Properties

Label 552.2.a.g.1.1
Level $552$
Weight $2$
Character 552.1
Self dual yes
Analytic conductor $4.408$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 552.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.42864 q^{5} +0.622216 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -4.42864 q^{5} +0.622216 q^{7} +1.00000 q^{9} +5.80642 q^{11} +2.00000 q^{13} -4.42864 q^{15} +1.37778 q^{17} +6.42864 q^{19} +0.622216 q^{21} -1.00000 q^{23} +14.6128 q^{25} +1.00000 q^{27} -0.755569 q^{29} -1.24443 q^{31} +5.80642 q^{33} -2.75557 q^{35} +9.05086 q^{37} +2.00000 q^{39} -9.61285 q^{41} -5.18421 q^{43} -4.42864 q^{45} -5.24443 q^{47} -6.61285 q^{49} +1.37778 q^{51} +12.0415 q^{53} -25.7146 q^{55} +6.42864 q^{57} -12.8573 q^{59} +2.94914 q^{61} +0.622216 q^{63} -8.85728 q^{65} +5.18421 q^{67} -1.00000 q^{69} +6.10171 q^{71} +2.00000 q^{73} +14.6128 q^{75} +3.61285 q^{77} -8.62222 q^{79} +1.00000 q^{81} +11.9081 q^{83} -6.10171 q^{85} -0.755569 q^{87} +6.23506 q^{89} +1.24443 q^{91} -1.24443 q^{93} -28.4701 q^{95} -2.85728 q^{97} +5.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 2q^{7} + 3q^{9} + 4q^{11} + 6q^{13} + 4q^{17} + 6q^{19} + 2q^{21} - 3q^{23} + 17q^{25} + 3q^{27} - 2q^{29} - 4q^{31} + 4q^{33} - 8q^{35} + 14q^{37} + 6q^{39} - 2q^{41} - 2q^{43} - 16q^{47} + 7q^{49} + 4q^{51} - 4q^{53} - 24q^{55} + 6q^{57} - 12q^{59} + 22q^{61} + 2q^{63} + 2q^{67} - 3q^{69} - 8q^{71} + 6q^{73} + 17q^{75} - 16q^{77} - 26q^{79} + 3q^{81} - 4q^{83} + 8q^{85} - 2q^{87} - 8q^{89} + 4q^{91} - 4q^{93} - 32q^{95} + 18q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −4.42864 −1.98055 −0.990274 0.139132i \(-0.955569\pi\)
−0.990274 + 0.139132i \(0.955569\pi\)
\(6\) 0 0
\(7\) 0.622216 0.235175 0.117588 0.993063i \(-0.462484\pi\)
0.117588 + 0.993063i \(0.462484\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.80642 1.75070 0.875351 0.483487i \(-0.160630\pi\)
0.875351 + 0.483487i \(0.160630\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −4.42864 −1.14347
\(16\) 0 0
\(17\) 1.37778 0.334162 0.167081 0.985943i \(-0.446566\pi\)
0.167081 + 0.985943i \(0.446566\pi\)
\(18\) 0 0
\(19\) 6.42864 1.47483 0.737416 0.675439i \(-0.236046\pi\)
0.737416 + 0.675439i \(0.236046\pi\)
\(20\) 0 0
\(21\) 0.622216 0.135779
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 14.6128 2.92257
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.755569 −0.140306 −0.0701528 0.997536i \(-0.522349\pi\)
−0.0701528 + 0.997536i \(0.522349\pi\)
\(30\) 0 0
\(31\) −1.24443 −0.223506 −0.111753 0.993736i \(-0.535647\pi\)
−0.111753 + 0.993736i \(0.535647\pi\)
\(32\) 0 0
\(33\) 5.80642 1.01077
\(34\) 0 0
\(35\) −2.75557 −0.465776
\(36\) 0 0
\(37\) 9.05086 1.48795 0.743976 0.668207i \(-0.232938\pi\)
0.743976 + 0.668207i \(0.232938\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −9.61285 −1.50127 −0.750637 0.660715i \(-0.770253\pi\)
−0.750637 + 0.660715i \(0.770253\pi\)
\(42\) 0 0
\(43\) −5.18421 −0.790584 −0.395292 0.918555i \(-0.629357\pi\)
−0.395292 + 0.918555i \(0.629357\pi\)
\(44\) 0 0
\(45\) −4.42864 −0.660183
\(46\) 0 0
\(47\) −5.24443 −0.764979 −0.382489 0.923960i \(-0.624933\pi\)
−0.382489 + 0.923960i \(0.624933\pi\)
\(48\) 0 0
\(49\) −6.61285 −0.944693
\(50\) 0 0
\(51\) 1.37778 0.192928
\(52\) 0 0
\(53\) 12.0415 1.65403 0.827013 0.562183i \(-0.190038\pi\)
0.827013 + 0.562183i \(0.190038\pi\)
\(54\) 0 0
\(55\) −25.7146 −3.46735
\(56\) 0 0
\(57\) 6.42864 0.851494
\(58\) 0 0
\(59\) −12.8573 −1.67388 −0.836938 0.547298i \(-0.815656\pi\)
−0.836938 + 0.547298i \(0.815656\pi\)
\(60\) 0 0
\(61\) 2.94914 0.377599 0.188800 0.982016i \(-0.439540\pi\)
0.188800 + 0.982016i \(0.439540\pi\)
\(62\) 0 0
\(63\) 0.622216 0.0783918
\(64\) 0 0
\(65\) −8.85728 −1.09861
\(66\) 0 0
\(67\) 5.18421 0.633352 0.316676 0.948534i \(-0.397433\pi\)
0.316676 + 0.948534i \(0.397433\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 6.10171 0.724140 0.362070 0.932151i \(-0.382070\pi\)
0.362070 + 0.932151i \(0.382070\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 14.6128 1.68735
\(76\) 0 0
\(77\) 3.61285 0.411722
\(78\) 0 0
\(79\) −8.62222 −0.970075 −0.485038 0.874493i \(-0.661194\pi\)
−0.485038 + 0.874493i \(0.661194\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.9081 1.30709 0.653544 0.756889i \(-0.273282\pi\)
0.653544 + 0.756889i \(0.273282\pi\)
\(84\) 0 0
\(85\) −6.10171 −0.661823
\(86\) 0 0
\(87\) −0.755569 −0.0810055
\(88\) 0 0
\(89\) 6.23506 0.660915 0.330458 0.943821i \(-0.392797\pi\)
0.330458 + 0.943821i \(0.392797\pi\)
\(90\) 0 0
\(91\) 1.24443 0.130452
\(92\) 0 0
\(93\) −1.24443 −0.129042
\(94\) 0 0
\(95\) −28.4701 −2.92097
\(96\) 0 0
\(97\) −2.85728 −0.290113 −0.145056 0.989423i \(-0.546336\pi\)
−0.145056 + 0.989423i \(0.546336\pi\)
\(98\) 0 0
\(99\) 5.80642 0.583568
\(100\) 0 0
\(101\) 15.7146 1.56366 0.781828 0.623494i \(-0.214287\pi\)
0.781828 + 0.623494i \(0.214287\pi\)
\(102\) 0 0
\(103\) 5.47949 0.539911 0.269955 0.962873i \(-0.412991\pi\)
0.269955 + 0.962873i \(0.412991\pi\)
\(104\) 0 0
\(105\) −2.75557 −0.268916
\(106\) 0 0
\(107\) −0.295286 −0.0285464 −0.0142732 0.999898i \(-0.504543\pi\)
−0.0142732 + 0.999898i \(0.504543\pi\)
\(108\) 0 0
\(109\) −9.31756 −0.892461 −0.446230 0.894918i \(-0.647234\pi\)
−0.446230 + 0.894918i \(0.647234\pi\)
\(110\) 0 0
\(111\) 9.05086 0.859069
\(112\) 0 0
\(113\) −11.4795 −1.07990 −0.539950 0.841697i \(-0.681557\pi\)
−0.539950 + 0.841697i \(0.681557\pi\)
\(114\) 0 0
\(115\) 4.42864 0.412973
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0.857279 0.0785866
\(120\) 0 0
\(121\) 22.7146 2.06496
\(122\) 0 0
\(123\) −9.61285 −0.866761
\(124\) 0 0
\(125\) −42.5718 −3.80774
\(126\) 0 0
\(127\) 8.47013 0.751602 0.375801 0.926700i \(-0.377368\pi\)
0.375801 + 0.926700i \(0.377368\pi\)
\(128\) 0 0
\(129\) −5.18421 −0.456444
\(130\) 0 0
\(131\) −7.61285 −0.665138 −0.332569 0.943079i \(-0.607915\pi\)
−0.332569 + 0.943079i \(0.607915\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −4.42864 −0.381157
\(136\) 0 0
\(137\) −5.37778 −0.459455 −0.229728 0.973255i \(-0.573784\pi\)
−0.229728 + 0.973255i \(0.573784\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −5.24443 −0.441661
\(142\) 0 0
\(143\) 11.6128 0.971115
\(144\) 0 0
\(145\) 3.34614 0.277882
\(146\) 0 0
\(147\) −6.61285 −0.545418
\(148\) 0 0
\(149\) 13.2859 1.08842 0.544212 0.838947i \(-0.316829\pi\)
0.544212 + 0.838947i \(0.316829\pi\)
\(150\) 0 0
\(151\) −2.48886 −0.202541 −0.101270 0.994859i \(-0.532291\pi\)
−0.101270 + 0.994859i \(0.532291\pi\)
\(152\) 0 0
\(153\) 1.37778 0.111387
\(154\) 0 0
\(155\) 5.51114 0.442665
\(156\) 0 0
\(157\) −11.1526 −0.890072 −0.445036 0.895513i \(-0.646809\pi\)
−0.445036 + 0.895513i \(0.646809\pi\)
\(158\) 0 0
\(159\) 12.0415 0.954952
\(160\) 0 0
\(161\) −0.622216 −0.0490375
\(162\) 0 0
\(163\) −8.85728 −0.693756 −0.346878 0.937910i \(-0.612758\pi\)
−0.346878 + 0.937910i \(0.612758\pi\)
\(164\) 0 0
\(165\) −25.7146 −2.00188
\(166\) 0 0
\(167\) 1.89829 0.146894 0.0734470 0.997299i \(-0.476600\pi\)
0.0734470 + 0.997299i \(0.476600\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.42864 0.491610
\(172\) 0 0
\(173\) −16.7556 −1.27390 −0.636951 0.770904i \(-0.719805\pi\)
−0.636951 + 0.770904i \(0.719805\pi\)
\(174\) 0 0
\(175\) 9.09234 0.687316
\(176\) 0 0
\(177\) −12.8573 −0.966412
\(178\) 0 0
\(179\) −18.9590 −1.41706 −0.708531 0.705680i \(-0.750642\pi\)
−0.708531 + 0.705680i \(0.750642\pi\)
\(180\) 0 0
\(181\) 1.70471 0.126710 0.0633552 0.997991i \(-0.479820\pi\)
0.0633552 + 0.997991i \(0.479820\pi\)
\(182\) 0 0
\(183\) 2.94914 0.218007
\(184\) 0 0
\(185\) −40.0830 −2.94696
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0.622216 0.0452595
\(190\) 0 0
\(191\) −10.3684 −0.750232 −0.375116 0.926978i \(-0.622397\pi\)
−0.375116 + 0.926978i \(0.622397\pi\)
\(192\) 0 0
\(193\) 1.61285 0.116095 0.0580477 0.998314i \(-0.481512\pi\)
0.0580477 + 0.998314i \(0.481512\pi\)
\(194\) 0 0
\(195\) −8.85728 −0.634283
\(196\) 0 0
\(197\) 5.34614 0.380897 0.190448 0.981697i \(-0.439006\pi\)
0.190448 + 0.981697i \(0.439006\pi\)
\(198\) 0 0
\(199\) −21.3590 −1.51410 −0.757051 0.653355i \(-0.773361\pi\)
−0.757051 + 0.653355i \(0.773361\pi\)
\(200\) 0 0
\(201\) 5.18421 0.365666
\(202\) 0 0
\(203\) −0.470127 −0.0329964
\(204\) 0 0
\(205\) 42.5718 2.97335
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 37.3274 2.58199
\(210\) 0 0
\(211\) −6.36842 −0.438420 −0.219210 0.975678i \(-0.570348\pi\)
−0.219210 + 0.975678i \(0.570348\pi\)
\(212\) 0 0
\(213\) 6.10171 0.418082
\(214\) 0 0
\(215\) 22.9590 1.56579
\(216\) 0 0
\(217\) −0.774305 −0.0525632
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 2.75557 0.185360
\(222\) 0 0
\(223\) −7.22570 −0.483868 −0.241934 0.970293i \(-0.577782\pi\)
−0.241934 + 0.970293i \(0.577782\pi\)
\(224\) 0 0
\(225\) 14.6128 0.974190
\(226\) 0 0
\(227\) 15.0509 0.998960 0.499480 0.866325i \(-0.333524\pi\)
0.499480 + 0.866325i \(0.333524\pi\)
\(228\) 0 0
\(229\) −6.29529 −0.416004 −0.208002 0.978128i \(-0.566696\pi\)
−0.208002 + 0.978128i \(0.566696\pi\)
\(230\) 0 0
\(231\) 3.61285 0.237708
\(232\) 0 0
\(233\) 27.7146 1.81564 0.907821 0.419359i \(-0.137745\pi\)
0.907821 + 0.419359i \(0.137745\pi\)
\(234\) 0 0
\(235\) 23.2257 1.51508
\(236\) 0 0
\(237\) −8.62222 −0.560073
\(238\) 0 0
\(239\) 15.2257 0.984868 0.492434 0.870350i \(-0.336107\pi\)
0.492434 + 0.870350i \(0.336107\pi\)
\(240\) 0 0
\(241\) 5.14272 0.331272 0.165636 0.986187i \(-0.447032\pi\)
0.165636 + 0.986187i \(0.447032\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 29.2859 1.87101
\(246\) 0 0
\(247\) 12.8573 0.818089
\(248\) 0 0
\(249\) 11.9081 0.754647
\(250\) 0 0
\(251\) 11.4380 0.721961 0.360980 0.932573i \(-0.382442\pi\)
0.360980 + 0.932573i \(0.382442\pi\)
\(252\) 0 0
\(253\) −5.80642 −0.365047
\(254\) 0 0
\(255\) −6.10171 −0.382104
\(256\) 0 0
\(257\) −4.10171 −0.255858 −0.127929 0.991783i \(-0.540833\pi\)
−0.127929 + 0.991783i \(0.540833\pi\)
\(258\) 0 0
\(259\) 5.63158 0.349930
\(260\) 0 0
\(261\) −0.755569 −0.0467685
\(262\) 0 0
\(263\) −24.4701 −1.50889 −0.754446 0.656362i \(-0.772095\pi\)
−0.754446 + 0.656362i \(0.772095\pi\)
\(264\) 0 0
\(265\) −53.3274 −3.27588
\(266\) 0 0
\(267\) 6.23506 0.381580
\(268\) 0 0
\(269\) −14.8573 −0.905864 −0.452932 0.891545i \(-0.649622\pi\)
−0.452932 + 0.891545i \(0.649622\pi\)
\(270\) 0 0
\(271\) −17.2444 −1.04752 −0.523762 0.851864i \(-0.675472\pi\)
−0.523762 + 0.851864i \(0.675472\pi\)
\(272\) 0 0
\(273\) 1.24443 0.0753164
\(274\) 0 0
\(275\) 84.8484 5.11655
\(276\) 0 0
\(277\) 14.5906 0.876663 0.438331 0.898813i \(-0.355570\pi\)
0.438331 + 0.898813i \(0.355570\pi\)
\(278\) 0 0
\(279\) −1.24443 −0.0745022
\(280\) 0 0
\(281\) 7.00937 0.418144 0.209072 0.977900i \(-0.432956\pi\)
0.209072 + 0.977900i \(0.432956\pi\)
\(282\) 0 0
\(283\) 7.08250 0.421011 0.210505 0.977593i \(-0.432489\pi\)
0.210505 + 0.977593i \(0.432489\pi\)
\(284\) 0 0
\(285\) −28.4701 −1.68642
\(286\) 0 0
\(287\) −5.98126 −0.353063
\(288\) 0 0
\(289\) −15.1017 −0.888336
\(290\) 0 0
\(291\) −2.85728 −0.167497
\(292\) 0 0
\(293\) −5.67307 −0.331424 −0.165712 0.986174i \(-0.552992\pi\)
−0.165712 + 0.986174i \(0.552992\pi\)
\(294\) 0 0
\(295\) 56.9403 3.31519
\(296\) 0 0
\(297\) 5.80642 0.336923
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −3.22570 −0.185926
\(302\) 0 0
\(303\) 15.7146 0.902778
\(304\) 0 0
\(305\) −13.0607 −0.747853
\(306\) 0 0
\(307\) −29.0607 −1.65858 −0.829291 0.558817i \(-0.811255\pi\)
−0.829291 + 0.558817i \(0.811255\pi\)
\(308\) 0 0
\(309\) 5.47949 0.311718
\(310\) 0 0
\(311\) 14.3684 0.814758 0.407379 0.913259i \(-0.366443\pi\)
0.407379 + 0.913259i \(0.366443\pi\)
\(312\) 0 0
\(313\) −29.8163 −1.68532 −0.842658 0.538450i \(-0.819010\pi\)
−0.842658 + 0.538450i \(0.819010\pi\)
\(314\) 0 0
\(315\) −2.75557 −0.155259
\(316\) 0 0
\(317\) 20.7556 1.16575 0.582874 0.812562i \(-0.301928\pi\)
0.582874 + 0.812562i \(0.301928\pi\)
\(318\) 0 0
\(319\) −4.38715 −0.245633
\(320\) 0 0
\(321\) −0.295286 −0.0164813
\(322\) 0 0
\(323\) 8.85728 0.492832
\(324\) 0 0
\(325\) 29.2257 1.62115
\(326\) 0 0
\(327\) −9.31756 −0.515262
\(328\) 0 0
\(329\) −3.26317 −0.179904
\(330\) 0 0
\(331\) −8.85728 −0.486840 −0.243420 0.969921i \(-0.578269\pi\)
−0.243420 + 0.969921i \(0.578269\pi\)
\(332\) 0 0
\(333\) 9.05086 0.495984
\(334\) 0 0
\(335\) −22.9590 −1.25438
\(336\) 0 0
\(337\) −19.9813 −1.08845 −0.544224 0.838940i \(-0.683176\pi\)
−0.544224 + 0.838940i \(0.683176\pi\)
\(338\) 0 0
\(339\) −11.4795 −0.623481
\(340\) 0 0
\(341\) −7.22570 −0.391293
\(342\) 0 0
\(343\) −8.47013 −0.457344
\(344\) 0 0
\(345\) 4.42864 0.238430
\(346\) 0 0
\(347\) 3.73329 0.200414 0.100207 0.994967i \(-0.468050\pi\)
0.100207 + 0.994967i \(0.468050\pi\)
\(348\) 0 0
\(349\) −4.10171 −0.219560 −0.109780 0.993956i \(-0.535015\pi\)
−0.109780 + 0.993956i \(0.535015\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 5.73329 0.305152 0.152576 0.988292i \(-0.451243\pi\)
0.152576 + 0.988292i \(0.451243\pi\)
\(354\) 0 0
\(355\) −27.0223 −1.43419
\(356\) 0 0
\(357\) 0.857279 0.0453720
\(358\) 0 0
\(359\) −20.8573 −1.10081 −0.550403 0.834899i \(-0.685526\pi\)
−0.550403 + 0.834899i \(0.685526\pi\)
\(360\) 0 0
\(361\) 22.3274 1.17513
\(362\) 0 0
\(363\) 22.7146 1.19221
\(364\) 0 0
\(365\) −8.85728 −0.463611
\(366\) 0 0
\(367\) −15.8479 −0.827254 −0.413627 0.910446i \(-0.635738\pi\)
−0.413627 + 0.910446i \(0.635738\pi\)
\(368\) 0 0
\(369\) −9.61285 −0.500425
\(370\) 0 0
\(371\) 7.49240 0.388986
\(372\) 0 0
\(373\) 25.0509 1.29708 0.648542 0.761179i \(-0.275379\pi\)
0.648542 + 0.761179i \(0.275379\pi\)
\(374\) 0 0
\(375\) −42.5718 −2.19840
\(376\) 0 0
\(377\) −1.51114 −0.0778275
\(378\) 0 0
\(379\) 9.45091 0.485461 0.242730 0.970094i \(-0.421957\pi\)
0.242730 + 0.970094i \(0.421957\pi\)
\(380\) 0 0
\(381\) 8.47013 0.433938
\(382\) 0 0
\(383\) 37.9813 1.94075 0.970376 0.241600i \(-0.0776721\pi\)
0.970376 + 0.241600i \(0.0776721\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) −5.18421 −0.263528
\(388\) 0 0
\(389\) −35.0005 −1.77459 −0.887297 0.461198i \(-0.847420\pi\)
−0.887297 + 0.461198i \(0.847420\pi\)
\(390\) 0 0
\(391\) −1.37778 −0.0696776
\(392\) 0 0
\(393\) −7.61285 −0.384017
\(394\) 0 0
\(395\) 38.1847 1.92128
\(396\) 0 0
\(397\) 33.4291 1.67776 0.838880 0.544317i \(-0.183211\pi\)
0.838880 + 0.544317i \(0.183211\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 32.0701 1.60150 0.800751 0.598997i \(-0.204434\pi\)
0.800751 + 0.598997i \(0.204434\pi\)
\(402\) 0 0
\(403\) −2.48886 −0.123979
\(404\) 0 0
\(405\) −4.42864 −0.220061
\(406\) 0 0
\(407\) 52.5531 2.60496
\(408\) 0 0
\(409\) 11.3274 0.560104 0.280052 0.959985i \(-0.409648\pi\)
0.280052 + 0.959985i \(0.409648\pi\)
\(410\) 0 0
\(411\) −5.37778 −0.265267
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −52.7368 −2.58875
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −11.3176 −0.552899 −0.276450 0.961028i \(-0.589158\pi\)
−0.276450 + 0.961028i \(0.589158\pi\)
\(420\) 0 0
\(421\) 0.580728 0.0283030 0.0141515 0.999900i \(-0.495495\pi\)
0.0141515 + 0.999900i \(0.495495\pi\)
\(422\) 0 0
\(423\) −5.24443 −0.254993
\(424\) 0 0
\(425\) 20.1334 0.976611
\(426\) 0 0
\(427\) 1.83500 0.0888021
\(428\) 0 0
\(429\) 11.6128 0.560674
\(430\) 0 0
\(431\) −32.3497 −1.55823 −0.779115 0.626881i \(-0.784331\pi\)
−0.779115 + 0.626881i \(0.784331\pi\)
\(432\) 0 0
\(433\) 5.61285 0.269736 0.134868 0.990864i \(-0.456939\pi\)
0.134868 + 0.990864i \(0.456939\pi\)
\(434\) 0 0
\(435\) 3.34614 0.160435
\(436\) 0 0
\(437\) −6.42864 −0.307524
\(438\) 0 0
\(439\) −10.4889 −0.500606 −0.250303 0.968168i \(-0.580530\pi\)
−0.250303 + 0.968168i \(0.580530\pi\)
\(440\) 0 0
\(441\) −6.61285 −0.314898
\(442\) 0 0
\(443\) −20.9403 −0.994901 −0.497451 0.867492i \(-0.665730\pi\)
−0.497451 + 0.867492i \(0.665730\pi\)
\(444\) 0 0
\(445\) −27.6128 −1.30897
\(446\) 0 0
\(447\) 13.2859 0.628402
\(448\) 0 0
\(449\) −4.87601 −0.230113 −0.115057 0.993359i \(-0.536705\pi\)
−0.115057 + 0.993359i \(0.536705\pi\)
\(450\) 0 0
\(451\) −55.8163 −2.62829
\(452\) 0 0
\(453\) −2.48886 −0.116937
\(454\) 0 0
\(455\) −5.51114 −0.258366
\(456\) 0 0
\(457\) 22.8573 1.06922 0.534609 0.845099i \(-0.320459\pi\)
0.534609 + 0.845099i \(0.320459\pi\)
\(458\) 0 0
\(459\) 1.37778 0.0643095
\(460\) 0 0
\(461\) 23.5941 1.09889 0.549444 0.835531i \(-0.314840\pi\)
0.549444 + 0.835531i \(0.314840\pi\)
\(462\) 0 0
\(463\) 10.4889 0.487459 0.243729 0.969843i \(-0.421629\pi\)
0.243729 + 0.969843i \(0.421629\pi\)
\(464\) 0 0
\(465\) 5.51114 0.255573
\(466\) 0 0
\(467\) −15.0509 −0.696471 −0.348235 0.937407i \(-0.613219\pi\)
−0.348235 + 0.937407i \(0.613219\pi\)
\(468\) 0 0
\(469\) 3.22570 0.148949
\(470\) 0 0
\(471\) −11.1526 −0.513883
\(472\) 0 0
\(473\) −30.1017 −1.38408
\(474\) 0 0
\(475\) 93.9407 4.31030
\(476\) 0 0
\(477\) 12.0415 0.551342
\(478\) 0 0
\(479\) −33.7146 −1.54046 −0.770229 0.637768i \(-0.779858\pi\)
−0.770229 + 0.637768i \(0.779858\pi\)
\(480\) 0 0
\(481\) 18.1017 0.825367
\(482\) 0 0
\(483\) −0.622216 −0.0283118
\(484\) 0 0
\(485\) 12.6539 0.574582
\(486\) 0 0
\(487\) −12.2034 −0.552990 −0.276495 0.961015i \(-0.589173\pi\)
−0.276495 + 0.961015i \(0.589173\pi\)
\(488\) 0 0
\(489\) −8.85728 −0.400540
\(490\) 0 0
\(491\) −6.75557 −0.304875 −0.152437 0.988313i \(-0.548712\pi\)
−0.152437 + 0.988313i \(0.548712\pi\)
\(492\) 0 0
\(493\) −1.04101 −0.0468848
\(494\) 0 0
\(495\) −25.7146 −1.15578
\(496\) 0 0
\(497\) 3.79658 0.170300
\(498\) 0 0
\(499\) 37.5941 1.68294 0.841472 0.540301i \(-0.181690\pi\)
0.841472 + 0.540301i \(0.181690\pi\)
\(500\) 0 0
\(501\) 1.89829 0.0848093
\(502\) 0 0
\(503\) −8.59057 −0.383035 −0.191517 0.981489i \(-0.561341\pi\)
−0.191517 + 0.981489i \(0.561341\pi\)
\(504\) 0 0
\(505\) −69.5941 −3.09690
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −3.77784 −0.167450 −0.0837250 0.996489i \(-0.526682\pi\)
−0.0837250 + 0.996489i \(0.526682\pi\)
\(510\) 0 0
\(511\) 1.24443 0.0550504
\(512\) 0 0
\(513\) 6.42864 0.283831
\(514\) 0 0
\(515\) −24.2667 −1.06932
\(516\) 0 0
\(517\) −30.4514 −1.33925
\(518\) 0 0
\(519\) −16.7556 −0.735488
\(520\) 0 0
\(521\) 30.8256 1.35050 0.675248 0.737591i \(-0.264037\pi\)
0.675248 + 0.737591i \(0.264037\pi\)
\(522\) 0 0
\(523\) 6.89877 0.301662 0.150831 0.988560i \(-0.451805\pi\)
0.150831 + 0.988560i \(0.451805\pi\)
\(524\) 0 0
\(525\) 9.09234 0.396822
\(526\) 0 0
\(527\) −1.71456 −0.0746873
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.8573 −0.557958
\(532\) 0 0
\(533\) −19.2257 −0.832757
\(534\) 0 0
\(535\) 1.30772 0.0565375
\(536\) 0 0
\(537\) −18.9590 −0.818141
\(538\) 0 0
\(539\) −38.3970 −1.65388
\(540\) 0 0
\(541\) −30.5906 −1.31519 −0.657596 0.753371i \(-0.728426\pi\)
−0.657596 + 0.753371i \(0.728426\pi\)
\(542\) 0 0
\(543\) 1.70471 0.0731563
\(544\) 0 0
\(545\) 41.2641 1.76756
\(546\) 0 0
\(547\) −27.3461 −1.16924 −0.584618 0.811308i \(-0.698756\pi\)
−0.584618 + 0.811308i \(0.698756\pi\)
\(548\) 0 0
\(549\) 2.94914 0.125866
\(550\) 0 0
\(551\) −4.85728 −0.206927
\(552\) 0 0
\(553\) −5.36488 −0.228138
\(554\) 0 0
\(555\) −40.0830 −1.70143
\(556\) 0 0
\(557\) 34.1432 1.44669 0.723347 0.690485i \(-0.242603\pi\)
0.723347 + 0.690485i \(0.242603\pi\)
\(558\) 0 0
\(559\) −10.3684 −0.438537
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 31.0509 1.30864 0.654319 0.756219i \(-0.272955\pi\)
0.654319 + 0.756219i \(0.272955\pi\)
\(564\) 0 0
\(565\) 50.8385 2.13879
\(566\) 0 0
\(567\) 0.622216 0.0261306
\(568\) 0 0
\(569\) −21.8479 −0.915912 −0.457956 0.888975i \(-0.651418\pi\)
−0.457956 + 0.888975i \(0.651418\pi\)
\(570\) 0 0
\(571\) 18.1619 0.760053 0.380027 0.924976i \(-0.375915\pi\)
0.380027 + 0.924976i \(0.375915\pi\)
\(572\) 0 0
\(573\) −10.3684 −0.433147
\(574\) 0 0
\(575\) −14.6128 −0.609398
\(576\) 0 0
\(577\) −39.3274 −1.63722 −0.818611 0.574349i \(-0.805255\pi\)
−0.818611 + 0.574349i \(0.805255\pi\)
\(578\) 0 0
\(579\) 1.61285 0.0670277
\(580\) 0 0
\(581\) 7.40943 0.307395
\(582\) 0 0
\(583\) 69.9180 2.89571
\(584\) 0 0
\(585\) −8.85728 −0.366203
\(586\) 0 0
\(587\) −38.8385 −1.60304 −0.801519 0.597969i \(-0.795975\pi\)
−0.801519 + 0.597969i \(0.795975\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 5.34614 0.219911
\(592\) 0 0
\(593\) 29.6128 1.21605 0.608027 0.793916i \(-0.291961\pi\)
0.608027 + 0.793916i \(0.291961\pi\)
\(594\) 0 0
\(595\) −3.79658 −0.155645
\(596\) 0 0
\(597\) −21.3590 −0.874168
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 0.672594 0.0274357 0.0137178 0.999906i \(-0.495633\pi\)
0.0137178 + 0.999906i \(0.495633\pi\)
\(602\) 0 0
\(603\) 5.18421 0.211117
\(604\) 0 0
\(605\) −100.595 −4.08975
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) −0.470127 −0.0190505
\(610\) 0 0
\(611\) −10.4889 −0.424334
\(612\) 0 0
\(613\) 2.94914 0.119115 0.0595574 0.998225i \(-0.481031\pi\)
0.0595574 + 0.998225i \(0.481031\pi\)
\(614\) 0 0
\(615\) 42.5718 1.71666
\(616\) 0 0
\(617\) 39.1753 1.57714 0.788569 0.614946i \(-0.210822\pi\)
0.788569 + 0.614946i \(0.210822\pi\)
\(618\) 0 0
\(619\) −23.3689 −0.939275 −0.469638 0.882859i \(-0.655615\pi\)
−0.469638 + 0.882859i \(0.655615\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 3.87955 0.155431
\(624\) 0 0
\(625\) 115.471 4.61884
\(626\) 0 0
\(627\) 37.3274 1.49071
\(628\) 0 0
\(629\) 12.4701 0.497217
\(630\) 0 0
\(631\) −22.7239 −0.904625 −0.452313 0.891859i \(-0.649401\pi\)
−0.452313 + 0.891859i \(0.649401\pi\)
\(632\) 0 0
\(633\) −6.36842 −0.253122
\(634\) 0 0
\(635\) −37.5111 −1.48858
\(636\) 0 0
\(637\) −13.2257 −0.524021
\(638\) 0 0
\(639\) 6.10171 0.241380
\(640\) 0 0
\(641\) 45.5812 1.80035 0.900175 0.435529i \(-0.143439\pi\)
0.900175 + 0.435529i \(0.143439\pi\)
\(642\) 0 0
\(643\) 13.8381 0.545720 0.272860 0.962054i \(-0.412030\pi\)
0.272860 + 0.962054i \(0.412030\pi\)
\(644\) 0 0
\(645\) 22.9590 0.904009
\(646\) 0 0
\(647\) 18.5718 0.730134 0.365067 0.930981i \(-0.381046\pi\)
0.365067 + 0.930981i \(0.381046\pi\)
\(648\) 0 0
\(649\) −74.6548 −2.93046
\(650\) 0 0
\(651\) −0.774305 −0.0303474
\(652\) 0 0
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) 33.7146 1.31734
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −33.2988 −1.29714 −0.648569 0.761156i \(-0.724632\pi\)
−0.648569 + 0.761156i \(0.724632\pi\)
\(660\) 0 0
\(661\) 7.68598 0.298950 0.149475 0.988766i \(-0.452242\pi\)
0.149475 + 0.988766i \(0.452242\pi\)
\(662\) 0 0
\(663\) 2.75557 0.107017
\(664\) 0 0
\(665\) −17.7146 −0.686941
\(666\) 0 0
\(667\) 0.755569 0.0292557
\(668\) 0 0
\(669\) −7.22570 −0.279362
\(670\) 0 0
\(671\) 17.1240 0.661064
\(672\) 0 0
\(673\) −31.3274 −1.20758 −0.603791 0.797142i \(-0.706344\pi\)
−0.603791 + 0.797142i \(0.706344\pi\)
\(674\) 0 0
\(675\) 14.6128 0.562449
\(676\) 0 0
\(677\) −3.89523 −0.149706 −0.0748529 0.997195i \(-0.523849\pi\)
−0.0748529 + 0.997195i \(0.523849\pi\)
\(678\) 0 0
\(679\) −1.77784 −0.0682274
\(680\) 0 0
\(681\) 15.0509 0.576750
\(682\) 0 0
\(683\) −10.8760 −0.416159 −0.208080 0.978112i \(-0.566721\pi\)
−0.208080 + 0.978112i \(0.566721\pi\)
\(684\) 0 0
\(685\) 23.8163 0.909973
\(686\) 0 0
\(687\) −6.29529 −0.240180
\(688\) 0 0
\(689\) 24.0830 0.917488
\(690\) 0 0
\(691\) 10.5718 0.402172 0.201086 0.979574i \(-0.435553\pi\)
0.201086 + 0.979574i \(0.435553\pi\)
\(692\) 0 0
\(693\) 3.61285 0.137241
\(694\) 0 0
\(695\) 17.7146 0.671951
\(696\) 0 0
\(697\) −13.2444 −0.501669
\(698\) 0 0
\(699\) 27.7146 1.04826
\(700\) 0 0
\(701\) 18.2065 0.687649 0.343825 0.939034i \(-0.388277\pi\)
0.343825 + 0.939034i \(0.388277\pi\)
\(702\) 0 0
\(703\) 58.1847 2.19448
\(704\) 0 0
\(705\) 23.2257 0.874730
\(706\) 0 0
\(707\) 9.77784 0.367734
\(708\) 0 0
\(709\) −17.1338 −0.643474 −0.321737 0.946829i \(-0.604267\pi\)
−0.321737 + 0.946829i \(0.604267\pi\)
\(710\) 0 0
\(711\) −8.62222 −0.323358
\(712\) 0 0
\(713\) 1.24443 0.0466043
\(714\) 0 0
\(715\) −51.4291 −1.92334
\(716\) 0 0
\(717\) 15.2257 0.568614
\(718\) 0 0
\(719\) 3.34614 0.124790 0.0623950 0.998052i \(-0.480126\pi\)
0.0623950 + 0.998052i \(0.480126\pi\)
\(720\) 0 0
\(721\) 3.40943 0.126974
\(722\) 0 0
\(723\) 5.14272 0.191260
\(724\) 0 0
\(725\) −11.0410 −0.410053
\(726\) 0 0
\(727\) −9.09234 −0.337216 −0.168608 0.985683i \(-0.553927\pi\)
−0.168608 + 0.985683i \(0.553927\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.14272 −0.264183
\(732\) 0 0
\(733\) −40.3595 −1.49071 −0.745357 0.666666i \(-0.767721\pi\)
−0.745357 + 0.666666i \(0.767721\pi\)
\(734\) 0 0
\(735\) 29.2859 1.08023
\(736\) 0 0
\(737\) 30.1017 1.10881
\(738\) 0 0
\(739\) 41.9180 1.54198 0.770989 0.636849i \(-0.219762\pi\)
0.770989 + 0.636849i \(0.219762\pi\)
\(740\) 0 0
\(741\) 12.8573 0.472324
\(742\) 0 0
\(743\) 34.9590 1.28252 0.641260 0.767323i \(-0.278412\pi\)
0.641260 + 0.767323i \(0.278412\pi\)
\(744\) 0 0
\(745\) −58.8385 −2.15568
\(746\) 0 0
\(747\) 11.9081 0.435696
\(748\) 0 0
\(749\) −0.183732 −0.00671341
\(750\) 0 0
\(751\) 25.7462 0.939492 0.469746 0.882802i \(-0.344345\pi\)
0.469746 + 0.882802i \(0.344345\pi\)
\(752\) 0 0
\(753\) 11.4380 0.416824
\(754\) 0 0
\(755\) 11.0223 0.401142
\(756\) 0 0
\(757\) 40.9304 1.48764 0.743821 0.668379i \(-0.233012\pi\)
0.743821 + 0.668379i \(0.233012\pi\)
\(758\) 0 0
\(759\) −5.80642 −0.210760
\(760\) 0 0
\(761\) −2.26671 −0.0821680 −0.0410840 0.999156i \(-0.513081\pi\)
−0.0410840 + 0.999156i \(0.513081\pi\)
\(762\) 0 0
\(763\) −5.79753 −0.209885
\(764\) 0 0
\(765\) −6.10171 −0.220608
\(766\) 0 0
\(767\) −25.7146 −0.928499
\(768\) 0 0
\(769\) −41.6128 −1.50060 −0.750299 0.661099i \(-0.770090\pi\)
−0.750299 + 0.661099i \(0.770090\pi\)
\(770\) 0 0
\(771\) −4.10171 −0.147719
\(772\) 0 0
\(773\) 2.44738 0.0880260 0.0440130 0.999031i \(-0.485986\pi\)
0.0440130 + 0.999031i \(0.485986\pi\)
\(774\) 0 0
\(775\) −18.1847 −0.653213
\(776\) 0 0
\(777\) 5.63158 0.202032
\(778\) 0 0
\(779\) −61.7975 −2.21413
\(780\) 0 0
\(781\) 35.4291 1.26775
\(782\) 0 0
\(783\) −0.755569 −0.0270018
\(784\) 0 0
\(785\) 49.3907 1.76283
\(786\) 0 0
\(787\) −36.4099 −1.29787 −0.648936 0.760843i \(-0.724786\pi\)
−0.648936 + 0.760843i \(0.724786\pi\)
\(788\) 0 0
\(789\) −24.4701 −0.871160
\(790\) 0 0
\(791\) −7.14272 −0.253966
\(792\) 0 0
\(793\) 5.89829 0.209454
\(794\) 0 0
\(795\) −53.3274 −1.89133
\(796\) 0 0
\(797\) 0.428639 0.0151832 0.00759159 0.999971i \(-0.497583\pi\)
0.00759159 + 0.999971i \(0.497583\pi\)
\(798\) 0 0
\(799\) −7.22570 −0.255627
\(800\) 0 0
\(801\) 6.23506 0.220305
\(802\) 0 0
\(803\) 11.6128 0.409808
\(804\) 0 0
\(805\) 2.75557 0.0971210
\(806\) 0 0
\(807\) −14.8573 −0.523001
\(808\) 0 0
\(809\) −35.9813 −1.26503 −0.632517 0.774547i \(-0.717978\pi\)
−0.632517 + 0.774547i \(0.717978\pi\)
\(810\) 0 0
\(811\) 4.12045 0.144688 0.0723442 0.997380i \(-0.476952\pi\)
0.0723442 + 0.997380i \(0.476952\pi\)
\(812\) 0 0
\(813\) −17.2444 −0.604789
\(814\) 0 0
\(815\) 39.2257 1.37402
\(816\) 0 0
\(817\) −33.3274 −1.16598
\(818\) 0 0
\(819\) 1.24443 0.0434839
\(820\) 0 0
\(821\) −24.4514 −0.853359 −0.426680 0.904403i \(-0.640317\pi\)
−0.426680 + 0.904403i \(0.640317\pi\)
\(822\) 0 0
\(823\) −12.2667 −0.427590 −0.213795 0.976878i \(-0.568583\pi\)
−0.213795 + 0.976878i \(0.568583\pi\)
\(824\) 0 0
\(825\) 84.8484 2.95404
\(826\) 0 0
\(827\) −21.5022 −0.747706 −0.373853 0.927488i \(-0.621964\pi\)
−0.373853 + 0.927488i \(0.621964\pi\)
\(828\) 0 0
\(829\) −2.38715 −0.0829092 −0.0414546 0.999140i \(-0.513199\pi\)
−0.0414546 + 0.999140i \(0.513199\pi\)
\(830\) 0 0
\(831\) 14.5906 0.506141
\(832\) 0 0
\(833\) −9.11108 −0.315680
\(834\) 0 0
\(835\) −8.40684 −0.290931
\(836\) 0 0
\(837\) −1.24443 −0.0430138
\(838\) 0 0
\(839\) 1.24443 0.0429625 0.0214813 0.999769i \(-0.493162\pi\)
0.0214813 + 0.999769i \(0.493162\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 0 0
\(843\) 7.00937 0.241415
\(844\) 0 0
\(845\) 39.8578 1.37115
\(846\) 0 0
\(847\) 14.1334 0.485628
\(848\) 0 0
\(849\) 7.08250 0.243071
\(850\) 0 0
\(851\) −9.05086 −0.310259
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) −28.4701 −0.973658
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −5.98126 −0.203841
\(862\) 0 0
\(863\) 40.8573 1.39080 0.695399 0.718624i \(-0.255228\pi\)
0.695399 + 0.718624i \(0.255228\pi\)
\(864\) 0 0
\(865\) 74.2044 2.52302
\(866\) 0 0
\(867\) −15.1017 −0.512881
\(868\) 0 0
\(869\) −50.0642 −1.69831
\(870\) 0 0
\(871\) 10.3684 0.351320
\(872\) 0 0
\(873\) −2.85728 −0.0967042
\(874\) 0 0
\(875\) −26.4889 −0.895487
\(876\) 0 0
\(877\) 7.89829 0.266706 0.133353 0.991069i \(-0.457426\pi\)
0.133353 + 0.991069i \(0.457426\pi\)
\(878\) 0 0
\(879\) −5.67307 −0.191348
\(880\) 0 0
\(881\) −48.9906 −1.65054 −0.825268 0.564741i \(-0.808976\pi\)
−0.825268 + 0.564741i \(0.808976\pi\)
\(882\) 0 0
\(883\) 8.85728 0.298071 0.149036 0.988832i \(-0.452383\pi\)
0.149036 + 0.988832i \(0.452383\pi\)
\(884\) 0 0
\(885\) 56.9403 1.91403
\(886\) 0 0
\(887\) 51.1624 1.71787 0.858933 0.512088i \(-0.171128\pi\)
0.858933 + 0.512088i \(0.171128\pi\)
\(888\) 0 0
\(889\) 5.27025 0.176758
\(890\) 0 0
\(891\) 5.80642 0.194523
\(892\) 0 0
\(893\) −33.7146 −1.12821
\(894\) 0 0
\(895\) 83.9625 2.80656
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) 0.940253 0.0313592
\(900\) 0 0
\(901\) 16.5906 0.552712
\(902\) 0 0
\(903\) −3.22570 −0.107344
\(904\) 0 0
\(905\) −7.54956 −0.250956
\(906\) 0 0
\(907\) 24.0228 0.797662 0.398831 0.917024i \(-0.369416\pi\)
0.398831 + 0.917024i \(0.369416\pi\)
\(908\) 0 0
\(909\) 15.7146 0.521219
\(910\) 0 0
\(911\) −34.8385 −1.15425 −0.577126 0.816655i \(-0.695826\pi\)
−0.577126 + 0.816655i \(0.695826\pi\)
\(912\) 0 0
\(913\) 69.1437 2.28832
\(914\) 0 0
\(915\) −13.0607 −0.431773
\(916\) 0 0
\(917\) −4.73683 −0.156424
\(918\) 0 0
\(919\) 38.6035 1.27341 0.636706 0.771107i \(-0.280297\pi\)
0.636706 + 0.771107i \(0.280297\pi\)
\(920\) 0 0
\(921\) −29.0607 −0.957583
\(922\) 0 0
\(923\) 12.2034 0.401680
\(924\) 0 0
\(925\) 132.259 4.34864
\(926\) 0 0
\(927\) 5.47949 0.179970
\(928\) 0 0
\(929\) −29.8163 −0.978240 −0.489120 0.872216i \(-0.662682\pi\)
−0.489120 + 0.872216i \(0.662682\pi\)
\(930\) 0 0
\(931\) −42.5116 −1.39326
\(932\) 0 0
\(933\) 14.3684 0.470401
\(934\) 0 0
\(935\) −35.4291 −1.15866
\(936\) 0 0
\(937\) −35.5111 −1.16010 −0.580049 0.814581i \(-0.696967\pi\)
−0.580049 + 0.814581i \(0.696967\pi\)
\(938\) 0 0
\(939\) −29.8163 −0.973017
\(940\) 0 0
\(941\) −21.2029 −0.691196 −0.345598 0.938383i \(-0.612324\pi\)
−0.345598 + 0.938383i \(0.612324\pi\)
\(942\) 0 0
\(943\) 9.61285 0.313037
\(944\) 0 0
\(945\) −2.75557 −0.0896387
\(946\) 0 0
\(947\) −37.2070 −1.20906 −0.604532 0.796581i \(-0.706640\pi\)
−0.604532 + 0.796581i \(0.706640\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 20.7556 0.673045
\(952\) 0 0
\(953\) 39.2958 1.27291 0.636457 0.771312i \(-0.280399\pi\)
0.636457 + 0.771312i \(0.280399\pi\)
\(954\) 0 0
\(955\) 45.9180 1.48587
\(956\) 0 0
\(957\) −4.38715 −0.141816
\(958\) 0 0
\(959\) −3.34614 −0.108053
\(960\) 0 0
\(961\) −29.4514 −0.950045
\(962\) 0 0
\(963\) −0.295286 −0.00951547
\(964\) 0 0
\(965\) −7.14272 −0.229932
\(966\) 0 0
\(967\) −23.9367 −0.769753 −0.384876 0.922968i \(-0.625756\pi\)
−0.384876 + 0.922968i \(0.625756\pi\)
\(968\) 0 0
\(969\) 8.85728 0.284537
\(970\) 0 0
\(971\) −51.8448 −1.66378 −0.831890 0.554940i \(-0.812741\pi\)
−0.831890 + 0.554940i \(0.812741\pi\)
\(972\) 0 0
\(973\) −2.48886 −0.0797893
\(974\) 0 0
\(975\) 29.2257 0.935971
\(976\) 0 0
\(977\) 29.1111 0.931346 0.465673 0.884957i \(-0.345812\pi\)
0.465673 + 0.884957i \(0.345812\pi\)
\(978\) 0 0
\(979\) 36.2034 1.15707
\(980\) 0 0
\(981\) −9.31756 −0.297487
\(982\) 0 0
\(983\) 1.36488 0.0435328 0.0217664 0.999763i \(-0.493071\pi\)
0.0217664 + 0.999763i \(0.493071\pi\)
\(984\) 0 0
\(985\) −23.6761 −0.754384
\(986\) 0 0
\(987\) −3.26317 −0.103868
\(988\) 0 0
\(989\) 5.18421 0.164848
\(990\) 0 0
\(991\) 38.3881 1.21944 0.609719 0.792618i \(-0.291282\pi\)
0.609719 + 0.792618i \(0.291282\pi\)
\(992\) 0 0
\(993\) −8.85728 −0.281077
\(994\) 0 0
\(995\) 94.5915 2.99875
\(996\) 0 0
\(997\) 12.3051 0.389707 0.194854 0.980832i \(-0.437577\pi\)
0.194854 + 0.980832i \(0.437577\pi\)
\(998\) 0 0
\(999\) 9.05086 0.286356
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 552.2.a.g.1.1 3
3.2 odd 2 1656.2.a.n.1.3 3
4.3 odd 2 1104.2.a.o.1.1 3
8.3 odd 2 4416.2.a.bs.1.3 3
8.5 even 2 4416.2.a.bp.1.3 3
12.11 even 2 3312.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.2.a.g.1.1 3 1.1 even 1 trivial
1104.2.a.o.1.1 3 4.3 odd 2
1656.2.a.n.1.3 3 3.2 odd 2
3312.2.a.bf.1.3 3 12.11 even 2
4416.2.a.bp.1.3 3 8.5 even 2
4416.2.a.bs.1.3 3 8.3 odd 2