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Properties

Label 600.8.a.d.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 120)
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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} +776.000 q^{7} +729.000 q^{9} +612.000 q^{11} +4506.00 q^{13} +31502.0 q^{17} +14812.0 q^{19} -20952.0 q^{21} +71768.0 q^{23} -19683.0 q^{27} +53142.0 q^{29} -13920.0 q^{31} -16524.0 q^{33} +66930.0 q^{37} -121662. q^{39} -145958. q^{41} +281404. q^{43} +635440. q^{47} -221367. q^{49} -850554. q^{51} +792770. q^{53} -399924. q^{57} +1.85068e6 q^{59} -1.73678e6 q^{61} +565704. q^{63} +661204. q^{67} -1.93774e6 q^{69} -3.67130e6 q^{71} +5.45274e6 q^{73} +474912. q^{77} -3.08571e6 q^{79} +531441. q^{81} -4.80899e6 q^{83} -1.43483e6 q^{87} -9.54377e6 q^{89} +3.49666e6 q^{91} +375840. q^{93} +1.00541e6 q^{97} +446148. q^{99} +O(q^{100})\)

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
\(7\) 776.000 0.855103 0.427552 − 0.903991i \(-0.359376\pi\)
0.427552 + 0.903991i \(0.359376\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 612.000 0.138636 0.0693182 − 0.997595i \(-0.477918\pi\)
0.0693182 + 0.997595i \(0.477918\pi\)
\(12\) 0 0
\(13\) 4506.00 0.568839 0.284420 − 0.958700i \(-0.408199\pi\)
0.284420 + 0.958700i \(0.408199\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 31502.0 1.55513 0.777565 − 0.628802i \(-0.216454\pi\)
0.777565 + 0.628802i \(0.216454\pi\)
\(18\) 0 0
\(19\) 14812.0 0.495423 0.247711 − 0.968834i \(-0.420322\pi\)
0.247711 + 0.968834i \(0.420322\pi\)
\(20\) 0 0
\(21\) −20952.0 −0.493694
\(22\) 0 0
\(23\) 71768.0 1.22994 0.614969 − 0.788551i \(-0.289168\pi\)
0.614969 + 0.788551i \(0.289168\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) 53142.0 0.404618 0.202309 − 0.979322i \(-0.435156\pi\)
0.202309 + 0.979322i \(0.435156\pi\)
\(30\) 0 0
\(31\) −13920.0 −0.0839215 −0.0419608 − 0.999119i \(-0.513360\pi\)
−0.0419608 + 0.999119i \(0.513360\pi\)
\(32\) 0 0
\(33\) −16524.0 −0.0800417
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 66930.0 0.217227 0.108614 − 0.994084i \(-0.465359\pi\)
0.108614 + 0.994084i \(0.465359\pi\)
\(38\) 0 0
\(39\) −121662. −0.328419
\(40\) 0 0
\(41\) −145958. −0.330738 −0.165369 − 0.986232i \(-0.552882\pi\)
−0.165369 + 0.986232i \(0.552882\pi\)
\(42\) 0 0
\(43\) 281404. 0.539748 0.269874 − 0.962896i \(-0.413018\pi\)
0.269874 + 0.962896i \(0.413018\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 635440. 0.892754 0.446377 − 0.894845i \(-0.352714\pi\)
0.446377 + 0.894845i \(0.352714\pi\)
\(48\) 0 0
\(49\) −221367. −0.268798
\(50\) 0 0
\(51\) −850554. −0.897855
\(52\) 0 0
\(53\) 792770. 0.731445 0.365722 − 0.930724i \(-0.380822\pi\)
0.365722 + 0.930724i \(0.380822\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −399924. −0.286033
\(58\) 0 0
\(59\) 1.85068e6 1.17314 0.586568 − 0.809900i \(-0.300479\pi\)
0.586568 + 0.809900i \(0.300479\pi\)
\(60\) 0 0
\(61\) −1.73678e6 −0.979693 −0.489846 − 0.871809i \(-0.662947\pi\)
−0.489846 + 0.871809i \(0.662947\pi\)
\(62\) 0 0
\(63\) 565704. 0.285034
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 661204. 0.268580 0.134290 − 0.990942i \(-0.457125\pi\)
0.134290 + 0.990942i \(0.457125\pi\)
\(68\) 0 0
\(69\) −1.93774e6 −0.710105
\(70\) 0 0
\(71\) −3.67130e6 −1.21735 −0.608676 − 0.793419i \(-0.708299\pi\)
−0.608676 + 0.793419i \(0.708299\pi\)
\(72\) 0 0
\(73\) 5.45274e6 1.64053 0.820266 − 0.571982i \(-0.193825\pi\)
0.820266 + 0.571982i \(0.193825\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 474912. 0.118548
\(78\) 0 0
\(79\) −3.08571e6 −0.704143 −0.352071 − 0.935973i \(-0.614523\pi\)
−0.352071 + 0.935973i \(0.614523\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −4.80899e6 −0.923167 −0.461584 − 0.887097i \(-0.652719\pi\)
−0.461584 + 0.887097i \(0.652719\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.43483e6 −0.233606
\(88\) 0 0
\(89\) −9.54377e6 −1.43501 −0.717505 − 0.696554i \(-0.754716\pi\)
−0.717505 + 0.696554i \(0.754716\pi\)
\(90\) 0 0
\(91\) 3.49666e6 0.486416
\(92\) 0 0
\(93\) 375840. 0.0484521
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00541e6 0.111851 0.0559256 − 0.998435i \(-0.482189\pi\)
0.0559256 + 0.998435i \(0.482189\pi\)
\(98\) 0 0
\(99\) 446148. 0.0462121
\(100\) 0 0
\(101\) 2.73307e6 0.263953 0.131976 − 0.991253i \(-0.457868\pi\)
0.131976 + 0.991253i \(0.457868\pi\)
\(102\) 0 0
\(103\) −1.49125e7 −1.34469 −0.672344 − 0.740239i \(-0.734712\pi\)
−0.672344 + 0.740239i \(0.734712\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.27451e7 −1.79492 −0.897459 − 0.441098i \(-0.854589\pi\)
−0.897459 + 0.441098i \(0.854589\pi\)
\(108\) 0 0
\(109\) 1.63981e7 1.21283 0.606416 − 0.795148i \(-0.292607\pi\)
0.606416 + 0.795148i \(0.292607\pi\)
\(110\) 0 0
\(111\) −1.80711e6 −0.125416
\(112\) 0 0
\(113\) 2.50597e6 0.163381 0.0816903 − 0.996658i \(-0.473968\pi\)
0.0816903 + 0.996658i \(0.473968\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.28487e6 0.189613
\(118\) 0 0
\(119\) 2.44456e7 1.32980
\(120\) 0 0
\(121\) −1.91126e7 −0.980780
\(122\) 0 0
\(123\) 3.94087e6 0.190952
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.74822e6 0.205692 0.102846 − 0.994697i \(-0.467205\pi\)
0.102846 + 0.994697i \(0.467205\pi\)
\(128\) 0 0
\(129\) −7.59791e6 −0.311623
\(130\) 0 0
\(131\) 2.61554e7 1.01651 0.508255 − 0.861207i \(-0.330291\pi\)
0.508255 + 0.861207i \(0.330291\pi\)
\(132\) 0 0
\(133\) 1.14941e7 0.423638
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.64822e7 −1.87668 −0.938338 − 0.345719i \(-0.887635\pi\)
−0.938338 + 0.345719i \(0.887635\pi\)
\(138\) 0 0
\(139\) 3.48240e7 1.09983 0.549917 − 0.835219i \(-0.314659\pi\)
0.549917 + 0.835219i \(0.314659\pi\)
\(140\) 0 0
\(141\) −1.71569e7 −0.515432
\(142\) 0 0
\(143\) 2.75767e6 0.0788618
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.97691e6 0.155191
\(148\) 0 0
\(149\) 7.85494e6 0.194532 0.0972660 − 0.995258i \(-0.468990\pi\)
0.0972660 + 0.995258i \(0.468990\pi\)
\(150\) 0 0
\(151\) 3.33888e7 0.789189 0.394595 − 0.918855i \(-0.370885\pi\)
0.394595 + 0.918855i \(0.370885\pi\)
\(152\) 0 0
\(153\) 2.29650e7 0.518377
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.18324e7 0.862708 0.431354 − 0.902183i \(-0.358036\pi\)
0.431354 + 0.902183i \(0.358036\pi\)
\(158\) 0 0
\(159\) −2.14048e7 −0.422300
\(160\) 0 0
\(161\) 5.56920e7 1.05172
\(162\) 0 0
\(163\) −6.82785e6 −0.123489 −0.0617444 − 0.998092i \(-0.519666\pi\)
−0.0617444 + 0.998092i \(0.519666\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.51865e7 −0.418465 −0.209233 − 0.977866i \(-0.567097\pi\)
−0.209233 + 0.977866i \(0.567097\pi\)
\(168\) 0 0
\(169\) −4.24445e7 −0.676422
\(170\) 0 0
\(171\) 1.07979e7 0.165141
\(172\) 0 0
\(173\) −1.26325e7 −0.185493 −0.0927463 − 0.995690i \(-0.529565\pi\)
−0.0927463 + 0.995690i \(0.529565\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.99683e7 −0.677310
\(178\) 0 0
\(179\) 8.00473e7 1.04318 0.521592 − 0.853195i \(-0.325338\pi\)
0.521592 + 0.853195i \(0.325338\pi\)
\(180\) 0 0
\(181\) −2.03813e7 −0.255480 −0.127740 − 0.991808i \(-0.540772\pi\)
−0.127740 + 0.991808i \(0.540772\pi\)
\(182\) 0 0
\(183\) 4.68930e7 0.565626
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.92792e7 0.215598
\(188\) 0 0
\(189\) −1.52740e7 −0.164565
\(190\) 0 0
\(191\) 6.61244e7 0.686665 0.343332 − 0.939214i \(-0.388444\pi\)
0.343332 + 0.939214i \(0.388444\pi\)
\(192\) 0 0
\(193\) 1.67406e8 1.67617 0.838087 − 0.545536i \(-0.183674\pi\)
0.838087 + 0.545536i \(0.183674\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.30179e7 0.121314 0.0606569 − 0.998159i \(-0.480680\pi\)
0.0606569 + 0.998159i \(0.480680\pi\)
\(198\) 0 0
\(199\) −1.77313e7 −0.159498 −0.0797488 − 0.996815i \(-0.525412\pi\)
−0.0797488 + 0.996815i \(0.525412\pi\)
\(200\) 0 0
\(201\) −1.78525e7 −0.155065
\(202\) 0 0
\(203\) 4.12382e7 0.345990
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.23189e7 0.409980
\(208\) 0 0
\(209\) 9.06494e6 0.0686836
\(210\) 0 0
\(211\) 7.35024e7 0.538658 0.269329 − 0.963048i \(-0.413198\pi\)
0.269329 + 0.963048i \(0.413198\pi\)
\(212\) 0 0
\(213\) 9.91252e7 0.702838
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.08019e7 −0.0717616
\(218\) 0 0
\(219\) −1.47224e8 −0.947162
\(220\) 0 0
\(221\) 1.41948e8 0.884619
\(222\) 0 0
\(223\) 2.90300e8 1.75300 0.876498 − 0.481406i \(-0.159874\pi\)
0.876498 + 0.481406i \(0.159874\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.47339e8 0.836042 0.418021 − 0.908437i \(-0.362724\pi\)
0.418021 + 0.908437i \(0.362724\pi\)
\(228\) 0 0
\(229\) −2.70407e8 −1.48797 −0.743985 − 0.668196i \(-0.767067\pi\)
−0.743985 + 0.668196i \(0.767067\pi\)
\(230\) 0 0
\(231\) −1.28226e7 −0.0684439
\(232\) 0 0
\(233\) −1.34158e8 −0.694816 −0.347408 − 0.937714i \(-0.612938\pi\)
−0.347408 + 0.937714i \(0.612938\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.33142e7 0.406537
\(238\) 0 0
\(239\) −8.57167e7 −0.406137 −0.203069 − 0.979165i \(-0.565091\pi\)
−0.203069 + 0.979165i \(0.565091\pi\)
\(240\) 0 0
\(241\) 8.19327e7 0.377049 0.188524 − 0.982069i \(-0.439630\pi\)
0.188524 + 0.982069i \(0.439630\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.67429e7 0.281816
\(248\) 0 0
\(249\) 1.29843e8 0.532991
\(250\) 0 0
\(251\) 2.66372e8 1.06324 0.531619 − 0.846984i \(-0.321584\pi\)
0.531619 + 0.846984i \(0.321584\pi\)
\(252\) 0 0
\(253\) 4.39220e7 0.170514
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.75459e8 −1.37974 −0.689868 − 0.723935i \(-0.742332\pi\)
−0.689868 + 0.723935i \(0.742332\pi\)
\(258\) 0 0
\(259\) 5.19377e7 0.185752
\(260\) 0 0
\(261\) 3.87405e7 0.134873
\(262\) 0 0
\(263\) 4.31735e8 1.46343 0.731716 − 0.681609i \(-0.238720\pi\)
0.731716 + 0.681609i \(0.238720\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.57682e8 0.828503
\(268\) 0 0
\(269\) 1.73237e7 0.0542634 0.0271317 − 0.999632i \(-0.491363\pi\)
0.0271317 + 0.999632i \(0.491363\pi\)
\(270\) 0 0
\(271\) 3.38848e8 1.03422 0.517110 − 0.855919i \(-0.327008\pi\)
0.517110 + 0.855919i \(0.327008\pi\)
\(272\) 0 0
\(273\) −9.44097e7 −0.280833
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.15276e8 −1.17397 −0.586986 − 0.809597i \(-0.699686\pi\)
−0.586986 + 0.809597i \(0.699686\pi\)
\(278\) 0 0
\(279\) −1.01477e7 −0.0279738
\(280\) 0 0
\(281\) 4.99229e8 1.34223 0.671116 − 0.741352i \(-0.265815\pi\)
0.671116 + 0.741352i \(0.265815\pi\)
\(282\) 0 0
\(283\) 2.79212e8 0.732287 0.366143 − 0.930558i \(-0.380678\pi\)
0.366143 + 0.930558i \(0.380678\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.13263e8 −0.282815
\(288\) 0 0
\(289\) 5.82037e8 1.41843
\(290\) 0 0
\(291\) −2.71460e7 −0.0645773
\(292\) 0 0
\(293\) 5.56612e8 1.29275 0.646377 − 0.763018i \(-0.276283\pi\)
0.646377 + 0.763018i \(0.276283\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.20460e7 −0.0266806
\(298\) 0 0
\(299\) 3.23387e8 0.699637
\(300\) 0 0
\(301\) 2.18370e8 0.461540
\(302\) 0 0
\(303\) −7.37929e7 −0.152393
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.52711e8 −0.498471 −0.249236 − 0.968443i \(-0.580179\pi\)
−0.249236 + 0.968443i \(0.580179\pi\)
\(308\) 0 0
\(309\) 4.02638e8 0.776355
\(310\) 0 0
\(311\) −1.70383e8 −0.321192 −0.160596 − 0.987020i \(-0.551342\pi\)
−0.160596 + 0.987020i \(0.551342\pi\)
\(312\) 0 0
\(313\) 9.44450e8 1.74090 0.870450 − 0.492257i \(-0.163828\pi\)
0.870450 + 0.492257i \(0.163828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.84654e8 −1.38347 −0.691736 − 0.722150i \(-0.743154\pi\)
−0.691736 + 0.722150i \(0.743154\pi\)
\(318\) 0 0
\(319\) 3.25229e7 0.0560947
\(320\) 0 0
\(321\) 6.14117e8 1.03630
\(322\) 0 0
\(323\) 4.66608e8 0.770447
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4.42748e8 −0.700229
\(328\) 0 0
\(329\) 4.93101e8 0.763397
\(330\) 0 0
\(331\) 5.09538e8 0.772287 0.386143 − 0.922439i \(-0.373807\pi\)
0.386143 + 0.922439i \(0.373807\pi\)
\(332\) 0 0
\(333\) 4.87920e7 0.0724092
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.15903e8 −1.01894 −0.509471 − 0.860488i \(-0.670159\pi\)
−0.509471 + 0.860488i \(0.670159\pi\)
\(338\) 0 0
\(339\) −6.76611e7 −0.0943279
\(340\) 0 0
\(341\) −8.51904e6 −0.0116346
\(342\) 0 0
\(343\) −8.10850e8 −1.08495
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.07047e9 1.37537 0.687687 − 0.726007i \(-0.258626\pi\)
0.687687 + 0.726007i \(0.258626\pi\)
\(348\) 0 0
\(349\) −4.06658e8 −0.512083 −0.256041 − 0.966666i \(-0.582418\pi\)
−0.256041 + 0.966666i \(0.582418\pi\)
\(350\) 0 0
\(351\) −8.86916e7 −0.109473
\(352\) 0 0
\(353\) −6.10099e8 −0.738226 −0.369113 − 0.929385i \(-0.620338\pi\)
−0.369113 + 0.929385i \(0.620338\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.60030e8 −0.767759
\(358\) 0 0
\(359\) −4.52380e8 −0.516028 −0.258014 − 0.966141i \(-0.583068\pi\)
−0.258014 + 0.966141i \(0.583068\pi\)
\(360\) 0 0
\(361\) −6.74476e8 −0.754556
\(362\) 0 0
\(363\) 5.16041e8 0.566254
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.35838e9 −1.43447 −0.717234 − 0.696832i \(-0.754592\pi\)
−0.717234 + 0.696832i \(0.754592\pi\)
\(368\) 0 0
\(369\) −1.06403e8 −0.110246
\(370\) 0 0
\(371\) 6.15190e8 0.625461
\(372\) 0 0
\(373\) 1.78714e9 1.78310 0.891552 − 0.452918i \(-0.149617\pi\)
0.891552 + 0.452918i \(0.149617\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.39458e8 0.230162
\(378\) 0 0
\(379\) 5.95959e8 0.562314 0.281157 − 0.959662i \(-0.409282\pi\)
0.281157 + 0.959662i \(0.409282\pi\)
\(380\) 0 0
\(381\) −1.28202e8 −0.118757
\(382\) 0 0
\(383\) 9.45999e8 0.860389 0.430195 − 0.902736i \(-0.358445\pi\)
0.430195 + 0.902736i \(0.358445\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.05144e8 0.179916
\(388\) 0 0
\(389\) −8.61009e8 −0.741625 −0.370812 − 0.928708i \(-0.620921\pi\)
−0.370812 + 0.928708i \(0.620921\pi\)
\(390\) 0 0
\(391\) 2.26084e9 1.91272
\(392\) 0 0
\(393\) −7.06195e8 −0.586882
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.16109e9 1.73343 0.866715 − 0.498803i \(-0.166227\pi\)
0.866715 + 0.498803i \(0.166227\pi\)
\(398\) 0 0
\(399\) −3.10341e8 −0.244587
\(400\) 0 0
\(401\) −9.34605e8 −0.723807 −0.361904 − 0.932216i \(-0.617873\pi\)
−0.361904 + 0.932216i \(0.617873\pi\)
\(402\) 0 0
\(403\) −6.27235e7 −0.0477378
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.09612e7 0.0301156
\(408\) 0 0
\(409\) 1.43416e9 1.03649 0.518245 − 0.855232i \(-0.326585\pi\)
0.518245 + 0.855232i \(0.326585\pi\)
\(410\) 0 0
\(411\) 1.52502e9 1.08350
\(412\) 0 0
\(413\) 1.43612e9 1.00315
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −9.40249e8 −0.634990
\(418\) 0 0
\(419\) −1.20979e9 −0.803452 −0.401726 − 0.915760i \(-0.631590\pi\)
−0.401726 + 0.915760i \(0.631590\pi\)
\(420\) 0 0
\(421\) −1.98061e9 −1.29363 −0.646816 − 0.762646i \(-0.723900\pi\)
−0.646816 + 0.762646i \(0.723900\pi\)
\(422\) 0 0
\(423\) 4.63236e8 0.297585
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.34774e9 −0.837738
\(428\) 0 0
\(429\) −7.44571e7 −0.0455309
\(430\) 0 0
\(431\) 1.82217e8 0.109627 0.0548135 − 0.998497i \(-0.482544\pi\)
0.0548135 + 0.998497i \(0.482544\pi\)
\(432\) 0 0
\(433\) −9.97172e8 −0.590286 −0.295143 − 0.955453i \(-0.595367\pi\)
−0.295143 + 0.955453i \(0.595367\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.06303e9 0.609340
\(438\) 0 0
\(439\) 3.83798e8 0.216509 0.108255 − 0.994123i \(-0.465474\pi\)
0.108255 + 0.994123i \(0.465474\pi\)
\(440\) 0 0
\(441\) −1.61377e8 −0.0895995
\(442\) 0 0
\(443\) 3.02192e9 1.65147 0.825734 − 0.564060i \(-0.190761\pi\)
0.825734 + 0.564060i \(0.190761\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.12083e8 −0.112313
\(448\) 0 0
\(449\) −1.24787e9 −0.650588 −0.325294 − 0.945613i \(-0.605463\pi\)
−0.325294 + 0.945613i \(0.605463\pi\)
\(450\) 0 0
\(451\) −8.93263e7 −0.0458523
\(452\) 0 0
\(453\) −9.01497e8 −0.455639
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.51650e8 −0.0743254 −0.0371627 − 0.999309i \(-0.511832\pi\)
−0.0371627 + 0.999309i \(0.511832\pi\)
\(458\) 0 0
\(459\) −6.20054e8 −0.299285
\(460\) 0 0
\(461\) −2.15143e9 −1.02276 −0.511380 − 0.859355i \(-0.670866\pi\)
−0.511380 + 0.859355i \(0.670866\pi\)
\(462\) 0 0
\(463\) 2.12703e9 0.995956 0.497978 − 0.867190i \(-0.334076\pi\)
0.497978 + 0.867190i \(0.334076\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.27041e8 −0.194026 −0.0970130 − 0.995283i \(-0.530929\pi\)
−0.0970130 + 0.995283i \(0.530929\pi\)
\(468\) 0 0
\(469\) 5.13094e8 0.229664
\(470\) 0 0
\(471\) −1.12947e9 −0.498084
\(472\) 0 0
\(473\) 1.72219e8 0.0748286
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5.77929e8 0.243815
\(478\) 0 0
\(479\) −3.71866e9 −1.54601 −0.773004 − 0.634401i \(-0.781247\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(480\) 0 0
\(481\) 3.01587e8 0.123567
\(482\) 0 0
\(483\) −1.50368e9 −0.607213
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.21912e9 1.65528 0.827639 − 0.561261i \(-0.189684\pi\)
0.827639 + 0.561261i \(0.189684\pi\)
\(488\) 0 0
\(489\) 1.84352e8 0.0712963
\(490\) 0 0
\(491\) 2.25020e9 0.857897 0.428948 − 0.903329i \(-0.358884\pi\)
0.428948 + 0.903329i \(0.358884\pi\)
\(492\) 0 0
\(493\) 1.67408e9 0.629233
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.84893e9 −1.04096
\(498\) 0 0
\(499\) 4.34769e9 1.56642 0.783208 − 0.621760i \(-0.213582\pi\)
0.783208 + 0.621760i \(0.213582\pi\)
\(500\) 0 0
\(501\) 6.80034e8 0.241601
\(502\) 0 0
\(503\) −3.62193e9 −1.26897 −0.634486 − 0.772935i \(-0.718788\pi\)
−0.634486 + 0.772935i \(0.718788\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.14600e9 0.390532
\(508\) 0 0
\(509\) 1.85018e9 0.621872 0.310936 − 0.950431i \(-0.399357\pi\)
0.310936 + 0.950431i \(0.399357\pi\)
\(510\) 0 0
\(511\) 4.23133e9 1.40283
\(512\) 0 0
\(513\) −2.91545e8 −0.0953442
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.88889e8 0.123768
\(518\) 0 0
\(519\) 3.41076e8 0.107094
\(520\) 0 0
\(521\) 1.76311e9 0.546195 0.273098 − 0.961986i \(-0.411952\pi\)
0.273098 + 0.961986i \(0.411952\pi\)
\(522\) 0 0
\(523\) 1.19055e8 0.0363909 0.0181954 − 0.999834i \(-0.494208\pi\)
0.0181954 + 0.999834i \(0.494208\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.38508e8 −0.130509
\(528\) 0 0
\(529\) 1.74582e9 0.512749
\(530\) 0 0
\(531\) 1.34914e9 0.391045
\(532\) 0 0
\(533\) −6.57687e8 −0.188137
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2.16128e9 −0.602283
\(538\) 0 0
\(539\) −1.35477e8 −0.0372652
\(540\) 0 0
\(541\) −5.07351e9 −1.37758 −0.688791 − 0.724960i \(-0.741858\pi\)
−0.688791 + 0.724960i \(0.741858\pi\)
\(542\) 0 0
\(543\) 5.50295e8 0.147502
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.01425e9 0.787452 0.393726 − 0.919228i \(-0.371186\pi\)
0.393726 + 0.919228i \(0.371186\pi\)
\(548\) 0 0
\(549\) −1.26611e9 −0.326564
\(550\) 0 0
\(551\) 7.87139e8 0.200457
\(552\) 0 0
\(553\) −2.39451e9 −0.602115
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.13732e8 0.224040 0.112020 − 0.993706i \(-0.464268\pi\)
0.112020 + 0.993706i \(0.464268\pi\)
\(558\) 0 0
\(559\) 1.26801e9 0.307030
\(560\) 0 0
\(561\) −5.20539e8 −0.124475
\(562\) 0 0
\(563\) −7.21616e9 −1.70423 −0.852113 − 0.523359i \(-0.824679\pi\)
−0.852113 + 0.523359i \(0.824679\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.12398e8 0.0950115
\(568\) 0 0
\(569\) 5.33850e9 1.21486 0.607430 − 0.794373i \(-0.292200\pi\)
0.607430 + 0.794373i \(0.292200\pi\)
\(570\) 0 0
\(571\) 1.79662e9 0.403860 0.201930 − 0.979400i \(-0.435279\pi\)
0.201930 + 0.979400i \(0.435279\pi\)
\(572\) 0 0
\(573\) −1.78536e9 −0.396446
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.83946e9 −1.26549 −0.632743 − 0.774362i \(-0.718071\pi\)
−0.632743 + 0.774362i \(0.718071\pi\)
\(578\) 0 0
\(579\) −4.51995e9 −0.967740
\(580\) 0 0
\(581\) −3.73177e9 −0.789403
\(582\) 0 0
\(583\) 4.85175e8 0.101405
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.96397e9 1.01297 0.506484 − 0.862249i \(-0.330945\pi\)
0.506484 + 0.862249i \(0.330945\pi\)
\(588\) 0 0
\(589\) −2.06183e8 −0.0415766
\(590\) 0 0
\(591\) −3.51484e8 −0.0700406
\(592\) 0 0
\(593\) −7.25284e9 −1.42829 −0.714146 − 0.699997i \(-0.753185\pi\)
−0.714146 + 0.699997i \(0.753185\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.78744e8 0.0920860
\(598\) 0 0
\(599\) −4.27870e9 −0.813426 −0.406713 − 0.913556i \(-0.633325\pi\)
−0.406713 + 0.913556i \(0.633325\pi\)
\(600\) 0 0
\(601\) −2.36601e9 −0.444586 −0.222293 − 0.974980i \(-0.571354\pi\)
−0.222293 + 0.974980i \(0.571354\pi\)
\(602\) 0 0
\(603\) 4.82018e8 0.0895267
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.36397e9 0.610508 0.305254 − 0.952271i \(-0.401259\pi\)
0.305254 + 0.952271i \(0.401259\pi\)
\(608\) 0 0
\(609\) −1.11343e9 −0.199757
\(610\) 0 0
\(611\) 2.86329e9 0.507834
\(612\) 0 0
\(613\) −5.05211e9 −0.885851 −0.442926 − 0.896558i \(-0.646059\pi\)
−0.442926 + 0.896558i \(0.646059\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.68410e9 −0.460045 −0.230023 − 0.973185i \(-0.573880\pi\)
−0.230023 + 0.973185i \(0.573880\pi\)
\(618\) 0 0
\(619\) −5.59615e9 −0.948357 −0.474178 − 0.880429i \(-0.657255\pi\)
−0.474178 + 0.880429i \(0.657255\pi\)
\(620\) 0 0
\(621\) −1.41261e9 −0.236702
\(622\) 0 0
\(623\) −7.40596e9 −1.22708
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.44753e8 −0.0396545
\(628\) 0 0
\(629\) 2.10843e9 0.337817
\(630\) 0 0
\(631\) −1.02273e10 −1.62054 −0.810269 − 0.586058i \(-0.800679\pi\)
−0.810269 + 0.586058i \(0.800679\pi\)
\(632\) 0 0
\(633\) −1.98456e9 −0.310994
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.97480e8 −0.152903
\(638\) 0 0
\(639\) −2.67638e9 −0.405784
\(640\) 0 0
\(641\) 1.98291e9 0.297371 0.148686 − 0.988885i \(-0.452496\pi\)
0.148686 + 0.988885i \(0.452496\pi\)
\(642\) 0 0
\(643\) 6.56084e9 0.973243 0.486621 − 0.873613i \(-0.338229\pi\)
0.486621 + 0.873613i \(0.338229\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.64785e9 1.25529 0.627643 − 0.778501i \(-0.284020\pi\)
0.627643 + 0.778501i \(0.284020\pi\)
\(648\) 0 0
\(649\) 1.13261e9 0.162639
\(650\) 0 0
\(651\) 2.91652e8 0.0414316
\(652\) 0 0
\(653\) −3.34035e9 −0.469458 −0.234729 − 0.972061i \(-0.575420\pi\)
−0.234729 + 0.972061i \(0.575420\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.97505e9 0.546844
\(658\) 0 0
\(659\) −3.52240e9 −0.479446 −0.239723 − 0.970841i \(-0.577057\pi\)
−0.239723 + 0.970841i \(0.577057\pi\)
\(660\) 0 0
\(661\) −1.17841e10 −1.58705 −0.793526 − 0.608536i \(-0.791757\pi\)
−0.793526 + 0.608536i \(0.791757\pi\)
\(662\) 0 0
\(663\) −3.83260e9 −0.510735
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.81390e9 0.497655
\(668\) 0 0
\(669\) −7.83811e9 −1.01209
\(670\) 0 0
\(671\) −1.06291e9 −0.135821
\(672\) 0 0
\(673\) 1.10399e9 0.139608 0.0698040 − 0.997561i \(-0.477763\pi\)
0.0698040 + 0.997561i \(0.477763\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.58398e10 −1.96196 −0.980980 − 0.194108i \(-0.937819\pi\)
−0.980980 + 0.194108i \(0.937819\pi\)
\(678\) 0 0
\(679\) 7.80195e8 0.0956443
\(680\) 0 0
\(681\) −3.97816e9 −0.482689
\(682\) 0 0
\(683\) 3.57076e8 0.0428833 0.0214416 − 0.999770i \(-0.493174\pi\)
0.0214416 + 0.999770i \(0.493174\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.30100e9 0.859080
\(688\) 0 0
\(689\) 3.57222e9 0.416075
\(690\) 0 0
\(691\) −1.43357e10 −1.65290 −0.826449 − 0.563012i \(-0.809643\pi\)
−0.826449 + 0.563012i \(0.809643\pi\)
\(692\) 0 0
\(693\) 3.46211e8 0.0395161
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.59797e9 −0.514341
\(698\) 0 0
\(699\) 3.62226e9 0.401152
\(700\) 0 0
\(701\) −1.26602e10 −1.38812 −0.694060 − 0.719917i \(-0.744180\pi\)
−0.694060 + 0.719917i \(0.744180\pi\)
\(702\) 0 0
\(703\) 9.91367e8 0.107619
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.12086e9 0.225707
\(708\) 0 0
\(709\) 1.36125e10 1.43442 0.717211 − 0.696856i \(-0.245418\pi\)
0.717211 + 0.696856i \(0.245418\pi\)
\(710\) 0 0
\(711\) −2.24948e9 −0.234714
\(712\) 0 0
\(713\) −9.99011e8 −0.103218
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.31435e9 0.234483
\(718\) 0 0
\(719\) 4.25635e9 0.427057 0.213529 − 0.976937i \(-0.431504\pi\)
0.213529 + 0.976937i \(0.431504\pi\)
\(720\) 0 0
\(721\) −1.15721e10 −1.14985
\(722\) 0 0
\(723\) −2.21218e9 −0.217689
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.57915e10 1.52424 0.762121 − 0.647434i \(-0.224158\pi\)
0.762121 + 0.647434i \(0.224158\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 8.86479e9 0.839378
\(732\) 0 0
\(733\) 1.07903e10 1.01198 0.505988 − 0.862541i \(-0.331128\pi\)
0.505988 + 0.862541i \(0.331128\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.04657e8 0.0372349
\(738\) 0 0
\(739\) −3.87504e9 −0.353200 −0.176600 − 0.984283i \(-0.556510\pi\)
−0.176600 + 0.984283i \(0.556510\pi\)
\(740\) 0 0
\(741\) −1.80206e9 −0.162707
\(742\) 0 0
\(743\) −2.15396e10 −1.92653 −0.963265 − 0.268552i \(-0.913455\pi\)
−0.963265 + 0.268552i \(0.913455\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.50575e9 −0.307722
\(748\) 0 0
\(749\) −1.76502e10 −1.53484
\(750\) 0 0
\(751\) −3.99904e9 −0.344521 −0.172261 − 0.985051i \(-0.555107\pi\)
−0.172261 + 0.985051i \(0.555107\pi\)
\(752\) 0 0
\(753\) −7.19204e9 −0.613860
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.89152e9 −0.326050 −0.163025 − 0.986622i \(-0.552125\pi\)
−0.163025 + 0.986622i \(0.552125\pi\)
\(758\) 0 0
\(759\) −1.18589e9 −0.0984464
\(760\) 0 0
\(761\) −1.13534e10 −0.933856 −0.466928 − 0.884295i \(-0.654639\pi\)
−0.466928 + 0.884295i \(0.654639\pi\)
\(762\) 0 0
\(763\) 1.27249e10 1.03710
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.33915e9 0.667326
\(768\) 0 0
\(769\) 1.12629e10 0.893114 0.446557 − 0.894755i \(-0.352650\pi\)
0.446557 + 0.894755i \(0.352650\pi\)
\(770\) 0 0
\(771\) 1.01374e10 0.796591
\(772\) 0 0
\(773\) −1.24048e10 −0.965962 −0.482981 − 0.875631i \(-0.660446\pi\)
−0.482981 + 0.875631i \(0.660446\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.40232e9 −0.107244
\(778\) 0 0
\(779\) −2.16193e9 −0.163855
\(780\) 0 0
\(781\) −2.24684e9 −0.168769
\(782\) 0 0
\(783\) −1.04599e9 −0.0778687
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.26109e9 0.238479 0.119240 − 0.992865i \(-0.461954\pi\)
0.119240 + 0.992865i \(0.461954\pi\)
\(788\) 0 0
\(789\) −1.16569e10 −0.844913
\(790\) 0 0
\(791\) 1.94463e9 0.139707
\(792\) 0 0
\(793\) −7.82592e9 −0.557288
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.43714e9 −0.170520 −0.0852601 − 0.996359i \(-0.527172\pi\)
−0.0852601 + 0.996359i \(0.527172\pi\)
\(798\) 0 0
\(799\) 2.00176e10 1.38835
\(800\) 0 0
\(801\) −6.95741e9 −0.478336
\(802\) 0 0
\(803\) 3.33708e9 0.227437
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.67740e8 −0.0313290
\(808\) 0 0
\(809\) −1.83409e10 −1.21787 −0.608933 − 0.793221i \(-0.708402\pi\)
−0.608933 + 0.793221i \(0.708402\pi\)
\(810\) 0 0
\(811\) 2.91983e10 1.92214 0.961070 − 0.276306i \(-0.0891103\pi\)
0.961070 + 0.276306i \(0.0891103\pi\)
\(812\) 0 0
\(813\) −9.14890e9 −0.597107
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.16816e9 0.267403
\(818\) 0 0
\(819\) 2.54906e9 0.162139
\(820\) 0 0
\(821\) −2.08184e10 −1.31294 −0.656471 − 0.754352i \(-0.727951\pi\)
−0.656471 + 0.754352i \(0.727951\pi\)
\(822\) 0 0
\(823\) −6.73804e9 −0.421341 −0.210671 − 0.977557i \(-0.567565\pi\)
−0.210671 + 0.977557i \(0.567565\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.90547e10 −1.17147 −0.585736 − 0.810502i \(-0.699194\pi\)
−0.585736 + 0.810502i \(0.699194\pi\)
\(828\) 0 0
\(829\) 2.56278e10 1.56232 0.781162 − 0.624329i \(-0.214627\pi\)
0.781162 + 0.624329i \(0.214627\pi\)
\(830\) 0 0
\(831\) 1.12125e10 0.677793
\(832\) 0 0
\(833\) −6.97350e9 −0.418017
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.73987e8 0.0161507
\(838\) 0 0
\(839\) 1.91353e9 0.111859 0.0559293 − 0.998435i \(-0.482188\pi\)
0.0559293 + 0.998435i \(0.482188\pi\)
\(840\) 0 0
\(841\) −1.44258e10 −0.836284
\(842\) 0 0
\(843\) −1.34792e10 −0.774938
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.48314e10 −0.838668
\(848\) 0 0
\(849\) −7.53872e9 −0.422786
\(850\) 0 0
\(851\) 4.80343e9 0.267176
\(852\) 0 0
\(853\) 6.28076e9 0.346490 0.173245 − 0.984879i \(-0.444575\pi\)
0.173245 + 0.984879i \(0.444575\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.09935e9 0.385288 0.192644 − 0.981269i \(-0.438294\pi\)
0.192644 + 0.981269i \(0.438294\pi\)
\(858\) 0 0
\(859\) 5.72657e9 0.308261 0.154130 − 0.988051i \(-0.450742\pi\)
0.154130 + 0.988051i \(0.450742\pi\)
\(860\) 0 0
\(861\) 3.05811e9 0.163284
\(862\) 0 0
\(863\) −8.14767e9 −0.431515 −0.215757 − 0.976447i \(-0.569222\pi\)
−0.215757 + 0.976447i \(0.569222\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.57150e10 −0.818932
\(868\) 0 0
\(869\) −1.88846e9 −0.0976197
\(870\) 0 0
\(871\) 2.97939e9 0.152779
\(872\) 0 0
\(873\) 7.32941e8 0.0372837
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.01644e10 −0.508844 −0.254422 − 0.967093i \(-0.581885\pi\)
−0.254422 + 0.967093i \(0.581885\pi\)
\(878\) 0 0
\(879\) −1.50285e10 −0.746372
\(880\) 0 0
\(881\) 3.15577e10 1.55485 0.777427 − 0.628974i \(-0.216525\pi\)
0.777427 + 0.628974i \(0.216525\pi\)
\(882\) 0 0
\(883\) 2.83065e10 1.38364 0.691822 − 0.722068i \(-0.256808\pi\)
0.691822 + 0.722068i \(0.256808\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.29779e10 0.624414 0.312207 − 0.950014i \(-0.398932\pi\)
0.312207 + 0.950014i \(0.398932\pi\)
\(888\) 0 0
\(889\) 3.68462e9 0.175888
\(890\) 0 0
\(891\) 3.25242e8 0.0154040
\(892\) 0 0
\(893\) 9.41214e9 0.442291
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −8.73144e9 −0.403936
\(898\) 0 0
\(899\) −7.39737e8 −0.0339561
\(900\) 0 0
\(901\) 2.49738e10 1.13749
\(902\) 0 0
\(903\) −5.89598e9 −0.266470
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.48257e10 1.54980 0.774898 − 0.632086i \(-0.217801\pi\)
0.774898 + 0.632086i \(0.217801\pi\)
\(908\) 0 0
\(909\) 1.99241e9 0.0879842
\(910\) 0 0
\(911\) 2.65554e10 1.16369 0.581846 − 0.813299i \(-0.302331\pi\)
0.581846 + 0.813299i \(0.302331\pi\)
\(912\) 0 0
\(913\) −2.94310e9 −0.127985
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.02966e10 0.869220
\(918\) 0 0
\(919\) 2.92706e10 1.24402 0.622011 − 0.783009i \(-0.286316\pi\)
0.622011 + 0.783009i \(0.286316\pi\)
\(920\) 0 0
\(921\) 6.82320e9 0.287792
\(922\) 0 0
\(923\) −1.65429e10 −0.692477
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.08712e10 −0.448229
\(928\) 0 0
\(929\) −2.90944e10 −1.19057 −0.595284 − 0.803515i \(-0.702960\pi\)
−0.595284 + 0.803515i \(0.702960\pi\)
\(930\) 0 0
\(931\) −3.27889e9 −0.133169
\(932\) 0 0
\(933\) 4.60034e9 0.185440
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.78798e10 −0.710027 −0.355013 − 0.934861i \(-0.615524\pi\)
−0.355013 + 0.934861i \(0.615524\pi\)
\(938\) 0 0
\(939\) −2.55002e10 −1.00511
\(940\) 0 0
\(941\) 2.83901e10 1.11072 0.555358 − 0.831611i \(-0.312581\pi\)
0.555358 + 0.831611i \(0.312581\pi\)
\(942\) 0 0
\(943\) −1.04751e10 −0.406788
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.12148e9 −0.310749 −0.155375 − 0.987856i \(-0.549658\pi\)
−0.155375 + 0.987856i \(0.549658\pi\)
\(948\) 0 0
\(949\) 2.45701e10 0.933199
\(950\) 0 0
\(951\) 2.11856e10 0.798748
\(952\) 0 0
\(953\) 5.32773e10 1.99396 0.996982 − 0.0776343i \(-0.0247366\pi\)
0.996982 + 0.0776343i \(0.0247366\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.78118e8 −0.0323863
\(958\) 0 0
\(959\) −4.38302e10 −1.60475
\(960\) 0 0
\(961\) −2.73188e10 −0.992957
\(962\) 0 0
\(963\) −1.65812e10 −0.598306
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 5.34246e9 0.189998 0.0949989 − 0.995477i \(-0.469715\pi\)
0.0949989 + 0.995477i \(0.469715\pi\)
\(968\) 0 0
\(969\) −1.25984e10 −0.444818
\(970\) 0 0
\(971\) −3.70155e10 −1.29753 −0.648764 − 0.760990i \(-0.724714\pi\)
−0.648764 + 0.760990i \(0.724714\pi\)
\(972\) 0 0
\(973\) 2.70235e10 0.940472
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.25193e10 0.772544 0.386272 − 0.922385i \(-0.373763\pi\)
0.386272 + 0.922385i \(0.373763\pi\)
\(978\) 0 0
\(979\) −5.84078e9 −0.198944
\(980\) 0 0
\(981\) 1.19542e10 0.404277
\(982\) 0 0
\(983\) 4.33313e10 1.45501 0.727503 − 0.686105i \(-0.240681\pi\)
0.727503 + 0.686105i \(0.240681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.33137e10 −0.440748
\(988\) 0 0
\(989\) 2.01958e10 0.663856
\(990\) 0 0
\(991\) −1.16405e10 −0.379937 −0.189969 − 0.981790i \(-0.560839\pi\)
−0.189969 + 0.981790i \(0.560839\pi\)
\(992\) 0 0
\(993\) −1.37575e10 −0.445880
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.18333e10 1.01730 0.508650 − 0.860974i \(-0.330145\pi\)
0.508650 + 0.860974i \(0.330145\pi\)
\(998\) 0 0
\(999\) −1.31738e9 −0.0418055
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.d.1.1 1
5.2 odd 4 600.8.f.b.49.2 2
5.3 odd 4 600.8.f.b.49.1 2
5.4 even 2 120.8.a.b.1.1 ✓ 1
15.14 odd 2 360.8.a.a.1.1 1
20.19 odd 2 240.8.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.8.a.b.1.1 ✓ 1 5.4 even 2
240.8.a.f.1.1 1 20.19 odd 2
360.8.a.a.1.1 1 15.14 odd 2
600.8.a.d.1.1 1 1.1 even 1 trivial
600.8.f.b.49.1 2 5.3 odd 4
600.8.f.b.49.2 2 5.2 odd 4