Properties

Label 1200.2.h.i.1151.1
Level $1200$
Weight $2$
Character 1200.1151
Analytic conductor $9.582$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1151
Dual form 1200.2.h.i.1151.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{9} +3.00000 q^{11} +2.00000 q^{13} +5.19615i q^{17} +5.19615i q^{19} +6.00000 q^{23} -5.19615i q^{27} -10.3923i q^{29} -3.46410i q^{31} +(4.50000 - 2.59808i) q^{33} -8.00000 q^{37} +(3.00000 - 1.73205i) q^{39} -5.19615i q^{41} +3.46410i q^{43} -6.00000 q^{47} +7.00000 q^{49} +(4.50000 + 7.79423i) q^{51} -10.3923i q^{53} +(4.50000 + 7.79423i) q^{57} +12.0000 q^{59} +8.00000 q^{61} +12.1244i q^{67} +(9.00000 - 5.19615i) q^{69} -6.00000 q^{71} -1.00000 q^{73} +6.92820i q^{79} +(-4.50000 - 7.79423i) q^{81} -9.00000 q^{83} +(-9.00000 - 15.5885i) q^{87} -5.19615i q^{89} +(-3.00000 - 5.19615i) q^{93} -10.0000 q^{97} +(4.50000 - 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 3q^{9} + 6q^{11} + 4q^{13} + 12q^{23} + 9q^{33} - 16q^{37} + 6q^{39} - 12q^{47} + 14q^{49} + 9q^{51} + 9q^{57} + 24q^{59} + 16q^{61} + 18q^{69} - 12q^{71} - 2q^{73} - 9q^{81} - 18q^{83} - 18q^{87} - 6q^{93} - 20q^{97} + 9q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.50000 0.866025i 0.866025 0.500000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 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
\(17\) 5.19615i 1.26025i 0.776493 + 0.630126i \(0.216997\pi\)
−0.776493 + 0.630126i \(0.783003\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i 0.802955 + 0.596040i \(0.203260\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 10.3923i 1.92980i −0.262613 0.964901i \(-0.584584\pi\)
0.262613 0.964901i \(-0.415416\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) 4.50000 2.59808i 0.783349 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 3.00000 1.73205i 0.480384 0.277350i
\(40\) 0 0
\(41\) 5.19615i 0.811503i −0.913984 0.405751i \(-0.867010\pi\)
0.913984 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.50000 + 7.79423i 0.630126 + 1.09141i
\(52\) 0 0
\(53\) 10.3923i 1.42749i −0.700404 0.713746i \(-0.746997\pi\)
0.700404 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.50000 + 7.79423i 0.596040 + 1.03237i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.1244i 1.48123i 0.671932 + 0.740613i \(0.265465\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 9.00000 5.19615i 1.08347 0.625543i
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.92820i 0.779484i 0.920924 + 0.389742i \(0.127436\pi\)
−0.920924 + 0.389742i \(0.872564\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.00000 15.5885i −0.964901 1.67126i
\(88\) 0 0
\(89\) 5.19615i 0.550791i −0.961331 0.275396i \(-0.911191\pi\)
0.961331 0.275396i \(-0.0888088\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.00000 5.19615i −0.311086 0.538816i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 4.50000 7.79423i 0.452267 0.783349i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 17.3205i 1.70664i 0.521387 + 0.853320i \(0.325415\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) −12.0000 + 6.92820i −1.13899 + 0.657596i
\(112\) 0 0
\(113\) 5.19615i 0.488813i −0.969673 0.244406i \(-0.921407\pi\)
0.969673 0.244406i \(-0.0785931\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.00000 5.19615i 0.277350 0.480384i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −4.50000 7.79423i −0.405751 0.702782i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.3923i 0.922168i 0.887357 + 0.461084i \(0.152539\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 3.00000 + 5.19615i 0.264135 + 0.457496i
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.19615i 0.443937i 0.975054 + 0.221969i \(0.0712483\pi\)
−0.975054 + 0.221969i \(0.928752\pi\)
\(138\) 0 0
\(139\) 8.66025i 0.734553i 0.930112 + 0.367277i \(0.119710\pi\)
−0.930112 + 0.367277i \(0.880290\pi\)
\(140\) 0 0
\(141\) −9.00000 + 5.19615i −0.757937 + 0.437595i
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 10.5000 6.06218i 0.866025 0.500000i
\(148\) 0 0
\(149\) 10.3923i 0.851371i 0.904871 + 0.425685i \(0.139967\pi\)
−0.904871 + 0.425685i \(0.860033\pi\)
\(150\) 0 0
\(151\) 20.7846i 1.69143i −0.533637 0.845714i \(-0.679175\pi\)
0.533637 0.845714i \(-0.320825\pi\)
\(152\) 0 0
\(153\) 13.5000 + 7.79423i 1.09141 + 0.630126i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −9.00000 15.5885i −0.713746 1.23625i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.1244i 0.949653i −0.880079 0.474826i \(-0.842511\pi\)
0.880079 0.474826i \(-0.157489\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 13.5000 + 7.79423i 1.03237 + 0.596040i
\(172\) 0 0
\(173\) 20.7846i 1.58022i 0.612962 + 0.790112i \(0.289978\pi\)
−0.612962 + 0.790112i \(0.710022\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 18.0000 10.3923i 1.35296 0.781133i
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 12.0000 6.92820i 0.887066 0.512148i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.5885i 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923i 0.740421i −0.928948 0.370211i \(-0.879286\pi\)
0.928948 0.370211i \(-0.120714\pi\)
\(198\) 0 0
\(199\) 6.92820i 0.491127i 0.969380 + 0.245564i \(0.0789730\pi\)
−0.969380 + 0.245564i \(0.921027\pi\)
\(200\) 0 0
\(201\) 10.5000 + 18.1865i 0.740613 + 1.28278i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.00000 15.5885i 0.625543 1.08347i
\(208\) 0 0
\(209\) 15.5885i 1.07828i
\(210\) 0 0
\(211\) 1.73205i 0.119239i 0.998221 + 0.0596196i \(0.0189888\pi\)
−0.998221 + 0.0596196i \(0.981011\pi\)
\(212\) 0 0
\(213\) −9.00000 + 5.19615i −0.616670 + 0.356034i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.50000 + 0.866025i −0.101361 + 0.0585206i
\(220\) 0 0
\(221\) 10.3923i 0.699062i
\(222\) 0 0
\(223\) 6.92820i 0.463947i −0.972722 0.231973i \(-0.925482\pi\)
0.972722 0.231973i \(-0.0745182\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.7846i 1.36165i 0.732448 + 0.680823i \(0.238378\pi\)
−0.732448 + 0.680823i \(0.761622\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.00000 + 10.3923i 0.389742 + 0.675053i
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 10.3923i 0.661247i
\(248\) 0 0
\(249\) −13.5000 + 7.79423i −0.855528 + 0.493939i
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.7846i 1.29651i −0.761424 0.648254i \(-0.775499\pi\)
0.761424 0.648254i \(-0.224501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −27.0000 15.5885i −1.67126 0.964901i
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4.50000 7.79423i −0.275396 0.476999i
\(268\) 0 0
\(269\) 20.7846i 1.26726i 0.773636 + 0.633630i \(0.218436\pi\)
−0.773636 + 0.633630i \(0.781564\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i −0.676435 0.736502i \(-0.736476\pi\)
0.676435 0.736502i \(-0.263524\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) −9.00000 5.19615i −0.538816 0.311086i
\(280\) 0 0
\(281\) 20.7846i 1.23991i 0.784639 + 0.619953i \(0.212848\pi\)
−0.784639 + 0.619953i \(0.787152\pi\)
\(282\) 0 0
\(283\) 12.1244i 0.720718i 0.932814 + 0.360359i \(0.117346\pi\)
−0.932814 + 0.360359i \(0.882654\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.0000 −0.588235
\(290\) 0 0
\(291\) −15.0000 + 8.66025i −0.879316 + 0.507673i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.5885i 0.904534i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.1244i 0.691974i 0.938239 + 0.345987i \(0.112456\pi\)
−0.938239 + 0.345987i \(0.887544\pi\)
\(308\) 0 0
\(309\) 15.0000 + 25.9808i 0.853320 + 1.47799i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 31.1769i 1.74557i
\(320\) 0 0
\(321\) 4.50000 2.59808i 0.251166 0.145010i
\(322\) 0 0
\(323\) −27.0000 −1.50232
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.0000 + 6.92820i −0.663602 + 0.383131i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.66025i 0.476011i 0.971264 + 0.238005i \(0.0764936\pi\)
−0.971264 + 0.238005i \(0.923506\pi\)
\(332\) 0 0
\(333\) −12.0000 + 20.7846i −0.657596 + 1.13899i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −25.0000 −1.36184 −0.680918 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) 0 0
\(339\) −4.50000 7.79423i −0.244406 0.423324i
\(340\) 0 0
\(341\) 10.3923i 0.562775i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.00000 0.483145 0.241573 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 0 0
\(351\) 10.3923i 0.554700i
\(352\) 0 0
\(353\) 20.7846i 1.10625i −0.833097 0.553127i \(-0.813435\pi\)
0.833097 0.553127i \(-0.186565\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) −3.00000 + 1.73205i −0.157459 + 0.0909091i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.92820i 0.361649i −0.983515 0.180825i \(-0.942123\pi\)
0.983515 0.180825i \(-0.0578766\pi\)
\(368\) 0 0
\(369\) −13.5000 7.79423i −0.702782 0.405751i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.7846i 1.07046i
\(378\) 0 0
\(379\) 19.0526i 0.978664i 0.872098 + 0.489332i \(0.162759\pi\)
−0.872098 + 0.489332i \(0.837241\pi\)
\(380\) 0 0
\(381\) 9.00000 + 15.5885i 0.461084 + 0.798621i
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.00000 + 5.19615i 0.457496 + 0.264135i
\(388\) 0 0
\(389\) 10.3923i 0.526911i 0.964672 + 0.263455i \(0.0848622\pi\)
−0.964672 + 0.263455i \(0.915138\pi\)
\(390\) 0 0
\(391\) 31.1769i 1.57668i
\(392\) 0 0
\(393\) 18.0000 10.3923i 0.907980 0.524222i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.19615i 0.259483i −0.991548 0.129742i \(-0.958585\pi\)
0.991548 0.129742i \(-0.0414148\pi\)
\(402\) 0 0
\(403\) 6.92820i 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 4.50000 + 7.79423i 0.221969 + 0.384461i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.50000 + 12.9904i 0.367277 + 0.636142i
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) −9.00000 + 15.5885i −0.437595 + 0.757937i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 9.00000 5.19615i 0.434524 0.250873i
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.1769i 1.49139i
\(438\) 0 0
\(439\) 17.3205i 0.826663i 0.910581 + 0.413331i \(0.135635\pi\)
−0.910581 + 0.413331i \(0.864365\pi\)
\(440\) 0 0
\(441\) 10.5000 18.1865i 0.500000 0.866025i
\(442\) 0 0
\(443\) 15.0000 0.712672 0.356336 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.00000 + 15.5885i 0.425685 + 0.737309i
\(448\) 0 0
\(449\) 5.19615i 0.245222i −0.992455 0.122611i \(-0.960873\pi\)
0.992455 0.122611i \(-0.0391267\pi\)
\(450\) 0 0
\(451\) 15.5885i 0.734032i
\(452\) 0 0
\(453\) −18.0000 31.1769i −0.845714 1.46482i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) 27.0000 1.26025
\(460\) 0 0
\(461\) 20.7846i 0.968036i −0.875058 0.484018i \(-0.839177\pi\)
0.875058 0.484018i \(-0.160823\pi\)
\(462\) 0 0
\(463\) 27.7128i 1.28792i 0.765058 + 0.643962i \(0.222710\pi\)
−0.765058 + 0.643962i \(0.777290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3.00000 + 1.73205i −0.138233 + 0.0798087i
\(472\) 0 0
\(473\) 10.3923i 0.477839i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −27.0000 15.5885i −1.23625 0.713746i
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.1051i 1.72671i −0.504599 0.863354i \(-0.668360\pi\)
0.504599 0.863354i \(-0.331640\pi\)
\(488\) 0 0
\(489\) −10.5000 18.1865i −0.474826 0.822423i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 54.0000 2.43204
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.1051i 1.70582i −0.522059 0.852910i \(-0.674836\pi\)
0.522059 0.852910i \(-0.325164\pi\)
\(500\) 0 0
\(501\) −9.00000 + 5.19615i −0.402090 + 0.232147i
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.5000 + 7.79423i −0.599556 + 0.346154i
\(508\) 0 0
\(509\) 31.1769i 1.38189i −0.722906 0.690946i \(-0.757194\pi\)
0.722906 0.690946i \(-0.242806\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 27.0000 1.19208
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) 18.0000 + 31.1769i 0.790112 + 1.36851i
\(520\) 0 0
\(521\) 5.19615i 0.227648i 0.993501 + 0.113824i \(0.0363099\pi\)
−0.993501 + 0.113824i \(0.963690\pi\)
\(522\) 0 0
\(523\) 25.9808i 1.13606i −0.823008 0.568030i \(-0.807706\pi\)
0.823008 0.568030i \(-0.192294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 18.0000 31.1769i 0.781133 1.35296i
\(532\) 0 0
\(533\) 10.3923i 0.450141i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −22.5000 + 12.9904i −0.970947 + 0.560576i
\(538\) 0 0
\(539\) 21.0000 0.904534
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) 15.0000 8.66025i 0.643712 0.371647i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19.0526i 0.814629i −0.913288 0.407314i \(-0.866465\pi\)
0.913288 0.407314i \(-0.133535\pi\)
\(548\) 0 0
\(549\) 12.0000 20.7846i 0.512148 0.887066i
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.1769i 1.32101i 0.750822 + 0.660504i \(0.229657\pi\)
−0.750822 + 0.660504i \(0.770343\pi\)
\(558\) 0 0
\(559\) 6.92820i 0.293032i
\(560\) 0 0
\(561\) 13.5000 + 23.3827i 0.569970 + 0.987218i
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.9808i 1.08917i −0.838706 0.544585i \(-0.816687\pi\)
0.838706 0.544585i \(-0.183313\pi\)
\(570\) 0 0
\(571\) 31.1769i 1.30471i 0.757912 + 0.652357i \(0.226220\pi\)
−0.757912 + 0.652357i \(0.773780\pi\)
\(572\) 0 0
\(573\) −27.0000 + 15.5885i −1.12794 + 0.651217i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000 0.291414 0.145707 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 0 0
\(579\) −34.5000 + 19.9186i −1.43377 + 0.827788i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 31.1769i 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) −9.00000 15.5885i −0.370211 0.641223i
\(592\) 0 0
\(593\) 15.5885i 0.640141i −0.947394 0.320071i \(-0.896293\pi\)
0.947394 0.320071i \(-0.103707\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.00000 + 10.3923i 0.245564 + 0.425329i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 31.5000 + 18.1865i 1.28278 + 0.740613i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 6.92820i 0.281207i 0.990066 + 0.140604i \(0.0449043\pi\)
−0.990066 + 0.140604i \(0.955096\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 46.0000 1.85792 0.928961 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.5692i 1.67351i 0.547575 + 0.836757i \(0.315551\pi\)
−0.547575 + 0.836757i \(0.684449\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i 0.977947 + 0.208851i \(0.0669724\pi\)
−0.977947 + 0.208851i \(0.933028\pi\)
\(620\) 0 0
\(621\) 31.1769i 1.25109i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 13.5000 + 23.3827i 0.539138 + 0.933815i
\(628\) 0 0
\(629\) 41.5692i 1.65747i
\(630\) 0 0
\(631\) 3.46410i 0.137904i 0.997620 + 0.0689519i \(0.0219655\pi\)
−0.997620 + 0.0689519i \(0.978035\pi\)
\(632\) 0 0
\(633\) 1.50000 + 2.59808i 0.0596196 + 0.103264i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.0000 0.554700
\(638\) 0 0
\(639\) −9.00000 + 15.5885i −0.356034 + 0.616670i
\(640\) 0 0
\(641\) 20.7846i 0.820943i −0.911873 0.410471i \(-0.865364\pi\)
0.911873 0.410471i \(-0.134636\pi\)
\(642\) 0 0
\(643\) 31.1769i 1.22950i 0.788723 + 0.614749i \(0.210743\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.7846i 0.813365i −0.913570 0.406682i \(-0.866686\pi\)
0.913570 0.406682i \(-0.133314\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.50000 + 2.59808i −0.0585206 + 0.101361i
\(658\) 0 0
\(659\) −9.00000 −0.350590 −0.175295 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) 9.00000 + 15.5885i 0.349531 + 0.605406i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 62.3538i 2.41435i
\(668\) 0 0
\(669\) −6.00000 10.3923i −0.231973 0.401790i
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 41.5692i 1.59763i −0.601574 0.798817i \(-0.705459\pi\)
0.601574 0.798817i \(-0.294541\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 36.0000 20.7846i 1.37952 0.796468i
\(682\) 0 0
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −30.0000 + 17.3205i −1.14457 + 0.660819i
\(688\) 0 0
\(689\) 20.7846i 0.791831i
\(690\) 0 0
\(691\) 5.19615i 0.197671i 0.995104 + 0.0988355i \(0.0315118\pi\)
−0.995104 + 0.0988355i \(0.968488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 0 0
\(699\) 18.0000 + 31.1769i 0.680823 + 1.17922i
\(700\) 0 0
\(701\) 31.1769i 1.17754i 0.808302 + 0.588768i \(0.200387\pi\)
−0.808302 + 0.588768i \(0.799613\pi\)
\(702\) 0 0
\(703\) 41.5692i 1.56781i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) 18.0000 + 10.3923i 0.675053 + 0.389742i
\(712\) 0 0
\(713\) 20.7846i 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −45.0000 + 25.9808i −1.68056 + 0.970269i
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −7.50000 + 4.33013i −0.278928 + 0.161039i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.3205i 0.642382i −0.947014 0.321191i \(-0.895917\pi\)
0.947014 0.321191i \(-0.104083\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.3731i 1.33982i
\(738\) 0 0
\(739\) 10.3923i 0.382287i −0.981562 0.191144i \(-0.938780\pi\)
0.981562 0.191144i \(-0.0612196\pi\)
\(740\) 0 0
\(741\) 9.00000 + 15.5885i 0.330623 + 0.572656i
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.5000 + 23.3827i −0.493939 + 0.855528i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 38.1051i 1.39048i 0.718780 + 0.695238i \(0.244701\pi\)
−0.718780 + 0.695238i \(0.755299\pi\)
\(752\) 0 0
\(753\) 31.5000 18.1865i 1.14792 0.662754i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 27.0000 15.5885i 0.980038 0.565825i
\(760\) 0 0
\(761\) 15.5885i 0.565081i −0.959255 0.282541i \(-0.908823\pi\)
0.959255 0.282541i \(-0.0911772\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 47.0000 1.69486 0.847432 0.530904i \(-0.178148\pi\)
0.847432 + 0.530904i \(0.178148\pi\)
\(770\) 0 0
\(771\) −18.0000 31.1769i −0.648254 1.12281i
\(772\) 0 0
\(773\) 10.3923i 0.373785i 0.982380 + 0.186893i \(0.0598416\pi\)
−0.982380 + 0.186893i \(0.940158\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.0000 0.967375
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) −54.0000 −1.92980
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.3923i 0.370446i −0.982697 0.185223i \(-0.940699\pi\)
0.982697 0.185223i \(-0.0593007\pi\)
\(788\) 0 0
\(789\) 9.00000 5.19615i 0.320408 0.184988i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 51.9615i 1.84057i −0.391247 0.920286i \(-0.627956\pi\)
0.391247 0.920286i \(-0.372044\pi\)
\(798\) 0 0
\(799\) 31.1769i 1.10296i
\(800\) 0 0
\(801\) −13.5000 7.79423i −0.476999 0.275396i
\(802\) 0 0
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0