Properties

Label 384.8.d.e.193.9
Level $384$
Weight $8$
Character 384.193
Analytic conductor $119.956$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,8,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.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: no
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.9
Root \(49.2636 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.8.d.e.193.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+27.0000i q^{3} -8.99895i q^{5} +616.197 q^{7} -729.000 q^{9} +O(q^{10})\) \(q+27.0000i q^{3} -8.99895i q^{5} +616.197 q^{7} -729.000 q^{9} -3035.60i q^{11} -12272.8i q^{13} +242.972 q^{15} -35454.2 q^{17} +46835.6i q^{19} +16637.3i q^{21} -89833.4 q^{23} +78044.0 q^{25} -19683.0i q^{27} +113126. i q^{29} +246616. q^{31} +81961.3 q^{33} -5545.13i q^{35} -383887. i q^{37} +331365. q^{39} +63822.4 q^{41} +697052. i q^{43} +6560.24i q^{45} +66539.5 q^{47} -443844. q^{49} -957264. i q^{51} +1.02190e6i q^{53} -27317.3 q^{55} -1.26456e6 q^{57} -1.66944e6i q^{59} +1.94570e6i q^{61} -449208. q^{63} -110442. q^{65} +50973.1i q^{67} -2.42550e6i q^{69} +5.04161e6 q^{71} +5.82121e6 q^{73} +2.10719e6i q^{75} -1.87053e6i q^{77} +2.30931e6 q^{79} +531441. q^{81} +3.20768e6i q^{83} +319051. i q^{85} -3.05441e6 q^{87} -5.21504e6 q^{89} -7.56246e6i q^{91} +6.65862e6i q^{93} +421471. q^{95} -3.08632e6 q^{97} +2.21296e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 27.0000i 0.577350i
\(4\) 0 0
\(5\) − 8.99895i − 0.0321956i −0.999870 0.0160978i \(-0.994876\pi\)
0.999870 0.0160978i \(-0.00512432\pi\)
\(6\) 0 0
\(7\) 616.197 0.679011 0.339505 0.940604i \(-0.389740\pi\)
0.339505 + 0.940604i \(0.389740\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) − 3035.60i − 0.687655i −0.939033 0.343828i \(-0.888276\pi\)
0.939033 0.343828i \(-0.111724\pi\)
\(12\) 0 0
\(13\) − 12272.8i − 1.54932i −0.632377 0.774660i \(-0.717921\pi\)
0.632377 0.774660i \(-0.282079\pi\)
\(14\) 0 0
\(15\) 242.972 0.0185882
\(16\) 0 0
\(17\) −35454.2 −1.75024 −0.875118 0.483909i \(-0.839217\pi\)
−0.875118 + 0.483909i \(0.839217\pi\)
\(18\) 0 0
\(19\) 46835.6i 1.56653i 0.621689 + 0.783264i \(0.286447\pi\)
−0.621689 + 0.783264i \(0.713553\pi\)
\(20\) 0 0
\(21\) 16637.3i 0.392027i
\(22\) 0 0
\(23\) −89833.4 −1.53954 −0.769769 0.638323i \(-0.779629\pi\)
−0.769769 + 0.638323i \(0.779629\pi\)
\(24\) 0 0
\(25\) 78044.0 0.998963
\(26\) 0 0
\(27\) − 19683.0i − 0.192450i
\(28\) 0 0
\(29\) 113126.i 0.861333i 0.902511 + 0.430666i \(0.141721\pi\)
−0.902511 + 0.430666i \(0.858279\pi\)
\(30\) 0 0
\(31\) 246616. 1.48681 0.743403 0.668843i \(-0.233210\pi\)
0.743403 + 0.668843i \(0.233210\pi\)
\(32\) 0 0
\(33\) 81961.3 0.397018
\(34\) 0 0
\(35\) − 5545.13i − 0.0218612i
\(36\) 0 0
\(37\) − 383887.i − 1.24594i −0.782245 0.622970i \(-0.785926\pi\)
0.782245 0.622970i \(-0.214074\pi\)
\(38\) 0 0
\(39\) 331365. 0.894501
\(40\) 0 0
\(41\) 63822.4 0.144620 0.0723102 0.997382i \(-0.476963\pi\)
0.0723102 + 0.997382i \(0.476963\pi\)
\(42\) 0 0
\(43\) 697052.i 1.33698i 0.743720 + 0.668491i \(0.233060\pi\)
−0.743720 + 0.668491i \(0.766940\pi\)
\(44\) 0 0
\(45\) 6560.24i 0.0107319i
\(46\) 0 0
\(47\) 66539.5 0.0934840 0.0467420 0.998907i \(-0.485116\pi\)
0.0467420 + 0.998907i \(0.485116\pi\)
\(48\) 0 0
\(49\) −443844. −0.538944
\(50\) 0 0
\(51\) − 957264.i − 1.01050i
\(52\) 0 0
\(53\) 1.02190e6i 0.942851i 0.881906 + 0.471426i \(0.156260\pi\)
−0.881906 + 0.471426i \(0.843740\pi\)
\(54\) 0 0
\(55\) −27317.3 −0.0221395
\(56\) 0 0
\(57\) −1.26456e6 −0.904435
\(58\) 0 0
\(59\) − 1.66944e6i − 1.05825i −0.848544 0.529125i \(-0.822520\pi\)
0.848544 0.529125i \(-0.177480\pi\)
\(60\) 0 0
\(61\) 1.94570e6i 1.09754i 0.835972 + 0.548772i \(0.184905\pi\)
−0.835972 + 0.548772i \(0.815095\pi\)
\(62\) 0 0
\(63\) −449208. −0.226337
\(64\) 0 0
\(65\) −110442. −0.0498814
\(66\) 0 0
\(67\) 50973.1i 0.0207052i 0.999946 + 0.0103526i \(0.00329539\pi\)
−0.999946 + 0.0103526i \(0.996705\pi\)
\(68\) 0 0
\(69\) − 2.42550e6i − 0.888852i
\(70\) 0 0
\(71\) 5.04161e6 1.67173 0.835863 0.548938i \(-0.184968\pi\)
0.835863 + 0.548938i \(0.184968\pi\)
\(72\) 0 0
\(73\) 5.82121e6 1.75139 0.875696 0.482863i \(-0.160403\pi\)
0.875696 + 0.482863i \(0.160403\pi\)
\(74\) 0 0
\(75\) 2.10719e6i 0.576752i
\(76\) 0 0
\(77\) − 1.87053e6i − 0.466925i
\(78\) 0 0
\(79\) 2.30931e6 0.526972 0.263486 0.964663i \(-0.415128\pi\)
0.263486 + 0.964663i \(0.415128\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 3.20768e6i 0.615768i 0.951424 + 0.307884i \(0.0996209\pi\)
−0.951424 + 0.307884i \(0.900379\pi\)
\(84\) 0 0
\(85\) 319051.i 0.0563500i
\(86\) 0 0
\(87\) −3.05441e6 −0.497291
\(88\) 0 0
\(89\) −5.21504e6 −0.784138 −0.392069 0.919936i \(-0.628240\pi\)
−0.392069 + 0.919936i \(0.628240\pi\)
\(90\) 0 0
\(91\) − 7.56246e6i − 1.05201i
\(92\) 0 0
\(93\) 6.65862e6i 0.858408i
\(94\) 0 0
\(95\) 421471. 0.0504354
\(96\) 0 0
\(97\) −3.08632e6 −0.343352 −0.171676 0.985153i \(-0.554918\pi\)
−0.171676 + 0.985153i \(0.554918\pi\)
\(98\) 0 0
\(99\) 2.21296e6i 0.229218i
\(100\) 0 0
\(101\) 1.44652e7i 1.39701i 0.715604 + 0.698507i \(0.246152\pi\)
−0.715604 + 0.698507i \(0.753848\pi\)
\(102\) 0 0
\(103\) 1.89906e6 0.171241 0.0856205 0.996328i \(-0.472713\pi\)
0.0856205 + 0.996328i \(0.472713\pi\)
\(104\) 0 0
\(105\) 149719. 0.0126216
\(106\) 0 0
\(107\) 1.54233e7i 1.21712i 0.793506 + 0.608562i \(0.208253\pi\)
−0.793506 + 0.608562i \(0.791747\pi\)
\(108\) 0 0
\(109\) 6.59792e6i 0.487994i 0.969776 + 0.243997i \(0.0784587\pi\)
−0.969776 + 0.243997i \(0.921541\pi\)
\(110\) 0 0
\(111\) 1.03650e7 0.719344
\(112\) 0 0
\(113\) −5.66038e6 −0.369038 −0.184519 0.982829i \(-0.559073\pi\)
−0.184519 + 0.982829i \(0.559073\pi\)
\(114\) 0 0
\(115\) 808407.i 0.0495664i
\(116\) 0 0
\(117\) 8.94686e6i 0.516440i
\(118\) 0 0
\(119\) −2.18468e7 −1.18843
\(120\) 0 0
\(121\) 1.02723e7 0.527130
\(122\) 0 0
\(123\) 1.72321e6i 0.0834967i
\(124\) 0 0
\(125\) − 1.40536e6i − 0.0643579i
\(126\) 0 0
\(127\) −1.21577e7 −0.526669 −0.263334 0.964705i \(-0.584822\pi\)
−0.263334 + 0.964705i \(0.584822\pi\)
\(128\) 0 0
\(129\) −1.88204e7 −0.771907
\(130\) 0 0
\(131\) − 3.25216e7i − 1.26393i −0.774997 0.631965i \(-0.782249\pi\)
0.774997 0.631965i \(-0.217751\pi\)
\(132\) 0 0
\(133\) 2.88599e7i 1.06369i
\(134\) 0 0
\(135\) −177126. −0.00619605
\(136\) 0 0
\(137\) 4.34472e7 1.44358 0.721789 0.692114i \(-0.243320\pi\)
0.721789 + 0.692114i \(0.243320\pi\)
\(138\) 0 0
\(139\) − 2.11016e6i − 0.0666444i −0.999445 0.0333222i \(-0.989391\pi\)
0.999445 0.0333222i \(-0.0106088\pi\)
\(140\) 0 0
\(141\) 1.79657e6i 0.0539730i
\(142\) 0 0
\(143\) −3.72553e7 −1.06540
\(144\) 0 0
\(145\) 1.01802e6 0.0277312
\(146\) 0 0
\(147\) − 1.19838e7i − 0.311160i
\(148\) 0 0
\(149\) 3.58625e7i 0.888156i 0.895988 + 0.444078i \(0.146469\pi\)
−0.895988 + 0.444078i \(0.853531\pi\)
\(150\) 0 0
\(151\) −7.80703e6 −0.184530 −0.0922649 0.995734i \(-0.529411\pi\)
−0.0922649 + 0.995734i \(0.529411\pi\)
\(152\) 0 0
\(153\) 2.58461e7 0.583412
\(154\) 0 0
\(155\) − 2.21928e6i − 0.0478687i
\(156\) 0 0
\(157\) 7.98173e7i 1.64607i 0.567991 + 0.823035i \(0.307721\pi\)
−0.567991 + 0.823035i \(0.692279\pi\)
\(158\) 0 0
\(159\) −2.75913e7 −0.544355
\(160\) 0 0
\(161\) −5.53551e7 −1.04536
\(162\) 0 0
\(163\) 4.10447e7i 0.742336i 0.928566 + 0.371168i \(0.121043\pi\)
−0.928566 + 0.371168i \(0.878957\pi\)
\(164\) 0 0
\(165\) − 737566.i − 0.0127822i
\(166\) 0 0
\(167\) 8.79088e7 1.46058 0.730289 0.683139i \(-0.239386\pi\)
0.730289 + 0.683139i \(0.239386\pi\)
\(168\) 0 0
\(169\) −8.78727e7 −1.40040
\(170\) 0 0
\(171\) − 3.41431e7i − 0.522176i
\(172\) 0 0
\(173\) 1.29158e8i 1.89653i 0.317478 + 0.948266i \(0.397164\pi\)
−0.317478 + 0.948266i \(0.602836\pi\)
\(174\) 0 0
\(175\) 4.80905e7 0.678307
\(176\) 0 0
\(177\) 4.50749e7 0.610981
\(178\) 0 0
\(179\) − 1.13096e7i − 0.147388i −0.997281 0.0736941i \(-0.976521\pi\)
0.997281 0.0736941i \(-0.0234789\pi\)
\(180\) 0 0
\(181\) 1.01595e8i 1.27349i 0.771074 + 0.636746i \(0.219720\pi\)
−0.771074 + 0.636746i \(0.780280\pi\)
\(182\) 0 0
\(183\) −5.25340e7 −0.633668
\(184\) 0 0
\(185\) −3.45458e6 −0.0401139
\(186\) 0 0
\(187\) 1.07625e8i 1.20356i
\(188\) 0 0
\(189\) − 1.21286e7i − 0.130676i
\(190\) 0 0
\(191\) −2.54258e7 −0.264032 −0.132016 0.991248i \(-0.542145\pi\)
−0.132016 + 0.991248i \(0.542145\pi\)
\(192\) 0 0
\(193\) 1.01605e7 0.101734 0.0508668 0.998705i \(-0.483802\pi\)
0.0508668 + 0.998705i \(0.483802\pi\)
\(194\) 0 0
\(195\) − 2.98194e6i − 0.0287990i
\(196\) 0 0
\(197\) 5.94284e7i 0.553812i 0.960897 + 0.276906i \(0.0893091\pi\)
−0.960897 + 0.276906i \(0.910691\pi\)
\(198\) 0 0
\(199\) −1.32802e8 −1.19459 −0.597293 0.802023i \(-0.703757\pi\)
−0.597293 + 0.802023i \(0.703757\pi\)
\(200\) 0 0
\(201\) −1.37627e6 −0.0119541
\(202\) 0 0
\(203\) 6.97082e7i 0.584854i
\(204\) 0 0
\(205\) − 574335.i − 0.00465615i
\(206\) 0 0
\(207\) 6.54885e7 0.513179
\(208\) 0 0
\(209\) 1.42174e8 1.07723
\(210\) 0 0
\(211\) − 2.57624e8i − 1.88798i −0.329971 0.943991i \(-0.607039\pi\)
0.329971 0.943991i \(-0.392961\pi\)
\(212\) 0 0
\(213\) 1.36123e8i 0.965171i
\(214\) 0 0
\(215\) 6.27274e6 0.0430450
\(216\) 0 0
\(217\) 1.51964e8 1.00956
\(218\) 0 0
\(219\) 1.57173e8i 1.01117i
\(220\) 0 0
\(221\) 4.35122e8i 2.71168i
\(222\) 0 0
\(223\) 1.96405e8 1.18600 0.593002 0.805201i \(-0.297943\pi\)
0.593002 + 0.805201i \(0.297943\pi\)
\(224\) 0 0
\(225\) −5.68941e7 −0.332988
\(226\) 0 0
\(227\) 8.67364e7i 0.492165i 0.969249 + 0.246082i \(0.0791434\pi\)
−0.969249 + 0.246082i \(0.920857\pi\)
\(228\) 0 0
\(229\) 1.22165e8i 0.672237i 0.941820 + 0.336118i \(0.109114\pi\)
−0.941820 + 0.336118i \(0.890886\pi\)
\(230\) 0 0
\(231\) 5.05044e7 0.269580
\(232\) 0 0
\(233\) −5.39072e7 −0.279191 −0.139595 0.990209i \(-0.544580\pi\)
−0.139595 + 0.990209i \(0.544580\pi\)
\(234\) 0 0
\(235\) − 598786.i − 0.00300978i
\(236\) 0 0
\(237\) 6.23514e7i 0.304247i
\(238\) 0 0
\(239\) −2.90175e7 −0.137489 −0.0687443 0.997634i \(-0.521899\pi\)
−0.0687443 + 0.997634i \(0.521899\pi\)
\(240\) 0 0
\(241\) −3.43173e8 −1.57926 −0.789630 0.613583i \(-0.789728\pi\)
−0.789630 + 0.613583i \(0.789728\pi\)
\(242\) 0 0
\(243\) 1.43489e7i 0.0641500i
\(244\) 0 0
\(245\) 3.99413e6i 0.0173517i
\(246\) 0 0
\(247\) 5.74803e8 2.42705
\(248\) 0 0
\(249\) −8.66072e7 −0.355514
\(250\) 0 0
\(251\) − 1.87903e8i − 0.750025i −0.927020 0.375012i \(-0.877638\pi\)
0.927020 0.375012i \(-0.122362\pi\)
\(252\) 0 0
\(253\) 2.72699e8i 1.05867i
\(254\) 0 0
\(255\) −8.61438e6 −0.0325337
\(256\) 0 0
\(257\) 5.70210e7 0.209541 0.104770 0.994496i \(-0.466589\pi\)
0.104770 + 0.994496i \(0.466589\pi\)
\(258\) 0 0
\(259\) − 2.36550e8i − 0.846007i
\(260\) 0 0
\(261\) − 8.24692e7i − 0.287111i
\(262\) 0 0
\(263\) −2.54738e8 −0.863474 −0.431737 0.902000i \(-0.642099\pi\)
−0.431737 + 0.902000i \(0.642099\pi\)
\(264\) 0 0
\(265\) 9.19604e6 0.0303557
\(266\) 0 0
\(267\) − 1.40806e8i − 0.452722i
\(268\) 0 0
\(269\) − 6.31643e8i − 1.97851i −0.146193 0.989256i \(-0.546702\pi\)
0.146193 0.989256i \(-0.453298\pi\)
\(270\) 0 0
\(271\) −2.74328e8 −0.837293 −0.418646 0.908149i \(-0.637495\pi\)
−0.418646 + 0.908149i \(0.637495\pi\)
\(272\) 0 0
\(273\) 2.04186e8 0.607376
\(274\) 0 0
\(275\) − 2.36911e8i − 0.686943i
\(276\) 0 0
\(277\) 3.83552e8i 1.08429i 0.840285 + 0.542145i \(0.182388\pi\)
−0.840285 + 0.542145i \(0.817612\pi\)
\(278\) 0 0
\(279\) −1.79783e8 −0.495602
\(280\) 0 0
\(281\) −4.03791e8 −1.08564 −0.542819 0.839850i \(-0.682643\pi\)
−0.542819 + 0.839850i \(0.682643\pi\)
\(282\) 0 0
\(283\) 4.57245e8i 1.19921i 0.800295 + 0.599607i \(0.204676\pi\)
−0.800295 + 0.599607i \(0.795324\pi\)
\(284\) 0 0
\(285\) 1.13797e7i 0.0291189i
\(286\) 0 0
\(287\) 3.93272e7 0.0981989
\(288\) 0 0
\(289\) 8.46664e8 2.06333
\(290\) 0 0
\(291\) − 8.33306e7i − 0.198234i
\(292\) 0 0
\(293\) 7.48249e8i 1.73784i 0.494953 + 0.868920i \(0.335185\pi\)
−0.494953 + 0.868920i \(0.664815\pi\)
\(294\) 0 0
\(295\) −1.50232e7 −0.0340711
\(296\) 0 0
\(297\) −5.97498e7 −0.132339
\(298\) 0 0
\(299\) 1.10251e9i 2.38524i
\(300\) 0 0
\(301\) 4.29522e8i 0.907826i
\(302\) 0 0
\(303\) −3.90561e8 −0.806566
\(304\) 0 0
\(305\) 1.75093e7 0.0353362
\(306\) 0 0
\(307\) 4.34755e7i 0.0857551i 0.999080 + 0.0428775i \(0.0136525\pi\)
−0.999080 + 0.0428775i \(0.986347\pi\)
\(308\) 0 0
\(309\) 5.12746e7i 0.0988661i
\(310\) 0 0
\(311\) 1.82116e8 0.343310 0.171655 0.985157i \(-0.445089\pi\)
0.171655 + 0.985157i \(0.445089\pi\)
\(312\) 0 0
\(313\) 5.21850e7 0.0961923 0.0480961 0.998843i \(-0.484685\pi\)
0.0480961 + 0.998843i \(0.484685\pi\)
\(314\) 0 0
\(315\) 4.04240e6i 0.00728706i
\(316\) 0 0
\(317\) − 6.76984e7i − 0.119363i −0.998217 0.0596816i \(-0.980991\pi\)
0.998217 0.0596816i \(-0.0190086\pi\)
\(318\) 0 0
\(319\) 3.43407e8 0.592300
\(320\) 0 0
\(321\) −4.16430e8 −0.702707
\(322\) 0 0
\(323\) − 1.66052e9i − 2.74179i
\(324\) 0 0
\(325\) − 9.57817e8i − 1.54771i
\(326\) 0 0
\(327\) −1.78144e8 −0.281743
\(328\) 0 0
\(329\) 4.10015e7 0.0634767
\(330\) 0 0
\(331\) 7.97772e8i 1.20915i 0.796548 + 0.604576i \(0.206657\pi\)
−0.796548 + 0.604576i \(0.793343\pi\)
\(332\) 0 0
\(333\) 2.79854e8i 0.415314i
\(334\) 0 0
\(335\) 458704. 0.000666617 0
\(336\) 0 0
\(337\) 1.52453e7 0.0216986 0.0108493 0.999941i \(-0.496546\pi\)
0.0108493 + 0.999941i \(0.496546\pi\)
\(338\) 0 0
\(339\) − 1.52830e8i − 0.213064i
\(340\) 0 0
\(341\) − 7.48627e8i − 1.02241i
\(342\) 0 0
\(343\) −7.80960e8 −1.04496
\(344\) 0 0
\(345\) −2.18270e7 −0.0286172
\(346\) 0 0
\(347\) 4.45017e8i 0.571773i 0.958263 + 0.285887i \(0.0922881\pi\)
−0.958263 + 0.285887i \(0.907712\pi\)
\(348\) 0 0
\(349\) − 1.38196e9i − 1.74023i −0.492846 0.870117i \(-0.664043\pi\)
0.492846 0.870117i \(-0.335957\pi\)
\(350\) 0 0
\(351\) −2.41565e8 −0.298167
\(352\) 0 0
\(353\) −3.14032e8 −0.379981 −0.189991 0.981786i \(-0.560846\pi\)
−0.189991 + 0.981786i \(0.560846\pi\)
\(354\) 0 0
\(355\) − 4.53692e7i − 0.0538223i
\(356\) 0 0
\(357\) − 5.89864e8i − 0.686140i
\(358\) 0 0
\(359\) −1.14356e9 −1.30445 −0.652224 0.758027i \(-0.726164\pi\)
−0.652224 + 0.758027i \(0.726164\pi\)
\(360\) 0 0
\(361\) −1.29970e9 −1.45401
\(362\) 0 0
\(363\) 2.77351e8i 0.304339i
\(364\) 0 0
\(365\) − 5.23848e7i − 0.0563872i
\(366\) 0 0
\(367\) 1.40602e9 1.48478 0.742388 0.669970i \(-0.233693\pi\)
0.742388 + 0.669970i \(0.233693\pi\)
\(368\) 0 0
\(369\) −4.65266e7 −0.0482068
\(370\) 0 0
\(371\) 6.29693e8i 0.640206i
\(372\) 0 0
\(373\) − 3.92421e8i − 0.391536i −0.980650 0.195768i \(-0.937280\pi\)
0.980650 0.195768i \(-0.0627199\pi\)
\(374\) 0 0
\(375\) 3.79447e7 0.0371571
\(376\) 0 0
\(377\) 1.38838e9 1.33448
\(378\) 0 0
\(379\) − 6.58729e8i − 0.621541i −0.950485 0.310770i \(-0.899413\pi\)
0.950485 0.310770i \(-0.100587\pi\)
\(380\) 0 0
\(381\) − 3.28257e8i − 0.304072i
\(382\) 0 0
\(383\) −1.37352e9 −1.24923 −0.624613 0.780935i \(-0.714743\pi\)
−0.624613 + 0.780935i \(0.714743\pi\)
\(384\) 0 0
\(385\) −1.68328e7 −0.0150330
\(386\) 0 0
\(387\) − 5.08151e8i − 0.445661i
\(388\) 0 0
\(389\) − 5.75082e7i − 0.0495343i −0.999693 0.0247671i \(-0.992116\pi\)
0.999693 0.0247671i \(-0.00788443\pi\)
\(390\) 0 0
\(391\) 3.18497e9 2.69456
\(392\) 0 0
\(393\) 8.78085e8 0.729730
\(394\) 0 0
\(395\) − 2.07814e7i − 0.0169662i
\(396\) 0 0
\(397\) − 4.45178e8i − 0.357082i −0.983933 0.178541i \(-0.942862\pi\)
0.983933 0.178541i \(-0.0571376\pi\)
\(398\) 0 0
\(399\) −7.79219e8 −0.614121
\(400\) 0 0
\(401\) 7.79495e8 0.603682 0.301841 0.953358i \(-0.402399\pi\)
0.301841 + 0.953358i \(0.402399\pi\)
\(402\) 0 0
\(403\) − 3.02666e9i − 2.30354i
\(404\) 0 0
\(405\) − 4.78241e6i − 0.00357729i
\(406\) 0 0
\(407\) −1.16533e9 −0.856778
\(408\) 0 0
\(409\) 1.35057e9 0.976077 0.488039 0.872822i \(-0.337712\pi\)
0.488039 + 0.872822i \(0.337712\pi\)
\(410\) 0 0
\(411\) 1.17307e9i 0.833450i
\(412\) 0 0
\(413\) − 1.02870e9i − 0.718564i
\(414\) 0 0
\(415\) 2.88657e7 0.0198250
\(416\) 0 0
\(417\) 5.69743e7 0.0384772
\(418\) 0 0
\(419\) − 5.86568e8i − 0.389556i −0.980847 0.194778i \(-0.937601\pi\)
0.980847 0.194778i \(-0.0623986\pi\)
\(420\) 0 0
\(421\) − 1.31673e9i − 0.860021i −0.902824 0.430011i \(-0.858510\pi\)
0.902824 0.430011i \(-0.141490\pi\)
\(422\) 0 0
\(423\) −4.85073e7 −0.0311613
\(424\) 0 0
\(425\) −2.76699e9 −1.74842
\(426\) 0 0
\(427\) 1.19894e9i 0.745245i
\(428\) 0 0
\(429\) − 1.00589e9i − 0.615108i
\(430\) 0 0
\(431\) 5.97794e8 0.359651 0.179825 0.983699i \(-0.442447\pi\)
0.179825 + 0.983699i \(0.442447\pi\)
\(432\) 0 0
\(433\) −1.45917e9 −0.863769 −0.431884 0.901929i \(-0.642151\pi\)
−0.431884 + 0.901929i \(0.642151\pi\)
\(434\) 0 0
\(435\) 2.74865e7i 0.0160106i
\(436\) 0 0
\(437\) − 4.20740e9i − 2.41173i
\(438\) 0 0
\(439\) 1.42889e9 0.806069 0.403035 0.915185i \(-0.367955\pi\)
0.403035 + 0.915185i \(0.367955\pi\)
\(440\) 0 0
\(441\) 3.23562e8 0.179648
\(442\) 0 0
\(443\) 3.28479e9i 1.79512i 0.440887 + 0.897562i \(0.354664\pi\)
−0.440887 + 0.897562i \(0.645336\pi\)
\(444\) 0 0
\(445\) 4.69299e7i 0.0252458i
\(446\) 0 0
\(447\) −9.68289e8 −0.512777
\(448\) 0 0
\(449\) −8.38143e8 −0.436974 −0.218487 0.975840i \(-0.570112\pi\)
−0.218487 + 0.975840i \(0.570112\pi\)
\(450\) 0 0
\(451\) − 1.93740e8i − 0.0994491i
\(452\) 0 0
\(453\) − 2.10790e8i − 0.106538i
\(454\) 0 0
\(455\) −6.80542e7 −0.0338700
\(456\) 0 0
\(457\) 1.45042e9 0.710865 0.355432 0.934702i \(-0.384334\pi\)
0.355432 + 0.934702i \(0.384334\pi\)
\(458\) 0 0
\(459\) 6.97846e8i 0.336833i
\(460\) 0 0
\(461\) − 1.19272e9i − 0.567001i −0.958972 0.283500i \(-0.908504\pi\)
0.958972 0.283500i \(-0.0914957\pi\)
\(462\) 0 0
\(463\) 1.74225e9 0.815786 0.407893 0.913030i \(-0.366264\pi\)
0.407893 + 0.913030i \(0.366264\pi\)
\(464\) 0 0
\(465\) 5.99206e7 0.0276370
\(466\) 0 0
\(467\) − 2.77787e9i − 1.26213i −0.775732 0.631063i \(-0.782619\pi\)
0.775732 0.631063i \(-0.217381\pi\)
\(468\) 0 0
\(469\) 3.14095e7i 0.0140590i
\(470\) 0 0
\(471\) −2.15507e9 −0.950359
\(472\) 0 0
\(473\) 2.11598e9 0.919383
\(474\) 0 0
\(475\) 3.65524e9i 1.56490i
\(476\) 0 0
\(477\) − 7.44966e8i − 0.314284i
\(478\) 0 0
\(479\) 3.35409e9 1.39444 0.697221 0.716857i \(-0.254420\pi\)
0.697221 + 0.716857i \(0.254420\pi\)
\(480\) 0 0
\(481\) −4.71136e9 −1.93036
\(482\) 0 0
\(483\) − 1.49459e9i − 0.603540i
\(484\) 0 0
\(485\) 2.77736e7i 0.0110544i
\(486\) 0 0
\(487\) 3.25553e9 1.27723 0.638617 0.769525i \(-0.279507\pi\)
0.638617 + 0.769525i \(0.279507\pi\)
\(488\) 0 0
\(489\) −1.10821e9 −0.428588
\(490\) 0 0
\(491\) − 1.37036e9i − 0.522457i −0.965277 0.261228i \(-0.915872\pi\)
0.965277 0.261228i \(-0.0841276\pi\)
\(492\) 0 0
\(493\) − 4.01081e9i − 1.50754i
\(494\) 0 0
\(495\) 1.99143e7 0.00737984
\(496\) 0 0
\(497\) 3.10663e9 1.13512
\(498\) 0 0
\(499\) 6.61371e7i 0.0238283i 0.999929 + 0.0119142i \(0.00379249\pi\)
−0.999929 + 0.0119142i \(0.996208\pi\)
\(500\) 0 0
\(501\) 2.37354e9i 0.843265i
\(502\) 0 0
\(503\) 8.57092e8 0.300289 0.150145 0.988664i \(-0.452026\pi\)
0.150145 + 0.988664i \(0.452026\pi\)
\(504\) 0 0
\(505\) 1.30172e8 0.0449777
\(506\) 0 0
\(507\) − 2.37256e9i − 0.808519i
\(508\) 0 0
\(509\) − 9.83151e8i − 0.330452i −0.986256 0.165226i \(-0.947165\pi\)
0.986256 0.165226i \(-0.0528353\pi\)
\(510\) 0 0
\(511\) 3.58702e9 1.18921
\(512\) 0 0
\(513\) 9.21864e8 0.301478
\(514\) 0 0
\(515\) − 1.70895e7i − 0.00551322i
\(516\) 0 0
\(517\) − 2.01988e8i − 0.0642848i
\(518\) 0 0
\(519\) −3.48727e9 −1.09496
\(520\) 0 0
\(521\) −3.95745e7 −0.0122598 −0.00612990 0.999981i \(-0.501951\pi\)
−0.00612990 + 0.999981i \(0.501951\pi\)
\(522\) 0 0
\(523\) 2.92926e9i 0.895370i 0.894191 + 0.447685i \(0.147751\pi\)
−0.894191 + 0.447685i \(0.852249\pi\)
\(524\) 0 0
\(525\) 1.29844e9i 0.391621i
\(526\) 0 0
\(527\) −8.74356e9 −2.60226
\(528\) 0 0
\(529\) 4.66521e9 1.37018
\(530\) 0 0
\(531\) 1.21702e9i 0.352750i
\(532\) 0 0
\(533\) − 7.83279e8i − 0.224064i
\(534\) 0 0
\(535\) 1.38794e8 0.0391861
\(536\) 0 0
\(537\) 3.05360e8 0.0850947
\(538\) 0 0
\(539\) 1.34733e9i 0.370608i
\(540\) 0 0
\(541\) − 3.59458e9i − 0.976018i −0.872839 0.488009i \(-0.837723\pi\)
0.872839 0.488009i \(-0.162277\pi\)
\(542\) 0 0
\(543\) −2.74306e9 −0.735250
\(544\) 0 0
\(545\) 5.93744e7 0.0157113
\(546\) 0 0
\(547\) − 2.96194e9i − 0.773786i −0.922125 0.386893i \(-0.873548\pi\)
0.922125 0.386893i \(-0.126452\pi\)
\(548\) 0 0
\(549\) − 1.41842e9i − 0.365848i
\(550\) 0 0
\(551\) −5.29834e9 −1.34930
\(552\) 0 0
\(553\) 1.42299e9 0.357820
\(554\) 0 0
\(555\) − 9.32737e7i − 0.0231598i
\(556\) 0 0
\(557\) 3.16328e9i 0.775611i 0.921741 + 0.387805i \(0.126767\pi\)
−0.921741 + 0.387805i \(0.873233\pi\)
\(558\) 0 0
\(559\) 8.55477e9 2.07142
\(560\) 0 0
\(561\) −2.90588e9 −0.694876
\(562\) 0 0
\(563\) − 3.08736e9i − 0.729135i −0.931177 0.364567i \(-0.881217\pi\)
0.931177 0.364567i \(-0.118783\pi\)
\(564\) 0 0
\(565\) 5.09375e7i 0.0118814i
\(566\) 0 0
\(567\) 3.27473e8 0.0754457
\(568\) 0 0
\(569\) −5.45313e9 −1.24095 −0.620473 0.784228i \(-0.713059\pi\)
−0.620473 + 0.784228i \(0.713059\pi\)
\(570\) 0 0
\(571\) − 4.86211e9i − 1.09294i −0.837477 0.546472i \(-0.815970\pi\)
0.837477 0.546472i \(-0.184030\pi\)
\(572\) 0 0
\(573\) − 6.86495e8i − 0.152439i
\(574\) 0 0
\(575\) −7.01096e9 −1.53794
\(576\) 0 0
\(577\) −1.41178e8 −0.0305950 −0.0152975 0.999883i \(-0.504870\pi\)
−0.0152975 + 0.999883i \(0.504870\pi\)
\(578\) 0 0
\(579\) 2.74333e8i 0.0587359i
\(580\) 0 0
\(581\) 1.97656e9i 0.418113i
\(582\) 0 0
\(583\) 3.10209e9 0.648357
\(584\) 0 0
\(585\) 8.05124e7 0.0166271
\(586\) 0 0
\(587\) 2.79450e9i 0.570257i 0.958489 + 0.285128i \(0.0920363\pi\)
−0.958489 + 0.285128i \(0.907964\pi\)
\(588\) 0 0
\(589\) 1.15504e10i 2.32912i
\(590\) 0 0
\(591\) −1.60457e9 −0.319743
\(592\) 0 0
\(593\) −7.71016e9 −1.51835 −0.759175 0.650887i \(-0.774397\pi\)
−0.759175 + 0.650887i \(0.774397\pi\)
\(594\) 0 0
\(595\) 1.96598e8i 0.0382623i
\(596\) 0 0
\(597\) − 3.58565e9i − 0.689695i
\(598\) 0 0
\(599\) 2.24833e9 0.427432 0.213716 0.976896i \(-0.431443\pi\)
0.213716 + 0.976896i \(0.431443\pi\)
\(600\) 0 0
\(601\) 1.75844e9 0.330421 0.165210 0.986258i \(-0.447170\pi\)
0.165210 + 0.986258i \(0.447170\pi\)
\(602\) 0 0
\(603\) − 3.71594e7i − 0.00690173i
\(604\) 0 0
\(605\) − 9.24397e7i − 0.0169713i
\(606\) 0 0
\(607\) 1.78653e9 0.324227 0.162114 0.986772i \(-0.448169\pi\)
0.162114 + 0.986772i \(0.448169\pi\)
\(608\) 0 0
\(609\) −1.88212e9 −0.337666
\(610\) 0 0
\(611\) − 8.16625e8i − 0.144837i
\(612\) 0 0
\(613\) 5.12242e9i 0.898180i 0.893487 + 0.449090i \(0.148252\pi\)
−0.893487 + 0.449090i \(0.851748\pi\)
\(614\) 0 0
\(615\) 1.55070e7 0.00268823
\(616\) 0 0
\(617\) −3.14822e8 −0.0539594 −0.0269797 0.999636i \(-0.508589\pi\)
−0.0269797 + 0.999636i \(0.508589\pi\)
\(618\) 0 0
\(619\) 4.60016e9i 0.779572i 0.920906 + 0.389786i \(0.127451\pi\)
−0.920906 + 0.389786i \(0.872549\pi\)
\(620\) 0 0
\(621\) 1.76819e9i 0.296284i
\(622\) 0 0
\(623\) −3.21349e9 −0.532438
\(624\) 0 0
\(625\) 6.08454e9 0.996891
\(626\) 0 0
\(627\) 3.83870e9i 0.621940i
\(628\) 0 0
\(629\) 1.36104e10i 2.18069i
\(630\) 0 0
\(631\) 9.57151e9 1.51662 0.758312 0.651892i \(-0.226024\pi\)
0.758312 + 0.651892i \(0.226024\pi\)
\(632\) 0 0
\(633\) 6.95585e9 1.09003
\(634\) 0 0
\(635\) 1.09406e8i 0.0169564i
\(636\) 0 0
\(637\) 5.44720e9i 0.834998i
\(638\) 0 0
\(639\) −3.67533e9 −0.557242
\(640\) 0 0
\(641\) 1.32428e10 1.98598 0.992992 0.118182i \(-0.0377066\pi\)
0.992992 + 0.118182i \(0.0377066\pi\)
\(642\) 0 0
\(643\) − 6.68707e8i − 0.0991967i −0.998769 0.0495984i \(-0.984206\pi\)
0.998769 0.0495984i \(-0.0157941\pi\)
\(644\) 0 0
\(645\) 1.69364e8i 0.0248520i
\(646\) 0 0
\(647\) −6.36051e9 −0.923266 −0.461633 0.887071i \(-0.652736\pi\)
−0.461633 + 0.887071i \(0.652736\pi\)
\(648\) 0 0
\(649\) −5.06776e9 −0.727712
\(650\) 0 0
\(651\) 4.10302e9i 0.582868i
\(652\) 0 0
\(653\) − 2.19579e9i − 0.308599i −0.988024 0.154299i \(-0.950688\pi\)
0.988024 0.154299i \(-0.0493121\pi\)
\(654\) 0 0
\(655\) −2.92661e8 −0.0406930
\(656\) 0 0
\(657\) −4.24366e9 −0.583797
\(658\) 0 0
\(659\) 8.29618e9i 1.12922i 0.825357 + 0.564611i \(0.190974\pi\)
−0.825357 + 0.564611i \(0.809026\pi\)
\(660\) 0 0
\(661\) − 6.60145e9i − 0.889067i −0.895762 0.444533i \(-0.853370\pi\)
0.895762 0.444533i \(-0.146630\pi\)
\(662\) 0 0
\(663\) −1.17483e10 −1.56559
\(664\) 0 0
\(665\) 2.59709e8 0.0342462
\(666\) 0 0
\(667\) − 1.01625e10i − 1.32605i
\(668\) 0 0
\(669\) 5.30294e9i 0.684740i
\(670\) 0 0
\(671\) 5.90639e9 0.754733
\(672\) 0 0
\(673\) −6.92354e9 −0.875539 −0.437769 0.899087i \(-0.644231\pi\)
−0.437769 + 0.899087i \(0.644231\pi\)
\(674\) 0 0
\(675\) − 1.53614e9i − 0.192251i
\(676\) 0 0
\(677\) − 2.27425e8i − 0.0281694i −0.999901 0.0140847i \(-0.995517\pi\)
0.999901 0.0140847i \(-0.00448344\pi\)
\(678\) 0 0
\(679\) −1.90178e9 −0.233140
\(680\) 0 0
\(681\) −2.34188e9 −0.284152
\(682\) 0 0
\(683\) − 2.09687e9i − 0.251825i −0.992041 0.125913i \(-0.959814\pi\)
0.992041 0.125913i \(-0.0401859\pi\)
\(684\) 0 0
\(685\) − 3.90980e8i − 0.0464769i
\(686\) 0 0
\(687\) −3.29845e9 −0.388116
\(688\) 0 0
\(689\) 1.25416e10 1.46078
\(690\) 0 0
\(691\) 6.49885e8i 0.0749313i 0.999298 + 0.0374656i \(0.0119285\pi\)
−0.999298 + 0.0374656i \(0.988072\pi\)
\(692\) 0 0
\(693\) 1.36362e9i 0.155642i
\(694\) 0 0
\(695\) −1.89892e7 −0.00214566
\(696\) 0 0
\(697\) −2.26277e9 −0.253120
\(698\) 0 0
\(699\) − 1.45549e9i − 0.161191i
\(700\) 0 0
\(701\) 1.22103e10i 1.33879i 0.742907 + 0.669394i \(0.233446\pi\)
−0.742907 + 0.669394i \(0.766554\pi\)
\(702\) 0 0
\(703\) 1.79796e10 1.95180
\(704\) 0 0
\(705\) 1.61672e7 0.00173770
\(706\) 0 0
\(707\) 8.91344e9i 0.948587i
\(708\) 0 0
\(709\) 1.06693e10i 1.12428i 0.827041 + 0.562141i \(0.190022\pi\)
−0.827041 + 0.562141i \(0.809978\pi\)
\(710\) 0 0
\(711\) −1.68349e9 −0.175657
\(712\) 0 0
\(713\) −2.21543e10 −2.28899
\(714\) 0 0
\(715\) 3.35259e8i 0.0343012i
\(716\) 0 0
\(717\) − 7.83471e8i − 0.0793791i
\(718\) 0 0
\(719\) 4.40260e9 0.441731 0.220866 0.975304i \(-0.429112\pi\)
0.220866 + 0.975304i \(0.429112\pi\)
\(720\) 0 0
\(721\) 1.17019e9 0.116275
\(722\) 0 0
\(723\) − 9.26568e9i − 0.911786i
\(724\) 0 0
\(725\) 8.82884e9i 0.860440i
\(726\) 0 0
\(727\) 1.63117e10 1.57445 0.787225 0.616666i \(-0.211517\pi\)
0.787225 + 0.616666i \(0.211517\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) − 2.47134e10i − 2.34004i
\(732\) 0 0
\(733\) 1.33690e10i 1.25382i 0.779091 + 0.626911i \(0.215681\pi\)
−0.779091 + 0.626911i \(0.784319\pi\)
\(734\) 0 0
\(735\) −1.07842e8 −0.0100180
\(736\) 0 0
\(737\) 1.54734e8 0.0142380
\(738\) 0 0
\(739\) 1.52697e10i 1.39180i 0.718141 + 0.695898i \(0.244993\pi\)
−0.718141 + 0.695898i \(0.755007\pi\)
\(740\) 0 0
\(741\) 1.55197e10i 1.40126i
\(742\) 0 0
\(743\) −1.40139e9 −0.125342 −0.0626712 0.998034i \(-0.519962\pi\)
−0.0626712 + 0.998034i \(0.519962\pi\)
\(744\) 0 0
\(745\) 3.22725e8 0.0285947
\(746\) 0 0
\(747\) − 2.33840e9i − 0.205256i
\(748\) 0 0
\(749\) 9.50382e9i 0.826441i
\(750\) 0 0
\(751\) 6.12628e9 0.527785 0.263893 0.964552i \(-0.414994\pi\)
0.263893 + 0.964552i \(0.414994\pi\)
\(752\) 0 0
\(753\) 5.07338e9 0.433027
\(754\) 0 0
\(755\) 7.02551e7i 0.00594106i
\(756\) 0 0
\(757\) 1.54406e10i 1.29369i 0.762623 + 0.646844i \(0.223911\pi\)
−0.762623 + 0.646844i \(0.776089\pi\)
\(758\) 0 0
\(759\) −7.36286e9 −0.611224
\(760\) 0 0
\(761\) 5.10270e9 0.419714 0.209857 0.977732i \(-0.432700\pi\)
0.209857 + 0.977732i \(0.432700\pi\)
\(762\) 0 0
\(763\) 4.06562e9i 0.331353i
\(764\) 0 0
\(765\) − 2.32588e8i − 0.0187833i
\(766\) 0 0
\(767\) −2.04887e10 −1.63957
\(768\) 0 0
\(769\) −7.19721e9 −0.570718 −0.285359 0.958421i \(-0.592113\pi\)
−0.285359 + 0.958421i \(0.592113\pi\)
\(770\) 0 0
\(771\) 1.53957e9i 0.120978i
\(772\) 0 0
\(773\) − 1.39957e10i − 1.08985i −0.838486 0.544924i \(-0.816559\pi\)
0.838486 0.544924i \(-0.183441\pi\)
\(774\) 0 0
\(775\) 1.92469e10 1.48527
\(776\) 0 0
\(777\) 6.38686e9 0.488443
\(778\) 0 0
\(779\) 2.98916e9i 0.226552i
\(780\) 0 0
\(781\) − 1.53043e10i − 1.14957i
\(782\) 0 0
\(783\) 2.22667e9 0.165764
\(784\) 0 0
\(785\) 7.18272e8 0.0529963
\(786\) 0 0
\(787\) 1.50580e10i 1.10117i 0.834778 + 0.550587i \(0.185596\pi\)
−0.834778 + 0.550587i \(0.814404\pi\)
\(788\) 0 0
\(789\) − 6.87794e9i − 0.498527i
\(790\) 0 0
\(791\) −3.48791e9 −0.250581
\(792\) 0 0
\(793\) 2.38792e10 1.70045
\(794\) 0 0
\(795\) 2.48293e8i 0.0175259i
\(796\) 0 0
\(797\) − 5.06926e9i − 0.354683i −0.984149 0.177342i \(-0.943250\pi\)
0.984149 0.177342i \(-0.0567498\pi\)
\(798\) 0 0
\(799\) −2.35911e9 −0.163619
\(800\) 0 0
\(801\) 3.80176e9 0.261379
\(802\) 0 0
\(803\) − 1.76709e10i − 1.20435i
\(804\) 0 0
\(805\) 4.98138e8i 0.0336561i
\(806\) 0 0
\(807\) 1.70544e10 1.14229
\(808\) 0 0
\(809\) 5.19545e9 0.344988 0.172494 0.985011i \(-0.444818\pi\)
0.172494 + 0.985011i \(0.444818\pi\)
\(810\) 0 0
\(811\) − 2.10691e10i − 1.38699i −0.720461 0.693496i \(-0.756070\pi\)
0.720461 0.693496i \(-0.243930\pi\)
\(812\) 0 0
\(813\) − 7.40685e9i − 0.483411i
\(814\) 0 0
\(815\) 3.69360e8 0.0239000
\(816\) 0 0
\(817\) −3.26468e10 −2.09442
\(818\) 0 0
\(819\) 5.51303e9i 0.350669i
\(820\) 0 0
\(821\) 5.98065e9i 0.377179i 0.982056 + 0.188590i \(0.0603916\pi\)
−0.982056 + 0.188590i \(0.939608\pi\)
\(822\) 0 0
\(823\) −7.62100e9 −0.476554 −0.238277 0.971197i \(-0.576583\pi\)
−0.238277 + 0.971197i \(0.576583\pi\)
\(824\) 0 0
\(825\) 6.39659e9 0.396606
\(826\) 0 0
\(827\) 1.30836e10i 0.804373i 0.915558 + 0.402187i \(0.131750\pi\)
−0.915558 + 0.402187i \(0.868250\pi\)
\(828\) 0 0
\(829\) 5.21207e9i 0.317738i 0.987300 + 0.158869i \(0.0507848\pi\)
−0.987300 + 0.158869i \(0.949215\pi\)
\(830\) 0 0
\(831\) −1.03559e10 −0.626015
\(832\) 0 0
\(833\) 1.57361e10 0.943280
\(834\) 0 0
\(835\) − 7.91087e8i − 0.0470242i
\(836\) 0 0
\(837\) − 4.85413e9i − 0.286136i
\(838\) 0 0
\(839\) −9.15426e9 −0.535126 −0.267563 0.963540i \(-0.586218\pi\)
−0.267563 + 0.963540i \(0.586218\pi\)
\(840\) 0 0
\(841\) 4.45229e9 0.258106
\(842\) 0 0
\(843\) − 1.09024e10i − 0.626793i
\(844\) 0 0
\(845\) 7.90763e8i 0.0450866i
\(846\) 0 0
\(847\) 6.32975e9 0.357927
\(848\) 0 0
\(849\) −1.23456e10 −0.692367
\(850\) 0 0
\(851\) 3.44859e10i 1.91817i
\(852\) 0 0
\(853\) 1.58918e10i 0.876699i 0.898805 + 0.438349i \(0.144437\pi\)
−0.898805 + 0.438349i \(0.855563\pi\)
\(854\) 0 0
\(855\) −3.07252e8 −0.0168118
\(856\) 0 0
\(857\) −3.04443e10 −1.65224 −0.826120 0.563495i \(-0.809457\pi\)
−0.826120 + 0.563495i \(0.809457\pi\)
\(858\) 0 0
\(859\) − 1.79263e10i − 0.964971i −0.875904 0.482486i \(-0.839734\pi\)
0.875904 0.482486i \(-0.160266\pi\)
\(860\) 0 0
\(861\) 1.06183e9i 0.0566952i
\(862\) 0 0
\(863\) 7.12123e9 0.377153 0.188576 0.982059i \(-0.439613\pi\)
0.188576 + 0.982059i \(0.439613\pi\)
\(864\) 0 0
\(865\) 1.16229e9 0.0610600
\(866\) 0 0
\(867\) 2.28599e10i 1.19126i
\(868\) 0 0
\(869\) − 7.01015e9i − 0.362375i
\(870\) 0 0
\(871\) 6.25582e8 0.0320790
\(872\) 0 0
\(873\) 2.24993e9 0.114451
\(874\) 0 0
\(875\) − 8.65978e8i − 0.0436997i
\(876\) 0 0
\(877\) − 5.49211e8i − 0.0274942i −0.999906 0.0137471i \(-0.995624\pi\)
0.999906 0.0137471i \(-0.00437597\pi\)
\(878\) 0 0
\(879\) −2.02027e10 −1.00334
\(880\) 0 0
\(881\) 2.12382e10 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(882\) 0 0
\(883\) 2.26798e10i 1.10861i 0.832315 + 0.554303i \(0.187015\pi\)
−0.832315 + 0.554303i \(0.812985\pi\)
\(884\) 0 0
\(885\) − 4.05627e8i − 0.0196709i
\(886\) 0 0
\(887\) −2.09480e10 −1.00788 −0.503942 0.863737i \(-0.668117\pi\)
−0.503942 + 0.863737i \(0.668117\pi\)
\(888\) 0 0
\(889\) −7.49153e9 −0.357614
\(890\) 0 0
\(891\) − 1.61324e9i − 0.0764062i
\(892\) 0 0
\(893\) 3.11642e9i 0.146445i
\(894\) 0 0
\(895\) −1.01775e8 −0.00474526
\(896\) 0 0
\(897\) −2.97677e10 −1.37712
\(898\) 0 0
\(899\) 2.78987e10i 1.28064i
\(900\) 0 0
\(901\) − 3.62307e10i − 1.65021i
\(902\) 0 0
\(903\) −1.15971e10 −0.524133
\(904\) 0 0
\(905\) 9.14246e8 0.0410009
\(906\) 0 0
\(907\) − 2.65970e10i − 1.18361i −0.806083 0.591803i \(-0.798416\pi\)
0.806083 0.591803i \(-0.201584\pi\)
\(908\) 0 0
\(909\) − 1.05452e10i − 0.465671i
\(910\) 0 0
\(911\) 1.28193e10 0.561760 0.280880 0.959743i \(-0.409374\pi\)
0.280880 + 0.959743i \(0.409374\pi\)
\(912\) 0 0
\(913\) 9.73723e9 0.423436
\(914\) 0 0
\(915\) 4.72751e8i 0.0204013i
\(916\) 0 0
\(917\) − 2.00398e10i − 0.858222i
\(918\) 0 0
\(919\) −3.23675e10 −1.37564 −0.687820 0.725881i \(-0.741432\pi\)
−0.687820 + 0.725881i \(0.741432\pi\)
\(920\) 0 0
\(921\) −1.17384e9 −0.0495107
\(922\) 0 0
\(923\) − 6.18746e10i − 2.59004i
\(924\) 0 0
\(925\) − 2.99601e10i − 1.24465i
\(926\) 0 0
\(927\) −1.38441e9 −0.0570804
\(928\) 0 0
\(929\) −1.31144e10 −0.536653 −0.268327 0.963328i \(-0.586471\pi\)
−0.268327 + 0.963328i \(0.586471\pi\)
\(930\) 0 0
\(931\) − 2.07877e10i − 0.844271i
\(932\) 0 0
\(933\) 4.91713e9i 0.198210i
\(934\) 0 0
\(935\) 9.68513e8 0.0387494
\(936\) 0 0
\(937\) 3.21245e10 1.27570 0.637849 0.770161i \(-0.279824\pi\)
0.637849 + 0.770161i \(0.279824\pi\)
\(938\) 0 0
\(939\) 1.40899e9i 0.0555366i
\(940\) 0 0
\(941\) 1.15677e10i 0.452566i 0.974062 + 0.226283i \(0.0726574\pi\)
−0.974062 + 0.226283i \(0.927343\pi\)
\(942\) 0 0
\(943\) −5.73338e9 −0.222649
\(944\) 0 0
\(945\) −1.09145e8 −0.00420719
\(946\) 0 0
\(947\) − 1.21746e10i − 0.465833i −0.972497 0.232916i \(-0.925173\pi\)
0.972497 0.232916i \(-0.0748269\pi\)
\(948\) 0 0
\(949\) − 7.14425e10i − 2.71347i
\(950\) 0 0
\(951\) 1.82786e9 0.0689144
\(952\) 0 0
\(953\) −2.00590e10 −0.750730 −0.375365 0.926877i \(-0.622483\pi\)
−0.375365 + 0.926877i \(0.622483\pi\)
\(954\) 0 0
\(955\) 2.28805e8i 0.00850069i
\(956\) 0 0
\(957\) 9.27199e9i 0.341965i
\(958\) 0 0
\(959\) 2.67721e10 0.980205
\(960\) 0 0
\(961\) 3.33066e10 1.21059
\(962\) 0 0
\(963\) − 1.12436e10i − 0.405708i
\(964\) 0 0
\(965\) − 9.14338e7i − 0.00327538i
\(966\) 0 0
\(967\) 3.05520e10 1.08654 0.543271 0.839558i \(-0.317186\pi\)
0.543271 + 0.839558i \(0.317186\pi\)
\(968\) 0 0
\(969\) 4.48340e10 1.58298
\(970\) 0 0
\(971\) 4.05142e10i 1.42017i 0.704117 + 0.710084i \(0.251343\pi\)
−0.704117 + 0.710084i \(0.748657\pi\)
\(972\) 0 0
\(973\) − 1.30028e9i − 0.0452523i
\(974\) 0 0
\(975\) 2.58611e10 0.893574
\(976\) 0 0
\(977\) 2.00689e10 0.688483 0.344242 0.938881i \(-0.388136\pi\)
0.344242 + 0.938881i \(0.388136\pi\)
\(978\) 0 0
\(979\) 1.58308e10i 0.539216i
\(980\) 0 0
\(981\) − 4.80988e9i − 0.162665i
\(982\) 0 0
\(983\) 3.88669e10 1.30510 0.652548 0.757748i \(-0.273700\pi\)
0.652548 + 0.757748i \(0.273700\pi\)
\(984\) 0 0
\(985\) 5.34794e8 0.0178303
\(986\) 0 0
\(987\) 1.10704e9i 0.0366483i
\(988\) 0 0
\(989\) − 6.26186e10i − 2.05834i
\(990\) 0 0
\(991\) −4.43097e10 −1.44624 −0.723122 0.690721i \(-0.757293\pi\)
−0.723122 + 0.690721i \(0.757293\pi\)
\(992\) 0 0
\(993\) −2.15398e10 −0.698104
\(994\) 0 0
\(995\) 1.19508e9i 0.0384605i
\(996\) 0 0
\(997\) 3.67500e10i 1.17442i 0.809433 + 0.587212i \(0.199774\pi\)
−0.809433 + 0.587212i \(0.800226\pi\)
\(998\) 0 0
\(999\) −7.55605e9 −0.239781
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 384.8.d.e.193.9 yes 12
4.3 odd 2 inner 384.8.d.e.193.3 12
8.3 odd 2 inner 384.8.d.e.193.10 yes 12
8.5 even 2 inner 384.8.d.e.193.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.e.193.3 12 4.3 odd 2 inner
384.8.d.e.193.4 yes 12 8.5 even 2 inner
384.8.d.e.193.9 yes 12 1.1 even 1 trivial
384.8.d.e.193.10 yes 12 8.3 odd 2 inner