↑

Properties

Label 23.31.b.a.22.1
Level $23$
Weight $31$
Character 23.22
Self dual yes
Analytic conductor $131.133$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,31,Mod(22,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, 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("23.22");
 
S:= CuspForms(chi, 31);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 23.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: \(131.132802376\)
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 22.1
Character \(\chi\) \(=\) 23.22

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-50407.0 q^{2} +1.97995e7 q^{3} +1.46712e9 q^{4} -9.98032e11 q^{6} -1.98292e13 q^{8} +1.86128e14 q^{9} +O(q^{10})\) \(q-50407.0 q^{2} +1.97995e7 q^{3} +1.46712e9 q^{4} -9.98032e11 q^{6} -1.98292e13 q^{8} +1.86128e14 q^{9} +2.90483e16 q^{12} +9.69545e16 q^{13} -5.75781e17 q^{16} -9.38217e18 q^{18} -2.66635e20 q^{23} -3.92608e20 q^{24} +9.31323e20 q^{25} -4.88718e21 q^{26} -3.91293e20 q^{27} -4.57452e21 q^{29} +3.15147e22 q^{31} +5.03149e22 q^{32} +2.73073e23 q^{36} +1.91965e24 q^{39} +3.10775e24 q^{41} +1.34403e25 q^{46} -1.77265e25 q^{47} -1.14002e25 q^{48} +2.25393e25 q^{49} -4.69452e25 q^{50} +1.42244e26 q^{52} +1.97239e25 q^{54} +2.30588e26 q^{58} +1.38622e26 q^{59} -1.58856e27 q^{62} -1.91798e27 q^{64} -5.27924e27 q^{69} -3.28650e27 q^{71} -3.69078e27 q^{72} -5.01398e27 q^{73} +1.84397e28 q^{75} -9.67637e28 q^{78} -4.60696e28 q^{81} -1.56652e29 q^{82} -9.05732e28 q^{87} -3.91187e29 q^{92} +6.23976e29 q^{93} +8.93541e29 q^{94} +9.96208e29 q^{96} -1.13614e30 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −50407.0 −1.53830 −0.769150 − 0.639069i \(-0.779320\pi\)
−0.769150 + 0.639069i \(0.779320\pi\)
\(3\) 1.97995e7 1.37986 0.689930 − 0.723876i \(-0.257641\pi\)
0.689930 + 0.723876i \(0.257641\pi\)
\(4\) 1.46712e9 1.36637
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −9.98032e11 −2.12264
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.98292e13 −0.563580
\(9\) 1.86128e14 0.904013
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2.90483e16 1.88539
\(13\) 9.69545e16 1.89416 0.947082 − 0.320992i \(-0.104016\pi\)
0.947082 + 0.320992i \(0.104016\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −5.75781e17 −0.499411
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −9.38217e18 −1.39064
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.66635e20 −1.00000
\(24\) −3.92608e20 −0.777661
\(25\) 9.31323e20 1.00000
\(26\) −4.88718e21 −2.91379
\(27\) −3.91293e20 −0.132448
\(28\) 0 0
\(29\) −4.57452e21 −0.530122 −0.265061 − 0.964232i \(-0.585392\pi\)
−0.265061 + 0.964232i \(0.585392\pi\)
\(30\) 0 0
\(31\) 3.15147e22 1.34304 0.671519 − 0.740987i \(-0.265642\pi\)
0.671519 + 0.740987i \(0.265642\pi\)
\(32\) 5.03149e22 1.33182
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.73073e23 1.23521
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.91965e24 2.61368
\(40\) 0 0
\(41\) 3.10775e24 1.99843 0.999214 − 0.0396489i \(-0.0126239\pi\)
0.999214 + 0.0396489i \(0.0126239\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.34403e25 1.53830
\(47\) −1.77265e25 −1.46945 −0.734726 − 0.678363i \(-0.762689\pi\)
−0.734726 + 0.678363i \(0.762689\pi\)
\(48\) −1.14002e25 −0.689117
\(49\) 2.25393e25 1.00000
\(50\) −4.69452e25 −1.53830
\(51\) 0 0
\(52\) 1.42244e26 2.58812
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.97239e25 0.203745
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.30588e26 0.815486
\(59\) 1.38622e26 0.379362 0.189681 − 0.981846i \(-0.439255\pi\)
0.189681 + 0.981846i \(0.439255\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −1.58856e27 −2.06599
\(63\) 0 0
\(64\) −1.91798e27 −1.54933
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −5.27924e27 −1.37986
\(70\) 0 0
\(71\) −3.28650e27 −0.559574 −0.279787 − 0.960062i \(-0.590264\pi\)
−0.279787 + 0.960062i \(0.590264\pi\)
\(72\) −3.69078e27 −0.509484
\(73\) −5.01398e27 −0.562782 −0.281391 − 0.959593i \(-0.590796\pi\)
−0.281391 + 0.959593i \(0.590796\pi\)
\(74\) 0 0
\(75\) 1.84397e28 1.37986
\(76\) 0 0
\(77\) 0 0
\(78\) −9.67637e28 −4.02062
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −4.60696e28 −1.08677
\(82\) −1.56652e29 −3.07418
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.05732e28 −0.731494
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.91187e29 −1.36637
\(93\) 6.23976e29 1.85320
\(94\) 8.93541e29 2.26046
\(95\) 0 0
\(96\) 9.96208e29 1.83773
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.13614e30 −1.53830
\(99\) 0 0
\(100\) 1.36637e30 1.36637
\(101\) 2.19604e30 1.89156 0.945780 − 0.324808i \(-0.105300\pi\)
0.945780 + 0.324808i \(0.105300\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −1.92253e30 −1.06751
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −5.74075e29 −0.180972
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.71139e30 −0.724340
\(117\) 1.80460e31 1.71235
\(118\) −6.98754e30 −0.583572
\(119\) 0 0
\(120\) 0 0
\(121\) 1.74494e31 1.00000
\(122\) 0 0
\(123\) 6.15319e31 2.75755
\(124\) 4.62360e31 1.83508
\(125\) 0 0
\(126\) 0 0
\(127\) −3.22895e31 −0.895377 −0.447689 − 0.894190i \(-0.647753\pi\)
−0.447689 + 0.894190i \(0.647753\pi\)
\(128\) 4.26545e31 1.05151
\(129\) 0 0
\(130\) 0 0
\(131\) −9.75120e31 −1.69820 −0.849099 − 0.528233i \(-0.822855\pi\)
−0.849099 + 0.528233i \(0.822855\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 2.66111e32 2.12264
\(139\) −8.53539e31 −0.610944 −0.305472 − 0.952201i \(-0.598814\pi\)
−0.305472 + 0.952201i \(0.598814\pi\)
\(140\) 0 0
\(141\) −3.50976e32 −2.02764
\(142\) 1.65662e32 0.860793
\(143\) 0 0
\(144\) −1.07169e32 −0.451474
\(145\) 0 0
\(146\) 2.52740e32 0.865727
\(147\) 4.46267e32 1.37986
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −9.29490e32 −2.12264
\(151\) −6.21274e32 −1.28419 −0.642094 − 0.766626i \(-0.721934\pi\)
−0.642094 + 0.766626i \(0.721934\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 2.81636e33 3.57124
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 2.32223e33 1.67178
\(163\) 1.89844e33 1.24618 0.623092 − 0.782149i \(-0.285876\pi\)
0.623092 + 0.782149i \(0.285876\pi\)
\(164\) 4.55946e33 2.73058
\(165\) 0 0
\(166\) 0 0
\(167\) 1.74967e33 0.798378 0.399189 − 0.916869i \(-0.369292\pi\)
0.399189 + 0.916869i \(0.369292\pi\)
\(168\) 0 0
\(169\) 6.78017e33 2.58786
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.07288e33 1.63194 0.815968 − 0.578098i \(-0.196205\pi\)
0.815968 + 0.578098i \(0.196205\pi\)
\(174\) 4.56552e33 1.12526
\(175\) 0 0
\(176\) 0 0
\(177\) 2.74465e33 0.523466
\(178\) 0 0
\(179\) −1.22270e34 −1.97026 −0.985131 − 0.171807i \(-0.945039\pi\)
−0.985131 + 0.171807i \(0.945039\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.28717e33 0.563580
\(185\) 0 0
\(186\) −3.14527e34 −2.85078
\(187\) 0 0
\(188\) −2.60070e34 −2.00781
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −3.79750e34 −2.13786
\(193\) 3.78886e34 1.97310 0.986548 − 0.163469i \(-0.0522685\pi\)
0.986548 + 0.163469i \(0.0522685\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.30680e34 1.36637
\(197\) 2.78877e34 1.06763 0.533814 − 0.845602i \(-0.320758\pi\)
0.533814 + 0.845602i \(0.320758\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.84674e34 −0.563580
\(201\) 0 0
\(202\) −1.10696e35 −2.90979
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.96284e34 −0.904013
\(208\) −5.58246e34 −0.945966
\(209\) 0 0
\(210\) 0 0
\(211\) 7.87865e34 1.07700 0.538499 − 0.842626i \(-0.318991\pi\)
0.538499 + 0.842626i \(0.318991\pi\)
\(212\) 0 0
\(213\) −6.50709e34 −0.772134
\(214\) 0 0
\(215\) 0 0
\(216\) 7.75902e33 0.0746450
\(217\) 0 0
\(218\) 0 0
\(219\) −9.92743e34 −0.776560
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 9.30682e34 0.554917 0.277459 − 0.960738i \(-0.410508\pi\)
0.277459 + 0.960738i \(0.410508\pi\)
\(224\) 0 0
\(225\) 1.73346e35 0.904013
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.07092e34 0.298766
\(233\) 6.47692e35 2.00000 0.999999 − 0.00163708i \(-0.000521099\pi\)
0.999999 + 0.00163708i \(0.000521099\pi\)
\(234\) −9.09643e35 −2.63411
\(235\) 0 0
\(236\) 2.03376e35 0.518347
\(237\) 0 0
\(238\) 0 0
\(239\) 6.83146e35 1.44061 0.720303 − 0.693659i \(-0.244003\pi\)
0.720303 + 0.693659i \(0.244003\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −8.79572e35 −1.53830
\(243\) −8.31590e35 −1.36715
\(244\) 0 0
\(245\) 0 0
\(246\) −3.10164e36 −4.24194
\(247\) 0 0
\(248\) −6.24912e35 −0.756909
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.62762e36 1.37736
\(255\) 0 0
\(256\) −9.06683e34 −0.0682113
\(257\) 1.62798e36 1.15518 0.577592 − 0.816326i \(-0.303993\pi\)
0.577592 + 0.816326i \(0.303993\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.51448e35 −0.479237
\(262\) 4.91529e36 2.61234
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.34392e36 −0.838799 −0.419399 − 0.907802i \(-0.637759\pi\)
−0.419399 + 0.907802i \(0.637759\pi\)
\(270\) 0 0
\(271\) 5.52835e36 1.77034 0.885168 − 0.465271i \(-0.154043\pi\)
0.885168 + 0.465271i \(0.154043\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −7.74530e36 −1.88539
\(277\) 3.66366e36 0.844732 0.422366 − 0.906425i \(-0.361200\pi\)
0.422366 + 0.906425i \(0.361200\pi\)
\(278\) 4.30243e36 0.939814
\(279\) 5.86579e36 1.21412
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 1.76916e37 3.11912
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −4.82170e36 −0.764583
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 9.36502e36 1.20399
\(289\) 8.19347e36 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −7.35613e36 −0.768966
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −2.24950e37 −2.12264
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.58515e37 −1.89416
\(300\) 2.70533e37 1.88539
\(301\) 0 0
\(302\) 3.13165e37 1.97547
\(303\) 4.34805e37 2.61009
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3.28287e37 −1.61875 −0.809373 − 0.587295i \(-0.800193\pi\)
−0.809373 + 0.587295i \(0.800193\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.91635e37 −1.59029 −0.795145 − 0.606420i \(-0.792605\pi\)
−0.795145 + 0.606420i \(0.792605\pi\)
\(312\) −3.80651e37 −1.47302
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.37522e37 −0.419258 −0.209629 − 0.977781i \(-0.567226\pi\)
−0.209629 + 0.977781i \(0.567226\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −6.75898e37 −1.48493
\(325\) 9.02959e37 1.89416
\(326\) −9.56945e37 −1.91700
\(327\) 0 0
\(328\) −6.16242e37 −1.12627
\(329\) 0 0
\(330\) 0 0
\(331\) 1.25408e38 1.99943 0.999715 − 0.0238892i \(-0.00760490\pi\)
0.999715 + 0.0238892i \(0.00760490\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −8.81955e37 −1.22814
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −3.41768e38 −3.98090
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −3.06116e38 −2.51041
\(347\) 1.07937e37 0.0847675 0.0423838 − 0.999101i \(-0.486505\pi\)
0.0423838 + 0.999101i \(0.486505\pi\)
\(348\) −1.32882e38 −0.999488
\(349\) 1.47429e38 1.06219 0.531094 − 0.847313i \(-0.321781\pi\)
0.531094 + 0.847313i \(0.321781\pi\)
\(350\) 0 0
\(351\) −3.79376e37 −0.250878
\(352\) 0 0
\(353\) 2.84302e38 1.72647 0.863236 − 0.504801i \(-0.168434\pi\)
0.863236 + 0.504801i \(0.168434\pi\)
\(354\) −1.38350e38 −0.805247
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 6.16325e38 3.03085
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 2.30467e38 1.00000
\(362\) 0 0
\(363\) 3.45489e38 1.37986
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 1.53524e38 0.499411
\(369\) 5.78441e38 1.80661
\(370\) 0 0
\(371\) 0 0
\(372\) 9.15449e38 2.53215
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.51503e38 0.828154
\(377\) −4.43520e38 −1.00414
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −6.39316e38 −1.23549
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 8.44537e38 1.45094
\(385\) 0 0
\(386\) −1.90985e39 −3.03521
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −4.46937e38 −0.563580
\(393\) −1.93069e39 −2.34328
\(394\) −1.40574e39 −1.64233
\(395\) 0 0
\(396\) 0 0
\(397\) −1.91286e39 −1.99446 −0.997232 − 0.0743501i \(-0.976312\pi\)
−0.997232 + 0.0743501i \(0.976312\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5.36238e38 −0.499411
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 3.05549e39 2.54393
\(404\) 3.22187e39 2.58456
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.99101e39 1.99512 0.997559 − 0.0698280i \(-0.0222450\pi\)
0.997559 + 0.0698280i \(0.0222450\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 2.50162e39 1.39064
\(415\) 0 0
\(416\) 4.87825e39 2.52269
\(417\) −1.68996e39 −0.843017
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −3.97139e39 −1.65675
\(423\) −3.29941e39 −1.32841
\(424\) 0 0
\(425\) 0 0
\(426\) 3.28003e39 1.18777
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 2.25299e38 0.0661459
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 5.00412e39 1.19458
\(439\) 5.17027e39 1.19274 0.596370 − 0.802710i \(-0.296609\pi\)
0.596370 + 0.802710i \(0.296609\pi\)
\(440\) 0 0
\(441\) 4.19521e39 0.904013
\(442\) 0 0
\(443\) −9.87647e39 −1.98859 −0.994296 − 0.106651i \(-0.965987\pi\)
−0.994296 + 0.106651i \(0.965987\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.69129e39 −0.853629
\(447\) 0 0
\(448\) 0 0
\(449\) −1.00740e40 −1.65770 −0.828851 − 0.559470i \(-0.811005\pi\)
−0.828851 + 0.559470i \(0.811005\pi\)
\(450\) −8.73783e39 −1.39064
\(451\) 0 0
\(452\) 0 0
\(453\) −1.23009e40 −1.77200
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.69581e38 0.0298659 0.0149330 − 0.999888i \(-0.495247\pi\)
0.0149330 + 0.999888i \(0.495247\pi\)
\(462\) 0 0
\(463\) −1.92602e40 −1.99961 −0.999807 − 0.0196538i \(-0.993744\pi\)
−0.999807 + 0.0196538i \(0.993744\pi\)
\(464\) 2.63393e39 0.264749
\(465\) 0 0
\(466\) −3.26482e40 −3.07659
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 2.64757e40 2.33970
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −2.74877e39 −0.213801
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −3.44354e40 −2.21608
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.56004e40 1.36637
\(485\) 0 0
\(486\) 4.19180e40 2.10308
\(487\) −2.20097e40 −1.07073 −0.535364 − 0.844621i \(-0.679826\pi\)
−0.535364 + 0.844621i \(0.679826\pi\)
\(488\) 0 0
\(489\) 3.75881e40 1.71956
\(490\) 0 0
\(491\) −4.64608e40 −1.99924 −0.999618 − 0.0276484i \(-0.991198\pi\)
−0.999618 + 0.0276484i \(0.991198\pi\)
\(492\) 9.02749e40 3.76782
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.81456e40 −0.670728
\(497\) 0 0
\(498\) 0 0
\(499\) 9.03231e38 0.0304993 0.0152497 − 0.999884i \(-0.495146\pi\)
0.0152497 + 0.999884i \(0.495146\pi\)
\(500\) 0 0
\(501\) 3.46425e40 1.10165
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.34244e41 3.57088
\(508\) −4.73727e40 −1.22341
\(509\) 7.35096e40 1.84322 0.921610 − 0.388116i \(-0.126874\pi\)
0.921610 + 0.388116i \(0.126874\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −4.12296e40 −0.946585
\(513\) 0 0
\(514\) −8.20614e40 −1.77702
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.20240e41 2.25184
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 4.29190e40 0.737211
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −1.43062e41 −2.32036
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.10943e40 1.00000
\(530\) 0 0
\(531\) 2.58016e40 0.342948
\(532\) 0 0
\(533\) 3.01310e41 3.78535
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2.42088e41 −2.71868
\(538\) 1.18150e41 1.29032
\(539\) 0 0
\(540\) 0 0
\(541\) −1.85504e41 −1.86378 −0.931889 − 0.362744i \(-0.881840\pi\)
−0.931889 + 0.362744i \(0.881840\pi\)
\(542\) −2.78667e41 −2.72331
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.13166e41 1.81514 0.907568 − 0.419906i \(-0.137937\pi\)
0.907568 + 0.419906i \(0.137937\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 1.04683e41 0.777661
\(553\) 0 0
\(554\) −1.84674e41 −1.29945
\(555\) 0 0
\(556\) −1.25225e41 −0.834772
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −2.95677e41 −1.86769
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) −5.14925e41 −2.77050
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 6.51686e40 0.315365
\(569\) 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
\(575\) −2.48323e41 −1.00000
\(576\) −3.56991e41 −1.40062
\(577\) 4.84416e41 1.85174 0.925872 − 0.377837i \(-0.123332\pi\)
0.925872 + 0.377837i \(0.123332\pi\)
\(578\) −4.13008e41 −1.53830
\(579\) 7.50174e41 2.72260
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 9.94233e40 0.317172
\(585\) 0 0
\(586\) 0 0
\(587\) −4.92640e41 −1.45532 −0.727659 − 0.685939i \(-0.759392\pi\)
−0.727659 + 0.685939i \(0.759392\pi\)
\(588\) 6.54729e41 1.88539
\(589\) 0 0
\(590\) 0 0
\(591\) 5.52163e41 1.47318
\(592\) 0 0
\(593\) −3.86083e41 −0.979174 −0.489587 − 0.871954i \(-0.662853\pi\)
−0.489587 + 0.871954i \(0.662853\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 1.30310e42 2.91379
\(599\) 8.56280e41 1.86730 0.933649 − 0.358190i \(-0.116606\pi\)
0.933649 + 0.358190i \(0.116606\pi\)
\(600\) −3.65645e41 −0.777661
\(601\) −9.58261e41 −1.98777 −0.993886 − 0.110407i \(-0.964784\pi\)
−0.993886 + 0.110407i \(0.964784\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.11485e41 −1.75467
\(605\) 0 0
\(606\) −2.19172e42 −4.01510
\(607\) 1.10794e42 1.98009 0.990045 − 0.140753i \(-0.0449523\pi\)
0.990045 + 0.140753i \(0.0449523\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.71867e42 −2.78338
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.65480e42 2.49012
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 1.04332e41 0.132448
\(622\) 1.97412e42 2.44634
\(623\) 0 0
\(624\) −1.10530e42 −1.30530
\(625\) 8.67362e41 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.55993e42 1.48611
\(634\) 6.93205e41 0.644945
\(635\) 0 0
\(636\) 0 0
\(637\) 2.18529e42 1.89416
\(638\) 0 0
\(639\) −6.11710e41 −0.505863
\(640\) 0 0
\(641\) 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
\(647\) 1.29530e42 0.888806 0.444403 − 0.895827i \(-0.353416\pi\)
0.444403 + 0.895827i \(0.353416\pi\)
\(648\) 9.13523e41 0.612484
\(649\) 0 0
\(650\) −4.55154e42 −2.91379
\(651\) 0 0
\(652\) 2.78524e42 1.70274
\(653\) −2.01510e42 −1.20393 −0.601963 − 0.798524i \(-0.705615\pi\)
−0.601963 + 0.798524i \(0.705615\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.78939e42 −0.998036
\(657\) −9.33244e41 −0.508762
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −6.32145e42 −3.07572
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.21973e42 0.530122
\(668\) 2.56698e42 1.09088
\(669\) 1.84270e42 0.765708
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −4.90642e42 −1.86440 −0.932199 − 0.361946i \(-0.882112\pi\)
−0.932199 + 0.361946i \(0.882112\pi\)
\(674\) 0 0
\(675\) −3.64420e41 −0.132448
\(676\) 9.94735e42 3.53596
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.29766e42 0.395229 0.197614 − 0.980280i \(-0.436681\pi\)
0.197614 + 0.980280i \(0.436681\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 7.51259e41 0.192141 0.0960703 − 0.995375i \(-0.469373\pi\)
0.0960703 + 0.995375i \(0.469373\pi\)
\(692\) 8.90966e42 2.22982
\(693\) 0 0
\(694\) −5.44080e41 −0.130398
\(695\) 0 0
\(696\) 1.79599e42 0.412255
\(697\) 0 0
\(698\) −7.43145e42 −1.63396
\(699\) 1.28240e43 2.75972
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 1.91232e42 0.385926
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −1.43308e43 −2.65583
\(707\) 0 0
\(708\) 4.02675e42 0.715246
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.40294e42 −1.34304
\(714\) 0 0
\(715\) 0 0
\(716\) −1.79385e43 −2.69210
\(717\) 1.35259e43 1.98783
\(718\) 0 0
\(719\) −3.05929e42 −0.431208 −0.215604 − 0.976481i \(-0.569172\pi\)
−0.215604 + 0.976481i \(0.569172\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.16171e43 −1.53830
\(723\) 0 0
\(724\) 0 0
\(725\) −4.26036e42 −0.530122
\(726\) −1.74151e43 −2.12264
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −6.97973e42 −0.799698
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.34157e43 −1.33182
\(737\) 0 0
\(738\) −2.91575e43 −2.77910
\(739\) 1.51177e43 1.41195 0.705974 − 0.708238i \(-0.250510\pi\)
0.705974 + 0.708238i \(0.250510\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −1.23729e43 −1.04443
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 1.02066e43 0.733861
\(753\) 0 0
\(754\) 2.23565e43 1.54466
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.79785e43 −1.68289 −0.841443 − 0.540346i \(-0.818293\pi\)
−0.841443 + 0.540346i \(0.818293\pi\)
\(762\) 3.22260e43 1.90056
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.34401e43 0.718573
\(768\) −1.79519e42 −0.0941220
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 3.22331e43 1.59399
\(772\) 5.55872e43 2.69597
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 2.93504e43 1.34304
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.78998e42 0.0702136
\(784\) −1.29777e43 −0.499411
\(785\) 0 0
\(786\) 9.73201e43 3.60466
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 4.09148e43 1.45877
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 9.64217e43 3.06808
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.68594e43 1.33182
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −1.54018e44 −3.91333
\(807\) −4.64083e43 −1.15742
\(808\) −4.35458e43 −1.06605
\(809\) −8.13020e43 −1.95377 −0.976885 − 0.213767i \(-0.931427\pi\)
−0.976885 + 0.213767i \(0.931427\pi\)
\(810\) 0 0
\(811\) −6.44199e43 −1.49179 −0.745895 − 0.666064i \(-0.767978\pi\)
−0.745895 + 0.666064i \(0.767978\pi\)
\(812\) 0 0
\(813\) 1.09458e44 2.44282
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.50768e44 −3.06909
\(819\) 0 0
\(820\) 0 0
\(821\) −7.11549e43 −1.37106 −0.685532 − 0.728043i \(-0.740430\pi\)
−0.685532 + 0.728043i \(0.740430\pi\)
\(822\) 0 0
\(823\) 8.41033e43 1.56248 0.781242 − 0.624228i \(-0.214586\pi\)
0.781242 + 0.624228i \(0.214586\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −7.28110e43 −1.23521
\(829\) −6.05468e43 −1.00873 −0.504363 − 0.863492i \(-0.668273\pi\)
−0.504363 + 0.863492i \(0.668273\pi\)
\(830\) 0 0
\(831\) 7.25386e43 1.16561
\(832\) −1.85957e44 −2.93469
\(833\) 0 0
\(834\) 8.51859e43 1.29681
\(835\) 0 0
\(836\) 0 0
\(837\) −1.23315e43 −0.177883
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −5.35366e43 −0.718971
\(842\) 0 0
\(843\) 0 0
\(844\) 1.15590e44 1.47157
\(845\) 0 0
\(846\) 1.66313e44 2.04349
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −9.54671e43 −1.05502
\(853\) −7.99399e43 −0.868017 −0.434008 − 0.900909i \(-0.642901\pi\)
−0.434008 + 0.900909i \(0.642901\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.70871e44 −1.72964 −0.864819 − 0.502084i \(-0.832567\pi\)
−0.864819 + 0.502084i \(0.832567\pi\)
\(858\) 0 0
\(859\) 2.02845e44 1.98275 0.991374 − 0.131064i \(-0.0418394\pi\)
0.991374 + 0.131064i \(0.0418394\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.19354e44 1.99979 0.999895 − 0.0145183i \(-0.00462147\pi\)
0.999895 + 0.0145183i \(0.00462147\pi\)
\(864\) −1.96878e43 −0.176397
\(865\) 0 0
\(866\) 0 0
\(867\) 1.62226e44 1.37986
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −1.45648e44 −1.06106
\(877\) −1.78848e44 −1.28082 −0.640412 − 0.768032i \(-0.721236\pi\)
−0.640412 + 0.768032i \(0.721236\pi\)
\(878\) −2.60618e44 −1.83479
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −2.11468e44 −1.39064
\(883\) −2.64485e44 −1.70998 −0.854988 − 0.518647i \(-0.826436\pi\)
−0.854988 + 0.518647i \(0.826436\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 4.97843e44 3.05905
\(887\) −2.99404e44 −1.80885 −0.904425 − 0.426632i \(-0.859700\pi\)
−0.904425 + 0.426632i \(0.859700\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 1.36543e44 0.758220
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.11846e44 −2.61368
\(898\) 5.07800e44 2.55004
\(899\) −1.44165e44 −0.711974
\(900\) 2.54319e44 1.23521
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 6.20051e44 2.72587
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 4.08746e44 1.71000
\(910\) 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
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −6.49992e44 −2.23364
\(922\) −1.35888e43 −0.0459428
\(923\) −3.18640e44 −1.05993
\(924\) 0 0
\(925\) 0 0
\(926\) 9.70849e44 3.07600
\(927\) 0 0
\(928\) −2.30166e44 −0.706029
\(929\) 2.42956e44 0.733317 0.366658 − 0.930356i \(-0.380502\pi\)
0.366658 + 0.930356i \(0.380502\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 9.50244e44 2.73273
\(933\) −7.75418e44 −2.19438
\(934\) 0 0
\(935\) 0 0
\(936\) −3.57837e44 −0.965046
\(937\) 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
\(943\) −8.28636e44 −1.99843
\(944\) −7.98162e43 −0.189457
\(945\) 0 0
\(946\) 0 0
\(947\) 8.62839e44 1.95290 0.976451 − 0.215741i \(-0.0692166\pi\)
0.976451 + 0.215741i \(0.0692166\pi\)
\(948\) 0 0
\(949\) −4.86128e44 −1.06600
\(950\) 0 0
\(951\) −2.72285e44 −0.578518
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.00226e45 1.96839
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.42560e44 0.803751
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.81489e44 −0.300229 −0.150115 − 0.988669i \(-0.547964\pi\)
−0.150115 + 0.988669i \(0.547964\pi\)
\(968\) −3.46008e44 −0.563580
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −1.22005e45 −1.86802
\(973\) 0 0
\(974\) 1.10944e45 1.64710
\(975\) 1.78781e45 2.61368
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) −1.89470e45 −2.64520
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 2.34195e45 3.07542
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −1.22013e45 −1.55410
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.78789e44 −0.204756 −0.102378 − 0.994746i \(-0.532645\pi\)
−0.102378 + 0.994746i \(0.532645\pi\)
\(992\) 1.58566e45 1.78869
\(993\) 2.48302e45 2.75893
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.88632e45 −1.97328 −0.986638 − 0.162926i \(-0.947907\pi\)
−0.986638 + 0.162926i \(0.947907\pi\)
\(998\) −4.55291e43 −0.0469171
\(999\) 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 23.31.b.a.22.1 ✓ 1
23.22 odd 2 CM 23.31.b.a.22.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.31.b.a.22.1 ✓ 1 1.1 even 1 trivial
23.31.b.a.22.1 ✓ 1 23.22 odd 2 CM