Properties

Label 2592.2.i.m.1729.1
Level $2592$
Weight $2$
Character 2592.1729
Analytic conductor $20.697$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
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 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1729.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2592.1729
Dual form 2592.2.i.m.865.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{11} +4.00000 q^{17} +6.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} +(1.00000 + 1.73205i) q^{29} +(-4.50000 + 7.79423i) q^{31} -3.00000 q^{35} -2.00000 q^{37} +(5.00000 - 8.66025i) q^{41} +(-3.00000 - 5.19615i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-1.00000 + 1.73205i) q^{49} +13.0000 q^{53} +3.00000 q^{55} +(6.00000 - 10.3923i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(-3.00000 + 5.19615i) q^{67} +12.0000 q^{71} +9.00000 q^{73} +(4.50000 - 7.79423i) q^{77} +(1.50000 + 2.59808i) q^{83} +(2.00000 - 3.46410i) q^{85} +14.0000 q^{89} +(3.00000 - 5.19615i) q^{95} +(4.50000 + 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 3 q^{7} + O(q^{10}) \) \( 2 q + q^{5} - 3 q^{7} + 3 q^{11} + 8 q^{17} + 12 q^{19} - 6 q^{23} + 4 q^{25} + 2 q^{29} - 9 q^{31} - 6 q^{35} - 4 q^{37} + 10 q^{41} - 6 q^{43} - 6 q^{47} - 2 q^{49} + 26 q^{53} + 6 q^{55} + 12 q^{59} - 8 q^{61} - 6 q^{67} + 24 q^{71} + 18 q^{73} + 9 q^{77} + 3 q^{83} + 4 q^{85} + 28 q^{89} + 6 q^{95} + 9 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(5\) 0.500000 0.866025i 0.223607 0.387298i −0.732294 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) −1.50000 2.59808i −0.566947 0.981981i −0.996866 0.0791130i \(-0.974791\pi\)
0.429919 0.902867i \(-0.358542\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.50000 + 2.59808i 0.452267 + 0.783349i 0.998526 0.0542666i \(-0.0172821\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) 0 0
\(13\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 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
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 + 1.73205i 0.185695 + 0.321634i 0.943811 0.330487i \(-0.107213\pi\)
−0.758115 + 0.652121i \(0.773880\pi\)
\(30\) 0 0
\(31\) −4.50000 + 7.79423i −0.808224 + 1.39988i 0.105869 + 0.994380i \(0.466238\pi\)
−0.914093 + 0.405505i \(0.867096\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 8.66025i 0.780869 1.35250i −0.150567 0.988600i \(-0.548110\pi\)
0.931436 0.363905i \(-0.118557\pi\)
\(42\) 0 0
\(43\) −3.00000 5.19615i −0.457496 0.792406i 0.541332 0.840809i \(-0.317920\pi\)
−0.998828 + 0.0484030i \(0.984587\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 5.19615i −0.437595 0.757937i 0.559908 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 10.3923i 0.781133 1.35296i −0.150148 0.988663i \(-0.547975\pi\)
0.931282 0.364299i \(-0.118692\pi\)
\(60\) 0 0
\(61\) −4.00000 6.92820i −0.512148 0.887066i −0.999901 0.0140840i \(-0.995517\pi\)
0.487753 0.872982i \(-0.337817\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 + 5.19615i −0.366508 + 0.634811i −0.989017 0.147802i \(-0.952780\pi\)
0.622509 + 0.782613i \(0.286114\pi\)
\(68\) 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
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.50000 7.79423i 0.512823 0.888235i
\(78\) 0 0
\(79\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.50000 + 2.59808i 0.164646 + 0.285176i 0.936530 0.350588i \(-0.114018\pi\)
−0.771883 + 0.635764i \(0.780685\pi\)
\(84\) 0 0
\(85\) 2.00000 3.46410i 0.216930 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 5.19615i 0.307794 0.533114i
\(96\) 0 0
\(97\) 4.50000 + 7.79423i 0.456906 + 0.791384i 0.998796 0.0490655i \(-0.0156243\pi\)
−0.541890 + 0.840450i \(0.682291\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.50000 + 6.06218i 0.348263 + 0.603209i 0.985941 0.167094i \(-0.0534383\pi\)
−0.637678 + 0.770303i \(0.720105\pi\)
\(102\) 0 0
\(103\) 6.00000 10.3923i 0.591198 1.02398i −0.402874 0.915255i \(-0.631989\pi\)
0.994071 0.108729i \(-0.0346780\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.0000 −1.45010 −0.725052 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.00000 8.66025i 0.470360 0.814688i −0.529065 0.848581i \(-0.677457\pi\)
0.999425 + 0.0338931i \(0.0107906\pi\)
\(114\) 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
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 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
\(133\) −9.00000 15.5885i −0.780399 1.35169i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.00000 + 1.73205i 0.0854358 + 0.147979i 0.905577 0.424182i \(-0.139438\pi\)
−0.820141 + 0.572161i \(0.806105\pi\)
\(138\) 0 0
\(139\) 6.00000 10.3923i 0.508913 0.881464i −0.491033 0.871141i \(-0.663381\pi\)
0.999947 0.0103230i \(-0.00328598\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.50000 14.7224i 0.696347 1.20611i −0.273377 0.961907i \(-0.588141\pi\)
0.969724 0.244202i \(-0.0785259\pi\)
\(150\) 0 0
\(151\) 1.50000 + 2.59808i 0.122068 + 0.211428i 0.920583 0.390547i \(-0.127714\pi\)
−0.798515 + 0.601975i \(0.794381\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.50000 + 7.79423i 0.361449 + 0.626048i
\(156\) 0 0
\(157\) −2.00000 + 3.46410i −0.159617 + 0.276465i −0.934731 0.355357i \(-0.884359\pi\)
0.775113 + 0.631822i \(0.217693\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) 0 0
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 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
\(169\) 6.50000 + 11.2583i 0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.50000 9.52628i −0.418157 0.724270i 0.577597 0.816322i \(-0.303991\pi\)
−0.995754 + 0.0920525i \(0.970657\pi\)
\(174\) 0 0
\(175\) 6.00000 10.3923i 0.453557 0.785584i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 + 1.73205i −0.0735215 + 0.127343i
\(186\) 0 0
\(187\) 6.00000 + 10.3923i 0.438763 + 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 + 20.7846i 0.868290 + 1.50392i 0.863743 + 0.503932i \(0.168114\pi\)
0.00454614 + 0.999990i \(0.498553\pi\)
\(192\) 0 0
\(193\) −6.50000 + 11.2583i −0.467880 + 0.810392i −0.999326 0.0366998i \(-0.988315\pi\)
0.531446 + 0.847092i \(0.321649\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.00000 0.356235 0.178118 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.00000 5.19615i 0.210559 0.364698i
\(204\) 0 0
\(205\) −5.00000 8.66025i −0.349215 0.604858i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.00000 + 15.5885i 0.622543 + 1.07828i
\(210\) 0 0
\(211\) −3.00000 + 5.19615i −0.206529 + 0.357718i −0.950619 0.310361i \(-0.899550\pi\)
0.744090 + 0.668079i \(0.232883\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 27.0000 1.83288
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.0000 + 20.7846i 0.803579 + 1.39184i 0.917246 + 0.398321i \(0.130407\pi\)
−0.113666 + 0.993519i \(0.536260\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.00000 10.3923i −0.398234 0.689761i 0.595274 0.803523i \(-0.297043\pi\)
−0.993508 + 0.113761i \(0.963710\pi\)
\(228\) 0 0
\(229\) 9.00000 15.5885i 0.594737 1.03011i −0.398847 0.917017i \(-0.630590\pi\)
0.993584 0.113097i \(-0.0360770\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.00000 5.19615i 0.194054 0.336111i −0.752536 0.658551i \(-0.771170\pi\)
0.946590 + 0.322440i \(0.104503\pi\)
\(240\) 0 0
\(241\) 9.00000 + 15.5885i 0.579741 + 1.00414i 0.995509 + 0.0946700i \(0.0301796\pi\)
−0.415768 + 0.909471i \(0.636487\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 + 1.73205i 0.0638877 + 0.110657i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.00000 6.92820i 0.249513 0.432169i −0.713878 0.700270i \(-0.753063\pi\)
0.963391 + 0.268101i \(0.0863961\pi\)
\(258\) 0 0
\(259\) 3.00000 + 5.19615i 0.186411 + 0.322873i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.00000 5.19615i −0.184988 0.320408i 0.758585 0.651575i \(-0.225891\pi\)
−0.943572 + 0.331166i \(0.892558\pi\)
\(264\) 0 0
\(265\) 6.50000 11.2583i 0.399292 0.691594i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 + 10.3923i −0.361814 + 0.626680i
\(276\) 0 0
\(277\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 17.3205i −0.596550 1.03325i −0.993326 0.115339i \(-0.963204\pi\)
0.396776 0.917915i \(-0.370129\pi\)
\(282\) 0 0
\(283\) 3.00000 5.19615i 0.178331 0.308879i −0.762978 0.646425i \(-0.776263\pi\)
0.941309 + 0.337546i \(0.109597\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −30.0000 −1.77084
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.00000 + 8.66025i −0.292103 + 0.505937i −0.974307 0.225225i \(-0.927688\pi\)
0.682204 + 0.731162i \(0.261022\pi\)
\(294\) 0 0
\(295\) −6.00000 10.3923i −0.349334 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −9.00000 + 15.5885i −0.518751 + 0.898504i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.00000 15.5885i 0.510343 0.883940i −0.489585 0.871956i \(-0.662852\pi\)
0.999928 0.0119847i \(-0.00381495\pi\)
\(312\) 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
\(317\) 3.50000 + 6.06218i 0.196580 + 0.340486i 0.947417 0.320001i \(-0.103683\pi\)
−0.750838 + 0.660487i \(0.770350\pi\)
\(318\) 0 0
\(319\) −3.00000 + 5.19615i −0.167968 + 0.290929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 + 15.5885i −0.496186 + 0.859419i
\(330\) 0 0
\(331\) −15.0000 25.9808i −0.824475 1.42803i −0.902320 0.431066i \(-0.858137\pi\)
0.0778456 0.996965i \(-0.475196\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 + 5.19615i 0.163908 + 0.283896i
\(336\) 0 0
\(337\) −9.00000 + 15.5885i −0.490261 + 0.849157i −0.999937 0.0112091i \(-0.996432\pi\)
0.509676 + 0.860366i \(0.329765\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −27.0000 −1.46213
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.50000 + 2.59808i −0.0805242 + 0.139472i −0.903475 0.428640i \(-0.858993\pi\)
0.822951 + 0.568112i \(0.192326\pi\)
\(348\) 0 0
\(349\) −13.0000 22.5167i −0.695874 1.20529i −0.969885 0.243563i \(-0.921684\pi\)
0.274011 0.961727i \(-0.411649\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 1.73205i −0.0532246 0.0921878i 0.838186 0.545385i \(-0.183617\pi\)
−0.891410 + 0.453197i \(0.850283\pi\)
\(354\) 0 0
\(355\) 6.00000 10.3923i 0.318447 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.50000 7.79423i 0.235541 0.407969i
\(366\) 0 0
\(367\) −10.5000 18.1865i −0.548096 0.949329i −0.998405 0.0564568i \(-0.982020\pi\)
0.450310 0.892873i \(-0.351314\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19.5000 33.7750i −1.01239 1.75351i
\(372\) 0 0
\(373\) 2.00000 3.46410i 0.103556 0.179364i −0.809591 0.586994i \(-0.800311\pi\)
0.913147 + 0.407630i \(0.133645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 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
\(385\) −4.50000 7.79423i −0.229341 0.397231i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.50000 6.06218i −0.177457 0.307365i 0.763552 0.645747i \(-0.223454\pi\)
−0.941009 + 0.338382i \(0.890120\pi\)
\(390\) 0 0
\(391\) −12.0000 + 20.7846i −0.606866 + 1.05112i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 + 24.2487i −0.699127 + 1.21092i 0.269643 + 0.962960i \(0.413094\pi\)
−0.968770 + 0.247962i \(0.920239\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.00000 5.19615i −0.148704 0.257564i
\(408\) 0 0
\(409\) 4.50000 7.79423i 0.222511 0.385400i −0.733059 0.680165i \(-0.761908\pi\)
0.955570 + 0.294765i \(0.0952414\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −36.0000 −1.77144
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 + 10.3923i −0.293119 + 0.507697i −0.974546 0.224189i \(-0.928027\pi\)
0.681426 + 0.731887i \(0.261360\pi\)
\(420\) 0 0
\(421\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.00000 + 13.8564i 0.388057 + 0.672134i
\(426\) 0 0
\(427\) −12.0000 + 20.7846i −0.580721 + 1.00584i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18.0000 + 31.1769i −0.861057 + 1.49139i
\(438\) 0 0
\(439\) −4.50000 7.79423i −0.214773 0.371998i 0.738429 0.674331i \(-0.235568\pi\)
−0.953202 + 0.302333i \(0.902235\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00000 + 10.3923i 0.285069 + 0.493753i 0.972626 0.232377i \(-0.0746503\pi\)
−0.687557 + 0.726130i \(0.741317\pi\)
\(444\) 0 0
\(445\) 7.00000 12.1244i 0.331832 0.574750i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.50000 + 7.79423i 0.210501 + 0.364599i 0.951871 0.306497i \(-0.0991571\pi\)
−0.741370 + 0.671096i \(0.765824\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.50000 6.06218i −0.163011 0.282344i 0.772936 0.634484i \(-0.218787\pi\)
−0.935947 + 0.352140i \(0.885454\pi\)
\(462\) 0 0
\(463\) −7.50000 + 12.9904i −0.348555 + 0.603714i −0.985993 0.166787i \(-0.946661\pi\)
0.637438 + 0.770501i \(0.279994\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 18.0000 0.831163
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.00000 15.5885i 0.413820 0.716758i
\(474\) 0 0
\(475\) 12.0000 + 20.7846i 0.550598 + 0.953663i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.0000 + 36.3731i 0.959514 + 1.66193i 0.723681 + 0.690134i \(0.242449\pi\)
0.235833 + 0.971794i \(0.424218\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.00000 0.408669
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.50000 + 12.9904i −0.338470 + 0.586248i −0.984145 0.177365i \(-0.943243\pi\)
0.645675 + 0.763612i \(0.276576\pi\)
\(492\) 0 0
\(493\) 4.00000 + 6.92820i 0.180151 + 0.312031i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.0000 31.1769i −0.807410 1.39848i
\(498\) 0 0
\(499\) −3.00000 + 5.19615i −0.134298 + 0.232612i −0.925329 0.379165i \(-0.876211\pi\)
0.791031 + 0.611776i \(0.209545\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 7.00000 0.311496
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.5000 + 32.0429i −0.819998 + 1.42028i 0.0856847 + 0.996322i \(0.472692\pi\)
−0.905683 + 0.423956i \(0.860641\pi\)
\(510\) 0 0
\(511\) −13.5000 23.3827i −0.597205 1.03439i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.00000 10.3923i −0.264392 0.457940i
\(516\) 0 0
\(517\) 9.00000 15.5885i 0.395820 0.685580i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −32.0000 −1.40195 −0.700973 0.713188i \(-0.747251\pi\)
−0.700973 + 0.713188i \(0.747251\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.0000 + 31.1769i −0.784092 + 1.35809i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −7.50000 + 12.9904i −0.324253 + 0.561623i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −36.0000 −1.54776 −0.773880 0.633332i \(-0.781687\pi\)
−0.773880 + 0.633332i \(0.781687\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.00000 + 15.5885i −0.385518 + 0.667736i
\(546\) 0 0
\(547\) −18.0000 31.1769i −0.769624 1.33303i −0.937767 0.347266i \(-0.887110\pi\)
0.168142 0.985763i \(-0.446223\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 + 10.3923i 0.255609 + 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −41.0000 −1.73723 −0.868613 0.495491i \(-0.834988\pi\)
−0.868613 + 0.495491i \(0.834988\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.50000 + 2.59808i −0.0632175 + 0.109496i −0.895902 0.444252i \(-0.853470\pi\)
0.832684 + 0.553748i \(0.186803\pi\)
\(564\) 0 0
\(565\) −5.00000 8.66025i −0.210352 0.364340i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 + 17.3205i 0.419222 + 0.726113i 0.995861 0.0908852i \(-0.0289696\pi\)
−0.576640 + 0.816999i \(0.695636\pi\)
\(570\) 0 0
\(571\) −12.0000 + 20.7846i −0.502184 + 0.869809i 0.497812 + 0.867285i \(0.334137\pi\)
−0.999997 + 0.00252413i \(0.999197\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.50000 7.79423i 0.186691 0.323359i
\(582\) 0 0
\(583\) 19.5000 + 33.7750i 0.807607 + 1.39882i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.5000 + 23.3827i 0.557205 + 0.965107i 0.997728 + 0.0673658i \(0.0214594\pi\)
−0.440524 + 0.897741i \(0.645207\pi\)
\(588\) 0 0
\(589\) −27.0000 + 46.7654i −1.11252 + 1.92693i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.00000 + 5.19615i −0.122577 + 0.212309i −0.920783 0.390075i \(-0.872449\pi\)
0.798206 + 0.602384i \(0.205782\pi\)
\(600\) 0 0
\(601\) −9.50000 16.4545i −0.387513 0.671192i 0.604601 0.796528i \(-0.293332\pi\)
−0.992114 + 0.125336i \(0.959999\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 1.73205i −0.0406558 0.0704179i
\(606\) 0 0
\(607\) 12.0000 20.7846i 0.487065 0.843621i −0.512824 0.858494i \(-0.671401\pi\)
0.999889 + 0.0148722i \(0.00473415\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.0000 + 22.5167i −0.523360 + 0.906487i 0.476270 + 0.879299i \(0.341988\pi\)
−0.999630 + 0.0271876i \(0.991345\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −21.0000 36.3731i −0.841347 1.45726i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.50000 + 2.59808i −0.0595257 + 0.103102i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.00000 + 1.73205i 0.0394976 + 0.0684119i 0.885098 0.465404i \(-0.154091\pi\)
−0.845601 + 0.533816i \(0.820758\pi\)
\(642\) 0 0
\(643\) 18.0000 31.1769i 0.709851 1.22950i −0.255062 0.966925i \(-0.582096\pi\)
0.964912 0.262573i \(-0.0845709\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −17.5000 + 30.3109i −0.684828 + 1.18616i 0.288663 + 0.957431i \(0.406789\pi\)
−0.973491 + 0.228726i \(0.926544\pi\)
\(654\) 0 0
\(655\) 4.50000 + 7.79423i 0.175830 + 0.304546i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.5000 + 18.1865i 0.409022 + 0.708447i 0.994780 0.102039i \(-0.0325366\pi\)
−0.585758 + 0.810486i \(0.699203\pi\)
\(660\) 0 0
\(661\) 7.00000 12.1244i 0.272268 0.471583i −0.697174 0.716902i \(-0.745559\pi\)
0.969442 + 0.245319i \(0.0788928\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −18.0000 −0.698010
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 20.7846i 0.463255 0.802381i
\(672\) 0 0
\(673\) 9.50000 + 16.4545i 0.366198 + 0.634274i 0.988968 0.148132i \(-0.0473259\pi\)
−0.622770 + 0.782405i \(0.713993\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.00000 1.73205i −0.0384331 0.0665681i 0.846169 0.532915i \(-0.178903\pi\)
−0.884602 + 0.466347i \(0.845570\pi\)
\(678\) 0 0
\(679\) 13.5000 23.3827i 0.518082 0.897345i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6.00000 10.3923i −0.228251 0.395342i 0.729039 0.684472i \(-0.239967\pi\)
−0.957290 + 0.289130i \(0.906634\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 10.3923i −0.227593 0.394203i
\(696\) 0 0
\(697\) 20.0000 34.6410i 0.757554 1.31212i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.0000 0.491003 0.245502 0.969396i \(-0.421047\pi\)
0.245502 + 0.969396i \(0.421047\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.5000 18.1865i 0.394893 0.683975i
\(708\) 0 0
\(709\) 18.0000 + 31.1769i 0.676004 + 1.17087i 0.976174 + 0.216988i \(0.0696232\pi\)
−0.300170 + 0.953886i \(0.597043\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.0000 46.7654i −1.01116 1.75138i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −36.0000 −1.34071
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 + 6.92820i −0.148556 + 0.257307i
\(726\) 0 0
\(727\) 10.5000 + 18.1865i 0.389423 + 0.674501i 0.992372 0.123279i \(-0.0393409\pi\)
−0.602949 + 0.797780i \(0.706008\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 20.7846i −0.443836 0.768747i
\(732\) 0 0
\(733\) 9.00000 15.5885i 0.332423 0.575773i −0.650564 0.759452i \(-0.725467\pi\)
0.982986 + 0.183679i \(0.0588007\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.0000 −0.663039
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.0000 + 20.7846i −0.440237 + 0.762513i −0.997707 0.0676840i \(-0.978439\pi\)
0.557470 + 0.830197i \(0.311772\pi\)
\(744\) 0 0
\(745\) −8.50000 14.7224i −0.311416 0.539388i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 22.5000 + 38.9711i 0.822132 + 1.42397i
\(750\) 0 0
\(751\) 13.5000 23.3827i 0.492622 0.853246i −0.507342 0.861745i \(-0.669372\pi\)
0.999964 + 0.00849853i \(0.00270520\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.00000 + 6.92820i −0.145000 + 0.251147i −0.929373 0.369142i \(-0.879652\pi\)
0.784373 + 0.620289i \(0.212985\pi\)
\(762\) 0 0
\(763\) 27.0000 + 46.7654i 0.977466 + 1.69302i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −20.5000 + 35.5070i −0.739249 + 1.28042i 0.213585 + 0.976924i \(0.431486\pi\)
−0.952834 + 0.303492i \(0.901847\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −50.0000 −1.79838 −0.899188 0.437564i \(-0.855842\pi\)
−0.899188 + 0.437564i \(0.855842\pi\)
\(774\) 0 0
\(775\) −36.0000 −1.29316
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.0000 51.9615i 1.07486 1.86171i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.00000 + 3.46410i 0.0713831 + 0.123639i
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.5000 32.0429i 0.655304 1.13502i −0.326514 0.945192i \(-0.605874\pi\)
0.981818 0.189827i \(-0.0607926\pi\)
\(798\) 0 0
\(799\) −12.0000 20.7846i −0.424529 0.735307i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.5000 + 23.3827i 0.476405 + 0.825157i
\(804\) 0 0