Properties

Label 384.10.d.f.193.1
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,10,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (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: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{175}\cdot 3^{32} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.1
Root \(255.957 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.f.193.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-81.0000i q^{3} -2358.11i q^{5} -1310.82 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q-81.0000i q^{3} -2358.11i q^{5} -1310.82 q^{7} -6561.00 q^{9} -12659.0i q^{11} -149414. i q^{13} -191007. q^{15} -242843. q^{17} -713999. i q^{19} +106177. i q^{21} -634045. q^{23} -3.60758e6 q^{25} +531441. i q^{27} +919977. i q^{29} +5.06417e6 q^{31} -1.02538e6 q^{33} +3.09107e6i q^{35} -1.14011e7i q^{37} -1.21025e7 q^{39} -8.44272e6 q^{41} -1.27628e7i q^{43} +1.54716e7i q^{45} -1.08809e6 q^{47} -3.86353e7 q^{49} +1.96703e7i q^{51} +9.16666e6i q^{53} -2.98514e7 q^{55} -5.78339e7 q^{57} +2.45165e6i q^{59} -4.46880e7i q^{61} +8.60032e6 q^{63} -3.52335e8 q^{65} -2.55851e8i q^{67} +5.13577e7i q^{69} +1.45406e8 q^{71} -3.89644e8 q^{73} +2.92214e8i q^{75} +1.65937e7i q^{77} -6.65328e8 q^{79} +4.30467e7 q^{81} +3.04812e8i q^{83} +5.72653e8i q^{85} +7.45182e7 q^{87} +3.38397e8 q^{89} +1.95855e8i q^{91} -4.10198e8i q^{93} -1.68369e9 q^{95} +1.28207e9 q^{97} +8.30558e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 131220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) 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
\(3\) − 81.0000i − 0.577350i
\(4\) 0 0
\(5\) − 2358.11i − 1.68733i −0.536870 0.843665i \(-0.680394\pi\)
0.536870 0.843665i \(-0.319606\pi\)
\(6\) 0 0
\(7\) −1310.82 −0.206349 −0.103175 0.994663i \(-0.532900\pi\)
−0.103175 + 0.994663i \(0.532900\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) − 12659.0i − 0.260695i −0.991468 0.130348i \(-0.958391\pi\)
0.991468 0.130348i \(-0.0416093\pi\)
\(12\) 0 0
\(13\) − 149414.i − 1.45093i −0.688261 0.725463i \(-0.741626\pi\)
0.688261 0.725463i \(-0.258374\pi\)
\(14\) 0 0
\(15\) −191007. −0.974180
\(16\) 0 0
\(17\) −242843. −0.705190 −0.352595 0.935776i \(-0.614701\pi\)
−0.352595 + 0.935776i \(0.614701\pi\)
\(18\) 0 0
\(19\) − 713999.i − 1.25692i −0.777844 0.628458i \(-0.783686\pi\)
0.777844 0.628458i \(-0.216314\pi\)
\(20\) 0 0
\(21\) 106177.i 0.119136i
\(22\) 0 0
\(23\) −634045. −0.472438 −0.236219 0.971700i \(-0.575908\pi\)
−0.236219 + 0.971700i \(0.575908\pi\)
\(24\) 0 0
\(25\) −3.60758e6 −1.84708
\(26\) 0 0
\(27\) 531441.i 0.192450i
\(28\) 0 0
\(29\) 919977.i 0.241538i 0.992681 + 0.120769i \(0.0385361\pi\)
−0.992681 + 0.120769i \(0.961464\pi\)
\(30\) 0 0
\(31\) 5.06417e6 0.984874 0.492437 0.870348i \(-0.336106\pi\)
0.492437 + 0.870348i \(0.336106\pi\)
\(32\) 0 0
\(33\) −1.02538e6 −0.150512
\(34\) 0 0
\(35\) 3.09107e6i 0.348179i
\(36\) 0 0
\(37\) − 1.14011e7i − 1.00009i −0.866000 0.500045i \(-0.833317\pi\)
0.866000 0.500045i \(-0.166683\pi\)
\(38\) 0 0
\(39\) −1.21025e7 −0.837693
\(40\) 0 0
\(41\) −8.44272e6 −0.466611 −0.233306 0.972403i \(-0.574954\pi\)
−0.233306 + 0.972403i \(0.574954\pi\)
\(42\) 0 0
\(43\) − 1.27628e7i − 0.569297i −0.958632 0.284649i \(-0.908123\pi\)
0.958632 0.284649i \(-0.0918769\pi\)
\(44\) 0 0
\(45\) 1.54716e7i 0.562443i
\(46\) 0 0
\(47\) −1.08809e6 −0.0325255 −0.0162628 0.999868i \(-0.505177\pi\)
−0.0162628 + 0.999868i \(0.505177\pi\)
\(48\) 0 0
\(49\) −3.86353e7 −0.957420
\(50\) 0 0
\(51\) 1.96703e7i 0.407142i
\(52\) 0 0
\(53\) 9.16666e6i 0.159577i 0.996812 + 0.0797884i \(0.0254245\pi\)
−0.996812 + 0.0797884i \(0.974576\pi\)
\(54\) 0 0
\(55\) −2.98514e7 −0.439879
\(56\) 0 0
\(57\) −5.78339e7 −0.725681
\(58\) 0 0
\(59\) 2.45165e6i 0.0263406i 0.999913 + 0.0131703i \(0.00419235\pi\)
−0.999913 + 0.0131703i \(0.995808\pi\)
\(60\) 0 0
\(61\) − 4.46880e7i − 0.413244i −0.978421 0.206622i \(-0.933753\pi\)
0.978421 0.206622i \(-0.0662471\pi\)
\(62\) 0 0
\(63\) 8.60032e6 0.0687831
\(64\) 0 0
\(65\) −3.52335e8 −2.44819
\(66\) 0 0
\(67\) − 2.55851e8i − 1.55114i −0.631264 0.775568i \(-0.717463\pi\)
0.631264 0.775568i \(-0.282537\pi\)
\(68\) 0 0
\(69\) 5.13577e7i 0.272762i
\(70\) 0 0
\(71\) 1.45406e8 0.679079 0.339539 0.940592i \(-0.389729\pi\)
0.339539 + 0.940592i \(0.389729\pi\)
\(72\) 0 0
\(73\) −3.89644e8 −1.60589 −0.802944 0.596054i \(-0.796734\pi\)
−0.802944 + 0.596054i \(0.796734\pi\)
\(74\) 0 0
\(75\) 2.92214e8i 1.06641i
\(76\) 0 0
\(77\) 1.65937e7i 0.0537943i
\(78\) 0 0
\(79\) −6.65328e8 −1.92183 −0.960913 0.276850i \(-0.910710\pi\)
−0.960913 + 0.276850i \(0.910710\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) 3.04812e8i 0.704986i 0.935814 + 0.352493i \(0.114666\pi\)
−0.935814 + 0.352493i \(0.885334\pi\)
\(84\) 0 0
\(85\) 5.72653e8i 1.18989i
\(86\) 0 0
\(87\) 7.45182e7 0.139452
\(88\) 0 0
\(89\) 3.38397e8 0.571705 0.285852 0.958274i \(-0.407723\pi\)
0.285852 + 0.958274i \(0.407723\pi\)
\(90\) 0 0
\(91\) 1.95855e8i 0.299398i
\(92\) 0 0
\(93\) − 4.10198e8i − 0.568617i
\(94\) 0 0
\(95\) −1.68369e9 −2.12083
\(96\) 0 0
\(97\) 1.28207e9 1.47041 0.735203 0.677847i \(-0.237087\pi\)
0.735203 + 0.677847i \(0.237087\pi\)
\(98\) 0 0
\(99\) 8.30558e7i 0.0868984i
\(100\) 0 0
\(101\) − 1.21088e9i − 1.15786i −0.815379 0.578928i \(-0.803471\pi\)
0.815379 0.578928i \(-0.196529\pi\)
\(102\) 0 0
\(103\) 1.71434e9 1.50082 0.750412 0.660970i \(-0.229855\pi\)
0.750412 + 0.660970i \(0.229855\pi\)
\(104\) 0 0
\(105\) 2.50377e8 0.201021
\(106\) 0 0
\(107\) − 2.28518e9i − 1.68536i −0.538412 0.842682i \(-0.680976\pi\)
0.538412 0.842682i \(-0.319024\pi\)
\(108\) 0 0
\(109\) − 1.16653e8i − 0.0791549i −0.999217 0.0395774i \(-0.987399\pi\)
0.999217 0.0395774i \(-0.0126012\pi\)
\(110\) 0 0
\(111\) −9.23489e8 −0.577402
\(112\) 0 0
\(113\) 2.33117e9 1.34500 0.672499 0.740098i \(-0.265221\pi\)
0.672499 + 0.740098i \(0.265221\pi\)
\(114\) 0 0
\(115\) 1.49515e9i 0.797159i
\(116\) 0 0
\(117\) 9.80303e8i 0.483642i
\(118\) 0 0
\(119\) 3.18325e8 0.145516
\(120\) 0 0
\(121\) 2.19770e9 0.932038
\(122\) 0 0
\(123\) 6.83860e8i 0.269398i
\(124\) 0 0
\(125\) 3.90140e9i 1.42931i
\(126\) 0 0
\(127\) −3.43019e9 −1.17004 −0.585021 0.811018i \(-0.698914\pi\)
−0.585021 + 0.811018i \(0.698914\pi\)
\(128\) 0 0
\(129\) −1.03379e9 −0.328684
\(130\) 0 0
\(131\) − 5.02477e9i − 1.49072i −0.666664 0.745358i \(-0.732279\pi\)
0.666664 0.745358i \(-0.267721\pi\)
\(132\) 0 0
\(133\) 9.35926e8i 0.259364i
\(134\) 0 0
\(135\) 1.25320e9 0.324727
\(136\) 0 0
\(137\) −1.35961e9 −0.329741 −0.164870 0.986315i \(-0.552721\pi\)
−0.164870 + 0.986315i \(0.552721\pi\)
\(138\) 0 0
\(139\) 7.29504e9i 1.65753i 0.559597 + 0.828765i \(0.310956\pi\)
−0.559597 + 0.828765i \(0.689044\pi\)
\(140\) 0 0
\(141\) 8.81353e7i 0.0187786i
\(142\) 0 0
\(143\) −1.89143e9 −0.378249
\(144\) 0 0
\(145\) 2.16941e9 0.407555
\(146\) 0 0
\(147\) 3.12946e9i 0.552767i
\(148\) 0 0
\(149\) 6.34307e8i 0.105429i 0.998610 + 0.0527146i \(0.0167874\pi\)
−0.998610 + 0.0527146i \(0.983213\pi\)
\(150\) 0 0
\(151\) −2.80634e9 −0.439283 −0.219642 0.975581i \(-0.570489\pi\)
−0.219642 + 0.975581i \(0.570489\pi\)
\(152\) 0 0
\(153\) 1.59330e9 0.235063
\(154\) 0 0
\(155\) − 1.19419e10i − 1.66181i
\(156\) 0 0
\(157\) − 4.70003e9i − 0.617380i −0.951163 0.308690i \(-0.900110\pi\)
0.951163 0.308690i \(-0.0998905\pi\)
\(158\) 0 0
\(159\) 7.42500e8 0.0921317
\(160\) 0 0
\(161\) 8.31122e8 0.0974874
\(162\) 0 0
\(163\) 9.00023e9i 0.998641i 0.866417 + 0.499320i \(0.166417\pi\)
−0.866417 + 0.499320i \(0.833583\pi\)
\(164\) 0 0
\(165\) 2.41796e9i 0.253964i
\(166\) 0 0
\(167\) 1.51957e10 1.51181 0.755904 0.654683i \(-0.227198\pi\)
0.755904 + 0.654683i \(0.227198\pi\)
\(168\) 0 0
\(169\) −1.17200e10 −1.10519
\(170\) 0 0
\(171\) 4.68454e9i 0.418972i
\(172\) 0 0
\(173\) 1.48179e10i 1.25771i 0.777524 + 0.628853i \(0.216475\pi\)
−0.777524 + 0.628853i \(0.783525\pi\)
\(174\) 0 0
\(175\) 4.72890e9 0.381144
\(176\) 0 0
\(177\) 1.98584e8 0.0152077
\(178\) 0 0
\(179\) − 2.46574e10i − 1.79518i −0.440832 0.897590i \(-0.645317\pi\)
0.440832 0.897590i \(-0.354683\pi\)
\(180\) 0 0
\(181\) 2.24515e10i 1.55486i 0.628969 + 0.777430i \(0.283477\pi\)
−0.628969 + 0.777430i \(0.716523\pi\)
\(182\) 0 0
\(183\) −3.61973e9 −0.238587
\(184\) 0 0
\(185\) −2.68851e10 −1.68748
\(186\) 0 0
\(187\) 3.07416e9i 0.183840i
\(188\) 0 0
\(189\) − 6.96626e8i − 0.0397120i
\(190\) 0 0
\(191\) −6.05951e8 −0.0329448 −0.0164724 0.999864i \(-0.505244\pi\)
−0.0164724 + 0.999864i \(0.505244\pi\)
\(192\) 0 0
\(193\) −1.24435e10 −0.645559 −0.322780 0.946474i \(-0.604617\pi\)
−0.322780 + 0.946474i \(0.604617\pi\)
\(194\) 0 0
\(195\) 2.85391e10i 1.41346i
\(196\) 0 0
\(197\) − 6.81666e9i − 0.322458i −0.986917 0.161229i \(-0.948454\pi\)
0.986917 0.161229i \(-0.0515458\pi\)
\(198\) 0 0
\(199\) 1.91661e10 0.866351 0.433175 0.901310i \(-0.357393\pi\)
0.433175 + 0.901310i \(0.357393\pi\)
\(200\) 0 0
\(201\) −2.07239e10 −0.895549
\(202\) 0 0
\(203\) − 1.20593e9i − 0.0498413i
\(204\) 0 0
\(205\) 1.99089e10i 0.787327i
\(206\) 0 0
\(207\) 4.15997e9 0.157479
\(208\) 0 0
\(209\) −9.03852e9 −0.327672
\(210\) 0 0
\(211\) − 9.48425e9i − 0.329406i −0.986343 0.164703i \(-0.947333\pi\)
0.986343 0.164703i \(-0.0526666\pi\)
\(212\) 0 0
\(213\) − 1.17779e10i − 0.392066i
\(214\) 0 0
\(215\) −3.00962e10 −0.960592
\(216\) 0 0
\(217\) −6.63824e9 −0.203228
\(218\) 0 0
\(219\) 3.15612e10i 0.927160i
\(220\) 0 0
\(221\) 3.62841e10i 1.02318i
\(222\) 0 0
\(223\) 4.82364e10 1.30618 0.653089 0.757281i \(-0.273473\pi\)
0.653089 + 0.757281i \(0.273473\pi\)
\(224\) 0 0
\(225\) 2.36693e10 0.615694
\(226\) 0 0
\(227\) − 3.93639e10i − 0.983970i −0.870604 0.491985i \(-0.836271\pi\)
0.870604 0.491985i \(-0.163729\pi\)
\(228\) 0 0
\(229\) 6.21477e10i 1.49336i 0.665182 + 0.746681i \(0.268354\pi\)
−0.665182 + 0.746681i \(0.731646\pi\)
\(230\) 0 0
\(231\) 1.34409e9 0.0310581
\(232\) 0 0
\(233\) 7.09062e10 1.57610 0.788048 0.615614i \(-0.211092\pi\)
0.788048 + 0.615614i \(0.211092\pi\)
\(234\) 0 0
\(235\) 2.56584e9i 0.0548813i
\(236\) 0 0
\(237\) 5.38916e10i 1.10957i
\(238\) 0 0
\(239\) 7.01178e10 1.39007 0.695036 0.718975i \(-0.255388\pi\)
0.695036 + 0.718975i \(0.255388\pi\)
\(240\) 0 0
\(241\) 7.33658e10 1.40093 0.700466 0.713686i \(-0.252975\pi\)
0.700466 + 0.713686i \(0.252975\pi\)
\(242\) 0 0
\(243\) − 3.48678e9i − 0.0641500i
\(244\) 0 0
\(245\) 9.11066e10i 1.61548i
\(246\) 0 0
\(247\) −1.06681e11 −1.82369
\(248\) 0 0
\(249\) 2.46898e10 0.407024
\(250\) 0 0
\(251\) − 5.19577e10i − 0.826263i −0.910671 0.413132i \(-0.864435\pi\)
0.910671 0.413132i \(-0.135565\pi\)
\(252\) 0 0
\(253\) 8.02639e9i 0.123162i
\(254\) 0 0
\(255\) 4.63849e10 0.686982
\(256\) 0 0
\(257\) 1.06918e11 1.52881 0.764405 0.644736i \(-0.223033\pi\)
0.764405 + 0.644736i \(0.223033\pi\)
\(258\) 0 0
\(259\) 1.49448e10i 0.206368i
\(260\) 0 0
\(261\) − 6.03597e9i − 0.0805128i
\(262\) 0 0
\(263\) −7.44592e10 −0.959660 −0.479830 0.877362i \(-0.659302\pi\)
−0.479830 + 0.877362i \(0.659302\pi\)
\(264\) 0 0
\(265\) 2.16160e10 0.269259
\(266\) 0 0
\(267\) − 2.74102e10i − 0.330074i
\(268\) 0 0
\(269\) 4.17901e10i 0.486618i 0.969949 + 0.243309i \(0.0782329\pi\)
−0.969949 + 0.243309i \(0.921767\pi\)
\(270\) 0 0
\(271\) −4.20098e10 −0.473139 −0.236570 0.971615i \(-0.576023\pi\)
−0.236570 + 0.971615i \(0.576023\pi\)
\(272\) 0 0
\(273\) 1.58643e10 0.172857
\(274\) 0 0
\(275\) 4.56684e10i 0.481525i
\(276\) 0 0
\(277\) 1.22601e11i 1.25122i 0.780136 + 0.625610i \(0.215150\pi\)
−0.780136 + 0.625610i \(0.784850\pi\)
\(278\) 0 0
\(279\) −3.32260e10 −0.328291
\(280\) 0 0
\(281\) −1.14643e11 −1.09690 −0.548452 0.836182i \(-0.684783\pi\)
−0.548452 + 0.836182i \(0.684783\pi\)
\(282\) 0 0
\(283\) 1.15293e11i 1.06848i 0.845334 + 0.534239i \(0.179402\pi\)
−0.845334 + 0.534239i \(0.820598\pi\)
\(284\) 0 0
\(285\) 1.36379e11i 1.22446i
\(286\) 0 0
\(287\) 1.10669e10 0.0962849
\(288\) 0 0
\(289\) −5.96149e10 −0.502707
\(290\) 0 0
\(291\) − 1.03847e11i − 0.848939i
\(292\) 0 0
\(293\) 1.62732e11i 1.28994i 0.764207 + 0.644971i \(0.223130\pi\)
−0.764207 + 0.644971i \(0.776870\pi\)
\(294\) 0 0
\(295\) 5.78128e9 0.0444452
\(296\) 0 0
\(297\) 6.72752e9 0.0501708
\(298\) 0 0
\(299\) 9.47351e10i 0.685473i
\(300\) 0 0
\(301\) 1.67298e10i 0.117474i
\(302\) 0 0
\(303\) −9.80812e10 −0.668488
\(304\) 0 0
\(305\) −1.05380e11 −0.697280
\(306\) 0 0
\(307\) − 4.86945e10i − 0.312865i −0.987689 0.156433i \(-0.950001\pi\)
0.987689 0.156433i \(-0.0499994\pi\)
\(308\) 0 0
\(309\) − 1.38862e11i − 0.866501i
\(310\) 0 0
\(311\) 1.73861e11 1.05386 0.526928 0.849910i \(-0.323344\pi\)
0.526928 + 0.849910i \(0.323344\pi\)
\(312\) 0 0
\(313\) −3.89931e10 −0.229635 −0.114818 0.993387i \(-0.536628\pi\)
−0.114818 + 0.993387i \(0.536628\pi\)
\(314\) 0 0
\(315\) − 2.02805e10i − 0.116060i
\(316\) 0 0
\(317\) 2.07976e11i 1.15677i 0.815764 + 0.578385i \(0.196317\pi\)
−0.815764 + 0.578385i \(0.803683\pi\)
\(318\) 0 0
\(319\) 1.16460e10 0.0629679
\(320\) 0 0
\(321\) −1.85100e11 −0.973045
\(322\) 0 0
\(323\) 1.73390e11i 0.886365i
\(324\) 0 0
\(325\) 5.39022e11i 2.67998i
\(326\) 0 0
\(327\) −9.44892e9 −0.0457001
\(328\) 0 0
\(329\) 1.42629e9 0.00671163
\(330\) 0 0
\(331\) − 1.26118e11i − 0.577498i −0.957405 0.288749i \(-0.906761\pi\)
0.957405 0.288749i \(-0.0932393\pi\)
\(332\) 0 0
\(333\) 7.48026e10i 0.333363i
\(334\) 0 0
\(335\) −6.03325e11 −2.61728
\(336\) 0 0
\(337\) −1.49166e11 −0.629991 −0.314996 0.949093i \(-0.602003\pi\)
−0.314996 + 0.949093i \(0.602003\pi\)
\(338\) 0 0
\(339\) − 1.88825e11i − 0.776535i
\(340\) 0 0
\(341\) − 6.41074e10i − 0.256752i
\(342\) 0 0
\(343\) 1.03541e11 0.403912
\(344\) 0 0
\(345\) 1.21107e11 0.460240
\(346\) 0 0
\(347\) 3.56632e11i 1.32050i 0.751046 + 0.660250i \(0.229550\pi\)
−0.751046 + 0.660250i \(0.770450\pi\)
\(348\) 0 0
\(349\) 5.37695e10i 0.194009i 0.995284 + 0.0970044i \(0.0309261\pi\)
−0.995284 + 0.0970044i \(0.969074\pi\)
\(350\) 0 0
\(351\) 7.94046e10 0.279231
\(352\) 0 0
\(353\) −1.75541e11 −0.601718 −0.300859 0.953669i \(-0.597273\pi\)
−0.300859 + 0.953669i \(0.597273\pi\)
\(354\) 0 0
\(355\) − 3.42884e11i − 1.14583i
\(356\) 0 0
\(357\) − 2.57843e10i − 0.0840134i
\(358\) 0 0
\(359\) 2.70333e11 0.858963 0.429482 0.903076i \(-0.358696\pi\)
0.429482 + 0.903076i \(0.358696\pi\)
\(360\) 0 0
\(361\) −1.87106e11 −0.579837
\(362\) 0 0
\(363\) − 1.78013e11i − 0.538112i
\(364\) 0 0
\(365\) 9.18826e11i 2.70966i
\(366\) 0 0
\(367\) −6.72371e11 −1.93469 −0.967345 0.253463i \(-0.918431\pi\)
−0.967345 + 0.253463i \(0.918431\pi\)
\(368\) 0 0
\(369\) 5.53927e10 0.155537
\(370\) 0 0
\(371\) − 1.20159e10i − 0.0329286i
\(372\) 0 0
\(373\) 5.33707e11i 1.42762i 0.700339 + 0.713811i \(0.253032\pi\)
−0.700339 + 0.713811i \(0.746968\pi\)
\(374\) 0 0
\(375\) 3.16013e11 0.825210
\(376\) 0 0
\(377\) 1.37457e11 0.350454
\(378\) 0 0
\(379\) − 2.28214e11i − 0.568155i −0.958801 0.284077i \(-0.908313\pi\)
0.958801 0.284077i \(-0.0916872\pi\)
\(380\) 0 0
\(381\) 2.77845e11i 0.675524i
\(382\) 0 0
\(383\) 7.21713e11 1.71384 0.856919 0.515451i \(-0.172376\pi\)
0.856919 + 0.515451i \(0.172376\pi\)
\(384\) 0 0
\(385\) 3.91299e10 0.0907687
\(386\) 0 0
\(387\) 8.37370e10i 0.189766i
\(388\) 0 0
\(389\) − 3.78958e11i − 0.839107i −0.907731 0.419554i \(-0.862187\pi\)
0.907731 0.419554i \(-0.137813\pi\)
\(390\) 0 0
\(391\) 1.53974e11 0.333159
\(392\) 0 0
\(393\) −4.07006e11 −0.860665
\(394\) 0 0
\(395\) 1.56892e12i 3.24275i
\(396\) 0 0
\(397\) − 7.33971e11i − 1.48293i −0.670989 0.741467i \(-0.734130\pi\)
0.670989 0.741467i \(-0.265870\pi\)
\(398\) 0 0
\(399\) 7.58100e10 0.149744
\(400\) 0 0
\(401\) −2.66124e11 −0.513967 −0.256983 0.966416i \(-0.582729\pi\)
−0.256983 + 0.966416i \(0.582729\pi\)
\(402\) 0 0
\(403\) − 7.56657e11i − 1.42898i
\(404\) 0 0
\(405\) − 1.01509e11i − 0.187481i
\(406\) 0 0
\(407\) −1.44327e11 −0.260718
\(408\) 0 0
\(409\) 6.86135e11 1.21242 0.606212 0.795303i \(-0.292688\pi\)
0.606212 + 0.795303i \(0.292688\pi\)
\(410\) 0 0
\(411\) 1.10129e11i 0.190376i
\(412\) 0 0
\(413\) − 3.21369e9i − 0.00543536i
\(414\) 0 0
\(415\) 7.18781e11 1.18954
\(416\) 0 0
\(417\) 5.90898e11 0.956975
\(418\) 0 0
\(419\) − 3.83589e11i − 0.608000i −0.952672 0.304000i \(-0.901678\pi\)
0.952672 0.304000i \(-0.0983223\pi\)
\(420\) 0 0
\(421\) 6.77263e11i 1.05072i 0.850879 + 0.525361i \(0.176070\pi\)
−0.850879 + 0.525361i \(0.823930\pi\)
\(422\) 0 0
\(423\) 7.13896e9 0.0108418
\(424\) 0 0
\(425\) 8.76077e11 1.30254
\(426\) 0 0
\(427\) 5.85782e10i 0.0852727i
\(428\) 0 0
\(429\) 1.53206e11i 0.218382i
\(430\) 0 0
\(431\) −3.46217e11 −0.483282 −0.241641 0.970366i \(-0.577686\pi\)
−0.241641 + 0.970366i \(0.577686\pi\)
\(432\) 0 0
\(433\) 7.60273e11 1.03938 0.519689 0.854355i \(-0.326048\pi\)
0.519689 + 0.854355i \(0.326048\pi\)
\(434\) 0 0
\(435\) − 1.75722e11i − 0.235302i
\(436\) 0 0
\(437\) 4.52708e11i 0.593815i
\(438\) 0 0
\(439\) −1.35206e11 −0.173742 −0.0868710 0.996220i \(-0.527687\pi\)
−0.0868710 + 0.996220i \(0.527687\pi\)
\(440\) 0 0
\(441\) 2.53487e11 0.319140
\(442\) 0 0
\(443\) − 9.04003e11i − 1.11520i −0.830110 0.557600i \(-0.811722\pi\)
0.830110 0.557600i \(-0.188278\pi\)
\(444\) 0 0
\(445\) − 7.97980e11i − 0.964655i
\(446\) 0 0
\(447\) 5.13789e10 0.0608696
\(448\) 0 0
\(449\) 5.50987e11 0.639784 0.319892 0.947454i \(-0.396353\pi\)
0.319892 + 0.947454i \(0.396353\pi\)
\(450\) 0 0
\(451\) 1.06877e11i 0.121643i
\(452\) 0 0
\(453\) 2.27314e11i 0.253620i
\(454\) 0 0
\(455\) 4.61849e11 0.505183
\(456\) 0 0
\(457\) 1.15885e12 1.24281 0.621405 0.783489i \(-0.286562\pi\)
0.621405 + 0.783489i \(0.286562\pi\)
\(458\) 0 0
\(459\) − 1.29057e11i − 0.135714i
\(460\) 0 0
\(461\) 6.46065e11i 0.666227i 0.942887 + 0.333113i \(0.108099\pi\)
−0.942887 + 0.333113i \(0.891901\pi\)
\(462\) 0 0
\(463\) −8.66388e11 −0.876189 −0.438094 0.898929i \(-0.644346\pi\)
−0.438094 + 0.898929i \(0.644346\pi\)
\(464\) 0 0
\(465\) −9.67294e11 −0.959445
\(466\) 0 0
\(467\) − 2.75888e11i − 0.268415i −0.990953 0.134207i \(-0.957151\pi\)
0.990953 0.134207i \(-0.0428488\pi\)
\(468\) 0 0
\(469\) 3.35375e11i 0.320076i
\(470\) 0 0
\(471\) −3.80702e11 −0.356444
\(472\) 0 0
\(473\) −1.61565e11 −0.148413
\(474\) 0 0
\(475\) 2.57581e12i 2.32163i
\(476\) 0 0
\(477\) − 6.01425e10i − 0.0531923i
\(478\) 0 0
\(479\) 2.20650e12 1.91511 0.957555 0.288249i \(-0.0930732\pi\)
0.957555 + 0.288249i \(0.0930732\pi\)
\(480\) 0 0
\(481\) −1.70348e12 −1.45106
\(482\) 0 0
\(483\) − 6.73209e10i − 0.0562844i
\(484\) 0 0
\(485\) − 3.02326e12i − 2.48106i
\(486\) 0 0
\(487\) −3.21730e11 −0.259186 −0.129593 0.991567i \(-0.541367\pi\)
−0.129593 + 0.991567i \(0.541367\pi\)
\(488\) 0 0
\(489\) 7.29018e11 0.576566
\(490\) 0 0
\(491\) − 3.80381e11i − 0.295360i −0.989035 0.147680i \(-0.952819\pi\)
0.989035 0.147680i \(-0.0471806\pi\)
\(492\) 0 0
\(493\) − 2.23410e11i − 0.170330i
\(494\) 0 0
\(495\) 1.95855e11 0.146626
\(496\) 0 0
\(497\) −1.90602e11 −0.140127
\(498\) 0 0
\(499\) 3.33496e11i 0.240790i 0.992726 + 0.120395i \(0.0384161\pi\)
−0.992726 + 0.120395i \(0.961584\pi\)
\(500\) 0 0
\(501\) − 1.23085e12i − 0.872843i
\(502\) 0 0
\(503\) −1.16862e12 −0.813990 −0.406995 0.913430i \(-0.633423\pi\)
−0.406995 + 0.913430i \(0.633423\pi\)
\(504\) 0 0
\(505\) −2.85539e12 −1.95368
\(506\) 0 0
\(507\) 9.49317e11i 0.638080i
\(508\) 0 0
\(509\) − 1.20618e12i − 0.796494i −0.917278 0.398247i \(-0.869619\pi\)
0.917278 0.398247i \(-0.130381\pi\)
\(510\) 0 0
\(511\) 5.10755e11 0.331374
\(512\) 0 0
\(513\) 3.79448e11 0.241894
\(514\) 0 0
\(515\) − 4.04261e12i − 2.53238i
\(516\) 0 0
\(517\) 1.37742e10i 0.00847925i
\(518\) 0 0
\(519\) 1.20025e12 0.726137
\(520\) 0 0
\(521\) −3.06301e12 −1.82129 −0.910645 0.413189i \(-0.864415\pi\)
−0.910645 + 0.413189i \(0.864415\pi\)
\(522\) 0 0
\(523\) 2.23351e12i 1.30536i 0.757633 + 0.652681i \(0.226356\pi\)
−0.757633 + 0.652681i \(0.773644\pi\)
\(524\) 0 0
\(525\) − 3.83041e11i − 0.220054i
\(526\) 0 0
\(527\) −1.22980e12 −0.694524
\(528\) 0 0
\(529\) −1.39914e12 −0.776802
\(530\) 0 0
\(531\) − 1.60853e10i − 0.00878019i
\(532\) 0 0
\(533\) 1.26146e12i 0.677018i
\(534\) 0 0
\(535\) −5.38872e12 −2.84376
\(536\) 0 0
\(537\) −1.99725e12 −1.03645
\(538\) 0 0
\(539\) 4.89085e11i 0.249595i
\(540\) 0 0
\(541\) − 5.90938e11i − 0.296588i −0.988943 0.148294i \(-0.952622\pi\)
0.988943 0.148294i \(-0.0473782\pi\)
\(542\) 0 0
\(543\) 1.81857e12 0.897699
\(544\) 0 0
\(545\) −2.75082e11 −0.133560
\(546\) 0 0
\(547\) − 3.09173e12i − 1.47659i −0.674479 0.738294i \(-0.735632\pi\)
0.674479 0.738294i \(-0.264368\pi\)
\(548\) 0 0
\(549\) 2.93198e11i 0.137748i
\(550\) 0 0
\(551\) 6.56862e11 0.303593
\(552\) 0 0
\(553\) 8.72128e11 0.396568
\(554\) 0 0
\(555\) 2.17769e12i 0.974267i
\(556\) 0 0
\(557\) − 2.94524e12i − 1.29650i −0.761428 0.648249i \(-0.775501\pi\)
0.761428 0.648249i \(-0.224499\pi\)
\(558\) 0 0
\(559\) −1.90694e12 −0.826009
\(560\) 0 0
\(561\) 2.49007e11 0.106140
\(562\) 0 0
\(563\) − 2.62039e12i − 1.09920i −0.835427 0.549602i \(-0.814779\pi\)
0.835427 0.549602i \(-0.185221\pi\)
\(564\) 0 0
\(565\) − 5.49718e12i − 2.26946i
\(566\) 0 0
\(567\) −5.64267e10 −0.0229277
\(568\) 0 0
\(569\) 9.57955e11 0.383124 0.191562 0.981480i \(-0.438645\pi\)
0.191562 + 0.981480i \(0.438645\pi\)
\(570\) 0 0
\(571\) 1.78103e12i 0.701148i 0.936535 + 0.350574i \(0.114014\pi\)
−0.936535 + 0.350574i \(0.885986\pi\)
\(572\) 0 0
\(573\) 4.90820e10i 0.0190207i
\(574\) 0 0
\(575\) 2.28737e12 0.872632
\(576\) 0 0
\(577\) 4.57175e12 1.71708 0.858542 0.512743i \(-0.171371\pi\)
0.858542 + 0.512743i \(0.171371\pi\)
\(578\) 0 0
\(579\) 1.00793e12i 0.372714i
\(580\) 0 0
\(581\) − 3.99555e11i − 0.145473i
\(582\) 0 0
\(583\) 1.16041e11 0.0416009
\(584\) 0 0
\(585\) 2.31167e12 0.816064
\(586\) 0 0
\(587\) − 2.32819e12i − 0.809368i −0.914456 0.404684i \(-0.867381\pi\)
0.914456 0.404684i \(-0.132619\pi\)
\(588\) 0 0
\(589\) − 3.61581e12i − 1.23790i
\(590\) 0 0
\(591\) −5.52149e11 −0.186171
\(592\) 0 0
\(593\) 7.75980e11 0.257694 0.128847 0.991664i \(-0.458872\pi\)
0.128847 + 0.991664i \(0.458872\pi\)
\(594\) 0 0
\(595\) − 7.50647e11i − 0.245533i
\(596\) 0 0
\(597\) − 1.55245e12i − 0.500188i
\(598\) 0 0
\(599\) −3.58904e12 −1.13909 −0.569544 0.821961i \(-0.692880\pi\)
−0.569544 + 0.821961i \(0.692880\pi\)
\(600\) 0 0
\(601\) −2.43528e12 −0.761400 −0.380700 0.924699i \(-0.624317\pi\)
−0.380700 + 0.924699i \(0.624317\pi\)
\(602\) 0 0
\(603\) 1.67864e12i 0.517045i
\(604\) 0 0
\(605\) − 5.18242e12i − 1.57266i
\(606\) 0 0
\(607\) −2.16109e11 −0.0646136 −0.0323068 0.999478i \(-0.510285\pi\)
−0.0323068 + 0.999478i \(0.510285\pi\)
\(608\) 0 0
\(609\) −9.76802e10 −0.0287759
\(610\) 0 0
\(611\) 1.62576e11i 0.0471922i
\(612\) 0 0
\(613\) − 5.34985e12i − 1.53028i −0.643866 0.765138i \(-0.722671\pi\)
0.643866 0.765138i \(-0.277329\pi\)
\(614\) 0 0
\(615\) 1.61262e12 0.454563
\(616\) 0 0
\(617\) 4.02003e12 1.11673 0.558363 0.829597i \(-0.311430\pi\)
0.558363 + 0.829597i \(0.311430\pi\)
\(618\) 0 0
\(619\) 2.33115e12i 0.638209i 0.947720 + 0.319104i \(0.103382\pi\)
−0.947720 + 0.319104i \(0.896618\pi\)
\(620\) 0 0
\(621\) − 3.36958e11i − 0.0909208i
\(622\) 0 0
\(623\) −4.43579e11 −0.117971
\(624\) 0 0
\(625\) 2.15389e12 0.564629
\(626\) 0 0
\(627\) 7.32120e11i 0.189181i
\(628\) 0 0
\(629\) 2.76868e12i 0.705253i
\(630\) 0 0
\(631\) 1.60959e12 0.404188 0.202094 0.979366i \(-0.435225\pi\)
0.202094 + 0.979366i \(0.435225\pi\)
\(632\) 0 0
\(633\) −7.68225e11 −0.190183
\(634\) 0 0
\(635\) 8.08878e12i 1.97425i
\(636\) 0 0
\(637\) 5.77265e12i 1.38915i
\(638\) 0 0
\(639\) −9.54010e11 −0.226360
\(640\) 0 0
\(641\) 9.07113e10 0.0212227 0.0106113 0.999944i \(-0.496622\pi\)
0.0106113 + 0.999944i \(0.496622\pi\)
\(642\) 0 0
\(643\) − 3.99830e12i − 0.922415i −0.887292 0.461207i \(-0.847416\pi\)
0.887292 0.461207i \(-0.152584\pi\)
\(644\) 0 0
\(645\) 2.43780e12i 0.554598i
\(646\) 0 0
\(647\) 7.74626e12 1.73789 0.868946 0.494907i \(-0.164798\pi\)
0.868946 + 0.494907i \(0.164798\pi\)
\(648\) 0 0
\(649\) 3.10355e10 0.00686685
\(650\) 0 0
\(651\) 5.37697e11i 0.117334i
\(652\) 0 0
\(653\) − 3.69315e12i − 0.794854i −0.917634 0.397427i \(-0.869903\pi\)
0.917634 0.397427i \(-0.130097\pi\)
\(654\) 0 0
\(655\) −1.18490e13 −2.51533
\(656\) 0 0
\(657\) 2.55646e12 0.535296
\(658\) 0 0
\(659\) − 9.56399e12i − 1.97540i −0.156363 0.987700i \(-0.549977\pi\)
0.156363 0.987700i \(-0.450023\pi\)
\(660\) 0 0
\(661\) − 5.90211e12i − 1.20254i −0.799044 0.601272i \(-0.794661\pi\)
0.799044 0.601272i \(-0.205339\pi\)
\(662\) 0 0
\(663\) 2.93902e12 0.590733
\(664\) 0 0
\(665\) 2.20702e12 0.437632
\(666\) 0 0
\(667\) − 5.83307e11i − 0.114112i
\(668\) 0 0
\(669\) − 3.90714e12i − 0.754123i
\(670\) 0 0
\(671\) −5.65707e11 −0.107731
\(672\) 0 0
\(673\) 2.04043e12 0.383402 0.191701 0.981453i \(-0.438600\pi\)
0.191701 + 0.981453i \(0.438600\pi\)
\(674\) 0 0
\(675\) − 1.91722e12i − 0.355471i
\(676\) 0 0
\(677\) − 5.46220e12i − 0.999352i −0.866212 0.499676i \(-0.833452\pi\)
0.866212 0.499676i \(-0.166548\pi\)
\(678\) 0 0
\(679\) −1.68056e12 −0.303417
\(680\) 0 0
\(681\) −3.18848e12 −0.568096
\(682\) 0 0
\(683\) 1.00775e13i 1.77198i 0.463703 + 0.885991i \(0.346520\pi\)
−0.463703 + 0.885991i \(0.653480\pi\)
\(684\) 0 0
\(685\) 3.20612e12i 0.556381i
\(686\) 0 0
\(687\) 5.03396e12 0.862193
\(688\) 0 0
\(689\) 1.36962e12 0.231534
\(690\) 0 0
\(691\) − 4.61451e12i − 0.769970i −0.922923 0.384985i \(-0.874207\pi\)
0.922923 0.384985i \(-0.125793\pi\)
\(692\) 0 0
\(693\) − 1.08872e11i − 0.0179314i
\(694\) 0 0
\(695\) 1.72025e13 2.79680
\(696\) 0 0
\(697\) 2.05026e12 0.329050
\(698\) 0 0
\(699\) − 5.74340e12i − 0.909959i
\(700\) 0 0
\(701\) − 1.75605e12i − 0.274666i −0.990525 0.137333i \(-0.956147\pi\)
0.990525 0.137333i \(-0.0438531\pi\)
\(702\) 0 0
\(703\) −8.14037e12 −1.25703
\(704\) 0 0
\(705\) 2.07833e11 0.0316857
\(706\) 0 0
\(707\) 1.58725e12i 0.238923i
\(708\) 0 0
\(709\) − 1.66308e12i − 0.247175i −0.992334 0.123587i \(-0.960560\pi\)
0.992334 0.123587i \(-0.0394399\pi\)
\(710\) 0 0
\(711\) 4.36522e12 0.640609
\(712\) 0 0
\(713\) −3.21091e12 −0.465292
\(714\) 0 0
\(715\) 4.46021e12i 0.638231i
\(716\) 0 0
\(717\) − 5.67954e12i − 0.802559i
\(718\) 0 0
\(719\) 2.86871e12 0.400319 0.200160 0.979763i \(-0.435854\pi\)
0.200160 + 0.979763i \(0.435854\pi\)
\(720\) 0 0
\(721\) −2.24720e12 −0.309694
\(722\) 0 0
\(723\) − 5.94263e12i − 0.808829i
\(724\) 0 0
\(725\) − 3.31889e12i − 0.446141i
\(726\) 0 0
\(727\) −3.23515e12 −0.429526 −0.214763 0.976666i \(-0.568898\pi\)
−0.214763 + 0.976666i \(0.568898\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) 3.09937e12i 0.401463i
\(732\) 0 0
\(733\) 3.30921e12i 0.423405i 0.977334 + 0.211703i \(0.0679008\pi\)
−0.977334 + 0.211703i \(0.932099\pi\)
\(734\) 0 0
\(735\) 7.37963e12 0.932700
\(736\) 0 0
\(737\) −3.23882e12 −0.404374
\(738\) 0 0
\(739\) 3.46011e12i 0.426766i 0.976969 + 0.213383i \(0.0684482\pi\)
−0.976969 + 0.213383i \(0.931552\pi\)
\(740\) 0 0
\(741\) 8.64118e12i 1.05291i
\(742\) 0 0
\(743\) −4.38595e12 −0.527976 −0.263988 0.964526i \(-0.585038\pi\)
−0.263988 + 0.964526i \(0.585038\pi\)
\(744\) 0 0
\(745\) 1.49577e12 0.177894
\(746\) 0 0
\(747\) − 1.99987e12i − 0.234995i
\(748\) 0 0
\(749\) 2.99547e12i 0.347774i
\(750\) 0 0
\(751\) 1.34891e12 0.154741 0.0773703 0.997002i \(-0.475348\pi\)
0.0773703 + 0.997002i \(0.475348\pi\)
\(752\) 0 0
\(753\) −4.20858e12 −0.477043
\(754\) 0 0
\(755\) 6.61768e12i 0.741216i
\(756\) 0 0
\(757\) − 8.21137e10i − 0.00908833i −0.999990 0.00454417i \(-0.998554\pi\)
0.999990 0.00454417i \(-0.00144646\pi\)
\(758\) 0 0
\(759\) 6.50138e11 0.0711078
\(760\) 0 0
\(761\) −1.70402e13 −1.84180 −0.920901 0.389796i \(-0.872545\pi\)
−0.920901 + 0.389796i \(0.872545\pi\)
\(762\) 0 0
\(763\) 1.52912e11i 0.0163336i
\(764\) 0 0
\(765\) − 3.75717e12i − 0.396629i
\(766\) 0 0
\(767\) 3.66311e11 0.0382182
\(768\) 0 0
\(769\) −1.31929e13 −1.36041 −0.680206 0.733021i \(-0.738109\pi\)
−0.680206 + 0.733021i \(0.738109\pi\)
\(770\) 0 0
\(771\) − 8.66040e12i − 0.882659i
\(772\) 0 0
\(773\) 9.36151e12i 0.943057i 0.881851 + 0.471529i \(0.156298\pi\)
−0.881851 + 0.471529i \(0.843702\pi\)
\(774\) 0 0
\(775\) −1.82694e13 −1.81914
\(776\) 0 0
\(777\) 1.21053e12 0.119147
\(778\) 0 0
\(779\) 6.02809e12i 0.586491i
\(780\) 0 0
\(781\) − 1.84070e12i − 0.177033i
\(782\) 0 0
\(783\) −4.88914e11 −0.0464841
\(784\) 0 0
\(785\) −1.10832e13 −1.04172
\(786\) 0 0
\(787\) − 4.69737e12i − 0.436484i −0.975895 0.218242i \(-0.929968\pi\)
0.975895 0.218242i \(-0.0700323\pi\)
\(788\) 0 0
\(789\) 6.03119e12i 0.554060i
\(790\) 0 0
\(791\) −3.05576e12 −0.277540
\(792\) 0 0
\(793\) −6.67701e12 −0.599587
\(794\) 0 0
\(795\) − 1.75090e12i − 0.155457i
\(796\) 0 0
\(797\) − 1.57832e12i − 0.138558i −0.997597 0.0692791i \(-0.977930\pi\)
0.997597 0.0692791i \(-0.0220699\pi\)
\(798\) 0 0
\(799\) 2.64236e11 0.0229367
\(800\) 0 0
\(801\) −2.22022e12 −0.190568
\(802\) 0 0
\(803\) 4.93251e12i 0.418647i
\(804\) 0 0
\(805\) − 1.95988e12i − 0.164493i
\(806\) 0 0
\(807\) 3.38500e12 0.280949
\(808\) 0 0
\(809\) 1.60915e13 1.32077 0.660387 0.750925i \(-0.270392\pi\)
0.660387 + 0.750925i \(0.270392\pi\)
\(810\) 0 0
\(811\) 2.29953e12i 0.186657i 0.995635 + 0.0933287i \(0.0297507\pi\)
−0.995635 + 0.0933287i \(0.970249\pi\)
\(812\) 0 0
\(813\) 3.40280e12i 0.273167i
\(814\) 0 0
\(815\) 2.12236e13 1.68504
\(816\) 0 0
\(817\) −9.11265e12 −0.715559
\(818\) 0 0
\(819\) − 1.28501e12i − 0.0997993i
\(820\) 0 0
\(821\) 1.01265e13i 0.777887i 0.921262 + 0.388944i \(0.127160\pi\)
−0.921262 + 0.388944i \(0.872840\pi\)
\(822\) 0 0
\(823\) −1.20614e13 −0.916426 −0.458213 0.888842i \(-0.651510\pi\)
−0.458213 + 0.888842i \(0.651510\pi\)
\(824\) 0 0
\(825\) 3.69914e12 0.278009
\(826\) 0 0
\(827\) 1.44216e13i 1.07211i 0.844183 + 0.536054i \(0.180086\pi\)
−0.844183 + 0.536054i \(0.819914\pi\)
\(828\) 0 0
\(829\) 2.01515e13i 1.48187i 0.671574 + 0.740937i \(0.265618\pi\)
−0.671574 + 0.740937i \(0.734382\pi\)
\(830\) 0 0
\(831\) 9.93065e12 0.722392
\(832\) 0 0
\(833\) 9.38234e12 0.675163
\(834\) 0 0
\(835\) − 3.58332e13i − 2.55092i
\(836\) 0 0
\(837\) 2.69131e12i 0.189539i
\(838\) 0 0
\(839\) −1.41876e13 −0.988507 −0.494253 0.869318i \(-0.664558\pi\)
−0.494253 + 0.869318i \(0.664558\pi\)
\(840\) 0 0
\(841\) 1.36608e13 0.941659
\(842\) 0 0
\(843\) 9.28607e12i 0.633298i
\(844\) 0 0
\(845\) 2.76370e13i 1.86482i
\(846\) 0 0
\(847\) −2.88079e12 −0.192325
\(848\) 0 0
\(849\) 9.33877e12 0.616886
\(850\) 0 0
\(851\) 7.22881e12i 0.472481i
\(852\) 0 0
\(853\) 1.03561e13i 0.669768i 0.942259 + 0.334884i \(0.108697\pi\)
−0.942259 + 0.334884i \(0.891303\pi\)
\(854\) 0 0
\(855\) 1.10467e13 0.706944
\(856\) 0 0
\(857\) 1.15824e13 0.733476 0.366738 0.930324i \(-0.380475\pi\)
0.366738 + 0.930324i \(0.380475\pi\)
\(858\) 0 0
\(859\) − 4.06490e12i − 0.254730i −0.991856 0.127365i \(-0.959348\pi\)
0.991856 0.127365i \(-0.0406520\pi\)
\(860\) 0 0
\(861\) − 8.96421e11i − 0.0555901i
\(862\) 0 0
\(863\) 3.08032e12 0.189037 0.0945186 0.995523i \(-0.469869\pi\)
0.0945186 + 0.995523i \(0.469869\pi\)
\(864\) 0 0
\(865\) 3.49423e13 2.12217
\(866\) 0 0
\(867\) 4.82881e12i 0.290238i
\(868\) 0 0
\(869\) 8.42240e12i 0.501011i
\(870\) 0 0
\(871\) −3.82276e13 −2.25058
\(872\) 0 0
\(873\) −8.41163e12 −0.490135
\(874\) 0 0
\(875\) − 5.11405e12i − 0.294936i
\(876\) 0 0
\(877\) − 2.13243e11i − 0.0121724i −0.999981 0.00608620i \(-0.998063\pi\)
0.999981 0.00608620i \(-0.00193731\pi\)
\(878\) 0 0
\(879\) 1.31813e13 0.744748
\(880\) 0 0
\(881\) −7.30543e10 −0.00408558 −0.00204279 0.999998i \(-0.500650\pi\)
−0.00204279 + 0.999998i \(0.500650\pi\)
\(882\) 0 0
\(883\) 1.02997e13i 0.570164i 0.958503 + 0.285082i \(0.0920209\pi\)
−0.958503 + 0.285082i \(0.907979\pi\)
\(884\) 0 0
\(885\) − 4.68284e11i − 0.0256605i
\(886\) 0 0
\(887\) 2.79649e13 1.51690 0.758450 0.651732i \(-0.225957\pi\)
0.758450 + 0.651732i \(0.225957\pi\)
\(888\) 0 0
\(889\) 4.49637e12 0.241437
\(890\) 0 0
\(891\) − 5.44929e11i − 0.0289661i
\(892\) 0 0
\(893\) 7.76895e11i 0.0408819i
\(894\) 0 0
\(895\) −5.81449e13 −3.02906
\(896\) 0 0
\(897\) 7.67354e12 0.395758
\(898\) 0 0
\(899\) 4.65892e12i 0.237885i
\(900\) 0 0
\(901\) − 2.22606e12i − 0.112532i
\(902\) 0 0
\(903\) 1.35512e12 0.0678237
\(904\) 0 0
\(905\) 5.29432e13 2.62356
\(906\) 0 0
\(907\) 2.31646e12i 0.113656i 0.998384 + 0.0568279i \(0.0180986\pi\)
−0.998384 + 0.0568279i \(0.981901\pi\)
\(908\) 0 0
\(909\) 7.94457e12i 0.385952i
\(910\) 0 0
\(911\) 1.61506e13 0.776883 0.388442 0.921473i \(-0.373014\pi\)
0.388442 + 0.921473i \(0.373014\pi\)
\(912\) 0 0
\(913\) 3.85862e12 0.183786
\(914\) 0 0
\(915\) 8.53574e12i 0.402575i
\(916\) 0 0
\(917\) 6.58658e12i 0.307608i
\(918\) 0 0
\(919\) −3.40206e13 −1.57334 −0.786670 0.617374i \(-0.788197\pi\)
−0.786670 + 0.617374i \(0.788197\pi\)
\(920\) 0 0
\(921\) −3.94425e12 −0.180633
\(922\) 0 0
\(923\) − 2.17257e13i − 0.985293i
\(924\) 0 0
\(925\) 4.11304e13i 1.84725i
\(926\) 0 0
\(927\) −1.12478e13 −0.500275
\(928\) 0 0
\(929\) −2.99915e13 −1.32108 −0.660538 0.750793i \(-0.729672\pi\)
−0.660538 + 0.750793i \(0.729672\pi\)
\(930\) 0 0
\(931\) 2.75856e13i 1.20340i
\(932\) 0 0
\(933\) − 1.40828e13i − 0.608444i
\(934\) 0 0
\(935\) 7.24922e12 0.310198
\(936\) 0 0
\(937\) −2.38419e13 −1.01044 −0.505222 0.862990i \(-0.668589\pi\)
−0.505222 + 0.862990i \(0.668589\pi\)
\(938\) 0 0
\(939\) 3.15844e12i 0.132580i
\(940\) 0 0
\(941\) 3.93491e12i 0.163599i 0.996649 + 0.0817996i \(0.0260667\pi\)
−0.996649 + 0.0817996i \(0.973933\pi\)
\(942\) 0 0
\(943\) 5.35307e12 0.220445
\(944\) 0 0
\(945\) −1.64272e12 −0.0670072
\(946\) 0 0
\(947\) − 2.29195e13i − 0.926042i −0.886347 0.463021i \(-0.846765\pi\)
0.886347 0.463021i \(-0.153235\pi\)
\(948\) 0 0
\(949\) 5.82182e13i 2.33003i
\(950\) 0 0
\(951\) 1.68461e13 0.667862
\(952\) 0 0
\(953\) −2.69689e13 −1.05912 −0.529561 0.848272i \(-0.677643\pi\)
−0.529561 + 0.848272i \(0.677643\pi\)
\(954\) 0 0
\(955\) 1.42890e12i 0.0555888i
\(956\) 0 0
\(957\) − 9.43326e11i − 0.0363545i
\(958\) 0 0
\(959\) 1.78221e12 0.0680418
\(960\) 0 0
\(961\) −7.93797e11 −0.0300230
\(962\) 0 0
\(963\) 1.49931e13i 0.561788i
\(964\) 0 0
\(965\) 2.93433e13i 1.08927i
\(966\) 0 0
\(967\) 1.79377e13 0.659703 0.329852 0.944033i \(-0.393001\pi\)
0.329852 + 0.944033i \(0.393001\pi\)
\(968\) 0 0
\(969\) 1.40446e13 0.511743
\(970\) 0 0
\(971\) − 3.36215e13i − 1.21375i −0.794796 0.606876i \(-0.792422\pi\)
0.794796 0.606876i \(-0.207578\pi\)
\(972\) 0 0
\(973\) − 9.56252e12i − 0.342030i
\(974\) 0 0
\(975\) 4.36608e13 1.54729
\(976\) 0 0
\(977\) −3.54989e13 −1.24649 −0.623245 0.782026i \(-0.714186\pi\)
−0.623245 + 0.782026i \(0.714186\pi\)
\(978\) 0 0
\(979\) − 4.28378e12i − 0.149041i
\(980\) 0 0
\(981\) 7.65362e11i 0.0263850i
\(982\) 0 0
\(983\) −7.84871e11 −0.0268107 −0.0134053 0.999910i \(-0.504267\pi\)
−0.0134053 + 0.999910i \(0.504267\pi\)
\(984\) 0 0
\(985\) −1.60745e13 −0.544093
\(986\) 0 0
\(987\) − 1.15530e11i − 0.00387496i
\(988\) 0 0
\(989\) 8.09222e12i 0.268958i
\(990\) 0 0
\(991\) −4.31644e13 −1.42165 −0.710827 0.703367i \(-0.751679\pi\)
−0.710827 + 0.703367i \(0.751679\pi\)
\(992\) 0 0
\(993\) −1.02155e13 −0.333419
\(994\) 0 0
\(995\) − 4.51958e13i − 1.46182i
\(996\) 0 0
\(997\) − 3.72331e12i − 0.119344i −0.998218 0.0596720i \(-0.980995\pi\)
0.998218 0.0596720i \(-0.0190055\pi\)
\(998\) 0 0
\(999\) 6.05901e12 0.192467
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.10.d.f.193.1 20
4.3 odd 2 inner 384.10.d.f.193.11 yes 20
8.3 odd 2 inner 384.10.d.f.193.10 yes 20
8.5 even 2 inner 384.10.d.f.193.20 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.f.193.1 20 1.1 even 1 trivial
384.10.d.f.193.10 yes 20 8.3 odd 2 inner
384.10.d.f.193.11 yes 20 4.3 odd 2 inner
384.10.d.f.193.20 yes 20 8.5 even 2 inner