Properties

Label 384.5.b.d.319.11
Level $384$
Weight $5$
Character 384.319
Analytic conductor $39.694$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,5,Mod(319,384)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.b (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: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + \cdots + 362673936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{84}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.11
Root \(-1.45110 + 3.51338i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.5.b.d.319.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.19615 q^{3} -19.1142i q^{5} +6.40010i q^{7} +27.0000 q^{9} +O(q^{10})\) \(q+5.19615 q^{3} -19.1142i q^{5} +6.40010i q^{7} +27.0000 q^{9} -67.7877 q^{11} +154.346i q^{13} -99.3205i q^{15} +284.063 q^{17} +434.548 q^{19} +33.2559i q^{21} -462.698i q^{23} +259.646 q^{25} +140.296 q^{27} -350.108i q^{29} +88.4523i q^{31} -352.235 q^{33} +122.333 q^{35} -1636.60i q^{37} +802.006i q^{39} +395.726 q^{41} -350.192 q^{43} -516.084i q^{45} -2459.53i q^{47} +2360.04 q^{49} +1476.04 q^{51} -3827.15i q^{53} +1295.71i q^{55} +2257.98 q^{57} -1754.25 q^{59} -2335.10i q^{61} +172.803i q^{63} +2950.21 q^{65} -2198.93 q^{67} -2404.25i q^{69} +4463.62i q^{71} +2592.13 q^{73} +1349.16 q^{75} -433.848i q^{77} +2040.82i q^{79} +729.000 q^{81} -5245.36 q^{83} -5429.65i q^{85} -1819.22i q^{87} +14320.4 q^{89} -987.830 q^{91} +459.612i q^{93} -8306.04i q^{95} +16844.8 q^{97} -1830.27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 432 q^{9} + 480 q^{17} + 144 q^{25} + 2880 q^{33} - 3552 q^{41} - 20080 q^{49} - 7488 q^{57} + 23040 q^{65} + 34400 q^{73} + 11664 q^{81} - 15648 q^{89} + 5088 q^{97}+O(q^{100}) \) 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
\(3\) 5.19615 0.577350
\(4\) 0 0
\(5\) − 19.1142i − 0.764569i −0.924045 0.382285i \(-0.875137\pi\)
0.924045 0.382285i \(-0.124863\pi\)
\(6\) 0 0
\(7\) 6.40010i 0.130614i 0.997865 + 0.0653071i \(0.0208027\pi\)
−0.997865 + 0.0653071i \(0.979197\pi\)
\(8\) 0 0
\(9\) 27.0000 0.333333
\(10\) 0 0
\(11\) −67.7877 −0.560229 −0.280114 0.959967i \(-0.590372\pi\)
−0.280114 + 0.959967i \(0.590372\pi\)
\(12\) 0 0
\(13\) 154.346i 0.913290i 0.889649 + 0.456645i \(0.150949\pi\)
−0.889649 + 0.456645i \(0.849051\pi\)
\(14\) 0 0
\(15\) − 99.3205i − 0.441424i
\(16\) 0 0
\(17\) 284.063 0.982918 0.491459 0.870901i \(-0.336464\pi\)
0.491459 + 0.870901i \(0.336464\pi\)
\(18\) 0 0
\(19\) 434.548 1.20373 0.601866 0.798597i \(-0.294424\pi\)
0.601866 + 0.798597i \(0.294424\pi\)
\(20\) 0 0
\(21\) 33.2559i 0.0754102i
\(22\) 0 0
\(23\) − 462.698i − 0.874666i −0.899300 0.437333i \(-0.855923\pi\)
0.899300 0.437333i \(-0.144077\pi\)
\(24\) 0 0
\(25\) 259.646 0.415434
\(26\) 0 0
\(27\) 140.296 0.192450
\(28\) 0 0
\(29\) − 350.108i − 0.416300i −0.978097 0.208150i \(-0.933256\pi\)
0.978097 0.208150i \(-0.0667442\pi\)
\(30\) 0 0
\(31\) 88.4523i 0.0920420i 0.998940 + 0.0460210i \(0.0146541\pi\)
−0.998940 + 0.0460210i \(0.985346\pi\)
\(32\) 0 0
\(33\) −352.235 −0.323448
\(34\) 0 0
\(35\) 122.333 0.0998637
\(36\) 0 0
\(37\) − 1636.60i − 1.19547i −0.801694 0.597734i \(-0.796068\pi\)
0.801694 0.597734i \(-0.203932\pi\)
\(38\) 0 0
\(39\) 802.006i 0.527288i
\(40\) 0 0
\(41\) 395.726 0.235411 0.117706 0.993049i \(-0.462446\pi\)
0.117706 + 0.993049i \(0.462446\pi\)
\(42\) 0 0
\(43\) −350.192 −0.189396 −0.0946978 0.995506i \(-0.530188\pi\)
−0.0946978 + 0.995506i \(0.530188\pi\)
\(44\) 0 0
\(45\) − 516.084i − 0.254856i
\(46\) 0 0
\(47\) − 2459.53i − 1.11341i −0.830709 0.556707i \(-0.812065\pi\)
0.830709 0.556707i \(-0.187935\pi\)
\(48\) 0 0
\(49\) 2360.04 0.982940
\(50\) 0 0
\(51\) 1476.04 0.567488
\(52\) 0 0
\(53\) − 3827.15i − 1.36246i −0.732069 0.681230i \(-0.761445\pi\)
0.732069 0.681230i \(-0.238555\pi\)
\(54\) 0 0
\(55\) 1295.71i 0.428334i
\(56\) 0 0
\(57\) 2257.98 0.694975
\(58\) 0 0
\(59\) −1754.25 −0.503951 −0.251975 0.967734i \(-0.581080\pi\)
−0.251975 + 0.967734i \(0.581080\pi\)
\(60\) 0 0
\(61\) − 2335.10i − 0.627546i −0.949498 0.313773i \(-0.898407\pi\)
0.949498 0.313773i \(-0.101593\pi\)
\(62\) 0 0
\(63\) 172.803i 0.0435381i
\(64\) 0 0
\(65\) 2950.21 0.698274
\(66\) 0 0
\(67\) −2198.93 −0.489848 −0.244924 0.969542i \(-0.578763\pi\)
−0.244924 + 0.969542i \(0.578763\pi\)
\(68\) 0 0
\(69\) − 2404.25i − 0.504989i
\(70\) 0 0
\(71\) 4463.62i 0.885463i 0.896654 + 0.442732i \(0.145991\pi\)
−0.896654 + 0.442732i \(0.854009\pi\)
\(72\) 0 0
\(73\) 2592.13 0.486420 0.243210 0.969974i \(-0.421800\pi\)
0.243210 + 0.969974i \(0.421800\pi\)
\(74\) 0 0
\(75\) 1349.16 0.239851
\(76\) 0 0
\(77\) − 433.848i − 0.0731738i
\(78\) 0 0
\(79\) 2040.82i 0.327002i 0.986543 + 0.163501i \(0.0522788\pi\)
−0.986543 + 0.163501i \(0.947721\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) −5245.36 −0.761411 −0.380706 0.924696i \(-0.624319\pi\)
−0.380706 + 0.924696i \(0.624319\pi\)
\(84\) 0 0
\(85\) − 5429.65i − 0.751509i
\(86\) 0 0
\(87\) − 1819.22i − 0.240351i
\(88\) 0 0
\(89\) 14320.4 1.80790 0.903952 0.427635i \(-0.140653\pi\)
0.903952 + 0.427635i \(0.140653\pi\)
\(90\) 0 0
\(91\) −987.830 −0.119289
\(92\) 0 0
\(93\) 459.612i 0.0531405i
\(94\) 0 0
\(95\) − 8306.04i − 0.920337i
\(96\) 0 0
\(97\) 16844.8 1.79029 0.895145 0.445776i \(-0.147072\pi\)
0.895145 + 0.445776i \(0.147072\pi\)
\(98\) 0 0
\(99\) −1830.27 −0.186743
\(100\) 0 0
\(101\) − 8909.06i − 0.873352i −0.899619 0.436676i \(-0.856156\pi\)
0.899619 0.436676i \(-0.143844\pi\)
\(102\) 0 0
\(103\) 19586.8i 1.84624i 0.384507 + 0.923122i \(0.374371\pi\)
−0.384507 + 0.923122i \(0.625629\pi\)
\(104\) 0 0
\(105\) 635.661 0.0576563
\(106\) 0 0
\(107\) 13789.3 1.20441 0.602207 0.798340i \(-0.294288\pi\)
0.602207 + 0.798340i \(0.294288\pi\)
\(108\) 0 0
\(109\) − 13179.9i − 1.10933i −0.832074 0.554665i \(-0.812847\pi\)
0.832074 0.554665i \(-0.187153\pi\)
\(110\) 0 0
\(111\) − 8504.00i − 0.690204i
\(112\) 0 0
\(113\) −2732.48 −0.213993 −0.106997 0.994259i \(-0.534123\pi\)
−0.106997 + 0.994259i \(0.534123\pi\)
\(114\) 0 0
\(115\) −8844.13 −0.668743
\(116\) 0 0
\(117\) 4167.34i 0.304430i
\(118\) 0 0
\(119\) 1818.03i 0.128383i
\(120\) 0 0
\(121\) −10045.8 −0.686144
\(122\) 0 0
\(123\) 2056.25 0.135915
\(124\) 0 0
\(125\) − 16909.3i − 1.08220i
\(126\) 0 0
\(127\) − 23919.4i − 1.48301i −0.670949 0.741503i \(-0.734113\pi\)
0.670949 0.741503i \(-0.265887\pi\)
\(128\) 0 0
\(129\) −1819.65 −0.109348
\(130\) 0 0
\(131\) 25059.5 1.46026 0.730129 0.683309i \(-0.239460\pi\)
0.730129 + 0.683309i \(0.239460\pi\)
\(132\) 0 0
\(133\) 2781.15i 0.157225i
\(134\) 0 0
\(135\) − 2681.65i − 0.147141i
\(136\) 0 0
\(137\) 528.551 0.0281609 0.0140804 0.999901i \(-0.495518\pi\)
0.0140804 + 0.999901i \(0.495518\pi\)
\(138\) 0 0
\(139\) −24051.5 −1.24484 −0.622419 0.782684i \(-0.713850\pi\)
−0.622419 + 0.782684i \(0.713850\pi\)
\(140\) 0 0
\(141\) − 12780.1i − 0.642830i
\(142\) 0 0
\(143\) − 10462.8i − 0.511651i
\(144\) 0 0
\(145\) −6692.06 −0.318290
\(146\) 0 0
\(147\) 12263.1 0.567501
\(148\) 0 0
\(149\) 5229.61i 0.235557i 0.993040 + 0.117779i \(0.0375773\pi\)
−0.993040 + 0.117779i \(0.962423\pi\)
\(150\) 0 0
\(151\) 26586.7i 1.16603i 0.812460 + 0.583017i \(0.198128\pi\)
−0.812460 + 0.583017i \(0.801872\pi\)
\(152\) 0 0
\(153\) 7669.71 0.327639
\(154\) 0 0
\(155\) 1690.70 0.0703725
\(156\) 0 0
\(157\) 10616.7i 0.430717i 0.976535 + 0.215359i \(0.0690920\pi\)
−0.976535 + 0.215359i \(0.930908\pi\)
\(158\) 0 0
\(159\) − 19886.5i − 0.786617i
\(160\) 0 0
\(161\) 2961.32 0.114244
\(162\) 0 0
\(163\) −13539.3 −0.509589 −0.254794 0.966995i \(-0.582008\pi\)
−0.254794 + 0.966995i \(0.582008\pi\)
\(164\) 0 0
\(165\) 6732.70i 0.247299i
\(166\) 0 0
\(167\) 31228.7i 1.11975i 0.828577 + 0.559876i \(0.189151\pi\)
−0.828577 + 0.559876i \(0.810849\pi\)
\(168\) 0 0
\(169\) 4738.29 0.165901
\(170\) 0 0
\(171\) 11732.8 0.401244
\(172\) 0 0
\(173\) 42767.2i 1.42895i 0.699659 + 0.714477i \(0.253335\pi\)
−0.699659 + 0.714477i \(0.746665\pi\)
\(174\) 0 0
\(175\) 1661.76i 0.0542615i
\(176\) 0 0
\(177\) −9115.36 −0.290956
\(178\) 0 0
\(179\) 16667.9 0.520206 0.260103 0.965581i \(-0.416244\pi\)
0.260103 + 0.965581i \(0.416244\pi\)
\(180\) 0 0
\(181\) 23689.5i 0.723101i 0.932353 + 0.361550i \(0.117752\pi\)
−0.932353 + 0.361550i \(0.882248\pi\)
\(182\) 0 0
\(183\) − 12133.5i − 0.362314i
\(184\) 0 0
\(185\) −31282.3 −0.914019
\(186\) 0 0
\(187\) −19256.0 −0.550659
\(188\) 0 0
\(189\) 897.909i 0.0251367i
\(190\) 0 0
\(191\) − 67099.3i − 1.83929i −0.392747 0.919647i \(-0.628475\pi\)
0.392747 0.919647i \(-0.371525\pi\)
\(192\) 0 0
\(193\) −11239.9 −0.301750 −0.150875 0.988553i \(-0.548209\pi\)
−0.150875 + 0.988553i \(0.548209\pi\)
\(194\) 0 0
\(195\) 15329.7 0.403149
\(196\) 0 0
\(197\) 14984.6i 0.386113i 0.981188 + 0.193056i \(0.0618400\pi\)
−0.981188 + 0.193056i \(0.938160\pi\)
\(198\) 0 0
\(199\) 71104.7i 1.79553i 0.440477 + 0.897764i \(0.354809\pi\)
−0.440477 + 0.897764i \(0.645191\pi\)
\(200\) 0 0
\(201\) −11426.0 −0.282814
\(202\) 0 0
\(203\) 2240.73 0.0543747
\(204\) 0 0
\(205\) − 7564.01i − 0.179988i
\(206\) 0 0
\(207\) − 12492.9i − 0.291555i
\(208\) 0 0
\(209\) −29457.0 −0.674365
\(210\) 0 0
\(211\) −60965.8 −1.36937 −0.684686 0.728838i \(-0.740061\pi\)
−0.684686 + 0.728838i \(0.740061\pi\)
\(212\) 0 0
\(213\) 23193.7i 0.511222i
\(214\) 0 0
\(215\) 6693.66i 0.144806i
\(216\) 0 0
\(217\) −566.104 −0.0120220
\(218\) 0 0
\(219\) 13469.1 0.280835
\(220\) 0 0
\(221\) 43844.0i 0.897689i
\(222\) 0 0
\(223\) 87167.1i 1.75284i 0.481545 + 0.876422i \(0.340076\pi\)
−0.481545 + 0.876422i \(0.659924\pi\)
\(224\) 0 0
\(225\) 7010.44 0.138478
\(226\) 0 0
\(227\) −14968.7 −0.290492 −0.145246 0.989396i \(-0.546397\pi\)
−0.145246 + 0.989396i \(0.546397\pi\)
\(228\) 0 0
\(229\) 78306.4i 1.49323i 0.665257 + 0.746615i \(0.268322\pi\)
−0.665257 + 0.746615i \(0.731678\pi\)
\(230\) 0 0
\(231\) − 2254.34i − 0.0422469i
\(232\) 0 0
\(233\) −54896.5 −1.01119 −0.505595 0.862771i \(-0.668727\pi\)
−0.505595 + 0.862771i \(0.668727\pi\)
\(234\) 0 0
\(235\) −47012.1 −0.851283
\(236\) 0 0
\(237\) 10604.4i 0.188795i
\(238\) 0 0
\(239\) − 66861.2i − 1.17052i −0.810846 0.585259i \(-0.800993\pi\)
0.810846 0.585259i \(-0.199007\pi\)
\(240\) 0 0
\(241\) −99876.4 −1.71961 −0.859803 0.510626i \(-0.829414\pi\)
−0.859803 + 0.510626i \(0.829414\pi\)
\(242\) 0 0
\(243\) 3788.00 0.0641500
\(244\) 0 0
\(245\) − 45110.3i − 0.751526i
\(246\) 0 0
\(247\) 67070.7i 1.09936i
\(248\) 0 0
\(249\) −27255.7 −0.439601
\(250\) 0 0
\(251\) 18469.6 0.293164 0.146582 0.989199i \(-0.453173\pi\)
0.146582 + 0.989199i \(0.453173\pi\)
\(252\) 0 0
\(253\) 31365.2i 0.490013i
\(254\) 0 0
\(255\) − 28213.3i − 0.433884i
\(256\) 0 0
\(257\) 12430.9 0.188208 0.0941040 0.995562i \(-0.470001\pi\)
0.0941040 + 0.995562i \(0.470001\pi\)
\(258\) 0 0
\(259\) 10474.4 0.156145
\(260\) 0 0
\(261\) − 9452.93i − 0.138767i
\(262\) 0 0
\(263\) − 50546.1i − 0.730762i −0.930858 0.365381i \(-0.880939\pi\)
0.930858 0.365381i \(-0.119061\pi\)
\(264\) 0 0
\(265\) −73153.1 −1.04170
\(266\) 0 0
\(267\) 74411.0 1.04379
\(268\) 0 0
\(269\) 88678.2i 1.22550i 0.790278 + 0.612748i \(0.209936\pi\)
−0.790278 + 0.612748i \(0.790064\pi\)
\(270\) 0 0
\(271\) − 18805.0i − 0.256056i −0.991771 0.128028i \(-0.959135\pi\)
0.991771 0.128028i \(-0.0408647\pi\)
\(272\) 0 0
\(273\) −5132.92 −0.0688714
\(274\) 0 0
\(275\) −17600.8 −0.232738
\(276\) 0 0
\(277\) − 37803.3i − 0.492686i −0.969183 0.246343i \(-0.920771\pi\)
0.969183 0.246343i \(-0.0792289\pi\)
\(278\) 0 0
\(279\) 2388.21i 0.0306807i
\(280\) 0 0
\(281\) 25583.9 0.324006 0.162003 0.986790i \(-0.448205\pi\)
0.162003 + 0.986790i \(0.448205\pi\)
\(282\) 0 0
\(283\) −42546.3 −0.531237 −0.265619 0.964078i \(-0.585576\pi\)
−0.265619 + 0.964078i \(0.585576\pi\)
\(284\) 0 0
\(285\) − 43159.5i − 0.531357i
\(286\) 0 0
\(287\) 2532.69i 0.0307481i
\(288\) 0 0
\(289\) −2829.08 −0.0338726
\(290\) 0 0
\(291\) 87528.3 1.03362
\(292\) 0 0
\(293\) − 101982.i − 1.18793i −0.804492 0.593963i \(-0.797562\pi\)
0.804492 0.593963i \(-0.202438\pi\)
\(294\) 0 0
\(295\) 33531.2i 0.385305i
\(296\) 0 0
\(297\) −9510.34 −0.107816
\(298\) 0 0
\(299\) 71415.7 0.798824
\(300\) 0 0
\(301\) − 2241.27i − 0.0247378i
\(302\) 0 0
\(303\) − 46292.9i − 0.504230i
\(304\) 0 0
\(305\) −44633.7 −0.479803
\(306\) 0 0
\(307\) −170124. −1.80505 −0.902523 0.430642i \(-0.858287\pi\)
−0.902523 + 0.430642i \(0.858287\pi\)
\(308\) 0 0
\(309\) 101776.i 1.06593i
\(310\) 0 0
\(311\) − 41592.7i − 0.430028i −0.976611 0.215014i \(-0.931020\pi\)
0.976611 0.215014i \(-0.0689797\pi\)
\(312\) 0 0
\(313\) 121687. 1.24210 0.621049 0.783772i \(-0.286707\pi\)
0.621049 + 0.783772i \(0.286707\pi\)
\(314\) 0 0
\(315\) 3302.99 0.0332879
\(316\) 0 0
\(317\) 126498.i 1.25882i 0.777073 + 0.629411i \(0.216704\pi\)
−0.777073 + 0.629411i \(0.783296\pi\)
\(318\) 0 0
\(319\) 23733.0i 0.233223i
\(320\) 0 0
\(321\) 71651.5 0.695368
\(322\) 0 0
\(323\) 123439. 1.18317
\(324\) 0 0
\(325\) 40075.3i 0.379411i
\(326\) 0 0
\(327\) − 68485.0i − 0.640471i
\(328\) 0 0
\(329\) 15741.3 0.145428
\(330\) 0 0
\(331\) 33699.5 0.307586 0.153793 0.988103i \(-0.450851\pi\)
0.153793 + 0.988103i \(0.450851\pi\)
\(332\) 0 0
\(333\) − 44188.1i − 0.398489i
\(334\) 0 0
\(335\) 42030.8i 0.374523i
\(336\) 0 0
\(337\) 92980.8 0.818717 0.409358 0.912374i \(-0.365753\pi\)
0.409358 + 0.912374i \(0.365753\pi\)
\(338\) 0 0
\(339\) −14198.4 −0.123549
\(340\) 0 0
\(341\) − 5995.98i − 0.0515645i
\(342\) 0 0
\(343\) 30471.1i 0.259000i
\(344\) 0 0
\(345\) −45955.4 −0.386099
\(346\) 0 0
\(347\) −6718.56 −0.0557978 −0.0278989 0.999611i \(-0.508882\pi\)
−0.0278989 + 0.999611i \(0.508882\pi\)
\(348\) 0 0
\(349\) − 79498.6i − 0.652693i −0.945250 0.326346i \(-0.894182\pi\)
0.945250 0.326346i \(-0.105818\pi\)
\(350\) 0 0
\(351\) 21654.2i 0.175763i
\(352\) 0 0
\(353\) 45108.7 0.362002 0.181001 0.983483i \(-0.442066\pi\)
0.181001 + 0.983483i \(0.442066\pi\)
\(354\) 0 0
\(355\) 85318.7 0.676998
\(356\) 0 0
\(357\) 9446.78i 0.0741220i
\(358\) 0 0
\(359\) − 171192.i − 1.32829i −0.747602 0.664147i \(-0.768795\pi\)
0.747602 0.664147i \(-0.231205\pi\)
\(360\) 0 0
\(361\) 58510.5 0.448972
\(362\) 0 0
\(363\) −52199.7 −0.396145
\(364\) 0 0
\(365\) − 49546.6i − 0.371902i
\(366\) 0 0
\(367\) 197479.i 1.46618i 0.680130 + 0.733091i \(0.261923\pi\)
−0.680130 + 0.733091i \(0.738077\pi\)
\(368\) 0 0
\(369\) 10684.6 0.0784704
\(370\) 0 0
\(371\) 24494.1 0.177957
\(372\) 0 0
\(373\) − 78469.2i − 0.564003i −0.959414 0.282002i \(-0.909002\pi\)
0.959414 0.282002i \(-0.0909984\pi\)
\(374\) 0 0
\(375\) − 87863.5i − 0.624807i
\(376\) 0 0
\(377\) 54037.9 0.380203
\(378\) 0 0
\(379\) −248236. −1.72817 −0.864086 0.503344i \(-0.832103\pi\)
−0.864086 + 0.503344i \(0.832103\pi\)
\(380\) 0 0
\(381\) − 124289.i − 0.856214i
\(382\) 0 0
\(383\) 75091.5i 0.511910i 0.966689 + 0.255955i \(0.0823898\pi\)
−0.966689 + 0.255955i \(0.917610\pi\)
\(384\) 0 0
\(385\) −8292.67 −0.0559465
\(386\) 0 0
\(387\) −9455.19 −0.0631319
\(388\) 0 0
\(389\) − 280557.i − 1.85405i −0.375000 0.927025i \(-0.622357\pi\)
0.375000 0.927025i \(-0.377643\pi\)
\(390\) 0 0
\(391\) − 131436.i − 0.859725i
\(392\) 0 0
\(393\) 130213. 0.843081
\(394\) 0 0
\(395\) 39008.8 0.250016
\(396\) 0 0
\(397\) − 225848.i − 1.43296i −0.697605 0.716482i \(-0.745751\pi\)
0.697605 0.716482i \(-0.254249\pi\)
\(398\) 0 0
\(399\) 14451.3i 0.0907737i
\(400\) 0 0
\(401\) 165029. 1.02629 0.513147 0.858301i \(-0.328480\pi\)
0.513147 + 0.858301i \(0.328480\pi\)
\(402\) 0 0
\(403\) −13652.3 −0.0840611
\(404\) 0 0
\(405\) − 13934.3i − 0.0849522i
\(406\) 0 0
\(407\) 110941.i 0.669735i
\(408\) 0 0
\(409\) −29698.0 −0.177534 −0.0887669 0.996052i \(-0.528293\pi\)
−0.0887669 + 0.996052i \(0.528293\pi\)
\(410\) 0 0
\(411\) 2746.43 0.0162587
\(412\) 0 0
\(413\) − 11227.4i − 0.0658231i
\(414\) 0 0
\(415\) 100261.i 0.582152i
\(416\) 0 0
\(417\) −124975. −0.718708
\(418\) 0 0
\(419\) 100933. 0.574918 0.287459 0.957793i \(-0.407190\pi\)
0.287459 + 0.957793i \(0.407190\pi\)
\(420\) 0 0
\(421\) 211728.i 1.19457i 0.802028 + 0.597287i \(0.203755\pi\)
−0.802028 + 0.597287i \(0.796245\pi\)
\(422\) 0 0
\(423\) − 66407.4i − 0.371138i
\(424\) 0 0
\(425\) 73755.9 0.408337
\(426\) 0 0
\(427\) 14944.9 0.0819665
\(428\) 0 0
\(429\) − 54366.1i − 0.295402i
\(430\) 0 0
\(431\) 147606.i 0.794603i 0.917688 + 0.397302i \(0.130053\pi\)
−0.917688 + 0.397302i \(0.869947\pi\)
\(432\) 0 0
\(433\) 98534.8 0.525550 0.262775 0.964857i \(-0.415362\pi\)
0.262775 + 0.964857i \(0.415362\pi\)
\(434\) 0 0
\(435\) −34772.9 −0.183765
\(436\) 0 0
\(437\) − 201064.i − 1.05286i
\(438\) 0 0
\(439\) − 27484.6i − 0.142613i −0.997454 0.0713067i \(-0.977283\pi\)
0.997454 0.0713067i \(-0.0227169\pi\)
\(440\) 0 0
\(441\) 63721.0 0.327647
\(442\) 0 0
\(443\) −53561.9 −0.272928 −0.136464 0.990645i \(-0.543574\pi\)
−0.136464 + 0.990645i \(0.543574\pi\)
\(444\) 0 0
\(445\) − 273724.i − 1.38227i
\(446\) 0 0
\(447\) 27173.8i 0.135999i
\(448\) 0 0
\(449\) 69100.4 0.342758 0.171379 0.985205i \(-0.445178\pi\)
0.171379 + 0.985205i \(0.445178\pi\)
\(450\) 0 0
\(451\) −26825.4 −0.131884
\(452\) 0 0
\(453\) 138149.i 0.673210i
\(454\) 0 0
\(455\) 18881.6i 0.0912045i
\(456\) 0 0
\(457\) 115106. 0.551146 0.275573 0.961280i \(-0.411132\pi\)
0.275573 + 0.961280i \(0.411132\pi\)
\(458\) 0 0
\(459\) 39853.0 0.189163
\(460\) 0 0
\(461\) 108198.i 0.509119i 0.967057 + 0.254559i \(0.0819304\pi\)
−0.967057 + 0.254559i \(0.918070\pi\)
\(462\) 0 0
\(463\) 63779.4i 0.297522i 0.988873 + 0.148761i \(0.0475285\pi\)
−0.988873 + 0.148761i \(0.952472\pi\)
\(464\) 0 0
\(465\) 8785.13 0.0406296
\(466\) 0 0
\(467\) −392929. −1.80169 −0.900844 0.434142i \(-0.857052\pi\)
−0.900844 + 0.434142i \(0.857052\pi\)
\(468\) 0 0
\(469\) − 14073.3i − 0.0639811i
\(470\) 0 0
\(471\) 55166.2i 0.248675i
\(472\) 0 0
\(473\) 23738.7 0.106105
\(474\) 0 0
\(475\) 112829. 0.500071
\(476\) 0 0
\(477\) − 103333.i − 0.454153i
\(478\) 0 0
\(479\) 249509.i 1.08746i 0.839259 + 0.543732i \(0.182989\pi\)
−0.839259 + 0.543732i \(0.817011\pi\)
\(480\) 0 0
\(481\) 252602. 1.09181
\(482\) 0 0
\(483\) 15387.4 0.0659587
\(484\) 0 0
\(485\) − 321976.i − 1.36880i
\(486\) 0 0
\(487\) − 401706.i − 1.69375i −0.531789 0.846877i \(-0.678480\pi\)
0.531789 0.846877i \(-0.321520\pi\)
\(488\) 0 0
\(489\) −70352.1 −0.294211
\(490\) 0 0
\(491\) 110571. 0.458645 0.229322 0.973351i \(-0.426349\pi\)
0.229322 + 0.973351i \(0.426349\pi\)
\(492\) 0 0
\(493\) − 99452.9i − 0.409189i
\(494\) 0 0
\(495\) 34984.1i 0.142778i
\(496\) 0 0
\(497\) −28567.6 −0.115654
\(498\) 0 0
\(499\) −186945. −0.750779 −0.375390 0.926867i \(-0.622491\pi\)
−0.375390 + 0.926867i \(0.622491\pi\)
\(500\) 0 0
\(501\) 162269.i 0.646489i
\(502\) 0 0
\(503\) − 439468.i − 1.73697i −0.495719 0.868483i \(-0.665095\pi\)
0.495719 0.868483i \(-0.334905\pi\)
\(504\) 0 0
\(505\) −170290. −0.667738
\(506\) 0 0
\(507\) 24620.9 0.0957828
\(508\) 0 0
\(509\) 339847.i 1.31174i 0.754874 + 0.655870i \(0.227698\pi\)
−0.754874 + 0.655870i \(0.772302\pi\)
\(510\) 0 0
\(511\) 16589.9i 0.0635334i
\(512\) 0 0
\(513\) 60965.3 0.231658
\(514\) 0 0
\(515\) 374387. 1.41158
\(516\) 0 0
\(517\) 166726.i 0.623767i
\(518\) 0 0
\(519\) 222225.i 0.825007i
\(520\) 0 0
\(521\) 107168. 0.394812 0.197406 0.980322i \(-0.436748\pi\)
0.197406 + 0.980322i \(0.436748\pi\)
\(522\) 0 0
\(523\) 372679. 1.36248 0.681242 0.732058i \(-0.261440\pi\)
0.681242 + 0.732058i \(0.261440\pi\)
\(524\) 0 0
\(525\) 8634.76i 0.0313279i
\(526\) 0 0
\(527\) 25126.1i 0.0904697i
\(528\) 0 0
\(529\) 65751.1 0.234959
\(530\) 0 0
\(531\) −47364.8 −0.167984
\(532\) 0 0
\(533\) 61078.8i 0.214999i
\(534\) 0 0
\(535\) − 263573.i − 0.920858i
\(536\) 0 0
\(537\) 86609.0 0.300341
\(538\) 0 0
\(539\) −159981. −0.550671
\(540\) 0 0
\(541\) 105010.i 0.358785i 0.983778 + 0.179393i \(0.0574132\pi\)
−0.983778 + 0.179393i \(0.942587\pi\)
\(542\) 0 0
\(543\) 123094.i 0.417482i
\(544\) 0 0
\(545\) −251924. −0.848159
\(546\) 0 0
\(547\) 556117. 1.85862 0.929311 0.369298i \(-0.120402\pi\)
0.929311 + 0.369298i \(0.120402\pi\)
\(548\) 0 0
\(549\) − 63047.7i − 0.209182i
\(550\) 0 0
\(551\) − 152139.i − 0.501114i
\(552\) 0 0
\(553\) −13061.5 −0.0427112
\(554\) 0 0
\(555\) −162548. −0.527709
\(556\) 0 0
\(557\) 336211.i 1.08368i 0.840481 + 0.541841i \(0.182273\pi\)
−0.840481 + 0.541841i \(0.817727\pi\)
\(558\) 0 0
\(559\) − 54050.8i − 0.172973i
\(560\) 0 0
\(561\) −100057. −0.317923
\(562\) 0 0
\(563\) −558606. −1.76234 −0.881168 0.472803i \(-0.843242\pi\)
−0.881168 + 0.472803i \(0.843242\pi\)
\(564\) 0 0
\(565\) 52229.2i 0.163613i
\(566\) 0 0
\(567\) 4665.67i 0.0145127i
\(568\) 0 0
\(569\) 190223. 0.587542 0.293771 0.955876i \(-0.405090\pi\)
0.293771 + 0.955876i \(0.405090\pi\)
\(570\) 0 0
\(571\) 72652.1 0.222831 0.111416 0.993774i \(-0.464462\pi\)
0.111416 + 0.993774i \(0.464462\pi\)
\(572\) 0 0
\(573\) − 348658.i − 1.06192i
\(574\) 0 0
\(575\) − 120138.i − 0.363366i
\(576\) 0 0
\(577\) −294340. −0.884091 −0.442046 0.896993i \(-0.645747\pi\)
−0.442046 + 0.896993i \(0.645747\pi\)
\(578\) 0 0
\(579\) −58404.2 −0.174216
\(580\) 0 0
\(581\) − 33570.8i − 0.0994512i
\(582\) 0 0
\(583\) 259434.i 0.763289i
\(584\) 0 0
\(585\) 79655.6 0.232758
\(586\) 0 0
\(587\) −189152. −0.548953 −0.274477 0.961594i \(-0.588505\pi\)
−0.274477 + 0.961594i \(0.588505\pi\)
\(588\) 0 0
\(589\) 38436.7i 0.110794i
\(590\) 0 0
\(591\) 77862.5i 0.222922i
\(592\) 0 0
\(593\) −50118.2 −0.142523 −0.0712617 0.997458i \(-0.522703\pi\)
−0.0712617 + 0.997458i \(0.522703\pi\)
\(594\) 0 0
\(595\) 34750.3 0.0981578
\(596\) 0 0
\(597\) 369471.i 1.03665i
\(598\) 0 0
\(599\) 142995.i 0.398536i 0.979945 + 0.199268i \(0.0638564\pi\)
−0.979945 + 0.199268i \(0.936144\pi\)
\(600\) 0 0
\(601\) −582870. −1.61370 −0.806849 0.590757i \(-0.798829\pi\)
−0.806849 + 0.590757i \(0.798829\pi\)
\(602\) 0 0
\(603\) −59371.0 −0.163283
\(604\) 0 0
\(605\) 192018.i 0.524605i
\(606\) 0 0
\(607\) − 44407.7i − 0.120526i −0.998183 0.0602630i \(-0.980806\pi\)
0.998183 0.0602630i \(-0.0191939\pi\)
\(608\) 0 0
\(609\) 11643.2 0.0313933
\(610\) 0 0
\(611\) 379619. 1.01687
\(612\) 0 0
\(613\) 108875.i 0.289738i 0.989451 + 0.144869i \(0.0462761\pi\)
−0.989451 + 0.144869i \(0.953724\pi\)
\(614\) 0 0
\(615\) − 39303.7i − 0.103916i
\(616\) 0 0
\(617\) −698860. −1.83578 −0.917888 0.396839i \(-0.870107\pi\)
−0.917888 + 0.396839i \(0.870107\pi\)
\(618\) 0 0
\(619\) 441728. 1.15285 0.576427 0.817149i \(-0.304447\pi\)
0.576427 + 0.817149i \(0.304447\pi\)
\(620\) 0 0
\(621\) − 64914.8i − 0.168330i
\(622\) 0 0
\(623\) 91652.0i 0.236138i
\(624\) 0 0
\(625\) −160930. −0.411981
\(626\) 0 0
\(627\) −153063. −0.389345
\(628\) 0 0
\(629\) − 464897.i − 1.17505i
\(630\) 0 0
\(631\) − 649021.i − 1.63005i −0.579428 0.815023i \(-0.696724\pi\)
0.579428 0.815023i \(-0.303276\pi\)
\(632\) 0 0
\(633\) −316788. −0.790608
\(634\) 0 0
\(635\) −457201. −1.13386
\(636\) 0 0
\(637\) 364263.i 0.897710i
\(638\) 0 0
\(639\) 120518.i 0.295154i
\(640\) 0 0
\(641\) −774491. −1.88495 −0.942477 0.334272i \(-0.891510\pi\)
−0.942477 + 0.334272i \(0.891510\pi\)
\(642\) 0 0
\(643\) 611676. 1.47945 0.739723 0.672911i \(-0.234956\pi\)
0.739723 + 0.672911i \(0.234956\pi\)
\(644\) 0 0
\(645\) 34781.3i 0.0836038i
\(646\) 0 0
\(647\) 518940.i 1.23968i 0.784729 + 0.619838i \(0.212802\pi\)
−0.784729 + 0.619838i \(0.787198\pi\)
\(648\) 0 0
\(649\) 118917. 0.282327
\(650\) 0 0
\(651\) −2941.56 −0.00694090
\(652\) 0 0
\(653\) 648450.i 1.52072i 0.649500 + 0.760362i \(0.274978\pi\)
−0.649500 + 0.760362i \(0.725022\pi\)
\(654\) 0 0
\(655\) − 478993.i − 1.11647i
\(656\) 0 0
\(657\) 69987.6 0.162140
\(658\) 0 0
\(659\) −611892. −1.40898 −0.704489 0.709715i \(-0.748824\pi\)
−0.704489 + 0.709715i \(0.748824\pi\)
\(660\) 0 0
\(661\) 157591.i 0.360684i 0.983604 + 0.180342i \(0.0577205\pi\)
−0.983604 + 0.180342i \(0.942280\pi\)
\(662\) 0 0
\(663\) 227820.i 0.518281i
\(664\) 0 0
\(665\) 53159.5 0.120209
\(666\) 0 0
\(667\) −161995. −0.364124
\(668\) 0 0
\(669\) 452934.i 1.01200i
\(670\) 0 0
\(671\) 158291.i 0.351569i
\(672\) 0 0
\(673\) −96816.9 −0.213757 −0.106879 0.994272i \(-0.534086\pi\)
−0.106879 + 0.994272i \(0.534086\pi\)
\(674\) 0 0
\(675\) 36427.3 0.0799502
\(676\) 0 0
\(677\) 50973.9i 0.111217i 0.998453 + 0.0556084i \(0.0177098\pi\)
−0.998453 + 0.0556084i \(0.982290\pi\)
\(678\) 0 0
\(679\) 107809.i 0.233837i
\(680\) 0 0
\(681\) −77779.9 −0.167715
\(682\) 0 0
\(683\) 179552. 0.384901 0.192451 0.981307i \(-0.438356\pi\)
0.192451 + 0.981307i \(0.438356\pi\)
\(684\) 0 0
\(685\) − 10102.9i − 0.0215309i
\(686\) 0 0
\(687\) 406892.i 0.862116i
\(688\) 0 0
\(689\) 590706. 1.24432
\(690\) 0 0
\(691\) −189721. −0.397338 −0.198669 0.980067i \(-0.563662\pi\)
−0.198669 + 0.980067i \(0.563662\pi\)
\(692\) 0 0
\(693\) − 11713.9i − 0.0243913i
\(694\) 0 0
\(695\) 459726.i 0.951765i
\(696\) 0 0
\(697\) 112411. 0.231390
\(698\) 0 0
\(699\) −285251. −0.583811
\(700\) 0 0
\(701\) 499395.i 1.01627i 0.861278 + 0.508133i \(0.169664\pi\)
−0.861278 + 0.508133i \(0.830336\pi\)
\(702\) 0 0
\(703\) − 711179.i − 1.43902i
\(704\) 0 0
\(705\) −244282. −0.491488
\(706\) 0 0
\(707\) 57018.9 0.114072
\(708\) 0 0
\(709\) 328533.i 0.653561i 0.945100 + 0.326781i \(0.105964\pi\)
−0.945100 + 0.326781i \(0.894036\pi\)
\(710\) 0 0
\(711\) 55102.2i 0.109001i
\(712\) 0 0
\(713\) 40926.8 0.0805060
\(714\) 0 0
\(715\) −199988. −0.391193
\(716\) 0 0
\(717\) − 347421.i − 0.675799i
\(718\) 0 0
\(719\) 70795.6i 0.136946i 0.997653 + 0.0684728i \(0.0218126\pi\)
−0.997653 + 0.0684728i \(0.978187\pi\)
\(720\) 0 0
\(721\) −125357. −0.241146
\(722\) 0 0
\(723\) −518973. −0.992815
\(724\) 0 0
\(725\) − 90904.2i − 0.172945i
\(726\) 0 0
\(727\) 264078.i 0.499646i 0.968291 + 0.249823i \(0.0803725\pi\)
−0.968291 + 0.249823i \(0.919627\pi\)
\(728\) 0 0
\(729\) 19683.0 0.0370370
\(730\) 0 0
\(731\) −99476.8 −0.186160
\(732\) 0 0
\(733\) − 216452.i − 0.402859i −0.979503 0.201430i \(-0.935441\pi\)
0.979503 0.201430i \(-0.0645587\pi\)
\(734\) 0 0
\(735\) − 234400.i − 0.433894i
\(736\) 0 0
\(737\) 149060. 0.274427
\(738\) 0 0
\(739\) 515590. 0.944094 0.472047 0.881573i \(-0.343515\pi\)
0.472047 + 0.881573i \(0.343515\pi\)
\(740\) 0 0
\(741\) 348510.i 0.634714i
\(742\) 0 0
\(743\) 622349.i 1.12734i 0.825999 + 0.563672i \(0.190612\pi\)
−0.825999 + 0.563672i \(0.809388\pi\)
\(744\) 0 0
\(745\) 99960.0 0.180100
\(746\) 0 0
\(747\) −141625. −0.253804
\(748\) 0 0
\(749\) 88253.1i 0.157314i
\(750\) 0 0
\(751\) − 583086.i − 1.03384i −0.856034 0.516919i \(-0.827079\pi\)
0.856034 0.516919i \(-0.172921\pi\)
\(752\) 0 0
\(753\) 95970.8 0.169258
\(754\) 0 0
\(755\) 508185. 0.891513
\(756\) 0 0
\(757\) 884570.i 1.54362i 0.635853 + 0.771810i \(0.280649\pi\)
−0.635853 + 0.771810i \(0.719351\pi\)
\(758\) 0 0
\(759\) 162979.i 0.282909i
\(760\) 0 0
\(761\) −343529. −0.593190 −0.296595 0.955003i \(-0.595851\pi\)
−0.296595 + 0.955003i \(0.595851\pi\)
\(762\) 0 0
\(763\) 84352.9 0.144894
\(764\) 0 0
\(765\) − 146601.i − 0.250503i
\(766\) 0 0
\(767\) − 270762.i − 0.460253i
\(768\) 0 0
\(769\) −694796. −1.17491 −0.587455 0.809257i \(-0.699870\pi\)
−0.587455 + 0.809257i \(0.699870\pi\)
\(770\) 0 0
\(771\) 64593.1 0.108662
\(772\) 0 0
\(773\) − 49585.9i − 0.0829850i −0.999139 0.0414925i \(-0.986789\pi\)
0.999139 0.0414925i \(-0.0132113\pi\)
\(774\) 0 0
\(775\) 22966.3i 0.0382373i
\(776\) 0 0
\(777\) 54426.5 0.0901505
\(778\) 0 0
\(779\) 171962. 0.283372
\(780\) 0 0
\(781\) − 302578.i − 0.496062i
\(782\) 0 0
\(783\) − 49118.9i − 0.0801170i
\(784\) 0 0
\(785\) 202931. 0.329313
\(786\) 0 0
\(787\) −557614. −0.900293 −0.450147 0.892955i \(-0.648628\pi\)
−0.450147 + 0.892955i \(0.648628\pi\)
\(788\) 0 0
\(789\) − 262645.i − 0.421906i
\(790\) 0 0
\(791\) − 17488.1i − 0.0279506i
\(792\) 0 0
\(793\) 360414. 0.573132
\(794\) 0 0
\(795\) −380114. −0.601423
\(796\) 0 0
\(797\) − 459192.i − 0.722899i −0.932392 0.361449i \(-0.882282\pi\)
0.932392 0.361449i \(-0.117718\pi\)
\(798\) 0 0
\(799\) − 698663.i − 1.09439i
\(800\) 0 0
\(801\) 386651. 0.602634
\(802\) 0 0
\(803\) −175715. −0.272506
\(804\) 0 0
\(805\) − 56603.3i − 0.0873474i
\(806\) 0 0
\(807\) 460785.i 0.707541i
\(808\) 0 0
\(809\) −594379. −0.908168 −0.454084 0.890959i \(-0.650033\pi\)
−0.454084 + 0.890959i \(0.650033\pi\)
\(810\) 0 0
\(811\) 1.17014e6 1.77908 0.889541 0.456855i \(-0.151024\pi\)
0.889541 + 0.456855i \(0.151024\pi\)
\(812\) 0 0
\(813\) − 97713.7i − 0.147834i
\(814\) 0 0
\(815\) 258793.i 0.389616i
\(816\) 0 0
\(817\) −152175. −0.227982
\(818\) 0 0
\(819\) −26671.4 −0.0397629
\(820\) 0 0
\(821\) − 63648.5i − 0.0944282i −0.998885 0.0472141i \(-0.984966\pi\)
0.998885 0.0472141i \(-0.0150343\pi\)
\(822\) 0 0
\(823\) 79834.3i 0.117866i 0.998262 + 0.0589331i \(0.0187699\pi\)
−0.998262 + 0.0589331i \(0.981230\pi\)
\(824\) 0 0
\(825\) −91456.4 −0.134371
\(826\) 0 0
\(827\) 242691. 0.354848 0.177424 0.984134i \(-0.443224\pi\)
0.177424 + 0.984134i \(0.443224\pi\)
\(828\) 0 0
\(829\) − 417062.i − 0.606864i −0.952853 0.303432i \(-0.901867\pi\)
0.952853 0.303432i \(-0.0981326\pi\)
\(830\) 0 0
\(831\) − 196432.i − 0.284452i
\(832\) 0 0
\(833\) 670400. 0.966149
\(834\) 0 0
\(835\) 596914. 0.856128
\(836\) 0 0
\(837\) 12409.5i 0.0177135i
\(838\) 0 0
\(839\) 31642.8i 0.0449522i 0.999747 + 0.0224761i \(0.00715497\pi\)
−0.999747 + 0.0224761i \(0.992845\pi\)
\(840\) 0 0
\(841\) 584705. 0.826694
\(842\) 0 0
\(843\) 132938. 0.187065
\(844\) 0 0
\(845\) − 90568.8i − 0.126843i
\(846\) 0 0
\(847\) − 64294.3i − 0.0896202i
\(848\) 0 0
\(849\) −221077. −0.306710
\(850\) 0 0
\(851\) −757251. −1.04564
\(852\) 0 0
\(853\) − 1.10273e6i − 1.51555i −0.652513 0.757777i \(-0.726285\pi\)
0.652513 0.757777i \(-0.273715\pi\)
\(854\) 0 0
\(855\) − 224263.i − 0.306779i
\(856\) 0 0
\(857\) 505940. 0.688870 0.344435 0.938810i \(-0.388071\pi\)
0.344435 + 0.938810i \(0.388071\pi\)
\(858\) 0 0
\(859\) 741942. 1.00550 0.502752 0.864431i \(-0.332321\pi\)
0.502752 + 0.864431i \(0.332321\pi\)
\(860\) 0 0
\(861\) 13160.2i 0.0177524i
\(862\) 0 0
\(863\) − 536150.i − 0.719888i −0.932974 0.359944i \(-0.882796\pi\)
0.932974 0.359944i \(-0.117204\pi\)
\(864\) 0 0
\(865\) 817462. 1.09253
\(866\) 0 0
\(867\) −14700.3 −0.0195564
\(868\) 0 0
\(869\) − 138343.i − 0.183196i
\(870\) 0 0
\(871\) − 339396.i − 0.447373i
\(872\) 0 0
\(873\) 454811. 0.596763
\(874\) 0 0
\(875\) 108221. 0.141350
\(876\) 0 0
\(877\) − 59941.4i − 0.0779341i −0.999240 0.0389671i \(-0.987593\pi\)
0.999240 0.0389671i \(-0.0124067\pi\)
\(878\) 0 0
\(879\) − 529916.i − 0.685850i
\(880\) 0 0
\(881\) −828350. −1.06724 −0.533620 0.845724i \(-0.679169\pi\)
−0.533620 + 0.845724i \(0.679169\pi\)
\(882\) 0 0
\(883\) −245690. −0.315113 −0.157556 0.987510i \(-0.550362\pi\)
−0.157556 + 0.987510i \(0.550362\pi\)
\(884\) 0 0
\(885\) 174233.i 0.222456i
\(886\) 0 0
\(887\) 758015.i 0.963453i 0.876322 + 0.481726i \(0.159990\pi\)
−0.876322 + 0.481726i \(0.840010\pi\)
\(888\) 0 0
\(889\) 153087. 0.193702
\(890\) 0 0
\(891\) −49417.2 −0.0622476
\(892\) 0 0
\(893\) − 1.06878e6i − 1.34025i
\(894\) 0 0
\(895\) − 318594.i − 0.397733i
\(896\) 0 0
\(897\) 371087. 0.461201
\(898\) 0 0
\(899\) 30967.9 0.0383171
\(900\) 0 0
\(901\) − 1.08715e6i − 1.33919i
\(902\) 0 0
\(903\) − 11646.0i − 0.0142824i
\(904\) 0 0
\(905\) 452807. 0.552861
\(906\) 0 0
\(907\) −520331. −0.632506 −0.316253 0.948675i \(-0.602425\pi\)
−0.316253 + 0.948675i \(0.602425\pi\)
\(908\) 0 0
\(909\) − 240545.i − 0.291117i
\(910\) 0 0
\(911\) − 1.10094e6i − 1.32656i −0.748370 0.663282i \(-0.769163\pi\)
0.748370 0.663282i \(-0.230837\pi\)
\(912\) 0 0
\(913\) 355571. 0.426564
\(914\) 0 0
\(915\) −231923. −0.277014
\(916\) 0 0
\(917\) 160383.i 0.190731i
\(918\) 0 0
\(919\) 990050.i 1.17227i 0.810215 + 0.586133i \(0.199350\pi\)
−0.810215 + 0.586133i \(0.800650\pi\)
\(920\) 0 0
\(921\) −883989. −1.04214
\(922\) 0 0
\(923\) −688942. −0.808685
\(924\) 0 0
\(925\) − 424936.i − 0.496638i
\(926\) 0 0
\(927\) 528844.i 0.615415i
\(928\) 0 0
\(929\) 112392. 0.130228 0.0651140 0.997878i \(-0.479259\pi\)
0.0651140 + 0.997878i \(0.479259\pi\)
\(930\) 0 0
\(931\) 1.02555e6 1.18320
\(932\) 0 0
\(933\) − 216122.i − 0.248277i
\(934\) 0 0
\(935\) 368063.i 0.421017i
\(936\) 0 0
\(937\) −293444. −0.334231 −0.167115 0.985937i \(-0.553445\pi\)
−0.167115 + 0.985937i \(0.553445\pi\)
\(938\) 0 0
\(939\) 632304. 0.717125
\(940\) 0 0
\(941\) − 531934.i − 0.600729i −0.953825 0.300364i \(-0.902892\pi\)
0.953825 0.300364i \(-0.0971083\pi\)
\(942\) 0 0
\(943\) − 183102.i − 0.205906i
\(944\) 0 0
\(945\) 17162.8 0.0192188
\(946\) 0 0
\(947\) −378760. −0.422342 −0.211171 0.977449i \(-0.567728\pi\)
−0.211171 + 0.977449i \(0.567728\pi\)
\(948\) 0 0
\(949\) 400085.i 0.444243i
\(950\) 0 0
\(951\) 657301.i 0.726781i
\(952\) 0 0
\(953\) 1.75746e6 1.93508 0.967539 0.252721i \(-0.0813256\pi\)
0.967539 + 0.252721i \(0.0813256\pi\)
\(954\) 0 0
\(955\) −1.28255e6 −1.40627
\(956\) 0 0
\(957\) 123320.i 0.134651i
\(958\) 0 0
\(959\) 3382.78i 0.00367821i
\(960\) 0 0
\(961\) 915697. 0.991528
\(962\) 0 0
\(963\) 372312. 0.401471
\(964\) 0 0
\(965\) 214842.i 0.230709i
\(966\) 0 0
\(967\) 59833.4i 0.0639868i 0.999488 + 0.0319934i \(0.0101856\pi\)
−0.999488 + 0.0319934i \(0.989814\pi\)
\(968\) 0 0
\(969\) 641408. 0.683104
\(970\) 0 0
\(971\) 687930. 0.729635 0.364817 0.931079i \(-0.381131\pi\)
0.364817 + 0.931079i \(0.381131\pi\)
\(972\) 0 0
\(973\) − 153932.i − 0.162594i
\(974\) 0 0
\(975\) 208238.i 0.219053i
\(976\) 0 0
\(977\) 1.20414e6 1.26150 0.630752 0.775985i \(-0.282747\pi\)
0.630752 + 0.775985i \(0.282747\pi\)
\(978\) 0 0
\(979\) −970746. −1.01284
\(980\) 0 0
\(981\) − 355858.i − 0.369776i
\(982\) 0 0
\(983\) 202530.i 0.209596i 0.994494 + 0.104798i \(0.0334196\pi\)
−0.994494 + 0.104798i \(0.966580\pi\)
\(984\) 0 0
\(985\) 286420. 0.295210
\(986\) 0 0
\(987\) 81793.9 0.0839628
\(988\) 0 0
\(989\) 162033.i 0.165658i
\(990\) 0 0
\(991\) 762037.i 0.775941i 0.921672 + 0.387970i \(0.126824\pi\)
−0.921672 + 0.387970i \(0.873176\pi\)
\(992\) 0 0
\(993\) 175108. 0.177585
\(994\) 0 0
\(995\) 1.35911e6 1.37281
\(996\) 0 0
\(997\) − 75700.5i − 0.0761567i −0.999275 0.0380784i \(-0.987876\pi\)
0.999275 0.0380784i \(-0.0121237\pi\)
\(998\) 0 0
\(999\) − 229608.i − 0.230068i
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 384.5.b.d.319.11 yes 16
3.2 odd 2 1152.5.b.m.703.12 16
4.3 odd 2 inner 384.5.b.d.319.3 16
8.3 odd 2 inner 384.5.b.d.319.14 yes 16
8.5 even 2 inner 384.5.b.d.319.6 yes 16
12.11 even 2 1152.5.b.m.703.11 16
16.3 odd 4 768.5.g.j.511.1 8
16.5 even 4 768.5.g.h.511.4 8
16.11 odd 4 768.5.g.h.511.8 8
16.13 even 4 768.5.g.j.511.5 8
24.5 odd 2 1152.5.b.m.703.6 16
24.11 even 2 1152.5.b.m.703.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.d.319.3 16 4.3 odd 2 inner
384.5.b.d.319.6 yes 16 8.5 even 2 inner
384.5.b.d.319.11 yes 16 1.1 even 1 trivial
384.5.b.d.319.14 yes 16 8.3 odd 2 inner
768.5.g.h.511.4 8 16.5 even 4
768.5.g.h.511.8 8 16.11 odd 4
768.5.g.j.511.1 8 16.3 odd 4
768.5.g.j.511.5 8 16.13 even 4
1152.5.b.m.703.5 16 24.11 even 2
1152.5.b.m.703.6 16 24.5 odd 2
1152.5.b.m.703.11 16 12.11 even 2
1152.5.b.m.703.12 16 3.2 odd 2