Properties

Label 324.9.g.a.269.1
Level $324$
Weight $9$
Character 324.269
Analytic conductor $131.991$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,9,Mod(53,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 324.g (of order \(6\), degree \(2\), not 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: \(131.990669660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 269.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 324.269
Dual form 324.9.g.a.53.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2017.00 + 3493.55i) q^{7} +O(q^{10})\) \(q+(-2017.00 + 3493.55i) q^{7} +(17903.0 + 31008.9i) q^{13} -258526. q^{19} +(-195312. + 338291. i) q^{25} +(904703. + 1.56699e6i) q^{31} +503522. q^{37} +(-1.74610e6 + 3.02433e6i) q^{43} +(-5.25418e6 - 9.10050e6i) q^{49} +(1.19133e7 - 2.06344e7i) q^{61} +(2.71070e6 + 4.69508e6i) q^{67} +1.61693e7 q^{73} +(9.44352e6 - 1.63567e7i) q^{79} -1.44441e8 q^{91} +(-8.84540e7 + 1.53207e8i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4034 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4034 q^{7} + 35806 q^{13} - 517052 q^{19} - 390625 q^{25} + 1809406 q^{31} + 1007044 q^{37} - 3492194 q^{43} - 10508355 q^{49} + 23826526 q^{61} + 5421406 q^{67} + 32338564 q^{73} + 18887038 q^{79} - 288882808 q^{91} - 176908034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) −2017.00 + 3493.55i −0.840067 + 1.45504i 0.0497709 + 0.998761i \(0.484151\pi\)
−0.889838 + 0.456277i \(0.849182\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 17903.0 + 31008.9i 0.626834 + 1.08571i 0.988183 + 0.153277i \(0.0489828\pi\)
−0.361349 + 0.932430i \(0.617684\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −258526. −1.98376 −0.991882 0.127165i \(-0.959412\pi\)
−0.991882 + 0.127165i \(0.959412\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −195312. + 338291.i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 904703. + 1.56699e6i 0.979624 + 1.69676i 0.663746 + 0.747958i \(0.268966\pi\)
0.315877 + 0.948800i \(0.397701\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 503522. 0.268665 0.134333 0.990936i \(-0.457111\pi\)
0.134333 + 0.990936i \(0.457111\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −1.74610e6 + 3.02433e6i −0.510734 + 0.884617i 0.489189 + 0.872178i \(0.337293\pi\)
−0.999923 + 0.0124389i \(0.996040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −5.25418e6 9.10050e6i −0.911424 1.57863i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 1.19133e7 2.06344e7i 0.860422 1.49029i −0.0111006 0.999938i \(-0.503533\pi\)
0.871522 0.490356i \(-0.163133\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.71070e6 + 4.69508e6i 0.134519 + 0.232993i 0.925414 0.378959i \(-0.123718\pi\)
−0.790895 + 0.611952i \(0.790384\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.61693e7 0.569376 0.284688 0.958620i \(-0.408110\pi\)
0.284688 + 0.958620i \(0.408110\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.44352e6 1.63567e7i 0.242452 0.419939i −0.718960 0.695051i \(-0.755382\pi\)
0.961412 + 0.275112i \(0.0887150\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.44441e8 −2.10633
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.84540e7 + 1.53207e8i −0.999150 + 1.73058i −0.463872 + 0.885902i \(0.653540\pi\)
−0.535278 + 0.844676i \(0.679793\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) −2.22245e7 3.84939e7i −0.197462 0.342014i 0.750243 0.661162i \(-0.229936\pi\)
−0.947705 + 0.319149i \(0.896603\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.03181e8 1.43938 0.719692 0.694293i \(-0.244283\pi\)
0.719692 + 0.694293i \(0.244283\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.07179e8 1.85640e8i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00562e8 −1.53977 −0.769883 0.638185i \(-0.779686\pi\)
−0.769883 + 0.638185i \(0.779686\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 5.21447e8 9.03173e8i 1.66649 2.88645i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) −3.54715e8 6.14385e8i −0.950213 1.64582i −0.744961 0.667108i \(-0.767532\pi\)
−0.205252 0.978709i \(-0.565801\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) −1.35117e8 + 2.34030e8i −0.259898 + 0.450157i −0.966214 0.257740i \(-0.917022\pi\)
0.706316 + 0.707896i \(0.250356\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.80867e7 3.13271e7i −0.0297688 0.0515611i 0.850757 0.525559i \(-0.176144\pi\)
−0.880526 + 0.473998i \(0.842810\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.17139e9 −1.65940 −0.829698 0.558212i \(-0.811488\pi\)
−0.829698 + 0.558212i \(0.811488\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0 0
\(169\) −2.33169e8 + 4.03861e8i −0.285841 + 0.495092i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) −7.87891e8 1.36467e9i −0.840067 1.45504i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.05268e9 −0.980801 −0.490400 0.871497i \(-0.663149\pi\)
−0.490400 + 0.871497i \(0.663149\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) −1.16335e9 2.01498e9i −0.838457 1.45225i −0.891185 0.453641i \(-0.850125\pi\)
0.0527278 0.998609i \(-0.483208\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.73472e9 1.10616 0.553080 0.833128i \(-0.313452\pi\)
0.553080 + 0.833128i \(0.313452\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.18553e7 + 1.59098e8i 0.0463420 + 0.0802667i 0.888266 0.459329i \(-0.151910\pi\)
−0.841924 + 0.539596i \(0.818577\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.29914e9 −3.29180
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.36366e9 4.09398e9i 0.955797 1.65549i 0.223263 0.974758i \(-0.428329\pi\)
0.732534 0.680731i \(-0.238338\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) −2.67939e9 4.64083e9i −0.974302 1.68754i −0.682220 0.731147i \(-0.738986\pi\)
−0.292081 0.956393i \(-0.594348\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 2.78289e9 4.82011e9i 0.824951 1.42886i −0.0770059 0.997031i \(-0.524536\pi\)
0.901957 0.431826i \(-0.142131\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.62839e9 8.01661e9i −1.24349 2.15379i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) −1.01560e9 + 1.75908e9i −0.225697 + 0.390918i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −2.98709e9 −0.553824 −0.276912 0.960895i \(-0.589311\pi\)
−0.276912 + 0.960895i \(0.589311\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.71404e9 + 6.43290e9i −0.630852 + 1.09267i 0.356526 + 0.934285i \(0.383961\pi\)
−0.987378 + 0.158382i \(0.949372\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 5.18486e9 + 8.98044e9i 0.808335 + 1.40008i 0.914016 + 0.405678i \(0.132964\pi\)
−0.105681 + 0.994400i \(0.533702\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −7.04376e9 1.22001e10i −0.858101 1.48627i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.41256e10 −1.59020 −0.795101 0.606478i \(-0.792582\pi\)
−0.795101 + 0.606478i \(0.792582\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) −5.85032e9 + 1.01331e10i −0.609541 + 1.05576i 0.381776 + 0.924255i \(0.375313\pi\)
−0.991316 + 0.131500i \(0.958021\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.39867e10 −1.25367
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.12475e9 1.40725e10i 0.676858 1.17235i −0.299064 0.954233i \(-0.596674\pi\)
0.975922 0.218120i \(-0.0699924\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.28845e10 + 2.23166e10i 0.998957 + 1.73024i 0.539014 + 0.842297i \(0.318797\pi\)
0.459943 + 0.887948i \(0.347870\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.91355e10 1.38249
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 1.48002e10 2.56347e10i 0.997621 1.72793i 0.439106 0.898435i \(-0.355295\pi\)
0.558514 0.829495i \(-0.311372\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.98521e10 2.93532
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.21528e10 + 2.10493e10i −0.669903 + 1.16031i 0.308028 + 0.951377i \(0.400331\pi\)
−0.977931 + 0.208929i \(0.933002\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.47234e9 4.28221e9i −0.127724 0.221224i 0.795070 0.606517i \(-0.207434\pi\)
−0.922794 + 0.385293i \(0.874100\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.49200e9 −0.169245 −0.0846227 0.996413i \(-0.526968\pi\)
−0.0846227 + 0.996413i \(0.526968\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.57611e10 −0.634491 −0.317245 0.948343i \(-0.602758\pi\)
−0.317245 + 0.948343i \(0.602758\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −3.23938e10 + 5.61077e10i −1.22812 + 2.12717i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.63405e10 + 4.56231e10i 0.941305 + 1.63039i 0.762985 + 0.646416i \(0.223733\pi\)
0.178320 + 0.983972i \(0.442934\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 5.80445e9 1.00536e10i 0.184771 0.320032i −0.758729 0.651407i \(-0.774179\pi\)
0.943499 + 0.331375i \(0.107512\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.80581e10 + 8.32391e10i 1.44562 + 2.50389i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 6.51683e10 1.85389 0.926947 0.375192i \(-0.122423\pi\)
0.926947 + 0.375192i \(0.122423\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −3.37747e10 + 5.84994e10i −0.909354 + 1.57505i −0.0943897 + 0.995535i \(0.530090\pi\)
−0.814964 + 0.579512i \(0.803243\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.36304e10 2.36085e10i 0.312495 0.541258i −0.666407 0.745589i \(-0.732168\pi\)
0.978902 + 0.204331i \(0.0655018\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 3.43926e10 + 5.95698e10i 0.748413 + 1.29629i 0.948583 + 0.316529i \(0.102517\pi\)
−0.200169 + 0.979761i \(0.564149\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −2.18700e10 −0.452019
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 5.04934e10 8.74571e10i 0.991882 1.71799i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 9.01455e9 + 1.56137e10i 0.168408 + 0.291692i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.09984e10 1.61777 0.808887 0.587965i \(-0.200071\pi\)
0.808887 + 0.587965i \(0.200071\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6.16651e10 1.06807e11i −0.994574 1.72265i −0.587381 0.809311i \(-0.699841\pi\)
−0.407193 0.913342i \(-0.633492\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) −3.26134e10 + 5.64881e10i −0.478314 + 0.828464i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 2.44747e10 0.327122 0.163561 0.986533i \(-0.447702\pi\)
0.163561 + 0.986533i \(0.447702\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.91555e10 + 6.78193e10i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6.27323e10 0.732322 0.366161 0.930551i \(-0.380672\pi\)
0.366161 + 0.930551i \(0.380672\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.09133e10 5.35434e10i 0.345300 0.598077i −0.640108 0.768285i \(-0.721111\pi\)
0.985408 + 0.170208i \(0.0544439\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3.80952e10 + 6.59827e10i 0.407351 + 0.705553i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.25041e11 −1.28058
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 9.74697e10 + 1.68822e11i 0.916907 + 1.58813i 0.804086 + 0.594513i \(0.202655\pi\)
0.112821 + 0.993615i \(0.464011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.54299e11 −1.39206 −0.696031 0.718012i \(-0.745052\pi\)
−0.696031 + 0.718012i \(0.745052\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) 0 0
\(589\) −2.33889e11 4.05108e11i −1.94334 3.36597i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 3.11303e10 5.39193e10i 0.238608 0.413282i −0.721707 0.692199i \(-0.756642\pi\)
0.960315 + 0.278917i \(0.0899754\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.32994e11 + 2.30353e11i 0.979666 + 1.69683i 0.663587 + 0.748099i \(0.269033\pi\)
0.316079 + 0.948733i \(0.397633\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.57998e10 −0.111894 −0.0559472 0.998434i \(-0.517818\pi\)
−0.0559472 + 0.998434i \(0.517818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(618\) 0 0
\(619\) −1.12767e11 + 1.95317e11i −0.768100 + 1.33039i 0.170493 + 0.985359i \(0.445464\pi\)
−0.938592 + 0.345029i \(0.887869\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.62939e10 1.32145e11i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00069e11 −1.26201 −0.631004 0.775780i \(-0.717357\pi\)
−0.631004 + 0.775780i \(0.717357\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.88131e11 3.25853e11i 1.14262 1.97908i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) −9.42718e10 1.63284e11i −0.551490 0.955210i −0.998167 0.0605142i \(-0.980726\pi\)
0.446677 0.894695i \(-0.352607\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) −1.78004e11 3.08313e11i −0.932449 1.61505i −0.779121 0.626874i \(-0.784334\pi\)
−0.153328 0.988175i \(-0.548999\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.14467e11 1.98263e11i 0.557983 0.966454i −0.439682 0.898153i \(-0.644909\pi\)
0.997665 0.0683008i \(-0.0217578\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) −3.56824e11 6.18036e11i −1.67870 2.90760i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −7.79006e10 + 1.34928e11i −0.341687 + 0.591819i −0.984746 0.173997i \(-0.944332\pi\)
0.643059 + 0.765817i \(0.277665\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −1.30174e11 −0.532968
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.43843e11 4.22348e11i 0.964995 1.67142i 0.255367 0.966844i \(-0.417804\pi\)
0.709628 0.704577i \(-0.248863\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 1.79307e11 0.663524
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.07502e10 + 1.05222e11i −0.217475 + 0.376678i −0.954035 0.299694i \(-0.903115\pi\)
0.736560 + 0.676372i \(0.236449\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.88665e11 + 4.99982e11i 0.999949 + 1.73196i 0.508714 + 0.860936i \(0.330121\pi\)
0.491235 + 0.871027i \(0.336546\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.35662e11 −1.46074 −0.730368 0.683054i \(-0.760651\pi\)
−0.730368 + 0.683054i \(0.760651\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.67915e11 + 4.64043e11i 0.842244 + 1.45881i 0.887993 + 0.459856i \(0.152099\pi\)
−0.0457494 + 0.998953i \(0.514568\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −4.59764e11 −1.40008 −0.700038 0.714106i \(-0.746833\pi\)
−0.700038 + 0.714106i \(0.746833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −4.09816e11 + 7.09822e11i −1.20918 + 2.09436i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.35250e11 5.80670e11i −0.958658 1.66044i −0.725767 0.687941i \(-0.758515\pi\)
−0.232890 0.972503i \(-0.574818\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −7.06799e11 −1.95925
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.37710e11 5.84931e11i −0.880329 1.52477i −0.850976 0.525205i \(-0.823989\pi\)
−0.0293527 0.999569i \(-0.509345\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.53133e11 2.15737
\(794\) 0 0
\(795\) 0