Properties

Label 600.8.a.a.1.1
Level $600$
Weight $8$
Character 600.1
Self dual yes
Analytic conductor $187.431$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -504.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} -504.000 q^{7} +729.000 q^{9} +3812.00 q^{11} -9574.00 q^{13} -26098.0 q^{17} -38308.0 q^{19} +13608.0 q^{21} +71128.0 q^{23} -19683.0 q^{27} +74262.0 q^{29} -275680. q^{31} -102924. q^{33} +266610. q^{37} +258498. q^{39} +684762. q^{41} -245956. q^{43} -478800. q^{47} -569527. q^{49} +704646. q^{51} +569410. q^{53} +1.03432e6 q^{57} -1.52532e6 q^{59} -2.64046e6 q^{61} -367416. q^{63} -1.41624e6 q^{67} -1.92046e6 q^{69} -3.51130e6 q^{71} -4.73862e6 q^{73} -1.92125e6 q^{77} +4.66149e6 q^{79} +531441. q^{81} +5.72925e6 q^{83} -2.00507e6 q^{87} +1.19935e7 q^{89} +4.82530e6 q^{91} +7.44336e6 q^{93} -7.15075e6 q^{97} +2.77895e6 q^{99} +O(q^{100})\)

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
\(7\) −504.000 −0.555376 −0.277688 0.960671i \(-0.589568\pi\)
−0.277688 + 0.960671i \(0.589568\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 3812.00 0.863532 0.431766 0.901986i \(-0.357891\pi\)
0.431766 + 0.901986i \(0.357891\pi\)
\(12\) 0 0
\(13\) −9574.00 −1.20863 −0.604313 0.796747i \(-0.706552\pi\)
−0.604313 + 0.796747i \(0.706552\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −26098.0 −1.28836 −0.644178 0.764875i \(-0.722800\pi\)
−0.644178 + 0.764875i \(0.722800\pi\)
\(18\) 0 0
\(19\) −38308.0 −1.28130 −0.640652 0.767832i \(-0.721336\pi\)
−0.640652 + 0.767832i \(0.721336\pi\)
\(20\) 0 0
\(21\) 13608.0 0.320647
\(22\) 0 0
\(23\) 71128.0 1.21897 0.609485 0.792797i \(-0.291376\pi\)
0.609485 + 0.792797i \(0.291376\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) 74262.0 0.565423 0.282712 0.959205i \(-0.408766\pi\)
0.282712 + 0.959205i \(0.408766\pi\)
\(30\) 0 0
\(31\) −275680. −1.66203 −0.831016 0.556249i \(-0.812240\pi\)
−0.831016 + 0.556249i \(0.812240\pi\)
\(32\) 0 0
\(33\) −102924. −0.498560
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 266610. 0.865307 0.432654 0.901560i \(-0.357577\pi\)
0.432654 + 0.901560i \(0.357577\pi\)
\(38\) 0 0
\(39\) 258498. 0.697800
\(40\) 0 0
\(41\) 684762. 1.55166 0.775829 0.630943i \(-0.217332\pi\)
0.775829 + 0.630943i \(0.217332\pi\)
\(42\) 0 0
\(43\) −245956. −0.471756 −0.235878 0.971783i \(-0.575797\pi\)
−0.235878 + 0.971783i \(0.575797\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −478800. −0.672685 −0.336342 0.941740i \(-0.609190\pi\)
−0.336342 + 0.941740i \(0.609190\pi\)
\(48\) 0 0
\(49\) −569527. −0.691557
\(50\) 0 0
\(51\) 704646. 0.743833
\(52\) 0 0
\(53\) 569410. 0.525363 0.262682 0.964883i \(-0.415393\pi\)
0.262682 + 0.964883i \(0.415393\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.03432e6 0.739761
\(58\) 0 0
\(59\) −1.52532e6 −0.966897 −0.483448 0.875373i \(-0.660616\pi\)
−0.483448 + 0.875373i \(0.660616\pi\)
\(60\) 0 0
\(61\) −2.64046e6 −1.48945 −0.744723 0.667374i \(-0.767418\pi\)
−0.744723 + 0.667374i \(0.767418\pi\)
\(62\) 0 0
\(63\) −367416. −0.185125
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.41624e6 −0.575273 −0.287636 0.957740i \(-0.592869\pi\)
−0.287636 + 0.957740i \(0.592869\pi\)
\(68\) 0 0
\(69\) −1.92046e6 −0.703773
\(70\) 0 0
\(71\) −3.51130e6 −1.16430 −0.582149 0.813082i \(-0.697788\pi\)
−0.582149 + 0.813082i \(0.697788\pi\)
\(72\) 0 0
\(73\) −4.73862e6 −1.42568 −0.712839 0.701327i \(-0.752591\pi\)
−0.712839 + 0.701327i \(0.752591\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.92125e6 −0.479585
\(78\) 0 0
\(79\) 4.66149e6 1.06373 0.531863 0.846830i \(-0.321492\pi\)
0.531863 + 0.846830i \(0.321492\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 5.72925e6 1.09983 0.549914 0.835221i \(-0.314661\pi\)
0.549914 + 0.835221i \(0.314661\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.00507e6 −0.326447
\(88\) 0 0
\(89\) 1.19935e7 1.80336 0.901678 0.432408i \(-0.142336\pi\)
0.901678 + 0.432408i \(0.142336\pi\)
\(90\) 0 0
\(91\) 4.82530e6 0.671242
\(92\) 0 0
\(93\) 7.44336e6 0.959575
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.15075e6 −0.795519 −0.397760 0.917490i \(-0.630212\pi\)
−0.397760 + 0.917490i \(0.630212\pi\)
\(98\) 0 0
\(99\) 2.77895e6 0.287844
\(100\) 0 0
\(101\) −8.78373e6 −0.848309 −0.424155 0.905590i \(-0.639429\pi\)
−0.424155 + 0.905590i \(0.639429\pi\)
\(102\) 0 0
\(103\) 8.01610e6 0.722825 0.361412 0.932406i \(-0.382295\pi\)
0.361412 + 0.932406i \(0.382295\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.15123e6 0.406507 0.203253 0.979126i \(-0.434849\pi\)
0.203253 + 0.979126i \(0.434849\pi\)
\(108\) 0 0
\(109\) −2.41280e7 −1.78455 −0.892274 0.451493i \(-0.850891\pi\)
−0.892274 + 0.451493i \(0.850891\pi\)
\(110\) 0 0
\(111\) −7.19847e6 −0.499585
\(112\) 0 0
\(113\) −2.04827e6 −0.133541 −0.0667703 0.997768i \(-0.521269\pi\)
−0.0667703 + 0.997768i \(0.521269\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.97945e6 −0.402875
\(118\) 0 0
\(119\) 1.31534e7 0.715523
\(120\) 0 0
\(121\) −4.95583e6 −0.254312
\(122\) 0 0
\(123\) −1.84886e7 −0.895850
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.36634e6 −0.0591895 −0.0295947 0.999562i \(-0.509422\pi\)
−0.0295947 + 0.999562i \(0.509422\pi\)
\(128\) 0 0
\(129\) 6.64081e6 0.272369
\(130\) 0 0
\(131\) 3.84645e7 1.49489 0.747447 0.664321i \(-0.231279\pi\)
0.747447 + 0.664321i \(0.231279\pi\)
\(132\) 0 0
\(133\) 1.93072e7 0.711605
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.62585e6 −0.253376 −0.126688 0.991943i \(-0.540435\pi\)
−0.126688 + 0.991943i \(0.540435\pi\)
\(138\) 0 0
\(139\) 5.32324e6 0.168122 0.0840609 0.996461i \(-0.473211\pi\)
0.0840609 + 0.996461i \(0.473211\pi\)
\(140\) 0 0
\(141\) 1.29276e7 0.388375
\(142\) 0 0
\(143\) −3.64961e7 −1.04369
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.53772e7 0.399271
\(148\) 0 0
\(149\) 7.61366e6 0.188557 0.0942783 0.995546i \(-0.469946\pi\)
0.0942783 + 0.995546i \(0.469946\pi\)
\(150\) 0 0
\(151\) −2.50221e7 −0.591432 −0.295716 0.955276i \(-0.595558\pi\)
−0.295716 + 0.955276i \(0.595558\pi\)
\(152\) 0 0
\(153\) −1.90254e7 −0.429452
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.93145e7 −0.810782 −0.405391 0.914143i \(-0.632865\pi\)
−0.405391 + 0.914143i \(0.632865\pi\)
\(158\) 0 0
\(159\) −1.53741e7 −0.303319
\(160\) 0 0
\(161\) −3.58485e7 −0.676987
\(162\) 0 0
\(163\) 6.28387e7 1.13650 0.568252 0.822855i \(-0.307620\pi\)
0.568252 + 0.822855i \(0.307620\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.04133e7 −0.173014 −0.0865072 0.996251i \(-0.527571\pi\)
−0.0865072 + 0.996251i \(0.527571\pi\)
\(168\) 0 0
\(169\) 2.89130e7 0.460775
\(170\) 0 0
\(171\) −2.79265e7 −0.427101
\(172\) 0 0
\(173\) 8.03551e7 1.17992 0.589959 0.807433i \(-0.299144\pi\)
0.589959 + 0.807433i \(0.299144\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.11837e7 0.558238
\(178\) 0 0
\(179\) −8.40084e7 −1.09481 −0.547403 0.836869i \(-0.684383\pi\)
−0.547403 + 0.836869i \(0.684383\pi\)
\(180\) 0 0
\(181\) 1.15469e8 1.44741 0.723703 0.690112i \(-0.242439\pi\)
0.723703 + 0.690112i \(0.242439\pi\)
\(182\) 0 0
\(183\) 7.12924e7 0.859932
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.94856e7 −1.11254
\(188\) 0 0
\(189\) 9.92023e6 0.106882
\(190\) 0 0
\(191\) 9.97154e7 1.03549 0.517744 0.855535i \(-0.326772\pi\)
0.517744 + 0.855535i \(0.326772\pi\)
\(192\) 0 0
\(193\) 1.86157e7 0.186393 0.0931965 0.995648i \(-0.470292\pi\)
0.0931965 + 0.995648i \(0.470292\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.30384e7 −0.867022 −0.433511 0.901148i \(-0.642726\pi\)
−0.433511 + 0.901148i \(0.642726\pi\)
\(198\) 0 0
\(199\) 7.39686e7 0.665367 0.332684 0.943038i \(-0.392046\pi\)
0.332684 + 0.943038i \(0.392046\pi\)
\(200\) 0 0
\(201\) 3.82384e7 0.332134
\(202\) 0 0
\(203\) −3.74280e7 −0.314023
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.18523e7 0.406323
\(208\) 0 0
\(209\) −1.46030e8 −1.10645
\(210\) 0 0
\(211\) 1.85163e8 1.35695 0.678476 0.734623i \(-0.262641\pi\)
0.678476 + 0.734623i \(0.262641\pi\)
\(212\) 0 0
\(213\) 9.48052e7 0.672208
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.38943e8 0.923053
\(218\) 0 0
\(219\) 1.27943e8 0.823116
\(220\) 0 0
\(221\) 2.49862e8 1.55714
\(222\) 0 0
\(223\) 1.20862e8 0.729830 0.364915 0.931041i \(-0.381098\pi\)
0.364915 + 0.931041i \(0.381098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.82315e8 1.60193 0.800966 0.598710i \(-0.204320\pi\)
0.800966 + 0.598710i \(0.204320\pi\)
\(228\) 0 0
\(229\) −8.91913e7 −0.490793 −0.245397 0.969423i \(-0.578918\pi\)
−0.245397 + 0.969423i \(0.578918\pi\)
\(230\) 0 0
\(231\) 5.18737e7 0.276889
\(232\) 0 0
\(233\) 2.32240e8 1.20279 0.601396 0.798951i \(-0.294611\pi\)
0.601396 + 0.798951i \(0.294611\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.25860e8 −0.614142
\(238\) 0 0
\(239\) 3.21986e8 1.52561 0.762807 0.646626i \(-0.223821\pi\)
0.762807 + 0.646626i \(0.223821\pi\)
\(240\) 0 0
\(241\) −2.00366e8 −0.922072 −0.461036 0.887381i \(-0.652522\pi\)
−0.461036 + 0.887381i \(0.652522\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.66761e8 1.54862
\(248\) 0 0
\(249\) −1.54690e8 −0.634986
\(250\) 0 0
\(251\) −8.70560e7 −0.347489 −0.173744 0.984791i \(-0.555587\pi\)
−0.173744 + 0.984791i \(0.555587\pi\)
\(252\) 0 0
\(253\) 2.71140e8 1.05262
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.22879e8 −1.92148 −0.960738 0.277457i \(-0.910509\pi\)
−0.960738 + 0.277457i \(0.910509\pi\)
\(258\) 0 0
\(259\) −1.34371e8 −0.480571
\(260\) 0 0
\(261\) 5.41370e7 0.188474
\(262\) 0 0
\(263\) 4.06215e8 1.37693 0.688464 0.725270i \(-0.258285\pi\)
0.688464 + 0.725270i \(0.258285\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.23825e8 −1.04117
\(268\) 0 0
\(269\) 3.82347e8 1.19764 0.598818 0.800885i \(-0.295637\pi\)
0.598818 + 0.800885i \(0.295637\pi\)
\(270\) 0 0
\(271\) 2.84165e8 0.867317 0.433658 0.901077i \(-0.357222\pi\)
0.433658 + 0.901077i \(0.357222\pi\)
\(272\) 0 0
\(273\) −1.30283e8 −0.387542
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.93752e8 −0.830427 −0.415213 0.909724i \(-0.636293\pi\)
−0.415213 + 0.909724i \(0.636293\pi\)
\(278\) 0 0
\(279\) −2.00971e8 −0.554011
\(280\) 0 0
\(281\) 4.15399e8 1.11685 0.558424 0.829556i \(-0.311406\pi\)
0.558424 + 0.829556i \(0.311406\pi\)
\(282\) 0 0
\(283\) −5.06429e7 −0.132821 −0.0664104 0.997792i \(-0.521155\pi\)
−0.0664104 + 0.997792i \(0.521155\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.45120e8 −0.861754
\(288\) 0 0
\(289\) 2.70767e8 0.659862
\(290\) 0 0
\(291\) 1.93070e8 0.459293
\(292\) 0 0
\(293\) −7.47714e7 −0.173660 −0.0868298 0.996223i \(-0.527674\pi\)
−0.0868298 + 0.996223i \(0.527674\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.50316e7 −0.166187
\(298\) 0 0
\(299\) −6.80979e8 −1.47328
\(300\) 0 0
\(301\) 1.23962e8 0.262002
\(302\) 0 0
\(303\) 2.37161e8 0.489772
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8.52577e7 −0.168170 −0.0840851 0.996459i \(-0.526797\pi\)
−0.0840851 + 0.996459i \(0.526797\pi\)
\(308\) 0 0
\(309\) −2.16435e8 −0.417323
\(310\) 0 0
\(311\) 9.39129e8 1.77037 0.885184 0.465240i \(-0.154032\pi\)
0.885184 + 0.465240i \(0.154032\pi\)
\(312\) 0 0
\(313\) 3.43040e8 0.632323 0.316162 0.948705i \(-0.397606\pi\)
0.316162 + 0.948705i \(0.397606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.03960e9 1.83298 0.916492 0.400054i \(-0.131009\pi\)
0.916492 + 0.400054i \(0.131009\pi\)
\(318\) 0 0
\(319\) 2.83087e8 0.488261
\(320\) 0 0
\(321\) −1.39083e8 −0.234697
\(322\) 0 0
\(323\) 9.99762e8 1.65077
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.51456e8 1.03031
\(328\) 0 0
\(329\) 2.41315e8 0.373593
\(330\) 0 0
\(331\) 1.10022e9 1.66756 0.833779 0.552098i \(-0.186173\pi\)
0.833779 + 0.552098i \(0.186173\pi\)
\(332\) 0 0
\(333\) 1.94359e8 0.288436
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.28272e9 −1.82569 −0.912847 0.408302i \(-0.866121\pi\)
−0.912847 + 0.408302i \(0.866121\pi\)
\(338\) 0 0
\(339\) 5.53034e7 0.0770997
\(340\) 0 0
\(341\) −1.05089e9 −1.43522
\(342\) 0 0
\(343\) 7.02107e8 0.939451
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.25822e9 −1.61660 −0.808301 0.588770i \(-0.799612\pi\)
−0.808301 + 0.588770i \(0.799612\pi\)
\(348\) 0 0
\(349\) 1.35371e8 0.170465 0.0852327 0.996361i \(-0.472837\pi\)
0.0852327 + 0.996361i \(0.472837\pi\)
\(350\) 0 0
\(351\) 1.88445e8 0.232600
\(352\) 0 0
\(353\) 8.49221e8 1.02756 0.513782 0.857921i \(-0.328244\pi\)
0.513782 + 0.857921i \(0.328244\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.55142e8 −0.413107
\(358\) 0 0
\(359\) 2.20121e8 0.251091 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(360\) 0 0
\(361\) 5.73631e8 0.641738
\(362\) 0 0
\(363\) 1.33807e8 0.146827
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.66505e8 −0.492635 −0.246317 0.969189i \(-0.579221\pi\)
−0.246317 + 0.969189i \(0.579221\pi\)
\(368\) 0 0
\(369\) 4.99191e8 0.517220
\(370\) 0 0
\(371\) −2.86983e8 −0.291774
\(372\) 0 0
\(373\) 2.98453e8 0.297780 0.148890 0.988854i \(-0.452430\pi\)
0.148890 + 0.988854i \(0.452430\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.10984e8 −0.683385
\(378\) 0 0
\(379\) 1.46218e9 1.37964 0.689818 0.723983i \(-0.257691\pi\)
0.689818 + 0.723983i \(0.257691\pi\)
\(380\) 0 0
\(381\) 3.68911e7 0.0341731
\(382\) 0 0
\(383\) −1.58702e9 −1.44340 −0.721700 0.692206i \(-0.756639\pi\)
−0.721700 + 0.692206i \(0.756639\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.79302e8 −0.157252
\(388\) 0 0
\(389\) −3.14439e8 −0.270840 −0.135420 0.990788i \(-0.543238\pi\)
−0.135420 + 0.990788i \(0.543238\pi\)
\(390\) 0 0
\(391\) −1.85630e9 −1.57047
\(392\) 0 0
\(393\) −1.03854e9 −0.863078
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.52757e8 0.684004 0.342002 0.939699i \(-0.388895\pi\)
0.342002 + 0.939699i \(0.388895\pi\)
\(398\) 0 0
\(399\) −5.21295e8 −0.410846
\(400\) 0 0
\(401\) 6.92522e8 0.536325 0.268163 0.963374i \(-0.413584\pi\)
0.268163 + 0.963374i \(0.413584\pi\)
\(402\) 0 0
\(403\) 2.63936e9 2.00877
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.01632e9 0.747221
\(408\) 0 0
\(409\) 6.17357e8 0.446174 0.223087 0.974799i \(-0.428387\pi\)
0.223087 + 0.974799i \(0.428387\pi\)
\(410\) 0 0
\(411\) 2.05898e8 0.146287
\(412\) 0 0
\(413\) 7.68763e8 0.536992
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.43727e8 −0.0970651
\(418\) 0 0
\(419\) −1.65512e9 −1.09921 −0.549604 0.835425i \(-0.685221\pi\)
−0.549604 + 0.835425i \(0.685221\pi\)
\(420\) 0 0
\(421\) 7.01472e7 0.0458166 0.0229083 0.999738i \(-0.492707\pi\)
0.0229083 + 0.999738i \(0.492707\pi\)
\(422\) 0 0
\(423\) −3.49045e8 −0.224228
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.33079e9 0.827203
\(428\) 0 0
\(429\) 9.85394e8 0.602573
\(430\) 0 0
\(431\) 1.81387e9 1.09128 0.545640 0.838020i \(-0.316287\pi\)
0.545640 + 0.838020i \(0.316287\pi\)
\(432\) 0 0
\(433\) 2.59970e9 1.53892 0.769460 0.638695i \(-0.220525\pi\)
0.769460 + 0.638695i \(0.220525\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.72477e9 −1.56187
\(438\) 0 0
\(439\) −1.67431e9 −0.944517 −0.472258 0.881460i \(-0.656561\pi\)
−0.472258 + 0.881460i \(0.656561\pi\)
\(440\) 0 0
\(441\) −4.15185e8 −0.230519
\(442\) 0 0
\(443\) 2.52711e8 0.138105 0.0690527 0.997613i \(-0.478002\pi\)
0.0690527 + 0.997613i \(0.478002\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.05569e8 −0.108863
\(448\) 0 0
\(449\) 7.55311e8 0.393789 0.196895 0.980425i \(-0.436914\pi\)
0.196895 + 0.980425i \(0.436914\pi\)
\(450\) 0 0
\(451\) 2.61031e9 1.33991
\(452\) 0 0
\(453\) 6.75597e8 0.341463
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.51584e8 0.0742928 0.0371464 0.999310i \(-0.488173\pi\)
0.0371464 + 0.999310i \(0.488173\pi\)
\(458\) 0 0
\(459\) 5.13687e8 0.247944
\(460\) 0 0
\(461\) −7.78405e8 −0.370043 −0.185022 0.982734i \(-0.559236\pi\)
−0.185022 + 0.982734i \(0.559236\pi\)
\(462\) 0 0
\(463\) −2.41052e9 −1.12870 −0.564349 0.825536i \(-0.690873\pi\)
−0.564349 + 0.825536i \(0.690873\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.76192e9 0.800527 0.400264 0.916400i \(-0.368919\pi\)
0.400264 + 0.916400i \(0.368919\pi\)
\(468\) 0 0
\(469\) 7.13783e8 0.319493
\(470\) 0 0
\(471\) 1.06149e9 0.468105
\(472\) 0 0
\(473\) −9.37584e8 −0.407377
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.15100e8 0.175121
\(478\) 0 0
\(479\) 6.43811e8 0.267661 0.133830 0.991004i \(-0.457272\pi\)
0.133830 + 0.991004i \(0.457272\pi\)
\(480\) 0 0
\(481\) −2.55252e9 −1.04583
\(482\) 0 0
\(483\) 9.67910e8 0.390859
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.16421e9 −1.24141 −0.620704 0.784045i \(-0.713153\pi\)
−0.620704 + 0.784045i \(0.713153\pi\)
\(488\) 0 0
\(489\) −1.69665e9 −0.656160
\(490\) 0 0
\(491\) 3.62406e9 1.38169 0.690844 0.723004i \(-0.257239\pi\)
0.690844 + 0.723004i \(0.257239\pi\)
\(492\) 0 0
\(493\) −1.93809e9 −0.728467
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.76970e9 0.646624
\(498\) 0 0
\(499\) −1.35483e9 −0.488128 −0.244064 0.969759i \(-0.578481\pi\)
−0.244064 + 0.969759i \(0.578481\pi\)
\(500\) 0 0
\(501\) 2.81160e8 0.0998899
\(502\) 0 0
\(503\) 4.66389e9 1.63403 0.817015 0.576616i \(-0.195627\pi\)
0.817015 + 0.576616i \(0.195627\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.80650e8 −0.266029
\(508\) 0 0
\(509\) −1.34292e9 −0.451376 −0.225688 0.974200i \(-0.572463\pi\)
−0.225688 + 0.974200i \(0.572463\pi\)
\(510\) 0 0
\(511\) 2.38826e9 0.791788
\(512\) 0 0
\(513\) 7.54016e8 0.246587
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.82519e9 −0.580885
\(518\) 0 0
\(519\) −2.16959e9 −0.681226
\(520\) 0 0
\(521\) −1.45400e9 −0.450435 −0.225217 0.974309i \(-0.572309\pi\)
−0.225217 + 0.974309i \(0.572309\pi\)
\(522\) 0 0
\(523\) −4.90309e8 −0.149870 −0.0749349 0.997188i \(-0.523875\pi\)
−0.0749349 + 0.997188i \(0.523875\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.19470e9 2.14129
\(528\) 0 0
\(529\) 1.65437e9 0.485889
\(530\) 0 0
\(531\) −1.11196e9 −0.322299
\(532\) 0 0
\(533\) −6.55591e9 −1.87537
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.26823e9 0.632086
\(538\) 0 0
\(539\) −2.17104e9 −0.597182
\(540\) 0 0
\(541\) −4.82889e9 −1.31116 −0.655582 0.755124i \(-0.727577\pi\)
−0.655582 + 0.755124i \(0.727577\pi\)
\(542\) 0 0
\(543\) −3.11766e9 −0.835660
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.08793e9 0.806698 0.403349 0.915046i \(-0.367846\pi\)
0.403349 + 0.915046i \(0.367846\pi\)
\(548\) 0 0
\(549\) −1.92489e9 −0.496482
\(550\) 0 0
\(551\) −2.84483e9 −0.724479
\(552\) 0 0
\(553\) −2.34939e9 −0.590768
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.09889e9 −1.00502 −0.502508 0.864573i \(-0.667589\pi\)
−0.502508 + 0.864573i \(0.667589\pi\)
\(558\) 0 0
\(559\) 2.35478e9 0.570177
\(560\) 0 0
\(561\) 2.68611e9 0.642324
\(562\) 0 0
\(563\) −4.97105e9 −1.17400 −0.587001 0.809587i \(-0.699691\pi\)
−0.587001 + 0.809587i \(0.699691\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.67846e8 −0.0617085
\(568\) 0 0
\(569\) 3.71316e9 0.844988 0.422494 0.906366i \(-0.361155\pi\)
0.422494 + 0.906366i \(0.361155\pi\)
\(570\) 0 0
\(571\) −2.36205e9 −0.530961 −0.265481 0.964116i \(-0.585531\pi\)
−0.265481 + 0.964116i \(0.585531\pi\)
\(572\) 0 0
\(573\) −2.69232e9 −0.597840
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.81146e8 0.0392566 0.0196283 0.999807i \(-0.493752\pi\)
0.0196283 + 0.999807i \(0.493752\pi\)
\(578\) 0 0
\(579\) −5.02625e8 −0.107614
\(580\) 0 0
\(581\) −2.88754e9 −0.610818
\(582\) 0 0
\(583\) 2.17059e9 0.453668
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.31976e9 −0.473380 −0.236690 0.971585i \(-0.576063\pi\)
−0.236690 + 0.971585i \(0.576063\pi\)
\(588\) 0 0
\(589\) 1.05607e10 2.12957
\(590\) 0 0
\(591\) 2.51204e9 0.500576
\(592\) 0 0
\(593\) −2.27806e9 −0.448615 −0.224308 0.974518i \(-0.572012\pi\)
−0.224308 + 0.974518i \(0.572012\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.99715e9 −0.384150
\(598\) 0 0
\(599\) −4.88253e9 −0.928220 −0.464110 0.885778i \(-0.653626\pi\)
−0.464110 + 0.885778i \(0.653626\pi\)
\(600\) 0 0
\(601\) −6.74758e9 −1.26791 −0.633954 0.773371i \(-0.718569\pi\)
−0.633954 + 0.773371i \(0.718569\pi\)
\(602\) 0 0
\(603\) −1.03244e9 −0.191758
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 9.05928e9 1.64412 0.822060 0.569401i \(-0.192825\pi\)
0.822060 + 0.569401i \(0.192825\pi\)
\(608\) 0 0
\(609\) 1.01056e9 0.181301
\(610\) 0 0
\(611\) 4.58403e9 0.813024
\(612\) 0 0
\(613\) 8.48777e9 1.48827 0.744135 0.668029i \(-0.232862\pi\)
0.744135 + 0.668029i \(0.232862\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.34407e9 −0.401766 −0.200883 0.979615i \(-0.564381\pi\)
−0.200883 + 0.979615i \(0.564381\pi\)
\(618\) 0 0
\(619\) −1.01541e9 −0.172077 −0.0860384 0.996292i \(-0.527421\pi\)
−0.0860384 + 0.996292i \(0.527421\pi\)
\(620\) 0 0
\(621\) −1.40001e9 −0.234591
\(622\) 0 0
\(623\) −6.04473e9 −1.00154
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.94281e9 0.638807
\(628\) 0 0
\(629\) −6.95799e9 −1.11482
\(630\) 0 0
\(631\) 7.01911e9 1.11219 0.556095 0.831119i \(-0.312299\pi\)
0.556095 + 0.831119i \(0.312299\pi\)
\(632\) 0 0
\(633\) −4.99939e9 −0.783437
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.45265e9 0.835833
\(638\) 0 0
\(639\) −2.55974e9 −0.388099
\(640\) 0 0
\(641\) −4.52776e9 −0.679016 −0.339508 0.940603i \(-0.610261\pi\)
−0.339508 + 0.940603i \(0.610261\pi\)
\(642\) 0 0
\(643\) 8.63094e9 1.28032 0.640162 0.768240i \(-0.278867\pi\)
0.640162 + 0.768240i \(0.278867\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.57401e9 −0.373632 −0.186816 0.982395i \(-0.559817\pi\)
−0.186816 + 0.982395i \(0.559817\pi\)
\(648\) 0 0
\(649\) −5.81454e9 −0.834946
\(650\) 0 0
\(651\) −3.75145e9 −0.532925
\(652\) 0 0
\(653\) 9.31827e9 1.30960 0.654800 0.755802i \(-0.272753\pi\)
0.654800 + 0.755802i \(0.272753\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.45445e9 −0.475226
\(658\) 0 0
\(659\) 1.04422e10 1.42133 0.710663 0.703532i \(-0.248395\pi\)
0.710663 + 0.703532i \(0.248395\pi\)
\(660\) 0 0
\(661\) −1.04761e10 −1.41090 −0.705449 0.708761i \(-0.749254\pi\)
−0.705449 + 0.708761i \(0.749254\pi\)
\(662\) 0 0
\(663\) −6.74628e9 −0.899015
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.28211e9 0.689234
\(668\) 0 0
\(669\) −3.26327e9 −0.421368
\(670\) 0 0
\(671\) −1.00654e10 −1.28618
\(672\) 0 0
\(673\) −1.38891e10 −1.75639 −0.878197 0.478299i \(-0.841254\pi\)
−0.878197 + 0.478299i \(0.841254\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.48893e8 0.0927598 0.0463799 0.998924i \(-0.485232\pi\)
0.0463799 + 0.998924i \(0.485232\pi\)
\(678\) 0 0
\(679\) 3.60398e9 0.441813
\(680\) 0 0
\(681\) −7.62251e9 −0.924875
\(682\) 0 0
\(683\) −1.15581e10 −1.38808 −0.694038 0.719938i \(-0.744170\pi\)
−0.694038 + 0.719938i \(0.744170\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.40817e9 0.283360
\(688\) 0 0
\(689\) −5.45153e9 −0.634967
\(690\) 0 0
\(691\) 3.34337e8 0.0385489 0.0192744 0.999814i \(-0.493864\pi\)
0.0192744 + 0.999814i \(0.493864\pi\)
\(692\) 0 0
\(693\) −1.40059e9 −0.159862
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.78709e10 −1.99909
\(698\) 0 0
\(699\) −6.27047e9 −0.694433
\(700\) 0 0
\(701\) −5.55383e9 −0.608948 −0.304474 0.952521i \(-0.598481\pi\)
−0.304474 + 0.952521i \(0.598481\pi\)
\(702\) 0 0
\(703\) −1.02133e10 −1.10872
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.42700e9 0.471131
\(708\) 0 0
\(709\) 1.30817e10 1.37849 0.689243 0.724530i \(-0.257943\pi\)
0.689243 + 0.724530i \(0.257943\pi\)
\(710\) 0 0
\(711\) 3.39822e9 0.354575
\(712\) 0 0
\(713\) −1.96086e10 −2.02597
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.69363e9 −0.880814
\(718\) 0 0
\(719\) −1.10847e10 −1.11217 −0.556085 0.831125i \(-0.687697\pi\)
−0.556085 + 0.831125i \(0.687697\pi\)
\(720\) 0 0
\(721\) −4.04012e9 −0.401440
\(722\) 0 0
\(723\) 5.40989e9 0.532358
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.79416e9 0.752314 0.376157 0.926556i \(-0.377245\pi\)
0.376157 + 0.926556i \(0.377245\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 6.41896e9 0.607790
\(732\) 0 0
\(733\) −6.83552e9 −0.641073 −0.320537 0.947236i \(-0.603863\pi\)
−0.320537 + 0.947236i \(0.603863\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.39869e9 −0.496767
\(738\) 0 0
\(739\) 1.73862e10 1.58471 0.792356 0.610059i \(-0.208854\pi\)
0.792356 + 0.610059i \(0.208854\pi\)
\(740\) 0 0
\(741\) −9.90254e9 −0.894094
\(742\) 0 0
\(743\) −2.25537e9 −0.201724 −0.100862 0.994900i \(-0.532160\pi\)
−0.100862 + 0.994900i \(0.532160\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.17662e9 0.366609
\(748\) 0 0
\(749\) −2.59622e9 −0.225764
\(750\) 0 0
\(751\) −2.05027e10 −1.76632 −0.883162 0.469068i \(-0.844590\pi\)
−0.883162 + 0.469068i \(0.844590\pi\)
\(752\) 0 0
\(753\) 2.35051e9 0.200623
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.57872e8 0.0216057 0.0108029 0.999942i \(-0.496561\pi\)
0.0108029 + 0.999942i \(0.496561\pi\)
\(758\) 0 0
\(759\) −7.32078e9 −0.607731
\(760\) 0 0
\(761\) −1.34452e10 −1.10591 −0.552957 0.833210i \(-0.686501\pi\)
−0.552957 + 0.833210i \(0.686501\pi\)
\(762\) 0 0
\(763\) 1.21605e10 0.991096
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.46035e10 1.16862
\(768\) 0 0
\(769\) 8.28541e9 0.657009 0.328505 0.944502i \(-0.393455\pi\)
0.328505 + 0.944502i \(0.393455\pi\)
\(770\) 0 0
\(771\) 1.41177e10 1.10936
\(772\) 0 0
\(773\) −1.43430e10 −1.11689 −0.558447 0.829540i \(-0.688603\pi\)
−0.558447 + 0.829540i \(0.688603\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.62803e9 0.277458
\(778\) 0 0
\(779\) −2.62319e10 −1.98814
\(780\) 0 0
\(781\) −1.33851e10 −1.00541
\(782\) 0 0
\(783\) −1.46170e9 −0.108816
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.83137e9 0.280184 0.140092 0.990139i \(-0.455260\pi\)
0.140092 + 0.990139i \(0.455260\pi\)
\(788\) 0 0
\(789\) −1.09678e10 −0.794970
\(790\) 0 0
\(791\) 1.03233e9 0.0741653
\(792\) 0 0
\(793\) 2.52797e10 1.80018
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.95859e9 0.137038 0.0685188 0.997650i \(-0.478173\pi\)
0.0685188 + 0.997650i \(0.478173\pi\)
\(798\) 0 0
\(799\) 1.24957e10 0.866658
\(800\) 0 0
\(801\) 8.74327e9 0.601119
\(802\) 0 0
\(803\) −1.80636e10 −1.23112
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.03234e10 −0.691455
\(808\) 0 0
\(809\) 2.07415e9 0.137727 0.0688637 0.997626i \(-0.478063\pi\)
0.0688637 + 0.997626i \(0.478063\pi\)
\(810\) 0 0
\(811\) −5.71508e9 −0.376227 −0.188113 0.982147i \(-0.560237\pi\)
−0.188113 + 0.982147i \(0.560237\pi\)
\(812\) 0 0
\(813\) −7.67245e9 −0.500746
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.42208e9 0.604463
\(818\) 0 0
\(819\) 3.51764e9 0.223747
\(820\) 0 0
\(821\) 2.82748e10 1.78319 0.891596 0.452832i \(-0.149586\pi\)
0.891596 + 0.452832i \(0.149586\pi\)
\(822\) 0 0
\(823\) 2.09283e9 0.130868 0.0654342 0.997857i \(-0.479157\pi\)
0.0654342 + 0.997857i \(0.479157\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.71453e9 0.228368 0.114184 0.993460i \(-0.463575\pi\)
0.114184 + 0.993460i \(0.463575\pi\)
\(828\) 0 0
\(829\) 3.37924e9 0.206005 0.103003 0.994681i \(-0.467155\pi\)
0.103003 + 0.994681i \(0.467155\pi\)
\(830\) 0 0
\(831\) 7.93130e9 0.479447
\(832\) 0 0
\(833\) 1.48635e10 0.890972
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.42621e9 0.319858
\(838\) 0 0
\(839\) 1.64907e10 0.963990 0.481995 0.876174i \(-0.339912\pi\)
0.481995 + 0.876174i \(0.339912\pi\)
\(840\) 0 0
\(841\) −1.17350e10 −0.680297
\(842\) 0 0
\(843\) −1.12158e10 −0.644812
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.49774e9 0.141239
\(848\) 0 0
\(849\) 1.36736e9 0.0766841
\(850\) 0 0
\(851\) 1.89634e10 1.05478
\(852\) 0 0
\(853\) 4.77028e9 0.263162 0.131581 0.991305i \(-0.457995\pi\)
0.131581 + 0.991305i \(0.457995\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.61514e10 −0.876554 −0.438277 0.898840i \(-0.644411\pi\)
−0.438277 + 0.898840i \(0.644411\pi\)
\(858\) 0 0
\(859\) −3.41593e8 −0.0183879 −0.00919397 0.999958i \(-0.502927\pi\)
−0.00919397 + 0.999958i \(0.502927\pi\)
\(860\) 0 0
\(861\) 9.31824e9 0.497534
\(862\) 0 0
\(863\) −6.07878e9 −0.321943 −0.160971 0.986959i \(-0.551463\pi\)
−0.160971 + 0.986959i \(0.551463\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −7.31071e9 −0.380972
\(868\) 0 0
\(869\) 1.77696e10 0.918562
\(870\) 0 0
\(871\) 1.35590e10 0.695289
\(872\) 0 0
\(873\) −5.21290e9 −0.265173
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.23852e10 −0.620020 −0.310010 0.950733i \(-0.600332\pi\)
−0.310010 + 0.950733i \(0.600332\pi\)
\(878\) 0 0
\(879\) 2.01883e9 0.100262
\(880\) 0 0
\(881\) −1.37801e10 −0.678949 −0.339475 0.940615i \(-0.610249\pi\)
−0.339475 + 0.940615i \(0.610249\pi\)
\(882\) 0 0
\(883\) −1.89296e9 −0.0925292 −0.0462646 0.998929i \(-0.514732\pi\)
−0.0462646 + 0.998929i \(0.514732\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.23912e9 −0.203959 −0.101979 0.994787i \(-0.532518\pi\)
−0.101979 + 0.994787i \(0.532518\pi\)
\(888\) 0 0
\(889\) 6.88633e8 0.0328724
\(890\) 0 0
\(891\) 2.02585e9 0.0959480
\(892\) 0 0
\(893\) 1.83419e10 0.861913
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.83864e10 0.850598
\(898\) 0 0
\(899\) −2.04725e10 −0.939751
\(900\) 0 0
\(901\) −1.48605e10 −0.676855
\(902\) 0 0
\(903\) −3.34697e9 −0.151267
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 9.51367e9 0.423372 0.211686 0.977338i \(-0.432105\pi\)
0.211686 + 0.977338i \(0.432105\pi\)
\(908\) 0 0
\(909\) −6.40334e9 −0.282770
\(910\) 0 0
\(911\) −1.16235e10 −0.509359 −0.254680 0.967025i \(-0.581970\pi\)
−0.254680 + 0.967025i \(0.581970\pi\)
\(912\) 0 0
\(913\) 2.18399e10 0.949736
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.93861e10 −0.830229
\(918\) 0 0
\(919\) −9.22943e9 −0.392257 −0.196128 0.980578i \(-0.562837\pi\)
−0.196128 + 0.980578i \(0.562837\pi\)
\(920\) 0 0
\(921\) 2.30196e9 0.0970931
\(922\) 0 0
\(923\) 3.36172e10 1.40720
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 5.84374e9 0.240942
\(928\) 0 0
\(929\) −4.09353e10 −1.67511 −0.837555 0.546353i \(-0.816016\pi\)
−0.837555 + 0.546353i \(0.816016\pi\)
\(930\) 0 0
\(931\) 2.18174e10 0.886094
\(932\) 0 0
\(933\) −2.53565e10 −1.02212
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.75085e9 0.0695281 0.0347641 0.999396i \(-0.488932\pi\)
0.0347641 + 0.999396i \(0.488932\pi\)
\(938\) 0 0
\(939\) −9.26207e9 −0.365072
\(940\) 0 0
\(941\) 5.91102e9 0.231259 0.115630 0.993292i \(-0.463111\pi\)
0.115630 + 0.993292i \(0.463111\pi\)
\(942\) 0 0
\(943\) 4.87058e10 1.89143
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.26089e10 0.865077 0.432538 0.901616i \(-0.357618\pi\)
0.432538 + 0.901616i \(0.357618\pi\)
\(948\) 0 0
\(949\) 4.53675e10 1.72311
\(950\) 0 0
\(951\) −2.80692e10 −1.05827
\(952\) 0 0
\(953\) −1.11773e10 −0.418322 −0.209161 0.977881i \(-0.567073\pi\)
−0.209161 + 0.977881i \(0.567073\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −7.64334e9 −0.281898
\(958\) 0 0
\(959\) 3.84343e9 0.140719
\(960\) 0 0
\(961\) 4.84868e10 1.76235
\(962\) 0 0
\(963\) 3.75525e9 0.135502
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.55518e10 −0.553078 −0.276539 0.961003i \(-0.589188\pi\)
−0.276539 + 0.961003i \(0.589188\pi\)
\(968\) 0 0
\(969\) −2.69936e10 −0.953075
\(970\) 0 0
\(971\) 8.34508e9 0.292525 0.146263 0.989246i \(-0.453276\pi\)
0.146263 + 0.989246i \(0.453276\pi\)
\(972\) 0 0
\(973\) −2.68291e9 −0.0933709
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.85180e9 0.337975 0.168988 0.985618i \(-0.445950\pi\)
0.168988 + 0.985618i \(0.445950\pi\)
\(978\) 0 0
\(979\) 4.57193e10 1.55726
\(980\) 0 0
\(981\) −1.75893e10 −0.594850
\(982\) 0 0
\(983\) −3.70884e10 −1.24538 −0.622688 0.782470i \(-0.713959\pi\)
−0.622688 + 0.782470i \(0.713959\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.51551e9 −0.215694
\(988\) 0 0
\(989\) −1.74944e10 −0.575057
\(990\) 0 0
\(991\) 6.43526e9 0.210043 0.105022 0.994470i \(-0.466509\pi\)
0.105022 + 0.994470i \(0.466509\pi\)
\(992\) 0 0
\(993\) −2.97059e10 −0.962765
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.23071e10 0.393299 0.196650 0.980474i \(-0.436994\pi\)
0.196650 + 0.980474i \(0.436994\pi\)
\(998\) 0 0
\(999\) −5.24768e9 −0.166528
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.a.1.1 1
5.2 odd 4 600.8.f.d.49.2 2
5.3 odd 4 600.8.f.d.49.1 2
5.4 even 2 24.8.a.c.1.1 1
15.14 odd 2 72.8.a.b.1.1 1
20.19 odd 2 48.8.a.c.1.1 1
40.19 odd 2 192.8.a.k.1.1 1
40.29 even 2 192.8.a.c.1.1 1
60.59 even 2 144.8.a.d.1.1 1
120.29 odd 2 576.8.a.t.1.1 1
120.59 even 2 576.8.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.8.a.c.1.1 1 5.4 even 2
48.8.a.c.1.1 1 20.19 odd 2
72.8.a.b.1.1 1 15.14 odd 2
144.8.a.d.1.1 1 60.59 even 2
192.8.a.c.1.1 1 40.29 even 2
192.8.a.k.1.1 1 40.19 odd 2
576.8.a.s.1.1 1 120.59 even 2
576.8.a.t.1.1 1 120.29 odd 2
600.8.a.a.1.1 1 1.1 even 1 trivial
600.8.f.d.49.1 2 5.3 odd 4
600.8.f.d.49.2 2 5.2 odd 4