Properties

Label 1200.2.o.f.1199.2
Level $1200$
Weight $2$
Character 1200.1199
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.o (of order \(2\), degree \(1\), not 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1199.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1199
Dual form 1200.2.o.f.1199.1

$q$-expansion

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

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
\(7\) 2.44949 0.925820 0.462910 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.89898i 1.12390i 0.827170 + 0.561951i \(0.189949\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) −3.00000 + 3.00000i −0.654654 + 0.654654i
\(22\) 0 0
\(23\) 2.44949i 0.510754i −0.966842 0.255377i \(-0.917800\pi\)
0.966842 0.255377i \(-0.0821996\pi\)
\(24\) 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
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −2.44949 2.44949i −0.392232 0.392232i
\(40\) 0 0
\(41\) 6.00000i 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 2.44949 0.373544 0.186772 0.982403i \(-0.440197\pi\)
0.186772 + 0.982403i \(0.440197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.2474i 1.78647i 0.449586 + 0.893237i \(0.351571\pi\)
−0.449586 + 0.893237i \(0.648429\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −7.34847 + 7.34847i −1.02899 + 1.02899i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 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
\(63\) 7.34847i 0.925820i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.34847 0.897758 0.448879 0.893592i \(-0.351823\pi\)
0.448879 + 0.893592i \(0.351823\pi\)
\(68\) 0 0
\(69\) 3.00000 + 3.00000i 0.361158 + 0.361158i
\(70\) 0 0
\(71\) −4.89898 −0.581402 −0.290701 0.956814i \(-0.593888\pi\)
−0.290701 + 0.956814i \(0.593888\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 4.89898i 0.551178i −0.961276 0.275589i \(-0.911127\pi\)
0.961276 0.275589i \(-0.0888729\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 7.34847i 0.806599i 0.915068 + 0.403300i \(0.132137\pi\)
−0.915068 + 0.403300i \(0.867863\pi\)
\(84\) 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
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 14.6969i 1.47710i
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) −12.2474 −1.20678 −0.603388 0.797447i \(-0.706183\pi\)
−0.603388 + 0.797447i \(0.706183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.44949i 0.236801i −0.992966 0.118401i \(-0.962223\pi\)
0.992966 0.118401i \(-0.0377767\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −2.44949 2.44949i −0.232495 0.232495i
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 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
\(127\) 7.34847 0.652071 0.326036 0.945357i \(-0.394287\pi\)
0.326036 + 0.945357i \(0.394287\pi\)
\(128\) 0 0
\(129\) −3.00000 + 3.00000i −0.264135 + 0.264135i
\(130\) 0 0
\(131\) −4.89898 −0.428026 −0.214013 0.976831i \(-0.568653\pi\)
−0.214013 + 0.976831i \(0.568653\pi\)
\(132\) 0 0
\(133\) 12.0000i 1.04053i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.89898i 0.415526i 0.978179 + 0.207763i \(0.0666183\pi\)
−0.978179 + 0.207763i \(0.933382\pi\)
\(140\) 0 0
\(141\) −15.0000 15.0000i −1.26323 1.26323i
\(142\) 0 0
\(143\) 9.79796i 0.819346i
\(144\) 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
\(153\) 18.0000i 1.45521i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 0 0
\(159\) 7.34847 7.34847i 0.582772 0.582772i
\(160\) 0 0
\(161\) 6.00000i 0.472866i
\(162\) 0 0
\(163\) 7.34847 0.575577 0.287788 0.957694i \(-0.407080\pi\)
0.287788 + 0.957694i \(0.407080\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.1464i 1.32683i −0.748251 0.663415i \(-0.769106\pi\)
0.748251 0.663415i \(-0.230894\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 14.6969 1.12390
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 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
\(187\) −29.3939 −2.14949
\(188\) 0 0
\(189\) 9.00000 + 9.00000i 0.654654 + 0.654654i
\(190\) 0 0
\(191\) 14.6969 1.06343 0.531717 0.846922i \(-0.321547\pi\)
0.531717 + 0.846922i \(0.321547\pi\)
\(192\) 0 0
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 14.6969i 1.04184i −0.853606 0.520919i \(-0.825589\pi\)
0.853606 0.520919i \(-0.174411\pi\)
\(200\) 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
\(207\) −7.34847 −0.510754
\(208\) 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
\(217\) 24.0000i 1.62923i
\(218\) 0 0
\(219\) −17.1464 17.1464i −1.15865 1.15865i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 2.44949 0.164030 0.0820150 0.996631i \(-0.473864\pi\)
0.0820150 + 0.996631i \(0.473864\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.44949i 0.162578i 0.996691 + 0.0812892i \(0.0259037\pi\)
−0.996691 + 0.0812892i \(0.974096\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 14.6969 14.6969i 0.966988 0.966988i
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 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
\(247\) −9.79796 −0.623429
\(248\) 0 0
\(249\) −9.00000 9.00000i −0.570352 0.570352i
\(250\) 0 0
\(251\) 24.4949 1.54610 0.773052 0.634343i \(-0.218729\pi\)
0.773052 + 0.634343i \(0.218729\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 4.89898i 0.304408i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.44949i 0.151042i −0.997144 0.0755210i \(-0.975938\pi\)
0.997144 0.0755210i \(-0.0240620\pi\)
\(264\) 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
\(277\) 2.00000i 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) 29.3939 1.75977
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) 26.9444 1.60168 0.800839 0.598880i \(-0.204387\pi\)
0.800839 + 0.598880i \(0.204387\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.6969i 0.867533i
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −12.2474 12.2474i −0.717958 0.717958i
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 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
\(307\) 2.44949 0.139800 0.0698999 0.997554i \(-0.477732\pi\)
0.0698999 + 0.997554i \(0.477732\pi\)
\(308\) 0 0
\(309\) 15.0000 15.0000i 0.853320 0.853320i
\(310\) 0 0
\(311\) 24.4949 1.38898 0.694489 0.719503i \(-0.255630\pi\)
0.694489 + 0.719503i \(0.255630\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.00000 + 3.00000i 0.167444 + 0.167444i
\(322\) 0 0
\(323\) 29.3939i 1.63552i
\(324\) 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
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 0 0
\(339\) −7.34847 + 7.34847i −0.399114 + 0.399114i
\(340\) 0 0
\(341\) 48.0000i 2.59935i
\(342\) 0 0
\(343\) −19.5959 −1.05808
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.34847i 0.394486i −0.980355 0.197243i \(-0.936801\pi\)
0.980355 0.197243i \(-0.0631989\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −7.34847 + 7.34847i −0.392232 + 0.392232i
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 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
\(367\) −7.34847 −0.383587 −0.191793 0.981435i \(-0.561430\pi\)
−0.191793 + 0.981435i \(0.561430\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) −14.6969 −0.763027
\(372\) 0 0
\(373\) 26.0000i 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 14.6969i 0.754931i 0.926024 + 0.377466i \(0.123204\pi\)
−0.926024 + 0.377466i \(0.876796\pi\)
\(380\) 0 0
\(381\) −9.00000 + 9.00000i −0.461084 + 0.461084i
\(382\) 0 0
\(383\) 2.44949i 0.125163i 0.998040 + 0.0625815i \(0.0199333\pi\)
−0.998040 + 0.0625815i \(0.980067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.34847i 0.373544i
\(388\) 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
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) −14.6969 14.6969i −0.735767 0.735767i
\(400\) 0 0
\(401\) 24.0000i 1.19850i 0.800561 + 0.599251i \(0.204535\pi\)
−0.800561 + 0.599251i \(0.795465\pi\)
\(402\) 0 0
\(403\) −19.5959 −0.976142
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 7.34847 7.34847i 0.362473 0.362473i
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 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
\(423\) 36.7423 1.78647
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 19.5959 0.948313
\(428\) 0 0
\(429\) 12.0000 + 12.0000i 0.579365 + 0.579365i
\(430\) 0 0
\(431\) −24.4949 −1.17988 −0.589939 0.807448i \(-0.700848\pi\)
−0.589939 + 0.807448i \(0.700848\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 24.4949i 1.16908i −0.811366 0.584539i \(-0.801275\pi\)
0.811366 0.584539i \(-0.198725\pi\)
\(440\) 0 0
\(441\) 3.00000i 0.142857i
\(442\) 0 0
\(443\) 17.1464i 0.814651i −0.913283 0.407326i \(-0.866461\pi\)
0.913283 0.407326i \(-0.133539\pi\)
\(444\) 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
\(457\) 22.0000i 1.02912i 0.857455 + 0.514558i \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) 0 0
\(459\) 22.0454 + 22.0454i 1.02899 + 1.02899i
\(460\) 0 0
\(461\) 12.0000i 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901514\pi\)
\(462\) 0 0
\(463\) 2.44949 0.113837 0.0569187 0.998379i \(-0.481872\pi\)
0.0569187 + 0.998379i \(0.481872\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.1464i 0.793442i 0.917939 + 0.396721i \(0.129852\pi\)
−0.917939 + 0.396721i \(0.870148\pi\)
\(468\) 0 0
\(469\) 18.0000 0.831163
\(470\) 0 0
\(471\) 12.2474 + 12.2474i 0.564333 + 0.564333i
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 18.0000i 0.824163i
\(478\) 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
\(487\) −22.0454 −0.998973 −0.499486 0.866322i \(-0.666478\pi\)
−0.499486 + 0.866322i \(0.666478\pi\)
\(488\) 0 0
\(489\) −9.00000 + 9.00000i −0.406994 + 0.406994i
\(490\) 0 0
\(491\) −14.6969 −0.663264 −0.331632 0.943409i \(-0.607599\pi\)
−0.331632 + 0.943409i \(0.607599\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 4.89898i 0.219308i −0.993970 0.109654i \(-0.965026\pi\)
0.993970 0.109654i \(-0.0349744\pi\)
\(500\) 0 0
\(501\) 21.0000 + 21.0000i 0.938211 + 0.938211i
\(502\) 0 0
\(503\) 36.7423i 1.63826i −0.573608 0.819130i \(-0.694457\pi\)
0.573608 0.819130i \(-0.305543\pi\)
\(504\) 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
\(517\) 60.0000i 2.63880i
\(518\) 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
\(523\) 2.44949 0.107109 0.0535544 0.998565i \(-0.482945\pi\)
0.0535544 + 0.998565i \(0.482945\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 58.7878i 2.56083i
\(528\) 0 0
\(529\) 17.0000 0.739130
\(530\) 0 0
\(531\) 29.3939i 1.27559i
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 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
\(547\) 41.6413 1.78045 0.890227 0.455517i \(-0.150545\pi\)
0.890227 + 0.455517i \(0.150545\pi\)
\(548\) 0 0
\(549\) 24.0000i 1.02430i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 12.0000i 0.510292i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 4.89898i 0.207205i
\(560\) 0 0
\(561\) 36.0000 36.0000i 1.51992 1.51992i
\(562\) 0 0
\(563\) 12.2474i 0.516168i −0.966122 0.258084i \(-0.916909\pi\)
0.966122 0.258084i \(-0.0830912\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −22.0454 −0.925820
\(568\) 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
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) −12.2474 12.2474i −0.508987 0.508987i
\(580\) 0 0
\(581\) 18.0000i 0.746766i
\(582\) 0 0
\(583\) 29.3939 1.21737
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.7423i 1.51652i −0.651953 0.758259i \(-0.726050\pi\)
0.651953 0.758259i \(-0.273950\pi\)
\(588\) 0 0
\(589\) −48.0000 −1.97781
\(590\) 0 0
\(591\) −22.0454 + 22.0454i −0.906827 + 0.906827i
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 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
\(603\) 22.0454i 0.897758i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 26.9444 1.09364 0.546819 0.837251i \(-0.315838\pi\)
0.546819 + 0.837251i \(0.315838\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.4949 −0.990957
\(612\) 0 0
\(613\) 38.0000i 1.53481i −0.641165 0.767403i \(-0.721549\pi\)
0.641165 0.767403i \(-0.278451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 4.89898i 0.196907i 0.995142 + 0.0984533i \(0.0313895\pi\)
−0.995142 + 0.0984533i \(0.968610\pi\)
\(620\) 0 0
\(621\) 9.00000 9.00000i 0.361158 0.361158i
\(622\) 0 0
\(623\) 29.3939i 1.17764i
\(624\) 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
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 14.6969i 0.581402i
\(640\) 0 0
\(641\) 18.0000i 0.710957i 0.934684 + 0.355479i \(0.115682\pi\)
−0.934684 + 0.355479i \(0.884318\pi\)
\(642\) 0 0
\(643\) −36.7423 −1.44898 −0.724488 0.689287i \(-0.757924\pi\)
−0.724488 + 0.689287i \(0.757924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.9444i 1.05929i 0.848218 + 0.529647i \(0.177675\pi\)
−0.848218 + 0.529647i \(0.822325\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) −29.3939 29.3939i −1.15204 1.15204i
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 42.0000 1.63858
\(658\) 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
\(673\) 26.0000i 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 24.4949i 0.940028i
\(680\) 0 0
\(681\) −3.00000 3.00000i −0.114960 0.114960i
\(682\) 0 0
\(683\) 31.8434i 1.21845i 0.792996 + 0.609226i \(0.208520\pi\)
−0.792996 + 0.609226i \(0.791480\pi\)
\(684\) 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
\(693\) 36.0000i 1.36753i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) 22.0454 22.0454i 0.833834 0.833834i
\(700\) 0 0
\(701\) 30.0000i 1.13308i 0.824033 + 0.566542i \(0.191719\pi\)
−0.824033 + 0.566542i \(0.808281\pi\)
\(702\) 0 0
\(703\) −9.79796 −0.369537
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29.3939i 1.10547i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −14.6969 −0.551178
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 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
\(727\) 26.9444 0.999312 0.499656 0.866224i \(-0.333460\pi\)
0.499656 + 0.866224i \(0.333460\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 14.6969 0.543586
\(732\) 0 0
\(733\) 34.0000i 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) 14.6969i 0.540636i 0.962771 + 0.270318i \(0.0871288\pi\)
−0.962771 + 0.270318i \(0.912871\pi\)
\(740\) 0 0
\(741\) 12.0000 12.0000i 0.440831 0.440831i
\(742\) 0 0
\(743\) 41.6413i 1.52767i 0.645410 + 0.763836i \(0.276686\pi\)
−0.645410 + 0.763836i \(0.723314\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22.0454 0.806599
\(748\) 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
\(757\) 2.00000i 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) −14.6969 14.6969i −0.533465 0.533465i
\(760\) 0 0
\(761\) 48.0000i 1.74000i −0.493053 0.869999i \(-0.664119\pi\)
0.493053 0.869999i \(-0.335881\pi\)
\(762\) 0 0
\(763\) −9.79796 −0.354710
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 19.5959i 0.707568i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −7.34847 + 7.34847i −0.264649 + 0.264649i
\(772\) 0 0
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 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
\(787\) −46.5403 −1.65898 −0.829491 0.558520i \(-0.811370\pi\)
−0.829491 + 0.558520i \(0.811370\pi\)
\(788\) 0 0
\(789\) 3.00000 + 3.00000i 0.106803 + 0.106803i
\(790\) 0 0
\(791\) 14.6969 0.522563
\(792\) 0 0
\(793\) 16.0000i 0.568177i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 73.4847i 2.59970i
\(800\) 0 0
\(801\) 36.0000 1.27200
\(802\) 0 0
\(803\) 68.5857i 2.42034i
\(804\) 0 0
\(805\) 0 0
\(806\) 0