Properties

Label 76.4.a.a.1.2
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 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.372281 q^{3} -16.8614 q^{5} -3.51087 q^{7} -26.8614 q^{9} +O(q^{10})\) \(q+0.372281 q^{3} -16.8614 q^{5} -3.51087 q^{7} -26.8614 q^{9} -21.1386 q^{11} +14.0951 q^{13} -6.27719 q^{15} -17.2337 q^{17} -19.0000 q^{19} -1.30703 q^{21} -171.965 q^{23} +159.307 q^{25} -20.0516 q^{27} +264.198 q^{29} +185.783 q^{31} -7.86950 q^{33} +59.1983 q^{35} +212.978 q^{37} +5.24734 q^{39} -157.168 q^{41} -258.557 q^{43} +452.921 q^{45} -293.562 q^{47} -330.674 q^{49} -6.41578 q^{51} +215.791 q^{53} +356.426 q^{55} -7.07335 q^{57} -537.030 q^{59} -280.149 q^{61} +94.3070 q^{63} -237.663 q^{65} +147.291 q^{67} -64.0192 q^{69} -913.446 q^{71} -678.826 q^{73} +59.3070 q^{75} +74.2150 q^{77} -608.277 q^{79} +717.793 q^{81} +282.440 q^{83} +290.584 q^{85} +98.3561 q^{87} -214.217 q^{89} -49.4861 q^{91} +69.1634 q^{93} +320.367 q^{95} +1670.50 q^{97} +567.812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{3} - 5q^{5} - 30q^{7} - 25q^{9} + O(q^{10}) \) \( 2q - 5q^{3} - 5q^{5} - 30q^{7} - 25q^{9} - 71q^{11} - 35q^{13} - 70q^{15} - 38q^{19} + 141q^{21} - 5q^{23} + 175q^{25} + 115q^{27} + 155q^{29} - 88q^{31} + 260q^{33} - 255q^{35} + 380q^{37} + 269q^{39} - 142q^{41} + 155q^{43} + 475q^{45} - 455q^{47} + 28q^{49} - 99q^{51} - 275q^{53} - 235q^{55} + 95q^{57} - 873q^{59} + 445q^{61} + 45q^{63} - 820q^{65} + 645q^{67} - 961q^{69} - 1712q^{71} - 990q^{73} - 25q^{75} + 1395q^{77} - 1274q^{79} - 58q^{81} - 90q^{83} + 495q^{85} + 685q^{87} - 888q^{89} + 1251q^{91} + 1540q^{93} + 95q^{95} + 710q^{97} + 475q^{99} + O(q^{100}) \)

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
\(3\) 0.372281 0.0716456 0.0358228 0.999358i \(-0.488595\pi\)
0.0358228 + 0.999358i \(0.488595\pi\)
\(4\) 0 0
\(5\) −16.8614 −1.50813 −0.754065 0.656800i \(-0.771910\pi\)
−0.754065 + 0.656800i \(0.771910\pi\)
\(6\) 0 0
\(7\) −3.51087 −0.189569 −0.0947847 0.995498i \(-0.530216\pi\)
−0.0947847 + 0.995498i \(0.530216\pi\)
\(8\) 0 0
\(9\) −26.8614 −0.994867
\(10\) 0 0
\(11\) −21.1386 −0.579411 −0.289706 0.957116i \(-0.593557\pi\)
−0.289706 + 0.957116i \(0.593557\pi\)
\(12\) 0 0
\(13\) 14.0951 0.300714 0.150357 0.988632i \(-0.451958\pi\)
0.150357 + 0.988632i \(0.451958\pi\)
\(14\) 0 0
\(15\) −6.27719 −0.108051
\(16\) 0 0
\(17\) −17.2337 −0.245870 −0.122935 0.992415i \(-0.539231\pi\)
−0.122935 + 0.992415i \(0.539231\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −1.30703 −0.0135818
\(22\) 0 0
\(23\) −171.965 −1.55900 −0.779502 0.626400i \(-0.784528\pi\)
−0.779502 + 0.626400i \(0.784528\pi\)
\(24\) 0 0
\(25\) 159.307 1.27446
\(26\) 0 0
\(27\) −20.0516 −0.142923
\(28\) 0 0
\(29\) 264.198 1.69174 0.845869 0.533391i \(-0.179083\pi\)
0.845869 + 0.533391i \(0.179083\pi\)
\(30\) 0 0
\(31\) 185.783 1.07637 0.538186 0.842826i \(-0.319110\pi\)
0.538186 + 0.842826i \(0.319110\pi\)
\(32\) 0 0
\(33\) −7.86950 −0.0415123
\(34\) 0 0
\(35\) 59.1983 0.285895
\(36\) 0 0
\(37\) 212.978 0.946308 0.473154 0.880980i \(-0.343115\pi\)
0.473154 + 0.880980i \(0.343115\pi\)
\(38\) 0 0
\(39\) 5.24734 0.0215448
\(40\) 0 0
\(41\) −157.168 −0.598673 −0.299336 0.954148i \(-0.596765\pi\)
−0.299336 + 0.954148i \(0.596765\pi\)
\(42\) 0 0
\(43\) −258.557 −0.916967 −0.458483 0.888703i \(-0.651607\pi\)
−0.458483 + 0.888703i \(0.651607\pi\)
\(44\) 0 0
\(45\) 452.921 1.50039
\(46\) 0 0
\(47\) −293.562 −0.911074 −0.455537 0.890217i \(-0.650553\pi\)
−0.455537 + 0.890217i \(0.650553\pi\)
\(48\) 0 0
\(49\) −330.674 −0.964063
\(50\) 0 0
\(51\) −6.41578 −0.0176155
\(52\) 0 0
\(53\) 215.791 0.559266 0.279633 0.960107i \(-0.409787\pi\)
0.279633 + 0.960107i \(0.409787\pi\)
\(54\) 0 0
\(55\) 356.426 0.873828
\(56\) 0 0
\(57\) −7.07335 −0.0164366
\(58\) 0 0
\(59\) −537.030 −1.18501 −0.592503 0.805568i \(-0.701860\pi\)
−0.592503 + 0.805568i \(0.701860\pi\)
\(60\) 0 0
\(61\) −280.149 −0.588024 −0.294012 0.955802i \(-0.594990\pi\)
−0.294012 + 0.955802i \(0.594990\pi\)
\(62\) 0 0
\(63\) 94.3070 0.188596
\(64\) 0 0
\(65\) −237.663 −0.453515
\(66\) 0 0
\(67\) 147.291 0.268574 0.134287 0.990942i \(-0.457126\pi\)
0.134287 + 0.990942i \(0.457126\pi\)
\(68\) 0 0
\(69\) −64.0192 −0.111696
\(70\) 0 0
\(71\) −913.446 −1.52685 −0.763423 0.645899i \(-0.776483\pi\)
−0.763423 + 0.645899i \(0.776483\pi\)
\(72\) 0 0
\(73\) −678.826 −1.08836 −0.544182 0.838967i \(-0.683160\pi\)
−0.544182 + 0.838967i \(0.683160\pi\)
\(74\) 0 0
\(75\) 59.3070 0.0913092
\(76\) 0 0
\(77\) 74.2150 0.109839
\(78\) 0 0
\(79\) −608.277 −0.866285 −0.433143 0.901325i \(-0.642595\pi\)
−0.433143 + 0.901325i \(0.642595\pi\)
\(80\) 0 0
\(81\) 717.793 0.984627
\(82\) 0 0
\(83\) 282.440 0.373516 0.186758 0.982406i \(-0.440202\pi\)
0.186758 + 0.982406i \(0.440202\pi\)
\(84\) 0 0
\(85\) 290.584 0.370803
\(86\) 0 0
\(87\) 98.3561 0.121206
\(88\) 0 0
\(89\) −214.217 −0.255135 −0.127567 0.991830i \(-0.540717\pi\)
−0.127567 + 0.991830i \(0.540717\pi\)
\(90\) 0 0
\(91\) −49.4861 −0.0570061
\(92\) 0 0
\(93\) 69.1634 0.0771173
\(94\) 0 0
\(95\) 320.367 0.345989
\(96\) 0 0
\(97\) 1670.50 1.74860 0.874299 0.485387i \(-0.161321\pi\)
0.874299 + 0.485387i \(0.161321\pi\)
\(98\) 0 0
\(99\) 567.812 0.576437
\(100\) 0 0
\(101\) 166.316 0.163852 0.0819258 0.996638i \(-0.473893\pi\)
0.0819258 + 0.996638i \(0.473893\pi\)
\(102\) 0 0
\(103\) 1219.78 1.16688 0.583440 0.812156i \(-0.301706\pi\)
0.583440 + 0.812156i \(0.301706\pi\)
\(104\) 0 0
\(105\) 22.0384 0.0204831
\(106\) 0 0
\(107\) −1736.73 −1.56912 −0.784561 0.620052i \(-0.787112\pi\)
−0.784561 + 0.620052i \(0.787112\pi\)
\(108\) 0 0
\(109\) −18.2367 −0.0160253 −0.00801266 0.999968i \(-0.502551\pi\)
−0.00801266 + 0.999968i \(0.502551\pi\)
\(110\) 0 0
\(111\) 79.2878 0.0677988
\(112\) 0 0
\(113\) −1586.59 −1.32083 −0.660414 0.750902i \(-0.729619\pi\)
−0.660414 + 0.750902i \(0.729619\pi\)
\(114\) 0 0
\(115\) 2899.57 2.35118
\(116\) 0 0
\(117\) −378.614 −0.299170
\(118\) 0 0
\(119\) 60.5053 0.0466094
\(120\) 0 0
\(121\) −884.160 −0.664282
\(122\) 0 0
\(123\) −58.5109 −0.0428923
\(124\) 0 0
\(125\) −578.465 −0.413916
\(126\) 0 0
\(127\) 2327.85 1.62648 0.813240 0.581928i \(-0.197701\pi\)
0.813240 + 0.581928i \(0.197701\pi\)
\(128\) 0 0
\(129\) −96.2559 −0.0656966
\(130\) 0 0
\(131\) 631.753 0.421347 0.210674 0.977556i \(-0.432434\pi\)
0.210674 + 0.977556i \(0.432434\pi\)
\(132\) 0 0
\(133\) 66.7066 0.0434902
\(134\) 0 0
\(135\) 338.098 0.215547
\(136\) 0 0
\(137\) 1098.25 0.684892 0.342446 0.939537i \(-0.388745\pi\)
0.342446 + 0.939537i \(0.388745\pi\)
\(138\) 0 0
\(139\) 1523.38 0.929577 0.464788 0.885422i \(-0.346130\pi\)
0.464788 + 0.885422i \(0.346130\pi\)
\(140\) 0 0
\(141\) −109.288 −0.0652744
\(142\) 0 0
\(143\) −297.950 −0.174237
\(144\) 0 0
\(145\) −4454.75 −2.55136
\(146\) 0 0
\(147\) −123.104 −0.0690709
\(148\) 0 0
\(149\) −3160.49 −1.73770 −0.868849 0.495077i \(-0.835140\pi\)
−0.868849 + 0.495077i \(0.835140\pi\)
\(150\) 0 0
\(151\) −1930.72 −1.04053 −0.520263 0.854006i \(-0.674166\pi\)
−0.520263 + 0.854006i \(0.674166\pi\)
\(152\) 0 0
\(153\) 462.921 0.244608
\(154\) 0 0
\(155\) −3132.55 −1.62331
\(156\) 0 0
\(157\) 1818.29 0.924303 0.462152 0.886801i \(-0.347078\pi\)
0.462152 + 0.886801i \(0.347078\pi\)
\(158\) 0 0
\(159\) 80.3348 0.0400690
\(160\) 0 0
\(161\) 603.746 0.295540
\(162\) 0 0
\(163\) −1259.17 −0.605068 −0.302534 0.953139i \(-0.597833\pi\)
−0.302534 + 0.953139i \(0.597833\pi\)
\(164\) 0 0
\(165\) 132.691 0.0626059
\(166\) 0 0
\(167\) −1315.17 −0.609405 −0.304703 0.952447i \(-0.598557\pi\)
−0.304703 + 0.952447i \(0.598557\pi\)
\(168\) 0 0
\(169\) −1998.33 −0.909571
\(170\) 0 0
\(171\) 510.367 0.228238
\(172\) 0 0
\(173\) 2407.00 1.05781 0.528904 0.848682i \(-0.322603\pi\)
0.528904 + 0.848682i \(0.322603\pi\)
\(174\) 0 0
\(175\) −559.307 −0.241598
\(176\) 0 0
\(177\) −199.926 −0.0849004
\(178\) 0 0
\(179\) 1082.32 0.451935 0.225968 0.974135i \(-0.427446\pi\)
0.225968 + 0.974135i \(0.427446\pi\)
\(180\) 0 0
\(181\) 4317.41 1.77299 0.886494 0.462741i \(-0.153134\pi\)
0.886494 + 0.462741i \(0.153134\pi\)
\(182\) 0 0
\(183\) −104.294 −0.0421293
\(184\) 0 0
\(185\) −3591.11 −1.42716
\(186\) 0 0
\(187\) 364.296 0.142460
\(188\) 0 0
\(189\) 70.3986 0.0270939
\(190\) 0 0
\(191\) 3208.27 1.21540 0.607702 0.794165i \(-0.292091\pi\)
0.607702 + 0.794165i \(0.292091\pi\)
\(192\) 0 0
\(193\) −4206.70 −1.56894 −0.784469 0.620168i \(-0.787064\pi\)
−0.784469 + 0.620168i \(0.787064\pi\)
\(194\) 0 0
\(195\) −88.4775 −0.0324924
\(196\) 0 0
\(197\) 4789.05 1.73201 0.866004 0.500037i \(-0.166680\pi\)
0.866004 + 0.500037i \(0.166680\pi\)
\(198\) 0 0
\(199\) −3504.05 −1.24822 −0.624110 0.781337i \(-0.714538\pi\)
−0.624110 + 0.781337i \(0.714538\pi\)
\(200\) 0 0
\(201\) 54.8336 0.0192421
\(202\) 0 0
\(203\) −927.567 −0.320702
\(204\) 0 0
\(205\) 2650.08 0.902877
\(206\) 0 0
\(207\) 4619.21 1.55100
\(208\) 0 0
\(209\) 401.633 0.132926
\(210\) 0 0
\(211\) −3066.90 −1.00063 −0.500317 0.865842i \(-0.666783\pi\)
−0.500317 + 0.865842i \(0.666783\pi\)
\(212\) 0 0
\(213\) −340.059 −0.109392
\(214\) 0 0
\(215\) 4359.63 1.38290
\(216\) 0 0
\(217\) −652.259 −0.204047
\(218\) 0 0
\(219\) −252.714 −0.0779765
\(220\) 0 0
\(221\) −242.910 −0.0739363
\(222\) 0 0
\(223\) −2117.06 −0.635733 −0.317867 0.948135i \(-0.602966\pi\)
−0.317867 + 0.948135i \(0.602966\pi\)
\(224\) 0 0
\(225\) −4279.21 −1.26791
\(226\) 0 0
\(227\) 6476.34 1.89361 0.946806 0.321804i \(-0.104289\pi\)
0.946806 + 0.321804i \(0.104289\pi\)
\(228\) 0 0
\(229\) 3686.92 1.06392 0.531962 0.846768i \(-0.321455\pi\)
0.531962 + 0.846768i \(0.321455\pi\)
\(230\) 0 0
\(231\) 27.6288 0.00786946
\(232\) 0 0
\(233\) 1119.42 0.314744 0.157372 0.987539i \(-0.449698\pi\)
0.157372 + 0.987539i \(0.449698\pi\)
\(234\) 0 0
\(235\) 4949.88 1.37402
\(236\) 0 0
\(237\) −226.450 −0.0620655
\(238\) 0 0
\(239\) −6621.48 −1.79208 −0.896041 0.443971i \(-0.853569\pi\)
−0.896041 + 0.443971i \(0.853569\pi\)
\(240\) 0 0
\(241\) 2784.49 0.744253 0.372127 0.928182i \(-0.378629\pi\)
0.372127 + 0.928182i \(0.378629\pi\)
\(242\) 0 0
\(243\) 808.614 0.213468
\(244\) 0 0
\(245\) 5575.62 1.45393
\(246\) 0 0
\(247\) −267.807 −0.0689884
\(248\) 0 0
\(249\) 105.147 0.0267608
\(250\) 0 0
\(251\) −4670.19 −1.17442 −0.587210 0.809434i \(-0.699774\pi\)
−0.587210 + 0.809434i \(0.699774\pi\)
\(252\) 0 0
\(253\) 3635.09 0.903305
\(254\) 0 0
\(255\) 108.179 0.0265664
\(256\) 0 0
\(257\) 5167.86 1.25433 0.627164 0.778887i \(-0.284216\pi\)
0.627164 + 0.778887i \(0.284216\pi\)
\(258\) 0 0
\(259\) −747.740 −0.179391
\(260\) 0 0
\(261\) −7096.74 −1.68305
\(262\) 0 0
\(263\) −3254.16 −0.762966 −0.381483 0.924376i \(-0.624587\pi\)
−0.381483 + 0.924376i \(0.624587\pi\)
\(264\) 0 0
\(265\) −3638.53 −0.843446
\(266\) 0 0
\(267\) −79.7492 −0.0182793
\(268\) 0 0
\(269\) −1547.77 −0.350814 −0.175407 0.984496i \(-0.556124\pi\)
−0.175407 + 0.984496i \(0.556124\pi\)
\(270\) 0 0
\(271\) −4969.53 −1.11394 −0.556969 0.830533i \(-0.688036\pi\)
−0.556969 + 0.830533i \(0.688036\pi\)
\(272\) 0 0
\(273\) −18.4228 −0.00408423
\(274\) 0 0
\(275\) −3367.53 −0.738435
\(276\) 0 0
\(277\) 279.862 0.0607049 0.0303525 0.999539i \(-0.490337\pi\)
0.0303525 + 0.999539i \(0.490337\pi\)
\(278\) 0 0
\(279\) −4990.38 −1.07085
\(280\) 0 0
\(281\) 2092.54 0.444236 0.222118 0.975020i \(-0.428703\pi\)
0.222118 + 0.975020i \(0.428703\pi\)
\(282\) 0 0
\(283\) −3939.01 −0.827384 −0.413692 0.910417i \(-0.635761\pi\)
−0.413692 + 0.910417i \(0.635761\pi\)
\(284\) 0 0
\(285\) 119.267 0.0247886
\(286\) 0 0
\(287\) 551.799 0.113490
\(288\) 0 0
\(289\) −4616.00 −0.939548
\(290\) 0 0
\(291\) 621.898 0.125279
\(292\) 0 0
\(293\) 965.927 0.192594 0.0962970 0.995353i \(-0.469300\pi\)
0.0962970 + 0.995353i \(0.469300\pi\)
\(294\) 0 0
\(295\) 9055.08 1.78714
\(296\) 0 0
\(297\) 423.863 0.0828114
\(298\) 0 0
\(299\) −2423.86 −0.468814
\(300\) 0 0
\(301\) 907.761 0.173829
\(302\) 0 0
\(303\) 61.9162 0.0117392
\(304\) 0 0
\(305\) 4723.71 0.886816
\(306\) 0 0
\(307\) −9718.56 −1.80673 −0.903367 0.428867i \(-0.858913\pi\)
−0.903367 + 0.428867i \(0.858913\pi\)
\(308\) 0 0
\(309\) 454.102 0.0836019
\(310\) 0 0
\(311\) 3449.04 0.628865 0.314433 0.949280i \(-0.398186\pi\)
0.314433 + 0.949280i \(0.398186\pi\)
\(312\) 0 0
\(313\) 23.3728 0.00422079 0.00211039 0.999998i \(-0.499328\pi\)
0.00211039 + 0.999998i \(0.499328\pi\)
\(314\) 0 0
\(315\) −1590.15 −0.284428
\(316\) 0 0
\(317\) 567.966 0.100631 0.0503157 0.998733i \(-0.483977\pi\)
0.0503157 + 0.998733i \(0.483977\pi\)
\(318\) 0 0
\(319\) −5584.78 −0.980212
\(320\) 0 0
\(321\) −646.552 −0.112421
\(322\) 0 0
\(323\) 327.440 0.0564064
\(324\) 0 0
\(325\) 2245.45 0.383246
\(326\) 0 0
\(327\) −6.78918 −0.00114814
\(328\) 0 0
\(329\) 1030.66 0.172712
\(330\) 0 0
\(331\) −553.324 −0.0918835 −0.0459418 0.998944i \(-0.514629\pi\)
−0.0459418 + 0.998944i \(0.514629\pi\)
\(332\) 0 0
\(333\) −5720.90 −0.941451
\(334\) 0 0
\(335\) −2483.53 −0.405044
\(336\) 0 0
\(337\) 7328.60 1.18461 0.592306 0.805713i \(-0.298218\pi\)
0.592306 + 0.805713i \(0.298218\pi\)
\(338\) 0 0
\(339\) −590.657 −0.0946315
\(340\) 0 0
\(341\) −3927.18 −0.623662
\(342\) 0 0
\(343\) 2365.18 0.372326
\(344\) 0 0
\(345\) 1079.45 0.168452
\(346\) 0 0
\(347\) 4828.56 0.747005 0.373502 0.927629i \(-0.378157\pi\)
0.373502 + 0.927629i \(0.378157\pi\)
\(348\) 0 0
\(349\) −10256.6 −1.57313 −0.786564 0.617508i \(-0.788142\pi\)
−0.786564 + 0.617508i \(0.788142\pi\)
\(350\) 0 0
\(351\) −282.629 −0.0429790
\(352\) 0 0
\(353\) 2003.83 0.302133 0.151067 0.988524i \(-0.451729\pi\)
0.151067 + 0.988524i \(0.451729\pi\)
\(354\) 0 0
\(355\) 15402.0 2.30268
\(356\) 0 0
\(357\) 22.5250 0.00333935
\(358\) 0 0
\(359\) 4841.62 0.711786 0.355893 0.934527i \(-0.384177\pi\)
0.355893 + 0.934527i \(0.384177\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −329.156 −0.0475929
\(364\) 0 0
\(365\) 11446.0 1.64139
\(366\) 0 0
\(367\) 7004.47 0.996268 0.498134 0.867100i \(-0.334019\pi\)
0.498134 + 0.867100i \(0.334019\pi\)
\(368\) 0 0
\(369\) 4221.77 0.595600
\(370\) 0 0
\(371\) −757.614 −0.106020
\(372\) 0 0
\(373\) −11718.1 −1.62665 −0.813324 0.581812i \(-0.802344\pi\)
−0.813324 + 0.581812i \(0.802344\pi\)
\(374\) 0 0
\(375\) −215.352 −0.0296552
\(376\) 0 0
\(377\) 3723.90 0.508728
\(378\) 0 0
\(379\) −4807.97 −0.651633 −0.325817 0.945433i \(-0.605639\pi\)
−0.325817 + 0.945433i \(0.605639\pi\)
\(380\) 0 0
\(381\) 866.614 0.116530
\(382\) 0 0
\(383\) 5998.09 0.800230 0.400115 0.916465i \(-0.368970\pi\)
0.400115 + 0.916465i \(0.368970\pi\)
\(384\) 0 0
\(385\) −1251.37 −0.165651
\(386\) 0 0
\(387\) 6945.20 0.912260
\(388\) 0 0
\(389\) 7480.25 0.974971 0.487486 0.873131i \(-0.337914\pi\)
0.487486 + 0.873131i \(0.337914\pi\)
\(390\) 0 0
\(391\) 2963.58 0.383312
\(392\) 0 0
\(393\) 235.190 0.0301877
\(394\) 0 0
\(395\) 10256.4 1.30647
\(396\) 0 0
\(397\) −10366.4 −1.31052 −0.655258 0.755406i \(-0.727440\pi\)
−0.655258 + 0.755406i \(0.727440\pi\)
\(398\) 0 0
\(399\) 24.8336 0.00311588
\(400\) 0 0
\(401\) 801.467 0.0998088 0.0499044 0.998754i \(-0.484108\pi\)
0.0499044 + 0.998754i \(0.484108\pi\)
\(402\) 0 0
\(403\) 2618.62 0.323680
\(404\) 0 0
\(405\) −12103.0 −1.48495
\(406\) 0 0
\(407\) −4502.06 −0.548302
\(408\) 0 0
\(409\) −9396.88 −1.13605 −0.568027 0.823010i \(-0.692293\pi\)
−0.568027 + 0.823010i \(0.692293\pi\)
\(410\) 0 0
\(411\) 408.860 0.0490695
\(412\) 0 0
\(413\) 1885.44 0.224641
\(414\) 0 0
\(415\) −4762.34 −0.563311
\(416\) 0 0
\(417\) 567.125 0.0666001
\(418\) 0 0
\(419\) −5721.15 −0.667056 −0.333528 0.942740i \(-0.608239\pi\)
−0.333528 + 0.942740i \(0.608239\pi\)
\(420\) 0 0
\(421\) 678.987 0.0786029 0.0393014 0.999227i \(-0.487487\pi\)
0.0393014 + 0.999227i \(0.487487\pi\)
\(422\) 0 0
\(423\) 7885.50 0.906398
\(424\) 0 0
\(425\) −2745.45 −0.313350
\(426\) 0 0
\(427\) 983.569 0.111471
\(428\) 0 0
\(429\) −110.921 −0.0124833
\(430\) 0 0
\(431\) −16238.9 −1.81485 −0.907423 0.420219i \(-0.861953\pi\)
−0.907423 + 0.420219i \(0.861953\pi\)
\(432\) 0 0
\(433\) −12712.1 −1.41087 −0.705433 0.708777i \(-0.749247\pi\)
−0.705433 + 0.708777i \(0.749247\pi\)
\(434\) 0 0
\(435\) −1658.42 −0.182794
\(436\) 0 0
\(437\) 3267.33 0.357660
\(438\) 0 0
\(439\) 15433.5 1.67791 0.838955 0.544200i \(-0.183167\pi\)
0.838955 + 0.544200i \(0.183167\pi\)
\(440\) 0 0
\(441\) 8882.36 0.959115
\(442\) 0 0
\(443\) −14185.4 −1.52137 −0.760685 0.649122i \(-0.775137\pi\)
−0.760685 + 0.649122i \(0.775137\pi\)
\(444\) 0 0
\(445\) 3612.01 0.384777
\(446\) 0 0
\(447\) −1176.59 −0.124498
\(448\) 0 0
\(449\) −3674.17 −0.386180 −0.193090 0.981181i \(-0.561851\pi\)
−0.193090 + 0.981181i \(0.561851\pi\)
\(450\) 0 0
\(451\) 3322.32 0.346878
\(452\) 0 0
\(453\) −718.770 −0.0745491
\(454\) 0 0
\(455\) 834.405 0.0859726
\(456\) 0 0
\(457\) −2247.26 −0.230027 −0.115013 0.993364i \(-0.536691\pi\)
−0.115013 + 0.993364i \(0.536691\pi\)
\(458\) 0 0
\(459\) 345.563 0.0351405
\(460\) 0 0
\(461\) 6336.44 0.640168 0.320084 0.947389i \(-0.396289\pi\)
0.320084 + 0.947389i \(0.396289\pi\)
\(462\) 0 0
\(463\) −14249.5 −1.43030 −0.715152 0.698969i \(-0.753643\pi\)
−0.715152 + 0.698969i \(0.753643\pi\)
\(464\) 0 0
\(465\) −1166.19 −0.116303
\(466\) 0 0
\(467\) 7043.41 0.697923 0.348961 0.937137i \(-0.386534\pi\)
0.348961 + 0.937137i \(0.386534\pi\)
\(468\) 0 0
\(469\) −517.120 −0.0509134
\(470\) 0 0
\(471\) 676.917 0.0662222
\(472\) 0 0
\(473\) 5465.53 0.531301
\(474\) 0 0
\(475\) −3026.83 −0.292380
\(476\) 0 0
\(477\) −5796.44 −0.556396
\(478\) 0 0
\(479\) −13038.5 −1.24372 −0.621860 0.783128i \(-0.713623\pi\)
−0.621860 + 0.783128i \(0.713623\pi\)
\(480\) 0 0
\(481\) 3001.95 0.284568
\(482\) 0 0
\(483\) 224.763 0.0211741
\(484\) 0 0
\(485\) −28167.1 −2.63711
\(486\) 0 0
\(487\) 18977.6 1.76583 0.882913 0.469536i \(-0.155579\pi\)
0.882913 + 0.469536i \(0.155579\pi\)
\(488\) 0 0
\(489\) −468.767 −0.0433505
\(490\) 0 0
\(491\) 3177.78 0.292080 0.146040 0.989279i \(-0.453347\pi\)
0.146040 + 0.989279i \(0.453347\pi\)
\(492\) 0 0
\(493\) −4553.11 −0.415947
\(494\) 0 0
\(495\) −9574.11 −0.869342
\(496\) 0 0
\(497\) 3206.99 0.289443
\(498\) 0 0
\(499\) −9180.49 −0.823597 −0.411799 0.911275i \(-0.635099\pi\)
−0.411799 + 0.911275i \(0.635099\pi\)
\(500\) 0 0
\(501\) −489.612 −0.0436612
\(502\) 0 0
\(503\) 10409.3 0.922723 0.461362 0.887212i \(-0.347361\pi\)
0.461362 + 0.887212i \(0.347361\pi\)
\(504\) 0 0
\(505\) −2804.32 −0.247110
\(506\) 0 0
\(507\) −743.940 −0.0651668
\(508\) 0 0
\(509\) −6173.36 −0.537582 −0.268791 0.963198i \(-0.586624\pi\)
−0.268791 + 0.963198i \(0.586624\pi\)
\(510\) 0 0
\(511\) 2383.27 0.206321
\(512\) 0 0
\(513\) 380.980 0.0327889
\(514\) 0 0
\(515\) −20567.2 −1.75981
\(516\) 0 0
\(517\) 6205.50 0.527887
\(518\) 0 0
\(519\) 896.081 0.0757873
\(520\) 0 0
\(521\) 660.989 0.0555825 0.0277912 0.999614i \(-0.491153\pi\)
0.0277912 + 0.999614i \(0.491153\pi\)
\(522\) 0 0
\(523\) 19526.4 1.63257 0.816283 0.577652i \(-0.196031\pi\)
0.816283 + 0.577652i \(0.196031\pi\)
\(524\) 0 0
\(525\) −208.220 −0.0173094
\(526\) 0 0
\(527\) −3201.72 −0.264647
\(528\) 0 0
\(529\) 17404.8 1.43049
\(530\) 0 0
\(531\) 14425.4 1.17892
\(532\) 0 0
\(533\) −2215.30 −0.180029
\(534\) 0 0
\(535\) 29283.7 2.36644
\(536\) 0 0
\(537\) 402.927 0.0323791
\(538\) 0 0
\(539\) 6989.98 0.558589
\(540\) 0 0
\(541\) −9259.01 −0.735815 −0.367907 0.929862i \(-0.619926\pi\)
−0.367907 + 0.929862i \(0.619926\pi\)
\(542\) 0 0
\(543\) 1607.29 0.127027
\(544\) 0 0
\(545\) 307.496 0.0241683
\(546\) 0 0
\(547\) 11947.9 0.933924 0.466962 0.884277i \(-0.345348\pi\)
0.466962 + 0.884277i \(0.345348\pi\)
\(548\) 0 0
\(549\) 7525.20 0.585005
\(550\) 0 0
\(551\) −5019.77 −0.388111
\(552\) 0 0
\(553\) 2135.58 0.164221
\(554\) 0 0
\(555\) −1336.90 −0.102249
\(556\) 0 0
\(557\) −3317.62 −0.252373 −0.126187 0.992007i \(-0.540274\pi\)
−0.126187 + 0.992007i \(0.540274\pi\)
\(558\) 0 0
\(559\) −3644.38 −0.275744
\(560\) 0 0
\(561\) 135.621 0.0102066
\(562\) 0 0
\(563\) 21580.3 1.61546 0.807729 0.589555i \(-0.200697\pi\)
0.807729 + 0.589555i \(0.200697\pi\)
\(564\) 0 0
\(565\) 26752.1 1.99198
\(566\) 0 0
\(567\) −2520.08 −0.186655
\(568\) 0 0
\(569\) 11040.1 0.813401 0.406700 0.913562i \(-0.366679\pi\)
0.406700 + 0.913562i \(0.366679\pi\)
\(570\) 0 0
\(571\) −19559.0 −1.43348 −0.716742 0.697338i \(-0.754368\pi\)
−0.716742 + 0.697338i \(0.754368\pi\)
\(572\) 0 0
\(573\) 1194.38 0.0870784
\(574\) 0 0
\(575\) −27395.2 −1.98688
\(576\) 0 0
\(577\) −4148.41 −0.299308 −0.149654 0.988738i \(-0.547816\pi\)
−0.149654 + 0.988738i \(0.547816\pi\)
\(578\) 0 0
\(579\) −1566.08 −0.112407
\(580\) 0 0
\(581\) −991.612 −0.0708072
\(582\) 0 0
\(583\) −4561.51 −0.324045
\(584\) 0 0
\(585\) 6383.97 0.451187
\(586\) 0 0
\(587\) −23530.2 −1.65451 −0.827253 0.561829i \(-0.810098\pi\)
−0.827253 + 0.561829i \(0.810098\pi\)
\(588\) 0 0
\(589\) −3529.87 −0.246937
\(590\) 0 0
\(591\) 1782.87 0.124091
\(592\) 0 0
\(593\) −3378.46 −0.233957 −0.116979 0.993134i \(-0.537321\pi\)
−0.116979 + 0.993134i \(0.537321\pi\)
\(594\) 0 0
\(595\) −1020.20 −0.0702930
\(596\) 0 0
\(597\) −1304.49 −0.0894294
\(598\) 0 0
\(599\) 15748.7 1.07424 0.537122 0.843504i \(-0.319511\pi\)
0.537122 + 0.843504i \(0.319511\pi\)
\(600\) 0 0
\(601\) 17980.4 1.22036 0.610180 0.792263i \(-0.291097\pi\)
0.610180 + 0.792263i \(0.291097\pi\)
\(602\) 0 0
\(603\) −3956.44 −0.267195
\(604\) 0 0
\(605\) 14908.2 1.00182
\(606\) 0 0
\(607\) −4314.26 −0.288485 −0.144243 0.989542i \(-0.546075\pi\)
−0.144243 + 0.989542i \(0.546075\pi\)
\(608\) 0 0
\(609\) −345.316 −0.0229769
\(610\) 0 0
\(611\) −4137.79 −0.273972
\(612\) 0 0
\(613\) −3891.03 −0.256374 −0.128187 0.991750i \(-0.540916\pi\)
−0.128187 + 0.991750i \(0.540916\pi\)
\(614\) 0 0
\(615\) 986.576 0.0646871
\(616\) 0 0
\(617\) 6001.52 0.391592 0.195796 0.980645i \(-0.437271\pi\)
0.195796 + 0.980645i \(0.437271\pi\)
\(618\) 0 0
\(619\) 16195.8 1.05164 0.525820 0.850596i \(-0.323759\pi\)
0.525820 + 0.850596i \(0.323759\pi\)
\(620\) 0 0
\(621\) 3448.16 0.222818
\(622\) 0 0
\(623\) 752.091 0.0483658
\(624\) 0 0
\(625\) −10159.6 −0.650217
\(626\) 0 0
\(627\) 149.521 0.00952357
\(628\) 0 0
\(629\) −3670.40 −0.232668
\(630\) 0 0
\(631\) −29691.7 −1.87323 −0.936615 0.350361i \(-0.886059\pi\)
−0.936615 + 0.350361i \(0.886059\pi\)
\(632\) 0 0
\(633\) −1141.75 −0.0716910
\(634\) 0 0
\(635\) −39250.8 −2.45294
\(636\) 0 0
\(637\) −4660.88 −0.289907
\(638\) 0 0
\(639\) 24536.4 1.51901
\(640\) 0 0
\(641\) 19562.7 1.20543 0.602716 0.797956i \(-0.294085\pi\)
0.602716 + 0.797956i \(0.294085\pi\)
\(642\) 0 0
\(643\) 20169.7 1.23704 0.618518 0.785770i \(-0.287733\pi\)
0.618518 + 0.785770i \(0.287733\pi\)
\(644\) 0 0
\(645\) 1623.01 0.0990790
\(646\) 0 0
\(647\) −19276.8 −1.17133 −0.585663 0.810555i \(-0.699166\pi\)
−0.585663 + 0.810555i \(0.699166\pi\)
\(648\) 0 0
\(649\) 11352.1 0.686606
\(650\) 0 0
\(651\) −242.824 −0.0146191
\(652\) 0 0
\(653\) −17256.6 −1.03416 −0.517078 0.855938i \(-0.672980\pi\)
−0.517078 + 0.855938i \(0.672980\pi\)
\(654\) 0 0
\(655\) −10652.2 −0.635446
\(656\) 0 0
\(657\) 18234.2 1.08278
\(658\) 0 0
\(659\) −15812.2 −0.934681 −0.467340 0.884078i \(-0.654788\pi\)
−0.467340 + 0.884078i \(0.654788\pi\)
\(660\) 0 0
\(661\) −21264.8 −1.25130 −0.625648 0.780106i \(-0.715165\pi\)
−0.625648 + 0.780106i \(0.715165\pi\)
\(662\) 0 0
\(663\) −90.4310 −0.00529721
\(664\) 0 0
\(665\) −1124.77 −0.0655889
\(666\) 0 0
\(667\) −45432.8 −2.63743
\(668\) 0 0
\(669\) −788.140 −0.0455475
\(670\) 0 0
\(671\) 5921.96 0.340708
\(672\) 0 0
\(673\) 10290.0 0.589374 0.294687 0.955594i \(-0.404785\pi\)
0.294687 + 0.955594i \(0.404785\pi\)
\(674\) 0 0
\(675\) −3194.36 −0.182150
\(676\) 0 0
\(677\) −3204.13 −0.181897 −0.0909487 0.995856i \(-0.528990\pi\)
−0.0909487 + 0.995856i \(0.528990\pi\)
\(678\) 0 0
\(679\) −5864.93 −0.331481
\(680\) 0 0
\(681\) 2411.02 0.135669
\(682\) 0 0
\(683\) 22703.2 1.27191 0.635954 0.771727i \(-0.280607\pi\)
0.635954 + 0.771727i \(0.280607\pi\)
\(684\) 0 0
\(685\) −18518.1 −1.03291
\(686\) 0 0
\(687\) 1372.57 0.0762254
\(688\) 0 0
\(689\) 3041.59 0.168179
\(690\) 0 0
\(691\) 14141.0 0.778509 0.389255 0.921130i \(-0.372733\pi\)
0.389255 + 0.921130i \(0.372733\pi\)
\(692\) 0 0
\(693\) −1993.52 −0.109275
\(694\) 0 0
\(695\) −25686.3 −1.40192
\(696\) 0 0
\(697\) 2708.59 0.147195
\(698\) 0 0
\(699\) 416.738 0.0225500
\(700\) 0 0
\(701\) −8699.08 −0.468701 −0.234351 0.972152i \(-0.575296\pi\)
−0.234351 + 0.972152i \(0.575296\pi\)
\(702\) 0 0
\(703\) −4046.59 −0.217098
\(704\) 0 0
\(705\) 1842.75 0.0984423
\(706\) 0 0
\(707\) −583.913 −0.0310613
\(708\) 0 0
\(709\) −19611.9 −1.03884 −0.519422 0.854518i \(-0.673853\pi\)
−0.519422 + 0.854518i \(0.673853\pi\)
\(710\) 0 0
\(711\) 16339.2 0.861838
\(712\) 0 0
\(713\) −31948.0 −1.67807
\(714\) 0 0
\(715\) 5023.86 0.262772
\(716\) 0 0
\(717\) −2465.05 −0.128395
\(718\) 0 0
\(719\) 3735.43 0.193752 0.0968762 0.995296i \(-0.469115\pi\)
0.0968762 + 0.995296i \(0.469115\pi\)
\(720\) 0 0
\(721\) −4282.50 −0.221205
\(722\) 0 0
\(723\) 1036.62 0.0533225
\(724\) 0 0
\(725\) 42088.6 2.15605
\(726\) 0 0
\(727\) −4091.05 −0.208705 −0.104353 0.994540i \(-0.533277\pi\)
−0.104353 + 0.994540i \(0.533277\pi\)
\(728\) 0 0
\(729\) −19079.4 −0.969333
\(730\) 0 0
\(731\) 4455.89 0.225454
\(732\) 0 0
\(733\) 21206.4 1.06859 0.534294 0.845299i \(-0.320577\pi\)
0.534294 + 0.845299i \(0.320577\pi\)
\(734\) 0 0
\(735\) 2075.70 0.104168
\(736\) 0 0
\(737\) −3113.52 −0.155615
\(738\) 0 0
\(739\) 8939.54 0.444988 0.222494 0.974934i \(-0.428580\pi\)
0.222494 + 0.974934i \(0.428580\pi\)
\(740\) 0 0
\(741\) −99.6995 −0.00494271
\(742\) 0 0
\(743\) 26137.1 1.29055 0.645273 0.763952i \(-0.276743\pi\)
0.645273 + 0.763952i \(0.276743\pi\)
\(744\) 0 0
\(745\) 53290.2 2.62068
\(746\) 0 0
\(747\) −7586.74 −0.371599
\(748\) 0 0
\(749\) 6097.44 0.297458
\(750\) 0 0
\(751\) −11466.6 −0.557155 −0.278578 0.960414i \(-0.589863\pi\)
−0.278578 + 0.960414i \(0.589863\pi\)
\(752\) 0 0
\(753\) −1738.62 −0.0841421
\(754\) 0 0
\(755\) 32554.6 1.56925
\(756\) 0 0
\(757\) −8936.66 −0.429073 −0.214537 0.976716i \(-0.568824\pi\)
−0.214537 + 0.976716i \(0.568824\pi\)
\(758\) 0 0
\(759\) 1353.28 0.0647178
\(760\) 0 0
\(761\) −9971.09 −0.474969 −0.237485 0.971391i \(-0.576323\pi\)
−0.237485 + 0.971391i \(0.576323\pi\)
\(762\) 0 0
\(763\) 64.0268 0.00303791
\(764\) 0 0
\(765\) −7805.50 −0.368900
\(766\) 0 0
\(767\) −7569.49 −0.356347
\(768\) 0 0
\(769\) 25560.0 1.19859 0.599296 0.800527i \(-0.295447\pi\)
0.599296 + 0.800527i \(0.295447\pi\)
\(770\) 0 0
\(771\) 1923.90 0.0898670
\(772\) 0 0
\(773\) −15918.8 −0.740698 −0.370349 0.928893i \(-0.620762\pi\)
−0.370349 + 0.928893i \(0.620762\pi\)
\(774\) 0 0
\(775\) 29596.5 1.37179
\(776\) 0 0
\(777\) −278.370 −0.0128526
\(778\) 0 0
\(779\) 2986.20 0.137345
\(780\) 0 0
\(781\) 19309.0 0.884672
\(782\) 0 0
\(783\) −5297.60 −0.241789
\(784\) 0 0
\(785\) −30659.0 −1.39397
\(786\) 0 0
\(787\) −11278.0 −0.510822 −0.255411 0.966833i \(-0.582211\pi\)
−0.255411 + 0.966833i \(0.582211\pi\)
\(788\) 0 0
\(789\) −1211.46 −0.0546631
\(790\) 0 0
\(791\) 5570.31 0.250389
\(792\) 0 0
\(793\) −3948.73 −0.176827
\(794\) 0 0
\(795\) −1354.56 −0.0604292
\(796\) 0 0
\(797\) −23845.2 −1.05978 −0.529888 0.848068i \(-0.677766\pi\)
−0.529888 + 0.848068i \(0.677766\pi\)
\(798\) 0 0
\(799\) 5059.16 0.224005
\(800\) 0 0
\(801\) 5754.18 0.253825
\(802\) 0 0
\(803\) 14349.4 0.630611
\(804\) 0 0
\(805\) −10180.0 −0.445712
\(806\) 0 0
\(807\) −576.204 −0.0251343
\(808\) 0 0
\(809\) 12729.7 0.553216 0.276608 0.960983i \(-0.410790\pi\)
0.276608 + 0.960983i \(0.410790\pi\)
\(810\) 0 0
\(811\) −5859.98 −0.253726 −0.126863 0.991920i \(-0.540491\pi\)
−0.126863 + 0.991920i \(0.540491\pi\)
\(812\) 0 0
\(813\) −1850.06 −0.0798088
\(814\) 0 0
\(815\) 21231.5 0.912522
\(816\) 0 0
\(817\) 4912.58 0.210367
\(818\) 0 0
\(819\) 1329.27 0.0567135
\(820\) 0 0
\(821\) 12778.2 0.543195 0.271598 0.962411i \(-0.412448\pi\)
0.271598 + 0.962411i \(0.412448\pi\)
\(822\) 0 0
\(823\) −15511.0 −0.656963 −0.328482 0.944510i \(-0.606537\pi\)
−0.328482 + 0.944510i \(0.606537\pi\)
\(824\) 0 0
\(825\) −1253.67 −0.0529056
\(826\) 0 0
\(827\) −12501.0 −0.525639 −0.262819 0.964845i \(-0.584652\pi\)
−0.262819 + 0.964845i \(0.584652\pi\)
\(828\) 0 0
\(829\) −2013.80 −0.0843694 −0.0421847 0.999110i \(-0.513432\pi\)
−0.0421847 + 0.999110i \(0.513432\pi\)
\(830\) 0 0
\(831\) 104.187 0.00434924
\(832\) 0 0
\(833\) 5698.73 0.237034
\(834\) 0 0
\(835\) 22175.6 0.919063
\(836\) 0 0
\(837\) −3725.24 −0.153839
\(838\) 0 0
\(839\) 16805.2 0.691513 0.345757 0.938324i \(-0.387622\pi\)
0.345757 + 0.938324i \(0.387622\pi\)
\(840\) 0 0
\(841\) 45411.7 1.86198
\(842\) 0 0
\(843\) 779.013 0.0318275
\(844\) 0 0
\(845\) 33694.6 1.37175
\(846\) 0 0
\(847\) 3104.17 0.125928
\(848\) 0 0
\(849\) −1466.42 −0.0592784
\(850\) 0 0
\(851\) −36624.7 −1.47530
\(852\) 0 0
\(853\) −16648.7 −0.668279 −0.334139 0.942524i \(-0.608446\pi\)
−0.334139 + 0.942524i \(0.608446\pi\)
\(854\) 0 0
\(855\) −8605.50 −0.344213
\(856\) 0 0
\(857\) 12094.5 0.482078 0.241039 0.970515i \(-0.422512\pi\)
0.241039 + 0.970515i \(0.422512\pi\)
\(858\) 0 0
\(859\) 10296.1 0.408962 0.204481 0.978871i \(-0.434449\pi\)
0.204481 + 0.978871i \(0.434449\pi\)
\(860\) 0 0
\(861\) 205.424 0.00813106
\(862\) 0 0
\(863\) 27367.0 1.07947 0.539736 0.841834i \(-0.318524\pi\)
0.539736 + 0.841834i \(0.318524\pi\)
\(864\) 0 0
\(865\) −40585.4 −1.59531
\(866\) 0 0
\(867\) −1718.45 −0.0673145
\(868\) 0 0
\(869\) 12858.1 0.501936
\(870\) 0 0
\(871\) 2076.08 0.0807638
\(872\) 0 0
\(873\) −44872.1 −1.73962
\(874\) 0 0
\(875\) 2030.92 0.0784658
\(876\) 0 0
\(877\) −29880.2 −1.15049 −0.575247 0.817979i \(-0.695094\pi\)
−0.575247 + 0.817979i \(0.695094\pi\)
\(878\) 0 0
\(879\) 359.597 0.0137985
\(880\) 0 0
\(881\) −28639.8 −1.09523 −0.547617 0.836729i \(-0.684465\pi\)
−0.547617 + 0.836729i \(0.684465\pi\)
\(882\) 0 0
\(883\) −16394.7 −0.624832 −0.312416 0.949945i \(-0.601138\pi\)
−0.312416 + 0.949945i \(0.601138\pi\)
\(884\) 0 0
\(885\) 3371.04 0.128041
\(886\) 0 0
\(887\) −16041.2 −0.607226 −0.303613 0.952795i \(-0.598193\pi\)
−0.303613 + 0.952795i \(0.598193\pi\)
\(888\) 0 0
\(889\) −8172.78 −0.308331
\(890\) 0 0
\(891\) −15173.1 −0.570504
\(892\) 0 0
\(893\) 5577.69 0.209015
\(894\) 0 0
\(895\) −18249.4 −0.681577
\(896\) 0 0
\(897\) −902.357 −0.0335884
\(898\) 0 0
\(899\) 49083.4 1.82094
\(900\) 0 0
\(901\) −3718.87 −0.137507
\(902\) 0 0
\(903\) 337.942 0.0124541
\(904\) 0 0
\(905\) −72797.6 −2.67390
\(906\) 0 0
\(907\) −10953.4 −0.400994 −0.200497 0.979694i \(-0.564256\pi\)
−0.200497 + 0.979694i \(0.564256\pi\)
\(908\) 0 0
\(909\) −4467.47 −0.163011
\(910\) 0 0
\(911\) 12739.1 0.463299 0.231649 0.972799i \(-0.425588\pi\)
0.231649 + 0.972799i \(0.425588\pi\)
\(912\) 0 0
\(913\) −5970.39 −0.216419
\(914\) 0 0
\(915\) 1758.55 0.0635364
\(916\) 0 0
\(917\) −2218.00 −0.0798745
\(918\) 0 0
\(919\) −10544.1 −0.378475 −0.189238 0.981931i \(-0.560602\pi\)
−0.189238 + 0.981931i \(0.560602\pi\)
\(920\) 0 0
\(921\) −3618.04 −0.129445
\(922\) 0 0
\(923\) −12875.1 −0.459143
\(924\) 0 0
\(925\) 33928.9 1.20603
\(926\) 0 0
\(927\) −32765.1 −1.16089
\(928\) 0 0
\(929\) −28682.8 −1.01297 −0.506487 0.862247i \(-0.669056\pi\)
−0.506487 + 0.862247i \(0.669056\pi\)
\(930\) 0 0
\(931\) 6282.80 0.221171
\(932\) 0 0
\(933\) 1284.01 0.0450554
\(934\) 0 0
\(935\) −6142.54 −0.214848
\(936\) 0 0
\(937\) −39763.3 −1.38635 −0.693175 0.720769i \(-0.743789\pi\)
−0.693175 + 0.720769i \(0.743789\pi\)
\(938\) 0 0
\(939\) 8.70124 0.000302401 0
\(940\) 0 0
\(941\) 27875.7 0.965696 0.482848 0.875704i \(-0.339602\pi\)
0.482848 + 0.875704i \(0.339602\pi\)
\(942\) 0 0
\(943\) 27027.4 0.933333
\(944\) 0 0
\(945\) −1187.02 −0.0408611
\(946\) 0 0
\(947\) 54572.7 1.87262 0.936312 0.351169i \(-0.114216\pi\)
0.936312 + 0.351169i \(0.114216\pi\)
\(948\) 0 0
\(949\) −9568.12 −0.327286
\(950\) 0 0
\(951\) 211.443 0.00720979
\(952\) 0 0
\(953\) −8065.62 −0.274157 −0.137078 0.990560i \(-0.543771\pi\)
−0.137078 + 0.990560i \(0.543771\pi\)
\(954\) 0 0
\(955\) −54096.0 −1.83299
\(956\) 0 0
\(957\) −2079.11 −0.0702279
\(958\) 0 0
\(959\) −3855.83 −0.129835
\(960\) 0 0
\(961\) 4724.14 0.158576
\(962\) 0 0
\(963\) 46651.0 1.56107
\(964\) 0 0
\(965\) 70930.9 2.36616
\(966\) 0 0
\(967\) 14289.8 0.475210 0.237605 0.971362i \(-0.423638\pi\)
0.237605 + 0.971362i \(0.423638\pi\)
\(968\) 0 0
\(969\) 121.900 0.00404127
\(970\) 0 0
\(971\) −14397.3 −0.475832 −0.237916 0.971286i \(-0.576464\pi\)
−0.237916 + 0.971286i \(0.576464\pi\)
\(972\) 0 0
\(973\) −5348.39 −0.176219
\(974\) 0 0
\(975\) 835.938 0.0274579
\(976\) 0 0
\(977\) −41896.2 −1.37193 −0.685966 0.727634i \(-0.740620\pi\)
−0.685966 + 0.727634i \(0.740620\pi\)
\(978\) 0 0
\(979\) 4528.26 0.147828
\(980\) 0 0
\(981\) 489.863 0.0159431
\(982\) 0 0
\(983\) 19373.7 0.628612 0.314306 0.949322i \(-0.398228\pi\)
0.314306 + 0.949322i \(0.398228\pi\)
\(984\) 0 0
\(985\) −80750.1 −2.61209
\(986\) 0 0
\(987\) 383.696 0.0123740
\(988\) 0 0
\(989\) 44462.6 1.42955
\(990\) 0 0
\(991\) −13250.6 −0.424741 −0.212371 0.977189i \(-0.568118\pi\)
−0.212371 + 0.977189i \(0.568118\pi\)
\(992\) 0 0
\(993\) −205.992 −0.00658305
\(994\) 0 0
\(995\) 59083.3 1.88248
\(996\) 0 0
\(997\) −4129.59 −0.131179 −0.0655895 0.997847i \(-0.520893\pi\)
−0.0655895 + 0.997847i \(0.520893\pi\)
\(998\) 0 0
\(999\) −4270.55 −0.135250
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.1.2 2
3.2 odd 2 684.4.a.g.1.2 2
4.3 odd 2 304.4.a.f.1.1 2
5.2 odd 4 1900.4.c.b.1749.2 4
5.3 odd 4 1900.4.c.b.1749.3 4
5.4 even 2 1900.4.a.b.1.1 2
8.3 odd 2 1216.4.a.h.1.2 2
8.5 even 2 1216.4.a.o.1.1 2
19.18 odd 2 1444.4.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.a.a.1.2 2 1.1 even 1 trivial
304.4.a.f.1.1 2 4.3 odd 2
684.4.a.g.1.2 2 3.2 odd 2
1216.4.a.h.1.2 2 8.3 odd 2
1216.4.a.o.1.1 2 8.5 even 2
1444.4.a.d.1.1 2 19.18 odd 2
1900.4.a.b.1.1 2 5.4 even 2
1900.4.c.b.1749.2 4 5.2 odd 4
1900.4.c.b.1749.3 4 5.3 odd 4