Properties

Label 576.7.h.a.161.14
Level $576$
Weight $7$
Character 576.161
Analytic conductor $132.511$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,7,Mod(161,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.h (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: \(132.511152165\)
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: \( x^{16} - 8 x^{15} + 964 x^{14} - 6608 x^{13} + 404616 x^{12} - 2342156 x^{11} + 95125730 x^{10} - 454315424 x^{9} + 13646460877 x^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{36} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.14
Root \(3.36747 + 7.23051i\) of defining polynomial
Character \(\chi\) \(=\) 576.161
Dual form 576.7.h.a.161.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+168.989 q^{5} -13.9908 q^{7} +O(q^{10})\) \(q+168.989 q^{5} -13.9908 q^{7} +1500.86 q^{11} +744.941i q^{13} -2963.54i q^{17} +6311.61i q^{19} -9997.60i q^{23} +12932.2 q^{25} +20393.1 q^{29} +39212.3 q^{31} -2364.29 q^{35} +89287.8i q^{37} +20087.0i q^{41} -55492.9i q^{43} -108404. i q^{47} -117453. q^{49} +143755. q^{53} +253629. q^{55} -11065.7 q^{59} +175980. i q^{61} +125887. i q^{65} +42074.2i q^{67} -103977. i q^{71} +539416. q^{73} -20998.3 q^{77} -479208. q^{79} +485444. q^{83} -500805. i q^{85} -310114. i q^{89} -10422.3i q^{91} +1.06659e6i q^{95} -199839. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 19888 q^{25} + 388784 q^{49} + 465728 q^{73} + 6555136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 168.989 1.35191 0.675955 0.736943i \(-0.263731\pi\)
0.675955 + 0.736943i \(0.263731\pi\)
\(6\) 0 0
\(7\) −13.9908 −0.0407895 −0.0203948 0.999792i \(-0.506492\pi\)
−0.0203948 + 0.999792i \(0.506492\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1500.86 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(12\) 0 0
\(13\) 744.941i 0.339072i 0.985524 + 0.169536i \(0.0542269\pi\)
−0.985524 + 0.169536i \(0.945773\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2963.54i − 0.603203i −0.953434 0.301602i \(-0.902479\pi\)
0.953434 0.301602i \(-0.0975213\pi\)
\(18\) 0 0
\(19\) 6311.61i 0.920194i 0.887869 + 0.460097i \(0.152185\pi\)
−0.887869 + 0.460097i \(0.847815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 9997.60i − 0.821698i −0.911703 0.410849i \(-0.865232\pi\)
0.911703 0.410849i \(-0.134768\pi\)
\(24\) 0 0
\(25\) 12932.2 0.827662
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 20393.1 0.836161 0.418081 0.908410i \(-0.362703\pi\)
0.418081 + 0.908410i \(0.362703\pi\)
\(30\) 0 0
\(31\) 39212.3 1.31625 0.658123 0.752910i \(-0.271350\pi\)
0.658123 + 0.752910i \(0.271350\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2364.29 −0.0551438
\(36\) 0 0
\(37\) 89287.8i 1.76273i 0.472432 + 0.881367i \(0.343376\pi\)
−0.472432 + 0.881367i \(0.656624\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 20087.0i 0.291450i 0.989325 + 0.145725i \(0.0465515\pi\)
−0.989325 + 0.145725i \(0.953448\pi\)
\(42\) 0 0
\(43\) − 55492.9i − 0.697962i −0.937130 0.348981i \(-0.886528\pi\)
0.937130 0.348981i \(-0.113472\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 108404.i − 1.04412i −0.852909 0.522060i \(-0.825164\pi\)
0.852909 0.522060i \(-0.174836\pi\)
\(48\) 0 0
\(49\) −117453. −0.998336
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 143755. 0.965596 0.482798 0.875732i \(-0.339621\pi\)
0.482798 + 0.875732i \(0.339621\pi\)
\(54\) 0 0
\(55\) 253629. 1.52444
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11065.7 −0.0538792 −0.0269396 0.999637i \(-0.508576\pi\)
−0.0269396 + 0.999637i \(0.508576\pi\)
\(60\) 0 0
\(61\) 175980.i 0.775305i 0.921806 + 0.387653i \(0.126714\pi\)
−0.921806 + 0.387653i \(0.873286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 125887.i 0.458395i
\(66\) 0 0
\(67\) 42074.2i 0.139892i 0.997551 + 0.0699458i \(0.0222826\pi\)
−0.997551 + 0.0699458i \(0.977717\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 103977.i − 0.290511i −0.989394 0.145256i \(-0.953600\pi\)
0.989394 0.145256i \(-0.0464005\pi\)
\(72\) 0 0
\(73\) 539416. 1.38661 0.693307 0.720643i \(-0.256153\pi\)
0.693307 + 0.720643i \(0.256153\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −20998.3 −0.0459951
\(78\) 0 0
\(79\) −479208. −0.971946 −0.485973 0.873974i \(-0.661535\pi\)
−0.485973 + 0.873974i \(0.661535\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 485444. 0.848995 0.424498 0.905429i \(-0.360451\pi\)
0.424498 + 0.905429i \(0.360451\pi\)
\(84\) 0 0
\(85\) − 500805.i − 0.815477i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 310114.i − 0.439898i −0.975511 0.219949i \(-0.929411\pi\)
0.975511 0.219949i \(-0.0705891\pi\)
\(90\) 0 0
\(91\) − 10422.3i − 0.0138306i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.06659e6i 1.24402i
\(96\) 0 0
\(97\) −199839. −0.218960 −0.109480 0.993989i \(-0.534919\pi\)
−0.109480 + 0.993989i \(0.534919\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.20189e6 −1.16654 −0.583270 0.812279i \(-0.698227\pi\)
−0.583270 + 0.812279i \(0.698227\pi\)
\(102\) 0 0
\(103\) −416228. −0.380907 −0.190454 0.981696i \(-0.560996\pi\)
−0.190454 + 0.981696i \(0.560996\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 664061. 0.542071 0.271036 0.962569i \(-0.412634\pi\)
0.271036 + 0.962569i \(0.412634\pi\)
\(108\) 0 0
\(109\) − 1.42279e6i − 1.09865i −0.835608 0.549326i \(-0.814885\pi\)
0.835608 0.549326i \(-0.185115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.58232e6i 1.78968i 0.446390 + 0.894838i \(0.352709\pi\)
−0.446390 + 0.894838i \(0.647291\pi\)
\(114\) 0 0
\(115\) − 1.68948e6i − 1.11086i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 41462.3i 0.0246044i
\(120\) 0 0
\(121\) 481022. 0.271524
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −455049. −0.232985
\(126\) 0 0
\(127\) −3.13676e6 −1.53134 −0.765668 0.643237i \(-0.777591\pi\)
−0.765668 + 0.643237i \(0.777591\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 418255. 0.186049 0.0930245 0.995664i \(-0.470347\pi\)
0.0930245 + 0.995664i \(0.470347\pi\)
\(132\) 0 0
\(133\) − 88304.6i − 0.0375343i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.05135e6i 0.797772i 0.917000 + 0.398886i \(0.130603\pi\)
−0.917000 + 0.398886i \(0.869397\pi\)
\(138\) 0 0
\(139\) − 2.00434e6i − 0.746322i −0.927767 0.373161i \(-0.878274\pi\)
0.927767 0.373161i \(-0.121726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.11805e6i 0.382344i
\(144\) 0 0
\(145\) 3.44621e6 1.13042
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.32337e6 0.702361 0.351180 0.936308i \(-0.385780\pi\)
0.351180 + 0.936308i \(0.385780\pi\)
\(150\) 0 0
\(151\) 6.44651e6 1.87238 0.936190 0.351493i \(-0.114326\pi\)
0.936190 + 0.351493i \(0.114326\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.62644e6 1.77945
\(156\) 0 0
\(157\) 2.70983e6i 0.700233i 0.936706 + 0.350117i \(0.113858\pi\)
−0.936706 + 0.350117i \(0.886142\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 139875.i 0.0335167i
\(162\) 0 0
\(163\) − 940013.i − 0.217056i −0.994093 0.108528i \(-0.965386\pi\)
0.994093 0.108528i \(-0.0346136\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 8.67355e6i − 1.86229i −0.364647 0.931146i \(-0.618810\pi\)
0.364647 0.931146i \(-0.381190\pi\)
\(168\) 0 0
\(169\) 4.27187e6 0.885030
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.50876e6 0.291395 0.145697 0.989329i \(-0.453457\pi\)
0.145697 + 0.989329i \(0.453457\pi\)
\(174\) 0 0
\(175\) −180932. −0.0337600
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.07397e7 1.87255 0.936277 0.351262i \(-0.114247\pi\)
0.936277 + 0.351262i \(0.114247\pi\)
\(180\) 0 0
\(181\) 6.45716e6i 1.08894i 0.838779 + 0.544472i \(0.183270\pi\)
−0.838779 + 0.544472i \(0.816730\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.50886e7i 2.38306i
\(186\) 0 0
\(187\) − 4.44786e6i − 0.680183i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.54231e6i 1.08244i 0.840881 + 0.541220i \(0.182038\pi\)
−0.840881 + 0.541220i \(0.817962\pi\)
\(192\) 0 0
\(193\) 8.76476e6 1.21918 0.609591 0.792716i \(-0.291334\pi\)
0.609591 + 0.792716i \(0.291334\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.83274e6 0.501315 0.250657 0.968076i \(-0.419353\pi\)
0.250657 + 0.968076i \(0.419353\pi\)
\(198\) 0 0
\(199\) 1.16120e7 1.47349 0.736744 0.676171i \(-0.236362\pi\)
0.736744 + 0.676171i \(0.236362\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −285317. −0.0341066
\(204\) 0 0
\(205\) 3.39449e6i 0.394015i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.47285e6i 1.03763i
\(210\) 0 0
\(211\) − 1.50589e7i − 1.60305i −0.597962 0.801524i \(-0.704023\pi\)
0.597962 0.801524i \(-0.295977\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 9.37768e6i − 0.943583i
\(216\) 0 0
\(217\) −548612. −0.0536891
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.20766e6 0.204529
\(222\) 0 0
\(223\) −1.46388e7 −1.32005 −0.660027 0.751242i \(-0.729455\pi\)
−0.660027 + 0.751242i \(0.729455\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.60373e7 1.37105 0.685526 0.728048i \(-0.259572\pi\)
0.685526 + 0.728048i \(0.259572\pi\)
\(228\) 0 0
\(229\) − 1.16093e7i − 0.966721i −0.875421 0.483360i \(-0.839416\pi\)
0.875421 0.483360i \(-0.160584\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.35696e7i 1.07275i 0.843980 + 0.536375i \(0.180207\pi\)
−0.843980 + 0.536375i \(0.819793\pi\)
\(234\) 0 0
\(235\) − 1.83190e7i − 1.41156i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.26081e6i 0.312103i 0.987749 + 0.156052i \(0.0498766\pi\)
−0.987749 + 0.156052i \(0.950123\pi\)
\(240\) 0 0
\(241\) 1.40563e7 1.00420 0.502101 0.864809i \(-0.332561\pi\)
0.502101 + 0.864809i \(0.332561\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.98483e7 −1.34966
\(246\) 0 0
\(247\) −4.70178e6 −0.312012
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.96750e7 1.87659 0.938296 0.345833i \(-0.112404\pi\)
0.938296 + 0.345833i \(0.112404\pi\)
\(252\) 0 0
\(253\) − 1.50050e7i − 0.926562i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.40873e7i 1.41902i 0.704696 + 0.709509i \(0.251083\pi\)
−0.704696 + 0.709509i \(0.748917\pi\)
\(258\) 0 0
\(259\) − 1.24921e6i − 0.0719011i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.72669e7i 1.49888i 0.662070 + 0.749442i \(0.269678\pi\)
−0.662070 + 0.749442i \(0.730322\pi\)
\(264\) 0 0
\(265\) 2.42930e7 1.30540
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.01871e6 −0.206457 −0.103229 0.994658i \(-0.532917\pi\)
−0.103229 + 0.994658i \(0.532917\pi\)
\(270\) 0 0
\(271\) −2.31382e7 −1.16258 −0.581288 0.813698i \(-0.697451\pi\)
−0.581288 + 0.813698i \(0.697451\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.94095e7 0.933288
\(276\) 0 0
\(277\) − 6.24314e6i − 0.293740i −0.989156 0.146870i \(-0.953080\pi\)
0.989156 0.146870i \(-0.0469200\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.15373e6i 0.232275i 0.993233 + 0.116137i \(0.0370513\pi\)
−0.993233 + 0.116137i \(0.962949\pi\)
\(282\) 0 0
\(283\) 1.21978e7i 0.538174i 0.963116 + 0.269087i \(0.0867218\pi\)
−0.963116 + 0.269087i \(0.913278\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 281034.i − 0.0118881i
\(288\) 0 0
\(289\) 1.53550e7 0.636146
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.58367e6 0.381003 0.190502 0.981687i \(-0.438989\pi\)
0.190502 + 0.981687i \(0.438989\pi\)
\(294\) 0 0
\(295\) −1.86997e6 −0.0728399
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.44761e6 0.278614
\(300\) 0 0
\(301\) 776391.i 0.0284696i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.97386e7i 1.04814i
\(306\) 0 0
\(307\) − 3.65639e7i − 1.26368i −0.775098 0.631841i \(-0.782300\pi\)
0.775098 0.631841i \(-0.217700\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 2.14969e7i − 0.714653i −0.933980 0.357327i \(-0.883688\pi\)
0.933980 0.357327i \(-0.116312\pi\)
\(312\) 0 0
\(313\) 2.15571e7 0.703004 0.351502 0.936187i \(-0.385671\pi\)
0.351502 + 0.936187i \(0.385671\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.84431e7 −1.20681 −0.603407 0.797433i \(-0.706190\pi\)
−0.603407 + 0.797433i \(0.706190\pi\)
\(318\) 0 0
\(319\) 3.06073e7 0.942871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.87047e7 0.555064
\(324\) 0 0
\(325\) 9.63374e6i 0.280637i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.51665e6i 0.0425892i
\(330\) 0 0
\(331\) − 4.00607e7i − 1.10467i −0.833621 0.552337i \(-0.813736\pi\)
0.833621 0.552337i \(-0.186264\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.11007e6i 0.189121i
\(336\) 0 0
\(337\) −3.98363e7 −1.04085 −0.520426 0.853907i \(-0.674227\pi\)
−0.520426 + 0.853907i \(0.674227\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.88522e7 1.48422
\(342\) 0 0
\(343\) 3.28927e6 0.0815112
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.80254e6 0.0910092 0.0455046 0.998964i \(-0.485510\pi\)
0.0455046 + 0.998964i \(0.485510\pi\)
\(348\) 0 0
\(349\) 2.37735e7i 0.559263i 0.960107 + 0.279632i \(0.0902123\pi\)
−0.960107 + 0.279632i \(0.909788\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.78223e7i 0.632512i 0.948674 + 0.316256i \(0.102426\pi\)
−0.948674 + 0.316256i \(0.897574\pi\)
\(354\) 0 0
\(355\) − 1.75710e7i − 0.392745i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 2.62940e7i − 0.568293i −0.958781 0.284147i \(-0.908290\pi\)
0.958781 0.284147i \(-0.0917103\pi\)
\(360\) 0 0
\(361\) 7.20942e6 0.153242
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.11553e7 1.87458
\(366\) 0 0
\(367\) 6.54421e7 1.32391 0.661956 0.749542i \(-0.269726\pi\)
0.661956 + 0.749542i \(0.269726\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.01125e6 −0.0393862
\(372\) 0 0
\(373\) 6.34853e7i 1.22334i 0.791114 + 0.611669i \(0.209502\pi\)
−0.791114 + 0.611669i \(0.790498\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.51917e7i 0.283519i
\(378\) 0 0
\(379\) 8.56684e7i 1.57363i 0.617188 + 0.786816i \(0.288272\pi\)
−0.617188 + 0.786816i \(0.711728\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 9.73160e7i − 1.73216i −0.499906 0.866080i \(-0.666632\pi\)
0.499906 0.866080i \(-0.333368\pi\)
\(384\) 0 0
\(385\) −3.54847e6 −0.0621812
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.00954e8 −1.71504 −0.857519 0.514452i \(-0.827995\pi\)
−0.857519 + 0.514452i \(0.827995\pi\)
\(390\) 0 0
\(391\) −2.96283e7 −0.495651
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.09807e7 −1.31398
\(396\) 0 0
\(397\) 2.94102e7i 0.470031i 0.971992 + 0.235015i \(0.0755140\pi\)
−0.971992 + 0.235015i \(0.924486\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 9.81649e7i − 1.52238i −0.648529 0.761190i \(-0.724616\pi\)
0.648529 0.761190i \(-0.275384\pi\)
\(402\) 0 0
\(403\) 2.92108e7i 0.446302i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.34009e8i 1.98769i
\(408\) 0 0
\(409\) 2.36974e7 0.346363 0.173181 0.984890i \(-0.444595\pi\)
0.173181 + 0.984890i \(0.444595\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 154818. 0.00219771
\(414\) 0 0
\(415\) 8.20347e7 1.14777
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.91039e7 1.21131 0.605654 0.795728i \(-0.292911\pi\)
0.605654 + 0.795728i \(0.292911\pi\)
\(420\) 0 0
\(421\) − 8.70306e7i − 1.16634i −0.812350 0.583170i \(-0.801812\pi\)
0.812350 0.583170i \(-0.198188\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 3.83251e7i − 0.499249i
\(426\) 0 0
\(427\) − 2.46210e6i − 0.0316244i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 3.98013e7i − 0.497125i −0.968616 0.248563i \(-0.920042\pi\)
0.968616 0.248563i \(-0.0799582\pi\)
\(432\) 0 0
\(433\) 2.08244e7 0.256512 0.128256 0.991741i \(-0.459062\pi\)
0.128256 + 0.991741i \(0.459062\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.31010e7 0.756122
\(438\) 0 0
\(439\) −1.38090e8 −1.63218 −0.816090 0.577926i \(-0.803862\pi\)
−0.816090 + 0.577926i \(0.803862\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.08013e7 −1.04443 −0.522217 0.852813i \(-0.674895\pi\)
−0.522217 + 0.852813i \(0.674895\pi\)
\(444\) 0 0
\(445\) − 5.24058e7i − 0.594702i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 8.73485e7i − 0.964976i −0.875903 0.482488i \(-0.839733\pi\)
0.875903 0.482488i \(-0.160267\pi\)
\(450\) 0 0
\(451\) 3.01479e7i 0.328645i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.76126e6i − 0.0186977i
\(456\) 0 0
\(457\) −1.75726e8 −1.84114 −0.920570 0.390578i \(-0.872275\pi\)
−0.920570 + 0.390578i \(0.872275\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.37345e8 −1.40187 −0.700936 0.713224i \(-0.747234\pi\)
−0.700936 + 0.713224i \(0.747234\pi\)
\(462\) 0 0
\(463\) −3.30418e7 −0.332905 −0.166452 0.986049i \(-0.553231\pi\)
−0.166452 + 0.986049i \(0.553231\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.48968e8 −1.46266 −0.731330 0.682024i \(-0.761100\pi\)
−0.731330 + 0.682024i \(0.761100\pi\)
\(468\) 0 0
\(469\) − 588652.i − 0.00570611i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 8.32871e7i − 0.787036i
\(474\) 0 0
\(475\) 8.16232e7i 0.761610i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.47748e8i 1.34436i 0.740389 + 0.672178i \(0.234641\pi\)
−0.740389 + 0.672178i \(0.765359\pi\)
\(480\) 0 0
\(481\) −6.65141e7 −0.597693
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.37705e7 −0.296014
\(486\) 0 0
\(487\) −1.43723e8 −1.24434 −0.622170 0.782882i \(-0.713749\pi\)
−0.622170 + 0.782882i \(0.713749\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.74135e8 −1.47110 −0.735550 0.677471i \(-0.763076\pi\)
−0.735550 + 0.677471i \(0.763076\pi\)
\(492\) 0 0
\(493\) − 6.04358e7i − 0.504375i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.45473e6i 0.0118498i
\(498\) 0 0
\(499\) 2.57033e7i 0.206865i 0.994636 + 0.103433i \(0.0329826\pi\)
−0.994636 + 0.103433i \(0.967017\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.86336e6i 0.0539303i 0.999636 + 0.0269651i \(0.00858431\pi\)
−0.999636 + 0.0269651i \(0.991416\pi\)
\(504\) 0 0
\(505\) −2.03105e8 −1.57706
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.26759e6 0.0247784 0.0123892 0.999923i \(-0.496056\pi\)
0.0123892 + 0.999923i \(0.496056\pi\)
\(510\) 0 0
\(511\) −7.54687e6 −0.0565593
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.03378e7 −0.514953
\(516\) 0 0
\(517\) − 1.62699e8i − 1.17737i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 7.16898e7i − 0.506926i −0.967345 0.253463i \(-0.918430\pi\)
0.967345 0.253463i \(-0.0815695\pi\)
\(522\) 0 0
\(523\) 1.78584e8i 1.24835i 0.781284 + 0.624176i \(0.214565\pi\)
−0.781284 + 0.624176i \(0.785435\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.16207e8i − 0.793964i
\(528\) 0 0
\(529\) 4.80840e7 0.324813
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.49637e7 −0.0988226
\(534\) 0 0
\(535\) 1.12219e8 0.732832
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.76281e8 −1.12574
\(540\) 0 0
\(541\) − 5.04327e7i − 0.318508i −0.987238 0.159254i \(-0.949091\pi\)
0.987238 0.159254i \(-0.0509089\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 2.40435e8i − 1.48528i
\(546\) 0 0
\(547\) − 5.13884e7i − 0.313981i −0.987600 0.156990i \(-0.949821\pi\)
0.987600 0.156990i \(-0.0501791\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.28714e8i 0.769431i
\(552\) 0 0
\(553\) 6.70450e6 0.0396453
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.14558e8 0.662920 0.331460 0.943469i \(-0.392459\pi\)
0.331460 + 0.943469i \(0.392459\pi\)
\(558\) 0 0
\(559\) 4.13389e7 0.236659
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.54440e7 0.478803 0.239401 0.970921i \(-0.423049\pi\)
0.239401 + 0.970921i \(0.423049\pi\)
\(564\) 0 0
\(565\) 4.36383e8i 2.41948i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 3.26748e8i − 1.77368i −0.462074 0.886841i \(-0.652895\pi\)
0.462074 0.886841i \(-0.347105\pi\)
\(570\) 0 0
\(571\) 2.77811e8i 1.49225i 0.665806 + 0.746125i \(0.268088\pi\)
−0.665806 + 0.746125i \(0.731912\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 1.29291e8i − 0.680088i
\(576\) 0 0
\(577\) −1.83022e8 −0.952744 −0.476372 0.879244i \(-0.658048\pi\)
−0.476372 + 0.879244i \(0.658048\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.79176e6 −0.0346301
\(582\) 0 0
\(583\) 2.15756e8 1.08882
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.57537e8 −1.27328 −0.636641 0.771160i \(-0.719677\pi\)
−0.636641 + 0.771160i \(0.719677\pi\)
\(588\) 0 0
\(589\) 2.47493e8i 1.21120i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.96060e8i − 0.940210i −0.882610 0.470105i \(-0.844216\pi\)
0.882610 0.470105i \(-0.155784\pi\)
\(594\) 0 0
\(595\) 7.00667e6i 0.0332629i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1.79045e8i − 0.833072i −0.909119 0.416536i \(-0.863244\pi\)
0.909119 0.416536i \(-0.136756\pi\)
\(600\) 0 0
\(601\) 1.01945e8 0.469617 0.234808 0.972042i \(-0.424554\pi\)
0.234808 + 0.972042i \(0.424554\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.12873e7 0.367077
\(606\) 0 0
\(607\) −5.68302e7 −0.254105 −0.127052 0.991896i \(-0.540552\pi\)
−0.127052 + 0.991896i \(0.540552\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.07542e7 0.354031
\(612\) 0 0
\(613\) − 1.94135e8i − 0.842797i −0.906876 0.421399i \(-0.861539\pi\)
0.906876 0.421399i \(-0.138461\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.21008e8i 0.515180i 0.966254 + 0.257590i \(0.0829284\pi\)
−0.966254 + 0.257590i \(0.917072\pi\)
\(618\) 0 0
\(619\) 3.38818e8i 1.42855i 0.699867 + 0.714273i \(0.253243\pi\)
−0.699867 + 0.714273i \(0.746757\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.33875e6i 0.0179432i
\(624\) 0 0
\(625\) −2.78964e8 −1.14264
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.64608e8 1.06329
\(630\) 0 0
\(631\) 2.31097e8 0.919827 0.459913 0.887964i \(-0.347880\pi\)
0.459913 + 0.887964i \(0.347880\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.30078e8 −2.07023
\(636\) 0 0
\(637\) − 8.74957e7i − 0.338508i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.38832e8i 0.527125i 0.964642 + 0.263563i \(0.0848976\pi\)
−0.964642 + 0.263563i \(0.915102\pi\)
\(642\) 0 0
\(643\) − 1.50268e8i − 0.565242i −0.959232 0.282621i \(-0.908796\pi\)
0.959232 0.282621i \(-0.0912039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2.22339e8i − 0.820922i −0.911878 0.410461i \(-0.865368\pi\)
0.911878 0.410461i \(-0.134632\pi\)
\(648\) 0 0
\(649\) −1.66080e7 −0.0607552
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.97454e7 0.322309 0.161155 0.986929i \(-0.448478\pi\)
0.161155 + 0.986929i \(0.448478\pi\)
\(654\) 0 0
\(655\) 7.06804e7 0.251521
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.46917e8 0.862770 0.431385 0.902168i \(-0.358025\pi\)
0.431385 + 0.902168i \(0.358025\pi\)
\(660\) 0 0
\(661\) 1.72712e8i 0.598024i 0.954249 + 0.299012i \(0.0966571\pi\)
−0.954249 + 0.299012i \(0.903343\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.49225e7i − 0.0507430i
\(666\) 0 0
\(667\) − 2.03882e8i − 0.687072i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.64121e8i 0.874249i
\(672\) 0 0
\(673\) −1.66236e8 −0.545356 −0.272678 0.962105i \(-0.587909\pi\)
−0.272678 + 0.962105i \(0.587909\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.38587e8 −1.09120 −0.545601 0.838045i \(-0.683698\pi\)
−0.545601 + 0.838045i \(0.683698\pi\)
\(678\) 0 0
\(679\) 2.79591e6 0.00893127
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.36588e7 −0.0428698 −0.0214349 0.999770i \(-0.506823\pi\)
−0.0214349 + 0.999770i \(0.506823\pi\)
\(684\) 0 0
\(685\) 3.46656e8i 1.07852i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.07089e8i 0.327406i
\(690\) 0 0
\(691\) − 2.87568e8i − 0.871577i −0.900049 0.435789i \(-0.856470\pi\)
0.900049 0.435789i \(-0.143530\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 3.38711e8i − 1.00896i
\(696\) 0 0
\(697\) 5.95287e7 0.175804
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3.04539e8 −0.884075 −0.442037 0.896997i \(-0.645744\pi\)
−0.442037 + 0.896997i \(0.645744\pi\)
\(702\) 0 0
\(703\) −5.63550e8 −1.62206
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.68154e7 0.0475826
\(708\) 0 0
\(709\) − 5.94975e8i − 1.66940i −0.550706 0.834699i \(-0.685642\pi\)
0.550706 0.834699i \(-0.314358\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 3.92029e8i − 1.08156i
\(714\) 0 0
\(715\) 1.88938e8i 0.516894i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.96990e8i 1.87517i 0.347760 + 0.937583i \(0.386942\pi\)
−0.347760 + 0.937583i \(0.613058\pi\)
\(720\) 0 0
\(721\) 5.82336e6 0.0155370
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.63729e8 0.692059
\(726\) 0 0
\(727\) −7.97034e7 −0.207431 −0.103716 0.994607i \(-0.533073\pi\)
−0.103716 + 0.994607i \(0.533073\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.64455e8 −0.421013
\(732\) 0 0
\(733\) − 5.90402e8i − 1.49912i −0.661937 0.749559i \(-0.730266\pi\)
0.661937 0.749559i \(-0.269734\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.31475e7i 0.157744i
\(738\) 0 0
\(739\) 5.48328e7i 0.135865i 0.997690 + 0.0679324i \(0.0216402\pi\)
−0.997690 + 0.0679324i \(0.978360\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.92105e8i 0.712153i 0.934457 + 0.356076i \(0.115886\pi\)
−0.934457 + 0.356076i \(0.884114\pi\)
\(744\) 0 0
\(745\) 3.92624e8 0.949529
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.29075e6 −0.0221108
\(750\) 0 0
\(751\) −4.91813e8 −1.16113 −0.580564 0.814215i \(-0.697168\pi\)
−0.580564 + 0.814215i \(0.697168\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.08939e9 2.53129
\(756\) 0 0
\(757\) 1.26268e8i 0.291076i 0.989353 + 0.145538i \(0.0464912\pi\)
−0.989353 + 0.145538i \(0.953509\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.59472e8i 1.26947i 0.772728 + 0.634737i \(0.218892\pi\)
−0.772728 + 0.634737i \(0.781108\pi\)
\(762\) 0 0
\(763\) 1.99059e7i 0.0448135i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 8.24326e6i − 0.0182689i
\(768\) 0 0
\(769\) 7.51300e8 1.65209 0.826046 0.563603i \(-0.190585\pi\)
0.826046 + 0.563603i \(0.190585\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.22477e8 −1.56417 −0.782087 0.623169i \(-0.785845\pi\)
−0.782087 + 0.623169i \(0.785845\pi\)
\(774\) 0 0
\(775\) 5.07102e8 1.08941
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.26782e8 −0.268191
\(780\) 0 0
\(781\) − 1.56055e8i − 0.327586i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.57931e8i 0.946653i
\(786\) 0 0
\(787\) − 4.69415e8i − 0.963014i −0.876442 0.481507i \(-0.840090\pi\)
0.876442 0.481507i \(-0.159910\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 3.61288e7i − 0.0730001i
\(792\) 0 0
\(793\) −1.31094e8 −0.262884
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.15029e8 −1.01732 −0.508659 0.860968i \(-0.669859\pi\)
−0.508659 + 0.860968i \(0.669859\pi\)
\(798\) 0 0
\(799\) −3.21258e8 −0.629816
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.09588e8 1.56357
\(804\) 0 0
\(805\) 2.36372e7i 0.0453116i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.78752e8i 0.715335i 0.933849 + 0.357668i \(0.116428\pi\)
−0.933849 + 0.357668i \(0.883572\pi\)
\(810\) 0 0
\(811\) 8.01790e8i 1.50314i 0.659656 + 0.751568i \(0.270702\pi\)
−0.659656 + 0.751568i \(0.729298\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1.58852e8i − 0.293440i
\(816\) 0 0
\(817\) 3.50250e8 0.642261
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.04505e9 1.88845 0.944226 0.329298i \(-0.106812\pi\)
0.944226 + 0.329298i \(0.106812\pi\)
\(822\) 0 0
\(823\) −5.24543e7 −0.0940982 −0.0470491 0.998893i \(-0.514982\pi\)
−0.0470491 + 0.998893i \(0.514982\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.55035e8 −0.627703 −0.313851 0.949472i \(-0.601619\pi\)
−0.313851 + 0.949472i \(0.601619\pi\)
\(828\) 0 0
\(829\) − 3.06153e8i − 0.537372i −0.963228 0.268686i \(-0.913411\pi\)
0.963228 0.268686i \(-0.0865894\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.48077e8i 0.602200i
\(834\) 0 0
\(835\) − 1.46573e9i − 2.51765i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 5.16944e8i − 0.875301i −0.899145 0.437650i \(-0.855811\pi\)
0.899145 0.437650i \(-0.144189\pi\)
\(840\) 0 0
\(841\) −1.78943e8 −0.300834
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.21899e8 1.19648
\(846\) 0 0
\(847\) −6.72989e6 −0.0110754
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.92663e8 1.44844
\(852\) 0 0
\(853\) 7.67664e8i 1.23687i 0.785836 + 0.618435i \(0.212233\pi\)
−0.785836 + 0.618435i \(0.787767\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.58878e8i 0.729045i 0.931195 + 0.364523i \(0.118768\pi\)
−0.931195 + 0.364523i \(0.881232\pi\)
\(858\) 0 0
\(859\) − 7.79347e8i − 1.22957i −0.788697 0.614783i \(-0.789244\pi\)
0.788697 0.614783i \(-0.210756\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.69372e8i 0.263518i 0.991282 + 0.131759i \(0.0420625\pi\)
−0.991282 + 0.131759i \(0.957937\pi\)
\(864\) 0 0
\(865\) 2.54963e8 0.393940
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7.19224e8 −1.09599
\(870\) 0 0
\(871\) −3.13428e7 −0.0474333
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.36650e6 0.00950335
\(876\) 0 0
\(877\) 6.85516e8i 1.01629i 0.861271 + 0.508146i \(0.169669\pi\)
−0.861271 + 0.508146i \(0.830331\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.99510e7i 0.131546i 0.997835 + 0.0657731i \(0.0209513\pi\)
−0.997835 + 0.0657731i \(0.979049\pi\)
\(882\) 0 0
\(883\) − 7.65510e8i − 1.11191i −0.831213 0.555954i \(-0.812353\pi\)
0.831213 0.555954i \(-0.187647\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.43357e8i 0.921894i 0.887428 + 0.460947i \(0.152490\pi\)
−0.887428 + 0.460947i \(0.847510\pi\)
\(888\) 0 0
\(889\) 4.38858e7 0.0624625
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.84202e8 0.960793
\(894\) 0 0
\(895\) 1.81489e9 2.53153
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7.99662e8 1.10059
\(900\) 0 0
\(901\) − 4.26023e8i − 0.582450i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.09119e9i 1.47216i
\(906\) 0 0
\(907\) 2.38011e8i 0.318988i 0.987199 + 0.159494i \(0.0509863\pi\)
−0.987199 + 0.159494i \(0.949014\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1.29557e9i − 1.71358i −0.515664 0.856791i \(-0.672455\pi\)
0.515664 0.856791i \(-0.327545\pi\)
\(912\) 0 0
\(913\) 7.28584e8 0.957343
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.85173e6 −0.00758885
\(918\) 0 0
\(919\) 1.83739e8 0.236731 0.118366 0.992970i \(-0.462234\pi\)
0.118366 + 0.992970i \(0.462234\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.74568e7 0.0985042
\(924\) 0 0
\(925\) 1.15469e9i 1.45895i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.72117e8i 1.08775i 0.839167 + 0.543873i \(0.183043\pi\)
−0.839167 + 0.543873i \(0.816957\pi\)
\(930\) 0 0
\(931\) − 7.41319e8i − 0.918663i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 7.51638e8i − 0.919547i
\(936\) 0 0
\(937\) 7.39899e8 0.899402 0.449701 0.893179i \(-0.351531\pi\)
0.449701 + 0.893179i \(0.351531\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5.86767e8 −0.704201 −0.352101 0.935962i \(-0.614532\pi\)
−0.352101 + 0.935962i \(0.614532\pi\)
\(942\) 0 0
\(943\) 2.00822e8 0.239484
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.35408e9 1.59439 0.797195 0.603722i \(-0.206316\pi\)
0.797195 + 0.603722i \(0.206316\pi\)
\(948\) 0 0
\(949\) 4.01833e8i 0.470161i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.06231e8i 0.931495i 0.884918 + 0.465748i \(0.154215\pi\)
−0.884918 + 0.465748i \(0.845785\pi\)
\(954\) 0 0
\(955\) 1.27457e9i 1.46336i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 2.87001e7i − 0.0325408i
\(960\) 0 0
\(961\) 6.50102e8 0.732506
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.48115e9 1.64822
\(966\) 0 0
\(967\) −6.61478e8 −0.731536 −0.365768 0.930706i \(-0.619194\pi\)
−0.365768 + 0.930706i \(0.619194\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.01203e8 −0.438234 −0.219117 0.975699i \(-0.570318\pi\)
−0.219117 + 0.975699i \(0.570318\pi\)
\(972\) 0 0
\(973\) 2.80423e7i 0.0304422i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7.93550e8i − 0.850924i −0.904976 0.425462i \(-0.860112\pi\)
0.904976 0.425462i \(-0.139888\pi\)
\(978\) 0 0
\(979\) − 4.65438e8i − 0.496037i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.47001e8i 0.681153i 0.940217 + 0.340576i \(0.110622\pi\)
−0.940217 + 0.340576i \(0.889378\pi\)
\(984\) 0 0
\(985\) 6.47690e8 0.677733
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.54796e8 −0.573514
\(990\) 0 0
\(991\) −2.18163e8 −0.224161 −0.112081 0.993699i \(-0.535752\pi\)
−0.112081 + 0.993699i \(0.535752\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.96229e9 1.99203
\(996\) 0 0
\(997\) − 1.47959e9i − 1.49299i −0.665393 0.746494i \(-0.731736\pi\)
0.665393 0.746494i \(-0.268264\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 576.7.h.a.161.14 yes 16
3.2 odd 2 inner 576.7.h.a.161.2 yes 16
4.3 odd 2 inner 576.7.h.a.161.16 yes 16
8.3 odd 2 inner 576.7.h.a.161.3 yes 16
8.5 even 2 inner 576.7.h.a.161.1 16
12.11 even 2 inner 576.7.h.a.161.4 yes 16
24.5 odd 2 inner 576.7.h.a.161.13 yes 16
24.11 even 2 inner 576.7.h.a.161.15 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.7.h.a.161.1 16 8.5 even 2 inner
576.7.h.a.161.2 yes 16 3.2 odd 2 inner
576.7.h.a.161.3 yes 16 8.3 odd 2 inner
576.7.h.a.161.4 yes 16 12.11 even 2 inner
576.7.h.a.161.13 yes 16 24.5 odd 2 inner
576.7.h.a.161.14 yes 16 1.1 even 1 trivial
576.7.h.a.161.15 yes 16 24.11 even 2 inner
576.7.h.a.161.16 yes 16 4.3 odd 2 inner