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Properties

Label 31.31.b.a.30.1
Level $31$
Weight $31$
Character 31.30
Self dual yes
Analytic conductor $176.744$
Analytic rank $0$
Dimension $1$
CM discriminant -31
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,31,Mod(30,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 31, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.30");
 
S:= CuspForms(chi, 31);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 31.b (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: \(176.744211898\)
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 30.1
Character \(\chi\) \(=\) 31.30

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+13425.0 q^{2} -8.93511e8 q^{4} -3.81489e10 q^{5} +2.30998e12 q^{7} -2.64104e13 q^{8} +2.05891e14 q^{9} +O(q^{10})\) \(q+13425.0 q^{2} -8.93511e8 q^{4} -3.81489e10 q^{5} +2.30998e12 q^{7} -2.64104e13 q^{8} +2.05891e14 q^{9} -5.12149e14 q^{10} +3.10115e16 q^{14} +6.04841e17 q^{16} +2.76409e18 q^{18} -2.91241e19 q^{19} +3.40865e19 q^{20} +5.24017e20 q^{25} -2.06400e21 q^{28} -2.34653e22 q^{31} +3.64779e22 q^{32} -8.81234e22 q^{35} -1.83966e23 q^{36} -3.90991e23 q^{38} +1.00753e24 q^{40} -2.37186e24 q^{41} -7.85452e24 q^{45} -2.38182e25 q^{47} -1.72033e25 q^{49} +7.03493e24 q^{50} -6.10075e25 q^{56} +7.24898e26 q^{59} -3.15021e26 q^{62} +4.75605e26 q^{63} -1.59727e26 q^{64} -4.43135e27 q^{67} -1.18306e27 q^{70} +1.04706e28 q^{71} -5.43766e27 q^{72} +2.60227e28 q^{76} -2.30740e28 q^{80} +4.23912e28 q^{81} -3.18422e28 q^{82} -1.05447e29 q^{90} -3.19760e29 q^{94} +1.11105e30 q^{95} +9.68726e29 q^{97} -2.30955e29 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13425.0 0.409698 0.204849 − 0.978794i \(-0.434330\pi\)
0.204849 + 0.978794i \(0.434330\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −8.93511e8 −0.832147
\(5\) −3.81489e10 −1.25006 −0.625032 − 0.780599i \(-0.714914\pi\)
−0.625032 + 0.780599i \(0.714914\pi\)
\(6\) 0 0
\(7\) 2.30998e12 0.486562 0.243281 − 0.969956i \(-0.421776\pi\)
0.243281 + 0.969956i \(0.421776\pi\)
\(8\) −2.64104e13 −0.750628
\(9\) 2.05891e14 1.00000
\(10\) −5.12149e14 −0.512149
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 3.10115e16 0.199344
\(15\) 0 0
\(16\) 6.04841e17 0.524616
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 2.76409e18 0.409698
\(19\) −2.91241e19 −1.91844 −0.959221 − 0.282657i \(-0.908784\pi\)
−0.959221 + 0.282657i \(0.908784\pi\)
\(20\) 3.40865e19 1.04024
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 5.24017e20 0.562659
\(26\) 0 0
\(27\) 0 0
\(28\) −2.06400e21 −0.404891
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.34653e22 −1.00000
\(32\) 3.64779e22 0.965562
\(33\) 0 0
\(34\) 0 0
\(35\) −8.81234e22 −0.608234
\(36\) −1.83966e23 −0.832147
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −3.90991e23 −0.785983
\(39\) 0 0
\(40\) 1.00753e24 0.938333
\(41\) −2.37186e24 −1.52521 −0.762607 − 0.646862i \(-0.776081\pi\)
−0.762607 + 0.646862i \(0.776081\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −7.85452e24 −1.25006
\(46\) 0 0
\(47\) −2.38182e25 −1.97443 −0.987214 − 0.159398i \(-0.949045\pi\)
−0.987214 + 0.159398i \(0.949045\pi\)
\(48\) 0 0
\(49\) −1.72033e25 −0.763257
\(50\) 7.03493e24 0.230521
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.10075e25 −0.365227
\(57\) 0 0
\(58\) 0 0
\(59\) 7.24898e26 1.98379 0.991897 − 0.127046i \(-0.0405494\pi\)
0.991897 + 0.127046i \(0.0405494\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −3.15021e26 −0.409698
\(63\) 4.75605e26 0.486562
\(64\) −1.59727e26 −0.129027
\(65\) 0 0
\(66\) 0 0
\(67\) −4.43135e27 −1.80059 −0.900294 − 0.435282i \(-0.856649\pi\)
−0.900294 + 0.435282i \(0.856649\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.18306e27 −0.249192
\(71\) 1.04706e28 1.78277 0.891384 − 0.453250i \(-0.149735\pi\)
0.891384 + 0.453250i \(0.149735\pi\)
\(72\) −5.43766e27 −0.750628
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.60227e28 1.59643
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −2.30740e28 −0.655803
\(81\) 4.23912e28 1.00000
\(82\) −3.18422e28 −0.624878
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −1.05447e29 −0.512149
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −3.19760e29 −0.808920
\(95\) 1.11105e30 2.39817
\(96\) 0 0
\(97\) 9.68726e29 1.52977 0.764883 − 0.644170i \(-0.222797\pi\)
0.764883 + 0.644170i \(0.222797\pi\)
\(98\) −2.30955e29 −0.312705
\(99\) 0 0
\(100\) −4.68215e29 −0.468215
\(101\) 4.61696e29 0.397681 0.198841 − 0.980032i \(-0.436282\pi\)
0.198841 + 0.980032i \(0.436282\pi\)
\(102\) 0 0
\(103\) −3.01324e30 −1.93408 −0.967040 − 0.254623i \(-0.918049\pi\)
−0.967040 + 0.254623i \(0.918049\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.36821e30 0.495904 0.247952 − 0.968772i \(-0.420243\pi\)
0.247952 + 0.968772i \(0.420243\pi\)
\(108\) 0 0
\(109\) −2.74514e30 −0.753646 −0.376823 − 0.926285i \(-0.622984\pi\)
−0.376823 + 0.926285i \(0.622984\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.39717e30 0.255258
\(113\) 1.22941e31 1.96572 0.982860 − 0.184356i \(-0.0590199\pi\)
0.982860 + 0.184356i \(0.0590199\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 9.73175e30 0.812757
\(119\) 0 0
\(120\) 0 0
\(121\) 1.74494e31 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 2.09665e31 0.832147
\(125\) 1.55383e31 0.546704
\(126\) 6.38500e30 0.199344
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −4.13122e31 −1.01842
\(129\) 0 0
\(130\) 0 0
\(131\) −1.02543e32 −1.78581 −0.892907 − 0.450240i \(-0.851338\pi\)
−0.892907 + 0.450240i \(0.851338\pi\)
\(132\) 0 0
\(133\) −6.72762e31 −0.933441
\(134\) −5.94909e31 −0.737698
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 7.87392e31 0.506140
\(141\) 0 0
\(142\) 1.40567e32 0.730397
\(143\) 0 0
\(144\) 1.24531e32 0.524616
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.88637e32 −1.99105 −0.995527 − 0.0944796i \(-0.969881\pi\)
−0.995527 + 0.0944796i \(0.969881\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 7.69179e32 1.44004
\(153\) 0 0
\(154\) 0 0
\(155\) 8.95174e32 1.25006
\(156\) 0 0
\(157\) 1.70596e33 1.96551 0.982754 − 0.184915i \(-0.0592010\pi\)
0.982754 + 0.184915i \(0.0592010\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.39159e33 −1.20701
\(161\) 0 0
\(162\) 5.69101e32 0.409698
\(163\) −2.67223e33 −1.75412 −0.877060 − 0.480380i \(-0.840499\pi\)
−0.877060 + 0.480380i \(0.840499\pi\)
\(164\) 2.11928e33 1.26920
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.62000e33 1.00000
\(170\) 0 0
\(171\) −5.99640e33 −1.91844
\(172\) 0 0
\(173\) 3.18921e33 0.857021 0.428511 − 0.903537i \(-0.359038\pi\)
0.428511 + 0.903537i \(0.359038\pi\)
\(174\) 0 0
\(175\) 1.21047e33 0.273769
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 7.01810e33 1.04024
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 2.12818e34 1.64302
\(189\) 0 0
\(190\) 1.49159e34 0.982528
\(191\) 2.83833e34 1.72808 0.864038 − 0.503427i \(-0.167928\pi\)
0.864038 + 0.503427i \(0.167928\pi\)
\(192\) 0 0
\(193\) 2.99038e34 1.55728 0.778639 − 0.627472i \(-0.215910\pi\)
0.778639 + 0.627472i \(0.215910\pi\)
\(194\) 1.30051e34 0.626742
\(195\) 0 0
\(196\) 1.53714e34 0.635142
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.38395e34 −0.422348
\(201\) 0 0
\(202\) 6.19827e33 0.162929
\(203\) 0 0
\(204\) 0 0
\(205\) 9.04838e34 1.90662
\(206\) −4.04527e34 −0.792390
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 6.23143e34 0.851826 0.425913 − 0.904764i \(-0.359953\pi\)
0.425913 + 0.904764i \(0.359953\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.83683e34 0.203171
\(215\) 0 0
\(216\) 0 0
\(217\) −5.42044e34 −0.486562
\(218\) −3.68535e34 −0.308768
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 8.42634e34 0.469806
\(225\) 1.07890e35 0.562659
\(226\) 1.65049e35 0.805352
\(227\) 1.54285e35 0.704588 0.352294 − 0.935889i \(-0.385402\pi\)
0.352294 + 0.935889i \(0.385402\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.47065e35 1.38048 0.690242 − 0.723578i \(-0.257504\pi\)
0.690242 + 0.723578i \(0.257504\pi\)
\(234\) 0 0
\(235\) 9.08639e35 2.46816
\(236\) −6.47704e35 −1.65081
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 2.34258e35 0.409698
\(243\) 0 0
\(244\) 0 0
\(245\) 6.56288e35 0.954120
\(246\) 0 0
\(247\) 0 0
\(248\) 6.19726e35 0.750628
\(249\) 0 0
\(250\) 2.08601e35 0.223984
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −4.24958e35 −0.404891
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −3.83110e35 −0.288220
\(257\) −1.93725e36 −1.37464 −0.687319 − 0.726356i \(-0.741212\pi\)
−0.687319 + 0.726356i \(0.741212\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.37664e36 −0.731646
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −9.03183e35 −0.382429
\(267\) 0 0
\(268\) 3.95946e36 1.49835
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −4.83129e36 −1.00000
\(280\) 2.32737e36 0.456557
\(281\) −8.29154e36 −1.54184 −0.770922 − 0.636929i \(-0.780204\pi\)
−0.770922 + 0.636929i \(0.780204\pi\)
\(282\) 0 0
\(283\) 1.08561e37 1.81501 0.907504 − 0.420043i \(-0.137985\pi\)
0.907504 + 0.420043i \(0.137985\pi\)
\(284\) −9.35556e36 −1.48352
\(285\) 0 0
\(286\) 0 0
\(287\) −5.47895e36 −0.742112
\(288\) 7.51048e36 0.965562
\(289\) 8.19347e36 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.48435e37 −1.47409 −0.737043 − 0.675846i \(-0.763778\pi\)
−0.737043 + 0.675846i \(0.763778\pi\)
\(294\) 0 0
\(295\) −2.76541e37 −2.47987
\(296\) 0 0
\(297\) 0 0
\(298\) −1.05874e37 −0.815732
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.76155e37 −1.00645
\(305\) 0 0
\(306\) 0 0
\(307\) −2.76050e37 −1.36117 −0.680584 − 0.732670i \(-0.738274\pi\)
−0.680584 + 0.732670i \(0.738274\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.20177e37 0.512149
\(311\) −4.21120e37 −1.71002 −0.855008 − 0.518615i \(-0.826448\pi\)
−0.855008 + 0.518615i \(0.826448\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 2.29026e37 0.805266
\(315\) −1.81438e37 −0.608234
\(316\) 0 0
\(317\) 2.33772e37 0.712695 0.356348 − 0.934353i \(-0.384022\pi\)
0.356348 + 0.934353i \(0.384022\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 6.09342e36 0.161291
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −3.78770e37 −0.832147
\(325\) 0 0
\(326\) −3.58747e37 −0.718660
\(327\) 0 0
\(328\) 6.26417e37 1.14487
\(329\) −5.50197e37 −0.960682
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.69051e38 2.25085
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 3.51734e37 0.409698
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −8.05016e37 −0.785983
\(343\) −9.18049e37 −0.857934
\(344\) 0 0
\(345\) 0 0
\(346\) 4.28152e37 0.351120
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 2.64497e38 1.90563 0.952815 − 0.303550i \(-0.0981720\pi\)
0.952815 + 0.303550i \(0.0981720\pi\)
\(350\) 1.62506e37 0.112163
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −3.99440e38 −2.22857
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.22609e38 1.99307 0.996533 − 0.0832000i \(-0.0265140\pi\)
0.996533 + 0.0832000i \(0.0265140\pi\)
\(360\) 2.07441e38 0.938333
\(361\) 6.17747e38 2.68042
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −4.88345e38 −1.52521
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9.62582e37 0.255744 0.127872 − 0.991791i \(-0.459185\pi\)
0.127872 + 0.991791i \(0.459185\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.29048e38 1.48206
\(377\) 0 0
\(378\) 0 0
\(379\) 8.50222e38 1.77805 0.889026 − 0.457857i \(-0.151383\pi\)
0.889026 + 0.457857i \(0.151383\pi\)
\(380\) −9.92738e38 −1.99563
\(381\) 0 0
\(382\) 3.81046e38 0.707990
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.01458e38 0.638015
\(387\) 0 0
\(388\) −8.65567e38 −1.27299
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 4.54346e38 0.572922
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.38652e36 0.00353099 0.00176549 − 0.999998i \(-0.499438\pi\)
0.00176549 + 0.999998i \(0.499438\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.16947e38 0.295180
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −4.12530e38 −0.330929
\(405\) −1.61718e39 −1.25006
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 1.21475e39 0.781137
\(411\) 0 0
\(412\) 2.69236e39 1.60944
\(413\) 1.67450e39 0.965239
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.51375e38 0.348856 0.174428 − 0.984670i \(-0.444192\pi\)
0.174428 + 0.984670i \(0.444192\pi\)
\(420\) 0 0
\(421\) −1.38946e39 −0.600642 −0.300321 − 0.953838i \(-0.597094\pi\)
−0.300321 + 0.953838i \(0.597094\pi\)
\(422\) 8.36570e38 0.348992
\(423\) −4.90396e39 −1.97443
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.22251e39 −0.412665
\(429\) 0 0
\(430\) 0 0
\(431\) −1.89037e39 −0.574630 −0.287315 − 0.957836i \(-0.592763\pi\)
−0.287315 + 0.957836i \(0.592763\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −7.27694e38 −0.199344
\(435\) 0 0
\(436\) 2.45282e39 0.627145
\(437\) 0 0
\(438\) 0 0
\(439\) −4.34677e39 −1.00276 −0.501382 − 0.865226i \(-0.667175\pi\)
−0.501382 + 0.865226i \(0.667175\pi\)
\(440\) 0 0
\(441\) −3.54201e39 −0.763257
\(442\) 0 0
\(443\) −7.01750e39 −1.41295 −0.706475 − 0.707738i \(-0.749716\pi\)
−0.706475 + 0.707738i \(0.749716\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −3.68967e38 −0.0627795
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 1.44843e39 0.230521
\(451\) 0 0
\(452\) −1.09850e40 −1.63577
\(453\) 0 0
\(454\) 2.07127e39 0.288669
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00184e39 0.565582
\(467\) 7.56460e39 0.690292 0.345146 − 0.938549i \(-0.387829\pi\)
0.345146 + 0.938549i \(0.387829\pi\)
\(468\) 0 0
\(469\) −1.02364e40 −0.876098
\(470\) 1.21985e40 1.01120
\(471\) 0 0
\(472\) −1.91448e40 −1.48909
\(473\) 0 0
\(474\) 0 0
\(475\) −1.52615e40 −1.07943
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.18012e40 1.98340 0.991702 − 0.128557i \(-0.0410344\pi\)
0.991702 + 0.128557i \(0.0410344\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.55912e40 −0.832147
\(485\) −3.69558e40 −1.91230
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 8.81066e39 0.390902
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.41928e40 −0.524616
\(497\) 2.41868e40 0.867427
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.38836e40 −0.454938
\(501\) 0 0
\(502\) 0 0
\(503\) 6.37612e40 1.91002 0.955008 − 0.296580i \(-0.0958461\pi\)
0.955008 + 0.296580i \(0.0958461\pi\)
\(504\) −1.25609e40 −0.365227
\(505\) −1.76132e40 −0.497127
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3.92154e40 0.900341
\(513\) 0 0
\(514\) −2.60075e40 −0.563187
\(515\) 1.14952e41 2.41772
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.06577e41 −1.88407 −0.942033 − 0.335521i \(-0.891088\pi\)
−0.942033 + 0.335521i \(0.891088\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 9.16233e40 1.48606
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.10943e40 1.00000
\(530\) 0 0
\(531\) 1.49250e41 1.98379
\(532\) 6.01120e40 0.776760
\(533\) 0 0
\(534\) 0 0
\(535\) −5.21959e40 −0.619911
\(536\) 1.17034e41 1.35157
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6.72738e39 0.0675907 0.0337954 − 0.999429i \(-0.489241\pi\)
0.0337954 + 0.999429i \(0.489241\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.04724e41 0.942106
\(546\) 0 0
\(547\) 1.57254e41 1.33904 0.669518 − 0.742796i \(-0.266501\pi\)
0.669518 + 0.742796i \(0.266501\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −6.48601e40 −0.409698
\(559\) 0 0
\(560\) −5.33006e40 −0.319089
\(561\) 0 0
\(562\) −1.11314e41 −0.631691
\(563\) −3.15123e41 −1.74122 −0.870612 − 0.491970i \(-0.836277\pi\)
−0.870612 + 0.491970i \(0.836277\pi\)
\(564\) 0 0
\(565\) −4.69008e41 −2.45727
\(566\) 1.45744e41 0.743606
\(567\) 9.79229e40 0.486562
\(568\) −2.76531e41 −1.33819
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −7.35550e40 −0.304042
\(575\) 0 0
\(576\) −3.28864e40 −0.129027
\(577\) −2.62601e41 −1.00383 −0.501913 − 0.864918i \(-0.667370\pi\)
−0.501913 + 0.864918i \(0.667370\pi\)
\(578\) 1.09997e41 0.409698
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −1.99274e41 −0.603931
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 6.83405e41 1.91844
\(590\) −3.71256e41 −1.01600
\(591\) 0 0
\(592\) 0 0
\(593\) 2.25576e41 0.572101 0.286051 − 0.958215i \(-0.407657\pi\)
0.286051 + 0.958215i \(0.407657\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.04656e41 1.65685
\(597\) 0 0
\(598\) 0 0
\(599\) 4.55506e41 0.993325 0.496663 − 0.867944i \(-0.334559\pi\)
0.496663 + 0.867944i \(0.334559\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −9.12376e41 −1.80059
\(604\) 0 0
\(605\) −6.65676e41 −1.25006
\(606\) 0 0
\(607\) 9.50782e41 1.69922 0.849612 − 0.527408i \(-0.176836\pi\)
0.849612 + 0.527408i \(0.176836\pi\)
\(608\) −1.06239e42 −1.85238
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −3.70597e41 −0.557669
\(615\) 0 0
\(616\) 0 0
\(617\) −5.22596e41 −0.730952 −0.365476 − 0.930821i \(-0.619094\pi\)
−0.365476 + 0.930821i \(0.619094\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −7.99848e41 −1.04024
\(621\) 0 0
\(622\) −5.65354e41 −0.700591
\(623\) 0 0
\(624\) 0 0
\(625\) −1.08080e42 −1.24607
\(626\) 0 0
\(627\) 0 0
\(628\) −1.52430e42 −1.63559
\(629\) 0 0
\(630\) −2.43581e41 −0.249192
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 3.13839e41 0.291990
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.15580e42 1.78277
\(640\) 1.57602e42 1.27310
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −1.11957e42 −0.750628
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 2.38767e42 1.45969
\(653\) 1.91897e42 1.14649 0.573246 − 0.819383i \(-0.305684\pi\)
0.573246 + 0.819383i \(0.305684\pi\)
\(654\) 0 0
\(655\) 3.91190e42 2.23238
\(656\) −1.43460e42 −0.800152
\(657\) 0 0
\(658\) −7.38639e41 −0.393590
\(659\) −3.41604e42 −1.77927 −0.889635 − 0.456673i \(-0.849041\pi\)
−0.889635 + 0.456673i \(0.849041\pi\)
\(660\) 0 0
\(661\) −1.72582e41 −0.0858964 −0.0429482 − 0.999077i \(-0.513675\pi\)
−0.0429482 + 0.999077i \(0.513675\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.56651e42 1.16686
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 2.26951e42 0.922170
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −2.34100e42 −0.832147
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 2.23774e42 0.744326
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3.50844e41 −0.106857 −0.0534286 − 0.998572i \(-0.517015\pi\)
−0.0534286 + 0.998572i \(0.517015\pi\)
\(684\) 5.35785e42 1.59643
\(685\) 0 0
\(686\) −1.23248e42 −0.351494
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 3.26787e42 0.835785 0.417893 − 0.908496i \(-0.362769\pi\)
0.417893 + 0.908496i \(0.362769\pi\)
\(692\) −2.84960e42 −0.713168
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 3.55087e42 0.780734
\(699\) 0 0
\(700\) −1.08157e42 −0.227816
\(701\) −2.48170e42 −0.511658 −0.255829 − 0.966722i \(-0.582348\pi\)
−0.255829 + 0.966722i \(0.582348\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.06651e42 0.193497
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −5.36249e42 −0.913043
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 5.67352e42 0.816556
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −4.75074e42 −0.655803
\(721\) −6.96052e42 −0.941050
\(722\) 8.29325e42 1.09816
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.67506e43 1.99993 0.999965 − 0.00834012i \(-0.00265477\pi\)
0.999965 + 0.00834012i \(0.00265477\pi\)
\(728\) 0 0
\(729\) 8.72796e42 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −6.42462e42 −0.678091 −0.339045 − 0.940770i \(-0.610104\pi\)
−0.339045 + 0.940770i \(0.610104\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −6.55603e42 −0.624878
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 3.00856e43 2.48894
\(746\) 1.29227e42 0.104778
\(747\) 0 0
\(748\) 0 0
\(749\) 3.16055e42 0.241288
\(750\) 0 0
\(751\) −8.16570e42 −0.598955 −0.299478 − 0.954103i \(-0.596812\pi\)
−0.299478 + 0.954103i \(0.596812\pi\)
\(752\) −1.44062e43 −1.03582
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.14142e43 0.728465
\(759\) 0 0
\(760\) −2.93433e43 −1.80014
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −6.34123e42 −0.366696
\(764\) −2.53608e43 −1.43801
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.88811e43 −1.99914 −0.999571 − 0.0292787i \(-0.990679\pi\)
−0.999571 + 0.0292787i \(0.990679\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.67193e43 −1.29588
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.22962e43 −0.562659
\(776\) −2.55844e43 −1.14828
\(777\) 0 0
\(778\) 0 0
\(779\) 6.90783e43 2.92604
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.04053e43 −0.400417
\(785\) −6.50807e43 −2.45701
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.83993e43 0.956444
\(792\) 0 0
\(793\) 0 0
\(794\) 4.54640e40 0.00144664
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.91150e43 0.543282
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −1.21936e43 −0.298511
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −2.17106e43 −0.512149
\(811\) −8.60528e43 −1.99275 −0.996373 − 0.0850880i \(-0.972883\pi\)
−0.996373 + 0.0850880i \(0.972883\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.01943e44 2.19276
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −8.08483e43 −1.58658
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 7.95807e43 1.45178
\(825\) 0 0
\(826\) 2.24802e43 0.395457
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 1.00872e43 0.142926
\(839\) 7.39888e43 1.02976 0.514881 − 0.857262i \(-0.327836\pi\)
0.514881 + 0.857262i \(0.327836\pi\)
\(840\) 0 0
\(841\) 7.44629e43 1.00000
\(842\) −1.86535e43 −0.246082
\(843\) 0 0
\(844\) −5.56786e43 −0.708845
\(845\) −9.99500e43 −1.25006
\(846\) −6.58357e43 −0.808920
\(847\) 4.03078e43 0.486562
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.24145e44 1.34802 0.674008 − 0.738724i \(-0.264571\pi\)
0.674008 + 0.738724i \(0.264571\pi\)
\(854\) 0 0
\(855\) 2.28756e44 2.39817
\(856\) −3.61350e43 −0.372239
\(857\) 1.37374e44 1.39057 0.695285 − 0.718735i \(-0.255278\pi\)
0.695285 + 0.718735i \(0.255278\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.53783e43 −0.235425
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −1.21665e44 −1.07133
\(866\) 0 0
\(867\) 0 0
\(868\) 4.84322e43 0.404891
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 7.25002e43 0.565708
\(873\) 1.99452e44 1.52977
\(874\) 0 0
\(875\) 3.58931e43 0.266006
\(876\) 0 0
\(877\) 1.41900e44 1.01622 0.508110 − 0.861292i \(-0.330344\pi\)
0.508110 + 0.861292i \(0.330344\pi\)
\(878\) −5.83553e43 −0.410831
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −4.75515e43 −0.312705
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9.42100e43 −0.578883
\(887\) −1.81641e44 −1.09739 −0.548693 − 0.836024i \(-0.684875\pi\)
−0.548693 + 0.836024i \(0.684875\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.93684e44 3.78783
\(894\) 0 0
\(895\) 0 0
\(896\) −9.54305e43 −0.495527
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −9.64013e43 −0.468215
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −3.24693e44 −1.47552
\(905\) 0 0
\(906\) 0 0
\(907\) 3.82315e44 1.65315 0.826574 − 0.562828i \(-0.190287\pi\)
0.826574 + 0.562828i \(0.190287\pi\)
\(908\) −1.37855e44 −0.586321
\(909\) 9.50591e43 0.397681
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.36873e44 −0.868910
\(918\) 0 0
\(919\) −5.27854e44 −1.87405 −0.937024 − 0.349264i \(-0.886432\pi\)
−0.937024 + 0.349264i \(0.886432\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6.20398e44 −1.93408
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 5.01031e44 1.46426
\(932\) −3.99457e44 −1.14877
\(933\) 0 0
\(934\) 1.01555e44 0.282812
\(935\) 0 0
\(936\) 0 0
\(937\) 5.81274e44 1.54272 0.771360 − 0.636399i \(-0.219577\pi\)
0.771360 + 0.636399i \(0.219577\pi\)
\(938\) −1.37423e44 −0.358936
\(939\) 0 0
\(940\) −8.11879e44 −2.05387
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 4.38448e44 1.04073
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.04886e44 −0.442240
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.08279e45 −2.16020
\(956\) 0 0
\(957\) 0 0
\(958\) 4.26932e44 0.812598
\(959\) 0 0
\(960\) 0 0
\(961\) 5.50619e44 1.00000
\(962\) 0 0
\(963\) 2.81703e44 0.495904
\(964\) 0 0
\(965\) −1.14080e45 −1.94670
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −4.60845e44 −0.750628
\(969\) 0 0
\(970\) −4.96132e44 −0.783468
\(971\) −2.02815e44 −0.315364 −0.157682 − 0.987490i \(-0.550402\pi\)
−0.157682 + 0.987490i \(0.550402\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.01971e45 1.44563 0.722814 − 0.691042i \(-0.242848\pi\)
0.722814 + 0.691042i \(0.242848\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.86401e44 −0.793968
\(981\) −5.65201e44 −0.753646
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) −8.55964e44 −0.965562
\(993\) 0 0
\(994\) 3.24708e44 0.355383
\(995\) 0 0
\(996\) 0 0
\(997\) 8.70179e44 0.910293 0.455147 − 0.890417i \(-0.349587\pi\)
0.455147 + 0.890417i \(0.349587\pi\)
\(998\) 0 0
\(999\) 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 31.31.b.a.30.1 ✓ 1
31.30 odd 2 CM 31.31.b.a.30.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.31.b.a.30.1 ✓ 1 1.1 even 1 trivial
31.31.b.a.30.1 ✓ 1 31.30 odd 2 CM