Properties

Label 24.8.a.b.1.1
Level 24
Weight 8
Character 24.1
Self dual yes
Analytic conductor 7.497
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 24.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.49724061162\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 24.1

$q$-expansion

\(f(q)\) \(=\) \(q+27.0000 q^{3} -530.000 q^{5} +120.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q+27.0000 q^{3} -530.000 q^{5} +120.000 q^{7} +729.000 q^{9} -7196.00 q^{11} -9626.00 q^{13} -14310.0 q^{15} +18674.0 q^{17} +7004.00 q^{19} +3240.00 q^{21} -63704.0 q^{23} +202775. q^{25} +19683.0 q^{27} +29334.0 q^{29} +87968.0 q^{31} -194292. q^{33} -63600.0 q^{35} +227982. q^{37} -259902. q^{39} -160806. q^{41} +136132. q^{43} -386370. q^{45} -1.20696e6 q^{47} -809143. q^{49} +504198. q^{51} -398786. q^{53} +3.81388e6 q^{55} +189108. q^{57} +1.15244e6 q^{59} -2.07060e6 q^{61} +87480.0 q^{63} +5.10178e6 q^{65} -4.07343e6 q^{67} -1.72001e6 q^{69} -383752. q^{71} +3.00601e6 q^{73} +5.47493e6 q^{75} -863520. q^{77} -4.94811e6 q^{79} +531441. q^{81} -9.16349e6 q^{83} -9.89722e6 q^{85} +792018. q^{87} +7.30411e6 q^{89} -1.15512e6 q^{91} +2.37514e6 q^{93} -3.71212e6 q^{95} -690526. q^{97} -5.24588e6 q^{99} +O(q^{100})\)

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
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) −530.000 −1.89619 −0.948093 0.317994i \(-0.896991\pi\)
−0.948093 + 0.317994i \(0.896991\pi\)
\(6\) 0 0
\(7\) 120.000 0.132232 0.0661162 0.997812i \(-0.478939\pi\)
0.0661162 + 0.997812i \(0.478939\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −7196.00 −1.63011 −0.815055 0.579384i \(-0.803293\pi\)
−0.815055 + 0.579384i \(0.803293\pi\)
\(12\) 0 0
\(13\) −9626.00 −1.21519 −0.607595 0.794247i \(-0.707866\pi\)
−0.607595 + 0.794247i \(0.707866\pi\)
\(14\) 0 0
\(15\) −14310.0 −1.09476
\(16\) 0 0
\(17\) 18674.0 0.921862 0.460931 0.887436i \(-0.347515\pi\)
0.460931 + 0.887436i \(0.347515\pi\)
\(18\) 0 0
\(19\) 7004.00 0.234266 0.117133 0.993116i \(-0.462630\pi\)
0.117133 + 0.993116i \(0.462630\pi\)
\(20\) 0 0
\(21\) 3240.00 0.0763445
\(22\) 0 0
\(23\) −63704.0 −1.09174 −0.545870 0.837870i \(-0.683801\pi\)
−0.545870 + 0.837870i \(0.683801\pi\)
\(24\) 0 0
\(25\) 202775. 2.59552
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 29334.0 0.223346 0.111673 0.993745i \(-0.464379\pi\)
0.111673 + 0.993745i \(0.464379\pi\)
\(30\) 0 0
\(31\) 87968.0 0.530345 0.265173 0.964201i \(-0.414571\pi\)
0.265173 + 0.964201i \(0.414571\pi\)
\(32\) 0 0
\(33\) −194292. −0.941144
\(34\) 0 0
\(35\) −63600.0 −0.250737
\(36\) 0 0
\(37\) 227982. 0.739937 0.369968 0.929044i \(-0.379369\pi\)
0.369968 + 0.929044i \(0.379369\pi\)
\(38\) 0 0
\(39\) −259902. −0.701590
\(40\) 0 0
\(41\) −160806. −0.364384 −0.182192 0.983263i \(-0.558319\pi\)
−0.182192 + 0.983263i \(0.558319\pi\)
\(42\) 0 0
\(43\) 136132. 0.261108 0.130554 0.991441i \(-0.458324\pi\)
0.130554 + 0.991441i \(0.458324\pi\)
\(44\) 0 0
\(45\) −386370. −0.632062
\(46\) 0 0
\(47\) −1.20696e6 −1.69571 −0.847853 0.530232i \(-0.822105\pi\)
−0.847853 + 0.530232i \(0.822105\pi\)
\(48\) 0 0
\(49\) −809143. −0.982515
\(50\) 0 0
\(51\) 504198. 0.532238
\(52\) 0 0
\(53\) −398786. −0.367938 −0.183969 0.982932i \(-0.558895\pi\)
−0.183969 + 0.982932i \(0.558895\pi\)
\(54\) 0 0
\(55\) 3.81388e6 3.09099
\(56\) 0 0
\(57\) 189108. 0.135253
\(58\) 0 0
\(59\) 1.15244e6 0.730524 0.365262 0.930905i \(-0.380979\pi\)
0.365262 + 0.930905i \(0.380979\pi\)
\(60\) 0 0
\(61\) −2.07060e6 −1.16800 −0.583999 0.811754i \(-0.698513\pi\)
−0.583999 + 0.811754i \(0.698513\pi\)
\(62\) 0 0
\(63\) 87480.0 0.0440775
\(64\) 0 0
\(65\) 5.10178e6 2.30423
\(66\) 0 0
\(67\) −4.07343e6 −1.65462 −0.827310 0.561746i \(-0.810130\pi\)
−0.827310 + 0.561746i \(0.810130\pi\)
\(68\) 0 0
\(69\) −1.72001e6 −0.630316
\(70\) 0 0
\(71\) −383752. −0.127247 −0.0636233 0.997974i \(-0.520266\pi\)
−0.0636233 + 0.997974i \(0.520266\pi\)
\(72\) 0 0
\(73\) 3.00601e6 0.904400 0.452200 0.891917i \(-0.350639\pi\)
0.452200 + 0.891917i \(0.350639\pi\)
\(74\) 0 0
\(75\) 5.47493e6 1.49852
\(76\) 0 0
\(77\) −863520. −0.215553
\(78\) 0 0
\(79\) −4.94811e6 −1.12913 −0.564566 0.825388i \(-0.690956\pi\)
−0.564566 + 0.825388i \(0.690956\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −9.16349e6 −1.75909 −0.879544 0.475817i \(-0.842152\pi\)
−0.879544 + 0.475817i \(0.842152\pi\)
\(84\) 0 0
\(85\) −9.89722e6 −1.74802
\(86\) 0 0
\(87\) 792018. 0.128949
\(88\) 0 0
\(89\) 7.30411e6 1.09825 0.549126 0.835740i \(-0.314961\pi\)
0.549126 + 0.835740i \(0.314961\pi\)
\(90\) 0 0
\(91\) −1.15512e6 −0.160688
\(92\) 0 0
\(93\) 2.37514e6 0.306195
\(94\) 0 0
\(95\) −3.71212e6 −0.444211
\(96\) 0 0
\(97\) −690526. −0.0768208 −0.0384104 0.999262i \(-0.512229\pi\)
−0.0384104 + 0.999262i \(0.512229\pi\)
\(98\) 0 0
\(99\) −5.24588e6 −0.543370
\(100\) 0 0
\(101\) 1.32667e7 1.28126 0.640631 0.767849i \(-0.278673\pi\)
0.640631 + 0.767849i \(0.278673\pi\)
\(102\) 0 0
\(103\) 4.86825e6 0.438978 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(104\) 0 0
\(105\) −1.71720e6 −0.144763
\(106\) 0 0
\(107\) 7.91860e6 0.624892 0.312446 0.949936i \(-0.398852\pi\)
0.312446 + 0.949936i \(0.398852\pi\)
\(108\) 0 0
\(109\) −4.93030e6 −0.364654 −0.182327 0.983238i \(-0.558363\pi\)
−0.182327 + 0.983238i \(0.558363\pi\)
\(110\) 0 0
\(111\) 6.15551e6 0.427203
\(112\) 0 0
\(113\) 2.31707e6 0.151066 0.0755328 0.997143i \(-0.475934\pi\)
0.0755328 + 0.997143i \(0.475934\pi\)
\(114\) 0 0
\(115\) 3.37631e7 2.07014
\(116\) 0 0
\(117\) −7.01735e6 −0.405063
\(118\) 0 0
\(119\) 2.24088e6 0.121900
\(120\) 0 0
\(121\) 3.22952e7 1.65726
\(122\) 0 0
\(123\) −4.34176e6 −0.210377
\(124\) 0 0
\(125\) −6.60645e7 −3.02540
\(126\) 0 0
\(127\) −2.25119e7 −0.975214 −0.487607 0.873063i \(-0.662130\pi\)
−0.487607 + 0.873063i \(0.662130\pi\)
\(128\) 0 0
\(129\) 3.67556e6 0.150751
\(130\) 0 0
\(131\) −1.35895e7 −0.528147 −0.264073 0.964503i \(-0.585066\pi\)
−0.264073 + 0.964503i \(0.585066\pi\)
\(132\) 0 0
\(133\) 840480. 0.0309775
\(134\) 0 0
\(135\) −1.04320e7 −0.364921
\(136\) 0 0
\(137\) 714618. 0.0237439 0.0118719 0.999930i \(-0.496221\pi\)
0.0118719 + 0.999930i \(0.496221\pi\)
\(138\) 0 0
\(139\) 1.78816e7 0.564747 0.282373 0.959305i \(-0.408878\pi\)
0.282373 + 0.959305i \(0.408878\pi\)
\(140\) 0 0
\(141\) −3.25879e7 −0.979016
\(142\) 0 0
\(143\) 6.92687e7 1.98089
\(144\) 0 0
\(145\) −1.55470e7 −0.423506
\(146\) 0 0
\(147\) −2.18469e7 −0.567255
\(148\) 0 0
\(149\) −2.00391e7 −0.496279 −0.248139 0.968724i \(-0.579819\pi\)
−0.248139 + 0.968724i \(0.579819\pi\)
\(150\) 0 0
\(151\) −4.07634e7 −0.963499 −0.481749 0.876309i \(-0.659998\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(152\) 0 0
\(153\) 1.36133e7 0.307287
\(154\) 0 0
\(155\) −4.66230e7 −1.00563
\(156\) 0 0
\(157\) 5.54702e7 1.14396 0.571980 0.820268i \(-0.306176\pi\)
0.571980 + 0.820268i \(0.306176\pi\)
\(158\) 0 0
\(159\) −1.07672e7 −0.212429
\(160\) 0 0
\(161\) −7.64448e6 −0.144363
\(162\) 0 0
\(163\) −1.80344e6 −0.0326172 −0.0163086 0.999867i \(-0.505191\pi\)
−0.0163086 + 0.999867i \(0.505191\pi\)
\(164\) 0 0
\(165\) 1.02975e8 1.78458
\(166\) 0 0
\(167\) 5.99273e6 0.0995673 0.0497837 0.998760i \(-0.484147\pi\)
0.0497837 + 0.998760i \(0.484147\pi\)
\(168\) 0 0
\(169\) 2.99114e7 0.476686
\(170\) 0 0
\(171\) 5.10592e6 0.0780885
\(172\) 0 0
\(173\) 1.25631e8 1.84474 0.922368 0.386313i \(-0.126252\pi\)
0.922368 + 0.386313i \(0.126252\pi\)
\(174\) 0 0
\(175\) 2.43330e7 0.343212
\(176\) 0 0
\(177\) 3.11158e7 0.421769
\(178\) 0 0
\(179\) 4.86498e7 0.634009 0.317004 0.948424i \(-0.397323\pi\)
0.317004 + 0.948424i \(0.397323\pi\)
\(180\) 0 0
\(181\) −4.97548e7 −0.623677 −0.311838 0.950135i \(-0.600945\pi\)
−0.311838 + 0.950135i \(0.600945\pi\)
\(182\) 0 0
\(183\) −5.59063e7 −0.674344
\(184\) 0 0
\(185\) −1.20830e8 −1.40306
\(186\) 0 0
\(187\) −1.34378e8 −1.50274
\(188\) 0 0
\(189\) 2.36196e6 0.0254482
\(190\) 0 0
\(191\) 1.11324e8 1.15604 0.578021 0.816022i \(-0.303825\pi\)
0.578021 + 0.816022i \(0.303825\pi\)
\(192\) 0 0
\(193\) −1.34786e8 −1.34957 −0.674786 0.738014i \(-0.735764\pi\)
−0.674786 + 0.738014i \(0.735764\pi\)
\(194\) 0 0
\(195\) 1.37748e8 1.33035
\(196\) 0 0
\(197\) 3.97557e7 0.370482 0.185241 0.982693i \(-0.440693\pi\)
0.185241 + 0.982693i \(0.440693\pi\)
\(198\) 0 0
\(199\) −1.03321e8 −0.929398 −0.464699 0.885469i \(-0.653838\pi\)
−0.464699 + 0.885469i \(0.653838\pi\)
\(200\) 0 0
\(201\) −1.09983e8 −0.955295
\(202\) 0 0
\(203\) 3.52008e6 0.0295336
\(204\) 0 0
\(205\) 8.52272e7 0.690939
\(206\) 0 0
\(207\) −4.64402e7 −0.363913
\(208\) 0 0
\(209\) −5.04008e7 −0.381879
\(210\) 0 0
\(211\) −1.79475e8 −1.31527 −0.657634 0.753337i \(-0.728443\pi\)
−0.657634 + 0.753337i \(0.728443\pi\)
\(212\) 0 0
\(213\) −1.03613e7 −0.0734659
\(214\) 0 0
\(215\) −7.21500e7 −0.495110
\(216\) 0 0
\(217\) 1.05562e7 0.0701289
\(218\) 0 0
\(219\) 8.11623e7 0.522155
\(220\) 0 0
\(221\) −1.79756e8 −1.12024
\(222\) 0 0
\(223\) 2.85311e8 1.72286 0.861432 0.507872i \(-0.169568\pi\)
0.861432 + 0.507872i \(0.169568\pi\)
\(224\) 0 0
\(225\) 1.47823e8 0.865173
\(226\) 0 0
\(227\) −2.74798e7 −0.155928 −0.0779638 0.996956i \(-0.524842\pi\)
−0.0779638 + 0.996956i \(0.524842\pi\)
\(228\) 0 0
\(229\) −8.56562e7 −0.471341 −0.235670 0.971833i \(-0.575729\pi\)
−0.235670 + 0.971833i \(0.575729\pi\)
\(230\) 0 0
\(231\) −2.33150e7 −0.124450
\(232\) 0 0
\(233\) 1.04907e8 0.543322 0.271661 0.962393i \(-0.412427\pi\)
0.271661 + 0.962393i \(0.412427\pi\)
\(234\) 0 0
\(235\) 6.39689e8 3.21537
\(236\) 0 0
\(237\) −1.33599e8 −0.651905
\(238\) 0 0
\(239\) −2.77184e7 −0.131333 −0.0656667 0.997842i \(-0.520917\pi\)
−0.0656667 + 0.997842i \(0.520917\pi\)
\(240\) 0 0
\(241\) −2.95271e8 −1.35882 −0.679409 0.733760i \(-0.737764\pi\)
−0.679409 + 0.733760i \(0.737764\pi\)
\(242\) 0 0
\(243\) 1.43489e7 0.0641500
\(244\) 0 0
\(245\) 4.28846e8 1.86303
\(246\) 0 0
\(247\) −6.74205e7 −0.284677
\(248\) 0 0
\(249\) −2.47414e8 −1.01561
\(250\) 0 0
\(251\) −3.02885e8 −1.20898 −0.604491 0.796612i \(-0.706624\pi\)
−0.604491 + 0.796612i \(0.706624\pi\)
\(252\) 0 0
\(253\) 4.58414e8 1.77966
\(254\) 0 0
\(255\) −2.67225e8 −1.00922
\(256\) 0 0
\(257\) 4.14636e7 0.152370 0.0761852 0.997094i \(-0.475726\pi\)
0.0761852 + 0.997094i \(0.475726\pi\)
\(258\) 0 0
\(259\) 2.73578e7 0.0978436
\(260\) 0 0
\(261\) 2.13845e7 0.0744487
\(262\) 0 0
\(263\) 4.20007e8 1.42368 0.711839 0.702343i \(-0.247863\pi\)
0.711839 + 0.702343i \(0.247863\pi\)
\(264\) 0 0
\(265\) 2.11357e8 0.697678
\(266\) 0 0
\(267\) 1.97211e8 0.634076
\(268\) 0 0
\(269\) 1.56477e8 0.490138 0.245069 0.969506i \(-0.421189\pi\)
0.245069 + 0.969506i \(0.421189\pi\)
\(270\) 0 0
\(271\) −2.71491e8 −0.828633 −0.414316 0.910133i \(-0.635979\pi\)
−0.414316 + 0.910133i \(0.635979\pi\)
\(272\) 0 0
\(273\) −3.11882e7 −0.0927730
\(274\) 0 0
\(275\) −1.45917e9 −4.23098
\(276\) 0 0
\(277\) −9.80318e7 −0.277133 −0.138566 0.990353i \(-0.544249\pi\)
−0.138566 + 0.990353i \(0.544249\pi\)
\(278\) 0 0
\(279\) 6.41287e7 0.176782
\(280\) 0 0
\(281\) −4.04839e8 −1.08845 −0.544227 0.838938i \(-0.683177\pi\)
−0.544227 + 0.838938i \(0.683177\pi\)
\(282\) 0 0
\(283\) 2.41439e8 0.633220 0.316610 0.948556i \(-0.397455\pi\)
0.316610 + 0.948556i \(0.397455\pi\)
\(284\) 0 0
\(285\) −1.00227e8 −0.256465
\(286\) 0 0
\(287\) −1.92967e7 −0.0481833
\(288\) 0 0
\(289\) −6.16204e7 −0.150170
\(290\) 0 0
\(291\) −1.86442e7 −0.0443525
\(292\) 0 0
\(293\) −1.95052e8 −0.453016 −0.226508 0.974009i \(-0.572731\pi\)
−0.226508 + 0.974009i \(0.572731\pi\)
\(294\) 0 0
\(295\) −6.10791e8 −1.38521
\(296\) 0 0
\(297\) −1.41639e8 −0.313715
\(298\) 0 0
\(299\) 6.13215e8 1.32667
\(300\) 0 0
\(301\) 1.63358e7 0.0345270
\(302\) 0 0
\(303\) 3.58201e8 0.739737
\(304\) 0 0
\(305\) 1.09742e9 2.21474
\(306\) 0 0
\(307\) −1.02082e8 −0.201356 −0.100678 0.994919i \(-0.532101\pi\)
−0.100678 + 0.994919i \(0.532101\pi\)
\(308\) 0 0
\(309\) 1.31443e8 0.253444
\(310\) 0 0
\(311\) 3.04913e8 0.574797 0.287398 0.957811i \(-0.407210\pi\)
0.287398 + 0.957811i \(0.407210\pi\)
\(312\) 0 0
\(313\) 6.47441e8 1.19342 0.596712 0.802455i \(-0.296473\pi\)
0.596712 + 0.802455i \(0.296473\pi\)
\(314\) 0 0
\(315\) −4.63644e7 −0.0835791
\(316\) 0 0
\(317\) −1.48935e8 −0.262596 −0.131298 0.991343i \(-0.541914\pi\)
−0.131298 + 0.991343i \(0.541914\pi\)
\(318\) 0 0
\(319\) −2.11087e8 −0.364079
\(320\) 0 0
\(321\) 2.13802e8 0.360782
\(322\) 0 0
\(323\) 1.30793e8 0.215961
\(324\) 0 0
\(325\) −1.95191e9 −3.15405
\(326\) 0 0
\(327\) −1.33118e8 −0.210533
\(328\) 0 0
\(329\) −1.44835e8 −0.224227
\(330\) 0 0
\(331\) 9.68290e8 1.46760 0.733800 0.679366i \(-0.237745\pi\)
0.733800 + 0.679366i \(0.237745\pi\)
\(332\) 0 0
\(333\) 1.66199e8 0.246646
\(334\) 0 0
\(335\) 2.15892e9 3.13747
\(336\) 0 0
\(337\) 1.65424e8 0.235447 0.117723 0.993046i \(-0.462440\pi\)
0.117723 + 0.993046i \(0.462440\pi\)
\(338\) 0 0
\(339\) 6.25610e7 0.0872177
\(340\) 0 0
\(341\) −6.33018e8 −0.864521
\(342\) 0 0
\(343\) −1.95922e8 −0.262153
\(344\) 0 0
\(345\) 9.11604e8 1.19520
\(346\) 0 0
\(347\) −1.12455e9 −1.44486 −0.722431 0.691443i \(-0.756975\pi\)
−0.722431 + 0.691443i \(0.756975\pi\)
\(348\) 0 0
\(349\) −2.55383e8 −0.321590 −0.160795 0.986988i \(-0.551406\pi\)
−0.160795 + 0.986988i \(0.551406\pi\)
\(350\) 0 0
\(351\) −1.89469e8 −0.233863
\(352\) 0 0
\(353\) 3.45151e8 0.417635 0.208818 0.977955i \(-0.433038\pi\)
0.208818 + 0.977955i \(0.433038\pi\)
\(354\) 0 0
\(355\) 2.03389e8 0.241283
\(356\) 0 0
\(357\) 6.05038e7 0.0703791
\(358\) 0 0
\(359\) −8.11307e8 −0.925453 −0.462727 0.886501i \(-0.653129\pi\)
−0.462727 + 0.886501i \(0.653129\pi\)
\(360\) 0 0
\(361\) −8.44816e8 −0.945120
\(362\) 0 0
\(363\) 8.71972e8 0.956818
\(364\) 0 0
\(365\) −1.59319e9 −1.71491
\(366\) 0 0
\(367\) −7.76625e8 −0.820125 −0.410062 0.912057i \(-0.634493\pi\)
−0.410062 + 0.912057i \(0.634493\pi\)
\(368\) 0 0
\(369\) −1.17228e8 −0.121461
\(370\) 0 0
\(371\) −4.78543e7 −0.0486533
\(372\) 0 0
\(373\) 1.66790e9 1.66414 0.832071 0.554669i \(-0.187155\pi\)
0.832071 + 0.554669i \(0.187155\pi\)
\(374\) 0 0
\(375\) −1.78374e9 −1.74672
\(376\) 0 0
\(377\) −2.82369e8 −0.271408
\(378\) 0 0
\(379\) −1.25007e9 −1.17949 −0.589747 0.807588i \(-0.700772\pi\)
−0.589747 + 0.807588i \(0.700772\pi\)
\(380\) 0 0
\(381\) −6.07822e8 −0.563040
\(382\) 0 0
\(383\) 1.43374e9 1.30399 0.651995 0.758223i \(-0.273932\pi\)
0.651995 + 0.758223i \(0.273932\pi\)
\(384\) 0 0
\(385\) 4.57666e8 0.408729
\(386\) 0 0
\(387\) 9.92402e7 0.0870361
\(388\) 0 0
\(389\) 7.31613e8 0.630170 0.315085 0.949063i \(-0.397967\pi\)
0.315085 + 0.949063i \(0.397967\pi\)
\(390\) 0 0
\(391\) −1.18961e9 −1.00643
\(392\) 0 0
\(393\) −3.66917e8 −0.304926
\(394\) 0 0
\(395\) 2.62250e9 2.14104
\(396\) 0 0
\(397\) 1.19059e9 0.954983 0.477492 0.878636i \(-0.341546\pi\)
0.477492 + 0.878636i \(0.341546\pi\)
\(398\) 0 0
\(399\) 2.26930e7 0.0178849
\(400\) 0 0
\(401\) −2.44638e9 −1.89460 −0.947302 0.320343i \(-0.896202\pi\)
−0.947302 + 0.320343i \(0.896202\pi\)
\(402\) 0 0
\(403\) −8.46780e8 −0.644470
\(404\) 0 0
\(405\) −2.81664e8 −0.210687
\(406\) 0 0
\(407\) −1.64056e9 −1.20618
\(408\) 0 0
\(409\) 1.83663e9 1.32737 0.663684 0.748013i \(-0.268992\pi\)
0.663684 + 0.748013i \(0.268992\pi\)
\(410\) 0 0
\(411\) 1.92947e7 0.0137085
\(412\) 0 0
\(413\) 1.38292e8 0.0965991
\(414\) 0 0
\(415\) 4.85665e9 3.33556
\(416\) 0 0
\(417\) 4.82802e8 0.326057
\(418\) 0 0
\(419\) −2.77854e9 −1.84530 −0.922651 0.385636i \(-0.873982\pi\)
−0.922651 + 0.385636i \(0.873982\pi\)
\(420\) 0 0
\(421\) 1.16625e9 0.761736 0.380868 0.924629i \(-0.375625\pi\)
0.380868 + 0.924629i \(0.375625\pi\)
\(422\) 0 0
\(423\) −8.79874e8 −0.565235
\(424\) 0 0
\(425\) 3.78662e9 2.39271
\(426\) 0 0
\(427\) −2.48472e8 −0.154447
\(428\) 0 0
\(429\) 1.87025e9 1.14367
\(430\) 0 0
\(431\) −2.90479e8 −0.174761 −0.0873806 0.996175i \(-0.527850\pi\)
−0.0873806 + 0.996175i \(0.527850\pi\)
\(432\) 0 0
\(433\) 2.20651e9 1.30616 0.653082 0.757288i \(-0.273476\pi\)
0.653082 + 0.757288i \(0.273476\pi\)
\(434\) 0 0
\(435\) −4.19770e8 −0.244511
\(436\) 0 0
\(437\) −4.46183e8 −0.255757
\(438\) 0 0
\(439\) 9.97564e7 0.0562749 0.0281375 0.999604i \(-0.491042\pi\)
0.0281375 + 0.999604i \(0.491042\pi\)
\(440\) 0 0
\(441\) −5.89865e8 −0.327505
\(442\) 0 0
\(443\) −8.18948e8 −0.447552 −0.223776 0.974641i \(-0.571838\pi\)
−0.223776 + 0.974641i \(0.571838\pi\)
\(444\) 0 0
\(445\) −3.87118e9 −2.08249
\(446\) 0 0
\(447\) −5.41055e8 −0.286527
\(448\) 0 0
\(449\) −2.46241e9 −1.28380 −0.641900 0.766788i \(-0.721854\pi\)
−0.641900 + 0.766788i \(0.721854\pi\)
\(450\) 0 0
\(451\) 1.15716e9 0.593985
\(452\) 0 0
\(453\) −1.10061e9 −0.556276
\(454\) 0 0
\(455\) 6.12214e8 0.304693
\(456\) 0 0
\(457\) −2.76135e9 −1.35337 −0.676683 0.736274i \(-0.736583\pi\)
−0.676683 + 0.736274i \(0.736583\pi\)
\(458\) 0 0
\(459\) 3.67560e8 0.177413
\(460\) 0 0
\(461\) 1.48308e9 0.705036 0.352518 0.935805i \(-0.385326\pi\)
0.352518 + 0.935805i \(0.385326\pi\)
\(462\) 0 0
\(463\) −1.42591e9 −0.667666 −0.333833 0.942632i \(-0.608342\pi\)
−0.333833 + 0.942632i \(0.608342\pi\)
\(464\) 0 0
\(465\) −1.25882e9 −0.580603
\(466\) 0 0
\(467\) 1.99579e9 0.906788 0.453394 0.891310i \(-0.350213\pi\)
0.453394 + 0.891310i \(0.350213\pi\)
\(468\) 0 0
\(469\) −4.88811e8 −0.218794
\(470\) 0 0
\(471\) 1.49769e9 0.660465
\(472\) 0 0
\(473\) −9.79606e8 −0.425635
\(474\) 0 0
\(475\) 1.42024e9 0.608041
\(476\) 0 0
\(477\) −2.90715e8 −0.122646
\(478\) 0 0
\(479\) 5.66385e8 0.235471 0.117736 0.993045i \(-0.462436\pi\)
0.117736 + 0.993045i \(0.462436\pi\)
\(480\) 0 0
\(481\) −2.19455e9 −0.899163
\(482\) 0 0
\(483\) −2.06401e8 −0.0833483
\(484\) 0 0
\(485\) 3.65979e8 0.145667
\(486\) 0 0
\(487\) −1.23609e9 −0.484953 −0.242476 0.970157i \(-0.577960\pi\)
−0.242476 + 0.970157i \(0.577960\pi\)
\(488\) 0 0
\(489\) −4.86930e7 −0.0188315
\(490\) 0 0
\(491\) −7.49612e8 −0.285793 −0.142896 0.989738i \(-0.545642\pi\)
−0.142896 + 0.989738i \(0.545642\pi\)
\(492\) 0 0
\(493\) 5.47783e8 0.205894
\(494\) 0 0
\(495\) 2.78032e9 1.03033
\(496\) 0 0
\(497\) −4.60502e7 −0.0168261
\(498\) 0 0
\(499\) −3.65642e9 −1.31736 −0.658680 0.752423i \(-0.728885\pi\)
−0.658680 + 0.752423i \(0.728885\pi\)
\(500\) 0 0
\(501\) 1.61804e8 0.0574852
\(502\) 0 0
\(503\) −2.19390e9 −0.768649 −0.384324 0.923198i \(-0.625566\pi\)
−0.384324 + 0.923198i \(0.625566\pi\)
\(504\) 0 0
\(505\) −7.03135e9 −2.42951
\(506\) 0 0
\(507\) 8.07607e8 0.275215
\(508\) 0 0
\(509\) 4.24865e9 1.42804 0.714018 0.700127i \(-0.246873\pi\)
0.714018 + 0.700127i \(0.246873\pi\)
\(510\) 0 0
\(511\) 3.60721e8 0.119591
\(512\) 0 0
\(513\) 1.37860e8 0.0450844
\(514\) 0 0
\(515\) −2.58017e9 −0.832383
\(516\) 0 0
\(517\) 8.68528e9 2.76419
\(518\) 0 0
\(519\) 3.39203e9 1.06506
\(520\) 0 0
\(521\) −3.70617e8 −0.114814 −0.0574068 0.998351i \(-0.518283\pi\)
−0.0574068 + 0.998351i \(0.518283\pi\)
\(522\) 0 0
\(523\) 6.33645e9 1.93682 0.968412 0.249356i \(-0.0802188\pi\)
0.968412 + 0.249356i \(0.0802188\pi\)
\(524\) 0 0
\(525\) 6.56991e8 0.198154
\(526\) 0 0
\(527\) 1.64271e9 0.488905
\(528\) 0 0
\(529\) 6.53374e8 0.191897
\(530\) 0 0
\(531\) 8.40126e8 0.243508
\(532\) 0 0
\(533\) 1.54792e9 0.442795
\(534\) 0 0
\(535\) −4.19686e9 −1.18491
\(536\) 0 0
\(537\) 1.31354e9 0.366045
\(538\) 0 0
\(539\) 5.82259e9 1.60161
\(540\) 0 0
\(541\) −4.00792e9 −1.08825 −0.544125 0.839004i \(-0.683138\pi\)
−0.544125 + 0.839004i \(0.683138\pi\)
\(542\) 0 0
\(543\) −1.34338e9 −0.360080
\(544\) 0 0
\(545\) 2.61306e9 0.691451
\(546\) 0 0
\(547\) −3.66046e9 −0.956269 −0.478135 0.878287i \(-0.658687\pi\)
−0.478135 + 0.878287i \(0.658687\pi\)
\(548\) 0 0
\(549\) −1.50947e9 −0.389333
\(550\) 0 0
\(551\) 2.05455e8 0.0523223
\(552\) 0 0
\(553\) −5.93773e8 −0.149308
\(554\) 0 0
\(555\) −3.26242e9 −0.810055
\(556\) 0 0
\(557\) 5.25630e9 1.28880 0.644402 0.764687i \(-0.277106\pi\)
0.644402 + 0.764687i \(0.277106\pi\)
\(558\) 0 0
\(559\) −1.31041e9 −0.317296
\(560\) 0 0
\(561\) −3.62821e9 −0.867605
\(562\) 0 0
\(563\) −5.01985e9 −1.18553 −0.592764 0.805376i \(-0.701963\pi\)
−0.592764 + 0.805376i \(0.701963\pi\)
\(564\) 0 0
\(565\) −1.22805e9 −0.286448
\(566\) 0 0
\(567\) 6.37729e7 0.0146925
\(568\) 0 0
\(569\) 2.80495e9 0.638311 0.319155 0.947702i \(-0.396601\pi\)
0.319155 + 0.947702i \(0.396601\pi\)
\(570\) 0 0
\(571\) 6.10454e9 1.37223 0.686115 0.727493i \(-0.259315\pi\)
0.686115 + 0.727493i \(0.259315\pi\)
\(572\) 0 0
\(573\) 3.00576e9 0.667442
\(574\) 0 0
\(575\) −1.29176e10 −2.83363
\(576\) 0 0
\(577\) −3.02864e8 −0.0656346 −0.0328173 0.999461i \(-0.510448\pi\)
−0.0328173 + 0.999461i \(0.510448\pi\)
\(578\) 0 0
\(579\) −3.63924e9 −0.779175
\(580\) 0 0
\(581\) −1.09962e9 −0.232609
\(582\) 0 0
\(583\) 2.86966e9 0.599779
\(584\) 0 0
\(585\) 3.71920e9 0.768075
\(586\) 0 0
\(587\) −2.42370e9 −0.494590 −0.247295 0.968940i \(-0.579542\pi\)
−0.247295 + 0.968940i \(0.579542\pi\)
\(588\) 0 0
\(589\) 6.16128e8 0.124242
\(590\) 0 0
\(591\) 1.07340e9 0.213898
\(592\) 0 0
\(593\) −6.85538e9 −1.35002 −0.675010 0.737809i \(-0.735861\pi\)
−0.675010 + 0.737809i \(0.735861\pi\)
\(594\) 0 0
\(595\) −1.18767e9 −0.231145
\(596\) 0 0
\(597\) −2.78966e9 −0.536588
\(598\) 0 0
\(599\) 1.47203e8 0.0279847 0.0139924 0.999902i \(-0.495546\pi\)
0.0139924 + 0.999902i \(0.495546\pi\)
\(600\) 0 0
\(601\) 7.15884e9 1.34519 0.672593 0.740013i \(-0.265181\pi\)
0.672593 + 0.740013i \(0.265181\pi\)
\(602\) 0 0
\(603\) −2.96953e9 −0.551540
\(604\) 0 0
\(605\) −1.71165e10 −3.14247
\(606\) 0 0
\(607\) 4.08333e9 0.741062 0.370531 0.928820i \(-0.379176\pi\)
0.370531 + 0.928820i \(0.379176\pi\)
\(608\) 0 0
\(609\) 9.50422e7 0.0170512
\(610\) 0 0
\(611\) 1.16182e10 2.06060
\(612\) 0 0
\(613\) −6.81081e9 −1.19423 −0.597113 0.802157i \(-0.703686\pi\)
−0.597113 + 0.802157i \(0.703686\pi\)
\(614\) 0 0
\(615\) 2.30113e9 0.398914
\(616\) 0 0
\(617\) 3.95161e9 0.677292 0.338646 0.940914i \(-0.390031\pi\)
0.338646 + 0.940914i \(0.390031\pi\)
\(618\) 0 0
\(619\) −1.07835e10 −1.82744 −0.913722 0.406339i \(-0.866805\pi\)
−0.913722 + 0.406339i \(0.866805\pi\)
\(620\) 0 0
\(621\) −1.25389e9 −0.210105
\(622\) 0 0
\(623\) 8.76493e8 0.145225
\(624\) 0 0
\(625\) 1.91724e10 3.14120
\(626\) 0 0
\(627\) −1.36082e9 −0.220478
\(628\) 0 0
\(629\) 4.25734e9 0.682120
\(630\) 0 0
\(631\) 1.10443e10 1.75000 0.874999 0.484125i \(-0.160862\pi\)
0.874999 + 0.484125i \(0.160862\pi\)
\(632\) 0 0
\(633\) −4.84582e9 −0.759371
\(634\) 0 0
\(635\) 1.19313e10 1.84919
\(636\) 0 0
\(637\) 7.78881e9 1.19394
\(638\) 0 0
\(639\) −2.79755e8 −0.0424156
\(640\) 0 0
\(641\) −5.93797e9 −0.890502 −0.445251 0.895406i \(-0.646886\pi\)
−0.445251 + 0.895406i \(0.646886\pi\)
\(642\) 0 0
\(643\) −8.24945e9 −1.22373 −0.611866 0.790961i \(-0.709581\pi\)
−0.611866 + 0.790961i \(0.709581\pi\)
\(644\) 0 0
\(645\) −1.94805e9 −0.285852
\(646\) 0 0
\(647\) −6.49145e9 −0.942272 −0.471136 0.882061i \(-0.656156\pi\)
−0.471136 + 0.882061i \(0.656156\pi\)
\(648\) 0 0
\(649\) −8.29293e9 −1.19083
\(650\) 0 0
\(651\) 2.85016e8 0.0404889
\(652\) 0 0
\(653\) −1.04406e9 −0.146734 −0.0733668 0.997305i \(-0.523374\pi\)
−0.0733668 + 0.997305i \(0.523374\pi\)
\(654\) 0 0
\(655\) 7.20245e9 1.00146
\(656\) 0 0
\(657\) 2.19138e9 0.301467
\(658\) 0 0
\(659\) 1.11126e10 1.51257 0.756285 0.654243i \(-0.227012\pi\)
0.756285 + 0.654243i \(0.227012\pi\)
\(660\) 0 0
\(661\) −1.70105e9 −0.229093 −0.114547 0.993418i \(-0.536541\pi\)
−0.114547 + 0.993418i \(0.536541\pi\)
\(662\) 0 0
\(663\) −4.85341e9 −0.646770
\(664\) 0 0
\(665\) −4.45454e8 −0.0587391
\(666\) 0 0
\(667\) −1.86869e9 −0.243836
\(668\) 0 0
\(669\) 7.70339e9 0.994696
\(670\) 0 0
\(671\) 1.49001e10 1.90397
\(672\) 0 0
\(673\) −7.45938e9 −0.943301 −0.471650 0.881786i \(-0.656342\pi\)
−0.471650 + 0.881786i \(0.656342\pi\)
\(674\) 0 0
\(675\) 3.99122e9 0.499508
\(676\) 0 0
\(677\) −5.80473e9 −0.718988 −0.359494 0.933147i \(-0.617051\pi\)
−0.359494 + 0.933147i \(0.617051\pi\)
\(678\) 0 0
\(679\) −8.28631e7 −0.0101582
\(680\) 0 0
\(681\) −7.41954e8 −0.0900248
\(682\) 0 0
\(683\) −7.26903e9 −0.872979 −0.436490 0.899709i \(-0.643778\pi\)
−0.436490 + 0.899709i \(0.643778\pi\)
\(684\) 0 0
\(685\) −3.78748e8 −0.0450228
\(686\) 0 0
\(687\) −2.31272e9 −0.272129
\(688\) 0 0
\(689\) 3.83871e9 0.447114
\(690\) 0 0
\(691\) 1.17298e10 1.35244 0.676219 0.736701i \(-0.263617\pi\)
0.676219 + 0.736701i \(0.263617\pi\)
\(692\) 0 0
\(693\) −6.29506e8 −0.0718511
\(694\) 0 0
\(695\) −9.47723e9 −1.07087
\(696\) 0 0
\(697\) −3.00289e9 −0.335912
\(698\) 0 0
\(699\) 2.83248e9 0.313687
\(700\) 0 0
\(701\) −9.88743e9 −1.08410 −0.542051 0.840345i \(-0.682352\pi\)
−0.542051 + 0.840345i \(0.682352\pi\)
\(702\) 0 0
\(703\) 1.59679e9 0.173342
\(704\) 0 0
\(705\) 1.72716e10 1.85640
\(706\) 0 0
\(707\) 1.59200e9 0.169425
\(708\) 0 0
\(709\) −1.43284e10 −1.50986 −0.754931 0.655804i \(-0.772330\pi\)
−0.754931 + 0.655804i \(0.772330\pi\)
\(710\) 0 0
\(711\) −3.60717e9 −0.376377
\(712\) 0 0
\(713\) −5.60391e9 −0.578999
\(714\) 0 0
\(715\) −3.67124e10 −3.75614
\(716\) 0 0
\(717\) −7.48396e8 −0.0758254
\(718\) 0 0
\(719\) −1.73607e10 −1.74187 −0.870936 0.491396i \(-0.836487\pi\)
−0.870936 + 0.491396i \(0.836487\pi\)
\(720\) 0 0
\(721\) 5.84190e8 0.0580471
\(722\) 0 0
\(723\) −7.97232e9 −0.784514
\(724\) 0 0
\(725\) 5.94820e9 0.579699
\(726\) 0 0
\(727\) −1.27740e10 −1.23298 −0.616489 0.787364i \(-0.711446\pi\)
−0.616489 + 0.787364i \(0.711446\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 2.54213e9 0.240706
\(732\) 0 0
\(733\) 6.12419e9 0.574361 0.287180 0.957877i \(-0.407282\pi\)
0.287180 + 0.957877i \(0.407282\pi\)
\(734\) 0 0
\(735\) 1.15788e10 1.07562
\(736\) 0 0
\(737\) 2.93124e10 2.69721
\(738\) 0 0
\(739\) 1.05335e10 0.960104 0.480052 0.877240i \(-0.340618\pi\)
0.480052 + 0.877240i \(0.340618\pi\)
\(740\) 0 0
\(741\) −1.82035e9 −0.164358
\(742\) 0 0
\(743\) 7.55392e9 0.675634 0.337817 0.941212i \(-0.390312\pi\)
0.337817 + 0.941212i \(0.390312\pi\)
\(744\) 0 0
\(745\) 1.06207e10 0.941037
\(746\) 0 0
\(747\) −6.68019e9 −0.586363
\(748\) 0 0
\(749\) 9.50232e8 0.0826310
\(750\) 0 0
\(751\) −1.31273e10 −1.13093 −0.565466 0.824772i \(-0.691304\pi\)
−0.565466 + 0.824772i \(0.691304\pi\)
\(752\) 0 0
\(753\) −8.17790e9 −0.698007
\(754\) 0 0
\(755\) 2.16046e10 1.82697
\(756\) 0 0
\(757\) −5.72126e9 −0.479353 −0.239677 0.970853i \(-0.577041\pi\)
−0.239677 + 0.970853i \(0.577041\pi\)
\(758\) 0 0
\(759\) 1.23772e10 1.02748
\(760\) 0 0
\(761\) 1.63771e10 1.34708 0.673538 0.739153i \(-0.264774\pi\)
0.673538 + 0.739153i \(0.264774\pi\)
\(762\) 0 0
\(763\) −5.91636e8 −0.0482190
\(764\) 0 0
\(765\) −7.21507e9 −0.582674
\(766\) 0 0
\(767\) −1.10933e10 −0.887726
\(768\) 0 0
\(769\) −6.59634e9 −0.523071 −0.261536 0.965194i \(-0.584229\pi\)
−0.261536 + 0.965194i \(0.584229\pi\)
\(770\) 0 0
\(771\) 1.11952e9 0.0879711
\(772\) 0 0
\(773\) −6.24126e9 −0.486009 −0.243004 0.970025i \(-0.578133\pi\)
−0.243004 + 0.970025i \(0.578133\pi\)
\(774\) 0 0
\(775\) 1.78377e10 1.37652
\(776\) 0 0
\(777\) 7.38662e8 0.0564901
\(778\) 0 0
\(779\) −1.12629e9 −0.0853625
\(780\) 0 0
\(781\) 2.76148e9 0.207426
\(782\) 0 0
\(783\) 5.77381e8 0.0429830
\(784\) 0 0
\(785\) −2.93992e10 −2.16916
\(786\) 0 0
\(787\) 1.69328e10 1.23827 0.619136 0.785284i \(-0.287483\pi\)
0.619136 + 0.785284i \(0.287483\pi\)
\(788\) 0 0
\(789\) 1.13402e10 0.821960
\(790\) 0 0
\(791\) 2.78049e8 0.0199758
\(792\) 0 0
\(793\) 1.99316e10 1.41934
\(794\) 0 0
\(795\) 5.70663e9 0.402805
\(796\) 0 0
\(797\) 1.25579e10 0.878641 0.439321 0.898330i \(-0.355219\pi\)
0.439321 + 0.898330i \(0.355219\pi\)
\(798\) 0 0
\(799\) −2.25388e10 −1.56321
\(800\) 0 0
\(801\) 5.32469e9 0.366084
\(802\) 0 0
\(803\) −2.16312e10 −1.47427
\(804\) 0 0
\(805\) 4.05157e9 0.273740
\(806\) 0 0
\(807\) 4.22489e9 0.282981
\(808\) 0 0
\(809\) −2.51860e9 −0.167239 −0.0836197 0.996498i \(-0.526648\pi\)
−0.0836197 + 0.996498i \(0.526648\pi\)
\(810\) 0 0
\(811\) 2.70760e10 1.78243 0.891213 0.453585i \(-0.149855\pi\)
0.891213 + 0.453585i \(0.149855\pi\)
\(812\) 0 0
\(813\) −7.33024e9 −0.478411
\(814\) 0 0
\(815\) 9.55825e8 0.0618482
\(816\) 0 0
\(817\) 9.53469e8 0.0611687
\(818\) 0 0
\(819\) −8.42082e8 −0.0535625
\(820\) 0 0
\(821\) 2.70089e10 1.70336 0.851678 0.524066i \(-0.175585\pi\)
0.851678 + 0.524066i \(0.175585\pi\)
\(822\) 0 0
\(823\) 1.23883e10 0.774662 0.387331 0.921941i \(-0.373397\pi\)
0.387331 + 0.921941i \(0.373397\pi\)
\(824\) 0 0
\(825\) −3.93976e10 −2.44276
\(826\) 0 0
\(827\) −2.28562e10 −1.40519 −0.702596 0.711589i \(-0.747976\pi\)
−0.702596 + 0.711589i \(0.747976\pi\)
\(828\) 0 0
\(829\) −8.68742e9 −0.529602 −0.264801 0.964303i \(-0.585306\pi\)
−0.264801 + 0.964303i \(0.585306\pi\)
\(830\) 0 0
\(831\) −2.64686e9 −0.160003
\(832\) 0 0
\(833\) −1.51099e10 −0.905743
\(834\) 0 0
\(835\) −3.17615e9 −0.188798
\(836\) 0 0
\(837\) 1.73147e9 0.102065
\(838\) 0 0
\(839\) 1.16792e10 0.682724 0.341362 0.939932i \(-0.389112\pi\)
0.341362 + 0.939932i \(0.389112\pi\)
\(840\) 0 0
\(841\) −1.63894e10 −0.950117
\(842\) 0 0
\(843\) −1.09307e10 −0.628419
\(844\) 0 0
\(845\) −1.58530e10 −0.903886
\(846\) 0 0
\(847\) 3.87543e9 0.219143
\(848\) 0 0
\(849\) 6.51884e9 0.365589
\(850\) 0 0
\(851\) −1.45234e10 −0.807819
\(852\) 0 0
\(853\) 2.25452e9 0.124375 0.0621875 0.998064i \(-0.480192\pi\)
0.0621875 + 0.998064i \(0.480192\pi\)
\(854\) 0 0
\(855\) −2.70614e9 −0.148070
\(856\) 0 0
\(857\) −4.96605e9 −0.269512 −0.134756 0.990879i \(-0.543025\pi\)
−0.134756 + 0.990879i \(0.543025\pi\)
\(858\) 0 0
\(859\) 1.54039e10 0.829192 0.414596 0.910006i \(-0.363923\pi\)
0.414596 + 0.910006i \(0.363923\pi\)
\(860\) 0 0
\(861\) −5.21011e8 −0.0278187
\(862\) 0 0
\(863\) 4.61225e9 0.244273 0.122136 0.992513i \(-0.461025\pi\)
0.122136 + 0.992513i \(0.461025\pi\)
\(864\) 0 0
\(865\) −6.65842e10 −3.49796
\(866\) 0 0
\(867\) −1.66375e9 −0.0867005
\(868\) 0 0
\(869\) 3.56066e10 1.84061
\(870\) 0 0
\(871\) 3.92108e10 2.01068
\(872\) 0 0
\(873\) −5.03393e8 −0.0256069
\(874\) 0 0
\(875\) −7.92774e9 −0.400056
\(876\) 0 0
\(877\) 3.57505e10 1.78971 0.894856 0.446355i \(-0.147278\pi\)
0.894856 + 0.446355i \(0.147278\pi\)
\(878\) 0 0
\(879\) −5.26640e9 −0.261549
\(880\) 0 0
\(881\) −2.48309e10 −1.22342 −0.611711 0.791082i \(-0.709518\pi\)
−0.611711 + 0.791082i \(0.709518\pi\)
\(882\) 0 0
\(883\) 9.37996e9 0.458499 0.229249 0.973368i \(-0.426373\pi\)
0.229249 + 0.973368i \(0.426373\pi\)
\(884\) 0 0
\(885\) −1.64914e10 −0.799751
\(886\) 0 0
\(887\) 5.98940e9 0.288171 0.144086 0.989565i \(-0.453976\pi\)
0.144086 + 0.989565i \(0.453976\pi\)
\(888\) 0 0
\(889\) −2.70143e9 −0.128955
\(890\) 0 0
\(891\) −3.82425e9 −0.181123
\(892\) 0 0
\(893\) −8.45355e9 −0.397245
\(894\) 0 0
\(895\) −2.57844e10 −1.20220
\(896\) 0 0
\(897\) 1.65568e10 0.765954
\(898\) 0 0
\(899\) 2.58045e9 0.118451
\(900\) 0 0
\(901\) −7.44693e9 −0.339188
\(902\) 0 0
\(903\) 4.41068e8 0.0199342
\(904\) 0 0
\(905\) 2.63700e10 1.18261
\(906\) 0 0
\(907\) 1.76568e9 0.0785755 0.0392878 0.999228i \(-0.487491\pi\)
0.0392878 + 0.999228i \(0.487491\pi\)
\(908\) 0 0
\(909\) 9.67143e9 0.427088
\(910\) 0 0
\(911\) −4.18232e10 −1.83275 −0.916373 0.400325i \(-0.868897\pi\)
−0.916373 + 0.400325i \(0.868897\pi\)
\(912\) 0 0
\(913\) 6.59405e10 2.86751
\(914\) 0 0
\(915\) 2.96303e10 1.27868
\(916\) 0 0
\(917\) −1.63074e9 −0.0698382
\(918\) 0 0
\(919\) −3.71658e10 −1.57957 −0.789785 0.613384i \(-0.789808\pi\)
−0.789785 + 0.613384i \(0.789808\pi\)
\(920\) 0 0
\(921\) −2.75622e9 −0.116253
\(922\) 0 0
\(923\) 3.69400e9 0.154629
\(924\) 0 0
\(925\) 4.62291e10 1.92052
\(926\) 0 0
\(927\) 3.54895e9 0.146326
\(928\) 0 0
\(929\) 2.56580e10 1.04995 0.524974 0.851118i \(-0.324075\pi\)
0.524974 + 0.851118i \(0.324075\pi\)
\(930\) 0 0
\(931\) −5.66724e9 −0.230169
\(932\) 0 0
\(933\) 8.23264e9 0.331859
\(934\) 0 0
\(935\) 7.12204e10 2.84947
\(936\) 0 0
\(937\) −5.75353e9 −0.228479 −0.114239 0.993453i \(-0.536443\pi\)
−0.114239 + 0.993453i \(0.536443\pi\)
\(938\) 0 0
\(939\) 1.74809e10 0.689024
\(940\) 0 0
\(941\) −4.60599e10 −1.80202 −0.901010 0.433799i \(-0.857173\pi\)
−0.901010 + 0.433799i \(0.857173\pi\)
\(942\) 0 0
\(943\) 1.02440e10 0.397812
\(944\) 0 0
\(945\) −1.25184e9 −0.0482544
\(946\) 0 0
\(947\) −9.34709e7 −0.00357645 −0.00178822 0.999998i \(-0.500569\pi\)
−0.00178822 + 0.999998i \(0.500569\pi\)
\(948\) 0 0
\(949\) −2.89359e10 −1.09902
\(950\) 0 0
\(951\) −4.02123e9 −0.151610
\(952\) 0 0
\(953\) 1.17289e10 0.438968 0.219484 0.975616i \(-0.429563\pi\)
0.219484 + 0.975616i \(0.429563\pi\)
\(954\) 0 0
\(955\) −5.90020e10 −2.19207
\(956\) 0 0
\(957\) −5.69936e9 −0.210201
\(958\) 0 0
\(959\) 8.57542e7 0.00313971
\(960\) 0 0
\(961\) −1.97742e10 −0.718734
\(962\) 0 0
\(963\) 5.77266e9 0.208297
\(964\) 0 0
\(965\) 7.14368e10 2.55904
\(966\) 0 0
\(967\) −1.64786e10 −0.586040 −0.293020 0.956106i \(-0.594660\pi\)
−0.293020 + 0.956106i \(0.594660\pi\)
\(968\) 0 0
\(969\) 3.53140e9 0.124685
\(970\) 0 0
\(971\) −1.88436e10 −0.660536 −0.330268 0.943887i \(-0.607139\pi\)
−0.330268 + 0.943887i \(0.607139\pi\)
\(972\) 0 0
\(973\) 2.14579e9 0.0746779
\(974\) 0 0
\(975\) −5.27016e10 −1.82099
\(976\) 0 0
\(977\) −4.26661e10 −1.46370 −0.731849 0.681467i \(-0.761343\pi\)
−0.731849 + 0.681467i \(0.761343\pi\)
\(978\) 0 0
\(979\) −5.25603e10 −1.79027
\(980\) 0 0
\(981\) −3.59419e9 −0.121551
\(982\) 0 0
\(983\) 4.34527e9 0.145908 0.0729541 0.997335i \(-0.476757\pi\)
0.0729541 + 0.997335i \(0.476757\pi\)
\(984\) 0 0
\(985\) −2.10705e10 −0.702503
\(986\) 0 0
\(987\) −3.91055e9 −0.129458
\(988\) 0 0
\(989\) −8.67215e9 −0.285062
\(990\) 0 0
\(991\) 1.36239e10 0.444675 0.222338 0.974970i \(-0.428631\pi\)
0.222338 + 0.974970i \(0.428631\pi\)
\(992\) 0 0
\(993\) 2.61438e10 0.847319
\(994\) 0 0
\(995\) 5.47600e10 1.76231
\(996\) 0 0
\(997\) −4.32788e10 −1.38306 −0.691531 0.722346i \(-0.743064\pi\)
−0.691531 + 0.722346i \(0.743064\pi\)
\(998\) 0 0
\(999\) 4.48737e9 0.142401
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 24.8.a.b.1.1 1
3.2 odd 2 72.8.a.e.1.1 1
4.3 odd 2 48.8.a.a.1.1 1
5.2 odd 4 600.8.f.a.49.1 2
5.3 odd 4 600.8.f.a.49.2 2
5.4 even 2 600.8.a.b.1.1 1
8.3 odd 2 192.8.a.p.1.1 1
8.5 even 2 192.8.a.h.1.1 1
12.11 even 2 144.8.a.k.1.1 1
24.5 odd 2 576.8.a.c.1.1 1
24.11 even 2 576.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.8.a.b.1.1 1 1.1 even 1 trivial
48.8.a.a.1.1 1 4.3 odd 2
72.8.a.e.1.1 1 3.2 odd 2
144.8.a.k.1.1 1 12.11 even 2
192.8.a.h.1.1 1 8.5 even 2
192.8.a.p.1.1 1 8.3 odd 2
576.8.a.b.1.1 1 24.11 even 2
576.8.a.c.1.1 1 24.5 odd 2
600.8.a.b.1.1 1 5.4 even 2
600.8.f.a.49.1 2 5.2 odd 4
600.8.f.a.49.2 2 5.3 odd 4