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Properties

Label 8.53.d.a.3.1
Level $8$
Weight $53$
Character 8.3
Self dual yes
Analytic conductor $137.003$
Analytic rank $0$
Dimension $1$
CM discriminant -8
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] = [8,53,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 53, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 53);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 53 \)
Character orbit: \([\chi]\) \(=\) 8.d (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: \(137.003123821\)
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 3.1
Character \(\chi\) \(=\) 8.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.71089e7 q^{2} +4.22705e12 q^{3} +4.50360e15 q^{4} +2.83673e20 q^{6} +3.02231e23 q^{8} +1.14069e25 q^{9} +O(q^{10})\) \(q+6.71089e7 q^{2} +4.22705e12 q^{3} +4.50360e15 q^{4} +2.83673e20 q^{6} +3.02231e23 q^{8} +1.14069e25 q^{9} -1.45275e27 q^{11} +1.90370e28 q^{12} +2.02824e31 q^{16} -7.94421e30 q^{17} +7.65504e32 q^{18} +3.05248e33 q^{19} -9.74925e34 q^{22} +1.27755e36 q^{24} +2.22045e36 q^{25} +2.09063e37 q^{27} +1.36113e39 q^{32} -6.14086e39 q^{33} -5.33127e38 q^{34} +5.13721e40 q^{36} +2.04848e41 q^{38} +1.66913e42 q^{41} +2.62627e42 q^{43} -6.54261e42 q^{44} +8.57348e43 q^{48} +8.81248e43 q^{49} +1.49012e44 q^{50} -3.35806e43 q^{51} +1.40300e45 q^{54} +1.29030e46 q^{57} -9.47669e45 q^{59} +9.13439e46 q^{64} -4.12106e47 q^{66} -6.00066e47 q^{67} -3.57775e46 q^{68} +3.44753e48 q^{72} +5.49675e48 q^{73} +9.38595e48 q^{75} +1.37471e49 q^{76} +1.46709e49 q^{81} +1.12014e50 q^{82} -3.90620e49 q^{83} +1.76246e50 q^{86} -4.39067e50 q^{88} -9.58787e50 q^{89} +5.75357e51 q^{96} -7.55145e51 q^{97} +5.91395e51 q^{98} -1.65714e52 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.71089e7 1.00000
\(3\) 4.22705e12 1.66297 0.831486 − 0.555545i \(-0.187490\pi\)
0.831486 + 0.555545i \(0.187490\pi\)
\(4\) 4.50360e15 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 2.83673e20 1.66297
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 3.02231e23 1.00000
\(9\) 1.14069e25 1.76548
\(10\) 0 0
\(11\) −1.45275e27 −1.21894 −0.609469 − 0.792810i \(-0.708617\pi\)
−0.609469 + 0.792810i \(0.708617\pi\)
\(12\) 1.90370e28 1.66297
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.02824e31 1.00000
\(17\) −7.94421e30 −0.0809802 −0.0404901 − 0.999180i \(-0.512892\pi\)
−0.0404901 + 0.999180i \(0.512892\pi\)
\(18\) 7.65504e32 1.76548
\(19\) 3.05248e33 1.72607 0.863036 − 0.505143i \(-0.168560\pi\)
0.863036 + 0.505143i \(0.168560\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −9.74925e34 −1.21894
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.27755e36 1.66297
\(25\) 2.22045e36 1.00000
\(26\) 0 0
\(27\) 2.09063e37 1.27297
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.36113e39 1.00000
\(33\) −6.14086e39 −2.02706
\(34\) −5.33127e38 −0.0809802
\(35\) 0 0
\(36\) 5.13721e40 1.76548
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 2.04848e41 1.72607
\(39\) 0 0
\(40\) 0 0
\(41\) 1.66913e42 1.95034 0.975171 − 0.221452i \(-0.0710797\pi\)
0.975171 + 0.221452i \(0.0710797\pi\)
\(42\) 0 0
\(43\) 2.62627e42 0.889527 0.444763 − 0.895648i \(-0.353288\pi\)
0.444763 + 0.895648i \(0.353288\pi\)
\(44\) −6.54261e42 −1.21894
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 8.57348e43 1.66297
\(49\) 8.81248e43 1.00000
\(50\) 1.49012e44 1.00000
\(51\) −3.35806e43 −0.134668
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.40300e45 1.27297
\(55\) 0 0
\(56\) 0 0
\(57\) 1.29030e46 2.87041
\(58\) 0 0
\(59\) −9.47669e45 −0.860011 −0.430006 − 0.902826i \(-0.641488\pi\)
−0.430006 + 0.902826i \(0.641488\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 9.13439e46 1.00000
\(65\) 0 0
\(66\) −4.12106e47 −2.02706
\(67\) −6.00066e47 −1.99643 −0.998215 − 0.0597301i \(-0.980976\pi\)
−0.998215 + 0.0597301i \(0.980976\pi\)
\(68\) −3.57775e46 −0.0809802
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 3.44753e48 1.76548
\(73\) 5.49675e48 1.96658 0.983291 − 0.182043i \(-0.0582711\pi\)
0.983291 + 0.182043i \(0.0582711\pi\)
\(74\) 0 0
\(75\) 9.38595e48 1.66297
\(76\) 1.37471e49 1.72607
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.46709e49 0.351437
\(82\) 1.12014e50 1.95034
\(83\) −3.90620e49 −0.496277 −0.248138 − 0.968725i \(-0.579819\pi\)
−0.248138 + 0.968725i \(0.579819\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.76246e50 0.889527
\(87\) 0 0
\(88\) −4.39067e50 −1.21894
\(89\) −9.58787e50 −1.98418 −0.992092 − 0.125512i \(-0.959943\pi\)
−0.992092 + 0.125512i \(0.959943\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 5.75357e51 1.66297
\(97\) −7.55145e51 −1.66711 −0.833557 − 0.552434i \(-0.813699\pi\)
−0.833557 + 0.552434i \(0.813699\pi\)
\(98\) 5.91395e51 1.00000
\(99\) −1.65714e52 −2.15201
\(100\) 1.00000e52 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −2.25356e51 −0.134668
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.06658e53 −1.83660 −0.918301 − 0.395883i \(-0.870438\pi\)
−0.918301 + 0.395883i \(0.870438\pi\)
\(108\) 9.41534e52 1.27297
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.96384e53 −1.23542 −0.617711 − 0.786406i \(-0.711940\pi\)
−0.617711 + 0.786406i \(0.711940\pi\)
\(114\) 8.65905e53 2.87041
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −6.35970e53 −0.860011
\(119\) 0 0
\(120\) 0 0
\(121\) 6.90057e53 0.485809
\(122\) 0 0
\(123\) 7.05552e54 3.24337
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 6.12998e54 1.00000
\(129\) 1.11014e55 1.47926
\(130\) 0 0
\(131\) −2.23342e55 −1.99488 −0.997438 − 0.0715312i \(-0.977211\pi\)
−0.997438 + 0.0715312i \(0.977211\pi\)
\(132\) −2.76560e55 −2.02706
\(133\) 0 0
\(134\) −4.02697e55 −1.99643
\(135\) 0 0
\(136\) −2.40099e54 −0.0809802
\(137\) 4.39156e55 1.22429 0.612145 − 0.790746i \(-0.290307\pi\)
0.612145 + 0.790746i \(0.290307\pi\)
\(138\) 0 0
\(139\) 1.00135e56 1.91514 0.957568 − 0.288207i \(-0.0930590\pi\)
0.957568 + 0.288207i \(0.0930590\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.31359e56 1.76548
\(145\) 0 0
\(146\) 3.68881e56 1.96658
\(147\) 3.72508e56 1.66297
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 6.29880e56 1.66297
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 9.22554e56 1.72607
\(153\) −9.06189e55 −0.142969
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 9.84550e56 0.351437
\(163\) −3.90162e57 −1.18677 −0.593386 − 0.804918i \(-0.702209\pi\)
−0.593386 + 0.804918i \(0.702209\pi\)
\(164\) 7.51711e57 1.95034
\(165\) 0 0
\(166\) −2.62141e57 −0.496277
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.41500e57 1.00000
\(170\) 0 0
\(171\) 3.48193e58 3.04734
\(172\) 1.18277e58 0.889527
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.94653e58 −1.21894
\(177\) −4.00585e58 −1.43018
\(178\) −6.43431e58 −1.98418
\(179\) 5.88722e58 1.56939 0.784695 − 0.619882i \(-0.212819\pi\)
0.784695 + 0.619882i \(0.212819\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.15410e58 0.0987098
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 3.86115e59 1.66297
\(193\) −4.91425e59 −1.84913 −0.924564 − 0.381026i \(-0.875571\pi\)
−0.924564 + 0.381026i \(0.875571\pi\)
\(194\) −5.06769e59 −1.66711
\(195\) 0 0
\(196\) 3.96879e59 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −1.11209e60 −2.15201
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 6.71089e59 1.00000
\(201\) −2.53651e60 −3.32001
\(202\) 0 0
\(203\) 0 0
\(204\) −1.51234e59 −0.134668
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.43449e60 −2.10397
\(210\) 0 0
\(211\) 2.85367e60 1.05696 0.528481 − 0.848945i \(-0.322762\pi\)
0.528481 + 0.848945i \(0.322762\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −7.15769e60 −1.83660
\(215\) 0 0
\(216\) 6.31853e60 1.27297
\(217\) 0 0
\(218\) 0 0
\(219\) 2.32351e61 3.27037
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.53284e61 1.76548
\(226\) −1.98900e61 −1.23542
\(227\) 6.31547e60 0.349730 0.174865 − 0.984592i \(-0.444051\pi\)
0.174865 + 0.984592i \(0.444051\pi\)
\(228\) 5.81099e61 2.87041
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.86528e61 −1.92932 −0.964661 − 0.263496i \(-0.915125\pi\)
−0.964661 + 0.263496i \(0.915125\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.26792e61 −0.860011
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 3.82712e61 0.447125 0.223562 − 0.974690i \(-0.428231\pi\)
0.223562 + 0.974690i \(0.428231\pi\)
\(242\) 4.63090e61 0.485809
\(243\) −7.30622e61 −0.688541
\(244\) 0 0
\(245\) 0 0
\(246\) 4.73488e62 3.24337
\(247\) 0 0
\(248\) 0 0
\(249\) −1.65117e62 −0.825295
\(250\) 0 0
\(251\) −4.36606e62 −1.77245 −0.886223 − 0.463259i \(-0.846680\pi\)
−0.886223 + 0.463259i \(0.846680\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.11376e62 1.00000
\(257\) 9.06267e62 1.99065 0.995325 − 0.0965843i \(-0.0307917\pi\)
0.995325 + 0.0965843i \(0.0307917\pi\)
\(258\) 7.45000e62 1.47926
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.49882e63 −1.99488
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −1.85596e63 −2.02706
\(265\) 0 0
\(266\) 0 0
\(267\) −4.05284e63 −3.29964
\(268\) −2.70246e63 −1.99643
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.61128e62 −0.0809802
\(273\) 0 0
\(274\) 2.94713e63 1.22429
\(275\) −3.22576e63 −1.21894
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 6.71992e63 1.91514
\(279\) 0 0
\(280\) 0 0
\(281\) −3.38925e63 −0.730701 −0.365350 − 0.930870i \(-0.619051\pi\)
−0.365350 + 0.930870i \(0.619051\pi\)
\(282\) 0 0
\(283\) 8.95756e63 1.60599 0.802994 − 0.595987i \(-0.203239\pi\)
0.802994 + 0.595987i \(0.203239\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.55263e64 1.76548
\(289\) −9.56063e63 −0.993442
\(290\) 0 0
\(291\) −3.19204e64 −2.77236
\(292\) 2.47552e64 1.96658
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 2.49986e64 1.66297
\(295\) 0 0
\(296\) 0 0
\(297\) −3.03716e64 −1.55167
\(298\) 0 0
\(299\) 0 0
\(300\) 4.22705e64 1.66297
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 6.19116e64 1.72607
\(305\) 0 0
\(306\) −6.08133e63 −0.142969
\(307\) 9.15452e64 1.97714 0.988571 − 0.150754i \(-0.0481702\pi\)
0.988571 + 0.150754i \(0.0481702\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.05737e65 −1.38063 −0.690313 − 0.723511i \(-0.742527\pi\)
−0.690313 + 0.723511i \(0.742527\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4.50849e65 −3.05422
\(322\) 0 0
\(323\) −2.42495e64 −0.139778
\(324\) 6.60720e64 0.351437
\(325\) 0 0
\(326\) −2.61833e65 −1.18677
\(327\) 0 0
\(328\) 5.04465e65 1.95034
\(329\) 0 0
\(330\) 0 0
\(331\) −5.58848e65 −1.70516 −0.852582 − 0.522593i \(-0.824965\pi\)
−0.852582 + 0.522593i \(0.824965\pi\)
\(332\) −1.75920e65 −0.496277
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.04783e64 0.153922 0.0769611 − 0.997034i \(-0.475478\pi\)
0.0769611 + 0.997034i \(0.475478\pi\)
\(338\) 5.64721e65 1.00000
\(339\) −1.25283e66 −2.05447
\(340\) 0 0
\(341\) 0 0
\(342\) 2.33668e66 3.04734
\(343\) 0 0
\(344\) 7.93740e65 0.889527
\(345\) 0 0
\(346\) 0 0
\(347\) −1.18270e66 −1.05757 −0.528784 − 0.848756i \(-0.677352\pi\)
−0.528784 + 0.848756i \(0.677352\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.97738e66 −1.21894
\(353\) −2.01917e66 −1.15619 −0.578097 − 0.815968i \(-0.696205\pi\)
−0.578097 + 0.815968i \(0.696205\pi\)
\(354\) −2.68828e66 −1.43018
\(355\) 0 0
\(356\) −4.31799e66 −1.98418
\(357\) 0 0
\(358\) 3.95084e66 1.56939
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 6.19019e66 1.97932
\(362\) 0 0
\(363\) 2.91691e66 0.807887
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 1.90396e67 3.44329
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 7.74501e65 0.0987098
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 9.20834e66 0.830925 0.415462 − 0.909610i \(-0.363620\pi\)
0.415462 + 0.909610i \(0.363620\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 2.59118e67 1.66297
\(385\) 0 0
\(386\) −3.29790e67 −1.84913
\(387\) 2.99576e67 1.57044
\(388\) −3.40087e67 −1.66711
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 2.66341e67 1.00000
\(393\) −9.44077e67 −3.31743
\(394\) 0 0
\(395\) 0 0
\(396\) −7.46309e67 −2.15201
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.50360e67 1.00000
\(401\) 3.29359e66 0.0685354 0.0342677 − 0.999413i \(-0.489090\pi\)
0.0342677 + 0.999413i \(0.489090\pi\)
\(402\) −1.70222e68 −3.32001
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −1.01491e67 −0.134668
\(409\) 9.60652e66 0.119608 0.0598040 − 0.998210i \(-0.480952\pi\)
0.0598040 + 0.998210i \(0.480952\pi\)
\(410\) 0 0
\(411\) 1.85634e68 2.03596
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.23275e68 3.18482
\(418\) −2.97594e68 −2.10397
\(419\) 3.32209e67 0.220723 0.110361 − 0.993892i \(-0.464799\pi\)
0.110361 + 0.993892i \(0.464799\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 1.91506e68 1.05696
\(423\) 0 0
\(424\) 0 0
\(425\) −1.76397e67 −0.0809802
\(426\) 0 0
\(427\) 0 0
\(428\) −4.80345e68 −1.83660
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 4.24029e68 1.27297
\(433\) −6.99513e68 −1.97747 −0.988737 − 0.149665i \(-0.952181\pi\)
−0.988737 + 0.149665i \(0.952181\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.55928e69 3.27037
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.00523e69 1.76548
\(442\) 0 0
\(443\) −1.20107e69 −1.87530 −0.937650 − 0.347581i \(-0.887003\pi\)
−0.937650 + 0.347581i \(0.887003\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41466e69 1.55685 0.778425 − 0.627738i \(-0.216019\pi\)
0.778425 + 0.627738i \(0.216019\pi\)
\(450\) 1.69976e69 1.76548
\(451\) −2.42484e69 −2.37735
\(452\) −1.33479e69 −1.23542
\(453\) 0 0
\(454\) 4.23824e68 0.349730
\(455\) 0 0
\(456\) 3.89969e69 2.87041
\(457\) −2.42785e69 −1.68811 −0.844054 − 0.536258i \(-0.819837\pi\)
−0.844054 + 0.536258i \(0.819837\pi\)
\(458\) 0 0
\(459\) −1.66084e68 −0.103085
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −4.60721e69 −1.92932
\(467\) −2.95098e69 −1.16877 −0.584384 − 0.811477i \(-0.698664\pi\)
−0.584384 + 0.811477i \(0.698664\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −2.86415e69 −0.860011
\(473\) −3.81531e69 −1.08428
\(474\) 0 0
\(475\) 6.77786e69 1.72607
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.56833e69 0.447125
\(483\) 0 0
\(484\) 3.10774e69 0.485809
\(485\) 0 0
\(486\) −4.90312e69 −0.688541
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −1.64923e70 −1.97357
\(490\) 0 0
\(491\) −7.05719e69 −0.759474 −0.379737 − 0.925094i \(-0.623986\pi\)
−0.379737 + 0.925094i \(0.623986\pi\)
\(492\) 3.17752e70 3.24337
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.10808e70 −0.825295
\(499\) −1.83264e70 −1.29557 −0.647787 − 0.761822i \(-0.724305\pi\)
−0.647787 + 0.761822i \(0.724305\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.93001e70 −1.77245
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.55707e70 1.66297
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2.76070e70 1.00000
\(513\) 6.38159e70 2.19724
\(514\) 6.08185e70 1.99065
\(515\) 0 0
\(516\) 4.99961e70 1.47926
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.30730e70 1.68258 0.841290 − 0.540583i \(-0.181796\pi\)
0.841290 + 0.540583i \(0.181796\pi\)
\(522\) 0 0
\(523\) 7.18932e70 1.49846 0.749228 − 0.662312i \(-0.230425\pi\)
0.749228 + 0.662312i \(0.230425\pi\)
\(524\) −1.00584e71 −1.99488
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −1.24551e71 −2.02706
\(529\) 6.45428e70 1.00000
\(530\) 0 0
\(531\) −1.08100e71 −1.51833
\(532\) 0 0
\(533\) 0 0
\(534\) −2.71982e71 −3.29964
\(535\) 0 0
\(536\) −1.81359e71 −1.99643
\(537\) 2.48856e71 2.60985
\(538\) 0 0
\(539\) −1.28023e71 −1.21894
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.08131e70 −0.0809802
\(545\) 0 0
\(546\) 0 0
\(547\) −3.07540e71 −1.99632 −0.998161 − 0.0606160i \(-0.980693\pi\)
−0.998161 + 0.0606160i \(0.980693\pi\)
\(548\) 1.97778e71 1.22429
\(549\) 0 0
\(550\) −2.16477e71 −1.21894
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 4.50967e71 1.91514
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.87843e70 0.164152
\(562\) −2.27449e71 −0.730701
\(563\) −3.36207e71 −1.03131 −0.515655 − 0.856796i \(-0.672451\pi\)
−0.515655 + 0.856796i \(0.672451\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.01131e71 1.60599
\(567\) 0 0
\(568\) 0 0
\(569\) 7.25999e71 1.69051 0.845253 − 0.534366i \(-0.179450\pi\)
0.845253 + 0.534366i \(0.179450\pi\)
\(570\) 0 0
\(571\) 9.09972e71 1.93415 0.967073 − 0.254500i \(-0.0819109\pi\)
0.967073 + 0.254500i \(0.0819109\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.04195e72 1.76548
\(577\) −1.23356e72 −1.99797 −0.998985 − 0.0450454i \(-0.985657\pi\)
−0.998985 + 0.0450454i \(0.985657\pi\)
\(578\) −6.41603e71 −0.993442
\(579\) −2.07728e72 −3.07505
\(580\) 0 0
\(581\) 0 0
\(582\) −2.14214e72 −2.77236
\(583\) 0 0
\(584\) 1.66129e72 1.96658
\(585\) 0 0
\(586\) 0 0
\(587\) 1.90174e72 1.97042 0.985212 − 0.171342i \(-0.0548104\pi\)
0.985212 + 0.171342i \(0.0548104\pi\)
\(588\) 1.67763e72 1.66297
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.68632e72 1.34128 0.670639 − 0.741784i \(-0.266020\pi\)
0.670639 + 0.741784i \(0.266020\pi\)
\(594\) −2.03820e72 −1.55167
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 2.83673e72 1.66297
\(601\) 3.17472e72 1.78225 0.891127 − 0.453754i \(-0.149915\pi\)
0.891127 + 0.453754i \(0.149915\pi\)
\(602\) 0 0
\(603\) −6.84489e72 −3.52465
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 4.15482e72 1.72607
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −4.08111e71 −0.142969
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 6.14350e72 1.97714
\(615\) 0 0
\(616\) 0 0
\(617\) −4.69271e72 −1.33048 −0.665242 − 0.746628i \(-0.731672\pi\)
−0.665242 + 0.746628i \(0.731672\pi\)
\(618\) 0 0
\(619\) −7.29557e72 −1.90153 −0.950763 − 0.309919i \(-0.899698\pi\)
−0.950763 + 0.309919i \(0.899698\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.93038e72 1.00000
\(626\) −7.09592e72 −1.38063
\(627\) −1.87448e73 −3.49885
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.20626e73 1.75770
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.57358e73 1.65417 0.827087 − 0.562073i \(-0.189996\pi\)
0.827087 + 0.562073i \(0.189996\pi\)
\(642\) −3.02560e73 −3.05422
\(643\) −1.59981e73 −1.55090 −0.775451 − 0.631408i \(-0.782477\pi\)
−0.775451 + 0.631408i \(0.782477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.62736e72 −0.139778
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 4.43402e72 0.351437
\(649\) 1.37673e73 1.04830
\(650\) 0 0
\(651\) 0 0
\(652\) −1.75713e73 −1.18677
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.38541e73 1.95034
\(657\) 6.27009e73 3.47196
\(658\) 0 0
\(659\) 3.81200e73 1.95044 0.975218 − 0.221245i \(-0.0710120\pi\)
0.975218 + 0.221245i \(0.0710120\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −3.75037e73 −1.70516
\(663\) 0 0
\(664\) −1.18058e73 −0.496277
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.33143e73 −1.57925 −0.789627 − 0.613588i \(-0.789726\pi\)
−0.789627 + 0.613588i \(0.789726\pi\)
\(674\) 5.40081e72 0.153922
\(675\) 4.64212e73 1.27297
\(676\) 3.78978e73 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −8.40760e73 −2.05447
\(679\) 0 0
\(680\) 0 0
\(681\) 2.66958e73 0.581592
\(682\) 0 0
\(683\) −1.64770e73 −0.332614 −0.166307 − 0.986074i \(-0.553184\pi\)
−0.166307 + 0.986074i \(0.553184\pi\)
\(684\) 1.56812e74 3.04734
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 5.32670e73 0.889527
\(689\) 0 0
\(690\) 0 0
\(691\) 5.47614e73 0.816667 0.408333 − 0.912833i \(-0.366110\pi\)
0.408333 + 0.912833i \(0.366110\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −7.93697e73 −1.05757
\(695\) 0 0
\(696\) 0 0
\(697\) −1.32600e73 −0.157939
\(698\) 0 0
\(699\) −2.90199e74 −3.20841
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.32700e74 −1.21894
\(705\) 0 0
\(706\) −1.35504e74 −1.15619
\(707\) 0 0
\(708\) −1.80407e74 −1.43018
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2.89776e74 −1.98418
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 2.65137e74 1.56939
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 4.15416e74 1.97932
\(723\) 1.61774e74 0.743556
\(724\) 0 0
\(725\) 0 0
\(726\) 1.95750e74 0.807887
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −4.03628e74 −1.49646
\(730\) 0 0
\(731\) −2.08636e73 −0.0720340
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.71746e74 2.43352
\(738\) 1.27773e75 3.44329
\(739\) −6.45973e74 −1.68058 −0.840289 − 0.542138i \(-0.817615\pi\)
−0.840289 + 0.542138i \(0.817615\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4.45576e74 −0.876166
\(748\) 5.19759e73 0.0987098
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −1.84556e75 −2.94753
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 6.17961e74 0.830925
\(759\) 0 0
\(760\) 0 0
\(761\) −4.17734e74 −0.506873 −0.253436 − 0.967352i \(-0.581561\pi\)
−0.253436 + 0.967352i \(0.581561\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.73891e75 1.66297
\(769\) −1.53632e75 −1.42035 −0.710176 − 0.704024i \(-0.751385\pi\)
−0.710176 + 0.704024i \(0.751385\pi\)
\(770\) 0 0
\(771\) 3.83084e75 3.31040
\(772\) −2.21318e75 −1.84913
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 2.01042e75 1.57044
\(775\) 0 0
\(776\) −2.28229e75 −1.66711
\(777\) 0 0
\(778\) 0 0
\(779\) 5.09499e75 3.36643
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.78738e75 1.00000
\(785\) 0 0
\(786\) −6.33560e75 −3.31743
\(787\) 3.15437e75 1.59797 0.798987 − 0.601349i \(-0.205370\pi\)
0.798987 + 0.601349i \(0.205370\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −5.00840e75 −2.15201
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.02231e75 1.00000
\(801\) −1.09368e76 −3.50304
\(802\) 2.21029e74 0.0685354
\(803\) −7.98542e75 −2.39714
\(804\) −1.14234e76 −3.32001
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.70051e75 1.41011 0.705054 − 0.709153i \(-0.250923\pi\)
0.705054 + 0.709153i \(0.250923\pi\)
\(810\) 0 0
\(811\) 8.27939e75 1.92069 0.960343 − 0.278822i \(-0.0899438\pi\)
0.960343 + 0.278822i \(0.0899438\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −6.81096e74 −0.134668
\(817\) 8.01662e75 1.53539
\(818\) 6.44683e74 0.119608
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 1.24577e76 2.03596
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −1.36354e76 −2.02706
\(826\) 0 0
\(827\) 1.20214e76 1.67808 0.839038 − 0.544073i \(-0.183119\pi\)
0.839038 + 0.544073i \(0.183119\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.00082e74 −0.0809802
\(834\) 2.84055e76 3.18482
\(835\) 0 0
\(836\) −1.99712e76 −2.10397
\(837\) 0 0
\(838\) 2.22942e75 0.220723
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.10840e76 1.00000
\(842\) 0 0
\(843\) −1.43266e76 −1.21514
\(844\) 1.28518e76 1.05696
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 3.78641e76 2.67071
\(850\) −1.18378e75 −0.0809802
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.22354e76 −1.83660
\(857\) −7.65799e75 −0.423266 −0.211633 − 0.977349i \(-0.567878\pi\)
−0.211633 + 0.977349i \(0.567878\pi\)
\(858\) 0 0
\(859\) 1.13644e76 0.591186 0.295593 − 0.955314i \(-0.404483\pi\)
0.295593 + 0.955314i \(0.404483\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 2.84561e76 1.27297
\(865\) 0 0
\(866\) −4.69435e76 −1.97747
\(867\) −4.04133e76 −1.65207
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −8.61386e76 −2.94325
\(874\) 0 0
\(875\) 0 0
\(876\) 1.04641e77 3.27037
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7.06280e76 −1.90372 −0.951860 − 0.306532i \(-0.900831\pi\)
−0.951860 + 0.306532i \(0.900831\pi\)
\(882\) 6.74599e76 1.76548
\(883\) −1.40499e76 −0.357023 −0.178512 − 0.983938i \(-0.557128\pi\)
−0.178512 + 0.983938i \(0.557128\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −8.06023e76 −1.87530
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.13132e76 −0.428379
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 9.49361e76 1.55685
\(899\) 0 0
\(900\) 1.14069e77 1.76548
\(901\) 0 0
\(902\) −1.62728e77 −2.37735
\(903\) 0 0
\(904\) −8.95765e76 −1.23542
\(905\) 0 0
\(906\) 0 0
\(907\) −1.15493e77 −1.46139 −0.730693 − 0.682706i \(-0.760803\pi\)
−0.730693 + 0.682706i \(0.760803\pi\)
\(908\) 2.84423e76 0.349730
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 2.61704e77 2.87041
\(913\) 5.67473e76 0.604930
\(914\) −1.62930e77 −1.68811
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −1.11457e76 −0.103085
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 3.86967e77 3.28793
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.94465e77 1.99814 0.999070 − 0.0431269i \(-0.0137320\pi\)
0.999070 + 0.0431269i \(0.0137320\pi\)
\(930\) 0 0
\(931\) 2.68999e77 1.72607
\(932\) −3.09185e77 −1.92932
\(933\) 0 0
\(934\) −1.98037e77 −1.16877
\(935\) 0 0
\(936\) 0 0
\(937\) 2.57111e77 1.39602 0.698009 − 0.716089i \(-0.254069\pi\)
0.698009 + 0.716089i \(0.254069\pi\)
\(938\) 0 0
\(939\) −4.46958e77 −2.29594
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.92210e77 −0.860011
\(945\) 0 0
\(946\) −2.56041e77 −1.08428
\(947\) 1.09619e77 0.451633 0.225817 − 0.974170i \(-0.427495\pi\)
0.225817 + 0.974170i \(0.427495\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.54854e77 1.72607
\(951\) 0 0
\(952\) 0 0
\(953\) 2.05822e77 0.719577 0.359788 − 0.933034i \(-0.382849\pi\)
0.359788 + 0.933034i \(0.382849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 3.55474e77 1.00000
\(962\) 0 0
\(963\) −1.21664e78 −3.24248
\(964\) 1.72358e77 0.447125
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 2.08557e77 0.485809
\(969\) −1.02504e77 −0.232446
\(970\) 0 0
\(971\) −4.27383e77 −0.918581 −0.459291 − 0.888286i \(-0.651896\pi\)
−0.459291 + 0.888286i \(0.651896\pi\)
\(972\) −3.29043e77 −0.688541
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.08315e77 0.564594 0.282297 − 0.959327i \(-0.408904\pi\)
0.282297 + 0.959327i \(0.408904\pi\)
\(978\) −1.10678e78 −1.97357
\(979\) 1.39288e78 2.41860
\(980\) 0 0
\(981\) 0 0
\(982\) −4.73600e77 −0.759474
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 2.13240e78 3.24337
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −2.36228e78 −2.83564
\(994\) 0 0
\(995\) 0 0
\(996\) −7.43621e77 −0.825295
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −1.22986e78 −1.29557
\(999\) 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 8.53.d.a.3.1 ✓ 1
8.3 odd 2 CM 8.53.d.a.3.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.53.d.a.3.1 ✓ 1 1.1 even 1 trivial
8.53.d.a.3.1 ✓ 1 8.3 odd 2 CM