Properties

Label 384.9.g.a.127.16
Level $384$
Weight $9$
Character 384.127
Analytic conductor $156.433$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,9,Mod(127,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.16
Character \(\chi\) \(=\) 384.127
Dual form 384.9.g.a.127.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+46.7654i q^{3} -19.7383 q^{5} +759.796i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q+46.7654i q^{3} -19.7383 q^{5} +759.796i q^{7} -2187.00 q^{9} -15847.0i q^{11} -47667.3 q^{13} -923.068i q^{15} -117951. q^{17} +196795. i q^{19} -35532.1 q^{21} -143587. i q^{23} -390235. q^{25} -102276. i q^{27} +1.30803e6 q^{29} +1.08127e6i q^{31} +741092. q^{33} -14997.1i q^{35} -2.61456e6 q^{37} -2.22918e6i q^{39} +956143. q^{41} -4.78185e6i q^{43} +43167.6 q^{45} -2.03435e6i q^{47} +5.18751e6 q^{49} -5.51603e6i q^{51} +4.50474e6 q^{53} +312793. i q^{55} -9.20320e6 q^{57} +1.40186e7i q^{59} -2.05420e7 q^{61} -1.66167e6i q^{63} +940872. q^{65} +2.77586e7i q^{67} +6.71488e6 q^{69} -3.22968e7i q^{71} +2.34885e7 q^{73} -1.82495e7i q^{75} +1.20405e7 q^{77} +1.08547e7i q^{79} +4.78297e6 q^{81} -7.99186e7i q^{83} +2.32815e6 q^{85} +6.11706e7i q^{87} +7.46098e7 q^{89} -3.62174e7i q^{91} -5.05661e7 q^{93} -3.88440e6i q^{95} -5.73670e7 q^{97} +3.46575e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 46.7654i 0.577350i
\(4\) 0 0
\(5\) −19.7383 −0.0315813 −0.0157906 0.999875i \(-0.505027\pi\)
−0.0157906 + 0.999875i \(0.505027\pi\)
\(6\) 0 0
\(7\) 759.796i 0.316450i 0.987403 + 0.158225i \(0.0505771\pi\)
−0.987403 + 0.158225i \(0.949423\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) − 15847.0i − 1.08237i −0.840902 0.541187i \(-0.817975\pi\)
0.840902 0.541187i \(-0.182025\pi\)
\(12\) 0 0
\(13\) −47667.3 −1.66897 −0.834483 0.551034i \(-0.814234\pi\)
−0.834483 + 0.551034i \(0.814234\pi\)
\(14\) 0 0
\(15\) − 923.068i − 0.0182335i
\(16\) 0 0
\(17\) −117951. −1.41223 −0.706117 0.708095i \(-0.749555\pi\)
−0.706117 + 0.708095i \(0.749555\pi\)
\(18\) 0 0
\(19\) 196795.i 1.51008i 0.655678 + 0.755041i \(0.272383\pi\)
−0.655678 + 0.755041i \(0.727617\pi\)
\(20\) 0 0
\(21\) −35532.1 −0.182702
\(22\) 0 0
\(23\) − 143587.i − 0.513101i −0.966531 0.256550i \(-0.917414\pi\)
0.966531 0.256550i \(-0.0825859\pi\)
\(24\) 0 0
\(25\) −390235. −0.999003
\(26\) 0 0
\(27\) − 102276.i − 0.192450i
\(28\) 0 0
\(29\) 1.30803e6 1.84938 0.924690 0.380720i \(-0.124324\pi\)
0.924690 + 0.380720i \(0.124324\pi\)
\(30\) 0 0
\(31\) 1.08127e6i 1.17081i 0.810740 + 0.585407i \(0.199065\pi\)
−0.810740 + 0.585407i \(0.800935\pi\)
\(32\) 0 0
\(33\) 741092. 0.624909
\(34\) 0 0
\(35\) − 14997.1i − 0.00999388i
\(36\) 0 0
\(37\) −2.61456e6 −1.39506 −0.697528 0.716558i \(-0.745717\pi\)
−0.697528 + 0.716558i \(0.745717\pi\)
\(38\) 0 0
\(39\) − 2.22918e6i − 0.963578i
\(40\) 0 0
\(41\) 956143. 0.338367 0.169183 0.985585i \(-0.445887\pi\)
0.169183 + 0.985585i \(0.445887\pi\)
\(42\) 0 0
\(43\) − 4.78185e6i − 1.39869i −0.714783 0.699346i \(-0.753475\pi\)
0.714783 0.699346i \(-0.246525\pi\)
\(44\) 0 0
\(45\) 43167.6 0.0105271
\(46\) 0 0
\(47\) − 2.03435e6i − 0.416903i −0.978033 0.208451i \(-0.933158\pi\)
0.978033 0.208451i \(-0.0668423\pi\)
\(48\) 0 0
\(49\) 5.18751e6 0.899860
\(50\) 0 0
\(51\) − 5.51603e6i − 0.815353i
\(52\) 0 0
\(53\) 4.50474e6 0.570908 0.285454 0.958392i \(-0.407856\pi\)
0.285454 + 0.958392i \(0.407856\pi\)
\(54\) 0 0
\(55\) 312793.i 0.0341827i
\(56\) 0 0
\(57\) −9.20320e6 −0.871846
\(58\) 0 0
\(59\) 1.40186e7i 1.15690i 0.815718 + 0.578449i \(0.196342\pi\)
−0.815718 + 0.578449i \(0.803658\pi\)
\(60\) 0 0
\(61\) −2.05420e7 −1.48362 −0.741812 0.670608i \(-0.766033\pi\)
−0.741812 + 0.670608i \(0.766033\pi\)
\(62\) 0 0
\(63\) − 1.66167e6i − 0.105483i
\(64\) 0 0
\(65\) 940872. 0.0527080
\(66\) 0 0
\(67\) 2.77586e7i 1.37752i 0.724989 + 0.688760i \(0.241845\pi\)
−0.724989 + 0.688760i \(0.758155\pi\)
\(68\) 0 0
\(69\) 6.71488e6 0.296239
\(70\) 0 0
\(71\) − 3.22968e7i − 1.27094i −0.772125 0.635471i \(-0.780806\pi\)
0.772125 0.635471i \(-0.219194\pi\)
\(72\) 0 0
\(73\) 2.34885e7 0.827112 0.413556 0.910479i \(-0.364287\pi\)
0.413556 + 0.910479i \(0.364287\pi\)
\(74\) 0 0
\(75\) − 1.82495e7i − 0.576774i
\(76\) 0 0
\(77\) 1.20405e7 0.342517
\(78\) 0 0
\(79\) 1.08547e7i 0.278683i 0.990244 + 0.139341i \(0.0444985\pi\)
−0.990244 + 0.139341i \(0.955502\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) − 7.99186e7i − 1.68398i −0.539497 0.841988i \(-0.681386\pi\)
0.539497 0.841988i \(-0.318614\pi\)
\(84\) 0 0
\(85\) 2.32815e6 0.0446001
\(86\) 0 0
\(87\) 6.11706e7i 1.06774i
\(88\) 0 0
\(89\) 7.46098e7 1.18915 0.594574 0.804041i \(-0.297321\pi\)
0.594574 + 0.804041i \(0.297321\pi\)
\(90\) 0 0
\(91\) − 3.62174e7i − 0.528144i
\(92\) 0 0
\(93\) −5.05661e7 −0.675970
\(94\) 0 0
\(95\) − 3.88440e6i − 0.0476903i
\(96\) 0 0
\(97\) −5.73670e7 −0.648001 −0.324000 0.946057i \(-0.605028\pi\)
−0.324000 + 0.946057i \(0.605028\pi\)
\(98\) 0 0
\(99\) 3.46575e7i 0.360791i
\(100\) 0 0
\(101\) −9.65501e7 −0.927827 −0.463914 0.885881i \(-0.653555\pi\)
−0.463914 + 0.885881i \(0.653555\pi\)
\(102\) 0 0
\(103\) − 1.41970e8i − 1.26138i −0.776033 0.630692i \(-0.782771\pi\)
0.776033 0.630692i \(-0.217229\pi\)
\(104\) 0 0
\(105\) 701343. 0.00576997
\(106\) 0 0
\(107\) 6.39218e7i 0.487656i 0.969818 + 0.243828i \(0.0784033\pi\)
−0.969818 + 0.243828i \(0.921597\pi\)
\(108\) 0 0
\(109\) 1.31267e8 0.929932 0.464966 0.885329i \(-0.346067\pi\)
0.464966 + 0.885329i \(0.346067\pi\)
\(110\) 0 0
\(111\) − 1.22271e8i − 0.805436i
\(112\) 0 0
\(113\) 1.49881e8 0.919246 0.459623 0.888114i \(-0.347985\pi\)
0.459623 + 0.888114i \(0.347985\pi\)
\(114\) 0 0
\(115\) 2.83415e6i 0.0162044i
\(116\) 0 0
\(117\) 1.04248e8 0.556322
\(118\) 0 0
\(119\) − 8.96188e7i − 0.446901i
\(120\) 0 0
\(121\) −3.67695e7 −0.171533
\(122\) 0 0
\(123\) 4.47144e7i 0.195356i
\(124\) 0 0
\(125\) 1.54128e7 0.0631310
\(126\) 0 0
\(127\) − 1.99191e8i − 0.765695i −0.923812 0.382848i \(-0.874943\pi\)
0.923812 0.382848i \(-0.125057\pi\)
\(128\) 0 0
\(129\) 2.23625e8 0.807536
\(130\) 0 0
\(131\) 3.03185e8i 1.02949i 0.857343 + 0.514746i \(0.172114\pi\)
−0.857343 + 0.514746i \(0.827886\pi\)
\(132\) 0 0
\(133\) −1.49524e8 −0.477865
\(134\) 0 0
\(135\) 2.01875e6i 0.00607782i
\(136\) 0 0
\(137\) −3.76126e8 −1.06770 −0.533852 0.845578i \(-0.679256\pi\)
−0.533852 + 0.845578i \(0.679256\pi\)
\(138\) 0 0
\(139\) − 2.37171e8i − 0.635334i −0.948202 0.317667i \(-0.897101\pi\)
0.948202 0.317667i \(-0.102899\pi\)
\(140\) 0 0
\(141\) 9.51372e7 0.240699
\(142\) 0 0
\(143\) 7.55386e8i 1.80644i
\(144\) 0 0
\(145\) −2.58183e7 −0.0584058
\(146\) 0 0
\(147\) 2.42596e8i 0.519534i
\(148\) 0 0
\(149\) 8.73173e8 1.77156 0.885779 0.464108i \(-0.153625\pi\)
0.885779 + 0.464108i \(0.153625\pi\)
\(150\) 0 0
\(151\) 1.26207e6i 0.00242758i 0.999999 + 0.00121379i \(0.000386362\pi\)
−0.999999 + 0.00121379i \(0.999614\pi\)
\(152\) 0 0
\(153\) 2.57959e8 0.470745
\(154\) 0 0
\(155\) − 2.13424e7i − 0.0369758i
\(156\) 0 0
\(157\) 5.79730e8 0.954174 0.477087 0.878856i \(-0.341693\pi\)
0.477087 + 0.878856i \(0.341693\pi\)
\(158\) 0 0
\(159\) 2.10666e8i 0.329614i
\(160\) 0 0
\(161\) 1.09096e8 0.162370
\(162\) 0 0
\(163\) 1.36798e9i 1.93790i 0.247264 + 0.968948i \(0.420468\pi\)
−0.247264 + 0.968948i \(0.579532\pi\)
\(164\) 0 0
\(165\) −1.46279e7 −0.0197354
\(166\) 0 0
\(167\) − 9.31645e8i − 1.19780i −0.800824 0.598900i \(-0.795605\pi\)
0.800824 0.598900i \(-0.204395\pi\)
\(168\) 0 0
\(169\) 1.45644e9 1.78545
\(170\) 0 0
\(171\) − 4.30391e8i − 0.503360i
\(172\) 0 0
\(173\) −6.08081e8 −0.678856 −0.339428 0.940632i \(-0.610233\pi\)
−0.339428 + 0.940632i \(0.610233\pi\)
\(174\) 0 0
\(175\) − 2.96499e8i − 0.316134i
\(176\) 0 0
\(177\) −6.55583e8 −0.667936
\(178\) 0 0
\(179\) 1.47872e8i 0.144037i 0.997403 + 0.0720184i \(0.0229440\pi\)
−0.997403 + 0.0720184i \(0.977056\pi\)
\(180\) 0 0
\(181\) 6.89805e8 0.642705 0.321353 0.946960i \(-0.395863\pi\)
0.321353 + 0.946960i \(0.395863\pi\)
\(182\) 0 0
\(183\) − 9.60655e8i − 0.856571i
\(184\) 0 0
\(185\) 5.16069e7 0.0440576
\(186\) 0 0
\(187\) 1.86918e9i 1.52856i
\(188\) 0 0
\(189\) 7.77088e7 0.0609008
\(190\) 0 0
\(191\) − 1.82722e9i − 1.37296i −0.727148 0.686481i \(-0.759155\pi\)
0.727148 0.686481i \(-0.240845\pi\)
\(192\) 0 0
\(193\) 4.57534e8 0.329757 0.164879 0.986314i \(-0.447277\pi\)
0.164879 + 0.986314i \(0.447277\pi\)
\(194\) 0 0
\(195\) 4.40002e7i 0.0304310i
\(196\) 0 0
\(197\) 4.22625e8 0.280601 0.140301 0.990109i \(-0.455193\pi\)
0.140301 + 0.990109i \(0.455193\pi\)
\(198\) 0 0
\(199\) 4.66840e8i 0.297684i 0.988861 + 0.148842i \(0.0475546\pi\)
−0.988861 + 0.148842i \(0.952445\pi\)
\(200\) 0 0
\(201\) −1.29814e9 −0.795312
\(202\) 0 0
\(203\) 9.93837e8i 0.585236i
\(204\) 0 0
\(205\) −1.88726e7 −0.0106860
\(206\) 0 0
\(207\) 3.14024e8i 0.171034i
\(208\) 0 0
\(209\) 3.11862e9 1.63447
\(210\) 0 0
\(211\) − 1.16847e9i − 0.589505i −0.955574 0.294753i \(-0.904763\pi\)
0.955574 0.294753i \(-0.0952373\pi\)
\(212\) 0 0
\(213\) 1.51037e9 0.733779
\(214\) 0 0
\(215\) 9.43856e7i 0.0441725i
\(216\) 0 0
\(217\) −8.21545e8 −0.370504
\(218\) 0 0
\(219\) 1.09845e9i 0.477533i
\(220\) 0 0
\(221\) 5.62242e9 2.35697
\(222\) 0 0
\(223\) 5.79628e8i 0.234385i 0.993109 + 0.117192i \(0.0373894\pi\)
−0.993109 + 0.117192i \(0.962611\pi\)
\(224\) 0 0
\(225\) 8.53445e8 0.333001
\(226\) 0 0
\(227\) 4.62864e9i 1.74321i 0.490207 + 0.871606i \(0.336921\pi\)
−0.490207 + 0.871606i \(0.663079\pi\)
\(228\) 0 0
\(229\) 3.38866e9 1.23221 0.616107 0.787663i \(-0.288709\pi\)
0.616107 + 0.787663i \(0.288709\pi\)
\(230\) 0 0
\(231\) 5.63079e8i 0.197752i
\(232\) 0 0
\(233\) 2.20266e9 0.747351 0.373675 0.927560i \(-0.378097\pi\)
0.373675 + 0.927560i \(0.378097\pi\)
\(234\) 0 0
\(235\) 4.01546e7i 0.0131663i
\(236\) 0 0
\(237\) −5.07624e8 −0.160897
\(238\) 0 0
\(239\) − 4.26194e9i − 1.30622i −0.757264 0.653108i \(-0.773465\pi\)
0.757264 0.653108i \(-0.226535\pi\)
\(240\) 0 0
\(241\) 4.80122e9 1.42326 0.711628 0.702556i \(-0.247958\pi\)
0.711628 + 0.702556i \(0.247958\pi\)
\(242\) 0 0
\(243\) 2.23677e8i 0.0641500i
\(244\) 0 0
\(245\) −1.02393e8 −0.0284187
\(246\) 0 0
\(247\) − 9.38071e9i − 2.52027i
\(248\) 0 0
\(249\) 3.73742e9 0.972244
\(250\) 0 0
\(251\) − 1.15369e9i − 0.290667i −0.989383 0.145333i \(-0.953575\pi\)
0.989383 0.145333i \(-0.0464255\pi\)
\(252\) 0 0
\(253\) −2.27542e9 −0.555366
\(254\) 0 0
\(255\) 1.08877e8i 0.0257499i
\(256\) 0 0
\(257\) 6.22177e8 0.142620 0.0713102 0.997454i \(-0.477282\pi\)
0.0713102 + 0.997454i \(0.477282\pi\)
\(258\) 0 0
\(259\) − 1.98653e9i − 0.441465i
\(260\) 0 0
\(261\) −2.86067e9 −0.616460
\(262\) 0 0
\(263\) − 2.82148e9i − 0.589732i −0.955539 0.294866i \(-0.904725\pi\)
0.955539 0.294866i \(-0.0952750\pi\)
\(264\) 0 0
\(265\) −8.89159e7 −0.0180300
\(266\) 0 0
\(267\) 3.48915e9i 0.686555i
\(268\) 0 0
\(269\) −2.33848e9 −0.446605 −0.223303 0.974749i \(-0.571684\pi\)
−0.223303 + 0.974749i \(0.571684\pi\)
\(270\) 0 0
\(271\) − 1.92807e9i − 0.357475i −0.983897 0.178737i \(-0.942799\pi\)
0.983897 0.178737i \(-0.0572012\pi\)
\(272\) 0 0
\(273\) 1.69372e9 0.304924
\(274\) 0 0
\(275\) 6.18407e9i 1.08129i
\(276\) 0 0
\(277\) 3.15707e9 0.536248 0.268124 0.963384i \(-0.413596\pi\)
0.268124 + 0.963384i \(0.413596\pi\)
\(278\) 0 0
\(279\) − 2.36474e9i − 0.390271i
\(280\) 0 0
\(281\) 2.57086e9 0.412338 0.206169 0.978516i \(-0.433900\pi\)
0.206169 + 0.978516i \(0.433900\pi\)
\(282\) 0 0
\(283\) − 1.59550e9i − 0.248743i −0.992236 0.124372i \(-0.960308\pi\)
0.992236 0.124372i \(-0.0396915\pi\)
\(284\) 0 0
\(285\) 1.81656e8 0.0275340
\(286\) 0 0
\(287\) 7.26473e8i 0.107076i
\(288\) 0 0
\(289\) 6.93672e9 0.994404
\(290\) 0 0
\(291\) − 2.68279e9i − 0.374123i
\(292\) 0 0
\(293\) 4.12015e9 0.559039 0.279520 0.960140i \(-0.409825\pi\)
0.279520 + 0.960140i \(0.409825\pi\)
\(294\) 0 0
\(295\) − 2.76702e8i − 0.0365363i
\(296\) 0 0
\(297\) −1.62077e9 −0.208303
\(298\) 0 0
\(299\) 6.84439e9i 0.856347i
\(300\) 0 0
\(301\) 3.63323e9 0.442616
\(302\) 0 0
\(303\) − 4.51520e9i − 0.535681i
\(304\) 0 0
\(305\) 4.05464e8 0.0468547
\(306\) 0 0
\(307\) 1.53448e10i 1.72746i 0.503954 + 0.863731i \(0.331878\pi\)
−0.503954 + 0.863731i \(0.668122\pi\)
\(308\) 0 0
\(309\) 6.63928e9 0.728261
\(310\) 0 0
\(311\) 8.41711e9i 0.899749i 0.893092 + 0.449875i \(0.148531\pi\)
−0.893092 + 0.449875i \(0.851469\pi\)
\(312\) 0 0
\(313\) 5.18210e9 0.539918 0.269959 0.962872i \(-0.412990\pi\)
0.269959 + 0.962872i \(0.412990\pi\)
\(314\) 0 0
\(315\) 3.27986e7i 0.00333129i
\(316\) 0 0
\(317\) −4.31162e9 −0.426976 −0.213488 0.976946i \(-0.568482\pi\)
−0.213488 + 0.976946i \(0.568482\pi\)
\(318\) 0 0
\(319\) − 2.07284e10i − 2.00172i
\(320\) 0 0
\(321\) −2.98933e9 −0.281548
\(322\) 0 0
\(323\) − 2.32122e10i − 2.13259i
\(324\) 0 0
\(325\) 1.86015e10 1.66730
\(326\) 0 0
\(327\) 6.13877e9i 0.536896i
\(328\) 0 0
\(329\) 1.54569e9 0.131929
\(330\) 0 0
\(331\) 7.84383e9i 0.653456i 0.945118 + 0.326728i \(0.105946\pi\)
−0.945118 + 0.326728i \(0.894054\pi\)
\(332\) 0 0
\(333\) 5.71804e9 0.465018
\(334\) 0 0
\(335\) − 5.47907e8i − 0.0435039i
\(336\) 0 0
\(337\) 1.60451e10 1.24401 0.622005 0.783013i \(-0.286318\pi\)
0.622005 + 0.783013i \(0.286318\pi\)
\(338\) 0 0
\(339\) 7.00922e9i 0.530727i
\(340\) 0 0
\(341\) 1.71349e10 1.26726
\(342\) 0 0
\(343\) 8.32152e9i 0.601210i
\(344\) 0 0
\(345\) −1.32540e8 −0.00935559
\(346\) 0 0
\(347\) 1.36821e10i 0.943700i 0.881679 + 0.471850i \(0.156414\pi\)
−0.881679 + 0.471850i \(0.843586\pi\)
\(348\) 0 0
\(349\) −2.11356e10 −1.42466 −0.712332 0.701843i \(-0.752361\pi\)
−0.712332 + 0.701843i \(0.752361\pi\)
\(350\) 0 0
\(351\) 4.87522e9i 0.321193i
\(352\) 0 0
\(353\) 1.47373e10 0.949112 0.474556 0.880225i \(-0.342609\pi\)
0.474556 + 0.880225i \(0.342609\pi\)
\(354\) 0 0
\(355\) 6.37483e8i 0.0401379i
\(356\) 0 0
\(357\) 4.19106e9 0.258018
\(358\) 0 0
\(359\) − 2.49699e10i − 1.50328i −0.659575 0.751638i \(-0.729264\pi\)
0.659575 0.751638i \(-0.270736\pi\)
\(360\) 0 0
\(361\) −2.17448e10 −1.28034
\(362\) 0 0
\(363\) − 1.71954e9i − 0.0990344i
\(364\) 0 0
\(365\) −4.63623e8 −0.0261212
\(366\) 0 0
\(367\) 1.11559e10i 0.614951i 0.951556 + 0.307475i \(0.0994842\pi\)
−0.951556 + 0.307475i \(0.900516\pi\)
\(368\) 0 0
\(369\) −2.09108e9 −0.112789
\(370\) 0 0
\(371\) 3.42268e9i 0.180664i
\(372\) 0 0
\(373\) −3.65556e10 −1.88851 −0.944253 0.329221i \(-0.893214\pi\)
−0.944253 + 0.329221i \(0.893214\pi\)
\(374\) 0 0
\(375\) 7.20788e8i 0.0364487i
\(376\) 0 0
\(377\) −6.23504e10 −3.08655
\(378\) 0 0
\(379\) 2.16949e9i 0.105148i 0.998617 + 0.0525740i \(0.0167425\pi\)
−0.998617 + 0.0525740i \(0.983257\pi\)
\(380\) 0 0
\(381\) 9.31526e9 0.442074
\(382\) 0 0
\(383\) − 8.10758e9i − 0.376787i −0.982094 0.188394i \(-0.939672\pi\)
0.982094 0.188394i \(-0.0603280\pi\)
\(384\) 0 0
\(385\) −2.37659e8 −0.0108171
\(386\) 0 0
\(387\) 1.04579e10i 0.466231i
\(388\) 0 0
\(389\) 1.73636e10 0.758300 0.379150 0.925335i \(-0.376217\pi\)
0.379150 + 0.925335i \(0.376217\pi\)
\(390\) 0 0
\(391\) 1.69362e10i 0.724618i
\(392\) 0 0
\(393\) −1.41786e10 −0.594377
\(394\) 0 0
\(395\) − 2.14253e8i − 0.00880115i
\(396\) 0 0
\(397\) −7.49508e9 −0.301727 −0.150864 0.988555i \(-0.548205\pi\)
−0.150864 + 0.988555i \(0.548205\pi\)
\(398\) 0 0
\(399\) − 6.99255e9i − 0.275895i
\(400\) 0 0
\(401\) 3.17286e10 1.22708 0.613541 0.789663i \(-0.289745\pi\)
0.613541 + 0.789663i \(0.289745\pi\)
\(402\) 0 0
\(403\) − 5.15413e10i − 1.95405i
\(404\) 0 0
\(405\) −9.44076e7 −0.00350903
\(406\) 0 0
\(407\) 4.14330e10i 1.50997i
\(408\) 0 0
\(409\) 3.02548e10 1.08119 0.540595 0.841283i \(-0.318199\pi\)
0.540595 + 0.841283i \(0.318199\pi\)
\(410\) 0 0
\(411\) − 1.75897e10i − 0.616440i
\(412\) 0 0
\(413\) −1.06512e10 −0.366100
\(414\) 0 0
\(415\) 1.57746e9i 0.0531821i
\(416\) 0 0
\(417\) 1.10914e10 0.366810
\(418\) 0 0
\(419\) 3.95924e10i 1.28456i 0.766468 + 0.642282i \(0.222012\pi\)
−0.766468 + 0.642282i \(0.777988\pi\)
\(420\) 0 0
\(421\) 3.41632e10 1.08750 0.543751 0.839247i \(-0.317004\pi\)
0.543751 + 0.839247i \(0.317004\pi\)
\(422\) 0 0
\(423\) 4.44913e9i 0.138968i
\(424\) 0 0
\(425\) 4.60287e10 1.41083
\(426\) 0 0
\(427\) − 1.56077e10i − 0.469492i
\(428\) 0 0
\(429\) −3.53259e10 −1.04295
\(430\) 0 0
\(431\) 4.74352e10i 1.37465i 0.726351 + 0.687324i \(0.241215\pi\)
−0.726351 + 0.687324i \(0.758785\pi\)
\(432\) 0 0
\(433\) −3.46150e10 −0.984720 −0.492360 0.870392i \(-0.663866\pi\)
−0.492360 + 0.870392i \(0.663866\pi\)
\(434\) 0 0
\(435\) − 1.20740e9i − 0.0337206i
\(436\) 0 0
\(437\) 2.82572e10 0.774823
\(438\) 0 0
\(439\) − 3.47839e10i − 0.936525i −0.883589 0.468263i \(-0.844880\pi\)
0.883589 0.468263i \(-0.155120\pi\)
\(440\) 0 0
\(441\) −1.13451e10 −0.299953
\(442\) 0 0
\(443\) 2.71982e10i 0.706196i 0.935586 + 0.353098i \(0.114872\pi\)
−0.935586 + 0.353098i \(0.885128\pi\)
\(444\) 0 0
\(445\) −1.47267e9 −0.0375548
\(446\) 0 0
\(447\) 4.08343e10i 1.02281i
\(448\) 0 0
\(449\) −6.44839e10 −1.58659 −0.793297 0.608834i \(-0.791637\pi\)
−0.793297 + 0.608834i \(0.791637\pi\)
\(450\) 0 0
\(451\) − 1.51520e10i − 0.366239i
\(452\) 0 0
\(453\) −5.90210e7 −0.00140157
\(454\) 0 0
\(455\) 7.14870e8i 0.0166794i
\(456\) 0 0
\(457\) −7.83314e10 −1.79586 −0.897928 0.440143i \(-0.854928\pi\)
−0.897928 + 0.440143i \(0.854928\pi\)
\(458\) 0 0
\(459\) 1.20636e10i 0.271784i
\(460\) 0 0
\(461\) −7.86144e10 −1.74060 −0.870299 0.492525i \(-0.836074\pi\)
−0.870299 + 0.492525i \(0.836074\pi\)
\(462\) 0 0
\(463\) − 3.70967e10i − 0.807257i −0.914923 0.403628i \(-0.867749\pi\)
0.914923 0.403628i \(-0.132251\pi\)
\(464\) 0 0
\(465\) 9.98088e8 0.0213480
\(466\) 0 0
\(467\) − 4.80694e10i − 1.01065i −0.862929 0.505325i \(-0.831373\pi\)
0.862929 0.505325i \(-0.168627\pi\)
\(468\) 0 0
\(469\) −2.10909e10 −0.435916
\(470\) 0 0
\(471\) 2.71113e10i 0.550892i
\(472\) 0 0
\(473\) −7.57782e10 −1.51391
\(474\) 0 0
\(475\) − 7.67965e10i − 1.50857i
\(476\) 0 0
\(477\) −9.85187e9 −0.190303
\(478\) 0 0
\(479\) 6.39132e10i 1.21408i 0.794670 + 0.607042i \(0.207644\pi\)
−0.794670 + 0.607042i \(0.792356\pi\)
\(480\) 0 0
\(481\) 1.24629e11 2.32830
\(482\) 0 0
\(483\) 5.10194e9i 0.0937447i
\(484\) 0 0
\(485\) 1.13233e9 0.0204647
\(486\) 0 0
\(487\) − 2.04932e10i − 0.364329i −0.983268 0.182164i \(-0.941690\pi\)
0.983268 0.182164i \(-0.0583103\pi\)
\(488\) 0 0
\(489\) −6.39743e10 −1.11884
\(490\) 0 0
\(491\) − 9.60306e10i − 1.65228i −0.563464 0.826140i \(-0.690532\pi\)
0.563464 0.826140i \(-0.309468\pi\)
\(492\) 0 0
\(493\) −1.54284e11 −2.61176
\(494\) 0 0
\(495\) − 6.84079e8i − 0.0113942i
\(496\) 0 0
\(497\) 2.45389e10 0.402189
\(498\) 0 0
\(499\) 1.03307e9i 0.0166620i 0.999965 + 0.00833099i \(0.00265187\pi\)
−0.999965 + 0.00833099i \(0.997348\pi\)
\(500\) 0 0
\(501\) 4.35687e10 0.691550
\(502\) 0 0
\(503\) − 2.88572e10i − 0.450798i −0.974267 0.225399i \(-0.927632\pi\)
0.974267 0.225399i \(-0.0723685\pi\)
\(504\) 0 0
\(505\) 1.90573e9 0.0293020
\(506\) 0 0
\(507\) 6.81111e10i 1.03083i
\(508\) 0 0
\(509\) −4.12139e10 −0.614005 −0.307003 0.951709i \(-0.599326\pi\)
−0.307003 + 0.951709i \(0.599326\pi\)
\(510\) 0 0
\(511\) 1.78465e10i 0.261739i
\(512\) 0 0
\(513\) 2.01274e10 0.290615
\(514\) 0 0
\(515\) 2.80224e9i 0.0398361i
\(516\) 0 0
\(517\) −3.22384e10 −0.451244
\(518\) 0 0
\(519\) − 2.84372e10i − 0.391937i
\(520\) 0 0
\(521\) −7.02598e10 −0.953577 −0.476789 0.879018i \(-0.658199\pi\)
−0.476789 + 0.879018i \(0.658199\pi\)
\(522\) 0 0
\(523\) − 1.04627e11i − 1.39842i −0.714918 0.699208i \(-0.753536\pi\)
0.714918 0.699208i \(-0.246464\pi\)
\(524\) 0 0
\(525\) 1.38659e10 0.182520
\(526\) 0 0
\(527\) − 1.27537e11i − 1.65346i
\(528\) 0 0
\(529\) 5.76939e10 0.736728
\(530\) 0 0
\(531\) − 3.06586e10i − 0.385633i
\(532\) 0 0
\(533\) −4.55768e10 −0.564722
\(534\) 0 0
\(535\) − 1.26171e9i − 0.0154008i
\(536\) 0 0
\(537\) −6.91529e9 −0.0831597
\(538\) 0 0
\(539\) − 8.22067e10i − 0.973984i
\(540\) 0 0
\(541\) −1.70512e10 −0.199052 −0.0995260 0.995035i \(-0.531733\pi\)
−0.0995260 + 0.995035i \(0.531733\pi\)
\(542\) 0 0
\(543\) 3.22590e10i 0.371066i
\(544\) 0 0
\(545\) −2.59099e9 −0.0293684
\(546\) 0 0
\(547\) − 8.74356e10i − 0.976650i −0.872662 0.488325i \(-0.837608\pi\)
0.872662 0.488325i \(-0.162392\pi\)
\(548\) 0 0
\(549\) 4.49254e10 0.494541
\(550\) 0 0
\(551\) 2.57415e11i 2.79271i
\(552\) 0 0
\(553\) −8.24736e9 −0.0881890
\(554\) 0 0
\(555\) 2.41342e9i 0.0254367i
\(556\) 0 0
\(557\) −3.25190e10 −0.337844 −0.168922 0.985629i \(-0.554029\pi\)
−0.168922 + 0.985629i \(0.554029\pi\)
\(558\) 0 0
\(559\) 2.27938e11i 2.33437i
\(560\) 0 0
\(561\) −8.74127e10 −0.882517
\(562\) 0 0
\(563\) − 2.12615e10i − 0.211621i −0.994386 0.105811i \(-0.966256\pi\)
0.994386 0.105811i \(-0.0337438\pi\)
\(564\) 0 0
\(565\) −2.95839e9 −0.0290309
\(566\) 0 0
\(567\) 3.63408e9i 0.0351611i
\(568\) 0 0
\(569\) −1.50487e11 −1.43566 −0.717830 0.696219i \(-0.754864\pi\)
−0.717830 + 0.696219i \(0.754864\pi\)
\(570\) 0 0
\(571\) 5.23675e9i 0.0492626i 0.999697 + 0.0246313i \(0.00784118\pi\)
−0.999697 + 0.0246313i \(0.992159\pi\)
\(572\) 0 0
\(573\) 8.54508e10 0.792679
\(574\) 0 0
\(575\) 5.60326e10i 0.512589i
\(576\) 0 0
\(577\) 7.74601e9 0.0698835 0.0349418 0.999389i \(-0.488875\pi\)
0.0349418 + 0.999389i \(0.488875\pi\)
\(578\) 0 0
\(579\) 2.13968e10i 0.190385i
\(580\) 0 0
\(581\) 6.07218e10 0.532893
\(582\) 0 0
\(583\) − 7.13868e10i − 0.617936i
\(584\) 0 0
\(585\) −2.05769e9 −0.0175693
\(586\) 0 0
\(587\) 1.50562e11i 1.26813i 0.773279 + 0.634066i \(0.218615\pi\)
−0.773279 + 0.634066i \(0.781385\pi\)
\(588\) 0 0
\(589\) −2.12789e11 −1.76802
\(590\) 0 0
\(591\) 1.97642e10i 0.162005i
\(592\) 0 0
\(593\) 4.15664e10 0.336143 0.168071 0.985775i \(-0.446246\pi\)
0.168071 + 0.985775i \(0.446246\pi\)
\(594\) 0 0
\(595\) 1.76892e9i 0.0141137i
\(596\) 0 0
\(597\) −2.18320e10 −0.171868
\(598\) 0 0
\(599\) 6.23752e10i 0.484512i 0.970212 + 0.242256i \(0.0778874\pi\)
−0.970212 + 0.242256i \(0.922113\pi\)
\(600\) 0 0
\(601\) −1.01687e11 −0.779414 −0.389707 0.920939i \(-0.627424\pi\)
−0.389707 + 0.920939i \(0.627424\pi\)
\(602\) 0 0
\(603\) − 6.07080e10i − 0.459174i
\(604\) 0 0
\(605\) 7.25768e8 0.00541722
\(606\) 0 0
\(607\) − 2.50543e11i − 1.84555i −0.385334 0.922777i \(-0.625914\pi\)
0.385334 0.922777i \(-0.374086\pi\)
\(608\) 0 0
\(609\) −4.64772e10 −0.337886
\(610\) 0 0
\(611\) 9.69721e10i 0.695796i
\(612\) 0 0
\(613\) −1.28165e11 −0.907667 −0.453833 0.891087i \(-0.649944\pi\)
−0.453833 + 0.891087i \(0.649944\pi\)
\(614\) 0 0
\(615\) − 8.82585e8i − 0.00616959i
\(616\) 0 0
\(617\) −1.17897e11 −0.813510 −0.406755 0.913537i \(-0.633340\pi\)
−0.406755 + 0.913537i \(0.633340\pi\)
\(618\) 0 0
\(619\) − 4.79933e10i − 0.326902i −0.986551 0.163451i \(-0.947737\pi\)
0.986551 0.163451i \(-0.0522626\pi\)
\(620\) 0 0
\(621\) −1.46854e10 −0.0987462
\(622\) 0 0
\(623\) 5.66882e10i 0.376305i
\(624\) 0 0
\(625\) 1.52131e11 0.997009
\(626\) 0 0
\(627\) 1.45843e11i 0.943663i
\(628\) 0 0
\(629\) 3.08390e11 1.97014
\(630\) 0 0
\(631\) − 1.90637e11i − 1.20251i −0.799057 0.601256i \(-0.794667\pi\)
0.799057 0.601256i \(-0.205333\pi\)
\(632\) 0 0
\(633\) 5.46439e10 0.340351
\(634\) 0 0
\(635\) 3.93170e9i 0.0241816i
\(636\) 0 0
\(637\) −2.47275e11 −1.50183
\(638\) 0 0
\(639\) 7.06330e10i 0.423647i
\(640\) 0 0
\(641\) 2.61699e10 0.155014 0.0775069 0.996992i \(-0.475304\pi\)
0.0775069 + 0.996992i \(0.475304\pi\)
\(642\) 0 0
\(643\) − 6.78239e10i − 0.396770i −0.980124 0.198385i \(-0.936430\pi\)
0.980124 0.198385i \(-0.0635697\pi\)
\(644\) 0 0
\(645\) −4.41398e9 −0.0255030
\(646\) 0 0
\(647\) − 2.04734e11i − 1.16835i −0.811628 0.584174i \(-0.801419\pi\)
0.811628 0.584174i \(-0.198581\pi\)
\(648\) 0 0
\(649\) 2.22153e11 1.25220
\(650\) 0 0
\(651\) − 3.84199e10i − 0.213910i
\(652\) 0 0
\(653\) 1.51233e10 0.0831753 0.0415876 0.999135i \(-0.486758\pi\)
0.0415876 + 0.999135i \(0.486758\pi\)
\(654\) 0 0
\(655\) − 5.98436e9i − 0.0325126i
\(656\) 0 0
\(657\) −5.13694e10 −0.275704
\(658\) 0 0
\(659\) 4.16908e10i 0.221054i 0.993873 + 0.110527i \(0.0352539\pi\)
−0.993873 + 0.110527i \(0.964746\pi\)
\(660\) 0 0
\(661\) 3.51774e11 1.84271 0.921357 0.388719i \(-0.127082\pi\)
0.921357 + 0.388719i \(0.127082\pi\)
\(662\) 0 0
\(663\) 2.62934e11i 1.36080i
\(664\) 0 0
\(665\) 2.95135e9 0.0150916
\(666\) 0 0
\(667\) − 1.87816e11i − 0.948918i
\(668\) 0 0
\(669\) −2.71065e10 −0.135322
\(670\) 0 0
\(671\) 3.25530e11i 1.60584i
\(672\) 0 0
\(673\) 1.26006e11 0.614229 0.307115 0.951673i \(-0.400637\pi\)
0.307115 + 0.951673i \(0.400637\pi\)
\(674\) 0 0
\(675\) 3.99117e10i 0.192258i
\(676\) 0 0
\(677\) 1.43623e11 0.683708 0.341854 0.939753i \(-0.388945\pi\)
0.341854 + 0.939753i \(0.388945\pi\)
\(678\) 0 0
\(679\) − 4.35872e10i − 0.205060i
\(680\) 0 0
\(681\) −2.16460e11 −1.00644
\(682\) 0 0
\(683\) 2.24502e11i 1.03166i 0.856690 + 0.515831i \(0.172517\pi\)
−0.856690 + 0.515831i \(0.827483\pi\)
\(684\) 0 0
\(685\) 7.42409e9 0.0337195
\(686\) 0 0
\(687\) 1.58472e11i 0.711419i
\(688\) 0 0
\(689\) −2.14729e11 −0.952826
\(690\) 0 0
\(691\) − 1.36709e11i − 0.599634i −0.953997 0.299817i \(-0.903074\pi\)
0.953997 0.299817i \(-0.0969256\pi\)
\(692\) 0 0
\(693\) −2.63326e10 −0.114172
\(694\) 0 0
\(695\) 4.68135e9i 0.0200647i
\(696\) 0 0
\(697\) −1.12778e11 −0.477853
\(698\) 0 0
\(699\) 1.03008e11i 0.431483i
\(700\) 0 0
\(701\) 3.04497e11 1.26099 0.630493 0.776195i \(-0.282853\pi\)
0.630493 + 0.776195i \(0.282853\pi\)
\(702\) 0 0
\(703\) − 5.14533e11i − 2.10665i
\(704\) 0 0
\(705\) −1.87785e9 −0.00760157
\(706\) 0 0
\(707\) − 7.33583e10i − 0.293611i
\(708\) 0 0
\(709\) −1.06434e11 −0.421206 −0.210603 0.977572i \(-0.567543\pi\)
−0.210603 + 0.977572i \(0.567543\pi\)
\(710\) 0 0
\(711\) − 2.37392e10i − 0.0928942i
\(712\) 0 0
\(713\) 1.55256e11 0.600745
\(714\) 0 0
\(715\) − 1.49100e10i − 0.0570498i
\(716\) 0 0
\(717\) 1.99311e11 0.754145
\(718\) 0 0
\(719\) − 6.97035e10i − 0.260819i −0.991460 0.130410i \(-0.958371\pi\)
0.991460 0.130410i \(-0.0416292\pi\)
\(720\) 0 0
\(721\) 1.07868e11 0.399165
\(722\) 0 0
\(723\) 2.24531e11i 0.821717i
\(724\) 0 0
\(725\) −5.10440e11 −1.84754
\(726\) 0 0
\(727\) 2.36941e11i 0.848207i 0.905614 + 0.424104i \(0.139411\pi\)
−0.905614 + 0.424104i \(0.860589\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) 5.64025e11i 1.97528i
\(732\) 0 0
\(733\) 3.77173e11 1.30655 0.653274 0.757122i \(-0.273395\pi\)
0.653274 + 0.757122i \(0.273395\pi\)
\(734\) 0 0
\(735\) − 4.78843e9i − 0.0164075i
\(736\) 0 0
\(737\) 4.39891e11 1.49099
\(738\) 0 0
\(739\) − 2.53030e11i − 0.848387i −0.905572 0.424193i \(-0.860558\pi\)
0.905572 0.424193i \(-0.139442\pi\)
\(740\) 0 0
\(741\) 4.38692e11 1.45508
\(742\) 0 0
\(743\) 1.74065e10i 0.0571159i 0.999592 + 0.0285580i \(0.00909152\pi\)
−0.999592 + 0.0285580i \(0.990908\pi\)
\(744\) 0 0
\(745\) −1.72349e10 −0.0559480
\(746\) 0 0
\(747\) 1.74782e11i 0.561325i
\(748\) 0 0
\(749\) −4.85675e10 −0.154319
\(750\) 0 0
\(751\) − 4.54672e11i − 1.42935i −0.699457 0.714675i \(-0.746575\pi\)
0.699457 0.714675i \(-0.253425\pi\)
\(752\) 0 0
\(753\) 5.39529e10 0.167817
\(754\) 0 0
\(755\) − 2.49110e7i 0 7.66661e-5i
\(756\) 0 0
\(757\) −5.54282e11 −1.68790 −0.843952 0.536419i \(-0.819777\pi\)
−0.843952 + 0.536419i \(0.819777\pi\)
\(758\) 0 0
\(759\) − 1.06411e11i − 0.320641i
\(760\) 0 0
\(761\) 8.46497e10 0.252399 0.126199 0.992005i \(-0.459722\pi\)
0.126199 + 0.992005i \(0.459722\pi\)
\(762\) 0 0
\(763\) 9.97364e10i 0.294277i
\(764\) 0 0
\(765\) −5.09167e9 −0.0148667
\(766\) 0 0
\(767\) − 6.68227e11i − 1.93082i
\(768\) 0 0
\(769\) 1.61130e11 0.460756 0.230378 0.973101i \(-0.426004\pi\)
0.230378 + 0.973101i \(0.426004\pi\)
\(770\) 0 0
\(771\) 2.90963e10i 0.0823419i
\(772\) 0 0
\(773\) 5.45092e11 1.52669 0.763346 0.645990i \(-0.223555\pi\)
0.763346 + 0.645990i \(0.223555\pi\)
\(774\) 0 0
\(775\) − 4.21950e11i − 1.16965i
\(776\) 0 0
\(777\) 9.29008e10 0.254880
\(778\) 0 0
\(779\) 1.88164e11i 0.510961i
\(780\) 0 0
\(781\) −5.11808e11 −1.37563
\(782\) 0 0
\(783\) − 1.33780e11i − 0.355914i
\(784\) 0 0
\(785\) −1.14429e10 −0.0301340
\(786\) 0 0
\(787\) 4.18854e11i 1.09185i 0.837833 + 0.545926i \(0.183822\pi\)
−0.837833 + 0.545926i \(0.816178\pi\)
\(788\) 0 0
\(789\) 1.31948e11 0.340482
\(790\) 0 0
\(791\) 1.13879e11i 0.290895i
\(792\) 0 0
\(793\) 9.79183e11 2.47612
\(794\) 0 0
\(795\) − 4.15818e9i − 0.0104096i
\(796\) 0 0
\(797\) 5.31554e11 1.31739 0.658694 0.752411i \(-0.271109\pi\)
0.658694 + 0.752411i \(0.271109\pi\)
\(798\) 0 0
\(799\) 2.39954e11i 0.588764i
\(800\) 0 0
\(801\) −1.63172e11 −0.396382
\(802\) 0 0
\(803\) − 3.72223e11i − 0.895244i
\(804\) 0 0
\(805\) −2.15338e9 −0.00512787
\(806\) 0 0
\(807\) − 1.09360e11i − 0.257848i
\(808\) 0 0
\(809\) −3.98989e11 −0.931466 −0.465733 0.884925i \(-0.654209\pi\)
−0.465733 + 0.884925i \(0.654209\pi\)
\(810\) 0 0
\(811\) − 1.83588e10i − 0.0424385i −0.999775 0.0212192i \(-0.993245\pi\)
0.999775 0.0212192i \(-0.00675480\pi\)
\(812\) 0 0
\(813\) 9.01668e10 0.206388
\(814\) 0 0
\(815\) − 2.70017e10i − 0.0612012i
\(816\) 0 0
\(817\) 9.41046e11 2.11214
\(818\) 0 0
\(819\) 7.92075e10i 0.176048i
\(820\) 0 0
\(821\) 1.50902e11 0.332141 0.166070 0.986114i \(-0.446892\pi\)
0.166070 + 0.986114i \(0.446892\pi\)
\(822\) 0 0
\(823\) − 5.25782e11i − 1.14606i −0.819535 0.573029i \(-0.805768\pi\)
0.819535 0.573029i \(-0.194232\pi\)
\(824\) 0 0
\(825\) −2.89200e11 −0.624285
\(826\) 0 0
\(827\) − 1.65875e11i − 0.354617i −0.984155 0.177309i \(-0.943261\pi\)
0.984155 0.177309i \(-0.0567390\pi\)
\(828\) 0 0
\(829\) −1.36343e11 −0.288678 −0.144339 0.989528i \(-0.546106\pi\)
−0.144339 + 0.989528i \(0.546106\pi\)
\(830\) 0 0
\(831\) 1.47642e11i 0.309603i
\(832\) 0 0
\(833\) −6.11873e11 −1.27081
\(834\) 0 0
\(835\) 1.83891e10i 0.0378281i
\(836\) 0 0
\(837\) 1.10588e11 0.225323
\(838\) 0 0
\(839\) 6.41586e11i 1.29481i 0.762145 + 0.647406i \(0.224146\pi\)
−0.762145 + 0.647406i \(0.775854\pi\)
\(840\) 0 0
\(841\) 1.21070e12 2.42021
\(842\) 0 0
\(843\) 1.20227e11i 0.238063i
\(844\) 0 0
\(845\) −2.87477e10 −0.0563867
\(846\) 0 0
\(847\) − 2.79373e10i − 0.0542814i
\(848\) 0 0
\(849\) 7.46142e10 0.143612
\(850\) 0 0
\(851\) 3.75415e11i 0.715804i
\(852\) 0 0
\(853\) −7.03535e9 −0.0132889 −0.00664446 0.999978i \(-0.502115\pi\)
−0.00664446 + 0.999978i \(0.502115\pi\)
\(854\) 0 0
\(855\) 8.49519e9i 0.0158968i
\(856\) 0 0
\(857\) 4.30808e11 0.798657 0.399328 0.916808i \(-0.369243\pi\)
0.399328 + 0.916808i \(0.369243\pi\)
\(858\) 0 0
\(859\) − 7.05962e11i − 1.29661i −0.761382 0.648304i \(-0.775479\pi\)
0.761382 0.648304i \(-0.224521\pi\)
\(860\) 0 0
\(861\) −3.39738e10 −0.0618203
\(862\) 0 0
\(863\) − 3.49784e11i − 0.630604i −0.948991 0.315302i \(-0.897894\pi\)
0.948991 0.315302i \(-0.102106\pi\)
\(864\) 0 0
\(865\) 1.20025e10 0.0214391
\(866\) 0 0
\(867\) 3.24398e11i 0.574119i
\(868\) 0 0
\(869\) 1.72015e11 0.301639
\(870\) 0 0
\(871\) − 1.32318e12i − 2.29904i
\(872\) 0 0
\(873\) 1.25462e11 0.216000
\(874\) 0 0
\(875\) 1.17106e10i 0.0199778i
\(876\) 0 0
\(877\) 5.02015e11 0.848631 0.424315 0.905514i \(-0.360515\pi\)
0.424315 + 0.905514i \(0.360515\pi\)
\(878\) 0 0
\(879\) 1.92680e11i 0.322761i
\(880\) 0 0
\(881\) −1.95986e11 −0.325328 −0.162664 0.986682i \(-0.552009\pi\)
−0.162664 + 0.986682i \(0.552009\pi\)
\(882\) 0 0
\(883\) − 6.69240e11i − 1.10088i −0.834876 0.550439i \(-0.814460\pi\)
0.834876 0.550439i \(-0.185540\pi\)
\(884\) 0 0
\(885\) 1.29401e10 0.0210943
\(886\) 0 0
\(887\) 5.04570e11i 0.815130i 0.913176 + 0.407565i \(0.133622\pi\)
−0.913176 + 0.407565i \(0.866378\pi\)
\(888\) 0 0
\(889\) 1.51345e11 0.242304
\(890\) 0 0
\(891\) − 7.57959e10i − 0.120264i
\(892\) 0 0
\(893\) 4.00351e11 0.629557
\(894\) 0 0
\(895\) − 2.91874e9i − 0.00454887i
\(896\) 0 0
\(897\) −3.20080e11 −0.494412
\(898\) 0 0
\(899\) 1.41434e12i 2.16528i
\(900\) 0 0
\(901\) −5.31340e11 −0.806256
\(902\) 0 0
\(903\) 1.69909e11i 0.255544i
\(904\) 0 0
\(905\) −1.36156e10 −0.0202974
\(906\) 0 0
\(907\) − 1.07760e11i − 0.159232i −0.996826 0.0796158i \(-0.974631\pi\)
0.996826 0.0796158i \(-0.0253693\pi\)
\(908\) 0 0
\(909\) 2.11155e11 0.309276
\(910\) 0 0
\(911\) − 3.44029e11i − 0.499485i −0.968312 0.249742i \(-0.919654\pi\)
0.968312 0.249742i \(-0.0803459\pi\)
\(912\) 0 0
\(913\) −1.26647e12 −1.82269
\(914\) 0 0
\(915\) 1.89617e10i 0.0270516i
\(916\) 0 0
\(917\) −2.30359e11 −0.325782
\(918\) 0 0
\(919\) − 6.25848e11i − 0.877419i −0.898629 0.438709i \(-0.855436\pi\)
0.898629 0.438709i \(-0.144564\pi\)
\(920\) 0 0
\(921\) −7.17606e11 −0.997350
\(922\) 0 0
\(923\) 1.53950e12i 2.12116i
\(924\) 0 0
\(925\) 1.02029e12 1.39366
\(926\) 0 0
\(927\) 3.10488e11i 0.420461i
\(928\) 0 0
\(929\) −1.19959e11 −0.161054 −0.0805269 0.996752i \(-0.525660\pi\)
−0.0805269 + 0.996752i \(0.525660\pi\)
\(930\) 0 0
\(931\) 1.02088e12i 1.35886i
\(932\) 0 0
\(933\) −3.93629e11 −0.519470
\(934\) 0 0
\(935\) − 3.68943e10i − 0.0482740i
\(936\) 0 0
\(937\) −1.81236e11 −0.235118 −0.117559 0.993066i \(-0.537507\pi\)
−0.117559 + 0.993066i \(0.537507\pi\)
\(938\) 0 0
\(939\) 2.42343e11i 0.311722i
\(940\) 0 0
\(941\) 3.59055e11 0.457934 0.228967 0.973434i \(-0.426465\pi\)
0.228967 + 0.973434i \(0.426465\pi\)
\(942\) 0 0
\(943\) − 1.37289e11i − 0.173616i
\(944\) 0 0
\(945\) −1.53384e9 −0.00192332
\(946\) 0 0
\(947\) − 1.42565e12i − 1.77260i −0.463109 0.886302i \(-0.653266\pi\)
0.463109 0.886302i \(-0.346734\pi\)
\(948\) 0 0
\(949\) −1.11963e12 −1.38042
\(950\) 0 0
\(951\) − 2.01635e11i − 0.246515i
\(952\) 0 0
\(953\) 1.01220e12 1.22715 0.613573 0.789638i \(-0.289732\pi\)
0.613573 + 0.789638i \(0.289732\pi\)
\(954\) 0 0
\(955\) 3.60663e10i 0.0433598i
\(956\) 0 0
\(957\) 9.69372e11 1.15569
\(958\) 0 0
\(959\) − 2.85779e11i − 0.337875i
\(960\) 0 0
\(961\) −3.16257e11 −0.370805
\(962\) 0 0
\(963\) − 1.39797e11i − 0.162552i
\(964\) 0 0
\(965\) −9.03094e9 −0.0104141
\(966\) 0 0
\(967\) 1.29366e12i 1.47950i 0.672881 + 0.739750i \(0.265056\pi\)
−0.672881 + 0.739750i \(0.734944\pi\)
\(968\) 0 0
\(969\) 1.08553e12 1.23125
\(970\) 0 0
\(971\) − 3.23711e11i − 0.364150i −0.983285 0.182075i \(-0.941719\pi\)
0.983285 0.182075i \(-0.0582813\pi\)
\(972\) 0 0
\(973\) 1.80201e11 0.201051
\(974\) 0 0
\(975\) 8.69905e11i 0.962617i
\(976\) 0 0
\(977\) 3.69052e11 0.405050 0.202525 0.979277i \(-0.435085\pi\)
0.202525 + 0.979277i \(0.435085\pi\)
\(978\) 0 0
\(979\) − 1.18234e12i − 1.28710i
\(980\) 0 0
\(981\) −2.87082e11 −0.309977
\(982\) 0 0
\(983\) 5.92111e11i 0.634146i 0.948401 + 0.317073i \(0.102700\pi\)
−0.948401 + 0.317073i \(0.897300\pi\)
\(984\) 0 0
\(985\) −8.34189e9 −0.00886175
\(986\) 0 0
\(987\) 7.22849e10i 0.0761691i
\(988\) 0 0
\(989\) −6.86610e11 −0.717670
\(990\) 0 0
\(991\) 2.22630e11i 0.230828i 0.993317 + 0.115414i \(0.0368195\pi\)
−0.993317 + 0.115414i \(0.963181\pi\)
\(992\) 0 0
\(993\) −3.66820e11 −0.377273
\(994\) 0 0
\(995\) − 9.21463e9i − 0.00940125i
\(996\) 0 0
\(997\) −2.32972e11 −0.235788 −0.117894 0.993026i \(-0.537614\pi\)
−0.117894 + 0.993026i \(0.537614\pi\)
\(998\) 0 0
\(999\) 2.67406e11i 0.268479i
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 384.9.g.a.127.16 yes 32
4.3 odd 2 inner 384.9.g.a.127.15 32
8.3 odd 2 384.9.g.b.127.18 yes 32
8.5 even 2 384.9.g.b.127.17 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.9.g.a.127.15 32 4.3 odd 2 inner
384.9.g.a.127.16 yes 32 1.1 even 1 trivial
384.9.g.b.127.17 yes 32 8.5 even 2
384.9.g.b.127.18 yes 32 8.3 odd 2