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Properties

Label 20.11.d.a.19.1
Level $20$
Weight $11$
Character 20.19
Self dual yes
Analytic conductor $12.707$
Analytic rank $0$
Dimension $1$
CM discriminant -20
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] = [20,11,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.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: \(12.7071450535\)
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 19.1
Character \(\chi\) \(=\) 20.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{2} -236.000 q^{3} +1024.00 q^{4} -3125.00 q^{5} +7552.00 q^{6} -33364.0 q^{7} -32768.0 q^{8} -3353.00 q^{9} +O(q^{10})\) \(q-32.0000 q^{2} -236.000 q^{3} +1024.00 q^{4} -3125.00 q^{5} +7552.00 q^{6} -33364.0 q^{7} -32768.0 q^{8} -3353.00 q^{9} +100000. q^{10} -241664. q^{12} +1.06765e6 q^{14} +737500. q^{15} +1.04858e6 q^{16} +107296. q^{18} -3.20000e6 q^{20} +7.87390e6 q^{21} +1.16956e6 q^{23} +7.73325e6 q^{24} +9.76562e6 q^{25} +1.47269e7 q^{27} -3.41647e7 q^{28} -3.81797e7 q^{29} -2.36000e7 q^{30} -3.35544e7 q^{32} +1.04262e8 q^{35} -3.43347e6 q^{36} +1.02400e8 q^{40} -2.11028e8 q^{41} -2.51965e8 q^{42} +2.23663e8 q^{43} +1.04781e7 q^{45} -3.74260e7 q^{46} -9.68878e7 q^{47} -2.47464e8 q^{48} +8.30681e8 q^{49} -3.12500e8 q^{50} -4.71260e8 q^{54} +1.09327e9 q^{56} +1.22175e9 q^{58} +7.55200e8 q^{60} -1.04159e9 q^{61} +1.11869e8 q^{63} +1.07374e9 q^{64} -2.34324e9 q^{67} -2.76017e8 q^{69} -3.33640e9 q^{70} +1.09871e8 q^{72} -2.30469e9 q^{75} -3.27680e9 q^{80} -3.27755e9 q^{81} +6.75290e9 q^{82} -5.44916e9 q^{83} +8.06288e9 q^{84} -7.15723e9 q^{86} +9.01041e9 q^{87} +1.11182e10 q^{89} -3.35300e8 q^{90} +1.19763e9 q^{92} +3.10041e9 q^{94} +7.91885e9 q^{96} -2.65818e10 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32.0000 −1.00000
\(3\) −236.000 −0.971193 −0.485597 − 0.874183i \(-0.661398\pi\)
−0.485597 + 0.874183i \(0.661398\pi\)
\(4\) 1024.00 1.00000
\(5\) −3125.00 −1.00000
\(6\) 7552.00 0.971193
\(7\) −33364.0 −1.98513 −0.992563 − 0.121735i \(-0.961154\pi\)
−0.992563 + 0.121735i \(0.961154\pi\)
\(8\) −32768.0 −1.00000
\(9\) −3353.00 −0.0567833
\(10\) 100000. 1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −241664. −0.971193
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.06765e6 1.98513
\(15\) 737500. 0.971193
\(16\) 1.04858e6 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 107296. 0.0567833
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −3.20000e6 −1.00000
\(21\) 7.87390e6 1.92794
\(22\) 0 0
\(23\) 1.16956e6 0.181713 0.0908563 − 0.995864i \(-0.471040\pi\)
0.0908563 + 0.995864i \(0.471040\pi\)
\(24\) 7.73325e6 0.971193
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 1.47269e7 1.02634
\(28\) −3.41647e7 −1.98513
\(29\) −3.81797e7 −1.86141 −0.930706 − 0.365768i \(-0.880806\pi\)
−0.930706 + 0.365768i \(0.880806\pi\)
\(30\) −2.36000e7 −0.971193
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.35544e7 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 1.04262e8 1.98513
\(36\) −3.43347e6 −0.0567833
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.02400e8 1.00000
\(41\) −2.11028e8 −1.82147 −0.910733 − 0.412996i \(-0.864482\pi\)
−0.910733 + 0.412996i \(0.864482\pi\)
\(42\) −2.51965e8 −1.92794
\(43\) 2.23663e8 1.52143 0.760716 − 0.649085i \(-0.224848\pi\)
0.760716 + 0.649085i \(0.224848\pi\)
\(44\) 0 0
\(45\) 1.04781e7 0.0567833
\(46\) −3.74260e7 −0.181713
\(47\) −9.68878e7 −0.422454 −0.211227 − 0.977437i \(-0.567746\pi\)
−0.211227 + 0.977437i \(0.567746\pi\)
\(48\) −2.47464e8 −0.971193
\(49\) 8.30681e8 2.94072
\(50\) −3.12500e8 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −4.71260e8 −1.02634
\(55\) 0 0
\(56\) 1.09327e9 1.98513
\(57\) 0 0
\(58\) 1.22175e9 1.86141
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 7.55200e8 0.971193
\(61\) −1.04159e9 −1.23324 −0.616621 − 0.787260i \(-0.711499\pi\)
−0.616621 + 0.787260i \(0.711499\pi\)
\(62\) 0 0
\(63\) 1.11869e8 0.112722
\(64\) 1.07374e9 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.34324e9 −1.73558 −0.867788 − 0.496935i \(-0.834459\pi\)
−0.867788 + 0.496935i \(0.834459\pi\)
\(68\) 0 0
\(69\) −2.76017e8 −0.176478
\(70\) −3.33640e9 −1.98513
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.09871e8 0.0567833
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −2.30469e9 −0.971193
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −3.27680e9 −1.00000
\(81\) −3.27755e9 −0.939992
\(82\) 6.75290e9 1.82147
\(83\) −5.44916e9 −1.38337 −0.691686 − 0.722198i \(-0.743132\pi\)
−0.691686 + 0.722198i \(0.743132\pi\)
\(84\) 8.06288e9 1.92794
\(85\) 0 0
\(86\) −7.15723e9 −1.52143
\(87\) 9.01041e9 1.80779
\(88\) 0 0
\(89\) 1.11182e10 1.99106 0.995529 − 0.0944520i \(-0.0301099\pi\)
0.995529 + 0.0944520i \(0.0301099\pi\)
\(90\) −3.35300e8 −0.0567833
\(91\) 0 0
\(92\) 1.19763e9 0.181713
\(93\) 0 0
\(94\) 3.10041e9 0.422454
\(95\) 0 0
\(96\) 7.91885e9 0.971193
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −2.65818e10 −2.94072
\(99\) 0 0
\(100\) 1.00000e10 1.00000
\(101\) −2.11340e9 −0.201083 −0.100541 − 0.994933i \(-0.532057\pi\)
−0.100541 + 0.994933i \(0.532057\pi\)
\(102\) 0 0
\(103\) 2.04075e10 1.76037 0.880186 − 0.474630i \(-0.157418\pi\)
0.880186 + 0.474630i \(0.157418\pi\)
\(104\) 0 0
\(105\) −2.46060e10 −1.92794
\(106\) 0 0
\(107\) −1.44091e9 −0.102735 −0.0513675 − 0.998680i \(-0.516358\pi\)
−0.0513675 + 0.998680i \(0.516358\pi\)
\(108\) 1.50803e10 1.02634
\(109\) 2.36396e10 1.53641 0.768206 − 0.640202i \(-0.221149\pi\)
0.768206 + 0.640202i \(0.221149\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.49847e10 −1.98513
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −3.65489e9 −0.181713
\(116\) −3.90960e10 −1.86141
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −2.41664e10 −0.971193
\(121\) 2.59374e10 1.00000
\(122\) 3.33309e10 1.23324
\(123\) 4.98026e10 1.76900
\(124\) 0 0
\(125\) −3.05176e10 −1.00000
\(126\) −3.57982e9 −0.112722
\(127\) −2.09654e10 −0.634576 −0.317288 − 0.948329i \(-0.602772\pi\)
−0.317288 + 0.948329i \(0.602772\pi\)
\(128\) −3.43597e10 −1.00000
\(129\) −5.27846e10 −1.47760
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 7.49838e10 1.73558
\(135\) −4.60215e10 −1.02634
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 8.83255e9 0.176478
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1.06765e11 1.98513
\(141\) 2.28655e10 0.410285
\(142\) 0 0
\(143\) 0 0
\(144\) −3.51588e9 −0.0567833
\(145\) 1.19312e11 1.86141
\(146\) 0 0
\(147\) −1.96041e11 −2.85601
\(148\) 0 0
\(149\) −3.93847e10 −0.536286 −0.268143 − 0.963379i \(-0.586410\pi\)
−0.268143 + 0.963379i \(0.586410\pi\)
\(150\) 7.37500e10 0.971193
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.04858e11 1.00000
\(161\) −3.90213e10 −0.360722
\(162\) 1.04882e11 0.939992
\(163\) 1.11513e11 0.969138 0.484569 − 0.874753i \(-0.338976\pi\)
0.484569 + 0.874753i \(0.338976\pi\)
\(164\) −2.16093e11 −1.82147
\(165\) 0 0
\(166\) 1.74373e11 1.38337
\(167\) 2.11909e11 1.63143 0.815713 − 0.578456i \(-0.196345\pi\)
0.815713 + 0.578456i \(0.196345\pi\)
\(168\) −2.58012e11 −1.92794
\(169\) 1.37858e11 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 2.29031e11 1.52143
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) −2.88333e11 −1.80779
\(175\) −3.25820e11 −1.98513
\(176\) 0 0
\(177\) 0 0
\(178\) −3.55782e11 −1.99106
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 1.07296e10 0.0567833
\(181\) −2.85871e11 −1.47156 −0.735779 − 0.677222i \(-0.763184\pi\)
−0.735779 + 0.677222i \(0.763184\pi\)
\(182\) 0 0
\(183\) 2.45816e11 1.19772
\(184\) −3.83243e10 −0.181713
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −9.92131e10 −0.422454
\(189\) −4.91347e11 −2.03742
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −2.53403e11 −0.971193
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 8.50618e11 2.94072
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −3.20000e11 −1.00000
\(201\) 5.53006e11 1.68558
\(202\) 6.76288e10 0.201083
\(203\) 1.27383e12 3.69514
\(204\) 0 0
\(205\) 6.59463e11 1.82147
\(206\) −6.53041e11 −1.76037
\(207\) −3.92155e9 −0.0103182
\(208\) 0 0
\(209\) 0 0
\(210\) 7.87390e11 1.92794
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 4.61092e10 0.102735
\(215\) −6.98948e11 −1.52143
\(216\) −4.82570e11 −1.02634
\(217\) 0 0
\(218\) −7.56468e11 −1.53641
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −5.38432e11 −0.976353 −0.488176 − 0.872745i \(-0.662338\pi\)
−0.488176 + 0.872745i \(0.662338\pi\)
\(224\) 1.11951e12 1.98513
\(225\) −3.27441e10 −0.0567833
\(226\) 0 0
\(227\) −1.18003e12 −1.95778 −0.978891 − 0.204385i \(-0.934481\pi\)
−0.978891 + 0.204385i \(0.934481\pi\)
\(228\) 0 0
\(229\) −1.21435e11 −0.192826 −0.0964129 − 0.995341i \(-0.530737\pi\)
−0.0964129 + 0.995341i \(0.530737\pi\)
\(230\) 1.16956e11 0.181713
\(231\) 0 0
\(232\) 1.25107e12 1.86141
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 3.02774e11 0.422454
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 7.73325e11 0.971193
\(241\) −3.92876e11 −0.483249 −0.241624 − 0.970370i \(-0.577680\pi\)
−0.241624 + 0.970370i \(0.577680\pi\)
\(242\) −8.29998e11 −1.00000
\(243\) −9.61051e10 −0.113427
\(244\) −1.06659e12 −1.23324
\(245\) −2.59588e12 −2.94072
\(246\) −1.59368e12 −1.76900
\(247\) 0 0
\(248\) 0 0
\(249\) 1.28600e12 1.34352
\(250\) 9.76562e11 1.00000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.14554e11 0.112722
\(253\) 0 0
\(254\) 6.70892e11 0.634576
\(255\) 0 0
\(256\) 1.09951e12 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 1.68911e12 1.47760
\(259\) 0 0
\(260\) 0 0
\(261\) 1.28017e11 0.105697
\(262\) 0 0
\(263\) 7.19278e11 0.571634 0.285817 − 0.958284i \(-0.407735\pi\)
0.285817 + 0.958284i \(0.407735\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.62389e12 −1.93370
\(268\) −2.39948e12 −1.73558
\(269\) 9.75088e11 0.692281 0.346140 − 0.938183i \(-0.387492\pi\)
0.346140 + 0.938183i \(0.387492\pi\)
\(270\) 1.47269e12 1.02634
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −2.82642e11 −0.176478
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −3.41647e12 −1.98513
\(281\) 3.04272e12 1.73672 0.868361 − 0.495932i \(-0.165174\pi\)
0.868361 + 0.495932i \(0.165174\pi\)
\(282\) −7.31696e11 −0.410285
\(283\) −6.49577e11 −0.357848 −0.178924 − 0.983863i \(-0.557262\pi\)
−0.178924 + 0.983863i \(0.557262\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.04074e12 3.61584
\(288\) 1.12508e11 0.0567833
\(289\) 2.01599e12 1.00000
\(290\) −3.81797e12 −1.86141
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 6.27330e12 2.85601
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 1.26031e12 0.536286
\(299\) 0 0
\(300\) −2.36000e12 −0.971193
\(301\) −7.46230e12 −3.02023
\(302\) 0 0
\(303\) 4.98763e11 0.195290
\(304\) 0 0
\(305\) 3.25497e12 1.23324
\(306\) 0 0
\(307\) 1.90661e12 0.699150 0.349575 − 0.936908i \(-0.386326\pi\)
0.349575 + 0.936908i \(0.386326\pi\)
\(308\) 0 0
\(309\) −4.81618e12 −1.70966
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −3.49592e11 −0.112722
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.35544e12 −1.00000
\(321\) 3.40055e11 0.0997756
\(322\) 1.24868e12 0.360722
\(323\) 0 0
\(324\) −3.35621e12 −0.939992
\(325\) 0 0
\(326\) −3.56840e12 −0.969138
\(327\) −5.57895e12 −1.49215
\(328\) 6.91497e12 1.82147
\(329\) 3.23256e12 0.838625
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −5.57994e12 −1.38337
\(333\) 0 0
\(334\) −6.78110e12 −1.63143
\(335\) 7.32264e12 1.73558
\(336\) 8.25639e12 1.92794
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −4.41147e12 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.82903e13 −3.85258
\(344\) −7.32900e12 −1.52143
\(345\) 8.62553e11 0.176478
\(346\) 0 0
\(347\) 7.49030e12 1.48885 0.744427 − 0.667704i \(-0.232723\pi\)
0.744427 + 0.667704i \(0.232723\pi\)
\(348\) 9.22666e12 1.80779
\(349\) −1.62542e12 −0.313935 −0.156967 − 0.987604i \(-0.550172\pi\)
−0.156967 + 0.987604i \(0.550172\pi\)
\(350\) 1.04262e13 1.98513
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.13850e13 1.99106
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −3.43347e11 −0.0567833
\(361\) 6.13107e12 1.00000
\(362\) 9.14787e12 1.47156
\(363\) −6.12123e12 −0.971193
\(364\) 0 0
\(365\) 0 0
\(366\) −7.86610e12 −1.19772
\(367\) −9.02278e12 −1.35522 −0.677611 − 0.735421i \(-0.736984\pi\)
−0.677611 + 0.735421i \(0.736984\pi\)
\(368\) 1.22638e12 0.181713
\(369\) 7.07577e11 0.103429
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 7.20215e12 0.971193
\(376\) 3.17482e12 0.422454
\(377\) 0 0
\(378\) 1.57231e13 2.03742
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 4.94783e12 0.616296
\(382\) 0 0
\(383\) 4.67157e12 0.566851 0.283426 − 0.958994i \(-0.408529\pi\)
0.283426 + 0.958994i \(0.408529\pi\)
\(384\) 8.10890e12 0.971193
\(385\) 0 0
\(386\) 0 0
\(387\) −7.49943e11 −0.0863920
\(388\) 0 0
\(389\) −1.72271e13 −1.93403 −0.967017 − 0.254711i \(-0.918020\pi\)
−0.967017 + 0.254711i \(0.918020\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.72198e13 −2.94072
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.02400e13 1.00000
\(401\) 1.05759e12 0.101999 0.0509995 − 0.998699i \(-0.483759\pi\)
0.0509995 + 0.998699i \(0.483759\pi\)
\(402\) −1.76962e13 −1.68558
\(403\) 0 0
\(404\) −2.16412e12 −0.201083
\(405\) 1.02423e13 0.939992
\(406\) −4.07625e13 −3.69514
\(407\) 0 0
\(408\) 0 0
\(409\) −1.25489e13 −1.09645 −0.548225 − 0.836331i \(-0.684696\pi\)
−0.548225 + 0.836331i \(0.684696\pi\)
\(410\) −2.11028e13 −1.82147
\(411\) 0 0
\(412\) 2.08973e13 1.76037
\(413\) 0 0
\(414\) 1.25490e11 0.0103182
\(415\) 1.70286e13 1.38337
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −2.51965e13 −1.92794
\(421\) 1.00869e13 0.762686 0.381343 − 0.924434i \(-0.375462\pi\)
0.381343 + 0.924434i \(0.375462\pi\)
\(422\) 0 0
\(423\) 3.24865e11 0.0239884
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.47517e13 2.44814
\(428\) −1.47549e12 −0.102735
\(429\) 0 0
\(430\) 2.23663e13 1.52143
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.54422e13 1.02634
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −2.81575e13 −1.80779
\(436\) 2.42070e13 1.53641
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −2.78527e12 −0.166984
\(442\) 0 0
\(443\) 2.20452e13 1.29210 0.646048 − 0.763297i \(-0.276420\pi\)
0.646048 + 0.763297i \(0.276420\pi\)
\(444\) 0 0
\(445\) −3.47443e13 −1.99106
\(446\) 1.72298e13 0.976353
\(447\) 9.29480e12 0.520838
\(448\) −3.58243e13 −1.98513
\(449\) 2.73166e13 1.49691 0.748453 − 0.663188i \(-0.230797\pi\)
0.748453 + 0.663188i \(0.230797\pi\)
\(450\) 1.04781e12 0.0567833
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 3.77610e13 1.95778
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 3.88591e12 0.192826
\(459\) 0 0
\(460\) −3.74260e12 −0.181713
\(461\) −2.09598e13 −1.00666 −0.503329 − 0.864095i \(-0.667892\pi\)
−0.503329 + 0.864095i \(0.667892\pi\)
\(462\) 0 0
\(463\) −4.21425e13 −1.98068 −0.990341 − 0.138652i \(-0.955723\pi\)
−0.990341 + 0.138652i \(0.955723\pi\)
\(464\) −4.00343e13 −1.86141
\(465\) 0 0
\(466\) 0 0
\(467\) −2.74392e13 −1.23534 −0.617671 − 0.786437i \(-0.711923\pi\)
−0.617671 + 0.786437i \(0.711923\pi\)
\(468\) 0 0
\(469\) 7.81800e13 3.44533
\(470\) −9.68878e12 −0.422454
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −2.47464e13 −0.971193
\(481\) 0 0
\(482\) 1.25720e13 0.483249
\(483\) 9.20903e12 0.350331
\(484\) 2.65599e13 1.00000
\(485\) 0 0
\(486\) 3.07536e12 0.113427
\(487\) −2.82318e13 −1.03061 −0.515305 − 0.857007i \(-0.672321\pi\)
−0.515305 + 0.857007i \(0.672321\pi\)
\(488\) 3.41309e13 1.23324
\(489\) −2.63170e13 −0.941221
\(490\) 8.30681e13 2.94072
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 5.09979e13 1.76900
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −4.11520e13 −1.34352
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −3.12500e13 −1.00000
\(501\) −5.00106e13 −1.58443
\(502\) 0 0
\(503\) 3.42225e13 1.06285 0.531425 − 0.847105i \(-0.321657\pi\)
0.531425 + 0.847105i \(0.321657\pi\)
\(504\) −3.66574e12 −0.112722
\(505\) 6.60438e12 0.201083
\(506\) 0 0
\(507\) −3.25346e13 −0.971193
\(508\) −2.14685e13 −0.634576
\(509\) −1.60513e13 −0.469808 −0.234904 − 0.972019i \(-0.575477\pi\)
−0.234904 + 0.972019i \(0.575477\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.51844e13 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −6.37735e13 −1.76037
\(516\) −5.40514e13 −1.47760
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.86275e13 −1.26676 −0.633379 − 0.773842i \(-0.718333\pi\)
−0.633379 + 0.773842i \(0.718333\pi\)
\(522\) −4.09653e12 −0.105697
\(523\) 5.54369e13 1.41674 0.708370 − 0.705841i \(-0.249431\pi\)
0.708370 + 0.705841i \(0.249431\pi\)
\(524\) 0 0
\(525\) 7.68936e13 1.92794
\(526\) −2.30169e13 −0.571634
\(527\) 0 0
\(528\) 0 0
\(529\) −4.00586e13 −0.966981
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 8.39646e13 1.93370
\(535\) 4.50285e12 0.102735
\(536\) 7.67834e13 1.73558
\(537\) 0 0
\(538\) −3.12028e13 −0.692281
\(539\) 0 0
\(540\) −4.71260e13 −1.02634
\(541\) 9.18441e13 1.98182 0.990911 − 0.134516i \(-0.0429479\pi\)
0.990911 + 0.134516i \(0.0429479\pi\)
\(542\) 0 0
\(543\) 6.74656e13 1.42917
\(544\) 0 0
\(545\) −7.38738e13 −1.53641
\(546\) 0 0
\(547\) −7.63679e13 −1.55946 −0.779730 − 0.626116i \(-0.784644\pi\)
−0.779730 + 0.626116i \(0.784644\pi\)
\(548\) 0 0
\(549\) 3.49246e12 0.0700276
\(550\) 0 0
\(551\) 0 0
\(552\) 9.04453e12 0.176478
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.09327e14 1.98513
\(561\) 0 0
\(562\) −9.73670e13 −1.73672
\(563\) 1.08141e14 1.91182 0.955910 − 0.293661i \(-0.0948736\pi\)
0.955910 + 0.293661i \(0.0948736\pi\)
\(564\) 2.34143e13 0.410285
\(565\) 0 0
\(566\) 2.07865e13 0.357848
\(567\) 1.09352e14 1.86600
\(568\) 0 0
\(569\) 7.65221e13 1.28300 0.641499 − 0.767124i \(-0.278313\pi\)
0.641499 + 0.767124i \(0.278313\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2.25304e14 −3.61584
\(575\) 1.14215e13 0.181713
\(576\) −3.60026e12 −0.0567833
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −6.45118e13 −1.00000
\(579\) 0 0
\(580\) 1.22175e14 1.86141
\(581\) 1.81806e14 2.74617
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.51888e13 −0.935367 −0.467684 − 0.883896i \(-0.654911\pi\)
−0.467684 + 0.883896i \(0.654911\pi\)
\(588\) −2.00746e14 −2.85601
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.03300e13 −0.536286
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 7.55200e13 0.971193
\(601\) −1.53534e14 −1.95809 −0.979046 − 0.203641i \(-0.934723\pi\)
−0.979046 + 0.203641i \(0.934723\pi\)
\(602\) 2.38794e14 3.02023
\(603\) 7.85690e12 0.0985518
\(604\) 0 0
\(605\) −8.10545e13 −1.00000
\(606\) −1.59604e13 −0.195290
\(607\) 1.63520e14 1.98439 0.992196 − 0.124690i \(-0.0397937\pi\)
0.992196 + 0.124690i \(0.0397937\pi\)
\(608\) 0 0
\(609\) −3.00623e14 −3.58869
\(610\) −1.04159e14 −1.23324
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −6.10116e13 −0.699150
\(615\) −1.55633e14 −1.76900
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 1.54118e14 1.70966
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 1.72240e13 0.186499
\(622\) 0 0
\(623\) −3.70947e14 −3.95250
\(624\) 0 0
\(625\) 9.53674e13 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 1.11869e13 0.112722
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.55168e13 0.634576
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 1.07374e14 1.00000
\(641\) 1.58512e14 1.46478 0.732389 − 0.680886i \(-0.238405\pi\)
0.732389 + 0.680886i \(0.238405\pi\)
\(642\) −1.08818e13 −0.0997756
\(643\) 2.19749e14 1.99927 0.999635 − 0.0270190i \(-0.00860148\pi\)
0.999635 + 0.0270190i \(0.00860148\pi\)
\(644\) −3.99578e13 −0.360722
\(645\) 1.64952e14 1.47760
\(646\) 0 0
\(647\) −1.92598e14 −1.69876 −0.849378 − 0.527785i \(-0.823023\pi\)
−0.849378 + 0.527785i \(0.823023\pi\)
\(648\) 1.07399e14 0.939992
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.14189e14 0.969138
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 1.78526e14 1.49215
\(655\) 0 0
\(656\) −2.21279e14 −1.82147
\(657\) 0 0
\(658\) −1.03442e14 −0.838625
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −2.28979e14 −1.81463 −0.907314 − 0.420454i \(-0.861871\pi\)
−0.907314 + 0.420454i \(0.861871\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.78558e14 1.38337
\(665\) 0 0
\(666\) 0 0
\(667\) −4.46536e13 −0.338242
\(668\) 2.16995e14 1.63143
\(669\) 1.27070e14 0.948227
\(670\) −2.34324e14 −1.73558
\(671\) 0 0
\(672\) −2.64204e14 −1.92794
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.43817e14 1.02634
\(676\) 1.41167e14 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 2.78487e14 1.90138
\(682\) 0 0
\(683\) −2.57196e14 −1.73045 −0.865227 − 0.501380i \(-0.832826\pi\)
−0.865227 + 0.501380i \(0.832826\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.85291e14 3.85258
\(687\) 2.86586e13 0.187271
\(688\) 2.34528e14 1.52143
\(689\) 0 0
\(690\) −2.76017e13 −0.176478
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −2.39690e14 −1.48885
\(695\) 0 0
\(696\) −2.95253e14 −1.80779
\(697\) 0 0
\(698\) 5.20136e13 0.313935
\(699\) 0 0
\(700\) −3.33640e14 −1.98513
\(701\) −1.16992e14 −0.691140 −0.345570 − 0.938393i \(-0.612314\pi\)
−0.345570 + 0.938393i \(0.612314\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −7.14547e13 −0.410285
\(706\) 0 0
\(707\) 7.05115e13 0.399175
\(708\) 0 0
\(709\) 1.92835e14 1.07635 0.538176 − 0.842832i \(-0.319114\pi\)
0.538176 + 0.842832i \(0.319114\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.64321e14 −1.99106
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.09871e13 0.0567833
\(721\) −6.80877e14 −3.49456
\(722\) −1.96194e14 −1.00000
\(723\) 9.27188e13 0.469328
\(724\) −2.92732e14 −1.47156
\(725\) −3.72849e14 −1.86141
\(726\) 1.95879e14 0.971193
\(727\) 2.86578e14 1.41114 0.705571 − 0.708639i \(-0.250691\pi\)
0.705571 + 0.708639i \(0.250691\pi\)
\(728\) 0 0
\(729\) 2.16217e14 1.05015
\(730\) 0 0
\(731\) 0 0
\(732\) 2.51715e14 1.19772
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 2.88729e14 1.35522
\(735\) 6.12627e14 2.85601
\(736\) −3.92441e13 −0.181713
\(737\) 0 0
\(738\) −2.26425e13 −0.103429
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.93558e14 0.854803 0.427401 − 0.904062i \(-0.359429\pi\)
0.427401 + 0.904062i \(0.359429\pi\)
\(744\) 0 0
\(745\) 1.23077e14 0.536286
\(746\) 0 0
\(747\) 1.82710e13 0.0785525
\(748\) 0 0
\(749\) 4.80746e13 0.203942
\(750\) −2.30469e14 −0.971193
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −1.01594e14 −0.422454
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −5.03140e14 −2.03742
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.65603e14 1.43247 0.716237 − 0.697857i \(-0.245863\pi\)
0.716237 + 0.697857i \(0.245863\pi\)
\(762\) −1.58330e14 −0.616296
\(763\) −7.88712e14 −3.04997
\(764\) 0 0
\(765\) 0 0
\(766\) −1.49490e14 −0.566851
\(767\) 0 0
\(768\) −2.59485e14 −0.971193
\(769\) 4.47005e14 1.66219 0.831096 − 0.556130i \(-0.187714\pi\)
0.831096 + 0.556130i \(0.187714\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 2.39982e13 0.0863920
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 5.51267e14 1.93403
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5.62268e14 −1.91044
\(784\) 8.71032e14 2.94072
\(785\) 0 0
\(786\) 0 0
\(787\) 2.17686e14 0.721034 0.360517 − 0.932753i \(-0.382600\pi\)
0.360517 + 0.932753i \(0.382600\pi\)
\(788\) 0 0
\(789\) −1.69750e14 −0.555167
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −3.27680e14 −1.00000
\(801\) −3.72793e13 −0.113059
\(802\) −3.38429e13 −0.101999
\(803\) 0 0
\(804\) 5.66278e14 1.68558
\(805\) 1.21942e14 0.360722
\(806\) 0 0
\(807\) −2.30121e14 −0.672339
\(808\) 6.92519e13 0.201083
\(809\) −6.91924e14 −1.99671 −0.998357 − 0.0572940i \(-0.981753\pi\)
−0.998357 + 0.0572940i \(0.981753\pi\)
\(810\) −3.27755e14 −0.939992
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1.30440e15 3.69514
\(813\) 0 0
\(814\) 0 0
\(815\) −3.48477e14 −0.969138
\(816\) 0 0
\(817\) 0 0
\(818\) 4.01565e14 1.09645
\(819\) 0 0
\(820\) 6.75290e14 1.82147
\(821\) 6.53223e14 1.75124 0.875620 − 0.483000i \(-0.160453\pi\)
0.875620 + 0.483000i \(0.160453\pi\)
\(822\) 0 0
\(823\) 7.09435e14 1.87894 0.939471 − 0.342628i \(-0.111317\pi\)
0.939471 + 0.342628i \(0.111317\pi\)
\(824\) −6.68714e14 −1.76037
\(825\) 0 0
\(826\) 0 0
\(827\) 7.45155e14 1.92628 0.963139 − 0.269003i \(-0.0866942\pi\)
0.963139 + 0.269003i \(0.0866942\pi\)
\(828\) −4.01567e12 −0.0103182
\(829\) −5.51092e14 −1.40751 −0.703755 − 0.710442i \(-0.748495\pi\)
−0.703755 + 0.710442i \(0.748495\pi\)
\(830\) −5.44916e14 −1.38337
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −6.62216e14 −1.63143
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 8.06288e14 1.92794
\(841\) 1.03698e15 2.46486
\(842\) −3.22780e14 −0.762686
\(843\) −7.18082e14 −1.68669
\(844\) 0 0
\(845\) −4.30808e14 −1.00000
\(846\) −1.03957e13 −0.0239884
\(847\) −8.65376e14 −1.98513
\(848\) 0 0
\(849\) 1.53300e14 0.347540
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) −1.11205e15 −2.44814
\(855\) 0 0
\(856\) 4.72158e13 0.102735
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −7.15723e14 −1.52143
\(861\) −1.66161e15 −3.51168
\(862\) 0 0
\(863\) 4.87873e13 0.101918 0.0509592 − 0.998701i \(-0.483772\pi\)
0.0509592 + 0.998701i \(0.483772\pi\)
\(864\) −4.94152e14 −1.02634
\(865\) 0 0
\(866\) 0 0
\(867\) −4.75775e14 −0.971193
\(868\) 0 0
\(869\) 0 0
\(870\) 9.01041e14 1.80779
\(871\) 0 0
\(872\) −7.74623e14 −1.53641
\(873\) 0 0
\(874\) 0 0
\(875\) 1.01819e15 1.98513
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.75687e14 0.896276 0.448138 − 0.893964i \(-0.352087\pi\)
0.448138 + 0.893964i \(0.352087\pi\)
\(882\) 8.91288e13 0.166984
\(883\) 8.38106e14 1.56133 0.780666 − 0.624948i \(-0.214880\pi\)
0.780666 + 0.624948i \(0.214880\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −7.05446e14 −1.29210
\(887\) −2.92007e14 −0.531833 −0.265917 − 0.963996i \(-0.585675\pi\)
−0.265917 + 0.963996i \(0.585675\pi\)
\(888\) 0 0
\(889\) 6.99488e14 1.25971
\(890\) 1.11182e15 1.99106
\(891\) 0 0
\(892\) −5.51355e14 −0.976353
\(893\) 0 0
\(894\) −2.97434e14 −0.520838
\(895\) 0 0
\(896\) 1.14638e15 1.98513
\(897\) 0 0
\(898\) −8.74131e14 −1.49691
\(899\) 0 0
\(900\) −3.35300e13 −0.0567833
\(901\) 0 0
\(902\) 0 0
\(903\) 1.76110e15 2.93323
\(904\) 0 0
\(905\) 8.93347e14 1.47156
\(906\) 0 0
\(907\) 9.35009e14 1.52328 0.761640 − 0.648001i \(-0.224395\pi\)
0.761640 + 0.648001i \(0.224395\pi\)
\(908\) −1.20835e15 −1.95778
\(909\) 7.08623e12 0.0114182
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −7.68174e14 −1.19772
\(916\) −1.24349e14 −0.192826
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 1.19763e14 0.181713
\(921\) −4.49960e14 −0.679010
\(922\) 6.70713e14 1.00666
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1.34856e15 1.98068
\(927\) −6.84264e13 −0.0999598
\(928\) 1.28110e15 1.86141
\(929\) −1.38310e15 −1.99882 −0.999410 − 0.0343444i \(-0.989066\pi\)
−0.999410 + 0.0343444i \(0.989066\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 8.78054e14 1.23534
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −2.50176e15 −3.44533
\(939\) 0 0
\(940\) 3.10041e14 0.422454
\(941\) −4.55354e14 −0.617164 −0.308582 − 0.951198i \(-0.599854\pi\)
−0.308582 + 0.951198i \(0.599854\pi\)
\(942\) 0 0
\(943\) −2.46811e14 −0.330983
\(944\) 0 0
\(945\) 1.53546e15 2.03742
\(946\) 0 0
\(947\) −3.53721e13 −0.0464421 −0.0232210 − 0.999730i \(-0.507392\pi\)
−0.0232210 + 0.999730i \(0.507392\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 7.91885e14 0.971193
\(961\) 8.19628e14 1.00000
\(962\) 0 0
\(963\) 4.83138e12 0.00583364
\(964\) −4.02305e14 −0.483249
\(965\) 0 0
\(966\) −2.94689e14 −0.350331
\(967\) −9.99703e14 −1.18233 −0.591165 − 0.806551i \(-0.701332\pi\)
−0.591165 + 0.806551i \(0.701332\pi\)
\(968\) −8.49918e14 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −9.84117e13 −0.113427
\(973\) 0 0
\(974\) 9.03419e14 1.03061
\(975\) 0 0
\(976\) −1.09219e15 −1.23324
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 8.42143e14 0.941221
\(979\) 0 0
\(980\) −2.65818e15 −2.94072
\(981\) −7.92636e13 −0.0872427
\(982\) 0 0
\(983\) 1.21705e15 1.32599 0.662997 − 0.748622i \(-0.269284\pi\)
0.662997 + 0.748622i \(0.269284\pi\)
\(984\) −1.63193e15 −1.76900
\(985\) 0 0
\(986\) 0 0
\(987\) −7.62885e14 −0.814467
\(988\) 0 0
\(989\) 2.61589e14 0.276463
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 1.31687e15 1.34352
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 20.11.d.a.19.1 ✓ 1
4.3 odd 2 20.11.d.b.19.1 yes 1
5.2 odd 4 100.11.b.c.51.1 2
5.3 odd 4 100.11.b.c.51.2 2
5.4 even 2 20.11.d.b.19.1 yes 1
20.3 even 4 100.11.b.c.51.1 2
20.7 even 4 100.11.b.c.51.2 2
20.19 odd 2 CM 20.11.d.a.19.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.11.d.a.19.1 ✓ 1 1.1 even 1 trivial
20.11.d.a.19.1 ✓ 1 20.19 odd 2 CM
20.11.d.b.19.1 yes 1 4.3 odd 2
20.11.d.b.19.1 yes 1 5.4 even 2
100.11.b.c.51.1 2 5.2 odd 4
100.11.b.c.51.1 2 20.3 even 4
100.11.b.c.51.2 2 5.3 odd 4
100.11.b.c.51.2 2 20.7 even 4