Properties

Label 76.4.a.a.1.1
Level $76$
Weight $4$
Character 76.1
Self dual yes
Analytic conductor $4.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.48414516044\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.37228 q^{3} +11.8614 q^{5} -26.4891 q^{7} +1.86141 q^{9} +O(q^{10})\) \(q-5.37228 q^{3} +11.8614 q^{5} -26.4891 q^{7} +1.86141 q^{9} -49.8614 q^{11} -49.0951 q^{13} -63.7228 q^{15} +17.2337 q^{17} -19.0000 q^{19} +142.307 q^{21} +166.965 q^{23} +15.6930 q^{25} +135.052 q^{27} -109.198 q^{29} -273.783 q^{31} +267.870 q^{33} -314.198 q^{35} +167.022 q^{37} +263.753 q^{39} +15.1684 q^{41} +413.557 q^{43} +22.0789 q^{45} -161.438 q^{47} +358.674 q^{49} -92.5842 q^{51} -490.791 q^{53} -591.426 q^{55} +102.073 q^{57} -335.970 q^{59} +725.149 q^{61} -49.3070 q^{63} -582.337 q^{65} +497.709 q^{67} -896.981 q^{69} -798.554 q^{71} -311.174 q^{73} -84.3070 q^{75} +1320.79 q^{77} -665.723 q^{79} -775.793 q^{81} -372.440 q^{83} +204.416 q^{85} +586.644 q^{87} -673.783 q^{89} +1300.49 q^{91} +1470.84 q^{93} -225.367 q^{95} -960.505 q^{97} -92.8124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{3} - 5q^{5} - 30q^{7} - 25q^{9} + O(q^{10}) \) \( 2q - 5q^{3} - 5q^{5} - 30q^{7} - 25q^{9} - 71q^{11} - 35q^{13} - 70q^{15} - 38q^{19} + 141q^{21} - 5q^{23} + 175q^{25} + 115q^{27} + 155q^{29} - 88q^{31} + 260q^{33} - 255q^{35} + 380q^{37} + 269q^{39} - 142q^{41} + 155q^{43} + 475q^{45} - 455q^{47} + 28q^{49} - 99q^{51} - 275q^{53} - 235q^{55} + 95q^{57} - 873q^{59} + 445q^{61} + 45q^{63} - 820q^{65} + 645q^{67} - 961q^{69} - 1712q^{71} - 990q^{73} - 25q^{75} + 1395q^{77} - 1274q^{79} - 58q^{81} - 90q^{83} + 495q^{85} + 685q^{87} - 888q^{89} + 1251q^{91} + 1540q^{93} + 95q^{95} + 710q^{97} + 475q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −5.37228 −1.03390 −0.516948 0.856017i \(-0.672932\pi\)
−0.516948 + 0.856017i \(0.672932\pi\)
\(4\) 0 0
\(5\) 11.8614 1.06092 0.530458 0.847711i \(-0.322020\pi\)
0.530458 + 0.847711i \(0.322020\pi\)
\(6\) 0 0
\(7\) −26.4891 −1.43028 −0.715139 0.698982i \(-0.753637\pi\)
−0.715139 + 0.698982i \(0.753637\pi\)
\(8\) 0 0
\(9\) 1.86141 0.0689410
\(10\) 0 0
\(11\) −49.8614 −1.36671 −0.683354 0.730088i \(-0.739479\pi\)
−0.683354 + 0.730088i \(0.739479\pi\)
\(12\) 0 0
\(13\) −49.0951 −1.04743 −0.523713 0.851895i \(-0.675453\pi\)
−0.523713 + 0.851895i \(0.675453\pi\)
\(14\) 0 0
\(15\) −63.7228 −1.09688
\(16\) 0 0
\(17\) 17.2337 0.245870 0.122935 0.992415i \(-0.460769\pi\)
0.122935 + 0.992415i \(0.460769\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) 142.307 1.47876
\(22\) 0 0
\(23\) 166.965 1.51368 0.756838 0.653603i \(-0.226743\pi\)
0.756838 + 0.653603i \(0.226743\pi\)
\(24\) 0 0
\(25\) 15.6930 0.125544
\(26\) 0 0
\(27\) 135.052 0.962618
\(28\) 0 0
\(29\) −109.198 −0.699228 −0.349614 0.936894i \(-0.613687\pi\)
−0.349614 + 0.936894i \(0.613687\pi\)
\(30\) 0 0
\(31\) −273.783 −1.58622 −0.793110 0.609079i \(-0.791539\pi\)
−0.793110 + 0.609079i \(0.791539\pi\)
\(32\) 0 0
\(33\) 267.870 1.41303
\(34\) 0 0
\(35\) −314.198 −1.51741
\(36\) 0 0
\(37\) 167.022 0.742114 0.371057 0.928610i \(-0.378995\pi\)
0.371057 + 0.928610i \(0.378995\pi\)
\(38\) 0 0
\(39\) 263.753 1.08293
\(40\) 0 0
\(41\) 15.1684 0.0577783 0.0288892 0.999583i \(-0.490803\pi\)
0.0288892 + 0.999583i \(0.490803\pi\)
\(42\) 0 0
\(43\) 413.557 1.46667 0.733335 0.679867i \(-0.237963\pi\)
0.733335 + 0.679867i \(0.237963\pi\)
\(44\) 0 0
\(45\) 22.0789 0.0731406
\(46\) 0 0
\(47\) −161.438 −0.501023 −0.250512 0.968114i \(-0.580599\pi\)
−0.250512 + 0.968114i \(0.580599\pi\)
\(48\) 0 0
\(49\) 358.674 1.04570
\(50\) 0 0
\(51\) −92.5842 −0.254204
\(52\) 0 0
\(53\) −490.791 −1.27199 −0.635993 0.771695i \(-0.719409\pi\)
−0.635993 + 0.771695i \(0.719409\pi\)
\(54\) 0 0
\(55\) −591.426 −1.44996
\(56\) 0 0
\(57\) 102.073 0.237192
\(58\) 0 0
\(59\) −335.970 −0.741349 −0.370674 0.928763i \(-0.620873\pi\)
−0.370674 + 0.928763i \(0.620873\pi\)
\(60\) 0 0
\(61\) 725.149 1.52206 0.761032 0.648715i \(-0.224693\pi\)
0.761032 + 0.648715i \(0.224693\pi\)
\(62\) 0 0
\(63\) −49.3070 −0.0986048
\(64\) 0 0
\(65\) −582.337 −1.11123
\(66\) 0 0
\(67\) 497.709 0.907535 0.453768 0.891120i \(-0.350080\pi\)
0.453768 + 0.891120i \(0.350080\pi\)
\(68\) 0 0
\(69\) −896.981 −1.56498
\(70\) 0 0
\(71\) −798.554 −1.33480 −0.667401 0.744698i \(-0.732593\pi\)
−0.667401 + 0.744698i \(0.732593\pi\)
\(72\) 0 0
\(73\) −311.174 −0.498906 −0.249453 0.968387i \(-0.580251\pi\)
−0.249453 + 0.968387i \(0.580251\pi\)
\(74\) 0 0
\(75\) −84.3070 −0.129799
\(76\) 0 0
\(77\) 1320.79 1.95477
\(78\) 0 0
\(79\) −665.723 −0.948097 −0.474049 0.880499i \(-0.657208\pi\)
−0.474049 + 0.880499i \(0.657208\pi\)
\(80\) 0 0
\(81\) −775.793 −1.06419
\(82\) 0 0
\(83\) −372.440 −0.492537 −0.246269 0.969202i \(-0.579205\pi\)
−0.246269 + 0.969202i \(0.579205\pi\)
\(84\) 0 0
\(85\) 204.416 0.260847
\(86\) 0 0
\(87\) 586.644 0.722929
\(88\) 0 0
\(89\) −673.783 −0.802481 −0.401240 0.915973i \(-0.631421\pi\)
−0.401240 + 0.915973i \(0.631421\pi\)
\(90\) 0 0
\(91\) 1300.49 1.49811
\(92\) 0 0
\(93\) 1470.84 1.63999
\(94\) 0 0
\(95\) −225.367 −0.243391
\(96\) 0 0
\(97\) −960.505 −1.00541 −0.502704 0.864459i \(-0.667661\pi\)
−0.502704 + 0.864459i \(0.667661\pi\)
\(98\) 0 0
\(99\) −92.8124 −0.0942221
\(100\) 0 0
\(101\) 1889.68 1.86169 0.930845 0.365415i \(-0.119073\pi\)
0.930845 + 0.365415i \(0.119073\pi\)
\(102\) 0 0
\(103\) 760.217 0.727247 0.363624 0.931546i \(-0.381539\pi\)
0.363624 + 0.931546i \(0.381539\pi\)
\(104\) 0 0
\(105\) 1687.96 1.56884
\(106\) 0 0
\(107\) 681.730 0.615938 0.307969 0.951396i \(-0.400351\pi\)
0.307969 + 0.951396i \(0.400351\pi\)
\(108\) 0 0
\(109\) −1310.76 −1.15182 −0.575910 0.817513i \(-0.695352\pi\)
−0.575910 + 0.817513i \(0.695352\pi\)
\(110\) 0 0
\(111\) −897.288 −0.767268
\(112\) 0 0
\(113\) −713.413 −0.593914 −0.296957 0.954891i \(-0.595972\pi\)
−0.296957 + 0.954891i \(0.595972\pi\)
\(114\) 0 0
\(115\) 1980.43 1.60588
\(116\) 0 0
\(117\) −91.3859 −0.0722105
\(118\) 0 0
\(119\) −456.505 −0.351662
\(120\) 0 0
\(121\) 1155.16 0.867889
\(122\) 0 0
\(123\) −81.4891 −0.0597368
\(124\) 0 0
\(125\) −1296.54 −0.927725
\(126\) 0 0
\(127\) −107.847 −0.0753535 −0.0376768 0.999290i \(-0.511996\pi\)
−0.0376768 + 0.999290i \(0.511996\pi\)
\(128\) 0 0
\(129\) −2221.74 −1.51638
\(130\) 0 0
\(131\) 373.247 0.248937 0.124469 0.992224i \(-0.460277\pi\)
0.124469 + 0.992224i \(0.460277\pi\)
\(132\) 0 0
\(133\) 503.293 0.328128
\(134\) 0 0
\(135\) 1601.90 1.02126
\(136\) 0 0
\(137\) −3118.25 −1.94460 −0.972301 0.233732i \(-0.924906\pi\)
−0.972301 + 0.233732i \(0.924906\pi\)
\(138\) 0 0
\(139\) −56.3774 −0.0344019 −0.0172010 0.999852i \(-0.505476\pi\)
−0.0172010 + 0.999852i \(0.505476\pi\)
\(140\) 0 0
\(141\) 867.288 0.518006
\(142\) 0 0
\(143\) 2447.95 1.43152
\(144\) 0 0
\(145\) −1295.25 −0.741823
\(146\) 0 0
\(147\) −1926.90 −1.08114
\(148\) 0 0
\(149\) −1810.51 −0.995457 −0.497728 0.867333i \(-0.665832\pi\)
−0.497728 + 0.867333i \(0.665832\pi\)
\(150\) 0 0
\(151\) 2894.72 1.56006 0.780029 0.625744i \(-0.215204\pi\)
0.780029 + 0.625744i \(0.215204\pi\)
\(152\) 0 0
\(153\) 32.0789 0.0169505
\(154\) 0 0
\(155\) −3247.45 −1.68285
\(156\) 0 0
\(157\) 1381.71 0.702371 0.351185 0.936306i \(-0.385779\pi\)
0.351185 + 0.936306i \(0.385779\pi\)
\(158\) 0 0
\(159\) 2636.67 1.31510
\(160\) 0 0
\(161\) −4422.75 −2.16498
\(162\) 0 0
\(163\) −3740.83 −1.79757 −0.898785 0.438389i \(-0.855549\pi\)
−0.898785 + 0.438389i \(0.855549\pi\)
\(164\) 0 0
\(165\) 3177.31 1.49911
\(166\) 0 0
\(167\) 3085.17 1.42957 0.714783 0.699347i \(-0.246526\pi\)
0.714783 + 0.699347i \(0.246526\pi\)
\(168\) 0 0
\(169\) 213.328 0.0970998
\(170\) 0 0
\(171\) −35.3667 −0.0158161
\(172\) 0 0
\(173\) 293.000 0.128765 0.0643827 0.997925i \(-0.479492\pi\)
0.0643827 + 0.997925i \(0.479492\pi\)
\(174\) 0 0
\(175\) −415.693 −0.179562
\(176\) 0 0
\(177\) 1804.93 0.766478
\(178\) 0 0
\(179\) −2996.32 −1.25115 −0.625574 0.780165i \(-0.715135\pi\)
−0.625574 + 0.780165i \(0.715135\pi\)
\(180\) 0 0
\(181\) −3265.41 −1.34097 −0.670487 0.741922i \(-0.733915\pi\)
−0.670487 + 0.741922i \(0.733915\pi\)
\(182\) 0 0
\(183\) −3895.71 −1.57365
\(184\) 0 0
\(185\) 1981.11 0.787321
\(186\) 0 0
\(187\) −859.296 −0.336032
\(188\) 0 0
\(189\) −3577.40 −1.37681
\(190\) 0 0
\(191\) −1502.27 −0.569113 −0.284556 0.958659i \(-0.591846\pi\)
−0.284556 + 0.958659i \(0.591846\pi\)
\(192\) 0 0
\(193\) −4103.30 −1.53037 −0.765186 0.643809i \(-0.777353\pi\)
−0.765186 + 0.643809i \(0.777353\pi\)
\(194\) 0 0
\(195\) 3128.48 1.14890
\(196\) 0 0
\(197\) 5420.95 1.96054 0.980271 0.197658i \(-0.0633337\pi\)
0.980271 + 0.197658i \(0.0633337\pi\)
\(198\) 0 0
\(199\) 1666.05 0.593484 0.296742 0.954958i \(-0.404100\pi\)
0.296742 + 0.954958i \(0.404100\pi\)
\(200\) 0 0
\(201\) −2673.83 −0.938297
\(202\) 0 0
\(203\) 2892.57 1.00009
\(204\) 0 0
\(205\) 179.919 0.0612980
\(206\) 0 0
\(207\) 310.789 0.104354
\(208\) 0 0
\(209\) 947.367 0.313544
\(210\) 0 0
\(211\) 5865.90 1.91386 0.956931 0.290314i \(-0.0937598\pi\)
0.956931 + 0.290314i \(0.0937598\pi\)
\(212\) 0 0
\(213\) 4290.06 1.38005
\(214\) 0 0
\(215\) 4905.37 1.55602
\(216\) 0 0
\(217\) 7252.26 2.26873
\(218\) 0 0
\(219\) 1671.71 0.515817
\(220\) 0 0
\(221\) −846.090 −0.257530
\(222\) 0 0
\(223\) −5402.94 −1.62246 −0.811229 0.584729i \(-0.801201\pi\)
−0.811229 + 0.584729i \(0.801201\pi\)
\(224\) 0 0
\(225\) 29.2110 0.00865511
\(226\) 0 0
\(227\) −2571.34 −0.751833 −0.375916 0.926654i \(-0.622672\pi\)
−0.375916 + 0.926654i \(0.622672\pi\)
\(228\) 0 0
\(229\) 3256.08 0.939597 0.469799 0.882774i \(-0.344327\pi\)
0.469799 + 0.882774i \(0.344327\pi\)
\(230\) 0 0
\(231\) −7095.63 −2.02103
\(232\) 0 0
\(233\) 1205.58 0.338972 0.169486 0.985533i \(-0.445789\pi\)
0.169486 + 0.985533i \(0.445789\pi\)
\(234\) 0 0
\(235\) −1914.88 −0.531544
\(236\) 0 0
\(237\) 3576.45 0.980234
\(238\) 0 0
\(239\) −3404.52 −0.921424 −0.460712 0.887550i \(-0.652406\pi\)
−0.460712 + 0.887550i \(0.652406\pi\)
\(240\) 0 0
\(241\) 3301.51 0.882443 0.441221 0.897398i \(-0.354545\pi\)
0.441221 + 0.897398i \(0.354545\pi\)
\(242\) 0 0
\(243\) 521.386 0.137642
\(244\) 0 0
\(245\) 4254.38 1.10940
\(246\) 0 0
\(247\) 932.807 0.240296
\(248\) 0 0
\(249\) 2000.85 0.509233
\(250\) 0 0
\(251\) −5330.81 −1.34055 −0.670275 0.742113i \(-0.733824\pi\)
−0.670275 + 0.742113i \(0.733824\pi\)
\(252\) 0 0
\(253\) −8325.09 −2.06875
\(254\) 0 0
\(255\) −1098.18 −0.269689
\(256\) 0 0
\(257\) 3812.14 0.925272 0.462636 0.886548i \(-0.346904\pi\)
0.462636 + 0.886548i \(0.346904\pi\)
\(258\) 0 0
\(259\) −4424.26 −1.06143
\(260\) 0 0
\(261\) −203.262 −0.0482055
\(262\) 0 0
\(263\) 899.159 0.210816 0.105408 0.994429i \(-0.466385\pi\)
0.105408 + 0.994429i \(0.466385\pi\)
\(264\) 0 0
\(265\) −5821.47 −1.34947
\(266\) 0 0
\(267\) 3619.75 0.829682
\(268\) 0 0
\(269\) 2645.77 0.599685 0.299842 0.953989i \(-0.403066\pi\)
0.299842 + 0.953989i \(0.403066\pi\)
\(270\) 0 0
\(271\) 516.529 0.115782 0.0578909 0.998323i \(-0.481562\pi\)
0.0578909 + 0.998323i \(0.481562\pi\)
\(272\) 0 0
\(273\) −6986.58 −1.54889
\(274\) 0 0
\(275\) −782.473 −0.171582
\(276\) 0 0
\(277\) 2365.14 0.513023 0.256512 0.966541i \(-0.417427\pi\)
0.256512 + 0.966541i \(0.417427\pi\)
\(278\) 0 0
\(279\) −509.621 −0.109356
\(280\) 0 0
\(281\) −1526.54 −0.324077 −0.162038 0.986784i \(-0.551807\pi\)
−0.162038 + 0.986784i \(0.551807\pi\)
\(282\) 0 0
\(283\) 4764.01 1.00067 0.500337 0.865831i \(-0.333209\pi\)
0.500337 + 0.865831i \(0.333209\pi\)
\(284\) 0 0
\(285\) 1210.73 0.251641
\(286\) 0 0
\(287\) −401.799 −0.0826391
\(288\) 0 0
\(289\) −4616.00 −0.939548
\(290\) 0 0
\(291\) 5160.10 1.03949
\(292\) 0 0
\(293\) 3189.07 0.635862 0.317931 0.948114i \(-0.397012\pi\)
0.317931 + 0.948114i \(0.397012\pi\)
\(294\) 0 0
\(295\) −3985.08 −0.786509
\(296\) 0 0
\(297\) −6733.86 −1.31562
\(298\) 0 0
\(299\) −8197.14 −1.58546
\(300\) 0 0
\(301\) −10954.8 −2.09775
\(302\) 0 0
\(303\) −10151.9 −1.92479
\(304\) 0 0
\(305\) 8601.29 1.61478
\(306\) 0 0
\(307\) 5998.56 1.11517 0.557583 0.830121i \(-0.311729\pi\)
0.557583 + 0.830121i \(0.311729\pi\)
\(308\) 0 0
\(309\) −4084.10 −0.751898
\(310\) 0 0
\(311\) 5114.96 0.932614 0.466307 0.884623i \(-0.345584\pi\)
0.466307 + 0.884623i \(0.345584\pi\)
\(312\) 0 0
\(313\) 2131.63 0.384942 0.192471 0.981303i \(-0.438350\pi\)
0.192471 + 0.981303i \(0.438350\pi\)
\(314\) 0 0
\(315\) −584.851 −0.104611
\(316\) 0 0
\(317\) 4457.03 0.789691 0.394845 0.918748i \(-0.370798\pi\)
0.394845 + 0.918748i \(0.370798\pi\)
\(318\) 0 0
\(319\) 5444.78 0.955640
\(320\) 0 0
\(321\) −3662.45 −0.636816
\(322\) 0 0
\(323\) −327.440 −0.0564064
\(324\) 0 0
\(325\) −770.448 −0.131498
\(326\) 0 0
\(327\) 7041.79 1.19086
\(328\) 0 0
\(329\) 4276.34 0.716602
\(330\) 0 0
\(331\) −4143.68 −0.688088 −0.344044 0.938954i \(-0.611797\pi\)
−0.344044 + 0.938954i \(0.611797\pi\)
\(332\) 0 0
\(333\) 310.895 0.0511620
\(334\) 0 0
\(335\) 5903.53 0.962819
\(336\) 0 0
\(337\) −5148.60 −0.832231 −0.416115 0.909312i \(-0.636609\pi\)
−0.416115 + 0.909312i \(0.636609\pi\)
\(338\) 0 0
\(339\) 3832.66 0.614045
\(340\) 0 0
\(341\) 13651.2 2.16790
\(342\) 0 0
\(343\) −415.184 −0.0653581
\(344\) 0 0
\(345\) −10639.5 −1.66032
\(346\) 0 0
\(347\) −5873.56 −0.908672 −0.454336 0.890830i \(-0.650123\pi\)
−0.454336 + 0.890830i \(0.650123\pi\)
\(348\) 0 0
\(349\) 5167.58 0.792590 0.396295 0.918123i \(-0.370296\pi\)
0.396295 + 0.918123i \(0.370296\pi\)
\(350\) 0 0
\(351\) −6630.37 −1.00827
\(352\) 0 0
\(353\) 7191.17 1.08427 0.542135 0.840291i \(-0.317616\pi\)
0.542135 + 0.840291i \(0.317616\pi\)
\(354\) 0 0
\(355\) −9471.98 −1.41611
\(356\) 0 0
\(357\) 2452.47 0.363582
\(358\) 0 0
\(359\) −7049.62 −1.03639 −0.518196 0.855262i \(-0.673396\pi\)
−0.518196 + 0.855262i \(0.673396\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −6205.84 −0.897307
\(364\) 0 0
\(365\) −3690.96 −0.529298
\(366\) 0 0
\(367\) 6935.53 0.986463 0.493231 0.869898i \(-0.335816\pi\)
0.493231 + 0.869898i \(0.335816\pi\)
\(368\) 0 0
\(369\) 28.2346 0.00398330
\(370\) 0 0
\(371\) 13000.6 1.81929
\(372\) 0 0
\(373\) −6346.92 −0.881048 −0.440524 0.897741i \(-0.645207\pi\)
−0.440524 + 0.897741i \(0.645207\pi\)
\(374\) 0 0
\(375\) 6965.35 0.959171
\(376\) 0 0
\(377\) 5361.10 0.732389
\(378\) 0 0
\(379\) 11363.0 1.54004 0.770022 0.638017i \(-0.220245\pi\)
0.770022 + 0.638017i \(0.220245\pi\)
\(380\) 0 0
\(381\) 579.386 0.0779077
\(382\) 0 0
\(383\) −3848.09 −0.513390 −0.256695 0.966493i \(-0.582634\pi\)
−0.256695 + 0.966493i \(0.582634\pi\)
\(384\) 0 0
\(385\) 15666.4 2.07385
\(386\) 0 0
\(387\) 769.798 0.101114
\(388\) 0 0
\(389\) 1936.75 0.252435 0.126217 0.992003i \(-0.459716\pi\)
0.126217 + 0.992003i \(0.459716\pi\)
\(390\) 0 0
\(391\) 2877.42 0.372167
\(392\) 0 0
\(393\) −2005.19 −0.257375
\(394\) 0 0
\(395\) −7896.41 −1.00585
\(396\) 0 0
\(397\) 3851.40 0.486892 0.243446 0.969914i \(-0.421722\pi\)
0.243446 + 0.969914i \(0.421722\pi\)
\(398\) 0 0
\(399\) −2703.83 −0.339251
\(400\) 0 0
\(401\) −1381.47 −0.172038 −0.0860189 0.996294i \(-0.527415\pi\)
−0.0860189 + 0.996294i \(0.527415\pi\)
\(402\) 0 0
\(403\) 13441.4 1.66145
\(404\) 0 0
\(405\) −9202.00 −1.12901
\(406\) 0 0
\(407\) −8327.94 −1.01425
\(408\) 0 0
\(409\) 368.878 0.0445962 0.0222981 0.999751i \(-0.492902\pi\)
0.0222981 + 0.999751i \(0.492902\pi\)
\(410\) 0 0
\(411\) 16752.1 2.01052
\(412\) 0 0
\(413\) 8899.56 1.06034
\(414\) 0 0
\(415\) −4417.66 −0.522541
\(416\) 0 0
\(417\) 302.875 0.0355680
\(418\) 0 0
\(419\) −1814.85 −0.211602 −0.105801 0.994387i \(-0.533741\pi\)
−0.105801 + 0.994387i \(0.533741\pi\)
\(420\) 0 0
\(421\) 4614.01 0.534141 0.267070 0.963677i \(-0.413944\pi\)
0.267070 + 0.963677i \(0.413944\pi\)
\(422\) 0 0
\(423\) −300.501 −0.0345410
\(424\) 0 0
\(425\) 270.448 0.0308674
\(426\) 0 0
\(427\) −19208.6 −2.17697
\(428\) 0 0
\(429\) −13151.1 −1.48005
\(430\) 0 0
\(431\) −2739.14 −0.306125 −0.153062 0.988217i \(-0.548914\pi\)
−0.153062 + 0.988217i \(0.548914\pi\)
\(432\) 0 0
\(433\) −10827.9 −1.20174 −0.600872 0.799345i \(-0.705180\pi\)
−0.600872 + 0.799345i \(0.705180\pi\)
\(434\) 0 0
\(435\) 6958.42 0.766967
\(436\) 0 0
\(437\) −3172.33 −0.347261
\(438\) 0 0
\(439\) −10359.5 −1.12627 −0.563137 0.826364i \(-0.690406\pi\)
−0.563137 + 0.826364i \(0.690406\pi\)
\(440\) 0 0
\(441\) 667.638 0.0720913
\(442\) 0 0
\(443\) 8040.36 0.862322 0.431161 0.902275i \(-0.358104\pi\)
0.431161 + 0.902275i \(0.358104\pi\)
\(444\) 0 0
\(445\) −7992.01 −0.851365
\(446\) 0 0
\(447\) 9726.59 1.02920
\(448\) 0 0
\(449\) −14071.8 −1.47904 −0.739522 0.673132i \(-0.764948\pi\)
−0.739522 + 0.673132i \(0.764948\pi\)
\(450\) 0 0
\(451\) −756.320 −0.0789661
\(452\) 0 0
\(453\) −15551.2 −1.61294
\(454\) 0 0
\(455\) 15425.6 1.58937
\(456\) 0 0
\(457\) −4372.74 −0.447589 −0.223795 0.974636i \(-0.571845\pi\)
−0.223795 + 0.974636i \(0.571845\pi\)
\(458\) 0 0
\(459\) 2327.44 0.236679
\(460\) 0 0
\(461\) −17819.4 −1.80029 −0.900146 0.435589i \(-0.856540\pi\)
−0.900146 + 0.435589i \(0.856540\pi\)
\(462\) 0 0
\(463\) 7114.51 0.714124 0.357062 0.934081i \(-0.383778\pi\)
0.357062 + 0.934081i \(0.383778\pi\)
\(464\) 0 0
\(465\) 17446.2 1.73989
\(466\) 0 0
\(467\) −8208.41 −0.813361 −0.406681 0.913570i \(-0.633314\pi\)
−0.406681 + 0.913570i \(0.633314\pi\)
\(468\) 0 0
\(469\) −13183.9 −1.29803
\(470\) 0 0
\(471\) −7422.92 −0.726178
\(472\) 0 0
\(473\) −20620.5 −2.00451
\(474\) 0 0
\(475\) −298.166 −0.0288017
\(476\) 0 0
\(477\) −913.561 −0.0876920
\(478\) 0 0
\(479\) −11889.5 −1.13413 −0.567064 0.823674i \(-0.691921\pi\)
−0.567064 + 0.823674i \(0.691921\pi\)
\(480\) 0 0
\(481\) −8199.95 −0.777309
\(482\) 0 0
\(483\) 23760.2 2.23836
\(484\) 0 0
\(485\) −11392.9 −1.06665
\(486\) 0 0
\(487\) 13922.4 1.29545 0.647725 0.761875i \(-0.275721\pi\)
0.647725 + 0.761875i \(0.275721\pi\)
\(488\) 0 0
\(489\) 20096.8 1.85850
\(490\) 0 0
\(491\) 2718.22 0.249840 0.124920 0.992167i \(-0.460133\pi\)
0.124920 + 0.992167i \(0.460133\pi\)
\(492\) 0 0
\(493\) −1881.89 −0.171919
\(494\) 0 0
\(495\) −1100.89 −0.0999618
\(496\) 0 0
\(497\) 21153.0 1.90914
\(498\) 0 0
\(499\) −7830.51 −0.702489 −0.351244 0.936284i \(-0.614241\pi\)
−0.351244 + 0.936284i \(0.614241\pi\)
\(500\) 0 0
\(501\) −16574.4 −1.47802
\(502\) 0 0
\(503\) −6554.35 −0.581002 −0.290501 0.956875i \(-0.593822\pi\)
−0.290501 + 0.956875i \(0.593822\pi\)
\(504\) 0 0
\(505\) 22414.3 1.97510
\(506\) 0 0
\(507\) −1146.06 −0.100391
\(508\) 0 0
\(509\) 12611.4 1.09821 0.549105 0.835753i \(-0.314969\pi\)
0.549105 + 0.835753i \(0.314969\pi\)
\(510\) 0 0
\(511\) 8242.73 0.713575
\(512\) 0 0
\(513\) −2565.98 −0.220840
\(514\) 0 0
\(515\) 9017.25 0.771548
\(516\) 0 0
\(517\) 8049.50 0.684752
\(518\) 0 0
\(519\) −1574.08 −0.133130
\(520\) 0 0
\(521\) 1695.01 0.142533 0.0712666 0.997457i \(-0.477296\pi\)
0.0712666 + 0.997457i \(0.477296\pi\)
\(522\) 0 0
\(523\) −21001.4 −1.75589 −0.877944 0.478764i \(-0.841085\pi\)
−0.877944 + 0.478764i \(0.841085\pi\)
\(524\) 0 0
\(525\) 2233.22 0.185649
\(526\) 0 0
\(527\) −4718.28 −0.390003
\(528\) 0 0
\(529\) 15710.2 1.29121
\(530\) 0 0
\(531\) −625.377 −0.0511093
\(532\) 0 0
\(533\) −744.696 −0.0605185
\(534\) 0 0
\(535\) 8086.28 0.653459
\(536\) 0 0
\(537\) 16097.1 1.29356
\(538\) 0 0
\(539\) −17884.0 −1.42916
\(540\) 0 0
\(541\) −555.994 −0.0441849 −0.0220925 0.999756i \(-0.507033\pi\)
−0.0220925 + 0.999756i \(0.507033\pi\)
\(542\) 0 0
\(543\) 17542.7 1.38643
\(544\) 0 0
\(545\) −15547.5 −1.22198
\(546\) 0 0
\(547\) −2987.93 −0.233555 −0.116778 0.993158i \(-0.537256\pi\)
−0.116778 + 0.993158i \(0.537256\pi\)
\(548\) 0 0
\(549\) 1349.80 0.104933
\(550\) 0 0
\(551\) 2074.77 0.160414
\(552\) 0 0
\(553\) 17634.4 1.35604
\(554\) 0 0
\(555\) −10643.1 −0.814008
\(556\) 0 0
\(557\) −4357.38 −0.331469 −0.165735 0.986170i \(-0.552999\pi\)
−0.165735 + 0.986170i \(0.552999\pi\)
\(558\) 0 0
\(559\) −20303.6 −1.53623
\(560\) 0 0
\(561\) 4616.38 0.347422
\(562\) 0 0
\(563\) 1669.67 0.124988 0.0624941 0.998045i \(-0.480095\pi\)
0.0624941 + 0.998045i \(0.480095\pi\)
\(564\) 0 0
\(565\) −8462.08 −0.630093
\(566\) 0 0
\(567\) 20550.1 1.52209
\(568\) 0 0
\(569\) −20440.1 −1.50596 −0.752982 0.658041i \(-0.771385\pi\)
−0.752982 + 0.658041i \(0.771385\pi\)
\(570\) 0 0
\(571\) −6920.98 −0.507240 −0.253620 0.967304i \(-0.581621\pi\)
−0.253620 + 0.967304i \(0.581621\pi\)
\(572\) 0 0
\(573\) 8070.62 0.588403
\(574\) 0 0
\(575\) 2620.17 0.190032
\(576\) 0 0
\(577\) −5021.59 −0.362307 −0.181154 0.983455i \(-0.557983\pi\)
−0.181154 + 0.983455i \(0.557983\pi\)
\(578\) 0 0
\(579\) 22044.1 1.58225
\(580\) 0 0
\(581\) 9865.61 0.704466
\(582\) 0 0
\(583\) 24471.5 1.73843
\(584\) 0 0
\(585\) −1083.97 −0.0766093
\(586\) 0 0
\(587\) −6784.80 −0.477068 −0.238534 0.971134i \(-0.576667\pi\)
−0.238534 + 0.971134i \(0.576667\pi\)
\(588\) 0 0
\(589\) 5201.87 0.363904
\(590\) 0 0
\(591\) −29122.9 −2.02700
\(592\) 0 0
\(593\) 22058.5 1.52754 0.763771 0.645487i \(-0.223346\pi\)
0.763771 + 0.645487i \(0.223346\pi\)
\(594\) 0 0
\(595\) −5414.80 −0.373084
\(596\) 0 0
\(597\) −8950.51 −0.613601
\(598\) 0 0
\(599\) −15616.7 −1.06524 −0.532620 0.846354i \(-0.678793\pi\)
−0.532620 + 0.846354i \(0.678793\pi\)
\(600\) 0 0
\(601\) 26769.6 1.81690 0.908448 0.417998i \(-0.137268\pi\)
0.908448 + 0.417998i \(0.137268\pi\)
\(602\) 0 0
\(603\) 926.439 0.0625664
\(604\) 0 0
\(605\) 13701.8 0.920757
\(606\) 0 0
\(607\) 5164.26 0.345323 0.172661 0.984981i \(-0.444763\pi\)
0.172661 + 0.984981i \(0.444763\pi\)
\(608\) 0 0
\(609\) −15539.7 −1.03399
\(610\) 0 0
\(611\) 7925.79 0.524784
\(612\) 0 0
\(613\) 4226.03 0.278447 0.139223 0.990261i \(-0.455539\pi\)
0.139223 + 0.990261i \(0.455539\pi\)
\(614\) 0 0
\(615\) −966.576 −0.0633758
\(616\) 0 0
\(617\) 14233.5 0.928717 0.464358 0.885647i \(-0.346285\pi\)
0.464358 + 0.885647i \(0.346285\pi\)
\(618\) 0 0
\(619\) −3737.82 −0.242707 −0.121353 0.992609i \(-0.538723\pi\)
−0.121353 + 0.992609i \(0.538723\pi\)
\(620\) 0 0
\(621\) 22548.8 1.45709
\(622\) 0 0
\(623\) 17847.9 1.14777
\(624\) 0 0
\(625\) −17340.4 −1.10978
\(626\) 0 0
\(627\) −5089.52 −0.324172
\(628\) 0 0
\(629\) 2878.40 0.182463
\(630\) 0 0
\(631\) −2893.31 −0.182537 −0.0912684 0.995826i \(-0.529092\pi\)
−0.0912684 + 0.995826i \(0.529092\pi\)
\(632\) 0 0
\(633\) −31513.3 −1.97874
\(634\) 0 0
\(635\) −1279.22 −0.0799438
\(636\) 0 0
\(637\) −17609.1 −1.09529
\(638\) 0 0
\(639\) −1486.43 −0.0920226
\(640\) 0 0
\(641\) 6867.26 0.423152 0.211576 0.977362i \(-0.432140\pi\)
0.211576 + 0.977362i \(0.432140\pi\)
\(642\) 0 0
\(643\) 4975.32 0.305144 0.152572 0.988292i \(-0.451244\pi\)
0.152572 + 0.988292i \(0.451244\pi\)
\(644\) 0 0
\(645\) −26353.0 −1.60876
\(646\) 0 0
\(647\) −24033.2 −1.46035 −0.730174 0.683262i \(-0.760561\pi\)
−0.730174 + 0.683262i \(0.760561\pi\)
\(648\) 0 0
\(649\) 16751.9 1.01321
\(650\) 0 0
\(651\) −38961.2 −2.34564
\(652\) 0 0
\(653\) −25718.4 −1.54125 −0.770626 0.637288i \(-0.780056\pi\)
−0.770626 + 0.637288i \(0.780056\pi\)
\(654\) 0 0
\(655\) 4427.24 0.264102
\(656\) 0 0
\(657\) −579.221 −0.0343951
\(658\) 0 0
\(659\) −7970.84 −0.471168 −0.235584 0.971854i \(-0.575700\pi\)
−0.235584 + 0.971854i \(0.575700\pi\)
\(660\) 0 0
\(661\) 14207.8 0.836038 0.418019 0.908438i \(-0.362725\pi\)
0.418019 + 0.908438i \(0.362725\pi\)
\(662\) 0 0
\(663\) 4545.43 0.266259
\(664\) 0 0
\(665\) 5969.77 0.348117
\(666\) 0 0
\(667\) −18232.2 −1.05840
\(668\) 0 0
\(669\) 29026.1 1.67745
\(670\) 0 0
\(671\) −36157.0 −2.08021
\(672\) 0 0
\(673\) 17080.0 0.978287 0.489143 0.872203i \(-0.337309\pi\)
0.489143 + 0.872203i \(0.337309\pi\)
\(674\) 0 0
\(675\) 2119.36 0.120851
\(676\) 0 0
\(677\) −16410.9 −0.931641 −0.465821 0.884879i \(-0.654241\pi\)
−0.465821 + 0.884879i \(0.654241\pi\)
\(678\) 0 0
\(679\) 25442.9 1.43801
\(680\) 0 0
\(681\) 13814.0 0.777317
\(682\) 0 0
\(683\) −183.169 −0.0102617 −0.00513086 0.999987i \(-0.501633\pi\)
−0.00513086 + 0.999987i \(0.501633\pi\)
\(684\) 0 0
\(685\) −36986.9 −2.06306
\(686\) 0 0
\(687\) −17492.6 −0.971446
\(688\) 0 0
\(689\) 24095.4 1.33231
\(690\) 0 0
\(691\) 14974.0 0.824366 0.412183 0.911101i \(-0.364766\pi\)
0.412183 + 0.911101i \(0.364766\pi\)
\(692\) 0 0
\(693\) 2458.52 0.134764
\(694\) 0 0
\(695\) −668.715 −0.0364976
\(696\) 0 0
\(697\) 261.408 0.0142059
\(698\) 0 0
\(699\) −6476.74 −0.350462
\(700\) 0 0
\(701\) 4341.08 0.233895 0.116947 0.993138i \(-0.462689\pi\)
0.116947 + 0.993138i \(0.462689\pi\)
\(702\) 0 0
\(703\) −3173.41 −0.170253
\(704\) 0 0
\(705\) 10287.3 0.549561
\(706\) 0 0
\(707\) −50056.1 −2.66273
\(708\) 0 0
\(709\) 23931.9 1.26767 0.633837 0.773467i \(-0.281479\pi\)
0.633837 + 0.773467i \(0.281479\pi\)
\(710\) 0 0
\(711\) −1239.18 −0.0653627
\(712\) 0 0
\(713\) −45712.0 −2.40102
\(714\) 0 0
\(715\) 29036.1 1.51873
\(716\) 0 0
\(717\) 18290.1 0.952656
\(718\) 0 0
\(719\) 16258.6 0.843314 0.421657 0.906756i \(-0.361449\pi\)
0.421657 + 0.906756i \(0.361449\pi\)
\(720\) 0 0
\(721\) −20137.5 −1.04017
\(722\) 0 0
\(723\) −17736.6 −0.912354
\(724\) 0 0
\(725\) −1713.65 −0.0877837
\(726\) 0 0
\(727\) −2608.95 −0.133096 −0.0665479 0.997783i \(-0.521199\pi\)
−0.0665479 + 0.997783i \(0.521199\pi\)
\(728\) 0 0
\(729\) 18145.4 0.921881
\(730\) 0 0
\(731\) 7127.11 0.360610
\(732\) 0 0
\(733\) 9843.63 0.496020 0.248010 0.968757i \(-0.420223\pi\)
0.248010 + 0.968757i \(0.420223\pi\)
\(734\) 0 0
\(735\) −22855.7 −1.14700
\(736\) 0 0
\(737\) −24816.5 −1.24033
\(738\) 0 0
\(739\) 2419.46 0.120435 0.0602174 0.998185i \(-0.480821\pi\)
0.0602174 + 0.998185i \(0.480821\pi\)
\(740\) 0 0
\(741\) −5011.30 −0.248441
\(742\) 0 0
\(743\) 10442.9 0.515631 0.257816 0.966194i \(-0.416997\pi\)
0.257816 + 0.966194i \(0.416997\pi\)
\(744\) 0 0
\(745\) −21475.2 −1.05610
\(746\) 0 0
\(747\) −693.262 −0.0339560
\(748\) 0 0
\(749\) −18058.4 −0.880963
\(750\) 0 0
\(751\) 29434.6 1.43021 0.715103 0.699019i \(-0.246380\pi\)
0.715103 + 0.699019i \(0.246380\pi\)
\(752\) 0 0
\(753\) 28638.6 1.38599
\(754\) 0 0
\(755\) 34335.4 1.65509
\(756\) 0 0
\(757\) −20638.3 −0.990902 −0.495451 0.868636i \(-0.664997\pi\)
−0.495451 + 0.868636i \(0.664997\pi\)
\(758\) 0 0
\(759\) 44724.7 2.13887
\(760\) 0 0
\(761\) 4103.09 0.195449 0.0977246 0.995213i \(-0.468844\pi\)
0.0977246 + 0.995213i \(0.468844\pi\)
\(762\) 0 0
\(763\) 34721.0 1.64742
\(764\) 0 0
\(765\) 380.501 0.0179831
\(766\) 0 0
\(767\) 16494.5 0.776508
\(768\) 0 0
\(769\) −14020.0 −0.657444 −0.328722 0.944427i \(-0.606618\pi\)
−0.328722 + 0.944427i \(0.606618\pi\)
\(770\) 0 0
\(771\) −20479.9 −0.956635
\(772\) 0 0
\(773\) 32823.8 1.52728 0.763642 0.645640i \(-0.223409\pi\)
0.763642 + 0.645640i \(0.223409\pi\)
\(774\) 0 0
\(775\) −4296.46 −0.199140
\(776\) 0 0
\(777\) 23768.4 1.09741
\(778\) 0 0
\(779\) −288.200 −0.0132553
\(780\) 0 0
\(781\) 39817.0 1.82428
\(782\) 0 0
\(783\) −14747.4 −0.673090
\(784\) 0 0
\(785\) 16389.0 0.745157
\(786\) 0 0
\(787\) 33903.0 1.53559 0.767796 0.640694i \(-0.221353\pi\)
0.767796 + 0.640694i \(0.221353\pi\)
\(788\) 0 0
\(789\) −4830.54 −0.217962
\(790\) 0 0
\(791\) 18897.7 0.849462
\(792\) 0 0
\(793\) −35601.3 −1.59425
\(794\) 0 0
\(795\) 31274.6 1.39521
\(796\) 0 0
\(797\) 20830.2 0.925778 0.462889 0.886416i \(-0.346813\pi\)
0.462889 + 0.886416i \(0.346813\pi\)
\(798\) 0 0
\(799\) −2782.16 −0.123186
\(800\) 0 0
\(801\) −1254.18 −0.0553238
\(802\) 0 0
\(803\) 15515.6 0.681859
\(804\) 0 0
\(805\) −52460.0 −2.29686
\(806\) 0 0
\(807\) −14213.8 −0.620012
\(808\) 0 0
\(809\) −26505.7 −1.15190 −0.575952 0.817484i \(-0.695368\pi\)
−0.575952 + 0.817484i \(0.695368\pi\)
\(810\) 0 0
\(811\) −10829.0 −0.468876 −0.234438 0.972131i \(-0.575325\pi\)
−0.234438 + 0.972131i \(0.575325\pi\)
\(812\) 0 0
\(813\) −2774.94 −0.119706
\(814\) 0 0
\(815\) −44371.5 −1.90707
\(816\) 0 0
\(817\) −7857.58 −0.336477
\(818\) 0 0
\(819\) 2420.73 0.103281
\(820\) 0 0
\(821\) 42046.8 1.78738 0.893692 0.448681i \(-0.148106\pi\)
0.893692 + 0.448681i \(0.148106\pi\)
\(822\) 0 0
\(823\) −12949.0 −0.548448 −0.274224 0.961666i \(-0.588421\pi\)
−0.274224 + 0.961666i \(0.588421\pi\)
\(824\) 0 0
\(825\) 4203.67 0.177397
\(826\) 0 0
\(827\) −13874.0 −0.583368 −0.291684 0.956515i \(-0.594216\pi\)
−0.291684 + 0.956515i \(0.594216\pi\)
\(828\) 0 0
\(829\) −2387.20 −0.100013 −0.0500065 0.998749i \(-0.515924\pi\)
−0.0500065 + 0.998749i \(0.515924\pi\)
\(830\) 0 0
\(831\) −12706.2 −0.530412
\(832\) 0 0
\(833\) 6181.27 0.257105
\(834\) 0 0
\(835\) 36594.4 1.51665
\(836\) 0 0
\(837\) −36974.8 −1.52692
\(838\) 0 0
\(839\) 47308.8 1.94670 0.973350 0.229324i \(-0.0736515\pi\)
0.973350 + 0.229324i \(0.0736515\pi\)
\(840\) 0 0
\(841\) −12464.7 −0.511080
\(842\) 0 0
\(843\) 8200.99 0.335062
\(844\) 0 0
\(845\) 2530.37 0.103015
\(846\) 0 0
\(847\) −30599.2 −1.24132
\(848\) 0 0
\(849\) −25593.6 −1.03459
\(850\) 0 0
\(851\) 27886.7 1.12332
\(852\) 0 0
\(853\) −18211.3 −0.730998 −0.365499 0.930812i \(-0.619102\pi\)
−0.365499 + 0.930812i \(0.619102\pi\)
\(854\) 0 0
\(855\) −419.499 −0.0167796
\(856\) 0 0
\(857\) −14824.5 −0.590893 −0.295447 0.955359i \(-0.595468\pi\)
−0.295447 + 0.955359i \(0.595468\pi\)
\(858\) 0 0
\(859\) −44191.1 −1.75527 −0.877637 0.479326i \(-0.840881\pi\)
−0.877637 + 0.479326i \(0.840881\pi\)
\(860\) 0 0
\(861\) 2158.58 0.0854403
\(862\) 0 0
\(863\) −34927.0 −1.37767 −0.688835 0.724918i \(-0.741878\pi\)
−0.688835 + 0.724918i \(0.741878\pi\)
\(864\) 0 0
\(865\) 3475.40 0.136609
\(866\) 0 0
\(867\) 24798.5 0.971395
\(868\) 0 0
\(869\) 33193.9 1.29577
\(870\) 0 0
\(871\) −24435.1 −0.950575
\(872\) 0 0
\(873\) −1787.89 −0.0693138
\(874\) 0 0
\(875\) 34344.1 1.32691
\(876\) 0 0
\(877\) −16914.8 −0.651278 −0.325639 0.945494i \(-0.605579\pi\)
−0.325639 + 0.945494i \(0.605579\pi\)
\(878\) 0 0
\(879\) −17132.6 −0.657415
\(880\) 0 0
\(881\) 12634.8 0.483177 0.241588 0.970379i \(-0.422332\pi\)
0.241588 + 0.970379i \(0.422332\pi\)
\(882\) 0 0
\(883\) −47260.3 −1.80117 −0.900586 0.434678i \(-0.856862\pi\)
−0.900586 + 0.434678i \(0.856862\pi\)
\(884\) 0 0
\(885\) 21409.0 0.813169
\(886\) 0 0
\(887\) −4758.84 −0.180142 −0.0900711 0.995935i \(-0.528709\pi\)
−0.0900711 + 0.995935i \(0.528709\pi\)
\(888\) 0 0
\(889\) 2856.78 0.107777
\(890\) 0 0
\(891\) 38682.1 1.45443
\(892\) 0 0
\(893\) 3067.31 0.114943
\(894\) 0 0
\(895\) −35540.6 −1.32736
\(896\) 0 0
\(897\) 44037.4 1.63920
\(898\) 0 0
\(899\) 29896.6 1.10913
\(900\) 0 0
\(901\) −8458.13 −0.312743
\(902\) 0 0
\(903\) 58852.1 2.16885
\(904\) 0 0
\(905\) −38732.4 −1.42266
\(906\) 0 0
\(907\) −4111.61 −0.150522 −0.0752612 0.997164i \(-0.523979\pi\)
−0.0752612 + 0.997164i \(0.523979\pi\)
\(908\) 0 0
\(909\) 3517.47 0.128347
\(910\) 0 0
\(911\) −35113.1 −1.27700 −0.638501 0.769621i \(-0.720445\pi\)
−0.638501 + 0.769621i \(0.720445\pi\)
\(912\) 0 0
\(913\) 18570.4 0.673155
\(914\) 0 0
\(915\) −46208.5 −1.66952
\(916\) 0 0
\(917\) −9887.00 −0.356049
\(918\) 0 0
\(919\) 37509.1 1.34637 0.673184 0.739475i \(-0.264926\pi\)
0.673184 + 0.739475i \(0.264926\pi\)
\(920\) 0 0
\(921\) −32226.0 −1.15297
\(922\) 0 0
\(923\) 39205.1 1.39811
\(924\) 0 0
\(925\) 2621.07 0.0931677
\(926\) 0 0
\(927\) 1415.07 0.0501371
\(928\) 0 0
\(929\) 18393.8 0.649604 0.324802 0.945782i \(-0.394702\pi\)
0.324802 + 0.945782i \(0.394702\pi\)
\(930\) 0 0
\(931\) −6814.80 −0.239899
\(932\) 0 0
\(933\) −27479.0 −0.964226
\(934\) 0 0
\(935\) −10192.5 −0.356502
\(936\) 0 0
\(937\) −17106.7 −0.596427 −0.298214 0.954499i \(-0.596391\pi\)
−0.298214 + 0.954499i \(0.596391\pi\)
\(938\) 0 0
\(939\) −11451.7 −0.397990
\(940\) 0 0
\(941\) −38100.7 −1.31992 −0.659961 0.751300i \(-0.729427\pi\)
−0.659961 + 0.751300i \(0.729427\pi\)
\(942\) 0 0
\(943\) 2532.59 0.0874576
\(944\) 0 0
\(945\) −42433.0 −1.46068
\(946\) 0 0
\(947\) −52712.7 −1.80880 −0.904400 0.426686i \(-0.859681\pi\)
−0.904400 + 0.426686i \(0.859681\pi\)
\(948\) 0 0
\(949\) 15277.1 0.522567
\(950\) 0 0
\(951\) −23944.4 −0.816458
\(952\) 0 0
\(953\) −6744.38 −0.229246 −0.114623 0.993409i \(-0.536566\pi\)
−0.114623 + 0.993409i \(0.536566\pi\)
\(954\) 0 0
\(955\) −17819.0 −0.603781
\(956\) 0 0
\(957\) −29250.9 −0.988032
\(958\) 0 0
\(959\) 82599.8 2.78132
\(960\) 0 0
\(961\) 45165.9 1.51609
\(962\) 0 0
\(963\) 1268.98 0.0424634
\(964\) 0 0
\(965\) −48670.9 −1.62360
\(966\) 0 0
\(967\) −28449.8 −0.946104 −0.473052 0.881034i \(-0.656848\pi\)
−0.473052 + 0.881034i \(0.656848\pi\)
\(968\) 0 0
\(969\) 1759.10 0.0583183
\(970\) 0 0
\(971\) −13018.7 −0.430266 −0.215133 0.976585i \(-0.569019\pi\)
−0.215133 + 0.976585i \(0.569019\pi\)
\(972\) 0 0
\(973\) 1493.39 0.0492043
\(974\) 0 0
\(975\) 4139.06 0.135955
\(976\) 0 0
\(977\) −20043.8 −0.656355 −0.328178 0.944616i \(-0.606434\pi\)
−0.328178 + 0.944616i \(0.606434\pi\)
\(978\) 0 0
\(979\) 33595.7 1.09676
\(980\) 0 0
\(981\) −2439.86 −0.0794076
\(982\) 0 0
\(983\) −1823.72 −0.0591735 −0.0295868 0.999562i \(-0.509419\pi\)
−0.0295868 + 0.999562i \(0.509419\pi\)
\(984\) 0 0
\(985\) 64300.1 2.07997
\(986\) 0 0
\(987\) −22973.7 −0.740892
\(988\) 0 0
\(989\) 69049.4 2.22006
\(990\) 0 0
\(991\) −26635.4 −0.853786 −0.426893 0.904302i \(-0.640392\pi\)
−0.426893 + 0.904302i \(0.640392\pi\)
\(992\) 0 0
\(993\) 22261.0 0.711411
\(994\) 0 0
\(995\) 19761.7 0.629637
\(996\) 0 0
\(997\) 50874.6 1.61606 0.808031 0.589139i \(-0.200533\pi\)
0.808031 + 0.589139i \(0.200533\pi\)
\(998\) 0 0
\(999\) 22556.6 0.714372
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 76.4.a.a.1.1 2
3.2 odd 2 684.4.a.g.1.1 2
4.3 odd 2 304.4.a.f.1.2 2
5.2 odd 4 1900.4.c.b.1749.4 4
5.3 odd 4 1900.4.c.b.1749.1 4
5.4 even 2 1900.4.a.b.1.2 2
8.3 odd 2 1216.4.a.h.1.1 2
8.5 even 2 1216.4.a.o.1.2 2
19.18 odd 2 1444.4.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.a.a.1.1 2 1.1 even 1 trivial
304.4.a.f.1.2 2 4.3 odd 2
684.4.a.g.1.1 2 3.2 odd 2
1216.4.a.h.1.1 2 8.3 odd 2
1216.4.a.o.1.2 2 8.5 even 2
1444.4.a.d.1.2 2 19.18 odd 2
1900.4.a.b.1.2 2 5.4 even 2
1900.4.c.b.1749.1 4 5.3 odd 4
1900.4.c.b.1749.4 4 5.2 odd 4