Properties

Label 1849.4.a.j.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

\(f(q)\) \(=\) \(q+4.52050 q^{2} +1.75128 q^{3} +12.4349 q^{4} +13.0472 q^{5} +7.91667 q^{6} -4.32755 q^{7} +20.0482 q^{8} -23.9330 q^{9} +O(q^{10})\) \(q+4.52050 q^{2} +1.75128 q^{3} +12.4349 q^{4} +13.0472 q^{5} +7.91667 q^{6} -4.32755 q^{7} +20.0482 q^{8} -23.9330 q^{9} +58.9799 q^{10} -55.2516 q^{11} +21.7771 q^{12} -37.7658 q^{13} -19.5627 q^{14} +22.8493 q^{15} -8.85170 q^{16} -24.9093 q^{17} -108.189 q^{18} -39.7201 q^{19} +162.241 q^{20} -7.57876 q^{21} -249.765 q^{22} -2.55110 q^{23} +35.1100 q^{24} +45.2293 q^{25} -170.720 q^{26} -89.1980 q^{27} -53.8129 q^{28} +238.086 q^{29} +103.290 q^{30} +263.063 q^{31} -200.400 q^{32} -96.7611 q^{33} -112.603 q^{34} -56.4624 q^{35} -297.606 q^{36} -353.428 q^{37} -179.555 q^{38} -66.1386 q^{39} +261.573 q^{40} +254.188 q^{41} -34.2598 q^{42} -687.051 q^{44} -312.259 q^{45} -11.5323 q^{46} -386.866 q^{47} -15.5018 q^{48} -324.272 q^{49} +204.459 q^{50} -43.6232 q^{51} -469.616 q^{52} -35.7551 q^{53} -403.220 q^{54} -720.878 q^{55} -86.7595 q^{56} -69.5611 q^{57} +1076.27 q^{58} -87.4154 q^{59} +284.130 q^{60} -890.643 q^{61} +1189.18 q^{62} +103.571 q^{63} -835.093 q^{64} -492.738 q^{65} -437.409 q^{66} -998.120 q^{67} -309.746 q^{68} -4.46769 q^{69} -255.238 q^{70} -443.994 q^{71} -479.813 q^{72} +70.5226 q^{73} -1597.67 q^{74} +79.2092 q^{75} -493.917 q^{76} +239.104 q^{77} -298.980 q^{78} +975.256 q^{79} -115.490 q^{80} +489.980 q^{81} +1149.06 q^{82} +836.712 q^{83} -94.2415 q^{84} -324.997 q^{85} +416.956 q^{87} -1107.69 q^{88} +1086.38 q^{89} -1411.57 q^{90} +163.434 q^{91} -31.7228 q^{92} +460.697 q^{93} -1748.83 q^{94} -518.236 q^{95} -350.956 q^{96} -1132.50 q^{97} -1465.87 q^{98} +1322.34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.52050 1.59824 0.799120 0.601172i \(-0.205299\pi\)
0.799120 + 0.601172i \(0.205299\pi\)
\(3\) 1.75128 0.337034 0.168517 0.985699i \(-0.446102\pi\)
0.168517 + 0.985699i \(0.446102\pi\)
\(4\) 12.4349 1.55437
\(5\) 13.0472 1.16698 0.583488 0.812122i \(-0.301687\pi\)
0.583488 + 0.812122i \(0.301687\pi\)
\(6\) 7.91667 0.538661
\(7\) −4.32755 −0.233666 −0.116833 0.993152i \(-0.537274\pi\)
−0.116833 + 0.993152i \(0.537274\pi\)
\(8\) 20.0482 0.886013
\(9\) −23.9330 −0.886408
\(10\) 58.9799 1.86511
\(11\) −55.2516 −1.51445 −0.757227 0.653152i \(-0.773446\pi\)
−0.757227 + 0.653152i \(0.773446\pi\)
\(12\) 21.7771 0.523875
\(13\) −37.7658 −0.805719 −0.402860 0.915262i \(-0.631984\pi\)
−0.402860 + 0.915262i \(0.631984\pi\)
\(14\) −19.5627 −0.373454
\(15\) 22.8493 0.393311
\(16\) −8.85170 −0.138308
\(17\) −24.9093 −0.355376 −0.177688 0.984087i \(-0.556862\pi\)
−0.177688 + 0.984087i \(0.556862\pi\)
\(18\) −108.189 −1.41669
\(19\) −39.7201 −0.479601 −0.239800 0.970822i \(-0.577082\pi\)
−0.239800 + 0.970822i \(0.577082\pi\)
\(20\) 162.241 1.81391
\(21\) −7.57876 −0.0787534
\(22\) −249.765 −2.42046
\(23\) −2.55110 −0.0231279 −0.0115639 0.999933i \(-0.503681\pi\)
−0.0115639 + 0.999933i \(0.503681\pi\)
\(24\) 35.1100 0.298617
\(25\) 45.2293 0.361834
\(26\) −170.720 −1.28773
\(27\) −89.1980 −0.635784
\(28\) −53.8129 −0.363203
\(29\) 238.086 1.52454 0.762268 0.647262i \(-0.224086\pi\)
0.762268 + 0.647262i \(0.224086\pi\)
\(30\) 103.290 0.628605
\(31\) 263.063 1.52411 0.762056 0.647512i \(-0.224190\pi\)
0.762056 + 0.647512i \(0.224190\pi\)
\(32\) −200.400 −1.10706
\(33\) −96.7611 −0.510423
\(34\) −112.603 −0.567977
\(35\) −56.4624 −0.272682
\(36\) −297.606 −1.37780
\(37\) −353.428 −1.57036 −0.785178 0.619270i \(-0.787429\pi\)
−0.785178 + 0.619270i \(0.787429\pi\)
\(38\) −179.555 −0.766517
\(39\) −66.1386 −0.271555
\(40\) 261.573 1.03396
\(41\) 254.188 0.968233 0.484117 0.875004i \(-0.339141\pi\)
0.484117 + 0.875004i \(0.339141\pi\)
\(42\) −34.2598 −0.125867
\(43\) 0 0
\(44\) −687.051 −2.35402
\(45\) −312.259 −1.03442
\(46\) −11.5323 −0.0369639
\(47\) −386.866 −1.20064 −0.600321 0.799759i \(-0.704961\pi\)
−0.600321 + 0.799759i \(0.704961\pi\)
\(48\) −15.5018 −0.0466145
\(49\) −324.272 −0.945400
\(50\) 204.459 0.578297
\(51\) −43.6232 −0.119774
\(52\) −469.616 −1.25238
\(53\) −35.7551 −0.0926668 −0.0463334 0.998926i \(-0.514754\pi\)
−0.0463334 + 0.998926i \(0.514754\pi\)
\(54\) −403.220 −1.01614
\(55\) −720.878 −1.76733
\(56\) −86.7595 −0.207031
\(57\) −69.5611 −0.161642
\(58\) 1076.27 2.43657
\(59\) −87.4154 −0.192890 −0.0964451 0.995338i \(-0.530747\pi\)
−0.0964451 + 0.995338i \(0.530747\pi\)
\(60\) 284.130 0.611350
\(61\) −890.643 −1.86943 −0.934714 0.355400i \(-0.884345\pi\)
−0.934714 + 0.355400i \(0.884345\pi\)
\(62\) 1189.18 2.43589
\(63\) 103.571 0.207123
\(64\) −835.093 −1.63104
\(65\) −492.738 −0.940256
\(66\) −437.409 −0.815777
\(67\) −998.120 −1.82000 −0.909998 0.414612i \(-0.863917\pi\)
−0.909998 + 0.414612i \(0.863917\pi\)
\(68\) −309.746 −0.552386
\(69\) −4.46769 −0.00779489
\(70\) −255.238 −0.435812
\(71\) −443.994 −0.742146 −0.371073 0.928604i \(-0.621010\pi\)
−0.371073 + 0.928604i \(0.621010\pi\)
\(72\) −479.813 −0.785369
\(73\) 70.5226 0.113069 0.0565346 0.998401i \(-0.481995\pi\)
0.0565346 + 0.998401i \(0.481995\pi\)
\(74\) −1597.67 −2.50980
\(75\) 79.2092 0.121951
\(76\) −493.917 −0.745476
\(77\) 239.104 0.353876
\(78\) −298.980 −0.434010
\(79\) 975.256 1.38892 0.694461 0.719530i \(-0.255643\pi\)
0.694461 + 0.719530i \(0.255643\pi\)
\(80\) −115.490 −0.161402
\(81\) 489.980 0.672127
\(82\) 1149.06 1.54747
\(83\) 836.712 1.10652 0.553260 0.833009i \(-0.313384\pi\)
0.553260 + 0.833009i \(0.313384\pi\)
\(84\) −94.2415 −0.122412
\(85\) −324.997 −0.414716
\(86\) 0 0
\(87\) 416.956 0.513821
\(88\) −1107.69 −1.34182
\(89\) 1086.38 1.29389 0.646944 0.762537i \(-0.276047\pi\)
0.646944 + 0.762537i \(0.276047\pi\)
\(90\) −1411.57 −1.65325
\(91\) 163.434 0.188269
\(92\) −31.7228 −0.0359492
\(93\) 460.697 0.513678
\(94\) −1748.83 −1.91891
\(95\) −518.236 −0.559683
\(96\) −350.956 −0.373118
\(97\) −1132.50 −1.18544 −0.592720 0.805408i \(-0.701946\pi\)
−0.592720 + 0.805408i \(0.701946\pi\)
\(98\) −1465.87 −1.51098
\(99\) 1322.34 1.34242
\(100\) 562.423 0.562423
\(101\) 1437.96 1.41665 0.708326 0.705885i \(-0.249450\pi\)
0.708326 + 0.705885i \(0.249450\pi\)
\(102\) −197.199 −0.191428
\(103\) 1763.53 1.68704 0.843521 0.537096i \(-0.180479\pi\)
0.843521 + 0.537096i \(0.180479\pi\)
\(104\) −757.136 −0.713878
\(105\) −98.8816 −0.0919033
\(106\) −161.631 −0.148104
\(107\) 166.103 0.150073 0.0750365 0.997181i \(-0.476093\pi\)
0.0750365 + 0.997181i \(0.476093\pi\)
\(108\) −1109.17 −0.988243
\(109\) −441.796 −0.388224 −0.194112 0.980979i \(-0.562182\pi\)
−0.194112 + 0.980979i \(0.562182\pi\)
\(110\) −3258.73 −2.82462
\(111\) −618.951 −0.529264
\(112\) 38.3062 0.0323178
\(113\) 971.442 0.808722 0.404361 0.914599i \(-0.367494\pi\)
0.404361 + 0.914599i \(0.367494\pi\)
\(114\) −314.451 −0.258342
\(115\) −33.2847 −0.0269897
\(116\) 2960.59 2.36969
\(117\) 903.850 0.714196
\(118\) −395.162 −0.308285
\(119\) 107.796 0.0830393
\(120\) 458.087 0.348479
\(121\) 1721.74 1.29357
\(122\) −4026.15 −2.98779
\(123\) 445.155 0.326328
\(124\) 3271.17 2.36903
\(125\) −1040.78 −0.744725
\(126\) 468.195 0.331032
\(127\) −1595.77 −1.11498 −0.557489 0.830185i \(-0.688235\pi\)
−0.557489 + 0.830185i \(0.688235\pi\)
\(128\) −2171.84 −1.49973
\(129\) 0 0
\(130\) −2227.42 −1.50275
\(131\) −55.8044 −0.0372187 −0.0186094 0.999827i \(-0.505924\pi\)
−0.0186094 + 0.999827i \(0.505924\pi\)
\(132\) −1203.22 −0.793385
\(133\) 171.891 0.112066
\(134\) −4512.00 −2.90879
\(135\) −1163.78 −0.741945
\(136\) −499.387 −0.314868
\(137\) −137.795 −0.0859316 −0.0429658 0.999077i \(-0.513681\pi\)
−0.0429658 + 0.999077i \(0.513681\pi\)
\(138\) −20.1962 −0.0124581
\(139\) 1016.38 0.620206 0.310103 0.950703i \(-0.399637\pi\)
0.310103 + 0.950703i \(0.399637\pi\)
\(140\) −702.107 −0.423849
\(141\) −677.511 −0.404658
\(142\) −2007.08 −1.18613
\(143\) 2086.62 1.22022
\(144\) 211.848 0.122597
\(145\) 3106.36 1.77910
\(146\) 318.798 0.180712
\(147\) −567.892 −0.318632
\(148\) −4394.85 −2.44091
\(149\) 2725.90 1.49876 0.749378 0.662142i \(-0.230352\pi\)
0.749378 + 0.662142i \(0.230352\pi\)
\(150\) 358.065 0.194906
\(151\) −1350.49 −0.727822 −0.363911 0.931434i \(-0.618559\pi\)
−0.363911 + 0.931434i \(0.618559\pi\)
\(152\) −796.315 −0.424932
\(153\) 596.155 0.315008
\(154\) 1080.87 0.565578
\(155\) 3432.23 1.77860
\(156\) −822.430 −0.422097
\(157\) −2623.54 −1.33364 −0.666818 0.745220i \(-0.732344\pi\)
−0.666818 + 0.745220i \(0.732344\pi\)
\(158\) 4408.65 2.21983
\(159\) −62.6172 −0.0312319
\(160\) −2614.65 −1.29192
\(161\) 11.0400 0.00540419
\(162\) 2214.96 1.07422
\(163\) −407.303 −0.195721 −0.0978603 0.995200i \(-0.531200\pi\)
−0.0978603 + 0.995200i \(0.531200\pi\)
\(164\) 3160.82 1.50499
\(165\) −1262.46 −0.595651
\(166\) 3782.36 1.76848
\(167\) 1743.79 0.808015 0.404008 0.914756i \(-0.367617\pi\)
0.404008 + 0.914756i \(0.367617\pi\)
\(168\) −151.940 −0.0697765
\(169\) −770.743 −0.350816
\(170\) −1469.15 −0.662815
\(171\) 950.621 0.425122
\(172\) 0 0
\(173\) 1182.29 0.519584 0.259792 0.965665i \(-0.416346\pi\)
0.259792 + 0.965665i \(0.416346\pi\)
\(174\) 1884.85 0.821208
\(175\) −195.732 −0.0845483
\(176\) 489.071 0.209461
\(177\) −153.089 −0.0650106
\(178\) 4910.99 2.06794
\(179\) −4505.20 −1.88120 −0.940600 0.339517i \(-0.889737\pi\)
−0.940600 + 0.339517i \(0.889737\pi\)
\(180\) −3882.92 −1.60787
\(181\) −2610.79 −1.07215 −0.536073 0.844172i \(-0.680093\pi\)
−0.536073 + 0.844172i \(0.680093\pi\)
\(182\) 738.802 0.300899
\(183\) −1559.77 −0.630061
\(184\) −51.1449 −0.0204916
\(185\) −4611.24 −1.83257
\(186\) 2082.58 0.820980
\(187\) 1376.28 0.538201
\(188\) −4810.66 −1.86624
\(189\) 386.009 0.148561
\(190\) −2342.69 −0.894507
\(191\) −2992.98 −1.13384 −0.566922 0.823771i \(-0.691866\pi\)
−0.566922 + 0.823771i \(0.691866\pi\)
\(192\) −1462.48 −0.549717
\(193\) −842.048 −0.314051 −0.157026 0.987595i \(-0.550191\pi\)
−0.157026 + 0.987595i \(0.550191\pi\)
\(194\) −5119.46 −1.89462
\(195\) −862.923 −0.316898
\(196\) −4032.31 −1.46950
\(197\) 1391.04 0.503082 0.251541 0.967847i \(-0.419063\pi\)
0.251541 + 0.967847i \(0.419063\pi\)
\(198\) 5977.63 2.14551
\(199\) −989.587 −0.352512 −0.176256 0.984344i \(-0.556399\pi\)
−0.176256 + 0.984344i \(0.556399\pi\)
\(200\) 906.764 0.320590
\(201\) −1747.99 −0.613401
\(202\) 6500.28 2.26415
\(203\) −1030.33 −0.356232
\(204\) −542.453 −0.186173
\(205\) 3316.45 1.12991
\(206\) 7972.02 2.69630
\(207\) 61.0555 0.0205007
\(208\) 334.292 0.111437
\(209\) 2194.60 0.726333
\(210\) −446.994 −0.146884
\(211\) 943.172 0.307728 0.153864 0.988092i \(-0.450828\pi\)
0.153864 + 0.988092i \(0.450828\pi\)
\(212\) −444.612 −0.144038
\(213\) −777.558 −0.250129
\(214\) 750.871 0.239853
\(215\) 0 0
\(216\) −1788.26 −0.563313
\(217\) −1138.42 −0.356133
\(218\) −1997.14 −0.620474
\(219\) 123.505 0.0381082
\(220\) −8964.08 −2.74708
\(221\) 940.721 0.286334
\(222\) −2797.97 −0.845890
\(223\) −4371.74 −1.31280 −0.656398 0.754415i \(-0.727921\pi\)
−0.656398 + 0.754415i \(0.727921\pi\)
\(224\) 867.240 0.258682
\(225\) −1082.47 −0.320733
\(226\) 4391.41 1.29253
\(227\) 1293.65 0.378249 0.189124 0.981953i \(-0.439435\pi\)
0.189124 + 0.981953i \(0.439435\pi\)
\(228\) −864.988 −0.251251
\(229\) 2720.83 0.785141 0.392570 0.919722i \(-0.371586\pi\)
0.392570 + 0.919722i \(0.371586\pi\)
\(230\) −150.464 −0.0431360
\(231\) 418.739 0.119268
\(232\) 4773.20 1.35076
\(233\) 1824.91 0.513108 0.256554 0.966530i \(-0.417413\pi\)
0.256554 + 0.966530i \(0.417413\pi\)
\(234\) 4085.86 1.14146
\(235\) −5047.52 −1.40112
\(236\) −1087.01 −0.299822
\(237\) 1707.95 0.468114
\(238\) 487.294 0.132717
\(239\) 2517.89 0.681460 0.340730 0.940161i \(-0.389326\pi\)
0.340730 + 0.940161i \(0.389326\pi\)
\(240\) −202.255 −0.0543980
\(241\) 2023.30 0.540797 0.270398 0.962749i \(-0.412845\pi\)
0.270398 + 0.962749i \(0.412845\pi\)
\(242\) 7783.13 2.06743
\(243\) 3266.44 0.862314
\(244\) −11075.1 −2.90578
\(245\) −4230.84 −1.10326
\(246\) 2012.33 0.521550
\(247\) 1500.06 0.386424
\(248\) 5273.93 1.35038
\(249\) 1465.32 0.372935
\(250\) −4704.87 −1.19025
\(251\) −2200.46 −0.553355 −0.276677 0.960963i \(-0.589233\pi\)
−0.276677 + 0.960963i \(0.589233\pi\)
\(252\) 1287.90 0.321946
\(253\) 140.952 0.0350261
\(254\) −7213.71 −1.78200
\(255\) −569.161 −0.139773
\(256\) −3137.08 −0.765890
\(257\) −869.760 −0.211106 −0.105553 0.994414i \(-0.533661\pi\)
−0.105553 + 0.994414i \(0.533661\pi\)
\(258\) 0 0
\(259\) 1529.48 0.366938
\(260\) −6127.17 −1.46150
\(261\) −5698.12 −1.35136
\(262\) −252.264 −0.0594844
\(263\) 4849.52 1.13701 0.568506 0.822679i \(-0.307522\pi\)
0.568506 + 0.822679i \(0.307522\pi\)
\(264\) −1939.88 −0.452241
\(265\) −466.503 −0.108140
\(266\) 777.032 0.179109
\(267\) 1902.56 0.436085
\(268\) −12411.6 −2.82894
\(269\) 432.453 0.0980191 0.0490096 0.998798i \(-0.484394\pi\)
0.0490096 + 0.998798i \(0.484394\pi\)
\(270\) −5260.89 −1.18581
\(271\) 861.175 0.193036 0.0965179 0.995331i \(-0.469230\pi\)
0.0965179 + 0.995331i \(0.469230\pi\)
\(272\) 220.490 0.0491514
\(273\) 286.218 0.0634531
\(274\) −622.903 −0.137339
\(275\) −2498.99 −0.547981
\(276\) −55.5555 −0.0121161
\(277\) 2228.63 0.483412 0.241706 0.970350i \(-0.422293\pi\)
0.241706 + 0.970350i \(0.422293\pi\)
\(278\) 4594.57 0.991237
\(279\) −6295.88 −1.35098
\(280\) −1131.97 −0.241600
\(281\) −5991.67 −1.27200 −0.636002 0.771688i \(-0.719413\pi\)
−0.636002 + 0.771688i \(0.719413\pi\)
\(282\) −3062.69 −0.646740
\(283\) 1125.62 0.236434 0.118217 0.992988i \(-0.462282\pi\)
0.118217 + 0.992988i \(0.462282\pi\)
\(284\) −5521.04 −1.15357
\(285\) −907.577 −0.188632
\(286\) 9432.58 1.95021
\(287\) −1100.01 −0.226243
\(288\) 4796.17 0.981308
\(289\) −4292.53 −0.873708
\(290\) 14042.3 2.84342
\(291\) −1983.32 −0.399534
\(292\) 876.945 0.175751
\(293\) 6457.05 1.28746 0.643729 0.765254i \(-0.277387\pi\)
0.643729 + 0.765254i \(0.277387\pi\)
\(294\) −2567.16 −0.509251
\(295\) −1140.53 −0.225098
\(296\) −7085.58 −1.39136
\(297\) 4928.33 0.962865
\(298\) 12322.5 2.39537
\(299\) 96.3444 0.0186346
\(300\) 984.962 0.189556
\(301\) 0 0
\(302\) −6104.88 −1.16323
\(303\) 2518.27 0.477461
\(304\) 351.590 0.0663325
\(305\) −11620.4 −2.18158
\(306\) 2694.92 0.503459
\(307\) 7900.21 1.46869 0.734347 0.678775i \(-0.237489\pi\)
0.734347 + 0.678775i \(0.237489\pi\)
\(308\) 2973.25 0.550053
\(309\) 3088.43 0.568591
\(310\) 15515.4 2.84263
\(311\) 3923.50 0.715374 0.357687 0.933841i \(-0.383565\pi\)
0.357687 + 0.933841i \(0.383565\pi\)
\(312\) −1325.96 −0.240601
\(313\) 7259.87 1.31103 0.655515 0.755182i \(-0.272452\pi\)
0.655515 + 0.755182i \(0.272452\pi\)
\(314\) −11859.7 −2.13147
\(315\) 1351.32 0.241708
\(316\) 12127.3 2.15890
\(317\) −5967.29 −1.05728 −0.528638 0.848847i \(-0.677297\pi\)
−0.528638 + 0.848847i \(0.677297\pi\)
\(318\) −283.061 −0.0499160
\(319\) −13154.7 −2.30884
\(320\) −10895.6 −1.90339
\(321\) 290.894 0.0505798
\(322\) 49.9064 0.00863719
\(323\) 989.401 0.170439
\(324\) 6092.88 1.04473
\(325\) −1708.12 −0.291537
\(326\) −1841.22 −0.312808
\(327\) −773.709 −0.130845
\(328\) 5096.01 0.857867
\(329\) 1674.18 0.280549
\(330\) −5706.96 −0.951993
\(331\) −7412.74 −1.23094 −0.615470 0.788161i \(-0.711034\pi\)
−0.615470 + 0.788161i \(0.711034\pi\)
\(332\) 10404.5 1.71994
\(333\) 8458.59 1.39198
\(334\) 7882.81 1.29140
\(335\) −13022.7 −2.12389
\(336\) 67.0849 0.0108922
\(337\) 12010.9 1.94147 0.970733 0.240160i \(-0.0771998\pi\)
0.970733 + 0.240160i \(0.0771998\pi\)
\(338\) −3484.15 −0.560688
\(339\) 1701.27 0.272567
\(340\) −4041.32 −0.644621
\(341\) −14534.6 −2.30819
\(342\) 4297.29 0.679446
\(343\) 2887.66 0.454574
\(344\) 0 0
\(345\) −58.2909 −0.00909645
\(346\) 5344.56 0.830419
\(347\) 2376.88 0.367716 0.183858 0.982953i \(-0.441141\pi\)
0.183858 + 0.982953i \(0.441141\pi\)
\(348\) 5184.83 0.798666
\(349\) 2443.29 0.374747 0.187373 0.982289i \(-0.440003\pi\)
0.187373 + 0.982289i \(0.440003\pi\)
\(350\) −884.807 −0.135128
\(351\) 3368.64 0.512264
\(352\) 11072.4 1.67659
\(353\) −6845.61 −1.03217 −0.516084 0.856538i \(-0.672611\pi\)
−0.516084 + 0.856538i \(0.672611\pi\)
\(354\) −692.039 −0.103902
\(355\) −5792.88 −0.866067
\(356\) 13509.1 2.01118
\(357\) 188.782 0.0279871
\(358\) −20365.8 −3.00661
\(359\) −8886.04 −1.30637 −0.653185 0.757198i \(-0.726568\pi\)
−0.653185 + 0.757198i \(0.726568\pi\)
\(360\) −6260.22 −0.916507
\(361\) −5281.31 −0.769983
\(362\) −11802.1 −1.71354
\(363\) 3015.25 0.435977
\(364\) 2032.29 0.292639
\(365\) 920.122 0.131949
\(366\) −7050.93 −1.00699
\(367\) 1772.16 0.252060 0.126030 0.992026i \(-0.459776\pi\)
0.126030 + 0.992026i \(0.459776\pi\)
\(368\) 22.5816 0.00319877
\(369\) −6083.49 −0.858249
\(370\) −20845.1 −2.92888
\(371\) 154.732 0.0216531
\(372\) 5728.74 0.798444
\(373\) −2089.79 −0.290095 −0.145047 0.989425i \(-0.546333\pi\)
−0.145047 + 0.989425i \(0.546333\pi\)
\(374\) 6221.48 0.860174
\(375\) −1822.71 −0.250998
\(376\) −7755.96 −1.06378
\(377\) −8991.53 −1.22835
\(378\) 1744.96 0.237436
\(379\) −12958.0 −1.75622 −0.878108 0.478462i \(-0.841194\pi\)
−0.878108 + 0.478462i \(0.841194\pi\)
\(380\) −6444.23 −0.869953
\(381\) −2794.65 −0.375786
\(382\) −13529.8 −1.81215
\(383\) 2891.35 0.385747 0.192874 0.981224i \(-0.438219\pi\)
0.192874 + 0.981224i \(0.438219\pi\)
\(384\) −3803.51 −0.505461
\(385\) 3119.64 0.412965
\(386\) −3806.48 −0.501929
\(387\) 0 0
\(388\) −14082.5 −1.84261
\(389\) 8007.52 1.04370 0.521848 0.853039i \(-0.325243\pi\)
0.521848 + 0.853039i \(0.325243\pi\)
\(390\) −3900.84 −0.506479
\(391\) 63.5462 0.00821910
\(392\) −6501.07 −0.837637
\(393\) −97.7292 −0.0125440
\(394\) 6288.18 0.804046
\(395\) 12724.4 1.62084
\(396\) 16443.2 2.08662
\(397\) −5407.44 −0.683606 −0.341803 0.939772i \(-0.611038\pi\)
−0.341803 + 0.939772i \(0.611038\pi\)
\(398\) −4473.43 −0.563399
\(399\) 301.029 0.0377702
\(400\) −400.356 −0.0500445
\(401\) −11696.4 −1.45658 −0.728290 0.685269i \(-0.759684\pi\)
−0.728290 + 0.685269i \(0.759684\pi\)
\(402\) −7901.79 −0.980362
\(403\) −9934.78 −1.22801
\(404\) 17880.9 2.20200
\(405\) 6392.87 0.784356
\(406\) −4657.61 −0.569344
\(407\) 19527.4 2.37823
\(408\) −874.567 −0.106121
\(409\) 4629.41 0.559681 0.279840 0.960047i \(-0.409718\pi\)
0.279840 + 0.960047i \(0.409718\pi\)
\(410\) 14992.0 1.80586
\(411\) −241.318 −0.0289619
\(412\) 21929.4 2.62229
\(413\) 378.295 0.0450718
\(414\) 276.002 0.0327651
\(415\) 10916.7 1.29128
\(416\) 7568.25 0.891981
\(417\) 1779.97 0.209031
\(418\) 9920.69 1.16085
\(419\) −3307.85 −0.385678 −0.192839 0.981230i \(-0.561770\pi\)
−0.192839 + 0.981230i \(0.561770\pi\)
\(420\) −1229.59 −0.142852
\(421\) 242.081 0.0280245 0.0140122 0.999902i \(-0.495540\pi\)
0.0140122 + 0.999902i \(0.495540\pi\)
\(422\) 4263.61 0.491823
\(423\) 9258.87 1.06426
\(424\) −716.824 −0.0821039
\(425\) −1126.63 −0.128587
\(426\) −3514.95 −0.399766
\(427\) 3854.30 0.436821
\(428\) 2065.49 0.233269
\(429\) 3654.26 0.411257
\(430\) 0 0
\(431\) −16410.0 −1.83398 −0.916989 0.398913i \(-0.869387\pi\)
−0.916989 + 0.398913i \(0.869387\pi\)
\(432\) 789.555 0.0879339
\(433\) −6230.06 −0.691450 −0.345725 0.938336i \(-0.612367\pi\)
−0.345725 + 0.938336i \(0.612367\pi\)
\(434\) −5146.22 −0.569185
\(435\) 5440.11 0.599617
\(436\) −5493.71 −0.603442
\(437\) 101.330 0.0110921
\(438\) 558.305 0.0609060
\(439\) −4436.43 −0.482322 −0.241161 0.970485i \(-0.577528\pi\)
−0.241161 + 0.970485i \(0.577528\pi\)
\(440\) −14452.3 −1.56588
\(441\) 7760.81 0.838010
\(442\) 4252.53 0.457630
\(443\) −13607.7 −1.45941 −0.729707 0.683760i \(-0.760343\pi\)
−0.729707 + 0.683760i \(0.760343\pi\)
\(444\) −7696.63 −0.822671
\(445\) 14174.2 1.50994
\(446\) −19762.5 −2.09816
\(447\) 4773.82 0.505132
\(448\) 3613.91 0.381119
\(449\) 3677.16 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(450\) −4893.32 −0.512607
\(451\) −14044.3 −1.46634
\(452\) 12079.8 1.25705
\(453\) −2365.08 −0.245301
\(454\) 5847.94 0.604532
\(455\) 2132.35 0.219706
\(456\) −1394.57 −0.143217
\(457\) −9654.94 −0.988270 −0.494135 0.869385i \(-0.664515\pi\)
−0.494135 + 0.869385i \(0.664515\pi\)
\(458\) 12299.5 1.25484
\(459\) 2221.86 0.225943
\(460\) −413.893 −0.0419519
\(461\) 6387.32 0.645308 0.322654 0.946517i \(-0.395425\pi\)
0.322654 + 0.946517i \(0.395425\pi\)
\(462\) 1892.91 0.190619
\(463\) −8369.16 −0.840060 −0.420030 0.907510i \(-0.637981\pi\)
−0.420030 + 0.907510i \(0.637981\pi\)
\(464\) −2107.47 −0.210855
\(465\) 6010.80 0.599450
\(466\) 8249.53 0.820069
\(467\) 11897.9 1.17895 0.589475 0.807786i \(-0.299334\pi\)
0.589475 + 0.807786i \(0.299334\pi\)
\(468\) 11239.3 1.11012
\(469\) 4319.41 0.425271
\(470\) −22817.3 −2.23933
\(471\) −4594.55 −0.449481
\(472\) −1752.52 −0.170903
\(473\) 0 0
\(474\) 7720.78 0.748159
\(475\) −1796.51 −0.173536
\(476\) 1340.44 0.129074
\(477\) 855.727 0.0821405
\(478\) 11382.1 1.08914
\(479\) −5313.81 −0.506877 −0.253438 0.967352i \(-0.581562\pi\)
−0.253438 + 0.967352i \(0.581562\pi\)
\(480\) −4578.99 −0.435420
\(481\) 13347.5 1.26527
\(482\) 9146.32 0.864322
\(483\) 19.3342 0.00182140
\(484\) 21409.7 2.01068
\(485\) −14775.9 −1.38338
\(486\) 14766.0 1.37818
\(487\) 10792.5 1.00422 0.502110 0.864804i \(-0.332557\pi\)
0.502110 + 0.864804i \(0.332557\pi\)
\(488\) −17855.8 −1.65634
\(489\) −713.303 −0.0659645
\(490\) −19125.5 −1.76327
\(491\) 2458.96 0.226011 0.113006 0.993594i \(-0.463952\pi\)
0.113006 + 0.993594i \(0.463952\pi\)
\(492\) 5535.48 0.507233
\(493\) −5930.57 −0.541784
\(494\) 6781.03 0.617597
\(495\) 17252.8 1.56658
\(496\) −2328.55 −0.210797
\(497\) 1921.41 0.173414
\(498\) 6623.98 0.596039
\(499\) 8744.82 0.784513 0.392256 0.919856i \(-0.371695\pi\)
0.392256 + 0.919856i \(0.371695\pi\)
\(500\) −12942.1 −1.15758
\(501\) 3053.87 0.272329
\(502\) −9947.20 −0.884393
\(503\) −12931.7 −1.14631 −0.573157 0.819446i \(-0.694281\pi\)
−0.573157 + 0.819446i \(0.694281\pi\)
\(504\) 2076.42 0.183514
\(505\) 18761.3 1.65320
\(506\) 637.175 0.0559800
\(507\) −1349.79 −0.118237
\(508\) −19843.4 −1.73309
\(509\) 19155.2 1.66805 0.834026 0.551726i \(-0.186030\pi\)
0.834026 + 0.551726i \(0.186030\pi\)
\(510\) −2572.89 −0.223391
\(511\) −305.190 −0.0264204
\(512\) 3193.56 0.275658
\(513\) 3542.95 0.304923
\(514\) −3931.75 −0.337397
\(515\) 23009.1 1.96874
\(516\) 0 0
\(517\) 21375.0 1.81832
\(518\) 6914.00 0.586455
\(519\) 2070.53 0.175118
\(520\) −9878.50 −0.833079
\(521\) 2030.86 0.170775 0.0853875 0.996348i \(-0.472787\pi\)
0.0853875 + 0.996348i \(0.472787\pi\)
\(522\) −25758.4 −2.15980
\(523\) −13361.0 −1.11708 −0.558542 0.829477i \(-0.688639\pi\)
−0.558542 + 0.829477i \(0.688639\pi\)
\(524\) −693.925 −0.0578516
\(525\) −342.782 −0.0284957
\(526\) 21922.2 1.81722
\(527\) −6552.71 −0.541633
\(528\) 856.501 0.0705955
\(529\) −12160.5 −0.999465
\(530\) −2108.83 −0.172833
\(531\) 2092.11 0.170979
\(532\) 2137.45 0.174192
\(533\) −9599.63 −0.780124
\(534\) 8600.52 0.696968
\(535\) 2167.18 0.175132
\(536\) −20010.5 −1.61254
\(537\) −7889.88 −0.634029
\(538\) 1954.91 0.156658
\(539\) 17916.6 1.43176
\(540\) −14471.6 −1.15326
\(541\) −1543.79 −0.122685 −0.0613425 0.998117i \(-0.519538\pi\)
−0.0613425 + 0.998117i \(0.519538\pi\)
\(542\) 3892.95 0.308517
\(543\) −4572.22 −0.361350
\(544\) 4991.82 0.393424
\(545\) −5764.20 −0.453048
\(546\) 1293.85 0.101413
\(547\) −15805.6 −1.23546 −0.617731 0.786390i \(-0.711948\pi\)
−0.617731 + 0.786390i \(0.711948\pi\)
\(548\) −1713.47 −0.133569
\(549\) 21315.8 1.65708
\(550\) −11296.7 −0.875804
\(551\) −9456.81 −0.731168
\(552\) −89.5691 −0.00690637
\(553\) −4220.47 −0.324544
\(554\) 10074.5 0.772608
\(555\) −8075.58 −0.617638
\(556\) 12638.7 0.964028
\(557\) 552.342 0.0420170 0.0210085 0.999779i \(-0.493312\pi\)
0.0210085 + 0.999779i \(0.493312\pi\)
\(558\) −28460.6 −2.15920
\(559\) 0 0
\(560\) 499.788 0.0377141
\(561\) 2410.25 0.181392
\(562\) −27085.4 −2.03297
\(563\) 12098.2 0.905644 0.452822 0.891601i \(-0.350417\pi\)
0.452822 + 0.891601i \(0.350417\pi\)
\(564\) −8424.81 −0.628987
\(565\) 12674.6 0.943760
\(566\) 5088.35 0.377878
\(567\) −2120.42 −0.157053
\(568\) −8901.27 −0.657551
\(569\) 519.978 0.0383104 0.0191552 0.999817i \(-0.493902\pi\)
0.0191552 + 0.999817i \(0.493902\pi\)
\(570\) −4102.70 −0.301479
\(571\) −13553.9 −0.993365 −0.496682 0.867932i \(-0.665449\pi\)
−0.496682 + 0.867932i \(0.665449\pi\)
\(572\) 25947.0 1.89668
\(573\) −5241.55 −0.382144
\(574\) −4972.61 −0.361590
\(575\) −115.384 −0.00836845
\(576\) 19986.3 1.44577
\(577\) −6324.19 −0.456290 −0.228145 0.973627i \(-0.573266\pi\)
−0.228145 + 0.973627i \(0.573266\pi\)
\(578\) −19404.4 −1.39639
\(579\) −1474.66 −0.105846
\(580\) 38627.4 2.76537
\(581\) −3620.92 −0.258556
\(582\) −8965.61 −0.638551
\(583\) 1975.53 0.140339
\(584\) 1413.85 0.100181
\(585\) 11792.7 0.833450
\(586\) 29189.1 2.05766
\(587\) −21394.9 −1.50437 −0.752183 0.658954i \(-0.770999\pi\)
−0.752183 + 0.658954i \(0.770999\pi\)
\(588\) −7061.71 −0.495272
\(589\) −10448.9 −0.730965
\(590\) −5155.75 −0.359761
\(591\) 2436.10 0.169556
\(592\) 3128.44 0.217193
\(593\) 17243.2 1.19409 0.597043 0.802209i \(-0.296342\pi\)
0.597043 + 0.802209i \(0.296342\pi\)
\(594\) 22278.5 1.53889
\(595\) 1406.44 0.0969049
\(596\) 33896.5 2.32962
\(597\) −1733.04 −0.118809
\(598\) 435.525 0.0297825
\(599\) 1214.43 0.0828387 0.0414193 0.999142i \(-0.486812\pi\)
0.0414193 + 0.999142i \(0.486812\pi\)
\(600\) 1588.00 0.108050
\(601\) 3251.39 0.220677 0.110338 0.993894i \(-0.464807\pi\)
0.110338 + 0.993894i \(0.464807\pi\)
\(602\) 0 0
\(603\) 23888.0 1.61326
\(604\) −16793.2 −1.13130
\(605\) 22463.9 1.50956
\(606\) 11383.8 0.763096
\(607\) 13945.8 0.932526 0.466263 0.884646i \(-0.345600\pi\)
0.466263 + 0.884646i \(0.345600\pi\)
\(608\) 7959.89 0.530948
\(609\) −1804.40 −0.120062
\(610\) −52530.0 −3.48668
\(611\) 14610.3 0.967381
\(612\) 7413.16 0.489639
\(613\) 8980.34 0.591701 0.295850 0.955234i \(-0.404397\pi\)
0.295850 + 0.955234i \(0.404397\pi\)
\(614\) 35712.9 2.34732
\(615\) 5808.03 0.380817
\(616\) 4793.60 0.313539
\(617\) −9450.64 −0.616642 −0.308321 0.951282i \(-0.599767\pi\)
−0.308321 + 0.951282i \(0.599767\pi\)
\(618\) 13961.3 0.908745
\(619\) −9340.01 −0.606473 −0.303237 0.952915i \(-0.598067\pi\)
−0.303237 + 0.952915i \(0.598067\pi\)
\(620\) 42679.6 2.76460
\(621\) 227.553 0.0147043
\(622\) 17736.2 1.14334
\(623\) −4701.37 −0.302337
\(624\) 585.439 0.0375582
\(625\) −19233.0 −1.23091
\(626\) 32818.3 2.09534
\(627\) 3843.36 0.244799
\(628\) −32623.5 −2.07296
\(629\) 8803.65 0.558067
\(630\) 6108.62 0.386307
\(631\) −9633.83 −0.607792 −0.303896 0.952705i \(-0.598288\pi\)
−0.303896 + 0.952705i \(0.598288\pi\)
\(632\) 19552.1 1.23060
\(633\) 1651.76 0.103715
\(634\) −26975.2 −1.68978
\(635\) −20820.4 −1.30115
\(636\) −778.642 −0.0485458
\(637\) 12246.4 0.761727
\(638\) −59465.6 −3.69007
\(639\) 10626.1 0.657844
\(640\) −28336.5 −1.75015
\(641\) −10806.5 −0.665885 −0.332943 0.942947i \(-0.608042\pi\)
−0.332943 + 0.942947i \(0.608042\pi\)
\(642\) 1314.99 0.0808386
\(643\) 5396.16 0.330955 0.165477 0.986214i \(-0.447084\pi\)
0.165477 + 0.986214i \(0.447084\pi\)
\(644\) 137.282 0.00840011
\(645\) 0 0
\(646\) 4472.59 0.272402
\(647\) 13276.4 0.806722 0.403361 0.915041i \(-0.367842\pi\)
0.403361 + 0.915041i \(0.367842\pi\)
\(648\) 9823.22 0.595513
\(649\) 4829.84 0.292123
\(650\) −7721.56 −0.465945
\(651\) −1993.69 −0.120029
\(652\) −5064.79 −0.304222
\(653\) 7560.53 0.453088 0.226544 0.974001i \(-0.427257\pi\)
0.226544 + 0.974001i \(0.427257\pi\)
\(654\) −3497.55 −0.209121
\(655\) −728.091 −0.0434334
\(656\) −2250.00 −0.133914
\(657\) −1687.82 −0.100225
\(658\) 7568.15 0.448385
\(659\) 17888.9 1.05744 0.528718 0.848797i \(-0.322673\pi\)
0.528718 + 0.848797i \(0.322673\pi\)
\(660\) −15698.6 −0.925861
\(661\) 28643.1 1.68546 0.842729 0.538337i \(-0.180947\pi\)
0.842729 + 0.538337i \(0.180947\pi\)
\(662\) −33509.3 −1.96733
\(663\) 1647.47 0.0965043
\(664\) 16774.6 0.980390
\(665\) 2242.69 0.130779
\(666\) 38237.1 2.22471
\(667\) −607.382 −0.0352593
\(668\) 21683.9 1.25595
\(669\) −7656.15 −0.442457
\(670\) −58869.0 −3.39449
\(671\) 49209.4 2.83116
\(672\) 1518.78 0.0871849
\(673\) −26681.2 −1.52821 −0.764105 0.645091i \(-0.776819\pi\)
−0.764105 + 0.645091i \(0.776819\pi\)
\(674\) 54295.2 3.10293
\(675\) −4034.36 −0.230048
\(676\) −9584.15 −0.545297
\(677\) −24196.9 −1.37365 −0.686826 0.726821i \(-0.740997\pi\)
−0.686826 + 0.726821i \(0.740997\pi\)
\(678\) 7690.59 0.435627
\(679\) 4900.94 0.276997
\(680\) −6515.60 −0.367444
\(681\) 2265.54 0.127483
\(682\) −65703.8 −3.68905
\(683\) 33758.8 1.89128 0.945641 0.325213i \(-0.105436\pi\)
0.945641 + 0.325213i \(0.105436\pi\)
\(684\) 11820.9 0.660796
\(685\) −1797.84 −0.100280
\(686\) 13053.7 0.726517
\(687\) 4764.93 0.264619
\(688\) 0 0
\(689\) 1350.32 0.0746634
\(690\) −263.504 −0.0145383
\(691\) 18145.5 0.998970 0.499485 0.866322i \(-0.333523\pi\)
0.499485 + 0.866322i \(0.333523\pi\)
\(692\) 14701.7 0.807625
\(693\) −5722.48 −0.313678
\(694\) 10744.7 0.587698
\(695\) 13261.0 0.723765
\(696\) 8359.21 0.455252
\(697\) −6331.66 −0.344087
\(698\) 11044.9 0.598935
\(699\) 3195.94 0.172935
\(700\) −2433.92 −0.131419
\(701\) −21944.9 −1.18238 −0.591189 0.806533i \(-0.701341\pi\)
−0.591189 + 0.806533i \(0.701341\pi\)
\(702\) 15227.9 0.818720
\(703\) 14038.2 0.753144
\(704\) 46140.2 2.47014
\(705\) −8839.62 −0.472226
\(706\) −30945.6 −1.64965
\(707\) −6222.83 −0.331023
\(708\) −1903.65 −0.101050
\(709\) −36958.0 −1.95767 −0.978834 0.204656i \(-0.934392\pi\)
−0.978834 + 0.204656i \(0.934392\pi\)
\(710\) −26186.7 −1.38418
\(711\) −23340.8 −1.23115
\(712\) 21779.9 1.14640
\(713\) −671.099 −0.0352494
\(714\) 853.389 0.0447301
\(715\) 27224.6 1.42397
\(716\) −56022.0 −2.92408
\(717\) 4409.54 0.229675
\(718\) −40169.4 −2.08789
\(719\) 5395.23 0.279845 0.139922 0.990162i \(-0.455315\pi\)
0.139922 + 0.990162i \(0.455315\pi\)
\(720\) 2764.02 0.143068
\(721\) −7631.75 −0.394204
\(722\) −23874.2 −1.23062
\(723\) 3543.36 0.182267
\(724\) −32465.0 −1.66651
\(725\) 10768.5 0.551629
\(726\) 13630.4 0.696795
\(727\) −9561.59 −0.487785 −0.243892 0.969802i \(-0.578424\pi\)
−0.243892 + 0.969802i \(0.578424\pi\)
\(728\) 3276.54 0.166809
\(729\) −7509.02 −0.381497
\(730\) 4159.42 0.210886
\(731\) 0 0
\(732\) −19395.6 −0.979347
\(733\) 13308.7 0.670623 0.335312 0.942107i \(-0.391158\pi\)
0.335312 + 0.942107i \(0.391158\pi\)
\(734\) 8011.05 0.402852
\(735\) −7409.40 −0.371836
\(736\) 511.239 0.0256040
\(737\) 55147.7 2.75630
\(738\) −27500.5 −1.37169
\(739\) −16167.3 −0.804768 −0.402384 0.915471i \(-0.631818\pi\)
−0.402384 + 0.915471i \(0.631818\pi\)
\(740\) −57340.5 −2.84849
\(741\) 2627.03 0.130238
\(742\) 699.466 0.0346068
\(743\) −25680.2 −1.26799 −0.633994 0.773338i \(-0.718585\pi\)
−0.633994 + 0.773338i \(0.718585\pi\)
\(744\) 9236.13 0.455125
\(745\) 35565.4 1.74901
\(746\) −9446.91 −0.463641
\(747\) −20025.0 −0.980828
\(748\) 17114.0 0.836562
\(749\) −718.821 −0.0350669
\(750\) −8239.55 −0.401154
\(751\) −12689.2 −0.616559 −0.308279 0.951296i \(-0.599753\pi\)
−0.308279 + 0.951296i \(0.599753\pi\)
\(752\) 3424.42 0.166058
\(753\) −3853.63 −0.186500
\(754\) −40646.2 −1.96319
\(755\) −17620.1 −0.849351
\(756\) 4800.00 0.230918
\(757\) 17431.3 0.836926 0.418463 0.908234i \(-0.362569\pi\)
0.418463 + 0.908234i \(0.362569\pi\)
\(758\) −58576.5 −2.80685
\(759\) 246.847 0.0118050
\(760\) −10389.7 −0.495886
\(761\) 16997.5 0.809670 0.404835 0.914390i \(-0.367329\pi\)
0.404835 + 0.914390i \(0.367329\pi\)
\(762\) −12633.2 −0.600595
\(763\) 1911.89 0.0907146
\(764\) −37217.5 −1.76241
\(765\) 7778.15 0.367607
\(766\) 13070.4 0.616516
\(767\) 3301.31 0.155415
\(768\) −5493.92 −0.258131
\(769\) −29896.1 −1.40193 −0.700963 0.713198i \(-0.747246\pi\)
−0.700963 + 0.713198i \(0.747246\pi\)
\(770\) 14102.3 0.660017
\(771\) −1523.19 −0.0711498
\(772\) −10470.8 −0.488152
\(773\) 18464.3 0.859138 0.429569 0.903034i \(-0.358665\pi\)
0.429569 + 0.903034i \(0.358665\pi\)
\(774\) 0 0
\(775\) 11898.1 0.551475
\(776\) −22704.5 −1.05032
\(777\) 2678.54 0.123671
\(778\) 36198.0 1.66807
\(779\) −10096.4 −0.464365
\(780\) −10730.4 −0.492577
\(781\) 24531.4 1.12395
\(782\) 287.261 0.0131361
\(783\) −21236.8 −0.969275
\(784\) 2870.36 0.130756
\(785\) −34229.8 −1.55632
\(786\) −441.785 −0.0200483
\(787\) −7383.94 −0.334446 −0.167223 0.985919i \(-0.553480\pi\)
−0.167223 + 0.985919i \(0.553480\pi\)
\(788\) 17297.5 0.781975
\(789\) 8492.87 0.383212
\(790\) 57520.5 2.59049
\(791\) −4203.97 −0.188971
\(792\) 26510.5 1.18940
\(793\) 33635.9 1.50623
\(794\) −24444.3 −1.09257
\(795\) −816.979 −0.0364469
\(796\) −12305.5 −0.547934
\(797\) 23799.0 1.05772 0.528860 0.848709i \(-0.322620\pi\)
0.528860 + 0.848709i \(0.322620\pi\)
\(798\) 1360.80 0.0603658
\(799\) 9636.57 0.426680
\(800\) −9063.93 −0.400573
\(801\) −26000.4 −1.14691
\(802\) −52873.4 −2.32796
\(803\) −3896.49 −0.171238
\(804\) −21736.1 −0.953451
\(805\) 144.041 0.00630657
\(806\) −44910.2 −1.96265
\(807\) 757.348 0.0330358
\(808\) 28828.4 1.25517
\(809\) 2665.36 0.115833 0.0579166 0.998321i \(-0.481554\pi\)
0.0579166 + 0.998321i \(0.481554\pi\)
\(810\) 28899.0 1.25359
\(811\) −36736.8 −1.59063 −0.795317 0.606194i \(-0.792695\pi\)
−0.795317 + 0.606194i \(0.792695\pi\)
\(812\) −12812.1 −0.553715
\(813\) 1508.16 0.0650597
\(814\) 88273.9 3.80098
\(815\) −5314.16 −0.228401
\(816\) 386.140 0.0165657
\(817\) 0 0
\(818\) 20927.2 0.894504
\(819\) −3911.46 −0.166883
\(820\) 41239.8 1.75629
\(821\) −13460.1 −0.572180 −0.286090 0.958203i \(-0.592356\pi\)
−0.286090 + 0.958203i \(0.592356\pi\)
\(822\) −1090.88 −0.0462880
\(823\) −25589.3 −1.08382 −0.541912 0.840435i \(-0.682300\pi\)
−0.541912 + 0.840435i \(0.682300\pi\)
\(824\) 35355.5 1.49474
\(825\) −4376.43 −0.184688
\(826\) 1710.08 0.0720355
\(827\) 24201.4 1.01761 0.508807 0.860881i \(-0.330087\pi\)
0.508807 + 0.860881i \(0.330087\pi\)
\(828\) 759.222 0.0318657
\(829\) −15933.9 −0.667560 −0.333780 0.942651i \(-0.608324\pi\)
−0.333780 + 0.942651i \(0.608324\pi\)
\(830\) 49349.2 2.06378
\(831\) 3902.95 0.162926
\(832\) 31538.0 1.31416
\(833\) 8077.41 0.335973
\(834\) 8046.38 0.334081
\(835\) 22751.6 0.942935
\(836\) 27289.7 1.12899
\(837\) −23464.7 −0.969006
\(838\) −14953.2 −0.616406
\(839\) −23536.8 −0.968512 −0.484256 0.874926i \(-0.660910\pi\)
−0.484256 + 0.874926i \(0.660910\pi\)
\(840\) −1982.40 −0.0814275
\(841\) 32296.1 1.32421
\(842\) 1094.33 0.0447898
\(843\) −10493.1 −0.428709
\(844\) 11728.3 0.478323
\(845\) −10056.0 −0.409394
\(846\) 41854.7 1.70094
\(847\) −7450.91 −0.302263
\(848\) 316.493 0.0128165
\(849\) 1971.27 0.0796864
\(850\) −5092.94 −0.205513
\(851\) 901.629 0.0363190
\(852\) −9668.90 −0.388792
\(853\) 14346.0 0.575845 0.287923 0.957654i \(-0.407035\pi\)
0.287923 + 0.957654i \(0.407035\pi\)
\(854\) 17423.4 0.698145
\(855\) 12402.9 0.496107
\(856\) 3330.07 0.132967
\(857\) −25049.7 −0.998462 −0.499231 0.866469i \(-0.666384\pi\)
−0.499231 + 0.866469i \(0.666384\pi\)
\(858\) 16519.1 0.657288
\(859\) −26179.1 −1.03983 −0.519917 0.854217i \(-0.674037\pi\)
−0.519917 + 0.854217i \(0.674037\pi\)
\(860\) 0 0
\(861\) −1926.43 −0.0762516
\(862\) −74181.7 −2.93113
\(863\) 10987.8 0.433404 0.216702 0.976238i \(-0.430470\pi\)
0.216702 + 0.976238i \(0.430470\pi\)
\(864\) 17875.3 0.703852
\(865\) 15425.6 0.606342
\(866\) −28163.0 −1.10510
\(867\) −7517.42 −0.294469
\(868\) −14156.2 −0.553561
\(869\) −53884.4 −2.10346
\(870\) 24592.0 0.958331
\(871\) 37694.8 1.46641
\(872\) −8857.20 −0.343971
\(873\) 27104.1 1.05078
\(874\) 458.062 0.0177279
\(875\) 4504.05 0.174017
\(876\) 1535.78 0.0592341
\(877\) 43452.2 1.67306 0.836531 0.547919i \(-0.184580\pi\)
0.836531 + 0.547919i \(0.184580\pi\)
\(878\) −20054.9 −0.770865
\(879\) 11308.1 0.433917
\(880\) 6381.00 0.244436
\(881\) −10548.4 −0.403389 −0.201694 0.979449i \(-0.564645\pi\)
−0.201694 + 0.979449i \(0.564645\pi\)
\(882\) 35082.8 1.33934
\(883\) −12217.7 −0.465637 −0.232818 0.972520i \(-0.574795\pi\)
−0.232818 + 0.972520i \(0.574795\pi\)
\(884\) 11697.8 0.445068
\(885\) −1997.38 −0.0758658
\(886\) −61513.5 −2.33249
\(887\) −39821.5 −1.50741 −0.753706 0.657212i \(-0.771736\pi\)
−0.753706 + 0.657212i \(0.771736\pi\)
\(888\) −12408.9 −0.468934
\(889\) 6905.80 0.260532
\(890\) 64074.6 2.41324
\(891\) −27072.2 −1.01790
\(892\) −54362.4 −2.04057
\(893\) 15366.4 0.575829
\(894\) 21580.1 0.807322
\(895\) −58780.3 −2.19532
\(896\) 9398.77 0.350436
\(897\) 168.726 0.00628049
\(898\) 16622.6 0.617710
\(899\) 62631.6 2.32356
\(900\) −13460.5 −0.498537
\(901\) 890.635 0.0329316
\(902\) −63487.4 −2.34357
\(903\) 0 0
\(904\) 19475.6 0.716538
\(905\) −34063.4 −1.25117
\(906\) −10691.4 −0.392049
\(907\) −32895.0 −1.20426 −0.602128 0.798399i \(-0.705680\pi\)
−0.602128 + 0.798399i \(0.705680\pi\)
\(908\) 16086.5 0.587938
\(909\) −34414.6 −1.25573
\(910\) 9639.29 0.351142
\(911\) −14491.2 −0.527019 −0.263509 0.964657i \(-0.584880\pi\)
−0.263509 + 0.964657i \(0.584880\pi\)
\(912\) 615.734 0.0223563
\(913\) −46229.7 −1.67577
\(914\) −43645.2 −1.57949
\(915\) −20350.6 −0.735267
\(916\) 33833.3 1.22040
\(917\) 241.496 0.00869675
\(918\) 10043.9 0.361110
\(919\) 3819.84 0.137111 0.0685554 0.997647i \(-0.478161\pi\)
0.0685554 + 0.997647i \(0.478161\pi\)
\(920\) −667.298 −0.0239132
\(921\) 13835.5 0.495000
\(922\) 28873.9 1.03136
\(923\) 16767.8 0.597962
\(924\) 5206.99 0.185387
\(925\) −15985.3 −0.568208
\(926\) −37832.8 −1.34262
\(927\) −42206.5 −1.49541
\(928\) −47712.4 −1.68775
\(929\) 6597.01 0.232983 0.116491 0.993192i \(-0.462835\pi\)
0.116491 + 0.993192i \(0.462835\pi\)
\(930\) 27171.8 0.958064
\(931\) 12880.1 0.453415
\(932\) 22692.7 0.797559
\(933\) 6871.16 0.241106
\(934\) 53784.6 1.88425
\(935\) 17956.6 0.628068
\(936\) 18120.5 0.632787
\(937\) −22841.5 −0.796370 −0.398185 0.917305i \(-0.630360\pi\)
−0.398185 + 0.917305i \(0.630360\pi\)
\(938\) 19525.9 0.679685
\(939\) 12714.1 0.441862
\(940\) −62765.6 −2.17786
\(941\) 36677.3 1.27061 0.635306 0.772261i \(-0.280874\pi\)
0.635306 + 0.772261i \(0.280874\pi\)
\(942\) −20769.7 −0.718378
\(943\) −648.460 −0.0223932
\(944\) 773.775 0.0266782
\(945\) 5036.34 0.173367
\(946\) 0 0
\(947\) −19733.4 −0.677138 −0.338569 0.940942i \(-0.609943\pi\)
−0.338569 + 0.940942i \(0.609943\pi\)
\(948\) 21238.2 0.727622
\(949\) −2663.34 −0.0911020
\(950\) −8121.13 −0.277352
\(951\) −10450.4 −0.356338
\(952\) 2161.12 0.0735739
\(953\) −12496.6 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(954\) 3868.32 0.131280
\(955\) −39050.0 −1.32317
\(956\) 31309.9 1.05924
\(957\) −23037.5 −0.778157
\(958\) −24021.1 −0.810111
\(959\) 596.315 0.0200793
\(960\) −19081.3 −0.641507
\(961\) 39411.0 1.32291
\(962\) 60337.4 2.02220
\(963\) −3975.35 −0.133026
\(964\) 25159.6 0.840597
\(965\) −10986.4 −0.366491
\(966\) 87.4002 0.00291103
\(967\) −1702.88 −0.0566297 −0.0283149 0.999599i \(-0.509014\pi\)
−0.0283149 + 0.999599i \(0.509014\pi\)
\(968\) 34517.7 1.14612
\(969\) 1732.72 0.0574437
\(970\) −66794.6 −2.21097
\(971\) −3923.91 −0.129685 −0.0648426 0.997896i \(-0.520655\pi\)
−0.0648426 + 0.997896i \(0.520655\pi\)
\(972\) 40618.0 1.34035
\(973\) −4398.45 −0.144921
\(974\) 48787.5 1.60498
\(975\) −2991.40 −0.0982579
\(976\) 7883.71 0.258557
\(977\) −24540.8 −0.803614 −0.401807 0.915724i \(-0.631618\pi\)
−0.401807 + 0.915724i \(0.631618\pi\)
\(978\) −3224.49 −0.105427
\(979\) −60024.2 −1.95953
\(980\) −52610.3 −1.71487
\(981\) 10573.5 0.344124
\(982\) 11115.8 0.361220
\(983\) 10646.8 0.345453 0.172727 0.984970i \(-0.444742\pi\)
0.172727 + 0.984970i \(0.444742\pi\)
\(984\) 8924.56 0.289131
\(985\) 18149.1 0.587085
\(986\) −26809.2 −0.865900
\(987\) 2931.96 0.0945547
\(988\) 18653.2 0.600645
\(989\) 0 0
\(990\) 77991.3 2.50376
\(991\) 68.7730 0.00220449 0.00110224 0.999999i \(-0.499649\pi\)
0.00110224 + 0.999999i \(0.499649\pi\)
\(992\) −52717.6 −1.68729
\(993\) −12981.8 −0.414869
\(994\) 8685.72 0.277157
\(995\) −12911.3 −0.411373
\(996\) 18221.2 0.579678
\(997\) 56204.3 1.78536 0.892682 0.450686i \(-0.148821\pi\)
0.892682 + 0.450686i \(0.148821\pi\)
\(998\) 39531.0 1.25384
\(999\) 31525.1 0.998407
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.1.43 yes 50
43.42 odd 2 1849.4.a.i.1.8 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.8 50 43.42 odd 2
1849.4.a.j.1.43 yes 50 1.1 even 1 trivial