Properties

Label 588.4.a.g.1.1
Level $588$
Weight $4$
Character 588.1
Self dual yes
Analytic conductor $34.693$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.6931230834\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -12.8248 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -12.8248 q^{5} +9.00000 q^{9} -36.8248 q^{11} +87.1238 q^{13} +38.4743 q^{15} +102.598 q^{17} +95.8248 q^{19} -96.0000 q^{23} +39.4743 q^{25} -27.0000 q^{27} -212.021 q^{29} -159.248 q^{31} +110.474 q^{33} +128.670 q^{37} -261.371 q^{39} -298.042 q^{41} -33.3297 q^{43} -115.423 q^{45} +271.196 q^{47} -307.794 q^{51} +448.268 q^{53} +472.268 q^{55} -287.474 q^{57} -668.474 q^{59} -243.691 q^{61} -1117.34 q^{65} -335.577 q^{67} +288.000 q^{69} -339.608 q^{71} -918.320 q^{73} -118.423 q^{75} -136.299 q^{79} +81.0000 q^{81} +287.464 q^{83} -1315.79 q^{85} +636.062 q^{87} +161.855 q^{89} +477.743 q^{93} -1228.93 q^{95} +182.680 q^{97} -331.423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 3 q^{5} + 18 q^{9} + O(q^{10}) \) \( 2 q - 6 q^{3} - 3 q^{5} + 18 q^{9} - 51 q^{11} + 61 q^{13} + 9 q^{15} + 24 q^{17} + 169 q^{19} - 192 q^{23} + 11 q^{25} - 54 q^{27} - 39 q^{29} - 92 q^{31} + 153 q^{33} - 173 q^{37} - 183 q^{39} + 174 q^{41} - 497 q^{43} - 27 q^{45} + 180 q^{47} - 72 q^{51} + 285 q^{53} + 333 q^{55} - 507 q^{57} - 1269 q^{59} + 328 q^{61} - 1374 q^{65} - 875 q^{67} + 576 q^{69} - 1404 q^{71} - 1361 q^{73} - 33 q^{75} - 182 q^{79} + 162 q^{81} - 399 q^{83} - 2088 q^{85} + 117 q^{87} + 822 q^{89} + 276 q^{93} - 510 q^{95} + 841 q^{97} - 459 q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −12.8248 −1.14708 −0.573540 0.819177i \(-0.694430\pi\)
−0.573540 + 0.819177i \(0.694430\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −36.8248 −1.00937 −0.504685 0.863303i \(-0.668392\pi\)
−0.504685 + 0.863303i \(0.668392\pi\)
\(12\) 0 0
\(13\) 87.1238 1.85875 0.929376 0.369134i \(-0.120346\pi\)
0.929376 + 0.369134i \(0.120346\pi\)
\(14\) 0 0
\(15\) 38.4743 0.662267
\(16\) 0 0
\(17\) 102.598 1.46375 0.731873 0.681441i \(-0.238647\pi\)
0.731873 + 0.681441i \(0.238647\pi\)
\(18\) 0 0
\(19\) 95.8248 1.15704 0.578519 0.815669i \(-0.303631\pi\)
0.578519 + 0.815669i \(0.303631\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −96.0000 −0.870321 −0.435161 0.900353i \(-0.643308\pi\)
−0.435161 + 0.900353i \(0.643308\pi\)
\(24\) 0 0
\(25\) 39.4743 0.315794
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −212.021 −1.35763 −0.678815 0.734309i \(-0.737506\pi\)
−0.678815 + 0.734309i \(0.737506\pi\)
\(30\) 0 0
\(31\) −159.248 −0.922635 −0.461318 0.887235i \(-0.652623\pi\)
−0.461318 + 0.887235i \(0.652623\pi\)
\(32\) 0 0
\(33\) 110.474 0.582761
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 128.670 0.571710 0.285855 0.958273i \(-0.407722\pi\)
0.285855 + 0.958273i \(0.407722\pi\)
\(38\) 0 0
\(39\) −261.371 −1.07315
\(40\) 0 0
\(41\) −298.042 −1.13527 −0.567637 0.823279i \(-0.692142\pi\)
−0.567637 + 0.823279i \(0.692142\pi\)
\(42\) 0 0
\(43\) −33.3297 −0.118203 −0.0591016 0.998252i \(-0.518824\pi\)
−0.0591016 + 0.998252i \(0.518824\pi\)
\(44\) 0 0
\(45\) −115.423 −0.382360
\(46\) 0 0
\(47\) 271.196 0.841660 0.420830 0.907140i \(-0.361739\pi\)
0.420830 + 0.907140i \(0.361739\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −307.794 −0.845094
\(52\) 0 0
\(53\) 448.268 1.16178 0.580890 0.813982i \(-0.302704\pi\)
0.580890 + 0.813982i \(0.302704\pi\)
\(54\) 0 0
\(55\) 472.268 1.15783
\(56\) 0 0
\(57\) −287.474 −0.668016
\(58\) 0 0
\(59\) −668.474 −1.47505 −0.737525 0.675320i \(-0.764006\pi\)
−0.737525 + 0.675320i \(0.764006\pi\)
\(60\) 0 0
\(61\) −243.691 −0.511499 −0.255750 0.966743i \(-0.582322\pi\)
−0.255750 + 0.966743i \(0.582322\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1117.34 −2.13214
\(66\) 0 0
\(67\) −335.577 −0.611900 −0.305950 0.952048i \(-0.598974\pi\)
−0.305950 + 0.952048i \(0.598974\pi\)
\(68\) 0 0
\(69\) 288.000 0.502480
\(70\) 0 0
\(71\) −339.608 −0.567663 −0.283831 0.958874i \(-0.591606\pi\)
−0.283831 + 0.958874i \(0.591606\pi\)
\(72\) 0 0
\(73\) −918.320 −1.47235 −0.736173 0.676794i \(-0.763369\pi\)
−0.736173 + 0.676794i \(0.763369\pi\)
\(74\) 0 0
\(75\) −118.423 −0.182324
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −136.299 −0.194112 −0.0970559 0.995279i \(-0.530943\pi\)
−0.0970559 + 0.995279i \(0.530943\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 287.464 0.380160 0.190080 0.981769i \(-0.439125\pi\)
0.190080 + 0.981769i \(0.439125\pi\)
\(84\) 0 0
\(85\) −1315.79 −1.67903
\(86\) 0 0
\(87\) 636.062 0.783828
\(88\) 0 0
\(89\) 161.855 0.192771 0.0963856 0.995344i \(-0.469272\pi\)
0.0963856 + 0.995344i \(0.469272\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 477.743 0.532684
\(94\) 0 0
\(95\) −1228.93 −1.32721
\(96\) 0 0
\(97\) 182.680 0.191220 0.0956101 0.995419i \(-0.469520\pi\)
0.0956101 + 0.995419i \(0.469520\pi\)
\(98\) 0 0
\(99\) −331.423 −0.336457
\(100\) 0 0
\(101\) −1532.10 −1.50941 −0.754703 0.656067i \(-0.772219\pi\)
−0.754703 + 0.656067i \(0.772219\pi\)
\(102\) 0 0
\(103\) 487.908 0.466747 0.233374 0.972387i \(-0.425023\pi\)
0.233374 + 0.972387i \(0.425023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 492.351 0.444836 0.222418 0.974951i \(-0.428605\pi\)
0.222418 + 0.974951i \(0.428605\pi\)
\(108\) 0 0
\(109\) 848.072 0.745235 0.372617 0.927985i \(-0.378460\pi\)
0.372617 + 0.927985i \(0.378460\pi\)
\(110\) 0 0
\(111\) −386.011 −0.330077
\(112\) 0 0
\(113\) −736.350 −0.613009 −0.306505 0.951869i \(-0.599159\pi\)
−0.306505 + 0.951869i \(0.599159\pi\)
\(114\) 0 0
\(115\) 1231.18 0.998328
\(116\) 0 0
\(117\) 784.114 0.619584
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0623 0.0188297
\(122\) 0 0
\(123\) 894.125 0.655451
\(124\) 0 0
\(125\) 1096.85 0.784839
\(126\) 0 0
\(127\) −2511.37 −1.75471 −0.877355 0.479841i \(-0.840694\pi\)
−0.877355 + 0.479841i \(0.840694\pi\)
\(128\) 0 0
\(129\) 99.9892 0.0682446
\(130\) 0 0
\(131\) −679.423 −0.453141 −0.226570 0.973995i \(-0.572751\pi\)
−0.226570 + 0.973995i \(0.572751\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 346.268 0.220756
\(136\) 0 0
\(137\) −164.537 −0.102608 −0.0513040 0.998683i \(-0.516338\pi\)
−0.0513040 + 0.998683i \(0.516338\pi\)
\(138\) 0 0
\(139\) −521.991 −0.318523 −0.159261 0.987236i \(-0.550911\pi\)
−0.159261 + 0.987236i \(0.550911\pi\)
\(140\) 0 0
\(141\) −813.588 −0.485932
\(142\) 0 0
\(143\) −3208.31 −1.87617
\(144\) 0 0
\(145\) 2719.11 1.55731
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2412.12 −1.32623 −0.663117 0.748516i \(-0.730767\pi\)
−0.663117 + 0.748516i \(0.730767\pi\)
\(150\) 0 0
\(151\) −1574.58 −0.848591 −0.424296 0.905524i \(-0.639478\pi\)
−0.424296 + 0.905524i \(0.639478\pi\)
\(152\) 0 0
\(153\) 923.382 0.487915
\(154\) 0 0
\(155\) 2042.31 1.05834
\(156\) 0 0
\(157\) 2078.74 1.05670 0.528349 0.849027i \(-0.322811\pi\)
0.528349 + 0.849027i \(0.322811\pi\)
\(158\) 0 0
\(159\) −1344.80 −0.670754
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3179.43 −1.52780 −0.763901 0.645333i \(-0.776719\pi\)
−0.763901 + 0.645333i \(0.776719\pi\)
\(164\) 0 0
\(165\) −1416.80 −0.668473
\(166\) 0 0
\(167\) −2979.28 −1.38050 −0.690250 0.723571i \(-0.742500\pi\)
−0.690250 + 0.723571i \(0.742500\pi\)
\(168\) 0 0
\(169\) 5393.55 2.45496
\(170\) 0 0
\(171\) 862.423 0.385679
\(172\) 0 0
\(173\) 16.3704 0.00719431 0.00359716 0.999994i \(-0.498855\pi\)
0.00359716 + 0.999994i \(0.498855\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2005.42 0.851620
\(178\) 0 0
\(179\) 2698.29 1.12670 0.563351 0.826218i \(-0.309512\pi\)
0.563351 + 0.826218i \(0.309512\pi\)
\(180\) 0 0
\(181\) 31.3297 0.0128659 0.00643293 0.999979i \(-0.497952\pi\)
0.00643293 + 0.999979i \(0.497952\pi\)
\(182\) 0 0
\(183\) 731.073 0.295314
\(184\) 0 0
\(185\) −1650.16 −0.655797
\(186\) 0 0
\(187\) −3778.15 −1.47746
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1545.17 −0.585366 −0.292683 0.956210i \(-0.594548\pi\)
−0.292683 + 0.956210i \(0.594548\pi\)
\(192\) 0 0
\(193\) −1830.20 −0.682593 −0.341297 0.939956i \(-0.610866\pi\)
−0.341297 + 0.939956i \(0.610866\pi\)
\(194\) 0 0
\(195\) 3352.02 1.23099
\(196\) 0 0
\(197\) 4728.45 1.71009 0.855047 0.518551i \(-0.173528\pi\)
0.855047 + 0.518551i \(0.173528\pi\)
\(198\) 0 0
\(199\) −328.249 −0.116930 −0.0584648 0.998289i \(-0.518621\pi\)
−0.0584648 + 0.998289i \(0.518621\pi\)
\(200\) 0 0
\(201\) 1006.73 0.353280
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3822.31 1.30225
\(206\) 0 0
\(207\) −864.000 −0.290107
\(208\) 0 0
\(209\) −3528.72 −1.16788
\(210\) 0 0
\(211\) −4935.76 −1.61039 −0.805193 0.593013i \(-0.797938\pi\)
−0.805193 + 0.593013i \(0.797938\pi\)
\(212\) 0 0
\(213\) 1018.82 0.327740
\(214\) 0 0
\(215\) 427.445 0.135589
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2754.96 0.850059
\(220\) 0 0
\(221\) 8938.72 2.72074
\(222\) 0 0
\(223\) 3446.00 1.03480 0.517402 0.855742i \(-0.326899\pi\)
0.517402 + 0.855742i \(0.326899\pi\)
\(224\) 0 0
\(225\) 355.268 0.105265
\(226\) 0 0
\(227\) 5724.75 1.67385 0.836927 0.547315i \(-0.184350\pi\)
0.836927 + 0.547315i \(0.184350\pi\)
\(228\) 0 0
\(229\) 3017.15 0.870649 0.435325 0.900274i \(-0.356634\pi\)
0.435325 + 0.900274i \(0.356634\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 381.713 0.107325 0.0536627 0.998559i \(-0.482910\pi\)
0.0536627 + 0.998559i \(0.482910\pi\)
\(234\) 0 0
\(235\) −3478.02 −0.965452
\(236\) 0 0
\(237\) 408.897 0.112071
\(238\) 0 0
\(239\) −1377.38 −0.372785 −0.186392 0.982475i \(-0.559680\pi\)
−0.186392 + 0.982475i \(0.559680\pi\)
\(240\) 0 0
\(241\) 5806.72 1.55205 0.776025 0.630702i \(-0.217233\pi\)
0.776025 + 0.630702i \(0.217233\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8348.61 2.15065
\(248\) 0 0
\(249\) −862.393 −0.219486
\(250\) 0 0
\(251\) −4348.52 −1.09353 −0.546765 0.837286i \(-0.684141\pi\)
−0.546765 + 0.837286i \(0.684141\pi\)
\(252\) 0 0
\(253\) 3535.18 0.878477
\(254\) 0 0
\(255\) 3947.38 0.969391
\(256\) 0 0
\(257\) −4691.11 −1.13861 −0.569307 0.822125i \(-0.692788\pi\)
−0.569307 + 0.822125i \(0.692788\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1908.19 −0.452543
\(262\) 0 0
\(263\) −7581.61 −1.77758 −0.888788 0.458319i \(-0.848452\pi\)
−0.888788 + 0.458319i \(0.848452\pi\)
\(264\) 0 0
\(265\) −5748.93 −1.33266
\(266\) 0 0
\(267\) −485.566 −0.111297
\(268\) 0 0
\(269\) −3601.90 −0.816400 −0.408200 0.912893i \(-0.633843\pi\)
−0.408200 + 0.912893i \(0.633843\pi\)
\(270\) 0 0
\(271\) −3836.27 −0.859914 −0.429957 0.902849i \(-0.641471\pi\)
−0.429957 + 0.902849i \(0.641471\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1453.63 −0.318753
\(276\) 0 0
\(277\) 5403.04 1.17198 0.585988 0.810320i \(-0.300706\pi\)
0.585988 + 0.810320i \(0.300706\pi\)
\(278\) 0 0
\(279\) −1433.23 −0.307545
\(280\) 0 0
\(281\) 150.842 0.0320230 0.0160115 0.999872i \(-0.494903\pi\)
0.0160115 + 0.999872i \(0.494903\pi\)
\(282\) 0 0
\(283\) 1817.14 0.381689 0.190844 0.981620i \(-0.438877\pi\)
0.190844 + 0.981620i \(0.438877\pi\)
\(284\) 0 0
\(285\) 3686.79 0.766268
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5613.35 1.14255
\(290\) 0 0
\(291\) −548.041 −0.110401
\(292\) 0 0
\(293\) −2817.59 −0.561792 −0.280896 0.959738i \(-0.590632\pi\)
−0.280896 + 0.959738i \(0.590632\pi\)
\(294\) 0 0
\(295\) 8573.02 1.69200
\(296\) 0 0
\(297\) 994.268 0.194254
\(298\) 0 0
\(299\) −8363.88 −1.61771
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4596.31 0.871456
\(304\) 0 0
\(305\) 3125.28 0.586731
\(306\) 0 0
\(307\) 8589.21 1.59678 0.798391 0.602139i \(-0.205685\pi\)
0.798391 + 0.602139i \(0.205685\pi\)
\(308\) 0 0
\(309\) −1463.72 −0.269477
\(310\) 0 0
\(311\) −5998.29 −1.09367 −0.546836 0.837240i \(-0.684168\pi\)
−0.546836 + 0.837240i \(0.684168\pi\)
\(312\) 0 0
\(313\) −4963.27 −0.896297 −0.448148 0.893959i \(-0.647916\pi\)
−0.448148 + 0.893959i \(0.647916\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3904.60 −0.691812 −0.345906 0.938269i \(-0.612428\pi\)
−0.345906 + 0.938269i \(0.612428\pi\)
\(318\) 0 0
\(319\) 7807.61 1.37035
\(320\) 0 0
\(321\) −1477.05 −0.256826
\(322\) 0 0
\(323\) 9831.43 1.69361
\(324\) 0 0
\(325\) 3439.15 0.586983
\(326\) 0 0
\(327\) −2544.22 −0.430262
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2449.73 −0.406795 −0.203397 0.979096i \(-0.565198\pi\)
−0.203397 + 0.979096i \(0.565198\pi\)
\(332\) 0 0
\(333\) 1158.03 0.190570
\(334\) 0 0
\(335\) 4303.69 0.701898
\(336\) 0 0
\(337\) −1770.59 −0.286203 −0.143101 0.989708i \(-0.545707\pi\)
−0.143101 + 0.989708i \(0.545707\pi\)
\(338\) 0 0
\(339\) 2209.05 0.353921
\(340\) 0 0
\(341\) 5864.25 0.931281
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3693.53 −0.576385
\(346\) 0 0
\(347\) −4034.72 −0.624194 −0.312097 0.950050i \(-0.601031\pi\)
−0.312097 + 0.950050i \(0.601031\pi\)
\(348\) 0 0
\(349\) 6791.53 1.04167 0.520834 0.853658i \(-0.325621\pi\)
0.520834 + 0.853658i \(0.325621\pi\)
\(350\) 0 0
\(351\) −2352.34 −0.357717
\(352\) 0 0
\(353\) −10156.8 −1.53142 −0.765712 0.643184i \(-0.777613\pi\)
−0.765712 + 0.643184i \(0.777613\pi\)
\(354\) 0 0
\(355\) 4355.39 0.651155
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12842.7 −1.88806 −0.944029 0.329861i \(-0.892998\pi\)
−0.944029 + 0.329861i \(0.892998\pi\)
\(360\) 0 0
\(361\) 2323.38 0.338735
\(362\) 0 0
\(363\) −75.1870 −0.0108713
\(364\) 0 0
\(365\) 11777.2 1.68890
\(366\) 0 0
\(367\) −1914.82 −0.272350 −0.136175 0.990685i \(-0.543481\pi\)
−0.136175 + 0.990685i \(0.543481\pi\)
\(368\) 0 0
\(369\) −2682.37 −0.378425
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5714.86 0.793309 0.396654 0.917968i \(-0.370171\pi\)
0.396654 + 0.917968i \(0.370171\pi\)
\(374\) 0 0
\(375\) −3290.54 −0.453127
\(376\) 0 0
\(377\) −18472.0 −2.52350
\(378\) 0 0
\(379\) −11570.3 −1.56815 −0.784075 0.620666i \(-0.786862\pi\)
−0.784075 + 0.620666i \(0.786862\pi\)
\(380\) 0 0
\(381\) 7534.12 1.01308
\(382\) 0 0
\(383\) 6118.02 0.816231 0.408115 0.912930i \(-0.366186\pi\)
0.408115 + 0.912930i \(0.366186\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −299.967 −0.0394010
\(388\) 0 0
\(389\) −3258.46 −0.424705 −0.212353 0.977193i \(-0.568113\pi\)
−0.212353 + 0.977193i \(0.568113\pi\)
\(390\) 0 0
\(391\) −9849.41 −1.27393
\(392\) 0 0
\(393\) 2038.27 0.261621
\(394\) 0 0
\(395\) 1748.00 0.222662
\(396\) 0 0
\(397\) 723.926 0.0915184 0.0457592 0.998952i \(-0.485429\pi\)
0.0457592 + 0.998952i \(0.485429\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11029.8 1.37357 0.686787 0.726859i \(-0.259021\pi\)
0.686787 + 0.726859i \(0.259021\pi\)
\(402\) 0 0
\(403\) −13874.2 −1.71495
\(404\) 0 0
\(405\) −1038.80 −0.127453
\(406\) 0 0
\(407\) −4738.25 −0.577067
\(408\) 0 0
\(409\) 2683.56 0.324434 0.162217 0.986755i \(-0.448136\pi\)
0.162217 + 0.986755i \(0.448136\pi\)
\(410\) 0 0
\(411\) 493.610 0.0592408
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3686.66 −0.436075
\(416\) 0 0
\(417\) 1565.97 0.183899
\(418\) 0 0
\(419\) 10024.9 1.16885 0.584427 0.811446i \(-0.301319\pi\)
0.584427 + 0.811446i \(0.301319\pi\)
\(420\) 0 0
\(421\) −5560.68 −0.643731 −0.321866 0.946785i \(-0.604310\pi\)
−0.321866 + 0.946785i \(0.604310\pi\)
\(422\) 0 0
\(423\) 2440.76 0.280553
\(424\) 0 0
\(425\) 4049.98 0.462242
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 9624.93 1.08321
\(430\) 0 0
\(431\) 11526.2 1.28816 0.644081 0.764957i \(-0.277240\pi\)
0.644081 + 0.764957i \(0.277240\pi\)
\(432\) 0 0
\(433\) 2228.79 0.247365 0.123683 0.992322i \(-0.460530\pi\)
0.123683 + 0.992322i \(0.460530\pi\)
\(434\) 0 0
\(435\) −8157.34 −0.899114
\(436\) 0 0
\(437\) −9199.18 −1.00699
\(438\) 0 0
\(439\) 4609.26 0.501112 0.250556 0.968102i \(-0.419387\pi\)
0.250556 + 0.968102i \(0.419387\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1063.32 0.114041 0.0570203 0.998373i \(-0.481840\pi\)
0.0570203 + 0.998373i \(0.481840\pi\)
\(444\) 0 0
\(445\) −2075.76 −0.221124
\(446\) 0 0
\(447\) 7236.37 0.765702
\(448\) 0 0
\(449\) −12265.9 −1.28923 −0.644613 0.764509i \(-0.722982\pi\)
−0.644613 + 0.764509i \(0.722982\pi\)
\(450\) 0 0
\(451\) 10975.3 1.14591
\(452\) 0 0
\(453\) 4723.73 0.489934
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17791.2 −1.82109 −0.910543 0.413415i \(-0.864336\pi\)
−0.910543 + 0.413415i \(0.864336\pi\)
\(458\) 0 0
\(459\) −2770.15 −0.281698
\(460\) 0 0
\(461\) 15368.9 1.55272 0.776358 0.630293i \(-0.217065\pi\)
0.776358 + 0.630293i \(0.217065\pi\)
\(462\) 0 0
\(463\) −4104.98 −0.412040 −0.206020 0.978548i \(-0.566051\pi\)
−0.206020 + 0.978548i \(0.566051\pi\)
\(464\) 0 0
\(465\) −6126.93 −0.611031
\(466\) 0 0
\(467\) 3807.36 0.377267 0.188634 0.982048i \(-0.439594\pi\)
0.188634 + 0.982048i \(0.439594\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6236.23 −0.610085
\(472\) 0 0
\(473\) 1227.36 0.119311
\(474\) 0 0
\(475\) 3782.61 0.365385
\(476\) 0 0
\(477\) 4034.41 0.387260
\(478\) 0 0
\(479\) 9874.65 0.941930 0.470965 0.882152i \(-0.343906\pi\)
0.470965 + 0.882152i \(0.343906\pi\)
\(480\) 0 0
\(481\) 11210.2 1.06267
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2342.83 −0.219345
\(486\) 0 0
\(487\) −6763.72 −0.629350 −0.314675 0.949199i \(-0.601896\pi\)
−0.314675 + 0.949199i \(0.601896\pi\)
\(488\) 0 0
\(489\) 9538.28 0.882077
\(490\) 0 0
\(491\) −5574.29 −0.512351 −0.256175 0.966630i \(-0.582462\pi\)
−0.256175 + 0.966630i \(0.582462\pi\)
\(492\) 0 0
\(493\) −21752.9 −1.98722
\(494\) 0 0
\(495\) 4250.41 0.385943
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5787.46 0.519203 0.259601 0.965716i \(-0.416409\pi\)
0.259601 + 0.965716i \(0.416409\pi\)
\(500\) 0 0
\(501\) 8937.84 0.797032
\(502\) 0 0
\(503\) 8296.10 0.735397 0.367699 0.929945i \(-0.380146\pi\)
0.367699 + 0.929945i \(0.380146\pi\)
\(504\) 0 0
\(505\) 19648.8 1.73141
\(506\) 0 0
\(507\) −16180.6 −1.41737
\(508\) 0 0
\(509\) −9760.38 −0.849943 −0.424972 0.905207i \(-0.639716\pi\)
−0.424972 + 0.905207i \(0.639716\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2587.27 −0.222672
\(514\) 0 0
\(515\) −6257.30 −0.535397
\(516\) 0 0
\(517\) −9986.73 −0.849547
\(518\) 0 0
\(519\) −49.1111 −0.00415364
\(520\) 0 0
\(521\) −6537.98 −0.549778 −0.274889 0.961476i \(-0.588641\pi\)
−0.274889 + 0.961476i \(0.588641\pi\)
\(522\) 0 0
\(523\) −10343.8 −0.864826 −0.432413 0.901676i \(-0.642338\pi\)
−0.432413 + 0.901676i \(0.642338\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16338.5 −1.35050
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) −6016.27 −0.491683
\(532\) 0 0
\(533\) −25966.5 −2.11020
\(534\) 0 0
\(535\) −6314.28 −0.510262
\(536\) 0 0
\(537\) −8094.87 −0.650502
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6843.19 −0.543829 −0.271915 0.962321i \(-0.587657\pi\)
−0.271915 + 0.962321i \(0.587657\pi\)
\(542\) 0 0
\(543\) −93.9892 −0.00742810
\(544\) 0 0
\(545\) −10876.3 −0.854844
\(546\) 0 0
\(547\) −18402.1 −1.43842 −0.719211 0.694791i \(-0.755497\pi\)
−0.719211 + 0.694791i \(0.755497\pi\)
\(548\) 0 0
\(549\) −2193.22 −0.170500
\(550\) 0 0
\(551\) −20316.8 −1.57083
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4950.49 0.378625
\(556\) 0 0
\(557\) 646.478 0.0491780 0.0245890 0.999698i \(-0.492172\pi\)
0.0245890 + 0.999698i \(0.492172\pi\)
\(558\) 0 0
\(559\) −2903.81 −0.219710
\(560\) 0 0
\(561\) 11334.4 0.853013
\(562\) 0 0
\(563\) 6174.80 0.462232 0.231116 0.972926i \(-0.425762\pi\)
0.231116 + 0.972926i \(0.425762\pi\)
\(564\) 0 0
\(565\) 9443.51 0.703171
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6837.17 −0.503742 −0.251871 0.967761i \(-0.581046\pi\)
−0.251871 + 0.967761i \(0.581046\pi\)
\(570\) 0 0
\(571\) −5885.54 −0.431353 −0.215676 0.976465i \(-0.569196\pi\)
−0.215676 + 0.976465i \(0.569196\pi\)
\(572\) 0 0
\(573\) 4635.52 0.337961
\(574\) 0 0
\(575\) −3789.53 −0.274842
\(576\) 0 0
\(577\) −12003.2 −0.866033 −0.433017 0.901386i \(-0.642551\pi\)
−0.433017 + 0.901386i \(0.642551\pi\)
\(578\) 0 0
\(579\) 5490.59 0.394095
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −16507.4 −1.17267
\(584\) 0 0
\(585\) −10056.1 −0.710713
\(586\) 0 0
\(587\) −12719.4 −0.894358 −0.447179 0.894445i \(-0.647571\pi\)
−0.447179 + 0.894445i \(0.647571\pi\)
\(588\) 0 0
\(589\) −15259.9 −1.06752
\(590\) 0 0
\(591\) −14185.4 −0.987323
\(592\) 0 0
\(593\) 19160.4 1.32685 0.663426 0.748242i \(-0.269102\pi\)
0.663426 + 0.748242i \(0.269102\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 984.748 0.0675093
\(598\) 0 0
\(599\) 1067.97 0.0728479 0.0364240 0.999336i \(-0.488403\pi\)
0.0364240 + 0.999336i \(0.488403\pi\)
\(600\) 0 0
\(601\) 7554.96 0.512768 0.256384 0.966575i \(-0.417469\pi\)
0.256384 + 0.966575i \(0.417469\pi\)
\(602\) 0 0
\(603\) −3020.20 −0.203967
\(604\) 0 0
\(605\) −321.418 −0.0215992
\(606\) 0 0
\(607\) −11777.4 −0.787529 −0.393765 0.919211i \(-0.628828\pi\)
−0.393765 + 0.919211i \(0.628828\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 23627.6 1.56444
\(612\) 0 0
\(613\) 26852.9 1.76930 0.884649 0.466258i \(-0.154398\pi\)
0.884649 + 0.466258i \(0.154398\pi\)
\(614\) 0 0
\(615\) −11466.9 −0.751855
\(616\) 0 0
\(617\) −6816.72 −0.444782 −0.222391 0.974958i \(-0.571386\pi\)
−0.222391 + 0.974958i \(0.571386\pi\)
\(618\) 0 0
\(619\) −16713.5 −1.08525 −0.542627 0.839974i \(-0.682570\pi\)
−0.542627 + 0.839974i \(0.682570\pi\)
\(620\) 0 0
\(621\) 2592.00 0.167493
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19001.1 −1.21607
\(626\) 0 0
\(627\) 10586.2 0.674276
\(628\) 0 0
\(629\) 13201.3 0.836838
\(630\) 0 0
\(631\) 592.225 0.0373631 0.0186815 0.999825i \(-0.494053\pi\)
0.0186815 + 0.999825i \(0.494053\pi\)
\(632\) 0 0
\(633\) 14807.3 0.929757
\(634\) 0 0
\(635\) 32207.7 2.01279
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3056.47 −0.189221
\(640\) 0 0
\(641\) 6963.91 0.429108 0.214554 0.976712i \(-0.431170\pi\)
0.214554 + 0.976712i \(0.431170\pi\)
\(642\) 0 0
\(643\) 5466.06 0.335242 0.167621 0.985852i \(-0.446392\pi\)
0.167621 + 0.985852i \(0.446392\pi\)
\(644\) 0 0
\(645\) −1282.34 −0.0782821
\(646\) 0 0
\(647\) −1237.27 −0.0751808 −0.0375904 0.999293i \(-0.511968\pi\)
−0.0375904 + 0.999293i \(0.511968\pi\)
\(648\) 0 0
\(649\) 24616.4 1.48887
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27151.8 1.62716 0.813578 0.581455i \(-0.197516\pi\)
0.813578 + 0.581455i \(0.197516\pi\)
\(654\) 0 0
\(655\) 8713.43 0.519789
\(656\) 0 0
\(657\) −8264.88 −0.490782
\(658\) 0 0
\(659\) 5900.66 0.348797 0.174398 0.984675i \(-0.444202\pi\)
0.174398 + 0.984675i \(0.444202\pi\)
\(660\) 0 0
\(661\) 3618.98 0.212953 0.106477 0.994315i \(-0.466043\pi\)
0.106477 + 0.994315i \(0.466043\pi\)
\(662\) 0 0
\(663\) −26816.2 −1.57082
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20354.0 1.18157
\(668\) 0 0
\(669\) −10338.0 −0.597445
\(670\) 0 0
\(671\) 8973.86 0.516292
\(672\) 0 0
\(673\) −13952.5 −0.799151 −0.399576 0.916700i \(-0.630842\pi\)
−0.399576 + 0.916700i \(0.630842\pi\)
\(674\) 0 0
\(675\) −1065.80 −0.0607746
\(676\) 0 0
\(677\) −28918.9 −1.64172 −0.820859 0.571131i \(-0.806505\pi\)
−0.820859 + 0.571131i \(0.806505\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −17174.2 −0.966400
\(682\) 0 0
\(683\) −11130.1 −0.623547 −0.311773 0.950156i \(-0.600923\pi\)
−0.311773 + 0.950156i \(0.600923\pi\)
\(684\) 0 0
\(685\) 2110.14 0.117700
\(686\) 0 0
\(687\) −9051.44 −0.502670
\(688\) 0 0
\(689\) 39054.8 2.15946
\(690\) 0 0
\(691\) −14025.3 −0.772137 −0.386069 0.922470i \(-0.626167\pi\)
−0.386069 + 0.922470i \(0.626167\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6694.40 0.365372
\(696\) 0 0
\(697\) −30578.5 −1.66175
\(698\) 0 0
\(699\) −1145.14 −0.0619644
\(700\) 0 0
\(701\) 30366.7 1.63614 0.818070 0.575119i \(-0.195044\pi\)
0.818070 + 0.575119i \(0.195044\pi\)
\(702\) 0 0
\(703\) 12329.8 0.661490
\(704\) 0 0
\(705\) 10434.1 0.557404
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 16891.0 0.894716 0.447358 0.894355i \(-0.352365\pi\)
0.447358 + 0.894355i \(0.352365\pi\)
\(710\) 0 0
\(711\) −1226.69 −0.0647040
\(712\) 0 0
\(713\) 15287.8 0.802989
\(714\) 0 0
\(715\) 41145.8 2.15212
\(716\) 0 0
\(717\) 4132.15 0.215227
\(718\) 0 0
\(719\) 11007.2 0.570932 0.285466 0.958389i \(-0.407852\pi\)
0.285466 + 0.958389i \(0.407852\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −17420.2 −0.896076
\(724\) 0 0
\(725\) −8369.36 −0.428731
\(726\) 0 0
\(727\) −14618.0 −0.745737 −0.372869 0.927884i \(-0.621626\pi\)
−0.372869 + 0.927884i \(0.621626\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −3419.56 −0.173019
\(732\) 0 0
\(733\) −28522.2 −1.43723 −0.718617 0.695406i \(-0.755225\pi\)
−0.718617 + 0.695406i \(0.755225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12357.5 0.617634
\(738\) 0 0
\(739\) 20239.5 1.00747 0.503735 0.863858i \(-0.331959\pi\)
0.503735 + 0.863858i \(0.331959\pi\)
\(740\) 0 0
\(741\) −25045.8 −1.24168
\(742\) 0 0
\(743\) 13977.6 0.690159 0.345079 0.938573i \(-0.387852\pi\)
0.345079 + 0.938573i \(0.387852\pi\)
\(744\) 0 0
\(745\) 30934.9 1.52130
\(746\) 0 0
\(747\) 2587.18 0.126720
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 15027.8 0.730190 0.365095 0.930970i \(-0.381037\pi\)
0.365095 + 0.930970i \(0.381037\pi\)
\(752\) 0 0
\(753\) 13045.6 0.631350
\(754\) 0 0
\(755\) 20193.6 0.973403
\(756\) 0 0
\(757\) 20769.4 0.997196 0.498598 0.866833i \(-0.333848\pi\)
0.498598 + 0.866833i \(0.333848\pi\)
\(758\) 0 0
\(759\) −10605.5 −0.507189
\(760\) 0 0
\(761\) 11211.1 0.534039 0.267019 0.963691i \(-0.413961\pi\)
0.267019 + 0.963691i \(0.413961\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −11842.1 −0.559678
\(766\) 0 0
\(767\) −58240.0 −2.74175
\(768\) 0 0
\(769\) 4305.86 0.201916 0.100958 0.994891i \(-0.467809\pi\)
0.100958 + 0.994891i \(0.467809\pi\)
\(770\) 0 0
\(771\) 14073.3 0.657379
\(772\) 0 0
\(773\) 16640.5 0.774277 0.387139 0.922022i \(-0.373464\pi\)
0.387139 + 0.922022i \(0.373464\pi\)
\(774\) 0 0
\(775\) −6286.18 −0.291363
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28559.8 −1.31356
\(780\) 0 0
\(781\) 12506.0 0.572982
\(782\) 0 0
\(783\) 5724.56 0.261276
\(784\) 0 0
\(785\) −26659.4 −1.21212
\(786\) 0 0
\(787\) −9767.05 −0.442386 −0.221193 0.975230i \(-0.570995\pi\)
−0.221193 + 0.975230i \(0.570995\pi\)
\(788\) 0 0
\(789\) 22744.8 1.02628
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −21231.3 −0.950750
\(794\) 0 0
\(795\) 17246.8 0.769410
\(796\) 0 0
\(797\) 23118.5 1.02748 0.513738 0.857947i \(-0.328260\pi\)
0.513738 + 0.857947i \(0.328260\pi\)
\(798\) 0 0
\(799\) 27824.2 1.23198
\(800\) 0 0
\(801\) 1456.70 0.0642571
\(802\) 0 0
\(803\) 33816.9 1.48614
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10805.7 0.471349
\(808\) 0 0
\(809\) 10460.7 0.454607 0.227304 0.973824i \(-0.427009\pi\)
0.227304 + 0.973824i \(0.427009\pi\)
\(810\) 0 0
\(811\) −9167.55 −0.396937 −0.198469 0.980107i \(-0.563597\pi\)
−0.198469 + 0.980107i \(0.563597\pi\)
\(812\) 0 0
\(813\) 11508.8 0.496472
\(814\) 0 0
\(815\) 40775.3 1.75251
\(816\) 0 0
\(817\) −3193.81 −0.136765
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28827.3 1.22543 0.612717 0.790303i \(-0.290077\pi\)
0.612717 + 0.790303i \(0.290077\pi\)
\(822\) 0 0
\(823\) 42794.3 1.81254 0.906268 0.422704i \(-0.138919\pi\)
0.906268 + 0.422704i \(0.138919\pi\)
\(824\) 0 0
\(825\) 4360.89 0.184032
\(826\) 0 0
\(827\) −6028.87 −0.253500 −0.126750 0.991935i \(-0.540455\pi\)
−0.126750 + 0.991935i \(0.540455\pi\)
\(828\) 0 0
\(829\) −19407.4 −0.813085 −0.406542 0.913632i \(-0.633266\pi\)
−0.406542 + 0.913632i \(0.633266\pi\)
\(830\) 0 0
\(831\) −16209.1 −0.676641
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 38208.5 1.58355
\(836\) 0 0
\(837\) 4299.68 0.177561
\(838\) 0 0
\(839\) −10599.5 −0.436156 −0.218078 0.975931i \(-0.569979\pi\)
−0.218078 + 0.975931i \(0.569979\pi\)
\(840\) 0 0
\(841\) 20563.8 0.843159
\(842\) 0 0
\(843\) −452.526 −0.0184885
\(844\) 0 0
\(845\) −69170.9 −2.81604
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −5451.43 −0.220368
\(850\) 0 0
\(851\) −12352.3 −0.497571
\(852\) 0 0
\(853\) −34766.1 −1.39551 −0.697754 0.716338i \(-0.745817\pi\)
−0.697754 + 0.716338i \(0.745817\pi\)
\(854\) 0 0
\(855\) −11060.4 −0.442405
\(856\) 0 0
\(857\) 30004.0 1.19594 0.597968 0.801520i \(-0.295975\pi\)
0.597968 + 0.801520i \(0.295975\pi\)
\(858\) 0 0
\(859\) −26130.6 −1.03791 −0.518955 0.854802i \(-0.673679\pi\)
−0.518955 + 0.854802i \(0.673679\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 44668.3 1.76191 0.880955 0.473201i \(-0.156901\pi\)
0.880955 + 0.473201i \(0.156901\pi\)
\(864\) 0 0
\(865\) −209.946 −0.00825245
\(866\) 0 0
\(867\) −16840.1 −0.659652
\(868\) 0 0
\(869\) 5019.18 0.195931
\(870\) 0 0
\(871\) −29236.7 −1.13737
\(872\) 0 0
\(873\) 1644.12 0.0637401
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3678.56 0.141637 0.0708187 0.997489i \(-0.477439\pi\)
0.0708187 + 0.997489i \(0.477439\pi\)
\(878\) 0 0
\(879\) 8452.76 0.324351
\(880\) 0 0
\(881\) −33443.6 −1.27894 −0.639468 0.768817i \(-0.720846\pi\)
−0.639468 + 0.768817i \(0.720846\pi\)
\(882\) 0 0
\(883\) −21095.4 −0.803983 −0.401991 0.915644i \(-0.631682\pi\)
−0.401991 + 0.915644i \(0.631682\pi\)
\(884\) 0 0
\(885\) −25719.0 −0.976877
\(886\) 0 0
\(887\) 14453.3 0.547119 0.273560 0.961855i \(-0.411799\pi\)
0.273560 + 0.961855i \(0.411799\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2982.80 −0.112152
\(892\) 0 0
\(893\) 25987.3 0.973832
\(894\) 0 0
\(895\) −34604.9 −1.29242
\(896\) 0 0
\(897\) 25091.6 0.933986
\(898\) 0 0
\(899\) 33763.8 1.25260
\(900\) 0 0
\(901\) 45991.4 1.70055
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −401.796 −0.0147582
\(906\) 0 0
\(907\) −5204.09 −0.190517 −0.0952584 0.995453i \(-0.530368\pi\)
−0.0952584 + 0.995453i \(0.530368\pi\)
\(908\) 0 0
\(909\) −13788.9 −0.503135
\(910\) 0 0
\(911\) 31584.8 1.14868 0.574342 0.818616i \(-0.305258\pi\)
0.574342 + 0.818616i \(0.305258\pi\)
\(912\) 0 0
\(913\) −10585.8 −0.383723
\(914\) 0 0
\(915\) −9375.83 −0.338749
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −46630.6 −1.67378 −0.836889 0.547373i \(-0.815628\pi\)
−0.836889 + 0.547373i \(0.815628\pi\)
\(920\) 0 0
\(921\) −25767.6 −0.921902
\(922\) 0 0
\(923\) −29587.9 −1.05514
\(924\) 0 0
\(925\) 5079.16 0.180543
\(926\) 0 0
\(927\) 4391.17 0.155582
\(928\) 0 0
\(929\) −38492.6 −1.35942 −0.679711 0.733480i \(-0.737895\pi\)
−0.679711 + 0.733480i \(0.737895\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 17994.9 0.631432
\(934\) 0 0
\(935\) 48453.8 1.69477
\(936\) 0 0
\(937\) 39893.8 1.39090 0.695450 0.718574i \(-0.255205\pi\)
0.695450 + 0.718574i \(0.255205\pi\)
\(938\) 0 0
\(939\) 14889.8 0.517477
\(940\) 0 0
\(941\) −32630.7 −1.13043 −0.565213 0.824945i \(-0.691206\pi\)
−0.565213 + 0.824945i \(0.691206\pi\)
\(942\) 0 0
\(943\) 28612.0 0.988054
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30029.3 1.03043 0.515217 0.857060i \(-0.327711\pi\)
0.515217 + 0.857060i \(0.327711\pi\)
\(948\) 0 0
\(949\) −80007.5 −2.73673
\(950\) 0 0
\(951\) 11713.8 0.399418
\(952\) 0 0
\(953\) −34963.6 −1.18844 −0.594220 0.804303i \(-0.702539\pi\)
−0.594220 + 0.804303i \(0.702539\pi\)
\(954\) 0 0
\(955\) 19816.5 0.671462
\(956\) 0 0
\(957\) −23422.8 −0.791173
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −4431.23 −0.148744
\(962\) 0 0
\(963\) 4431.16 0.148279
\(964\) 0 0
\(965\) 23471.8 0.782990
\(966\) 0 0
\(967\) 20520.5 0.682414 0.341207 0.939988i \(-0.389164\pi\)
0.341207 + 0.939988i \(0.389164\pi\)
\(968\) 0 0
\(969\) −29494.3 −0.977805
\(970\) 0 0
\(971\) 39775.3 1.31457 0.657286 0.753641i \(-0.271704\pi\)
0.657286 + 0.753641i \(0.271704\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −10317.4 −0.338895
\(976\) 0 0
\(977\) −7172.04 −0.234856 −0.117428 0.993081i \(-0.537465\pi\)
−0.117428 + 0.993081i \(0.537465\pi\)
\(978\) 0 0
\(979\) −5960.29 −0.194578
\(980\) 0 0
\(981\) 7632.65 0.248412
\(982\) 0 0
\(983\) 48348.3 1.56874 0.784369 0.620294i \(-0.212987\pi\)
0.784369 + 0.620294i \(0.212987\pi\)
\(984\) 0 0
\(985\) −60641.2 −1.96161
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3199.65 0.102875
\(990\) 0 0
\(991\) 36944.6 1.18424 0.592121 0.805849i \(-0.298291\pi\)
0.592121 + 0.805849i \(0.298291\pi\)
\(992\) 0 0
\(993\) 7349.18 0.234863
\(994\) 0 0
\(995\) 4209.72 0.134128
\(996\) 0 0
\(997\) 56589.6 1.79760 0.898802 0.438355i \(-0.144439\pi\)
0.898802 + 0.438355i \(0.144439\pi\)
\(998\) 0 0
\(999\) −3474.10 −0.110026
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 588.4.a.g.1.1 2
3.2 odd 2 1764.4.a.x.1.2 2
4.3 odd 2 2352.4.a.cb.1.1 2
7.2 even 3 84.4.i.b.25.2 4
7.3 odd 6 588.4.i.i.373.1 4
7.4 even 3 84.4.i.b.37.2 yes 4
7.5 odd 6 588.4.i.i.361.1 4
7.6 odd 2 588.4.a.h.1.2 2
21.2 odd 6 252.4.k.d.109.1 4
21.5 even 6 1764.4.k.z.361.2 4
21.11 odd 6 252.4.k.d.37.1 4
21.17 even 6 1764.4.k.z.1549.2 4
21.20 even 2 1764.4.a.p.1.1 2
28.11 odd 6 336.4.q.h.289.2 4
28.23 odd 6 336.4.q.h.193.2 4
28.27 even 2 2352.4.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.b.25.2 4 7.2 even 3
84.4.i.b.37.2 yes 4 7.4 even 3
252.4.k.d.37.1 4 21.11 odd 6
252.4.k.d.109.1 4 21.2 odd 6
336.4.q.h.193.2 4 28.23 odd 6
336.4.q.h.289.2 4 28.11 odd 6
588.4.a.g.1.1 2 1.1 even 1 trivial
588.4.a.h.1.2 2 7.6 odd 2
588.4.i.i.361.1 4 7.5 odd 6
588.4.i.i.373.1 4 7.3 odd 6
1764.4.a.p.1.1 2 21.20 even 2
1764.4.a.x.1.2 2 3.2 odd 2
1764.4.k.z.361.2 4 21.5 even 6
1764.4.k.z.1549.2 4 21.17 even 6
2352.4.a.bp.1.2 2 28.27 even 2
2352.4.a.cb.1.1 2 4.3 odd 2