Properties

Label 1849.4.a.j.1.13
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 1.13
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.44476 q^{2} -8.05562 q^{3} -2.02317 q^{4} -13.2954 q^{5} +19.6940 q^{6} +1.50720 q^{7} +24.5042 q^{8} +37.8930 q^{9} +O(q^{10})\) \(q-2.44476 q^{2} -8.05562 q^{3} -2.02317 q^{4} -13.2954 q^{5} +19.6940 q^{6} +1.50720 q^{7} +24.5042 q^{8} +37.8930 q^{9} +32.5040 q^{10} -19.9849 q^{11} +16.2979 q^{12} -49.4213 q^{13} -3.68472 q^{14} +107.103 q^{15} -43.7214 q^{16} -96.4429 q^{17} -92.6392 q^{18} +105.153 q^{19} +26.8989 q^{20} -12.1414 q^{21} +48.8582 q^{22} -171.786 q^{23} -197.397 q^{24} +51.7679 q^{25} +120.823 q^{26} -87.7502 q^{27} -3.04932 q^{28} -218.628 q^{29} -261.840 q^{30} +91.9262 q^{31} -89.1455 q^{32} +160.991 q^{33} +235.779 q^{34} -20.0388 q^{35} -76.6642 q^{36} +104.603 q^{37} -257.073 q^{38} +398.119 q^{39} -325.793 q^{40} -407.432 q^{41} +29.6827 q^{42} +40.4329 q^{44} -503.804 q^{45} +419.975 q^{46} +482.077 q^{47} +352.203 q^{48} -340.728 q^{49} -126.560 q^{50} +776.908 q^{51} +99.9878 q^{52} -698.058 q^{53} +214.528 q^{54} +265.708 q^{55} +36.9326 q^{56} -847.073 q^{57} +534.493 q^{58} +338.239 q^{59} -216.687 q^{60} -330.219 q^{61} -224.737 q^{62} +57.1122 q^{63} +567.710 q^{64} +657.076 q^{65} -393.583 q^{66} +307.046 q^{67} +195.121 q^{68} +1383.84 q^{69} +48.9899 q^{70} -49.0861 q^{71} +928.539 q^{72} +919.255 q^{73} -255.728 q^{74} -417.023 q^{75} -212.743 q^{76} -30.1212 q^{77} -973.304 q^{78} +682.335 q^{79} +581.294 q^{80} -316.229 q^{81} +996.070 q^{82} +411.921 q^{83} +24.5641 q^{84} +1282.25 q^{85} +1761.19 q^{87} -489.714 q^{88} -764.419 q^{89} +1231.68 q^{90} -74.4875 q^{91} +347.553 q^{92} -740.523 q^{93} -1178.56 q^{94} -1398.05 q^{95} +718.123 q^{96} +1512.74 q^{97} +832.997 q^{98} -757.289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44476 −0.864351 −0.432176 0.901789i \(-0.642254\pi\)
−0.432176 + 0.901789i \(0.642254\pi\)
\(3\) −8.05562 −1.55031 −0.775153 0.631774i \(-0.782327\pi\)
−0.775153 + 0.631774i \(0.782327\pi\)
\(4\) −2.02317 −0.252897
\(5\) −13.2954 −1.18918 −0.594589 0.804030i \(-0.702685\pi\)
−0.594589 + 0.804030i \(0.702685\pi\)
\(6\) 19.6940 1.34001
\(7\) 1.50720 0.0813809 0.0406904 0.999172i \(-0.487044\pi\)
0.0406904 + 0.999172i \(0.487044\pi\)
\(8\) 24.5042 1.08294
\(9\) 37.8930 1.40345
\(10\) 32.5040 1.02787
\(11\) −19.9849 −0.547789 −0.273894 0.961760i \(-0.588312\pi\)
−0.273894 + 0.961760i \(0.588312\pi\)
\(12\) 16.2979 0.392067
\(13\) −49.4213 −1.05438 −0.527192 0.849746i \(-0.676755\pi\)
−0.527192 + 0.849746i \(0.676755\pi\)
\(14\) −3.68472 −0.0703417
\(15\) 107.103 1.84359
\(16\) −43.7214 −0.683147
\(17\) −96.4429 −1.37593 −0.687966 0.725743i \(-0.741496\pi\)
−0.687966 + 0.725743i \(0.741496\pi\)
\(18\) −92.6392 −1.21307
\(19\) 105.153 1.26967 0.634836 0.772647i \(-0.281068\pi\)
0.634836 + 0.772647i \(0.281068\pi\)
\(20\) 26.8989 0.300739
\(21\) −12.1414 −0.126165
\(22\) 48.8582 0.473482
\(23\) −171.786 −1.55739 −0.778693 0.627405i \(-0.784117\pi\)
−0.778693 + 0.627405i \(0.784117\pi\)
\(24\) −197.397 −1.67889
\(25\) 51.7679 0.414143
\(26\) 120.823 0.911359
\(27\) −87.7502 −0.625465
\(28\) −3.04932 −0.0205809
\(29\) −218.628 −1.39994 −0.699970 0.714172i \(-0.746803\pi\)
−0.699970 + 0.714172i \(0.746803\pi\)
\(30\) −261.840 −1.59351
\(31\) 91.9262 0.532595 0.266297 0.963891i \(-0.414200\pi\)
0.266297 + 0.963891i \(0.414200\pi\)
\(32\) −89.1455 −0.492464
\(33\) 160.991 0.849240
\(34\) 235.779 1.18929
\(35\) −20.0388 −0.0967763
\(36\) −76.6642 −0.354927
\(37\) 104.603 0.464773 0.232387 0.972623i \(-0.425347\pi\)
0.232387 + 0.972623i \(0.425347\pi\)
\(38\) −257.073 −1.09744
\(39\) 398.119 1.63462
\(40\) −325.793 −1.28781
\(41\) −407.432 −1.55195 −0.775977 0.630761i \(-0.782743\pi\)
−0.775977 + 0.630761i \(0.782743\pi\)
\(42\) 29.6827 0.109051
\(43\) 0 0
\(44\) 40.4329 0.138534
\(45\) −503.804 −1.66895
\(46\) 419.975 1.34613
\(47\) 482.077 1.49613 0.748066 0.663624i \(-0.230983\pi\)
0.748066 + 0.663624i \(0.230983\pi\)
\(48\) 352.203 1.05909
\(49\) −340.728 −0.993377
\(50\) −126.560 −0.357965
\(51\) 776.908 2.13311
\(52\) 99.9878 0.266650
\(53\) −698.058 −1.80916 −0.904581 0.426302i \(-0.859816\pi\)
−0.904581 + 0.426302i \(0.859816\pi\)
\(54\) 214.528 0.540621
\(55\) 265.708 0.651418
\(56\) 36.9326 0.0881309
\(57\) −847.073 −1.96838
\(58\) 534.493 1.21004
\(59\) 338.239 0.746355 0.373177 0.927760i \(-0.378268\pi\)
0.373177 + 0.927760i \(0.378268\pi\)
\(60\) −216.687 −0.466237
\(61\) −330.219 −0.693119 −0.346559 0.938028i \(-0.612650\pi\)
−0.346559 + 0.938028i \(0.612650\pi\)
\(62\) −224.737 −0.460349
\(63\) 57.1122 0.114214
\(64\) 567.710 1.10881
\(65\) 657.076 1.25385
\(66\) −393.583 −0.734042
\(67\) 307.046 0.559875 0.279937 0.960018i \(-0.409686\pi\)
0.279937 + 0.960018i \(0.409686\pi\)
\(68\) 195.121 0.347968
\(69\) 1383.84 2.41442
\(70\) 48.9899 0.0836488
\(71\) −49.0861 −0.0820486 −0.0410243 0.999158i \(-0.513062\pi\)
−0.0410243 + 0.999158i \(0.513062\pi\)
\(72\) 928.539 1.51985
\(73\) 919.255 1.47384 0.736922 0.675978i \(-0.236278\pi\)
0.736922 + 0.675978i \(0.236278\pi\)
\(74\) −255.728 −0.401727
\(75\) −417.023 −0.642048
\(76\) −212.743 −0.321096
\(77\) −30.1212 −0.0445796
\(78\) −973.304 −1.41288
\(79\) 682.335 0.971755 0.485878 0.874027i \(-0.338500\pi\)
0.485878 + 0.874027i \(0.338500\pi\)
\(80\) 581.294 0.812383
\(81\) −316.229 −0.433785
\(82\) 996.070 1.34143
\(83\) 411.921 0.544750 0.272375 0.962191i \(-0.412191\pi\)
0.272375 + 0.962191i \(0.412191\pi\)
\(84\) 24.5641 0.0319067
\(85\) 1282.25 1.63623
\(86\) 0 0
\(87\) 1761.19 2.17033
\(88\) −489.714 −0.593224
\(89\) −764.419 −0.910430 −0.455215 0.890382i \(-0.650438\pi\)
−0.455215 + 0.890382i \(0.650438\pi\)
\(90\) 1231.68 1.44256
\(91\) −74.4875 −0.0858067
\(92\) 347.553 0.393858
\(93\) −740.523 −0.825684
\(94\) −1178.56 −1.29318
\(95\) −1398.05 −1.50987
\(96\) 718.123 0.763470
\(97\) 1512.74 1.58346 0.791729 0.610873i \(-0.209181\pi\)
0.791729 + 0.610873i \(0.209181\pi\)
\(98\) 832.997 0.858627
\(99\) −757.289 −0.768792
\(100\) −104.735 −0.104735
\(101\) 833.158 0.820815 0.410408 0.911902i \(-0.365386\pi\)
0.410408 + 0.911902i \(0.365386\pi\)
\(102\) −1899.35 −1.84376
\(103\) 609.451 0.583019 0.291510 0.956568i \(-0.405842\pi\)
0.291510 + 0.956568i \(0.405842\pi\)
\(104\) −1211.03 −1.14184
\(105\) 161.425 0.150033
\(106\) 1706.58 1.56375
\(107\) 1522.90 1.37593 0.687964 0.725745i \(-0.258505\pi\)
0.687964 + 0.725745i \(0.258505\pi\)
\(108\) 177.534 0.158178
\(109\) 593.526 0.521555 0.260778 0.965399i \(-0.416021\pi\)
0.260778 + 0.965399i \(0.416021\pi\)
\(110\) −649.590 −0.563054
\(111\) −842.641 −0.720540
\(112\) −65.8967 −0.0555951
\(113\) 721.193 0.600391 0.300195 0.953878i \(-0.402948\pi\)
0.300195 + 0.953878i \(0.402948\pi\)
\(114\) 2070.89 1.70137
\(115\) 2283.97 1.85201
\(116\) 442.323 0.354040
\(117\) −1872.72 −1.47977
\(118\) −826.911 −0.645113
\(119\) −145.358 −0.111975
\(120\) 2624.47 1.99650
\(121\) −931.603 −0.699927
\(122\) 807.305 0.599098
\(123\) 3282.11 2.40600
\(124\) −185.983 −0.134691
\(125\) 973.651 0.696688
\(126\) −139.625 −0.0987208
\(127\) −488.453 −0.341285 −0.170643 0.985333i \(-0.554584\pi\)
−0.170643 + 0.985333i \(0.554584\pi\)
\(128\) −674.748 −0.465936
\(129\) 0 0
\(130\) −1606.39 −1.08377
\(131\) −197.970 −0.132036 −0.0660180 0.997818i \(-0.521029\pi\)
−0.0660180 + 0.997818i \(0.521029\pi\)
\(132\) −325.712 −0.214770
\(133\) 158.486 0.103327
\(134\) −750.651 −0.483928
\(135\) 1166.68 0.743788
\(136\) −2363.26 −1.49006
\(137\) 2218.38 1.38342 0.691712 0.722174i \(-0.256857\pi\)
0.691712 + 0.722174i \(0.256857\pi\)
\(138\) −3383.16 −2.08691
\(139\) 1426.45 0.870428 0.435214 0.900327i \(-0.356673\pi\)
0.435214 + 0.900327i \(0.356673\pi\)
\(140\) 40.5419 0.0244744
\(141\) −3883.43 −2.31946
\(142\) 120.004 0.0709189
\(143\) 987.680 0.577580
\(144\) −1656.74 −0.958760
\(145\) 2906.75 1.66478
\(146\) −2247.35 −1.27392
\(147\) 2744.78 1.54004
\(148\) −211.630 −0.117540
\(149\) −324.928 −0.178652 −0.0893259 0.996002i \(-0.528471\pi\)
−0.0893259 + 0.996002i \(0.528471\pi\)
\(150\) 1019.52 0.554955
\(151\) 1010.13 0.544391 0.272196 0.962242i \(-0.412250\pi\)
0.272196 + 0.962242i \(0.412250\pi\)
\(152\) 2576.69 1.37498
\(153\) −3654.52 −1.93105
\(154\) 73.6389 0.0385324
\(155\) −1222.20 −0.633350
\(156\) −805.464 −0.413389
\(157\) −2780.79 −1.41358 −0.706788 0.707425i \(-0.749857\pi\)
−0.706788 + 0.707425i \(0.749857\pi\)
\(158\) −1668.14 −0.839938
\(159\) 5623.29 2.80475
\(160\) 1185.23 0.585627
\(161\) −258.915 −0.126741
\(162\) 773.103 0.374943
\(163\) −261.449 −0.125634 −0.0628168 0.998025i \(-0.520008\pi\)
−0.0628168 + 0.998025i \(0.520008\pi\)
\(164\) 824.304 0.392484
\(165\) −2140.44 −1.00990
\(166\) −1007.05 −0.470855
\(167\) −1042.23 −0.482937 −0.241469 0.970409i \(-0.577629\pi\)
−0.241469 + 0.970409i \(0.577629\pi\)
\(168\) −297.515 −0.136630
\(169\) 245.462 0.111726
\(170\) −3134.78 −1.41428
\(171\) 3984.57 1.78192
\(172\) 0 0
\(173\) −676.020 −0.297092 −0.148546 0.988906i \(-0.547459\pi\)
−0.148546 + 0.988906i \(0.547459\pi\)
\(174\) −4305.67 −1.87593
\(175\) 78.0243 0.0337033
\(176\) 873.768 0.374220
\(177\) −2724.72 −1.15708
\(178\) 1868.82 0.786931
\(179\) −1330.90 −0.555731 −0.277866 0.960620i \(-0.589627\pi\)
−0.277866 + 0.960620i \(0.589627\pi\)
\(180\) 1019.28 0.422071
\(181\) −1960.59 −0.805137 −0.402568 0.915390i \(-0.631882\pi\)
−0.402568 + 0.915390i \(0.631882\pi\)
\(182\) 182.104 0.0741672
\(183\) 2660.12 1.07455
\(184\) −4209.48 −1.68656
\(185\) −1390.74 −0.552698
\(186\) 1810.40 0.713681
\(187\) 1927.40 0.753720
\(188\) −975.326 −0.378367
\(189\) −132.257 −0.0509009
\(190\) 3417.90 1.30505
\(191\) −1864.27 −0.706249 −0.353125 0.935576i \(-0.614881\pi\)
−0.353125 + 0.935576i \(0.614881\pi\)
\(192\) −4573.26 −1.71899
\(193\) −1508.73 −0.562697 −0.281348 0.959606i \(-0.590782\pi\)
−0.281348 + 0.959606i \(0.590782\pi\)
\(194\) −3698.28 −1.36866
\(195\) −5293.16 −1.94385
\(196\) 689.352 0.251222
\(197\) −1352.69 −0.489216 −0.244608 0.969622i \(-0.578659\pi\)
−0.244608 + 0.969622i \(0.578659\pi\)
\(198\) 1851.39 0.664507
\(199\) −1294.26 −0.461045 −0.230522 0.973067i \(-0.574043\pi\)
−0.230522 + 0.973067i \(0.574043\pi\)
\(200\) 1268.53 0.448493
\(201\) −2473.44 −0.867976
\(202\) −2036.87 −0.709473
\(203\) −329.516 −0.113928
\(204\) −1571.82 −0.539457
\(205\) 5416.97 1.84555
\(206\) −1489.96 −0.503933
\(207\) −6509.50 −2.18571
\(208\) 2160.77 0.720299
\(209\) −2101.47 −0.695512
\(210\) −394.644 −0.129681
\(211\) 3425.33 1.11758 0.558790 0.829309i \(-0.311266\pi\)
0.558790 + 0.829309i \(0.311266\pi\)
\(212\) 1412.29 0.457531
\(213\) 395.419 0.127200
\(214\) −3723.12 −1.18929
\(215\) 0 0
\(216\) −2150.25 −0.677342
\(217\) 138.551 0.0433430
\(218\) −1451.03 −0.450807
\(219\) −7405.17 −2.28491
\(220\) −537.572 −0.164741
\(221\) 4766.33 1.45076
\(222\) 2060.05 0.622800
\(223\) −3651.82 −1.09661 −0.548305 0.836278i \(-0.684727\pi\)
−0.548305 + 0.836278i \(0.684727\pi\)
\(224\) −134.360 −0.0400772
\(225\) 1961.64 0.581228
\(226\) −1763.14 −0.518949
\(227\) 4830.81 1.41248 0.706239 0.707974i \(-0.250391\pi\)
0.706239 + 0.707974i \(0.250391\pi\)
\(228\) 1713.78 0.497796
\(229\) −833.344 −0.240476 −0.120238 0.992745i \(-0.538366\pi\)
−0.120238 + 0.992745i \(0.538366\pi\)
\(230\) −5583.74 −1.60079
\(231\) 242.645 0.0691119
\(232\) −5357.31 −1.51605
\(233\) 2172.62 0.610872 0.305436 0.952213i \(-0.401198\pi\)
0.305436 + 0.952213i \(0.401198\pi\)
\(234\) 4578.35 1.27904
\(235\) −6409.42 −1.77917
\(236\) −684.315 −0.188751
\(237\) −5496.63 −1.50652
\(238\) 355.365 0.0967854
\(239\) −1933.97 −0.523423 −0.261712 0.965146i \(-0.584287\pi\)
−0.261712 + 0.965146i \(0.584287\pi\)
\(240\) −4682.68 −1.25944
\(241\) 490.620 0.131135 0.0655677 0.997848i \(-0.479114\pi\)
0.0655677 + 0.997848i \(0.479114\pi\)
\(242\) 2277.54 0.604983
\(243\) 4916.68 1.29796
\(244\) 668.091 0.175287
\(245\) 4530.12 1.18130
\(246\) −8023.97 −2.07963
\(247\) −5196.80 −1.33872
\(248\) 2252.58 0.576770
\(249\) −3318.28 −0.844528
\(250\) −2380.34 −0.602183
\(251\) −4205.04 −1.05745 −0.528725 0.848793i \(-0.677330\pi\)
−0.528725 + 0.848793i \(0.677330\pi\)
\(252\) −115.548 −0.0288843
\(253\) 3433.13 0.853119
\(254\) 1194.15 0.294990
\(255\) −10329.3 −2.53665
\(256\) −2892.09 −0.706076
\(257\) 4274.70 1.03754 0.518772 0.854913i \(-0.326389\pi\)
0.518772 + 0.854913i \(0.326389\pi\)
\(258\) 0 0
\(259\) 157.657 0.0378237
\(260\) −1329.38 −0.317094
\(261\) −8284.49 −1.96474
\(262\) 483.988 0.114125
\(263\) 7460.19 1.74911 0.874553 0.484931i \(-0.161155\pi\)
0.874553 + 0.484931i \(0.161155\pi\)
\(264\) 3944.95 0.919679
\(265\) 9280.96 2.15141
\(266\) −387.460 −0.0893109
\(267\) 6157.87 1.41144
\(268\) −621.206 −0.141590
\(269\) 2275.62 0.515789 0.257894 0.966173i \(-0.416971\pi\)
0.257894 + 0.966173i \(0.416971\pi\)
\(270\) −2852.24 −0.642895
\(271\) 6143.57 1.37710 0.688552 0.725187i \(-0.258247\pi\)
0.688552 + 0.725187i \(0.258247\pi\)
\(272\) 4216.62 0.939963
\(273\) 600.043 0.133027
\(274\) −5423.40 −1.19576
\(275\) −1034.58 −0.226863
\(276\) −2799.75 −0.610599
\(277\) 8485.97 1.84070 0.920348 0.391101i \(-0.127906\pi\)
0.920348 + 0.391101i \(0.127906\pi\)
\(278\) −3487.31 −0.752356
\(279\) 3483.36 0.747468
\(280\) −491.034 −0.104803
\(281\) −5614.23 −1.19187 −0.595937 0.803031i \(-0.703219\pi\)
−0.595937 + 0.803031i \(0.703219\pi\)
\(282\) 9494.04 2.00483
\(283\) −3413.12 −0.716921 −0.358461 0.933545i \(-0.616698\pi\)
−0.358461 + 0.933545i \(0.616698\pi\)
\(284\) 99.3097 0.0207498
\(285\) 11262.2 2.34075
\(286\) −2414.64 −0.499232
\(287\) −614.079 −0.126299
\(288\) −3378.00 −0.691147
\(289\) 4388.23 0.893188
\(290\) −7106.30 −1.43895
\(291\) −12186.1 −2.45484
\(292\) −1859.81 −0.372730
\(293\) 9178.45 1.83007 0.915036 0.403373i \(-0.132162\pi\)
0.915036 + 0.403373i \(0.132162\pi\)
\(294\) −6710.31 −1.33113
\(295\) −4497.02 −0.887548
\(296\) 2563.21 0.503323
\(297\) 1753.68 0.342623
\(298\) 794.369 0.154418
\(299\) 8489.89 1.64208
\(300\) 843.709 0.162372
\(301\) 0 0
\(302\) −2469.52 −0.470545
\(303\) −6711.61 −1.27251
\(304\) −4597.44 −0.867372
\(305\) 4390.40 0.824241
\(306\) 8934.40 1.66910
\(307\) −7166.19 −1.33223 −0.666117 0.745847i \(-0.732045\pi\)
−0.666117 + 0.745847i \(0.732045\pi\)
\(308\) 60.9403 0.0112740
\(309\) −4909.50 −0.903857
\(310\) 2987.97 0.547437
\(311\) −6648.99 −1.21231 −0.606157 0.795345i \(-0.707290\pi\)
−0.606157 + 0.795345i \(0.707290\pi\)
\(312\) 9755.59 1.77020
\(313\) −2436.83 −0.440057 −0.220029 0.975493i \(-0.570615\pi\)
−0.220029 + 0.975493i \(0.570615\pi\)
\(314\) 6798.36 1.22183
\(315\) −759.330 −0.135820
\(316\) −1380.48 −0.245754
\(317\) 2411.80 0.427318 0.213659 0.976908i \(-0.431462\pi\)
0.213659 + 0.976908i \(0.431462\pi\)
\(318\) −13747.6 −2.42429
\(319\) 4369.27 0.766872
\(320\) −7547.94 −1.31857
\(321\) −12267.9 −2.13311
\(322\) 632.984 0.109549
\(323\) −10141.3 −1.74698
\(324\) 639.787 0.109703
\(325\) −2558.44 −0.436666
\(326\) 639.179 0.108592
\(327\) −4781.22 −0.808570
\(328\) −9983.78 −1.68068
\(329\) 726.585 0.121757
\(330\) 5232.85 0.872906
\(331\) 11124.7 1.84734 0.923669 0.383190i \(-0.125175\pi\)
0.923669 + 0.383190i \(0.125175\pi\)
\(332\) −833.387 −0.137765
\(333\) 3963.72 0.652284
\(334\) 2548.01 0.417428
\(335\) −4082.30 −0.665790
\(336\) 530.839 0.0861894
\(337\) 718.615 0.116159 0.0580793 0.998312i \(-0.481502\pi\)
0.0580793 + 0.998312i \(0.481502\pi\)
\(338\) −600.095 −0.0965706
\(339\) −5809.66 −0.930789
\(340\) −2594.21 −0.413796
\(341\) −1837.14 −0.291749
\(342\) −9741.30 −1.54020
\(343\) −1030.51 −0.162223
\(344\) 0 0
\(345\) −18398.8 −2.87118
\(346\) 1652.70 0.256792
\(347\) 3904.37 0.604028 0.302014 0.953303i \(-0.402341\pi\)
0.302014 + 0.953303i \(0.402341\pi\)
\(348\) −3563.18 −0.548870
\(349\) 6023.61 0.923887 0.461943 0.886909i \(-0.347152\pi\)
0.461943 + 0.886909i \(0.347152\pi\)
\(350\) −190.750 −0.0291315
\(351\) 4336.73 0.659480
\(352\) 1781.57 0.269766
\(353\) −3398.18 −0.512371 −0.256186 0.966628i \(-0.582466\pi\)
−0.256186 + 0.966628i \(0.582466\pi\)
\(354\) 6661.28 1.00012
\(355\) 652.620 0.0975704
\(356\) 1546.55 0.230245
\(357\) 1170.95 0.173595
\(358\) 3253.72 0.480347
\(359\) 10550.4 1.55106 0.775531 0.631310i \(-0.217482\pi\)
0.775531 + 0.631310i \(0.217482\pi\)
\(360\) −12345.3 −1.80737
\(361\) 4198.17 0.612067
\(362\) 4793.17 0.695921
\(363\) 7504.64 1.08510
\(364\) 150.701 0.0217002
\(365\) −12221.9 −1.75266
\(366\) −6503.35 −0.928785
\(367\) 7171.06 1.01996 0.509981 0.860186i \(-0.329652\pi\)
0.509981 + 0.860186i \(0.329652\pi\)
\(368\) 7510.73 1.06392
\(369\) −15438.8 −2.17808
\(370\) 3400.01 0.477725
\(371\) −1052.11 −0.147231
\(372\) 1498.21 0.208813
\(373\) 373.809 0.0518903 0.0259452 0.999663i \(-0.491740\pi\)
0.0259452 + 0.999663i \(0.491740\pi\)
\(374\) −4712.03 −0.651479
\(375\) −7843.36 −1.08008
\(376\) 11812.9 1.62023
\(377\) 10804.9 1.47607
\(378\) 323.335 0.0439962
\(379\) −2707.62 −0.366969 −0.183484 0.983023i \(-0.558738\pi\)
−0.183484 + 0.983023i \(0.558738\pi\)
\(380\) 2828.50 0.381840
\(381\) 3934.80 0.529096
\(382\) 4557.68 0.610448
\(383\) −4187.25 −0.558638 −0.279319 0.960198i \(-0.590109\pi\)
−0.279319 + 0.960198i \(0.590109\pi\)
\(384\) 5435.52 0.722344
\(385\) 400.473 0.0530130
\(386\) 3688.47 0.486368
\(387\) 0 0
\(388\) −3060.53 −0.400451
\(389\) −3278.56 −0.427326 −0.213663 0.976907i \(-0.568539\pi\)
−0.213663 + 0.976907i \(0.568539\pi\)
\(390\) 12940.5 1.68017
\(391\) 16567.6 2.14286
\(392\) −8349.28 −1.07577
\(393\) 1594.77 0.204696
\(394\) 3307.01 0.422854
\(395\) −9071.92 −1.15559
\(396\) 1532.13 0.194425
\(397\) −10583.0 −1.33790 −0.668951 0.743306i \(-0.733257\pi\)
−0.668951 + 0.743306i \(0.733257\pi\)
\(398\) 3164.16 0.398505
\(399\) −1276.70 −0.160188
\(400\) −2263.36 −0.282921
\(401\) 6969.95 0.867986 0.433993 0.900916i \(-0.357104\pi\)
0.433993 + 0.900916i \(0.357104\pi\)
\(402\) 6046.96 0.750237
\(403\) −4543.11 −0.561559
\(404\) −1685.62 −0.207581
\(405\) 4204.40 0.515848
\(406\) 805.585 0.0984741
\(407\) −2090.48 −0.254598
\(408\) 19037.5 2.31004
\(409\) 7908.92 0.956163 0.478082 0.878315i \(-0.341332\pi\)
0.478082 + 0.878315i \(0.341332\pi\)
\(410\) −13243.2 −1.59520
\(411\) −17870.4 −2.14473
\(412\) −1233.02 −0.147444
\(413\) 509.792 0.0607390
\(414\) 15914.1 1.88922
\(415\) −5476.66 −0.647804
\(416\) 4405.69 0.519246
\(417\) −11490.9 −1.34943
\(418\) 5137.59 0.601167
\(419\) 9047.57 1.05490 0.527449 0.849587i \(-0.323148\pi\)
0.527449 + 0.849587i \(0.323148\pi\)
\(420\) −326.590 −0.0379428
\(421\) −4816.23 −0.557551 −0.278775 0.960356i \(-0.589928\pi\)
−0.278775 + 0.960356i \(0.589928\pi\)
\(422\) −8374.09 −0.965982
\(423\) 18267.4 2.09974
\(424\) −17105.3 −1.95922
\(425\) −4992.65 −0.569833
\(426\) −966.704 −0.109946
\(427\) −497.705 −0.0564066
\(428\) −3081.09 −0.347968
\(429\) −7956.38 −0.895425
\(430\) 0 0
\(431\) −3817.94 −0.426691 −0.213346 0.976977i \(-0.568436\pi\)
−0.213346 + 0.976977i \(0.568436\pi\)
\(432\) 3836.56 0.427284
\(433\) −10734.2 −1.19134 −0.595672 0.803228i \(-0.703114\pi\)
−0.595672 + 0.803228i \(0.703114\pi\)
\(434\) −338.723 −0.0374636
\(435\) −23415.7 −2.58091
\(436\) −1200.81 −0.131900
\(437\) −18063.8 −1.97737
\(438\) 18103.8 1.97496
\(439\) −10079.6 −1.09584 −0.547920 0.836531i \(-0.684580\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(440\) 6510.95 0.705449
\(441\) −12911.2 −1.39415
\(442\) −11652.5 −1.25397
\(443\) −3418.64 −0.366646 −0.183323 0.983053i \(-0.558685\pi\)
−0.183323 + 0.983053i \(0.558685\pi\)
\(444\) 1704.81 0.182222
\(445\) 10163.3 1.08266
\(446\) 8927.81 0.947856
\(447\) 2617.50 0.276965
\(448\) 855.650 0.0902358
\(449\) −3683.37 −0.387147 −0.193573 0.981086i \(-0.562008\pi\)
−0.193573 + 0.981086i \(0.562008\pi\)
\(450\) −4795.74 −0.502385
\(451\) 8142.48 0.850143
\(452\) −1459.10 −0.151837
\(453\) −8137.21 −0.843972
\(454\) −11810.2 −1.22088
\(455\) 990.342 0.102039
\(456\) −20756.9 −2.13164
\(457\) 3799.32 0.388894 0.194447 0.980913i \(-0.437709\pi\)
0.194447 + 0.980913i \(0.437709\pi\)
\(458\) 2037.32 0.207855
\(459\) 8462.89 0.860597
\(460\) −4620.86 −0.468367
\(461\) −4025.57 −0.406701 −0.203351 0.979106i \(-0.565183\pi\)
−0.203351 + 0.979106i \(0.565183\pi\)
\(462\) −593.207 −0.0597370
\(463\) −4497.86 −0.451476 −0.225738 0.974188i \(-0.572479\pi\)
−0.225738 + 0.974188i \(0.572479\pi\)
\(464\) 9558.73 0.956364
\(465\) 9845.55 0.981885
\(466\) −5311.52 −0.528008
\(467\) 6088.74 0.603326 0.301663 0.953415i \(-0.402458\pi\)
0.301663 + 0.953415i \(0.402458\pi\)
\(468\) 3788.84 0.374229
\(469\) 462.778 0.0455631
\(470\) 15669.5 1.53783
\(471\) 22401.0 2.19148
\(472\) 8288.27 0.808260
\(473\) 0 0
\(474\) 13437.9 1.30216
\(475\) 5443.55 0.525826
\(476\) 294.085 0.0283180
\(477\) −26451.5 −2.53906
\(478\) 4728.09 0.452422
\(479\) −18228.9 −1.73883 −0.869414 0.494084i \(-0.835503\pi\)
−0.869414 + 0.494084i \(0.835503\pi\)
\(480\) −9547.73 −0.907901
\(481\) −5169.61 −0.490050
\(482\) −1199.45 −0.113347
\(483\) 2085.72 0.196488
\(484\) 1884.79 0.177009
\(485\) −20112.5 −1.88301
\(486\) −12020.1 −1.12190
\(487\) −507.709 −0.0472412 −0.0236206 0.999721i \(-0.507519\pi\)
−0.0236206 + 0.999721i \(0.507519\pi\)
\(488\) −8091.76 −0.750608
\(489\) 2106.13 0.194770
\(490\) −11075.0 −1.02106
\(491\) −909.154 −0.0835632 −0.0417816 0.999127i \(-0.513303\pi\)
−0.0417816 + 0.999127i \(0.513303\pi\)
\(492\) −6640.28 −0.608470
\(493\) 21085.1 1.92622
\(494\) 12704.9 1.15713
\(495\) 10068.5 0.914231
\(496\) −4019.14 −0.363840
\(497\) −73.9824 −0.00667719
\(498\) 8112.38 0.729969
\(499\) 18290.1 1.64084 0.820420 0.571762i \(-0.193740\pi\)
0.820420 + 0.571762i \(0.193740\pi\)
\(500\) −1969.86 −0.176190
\(501\) 8395.85 0.748700
\(502\) 10280.3 0.914008
\(503\) −3307.69 −0.293206 −0.146603 0.989195i \(-0.546834\pi\)
−0.146603 + 0.989195i \(0.546834\pi\)
\(504\) 1399.49 0.123687
\(505\) −11077.2 −0.976095
\(506\) −8393.16 −0.737395
\(507\) −1977.35 −0.173210
\(508\) 988.225 0.0863099
\(509\) −3239.45 −0.282094 −0.141047 0.990003i \(-0.545047\pi\)
−0.141047 + 0.990003i \(0.545047\pi\)
\(510\) 25252.6 2.19256
\(511\) 1385.50 0.119943
\(512\) 12468.4 1.07623
\(513\) −9227.21 −0.794135
\(514\) −10450.6 −0.896802
\(515\) −8102.90 −0.693313
\(516\) 0 0
\(517\) −9634.28 −0.819565
\(518\) −385.433 −0.0326929
\(519\) 5445.76 0.460583
\(520\) 16101.1 1.35785
\(521\) −12768.2 −1.07367 −0.536837 0.843686i \(-0.680381\pi\)
−0.536837 + 0.843686i \(0.680381\pi\)
\(522\) 20253.6 1.69823
\(523\) −5646.14 −0.472062 −0.236031 0.971746i \(-0.575847\pi\)
−0.236031 + 0.971746i \(0.575847\pi\)
\(524\) 400.527 0.0333914
\(525\) −628.535 −0.0522505
\(526\) −18238.3 −1.51184
\(527\) −8865.63 −0.732814
\(528\) −7038.75 −0.580156
\(529\) 17343.5 1.42545
\(530\) −22689.7 −1.85958
\(531\) 12816.9 1.04747
\(532\) −320.645 −0.0261311
\(533\) 20135.8 1.63636
\(534\) −15054.5 −1.21998
\(535\) −20247.6 −1.63622
\(536\) 7523.91 0.606312
\(537\) 10721.2 0.861553
\(538\) −5563.34 −0.445823
\(539\) 6809.43 0.544161
\(540\) −2360.39 −0.188102
\(541\) 8682.75 0.690020 0.345010 0.938599i \(-0.387876\pi\)
0.345010 + 0.938599i \(0.387876\pi\)
\(542\) −15019.5 −1.19030
\(543\) 15793.8 1.24821
\(544\) 8597.45 0.677597
\(545\) −7891.18 −0.620222
\(546\) −1466.96 −0.114982
\(547\) 18179.6 1.42103 0.710516 0.703681i \(-0.248461\pi\)
0.710516 + 0.703681i \(0.248461\pi\)
\(548\) −4488.17 −0.349863
\(549\) −12513.0 −0.972755
\(550\) 2529.29 0.196089
\(551\) −22989.4 −1.77746
\(552\) 33910.0 2.61468
\(553\) 1028.41 0.0790823
\(554\) −20746.1 −1.59101
\(555\) 11203.3 0.856850
\(556\) −2885.95 −0.220128
\(557\) 21995.2 1.67319 0.836596 0.547821i \(-0.184542\pi\)
0.836596 + 0.547821i \(0.184542\pi\)
\(558\) −8515.97 −0.646075
\(559\) 0 0
\(560\) 876.123 0.0661124
\(561\) −15526.4 −1.16850
\(562\) 13725.4 1.03020
\(563\) 5894.15 0.441224 0.220612 0.975362i \(-0.429195\pi\)
0.220612 + 0.975362i \(0.429195\pi\)
\(564\) 7856.86 0.586584
\(565\) −9588.56 −0.713971
\(566\) 8344.24 0.619672
\(567\) −476.619 −0.0353018
\(568\) −1202.82 −0.0888540
\(569\) −23140.1 −1.70489 −0.852446 0.522815i \(-0.824882\pi\)
−0.852446 + 0.522815i \(0.824882\pi\)
\(570\) −27533.3 −2.02323
\(571\) 16717.7 1.22524 0.612621 0.790377i \(-0.290115\pi\)
0.612621 + 0.790377i \(0.290115\pi\)
\(572\) −1998.25 −0.146068
\(573\) 15017.8 1.09490
\(574\) 1501.27 0.109167
\(575\) −8893.01 −0.644981
\(576\) 21512.3 1.55615
\(577\) −4191.15 −0.302391 −0.151196 0.988504i \(-0.548312\pi\)
−0.151196 + 0.988504i \(0.548312\pi\)
\(578\) −10728.2 −0.772029
\(579\) 12153.7 0.872352
\(580\) −5880.86 −0.421016
\(581\) 620.846 0.0443322
\(582\) 29791.9 2.12185
\(583\) 13950.6 0.991039
\(584\) 22525.6 1.59609
\(585\) 24898.6 1.75971
\(586\) −22439.1 −1.58182
\(587\) 13306.5 0.935636 0.467818 0.883825i \(-0.345040\pi\)
0.467818 + 0.883825i \(0.345040\pi\)
\(588\) −5553.16 −0.389470
\(589\) 9666.32 0.676220
\(590\) 10994.1 0.767154
\(591\) 10896.8 0.758434
\(592\) −4573.38 −0.317508
\(593\) 2964.50 0.205290 0.102645 0.994718i \(-0.467269\pi\)
0.102645 + 0.994718i \(0.467269\pi\)
\(594\) −4287.32 −0.296146
\(595\) 1932.60 0.133158
\(596\) 657.385 0.0451804
\(597\) 10426.1 0.714760
\(598\) −20755.7 −1.41934
\(599\) −93.2120 −0.00635816 −0.00317908 0.999995i \(-0.501012\pi\)
−0.00317908 + 0.999995i \(0.501012\pi\)
\(600\) −10218.8 −0.695302
\(601\) 10239.7 0.694984 0.347492 0.937683i \(-0.387033\pi\)
0.347492 + 0.937683i \(0.387033\pi\)
\(602\) 0 0
\(603\) 11634.9 0.785754
\(604\) −2043.66 −0.137675
\(605\) 12386.0 0.832338
\(606\) 16408.2 1.09990
\(607\) 7367.72 0.492663 0.246331 0.969186i \(-0.420775\pi\)
0.246331 + 0.969186i \(0.420775\pi\)
\(608\) −9373.92 −0.625268
\(609\) 2654.45 0.176624
\(610\) −10733.5 −0.712434
\(611\) −23824.9 −1.57750
\(612\) 7393.72 0.488355
\(613\) 4128.99 0.272053 0.136026 0.990705i \(-0.456567\pi\)
0.136026 + 0.990705i \(0.456567\pi\)
\(614\) 17519.6 1.15152
\(615\) −43637.1 −2.86116
\(616\) −738.095 −0.0482771
\(617\) −16159.7 −1.05440 −0.527199 0.849742i \(-0.676758\pi\)
−0.527199 + 0.849742i \(0.676758\pi\)
\(618\) 12002.5 0.781251
\(619\) −15805.2 −1.02628 −0.513138 0.858306i \(-0.671517\pi\)
−0.513138 + 0.858306i \(0.671517\pi\)
\(620\) 2472.71 0.160172
\(621\) 15074.3 0.974090
\(622\) 16255.2 1.04787
\(623\) −1152.13 −0.0740916
\(624\) −17406.3 −1.11668
\(625\) −19416.1 −1.24263
\(626\) 5957.46 0.380364
\(627\) 16928.7 1.07826
\(628\) 5626.03 0.357489
\(629\) −10088.2 −0.639496
\(630\) 1856.38 0.117397
\(631\) −1609.55 −0.101545 −0.0507726 0.998710i \(-0.516168\pi\)
−0.0507726 + 0.998710i \(0.516168\pi\)
\(632\) 16720.1 1.05236
\(633\) −27593.2 −1.73259
\(634\) −5896.25 −0.369353
\(635\) 6494.19 0.405849
\(636\) −11376.9 −0.709312
\(637\) 16839.2 1.04740
\(638\) −10681.8 −0.662847
\(639\) −1860.02 −0.115151
\(640\) 8971.05 0.554081
\(641\) −8288.64 −0.510736 −0.255368 0.966844i \(-0.582197\pi\)
−0.255368 + 0.966844i \(0.582197\pi\)
\(642\) 29992.0 1.84376
\(643\) −18375.4 −1.12699 −0.563496 0.826119i \(-0.690544\pi\)
−0.563496 + 0.826119i \(0.690544\pi\)
\(644\) 523.830 0.0320525
\(645\) 0 0
\(646\) 24792.9 1.51001
\(647\) −22364.7 −1.35896 −0.679481 0.733693i \(-0.737795\pi\)
−0.679481 + 0.733693i \(0.737795\pi\)
\(648\) −7748.95 −0.469765
\(649\) −6759.67 −0.408845
\(650\) 6254.75 0.377433
\(651\) −1116.11 −0.0671949
\(652\) 528.957 0.0317723
\(653\) −9026.60 −0.540946 −0.270473 0.962728i \(-0.587180\pi\)
−0.270473 + 0.962728i \(0.587180\pi\)
\(654\) 11688.9 0.698888
\(655\) 2632.09 0.157014
\(656\) 17813.5 1.06021
\(657\) 34833.4 2.06846
\(658\) −1776.32 −0.105240
\(659\) −14302.0 −0.845413 −0.422707 0.906267i \(-0.638920\pi\)
−0.422707 + 0.906267i \(0.638920\pi\)
\(660\) 4330.48 0.255400
\(661\) 1869.11 0.109985 0.0549925 0.998487i \(-0.482487\pi\)
0.0549925 + 0.998487i \(0.482487\pi\)
\(662\) −27197.2 −1.59675
\(663\) −38395.8 −2.24912
\(664\) 10093.8 0.589933
\(665\) −2107.14 −0.122874
\(666\) −9690.33 −0.563803
\(667\) 37557.3 2.18025
\(668\) 2108.62 0.122133
\(669\) 29417.7 1.70008
\(670\) 9980.22 0.575477
\(671\) 6599.40 0.379683
\(672\) 1082.35 0.0621318
\(673\) −25729.1 −1.47367 −0.736837 0.676071i \(-0.763681\pi\)
−0.736837 + 0.676071i \(0.763681\pi\)
\(674\) −1756.84 −0.100402
\(675\) −4542.65 −0.259032
\(676\) −496.613 −0.0282552
\(677\) −18088.6 −1.02688 −0.513442 0.858124i \(-0.671630\pi\)
−0.513442 + 0.858124i \(0.671630\pi\)
\(678\) 14203.2 0.804529
\(679\) 2279.99 0.128863
\(680\) 31420.5 1.77194
\(681\) −38915.2 −2.18977
\(682\) 4491.35 0.252174
\(683\) 15040.0 0.842588 0.421294 0.906924i \(-0.361576\pi\)
0.421294 + 0.906924i \(0.361576\pi\)
\(684\) −8061.47 −0.450640
\(685\) −29494.3 −1.64514
\(686\) 2519.35 0.140218
\(687\) 6713.10 0.372811
\(688\) 0 0
\(689\) 34498.9 1.90755
\(690\) 44980.5 2.48171
\(691\) 20032.0 1.10283 0.551413 0.834232i \(-0.314089\pi\)
0.551413 + 0.834232i \(0.314089\pi\)
\(692\) 1367.71 0.0751335
\(693\) −1141.38 −0.0625650
\(694\) −9545.24 −0.522092
\(695\) −18965.2 −1.03509
\(696\) 43156.5 2.35035
\(697\) 39293.9 2.13538
\(698\) −14726.3 −0.798563
\(699\) −17501.8 −0.947037
\(700\) −157.857 −0.00852346
\(701\) 9706.13 0.522961 0.261480 0.965209i \(-0.415789\pi\)
0.261480 + 0.965209i \(0.415789\pi\)
\(702\) −10602.2 −0.570022
\(703\) 10999.3 0.590109
\(704\) −11345.6 −0.607393
\(705\) 51631.8 2.75825
\(706\) 8307.72 0.442869
\(707\) 1255.73 0.0667987
\(708\) 5512.59 0.292621
\(709\) −13899.8 −0.736275 −0.368138 0.929771i \(-0.620004\pi\)
−0.368138 + 0.929771i \(0.620004\pi\)
\(710\) −1595.50 −0.0843351
\(711\) 25855.7 1.36381
\(712\) −18731.5 −0.985944
\(713\) −15791.6 −0.829455
\(714\) −2862.69 −0.150047
\(715\) −13131.6 −0.686845
\(716\) 2692.63 0.140543
\(717\) 15579.3 0.811466
\(718\) −25793.2 −1.34066
\(719\) 23427.1 1.21514 0.607568 0.794268i \(-0.292145\pi\)
0.607568 + 0.794268i \(0.292145\pi\)
\(720\) 22027.0 1.14014
\(721\) 918.561 0.0474466
\(722\) −10263.5 −0.529041
\(723\) −3952.25 −0.203300
\(724\) 3966.62 0.203616
\(725\) −11317.9 −0.579776
\(726\) −18347.0 −0.937908
\(727\) 28409.8 1.44933 0.724663 0.689104i \(-0.241996\pi\)
0.724663 + 0.689104i \(0.241996\pi\)
\(728\) −1825.26 −0.0929238
\(729\) −31068.7 −1.57846
\(730\) 29879.5 1.51492
\(731\) 0 0
\(732\) −5381.89 −0.271749
\(733\) 7464.74 0.376148 0.188074 0.982155i \(-0.439775\pi\)
0.188074 + 0.982155i \(0.439775\pi\)
\(734\) −17531.5 −0.881606
\(735\) −36493.0 −1.83138
\(736\) 15314.0 0.766957
\(737\) −6136.28 −0.306693
\(738\) 37744.1 1.88263
\(739\) 16315.6 0.812150 0.406075 0.913840i \(-0.366897\pi\)
0.406075 + 0.913840i \(0.366897\pi\)
\(740\) 2813.70 0.139775
\(741\) 41863.4 2.07543
\(742\) 2572.15 0.127260
\(743\) 5142.26 0.253905 0.126952 0.991909i \(-0.459480\pi\)
0.126952 + 0.991909i \(0.459480\pi\)
\(744\) −18145.9 −0.894169
\(745\) 4320.05 0.212449
\(746\) −913.871 −0.0448515
\(747\) 15608.9 0.764527
\(748\) −3899.47 −0.190613
\(749\) 2295.31 0.111974
\(750\) 19175.1 0.933568
\(751\) −4595.89 −0.223311 −0.111656 0.993747i \(-0.535615\pi\)
−0.111656 + 0.993747i \(0.535615\pi\)
\(752\) −21077.1 −1.02208
\(753\) 33874.2 1.63937
\(754\) −26415.3 −1.27585
\(755\) −13430.1 −0.647378
\(756\) 267.578 0.0128727
\(757\) −26851.1 −1.28919 −0.644597 0.764523i \(-0.722975\pi\)
−0.644597 + 0.764523i \(0.722975\pi\)
\(758\) 6619.47 0.317190
\(759\) −27656.0 −1.32259
\(760\) −34258.2 −1.63510
\(761\) −19351.7 −0.921813 −0.460906 0.887449i \(-0.652476\pi\)
−0.460906 + 0.887449i \(0.652476\pi\)
\(762\) −9619.61 −0.457325
\(763\) 894.560 0.0424446
\(764\) 3771.73 0.178608
\(765\) 48588.3 2.29636
\(766\) 10236.8 0.482860
\(767\) −16716.2 −0.786945
\(768\) 23297.6 1.09463
\(769\) 9736.85 0.456593 0.228296 0.973592i \(-0.426684\pi\)
0.228296 + 0.973592i \(0.426684\pi\)
\(770\) −979.059 −0.0458219
\(771\) −34435.4 −1.60851
\(772\) 3052.41 0.142304
\(773\) −24615.9 −1.14537 −0.572685 0.819775i \(-0.694098\pi\)
−0.572685 + 0.819775i \(0.694098\pi\)
\(774\) 0 0
\(775\) 4758.83 0.220570
\(776\) 37068.5 1.71479
\(777\) −1270.02 −0.0586382
\(778\) 8015.28 0.369359
\(779\) −42842.7 −1.97047
\(780\) 10709.0 0.491593
\(781\) 980.982 0.0449453
\(782\) −40503.6 −1.85218
\(783\) 19184.7 0.875613
\(784\) 14897.1 0.678622
\(785\) 36971.8 1.68099
\(786\) −3898.82 −0.176929
\(787\) 16844.6 0.762955 0.381477 0.924378i \(-0.375415\pi\)
0.381477 + 0.924378i \(0.375415\pi\)
\(788\) 2736.73 0.123721
\(789\) −60096.4 −2.71165
\(790\) 22178.6 0.998836
\(791\) 1086.98 0.0488603
\(792\) −18556.8 −0.832558
\(793\) 16319.9 0.730814
\(794\) 25872.9 1.15642
\(795\) −74763.9 −3.33535
\(796\) 2618.52 0.116597
\(797\) 31705.8 1.40913 0.704566 0.709638i \(-0.251142\pi\)
0.704566 + 0.709638i \(0.251142\pi\)
\(798\) 3121.23 0.138459
\(799\) −46492.9 −2.05858
\(800\) −4614.88 −0.203951
\(801\) −28966.2 −1.27774
\(802\) −17039.8 −0.750245
\(803\) −18371.2 −0.807356
\(804\) 5004.20 0.219508
\(805\) 3442.38 0.150718
\(806\) 11106.8 0.485385
\(807\) −18331.6 −0.799630
\(808\) 20415.9 0.888896
\(809\) −9917.38 −0.430997 −0.215498 0.976504i \(-0.569138\pi\)
−0.215498 + 0.976504i \(0.569138\pi\)
\(810\) −10278.7 −0.445874
\(811\) −34830.2 −1.50808 −0.754041 0.656828i \(-0.771898\pi\)
−0.754041 + 0.656828i \(0.771898\pi\)
\(812\) 666.667 0.0288121
\(813\) −49490.2 −2.13493
\(814\) 5110.71 0.220062
\(815\) 3476.07 0.149401
\(816\) −33967.5 −1.45723
\(817\) 0 0
\(818\) −19335.4 −0.826461
\(819\) −2822.56 −0.120425
\(820\) −10959.5 −0.466733
\(821\) −11834.0 −0.503056 −0.251528 0.967850i \(-0.580933\pi\)
−0.251528 + 0.967850i \(0.580933\pi\)
\(822\) 43688.8 1.85380
\(823\) 1274.38 0.0539757 0.0269879 0.999636i \(-0.491408\pi\)
0.0269879 + 0.999636i \(0.491408\pi\)
\(824\) 14934.1 0.631376
\(825\) 8334.16 0.351707
\(826\) −1246.32 −0.0524999
\(827\) 28342.9 1.19175 0.595876 0.803076i \(-0.296805\pi\)
0.595876 + 0.803076i \(0.296805\pi\)
\(828\) 13169.8 0.552758
\(829\) 39168.9 1.64100 0.820501 0.571644i \(-0.193694\pi\)
0.820501 + 0.571644i \(0.193694\pi\)
\(830\) 13389.1 0.559930
\(831\) −68359.8 −2.85364
\(832\) −28057.0 −1.16911
\(833\) 32860.8 1.36682
\(834\) 28092.5 1.16638
\(835\) 13856.9 0.574298
\(836\) 4251.65 0.175893
\(837\) −8066.55 −0.333119
\(838\) −22119.1 −0.911803
\(839\) 8161.55 0.335838 0.167919 0.985801i \(-0.446295\pi\)
0.167919 + 0.985801i \(0.446295\pi\)
\(840\) 3955.59 0.162477
\(841\) 23409.3 0.959831
\(842\) 11774.5 0.481920
\(843\) 45226.1 1.84777
\(844\) −6930.03 −0.282632
\(845\) −3263.52 −0.132862
\(846\) −44659.3 −1.81491
\(847\) −1404.11 −0.0569607
\(848\) 30520.1 1.23592
\(849\) 27494.8 1.11145
\(850\) 12205.8 0.492536
\(851\) −17969.3 −0.723831
\(852\) −800.002 −0.0321685
\(853\) −1042.20 −0.0418337 −0.0209169 0.999781i \(-0.506659\pi\)
−0.0209169 + 0.999781i \(0.506659\pi\)
\(854\) 1216.77 0.0487551
\(855\) −52976.5 −2.11901
\(856\) 37317.5 1.49005
\(857\) 42411.7 1.69050 0.845249 0.534373i \(-0.179452\pi\)
0.845249 + 0.534373i \(0.179452\pi\)
\(858\) 19451.4 0.773962
\(859\) −23599.2 −0.937361 −0.468681 0.883368i \(-0.655270\pi\)
−0.468681 + 0.883368i \(0.655270\pi\)
\(860\) 0 0
\(861\) 4946.79 0.195803
\(862\) 9333.94 0.368811
\(863\) −37727.6 −1.48814 −0.744068 0.668104i \(-0.767106\pi\)
−0.744068 + 0.668104i \(0.767106\pi\)
\(864\) 7822.54 0.308019
\(865\) 8987.96 0.353295
\(866\) 26242.4 1.02974
\(867\) −35350.0 −1.38471
\(868\) −280.312 −0.0109613
\(869\) −13636.4 −0.532317
\(870\) 57245.7 2.23082
\(871\) −15174.6 −0.590323
\(872\) 14543.9 0.564815
\(873\) 57322.3 2.22230
\(874\) 44161.6 1.70914
\(875\) 1467.48 0.0566971
\(876\) 14981.9 0.577845
\(877\) −42264.0 −1.62731 −0.813656 0.581346i \(-0.802526\pi\)
−0.813656 + 0.581346i \(0.802526\pi\)
\(878\) 24642.2 0.947191
\(879\) −73938.1 −2.83717
\(880\) −11617.1 −0.445014
\(881\) 11404.2 0.436114 0.218057 0.975936i \(-0.430028\pi\)
0.218057 + 0.975936i \(0.430028\pi\)
\(882\) 31564.8 1.20504
\(883\) −11567.7 −0.440864 −0.220432 0.975402i \(-0.570747\pi\)
−0.220432 + 0.975402i \(0.570747\pi\)
\(884\) −9643.11 −0.366892
\(885\) 36226.3 1.37597
\(886\) 8357.73 0.316911
\(887\) −31451.6 −1.19058 −0.595288 0.803512i \(-0.702962\pi\)
−0.595288 + 0.803512i \(0.702962\pi\)
\(888\) −20648.3 −0.780304
\(889\) −736.194 −0.0277741
\(890\) −24846.7 −0.935801
\(891\) 6319.82 0.237623
\(892\) 7388.26 0.277329
\(893\) 50691.9 1.89960
\(894\) −6399.14 −0.239395
\(895\) 17694.8 0.660863
\(896\) −1016.98 −0.0379183
\(897\) −68391.3 −2.54573
\(898\) 9004.93 0.334631
\(899\) −20097.7 −0.745600
\(900\) −3968.74 −0.146990
\(901\) 67322.7 2.48928
\(902\) −19906.4 −0.734823
\(903\) 0 0
\(904\) 17672.3 0.650189
\(905\) 26066.9 0.957451
\(906\) 19893.5 0.729489
\(907\) 39084.0 1.43083 0.715414 0.698701i \(-0.246238\pi\)
0.715414 + 0.698701i \(0.246238\pi\)
\(908\) −9773.57 −0.357211
\(909\) 31570.9 1.15197
\(910\) −2421.14 −0.0881979
\(911\) −27906.3 −1.01490 −0.507452 0.861680i \(-0.669412\pi\)
−0.507452 + 0.861680i \(0.669412\pi\)
\(912\) 37035.2 1.34469
\(913\) −8232.21 −0.298408
\(914\) −9288.40 −0.336141
\(915\) −35367.4 −1.27783
\(916\) 1686.00 0.0608154
\(917\) −298.379 −0.0107452
\(918\) −20689.7 −0.743858
\(919\) −28865.1 −1.03609 −0.518047 0.855352i \(-0.673341\pi\)
−0.518047 + 0.855352i \(0.673341\pi\)
\(920\) 55966.8 2.00562
\(921\) 57728.1 2.06537
\(922\) 9841.52 0.351533
\(923\) 2425.90 0.0865108
\(924\) −490.912 −0.0174782
\(925\) 5415.07 0.192483
\(926\) 10996.2 0.390234
\(927\) 23093.9 0.818236
\(928\) 19489.7 0.689420
\(929\) −51121.7 −1.80544 −0.902718 0.430234i \(-0.858431\pi\)
−0.902718 + 0.430234i \(0.858431\pi\)
\(930\) −24070.0 −0.848694
\(931\) −35828.6 −1.26126
\(932\) −4395.58 −0.154487
\(933\) 53561.7 1.87946
\(934\) −14885.5 −0.521486
\(935\) −25625.6 −0.896307
\(936\) −45889.6 −1.60251
\(937\) 14721.1 0.513252 0.256626 0.966511i \(-0.417389\pi\)
0.256626 + 0.966511i \(0.417389\pi\)
\(938\) −1131.38 −0.0393825
\(939\) 19630.2 0.682223
\(940\) 12967.4 0.449945
\(941\) 9191.00 0.318404 0.159202 0.987246i \(-0.449108\pi\)
0.159202 + 0.987246i \(0.449108\pi\)
\(942\) −54765.0 −1.89420
\(943\) 69991.1 2.41699
\(944\) −14788.3 −0.509870
\(945\) 1758.41 0.0605302
\(946\) 0 0
\(947\) 29365.9 1.00767 0.503835 0.863800i \(-0.331922\pi\)
0.503835 + 0.863800i \(0.331922\pi\)
\(948\) 11120.6 0.380993
\(949\) −45430.7 −1.55400
\(950\) −13308.2 −0.454498
\(951\) −19428.5 −0.662474
\(952\) −3561.89 −0.121262
\(953\) 38650.9 1.31377 0.656887 0.753989i \(-0.271873\pi\)
0.656887 + 0.753989i \(0.271873\pi\)
\(954\) 64667.5 2.19464
\(955\) 24786.2 0.839856
\(956\) 3912.76 0.132372
\(957\) −35197.2 −1.18888
\(958\) 44565.1 1.50296
\(959\) 3343.53 0.112584
\(960\) 60803.3 2.04419
\(961\) −21340.6 −0.716343
\(962\) 12638.4 0.423575
\(963\) 57707.3 1.93104
\(964\) −992.610 −0.0331637
\(965\) 20059.1 0.669146
\(966\) −5099.08 −0.169835
\(967\) −37964.3 −1.26251 −0.631256 0.775574i \(-0.717460\pi\)
−0.631256 + 0.775574i \(0.717460\pi\)
\(968\) −22828.2 −0.757981
\(969\) 81694.2 2.70836
\(970\) 49170.1 1.62758
\(971\) −1349.75 −0.0446091 −0.0223045 0.999751i \(-0.507100\pi\)
−0.0223045 + 0.999751i \(0.507100\pi\)
\(972\) −9947.29 −0.328251
\(973\) 2149.93 0.0708362
\(974\) 1241.22 0.0408330
\(975\) 20609.8 0.676966
\(976\) 14437.6 0.473502
\(977\) −15711.0 −0.514471 −0.257236 0.966349i \(-0.582812\pi\)
−0.257236 + 0.966349i \(0.582812\pi\)
\(978\) −5148.98 −0.168350
\(979\) 15276.8 0.498723
\(980\) −9165.22 −0.298747
\(981\) 22490.5 0.731975
\(982\) 2222.66 0.0722280
\(983\) 20977.9 0.680663 0.340332 0.940305i \(-0.389461\pi\)
0.340332 + 0.940305i \(0.389461\pi\)
\(984\) 80425.6 2.60556
\(985\) 17984.6 0.581764
\(986\) −51548.0 −1.66493
\(987\) −5853.09 −0.188760
\(988\) 10514.0 0.338558
\(989\) 0 0
\(990\) −24614.9 −0.790216
\(991\) 48127.0 1.54269 0.771345 0.636417i \(-0.219584\pi\)
0.771345 + 0.636417i \(0.219584\pi\)
\(992\) −8194.81 −0.262284
\(993\) −89616.4 −2.86394
\(994\) 180.869 0.00577144
\(995\) 17207.8 0.548264
\(996\) 6713.45 0.213578
\(997\) −4405.84 −0.139954 −0.0699771 0.997549i \(-0.522293\pi\)
−0.0699771 + 0.997549i \(0.522293\pi\)
\(998\) −44714.9 −1.41826
\(999\) −9178.93 −0.290699
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.1.13 yes 50
43.42 odd 2 1849.4.a.i.1.38 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.38 50 43.42 odd 2
1849.4.a.j.1.13 yes 50 1.1 even 1 trivial