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Properties

Label 59.13.b.a.58.1
Level $59$
Weight $13$
Character 59.58
Self dual yes
Analytic conductor $53.926$
Analytic rank $0$
Dimension $1$
CM discriminant -59
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] = [59,13,Mod(58,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 59.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: \(53.9256352193\)
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 58.1
Character \(\chi\) \(=\) 59.58

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1433.00 q^{3} +4096.00 q^{4} +5231.00 q^{5} -211273. q^{7} +1.52205e6 q^{9} +O(q^{10})\) \(q-1433.00 q^{3} +4096.00 q^{4} +5231.00 q^{5} -211273. q^{7} +1.52205e6 q^{9} -5.86957e6 q^{12} -7.49602e6 q^{15} +1.67772e7 q^{16} -1.16726e7 q^{17} -9.38955e7 q^{19} +2.14262e7 q^{20} +3.02754e8 q^{21} -2.16777e8 q^{25} -1.41954e9 q^{27} -8.65374e8 q^{28} -6.37538e8 q^{29} -1.10517e9 q^{35} +6.23431e9 q^{36} +9.48395e9 q^{41} +7.96183e9 q^{45} -2.40418e10 q^{48} +3.07950e10 q^{49} +1.67269e10 q^{51} -7.91454e9 q^{53} +1.34552e11 q^{57} +4.21805e10 q^{59} -3.07037e10 q^{60} -3.21568e11 q^{63} +6.87195e10 q^{64} -4.78111e10 q^{68} +2.10865e11 q^{71} +3.10642e11 q^{75} -3.84596e11 q^{76} -4.02739e11 q^{79} +8.77616e10 q^{80} +1.22532e12 q^{81} +1.24008e12 q^{84} -6.10596e10 q^{85} +9.13591e11 q^{87} -4.91167e11 q^{95} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −1433.00 −1.96571 −0.982853 − 0.184390i \(-0.940969\pi\)
−0.982853 + 0.184390i \(0.940969\pi\)
\(4\) 4096.00 1.00000
\(5\) 5231.00 0.334784 0.167392 − 0.985890i \(-0.446465\pi\)
0.167392 + 0.985890i \(0.446465\pi\)
\(6\) 0 0
\(7\) −211273. −1.79579 −0.897895 − 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(8\) 0 0
\(9\) 1.52205e6 2.86400
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −5.86957e6 −1.96571
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −7.49602e6 −0.658087
\(16\) 1.67772e7 1.00000
\(17\) −1.16726e7 −0.483588 −0.241794 − 0.970328i \(-0.577736\pi\)
−0.241794 + 0.970328i \(0.577736\pi\)
\(18\) 0 0
\(19\) −9.38955e7 −1.99583 −0.997914 − 0.0645530i \(-0.979438\pi\)
−0.997914 + 0.0645530i \(0.979438\pi\)
\(20\) 2.14262e7 0.334784
\(21\) 3.02754e8 3.53000
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −2.16777e8 −0.887920
\(26\) 0 0
\(27\) −1.41954e9 −3.66408
\(28\) −8.65374e8 −1.79579
\(29\) −6.37538e8 −1.07181 −0.535905 − 0.844278i \(-0.680029\pi\)
−0.535905 + 0.844278i \(0.680029\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.10517e9 −0.601202
\(36\) 6.23431e9 2.86400
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.48395e9 1.99658 0.998288 − 0.0584854i \(-0.0186271\pi\)
0.998288 + 0.0584854i \(0.0186271\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 7.96183e9 0.958822
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −2.40418e10 −1.96571
\(49\) 3.07950e10 2.22486
\(50\) 0 0
\(51\) 1.67269e10 0.950592
\(52\) 0 0
\(53\) −7.91454e9 −0.357084 −0.178542 − 0.983932i \(-0.557138\pi\)
−0.178542 + 0.983932i \(0.557138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.34552e11 3.92321
\(58\) 0 0
\(59\) 4.21805e10 1.00000
\(60\) −3.07037e10 −0.658087
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −3.21568e11 −5.14315
\(64\) 6.87195e10 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −4.78111e10 −0.483588
\(69\) 0 0
\(70\) 0 0
\(71\) 2.10865e11 1.64609 0.823047 − 0.567974i \(-0.192272\pi\)
0.823047 + 0.567974i \(0.192272\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.10642e11 1.74539
\(76\) −3.84596e11 −1.99583
\(77\) 0 0
\(78\) 0 0
\(79\) −4.02739e11 −1.65677 −0.828383 − 0.560163i \(-0.810739\pi\)
−0.828383 + 0.560163i \(0.810739\pi\)
\(80\) 8.77616e10 0.334784
\(81\) 1.22532e12 4.33850
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.24008e12 3.53000
\(85\) −6.10596e10 −0.161898
\(86\) 0 0
\(87\) 9.13591e11 2.10686
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.91167e11 −0.668171
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.87920e11 −0.887920
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 1.58371e12 1.18179
\(106\) 0 0
\(107\) 2.80512e12 1.86917 0.934585 − 0.355739i \(-0.115771\pi\)
0.934585 + 0.355739i \(0.115771\pi\)
\(108\) −5.81444e12 −3.66408
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.54457e12 −1.79579
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.61135e12 −1.07181
\(117\) 0 0
\(118\) 0 0
\(119\) 2.46611e12 0.868423
\(120\) 0 0
\(121\) 3.13843e12 1.00000
\(122\) 0 0
\(123\) −1.35905e13 −3.92468
\(124\) 0 0
\(125\) −2.41106e12 −0.632045
\(126\) 0 0
\(127\) 3.32330e12 0.792040 0.396020 − 0.918242i \(-0.370391\pi\)
0.396020 + 0.918242i \(0.370391\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.98376e13 3.58409
\(134\) 0 0
\(135\) −7.42561e12 −1.22668
\(136\) 0 0
\(137\) 1.09912e13 1.66235 0.831174 − 0.556013i \(-0.187670\pi\)
0.831174 + 0.556013i \(0.187670\pi\)
\(138\) 0 0
\(139\) −1.20853e13 −1.67559 −0.837797 − 0.545982i \(-0.816157\pi\)
−0.837797 + 0.545982i \(0.816157\pi\)
\(140\) −4.52677e12 −0.601202
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.55357e13 2.86400
\(145\) −3.33496e12 −0.358825
\(146\) 0 0
\(147\) −4.41292e13 −4.37343
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1.77663e13 −1.38500
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 1.13415e13 0.701923
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.21620e13 1.18163 0.590816 − 0.806806i \(-0.298806\pi\)
0.590816 + 0.806806i \(0.298806\pi\)
\(164\) 3.88462e13 1.99658
\(165\) 0 0
\(166\) 0 0
\(167\) −4.03322e13 −1.85931 −0.929657 − 0.368426i \(-0.879897\pi\)
−0.929657 + 0.368426i \(0.879897\pi\)
\(168\) 0 0
\(169\) 2.32981e13 1.00000
\(170\) 0 0
\(171\) −1.42913e14 −5.71606
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.57992e13 1.59452
\(176\) 0 0
\(177\) −6.04447e13 −1.96571
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 3.26117e13 0.958822
\(181\) 5.36099e13 1.52466 0.762330 − 0.647188i \(-0.224055\pi\)
0.762330 + 0.647188i \(0.224055\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.99910e14 6.57992
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −9.84750e13 −1.96571
\(193\) 9.54988e12 0.184780 0.0923898 − 0.995723i \(-0.470549\pi\)
0.0923898 + 0.995723i \(0.470549\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.26136e14 2.22486
\(197\) −1.14922e14 −1.96610 −0.983050 − 0.183339i \(-0.941310\pi\)
−0.983050 + 0.183339i \(0.941310\pi\)
\(198\) 0 0
\(199\) 5.85212e13 0.942311 0.471156 − 0.882050i \(-0.343837\pi\)
0.471156 + 0.882050i \(0.343837\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.34694e14 1.92475
\(204\) 6.85133e13 0.950592
\(205\) 4.96105e13 0.668422
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −3.24180e13 −0.357084
\(213\) −3.02170e14 −3.23574
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00350e14 −0.815997 −0.407998 − 0.912983i \(-0.633773\pi\)
−0.407998 + 0.912983i \(0.633773\pi\)
\(224\) 0 0
\(225\) −3.29945e14 −2.54300
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 5.51126e14 3.92321
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.72771e14 1.00000
\(237\) 5.77125e14 3.25671
\(238\) 0 0
\(239\) 1.24924e14 0.670286 0.335143 − 0.942167i \(-0.391215\pi\)
0.335143 + 0.942167i \(0.391215\pi\)
\(240\) −1.25762e14 −0.658087
\(241\) −3.59601e14 −1.83535 −0.917676 − 0.397331i \(-0.869937\pi\)
−0.917676 + 0.397331i \(0.869937\pi\)
\(242\) 0 0
\(243\) −1.00148e15 −4.86415
\(244\) 0 0
\(245\) 1.61089e14 0.744849
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.08732e14 1.23464 0.617318 − 0.786714i \(-0.288219\pi\)
0.617318 + 0.786714i \(0.288219\pi\)
\(252\) −1.31714e15 −5.14315
\(253\) 0 0
\(254\) 0 0
\(255\) 8.74984e13 0.318243
\(256\) 2.81475e14 1.00000
\(257\) −5.24435e13 −0.182009 −0.0910045 − 0.995850i \(-0.529008\pi\)
−0.0910045 + 0.995850i \(0.529008\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −9.70363e14 −3.06967
\(262\) 0 0
\(263\) 5.54573e14 1.67581 0.837904 − 0.545817i \(-0.183781\pi\)
0.837904 + 0.545817i \(0.183781\pi\)
\(264\) 0 0
\(265\) −4.14010e13 −0.119546
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 3.45196e14 0.871466 0.435733 − 0.900076i \(-0.356489\pi\)
0.435733 + 0.900076i \(0.356489\pi\)
\(272\) −1.95834e14 −0.483588
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.96696e14 1.76366 0.881828 − 0.471571i \(-0.156313\pi\)
0.881828 + 0.471571i \(0.156313\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.78209e14 1.98698 0.993491 − 0.113915i \(-0.0363391\pi\)
0.993491 + 0.113915i \(0.0363391\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 8.63703e14 1.64609
\(285\) 7.03843e14 1.31343
\(286\) 0 0
\(287\) −2.00370e15 −3.58543
\(288\) 0 0
\(289\) −4.46372e14 −0.766143
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.11100e14 −1.28194 −0.640971 − 0.767565i \(-0.721468\pi\)
−0.640971 + 0.767565i \(0.721468\pi\)
\(294\) 0 0
\(295\) 2.20646e14 0.334784
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 1.27239e15 1.74539
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.57531e15 −1.99583
\(305\) 0 0
\(306\) 0 0
\(307\) 3.02818e14 0.361703 0.180851 − 0.983510i \(-0.442115\pi\)
0.180851 + 0.983510i \(0.442115\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.55858e15 −1.72253 −0.861266 − 0.508154i \(-0.830328\pi\)
−0.861266 + 0.508154i \(0.830328\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −1.68212e15 −1.72184
\(316\) −1.64962e15 −1.65677
\(317\) 2.02359e15 1.99419 0.997097 − 0.0761432i \(-0.0242606\pi\)
0.997097 + 0.0761432i \(0.0242606\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3.59472e14 0.334784
\(321\) −4.01974e15 −3.67424
\(322\) 0 0
\(323\) 1.09601e15 0.965159
\(324\) 5.01892e15 4.33850
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.53632e15 1.92857 0.964287 − 0.264859i \(-0.0853254\pi\)
0.964287 + 0.264859i \(0.0853254\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 5.07937e15 3.53000
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −2.50100e14 −0.161898
\(341\) 0 0
\(342\) 0 0
\(343\) −3.58186e15 −2.19960
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 3.74207e15 2.10686
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 1.10304e15 0.551086
\(356\) 0 0
\(357\) −3.53394e15 −1.70706
\(358\) 0 0
\(359\) −3.47759e15 −1.62447 −0.812235 − 0.583331i \(-0.801749\pi\)
−0.812235 + 0.583331i \(0.801749\pi\)
\(360\) 0 0
\(361\) 6.60305e15 2.98333
\(362\) 0 0
\(363\) −4.49737e15 −1.96571
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 1.44350e16 5.71820
\(370\) 0 0
\(371\) 1.67213e15 0.641248
\(372\) 0 0
\(373\) 9.81807e14 0.364563 0.182282 − 0.983246i \(-0.441652\pi\)
0.182282 + 0.983246i \(0.441652\pi\)
\(374\) 0 0
\(375\) 3.45505e15 1.24242
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.23433e15 −0.753897 −0.376949 − 0.926234i \(-0.623027\pi\)
−0.376949 + 0.926234i \(0.623027\pi\)
\(380\) −2.01182e15 −0.668171
\(381\) −4.76229e15 −1.55692
\(382\) 0 0
\(383\) 8.03549e14 0.254577 0.127289 − 0.991866i \(-0.459373\pi\)
0.127289 + 0.991866i \(0.459373\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.43722e15 −1.85781 −0.928904 − 0.370322i \(-0.879247\pi\)
−0.928904 + 0.370322i \(0.879247\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.10673e15 −0.554659
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) −2.84273e16 −7.04527
\(400\) −3.63692e15 −0.887920
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 6.40966e15 1.45246
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −1.57504e16 −3.26769
\(412\) 0 0
\(413\) −8.91161e15 −1.79579
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.73182e16 3.29373
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 6.48686e15 1.18179
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.53036e15 0.429387
\(426\) 0 0
\(427\) 0 0
\(428\) 1.14898e16 1.86917
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −2.38159e16 −3.66408
\(433\) −1.05325e16 −1.59809 −0.799047 − 0.601269i \(-0.794662\pi\)
−0.799047 + 0.601269i \(0.794662\pi\)
\(434\) 0 0
\(435\) 4.77900e15 0.705344
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.09145e16 1.52481 0.762407 − 0.647098i \(-0.224018\pi\)
0.762407 + 0.647098i \(0.224018\pi\)
\(440\) 0 0
\(441\) 4.68715e16 6.37202
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.45186e16 −1.79579
\(449\) −7.83092e15 −0.955729 −0.477865 − 0.878434i \(-0.658589\pi\)
−0.477865 + 0.878434i \(0.658589\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 1.65698e16 1.77191
\(460\) 0 0
\(461\) −1.90007e16 −1.97953 −0.989767 − 0.142692i \(-0.954424\pi\)
−0.989767 + 0.142692i \(0.954424\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −1.06961e16 −1.07181
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.03544e16 1.77214
\(476\) 1.01012e16 0.868423
\(477\) −1.20463e16 −1.02269
\(478\) 0 0
\(479\) 2.32252e16 1.92286 0.961428 − 0.275057i \(-0.0886968\pi\)
0.961428 + 0.275057i \(0.0886968\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.28550e16 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −1.74388e15 −0.130720 −0.0653601 − 0.997862i \(-0.520820\pi\)
−0.0653601 + 0.997862i \(0.520820\pi\)
\(488\) 0 0
\(489\) −3.17581e16 −2.32274
\(490\) 0 0
\(491\) −9.03525e15 −0.644839 −0.322419 − 0.946597i \(-0.604496\pi\)
−0.322419 + 0.946597i \(0.604496\pi\)
\(492\) −5.56667e16 −3.92468
\(493\) 7.44175e15 0.518314
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.45501e16 −2.95604
\(498\) 0 0
\(499\) −5.89900e15 −0.382099 −0.191049 − 0.981580i \(-0.561189\pi\)
−0.191049 + 0.981580i \(0.561189\pi\)
\(500\) −9.87571e15 −0.632045
\(501\) 5.77960e16 3.65487
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.33862e16 −1.96571
\(508\) 1.36122e16 0.792040
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.33288e17 7.31288
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.76845e16 −0.884233 −0.442116 − 0.896958i \(-0.645772\pi\)
−0.442116 + 0.896958i \(0.645772\pi\)
\(522\) 0 0
\(523\) 2.06136e16 1.00726 0.503632 − 0.863919i \(-0.331997\pi\)
0.503632 + 0.863919i \(0.331997\pi\)
\(524\) 0 0
\(525\) −6.56302e16 −3.13435
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 2.19146e16 1.00000
\(530\) 0 0
\(531\) 6.42008e16 2.86400
\(532\) 8.12548e16 3.58409
\(533\) 0 0
\(534\) 0 0
\(535\) 1.46736e16 0.625768
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −3.04153e16 −1.22668
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −7.68229e16 −2.99704
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.52924e16 −0.944206 −0.472103 − 0.881543i \(-0.656505\pi\)
−0.472103 + 0.881543i \(0.656505\pi\)
\(548\) 4.50200e16 1.66235
\(549\) 0 0
\(550\) 0 0
\(551\) 5.98619e16 2.13915
\(552\) 0 0
\(553\) 8.50878e16 2.97520
\(554\) 0 0
\(555\) 0 0
\(556\) −4.95014e16 −1.67559
\(557\) 1.94796e16 0.652301 0.326151 − 0.945318i \(-0.394248\pi\)
0.326151 + 0.945318i \(0.394248\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.85417e16 −0.601202
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.58877e17 −7.79105
\(568\) 0 0
\(569\) 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
\(574\) 0 0
\(575\) 0 0
\(576\) 1.04594e17 2.86400
\(577\) 1.18300e16 0.320575 0.160287 − 0.987070i \(-0.448758\pi\)
0.160287 + 0.987070i \(0.448758\pi\)
\(578\) 0 0
\(579\) −1.36850e16 −0.363222
\(580\) −1.36600e16 −0.358825
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −1.80753e17 −4.37343
\(589\) 0 0
\(590\) 0 0
\(591\) 1.64683e17 3.86477
\(592\) 0 0
\(593\) 5.08679e16 1.16981 0.584905 − 0.811102i \(-0.301132\pi\)
0.584905 + 0.811102i \(0.301132\pi\)
\(594\) 0 0
\(595\) 1.29002e16 0.290734
\(596\) 0 0
\(597\) −8.38608e16 −1.85231
\(598\) 0 0
\(599\) −6.39346e16 −1.38413 −0.692063 − 0.721838i \(-0.743298\pi\)
−0.692063 + 0.721838i \(0.743298\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.64171e16 0.334784
\(606\) 0 0
\(607\) −6.24965e16 −1.24946 −0.624731 − 0.780840i \(-0.714792\pi\)
−0.624731 + 0.780840i \(0.714792\pi\)
\(608\) 0 0
\(609\) −1.93017e17 −3.78349
\(610\) 0 0
\(611\) 0 0
\(612\) −7.27708e16 −1.38500
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −7.10919e16 −1.31392
\(616\) 0 0
\(617\) −6.71434e16 −1.21701 −0.608503 − 0.793552i \(-0.708230\pi\)
−0.608503 + 0.793552i \(0.708230\pi\)
\(618\) 0 0
\(619\) 9.45030e16 1.67997 0.839985 − 0.542609i \(-0.182563\pi\)
0.839985 + 0.542609i \(0.182563\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.03119e16 0.676321
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 2.51876e16 0.399035 0.199517 − 0.979894i \(-0.436063\pi\)
0.199517 + 0.979894i \(0.436063\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.73842e16 0.265162
\(636\) 4.64549e16 0.701923
\(637\) 0 0
\(638\) 0 0
\(639\) 3.20947e17 4.71442
\(640\) 0 0
\(641\) −8.55049e16 −1.23266 −0.616329 − 0.787489i \(-0.711381\pi\)
−0.616329 + 0.787489i \(0.711381\pi\)
\(642\) 0 0
\(643\) −1.10997e17 −1.57053 −0.785264 − 0.619161i \(-0.787473\pi\)
−0.785264 + 0.619161i \(0.787473\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.45563e17 1.98438 0.992189 − 0.124746i \(-0.0398116\pi\)
0.992189 + 0.124746i \(0.0398116\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 9.07753e16 1.18163
\(653\) −1.51113e17 −1.94905 −0.974525 − 0.224278i \(-0.927998\pi\)
−0.974525 + 0.224278i \(0.927998\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.59114e17 1.99658
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.82001e16 0.218205 0.109103 − 0.994030i \(-0.465202\pi\)
0.109103 + 0.994030i \(0.465202\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.03770e17 1.19990
\(666\) 0 0
\(667\) 0 0
\(668\) −1.65201e17 −1.85931
\(669\) 1.43802e17 1.60401
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 3.07724e17 3.25341
\(676\) 9.54290e16 1.00000
\(677\) 5.22923e16 0.543132 0.271566 − 0.962420i \(-0.412458\pi\)
0.271566 + 0.962420i \(0.412458\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −5.85374e17 −5.71606
\(685\) 5.74950e16 0.556527
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.32182e16 −0.560962
\(696\) 0 0
\(697\) −1.10703e17 −0.965520
\(698\) 0 0
\(699\) 0 0
\(700\) 1.87593e17 1.59452
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −2.47582e17 −1.96571
\(709\) −1.40150e16 −0.110336 −0.0551679 − 0.998477i \(-0.517569\pi\)
−0.0551679 + 0.998477i \(0.517569\pi\)
\(710\) 0 0
\(711\) −6.12988e17 −4.74498
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.79017e17 −1.31758
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.33577e17 0.958822
\(721\) 0 0
\(722\) 0 0
\(723\) 5.15309e17 3.60776
\(724\) 2.19586e17 1.52466
\(725\) 1.38204e17 0.951681
\(726\) 0 0
\(727\) 2.84723e17 1.92848 0.964242 − 0.265024i \(-0.0853800\pi\)
0.964242 + 0.265024i \(0.0853800\pi\)
\(728\) 0 0
\(729\) 7.83941e17 5.22298
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.65471e17 1.06684 0.533418 − 0.845852i \(-0.320907\pi\)
0.533418 + 0.845852i \(0.320907\pi\)
\(734\) 0 0
\(735\) −2.30840e17 −1.46415
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.37015e17 −0.814393 −0.407197 − 0.913341i \(-0.633494\pi\)
−0.407197 + 0.913341i \(0.633494\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.92646e17 −3.35664
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −4.42412e17 −2.42693
\(754\) 0 0
\(755\) 0 0
\(756\) 1.22843e18 6.57992
\(757\) −1.52503e17 −0.810407 −0.405204 − 0.914226i \(-0.632799\pi\)
−0.405204 + 0.914226i \(0.632799\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.96486e17 −1.52650 −0.763249 − 0.646105i \(-0.776397\pi\)
−0.763249 + 0.646105i \(0.776397\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −9.29356e16 −0.463675
\(766\) 0 0
\(767\) 0 0
\(768\) −4.03354e17 −1.96571
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 7.51516e16 0.357776
\(772\) 3.91163e16 0.184780
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.90500e17 −3.98482
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 9.05010e17 3.92720
\(784\) 5.16654e17 2.22486
\(785\) 0 0
\(786\) 0 0
\(787\) 2.56822e17 1.08089 0.540447 − 0.841378i \(-0.318255\pi\)
0.540447 + 0.841378i \(0.318255\pi\)
\(788\) −4.70720e17 −1.96610
\(789\) −7.94703e17 −3.29415
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 5.93276e16 0.234992
\(796\) 2.39703e17 0.942311
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 5.51709e17 1.92475
\(813\) −4.94666e17 −1.71305
\(814\) 0 0
\(815\) 1.15929e17 0.395592
\(816\) 2.80631e17 0.950592
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 2.03205e17 0.668422
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.34149e16 −0.0731913 −0.0365957 − 0.999330i \(-0.511651\pi\)
−0.0365957 + 0.999330i \(0.511651\pi\)
\(828\) 0 0
\(829\) −5.26429e17 −1.62186 −0.810929 − 0.585145i \(-0.801038\pi\)
−0.810929 + 0.585145i \(0.801038\pi\)
\(830\) 0 0
\(831\) −1.14166e18 −3.46683
\(832\) 0 0
\(833\) −3.59459e17 −1.07592
\(834\) 0 0
\(835\) −2.10978e17 −0.622469
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.26395e16 0.148777
\(842\) 0 0
\(843\) −1.40177e18 −3.90582
\(844\) 0 0
\(845\) 1.21872e17 0.334784
\(846\) 0 0
\(847\) −6.63065e17 −1.79579
\(848\) −1.32784e17 −0.357084
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −1.23769e18 −3.23574
\(853\) 3.30817e17 0.858802 0.429401 − 0.903114i \(-0.358725\pi\)
0.429401 + 0.903114i \(0.358725\pi\)
\(854\) 0 0
\(855\) −7.47580e17 −1.91364
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 2.87130e18 7.04791
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.39651e17 1.50601
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.09392e17 1.13502
\(876\) 0 0
\(877\) 5.55090e17 1.22002 0.610008 − 0.792395i \(-0.291166\pi\)
0.610008 + 0.792395i \(0.291166\pi\)
\(878\) 0 0
\(879\) 1.16231e18 2.51992
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 9.47898e17 1.99985 0.999924 − 0.0122949i \(-0.00391368\pi\)
0.999924 + 0.0122949i \(0.00391368\pi\)
\(884\) 0 0
\(885\) −3.16186e17 −0.658087
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −7.02124e17 −1.42234
\(890\) 0 0
\(891\) 0 0
\(892\) −4.11034e17 −0.815997
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.35146e18 −2.54300
\(901\) 9.23836e16 0.172682
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.80433e17 0.510432
\(906\) 0 0
\(907\) 3.30541e15 0.00593720 0.00296860 − 0.999996i \(-0.499055\pi\)
0.00296860 + 0.999996i \(0.499055\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 9.05913e17 1.58481 0.792403 − 0.609997i \(-0.208830\pi\)
0.792403 + 0.609997i \(0.208830\pi\)
\(912\) 2.25741e18 3.92321
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −4.33938e17 −0.711001
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −2.89151e18 −4.44045
\(932\) 0 0
\(933\) 2.23345e18 3.38599
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 7.07672e17 1.00000
\(945\) 1.56883e18 2.20285
\(946\) 0 0
\(947\) 2.87048e17 0.397974 0.198987 − 0.980002i \(-0.436235\pi\)
0.198987 + 0.980002i \(0.436235\pi\)
\(948\) 2.36390e18 3.25671
\(949\) 0 0
\(950\) 0 0
\(951\) −2.89981e18 −3.92000
\(952\) 0 0
\(953\) 5.87310e17 0.783989 0.391994 − 0.919968i \(-0.371785\pi\)
0.391994 + 0.919968i \(0.371785\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.11690e17 0.670286
\(957\) 0 0
\(958\) 0 0
\(959\) −2.32214e18 −2.98523
\(960\) −5.15123e17 −0.658087
\(961\) 7.87663e17 1.00000
\(962\) 0 0
\(963\) 4.26953e18 5.35331
\(964\) −1.47293e18 −1.83535
\(965\) 4.99554e16 0.0618612
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) −1.57058e18 −1.89722
\(970\) 0 0
\(971\) 3.69989e17 0.441442 0.220721 − 0.975337i \(-0.429159\pi\)
0.220721 + 0.975337i \(0.429159\pi\)
\(972\) −4.10208e18 −4.86415
\(973\) 2.55330e18 3.00902
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.59819e17 0.744849
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −6.01157e17 −0.658219
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −3.63455e18 −3.79101
\(994\) 0 0
\(995\) 3.06124e17 0.315471
\(996\) 0 0
\(997\) 1.43784e18 1.46400 0.732000 − 0.681305i \(-0.238587\pi\)
0.732000 + 0.681305i \(0.238587\pi\)
\(998\) 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 59.13.b.a.58.1 ✓ 1
59.58 odd 2 CM 59.13.b.a.58.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
59.13.b.a.58.1 ✓ 1 1.1 even 1 trivial
59.13.b.a.58.1 ✓ 1 59.58 odd 2 CM