Properties

Label 16.13.c.a.15.1
Level $16$
Weight $13$
Character 16.15
Analytic conductor $14.624$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,13,Mod(15,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.15");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 16.c (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: no
Analytic conductor: \(14.6239010764\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1155}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.1
Root \(0.500000 + 16.9926i\) of defining polynomial
Character \(\chi\) \(=\) 16.15
Dual form 16.13.c.a.15.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-815.647i q^{3} -270.000 q^{5} -181074. i q^{7} -133839. q^{9} +O(q^{10})\) \(q-815.647i q^{3} -270.000 q^{5} -181074. i q^{7} -133839. q^{9} +1.65169e6i q^{11} -4.38635e6 q^{13} +220225. i q^{15} -3.93836e7 q^{17} -5.42609e7i q^{19} -1.47692e8 q^{21} +2.40353e8i q^{23} -2.44068e8 q^{25} -3.24303e8i q^{27} +1.63561e8 q^{29} -2.31481e8i q^{31} +1.34719e9 q^{33} +4.88899e7i q^{35} +3.60002e9 q^{37} +3.57771e9i q^{39} +2.12486e9 q^{41} -2.60465e9i q^{43} +3.61365e7 q^{45} -1.75816e10i q^{47} -1.89464e10 q^{49} +3.21231e10i q^{51} -1.35853e10 q^{53} -4.45955e8i q^{55} -4.42578e10 q^{57} -2.48727e10i q^{59} +3.54966e10 q^{61} +2.42347e10i q^{63} +1.18431e9 q^{65} -1.21936e11i q^{67} +1.96043e11 q^{69} +1.24791e11i q^{71} -5.98227e9 q^{73} +1.99073e11i q^{75} +2.99077e11 q^{77} -1.21589e11i q^{79} -3.35644e11 q^{81} -3.97309e11i q^{83} +1.06336e10 q^{85} -1.33408e11i q^{87} -7.53989e11 q^{89} +7.94252e11i q^{91} -1.88806e11 q^{93} +1.46504e10i q^{95} -9.70604e11 q^{97} -2.21060e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 540 q^{5} - 267678 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 540 q^{5} - 267678 q^{9} - 8772700 q^{13} - 78767100 q^{17} - 295384320 q^{21} - 488135450 q^{25} + 327121956 q^{29} + 2694384000 q^{33} + 7200048100 q^{37} + 4249729476 q^{41} + 72273060 q^{45} - 37892744638 q^{49} - 27170502300 q^{53} - 88515504000 q^{57} + 70993108516 q^{61} + 2368629000 q^{65} + 392086759680 q^{69} - 11964538300 q^{73} + 598153248000 q^{77} - 671288381118 q^{81} + 21267117000 q^{85} - 1507978573884 q^{89} - 377612928000 q^{93} - 1941207691900 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 815.647i − 1.11886i −0.828879 0.559429i \(-0.811021\pi\)
0.828879 0.559429i \(-0.188979\pi\)
\(4\) 0 0
\(5\) −270.000 −0.0172800 −0.00864000 0.999963i \(-0.502750\pi\)
−0.00864000 + 0.999963i \(0.502750\pi\)
\(6\) 0 0
\(7\) − 181074.i − 1.53910i −0.638586 0.769550i \(-0.720480\pi\)
0.638586 0.769550i \(-0.279520\pi\)
\(8\) 0 0
\(9\) −133839. −0.251842
\(10\) 0 0
\(11\) 1.65169e6i 0.932333i 0.884697 + 0.466167i \(0.154365\pi\)
−0.884697 + 0.466167i \(0.845635\pi\)
\(12\) 0 0
\(13\) −4.38635e6 −0.908747 −0.454374 0.890811i \(-0.650137\pi\)
−0.454374 + 0.890811i \(0.650137\pi\)
\(14\) 0 0
\(15\) 220225.i 0.0193339i
\(16\) 0 0
\(17\) −3.93836e7 −1.63163 −0.815814 0.578314i \(-0.803711\pi\)
−0.815814 + 0.578314i \(0.803711\pi\)
\(18\) 0 0
\(19\) − 5.42609e7i − 1.15336i −0.816970 0.576681i \(-0.804348\pi\)
0.816970 0.576681i \(-0.195652\pi\)
\(20\) 0 0
\(21\) −1.47692e8 −1.72203
\(22\) 0 0
\(23\) 2.40353e8i 1.62361i 0.583925 + 0.811807i \(0.301516\pi\)
−0.583925 + 0.811807i \(0.698484\pi\)
\(24\) 0 0
\(25\) −2.44068e8 −0.999701
\(26\) 0 0
\(27\) − 3.24303e8i − 0.837082i
\(28\) 0 0
\(29\) 1.63561e8 0.274974 0.137487 0.990504i \(-0.456097\pi\)
0.137487 + 0.990504i \(0.456097\pi\)
\(30\) 0 0
\(31\) − 2.31481e8i − 0.260822i −0.991460 0.130411i \(-0.958370\pi\)
0.991460 0.130411i \(-0.0416297\pi\)
\(32\) 0 0
\(33\) 1.34719e9 1.04315
\(34\) 0 0
\(35\) 4.88899e7i 0.0265957i
\(36\) 0 0
\(37\) 3.60002e9 1.40312 0.701560 0.712610i \(-0.252487\pi\)
0.701560 + 0.712610i \(0.252487\pi\)
\(38\) 0 0
\(39\) 3.57771e9i 1.01676i
\(40\) 0 0
\(41\) 2.12486e9 0.447330 0.223665 0.974666i \(-0.428198\pi\)
0.223665 + 0.974666i \(0.428198\pi\)
\(42\) 0 0
\(43\) − 2.60465e9i − 0.412040i −0.978548 0.206020i \(-0.933949\pi\)
0.978548 0.206020i \(-0.0660512\pi\)
\(44\) 0 0
\(45\) 3.61365e7 0.00435182
\(46\) 0 0
\(47\) − 1.75816e10i − 1.63106i −0.578712 0.815532i \(-0.696444\pi\)
0.578712 0.815532i \(-0.303556\pi\)
\(48\) 0 0
\(49\) −1.89464e10 −1.36883
\(50\) 0 0
\(51\) 3.21231e10i 1.82556i
\(52\) 0 0
\(53\) −1.35853e10 −0.612932 −0.306466 0.951882i \(-0.599147\pi\)
−0.306466 + 0.951882i \(0.599147\pi\)
\(54\) 0 0
\(55\) − 4.45955e8i − 0.0161107i
\(56\) 0 0
\(57\) −4.42578e10 −1.29045
\(58\) 0 0
\(59\) − 2.48727e10i − 0.589673i −0.955548 0.294837i \(-0.904735\pi\)
0.955548 0.294837i \(-0.0952652\pi\)
\(60\) 0 0
\(61\) 3.54966e10 0.688981 0.344490 0.938790i \(-0.388052\pi\)
0.344490 + 0.938790i \(0.388052\pi\)
\(62\) 0 0
\(63\) 2.42347e10i 0.387610i
\(64\) 0 0
\(65\) 1.18431e9 0.0157032
\(66\) 0 0
\(67\) − 1.21936e11i − 1.34798i −0.738741 0.673989i \(-0.764579\pi\)
0.738741 0.673989i \(-0.235421\pi\)
\(68\) 0 0
\(69\) 1.96043e11 1.81659
\(70\) 0 0
\(71\) 1.24791e11i 0.974170i 0.873355 + 0.487085i \(0.161940\pi\)
−0.873355 + 0.487085i \(0.838060\pi\)
\(72\) 0 0
\(73\) −5.98227e9 −0.0395302 −0.0197651 0.999805i \(-0.506292\pi\)
−0.0197651 + 0.999805i \(0.506292\pi\)
\(74\) 0 0
\(75\) 1.99073e11i 1.11852i
\(76\) 0 0
\(77\) 2.99077e11 1.43495
\(78\) 0 0
\(79\) − 1.21589e11i − 0.500185i −0.968222 0.250093i \(-0.919539\pi\)
0.968222 0.250093i \(-0.0804611\pi\)
\(80\) 0 0
\(81\) −3.35644e11 −1.18842
\(82\) 0 0
\(83\) − 3.97309e11i − 1.21523i −0.794230 0.607617i \(-0.792126\pi\)
0.794230 0.607617i \(-0.207874\pi\)
\(84\) 0 0
\(85\) 1.06336e10 0.0281945
\(86\) 0 0
\(87\) − 1.33408e11i − 0.307657i
\(88\) 0 0
\(89\) −7.53989e11 −1.51714 −0.758569 0.651593i \(-0.774101\pi\)
−0.758569 + 0.651593i \(0.774101\pi\)
\(90\) 0 0
\(91\) 7.94252e11i 1.39865i
\(92\) 0 0
\(93\) −1.88806e11 −0.291823
\(94\) 0 0
\(95\) 1.46504e10i 0.0199301i
\(96\) 0 0
\(97\) −9.70604e11 −1.16523 −0.582615 0.812748i \(-0.697970\pi\)
−0.582615 + 0.812748i \(0.697970\pi\)
\(98\) 0 0
\(99\) − 2.21060e11i − 0.234800i
\(100\) 0 0
\(101\) 1.00753e12 0.949142 0.474571 0.880217i \(-0.342603\pi\)
0.474571 + 0.880217i \(0.342603\pi\)
\(102\) 0 0
\(103\) − 1.35534e9i − 0.00113507i −1.00000 0.000567536i \(-0.999819\pi\)
1.00000 0.000567536i \(-0.000180652\pi\)
\(104\) 0 0
\(105\) 3.98769e10 0.0297567
\(106\) 0 0
\(107\) 3.20319e11i 0.213442i 0.994289 + 0.106721i \(0.0340352\pi\)
−0.994289 + 0.106721i \(0.965965\pi\)
\(108\) 0 0
\(109\) 1.23486e12 0.736309 0.368154 0.929765i \(-0.379990\pi\)
0.368154 + 0.929765i \(0.379990\pi\)
\(110\) 0 0
\(111\) − 2.93635e12i − 1.56989i
\(112\) 0 0
\(113\) 3.02216e12 1.45160 0.725800 0.687905i \(-0.241470\pi\)
0.725800 + 0.687905i \(0.241470\pi\)
\(114\) 0 0
\(115\) − 6.48954e10i − 0.0280561i
\(116\) 0 0
\(117\) 5.87065e11 0.228860
\(118\) 0 0
\(119\) 7.13132e12i 2.51124i
\(120\) 0 0
\(121\) 4.10365e11 0.130755
\(122\) 0 0
\(123\) − 1.73314e12i − 0.500499i
\(124\) 0 0
\(125\) 1.31816e11 0.0345548
\(126\) 0 0
\(127\) − 9.15391e11i − 0.218165i −0.994033 0.109082i \(-0.965209\pi\)
0.994033 0.109082i \(-0.0347912\pi\)
\(128\) 0 0
\(129\) −2.12448e12 −0.461014
\(130\) 0 0
\(131\) − 7.42385e12i − 1.46893i −0.678646 0.734466i \(-0.737433\pi\)
0.678646 0.734466i \(-0.262567\pi\)
\(132\) 0 0
\(133\) −9.82522e12 −1.77514
\(134\) 0 0
\(135\) 8.75618e10i 0.0144648i
\(136\) 0 0
\(137\) 4.33044e12 0.654950 0.327475 0.944860i \(-0.393802\pi\)
0.327475 + 0.944860i \(0.393802\pi\)
\(138\) 0 0
\(139\) 7.77715e12i 1.07828i 0.842216 + 0.539140i \(0.181251\pi\)
−0.842216 + 0.539140i \(0.818749\pi\)
\(140\) 0 0
\(141\) −1.43404e13 −1.82493
\(142\) 0 0
\(143\) − 7.24487e12i − 0.847255i
\(144\) 0 0
\(145\) −4.41615e10 −0.00475155
\(146\) 0 0
\(147\) 1.54536e13i 1.53153i
\(148\) 0 0
\(149\) 7.12754e12 0.651361 0.325681 0.945480i \(-0.394407\pi\)
0.325681 + 0.945480i \(0.394407\pi\)
\(150\) 0 0
\(151\) − 8.27460e12i − 0.698048i −0.937114 0.349024i \(-0.886513\pi\)
0.937114 0.349024i \(-0.113487\pi\)
\(152\) 0 0
\(153\) 5.27105e12 0.410912
\(154\) 0 0
\(155\) 6.24998e10i 0.00450701i
\(156\) 0 0
\(157\) −5.93139e12 −0.396058 −0.198029 0.980196i \(-0.563454\pi\)
−0.198029 + 0.980196i \(0.563454\pi\)
\(158\) 0 0
\(159\) 1.10808e13i 0.685784i
\(160\) 0 0
\(161\) 4.35216e13 2.49891
\(162\) 0 0
\(163\) − 2.97063e13i − 1.58388i −0.610599 0.791940i \(-0.709071\pi\)
0.610599 0.791940i \(-0.290929\pi\)
\(164\) 0 0
\(165\) −3.63742e11 −0.0180256
\(166\) 0 0
\(167\) − 1.32583e13i − 0.611208i −0.952159 0.305604i \(-0.901142\pi\)
0.952159 0.305604i \(-0.0988584\pi\)
\(168\) 0 0
\(169\) −4.05802e12 −0.174178
\(170\) 0 0
\(171\) 7.26223e12i 0.290465i
\(172\) 0 0
\(173\) 2.75686e13 1.02834 0.514171 0.857688i \(-0.328100\pi\)
0.514171 + 0.857688i \(0.328100\pi\)
\(174\) 0 0
\(175\) 4.41942e13i 1.53864i
\(176\) 0 0
\(177\) −2.02874e13 −0.659760
\(178\) 0 0
\(179\) 3.86049e13i 1.17361i 0.809728 + 0.586805i \(0.199614\pi\)
−0.809728 + 0.586805i \(0.800386\pi\)
\(180\) 0 0
\(181\) −2.74367e13 −0.780297 −0.390148 0.920752i \(-0.627576\pi\)
−0.390148 + 0.920752i \(0.627576\pi\)
\(182\) 0 0
\(183\) − 2.89527e13i − 0.770871i
\(184\) 0 0
\(185\) −9.72006e11 −0.0242459
\(186\) 0 0
\(187\) − 6.50492e13i − 1.52122i
\(188\) 0 0
\(189\) −5.87227e13 −1.28835
\(190\) 0 0
\(191\) − 2.46619e12i − 0.0507956i −0.999677 0.0253978i \(-0.991915\pi\)
0.999677 0.0253978i \(-0.00808525\pi\)
\(192\) 0 0
\(193\) −7.51237e13 −1.45356 −0.726780 0.686871i \(-0.758984\pi\)
−0.726780 + 0.686871i \(0.758984\pi\)
\(194\) 0 0
\(195\) − 9.65983e11i − 0.0175696i
\(196\) 0 0
\(197\) 8.83661e13 1.51178 0.755890 0.654699i \(-0.227204\pi\)
0.755890 + 0.654699i \(0.227204\pi\)
\(198\) 0 0
\(199\) 4.96844e13i 0.800022i 0.916510 + 0.400011i \(0.130994\pi\)
−0.916510 + 0.400011i \(0.869006\pi\)
\(200\) 0 0
\(201\) −9.94567e13 −1.50820
\(202\) 0 0
\(203\) − 2.96166e13i − 0.423213i
\(204\) 0 0
\(205\) −5.73713e11 −0.00772986
\(206\) 0 0
\(207\) − 3.21686e13i − 0.408894i
\(208\) 0 0
\(209\) 8.96219e13 1.07532
\(210\) 0 0
\(211\) 1.28031e14i 1.45085i 0.688303 + 0.725423i \(0.258356\pi\)
−0.688303 + 0.725423i \(0.741644\pi\)
\(212\) 0 0
\(213\) 1.01786e14 1.08996
\(214\) 0 0
\(215\) 7.03256e11i 0.00712005i
\(216\) 0 0
\(217\) −4.19150e13 −0.401431
\(218\) 0 0
\(219\) 4.87942e12i 0.0442286i
\(220\) 0 0
\(221\) 1.72750e14 1.48274
\(222\) 0 0
\(223\) − 8.64588e13i − 0.703040i −0.936180 0.351520i \(-0.885665\pi\)
0.936180 0.351520i \(-0.114335\pi\)
\(224\) 0 0
\(225\) 3.26658e13 0.251766
\(226\) 0 0
\(227\) − 5.88647e12i − 0.0430229i −0.999769 0.0215115i \(-0.993152\pi\)
0.999769 0.0215115i \(-0.00684784\pi\)
\(228\) 0 0
\(229\) 9.06405e13 0.628506 0.314253 0.949339i \(-0.398246\pi\)
0.314253 + 0.949339i \(0.398246\pi\)
\(230\) 0 0
\(231\) − 2.43941e14i − 1.60551i
\(232\) 0 0
\(233\) −3.58282e13 −0.223918 −0.111959 0.993713i \(-0.535713\pi\)
−0.111959 + 0.993713i \(0.535713\pi\)
\(234\) 0 0
\(235\) 4.74703e12i 0.0281848i
\(236\) 0 0
\(237\) −9.91735e13 −0.559636
\(238\) 0 0
\(239\) 2.86133e14i 1.53525i 0.640897 + 0.767627i \(0.278562\pi\)
−0.640897 + 0.767627i \(0.721438\pi\)
\(240\) 0 0
\(241\) −5.33804e13 −0.272446 −0.136223 0.990678i \(-0.543496\pi\)
−0.136223 + 0.990678i \(0.543496\pi\)
\(242\) 0 0
\(243\) 1.01419e14i 0.492587i
\(244\) 0 0
\(245\) 5.11552e12 0.0236534
\(246\) 0 0
\(247\) 2.38007e14i 1.04811i
\(248\) 0 0
\(249\) −3.24064e14 −1.35967
\(250\) 0 0
\(251\) − 1.92417e14i − 0.769488i −0.923023 0.384744i \(-0.874290\pi\)
0.923023 0.384744i \(-0.125710\pi\)
\(252\) 0 0
\(253\) −3.96988e14 −1.51375
\(254\) 0 0
\(255\) − 8.67323e12i − 0.0315457i
\(256\) 0 0
\(257\) 1.57922e14 0.548079 0.274039 0.961719i \(-0.411640\pi\)
0.274039 + 0.961719i \(0.411640\pi\)
\(258\) 0 0
\(259\) − 6.51869e14i − 2.15954i
\(260\) 0 0
\(261\) −2.18908e13 −0.0692499
\(262\) 0 0
\(263\) 9.95937e13i 0.300952i 0.988614 + 0.150476i \(0.0480806\pi\)
−0.988614 + 0.150476i \(0.951919\pi\)
\(264\) 0 0
\(265\) 3.66802e12 0.0105915
\(266\) 0 0
\(267\) 6.14989e14i 1.69746i
\(268\) 0 0
\(269\) −5.38111e14 −1.42023 −0.710114 0.704087i \(-0.751357\pi\)
−0.710114 + 0.704087i \(0.751357\pi\)
\(270\) 0 0
\(271\) − 2.33208e14i − 0.588746i −0.955691 0.294373i \(-0.904889\pi\)
0.955691 0.294373i \(-0.0951108\pi\)
\(272\) 0 0
\(273\) 6.47830e14 1.56489
\(274\) 0 0
\(275\) − 4.03123e14i − 0.932055i
\(276\) 0 0
\(277\) 5.76832e14 1.27694 0.638470 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(278\) 0 0
\(279\) 3.09811e13i 0.0656859i
\(280\) 0 0
\(281\) 3.02774e13 0.0615008 0.0307504 0.999527i \(-0.490210\pi\)
0.0307504 + 0.999527i \(0.490210\pi\)
\(282\) 0 0
\(283\) 8.61277e14i 1.67658i 0.545225 + 0.838290i \(0.316444\pi\)
−0.545225 + 0.838290i \(0.683556\pi\)
\(284\) 0 0
\(285\) 1.19496e13 0.0222989
\(286\) 0 0
\(287\) − 3.84757e14i − 0.688486i
\(288\) 0 0
\(289\) 9.68442e14 1.66221
\(290\) 0 0
\(291\) 7.91670e14i 1.30373i
\(292\) 0 0
\(293\) −2.10405e14 −0.332545 −0.166273 0.986080i \(-0.553173\pi\)
−0.166273 + 0.986080i \(0.553173\pi\)
\(294\) 0 0
\(295\) 6.71564e12i 0.0101896i
\(296\) 0 0
\(297\) 5.35646e14 0.780440
\(298\) 0 0
\(299\) − 1.05427e15i − 1.47546i
\(300\) 0 0
\(301\) −4.71634e14 −0.634171
\(302\) 0 0
\(303\) − 8.21792e14i − 1.06195i
\(304\) 0 0
\(305\) −9.58407e12 −0.0119056
\(306\) 0 0
\(307\) − 7.57481e14i − 0.904777i −0.891821 0.452389i \(-0.850572\pi\)
0.891821 0.452389i \(-0.149428\pi\)
\(308\) 0 0
\(309\) −1.10548e12 −0.00126998
\(310\) 0 0
\(311\) 1.39394e15i 1.54057i 0.637702 + 0.770283i \(0.279885\pi\)
−0.637702 + 0.770283i \(0.720115\pi\)
\(312\) 0 0
\(313\) −8.80830e14 −0.936755 −0.468378 0.883528i \(-0.655161\pi\)
−0.468378 + 0.883528i \(0.655161\pi\)
\(314\) 0 0
\(315\) − 6.54337e12i − 0.00669789i
\(316\) 0 0
\(317\) −1.75062e15 −1.72518 −0.862592 0.505901i \(-0.831160\pi\)
−0.862592 + 0.505901i \(0.831160\pi\)
\(318\) 0 0
\(319\) 2.70151e14i 0.256367i
\(320\) 0 0
\(321\) 2.61267e14 0.238811
\(322\) 0 0
\(323\) 2.13699e15i 1.88186i
\(324\) 0 0
\(325\) 1.07057e15 0.908476
\(326\) 0 0
\(327\) − 1.00721e15i − 0.823824i
\(328\) 0 0
\(329\) −3.18356e15 −2.51037
\(330\) 0 0
\(331\) − 2.85677e14i − 0.217224i −0.994084 0.108612i \(-0.965359\pi\)
0.994084 0.108612i \(-0.0346406\pi\)
\(332\) 0 0
\(333\) −4.81824e14 −0.353364
\(334\) 0 0
\(335\) 3.29227e13i 0.0232931i
\(336\) 0 0
\(337\) 1.86299e15 1.27184 0.635918 0.771757i \(-0.280622\pi\)
0.635918 + 0.771757i \(0.280622\pi\)
\(338\) 0 0
\(339\) − 2.46502e15i − 1.62413i
\(340\) 0 0
\(341\) 3.82333e14 0.243173
\(342\) 0 0
\(343\) 9.24396e14i 0.567667i
\(344\) 0 0
\(345\) −5.29317e13 −0.0313907
\(346\) 0 0
\(347\) − 5.35266e14i − 0.306615i −0.988179 0.153307i \(-0.951008\pi\)
0.988179 0.153307i \(-0.0489925\pi\)
\(348\) 0 0
\(349\) 4.28497e14 0.237135 0.118567 0.992946i \(-0.462170\pi\)
0.118567 + 0.992946i \(0.462170\pi\)
\(350\) 0 0
\(351\) 1.42251e15i 0.760696i
\(352\) 0 0
\(353\) −4.03428e13 −0.0208506 −0.0104253 0.999946i \(-0.503319\pi\)
−0.0104253 + 0.999946i \(0.503319\pi\)
\(354\) 0 0
\(355\) − 3.36937e13i − 0.0168337i
\(356\) 0 0
\(357\) 5.81664e15 2.80972
\(358\) 0 0
\(359\) − 2.04103e15i − 0.953417i −0.879061 0.476709i \(-0.841830\pi\)
0.879061 0.476709i \(-0.158170\pi\)
\(360\) 0 0
\(361\) −7.30932e14 −0.330243
\(362\) 0 0
\(363\) − 3.34713e14i − 0.146296i
\(364\) 0 0
\(365\) 1.61521e12 0.000683082 0
\(366\) 0 0
\(367\) 3.14353e15i 1.28653i 0.765642 + 0.643266i \(0.222421\pi\)
−0.765642 + 0.643266i \(0.777579\pi\)
\(368\) 0 0
\(369\) −2.84390e14 −0.112656
\(370\) 0 0
\(371\) 2.45993e15i 0.943364i
\(372\) 0 0
\(373\) 2.60286e15 0.966492 0.483246 0.875485i \(-0.339458\pi\)
0.483246 + 0.875485i \(0.339458\pi\)
\(374\) 0 0
\(375\) − 1.07516e14i − 0.0386619i
\(376\) 0 0
\(377\) −7.17436e14 −0.249882
\(378\) 0 0
\(379\) 2.15054e15i 0.725624i 0.931862 + 0.362812i \(0.118183\pi\)
−0.931862 + 0.362812i \(0.881817\pi\)
\(380\) 0 0
\(381\) −7.46636e14 −0.244095
\(382\) 0 0
\(383\) − 5.24573e15i − 1.66193i −0.556324 0.830966i \(-0.687789\pi\)
0.556324 0.830966i \(-0.312211\pi\)
\(384\) 0 0
\(385\) −8.07507e13 −0.0247960
\(386\) 0 0
\(387\) 3.48604e14i 0.103769i
\(388\) 0 0
\(389\) −3.40137e15 −0.981649 −0.490824 0.871259i \(-0.663304\pi\)
−0.490824 + 0.871259i \(0.663304\pi\)
\(390\) 0 0
\(391\) − 9.46596e15i − 2.64914i
\(392\) 0 0
\(393\) −6.05524e15 −1.64352
\(394\) 0 0
\(395\) 3.28290e13i 0.00864320i
\(396\) 0 0
\(397\) 5.10617e14 0.130422 0.0652112 0.997871i \(-0.479228\pi\)
0.0652112 + 0.997871i \(0.479228\pi\)
\(398\) 0 0
\(399\) 8.01391e15i 1.98613i
\(400\) 0 0
\(401\) −8.14473e12 −0.00195889 −0.000979446 1.00000i \(-0.500312\pi\)
−0.000979446 1.00000i \(0.500312\pi\)
\(402\) 0 0
\(403\) 1.01535e15i 0.237021i
\(404\) 0 0
\(405\) 9.06239e13 0.0205359
\(406\) 0 0
\(407\) 5.94611e15i 1.30818i
\(408\) 0 0
\(409\) −6.52561e15 −1.39406 −0.697030 0.717042i \(-0.745496\pi\)
−0.697030 + 0.717042i \(0.745496\pi\)
\(410\) 0 0
\(411\) − 3.53211e15i − 0.732796i
\(412\) 0 0
\(413\) −4.50379e15 −0.907566
\(414\) 0 0
\(415\) 1.07273e14i 0.0209992i
\(416\) 0 0
\(417\) 6.34341e15 1.20644
\(418\) 0 0
\(419\) 8.75593e14i 0.161815i 0.996722 + 0.0809073i \(0.0257818\pi\)
−0.996722 + 0.0809073i \(0.974218\pi\)
\(420\) 0 0
\(421\) −7.20280e14 −0.129363 −0.0646813 0.997906i \(-0.520603\pi\)
−0.0646813 + 0.997906i \(0.520603\pi\)
\(422\) 0 0
\(423\) 2.35310e15i 0.410770i
\(424\) 0 0
\(425\) 9.61225e15 1.63114
\(426\) 0 0
\(427\) − 6.42749e15i − 1.06041i
\(428\) 0 0
\(429\) −5.90926e15 −0.947958
\(430\) 0 0
\(431\) − 3.23527e15i − 0.504717i −0.967634 0.252358i \(-0.918794\pi\)
0.967634 0.252358i \(-0.0812061\pi\)
\(432\) 0 0
\(433\) 1.88418e15 0.285888 0.142944 0.989731i \(-0.454343\pi\)
0.142944 + 0.989731i \(0.454343\pi\)
\(434\) 0 0
\(435\) 3.60202e13i 0.00531631i
\(436\) 0 0
\(437\) 1.30418e16 1.87261
\(438\) 0 0
\(439\) 6.20354e15i 0.866667i 0.901234 + 0.433333i \(0.142663\pi\)
−0.901234 + 0.433333i \(0.857337\pi\)
\(440\) 0 0
\(441\) 2.53576e15 0.344729
\(442\) 0 0
\(443\) − 6.29203e15i − 0.832470i −0.909257 0.416235i \(-0.863349\pi\)
0.909257 0.416235i \(-0.136651\pi\)
\(444\) 0 0
\(445\) 2.03577e14 0.0262161
\(446\) 0 0
\(447\) − 5.81355e15i − 0.728780i
\(448\) 0 0
\(449\) 6.72941e15 0.821295 0.410647 0.911794i \(-0.365303\pi\)
0.410647 + 0.911794i \(0.365303\pi\)
\(450\) 0 0
\(451\) 3.50961e15i 0.417061i
\(452\) 0 0
\(453\) −6.74915e15 −0.781016
\(454\) 0 0
\(455\) − 2.14448e14i − 0.0241687i
\(456\) 0 0
\(457\) 1.09987e16 1.20738 0.603690 0.797219i \(-0.293696\pi\)
0.603690 + 0.797219i \(0.293696\pi\)
\(458\) 0 0
\(459\) 1.27722e16i 1.36581i
\(460\) 0 0
\(461\) −1.49346e15 −0.155592 −0.0777961 0.996969i \(-0.524788\pi\)
−0.0777961 + 0.996969i \(0.524788\pi\)
\(462\) 0 0
\(463\) − 2.15660e15i − 0.218920i −0.993991 0.109460i \(-0.965088\pi\)
0.993991 0.109460i \(-0.0349121\pi\)
\(464\) 0 0
\(465\) 5.09777e13 0.00504270
\(466\) 0 0
\(467\) 3.54346e15i 0.341606i 0.985305 + 0.170803i \(0.0546362\pi\)
−0.985305 + 0.170803i \(0.945364\pi\)
\(468\) 0 0
\(469\) −2.20794e16 −2.07467
\(470\) 0 0
\(471\) 4.83792e15i 0.443132i
\(472\) 0 0
\(473\) 4.30207e15 0.384158
\(474\) 0 0
\(475\) 1.32433e16i 1.15302i
\(476\) 0 0
\(477\) 1.81824e15 0.154362
\(478\) 0 0
\(479\) − 8.95923e15i − 0.741750i −0.928683 0.370875i \(-0.879058\pi\)
0.928683 0.370875i \(-0.120942\pi\)
\(480\) 0 0
\(481\) −1.57910e16 −1.27508
\(482\) 0 0
\(483\) − 3.54983e16i − 2.79592i
\(484\) 0 0
\(485\) 2.62063e14 0.0201352
\(486\) 0 0
\(487\) − 1.24818e16i − 0.935631i −0.883826 0.467815i \(-0.845041\pi\)
0.883826 0.467815i \(-0.154959\pi\)
\(488\) 0 0
\(489\) −2.42298e16 −1.77214
\(490\) 0 0
\(491\) 1.05874e16i 0.755618i 0.925884 + 0.377809i \(0.123322\pi\)
−0.925884 + 0.377809i \(0.876678\pi\)
\(492\) 0 0
\(493\) −6.44161e15 −0.448656
\(494\) 0 0
\(495\) 5.96862e13i 0.00405735i
\(496\) 0 0
\(497\) 2.25964e16 1.49935
\(498\) 0 0
\(499\) 9.22201e15i 0.597341i 0.954356 + 0.298670i \(0.0965432\pi\)
−0.954356 + 0.298670i \(0.903457\pi\)
\(500\) 0 0
\(501\) −1.08141e16 −0.683855
\(502\) 0 0
\(503\) 2.52282e16i 1.55768i 0.627224 + 0.778839i \(0.284191\pi\)
−0.627224 + 0.778839i \(0.715809\pi\)
\(504\) 0 0
\(505\) −2.72034e14 −0.0164012
\(506\) 0 0
\(507\) 3.30991e15i 0.194881i
\(508\) 0 0
\(509\) 8.27357e14 0.0475758 0.0237879 0.999717i \(-0.492427\pi\)
0.0237879 + 0.999717i \(0.492427\pi\)
\(510\) 0 0
\(511\) 1.08323e15i 0.0608409i
\(512\) 0 0
\(513\) −1.75970e16 −0.965459
\(514\) 0 0
\(515\) 3.65941e11i 0 1.96141e-5i
\(516\) 0 0
\(517\) 2.90393e16 1.52070
\(518\) 0 0
\(519\) − 2.24862e16i − 1.15057i
\(520\) 0 0
\(521\) 3.03033e16 1.51518 0.757590 0.652731i \(-0.226377\pi\)
0.757590 + 0.652731i \(0.226377\pi\)
\(522\) 0 0
\(523\) 3.35359e15i 0.163870i 0.996638 + 0.0819350i \(0.0261100\pi\)
−0.996638 + 0.0819350i \(0.973890\pi\)
\(524\) 0 0
\(525\) 3.60469e16 1.72152
\(526\) 0 0
\(527\) 9.11653e15i 0.425565i
\(528\) 0 0
\(529\) −3.58550e16 −1.63612
\(530\) 0 0
\(531\) 3.32894e15i 0.148504i
\(532\) 0 0
\(533\) −9.32040e15 −0.406510
\(534\) 0 0
\(535\) − 8.64862e13i − 0.00368828i
\(536\) 0 0
\(537\) 3.14879e16 1.31310
\(538\) 0 0
\(539\) − 3.12934e16i − 1.27621i
\(540\) 0 0
\(541\) 3.49714e16 1.39486 0.697429 0.716654i \(-0.254327\pi\)
0.697429 + 0.716654i \(0.254327\pi\)
\(542\) 0 0
\(543\) 2.23786e16i 0.873041i
\(544\) 0 0
\(545\) −3.33413e14 −0.0127234
\(546\) 0 0
\(547\) − 3.12824e16i − 1.16782i −0.811818 0.583910i \(-0.801522\pi\)
0.811818 0.583910i \(-0.198478\pi\)
\(548\) 0 0
\(549\) −4.75082e15 −0.173514
\(550\) 0 0
\(551\) − 8.87497e15i − 0.317145i
\(552\) 0 0
\(553\) −2.20165e16 −0.769835
\(554\) 0 0
\(555\) 7.92814e14i 0.0271277i
\(556\) 0 0
\(557\) 2.39444e16 0.801811 0.400906 0.916119i \(-0.368695\pi\)
0.400906 + 0.916119i \(0.368695\pi\)
\(558\) 0 0
\(559\) 1.14249e16i 0.374440i
\(560\) 0 0
\(561\) −5.30572e16 −1.70203
\(562\) 0 0
\(563\) 2.34970e16i 0.737841i 0.929461 + 0.368920i \(0.120272\pi\)
−0.929461 + 0.368920i \(0.879728\pi\)
\(564\) 0 0
\(565\) −8.15984e14 −0.0250837
\(566\) 0 0
\(567\) 6.07763e16i 1.82909i
\(568\) 0 0
\(569\) −2.58337e16 −0.761224 −0.380612 0.924735i \(-0.624287\pi\)
−0.380612 + 0.924735i \(0.624287\pi\)
\(570\) 0 0
\(571\) 3.10927e15i 0.0897101i 0.998994 + 0.0448550i \(0.0142826\pi\)
−0.998994 + 0.0448550i \(0.985717\pi\)
\(572\) 0 0
\(573\) −2.01154e15 −0.0568331
\(574\) 0 0
\(575\) − 5.86625e16i − 1.62313i
\(576\) 0 0
\(577\) 6.04365e15 0.163774 0.0818869 0.996642i \(-0.473905\pi\)
0.0818869 + 0.996642i \(0.473905\pi\)
\(578\) 0 0
\(579\) 6.12744e16i 1.62633i
\(580\) 0 0
\(581\) −7.19422e16 −1.87037
\(582\) 0 0
\(583\) − 2.24386e16i − 0.571457i
\(584\) 0 0
\(585\) −1.58507e14 −0.00395471
\(586\) 0 0
\(587\) − 1.81607e16i − 0.443919i −0.975056 0.221959i \(-0.928755\pi\)
0.975056 0.221959i \(-0.0712452\pi\)
\(588\) 0 0
\(589\) −1.25604e16 −0.300822
\(590\) 0 0
\(591\) − 7.20756e16i − 1.69147i
\(592\) 0 0
\(593\) −3.44671e16 −0.792641 −0.396320 0.918112i \(-0.629713\pi\)
−0.396320 + 0.918112i \(0.629713\pi\)
\(594\) 0 0
\(595\) − 1.92546e15i − 0.0433942i
\(596\) 0 0
\(597\) 4.05250e16 0.895110
\(598\) 0 0
\(599\) − 6.45268e16i − 1.39694i −0.715637 0.698472i \(-0.753863\pi\)
0.715637 0.698472i \(-0.246137\pi\)
\(600\) 0 0
\(601\) 1.99244e16 0.422804 0.211402 0.977399i \(-0.432197\pi\)
0.211402 + 0.977399i \(0.432197\pi\)
\(602\) 0 0
\(603\) 1.63198e16i 0.339477i
\(604\) 0 0
\(605\) −1.10798e14 −0.00225944
\(606\) 0 0
\(607\) 7.82361e16i 1.56414i 0.623192 + 0.782069i \(0.285836\pi\)
−0.623192 + 0.782069i \(0.714164\pi\)
\(608\) 0 0
\(609\) −2.41567e16 −0.473515
\(610\) 0 0
\(611\) 7.71190e16i 1.48223i
\(612\) 0 0
\(613\) 3.30905e15 0.0623650 0.0311825 0.999514i \(-0.490073\pi\)
0.0311825 + 0.999514i \(0.490073\pi\)
\(614\) 0 0
\(615\) 4.67948e14i 0.00864862i
\(616\) 0 0
\(617\) 8.04322e16 1.45787 0.728936 0.684582i \(-0.240015\pi\)
0.728936 + 0.684582i \(0.240015\pi\)
\(618\) 0 0
\(619\) − 8.90694e16i − 1.58338i −0.610924 0.791689i \(-0.709202\pi\)
0.610924 0.791689i \(-0.290798\pi\)
\(620\) 0 0
\(621\) 7.79472e16 1.35910
\(622\) 0 0
\(623\) 1.36528e17i 2.33503i
\(624\) 0 0
\(625\) 5.95513e16 0.999104
\(626\) 0 0
\(627\) − 7.30999e16i − 1.20313i
\(628\) 0 0
\(629\) −1.41782e17 −2.28937
\(630\) 0 0
\(631\) − 1.93551e16i − 0.306634i −0.988177 0.153317i \(-0.951004\pi\)
0.988177 0.153317i \(-0.0489955\pi\)
\(632\) 0 0
\(633\) 1.04428e17 1.62329
\(634\) 0 0
\(635\) 2.47156e14i 0.00376988i
\(636\) 0 0
\(637\) 8.31054e16 1.24392
\(638\) 0 0
\(639\) − 1.67020e16i − 0.245337i
\(640\) 0 0
\(641\) −9.07240e15 −0.130790 −0.0653949 0.997859i \(-0.520831\pi\)
−0.0653949 + 0.997859i \(0.520831\pi\)
\(642\) 0 0
\(643\) 1.47136e16i 0.208186i 0.994568 + 0.104093i \(0.0331940\pi\)
−0.994568 + 0.104093i \(0.966806\pi\)
\(644\) 0 0
\(645\) 5.73609e14 0.00796632
\(646\) 0 0
\(647\) 1.15025e16i 0.156807i 0.996922 + 0.0784037i \(0.0249823\pi\)
−0.996922 + 0.0784037i \(0.975018\pi\)
\(648\) 0 0
\(649\) 4.10819e16 0.549772
\(650\) 0 0
\(651\) 3.41879e16i 0.449145i
\(652\) 0 0
\(653\) −1.02325e17 −1.31979 −0.659893 0.751359i \(-0.729399\pi\)
−0.659893 + 0.751359i \(0.729399\pi\)
\(654\) 0 0
\(655\) 2.00444e15i 0.0253831i
\(656\) 0 0
\(657\) 8.00661e14 0.00995535
\(658\) 0 0
\(659\) − 1.35977e17i − 1.66017i −0.557639 0.830084i \(-0.688293\pi\)
0.557639 0.830084i \(-0.311707\pi\)
\(660\) 0 0
\(661\) −1.02702e15 −0.0123132 −0.00615660 0.999981i \(-0.501960\pi\)
−0.00615660 + 0.999981i \(0.501960\pi\)
\(662\) 0 0
\(663\) − 1.40903e17i − 1.65897i
\(664\) 0 0
\(665\) 2.65281e15 0.0306744
\(666\) 0 0
\(667\) 3.93124e16i 0.446452i
\(668\) 0 0
\(669\) −7.05199e16 −0.786601
\(670\) 0 0
\(671\) 5.86291e16i 0.642360i
\(672\) 0 0
\(673\) −9.35963e16 −1.00732 −0.503661 0.863902i \(-0.668014\pi\)
−0.503661 + 0.863902i \(0.668014\pi\)
\(674\) 0 0
\(675\) 7.91519e16i 0.836832i
\(676\) 0 0
\(677\) −1.23322e17 −1.28088 −0.640442 0.768006i \(-0.721249\pi\)
−0.640442 + 0.768006i \(0.721249\pi\)
\(678\) 0 0
\(679\) 1.75751e17i 1.79341i
\(680\) 0 0
\(681\) −4.80128e15 −0.0481365
\(682\) 0 0
\(683\) 3.82799e16i 0.377091i 0.982064 + 0.188546i \(0.0603774\pi\)
−0.982064 + 0.188546i \(0.939623\pi\)
\(684\) 0 0
\(685\) −1.16922e15 −0.0113175
\(686\) 0 0
\(687\) − 7.39307e16i − 0.703209i
\(688\) 0 0
\(689\) 5.95897e16 0.557001
\(690\) 0 0
\(691\) 6.37695e16i 0.585794i 0.956144 + 0.292897i \(0.0946193\pi\)
−0.956144 + 0.292897i \(0.905381\pi\)
\(692\) 0 0
\(693\) −4.00281e16 −0.361381
\(694\) 0 0
\(695\) − 2.09983e15i − 0.0186327i
\(696\) 0 0
\(697\) −8.36847e16 −0.729877
\(698\) 0 0
\(699\) 2.92231e16i 0.250532i
\(700\) 0 0
\(701\) −1.51420e17 −1.27607 −0.638036 0.770007i \(-0.720253\pi\)
−0.638036 + 0.770007i \(0.720253\pi\)
\(702\) 0 0
\(703\) − 1.95341e17i − 1.61831i
\(704\) 0 0
\(705\) 3.87190e15 0.0315348
\(706\) 0 0
\(707\) − 1.82438e17i − 1.46083i
\(708\) 0 0
\(709\) 1.73160e17 1.36323 0.681615 0.731711i \(-0.261278\pi\)
0.681615 + 0.731711i \(0.261278\pi\)
\(710\) 0 0
\(711\) 1.62733e16i 0.125967i
\(712\) 0 0
\(713\) 5.56371e16 0.423475
\(714\) 0 0
\(715\) 1.95611e15i 0.0146406i
\(716\) 0 0
\(717\) 2.33383e17 1.71773
\(718\) 0 0
\(719\) 2.06527e16i 0.149487i 0.997203 + 0.0747437i \(0.0238139\pi\)
−0.997203 + 0.0747437i \(0.976186\pi\)
\(720\) 0 0
\(721\) −2.45416e14 −0.00174699
\(722\) 0 0
\(723\) 4.35396e16i 0.304828i
\(724\) 0 0
\(725\) −3.99200e16 −0.274892
\(726\) 0 0
\(727\) 2.17858e15i 0.0147559i 0.999973 + 0.00737797i \(0.00234850\pi\)
−0.999973 + 0.00737797i \(0.997651\pi\)
\(728\) 0 0
\(729\) −9.56527e16 −0.637283
\(730\) 0 0
\(731\) 1.02581e17i 0.672296i
\(732\) 0 0
\(733\) 1.40075e17 0.903105 0.451553 0.892245i \(-0.350870\pi\)
0.451553 + 0.892245i \(0.350870\pi\)
\(734\) 0 0
\(735\) − 4.17246e15i − 0.0264648i
\(736\) 0 0
\(737\) 2.01400e17 1.25677
\(738\) 0 0
\(739\) 9.13793e16i 0.561023i 0.959851 + 0.280512i \(0.0905042\pi\)
−0.959851 + 0.280512i \(0.909496\pi\)
\(740\) 0 0
\(741\) 1.94130e17 1.17269
\(742\) 0 0
\(743\) 1.19217e17i 0.708608i 0.935130 + 0.354304i \(0.115282\pi\)
−0.935130 + 0.354304i \(0.884718\pi\)
\(744\) 0 0
\(745\) −1.92443e15 −0.0112555
\(746\) 0 0
\(747\) 5.31755e16i 0.306047i
\(748\) 0 0
\(749\) 5.80014e16 0.328509
\(750\) 0 0
\(751\) − 3.29458e17i − 1.83637i −0.396148 0.918187i \(-0.629653\pi\)
0.396148 0.918187i \(-0.370347\pi\)
\(752\) 0 0
\(753\) −1.56945e17 −0.860947
\(754\) 0 0
\(755\) 2.23414e15i 0.0120623i
\(756\) 0 0
\(757\) −7.24096e16 −0.384787 −0.192394 0.981318i \(-0.561625\pi\)
−0.192394 + 0.981318i \(0.561625\pi\)
\(758\) 0 0
\(759\) 3.23802e17i 1.69367i
\(760\) 0 0
\(761\) 8.80979e16 0.453584 0.226792 0.973943i \(-0.427176\pi\)
0.226792 + 0.973943i \(0.427176\pi\)
\(762\) 0 0
\(763\) − 2.23601e17i − 1.13325i
\(764\) 0 0
\(765\) −1.42318e15 −0.00710056
\(766\) 0 0
\(767\) 1.09100e17i 0.535864i
\(768\) 0 0
\(769\) −2.19640e17 −1.06207 −0.531036 0.847349i \(-0.678197\pi\)
−0.531036 + 0.847349i \(0.678197\pi\)
\(770\) 0 0
\(771\) − 1.28808e17i − 0.613222i
\(772\) 0 0
\(773\) 3.36479e17 1.57718 0.788589 0.614921i \(-0.210812\pi\)
0.788589 + 0.614921i \(0.210812\pi\)
\(774\) 0 0
\(775\) 5.64969e16i 0.260744i
\(776\) 0 0
\(777\) −5.31695e17 −2.41622
\(778\) 0 0
\(779\) − 1.15297e17i − 0.515933i
\(780\) 0 0
\(781\) −2.06116e17 −0.908251
\(782\) 0 0
\(783\) − 5.30433e16i − 0.230176i
\(784\) 0 0
\(785\) 1.60147e15 0.00684388
\(786\) 0 0
\(787\) − 8.11217e16i − 0.341420i −0.985321 0.170710i \(-0.945394\pi\)
0.985321 0.170710i \(-0.0546061\pi\)
\(788\) 0 0
\(789\) 8.12333e16 0.336722
\(790\) 0 0
\(791\) − 5.47234e17i − 2.23416i
\(792\) 0 0
\(793\) −1.55700e17 −0.626110
\(794\) 0 0
\(795\) − 2.99181e15i − 0.0118503i
\(796\) 0 0
\(797\) −4.21727e17 −1.64544 −0.822719 0.568448i \(-0.807544\pi\)
−0.822719 + 0.568448i \(0.807544\pi\)
\(798\) 0 0
\(799\) 6.92426e17i 2.66129i
\(800\) 0 0
\(801\) 1.00913e17 0.382079
\(802\) 0 0
\(803\) − 9.88083e15i − 0.0368553i
\(804\) 0 0
\(805\) −1.17508e16 −0.0431811
\(806\) 0 0
\(807\) 4.38908e17i 1.58903i
\(808\) 0 0
\(809\) 1.89060e17 0.674385 0.337193 0.941436i \(-0.390523\pi\)
0.337193 + 0.941436i \(0.390523\pi\)
\(810\) 0 0
\(811\) 2.86016e17i 1.00523i 0.864510 + 0.502615i \(0.167629\pi\)
−0.864510 + 0.502615i \(0.832371\pi\)
\(812\) 0 0
\(813\) −1.90215e17 −0.658723
\(814\) 0 0
\(815\) 8.02069e15i 0.0273695i
\(816\) 0 0
\(817\) −1.41331e17 −0.475231
\(818\) 0 0
\(819\) − 1.06302e17i − 0.352239i
\(820\) 0 0
\(821\) 2.69622e17 0.880435 0.440217 0.897891i \(-0.354901\pi\)
0.440217 + 0.897891i \(0.354901\pi\)
\(822\) 0 0
\(823\) − 5.35943e17i − 1.72473i −0.506291 0.862363i \(-0.668984\pi\)
0.506291 0.862363i \(-0.331016\pi\)
\(824\) 0 0
\(825\) −3.28806e17 −1.04284
\(826\) 0 0
\(827\) 3.10780e17i 0.971449i 0.874112 + 0.485724i \(0.161444\pi\)
−0.874112 + 0.485724i \(0.838556\pi\)
\(828\) 0 0
\(829\) 4.99181e17 1.53791 0.768955 0.639303i \(-0.220777\pi\)
0.768955 + 0.639303i \(0.220777\pi\)
\(830\) 0 0
\(831\) − 4.70491e17i − 1.42871i
\(832\) 0 0
\(833\) 7.46175e17 2.23342
\(834\) 0 0
\(835\) 3.57974e15i 0.0105617i
\(836\) 0 0
\(837\) −7.50698e16 −0.218330
\(838\) 0 0
\(839\) − 5.10671e17i − 1.46409i −0.681254 0.732047i \(-0.738565\pi\)
0.681254 0.732047i \(-0.261435\pi\)
\(840\) 0 0
\(841\) −3.27063e17 −0.924389
\(842\) 0 0
\(843\) − 2.46957e16i − 0.0688107i
\(844\) 0 0
\(845\) 1.09567e15 0.00300980
\(846\) 0 0
\(847\) − 7.43062e16i − 0.201245i
\(848\) 0 0
\(849\) 7.02498e17 1.87585
\(850\) 0 0
\(851\) 8.65277e17i 2.27813i
\(852\) 0 0
\(853\) −6.26560e16 −0.162656 −0.0813278 0.996687i \(-0.525916\pi\)
−0.0813278 + 0.996687i \(0.525916\pi\)
\(854\) 0 0
\(855\) − 1.96080e15i − 0.00501923i
\(856\) 0 0
\(857\) 5.64927e17 1.42596 0.712980 0.701184i \(-0.247345\pi\)
0.712980 + 0.701184i \(0.247345\pi\)
\(858\) 0 0
\(859\) − 3.44043e17i − 0.856355i −0.903695 0.428178i \(-0.859156\pi\)
0.903695 0.428178i \(-0.140844\pi\)
\(860\) 0 0
\(861\) −3.13826e17 −0.770318
\(862\) 0 0
\(863\) 4.90262e17i 1.18676i 0.804922 + 0.593381i \(0.202207\pi\)
−0.804922 + 0.593381i \(0.797793\pi\)
\(864\) 0 0
\(865\) −7.44352e15 −0.0177698
\(866\) 0 0
\(867\) − 7.89907e17i − 1.85978i
\(868\) 0 0
\(869\) 2.00826e17 0.466339
\(870\) 0 0
\(871\) 5.34854e17i 1.22497i
\(872\) 0 0
\(873\) 1.29905e17 0.293453
\(874\) 0 0
\(875\) − 2.38684e16i − 0.0531834i
\(876\) 0 0
\(877\) −5.61799e17 −1.23476 −0.617382 0.786664i \(-0.711807\pi\)
−0.617382 + 0.786664i \(0.711807\pi\)
\(878\) 0 0
\(879\) 1.71616e17i 0.372071i
\(880\) 0 0
\(881\) 3.27518e17 0.700455 0.350228 0.936665i \(-0.386104\pi\)
0.350228 + 0.936665i \(0.386104\pi\)
\(882\) 0 0
\(883\) 5.81950e17i 1.22778i 0.789391 + 0.613891i \(0.210397\pi\)
−0.789391 + 0.613891i \(0.789603\pi\)
\(884\) 0 0
\(885\) 5.47759e15 0.0114007
\(886\) 0 0
\(887\) − 4.36465e16i − 0.0896205i −0.998996 0.0448103i \(-0.985732\pi\)
0.998996 0.0448103i \(-0.0142683\pi\)
\(888\) 0 0
\(889\) −1.65753e17 −0.335777
\(890\) 0 0
\(891\) − 5.54379e17i − 1.10800i
\(892\) 0 0
\(893\) −9.53993e17 −1.88121
\(894\) 0 0
\(895\) − 1.04233e16i − 0.0202800i
\(896\) 0 0
\(897\) −8.59915e17 −1.65082
\(898\) 0 0
\(899\) − 3.78612e16i − 0.0717193i
\(900\) 0 0
\(901\) 5.35035e17 1.00008
\(902\) 0 0
\(903\) 3.84687e17i 0.709547i
\(904\) 0 0
\(905\) 7.40790e15 0.0134835
\(906\) 0 0
\(907\) − 4.66428e17i − 0.837800i −0.908032 0.418900i \(-0.862416\pi\)
0.908032 0.418900i \(-0.137584\pi\)
\(908\) 0 0
\(909\) −1.34847e17 −0.239034
\(910\) 0 0
\(911\) 1.31660e16i 0.0230326i 0.999934 + 0.0115163i \(0.00366584\pi\)
−0.999934 + 0.0115163i \(0.996334\pi\)
\(912\) 0 0
\(913\) 6.56230e17 1.13300
\(914\) 0 0
\(915\) 7.81722e15i 0.0133207i
\(916\) 0 0
\(917\) −1.34426e18 −2.26083
\(918\) 0 0
\(919\) − 1.06905e18i − 1.77462i −0.461171 0.887311i \(-0.652571\pi\)
0.461171 0.887311i \(-0.347429\pi\)
\(920\) 0 0
\(921\) −6.17837e17 −1.01232
\(922\) 0 0
\(923\) − 5.47379e17i − 0.885274i
\(924\) 0 0
\(925\) −8.78650e17 −1.40270
\(926\) 0 0
\(927\) 1.81397e14i 0 0.000285859i
\(928\) 0 0
\(929\) 6.94106e17 1.07977 0.539886 0.841738i \(-0.318467\pi\)
0.539886 + 0.841738i \(0.318467\pi\)
\(930\) 0 0
\(931\) 1.02805e18i 1.57876i
\(932\) 0 0
\(933\) 1.13696e18 1.72367
\(934\) 0 0
\(935\) 1.75633e16i 0.0262867i
\(936\) 0 0
\(937\) −5.34686e17 −0.790062 −0.395031 0.918668i \(-0.629266\pi\)
−0.395031 + 0.918668i \(0.629266\pi\)
\(938\) 0 0
\(939\) 7.18447e17i 1.04810i
\(940\) 0 0
\(941\) 8.45210e17 1.21738 0.608691 0.793408i \(-0.291695\pi\)
0.608691 + 0.793408i \(0.291695\pi\)
\(942\) 0 0
\(943\) 5.10718e17i 0.726292i
\(944\) 0 0
\(945\) 1.58551e16 0.0222628
\(946\) 0 0
\(947\) − 2.47183e17i − 0.342703i −0.985210 0.171352i \(-0.945187\pi\)
0.985210 0.171352i \(-0.0548134\pi\)
\(948\) 0 0
\(949\) 2.62403e16 0.0359229
\(950\) 0 0
\(951\) 1.42788e18i 1.93023i
\(952\) 0 0
\(953\) −2.88377e17 −0.384949 −0.192475 0.981302i \(-0.561651\pi\)
−0.192475 + 0.981302i \(0.561651\pi\)
\(954\) 0 0
\(955\) 6.65871e14i 0 0.000877749i
\(956\) 0 0
\(957\) 2.20348e17 0.286839
\(958\) 0 0
\(959\) − 7.84128e17i − 1.00803i
\(960\) 0 0
\(961\) 7.34080e17 0.931972
\(962\) 0 0
\(963\) − 4.28712e16i − 0.0537536i
\(964\) 0 0
\(965\) 2.02834e16 0.0251175
\(966\) 0 0
\(967\) − 5.78010e17i − 0.706930i −0.935448 0.353465i \(-0.885003\pi\)
0.935448 0.353465i \(-0.114997\pi\)
\(968\) 0 0
\(969\) 1.74303e18 2.10553
\(970\) 0 0
\(971\) − 9.64225e17i − 1.15044i −0.818000 0.575219i \(-0.804917\pi\)
0.818000 0.575219i \(-0.195083\pi\)
\(972\) 0 0
\(973\) 1.40824e18 1.65958
\(974\) 0 0
\(975\) − 8.73204e17i − 1.01646i
\(976\) 0 0
\(977\) −1.54758e18 −1.77945 −0.889727 0.456494i \(-0.849105\pi\)
−0.889727 + 0.456494i \(0.849105\pi\)
\(978\) 0 0
\(979\) − 1.24535e18i − 1.41448i
\(980\) 0 0
\(981\) −1.65273e17 −0.185433
\(982\) 0 0
\(983\) 7.20190e17i 0.798227i 0.916902 + 0.399113i \(0.130682\pi\)
−0.916902 + 0.399113i \(0.869318\pi\)
\(984\) 0 0
\(985\) −2.38589e16 −0.0261236
\(986\) 0 0
\(987\) 2.59666e18i 2.80875i
\(988\) 0 0
\(989\) 6.26037e17 0.668994
\(990\) 0 0
\(991\) − 5.74291e17i − 0.606303i −0.952942 0.303152i \(-0.901961\pi\)
0.952942 0.303152i \(-0.0980388\pi\)
\(992\) 0 0
\(993\) −2.33012e17 −0.243043
\(994\) 0 0
\(995\) − 1.34148e16i − 0.0138244i
\(996\) 0 0
\(997\) 1.64429e18 1.67420 0.837099 0.547052i \(-0.184250\pi\)
0.837099 + 0.547052i \(0.184250\pi\)
\(998\) 0 0
\(999\) − 1.16750e18i − 1.17453i
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 16.13.c.a.15.1 2
3.2 odd 2 144.13.g.d.127.1 2
4.3 odd 2 inner 16.13.c.a.15.2 yes 2
8.3 odd 2 64.13.c.b.63.1 2
8.5 even 2 64.13.c.b.63.2 2
12.11 even 2 144.13.g.d.127.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.13.c.a.15.1 2 1.1 even 1 trivial
16.13.c.a.15.2 yes 2 4.3 odd 2 inner
64.13.c.b.63.1 2 8.3 odd 2
64.13.c.b.63.2 2 8.5 even 2
144.13.g.d.127.1 2 3.2 odd 2
144.13.g.d.127.2 2 12.11 even 2