Properties

Label 1849.4.a.j.1.12
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 1.12
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.07922 q^{2} -5.77516 q^{3} +1.48163 q^{4} -13.4757 q^{5} +17.7830 q^{6} +34.2458 q^{7} +20.0715 q^{8} +6.35250 q^{9} +O(q^{10})\) \(q-3.07922 q^{2} -5.77516 q^{3} +1.48163 q^{4} -13.4757 q^{5} +17.7830 q^{6} +34.2458 q^{7} +20.0715 q^{8} +6.35250 q^{9} +41.4947 q^{10} +24.2940 q^{11} -8.55663 q^{12} +11.8136 q^{13} -105.451 q^{14} +77.8243 q^{15} -73.6578 q^{16} +110.941 q^{17} -19.5608 q^{18} +45.6852 q^{19} -19.9659 q^{20} -197.775 q^{21} -74.8068 q^{22} -214.470 q^{23} -115.916 q^{24} +56.5941 q^{25} -36.3769 q^{26} +119.243 q^{27} +50.7395 q^{28} -57.4470 q^{29} -239.638 q^{30} -181.469 q^{31} +66.2366 q^{32} -140.302 q^{33} -341.613 q^{34} -461.486 q^{35} +9.41203 q^{36} +156.106 q^{37} -140.675 q^{38} -68.2257 q^{39} -270.478 q^{40} +0.775433 q^{41} +608.995 q^{42} +35.9947 q^{44} -85.6043 q^{45} +660.401 q^{46} -146.038 q^{47} +425.386 q^{48} +829.777 q^{49} -174.266 q^{50} -640.704 q^{51} +17.5034 q^{52} -282.368 q^{53} -367.175 q^{54} -327.379 q^{55} +687.367 q^{56} -263.840 q^{57} +176.892 q^{58} -698.452 q^{59} +115.306 q^{60} +98.7249 q^{61} +558.783 q^{62} +217.547 q^{63} +385.305 q^{64} -159.197 q^{65} +432.022 q^{66} -517.724 q^{67} +164.373 q^{68} +1238.60 q^{69} +1421.02 q^{70} -29.8847 q^{71} +127.505 q^{72} +75.4773 q^{73} -480.686 q^{74} -326.840 q^{75} +67.6884 q^{76} +831.970 q^{77} +210.082 q^{78} +882.904 q^{79} +992.589 q^{80} -860.163 q^{81} -2.38773 q^{82} +716.095 q^{83} -293.029 q^{84} -1495.01 q^{85} +331.766 q^{87} +487.619 q^{88} -142.143 q^{89} +263.595 q^{90} +404.568 q^{91} -317.764 q^{92} +1048.01 q^{93} +449.685 q^{94} -615.640 q^{95} -382.527 q^{96} -352.135 q^{97} -2555.07 q^{98} +154.328 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.07922 −1.08867 −0.544335 0.838868i \(-0.683218\pi\)
−0.544335 + 0.838868i \(0.683218\pi\)
\(3\) −5.77516 −1.11143 −0.555715 0.831373i \(-0.687556\pi\)
−0.555715 + 0.831373i \(0.687556\pi\)
\(4\) 1.48163 0.185203
\(5\) −13.4757 −1.20530 −0.602651 0.798005i \(-0.705889\pi\)
−0.602651 + 0.798005i \(0.705889\pi\)
\(6\) 17.7830 1.20998
\(7\) 34.2458 1.84910 0.924551 0.381059i \(-0.124441\pi\)
0.924551 + 0.381059i \(0.124441\pi\)
\(8\) 20.0715 0.887045
\(9\) 6.35250 0.235278
\(10\) 41.4947 1.31218
\(11\) 24.2940 0.665903 0.332951 0.942944i \(-0.391955\pi\)
0.332951 + 0.942944i \(0.391955\pi\)
\(12\) −8.55663 −0.205840
\(13\) 11.8136 0.252040 0.126020 0.992028i \(-0.459780\pi\)
0.126020 + 0.992028i \(0.459780\pi\)
\(14\) −105.451 −2.01306
\(15\) 77.8243 1.33961
\(16\) −73.6578 −1.15090
\(17\) 110.941 1.58278 0.791389 0.611313i \(-0.209359\pi\)
0.791389 + 0.611313i \(0.209359\pi\)
\(18\) −19.5608 −0.256140
\(19\) 45.6852 0.551627 0.275813 0.961211i \(-0.411053\pi\)
0.275813 + 0.961211i \(0.411053\pi\)
\(20\) −19.9659 −0.223226
\(21\) −197.775 −2.05515
\(22\) −74.8068 −0.724949
\(23\) −214.470 −1.94435 −0.972176 0.234254i \(-0.924735\pi\)
−0.972176 + 0.234254i \(0.924735\pi\)
\(24\) −115.916 −0.985889
\(25\) 56.5941 0.452753
\(26\) −36.3769 −0.274388
\(27\) 119.243 0.849936
\(28\) 50.7395 0.342459
\(29\) −57.4470 −0.367850 −0.183925 0.982940i \(-0.558880\pi\)
−0.183925 + 0.982940i \(0.558880\pi\)
\(30\) −239.638 −1.45839
\(31\) −181.469 −1.05138 −0.525689 0.850677i \(-0.676193\pi\)
−0.525689 + 0.850677i \(0.676193\pi\)
\(32\) 66.2366 0.365909
\(33\) −140.302 −0.740105
\(34\) −341.613 −1.72312
\(35\) −461.486 −2.22873
\(36\) 9.41203 0.0435742
\(37\) 156.106 0.693613 0.346806 0.937937i \(-0.387266\pi\)
0.346806 + 0.937937i \(0.387266\pi\)
\(38\) −140.675 −0.600540
\(39\) −68.2257 −0.280124
\(40\) −270.478 −1.06916
\(41\) 0.775433 0.00295371 0.00147686 0.999999i \(-0.499530\pi\)
0.00147686 + 0.999999i \(0.499530\pi\)
\(42\) 608.995 2.23738
\(43\) 0 0
\(44\) 35.9947 0.123327
\(45\) −85.6043 −0.283581
\(46\) 660.401 2.11676
\(47\) −146.038 −0.453232 −0.226616 0.973984i \(-0.572766\pi\)
−0.226616 + 0.973984i \(0.572766\pi\)
\(48\) 425.386 1.27915
\(49\) 829.777 2.41918
\(50\) −174.266 −0.492898
\(51\) −640.704 −1.75915
\(52\) 17.5034 0.0466785
\(53\) −282.368 −0.731815 −0.365908 0.930651i \(-0.619241\pi\)
−0.365908 + 0.930651i \(0.619241\pi\)
\(54\) −367.175 −0.925300
\(55\) −327.379 −0.802614
\(56\) 687.367 1.64024
\(57\) −263.840 −0.613095
\(58\) 176.892 0.400467
\(59\) −698.452 −1.54120 −0.770599 0.637320i \(-0.780043\pi\)
−0.770599 + 0.637320i \(0.780043\pi\)
\(60\) 115.306 0.248100
\(61\) 98.7249 0.207220 0.103610 0.994618i \(-0.466961\pi\)
0.103610 + 0.994618i \(0.466961\pi\)
\(62\) 558.783 1.14461
\(63\) 217.547 0.435053
\(64\) 385.305 0.752549
\(65\) −159.197 −0.303784
\(66\) 432.022 0.805730
\(67\) −517.724 −0.944031 −0.472015 0.881590i \(-0.656473\pi\)
−0.472015 + 0.881590i \(0.656473\pi\)
\(68\) 164.373 0.293135
\(69\) 1238.60 2.16101
\(70\) 1421.02 2.42635
\(71\) −29.8847 −0.0499529 −0.0249765 0.999688i \(-0.507951\pi\)
−0.0249765 + 0.999688i \(0.507951\pi\)
\(72\) 127.505 0.208702
\(73\) 75.4773 0.121013 0.0605065 0.998168i \(-0.480728\pi\)
0.0605065 + 0.998168i \(0.480728\pi\)
\(74\) −480.686 −0.755116
\(75\) −326.840 −0.503203
\(76\) 67.6884 0.102163
\(77\) 831.970 1.23132
\(78\) 210.082 0.304963
\(79\) 882.904 1.25740 0.628699 0.777649i \(-0.283588\pi\)
0.628699 + 0.777649i \(0.283588\pi\)
\(80\) 992.589 1.38719
\(81\) −860.163 −1.17992
\(82\) −2.38773 −0.00321562
\(83\) 716.095 0.947008 0.473504 0.880792i \(-0.342989\pi\)
0.473504 + 0.880792i \(0.342989\pi\)
\(84\) −293.029 −0.380620
\(85\) −1495.01 −1.90772
\(86\) 0 0
\(87\) 331.766 0.408839
\(88\) 487.619 0.590686
\(89\) −142.143 −0.169293 −0.0846466 0.996411i \(-0.526976\pi\)
−0.0846466 + 0.996411i \(0.526976\pi\)
\(90\) 263.595 0.308726
\(91\) 404.568 0.466047
\(92\) −317.764 −0.360100
\(93\) 1048.01 1.16853
\(94\) 449.685 0.493420
\(95\) −615.640 −0.664877
\(96\) −382.527 −0.406682
\(97\) −352.135 −0.368597 −0.184299 0.982870i \(-0.559001\pi\)
−0.184299 + 0.982870i \(0.559001\pi\)
\(98\) −2555.07 −2.63369
\(99\) 154.328 0.156672
\(100\) 83.8512 0.0838512
\(101\) 1101.97 1.08564 0.542820 0.839849i \(-0.317357\pi\)
0.542820 + 0.839849i \(0.317357\pi\)
\(102\) 1972.87 1.91513
\(103\) −909.666 −0.870214 −0.435107 0.900379i \(-0.643290\pi\)
−0.435107 + 0.900379i \(0.643290\pi\)
\(104\) 237.118 0.223570
\(105\) 2665.16 2.47707
\(106\) 869.474 0.796706
\(107\) −1868.64 −1.68830 −0.844150 0.536107i \(-0.819894\pi\)
−0.844150 + 0.536107i \(0.819894\pi\)
\(108\) 176.673 0.157411
\(109\) 1049.08 0.921869 0.460935 0.887434i \(-0.347514\pi\)
0.460935 + 0.887434i \(0.347514\pi\)
\(110\) 1008.07 0.873782
\(111\) −901.538 −0.770903
\(112\) −2522.47 −2.12814
\(113\) 739.032 0.615242 0.307621 0.951509i \(-0.400467\pi\)
0.307621 + 0.951509i \(0.400467\pi\)
\(114\) 812.421 0.667458
\(115\) 2890.13 2.34353
\(116\) −85.1149 −0.0681269
\(117\) 75.0462 0.0592993
\(118\) 2150.69 1.67786
\(119\) 3799.28 2.92672
\(120\) 1562.05 1.18829
\(121\) −740.799 −0.556573
\(122\) −303.996 −0.225594
\(123\) −4.47825 −0.00328285
\(124\) −268.869 −0.194719
\(125\) 921.817 0.659598
\(126\) −669.876 −0.473629
\(127\) 1516.44 1.05955 0.529773 0.848140i \(-0.322277\pi\)
0.529773 + 0.848140i \(0.322277\pi\)
\(128\) −1716.33 −1.18519
\(129\) 0 0
\(130\) 490.203 0.330720
\(131\) 1383.75 0.922893 0.461446 0.887168i \(-0.347331\pi\)
0.461446 + 0.887168i \(0.347331\pi\)
\(132\) −207.875 −0.137070
\(133\) 1564.53 1.02001
\(134\) 1594.19 1.02774
\(135\) −1606.88 −1.02443
\(136\) 2226.76 1.40399
\(137\) −1180.51 −0.736190 −0.368095 0.929788i \(-0.619990\pi\)
−0.368095 + 0.929788i \(0.619990\pi\)
\(138\) −3813.92 −2.35263
\(139\) −2967.86 −1.81101 −0.905505 0.424336i \(-0.860507\pi\)
−0.905505 + 0.424336i \(0.860507\pi\)
\(140\) −683.749 −0.412767
\(141\) 843.396 0.503736
\(142\) 92.0216 0.0543823
\(143\) 287.001 0.167834
\(144\) −467.911 −0.270782
\(145\) 774.138 0.443370
\(146\) −232.412 −0.131743
\(147\) −4792.10 −2.68875
\(148\) 231.291 0.128459
\(149\) 568.440 0.312540 0.156270 0.987714i \(-0.450053\pi\)
0.156270 + 0.987714i \(0.450053\pi\)
\(150\) 1006.41 0.547822
\(151\) 1740.05 0.937772 0.468886 0.883259i \(-0.344656\pi\)
0.468886 + 0.883259i \(0.344656\pi\)
\(152\) 916.973 0.489318
\(153\) 704.755 0.372393
\(154\) −2561.82 −1.34050
\(155\) 2445.42 1.26723
\(156\) −101.085 −0.0518799
\(157\) −1742.13 −0.885587 −0.442793 0.896624i \(-0.646012\pi\)
−0.442793 + 0.896624i \(0.646012\pi\)
\(158\) −2718.66 −1.36889
\(159\) 1630.72 0.813362
\(160\) −892.583 −0.441030
\(161\) −7344.70 −3.59530
\(162\) 2648.64 1.28455
\(163\) −1870.73 −0.898939 −0.449469 0.893296i \(-0.648387\pi\)
−0.449469 + 0.893296i \(0.648387\pi\)
\(164\) 1.14890 0.000547037 0
\(165\) 1890.67 0.892050
\(166\) −2205.02 −1.03098
\(167\) 295.044 0.136714 0.0683568 0.997661i \(-0.478224\pi\)
0.0683568 + 0.997661i \(0.478224\pi\)
\(168\) −3969.65 −1.82301
\(169\) −2057.44 −0.936476
\(170\) 4603.47 2.07688
\(171\) 290.216 0.129786
\(172\) 0 0
\(173\) −1615.88 −0.710132 −0.355066 0.934841i \(-0.615542\pi\)
−0.355066 + 0.934841i \(0.615542\pi\)
\(174\) −1021.58 −0.445091
\(175\) 1938.11 0.837186
\(176\) −1789.45 −0.766390
\(177\) 4033.67 1.71294
\(178\) 437.689 0.184305
\(179\) 2117.02 0.883987 0.441993 0.897018i \(-0.354271\pi\)
0.441993 + 0.897018i \(0.354271\pi\)
\(180\) −126.834 −0.0525201
\(181\) 297.604 0.122214 0.0611070 0.998131i \(-0.480537\pi\)
0.0611070 + 0.998131i \(0.480537\pi\)
\(182\) −1245.76 −0.507371
\(183\) −570.152 −0.230311
\(184\) −4304.74 −1.72473
\(185\) −2103.64 −0.836013
\(186\) −3227.06 −1.27215
\(187\) 2695.21 1.05398
\(188\) −216.374 −0.0839400
\(189\) 4083.56 1.57162
\(190\) 1895.69 0.723832
\(191\) −4542.23 −1.72075 −0.860377 0.509659i \(-0.829772\pi\)
−0.860377 + 0.509659i \(0.829772\pi\)
\(192\) −2225.20 −0.836406
\(193\) 3262.95 1.21696 0.608478 0.793571i \(-0.291781\pi\)
0.608478 + 0.793571i \(0.291781\pi\)
\(194\) 1084.30 0.401281
\(195\) 919.388 0.337635
\(196\) 1229.42 0.448039
\(197\) −868.209 −0.313997 −0.156998 0.987599i \(-0.550182\pi\)
−0.156998 + 0.987599i \(0.550182\pi\)
\(198\) −475.211 −0.170564
\(199\) 4215.53 1.50166 0.750832 0.660493i \(-0.229653\pi\)
0.750832 + 0.660493i \(0.229653\pi\)
\(200\) 1135.93 0.401612
\(201\) 2989.94 1.04922
\(202\) −3393.20 −1.18190
\(203\) −1967.32 −0.680191
\(204\) −949.283 −0.325800
\(205\) −10.4495 −0.00356012
\(206\) 2801.07 0.947377
\(207\) −1362.42 −0.457463
\(208\) −870.167 −0.290073
\(209\) 1109.88 0.367330
\(210\) −8206.62 −2.69672
\(211\) −2905.47 −0.947964 −0.473982 0.880534i \(-0.657184\pi\)
−0.473982 + 0.880534i \(0.657184\pi\)
\(212\) −418.363 −0.135535
\(213\) 172.589 0.0555192
\(214\) 5753.96 1.83800
\(215\) 0 0
\(216\) 2393.38 0.753931
\(217\) −6214.55 −1.94411
\(218\) −3230.36 −1.00361
\(219\) −435.894 −0.134498
\(220\) −485.053 −0.148647
\(221\) 1310.62 0.398922
\(222\) 2776.04 0.839259
\(223\) 2146.80 0.644667 0.322333 0.946626i \(-0.395533\pi\)
0.322333 + 0.946626i \(0.395533\pi\)
\(224\) 2268.33 0.676602
\(225\) 359.514 0.106523
\(226\) −2275.65 −0.669795
\(227\) 873.162 0.255303 0.127652 0.991819i \(-0.459256\pi\)
0.127652 + 0.991819i \(0.459256\pi\)
\(228\) −390.911 −0.113547
\(229\) 3489.37 1.00692 0.503459 0.864019i \(-0.332061\pi\)
0.503459 + 0.864019i \(0.332061\pi\)
\(230\) −8899.36 −2.55133
\(231\) −4804.76 −1.36853
\(232\) −1153.05 −0.326299
\(233\) −5132.13 −1.44299 −0.721496 0.692418i \(-0.756545\pi\)
−0.721496 + 0.692418i \(0.756545\pi\)
\(234\) −231.084 −0.0645574
\(235\) 1967.97 0.546281
\(236\) −1034.84 −0.285435
\(237\) −5098.91 −1.39751
\(238\) −11698.8 −3.18623
\(239\) 3497.81 0.946672 0.473336 0.880882i \(-0.343050\pi\)
0.473336 + 0.880882i \(0.343050\pi\)
\(240\) −5732.36 −1.54176
\(241\) −5168.03 −1.38134 −0.690669 0.723171i \(-0.742684\pi\)
−0.690669 + 0.723171i \(0.742684\pi\)
\(242\) 2281.09 0.605925
\(243\) 1748.03 0.461466
\(244\) 146.273 0.0383778
\(245\) −11181.8 −2.91584
\(246\) 13.7895 0.00357394
\(247\) 539.709 0.139032
\(248\) −3642.36 −0.932621
\(249\) −4135.57 −1.05253
\(250\) −2838.48 −0.718085
\(251\) 2060.88 0.518253 0.259126 0.965843i \(-0.416565\pi\)
0.259126 + 0.965843i \(0.416565\pi\)
\(252\) 322.323 0.0805731
\(253\) −5210.34 −1.29475
\(254\) −4669.46 −1.15350
\(255\) 8633.92 2.12030
\(256\) 2202.54 0.537728
\(257\) −2254.70 −0.547254 −0.273627 0.961836i \(-0.588223\pi\)
−0.273627 + 0.961836i \(0.588223\pi\)
\(258\) 0 0
\(259\) 5345.98 1.28256
\(260\) −235.870 −0.0562617
\(261\) −364.932 −0.0865469
\(262\) −4260.88 −1.00473
\(263\) −6481.38 −1.51962 −0.759808 0.650147i \(-0.774707\pi\)
−0.759808 + 0.650147i \(0.774707\pi\)
\(264\) −2816.08 −0.656506
\(265\) 3805.10 0.882059
\(266\) −4817.53 −1.11046
\(267\) 820.897 0.188158
\(268\) −767.073 −0.174837
\(269\) 2971.06 0.673415 0.336708 0.941609i \(-0.390687\pi\)
0.336708 + 0.941609i \(0.390687\pi\)
\(270\) 4947.93 1.11527
\(271\) −5738.03 −1.28620 −0.643100 0.765782i \(-0.722352\pi\)
−0.643100 + 0.765782i \(0.722352\pi\)
\(272\) −8171.69 −1.82162
\(273\) −2336.45 −0.517979
\(274\) 3635.06 0.801468
\(275\) 1374.90 0.301489
\(276\) 1835.14 0.400226
\(277\) 1591.55 0.345224 0.172612 0.984990i \(-0.444779\pi\)
0.172612 + 0.984990i \(0.444779\pi\)
\(278\) 9138.70 1.97159
\(279\) −1152.78 −0.247366
\(280\) −9262.74 −1.97698
\(281\) −1279.65 −0.271665 −0.135832 0.990732i \(-0.543371\pi\)
−0.135832 + 0.990732i \(0.543371\pi\)
\(282\) −2597.01 −0.548402
\(283\) 8416.02 1.76778 0.883888 0.467699i \(-0.154917\pi\)
0.883888 + 0.467699i \(0.154917\pi\)
\(284\) −44.2779 −0.00925144
\(285\) 3555.42 0.738964
\(286\) −883.741 −0.182716
\(287\) 26.5553 0.00546171
\(288\) 420.768 0.0860902
\(289\) 7394.97 1.50518
\(290\) −2383.74 −0.482684
\(291\) 2033.64 0.409670
\(292\) 111.829 0.0224120
\(293\) −3718.98 −0.741519 −0.370760 0.928729i \(-0.620903\pi\)
−0.370760 + 0.928729i \(0.620903\pi\)
\(294\) 14756.0 2.92716
\(295\) 9412.12 1.85761
\(296\) 3133.29 0.615266
\(297\) 2896.89 0.565974
\(298\) −1750.35 −0.340253
\(299\) −2533.67 −0.490053
\(300\) −484.254 −0.0931948
\(301\) 0 0
\(302\) −5358.02 −1.02092
\(303\) −6364.03 −1.20661
\(304\) −3365.07 −0.634869
\(305\) −1330.39 −0.249763
\(306\) −2170.10 −0.405413
\(307\) 6406.06 1.19092 0.595461 0.803384i \(-0.296969\pi\)
0.595461 + 0.803384i \(0.296969\pi\)
\(308\) 1232.67 0.228045
\(309\) 5253.47 0.967183
\(310\) −7529.98 −1.37959
\(311\) −3626.35 −0.661194 −0.330597 0.943772i \(-0.607250\pi\)
−0.330597 + 0.943772i \(0.607250\pi\)
\(312\) −1369.39 −0.248483
\(313\) 6160.29 1.11246 0.556230 0.831028i \(-0.312247\pi\)
0.556230 + 0.831028i \(0.312247\pi\)
\(314\) 5364.41 0.964112
\(315\) −2931.59 −0.524370
\(316\) 1308.13 0.232874
\(317\) −1508.62 −0.267294 −0.133647 0.991029i \(-0.542669\pi\)
−0.133647 + 0.991029i \(0.542669\pi\)
\(318\) −5021.36 −0.885483
\(319\) −1395.62 −0.244952
\(320\) −5192.25 −0.907049
\(321\) 10791.7 1.87643
\(322\) 22616.0 3.91410
\(323\) 5068.38 0.873102
\(324\) −1274.44 −0.218525
\(325\) 668.582 0.114112
\(326\) 5760.40 0.978648
\(327\) −6058.61 −1.02459
\(328\) 15.5641 0.00262008
\(329\) −5001.21 −0.838072
\(330\) −5821.79 −0.971148
\(331\) 2373.17 0.394082 0.197041 0.980395i \(-0.436867\pi\)
0.197041 + 0.980395i \(0.436867\pi\)
\(332\) 1060.98 0.175389
\(333\) 991.664 0.163192
\(334\) −908.506 −0.148836
\(335\) 6976.69 1.13784
\(336\) 14567.7 2.36528
\(337\) −6631.98 −1.07201 −0.536005 0.844215i \(-0.680067\pi\)
−0.536005 + 0.844215i \(0.680067\pi\)
\(338\) 6335.31 1.01951
\(339\) −4268.03 −0.683798
\(340\) −2215.04 −0.353317
\(341\) −4408.61 −0.700116
\(342\) −893.639 −0.141294
\(343\) 16670.1 2.62420
\(344\) 0 0
\(345\) −16691.0 −2.60467
\(346\) 4975.64 0.773099
\(347\) −4392.54 −0.679551 −0.339775 0.940507i \(-0.610351\pi\)
−0.339775 + 0.940507i \(0.610351\pi\)
\(348\) 491.553 0.0757183
\(349\) 1187.57 0.182147 0.0910736 0.995844i \(-0.470970\pi\)
0.0910736 + 0.995844i \(0.470970\pi\)
\(350\) −5967.88 −0.911419
\(351\) 1408.69 0.214217
\(352\) 1609.15 0.243660
\(353\) −6116.11 −0.922175 −0.461088 0.887355i \(-0.652541\pi\)
−0.461088 + 0.887355i \(0.652541\pi\)
\(354\) −12420.6 −1.86482
\(355\) 402.716 0.0602084
\(356\) −210.602 −0.0313536
\(357\) −21941.4 −3.25284
\(358\) −6518.79 −0.962370
\(359\) −867.123 −0.127479 −0.0637396 0.997967i \(-0.520303\pi\)
−0.0637396 + 0.997967i \(0.520303\pi\)
\(360\) −1718.21 −0.251549
\(361\) −4771.86 −0.695708
\(362\) −916.390 −0.133051
\(363\) 4278.24 0.618593
\(364\) 599.418 0.0863133
\(365\) −1017.11 −0.145857
\(366\) 1755.63 0.250733
\(367\) 7941.70 1.12957 0.564787 0.825237i \(-0.308959\pi\)
0.564787 + 0.825237i \(0.308959\pi\)
\(368\) 15797.4 2.23776
\(369\) 4.92594 0.000694943 0
\(370\) 6477.57 0.910143
\(371\) −9669.93 −1.35320
\(372\) 1552.76 0.216416
\(373\) 200.374 0.0278149 0.0139075 0.999903i \(-0.495573\pi\)
0.0139075 + 0.999903i \(0.495573\pi\)
\(374\) −8299.17 −1.14743
\(375\) −5323.64 −0.733098
\(376\) −2931.22 −0.402037
\(377\) −678.658 −0.0927127
\(378\) −12574.2 −1.71097
\(379\) 9566.94 1.29662 0.648312 0.761375i \(-0.275475\pi\)
0.648312 + 0.761375i \(0.275475\pi\)
\(380\) −912.147 −0.123137
\(381\) −8757.69 −1.17761
\(382\) 13986.5 1.87333
\(383\) 5373.45 0.716894 0.358447 0.933550i \(-0.383306\pi\)
0.358447 + 0.933550i \(0.383306\pi\)
\(384\) 9912.10 1.31725
\(385\) −11211.4 −1.48411
\(386\) −10047.4 −1.32486
\(387\) 0 0
\(388\) −521.733 −0.0682654
\(389\) 7774.81 1.01336 0.506682 0.862133i \(-0.330872\pi\)
0.506682 + 0.862133i \(0.330872\pi\)
\(390\) −2831.00 −0.367573
\(391\) −23793.6 −3.07747
\(392\) 16654.9 2.14592
\(393\) −7991.39 −1.02573
\(394\) 2673.41 0.341839
\(395\) −11897.7 −1.51554
\(396\) 228.656 0.0290162
\(397\) 974.462 0.123191 0.0615955 0.998101i \(-0.480381\pi\)
0.0615955 + 0.998101i \(0.480381\pi\)
\(398\) −12980.6 −1.63482
\(399\) −9035.41 −1.13367
\(400\) −4168.59 −0.521074
\(401\) 3270.33 0.407263 0.203631 0.979048i \(-0.434726\pi\)
0.203631 + 0.979048i \(0.434726\pi\)
\(402\) −9206.70 −1.14226
\(403\) −2143.81 −0.264989
\(404\) 1632.70 0.201064
\(405\) 11591.3 1.42216
\(406\) 6057.82 0.740504
\(407\) 3792.45 0.461879
\(408\) −12859.9 −1.56044
\(409\) −5108.88 −0.617647 −0.308824 0.951119i \(-0.599935\pi\)
−0.308824 + 0.951119i \(0.599935\pi\)
\(410\) 32.1763 0.00387579
\(411\) 6817.65 0.818224
\(412\) −1347.78 −0.161166
\(413\) −23919.1 −2.84983
\(414\) 4195.20 0.498026
\(415\) −9649.87 −1.14143
\(416\) 782.495 0.0922235
\(417\) 17139.9 2.01281
\(418\) −3417.57 −0.399901
\(419\) 4610.75 0.537590 0.268795 0.963197i \(-0.413375\pi\)
0.268795 + 0.963197i \(0.413375\pi\)
\(420\) 3948.76 0.458762
\(421\) 8130.00 0.941168 0.470584 0.882355i \(-0.344043\pi\)
0.470584 + 0.882355i \(0.344043\pi\)
\(422\) 8946.58 1.03202
\(423\) −927.710 −0.106635
\(424\) −5667.56 −0.649153
\(425\) 6278.62 0.716607
\(426\) −531.440 −0.0604421
\(427\) 3380.92 0.383171
\(428\) −2768.62 −0.312678
\(429\) −1657.48 −0.186536
\(430\) 0 0
\(431\) −10180.0 −1.13771 −0.568853 0.822439i \(-0.692613\pi\)
−0.568853 + 0.822439i \(0.692613\pi\)
\(432\) −8783.15 −0.978193
\(433\) 3809.91 0.422846 0.211423 0.977395i \(-0.432190\pi\)
0.211423 + 0.977395i \(0.432190\pi\)
\(434\) 19136.0 2.11649
\(435\) −4470.77 −0.492775
\(436\) 1554.35 0.170733
\(437\) −9798.11 −1.07256
\(438\) 1342.21 0.146423
\(439\) −13805.3 −1.50090 −0.750448 0.660930i \(-0.770162\pi\)
−0.750448 + 0.660930i \(0.770162\pi\)
\(440\) −6571.00 −0.711955
\(441\) 5271.16 0.569179
\(442\) −4035.69 −0.434295
\(443\) −3291.23 −0.352982 −0.176491 0.984302i \(-0.556475\pi\)
−0.176491 + 0.984302i \(0.556475\pi\)
\(444\) −1335.74 −0.142774
\(445\) 1915.47 0.204049
\(446\) −6610.49 −0.701830
\(447\) −3282.83 −0.347366
\(448\) 13195.1 1.39154
\(449\) −17505.7 −1.83997 −0.919986 0.391952i \(-0.871800\pi\)
−0.919986 + 0.391952i \(0.871800\pi\)
\(450\) −1107.02 −0.115968
\(451\) 18.8384 0.00196689
\(452\) 1094.97 0.113945
\(453\) −10049.1 −1.04227
\(454\) −2688.66 −0.277941
\(455\) −5451.83 −0.561727
\(456\) −5295.67 −0.543843
\(457\) −6024.93 −0.616706 −0.308353 0.951272i \(-0.599778\pi\)
−0.308353 + 0.951272i \(0.599778\pi\)
\(458\) −10744.6 −1.09620
\(459\) 13228.9 1.34526
\(460\) 4282.09 0.434029
\(461\) 9644.81 0.974411 0.487205 0.873287i \(-0.338016\pi\)
0.487205 + 0.873287i \(0.338016\pi\)
\(462\) 14794.9 1.48988
\(463\) 8578.11 0.861034 0.430517 0.902583i \(-0.358331\pi\)
0.430517 + 0.902583i \(0.358331\pi\)
\(464\) 4231.42 0.423359
\(465\) −14122.7 −1.40844
\(466\) 15803.0 1.57094
\(467\) −3847.04 −0.381199 −0.190599 0.981668i \(-0.561043\pi\)
−0.190599 + 0.981668i \(0.561043\pi\)
\(468\) 111.190 0.0109824
\(469\) −17729.9 −1.74561
\(470\) −6059.82 −0.594720
\(471\) 10061.1 0.984268
\(472\) −14019.0 −1.36711
\(473\) 0 0
\(474\) 15700.7 1.52143
\(475\) 2585.51 0.249750
\(476\) 5629.10 0.542037
\(477\) −1793.74 −0.172180
\(478\) −10770.5 −1.03061
\(479\) −3588.42 −0.342295 −0.171147 0.985245i \(-0.554747\pi\)
−0.171147 + 0.985245i \(0.554747\pi\)
\(480\) 5154.81 0.490175
\(481\) 1844.18 0.174818
\(482\) 15913.5 1.50382
\(483\) 42416.9 3.99593
\(484\) −1097.59 −0.103079
\(485\) 4745.27 0.444271
\(486\) −5382.58 −0.502384
\(487\) −14142.4 −1.31592 −0.657962 0.753051i \(-0.728581\pi\)
−0.657962 + 0.753051i \(0.728581\pi\)
\(488\) 1981.56 0.183814
\(489\) 10803.8 0.999108
\(490\) 34431.3 3.17439
\(491\) −7516.33 −0.690850 −0.345425 0.938446i \(-0.612265\pi\)
−0.345425 + 0.938446i \(0.612265\pi\)
\(492\) −6.63509 −0.000607993 0
\(493\) −6373.24 −0.582224
\(494\) −1661.88 −0.151360
\(495\) −2079.68 −0.188837
\(496\) 13366.6 1.21004
\(497\) −1023.43 −0.0923680
\(498\) 12734.3 1.14586
\(499\) −7452.63 −0.668588 −0.334294 0.942469i \(-0.608498\pi\)
−0.334294 + 0.942469i \(0.608498\pi\)
\(500\) 1365.79 0.122160
\(501\) −1703.93 −0.151948
\(502\) −6345.90 −0.564206
\(503\) −13585.0 −1.20422 −0.602112 0.798412i \(-0.705674\pi\)
−0.602112 + 0.798412i \(0.705674\pi\)
\(504\) 4366.50 0.385911
\(505\) −14849.7 −1.30852
\(506\) 16043.8 1.40955
\(507\) 11882.0 1.04083
\(508\) 2246.80 0.196231
\(509\) −3781.95 −0.329336 −0.164668 0.986349i \(-0.552655\pi\)
−0.164668 + 0.986349i \(0.552655\pi\)
\(510\) −26585.8 −2.30831
\(511\) 2584.78 0.223765
\(512\) 6948.56 0.599778
\(513\) 5447.63 0.468847
\(514\) 6942.72 0.595779
\(515\) 12258.4 1.04887
\(516\) 0 0
\(517\) −3547.87 −0.301808
\(518\) −16461.5 −1.39629
\(519\) 9331.94 0.789262
\(520\) −3195.33 −0.269470
\(521\) −5698.91 −0.479220 −0.239610 0.970869i \(-0.577020\pi\)
−0.239610 + 0.970869i \(0.577020\pi\)
\(522\) 1123.71 0.0942211
\(523\) −19344.4 −1.61734 −0.808671 0.588262i \(-0.799813\pi\)
−0.808671 + 0.588262i \(0.799813\pi\)
\(524\) 2050.20 0.170923
\(525\) −11192.9 −0.930474
\(526\) 19957.6 1.65436
\(527\) −20132.4 −1.66410
\(528\) 10334.3 0.851789
\(529\) 33830.4 2.78050
\(530\) −11716.8 −0.960271
\(531\) −4436.92 −0.362610
\(532\) 2318.04 0.188910
\(533\) 9.16068 0.000744452 0
\(534\) −2527.73 −0.204842
\(535\) 25181.2 2.03491
\(536\) −10391.5 −0.837398
\(537\) −12226.1 −0.982490
\(538\) −9148.56 −0.733127
\(539\) 20158.7 1.61094
\(540\) −2380.79 −0.189727
\(541\) 4375.41 0.347714 0.173857 0.984771i \(-0.444377\pi\)
0.173857 + 0.984771i \(0.444377\pi\)
\(542\) 17668.7 1.40025
\(543\) −1718.71 −0.135832
\(544\) 7348.37 0.579152
\(545\) −14137.1 −1.11113
\(546\) 7194.44 0.563908
\(547\) −22211.1 −1.73616 −0.868080 0.496424i \(-0.834646\pi\)
−0.868080 + 0.496424i \(0.834646\pi\)
\(548\) −1749.08 −0.136345
\(549\) 627.150 0.0487543
\(550\) −4233.62 −0.328222
\(551\) −2624.48 −0.202916
\(552\) 24860.6 1.91691
\(553\) 30235.8 2.32506
\(554\) −4900.74 −0.375835
\(555\) 12148.8 0.929170
\(556\) −4397.25 −0.335405
\(557\) 2299.41 0.174918 0.0874590 0.996168i \(-0.472125\pi\)
0.0874590 + 0.996168i \(0.472125\pi\)
\(558\) 3549.67 0.269300
\(559\) 0 0
\(560\) 33992.0 2.56505
\(561\) −15565.3 −1.17142
\(562\) 3940.34 0.295753
\(563\) 10335.3 0.773677 0.386839 0.922147i \(-0.373567\pi\)
0.386839 + 0.922147i \(0.373567\pi\)
\(564\) 1249.60 0.0932935
\(565\) −9958.96 −0.741552
\(566\) −25914.8 −1.92453
\(567\) −29457.0 −2.18180
\(568\) −599.831 −0.0443105
\(569\) 3082.61 0.227117 0.113559 0.993531i \(-0.463775\pi\)
0.113559 + 0.993531i \(0.463775\pi\)
\(570\) −10947.9 −0.804489
\(571\) 3788.95 0.277693 0.138846 0.990314i \(-0.455661\pi\)
0.138846 + 0.990314i \(0.455661\pi\)
\(572\) 425.228 0.0310834
\(573\) 26232.1 1.91250
\(574\) −81.7699 −0.00594601
\(575\) −12137.7 −0.880310
\(576\) 2447.65 0.177058
\(577\) 11757.4 0.848296 0.424148 0.905593i \(-0.360574\pi\)
0.424148 + 0.905593i \(0.360574\pi\)
\(578\) −22770.8 −1.63865
\(579\) −18844.1 −1.35256
\(580\) 1146.98 0.0821135
\(581\) 24523.3 1.75111
\(582\) −6262.03 −0.445996
\(583\) −6859.86 −0.487318
\(584\) 1514.95 0.107344
\(585\) −1011.30 −0.0714736
\(586\) 11451.6 0.807270
\(587\) −9782.37 −0.687839 −0.343919 0.938999i \(-0.611755\pi\)
−0.343919 + 0.938999i \(0.611755\pi\)
\(588\) −7100.09 −0.497964
\(589\) −8290.44 −0.579969
\(590\) −28982.0 −2.02232
\(591\) 5014.05 0.348985
\(592\) −11498.4 −0.798281
\(593\) −19483.1 −1.34920 −0.674600 0.738183i \(-0.735684\pi\)
−0.674600 + 0.738183i \(0.735684\pi\)
\(594\) −8920.17 −0.616160
\(595\) −51197.9 −3.52758
\(596\) 842.215 0.0578833
\(597\) −24345.4 −1.66900
\(598\) 7801.74 0.533507
\(599\) −4655.42 −0.317555 −0.158778 0.987314i \(-0.550755\pi\)
−0.158778 + 0.987314i \(0.550755\pi\)
\(600\) −6560.18 −0.446364
\(601\) 13673.1 0.928014 0.464007 0.885832i \(-0.346411\pi\)
0.464007 + 0.885832i \(0.346411\pi\)
\(602\) 0 0
\(603\) −3288.84 −0.222110
\(604\) 2578.11 0.173678
\(605\) 9982.78 0.670839
\(606\) 19596.3 1.31360
\(607\) 7757.17 0.518705 0.259352 0.965783i \(-0.416491\pi\)
0.259352 + 0.965783i \(0.416491\pi\)
\(608\) 3026.03 0.201845
\(609\) 11361.6 0.755986
\(610\) 4096.56 0.271909
\(611\) −1725.25 −0.114232
\(612\) 1044.18 0.0689683
\(613\) 14088.9 0.928297 0.464148 0.885757i \(-0.346360\pi\)
0.464148 + 0.885757i \(0.346360\pi\)
\(614\) −19725.7 −1.29652
\(615\) 60.3475 0.00395682
\(616\) 16698.9 1.09224
\(617\) 12906.3 0.842122 0.421061 0.907032i \(-0.361658\pi\)
0.421061 + 0.907032i \(0.361658\pi\)
\(618\) −16176.6 −1.05294
\(619\) −23723.2 −1.54041 −0.770206 0.637795i \(-0.779847\pi\)
−0.770206 + 0.637795i \(0.779847\pi\)
\(620\) 3623.19 0.234695
\(621\) −25574.0 −1.65257
\(622\) 11166.3 0.719823
\(623\) −4867.80 −0.313040
\(624\) 5025.35 0.322396
\(625\) −19496.4 −1.24777
\(626\) −18968.9 −1.21110
\(627\) −6409.73 −0.408262
\(628\) −2581.18 −0.164013
\(629\) 17318.6 1.09783
\(630\) 9027.03 0.570866
\(631\) 12214.1 0.770578 0.385289 0.922796i \(-0.374102\pi\)
0.385289 + 0.922796i \(0.374102\pi\)
\(632\) 17721.2 1.11537
\(633\) 16779.5 1.05360
\(634\) 4645.37 0.290995
\(635\) −20435.1 −1.27707
\(636\) 2416.12 0.150637
\(637\) 9802.69 0.609728
\(638\) 4297.43 0.266672
\(639\) −189.842 −0.0117528
\(640\) 23128.8 1.42851
\(641\) −16542.4 −1.01932 −0.509662 0.860375i \(-0.670229\pi\)
−0.509662 + 0.860375i \(0.670229\pi\)
\(642\) −33230.0 −2.04281
\(643\) 6165.28 0.378126 0.189063 0.981965i \(-0.439455\pi\)
0.189063 + 0.981965i \(0.439455\pi\)
\(644\) −10882.1 −0.665861
\(645\) 0 0
\(646\) −15606.7 −0.950520
\(647\) −1033.87 −0.0628217 −0.0314108 0.999507i \(-0.510000\pi\)
−0.0314108 + 0.999507i \(0.510000\pi\)
\(648\) −17264.8 −1.04664
\(649\) −16968.2 −1.02629
\(650\) −2058.71 −0.124230
\(651\) 35890.0 2.16074
\(652\) −2771.72 −0.166486
\(653\) 10363.9 0.621085 0.310543 0.950559i \(-0.399489\pi\)
0.310543 + 0.950559i \(0.399489\pi\)
\(654\) 18655.8 1.11544
\(655\) −18647.0 −1.11236
\(656\) −57.1166 −0.00339944
\(657\) 479.470 0.0284717
\(658\) 15399.8 0.912384
\(659\) 27813.2 1.64408 0.822039 0.569431i \(-0.192837\pi\)
0.822039 + 0.569431i \(0.192837\pi\)
\(660\) 2801.26 0.165210
\(661\) −12139.7 −0.714344 −0.357172 0.934039i \(-0.616259\pi\)
−0.357172 + 0.934039i \(0.616259\pi\)
\(662\) −7307.52 −0.429026
\(663\) −7569.05 −0.443375
\(664\) 14373.1 0.840039
\(665\) −21083.1 −1.22942
\(666\) −3053.56 −0.177662
\(667\) 12320.7 0.715229
\(668\) 437.144 0.0253198
\(669\) −12398.1 −0.716502
\(670\) −21482.8 −1.23873
\(671\) 2398.43 0.137988
\(672\) −13100.0 −0.751996
\(673\) 14488.3 0.829842 0.414921 0.909857i \(-0.363809\pi\)
0.414921 + 0.909857i \(0.363809\pi\)
\(674\) 20421.4 1.16706
\(675\) 6748.43 0.384811
\(676\) −3048.35 −0.173438
\(677\) 6394.53 0.363016 0.181508 0.983389i \(-0.441902\pi\)
0.181508 + 0.983389i \(0.441902\pi\)
\(678\) 13142.2 0.744431
\(679\) −12059.2 −0.681574
\(680\) −30007.1 −1.69224
\(681\) −5042.65 −0.283752
\(682\) 13575.1 0.762196
\(683\) 24918.6 1.39602 0.698011 0.716087i \(-0.254069\pi\)
0.698011 + 0.716087i \(0.254069\pi\)
\(684\) 429.991 0.0240367
\(685\) 15908.2 0.887331
\(686\) −51331.0 −2.85689
\(687\) −20151.7 −1.11912
\(688\) 0 0
\(689\) −3335.79 −0.184446
\(690\) 51395.2 2.83563
\(691\) 26390.3 1.45287 0.726434 0.687236i \(-0.241176\pi\)
0.726434 + 0.687236i \(0.241176\pi\)
\(692\) −2394.12 −0.131519
\(693\) 5285.09 0.289703
\(694\) 13525.6 0.739807
\(695\) 39993.9 2.18281
\(696\) 6659.05 0.362659
\(697\) 86.0275 0.00467507
\(698\) −3656.81 −0.198298
\(699\) 29638.9 1.60379
\(700\) 2871.55 0.155049
\(701\) −16215.7 −0.873693 −0.436846 0.899536i \(-0.643905\pi\)
−0.436846 + 0.899536i \(0.643905\pi\)
\(702\) −4337.67 −0.233212
\(703\) 7131.74 0.382615
\(704\) 9360.62 0.501125
\(705\) −11365.3 −0.607154
\(706\) 18832.9 1.00394
\(707\) 37737.7 2.00746
\(708\) 5976.39 0.317241
\(709\) −4007.85 −0.212296 −0.106148 0.994350i \(-0.533852\pi\)
−0.106148 + 0.994350i \(0.533852\pi\)
\(710\) −1240.05 −0.0655471
\(711\) 5608.65 0.295838
\(712\) −2853.02 −0.150171
\(713\) 38919.6 2.04425
\(714\) 67562.6 3.54127
\(715\) −3867.54 −0.202290
\(716\) 3136.63 0.163717
\(717\) −20200.4 −1.05216
\(718\) 2670.07 0.138783
\(719\) −27845.1 −1.44429 −0.722147 0.691740i \(-0.756844\pi\)
−0.722147 + 0.691740i \(0.756844\pi\)
\(720\) 6305.43 0.326374
\(721\) −31152.3 −1.60911
\(722\) 14693.6 0.757397
\(723\) 29846.2 1.53526
\(724\) 440.938 0.0226344
\(725\) −3251.16 −0.166545
\(726\) −13173.6 −0.673443
\(727\) 17757.5 0.905899 0.452949 0.891536i \(-0.350372\pi\)
0.452949 + 0.891536i \(0.350372\pi\)
\(728\) 8120.30 0.413405
\(729\) 13129.2 0.667035
\(730\) 3131.90 0.158790
\(731\) 0 0
\(732\) −844.752 −0.0426543
\(733\) 37573.6 1.89333 0.946667 0.322212i \(-0.104427\pi\)
0.946667 + 0.322212i \(0.104427\pi\)
\(734\) −24454.3 −1.22973
\(735\) 64576.8 3.24075
\(736\) −14205.7 −0.711455
\(737\) −12577.6 −0.628633
\(738\) −15.1681 −0.000756564 0
\(739\) 5146.38 0.256174 0.128087 0.991763i \(-0.459116\pi\)
0.128087 + 0.991763i \(0.459116\pi\)
\(740\) −3116.80 −0.154832
\(741\) −3116.91 −0.154524
\(742\) 29775.9 1.47319
\(743\) −14021.8 −0.692342 −0.346171 0.938171i \(-0.612518\pi\)
−0.346171 + 0.938171i \(0.612518\pi\)
\(744\) 21035.2 1.03654
\(745\) −7660.12 −0.376705
\(746\) −616.996 −0.0302813
\(747\) 4549.00 0.222810
\(748\) 3993.29 0.195200
\(749\) −63993.1 −3.12184
\(750\) 16392.7 0.798102
\(751\) 7633.53 0.370907 0.185454 0.982653i \(-0.440625\pi\)
0.185454 + 0.982653i \(0.440625\pi\)
\(752\) 10756.9 0.521626
\(753\) −11901.9 −0.576002
\(754\) 2089.74 0.100934
\(755\) −23448.4 −1.13030
\(756\) 6050.31 0.291068
\(757\) 9584.30 0.460168 0.230084 0.973171i \(-0.426100\pi\)
0.230084 + 0.973171i \(0.426100\pi\)
\(758\) −29458.7 −1.41160
\(759\) 30090.6 1.43902
\(760\) −12356.8 −0.589776
\(761\) 12340.8 0.587849 0.293925 0.955829i \(-0.405039\pi\)
0.293925 + 0.955829i \(0.405039\pi\)
\(762\) 26966.9 1.28203
\(763\) 35926.7 1.70463
\(764\) −6729.87 −0.318689
\(765\) −9497.05 −0.448845
\(766\) −16546.1 −0.780461
\(767\) −8251.26 −0.388443
\(768\) −12720.0 −0.597648
\(769\) −6270.07 −0.294024 −0.147012 0.989135i \(-0.546966\pi\)
−0.147012 + 0.989135i \(0.546966\pi\)
\(770\) 34522.3 1.61571
\(771\) 13021.2 0.608234
\(772\) 4834.47 0.225384
\(773\) −29116.0 −1.35476 −0.677379 0.735634i \(-0.736884\pi\)
−0.677379 + 0.735634i \(0.736884\pi\)
\(774\) 0 0
\(775\) −10270.1 −0.476015
\(776\) −7067.90 −0.326962
\(777\) −30873.9 −1.42548
\(778\) −23940.4 −1.10322
\(779\) 35.4258 0.00162935
\(780\) 1362.19 0.0625310
\(781\) −726.020 −0.0332638
\(782\) 73265.7 3.35036
\(783\) −6850.13 −0.312649
\(784\) −61119.6 −2.78424
\(785\) 23476.4 1.06740
\(786\) 24607.3 1.11668
\(787\) 28536.0 1.29250 0.646250 0.763126i \(-0.276336\pi\)
0.646250 + 0.763126i \(0.276336\pi\)
\(788\) −1286.36 −0.0581531
\(789\) 37431.0 1.68895
\(790\) 36635.8 1.64993
\(791\) 25308.8 1.13764
\(792\) 3097.60 0.138975
\(793\) 1166.30 0.0522277
\(794\) −3000.59 −0.134114
\(795\) −21975.1 −0.980347
\(796\) 6245.84 0.278113
\(797\) −16273.3 −0.723248 −0.361624 0.932324i \(-0.617778\pi\)
−0.361624 + 0.932324i \(0.617778\pi\)
\(798\) 27822.0 1.23420
\(799\) −16201.7 −0.717365
\(800\) 3748.60 0.165666
\(801\) −902.962 −0.0398310
\(802\) −10070.1 −0.443375
\(803\) 1833.65 0.0805829
\(804\) 4429.97 0.194320
\(805\) 98974.9 4.33342
\(806\) 6601.26 0.288486
\(807\) −17158.4 −0.748454
\(808\) 22118.1 0.963012
\(809\) −22873.9 −0.994073 −0.497037 0.867730i \(-0.665579\pi\)
−0.497037 + 0.867730i \(0.665579\pi\)
\(810\) −35692.2 −1.54827
\(811\) 1107.97 0.0479728 0.0239864 0.999712i \(-0.492364\pi\)
0.0239864 + 0.999712i \(0.492364\pi\)
\(812\) −2914.83 −0.125974
\(813\) 33138.0 1.42952
\(814\) −11677.8 −0.502834
\(815\) 25209.4 1.08349
\(816\) 47192.8 2.02461
\(817\) 0 0
\(818\) 15731.4 0.672414
\(819\) 2570.02 0.109651
\(820\) −15.4822 −0.000659345 0
\(821\) −24233.1 −1.03014 −0.515068 0.857149i \(-0.672233\pi\)
−0.515068 + 0.857149i \(0.672233\pi\)
\(822\) −20993.1 −0.890776
\(823\) −15663.8 −0.663435 −0.331717 0.943379i \(-0.607628\pi\)
−0.331717 + 0.943379i \(0.607628\pi\)
\(824\) −18258.4 −0.771920
\(825\) −7940.27 −0.335084
\(826\) 73652.2 3.10253
\(827\) −13112.5 −0.551349 −0.275674 0.961251i \(-0.588901\pi\)
−0.275674 + 0.961251i \(0.588901\pi\)
\(828\) −2018.60 −0.0847236
\(829\) −23726.4 −0.994030 −0.497015 0.867742i \(-0.665571\pi\)
−0.497015 + 0.867742i \(0.665571\pi\)
\(830\) 29714.1 1.24264
\(831\) −9191.47 −0.383692
\(832\) 4551.86 0.189672
\(833\) 92056.6 3.82902
\(834\) −52777.5 −2.19129
\(835\) −3975.92 −0.164781
\(836\) 1644.42 0.0680306
\(837\) −21638.8 −0.893604
\(838\) −14197.5 −0.585258
\(839\) −10199.7 −0.419704 −0.209852 0.977733i \(-0.567298\pi\)
−0.209852 + 0.977733i \(0.567298\pi\)
\(840\) 53493.8 2.19728
\(841\) −21088.8 −0.864687
\(842\) −25034.1 −1.02462
\(843\) 7390.21 0.301936
\(844\) −4304.81 −0.175566
\(845\) 27725.4 1.12874
\(846\) 2856.63 0.116091
\(847\) −25369.3 −1.02916
\(848\) 20798.6 0.842249
\(849\) −48603.9 −1.96476
\(850\) −19333.3 −0.780148
\(851\) −33480.1 −1.34863
\(852\) 255.712 0.0102823
\(853\) 23124.5 0.928214 0.464107 0.885779i \(-0.346375\pi\)
0.464107 + 0.885779i \(0.346375\pi\)
\(854\) −10410.6 −0.417147
\(855\) −3910.85 −0.156431
\(856\) −37506.5 −1.49760
\(857\) −2642.41 −0.105324 −0.0526621 0.998612i \(-0.516771\pi\)
−0.0526621 + 0.998612i \(0.516771\pi\)
\(858\) 5103.75 0.203076
\(859\) −20573.1 −0.817166 −0.408583 0.912721i \(-0.633977\pi\)
−0.408583 + 0.912721i \(0.633977\pi\)
\(860\) 0 0
\(861\) −153.361 −0.00607032
\(862\) 31346.4 1.23859
\(863\) 20572.4 0.811465 0.405732 0.913992i \(-0.367017\pi\)
0.405732 + 0.913992i \(0.367017\pi\)
\(864\) 7898.22 0.310999
\(865\) 21775.0 0.855923
\(866\) −11731.6 −0.460340
\(867\) −42707.1 −1.67291
\(868\) −9207.63 −0.360055
\(869\) 21449.3 0.837305
\(870\) 13766.5 0.536469
\(871\) −6116.20 −0.237933
\(872\) 21056.7 0.817740
\(873\) −2236.94 −0.0867228
\(874\) 30170.6 1.16766
\(875\) 31568.4 1.21966
\(876\) −645.831 −0.0249094
\(877\) −50009.6 −1.92555 −0.962773 0.270313i \(-0.912873\pi\)
−0.962773 + 0.270313i \(0.912873\pi\)
\(878\) 42509.8 1.63398
\(879\) 21477.7 0.824147
\(880\) 24114.0 0.923731
\(881\) 32839.4 1.25583 0.627915 0.778282i \(-0.283909\pi\)
0.627915 + 0.778282i \(0.283909\pi\)
\(882\) −16231.1 −0.619648
\(883\) 5095.20 0.194187 0.0970935 0.995275i \(-0.469045\pi\)
0.0970935 + 0.995275i \(0.469045\pi\)
\(884\) 1941.85 0.0738817
\(885\) −54356.5 −2.06460
\(886\) 10134.4 0.384281
\(887\) −41889.4 −1.58569 −0.792845 0.609423i \(-0.791401\pi\)
−0.792845 + 0.609423i \(0.791401\pi\)
\(888\) −18095.3 −0.683826
\(889\) 51931.8 1.95921
\(890\) −5898.16 −0.222143
\(891\) −20896.9 −0.785714
\(892\) 3180.76 0.119394
\(893\) −6671.80 −0.250015
\(894\) 10108.6 0.378167
\(895\) −28528.3 −1.06547
\(896\) −58777.3 −2.19153
\(897\) 14632.4 0.544660
\(898\) 53904.1 2.00312
\(899\) 10424.8 0.386749
\(900\) 532.665 0.0197283
\(901\) −31326.3 −1.15830
\(902\) −58.0077 −0.00214129
\(903\) 0 0
\(904\) 14833.5 0.545747
\(905\) −4010.42 −0.147305
\(906\) 30943.4 1.13469
\(907\) −7205.23 −0.263777 −0.131889 0.991265i \(-0.542104\pi\)
−0.131889 + 0.991265i \(0.542104\pi\)
\(908\) 1293.70 0.0472829
\(909\) 7000.24 0.255427
\(910\) 16787.4 0.611536
\(911\) −11977.2 −0.435591 −0.217796 0.975994i \(-0.569887\pi\)
−0.217796 + 0.975994i \(0.569887\pi\)
\(912\) 19433.8 0.705613
\(913\) 17396.9 0.630615
\(914\) 18552.1 0.671389
\(915\) 7683.19 0.277594
\(916\) 5169.94 0.186484
\(917\) 47387.7 1.70652
\(918\) −40734.9 −1.46454
\(919\) −54968.3 −1.97305 −0.986527 0.163597i \(-0.947690\pi\)
−0.986527 + 0.163597i \(0.947690\pi\)
\(920\) 58009.3 2.07882
\(921\) −36996.0 −1.32363
\(922\) −29698.5 −1.06081
\(923\) −353.047 −0.0125901
\(924\) −7118.86 −0.253456
\(925\) 8834.68 0.314035
\(926\) −26413.9 −0.937382
\(927\) −5778.66 −0.204742
\(928\) −3805.09 −0.134599
\(929\) 49645.6 1.75330 0.876652 0.481125i \(-0.159772\pi\)
0.876652 + 0.481125i \(0.159772\pi\)
\(930\) 43486.9 1.53332
\(931\) 37908.6 1.33448
\(932\) −7603.90 −0.267247
\(933\) 20942.8 0.734872
\(934\) 11845.9 0.415000
\(935\) −36319.8 −1.27036
\(936\) 1506.29 0.0526012
\(937\) 4460.11 0.155502 0.0777510 0.996973i \(-0.475226\pi\)
0.0777510 + 0.996973i \(0.475226\pi\)
\(938\) 54594.3 1.90039
\(939\) −35576.7 −1.23642
\(940\) 2915.79 0.101173
\(941\) −41928.2 −1.45252 −0.726259 0.687421i \(-0.758743\pi\)
−0.726259 + 0.687421i \(0.758743\pi\)
\(942\) −30980.3 −1.07154
\(943\) −166.307 −0.00574305
\(944\) 51446.4 1.77377
\(945\) −55028.8 −1.89427
\(946\) 0 0
\(947\) 37529.0 1.28778 0.643890 0.765118i \(-0.277320\pi\)
0.643890 + 0.765118i \(0.277320\pi\)
\(948\) −7554.68 −0.258823
\(949\) 891.661 0.0305001
\(950\) −7961.37 −0.271896
\(951\) 8712.50 0.297079
\(952\) 76257.3 2.59613
\(953\) −6351.12 −0.215879 −0.107940 0.994157i \(-0.534425\pi\)
−0.107940 + 0.994157i \(0.534425\pi\)
\(954\) 5523.34 0.187447
\(955\) 61209.6 2.07403
\(956\) 5182.44 0.175327
\(957\) 8059.94 0.272247
\(958\) 11049.6 0.372646
\(959\) −40427.6 −1.36129
\(960\) 29986.1 1.00812
\(961\) 3139.90 0.105398
\(962\) −5678.65 −0.190319
\(963\) −11870.5 −0.397220
\(964\) −7657.09 −0.255828
\(965\) −43970.5 −1.46680
\(966\) −130611. −4.35025
\(967\) −15622.6 −0.519535 −0.259768 0.965671i \(-0.583646\pi\)
−0.259768 + 0.965671i \(0.583646\pi\)
\(968\) −14869.0 −0.493706
\(969\) −29270.7 −0.970392
\(970\) −14611.7 −0.483665
\(971\) 5920.86 0.195684 0.0978422 0.995202i \(-0.468806\pi\)
0.0978422 + 0.995202i \(0.468806\pi\)
\(972\) 2589.93 0.0854650
\(973\) −101637. −3.34874
\(974\) 43547.8 1.43261
\(975\) −3861.17 −0.126827
\(976\) −7271.86 −0.238490
\(977\) −36549.5 −1.19685 −0.598424 0.801179i \(-0.704206\pi\)
−0.598424 + 0.801179i \(0.704206\pi\)
\(978\) −33267.3 −1.08770
\(979\) −3453.22 −0.112733
\(980\) −16567.3 −0.540022
\(981\) 6664.29 0.216896
\(982\) 23144.5 0.752108
\(983\) −52748.6 −1.71152 −0.855758 0.517377i \(-0.826909\pi\)
−0.855758 + 0.517377i \(0.826909\pi\)
\(984\) −89.8854 −0.00291203
\(985\) 11699.7 0.378461
\(986\) 19624.7 0.633850
\(987\) 28882.8 0.931459
\(988\) 799.646 0.0257491
\(989\) 0 0
\(990\) 6403.79 0.205582
\(991\) −56108.6 −1.79854 −0.899268 0.437397i \(-0.855900\pi\)
−0.899268 + 0.437397i \(0.855900\pi\)
\(992\) −12019.9 −0.384709
\(993\) −13705.4 −0.437995
\(994\) 3151.36 0.100558
\(995\) −56807.2 −1.80996
\(996\) −6127.36 −0.194933
\(997\) −23518.2 −0.747071 −0.373536 0.927616i \(-0.621855\pi\)
−0.373536 + 0.927616i \(0.621855\pi\)
\(998\) 22948.3 0.727872
\(999\) 18614.5 0.589526
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.1.12 yes 50
43.42 odd 2 1849.4.a.i.1.39 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.39 50 43.42 odd 2
1849.4.a.j.1.12 yes 50 1.1 even 1 trivial