Properties

Label 912.2.ch.c.575.1
Level $912$
Weight $2$
Character 912.575
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 575.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 912.575
Dual form 912.2.ch.c.479.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.70574 - 0.300767i) q^{3} +(3.93242 - 2.27038i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\) \(q+(-1.70574 - 0.300767i) q^{3} +(3.93242 - 2.27038i) q^{7} +(2.81908 + 1.02606i) q^{9} +(0.507274 + 2.87689i) q^{13} +(-3.50000 - 2.59808i) q^{19} +(-7.39053 + 2.68993i) q^{21} +(0.868241 + 4.92404i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(7.27244 - 4.19875i) q^{31} +7.02734 q^{37} -5.05980i q^{39} +(8.03596 - 9.57688i) q^{43} +(6.80928 - 11.7940i) q^{49} +(5.18866 + 5.48432i) q^{57} +(4.80406 - 4.03109i) q^{61} +(13.4153 - 2.36549i) q^{63} +(-5.59492 + 15.3719i) q^{67} +(2.87686 - 16.3155i) q^{73} -8.66025i q^{75} +(6.84137 + 1.20632i) q^{79} +(6.89440 + 5.78509i) q^{81} +(8.52646 + 10.1614i) q^{91} +(-13.6677 + 4.97464i) q^{93} +(4.69846 - 1.71010i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + O(q^{10}) \) \( 6 q + 21 q^{13} - 21 q^{19} - 27 q^{21} - 27 q^{27} + 15 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 15 q^{67} + 21 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(-1\) \(-1\)

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
\(3\) −1.70574 0.300767i −0.984808 0.173648i
\(4\) 0 0
\(5\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(6\) 0 0
\(7\) 3.93242 2.27038i 1.48631 0.858124i 0.486436 0.873716i \(-0.338297\pi\)
0.999878 + 0.0155920i \(0.00496330\pi\)
\(8\) 0 0
\(9\) 2.81908 + 1.02606i 0.939693 + 0.342020i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0.507274 + 2.87689i 0.140692 + 0.797907i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.830033 + 0.557714i \(0.811678\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(18\) 0 0
\(19\) −3.50000 2.59808i −0.802955 0.596040i
\(20\) 0 0
\(21\) −7.39053 + 2.68993i −1.61275 + 0.586991i
\(22\) 0 0
\(23\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(24\) 0 0
\(25\) 0.868241 + 4.92404i 0.173648 + 0.984808i
\(26\) 0 0
\(27\) −4.50000 2.59808i −0.866025 0.500000i
\(28\) 0 0
\(29\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(30\) 0 0
\(31\) 7.27244 4.19875i 1.30617 0.754117i 0.324714 0.945812i \(-0.394732\pi\)
0.981455 + 0.191695i \(0.0613985\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.02734 1.15529 0.577644 0.816289i \(-0.303972\pi\)
0.577644 + 0.816289i \(0.303972\pi\)
\(38\) 0 0
\(39\) 5.05980i 0.810216i
\(40\) 0 0
\(41\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(42\) 0 0
\(43\) 8.03596 9.57688i 1.22547 1.46046i 0.381246 0.924473i \(-0.375495\pi\)
0.844226 0.535988i \(-0.180061\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(48\) 0 0
\(49\) 6.80928 11.7940i 0.972754 1.68486i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.18866 + 5.48432i 0.687255 + 0.726416i
\(58\) 0 0
\(59\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(60\) 0 0
\(61\) 4.80406 4.03109i 0.615097 0.516128i −0.281161 0.959661i \(-0.590719\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 13.4153 2.36549i 1.69017 0.298023i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.59492 + 15.3719i −0.683529 + 1.87798i −0.305424 + 0.952217i \(0.598798\pi\)
−0.378105 + 0.925763i \(0.623424\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(72\) 0 0
\(73\) 2.87686 16.3155i 0.336711 1.90958i −0.0729331 0.997337i \(-0.523236\pi\)
0.409644 0.912245i \(-0.365653\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.84137 + 1.20632i 0.769714 + 0.135721i 0.544696 0.838633i \(-0.316645\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 6.89440 + 5.78509i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(90\) 0 0
\(91\) 8.52646 + 10.1614i 0.893816 + 1.06521i
\(92\) 0 0
\(93\) −13.6677 + 4.97464i −1.41728 + 0.515846i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.69846 1.71010i 0.477057 0.173634i −0.0922897 0.995732i \(-0.529419\pi\)
0.569346 + 0.822098i \(0.307196\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(102\) 0 0
\(103\) −13.4315 7.75470i −1.32345 0.764094i −0.339172 0.940724i \(-0.610147\pi\)
−0.984277 + 0.176631i \(0.943480\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(108\) 0 0
\(109\) −13.0228 10.9274i −1.24735 1.04665i −0.996912 0.0785223i \(-0.974980\pi\)
−0.250441 0.968132i \(-0.580576\pi\)
\(110\) 0 0
\(111\) −11.9868 2.11360i −1.13774 0.200614i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.52182 + 8.63068i −0.140692 + 0.797907i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −22.1746 + 3.90998i −1.96768 + 0.346954i −0.976134 + 0.217171i \(0.930317\pi\)
−0.991542 + 0.129783i \(0.958572\pi\)
\(128\) 0 0
\(129\) −16.5876 + 13.9187i −1.46046 + 1.22547i
\(130\) 0 0
\(131\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(132\) 0 0
\(133\) −19.6621 2.27038i −1.70492 0.196867i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(138\) 0 0
\(139\) 23.1951 4.08991i 1.96738 0.346902i 0.975417 0.220366i \(-0.0707252\pi\)
0.991962 0.126536i \(-0.0403860\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −15.1621 + 18.0695i −1.25055 + 1.49035i
\(148\) 0 0
\(149\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(150\) 0 0
\(151\) 8.66025i 0.704761i 0.935857 + 0.352381i \(0.114628\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.5424 + 14.7198i 1.40003 + 1.17477i 0.961085 + 0.276254i \(0.0890931\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.6336 6.71664i −0.911212 0.526088i −0.0303908 0.999538i \(-0.509675\pi\)
−0.880821 + 0.473450i \(0.843009\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(168\) 0 0
\(169\) 4.19681 1.52752i 0.322832 0.117501i
\(170\) 0 0
\(171\) −7.20099 10.9154i −0.550673 0.834721i
\(172\) 0 0
\(173\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(174\) 0 0
\(175\) 14.5937 + 17.3921i 1.10318 + 1.31472i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) −6.57785 2.39414i −0.488928 0.177955i 0.0857797 0.996314i \(-0.472662\pi\)
−0.574707 + 0.818359i \(0.694884\pi\)
\(182\) 0 0
\(183\) −9.40689 + 5.43107i −0.695377 + 0.401476i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −23.5945 −1.71625
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 1.31954 7.48351i 0.0949829 0.538675i −0.899770 0.436365i \(-0.856266\pi\)
0.994753 0.102310i \(-0.0326233\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) 0 0
\(199\) −4.38872 + 12.0579i −0.311108 + 0.854762i 0.681326 + 0.731980i \(0.261404\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) 14.1668 24.5377i 0.999252 1.73076i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.66503 + 26.5545i 0.665368 + 1.82808i 0.550743 + 0.834675i \(0.314345\pi\)
0.114625 + 0.993409i \(0.463433\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 19.0655 33.0225i 1.29425 2.24171i
\(218\) 0 0
\(219\) −9.81433 + 26.9647i −0.663191 + 1.82210i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.5646 + 19.7410i −1.10925 + 1.32195i −0.167412 + 0.985887i \(0.553541\pi\)
−0.941838 + 0.336066i \(0.890903\pi\)
\(224\) 0 0
\(225\) −2.60472 + 14.7721i −0.173648 + 0.984808i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −27.7820 −1.83589 −0.917943 0.396713i \(-0.870151\pi\)
−0.917943 + 0.396713i \(0.870151\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.3068 4.11532i −0.734452 0.267319i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 4.31702 + 24.4830i 0.278084 + 1.57709i 0.728993 + 0.684521i \(0.239989\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) −10.0201 11.9415i −0.642788 0.766044i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.69893 11.3871i 0.362614 0.724542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(258\) 0 0
\(259\) 27.6344 15.9548i 1.71712 0.991380i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(270\) 0 0
\(271\) −10.0201 + 11.9415i −0.608676 + 0.725391i −0.979079 0.203479i \(-0.934775\pi\)
0.370403 + 0.928871i \(0.379219\pi\)
\(272\) 0 0
\(273\) −11.4877 19.8972i −0.695266 1.20424i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −15.5000 + 26.8468i −0.931305 + 1.61307i −0.150210 + 0.988654i \(0.547995\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 24.8097 4.37463i 1.48532 0.261902i
\(280\) 0 0
\(281\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(282\) 0 0
\(283\) −7.70115 21.1587i −0.457786 1.25776i −0.927130 0.374741i \(-0.877732\pi\)
0.469344 0.883016i \(-0.344491\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.0228 10.9274i 0.766044 0.642788i
\(290\) 0 0
\(291\) −8.52869 + 1.50384i −0.499960 + 0.0881565i
\(292\) 0 0
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 9.85756 55.9050i 0.568181 3.22231i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.70574 + 0.300767i 0.0973516 + 0.0171657i 0.222112 0.975021i \(-0.428705\pi\)
−0.124760 + 0.992187i \(0.539816\pi\)
\(308\) 0 0
\(309\) 20.5783 + 17.2673i 1.17066 + 0.982300i
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −32.8892 11.9707i −1.85901 0.676624i −0.979731 0.200316i \(-0.935803\pi\)
−0.879279 0.476308i \(-0.841975\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −13.7255 + 4.99567i −0.761354 + 0.277110i
\(326\) 0 0
\(327\) 18.9268 + 22.5561i 1.04665 + 1.24735i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.61705 2.66565i −0.253776 0.146518i 0.367716 0.929938i \(-0.380140\pi\)
−0.621492 + 0.783420i \(0.713473\pi\)
\(332\) 0 0
\(333\) 19.8106 + 7.21048i 1.08562 + 0.395132i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.7750 + 22.4669i 1.45853 + 1.22385i 0.926049 + 0.377403i \(0.123183\pi\)
0.532476 + 0.846445i \(0.321262\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 30.0533i 1.62273i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(348\) 0 0
\(349\) 0.653886 + 1.13256i 0.0350017 + 0.0606247i 0.882996 0.469381i \(-0.155523\pi\)
−0.847994 + 0.530006i \(0.822190\pi\)
\(350\) 0 0
\(351\) 5.19166 14.2640i 0.277110 0.761354i
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(360\) 0 0
\(361\) 5.50000 + 18.1865i 0.289474 + 0.957186i
\(362\) 0 0
\(363\) −6.51636 17.9035i −0.342020 0.939693i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −36.6104 + 6.45540i −1.91105 + 0.336969i −0.997555 0.0698862i \(-0.977736\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −19.0000 32.9090i −0.983783 1.70396i −0.647225 0.762299i \(-0.724071\pi\)
−0.336557 0.941663i \(-0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.0479i 1.23526i 0.786469 + 0.617629i \(0.211907\pi\)
−0.786469 + 0.617629i \(0.788093\pi\)
\(380\) 0 0
\(381\) 39.0000 1.99803
\(382\) 0 0
\(383\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 32.4805 18.7526i 1.65107 0.953248i
\(388\) 0 0
\(389\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.0292 13.4775i 1.85844 0.676417i 0.878300 0.478110i \(-0.158678\pi\)
0.980140 0.198307i \(-0.0635442\pi\)
\(398\) 0 0
\(399\) 32.8555 + 9.78639i 1.64483 + 0.489933i
\(400\) 0 0
\(401\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(402\) 0 0
\(403\) 15.7685 + 18.7921i 0.785483 + 0.936102i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 35.7083 + 12.9968i 1.76566 + 0.642649i 1.00000 0.000593299i \(-0.000188853\pi\)
0.765663 + 0.643242i \(0.222411\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −40.7948 −1.99773
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −7.11958 + 40.3771i −0.346987 + 1.96786i −0.131927 + 0.991259i \(0.542117\pi\)
−0.215060 + 0.976601i \(0.568995\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.73947 26.7590i 0.471326 1.29496i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(432\) 0 0
\(433\) −31.0940 + 26.0910i −1.49428 + 1.25385i −0.605231 + 0.796050i \(0.706919\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −14.3241 39.3552i −0.683653 1.87832i −0.373425 0.927660i \(-0.621817\pi\)
−0.310228 0.950662i \(-0.600405\pi\)
\(440\) 0 0
\(441\) 31.2973 26.2615i 1.49035 1.25055i
\(442\) 0 0
\(443\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.60472 14.7721i 0.122381 0.694055i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −39.2012 −1.83376 −0.916878 0.399169i \(-0.869299\pi\)
−0.916878 + 0.399169i \(0.869299\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(462\) 0 0
\(463\) −36.4129 + 21.0230i −1.69225 + 0.977021i −0.739553 + 0.673098i \(0.764963\pi\)
−0.952697 + 0.303923i \(0.901704\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 12.8986 + 73.1515i 0.595601 + 3.37782i
\(470\) 0 0
\(471\) −25.4954 30.3843i −1.17477 1.40003i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 9.75418 19.4899i 0.447553 0.894258i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(480\) 0 0
\(481\) 3.56479 + 20.2169i 0.162540 + 0.921812i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 31.5000 18.1865i 1.42740 0.824110i 0.430486 0.902597i \(-0.358342\pi\)
0.996915 + 0.0784867i \(0.0250088\pi\)
\(488\) 0 0
\(489\) 17.8237 + 14.9558i 0.806014 + 0.676326i
\(490\) 0 0
\(491\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.13000 2.53844i 0.0953521 0.113636i −0.716258 0.697835i \(-0.754147\pi\)
0.811610 + 0.584199i \(0.198591\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.61809 + 1.34327i −0.338331 + 0.0596569i
\(508\) 0 0
\(509\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(510\) 0 0
\(511\) −25.7294 70.6909i −1.13820 3.12718i
\(512\) 0 0
\(513\) 9.00000 + 20.7846i 0.397360 + 0.917663i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 15.4093 42.3366i 0.673800 1.85125i 0.174908 0.984585i \(-0.444037\pi\)
0.498892 0.866664i \(-0.333741\pi\)
\(524\) 0 0
\(525\) −19.6621 34.0557i −0.858124 1.48631i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.99391 + 22.6506i −0.173648 + 0.984808i
\(530\) 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
\(539\) 0 0
\(540\) 0 0
\(541\) −1.09462 0.398408i −0.0470613 0.0171289i 0.318382 0.947962i \(-0.396860\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 0 0
\(543\) 10.5000 + 6.06218i 0.450598 + 0.260153i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −22.6143 26.9506i −0.966917 1.15233i −0.988295 0.152555i \(-0.951250\pi\)
0.0213785 0.999771i \(-0.493195\pi\)
\(548\) 0 0
\(549\) 17.6792 6.43469i 0.754528 0.274626i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 29.6419 10.7888i 1.26050 0.458785i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(558\) 0 0
\(559\) 31.6281 + 18.2605i 1.33773 + 0.772337i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 40.2460 + 7.09646i 1.69017 + 0.298023i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 36.8452i 1.54192i 0.636882 + 0.770961i \(0.280224\pi\)
−0.636882 + 0.770961i \(0.719776\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.50000 9.52628i −0.228968 0.396584i 0.728535 0.685009i \(-0.240202\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) −4.50159 + 12.3680i −0.187080 + 0.513998i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(588\) 0 0
\(589\) −36.3622 4.19875i −1.49828 0.173006i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.1126 19.2476i 0.454809 0.787753i
\(598\) 0 0
\(599\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(600\) 0 0
\(601\) −23.3187 40.3891i −0.951188 1.64751i −0.742859 0.669448i \(-0.766531\pi\)
−0.208329 0.978059i \(-0.566802\pi\)
\(602\) 0 0
\(603\) −31.5450 + 37.5939i −1.28461 + 1.53094i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11.8762i 0.482040i −0.970520 0.241020i \(-0.922518\pi\)
0.970520 0.241020i \(-0.0774820\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 36.0041 + 30.2110i 1.45419 + 1.22021i 0.929449 + 0.368950i \(0.120283\pi\)
0.524742 + 0.851261i \(0.324162\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(618\) 0 0
\(619\) 21.5614 + 12.4485i 0.866626 + 0.500347i 0.866226 0.499653i \(-0.166539\pi\)
0.000400419 1.00000i \(0.499873\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.4923 + 8.55050i −0.939693 + 0.342020i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 24.3200 + 28.9835i 0.968164 + 1.15381i 0.988069 + 0.154011i \(0.0492193\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) −8.49928 48.2018i −0.337816 1.91585i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 37.3843 + 13.6068i 1.48122 + 0.539120i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(642\) 0 0
\(643\) 33.4623 + 5.90030i 1.31962 + 0.232685i 0.788723 0.614749i \(-0.210743\pi\)
0.530901 + 0.847434i \(0.321854\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −42.4528 + 50.5933i −1.66386 + 1.98291i
\(652\) 0 0
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 24.8508 43.0428i 0.969520 1.67926i
\(658\) 0 0
\(659\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(660\) 0 0
\(661\) 29.1097 24.4259i 1.13224 0.950059i 0.133078 0.991106i \(-0.457514\pi\)
0.999157 + 0.0410470i \(0.0130693\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 34.1924 28.6908i 1.32195 1.10925i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −11.1642 + 19.3369i −0.430346 + 0.745382i −0.996903 0.0786409i \(-0.974942\pi\)
0.566557 + 0.824023i \(0.308275\pi\)
\(674\) 0 0
\(675\) 8.88594 24.4139i 0.342020 0.939693i
\(676\) 0 0
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 14.5937 17.3921i 0.560056 0.667449i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 47.3888 + 8.35592i 1.80799 + 0.318798i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −16.5000 + 9.52628i −0.627690 + 0.362397i −0.779857 0.625958i \(-0.784708\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(702\) 0 0
\(703\) −24.5957 18.2576i −0.927644 0.688597i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.48674 14.1030i −0.0933915 0.529649i −0.995228 0.0975728i \(-0.968892\pi\)
0.901837 0.432077i \(-0.142219\pi\)
\(710\) 0 0
\(711\) 18.0486 + 10.4204i 0.676875 + 0.390794i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(720\) 0 0
\(721\) −70.4246 −2.62275
\(722\) 0 0
\(723\) 43.0600i 1.60142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −33.5314 + 39.9611i −1.24361 + 1.48208i −0.427675 + 0.903933i \(0.640667\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 13.5000 + 23.3827i 0.500000 + 0.866025i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.50000 + 6.06218i −0.129275 + 0.223912i −0.923396 0.383849i \(-0.874598\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.30313 + 6.32780i 0.0847220 + 0.232772i 0.974818 0.223001i \(-0.0715853\pi\)
−0.890096 + 0.455773i \(0.849363\pi\)
\(740\) 0 0
\(741\) −13.1457 + 17.7093i −0.482921 + 0.650567i
\(742\) 0 0
\(743\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −6.89399 + 18.9411i −0.251565 + 0.691170i 0.748056 + 0.663636i \(0.230988\pi\)
−0.999621 + 0.0275338i \(0.991235\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7.12289 40.3959i 0.258886 1.46822i −0.527011 0.849858i \(-0.676688\pi\)
0.785897 0.618357i \(-0.212201\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −76.0203 13.4044i −2.75212 0.485273i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.9180 12.7091i −1.25918 0.458303i −0.375684 0.926748i \(-0.622592\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(774\) 0 0
\(775\) 26.9890 + 32.1643i 0.969474 + 1.15537i
\(776\) 0 0
\(777\) −51.9358 + 18.9031i −1.86319 + 0.678144i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −28.8754 16.6712i −1.02930 0.594265i −0.112514 0.993650i \(-0.535890\pi\)
−0.916783 + 0.399385i \(0.869224\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 14.0340 + 11.7759i 0.498361 + 0.418175i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0