Properties

Label 1840.2.m.c.1839.4
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1839,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1839");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (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: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1839.4
Root \(1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.c.1839.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.16228 q^{3} +2.23607i q^{5} +4.24264i q^{7} +7.00000 q^{9} +O(q^{10})\) \(q+3.16228 q^{3} +2.23607i q^{5} +4.24264i q^{7} +7.00000 q^{9} +7.07107i q^{15} +13.4164i q^{21} +(-4.74342 - 0.707107i) q^{23} -5.00000 q^{25} +12.6491 q^{27} -6.00000 q^{29} -9.48683 q^{35} +12.0000 q^{41} -12.7279i q^{43} +15.6525i q^{45} +9.48683 q^{47} -11.0000 q^{49} -13.4164i q^{61} +29.6985i q^{63} +4.24264i q^{67} +(-15.0000 - 2.23607i) q^{69} -15.8114 q^{75} +19.0000 q^{81} +15.5563i q^{83} -18.9737 q^{87} -17.8885i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{9} - 20 q^{25} - 24 q^{29} + 48 q^{41} - 44 q^{49} - 60 q^{69} + 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 3.16228 1.82574 0.912871 0.408248i \(-0.133860\pi\)
0.912871 + 0.408248i \(0.133860\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 7.00000 2.33333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 7.07107i 1.82574i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 13.4164i 2.92770i
\(22\) 0 0
\(23\) −4.74342 0.707107i −0.989071 0.147442i
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 12.6491 2.43432
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.48683 −1.60357
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 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
\(43\) 12.7279i 1.94099i −0.241121 0.970495i \(-0.577515\pi\)
0.241121 0.970495i \(-0.422485\pi\)
\(44\) 0 0
\(45\) 15.6525i 2.33333i
\(46\) 0 0
\(47\) 9.48683 1.38380 0.691898 0.721995i \(-0.256775\pi\)
0.691898 + 0.721995i \(0.256775\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 13.4164i 1.71780i −0.512148 0.858898i \(-0.671150\pi\)
0.512148 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 29.6985i 3.74166i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264i 0.518321i 0.965834 + 0.259161i \(0.0834459\pi\)
−0.965834 + 0.259161i \(0.916554\pi\)
\(68\) 0 0
\(69\) −15.0000 2.23607i −1.80579 0.269191i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −15.8114 −1.82574
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 19.0000 2.11111
\(82\) 0 0
\(83\) 15.5563i 1.70753i 0.520658 + 0.853766i \(0.325687\pi\)
−0.520658 + 0.853766i \(0.674313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.9737 −2.03419
\(88\) 0 0
\(89\) 17.8885i 1.89618i −0.317999 0.948091i \(-0.603011\pi\)
0.317999 0.948091i \(-0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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
\(103\) 12.7279i 1.25412i 0.778971 + 0.627060i \(0.215742\pi\)
−0.778971 + 0.627060i \(0.784258\pi\)
\(104\) 0 0
\(105\) −30.0000 −2.92770
\(106\) 0 0
\(107\) 18.3848i 1.77732i 0.458563 + 0.888662i \(0.348364\pi\)
−0.458563 + 0.888662i \(0.651636\pi\)
\(108\) 0 0
\(109\) 13.4164i 1.28506i 0.766261 + 0.642529i \(0.222115\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 1.58114 10.6066i 0.147442 0.989071i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 37.9473 3.42160
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 22.1359 1.96425 0.982124 0.188237i \(-0.0602772\pi\)
0.982124 + 0.188237i \(0.0602772\pi\)
\(128\) 0 0
\(129\) 40.2492i 3.54375i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 28.2843i 2.43432i
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 30.0000 2.52646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 13.4164i 1.11417i
\(146\) 0 0
\(147\) −34.7851 −2.86902
\(148\) 0 0
\(149\) 4.47214i 0.366372i −0.983078 0.183186i \(-0.941359\pi\)
0.983078 0.183186i \(-0.0586410\pi\)
\(150\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.00000 20.1246i 0.236433 1.58604i
\(162\) 0 0
\(163\) 22.1359 1.73382 0.866910 0.498464i \(-0.166102\pi\)
0.866910 + 0.498464i \(0.166102\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.48683 −0.734113 −0.367057 0.930199i \(-0.619634\pi\)
−0.367057 + 0.930199i \(0.619634\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 21.2132i 1.60357i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 26.8328i 1.99447i −0.0743294 0.997234i \(-0.523682\pi\)
0.0743294 0.997234i \(-0.476318\pi\)
\(182\) 0 0
\(183\) 42.4264i 3.13625i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 53.6656i 3.90360i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 13.4164i 0.946320i
\(202\) 0 0
\(203\) 25.4558i 1.78665i
\(204\) 0 0
\(205\) 26.8328i 1.87409i
\(206\) 0 0
\(207\) −33.2039 4.94975i −2.30783 0.344031i
\(208\) 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
\(215\) 28.4605 1.94099
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.16228 0.211762 0.105881 0.994379i \(-0.466234\pi\)
0.105881 + 0.994379i \(0.466234\pi\)
\(224\) 0 0
\(225\) −35.0000 −2.33333
\(226\) 0 0
\(227\) 9.89949i 0.657053i 0.944495 + 0.328526i \(0.106552\pi\)
−0.944495 + 0.328526i \(0.893448\pi\)
\(228\) 0 0
\(229\) 26.8328i 1.77316i −0.462573 0.886581i \(-0.653074\pi\)
0.462573 0.886581i \(-0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 21.2132i 1.38380i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 13.4164i 0.864227i −0.901819 0.432113i \(-0.857768\pi\)
0.901819 0.432113i \(-0.142232\pi\)
\(242\) 0 0
\(243\) 22.1359 1.42002
\(244\) 0 0
\(245\) 24.5967i 1.57143i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 49.1935i 3.11751i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −42.0000 −2.59973
\(262\) 0 0
\(263\) 15.5563i 0.959246i 0.877475 + 0.479623i \(0.159226\pi\)
−0.877475 + 0.479623i \(0.840774\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 56.5685i 3.46194i
\(268\) 0 0
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.3050i 1.86750i 0.357930 + 0.933748i \(0.383483\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 29.6985i 1.76539i −0.469945 0.882696i \(-0.655726\pi\)
0.469945 0.882696i \(-0.344274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 50.9117i 3.00522i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 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
\(303\) 56.9210 3.27003
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) −34.7851 −1.98529 −0.992644 0.121070i \(-0.961367\pi\)
−0.992644 + 0.121070i \(0.961367\pi\)
\(308\) 0 0
\(309\) 40.2492i 2.28970i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) −66.4078 −3.74166
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 58.1378i 3.24493i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 42.4264i 2.34619i
\(328\) 0 0
\(329\) 40.2492i 2.21901i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.48683 −0.518321
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 5.00000 33.5410i 0.269191 1.80579i
\(346\) 0 0
\(347\) −28.4605 −1.52784 −0.763920 0.645311i \(-0.776728\pi\)
−0.763920 + 0.645311i \(0.776728\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 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
\(363\) −34.7851 −1.82574
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 38.1838i 1.99318i −0.0825348 0.996588i \(-0.526302\pi\)
0.0825348 0.996588i \(-0.473698\pi\)
\(368\) 0 0
\(369\) 84.0000 4.37287
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 35.3553i 1.82574i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 70.0000 3.58621
\(382\) 0 0
\(383\) 26.8701i 1.37300i 0.727132 + 0.686498i \(0.240853\pi\)
−0.727132 + 0.686498i \(0.759147\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 89.0955i 4.52898i
\(388\) 0 0
\(389\) 31.3050i 1.58722i −0.608424 0.793612i \(-0.708198\pi\)
0.608424 0.793612i \(-0.291802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 35.7771i 1.78662i −0.449439 0.893311i \(-0.648376\pi\)
0.449439 0.893311i \(-0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 42.4853i 2.11111i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −34.7851 −1.70753
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 40.2492i 1.96163i −0.194948 0.980814i \(-0.562454\pi\)
0.194948 0.980814i \(-0.437546\pi\)
\(422\) 0 0
\(423\) 66.4078 3.22886
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 56.9210 2.75460
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 42.4264i 2.03419i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −77.0000 −3.66667
\(442\) 0 0
\(443\) 9.48683 0.450733 0.225367 0.974274i \(-0.427642\pi\)
0.225367 + 0.974274i \(0.427642\pi\)
\(444\) 0 0
\(445\) 40.0000 1.89618
\(446\) 0 0
\(447\) 14.1421i 0.668900i
\(448\) 0 0
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 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
\(463\) −41.1096 −1.91053 −0.955263 0.295758i \(-0.904428\pi\)
−0.955263 + 0.295758i \(0.904428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 32.5269i 1.50517i −0.658497 0.752583i \(-0.728808\pi\)
0.658497 0.752583i \(-0.271192\pi\)
\(468\) 0 0
\(469\) −18.0000 −0.831163
\(470\) 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
\(483\) 9.48683 63.6396i 0.431666 2.89570i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −22.1359 −1.00308 −0.501538 0.865136i \(-0.667232\pi\)
−0.501538 + 0.865136i \(0.667232\pi\)
\(488\) 0 0
\(489\) 70.0000 3.16551
\(490\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −30.0000 −1.34030
\(502\) 0 0
\(503\) 43.8406i 1.95476i 0.211498 + 0.977378i \(0.432166\pi\)
−0.211498 + 0.977378i \(0.567834\pi\)
\(504\) 0 0
\(505\) 40.2492i 1.79107i
\(506\) 0 0
\(507\) 41.1096 1.82574
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −28.4605 −1.25412
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.8885i 0.783711i 0.920027 + 0.391856i \(0.128167\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 29.6985i 1.29862i 0.760522 + 0.649312i \(0.224943\pi\)
−0.760522 + 0.649312i \(0.775057\pi\)
\(524\) 0 0
\(525\) 67.0820i 2.92770i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 22.0000 + 6.70820i 0.956522 + 0.291661i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −41.1096 −1.77732
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 84.8528i 3.64138i
\(544\) 0 0
\(545\) −30.0000 −1.28506
\(546\) 0 0
\(547\) 3.16228 0.135209 0.0676046 0.997712i \(-0.478464\pi\)
0.0676046 + 0.997712i \(0.478464\pi\)
\(548\) 0 0
\(549\) 93.9149i 4.00819i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.41421i 0.0596020i −0.999556 0.0298010i \(-0.990513\pi\)
0.999556 0.0298010i \(-0.00948736\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 80.6102i 3.38531i
\(568\) 0 0
\(569\) 31.3050i 1.31237i −0.754599 0.656186i \(-0.772169\pi\)
0.754599 0.656186i \(-0.227831\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.7171 + 3.53553i 0.989071 + 0.147442i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 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
\(587\) 47.4342 1.95782 0.978909 0.204298i \(-0.0654911\pi\)
0.978909 + 0.204298i \(0.0654911\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) 29.6985i 1.20942i
\(604\) 0 0
\(605\) 24.5967i 1.00000i
\(606\) 0 0
\(607\) 15.8114 0.641764 0.320882 0.947119i \(-0.396021\pi\)
0.320882 + 0.947119i \(0.396021\pi\)
\(608\) 0 0
\(609\) 80.4984i 3.26196i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 84.8528i 3.42160i
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −60.0000 8.94427i −2.40772 0.358921i
\(622\) 0 0
\(623\) 75.8947 3.04066
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 49.4975i 1.96425i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 49.1935i 1.94303i 0.236986 + 0.971513i \(0.423841\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 29.6985i 1.17119i −0.810602 0.585597i \(-0.800860\pi\)
0.810602 0.585597i \(-0.199140\pi\)
\(644\) 0 0
\(645\) 90.0000 3.54375
\(646\) 0 0
\(647\) −47.4342 −1.86483 −0.932415 0.361390i \(-0.882302\pi\)
−0.932415 + 0.361390i \(0.882302\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 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
\(661\) 40.2492i 1.56551i −0.622328 0.782757i \(-0.713813\pi\)
0.622328 0.782757i \(-0.286187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.4605 + 4.24264i 1.10199 + 0.164276i
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −63.2456 −2.43432
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 31.3050i 1.19961i
\(682\) 0 0
\(683\) 28.4605 1.08901 0.544505 0.838757i \(-0.316717\pi\)
0.544505 + 0.838757i \(0.316717\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 84.8528i 3.23734i
\(688\) 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
\(701\) 22.3607i 0.844551i 0.906467 + 0.422276i \(0.138769\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 67.0820i 2.52646i
\(706\) 0 0
\(707\) 76.3675i 2.87210i
\(708\) 0 0
\(709\) 26.8328i 1.00773i −0.863783 0.503864i \(-0.831911\pi\)
0.863783 0.503864i \(-0.168089\pi\)
\(710\) 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
\(721\) −54.0000 −2.01107
\(722\) 0 0
\(723\) 42.4264i 1.57786i
\(724\) 0 0
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) 4.24264i 0.157351i −0.996900 0.0786754i \(-0.974931\pi\)
0.996900 0.0786754i \(-0.0250691\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 77.7817i 2.86902i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 26.8701i 0.985767i 0.870095 + 0.492883i \(0.164057\pi\)
−0.870095 + 0.492883i \(0.835943\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 108.894i 3.98424i
\(748\) 0 0
\(749\) −78.0000 −2.85006
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 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
\(763\) −56.9210 −2.06068
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 53.6656i 1.93523i 0.252426 + 0.967616i \(0.418771\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 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
\(783\) −75.8947 −2.71225
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 38.1838i 1.36110i −0.732700 0.680552i \(-0.761740\pi\)
0.732700 0.680552i \(-0.238260\pi\)
\(788\) 0 0
\(789\) 49.1935i 1.75133i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 125.220i 4.42442i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 45.0000 + 6.70820i 1.58604 + 0.236433i
\(806\) 0