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Properties

Label 360.8.a.c.1.1
Level $360$
Weight $8$
Character 360.1
Self dual yes
Analytic conductor $112.459$
Analytic rank $1$
Dimension $1$
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] = [360,8,Mod(1,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("360.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 360.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: \(112.458609174\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 360.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+125.000 q^{5} -540.000 q^{7} +O(q^{10})\) \(q+125.000 q^{5} -540.000 q^{7} -3584.00 q^{11} +5994.00 q^{13} +24666.0 q^{17} -31276.0 q^{19} -5376.00 q^{23} +15625.0 q^{25} +194846. q^{29} -43592.0 q^{31} -67500.0 q^{35} -244358. q^{37} +73686.0 q^{41} -440268. q^{43} -465920. q^{47} -531943. q^{49} -47154.0 q^{53} -448000. q^{55} +2.28902e6 q^{59} +1.60648e6 q^{61} +749250. q^{65} -3.65323e6 q^{67} +1.99283e6 q^{71} -4.03707e6 q^{73} +1.93536e6 q^{77} -1.94247e6 q^{79} -1.10567e6 q^{83} +3.08325e6 q^{85} -14626.0 q^{89} -3.23676e6 q^{91} -3.90950e6 q^{95} +9.36787e6 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) −540.000 −0.595046 −0.297523 − 0.954715i \(-0.596161\pi\)
−0.297523 + 0.954715i \(0.596161\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3584.00 −0.811883 −0.405942 − 0.913899i \(-0.633056\pi\)
−0.405942 + 0.913899i \(0.633056\pi\)
\(12\) 0 0
\(13\) 5994.00 0.756685 0.378342 − 0.925666i \(-0.376494\pi\)
0.378342 + 0.925666i \(0.376494\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 24666.0 1.21766 0.608832 − 0.793299i \(-0.291638\pi\)
0.608832 + 0.793299i \(0.291638\pi\)
\(18\) 0 0
\(19\) −31276.0 −1.04610 −0.523050 − 0.852302i \(-0.675206\pi\)
−0.523050 + 0.852302i \(0.675206\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5376.00 −0.0921323 −0.0460661 − 0.998938i \(-0.514668\pi\)
−0.0460661 + 0.998938i \(0.514668\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 194846. 1.48354 0.741769 − 0.670656i \(-0.233987\pi\)
0.741769 + 0.670656i \(0.233987\pi\)
\(30\) 0 0
\(31\) −43592.0 −0.262809 −0.131405 − 0.991329i \(-0.541949\pi\)
−0.131405 + 0.991329i \(0.541949\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −67500.0 −0.266113
\(36\) 0 0
\(37\) −244358. −0.793086 −0.396543 − 0.918016i \(-0.629790\pi\)
−0.396543 + 0.918016i \(0.629790\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 73686.0 0.166971 0.0834856 − 0.996509i \(-0.473395\pi\)
0.0834856 + 0.996509i \(0.473395\pi\)
\(42\) 0 0
\(43\) −440268. −0.844457 −0.422228 − 0.906489i \(-0.638752\pi\)
−0.422228 + 0.906489i \(0.638752\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −465920. −0.654589 −0.327295 − 0.944922i \(-0.606137\pi\)
−0.327295 + 0.944922i \(0.606137\pi\)
\(48\) 0 0
\(49\) −531943. −0.645920
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −47154.0 −0.0435064 −0.0217532 − 0.999763i \(-0.506925\pi\)
−0.0217532 + 0.999763i \(0.506925\pi\)
\(54\) 0 0
\(55\) −448000. −0.363085
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.28902e6 1.45100 0.725501 − 0.688221i \(-0.241608\pi\)
0.725501 + 0.688221i \(0.241608\pi\)
\(60\) 0 0
\(61\) 1.60648e6 0.906192 0.453096 − 0.891462i \(-0.350319\pi\)
0.453096 + 0.891462i \(0.350319\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 749250. 0.338400
\(66\) 0 0
\(67\) −3.65323e6 −1.48394 −0.741968 − 0.670436i \(-0.766107\pi\)
−0.741968 + 0.670436i \(0.766107\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.99283e6 0.660795 0.330397 − 0.943842i \(-0.392817\pi\)
0.330397 + 0.943842i \(0.392817\pi\)
\(72\) 0 0
\(73\) −4.03707e6 −1.21461 −0.607304 − 0.794469i \(-0.707749\pi\)
−0.607304 + 0.794469i \(0.707749\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.93536e6 0.483108
\(78\) 0 0
\(79\) −1.94247e6 −0.443261 −0.221631 − 0.975131i \(-0.571138\pi\)
−0.221631 + 0.975131i \(0.571138\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.10567e6 −0.212252 −0.106126 − 0.994353i \(-0.533845\pi\)
−0.106126 + 0.994353i \(0.533845\pi\)
\(84\) 0 0
\(85\) 3.08325e6 0.544556
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14626.0 −0.00219918 −0.00109959 − 0.999999i \(-0.500350\pi\)
−0.00109959 + 0.999999i \(0.500350\pi\)
\(90\) 0 0
\(91\) −3.23676e6 −0.450262
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.90950e6 −0.467831
\(96\) 0 0
\(97\) 9.36787e6 1.04217 0.521087 − 0.853504i \(-0.325527\pi\)
0.521087 + 0.853504i \(0.325527\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.46520e7 −1.41506 −0.707528 − 0.706686i \(-0.750189\pi\)
−0.707528 + 0.706686i \(0.750189\pi\)
\(102\) 0 0
\(103\) −4.00105e6 −0.360781 −0.180391 − 0.983595i \(-0.557736\pi\)
−0.180391 + 0.983595i \(0.557736\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.72613e7 1.36217 0.681083 − 0.732206i \(-0.261509\pi\)
0.681083 + 0.732206i \(0.261509\pi\)
\(108\) 0 0
\(109\) 6.67294e6 0.493543 0.246771 − 0.969074i \(-0.420630\pi\)
0.246771 + 0.969074i \(0.420630\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.51498e7 −1.63968 −0.819841 − 0.572592i \(-0.805938\pi\)
−0.819841 + 0.572592i \(0.805938\pi\)
\(114\) 0 0
\(115\) −672000. −0.0412028
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.33196e7 −0.724566
\(120\) 0 0
\(121\) −6.64212e6 −0.340846
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.95312e6 0.0894427
\(126\) 0 0
\(127\) 3.65508e6 0.158338 0.0791689 − 0.996861i \(-0.474773\pi\)
0.0791689 + 0.996861i \(0.474773\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.25654e7 −1.26563 −0.632815 − 0.774303i \(-0.718101\pi\)
−0.632815 + 0.774303i \(0.718101\pi\)
\(132\) 0 0
\(133\) 1.68890e7 0.622478
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.40777e7 −1.46453 −0.732263 − 0.681022i \(-0.761536\pi\)
−0.732263 + 0.681022i \(0.761536\pi\)
\(138\) 0 0
\(139\) −5.02559e7 −1.58721 −0.793607 − 0.608430i \(-0.791800\pi\)
−0.793607 + 0.608430i \(0.791800\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.14825e7 −0.614340
\(144\) 0 0
\(145\) 2.43557e7 0.663458
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.40505e7 −1.33859 −0.669296 − 0.742996i \(-0.733404\pi\)
−0.669296 + 0.742996i \(0.733404\pi\)
\(150\) 0 0
\(151\) 3.56658e6 0.0843011 0.0421505 − 0.999111i \(-0.486579\pi\)
0.0421505 + 0.999111i \(0.486579\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.44900e6 −0.117532
\(156\) 0 0
\(157\) −2.07073e7 −0.427045 −0.213523 − 0.976938i \(-0.568494\pi\)
−0.213523 + 0.976938i \(0.568494\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.90304e6 0.0548230
\(162\) 0 0
\(163\) 1.61059e7 0.291291 0.145646 − 0.989337i \(-0.453474\pi\)
0.145646 + 0.989337i \(0.453474\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −569728. −0.00946586 −0.00473293 − 0.999989i \(-0.501507\pi\)
−0.00473293 + 0.999989i \(0.501507\pi\)
\(168\) 0 0
\(169\) −2.68205e7 −0.427428
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.81223e7 −0.559780 −0.279890 − 0.960032i \(-0.590298\pi\)
−0.279890 + 0.960032i \(0.590298\pi\)
\(174\) 0 0
\(175\) −8.43750e6 −0.119009
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.43251e7 −0.707970 −0.353985 − 0.935251i \(-0.615174\pi\)
−0.353985 + 0.935251i \(0.615174\pi\)
\(180\) 0 0
\(181\) −3.40094e7 −0.426308 −0.213154 − 0.977019i \(-0.568374\pi\)
−0.213154 + 0.977019i \(0.568374\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.05448e7 −0.354679
\(186\) 0 0
\(187\) −8.84029e7 −0.988601
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.78493e7 1.01611 0.508055 − 0.861324i \(-0.330365\pi\)
0.508055 + 0.861324i \(0.330365\pi\)
\(192\) 0 0
\(193\) 1.27215e8 1.27377 0.636883 − 0.770961i \(-0.280224\pi\)
0.636883 + 0.770961i \(0.280224\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.31465e7 −0.774840 −0.387420 − 0.921903i \(-0.626634\pi\)
−0.387420 + 0.921903i \(0.626634\pi\)
\(198\) 0 0
\(199\) 6.73202e6 0.0605564 0.0302782 − 0.999542i \(-0.490361\pi\)
0.0302782 + 0.999542i \(0.490361\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.05217e8 −0.882773
\(204\) 0 0
\(205\) 9.21075e6 0.0746718
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.12093e8 0.849312
\(210\) 0 0
\(211\) −2.32363e8 −1.70286 −0.851428 − 0.524472i \(-0.824263\pi\)
−0.851428 + 0.524472i \(0.824263\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.50335e7 −0.377653
\(216\) 0 0
\(217\) 2.35397e7 0.156384
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.47848e8 0.921388
\(222\) 0 0
\(223\) −1.64917e8 −0.995861 −0.497931 − 0.867217i \(-0.665907\pi\)
−0.497931 + 0.867217i \(0.665907\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.45397e7 0.139245 0.0696224 − 0.997573i \(-0.477821\pi\)
0.0696224 + 0.997573i \(0.477821\pi\)
\(228\) 0 0
\(229\) −1.47416e8 −0.811183 −0.405592 − 0.914054i \(-0.632934\pi\)
−0.405592 + 0.914054i \(0.632934\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.80364e7 −0.0934121 −0.0467060 − 0.998909i \(-0.514872\pi\)
−0.0467060 + 0.998909i \(0.514872\pi\)
\(234\) 0 0
\(235\) −5.82400e7 −0.292741
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.06434e8 −1.45192 −0.725962 − 0.687734i \(-0.758605\pi\)
−0.725962 + 0.687734i \(0.758605\pi\)
\(240\) 0 0
\(241\) −4.20718e8 −1.93611 −0.968057 − 0.250729i \(-0.919330\pi\)
−0.968057 + 0.250729i \(0.919330\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.64929e7 −0.288864
\(246\) 0 0
\(247\) −1.87468e8 −0.791569
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.60387e8 −0.640193 −0.320097 − 0.947385i \(-0.603715\pi\)
−0.320097 + 0.947385i \(0.603715\pi\)
\(252\) 0 0
\(253\) 1.92676e7 0.0748007
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.15916e8 0.425970 0.212985 − 0.977056i \(-0.431682\pi\)
0.212985 + 0.977056i \(0.431682\pi\)
\(258\) 0 0
\(259\) 1.31953e8 0.471923
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.29419e8 −1.45558 −0.727790 − 0.685800i \(-0.759453\pi\)
−0.727790 + 0.685800i \(0.759453\pi\)
\(264\) 0 0
\(265\) −5.89425e6 −0.0194566
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.43599e8 0.763030 0.381515 − 0.924363i \(-0.375402\pi\)
0.381515 + 0.924363i \(0.375402\pi\)
\(270\) 0 0
\(271\) 1.63200e8 0.498111 0.249056 − 0.968489i \(-0.419880\pi\)
0.249056 + 0.968489i \(0.419880\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.60000e7 −0.162377
\(276\) 0 0
\(277\) 1.69043e8 0.477879 0.238940 − 0.971034i \(-0.423200\pi\)
0.238940 + 0.971034i \(0.423200\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.40644e8 1.18472 0.592360 − 0.805673i \(-0.298196\pi\)
0.592360 + 0.805673i \(0.298196\pi\)
\(282\) 0 0
\(283\) 7.37145e8 1.93330 0.966652 − 0.256092i \(-0.0824350\pi\)
0.966652 + 0.256092i \(0.0824350\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.97904e7 −0.0993555
\(288\) 0 0
\(289\) 1.98073e8 0.482706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.95918e8 −1.61630 −0.808149 − 0.588978i \(-0.799530\pi\)
−0.808149 + 0.588978i \(0.799530\pi\)
\(294\) 0 0
\(295\) 2.86128e8 0.648908
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.22237e7 −0.0697151
\(300\) 0 0
\(301\) 2.37745e8 0.502491
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00810e8 0.405262
\(306\) 0 0
\(307\) −3.73398e8 −0.736526 −0.368263 − 0.929722i \(-0.620047\pi\)
−0.368263 + 0.929722i \(0.620047\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.22066e8 0.607132 0.303566 − 0.952810i \(-0.401823\pi\)
0.303566 + 0.952810i \(0.401823\pi\)
\(312\) 0 0
\(313\) 6.28126e8 1.15782 0.578911 − 0.815391i \(-0.303478\pi\)
0.578911 + 0.815391i \(0.303478\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.79960e8 0.317299 0.158649 − 0.987335i \(-0.449286\pi\)
0.158649 + 0.987335i \(0.449286\pi\)
\(318\) 0 0
\(319\) −6.98328e8 −1.20446
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.71454e8 −1.27380
\(324\) 0 0
\(325\) 9.36562e7 0.151337
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.51597e8 0.389511
\(330\) 0 0
\(331\) −5.17622e8 −0.784539 −0.392270 − 0.919850i \(-0.628310\pi\)
−0.392270 + 0.919850i \(0.628310\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.56654e8 −0.663636
\(336\) 0 0
\(337\) −2.06359e8 −0.293710 −0.146855 − 0.989158i \(-0.546915\pi\)
−0.146855 + 0.989158i \(0.546915\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.56234e8 0.213371
\(342\) 0 0
\(343\) 7.31962e8 0.979398
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.56243e7 0.110013 0.0550065 − 0.998486i \(-0.482482\pi\)
0.0550065 + 0.998486i \(0.482482\pi\)
\(348\) 0 0
\(349\) 1.44884e9 1.82445 0.912226 − 0.409688i \(-0.134362\pi\)
0.912226 + 0.409688i \(0.134362\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.63738e8 0.803129 0.401564 − 0.915831i \(-0.368467\pi\)
0.401564 + 0.915831i \(0.368467\pi\)
\(354\) 0 0
\(355\) 2.49104e8 0.295516
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.11116e8 0.925236 0.462618 − 0.886558i \(-0.346910\pi\)
0.462618 + 0.886558i \(0.346910\pi\)
\(360\) 0 0
\(361\) 8.43164e7 0.0943272
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.04634e8 −0.543189
\(366\) 0 0
\(367\) −1.25253e9 −1.32268 −0.661342 − 0.750084i \(-0.730013\pi\)
−0.661342 + 0.750084i \(0.730013\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.54632e7 0.0258883
\(372\) 0 0
\(373\) 1.61555e9 1.61190 0.805950 − 0.591983i \(-0.201655\pi\)
0.805950 + 0.591983i \(0.201655\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.16791e9 1.12257
\(378\) 0 0
\(379\) −8.94292e8 −0.843805 −0.421903 − 0.906641i \(-0.638638\pi\)
−0.421903 + 0.906641i \(0.638638\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.86145e9 1.69299 0.846496 − 0.532396i \(-0.178708\pi\)
0.846496 + 0.532396i \(0.178708\pi\)
\(384\) 0 0
\(385\) 2.41920e8 0.216052
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.69370e8 0.232020 0.116010 − 0.993248i \(-0.462990\pi\)
0.116010 + 0.993248i \(0.462990\pi\)
\(390\) 0 0
\(391\) −1.32604e8 −0.112186
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.42809e8 −0.198233
\(396\) 0 0
\(397\) 1.33926e9 1.07423 0.537116 − 0.843509i \(-0.319514\pi\)
0.537116 + 0.843509i \(0.319514\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.46336e9 1.13331 0.566653 − 0.823957i \(-0.308238\pi\)
0.566653 + 0.823957i \(0.308238\pi\)
\(402\) 0 0
\(403\) −2.61290e8 −0.198864
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.75779e8 0.643894
\(408\) 0 0
\(409\) −9.30881e8 −0.672763 −0.336382 − 0.941726i \(-0.609203\pi\)
−0.336382 + 0.941726i \(0.609203\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.23607e9 −0.863414
\(414\) 0 0
\(415\) −1.38208e8 −0.0949219
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.16338e7 0.0143676 0.00718379 − 0.999974i \(-0.497713\pi\)
0.00718379 + 0.999974i \(0.497713\pi\)
\(420\) 0 0
\(421\) −3.29561e8 −0.215253 −0.107626 − 0.994191i \(-0.534325\pi\)
−0.107626 + 0.994191i \(0.534325\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.85406e8 0.243533
\(426\) 0 0
\(427\) −8.67498e8 −0.539226
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.61775e9 0.973285 0.486643 − 0.873601i \(-0.338221\pi\)
0.486643 + 0.873601i \(0.338221\pi\)
\(432\) 0 0
\(433\) −3.13091e8 −0.185337 −0.0926687 − 0.995697i \(-0.529540\pi\)
−0.0926687 + 0.995697i \(0.529540\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.68140e8 0.0963797
\(438\) 0 0
\(439\) −3.41644e9 −1.92729 −0.963646 − 0.267181i \(-0.913908\pi\)
−0.963646 + 0.267181i \(0.913908\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.41875e9 0.775340 0.387670 − 0.921798i \(-0.373280\pi\)
0.387670 + 0.921798i \(0.373280\pi\)
\(444\) 0 0
\(445\) −1.82825e6 −0.000983503 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.05549e8 −0.315709 −0.157854 − 0.987462i \(-0.550458\pi\)
−0.157854 + 0.987462i \(0.550458\pi\)
\(450\) 0 0
\(451\) −2.64091e8 −0.135561
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.04595e8 −0.201363
\(456\) 0 0
\(457\) 2.29699e9 1.12578 0.562890 − 0.826532i \(-0.309690\pi\)
0.562890 + 0.826532i \(0.309690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.03718e9 1.91922 0.959609 − 0.281337i \(-0.0907778\pi\)
0.959609 + 0.281337i \(0.0907778\pi\)
\(462\) 0 0
\(463\) 7.00365e8 0.327937 0.163969 − 0.986466i \(-0.447570\pi\)
0.163969 + 0.986466i \(0.447570\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.08634e9 1.85663 0.928316 − 0.371793i \(-0.121257\pi\)
0.928316 + 0.371793i \(0.121257\pi\)
\(468\) 0 0
\(469\) 1.97274e9 0.883010
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.57792e9 0.685601
\(474\) 0 0
\(475\) −4.88687e8 −0.209220
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.33732e8 0.0555982 0.0277991 − 0.999614i \(-0.491150\pi\)
0.0277991 + 0.999614i \(0.491150\pi\)
\(480\) 0 0
\(481\) −1.46468e9 −0.600116
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.17098e9 0.466074
\(486\) 0 0
\(487\) 6.73233e8 0.264128 0.132064 − 0.991241i \(-0.457840\pi\)
0.132064 + 0.991241i \(0.457840\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.32002e9 −1.26577 −0.632886 − 0.774245i \(-0.718130\pi\)
−0.632886 + 0.774245i \(0.718130\pi\)
\(492\) 0 0
\(493\) 4.80607e9 1.80645
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.07613e9 −0.393203
\(498\) 0 0
\(499\) 3.01858e8 0.108755 0.0543776 − 0.998520i \(-0.482683\pi\)
0.0543776 + 0.998520i \(0.482683\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.37577e9 −1.18273 −0.591363 − 0.806405i \(-0.701410\pi\)
−0.591363 + 0.806405i \(0.701410\pi\)
\(504\) 0 0
\(505\) −1.83151e9 −0.632832
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.50975e9 −1.51580 −0.757898 − 0.652373i \(-0.773773\pi\)
−0.757898 + 0.652373i \(0.773773\pi\)
\(510\) 0 0
\(511\) 2.18002e9 0.722748
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.00132e8 −0.161346
\(516\) 0 0
\(517\) 1.66986e9 0.531450
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.68799e7 0.0176209 0.00881043 − 0.999961i \(-0.497196\pi\)
0.00881043 + 0.999961i \(0.497196\pi\)
\(522\) 0 0
\(523\) −5.69608e9 −1.74108 −0.870542 − 0.492094i \(-0.836232\pi\)
−0.870542 + 0.492094i \(0.836232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.07524e9 −0.320014
\(528\) 0 0
\(529\) −3.37592e9 −0.991512
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.41674e8 0.126345
\(534\) 0 0
\(535\) 2.15766e9 0.609179
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.90648e9 0.524412
\(540\) 0 0
\(541\) 5.67986e9 1.54222 0.771111 − 0.636701i \(-0.219701\pi\)
0.771111 + 0.636701i \(0.219701\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.34118e8 0.220719
\(546\) 0 0
\(547\) 4.04334e9 1.05629 0.528146 − 0.849153i \(-0.322887\pi\)
0.528146 + 0.849153i \(0.322887\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.09400e9 −1.55193
\(552\) 0 0
\(553\) 1.04893e9 0.263761
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.35076e9 −1.80235 −0.901175 − 0.433456i \(-0.857294\pi\)
−0.901175 + 0.433456i \(0.857294\pi\)
\(558\) 0 0
\(559\) −2.63897e9 −0.638988
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.95680e8 −0.0698301 −0.0349151 − 0.999390i \(-0.511116\pi\)
−0.0349151 + 0.999390i \(0.511116\pi\)
\(564\) 0 0
\(565\) −3.14372e9 −0.733288
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.95363e8 −0.226511 −0.113255 − 0.993566i \(-0.536128\pi\)
−0.113255 + 0.993566i \(0.536128\pi\)
\(570\) 0 0
\(571\) 6.41612e9 1.44227 0.721134 − 0.692795i \(-0.243621\pi\)
0.721134 + 0.692795i \(0.243621\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.40000e7 −0.0184265
\(576\) 0 0
\(577\) 1.59733e9 0.346163 0.173081 − 0.984908i \(-0.444628\pi\)
0.173081 + 0.984908i \(0.444628\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.97061e8 0.126300
\(582\) 0 0
\(583\) 1.69000e8 0.0353221
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.67543e9 −0.954087 −0.477043 − 0.878880i \(-0.658292\pi\)
−0.477043 + 0.878880i \(0.658292\pi\)
\(588\) 0 0
\(589\) 1.36338e9 0.274925
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.43350e9 1.26694 0.633469 − 0.773768i \(-0.281630\pi\)
0.633469 + 0.773768i \(0.281630\pi\)
\(594\) 0 0
\(595\) −1.66495e9 −0.324036
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.31872e9 −1.01115 −0.505573 − 0.862784i \(-0.668719\pi\)
−0.505573 + 0.862784i \(0.668719\pi\)
\(600\) 0 0
\(601\) −8.95968e9 −1.68357 −0.841786 − 0.539811i \(-0.818496\pi\)
−0.841786 + 0.539811i \(0.818496\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.30264e8 −0.152431
\(606\) 0 0
\(607\) −8.65725e9 −1.57116 −0.785579 − 0.618762i \(-0.787635\pi\)
−0.785579 + 0.618762i \(0.787635\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.79272e9 −0.495318
\(612\) 0 0
\(613\) −1.66931e9 −0.292703 −0.146351 − 0.989233i \(-0.546753\pi\)
−0.146351 + 0.989233i \(0.546753\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00732e9 −0.172651 −0.0863255 − 0.996267i \(-0.527513\pi\)
−0.0863255 + 0.996267i \(0.527513\pi\)
\(618\) 0 0
\(619\) 2.91303e9 0.493660 0.246830 − 0.969059i \(-0.420611\pi\)
0.246830 + 0.969059i \(0.420611\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.89804e6 0.00130861
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.02733e9 −0.965713
\(630\) 0 0
\(631\) 4.06191e9 0.643617 0.321809 − 0.946805i \(-0.395709\pi\)
0.321809 + 0.946805i \(0.395709\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.56886e8 0.0708108
\(636\) 0 0
\(637\) −3.18847e9 −0.488758
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.55612e7 −0.00683269 −0.00341635 − 0.999994i \(-0.501087\pi\)
−0.00341635 + 0.999994i \(0.501087\pi\)
\(642\) 0 0
\(643\) 2.86005e9 0.424263 0.212131 − 0.977241i \(-0.431960\pi\)
0.212131 + 0.977241i \(0.431960\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.57059e7 0.0124407 0.00622037 − 0.999981i \(-0.498020\pi\)
0.00622037 + 0.999981i \(0.498020\pi\)
\(648\) 0 0
\(649\) −8.20386e9 −1.17805
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.45177e9 1.32836 0.664182 − 0.747571i \(-0.268780\pi\)
0.664182 + 0.747571i \(0.268780\pi\)
\(654\) 0 0
\(655\) −4.07067e9 −0.566007
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.09489e9 0.829597 0.414798 − 0.909913i \(-0.363852\pi\)
0.414798 + 0.909913i \(0.363852\pi\)
\(660\) 0 0
\(661\) −5.32644e9 −0.717352 −0.358676 − 0.933462i \(-0.616772\pi\)
−0.358676 + 0.933462i \(0.616772\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.11113e9 0.278381
\(666\) 0 0
\(667\) −1.04749e9 −0.136682
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.75762e9 −0.735722
\(672\) 0 0
\(673\) −5.98275e9 −0.756569 −0.378284 − 0.925689i \(-0.623486\pi\)
−0.378284 + 0.925689i \(0.623486\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.24611e10 −1.54347 −0.771733 − 0.635946i \(-0.780610\pi\)
−0.771733 + 0.635946i \(0.780610\pi\)
\(678\) 0 0
\(679\) −5.05865e9 −0.620141
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.51389e9 0.662195 0.331098 − 0.943597i \(-0.392581\pi\)
0.331098 + 0.943597i \(0.392581\pi\)
\(684\) 0 0
\(685\) −5.50971e9 −0.654956
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.82641e8 −0.0329206
\(690\) 0 0
\(691\) 1.52984e10 1.76389 0.881946 − 0.471350i \(-0.156233\pi\)
0.881946 + 0.471350i \(0.156233\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.28199e9 −0.709824
\(696\) 0 0
\(697\) 1.81754e9 0.203315
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.44144e10 −1.58046 −0.790232 − 0.612808i \(-0.790040\pi\)
−0.790232 + 0.612808i \(0.790040\pi\)
\(702\) 0 0
\(703\) 7.64254e9 0.829648
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.91210e9 0.842023
\(708\) 0 0
\(709\) −1.54497e10 −1.62802 −0.814009 − 0.580853i \(-0.802719\pi\)
−0.814009 + 0.580853i \(0.802719\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.34351e8 0.0242132
\(714\) 0 0
\(715\) −2.68531e9 −0.274741
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.12386e10 −1.12761 −0.563806 − 0.825907i \(-0.690664\pi\)
−0.563806 + 0.825907i \(0.690664\pi\)
\(720\) 0 0
\(721\) 2.16057e9 0.214681
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.04447e9 0.296707
\(726\) 0 0
\(727\) −5.48758e9 −0.529676 −0.264838 − 0.964293i \(-0.585319\pi\)
−0.264838 + 0.964293i \(0.585319\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.08597e10 −1.02826
\(732\) 0 0
\(733\) −7.65914e9 −0.718316 −0.359158 − 0.933277i \(-0.616936\pi\)
−0.359158 + 0.933277i \(0.616936\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.30932e10 1.20478
\(738\) 0 0
\(739\) −8.88152e9 −0.809528 −0.404764 − 0.914421i \(-0.632646\pi\)
−0.404764 + 0.914421i \(0.632646\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.46556e10 1.31082 0.655409 − 0.755274i \(-0.272496\pi\)
0.655409 + 0.755274i \(0.272496\pi\)
\(744\) 0 0
\(745\) −6.75632e9 −0.598636
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.32109e9 −0.810551
\(750\) 0 0
\(751\) 2.03087e10 1.74961 0.874805 − 0.484475i \(-0.160989\pi\)
0.874805 + 0.484475i \(0.160989\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.45823e8 0.0377006
\(756\) 0 0
\(757\) 1.26420e10 1.05921 0.529603 − 0.848246i \(-0.322341\pi\)
0.529603 + 0.848246i \(0.322341\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.15079e9 0.259163 0.129582 − 0.991569i \(-0.458637\pi\)
0.129582 + 0.991569i \(0.458637\pi\)
\(762\) 0 0
\(763\) −3.60339e9 −0.293681
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.37204e10 1.09795
\(768\) 0 0
\(769\) −9.85574e9 −0.781532 −0.390766 − 0.920490i \(-0.627790\pi\)
−0.390766 + 0.920490i \(0.627790\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.51998e9 0.351972 0.175986 − 0.984393i \(-0.443689\pi\)
0.175986 + 0.984393i \(0.443689\pi\)
\(774\) 0 0
\(775\) −6.81125e8 −0.0525619
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.30460e9 −0.174669
\(780\) 0 0
\(781\) −7.14231e9 −0.536488
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.58841e9 −0.190980
\(786\) 0 0
\(787\) 2.14337e9 0.156742 0.0783712 − 0.996924i \(-0.475028\pi\)
0.0783712 + 0.996924i \(0.475028\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.35809e10 0.975686
\(792\) 0 0
\(793\) 9.62923e9 0.685702
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.74444e10 −1.22054 −0.610270 − 0.792193i \(-0.708939\pi\)
−0.610270 + 0.792193i \(0.708939\pi\)
\(798\) 0 0
\(799\) −1.14924e10 −0.797070
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.44689e10 0.986120
\(804\) 0 0
\(805\) 3.62880e8 0.0245176
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.77807e10 −1.18067 −0.590334 − 0.807159i \(-0.701004\pi\)
−0.590334 + 0.807159i \(0.701004\pi\)
\(810\) 0 0
\(811\) 2.38186e10 1.56799 0.783995 − 0.620767i \(-0.213179\pi\)
0.783995 + 0.620767i \(0.213179\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.01323e9 0.130269
\(816\) 0 0
\(817\) 1.37698e10 0.883387
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.17123e10 1.36932 0.684659 − 0.728863i \(-0.259951\pi\)
0.684659 + 0.728863i \(0.259951\pi\)
\(822\) 0 0
\(823\) 5.61245e9 0.350957 0.175478 − 0.984483i \(-0.443853\pi\)
0.175478 + 0.984483i \(0.443853\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.18286e9 −0.503079 −0.251539 − 0.967847i \(-0.580937\pi\)
−0.251539 + 0.967847i \(0.580937\pi\)
\(828\) 0 0
\(829\) 1.12415e10 0.685301 0.342651 − 0.939463i \(-0.388675\pi\)
0.342651 + 0.939463i \(0.388675\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.31209e10 −0.786514
\(834\) 0 0
\(835\) −7.12160e7 −0.00423326
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6.13514e9 −0.358639 −0.179320 − 0.983791i \(-0.557390\pi\)
−0.179320 + 0.983791i \(0.557390\pi\)
\(840\) 0 0
\(841\) 2.07151e10 1.20088
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.35256e9 −0.191152
\(846\) 0 0
\(847\) 3.58674e9 0.202819
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.31367e9 0.0730689
\(852\) 0 0
\(853\) 2.19770e10 1.21240 0.606200 − 0.795312i \(-0.292693\pi\)
0.606200 + 0.795312i \(0.292693\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.11557e10 −1.69085 −0.845424 − 0.534096i \(-0.820652\pi\)
−0.845424 + 0.534096i \(0.820652\pi\)
\(858\) 0 0
\(859\) −2.38146e10 −1.28194 −0.640970 − 0.767566i \(-0.721468\pi\)
−0.640970 + 0.767566i \(0.721468\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.48563e9 −0.290528 −0.145264 − 0.989393i \(-0.546403\pi\)
−0.145264 + 0.989393i \(0.546403\pi\)
\(864\) 0 0
\(865\) −4.76529e9 −0.250341
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.96182e9 0.359877
\(870\) 0 0
\(871\) −2.18974e10 −1.12287
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.05469e9 −0.0532225
\(876\) 0 0
\(877\) −2.32316e10 −1.16300 −0.581501 − 0.813545i \(-0.697534\pi\)
−0.581501 + 0.813545i \(0.697534\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.50127e9 −0.468129 −0.234065 − 0.972221i \(-0.575203\pi\)
−0.234065 + 0.972221i \(0.575203\pi\)
\(882\) 0 0
\(883\) 3.56277e10 1.74151 0.870754 − 0.491719i \(-0.163631\pi\)
0.870754 + 0.491719i \(0.163631\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.18377e10 1.05069 0.525345 − 0.850889i \(-0.323936\pi\)
0.525345 + 0.850889i \(0.323936\pi\)
\(888\) 0 0
\(889\) −1.97375e9 −0.0942182
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.45721e10 0.684766
\(894\) 0 0
\(895\) −6.79064e9 −0.316614
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.49373e9 −0.389888
\(900\) 0 0
\(901\) −1.16310e9 −0.0529762
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.25117e9 −0.190651
\(906\) 0 0
\(907\) 2.40770e10 1.07146 0.535731 − 0.844389i \(-0.320036\pi\)
0.535731 + 0.844389i \(0.320036\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.93946e10 0.849899 0.424949 − 0.905217i \(-0.360292\pi\)
0.424949 + 0.905217i \(0.360292\pi\)
\(912\) 0 0
\(913\) 3.96271e9 0.172324
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.75853e10 0.753108
\(918\) 0 0
\(919\) 8.23845e9 0.350140 0.175070 − 0.984556i \(-0.443985\pi\)
0.175070 + 0.984556i \(0.443985\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.19450e10 0.500013
\(924\) 0 0
\(925\) −3.81809e9 −0.158617
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.77268e9 0.195302 0.0976512 − 0.995221i \(-0.468867\pi\)
0.0976512 + 0.995221i \(0.468867\pi\)
\(930\) 0 0
\(931\) 1.66370e10 0.675698
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.10504e10 −0.442116
\(936\) 0 0
\(937\) 3.00758e10 1.19434 0.597171 − 0.802114i \(-0.296291\pi\)
0.597171 + 0.802114i \(0.296291\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.31601e8 −0.00514869 −0.00257435 − 0.999997i \(-0.500819\pi\)
−0.00257435 + 0.999997i \(0.500819\pi\)
\(942\) 0 0
\(943\) −3.96136e8 −0.0153834
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.71764e10 1.42247 0.711234 − 0.702955i \(-0.248137\pi\)
0.711234 + 0.702955i \(0.248137\pi\)
\(948\) 0 0
\(949\) −2.41982e10 −0.919076
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.65884e10 0.620840 0.310420 − 0.950600i \(-0.399530\pi\)
0.310420 + 0.950600i \(0.399530\pi\)
\(954\) 0 0
\(955\) 1.22312e10 0.454419
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.38020e10 0.871461
\(960\) 0 0
\(961\) −2.56124e10 −0.930931
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.59019e10 0.569645
\(966\) 0 0
\(967\) 2.95263e10 1.05007 0.525033 − 0.851082i \(-0.324053\pi\)
0.525033 + 0.851082i \(0.324053\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.85110e10 1.34995 0.674975 − 0.737841i \(-0.264155\pi\)
0.674975 + 0.737841i \(0.264155\pi\)
\(972\) 0 0
\(973\) 2.71382e10 0.944466
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.65181e9 0.296808 0.148404 − 0.988927i \(-0.452586\pi\)
0.148404 + 0.988927i \(0.452586\pi\)
\(978\) 0 0
\(979\) 5.24196e7 0.00178548
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.83575e10 −0.952205 −0.476102 − 0.879390i \(-0.657951\pi\)
−0.476102 + 0.879390i \(0.657951\pi\)
\(984\) 0 0
\(985\) −1.03933e10 −0.346519
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.36688e9 0.0778017
\(990\) 0 0
\(991\) −5.25975e10 −1.71675 −0.858375 − 0.513023i \(-0.828526\pi\)
−0.858375 + 0.513023i \(0.828526\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.41503e8 0.0270816
\(996\) 0 0
\(997\) −3.11108e10 −0.994210 −0.497105 − 0.867690i \(-0.665604\pi\)
−0.497105 + 0.867690i \(0.665604\pi\)
\(998\) 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 360.8.a.c.1.1 1
3.2 odd 2 120.8.a.a.1.1 ✓ 1
12.11 even 2 240.8.a.d.1.1 1
15.2 even 4 600.8.f.c.49.1 2
15.8 even 4 600.8.f.c.49.2 2
15.14 odd 2 600.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.8.a.a.1.1 ✓ 1 3.2 odd 2
240.8.a.d.1.1 1 12.11 even 2
360.8.a.c.1.1 1 1.1 even 1 trivial
600.8.a.c.1.1 1 15.14 odd 2
600.8.f.c.49.1 2 15.2 even 4
600.8.f.c.49.2 2 15.8 even 4