.0547087 + 0.998502i \(0.482577\pi\)
\(198\) −423.050 −0.151843
\(199\) −1003.68 −0.357533 −0.178766 0.983892i \(-0.557211\pi\)
−0.178766 + 0.983892i \(0.557211\pi\)
\(200\) 1325.78 0.468733
\(201\) −2531.27 −0.888268
\(202\) 553.373 0.192749
\(203\) 613.658 0.212169
\(204\) −786.026 −0.269769
\(205\) −899.157 −0.306341
\(206\) −545.502 −0.184500
\(207\) −1253.53 −0.420900
\(208\) −833.320 −0.277790
\(209\) 2825.49 0.935136
\(210\) −1070.61 −0.351806
\(211\) −4888.64 −1.59501 −0.797506 0.603311i \(-0.793848\pi\)
−0.797506 + 0.603311i \(0.793848\pi\)
\(212\) 3547.85 1.14937
\(213\) −1031.70 −0.331882
\(214\) −1319.90 −0.421618
\(215\) 0 0
\(216\) −2380.04 −0.749729
\(217\) −3941.53 −1.23303
\(218\) 817.439 0.253963
\(219\) 2577.09 0.795176
\(220\) 6227.52 1.90845
\(221\) −543.726 −0.165498
\(222\) −526.802 −0.159264
\(223\) −3321.24 −0.997340 −0.498670 0.866792i \(-0.666178\pi\)
−0.498670 + 0.866792i \(0.666178\pi\)
\(224\) −2571.08 −0.766907
\(225\) −542.432 −0.160721
\(226\) −68.1049 −0.0200455
\(227\) 3632.36 1.06206 0.531032 0.847352i \(-0.321805\pi\)
0.531032 + 0.847352i \(0.321805\pi\)
\(228\) 1413.30 0.410517
\(229\) 3958.77 1.14237 0.571185 0.820821i \(-0.306484\pi\)
0.571185 + 0.820821i \(0.306484\pi\)
\(230\) −2978.24 −0.853824
\(231\) −4382.72 −1.24832
\(232\) 622.855 0.176260
\(233\) 4538.90 1.27620 0.638098 0.769955i \(-0.279722\pi\)
0.638098 + 0.769955i \(0.279722\pi\)
\(234\) −146.378 −0.0408932
\(235\) 8482.53 2.35463
\(236\) −1723.29 −0.475324
\(237\) −6062.19 −1.66152
\(238\) −410.232 −0.111729
\(239\) −6078.04 −1.64500 −0.822502 0.568762i \(-0.807422\pi\)
−0.822502 + 0.568762i \(0.807422\pi\)
\(240\) 2531.07 0.680750
\(241\) −4552.97 −1.21694 −0.608471 0.793576i \(-0.708217\pi\)
−0.608471 + 0.793576i \(0.708217\pi\)
\(242\) −2711.21 −0.720179
\(243\) 1760.43 0.464739
\(244\) 5594.37 1.46780
\(245\) −1502.12 −0.391702
\(246\) 297.159 0.0770171
\(247\) 977.635 0.251844
\(248\) −4000.60 −1.02435
\(249\) −73.8292 −0.0187901
\(250\) 618.722 0.156526
\(251\) −5848.23 −1.47067 −0.735333 0.677706i \(-0.762974\pi\)
−0.735333 + 0.677706i \(0.762974\pi\)
\(252\) 684.260 0.171049
\(253\) −12191.9 −3.02964
\(254\) 244.614 0.0604270
\(255\) 1651.48 0.405567
\(256\) −485.064 −0.118424
\(257\) −1935.75 −0.469840 −0.234920 0.972015i \(-0.575483\pi\)
−0.234920 + 0.972015i \(0.575483\pi\)
\(258\) 0 0
\(259\) 1703.48 0.408684
\(260\) 2154.75 0.513970
\(261\) −254.837 −0.0604368
\(262\) 399.267 0.0941480
\(263\) 6747.04 1.58190 0.790951 0.611879i \(-0.209586\pi\)
0.790951 + 0.611879i \(0.209586\pi\)
\(264\) −4448.40 −1.03705
\(265\) −7454.20 −1.72795
\(266\) 737.609 0.170021
\(267\) −3894.66 −0.892694
\(268\) 3843.74 0.876095
\(269\) −198.559 −0.0450051 −0.0225025 0.999747i \(-0.507163\pi\)
−0.0225025 + 0.999747i \(0.507163\pi\)
\(270\) 2313.59 0.521483
\(271\) 2023.47 0.453570 0.226785 0.973945i \(-0.427179\pi\)
0.226785 + 0.973945i \(0.427179\pi\)
\(272\) 969.845 0.216197
\(273\) −1516.44 −0.336188
\(274\) −200.154 −0.0441304
\(275\) −5275.74 −1.15687
\(276\) −6098.34 −1.32999
\(277\) −8288.84 −1.79794 −0.898968 0.438014i \(-0.855682\pi\)
−0.898968 + 0.438014i \(0.855682\pi\)
\(278\) −2038.50 −0.439787
\(279\) 1636.82 0.351232
\(280\) 3513.84 0.749972
\(281\) 5413.58 1.14928 0.574639 0.818407i \(-0.305142\pi\)
0.574639 + 0.818407i \(0.305142\pi\)
\(282\) −2803.36 −0.591978
\(283\) −7149.31 −1.50170 −0.750852 0.660471i \(-0.770357\pi\)
−0.750852 + 0.660471i \(0.770357\pi\)
\(284\) 1566.64 0.327334
\(285\) −2969.41 −0.617167
\(286\) −1423.68 −0.294349
\(287\) −960.903 −0.197632
\(288\) 1067.70 0.218455
\(289\) −4280.19 −0.871198
\(290\) −605.463 −0.122600
\(291\) −1418.87 −0.285827
\(292\) −3913.31 −0.784278
\(293\) 4475.99 0.892458 0.446229 0.894919i \(-0.352767\pi\)
0.446229 + 0.894919i \(0.352767\pi\)
\(294\) 496.431 0.0984776
\(295\) 3620.71 0.714596
\(296\) 1729.01 0.339516
\(297\) 9471.05 1.85039
\(298\) 3320.52 0.645479
\(299\) −4218.47 −0.815920
\(300\) −2638.90 −0.507856
\(301\) 0 0
\(302\) 2199.33 0.419063
\(303\) −2380.71 −0.451380
\(304\) −1743.81 −0.328995
\(305\) −11754.0 −2.20667
\(306\) 170.359 0.0318261
\(307\) −3974.32 −0.738848 −0.369424 0.929261i \(-0.620445\pi\)
−0.369424 + 0.929261i \(0.620445\pi\)
\(308\) 6655.17 1.23121
\(309\) 2346.84 0.432062
\(310\) 3888.90 0.712498
\(311\) −5015.11 −0.914408 −0.457204 0.889362i \(-0.651149\pi\)
−0.457204 + 0.889362i \(0.651149\pi\)
\(312\) −1539.17 −0.279290
\(313\) −466.135 −0.0841774 −0.0420887 0.999114i \(-0.513401\pi\)
−0.0420887 + 0.999114i \(0.513401\pi\)
\(314\) −3703.04 −0.665524
\(315\) −1437.66 −0.257153
\(316\) 9205.44 1.63875
\(317\) −1289.75 −0.228515 −0.114258 0.993451i \(-0.536449\pi\)
−0.114258 + 0.993451i \(0.536449\pi\)
\(318\) 2463.51 0.434425
\(319\) −2478.56 −0.435025
\(320\) −1927.03 −0.336638
\(321\) 5678.42 0.987347
\(322\) −3182.76 −0.550833
\(323\) −1137.80 −0.196003
\(324\) 3542.84 0.607483
\(325\) −1825.43 −0.311559
\(326\) 2853.49 0.484786
\(327\) −3516.76 −0.594732
\(328\) −975.303 −0.164183
\(329\) 9065.03 1.51906
\(330\) 4324.19 0.721330
\(331\) −6084.71 −1.01041 −0.505205 0.862999i \(-0.668583\pi\)
−0.505205 + 0.862999i \(0.668583\pi\)
\(332\) 112.110 0.0185326
\(333\) −707.413 −0.116414
\(334\) −1068.08 −0.174978
\(335\) −8075.88 −1.31711
\(336\) 2704.89 0.439177
\(337\) −2709.51 −0.437972 −0.218986 0.975728i \(-0.570275\pi\)
−0.218986 + 0.975728i \(0.570275\pi\)
\(338\) 1823.92 0.293515
\(339\) 292.999 0.0469425
\(340\) −2507.77 −0.400009
\(341\) 15919.8 2.52817
\(342\) −306.310 −0.0484309
\(343\) −6910.24 −1.08781
\(344\) 0 0
\(345\) 12812.9 1.99949
\(346\) −480.967 −0.0747311
\(347\) −817.917 −0.126536 −0.0632681 0.997997i \(-0.520152\pi\)
−0.0632681 + 0.997997i \(0.520152\pi\)
\(348\) −1239.76 −0.190972
\(349\) −2334.83 −0.358111 −0.179055 0.983839i \(-0.557304\pi\)
−0.179055 + 0.983839i \(0.557304\pi\)
\(350\) −1377.26 −0.210336
\(351\) 3277.03 0.498333
\(352\) 10384.6 1.57244
\(353\) −6169.22 −0.930182 −0.465091 0.885263i \(-0.653978\pi\)
−0.465091 + 0.885263i \(0.653978\pi\)
\(354\) −1196.60 −0.179656
\(355\) −3291.58 −0.492110
\(356\) 5914.05 0.880460
\(357\) 1764.89 0.261647
\(358\) 2209.61 0.326205
\(359\) 183.060 0.0269123 0.0134562 0.999909i \(-0.495717\pi\)
0.0134562 + 0.999909i \(0.495717\pi\)
\(360\) −1459.21 −0.213631
\(361\) −4813.19 −0.701734
\(362\) −3287.29 −0.477283
\(363\) 11664.1 1.68652
\(364\) 2302.72 0.331581
\(365\) 8222.06 1.17907
\(366\) 3884.56 0.554779
\(367\) 4878.74 0.693919 0.346960 0.937880i \(-0.387214\pi\)
0.346960 + 0.937880i \(0.387214\pi\)
\(368\) 7524.49 1.06587
\(369\) 399.039 0.0562958
\(370\) −1680.73 −0.236154
\(371\) −7966.09 −1.11477
\(372\) 7963.02 1.10985
\(373\) 2998.62 0.416254 0.208127 0.978102i \(-0.433263\pi\)
0.208127 + 0.978102i \(0.433263\pi\)
\(374\) 1656.93 0.229084
\(375\) −2661.85 −0.366553
\(376\) 9200.88 1.26197
\(377\) −857.596 −0.117158
\(378\) 2472.46 0.336428
\(379\) 10795.8 1.46317 0.731585 0.681750i \(-0.238781\pi\)
0.731585 + 0.681750i \(0.238781\pi\)
\(380\) 4509.05 0.608709
\(381\) −1052.37 −0.141508
\(382\) −3234.93 −0.433281
\(383\) 9850.38 1.31418 0.657090 0.753812i \(-0.271787\pi\)
0.657090 + 0.753812i \(0.271787\pi\)
\(384\) 6669.52 0.886335
\(385\) −13982.8 −1.85099
\(386\) 2312.38 0.304914
\(387\) 0 0
\(388\) 2154.56 0.281910
\(389\) −547.776 −0.0713967 −0.0356984 0.999363i \(-0.511366\pi\)
−0.0356984 + 0.999363i \(0.511366\pi\)
\(390\) 1496.19 0.194263
\(391\) 4909.59 0.635010
\(392\) −1629.33 −0.209933
\(393\) −1717.71 −0.220476
\(394\) −319.000 −0.0407894
\(395\) −19341.1 −2.46368
\(396\) −2763.72 −0.350713
\(397\) 3433.56 0.434069 0.217035 0.976164i \(-0.430362\pi\)
0.217035 + 0.976164i \(0.430362\pi\)
\(398\) 1058.28 0.133283
\(399\) −3173.32 −0.398157
\(400\) 3256.03 0.407004
\(401\) 2541.81 0.316539 0.158269 0.987396i \(-0.449409\pi\)
0.158269 + 0.987396i \(0.449409\pi\)
\(402\) 2668.97 0.331135
\(403\) 5508.34 0.680869
\(404\) 3615.11 0.445194
\(405\) −7443.69 −0.913284
\(406\) −647.041 −0.0790939
\(407\) −6880.35 −0.837952
\(408\) 1791.34 0.217364
\(409\) −2769.69 −0.334847 −0.167423 0.985885i \(-0.553545\pi\)
−0.167423 + 0.985885i \(0.553545\pi\)
\(410\) 948.071 0.114200
\(411\) 861.096 0.103345
\(412\) −3563.68 −0.426141
\(413\) 3869.35 0.461013
\(414\) 1321.72 0.156906
\(415\) −235.548 −0.0278617
\(416\) 3593.11 0.423478
\(417\) 8769.96 1.02990
\(418\) −2979.20 −0.348606
\(419\) −3091.89 −0.360498 −0.180249 0.983621i \(-0.557690\pi\)
−0.180249 + 0.983621i \(0.557690\pi\)
\(420\) −6994.15 −0.812570
\(421\) 7883.68 0.912653 0.456327 0.889812i \(-0.349165\pi\)
0.456327 + 0.889812i \(0.349165\pi\)
\(422\) 5154.58 0.594599
\(423\) −3764.48 −0.432707
\(424\) −8085.47 −0.926097
\(425\) 2124.50 0.242479
\(426\) 1087.82 0.123721
\(427\) −12561.2 −1.42361
\(428\) −8622.68 −0.973815
\(429\) 6124.91 0.689309
\(430\) 0 0
\(431\) −12359.0 −1.38124 −0.690618 0.723219i \(-0.742661\pi\)
−0.690618 + 0.723219i \(0.742661\pi\)
\(432\) −5845.25 −0.650994
\(433\) −14075.5 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(434\) 4155.95 0.459659
\(435\) 2604.81 0.287106
\(436\) 5340.21 0.586582
\(437\) −8827.59 −0.966318
\(438\) −2717.28 −0.296431
\(439\) 3314.21 0.360316 0.180158 0.983638i \(-0.442339\pi\)
0.180158 + 0.983638i \(0.442339\pi\)
\(440\) −14192.4 −1.53772
\(441\) 666.629 0.0719824
\(442\) 573.305 0.0616953
\(443\) −6141.24 −0.658643 −0.329322 0.944218i \(-0.606820\pi\)
−0.329322 + 0.944218i \(0.606820\pi\)
\(444\) −3441.52 −0.367854
\(445\) −12425.7 −1.32367
\(446\) 3501.92 0.371795
\(447\) −14285.5 −1.51159
\(448\) −2059.36 −0.217177
\(449\) −4718.97 −0.495995 −0.247998 0.968761i \(-0.579773\pi\)
−0.247998 + 0.968761i \(0.579773\pi\)
\(450\) 571.941 0.0599145
\(451\) 3881.08 0.405217
\(452\) −444.919 −0.0462992
\(453\) −9461.88 −0.981364
\(454\) −3829.96 −0.395923
\(455\) −4838.13 −0.498495
\(456\) −3220.88 −0.330771
\(457\) −4042.78 −0.413815 −0.206907 0.978361i \(-0.566340\pi\)
−0.206907 + 0.978361i \(0.566340\pi\)
\(458\) −4174.13 −0.425861
\(459\) −3813.92 −0.387840
\(460\) −19456.4 −1.97209
\(461\) 17986.7 1.81719 0.908595 0.417678i \(-0.137156\pi\)
0.908595 + 0.417678i \(0.137156\pi\)
\(462\) 4621.14 0.465357
\(463\) 5977.08 0.599953 0.299977 0.953947i \(-0.403021\pi\)
0.299977 + 0.953947i \(0.403021\pi\)
\(464\) 1529.69 0.153048
\(465\) −16730.7 −1.66853
\(466\) −4785.82 −0.475749
\(467\) 7804.47 0.773336 0.386668 0.922219i \(-0.373626\pi\)
0.386668 + 0.922219i \(0.373626\pi\)
\(468\) −956.263 −0.0944514
\(469\) −8630.46 −0.849718
\(470\) −8943.98 −0.877776
\(471\) 15931.1 1.55853
\(472\) 3927.34 0.382988
\(473\) 0 0
\(474\) 6391.97 0.619394
\(475\) −3819.91 −0.368989
\(476\) −2679.99 −0.258061
\(477\) 3308.12 0.317544
\(478\) 6408.69 0.613236
\(479\) −271.734 −0.0259203 −0.0129602 0.999916i \(-0.504125\pi\)
−0.0129602 + 0.999916i \(0.504125\pi\)
\(480\) −10913.5 −1.03777
\(481\) −2380.64 −0.225671
\(482\) 4800.66 0.453659
\(483\) 13692.8 1.28994
\(484\) −17711.9 −1.66341
\(485\) −4526.84 −0.423821
\(486\) −1856.20 −0.173248
\(487\) −259.217 −0.0241196 −0.0120598 0.999927i \(-0.503839\pi\)
−0.0120598 + 0.999927i \(0.503839\pi\)
\(488\) −12749.5 −1.18267
\(489\) −12276.2 −1.13528
\(490\) 1583.84 0.146021
\(491\) −16271.5 −1.49557 −0.747784 0.663942i \(-0.768882\pi\)
−0.747784 + 0.663942i \(0.768882\pi\)
\(492\) 1941.30 0.177887
\(493\) 998.098 0.0911807
\(494\) −1030.82 −0.0938841
\(495\) 5806.72 0.527258
\(496\) −9825.24 −0.889448
\(497\) −3517.62 −0.317478
\(498\) 77.8455 0.00700470
\(499\) 8634.67 0.774631 0.387316 0.921947i \(-0.373402\pi\)
0.387316 + 0.921947i \(0.373402\pi\)
\(500\) 4042.02 0.361529
\(501\) 4595.07 0.409766
\(502\) 6166.38 0.548245
\(503\) 10084.7 0.893943 0.446971 0.894548i \(-0.352503\pi\)
0.446971 + 0.894548i \(0.352503\pi\)
\(504\) −1559.42 −0.137821
\(505\) −7595.52 −0.669299
\(506\) 12855.2 1.12941
\(507\) −7846.80 −0.687354
\(508\) 1598.03 0.139569
\(509\) −2063.68 −0.179707 −0.0898536 0.995955i \(-0.528640\pi\)
−0.0898536 + 0.995955i \(0.528640\pi\)
\(510\) −1741.32 −0.151190
\(511\) 8786.68 0.760665
\(512\) −11250.8 −0.971135
\(513\) 6857.54 0.590190
\(514\) 2041.06 0.175150
\(515\) 7487.48 0.640656
\(516\) 0 0
\(517\) −36613.6 −3.11463
\(518\) −1796.15 −0.152352
\(519\) 2069.20 0.175006
\(520\) −4910.64 −0.414127
\(521\) 12140.1 1.02086 0.510430 0.859919i \(-0.329486\pi\)
0.510430 + 0.859919i \(0.329486\pi\)
\(522\) 268.700 0.0225300
\(523\) −5578.83 −0.466434 −0.233217 0.972425i \(-0.574925\pi\)
−0.233217 + 0.972425i \(0.574925\pi\)
\(524\) 2608.35 0.217455
\(525\) 5925.20 0.492566
\(526\) −7114.08 −0.589712
\(527\) −6410.79 −0.529902
\(528\) −10925.0 −0.900474
\(529\) 25923.8 2.13066
\(530\) 7859.71 0.644158
\(531\) −1606.84 −0.131320
\(532\) 4818.69 0.392701
\(533\) 1342.88 0.109130
\(534\) 4106.53 0.332784
\(535\) 18116.7 1.46402
\(536\) −8759.80 −0.705906
\(537\) −9506.12 −0.763909
\(538\) 209.361 0.0167773
\(539\) 6483.69 0.518130
\(540\) 15114.3 1.20448
\(541\) 5469.14 0.434633 0.217317 0.976101i \(-0.430270\pi\)
0.217317 + 0.976101i \(0.430270\pi\)
\(542\) −2133.55 −0.169085
\(543\) 14142.5 1.11770
\(544\) −4181.78 −0.329582
\(545\) −11220.0 −0.881860
\(546\) 1598.94 0.125327
\(547\) −11040.8 −0.863019 −0.431510 0.902108i \(-0.642019\pi\)
−0.431510 + 0.902108i \(0.642019\pi\)
\(548\) −1307.58 −0.101929
\(549\) 5216.35 0.405517
\(550\) 5562.74 0.431266
\(551\) −1794.61 −0.138753
\(552\) 13898.0 1.07163
\(553\) −20669.3 −1.58941
\(554\) 8739.76 0.670247
\(555\) 7230.80 0.553028
\(556\) −13317.2 −1.01578
\(557\) 18238.0 1.38738 0.693688 0.720275i \(-0.255984\pi\)
0.693688 + 0.720275i \(0.255984\pi\)
\(558\) −1725.86 −0.130935
\(559\) 0 0
\(560\) 8629.79 0.651206
\(561\) −7128.38 −0.536471
\(562\) −5708.08 −0.428436
\(563\) 20112.6 1.50559 0.752793 0.658258i \(-0.228706\pi\)
0.752793 + 0.658258i \(0.228706\pi\)
\(564\) −18314.0 −1.36730
\(565\) 934.798 0.0696057
\(566\) 7538.23 0.559815
\(567\) −7954.86 −0.589193
\(568\) −3570.33 −0.263746
\(569\) 11537.1 0.850019 0.425010 0.905189i \(-0.360271\pi\)
0.425010 + 0.905189i \(0.360271\pi\)
\(570\) 3130.94 0.230072
\(571\) 2585.01 0.189456 0.0947281 0.995503i \(-0.469802\pi\)
0.0947281 + 0.995503i \(0.469802\pi\)
\(572\) −9300.69 −0.679862
\(573\) 13917.2 1.01466
\(574\) 1013.18 0.0736745
\(575\) 16482.8 1.19544
\(576\) 855.199 0.0618634
\(577\) 9197.18 0.663576 0.331788 0.943354i \(-0.392348\pi\)
0.331788 + 0.943354i \(0.392348\pi\)
\(578\) 4513.04 0.324771
\(579\) −9948.25 −0.714050
\(580\) −3955.40 −0.283171
\(581\) −251.723 −0.0179746
\(582\) 1496.06 0.106553
\(583\) 32175.0 2.28568
\(584\) 8918.36 0.631925
\(585\) 2009.16 0.141997
\(586\) −4719.49 −0.332697
\(587\) −11366.3 −0.799210 −0.399605 0.916687i \(-0.630853\pi\)
−0.399605 + 0.916687i \(0.630853\pi\)
\(588\) 3243.11 0.227455
\(589\) 11526.8 0.806372
\(590\) −3817.68 −0.266392
\(591\) 1372.39 0.0955208
\(592\) 4246.35 0.294804
\(593\) −8866.85 −0.614027 −0.307013 0.951705i \(-0.599330\pi\)
−0.307013 + 0.951705i \(0.599330\pi\)
\(594\) −9986.27 −0.689801
\(595\) 5630.78 0.387966
\(596\) 21692.5 1.49087
\(597\) −4552.90 −0.312124
\(598\) 4447.95 0.304164
\(599\) −15941.8 −1.08742 −0.543709 0.839274i \(-0.682980\pi\)
−0.543709 + 0.839274i \(0.682980\pi\)
\(600\) 6014.00 0.409201
\(601\) 590.214 0.0400588 0.0200294 0.999799i \(-0.493624\pi\)
0.0200294 + 0.999799i \(0.493624\pi\)
\(602\) 0 0
\(603\) 3584.01 0.242044
\(604\) 14367.9 0.967915
\(605\) 37213.7 2.50074
\(606\) 2510.22 0.168268
\(607\) −24494.4 −1.63788 −0.818942 0.573876i \(-0.805439\pi\)
−0.818942 + 0.573876i \(0.805439\pi\)
\(608\) 7518.97 0.501537
\(609\) 2783.68 0.185223
\(610\) 12393.5 0.822618
\(611\) −12668.5 −0.838810
\(612\) 1112.93 0.0735091
\(613\) 18011.0 1.18672 0.593358 0.804939i \(-0.297802\pi\)
0.593358 + 0.804939i \(0.297802\pi\)
\(614\) 4190.52 0.275433
\(615\) −4078.76 −0.267434
\(616\) −15167.0 −0.992038
\(617\) −3697.23 −0.241240 −0.120620 0.992699i \(-0.538488\pi\)
−0.120620 + 0.992699i \(0.538488\pi\)
\(618\) −2474.51 −0.161067
\(619\) −776.148 −0.0503974 −0.0251987 0.999682i \(-0.508022\pi\)
−0.0251987 + 0.999682i \(0.508022\pi\)
\(620\) 25405.6 1.64567
\(621\) −29590.1 −1.91209
\(622\) 5287.93 0.340879
\(623\) −13279.0 −0.853951
\(624\) −3780.11 −0.242509
\(625\) −19049.3 −1.21915
\(626\) 491.493 0.0313802
\(627\) 12817.0 0.816368
\(628\) −24191.4 −1.53717
\(629\) 2770.67 0.175634
\(630\) 1515.87 0.0958632
\(631\) 2827.64 0.178394 0.0891968 0.996014i \(-0.471570\pi\)
0.0891968 + 0.996014i \(0.471570\pi\)
\(632\) −20979.0 −1.32041
\(633\) −22175.9 −1.39244
\(634\) 1359.91 0.0851875
\(635\) −3357.54 −0.209826
\(636\) 16093.8 1.00340
\(637\) 2243.39 0.139539
\(638\) 2613.40 0.162171
\(639\) 1460.78 0.0904343
\(640\) 21278.8 1.31424
\(641\) −12066.0 −0.743491 −0.371746 0.928335i \(-0.621241\pi\)
−0.371746 + 0.928335i \(0.621241\pi\)
\(642\) −5987.32 −0.368070
\(643\) −6253.55 −0.383540 −0.191770 0.981440i \(-0.561423\pi\)
−0.191770 + 0.981440i \(0.561423\pi\)
\(644\) −20792.5 −1.27227
\(645\) 0 0
\(646\) 1199.70 0.0730675
\(647\) 8121.27 0.493478 0.246739 0.969082i \(-0.420641\pi\)
0.246739 + 0.969082i \(0.420641\pi\)
\(648\) −8074.07 −0.489474
\(649\) −15628.3 −0.945245
\(650\) 1924.74 0.116145
\(651\) −17879.6 −1.07643
\(652\) 18641.4 1.11972
\(653\) 10001.3 0.599356 0.299678 0.954040i \(-0.403121\pi\)
0.299678 + 0.954040i \(0.403121\pi\)
\(654\) 3708.07 0.221708
\(655\) −5480.27 −0.326919
\(656\) −2395.29 −0.142561
\(657\) −3648.89 −0.216677
\(658\) −9558.17 −0.566286
\(659\) 8841.69 0.522646 0.261323 0.965251i \(-0.415841\pi\)
0.261323 + 0.965251i \(0.415841\pi\)
\(660\) 28249.3 1.66607
\(661\) 1408.36 0.0828728 0.0414364 0.999141i \(-0.486807\pi\)
0.0414364 + 0.999141i \(0.486807\pi\)
\(662\) 6415.72 0.376668
\(663\) −2466.46 −0.144478
\(664\) −255.496 −0.0149325
\(665\) −10124.3 −0.590382
\(666\) 745.896 0.0433977
\(667\) 7743.69 0.449530
\(668\) −6977.62 −0.404150
\(669\) −15065.9 −0.870672
\(670\) 8515.21 0.491002
\(671\) 50734.7 2.91891
\(672\) −11662.9 −0.669505
\(673\) −20128.9 −1.15291 −0.576457 0.817127i \(-0.695565\pi\)
−0.576457 + 0.817127i \(0.695565\pi\)
\(674\) 2856.91 0.163270
\(675\) −12804.3 −0.730132
\(676\) 11915.4 0.677935
\(677\) −2419.42 −0.137350 −0.0686750 0.997639i \(-0.521877\pi\)
−0.0686750 + 0.997639i \(0.521877\pi\)
\(678\) −308.938 −0.0174996
\(679\) −4837.70 −0.273422
\(680\) 5715.17 0.322304
\(681\) 16477.2 0.927175
\(682\) −16785.9 −0.942469
\(683\) 12041.6 0.674611 0.337306 0.941395i \(-0.390484\pi\)
0.337306 + 0.941395i \(0.390484\pi\)
\(684\) −2001.08 −0.111861
\(685\) 2747.28 0.153238
\(686\) 7286.16 0.405520
\(687\) 17957.8 0.997283
\(688\) 0 0
\(689\) 11132.7 0.615563
\(690\) −13509.9 −0.745383
\(691\) −1693.27 −0.0932198 −0.0466099 0.998913i \(-0.514842\pi\)
−0.0466099 + 0.998913i \(0.514842\pi\)
\(692\) −3142.09 −0.172607
\(693\) 6205.47 0.340154
\(694\) 862.411 0.0471710
\(695\) 27980.1 1.52712
\(696\) 2825.40 0.153874
\(697\) −1562.88 −0.0849331
\(698\) 2461.85 0.133499
\(699\) 20589.4 1.11411
\(700\) −8997.43 −0.485815
\(701\) −26393.7 −1.42208 −0.711039 0.703152i \(-0.751775\pi\)
−0.711039 + 0.703152i \(0.751775\pi\)
\(702\) −3455.30 −0.185772
\(703\) −4981.74 −0.267269
\(704\) 8317.73 0.445293
\(705\) 38478.5 2.05558
\(706\) 6504.82 0.346760
\(707\) −8117.11 −0.431790
\(708\) −7817.19 −0.414955
\(709\) 21874.3 1.15868 0.579341 0.815085i \(-0.303310\pi\)
0.579341 + 0.815085i \(0.303310\pi\)
\(710\) 3470.64 0.183452
\(711\) 8583.42 0.452747
\(712\) −13478.0 −0.709423
\(713\) −49737.8 −2.61247
\(714\) −1860.90 −0.0975383
\(715\) 19541.2 1.02210
\(716\) 14435.1 0.753440
\(717\) −27571.3 −1.43608
\(718\) −193.018 −0.0100326
\(719\) 17951.1 0.931103 0.465552 0.885021i \(-0.345856\pi\)
0.465552 + 0.885021i \(0.345856\pi\)
\(720\) −3583.73 −0.185497
\(721\) 8001.65 0.413311
\(722\) 5075.03 0.261597
\(723\) −20653.2 −1.06238
\(724\) −21475.4 −1.10239
\(725\) 3350.88 0.171653
\(726\) −12298.6 −0.628712
\(727\) −21470.8 −1.09534 −0.547668 0.836696i \(-0.684484\pi\)
−0.547668 + 0.836696i \(0.684484\pi\)
\(728\) −5247.86 −0.267168
\(729\) 21872.6 1.11125
\(730\) −8669.34 −0.439543
\(731\) 0 0
\(732\) 25377.2 1.28138
\(733\) 32219.7 1.62355 0.811775 0.583970i \(-0.198501\pi\)
0.811775 + 0.583970i \(0.198501\pi\)
\(734\) −5144.15 −0.258684
\(735\) −6813.93 −0.341953
\(736\) −32444.1 −1.62487
\(737\) 34858.4 1.74223
\(738\) −420.747 −0.0209863
\(739\) 14452.3 0.719402 0.359701 0.933068i \(-0.382879\pi\)
0.359701 + 0.933068i \(0.382879\pi\)
\(740\) −10980.0 −0.545449
\(741\) 4434.76 0.219858
\(742\) 8399.45 0.415571
\(743\) −24465.7 −1.20802 −0.604011 0.796976i \(-0.706432\pi\)
−0.604011 + 0.796976i \(0.706432\pi\)
\(744\) −18147.6 −0.894250
\(745\) −45577.0 −2.24136
\(746\) −3161.75 −0.155174
\(747\) 104.534 0.00512010
\(748\) 10824.4 0.529119
\(749\) 19360.8 0.944496
\(750\) 2806.65 0.136646
\(751\) 38635.2 1.87725 0.938626 0.344935i \(-0.112099\pi\)
0.938626 + 0.344935i \(0.112099\pi\)
\(752\) 22596.8 1.09577
\(753\) −26528.8 −1.28388
\(754\) 904.249 0.0436748
\(755\) −30187.6 −1.45515
\(756\) 16152.2 0.777052
\(757\) −6833.97 −0.328118 −0.164059 0.986451i \(-0.552459\pi\)
−0.164059 + 0.986451i \(0.552459\pi\)
\(758\) −11383.1 −0.545450
\(759\) −55305.1 −2.64486
\(760\) −10276.0 −0.490462
\(761\) 30885.3 1.47121 0.735606 0.677410i \(-0.236897\pi\)
0.735606 + 0.677410i \(0.236897\pi\)
\(762\) 1109.62 0.0527524
\(763\) −11990.5 −0.568921
\(764\) −21133.3 −1.00075
\(765\) −2338.32 −0.110513
\(766\) −10386.2 −0.489909
\(767\) −5407.47 −0.254566
\(768\) −2200.35 −0.103383
\(769\) −31913.9 −1.49654 −0.748272 0.663392i \(-0.769116\pi\)
−0.748272 + 0.663392i \(0.769116\pi\)
\(770\) 14743.5 0.690025
\(771\) −8780.97 −0.410167
\(772\) 15106.4 0.704265
\(773\) 11641.0 0.541651 0.270825 0.962629i \(-0.412703\pi\)
0.270825 + 0.962629i \(0.412703\pi\)
\(774\) 0 0
\(775\) −21522.7 −0.997574
\(776\) −4910.20 −0.227147
\(777\) 7727.35 0.356779
\(778\) 577.575 0.0266157
\(779\) 2810.11 0.129246
\(780\) 9774.41 0.448692
\(781\) 14207.6 0.650947
\(782\) −5176.67 −0.236723
\(783\) −6015.53 −0.274556
\(784\) −4001.54 −0.182286
\(785\) 50827.4 2.31096
\(786\) 1811.16 0.0821907
\(787\) 29770.6 1.34842 0.674211 0.738539i \(-0.264484\pi\)
0.674211 + 0.738539i \(0.264484\pi\)
\(788\) −2083.98 −0.0942117
\(789\) 30606.0 1.38099
\(790\) 20393.2 0.918429
\(791\) 998.991 0.0449052
\(792\) 6298.47 0.282584
\(793\) 17554.5 0.786100
\(794\) −3620.35 −0.161815
\(795\) −33813.8 −1.50849
\(796\) 6913.59 0.307846
\(797\) 12956.7 0.575845 0.287923 0.957654i \(-0.407035\pi\)
0.287923 + 0.957654i \(0.407035\pi\)
\(798\) 3345.95 0.148428
\(799\) 14744.0 0.652824
\(800\) −14039.4 −0.620458
\(801\) 5514.43 0.243249
\(802\) −2680.09 −0.118001
\(803\) −35489.4 −1.55964
\(804\) 17436.0 0.764826
\(805\) 43686.1 1.91271
\(806\) −5808.00 −0.253819
\(807\) −900.706 −0.0392892
\(808\) −8238.75 −0.358711
\(809\) −20078.1 −0.872571 −0.436285 0.899808i \(-0.643706\pi\)
−0.436285 + 0.899808i \(0.643706\pi\)
\(810\) 7848.63 0.340460
\(811\) 11737.7 0.508221 0.254111 0.967175i \(-0.418217\pi\)
0.254111 + 0.967175i \(0.418217\pi\)
\(812\) −4227.03 −0.182684
\(813\) 9178.91 0.395963
\(814\) 7254.64 0.312377
\(815\) −39166.6 −1.68337
\(816\) 4399.42 0.188738
\(817\) 0 0
\(818\) 2920.36 0.124826
\(819\) 2147.13 0.0916077
\(820\) 6193.61 0.263769
\(821\) 480.443 0.0204234 0.0102117 0.999948i \(-0.496749\pi\)
0.0102117 + 0.999948i \(0.496749\pi\)
\(822\) −907.940 −0.0385256
\(823\) 6403.13 0.271202 0.135601 0.990764i \(-0.456704\pi\)
0.135601 + 0.990764i \(0.456704\pi\)
\(824\) 8121.56 0.343359
\(825\) −23931.9 −1.00994
\(826\) −4079.84 −0.171859
\(827\) −13707.5 −0.576370 −0.288185 0.957575i \(-0.593052\pi\)
−0.288185 + 0.957575i \(0.593052\pi\)
\(828\) 8634.61 0.362407
\(829\) 32923.0 1.37933 0.689665 0.724128i \(-0.257758\pi\)
0.689665 + 0.724128i \(0.257758\pi\)
\(830\) 248.362 0.0103865
\(831\) −37599.9 −1.56959
\(832\) 2877.98 0.119923
\(833\) −2610.93 −0.108600
\(834\) −9247.05 −0.383932
\(835\) 14660.3 0.607594
\(836\) −19462.7 −0.805180
\(837\) 38637.8 1.59560
\(838\) 3260.09 0.134389
\(839\) 4172.23 0.171682 0.0858410 0.996309i \(-0.472642\pi\)
0.0858410 + 0.996309i \(0.472642\pi\)
\(840\) 15939.5 0.654721
\(841\) −22814.7 −0.935452
\(842\) −8312.55 −0.340225
\(843\) 24557.1 1.00331
\(844\) 33674.1 1.37335
\(845\) −25034.8 −1.01920
\(846\) 3969.27 0.161308
\(847\) 39769.2 1.61332
\(848\) −19857.5 −0.804136
\(849\) −32430.7 −1.31098
\(850\) −2240.07 −0.0903928
\(851\) 21496.0 0.865893
\(852\) 7106.60 0.285761
\(853\) 39769.7 1.59635 0.798177 0.602424i \(-0.205798\pi\)
0.798177 + 0.602424i \(0.205798\pi\)
\(854\) 13244.5 0.530701
\(855\) 4204.37 0.168171
\(856\) 19650.9 0.784643
\(857\) 20869.8 0.831855 0.415927 0.909398i \(-0.363457\pi\)
0.415927 + 0.909398i \(0.363457\pi\)
\(858\) −6458.11 −0.256965
\(859\) 10297.0 0.408999 0.204499 0.978867i \(-0.434443\pi\)
0.204499 + 0.978867i \(0.434443\pi\)
\(860\) 0 0
\(861\) −4358.86 −0.172531
\(862\) 13031.3 0.514907
\(863\) −24683.3 −0.973615 −0.486808 0.873509i \(-0.661839\pi\)
−0.486808 + 0.873509i \(0.661839\pi\)
\(864\) 25203.6 0.992411
\(865\) 6601.68 0.259496
\(866\) 14841.3 0.582363
\(867\) −19415.9 −0.760550
\(868\) 27150.2 1.06168
\(869\) 83483.0 3.25888
\(870\) −2746.51 −0.107029
\(871\) 12061.2 0.469205
\(872\) −12170.2 −0.472633
\(873\) 2008.97 0.0778849
\(874\) 9307.81 0.360230
\(875\) −9075.68 −0.350644
\(876\) −17751.6 −0.684670
\(877\) −12580.6 −0.484398 −0.242199 0.970227i \(-0.577869\pi\)
−0.242199 + 0.970227i \(0.577869\pi\)
\(878\) −3494.50 −0.134321
\(879\) 20304.0 0.779111
\(880\) −34855.7 −1.33521
\(881\) −8858.61 −0.338768 −0.169384 0.985550i \(-0.554178\pi\)
−0.169384 + 0.985550i \(0.554178\pi\)
\(882\) −702.894 −0.0268341
\(883\) 6626.72 0.252556 0.126278 0.991995i \(-0.459697\pi\)
0.126278 + 0.991995i \(0.459697\pi\)
\(884\) 3745.32 0.142498
\(885\) 16424.3 0.623838
\(886\) 6475.32 0.245534
\(887\) 11414.3 0.432079 0.216039 0.976385i \(-0.430686\pi\)
0.216039 + 0.976385i \(0.430686\pi\)
\(888\) 7843.15 0.296395
\(889\) −3588.10 −0.135367
\(890\) 13101.7 0.493448
\(891\) 32129.6 1.20806
\(892\) 22877.5 0.858740
\(893\) −26510.2 −0.993426
\(894\) 15062.6 0.563500
\(895\) −30328.8 −1.13271
\(896\) 22740.0 0.847868
\(897\) −19135.9 −0.712294
\(898\) 4975.68 0.184900
\(899\) −10111.5 −0.375124
\(900\) 3736.40 0.138385
\(901\) −12956.6 −0.479076
\(902\) −4092.21 −0.151060
\(903\) 0 0
\(904\) 1013.96 0.0373052
\(905\) 45120.9 1.65731
\(906\) 9976.61 0.365840
\(907\) 11407.7 0.417626 0.208813 0.977956i \(-0.433040\pi\)
0.208813 + 0.977956i \(0.433040\pi\)
\(908\) −25020.6 −0.914469
\(909\) 3370.83 0.122996
\(910\) 5101.33 0.185832
\(911\) 49077.0 1.78484 0.892422 0.451201i \(-0.149004\pi\)
0.892422 + 0.451201i \(0.149004\pi\)
\(912\) −7910.29 −0.287210
\(913\) 1016.71 0.0368545
\(914\) 4262.71 0.154265
\(915\) −53318.8 −1.92641
\(916\) −27269.0 −0.983615
\(917\) −5856.61 −0.210908
\(918\) 4021.40 0.144582
\(919\) −2643.59 −0.0948901 −0.0474451 0.998874i \(-0.515108\pi\)
−0.0474451 + 0.998874i \(0.515108\pi\)
\(920\) 44340.8 1.58899
\(921\) −18028.3 −0.645010
\(922\) −18965.2 −0.677424
\(923\) 4915.92 0.175308
\(924\) 30189.2 1.07484
\(925\) 9301.86 0.330642
\(926\) −6302.23 −0.223655
\(927\) −3322.88 −0.117732
\(928\) −6595.75 −0.233315
\(929\) −5048.62 −0.178299 −0.0891495 0.996018i \(-0.528415\pi\)
−0.0891495 + 0.996018i \(0.528415\pi\)
\(930\) 17640.8 0.622007
\(931\) 4694.53 0.165260
\(932\) −31265.1 −1.09884
\(933\) −22749.6 −0.798272
\(934\) −8229.03 −0.288289
\(935\) −22742.7 −0.795471
\(936\) 2179.30 0.0761034
\(937\) 25866.9 0.901850 0.450925 0.892562i \(-0.351094\pi\)
0.450925 + 0.892562i \(0.351094\pi\)
\(938\) 9099.96 0.316763
\(939\) −2114.49 −0.0734864
\(940\) −58429.7 −2.02741
\(941\) 19865.7 0.688207 0.344103 0.938932i \(-0.388183\pi\)
0.344103 + 0.938932i \(0.388183\pi\)
\(942\) −16797.8 −0.580999
\(943\) −12125.5 −0.418729
\(944\) 9645.31 0.332551
\(945\) −33936.7 −1.16821
\(946\) 0 0
\(947\) −15772.5 −0.541223 −0.270611 0.962689i \(-0.587226\pi\)
−0.270611 + 0.962689i \(0.587226\pi\)
\(948\) 41757.8 1.43062
\(949\) −12279.5 −0.420031
\(950\) 4027.72 0.137554
\(951\) −5850.56 −0.199492
\(952\) 6107.63 0.207930
\(953\) 5260.78 0.178818 0.0894089 0.995995i \(-0.471502\pi\)
0.0894089 + 0.995995i \(0.471502\pi\)
\(954\) −3488.08 −0.118376
\(955\) 44402.1 1.50452
\(956\) 41867.0 1.41640
\(957\) −11243.3 −0.379774
\(958\) 286.516 0.00966276
\(959\) 2935.94 0.0988597
\(960\) −8741.39 −0.293883
\(961\) 35155.0 1.18006
\(962\) 2510.14 0.0841271
\(963\) −8040.04 −0.269041
\(964\) 31362.0 1.04782
\(965\) −31739.3 −1.05878
\(966\) −14437.7 −0.480874
\(967\) −22150.5 −0.736619 −0.368310 0.929703i \(-0.620063\pi\)
−0.368310 + 0.929703i \(0.620063\pi\)
\(968\) 40365.2 1.34027
\(969\) −5161.32 −0.171110
\(970\) 4773.10 0.157995
\(971\) −32180.2 −1.06355 −0.531777 0.846884i \(-0.678476\pi\)
−0.531777 + 0.846884i \(0.678476\pi\)
\(972\) −12126.3 −0.400154
\(973\) 29901.5 0.985199
\(974\) 273.318 0.00899147
\(975\) −8280.55 −0.271990
\(976\) −31311.9 −1.02692
\(977\) −21321.9 −0.698207 −0.349103 0.937084i \(-0.613514\pi\)
−0.349103 + 0.937084i \(0.613514\pi\)
\(978\) 12944.0 0.423216
\(979\) 53633.8 1.75091
\(980\) 10347.0 0.337267
\(981\) 4979.37 0.162058
\(982\) 17156.7 0.557528
\(983\) −14512.5 −0.470881 −0.235440 0.971889i \(-0.575653\pi\)
−0.235440 + 0.971889i \(0.575653\pi\)
\(984\) −4424.18 −0.143331
\(985\) 4378.55 0.141637
\(986\) −1052.39 −0.0339910
\(987\) 41120.9 1.32613
\(988\) −6734.19 −0.216845
\(989\) 0 0
\(990\) −6122.61 −0.196555
\(991\) −30878.2 −0.989785 −0.494893 0.868954i \(-0.664793\pi\)
−0.494893 + 0.868954i \(0.664793\pi\)
\(992\) 42364.5 1.35592
\(993\) −27601.5 −0.882082
\(994\) 3708.98 0.118352
\(995\) −14525.8 −0.462813
\(996\) 508.553 0.0161788
\(997\) −25585.2 −0.812731 −0.406366 0.913711i \(-0.633204\pi\)
−0.406366 + 0.913711i \(0.633204\pi\)
\(998\) −9104.40 −0.288772
\(999\) −16698.8 −0.528855
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.1.22 yes 50
43.42 odd 2 1849.4.a.i.1.29 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.29 50 43.42 odd 2
1849.4.a.j.1.22 yes 50 1.1 even 1 trivial