Properties

Label 1200.2.h.k.1151.2
Level $1200$
Weight $2$
Character 1200.1151
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.2
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1151
Dual form 1200.2.h.k.1151.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +2.44949i q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +2.44949i q^{7} -3.00000i q^{9} +4.89898 q^{11} +2.00000 q^{13} -6.00000i q^{17} -4.89898i q^{19} +(-3.00000 - 3.00000i) q^{21} +2.44949 q^{23} +(3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(-6.00000 + 6.00000i) q^{33} -2.00000 q^{37} +(-2.44949 + 2.44949i) q^{39} +6.00000i q^{41} -2.44949i q^{43} +12.2474 q^{47} +1.00000 q^{49} +(7.34847 + 7.34847i) q^{51} -6.00000i q^{53} +(6.00000 + 6.00000i) q^{57} -9.79796 q^{59} +8.00000 q^{61} +7.34847 q^{63} +7.34847i q^{67} +(-3.00000 + 3.00000i) q^{69} +4.89898 q^{71} +14.0000 q^{73} +12.0000i q^{77} +4.89898i q^{79} -9.00000 q^{81} -7.34847 q^{83} +12.0000i q^{89} +4.89898i q^{91} +(-12.0000 - 12.0000i) q^{93} -10.0000 q^{97} -14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{13} - 12q^{21} - 24q^{33} - 8q^{37} + 4q^{49} + 24q^{57} + 32q^{61} - 12q^{69} + 56q^{73} - 36q^{81} - 48q^{93} - 40q^{97} + O(q^{100}) \)

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
\(3\) −1.22474 + 1.22474i −0.707107 + 0.707107i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949i 0.925820i 0.886405 + 0.462910i \(0.153195\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 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
\(17\) 6.00000i 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 4.89898i 1.12390i −0.827170 0.561951i \(-0.810051\pi\)
0.827170 0.561951i \(-0.189949\pi\)
\(20\) 0 0
\(21\) −3.00000 3.00000i −0.654654 0.654654i
\(22\) 0 0
\(23\) 2.44949 0.510754 0.255377 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.707107 + 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 9.79796i 1.75977i 0.475191 + 0.879883i \(0.342379\pi\)
−0.475191 + 0.879883i \(0.657621\pi\)
\(32\) 0 0
\(33\) −6.00000 + 6.00000i −1.04447 + 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −2.44949 + 2.44949i −0.392232 + 0.392232i
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) 2.44949i 0.373544i −0.982403 0.186772i \(-0.940197\pi\)
0.982403 0.186772i \(-0.0598025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.2474 1.78647 0.893237 0.449586i \(-0.148429\pi\)
0.893237 + 0.449586i \(0.148429\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 7.34847 + 7.34847i 1.02899 + 1.02899i
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 + 6.00000i 0.794719 + 0.794719i
\(58\) 0 0
\(59\) −9.79796 −1.27559 −0.637793 0.770208i \(-0.720152\pi\)
−0.637793 + 0.770208i \(0.720152\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 7.34847 0.925820
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.34847i 0.897758i 0.893592 + 0.448879i \(0.148177\pi\)
−0.893592 + 0.448879i \(0.851823\pi\)
\(68\) 0 0
\(69\) −3.00000 + 3.00000i −0.361158 + 0.361158i
\(70\) 0 0
\(71\) 4.89898 0.581402 0.290701 0.956814i \(-0.406112\pi\)
0.290701 + 0.956814i \(0.406112\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) 4.89898i 0.551178i 0.961276 + 0.275589i \(0.0888729\pi\)
−0.961276 + 0.275589i \(0.911127\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −7.34847 −0.806599 −0.403300 0.915068i \(-0.632137\pi\)
−0.403300 + 0.915068i \(0.632137\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000i 1.27200i 0.771690 + 0.635999i \(0.219412\pi\)
−0.771690 + 0.635999i \(0.780588\pi\)
\(90\) 0 0
\(91\) 4.89898i 0.513553i
\(92\) 0 0
\(93\) −12.0000 12.0000i −1.24434 1.24434i
\(94\) 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
\(99\) 14.6969i 1.47710i
\(100\) 0 0
\(101\) 12.0000i 1.19404i −0.802225 0.597022i \(-0.796350\pi\)
0.802225 0.597022i \(-0.203650\pi\)
\(102\) 0 0
\(103\) 12.2474i 1.20678i 0.797447 + 0.603388i \(0.206183\pi\)
−0.797447 + 0.603388i \(0.793817\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.44949 −0.236801 −0.118401 0.992966i \(-0.537777\pi\)
−0.118401 + 0.992966i \(0.537777\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 2.44949 2.44949i 0.232495 0.232495i
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 14.6969 1.34727
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) −7.34847 7.34847i −0.662589 0.662589i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.34847i 0.652071i 0.945357 + 0.326036i \(0.105713\pi\)
−0.945357 + 0.326036i \(0.894287\pi\)
\(128\) 0 0
\(129\) 3.00000 + 3.00000i 0.264135 + 0.264135i
\(130\) 0 0
\(131\) 4.89898 0.428026 0.214013 0.976831i \(-0.431347\pi\)
0.214013 + 0.976831i \(0.431347\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 4.89898i 0.415526i −0.978179 0.207763i \(-0.933382\pi\)
0.978179 0.207763i \(-0.0666183\pi\)
\(140\) 0 0
\(141\) −15.0000 + 15.0000i −1.26323 + 1.26323i
\(142\) 0 0
\(143\) 9.79796 0.819346
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.22474 + 1.22474i −0.101015 + 0.101015i
\(148\) 0 0
\(149\) 18.0000i 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) 19.5959i 1.59469i −0.603522 0.797347i \(-0.706236\pi\)
0.603522 0.797347i \(-0.293764\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 7.34847 + 7.34847i 0.582772 + 0.582772i
\(160\) 0 0
\(161\) 6.00000i 0.472866i
\(162\) 0 0
\(163\) 7.34847i 0.575577i −0.957694 0.287788i \(-0.907080\pi\)
0.957694 0.287788i \(-0.0929199\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.1464 −1.32683 −0.663415 0.748251i \(-0.730894\pi\)
−0.663415 + 0.748251i \(0.730894\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −14.6969 −1.12390
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 12.0000i 0.901975 0.901975i
\(178\) 0 0
\(179\) −9.79796 −0.732334 −0.366167 0.930549i \(-0.619330\pi\)
−0.366167 + 0.930549i \(0.619330\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −9.79796 + 9.79796i −0.724286 + 0.724286i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 29.3939i 2.14949i
\(188\) 0 0
\(189\) −9.00000 + 9.00000i −0.654654 + 0.654654i
\(190\) 0 0
\(191\) −14.6969 −1.06343 −0.531717 0.846922i \(-0.678453\pi\)
−0.531717 + 0.846922i \(0.678453\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 14.6969i 1.04184i 0.853606 + 0.520919i \(0.174411\pi\)
−0.853606 + 0.520919i \(0.825589\pi\)
\(200\) 0 0
\(201\) −9.00000 9.00000i −0.634811 0.634811i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.34847i 0.510754i
\(208\) 0 0
\(209\) 24.0000i 1.66011i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −6.00000 + 6.00000i −0.411113 + 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) −17.1464 + 17.1464i −1.15865 + 1.15865i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 2.44949i 0.164030i −0.996631 0.0820150i \(-0.973864\pi\)
0.996631 0.0820150i \(-0.0261355\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.44949 0.162578 0.0812892 0.996691i \(-0.474096\pi\)
0.0812892 + 0.996691i \(0.474096\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −14.6969 14.6969i −0.966988 0.966988i
\(232\) 0 0
\(233\) 18.0000i 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.00000 6.00000i −0.389742 0.389742i
\(238\) 0 0
\(239\) 9.79796 0.633777 0.316889 0.948463i \(-0.397362\pi\)
0.316889 + 0.948463i \(0.397362\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.707107 0.707107i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.79796i 0.623429i
\(248\) 0 0
\(249\) 9.00000 9.00000i 0.570352 0.570352i
\(250\) 0 0
\(251\) −24.4949 −1.54610 −0.773052 0.634343i \(-0.781271\pi\)
−0.773052 + 0.634343i \(0.781271\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000i 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) 4.89898i 0.304408i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.44949 0.151042 0.0755210 0.997144i \(-0.475938\pi\)
0.0755210 + 0.997144i \(0.475938\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −14.6969 14.6969i −0.899438 0.899438i
\(268\) 0 0
\(269\) 18.0000i 1.09748i 0.835993 + 0.548740i \(0.184892\pi\)
−0.835993 + 0.548740i \(0.815108\pi\)
\(270\) 0 0
\(271\) 9.79796i 0.595184i −0.954693 0.297592i \(-0.903817\pi\)
0.954693 0.297592i \(-0.0961834\pi\)
\(272\) 0 0
\(273\) −6.00000 6.00000i −0.363137 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 29.3939 1.75977
\(280\) 0 0
\(281\) 6.00000i 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 0 0
\(283\) 26.9444i 1.60168i −0.598880 0.800839i \(-0.704387\pi\)
0.598880 0.800839i \(-0.295613\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.6969 −0.867533
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 12.2474 12.2474i 0.717958 0.717958i
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.0000 + 18.0000i 1.04447 + 1.04447i
\(298\) 0 0
\(299\) 4.89898 0.283315
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) 14.6969 + 14.6969i 0.844317 + 0.844317i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.44949i 0.139800i 0.997554 + 0.0698999i \(0.0222680\pi\)
−0.997554 + 0.0698999i \(0.977732\pi\)
\(308\) 0 0
\(309\) −15.0000 15.0000i −0.853320 0.853320i
\(310\) 0 0
\(311\) −24.4949 −1.38898 −0.694489 0.719503i \(-0.744370\pi\)
−0.694489 + 0.719503i \(0.744370\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.00000 3.00000i 0.167444 0.167444i
\(322\) 0 0
\(323\) −29.3939 −1.63552
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4.89898 + 4.89898i −0.270914 + 0.270914i
\(328\) 0 0
\(329\) 30.0000i 1.65395i
\(330\) 0 0
\(331\) 19.5959i 1.07709i −0.842597 0.538545i \(-0.818974\pi\)
0.842597 0.538545i \(-0.181026\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −7.34847 7.34847i −0.399114 0.399114i
\(340\) 0 0
\(341\) 48.0000i 2.59935i
\(342\) 0 0
\(343\) 19.5959i 1.05808i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.34847 −0.394486 −0.197243 0.980355i \(-0.563199\pi\)
−0.197243 + 0.980355i \(0.563199\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 7.34847 + 7.34847i 0.392232 + 0.392232i
\(352\) 0 0
\(353\) 6.00000i 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −18.0000 + 18.0000i −0.952661 + 0.952661i
\(358\) 0 0
\(359\) 9.79796 0.517116 0.258558 0.965996i \(-0.416753\pi\)
0.258558 + 0.965996i \(0.416753\pi\)
\(360\) 0 0
\(361\) −5.00000 −0.263158
\(362\) 0 0
\(363\) −15.9217 + 15.9217i −0.835672 + 0.835672i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.34847i 0.383587i −0.981435 0.191793i \(-0.938570\pi\)
0.981435 0.191793i \(-0.0614304\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 14.6969 0.763027
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 14.6969i 0.754931i −0.926024 0.377466i \(-0.876796\pi\)
0.926024 0.377466i \(-0.123204\pi\)
\(380\) 0 0
\(381\) −9.00000 9.00000i −0.461084 0.461084i
\(382\) 0 0
\(383\) −2.44949 −0.125163 −0.0625815 0.998040i \(-0.519933\pi\)
−0.0625815 + 0.998040i \(0.519933\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.34847 −0.373544
\(388\) 0 0
\(389\) 18.0000i 0.912636i 0.889817 + 0.456318i \(0.150832\pi\)
−0.889817 + 0.456318i \(0.849168\pi\)
\(390\) 0 0
\(391\) 14.6969i 0.743256i
\(392\) 0 0
\(393\) −6.00000 + 6.00000i −0.302660 + 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −14.6969 + 14.6969i −0.735767 + 0.735767i
\(400\) 0 0
\(401\) 24.0000i 1.19850i −0.800561 0.599251i \(-0.795465\pi\)
0.800561 0.599251i \(-0.204535\pi\)
\(402\) 0 0
\(403\) 19.5959i 0.976142i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.79796 −0.485667
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 0 0
\(411\) −7.34847 7.34847i −0.362473 0.362473i
\(412\) 0 0
\(413\) 24.0000i 1.18096i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.00000 + 6.00000i 0.293821 + 0.293821i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 0 0
\(423\) 36.7423i 1.78647i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 19.5959i 0.948313i
\(428\) 0 0
\(429\) −12.0000 + 12.0000i −0.579365 + 0.579365i
\(430\) 0 0
\(431\) 24.4949 1.17988 0.589939 0.807448i \(-0.299152\pi\)
0.589939 + 0.807448i \(0.299152\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) 24.4949i 1.16908i 0.811366 + 0.584539i \(0.198725\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 0 0
\(441\) 3.00000i 0.142857i
\(442\) 0 0
\(443\) 17.1464 0.814651 0.407326 0.913283i \(-0.366461\pi\)
0.407326 + 0.913283i \(0.366461\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 22.0454 + 22.0454i 1.04271 + 1.04271i
\(448\) 0 0
\(449\) 6.00000i 0.283158i −0.989927 0.141579i \(-0.954782\pi\)
0.989927 0.141579i \(-0.0452178\pi\)
\(450\) 0 0
\(451\) 29.3939i 1.38410i
\(452\) 0 0
\(453\) 24.0000 + 24.0000i 1.12762 + 1.12762i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 22.0454 22.0454i 1.02899 1.02899i
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 0 0
\(463\) 2.44949i 0.113837i −0.998379 0.0569187i \(-0.981872\pi\)
0.998379 0.0569187i \(-0.0181276\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.1464 0.793442 0.396721 0.917939i \(-0.370148\pi\)
0.396721 + 0.917939i \(0.370148\pi\)
\(468\) 0 0
\(469\) −18.0000 −0.831163
\(470\) 0 0
\(471\) −12.2474 + 12.2474i −0.564333 + 0.564333i
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 0 0
\(479\) −29.3939 −1.34304 −0.671520 0.740986i \(-0.734358\pi\)
−0.671520 + 0.740986i \(0.734358\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) −7.34847 7.34847i −0.334367 0.334367i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0454i 0.998973i −0.866322 0.499486i \(-0.833522\pi\)
0.866322 0.499486i \(-0.166478\pi\)
\(488\) 0 0
\(489\) 9.00000 + 9.00000i 0.406994 + 0.406994i
\(490\) 0 0
\(491\) 14.6969 0.663264 0.331632 0.943409i \(-0.392401\pi\)
0.331632 + 0.943409i \(0.392401\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) 4.89898i 0.219308i 0.993970 + 0.109654i \(0.0349744\pi\)
−0.993970 + 0.109654i \(0.965026\pi\)
\(500\) 0 0
\(501\) 21.0000 21.0000i 0.938211 0.938211i
\(502\) 0 0
\(503\) 36.7423 1.63826 0.819130 0.573608i \(-0.194457\pi\)
0.819130 + 0.573608i \(0.194457\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 11.0227 11.0227i 0.489535 0.489535i
\(508\) 0 0
\(509\) 24.0000i 1.06378i −0.846813 0.531891i \(-0.821482\pi\)
0.846813 0.531891i \(-0.178518\pi\)
\(510\) 0 0
\(511\) 34.2929i 1.51703i
\(512\) 0 0
\(513\) 18.0000 18.0000i 0.794719 0.794719i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 60.0000 2.63880
\(518\) 0 0
\(519\) −22.0454 22.0454i −0.967686 0.967686i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 2.44949i 0.107109i −0.998565 0.0535544i \(-0.982945\pi\)
0.998565 0.0535544i \(-0.0170550\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 58.7878 2.56083
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) 29.3939i 1.27559i
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000 12.0000i 0.517838 0.517838i
\(538\) 0 0
\(539\) 4.89898 0.211014
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) −12.2474 + 12.2474i −0.525588 + 0.525588i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 41.6413i 1.78045i 0.455517 + 0.890227i \(0.349455\pi\)
−0.455517 + 0.890227i \(0.650545\pi\)
\(548\) 0 0
\(549\) 24.0000i 1.02430i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 4.89898i 0.207205i
\(560\) 0 0
\(561\) 36.0000 + 36.0000i 1.51992 + 1.51992i
\(562\) 0 0
\(563\) 12.2474 0.516168 0.258084 0.966122i \(-0.416909\pi\)
0.258084 + 0.966122i \(0.416909\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.0454i 0.925820i
\(568\) 0 0
\(569\) 30.0000i 1.25767i −0.777541 0.628833i \(-0.783533\pi\)
0.777541 0.628833i \(-0.216467\pi\)
\(570\) 0 0
\(571\) 9.79796i 0.410032i 0.978759 + 0.205016i \(0.0657246\pi\)
−0.978759 + 0.205016i \(0.934275\pi\)
\(572\) 0 0
\(573\) 18.0000 18.0000i 0.751961 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) −12.2474 + 12.2474i −0.508987 + 0.508987i
\(580\) 0 0
\(581\) 18.0000i 0.746766i
\(582\) 0 0
\(583\) 29.3939i 1.21737i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.7423 −1.51652 −0.758259 0.651953i \(-0.773950\pi\)
−0.758259 + 0.651953i \(0.773950\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) 22.0454 + 22.0454i 0.906827 + 0.906827i
\(592\) 0 0
\(593\) 42.0000i 1.72473i 0.506284 + 0.862367i \(0.331019\pi\)
−0.506284 + 0.862367i \(0.668981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.0000 18.0000i −0.736691 0.736691i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 22.0454 0.897758
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 26.9444i 1.09364i 0.837251 + 0.546819i \(0.184162\pi\)
−0.837251 + 0.546819i \(0.815838\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.4949 0.990957
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) 4.89898i 0.196907i −0.995142 0.0984533i \(-0.968610\pi\)
0.995142 0.0984533i \(-0.0313895\pi\)
\(620\) 0 0
\(621\) 9.00000 + 9.00000i 0.361158 + 0.361158i
\(622\) 0 0
\(623\) −29.3939 −1.17764
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 29.3939 + 29.3939i 1.17388 + 1.17388i
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 14.6969i 0.581402i
\(640\) 0 0
\(641\) 18.0000i 0.710957i −0.934684 0.355479i \(-0.884318\pi\)
0.934684 0.355479i \(-0.115682\pi\)
\(642\) 0 0
\(643\) 36.7423i 1.44898i 0.689287 + 0.724488i \(0.257924\pi\)
−0.689287 + 0.724488i \(0.742076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.9444 1.05929 0.529647 0.848218i \(-0.322325\pi\)
0.529647 + 0.848218i \(0.322325\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 29.3939 29.3939i 1.15204 1.15204i
\(652\) 0 0
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 42.0000i 1.63858i
\(658\) 0 0
\(659\) −29.3939 −1.14502 −0.572511 0.819897i \(-0.694031\pi\)
−0.572511 + 0.819897i \(0.694031\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) 14.6969 + 14.6969i 0.570782 + 0.570782i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.00000 + 3.00000i 0.115987 + 0.115987i
\(670\) 0 0
\(671\) 39.1918 1.51298
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000i 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) 24.4949i 0.940028i
\(680\) 0 0
\(681\) −3.00000 + 3.00000i −0.114960 + 0.114960i
\(682\) 0 0
\(683\) −31.8434 −1.21845 −0.609226 0.792996i \(-0.708520\pi\)
−0.609226 + 0.792996i \(0.708520\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −26.9444 + 26.9444i −1.02799 + 1.02799i
\(688\) 0 0
\(689\) 12.0000i 0.457164i
\(690\) 0 0
\(691\) 29.3939i 1.11820i 0.829102 + 0.559098i \(0.188852\pi\)
−0.829102 + 0.559098i \(0.811148\pi\)
\(692\) 0 0
\(693\) 36.0000 1.36753
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 22.0454 + 22.0454i 0.833834 + 0.833834i
\(700\) 0 0
\(701\) 30.0000i 1.13308i −0.824033 0.566542i \(-0.808281\pi\)
0.824033 0.566542i \(-0.191719\pi\)
\(702\) 0 0
\(703\) 9.79796i 0.369537i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29.3939 1.10547
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 14.6969 0.551178
\(712\) 0 0
\(713\) 24.0000i 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 + 12.0000i −0.448148 + 0.448148i
\(718\) 0 0
\(719\) 9.79796 0.365402 0.182701 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) −4.89898 + 4.89898i −0.182195 + 0.182195i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 26.9444i 0.999312i 0.866224 + 0.499656i \(0.166540\pi\)
−0.866224 + 0.499656i \(0.833460\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) −14.6969 −0.543586
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.0000i 1.32608i
\(738\) 0 0
\(739\) 14.6969i 0.540636i −0.962771 0.270318i \(-0.912871\pi\)
0.962771 0.270318i \(-0.0871288\pi\)
\(740\) 0 0
\(741\) 12.0000 + 12.0000i 0.440831 + 0.440831i
\(742\) 0 0
\(743\) −41.6413 −1.52767 −0.763836 0.645410i \(-0.776686\pi\)
−0.763836 + 0.645410i \(0.776686\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22.0454i 0.806599i
\(748\) 0 0
\(749\) 6.00000i 0.219235i
\(750\) 0 0
\(751\) 19.5959i 0.715065i 0.933901 + 0.357533i \(0.116382\pi\)
−0.933901 + 0.357533i \(0.883618\pi\)
\(752\) 0 0
\(753\) 30.0000 30.0000i 1.09326 1.09326i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) −14.6969 + 14.6969i −0.533465 + 0.533465i
\(760\) 0 0
\(761\) 48.0000i 1.74000i 0.493053 + 0.869999i \(0.335881\pi\)
−0.493053 + 0.869999i \(0.664119\pi\)
\(762\) 0 0
\(763\) 9.79796i 0.354710i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −19.5959 −0.707568
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 7.34847 + 7.34847i 0.264649 + 0.264649i
\(772\) 0 0
\(773\) 42.0000i 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 + 6.00000i 0.215249 + 0.215249i
\(778\) 0 0
\(779\) 29.3939 1.05314
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.5403i 1.65898i −0.558520 0.829491i \(-0.688630\pi\)
0.558520 0.829491i \(-0.311370\pi\)
\(788\) 0 0
\(789\) −3.00000 + 3.00000i −0.106803 + 0.106803i
\(790\) 0 0
\(791\) −14.6969 −0.522563
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 0 0
\(799\) 73.4847i 2.59970i
\(800\) 0 0
\(801\) 36.0000 1.27200
\(802\) 0 0
\(803\) 68.5857 2.42034
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\)