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Properties

Label 56.13.h.a.13.1
Level $56$
Weight $13$
Character 56.13
Self dual yes
Analytic conductor $51.184$
Analytic rank $0$
Dimension $1$
CM discriminant -56
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] = [56,13,Mod(13,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("56.13");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 56.h (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: \(51.1836537675\)
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 13.1
Character \(\chi\) \(=\) 56.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+64.0000 q^{2} -1430.00 q^{3} +4096.00 q^{4} +30602.0 q^{5} -91520.0 q^{6} +117649. q^{7} +262144. q^{8} +1.51346e6 q^{9} +O(q^{10})\) \(q+64.0000 q^{2} -1430.00 q^{3} +4096.00 q^{4} +30602.0 q^{5} -91520.0 q^{6} +117649. q^{7} +262144. q^{8} +1.51346e6 q^{9} +1.95853e6 q^{10} -5.85728e6 q^{12} -3.22927e6 q^{13} +7.52954e6 q^{14} -4.37609e7 q^{15} +1.67772e7 q^{16} +9.68614e7 q^{18} +2.04990e7 q^{19} +1.25346e8 q^{20} -1.68238e8 q^{21} -7.49549e7 q^{23} -3.74866e8 q^{24} +6.92342e8 q^{25} -2.06673e8 q^{26} -1.40429e9 q^{27} +4.81890e8 q^{28} -2.80070e9 q^{30} +1.07374e9 q^{32} +3.60029e9 q^{35} +6.19913e9 q^{36} +1.31194e9 q^{38} +4.61786e9 q^{39} +8.02213e9 q^{40} -1.07672e10 q^{42} +4.63149e10 q^{45} -4.79711e9 q^{46} -2.39914e10 q^{48} +1.38413e10 q^{49} +4.43099e10 q^{50} -1.32271e10 q^{52} -8.98743e10 q^{54} +3.08410e10 q^{56} -2.93136e10 q^{57} -3.96350e10 q^{59} -1.79244e11 q^{60} +9.41840e10 q^{61} +1.78057e11 q^{63} +6.87195e10 q^{64} -9.88221e10 q^{65} +1.07185e11 q^{69} +2.30419e11 q^{70} -1.45624e11 q^{71} +3.96744e11 q^{72} -9.90049e11 q^{75} +8.39641e10 q^{76} +2.95543e11 q^{78} -4.30011e11 q^{79} +5.13416e11 q^{80} +1.20381e12 q^{81} +3.94188e11 q^{83} -6.89103e11 q^{84} +2.96415e12 q^{90} -3.79920e11 q^{91} -3.07015e11 q^{92} +6.27312e11 q^{95} -1.53545e12 q^{96} +8.85842e11 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 64.0000 1.00000
\(3\) −1430.00 −1.96159 −0.980796 − 0.195038i \(-0.937517\pi\)
−0.980796 + 0.195038i \(0.937517\pi\)
\(4\) 4096.00 1.00000
\(5\) 30602.0 1.95853 0.979264 − 0.202588i \(-0.0649353\pi\)
0.979264 + 0.202588i \(0.0649353\pi\)
\(6\) −91520.0 −1.96159
\(7\) 117649. 1.00000
\(8\) 262144. 1.00000
\(9\) 1.51346e6 2.84784
\(10\) 1.95853e6 1.95853
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −5.85728e6 −1.96159
\(13\) −3.22927e6 −0.669028 −0.334514 − 0.942391i \(-0.608572\pi\)
−0.334514 + 0.942391i \(0.608572\pi\)
\(14\) 7.52954e6 1.00000
\(15\) −4.37609e7 −3.84183
\(16\) 1.67772e7 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 9.68614e7 2.84784
\(19\) 2.04990e7 0.435725 0.217862 − 0.975980i \(-0.430092\pi\)
0.217862 + 0.975980i \(0.430092\pi\)
\(20\) 1.25346e8 1.95853
\(21\) −1.68238e8 −1.96159
\(22\) 0 0
\(23\) −7.49549e7 −0.506329 −0.253165 − 0.967423i \(-0.581471\pi\)
−0.253165 + 0.967423i \(0.581471\pi\)
\(24\) −3.74866e8 −1.96159
\(25\) 6.92342e8 2.83583
\(26\) −2.06673e8 −0.669028
\(27\) −1.40429e9 −3.62471
\(28\) 4.81890e8 1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −2.80070e9 −3.84183
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.07374e9 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 3.60029e9 1.95853
\(36\) 6.19913e9 2.84784
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.31194e9 0.435725
\(39\) 4.61786e9 1.31236
\(40\) 8.02213e9 1.95853
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −1.07672e10 −1.96159
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 4.63149e10 5.57757
\(46\) −4.79711e9 −0.506329
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −2.39914e10 −1.96159
\(49\) 1.38413e10 1.00000
\(50\) 4.43099e10 2.83583
\(51\) 0 0
\(52\) −1.32271e10 −0.669028
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −8.98743e10 −3.62471
\(55\) 0 0
\(56\) 3.08410e10 1.00000
\(57\) −2.93136e10 −0.854714
\(58\) 0 0
\(59\) −3.96350e10 −0.939650 −0.469825 − 0.882759i \(-0.655683\pi\)
−0.469825 + 0.882759i \(0.655683\pi\)
\(60\) −1.79244e11 −3.84183
\(61\) 9.41840e10 1.82809 0.914046 − 0.405610i \(-0.132941\pi\)
0.914046 + 0.405610i \(0.132941\pi\)
\(62\) 0 0
\(63\) 1.78057e11 2.84784
\(64\) 6.87195e10 1.00000
\(65\) −9.88221e10 −1.31031
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 1.07185e11 0.993211
\(70\) 2.30419e11 1.95853
\(71\) −1.45624e11 −1.13680 −0.568400 − 0.822752i \(-0.692437\pi\)
−0.568400 + 0.822752i \(0.692437\pi\)
\(72\) 3.96744e11 2.84784
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −9.90049e11 −5.56274
\(76\) 8.39641e10 0.435725
\(77\) 0 0
\(78\) 2.95543e11 1.31236
\(79\) −4.30011e11 −1.76895 −0.884477 − 0.466583i \(-0.845485\pi\)
−0.884477 + 0.466583i \(0.845485\pi\)
\(80\) 5.13416e11 1.95853
\(81\) 1.20381e12 4.26235
\(82\) 0 0
\(83\) 3.94188e11 1.20569 0.602844 − 0.797859i \(-0.294034\pi\)
0.602844 + 0.797859i \(0.294034\pi\)
\(84\) −6.89103e11 −1.96159
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 2.96415e12 5.57757
\(91\) −3.79920e11 −0.669028
\(92\) −3.07015e11 −0.506329
\(93\) 0 0
\(94\) 0 0
\(95\) 6.27312e11 0.853379
\(96\) −1.53545e12 −1.96159
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 8.85842e11 1.00000
\(99\) 0 0
\(100\) 2.83583e12 2.83583
\(101\) 1.57042e12 1.47940 0.739701 − 0.672935i \(-0.234967\pi\)
0.739701 + 0.672935i \(0.234967\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −8.46534e11 −0.669028
\(105\) −5.14842e12 −3.84183
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −5.75195e12 −3.62471
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.97382e12 1.00000
\(113\) −3.20663e12 −1.54020 −0.770101 − 0.637922i \(-0.779794\pi\)
−0.770101 + 0.637922i \(0.779794\pi\)
\(114\) −1.87607e12 −0.854714
\(115\) −2.29377e12 −0.991660
\(116\) 0 0
\(117\) −4.88737e12 −1.90528
\(118\) −2.53664e12 −0.939650
\(119\) 0 0
\(120\) −1.14716e13 −3.84183
\(121\) 3.13843e12 1.00000
\(122\) 6.02778e12 1.82809
\(123\) 0 0
\(124\) 0 0
\(125\) 1.37159e13 3.59553
\(126\) 1.13956e13 2.84784
\(127\) 4.05680e12 0.966855 0.483428 − 0.875384i \(-0.339392\pi\)
0.483428 + 0.875384i \(0.339392\pi\)
\(128\) 4.39805e12 1.00000
\(129\) 0 0
\(130\) −6.32462e12 −1.31031
\(131\) 5.12762e12 1.01458 0.507292 − 0.861774i \(-0.330647\pi\)
0.507292 + 0.861774i \(0.330647\pi\)
\(132\) 0 0
\(133\) 2.41169e12 0.435725
\(134\) 0 0
\(135\) −4.29740e13 −7.09909
\(136\) 0 0
\(137\) −5.98573e12 −0.905302 −0.452651 − 0.891688i \(-0.649522\pi\)
−0.452651 + 0.891688i \(0.649522\pi\)
\(138\) 6.85987e12 0.993211
\(139\) 5.70065e12 0.790379 0.395190 − 0.918600i \(-0.370679\pi\)
0.395190 + 0.918600i \(0.370679\pi\)
\(140\) 1.47468e13 1.95853
\(141\) 0 0
\(142\) −9.31996e12 −1.13680
\(143\) 0 0
\(144\) 2.53916e13 2.84784
\(145\) 0 0
\(146\) 0 0
\(147\) −1.97930e13 −1.96159
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −6.33631e13 −5.56274
\(151\) 7.37278e12 0.621970 0.310985 − 0.950415i \(-0.399341\pi\)
0.310985 + 0.950415i \(0.399341\pi\)
\(152\) 5.37370e12 0.435725
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.89147e13 1.31236
\(157\) −2.97992e13 −1.98978 −0.994892 − 0.100944i \(-0.967814\pi\)
−0.994892 + 0.100944i \(0.967814\pi\)
\(158\) −2.75207e13 −1.76895
\(159\) 0 0
\(160\) 3.28586e13 1.95853
\(161\) −8.81837e12 −0.506329
\(162\) 7.70441e13 4.26235
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 2.52280e13 1.20569
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −4.41026e13 −1.96159
\(169\) −1.28699e13 −0.552402
\(170\) 0 0
\(171\) 3.10245e13 1.24087
\(172\) 0 0
\(173\) −1.94046e13 −0.723816 −0.361908 − 0.932214i \(-0.617875\pi\)
−0.361908 + 0.932214i \(0.617875\pi\)
\(174\) 0 0
\(175\) 8.14533e13 2.83583
\(176\) 0 0
\(177\) 5.66780e13 1.84321
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 1.89706e14 5.57757
\(181\) 1.17674e13 0.334665 0.167332 − 0.985901i \(-0.446485\pi\)
0.167332 + 0.985901i \(0.446485\pi\)
\(182\) −2.43149e13 −0.669028
\(183\) −1.34683e14 −3.58597
\(184\) −1.96490e13 −0.506329
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.65213e14 −3.62471
\(190\) 4.01480e13 0.853379
\(191\) 4.76326e13 0.981080 0.490540 − 0.871419i \(-0.336800\pi\)
0.490540 + 0.871419i \(0.336800\pi\)
\(192\) −9.82689e13 −1.96159
\(193\) −8.63130e13 −1.67006 −0.835031 − 0.550203i \(-0.814550\pi\)
−0.835031 + 0.550203i \(0.814550\pi\)
\(194\) 0 0
\(195\) 1.41316e14 2.57029
\(196\) 5.66939e13 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 1.81493e14 2.83583
\(201\) 0 0
\(202\) 1.00507e14 1.47940
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.13441e14 −1.44194
\(208\) −5.41782e13 −0.669028
\(209\) 0 0
\(210\) −3.29499e14 −3.84183
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 2.08243e14 2.22994
\(214\) 0 0
\(215\) 0 0
\(216\) −3.68125e14 −3.62471
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 1.26325e14 1.00000
\(225\) 1.04783e15 8.07600
\(226\) −2.05224e14 −1.54020
\(227\) 2.70478e14 1.97687 0.988433 − 0.151656i \(-0.0484606\pi\)
0.988433 + 0.151656i \(0.0484606\pi\)
\(228\) −1.20069e14 −0.854714
\(229\) −2.78615e14 −1.93193 −0.965966 − 0.258668i \(-0.916716\pi\)
−0.965966 + 0.258668i \(0.916716\pi\)
\(230\) −1.46801e14 −0.991660
\(231\) 0 0
\(232\) 0 0
\(233\) −2.30251e14 −1.43902 −0.719509 − 0.694483i \(-0.755633\pi\)
−0.719509 + 0.694483i \(0.755633\pi\)
\(234\) −3.12792e14 −1.90528
\(235\) 0 0
\(236\) −1.62345e14 −0.939650
\(237\) 6.14915e14 3.46997
\(238\) 0 0
\(239\) −3.64705e14 −1.95684 −0.978419 − 0.206632i \(-0.933750\pi\)
−0.978419 + 0.206632i \(0.933750\pi\)
\(240\) −7.34185e14 −3.84183
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 2.00859e14 1.00000
\(243\) −9.75160e14 −4.73629
\(244\) 3.85778e14 1.82809
\(245\) 4.23571e14 1.95853
\(246\) 0 0
\(247\) −6.61970e13 −0.291512
\(248\) 0 0
\(249\) −5.63689e14 −2.36507
\(250\) 8.77815e14 3.59553
\(251\) −4.51816e14 −1.80684 −0.903419 − 0.428758i \(-0.858951\pi\)
−0.903419 + 0.428758i \(0.858951\pi\)
\(252\) 7.29321e14 2.84784
\(253\) 0 0
\(254\) 2.59635e14 0.966855
\(255\) 0 0
\(256\) 2.81475e14 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.04775e14 −1.31031
\(261\) 0 0
\(262\) 3.28168e14 1.01458
\(263\) 6.54823e14 1.97874 0.989372 − 0.145408i \(-0.0464494\pi\)
0.989372 + 0.145408i \(0.0464494\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.54348e14 0.435725
\(267\) 0 0
\(268\) 0 0
\(269\) −1.91337e14 −0.504993 −0.252496 − 0.967598i \(-0.581252\pi\)
−0.252496 + 0.967598i \(0.581252\pi\)
\(270\) −2.75033e15 −7.09909
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 5.43286e14 1.31236
\(274\) −3.83087e14 −0.905302
\(275\) 0 0
\(276\) 4.39032e14 0.993211
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 3.64842e14 0.790379
\(279\) 0 0
\(280\) 9.43796e14 1.95853
\(281\) 7.33648e14 1.49022 0.745110 − 0.666942i \(-0.232397\pi\)
0.745110 + 0.666942i \(0.232397\pi\)
\(282\) 0 0
\(283\) −9.51687e14 −1.85257 −0.926287 − 0.376820i \(-0.877018\pi\)
−0.926287 + 0.376820i \(0.877018\pi\)
\(284\) −5.96477e14 −1.13680
\(285\) −8.97056e14 −1.67398
\(286\) 0 0
\(287\) 0 0
\(288\) 1.62506e15 2.84784
\(289\) 5.82622e14 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.46953e14 1.02251 0.511254 − 0.859430i \(-0.329181\pi\)
0.511254 + 0.859430i \(0.329181\pi\)
\(294\) −1.26675e15 −1.96159
\(295\) −1.21291e15 −1.84033
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.42050e14 0.338748
\(300\) −4.05524e15 −5.56274
\(301\) 0 0
\(302\) 4.71858e14 0.621970
\(303\) −2.24569e15 −2.90198
\(304\) 3.43917e14 0.435725
\(305\) 2.88222e15 3.58037
\(306\) 0 0
\(307\) 3.63819e14 0.434565 0.217283 − 0.976109i \(-0.430281\pi\)
0.217283 + 0.976109i \(0.430281\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 1.21054e15 1.31236
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −1.90715e15 −1.98978
\(315\) 5.44890e15 5.57757
\(316\) −1.76132e15 −1.76895
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.10295e15 1.95853
\(321\) 0 0
\(322\) −5.64375e14 −0.506329
\(323\) 0 0
\(324\) 4.93082e15 4.26235
\(325\) −2.23576e15 −1.89725
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 1.61459e15 1.20569
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −2.82257e15 −1.96159
\(337\) −2.85145e15 −1.94664 −0.973321 − 0.229449i \(-0.926308\pi\)
−0.973321 + 0.229449i \(0.926308\pi\)
\(338\) −8.23674e14 −0.552402
\(339\) 4.58547e15 3.02125
\(340\) 0 0
\(341\) 0 0
\(342\) 1.98557e15 1.24087
\(343\) 1.62841e15 1.00000
\(344\) 0 0
\(345\) 3.28009e15 1.94523
\(346\) −1.24189e15 −0.723816
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.79335e15 −0.992459 −0.496230 − 0.868191i \(-0.665283\pi\)
−0.496230 + 0.868191i \(0.665283\pi\)
\(350\) 5.21301e15 2.83583
\(351\) 4.53482e15 2.42503
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 3.62739e15 1.84321
\(355\) −4.45640e15 −2.22645
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.41668e15 −0.661769 −0.330884 − 0.943671i \(-0.607347\pi\)
−0.330884 + 0.943671i \(0.607347\pi\)
\(360\) 1.21412e16 5.57757
\(361\) −1.79310e15 −0.810144
\(362\) 7.53115e14 0.334665
\(363\) −4.48795e15 −1.96159
\(364\) −1.55615e15 −0.669028
\(365\) 0 0
\(366\) −8.61972e15 −3.58597
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −1.25753e15 −0.506329
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −1.96137e16 −7.05296
\(376\) 0 0
\(377\) 0 0
\(378\) −1.05736e16 −3.62471
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 2.56947e15 0.853379
\(381\) −5.80123e15 −1.89657
\(382\) 3.04849e15 0.981080
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −6.28921e15 −1.96159
\(385\) 0 0
\(386\) −5.52403e15 −1.67006
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 9.04420e15 2.57029
\(391\) 0 0
\(392\) 3.62841e15 1.00000
\(393\) −7.33250e15 −1.99020
\(394\) 0 0
\(395\) −1.31592e16 −3.46455
\(396\) 0 0
\(397\) −5.52071e15 −1.41011 −0.705053 − 0.709155i \(-0.749077\pi\)
−0.705053 + 0.709155i \(0.749077\pi\)
\(398\) 0 0
\(399\) −3.44872e15 −0.854714
\(400\) 1.16156e16 2.83583
\(401\) 1.47479e15 0.354702 0.177351 − 0.984148i \(-0.443247\pi\)
0.177351 + 0.984148i \(0.443247\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.43242e15 1.47940
\(405\) 3.68391e16 8.34794
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 8.55959e15 1.77583
\(412\) 0 0
\(413\) −4.66301e15 −0.939650
\(414\) −7.26023e15 −1.44194
\(415\) 1.20629e16 2.36137
\(416\) −3.46740e15 −0.669028
\(417\) −8.15193e15 −1.55040
\(418\) 0 0
\(419\) −8.27704e15 −1.52965 −0.764823 − 0.644241i \(-0.777173\pi\)
−0.764823 + 0.644241i \(0.777173\pi\)
\(420\) −2.10879e16 −3.84183
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 1.33275e16 2.22994
\(427\) 1.10807e16 1.82809
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.94389e15 1.23928 0.619640 − 0.784886i \(-0.287279\pi\)
0.619640 + 0.784886i \(0.287279\pi\)
\(432\) −2.35600e16 −3.62471
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.53650e15 −0.220620
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 2.09482e16 2.84784
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 8.08478e15 1.00000
\(449\) −1.63859e16 −1.99982 −0.999911 − 0.0133626i \(-0.995746\pi\)
−0.999911 + 0.0133626i \(0.995746\pi\)
\(450\) 6.70612e16 8.07600
\(451\) 0 0
\(452\) −1.31343e16 −1.54020
\(453\) −1.05431e16 −1.22005
\(454\) 1.73106e16 1.97687
\(455\) −1.16263e16 −1.31031
\(456\) −7.68440e15 −0.854714
\(457\) 1.48907e15 0.163462 0.0817309 − 0.996654i \(-0.473955\pi\)
0.0817309 + 0.996654i \(0.473955\pi\)
\(458\) −1.78314e16 −1.93193
\(459\) 0 0
\(460\) −9.39528e15 −0.991660
\(461\) −3.15507e15 −0.328703 −0.164352 − 0.986402i \(-0.552553\pi\)
−0.164352 + 0.986402i \(0.552553\pi\)
\(462\) 0 0
\(463\) −1.80008e15 −0.182728 −0.0913642 − 0.995818i \(-0.529123\pi\)
−0.0913642 + 0.995818i \(0.529123\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.47361e16 −1.43902
\(467\) −2.05857e16 −1.98456 −0.992280 − 0.124020i \(-0.960421\pi\)
−0.992280 + 0.124020i \(0.960421\pi\)
\(468\) −2.00187e16 −1.90528
\(469\) 0 0
\(470\) 0 0
\(471\) 4.26128e16 3.90314
\(472\) −1.03901e16 −0.939650
\(473\) 0 0
\(474\) 3.93546e16 3.46997
\(475\) 1.41923e16 1.23564
\(476\) 0 0
\(477\) 0 0
\(478\) −2.33411e16 −1.95684
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −4.69879e16 −3.84183
\(481\) 0 0
\(482\) 0 0
\(483\) 1.26103e16 0.993211
\(484\) 1.28550e16 1.00000
\(485\) 0 0
\(486\) −6.24102e16 −4.73629
\(487\) 2.64885e16 1.98556 0.992780 − 0.119946i \(-0.0382723\pi\)
0.992780 + 0.119946i \(0.0382723\pi\)
\(488\) 2.46898e16 1.82809
\(489\) 0 0
\(490\) 2.71085e16 1.95853
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −4.23661e15 −0.291512
\(495\) 0 0
\(496\) 0 0
\(497\) −1.71326e16 −1.13680
\(498\) −3.60761e16 −2.36507
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 5.61801e16 3.59553
\(501\) 0 0
\(502\) −2.89162e16 −1.80684
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 4.66766e16 2.84784
\(505\) 4.80579e16 2.89745
\(506\) 0 0
\(507\) 1.84040e16 1.08359
\(508\) 1.66167e16 0.966855
\(509\) 2.40721e16 1.38422 0.692112 − 0.721790i \(-0.256680\pi\)
0.692112 + 0.721790i \(0.256680\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.80144e16 1.00000
\(513\) −2.87865e16 −1.57937
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.77486e16 1.41983
\(520\) −2.59056e16 −1.31031
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 3.19589e16 1.56164 0.780822 − 0.624753i \(-0.214800\pi\)
0.780822 + 0.624753i \(0.214800\pi\)
\(524\) 2.10027e16 1.01458
\(525\) −1.16478e17 −5.56274
\(526\) 4.19087e16 1.97874
\(527\) 0 0
\(528\) 0 0
\(529\) −1.62964e16 −0.743631
\(530\) 0 0
\(531\) −5.99859e16 −2.67597
\(532\) 9.87829e15 0.435725
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −1.22456e16 −0.504993
\(539\) 0 0
\(540\) −1.76021e17 −7.09909
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −1.68274e16 −0.656475
\(544\) 0 0
\(545\) 0 0
\(546\) 3.47703e16 1.31236
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −2.45175e16 −0.905302
\(549\) 1.42544e17 5.20612
\(550\) 0 0
\(551\) 0 0
\(552\) 2.80980e16 0.993211
\(553\) −5.05903e16 −1.76895
\(554\) 0 0
\(555\) 0 0
\(556\) 2.33499e16 0.790379
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 6.04029e16 1.95853
\(561\) 0 0
\(562\) 4.69535e16 1.49022
\(563\) −3.60382e16 −1.13165 −0.565826 − 0.824525i \(-0.691443\pi\)
−0.565826 + 0.824525i \(0.691443\pi\)
\(564\) 0 0
\(565\) −9.81291e16 −3.01653
\(566\) −6.09079e16 −1.85257
\(567\) 1.41628e17 4.26235
\(568\) −3.81746e16 −1.13680
\(569\) 5.13907e16 1.51430 0.757148 − 0.653244i \(-0.226592\pi\)
0.757148 + 0.653244i \(0.226592\pi\)
\(570\) −5.74116e16 −1.67398
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −6.81147e16 −1.92448
\(574\) 0 0
\(575\) −5.18944e16 −1.43586
\(576\) 1.04004e17 2.84784
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 3.72878e16 1.00000
\(579\) 1.23428e17 3.27598
\(580\) 0 0
\(581\) 4.63758e16 1.20569
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.49563e17 −3.73155
\(586\) 4.14050e16 1.02251
\(587\) −5.45447e16 −1.33329 −0.666644 − 0.745376i \(-0.732270\pi\)
−0.666644 + 0.745376i \(0.732270\pi\)
\(588\) −8.10723e16 −1.96159
\(589\) 0 0
\(590\) −7.76262e16 −1.84033
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 1.54912e16 0.338748
\(599\) −8.70111e16 −1.88371 −0.941854 − 0.336022i \(-0.890918\pi\)
−0.941854 + 0.336022i \(0.890918\pi\)
\(600\) −2.59535e17 −5.56274
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3.01989e16 0.621970
\(605\) 9.60422e16 1.95853
\(606\) −1.43724e17 −2.90198
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 2.20107e16 0.435725
\(609\) 0 0
\(610\) 1.84462e17 3.58037
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 2.32844e16 0.434565
\(615\) 0 0
\(616\) 0 0
\(617\) −1.04658e17 −1.89698 −0.948491 − 0.316805i \(-0.897390\pi\)
−0.948491 + 0.316805i \(0.897390\pi\)
\(618\) 0 0
\(619\) 1.03344e17 1.83713 0.918567 − 0.395264i \(-0.129347\pi\)
0.918567 + 0.395264i \(0.129347\pi\)
\(620\) 0 0
\(621\) 1.05258e17 1.83529
\(622\) 0 0
\(623\) 0 0
\(624\) 7.74748e16 1.31236
\(625\) 2.50704e17 4.20611
\(626\) 0 0
\(627\) 0 0
\(628\) −1.22057e17 −1.98978
\(629\) 0 0
\(630\) 3.48729e17 5.57757
\(631\) −1.09327e17 −1.73201 −0.866003 − 0.500039i \(-0.833319\pi\)
−0.866003 + 0.500039i \(0.833319\pi\)
\(632\) −1.12725e17 −1.76895
\(633\) 0 0
\(634\) 0 0
\(635\) 1.24146e17 1.89361
\(636\) 0 0
\(637\) −4.46973e16 −0.669028
\(638\) 0 0
\(639\) −2.20397e17 −3.23742
\(640\) 1.34589e17 1.95853
\(641\) −1.07441e17 −1.54889 −0.774444 − 0.632642i \(-0.781970\pi\)
−0.774444 + 0.632642i \(0.781970\pi\)
\(642\) 0 0
\(643\) −1.22170e17 −1.72862 −0.864311 − 0.502958i \(-0.832245\pi\)
−0.864311 + 0.502958i \(0.832245\pi\)
\(644\) −3.61200e16 −0.506329
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 3.15573e17 4.26235
\(649\) 0 0
\(650\) −1.43089e17 −1.89725
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 1.56915e17 1.98709
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.45005e16 0.413635 0.206817 − 0.978380i \(-0.433689\pi\)
0.206817 + 0.978380i \(0.433689\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.03334e17 1.20569
\(665\) 7.38026e16 0.853379
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −1.80644e17 −1.96159
\(673\) 1.85810e17 1.99976 0.999881 − 0.0154297i \(-0.00491162\pi\)
0.999881 + 0.0154297i \(0.00491162\pi\)
\(674\) −1.82493e17 −1.94664
\(675\) −9.72246e17 −10.2791
\(676\) −5.27151e16 −0.552402
\(677\) −1.12020e17 −1.16349 −0.581745 − 0.813371i \(-0.697630\pi\)
−0.581745 + 0.813371i \(0.697630\pi\)
\(678\) 2.93470e17 3.02125
\(679\) 0 0
\(680\) 0 0
\(681\) −3.86784e17 −3.87780
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 1.27076e17 1.24087
\(685\) −1.83175e17 −1.77306
\(686\) 1.04218e17 1.00000
\(687\) 3.98420e17 3.78966
\(688\) 0 0
\(689\) 0 0
\(690\) 2.09926e17 1.94523
\(691\) 1.03756e17 0.953114 0.476557 − 0.879144i \(-0.341885\pi\)
0.476557 + 0.879144i \(0.341885\pi\)
\(692\) −7.94813e16 −0.723816
\(693\) 0 0
\(694\) 0 0
\(695\) 1.74451e17 1.54798
\(696\) 0 0
\(697\) 0 0
\(698\) −1.14774e17 −0.992459
\(699\) 3.29259e17 2.82277
\(700\) 3.33633e17 2.83583
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 2.90228e17 2.42503
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.84758e17 1.47940
\(708\) 2.32153e17 1.84321
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −2.85209e17 −2.22645
\(711\) −6.50803e17 −5.03770
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.21529e17 3.83852
\(718\) −9.06678e16 −0.661769
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 7.77035e17 5.57757
\(721\) 0 0
\(722\) −1.14759e17 −0.810144
\(723\) 0 0
\(724\) 4.81993e16 0.334665
\(725\) 0 0
\(726\) −2.87229e17 −1.96159
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −9.95938e16 −0.669028
\(729\) 7.54722e17 5.02831
\(730\) 0 0
\(731\) 0 0
\(732\) −5.51662e17 −3.58597
\(733\) −4.90345e15 −0.0316139 −0.0158070 − 0.999875i \(-0.505032\pi\)
−0.0158070 + 0.999875i \(0.505032\pi\)
\(734\) 0 0
\(735\) −6.05707e17 −3.84183
\(736\) −8.04822e16 −0.506329
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 9.46617e16 0.571827
\(742\) 0 0
\(743\) −2.74393e16 −0.163095 −0.0815475 − 0.996669i \(-0.525986\pi\)
−0.0815475 + 0.996669i \(0.525986\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.96587e17 3.43361
\(748\) 0 0
\(749\) 0 0
\(750\) −1.25527e18 −7.05296
\(751\) 1.23665e17 0.689297 0.344648 − 0.938732i \(-0.387998\pi\)
0.344648 + 0.938732i \(0.387998\pi\)
\(752\) 0 0
\(753\) 6.46097e17 3.54428
\(754\) 0 0
\(755\) 2.25622e17 1.21815
\(756\) −6.76712e17 −3.62471
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 1.64446e17 0.853379
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) −3.71279e17 −1.89657
\(763\) 0 0
\(764\) 1.95103e17 0.981080
\(765\) 0 0
\(766\) 0 0
\(767\) 1.27992e17 0.628652
\(768\) −4.02509e17 −1.96159
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.53538e17 −1.67006
\(773\) −3.81699e17 −1.78914 −0.894569 − 0.446930i \(-0.852517\pi\)
−0.894569 + 0.446930i \(0.852517\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 5.78829e17 2.57029
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.32218e17 1.00000
\(785\) −9.11914e17 −3.89705
\(786\) −4.69280e17 −1.99020
\(787\) 1.18644e17 0.499341 0.249671 − 0.968331i \(-0.419678\pi\)
0.249671 + 0.968331i \(0.419678\pi\)
\(788\) 0 0
\(789\) −9.36397e17 −3.88149
\(790\) −8.42188e17 −3.46455
\(791\) −3.77256e17 −1.54020
\(792\) 0 0
\(793\) −3.04146e17 −1.22304
\(794\) −3.53325e17 −1.41011
\(795\) 0 0
\(796\) 0 0
\(797\) −7.42210e16 −0.289586 −0.144793 − 0.989462i \(-0.546252\pi\)
−0.144793 + 0.989462i \(0.546252\pi\)
\(798\) −2.20718e17 −0.854714
\(799\) 0 0
\(800\) 7.43396e17 2.83583
\(801\) 0 0
\(802\) 9.43865e16 0.354702
\(803\) 0 0
\(804\) 0 0
\(805\) −2.69860e17 −0.991660
\(806\) 0 0
\(807\) 2.73612e17 0.990589
\(808\) 4.11675e17 1.47940
\(809\) 4.42541e17 1.57856 0.789282 − 0.614031i \(-0.210453\pi\)
0.789282 + 0.614031i \(0.210453\pi\)
\(810\) 2.35770e18 8.34794
\(811\) 5.32267e17 1.87070 0.935351 − 0.353720i \(-0.115084\pi\)
0.935351 + 0.353720i \(0.115084\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −5.74994e17 −1.90528
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 5.47814e17 1.77583
\(823\) 5.35582e17 1.72356 0.861780 − 0.507282i \(-0.169350\pi\)
0.861780 + 0.507282i \(0.169350\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −2.98433e17 −0.939650
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −4.64655e17 −1.44194
\(829\) 6.24518e17 1.92406 0.962028 − 0.272950i \(-0.0879993\pi\)
0.962028 + 0.272950i \(0.0879993\pi\)
\(830\) 7.72028e17 2.36137
\(831\) 0 0
\(832\) −2.21914e17 −0.669028
\(833\) 0 0
\(834\) −5.21723e17 −1.55040
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −5.29730e17 −1.52965
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) −1.34963e18 −3.84183
\(841\) 3.53815e17 1.00000
\(842\) 0 0
\(843\) −1.04912e18 −2.92320
\(844\) 0 0
\(845\) −3.93845e17 −1.08189
\(846\) 0 0
\(847\) 3.69233e17 1.00000
\(848\) 0 0
\(849\) 1.36091e18 3.63399
\(850\) 0 0
\(851\) 0 0
\(852\) 8.52963e17 2.22994
\(853\) −2.29428e16 −0.0595597 −0.0297799 − 0.999556i \(-0.509481\pi\)
−0.0297799 + 0.999556i \(0.509481\pi\)
\(854\) 7.09162e17 1.82809
\(855\) 9.49411e17 2.43029
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −7.23217e16 −0.180015 −0.0900077 − 0.995941i \(-0.528689\pi\)
−0.0900077 + 0.995941i \(0.528689\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 5.08409e17 1.23928
\(863\) −8.17966e17 −1.98002 −0.990012 − 0.140986i \(-0.954973\pi\)
−0.990012 + 0.140986i \(0.954973\pi\)
\(864\) −1.50784e18 −3.62471
\(865\) −5.93820e17 −1.41761
\(866\) 0 0
\(867\) −8.33150e17 −1.96159
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) −9.83362e16 −0.220620
\(875\) 1.61366e18 3.59553
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −9.25142e17 −2.00574
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.34069e18 2.84784
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 1.73446e18 3.60998
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 4.77279e17 0.966855
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 5.17426e17 1.00000
\(897\) −3.46131e17 −0.664486
\(898\) −1.04870e18 −1.99982
\(899\) 0 0
\(900\) 4.29192e18 8.07600
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −8.40598e17 −1.54020
\(905\) 3.60106e17 0.655450
\(906\) −6.74757e17 −1.22005
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1.10788e18 1.97687
\(909\) 2.37676e18 4.21310
\(910\) −7.44085e17 −1.31031
\(911\) 1.14224e18 1.99824 0.999122 − 0.0419004i \(-0.0133412\pi\)
0.999122 + 0.0419004i \(0.0133412\pi\)
\(912\) −4.91801e17 −0.854714
\(913\) 0 0
\(914\) 9.53002e16 0.163462
\(915\) −4.12157e18 −7.02322
\(916\) −1.14121e18 −1.93193
\(917\) 6.03259e17 1.01458
\(918\) 0 0
\(919\) −1.16348e18 −1.93137 −0.965684 − 0.259720i \(-0.916370\pi\)
−0.965684 + 0.259720i \(0.916370\pi\)
\(920\) −6.01298e17 −0.991660
\(921\) −5.20261e17 −0.852439
\(922\) −2.01925e17 −0.328703
\(923\) 4.70260e17 0.760551
\(924\) 0 0
\(925\) 0 0
\(926\) −1.15205e17 −0.182728
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 2.83733e17 0.435725
\(932\) −9.43109e17 −1.43902
\(933\) 0 0
\(934\) −1.31748e18 −1.98456
\(935\) 0 0
\(936\) −1.28119e18 −1.90528
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.10586e18 −1.59281 −0.796406 − 0.604763i \(-0.793268\pi\)
−0.796406 + 0.604763i \(0.793268\pi\)
\(942\) 2.72722e18 3.90314
\(943\) 0 0
\(944\) −6.64964e17 −0.939650
\(945\) −5.05584e18 −7.09909
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 2.51869e18 3.46997
\(949\) 0 0
\(950\) 9.08310e17 1.23564
\(951\) 0 0
\(952\) 0 0
\(953\) 1.46555e18 1.95634 0.978168 − 0.207817i \(-0.0666358\pi\)
0.978168 + 0.207817i \(0.0666358\pi\)
\(954\) 0 0
\(955\) 1.45765e18 1.92147
\(956\) −1.49383e18 −1.95684
\(957\) 0 0
\(958\) 0 0
\(959\) −7.04215e17 −0.905302
\(960\) −3.00722e18 −3.84183
\(961\) 7.87663e17 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.64135e18 −3.27086
\(966\) 8.07057e17 0.993211
\(967\) −4.52992e17 −0.554028 −0.277014 − 0.960866i \(-0.589345\pi\)
−0.277014 + 0.960866i \(0.589345\pi\)
\(968\) 8.22720e17 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 1.67570e18 1.99931 0.999656 − 0.0262132i \(-0.00834487\pi\)
0.999656 + 0.0262132i \(0.00834487\pi\)
\(972\) −3.99425e18 −4.73629
\(973\) 6.70676e17 0.790379
\(974\) 1.69526e18 1.98556
\(975\) 3.19713e18 3.72163
\(976\) 1.58015e18 1.82809
\(977\) −1.62589e18 −1.86949 −0.934746 − 0.355315i \(-0.884373\pi\)
−0.934746 + 0.355315i \(0.884373\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.73495e18 1.95853
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.71143e17 −0.291512
\(989\) 0 0
\(990\) 0 0
\(991\) 5.71213e17 0.603053 0.301527 − 0.953458i \(-0.402504\pi\)
0.301527 + 0.953458i \(0.402504\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −1.09648e18 −1.13680
\(995\) 0 0
\(996\) −2.30887e18 −2.36507
\(997\) −1.39310e18 −1.41844 −0.709219 − 0.704989i \(-0.750952\pi\)
−0.709219 + 0.704989i \(0.750952\pi\)
\(998\) 0 0
\(999\) 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 56.13.h.a.13.1 ✓ 1
4.3 odd 2 224.13.h.b.209.1 1
7.6 odd 2 56.13.h.b.13.1 yes 1
8.3 odd 2 224.13.h.a.209.1 1
8.5 even 2 56.13.h.b.13.1 yes 1
28.27 even 2 224.13.h.a.209.1 1
56.13 odd 2 CM 56.13.h.a.13.1 ✓ 1
56.27 even 2 224.13.h.b.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.13.h.a.13.1 ✓ 1 1.1 even 1 trivial
56.13.h.a.13.1 ✓ 1 56.13 odd 2 CM
56.13.h.b.13.1 yes 1 7.6 odd 2
56.13.h.b.13.1 yes 1 8.5 even 2
224.13.h.a.209.1 1 8.3 odd 2
224.13.h.a.209.1 1 28.27 even 2
224.13.h.b.209.1 1 4.3 odd 2
224.13.h.b.209.1 1 56.27 even 2