Properties

Label 2736.2.f.b.1025.1
Level $2736$
Weight $2$
Character 2736.1025
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(1025,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2736.1025");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.f (of order \(2\), degree \(1\), 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 342)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1025
Dual form 2736.2.f.b.1025.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{5} -2.00000 q^{7} +O(q^{10})\) \(q-1.41421i q^{5} -2.00000 q^{7} +1.41421i q^{11} -1.41421i q^{17} +(-1.00000 + 4.24264i) q^{19} +1.41421i q^{23} +3.00000 q^{25} +6.00000 q^{29} +2.82843i q^{35} -8.48528i q^{37} +6.00000 q^{41} +4.00000 q^{43} -7.07107i q^{47} -3.00000 q^{49} +6.00000 q^{53} +2.00000 q^{55} -4.00000 q^{61} -8.48528i q^{67} +12.0000 q^{71} -10.0000 q^{73} -2.82843i q^{77} -8.48528i q^{79} -15.5563i q^{83} -2.00000 q^{85} +6.00000 q^{89} +(6.00000 + 1.41421i) q^{95} +8.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 2 q^{19} + 6 q^{25} + 12 q^{29} + 12 q^{41} + 8 q^{43} - 6 q^{49} + 12 q^{53} + 4 q^{55} - 8 q^{61} + 24 q^{71} - 20 q^{73} - 4 q^{85} + 12 q^{89} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.41421i 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i −0.985184 0.171499i \(-0.945139\pi\)
0.985184 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) −1.00000 + 4.24264i −0.229416 + 0.973329i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843i 0.478091i
\(36\) 0 0
\(37\) 8.48528i 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.07107i 1.03142i −0.856763 0.515711i \(-0.827528\pi\)
0.856763 0.515711i \(-0.172472\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528i 1.03664i −0.855186 0.518321i \(-0.826557\pi\)
0.855186 0.518321i \(-0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.82843i 0.322329i
\(78\) 0 0
\(79\) 8.48528i 0.954669i −0.878722 0.477334i \(-0.841603\pi\)
0.878722 0.477334i \(-0.158397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.5563i 1.70753i −0.520658 0.853766i \(-0.674313\pi\)
0.520658 0.853766i \(-0.325687\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 + 1.41421i 0.615587 + 0.145095i
\(96\) 0 0
\(97\) 8.48528i 0.861550i 0.902459 + 0.430775i \(0.141760\pi\)
−0.902459 + 0.430775i \(0.858240\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.5563i 1.54791i 0.633238 + 0.773957i \(0.281726\pi\)
−0.633238 + 0.773957i \(0.718274\pi\)
\(102\) 0 0
\(103\) 8.48528i 0.836080i −0.908429 0.418040i \(-0.862717\pi\)
0.908429 0.418040i \(-0.137283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 16.9706i 1.62549i −0.582623 0.812743i \(-0.697974\pi\)
0.582623 0.812743i \(-0.302026\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.82843i 0.259281i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 16.9706i 1.50589i −0.658081 0.752947i \(-0.728632\pi\)
0.658081 0.752947i \(-0.271368\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.89949i 0.864923i 0.901652 + 0.432461i \(0.142355\pi\)
−0.901652 + 0.432461i \(0.857645\pi\)
\(132\) 0 0
\(133\) 2.00000 8.48528i 0.173422 0.735767i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.07107i 0.604122i 0.953289 + 0.302061i \(0.0976746\pi\)
−0.953289 + 0.302061i \(0.902325\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 8.48528i 0.704664i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.3848i 1.50614i −0.657941 0.753070i \(-0.728572\pi\)
0.657941 0.753070i \(-0.271428\pi\)
\(150\) 0 0
\(151\) 8.48528i 0.690522i 0.938507 + 0.345261i \(0.112210\pi\)
−0.938507 + 0.345261i \(0.887790\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.82843i 0.222911i
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −6.00000 −0.453557
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.41421i 0.102329i 0.998690 + 0.0511645i \(0.0162933\pi\)
−0.998690 + 0.0511645i \(0.983707\pi\)
\(192\) 0 0
\(193\) 16.9706i 1.22157i 0.791797 + 0.610784i \(0.209146\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.07107i 0.503793i 0.967754 + 0.251896i \(0.0810542\pi\)
−0.967754 + 0.251896i \(0.918946\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 8.48528i 0.592638i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 1.41421i −0.415029 0.0978232i
\(210\) 0 0
\(211\) 25.4558i 1.75245i −0.481900 0.876226i \(-0.660053\pi\)
0.481900 0.876226i \(-0.339947\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.65685i 0.385794i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 25.4558i 1.70465i 0.523013 + 0.852325i \(0.324808\pi\)
−0.523013 + 0.852325i \(0.675192\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.89949i 0.648537i −0.945965 0.324269i \(-0.894882\pi\)
0.945965 0.324269i \(-0.105118\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.3848i 1.18921i 0.804017 + 0.594606i \(0.202692\pi\)
−0.804017 + 0.594606i \(0.797308\pi\)
\(240\) 0 0
\(241\) 25.4558i 1.63976i 0.572539 + 0.819878i \(0.305959\pi\)
−0.572539 + 0.819878i \(0.694041\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.24264i 0.271052i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.07107i 0.446322i −0.974782 0.223161i \(-0.928362\pi\)
0.974782 0.223161i \(-0.0716375\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 16.9706i 1.05450i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.5563i 0.959246i −0.877475 0.479623i \(-0.840774\pi\)
0.877475 0.479623i \(-0.159226\pi\)
\(264\) 0 0
\(265\) 8.48528i 0.521247i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.24264i 0.255841i
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.65685i 0.323911i
\(306\) 0 0
\(307\) 16.9706i 0.968561i −0.874913 0.484281i \(-0.839081\pi\)
0.874913 0.484281i \(-0.160919\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0416i 1.36328i −0.731690 0.681638i \(-0.761268\pi\)
0.731690 0.681638i \(-0.238732\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 8.48528i 0.475085i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 + 1.41421i 0.333849 + 0.0786889i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.1421i 0.779681i
\(330\) 0 0
\(331\) 16.9706i 0.932786i 0.884577 + 0.466393i \(0.154447\pi\)
−0.884577 + 0.466393i \(0.845553\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 8.48528i 0.462223i −0.972927 0.231111i \(-0.925764\pi\)
0.972927 0.231111i \(-0.0742362\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.41421i 0.0759190i 0.999279 + 0.0379595i \(0.0120858\pi\)
−0.999279 + 0.0379595i \(0.987914\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.5563i 0.827981i 0.910281 + 0.413990i \(0.135865\pi\)
−0.910281 + 0.413990i \(0.864135\pi\)
\(354\) 0 0
\(355\) 16.9706i 0.900704i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.3848i 0.970311i 0.874428 + 0.485156i \(0.161237\pi\)
−0.874428 + 0.485156i \(0.838763\pi\)
\(360\) 0 0
\(361\) −17.0000 8.48528i −0.894737 0.446594i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.1421i 0.740233i
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 25.4558i 1.31805i −0.752119 0.659027i \(-0.770968\pi\)
0.752119 0.659027i \(-0.229032\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.9706i 0.871719i 0.900015 + 0.435860i \(0.143556\pi\)
−0.900015 + 0.435860i \(0.856444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 35.3553i 1.79259i −0.443461 0.896293i \(-0.646250\pi\)
0.443461 0.896293i \(-0.353750\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 25.4558i 1.25871i 0.777118 + 0.629355i \(0.216681\pi\)
−0.777118 + 0.629355i \(0.783319\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −22.0000 −1.07994
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.41421i 0.0690889i 0.999403 + 0.0345444i \(0.0109980\pi\)
−0.999403 + 0.0345444i \(0.989002\pi\)
\(420\) 0 0
\(421\) 16.9706i 0.827095i 0.910483 + 0.413547i \(0.135710\pi\)
−0.910483 + 0.413547i \(0.864290\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24264i 0.205798i
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 1.41421i −0.287019 0.0676510i
\(438\) 0 0
\(439\) 8.48528i 0.404980i 0.979284 + 0.202490i \(0.0649034\pi\)
−0.979284 + 0.202490i \(0.935097\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.89949i 0.470339i 0.971954 + 0.235170i \(0.0755646\pi\)
−0.971954 + 0.235170i \(0.924435\pi\)
\(444\) 0 0
\(445\) 8.48528i 0.402241i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 8.48528i 0.399556i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.3848i 0.856264i −0.903716 0.428132i \(-0.859172\pi\)
0.903716 0.428132i \(-0.140828\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.41421i 0.0654420i 0.999465 + 0.0327210i \(0.0104173\pi\)
−0.999465 + 0.0327210i \(0.989583\pi\)
\(468\) 0 0
\(469\) 16.9706i 0.783628i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.65685i 0.260102i
\(474\) 0 0
\(475\) −3.00000 + 12.7279i −0.137649 + 0.583997i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.89949i 0.452319i 0.974090 + 0.226160i \(0.0726171\pi\)
−0.974090 + 0.226160i \(0.927383\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0416i 1.08498i −0.840061 0.542492i \(-0.817481\pi\)
0.840061 0.542492i \(-0.182519\pi\)
\(492\) 0 0
\(493\) 8.48528i 0.382158i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.3848i 0.819737i 0.912145 + 0.409868i \(0.134425\pi\)
−0.912145 + 0.409868i \(0.865575\pi\)
\(504\) 0 0
\(505\) 22.0000 0.978987
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 8.48528i 0.371035i 0.982641 + 0.185518i \(0.0593962\pi\)
−0.982641 + 0.185518i \(0.940604\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 16.9706i 0.733701i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.24264i 0.182743i
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 + 25.4558i −0.255609 + 1.08446i
\(552\) 0 0
\(553\) 16.9706i 0.721662i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.5563i 0.659144i 0.944131 + 0.329572i \(0.106904\pi\)
−0.944131 + 0.329572i \(0.893096\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 8.48528i 0.356978i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.24264i 0.176930i
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 31.1127i 1.29077i
\(582\) 0 0
\(583\) 8.48528i 0.351424i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.3553i 1.45927i 0.683836 + 0.729636i \(0.260310\pi\)
−0.683836 + 0.729636i \(0.739690\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.5269i 1.33572i 0.744287 + 0.667860i \(0.232790\pi\)
−0.744287 + 0.667860i \(0.767210\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 33.9411i 1.38449i 0.721664 + 0.692244i \(0.243378\pi\)
−0.721664 + 0.692244i \(0.756622\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.7279i 0.517464i
\(606\) 0 0
\(607\) 16.9706i 0.688814i 0.938820 + 0.344407i \(0.111920\pi\)
−0.938820 + 0.344407i \(0.888080\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 43.8406i 1.76496i −0.470353 0.882478i \(-0.655873\pi\)
0.470353 0.882478i \(-0.344127\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −26.0000 −1.03504 −0.517522 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 41.0122i 1.61236i −0.591673 0.806178i \(-0.701532\pi\)
0.591673 0.806178i \(-0.298468\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.07107i 0.276712i 0.990383 + 0.138356i \(0.0441819\pi\)
−0.990383 + 0.138356i \(0.955818\pi\)
\(654\) 0 0
\(655\) 14.0000 0.547025
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 8.48528i 0.330039i −0.986290 0.165020i \(-0.947231\pi\)
0.986290 0.165020i \(-0.0527687\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 2.82843i −0.465340 0.109682i
\(666\) 0 0
\(667\) 8.48528i 0.328551i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.65685i 0.218380i
\(672\) 0 0
\(673\) 33.9411i 1.30833i 0.756350 + 0.654167i \(0.226981\pi\)
−0.756350 + 0.654167i \(0.773019\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.7990i 0.751018i
\(696\) 0 0
\(697\) 8.48528i 0.321403i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 7.07107i 0.267071i 0.991044 + 0.133535i \(0.0426329\pi\)
−0.991044 + 0.133535i \(0.957367\pi\)
\(702\) 0 0
\(703\) 36.0000 + 8.48528i 1.35777 + 0.320028i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 31.1127i 1.17011i
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 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
\(719\) 32.5269i 1.21305i −0.795065 0.606525i \(-0.792563\pi\)
0.795065 0.606525i \(-0.207437\pi\)
\(720\) 0 0
\(721\) 16.9706i 0.632017i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 18.0000 0.668503
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.65685i 0.209226i
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −26.0000 −0.952566
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 8.48528i 0.309632i −0.987943 0.154816i \(-0.950521\pi\)
0.987943 0.154816i \(-0.0494785\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.4975i 1.79428i 0.441744 + 0.897141i \(0.354360\pi\)
−0.441744 + 0.897141i \(0.645640\pi\)
\(762\) 0 0
\(763\) 33.9411i 1.22875i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 + 25.4558i −0.214972 + 0.912050i
\(780\) 0 0
\(781\) 16.9706i 0.607254i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.65685i 0.201902i
\(786\) 0 0
\(787\) 25.4558i 0.907403i −0.891154 0.453701i \(-0.850103\pi\)
0.891154 0.453701i \(-0.149897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −10.0000 −0.353775
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.1421i 0.499065i
\(804\) 0 0