Properties

Label 1900.1.j.a.607.3
Level $1900$
Weight $1$
Character 1900.607
Analytic conductor $0.948$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -95
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.4170272000.1

Embedding invariants

Embedding label 607.3
Root \(-0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1900.607
Dual form 1900.1.j.a.1443.3

$q$-expansion

\(f(q)\) \(=\) \(q+(0.382683 + 0.923880i) q^{2} +(0.541196 - 0.541196i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(0.707107 + 0.292893i) q^{6} +(-0.923880 - 0.382683i) q^{8} +0.414214i q^{9} +O(q^{10})\) \(q+(0.382683 + 0.923880i) q^{2} +(0.541196 - 0.541196i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(0.707107 + 0.292893i) q^{6} +(-0.923880 - 0.382683i) q^{8} +0.414214i q^{9} +1.41421i q^{11} +0.765367i q^{12} +(0.541196 + 0.541196i) q^{13} -1.00000i q^{16} +(-0.382683 + 0.158513i) q^{18} -1.00000 q^{19} +(-1.30656 + 0.541196i) q^{22} +(-0.707107 + 0.292893i) q^{24} +(-0.292893 + 0.707107i) q^{26} +(0.765367 + 0.765367i) q^{27} +(0.923880 - 0.382683i) q^{32} +(0.765367 + 0.765367i) q^{33} +(-0.292893 - 0.292893i) q^{36} +(1.30656 - 1.30656i) q^{37} +(-0.382683 - 0.923880i) q^{38} +0.585786 q^{39} +(-1.00000 - 1.00000i) q^{44} +(-0.541196 - 0.541196i) q^{48} +1.00000i q^{49} -0.765367 q^{52} +(1.30656 + 1.30656i) q^{53} +(-0.414214 + 1.00000i) q^{54} +(-0.541196 + 0.541196i) q^{57} -1.41421 q^{61} +(0.707107 + 0.707107i) q^{64} +(-0.414214 + 1.00000i) q^{66} +(-1.30656 - 1.30656i) q^{67} +(0.158513 - 0.382683i) q^{72} +(1.70711 + 0.707107i) q^{74} +(0.707107 - 0.707107i) q^{76} +(0.224171 + 0.541196i) q^{78} +0.414214 q^{81} +(0.541196 - 1.30656i) q^{88} +(0.292893 - 0.707107i) q^{96} +(1.30656 - 1.30656i) q^{97} +(-0.923880 + 0.382683i) q^{98} -0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 8 q^{19} - 8 q^{26} - 8 q^{36} + 16 q^{39} - 8 q^{44} + 8 q^{54} + 8 q^{66} + 8 q^{74} - 8 q^{81} + 8 q^{96} - 16 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(3\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(4\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(5\) 0 0
\(6\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.923880 0.382683i −0.923880 0.382683i
\(9\) 0.414214i 0.414214i
\(10\) 0 0
\(11\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0.765367i 0.765367i
\(13\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000i 1.00000i
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) −0.382683 + 0.158513i −0.382683 + 0.158513i
\(19\) −1.00000 −1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(25\) 0 0
\(26\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(27\) 0.765367 + 0.765367i 0.765367 + 0.765367i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0.923880 0.382683i 0.923880 0.382683i
\(33\) 0.765367 + 0.765367i 0.765367 + 0.765367i
\(34\) 0 0
\(35\) 0 0
\(36\) −0.292893 0.292893i −0.292893 0.292893i
\(37\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(38\) −0.382683 0.923880i −0.382683 0.923880i
\(39\) 0.585786 0.585786
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) −1.00000 1.00000i −1.00000 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −0.541196 0.541196i −0.541196 0.541196i
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.765367 −0.765367
\(53\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(54\) −0.414214 + 1.00000i −0.414214 + 1.00000i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(65\) 0 0
\(66\) −0.414214 + 1.00000i −0.414214 + 1.00000i
\(67\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.158513 0.382683i 0.158513 0.382683i
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(75\) 0 0
\(76\) 0.707107 0.707107i 0.707107 0.707107i
\(77\) 0 0
\(78\) 0.224171 + 0.541196i 0.224171 + 0.541196i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0.414214 0.414214
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.541196 1.30656i 0.541196 1.30656i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.292893 0.707107i 0.292893 0.707107i
\(97\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(98\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(99\) −0.585786 −0.585786
\(100\) 0 0
\(101\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(102\) 0 0
\(103\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(104\) −0.292893 0.707107i −0.292893 0.707107i
\(105\) 0 0
\(106\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(107\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(108\) −1.08239 −1.08239
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 1.41421i 1.41421i
\(112\) 0 0
\(113\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(114\) −0.707107 0.292893i −0.707107 0.292893i
\(115\) 0 0
\(116\) 0 0
\(117\) −0.224171 + 0.224171i −0.224171 + 0.224171i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) −0.541196 1.30656i −0.541196 1.30656i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(128\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) −1.08239 −1.08239
\(133\) 0 0
\(134\) 0.707107 1.70711i 0.707107 1.70711i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.765367 + 0.765367i −0.765367 + 0.765367i
\(144\) 0.414214 0.414214
\(145\) 0 0
\(146\) 0 0
\(147\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(148\) 1.84776i 1.84776i
\(149\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −0.414214 + 0.414214i −0.414214 + 0.414214i
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 1.41421 1.41421
\(160\) 0 0
\(161\) 0 0
\(162\) 0.158513 + 0.382683i 0.158513 + 0.382683i
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(168\) 0 0
\(169\) 0.414214i 0.414214i
\(170\) 0 0
\(171\) 0.414214i 0.414214i
\(172\) 0 0
\(173\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421 1.41421
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −0.765367 + 0.765367i −0.765367 + 0.765367i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(192\) 0.765367 0.765367
\(193\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(194\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(195\) 0 0
\(196\) −0.707107 0.707107i −0.707107 0.707107i
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) −0.224171 0.541196i −0.224171 0.541196i
\(199\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(200\) 0 0
\(201\) −1.41421 −1.41421
\(202\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(207\) 0 0
\(208\) 0.541196 0.541196i 0.541196 0.541196i
\(209\) 1.41421i 1.41421i
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −1.84776 −1.84776
\(213\) 0 0
\(214\) 0.707107 1.70711i 0.707107 1.70711i
\(215\) 0 0
\(216\) −0.414214 1.00000i −0.414214 1.00000i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 1.30656 0.541196i 1.30656 0.541196i
\(223\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.292893 0.707107i 0.292893 0.707107i
\(227\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(228\) 0.765367i 0.765367i
\(229\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) −0.292893 0.121320i −0.292893 0.121320i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −0.382683 0.923880i −0.382683 0.923880i
\(243\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(244\) 1.00000 1.00000i 1.00000 1.00000i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.541196 0.541196i −0.541196 0.541196i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.292893 0.707107i 0.292893 0.707107i
\(255\) 0 0
\(256\) −1.00000 −1.00000
\(257\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) −0.414214 1.00000i −0.414214 1.00000i
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.84776 1.84776
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) −0.541196 1.30656i −0.541196 1.30656i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.00000 0.414214i −1.00000 0.414214i
\(287\) 0 0
\(288\) 0.158513 + 0.382683i 0.158513 + 0.382683i
\(289\) 1.00000i 1.00000i
\(290\) 0 0
\(291\) 1.41421i 1.41421i
\(292\) 0 0
\(293\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(294\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(295\) 0 0
\(296\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(297\) −1.08239 + 1.08239i −1.08239 + 1.08239i
\(298\) 1.30656 0.541196i 1.30656 0.541196i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0.765367 0.765367i 0.765367 0.765367i
\(304\) 1.00000i 1.00000i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(308\) 0 0
\(309\) 1.41421i 1.41421i
\(310\) 0 0
\(311\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(312\) −0.541196 0.224171i −0.541196 0.224171i
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(318\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(319\) 0 0
\(320\) 0 0
\(321\) −1.41421 −1.41421
\(322\) 0 0
\(323\) 0 0
\(324\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(334\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(335\) 0 0
\(336\) 0 0
\(337\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(338\) 0.382683 0.158513i 0.382683 0.158513i
\(339\) −0.585786 −0.585786
\(340\) 0 0
\(341\) 0 0
\(342\) 0.382683 0.158513i 0.382683 0.158513i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0.828427i 0.828427i
\(352\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(364\) 0 0
\(365\) 0 0
\(366\) −1.00000 0.414214i −1.00000 0.414214i
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −0.585786 −0.585786
\(382\) 1.84776 0.765367i 1.84776 0.765367i
\(383\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(384\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(385\) 0 0
\(386\) 0.707107 1.70711i 0.707107 1.70711i
\(387\) 0 0
\(388\) 1.84776i 1.84776i
\(389\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.382683 0.923880i 0.382683 0.923880i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0.414214 0.414214i 0.414214 0.414214i
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0.765367 + 1.84776i 0.765367 + 1.84776i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −0.541196 1.30656i −0.541196 1.30656i
\(403\) 0 0
\(404\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(405\) 0 0
\(406\) 0 0
\(407\) 1.84776 + 1.84776i 1.84776 + 1.84776i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.84776i 1.84776i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(417\) −0.765367 + 0.765367i −0.765367 + 0.765367i
\(418\) 1.30656 0.541196i 1.30656 0.541196i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.707107 1.70711i −0.707107 1.70711i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.84776 1.84776
\(429\) 0.828427i 0.828427i
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0.765367 0.765367i 0.765367 0.765367i
\(433\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −0.414214 −0.414214
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(445\) 0 0
\(446\) −0.707107 0.292893i −0.707107 0.292893i
\(447\) −0.765367 0.765367i −0.765367 0.765367i
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.765367 0.765367
\(453\) 0 0
\(454\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(455\) 0 0
\(456\) 0.707107 0.292893i 0.707107 0.292893i
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0.317025i 0.317025i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(478\) 0 0
\(479\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(480\) 0 0
\(481\) 1.41421 1.41421
\(482\) 0 0
\(483\) 0 0
\(484\) 0.707107 0.707107i 0.707107 0.707107i
\(485\) 0 0
\(486\) −0.707107 0.292893i −0.707107 0.292893i
\(487\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(488\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0.292893 0.707107i 0.292893 0.707107i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 0 0
\(501\) 0.585786 0.585786
\(502\) −1.84776 + 0.765367i −1.84776 + 0.765367i
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.224171 0.224171i −0.224171 0.224171i
\(508\) 0.765367 0.765367
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.382683 0.923880i −0.382683 0.923880i
\(513\) −0.765367 0.765367i −0.765367 0.765367i
\(514\) −1.70711 0.707107i −1.70711 0.707107i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0.585786 0.585786
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.765367 0.765367i 0.765367 0.765367i
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(537\) 0 0
\(538\) 0 0
\(539\) −1.41421 −1.41421
\(540\) 0 0
\(541\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 1.30656 0.541196i 1.30656 0.541196i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(548\) 0 0
\(549\) 0.585786i 0.585786i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.00000 1.00000i 1.00000 1.00000i
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 1.08239i 1.08239i
\(573\) −1.08239 1.08239i −1.08239 1.08239i
\(574\) 0 0
\(575\) 0 0
\(576\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 0.923880 0.382683i 0.923880 0.382683i
\(579\) −1.41421 −1.41421
\(580\) 0 0
\(581\) 0 0
\(582\) 1.30656 0.541196i 1.30656 0.541196i
\(583\) −1.84776 + 1.84776i −1.84776 + 1.84776i
\(584\) 0 0
\(585\) 0 0
\(586\) 0.707107 1.70711i 0.707107 1.70711i
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) −0.765367 −0.765367
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.30656 1.30656i −1.30656 1.30656i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) −1.41421 0.585786i −1.41421 0.585786i
\(595\) 0 0
\(596\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(597\) 1.08239 1.08239i 1.08239 1.08239i
\(598\) 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
\(603\) 0.541196 0.541196i 0.541196 0.541196i
\(604\) 0 0
\(605\) 0 0
\(606\) 1.00000 + 0.414214i 1.00000 + 0.414214i
\(607\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(608\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 1.30656 0.541196i 1.30656 0.541196i
\(619\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(623\) 0 0
\(624\) 0.585786i 0.585786i
\(625\) 0 0
\(626\) 0 0
\(627\) −0.765367 0.765367i −0.765367 0.765367i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.707107 0.292893i −0.707107 0.292893i
\(635\) 0 0
\(636\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(637\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −0.541196 1.30656i −0.541196 1.30656i
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) −0.382683 0.158513i −0.382683 0.158513i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(667\) 0 0
\(668\) −0.765367 −0.765367
\(669\) 0.585786i 0.585786i
\(670\) 0 0
\(671\) 2.00000i 2.00000i
\(672\) 0 0
\(673\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(674\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(675\) 0 0
\(676\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(677\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(678\) −0.224171 0.541196i −0.224171 0.541196i
\(679\) 0 0
\(680\) 0 0
\(681\) 1.41421 1.41421
\(682\) 0 0
\(683\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(684\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(685\) 0 0
\(686\) 0 0
\(687\) 0.765367 + 0.765367i 0.765367 + 0.765367i
\(688\) 0 0
\(689\) 1.41421i 1.41421i
\(690\) 0 0
\(691\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(692\) −0.765367 −0.765367
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(702\) −0.765367 + 0.317025i −0.765367 + 0.317025i
\(703\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(704\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(719\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −0.707107 0.292893i −0.707107 0.292893i
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 1.08239i 1.08239i
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.84776 1.84776i 1.84776 1.84776i
\(738\) 0 0
\(739\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) −0.585786 −0.585786
\(742\) 0 0
\(743\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.707107 1.70711i 0.707107 1.70711i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 1.08239 + 1.08239i 1.08239 + 1.08239i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) −0.224171 0.541196i −0.224171 0.541196i
\(763\) 0 0
\(764\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(765\) 0 0
\(766\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(767\) 0 0
\(768\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(769\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(770\) 0 0
\(771\) 1.41421i 1.41421i
\(772\) 1.84776 1.84776
\(773\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(777\) 0 0
\(778\) −1.84776 + 0.765367i −1.84776 + 0.765367i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.541196 + 0.224171i 0.541196 + 0.224171i
\(793\) −0.765367 0.765367i −0.765367 0.765367i
\(794\) 0 0
\(795\) 0 0
\(796\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(797\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.00000 1.00000i 1.00000 1.00000i
\(805\) 0 0