Properties

Label 1620.2.i.h.1081.1
Level 1620
Weight 2
Character 1620.1081
Analytic conductor 12.936
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1081.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1620.1081
Dual form 1620.2.i.h.541.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-1.00000 + 1.73205i) q^{13} -6.00000 q^{17} -4.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(2.00000 - 3.46410i) q^{31} -2.00000 q^{35} +2.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(5.00000 + 8.66025i) q^{43} +(3.00000 + 5.19615i) q^{47} +(1.50000 - 2.59808i) q^{49} -6.00000 q^{53} +(-6.00000 + 10.3923i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-1.00000 + 1.73205i) q^{67} -12.0000 q^{71} +2.00000 q^{73} +(-4.00000 - 6.92820i) q^{79} +(-3.00000 - 5.19615i) q^{83} +(-3.00000 + 5.19615i) q^{85} -6.00000 q^{89} +4.00000 q^{91} +(-2.00000 + 3.46410i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{5} - 2q^{7} - 2q^{13} - 12q^{17} - 8q^{19} - 6q^{23} - q^{25} - 6q^{29} + 4q^{31} - 4q^{35} + 4q^{37} - 6q^{41} + 10q^{43} + 6q^{47} + 3q^{49} - 12q^{53} - 12q^{59} - 2q^{61} + 2q^{65} - 2q^{67} - 24q^{71} + 4q^{73} - 8q^{79} - 6q^{83} - 6q^{85} - 12q^{89} + 8q^{91} - 4q^{95} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −1.00000 1.73205i −0.377964 0.654654i 0.612801 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 5.19615i −0.557086 0.964901i −0.997738 0.0672232i \(-0.978586\pi\)
0.440652 0.897678i \(-0.354747\pi\)
\(30\) 0 0
\(31\) 2.00000 3.46410i 0.359211 0.622171i −0.628619 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) 5.00000 + 8.66025i 0.762493 + 1.32068i 0.941562 + 0.336840i \(0.109358\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) −1.00000 1.73205i −0.128037 0.221766i 0.794879 0.606768i \(-0.207534\pi\)
−0.922916 + 0.385002i \(0.874201\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 + 1.73205i 0.124035 + 0.214834i
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 6.92820i −0.450035 0.779484i 0.548352 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 5.19615i −0.329293 0.570352i 0.653079 0.757290i \(-0.273477\pi\)
−0.982372 + 0.186938i \(0.940144\pi\)
\(84\) 0 0
\(85\) −3.00000 + 5.19615i −0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 + 3.46410i −0.205196 + 0.355409i
\(96\) 0 0
\(97\) −1.00000 1.73205i −0.101535 0.175863i 0.810782 0.585348i \(-0.199042\pi\)
−0.912317 + 0.409484i \(0.865709\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) −7.00000 + 12.1244i −0.689730 + 1.19465i 0.282194 + 0.959357i \(0.408938\pi\)
−0.971925 + 0.235291i \(0.924396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.00000 5.19615i 0.282216 0.488813i −0.689714 0.724082i \(-0.742264\pi\)
0.971930 + 0.235269i \(0.0755971\pi\)
\(114\) 0 0
\(115\) 3.00000 + 5.19615i 0.279751 + 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 + 10.3923i 0.550019 + 0.952661i
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 4.00000 + 6.92820i 0.346844 + 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 15.5885i −0.768922 1.33181i −0.938148 0.346235i \(-0.887460\pi\)
0.169226 0.985577i \(-0.445873\pi\)
\(138\) 0 0
\(139\) 2.00000 3.46410i 0.169638 0.293821i −0.768655 0.639664i \(-0.779074\pi\)
0.938293 + 0.345843i \(0.112407\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 5.19615i 0.245770 0.425685i −0.716578 0.697507i \(-0.754293\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) −10.0000 17.3205i −0.813788 1.40952i −0.910195 0.414181i \(-0.864068\pi\)
0.0964061 0.995342i \(-0.469265\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 3.46410i −0.160644 0.278243i
\(156\) 0 0
\(157\) 11.0000 19.0526i 0.877896 1.52056i 0.0242497 0.999706i \(-0.492280\pi\)
0.853646 0.520854i \(-0.174386\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.00000 + 15.5885i −0.696441 + 1.20627i 0.273252 + 0.961943i \(0.411901\pi\)
−0.969693 + 0.244328i \(0.921432\pi\)
\(168\) 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 0.957241 0.289292i \(-0.0934200\pi\)
−0.729155 + 0.684349i \(0.760087\pi\)
\(174\) 0 0
\(175\) −1.00000 + 1.73205i −0.0755929 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 1.73205i 0.0735215 0.127343i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 + 10.3923i 0.434145 + 0.751961i 0.997225 0.0744412i \(-0.0237173\pi\)
−0.563081 + 0.826402i \(0.690384\pi\)
\(192\) 0 0
\(193\) −13.0000 + 22.5167i −0.935760 + 1.62078i −0.162488 + 0.986710i \(0.551952\pi\)
−0.773272 + 0.634074i \(0.781381\pi\)
\(194\) 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
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 + 10.3923i −0.421117 + 0.729397i
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 13.8564i 0.550743 0.953914i −0.447478 0.894295i \(-0.647678\pi\)
0.998221 0.0596196i \(-0.0189888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 10.3923i 0.403604 0.699062i
\(222\) 0 0
\(223\) 5.00000 + 8.66025i 0.334825 + 0.579934i 0.983451 0.181173i \(-0.0579895\pi\)
−0.648626 + 0.761107i \(0.724656\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.00000 + 5.19615i 0.199117 + 0.344881i 0.948242 0.317547i \(-0.102859\pi\)
−0.749125 + 0.662428i \(0.769526\pi\)
\(228\) 0 0
\(229\) −7.00000 + 12.1244i −0.462573 + 0.801200i −0.999088 0.0426906i \(-0.986407\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 20.7846i 0.776215 1.34444i −0.157893 0.987456i \(-0.550470\pi\)
0.934109 0.356988i \(-0.116196\pi\)
\(240\) 0 0
\(241\) −7.00000 12.1244i −0.450910 0.780998i 0.547533 0.836784i \(-0.315567\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.50000 2.59808i −0.0958315 0.165985i
\(246\) 0 0
\(247\) 4.00000 6.92820i 0.254514 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(257\) 3.00000 5.19615i 0.187135 0.324127i −0.757159 0.653231i \(-0.773413\pi\)
0.944294 + 0.329104i \(0.106747\pi\)
\(258\) 0 0
\(259\) −2.00000 3.46410i −0.124274 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000 + 15.5885i 0.554964 + 0.961225i 0.997906 + 0.0646755i \(0.0206012\pi\)
−0.442943 + 0.896550i \(0.646065\pi\)
\(264\) 0 0
\(265\) −3.00000 + 5.19615i −0.184289 + 0.319197i
\(266\) 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
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.0000 22.5167i −0.781094 1.35290i −0.931305 0.364241i \(-0.881328\pi\)
0.150210 0.988654i \(-0.452005\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.00000 5.19615i −0.178965 0.309976i 0.762561 0.646916i \(-0.223942\pi\)
−0.941526 + 0.336939i \(0.890608\pi\)
\(282\) 0 0
\(283\) −7.00000 + 12.1244i −0.416107 + 0.720718i −0.995544 0.0942988i \(-0.969939\pi\)
0.579437 + 0.815017i \(0.303272\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.0000 25.9808i 0.876309 1.51781i 0.0209480 0.999781i \(-0.493332\pi\)
0.855361 0.518032i \(-0.173335\pi\)
\(294\) 0 0
\(295\) 6.00000 + 10.3923i 0.349334 + 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 10.3923i −0.346989 0.601003i
\(300\) 0 0
\(301\) 10.0000 17.3205i 0.576390 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 + 10.3923i −0.340229 + 0.589294i −0.984475 0.175525i \(-0.943838\pi\)
0.644246 + 0.764818i \(0.277171\pi\)
\(312\) 0 0
\(313\) 11.0000 + 19.0526i 0.621757 + 1.07691i 0.989158 + 0.146852i \(0.0469141\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 + 5.19615i 0.168497 + 0.291845i 0.937892 0.346929i \(-0.112775\pi\)
−0.769395 + 0.638774i \(0.779442\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.00000 10.3923i 0.330791 0.572946i
\(330\) 0 0
\(331\) −4.00000 6.92820i −0.219860 0.380808i 0.734905 0.678170i \(-0.237227\pi\)
−0.954765 + 0.297361i \(0.903893\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 + 1.73205i 0.0546358 + 0.0946320i
\(336\) 0 0
\(337\) −1.00000 + 1.73205i −0.0544735 + 0.0943508i −0.891976 0.452082i \(-0.850681\pi\)
0.837503 + 0.546433i \(0.184015\pi\)
\(338\) 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
\(347\) 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
\(349\) 5.00000 + 8.66025i 0.267644 + 0.463573i 0.968253 0.249973i \(-0.0804216\pi\)
−0.700609 + 0.713545i \(0.747088\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.00000 15.5885i −0.479022 0.829690i 0.520689 0.853746i \(-0.325675\pi\)
−0.999711 + 0.0240566i \(0.992342\pi\)
\(354\) 0 0
\(355\) −6.00000 + 10.3923i −0.318447 + 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000 1.73205i 0.0523424 0.0906597i
\(366\) 0 0
\(367\) 11.0000 + 19.0526i 0.574195 + 0.994535i 0.996129 + 0.0879086i \(0.0280183\pi\)
−0.421933 + 0.906627i \(0.638648\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 + 10.3923i 0.311504 + 0.539542i
\(372\) 0 0
\(373\) −13.0000 + 22.5167i −0.673114 + 1.16587i 0.303902 + 0.952703i \(0.401711\pi\)
−0.977016 + 0.213165i \(0.931623\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.00000 + 5.19615i −0.153293 + 0.265511i −0.932436 0.361335i \(-0.882321\pi\)
0.779143 + 0.626846i \(0.215654\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.00000 + 5.19615i 0.152106 + 0.263455i 0.932002 0.362454i \(-0.118061\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(390\) 0 0
\(391\) 18.0000 31.1769i 0.910299 1.57668i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 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
\(401\) 15.0000 25.9808i 0.749064 1.29742i −0.199207 0.979957i \(-0.563837\pi\)
0.948272 0.317460i \(-0.102830\pi\)
\(402\) 0 0
\(403\) 4.00000 + 6.92820i 0.199254 + 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 17.0000 29.4449i 0.840596 1.45595i −0.0487958 0.998809i \(-0.515538\pi\)
0.889392 0.457146i \(-0.151128\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.0000 + 31.1769i −0.879358 + 1.52309i −0.0273103 + 0.999627i \(0.508694\pi\)
−0.852047 + 0.523465i \(0.824639\pi\)
\(420\) 0 0
\(421\) −13.0000 22.5167i −0.633581 1.09739i −0.986814 0.161859i \(-0.948251\pi\)
0.353233 0.935536i \(-0.385082\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 + 5.19615i 0.145521 + 0.252050i
\(426\) 0 0
\(427\) −2.00000 + 3.46410i −0.0967868 + 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000 20.7846i 0.574038 0.994263i
\(438\) 0 0
\(439\) −4.00000 6.92820i −0.190910 0.330665i 0.754642 0.656136i \(-0.227810\pi\)
−0.945552 + 0.325471i \(0.894477\pi\)
\(440\) 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
\(445\) −3.00000 + 5.19615i −0.142214 + 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 3.46410i 0.0937614 0.162400i
\(456\) 0 0
\(457\) −13.0000 22.5167i −0.608114 1.05328i −0.991551 0.129718i \(-0.958593\pi\)
0.383437 0.923567i \(-0.374740\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −15.0000 25.9808i −0.698620 1.21004i −0.968945 0.247276i \(-0.920465\pi\)
0.270326 0.962769i \(-0.412869\pi\)
\(462\) 0 0
\(463\) −7.00000 + 12.1244i −0.325318 + 0.563467i −0.981577 0.191069i \(-0.938805\pi\)
0.656259 + 0.754536i \(0.272138\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.00000 + 3.46410i 0.0917663 + 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 + 20.7846i 0.548294 + 0.949673i 0.998392 + 0.0566937i \(0.0180558\pi\)
−0.450098 + 0.892979i \(0.648611\pi\)
\(480\) 0 0
\(481\) −2.00000 + 3.46410i −0.0911922 + 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 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
\(493\) 18.0000 + 31.1769i 0.810679 + 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 + 20.7846i 0.538274 + 0.932317i
\(498\) 0 0
\(499\) 2.00000 3.46410i 0.0895323 0.155074i −0.817781 0.575529i \(-0.804796\pi\)
0.907314 + 0.420455i \(0.138129\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.00000 + 5.19615i −0.132973 + 0.230315i −0.924821 0.380402i \(-0.875786\pi\)
0.791849 + 0.610718i \(0.209119\pi\)
\(510\) 0 0
\(511\) −2.00000 3.46410i −0.0884748 0.153243i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.00000 + 12.1244i 0.308457 + 0.534263i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 + 20.7846i −0.522728 + 0.905392i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 10.3923i −0.259889 0.450141i
\(534\) 0 0
\(535\) −3.00000 + 5.19615i −0.129701 + 0.224649i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.00000 1.73205i 0.0428353 0.0741929i
\(546\) 0 0
\(547\) −13.0000 22.5167i −0.555840 0.962743i −0.997838 0.0657267i \(-0.979063\pi\)
0.441998 0.897016i \(-0.354270\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 + 20.7846i 0.511217 + 0.885454i
\(552\) 0 0
\(553\) −8.00000 + 13.8564i −0.340195 + 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.00000 15.5885i 0.379305 0.656975i −0.611656 0.791123i \(-0.709497\pi\)
0.990961 + 0.134148i \(0.0428299\pi\)
\(564\) 0 0
\(565\) −3.00000 5.19615i −0.126211 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.0000 25.9808i −0.628833 1.08917i −0.987786 0.155815i \(-0.950200\pi\)
0.358954 0.933355i \(-0.383134\pi\)
\(570\) 0 0
\(571\) −4.00000 + 6.92820i −0.167395 + 0.289936i −0.937503 0.347977i \(-0.886869\pi\)
0.770108 + 0.637913i \(0.220202\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 + 10.3923i −0.248922 + 0.431145i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000 + 5.19615i 0.123823 + 0.214468i 0.921272 0.388918i \(-0.127151\pi\)
−0.797449 + 0.603386i \(0.793818\pi\)
\(588\) 0 0
\(589\) −8.00000 + 13.8564i −0.329634 + 0.570943i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 5.00000 + 8.66025i 0.203954 + 0.353259i 0.949799 0.312861i \(-0.101287\pi\)
−0.745845 + 0.666120i \(0.767954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.50000 9.52628i −0.223607 0.387298i
\(606\) 0 0
\(607\) 11.0000 19.0526i 0.446476 0.773320i −0.551678 0.834058i \(-0.686012\pi\)
0.998154 + 0.0607380i \(0.0193454\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 5.19615i 0.120775 0.209189i −0.799298 0.600935i \(-0.794795\pi\)
0.920074 + 0.391745i \(0.128129\pi\)
\(618\) 0 0
\(619\) −10.0000 17.3205i −0.401934 0.696170i 0.592025 0.805919i \(-0.298329\pi\)
−0.993959 + 0.109749i \(0.964995\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 + 10.3923i 0.240385 + 0.416359i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00000 1.73205i 0.0396838 0.0687343i
\(636\) 0 0
\(637\) 3.00000 + 5.19615i 0.118864 + 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 + 15.5885i 0.355479 + 0.615707i 0.987200 0.159489i \(-0.0509845\pi\)
−0.631721 + 0.775196i \(0.717651\pi\)
\(642\) 0 0
\(643\) −7.00000 + 12.1244i −0.276053 + 0.478138i −0.970400 0.241502i \(-0.922360\pi\)
0.694347 + 0.719640i \(0.255693\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.0000 + 36.3731i −0.821794 + 1.42339i 0.0825519 + 0.996587i \(0.473693\pi\)
−0.904345 + 0.426801i \(0.859640\pi\)
\(654\) 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
\(661\) 11.0000 19.0526i 0.427850 0.741059i −0.568831 0.822454i \(-0.692604\pi\)
0.996682 + 0.0813955i \(0.0259377\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 23.0000 + 39.8372i 0.886585 + 1.53561i 0.843886 + 0.536522i \(0.180262\pi\)
0.0426985 + 0.999088i \(0.486405\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.00000 15.5885i −0.345898 0.599113i 0.639618 0.768693i \(-0.279092\pi\)
−0.985517 + 0.169580i \(0.945759\pi\)
\(678\) 0 0
\(679\) −2.00000 + 3.46410i −0.0767530 + 0.132940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −42.0000 −1.60709 −0.803543 0.595247i \(-0.797054\pi\)
−0.803543 + 0.595247i \(0.797054\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 10.3923i 0.228582 0.395915i
\(690\) 0 0
\(691\) −4.00000 6.92820i −0.152167 0.263561i 0.779857 0.625958i \(-0.215292\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.00000 3.46410i −0.0758643 0.131401i
\(696\) 0 0
\(697\) 18.0000 31.1769i 0.681799 1.18091i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 + 10.3923i −0.225653 + 0.390843i
\(708\) 0 0
\(709\) 17.0000 + 29.4449i 0.638448 + 1.10583i 0.985773 + 0.168080i \(0.0537568\pi\)
−0.347325 + 0.937745i \(0.612910\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000 + 20.7846i 0.449404 + 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 + 5.19615i −0.111417 + 0.192980i
\(726\) 0 0
\(727\) 23.0000 + 39.8372i 0.853023 + 1.47748i 0.878467 + 0.477803i \(0.158567\pi\)
−0.0254445 + 0.999676i \(0.508100\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −30.0000 51.9615i −1.10959 1.92187i
\(732\) 0 0
\(733\) 11.0000 19.0526i 0.406294 0.703722i −0.588177 0.808732i \(-0.700154\pi\)
0.994471 + 0.105010i \(0.0334875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.00000 + 5.19615i −0.110059 + 0.190628i −0.915794 0.401648i \(-0.868437\pi\)
0.805735 + 0.592277i \(0.201771\pi\)
\(744\) 0 0
\(745\) −3.00000 5.19615i −0.109911 0.190372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.00000 + 10.3923i 0.219235 + 0.379727i
\(750\) 0 0
\(751\) 2.00000 3.46410i 0.0729810 0.126407i −0.827225 0.561870i \(-0.810082\pi\)
0.900207 + 0.435463i \(0.143415\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.0000 + 36.3731i −0.761249 + 1.31852i 0.180957 + 0.983491i \(0.442080\pi\)
−0.942207 + 0.335032i \(0.891253\pi\)
\(762\) 0 0
\(763\) −2.00000 3.46410i −0.0724049 0.125409i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 20.7846i −0.433295 0.750489i
\(768\) 0 0
\(769\) −1.00000 + 1.73205i −0.0360609 + 0.0624593i −0.883493 0.468445i \(-0.844814\pi\)
0.847432 + 0.530904i \(0.178148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 20.7846i 0.429945 0.744686i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11.0000 19.0526i −0.392607 0.680015i
\(786\) 0 0
\(787\) −13.0000 + 22.5167i −0.463400 + 0.802632i −0.999128 0.0417585i \(-0.986704\pi\)
0.535728 + 0.844391i \(0.320037\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.0000 + 36.3731i −0.743858 + 1.28840i 0.206868 + 0.978369i \(0.433673\pi\)
−0.950726 + 0.310031i \(0.899660\pi\)
\(798\) 0 0
\(799\) −18.0000 31.1769i −0.636794 1.10296i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 6.00000 10.3923i 0.211472 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.00000 + 8.66025i −0.175142 + 0.303355i
\(816\) 0 0
\(817\) −20.0000 34.6410i −0.699711 1.21194i