Properties

Label 840.2.t.d.169.3
Level $840$
Weight $2$
Character 840.169
Analytic conductor $6.707$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 169.3
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 840.169
Dual form 840.2.t.d.169.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +6.31265 q^{11} +6.96239i q^{13} +(-1.67513 - 1.48119i) q^{15} -6.57452i q^{17} +3.73813 q^{19} +1.00000 q^{21} +5.73813i q^{23} +(-0.612127 - 4.96239i) q^{25} +1.00000i q^{27} +2.00000 q^{29} -1.03761 q^{31} -6.31265i q^{33} +(1.67513 + 1.48119i) q^{35} -10.7005i q^{37} +6.96239 q^{39} -6.96239 q^{41} -5.92478i q^{43} +(-1.48119 + 1.67513i) q^{45} -1.00000 q^{49} -6.57452 q^{51} +1.03761i q^{53} +(9.35026 - 10.5745i) q^{55} -3.73813i q^{57} +3.22425 q^{59} -13.8496 q^{61} -1.00000i q^{63} +(11.6629 + 10.3127i) q^{65} +4.77575i q^{67} +5.73813 q^{69} +8.23743 q^{71} +4.26187i q^{73} +(-4.96239 + 0.612127i) q^{75} +6.31265i q^{77} +5.92478 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-11.0132 - 9.73813i) q^{85} -2.00000i q^{87} -2.18664 q^{89} -6.96239 q^{91} +1.03761i q^{93} +(5.53690 - 6.26187i) q^{95} -3.73813i q^{97} -6.31265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} - 28 q^{31} + 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} - 16 q^{51} + 36 q^{55} + 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} - 36 q^{71} - 8 q^{75} - 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} - 20 q^{91} - 12 q^{95} + 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.48119 1.67513i 0.662410 0.749141i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.31265 1.90334 0.951668 0.307129i \(-0.0993684\pi\)
0.951668 + 0.307129i \(0.0993684\pi\)
\(12\) 0 0
\(13\) 6.96239i 1.93102i 0.260368 + 0.965510i \(0.416156\pi\)
−0.260368 + 0.965510i \(0.583844\pi\)
\(14\) 0 0
\(15\) −1.67513 1.48119i −0.432517 0.382443i
\(16\) 0 0
\(17\) 6.57452i 1.59455i −0.603613 0.797277i \(-0.706273\pi\)
0.603613 0.797277i \(-0.293727\pi\)
\(18\) 0 0
\(19\) 3.73813 0.857587 0.428793 0.903403i \(-0.358939\pi\)
0.428793 + 0.903403i \(0.358939\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 5.73813i 1.19648i 0.801316 + 0.598242i \(0.204134\pi\)
−0.801316 + 0.598242i \(0.795866\pi\)
\(24\) 0 0
\(25\) −0.612127 4.96239i −0.122425 0.992478i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.03761 −0.186361 −0.0931803 0.995649i \(-0.529703\pi\)
−0.0931803 + 0.995649i \(0.529703\pi\)
\(32\) 0 0
\(33\) 6.31265i 1.09889i
\(34\) 0 0
\(35\) 1.67513 + 1.48119i 0.283149 + 0.250368i
\(36\) 0 0
\(37\) 10.7005i 1.75916i −0.475755 0.879578i \(-0.657825\pi\)
0.475755 0.879578i \(-0.342175\pi\)
\(38\) 0 0
\(39\) 6.96239 1.11487
\(40\) 0 0
\(41\) −6.96239 −1.08734 −0.543671 0.839298i \(-0.682966\pi\)
−0.543671 + 0.839298i \(0.682966\pi\)
\(42\) 0 0
\(43\) 5.92478i 0.903520i −0.892139 0.451760i \(-0.850796\pi\)
0.892139 0.451760i \(-0.149204\pi\)
\(44\) 0 0
\(45\) −1.48119 + 1.67513i −0.220803 + 0.249714i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.57452 −0.920616
\(52\) 0 0
\(53\) 1.03761i 0.142527i 0.997458 + 0.0712634i \(0.0227031\pi\)
−0.997458 + 0.0712634i \(0.977297\pi\)
\(54\) 0 0
\(55\) 9.35026 10.5745i 1.26079 1.42587i
\(56\) 0 0
\(57\) 3.73813i 0.495128i
\(58\) 0 0
\(59\) 3.22425 0.419762 0.209881 0.977727i \(-0.432692\pi\)
0.209881 + 0.977727i \(0.432692\pi\)
\(60\) 0 0
\(61\) −13.8496 −1.77325 −0.886627 0.462485i \(-0.846958\pi\)
−0.886627 + 0.462485i \(0.846958\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 11.6629 + 10.3127i 1.44661 + 1.27913i
\(66\) 0 0
\(67\) 4.77575i 0.583450i 0.956502 + 0.291725i \(0.0942293\pi\)
−0.956502 + 0.291725i \(0.905771\pi\)
\(68\) 0 0
\(69\) 5.73813 0.690790
\(70\) 0 0
\(71\) 8.23743 0.977603 0.488801 0.872395i \(-0.337434\pi\)
0.488801 + 0.872395i \(0.337434\pi\)
\(72\) 0 0
\(73\) 4.26187i 0.498814i 0.968399 + 0.249407i \(0.0802356\pi\)
−0.968399 + 0.249407i \(0.919764\pi\)
\(74\) 0 0
\(75\) −4.96239 + 0.612127i −0.573007 + 0.0706823i
\(76\) 0 0
\(77\) 6.31265i 0.719393i
\(78\) 0 0
\(79\) 5.92478 0.666590 0.333295 0.942823i \(-0.391840\pi\)
0.333295 + 0.942823i \(0.391840\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.22425i 0.353908i 0.984219 + 0.176954i \(0.0566244\pi\)
−0.984219 + 0.176954i \(0.943376\pi\)
\(84\) 0 0
\(85\) −11.0132 9.73813i −1.19455 1.05625i
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −2.18664 −0.231784 −0.115892 0.993262i \(-0.536973\pi\)
−0.115892 + 0.993262i \(0.536973\pi\)
\(90\) 0 0
\(91\) −6.96239 −0.729857
\(92\) 0 0
\(93\) 1.03761i 0.107595i
\(94\) 0 0
\(95\) 5.53690 6.26187i 0.568074 0.642454i
\(96\) 0 0
\(97\) 3.73813i 0.379550i −0.981828 0.189775i \(-0.939224\pi\)
0.981828 0.189775i \(-0.0607759\pi\)
\(98\) 0 0
\(99\) −6.31265 −0.634445
\(100\) 0 0
\(101\) 9.66291 0.961496 0.480748 0.876859i \(-0.340365\pi\)
0.480748 + 0.876859i \(0.340365\pi\)
\(102\) 0 0
\(103\) 1.29948i 0.128041i 0.997949 + 0.0640206i \(0.0203923\pi\)
−0.997949 + 0.0640206i \(0.979608\pi\)
\(104\) 0 0
\(105\) 1.48119 1.67513i 0.144550 0.163476i
\(106\) 0 0
\(107\) 3.66291i 0.354107i 0.984201 + 0.177054i \(0.0566566\pi\)
−0.984201 + 0.177054i \(0.943343\pi\)
\(108\) 0 0
\(109\) −6.77575 −0.648999 −0.324499 0.945886i \(-0.605196\pi\)
−0.324499 + 0.945886i \(0.605196\pi\)
\(110\) 0 0
\(111\) −10.7005 −1.01565
\(112\) 0 0
\(113\) 8.88717i 0.836034i −0.908439 0.418017i \(-0.862725\pi\)
0.908439 0.418017i \(-0.137275\pi\)
\(114\) 0 0
\(115\) 9.61213 + 8.49929i 0.896335 + 0.792563i
\(116\) 0 0
\(117\) 6.96239i 0.643673i
\(118\) 0 0
\(119\) 6.57452 0.602685
\(120\) 0 0
\(121\) 28.8496 2.62269
\(122\) 0 0
\(123\) 6.96239i 0.627777i
\(124\) 0 0
\(125\) −9.21933 6.32487i −0.824602 0.565713i
\(126\) 0 0
\(127\) 2.70052i 0.239633i 0.992796 + 0.119816i \(0.0382306\pi\)
−0.992796 + 0.119816i \(0.961769\pi\)
\(128\) 0 0
\(129\) −5.92478 −0.521648
\(130\) 0 0
\(131\) −9.40105 −0.821373 −0.410687 0.911777i \(-0.634711\pi\)
−0.410687 + 0.911777i \(0.634711\pi\)
\(132\) 0 0
\(133\) 3.73813i 0.324137i
\(134\) 0 0
\(135\) 1.67513 + 1.48119i 0.144172 + 0.127481i
\(136\) 0 0
\(137\) 9.66291i 0.825558i 0.910831 + 0.412779i \(0.135442\pi\)
−0.910831 + 0.412779i \(0.864558\pi\)
\(138\) 0 0
\(139\) −20.8872 −1.77163 −0.885813 0.464042i \(-0.846399\pi\)
−0.885813 + 0.464042i \(0.846399\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 43.9511i 3.67538i
\(144\) 0 0
\(145\) 2.96239 3.35026i 0.246013 0.278224i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −0.0752228 −0.00616249 −0.00308125 0.999995i \(-0.500981\pi\)
−0.00308125 + 0.999995i \(0.500981\pi\)
\(150\) 0 0
\(151\) 6.70052 0.545281 0.272640 0.962116i \(-0.412103\pi\)
0.272640 + 0.962116i \(0.412103\pi\)
\(152\) 0 0
\(153\) 6.57452i 0.531518i
\(154\) 0 0
\(155\) −1.53690 + 1.73813i −0.123447 + 0.139610i
\(156\) 0 0
\(157\) 5.66291i 0.451950i 0.974133 + 0.225975i \(0.0725567\pi\)
−0.974133 + 0.225975i \(0.927443\pi\)
\(158\) 0 0
\(159\) 1.03761 0.0822879
\(160\) 0 0
\(161\) −5.73813 −0.452228
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) −10.5745 9.35026i −0.823225 0.727917i
\(166\) 0 0
\(167\) 20.6253i 1.59603i 0.602635 + 0.798017i \(0.294117\pi\)
−0.602635 + 0.798017i \(0.705883\pi\)
\(168\) 0 0
\(169\) −35.4749 −2.72884
\(170\) 0 0
\(171\) −3.73813 −0.285862
\(172\) 0 0
\(173\) 10.5745i 0.803966i 0.915647 + 0.401983i \(0.131679\pi\)
−0.915647 + 0.401983i \(0.868321\pi\)
\(174\) 0 0
\(175\) 4.96239 0.612127i 0.375121 0.0462724i
\(176\) 0 0
\(177\) 3.22425i 0.242350i
\(178\) 0 0
\(179\) −0.911603 −0.0681364 −0.0340682 0.999420i \(-0.510846\pi\)
−0.0340682 + 0.999420i \(0.510846\pi\)
\(180\) 0 0
\(181\) −13.3258 −0.990501 −0.495250 0.868750i \(-0.664924\pi\)
−0.495250 + 0.868750i \(0.664924\pi\)
\(182\) 0 0
\(183\) 13.8496i 1.02379i
\(184\) 0 0
\(185\) −17.9248 15.8496i −1.31786 1.16528i
\(186\) 0 0
\(187\) 41.5026i 3.03497i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −25.7889 −1.86602 −0.933010 0.359849i \(-0.882828\pi\)
−0.933010 + 0.359849i \(0.882828\pi\)
\(192\) 0 0
\(193\) 15.3258i 1.10318i 0.834116 + 0.551588i \(0.185978\pi\)
−0.834116 + 0.551588i \(0.814022\pi\)
\(194\) 0 0
\(195\) 10.3127 11.6629i 0.738504 0.835198i
\(196\) 0 0
\(197\) 0.513881i 0.0366125i −0.999832 0.0183063i \(-0.994173\pi\)
0.999832 0.0183063i \(-0.00582739\pi\)
\(198\) 0 0
\(199\) −3.73813 −0.264989 −0.132495 0.991184i \(-0.542299\pi\)
−0.132495 + 0.991184i \(0.542299\pi\)
\(200\) 0 0
\(201\) 4.77575 0.336855
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) −10.3127 + 11.6629i −0.720267 + 0.814573i
\(206\) 0 0
\(207\) 5.73813i 0.398828i
\(208\) 0 0
\(209\) 23.5975 1.63228
\(210\) 0 0
\(211\) −3.22425 −0.221967 −0.110983 0.993822i \(-0.535400\pi\)
−0.110983 + 0.993822i \(0.535400\pi\)
\(212\) 0 0
\(213\) 8.23743i 0.564419i
\(214\) 0 0
\(215\) −9.92478 8.77575i −0.676864 0.598501i
\(216\) 0 0
\(217\) 1.03761i 0.0704377i
\(218\) 0 0
\(219\) 4.26187 0.287990
\(220\) 0 0
\(221\) 45.7743 3.07911
\(222\) 0 0
\(223\) 11.8496i 0.793505i −0.917926 0.396752i \(-0.870137\pi\)
0.917926 0.396752i \(-0.129863\pi\)
\(224\) 0 0
\(225\) 0.612127 + 4.96239i 0.0408085 + 0.330826i
\(226\) 0 0
\(227\) 14.7005i 0.975708i −0.872925 0.487854i \(-0.837780\pi\)
0.872925 0.487854i \(-0.162220\pi\)
\(228\) 0 0
\(229\) −9.47627 −0.626210 −0.313105 0.949719i \(-0.601369\pi\)
−0.313105 + 0.949719i \(0.601369\pi\)
\(230\) 0 0
\(231\) 6.31265 0.415342
\(232\) 0 0
\(233\) 29.5125i 1.93343i 0.255863 + 0.966713i \(0.417640\pi\)
−0.255863 + 0.966713i \(0.582360\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.92478i 0.384856i
\(238\) 0 0
\(239\) 25.0132 1.61797 0.808984 0.587831i \(-0.200018\pi\)
0.808984 + 0.587831i \(0.200018\pi\)
\(240\) 0 0
\(241\) 7.92478 0.510480 0.255240 0.966878i \(-0.417846\pi\)
0.255240 + 0.966878i \(0.417846\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −1.48119 + 1.67513i −0.0946300 + 0.107020i
\(246\) 0 0
\(247\) 26.0263i 1.65602i
\(248\) 0 0
\(249\) 3.22425 0.204329
\(250\) 0 0
\(251\) −15.0738 −0.951450 −0.475725 0.879594i \(-0.657814\pi\)
−0.475725 + 0.879594i \(0.657814\pi\)
\(252\) 0 0
\(253\) 36.2228i 2.27731i
\(254\) 0 0
\(255\) −9.73813 + 11.0132i −0.609826 + 0.689672i
\(256\) 0 0
\(257\) 22.0508i 1.37549i −0.725952 0.687745i \(-0.758601\pi\)
0.725952 0.687745i \(-0.241399\pi\)
\(258\) 0 0
\(259\) 10.7005 0.664898
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 11.8134i 0.728443i 0.931312 + 0.364221i \(0.118665\pi\)
−0.931312 + 0.364221i \(0.881335\pi\)
\(264\) 0 0
\(265\) 1.73813 + 1.53690i 0.106773 + 0.0944113i
\(266\) 0 0
\(267\) 2.18664i 0.133820i
\(268\) 0 0
\(269\) 27.0640 1.65012 0.825059 0.565046i \(-0.191142\pi\)
0.825059 + 0.565046i \(0.191142\pi\)
\(270\) 0 0
\(271\) −10.8119 −0.656779 −0.328389 0.944542i \(-0.606506\pi\)
−0.328389 + 0.944542i \(0.606506\pi\)
\(272\) 0 0
\(273\) 6.96239i 0.421383i
\(274\) 0 0
\(275\) −3.86414 31.3258i −0.233017 1.88902i
\(276\) 0 0
\(277\) 22.5501i 1.35490i 0.735568 + 0.677451i \(0.236916\pi\)
−0.735568 + 0.677451i \(0.763084\pi\)
\(278\) 0 0
\(279\) 1.03761 0.0621202
\(280\) 0 0
\(281\) −10.3733 −0.618818 −0.309409 0.950929i \(-0.600131\pi\)
−0.309409 + 0.950929i \(0.600131\pi\)
\(282\) 0 0
\(283\) 10.7005i 0.636080i −0.948077 0.318040i \(-0.896975\pi\)
0.948077 0.318040i \(-0.103025\pi\)
\(284\) 0 0
\(285\) −6.26187 5.53690i −0.370921 0.327978i
\(286\) 0 0
\(287\) 6.96239i 0.410977i
\(288\) 0 0
\(289\) −26.2243 −1.54260
\(290\) 0 0
\(291\) −3.73813 −0.219133
\(292\) 0 0
\(293\) 10.5745i 0.617770i −0.951099 0.308885i \(-0.900044\pi\)
0.951099 0.308885i \(-0.0999558\pi\)
\(294\) 0 0
\(295\) 4.77575 5.40105i 0.278055 0.314461i
\(296\) 0 0
\(297\) 6.31265i 0.366297i
\(298\) 0 0
\(299\) −39.9511 −2.31043
\(300\) 0 0
\(301\) 5.92478 0.341498
\(302\) 0 0
\(303\) 9.66291i 0.555120i
\(304\) 0 0
\(305\) −20.5139 + 23.1998i −1.17462 + 1.32842i
\(306\) 0 0
\(307\) 6.55008i 0.373833i −0.982376 0.186916i \(-0.940151\pi\)
0.982376 0.186916i \(-0.0598493\pi\)
\(308\) 0 0
\(309\) 1.29948 0.0739246
\(310\) 0 0
\(311\) −17.2995 −0.980963 −0.490482 0.871452i \(-0.663179\pi\)
−0.490482 + 0.871452i \(0.663179\pi\)
\(312\) 0 0
\(313\) 20.2130i 1.14251i −0.820774 0.571253i \(-0.806458\pi\)
0.820774 0.571253i \(-0.193542\pi\)
\(314\) 0 0
\(315\) −1.67513 1.48119i −0.0943829 0.0834558i
\(316\) 0 0
\(317\) 19.3357i 1.08600i −0.839733 0.543000i \(-0.817288\pi\)
0.839733 0.543000i \(-0.182712\pi\)
\(318\) 0 0
\(319\) 12.6253 0.706881
\(320\) 0 0
\(321\) 3.66291 0.204444
\(322\) 0 0
\(323\) 24.5764i 1.36747i
\(324\) 0 0
\(325\) 34.5501 4.26187i 1.91649 0.236406i
\(326\) 0 0
\(327\) 6.77575i 0.374700i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 24.6253 1.35353 0.676764 0.736200i \(-0.263382\pi\)
0.676764 + 0.736200i \(0.263382\pi\)
\(332\) 0 0
\(333\) 10.7005i 0.586385i
\(334\) 0 0
\(335\) 8.00000 + 7.07381i 0.437087 + 0.386483i
\(336\) 0 0
\(337\) 6.44851i 0.351273i −0.984455 0.175636i \(-0.943802\pi\)
0.984455 0.175636i \(-0.0561983\pi\)
\(338\) 0 0
\(339\) −8.88717 −0.482685
\(340\) 0 0
\(341\) −6.55008 −0.354707
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 8.49929 9.61213i 0.457587 0.517500i
\(346\) 0 0
\(347\) 11.6629i 0.626098i 0.949737 + 0.313049i \(0.101350\pi\)
−0.949737 + 0.313049i \(0.898650\pi\)
\(348\) 0 0
\(349\) −12.4485 −0.666353 −0.333177 0.942864i \(-0.608121\pi\)
−0.333177 + 0.942864i \(0.608121\pi\)
\(350\) 0 0
\(351\) −6.96239 −0.371625
\(352\) 0 0
\(353\) 13.0230i 0.693146i −0.938023 0.346573i \(-0.887345\pi\)
0.938023 0.346573i \(-0.112655\pi\)
\(354\) 0 0
\(355\) 12.2012 13.7988i 0.647574 0.732363i
\(356\) 0 0
\(357\) 6.57452i 0.347960i
\(358\) 0 0
\(359\) 20.3879 1.07603 0.538015 0.842935i \(-0.319174\pi\)
0.538015 + 0.842935i \(0.319174\pi\)
\(360\) 0 0
\(361\) −5.02635 −0.264545
\(362\) 0 0
\(363\) 28.8496i 1.51421i
\(364\) 0 0
\(365\) 7.13918 + 6.31265i 0.373682 + 0.330419i
\(366\) 0 0
\(367\) 19.8496i 1.03614i 0.855339 + 0.518069i \(0.173349\pi\)
−0.855339 + 0.518069i \(0.826651\pi\)
\(368\) 0 0
\(369\) 6.96239 0.362447
\(370\) 0 0
\(371\) −1.03761 −0.0538701
\(372\) 0 0
\(373\) 2.85097i 0.147618i 0.997272 + 0.0738088i \(0.0235155\pi\)
−0.997272 + 0.0738088i \(0.976485\pi\)
\(374\) 0 0
\(375\) −6.32487 + 9.21933i −0.326615 + 0.476084i
\(376\) 0 0
\(377\) 13.9248i 0.717163i
\(378\) 0 0
\(379\) −5.29948 −0.272216 −0.136108 0.990694i \(-0.543459\pi\)
−0.136108 + 0.990694i \(0.543459\pi\)
\(380\) 0 0
\(381\) 2.70052 0.138352
\(382\) 0 0
\(383\) 12.2520i 0.626049i −0.949745 0.313024i \(-0.898658\pi\)
0.949745 0.313024i \(-0.101342\pi\)
\(384\) 0 0
\(385\) 10.5745 + 9.35026i 0.538927 + 0.476533i
\(386\) 0 0
\(387\) 5.92478i 0.301173i
\(388\) 0 0
\(389\) −16.4485 −0.833972 −0.416986 0.908913i \(-0.636914\pi\)
−0.416986 + 0.908913i \(0.636914\pi\)
\(390\) 0 0
\(391\) 37.7255 1.90786
\(392\) 0 0
\(393\) 9.40105i 0.474220i
\(394\) 0 0
\(395\) 8.77575 9.92478i 0.441556 0.499370i
\(396\) 0 0
\(397\) 3.58769i 0.180061i 0.995939 + 0.0900305i \(0.0286964\pi\)
−0.995939 + 0.0900305i \(0.971304\pi\)
\(398\) 0 0
\(399\) 3.73813 0.187141
\(400\) 0 0
\(401\) 19.9248 0.994996 0.497498 0.867465i \(-0.334252\pi\)
0.497498 + 0.867465i \(0.334252\pi\)
\(402\) 0 0
\(403\) 7.22425i 0.359866i
\(404\) 0 0
\(405\) 1.48119 1.67513i 0.0736011 0.0832379i
\(406\) 0 0
\(407\) 67.5487i 3.34826i
\(408\) 0 0
\(409\) 25.1754 1.24484 0.622421 0.782682i \(-0.286149\pi\)
0.622421 + 0.782682i \(0.286149\pi\)
\(410\) 0 0
\(411\) 9.66291 0.476636
\(412\) 0 0
\(413\) 3.22425i 0.158655i
\(414\) 0 0
\(415\) 5.40105 + 4.77575i 0.265127 + 0.234432i
\(416\) 0 0
\(417\) 20.8872i 1.02285i
\(418\) 0 0
\(419\) −1.14903 −0.0561338 −0.0280669 0.999606i \(-0.508935\pi\)
−0.0280669 + 0.999606i \(0.508935\pi\)
\(420\) 0 0
\(421\) −0.176793 −0.00861637 −0.00430819 0.999991i \(-0.501371\pi\)
−0.00430819 + 0.999991i \(0.501371\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −32.6253 + 4.02444i −1.58256 + 0.195214i
\(426\) 0 0
\(427\) 13.8496i 0.670227i
\(428\) 0 0
\(429\) 43.9511 2.12198
\(430\) 0 0
\(431\) 12.3879 0.596703 0.298351 0.954456i \(-0.403563\pi\)
0.298351 + 0.954456i \(0.403563\pi\)
\(432\) 0 0
\(433\) 1.41090i 0.0678033i −0.999425 0.0339017i \(-0.989207\pi\)
0.999425 0.0339017i \(-0.0107933\pi\)
\(434\) 0 0
\(435\) −3.35026 2.96239i −0.160633 0.142036i
\(436\) 0 0
\(437\) 21.4499i 1.02609i
\(438\) 0 0
\(439\) −31.5877 −1.50760 −0.753799 0.657105i \(-0.771781\pi\)
−0.753799 + 0.657105i \(0.771781\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 28.0362i 1.33204i 0.745934 + 0.666020i \(0.232003\pi\)
−0.745934 + 0.666020i \(0.767997\pi\)
\(444\) 0 0
\(445\) −3.23884 + 3.66291i −0.153536 + 0.173639i
\(446\) 0 0
\(447\) 0.0752228i 0.00355792i
\(448\) 0 0
\(449\) 19.4010 0.915592 0.457796 0.889057i \(-0.348639\pi\)
0.457796 + 0.889057i \(0.348639\pi\)
\(450\) 0 0
\(451\) −43.9511 −2.06958
\(452\) 0 0
\(453\) 6.70052i 0.314818i
\(454\) 0 0
\(455\) −10.3127 + 11.6629i −0.483465 + 0.546766i
\(456\) 0 0
\(457\) 19.3258i 0.904024i −0.892012 0.452012i \(-0.850706\pi\)
0.892012 0.452012i \(-0.149294\pi\)
\(458\) 0 0
\(459\) 6.57452 0.306872
\(460\) 0 0
\(461\) 2.81194 0.130965 0.0654826 0.997854i \(-0.479141\pi\)
0.0654826 + 0.997854i \(0.479141\pi\)
\(462\) 0 0
\(463\) 24.1016i 1.12009i 0.828461 + 0.560047i \(0.189217\pi\)
−0.828461 + 0.560047i \(0.810783\pi\)
\(464\) 0 0
\(465\) 1.73813 + 1.53690i 0.0806041 + 0.0712722i
\(466\) 0 0
\(467\) 21.1490i 0.978660i 0.872099 + 0.489330i \(0.162759\pi\)
−0.872099 + 0.489330i \(0.837241\pi\)
\(468\) 0 0
\(469\) −4.77575 −0.220523
\(470\) 0 0
\(471\) 5.66291 0.260933
\(472\) 0 0
\(473\) 37.4010i 1.71970i
\(474\) 0 0
\(475\) −2.28821 18.5501i −0.104990 0.851136i
\(476\) 0 0
\(477\) 1.03761i 0.0475090i
\(478\) 0 0
\(479\) 17.2995 0.790433 0.395217 0.918588i \(-0.370670\pi\)
0.395217 + 0.918588i \(0.370670\pi\)
\(480\) 0 0
\(481\) 74.5012 3.39696
\(482\) 0 0
\(483\) 5.73813i 0.261094i
\(484\) 0 0
\(485\) −6.26187 5.53690i −0.284337 0.251418i
\(486\) 0 0
\(487\) 40.4749i 1.83409i −0.398783 0.917045i \(-0.630567\pi\)
0.398783 0.917045i \(-0.369433\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −19.9854 −0.901929 −0.450964 0.892542i \(-0.648920\pi\)
−0.450964 + 0.892542i \(0.648920\pi\)
\(492\) 0 0
\(493\) 13.1490i 0.592203i
\(494\) 0 0
\(495\) −9.35026 + 10.5745i −0.420263 + 0.475289i
\(496\) 0 0
\(497\) 8.23743i 0.369499i
\(498\) 0 0
\(499\) −14.8510 −0.664821 −0.332410 0.943135i \(-0.607862\pi\)
−0.332410 + 0.943135i \(0.607862\pi\)
\(500\) 0 0
\(501\) 20.6253 0.921470
\(502\) 0 0
\(503\) 22.5501i 1.00546i −0.864444 0.502729i \(-0.832329\pi\)
0.864444 0.502729i \(-0.167671\pi\)
\(504\) 0 0
\(505\) 14.3127 16.1866i 0.636905 0.720296i
\(506\) 0 0
\(507\) 35.4749i 1.57549i
\(508\) 0 0
\(509\) −17.5125 −0.776226 −0.388113 0.921612i \(-0.626873\pi\)
−0.388113 + 0.921612i \(0.626873\pi\)
\(510\) 0 0
\(511\) −4.26187 −0.188534
\(512\) 0 0
\(513\) 3.73813i 0.165043i
\(514\) 0 0
\(515\) 2.17679 + 1.92478i 0.0959210 + 0.0848158i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 10.5745 0.464170
\(520\) 0 0
\(521\) 26.4387 1.15830 0.579149 0.815221i \(-0.303385\pi\)
0.579149 + 0.815221i \(0.303385\pi\)
\(522\) 0 0
\(523\) 31.8496i 1.39268i 0.717710 + 0.696342i \(0.245190\pi\)
−0.717710 + 0.696342i \(0.754810\pi\)
\(524\) 0 0
\(525\) −0.612127 4.96239i −0.0267154 0.216576i
\(526\) 0 0
\(527\) 6.82179i 0.297162i
\(528\) 0 0
\(529\) −9.92619 −0.431574
\(530\) 0 0
\(531\) −3.22425 −0.139921
\(532\) 0 0
\(533\) 48.4749i 2.09968i
\(534\) 0 0
\(535\) 6.13586 + 5.42548i 0.265276 + 0.234564i
\(536\) 0 0
\(537\) 0.911603i 0.0393386i
\(538\) 0 0
\(539\) −6.31265 −0.271905
\(540\) 0 0
\(541\) 24.1768 1.03944 0.519721 0.854336i \(-0.326036\pi\)
0.519721 + 0.854336i \(0.326036\pi\)
\(542\) 0 0
\(543\) 13.3258i 0.571866i
\(544\) 0 0
\(545\) −10.0362 + 11.3503i −0.429903 + 0.486192i
\(546\) 0 0
\(547\) 34.1768i 1.46129i −0.682755 0.730647i \(-0.739219\pi\)
0.682755 0.730647i \(-0.260781\pi\)
\(548\) 0 0
\(549\) 13.8496 0.591085
\(550\) 0 0
\(551\) 7.47627 0.318500
\(552\) 0 0
\(553\) 5.92478i 0.251947i
\(554\) 0 0
\(555\) −15.8496 + 17.9248i −0.672776 + 0.760864i
\(556\) 0 0
\(557\) 24.7367i 1.04813i −0.851679 0.524064i \(-0.824415\pi\)
0.851679 0.524064i \(-0.175585\pi\)
\(558\) 0 0
\(559\) 41.2506 1.74471
\(560\) 0 0
\(561\) −41.5026 −1.75224
\(562\) 0 0
\(563\) 7.47627i 0.315087i −0.987512 0.157544i \(-0.949643\pi\)
0.987512 0.157544i \(-0.0503575\pi\)
\(564\) 0 0
\(565\) −14.8872 13.1636i −0.626308 0.553798i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −17.3258 −0.726336 −0.363168 0.931724i \(-0.618305\pi\)
−0.363168 + 0.931724i \(0.618305\pi\)
\(570\) 0 0
\(571\) 31.8496 1.33286 0.666431 0.745567i \(-0.267821\pi\)
0.666431 + 0.745567i \(0.267821\pi\)
\(572\) 0 0
\(573\) 25.7889i 1.07735i
\(574\) 0 0
\(575\) 28.4749 3.51247i 1.18748 0.146480i
\(576\) 0 0
\(577\) 20.5139i 0.854004i 0.904251 + 0.427002i \(0.140430\pi\)
−0.904251 + 0.427002i \(0.859570\pi\)
\(578\) 0 0
\(579\) 15.3258 0.636920
\(580\) 0 0
\(581\) −3.22425 −0.133765
\(582\) 0 0
\(583\) 6.55008i 0.271277i
\(584\) 0 0
\(585\) −11.6629 10.3127i −0.482202 0.426376i
\(586\) 0 0
\(587\) 20.8773i 0.861699i −0.902424 0.430850i \(-0.858214\pi\)
0.902424 0.430850i \(-0.141786\pi\)
\(588\) 0 0
\(589\) −3.87873 −0.159820
\(590\) 0 0
\(591\) −0.513881 −0.0211382
\(592\) 0 0
\(593\) 5.57593i 0.228976i −0.993425 0.114488i \(-0.963477\pi\)
0.993425 0.114488i \(-0.0365227\pi\)
\(594\) 0 0
\(595\) 9.73813 11.0132i 0.399225 0.451496i
\(596\) 0 0
\(597\) 3.73813i 0.152992i
\(598\) 0 0
\(599\) −1.01317 −0.0413972 −0.0206986 0.999786i \(-0.506589\pi\)
−0.0206986 + 0.999786i \(0.506589\pi\)
\(600\) 0 0
\(601\) −20.2981 −0.827975 −0.413988 0.910283i \(-0.635864\pi\)
−0.413988 + 0.910283i \(0.635864\pi\)
\(602\) 0 0
\(603\) 4.77575i 0.194483i
\(604\) 0 0
\(605\) 42.7318 48.3268i 1.73729 1.96476i
\(606\) 0 0
\(607\) 14.9525i 0.606905i 0.952847 + 0.303452i \(0.0981393\pi\)
−0.952847 + 0.303452i \(0.901861\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 19.0738i 0.770384i 0.922836 + 0.385192i \(0.125865\pi\)
−0.922836 + 0.385192i \(0.874135\pi\)
\(614\) 0 0
\(615\) 11.6629 + 10.3127i 0.470294 + 0.415846i
\(616\) 0 0
\(617\) 15.8886i 0.639650i 0.947477 + 0.319825i \(0.103624\pi\)
−0.947477 + 0.319825i \(0.896376\pi\)
\(618\) 0 0
\(619\) −32.7367 −1.31580 −0.657900 0.753105i \(-0.728555\pi\)
−0.657900 + 0.753105i \(0.728555\pi\)
\(620\) 0 0
\(621\) −5.73813 −0.230263
\(622\) 0 0
\(623\) 2.18664i 0.0876060i
\(624\) 0 0
\(625\) −24.2506 + 6.07522i −0.970024 + 0.243009i
\(626\) 0 0
\(627\) 23.5975i 0.942395i
\(628\) 0 0
\(629\) −70.3508 −2.80507
\(630\) 0 0
\(631\) −40.7269 −1.62131 −0.810656 0.585523i \(-0.800889\pi\)
−0.810656 + 0.585523i \(0.800889\pi\)
\(632\) 0 0
\(633\) 3.22425i 0.128153i
\(634\) 0 0
\(635\) 4.52373 + 4.00000i 0.179519 + 0.158735i
\(636\) 0 0
\(637\) 6.96239i 0.275860i
\(638\) 0 0
\(639\) −8.23743 −0.325868
\(640\) 0 0
\(641\) −37.1754 −1.46834 −0.734170 0.678966i \(-0.762428\pi\)
−0.734170 + 0.678966i \(0.762428\pi\)
\(642\) 0 0
\(643\) 18.3996i 0.725611i 0.931865 + 0.362805i \(0.118181\pi\)
−0.931865 + 0.362805i \(0.881819\pi\)
\(644\) 0 0
\(645\) −8.77575 + 9.92478i −0.345545 + 0.390788i
\(646\) 0 0
\(647\) 15.2243i 0.598527i −0.954170 0.299264i \(-0.903259\pi\)
0.954170 0.299264i \(-0.0967410\pi\)
\(648\) 0 0
\(649\) 20.3536 0.798948
\(650\) 0 0
\(651\) −1.03761 −0.0406672
\(652\) 0 0
\(653\) 0.785595i 0.0307427i −0.999882 0.0153714i \(-0.995107\pi\)
0.999882 0.0153714i \(-0.00489305\pi\)
\(654\) 0 0
\(655\) −13.9248 + 15.7480i −0.544086 + 0.615325i
\(656\) 0 0
\(657\) 4.26187i 0.166271i
\(658\) 0 0
\(659\) 21.2652 0.828374 0.414187 0.910192i \(-0.364066\pi\)
0.414187 + 0.910192i \(0.364066\pi\)
\(660\) 0 0
\(661\) −32.4485 −1.26210 −0.631050 0.775742i \(-0.717376\pi\)
−0.631050 + 0.775742i \(0.717376\pi\)
\(662\) 0 0
\(663\) 45.7743i 1.77773i
\(664\) 0 0
\(665\) 6.26187 + 5.53690i 0.242825 + 0.214712i
\(666\) 0 0
\(667\) 11.4763i 0.444363i
\(668\) 0 0
\(669\) −11.8496 −0.458130
\(670\) 0 0
\(671\) −87.4274 −3.37510
\(672\) 0 0
\(673\) 5.55149i 0.213994i −0.994259 0.106997i \(-0.965876\pi\)
0.994259 0.106997i \(-0.0341236\pi\)
\(674\) 0 0
\(675\) 4.96239 0.612127i 0.191002 0.0235608i
\(676\) 0 0
\(677\) 7.72355i 0.296840i 0.988924 + 0.148420i \(0.0474187\pi\)
−0.988924 + 0.148420i \(0.952581\pi\)
\(678\) 0 0
\(679\) 3.73813 0.143456
\(680\) 0 0
\(681\) −14.7005 −0.563325
\(682\) 0 0
\(683\) 15.5125i 0.593568i −0.954945 0.296784i \(-0.904086\pi\)
0.954945 0.296784i \(-0.0959141\pi\)
\(684\) 0 0
\(685\) 16.1866 + 14.3127i 0.618460 + 0.546858i
\(686\) 0 0
\(687\) 9.47627i 0.361542i
\(688\) 0 0
\(689\) −7.22425 −0.275222
\(690\) 0 0
\(691\) −24.7367 −0.941029 −0.470515 0.882392i \(-0.655932\pi\)
−0.470515 + 0.882392i \(0.655932\pi\)
\(692\) 0 0
\(693\) 6.31265i 0.239798i
\(694\) 0 0
\(695\) −30.9380 + 34.9887i −1.17354 + 1.32720i
\(696\) 0 0
\(697\) 45.7743i 1.73383i
\(698\) 0 0
\(699\) 29.5125 1.11626
\(700\) 0 0
\(701\) −3.55149 −0.134138 −0.0670690 0.997748i \(-0.521365\pi\)
−0.0670690 + 0.997748i \(0.521365\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.66291i 0.363411i
\(708\) 0 0
\(709\) −17.6991 −0.664704 −0.332352 0.943155i \(-0.607842\pi\)
−0.332352 + 0.943155i \(0.607842\pi\)
\(710\) 0 0
\(711\) −5.92478 −0.222197
\(712\) 0 0
\(713\) 5.95395i 0.222977i
\(714\) 0 0
\(715\) 73.6239 + 65.1002i 2.75338 + 2.43461i
\(716\) 0 0
\(717\) 25.0132i 0.934134i
\(718\) 0 0
\(719\) 12.3733 0.461446 0.230723 0.973020i \(-0.425891\pi\)
0.230723 + 0.973020i \(0.425891\pi\)
\(720\) 0 0
\(721\) −1.29948 −0.0483950
\(722\) 0 0
\(723\) 7.92478i 0.294726i
\(724\) 0 0
\(725\) −1.22425 9.92478i −0.0454676 0.368597i
\(726\) 0 0
\(727\) 47.9511i 1.77841i 0.457510 + 0.889204i \(0.348741\pi\)
−0.457510 + 0.889204i \(0.651259\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −38.9525 −1.44071
\(732\) 0 0
\(733\) 37.9149i 1.40042i 0.713937 + 0.700210i \(0.246910\pi\)
−0.713937 + 0.700210i \(0.753090\pi\)
\(734\) 0 0
\(735\) 1.67513 + 1.48119i 0.0617881 + 0.0546347i
\(736\) 0 0
\(737\) 30.1476i 1.11050i
\(738\) 0 0
\(739\) −41.4010 −1.52296 −0.761481 0.648187i \(-0.775527\pi\)
−0.761481 + 0.648187i \(0.775527\pi\)
\(740\) 0 0
\(741\) 26.0263 0.956102
\(742\) 0 0
\(743\) 7.28963i 0.267430i 0.991020 + 0.133715i \(0.0426908\pi\)
−0.991020 + 0.133715i \(0.957309\pi\)
\(744\) 0 0
\(745\) −0.111420 + 0.126008i −0.00408210 + 0.00461658i
\(746\) 0 0
\(747\) 3.22425i 0.117969i
\(748\) 0 0
\(749\) −3.66291 −0.133840
\(750\) 0 0
\(751\) 4.12127 0.150387 0.0751936 0.997169i \(-0.476043\pi\)
0.0751936 + 0.997169i \(0.476043\pi\)
\(752\) 0 0
\(753\) 15.0738i 0.549320i
\(754\) 0 0
\(755\) 9.92478 11.2243i 0.361200 0.408492i
\(756\) 0 0
\(757\) 23.4471i 0.852199i 0.904676 + 0.426100i \(0.140113\pi\)
−0.904676 + 0.426100i \(0.859887\pi\)
\(758\) 0 0
\(759\) 36.2228 1.31481
\(760\) 0 0
\(761\) 10.7104 0.388251 0.194125 0.980977i \(-0.437813\pi\)
0.194125 + 0.980977i \(0.437813\pi\)
\(762\) 0 0
\(763\) 6.77575i 0.245298i
\(764\) 0 0
\(765\) 11.0132 + 9.73813i 0.398182 + 0.352083i
\(766\) 0 0
\(767\) 22.4485i 0.810569i
\(768\) 0 0
\(769\) 21.8496 0.787915 0.393958 0.919129i \(-0.371106\pi\)
0.393958 + 0.919129i \(0.371106\pi\)
\(770\) 0 0
\(771\) −22.0508 −0.794140
\(772\) 0 0
\(773\) 31.9756i 1.15008i −0.818125 0.575041i \(-0.804986\pi\)
0.818125 0.575041i \(-0.195014\pi\)
\(774\) 0 0
\(775\) 0.635150 + 5.14903i 0.0228153 + 0.184959i
\(776\) 0 0
\(777\) 10.7005i 0.383879i
\(778\) 0 0
\(779\) −26.0263 −0.932491
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) 9.48612 + 8.38787i 0.338574 + 0.299376i
\(786\) 0 0
\(787\) 29.2506i 1.04267i 0.853352 + 0.521336i \(0.174566\pi\)
−0.853352 + 0.521336i \(0.825434\pi\)
\(788\) 0 0
\(789\) 11.8134 0.420567
\(790\) 0 0
\(791\) 8.88717 0.315991
\(792\) 0 0
\(793\) 96.4260i 3.42419i
\(794\) 0 0
\(795\) 1.53690 1.73813i 0.0545084 0.0616453i
\(796\) 0 0
\(797\) 29.1246i 1.03165i 0.856695 + 0.515823i \(0.172514\pi\)
−0.856695 + 0.515823i \(0.827486\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.18664 0.0772612
\(802\) 0 0
\(803\) 26.9037i