Properties

Label 68.9.d.a.67.1
Level $68$
Weight $9$
Character 68.67
Self dual yes
Analytic conductor $27.702$
Analytic rank $0$
Dimension $1$
CM discriminant -68
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,9,Mod(67,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.67");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 68.d (of order \(2\), degree \(1\), minimal)

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: \(27.7017454842\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 67.1
Character \(\chi\) \(=\) 68.67

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} -94.0000 q^{3} +256.000 q^{4} -1504.00 q^{6} +706.000 q^{7} +4096.00 q^{8} +2275.00 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} -94.0000 q^{3} +256.000 q^{4} -1504.00 q^{6} +706.000 q^{7} +4096.00 q^{8} +2275.00 q^{9} -13982.0 q^{11} -24064.0 q^{12} +17954.0 q^{13} +11296.0 q^{14} +65536.0 q^{16} +83521.0 q^{17} +36400.0 q^{18} -66364.0 q^{21} -223712. q^{22} +190018. q^{23} -385024. q^{24} +390625. q^{25} +287264. q^{26} +402884. q^{27} +180736. q^{28} +1.78151e6 q^{31} +1.04858e6 q^{32} +1.31431e6 q^{33} +1.33634e6 q^{34} +582400. q^{36} -1.68768e6 q^{39} -1.06182e6 q^{42} -3.57939e6 q^{44} +3.04029e6 q^{46} -6.16038e6 q^{48} -5.26636e6 q^{49} +6.25000e6 q^{50} -7.85097e6 q^{51} +4.59622e6 q^{52} +1.64131e6 q^{53} +6.44614e6 q^{54} +2.89178e6 q^{56} +2.85041e7 q^{62} +1.60615e6 q^{63} +1.67772e7 q^{64} +2.10289e7 q^{66} +2.13814e7 q^{68} -1.78617e7 q^{69} -1.16499e7 q^{71} +9.31840e6 q^{72} -3.67188e7 q^{75} -9.87129e6 q^{77} -2.70028e7 q^{78} +7.08458e7 q^{79} -5.27974e7 q^{81} -1.69892e7 q^{84} -5.72703e7 q^{88} -1.25191e8 q^{89} +1.26755e7 q^{91} +4.86446e7 q^{92} -1.67462e8 q^{93} -9.85661e7 q^{96} -8.42618e7 q^{98} -3.18091e7 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 1.00000
\(3\) −94.0000 −1.16049 −0.580247 0.814441i \(-0.697044\pi\)
−0.580247 + 0.814441i \(0.697044\pi\)
\(4\) 256.000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1504.00 −1.16049
\(7\) 706.000 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(8\) 4096.00 1.00000
\(9\) 2275.00 0.346746
\(10\) 0 0
\(11\) −13982.0 −0.954989 −0.477495 0.878635i \(-0.658455\pi\)
−0.477495 + 0.878635i \(0.658455\pi\)
\(12\) −24064.0 −1.16049
\(13\) 17954.0 0.628619 0.314310 0.949320i \(-0.398227\pi\)
0.314310 + 0.949320i \(0.398227\pi\)
\(14\) 11296.0 0.294044
\(15\) 0 0
\(16\) 65536.0 1.00000
\(17\) 83521.0 1.00000
\(18\) 36400.0 0.346746
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −66364.0 −0.341236
\(22\) −223712. −0.954989
\(23\) 190018. 0.679021 0.339511 0.940602i \(-0.389739\pi\)
0.339511 + 0.940602i \(0.389739\pi\)
\(24\) −385024. −1.16049
\(25\) 390625. 1.00000
\(26\) 287264. 0.628619
\(27\) 402884. 0.758097
\(28\) 180736. 0.294044
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.78151e6 1.92904 0.964518 0.264016i \(-0.0850471\pi\)
0.964518 + 0.264016i \(0.0850471\pi\)
\(32\) 1.04858e6 1.00000
\(33\) 1.31431e6 1.10826
\(34\) 1.33634e6 1.00000
\(35\) 0 0
\(36\) 582400. 0.346746
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −1.68768e6 −0.729509
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −1.06182e6 −0.341236
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −3.57939e6 −0.954989
\(45\) 0 0
\(46\) 3.04029e6 0.679021
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −6.16038e6 −1.16049
\(49\) −5.26636e6 −0.913538
\(50\) 6.25000e6 1.00000
\(51\) −7.85097e6 −1.16049
\(52\) 4.59622e6 0.628619
\(53\) 1.64131e6 0.208012 0.104006 0.994577i \(-0.466834\pi\)
0.104006 + 0.994577i \(0.466834\pi\)
\(54\) 6.44614e6 0.758097
\(55\) 0 0
\(56\) 2.89178e6 0.294044
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.85041e7 1.92904
\(63\) 1.60615e6 0.101959
\(64\) 1.67772e7 1.00000
\(65\) 0 0
\(66\) 2.10289e7 1.10826
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.13814e7 1.00000
\(69\) −1.78617e7 −0.788000
\(70\) 0 0
\(71\) −1.16499e7 −0.458445 −0.229222 0.973374i \(-0.573618\pi\)
−0.229222 + 0.973374i \(0.573618\pi\)
\(72\) 9.31840e6 0.346746
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −3.67188e7 −1.16049
\(76\) 0 0
\(77\) −9.87129e6 −0.280809
\(78\) −2.70028e7 −0.729509
\(79\) 7.08458e7 1.81889 0.909444 0.415827i \(-0.136508\pi\)
0.909444 + 0.415827i \(0.136508\pi\)
\(80\) 0 0
\(81\) −5.27974e7 −1.22651
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.69892e7 −0.341236
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −5.72703e7 −0.954989
\(89\) −1.25191e8 −1.99532 −0.997659 0.0683856i \(-0.978215\pi\)
−0.997659 + 0.0683856i \(0.978215\pi\)
\(90\) 0 0
\(91\) 1.26755e7 0.184842
\(92\) 4.86446e7 0.679021
\(93\) −1.67462e8 −2.23864
\(94\) 0 0
\(95\) 0 0
\(96\) −9.85661e7 −1.16049
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −8.42618e7 −0.913538
\(99\) −3.18091e7 −0.331139
\(100\) 1.00000e8 1.00000
\(101\) −1.16229e8 −1.11694 −0.558471 0.829524i \(-0.688612\pi\)
−0.558471 + 0.829524i \(0.688612\pi\)
\(102\) −1.25616e8 −1.16049
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 7.35396e7 0.628619
\(105\) 0 0
\(106\) 2.62610e7 0.208012
\(107\) 1.66589e8 1.27090 0.635450 0.772142i \(-0.280815\pi\)
0.635450 + 0.772142i \(0.280815\pi\)
\(108\) 1.03138e8 0.758097
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.62684e7 0.294044
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.08454e7 0.217971
\(118\) 0 0
\(119\) 5.89658e7 0.294044
\(120\) 0 0
\(121\) −1.88626e7 −0.0879952
\(122\) 0 0
\(123\) 0 0
\(124\) 4.56066e8 1.92904
\(125\) 0 0
\(126\) 2.56984e7 0.101959
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 2.68435e8 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −4.77414e8 −1.62110 −0.810551 0.585668i \(-0.800832\pi\)
−0.810551 + 0.585668i \(0.800832\pi\)
\(132\) 3.36463e8 1.10826
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 3.42102e8 1.00000
\(137\) 6.28721e8 1.78474 0.892372 0.451300i \(-0.149040\pi\)
0.892372 + 0.451300i \(0.149040\pi\)
\(138\) −2.85787e8 −0.788000
\(139\) −5.83021e8 −1.56180 −0.780900 0.624657i \(-0.785239\pi\)
−0.780900 + 0.624657i \(0.785239\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.86398e8 −0.458445
\(143\) −2.51033e8 −0.600325
\(144\) 1.49094e8 0.346746
\(145\) 0 0
\(146\) 0 0
\(147\) 4.95038e8 1.06016
\(148\) 0 0
\(149\) 9.26194e8 1.87913 0.939565 0.342369i \(-0.111229\pi\)
0.939565 + 0.342369i \(0.111229\pi\)
\(150\) −5.87500e8 −1.16049
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1.90010e8 0.346746
\(154\) −1.57941e8 −0.280809
\(155\) 0 0
\(156\) −4.32045e8 −0.729509
\(157\) 3.45578e8 0.568784 0.284392 0.958708i \(-0.408208\pi\)
0.284392 + 0.958708i \(0.408208\pi\)
\(158\) 1.13353e9 1.81889
\(159\) −1.54284e8 −0.241397
\(160\) 0 0
\(161\) 1.34153e8 0.199662
\(162\) −8.44758e8 −1.22651
\(163\) 5.99802e8 0.849683 0.424842 0.905268i \(-0.360330\pi\)
0.424842 + 0.905268i \(0.360330\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.18202e9 1.51971 0.759853 0.650095i \(-0.225271\pi\)
0.759853 + 0.650095i \(0.225271\pi\)
\(168\) −2.71827e8 −0.341236
\(169\) −4.93385e8 −0.604838
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 2.75781e8 0.294044
\(176\) −9.16324e8 −0.954989
\(177\) 0 0
\(178\) −2.00305e9 −1.99532
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 2.02808e8 0.184842
\(183\) 0 0
\(184\) 7.78314e8 0.679021
\(185\) 0 0
\(186\) −2.67939e9 −2.23864
\(187\) −1.16779e9 −0.954989
\(188\) 0 0
\(189\) 2.84436e8 0.222914
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.57706e9 −1.16049
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.34819e9 −0.913538
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −5.08945e8 −0.331139
\(199\) 2.65036e9 1.69003 0.845013 0.534746i \(-0.179593\pi\)
0.845013 + 0.534746i \(0.179593\pi\)
\(200\) 1.60000e9 1.00000
\(201\) 0 0
\(202\) −1.85967e9 −1.11694
\(203\) 0 0
\(204\) −2.00985e9 −1.16049
\(205\) 0 0
\(206\) 0 0
\(207\) 4.32291e8 0.235448
\(208\) 1.17663e9 0.628619
\(209\) 0 0
\(210\) 0 0
\(211\) −3.67033e9 −1.85172 −0.925859 0.377869i \(-0.876657\pi\)
−0.925859 + 0.377869i \(0.876657\pi\)
\(212\) 4.20176e8 0.208012
\(213\) 1.09509e9 0.532022
\(214\) 2.66542e9 1.27090
\(215\) 0 0
\(216\) 1.65021e9 0.758097
\(217\) 1.25774e9 0.567222
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.49954e9 0.628619
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 7.40295e8 0.294044
\(225\) 8.88672e8 0.346746
\(226\) 0 0
\(227\) −5.30347e9 −1.99736 −0.998681 0.0513528i \(-0.983647\pi\)
−0.998681 + 0.0513528i \(0.983647\pi\)
\(228\) 0 0
\(229\) 1.08994e8 0.0396334 0.0198167 0.999804i \(-0.493692\pi\)
0.0198167 + 0.999804i \(0.493692\pi\)
\(230\) 0 0
\(231\) 9.27901e8 0.325877
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 6.53526e8 0.217971
\(235\) 0 0
\(236\) 0 0
\(237\) −6.65951e9 −2.11081
\(238\) 9.43453e8 0.294044
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −3.01801e8 −0.0879952
\(243\) 2.31963e9 0.665264
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 7.29705e9 1.92904
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 4.11174e8 0.101959
\(253\) −2.65683e9 −0.648458
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) −8.33461e8 −0.191053 −0.0955263 0.995427i \(-0.530453\pi\)
−0.0955263 + 0.995427i \(0.530453\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −7.63863e9 −1.62110
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 5.38341e9 1.10826
\(265\) 0 0
\(266\) 0 0
\(267\) 1.17679e10 2.31555
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 5.47363e9 1.00000
\(273\) −1.19150e9 −0.214508
\(274\) 1.00595e10 1.78474
\(275\) −5.46172e9 −0.954989
\(276\) −4.57259e9 −0.788000
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −9.32834e9 −1.56180
\(279\) 4.05293e9 0.668886
\(280\) 0 0
\(281\) 1.63283e9 0.261889 0.130944 0.991390i \(-0.458199\pi\)
0.130944 + 0.991390i \(0.458199\pi\)
\(282\) 0 0
\(283\) −1.26845e10 −1.97756 −0.988778 0.149390i \(-0.952269\pi\)
−0.988778 + 0.149390i \(0.952269\pi\)
\(284\) −2.98236e9 −0.458445
\(285\) 0 0
\(286\) −4.01653e9 −0.600325
\(287\) 0 0
\(288\) 2.38551e9 0.346746
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.39620e9 −1.27492 −0.637458 0.770485i \(-0.720014\pi\)
−0.637458 + 0.770485i \(0.720014\pi\)
\(294\) 7.92061e9 1.06016
\(295\) 0 0
\(296\) 0 0
\(297\) −5.63312e9 −0.723975
\(298\) 1.48191e10 1.87913
\(299\) 3.41158e9 0.426846
\(300\) −9.40000e9 −1.16049
\(301\) 0 0
\(302\) 0 0
\(303\) 1.09256e10 1.29620
\(304\) 0 0
\(305\) 0 0
\(306\) 3.04016e9 0.346746
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −2.52705e9 −0.280809
\(309\) 0 0
\(310\) 0 0
\(311\) 2.71676e9 0.290409 0.145204 0.989402i \(-0.453616\pi\)
0.145204 + 0.989402i \(0.453616\pi\)
\(312\) −6.91272e9 −0.729509
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 5.52924e9 0.568784
\(315\) 0 0
\(316\) 1.81365e10 1.81889
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) −2.46854e9 −0.241397
\(319\) 0 0
\(320\) 0 0
\(321\) −1.56594e10 −1.47487
\(322\) 2.14644e9 0.199662
\(323\) 0 0
\(324\) −1.35161e10 −1.22651
\(325\) 7.01328e9 0.628619
\(326\) 9.59682e9 0.849683
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.89123e10 1.51971
\(335\) 0 0
\(336\) −4.34923e9 −0.341236
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −7.89415e9 −0.604838
\(339\) 0 0
\(340\) 0 0
\(341\) −2.49090e10 −1.84221
\(342\) 0 0
\(343\) −7.78800e9 −0.562665
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.89187e10 −1.99462 −0.997310 0.0732981i \(-0.976648\pi\)
−0.997310 + 0.0732981i \(0.976648\pi\)
\(348\) 0 0
\(349\) 2.61223e10 1.76080 0.880400 0.474233i \(-0.157274\pi\)
0.880400 + 0.474233i \(0.157274\pi\)
\(350\) 4.41250e9 0.294044
\(351\) 7.23338e9 0.476555
\(352\) −1.46612e10 −0.954989
\(353\) −2.03852e10 −1.31285 −0.656425 0.754391i \(-0.727932\pi\)
−0.656425 + 0.754391i \(0.727932\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.20488e10 −1.99532
\(357\) −5.54279e9 −0.341236
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.69836e10 1.00000
\(362\) 0 0
\(363\) 1.77308e9 0.102118
\(364\) 3.24493e9 0.184842
\(365\) 0 0
\(366\) 0 0
\(367\) −3.62639e10 −1.99899 −0.999495 0.0317713i \(-0.989885\pi\)
−0.999495 + 0.0317713i \(0.989885\pi\)
\(368\) 1.24530e10 0.679021
\(369\) 0 0
\(370\) 0 0
\(371\) 1.15877e9 0.0611647
\(372\) −4.28702e10 −2.23864
\(373\) 1.35835e10 0.701742 0.350871 0.936424i \(-0.385886\pi\)
0.350871 + 0.936424i \(0.385886\pi\)
\(374\) −1.86846e10 −0.954989
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 4.55098e9 0.222914
\(379\) 3.58744e10 1.73871 0.869356 0.494186i \(-0.164534\pi\)
0.869356 + 0.494186i \(0.164534\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −2.52329e10 −1.16049
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.94188e10 −1.72149 −0.860746 0.509035i \(-0.830002\pi\)
−0.860746 + 0.509035i \(0.830002\pi\)
\(390\) 0 0
\(391\) 1.58705e10 0.679021
\(392\) −2.15710e10 −0.913538
\(393\) 4.48770e10 1.88128
\(394\) 0 0
\(395\) 0 0
\(396\) −8.14312e9 −0.331139
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 4.24058e10 1.69003
\(399\) 0 0
\(400\) 2.56000e10 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 3.19852e10 1.21263
\(404\) −2.97547e10 −1.11694
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −3.21576e10 −1.16049
\(409\) 2.93717e10 1.04963 0.524815 0.851216i \(-0.324134\pi\)
0.524815 + 0.851216i \(0.324134\pi\)
\(410\) 0 0
\(411\) −5.90998e10 −2.07118
\(412\) 0 0
\(413\) 0 0
\(414\) 6.91666e9 0.235448
\(415\) 0 0
\(416\) 1.88261e10 0.628619
\(417\) 5.48040e10 1.81246
\(418\) 0 0
\(419\) 4.23423e10 1.37378 0.686892 0.726759i \(-0.258974\pi\)
0.686892 + 0.726759i \(0.258974\pi\)
\(420\) 0 0
\(421\) 4.24676e10 1.35185 0.675926 0.736969i \(-0.263744\pi\)
0.675926 + 0.736969i \(0.263744\pi\)
\(422\) −5.87252e10 −1.85172
\(423\) 0 0
\(424\) 6.72282e9 0.208012
\(425\) 3.26254e10 1.00000
\(426\) 1.75214e10 0.532022
\(427\) 0 0
\(428\) 4.26468e10 1.27090
\(429\) 2.35971e10 0.696673
\(430\) 0 0
\(431\) 6.50925e10 1.88635 0.943175 0.332298i \(-0.107824\pi\)
0.943175 + 0.332298i \(0.107824\pi\)
\(432\) 2.64034e10 0.758097
\(433\) 7.01123e10 1.99454 0.997270 0.0738399i \(-0.0235254\pi\)
0.997270 + 0.0738399i \(0.0235254\pi\)
\(434\) 2.01239e10 0.567222
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 3.25623e10 0.876711 0.438355 0.898802i \(-0.355561\pi\)
0.438355 + 0.898802i \(0.355561\pi\)
\(440\) 0 0
\(441\) −1.19810e10 −0.316766
\(442\) 2.39926e10 0.628619
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −8.70623e10 −2.18072
\(448\) 1.18447e10 0.294044
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 1.42188e10 0.346746
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −8.48555e10 −1.99736
\(455\) 0 0
\(456\) 0 0
\(457\) −6.53892e10 −1.49914 −0.749568 0.661927i \(-0.769739\pi\)
−0.749568 + 0.661927i \(0.769739\pi\)
\(458\) 1.74391e9 0.0396334
\(459\) 3.36493e10 0.758097
\(460\) 0 0
\(461\) −5.78432e10 −1.28070 −0.640351 0.768082i \(-0.721211\pi\)
−0.640351 + 0.768082i \(0.721211\pi\)
\(462\) 1.48464e10 0.325877
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.04564e10 0.217971
\(469\) 0 0
\(470\) 0 0
\(471\) −3.24843e10 −0.660070
\(472\) 0 0
\(473\) 0 0
\(474\) −1.06552e11 −2.11081
\(475\) 0 0
\(476\) 1.50953e10 0.294044
\(477\) 3.73399e9 0.0721273
\(478\) 0 0
\(479\) −7.31624e10 −1.38978 −0.694890 0.719116i \(-0.744547\pi\)
−0.694890 + 0.719116i \(0.744547\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.26104e10 −0.231707
\(484\) −4.82881e9 −0.0879952
\(485\) 0 0
\(486\) 3.71141e10 0.665264
\(487\) −1.12072e11 −1.99241 −0.996207 0.0870146i \(-0.972267\pi\)
−0.996207 + 0.0870146i \(0.972267\pi\)
\(488\) 0 0
\(489\) −5.63813e10 −0.986052
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.16753e11 1.92904
\(497\) −8.22480e9 −0.134803
\(498\) 0 0
\(499\) 1.04363e11 1.68323 0.841614 0.540080i \(-0.181606\pi\)
0.841614 + 0.540080i \(0.181606\pi\)
\(500\) 0 0
\(501\) −1.11110e11 −1.76361
\(502\) 0 0
\(503\) −1.10148e11 −1.72070 −0.860350 0.509704i \(-0.829755\pi\)
−0.860350 + 0.509704i \(0.829755\pi\)
\(504\) 6.57879e9 0.101959
\(505\) 0 0
\(506\) −4.25093e10 −0.648458
\(507\) 4.63782e10 0.701910
\(508\) 0 0
\(509\) −1.32775e11 −1.97808 −0.989042 0.147633i \(-0.952835\pi\)
−0.989042 + 0.147633i \(0.952835\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6.87195e10 1.00000
\(513\) 0 0
\(514\) −1.33354e10 −0.191053
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −1.22218e11 −1.62110
\(525\) −2.59234e10 −0.341236
\(526\) 0 0
\(527\) 1.48793e11 1.92904
\(528\) 8.61345e10 1.10826
\(529\) −4.22041e10 −0.538930
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 1.88287e11 2.31555
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7.36343e10 0.872419
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 8.75781e10 1.00000
\(545\) 0 0
\(546\) −1.90640e10 −0.214508
\(547\) 7.07426e10 0.790190 0.395095 0.918640i \(-0.370712\pi\)
0.395095 + 0.918640i \(0.370712\pi\)
\(548\) 1.60953e11 1.78474
\(549\) 0 0
\(550\) −8.73875e10 −0.954989
\(551\) 0 0
\(552\) −7.31615e10 −0.788000
\(553\) 5.00172e10 0.534833
\(554\) 0 0
\(555\) 0 0
\(556\) −1.49253e11 −1.56180
\(557\) −1.91461e11 −1.98912 −0.994558 0.104181i \(-0.966778\pi\)
−0.994558 + 0.104181i \(0.966778\pi\)
\(558\) 6.48468e10 0.668886
\(559\) 0 0
\(560\) 0 0
\(561\) 1.09772e11 1.10826
\(562\) 2.61253e10 0.261889
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.02953e11 −1.97756
\(567\) −3.72749e10 −0.360649
\(568\) −4.77178e10 −0.458445
\(569\) −9.96321e10 −0.950496 −0.475248 0.879852i \(-0.657642\pi\)
−0.475248 + 0.879852i \(0.657642\pi\)
\(570\) 0 0
\(571\) −2.10436e11 −1.97959 −0.989793 0.142509i \(-0.954483\pi\)
−0.989793 + 0.142509i \(0.954483\pi\)
\(572\) −6.42644e10 −0.600325
\(573\) 0 0
\(574\) 0 0
\(575\) 7.42258e10 0.679021
\(576\) 3.81682e10 0.346746
\(577\) 1.07152e11 0.966714 0.483357 0.875423i \(-0.339417\pi\)
0.483357 + 0.875423i \(0.339417\pi\)
\(578\) 1.11612e11 1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.29489e10 −0.198649
\(584\) 0 0
\(585\) 0 0
\(586\) −1.50339e11 −1.27492
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 1.26730e11 1.06016
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.14859e10 0.416361 0.208180 0.978090i \(-0.433246\pi\)
0.208180 + 0.978090i \(0.433246\pi\)
\(594\) −9.01300e10 −0.723975
\(595\) 0 0
\(596\) 2.37106e11 1.87913
\(597\) −2.49134e11 −1.96126
\(598\) 5.45853e10 0.426846
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −1.50400e11 −1.16049
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 1.74809e11 1.29620
\(607\) 2.70040e11 1.98917 0.994587 0.103906i \(-0.0331341\pi\)
0.994587 + 0.103906i \(0.0331341\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 4.86426e10 0.346746
\(613\) 2.67782e11 1.89644 0.948222 0.317607i \(-0.102879\pi\)
0.948222 + 0.317607i \(0.102879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −4.04328e10 −0.280809
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −6.03146e10 −0.410828 −0.205414 0.978675i \(-0.565854\pi\)
−0.205414 + 0.978675i \(0.565854\pi\)
\(620\) 0 0
\(621\) 7.65552e10 0.514764
\(622\) 4.34682e10 0.290409
\(623\) −8.83846e10 −0.586712
\(624\) −1.10604e11 −0.729509
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 8.84679e10 0.568784
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 2.90185e11 1.81889
\(633\) 3.45011e11 2.14891
\(634\) 0 0
\(635\) 0 0
\(636\) −3.94966e10 −0.241397
\(637\) −9.45523e10 −0.574268
\(638\) 0 0
\(639\) −2.65034e10 −0.158964
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −2.50550e11 −1.47487
\(643\) −3.59809e10 −0.210488 −0.105244 0.994446i \(-0.533562\pi\)
−0.105244 + 0.994446i \(0.533562\pi\)
\(644\) 3.43431e10 0.199662
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −2.16258e11 −1.22651
\(649\) 0 0
\(650\) 1.12212e11 0.628619
\(651\) −1.18228e11 −0.658258
\(652\) 1.53549e11 0.849683
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −2.26537e11 −1.18668 −0.593340 0.804952i \(-0.702191\pi\)
−0.593340 + 0.804952i \(0.702191\pi\)
\(662\) 0 0
\(663\) −1.40956e11 −0.729509
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 3.02597e11 1.51971
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −6.95877e10 −0.341236
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.57377e11 0.758097
\(676\) −1.26306e11 −0.604838
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4.98526e11 2.31793
\(682\) −3.98544e11 −1.84221
\(683\) −3.31844e11 −1.52493 −0.762467 0.647028i \(-0.776012\pi\)
−0.762467 + 0.647028i \(0.776012\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.24608e11 −0.562665
\(687\) −1.02455e10 −0.0459944
\(688\) 0 0
\(689\) 2.94682e10 0.130760
\(690\) 0 0
\(691\) 3.85775e10 0.169208 0.0846042 0.996415i \(-0.473037\pi\)
0.0846042 + 0.996415i \(0.473037\pi\)
\(692\) 0 0
\(693\) −2.24572e10 −0.0973694
\(694\) −4.62698e11 −1.99462
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 4.17957e11 1.76080
\(699\) 0 0
\(700\) 7.06000e10 0.294044
\(701\) 2.31248e11 0.957646 0.478823 0.877911i \(-0.341064\pi\)
0.478823 + 0.877911i \(0.341064\pi\)
\(702\) 1.15734e11 0.476555
\(703\) 0 0
\(704\) −2.34579e11 −0.954989
\(705\) 0 0
\(706\) −3.26162e11 −1.31285
\(707\) −8.20580e10 −0.328430
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 1.61174e11 0.630692
\(712\) −5.12781e11 −1.99532
\(713\) 3.38518e11 1.30986
\(714\) −8.86846e10 −0.341236
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.80421e11 1.79765 0.898827 0.438304i \(-0.144421\pi\)
0.898827 + 0.438304i \(0.144421\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.71737e11 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 2.83693e10 0.102118
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 5.19189e10 0.184842
\(729\) 1.28358e11 0.454479
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.65203e11 −1.95789 −0.978946 0.204121i \(-0.934566\pi\)
−0.978946 + 0.204121i \(0.934566\pi\)
\(734\) −5.80223e11 −1.99899
\(735\) 0 0
\(736\) 1.99248e11 0.679021
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.85403e10 0.0611647
\(743\) −1.85977e11 −0.610243 −0.305122 0.952313i \(-0.598697\pi\)
−0.305122 + 0.952313i \(0.598697\pi\)
\(744\) −6.85923e11 −2.23864
\(745\) 0 0
\(746\) 2.17336e11 0.701742
\(747\) 0 0
\(748\) −2.98954e11 −0.954989
\(749\) 1.17612e11 0.373701
\(750\) 0 0
\(751\) −8.07947e10 −0.253994 −0.126997 0.991903i \(-0.540534\pi\)
−0.126997 + 0.991903i \(0.540534\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 7.28156e10 0.222914
\(757\) −1.59460e11 −0.485590 −0.242795 0.970078i \(-0.578064\pi\)
−0.242795 + 0.970078i \(0.578064\pi\)
\(758\) 5.73990e11 1.73871
\(759\) 2.49742e11 0.752532
\(760\) 0 0
\(761\) −5.68097e11 −1.69389 −0.846943 0.531684i \(-0.821559\pi\)
−0.846943 + 0.531684i \(0.821559\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −4.03727e11 −1.16049
\(769\) 4.65460e11 1.33100 0.665498 0.746399i \(-0.268219\pi\)
0.665498 + 0.746399i \(0.268219\pi\)
\(770\) 0 0
\(771\) 7.83454e10 0.221715
\(772\) 0 0
\(773\) −6.44277e11 −1.80449 −0.902245 0.431223i \(-0.858082\pi\)
−0.902245 + 0.431223i \(0.858082\pi\)
\(774\) 0 0
\(775\) 6.95901e11 1.92904
\(776\) 0 0
\(777\) 0 0
\(778\) −6.30701e11 −1.72149
\(779\) 0 0
\(780\) 0 0
\(781\) 1.62888e11 0.437810
\(782\) 2.53928e11 0.679021
\(783\) 0 0
\(784\) −3.45136e11 −0.913538
\(785\) 0 0
\(786\) 7.18031e11 1.88128
\(787\) 1.93787e11 0.505157 0.252578 0.967576i \(-0.418721\pi\)
0.252578 + 0.967576i \(0.418721\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −1.30290e11 −0.331139
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 6.78493e11 1.69003
\(797\) −5.79198e11 −1.43547 −0.717735 0.696317i \(-0.754821\pi\)
−0.717735 + 0.696317i \(0.754821\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.09600e11 1.00000
\(801\) −2.84809e11 −0.691868
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 5.11763e11 1.21263
\(807\) 0 0
\(808\) −4.76076e11 −1.11694
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −8.18198e11 −1.89136 −0.945682 0.325093i \(-0.894604\pi\)
−0.945682 + 0.325093i \(0.894604\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −5.14521e11 −1.16049
\(817\) 0 0
\(818\) 4.69948e11 1.04963
\(819\) 2.88368e10 0.0640932
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −9.45597e11 −2.07118
\(823\) −8.51393e11 −1.85580 −0.927900 0.372830i \(-0.878387\pi\)
−0.927900 + 0.372830i \(0.878387\pi\)
\(824\) 0 0
\(825\) 5.13402e11 1.10826
\(826\) 0 0
\(827\) 1.64971e11 0.352684 0.176342 0.984329i \(-0.443574\pi\)
0.176342 + 0.984329i \(0.443574\pi\)
\(828\) 1.10666e11 0.235448
\(829\) −8.65995e11 −1.83357 −0.916785 0.399382i \(-0.869225\pi\)
−0.916785 + 0.399382i \(0.869225\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.01218e11 0.628619
\(833\) −4.39852e11 −0.913538
\(834\) 8.76864e11 1.81246
\(835\) 0 0
\(836\) 0 0
\(837\) 7.17740e11 1.46240
\(838\) 6.77477e11 1.37378
\(839\) −9.48615e11 −1.91444 −0.957221 0.289359i \(-0.906558\pi\)
−0.957221 + 0.289359i \(0.906558\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 6.79482e11 1.35185
\(843\) −1.53486e11 −0.303920
\(844\) −9.39604e11 −1.85172
\(845\) 0 0
\(846\) 0 0
\(847\) −1.33170e10 −0.0258745
\(848\) 1.07565e11 0.208012
\(849\) 1.19235e12 2.29494
\(850\) 5.22006e11 1.00000
\(851\) 0 0
\(852\) 2.80342e11 0.532022
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.82349e11 1.27090
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 3.77553e11 0.696673
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.04148e12 1.88635
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 4.22454e11 0.758097
\(865\) 0 0
\(866\) 1.12180e12 1.99454
\(867\) −6.55721e11 −1.16049
\(868\) 3.21982e11 0.567222
\(869\) −9.90566e11 −1.73702
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 5.20996e11 0.876711
\(879\) 8.83243e11 1.47953
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −1.91696e11 −0.316766
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 3.83881e11 0.628619
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00722e11 0.970463 0.485231 0.874386i \(-0.338735\pi\)
0.485231 + 0.874386i \(0.338735\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 7.38213e11 1.17131
\(892\) 0 0
\(893\) 0 0
\(894\) −1.39300e12 −2.18072
\(895\) 0 0
\(896\) 1.89515e11 0.294044
\(897\) −3.20689e11 −0.495352
\(898\) 0 0
\(899\) 0 0
\(900\) 2.27500e11 0.346746
\(901\) 1.37084e11 0.208012
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.89048e10 0.0279347 0.0139673 0.999902i \(-0.495554\pi\)
0.0139673 + 0.999902i \(0.495554\pi\)
\(908\) −1.35769e12 −1.99736
\(909\) −2.64422e11 −0.387295
\(910\) 0 0
\(911\) 1.37408e12 1.99498 0.997489 0.0708249i \(-0.0225632\pi\)
0.997489 + 0.0708249i \(0.0225632\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.04623e12 −1.49914
\(915\) 0 0
\(916\) 2.79025e10 0.0396334
\(917\) −3.37055e11 −0.476676
\(918\) 5.38388e11 0.758097
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9.25491e11 −1.28070
\(923\) −2.09161e11 −0.288187
\(924\) 2.37543e11 0.325877
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.55375e11 −0.337018
\(934\) 0 0
\(935\) 0 0
\(936\) 1.67303e11 0.217971
\(937\) −1.51499e12 −1.96541 −0.982704 0.185185i \(-0.940712\pi\)
−0.982704 + 0.185185i \(0.940712\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) −5.19749e11 −0.660070
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.91404e11 −1.23268 −0.616341 0.787480i \(-0.711386\pi\)
−0.616341 + 0.787480i \(0.711386\pi\)
\(948\) −1.70483e12 −2.11081
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 2.41524e11 0.294044
\(953\) −4.45589e11 −0.540210 −0.270105 0.962831i \(-0.587058\pi\)
−0.270105 + 0.962831i \(0.587058\pi\)
\(954\) 5.97438e10 0.0721273
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −1.17060e12 −1.38978
\(959\) 4.43877e11 0.524794
\(960\) 0 0
\(961\) 2.32087e12 2.72118
\(962\) 0 0
\(963\) 3.78990e11 0.440679
\(964\) 0 0
\(965\) 0 0
\(966\) −2.01766e11 −0.231707
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −7.72610e10 −0.0879952
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 5.93826e11 0.665264
\(973\) −4.11613e11 −0.459238
\(974\) −1.79315e12 −1.99241
\(975\) −6.59248e11 −0.729509
\(976\) 0 0
\(977\) −1.82215e12 −1.99989 −0.999943 0.0106585i \(-0.996607\pi\)
−0.999943 + 0.0106585i \(0.996607\pi\)
\(978\) −9.02101e11 −0.986052
\(979\) 1.75042e12 1.90551
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.23434e12 −1.32197 −0.660984 0.750400i \(-0.729861\pi\)
−0.660984 + 0.750400i \(0.729861\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.67034e12 −1.73185 −0.865925 0.500173i \(-0.833270\pi\)
−0.865925 + 0.500173i \(0.833270\pi\)
\(992\) 1.86804e12 1.92904
\(993\) 0 0
\(994\) −1.31597e11 −0.134803
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 1.66980e12 1.68323
\(999\) 0 0
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 68.9.d.a.67.1 1
4.3 odd 2 68.9.d.b.67.1 yes 1
17.16 even 2 68.9.d.b.67.1 yes 1
68.67 odd 2 CM 68.9.d.a.67.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.9.d.a.67.1 1 1.1 even 1 trivial
68.9.d.a.67.1 1 68.67 odd 2 CM
68.9.d.b.67.1 yes 1 4.3 odd 2
68.9.d.b.67.1 yes 1 17.16 even 2