Properties

Label 16.16.a.a.1.1
Level $16$
Weight $16$
Character 16.1
Self dual yes
Analytic conductor $22.831$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,16,Mod(1,16)]
 
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([0, 0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

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: \(22.8309608160\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 16.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6252.00 q^{3} +90510.0 q^{5} -56.0000 q^{7} +2.47386e7 q^{9} +O(q^{10})\) \(q-6252.00 q^{3} +90510.0 q^{5} -56.0000 q^{7} +2.47386e7 q^{9} +9.58899e7 q^{11} -5.97821e7 q^{13} -5.65869e8 q^{15} -1.35581e9 q^{17} -3.78359e9 q^{19} +350112. q^{21} +1.16088e10 q^{23} -2.23255e10 q^{25} -6.49563e10 q^{27} -2.89591e10 q^{29} -2.53685e11 q^{31} -5.99504e11 q^{33} -5.06856e6 q^{35} +8.17641e11 q^{37} +3.73758e11 q^{39} -6.82333e11 q^{41} -3.66946e11 q^{43} +2.23909e12 q^{45} -6.95742e11 q^{47} -4.74756e12 q^{49} +8.47655e12 q^{51} +1.29934e13 q^{53} +8.67900e12 q^{55} +2.36550e13 q^{57} -9.20904e12 q^{59} -4.23386e13 q^{61} -1.38536e9 q^{63} -5.41088e12 q^{65} -3.00298e13 q^{67} -7.25785e13 q^{69} -1.15329e14 q^{71} +4.37873e13 q^{73} +1.39579e14 q^{75} -5.36984e9 q^{77} -7.96038e13 q^{79} +5.11352e13 q^{81} +3.41707e12 q^{83} -1.22715e14 q^{85} +1.81052e14 q^{87} -3.77306e14 q^{89} +3.34780e9 q^{91} +1.58604e15 q^{93} -3.42453e14 q^{95} -1.66982e14 q^{97} +2.37218e15 q^{99} +O(q^{100})\)

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
\(3\) −6252.00 −1.65048 −0.825239 0.564784i \(-0.808959\pi\)
−0.825239 + 0.564784i \(0.808959\pi\)
\(4\) 0 0
\(5\) 90510.0 0.518109 0.259055 0.965863i \(-0.416589\pi\)
0.259055 + 0.965863i \(0.416589\pi\)
\(6\) 0 0
\(7\) −56.0000 −2.57012e−5 0 −1.28506e−5 1.00000i \(-0.500004\pi\)
−1.28506e−5 1.00000i \(0.500004\pi\)
\(8\) 0 0
\(9\) 2.47386e7 1.72408
\(10\) 0 0
\(11\) 9.58899e7 1.48364 0.741819 0.670600i \(-0.233963\pi\)
0.741819 + 0.670600i \(0.233963\pi\)
\(12\) 0 0
\(13\) −5.97821e7 −0.264239 −0.132119 0.991234i \(-0.542178\pi\)
−0.132119 + 0.991234i \(0.542178\pi\)
\(14\) 0 0
\(15\) −5.65869e8 −0.855128
\(16\) 0 0
\(17\) −1.35581e9 −0.801370 −0.400685 0.916216i \(-0.631228\pi\)
−0.400685 + 0.916216i \(0.631228\pi\)
\(18\) 0 0
\(19\) −3.78359e9 −0.971074 −0.485537 0.874216i \(-0.661376\pi\)
−0.485537 + 0.874216i \(0.661376\pi\)
\(20\) 0 0
\(21\) 350112. 4.24192e−5 0
\(22\) 0 0
\(23\) 1.16088e10 0.710936 0.355468 0.934689i \(-0.384322\pi\)
0.355468 + 0.934689i \(0.384322\pi\)
\(24\) 0 0
\(25\) −2.23255e10 −0.731563
\(26\) 0 0
\(27\) −6.49563e10 −1.19507
\(28\) 0 0
\(29\) −2.89591e10 −0.311745 −0.155873 0.987777i \(-0.549819\pi\)
−0.155873 + 0.987777i \(0.549819\pi\)
\(30\) 0 0
\(31\) −2.53685e11 −1.65609 −0.828043 0.560665i \(-0.810546\pi\)
−0.828043 + 0.560665i \(0.810546\pi\)
\(32\) 0 0
\(33\) −5.99504e11 −2.44871
\(34\) 0 0
\(35\) −5.06856e6 −1.33160e−5 0
\(36\) 0 0
\(37\) 8.17641e11 1.41596 0.707978 0.706234i \(-0.249607\pi\)
0.707978 + 0.706234i \(0.249607\pi\)
\(38\) 0 0
\(39\) 3.73758e11 0.436120
\(40\) 0 0
\(41\) −6.82333e11 −0.547164 −0.273582 0.961849i \(-0.588209\pi\)
−0.273582 + 0.961849i \(0.588209\pi\)
\(42\) 0 0
\(43\) −3.66946e11 −0.205868 −0.102934 0.994688i \(-0.532823\pi\)
−0.102934 + 0.994688i \(0.532823\pi\)
\(44\) 0 0
\(45\) 2.23909e12 0.893260
\(46\) 0 0
\(47\) −6.95742e11 −0.200315 −0.100158 0.994972i \(-0.531935\pi\)
−0.100158 + 0.994972i \(0.531935\pi\)
\(48\) 0 0
\(49\) −4.74756e12 −1.00000
\(50\) 0 0
\(51\) 8.47655e12 1.32264
\(52\) 0 0
\(53\) 1.29934e13 1.51933 0.759666 0.650313i \(-0.225362\pi\)
0.759666 + 0.650313i \(0.225362\pi\)
\(54\) 0 0
\(55\) 8.67900e12 0.768687
\(56\) 0 0
\(57\) 2.36550e13 1.60274
\(58\) 0 0
\(59\) −9.20904e12 −0.481753 −0.240876 0.970556i \(-0.577435\pi\)
−0.240876 + 0.970556i \(0.577435\pi\)
\(60\) 0 0
\(61\) −4.23386e13 −1.72490 −0.862448 0.506145i \(-0.831070\pi\)
−0.862448 + 0.506145i \(0.831070\pi\)
\(62\) 0 0
\(63\) −1.38536e9 −4.43107e−5 0
\(64\) 0 0
\(65\) −5.41088e12 −0.136905
\(66\) 0 0
\(67\) −3.00298e13 −0.605329 −0.302664 0.953097i \(-0.597876\pi\)
−0.302664 + 0.953097i \(0.597876\pi\)
\(68\) 0 0
\(69\) −7.25785e13 −1.17338
\(70\) 0 0
\(71\) −1.15329e14 −1.50487 −0.752436 0.658665i \(-0.771121\pi\)
−0.752436 + 0.658665i \(0.771121\pi\)
\(72\) 0 0
\(73\) 4.37873e13 0.463903 0.231951 0.972727i \(-0.425489\pi\)
0.231951 + 0.972727i \(0.425489\pi\)
\(74\) 0 0
\(75\) 1.39579e14 1.20743
\(76\) 0 0
\(77\) −5.36984e9 −3.81312e−5 0
\(78\) 0 0
\(79\) −7.96038e13 −0.466370 −0.233185 0.972432i \(-0.574915\pi\)
−0.233185 + 0.972432i \(0.574915\pi\)
\(80\) 0 0
\(81\) 5.11352e13 0.248360
\(82\) 0 0
\(83\) 3.41707e12 0.0138219 0.00691095 0.999976i \(-0.497800\pi\)
0.00691095 + 0.999976i \(0.497800\pi\)
\(84\) 0 0
\(85\) −1.22715e14 −0.415198
\(86\) 0 0
\(87\) 1.81052e14 0.514529
\(88\) 0 0
\(89\) −3.77306e14 −0.904209 −0.452105 0.891965i \(-0.649327\pi\)
−0.452105 + 0.891965i \(0.649327\pi\)
\(90\) 0 0
\(91\) 3.34780e9 6.79124e−6 0
\(92\) 0 0
\(93\) 1.58604e15 2.73333
\(94\) 0 0
\(95\) −3.42453e14 −0.503123
\(96\) 0 0
\(97\) −1.66982e14 −0.209837 −0.104919 0.994481i \(-0.533458\pi\)
−0.104919 + 0.994481i \(0.533458\pi\)
\(98\) 0 0
\(99\) 2.37218e15 2.55790
\(100\) 0 0
\(101\) 6.06802e13 0.0563167 0.0281583 0.999603i \(-0.491036\pi\)
0.0281583 + 0.999603i \(0.491036\pi\)
\(102\) 0 0
\(103\) −5.87676e14 −0.470825 −0.235412 0.971896i \(-0.575644\pi\)
−0.235412 + 0.971896i \(0.575644\pi\)
\(104\) 0 0
\(105\) 3.16886e10 2.19778e−5 0
\(106\) 0 0
\(107\) −6.56968e14 −0.395518 −0.197759 0.980251i \(-0.563366\pi\)
−0.197759 + 0.980251i \(0.563366\pi\)
\(108\) 0 0
\(109\) −2.09023e15 −1.09520 −0.547602 0.836739i \(-0.684459\pi\)
−0.547602 + 0.836739i \(0.684459\pi\)
\(110\) 0 0
\(111\) −5.11189e15 −2.33700
\(112\) 0 0
\(113\) −1.63296e15 −0.652959 −0.326480 0.945204i \(-0.605862\pi\)
−0.326480 + 0.945204i \(0.605862\pi\)
\(114\) 0 0
\(115\) 1.05072e15 0.368342
\(116\) 0 0
\(117\) −1.47893e15 −0.455567
\(118\) 0 0
\(119\) 7.59256e10 2.05962e−5 0
\(120\) 0 0
\(121\) 5.01763e15 1.20118
\(122\) 0 0
\(123\) 4.26595e15 0.903082
\(124\) 0 0
\(125\) −4.78283e15 −0.897139
\(126\) 0 0
\(127\) −6.51334e15 −1.08462 −0.542308 0.840180i \(-0.682449\pi\)
−0.542308 + 0.840180i \(0.682449\pi\)
\(128\) 0 0
\(129\) 2.29414e15 0.339780
\(130\) 0 0
\(131\) 1.12636e16 1.48642 0.743212 0.669056i \(-0.233301\pi\)
0.743212 + 0.669056i \(0.233301\pi\)
\(132\) 0 0
\(133\) 2.11881e11 2.49577e−5 0
\(134\) 0 0
\(135\) −5.87920e15 −0.619177
\(136\) 0 0
\(137\) 5.10328e15 0.481332 0.240666 0.970608i \(-0.422634\pi\)
0.240666 + 0.970608i \(0.422634\pi\)
\(138\) 0 0
\(139\) −2.76682e15 −0.234083 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(140\) 0 0
\(141\) 4.34978e15 0.330616
\(142\) 0 0
\(143\) −5.73251e15 −0.392034
\(144\) 0 0
\(145\) −2.62109e15 −0.161518
\(146\) 0 0
\(147\) 2.96818e16 1.65048
\(148\) 0 0
\(149\) 2.68469e16 1.34896 0.674478 0.738295i \(-0.264369\pi\)
0.674478 + 0.738295i \(0.264369\pi\)
\(150\) 0 0
\(151\) 3.65479e16 1.66164 0.830818 0.556544i \(-0.187873\pi\)
0.830818 + 0.556544i \(0.187873\pi\)
\(152\) 0 0
\(153\) −3.35409e16 −1.38162
\(154\) 0 0
\(155\) −2.29611e16 −0.858033
\(156\) 0 0
\(157\) 2.99685e16 1.01723 0.508613 0.860995i \(-0.330158\pi\)
0.508613 + 0.860995i \(0.330158\pi\)
\(158\) 0 0
\(159\) −8.12346e16 −2.50762
\(160\) 0 0
\(161\) −6.50095e11 −1.82719e−5 0
\(162\) 0 0
\(163\) 4.29497e16 1.10041 0.550203 0.835031i \(-0.314550\pi\)
0.550203 + 0.835031i \(0.314550\pi\)
\(164\) 0 0
\(165\) −5.42611e16 −1.26870
\(166\) 0 0
\(167\) −7.10064e15 −0.151678 −0.0758392 0.997120i \(-0.524164\pi\)
−0.0758392 + 0.997120i \(0.524164\pi\)
\(168\) 0 0
\(169\) −4.76120e16 −0.930178
\(170\) 0 0
\(171\) −9.36008e16 −1.67421
\(172\) 0 0
\(173\) −5.69309e16 −0.933259 −0.466629 0.884453i \(-0.654532\pi\)
−0.466629 + 0.884453i \(0.654532\pi\)
\(174\) 0 0
\(175\) 1.25023e12 1.88020e−5 0
\(176\) 0 0
\(177\) 5.75749e16 0.795122
\(178\) 0 0
\(179\) 2.65655e15 0.0337225 0.0168613 0.999858i \(-0.494633\pi\)
0.0168613 + 0.999858i \(0.494633\pi\)
\(180\) 0 0
\(181\) 5.17648e16 0.604569 0.302284 0.953218i \(-0.402251\pi\)
0.302284 + 0.953218i \(0.402251\pi\)
\(182\) 0 0
\(183\) 2.64701e17 2.84690
\(184\) 0 0
\(185\) 7.40047e16 0.733620
\(186\) 0 0
\(187\) −1.30009e17 −1.18894
\(188\) 0 0
\(189\) 3.63756e12 3.07147e−5 0
\(190\) 0 0
\(191\) −5.36753e16 −0.418817 −0.209408 0.977828i \(-0.567154\pi\)
−0.209408 + 0.977828i \(0.567154\pi\)
\(192\) 0 0
\(193\) 1.14450e17 0.825913 0.412957 0.910751i \(-0.364496\pi\)
0.412957 + 0.910751i \(0.364496\pi\)
\(194\) 0 0
\(195\) 3.38288e16 0.225958
\(196\) 0 0
\(197\) −2.07130e17 −1.28158 −0.640790 0.767716i \(-0.721393\pi\)
−0.640790 + 0.767716i \(0.721393\pi\)
\(198\) 0 0
\(199\) 3.59838e16 0.206399 0.103200 0.994661i \(-0.467092\pi\)
0.103200 + 0.994661i \(0.467092\pi\)
\(200\) 0 0
\(201\) 1.87746e17 0.999081
\(202\) 0 0
\(203\) 1.62171e12 8.01222e−6 0
\(204\) 0 0
\(205\) −6.17580e16 −0.283491
\(206\) 0 0
\(207\) 2.87187e17 1.22571
\(208\) 0 0
\(209\) −3.62809e17 −1.44072
\(210\) 0 0
\(211\) −9.02699e16 −0.333753 −0.166876 0.985978i \(-0.553368\pi\)
−0.166876 + 0.985978i \(0.553368\pi\)
\(212\) 0 0
\(213\) 7.21035e17 2.48376
\(214\) 0 0
\(215\) −3.32122e16 −0.106662
\(216\) 0 0
\(217\) 1.42064e13 4.25633e−5 0
\(218\) 0 0
\(219\) −2.73758e17 −0.765661
\(220\) 0 0
\(221\) 8.10535e16 0.211753
\(222\) 0 0
\(223\) 3.22894e17 0.788449 0.394225 0.919014i \(-0.371013\pi\)
0.394225 + 0.919014i \(0.371013\pi\)
\(224\) 0 0
\(225\) −5.52302e17 −1.26127
\(226\) 0 0
\(227\) −7.84856e17 −1.67724 −0.838621 0.544715i \(-0.816638\pi\)
−0.838621 + 0.544715i \(0.816638\pi\)
\(228\) 0 0
\(229\) −7.84092e17 −1.56892 −0.784460 0.620180i \(-0.787060\pi\)
−0.784460 + 0.620180i \(0.787060\pi\)
\(230\) 0 0
\(231\) 3.35722e13 6.29347e−5 0
\(232\) 0 0
\(233\) −7.17048e17 −1.26002 −0.630012 0.776586i \(-0.716950\pi\)
−0.630012 + 0.776586i \(0.716950\pi\)
\(234\) 0 0
\(235\) −6.29716e16 −0.103785
\(236\) 0 0
\(237\) 4.97683e17 0.769733
\(238\) 0 0
\(239\) −6.93110e17 −1.00651 −0.503255 0.864138i \(-0.667864\pi\)
−0.503255 + 0.864138i \(0.667864\pi\)
\(240\) 0 0
\(241\) 9.65566e17 1.31721 0.658604 0.752490i \(-0.271147\pi\)
0.658604 + 0.752490i \(0.271147\pi\)
\(242\) 0 0
\(243\) 6.12355e17 0.785157
\(244\) 0 0
\(245\) −4.29702e17 −0.518109
\(246\) 0 0
\(247\) 2.26191e17 0.256595
\(248\) 0 0
\(249\) −2.13635e16 −0.0228127
\(250\) 0 0
\(251\) 8.46641e17 0.851425 0.425712 0.904859i \(-0.360024\pi\)
0.425712 + 0.904859i \(0.360024\pi\)
\(252\) 0 0
\(253\) 1.11317e18 1.05477
\(254\) 0 0
\(255\) 7.67213e17 0.685274
\(256\) 0 0
\(257\) 1.40156e18 1.18063 0.590314 0.807174i \(-0.299004\pi\)
0.590314 + 0.807174i \(0.299004\pi\)
\(258\) 0 0
\(259\) −4.57879e13 −3.63917e−5 0
\(260\) 0 0
\(261\) −7.16408e17 −0.537473
\(262\) 0 0
\(263\) −1.38804e18 −0.983412 −0.491706 0.870761i \(-0.663626\pi\)
−0.491706 + 0.870761i \(0.663626\pi\)
\(264\) 0 0
\(265\) 1.17603e18 0.787181
\(266\) 0 0
\(267\) 2.35892e18 1.49238
\(268\) 0 0
\(269\) −2.85472e18 −1.70774 −0.853870 0.520486i \(-0.825751\pi\)
−0.853870 + 0.520486i \(0.825751\pi\)
\(270\) 0 0
\(271\) 1.76158e18 0.996858 0.498429 0.866931i \(-0.333910\pi\)
0.498429 + 0.866931i \(0.333910\pi\)
\(272\) 0 0
\(273\) −2.09304e13 −1.12088e−5 0
\(274\) 0 0
\(275\) −2.14079e18 −1.08537
\(276\) 0 0
\(277\) 3.96897e17 0.190581 0.0952905 0.995450i \(-0.469622\pi\)
0.0952905 + 0.995450i \(0.469622\pi\)
\(278\) 0 0
\(279\) −6.27582e18 −2.85522
\(280\) 0 0
\(281\) 3.36756e17 0.145217 0.0726086 0.997361i \(-0.476868\pi\)
0.0726086 + 0.997361i \(0.476868\pi\)
\(282\) 0 0
\(283\) 4.56529e18 1.86668 0.933340 0.358992i \(-0.116880\pi\)
0.933340 + 0.358992i \(0.116880\pi\)
\(284\) 0 0
\(285\) 2.14102e18 0.830393
\(286\) 0 0
\(287\) 3.82107e13 1.40628e−5 0
\(288\) 0 0
\(289\) −1.02419e18 −0.357805
\(290\) 0 0
\(291\) 1.04397e18 0.346331
\(292\) 0 0
\(293\) 8.51076e17 0.268201 0.134101 0.990968i \(-0.457185\pi\)
0.134101 + 0.990968i \(0.457185\pi\)
\(294\) 0 0
\(295\) −8.33510e17 −0.249601
\(296\) 0 0
\(297\) −6.22866e18 −1.77305
\(298\) 0 0
\(299\) −6.94002e17 −0.187857
\(300\) 0 0
\(301\) 2.05490e13 5.29104e−6 0
\(302\) 0 0
\(303\) −3.79373e17 −0.0929494
\(304\) 0 0
\(305\) −3.83207e18 −0.893685
\(306\) 0 0
\(307\) −3.96999e18 −0.881560 −0.440780 0.897615i \(-0.645298\pi\)
−0.440780 + 0.897615i \(0.645298\pi\)
\(308\) 0 0
\(309\) 3.67415e18 0.777085
\(310\) 0 0
\(311\) −3.18666e18 −0.642144 −0.321072 0.947055i \(-0.604043\pi\)
−0.321072 + 0.947055i \(0.604043\pi\)
\(312\) 0 0
\(313\) 1.59866e18 0.307024 0.153512 0.988147i \(-0.450942\pi\)
0.153512 + 0.988147i \(0.450942\pi\)
\(314\) 0 0
\(315\) −1.25389e14 −2.29578e−5 0
\(316\) 0 0
\(317\) −4.92262e18 −0.859512 −0.429756 0.902945i \(-0.641400\pi\)
−0.429756 + 0.902945i \(0.641400\pi\)
\(318\) 0 0
\(319\) −2.77689e18 −0.462517
\(320\) 0 0
\(321\) 4.10736e18 0.652793
\(322\) 0 0
\(323\) 5.12985e18 0.778190
\(324\) 0 0
\(325\) 1.33467e18 0.193307
\(326\) 0 0
\(327\) 1.30681e19 1.80761
\(328\) 0 0
\(329\) 3.89615e13 5.14833e−6 0
\(330\) 0 0
\(331\) 6.42897e18 0.811767 0.405883 0.913925i \(-0.366964\pi\)
0.405883 + 0.913925i \(0.366964\pi\)
\(332\) 0 0
\(333\) 2.02273e19 2.44122
\(334\) 0 0
\(335\) −2.71800e18 −0.313627
\(336\) 0 0
\(337\) 1.44912e19 1.59912 0.799560 0.600586i \(-0.205066\pi\)
0.799560 + 0.600586i \(0.205066\pi\)
\(338\) 0 0
\(339\) 1.02092e19 1.07769
\(340\) 0 0
\(341\) −2.43259e19 −2.45703
\(342\) 0 0
\(343\) 5.31727e14 5.14023e−5 0
\(344\) 0 0
\(345\) −6.56908e18 −0.607941
\(346\) 0 0
\(347\) −2.11500e19 −1.87430 −0.937149 0.348929i \(-0.886546\pi\)
−0.937149 + 0.348929i \(0.886546\pi\)
\(348\) 0 0
\(349\) −1.04783e19 −0.889409 −0.444705 0.895677i \(-0.646691\pi\)
−0.444705 + 0.895677i \(0.646691\pi\)
\(350\) 0 0
\(351\) 3.88323e18 0.315784
\(352\) 0 0
\(353\) 1.39442e19 1.08663 0.543316 0.839528i \(-0.317168\pi\)
0.543316 + 0.839528i \(0.317168\pi\)
\(354\) 0 0
\(355\) −1.04384e19 −0.779689
\(356\) 0 0
\(357\) −4.74687e14 −3.39935e−5 0
\(358\) 0 0
\(359\) 1.03027e19 0.707528 0.353764 0.935335i \(-0.384902\pi\)
0.353764 + 0.935335i \(0.384902\pi\)
\(360\) 0 0
\(361\) −8.65550e17 −0.0570148
\(362\) 0 0
\(363\) −3.13702e19 −1.98252
\(364\) 0 0
\(365\) 3.96319e18 0.240353
\(366\) 0 0
\(367\) 1.17900e19 0.686310 0.343155 0.939279i \(-0.388504\pi\)
0.343155 + 0.939279i \(0.388504\pi\)
\(368\) 0 0
\(369\) −1.68800e19 −0.943352
\(370\) 0 0
\(371\) −7.27629e14 −3.90486e−5 0
\(372\) 0 0
\(373\) −3.32551e19 −1.71413 −0.857063 0.515212i \(-0.827713\pi\)
−0.857063 + 0.515212i \(0.827713\pi\)
\(374\) 0 0
\(375\) 2.99022e19 1.48071
\(376\) 0 0
\(377\) 1.73124e18 0.0823752
\(378\) 0 0
\(379\) 6.27237e18 0.286839 0.143419 0.989662i \(-0.454190\pi\)
0.143419 + 0.989662i \(0.454190\pi\)
\(380\) 0 0
\(381\) 4.07214e19 1.79013
\(382\) 0 0
\(383\) −3.27919e19 −1.38604 −0.693020 0.720918i \(-0.743720\pi\)
−0.693020 + 0.720918i \(0.743720\pi\)
\(384\) 0 0
\(385\) −4.86024e14 −1.97561e−5 0
\(386\) 0 0
\(387\) −9.07772e18 −0.354931
\(388\) 0 0
\(389\) 2.69074e19 1.01216 0.506081 0.862486i \(-0.331094\pi\)
0.506081 + 0.862486i \(0.331094\pi\)
\(390\) 0 0
\(391\) −1.57394e19 −0.569723
\(392\) 0 0
\(393\) −7.04201e19 −2.45331
\(394\) 0 0
\(395\) −7.20494e18 −0.241631
\(396\) 0 0
\(397\) 6.98326e18 0.225491 0.112746 0.993624i \(-0.464035\pi\)
0.112746 + 0.993624i \(0.464035\pi\)
\(398\) 0 0
\(399\) −1.32468e15 −4.11922e−5 0
\(400\) 0 0
\(401\) 1.22571e19 0.367118 0.183559 0.983009i \(-0.441238\pi\)
0.183559 + 0.983009i \(0.441238\pi\)
\(402\) 0 0
\(403\) 1.51659e19 0.437602
\(404\) 0 0
\(405\) 4.62825e18 0.128678
\(406\) 0 0
\(407\) 7.84036e19 2.10077
\(408\) 0 0
\(409\) 4.13285e19 1.06739 0.533697 0.845676i \(-0.320802\pi\)
0.533697 + 0.845676i \(0.320802\pi\)
\(410\) 0 0
\(411\) −3.19057e19 −0.794428
\(412\) 0 0
\(413\) 5.15706e14 1.23816e−5 0
\(414\) 0 0
\(415\) 3.09279e17 0.00716126
\(416\) 0 0
\(417\) 1.72982e19 0.386349
\(418\) 0 0
\(419\) 6.58312e19 1.41849 0.709246 0.704961i \(-0.249036\pi\)
0.709246 + 0.704961i \(0.249036\pi\)
\(420\) 0 0
\(421\) −6.81572e18 −0.141709 −0.0708543 0.997487i \(-0.522573\pi\)
−0.0708543 + 0.997487i \(0.522573\pi\)
\(422\) 0 0
\(423\) −1.72117e19 −0.345359
\(424\) 0 0
\(425\) 3.02693e19 0.586253
\(426\) 0 0
\(427\) 2.37096e15 4.43319e−5 0
\(428\) 0 0
\(429\) 3.58396e19 0.647044
\(430\) 0 0
\(431\) 2.43444e19 0.424444 0.212222 0.977222i \(-0.431930\pi\)
0.212222 + 0.977222i \(0.431930\pi\)
\(432\) 0 0
\(433\) 3.73209e19 0.628483 0.314241 0.949343i \(-0.398250\pi\)
0.314241 + 0.949343i \(0.398250\pi\)
\(434\) 0 0
\(435\) 1.63870e19 0.266582
\(436\) 0 0
\(437\) −4.39231e19 −0.690371
\(438\) 0 0
\(439\) −8.88153e19 −1.34898 −0.674488 0.738286i \(-0.735635\pi\)
−0.674488 + 0.738286i \(0.735635\pi\)
\(440\) 0 0
\(441\) −1.17448e20 −1.72408
\(442\) 0 0
\(443\) 3.24463e19 0.460402 0.230201 0.973143i \(-0.426062\pi\)
0.230201 + 0.973143i \(0.426062\pi\)
\(444\) 0 0
\(445\) −3.41500e19 −0.468479
\(446\) 0 0
\(447\) −1.67847e20 −2.22642
\(448\) 0 0
\(449\) 1.40615e20 1.80378 0.901892 0.431961i \(-0.142178\pi\)
0.901892 + 0.431961i \(0.142178\pi\)
\(450\) 0 0
\(451\) −6.54289e19 −0.811794
\(452\) 0 0
\(453\) −2.28498e20 −2.74249
\(454\) 0 0
\(455\) 3.03009e14 3.51861e−6 0
\(456\) 0 0
\(457\) −1.10893e20 −1.24605 −0.623023 0.782204i \(-0.714096\pi\)
−0.623023 + 0.782204i \(0.714096\pi\)
\(458\) 0 0
\(459\) 8.80687e19 0.957694
\(460\) 0 0
\(461\) 2.96124e19 0.311686 0.155843 0.987782i \(-0.450191\pi\)
0.155843 + 0.987782i \(0.450191\pi\)
\(462\) 0 0
\(463\) 1.33021e20 1.35538 0.677692 0.735346i \(-0.262980\pi\)
0.677692 + 0.735346i \(0.262980\pi\)
\(464\) 0 0
\(465\) 1.43553e20 1.41616
\(466\) 0 0
\(467\) 7.07393e18 0.0675747 0.0337873 0.999429i \(-0.489243\pi\)
0.0337873 + 0.999429i \(0.489243\pi\)
\(468\) 0 0
\(469\) 1.68167e15 1.55576e−5 0
\(470\) 0 0
\(471\) −1.87363e20 −1.67891
\(472\) 0 0
\(473\) −3.51864e19 −0.305433
\(474\) 0 0
\(475\) 8.44707e19 0.710402
\(476\) 0 0
\(477\) 3.21438e20 2.61944
\(478\) 0 0
\(479\) −1.30964e19 −0.103427 −0.0517136 0.998662i \(-0.516468\pi\)
−0.0517136 + 0.998662i \(0.516468\pi\)
\(480\) 0 0
\(481\) −4.88803e19 −0.374150
\(482\) 0 0
\(483\) 4.06440e15 3.01573e−5 0
\(484\) 0 0
\(485\) −1.51136e19 −0.108719
\(486\) 0 0
\(487\) −8.82232e19 −0.615340 −0.307670 0.951493i \(-0.599549\pi\)
−0.307670 + 0.951493i \(0.599549\pi\)
\(488\) 0 0
\(489\) −2.68521e20 −1.81619
\(490\) 0 0
\(491\) 2.32523e20 1.52530 0.762651 0.646811i \(-0.223898\pi\)
0.762651 + 0.646811i \(0.223898\pi\)
\(492\) 0 0
\(493\) 3.92632e19 0.249824
\(494\) 0 0
\(495\) 2.14706e20 1.32527
\(496\) 0 0
\(497\) 6.45841e15 3.86770e−5 0
\(498\) 0 0
\(499\) −1.91680e20 −1.11384 −0.556919 0.830567i \(-0.688017\pi\)
−0.556919 + 0.830567i \(0.688017\pi\)
\(500\) 0 0
\(501\) 4.43932e19 0.250342
\(502\) 0 0
\(503\) −2.64372e20 −1.44696 −0.723478 0.690348i \(-0.757458\pi\)
−0.723478 + 0.690348i \(0.757458\pi\)
\(504\) 0 0
\(505\) 5.49217e18 0.0291782
\(506\) 0 0
\(507\) 2.97670e20 1.53524
\(508\) 0 0
\(509\) 1.35342e20 0.677718 0.338859 0.940837i \(-0.389959\pi\)
0.338859 + 0.940837i \(0.389959\pi\)
\(510\) 0 0
\(511\) −2.45209e15 −1.19228e−5 0
\(512\) 0 0
\(513\) 2.45768e20 1.16050
\(514\) 0 0
\(515\) −5.31906e19 −0.243939
\(516\) 0 0
\(517\) −6.67146e19 −0.297195
\(518\) 0 0
\(519\) 3.55932e20 1.54032
\(520\) 0 0
\(521\) 1.16387e20 0.489352 0.244676 0.969605i \(-0.421318\pi\)
0.244676 + 0.969605i \(0.421318\pi\)
\(522\) 0 0
\(523\) −2.37253e20 −0.969280 −0.484640 0.874714i \(-0.661049\pi\)
−0.484640 + 0.874714i \(0.661049\pi\)
\(524\) 0 0
\(525\) −7.81643e15 −3.10323e−5 0
\(526\) 0 0
\(527\) 3.43950e20 1.32714
\(528\) 0 0
\(529\) −1.31870e20 −0.494571
\(530\) 0 0
\(531\) −2.27819e20 −0.830578
\(532\) 0 0
\(533\) 4.07913e19 0.144582
\(534\) 0 0
\(535\) −5.94622e19 −0.204921
\(536\) 0 0
\(537\) −1.66087e19 −0.0556583
\(538\) 0 0
\(539\) −4.55243e20 −1.48364
\(540\) 0 0
\(541\) −2.01502e20 −0.638706 −0.319353 0.947636i \(-0.603466\pi\)
−0.319353 + 0.947636i \(0.603466\pi\)
\(542\) 0 0
\(543\) −3.23634e20 −0.997827
\(544\) 0 0
\(545\) −1.89187e20 −0.567435
\(546\) 0 0
\(547\) 2.12934e20 0.621355 0.310677 0.950515i \(-0.399444\pi\)
0.310677 + 0.950515i \(0.399444\pi\)
\(548\) 0 0
\(549\) −1.04740e21 −2.97385
\(550\) 0 0
\(551\) 1.09569e20 0.302728
\(552\) 0 0
\(553\) 4.45781e15 1.19863e−5 0
\(554\) 0 0
\(555\) −4.62677e20 −1.21082
\(556\) 0 0
\(557\) 2.73616e20 0.696991 0.348495 0.937310i \(-0.386693\pi\)
0.348495 + 0.937310i \(0.386693\pi\)
\(558\) 0 0
\(559\) 2.19368e19 0.0543982
\(560\) 0 0
\(561\) 8.12816e20 1.96232
\(562\) 0 0
\(563\) 2.23411e20 0.525159 0.262580 0.964910i \(-0.415427\pi\)
0.262580 + 0.964910i \(0.415427\pi\)
\(564\) 0 0
\(565\) −1.47799e20 −0.338304
\(566\) 0 0
\(567\) −2.86357e15 −6.38315e−6 0
\(568\) 0 0
\(569\) −5.24953e20 −1.13967 −0.569834 0.821760i \(-0.692993\pi\)
−0.569834 + 0.821760i \(0.692993\pi\)
\(570\) 0 0
\(571\) 1.54331e20 0.326348 0.163174 0.986597i \(-0.447827\pi\)
0.163174 + 0.986597i \(0.447827\pi\)
\(572\) 0 0
\(573\) 3.35578e20 0.691248
\(574\) 0 0
\(575\) −2.59173e20 −0.520094
\(576\) 0 0
\(577\) 7.19413e20 1.40656 0.703282 0.710911i \(-0.251717\pi\)
0.703282 + 0.710911i \(0.251717\pi\)
\(578\) 0 0
\(579\) −7.15539e20 −1.36315
\(580\) 0 0
\(581\) −1.91356e14 −3.55239e−7 0
\(582\) 0 0
\(583\) 1.24593e21 2.25414
\(584\) 0 0
\(585\) −1.33858e20 −0.236034
\(586\) 0 0
\(587\) 4.31990e20 0.742485 0.371243 0.928536i \(-0.378932\pi\)
0.371243 + 0.928536i \(0.378932\pi\)
\(588\) 0 0
\(589\) 9.59842e20 1.60818
\(590\) 0 0
\(591\) 1.29498e21 2.11522
\(592\) 0 0
\(593\) −4.19124e20 −0.667471 −0.333736 0.942667i \(-0.608309\pi\)
−0.333736 + 0.942667i \(0.608309\pi\)
\(594\) 0 0
\(595\) 6.87203e15 1.06711e−5 0
\(596\) 0 0
\(597\) −2.24970e20 −0.340658
\(598\) 0 0
\(599\) −3.18808e20 −0.470791 −0.235396 0.971900i \(-0.575639\pi\)
−0.235396 + 0.971900i \(0.575639\pi\)
\(600\) 0 0
\(601\) −9.85641e20 −1.41958 −0.709791 0.704413i \(-0.751210\pi\)
−0.709791 + 0.704413i \(0.751210\pi\)
\(602\) 0 0
\(603\) −7.42895e20 −1.04363
\(604\) 0 0
\(605\) 4.54146e20 0.622344
\(606\) 0 0
\(607\) −1.08297e21 −1.44777 −0.723886 0.689920i \(-0.757646\pi\)
−0.723886 + 0.689920i \(0.757646\pi\)
\(608\) 0 0
\(609\) −1.01389e16 −1.32240e−5 0
\(610\) 0 0
\(611\) 4.15929e19 0.0529310
\(612\) 0 0
\(613\) 1.47488e21 1.83148 0.915742 0.401767i \(-0.131604\pi\)
0.915742 + 0.401767i \(0.131604\pi\)
\(614\) 0 0
\(615\) 3.86111e20 0.467895
\(616\) 0 0
\(617\) −1.86048e20 −0.220032 −0.110016 0.993930i \(-0.535090\pi\)
−0.110016 + 0.993930i \(0.535090\pi\)
\(618\) 0 0
\(619\) −5.44499e20 −0.628517 −0.314258 0.949337i \(-0.601756\pi\)
−0.314258 + 0.949337i \(0.601756\pi\)
\(620\) 0 0
\(621\) −7.54068e20 −0.849618
\(622\) 0 0
\(623\) 2.11291e16 2.32392e−5 0
\(624\) 0 0
\(625\) 2.48427e20 0.266746
\(626\) 0 0
\(627\) 2.26828e21 2.37788
\(628\) 0 0
\(629\) −1.10857e21 −1.13471
\(630\) 0 0
\(631\) 1.34412e20 0.134344 0.0671721 0.997741i \(-0.478602\pi\)
0.0671721 + 0.997741i \(0.478602\pi\)
\(632\) 0 0
\(633\) 5.64367e20 0.550851
\(634\) 0 0
\(635\) −5.89522e20 −0.561949
\(636\) 0 0
\(637\) 2.83819e20 0.264239
\(638\) 0 0
\(639\) −2.85307e21 −2.59451
\(640\) 0 0
\(641\) −2.15011e20 −0.190997 −0.0954983 0.995430i \(-0.530444\pi\)
−0.0954983 + 0.995430i \(0.530444\pi\)
\(642\) 0 0
\(643\) 1.36705e21 1.18632 0.593160 0.805085i \(-0.297880\pi\)
0.593160 + 0.805085i \(0.297880\pi\)
\(644\) 0 0
\(645\) 2.07643e20 0.176043
\(646\) 0 0
\(647\) −1.14998e20 −0.0952597 −0.0476298 0.998865i \(-0.515167\pi\)
−0.0476298 + 0.998865i \(0.515167\pi\)
\(648\) 0 0
\(649\) −8.83054e20 −0.714746
\(650\) 0 0
\(651\) −8.88183e16 −7.02498e−5 0
\(652\) 0 0
\(653\) −1.23391e21 −0.953751 −0.476876 0.878971i \(-0.658231\pi\)
−0.476876 + 0.878971i \(0.658231\pi\)
\(654\) 0 0
\(655\) 1.01947e21 0.770131
\(656\) 0 0
\(657\) 1.08324e21 0.799804
\(658\) 0 0
\(659\) −1.63082e21 −1.17697 −0.588485 0.808508i \(-0.700275\pi\)
−0.588485 + 0.808508i \(0.700275\pi\)
\(660\) 0 0
\(661\) −1.38740e21 −0.978793 −0.489397 0.872061i \(-0.662783\pi\)
−0.489397 + 0.872061i \(0.662783\pi\)
\(662\) 0 0
\(663\) −5.06746e20 −0.349494
\(664\) 0 0
\(665\) 1.91774e16 1.29308e−5 0
\(666\) 0 0
\(667\) −3.36182e20 −0.221631
\(668\) 0 0
\(669\) −2.01873e21 −1.30132
\(670\) 0 0
\(671\) −4.05985e21 −2.55912
\(672\) 0 0
\(673\) −1.33937e21 −0.825631 −0.412816 0.910815i \(-0.635455\pi\)
−0.412816 + 0.910815i \(0.635455\pi\)
\(674\) 0 0
\(675\) 1.45018e21 0.874268
\(676\) 0 0
\(677\) 1.99408e21 1.17579 0.587893 0.808938i \(-0.299957\pi\)
0.587893 + 0.808938i \(0.299957\pi\)
\(678\) 0 0
\(679\) 9.35100e15 5.39306e−6 0
\(680\) 0 0
\(681\) 4.90692e21 2.76825
\(682\) 0 0
\(683\) 1.25180e21 0.690845 0.345422 0.938447i \(-0.387736\pi\)
0.345422 + 0.938447i \(0.387736\pi\)
\(684\) 0 0
\(685\) 4.61898e20 0.249383
\(686\) 0 0
\(687\) 4.90214e21 2.58947
\(688\) 0 0
\(689\) −7.76772e20 −0.401466
\(690\) 0 0
\(691\) 1.46431e21 0.740539 0.370269 0.928924i \(-0.379265\pi\)
0.370269 + 0.928924i \(0.379265\pi\)
\(692\) 0 0
\(693\) −1.32842e17 −6.57411e−5 0
\(694\) 0 0
\(695\) −2.50425e20 −0.121281
\(696\) 0 0
\(697\) 9.25117e20 0.438481
\(698\) 0 0
\(699\) 4.48298e21 2.07964
\(700\) 0 0
\(701\) −2.97987e20 −0.135305 −0.0676523 0.997709i \(-0.521551\pi\)
−0.0676523 + 0.997709i \(0.521551\pi\)
\(702\) 0 0
\(703\) −3.09362e21 −1.37500
\(704\) 0 0
\(705\) 3.93698e20 0.171295
\(706\) 0 0
\(707\) −3.39809e15 −1.44740e−6 0
\(708\) 0 0
\(709\) 1.21881e21 0.508264 0.254132 0.967169i \(-0.418210\pi\)
0.254132 + 0.967169i \(0.418210\pi\)
\(710\) 0 0
\(711\) −1.96929e21 −0.804057
\(712\) 0 0
\(713\) −2.94499e21 −1.17737
\(714\) 0 0
\(715\) −5.18849e20 −0.203117
\(716\) 0 0
\(717\) 4.33333e21 1.66122
\(718\) 0 0
\(719\) 3.01856e21 1.13327 0.566634 0.823970i \(-0.308245\pi\)
0.566634 + 0.823970i \(0.308245\pi\)
\(720\) 0 0
\(721\) 3.29099e16 1.21007e−5 0
\(722\) 0 0
\(723\) −6.03672e21 −2.17402
\(724\) 0 0
\(725\) 6.46527e20 0.228061
\(726\) 0 0
\(727\) −8.97710e20 −0.310190 −0.155095 0.987900i \(-0.549568\pi\)
−0.155095 + 0.987900i \(0.549568\pi\)
\(728\) 0 0
\(729\) −4.56218e21 −1.54424
\(730\) 0 0
\(731\) 4.97510e20 0.164976
\(732\) 0 0
\(733\) −8.52857e20 −0.277074 −0.138537 0.990357i \(-0.544240\pi\)
−0.138537 + 0.990357i \(0.544240\pi\)
\(734\) 0 0
\(735\) 2.68650e21 0.855128
\(736\) 0 0
\(737\) −2.87955e21 −0.898089
\(738\) 0 0
\(739\) −2.19196e21 −0.669883 −0.334941 0.942239i \(-0.608716\pi\)
−0.334941 + 0.942239i \(0.608716\pi\)
\(740\) 0 0
\(741\) −1.41415e21 −0.423505
\(742\) 0 0
\(743\) −1.89929e21 −0.557410 −0.278705 0.960377i \(-0.589905\pi\)
−0.278705 + 0.960377i \(0.589905\pi\)
\(744\) 0 0
\(745\) 2.42992e21 0.698906
\(746\) 0 0
\(747\) 8.45335e19 0.0238300
\(748\) 0 0
\(749\) 3.67902e16 1.01653e−5 0
\(750\) 0 0
\(751\) 4.40128e21 1.19201 0.596005 0.802981i \(-0.296754\pi\)
0.596005 + 0.802981i \(0.296754\pi\)
\(752\) 0 0
\(753\) −5.29320e21 −1.40526
\(754\) 0 0
\(755\) 3.30795e21 0.860909
\(756\) 0 0
\(757\) 3.32600e21 0.848601 0.424300 0.905521i \(-0.360520\pi\)
0.424300 + 0.905521i \(0.360520\pi\)
\(758\) 0 0
\(759\) −6.95955e21 −1.74088
\(760\) 0 0
\(761\) 6.66362e21 1.63428 0.817138 0.576443i \(-0.195560\pi\)
0.817138 + 0.576443i \(0.195560\pi\)
\(762\) 0 0
\(763\) 1.17053e17 2.81480e−5 0
\(764\) 0 0
\(765\) −3.03579e21 −0.715832
\(766\) 0 0
\(767\) 5.50536e20 0.127298
\(768\) 0 0
\(769\) 1.77725e21 0.402997 0.201499 0.979489i \(-0.435419\pi\)
0.201499 + 0.979489i \(0.435419\pi\)
\(770\) 0 0
\(771\) −8.76255e21 −1.94860
\(772\) 0 0
\(773\) −4.17145e21 −0.909790 −0.454895 0.890545i \(-0.650323\pi\)
−0.454895 + 0.890545i \(0.650323\pi\)
\(774\) 0 0
\(775\) 5.66366e21 1.21153
\(776\) 0 0
\(777\) 2.86266e17 6.00637e−5 0
\(778\) 0 0
\(779\) 2.58167e21 0.531337
\(780\) 0 0
\(781\) −1.10589e22 −2.23269
\(782\) 0 0
\(783\) 1.88108e21 0.372558
\(784\) 0 0
\(785\) 2.71245e21 0.527035
\(786\) 0 0
\(787\) −4.96836e21 −0.947114 −0.473557 0.880763i \(-0.657030\pi\)
−0.473557 + 0.880763i \(0.657030\pi\)
\(788\) 0 0
\(789\) 8.67805e21 1.62310
\(790\) 0 0
\(791\) 9.14455e16 1.67818e−5 0
\(792\) 0 0
\(793\) 2.53109e21 0.455784
\(794\) 0 0
\(795\) −7.35254e21 −1.29922
\(796\) 0 0
\(797\) 4.55056e21 0.789091 0.394546 0.918876i \(-0.370902\pi\)
0.394546 + 0.918876i \(0.370902\pi\)
\(798\) 0 0
\(799\) 9.43296e20 0.160527
\(800\) 0 0
\(801\) −9.33403e21 −1.55892
\(802\) 0 0
\(803\) 4.19877e21 0.688264
\(804\) 0 0
\(805\) −5.88401e16 −9.46683e−6 0
\(806\) 0 0
\(807\) 1.78477e22 2.81859
\(808\) 0 0
\(809\) 1.12040e22 1.73683 0.868417 0.495834i \(-0.165138\pi\)
0.868417 + 0.495834i \(0.165138\pi\)
\(810\) 0 0
\(811\) −1.56431e21 −0.238048 −0.119024 0.992891i \(-0.537977\pi\)
−0.119024 + 0.992891i \(0.537977\pi\)
\(812\) 0 0
\(813\) −1.10134e22 −1.64529
\(814\) 0 0
\(815\) 3.88737e21 0.570130
\(816\) 0 0
\(817\) 1.38837e21 0.199913
\(818\) 0 0
\(819\) 8.28199e16 1.17086e−5 0
\(820\) 0 0
\(821\) 5.31534e21 0.737831 0.368916 0.929463i \(-0.379729\pi\)
0.368916 + 0.929463i \(0.379729\pi\)
\(822\) 0 0
\(823\) −2.48380e21 −0.338546 −0.169273 0.985569i \(-0.554142\pi\)
−0.169273 + 0.985569i \(0.554142\pi\)
\(824\) 0 0
\(825\) 1.33842e22 1.79139
\(826\) 0 0
\(827\) 6.01420e21 0.790473 0.395236 0.918580i \(-0.370663\pi\)
0.395236 + 0.918580i \(0.370663\pi\)
\(828\) 0 0
\(829\) −4.33653e21 −0.559736 −0.279868 0.960039i \(-0.590291\pi\)
−0.279868 + 0.960039i \(0.590291\pi\)
\(830\) 0 0
\(831\) −2.48140e21 −0.314550
\(832\) 0 0
\(833\) 6.43681e21 0.801370
\(834\) 0 0
\(835\) −6.42679e20 −0.0785861
\(836\) 0 0
\(837\) 1.64785e22 1.97914
\(838\) 0 0
\(839\) −7.92224e21 −0.934616 −0.467308 0.884095i \(-0.654776\pi\)
−0.467308 + 0.884095i \(0.654776\pi\)
\(840\) 0 0
\(841\) −7.79056e21 −0.902815
\(842\) 0 0
\(843\) −2.10540e21 −0.239678
\(844\) 0 0
\(845\) −4.30936e21 −0.481934
\(846\) 0 0
\(847\) −2.80988e17 −3.08718e−5 0
\(848\) 0 0
\(849\) −2.85422e22 −3.08091
\(850\) 0 0
\(851\) 9.49187e21 1.00665
\(852\) 0 0
\(853\) 4.77976e21 0.498067 0.249034 0.968495i \(-0.419887\pi\)
0.249034 + 0.968495i \(0.419887\pi\)
\(854\) 0 0
\(855\) −8.47181e21 −0.867422
\(856\) 0 0
\(857\) −1.38506e22 −1.39352 −0.696758 0.717306i \(-0.745375\pi\)
−0.696758 + 0.717306i \(0.745375\pi\)
\(858\) 0 0
\(859\) −1.13389e22 −1.12104 −0.560522 0.828140i \(-0.689399\pi\)
−0.560522 + 0.828140i \(0.689399\pi\)
\(860\) 0 0
\(861\) −2.38893e17 −2.32103e−5 0
\(862\) 0 0
\(863\) −1.34506e22 −1.28428 −0.642142 0.766586i \(-0.721954\pi\)
−0.642142 + 0.766586i \(0.721954\pi\)
\(864\) 0 0
\(865\) −5.15281e21 −0.483530
\(866\) 0 0
\(867\) 6.40324e21 0.590550
\(868\) 0 0
\(869\) −7.63321e21 −0.691924
\(870\) 0 0
\(871\) 1.79524e21 0.159951
\(872\) 0 0
\(873\) −4.13091e21 −0.361775
\(874\) 0 0
\(875\) 2.67838e17 2.30575e−5 0
\(876\) 0 0
\(877\) 8.71264e21 0.737315 0.368657 0.929565i \(-0.379818\pi\)
0.368657 + 0.929565i \(0.379818\pi\)
\(878\) 0 0
\(879\) −5.32093e21 −0.442660
\(880\) 0 0
\(881\) 1.84288e22 1.50722 0.753611 0.657321i \(-0.228310\pi\)
0.753611 + 0.657321i \(0.228310\pi\)
\(882\) 0 0
\(883\) −6.38696e21 −0.513558 −0.256779 0.966470i \(-0.582661\pi\)
−0.256779 + 0.966470i \(0.582661\pi\)
\(884\) 0 0
\(885\) 5.21110e21 0.411960
\(886\) 0 0
\(887\) 1.26761e22 0.985279 0.492639 0.870234i \(-0.336032\pi\)
0.492639 + 0.870234i \(0.336032\pi\)
\(888\) 0 0
\(889\) 3.64747e17 2.78759e−5 0
\(890\) 0 0
\(891\) 4.90335e21 0.368477
\(892\) 0 0
\(893\) 2.63240e21 0.194521
\(894\) 0 0
\(895\) 2.40444e20 0.0174720
\(896\) 0 0
\(897\) 4.33890e21 0.310053
\(898\) 0 0
\(899\) 7.34650e21 0.516277
\(900\) 0 0
\(901\) −1.76166e22 −1.21755
\(902\) 0 0
\(903\) −1.28472e17 −8.73274e−6 0
\(904\) 0 0
\(905\) 4.68524e21 0.313233
\(906\) 0 0
\(907\) −1.29670e21 −0.0852676 −0.0426338 0.999091i \(-0.513575\pi\)
−0.0426338 + 0.999091i \(0.513575\pi\)
\(908\) 0 0
\(909\) 1.50114e21 0.0970942
\(910\) 0 0
\(911\) −2.26280e22 −1.43966 −0.719828 0.694153i \(-0.755779\pi\)
−0.719828 + 0.694153i \(0.755779\pi\)
\(912\) 0 0
\(913\) 3.27663e20 0.0205067
\(914\) 0 0
\(915\) 2.39581e22 1.47501
\(916\) 0 0
\(917\) −6.30762e17 −3.82028e−5 0
\(918\) 0 0
\(919\) 4.37047e21 0.260412 0.130206 0.991487i \(-0.458436\pi\)
0.130206 + 0.991487i \(0.458436\pi\)
\(920\) 0 0
\(921\) 2.48204e22 1.45499
\(922\) 0 0
\(923\) 6.89460e21 0.397645
\(924\) 0 0
\(925\) −1.82543e22 −1.03586
\(926\) 0 0
\(927\) −1.45383e22 −0.811737
\(928\) 0 0
\(929\) −2.43423e22 −1.33734 −0.668672 0.743557i \(-0.733137\pi\)
−0.668672 + 0.743557i \(0.733137\pi\)
\(930\) 0 0
\(931\) 1.79628e22 0.971074
\(932\) 0 0
\(933\) 1.99230e22 1.05984
\(934\) 0 0
\(935\) −1.17671e22 −0.616003
\(936\) 0 0
\(937\) 1.64776e22 0.848879 0.424439 0.905456i \(-0.360471\pi\)
0.424439 + 0.905456i \(0.360471\pi\)
\(938\) 0 0
\(939\) −9.99480e21 −0.506736
\(940\) 0 0
\(941\) −2.09383e22 −1.04477 −0.522384 0.852710i \(-0.674957\pi\)
−0.522384 + 0.852710i \(0.674957\pi\)
\(942\) 0 0
\(943\) −7.92110e21 −0.388998
\(944\) 0 0
\(945\) 3.29235e17 1.59136e−5 0
\(946\) 0 0
\(947\) 3.36036e22 1.59868 0.799338 0.600881i \(-0.205183\pi\)
0.799338 + 0.600881i \(0.205183\pi\)
\(948\) 0 0
\(949\) −2.61770e21 −0.122581
\(950\) 0 0
\(951\) 3.07762e22 1.41860
\(952\) 0 0
\(953\) 3.45292e22 1.56671 0.783357 0.621572i \(-0.213506\pi\)
0.783357 + 0.621572i \(0.213506\pi\)
\(954\) 0 0
\(955\) −4.85815e21 −0.216993
\(956\) 0 0
\(957\) 1.73611e22 0.763375
\(958\) 0 0
\(959\) −2.85784e17 −1.23708e−5 0
\(960\) 0 0
\(961\) 4.08910e22 1.74262
\(962\) 0 0
\(963\) −1.62525e22 −0.681902
\(964\) 0 0
\(965\) 1.03588e22 0.427913
\(966\) 0 0
\(967\) −1.34537e21 −0.0547197 −0.0273598 0.999626i \(-0.508710\pi\)
−0.0273598 + 0.999626i \(0.508710\pi\)
\(968\) 0 0
\(969\) −3.20718e22 −1.28439
\(970\) 0 0
\(971\) −1.50334e21 −0.0592807 −0.0296403 0.999561i \(-0.509436\pi\)
−0.0296403 + 0.999561i \(0.509436\pi\)
\(972\) 0 0
\(973\) 1.54942e17 6.01620e−6 0
\(974\) 0 0
\(975\) −8.34434e21 −0.319049
\(976\) 0 0
\(977\) −4.36133e22 −1.64214 −0.821068 0.570830i \(-0.806622\pi\)
−0.821068 + 0.570830i \(0.806622\pi\)
\(978\) 0 0
\(979\) −3.61799e22 −1.34152
\(980\) 0 0
\(981\) −5.17093e22 −1.88821
\(982\) 0 0
\(983\) −3.21986e21 −0.115794 −0.0578969 0.998323i \(-0.518439\pi\)
−0.0578969 + 0.998323i \(0.518439\pi\)
\(984\) 0 0
\(985\) −1.87473e22 −0.663999
\(986\) 0 0
\(987\) −2.43587e17 −8.49721e−6 0
\(988\) 0 0
\(989\) −4.25981e21 −0.146359
\(990\) 0 0
\(991\) 4.85010e22 1.64134 0.820670 0.571402i \(-0.193600\pi\)
0.820670 + 0.571402i \(0.193600\pi\)
\(992\) 0 0
\(993\) −4.01939e22 −1.33980
\(994\) 0 0
\(995\) 3.25689e21 0.106937
\(996\) 0 0
\(997\) 5.35543e22 1.73213 0.866065 0.499931i \(-0.166641\pi\)
0.866065 + 0.499931i \(0.166641\pi\)
\(998\) 0 0
\(999\) −5.31110e22 −1.69217
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 16.16.a.a.1.1 1
3.2 odd 2 144.16.a.d.1.1 1
4.3 odd 2 2.16.a.a.1.1 1
8.3 odd 2 64.16.a.a.1.1 1
8.5 even 2 64.16.a.k.1.1 1
12.11 even 2 18.16.a.e.1.1 1
20.3 even 4 50.16.b.a.49.2 2
20.7 even 4 50.16.b.a.49.1 2
20.19 odd 2 50.16.a.b.1.1 1
28.3 even 6 98.16.c.d.79.1 2
28.11 odd 6 98.16.c.a.79.1 2
28.19 even 6 98.16.c.d.67.1 2
28.23 odd 6 98.16.c.a.67.1 2
28.27 even 2 98.16.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.16.a.a.1.1 1 4.3 odd 2
16.16.a.a.1.1 1 1.1 even 1 trivial
18.16.a.e.1.1 1 12.11 even 2
50.16.a.b.1.1 1 20.19 odd 2
50.16.b.a.49.1 2 20.7 even 4
50.16.b.a.49.2 2 20.3 even 4
64.16.a.a.1.1 1 8.3 odd 2
64.16.a.k.1.1 1 8.5 even 2
98.16.a.a.1.1 1 28.27 even 2
98.16.c.a.67.1 2 28.23 odd 6
98.16.c.a.79.1 2 28.11 odd 6
98.16.c.d.67.1 2 28.19 even 6
98.16.c.d.79.1 2 28.3 even 6
144.16.a.d.1.1 1 3.2 odd 2