Properties

Label 522.2.a.b.1.1
Level $522$
Weight $2$
Character 522.1
Self dual yes
Analytic conductor $4.168$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.16819098551\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 522.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{20} -3.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} -2.00000 q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +8.00000 q^{34} +2.00000 q^{35} +8.00000 q^{37} +1.00000 q^{40} -2.00000 q^{41} -11.0000 q^{43} +3.00000 q^{44} +4.00000 q^{46} -13.0000 q^{47} -3.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} +11.0000 q^{53} -3.00000 q^{55} +2.00000 q^{56} -1.00000 q^{58} -8.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -12.0000 q^{67} -8.00000 q^{68} -2.00000 q^{70} -2.00000 q^{71} +4.00000 q^{73} -8.00000 q^{74} -6.00000 q^{77} +15.0000 q^{79} -1.00000 q^{80} +2.00000 q^{82} -4.00000 q^{83} +8.00000 q^{85} +11.0000 q^{86} -3.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{92} +13.0000 q^{94} -2.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −13.0000 −1.89624 −0.948122 0.317905i \(-0.897021\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.00000 −0.0877058
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −15.0000 −1.19334
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 9.00000 0.704934 0.352467 0.935824i \(-0.385343\pi\)
0.352467 + 0.935824i \(0.385343\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) −13.0000 −0.948122
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 11.0000 0.755483
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) 11.0000 0.750194
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −5.00000 −0.338643
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 1.00000 0.0655122 0.0327561 0.999463i \(-0.489572\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) 0 0
\(235\) 13.0000 0.848026
\(236\) 0 0
\(237\) 0 0
\(238\) −16.0000 −1.03713
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.0000 −0.810918 −0.405459 0.914113i \(-0.632888\pi\)
−0.405459 + 0.914113i \(0.632888\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 1.00000 0.0587220
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 22.0000 1.26806
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) −3.00000 −0.170389
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 15.0000 0.843816
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −9.00000 −0.498464
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 26.0000 1.43343
\(330\) 0 0
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −2.00000 −0.109435
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) −9.00000 −0.487377
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −7.00000 −0.367912
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −22.0000 −1.14218
\(372\) 0 0
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 13.0000 0.670424
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) −15.0000 −0.754732
\(396\) 0 0
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 3.00000 0.149441
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 3.00000 0.146038
\(423\) 0 0
\(424\) −11.0000 −0.534207
\(425\) 32.0000 1.55223
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) −11.0000 −0.530467
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) 26.0000 1.23114
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −1.00000 −0.0463241
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) −13.0000 −0.599645
\(471\) 0 0
\(472\) 0 0
\(473\) −33.0000 −1.51734
\(474\) 0 0
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 0 0
\(478\) 0 0
\(479\) 5.00000 0.228456 0.114228 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −17.0000 −0.774329
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) 27.0000 1.20507
\(503\) −19.0000 −0.847168 −0.423584 0.905857i \(-0.639228\pi\)
−0.423584 + 0.905857i \(0.639228\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 13.0000 0.573405
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) −39.0000 −1.71522
\(518\) 16.0000 0.703000
\(519\) 0 0
\(520\) −1.00000 −0.0438529
\(521\) 13.0000 0.569540 0.284770 0.958596i \(-0.408083\pi\)
0.284770 + 0.958596i \(0.408083\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 11.0000 0.477809
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 13.0000 0.558398
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 12.0000 0.511682
\(551\) 0 0
\(552\) 0 0
\(553\) −30.0000 −1.27573
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 11.0000 0.465250
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −3.00000 −0.125436
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) −47.0000 −1.95494
\(579\) 0 0
\(580\) −1.00000 −0.0415227
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 33.0000 1.36672
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) −15.0000 −0.614424
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −22.0000 −0.896653
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 3.00000 0.121766 0.0608831 0.998145i \(-0.480608\pi\)
0.0608831 + 0.998145i \(0.480608\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) 13.0000 0.525924
\(612\) 0 0
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 3.00000 0.120483
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −9.00000 −0.359712
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −64.0000 −2.55185
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) −15.0000 −0.596668
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) −3.00000 −0.118771
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 9.00000 0.352467
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −26.0000 −1.01359
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 23.0000 0.893920
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) 2.00000 0.0773823
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) 9.00000 0.344628
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −11.0000 −0.419371
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −2.00000 −0.0759190
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 15.0000 0.567758
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 15.0000 0.563337 0.281668 0.959512i \(-0.409112\pi\)
0.281668 + 0.959512i \(0.409112\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) −25.0000 −0.932992
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) 88.0000 3.25480
\(732\) 0 0
\(733\) 24.0000 0.886460 0.443230 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 22.0000 0.807645
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) 21.0000 0.768865
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −13.0000 −0.474061
\(753\) 0 0
\(754\) 1.00000 0.0364179
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 14.0000 0.505841
\(767\) 0 0
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) −6.00000 −0.216225
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 15.0000 0.533676
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 17.0000 0.603307
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 32.0000 1.13350 0.566749 0.823890i \(-0.308201\pi\)
0.566749 + 0.823890i \(0.308201\pi\)
\(798\) 0 0
\(799\) 104.000 3.67926
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 27.0000 0.953403
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −3.00000 −0.105670
\(807\) 0 0
\(808\) −8.00000 −0.281439
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) −9.00000 −0.315256
\(816\) 0 0
\(817\) 0 0
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 0 0
\(827\) −13.0000 −0.452054 −0.226027 0.974121i \(-0.572574\pi\)
−0.226027 + 0.974121i \(0.572574\pi\)
\(828\) 0 0
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 24.0000 0.831551
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 0 0
\(838\) −10.0000 −0.345444
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −32.0000 −1.10279
\(843\) 0 0
\(844\) −3.00000 −0.103264
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 11.0000 0.377742
\(849\) 0 0
\(850\) −32.0000 −1.09759
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 27.0000 0.922302 0.461151 0.887322i \(-0.347437\pi\)
0.461151 + 0.887322i \(0.347437\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 11.0000 0.375097
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 45.0000 1.52652
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) −5.00000 −0.169321
\(873\) 0 0
\(874\) 0 0
\(875\) −18.0000 −0.608511
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −33.0000 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) −26.0000 −0.870544
\(893\) 0 0
\(894\) 0 0
\(895\) −10.0000 −0.334263
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −88.0000 −2.93171
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −18.0000 −0.597351
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) 13.0000 0.430709 0.215355 0.976536i \(-0.430909\pi\)
0.215355 + 0.976536i \(0.430909\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 30.0000 0.989609 0.494804 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 2.00000 0.0658665
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.00000 0.0327561
\(933\) 0 0
\(934\) −27.0000 −0.883467
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −24.0000 −0.783628
\(939\) 0 0
\(940\) 13.0000 0.424013
\(941\) −37.0000 −1.20617 −0.603083 0.797679i \(-0.706061\pi\)
−0.603083 + 0.797679i \(0.706061\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 33.0000 1.07292
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) −16.0000 −0.518563
\(953\) 1.00000 0.0323932 0.0161966 0.999869i \(-0.494844\pi\)
0.0161966 + 0.999869i \(0.494844\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) 0 0
\(958\) −5.00000 −0.161543
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) 17.0000 0.547533
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 40.0000 1.28234
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) −33.0000 −1.05307
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) 44.0000 1.39912
\(990\) 0 0
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) 3.00000 0.0952501
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 522.2.a.b.1.1 1
3.2 odd 2 58.2.a.b.1.1 1
4.3 odd 2 4176.2.a.n.1.1 1
12.11 even 2 464.2.a.e.1.1 1
15.2 even 4 1450.2.b.b.349.2 2
15.8 even 4 1450.2.b.b.349.1 2
15.14 odd 2 1450.2.a.c.1.1 1
21.20 even 2 2842.2.a.e.1.1 1
24.5 odd 2 1856.2.a.k.1.1 1
24.11 even 2 1856.2.a.f.1.1 1
33.32 even 2 7018.2.a.a.1.1 1
39.38 odd 2 9802.2.a.a.1.1 1
87.17 even 4 1682.2.b.a.1681.1 2
87.41 even 4 1682.2.b.a.1681.2 2
87.86 odd 2 1682.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 3.2 odd 2
464.2.a.e.1.1 1 12.11 even 2
522.2.a.b.1.1 1 1.1 even 1 trivial
1450.2.a.c.1.1 1 15.14 odd 2
1450.2.b.b.349.1 2 15.8 even 4
1450.2.b.b.349.2 2 15.2 even 4
1682.2.a.d.1.1 1 87.86 odd 2
1682.2.b.a.1681.1 2 87.17 even 4
1682.2.b.a.1681.2 2 87.41 even 4
1856.2.a.f.1.1 1 24.11 even 2
1856.2.a.k.1.1 1 24.5 odd 2
2842.2.a.e.1.1 1 21.20 even 2
4176.2.a.n.1.1 1 4.3 odd 2
7018.2.a.a.1.1 1 33.32 even 2
9802.2.a.a.1.1 1 39.38 odd 2