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Properties

Label 55.17.d.b.54.1
Level $55$
Weight $17$
Character 55.54
Self dual yes
Analytic conductor $89.278$
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] = [55,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.54");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 17 \)
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: \(89.2784991211\)
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)
 
\(f(q)\) \(=\) \(q+17.0000 q^{2} -65247.0 q^{4} +390625. q^{5} -9.88608e6 q^{7} -2.22331e6 q^{8} +4.30467e7 q^{9} +O(q^{10})\) \(q+17.0000 q^{2} -65247.0 q^{4} +390625. q^{5} -9.88608e6 q^{7} -2.22331e6 q^{8} +4.30467e7 q^{9} +6.64062e6 q^{10} +2.14359e8 q^{11} -6.78011e8 q^{13} -1.68063e8 q^{14} +4.23823e9 q^{16} -1.39219e10 q^{17} +7.31794e8 q^{18} -2.54871e10 q^{20} +3.64410e9 q^{22} +1.52588e11 q^{25} -1.15262e10 q^{26} +6.45037e11 q^{28} -1.20655e12 q^{31} +2.17757e11 q^{32} -2.36673e11 q^{34} -3.86175e12 q^{35} -2.80867e12 q^{36} -8.68481e11 q^{40} +7.61377e12 q^{43} -1.39863e13 q^{44} +1.68151e13 q^{45} +6.45016e13 q^{49} +2.59399e12 q^{50} +4.42382e13 q^{52} +8.37339e13 q^{55} +2.19798e13 q^{56} -1.40214e14 q^{59} -2.05114e13 q^{62} -4.25563e14 q^{63} -2.74055e14 q^{64} -2.64848e14 q^{65} +9.08365e14 q^{68} -6.56497e13 q^{70} -7.77262e13 q^{71} -9.57062e13 q^{72} +1.56472e15 q^{73} -2.11917e15 q^{77} +1.65556e15 q^{80} +1.85302e15 q^{81} +3.61302e15 q^{83} -5.43826e15 q^{85} +1.29434e14 q^{86} -4.76586e14 q^{88} +7.84139e15 q^{89} +2.85857e14 q^{90} +6.70287e15 q^{91} +1.09653e15 q^{98} +9.22745e15 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.0000 0.0664062 0.0332031 − 0.999449i \(-0.489429\pi\)
0.0332031 + 0.999449i \(0.489429\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −65247.0 −0.995590
\(5\) 390625. 1.00000
\(6\) 0 0
\(7\) −9.88608e6 −1.71490 −0.857452 − 0.514564i \(-0.827954\pi\)
−0.857452 + 0.514564i \(0.827954\pi\)
\(8\) −2.22331e6 −0.132520
\(9\) 4.30467e7 1.00000
\(10\) 6.64062e6 0.0664062
\(11\) 2.14359e8 1.00000
\(12\) 0 0
\(13\) −6.78011e8 −0.831170 −0.415585 − 0.909554i \(-0.636423\pi\)
−0.415585 + 0.909554i \(0.636423\pi\)
\(14\) −1.68063e8 −0.113880
\(15\) 0 0
\(16\) 4.23823e9 0.986790
\(17\) −1.39219e10 −1.99576 −0.997880 − 0.0650749i \(-0.979271\pi\)
−0.997880 + 0.0650749i \(0.979271\pi\)
\(18\) 7.31794e8 0.0664062
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −2.54871e10 −0.995590
\(21\) 0 0
\(22\) 3.64410e9 0.0664062
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.52588e11 1.00000
\(26\) −1.15262e10 −0.0551949
\(27\) 0 0
\(28\) 6.45037e11 1.70734
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.20655e12 −1.41466 −0.707331 − 0.706883i \(-0.750101\pi\)
−0.707331 + 0.706883i \(0.750101\pi\)
\(32\) 2.17757e11 0.198049
\(33\) 0 0
\(34\) −2.36673e11 −0.132531
\(35\) −3.86175e12 −1.71490
\(36\) −2.80867e12 −0.995590
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −8.68481e11 −0.132520
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 7.61377e12 0.651407 0.325703 − 0.945472i \(-0.394399\pi\)
0.325703 + 0.945472i \(0.394399\pi\)
\(44\) −1.39863e13 −0.995590
\(45\) 1.68151e13 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 6.45016e13 1.94089
\(50\) 2.59399e12 0.0664062
\(51\) 0 0
\(52\) 4.42382e13 0.827504
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 8.37339e13 1.00000
\(56\) 2.19798e13 0.227258
\(57\) 0 0
\(58\) 0 0
\(59\) −1.40214e14 −0.954940 −0.477470 − 0.878648i \(-0.658446\pi\)
−0.477470 + 0.878648i \(0.658446\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −2.05114e13 −0.0939424
\(63\) −4.25563e14 −1.71490
\(64\) −2.74055e14 −0.973638
\(65\) −2.64848e14 −0.831170
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 9.08365e14 1.98696
\(69\) 0 0
\(70\) −6.56497e13 −0.113880
\(71\) −7.77262e13 −0.120365 −0.0601826 − 0.998187i \(-0.519168\pi\)
−0.0601826 + 0.998187i \(0.519168\pi\)
\(72\) −9.57062e13 −0.132520
\(73\) 1.56472e15 1.94023 0.970116 − 0.242641i \(-0.0780136\pi\)
0.970116 + 0.242641i \(0.0780136\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.11917e15 −1.71490
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.65556e15 0.986790
\(81\) 1.85302e15 1.00000
\(82\) 0 0
\(83\) 3.61302e15 1.60415 0.802077 − 0.597221i \(-0.203728\pi\)
0.802077 + 0.597221i \(0.203728\pi\)
\(84\) 0 0
\(85\) −5.43826e15 −1.99576
\(86\) 1.29434e14 0.0432575
\(87\) 0 0
\(88\) −4.76586e14 −0.132520
\(89\) 7.84139e15 1.99193 0.995963 − 0.0897685i \(-0.0286127\pi\)
0.995963 + 0.0897685i \(0.0286127\pi\)
\(90\) 2.85857e14 0.0664062
\(91\) 6.70287e15 1.42538
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.09653e15 0.128888
\(99\) 9.22745e15 1.00000
\(100\) −9.95590e15 −0.995590
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 1.50743e15 0.110146
\(105\) 0 0
\(106\) 0 0
\(107\) 3.30247e16 1.92207 0.961034 − 0.276431i \(-0.0891518\pi\)
0.961034 + 0.276431i \(0.0891518\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 1.42348e15 0.0664062
\(111\) 0 0
\(112\) −4.18995e16 −1.69225
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.91861e16 −0.831170
\(118\) −2.38364e15 −0.0634140
\(119\) 1.37633e17 3.42254
\(120\) 0 0
\(121\) 4.59497e16 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 7.87239e16 1.40842
\(125\) 5.96046e16 1.00000
\(126\) −7.23458e15 −0.113880
\(127\) 7.44355e16 1.09989 0.549947 − 0.835200i \(-0.314648\pi\)
0.549947 + 0.835200i \(0.314648\pi\)
\(128\) −1.89298e16 −0.262704
\(129\) 0 0
\(130\) −4.50241e15 −0.0551949
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 3.09528e16 0.264478
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 2.51968e17 1.70734
\(141\) 0 0
\(142\) −1.32135e15 −0.00799300
\(143\) −1.45338e17 −0.831170
\(144\) 1.82442e17 0.986790
\(145\) 0 0
\(146\) 2.66002e16 0.128844
\(147\) 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
\(153\) −5.99294e17 −1.99576
\(154\) −3.60259e16 −0.113880
\(155\) −4.71309e17 −1.41466
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 8.50613e16 0.198049
\(161\) 0 0
\(162\) 3.15013e16 0.0664062
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.14214e16 0.106526
\(167\) 1.02631e18 1.69646 0.848232 − 0.529625i \(-0.177667\pi\)
0.848232 + 0.529625i \(0.177667\pi\)
\(168\) 0 0
\(169\) −2.05718e17 −0.309157
\(170\) −9.24504e16 −0.132531
\(171\) 0 0
\(172\) −4.96776e17 −0.648534
\(173\) 1.55643e18 1.93982 0.969910 − 0.243465i \(-0.0782841\pi\)
0.969910 + 0.243465i \(0.0782841\pi\)
\(174\) 0 0
\(175\) −1.50850e18 −1.71490
\(176\) 9.08502e17 0.986790
\(177\) 0 0
\(178\) 1.33304e17 0.132276
\(179\) 1.38998e18 1.31882 0.659408 − 0.751785i \(-0.270807\pi\)
0.659408 + 0.751785i \(0.270807\pi\)
\(180\) −1.09714e18 −0.995590
\(181\) −2.05416e18 −1.78322 −0.891612 − 0.452801i \(-0.850425\pi\)
−0.891612 + 0.452801i \(0.850425\pi\)
\(182\) 1.13949e17 0.0946539
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.98429e18 −1.99576
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.62554e18 −1.48236 −0.741178 − 0.671309i \(-0.765732\pi\)
−0.741178 + 0.671309i \(0.765732\pi\)
\(192\) 0 0
\(193\) 3.06623e18 1.59275 0.796374 − 0.604805i \(-0.206749\pi\)
0.796374 + 0.604805i \(0.206749\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −4.20854e18 −1.93234
\(197\) −7.59320e17 −0.334730 −0.167365 − 0.985895i \(-0.553526\pi\)
−0.167365 + 0.985895i \(0.553526\pi\)
\(198\) 1.56867e17 0.0664062
\(199\) −4.14695e18 −1.68618 −0.843091 − 0.537770i \(-0.819267\pi\)
−0.843091 + 0.537770i \(0.819267\pi\)
\(200\) −3.39250e17 −0.132520
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −2.87357e18 −0.820190
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 5.61420e17 0.127637
\(215\) 2.97413e18 0.651407
\(216\) 0 0
\(217\) 1.19281e19 2.42601
\(218\) 0 0
\(219\) 0 0
\(220\) −5.46339e18 −0.995590
\(221\) 9.43922e18 1.65882
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −2.15276e18 −0.339634
\(225\) 6.56841e18 1.00000
\(226\) 0 0
\(227\) −1.32091e19 −1.87356 −0.936778 − 0.349924i \(-0.886208\pi\)
−0.936778 + 0.349924i \(0.886208\pi\)
\(228\) 0 0
\(229\) −1.51056e19 −1.99735 −0.998675 − 0.0514670i \(-0.983610\pi\)
−0.998675 + 0.0514670i \(0.983610\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.24121e19 −1.42889 −0.714446 − 0.699690i \(-0.753321\pi\)
−0.714446 + 0.699690i \(0.753321\pi\)
\(234\) −4.96164e17 −0.0551949
\(235\) 0 0
\(236\) 9.14856e18 0.950729
\(237\) 0 0
\(238\) 2.33977e18 0.227278
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 7.81145e17 0.0664062
\(243\) 0 0
\(244\) 0 0
\(245\) 2.51959e19 1.94089
\(246\) 0 0
\(247\) 0 0
\(248\) 2.68254e18 0.187470
\(249\) 0 0
\(250\) 1.01328e18 0.0664062
\(251\) 1.87791e18 0.119202 0.0596012 − 0.998222i \(-0.481017\pi\)
0.0596012 + 0.998222i \(0.481017\pi\)
\(252\) 2.77667e19 1.70734
\(253\) 0 0
\(254\) 1.26540e18 0.0730398
\(255\) 0 0
\(256\) 1.76387e19 0.956193
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.72805e19 0.827504
\(261\) 0 0
\(262\) 0 0
\(263\) −2.87221e19 −1.25479 −0.627393 − 0.778703i \(-0.715878\pi\)
−0.627393 + 0.778703i \(0.715878\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.92896e19 1.43304 0.716522 − 0.697564i \(-0.245733\pi\)
0.716522 + 0.697564i \(0.245733\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −5.90044e19 −1.96940
\(273\) 0 0
\(274\) 0 0
\(275\) 3.27086e19 1.00000
\(276\) 0 0
\(277\) 6.35935e19 1.83474 0.917370 − 0.398035i \(-0.130308\pi\)
0.917370 + 0.398035i \(0.130308\pi\)
\(278\) 0 0
\(279\) −5.19381e19 −1.41466
\(280\) 8.58587e18 0.227258
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −5.11485e19 −1.24320 −0.621600 − 0.783334i \(-0.713517\pi\)
−0.621600 + 0.783334i \(0.713517\pi\)
\(284\) 5.07140e18 0.119834
\(285\) 0 0
\(286\) −2.47074e18 −0.0551949
\(287\) 0 0
\(288\) 9.37372e18 0.198049
\(289\) 1.45159e20 2.98306
\(290\) 0 0
\(291\) 0 0
\(292\) −1.02093e20 −1.93168
\(293\) 1.06858e20 1.96728 0.983641 − 0.180139i \(-0.0576549\pi\)
0.983641 + 0.180139i \(0.0576549\pi\)
\(294\) 0 0
\(295\) −5.47712e19 −0.954940
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −7.52704e19 −1.11710
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −1.01880e19 −0.132531
\(307\) 1.02268e20 1.29608 0.648040 − 0.761606i \(-0.275589\pi\)
0.648040 + 0.761606i \(0.275589\pi\)
\(308\) 1.38269e20 1.70734
\(309\) 0 0
\(310\) −8.01226e18 −0.0939424
\(311\) 1.38127e20 1.57832 0.789160 − 0.614188i \(-0.210516\pi\)
0.789160 + 0.614188i \(0.210516\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −1.66236e20 −1.71490
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.07053e20 −0.973638
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.20904e20 −0.995590
\(325\) −1.03456e20 −0.831170
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.81937e20 −1.95672 −0.978359 − 0.206914i \(-0.933658\pi\)
−0.978359 + 0.206914i \(0.933658\pi\)
\(332\) −2.35739e20 −1.59708
\(333\) 0 0
\(334\) 1.74472e19 0.112656
\(335\) 0 0
\(336\) 0 0
\(337\) 1.11387e20 0.669569 0.334785 − 0.942295i \(-0.391336\pi\)
0.334785 + 0.942295i \(0.391336\pi\)
\(338\) −3.49721e18 −0.0205300
\(339\) 0 0
\(340\) 3.54830e20 1.98696
\(341\) −2.58635e20 −1.41466
\(342\) 0 0
\(343\) −3.09125e20 −1.61354
\(344\) −1.69278e19 −0.0863242
\(345\) 0 0
\(346\) 2.64593e19 0.128816
\(347\) 1.48818e20 0.707976 0.353988 − 0.935250i \(-0.384825\pi\)
0.353988 + 0.935250i \(0.384825\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −2.56444e19 −0.113880
\(351\) 0 0
\(352\) 4.66781e19 0.198049
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −3.03618e19 −0.120365
\(356\) −5.11627e20 −1.98314
\(357\) 0 0
\(358\) 2.36297e19 0.0875777
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −3.73853e19 −0.132520
\(361\) 2.88441e20 1.00000
\(362\) −3.49207e19 −0.118417
\(363\) 0 0
\(364\) −4.37342e20 −1.41909
\(365\) 6.11219e20 1.94023
\(366\) 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
\(373\) −6.94739e20 −1.85418 −0.927088 − 0.374843i \(-0.877697\pi\)
−0.927088 + 0.374843i \(0.877697\pi\)
\(374\) −5.07330e19 −0.132531
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.51034e20 1.99909 0.999547 − 0.0300986i \(-0.00958212\pi\)
0.999547 + 0.0300986i \(0.00958212\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.46342e19 −0.0984377
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −8.27800e20 −1.71490
\(386\) 5.21260e19 0.105768
\(387\) 3.27748e20 0.651407
\(388\) 0 0
\(389\) 5.20070e20 0.991894 0.495947 − 0.868353i \(-0.334821\pi\)
0.495947 + 0.868353i \(0.334821\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.43407e20 −0.257207
\(393\) 0 0
\(394\) −1.29084e19 −0.0222282
\(395\) 0 0
\(396\) −6.02063e20 −0.995590
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −7.04982e19 −0.111973
\(399\) 0 0
\(400\) 6.46703e20 0.986790
\(401\) −2.91235e20 −0.435600 −0.217800 − 0.975993i \(-0.569888\pi\)
−0.217800 + 0.975993i \(0.569888\pi\)
\(402\) 0 0
\(403\) 8.18055e20 1.17582
\(404\) 0 0
\(405\) 7.23836e20 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
\(413\) 1.38617e21 1.63763
\(414\) 0 0
\(415\) 1.41134e21 1.60415
\(416\) −1.47641e20 −0.164612
\(417\) 0 0
\(418\) 0 0
\(419\) −5.02478e20 −0.528938 −0.264469 − 0.964394i \(-0.585197\pi\)
−0.264469 + 0.964394i \(0.585197\pi\)
\(420\) 0 0
\(421\) 4.95361e20 0.501956 0.250978 − 0.967993i \(-0.419248\pi\)
0.250978 + 0.967993i \(0.419248\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.12432e21 −1.99576
\(426\) 0 0
\(427\) 0 0
\(428\) −2.15476e21 −1.91359
\(429\) 0 0
\(430\) 5.05602e19 0.0432575
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 2.02777e20 0.161102
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) −1.86167e20 −0.132520
\(441\) 2.77658e21 1.94089
\(442\) 1.60467e20 0.110156
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 3.06304e21 1.99193
\(446\) 0 0
\(447\) 0 0
\(448\) 2.70933e21 1.66970
\(449\) 1.23132e21 0.745419 0.372709 − 0.927948i \(-0.378429\pi\)
0.372709 + 0.927948i \(0.378429\pi\)
\(450\) 1.11663e20 0.0664062
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −2.24555e20 −0.124416
\(455\) 2.61831e21 1.42538
\(456\) 0 0
\(457\) −2.84572e21 −1.49576 −0.747882 − 0.663832i \(-0.768929\pi\)
−0.747882 + 0.663832i \(0.768929\pi\)
\(458\) −2.56795e20 −0.132636
\(459\) 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
\(466\) −2.11007e20 −0.0948874
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.90431e21 0.827504
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 3.11740e20 0.126548
\(473\) 1.63208e21 0.651407
\(474\) 0 0
\(475\) 0 0
\(476\) −8.98017e21 −3.40744
\(477\) 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
\(484\) −2.99808e21 −0.995590
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 4.28331e20 0.128888
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 3.60447e21 1.00000
\(496\) −5.11365e21 −1.39597
\(497\) 7.68407e20 0.206415
\(498\) 0 0
\(499\) 1.04513e21 0.271873 0.135937 − 0.990718i \(-0.456596\pi\)
0.135937 + 0.990718i \(0.456596\pi\)
\(500\) −3.88902e21 −0.995590
\(501\) 0 0
\(502\) 3.19245e19 0.00791579
\(503\) −7.89671e21 −1.92709 −0.963545 − 0.267546i \(-0.913787\pi\)
−0.963545 + 0.267546i \(0.913787\pi\)
\(504\) 9.46159e20 0.227258
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −4.85669e21 −1.09504
\(509\) 4.18479e21 0.928819 0.464409 − 0.885621i \(-0.346267\pi\)
0.464409 + 0.885621i \(0.346267\pi\)
\(510\) 0 0
\(511\) −1.54689e22 −3.32731
\(512\) 1.54044e21 0.326202
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 5.88839e20 0.110146
\(521\) −3.58135e21 −0.659697 −0.329849 − 0.944034i \(-0.606998\pi\)
−0.329849 + 0.944034i \(0.606998\pi\)
\(522\) 0 0
\(523\) −4.11305e21 −0.734768 −0.367384 − 0.930069i \(-0.619747\pi\)
−0.367384 + 0.930069i \(0.619747\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −4.88275e20 −0.0833256
\(527\) 1.67976e22 2.82333
\(528\) 0 0
\(529\) 6.13261e21 1.00000
\(530\) 0 0
\(531\) −6.03576e21 −0.954940
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.29003e22 1.92207
\(536\) 0 0
\(537\) 0 0
\(538\) 6.67924e20 0.0951631
\(539\) 1.38265e22 1.94089
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −3.03160e21 −0.395258
\(545\) 0 0
\(546\) 0 0
\(547\) −1.54629e22 −1.92927 −0.964634 − 0.263593i \(-0.915092\pi\)
−0.964634 + 0.263593i \(0.915092\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 5.56046e20 0.0664062
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.08109e21 0.121838
\(555\) 0 0
\(556\) 0 0
\(557\) −3.79906e21 −0.410048 −0.205024 − 0.978757i \(-0.565727\pi\)
−0.205024 + 0.978757i \(0.565727\pi\)
\(558\) −8.82948e20 −0.0939424
\(559\) −5.16222e21 −0.541430
\(560\) −1.63670e22 −1.69225
\(561\) 0 0
\(562\) 0 0
\(563\) 2.00064e22 1.98199 0.990994 − 0.133906i \(-0.0427521\pi\)
0.990994 + 0.133906i \(0.0427521\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.69524e20 −0.0825563
\(567\) −1.83191e22 −1.71490
\(568\) 1.72810e20 0.0159507
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 9.48284e21 0.827504
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −1.17972e22 −0.973638
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 2.46771e21 0.198094
\(579\) 0 0
\(580\) 0 0
\(581\) −3.57186e22 −2.75097
\(582\) 0 0
\(583\) 0 0
\(584\) −3.47886e21 −0.257119
\(585\) −1.14008e22 −0.831170
\(586\) 1.81659e21 0.130640
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −9.31110e20 −0.0634140
\(591\) 0 0
\(592\) 0 0
\(593\) 2.22663e22 1.45616 0.728082 − 0.685490i \(-0.240412\pi\)
0.728082 + 0.685490i \(0.240412\pi\)
\(594\) 0 0
\(595\) 5.37631e22 3.42254
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.03274e22 1.82987 0.914935 − 0.403602i \(-0.132242\pi\)
0.914935 + 0.403602i \(0.132242\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −1.27960e21 −0.0741824
\(603\) 0 0
\(604\) 0 0
\(605\) 1.79491e22 1.00000
\(606\) 0 0
\(607\) −1.70301e22 −0.924074 −0.462037 − 0.886861i \(-0.652881\pi\)
−0.462037 + 0.886861i \(0.652881\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 3.91021e22 1.98696
\(613\) 2.93541e22 1.47226 0.736130 − 0.676840i \(-0.236651\pi\)
0.736130 + 0.676840i \(0.236651\pi\)
\(614\) 1.73855e21 0.0860678
\(615\) 0 0
\(616\) 4.71157e21 0.227258
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 3.97139e22 1.84254 0.921270 − 0.388923i \(-0.127153\pi\)
0.921270 + 0.388923i \(0.127153\pi\)
\(620\) 3.07515e22 1.40842
\(621\) 0 0
\(622\) 2.34816e21 0.104810
\(623\) −7.75206e22 −3.41596
\(624\) 0 0
\(625\) 2.32831e22 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −2.82601e21 −0.113880
\(631\) 4.15824e21 0.165453 0.0827265 − 0.996572i \(-0.473637\pi\)
0.0827265 + 0.996572i \(0.473637\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.90764e22 1.09989
\(636\) 0 0
\(637\) −4.37328e22 −1.61321
\(638\) 0 0
\(639\) −3.34586e21 −0.120365
\(640\) −7.39447e21 −0.262704
\(641\) −1.85596e22 −0.651187 −0.325593 − 0.945510i \(-0.605564\pi\)
−0.325593 + 0.945510i \(0.605564\pi\)
\(642\) 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 \(\pi\)
\(648\) −4.11984e21 −0.132520
\(649\) −3.00562e22 −0.954940
\(650\) −1.75876e21 −0.0551949
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.73561e22 1.94023
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −6.67182e22 −1.83077 −0.915383 − 0.402584i \(-0.868112\pi\)
−0.915383 + 0.402584i \(0.868112\pi\)
\(662\) −4.79293e21 −0.129938
\(663\) 0 0
\(664\) −8.03287e21 −0.212582
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −6.69633e22 −1.68898
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.38836e22 1.99323 0.996613 − 0.0822304i \(-0.0262043\pi\)
0.996613 + 0.0822304i \(0.0262043\pi\)
\(674\) 1.89358e21 0.0444636
\(675\) 0 0
\(676\) 1.34225e22 0.307794
\(677\) 5.73194e22 1.29895 0.649475 − 0.760383i \(-0.274989\pi\)
0.649475 + 0.760383i \(0.274989\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.20909e22 0.264478
\(681\) 0 0
\(682\) −4.39680e21 −0.0939424
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −5.25512e21 −0.107149
\(687\) 0 0
\(688\) 3.22689e22 0.642802
\(689\) 0 0
\(690\) 0 0
\(691\) −9.41285e22 −1.81091 −0.905455 − 0.424443i \(-0.860470\pi\)
−0.905455 + 0.424443i \(0.860470\pi\)
\(692\) −1.01553e23 −1.93127
\(693\) −9.12233e22 −1.71490
\(694\) 2.52990e21 0.0470140
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 9.84248e22 1.70734
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −5.87461e22 −0.973638
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.85629e22 0.447334 0.223667 − 0.974666i \(-0.428197\pi\)
0.223667 + 0.974666i \(0.428197\pi\)
\(710\) −5.16151e20 −0.00799300
\(711\) 0 0
\(712\) −1.74339e22 −0.263969
\(713\) 0 0
\(714\) 0 0
\(715\) −5.67725e22 −0.831170
\(716\) −9.06921e22 −1.31300
\(717\) 0 0
\(718\) 0 0
\(719\) −7.22149e22 −1.01110 −0.505552 − 0.862796i \(-0.668711\pi\)
−0.505552 + 0.862796i \(0.668711\pi\)
\(720\) 7.12664e22 0.986790
\(721\) 0 0
\(722\) 4.90350e21 0.0664062
\(723\) 0 0
\(724\) 1.34028e23 1.77536
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −1.49026e22 −0.188890
\(729\) 7.97664e22 1.00000
\(730\) 1.03907e22 0.128844
\(731\) −1.05999e23 −1.30005
\(732\) 0 0
\(733\) −1.63330e23 −1.95990 −0.979949 − 0.199248i \(-0.936150\pi\)
−0.979949 + 0.199248i \(0.936150\pi\)
\(734\) 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
\(743\) 1.30627e23 1.40644 0.703222 − 0.710971i \(-0.251744\pi\)
0.703222 + 0.710971i \(0.251744\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.18106e22 −0.123129
\(747\) 1.55529e23 1.60415
\(748\) 1.94716e23 1.98696
\(749\) −3.26485e23 −3.29616
\(750\) 0 0
\(751\) −1.72464e23 −1.70443 −0.852214 − 0.523193i \(-0.824741\pi\)
−0.852214 + 0.523193i \(0.824741\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.44676e22 0.132752
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.71309e23 1.47582
\(765\) −2.34099e23 −1.99576
\(766\) 0 0
\(767\) 9.50667e22 0.793717
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) −1.40726e22 −0.113880
\(771\) 0 0
\(772\) −2.00063e23 −1.58572
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 5.57172e21 0.0432575
\(775\) −1.84105e23 −1.41466
\(776\) 0 0
\(777\) 0 0
\(778\) 8.84119e21 0.0658679
\(779\) 0 0
\(780\) 0 0
\(781\) −1.66613e22 −0.120365
\(782\) 0 0
\(783\) 0 0
\(784\) 2.73373e23 1.91526
\(785\) 0 0
\(786\) 0 0
\(787\) −2.92662e23 −1.98869 −0.994347 − 0.106179i \(-0.966138\pi\)
−0.994347 + 0.106179i \(0.966138\pi\)
\(788\) 4.95433e22 0.333254
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −2.05155e22 −0.132520
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 2.70576e23 1.67875
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.32271e22 0.198049
\(801\) 3.37546e23 1.99193
\(802\) −4.95099e21 −0.0289266
\(803\) 3.35412e23 1.94023
\(804\) 0 0
\(805\) 0 0
\(806\) 1.39069e22 0.0780820
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 1.23052e22 0.0664062
\(811\) 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
\(819\) 2.88536e23 1.42538
\(820\) 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
\(826\) 2.35649e22 0.108749
\(827\) 4.36309e23 1.99412 0.997058 − 0.0766567i \(-0.0244245\pi\)
0.997058 + 0.0766567i \(0.0244245\pi\)
\(828\) 0 0
\(829\) 4.44014e23 1.99049 0.995245 − 0.0973987i \(-0.0310522\pi\)
0.995245 + 0.0973987i \(0.0310522\pi\)
\(830\) 2.39927e22 0.106526
\(831\) 0 0
\(832\) 1.85812e23 0.809259
\(833\) −8.97988e23 −3.87356
\(834\) 0 0
\(835\) 4.00900e23 1.69646
\(836\) 0 0
\(837\) 0 0
\(838\) −8.54213e21 −0.0351248
\(839\) −2.64857e23 −1.07874 −0.539369 − 0.842070i \(-0.681337\pi\)
−0.539369 + 0.842070i \(0.681337\pi\)
\(840\) 0 0
\(841\) 2.50246e23 1.00000
\(842\) 8.42114e21 0.0333330
\(843\) 0 0
\(844\) 0 0
\(845\) −8.03587e22 −0.309157
\(846\) 0 0
\(847\) −4.54263e23 −1.71490
\(848\) 0 0
\(849\) 0 0
\(850\) −3.61134e22 −0.132531
\(851\) 0 0
\(852\) 0 0
\(853\) 3.07173e23 1.09595 0.547975 − 0.836495i \(-0.315399\pi\)
0.547975 + 0.836495i \(0.315399\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.34242e22 −0.254712
\(857\) 3.63318e23 1.24865 0.624325 − 0.781165i \(-0.285374\pi\)
0.624325 + 0.781165i \(0.285374\pi\)
\(858\) 0 0
\(859\) −2.79842e23 −0.943989 −0.471995 − 0.881601i \(-0.656466\pi\)
−0.471995 + 0.881601i \(0.656466\pi\)
\(860\) −1.94053e23 −0.648534
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 6.07981e23 1.93982
\(866\) 0 0
\(867\) 0 0
\(868\) −7.78271e23 −2.41531
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.89256e23 −1.71490
\(876\) 0 0
\(877\) 3.27168e23 0.934920 0.467460 − 0.884014i \(-0.345169\pi\)
0.467460 + 0.884014i \(0.345169\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 3.54884e23 0.986790
\(881\) 6.22869e23 1.71628 0.858142 − 0.513412i \(-0.171619\pi\)
0.858142 + 0.513412i \(0.171619\pi\)
\(882\) 4.72019e22 0.128888
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −6.15881e23 −1.65150
\(885\) 0 0
\(886\) 0 0
\(887\) −6.11015e23 −1.59464 −0.797320 − 0.603557i \(-0.793750\pi\)
−0.797320 + 0.603557i \(0.793750\pi\)
\(888\) 0 0
\(889\) −7.35875e23 −1.88621
\(890\) 5.20717e22 0.132276
\(891\) 3.97211e23 1.00000
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 5.42961e23 1.31882
\(896\) 1.87142e23 0.450513
\(897\) 0 0
\(898\) 2.09324e22 0.0495005
\(899\) 0 0
\(900\) −4.28569e23 −0.995590
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.02406e23 −1.78322
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 8.61855e23 1.86529
\(909\) 0 0
\(910\) 4.45112e22 0.0946539
\(911\) −6.86330e23 −1.44673 −0.723363 − 0.690468i \(-0.757405\pi\)
−0.723363 + 0.690468i \(0.757405\pi\)
\(912\) 0 0
\(913\) 7.74483e23 1.60415
\(914\) −4.83773e22 −0.0993280
\(915\) 0 0
\(916\) 9.85595e23 1.98854
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.26992e22 0.100044
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.36477e23 −1.68800 −0.843998 − 0.536347i \(-0.819804\pi\)
−0.843998 + 0.536347i \(0.819804\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.09855e23 1.42259
\(933\) 0 0
\(934\) 0 0
\(935\) −1.16574e24 −1.99576
\(936\) 6.48898e22 0.110146
\(937\) −8.40044e23 −1.41379 −0.706895 − 0.707318i \(-0.749905\pi\)
−0.706895 + 0.707318i \(0.749905\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −5.94260e23 −0.942325
\(945\) 0 0
\(946\) 2.77454e22 0.0432575
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.06090e24 −1.61266
\(950\) 0 0
\(951\) 0 0
\(952\) −3.06002e23 −0.453553
\(953\) −1.32063e24 −1.94105 −0.970525 − 0.241000i \(-0.922525\pi\)
−0.970525 + 0.241000i \(0.922525\pi\)
\(954\) 0 0
\(955\) −1.02560e24 −1.48236
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.28345e23 1.00127
\(962\) 0 0
\(963\) 1.42160e24 1.92207
\(964\) 0 0
\(965\) 1.19775e24 1.59275
\(966\) 0 0
\(967\) −9.96649e23 −1.30356 −0.651779 − 0.758409i \(-0.725977\pi\)
−0.651779 + 0.758409i \(0.725977\pi\)
\(968\) −1.02161e23 −0.132520
\(969\) 0 0
\(970\) 0 0
\(971\) −8.57782e22 −0.108548 −0.0542742 − 0.998526i \(-0.517284\pi\)
−0.0542742 + 0.998526i \(0.517284\pi\)
\(972\) 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
\(979\) 1.68087e24 1.99193
\(980\) −1.64396e24 −1.93234
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −2.96609e23 −0.334730
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 6.12760e22 0.0664062
\(991\) −2.19089e23 −0.235522 −0.117761 − 0.993042i \(-0.537572\pi\)
−0.117761 + 0.993042i \(0.537572\pi\)
\(992\) −2.62735e23 −0.280172
\(993\) 0 0
\(994\) 1.30629e22 0.0137072
\(995\) −1.61990e24 −1.68618
\(996\) 0 0
\(997\) 1.91065e24 1.95713 0.978567 − 0.205927i \(-0.0660210\pi\)
0.978567 + 0.205927i \(0.0660210\pi\)
\(998\) 1.77672e22 0.0180541
\(999\) 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 55.17.d.b.54.1 yes 1
5.4 even 2 55.17.d.a.54.1 ✓ 1
11.10 odd 2 55.17.d.a.54.1 ✓ 1
55.54 odd 2 CM 55.17.d.b.54.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.17.d.a.54.1 ✓ 1 5.4 even 2
55.17.d.a.54.1 ✓ 1 11.10 odd 2
55.17.d.b.54.1 yes 1 1.1 even 1 trivial
55.17.d.b.54.1 yes 1 55.54 odd 2 CM