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Properties

Label 23.37.b.a.22.1
Level $23$
Weight $37$
Character 23.22
Self dual yes
Analytic conductor $188.810$
Analytic rank $0$
Dimension $1$
CM discriminant -23
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] = [23,37,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, 37, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 37);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 37 \)
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: \(188.809894917\)
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)
 
\(f(q)\) \(=\) \(q+477713. q^{2} +2.23178e7 q^{3} +1.59490e11 q^{4} +1.06615e13 q^{6} +4.33624e16 q^{8} -1.49597e17 q^{9} +O(q^{10})\) \(q+477713. q^{2} +2.23178e7 q^{3} +1.59490e11 q^{4} +1.06615e13 q^{6} +4.33624e16 q^{8} -1.49597e17 q^{9} +3.55947e18 q^{12} -1.75192e19 q^{13} +9.75468e21 q^{16} -7.14642e22 q^{18} +3.24415e24 q^{23} +9.67752e23 q^{24} +1.45519e25 q^{25} -8.36914e24 q^{26} -6.68844e24 q^{27} +2.44757e26 q^{29} +7.53697e26 q^{31} +1.68010e27 q^{32} -2.38592e28 q^{36} -3.90989e26 q^{39} +7.58638e28 q^{41} +1.54977e30 q^{46} +1.56757e30 q^{47} +2.17703e29 q^{48} +2.65173e30 q^{49} +6.95164e30 q^{50} -2.79414e30 q^{52} -3.19516e30 q^{54} +1.16924e32 q^{58} +9.82318e31 q^{59} +3.60051e32 q^{62} +1.32269e32 q^{64} +7.24022e31 q^{69} -2.55916e33 q^{71} -6.48686e33 q^{72} -4.23050e33 q^{73} +3.24766e32 q^{75} -1.86781e32 q^{78} +2.23044e34 q^{81} +3.62411e34 q^{82} +5.46243e33 q^{87} +5.17410e35 q^{92} +1.68208e34 q^{93} +7.48849e35 q^{94} +3.74961e34 q^{96} +1.26677e36 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 477713. 1.82233 0.911165 − 0.412041i \(-0.135184\pi\)
0.911165 + 0.412041i \(0.135184\pi\)
\(3\) 2.23178e7 0.0576061 0.0288030 − 0.999585i \(-0.490830\pi\)
0.0288030 + 0.999585i \(0.490830\pi\)
\(4\) 1.59490e11 2.32089
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 1.06615e13 0.104977
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 4.33624e16 2.40710
\(9\) −1.49597e17 −0.996682
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 3.55947e18 0.133697
\(13\) −1.75192e19 −0.155788 −0.0778939 − 0.996962i \(-0.524820\pi\)
−0.0778939 + 0.996962i \(0.524820\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 9.75468e21 2.06563
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −7.14642e22 −1.81628
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.24415e24 1.00000
\(24\) 9.67752e23 0.138663
\(25\) 1.45519e25 1.00000
\(26\) −8.36914e24 −0.283897
\(27\) −6.68844e24 −0.115021
\(28\) 0 0
\(29\) 2.44757e26 1.16298 0.581489 − 0.813555i \(-0.302471\pi\)
0.581489 + 0.813555i \(0.302471\pi\)
\(30\) 0 0
\(31\) 7.53697e26 1.07817 0.539084 − 0.842252i \(-0.318771\pi\)
0.539084 + 0.842252i \(0.318771\pi\)
\(32\) 1.68010e27 1.35717
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.38592e28 −2.31319
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −3.90989e26 −0.00897432
\(40\) 0 0
\(41\) 7.58638e28 0.707824 0.353912 − 0.935279i \(-0.384851\pi\)
0.353912 + 0.935279i \(0.384851\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.54977e30 1.82233
\(47\) 1.56757e30 1.25160 0.625800 − 0.779984i \(-0.284773\pi\)
0.625800 + 0.779984i \(0.284773\pi\)
\(48\) 2.17703e29 0.118993
\(49\) 2.65173e30 1.00000
\(50\) 6.95164e30 1.82233
\(51\) 0 0
\(52\) −2.79414e30 −0.361566
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −3.19516e30 −0.209606
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.16924e32 2.11933
\(59\) 9.82318e31 1.30893 0.654464 − 0.756093i \(-0.272894\pi\)
0.654464 + 0.756093i \(0.272894\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 3.60051e32 1.96478
\(63\) 0 0
\(64\) 1.32269e32 0.407584
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 7.24022e31 0.0576061
\(70\) 0 0
\(71\) −2.55916e33 −1.21744 −0.608720 − 0.793385i \(-0.708317\pi\)
−0.608720 + 0.793385i \(0.708317\pi\)
\(72\) −6.48686e33 −2.39911
\(73\) −4.23050e33 −1.22062 −0.610309 − 0.792163i \(-0.708955\pi\)
−0.610309 + 0.792163i \(0.708955\pi\)
\(74\) 0 0
\(75\) 3.24766e32 0.0576061
\(76\) 0 0
\(77\) 0 0
\(78\) −1.86781e32 −0.0163542
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 2.23044e34 0.990056
\(82\) 3.62411e34 1.28989
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.46243e33 0.0669946
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.17410e35 2.32089
\(93\) 1.68208e34 0.0621090
\(94\) 7.48849e35 2.28083
\(95\) 0 0
\(96\) 3.74961e34 0.0781814
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.26677e36 1.82233
\(99\) 0 0
\(100\) 2.32089e36 2.32089
\(101\) −1.97883e35 −0.165433 −0.0827167 − 0.996573i \(-0.526360\pi\)
−0.0827167 + 0.996573i \(0.526360\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −7.59673e35 −0.374996
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.06674e36 −0.266951
\(109\) 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
\(116\) 3.90364e37 2.69914
\(117\) 2.62081e36 0.155271
\(118\) 4.69266e37 2.38530
\(119\) 0 0
\(120\) 0 0
\(121\) 3.09127e37 1.00000
\(122\) 0 0
\(123\) 1.69311e36 0.0407750
\(124\) 1.20207e38 2.50231
\(125\) 0 0
\(126\) 0 0
\(127\) 5.55484e37 0.751977 0.375989 − 0.926624i \(-0.377303\pi\)
0.375989 + 0.926624i \(0.377303\pi\)
\(128\) −5.22691e37 −0.614421
\(129\) 0 0
\(130\) 0 0
\(131\) −7.00209e37 −0.542431 −0.271215 − 0.962519i \(-0.587426\pi\)
−0.271215 + 0.962519i \(0.587426\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 3.45875e37 0.104977
\(139\) 4.37699e37 0.116657 0.0583283 − 0.998297i \(-0.481423\pi\)
0.0583283 + 0.998297i \(0.481423\pi\)
\(140\) 0 0
\(141\) 3.49847e37 0.0720998
\(142\) −1.22254e39 −2.21858
\(143\) 0 0
\(144\) −1.45927e39 −2.05878
\(145\) 0 0
\(146\) −2.02096e39 −2.22437
\(147\) 5.91807e37 0.0576061
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 1.55145e38 0.104977
\(151\) −3.04183e39 −1.82621 −0.913103 − 0.407729i \(-0.866321\pi\)
−0.913103 + 0.407729i \(0.866321\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −6.23589e37 −0.0208284
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 1.06551e40 1.80421
\(163\) 1.77918e39 0.269676 0.134838 − 0.990868i \(-0.456949\pi\)
0.134838 + 0.990868i \(0.456949\pi\)
\(164\) 1.20995e40 1.64278
\(165\) 0 0
\(166\) 0 0
\(167\) 8.87419e39 0.869425 0.434713 − 0.900569i \(-0.356850\pi\)
0.434713 + 0.900569i \(0.356850\pi\)
\(168\) 0 0
\(169\) −1.23393e40 −0.975730
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.59070e40 −0.825580 −0.412790 − 0.910826i \(-0.635446\pi\)
−0.412790 + 0.910826i \(0.635446\pi\)
\(174\) 2.60948e39 0.122086
\(175\) 0 0
\(176\) 0 0
\(177\) 2.19231e39 0.0754022
\(178\) 0 0
\(179\) 3.54538e40 0.996113 0.498057 − 0.867144i \(-0.334047\pi\)
0.498057 + 0.867144i \(0.334047\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.40674e41 2.40710
\(185\) 0 0
\(186\) 8.03554e39 0.113183
\(187\) 0 0
\(188\) 2.50012e41 2.90482
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2.95194e39 0.0234793
\(193\) 3.22603e40 0.233688 0.116844 − 0.993150i \(-0.462722\pi\)
0.116844 + 0.993150i \(0.462722\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.22925e41 2.32089
\(197\) 1.41288e41 0.707479 0.353739 − 0.935344i \(-0.384910\pi\)
0.353739 + 0.935344i \(0.384910\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 6.31006e41 2.40710
\(201\) 0 0
\(202\) −9.45312e40 −0.301474
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.85314e41 −0.996682
\(208\) −1.70894e41 −0.321800
\(209\) 0 0
\(210\) 0 0
\(211\) 3.53454e41 0.514338 0.257169 − 0.966366i \(-0.417210\pi\)
0.257169 + 0.966366i \(0.417210\pi\)
\(212\) 0 0
\(213\) −5.71148e40 −0.0701319
\(214\) 0 0
\(215\) 0 0
\(216\) −2.90027e41 −0.276867
\(217\) 0 0
\(218\) 0 0
\(219\) −9.44153e40 −0.0703151
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.51008e42 −1.88725 −0.943626 − 0.331013i \(-0.892609\pi\)
−0.943626 + 0.331013i \(0.892609\pi\)
\(224\) 0 0
\(225\) −2.17692e42 −0.996682
\(226\) 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
\(232\) 1.06132e43 2.79940
\(233\) 2.51643e42 0.614296 0.307148 − 0.951662i \(-0.400625\pi\)
0.307148 + 0.951662i \(0.400625\pi\)
\(234\) 1.25199e42 0.282955
\(235\) 0 0
\(236\) 1.56670e43 3.03788
\(237\) 0 0
\(238\) 0 0
\(239\) 7.84532e42 1.21185 0.605925 − 0.795522i \(-0.292803\pi\)
0.605925 + 0.795522i \(0.292803\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.47674e43 1.82233
\(243\) 1.50168e42 0.172054
\(244\) 0 0
\(245\) 0 0
\(246\) 8.08821e41 0.0743054
\(247\) 0 0
\(248\) 3.26821e43 2.59525
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.65362e43 1.37035
\(255\) 0 0
\(256\) −3.40591e43 −1.52726
\(257\) 9.67137e42 0.404289 0.202144 − 0.979356i \(-0.435209\pi\)
0.202144 + 0.979356i \(0.435209\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.66148e43 −1.15912
\(262\) −3.34499e43 −0.988488
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.21644e43 0.407489 0.203744 − 0.979024i \(-0.434689\pi\)
0.203744 + 0.979024i \(0.434689\pi\)
\(270\) 0 0
\(271\) 1.03924e44 1.67213 0.836063 − 0.548633i \(-0.184852\pi\)
0.836063 + 0.548633i \(0.184852\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 1.15475e43 0.133697
\(277\) −1.36965e44 −1.48584 −0.742921 − 0.669379i \(-0.766560\pi\)
−0.742921 + 0.669379i \(0.766560\pi\)
\(278\) 2.09095e43 0.212587
\(279\) −1.12750e44 −1.07459
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 1.67126e43 0.131390
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −4.08161e44 −2.82554
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2.51337e44 −1.35267
\(289\) 1.97770e44 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −6.74723e44 −2.83292
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 2.82714e43 0.104977
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.68348e43 −0.155788
\(300\) 5.17971e43 0.133697
\(301\) 0 0
\(302\) −1.45312e45 −3.32795
\(303\) −4.41630e42 −0.00952997
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.16445e45 −1.98441 −0.992206 − 0.124612i \(-0.960231\pi\)
−0.992206 + 0.124612i \(0.960231\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.31769e45 1.77879 0.889395 − 0.457140i \(-0.151126\pi\)
0.889395 + 0.457140i \(0.151126\pi\)
\(312\) −1.69542e43 −0.0216020
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.26822e44 −0.121374 −0.0606872 − 0.998157i \(-0.519329\pi\)
−0.0606872 + 0.998157i \(0.519329\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.55733e45 2.29781
\(325\) −2.54938e44 −0.155788
\(326\) 8.49937e44 0.491439
\(327\) 0 0
\(328\) 3.28963e45 1.70380
\(329\) 0 0
\(330\) 0 0
\(331\) 4.54732e45 1.99918 0.999589 − 0.0286659i \(-0.00912588\pi\)
0.999589 + 0.0286659i \(0.00912588\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 4.23932e45 1.58438
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −5.89464e45 −1.77810
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −7.59899e45 −1.50448
\(347\) −1.06267e46 −1.99741 −0.998706 − 0.0508538i \(-0.983806\pi\)
−0.998706 + 0.0508538i \(0.983806\pi\)
\(348\) 8.71205e44 0.155487
\(349\) −9.23550e45 −1.56532 −0.782659 − 0.622450i \(-0.786137\pi\)
−0.782659 + 0.622450i \(0.786137\pi\)
\(350\) 0 0
\(351\) 1.17176e44 0.0179189
\(352\) 0 0
\(353\) −1.44866e46 −1.99996 −0.999978 − 0.00666273i \(-0.997879\pi\)
−0.999978 + 0.00666273i \(0.997879\pi\)
\(354\) 1.04730e45 0.137408
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.69368e46 1.81525
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.08425e46 1.00000
\(362\) 0 0
\(363\) 6.89902e44 0.0576061
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 3.16457e46 2.06563
\(369\) −1.13490e46 −0.705475
\(370\) 0 0
\(371\) 0 0
\(372\) 2.68276e45 0.144148
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.79736e46 3.01272
\(377\) −4.28794e45 −0.181178
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 1.23972e45 0.0433185
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.16653e45 −0.0353944
\(385\) 0 0
\(386\) 1.54112e46 0.425857
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.14985e47 2.40710
\(393\) −1.56271e45 −0.0312473
\(394\) 6.74951e46 1.28926
\(395\) 0 0
\(396\) 0 0
\(397\) −1.03004e47 −1.71643 −0.858214 − 0.513292i \(-0.828426\pi\)
−0.858214 + 0.513292i \(0.828426\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.41949e47 2.06563
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.32041e46 −0.167965
\(404\) −3.15604e46 −0.383952
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.75478e47 −1.71082 −0.855409 − 0.517953i \(-0.826694\pi\)
−0.855409 + 0.517953i \(0.826694\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −2.31841e47 −1.81628
\(415\) 0 0
\(416\) −2.94339e46 −0.211431
\(417\) 9.76848e44 0.00672013
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 1.68850e47 0.937294
\(423\) −2.34503e47 −1.24745
\(424\) 0 0
\(425\) 0 0
\(426\) −2.72845e46 −0.127804
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −6.52436e46 −0.237591
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −4.51034e46 −0.128137
\(439\) 1.28220e47 0.349619 0.174809 − 0.984602i \(-0.444069\pi\)
0.174809 + 0.984602i \(0.444069\pi\)
\(440\) 0 0
\(441\) −3.96690e47 −0.996682
\(442\) 0 0
\(443\) −7.57815e47 −1.75508 −0.877538 − 0.479507i \(-0.840815\pi\)
−0.877538 + 0.479507i \(0.840815\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.67681e48 −3.43920
\(447\) 0 0
\(448\) 0 0
\(449\) −1.09629e48 −1.99292 −0.996459 − 0.0840769i \(-0.973206\pi\)
−0.996459 + 0.0840769i \(0.973206\pi\)
\(450\) −1.03994e48 −1.81628
\(451\) 0 0
\(452\) 0 0
\(453\) −6.78869e46 −0.105201
\(454\) 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
\(461\) −5.76603e47 −0.652019 −0.326010 − 0.945366i \(-0.605704\pi\)
−0.326010 + 0.945366i \(0.605704\pi\)
\(462\) 0 0
\(463\) 6.33562e47 0.662721 0.331361 − 0.943504i \(-0.392492\pi\)
0.331361 + 0.943504i \(0.392492\pi\)
\(464\) 2.38753e48 2.40229
\(465\) 0 0
\(466\) 1.20213e48 1.11945
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 4.17993e47 0.360366
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 4.25956e48 3.15071
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 3.74781e48 2.20839
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 4.93027e48 2.32089
\(485\) 0 0
\(486\) 7.17374e47 0.313540
\(487\) 1.24160e48 0.522952 0.261476 − 0.965210i \(-0.415791\pi\)
0.261476 + 0.965210i \(0.415791\pi\)
\(488\) 0 0
\(489\) 3.97073e46 0.0155350
\(490\) 0 0
\(491\) −4.34123e48 −1.57814 −0.789071 − 0.614302i \(-0.789438\pi\)
−0.789071 + 0.614302i \(0.789438\pi\)
\(492\) 2.70035e47 0.0946341
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 7.35208e48 2.22710
\(497\) 0 0
\(498\) 0 0
\(499\) −7.35812e48 −1.99967 −0.999833 − 0.0182993i \(-0.994175\pi\)
−0.999833 + 0.0182993i \(0.994175\pi\)
\(500\) 0 0
\(501\) 1.98052e47 0.0500842
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.75386e47 −0.0562080
\(508\) 8.85943e48 1.74526
\(509\) −4.70892e48 −0.895368 −0.447684 − 0.894192i \(-0.647751\pi\)
−0.447684 + 0.894192i \(0.647751\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.26786e49 −2.16875
\(513\) 0 0
\(514\) 4.62014e48 0.736748
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −3.55009e47 −0.0475584
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.74914e49 −2.11230
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −1.11676e49 −1.25892
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.05245e49 1.00000
\(530\) 0 0
\(531\) −1.46951e49 −1.30458
\(532\) 0 0
\(533\) −1.32907e48 −0.110270
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.91251e47 0.0573822
\(538\) 1.05882e49 0.742579
\(539\) 0 0
\(540\) 0 0
\(541\) 2.84438e49 1.80483 0.902416 − 0.430867i \(-0.141792\pi\)
0.902416 + 0.430867i \(0.141792\pi\)
\(542\) 4.96459e49 3.04717
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.84752e49 1.48148 0.740739 − 0.671793i \(-0.234476\pi\)
0.740739 + 0.671793i \(0.234476\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 3.13953e48 0.138663
\(553\) 0 0
\(554\) −6.54298e49 −2.70770
\(555\) 0 0
\(556\) 6.98088e48 0.270747
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −5.38624e49 −1.95826
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 5.57972e48 0.167336
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −1.10971e50 −2.93049
\(569\) 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
\(575\) 4.72086e49 1.00000
\(576\) −1.97869e49 −0.406232
\(577\) −1.50981e49 −0.300441 −0.150220 − 0.988653i \(-0.547998\pi\)
−0.150220 + 0.988653i \(0.547998\pi\)
\(578\) 9.44775e49 1.82233
\(579\) 7.19979e47 0.0134619
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −1.83444e50 −2.93815
\(585\) 0 0
\(586\) 0 0
\(587\) 8.43597e49 1.23211 0.616054 − 0.787704i \(-0.288730\pi\)
0.616054 + 0.787704i \(0.288730\pi\)
\(588\) 9.43875e48 0.133697
\(589\) 0 0
\(590\) 0 0
\(591\) 3.15324e48 0.0407551
\(592\) 0 0
\(593\) 5.30590e49 0.645320 0.322660 − 0.946515i \(-0.395423\pi\)
0.322660 + 0.946515i \(0.395423\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −2.71507e49 −0.283897
\(599\) 1.34900e50 1.36876 0.684381 − 0.729125i \(-0.260073\pi\)
0.684381 + 0.729125i \(0.260073\pi\)
\(600\) 1.40826e49 0.138663
\(601\) 2.07459e50 1.98240 0.991200 − 0.132370i \(-0.0422587\pi\)
0.991200 + 0.132370i \(0.0422587\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.85142e50 −4.23842
\(605\) 0 0
\(606\) −2.10973e48 −0.0173668
\(607\) −2.24392e50 −1.79312 −0.896561 − 0.442919i \(-0.853943\pi\)
−0.896561 + 0.442919i \(0.853943\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.74625e49 −0.194984
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −5.56275e50 −3.61625
\(615\) 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
\(621\) −2.16983e49 −0.115021
\(622\) 6.29475e50 3.24154
\(623\) 0 0
\(624\) −3.81397e48 −0.0185377
\(625\) 2.11758e50 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
\(633\) 7.88831e48 0.0296290
\(634\) −6.05844e49 −0.221184
\(635\) 0 0
\(636\) 0 0
\(637\) −4.64561e49 −0.155788
\(638\) 0 0
\(639\) 3.82842e50 1.21340
\(640\) 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
\(647\) 7.87157e50 1.99427 0.997135 − 0.0756370i \(-0.0240990\pi\)
0.997135 + 0.0756370i \(0.0240990\pi\)
\(648\) 9.67170e50 2.38316
\(649\) 0 0
\(650\) −1.21787e50 −0.283897
\(651\) 0 0
\(652\) 2.83762e50 0.625889
\(653\) −6.65822e50 −1.42863 −0.714317 − 0.699822i \(-0.753263\pi\)
−0.714317 + 0.699822i \(0.753263\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7.40027e50 1.46211
\(657\) 6.32868e50 1.21657
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 2.17231e51 3.64316
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.94029e50 1.16298
\(668\) 1.41535e51 2.01784
\(669\) −7.83373e49 −0.108717
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.57731e51 −1.96628 −0.983139 − 0.182858i \(-0.941465\pi\)
−0.983139 + 0.182858i \(0.941465\pi\)
\(674\) 0 0
\(675\) −9.73296e49 −0.115021
\(676\) −1.96800e51 −2.26456
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.57708e50 −0.150758 −0.0753792 − 0.997155i \(-0.524017\pi\)
−0.0753792 + 0.997155i \(0.524017\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.07468e51 −0.833054 −0.416527 − 0.909123i \(-0.636753\pi\)
−0.416527 + 0.909123i \(0.636753\pi\)
\(692\) −2.53702e51 −1.91608
\(693\) 0 0
\(694\) −5.07651e51 −3.63995
\(695\) 0 0
\(696\) 2.36864e50 0.161262
\(697\) 0 0
\(698\) −4.41192e51 −2.85253
\(699\) 5.61611e49 0.0353872
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 5.59765e49 0.0326541
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −6.92041e51 −3.64458
\(707\) 0 0
\(708\) 3.49653e50 0.175000
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.44511e51 1.07817
\(714\) 0 0
\(715\) 0 0
\(716\) 5.65454e51 2.31187
\(717\) 1.75090e50 0.0698099
\(718\) 0 0
\(719\) −2.86795e51 −1.08755 −0.543777 − 0.839230i \(-0.683006\pi\)
−0.543777 + 0.839230i \(0.683006\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5.17961e51 1.82233
\(723\) 0 0
\(724\) 0 0
\(725\) 3.56168e51 1.16298
\(726\) 3.29575e50 0.104977
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −3.31425e51 −0.980144
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 5.45050e51 1.35717
\(737\) 0 0
\(738\) −5.42155e51 −1.28561
\(739\) −5.09324e51 −1.17868 −0.589338 − 0.807886i \(-0.700611\pi\)
−0.589338 + 0.807886i \(0.700611\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 7.29392e50 0.149502
\(745\) 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 \(\pi\)
\(752\) 1.52912e52 2.58535
\(753\) 0 0
\(754\) −2.04840e51 −0.330165
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.73125e51 −0.509250 −0.254625 − 0.967040i \(-0.581952\pi\)
−0.254625 + 0.967040i \(0.581952\pi\)
\(762\) 5.92229e50 0.0789406
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.72094e51 −0.203915
\(768\) −7.60123e50 −0.0879795
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 2.15843e50 0.0232895
\(772\) 5.14521e51 0.542365
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 1.09677e52 1.07817
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.63704e51 −0.133767
\(784\) 2.58668e52 2.06563
\(785\) 0 0
\(786\) −7.46527e50 −0.0569429
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 2.25341e52 1.64198
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −4.92064e52 −3.12790
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 2.44487e52 1.35717
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −6.30779e51 −0.306088
\(807\) 4.94661e50 0.0234738
\(808\) −8.58066e51 −0.398214
\(809\) 2.45072e51 0.111229 0.0556147 − 0.998452i \(-0.482288\pi\)
0.0556147 + 0.998452i \(0.482288\pi\)
\(810\) 0 0
\(811\) 2.95404e52 1.28245 0.641225 − 0.767353i \(-0.278427\pi\)
0.641225 + 0.767353i \(0.278427\pi\)
\(812\) 0 0
\(813\) 2.31936e51 0.0963246
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −8.38282e52 −3.11768
\(819\) 0 0
\(820\) 0 0
\(821\) 5.52324e52 1.92317 0.961584 − 0.274512i \(-0.0885163\pi\)
0.961584 + 0.274512i \(0.0885163\pi\)
\(822\) 0 0
\(823\) −7.98797e51 −0.266219 −0.133110 − 0.991101i \(-0.542496\pi\)
−0.133110 + 0.991101i \(0.542496\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −7.74028e52 −2.31319
\(829\) 2.15280e52 0.629538 0.314769 − 0.949168i \(-0.398073\pi\)
0.314769 + 0.949168i \(0.398073\pi\)
\(830\) 0 0
\(831\) −3.05675e51 −0.0855936
\(832\) −2.31724e51 −0.0634966
\(833\) 0 0
\(834\) 4.66653e50 0.0122463
\(835\) 0 0
\(836\) 0 0
\(837\) −5.04106e51 −0.124012
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.56137e52 0.352516
\(842\) 0 0
\(843\) 0 0
\(844\) 5.63724e52 1.19372
\(845\) 0 0
\(846\) −1.12025e53 −2.27326
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −9.10925e51 −0.162768
\(853\) 1.13859e53 1.99198 0.995989 − 0.0894738i \(-0.0285185\pi\)
0.995989 + 0.0894738i \(0.0285185\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.00821e53 1.62142 0.810712 − 0.585445i \(-0.199080\pi\)
0.810712 + 0.585445i \(0.199080\pi\)
\(858\) 0 0
\(859\) 1.28080e53 1.97517 0.987586 − 0.157078i \(-0.0502073\pi\)
0.987586 + 0.157078i \(0.0502073\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.59016e52 0.651078 0.325539 − 0.945529i \(-0.394454\pi\)
0.325539 + 0.945529i \(0.394454\pi\)
\(864\) −1.12372e52 −0.156103
\(865\) 0 0
\(866\) 0 0
\(867\) 4.41379e51 0.0576061
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −1.50583e52 −0.163193
\(877\) −1.26567e53 −1.34378 −0.671888 − 0.740653i \(-0.734516\pi\)
−0.671888 + 0.740653i \(0.734516\pi\)
\(878\) 6.12524e52 0.637121
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −1.89504e53 −1.81628
\(883\) 1.68985e53 1.58692 0.793461 − 0.608621i \(-0.208277\pi\)
0.793461 + 0.608621i \(0.208277\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.62018e53 −3.19833
\(887\) 1.99454e53 1.72671 0.863353 − 0.504601i \(-0.168360\pi\)
0.863353 + 0.504601i \(0.168360\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −5.59824e53 −4.38010
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.26843e51 −0.00897432
\(898\) −5.23710e53 −3.63176
\(899\) 1.84473e53 1.25388
\(900\) −3.47197e53 −2.31319
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −3.24304e52 −0.191710
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 2.96026e52 0.164884
\(910\) 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
\(921\) −2.59880e52 −0.114314
\(922\) −2.75451e53 −1.18819
\(923\) 4.48344e52 0.189662
\(924\) 0 0
\(925\) 0 0
\(926\) 3.02661e53 1.20770
\(927\) 0 0
\(928\) 4.11216e53 1.57836
\(929\) 5.22890e53 1.96846 0.984231 − 0.176885i \(-0.0566022\pi\)
0.984231 + 0.176885i \(0.0566022\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.01346e53 1.42571
\(933\) 2.94078e52 0.102469
\(934\) 0 0
\(935\) 0 0
\(936\) 1.13644e53 0.373751
\(937\) 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
\(943\) 2.46114e53 0.707824
\(944\) 9.58220e53 2.70377
\(945\) 0 0
\(946\) 0 0
\(947\) −4.72783e53 −1.25998 −0.629988 − 0.776605i \(-0.716940\pi\)
−0.629988 + 0.776605i \(0.716940\pi\)
\(948\) 0 0
\(949\) 7.41148e52 0.190157
\(950\) 0 0
\(951\) −2.83038e51 −0.00699190
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.25125e54 2.81257
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.93833e52 0.162446
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.45336e53 −0.265887 −0.132943 − 0.991124i \(-0.542443\pi\)
−0.132943 + 0.991124i \(0.542443\pi\)
\(968\) 1.34045e54 2.40710
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 2.39504e53 0.399319
\(973\) 0 0
\(974\) 5.93130e53 0.952991
\(975\) −5.68964e51 −0.00897432
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 1.89687e52 0.0283099
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −2.07386e54 −2.87590
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 7.34173e52 0.0981492
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −3.22810e53 −0.379858 −0.189929 − 0.981798i \(-0.560826\pi\)
−0.189929 + 0.981798i \(0.560826\pi\)
\(992\) 1.26629e54 1.46326
\(993\) 1.01486e53 0.115165
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.72070e54 −1.81632 −0.908159 − 0.418626i \(-0.862512\pi\)
−0.908159 + 0.418626i \(0.862512\pi\)
\(998\) −3.51507e54 −3.64405
\(999\) 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 23.37.b.a.22.1 ✓ 1
23.22 odd 2 CM 23.37.b.a.22.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.37.b.a.22.1 ✓ 1 1.1 even 1 trivial
23.37.b.a.22.1 ✓ 1 23.22 odd 2 CM