Properties

Label 1040.2.k.c.961.5
Level $1040$
Weight $2$
Character 1040.961
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Defining polynomial: \(x^{6} + 12 x^{4} + 36 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.5
Root \(2.26180i\) of defining polynomial
Character \(\chi\) \(=\) 1040.961
Dual form 1040.2.k.c.961.6

$q$-expansion

\(f(q)\) \(=\) \(q+2.26180 q^{3} -1.00000i q^{5} -1.11575i q^{7} +2.11575 q^{9} +O(q^{10})\) \(q+2.26180 q^{3} -1.00000i q^{5} -1.11575i q^{7} +2.11575 q^{9} -5.37755i q^{11} +(-3.37755 + 1.26180i) q^{13} -2.26180i q^{15} -7.90116i q^{19} -2.52360i q^{21} +6.49330 q^{23} -1.00000 q^{25} -2.00000 q^{27} +3.63935 q^{29} +3.14605i q^{31} -12.1630i q^{33} -1.11575 q^{35} +7.40786i q^{37} +(-7.63935 + 2.85395i) q^{39} -4.75510i q^{41} +4.78541 q^{43} -2.11575i q^{45} +6.16296i q^{47} +5.75510 q^{49} +0.292106 q^{53} -5.37755 q^{55} -17.8709i q^{57} +11.3776i q^{59} -5.34725 q^{61} -2.36065i q^{63} +(1.26180 + 3.37755i) q^{65} -11.8709i q^{67} +14.6866 q^{69} -3.43816i q^{71} +4.59214i q^{73} -2.26180 q^{75} -6.00000 q^{77} +10.8157 q^{79} -10.8709 q^{81} +7.40786i q^{83} +8.23150 q^{87} +14.8157i q^{89} +(1.40786 + 3.76850i) q^{91} +7.11575i q^{93} -7.90116 q^{95} -3.76850i q^{97} -11.3776i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{9} + 12q^{23} - 6q^{25} - 12q^{27} - 12q^{29} - 12q^{39} - 12q^{43} - 6q^{49} - 12q^{53} - 12q^{55} - 12q^{61} - 6q^{65} - 36q^{77} + 24q^{79} - 18q^{81} + 36q^{87} - 12q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 2.26180 1.30585 0.652926 0.757422i \(-0.273541\pi\)
0.652926 + 0.757422i \(0.273541\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.11575i 0.421714i −0.977517 0.210857i \(-0.932375\pi\)
0.977517 0.210857i \(-0.0676254\pi\)
\(8\) 0 0
\(9\) 2.11575 0.705250
\(10\) 0 0
\(11\) 5.37755i 1.62139i −0.585467 0.810696i \(-0.699089\pi\)
0.585467 0.810696i \(-0.300911\pi\)
\(12\) 0 0
\(13\) −3.37755 + 1.26180i −0.936764 + 0.349961i
\(14\) 0 0
\(15\) 2.26180i 0.583995i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 7.90116i 1.81265i −0.422582 0.906325i \(-0.638876\pi\)
0.422582 0.906325i \(-0.361124\pi\)
\(20\) 0 0
\(21\) 2.52360i 0.550696i
\(22\) 0 0
\(23\) 6.49330 1.35395 0.676973 0.736007i \(-0.263291\pi\)
0.676973 + 0.736007i \(0.263291\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −2.00000 −0.384900
\(28\) 0 0
\(29\) 3.63935 0.675811 0.337906 0.941180i \(-0.390282\pi\)
0.337906 + 0.941180i \(0.390282\pi\)
\(30\) 0 0
\(31\) 3.14605i 0.565048i 0.959260 + 0.282524i \(0.0911716\pi\)
−0.959260 + 0.282524i \(0.908828\pi\)
\(32\) 0 0
\(33\) 12.1630i 2.11730i
\(34\) 0 0
\(35\) −1.11575 −0.188596
\(36\) 0 0
\(37\) 7.40786i 1.21784i 0.793230 + 0.608922i \(0.208398\pi\)
−0.793230 + 0.608922i \(0.791602\pi\)
\(38\) 0 0
\(39\) −7.63935 + 2.85395i −1.22328 + 0.456997i
\(40\) 0 0
\(41\) 4.75510i 0.742622i −0.928508 0.371311i \(-0.878908\pi\)
0.928508 0.371311i \(-0.121092\pi\)
\(42\) 0 0
\(43\) 4.78541 0.729768 0.364884 0.931053i \(-0.381109\pi\)
0.364884 + 0.931053i \(0.381109\pi\)
\(44\) 0 0
\(45\) 2.11575i 0.315397i
\(46\) 0 0
\(47\) 6.16296i 0.898960i 0.893290 + 0.449480i \(0.148391\pi\)
−0.893290 + 0.449480i \(0.851609\pi\)
\(48\) 0 0
\(49\) 5.75510 0.822158
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.292106 0.0401238 0.0200619 0.999799i \(-0.493614\pi\)
0.0200619 + 0.999799i \(0.493614\pi\)
\(54\) 0 0
\(55\) −5.37755 −0.725109
\(56\) 0 0
\(57\) 17.8709i 2.36705i
\(58\) 0 0
\(59\) 11.3776i 1.48123i 0.671929 + 0.740616i \(0.265466\pi\)
−0.671929 + 0.740616i \(0.734534\pi\)
\(60\) 0 0
\(61\) −5.34725 −0.684645 −0.342322 0.939583i \(-0.611214\pi\)
−0.342322 + 0.939583i \(0.611214\pi\)
\(62\) 0 0
\(63\) 2.36065i 0.297413i
\(64\) 0 0
\(65\) 1.26180 + 3.37755i 0.156507 + 0.418934i
\(66\) 0 0
\(67\) 11.8709i 1.45026i −0.688614 0.725128i \(-0.741781\pi\)
0.688614 0.725128i \(-0.258219\pi\)
\(68\) 0 0
\(69\) 14.6866 1.76805
\(70\) 0 0
\(71\) 3.43816i 0.408034i −0.978967 0.204017i \(-0.934600\pi\)
0.978967 0.204017i \(-0.0653998\pi\)
\(72\) 0 0
\(73\) 4.59214i 0.537470i 0.963214 + 0.268735i \(0.0866056\pi\)
−0.963214 + 0.268735i \(0.913394\pi\)
\(74\) 0 0
\(75\) −2.26180 −0.261170
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 10.8157 1.21686 0.608431 0.793607i \(-0.291799\pi\)
0.608431 + 0.793607i \(0.291799\pi\)
\(80\) 0 0
\(81\) −10.8709 −1.20787
\(82\) 0 0
\(83\) 7.40786i 0.813118i 0.913625 + 0.406559i \(0.133271\pi\)
−0.913625 + 0.406559i \(0.866729\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.23150 0.882509
\(88\) 0 0
\(89\) 14.8157i 1.57046i 0.619203 + 0.785231i \(0.287456\pi\)
−0.619203 + 0.785231i \(0.712544\pi\)
\(90\) 0 0
\(91\) 1.40786 + 3.76850i 0.147583 + 0.395046i
\(92\) 0 0
\(93\) 7.11575i 0.737869i
\(94\) 0 0
\(95\) −7.90116 −0.810642
\(96\) 0 0
\(97\) 3.76850i 0.382633i −0.981528 0.191317i \(-0.938724\pi\)
0.981528 0.191317i \(-0.0612757\pi\)
\(98\) 0 0
\(99\) 11.3776i 1.14349i
\(100\) 0 0
\(101\) −0.292106 −0.0290656 −0.0145328 0.999894i \(-0.504626\pi\)
−0.0145328 + 0.999894i \(0.504626\pi\)
\(102\) 0 0
\(103\) −4.03030 −0.397118 −0.198559 0.980089i \(-0.563626\pi\)
−0.198559 + 0.980089i \(0.563626\pi\)
\(104\) 0 0
\(105\) −2.52360 −0.246279
\(106\) 0 0
\(107\) −5.50670 −0.532353 −0.266176 0.963924i \(-0.585760\pi\)
−0.266176 + 0.963924i \(0.585760\pi\)
\(108\) 0 0
\(109\) 1.27871i 0.122478i 0.998123 + 0.0612390i \(0.0195052\pi\)
−0.998123 + 0.0612390i \(0.980495\pi\)
\(110\) 0 0
\(111\) 16.7551i 1.59032i
\(112\) 0 0
\(113\) 19.2787 1.81359 0.906794 0.421574i \(-0.138522\pi\)
0.906794 + 0.421574i \(0.138522\pi\)
\(114\) 0 0
\(115\) 6.49330i 0.605503i
\(116\) 0 0
\(117\) −7.14605 + 2.66966i −0.660653 + 0.246810i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −17.9181 −1.62891
\(122\) 0 0
\(123\) 10.7551i 0.969755i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −7.96970 −0.707196 −0.353598 0.935397i \(-0.615042\pi\)
−0.353598 + 0.935397i \(0.615042\pi\)
\(128\) 0 0
\(129\) 10.8236 0.952969
\(130\) 0 0
\(131\) −7.27871 −0.635944 −0.317972 0.948100i \(-0.603002\pi\)
−0.317972 + 0.948100i \(0.603002\pi\)
\(132\) 0 0
\(133\) −8.81571 −0.764419
\(134\) 0 0
\(135\) 2.00000i 0.172133i
\(136\) 0 0
\(137\) 4.75510i 0.406256i 0.979152 + 0.203128i \(0.0651107\pi\)
−0.979152 + 0.203128i \(0.934889\pi\)
\(138\) 0 0
\(139\) 16.3259 1.38475 0.692373 0.721540i \(-0.256565\pi\)
0.692373 + 0.721540i \(0.256565\pi\)
\(140\) 0 0
\(141\) 13.9394i 1.17391i
\(142\) 0 0
\(143\) 6.78541 + 18.1630i 0.567424 + 1.51886i
\(144\) 0 0
\(145\) 3.63935i 0.302232i
\(146\) 0 0
\(147\) 13.0169 1.07362
\(148\) 0 0
\(149\) 8.81571i 0.722211i −0.932525 0.361106i \(-0.882399\pi\)
0.932525 0.361106i \(-0.117601\pi\)
\(150\) 0 0
\(151\) 19.9012i 1.61953i −0.586752 0.809767i \(-0.699594\pi\)
0.586752 0.809767i \(-0.300406\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.14605 0.252697
\(156\) 0 0
\(157\) 18.7551 1.49682 0.748410 0.663236i \(-0.230818\pi\)
0.748410 + 0.663236i \(0.230818\pi\)
\(158\) 0 0
\(159\) 0.660685 0.0523958
\(160\) 0 0
\(161\) 7.24490i 0.570978i
\(162\) 0 0
\(163\) 13.4417i 1.05283i −0.850227 0.526416i \(-0.823535\pi\)
0.850227 0.526416i \(-0.176465\pi\)
\(164\) 0 0
\(165\) −12.1630 −0.946885
\(166\) 0 0
\(167\) 4.91806i 0.380571i 0.981729 + 0.190286i \(0.0609414\pi\)
−0.981729 + 0.190286i \(0.939059\pi\)
\(168\) 0 0
\(169\) 9.81571 8.52360i 0.755055 0.655662i
\(170\) 0 0
\(171\) 16.7169i 1.27837i
\(172\) 0 0
\(173\) −19.5708 −1.48794 −0.743971 0.668212i \(-0.767060\pi\)
−0.743971 + 0.668212i \(0.767060\pi\)
\(174\) 0 0
\(175\) 1.11575i 0.0843427i
\(176\) 0 0
\(177\) 25.7338i 1.93427i
\(178\) 0 0
\(179\) 12.9866 0.970664 0.485332 0.874330i \(-0.338699\pi\)
0.485332 + 0.874330i \(0.338699\pi\)
\(180\) 0 0
\(181\) 9.40786 0.699280 0.349640 0.936884i \(-0.386304\pi\)
0.349640 + 0.936884i \(0.386304\pi\)
\(182\) 0 0
\(183\) −12.0944 −0.894045
\(184\) 0 0
\(185\) 7.40786 0.544636
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.23150i 0.162318i
\(190\) 0 0
\(191\) −26.5574 −1.92163 −0.960814 0.277195i \(-0.910595\pi\)
−0.960814 + 0.277195i \(0.910595\pi\)
\(192\) 0 0
\(193\) 21.8023i 1.56936i 0.619898 + 0.784682i \(0.287174\pi\)
−0.619898 + 0.784682i \(0.712826\pi\)
\(194\) 0 0
\(195\) 2.85395 + 7.63935i 0.204375 + 0.547065i
\(196\) 0 0
\(197\) 7.57081i 0.539398i −0.962945 0.269699i \(-0.913076\pi\)
0.962945 0.269699i \(-0.0869242\pi\)
\(198\) 0 0
\(199\) −17.5708 −1.24556 −0.622781 0.782396i \(-0.713997\pi\)
−0.622781 + 0.782396i \(0.713997\pi\)
\(200\) 0 0
\(201\) 26.8495i 1.89382i
\(202\) 0 0
\(203\) 4.06061i 0.284999i
\(204\) 0 0
\(205\) −4.75510 −0.332111
\(206\) 0 0
\(207\) 13.7382 0.954871
\(208\) 0 0
\(209\) −42.4889 −2.93902
\(210\) 0 0
\(211\) 0.0606069 0.00417235 0.00208618 0.999998i \(-0.499336\pi\)
0.00208618 + 0.999998i \(0.499336\pi\)
\(212\) 0 0
\(213\) 7.77643i 0.532833i
\(214\) 0 0
\(215\) 4.78541i 0.326362i
\(216\) 0 0
\(217\) 3.51021 0.238288
\(218\) 0 0
\(219\) 10.3865i 0.701856i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 22.6260i 1.51515i 0.652750 + 0.757573i \(0.273615\pi\)
−0.652750 + 0.757573i \(0.726385\pi\)
\(224\) 0 0
\(225\) −2.11575 −0.141050
\(226\) 0 0
\(227\) 4.59214i 0.304791i −0.988320 0.152396i \(-0.951301\pi\)
0.988320 0.152396i \(-0.0486988\pi\)
\(228\) 0 0
\(229\) 0.986602i 0.0651965i −0.999469 0.0325983i \(-0.989622\pi\)
0.999469 0.0325983i \(-0.0103782\pi\)
\(230\) 0 0
\(231\) −13.5708 −0.892894
\(232\) 0 0
\(233\) 13.2787 0.869917 0.434959 0.900450i \(-0.356763\pi\)
0.434959 + 0.900450i \(0.356763\pi\)
\(234\) 0 0
\(235\) 6.16296 0.402027
\(236\) 0 0
\(237\) 24.4630 1.58904
\(238\) 0 0
\(239\) 25.6429i 1.65870i 0.558730 + 0.829349i \(0.311289\pi\)
−0.558730 + 0.829349i \(0.688711\pi\)
\(240\) 0 0
\(241\) 14.5236i 0.935548i 0.883848 + 0.467774i \(0.154944\pi\)
−0.883848 + 0.467774i \(0.845056\pi\)
\(242\) 0 0
\(243\) −18.5877 −1.19240
\(244\) 0 0
\(245\) 5.75510i 0.367680i
\(246\) 0 0
\(247\) 9.96970 + 26.6866i 0.634357 + 1.69803i
\(248\) 0 0
\(249\) 16.7551i 1.06181i
\(250\) 0 0
\(251\) −20.2653 −1.27914 −0.639568 0.768735i \(-0.720887\pi\)
−0.639568 + 0.768735i \(0.720887\pi\)
\(252\) 0 0
\(253\) 34.9181i 2.19528i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.58421 0.410712 0.205356 0.978687i \(-0.434165\pi\)
0.205356 + 0.978687i \(0.434165\pi\)
\(258\) 0 0
\(259\) 8.26531 0.513581
\(260\) 0 0
\(261\) 7.69996 0.476616
\(262\) 0 0
\(263\) 18.7854 1.15836 0.579179 0.815200i \(-0.303373\pi\)
0.579179 + 0.815200i \(0.303373\pi\)
\(264\) 0 0
\(265\) 0.292106i 0.0179439i
\(266\) 0 0
\(267\) 33.5102i 2.05079i
\(268\) 0 0
\(269\) −14.8495 −0.905391 −0.452696 0.891665i \(-0.649538\pi\)
−0.452696 + 0.891665i \(0.649538\pi\)
\(270\) 0 0
\(271\) 20.8878i 1.26884i 0.772988 + 0.634420i \(0.218761\pi\)
−0.772988 + 0.634420i \(0.781239\pi\)
\(272\) 0 0
\(273\) 3.18429 + 8.52360i 0.192722 + 0.515872i
\(274\) 0 0
\(275\) 5.37755i 0.324279i
\(276\) 0 0
\(277\) 22.2653 1.33779 0.668896 0.743356i \(-0.266767\pi\)
0.668896 + 0.743356i \(0.266767\pi\)
\(278\) 0 0
\(279\) 6.65626i 0.398500i
\(280\) 0 0
\(281\) 7.57081i 0.451637i 0.974169 + 0.225818i \(0.0725056\pi\)
−0.974169 + 0.225818i \(0.927494\pi\)
\(282\) 0 0
\(283\) −0.724800 −0.0430849 −0.0215424 0.999768i \(-0.506858\pi\)
−0.0215424 + 0.999768i \(0.506858\pi\)
\(284\) 0 0
\(285\) −17.8709 −1.05858
\(286\) 0 0
\(287\) −5.30550 −0.313174
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 8.52360i 0.499663i
\(292\) 0 0
\(293\) 10.2236i 0.597267i 0.954368 + 0.298634i \(0.0965308\pi\)
−0.954368 + 0.298634i \(0.903469\pi\)
\(294\) 0 0
\(295\) 11.3776 0.662427
\(296\) 0 0
\(297\) 10.7551i 0.624074i
\(298\) 0 0
\(299\) −21.9315 + 8.19326i −1.26833 + 0.473829i
\(300\) 0 0
\(301\) 5.33931i 0.307753i
\(302\) 0 0
\(303\) −0.660685 −0.0379554
\(304\) 0 0
\(305\) 5.34725i 0.306183i
\(306\) 0 0
\(307\) 21.6394i 1.23502i −0.786562 0.617512i \(-0.788141\pi\)
0.786562 0.617512i \(-0.211859\pi\)
\(308\) 0 0
\(309\) −9.11575 −0.518577
\(310\) 0 0
\(311\) −0.986602 −0.0559451 −0.0279725 0.999609i \(-0.508905\pi\)
−0.0279725 + 0.999609i \(0.508905\pi\)
\(312\) 0 0
\(313\) 25.8361 1.46034 0.730172 0.683263i \(-0.239440\pi\)
0.730172 + 0.683263i \(0.239440\pi\)
\(314\) 0 0
\(315\) −2.36065 −0.133007
\(316\) 0 0
\(317\) 26.4283i 1.48436i 0.670201 + 0.742180i \(0.266208\pi\)
−0.670201 + 0.742180i \(0.733792\pi\)
\(318\) 0 0
\(319\) 19.5708i 1.09576i
\(320\) 0 0
\(321\) −12.4551 −0.695174
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.37755 1.26180i 0.187353 0.0699922i
\(326\) 0 0
\(327\) 2.89218i 0.159938i
\(328\) 0 0
\(329\) 6.87632 0.379104
\(330\) 0 0
\(331\) 14.8878i 0.818305i −0.912466 0.409153i \(-0.865824\pi\)
0.912466 0.409153i \(-0.134176\pi\)
\(332\) 0 0
\(333\) 15.6732i 0.858884i
\(334\) 0 0
\(335\) −11.8709 −0.648574
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) 43.6046 2.36828
\(340\) 0 0
\(341\) 16.9181 0.916164
\(342\) 0 0
\(343\) 14.2315i 0.768429i
\(344\) 0 0
\(345\) 14.6866i 0.790698i
\(346\) 0 0
\(347\) 26.0641 1.39919 0.699597 0.714537i \(-0.253363\pi\)
0.699597 + 0.714537i \(0.253363\pi\)
\(348\) 0 0
\(349\) 10.7551i 0.575707i −0.957674 0.287854i \(-0.907058\pi\)
0.957674 0.287854i \(-0.0929417\pi\)
\(350\) 0 0
\(351\) 6.75510 2.52360i 0.360561 0.134700i
\(352\) 0 0
\(353\) 35.6126i 1.89547i 0.319066 + 0.947733i \(0.396631\pi\)
−0.319066 + 0.947733i \(0.603369\pi\)
\(354\) 0 0
\(355\) −3.43816 −0.182479
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.07205i 0.320470i −0.987079 0.160235i \(-0.948775\pi\)
0.987079 0.160235i \(-0.0512253\pi\)
\(360\) 0 0
\(361\) −43.4283 −2.28570
\(362\) 0 0
\(363\) −40.5271 −2.12712
\(364\) 0 0
\(365\) 4.59214 0.240364
\(366\) 0 0
\(367\) −26.7854 −1.39819 −0.699093 0.715030i \(-0.746413\pi\)
−0.699093 + 0.715030i \(0.746413\pi\)
\(368\) 0 0
\(369\) 10.0606i 0.523734i
\(370\) 0 0
\(371\) 0.325917i 0.0169208i
\(372\) 0 0
\(373\) 10.8763 0.563154 0.281577 0.959539i \(-0.409142\pi\)
0.281577 + 0.959539i \(0.409142\pi\)
\(374\) 0 0
\(375\) 2.26180i 0.116799i
\(376\) 0 0
\(377\) −12.2921 + 4.59214i −0.633076 + 0.236507i
\(378\) 0 0
\(379\) 5.96176i 0.306235i 0.988208 + 0.153118i \(0.0489313\pi\)
−0.988208 + 0.153118i \(0.951069\pi\)
\(380\) 0 0
\(381\) −18.0259 −0.923494
\(382\) 0 0
\(383\) 19.4079i 0.991695i −0.868410 0.495848i \(-0.834857\pi\)
0.868410 0.495848i \(-0.165143\pi\)
\(384\) 0 0
\(385\) 6.00000i 0.305788i
\(386\) 0 0
\(387\) 10.1247 0.514669
\(388\) 0 0
\(389\) −8.55742 −0.433878 −0.216939 0.976185i \(-0.569607\pi\)
−0.216939 + 0.976185i \(0.569607\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −16.4630 −0.830448
\(394\) 0 0
\(395\) 10.8157i 0.544197i
\(396\) 0 0
\(397\) 14.9449i 0.750061i 0.927012 + 0.375030i \(0.122368\pi\)
−0.927012 + 0.375030i \(0.877632\pi\)
\(398\) 0 0
\(399\) −19.9394 −0.998218
\(400\) 0 0
\(401\) 27.1416i 1.35539i 0.735344 + 0.677694i \(0.237021\pi\)
−0.735344 + 0.677694i \(0.762979\pi\)
\(402\) 0 0
\(403\) −3.96970 10.6260i −0.197745 0.529317i
\(404\) 0 0
\(405\) 10.8709i 0.540177i
\(406\) 0 0
\(407\) 39.8361 1.97460
\(408\) 0 0
\(409\) 33.2181i 1.64253i −0.570547 0.821265i \(-0.693269\pi\)
0.570547 0.821265i \(-0.306731\pi\)
\(410\) 0 0
\(411\) 10.7551i 0.530510i
\(412\) 0 0
\(413\) 12.6945 0.624655
\(414\) 0 0
\(415\) 7.40786 0.363637
\(416\) 0 0
\(417\) 36.9260 1.80827
\(418\) 0 0
\(419\) 17.7079 0.865087 0.432544 0.901613i \(-0.357616\pi\)
0.432544 + 0.901613i \(0.357616\pi\)
\(420\) 0 0
\(421\) 22.2047i 1.08219i −0.840961 0.541096i \(-0.818010\pi\)
0.840961 0.541096i \(-0.181990\pi\)
\(422\) 0 0
\(423\) 13.0393i 0.633991i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.96619i 0.288724i
\(428\) 0 0
\(429\) 15.3472 + 41.0810i 0.740972 + 1.98341i
\(430\) 0 0
\(431\) 10.6831i 0.514585i 0.966334 + 0.257292i \(0.0828303\pi\)
−0.966334 + 0.257292i \(0.917170\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 8.23150i 0.394670i
\(436\) 0 0
\(437\) 51.3046i 2.45423i
\(438\) 0 0
\(439\) 5.24490 0.250325 0.125163 0.992136i \(-0.460055\pi\)
0.125163 + 0.992136i \(0.460055\pi\)
\(440\) 0 0
\(441\) 12.1764 0.579826
\(442\) 0 0
\(443\) −25.4799 −1.21059 −0.605293 0.796002i \(-0.706944\pi\)
−0.605293 + 0.796002i \(0.706944\pi\)
\(444\) 0 0
\(445\) 14.8157 0.702332
\(446\) 0 0
\(447\) 19.9394i 0.943101i
\(448\) 0 0
\(449\) 13.9394i 0.657841i 0.944358 + 0.328920i \(0.106685\pi\)
−0.944358 + 0.328920i \(0.893315\pi\)
\(450\) 0 0
\(451\) −25.5708 −1.20408
\(452\) 0 0
\(453\) 45.0125i 2.11487i
\(454\) 0 0
\(455\) 3.76850 1.40786i 0.176670 0.0660013i
\(456\) 0 0
\(457\) 12.6945i 0.593823i −0.954905 0.296912i \(-0.904043\pi\)
0.954905 0.296912i \(-0.0959567\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.0810i 1.07499i 0.843267 + 0.537495i \(0.180629\pi\)
−0.843267 + 0.537495i \(0.819371\pi\)
\(462\) 0 0
\(463\) 6.16296i 0.286417i 0.989693 + 0.143208i \(0.0457419\pi\)
−0.989693 + 0.143208i \(0.954258\pi\)
\(464\) 0 0
\(465\) 7.11575 0.329985
\(466\) 0 0
\(467\) 23.5067 1.08776 0.543880 0.839163i \(-0.316955\pi\)
0.543880 + 0.839163i \(0.316955\pi\)
\(468\) 0 0
\(469\) −13.2449 −0.611593
\(470\) 0 0
\(471\) 42.4203 1.95463
\(472\) 0 0
\(473\) 25.7338i 1.18324i
\(474\) 0 0
\(475\) 7.90116i 0.362530i
\(476\) 0 0
\(477\) 0.618022 0.0282973
\(478\) 0 0
\(479\) 33.2137i 1.51757i −0.651340 0.758786i \(-0.725793\pi\)
0.651340 0.758786i \(-0.274207\pi\)
\(480\) 0 0
\(481\) −9.34725 25.0204i −0.426198 1.14083i
\(482\) 0 0
\(483\) 16.3865i 0.745613i
\(484\) 0 0
\(485\) −3.76850 −0.171119
\(486\) 0 0
\(487\) 18.7472i 0.849515i −0.905307 0.424758i \(-0.860359\pi\)
0.905307 0.424758i \(-0.139641\pi\)
\(488\) 0 0
\(489\) 30.4024i 1.37484i
\(490\) 0 0
\(491\) −26.5574 −1.19852 −0.599260 0.800555i \(-0.704538\pi\)
−0.599260 + 0.800555i \(0.704538\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −11.3776 −0.511383
\(496\) 0 0
\(497\) −3.83612 −0.172074
\(498\) 0 0
\(499\) 26.8539i 1.20215i −0.799193 0.601074i \(-0.794740\pi\)
0.799193 0.601074i \(-0.205260\pi\)
\(500\) 0 0
\(501\) 11.1237i 0.496970i
\(502\) 0 0
\(503\) 20.4665 0.912556 0.456278 0.889837i \(-0.349182\pi\)
0.456278 + 0.889837i \(0.349182\pi\)
\(504\) 0 0
\(505\) 0.292106i 0.0129985i
\(506\) 0 0
\(507\) 22.2012 19.2787i 0.985990 0.856197i
\(508\) 0 0
\(509\) 26.9598i 1.19497i −0.801879 0.597486i \(-0.796166\pi\)
0.801879 0.597486i \(-0.203834\pi\)
\(510\) 0 0
\(511\) 5.12368 0.226658
\(512\) 0 0
\(513\) 15.8023i 0.697689i
\(514\) 0 0
\(515\) 4.03030i 0.177596i
\(516\) 0 0
\(517\) 33.1416 1.45757
\(518\) 0 0
\(519\) −44.2653 −1.94303
\(520\) 0 0
\(521\) −8.36065 −0.366287 −0.183143 0.983086i \(-0.558627\pi\)
−0.183143 + 0.983086i \(0.558627\pi\)
\(522\) 0 0
\(523\) 18.7248 0.818778 0.409389 0.912360i \(-0.365742\pi\)
0.409389 + 0.912360i \(0.365742\pi\)
\(524\) 0 0
\(525\) 2.52360i 0.110139i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.1630 0.833172
\(530\) 0 0
\(531\) 24.0720i 1.04464i
\(532\) 0 0
\(533\) 6.00000 + 16.0606i 0.259889 + 0.695662i
\(534\) 0 0
\(535\) 5.50670i 0.238075i
\(536\) 0 0
\(537\) 29.3731 1.26754
\(538\) 0 0
\(539\) 30.9484i 1.33304i
\(540\) 0 0
\(541\) 42.9439i 1.84630i −0.384435 0.923152i \(-0.625604\pi\)
0.384435 0.923152i \(-0.374396\pi\)
\(542\) 0 0
\(543\) 21.2787 0.913157
\(544\) 0 0
\(545\) 1.27871 0.0547738
\(546\) 0 0
\(547\) −14.0909 −0.602484 −0.301242 0.953548i \(-0.597401\pi\)
−0.301242 + 0.953548i \(0.597401\pi\)
\(548\) 0 0
\(549\) −11.3134 −0.482846
\(550\) 0 0
\(551\) 28.7551i 1.22501i
\(552\) 0 0
\(553\) 12.0676i 0.513167i
\(554\) 0 0
\(555\) 16.7551 0.711215
\(556\) 0 0
\(557\) 0.713359i 0.0302260i −0.999886 0.0151130i \(-0.995189\pi\)
0.999886 0.0151130i \(-0.00481080\pi\)
\(558\) 0 0
\(559\) −16.1630 + 6.03824i −0.683620 + 0.255390i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.2012 0.514219 0.257110 0.966382i \(-0.417230\pi\)
0.257110 + 0.966382i \(0.417230\pi\)
\(564\) 0 0
\(565\) 19.2787i 0.811061i
\(566\) 0 0
\(567\) 12.1291i 0.509376i
\(568\) 0 0
\(569\) 16.6260 0.696996 0.348498 0.937309i \(-0.386692\pi\)
0.348498 + 0.937309i \(0.386692\pi\)
\(570\) 0 0
\(571\) 20.3259 0.850613 0.425307 0.905049i \(-0.360166\pi\)
0.425307 + 0.905049i \(0.360166\pi\)
\(572\) 0 0
\(573\) −60.0676 −2.50936
\(574\) 0 0
\(575\) −6.49330 −0.270789
\(576\) 0 0
\(577\) 19.1496i 0.797207i −0.917123 0.398603i \(-0.869495\pi\)
0.917123 0.398603i \(-0.130505\pi\)
\(578\) 0 0
\(579\) 49.3125i 2.04936i
\(580\) 0 0
\(581\) 8.26531 0.342903
\(582\) 0 0
\(583\) 1.57081i 0.0650564i
\(584\) 0 0
\(585\) 2.66966 + 7.14605i 0.110377 + 0.295453i
\(586\) 0 0
\(587\) 30.4889i 1.25841i 0.777239 + 0.629205i \(0.216620\pi\)
−0.777239 + 0.629205i \(0.783380\pi\)
\(588\) 0 0
\(589\) 24.8575 1.02423
\(590\) 0 0
\(591\) 17.1237i 0.704374i
\(592\) 0 0
\(593\) 42.3259i 1.73812i −0.494709 0.869059i \(-0.664725\pi\)
0.494709 0.869059i \(-0.335275\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −39.7417 −1.62652
\(598\) 0 0
\(599\) −15.5440 −0.635111 −0.317556 0.948240i \(-0.602862\pi\)
−0.317556 + 0.948240i \(0.602862\pi\)
\(600\) 0 0
\(601\) 25.1416 1.02555 0.512774 0.858524i \(-0.328618\pi\)
0.512774 + 0.858524i \(0.328618\pi\)
\(602\) 0 0
\(603\) 25.1157i 1.02279i
\(604\) 0 0
\(605\) 17.9181i 0.728473i
\(606\) 0 0
\(607\) 10.0909 0.409577 0.204789 0.978806i \(-0.434349\pi\)
0.204789 + 0.978806i \(0.434349\pi\)
\(608\) 0 0
\(609\) 9.18429i 0.372166i
\(610\) 0 0
\(611\) −7.77643 20.8157i −0.314601 0.842113i
\(612\) 0 0
\(613\) 37.0204i 1.49524i −0.664127 0.747620i \(-0.731196\pi\)
0.664127 0.747620i \(-0.268804\pi\)
\(614\) 0 0
\(615\) −10.7551 −0.433688
\(616\) 0 0
\(617\) 8.63389i 0.347587i −0.984782 0.173794i \(-0.944397\pi\)
0.984782 0.173794i \(-0.0556026\pi\)
\(618\) 0 0
\(619\) 9.47197i 0.380711i 0.981715 + 0.190355i \(0.0609640\pi\)
−0.981715 + 0.190355i \(0.939036\pi\)
\(620\) 0 0
\(621\) −12.9866 −0.521134
\(622\) 0 0
\(623\) 16.5306 0.662285
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −96.1014 −3.83792
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 18.0382i 0.718091i −0.933320 0.359045i \(-0.883102\pi\)
0.933320 0.359045i \(-0.116898\pi\)
\(632\) 0 0
\(633\) 0.137081 0.00544847
\(634\) 0 0
\(635\) 7.96970i 0.316268i
\(636\) 0 0
\(637\) −19.4382 + 7.26180i −0.770168 + 0.287723i
\(638\) 0 0
\(639\) 7.27428i 0.287766i
\(640\) 0 0
\(641\) 17.4158 0.687882 0.343941 0.938991i \(-0.388238\pi\)
0.343941 + 0.938991i \(0.388238\pi\)
\(642\) 0 0
\(643\) 15.3472i 0.605236i 0.953112 + 0.302618i \(0.0978607\pi\)
−0.953112 + 0.302618i \(0.902139\pi\)
\(644\) 0 0
\(645\) 10.8236i 0.426181i
\(646\) 0 0
\(647\) 3.05072 0.119936 0.0599680 0.998200i \(-0.480900\pi\)
0.0599680 + 0.998200i \(0.480900\pi\)
\(648\) 0 0
\(649\) 61.1834 2.40166
\(650\) 0 0
\(651\) 7.93939 0.311169
\(652\) 0 0
\(653\) −25.5708 −1.00066 −0.500332 0.865834i \(-0.666789\pi\)
−0.500332 + 0.865834i \(0.666789\pi\)
\(654\) 0 0
\(655\) 7.27871i 0.284403i
\(656\) 0 0
\(657\) 9.71583i 0.379051i
\(658\) 0 0
\(659\) −6.29211 −0.245106 −0.122553 0.992462i \(-0.539108\pi\)
−0.122553 + 0.992462i \(0.539108\pi\)
\(660\) 0 0
\(661\) 39.5102i 1.53677i −0.639989 0.768384i \(-0.721061\pi\)
0.639989 0.768384i \(-0.278939\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.81571i 0.341859i
\(666\) 0 0
\(667\) 23.6314 0.915012
\(668\) 0 0
\(669\) 51.1754i 1.97856i
\(670\) 0 0
\(671\) 28.7551i 1.11008i
\(672\) 0 0
\(673\) 1.30550 0.0503235 0.0251617 0.999683i \(-0.491990\pi\)
0.0251617 + 0.999683i \(0.491990\pi\)
\(674\) 0 0
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 5.01340 0.192681 0.0963403 0.995348i \(-0.469286\pi\)
0.0963403 + 0.995348i \(0.469286\pi\)
\(678\) 0 0
\(679\) −4.20470 −0.161362
\(680\) 0 0
\(681\) 10.3865i 0.398012i
\(682\) 0 0
\(683\) 19.4079i 0.742621i 0.928509 + 0.371310i \(0.121091\pi\)
−0.928509 + 0.371310i \(0.878909\pi\)
\(684\) 0 0
\(685\) 4.75510 0.181683
\(686\) 0 0
\(687\) 2.23150i 0.0851370i
\(688\) 0 0
\(689\) −0.986602 + 0.368580i −0.0375865 + 0.0140418i
\(690\) 0 0
\(691\) 2.56184i 0.0974570i −0.998812 0.0487285i \(-0.984483\pi\)
0.998812 0.0487285i \(-0.0155169\pi\)
\(692\) 0 0
\(693\) −12.6945 −0.482224
\(694\) 0 0
\(695\) 16.3259i 0.619277i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 30.0338 1.13598
\(700\) 0 0
\(701\) 38.8495 1.46733 0.733663 0.679513i \(-0.237809\pi\)
0.733663 + 0.679513i \(0.237809\pi\)
\(702\) 0 0
\(703\) 58.5306 2.20752
\(704\) 0 0
\(705\) 13.9394 0.524988
\(706\) 0 0
\(707\) 0.325917i 0.0122574i
\(708\) 0 0
\(709\) 35.3393i 1.32720i 0.748089 + 0.663598i \(0.230971\pi\)
−0.748089 + 0.663598i \(0.769029\pi\)
\(710\) 0 0
\(711\) 22.8833 0.858192
\(712\) 0 0
\(713\) 20.4283i 0.765045i
\(714\) 0 0
\(715\) 18.1630 6.78541i 0.679256 0.253760i
\(716\) 0 0
\(717\) 57.9991i 2.16602i
\(718\) 0 0
\(719\) −6.11028 −0.227875 −0.113938 0.993488i \(-0.536346\pi\)
−0.113938 + 0.993488i \(0.536346\pi\)
\(720\) 0 0
\(721\) 4.49681i 0.167470i
\(722\) 0 0
\(723\) 32.8495i 1.22169i
\(724\) 0 0
\(725\) −3.63935 −0.135162
\(726\) 0 0
\(727\) −1.05072 −0.0389689 −0.0194845 0.999810i \(-0.506202\pi\)
−0.0194845 + 0.999810i \(0.506202\pi\)
\(728\) 0 0
\(729\) −9.42919 −0.349229
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 31.5708i 1.16609i 0.812438 + 0.583047i \(0.198140\pi\)
−0.812438 + 0.583047i \(0.801860\pi\)
\(734\) 0 0
\(735\) 13.0169i 0.480136i
\(736\) 0 0
\(737\) −63.8361 −2.35143
\(738\) 0 0
\(739\) 31.8673i 1.17226i 0.810217 + 0.586130i \(0.199349\pi\)
−0.810217 + 0.586130i \(0.800651\pi\)
\(740\) 0 0
\(741\) 22.5495 + 60.3597i 0.828376 + 2.21737i
\(742\) 0 0
\(743\) 39.8550i 1.46214i 0.682304 + 0.731069i \(0.260978\pi\)
−0.682304 + 0.731069i \(0.739022\pi\)
\(744\) 0 0
\(745\) −8.81571 −0.322983
\(746\) 0 0
\(747\) 15.6732i 0.573451i
\(748\) 0 0
\(749\) 6.14410i 0.224500i
\(750\) 0 0
\(751\) −9.24490 −0.337351 −0.168676 0.985672i \(-0.553949\pi\)
−0.168676 + 0.985672i \(0.553949\pi\)
\(752\) 0 0
\(753\) −45.8361 −1.67036
\(754\) 0 0
\(755\) −19.9012 −0.724277
\(756\) 0 0
\(757\) −22.5912 −0.821092 −0.410546 0.911840i \(-0.634662\pi\)
−0.410546 + 0.911840i \(0.634662\pi\)
\(758\) 0 0
\(759\) 78.9778i 2.86671i
\(760\) 0 0
\(761\) 16.2047i 0.587420i −0.955895 0.293710i \(-0.905110\pi\)
0.955895 0.293710i \(-0.0948900\pi\)
\(762\) 0 0
\(763\) 1.42672 0.0516506
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.3562 38.4283i −0.518373 1.38756i
\(768\) 0 0
\(769\) 16.7889i 0.605424i 0.953082 + 0.302712i \(0.0978920\pi\)
−0.953082 + 0.302712i \(0.902108\pi\)
\(770\) 0 0
\(771\) 14.8922 0.536329
\(772\) 0 0
\(773\) 16.5921i 0.596778i 0.954444 + 0.298389i \(0.0964493\pi\)
−0.954444 + 0.298389i \(0.903551\pi\)
\(774\) 0 0
\(775\) 3.14605i 0.113010i
\(776\) 0 0
\(777\) 18.6945 0.670661
\(778\) 0 0
\(779\) −37.5708 −1.34611
\(780\) 0 0
\(781\) −18.4889 −0.661584
\(782\) 0 0
\(783\) −7.27871 −0.260120
\(784\) 0 0
\(785\) 18.7551i 0.669398i
\(786\) 0 0
\(787\) 31.1496i 1.11036i −0.831730 0.555181i \(-0.812649\pi\)
0.831730 0.555181i \(-0.187351\pi\)
\(788\) 0 0
\(789\) 42.4889 1.51264
\(790\) 0 0
\(791\) 21.5102i 0.764815i
\(792\) 0 0
\(793\) 18.0606 6.74717i 0.641351 0.239599i
\(794\) 0 0
\(795\) 0.660685i 0.0234321i
\(796\) 0 0
\(797\) −0.402391 −0.0142534 −0.00712670 0.999975i \(-0.502269\pi\)
−0.00712670 + 0.999975i \(0.502269\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 31.3463i 1.10757i
\(802\) 0 0
\(803\) 24.6945 0.871450
\(804\) 0 0
\(805\) −7.24490 −0.255349