Properties

Label 1600.2.n.b.1407.1
Level $1600$
Weight $2$
Character 1600.1407
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1343,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (of order \(4\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 1407.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1407
Dual form 1600.2.n.b.1343.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{3} +(-3.00000 + 3.00000i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.00000i) q^{3} +(-3.00000 + 3.00000i) q^{7} -1.00000i q^{9} +6.00000 q^{21} +(1.00000 + 1.00000i) q^{23} +(-4.00000 + 4.00000i) q^{27} -6.00000i q^{29} +12.0000 q^{41} +(9.00000 + 9.00000i) q^{43} +(7.00000 - 7.00000i) q^{47} -11.0000i q^{49} +8.00000 q^{61} +(3.00000 + 3.00000i) q^{63} +(3.00000 - 3.00000i) q^{67} -2.00000i q^{69} +5.00000 q^{81} +(-11.0000 - 11.0000i) q^{83} +(-6.00000 + 6.00000i) q^{87} +6.00000i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 6 q^{7} + 12 q^{21} + 2 q^{23} - 8 q^{27} + 24 q^{41} + 18 q^{43} + 14 q^{47} + 16 q^{61} + 6 q^{63} + 6 q^{67} + 10 q^{81} - 22 q^{83} - 12 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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
\(3\) −1.00000 1.00000i −0.577350 0.577350i 0.356822 0.934172i \(-0.383860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.00000 + 3.00000i −1.13389 + 1.13389i −0.144370 + 0.989524i \(0.546115\pi\)
−0.989524 + 0.144370i \(0.953885\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 1.00000 + 1.00000i 0.208514 + 0.208514i 0.803636 0.595121i \(-0.202896\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) 0 0
\(29\) 6.00000i 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 9.00000 + 9.00000i 1.37249 + 1.37249i 0.856742 + 0.515745i \(0.172485\pi\)
0.515745 + 0.856742i \(0.327515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00000 7.00000i 1.02105 1.02105i 0.0212814 0.999774i \(-0.493225\pi\)
0.999774 0.0212814i \(-0.00677460\pi\)
\(48\) 0 0
\(49\) 11.0000i 1.57143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 3.00000 + 3.00000i 0.377964 + 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000 3.00000i 0.366508 0.366508i −0.499694 0.866202i \(-0.666554\pi\)
0.866202 + 0.499694i \(0.166554\pi\)
\(68\) 0 0
\(69\) 2.00000i 0.240772i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) −11.0000 11.0000i −1.20741 1.20741i −0.971864 0.235543i \(-0.924313\pi\)
−0.235543 0.971864i \(-0.575687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 + 6.00000i −0.643268 + 0.643268i
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −9.00000 9.00000i −0.886796 0.886796i 0.107418 0.994214i \(-0.465742\pi\)
−0.994214 + 0.107418i \(0.965742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.0000 13.0000i 1.25676 1.25676i 0.304125 0.952632i \(-0.401636\pi\)
0.952632 0.304125i \(-0.0983642\pi\)
\(108\) 0 0
\(109\) 16.0000i 1.53252i −0.642529 0.766261i \(-0.722115\pi\)
0.642529 0.766261i \(-0.277885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −12.0000 12.0000i −1.08200 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.00000 + 3.00000i −0.266207 + 0.266207i −0.827570 0.561363i \(-0.810277\pi\)
0.561363 + 0.827570i \(0.310277\pi\)
\(128\) 0 0
\(129\) 18.0000i 1.58481i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −14.0000 −1.17901
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −11.0000 + 11.0000i −0.907265 + 0.907265i
\(148\) 0 0
\(149\) 24.0000i 1.96616i 0.183186 + 0.983078i \(0.441359\pi\)
−0.183186 + 0.983078i \(0.558641\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 9.00000 + 9.00000i 0.704934 + 0.704934i 0.965465 0.260531i \(-0.0838976\pi\)
−0.260531 + 0.965465i \(0.583898\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.0000 17.0000i 1.31550 1.31550i 0.398202 0.917298i \(-0.369634\pi\)
0.917298 0.398202i \(-0.130366\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −8.00000 8.00000i −0.591377 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 24.0000i 1.74574i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) 18.0000 + 18.0000i 1.26335 + 1.26335i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 1.00000i 0.0695048 0.0695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 21.0000 + 21.0000i 1.40626 + 1.40626i 0.778001 + 0.628263i \(0.216234\pi\)
0.628263 + 0.778001i \(0.283766\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.00000 + 7.00000i −0.464606 + 0.464606i −0.900162 0.435556i \(-0.856552\pi\)
0.435556 + 0.900162i \(0.356552\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) 7.00000 + 7.00000i 0.449050 + 0.449050i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 22.0000i 1.39419i
\(250\) 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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 11.0000 + 11.0000i 0.678289 + 0.678289i 0.959613 0.281324i \(-0.0907735\pi\)
−0.281324 + 0.959613i \(0.590774\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 6.00000i 0.367194 0.367194i
\(268\) 0 0
\(269\) 24.0000i 1.46331i 0.681677 + 0.731653i \(0.261251\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −21.0000 21.0000i −1.24832 1.24832i −0.956461 0.291859i \(-0.905726\pi\)
−0.291859 0.956461i \(-0.594274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −36.0000 + 36.0000i −2.12501 + 2.12501i
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −54.0000 −3.11251
\(302\) 0 0
\(303\) −18.0000 18.0000i −1.03407 1.03407i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.00000 3.00000i 0.171219 0.171219i −0.616296 0.787515i \(-0.711367\pi\)
0.787515 + 0.616296i \(0.211367\pi\)
\(308\) 0 0
\(309\) 18.0000i 1.02398i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −26.0000 −1.45118
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −16.0000 + 16.0000i −0.884802 + 0.884802i
\(328\) 0 0
\(329\) 42.0000i 2.31553i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 12.0000 + 12.0000i 0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −17.0000 + 17.0000i −0.912608 + 0.912608i −0.996477 0.0838690i \(-0.973272\pi\)
0.0838690 + 0.996477i \(0.473272\pi\)
\(348\) 0 0
\(349\) 26.0000i 1.39175i −0.718164 0.695874i \(-0.755017\pi\)
0.718164 0.695874i \(-0.244983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −11.0000 11.0000i −0.577350 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 27.0000 27.0000i 1.40939 1.40939i 0.646333 0.763055i \(-0.276302\pi\)
0.763055 0.646333i \(-0.223698\pi\)
\(368\) 0 0
\(369\) 12.0000i 0.624695i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 0 0
\(383\) −19.0000 19.0000i −0.970855 0.970855i 0.0287325 0.999587i \(-0.490853\pi\)
−0.999587 + 0.0287325i \(0.990853\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.00000 9.00000i 0.457496 0.457496i
\(388\) 0 0
\(389\) 24.0000i 1.21685i 0.793612 + 0.608424i \(0.208198\pi\)
−0.793612 + 0.608424i \(0.791802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000i 0.197787i −0.995098 0.0988936i \(-0.968470\pi\)
0.995098 0.0988936i \(-0.0315304\pi\)
\(410\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) −7.00000 7.00000i −0.340352 0.340352i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −24.0000 + 24.0000i −1.16144 + 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −11.0000 −0.523810
\(442\) 0 0
\(443\) 29.0000 + 29.0000i 1.37783 + 1.37783i 0.848274 + 0.529558i \(0.177642\pi\)
0.529558 + 0.848274i \(0.322358\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 24.0000 24.0000i 1.13516 1.13516i
\(448\) 0 0
\(449\) 36.0000i 1.69895i 0.527633 + 0.849473i \(0.323080\pi\)
−0.527633 + 0.849473i \(0.676920\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −9.00000 9.00000i −0.418265 0.418265i 0.466340 0.884606i \(-0.345572\pi\)
−0.884606 + 0.466340i \(0.845572\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.0000 23.0000i 1.06431 1.06431i 0.0665285 0.997785i \(-0.478808\pi\)
0.997785 0.0665285i \(-0.0211923\pi\)
\(468\) 0 0
\(469\) 18.0000i 0.831163i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 6.00000 + 6.00000i 0.273009 + 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 27.0000 27.0000i 1.22349 1.22349i 0.257103 0.966384i \(-0.417232\pi\)
0.966384 0.257103i \(-0.0827679\pi\)
\(488\) 0 0
\(489\) 18.0000i 0.813988i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −34.0000 −1.51901
\(502\) 0 0
\(503\) 31.0000 + 31.0000i 1.38222 + 1.38222i 0.840663 + 0.541559i \(0.182166\pi\)
0.541559 + 0.840663i \(0.317834\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 13.0000 13.0000i 0.577350 0.577350i
\(508\) 0 0
\(509\) 6.00000i 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −21.0000 21.0000i −0.918266 0.918266i 0.0786374 0.996903i \(-0.474943\pi\)
−0.996903 + 0.0786374i \(0.974943\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.0000i 0.913043i
\(530\) 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
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 2.00000 + 2.00000i 0.0858282 + 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 33.0000 33.0000i 1.41098 1.41098i 0.657685 0.753293i \(-0.271536\pi\)
0.753293 0.657685i \(-0.228464\pi\)
\(548\) 0 0
\(549\) 8.00000i 0.341432i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.00000 1.00000i −0.0421450 0.0421450i 0.685720 0.727865i \(-0.259487\pi\)
−0.727865 + 0.685720i \(0.759487\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −15.0000 + 15.0000i −0.629941 + 0.629941i
\(568\) 0 0
\(569\) 36.0000i 1.50920i 0.656186 + 0.754599i \(0.272169\pi\)
−0.656186 + 0.754599i \(0.727831\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 66.0000 2.73814
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.00000 + 7.00000i −0.288921 + 0.288921i −0.836653 0.547733i \(-0.815491\pi\)
0.547733 + 0.836653i \(0.315491\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) −3.00000 3.00000i −0.122169 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −33.0000 + 33.0000i −1.33943 + 1.33943i −0.442816 + 0.896612i \(0.646021\pi\)
−0.896612 + 0.442816i \(0.853979\pi\)
\(608\) 0 0
\(609\) 36.0000i 1.45879i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) −18.0000 18.0000i −0.721155 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) −21.0000 21.0000i −0.828159 0.828159i 0.159103 0.987262i \(-0.449140\pi\)
−0.987262 + 0.159103i \(0.949140\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.0000 + 13.0000i −0.511083 + 0.511083i −0.914858 0.403775i \(-0.867698\pi\)
0.403775 + 0.914858i \(0.367698\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000 6.00000i 0.232321 0.232321i
\(668\) 0 0
\(669\) 42.0000i 1.62381i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) −31.0000 31.0000i −1.18618 1.18618i −0.978114 0.208068i \(-0.933283\pi\)
−0.208068 0.978114i \(-0.566717\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000 14.0000i 0.534133 0.534133i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 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
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −54.0000 + 54.0000i −2.03088 + 2.03088i
\(708\) 0 0
\(709\) 46.0000i 1.72757i −0.503864 0.863783i \(-0.668089\pi\)
0.503864 0.863783i \(-0.331911\pi\)
\(710\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 54.0000 2.01107
\(722\) 0 0
\(723\) 28.0000 + 28.0000i 1.04133 + 1.04133i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −3.00000 + 3.00000i −0.111264 + 0.111264i −0.760547 0.649283i \(-0.775069\pi\)
0.649283 + 0.760547i \(0.275069\pi\)
\(728\) 0 0
\(729\) 29.0000i 1.07407i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19.0000 19.0000i −0.697042 0.697042i 0.266729 0.963772i \(-0.414057\pi\)
−0.963772 + 0.266729i \(0.914057\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.0000 + 11.0000i −0.402469 + 0.402469i
\(748\) 0 0
\(749\) 78.0000i 2.85006i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 48.0000 + 48.0000i 1.73772 + 1.73772i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000i 0.504853i −0.967616 0.252426i \(-0.918771\pi\)
0.967616 0.252426i \(-0.0812286\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 + 24.0000i 0.857690 + 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −27.0000 + 27.0000i −0.962446 + 0.962446i −0.999320 0.0368739i \(-0.988260\pi\)
0.0368739 + 0.999320i \(0.488260\pi\)
\(788\) 0 0
\(789\) 22.0000i 0.783221i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0