Properties

Label 531.4.a.e.1.6
Level $531$
Weight $4$
Character 531.1
Self dual yes
Analytic conductor $31.330$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3300142130\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.26905\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.26905 q^{2} -6.38952 q^{4} -14.8820 q^{5} +22.4903 q^{7} -18.2610 q^{8} +O(q^{10})\) \(q+1.26905 q^{2} -6.38952 q^{4} -14.8820 q^{5} +22.4903 q^{7} -18.2610 q^{8} -18.8859 q^{10} +70.2303 q^{11} -0.125351 q^{13} +28.5413 q^{14} +27.9422 q^{16} -57.2801 q^{17} +40.8191 q^{19} +95.0887 q^{20} +89.1255 q^{22} -190.065 q^{23} +96.4733 q^{25} -0.159076 q^{26} -143.702 q^{28} -133.399 q^{29} +129.061 q^{31} +181.548 q^{32} -72.6911 q^{34} -334.701 q^{35} -364.329 q^{37} +51.8013 q^{38} +271.759 q^{40} -195.632 q^{41} +31.6882 q^{43} -448.738 q^{44} -241.201 q^{46} -479.330 q^{47} +162.815 q^{49} +122.429 q^{50} +0.800930 q^{52} +402.892 q^{53} -1045.17 q^{55} -410.695 q^{56} -169.290 q^{58} -59.0000 q^{59} -209.849 q^{61} +163.785 q^{62} +6.85497 q^{64} +1.86547 q^{65} -455.024 q^{67} +365.992 q^{68} -424.750 q^{70} -203.497 q^{71} -177.712 q^{73} -462.350 q^{74} -260.815 q^{76} +1579.50 q^{77} -491.575 q^{79} -415.835 q^{80} -248.266 q^{82} +717.862 q^{83} +852.441 q^{85} +40.2137 q^{86} -1282.47 q^{88} -1306.23 q^{89} -2.81918 q^{91} +1214.42 q^{92} -608.291 q^{94} -607.469 q^{95} +538.879 q^{97} +206.620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{2} + 34q^{4} - 42q^{5} + 53q^{7} - 51q^{8} + O(q^{10}) \) \( 8q - 6q^{2} + 34q^{4} - 42q^{5} + 53q^{7} - 51q^{8} + 21q^{10} - 67q^{11} + 33q^{13} - 79q^{14} - 30q^{16} - 139q^{17} + 64q^{19} - 117q^{20} - 84q^{22} - 226q^{23} + 96q^{25} - 24q^{26} + 34q^{28} - 456q^{29} + 124q^{31} - 174q^{32} - 114q^{34} - 556q^{35} + 127q^{37} - 237q^{38} - 188q^{40} - 425q^{41} - 115q^{43} - 510q^{44} - 711q^{46} - 420q^{47} + 171q^{49} + 137q^{50} - 922q^{52} - 98q^{53} - 616q^{55} + 412q^{56} - 1548q^{58} - 472q^{59} - 1254q^{61} + 766q^{62} - 2019q^{64} + 734q^{65} - 1010q^{67} + 503q^{68} - 2956q^{70} + 17q^{71} - 1180q^{73} + 1228q^{74} - 2008q^{76} - 441q^{77} - 873q^{79} + 865q^{80} - 3645q^{82} - 759q^{83} - 850q^{85} + 1226q^{86} - 3047q^{88} - 988q^{89} - 2111q^{91} + 1062q^{92} - 2240q^{94} - 1822q^{95} - 668q^{97} + 1368q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.26905 0.448676 0.224338 0.974511i \(-0.427978\pi\)
0.224338 + 0.974511i \(0.427978\pi\)
\(3\) 0 0
\(4\) −6.38952 −0.798690
\(5\) −14.8820 −1.33108 −0.665542 0.746360i \(-0.731800\pi\)
−0.665542 + 0.746360i \(0.731800\pi\)
\(6\) 0 0
\(7\) 22.4903 1.21436 0.607182 0.794563i \(-0.292300\pi\)
0.607182 + 0.794563i \(0.292300\pi\)
\(8\) −18.2610 −0.807028
\(9\) 0 0
\(10\) −18.8859 −0.597225
\(11\) 70.2303 1.92502 0.962510 0.271244i \(-0.0874352\pi\)
0.962510 + 0.271244i \(0.0874352\pi\)
\(12\) 0 0
\(13\) −0.125351 −0.00267431 −0.00133715 0.999999i \(-0.500426\pi\)
−0.00133715 + 0.999999i \(0.500426\pi\)
\(14\) 28.5413 0.544855
\(15\) 0 0
\(16\) 27.9422 0.436596
\(17\) −57.2801 −0.817203 −0.408602 0.912713i \(-0.633983\pi\)
−0.408602 + 0.912713i \(0.633983\pi\)
\(18\) 0 0
\(19\) 40.8191 0.492871 0.246435 0.969159i \(-0.420741\pi\)
0.246435 + 0.969159i \(0.420741\pi\)
\(20\) 95.0887 1.06312
\(21\) 0 0
\(22\) 89.1255 0.863710
\(23\) −190.065 −1.72310 −0.861550 0.507673i \(-0.830506\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(24\) 0 0
\(25\) 96.4733 0.771787
\(26\) −0.159076 −0.00119990
\(27\) 0 0
\(28\) −143.702 −0.969900
\(29\) −133.399 −0.854193 −0.427096 0.904206i \(-0.640463\pi\)
−0.427096 + 0.904206i \(0.640463\pi\)
\(30\) 0 0
\(31\) 129.061 0.747745 0.373872 0.927480i \(-0.378030\pi\)
0.373872 + 0.927480i \(0.378030\pi\)
\(32\) 181.548 1.00292
\(33\) 0 0
\(34\) −72.6911 −0.366659
\(35\) −334.701 −1.61642
\(36\) 0 0
\(37\) −364.329 −1.61879 −0.809396 0.587263i \(-0.800205\pi\)
−0.809396 + 0.587263i \(0.800205\pi\)
\(38\) 51.8013 0.221139
\(39\) 0 0
\(40\) 271.759 1.07422
\(41\) −195.632 −0.745184 −0.372592 0.927995i \(-0.621531\pi\)
−0.372592 + 0.927995i \(0.621531\pi\)
\(42\) 0 0
\(43\) 31.6882 0.112381 0.0561907 0.998420i \(-0.482105\pi\)
0.0561907 + 0.998420i \(0.482105\pi\)
\(44\) −448.738 −1.53750
\(45\) 0 0
\(46\) −241.201 −0.773113
\(47\) −479.330 −1.48760 −0.743802 0.668400i \(-0.766980\pi\)
−0.743802 + 0.668400i \(0.766980\pi\)
\(48\) 0 0
\(49\) 162.815 0.474679
\(50\) 122.429 0.346282
\(51\) 0 0
\(52\) 0.800930 0.00213594
\(53\) 402.892 1.04418 0.522090 0.852891i \(-0.325153\pi\)
0.522090 + 0.852891i \(0.325153\pi\)
\(54\) 0 0
\(55\) −1045.17 −2.56237
\(56\) −410.695 −0.980026
\(57\) 0 0
\(58\) −169.290 −0.383255
\(59\) −59.0000 −0.130189
\(60\) 0 0
\(61\) −209.849 −0.440465 −0.220232 0.975447i \(-0.570682\pi\)
−0.220232 + 0.975447i \(0.570682\pi\)
\(62\) 163.785 0.335495
\(63\) 0 0
\(64\) 6.85497 0.0133886
\(65\) 1.86547 0.00355973
\(66\) 0 0
\(67\) −455.024 −0.829702 −0.414851 0.909889i \(-0.636166\pi\)
−0.414851 + 0.909889i \(0.636166\pi\)
\(68\) 365.992 0.652692
\(69\) 0 0
\(70\) −424.750 −0.725248
\(71\) −203.497 −0.340151 −0.170075 0.985431i \(-0.554401\pi\)
−0.170075 + 0.985431i \(0.554401\pi\)
\(72\) 0 0
\(73\) −177.712 −0.284927 −0.142463 0.989800i \(-0.545502\pi\)
−0.142463 + 0.989800i \(0.545502\pi\)
\(74\) −462.350 −0.726313
\(75\) 0 0
\(76\) −260.815 −0.393651
\(77\) 1579.50 2.33768
\(78\) 0 0
\(79\) −491.575 −0.700082 −0.350041 0.936734i \(-0.613832\pi\)
−0.350041 + 0.936734i \(0.613832\pi\)
\(80\) −415.835 −0.581147
\(81\) 0 0
\(82\) −248.266 −0.334346
\(83\) 717.862 0.949345 0.474673 0.880162i \(-0.342567\pi\)
0.474673 + 0.880162i \(0.342567\pi\)
\(84\) 0 0
\(85\) 852.441 1.08777
\(86\) 40.2137 0.0504228
\(87\) 0 0
\(88\) −1282.47 −1.55355
\(89\) −1306.23 −1.55573 −0.777865 0.628431i \(-0.783697\pi\)
−0.777865 + 0.628431i \(0.783697\pi\)
\(90\) 0 0
\(91\) −2.81918 −0.00324758
\(92\) 1214.42 1.37622
\(93\) 0 0
\(94\) −608.291 −0.667452
\(95\) −607.469 −0.656053
\(96\) 0 0
\(97\) 538.879 0.564071 0.282036 0.959404i \(-0.408990\pi\)
0.282036 + 0.959404i \(0.408990\pi\)
\(98\) 206.620 0.212977
\(99\) 0 0
\(100\) −616.419 −0.616419
\(101\) 1585.19 1.56171 0.780854 0.624714i \(-0.214784\pi\)
0.780854 + 0.624714i \(0.214784\pi\)
\(102\) 0 0
\(103\) −512.174 −0.489961 −0.244980 0.969528i \(-0.578782\pi\)
−0.244980 + 0.969528i \(0.578782\pi\)
\(104\) 2.28902 0.00215824
\(105\) 0 0
\(106\) 511.289 0.468498
\(107\) −90.3234 −0.0816065 −0.0408033 0.999167i \(-0.512992\pi\)
−0.0408033 + 0.999167i \(0.512992\pi\)
\(108\) 0 0
\(109\) −2241.55 −1.96974 −0.984870 0.173297i \(-0.944558\pi\)
−0.984870 + 0.173297i \(0.944558\pi\)
\(110\) −1326.36 −1.14967
\(111\) 0 0
\(112\) 628.428 0.530187
\(113\) 1538.31 1.28064 0.640319 0.768109i \(-0.278802\pi\)
0.640319 + 0.768109i \(0.278802\pi\)
\(114\) 0 0
\(115\) 2828.54 2.29359
\(116\) 852.356 0.682235
\(117\) 0 0
\(118\) −74.8737 −0.0584126
\(119\) −1288.25 −0.992382
\(120\) 0 0
\(121\) 3601.29 2.70571
\(122\) −266.308 −0.197626
\(123\) 0 0
\(124\) −824.640 −0.597216
\(125\) 424.533 0.303771
\(126\) 0 0
\(127\) 1230.53 0.859780 0.429890 0.902881i \(-0.358552\pi\)
0.429890 + 0.902881i \(0.358552\pi\)
\(128\) −1443.68 −0.996911
\(129\) 0 0
\(130\) 2.36736 0.00159716
\(131\) −972.455 −0.648578 −0.324289 0.945958i \(-0.605125\pi\)
−0.324289 + 0.945958i \(0.605125\pi\)
\(132\) 0 0
\(133\) 918.035 0.598525
\(134\) −577.447 −0.372267
\(135\) 0 0
\(136\) 1045.99 0.659506
\(137\) 335.709 0.209355 0.104677 0.994506i \(-0.466619\pi\)
0.104677 + 0.994506i \(0.466619\pi\)
\(138\) 0 0
\(139\) −2222.42 −1.35614 −0.678068 0.734999i \(-0.737183\pi\)
−0.678068 + 0.734999i \(0.737183\pi\)
\(140\) 2138.58 1.29102
\(141\) 0 0
\(142\) −258.248 −0.152617
\(143\) −8.80341 −0.00514810
\(144\) 0 0
\(145\) 1985.24 1.13700
\(146\) −225.525 −0.127840
\(147\) 0 0
\(148\) 2327.89 1.29291
\(149\) −1467.58 −0.806906 −0.403453 0.915000i \(-0.632190\pi\)
−0.403453 + 0.915000i \(0.632190\pi\)
\(150\) 0 0
\(151\) −2400.31 −1.29361 −0.646804 0.762656i \(-0.723895\pi\)
−0.646804 + 0.762656i \(0.723895\pi\)
\(152\) −745.397 −0.397761
\(153\) 0 0
\(154\) 2004.46 1.04886
\(155\) −1920.69 −0.995312
\(156\) 0 0
\(157\) −3571.25 −1.81539 −0.907697 0.419627i \(-0.862161\pi\)
−0.907697 + 0.419627i \(0.862161\pi\)
\(158\) −623.831 −0.314110
\(159\) 0 0
\(160\) −2701.79 −1.33497
\(161\) −4274.62 −2.09247
\(162\) 0 0
\(163\) −1305.57 −0.627365 −0.313683 0.949528i \(-0.601563\pi\)
−0.313683 + 0.949528i \(0.601563\pi\)
\(164\) 1249.99 0.595172
\(165\) 0 0
\(166\) 911.000 0.425948
\(167\) −156.555 −0.0725423 −0.0362711 0.999342i \(-0.511548\pi\)
−0.0362711 + 0.999342i \(0.511548\pi\)
\(168\) 0 0
\(169\) −2196.98 −0.999993
\(170\) 1081.79 0.488054
\(171\) 0 0
\(172\) −202.472 −0.0897579
\(173\) −357.266 −0.157008 −0.0785042 0.996914i \(-0.525014\pi\)
−0.0785042 + 0.996914i \(0.525014\pi\)
\(174\) 0 0
\(175\) 2169.72 0.937230
\(176\) 1962.39 0.840457
\(177\) 0 0
\(178\) −1657.66 −0.698018
\(179\) −1581.04 −0.660183 −0.330092 0.943949i \(-0.607080\pi\)
−0.330092 + 0.943949i \(0.607080\pi\)
\(180\) 0 0
\(181\) 1674.78 0.687766 0.343883 0.939012i \(-0.388258\pi\)
0.343883 + 0.939012i \(0.388258\pi\)
\(182\) −3.57766 −0.00145711
\(183\) 0 0
\(184\) 3470.77 1.39059
\(185\) 5421.94 2.15475
\(186\) 0 0
\(187\) −4022.80 −1.57313
\(188\) 3062.69 1.18814
\(189\) 0 0
\(190\) −770.907 −0.294355
\(191\) −5015.44 −1.90002 −0.950011 0.312217i \(-0.898929\pi\)
−0.950011 + 0.312217i \(0.898929\pi\)
\(192\) 0 0
\(193\) 1296.37 0.483496 0.241748 0.970339i \(-0.422279\pi\)
0.241748 + 0.970339i \(0.422279\pi\)
\(194\) 683.863 0.253085
\(195\) 0 0
\(196\) −1040.31 −0.379121
\(197\) 2894.70 1.04690 0.523449 0.852057i \(-0.324645\pi\)
0.523449 + 0.852057i \(0.324645\pi\)
\(198\) 0 0
\(199\) −2835.15 −1.00994 −0.504971 0.863136i \(-0.668497\pi\)
−0.504971 + 0.863136i \(0.668497\pi\)
\(200\) −1761.70 −0.622854
\(201\) 0 0
\(202\) 2011.68 0.700700
\(203\) −3000.19 −1.03730
\(204\) 0 0
\(205\) 2911.39 0.991904
\(206\) −649.972 −0.219833
\(207\) 0 0
\(208\) −3.50257 −0.00116759
\(209\) 2866.74 0.948787
\(210\) 0 0
\(211\) 894.249 0.291766 0.145883 0.989302i \(-0.453398\pi\)
0.145883 + 0.989302i \(0.453398\pi\)
\(212\) −2574.29 −0.833976
\(213\) 0 0
\(214\) −114.625 −0.0366148
\(215\) −471.583 −0.149589
\(216\) 0 0
\(217\) 2902.63 0.908034
\(218\) −2844.63 −0.883774
\(219\) 0 0
\(220\) 6678.11 2.04654
\(221\) 7.18009 0.00218545
\(222\) 0 0
\(223\) 2418.78 0.726338 0.363169 0.931723i \(-0.381695\pi\)
0.363169 + 0.931723i \(0.381695\pi\)
\(224\) 4083.07 1.21791
\(225\) 0 0
\(226\) 1952.19 0.574591
\(227\) 1116.89 0.326565 0.163283 0.986579i \(-0.447792\pi\)
0.163283 + 0.986579i \(0.447792\pi\)
\(228\) 0 0
\(229\) −4003.70 −1.15534 −0.577668 0.816272i \(-0.696037\pi\)
−0.577668 + 0.816272i \(0.696037\pi\)
\(230\) 3589.55 1.02908
\(231\) 0 0
\(232\) 2436.00 0.689358
\(233\) 3831.54 1.07731 0.538654 0.842527i \(-0.318933\pi\)
0.538654 + 0.842527i \(0.318933\pi\)
\(234\) 0 0
\(235\) 7133.37 1.98013
\(236\) 376.982 0.103981
\(237\) 0 0
\(238\) −1634.85 −0.445258
\(239\) 665.336 0.180071 0.0900356 0.995939i \(-0.471302\pi\)
0.0900356 + 0.995939i \(0.471302\pi\)
\(240\) 0 0
\(241\) 6021.52 1.60946 0.804730 0.593640i \(-0.202310\pi\)
0.804730 + 0.593640i \(0.202310\pi\)
\(242\) 4570.21 1.21398
\(243\) 0 0
\(244\) 1340.83 0.351795
\(245\) −2423.01 −0.631838
\(246\) 0 0
\(247\) −5.11670 −0.00131809
\(248\) −2356.78 −0.603451
\(249\) 0 0
\(250\) 538.752 0.136295
\(251\) 1854.47 0.466347 0.233173 0.972435i \(-0.425089\pi\)
0.233173 + 0.972435i \(0.425089\pi\)
\(252\) 0 0
\(253\) −13348.3 −3.31700
\(254\) 1561.60 0.385762
\(255\) 0 0
\(256\) −1886.94 −0.460678
\(257\) 3407.16 0.826976 0.413488 0.910510i \(-0.364310\pi\)
0.413488 + 0.910510i \(0.364310\pi\)
\(258\) 0 0
\(259\) −8193.88 −1.96580
\(260\) −11.9194 −0.00284312
\(261\) 0 0
\(262\) −1234.09 −0.291001
\(263\) 2085.60 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(264\) 0 0
\(265\) −5995.83 −1.38989
\(266\) 1165.03 0.268543
\(267\) 0 0
\(268\) 2907.39 0.662675
\(269\) 6495.06 1.47216 0.736080 0.676895i \(-0.236675\pi\)
0.736080 + 0.676895i \(0.236675\pi\)
\(270\) 0 0
\(271\) −3252.61 −0.729085 −0.364543 0.931187i \(-0.618775\pi\)
−0.364543 + 0.931187i \(0.618775\pi\)
\(272\) −1600.53 −0.356788
\(273\) 0 0
\(274\) 426.031 0.0939323
\(275\) 6775.35 1.48571
\(276\) 0 0
\(277\) 5905.21 1.28090 0.640451 0.767999i \(-0.278747\pi\)
0.640451 + 0.767999i \(0.278747\pi\)
\(278\) −2820.35 −0.608465
\(279\) 0 0
\(280\) 6111.96 1.30450
\(281\) −6504.05 −1.38078 −0.690390 0.723437i \(-0.742561\pi\)
−0.690390 + 0.723437i \(0.742561\pi\)
\(282\) 0 0
\(283\) 6587.83 1.38377 0.691883 0.722010i \(-0.256782\pi\)
0.691883 + 0.722010i \(0.256782\pi\)
\(284\) 1300.25 0.271675
\(285\) 0 0
\(286\) −11.1719 −0.00230983
\(287\) −4399.82 −0.904925
\(288\) 0 0
\(289\) −1631.99 −0.332179
\(290\) 2519.36 0.510145
\(291\) 0 0
\(292\) 1135.50 0.227568
\(293\) −4655.68 −0.928285 −0.464142 0.885761i \(-0.653637\pi\)
−0.464142 + 0.885761i \(0.653637\pi\)
\(294\) 0 0
\(295\) 878.037 0.173292
\(296\) 6653.00 1.30641
\(297\) 0 0
\(298\) −1862.43 −0.362039
\(299\) 23.8248 0.00460810
\(300\) 0 0
\(301\) 712.677 0.136472
\(302\) −3046.11 −0.580410
\(303\) 0 0
\(304\) 1140.57 0.215186
\(305\) 3122.96 0.586296
\(306\) 0 0
\(307\) −1381.12 −0.256758 −0.128379 0.991725i \(-0.540977\pi\)
−0.128379 + 0.991725i \(0.540977\pi\)
\(308\) −10092.3 −1.86708
\(309\) 0 0
\(310\) −2437.44 −0.446572
\(311\) −2269.90 −0.413873 −0.206936 0.978354i \(-0.566349\pi\)
−0.206936 + 0.978354i \(0.566349\pi\)
\(312\) 0 0
\(313\) −487.625 −0.0880581 −0.0440290 0.999030i \(-0.514019\pi\)
−0.0440290 + 0.999030i \(0.514019\pi\)
\(314\) −4532.08 −0.814523
\(315\) 0 0
\(316\) 3140.93 0.559149
\(317\) −8517.00 −1.50903 −0.754515 0.656283i \(-0.772128\pi\)
−0.754515 + 0.656283i \(0.772128\pi\)
\(318\) 0 0
\(319\) −9368.66 −1.64434
\(320\) −102.015 −0.0178214
\(321\) 0 0
\(322\) −5424.69 −0.938840
\(323\) −2338.12 −0.402776
\(324\) 0 0
\(325\) −12.0930 −0.00206400
\(326\) −1656.83 −0.281483
\(327\) 0 0
\(328\) 3572.43 0.601385
\(329\) −10780.3 −1.80649
\(330\) 0 0
\(331\) 10338.9 1.71685 0.858423 0.512942i \(-0.171444\pi\)
0.858423 + 0.512942i \(0.171444\pi\)
\(332\) −4586.80 −0.758233
\(333\) 0 0
\(334\) −198.675 −0.0325480
\(335\) 6771.66 1.10440
\(336\) 0 0
\(337\) −344.378 −0.0556660 −0.0278330 0.999613i \(-0.508861\pi\)
−0.0278330 + 0.999613i \(0.508861\pi\)
\(338\) −2788.07 −0.448672
\(339\) 0 0
\(340\) −5446.69 −0.868789
\(341\) 9064.01 1.43942
\(342\) 0 0
\(343\) −4052.42 −0.637931
\(344\) −578.656 −0.0906949
\(345\) 0 0
\(346\) −453.387 −0.0704458
\(347\) −4032.71 −0.623883 −0.311942 0.950101i \(-0.600979\pi\)
−0.311942 + 0.950101i \(0.600979\pi\)
\(348\) 0 0
\(349\) 6299.32 0.966175 0.483087 0.875572i \(-0.339515\pi\)
0.483087 + 0.875572i \(0.339515\pi\)
\(350\) 2753.47 0.420512
\(351\) 0 0
\(352\) 12750.1 1.93064
\(353\) 8240.96 1.24256 0.621278 0.783590i \(-0.286614\pi\)
0.621278 + 0.783590i \(0.286614\pi\)
\(354\) 0 0
\(355\) 3028.44 0.452769
\(356\) 8346.18 1.24255
\(357\) 0 0
\(358\) −2006.42 −0.296208
\(359\) 6605.66 0.971124 0.485562 0.874202i \(-0.338615\pi\)
0.485562 + 0.874202i \(0.338615\pi\)
\(360\) 0 0
\(361\) −5192.80 −0.757078
\(362\) 2125.38 0.308584
\(363\) 0 0
\(364\) 18.0132 0.00259381
\(365\) 2644.71 0.379262
\(366\) 0 0
\(367\) 10563.8 1.50253 0.751263 0.660003i \(-0.229445\pi\)
0.751263 + 0.660003i \(0.229445\pi\)
\(368\) −5310.83 −0.752299
\(369\) 0 0
\(370\) 6880.69 0.966784
\(371\) 9061.18 1.26801
\(372\) 0 0
\(373\) −9588.55 −1.33104 −0.665518 0.746382i \(-0.731789\pi\)
−0.665518 + 0.746382i \(0.731789\pi\)
\(374\) −5105.11 −0.705827
\(375\) 0 0
\(376\) 8753.02 1.20054
\(377\) 16.7217 0.00228437
\(378\) 0 0
\(379\) 8045.04 1.09036 0.545180 0.838319i \(-0.316461\pi\)
0.545180 + 0.838319i \(0.316461\pi\)
\(380\) 3881.44 0.523983
\(381\) 0 0
\(382\) −6364.82 −0.852493
\(383\) 13181.7 1.75863 0.879314 0.476242i \(-0.158001\pi\)
0.879314 + 0.476242i \(0.158001\pi\)
\(384\) 0 0
\(385\) −23506.1 −3.11164
\(386\) 1645.15 0.216933
\(387\) 0 0
\(388\) −3443.18 −0.450518
\(389\) −1629.56 −0.212396 −0.106198 0.994345i \(-0.533868\pi\)
−0.106198 + 0.994345i \(0.533868\pi\)
\(390\) 0 0
\(391\) 10886.9 1.40812
\(392\) −2973.16 −0.383079
\(393\) 0 0
\(394\) 3673.51 0.469718
\(395\) 7315.61 0.931869
\(396\) 0 0
\(397\) 2405.86 0.304148 0.152074 0.988369i \(-0.451405\pi\)
0.152074 + 0.988369i \(0.451405\pi\)
\(398\) −3597.94 −0.453136
\(399\) 0 0
\(400\) 2695.67 0.336959
\(401\) −4383.33 −0.545868 −0.272934 0.962033i \(-0.587994\pi\)
−0.272934 + 0.962033i \(0.587994\pi\)
\(402\) 0 0
\(403\) −16.1779 −0.00199970
\(404\) −10128.6 −1.24732
\(405\) 0 0
\(406\) −3807.38 −0.465411
\(407\) −25586.9 −3.11621
\(408\) 0 0
\(409\) −8567.44 −1.03578 −0.517888 0.855448i \(-0.673282\pi\)
−0.517888 + 0.855448i \(0.673282\pi\)
\(410\) 3694.69 0.445043
\(411\) 0 0
\(412\) 3272.54 0.391327
\(413\) −1326.93 −0.158097
\(414\) 0 0
\(415\) −10683.2 −1.26366
\(416\) −22.7571 −0.00268211
\(417\) 0 0
\(418\) 3638.02 0.425698
\(419\) −7908.61 −0.922102 −0.461051 0.887374i \(-0.652528\pi\)
−0.461051 + 0.887374i \(0.652528\pi\)
\(420\) 0 0
\(421\) 4262.15 0.493407 0.246703 0.969091i \(-0.420653\pi\)
0.246703 + 0.969091i \(0.420653\pi\)
\(422\) 1134.84 0.130908
\(423\) 0 0
\(424\) −7357.20 −0.842682
\(425\) −5526.00 −0.630707
\(426\) 0 0
\(427\) −4719.56 −0.534885
\(428\) 577.124 0.0651783
\(429\) 0 0
\(430\) −598.460 −0.0671170
\(431\) −12851.4 −1.43627 −0.718134 0.695905i \(-0.755003\pi\)
−0.718134 + 0.695905i \(0.755003\pi\)
\(432\) 0 0
\(433\) −10952.4 −1.21556 −0.607782 0.794104i \(-0.707940\pi\)
−0.607782 + 0.794104i \(0.707940\pi\)
\(434\) 3683.57 0.407413
\(435\) 0 0
\(436\) 14322.4 1.57321
\(437\) −7758.28 −0.849266
\(438\) 0 0
\(439\) −12153.2 −1.32128 −0.660638 0.750705i \(-0.729714\pi\)
−0.660638 + 0.750705i \(0.729714\pi\)
\(440\) 19085.7 2.06790
\(441\) 0 0
\(442\) 9.11187 0.000980560 0
\(443\) −11624.3 −1.24670 −0.623348 0.781944i \(-0.714228\pi\)
−0.623348 + 0.781944i \(0.714228\pi\)
\(444\) 0 0
\(445\) 19439.3 2.07081
\(446\) 3069.54 0.325890
\(447\) 0 0
\(448\) 154.170 0.0162586
\(449\) 9152.69 0.962009 0.481005 0.876718i \(-0.340272\pi\)
0.481005 + 0.876718i \(0.340272\pi\)
\(450\) 0 0
\(451\) −13739.3 −1.43450
\(452\) −9829.08 −1.02283
\(453\) 0 0
\(454\) 1417.38 0.146522
\(455\) 41.9549 0.00432281
\(456\) 0 0
\(457\) −1864.25 −0.190822 −0.0954112 0.995438i \(-0.530417\pi\)
−0.0954112 + 0.995438i \(0.530417\pi\)
\(458\) −5080.88 −0.518371
\(459\) 0 0
\(460\) −18073.0 −1.83187
\(461\) 19215.7 1.94136 0.970679 0.240377i \(-0.0772712\pi\)
0.970679 + 0.240377i \(0.0772712\pi\)
\(462\) 0 0
\(463\) 7787.00 0.781626 0.390813 0.920470i \(-0.372194\pi\)
0.390813 + 0.920470i \(0.372194\pi\)
\(464\) −3727.46 −0.372937
\(465\) 0 0
\(466\) 4862.41 0.483362
\(467\) 9537.66 0.945076 0.472538 0.881310i \(-0.343338\pi\)
0.472538 + 0.881310i \(0.343338\pi\)
\(468\) 0 0
\(469\) −10233.6 −1.00756
\(470\) 9052.58 0.888435
\(471\) 0 0
\(472\) 1077.40 0.105066
\(473\) 2225.47 0.216336
\(474\) 0 0
\(475\) 3937.96 0.380391
\(476\) 8231.29 0.792606
\(477\) 0 0
\(478\) 844.343 0.0807936
\(479\) 2560.30 0.244223 0.122112 0.992516i \(-0.461033\pi\)
0.122112 + 0.992516i \(0.461033\pi\)
\(480\) 0 0
\(481\) 45.6689 0.00432915
\(482\) 7641.58 0.722126
\(483\) 0 0
\(484\) −23010.5 −2.16102
\(485\) −8019.59 −0.750827
\(486\) 0 0
\(487\) 4855.22 0.451768 0.225884 0.974154i \(-0.427473\pi\)
0.225884 + 0.974154i \(0.427473\pi\)
\(488\) 3832.04 0.355468
\(489\) 0 0
\(490\) −3074.91 −0.283490
\(491\) −18773.3 −1.72551 −0.862755 0.505622i \(-0.831263\pi\)
−0.862755 + 0.505622i \(0.831263\pi\)
\(492\) 0 0
\(493\) 7641.11 0.698049
\(494\) −6.49333 −0.000591394 0
\(495\) 0 0
\(496\) 3606.25 0.326463
\(497\) −4576.72 −0.413067
\(498\) 0 0
\(499\) −8138.08 −0.730081 −0.365040 0.930992i \(-0.618945\pi\)
−0.365040 + 0.930992i \(0.618945\pi\)
\(500\) −2712.56 −0.242619
\(501\) 0 0
\(502\) 2353.41 0.209238
\(503\) −7732.32 −0.685422 −0.342711 0.939441i \(-0.611345\pi\)
−0.342711 + 0.939441i \(0.611345\pi\)
\(504\) 0 0
\(505\) −23590.8 −2.07877
\(506\) −16939.6 −1.48826
\(507\) 0 0
\(508\) −7862.51 −0.686698
\(509\) −1365.77 −0.118933 −0.0594665 0.998230i \(-0.518940\pi\)
−0.0594665 + 0.998230i \(0.518940\pi\)
\(510\) 0 0
\(511\) −3996.81 −0.346005
\(512\) 9154.84 0.790216
\(513\) 0 0
\(514\) 4323.84 0.371044
\(515\) 7622.16 0.652179
\(516\) 0 0
\(517\) −33663.5 −2.86367
\(518\) −10398.4 −0.882008
\(519\) 0 0
\(520\) −34.0652 −0.00287280
\(521\) 5123.71 0.430852 0.215426 0.976520i \(-0.430886\pi\)
0.215426 + 0.976520i \(0.430886\pi\)
\(522\) 0 0
\(523\) 20522.1 1.71581 0.857907 0.513805i \(-0.171765\pi\)
0.857907 + 0.513805i \(0.171765\pi\)
\(524\) 6213.52 0.518013
\(525\) 0 0
\(526\) 2646.72 0.219396
\(527\) −7392.64 −0.611060
\(528\) 0 0
\(529\) 23957.7 1.96907
\(530\) −7608.99 −0.623610
\(531\) 0 0
\(532\) −5865.81 −0.478036
\(533\) 24.5226 0.00199285
\(534\) 0 0
\(535\) 1344.19 0.108625
\(536\) 8309.18 0.669593
\(537\) 0 0
\(538\) 8242.53 0.660522
\(539\) 11434.5 0.913767
\(540\) 0 0
\(541\) −17106.3 −1.35944 −0.679720 0.733471i \(-0.737899\pi\)
−0.679720 + 0.733471i \(0.737899\pi\)
\(542\) −4127.71 −0.327123
\(543\) 0 0
\(544\) −10399.1 −0.819588
\(545\) 33358.7 2.62189
\(546\) 0 0
\(547\) 4023.62 0.314511 0.157255 0.987558i \(-0.449735\pi\)
0.157255 + 0.987558i \(0.449735\pi\)
\(548\) −2145.02 −0.167209
\(549\) 0 0
\(550\) 8598.23 0.666600
\(551\) −5445.23 −0.421007
\(552\) 0 0
\(553\) −11055.7 −0.850154
\(554\) 7493.99 0.574709
\(555\) 0 0
\(556\) 14200.2 1.08313
\(557\) −18019.0 −1.37072 −0.685360 0.728204i \(-0.740355\pi\)
−0.685360 + 0.728204i \(0.740355\pi\)
\(558\) 0 0
\(559\) −3.97213 −0.000300542 0
\(560\) −9352.26 −0.705723
\(561\) 0 0
\(562\) −8253.94 −0.619522
\(563\) −25706.0 −1.92430 −0.962148 0.272526i \(-0.912141\pi\)
−0.962148 + 0.272526i \(0.912141\pi\)
\(564\) 0 0
\(565\) −22893.1 −1.70464
\(566\) 8360.25 0.620862
\(567\) 0 0
\(568\) 3716.06 0.274511
\(569\) 3090.70 0.227713 0.113857 0.993497i \(-0.463680\pi\)
0.113857 + 0.993497i \(0.463680\pi\)
\(570\) 0 0
\(571\) 7296.29 0.534746 0.267373 0.963593i \(-0.413844\pi\)
0.267373 + 0.963593i \(0.413844\pi\)
\(572\) 56.2496 0.00411174
\(573\) 0 0
\(574\) −5583.58 −0.406018
\(575\) −18336.2 −1.32987
\(576\) 0 0
\(577\) −1681.44 −0.121316 −0.0606579 0.998159i \(-0.519320\pi\)
−0.0606579 + 0.998159i \(0.519320\pi\)
\(578\) −2071.07 −0.149040
\(579\) 0 0
\(580\) −12684.7 −0.908113
\(581\) 16145.0 1.15285
\(582\) 0 0
\(583\) 28295.2 2.01007
\(584\) 3245.20 0.229944
\(585\) 0 0
\(586\) −5908.27 −0.416499
\(587\) 6885.79 0.484169 0.242084 0.970255i \(-0.422169\pi\)
0.242084 + 0.970255i \(0.422169\pi\)
\(588\) 0 0
\(589\) 5268.17 0.368542
\(590\) 1114.27 0.0777521
\(591\) 0 0
\(592\) −10180.1 −0.706759
\(593\) 24880.0 1.72293 0.861467 0.507813i \(-0.169546\pi\)
0.861467 + 0.507813i \(0.169546\pi\)
\(594\) 0 0
\(595\) 19171.7 1.32094
\(596\) 9377.15 0.644468
\(597\) 0 0
\(598\) 30.2347 0.00206754
\(599\) −5068.39 −0.345724 −0.172862 0.984946i \(-0.555302\pi\)
−0.172862 + 0.984946i \(0.555302\pi\)
\(600\) 0 0
\(601\) −5800.47 −0.393687 −0.196844 0.980435i \(-0.563069\pi\)
−0.196844 + 0.980435i \(0.563069\pi\)
\(602\) 904.420 0.0612316
\(603\) 0 0
\(604\) 15336.9 1.03319
\(605\) −53594.4 −3.60152
\(606\) 0 0
\(607\) 8659.84 0.579064 0.289532 0.957168i \(-0.406500\pi\)
0.289532 + 0.957168i \(0.406500\pi\)
\(608\) 7410.61 0.494309
\(609\) 0 0
\(610\) 3963.18 0.263057
\(611\) 60.0843 0.00397831
\(612\) 0 0
\(613\) −3303.27 −0.217647 −0.108824 0.994061i \(-0.534708\pi\)
−0.108824 + 0.994061i \(0.534708\pi\)
\(614\) −1752.70 −0.115201
\(615\) 0 0
\(616\) −28843.2 −1.88657
\(617\) 9356.43 0.610495 0.305248 0.952273i \(-0.401261\pi\)
0.305248 + 0.952273i \(0.401261\pi\)
\(618\) 0 0
\(619\) 1106.92 0.0718756 0.0359378 0.999354i \(-0.488558\pi\)
0.0359378 + 0.999354i \(0.488558\pi\)
\(620\) 12272.3 0.794946
\(621\) 0 0
\(622\) −2880.61 −0.185695
\(623\) −29377.5 −1.88922
\(624\) 0 0
\(625\) −18377.1 −1.17613
\(626\) −618.818 −0.0395095
\(627\) 0 0
\(628\) 22818.6 1.44994
\(629\) 20868.8 1.32288
\(630\) 0 0
\(631\) 23600.6 1.48895 0.744473 0.667653i \(-0.232701\pi\)
0.744473 + 0.667653i \(0.232701\pi\)
\(632\) 8976.63 0.564986
\(633\) 0 0
\(634\) −10808.5 −0.677065
\(635\) −18312.7 −1.14444
\(636\) 0 0
\(637\) −20.4089 −0.00126944
\(638\) −11889.3 −0.737774
\(639\) 0 0
\(640\) 21484.8 1.32697
\(641\) 3922.67 0.241710 0.120855 0.992670i \(-0.461436\pi\)
0.120855 + 0.992670i \(0.461436\pi\)
\(642\) 0 0
\(643\) −11078.8 −0.679482 −0.339741 0.940519i \(-0.610339\pi\)
−0.339741 + 0.940519i \(0.610339\pi\)
\(644\) 27312.8 1.67123
\(645\) 0 0
\(646\) −2967.18 −0.180716
\(647\) 18371.0 1.11629 0.558143 0.829745i \(-0.311514\pi\)
0.558143 + 0.829745i \(0.311514\pi\)
\(648\) 0 0
\(649\) −4143.59 −0.250616
\(650\) −15.3466 −0.000926064 0
\(651\) 0 0
\(652\) 8342.00 0.501070
\(653\) −12179.0 −0.729862 −0.364931 0.931035i \(-0.618907\pi\)
−0.364931 + 0.931035i \(0.618907\pi\)
\(654\) 0 0
\(655\) 14472.1 0.863313
\(656\) −5466.38 −0.325345
\(657\) 0 0
\(658\) −13680.7 −0.810529
\(659\) 18245.7 1.07853 0.539266 0.842135i \(-0.318702\pi\)
0.539266 + 0.842135i \(0.318702\pi\)
\(660\) 0 0
\(661\) 20360.6 1.19809 0.599043 0.800717i \(-0.295548\pi\)
0.599043 + 0.800717i \(0.295548\pi\)
\(662\) 13120.5 0.770307
\(663\) 0 0
\(664\) −13108.9 −0.766148
\(665\) −13662.2 −0.796687
\(666\) 0 0
\(667\) 25354.5 1.47186
\(668\) 1000.31 0.0579388
\(669\) 0 0
\(670\) 8593.55 0.495519
\(671\) −14737.7 −0.847904
\(672\) 0 0
\(673\) −3894.49 −0.223063 −0.111531 0.993761i \(-0.535576\pi\)
−0.111531 + 0.993761i \(0.535576\pi\)
\(674\) −437.031 −0.0249760
\(675\) 0 0
\(676\) 14037.7 0.798685
\(677\) −23571.6 −1.33815 −0.669077 0.743194i \(-0.733310\pi\)
−0.669077 + 0.743194i \(0.733310\pi\)
\(678\) 0 0
\(679\) 12119.6 0.684987
\(680\) −15566.4 −0.877859
\(681\) 0 0
\(682\) 11502.6 0.645834
\(683\) 7789.31 0.436383 0.218192 0.975906i \(-0.429984\pi\)
0.218192 + 0.975906i \(0.429984\pi\)
\(684\) 0 0
\(685\) −4996.02 −0.278669
\(686\) −5142.71 −0.286224
\(687\) 0 0
\(688\) 885.436 0.0490653
\(689\) −50.5028 −0.00279246
\(690\) 0 0
\(691\) −8889.84 −0.489414 −0.244707 0.969597i \(-0.578692\pi\)
−0.244707 + 0.969597i \(0.578692\pi\)
\(692\) 2282.76 0.125401
\(693\) 0 0
\(694\) −5117.70 −0.279921
\(695\) 33074.0 1.80513
\(696\) 0 0
\(697\) 11205.8 0.608967
\(698\) 7994.13 0.433499
\(699\) 0 0
\(700\) −13863.5 −0.748556
\(701\) −32472.7 −1.74961 −0.874804 0.484477i \(-0.839010\pi\)
−0.874804 + 0.484477i \(0.839010\pi\)
\(702\) 0 0
\(703\) −14871.6 −0.797856
\(704\) 481.426 0.0257734
\(705\) 0 0
\(706\) 10458.2 0.557504
\(707\) 35651.5 1.89648
\(708\) 0 0
\(709\) 4596.06 0.243454 0.121727 0.992564i \(-0.461157\pi\)
0.121727 + 0.992564i \(0.461157\pi\)
\(710\) 3843.24 0.203147
\(711\) 0 0
\(712\) 23853.0 1.25552
\(713\) −24530.0 −1.28844
\(714\) 0 0
\(715\) 131.012 0.00685256
\(716\) 10102.1 0.527282
\(717\) 0 0
\(718\) 8382.89 0.435720
\(719\) 81.6045 0.00423273 0.00211637 0.999998i \(-0.499326\pi\)
0.00211637 + 0.999998i \(0.499326\pi\)
\(720\) 0 0
\(721\) −11519.0 −0.594991
\(722\) −6589.90 −0.339682
\(723\) 0 0
\(724\) −10701.1 −0.549312
\(725\) −12869.5 −0.659254
\(726\) 0 0
\(727\) −5240.27 −0.267333 −0.133666 0.991026i \(-0.542675\pi\)
−0.133666 + 0.991026i \(0.542675\pi\)
\(728\) 51.4809 0.00262089
\(729\) 0 0
\(730\) 3356.26 0.170166
\(731\) −1815.10 −0.0918384
\(732\) 0 0
\(733\) 27362.1 1.37877 0.689386 0.724394i \(-0.257880\pi\)
0.689386 + 0.724394i \(0.257880\pi\)
\(734\) 13406.0 0.674147
\(735\) 0 0
\(736\) −34505.8 −1.72813
\(737\) −31956.5 −1.59719
\(738\) 0 0
\(739\) 15458.7 0.769497 0.384748 0.923021i \(-0.374288\pi\)
0.384748 + 0.923021i \(0.374288\pi\)
\(740\) −34643.6 −1.72098
\(741\) 0 0
\(742\) 11499.1 0.568927
\(743\) 33376.7 1.64801 0.824006 0.566581i \(-0.191734\pi\)
0.824006 + 0.566581i \(0.191734\pi\)
\(744\) 0 0
\(745\) 21840.5 1.07406
\(746\) −12168.3 −0.597203
\(747\) 0 0
\(748\) 25703.7 1.25645
\(749\) −2031.40 −0.0991000
\(750\) 0 0
\(751\) −8832.20 −0.429150 −0.214575 0.976708i \(-0.568837\pi\)
−0.214575 + 0.976708i \(0.568837\pi\)
\(752\) −13393.5 −0.649483
\(753\) 0 0
\(754\) 21.2205 0.00102494
\(755\) 35721.4 1.72190
\(756\) 0 0
\(757\) −4971.26 −0.238684 −0.119342 0.992853i \(-0.538078\pi\)
−0.119342 + 0.992853i \(0.538078\pi\)
\(758\) 10209.5 0.489218
\(759\) 0 0
\(760\) 11093.0 0.529453
\(761\) −25414.3 −1.21060 −0.605300 0.795997i \(-0.706947\pi\)
−0.605300 + 0.795997i \(0.706947\pi\)
\(762\) 0 0
\(763\) −50413.2 −2.39198
\(764\) 32046.2 1.51753
\(765\) 0 0
\(766\) 16728.2 0.789054
\(767\) 7.39569 0.000348165 0
\(768\) 0 0
\(769\) 30483.0 1.42945 0.714725 0.699406i \(-0.246552\pi\)
0.714725 + 0.699406i \(0.246552\pi\)
\(770\) −29830.4 −1.39612
\(771\) 0 0
\(772\) −8283.19 −0.386164
\(773\) −18113.2 −0.842802 −0.421401 0.906874i \(-0.638462\pi\)
−0.421401 + 0.906874i \(0.638462\pi\)
\(774\) 0 0
\(775\) 12451.0 0.577099
\(776\) −9840.46 −0.455221
\(777\) 0 0
\(778\) −2067.99 −0.0952969
\(779\) −7985.52 −0.367280
\(780\) 0 0
\(781\) −14291.7 −0.654797
\(782\) 13816.0 0.631790
\(783\) 0 0
\(784\) 4549.40 0.207243
\(785\) 53147.3 2.41644
\(786\) 0 0
\(787\) 3681.38 0.166743 0.0833717 0.996519i \(-0.473431\pi\)
0.0833717 + 0.996519i \(0.473431\pi\)
\(788\) −18495.8 −0.836148
\(789\) 0 0
\(790\) 9283.84 0.418107
\(791\) 34597.1 1.55516
\(792\) 0 0
\(793\) 26.3046 0.00117794
\(794\) 3053.15 0.136464
\(795\) 0 0
\(796\) 18115.3 0.806631
\(797\) 40435.4 1.79711 0.898554 0.438863i \(-0.144619\pi\)
0.898554 + 0.438863i \(0.144619\pi\)
\(798\) 0 0
\(799\) 27456.0 1.21568
\(800\) 17514.5 0.774039
\(801\) 0 0
\(802\) −5562.65 −0.244918
\(803\) −12480.8 −0.548490
\(804\) 0 0
\(805\) 63614.9 2.78525
\(806\) −20.5305 −0.000897216 0
\(807\) 0 0
\(808\) −28947.1 −1.26034
\(809\) 36431.5 1.58327 0.791633 0.610997i \(-0.209231\pi\)
0.791633 + 0.610997i \(0.209231\pi\)
\(810\) 0 0
\(811\) −30923.2 −1.33891 −0.669457 0.742851i \(-0.733473\pi\)
−0.669457 + 0.742851i \(0.733473\pi\)
\(812\) 19169.8 0.828482
\(813\) 0 0
\(814\) −32471.0 −1.39817
\(815\) 19429.5 0.835076
\(816\) 0 0
\(817\) 1293.48 0.0553895
\(818\) −10872.5 −0.464728
\(819\) 0 0
\(820\) −18602.4 −0.792224
\(821\) −3516.20 −0.149472 −0.0747359 0.997203i \(-0.523811\pi\)
−0.0747359 + 0.997203i \(0.523811\pi\)
\(822\) 0 0
\(823\) 39234.9 1.66178 0.830889 0.556439i \(-0.187833\pi\)
0.830889 + 0.556439i \(0.187833\pi\)
\(824\) 9352.78 0.395412
\(825\) 0 0
\(826\) −1683.93 −0.0709341
\(827\) −20321.9 −0.854486 −0.427243 0.904137i \(-0.640515\pi\)
−0.427243 + 0.904137i \(0.640515\pi\)
\(828\) 0 0
\(829\) −36098.3 −1.51236 −0.756180 0.654364i \(-0.772936\pi\)
−0.756180 + 0.654364i \(0.772936\pi\)
\(830\) −13557.5 −0.566973
\(831\) 0 0
\(832\) −0.859274 −3.58053e−5 0
\(833\) −9326.04 −0.387909
\(834\) 0 0
\(835\) 2329.84 0.0965600
\(836\) −18317.1 −0.757787
\(837\) 0 0
\(838\) −10036.4 −0.413725
\(839\) −19751.4 −0.812746 −0.406373 0.913707i \(-0.633207\pi\)
−0.406373 + 0.913707i \(0.633207\pi\)
\(840\) 0 0
\(841\) −6593.69 −0.270355
\(842\) 5408.86 0.221380
\(843\) 0 0
\(844\) −5713.82 −0.233031
\(845\) 32695.5 1.33108
\(846\) 0 0
\(847\) 80994.3 3.28571
\(848\) 11257.7 0.455885
\(849\) 0 0
\(850\) −7012.75 −0.282983
\(851\) 69246.2 2.78934
\(852\) 0 0
\(853\) 13812.9 0.554448 0.277224 0.960805i \(-0.410586\pi\)
0.277224 + 0.960805i \(0.410586\pi\)
\(854\) −5989.34 −0.239990
\(855\) 0 0
\(856\) 1649.39 0.0658588
\(857\) −10194.0 −0.406323 −0.203162 0.979145i \(-0.565122\pi\)
−0.203162 + 0.979145i \(0.565122\pi\)
\(858\) 0 0
\(859\) −11608.7 −0.461099 −0.230549 0.973061i \(-0.574052\pi\)
−0.230549 + 0.973061i \(0.574052\pi\)
\(860\) 3013.19 0.119475
\(861\) 0 0
\(862\) −16309.0 −0.644418
\(863\) 45095.9 1.77877 0.889387 0.457155i \(-0.151132\pi\)
0.889387 + 0.457155i \(0.151132\pi\)
\(864\) 0 0
\(865\) 5316.83 0.208991
\(866\) −13899.1 −0.545393
\(867\) 0 0
\(868\) −18546.4 −0.725238
\(869\) −34523.5 −1.34767
\(870\) 0 0
\(871\) 57.0376 0.00221888
\(872\) 40932.9 1.58964
\(873\) 0 0
\(874\) −9845.62 −0.381045
\(875\) 9547.89 0.368889
\(876\) 0 0
\(877\) 38676.9 1.48920 0.744599 0.667512i \(-0.232641\pi\)
0.744599 + 0.667512i \(0.232641\pi\)
\(878\) −15423.0 −0.592824
\(879\) 0 0
\(880\) −29204.2 −1.11872
\(881\) 34130.6 1.30521 0.652604 0.757699i \(-0.273677\pi\)
0.652604 + 0.757699i \(0.273677\pi\)
\(882\) 0 0
\(883\) −39285.3 −1.49723 −0.748615 0.663005i \(-0.769281\pi\)
−0.748615 + 0.663005i \(0.769281\pi\)
\(884\) −45.8774 −0.00174550
\(885\) 0 0
\(886\) −14751.8 −0.559362
\(887\) 27255.1 1.03172 0.515860 0.856673i \(-0.327472\pi\)
0.515860 + 0.856673i \(0.327472\pi\)
\(888\) 0 0
\(889\) 27675.1 1.04409
\(890\) 24669.3 0.929121
\(891\) 0 0
\(892\) −15454.8 −0.580119
\(893\) −19565.8 −0.733197
\(894\) 0 0
\(895\) 23529.1 0.878760
\(896\) −32468.9 −1.21061
\(897\) 0 0
\(898\) 11615.2 0.431630
\(899\) −17216.6 −0.638718
\(900\) 0 0
\(901\) −23077.7 −0.853307
\(902\) −17435.8 −0.643623
\(903\) 0 0
\(904\) −28091.1 −1.03351
\(905\) −24924.1 −0.915475
\(906\) 0 0
\(907\) −40763.1 −1.49230 −0.746151 0.665777i \(-0.768100\pi\)
−0.746151 + 0.665777i \(0.768100\pi\)
\(908\) −7136.36 −0.260824
\(909\) 0 0
\(910\) 53.2427 0.00193954
\(911\) −9175.77 −0.333707 −0.166853 0.985982i \(-0.553361\pi\)
−0.166853 + 0.985982i \(0.553361\pi\)
\(912\) 0 0
\(913\) 50415.7 1.82751
\(914\) −2365.82 −0.0856174
\(915\) 0 0
\(916\) 25581.7 0.922756
\(917\) −21870.8 −0.787610
\(918\) 0 0
\(919\) −33814.3 −1.21374 −0.606871 0.794800i \(-0.707576\pi\)
−0.606871 + 0.794800i \(0.707576\pi\)
\(920\) −51651.9 −1.85099
\(921\) 0 0
\(922\) 24385.7 0.871040
\(923\) 25.5085 0.000909668 0
\(924\) 0 0
\(925\) −35148.0 −1.24936
\(926\) 9882.07 0.350696
\(927\) 0 0
\(928\) −24218.3 −0.856685
\(929\) −428.503 −0.0151332 −0.00756659 0.999971i \(-0.502409\pi\)
−0.00756659 + 0.999971i \(0.502409\pi\)
\(930\) 0 0
\(931\) 6645.96 0.233955
\(932\) −24481.7 −0.860435
\(933\) 0 0
\(934\) 12103.7 0.424032
\(935\) 59867.2 2.09397
\(936\) 0 0
\(937\) 32339.2 1.12751 0.563755 0.825942i \(-0.309356\pi\)
0.563755 + 0.825942i \(0.309356\pi\)
\(938\) −12987.0 −0.452068
\(939\) 0 0
\(940\) −45578.8 −1.58151
\(941\) −3308.12 −0.114603 −0.0573016 0.998357i \(-0.518250\pi\)
−0.0573016 + 0.998357i \(0.518250\pi\)
\(942\) 0 0
\(943\) 37182.8 1.28403
\(944\) −1648.59 −0.0568400
\(945\) 0 0
\(946\) 2824.22 0.0970649
\(947\) −2319.50 −0.0795921 −0.0397961 0.999208i \(-0.512671\pi\)
−0.0397961 + 0.999208i \(0.512671\pi\)
\(948\) 0 0
\(949\) 22.2764 0.000761982 0
\(950\) 4997.45 0.170672
\(951\) 0 0
\(952\) 23524.6 0.800880
\(953\) −8790.44 −0.298793 −0.149397 0.988777i \(-0.547733\pi\)
−0.149397 + 0.988777i \(0.547733\pi\)
\(954\) 0 0
\(955\) 74639.6 2.52909
\(956\) −4251.18 −0.143821
\(957\) 0 0
\(958\) 3249.13 0.109577
\(959\) 7550.21 0.254233
\(960\) 0 0
\(961\) −13134.2 −0.440878
\(962\) 57.9559 0.00194238
\(963\) 0 0
\(964\) −38474.6 −1.28546
\(965\) −19292.6 −0.643575
\(966\) 0 0
\(967\) −20867.5 −0.693953 −0.346976 0.937874i \(-0.612792\pi\)
−0.346976 + 0.937874i \(0.612792\pi\)
\(968\) −65763.1 −2.18358
\(969\) 0 0
\(970\) −10177.2 −0.336877
\(971\) −2174.66 −0.0718726 −0.0359363 0.999354i \(-0.511441\pi\)
−0.0359363 + 0.999354i \(0.511441\pi\)
\(972\) 0 0
\(973\) −49982.9 −1.64684
\(974\) 6161.49 0.202697
\(975\) 0 0
\(976\) −5863.62 −0.192305
\(977\) 6656.04 0.217959 0.108979 0.994044i \(-0.465242\pi\)
0.108979 + 0.994044i \(0.465242\pi\)
\(978\) 0 0
\(979\) −91736.8 −2.99481
\(980\) 15481.9 0.504643
\(981\) 0 0
\(982\) −23824.1 −0.774194
\(983\) −52028.5 −1.68815 −0.844075 0.536224i \(-0.819850\pi\)
−0.844075 + 0.536224i \(0.819850\pi\)
\(984\) 0 0
\(985\) −43078.9 −1.39351
\(986\) 9696.92 0.313198
\(987\) 0 0
\(988\) 32.6933 0.00105274
\(989\) −6022.81 −0.193644
\(990\) 0 0
\(991\) 9152.99 0.293395 0.146697 0.989181i \(-0.453136\pi\)
0.146697 + 0.989181i \(0.453136\pi\)
\(992\) 23430.8 0.749927
\(993\) 0 0
\(994\) −5808.07 −0.185333
\(995\) 42192.7 1.34432
\(996\) 0 0
\(997\) −43711.9 −1.38854 −0.694268 0.719717i \(-0.744272\pi\)
−0.694268 + 0.719717i \(0.744272\pi\)
\(998\) −10327.6 −0.327569
\(999\) 0 0
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 531.4.a.e.1.6 8
3.2 odd 2 177.4.a.d.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.d.1.3 8 3.2 odd 2
531.4.a.e.1.6 8 1.1 even 1 trivial