Properties

Label 3528.2.s.x.3313.1
Level $3528$
Weight $2$
Character 3528.3313
Analytic conductor $28.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3313.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3528.3313
Dual form 3528.2.s.x.361.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{5} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{5} +(1.00000 + 1.73205i) q^{11} +2.00000 q^{13} +(3.00000 + 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} +(3.00000 - 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{25} +(2.00000 + 3.46410i) q^{31} +(-5.00000 + 8.66025i) q^{37} -2.00000 q^{41} -4.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(-6.00000 - 10.3923i) q^{53} +4.00000 q^{55} +(6.00000 + 10.3923i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(2.00000 - 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} +14.0000 q^{71} +(1.00000 + 1.73205i) q^{73} +(4.00000 - 6.92820i) q^{79} +16.0000 q^{83} +12.0000 q^{85} +(-3.00000 + 5.19615i) q^{89} +(-4.00000 - 6.92820i) q^{95} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{11} + 4q^{13} + 6q^{17} + 4q^{19} + 6q^{23} + q^{25} + 4q^{31} - 10q^{37} - 4q^{41} - 8q^{43} + 4q^{47} - 12q^{53} + 8q^{55} + 12q^{59} - 6q^{61} + 4q^{65} + 4q^{67} + 28q^{71} + 2q^{73} + 8q^{79} + 32q^{83} + 24q^{85} - 6q^{89} - 8q^{95} - 36q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(5\) 1.00000 1.73205i 0.447214 0.774597i −0.550990 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.727607 + 1.26025i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) 2.00000 3.46410i 0.458831 0.794719i −0.540068 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 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
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 3.46410i 0.291730 0.505291i −0.682489 0.730896i \(-0.739102\pi\)
0.974219 + 0.225605i \(0.0724358\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 10.3923i −0.824163 1.42749i −0.902557 0.430570i \(-0.858312\pi\)
0.0783936 0.996922i \(-0.475021\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) −3.00000 + 5.19615i −0.384111 + 0.665299i −0.991645 0.128994i \(-0.958825\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 3.46410i 0.248069 0.429669i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 1.00000 + 1.73205i 0.117041 + 0.202721i 0.918594 0.395203i \(-0.129326\pi\)
−0.801553 + 0.597924i \(0.795992\pi\)
\(74\) 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 \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 6.92820i −0.410391 0.710819i
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.00000 + 12.1244i 0.696526 + 1.20642i 0.969664 + 0.244443i \(0.0786053\pi\)
−0.273138 + 0.961975i \(0.588061\pi\)
\(102\) 0 0
\(103\) 2.00000 3.46410i 0.197066 0.341328i −0.750510 0.660859i \(-0.770192\pi\)
0.947576 + 0.319531i \(0.103525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.00000 + 5.19615i −0.290021 + 0.502331i −0.973814 0.227345i \(-0.926996\pi\)
0.683793 + 0.729676i \(0.260329\pi\)
\(108\) 0 0
\(109\) −1.00000 1.73205i −0.0957826 0.165900i 0.814152 0.580651i \(-0.197202\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) −6.00000 10.3923i −0.559503 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 6.92820i 0.349482 0.605320i −0.636676 0.771132i \(-0.719691\pi\)
0.986157 + 0.165812i \(0.0530244\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 10.3923i −0.512615 0.887875i −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 + 3.46410i 0.167248 + 0.289683i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 10.3923i 0.491539 0.851371i −0.508413 0.861113i \(-0.669768\pi\)
0.999953 + 0.00974235i \(0.00310113\pi\)
\(150\) 0 0
\(151\) −12.0000 20.7846i −0.976546 1.69143i −0.674735 0.738060i \(-0.735742\pi\)
−0.301811 0.953368i \(-0.597591\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −3.00000 5.19615i −0.239426 0.414698i 0.721124 0.692806i \(-0.243626\pi\)
−0.960550 + 0.278108i \(0.910293\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 10.3923i 0.469956 0.813988i −0.529454 0.848339i \(-0.677603\pi\)
0.999410 + 0.0343508i \(0.0109363\pi\)
\(164\) 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
\(173\) 3.00000 5.19615i 0.228086 0.395056i −0.729155 0.684349i \(-0.760087\pi\)
0.957241 + 0.289292i \(0.0934200\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.00000 8.66025i −0.373718 0.647298i 0.616417 0.787420i \(-0.288584\pi\)
−0.990134 + 0.140122i \(0.955250\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.0000 + 17.3205i 0.735215 + 1.27343i
\(186\) 0 0
\(187\) −6.00000 + 10.3923i −0.438763 + 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.0000 22.5167i 0.940647 1.62925i 0.176406 0.984317i \(-0.443553\pi\)
0.764241 0.644931i \(-0.223114\pi\)
\(192\) 0 0
\(193\) 13.0000 + 22.5167i 0.935760 + 1.62078i 0.773272 + 0.634074i \(0.218619\pi\)
0.162488 + 0.986710i \(0.448048\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 + 3.46410i −0.139686 + 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 + 6.92820i −0.272798 + 0.472500i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0000 + 24.2487i 0.929213 + 1.60944i 0.784642 + 0.619949i \(0.212847\pi\)
0.144571 + 0.989494i \(0.453820\pi\)
\(228\) 0 0
\(229\) −5.00000 + 8.66025i −0.330409 + 0.572286i −0.982592 0.185776i \(-0.940520\pi\)
0.652183 + 0.758062i \(0.273853\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.00000 + 3.46410i −0.131024 + 0.226941i −0.924072 0.382219i \(-0.875160\pi\)
0.793047 + 0.609160i \(0.208493\pi\)
\(234\) 0 0
\(235\) −4.00000 6.92820i −0.260931 0.451946i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) 13.0000 + 22.5167i 0.837404 + 1.45043i 0.892058 + 0.451920i \(0.149261\pi\)
−0.0546547 + 0.998505i \(0.517406\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 6.92820i 0.254514 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.0000 22.5167i 0.810918 1.40455i −0.101305 0.994855i \(-0.532302\pi\)
0.912222 0.409695i \(-0.134365\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 15.5885i −0.554964 0.961225i −0.997906 0.0646755i \(-0.979399\pi\)
0.442943 0.896550i \(-0.353935\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.00000 + 8.66025i 0.304855 + 0.528025i 0.977229 0.212187i \(-0.0680585\pi\)
−0.672374 + 0.740212i \(0.734725\pi\)
\(270\) 0 0
\(271\) 6.00000 10.3923i 0.364474 0.631288i −0.624218 0.781251i \(-0.714582\pi\)
0.988692 + 0.149963i \(0.0479155\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 + 1.73205i −0.0603023 + 0.104447i
\(276\) 0 0
\(277\) 1.00000 + 1.73205i 0.0600842 + 0.104069i 0.894503 0.447062i \(-0.147530\pi\)
−0.834419 + 0.551131i \(0.814196\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −14.0000 24.2487i −0.832214 1.44144i −0.896279 0.443491i \(-0.853740\pi\)
0.0640654 0.997946i \(-0.479593\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 10.3923i 0.346989 0.601003i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 + 10.3923i 0.343559 + 0.595062i
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.0000 + 17.3205i 0.567048 + 0.982156i 0.996856 + 0.0792356i \(0.0252479\pi\)
−0.429808 + 0.902920i \(0.641419\pi\)
\(312\) 0 0
\(313\) 7.00000 12.1244i 0.395663 0.685309i −0.597522 0.801852i \(-0.703848\pi\)
0.993186 + 0.116543i \(0.0371814\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 10.3923i 0.336994 0.583690i −0.646872 0.762598i \(-0.723923\pi\)
0.983866 + 0.178908i \(0.0572566\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 1.00000 + 1.73205i 0.0554700 + 0.0960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 + 17.3205i −0.549650 + 0.952021i 0.448649 + 0.893708i \(0.351905\pi\)
−0.998298 + 0.0583130i \(0.981428\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.00000 + 6.92820i −0.216612 + 0.375183i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.00000 + 5.19615i 0.161048 + 0.278944i 0.935245 0.354001i \(-0.115179\pi\)
−0.774197 + 0.632945i \(0.781846\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.0000 22.5167i −0.691920 1.19844i −0.971208 0.238233i \(-0.923432\pi\)
0.279288 0.960207i \(-0.409902\pi\)
\(354\) 0 0
\(355\) 14.0000 24.2487i 0.743043 1.28699i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.00000 + 5.19615i −0.158334 + 0.274242i −0.934268 0.356572i \(-0.883946\pi\)
0.775934 + 0.630814i \(0.217279\pi\)
\(360\) 0 0
\(361\) 1.50000 + 2.59808i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 16.0000 + 27.7128i 0.835193 + 1.44660i 0.893873 + 0.448320i \(0.147978\pi\)
−0.0586798 + 0.998277i \(0.518689\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 + 19.0526i −0.569558 + 0.986504i 0.427051 + 0.904227i \(0.359552\pi\)
−0.996610 + 0.0822766i \(0.973781\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 + 20.7846i −0.613171 + 1.06204i 0.377531 + 0.925997i \(0.376773\pi\)
−0.990702 + 0.136047i \(0.956560\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.00000 6.92820i −0.202808 0.351274i 0.746624 0.665246i \(-0.231673\pi\)
−0.949432 + 0.313972i \(0.898340\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 13.8564i −0.402524 0.697191i
\(396\) 0 0
\(397\) −11.0000 + 19.0526i −0.552074 + 0.956221i 0.446051 + 0.895008i \(0.352830\pi\)
−0.998125 + 0.0612128i \(0.980503\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 10.3923i 0.299626 0.518967i −0.676425 0.736512i \(-0.736472\pi\)
0.976050 + 0.217545i \(0.0698049\pi\)
\(402\) 0 0
\(403\) 4.00000 + 6.92820i 0.199254 + 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) 1.00000 + 1.73205i 0.0494468 + 0.0856444i 0.889689 0.456566i \(-0.150921\pi\)
−0.840243 + 0.542211i \(0.817588\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 16.0000 27.7128i 0.785409 1.36037i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 + 5.19615i −0.145521 + 0.252050i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.00000 + 8.66025i 0.240842 + 0.417150i 0.960954 0.276707i \(-0.0892433\pi\)
−0.720113 + 0.693857i \(0.755910\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0000 20.7846i −0.574038 0.994263i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.00000 + 15.5885i −0.427603 + 0.740630i −0.996660 0.0816684i \(-0.973975\pi\)
0.569057 + 0.822298i \(0.307309\pi\)
\(444\) 0 0
\(445\) 6.00000 + 10.3923i 0.284427 + 0.492642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 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
\(457\) −11.0000 + 19.0526i −0.514558 + 0.891241i 0.485299 + 0.874348i \(0.338711\pi\)
−0.999857 + 0.0168929i \(0.994623\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.00000 3.46410i 0.0925490 0.160300i −0.816034 0.578004i \(-0.803832\pi\)
0.908583 + 0.417704i \(0.137165\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.00000 6.92820i −0.183920 0.318559i
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.0000 31.1769i −0.822441 1.42451i −0.903859 0.427830i \(-0.859278\pi\)
0.0814184 0.996680i \(-0.474055\pi\)
\(480\) 0 0
\(481\) −10.0000 + 17.3205i −0.455961 + 0.789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −18.0000 + 31.1769i −0.817338 + 1.41567i
\(486\) 0 0
\(487\) −4.00000 6.92820i −0.181257 0.313947i 0.761052 0.648691i \(-0.224683\pi\)
−0.942309 + 0.334744i \(0.891350\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 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
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.00000 + 12.1244i −0.310270 + 0.537403i −0.978421 0.206623i \(-0.933753\pi\)
0.668151 + 0.744026i \(0.267086\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 6.92820i −0.176261 0.305293i
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 + 5.19615i 0.131432 + 0.227648i 0.924229 0.381839i \(-0.124709\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(522\) 0 0
\(523\) −8.00000 + 13.8564i −0.349816 + 0.605898i −0.986216 0.165460i \(-0.947089\pi\)
0.636401 + 0.771358i \(0.280422\pi\)
\(524\) 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
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 6.00000 + 10.3923i 0.259403 + 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.00000 8.66025i 0.214967 0.372333i −0.738296 0.674477i \(-0.764369\pi\)
0.953262 + 0.302144i \(0.0977023\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.0000 + 17.3205i 0.423714 + 0.733893i 0.996299 0.0859514i \(-0.0273930\pi\)
−0.572586 + 0.819845i \(0.694060\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.0000 24.2487i −0.590030 1.02196i −0.994228 0.107290i \(-0.965783\pi\)
0.404198 0.914671i \(-0.367551\pi\)
\(564\) 0 0
\(565\) 16.0000 27.7128i 0.673125 1.16589i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.00000 + 3.46410i −0.0838444 + 0.145223i −0.904898 0.425628i \(-0.860053\pi\)
0.821054 + 0.570851i \(0.193387\pi\)
\(570\) 0 0
\(571\) −2.00000 3.46410i −0.0836974 0.144968i 0.821138 0.570730i \(-0.193340\pi\)
−0.904835 + 0.425762i \(0.860006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −17.0000 29.4449i −0.707719 1.22581i −0.965701 0.259656i \(-0.916391\pi\)
0.257982 0.966150i \(-0.416942\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 20.7846i 0.496989 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13.0000 + 22.5167i −0.533846 + 0.924648i 0.465372 + 0.885115i \(0.345920\pi\)
−0.999218 + 0.0395334i \(0.987413\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.00000 + 15.5885i 0.367730 + 0.636927i 0.989210 0.146503i \(-0.0468017\pi\)
−0.621480 + 0.783430i \(0.713468\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 12.1244i −0.284590 0.492925i
\(606\) 0 0
\(607\) 20.0000 34.6410i 0.811775 1.40604i −0.0998457 0.995003i \(-0.531835\pi\)
0.911621 0.411033i \(-0.134832\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 6.92820i 0.161823 0.280285i
\(612\) 0 0
\(613\) −3.00000 5.19615i −0.121169 0.209871i 0.799060 0.601251i \(-0.205331\pi\)
−0.920229 + 0.391381i \(0.871998\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) −20.0000 34.6410i −0.803868 1.39234i −0.917053 0.398766i \(-0.869439\pi\)
0.113185 0.993574i \(-0.463895\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 + 13.8564i −0.317470 + 0.549875i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.00000 + 10.3923i 0.236986 + 0.410471i 0.959848 0.280521i \(-0.0905072\pi\)
−0.722862 + 0.690992i \(0.757174\pi\)
\(642\) 0 0
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.0000 17.3205i −0.393141 0.680939i 0.599721 0.800209i \(-0.295278\pi\)
−0.992862 + 0.119269i \(0.961945\pi\)
\(648\) 0 0
\(649\) −12.0000 + 20.7846i −0.471041 + 0.815867i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −16.0000 + 27.7128i −0.626128 + 1.08449i 0.362193 + 0.932103i \(0.382028\pi\)
−0.988322 + 0.152383i \(0.951305\pi\)
\(654\) 0 0
\(655\) −8.00000 13.8564i −0.312586 0.541415i
\(656\) 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
\(661\) 5.00000 + 8.66025i 0.194477 + 0.336845i 0.946729 0.322031i \(-0.104366\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.0000 25.9808i 0.576497 0.998522i −0.419380 0.907811i \(-0.637753\pi\)
0.995877 0.0907112i \(-0.0289140\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0000 + 25.9808i 0.573959 + 0.994126i 0.996154 + 0.0876211i \(0.0279265\pi\)
−0.422195 + 0.906505i \(0.638740\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 20.7846i −0.457164 0.791831i
\(690\) 0 0
\(691\) −20.0000 + 34.6410i −0.760836 + 1.31781i 0.181584 + 0.983375i \(0.441877\pi\)
−0.942420 + 0.334431i \(0.891456\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000 27.7128i 0.606915 1.05121i
\(696\) 0 0
\(697\) −6.00000 10.3923i −0.227266 0.393637i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 20.0000 + 34.6410i 0.754314 + 1.30651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.00000 5.19615i 0.112667 0.195146i −0.804178 0.594389i \(-0.797394\pi\)
0.916845 + 0.399244i \(0.130727\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.0000 27.7128i 0.596699 1.03351i −0.396605 0.917989i \(-0.629812\pi\)
0.993305 0.115524i \(-0.0368548\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 20.7846i −0.443836 0.768747i
\(732\) 0 0
\(733\) −13.0000 + 22.5167i −0.480166 + 0.831672i −0.999741 0.0227529i \(-0.992757\pi\)
0.519575 + 0.854425i \(0.326090\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.00000 + 6.92820i −0.147342 + 0.255204i
\(738\) 0 0
\(739\) 22.0000 + 38.1051i 0.809283 + 1.40172i 0.913361 + 0.407150i \(0.133477\pi\)
−0.104078 + 0.994569i \(0.533189\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.00000 −0.0733729 −0.0366864 0.999327i \(-0.511680\pi\)
−0.0366864 + 0.999327i \(0.511680\pi\)
\(744\) 0 0
\(745\) −12.0000 20.7846i −0.439646 0.761489i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 6.92820i 0.145962 0.252814i −0.783769 0.621052i \(-0.786706\pi\)
0.929731 + 0.368238i \(0.120039\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −48.0000 −1.74690
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 + 25.9808i −0.543750 + 0.941802i 0.454935 + 0.890525i \(0.349663\pi\)
−0.998684 + 0.0512772i \(0.983671\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 + 20.7846i 0.433295 + 0.750489i
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.00000 8.66025i −0.179838 0.311488i 0.761987 0.647592i \(-0.224224\pi\)
−0.941825 + 0.336104i \(0.890891\pi\)
\(774\) 0 0
\(775\) −2.00000 + 3.46410i −0.0718421 + 0.124434i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.00000 + 6.92820i −0.143315 + 0.248229i
\(780\) 0 0
\(781\) 14.0000 + 24.2487i 0.500959 + 0.867687i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) −16.0000 27.7128i −0.570338 0.987855i −0.996531 0.0832226i \(-0.973479\pi\)
0.426193 0.904632i \(-0.359855\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.00000 + 10.3923i −0.213066 + 0.369042i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.00000 + 3.46410i −0.0705785 + 0.122245i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\)