Properties

Label 288.8.a.h.1.1
Level $288$
Weight $8$
Character 288.1
Self dual yes
Analytic conductor $89.967$
Analytic rank $1$
Dimension $2$
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] = [288,8,Mod(1,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.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: \(89.9668873394\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{46}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.78233\) of defining polynomial
Character \(\chi\) \(=\) 288.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-423.552 q^{5} +1228.66 q^{7} +O(q^{10})\) \(q-423.552 q^{5} +1228.66 q^{7} -2781.31 q^{11} +3729.10 q^{13} +11630.2 q^{17} -45178.2 q^{19} -27729.3 q^{23} +101271. q^{25} +210915. q^{29} +172577. q^{31} -520399. q^{35} -269522. q^{37} +379673. q^{41} +1.03330e6 q^{43} +624148. q^{47} +686051. q^{49} -1.84130e6 q^{53} +1.17803e6 q^{55} -2.74738e6 q^{59} -2.70188e6 q^{61} -1.57947e6 q^{65} +989057. q^{67} -3.30668e6 q^{71} -1.10275e6 q^{73} -3.41727e6 q^{77} -8.02881e6 q^{79} +3.55645e6 q^{83} -4.92600e6 q^{85} +5.21527e6 q^{89} +4.58178e6 q^{91} +1.91353e7 q^{95} -7.07452e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 196 q^{5} + 504 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 196 q^{5} + 504 q^{7} - 1656 q^{11} + 6156 q^{13} - 17108 q^{17} - 504 q^{19} - 51552 q^{23} + 74926 q^{25} + 199804 q^{29} + 257256 q^{31} - 685296 q^{35} - 468724 q^{37} + 106940 q^{41} + 1617336 q^{43} - 646416 q^{47} + 387634 q^{49} - 1469492 q^{53} + 1434096 q^{55} - 4541544 q^{59} - 481412 q^{61} - 1027224 q^{65} + 4775256 q^{67} + 1094400 q^{71} - 5731884 q^{73} - 4232736 q^{77} - 10402776 q^{79} - 2212200 q^{83} - 11465432 q^{85} + 3604364 q^{89} + 2823120 q^{91} + 29300976 q^{95} - 7156188 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −423.552 −1.51535 −0.757673 0.652635i \(-0.773664\pi\)
−0.757673 + 0.652635i \(0.773664\pi\)
\(6\) 0 0
\(7\) 1228.66 1.35390 0.676951 0.736028i \(-0.263301\pi\)
0.676951 + 0.736028i \(0.263301\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2781.31 −0.630050 −0.315025 0.949083i \(-0.602013\pi\)
−0.315025 + 0.949083i \(0.602013\pi\)
\(12\) 0 0
\(13\) 3729.10 0.470763 0.235382 0.971903i \(-0.424366\pi\)
0.235382 + 0.971903i \(0.424366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11630.2 0.574138 0.287069 0.957910i \(-0.407319\pi\)
0.287069 + 0.957910i \(0.407319\pi\)
\(18\) 0 0
\(19\) −45178.2 −1.51109 −0.755546 0.655096i \(-0.772628\pi\)
−0.755546 + 0.655096i \(0.772628\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −27729.3 −0.475217 −0.237608 0.971361i \(-0.576363\pi\)
−0.237608 + 0.971361i \(0.576363\pi\)
\(24\) 0 0
\(25\) 101271. 1.29627
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 210915. 1.60589 0.802943 0.596056i \(-0.203266\pi\)
0.802943 + 0.596056i \(0.203266\pi\)
\(30\) 0 0
\(31\) 172577. 1.04044 0.520221 0.854031i \(-0.325849\pi\)
0.520221 + 0.854031i \(0.325849\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −520399. −2.05163
\(36\) 0 0
\(37\) −269522. −0.874757 −0.437379 0.899277i \(-0.644093\pi\)
−0.437379 + 0.899277i \(0.644093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 379673. 0.860332 0.430166 0.902750i \(-0.358455\pi\)
0.430166 + 0.902750i \(0.358455\pi\)
\(42\) 0 0
\(43\) 1.03330e6 1.98192 0.990960 0.134154i \(-0.0428318\pi\)
0.990960 + 0.134154i \(0.0428318\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 624148. 0.876890 0.438445 0.898758i \(-0.355529\pi\)
0.438445 + 0.898758i \(0.355529\pi\)
\(48\) 0 0
\(49\) 686051. 0.833049
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.84130e6 −1.69886 −0.849431 0.527699i \(-0.823055\pi\)
−0.849431 + 0.527699i \(0.823055\pi\)
\(54\) 0 0
\(55\) 1.17803e6 0.954744
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.74738e6 −1.74155 −0.870776 0.491679i \(-0.836383\pi\)
−0.870776 + 0.491679i \(0.836383\pi\)
\(60\) 0 0
\(61\) −2.70188e6 −1.52409 −0.762046 0.647523i \(-0.775805\pi\)
−0.762046 + 0.647523i \(0.775805\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.57947e6 −0.713369
\(66\) 0 0
\(67\) 989057. 0.401753 0.200877 0.979617i \(-0.435621\pi\)
0.200877 + 0.979617i \(0.435621\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.30668e6 −1.09645 −0.548224 0.836331i \(-0.684696\pi\)
−0.548224 + 0.836331i \(0.684696\pi\)
\(72\) 0 0
\(73\) −1.10275e6 −0.331779 −0.165889 0.986144i \(-0.553049\pi\)
−0.165889 + 0.986144i \(0.553049\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.41727e6 −0.853026
\(78\) 0 0
\(79\) −8.02881e6 −1.83213 −0.916065 0.401031i \(-0.868652\pi\)
−0.916065 + 0.401031i \(0.868652\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.55645e6 0.682722 0.341361 0.939932i \(-0.389112\pi\)
0.341361 + 0.939932i \(0.389112\pi\)
\(84\) 0 0
\(85\) −4.92600e6 −0.870018
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.21527e6 0.784172 0.392086 0.919928i \(-0.371753\pi\)
0.392086 + 0.919928i \(0.371753\pi\)
\(90\) 0 0
\(91\) 4.58178e6 0.637367
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.91353e7 2.28983
\(96\) 0 0
\(97\) −7.07452e6 −0.787038 −0.393519 0.919316i \(-0.628743\pi\)
−0.393519 + 0.919316i \(0.628743\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.27236e6 0.605768 0.302884 0.953027i \(-0.402051\pi\)
0.302884 + 0.953027i \(0.402051\pi\)
\(102\) 0 0
\(103\) −1.22232e7 −1.10219 −0.551093 0.834444i \(-0.685789\pi\)
−0.551093 + 0.834444i \(0.685789\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.74065e6 −0.689764 −0.344882 0.938646i \(-0.612081\pi\)
−0.344882 + 0.938646i \(0.612081\pi\)
\(108\) 0 0
\(109\) 5.57044e6 0.412000 0.206000 0.978552i \(-0.433955\pi\)
0.206000 + 0.978552i \(0.433955\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.25206e7 −0.816302 −0.408151 0.912914i \(-0.633826\pi\)
−0.408151 + 0.912914i \(0.633826\pi\)
\(114\) 0 0
\(115\) 1.17448e7 0.720117
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.42895e7 0.777327
\(120\) 0 0
\(121\) −1.17515e7 −0.603037
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.80360e6 −0.448953
\(126\) 0 0
\(127\) −1.60536e7 −0.695438 −0.347719 0.937599i \(-0.613044\pi\)
−0.347719 + 0.937599i \(0.613044\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.53592e7 0.596923 0.298462 0.954422i \(-0.403526\pi\)
0.298462 + 0.954422i \(0.403526\pi\)
\(132\) 0 0
\(133\) −5.55084e7 −2.04587
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.09949e7 −0.697576 −0.348788 0.937202i \(-0.613407\pi\)
−0.348788 + 0.937202i \(0.613407\pi\)
\(138\) 0 0
\(139\) −3.99151e7 −1.26062 −0.630312 0.776342i \(-0.717073\pi\)
−0.630312 + 0.776342i \(0.717073\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.03718e7 −0.296605
\(144\) 0 0
\(145\) −8.93335e7 −2.43347
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.79531e7 −1.43524 −0.717621 0.696434i \(-0.754769\pi\)
−0.717621 + 0.696434i \(0.754769\pi\)
\(150\) 0 0
\(151\) 1.00182e7 0.236793 0.118397 0.992966i \(-0.462225\pi\)
0.118397 + 0.992966i \(0.462225\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.30955e7 −1.57663
\(156\) 0 0
\(157\) 1.88494e7 0.388732 0.194366 0.980929i \(-0.437735\pi\)
0.194366 + 0.980929i \(0.437735\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.40698e7 −0.643396
\(162\) 0 0
\(163\) 7.98770e7 1.44466 0.722329 0.691550i \(-0.243072\pi\)
0.722329 + 0.691550i \(0.243072\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.20039e7 −0.365587 −0.182794 0.983151i \(-0.558514\pi\)
−0.182794 + 0.983151i \(0.558514\pi\)
\(168\) 0 0
\(169\) −4.88423e7 −0.778382
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.83727e7 −1.00397 −0.501986 0.864876i \(-0.667397\pi\)
−0.501986 + 0.864876i \(0.667397\pi\)
\(174\) 0 0
\(175\) 1.24427e8 1.75502
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.95976e7 −0.776682 −0.388341 0.921516i \(-0.626952\pi\)
−0.388341 + 0.921516i \(0.626952\pi\)
\(180\) 0 0
\(181\) −1.51283e8 −1.89634 −0.948168 0.317769i \(-0.897066\pi\)
−0.948168 + 0.317769i \(0.897066\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.14156e8 1.32556
\(186\) 0 0
\(187\) −3.23472e7 −0.361736
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.66555e6 0.0899869 0.0449935 0.998987i \(-0.485673\pi\)
0.0449935 + 0.998987i \(0.485673\pi\)
\(192\) 0 0
\(193\) 1.51312e8 1.51504 0.757520 0.652812i \(-0.226411\pi\)
0.757520 + 0.652812i \(0.226411\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.77091e7 −0.910549 −0.455274 0.890351i \(-0.650459\pi\)
−0.455274 + 0.890351i \(0.650459\pi\)
\(198\) 0 0
\(199\) 3.19522e7 0.287419 0.143709 0.989620i \(-0.454097\pi\)
0.143709 + 0.989620i \(0.454097\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.59142e8 2.17421
\(204\) 0 0
\(205\) −1.60811e8 −1.30370
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.25654e8 0.952064
\(210\) 0 0
\(211\) −3.47789e7 −0.254875 −0.127437 0.991847i \(-0.540675\pi\)
−0.127437 + 0.991847i \(0.540675\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.37656e8 −3.00329
\(216\) 0 0
\(217\) 2.12038e8 1.40866
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.33703e7 0.270283
\(222\) 0 0
\(223\) 2.06077e7 0.124441 0.0622204 0.998062i \(-0.480182\pi\)
0.0622204 + 0.998062i \(0.480182\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.38134e8 1.35124 0.675619 0.737251i \(-0.263877\pi\)
0.675619 + 0.737251i \(0.263877\pi\)
\(228\) 0 0
\(229\) −5.34105e7 −0.293902 −0.146951 0.989144i \(-0.546946\pi\)
−0.146951 + 0.989144i \(0.546946\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.32484e8 1.20406 0.602028 0.798475i \(-0.294360\pi\)
0.602028 + 0.798475i \(0.294360\pi\)
\(234\) 0 0
\(235\) −2.64359e8 −1.32879
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.35251e8 1.11465 0.557326 0.830294i \(-0.311827\pi\)
0.557326 + 0.830294i \(0.311827\pi\)
\(240\) 0 0
\(241\) −3.44508e8 −1.58540 −0.792701 0.609610i \(-0.791326\pi\)
−0.792701 + 0.609610i \(0.791326\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.90578e8 −1.26236
\(246\) 0 0
\(247\) −1.68474e8 −0.711367
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.57037e7 0.142513 0.0712567 0.997458i \(-0.477299\pi\)
0.0712567 + 0.997458i \(0.477299\pi\)
\(252\) 0 0
\(253\) 7.71238e7 0.299410
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.77388e8 −0.651866 −0.325933 0.945393i \(-0.605678\pi\)
−0.325933 + 0.945393i \(0.605678\pi\)
\(258\) 0 0
\(259\) −3.31149e8 −1.18433
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.51237e8 0.851604 0.425802 0.904816i \(-0.359992\pi\)
0.425802 + 0.904816i \(0.359992\pi\)
\(264\) 0 0
\(265\) 7.79885e8 2.57436
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.42878e8 −1.07400 −0.537002 0.843581i \(-0.680443\pi\)
−0.537002 + 0.843581i \(0.680443\pi\)
\(270\) 0 0
\(271\) −4.49518e8 −1.37200 −0.686000 0.727601i \(-0.740635\pi\)
−0.686000 + 0.727601i \(0.740635\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.81667e8 −0.816716
\(276\) 0 0
\(277\) −2.29030e8 −0.647461 −0.323730 0.946149i \(-0.604937\pi\)
−0.323730 + 0.946149i \(0.604937\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.49209e8 0.401165 0.200582 0.979677i \(-0.435717\pi\)
0.200582 + 0.979677i \(0.435717\pi\)
\(282\) 0 0
\(283\) 4.05047e8 1.06231 0.531157 0.847273i \(-0.321757\pi\)
0.531157 + 0.847273i \(0.321757\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.66487e8 1.16480
\(288\) 0 0
\(289\) −2.75077e8 −0.670365
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.59019e7 −0.129834 −0.0649172 0.997891i \(-0.520678\pi\)
−0.0649172 + 0.997891i \(0.520678\pi\)
\(294\) 0 0
\(295\) 1.16366e9 2.63905
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.03405e8 −0.223715
\(300\) 0 0
\(301\) 1.26957e9 2.68333
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.14439e9 2.30953
\(306\) 0 0
\(307\) 7.13287e8 1.40695 0.703477 0.710718i \(-0.251630\pi\)
0.703477 + 0.710718i \(0.251630\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.32785e8 −0.438827 −0.219414 0.975632i \(-0.570414\pi\)
−0.219414 + 0.975632i \(0.570414\pi\)
\(312\) 0 0
\(313\) 1.35376e8 0.249538 0.124769 0.992186i \(-0.460181\pi\)
0.124769 + 0.992186i \(0.460181\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.29613e8 0.933794 0.466897 0.884312i \(-0.345372\pi\)
0.466897 + 0.884312i \(0.345372\pi\)
\(318\) 0 0
\(319\) −5.86621e8 −1.01179
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.25432e8 −0.867576
\(324\) 0 0
\(325\) 3.77651e8 0.610237
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.66863e8 1.18722
\(330\) 0 0
\(331\) −9.24935e8 −1.40189 −0.700944 0.713217i \(-0.747238\pi\)
−0.700944 + 0.713217i \(0.747238\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.18917e8 −0.608795
\(336\) 0 0
\(337\) 1.02943e9 1.46518 0.732592 0.680668i \(-0.238311\pi\)
0.732592 + 0.680668i \(0.238311\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.79992e8 −0.655531
\(342\) 0 0
\(343\) −1.68930e8 −0.226036
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.00011e8 −0.385464 −0.192732 0.981251i \(-0.561735\pi\)
−0.192732 + 0.981251i \(0.561735\pi\)
\(348\) 0 0
\(349\) −5.43096e8 −0.683893 −0.341946 0.939719i \(-0.611086\pi\)
−0.341946 + 0.939719i \(0.611086\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.10616e8 0.375849 0.187924 0.982184i \(-0.439824\pi\)
0.187924 + 0.982184i \(0.439824\pi\)
\(354\) 0 0
\(355\) 1.40055e9 1.66150
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.00225e8 0.228395 0.114198 0.993458i \(-0.463570\pi\)
0.114198 + 0.993458i \(0.463570\pi\)
\(360\) 0 0
\(361\) 1.14719e9 1.28340
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.67073e8 0.502759
\(366\) 0 0
\(367\) −7.09899e8 −0.749662 −0.374831 0.927093i \(-0.622299\pi\)
−0.374831 + 0.927093i \(0.622299\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.26232e9 −2.30009
\(372\) 0 0
\(373\) 1.90313e8 0.189883 0.0949416 0.995483i \(-0.469734\pi\)
0.0949416 + 0.995483i \(0.469734\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.86525e8 0.755993
\(378\) 0 0
\(379\) 1.52757e9 1.44133 0.720665 0.693284i \(-0.243837\pi\)
0.720665 + 0.693284i \(0.243837\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.22576e8 −0.384334 −0.192167 0.981362i \(-0.561552\pi\)
−0.192167 + 0.981362i \(0.561552\pi\)
\(384\) 0 0
\(385\) 1.44739e9 1.29263
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.37610e8 −0.721470 −0.360735 0.932668i \(-0.617474\pi\)
−0.360735 + 0.932668i \(0.617474\pi\)
\(390\) 0 0
\(391\) −3.22498e8 −0.272840
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.40062e9 2.77631
\(396\) 0 0
\(397\) −1.46607e9 −1.17595 −0.587973 0.808881i \(-0.700074\pi\)
−0.587973 + 0.808881i \(0.700074\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.61531e9 −1.25098 −0.625489 0.780233i \(-0.715101\pi\)
−0.625489 + 0.780233i \(0.715101\pi\)
\(402\) 0 0
\(403\) 6.43559e8 0.489802
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.49623e8 0.551141
\(408\) 0 0
\(409\) −1.00624e9 −0.727230 −0.363615 0.931549i \(-0.618458\pi\)
−0.363615 + 0.931549i \(0.618458\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.37558e9 −2.35789
\(414\) 0 0
\(415\) −1.50634e9 −1.03456
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.49577e8 −0.232164 −0.116082 0.993240i \(-0.537033\pi\)
−0.116082 + 0.993240i \(0.537033\pi\)
\(420\) 0 0
\(421\) −7.71712e8 −0.504043 −0.252022 0.967722i \(-0.581095\pi\)
−0.252022 + 0.967722i \(0.581095\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.17781e9 0.744239
\(426\) 0 0
\(427\) −3.31968e9 −2.06347
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.24587e9 1.35118 0.675592 0.737276i \(-0.263888\pi\)
0.675592 + 0.737276i \(0.263888\pi\)
\(432\) 0 0
\(433\) −1.16044e8 −0.0686937 −0.0343468 0.999410i \(-0.510935\pi\)
−0.0343468 + 0.999410i \(0.510935\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.25276e9 0.718096
\(438\) 0 0
\(439\) 2.46503e9 1.39058 0.695290 0.718729i \(-0.255276\pi\)
0.695290 + 0.718729i \(0.255276\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.71781e9 1.48527 0.742635 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(444\) 0 0
\(445\) −2.20894e9 −1.18829
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.31427e9 1.72793 0.863964 0.503553i \(-0.167974\pi\)
0.863964 + 0.503553i \(0.167974\pi\)
\(450\) 0 0
\(451\) −1.05599e9 −0.542052
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.94062e9 −0.965831
\(456\) 0 0
\(457\) −1.13520e9 −0.556374 −0.278187 0.960527i \(-0.589733\pi\)
−0.278187 + 0.960527i \(0.589733\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.89698e9 −0.901799 −0.450900 0.892575i \(-0.648897\pi\)
−0.450900 + 0.892575i \(0.648897\pi\)
\(462\) 0 0
\(463\) 1.13818e9 0.532940 0.266470 0.963843i \(-0.414143\pi\)
0.266470 + 0.963843i \(0.414143\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.10626e9 −0.956978 −0.478489 0.878094i \(-0.658815\pi\)
−0.478489 + 0.878094i \(0.658815\pi\)
\(468\) 0 0
\(469\) 1.21521e9 0.543935
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.87393e9 −1.24871
\(474\) 0 0
\(475\) −4.57524e9 −1.95878
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.63874e8 0.0681295 0.0340648 0.999420i \(-0.489155\pi\)
0.0340648 + 0.999420i \(0.489155\pi\)
\(480\) 0 0
\(481\) −1.00507e9 −0.411804
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.99643e9 1.19263
\(486\) 0 0
\(487\) −4.01594e8 −0.157556 −0.0787781 0.996892i \(-0.525102\pi\)
−0.0787781 + 0.996892i \(0.525102\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.05766e9 −0.784490 −0.392245 0.919861i \(-0.628301\pi\)
−0.392245 + 0.919861i \(0.628301\pi\)
\(492\) 0 0
\(493\) 2.45299e9 0.922001
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.06277e9 −1.48448
\(498\) 0 0
\(499\) −8.05327e8 −0.290148 −0.145074 0.989421i \(-0.546342\pi\)
−0.145074 + 0.989421i \(0.546342\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.32490e9 −0.464189 −0.232094 0.972693i \(-0.574558\pi\)
−0.232094 + 0.972693i \(0.574558\pi\)
\(504\) 0 0
\(505\) −2.65667e9 −0.917948
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.83357e9 0.952406 0.476203 0.879335i \(-0.342013\pi\)
0.476203 + 0.879335i \(0.342013\pi\)
\(510\) 0 0
\(511\) −1.35490e9 −0.449195
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.17716e9 1.67019
\(516\) 0 0
\(517\) −1.73595e9 −0.552485
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.08648e8 0.0336580 0.0168290 0.999858i \(-0.494643\pi\)
0.0168290 + 0.999858i \(0.494643\pi\)
\(522\) 0 0
\(523\) 5.09356e9 1.55692 0.778458 0.627696i \(-0.216002\pi\)
0.778458 + 0.627696i \(0.216002\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00711e9 0.597358
\(528\) 0 0
\(529\) −2.63591e9 −0.774169
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.41584e9 0.405013
\(534\) 0 0
\(535\) 3.70212e9 1.04523
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.90812e9 −0.524862
\(540\) 0 0
\(541\) −3.98616e9 −1.08234 −0.541171 0.840913i \(-0.682019\pi\)
−0.541171 + 0.840913i \(0.682019\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.35937e9 −0.624322
\(546\) 0 0
\(547\) 4.49694e9 1.17479 0.587397 0.809299i \(-0.300153\pi\)
0.587397 + 0.809299i \(0.300153\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.52876e9 −2.42664
\(552\) 0 0
\(553\) −9.86464e9 −2.48052
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.10617e9 −1.49719 −0.748593 0.663030i \(-0.769270\pi\)
−0.748593 + 0.663030i \(0.769270\pi\)
\(558\) 0 0
\(559\) 3.85328e9 0.933016
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.38158e9 0.326284 0.163142 0.986603i \(-0.447837\pi\)
0.163142 + 0.986603i \(0.447837\pi\)
\(564\) 0 0
\(565\) 5.30313e9 1.23698
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.73612e9 −1.98804 −0.994021 0.109188i \(-0.965175\pi\)
−0.994021 + 0.109188i \(0.965175\pi\)
\(570\) 0 0
\(571\) 1.33535e9 0.300170 0.150085 0.988673i \(-0.452045\pi\)
0.150085 + 0.988673i \(0.452045\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.80818e9 −0.616009
\(576\) 0 0
\(577\) 5.50143e8 0.119223 0.0596115 0.998222i \(-0.481014\pi\)
0.0596115 + 0.998222i \(0.481014\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.36966e9 0.924338
\(582\) 0 0
\(583\) 5.12122e9 1.07037
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.96942e9 −0.810017 −0.405008 0.914313i \(-0.632731\pi\)
−0.405008 + 0.914313i \(0.632731\pi\)
\(588\) 0 0
\(589\) −7.79673e9 −1.57220
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.66508e9 −0.524830 −0.262415 0.964955i \(-0.584519\pi\)
−0.262415 + 0.964955i \(0.584519\pi\)
\(594\) 0 0
\(595\) −6.05236e9 −1.17792
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.20668e7 −0.00229403 −0.00114701 0.999999i \(-0.500365\pi\)
−0.00114701 + 0.999999i \(0.500365\pi\)
\(600\) 0 0
\(601\) 1.12678e9 0.211728 0.105864 0.994381i \(-0.466239\pi\)
0.105864 + 0.994381i \(0.466239\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.97736e9 0.913809
\(606\) 0 0
\(607\) −3.33232e9 −0.604765 −0.302382 0.953187i \(-0.597782\pi\)
−0.302382 + 0.953187i \(0.597782\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.32751e9 0.412808
\(612\) 0 0
\(613\) 4.28653e9 0.751612 0.375806 0.926698i \(-0.377366\pi\)
0.375806 + 0.926698i \(0.377366\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.82726e9 −0.655978 −0.327989 0.944681i \(-0.606371\pi\)
−0.327989 + 0.944681i \(0.606371\pi\)
\(618\) 0 0
\(619\) −1.89529e9 −0.321187 −0.160594 0.987021i \(-0.551341\pi\)
−0.160594 + 0.987021i \(0.551341\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.40777e9 1.06169
\(624\) 0 0
\(625\) −3.75948e9 −0.615953
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.13459e9 −0.502232
\(630\) 0 0
\(631\) −1.20090e10 −1.90285 −0.951424 0.307882i \(-0.900380\pi\)
−0.951424 + 0.307882i \(0.900380\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.79951e9 1.05383
\(636\) 0 0
\(637\) 2.55836e9 0.392169
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.33728e9 −0.500484 −0.250242 0.968183i \(-0.580510\pi\)
−0.250242 + 0.968183i \(0.580510\pi\)
\(642\) 0 0
\(643\) 7.80682e9 1.15807 0.579037 0.815302i \(-0.303429\pi\)
0.579037 + 0.815302i \(0.303429\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.81228e9 0.698531 0.349265 0.937024i \(-0.386431\pi\)
0.349265 + 0.937024i \(0.386431\pi\)
\(648\) 0 0
\(649\) 7.64132e9 1.09727
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.16908e9 0.585927 0.292964 0.956124i \(-0.405359\pi\)
0.292964 + 0.956124i \(0.405359\pi\)
\(654\) 0 0
\(655\) −6.50541e9 −0.904545
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.02165e10 −1.39060 −0.695299 0.718721i \(-0.744728\pi\)
−0.695299 + 0.718721i \(0.744728\pi\)
\(660\) 0 0
\(661\) 1.41924e9 0.191139 0.0955697 0.995423i \(-0.469533\pi\)
0.0955697 + 0.995423i \(0.469533\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.35107e10 3.10020
\(666\) 0 0
\(667\) −5.84853e9 −0.763144
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.51476e9 0.960255
\(672\) 0 0
\(673\) 8.90621e9 1.12626 0.563132 0.826367i \(-0.309596\pi\)
0.563132 + 0.826367i \(0.309596\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.21191e9 0.397834 0.198917 0.980016i \(-0.436258\pi\)
0.198917 + 0.980016i \(0.436258\pi\)
\(678\) 0 0
\(679\) −8.69215e9 −1.06557
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.49063e10 −1.79018 −0.895091 0.445883i \(-0.852890\pi\)
−0.895091 + 0.445883i \(0.852890\pi\)
\(684\) 0 0
\(685\) 8.89242e9 1.05707
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.86639e9 −0.799762
\(690\) 0 0
\(691\) 3.38129e9 0.389860 0.194930 0.980817i \(-0.437552\pi\)
0.194930 + 0.980817i \(0.437552\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.69061e10 1.91028
\(696\) 0 0
\(697\) 4.41568e9 0.493950
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.19330e10 1.30839 0.654195 0.756326i \(-0.273008\pi\)
0.654195 + 0.756326i \(0.273008\pi\)
\(702\) 0 0
\(703\) 1.21765e10 1.32184
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.70657e9 0.820150
\(708\) 0 0
\(709\) 1.92709e9 0.203068 0.101534 0.994832i \(-0.467625\pi\)
0.101534 + 0.994832i \(0.467625\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.78546e9 −0.494436
\(714\) 0 0
\(715\) 4.39299e9 0.449458
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5.16235e9 −0.517960 −0.258980 0.965883i \(-0.583386\pi\)
−0.258980 + 0.965883i \(0.583386\pi\)
\(720\) 0 0
\(721\) −1.50181e10 −1.49225
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.13596e10 2.08166
\(726\) 0 0
\(727\) −1.38758e10 −1.33933 −0.669667 0.742662i \(-0.733563\pi\)
−0.669667 + 0.742662i \(0.733563\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.20175e10 1.13790
\(732\) 0 0
\(733\) 3.26388e9 0.306104 0.153052 0.988218i \(-0.451090\pi\)
0.153052 + 0.988218i \(0.451090\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.75088e9 −0.253125
\(738\) 0 0
\(739\) −5.49185e8 −0.0500568 −0.0250284 0.999687i \(-0.507968\pi\)
−0.0250284 + 0.999687i \(0.507968\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.11208e10 0.994663 0.497332 0.867561i \(-0.334313\pi\)
0.497332 + 0.867561i \(0.334313\pi\)
\(744\) 0 0
\(745\) 2.45462e10 2.17489
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.07392e10 −0.933872
\(750\) 0 0
\(751\) 5.06853e9 0.436659 0.218329 0.975875i \(-0.429939\pi\)
0.218329 + 0.975875i \(0.429939\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.24322e9 −0.358824
\(756\) 0 0
\(757\) 5.64424e9 0.472901 0.236450 0.971644i \(-0.424016\pi\)
0.236450 + 0.971644i \(0.424016\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.57769e10 −1.29770 −0.648850 0.760916i \(-0.724750\pi\)
−0.648850 + 0.760916i \(0.724750\pi\)
\(762\) 0 0
\(763\) 6.84416e9 0.557807
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.02453e10 −0.819859
\(768\) 0 0
\(769\) 1.05126e10 0.833616 0.416808 0.908994i \(-0.363149\pi\)
0.416808 + 0.908994i \(0.363149\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.70533e9 0.366405 0.183203 0.983075i \(-0.441354\pi\)
0.183203 + 0.983075i \(0.441354\pi\)
\(774\) 0 0
\(775\) 1.74771e10 1.34870
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.71529e10 −1.30004
\(780\) 0 0
\(781\) 9.19691e9 0.690818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.98372e9 −0.589063
\(786\) 0 0
\(787\) −2.07916e10 −1.52046 −0.760232 0.649652i \(-0.774915\pi\)
−0.760232 + 0.649652i \(0.774915\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.53835e10 −1.10519
\(792\) 0 0
\(793\) −1.00756e10 −0.717487
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.09035e10 −0.762890 −0.381445 0.924392i \(-0.624573\pi\)
−0.381445 + 0.924392i \(0.624573\pi\)
\(798\) 0 0
\(799\) 7.25897e9 0.503456
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.06710e9 0.209037
\(804\) 0 0
\(805\) 1.44303e10 0.974968
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.20873e10 0.802620 0.401310 0.915942i \(-0.368555\pi\)
0.401310 + 0.915942i \(0.368555\pi\)
\(810\) 0 0
\(811\) 1.39636e10 0.919229 0.459615 0.888118i \(-0.347988\pi\)
0.459615 + 0.888118i \(0.347988\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.38320e10 −2.18915
\(816\) 0 0
\(817\) −4.66825e10 −2.99486
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.90599e9 −0.435537 −0.217769 0.976000i \(-0.569878\pi\)
−0.217769 + 0.976000i \(0.569878\pi\)
\(822\) 0 0
\(823\) −1.31839e9 −0.0824413 −0.0412206 0.999150i \(-0.513125\pi\)
−0.0412206 + 0.999150i \(0.513125\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.16284e10 0.714907 0.357454 0.933931i \(-0.383645\pi\)
0.357454 + 0.933931i \(0.383645\pi\)
\(828\) 0 0
\(829\) 1.53849e10 0.937893 0.468947 0.883226i \(-0.344634\pi\)
0.468947 + 0.883226i \(0.344634\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.97892e9 0.478285
\(834\) 0 0
\(835\) 9.31978e9 0.553991
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.92329e9 −0.287799 −0.143899 0.989592i \(-0.545964\pi\)
−0.143899 + 0.989592i \(0.545964\pi\)
\(840\) 0 0
\(841\) 2.72353e10 1.57887
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.06872e10 1.17952
\(846\) 0 0
\(847\) −1.44385e10 −0.816452
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7.47365e9 0.415699
\(852\) 0 0
\(853\) −7.54820e9 −0.416411 −0.208205 0.978085i \(-0.566762\pi\)
−0.208205 + 0.978085i \(0.566762\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.27814e10 0.693656 0.346828 0.937929i \(-0.387259\pi\)
0.346828 + 0.937929i \(0.387259\pi\)
\(858\) 0 0
\(859\) 1.33691e10 0.719657 0.359829 0.933018i \(-0.382835\pi\)
0.359829 + 0.933018i \(0.382835\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.25958e9 −0.490404 −0.245202 0.969472i \(-0.578854\pi\)
−0.245202 + 0.969472i \(0.578854\pi\)
\(864\) 0 0
\(865\) 2.89594e10 1.52136
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.23306e10 1.15433
\(870\) 0 0
\(871\) 3.68830e9 0.189131
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.20452e10 −0.607837
\(876\) 0 0
\(877\) 1.40971e10 0.705717 0.352859 0.935677i \(-0.385210\pi\)
0.352859 + 0.935677i \(0.385210\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.97871e10 −0.974914 −0.487457 0.873147i \(-0.662075\pi\)
−0.487457 + 0.873147i \(0.662075\pi\)
\(882\) 0 0
\(883\) −3.39322e10 −1.65863 −0.829314 0.558782i \(-0.811269\pi\)
−0.829314 + 0.558782i \(0.811269\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.05047e10 1.46769 0.733846 0.679316i \(-0.237724\pi\)
0.733846 + 0.679316i \(0.237724\pi\)
\(888\) 0 0
\(889\) −1.97243e10 −0.941554
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.81978e10 −1.32506
\(894\) 0 0
\(895\) 2.52427e10 1.17694
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.63992e10 1.67083
\(900\) 0 0
\(901\) −2.14147e10 −0.975382
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.40762e10 2.87360
\(906\) 0 0
\(907\) 6.76417e9 0.301016 0.150508 0.988609i \(-0.451909\pi\)
0.150508 + 0.988609i \(0.451909\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.14881e10 −0.503424 −0.251712 0.967802i \(-0.580994\pi\)
−0.251712 + 0.967802i \(0.580994\pi\)
\(912\) 0 0
\(913\) −9.89160e9 −0.430149
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.88711e10 0.808175
\(918\) 0 0
\(919\) 3.87986e10 1.64897 0.824483 0.565887i \(-0.191466\pi\)
0.824483 + 0.565887i \(0.191466\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.23310e10 −0.516168
\(924\) 0 0
\(925\) −2.72948e10 −1.13392
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.14694e10 −1.69696 −0.848482 0.529224i \(-0.822483\pi\)
−0.848482 + 0.529224i \(0.822483\pi\)
\(930\) 0 0
\(931\) −3.09945e10 −1.25881
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.37007e10 0.548155
\(936\) 0 0
\(937\) 1.25279e10 0.497497 0.248748 0.968568i \(-0.419981\pi\)
0.248748 + 0.968568i \(0.419981\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.90647e10 −1.52834 −0.764171 0.645014i \(-0.776852\pi\)
−0.764171 + 0.645014i \(0.776852\pi\)
\(942\) 0 0
\(943\) −1.05281e10 −0.408844
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.55171e10 0.593726 0.296863 0.954920i \(-0.404059\pi\)
0.296863 + 0.954920i \(0.404059\pi\)
\(948\) 0 0
\(949\) −4.11228e9 −0.156189
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.71024e10 1.38860 0.694299 0.719687i \(-0.255715\pi\)
0.694299 + 0.719687i \(0.255715\pi\)
\(954\) 0 0
\(955\) −3.67031e9 −0.136361
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.57955e10 −0.944449
\(960\) 0 0
\(961\) 2.27038e9 0.0825214
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.40887e10 −2.29581
\(966\) 0 0
\(967\) −2.56321e10 −0.911574 −0.455787 0.890089i \(-0.650642\pi\)
−0.455787 + 0.890089i \(0.650642\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.08869e10 1.78377 0.891884 0.452264i \(-0.149384\pi\)
0.891884 + 0.452264i \(0.149384\pi\)
\(972\) 0 0
\(973\) −4.90420e10 −1.70676
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.68684e10 −0.578686 −0.289343 0.957225i \(-0.593437\pi\)
−0.289343 + 0.957225i \(0.593437\pi\)
\(978\) 0 0
\(979\) −1.45053e10 −0.494068
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.61062e10 0.540825 0.270412 0.962745i \(-0.412840\pi\)
0.270412 + 0.962745i \(0.412840\pi\)
\(984\) 0 0
\(985\) 4.13849e10 1.37980
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.86527e10 −0.941842
\(990\) 0 0
\(991\) −3.98303e9 −0.130004 −0.0650019 0.997885i \(-0.520705\pi\)
−0.0650019 + 0.997885i \(0.520705\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.35334e10 −0.435538
\(996\) 0 0
\(997\) −4.19886e10 −1.34183 −0.670917 0.741533i \(-0.734099\pi\)
−0.670917 + 0.741533i \(0.734099\pi\)
\(998\) 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 288.8.a.h.1.1 2
3.2 odd 2 96.8.a.h.1.2 yes 2
4.3 odd 2 288.8.a.g.1.1 2
8.3 odd 2 576.8.a.bn.1.2 2
8.5 even 2 576.8.a.bo.1.2 2
12.11 even 2 96.8.a.e.1.2 2
24.5 odd 2 192.8.a.q.1.1 2
24.11 even 2 192.8.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.8.a.e.1.2 2 12.11 even 2
96.8.a.h.1.2 yes 2 3.2 odd 2
192.8.a.q.1.1 2 24.5 odd 2
192.8.a.t.1.1 2 24.11 even 2
288.8.a.g.1.1 2 4.3 odd 2
288.8.a.h.1.1 2 1.1 even 1 trivial
576.8.a.bn.1.2 2 8.3 odd 2
576.8.a.bo.1.2 2 8.5 even 2