Properties

Label 600.8.a.b.1.1
Level $600$
Weight $8$
Character 600.1
Self dual yes
Analytic conductor $187.431$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,8,Mod(1,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(187.431015290\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -120.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} -120.000 q^{7} +729.000 q^{9} -7196.00 q^{11} +9626.00 q^{13} -18674.0 q^{17} +7004.00 q^{19} +3240.00 q^{21} +63704.0 q^{23} -19683.0 q^{27} +29334.0 q^{29} +87968.0 q^{31} +194292. q^{33} -227982. q^{37} -259902. q^{39} -160806. q^{41} -136132. q^{43} +1.20696e6 q^{47} -809143. q^{49} +504198. q^{51} +398786. q^{53} -189108. q^{57} +1.15244e6 q^{59} -2.07060e6 q^{61} -87480.0 q^{63} +4.07343e6 q^{67} -1.72001e6 q^{69} -383752. q^{71} -3.00601e6 q^{73} +863520. q^{77} -4.94811e6 q^{79} +531441. q^{81} +9.16349e6 q^{83} -792018. q^{87} +7.30411e6 q^{89} -1.15512e6 q^{91} -2.37514e6 q^{93} +690526. q^{97} -5.24588e6 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −120.000 −0.132232 −0.0661162 0.997812i \(-0.521061\pi\)
−0.0661162 + 0.997812i \(0.521061\pi\)
\(8\) 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 \(-0.292134\pi\)
0.607595 + 0.794247i \(0.292134\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −18674.0 −0.921862 −0.460931 0.887436i \(-0.652485\pi\)
−0.460931 + 0.887436i \(0.652485\pi\)
\(18\) 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 \(-0.316199\pi\)
0.545870 + 0.837870i \(0.316199\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) −227982. −0.739937 −0.369968 0.929044i \(-0.620631\pi\)
−0.369968 + 0.929044i \(0.620631\pi\)
\(38\) 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 \(-0.541676\pi\)
−0.130554 + 0.991441i \(0.541676\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.20696e6 1.69571 0.847853 0.530232i \(-0.177895\pi\)
0.847853 + 0.530232i \(0.177895\pi\)
\(48\) 0 0
\(49\) −809143. −0.982515
\(50\) 0 0
\(51\) 504198. 0.532238
\(52\) 0 0
\(53\) 398786. 0.367938 0.183969 0.982932i \(-0.441105\pi\)
0.183969 + 0.982932i \(0.441105\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) 4.07343e6 1.65462 0.827310 0.561746i \(-0.189870\pi\)
0.827310 + 0.561746i \(0.189870\pi\)
\(68\) 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 \(-0.649361\pi\)
−0.452200 + 0.891917i \(0.649361\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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 \(-0.157848\pi\)
0.879544 + 0.475817i \(0.157848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) 690526. 0.0768208 0.0384104 0.999262i \(-0.487771\pi\)
0.0384104 + 0.999262i \(0.487771\pi\)
\(98\) 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 \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.91860e6 −0.624892 −0.312446 0.949936i \(-0.601148\pi\)
−0.312446 + 0.949936i \(0.601148\pi\)
\(108\) 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 \(-0.524066\pi\)
−0.0755328 + 0.997143i \(0.524066\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 2.25119e7 0.975214 0.487607 0.873063i \(-0.337870\pi\)
0.487607 + 0.873063i \(0.337870\pi\)
\(128\) 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
\(135\) 0 0
\(136\) 0 0
\(137\) −714618. −0.0237439 −0.0118719 0.999930i \(-0.503779\pi\)
−0.0118719 + 0.999930i \(0.503779\pi\)
\(138\) 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
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) −5.54702e7 −1.14396 −0.571980 0.820268i \(-0.693824\pi\)
−0.571980 + 0.820268i \(0.693824\pi\)
\(158\) 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 \(-0.494809\pi\)
0.0163086 + 0.999867i \(0.494809\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.99273e6 −0.0995673 −0.0497837 0.998760i \(-0.515853\pi\)
−0.0497837 + 0.998760i \(0.515853\pi\)
\(168\) 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 \(-0.873748\pi\)
−0.922368 + 0.386313i \(0.873748\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 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 \(-0.264236\pi\)
0.674786 + 0.738014i \(0.264236\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.97557e7 −0.370482 −0.185241 0.982693i \(-0.559307\pi\)
−0.185241 + 0.982693i \(0.559307\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 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 \(-0.830432\pi\)
−0.861432 + 0.507872i \(0.830432\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.74798e7 0.155928 0.0779638 0.996956i \(-0.475158\pi\)
0.0779638 + 0.996956i \(0.475158\pi\)
\(228\) 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 \(-0.587573\pi\)
−0.271661 + 0.962393i \(0.587573\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 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
\(255\) 0 0
\(256\) 0 0
\(257\) −4.14636e7 −0.152370 −0.0761852 0.997094i \(-0.524274\pi\)
−0.0761852 + 0.997094i \(0.524274\pi\)
\(258\) 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 \(-0.752137\pi\)
−0.711839 + 0.702343i \(0.752137\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) 9.80318e7 0.277133 0.138566 0.990353i \(-0.455751\pi\)
0.138566 + 0.990353i \(0.455751\pi\)
\(278\) 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 \(-0.602545\pi\)
−0.316610 + 0.948556i \(0.602545\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 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 \(-0.427269\pi\)
0.226508 + 0.974009i \(0.427269\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 1.02082e8 0.201356 0.100678 0.994919i \(-0.467899\pi\)
0.100678 + 0.994919i \(0.467899\pi\)
\(308\) 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 \(-0.703527\pi\)
−0.596712 + 0.802455i \(0.703527\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.48935e8 0.262596 0.131298 0.991343i \(-0.458086\pi\)
0.131298 + 0.991343i \(0.458086\pi\)
\(318\) 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
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) −1.65424e8 −0.235447 −0.117723 0.993046i \(-0.537560\pi\)
−0.117723 + 0.993046i \(0.537560\pi\)
\(338\) 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
\(345\) 0 0
\(346\) 0 0
\(347\) 1.12455e9 1.44486 0.722431 0.691443i \(-0.243025\pi\)
0.722431 + 0.691443i \(0.243025\pi\)
\(348\) 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 \(-0.566962\pi\)
−0.208818 + 0.977955i \(0.566962\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 7.76625e8 0.820125 0.410062 0.912057i \(-0.365507\pi\)
0.410062 + 0.912057i \(0.365507\pi\)
\(368\) 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 \(-0.812845\pi\)
−0.832071 + 0.554669i \(0.812845\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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 \(-0.726068\pi\)
−0.651995 + 0.758223i \(0.726068\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) −1.19059e9 −0.954983 −0.477492 0.878636i \(-0.658454\pi\)
−0.477492 + 0.878636i \(0.658454\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 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
\(415\) 0 0
\(416\) 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
\(425\) 0 0
\(426\) 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 \(-0.726524\pi\)
−0.653082 + 0.757288i \(0.726524\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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 \(-0.428162\pi\)
0.223776 + 0.974641i \(0.428162\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 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
\(455\) 0 0
\(456\) 0 0
\(457\) 2.76135e9 1.35337 0.676683 0.736274i \(-0.263417\pi\)
0.676683 + 0.736274i \(0.263417\pi\)
\(458\) 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 \(-0.391658\pi\)
0.333833 + 0.942632i \(0.391658\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.99579e9 −0.906788 −0.453394 0.891310i \(-0.649787\pi\)
−0.453394 + 0.891310i \(0.649787\pi\)
\(468\) 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
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 1.23609e9 0.484953 0.242476 0.970157i \(-0.422040\pi\)
0.242476 + 0.970157i \(0.422040\pi\)
\(488\) 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
\(495\) 0 0
\(496\) 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 \(-0.374434\pi\)
0.384324 + 0.923198i \(0.374434\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 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 \(-0.919781\pi\)
−0.968412 + 0.249356i \(0.919781\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 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
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 3.66046e9 0.956269 0.478135 0.878287i \(-0.341313\pi\)
0.478135 + 0.878287i \(0.341313\pi\)
\(548\) 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
\(555\) 0 0
\(556\) 0 0
\(557\) −5.25630e9 −1.28880 −0.644402 0.764687i \(-0.722894\pi\)
−0.644402 + 0.764687i \(0.722894\pi\)
\(558\) 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 \(-0.298037\pi\)
0.592764 + 0.805376i \(0.298037\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) 3.02864e8 0.0656346 0.0328173 0.999461i \(-0.489552\pi\)
0.0328173 + 0.999461i \(0.489552\pi\)
\(578\) 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
\(585\) 0 0
\(586\) 0 0
\(587\) 2.42370e9 0.494590 0.247295 0.968940i \(-0.420458\pi\)
0.247295 + 0.968940i \(0.420458\pi\)
\(588\) 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 \(-0.264139\pi\)
0.675010 + 0.737809i \(0.264139\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) −4.08333e9 −0.741062 −0.370531 0.928820i \(-0.620824\pi\)
−0.370531 + 0.928820i \(0.620824\pi\)
\(608\) 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 \(-0.296314\pi\)
0.597113 + 0.802157i \(0.296314\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.95161e9 −0.677292 −0.338646 0.940914i \(-0.609969\pi\)
−0.338646 + 0.940914i \(0.609969\pi\)
\(618\) 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
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 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 \(-0.290419\pi\)
0.611866 + 0.790961i \(0.290419\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.49145e9 0.942272 0.471136 0.882061i \(-0.343844\pi\)
0.471136 + 0.882061i \(0.343844\pi\)
\(648\) 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 \(-0.476626\pi\)
0.0733668 + 0.997305i \(0.476626\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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
\(665\) 0 0
\(666\) 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 \(-0.343658\pi\)
0.471650 + 0.881786i \(0.343658\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.80473e9 0.718988 0.359494 0.933147i \(-0.382949\pi\)
0.359494 + 0.933147i \(0.382949\pi\)
\(678\) 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 \(-0.356222\pi\)
0.436490 + 0.899709i \(0.356222\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 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
\(705\) 0 0
\(706\) 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
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) 1.27740e10 1.23298 0.616489 0.787364i \(-0.288554\pi\)
0.616489 + 0.787364i \(0.288554\pi\)
\(728\) 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 \(-0.592718\pi\)
−0.287180 + 0.957877i \(0.592718\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 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 \(-0.609688\pi\)
−0.337817 + 0.941212i \(0.609688\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) 5.72126e9 0.479353 0.239677 0.970853i \(-0.422959\pi\)
0.239677 + 0.970853i \(0.422959\pi\)
\(758\) 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
\(765\) 0 0
\(766\) 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 \(-0.421867\pi\)
0.243004 + 0.970025i \(0.421867\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) −1.69328e10 −1.23827 −0.619136 0.785284i \(-0.712517\pi\)
−0.619136 + 0.785284i \(0.712517\pi\)
\(788\) 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
\(795\) 0 0
\(796\) 0 0
\(797\) −1.25579e10 −0.878641 −0.439321 0.898330i \(-0.644781\pi\)
−0.439321 + 0.898330i \(0.644781\pi\)
\(798\) 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
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 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 \(-0.626603\pi\)
−0.387331 + 0.921941i \(0.626603\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.28562e10 1.40519 0.702596 0.711589i \(-0.252024\pi\)
0.702596 + 0.711589i \(0.252024\pi\)
\(828\) 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
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 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 \(-0.519808\pi\)
−0.0621875 + 0.998064i \(0.519808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.96605e9 0.269512 0.134756 0.990879i \(-0.456975\pi\)
0.134756 + 0.990879i \(0.456975\pi\)
\(858\) 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 \(-0.538975\pi\)
−0.122136 + 0.992513i \(0.538975\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) −3.57505e10 −1.78971 −0.894856 0.446355i \(-0.852722\pi\)
−0.894856 + 0.446355i \(0.852722\pi\)
\(878\) 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 \(-0.573627\pi\)
−0.229249 + 0.973368i \(0.573627\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.98940e9 −0.288171 −0.144086 0.989565i \(-0.546024\pi\)
−0.144086 + 0.989565i \(0.546024\pi\)
\(888\) 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
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) −1.76568e9 −0.0785755 −0.0392878 0.999228i \(-0.512509\pi\)
−0.0392878 + 0.999228i \(0.512509\pi\)
\(908\) 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
\(915\) 0 0
\(916\) 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
\(925\) 0 0
\(926\) 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
\(935\) 0 0
\(936\) 0 0
\(937\) 5.75353e9 0.228479 0.114239 0.993453i \(-0.463557\pi\)
0.114239 + 0.993453i \(0.463557\pi\)
\(938\) 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
\(945\) 0 0
\(946\) 0 0
\(947\) 9.34709e7 0.00357645 0.00178822 0.999998i \(-0.499431\pi\)
0.00178822 + 0.999998i \(0.499431\pi\)
\(948\) 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 \(-0.570437\pi\)
−0.219484 + 0.975616i \(0.570437\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) 1.64786e10 0.586040 0.293020 0.956106i \(-0.405340\pi\)
0.293020 + 0.956106i \(0.405340\pi\)
\(968\) 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
\(975\) 0 0
\(976\) 0 0
\(977\) 4.26661e10 1.46370 0.731849 0.681467i \(-0.238657\pi\)
0.731849 + 0.681467i \(0.238657\pi\)
\(978\) 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 \(-0.523243\pi\)
−0.0729541 + 0.997335i \(0.523243\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) 4.32788e10 1.38306 0.691531 0.722346i \(-0.256936\pi\)
0.691531 + 0.722346i \(0.256936\pi\)
\(998\) 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 600.8.a.b.1.1 1
5.2 odd 4 600.8.f.a.49.2 2
5.3 odd 4 600.8.f.a.49.1 2
5.4 even 2 24.8.a.b.1.1 1
15.14 odd 2 72.8.a.e.1.1 1
20.19 odd 2 48.8.a.a.1.1 1
40.19 odd 2 192.8.a.p.1.1 1
40.29 even 2 192.8.a.h.1.1 1
60.59 even 2 144.8.a.k.1.1 1
120.29 odd 2 576.8.a.c.1.1 1
120.59 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 5.4 even 2
48.8.a.a.1.1 1 20.19 odd 2
72.8.a.e.1.1 1 15.14 odd 2
144.8.a.k.1.1 1 60.59 even 2
192.8.a.h.1.1 1 40.29 even 2
192.8.a.p.1.1 1 40.19 odd 2
576.8.a.b.1.1 1 120.59 even 2
576.8.a.c.1.1 1 120.29 odd 2
600.8.a.b.1.1 1 1.1 even 1 trivial
600.8.f.a.49.1 2 5.3 odd 4
600.8.f.a.49.2 2 5.2 odd 4