Properties

Label 2736.2.s.m.1873.1
Level $2736$
Weight $2$
Character 2736.1873
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1873.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1873
Dual form 2736.2.s.m.577.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{7} +O(q^{10})\) \(q+4.00000 q^{7} +3.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(-3.00000 + 5.19615i) q^{17} +(3.50000 + 2.59808i) q^{19} +(3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} -2.00000 q^{31} -10.0000 q^{37} +(4.50000 - 7.79423i) q^{41} +(-2.00000 + 3.46410i) q^{43} +9.00000 q^{49} +(3.00000 + 5.19615i) q^{53} +(4.50000 - 7.79423i) q^{59} +(2.00000 + 3.46410i) q^{61} +(-3.50000 - 6.06218i) q^{67} +(3.00000 - 5.19615i) q^{71} +(0.500000 - 0.866025i) q^{73} +12.0000 q^{77} +(-2.00000 + 3.46410i) q^{79} +3.00000 q^{83} +(3.00000 + 5.19615i) q^{89} +(-4.00000 - 6.92820i) q^{91} +(-8.50000 + 14.7224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{7} + 6 q^{11} - 2 q^{13} - 6 q^{17} + 7 q^{19} + 6 q^{23} + 5 q^{25} - 4 q^{31} - 20 q^{37} + 9 q^{41} - 4 q^{43} + 18 q^{49} + 6 q^{53} + 9 q^{59} + 4 q^{61} - 7 q^{67} + 6 q^{71} + q^{73} + 24 q^{77} - 4 q^{79} + 6 q^{83} + 6 q^{89} - 8 q^{91} - 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −1.00000 1.73205i −0.277350 0.480384i 0.693375 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) 3.50000 + 2.59808i 0.802955 + 0.596040i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 7.79423i 0.702782 1.21725i −0.264704 0.964330i \(-0.585274\pi\)
0.967486 0.252924i \(-0.0813924\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.50000 7.79423i 0.585850 1.01472i −0.408919 0.912571i \(-0.634094\pi\)
0.994769 0.102151i \(-0.0325726\pi\)
\(60\) 0 0
\(61\) 2.00000 + 3.46410i 0.256074 + 0.443533i 0.965187 0.261562i \(-0.0842377\pi\)
−0.709113 + 0.705095i \(0.750904\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.50000 6.06218i −0.427593 0.740613i 0.569066 0.822292i \(-0.307305\pi\)
−0.996659 + 0.0816792i \(0.973972\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 5.19615i 0.356034 0.616670i −0.631260 0.775571i \(-0.717462\pi\)
0.987294 + 0.158901i \(0.0507952\pi\)
\(72\) 0 0
\(73\) 0.500000 0.866025i 0.0585206 0.101361i −0.835281 0.549823i \(-0.814695\pi\)
0.893801 + 0.448463i \(0.148028\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) −4.00000 6.92820i −0.419314 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.50000 + 14.7224i −0.863044 + 1.49484i 0.00593185 + 0.999982i \(0.498112\pi\)
−0.868976 + 0.494854i \(0.835222\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 8.00000 13.8564i 0.766261 1.32720i −0.173316 0.984866i \(-0.555448\pi\)
0.939577 0.342337i \(-0.111218\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 + 20.7846i −1.10004 + 1.90532i
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 + 1.73205i 0.0887357 + 0.153695i 0.906977 0.421180i \(-0.138384\pi\)
−0.818241 + 0.574875i \(0.805051\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.50000 + 7.79423i −0.393167 + 0.680985i −0.992865 0.119241i \(-0.961954\pi\)
0.599699 + 0.800226i \(0.295287\pi\)
\(132\) 0 0
\(133\) 14.0000 + 10.3923i 1.21395 + 0.901127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.50000 + 7.79423i 0.384461 + 0.665906i 0.991694 0.128618i \(-0.0410540\pi\)
−0.607233 + 0.794524i \(0.707721\pi\)
\(138\) 0 0
\(139\) 5.50000 + 9.52628i 0.466504 + 0.808008i 0.999268 0.0382553i \(-0.0121800\pi\)
−0.532764 + 0.846264i \(0.678847\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 5.19615i −0.250873 0.434524i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 15.5885i 0.737309 1.27706i −0.216394 0.976306i \(-0.569430\pi\)
0.953703 0.300750i \(-0.0972370\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.00000 13.8564i 0.638470 1.10586i −0.347299 0.937754i \(-0.612901\pi\)
0.985769 0.168107i \(-0.0537655\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 + 20.7846i 0.945732 + 1.63806i
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 + 20.7846i 0.928588 + 1.60836i 0.785687 + 0.618624i \(0.212310\pi\)
0.142901 + 0.989737i \(0.454357\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.00000 5.19615i 0.228086 0.395056i −0.729155 0.684349i \(-0.760087\pi\)
0.957241 + 0.289292i \(0.0934200\pi\)
\(174\) 0 0
\(175\) 10.0000 + 17.3205i 0.755929 + 1.30931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −1.00000 1.73205i −0.0743294 0.128742i 0.826465 0.562988i \(-0.190348\pi\)
−0.900794 + 0.434246i \(0.857015\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.00000 + 15.5885i −0.658145 + 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.73205i −0.0719816 + 0.124676i −0.899770 0.436365i \(-0.856266\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −5.00000 8.66025i −0.354441 0.613909i 0.632581 0.774494i \(-0.281995\pi\)
−0.987022 + 0.160585i \(0.948662\pi\)
\(200\) 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
\(209\) 10.5000 + 7.79423i 0.726300 + 0.539138i
\(210\) 0 0
\(211\) 10.0000 17.3205i 0.688428 1.19239i −0.283918 0.958849i \(-0.591634\pi\)
0.972346 0.233544i \(-0.0750324\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 7.00000 12.1244i 0.468755 0.811907i −0.530607 0.847618i \(-0.678036\pi\)
0.999362 + 0.0357107i \(0.0113695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.50000 2.59808i 0.0982683 0.170206i −0.812700 0.582683i \(-0.802003\pi\)
0.910968 + 0.412477i \(0.135336\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −2.50000 4.33013i −0.161039 0.278928i 0.774202 0.632938i \(-0.218151\pi\)
−0.935242 + 0.354010i \(0.884818\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 8.66025i 0.0636285 0.551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.50000 + 2.59808i 0.0946792 + 0.163989i 0.909475 0.415759i \(-0.136484\pi\)
−0.814795 + 0.579748i \(0.803151\pi\)
\(252\) 0 0
\(253\) 9.00000 + 15.5885i 0.565825 + 0.980038i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 + 2.59808i 0.0935674 + 0.162064i 0.909010 0.416775i \(-0.136840\pi\)
−0.815442 + 0.578838i \(0.803506\pi\)
\(258\) 0 0
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 10.3923i 0.369976 0.640817i −0.619586 0.784929i \(-0.712699\pi\)
0.989561 + 0.144112i \(0.0460326\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 10.3923i 0.365826 0.633630i −0.623082 0.782157i \(-0.714120\pi\)
0.988908 + 0.148527i \(0.0474530\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.50000 + 12.9904i 0.452267 + 0.783349i
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.5000 + 23.3827i 0.805342 + 1.39489i 0.916060 + 0.401042i \(0.131352\pi\)
−0.110717 + 0.993852i \(0.535315\pi\)
\(282\) 0 0
\(283\) 2.50000 4.33013i 0.148610 0.257399i −0.782104 0.623148i \(-0.785854\pi\)
0.930714 + 0.365748i \(0.119187\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.0000 31.1769i 1.06251 1.84032i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 10.3923i 0.346989 0.601003i
\(300\) 0 0
\(301\) −8.00000 + 13.8564i −0.461112 + 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3.50000 + 6.06218i −0.199756 + 0.345987i −0.948449 0.316929i \(-0.897348\pi\)
0.748694 + 0.662916i \(0.230681\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 9.50000 + 16.4545i 0.536972 + 0.930062i 0.999065 + 0.0432311i \(0.0137652\pi\)
−0.462093 + 0.886831i \(0.652902\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.00000 15.5885i −0.505490 0.875535i −0.999980 0.00635137i \(-0.997978\pi\)
0.494489 0.869184i \(-0.335355\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 + 10.3923i −1.33540 + 0.578243i
\(324\) 0 0
\(325\) 5.00000 8.66025i 0.277350 0.480384i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5.50000 + 9.52628i −0.299604 + 0.518930i −0.976045 0.217567i \(-0.930188\pi\)
0.676441 + 0.736497i \(0.263521\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.50000 7.79423i 0.241573 0.418416i −0.719590 0.694399i \(-0.755670\pi\)
0.961162 + 0.275983i \(0.0890035\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 5.19615i 0.158334 0.274242i −0.775934 0.630814i \(-0.782721\pi\)
0.934268 + 0.356572i \(0.116054\pi\)
\(360\) 0 0
\(361\) 5.50000 + 18.1865i 0.289474 + 0.957186i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −11.0000 19.0526i −0.574195 0.994535i −0.996129 0.0879086i \(-0.971982\pi\)
0.421933 0.906627i \(-0.361352\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 + 20.7846i 0.623009 + 1.07908i
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.0000 31.1769i 0.919757 1.59307i 0.119974 0.992777i \(-0.461719\pi\)
0.799783 0.600289i \(-0.204948\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 31.1769i −0.912636 1.58073i −0.810326 0.585980i \(-0.800710\pi\)
−0.102311 0.994753i \(-0.532624\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.00000 8.66025i 0.250943 0.434646i −0.712843 0.701324i \(-0.752593\pi\)
0.963786 + 0.266678i \(0.0859261\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.5000 + 23.3827i −0.674158 + 1.16768i 0.302556 + 0.953131i \(0.402160\pi\)
−0.976714 + 0.214544i \(0.931173\pi\)
\(402\) 0 0
\(403\) 2.00000 + 3.46410i 0.0996271 + 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −2.50000 4.33013i −0.123617 0.214111i 0.797574 0.603220i \(-0.206116\pi\)
−0.921192 + 0.389109i \(0.872783\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 18.0000 31.1769i 0.885722 1.53412i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 5.00000 8.66025i 0.243685 0.422075i −0.718076 0.695965i \(-0.754977\pi\)
0.961761 + 0.273890i \(0.0883103\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 8.00000 + 13.8564i 0.387147 + 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 + 25.9808i 0.722525 + 1.25145i 0.959985 + 0.280052i \(0.0903517\pi\)
−0.237460 + 0.971397i \(0.576315\pi\)
\(432\) 0 0
\(433\) −13.0000 22.5167i −0.624740 1.08208i −0.988591 0.150624i \(-0.951872\pi\)
0.363851 0.931457i \(-0.381462\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.00000 + 25.9808i −0.143509 + 1.24283i
\(438\) 0 0
\(439\) 7.00000 12.1244i 0.334092 0.578664i −0.649218 0.760602i \(-0.724904\pi\)
0.983310 + 0.181938i \(0.0582371\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.50000 7.79423i −0.213801 0.370315i 0.739100 0.673596i \(-0.235251\pi\)
−0.952901 + 0.303281i \(0.901918\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 13.5000 23.3827i 0.635690 1.10105i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 5.19615i 0.139724 0.242009i −0.787668 0.616100i \(-0.788712\pi\)
0.927392 + 0.374091i \(0.122045\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) −14.0000 24.2487i −0.646460 1.11970i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 + 10.3923i −0.275880 + 0.477839i
\(474\) 0 0
\(475\) −2.50000 + 21.6506i −0.114708 + 0.993399i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.0000 31.1769i −0.822441 1.42451i −0.903859 0.427830i \(-0.859278\pi\)
0.0814184 0.996680i \(-0.474055\pi\)
\(480\) 0 0
\(481\) 10.0000 + 17.3205i 0.455961 + 0.789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 20.7846i 0.538274 0.932317i
\(498\) 0 0
\(499\) −12.5000 + 21.6506i −0.559577 + 0.969216i 0.437955 + 0.898997i \(0.355703\pi\)
−0.997532 + 0.0702185i \(0.977630\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.00000 5.19615i −0.133763 0.231685i 0.791361 0.611349i \(-0.209373\pi\)
−0.925124 + 0.379664i \(0.876040\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.0000 20.7846i −0.531891 0.921262i −0.999307 0.0372243i \(-0.988148\pi\)
0.467416 0.884037i \(-0.345185\pi\)
\(510\) 0 0
\(511\) 2.00000 3.46410i 0.0884748 0.153243i
\(512\) 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
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) −14.0000 24.2487i −0.612177 1.06032i −0.990873 0.134801i \(-0.956961\pi\)
0.378695 0.925521i \(-0.376373\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 27.0000 1.16297
\(540\) 0 0
\(541\) −22.0000 38.1051i −0.945854 1.63827i −0.754032 0.656837i \(-0.771894\pi\)
−0.191821 0.981430i \(-0.561439\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 3.46410i −0.0855138 0.148114i 0.820096 0.572226i \(-0.193920\pi\)
−0.905610 + 0.424111i \(0.860587\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 + 13.8564i −0.340195 + 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.0000 + 20.7846i 0.508456 + 0.880672i 0.999952 + 0.00979220i \(0.00311700\pi\)
−0.491496 + 0.870880i \(0.663550\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15.0000 + 25.9808i −0.625543 + 1.08347i
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 9.00000 + 15.5885i 0.372742 + 0.645608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 10.3923i 0.247647 0.428936i −0.715226 0.698893i \(-0.753676\pi\)
0.962872 + 0.269957i \(0.0870095\pi\)
\(588\) 0 0
\(589\) −7.00000 5.19615i −0.288430 0.214104i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −10.5000 18.1865i −0.431183 0.746831i 0.565792 0.824548i \(-0.308570\pi\)
−0.996976 + 0.0777165i \(0.975237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.00000 + 5.19615i 0.122577 + 0.212309i 0.920783 0.390075i \(-0.127551\pi\)
−0.798206 + 0.602384i \(0.794218\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 + 1.73205i −0.0403896 + 0.0699569i −0.885514 0.464614i \(-0.846193\pi\)
0.845124 + 0.534570i \(0.179527\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.50000 + 2.59808i 0.0603877 + 0.104595i 0.894639 0.446790i \(-0.147433\pi\)
−0.834251 + 0.551385i \(0.814100\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000 + 20.7846i 0.480770 + 0.832718i
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 30.0000 51.9615i 1.19618 2.07184i
\(630\) 0 0
\(631\) −14.0000 24.2487i −0.557331 0.965326i −0.997718 0.0675178i \(-0.978492\pi\)
0.440387 0.897808i \(-0.354841\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.00000 15.5885i −0.356593 0.617637i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −19.5000 + 33.7750i −0.770204 + 1.33403i 0.167247 + 0.985915i \(0.446512\pi\)
−0.937451 + 0.348117i \(0.886821\pi\)
\(642\) 0 0
\(643\) −21.5000 + 37.2391i −0.847877 + 1.46857i 0.0352216 + 0.999380i \(0.488786\pi\)
−0.883099 + 0.469187i \(0.844547\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 13.5000 23.3827i 0.529921 0.917851i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18.0000 31.1769i −0.701180 1.21448i −0.968052 0.250748i \(-0.919323\pi\)
0.266872 0.963732i \(-0.414010\pi\)
\(660\) 0 0
\(661\) 20.0000 + 34.6410i 0.777910 + 1.34738i 0.933144 + 0.359502i \(0.117053\pi\)
−0.155235 + 0.987878i \(0.549613\pi\)
\(662\) 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
\(671\) 6.00000 + 10.3923i 0.231627 + 0.401190i
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −34.0000 + 58.8897i −1.30480 + 2.25998i
\(680\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 10.3923i 0.228582 0.395915i
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 27.0000 + 46.7654i 1.02270 + 1.77136i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.0000 + 20.7846i −0.453234 + 0.785024i −0.998585 0.0531839i \(-0.983063\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(702\) 0 0
\(703\) −35.0000 25.9808i −1.32005 0.979883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.00000 12.1244i −0.262891 0.455340i 0.704118 0.710083i \(-0.251342\pi\)
−0.967009 + 0.254743i \(0.918009\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.00000 10.3923i −0.224702 0.389195i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.0000 + 25.9808i −0.559406 + 0.968919i 0.438141 + 0.898906i \(0.355637\pi\)
−0.997546 + 0.0700124i \(0.977696\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 16.0000 27.7128i 0.593407 1.02781i −0.400362 0.916357i \(-0.631116\pi\)
0.993770 0.111454i \(-0.0355509\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 20.7846i −0.443836 0.768747i
\(732\) 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
\(737\) −10.5000 18.1865i −0.386772 0.669910i
\(738\) 0 0
\(739\) 17.5000 30.3109i 0.643748 1.11500i −0.340841 0.940121i \(-0.610712\pi\)
0.984589 0.174883i \(-0.0559548\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 15.5885i 0.330178 0.571885i −0.652369 0.757902i \(-0.726225\pi\)
0.982547 + 0.186017i \(0.0595579\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19.0000 + 32.9090i 0.693320 + 1.20087i 0.970744 + 0.240118i \(0.0771860\pi\)
−0.277424 + 0.960748i \(0.589481\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.00000 8.66025i 0.181728 0.314762i −0.760741 0.649056i \(-0.775164\pi\)
0.942469 + 0.334293i \(0.108498\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39.0000 1.41375 0.706874 0.707339i \(-0.250105\pi\)
0.706874 + 0.707339i \(0.250105\pi\)
\(762\) 0 0
\(763\) 32.0000 55.4256i 1.15848 2.00654i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.0000 −0.649942
\(768\) 0 0
\(769\) −1.00000 1.73205i −0.0360609 0.0624593i 0.847432 0.530904i \(-0.178148\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24.0000 41.5692i −0.863220 1.49514i −0.868804 0.495156i \(-0.835111\pi\)
0.00558380 0.999984i \(-0.498223\pi\)
\(774\) 0 0
\(775\) −5.00000 8.66025i −0.179605 0.311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.0000 15.5885i 1.28983 0.558514i
\(780\) 0 0
\(781\) 9.00000 15.5885i 0.322045 0.557799i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −60.0000 −2.13335
\(792\) 0 0
\(793\) 4.00000 6.92820i 0.142044 0.246028i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.50000 2.59808i 0.0529339 0.0916841i
\(804\)