Properties

Label 3744.1.eh.a.883.2
Level $3744$
Weight $1$
Character 3744.883
Analytic conductor $1.868$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.eh (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

Embedding invariants

Embedding label 883.2
Root \(-0.195090 + 0.980785i\) of defining polynomial
Character \(\chi\) \(=\) 3744.883
Dual form 3744.1.eh.a.1819.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.555570 + 0.831470i) q^{2} +(-0.382683 - 0.923880i) q^{4} +(-0.636379 - 1.53636i) q^{5} +(0.980785 + 0.195090i) q^{8} +O(q^{10})\) \(q+(-0.555570 + 0.831470i) q^{2} +(-0.382683 - 0.923880i) q^{4} +(-0.636379 - 1.53636i) q^{5} +(0.980785 + 0.195090i) q^{8} +(1.63099 + 0.324423i) q^{10} +(-0.750661 - 1.81225i) q^{11} +(-0.923880 - 0.382683i) q^{13} +(-0.707107 + 0.707107i) q^{16} +(-1.17588 + 1.17588i) q^{20} +(1.92388 + 0.382683i) q^{22} +(-1.24830 + 1.24830i) q^{25} +(0.831470 - 0.555570i) q^{26} +(-0.195090 - 0.980785i) q^{32} +(-0.324423 - 1.63099i) q^{40} +(0.785695 - 0.785695i) q^{41} +(0.707107 - 0.292893i) q^{43} +(-1.38704 + 1.38704i) q^{44} -0.390181i q^{47} +1.00000i q^{49} +(-0.344406 - 1.73145i) q^{50} +1.00000i q^{52} +(-2.30656 + 2.30656i) q^{55} +(-1.02656 + 0.425215i) q^{59} +(-0.541196 + 1.30656i) q^{61} +(0.923880 + 0.382683i) q^{64} +1.66294i q^{65} +(-1.38704 + 1.38704i) q^{71} +0.765367 q^{79} +(1.53636 + 0.636379i) q^{80} +(0.216773 + 1.08979i) q^{82} +(-1.53636 - 0.636379i) q^{83} +(-0.149316 + 0.750661i) q^{86} +(-0.382683 - 1.92388i) q^{88} +(1.38704 + 1.38704i) q^{89} +(0.324423 + 0.216773i) q^{94} +(-0.831470 - 0.555570i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 16q^{22} - 16q^{55} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(3\) 0 0
\(4\) −0.382683 0.923880i −0.382683 0.923880i
\(5\) −0.636379 1.53636i −0.636379 1.53636i −0.831470 0.555570i \(-0.812500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(9\) 0 0
\(10\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(11\) −0.750661 1.81225i −0.750661 1.81225i −0.555570 0.831470i \(-0.687500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(12\) 0 0
\(13\) −0.923880 0.382683i −0.923880 0.382683i
\(14\) 0 0
\(15\) 0 0
\(16\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(20\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(21\) 0 0
\(22\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) −1.24830 + 1.24830i −1.24830 + 1.24830i
\(26\) 0.831470 0.555570i 0.831470 0.555570i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −0.195090 0.980785i −0.195090 0.980785i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.324423 1.63099i −0.324423 1.63099i
\(41\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(42\) 0 0
\(43\) 0.707107 0.292893i 0.707107 0.292893i 1.00000i \(-0.5\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(45\) 0 0
\(46\) 0 0
\(47\) 0.390181i 0.390181i −0.980785 0.195090i \(-0.937500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) −0.344406 1.73145i −0.344406 1.73145i
\(51\) 0 0
\(52\) 1.00000i 1.00000i
\(53\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(54\) 0 0
\(55\) −2.30656 + 2.30656i −2.30656 + 2.30656i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.02656 + 0.425215i −1.02656 + 0.425215i −0.831470 0.555570i \(-0.812500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(60\) 0 0
\(61\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(65\) 1.66294i 1.66294i
\(66\) 0 0
\(67\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(80\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(81\) 0 0
\(82\) 0.216773 + 1.08979i 0.216773 + 1.08979i
\(83\) −1.53636 0.636379i −1.53636 0.636379i −0.555570 0.831470i \(-0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.149316 + 0.750661i −0.149316 + 0.750661i
\(87\) 0 0
\(88\) −0.382683 1.92388i −0.382683 1.92388i
\(89\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.831470 0.555570i −0.831470 0.555570i
\(99\) 0 0
\(100\) 1.63099 + 0.675577i 1.63099 + 0.675577i
\(101\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(102\) 0 0
\(103\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(104\) −0.831470 0.555570i −0.831470 0.555570i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(108\) 0 0
\(109\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(110\) −0.636379 3.19929i −0.636379 3.19929i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0.216773 1.08979i 0.216773 1.08979i
\(119\) 0 0
\(120\) 0 0
\(121\) −2.01367 + 2.01367i −2.01367 + 2.01367i
\(122\) −0.785695 1.17588i −0.785695 1.17588i
\(123\) 0 0
\(124\) 0 0
\(125\) 1.17588 + 0.487064i 1.17588 + 0.487064i
\(126\) 0 0
\(127\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(128\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(129\) 0 0
\(130\) −1.38268 0.923880i −1.38268 0.923880i
\(131\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(138\) 0 0
\(139\) 1.70711 0.707107i 1.70711 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
1.00000 \(0\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.382683 1.92388i −0.382683 1.92388i
\(143\) 1.96157i 1.96157i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.81225 + 0.750661i −1.81225 + 0.750661i −0.831470 + 0.555570i \(0.812500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(150\) 0 0
\(151\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.707107 1.70711i 0.707107 1.70711i 1.00000i \(-0.5\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(158\) −0.425215 + 0.636379i −0.425215 + 0.636379i
\(159\) 0 0
\(160\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(164\) −1.02656 0.425215i −1.02656 0.425215i
\(165\) 0 0
\(166\) 1.38268 0.923880i 1.38268 0.923880i
\(167\) −1.17588 + 1.17588i −1.17588 + 1.17588i −0.195090 + 0.980785i \(0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(168\) 0 0
\(169\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(170\) 0 0
\(171\) 0 0
\(172\) −0.541196 0.541196i −0.541196 0.541196i
\(173\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.81225 + 0.750661i 1.81225 + 0.750661i
\(177\) 0 0
\(178\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(179\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(180\) 0 0
\(181\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −0.360480 + 0.149316i −0.360480 + 0.149316i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.923880 0.382683i 0.923880 0.382683i
\(197\) −0.149316 0.360480i −0.149316 0.360480i 0.831470 0.555570i \(-0.187500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(198\) 0 0
\(199\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(200\) −1.46785 + 0.980785i −1.46785 + 0.980785i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.70711 0.707107i −1.70711 0.707107i
\(206\) 1.96157 0.390181i 1.96157 0.390181i
\(207\) 0 0
\(208\) 0.923880 0.382683i 0.923880 0.382683i
\(209\) 0 0
\(210\) 0 0
\(211\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.899976 0.899976i −0.899976 0.899976i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 3.01367 + 1.24830i 3.01367 + 1.24830i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.149316 + 0.360480i −0.149316 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(228\) 0 0
\(229\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) −0.599456 + 0.248303i −0.599456 + 0.248303i
\(236\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(237\) 0 0
\(238\) 0 0
\(239\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −0.555570 2.79304i −0.555570 2.79304i
\(243\) 0 0
\(244\) 1.41421 1.41421
\(245\) 1.53636 0.636379i 1.53636 0.636379i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.05826 + 0.707107i −1.05826 + 0.707107i
\(251\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(255\) 0 0
\(256\) 1.00000i 1.00000i
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.53636 0.636379i 1.53636 0.636379i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.382683 + 1.92388i 0.382683 + 1.92388i
\(275\) 3.19929 + 1.32519i 3.19929 + 1.32519i
\(276\) 0 0
\(277\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(278\) −0.360480 + 1.81225i −0.360480 + 1.81225i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(282\) 0 0
\(283\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(284\) 1.81225 + 0.750661i 1.81225 + 0.750661i
\(285\) 0 0
\(286\) −1.63099 1.08979i −1.63099 1.08979i
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.750661 + 1.81225i 0.750661 + 1.81225i 0.555570 + 0.831470i \(0.312500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(294\) 0 0
\(295\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(296\) 0 0
\(297\) 0 0
\(298\) 0.382683 1.92388i 0.382683 1.92388i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.35175 2.35175
\(306\) 0 0
\(307\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(312\) 0 0
\(313\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(314\) 1.02656 + 1.53636i 1.02656 + 1.53636i
\(315\) 0 0
\(316\) −0.292893 0.707107i −0.292893 0.707107i
\(317\) 1.81225 + 0.750661i 1.81225 + 0.750661i 0.980785 + 0.195090i \(0.0625000\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.66294i 1.66294i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.63099 0.675577i 1.63099 0.675577i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.923880 0.617317i 0.923880 0.617317i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(332\) 1.66294i 1.66294i
\(333\) 0 0
\(334\) −0.324423 1.63099i −0.324423 1.63099i
\(335\) 0 0
\(336\) 0 0
\(337\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(338\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0.750661 0.149316i 0.750661 0.149316i
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(348\) 0 0
\(349\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(353\) 1.11114i 1.11114i −0.831470 0.555570i \(-0.812500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(354\) 0 0
\(355\) 3.01367 + 1.24830i 3.01367 + 1.24830i
\(356\) 0.750661 1.81225i 0.750661 1.81225i
\(357\) 0 0
\(358\) 0 0
\(359\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(360\) 0 0
\(361\) −0.707107 0.707107i −0.707107 0.707107i
\(362\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.0761205 0.382683i 0.0761205 0.382683i
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.66294 −1.66294 −0.831470 0.555570i \(-0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(393\) 0 0
\(394\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(395\) −0.487064 1.17588i −0.487064 1.17588i
\(396\) 0 0
\(397\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(398\) 1.81225 0.360480i 1.81225 0.360480i
\(399\) 0 0
\(400\) 1.76537i 1.76537i
\(401\) 0.390181 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(410\) 1.53636 1.02656i 1.53636 1.02656i
\(411\) 0 0
\(412\) −0.765367 + 1.84776i −0.765367 + 1.84776i
\(413\) 0 0
\(414\) 0 0
\(415\) 2.76537i 2.76537i
\(416\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(420\) 0 0
\(421\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(422\) −0.785695 1.17588i −0.785695 1.17588i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 1.24830 0.248303i 1.24830 0.248303i
\(431\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(432\) 0 0
\(433\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(440\) −2.71223 + 1.81225i −2.71223 + 1.81225i
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(444\) 0 0
\(445\) 1.24830 3.01367i 1.24830 3.01367i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.96157i 1.96157i 0.195090 + 0.980785i \(0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(450\) 0 0
\(451\) −2.01367 0.834089i −2.01367 0.834089i
\(452\) 0 0
\(453\) 0 0
\(454\) −0.216773 0.324423i −0.216773 0.324423i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.149316 + 0.360480i −0.149316 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.126584 0.636379i 0.126584 0.636379i
\(471\) 0 0
\(472\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(473\) −1.06159 1.06159i −1.06159 1.06159i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(479\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.63099 + 1.08979i 2.63099 + 1.08979i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) −0.785695 + 1.17588i −0.785695 + 1.17588i
\(489\) 0 0
\(490\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(491\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(500\) 1.27276i 1.27276i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(509\) −1.02656 0.425215i −1.02656 0.425215i −0.195090 0.980785i \(-0.562500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(513\) 0 0
\(514\) 0 0
\(515\) −1.27276 + 3.07271i −1.27276 + 3.07271i
\(516\) 0 0
\(517\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(518\) 0 0
\(519\) 0 0
\(520\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(521\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.02656 + 0.425215i −1.02656 + 0.425215i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.81225 0.750661i 1.81225 0.750661i
\(540\) 0 0
\(541\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.70711 + 0.707107i 1.70711 + 0.707107i 1.00000 \(0\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) −1.81225 0.750661i −1.81225 0.750661i
\(549\) 0 0
\(550\) −2.87929 + 1.92388i −2.87929 + 1.92388i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(555\) 0 0
\(556\) −1.30656 1.30656i −1.30656 1.30656i
\(557\) 0.425215 1.02656i 0.425215 1.02656i −0.555570 0.831470i \(-0.687500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(558\) 0 0
\(559\) −0.765367 −0.765367
\(560\) 0 0
\(561\) 0 0
\(562\) 1.63099 0.324423i 1.63099 0.324423i
\(563\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.38704 + 0.275899i 1.38704 + 0.275899i
\(567\) 0 0
\(568\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(569\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(570\) 0 0
\(571\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(572\) 1.81225 0.750661i 1.81225 0.750661i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −1.92388 0.382683i −1.92388 0.382683i
\(587\) −0.425215 1.02656i −0.425215 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.81225 + 0.360480i −1.81225 + 0.360480i
\(591\) 0 0
\(592\) 0 0
\(593\) −1.66294 −1.66294 −0.831470 0.555570i \(-0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(600\) 0 0
\(601\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.37517 + 1.81225i 4.37517 + 1.81225i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −1.30656 + 1.95541i −1.30656 + 1.95541i
\(611\) −0.149316 + 0.360480i −0.149316 + 0.360480i
\(612\) 0 0
\(613\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.17588 1.17588i 1.17588 1.17588i 0.195090 0.980785i \(-0.437500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(618\) 0 0
\(619\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.351153i 0.351153i
\(626\) 0.360480 + 1.81225i 0.360480 + 1.81225i
\(627\) 0 0
\(628\) −1.84776 −1.84776
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(633\) 0 0
\(634\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(635\) −1.17588 + 0.487064i −1.17588 + 0.487064i
\(636\) 0 0
\(637\) 0.382683 0.923880i 0.382683 0.923880i
\(638\) 0 0
\(639\) 0 0
\(640\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 1.54120 + 1.54120i 1.54120 + 1.54120i
\(650\) −0.344406 + 1.73145i −0.344406 + 1.73145i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.11114i 1.11114i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1.38268 0.923880i −1.38268 0.923880i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(669\) 0 0
\(670\) 0 0
\(671\) 2.77408 2.77408
\(672\) 0 0
\(673\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 1.53636 + 1.02656i 1.53636 + 1.02656i
\(675\) 0 0
\(676\) 0.382683 0.923880i 0.382683 0.923880i
\(677\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.425215 + 1.02656i 0.425215 + 1.02656i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(684\) 0 0
\(685\) −3.01367 1.24830i −3.01367 1.24830i
\(686\) 0 0
\(687\) 0 0
\(688\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.17273 2.17273i −2.17273 2.17273i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.96157i 1.96157i
\(705\) 0 0
\(706\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(710\) −2.71223 + 1.81225i −2.71223 + 1.81225i
\(711\) 0 0
\(712\) 1.08979 + 1.63099i 1.08979 + 1.63099i
\(713\) 0 0
\(714\) 0 0
\(715\) 3.01367 1.24830i 3.01367 1.24830i
\(716\) 0 0
\(717\) 0 0
\(718\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.980785 0.195090i 0.980785 0.195090i
\(723\) 0 0
\(724\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(734\) 0.785695 1.17588i 0.785695 1.17588i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.17588 + 1.17588i −1.17588 + 1.17588i −0.195090 + 0.980785i \(0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(744\) 0 0
\(745\) 2.30656 + 2.30656i 2.30656 + 2.30656i
\(746\) −1.38704 0.275899i −1.38704 0.275899i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) 0.275899 + 0.275899i 0.275899 + 0.275899i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.923880 1.38268i 0.923880 1.38268i
\(767\) 1.11114 1.11114
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.425215 1.02656i −0.425215 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 3.55487 + 1.47247i 3.55487 + 1.47247i
\(782\) 0 0
\(783\) 0 0
\(784\) −0.707107 0.707107i −0.707107 0.707107i
\(785\) −3.07271 −3.07271
\(786\) 0 0
\(787\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(788\) −0.275899 + 0.275899i −0.275899 + 0.275899i
\(789\) 0 0
\(790\) 1.24830 + 0.248303i 1.24830 + 0.248303i
\(791\) 0 0
\(792\) 0 0
\(793\) 1.00000 1.00000i 1.00000 1.00000i
\(794\) 0 0
\(795\) 0 0
\(796\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(797\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.46785 + 0.980785i 1.46785 + 0.980785i
\(801\) 0 0
\(802\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0