Properties

Label 1900.1.bb.a.341.1
Level $1900$
Weight $1$
Character 1900.341
Analytic conductor $0.948$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1900.bb (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \(x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\)

Embedding invariants

Embedding label 341.1
Root \(-0.978148 - 0.207912i\) of defining polynomial
Character \(\chi\) \(=\) 1900.341
Dual form 1900.1.bb.a.1861.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.104528 - 0.994522i) q^{5} +1.82709 q^{7} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\) \(q+(-0.104528 - 0.994522i) q^{5} +1.82709 q^{7} +(-0.809017 + 0.587785i) q^{9} +(-1.47815 - 1.07394i) q^{11} +(0.413545 - 1.27276i) q^{17} +(0.309017 - 0.951057i) q^{19} +(1.30902 + 0.951057i) q^{23} +(-0.978148 + 0.207912i) q^{25} +(-0.190983 - 1.81708i) q^{35} -1.95630 q^{43} +(0.669131 + 0.743145i) q^{45} +(-0.309017 - 0.951057i) q^{47} +2.33826 q^{49} +(-0.913545 + 1.58231i) q^{55} +(1.58268 + 1.14988i) q^{61} +(-1.47815 + 1.07394i) q^{63} +(0.809017 + 0.587785i) q^{73} +(-2.70071 - 1.96218i) q^{77} +(0.309017 - 0.951057i) q^{81} +(0.190983 - 0.587785i) q^{83} +(-1.30902 - 0.278240i) q^{85} +(-0.978148 - 0.207912i) q^{95} +1.82709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{5} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 8q + q^{5} + 2q^{7} - 2q^{9} - 3q^{11} - 3q^{17} - 2q^{19} + 6q^{23} + q^{25} - 6q^{35} + 2q^{43} + q^{45} + 2q^{47} + 10q^{49} - q^{55} + 2q^{61} - 3q^{63} + 2q^{73} - 2q^{77} - 2q^{81} + 6q^{83} - 6q^{85} + q^{95} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(1\)

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 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(4\) 0 0
\(5\) −0.104528 0.994522i −0.104528 0.994522i
\(6\) 0 0
\(7\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(8\) 0 0
\(9\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(10\) 0 0
\(11\) −1.47815 1.07394i −1.47815 1.07394i −0.978148 0.207912i \(-0.933333\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(12\) 0 0
\(13\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.413545 1.27276i 0.413545 1.27276i −0.500000 0.866025i \(-0.666667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(18\) 0 0
\(19\) 0.309017 0.951057i 0.309017 0.951057i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(24\) 0 0
\(25\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.190983 1.81708i −0.190983 1.81708i
\(36\) 0 0
\(37\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(42\) 0 0
\(43\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(44\) 0 0
\(45\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(46\) 0 0
\(47\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(48\) 0 0
\(49\) 2.33826 2.33826
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(54\) 0 0
\(55\) −0.913545 + 1.58231i −0.913545 + 1.58231i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) 0 0
\(61\) 1.58268 + 1.14988i 1.58268 + 1.14988i 0.913545 + 0.406737i \(0.133333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(62\) 0 0
\(63\) −1.47815 + 1.07394i −1.47815 + 1.07394i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(72\) 0 0
\(73\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.70071 1.96218i −2.70071 1.96218i
\(78\) 0 0
\(79\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(80\) 0 0
\(81\) 0.309017 0.951057i 0.309017 0.951057i
\(82\) 0 0
\(83\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(84\) 0 0
\(85\) −1.30902 0.278240i −1.30902 0.278240i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.978148 0.207912i −0.978148 0.207912i
\(96\) 0 0
\(97\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(98\) 0 0
\(99\) 1.82709 1.82709
\(100\) 0 0
\(101\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(114\) 0 0
\(115\) 0.809017 1.40126i 0.809017 1.40126i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.755585 2.32545i 0.755585 2.32545i
\(120\) 0 0
\(121\) 0.722562 + 2.22382i 0.722562 + 2.22382i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(126\) 0 0
\(127\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.604528 + 1.86055i −0.604528 + 1.86055i −0.104528 + 0.994522i \(0.533333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0.564602 1.73767i 0.564602 1.73767i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.169131 0.122881i 0.169131 0.122881i −0.500000 0.866025i \(-0.666667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(138\) 0 0
\(139\) 0.169131 + 0.122881i 0.169131 + 0.122881i 0.669131 0.743145i \(-0.266667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0.413545 + 1.27276i 0.413545 + 1.27276i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.39169 + 1.73767i 2.39169 + 1.73767i
\(162\) 0 0
\(163\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(168\) 0 0
\(169\) 0.309017 0.951057i 0.309017 0.951057i
\(170\) 0 0
\(171\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(172\) 0 0
\(173\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(174\) 0 0
\(175\) −1.78716 + 0.379874i −1.78716 + 0.379874i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(180\) 0 0
\(181\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.97815 + 1.43721i −1.97815 + 1.43721i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.08268 + 0.786610i −1.08268 + 0.786610i −0.978148 0.207912i \(-0.933333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(198\) 0 0
\(199\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.61803 −1.61803
\(208\) 0 0
\(209\) −1.47815 + 1.07394i −1.47815 + 1.07394i
\(210\) 0 0
\(211\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.204489 + 1.94558i 0.204489 + 1.94558i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(224\) 0 0
\(225\) 0.669131 0.743145i 0.669131 0.743145i
\(226\) 0 0
\(227\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(228\) 0 0
\(229\) 0.564602 + 1.73767i 0.564602 + 1.73767i 0.669131 + 0.743145i \(0.266667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.604528 + 1.86055i −0.604528 + 1.86055i −0.104528 + 0.994522i \(0.533333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.58268 + 1.14988i 1.58268 + 1.14988i 0.913545 + 0.406737i \(0.133333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(240\) 0 0
\(241\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.244415 2.32545i −0.244415 2.32545i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(252\) 0 0
\(253\) −0.913545 2.81160i −0.913545 2.81160i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.47815 + 1.07394i −1.47815 + 1.07394i −0.500000 + 0.866025i \(0.666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(270\) 0 0
\(271\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.66913 + 0.743145i 1.66913 + 0.743145i
\(276\) 0 0
\(277\) 1.58268 + 1.14988i 1.58268 + 1.14988i 0.913545 + 0.406737i \(0.133333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(282\) 0 0
\(283\) −0.0646021 + 0.198825i −0.0646021 + 0.198825i −0.978148 0.207912i \(-0.933333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.639886 0.464905i −0.639886 0.464905i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −3.57433 −3.57433
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.978148 1.69420i 0.978148 1.69420i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.08268 0.786610i −1.08268 0.786610i −0.104528 0.994522i \(-0.533333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(314\) 0 0
\(315\) 1.22256 + 1.35779i 1.22256 + 1.35779i
\(316\) 0 0
\(317\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.08268 0.786610i −1.08268 0.786610i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.564602 1.73767i −0.564602 1.73767i
\(330\) 0 0
\(331\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.44512 2.44512
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(348\) 0 0
\(349\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.169131 0.122881i 0.169131 0.122881i −0.500000 0.866025i \(-0.666667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(360\) 0 0
\(361\) −0.809017 0.587785i −0.809017 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.500000 0.866025i 0.500000 0.866025i
\(366\) 0 0
\(367\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(384\) 0 0
\(385\) −1.66913 + 2.89102i −1.66913 + 2.89102i
\(386\) 0 0
\(387\) 1.58268 1.14988i 1.58268 1.14988i
\(388\) 0 0
\(389\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(390\) 0 0
\(391\) 1.75181 1.27276i 1.75181 1.27276i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.0646021 0.198825i −0.0646021 0.198825i 0.913545 0.406737i \(-0.133333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.978148 0.207912i −0.978148 0.207912i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −0.604528 0.128496i −0.604528 0.128496i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0 0
\(421\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(424\) 0 0
\(425\) −0.139886 + 1.33093i −0.139886 + 1.33093i
\(426\) 0 0
\(427\) 2.89169 + 2.10094i 2.89169 + 2.10094i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(432\) 0 0
\(433\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.30902 0.951057i 1.30902 0.951057i
\(438\) 0 0
\(439\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(440\) 0 0
\(441\) −1.89169 + 1.37440i −1.89169 + 1.37440i
\(442\) 0 0
\(443\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.08268 0.786610i −1.08268 0.786610i −0.104528 0.994522i \(-0.533333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(462\) 0 0
\(463\) 0.169131 0.122881i 0.169131 0.122881i −0.500000 0.866025i \(-0.666667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.604528 + 1.86055i −0.604528 + 1.86055i −0.104528 + 0.994522i \(0.533333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.89169 + 2.10094i 2.89169 + 2.10094i
\(474\) 0 0
\(475\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −0.190983 1.81708i −0.190983 1.81708i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(504\) 0 0
\(505\) −0.0646021 0.614648i −0.0646021 0.614648i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(510\) 0 0
\(511\) 1.47815 + 1.07394i 1.47815 + 1.07394i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.564602 + 1.73767i −0.564602 + 1.73767i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(522\) 0 0
\(523\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.45630 2.51115i −3.45630 2.51115i
\(540\) 0 0
\(541\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(548\) 0 0
\(549\) −1.95630 −1.95630
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.564602 1.73767i 0.564602 1.73767i
\(568\) 0 0
\(569\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(570\) 0 0
\(571\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.47815 0.658114i −1.47815 0.658114i
\(576\) 0 0
\(577\) −1.47815 1.07394i −1.47815 1.07394i −0.978148 0.207912i \(-0.933333\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.348943 1.07394i 0.348943 1.07394i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(594\) 0 0
\(595\) −2.39169 0.508370i −2.39169 0.508370i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.13611 0.951057i 2.13611 0.951057i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.169131 0.122881i 0.169131 0.122881i −0.500000 0.866025i \(-0.666667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.413545 1.27276i 0.413545 1.27276i −0.500000 0.866025i \(-0.666667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(618\) 0 0
\(619\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.913545 0.406737i 0.913545 0.406737i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.413545 1.27276i 0.413545 1.27276i −0.500000 0.866025i \(-0.666667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(642\) 0 0
\(643\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.0646021 0.198825i −0.0646021 0.198825i 0.913545 0.406737i \(-0.133333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(654\) 0 0
\(655\) 1.91355 + 0.406737i 1.91355 + 0.406737i
\(656\) 0 0
\(657\) −1.00000 −1.00000
\(658\) 0 0
\(659\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.78716 0.379874i −1.78716 0.379874i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.10453 3.39939i −1.10453 3.39939i
\(672\) 0 0
\(673\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(684\) 0 0
\(685\) −0.139886 0.155360i −0.139886 0.155360i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.58268 1.14988i 1.58268 1.14988i 0.669131 0.743145i \(-0.266667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(692\) 0 0
\(693\) 3.33826 3.33826
\(694\) 0 0
\(695\) 0.104528 0.181049i 0.104528 0.181049i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.12920 1.12920
\(708\) 0 0
\(709\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.0646021 + 0.198825i −0.0646021 + 0.198825i −0.978148 0.207912i \(-0.933333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.08268 0.786610i −1.08268 0.786610i −0.104528 0.994522i \(-0.533333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(728\) 0 0
\(729\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(730\) 0 0
\(731\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(732\) 0 0
\(733\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.08268 0.786610i −1.08268 0.786610i −0.104528 0.994522i \(-0.533333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0.0218524 + 0.207912i 0.0218524 + 0.207912i
\(746\) 0 0
\(747\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.169131 + 0.122881i 0.169131 + 0.122881i 0.669131 0.743145i \(-0.266667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.22256 0.544320i 1.22256 0.544320i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.0646021 + 0.198825i −0.0646021 + 0.198825i −0.978148 0.207912i \(-0.933333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.0646021 0.614648i −0.0646021 0.614648i
\(786\) 0 0
\(787\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(798\) 0 0
\(799\) −1.33826 −1.33826
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.564602 1.73767i −0.564602 1.73767i
\(804\) 0 0
\(805\) 1.47815 2.56023i 1.47815 2.56023i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.58268 1.14988i 1.58268 1.14988i 0.669131 0.743145i \(-0.266667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(810\) 0