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Properties

Label 59.19.b.a.58.1
Level $59$
Weight $19$
Character 59.58
Self dual yes
Analytic conductor $121.178$
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] = [59,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 19 \)
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: \(121.177821249\)
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)
 
\(f(q)\) \(=\) \(q+10810.0 q^{3} +262144. q^{4} -1.98525e6 q^{5} -5.09829e7 q^{7} -2.70564e8 q^{9} +O(q^{10})\) \(q+10810.0 q^{3} +262144. q^{4} -1.98525e6 q^{5} -5.09829e7 q^{7} -2.70564e8 q^{9} +2.83378e9 q^{12} -2.14606e10 q^{15} +6.87195e10 q^{16} -2.16652e11 q^{17} -6.24370e10 q^{19} -5.20422e11 q^{20} -5.51125e11 q^{21} +1.26536e11 q^{25} -7.11282e12 q^{27} -1.33649e13 q^{28} +2.89568e13 q^{29} +1.01214e14 q^{35} -7.09268e13 q^{36} +6.52243e14 q^{41} +5.37139e14 q^{45} +7.42858e14 q^{48} +9.70844e14 q^{49} -2.34201e15 q^{51} +5.73981e15 q^{53} -6.74944e14 q^{57} -8.66300e15 q^{59} -5.62577e15 q^{60} +1.37942e16 q^{63} +1.80144e16 q^{64} -5.67940e16 q^{68} -5.65633e16 q^{71} +1.36786e15 q^{75} -1.63675e16 q^{76} +1.86549e17 q^{79} -1.36426e17 q^{80} +2.79326e16 q^{81} -1.44474e17 q^{84} +4.30109e17 q^{85} +3.13023e17 q^{87} +1.23953e17 q^{95} +O(q^{100})\)

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 \(\pi\)
\(3\) 10810.0 0.549205 0.274602 − 0.961558i \(-0.411454\pi\)
0.274602 + 0.961558i \(0.411454\pi\)
\(4\) 262144. 1.00000
\(5\) −1.98525e6 −1.01645 −0.508225 − 0.861224i \(-0.669698\pi\)
−0.508225 + 0.861224i \(0.669698\pi\)
\(6\) 0 0
\(7\) −5.09829e7 −1.26340 −0.631702 − 0.775211i \(-0.717643\pi\)
−0.631702 + 0.775211i \(0.717643\pi\)
\(8\) 0 0
\(9\) −2.70564e8 −0.698374
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2.83378e9 0.549205
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −2.14606e10 −0.558239
\(16\) 6.87195e10 1.00000
\(17\) −2.16652e11 −1.82693 −0.913465 − 0.406917i \(-0.866604\pi\)
−0.913465 + 0.406917i \(0.866604\pi\)
\(18\) 0 0
\(19\) −6.24370e10 −0.193491 −0.0967453 − 0.995309i \(-0.530843\pi\)
−0.0967453 + 0.995309i \(0.530843\pi\)
\(20\) −5.20422e11 −1.01645
\(21\) −5.51125e11 −0.693868
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.26536e11 0.0331707
\(26\) 0 0
\(27\) −7.11282e12 −0.932755
\(28\) −1.33649e13 −1.26340
\(29\) 2.89568e13 1.99604 0.998018 − 0.0629293i \(-0.0200442\pi\)
0.998018 + 0.0629293i \(0.0200442\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.01214e14 1.28419
\(36\) −7.09268e13 −0.698374
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.52243e14 1.99230 0.996150 − 0.0876656i \(-0.0279407\pi\)
0.996150 + 0.0876656i \(0.0279407\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 5.37139e14 0.709862
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 7.42858e14 0.549205
\(49\) 9.70844e14 0.596190
\(50\) 0 0
\(51\) −2.34201e15 −1.00336
\(52\) 0 0
\(53\) 5.73981e15 1.73946 0.869730 − 0.493528i \(-0.164293\pi\)
0.869730 + 0.493528i \(0.164293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.74944e14 −0.106266
\(58\) 0 0
\(59\) −8.66300e15 −1.00000
\(60\) −5.62577e15 −0.558239
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.37942e16 0.882329
\(64\) 1.80144e16 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −5.67940e16 −1.82693
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65633e16 −1.23370 −0.616850 − 0.787081i \(-0.711591\pi\)
−0.616850 + 0.787081i \(0.711591\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.36786e15 0.0182175
\(76\) −1.63675e16 −0.193491
\(77\) 0 0
\(78\) 0 0
\(79\) 1.86549e17 1.55650 0.778249 − 0.627955i \(-0.216108\pi\)
0.778249 + 0.627955i \(0.216108\pi\)
\(80\) −1.36426e17 −1.01645
\(81\) 2.79326e16 0.186100
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.44474e17 −0.693868
\(85\) 4.30109e17 1.85698
\(86\) 0 0
\(87\) 3.13023e17 1.09623
\(88\) 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
\(95\) 1.23953e17 0.196674
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3.31707e16 0.0331707
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 1.09412e18 0.705282
\(106\) 0 0
\(107\) −3.14317e18 −1.70968 −0.854838 − 0.518894i \(-0.826344\pi\)
−0.854838 + 0.518894i \(0.826344\pi\)
\(108\) −1.86458e18 −0.932755
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.50352e18 −1.26340
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.59085e18 1.99604
\(117\) 0 0
\(118\) 0 0
\(119\) 1.10455e19 2.30815
\(120\) 0 0
\(121\) 5.55992e18 1.00000
\(122\) 0 0
\(123\) 7.05075e18 1.09418
\(124\) 0 0
\(125\) 7.32194e18 0.982734
\(126\) 0 0
\(127\) 2.98658e18 0.347488 0.173744 − 0.984791i \(-0.444413\pi\)
0.173744 + 0.984791i \(0.444413\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 3.18322e18 0.244457
\(134\) 0 0
\(135\) 1.41207e19 0.948099
\(136\) 0 0
\(137\) −2.15502e19 −1.26755 −0.633775 − 0.773517i \(-0.718496\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(138\) 0 0
\(139\) 2.95187e19 1.52393 0.761964 − 0.647619i \(-0.224235\pi\)
0.761964 + 0.647619i \(0.224235\pi\)
\(140\) 2.65326e19 1.28419
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.85930e19 −0.698374
\(145\) −5.74866e19 −2.02887
\(146\) 0 0
\(147\) 1.04948e19 0.327430
\(148\) 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
\(153\) 5.86182e19 1.27588
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 6.20473e19 0.955320
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.63152e19 0.323980 0.161990 − 0.986792i \(-0.448209\pi\)
0.161990 + 0.986792i \(0.448209\pi\)
\(164\) 1.70982e20 1.99230
\(165\) 0 0
\(166\) 0 0
\(167\) 1.08352e20 1.07247 0.536237 − 0.844067i \(-0.319845\pi\)
0.536237 + 0.844067i \(0.319845\pi\)
\(168\) 0 0
\(169\) 1.12455e20 1.00000
\(170\) 0 0
\(171\) 1.68932e19 0.135129
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −6.45118e18 −0.0419080
\(176\) 0 0
\(177\) −9.36470e19 −0.549205
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 1.40808e20 0.709862
\(181\) 2.05373e20 0.985002 0.492501 − 0.870312i \(-0.336083\pi\)
0.492501 + 0.870312i \(0.336083\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.62632e20 1.17845
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.94736e20 0.549205
\(193\) 4.47708e20 1.20498 0.602489 − 0.798127i \(-0.294176\pi\)
0.602489 + 0.798127i \(0.294176\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.54501e20 0.596190
\(197\) 2.44053e20 0.546120 0.273060 − 0.961997i \(-0.411964\pi\)
0.273060 + 0.961997i \(0.411964\pi\)
\(198\) 0 0
\(199\) −4.84298e19 −0.0989543 −0.0494772 − 0.998775i \(-0.515755\pi\)
−0.0494772 + 0.998775i \(0.515755\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.47630e21 −2.52180
\(204\) −6.13943e20 −1.00336
\(205\) −1.29487e21 −2.02507
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.50466e21 1.73946
\(213\) −6.11449e20 −0.677554
\(214\) 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
\(223\) −2.69485e21 −1.97602 −0.988010 − 0.154387i \(-0.950660\pi\)
−0.988010 + 0.154387i \(0.950660\pi\)
\(224\) 0 0
\(225\) −3.42362e19 −0.0231656
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −1.76933e20 −0.106266
\(229\) 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
\(236\) −2.27095e21 −1.00000
\(237\) 2.01659e21 0.854837
\(238\) 0 0
\(239\) 1.37087e21 0.538786 0.269393 − 0.963030i \(-0.413177\pi\)
0.269393 + 0.963030i \(0.413177\pi\)
\(240\) −1.47476e21 −0.558239
\(241\) −3.15529e21 −1.15050 −0.575249 − 0.817978i \(-0.695095\pi\)
−0.575249 + 0.817978i \(0.695095\pi\)
\(242\) 0 0
\(243\) 3.05760e21 1.03496
\(244\) 0 0
\(245\) −1.92737e21 −0.605997
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.66867e21 −0.421994 −0.210997 − 0.977487i \(-0.567671\pi\)
−0.210997 + 0.977487i \(0.567671\pi\)
\(252\) 3.61606e21 0.882329
\(253\) 0 0
\(254\) 0 0
\(255\) 4.64948e21 1.01986
\(256\) 4.72237e21 1.00000
\(257\) −7.79498e21 −1.59374 −0.796869 − 0.604153i \(-0.793512\pi\)
−0.796869 + 0.604153i \(0.793512\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.83467e21 −1.39398
\(262\) 0 0
\(263\) −7.80013e21 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(264\) 0 0
\(265\) −1.13950e22 −1.76807
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.71709e21 −0.217806 −0.108903 − 0.994052i \(-0.534734\pi\)
−0.108903 + 0.994052i \(0.534734\pi\)
\(272\) −1.48882e22 −1.82693
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.42240e22 1.48150 0.740751 − 0.671779i \(-0.234470\pi\)
0.740751 + 0.671779i \(0.234470\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.15272e22 1.97075 0.985374 − 0.170408i \(-0.0545086\pi\)
0.985374 + 0.170408i \(0.0545086\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −1.48277e22 −1.23370
\(285\) 1.33994e21 0.108014
\(286\) 0 0
\(287\) −3.32532e22 −2.51708
\(288\) 0 0
\(289\) 3.28749e22 2.33767
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.07746e22 −1.93368 −0.966839 − 0.255386i \(-0.917798\pi\)
−0.966839 + 0.255386i \(0.917798\pi\)
\(294\) 0 0
\(295\) 1.71982e22 1.01645
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 3.58575e20 0.0182175
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −4.29064e21 −0.193491
\(305\) 0 0
\(306\) 0 0
\(307\) 2.37619e22 0.980925 0.490462 − 0.871462i \(-0.336828\pi\)
0.490462 + 0.871462i \(0.336828\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.90322e22 1.43410 0.717049 − 0.697022i \(-0.245492\pi\)
0.717049 + 0.697022i \(0.245492\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −2.73849e22 −0.896843
\(316\) 4.89027e22 1.55650
\(317\) 6.42272e22 1.98694 0.993472 − 0.114077i \(-0.0363909\pi\)
0.993472 + 0.114077i \(0.0363909\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.57632e22 −1.01645
\(321\) −3.39777e22 −0.938963
\(322\) 0 0
\(323\) 1.35271e22 0.353494
\(324\) 7.32237e21 0.186100
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.77781e22 1.84049 0.920246 − 0.391339i \(-0.127988\pi\)
0.920246 + 0.391339i \(0.127988\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −3.78730e22 −0.693868
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1.12750e23 1.85698
\(341\) 0 0
\(342\) 0 0
\(343\) 3.35248e22 0.510175
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 8.20571e22 1.09623
\(349\) 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
\(355\) 1.12292e23 1.25399
\(356\) 0 0
\(357\) 1.19402e23 1.26765
\(358\) 0 0
\(359\) −1.59299e23 −1.60829 −0.804145 − 0.594433i \(-0.797376\pi\)
−0.804145 + 0.594433i \(0.797376\pi\)
\(360\) 0 0
\(361\) −1.00229e23 −0.962561
\(362\) 0 0
\(363\) 6.01027e22 0.549205
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.76474e23 −1.39137
\(370\) 0 0
\(371\) −2.92632e23 −2.19764
\(372\) 0 0
\(373\) 1.36561e23 0.977120 0.488560 − 0.872530i \(-0.337522\pi\)
0.488560 + 0.872530i \(0.337522\pi\)
\(374\) 0 0
\(375\) 7.91501e22 0.539722
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.15889e23 −1.95786 −0.978929 − 0.204202i \(-0.934540\pi\)
−0.978929 + 0.204202i \(0.934540\pi\)
\(380\) 3.24936e22 0.196674
\(381\) 3.22849e22 0.190842
\(382\) 0 0
\(383\) 1.98483e23 1.11927 0.559635 − 0.828739i \(-0.310941\pi\)
0.559635 + 0.828739i \(0.310941\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.19795e23 1.07764 0.538818 − 0.842422i \(-0.318871\pi\)
0.538818 + 0.842422i \(0.318871\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.70347e23 −1.58210
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 3.44106e22 0.134257
\(400\) 8.69550e21 0.0331707
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −5.54534e22 −0.189162
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −2.32957e23 −0.696145
\(412\) 0 0
\(413\) 4.41665e23 1.26340
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.19098e23 0.836949
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 2.86818e23 0.705282
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.74143e22 −0.0606005
\(426\) 0 0
\(427\) 0 0
\(428\) −8.23963e23 −1.70968
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −4.88789e23 −0.932755
\(433\) −8.81268e23 −1.64709 −0.823545 − 0.567252i \(-0.808007\pi\)
−0.823545 + 0.567252i \(0.808007\pi\)
\(434\) 0 0
\(435\) −6.21430e23 −1.11427
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −5.96697e23 −0.985310 −0.492655 − 0.870225i \(-0.663974\pi\)
−0.492655 + 0.870225i \(0.663974\pi\)
\(440\) 0 0
\(441\) −2.62676e23 −0.416363
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −9.18426e23 −1.26340
\(449\) −1.48229e24 −1.99855 −0.999276 − 0.0380557i \(-0.987884\pi\)
−0.999276 + 0.0380557i \(0.987884\pi\)
\(450\) 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
\(459\) 1.54100e24 1.70408
\(460\) 0 0
\(461\) −4.00839e23 −0.426247 −0.213124 − 0.977025i \(-0.568364\pi\)
−0.213124 + 0.977025i \(0.568364\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.98990e24 1.99604
\(465\) 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
\(475\) −7.90054e21 −0.00641822
\(476\) 2.89552e24 2.30815
\(477\) −1.55299e24 −1.21479
\(478\) 0 0
\(479\) 2.42636e24 1.82783 0.913913 − 0.405909i \(-0.133045\pi\)
0.913913 + 0.405909i \(0.133045\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.45750e24 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −2.38206e24 −1.54594 −0.772969 − 0.634443i \(-0.781229\pi\)
−0.772969 + 0.634443i \(0.781229\pi\)
\(488\) 0 0
\(489\) 2.84467e23 0.177931
\(490\) 0 0
\(491\) −3.17579e24 −1.91478 −0.957390 − 0.288799i \(-0.906744\pi\)
−0.957390 + 0.288799i \(0.906744\pi\)
\(492\) 1.84831e24 1.09418
\(493\) −6.27354e24 −3.64662
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.88376e24 1.55866
\(498\) 0 0
\(499\) 3.37225e24 1.75798 0.878992 − 0.476836i \(-0.158216\pi\)
0.878992 + 0.476836i \(0.158216\pi\)
\(500\) 1.91940e24 0.982734
\(501\) 1.17128e24 0.589008
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.21564e24 0.549205
\(508\) 7.82913e23 0.347488
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.44103e23 0.180479
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.62938e24 −1.99031 −0.995156 − 0.0983038i \(-0.968658\pi\)
−0.995156 + 0.0983038i \(0.968658\pi\)
\(522\) 0 0
\(523\) −3.68744e22 −0.0125953 −0.00629767 − 0.999980i \(-0.502005\pi\)
−0.00629767 + 0.999980i \(0.502005\pi\)
\(524\) 0 0
\(525\) −6.97373e22 −0.0230161
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24415e24 1.00000
\(530\) 0 0
\(531\) 2.34390e24 0.698374
\(532\) 8.34463e23 0.244457
\(533\) 0 0
\(534\) 0 0
\(535\) 6.23999e24 1.73780
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 3.70167e24 0.948099
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 2.22009e24 0.540968
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.75826e24 1.99771 0.998854 − 0.0478677i \(-0.0152426\pi\)
0.998854 + 0.0478677i \(0.0152426\pi\)
\(548\) −5.64924e24 −1.26755
\(549\) 0 0
\(550\) 0 0
\(551\) −1.80798e24 −0.386214
\(552\) 0 0
\(553\) −9.51080e24 −1.96649
\(554\) 0 0
\(555\) 0 0
\(556\) 7.73816e24 1.52393
\(557\) −2.92221e24 −0.566258 −0.283129 − 0.959082i \(-0.591372\pi\)
−0.283129 + 0.959082i \(0.591372\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 6.95537e24 1.28419
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.42409e24 −0.235120
\(568\) 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
\(576\) −4.87405e24 −0.698374
\(577\) 7.33705e24 1.03500 0.517499 − 0.855684i \(-0.326863\pi\)
0.517499 + 0.855684i \(0.326863\pi\)
\(578\) 0 0
\(579\) 4.83972e24 0.661779
\(580\) −1.50698e25 −2.02887
\(581\) 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 \(\pi\)
\(588\) 2.75115e24 0.327430
\(589\) 0 0
\(590\) 0 0
\(591\) 2.63821e24 0.299932
\(592\) 0 0
\(593\) 2.74141e24 0.302331 0.151165 − 0.988509i \(-0.451697\pi\)
0.151165 + 0.988509i \(0.451697\pi\)
\(594\) 0 0
\(595\) −2.19282e25 −2.34612
\(596\) 0 0
\(597\) −5.23526e23 −0.0543462
\(598\) 0 0
\(599\) 1.85745e25 1.87101 0.935503 − 0.353318i \(-0.114947\pi\)
0.935503 + 0.353318i \(0.114947\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.10378e25 −1.01645
\(606\) 0 0
\(607\) −2.18003e25 −1.94879 −0.974395 − 0.224844i \(-0.927813\pi\)
−0.974395 + 0.224844i \(0.927813\pi\)
\(608\) 0 0
\(609\) −1.59588e25 −1.38498
\(610\) 0 0
\(611\) 0 0
\(612\) 1.53664e25 1.27588
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −1.39975e25 −1.11218
\(616\) 0 0
\(617\) −2.54222e25 −1.96176 −0.980881 − 0.194611i \(-0.937656\pi\)
−0.980881 + 0.194611i \(0.937656\pi\)
\(618\) 0 0
\(619\) 1.74032e25 1.30440 0.652202 − 0.758045i \(-0.273845\pi\)
0.652202 + 0.758045i \(0.273845\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.50186e25 −1.03207
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.47615e25 −0.930824 −0.465412 − 0.885094i \(-0.654094\pi\)
−0.465412 + 0.885094i \(0.654094\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.92911e24 −0.353204
\(636\) 1.62653e25 0.955320
\(637\) 0 0
\(638\) 0 0
\(639\) 1.53040e25 0.861584
\(640\) 0 0
\(641\) −3.57305e25 −1.95576 −0.977882 − 0.209155i \(-0.932929\pi\)
−0.977882 + 0.209155i \(0.932929\pi\)
\(642\) 0 0
\(643\) 3.16511e25 1.68457 0.842286 − 0.539031i \(-0.181210\pi\)
0.842286 + 0.539031i \(0.181210\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.90374e25 −1.96491 −0.982453 − 0.186509i \(-0.940283\pi\)
−0.982453 + 0.186509i \(0.940283\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 6.89837e24 0.323980
\(653\) 1.43705e25 0.665660 0.332830 − 0.942987i \(-0.391997\pi\)
0.332830 + 0.942987i \(0.391997\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.48218e25 1.99230
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −2.80483e25 −1.16438 −0.582188 − 0.813054i \(-0.697803\pi\)
−0.582188 + 0.813054i \(0.697803\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.31950e24 −0.248478
\(666\) 0 0
\(667\) 0 0
\(668\) 2.84037e25 1.07247
\(669\) −2.91314e25 −1.08524
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −9.00029e23 −0.0309401
\(676\) 2.94795e25 1.00000
\(677\) 2.17657e25 0.728576 0.364288 − 0.931286i \(-0.381312\pi\)
0.364288 + 0.931286i \(0.381312\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 4.42846e24 0.135129
\(685\) 4.27825e25 1.28840
\(686\) 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
\(695\) −5.86022e25 −1.54900
\(696\) 0 0
\(697\) −1.41310e26 −3.63979
\(698\) 0 0
\(699\) 0 0
\(700\) −1.69114e24 −0.0419080
\(701\) 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
\(708\) −2.45490e25 −0.549205
\(709\) 6.90976e25 1.52632 0.763162 − 0.646207i \(-0.223646\pi\)
0.763162 + 0.646207i \(0.223646\pi\)
\(710\) 0 0
\(711\) −5.04735e25 −1.08702
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.48192e25 0.295904
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 3.69119e25 0.709862
\(721\) 0 0
\(722\) 0 0
\(723\) −3.41087e25 −0.631860
\(724\) 5.38374e25 0.985002
\(725\) 3.66408e24 0.0662099
\(726\) 0 0
\(727\) 1.04399e26 1.84029 0.920146 − 0.391576i \(-0.128070\pi\)
0.920146 + 0.391576i \(0.128070\pi\)
\(728\) 0 0
\(729\) 2.22310e25 0.382306
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.14983e24 −0.117047 −0.0585235 − 0.998286i \(-0.518639\pi\)
−0.0585235 + 0.998286i \(0.518639\pi\)
\(734\) 0 0
\(735\) −2.08349e25 −0.332817
\(736\) 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
\(743\) 1.36333e26 1.97561 0.987806 − 0.155688i \(-0.0497596\pi\)
0.987806 + 0.155688i \(0.0497596\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.60248e26 2.16001
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −1.80383e25 −0.231761
\(754\) 0 0
\(755\) 0 0
\(756\) 9.50618e25 1.17845
\(757\) 1.61190e26 1.97458 0.987292 − 0.158919i \(-0.0508009\pi\)
0.987292 + 0.158919i \(0.0508009\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.48813e26 1.73852 0.869261 − 0.494354i \(-0.164595\pi\)
0.869261 + 0.494354i \(0.164595\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.16372e26 −1.29687
\(766\) 0 0
\(767\) 0 0
\(768\) 5.10488e25 0.549205
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −8.42637e25 −0.875288
\(772\) 1.17364e26 1.20498
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.07241e25 −0.385491
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −2.05964e26 −1.86181
\(784\) 6.67159e25 0.596190
\(785\) 0 0
\(786\) 0 0
\(787\) 1.64448e25 0.141989 0.0709945 − 0.997477i \(-0.477383\pi\)
0.0709945 + 0.997477i \(0.477383\pi\)
\(788\) 6.39770e25 0.546120
\(789\) −8.43194e25 −0.711597
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −1.23180e26 −0.971035
\(796\) −1.26956e25 −0.0989543
\(797\) 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 \(\pi\)
\(812\) −3.87003e26 −2.52180
\(813\) −1.85617e25 −0.119620
\(814\) 0 0
\(815\) −5.22423e25 −0.329309
\(816\) −1.60941e26 −1.00336
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −3.39442e26 −2.02507
\(821\) 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
\(827\) −2.69554e26 −1.48969 −0.744846 − 0.667236i \(-0.767477\pi\)
−0.744846 + 0.667236i \(0.767477\pi\)
\(828\) 0 0
\(829\) 2.98145e26 1.61227 0.806133 − 0.591734i \(-0.201556\pi\)
0.806133 + 0.591734i \(0.201556\pi\)
\(830\) 0 0
\(831\) 1.53761e26 0.813649
\(832\) 0 0
\(833\) −2.10335e26 −1.08920
\(834\) 0 0
\(835\) −2.15105e26 −1.09012
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 6.28038e26 2.98416
\(842\) 0 0
\(843\) 2.32709e26 1.08234
\(844\) 0 0
\(845\) −2.23253e26 −1.01645
\(846\) 0 0
\(847\) −2.83461e26 −1.26340
\(848\) 3.94437e26 1.73946
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −1.60288e26 −0.677554
\(853\) 5.70771e25 0.238737 0.119369 − 0.992850i \(-0.461913\pi\)
0.119369 + 0.992850i \(0.461913\pi\)
\(854\) 0 0
\(855\) −3.35374e25 −0.137352
\(856\) 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
\(861\) −3.59468e26 −1.38239
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.55378e26 1.28386
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.73294e26 −1.24159
\(876\) 0 0
\(877\) 1.21166e26 0.394807 0.197403 − 0.980322i \(-0.436749\pi\)
0.197403 + 0.980322i \(0.436749\pi\)
\(878\) 0 0
\(879\) −3.32674e26 −1.06199
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 6.52533e26 1.99966 0.999830 − 0.0184417i \(-0.00587052\pi\)
0.999830 + 0.0184417i \(0.00587052\pi\)
\(884\) 0 0
\(885\) 1.85913e26 0.558239
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.52264e26 −0.439018
\(890\) 0 0
\(891\) 0 0
\(892\) −7.06440e26 −1.97602
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −8.97481e24 −0.0231656
\(901\) −1.24354e27 −3.17787
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.07719e26 −1.00121
\(906\) 0 0
\(907\) 5.84841e26 1.40790 0.703951 − 0.710248i \(-0.251417\pi\)
0.703951 + 0.710248i \(0.251417\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.78532e26 1.10725 0.553624 − 0.832766i \(-0.313244\pi\)
0.553624 + 0.832766i \(0.313244\pi\)
\(912\) −4.63818e25 −0.106266
\(913\) 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
\(921\) 2.56866e26 0.538729
\(922\) 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
\(931\) −6.06166e25 −0.115357
\(932\) 0 0
\(933\) 4.21938e26 0.787614
\(934\) 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
\(944\) −5.95317e26 −1.00000
\(945\) −7.19917e26 −1.19783
\(946\) 0 0
\(947\) 5.71067e26 0.932260 0.466130 − 0.884716i \(-0.345648\pi\)
0.466130 + 0.884716i \(0.345648\pi\)
\(948\) 5.28638e26 0.854837
\(949\) 0 0
\(950\) 0 0
\(951\) 6.94296e26 1.09124
\(952\) 0 0
\(953\) 2.33693e26 0.360421 0.180210 − 0.983628i \(-0.442322\pi\)
0.180210 + 0.983628i \(0.442322\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.59367e26 0.538786
\(957\) 0 0
\(958\) 0 0
\(959\) 1.09869e27 1.60143
\(960\) −3.86600e26 −0.558239
\(961\) 6.99054e26 1.00000
\(962\) 0 0
\(963\) 8.50430e26 1.19399
\(964\) −8.27141e26 −1.15050
\(965\) −8.88814e26 −1.22480
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 1.46228e26 0.194141
\(970\) 0 0
\(971\) 6.69676e26 0.872753 0.436377 − 0.899764i \(-0.356262\pi\)
0.436377 + 0.899764i \(0.356262\pi\)
\(972\) 8.01532e26 1.03496
\(973\) −1.50495e27 −1.92534
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.05249e26 −0.605997
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −4.84507e26 −0.555103
\(986\) 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
\(993\) 9.48881e26 1.01081
\(994\) 0 0
\(995\) 9.61454e25 0.100582
\(996\) 0 0
\(997\) 8.40549e26 0.863588 0.431794 − 0.901972i \(-0.357881\pi\)
0.431794 + 0.901972i \(0.357881\pi\)
\(998\) 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 59.19.b.a.58.1 ✓ 1
59.58 odd 2 CM 59.19.b.a.58.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
59.19.b.a.58.1 ✓ 1 1.1 even 1 trivial
59.19.b.a.58.1 ✓ 1 59.58 odd 2 CM