Properties

Label 55.13.d.b.54.1
Level $55$
Weight $13$
Character 55.54
Self dual yes
Analytic conductor $50.270$
Analytic rank $0$
Dimension $1$
CM discriminant -55
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] = [55,13,Mod(54,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.54");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 55.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: \(50.2696599502\)
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 54.1
Character \(\chi\) \(=\) 55.54

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+117.000 q^{2} +9593.00 q^{4} +15625.0 q^{5} +87282.0 q^{7} +643149. q^{8} +531441. q^{9} +O(q^{10})\) \(q+117.000 q^{2} +9593.00 q^{4} +15625.0 q^{5} +87282.0 q^{7} +643149. q^{8} +531441. q^{9} +1.82812e6 q^{10} +1.77156e6 q^{11} -9.62912e6 q^{13} +1.02120e7 q^{14} +3.59555e7 q^{16} -3.57615e7 q^{17} +6.21786e7 q^{18} +1.49891e8 q^{20} +2.07273e8 q^{22} +2.44141e8 q^{25} -1.12661e9 q^{26} +8.37296e8 q^{28} +3.46700e8 q^{31} +1.57246e9 q^{32} -4.18410e9 q^{34} +1.36378e9 q^{35} +5.09811e9 q^{36} +1.00492e10 q^{40} +7.56554e9 q^{43} +1.69946e10 q^{44} +8.30377e9 q^{45} -6.22314e9 q^{49} +2.85645e10 q^{50} -9.23721e10 q^{52} +2.76806e10 q^{55} +5.61353e10 q^{56} -8.43452e10 q^{59} +4.05640e10 q^{62} +4.63852e10 q^{63} +3.67036e10 q^{64} -1.50455e11 q^{65} -3.43060e11 q^{68} +1.59562e11 q^{70} +2.40884e11 q^{71} +3.41796e11 q^{72} -2.97570e11 q^{73} +1.54625e11 q^{77} +5.61805e11 q^{80} +2.82430e11 q^{81} +5.79982e11 q^{83} -5.58774e11 q^{85} +8.85168e11 q^{86} +1.13938e12 q^{88} -9.91705e11 q^{89} +9.71541e11 q^{90} -8.40449e11 q^{91} -7.28107e11 q^{98} +9.41480e11 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 117.000 1.82812 0.914062 0.405573i \(-0.132928\pi\)
0.914062 + 0.405573i \(0.132928\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 9593.00 2.34204
\(5\) 15625.0 1.00000
\(6\) 0 0
\(7\) 87282.0 0.741885 0.370942 0.928656i \(-0.379035\pi\)
0.370942 + 0.928656i \(0.379035\pi\)
\(8\) 643149. 2.45342
\(9\) 531441. 1.00000
\(10\) 1.82812e6 1.82812
\(11\) 1.77156e6 1.00000
\(12\) 0 0
\(13\) −9.62912e6 −1.99492 −0.997462 0.0711996i \(-0.977317\pi\)
−0.997462 + 0.0711996i \(0.977317\pi\)
\(14\) 1.02120e7 1.35626
\(15\) 0 0
\(16\) 3.59555e7 2.14312
\(17\) −3.57615e7 −1.48157 −0.740785 0.671742i \(-0.765546\pi\)
−0.740785 + 0.671742i \(0.765546\pi\)
\(18\) 6.21786e7 1.82812
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.49891e8 2.34204
\(21\) 0 0
\(22\) 2.07273e8 1.82812
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.44141e8 1.00000
\(26\) −1.12661e9 −3.64697
\(27\) 0 0
\(28\) 8.37296e8 1.73752
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 3.46700e8 0.390647 0.195323 0.980739i \(-0.437424\pi\)
0.195323 + 0.980739i \(0.437424\pi\)
\(32\) 1.57246e9 1.46446
\(33\) 0 0
\(34\) −4.18410e9 −2.70850
\(35\) 1.36378e9 0.741885
\(36\) 5.09811e9 2.34204
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00492e10 2.45342
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 7.56554e9 1.19682 0.598411 0.801190i \(-0.295799\pi\)
0.598411 + 0.801190i \(0.295799\pi\)
\(44\) 1.69946e10 2.34204
\(45\) 8.30377e9 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −6.22314e9 −0.449607
\(50\) 2.85645e10 1.82812
\(51\) 0 0
\(52\) −9.23721e10 −4.67219
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 2.76806e10 1.00000
\(56\) 5.61353e10 1.82015
\(57\) 0 0
\(58\) 0 0
\(59\) −8.43452e10 −1.99962 −0.999812 0.0193682i \(-0.993835\pi\)
−0.999812 + 0.0193682i \(0.993835\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 4.05640e10 0.714151
\(63\) 4.63852e10 0.741885
\(64\) 3.67036e10 0.534107
\(65\) −1.50455e11 −1.99492
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −3.43060e11 −3.46990
\(69\) 0 0
\(70\) 1.59562e11 1.35626
\(71\) 2.40884e11 1.88043 0.940215 0.340581i \(-0.110624\pi\)
0.940215 + 0.340581i \(0.110624\pi\)
\(72\) 3.41796e11 2.45342
\(73\) −2.97570e11 −1.96631 −0.983154 0.182782i \(-0.941490\pi\)
−0.983154 + 0.182782i \(0.941490\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.54625e11 0.741885
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 5.61805e11 2.14312
\(81\) 2.82430e11 1.00000
\(82\) 0 0
\(83\) 5.79982e11 1.77397 0.886984 0.461800i \(-0.152796\pi\)
0.886984 + 0.461800i \(0.152796\pi\)
\(84\) 0 0
\(85\) −5.58774e11 −1.48157
\(86\) 8.85168e11 2.18794
\(87\) 0 0
\(88\) 1.13938e12 2.45342
\(89\) −9.91705e11 −1.99546 −0.997728 0.0673661i \(-0.978540\pi\)
−0.997728 + 0.0673661i \(0.978540\pi\)
\(90\) 9.71541e11 1.82812
\(91\) −8.40449e11 −1.48000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −7.28107e11 −0.821938
\(99\) 9.41480e11 1.00000
\(100\) 2.34204e12 2.34204
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −6.19296e12 −4.89438
\(105\) 0 0
\(106\) 0 0
\(107\) −6.25857e11 −0.417035 −0.208518 0.978019i \(-0.566864\pi\)
−0.208518 + 0.978019i \(0.566864\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 3.23863e12 1.82812
\(111\) 0 0
\(112\) 3.13827e12 1.58994
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.11731e12 −1.99492
\(118\) −9.86839e12 −3.65556
\(119\) −3.12134e12 −1.09915
\(120\) 0 0
\(121\) 3.13843e12 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 3.32590e12 0.914911
\(125\) 3.81470e12 1.00000
\(126\) 5.42707e12 1.35626
\(127\) −6.18927e12 −1.47509 −0.737543 0.675301i \(-0.764014\pi\)
−0.737543 + 0.675301i \(0.764014\pi\)
\(128\) −2.14646e12 −0.488049
\(129\) 0 0
\(130\) −1.76032e13 −3.64697
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −2.30000e13 −3.63491
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1.30828e13 1.73752
\(141\) 0 0
\(142\) 2.81834e13 3.43766
\(143\) −1.70586e13 −1.99492
\(144\) 1.91082e13 2.14312
\(145\) 0 0
\(146\) −3.48156e13 −3.59466
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1.90051e13 −1.48157
\(154\) 1.80912e13 1.35626
\(155\) 5.41720e12 0.390647
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 2.45696e13 1.46446
\(161\) 0 0
\(162\) 3.30443e13 1.82812
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.78579e13 3.24303
\(167\) 1.76355e13 0.812997 0.406499 0.913651i \(-0.366750\pi\)
0.406499 + 0.913651i \(0.366750\pi\)
\(168\) 0 0
\(169\) 6.94218e13 2.97972
\(170\) −6.53765e13 −2.70850
\(171\) 0 0
\(172\) 7.25762e13 2.80300
\(173\) 9.83390e12 0.366817 0.183408 0.983037i \(-0.441287\pi\)
0.183408 + 0.983037i \(0.441287\pi\)
\(174\) 0 0
\(175\) 2.13091e13 0.741885
\(176\) 6.36974e13 2.14312
\(177\) 0 0
\(178\) −1.16029e14 −3.64794
\(179\) −5.28440e13 −1.60649 −0.803245 0.595649i \(-0.796895\pi\)
−0.803245 + 0.595649i \(0.796895\pi\)
\(180\) 7.96580e13 2.34204
\(181\) −2.95056e13 −0.839137 −0.419568 0.907724i \(-0.637819\pi\)
−0.419568 + 0.907724i \(0.637819\pi\)
\(182\) −9.83325e13 −2.70563
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.33537e13 −1.48157
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.44692e13 −1.94576 −0.972881 0.231304i \(-0.925701\pi\)
−0.972881 + 0.231304i \(0.925701\pi\)
\(192\) 0 0
\(193\) −9.13411e13 −1.76735 −0.883675 0.468101i \(-0.844938\pi\)
−0.883675 + 0.468101i \(0.844938\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.96986e13 −1.05300
\(197\) −3.07965e13 −0.526871 −0.263435 0.964677i \(-0.584856\pi\)
−0.263435 + 0.964677i \(0.584856\pi\)
\(198\) 1.10153e14 1.82812
\(199\) 1.16265e14 1.87211 0.936053 0.351858i \(-0.114450\pi\)
0.936053 + 0.351858i \(0.114450\pi\)
\(200\) 1.57019e14 2.45342
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3.46220e14 −4.27535
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −7.32253e13 −0.762392
\(215\) 1.18212e14 1.19682
\(216\) 0 0
\(217\) 3.02607e13 0.289815
\(218\) 0 0
\(219\) 0 0
\(220\) 2.65540e14 2.34204
\(221\) 3.44352e14 2.95562
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 1.37247e14 1.08646
\(225\) 1.29746e14 1.00000
\(226\) 0 0
\(227\) 2.37842e14 1.73833 0.869167 0.494519i \(-0.164656\pi\)
0.869167 + 0.494519i \(0.164656\pi\)
\(228\) 0 0
\(229\) −2.11674e14 −1.46776 −0.733879 0.679280i \(-0.762292\pi\)
−0.733879 + 0.679280i \(0.762292\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.13364e14 1.95846 0.979229 0.202760i \(-0.0649910\pi\)
0.979229 + 0.202760i \(0.0649910\pi\)
\(234\) −5.98725e14 −3.64697
\(235\) 0 0
\(236\) −8.09124e14 −4.68320
\(237\) 0 0
\(238\) −3.65196e14 −2.00939
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 3.67196e14 1.82812
\(243\) 0 0
\(244\) 0 0
\(245\) −9.72366e13 −0.449607
\(246\) 0 0
\(247\) 0 0
\(248\) 2.22980e14 0.958420
\(249\) 0 0
\(250\) 4.46320e14 1.82812
\(251\) 4.53029e14 1.81169 0.905845 0.423610i \(-0.139237\pi\)
0.905845 + 0.423610i \(0.139237\pi\)
\(252\) 4.44974e14 1.73752
\(253\) 0 0
\(254\) −7.24145e14 −2.69664
\(255\) 0 0
\(256\) −4.01474e14 −1.42632
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.44331e15 −4.67219
\(261\) 0 0
\(262\) 0 0
\(263\) −7.65714e13 −0.231383 −0.115692 0.993285i \(-0.536908\pi\)
−0.115692 + 0.993285i \(0.536908\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.14644e14 1.09436 0.547182 0.837013i \(-0.315700\pi\)
0.547182 + 0.837013i \(0.315700\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.28582e15 −3.17518
\(273\) 0 0
\(274\) 0 0
\(275\) 4.32510e14 1.00000
\(276\) 0 0
\(277\) 2.73051e14 0.604458 0.302229 0.953235i \(-0.402269\pi\)
0.302229 + 0.953235i \(0.402269\pi\)
\(278\) 0 0
\(279\) 1.84251e14 0.390647
\(280\) 8.77115e14 1.82015
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.13186e14 0.220330 0.110165 0.993913i \(-0.464862\pi\)
0.110165 + 0.993913i \(0.464862\pi\)
\(284\) 2.31080e15 4.40404
\(285\) 0 0
\(286\) −1.99585e15 −3.64697
\(287\) 0 0
\(288\) 8.35667e14 1.46446
\(289\) 6.96264e14 1.19505
\(290\) 0 0
\(291\) 0 0
\(292\) −2.85458e15 −4.60517
\(293\) 1.25376e15 1.98157 0.990787 0.135429i \(-0.0432411\pi\)
0.990787 + 0.135429i \(0.0432411\pi\)
\(294\) 0 0
\(295\) −1.31789e15 −1.99962
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 6.60336e14 0.887903
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −2.22360e15 −2.70850
\(307\) −1.01245e15 −1.20932 −0.604662 0.796482i \(-0.706692\pi\)
−0.604662 + 0.796482i \(0.706692\pi\)
\(308\) 1.48332e15 1.73752
\(309\) 0 0
\(310\) 6.33812e14 0.714151
\(311\) 8.61256e14 0.951853 0.475927 0.879485i \(-0.342113\pi\)
0.475927 + 0.879485i \(0.342113\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 7.24769e14 0.741885
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.73493e14 0.534107
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 2.70935e15 2.34204
\(325\) −2.35086e15 −1.99492
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.54765e15 −1.17681 −0.588404 0.808567i \(-0.700243\pi\)
−0.588404 + 0.808567i \(0.700243\pi\)
\(332\) 5.56376e15 4.15471
\(333\) 0 0
\(334\) 2.06335e15 1.48626
\(335\) 0 0
\(336\) 0 0
\(337\) 2.33446e15 1.59370 0.796851 0.604176i \(-0.206498\pi\)
0.796851 + 0.604176i \(0.206498\pi\)
\(338\) 8.12235e15 5.44731
\(339\) 0 0
\(340\) −5.36032e15 −3.46990
\(341\) 6.14201e14 0.390647
\(342\) 0 0
\(343\) −1.75126e15 −1.07544
\(344\) 4.86577e15 2.93630
\(345\) 0 0
\(346\) 1.15057e15 0.670587
\(347\) −2.74948e15 −1.57498 −0.787488 0.616329i \(-0.788619\pi\)
−0.787488 + 0.616329i \(0.788619\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 2.49316e15 1.35626
\(351\) 0 0
\(352\) 2.78570e15 1.46446
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 3.76381e15 1.88043
\(356\) −9.51342e15 −4.67344
\(357\) 0 0
\(358\) −6.18275e15 −2.93686
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 5.34056e15 2.45342
\(361\) 2.21331e15 1.00000
\(362\) −3.45215e15 −1.53405
\(363\) 0 0
\(364\) −8.06242e15 −3.46623
\(365\) −4.64952e15 −1.96631
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.56917e15 −0.953981 −0.476991 0.878908i \(-0.658272\pi\)
−0.476991 + 0.878908i \(0.658272\pi\)
\(374\) −7.41238e15 −2.70850
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.33814e14 −0.0451508 −0.0225754 0.999745i \(-0.507187\pi\)
−0.0225754 + 0.999745i \(0.507187\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.10529e16 −3.55710
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 2.41602e15 0.741885
\(386\) −1.06869e16 −3.23094
\(387\) 4.02064e15 1.19682
\(388\) 0 0
\(389\) −4.88298e15 −1.40925 −0.704624 0.709581i \(-0.748884\pi\)
−0.704624 + 0.709581i \(0.748884\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −4.00241e15 −1.10307
\(393\) 0 0
\(394\) −3.60319e15 −0.963186
\(395\) 0 0
\(396\) 9.03162e15 2.34204
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 1.36030e16 3.42244
\(399\) 0 0
\(400\) 8.77820e15 2.14312
\(401\) 1.87982e15 0.452116 0.226058 0.974114i \(-0.427416\pi\)
0.226058 + 0.974114i \(0.427416\pi\)
\(402\) 0 0
\(403\) −3.33842e15 −0.779311
\(404\) 0 0
\(405\) 4.41296e15 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.36182e15 −1.48349
\(414\) 0 0
\(415\) 9.06221e15 1.77397
\(416\) −1.51414e16 −2.92149
\(417\) 0 0
\(418\) 0 0
\(419\) 1.06234e16 1.96326 0.981632 0.190782i \(-0.0611024\pi\)
0.981632 + 0.190782i \(0.0611024\pi\)
\(420\) 0 0
\(421\) −6.13031e15 −1.10101 −0.550503 0.834833i \(-0.685564\pi\)
−0.550503 + 0.834833i \(0.685564\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.73084e15 −1.48157
\(426\) 0 0
\(427\) 0 0
\(428\) −6.00385e15 −0.976713
\(429\) 0 0
\(430\) 1.38308e16 2.18794
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 3.54050e15 0.529818
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 1.78028e16 2.45342
\(441\) −3.30723e15 −0.449607
\(442\) 4.02892e16 5.40325
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −1.54954e16 −1.99546
\(446\) 0 0
\(447\) 0 0
\(448\) 3.20356e15 0.396246
\(449\) −1.27511e16 −1.55622 −0.778110 0.628128i \(-0.783822\pi\)
−0.778110 + 0.628128i \(0.783822\pi\)
\(450\) 1.51803e16 1.82812
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 2.78275e16 3.17789
\(455\) −1.31320e16 −1.48000
\(456\) 0 0
\(457\) 1.76928e16 1.94222 0.971112 0.238625i \(-0.0766967\pi\)
0.971112 + 0.238625i \(0.0766967\pi\)
\(458\) −2.47659e16 −2.68325
\(459\) 0 0
\(460\) 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
\(466\) 3.66636e16 3.58030
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −4.90903e16 −4.67219
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −5.42466e16 −4.90592
\(473\) 1.34028e16 1.19682
\(474\) 0 0
\(475\) 0 0
\(476\) −2.99430e16 −2.57427
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 3.01069e16 2.34204
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −1.13767e16 −0.821938
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.47106e16 1.00000
\(496\) 1.24658e16 0.837201
\(497\) 2.10248e16 1.39506
\(498\) 0 0
\(499\) 1.46666e16 0.950007 0.475004 0.879984i \(-0.342447\pi\)
0.475004 + 0.879984i \(0.342447\pi\)
\(500\) 3.65944e16 2.34204
\(501\) 0 0
\(502\) 5.30044e16 3.31200
\(503\) 1.78129e16 1.09983 0.549917 0.835220i \(-0.314659\pi\)
0.549917 + 0.835220i \(0.314659\pi\)
\(504\) 2.98326e16 1.82015
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −5.93737e16 −3.45471
\(509\) −2.38328e16 −1.37047 −0.685234 0.728323i \(-0.740300\pi\)
−0.685234 + 0.728323i \(0.740300\pi\)
\(510\) 0 0
\(511\) −2.59725e16 −1.45877
\(512\) −3.81805e16 −2.11944
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −9.67650e16 −4.89438
\(521\) 3.96050e16 1.98027 0.990133 0.140129i \(-0.0447517\pi\)
0.990133 + 0.140129i \(0.0447517\pi\)
\(522\) 0 0
\(523\) 4.51570e15 0.220656 0.110328 0.993895i \(-0.464810\pi\)
0.110328 + 0.993895i \(0.464810\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.95886e15 −0.422998
\(527\) −1.23985e16 −0.578771
\(528\) 0 0
\(529\) 2.19146e16 1.00000
\(530\) 0 0
\(531\) −4.48245e16 −1.99962
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −9.77902e15 −0.417035
\(536\) 0 0
\(537\) 0 0
\(538\) 4.85134e16 2.00064
\(539\) −1.10247e16 −0.449607
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −5.62334e16 −2.16971
\(545\) 0 0
\(546\) 0 0
\(547\) −2.95987e16 −1.10496 −0.552482 0.833525i \(-0.686319\pi\)
−0.552482 + 0.833525i \(0.686319\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 5.06037e16 1.82812
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 3.19470e16 1.10502
\(555\) 0 0
\(556\) 0 0
\(557\) 5.80445e16 1.94370 0.971851 0.235597i \(-0.0757045\pi\)
0.971851 + 0.235597i \(0.0757045\pi\)
\(558\) 2.15573e16 0.714151
\(559\) −7.28495e16 −2.38757
\(560\) 4.90354e16 1.58994
\(561\) 0 0
\(562\) 0 0
\(563\) −6.40493e15 −0.201124 −0.100562 0.994931i \(-0.532064\pi\)
−0.100562 + 0.994931i \(0.532064\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.32427e16 0.402791
\(567\) 2.46510e16 0.741885
\(568\) 1.54924e17 4.61348
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −1.63643e17 −4.67219
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.95058e16 0.534107
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 8.14629e16 2.18470
\(579\) 0 0
\(580\) 0 0
\(581\) 5.06220e16 1.31608
\(582\) 0 0
\(583\) 0 0
\(584\) −1.91382e17 −4.82417
\(585\) −7.99579e16 −1.99492
\(586\) 1.46690e17 3.62257
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.54194e17 −3.65556
\(591\) 0 0
\(592\) 0 0
\(593\) −7.33840e16 −1.68762 −0.843808 0.536646i \(-0.819691\pi\)
−0.843808 + 0.536646i \(0.819691\pi\)
\(594\) 0 0
\(595\) −4.87709e16 −1.09915
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.83218e16 0.613140 0.306570 0.951848i \(-0.400819\pi\)
0.306570 + 0.951848i \(0.400819\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 7.72593e16 1.62320
\(603\) 0 0
\(604\) 0 0
\(605\) 4.90379e16 1.00000
\(606\) 0 0
\(607\) 9.99846e16 1.99895 0.999473 0.0324755i \(-0.0103391\pi\)
0.999473 + 0.0324755i \(0.0103391\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.82316e17 −3.46990
\(613\) −5.61526e16 −1.05830 −0.529148 0.848530i \(-0.677488\pi\)
−0.529148 + 0.848530i \(0.677488\pi\)
\(614\) −1.18456e17 −2.21079
\(615\) 0 0
\(616\) 9.94472e16 1.82015
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −3.32043e16 −0.590270 −0.295135 0.955456i \(-0.595365\pi\)
−0.295135 + 0.955456i \(0.595365\pi\)
\(620\) 5.19672e16 0.914911
\(621\) 0 0
\(622\) 1.00767e17 1.74011
\(623\) −8.65580e16 −1.48040
\(624\) 0 0
\(625\) 5.96046e16 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 8.47980e16 1.35626
\(631\) −1.13409e17 −1.79668 −0.898342 0.439296i \(-0.855228\pi\)
−0.898342 + 0.439296i \(0.855228\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.67074e16 −1.47509
\(636\) 0 0
\(637\) 5.99233e16 0.896932
\(638\) 0 0
\(639\) 1.28015e17 1.88043
\(640\) −3.35384e16 −0.488049
\(641\) −1.99043e16 −0.286946 −0.143473 0.989654i \(-0.545827\pi\)
−0.143473 + 0.989654i \(0.545827\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.81644e17 2.45342
\(649\) −1.49423e17 −1.99962
\(650\) −2.75050e17 −3.64697
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.58141e17 −1.96631
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 7.62389e16 0.914045 0.457023 0.889455i \(-0.348916\pi\)
0.457023 + 0.889455i \(0.348916\pi\)
\(662\) −1.81075e17 −2.15135
\(663\) 0 0
\(664\) 3.73015e17 4.35229
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.69177e17 1.90407
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.14664e16 −0.123407 −0.0617033 0.998095i \(-0.519653\pi\)
−0.0617033 + 0.998095i \(0.519653\pi\)
\(674\) 2.73132e17 2.91349
\(675\) 0 0
\(676\) 6.65964e17 6.97863
\(677\) 1.53534e17 1.59467 0.797337 0.603535i \(-0.206242\pi\)
0.797337 + 0.603535i \(0.206242\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −3.59375e17 −3.63491
\(681\) 0 0
\(682\) 7.18615e16 0.714151
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.04898e17 −1.96604
\(687\) 0 0
\(688\) 2.72023e17 2.56493
\(689\) 0 0
\(690\) 0 0
\(691\) 1.95412e17 1.79508 0.897540 0.440933i \(-0.145352\pi\)
0.897540 + 0.440933i \(0.145352\pi\)
\(692\) 9.43366e16 0.859100
\(693\) 8.21743e16 0.741885
\(694\) −3.21690e17 −2.87925
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 2.04418e17 1.73752
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.50226e16 0.534107
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.35349e17 1.06556 0.532779 0.846255i \(-0.321148\pi\)
0.532779 + 0.846255i \(0.321148\pi\)
\(710\) 4.40365e17 3.43766
\(711\) 0 0
\(712\) −6.37814e17 −4.89569
\(713\) 0 0
\(714\) 0 0
\(715\) −2.66540e17 −1.99492
\(716\) −5.06933e17 −3.76246
\(717\) 0 0
\(718\) 0 0
\(719\) 2.76311e17 1.99998 0.999988 0.00481681i \(-0.00153324\pi\)
0.999988 + 0.00481681i \(0.00153324\pi\)
\(720\) 2.98566e17 2.14312
\(721\) 0 0
\(722\) 2.58958e17 1.82812
\(723\) 0 0
\(724\) −2.83047e17 −1.96529
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −5.40534e17 −3.63107
\(729\) 1.50095e17 1.00000
\(730\) −5.43994e17 −3.59466
\(731\) −2.70555e17 −1.77318
\(732\) 0 0
\(733\) 2.49748e17 1.61020 0.805098 0.593142i \(-0.202113\pi\)
0.805098 + 0.593142i \(0.202113\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 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
\(743\) −1.88088e17 −1.11796 −0.558982 0.829179i \(-0.688808\pi\)
−0.558982 + 0.829179i \(0.688808\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.00593e17 −1.74400
\(747\) 3.08226e17 1.77397
\(748\) −6.07752e17 −3.46990
\(749\) −5.46261e16 −0.309392
\(750\) 0 0
\(751\) 1.30573e17 0.727802 0.363901 0.931438i \(-0.381445\pi\)
0.363901 + 0.931438i \(0.381445\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −1.56562e16 −0.0825414
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −9.06243e17 −4.55706
\(765\) −2.96955e17 −1.48157
\(766\) 0 0
\(767\) 8.12170e17 3.98910
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 2.82675e17 1.35626
\(771\) 0 0
\(772\) −8.76236e17 −4.13921
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 4.70415e17 2.18794
\(775\) 8.46437e16 0.390647
\(776\) 0 0
\(777\) 0 0
\(778\) −5.71309e17 −2.57628
\(779\) 0 0
\(780\) 0 0
\(781\) 4.26740e17 1.88043
\(782\) 0 0
\(783\) 0 0
\(784\) −2.23756e17 −0.963560
\(785\) 0 0
\(786\) 0 0
\(787\) 3.61730e17 1.52243 0.761214 0.648501i \(-0.224604\pi\)
0.761214 + 0.648501i \(0.224604\pi\)
\(788\) −2.95431e17 −1.23395
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 6.05512e17 2.45342
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 1.11533e18 4.38455
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.83900e17 1.46446
\(801\) −5.27032e17 −1.99546
\(802\) 2.19939e17 0.826525
\(803\) −5.27163e17 −1.96631
\(804\) 0 0
\(805\) 0 0
\(806\) −3.90595e17 −1.42468
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 5.16316e17 1.82812
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −4.46649e17 −1.48000
\(820\) 0 0
\(821\) 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
\(826\) −8.61333e17 −2.71201
\(827\) 3.68011e16 0.115034 0.0575172 0.998345i \(-0.481682\pi\)
0.0575172 + 0.998345i \(0.481682\pi\)
\(828\) 0 0
\(829\) 4.74540e16 0.146199 0.0730997 0.997325i \(-0.476711\pi\)
0.0730997 + 0.997325i \(0.476711\pi\)
\(830\) 1.06028e18 3.24303
\(831\) 0 0
\(832\) −3.53423e17 −1.06550
\(833\) 2.22549e17 0.666125
\(834\) 0 0
\(835\) 2.75555e17 0.812997
\(836\) 0 0
\(837\) 0 0
\(838\) 1.24294e18 3.58909
\(839\) −6.97176e17 −1.99881 −0.999403 0.0345594i \(-0.988997\pi\)
−0.999403 + 0.0345594i \(0.988997\pi\)
\(840\) 0 0
\(841\) 3.53815e17 1.00000
\(842\) −7.17246e17 −2.01278
\(843\) 0 0
\(844\) 0 0
\(845\) 1.08472e18 2.97972
\(846\) 0 0
\(847\) 2.73928e17 0.741885
\(848\) 0 0
\(849\) 0 0
\(850\) −1.02151e18 −2.70850
\(851\) 0 0
\(852\) 0 0
\(853\) −5.21265e17 −1.35321 −0.676604 0.736347i \(-0.736549\pi\)
−0.676604 + 0.736347i \(0.736549\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.02519e17 −1.02316
\(857\) 4.93523e17 1.24573 0.622863 0.782331i \(-0.285969\pi\)
0.622863 + 0.782331i \(0.285969\pi\)
\(858\) 0 0
\(859\) −1.93105e16 −0.0480657 −0.0240329 0.999711i \(-0.507651\pi\)
−0.0240329 + 0.999711i \(0.507651\pi\)
\(860\) 1.13400e18 2.80300
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 1.53655e17 0.366817
\(866\) 0 0
\(867\) 0 0
\(868\) 2.90291e17 0.678758
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.32954e17 0.741885
\(876\) 0 0
\(877\) −6.25255e17 −1.37423 −0.687115 0.726549i \(-0.741123\pi\)
−0.687115 + 0.726549i \(0.741123\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 9.95271e17 2.14312
\(881\) 8.59740e17 1.83870 0.919352 0.393437i \(-0.128714\pi\)
0.919352 + 0.393437i \(0.128714\pi\)
\(882\) −3.86946e17 −0.821938
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 3.30337e18 6.92219
\(885\) 0 0
\(886\) 0 0
\(887\) −9.30688e17 −1.91101 −0.955504 0.294978i \(-0.904688\pi\)
−0.955504 + 0.294978i \(0.904688\pi\)
\(888\) 0 0
\(889\) −5.40212e17 −1.09434
\(890\) −1.81296e18 −3.64794
\(891\) 5.00341e17 1.00000
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −8.25688e17 −1.60649
\(896\) −1.87347e17 −0.362076
\(897\) 0 0
\(898\) −1.49188e18 −2.84497
\(899\) 0 0
\(900\) 1.24466e18 2.34204
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.61025e17 −0.839137
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 2.28162e18 4.07125
\(909\) 0 0
\(910\) −1.53645e18 −2.70563
\(911\) −2.42564e17 −0.424342 −0.212171 0.977233i \(-0.568053\pi\)
−0.212171 + 0.977233i \(0.568053\pi\)
\(912\) 0 0
\(913\) 1.02747e18 1.77397
\(914\) 2.07006e18 3.55063
\(915\) 0 0
\(916\) −2.03059e18 −3.43755
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.31950e18 −3.75132
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.20287e18 −1.87121 −0.935607 0.353043i \(-0.885147\pi\)
−0.935607 + 0.353043i \(0.885147\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.00610e18 4.58679
\(933\) 0 0
\(934\) 0 0
\(935\) −9.89902e17 −1.48157
\(936\) −3.29119e18 −4.89438
\(937\) −2.63763e17 −0.389741 −0.194870 0.980829i \(-0.562429\pi\)
−0.194870 + 0.980829i \(0.562429\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −3.03268e18 −4.28543
\(945\) 0 0
\(946\) 1.56813e18 2.18794
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 2.86533e18 3.92263
\(950\) 0 0
\(951\) 0 0
\(952\) −2.00748e18 −2.69669
\(953\) 1.23415e18 1.64744 0.823722 0.566993i \(-0.191893\pi\)
0.823722 + 0.566993i \(0.191893\pi\)
\(954\) 0 0
\(955\) −1.47608e18 −1.94576
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.67462e17 −0.847395
\(962\) 0 0
\(963\) −3.32606e17 −0.417035
\(964\) 0 0
\(965\) −1.42721e18 −1.76735
\(966\) 0 0
\(967\) −2.27780e17 −0.278584 −0.139292 0.990251i \(-0.544483\pi\)
−0.139292 + 0.990251i \(0.544483\pi\)
\(968\) 2.01848e18 2.45342
\(969\) 0 0
\(970\) 0 0
\(971\) −1.57351e18 −1.87739 −0.938694 0.344751i \(-0.887963\pi\)
−0.938694 + 0.344751i \(0.887963\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) −1.75687e18 −1.99546
\(980\) −9.32790e17 −1.05300
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −4.81196e17 −0.526871
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 1.72114e18 1.82812
\(991\) 5.67381e17 0.599008 0.299504 0.954095i \(-0.403179\pi\)
0.299504 + 0.954095i \(0.403179\pi\)
\(992\) 5.45171e17 0.572088
\(993\) 0 0
\(994\) 2.45990e18 2.55035
\(995\) 1.81664e18 1.87211
\(996\) 0 0
\(997\) 3.04327e17 0.309863 0.154932 0.987925i \(-0.450484\pi\)
0.154932 + 0.987925i \(0.450484\pi\)
\(998\) 1.71600e18 1.73673
\(999\) 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 55.13.d.b.54.1 yes 1
5.4 even 2 55.13.d.a.54.1 1
11.10 odd 2 55.13.d.a.54.1 1
55.54 odd 2 CM 55.13.d.b.54.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.13.d.a.54.1 1 5.4 even 2
55.13.d.a.54.1 1 11.10 odd 2
55.13.d.b.54.1 yes 1 1.1 even 1 trivial
55.13.d.b.54.1 yes 1 55.54 odd 2 CM