Properties

Label 384.10.a.d.1.2
Level $384$
Weight $10$
Character 384.1
Self dual yes
Analytic conductor $197.774$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,10,Mod(1,384)]
 
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, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(197.773761087\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 14124x^{2} - 170336x + 18391464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-103.810\) of defining polynomial
Character \(\chi\) \(=\) 384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-81.0000 q^{3} -397.737 q^{5} +7340.72 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q-81.0000 q^{3} -397.737 q^{5} +7340.72 q^{7} +6561.00 q^{9} +55385.5 q^{11} -3409.79 q^{13} +32216.7 q^{15} -155165. q^{17} -205264. q^{19} -594598. q^{21} +158366. q^{23} -1.79493e6 q^{25} -531441. q^{27} -1.82402e6 q^{29} -1.92219e6 q^{31} -4.48622e6 q^{33} -2.91968e6 q^{35} -4.17394e6 q^{37} +276193. q^{39} -1.70327e7 q^{41} +3.26916e7 q^{43} -2.60955e6 q^{45} -3.89556e7 q^{47} +1.35326e7 q^{49} +1.25683e7 q^{51} +4.64592e7 q^{53} -2.20288e7 q^{55} +1.66264e7 q^{57} +1.03738e8 q^{59} -4.66289e7 q^{61} +4.81625e7 q^{63} +1.35620e6 q^{65} -6.63363e7 q^{67} -1.28276e7 q^{69} -6.28975e7 q^{71} +1.04953e8 q^{73} +1.45389e8 q^{75} +4.06569e8 q^{77} +2.63880e8 q^{79} +4.30467e7 q^{81} -5.37032e8 q^{83} +6.17146e7 q^{85} +1.47746e8 q^{87} -2.14261e8 q^{89} -2.50303e7 q^{91} +1.55698e8 q^{93} +8.16410e7 q^{95} +1.21818e9 q^{97} +3.63384e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9} - 15824 q^{11} + 82440 q^{13} - 139968 q^{15} + 165912 q^{17} - 539904 q^{19} + 392040 q^{21} - 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} + 1850864 q^{29} - 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} + 4187992 q^{37} - 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} + 11337408 q^{45} - 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} + 29418288 q^{53} - 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} - 99764648 q^{61} - 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} + 59104080 q^{69} - 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} - 230564704 q^{77} - 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} + 1041391104 q^{85} - 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} + 259034760 q^{93} - 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −81.0000 −0.577350
\(4\) 0 0
\(5\) −397.737 −0.284597 −0.142299 0.989824i \(-0.545449\pi\)
−0.142299 + 0.989824i \(0.545449\pi\)
\(6\) 0 0
\(7\) 7340.72 1.15557 0.577787 0.816188i \(-0.303917\pi\)
0.577787 + 0.816188i \(0.303917\pi\)
\(8\) 0 0
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 55385.5 1.14059 0.570294 0.821441i \(-0.306829\pi\)
0.570294 + 0.821441i \(0.306829\pi\)
\(12\) 0 0
\(13\) −3409.79 −0.0331118 −0.0165559 0.999863i \(-0.505270\pi\)
−0.0165559 + 0.999863i \(0.505270\pi\)
\(14\) 0 0
\(15\) 32216.7 0.164312
\(16\) 0 0
\(17\) −155165. −0.450580 −0.225290 0.974292i \(-0.572333\pi\)
−0.225290 + 0.974292i \(0.572333\pi\)
\(18\) 0 0
\(19\) −205264. −0.361344 −0.180672 0.983543i \(-0.557827\pi\)
−0.180672 + 0.983543i \(0.557827\pi\)
\(20\) 0 0
\(21\) −594598. −0.667171
\(22\) 0 0
\(23\) 158366. 0.118001 0.0590006 0.998258i \(-0.481209\pi\)
0.0590006 + 0.998258i \(0.481209\pi\)
\(24\) 0 0
\(25\) −1.79493e6 −0.919004
\(26\) 0 0
\(27\) −531441. −0.192450
\(28\) 0 0
\(29\) −1.82402e6 −0.478893 −0.239446 0.970910i \(-0.576966\pi\)
−0.239446 + 0.970910i \(0.576966\pi\)
\(30\) 0 0
\(31\) −1.92219e6 −0.373826 −0.186913 0.982377i \(-0.559848\pi\)
−0.186913 + 0.982377i \(0.559848\pi\)
\(32\) 0 0
\(33\) −4.48622e6 −0.658519
\(34\) 0 0
\(35\) −2.91968e6 −0.328873
\(36\) 0 0
\(37\) −4.17394e6 −0.366132 −0.183066 0.983101i \(-0.558602\pi\)
−0.183066 + 0.983101i \(0.558602\pi\)
\(38\) 0 0
\(39\) 276193. 0.0191171
\(40\) 0 0
\(41\) −1.70327e7 −0.941361 −0.470681 0.882304i \(-0.655992\pi\)
−0.470681 + 0.882304i \(0.655992\pi\)
\(42\) 0 0
\(43\) 3.26916e7 1.45824 0.729119 0.684387i \(-0.239930\pi\)
0.729119 + 0.684387i \(0.239930\pi\)
\(44\) 0 0
\(45\) −2.60955e6 −0.0948658
\(46\) 0 0
\(47\) −3.89556e7 −1.16447 −0.582237 0.813019i \(-0.697822\pi\)
−0.582237 + 0.813019i \(0.697822\pi\)
\(48\) 0 0
\(49\) 1.35326e7 0.335350
\(50\) 0 0
\(51\) 1.25683e7 0.260143
\(52\) 0 0
\(53\) 4.64592e7 0.808779 0.404390 0.914587i \(-0.367484\pi\)
0.404390 + 0.914587i \(0.367484\pi\)
\(54\) 0 0
\(55\) −2.20288e7 −0.324608
\(56\) 0 0
\(57\) 1.66264e7 0.208622
\(58\) 0 0
\(59\) 1.03738e8 1.11457 0.557283 0.830323i \(-0.311844\pi\)
0.557283 + 0.830323i \(0.311844\pi\)
\(60\) 0 0
\(61\) −4.66289e7 −0.431193 −0.215596 0.976483i \(-0.569170\pi\)
−0.215596 + 0.976483i \(0.569170\pi\)
\(62\) 0 0
\(63\) 4.81625e7 0.385191
\(64\) 0 0
\(65\) 1.35620e6 0.00942352
\(66\) 0 0
\(67\) −6.63363e7 −0.402175 −0.201087 0.979573i \(-0.564448\pi\)
−0.201087 + 0.979573i \(0.564448\pi\)
\(68\) 0 0
\(69\) −1.28276e7 −0.0681280
\(70\) 0 0
\(71\) −6.28975e7 −0.293745 −0.146873 0.989155i \(-0.546921\pi\)
−0.146873 + 0.989155i \(0.546921\pi\)
\(72\) 0 0
\(73\) 1.04953e8 0.432557 0.216279 0.976332i \(-0.430608\pi\)
0.216279 + 0.976332i \(0.430608\pi\)
\(74\) 0 0
\(75\) 1.45389e8 0.530587
\(76\) 0 0
\(77\) 4.06569e8 1.31803
\(78\) 0 0
\(79\) 2.63880e8 0.762227 0.381113 0.924528i \(-0.375541\pi\)
0.381113 + 0.924528i \(0.375541\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) −5.37032e8 −1.24208 −0.621039 0.783780i \(-0.713289\pi\)
−0.621039 + 0.783780i \(0.713289\pi\)
\(84\) 0 0
\(85\) 6.17146e7 0.128234
\(86\) 0 0
\(87\) 1.47746e8 0.276489
\(88\) 0 0
\(89\) −2.14261e8 −0.361984 −0.180992 0.983485i \(-0.557931\pi\)
−0.180992 + 0.983485i \(0.557931\pi\)
\(90\) 0 0
\(91\) −2.50303e7 −0.0382631
\(92\) 0 0
\(93\) 1.55698e8 0.215828
\(94\) 0 0
\(95\) 8.16410e7 0.102838
\(96\) 0 0
\(97\) 1.21818e9 1.39714 0.698570 0.715542i \(-0.253820\pi\)
0.698570 + 0.715542i \(0.253820\pi\)
\(98\) 0 0
\(99\) 3.63384e8 0.380196
\(100\) 0 0
\(101\) 7.62080e8 0.728709 0.364355 0.931260i \(-0.381290\pi\)
0.364355 + 0.931260i \(0.381290\pi\)
\(102\) 0 0
\(103\) −1.91478e9 −1.67630 −0.838148 0.545443i \(-0.816361\pi\)
−0.838148 + 0.545443i \(0.816361\pi\)
\(104\) 0 0
\(105\) 2.36494e8 0.189875
\(106\) 0 0
\(107\) 5.38600e8 0.397228 0.198614 0.980078i \(-0.436356\pi\)
0.198614 + 0.980078i \(0.436356\pi\)
\(108\) 0 0
\(109\) −2.78761e8 −0.189153 −0.0945765 0.995518i \(-0.530150\pi\)
−0.0945765 + 0.995518i \(0.530150\pi\)
\(110\) 0 0
\(111\) 3.38089e8 0.211387
\(112\) 0 0
\(113\) −1.84040e8 −0.106184 −0.0530921 0.998590i \(-0.516908\pi\)
−0.0530921 + 0.998590i \(0.516908\pi\)
\(114\) 0 0
\(115\) −6.29880e7 −0.0335828
\(116\) 0 0
\(117\) −2.23716e7 −0.0110373
\(118\) 0 0
\(119\) −1.13902e9 −0.520679
\(120\) 0 0
\(121\) 7.09603e8 0.300941
\(122\) 0 0
\(123\) 1.37965e9 0.543495
\(124\) 0 0
\(125\) 1.49074e9 0.546143
\(126\) 0 0
\(127\) −4.29178e9 −1.46393 −0.731966 0.681341i \(-0.761397\pi\)
−0.731966 + 0.681341i \(0.761397\pi\)
\(128\) 0 0
\(129\) −2.64802e9 −0.841914
\(130\) 0 0
\(131\) 2.13152e9 0.632367 0.316184 0.948698i \(-0.397598\pi\)
0.316184 + 0.948698i \(0.397598\pi\)
\(132\) 0 0
\(133\) −1.50678e9 −0.417560
\(134\) 0 0
\(135\) 2.11374e8 0.0547708
\(136\) 0 0
\(137\) −8.67441e8 −0.210377 −0.105188 0.994452i \(-0.533545\pi\)
−0.105188 + 0.994452i \(0.533545\pi\)
\(138\) 0 0
\(139\) −1.94615e9 −0.442190 −0.221095 0.975252i \(-0.570963\pi\)
−0.221095 + 0.975252i \(0.570963\pi\)
\(140\) 0 0
\(141\) 3.15541e9 0.672310
\(142\) 0 0
\(143\) −1.88853e8 −0.0377669
\(144\) 0 0
\(145\) 7.25480e8 0.136292
\(146\) 0 0
\(147\) −1.09614e9 −0.193615
\(148\) 0 0
\(149\) −4.69461e9 −0.780299 −0.390149 0.920752i \(-0.627577\pi\)
−0.390149 + 0.920752i \(0.627577\pi\)
\(150\) 0 0
\(151\) 3.44172e9 0.538741 0.269370 0.963037i \(-0.413184\pi\)
0.269370 + 0.963037i \(0.413184\pi\)
\(152\) 0 0
\(153\) −1.01803e9 −0.150193
\(154\) 0 0
\(155\) 7.64526e8 0.106390
\(156\) 0 0
\(157\) −9.02010e9 −1.18485 −0.592424 0.805626i \(-0.701829\pi\)
−0.592424 + 0.805626i \(0.701829\pi\)
\(158\) 0 0
\(159\) −3.76319e9 −0.466949
\(160\) 0 0
\(161\) 1.16252e9 0.136359
\(162\) 0 0
\(163\) −1.13021e10 −1.25405 −0.627026 0.778999i \(-0.715728\pi\)
−0.627026 + 0.778999i \(0.715728\pi\)
\(164\) 0 0
\(165\) 1.78434e9 0.187413
\(166\) 0 0
\(167\) 1.59657e10 1.58841 0.794207 0.607647i \(-0.207886\pi\)
0.794207 + 0.607647i \(0.207886\pi\)
\(168\) 0 0
\(169\) −1.05929e10 −0.998904
\(170\) 0 0
\(171\) −1.34674e9 −0.120448
\(172\) 0 0
\(173\) 1.95986e10 1.66348 0.831740 0.555166i \(-0.187345\pi\)
0.831740 + 0.555166i \(0.187345\pi\)
\(174\) 0 0
\(175\) −1.31761e10 −1.06198
\(176\) 0 0
\(177\) −8.40281e9 −0.643495
\(178\) 0 0
\(179\) −9.19596e9 −0.669512 −0.334756 0.942305i \(-0.608654\pi\)
−0.334756 + 0.942305i \(0.608654\pi\)
\(180\) 0 0
\(181\) 2.44145e10 1.69080 0.845402 0.534130i \(-0.179361\pi\)
0.845402 + 0.534130i \(0.179361\pi\)
\(182\) 0 0
\(183\) 3.77694e9 0.248949
\(184\) 0 0
\(185\) 1.66013e9 0.104200
\(186\) 0 0
\(187\) −8.59386e9 −0.513927
\(188\) 0 0
\(189\) −3.90116e9 −0.222390
\(190\) 0 0
\(191\) −1.02163e10 −0.555446 −0.277723 0.960661i \(-0.589580\pi\)
−0.277723 + 0.960661i \(0.589580\pi\)
\(192\) 0 0
\(193\) −3.80724e9 −0.197516 −0.0987581 0.995111i \(-0.531487\pi\)
−0.0987581 + 0.995111i \(0.531487\pi\)
\(194\) 0 0
\(195\) −1.09852e8 −0.00544067
\(196\) 0 0
\(197\) −4.21835e9 −0.199547 −0.0997734 0.995010i \(-0.531812\pi\)
−0.0997734 + 0.995010i \(0.531812\pi\)
\(198\) 0 0
\(199\) 1.09330e9 0.0494198 0.0247099 0.999695i \(-0.492134\pi\)
0.0247099 + 0.999695i \(0.492134\pi\)
\(200\) 0 0
\(201\) 5.37324e9 0.232196
\(202\) 0 0
\(203\) −1.33896e10 −0.553396
\(204\) 0 0
\(205\) 6.77454e9 0.267909
\(206\) 0 0
\(207\) 1.03904e9 0.0393337
\(208\) 0 0
\(209\) −1.13686e10 −0.412145
\(210\) 0 0
\(211\) −1.95404e9 −0.0678677 −0.0339339 0.999424i \(-0.510804\pi\)
−0.0339339 + 0.999424i \(0.510804\pi\)
\(212\) 0 0
\(213\) 5.09470e9 0.169594
\(214\) 0 0
\(215\) −1.30027e10 −0.415011
\(216\) 0 0
\(217\) −1.41103e10 −0.431983
\(218\) 0 0
\(219\) −8.50123e9 −0.249737
\(220\) 0 0
\(221\) 5.29078e8 0.0149195
\(222\) 0 0
\(223\) 2.37625e8 0.00643457 0.00321729 0.999995i \(-0.498976\pi\)
0.00321729 + 0.999995i \(0.498976\pi\)
\(224\) 0 0
\(225\) −1.17765e10 −0.306335
\(226\) 0 0
\(227\) −5.44760e10 −1.36172 −0.680861 0.732412i \(-0.738394\pi\)
−0.680861 + 0.732412i \(0.738394\pi\)
\(228\) 0 0
\(229\) −6.04329e8 −0.0145216 −0.00726078 0.999974i \(-0.502311\pi\)
−0.00726078 + 0.999974i \(0.502311\pi\)
\(230\) 0 0
\(231\) −3.29321e10 −0.760967
\(232\) 0 0
\(233\) −7.72907e10 −1.71801 −0.859004 0.511968i \(-0.828917\pi\)
−0.859004 + 0.511968i \(0.828917\pi\)
\(234\) 0 0
\(235\) 1.54941e10 0.331406
\(236\) 0 0
\(237\) −2.13743e10 −0.440072
\(238\) 0 0
\(239\) −2.71496e10 −0.538235 −0.269118 0.963107i \(-0.586732\pi\)
−0.269118 + 0.963107i \(0.586732\pi\)
\(240\) 0 0
\(241\) −9.48987e10 −1.81211 −0.906053 0.423165i \(-0.860919\pi\)
−0.906053 + 0.423165i \(0.860919\pi\)
\(242\) 0 0
\(243\) −3.48678e9 −0.0641500
\(244\) 0 0
\(245\) −5.38241e9 −0.0954397
\(246\) 0 0
\(247\) 6.99906e8 0.0119647
\(248\) 0 0
\(249\) 4.34996e10 0.717114
\(250\) 0 0
\(251\) −3.17138e10 −0.504333 −0.252166 0.967684i \(-0.581143\pi\)
−0.252166 + 0.967684i \(0.581143\pi\)
\(252\) 0 0
\(253\) 8.77117e9 0.134591
\(254\) 0 0
\(255\) −4.99889e9 −0.0740359
\(256\) 0 0
\(257\) 1.11732e8 0.00159764 0.000798820 1.00000i \(-0.499746\pi\)
0.000798820 1.00000i \(0.499746\pi\)
\(258\) 0 0
\(259\) −3.06397e10 −0.423093
\(260\) 0 0
\(261\) −1.19674e10 −0.159631
\(262\) 0 0
\(263\) −1.46161e11 −1.88378 −0.941889 0.335924i \(-0.890952\pi\)
−0.941889 + 0.335924i \(0.890952\pi\)
\(264\) 0 0
\(265\) −1.84785e10 −0.230176
\(266\) 0 0
\(267\) 1.73552e10 0.208991
\(268\) 0 0
\(269\) 1.55093e11 1.80595 0.902977 0.429689i \(-0.141377\pi\)
0.902977 + 0.429689i \(0.141377\pi\)
\(270\) 0 0
\(271\) −3.67924e10 −0.414378 −0.207189 0.978301i \(-0.566431\pi\)
−0.207189 + 0.978301i \(0.566431\pi\)
\(272\) 0 0
\(273\) 2.02745e9 0.0220912
\(274\) 0 0
\(275\) −9.94131e10 −1.04821
\(276\) 0 0
\(277\) 1.16867e11 1.19270 0.596352 0.802723i \(-0.296616\pi\)
0.596352 + 0.802723i \(0.296616\pi\)
\(278\) 0 0
\(279\) −1.26115e10 −0.124609
\(280\) 0 0
\(281\) 2.69078e10 0.257454 0.128727 0.991680i \(-0.458911\pi\)
0.128727 + 0.991680i \(0.458911\pi\)
\(282\) 0 0
\(283\) 1.01819e10 0.0943603 0.0471801 0.998886i \(-0.484977\pi\)
0.0471801 + 0.998886i \(0.484977\pi\)
\(284\) 0 0
\(285\) −6.61292e9 −0.0593733
\(286\) 0 0
\(287\) −1.25032e11 −1.08781
\(288\) 0 0
\(289\) −9.45119e10 −0.796977
\(290\) 0 0
\(291\) −9.86729e10 −0.806639
\(292\) 0 0
\(293\) 6.27162e10 0.497136 0.248568 0.968614i \(-0.420040\pi\)
0.248568 + 0.968614i \(0.420040\pi\)
\(294\) 0 0
\(295\) −4.12606e10 −0.317202
\(296\) 0 0
\(297\) −2.94341e10 −0.219506
\(298\) 0 0
\(299\) −5.39994e8 −0.00390723
\(300\) 0 0
\(301\) 2.39980e11 1.68510
\(302\) 0 0
\(303\) −6.17285e10 −0.420721
\(304\) 0 0
\(305\) 1.85460e10 0.122716
\(306\) 0 0
\(307\) −8.29318e10 −0.532842 −0.266421 0.963857i \(-0.585841\pi\)
−0.266421 + 0.963857i \(0.585841\pi\)
\(308\) 0 0
\(309\) 1.55097e11 0.967810
\(310\) 0 0
\(311\) 3.17610e10 0.192518 0.0962592 0.995356i \(-0.469312\pi\)
0.0962592 + 0.995356i \(0.469312\pi\)
\(312\) 0 0
\(313\) 1.68887e11 0.994598 0.497299 0.867579i \(-0.334325\pi\)
0.497299 + 0.867579i \(0.334325\pi\)
\(314\) 0 0
\(315\) −1.91560e10 −0.109624
\(316\) 0 0
\(317\) 1.52554e11 0.848509 0.424254 0.905543i \(-0.360536\pi\)
0.424254 + 0.905543i \(0.360536\pi\)
\(318\) 0 0
\(319\) −1.01024e11 −0.546220
\(320\) 0 0
\(321\) −4.36266e10 −0.229339
\(322\) 0 0
\(323\) 3.18496e10 0.162815
\(324\) 0 0
\(325\) 6.12033e9 0.0304299
\(326\) 0 0
\(327\) 2.25797e10 0.109207
\(328\) 0 0
\(329\) −2.85962e11 −1.34564
\(330\) 0 0
\(331\) −3.26946e11 −1.49710 −0.748548 0.663081i \(-0.769249\pi\)
−0.748548 + 0.663081i \(0.769249\pi\)
\(332\) 0 0
\(333\) −2.73852e10 −0.122044
\(334\) 0 0
\(335\) 2.63844e10 0.114458
\(336\) 0 0
\(337\) −3.21830e11 −1.35923 −0.679614 0.733570i \(-0.737853\pi\)
−0.679614 + 0.733570i \(0.737853\pi\)
\(338\) 0 0
\(339\) 1.49073e10 0.0613055
\(340\) 0 0
\(341\) −1.06462e11 −0.426381
\(342\) 0 0
\(343\) −1.96886e11 −0.768052
\(344\) 0 0
\(345\) 5.10203e9 0.0193891
\(346\) 0 0
\(347\) −9.57562e10 −0.354556 −0.177278 0.984161i \(-0.556729\pi\)
−0.177278 + 0.984161i \(0.556729\pi\)
\(348\) 0 0
\(349\) 2.34178e11 0.844950 0.422475 0.906375i \(-0.361161\pi\)
0.422475 + 0.906375i \(0.361161\pi\)
\(350\) 0 0
\(351\) 1.81210e9 0.00637236
\(352\) 0 0
\(353\) 9.85742e9 0.0337891 0.0168946 0.999857i \(-0.494622\pi\)
0.0168946 + 0.999857i \(0.494622\pi\)
\(354\) 0 0
\(355\) 2.50166e10 0.0835991
\(356\) 0 0
\(357\) 9.22606e10 0.300614
\(358\) 0 0
\(359\) −1.59435e11 −0.506592 −0.253296 0.967389i \(-0.581515\pi\)
−0.253296 + 0.967389i \(0.581515\pi\)
\(360\) 0 0
\(361\) −2.80554e11 −0.869430
\(362\) 0 0
\(363\) −5.74779e10 −0.173748
\(364\) 0 0
\(365\) −4.17438e10 −0.123105
\(366\) 0 0
\(367\) −3.16207e11 −0.909859 −0.454930 0.890527i \(-0.650336\pi\)
−0.454930 + 0.890527i \(0.650336\pi\)
\(368\) 0 0
\(369\) −1.11752e11 −0.313787
\(370\) 0 0
\(371\) 3.41044e11 0.934604
\(372\) 0 0
\(373\) −3.44449e11 −0.921372 −0.460686 0.887563i \(-0.652397\pi\)
−0.460686 + 0.887563i \(0.652397\pi\)
\(374\) 0 0
\(375\) −1.20750e11 −0.315316
\(376\) 0 0
\(377\) 6.21952e9 0.0158570
\(378\) 0 0
\(379\) −5.90459e11 −1.46999 −0.734993 0.678075i \(-0.762814\pi\)
−0.734993 + 0.678075i \(0.762814\pi\)
\(380\) 0 0
\(381\) 3.47634e11 0.845202
\(382\) 0 0
\(383\) −1.38731e11 −0.329442 −0.164721 0.986340i \(-0.552672\pi\)
−0.164721 + 0.986340i \(0.552672\pi\)
\(384\) 0 0
\(385\) −1.61708e11 −0.375109
\(386\) 0 0
\(387\) 2.14490e11 0.486079
\(388\) 0 0
\(389\) 2.35069e10 0.0520501 0.0260251 0.999661i \(-0.491715\pi\)
0.0260251 + 0.999661i \(0.491715\pi\)
\(390\) 0 0
\(391\) −2.45728e10 −0.0531690
\(392\) 0 0
\(393\) −1.72653e11 −0.365097
\(394\) 0 0
\(395\) −1.04955e11 −0.216928
\(396\) 0 0
\(397\) −2.02054e11 −0.408234 −0.204117 0.978946i \(-0.565432\pi\)
−0.204117 + 0.978946i \(0.565432\pi\)
\(398\) 0 0
\(399\) 1.22050e11 0.241078
\(400\) 0 0
\(401\) −8.15838e11 −1.57563 −0.787815 0.615912i \(-0.788788\pi\)
−0.787815 + 0.615912i \(0.788788\pi\)
\(402\) 0 0
\(403\) 6.55427e9 0.0123780
\(404\) 0 0
\(405\) −1.71213e10 −0.0316219
\(406\) 0 0
\(407\) −2.31175e11 −0.417606
\(408\) 0 0
\(409\) 6.04841e11 1.06877 0.534387 0.845240i \(-0.320542\pi\)
0.534387 + 0.845240i \(0.320542\pi\)
\(410\) 0 0
\(411\) 7.02627e10 0.121461
\(412\) 0 0
\(413\) 7.61515e11 1.28796
\(414\) 0 0
\(415\) 2.13597e11 0.353492
\(416\) 0 0
\(417\) 1.57638e11 0.255298
\(418\) 0 0
\(419\) −3.44512e11 −0.546061 −0.273031 0.962005i \(-0.588026\pi\)
−0.273031 + 0.962005i \(0.588026\pi\)
\(420\) 0 0
\(421\) −4.38031e11 −0.679572 −0.339786 0.940503i \(-0.610355\pi\)
−0.339786 + 0.940503i \(0.610355\pi\)
\(422\) 0 0
\(423\) −2.55588e11 −0.388158
\(424\) 0 0
\(425\) 2.78509e11 0.414085
\(426\) 0 0
\(427\) −3.42290e11 −0.498275
\(428\) 0 0
\(429\) 1.52971e10 0.0218047
\(430\) 0 0
\(431\) 2.76492e11 0.385954 0.192977 0.981203i \(-0.438186\pi\)
0.192977 + 0.981203i \(0.438186\pi\)
\(432\) 0 0
\(433\) 7.56798e10 0.103463 0.0517315 0.998661i \(-0.483526\pi\)
0.0517315 + 0.998661i \(0.483526\pi\)
\(434\) 0 0
\(435\) −5.87639e10 −0.0786880
\(436\) 0 0
\(437\) −3.25068e10 −0.0426391
\(438\) 0 0
\(439\) −2.90618e11 −0.373450 −0.186725 0.982412i \(-0.559787\pi\)
−0.186725 + 0.982412i \(0.559787\pi\)
\(440\) 0 0
\(441\) 8.87873e10 0.111783
\(442\) 0 0
\(443\) −4.10685e11 −0.506632 −0.253316 0.967384i \(-0.581521\pi\)
−0.253316 + 0.967384i \(0.581521\pi\)
\(444\) 0 0
\(445\) 8.52197e10 0.103020
\(446\) 0 0
\(447\) 3.80263e11 0.450506
\(448\) 0 0
\(449\) −8.84583e11 −1.02714 −0.513571 0.858047i \(-0.671678\pi\)
−0.513571 + 0.858047i \(0.671678\pi\)
\(450\) 0 0
\(451\) −9.43365e11 −1.07371
\(452\) 0 0
\(453\) −2.78780e11 −0.311042
\(454\) 0 0
\(455\) 9.95547e9 0.0108896
\(456\) 0 0
\(457\) −4.66348e11 −0.500135 −0.250068 0.968228i \(-0.580453\pi\)
−0.250068 + 0.968228i \(0.580453\pi\)
\(458\) 0 0
\(459\) 8.24608e10 0.0867142
\(460\) 0 0
\(461\) −1.27386e11 −0.131361 −0.0656807 0.997841i \(-0.520922\pi\)
−0.0656807 + 0.997841i \(0.520922\pi\)
\(462\) 0 0
\(463\) 7.67555e11 0.776238 0.388119 0.921609i \(-0.373125\pi\)
0.388119 + 0.921609i \(0.373125\pi\)
\(464\) 0 0
\(465\) −6.19266e10 −0.0614242
\(466\) 0 0
\(467\) −6.96670e11 −0.677800 −0.338900 0.940822i \(-0.610055\pi\)
−0.338900 + 0.940822i \(0.610055\pi\)
\(468\) 0 0
\(469\) −4.86956e11 −0.464742
\(470\) 0 0
\(471\) 7.30628e11 0.684072
\(472\) 0 0
\(473\) 1.81064e12 1.66325
\(474\) 0 0
\(475\) 3.68434e11 0.332077
\(476\) 0 0
\(477\) 3.04819e11 0.269593
\(478\) 0 0
\(479\) 1.51269e11 0.131292 0.0656462 0.997843i \(-0.479089\pi\)
0.0656462 + 0.997843i \(0.479089\pi\)
\(480\) 0 0
\(481\) 1.42322e10 0.0121233
\(482\) 0 0
\(483\) −9.41641e10 −0.0787270
\(484\) 0 0
\(485\) −4.84516e11 −0.397622
\(486\) 0 0
\(487\) 2.04990e12 1.65140 0.825701 0.564108i \(-0.190780\pi\)
0.825701 + 0.564108i \(0.190780\pi\)
\(488\) 0 0
\(489\) 9.15471e11 0.724027
\(490\) 0 0
\(491\) 1.38298e12 1.07387 0.536933 0.843625i \(-0.319583\pi\)
0.536933 + 0.843625i \(0.319583\pi\)
\(492\) 0 0
\(493\) 2.83023e11 0.215780
\(494\) 0 0
\(495\) −1.44531e11 −0.108203
\(496\) 0 0
\(497\) −4.61713e11 −0.339444
\(498\) 0 0
\(499\) 1.12272e12 0.810626 0.405313 0.914178i \(-0.367163\pi\)
0.405313 + 0.914178i \(0.367163\pi\)
\(500\) 0 0
\(501\) −1.29322e12 −0.917072
\(502\) 0 0
\(503\) −6.76786e11 −0.471407 −0.235703 0.971825i \(-0.575739\pi\)
−0.235703 + 0.971825i \(0.575739\pi\)
\(504\) 0 0
\(505\) −3.03107e11 −0.207389
\(506\) 0 0
\(507\) 8.58023e11 0.576717
\(508\) 0 0
\(509\) 8.27029e11 0.546124 0.273062 0.961996i \(-0.411964\pi\)
0.273062 + 0.961996i \(0.411964\pi\)
\(510\) 0 0
\(511\) 7.70434e11 0.499852
\(512\) 0 0
\(513\) 1.09086e11 0.0695407
\(514\) 0 0
\(515\) 7.61577e11 0.477069
\(516\) 0 0
\(517\) −2.15758e12 −1.32819
\(518\) 0 0
\(519\) −1.58749e12 −0.960410
\(520\) 0 0
\(521\) 2.68612e12 1.59718 0.798592 0.601872i \(-0.205578\pi\)
0.798592 + 0.601872i \(0.205578\pi\)
\(522\) 0 0
\(523\) 1.94122e12 1.13453 0.567267 0.823534i \(-0.308001\pi\)
0.567267 + 0.823534i \(0.308001\pi\)
\(524\) 0 0
\(525\) 1.06726e12 0.613133
\(526\) 0 0
\(527\) 2.98256e11 0.168438
\(528\) 0 0
\(529\) −1.77607e12 −0.986076
\(530\) 0 0
\(531\) 6.80628e11 0.371522
\(532\) 0 0
\(533\) 5.80779e10 0.0311701
\(534\) 0 0
\(535\) −2.14221e11 −0.113050
\(536\) 0 0
\(537\) 7.44873e11 0.386543
\(538\) 0 0
\(539\) 7.49509e11 0.382496
\(540\) 0 0
\(541\) 3.54650e11 0.177997 0.0889983 0.996032i \(-0.471633\pi\)
0.0889983 + 0.996032i \(0.471633\pi\)
\(542\) 0 0
\(543\) −1.97757e12 −0.976187
\(544\) 0 0
\(545\) 1.10874e11 0.0538324
\(546\) 0 0
\(547\) 4.51666e11 0.215712 0.107856 0.994167i \(-0.465601\pi\)
0.107856 + 0.994167i \(0.465601\pi\)
\(548\) 0 0
\(549\) −3.05933e11 −0.143731
\(550\) 0 0
\(551\) 3.74405e11 0.173045
\(552\) 0 0
\(553\) 1.93707e12 0.880809
\(554\) 0 0
\(555\) −1.34470e11 −0.0601601
\(556\) 0 0
\(557\) −1.52104e12 −0.669563 −0.334781 0.942296i \(-0.608662\pi\)
−0.334781 + 0.942296i \(0.608662\pi\)
\(558\) 0 0
\(559\) −1.11472e11 −0.0482848
\(560\) 0 0
\(561\) 6.96103e11 0.296716
\(562\) 0 0
\(563\) 3.30988e12 1.38843 0.694217 0.719766i \(-0.255751\pi\)
0.694217 + 0.719766i \(0.255751\pi\)
\(564\) 0 0
\(565\) 7.31996e10 0.0302198
\(566\) 0 0
\(567\) 3.15994e11 0.128397
\(568\) 0 0
\(569\) 3.08233e12 1.23275 0.616374 0.787453i \(-0.288601\pi\)
0.616374 + 0.787453i \(0.288601\pi\)
\(570\) 0 0
\(571\) 1.13710e12 0.447648 0.223824 0.974630i \(-0.428146\pi\)
0.223824 + 0.974630i \(0.428146\pi\)
\(572\) 0 0
\(573\) 8.27518e11 0.320687
\(574\) 0 0
\(575\) −2.84256e11 −0.108444
\(576\) 0 0
\(577\) −2.48044e12 −0.931617 −0.465808 0.884886i \(-0.654236\pi\)
−0.465808 + 0.884886i \(0.654236\pi\)
\(578\) 0 0
\(579\) 3.08387e11 0.114036
\(580\) 0 0
\(581\) −3.94220e12 −1.43531
\(582\) 0 0
\(583\) 2.57316e12 0.922484
\(584\) 0 0
\(585\) 8.89802e9 0.00314117
\(586\) 0 0
\(587\) 1.17038e12 0.406869 0.203435 0.979089i \(-0.434790\pi\)
0.203435 + 0.979089i \(0.434790\pi\)
\(588\) 0 0
\(589\) 3.94556e11 0.135080
\(590\) 0 0
\(591\) 3.41687e11 0.115208
\(592\) 0 0
\(593\) 1.21383e12 0.403101 0.201550 0.979478i \(-0.435402\pi\)
0.201550 + 0.979478i \(0.435402\pi\)
\(594\) 0 0
\(595\) 4.53030e11 0.148184
\(596\) 0 0
\(597\) −8.85574e10 −0.0285325
\(598\) 0 0
\(599\) 5.28874e12 1.67854 0.839270 0.543716i \(-0.182983\pi\)
0.839270 + 0.543716i \(0.182983\pi\)
\(600\) 0 0
\(601\) −9.10430e11 −0.284650 −0.142325 0.989820i \(-0.545458\pi\)
−0.142325 + 0.989820i \(0.545458\pi\)
\(602\) 0 0
\(603\) −4.35233e11 −0.134058
\(604\) 0 0
\(605\) −2.82235e11 −0.0856470
\(606\) 0 0
\(607\) 5.57806e12 1.66776 0.833881 0.551945i \(-0.186114\pi\)
0.833881 + 0.551945i \(0.186114\pi\)
\(608\) 0 0
\(609\) 1.08456e12 0.319503
\(610\) 0 0
\(611\) 1.32830e11 0.0385578
\(612\) 0 0
\(613\) −5.74576e12 −1.64352 −0.821761 0.569832i \(-0.807008\pi\)
−0.821761 + 0.569832i \(0.807008\pi\)
\(614\) 0 0
\(615\) −5.48737e11 −0.154677
\(616\) 0 0
\(617\) 6.31735e12 1.75490 0.877449 0.479670i \(-0.159244\pi\)
0.877449 + 0.479670i \(0.159244\pi\)
\(618\) 0 0
\(619\) −4.23867e12 −1.16044 −0.580218 0.814461i \(-0.697033\pi\)
−0.580218 + 0.814461i \(0.697033\pi\)
\(620\) 0 0
\(621\) −8.41622e10 −0.0227093
\(622\) 0 0
\(623\) −1.57283e12 −0.418299
\(624\) 0 0
\(625\) 2.91280e12 0.763573
\(626\) 0 0
\(627\) 9.20859e11 0.237952
\(628\) 0 0
\(629\) 6.47647e11 0.164972
\(630\) 0 0
\(631\) 1.74768e12 0.438864 0.219432 0.975628i \(-0.429580\pi\)
0.219432 + 0.975628i \(0.429580\pi\)
\(632\) 0 0
\(633\) 1.58278e11 0.0391834
\(634\) 0 0
\(635\) 1.70700e12 0.416631
\(636\) 0 0
\(637\) −4.61433e10 −0.0111040
\(638\) 0 0
\(639\) −4.12670e11 −0.0979151
\(640\) 0 0
\(641\) −2.99186e11 −0.0699971 −0.0349986 0.999387i \(-0.511143\pi\)
−0.0349986 + 0.999387i \(0.511143\pi\)
\(642\) 0 0
\(643\) 1.75869e12 0.405732 0.202866 0.979207i \(-0.434974\pi\)
0.202866 + 0.979207i \(0.434974\pi\)
\(644\) 0 0
\(645\) 1.05322e12 0.239606
\(646\) 0 0
\(647\) 1.30030e12 0.291726 0.145863 0.989305i \(-0.453404\pi\)
0.145863 + 0.989305i \(0.453404\pi\)
\(648\) 0 0
\(649\) 5.74560e12 1.27126
\(650\) 0 0
\(651\) 1.14293e12 0.249406
\(652\) 0 0
\(653\) 7.41572e12 1.59604 0.798021 0.602630i \(-0.205881\pi\)
0.798021 + 0.602630i \(0.205881\pi\)
\(654\) 0 0
\(655\) −8.47786e11 −0.179970
\(656\) 0 0
\(657\) 6.88599e11 0.144186
\(658\) 0 0
\(659\) −2.67183e12 −0.551855 −0.275927 0.961178i \(-0.588985\pi\)
−0.275927 + 0.961178i \(0.588985\pi\)
\(660\) 0 0
\(661\) −8.73151e12 −1.77903 −0.889514 0.456908i \(-0.848957\pi\)
−0.889514 + 0.456908i \(0.848957\pi\)
\(662\) 0 0
\(663\) −4.28553e10 −0.00861378
\(664\) 0 0
\(665\) 5.99303e11 0.118836
\(666\) 0 0
\(667\) −2.88863e11 −0.0565100
\(668\) 0 0
\(669\) −1.92476e10 −0.00371500
\(670\) 0 0
\(671\) −2.58257e12 −0.491813
\(672\) 0 0
\(673\) −6.22078e12 −1.16890 −0.584449 0.811430i \(-0.698689\pi\)
−0.584449 + 0.811430i \(0.698689\pi\)
\(674\) 0 0
\(675\) 9.53900e11 0.176862
\(676\) 0 0
\(677\) −7.30526e12 −1.33655 −0.668277 0.743912i \(-0.732968\pi\)
−0.668277 + 0.743912i \(0.732968\pi\)
\(678\) 0 0
\(679\) 8.94235e12 1.61450
\(680\) 0 0
\(681\) 4.41255e12 0.786191
\(682\) 0 0
\(683\) 8.82808e12 1.55229 0.776145 0.630554i \(-0.217172\pi\)
0.776145 + 0.630554i \(0.217172\pi\)
\(684\) 0 0
\(685\) 3.45013e11 0.0598726
\(686\) 0 0
\(687\) 4.89506e10 0.00838403
\(688\) 0 0
\(689\) −1.58416e11 −0.0267801
\(690\) 0 0
\(691\) −8.73949e12 −1.45826 −0.729130 0.684375i \(-0.760075\pi\)
−0.729130 + 0.684375i \(0.760075\pi\)
\(692\) 0 0
\(693\) 2.66750e12 0.439344
\(694\) 0 0
\(695\) 7.74054e11 0.125846
\(696\) 0 0
\(697\) 2.64287e12 0.424159
\(698\) 0 0
\(699\) 6.26054e12 0.991893
\(700\) 0 0
\(701\) 2.40844e12 0.376708 0.188354 0.982101i \(-0.439685\pi\)
0.188354 + 0.982101i \(0.439685\pi\)
\(702\) 0 0
\(703\) 8.56758e11 0.132300
\(704\) 0 0
\(705\) −1.25502e12 −0.191337
\(706\) 0 0
\(707\) 5.59422e12 0.842077
\(708\) 0 0
\(709\) −1.05935e13 −1.57446 −0.787230 0.616660i \(-0.788485\pi\)
−0.787230 + 0.616660i \(0.788485\pi\)
\(710\) 0 0
\(711\) 1.73131e12 0.254076
\(712\) 0 0
\(713\) −3.04410e11 −0.0441119
\(714\) 0 0
\(715\) 7.51137e10 0.0107484
\(716\) 0 0
\(717\) 2.19911e12 0.310750
\(718\) 0 0
\(719\) 9.20874e11 0.128505 0.0642525 0.997934i \(-0.479534\pi\)
0.0642525 + 0.997934i \(0.479534\pi\)
\(720\) 0 0
\(721\) −1.40558e13 −1.93708
\(722\) 0 0
\(723\) 7.68679e12 1.04622
\(724\) 0 0
\(725\) 3.27399e12 0.440105
\(726\) 0 0
\(727\) −7.54020e12 −1.00110 −0.500551 0.865707i \(-0.666869\pi\)
−0.500551 + 0.865707i \(0.666869\pi\)
\(728\) 0 0
\(729\) 2.82430e11 0.0370370
\(730\) 0 0
\(731\) −5.07258e12 −0.657053
\(732\) 0 0
\(733\) −8.26906e12 −1.05801 −0.529003 0.848620i \(-0.677434\pi\)
−0.529003 + 0.848620i \(0.677434\pi\)
\(734\) 0 0
\(735\) 4.35975e11 0.0551022
\(736\) 0 0
\(737\) −3.67407e12 −0.458716
\(738\) 0 0
\(739\) −1.43637e13 −1.77160 −0.885802 0.464064i \(-0.846391\pi\)
−0.885802 + 0.464064i \(0.846391\pi\)
\(740\) 0 0
\(741\) −5.66924e10 −0.00690785
\(742\) 0 0
\(743\) −1.12328e13 −1.35220 −0.676099 0.736811i \(-0.736331\pi\)
−0.676099 + 0.736811i \(0.736331\pi\)
\(744\) 0 0
\(745\) 1.86722e12 0.222071
\(746\) 0 0
\(747\) −3.52347e12 −0.414026
\(748\) 0 0
\(749\) 3.95371e12 0.459026
\(750\) 0 0
\(751\) 1.15825e13 1.32869 0.664346 0.747425i \(-0.268710\pi\)
0.664346 + 0.747425i \(0.268710\pi\)
\(752\) 0 0
\(753\) 2.56882e12 0.291177
\(754\) 0 0
\(755\) −1.36890e12 −0.153324
\(756\) 0 0
\(757\) −6.19463e12 −0.685621 −0.342810 0.939405i \(-0.611379\pi\)
−0.342810 + 0.939405i \(0.611379\pi\)
\(758\) 0 0
\(759\) −7.10465e11 −0.0777060
\(760\) 0 0
\(761\) −1.57090e13 −1.69793 −0.848963 0.528453i \(-0.822772\pi\)
−0.848963 + 0.528453i \(0.822772\pi\)
\(762\) 0 0
\(763\) −2.04631e12 −0.218580
\(764\) 0 0
\(765\) 4.04910e11 0.0427446
\(766\) 0 0
\(767\) −3.53726e11 −0.0369052
\(768\) 0 0
\(769\) −4.65088e12 −0.479586 −0.239793 0.970824i \(-0.577080\pi\)
−0.239793 + 0.970824i \(0.577080\pi\)
\(770\) 0 0
\(771\) −9.05030e9 −0.000922398 0
\(772\) 0 0
\(773\) −3.78167e12 −0.380957 −0.190478 0.981691i \(-0.561004\pi\)
−0.190478 + 0.981691i \(0.561004\pi\)
\(774\) 0 0
\(775\) 3.45020e12 0.343547
\(776\) 0 0
\(777\) 2.48182e12 0.244273
\(778\) 0 0
\(779\) 3.49620e12 0.340155
\(780\) 0 0
\(781\) −3.48361e12 −0.335042
\(782\) 0 0
\(783\) 9.69359e11 0.0921630
\(784\) 0 0
\(785\) 3.58762e12 0.337205
\(786\) 0 0
\(787\) 3.94623e12 0.366688 0.183344 0.983049i \(-0.441308\pi\)
0.183344 + 0.983049i \(0.441308\pi\)
\(788\) 0 0
\(789\) 1.18390e13 1.08760
\(790\) 0 0
\(791\) −1.35099e12 −0.122704
\(792\) 0 0
\(793\) 1.58995e11 0.0142775
\(794\) 0 0
\(795\) 1.49676e12 0.132892
\(796\) 0 0
\(797\) 1.76742e13 1.55159 0.775797 0.630982i \(-0.217348\pi\)
0.775797 + 0.630982i \(0.217348\pi\)
\(798\) 0 0
\(799\) 6.04453e12 0.524689
\(800\) 0 0
\(801\) −1.40577e12 −0.120661
\(802\) 0 0
\(803\) 5.81289e12 0.493370
\(804\) 0 0
\(805\) −4.62377e11 −0.0388074
\(806\) 0 0
\(807\) −1.25625e13 −1.04267
\(808\) 0 0
\(809\) 8.53289e12 0.700370 0.350185 0.936681i \(-0.386119\pi\)
0.350185 + 0.936681i \(0.386119\pi\)
\(810\) 0 0
\(811\) −1.57738e13 −1.28039 −0.640197 0.768211i \(-0.721147\pi\)
−0.640197 + 0.768211i \(0.721147\pi\)
\(812\) 0 0
\(813\) 2.98018e12 0.239241
\(814\) 0 0
\(815\) 4.49526e12 0.356900
\(816\) 0 0
\(817\) −6.71041e12 −0.526926
\(818\) 0 0
\(819\) −1.64224e11 −0.0127544
\(820\) 0 0
\(821\) −4.11176e12 −0.315852 −0.157926 0.987451i \(-0.550481\pi\)
−0.157926 + 0.987451i \(0.550481\pi\)
\(822\) 0 0
\(823\) −4.47591e12 −0.340081 −0.170040 0.985437i \(-0.554390\pi\)
−0.170040 + 0.985437i \(0.554390\pi\)
\(824\) 0 0
\(825\) 8.05246e12 0.605182
\(826\) 0 0
\(827\) −8.03686e12 −0.597464 −0.298732 0.954337i \(-0.596564\pi\)
−0.298732 + 0.954337i \(0.596564\pi\)
\(828\) 0 0
\(829\) 1.67641e13 1.23277 0.616387 0.787443i \(-0.288596\pi\)
0.616387 + 0.787443i \(0.288596\pi\)
\(830\) 0 0
\(831\) −9.46621e12 −0.688607
\(832\) 0 0
\(833\) −2.09978e12 −0.151102
\(834\) 0 0
\(835\) −6.35015e12 −0.452058
\(836\) 0 0
\(837\) 1.02153e12 0.0719428
\(838\) 0 0
\(839\) −1.24921e13 −0.870375 −0.435187 0.900340i \(-0.643318\pi\)
−0.435187 + 0.900340i \(0.643318\pi\)
\(840\) 0 0
\(841\) −1.11801e13 −0.770662
\(842\) 0 0
\(843\) −2.17953e12 −0.148641
\(844\) 0 0
\(845\) 4.21318e12 0.284285
\(846\) 0 0
\(847\) 5.20900e12 0.347760
\(848\) 0 0
\(849\) −8.24732e11 −0.0544789
\(850\) 0 0
\(851\) −6.61009e11 −0.0432041
\(852\) 0 0
\(853\) 2.63321e13 1.70300 0.851500 0.524355i \(-0.175693\pi\)
0.851500 + 0.524355i \(0.175693\pi\)
\(854\) 0 0
\(855\) 5.35646e11 0.0342792
\(856\) 0 0
\(857\) 2.67767e13 1.69567 0.847837 0.530256i \(-0.177904\pi\)
0.847837 + 0.530256i \(0.177904\pi\)
\(858\) 0 0
\(859\) −1.65495e13 −1.03709 −0.518544 0.855051i \(-0.673526\pi\)
−0.518544 + 0.855051i \(0.673526\pi\)
\(860\) 0 0
\(861\) 1.01276e13 0.628049
\(862\) 0 0
\(863\) −2.48373e13 −1.52425 −0.762124 0.647432i \(-0.775843\pi\)
−0.762124 + 0.647432i \(0.775843\pi\)
\(864\) 0 0
\(865\) −7.79508e12 −0.473422
\(866\) 0 0
\(867\) 7.65546e12 0.460135
\(868\) 0 0
\(869\) 1.46151e13 0.869387
\(870\) 0 0
\(871\) 2.26193e11 0.0133167
\(872\) 0 0
\(873\) 7.99250e12 0.465713
\(874\) 0 0
\(875\) 1.09431e13 0.631109
\(876\) 0 0
\(877\) −2.75473e13 −1.57247 −0.786233 0.617930i \(-0.787971\pi\)
−0.786233 + 0.617930i \(0.787971\pi\)
\(878\) 0 0
\(879\) −5.08001e12 −0.287022
\(880\) 0 0
\(881\) 1.65762e13 0.927029 0.463514 0.886089i \(-0.346588\pi\)
0.463514 + 0.886089i \(0.346588\pi\)
\(882\) 0 0
\(883\) −2.97773e12 −0.164840 −0.0824198 0.996598i \(-0.526265\pi\)
−0.0824198 + 0.996598i \(0.526265\pi\)
\(884\) 0 0
\(885\) 3.34211e12 0.183137
\(886\) 0 0
\(887\) 5.03383e12 0.273050 0.136525 0.990637i \(-0.456407\pi\)
0.136525 + 0.990637i \(0.456407\pi\)
\(888\) 0 0
\(889\) −3.15048e13 −1.69168
\(890\) 0 0
\(891\) 2.38416e12 0.126732
\(892\) 0 0
\(893\) 7.99618e12 0.420776
\(894\) 0 0
\(895\) 3.65757e12 0.190541
\(896\) 0 0
\(897\) 4.37395e10 0.00225584
\(898\) 0 0
\(899\) 3.50612e12 0.179022
\(900\) 0 0
\(901\) −7.20881e12 −0.364420
\(902\) 0 0
\(903\) −1.94384e13 −0.972894
\(904\) 0 0
\(905\) −9.71053e12 −0.481198
\(906\) 0 0
\(907\) −3.38163e13 −1.65918 −0.829590 0.558373i \(-0.811426\pi\)
−0.829590 + 0.558373i \(0.811426\pi\)
\(908\) 0 0
\(909\) 5.00001e12 0.242903
\(910\) 0 0
\(911\) −3.91391e13 −1.88269 −0.941344 0.337449i \(-0.890436\pi\)
−0.941344 + 0.337449i \(0.890436\pi\)
\(912\) 0 0
\(913\) −2.97438e13 −1.41670
\(914\) 0 0
\(915\) −1.50223e12 −0.0708503
\(916\) 0 0
\(917\) 1.56469e13 0.730747
\(918\) 0 0
\(919\) 9.85085e12 0.455569 0.227784 0.973712i \(-0.426852\pi\)
0.227784 + 0.973712i \(0.426852\pi\)
\(920\) 0 0
\(921\) 6.71748e12 0.307637
\(922\) 0 0
\(923\) 2.14467e11 0.00972642
\(924\) 0 0
\(925\) 7.49193e12 0.336477
\(926\) 0 0
\(927\) −1.25628e13 −0.558765
\(928\) 0 0
\(929\) 1.01841e13 0.448593 0.224296 0.974521i \(-0.427992\pi\)
0.224296 + 0.974521i \(0.427992\pi\)
\(930\) 0 0
\(931\) −2.77775e12 −0.121177
\(932\) 0 0
\(933\) −2.57264e12 −0.111151
\(934\) 0 0
\(935\) 3.41809e12 0.146262
\(936\) 0 0
\(937\) −4.02950e13 −1.70775 −0.853873 0.520481i \(-0.825753\pi\)
−0.853873 + 0.520481i \(0.825753\pi\)
\(938\) 0 0
\(939\) −1.36799e13 −0.574232
\(940\) 0 0
\(941\) 4.92352e12 0.204702 0.102351 0.994748i \(-0.467363\pi\)
0.102351 + 0.994748i \(0.467363\pi\)
\(942\) 0 0
\(943\) −2.69740e12 −0.111082
\(944\) 0 0
\(945\) 1.55164e12 0.0632917
\(946\) 0 0
\(947\) −2.44185e13 −0.986606 −0.493303 0.869858i \(-0.664211\pi\)
−0.493303 + 0.869858i \(0.664211\pi\)
\(948\) 0 0
\(949\) −3.57869e11 −0.0143227
\(950\) 0 0
\(951\) −1.23569e13 −0.489887
\(952\) 0 0
\(953\) −1.80434e13 −0.708599 −0.354299 0.935132i \(-0.615281\pi\)
−0.354299 + 0.935132i \(0.615281\pi\)
\(954\) 0 0
\(955\) 4.06339e12 0.158079
\(956\) 0 0
\(957\) 8.18296e12 0.315360
\(958\) 0 0
\(959\) −6.36764e12 −0.243106
\(960\) 0 0
\(961\) −2.27448e13 −0.860254
\(962\) 0 0
\(963\) 3.53375e12 0.132409
\(964\) 0 0
\(965\) 1.51428e12 0.0562126
\(966\) 0 0
\(967\) 1.44115e13 0.530018 0.265009 0.964246i \(-0.414625\pi\)
0.265009 + 0.964246i \(0.414625\pi\)
\(968\) 0 0
\(969\) −2.57982e12 −0.0940010
\(970\) 0 0
\(971\) 1.58123e13 0.570831 0.285415 0.958404i \(-0.407868\pi\)
0.285415 + 0.958404i \(0.407868\pi\)
\(972\) 0 0
\(973\) −1.42861e13 −0.510983
\(974\) 0 0
\(975\) −4.95747e11 −0.0175687
\(976\) 0 0
\(977\) −4.64628e13 −1.63147 −0.815737 0.578423i \(-0.803668\pi\)
−0.815737 + 0.578423i \(0.803668\pi\)
\(978\) 0 0
\(979\) −1.18670e13 −0.412874
\(980\) 0 0
\(981\) −1.82895e12 −0.0630510
\(982\) 0 0
\(983\) −4.65220e13 −1.58916 −0.794580 0.607160i \(-0.792309\pi\)
−0.794580 + 0.607160i \(0.792309\pi\)
\(984\) 0 0
\(985\) 1.67779e12 0.0567905
\(986\) 0 0
\(987\) 2.31630e13 0.776903
\(988\) 0 0
\(989\) 5.17724e12 0.172074
\(990\) 0 0
\(991\) 3.96520e12 0.130597 0.0652985 0.997866i \(-0.479200\pi\)
0.0652985 + 0.997866i \(0.479200\pi\)
\(992\) 0 0
\(993\) 2.64826e13 0.864348
\(994\) 0 0
\(995\) −4.34846e11 −0.0140647
\(996\) 0 0
\(997\) −1.82677e12 −0.0585538 −0.0292769 0.999571i \(-0.509320\pi\)
−0.0292769 + 0.999571i \(0.509320\pi\)
\(998\) 0 0
\(999\) 2.21820e12 0.0704622
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 384.10.a.d.1.2 yes 4
4.3 odd 2 384.10.a.h.1.2 yes 4
8.3 odd 2 384.10.a.a.1.3 4
8.5 even 2 384.10.a.e.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.a.a.1.3 4 8.3 odd 2
384.10.a.d.1.2 yes 4 1.1 even 1 trivial
384.10.a.e.1.3 yes 4 8.5 even 2
384.10.a.h.1.2 yes 4 4.3 odd 2