Properties

Label 2016.2.s.f.865.1
Level $2016$
Weight $2$
Character 2016.865
Analytic conductor $16.098$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2016.865
Dual form 2016.2.s.f.289.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(-0.500000 + 2.59808i) q^{7} +(-1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(-1.00000 + 1.73205i) q^{17} +(1.50000 + 2.59808i) q^{19} +(-1.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} -8.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(2.50000 + 4.33013i) q^{37} -2.00000 q^{41} -7.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-1.00000 + 1.73205i) q^{53} +(-5.00000 + 8.66025i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-5.50000 + 9.52628i) q^{67} +12.0000 q^{71} +(1.50000 - 2.59808i) q^{73} +(-4.00000 - 3.46410i) q^{77} +(8.50000 + 14.7224i) q^{79} -16.0000 q^{83} +(6.00000 + 10.3923i) q^{89} +(-2.50000 + 12.9904i) q^{91} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} - 2q^{11} + 10q^{13} - 2q^{17} + 3q^{19} - 2q^{23} + 5q^{25} - 16q^{29} - q^{31} + 5q^{37} - 4q^{41} - 14q^{43} + 8q^{47} - 13q^{49} - 2q^{53} - 10q^{59} + 2q^{61} - 11q^{67} + 24q^{71} + 3q^{73} - 8q^{77} + 17q^{79} - 32q^{83} + 12q^{89} - 5q^{91} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 + 1.73205i −0.301511 + 0.522233i −0.976478 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) 1.50000 + 2.59808i 0.344124 + 0.596040i 0.985194 0.171442i \(-0.0548427\pi\)
−0.641071 + 0.767482i \(0.721509\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 1.73205i −0.208514 0.361158i 0.742732 0.669588i \(-0.233529\pi\)
−0.951247 + 0.308431i \(0.900196\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −0.500000 + 0.866025i −0.0898027 + 0.155543i −0.907428 0.420208i \(-0.861957\pi\)
0.817625 + 0.575751i \(0.195290\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.50000 + 4.33013i 0.410997 + 0.711868i 0.994999 0.0998840i \(-0.0318472\pi\)
−0.584002 + 0.811752i \(0.698514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 + 1.73205i −0.137361 + 0.237915i −0.926497 0.376303i \(-0.877195\pi\)
0.789136 + 0.614218i \(0.210529\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.00000 + 8.66025i −0.650945 + 1.12747i 0.331949 + 0.943297i \(0.392294\pi\)
−0.982894 + 0.184172i \(0.941040\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 + 9.52628i −0.671932 + 1.16382i 0.305424 + 0.952217i \(0.401202\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 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
\(73\) 1.50000 2.59808i 0.175562 0.304082i −0.764794 0.644275i \(-0.777159\pi\)
0.940356 + 0.340193i \(0.110493\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 3.46410i −0.455842 0.394771i
\(78\) 0 0
\(79\) 8.50000 + 14.7224i 0.956325 + 1.65640i 0.731307 + 0.682048i \(0.238911\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 + 10.3923i 0.635999 + 1.10158i 0.986303 + 0.164946i \(0.0527450\pi\)
−0.350304 + 0.936636i \(0.613922\pi\)
\(90\) 0 0
\(91\) −2.50000 + 12.9904i −0.262071 + 1.36176i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 15.5885i 0.895533 1.55111i 0.0623905 0.998052i \(-0.480128\pi\)
0.833143 0.553058i \(-0.186539\pi\)
\(102\) 0 0
\(103\) −0.500000 0.866025i −0.0492665 0.0853320i 0.840341 0.542059i \(-0.182355\pi\)
−0.889607 + 0.456727i \(0.849022\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.00000 1.73205i −0.0966736 0.167444i 0.813632 0.581380i \(-0.197487\pi\)
−0.910306 + 0.413936i \(0.864154\pi\)
\(108\) 0 0
\(109\) −0.500000 + 0.866025i −0.0478913 + 0.0829502i −0.888977 0.457951i \(-0.848583\pi\)
0.841086 + 0.540901i \(0.181917\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 3.46410i −0.366679 0.317554i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) −7.50000 + 2.59808i −0.650332 + 0.225282i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 3.46410i 0.170872 0.295958i −0.767853 0.640626i \(-0.778675\pi\)
0.938725 + 0.344668i \(0.112008\pi\)
\(138\) 0 0
\(139\) 1.00000 0.0848189 0.0424094 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.00000 + 8.66025i −0.418121 + 0.724207i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 + 10.3923i 0.491539 + 0.851371i 0.999953 0.00974235i \(-0.00310113\pi\)
−0.508413 + 0.861113i \(0.669768\pi\)
\(150\) 0 0
\(151\) 4.00000 6.92820i 0.325515 0.563809i −0.656101 0.754673i \(-0.727796\pi\)
0.981617 + 0.190864i \(0.0611289\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.00000 5.19615i 0.239426 0.414698i −0.721124 0.692806i \(-0.756374\pi\)
0.960550 + 0.278108i \(0.0897074\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.00000 1.73205i 0.394055 0.136505i
\(162\) 0 0
\(163\) 2.00000 + 3.46410i 0.156652 + 0.271329i 0.933659 0.358162i \(-0.116597\pi\)
−0.777007 + 0.629492i \(0.783263\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.00000 5.19615i −0.228086 0.395056i 0.729155 0.684349i \(-0.239913\pi\)
−0.957241 + 0.289292i \(0.906580\pi\)
\(174\) 0 0
\(175\) 10.0000 + 8.66025i 0.755929 + 0.654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) −15.0000 −1.11494 −0.557471 0.830197i \(-0.688228\pi\)
−0.557471 + 0.830197i \(0.688228\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.00000 3.46410i −0.146254 0.253320i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 10.3923i −0.434145 0.751961i 0.563081 0.826402i \(-0.309616\pi\)
−0.997225 + 0.0744412i \(0.976283\pi\)
\(192\) 0 0
\(193\) −11.5000 + 19.9186i −0.827788 + 1.43377i 0.0719816 + 0.997406i \(0.477068\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 2.00000 3.46410i 0.141776 0.245564i −0.786389 0.617731i \(-0.788052\pi\)
0.928166 + 0.372168i \(0.121385\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.00000 20.7846i 0.280745 1.45879i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 1.73205i −0.135769 0.117579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.00000 + 8.66025i −0.336336 + 0.582552i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.0000 + 22.5167i −0.862840 + 1.49448i 0.00633544 + 0.999980i \(0.497983\pi\)
−0.869176 + 0.494503i \(0.835350\pi\)
\(228\) 0 0
\(229\) −6.50000 11.2583i −0.429532 0.743971i 0.567300 0.823511i \(-0.307988\pi\)
−0.996832 + 0.0795401i \(0.974655\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 10.3923i −0.393073 0.680823i 0.599780 0.800165i \(-0.295255\pi\)
−0.992853 + 0.119342i \(0.961921\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) 11.0000 19.0526i 0.708572 1.22728i −0.256814 0.966461i \(-0.582673\pi\)
0.965387 0.260822i \(-0.0839937\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.50000 + 12.9904i 0.477214 + 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −22.0000 −1.38863 −0.694314 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.00000 13.8564i −0.499026 0.864339i 0.500973 0.865463i \(-0.332976\pi\)
−0.999999 + 0.00112398i \(0.999642\pi\)
\(258\) 0 0
\(259\) −12.5000 + 4.33013i −0.776712 + 0.269061i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.00000 + 6.92820i −0.246651 + 0.427211i −0.962594 0.270947i \(-0.912663\pi\)
0.715944 + 0.698158i \(0.245997\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.00000 + 15.5885i −0.548740 + 0.950445i 0.449622 + 0.893219i \(0.351559\pi\)
−0.998361 + 0.0572259i \(0.981774\pi\)
\(270\) 0 0
\(271\) −8.00000 13.8564i −0.485965 0.841717i 0.513905 0.857847i \(-0.328199\pi\)
−0.999870 + 0.0161307i \(0.994865\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.00000 + 8.66025i 0.301511 + 0.522233i
\(276\) 0 0
\(277\) −8.50000 + 14.7224i −0.510716 + 0.884585i 0.489207 + 0.872167i \(0.337286\pi\)
−0.999923 + 0.0124177i \(0.996047\pi\)
\(278\) 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
\(283\) 15.5000 26.8468i 0.921379 1.59588i 0.124096 0.992270i \(-0.460397\pi\)
0.797283 0.603606i \(-0.206270\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00000 5.19615i 0.0590281 0.306719i
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.00000 8.66025i −0.289157 0.500835i
\(300\) 0 0
\(301\) 3.50000 18.1865i 0.201737 1.04825i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 23.0000 1.31268 0.656340 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.0000 25.9808i 0.850572 1.47323i −0.0301210 0.999546i \(-0.509589\pi\)
0.880693 0.473688i \(-0.157077\pi\)
\(312\) 0 0
\(313\) 8.50000 + 14.7224i 0.480448 + 0.832161i 0.999748 0.0224310i \(-0.00714060\pi\)
−0.519300 + 0.854592i \(0.673807\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0000 + 17.3205i 0.561656 + 0.972817i 0.997352 + 0.0727229i \(0.0231689\pi\)
−0.435696 + 0.900094i \(0.643498\pi\)
\(318\) 0 0
\(319\) 8.00000 13.8564i 0.447914 0.775810i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 12.5000 21.6506i 0.693375 1.20096i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.0000 + 6.92820i −1.10264 + 0.381964i
\(330\) 0 0
\(331\) −7.50000 12.9904i −0.412237 0.714016i 0.582897 0.812546i \(-0.301919\pi\)
−0.995134 + 0.0985303i \(0.968586\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.00000 1.73205i −0.0541530 0.0937958i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.0000 25.9808i 0.805242 1.39472i −0.110885 0.993833i \(-0.535369\pi\)
0.916127 0.400887i \(-0.131298\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 20.7846i 0.638696 1.10625i −0.347024 0.937856i \(-0.612808\pi\)
0.985719 0.168397i \(-0.0538590\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.0000 27.7128i −0.844448 1.46263i −0.886100 0.463494i \(-0.846596\pi\)
0.0416523 0.999132i \(-0.486738\pi\)
\(360\) 0 0
\(361\) 5.00000 8.66025i 0.263158 0.455803i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.50000 14.7224i 0.443696 0.768505i −0.554264 0.832341i \(-0.687000\pi\)
0.997960 + 0.0638362i \(0.0203335\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 3.46410i −0.207670 0.179847i
\(372\) 0 0
\(373\) 10.5000 + 18.1865i 0.543669 + 0.941663i 0.998689 + 0.0511818i \(0.0162988\pi\)
−0.455020 + 0.890481i \(0.650368\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) −1.00000 −0.0513665 −0.0256833 0.999670i \(-0.508176\pi\)
−0.0256833 + 0.999670i \(0.508176\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.0000 + 19.0526i 0.562074 + 0.973540i 0.997315 + 0.0732266i \(0.0233296\pi\)
−0.435242 + 0.900314i \(0.643337\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.0000 + 20.7846i −0.608424 + 1.05382i 0.383076 + 0.923717i \(0.374865\pi\)
−0.991500 + 0.130105i \(0.958469\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.50000 + 11.2583i 0.326226 + 0.565039i 0.981760 0.190126i \(-0.0608897\pi\)
−0.655534 + 0.755166i \(0.727556\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.00000 + 8.66025i 0.249688 + 0.432472i 0.963439 0.267927i \(-0.0863386\pi\)
−0.713751 + 0.700399i \(0.753005\pi\)
\(402\) 0 0
\(403\) −2.50000 + 4.33013i −0.124534 + 0.215699i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 17.5000 30.3109i 0.865319 1.49878i −0.00141047 0.999999i \(-0.500449\pi\)
0.866730 0.498778i \(-0.166218\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −20.0000 17.3205i −0.984136 0.852286i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.00000 + 8.66025i 0.242536 + 0.420084i
\(426\) 0 0
\(427\) −5.00000 + 1.73205i −0.241967 + 0.0838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.0000 + 17.3205i −0.481683 + 0.834300i −0.999779 0.0210230i \(-0.993308\pi\)
0.518096 + 0.855323i \(0.326641\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.00000 5.19615i 0.143509 0.248566i
\(438\) 0 0
\(439\) −12.0000 20.7846i −0.572729 0.991995i −0.996284 0.0861252i \(-0.972552\pi\)
0.423556 0.905870i \(-0.360782\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.00000 5.19615i −0.142534 0.246877i 0.785916 0.618333i \(-0.212192\pi\)
−0.928450 + 0.371457i \(0.878858\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 2.00000 3.46410i 0.0941763 0.163118i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.5000 + 19.9186i 0.537947 + 0.931752i 0.999014 + 0.0443868i \(0.0141334\pi\)
−0.461067 + 0.887365i \(0.652533\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) −9.00000 −0.418265 −0.209133 0.977887i \(-0.567064\pi\)
−0.209133 + 0.977887i \(0.567064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.00000 10.3923i −0.277647 0.480899i 0.693153 0.720791i \(-0.256221\pi\)
−0.970799 + 0.239892i \(0.922888\pi\)
\(468\) 0 0
\(469\) −22.0000 19.0526i −1.01587 0.879765i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.00000 12.1244i 0.321860 0.557478i
\(474\) 0 0
\(475\) 15.0000 0.688247
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.00000 + 15.5885i −0.411220 + 0.712255i −0.995023 0.0996406i \(-0.968231\pi\)
0.583803 + 0.811895i \(0.301564\pi\)
\(480\) 0 0
\(481\) 12.5000 + 21.6506i 0.569951 + 0.987184i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 10.5000 18.1865i 0.475800 0.824110i −0.523815 0.851832i \(-0.675492\pi\)
0.999616 + 0.0277214i \(0.00882512\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 0 0
\(493\) 8.00000 13.8564i 0.360302 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 + 31.1769i −0.269137 + 1.39848i
\(498\) 0 0
\(499\) −20.5000 35.5070i −0.917706 1.58951i −0.802890 0.596127i \(-0.796706\pi\)
−0.114816 0.993387i \(-0.536628\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.00000 3.46410i −0.0886484 0.153544i 0.818292 0.574803i \(-0.194921\pi\)
−0.906940 + 0.421260i \(0.861588\pi\)
\(510\) 0 0
\(511\) 6.00000 + 5.19615i 0.265424 + 0.229864i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.0000 27.7128i 0.700973 1.21412i −0.267153 0.963654i \(-0.586083\pi\)
0.968125 0.250466i \(-0.0805839\pi\)
\(522\) 0 0
\(523\) −5.50000 9.52628i −0.240498 0.416555i 0.720358 0.693602i \(-0.243977\pi\)
−0.960856 + 0.277047i \(0.910644\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.00000 1.73205i −0.0435607 0.0754493i
\(528\) 0 0
\(529\) 9.50000 16.4545i 0.413043 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.0000 8.66025i 0.473804 0.373024i
\(540\) 0 0
\(541\) 13.5000 + 23.3827i 0.580410 + 1.00530i 0.995431 + 0.0954880i \(0.0304412\pi\)
−0.415020 + 0.909812i \(0.636226\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 20.7846i −0.511217 0.885454i
\(552\) 0 0
\(553\) −42.5000 + 14.7224i −1.80728 + 0.626061i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.00000 15.5885i 0.381342 0.660504i −0.609912 0.792469i \(-0.708795\pi\)
0.991254 + 0.131965i \(0.0421286\pi\)
\(558\) 0 0
\(559\) −35.0000 −1.48034
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.0000 17.3205i 0.421450 0.729972i −0.574632 0.818412i \(-0.694855\pi\)
0.996082 + 0.0884397i \(0.0281881\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.0000 + 36.3731i 0.880366 + 1.52484i 0.850935 + 0.525271i \(0.176036\pi\)
0.0294311 + 0.999567i \(0.490630\pi\)
\(570\) 0 0
\(571\) 11.5000 19.9186i 0.481260 0.833567i −0.518509 0.855072i \(-0.673513\pi\)
0.999769 + 0.0215055i \(0.00684595\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −10.0000 −0.417029
\(576\) 0 0
\(577\) −3.50000 + 6.06218i −0.145707 + 0.252372i −0.929636 0.368478i \(-0.879879\pi\)
0.783930 + 0.620850i \(0.213212\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 41.5692i 0.331896 1.72458i
\(582\) 0 0
\(583\) −2.00000 3.46410i −0.0828315 0.143468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.0000 + 25.9808i 0.615976 + 1.06690i 0.990212 + 0.139569i \(0.0445716\pi\)
−0.374236 + 0.927333i \(0.622095\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.0000 31.1769i 0.735460 1.27385i −0.219061 0.975711i \(-0.570299\pi\)
0.954521 0.298143i \(-0.0963673\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.5000 + 28.5788i 0.669714 + 1.15998i 0.977984 + 0.208680i \(0.0669168\pi\)
−0.308270 + 0.951299i \(0.599750\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.0000 + 34.6410i 0.809113 + 1.40143i
\(612\) 0 0
\(613\) −13.0000 + 22.5167i −0.525065 + 0.909439i 0.474509 + 0.880251i \(0.342626\pi\)
−0.999574 + 0.0291886i \(0.990708\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) 2.50000 4.33013i 0.100483 0.174042i −0.811400 0.584491i \(-0.801294\pi\)
0.911884 + 0.410448i \(0.134628\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −30.0000 + 10.3923i −1.20192 + 0.416359i
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −32.5000 12.9904i −1.28770 0.514698i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.0000 22.5167i 0.513469 0.889355i −0.486409 0.873731i \(-0.661693\pi\)
0.999878 0.0156233i \(-0.00497325\pi\)
\(642\) 0 0
\(643\) −11.0000 −0.433798 −0.216899 0.976194i \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000 15.5885i 0.353827 0.612845i −0.633090 0.774078i \(-0.718214\pi\)
0.986916 + 0.161233i \(0.0515470\pi\)
\(648\) 0 0
\(649\) −10.0000 17.3205i −0.392534 0.679889i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.00000 12.1244i −0.273931 0.474463i 0.695934 0.718106i \(-0.254991\pi\)
−0.969865 + 0.243643i \(0.921657\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −1.50000 + 2.59808i −0.0583432 + 0.101053i −0.893722 0.448622i \(-0.851915\pi\)
0.835379 + 0.549675i \(0.185248\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.00000 + 13.8564i 0.309761 + 0.536522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −21.0000 −0.809491 −0.404745 0.914429i \(-0.632640\pi\)
−0.404745 + 0.914429i \(0.632640\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.00000 10.3923i −0.230599 0.399409i 0.727386 0.686229i \(-0.240735\pi\)
−0.957984 + 0.286820i \(0.907402\pi\)
\(678\) 0 0
\(679\) 7.00000 36.3731i 0.268635 1.39587i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0000 25.9808i 0.573959 0.994126i −0.422195 0.906505i \(-0.638740\pi\)
0.996154 0.0876211i \(-0.0279265\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.00000 + 8.66025i −0.190485 + 0.329929i
\(690\) 0 0
\(691\) 2.50000 + 4.33013i 0.0951045 + 0.164726i 0.909652 0.415371i \(-0.136348\pi\)
−0.814548 + 0.580097i \(0.803015\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.00000 3.46410i 0.0757554 0.131212i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −7.50000 + 12.9904i −0.282868 + 0.489942i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000 + 31.1769i 1.35392 + 1.17253i
\(708\) 0 0
\(709\) 11.0000 + 19.0526i 0.413114 + 0.715534i 0.995228 0.0975728i \(-0.0311079\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.00000 + 5.19615i 0.111881 + 0.193784i 0.916529 0.399969i \(-0.130979\pi\)
−0.804648 + 0.593753i \(0.797646\pi\)
\(720\) 0 0
\(721\) 2.50000 0.866025i 0.0931049 0.0322525i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20.0000 + 34.6410i −0.742781 + 1.28654i
\(726\) 0 0
\(727\) −13.0000 −0.482143 −0.241072 0.970507i \(-0.577499\pi\)
−0.241072 + 0.970507i \(0.577499\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.00000 12.1244i 0.258904 0.448435i
\(732\) 0 0
\(733\) 9.50000 + 16.4545i 0.350891 + 0.607760i 0.986406 0.164328i \(-0.0525456\pi\)
−0.635515 + 0.772088i \(0.719212\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.0000 19.0526i −0.405190 0.701810i
\(738\) 0 0
\(739\) −8.50000 + 14.7224i −0.312678 + 0.541573i −0.978941 0.204143i \(-0.934559\pi\)
0.666264 + 0.745716i \(0.267893\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.0000 −0.513610 −0.256805 0.966463i \(-0.582670\pi\)
−0.256805 + 0.966463i \(0.582670\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.00000 1.73205i 0.182696 0.0632878i
\(750\) 0 0
\(751\) 13.5000 + 23.3827i 0.492622 + 0.853246i 0.999964 0.00849853i \(-0.00270520\pi\)
−0.507342 + 0.861745i \(0.669372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 + 25.9808i 0.543750 + 0.941802i 0.998684 + 0.0512772i \(0.0163292\pi\)
−0.454935 + 0.890525i \(0.650337\pi\)
\(762\) 0 0
\(763\) −2.00000 1.73205i −0.0724049 0.0627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25.0000 + 43.3013i −0.902698 + 1.56352i
\(768\) 0 0
\(769\) −9.00000 −0.324548 −0.162274 0.986746i \(-0.551883\pi\)
−0.162274 + 0.986746i \(0.551883\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.00000 15.5885i 0.323708 0.560678i −0.657542 0.753418i \(-0.728404\pi\)
0.981250 + 0.192740i \(0.0617373\pi\)
\(774\) 0 0
\(775\) 2.50000 + 4.33013i 0.0898027 + 0.155543i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.00000 5.19615i −0.107486 0.186171i
\(780\) 0 0
\(781\) −12.0000 + 20.7846i −0.429394 + 0.743732i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −20.0000 + 34.6410i −0.712923 + 1.23482i 0.250832 + 0.968031i \(0.419296\pi\)
−0.963755 + 0.266788i \(0.914038\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 + 31.1769i −0.213335 + 1.10852i
\(792\) 0 0
\(793\) 5.00000 + 8.66025i 0.177555 + 0.307535i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.00000 + 5.19615i 0.105868 + 0.183368i
\(804\) 0 0
\(805\) 0 0