Properties

Label 1512.2.s.g.865.1
Level $1512$
Weight $2$
Character 1512.865
Analytic conductor $12.073$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1512.865
Dual form 1512.2.s.g.1297.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(1.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} -2.00000 q^{13} +(-3.00000 + 5.19615i) q^{17} +(0.500000 + 0.866025i) q^{19} +(1.00000 + 1.73205i) q^{23} +(0.500000 - 0.866025i) q^{25} -6.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-4.00000 + 3.46410i) q^{35} +(-1.00000 - 1.73205i) q^{37} -2.00000 q^{41} +9.00000 q^{43} +(-1.00000 - 1.73205i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-3.00000 + 5.19615i) q^{53} +(-4.00000 + 6.92820i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(6.00000 - 10.3923i) q^{67} -4.00000 q^{71} +(-2.50000 + 4.33013i) q^{73} +(2.00000 + 3.46410i) q^{79} -4.00000 q^{83} -12.0000 q^{85} +(9.00000 + 15.5885i) q^{89} +(-1.00000 - 5.19615i) q^{91} +(-1.00000 + 1.73205i) q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + q^{7} - 4q^{13} - 6q^{17} + q^{19} + 2q^{23} + q^{25} - 12q^{29} - q^{31} - 8q^{35} - 2q^{37} - 4q^{41} + 18q^{43} - 2q^{47} - 13q^{49} - 6q^{53} - 8q^{59} - 11q^{61} - 4q^{65} + 12q^{67} - 8q^{71} - 5q^{73} + 4q^{79} - 8q^{83} - 24q^{85} + 18q^{89} - 2q^{91} - 2q^{95} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 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
\(19\) 0.500000 + 0.866025i 0.114708 + 0.198680i 0.917663 0.397360i \(-0.130073\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 + 1.73205i 0.208514 + 0.361158i 0.951247 0.308431i \(-0.0998038\pi\)
−0.742732 + 0.669588i \(0.766471\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(35\) −4.00000 + 3.46410i −0.676123 + 0.585540i
\(36\) 0 0
\(37\) −1.00000 1.73205i −0.164399 0.284747i 0.772043 0.635571i \(-0.219235\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) 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
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 1.73205i −0.145865 0.252646i 0.783830 0.620975i \(-0.213263\pi\)
−0.929695 + 0.368329i \(0.879930\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 + 6.92820i −0.520756 + 0.901975i 0.478953 + 0.877841i \(0.341016\pi\)
−0.999709 + 0.0241347i \(0.992317\pi\)
\(60\) 0 0
\(61\) −5.50000 9.52628i −0.704203 1.21972i −0.966978 0.254858i \(-0.917971\pi\)
0.262776 0.964857i \(-0.415362\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 3.46410i −0.248069 0.429669i
\(66\) 0 0
\(67\) 6.00000 10.3923i 0.733017 1.26962i −0.222571 0.974916i \(-0.571445\pi\)
0.955588 0.294706i \(-0.0952216\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −2.50000 + 4.33013i −0.292603 + 0.506803i −0.974424 0.224716i \(-0.927855\pi\)
0.681822 + 0.731519i \(0.261188\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 + 3.46410i 0.225018 + 0.389742i 0.956325 0.292306i \(-0.0944227\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 + 15.5885i 0.953998 + 1.65237i 0.736644 + 0.676280i \(0.236409\pi\)
0.217354 + 0.976093i \(0.430258\pi\)
\(90\) 0 0
\(91\) −1.00000 5.19615i −0.104828 0.544705i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 + 1.73205i −0.102598 + 0.177705i
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.00000 + 6.92820i −0.398015 + 0.689382i −0.993481 0.113998i \(-0.963634\pi\)
0.595466 + 0.803380i \(0.296967\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 + 13.8564i 0.773389 + 1.33955i 0.935695 + 0.352809i \(0.114773\pi\)
−0.162306 + 0.986740i \(0.551893\pi\)
\(108\) 0 0
\(109\) −2.50000 + 4.33013i −0.239457 + 0.414751i −0.960558 0.278078i \(-0.910303\pi\)
0.721102 + 0.692829i \(0.243636\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −2.00000 + 3.46410i −0.186501 + 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −15.0000 5.19615i −1.37505 0.476331i
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.00000 + 15.5885i 0.786334 + 1.36197i 0.928199 + 0.372084i \(0.121357\pi\)
−0.141865 + 0.989886i \(0.545310\pi\)
\(132\) 0 0
\(133\) −2.00000 + 1.73205i −0.173422 + 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0000 19.0526i 0.939793 1.62777i 0.173939 0.984757i \(-0.444351\pi\)
0.765855 0.643013i \(-0.222316\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 10.3923i −0.498273 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) −6.50000 + 11.2583i −0.528962 + 0.916190i 0.470467 + 0.882418i \(0.344085\pi\)
−0.999430 + 0.0337724i \(0.989248\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −1.00000 + 1.73205i −0.0798087 + 0.138233i −0.903167 0.429289i \(-0.858764\pi\)
0.823359 + 0.567521i \(0.192098\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 + 3.46410i −0.315244 + 0.273009i
\(162\) 0 0
\(163\) 0.500000 + 0.866025i 0.0391630 + 0.0678323i 0.884943 0.465700i \(-0.154198\pi\)
−0.845780 + 0.533533i \(0.820864\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.0000 19.0526i −0.836315 1.44854i −0.892956 0.450145i \(-0.851372\pi\)
0.0566411 0.998395i \(-0.481961\pi\)
\(174\) 0 0
\(175\) 2.50000 + 0.866025i 0.188982 + 0.0654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.00000 5.19615i 0.224231 0.388379i −0.731858 0.681457i \(-0.761346\pi\)
0.956088 + 0.293079i \(0.0946798\pi\)
\(180\) 0 0
\(181\) −3.00000 −0.222988 −0.111494 0.993765i \(-0.535564\pi\)
−0.111494 + 0.993765i \(0.535564\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.00000 3.46410i 0.147043 0.254686i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.00000 8.66025i −0.361787 0.626634i 0.626468 0.779447i \(-0.284500\pi\)
−0.988255 + 0.152813i \(0.951167\pi\)
\(192\) 0 0
\(193\) −3.00000 + 5.19615i −0.215945 + 0.374027i −0.953564 0.301189i \(-0.902616\pi\)
0.737620 + 0.675216i \(0.235950\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −7.50000 + 12.9904i −0.531661 + 0.920864i 0.467656 + 0.883911i \(0.345099\pi\)
−0.999317 + 0.0369532i \(0.988235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 15.5885i −0.210559 1.09410i
\(204\) 0 0
\(205\) −2.00000 3.46410i −0.139686 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.00000 + 15.5885i 0.613795 + 1.06312i
\(216\) 0 0
\(217\) −2.50000 0.866025i −0.169711 0.0587896i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 10.3923i 0.403604 0.699062i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.00000 15.5885i 0.597351 1.03464i −0.395860 0.918311i \(-0.629553\pi\)
0.993210 0.116331i \(-0.0371134\pi\)
\(228\) 0 0
\(229\) −4.50000 7.79423i −0.297368 0.515057i 0.678165 0.734910i \(-0.262776\pi\)
−0.975533 + 0.219853i \(0.929442\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 5.19615i −0.196537 0.340411i 0.750867 0.660454i \(-0.229636\pi\)
−0.947403 + 0.320043i \(0.896303\pi\)
\(234\) 0 0
\(235\) 2.00000 3.46410i 0.130466 0.225973i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 9.50000 16.4545i 0.611949 1.05993i −0.378963 0.925412i \(-0.623719\pi\)
0.990912 0.134515i \(-0.0429475\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −11.0000 8.66025i −0.702764 0.553283i
\(246\) 0 0
\(247\) −1.00000 1.73205i −0.0636285 0.110208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.0000 + 17.3205i 0.623783 + 1.08042i 0.988775 + 0.149413i \(0.0477384\pi\)
−0.364992 + 0.931011i \(0.618928\pi\)
\(258\) 0 0
\(259\) 4.00000 3.46410i 0.248548 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.0000 24.2487i 0.863277 1.49524i −0.00547092 0.999985i \(-0.501741\pi\)
0.868748 0.495255i \(-0.164925\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 + 17.3205i −0.609711 + 1.05605i 0.381577 + 0.924337i \(0.375381\pi\)
−0.991288 + 0.131713i \(0.957952\pi\)
\(270\) 0 0
\(271\) 11.5000 + 19.9186i 0.698575 + 1.20997i 0.968960 + 0.247216i \(0.0795156\pi\)
−0.270385 + 0.962752i \(0.587151\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.50000 16.4545i 0.570800 0.988654i −0.425684 0.904872i \(-0.639967\pi\)
0.996484 0.0837823i \(-0.0267000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 13.5000 23.3827i 0.802492 1.38996i −0.115480 0.993310i \(-0.536841\pi\)
0.917971 0.396647i \(-0.129826\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.00000 5.19615i −0.0590281 0.306719i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.0000 1.63578 0.817889 0.575376i \(-0.195144\pi\)
0.817889 + 0.575376i \(0.195144\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00000 3.46410i −0.115663 0.200334i
\(300\) 0 0
\(301\) 4.50000 + 23.3827i 0.259376 + 1.34776i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.0000 19.0526i 0.629858 1.09095i
\(306\) 0 0
\(307\) −13.0000 −0.741949 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 20.7846i 0.680458 1.17859i −0.294384 0.955687i \(-0.595114\pi\)
0.974841 0.222900i \(-0.0715523\pi\)
\(312\) 0 0
\(313\) 10.5000 + 18.1865i 0.593495 + 1.02796i 0.993757 + 0.111563i \(0.0355857\pi\)
−0.400262 + 0.916401i \(0.631081\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 + 3.46410i 0.112331 + 0.194563i 0.916710 0.399554i \(-0.130835\pi\)
−0.804379 + 0.594117i \(0.797502\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −1.00000 + 1.73205i −0.0554700 + 0.0960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 3.46410i 0.220527 0.190982i
\(330\) 0 0
\(331\) −6.50000 11.2583i −0.357272 0.618814i 0.630232 0.776407i \(-0.282960\pi\)
−0.987504 + 0.157593i \(0.949627\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.0000 24.2487i 0.751559 1.30174i −0.195507 0.980702i \(-0.562635\pi\)
0.947067 0.321037i \(-0.104031\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.00000 + 3.46410i −0.106449 + 0.184376i −0.914329 0.404971i \(-0.867282\pi\)
0.807880 + 0.589347i \(0.200615\pi\)
\(354\) 0 0
\(355\) −4.00000 6.92820i −0.212298 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 + 20.7846i 0.633336 + 1.09697i 0.986865 + 0.161546i \(0.0516481\pi\)
−0.353529 + 0.935423i \(0.615019\pi\)
\(360\) 0 0
\(361\) 9.00000 15.5885i 0.473684 0.820445i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −14.5000 + 25.1147i −0.756894 + 1.31098i 0.187533 + 0.982258i \(0.439951\pi\)
−0.944427 + 0.328720i \(0.893383\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.0000 5.19615i −0.778761 0.269771i
\(372\) 0 0
\(373\) 8.50000 + 14.7224i 0.440113 + 0.762299i 0.997697 0.0678218i \(-0.0216049\pi\)
−0.557584 + 0.830120i \(0.688272\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.0000 + 17.3205i 0.510976 + 0.885037i 0.999919 + 0.0127209i \(0.00404928\pi\)
−0.488943 + 0.872316i \(0.662617\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.0000 32.9090i 0.963338 1.66855i 0.249323 0.968420i \(-0.419792\pi\)
0.714015 0.700130i \(-0.246875\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 + 6.92820i −0.201262 + 0.348596i
\(396\) 0 0
\(397\) 8.50000 + 14.7224i 0.426603 + 0.738898i 0.996569 0.0827707i \(-0.0263769\pi\)
−0.569966 + 0.821668i \(0.693044\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 17.3205i −0.499376 0.864945i 0.500624 0.865665i \(-0.333104\pi\)
−1.00000 0.000720188i \(0.999771\pi\)
\(402\) 0 0
\(403\) 1.00000 1.73205i 0.0498135 0.0862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.00000 12.1244i 0.346128 0.599511i −0.639430 0.768849i \(-0.720830\pi\)
0.985558 + 0.169338i \(0.0541630\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −20.0000 6.92820i −0.984136 0.340915i
\(414\) 0 0
\(415\) −4.00000 6.92820i −0.196352 0.340092i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −3.00000 −0.146211 −0.0731055 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 + 5.19615i 0.145521 + 0.252050i
\(426\) 0 0
\(427\) 22.0000 19.0526i 1.06465 0.922018i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 20.7846i 0.578020 1.00116i −0.417687 0.908591i \(-0.637159\pi\)
0.995706 0.0925683i \(-0.0295076\pi\)
\(432\) 0 0
\(433\) 7.00000 0.336399 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.00000 + 1.73205i −0.0478365 + 0.0828552i
\(438\) 0 0
\(439\) 12.0000 + 20.7846i 0.572729 + 0.991995i 0.996284 + 0.0861252i \(0.0274485\pi\)
−0.423556 + 0.905870i \(0.639218\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11.0000 19.0526i −0.522626 0.905214i −0.999653 0.0263261i \(-0.991619\pi\)
0.477028 0.878888i \(-0.341714\pi\)
\(444\) 0 0
\(445\) −18.0000 + 31.1769i −0.853282 + 1.47793i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 6.92820i 0.375046 0.324799i
\(456\) 0 0
\(457\) 17.5000 + 30.3109i 0.818615 + 1.41788i 0.906702 + 0.421771i \(0.138591\pi\)
−0.0880870 + 0.996113i \(0.528075\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −7.00000 −0.325318 −0.162659 0.986682i \(-0.552007\pi\)
−0.162659 + 0.986682i \(0.552007\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.00000 3.46410i −0.0925490 0.160300i 0.816034 0.578004i \(-0.196168\pi\)
−0.908583 + 0.417704i \(0.862835\pi\)
\(468\) 0 0
\(469\) 30.0000 + 10.3923i 1.38527 + 0.479872i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.0000 + 25.9808i −0.685367 + 1.18709i 0.287954 + 0.957644i \(0.407025\pi\)
−0.973321 + 0.229447i \(0.926308\pi\)
\(480\) 0 0
\(481\) 2.00000 + 3.46410i 0.0911922 + 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00000 + 1.73205i 0.0454077 + 0.0786484i
\(486\) 0 0
\(487\) −18.5000 + 32.0429i −0.838315 + 1.45200i 0.0529875 + 0.998595i \(0.483126\pi\)
−0.891303 + 0.453409i \(0.850208\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 18.0000 31.1769i 0.810679 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.00000 10.3923i −0.0897123 0.466159i
\(498\) 0 0
\(499\) −18.5000 32.0429i −0.828174 1.43444i −0.899469 0.436984i \(-0.856047\pi\)
0.0712957 0.997455i \(-0.477287\pi\)
\(500\) 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
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.0000 + 31.1769i 0.797836 + 1.38189i 0.921023 + 0.389509i \(0.127355\pi\)
−0.123187 + 0.992384i \(0.539311\pi\)
\(510\) 0 0
\(511\) −12.5000 4.33013i −0.552967 0.191554i
\(512\) 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
\(521\) 12.0000 20.7846i 0.525730 0.910590i −0.473821 0.880621i \(-0.657126\pi\)
0.999551 0.0299693i \(-0.00954094\pi\)
\(522\) 0 0
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.00000 5.19615i −0.130682 0.226348i
\(528\) 0 0
\(529\) 9.50000 16.4545i 0.413043 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −16.0000 + 27.7128i −0.691740 + 1.19813i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.50000 + 9.52628i 0.236463 + 0.409567i 0.959697 0.281037i \(-0.0906783\pi\)
−0.723234 + 0.690604i \(0.757345\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 7.00000 0.299298 0.149649 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.00000 5.19615i −0.127804 0.221364i
\(552\) 0 0
\(553\) −8.00000 + 6.92820i −0.340195 + 0.294617i
\(554\) 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
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.00000 5.19615i 0.126435 0.218992i −0.795858 0.605483i \(-0.792980\pi\)
0.922293 + 0.386492i \(0.126313\pi\)
\(564\) 0 0
\(565\) −16.0000 27.7128i −0.673125 1.16589i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 + 31.1769i 0.754599 + 1.30700i 0.945573 + 0.325409i \(0.105502\pi\)
−0.190974 + 0.981595i \(0.561165\pi\)
\(570\) 0 0
\(571\) 2.50000 4.33013i 0.104622 0.181210i −0.808962 0.587861i \(-0.799970\pi\)
0.913584 + 0.406651i \(0.133303\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) 9.00000 15.5885i 0.374675 0.648956i −0.615603 0.788056i \(-0.711088\pi\)
0.990278 + 0.139100i \(0.0444210\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.00000 10.3923i −0.0829740 0.431145i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) −1.00000 −0.0412043
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.00000 + 10.3923i 0.246390 + 0.426761i 0.962522 0.271205i \(-0.0874221\pi\)
−0.716131 + 0.697966i \(0.754089\pi\)
\(594\) 0 0
\(595\) −6.00000 31.1769i −0.245976 1.27813i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15.0000 + 25.9808i −0.612883 + 1.06155i 0.377869 + 0.925859i \(0.376657\pi\)
−0.990752 + 0.135686i \(0.956676\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.0000 + 19.0526i −0.447214 + 0.774597i
\(606\) 0 0
\(607\) −6.50000 11.2583i −0.263827 0.456962i 0.703429 0.710766i \(-0.251651\pi\)
−0.967256 + 0.253804i \(0.918318\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.00000 + 3.46410i 0.0809113 + 0.140143i
\(612\) 0 0
\(613\) −5.50000 + 9.52628i −0.222143 + 0.384763i −0.955458 0.295126i \(-0.904638\pi\)
0.733316 + 0.679888i \(0.237972\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) −6.00000 + 10.3923i −0.241160 + 0.417702i −0.961045 0.276392i \(-0.910861\pi\)
0.719885 + 0.694094i \(0.244195\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −36.0000 + 31.1769i −1.44231 + 1.24908i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −23.0000 −0.915616 −0.457808 0.889051i \(-0.651365\pi\)
−0.457808 + 0.889051i \(0.651365\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 17.0000 + 29.4449i 0.674624 + 1.16848i
\(636\) 0 0
\(637\) 13.0000 5.19615i 0.515079 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.00000 + 8.66025i −0.197488 + 0.342059i −0.947713 0.319123i \(-0.896612\pi\)
0.750225 + 0.661182i \(0.229945\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.0000 + 38.1051i −0.864909 + 1.49807i 0.00222801 + 0.999998i \(0.499291\pi\)
−0.867137 + 0.498069i \(0.834043\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.00000 5.19615i −0.117399 0.203341i 0.801337 0.598213i \(-0.204122\pi\)
−0.918736 + 0.394872i \(0.870789\pi\)
\(654\) 0 0
\(655\) −18.0000 + 31.1769i −0.703318 + 1.21818i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 18.5000 32.0429i 0.719567 1.24633i −0.241605 0.970375i \(-0.577674\pi\)
0.961172 0.275951i \(-0.0889928\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.00000 1.73205i −0.193892 0.0671660i
\(666\) 0 0
\(667\) −6.00000 10.3923i −0.232321 0.402392i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.00000 + 3.46410i 0.0768662 + 0.133136i 0.901896 0.431953i \(-0.142175\pi\)
−0.825030 + 0.565089i \(0.808842\pi\)
\(678\) 0 0
\(679\) 0.500000 + 2.59808i 0.0191882 + 0.0997050i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.0000 27.7128i 0.612223 1.06040i −0.378642 0.925543i \(-0.623609\pi\)
0.990865 0.134858i \(-0.0430579\pi\)
\(684\) 0 0
\(685\) 44.0000 1.68115
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 10.3923i 0.228582 0.395915i
\(690\) 0 0
\(691\) −1.50000 2.59808i −0.0570627 0.0988355i 0.836083 0.548603i \(-0.184840\pi\)
−0.893146 + 0.449768i \(0.851507\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.0000 + 34.6410i 0.758643 + 1.31401i
\(696\) 0 0
\(697\) 6.00000 10.3923i 0.227266 0.393637i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) 1.00000 1.73205i 0.0377157 0.0653255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.0000 6.92820i −0.752177 0.260562i
\(708\) 0 0
\(709\) −16.5000 28.5788i −0.619671 1.07330i −0.989546 0.144219i \(-0.953933\pi\)
0.369875 0.929081i \(-0.379400\pi\)
\(710\) 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
\(719\) 4.00000 + 6.92820i 0.149175 + 0.258378i 0.930923 0.365216i \(-0.119005\pi\)
−0.781748 + 0.623595i \(0.785672\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 + 5.19615i −0.111417 + 0.192980i
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −27.0000 + 46.7654i −0.998631 + 1.72968i
\(732\) 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
\(737\) 0 0
\(738\) 0 0
\(739\) 5.50000 9.52628i 0.202321 0.350430i −0.746955 0.664875i \(-0.768485\pi\)
0.949276 + 0.314445i \(0.101818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 0 0
\(745\) −6.00000 + 10.3923i −0.219823 + 0.380745i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −32.0000 + 27.7128i −1.16925 + 1.01260i
\(750\) 0 0
\(751\) −7.50000 12.9904i −0.273679 0.474026i 0.696122 0.717923i \(-0.254907\pi\)
−0.969801 + 0.243898i \(0.921574\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −26.0000 −0.946237
\(756\) 0 0
\(757\) −7.00000 −0.254419 −0.127210 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.0000 + 41.5692i 0.869999 + 1.50688i 0.861996 + 0.506915i \(0.169214\pi\)
0.00800331 + 0.999968i \(0.497452\pi\)
\(762\) 0 0
\(763\) −12.5000 4.33013i −0.452530 0.156761i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 13.8564i 0.288863 0.500326i
\(768\) 0 0
\(769\) 19.0000 0.685158 0.342579 0.939489i \(-0.388700\pi\)
0.342579 + 0.939489i \(0.388700\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.00000 8.66025i 0.179838 0.311488i −0.761987 0.647592i \(-0.775776\pi\)
0.941825 + 0.336104i \(0.109109\pi\)
\(774\) 0 0
\(775\) 0.500000 + 0.866025i 0.0179605 + 0.0311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.00000 1.73205i −0.0358287 0.0620572i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −8.50000 + 14.7224i −0.302992 + 0.524798i −0.976812 0.214097i \(-0.931319\pi\)
0.673820 + 0.738896i \(0.264652\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.00000 41.5692i −0.284447 1.47803i
\(792\) 0 0
\(793\) 11.0000 + 19.0526i 0.390621 + 0.676576i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.0000 −1.27519 −0.637593 0.770374i \(-0.720070\pi\)
−0.637593 + 0.770374i \(0.720070\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −10.0000 3.46410i −0.352454