Properties

Label 3744.2.m.h.1585.4
Level $3744$
Weight $2$
Character 3744.1585
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1585.4
Root \(0.0783900 + 1.17295i\) of defining polynomial
Character \(\chi\) \(=\) 3744.1585
Dual form 3744.2.m.h.1585.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.68999 q^{5} +4.15163i q^{7} +O(q^{10})\) \(q-2.68999 q^{5} +4.15163i q^{7} -4.35250 q^{11} +(3.53159 + 0.726543i) q^{13} -5.87130 q^{17} +5.71423 q^{19} +3.62866 q^{23} +2.23607 q^{25} +3.08672i q^{29} +9.28334i q^{31} -11.1679i q^{35} -2.69790 q^{37} +11.1074i q^{41} -3.80423i q^{43} +4.91034i q^{47} -10.2361 q^{49} -1.17902i q^{53} +11.7082 q^{55} -2.29753 q^{59} -7.05342i q^{61} +(-9.49996 - 1.95440i) q^{65} -10.0795 q^{67} +2.08191i q^{71} -13.4350i q^{73} -18.0700i q^{77} -10.9443 q^{79} +9.73249 q^{83} +15.7938 q^{85} -12.1877i q^{89} +(-3.01634 + 14.6619i) q^{91} -15.3713 q^{95} -5.13170i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.68999 −1.20300 −0.601501 0.798872i \(-0.705430\pi\)
−0.601501 + 0.798872i \(0.705430\pi\)
\(6\) 0 0
\(7\) 4.15163i 1.56917i 0.620021 + 0.784585i \(0.287124\pi\)
−0.620021 + 0.784585i \(0.712876\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.35250 −1.31233 −0.656164 0.754618i \(-0.727822\pi\)
−0.656164 + 0.754618i \(0.727822\pi\)
\(12\) 0 0
\(13\) 3.53159 + 0.726543i 0.979487 + 0.201507i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.87130 −1.42400 −0.711999 0.702180i \(-0.752210\pi\)
−0.711999 + 0.702180i \(0.752210\pi\)
\(18\) 0 0
\(19\) 5.71423 1.31094 0.655468 0.755223i \(-0.272472\pi\)
0.655468 + 0.755223i \(0.272472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.62866 0.756628 0.378314 0.925677i \(-0.376504\pi\)
0.378314 + 0.925677i \(0.376504\pi\)
\(24\) 0 0
\(25\) 2.23607 0.447214
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.08672i 0.573190i 0.958052 + 0.286595i \(0.0925234\pi\)
−0.958052 + 0.286595i \(0.907477\pi\)
\(30\) 0 0
\(31\) 9.28334i 1.66734i 0.552266 + 0.833668i \(0.313763\pi\)
−0.552266 + 0.833668i \(0.686237\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.1679i 1.88771i
\(36\) 0 0
\(37\) −2.69790 −0.443531 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.1074i 1.73468i 0.497715 + 0.867340i \(0.334172\pi\)
−0.497715 + 0.867340i \(0.665828\pi\)
\(42\) 0 0
\(43\) 3.80423i 0.580139i −0.957006 0.290070i \(-0.906322\pi\)
0.957006 0.290070i \(-0.0936784\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.91034i 0.716247i 0.933674 + 0.358123i \(0.116583\pi\)
−0.933674 + 0.358123i \(0.883417\pi\)
\(48\) 0 0
\(49\) −10.2361 −1.46230
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.17902i 0.161951i −0.996716 0.0809757i \(-0.974196\pi\)
0.996716 0.0809757i \(-0.0258036\pi\)
\(54\) 0 0
\(55\) 11.7082 1.57873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.29753 −0.299113 −0.149556 0.988753i \(-0.547785\pi\)
−0.149556 + 0.988753i \(0.547785\pi\)
\(60\) 0 0
\(61\) 7.05342i 0.903098i −0.892246 0.451549i \(-0.850872\pi\)
0.892246 0.451549i \(-0.149128\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −9.49996 1.95440i −1.17832 0.242413i
\(66\) 0 0
\(67\) −10.0795 −1.23141 −0.615705 0.787977i \(-0.711129\pi\)
−0.615705 + 0.787977i \(0.711129\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.08191i 0.247078i 0.992340 + 0.123539i \(0.0394244\pi\)
−0.992340 + 0.123539i \(0.960576\pi\)
\(72\) 0 0
\(73\) 13.4350i 1.57244i −0.617944 0.786222i \(-0.712034\pi\)
0.617944 0.786222i \(-0.287966\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.0700i 2.05927i
\(78\) 0 0
\(79\) −10.9443 −1.23133 −0.615663 0.788009i \(-0.711112\pi\)
−0.615663 + 0.788009i \(0.711112\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.73249 1.06828 0.534140 0.845396i \(-0.320636\pi\)
0.534140 + 0.845396i \(0.320636\pi\)
\(84\) 0 0
\(85\) 15.7938 1.71307
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.1877i 1.29190i −0.763381 0.645949i \(-0.776462\pi\)
0.763381 0.645949i \(-0.223538\pi\)
\(90\) 0 0
\(91\) −3.01634 + 14.6619i −0.316198 + 1.53698i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.3713 −1.57706
\(96\) 0 0
\(97\) 5.13170i 0.521045i −0.965468 0.260523i \(-0.916105\pi\)
0.965468 0.260523i \(-0.0838949\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.9833i 1.49089i −0.666566 0.745446i \(-0.732237\pi\)
0.666566 0.745446i \(-0.267763\pi\)
\(102\) 0 0
\(103\) −1.70820 −0.168314 −0.0841572 0.996452i \(-0.526820\pi\)
−0.0841572 + 0.996452i \(0.526820\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.99442i 0.482829i −0.970422 0.241415i \(-0.922389\pi\)
0.970422 0.241415i \(-0.0776114\pi\)
\(108\) 0 0
\(109\) −8.73057 −0.836237 −0.418119 0.908392i \(-0.637310\pi\)
−0.418119 + 0.908392i \(0.637310\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.87130 −0.552325 −0.276163 0.961111i \(-0.589063\pi\)
−0.276163 + 0.961111i \(0.589063\pi\)
\(114\) 0 0
\(115\) −9.76108 −0.910225
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.3755i 2.23450i
\(120\) 0 0
\(121\) 7.94427 0.722207
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.43496 0.665003
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.80982i 0.769718i 0.922975 + 0.384859i \(0.125750\pi\)
−0.922975 + 0.384859i \(0.874250\pi\)
\(132\) 0 0
\(133\) 23.7234i 2.05708i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.4397i 0.891922i −0.895052 0.445961i \(-0.852862\pi\)
0.895052 0.445961i \(-0.147138\pi\)
\(138\) 0 0
\(139\) 14.3188i 1.21451i −0.794507 0.607254i \(-0.792271\pi\)
0.794507 0.607254i \(-0.207729\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −15.3713 3.16228i −1.28541 0.264443i
\(144\) 0 0
\(145\) 8.30327i 0.689549i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.635021 0.0520230 0.0260115 0.999662i \(-0.491719\pi\)
0.0260115 + 0.999662i \(0.491719\pi\)
\(150\) 0 0
\(151\) 4.15163i 0.337855i −0.985628 0.168928i \(-0.945970\pi\)
0.985628 0.168928i \(-0.0540304\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 24.9721i 2.00581i
\(156\) 0 0
\(157\) 16.1150i 1.28611i −0.765818 0.643057i \(-0.777666\pi\)
0.765818 0.643057i \(-0.222334\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.0649i 1.18728i
\(162\) 0 0
\(163\) −10.0795 −0.789489 −0.394745 0.918791i \(-0.629167\pi\)
−0.394745 + 0.918791i \(0.629167\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.2967i 1.88013i 0.340992 + 0.940066i \(0.389237\pi\)
−0.340992 + 0.940066i \(0.610763\pi\)
\(168\) 0 0
\(169\) 11.9443 + 5.13170i 0.918790 + 0.394746i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.17902i 0.0896395i −0.998995 0.0448198i \(-0.985729\pi\)
0.998995 0.0448198i \(-0.0142714\pi\)
\(174\) 0 0
\(175\) 9.28334i 0.701754i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.7987i 1.40508i −0.711645 0.702539i \(-0.752049\pi\)
0.711645 0.702539i \(-0.247951\pi\)
\(180\) 0 0
\(181\) 13.2088i 0.981802i 0.871215 + 0.490901i \(0.163332\pi\)
−0.871215 + 0.490901i \(0.836668\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.25732 0.533569
\(186\) 0 0
\(187\) 25.5548 1.86875
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.3713 −1.11223 −0.556113 0.831107i \(-0.687708\pi\)
−0.556113 + 0.831107i \(0.687708\pi\)
\(192\) 0 0
\(193\) 13.4350i 0.967070i −0.875325 0.483535i \(-0.839353\pi\)
0.875325 0.483535i \(-0.160647\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.6101 −0.755936 −0.377968 0.925819i \(-0.623377\pi\)
−0.377968 + 0.925819i \(0.623377\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.8149 −0.899433
\(204\) 0 0
\(205\) 29.8788i 2.08682i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −24.8712 −1.72038
\(210\) 0 0
\(211\) 17.0130i 1.17122i 0.810591 + 0.585612i \(0.199146\pi\)
−0.810591 + 0.585612i \(0.800854\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.2333i 0.697908i
\(216\) 0 0
\(217\) −38.5410 −2.61633
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −20.7350 4.26575i −1.39479 0.286945i
\(222\) 0 0
\(223\) 9.28334i 0.621658i −0.950466 0.310829i \(-0.899393\pi\)
0.950466 0.310829i \(-0.100607\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.02749 −0.0681967 −0.0340983 0.999418i \(-0.510856\pi\)
−0.0340983 + 0.999418i \(0.510856\pi\)
\(228\) 0 0
\(229\) −18.4917 −1.22196 −0.610981 0.791645i \(-0.709225\pi\)
−0.610981 + 0.791645i \(0.709225\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.25732 0.475443 0.237722 0.971333i \(-0.423599\pi\)
0.237722 + 0.971333i \(0.423599\pi\)
\(234\) 0 0
\(235\) 13.2088i 0.861646i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.07107i 0.457389i 0.973498 + 0.228695i \(0.0734457\pi\)
−0.973498 + 0.228695i \(0.926554\pi\)
\(240\) 0 0
\(241\) 18.5667i 1.19598i 0.801502 + 0.597992i \(0.204035\pi\)
−0.801502 + 0.597992i \(0.795965\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 27.5350 1.75914
\(246\) 0 0
\(247\) 20.1803 + 4.15163i 1.28404 + 0.264162i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.35805i 0.148839i −0.997227 0.0744193i \(-0.976290\pi\)
0.997227 0.0744193i \(-0.0237103\pi\)
\(252\) 0 0
\(253\) −15.7938 −0.992945
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.2572 1.63788 0.818941 0.573878i \(-0.194562\pi\)
0.818941 + 0.573878i \(0.194562\pi\)
\(258\) 0 0
\(259\) 11.2007i 0.695976i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.25732 0.447506 0.223753 0.974646i \(-0.428169\pi\)
0.223753 + 0.974646i \(0.428169\pi\)
\(264\) 0 0
\(265\) 3.17157i 0.194828i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 19.2490i 1.17363i −0.809720 0.586817i \(-0.800381\pi\)
0.809720 0.586817i \(-0.199619\pi\)
\(270\) 0 0
\(271\) 9.28334i 0.563923i −0.959426 0.281961i \(-0.909015\pi\)
0.959426 0.281961i \(-0.0909850\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.73249 −0.586891
\(276\) 0 0
\(277\) 21.7153i 1.30475i 0.757898 + 0.652373i \(0.226226\pi\)
−0.757898 + 0.652373i \(0.773774\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.35931i 0.558330i 0.960243 + 0.279165i \(0.0900576\pi\)
−0.960243 + 0.279165i \(0.909942\pi\)
\(282\) 0 0
\(283\) 12.3107i 0.731797i 0.930655 + 0.365899i \(0.119238\pi\)
−0.930655 + 0.365899i \(0.880762\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −46.1138 −2.72201
\(288\) 0 0
\(289\) 17.4721 1.02777
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −25.4800 −1.48856 −0.744278 0.667869i \(-0.767206\pi\)
−0.744278 + 0.667869i \(0.767206\pi\)
\(294\) 0 0
\(295\) 6.18034 0.359833
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.8149 + 2.63638i 0.741108 + 0.152466i
\(300\) 0 0
\(301\) 15.7938 0.910337
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.9737i 1.08643i
\(306\) 0 0
\(307\) −4.04684 −0.230966 −0.115483 0.993309i \(-0.536842\pi\)
−0.115483 + 0.993309i \(0.536842\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.8860 0.617288 0.308644 0.951178i \(-0.400125\pi\)
0.308644 + 0.951178i \(0.400125\pi\)
\(312\) 0 0
\(313\) −2.47214 −0.139733 −0.0698667 0.997556i \(-0.522257\pi\)
−0.0698667 + 0.997556i \(0.522257\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.2349 −0.799512 −0.399756 0.916622i \(-0.630905\pi\)
−0.399756 + 0.916622i \(0.630905\pi\)
\(318\) 0 0
\(319\) 13.4350i 0.752214i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −33.5500 −1.86677
\(324\) 0 0
\(325\) 7.89688 + 1.62460i 0.438040 + 0.0901165i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.3859 −1.12391
\(330\) 0 0
\(331\) 3.01634 0.165793 0.0828965 0.996558i \(-0.473583\pi\)
0.0828965 + 0.996558i \(0.473583\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 27.1139 1.48139
\(336\) 0 0
\(337\) 14.4721 0.788347 0.394174 0.919036i \(-0.371031\pi\)
0.394174 + 0.919036i \(0.371031\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 40.4057i 2.18809i
\(342\) 0 0
\(343\) 13.4350i 0.725420i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.5092i 1.53045i 0.643761 + 0.765227i \(0.277373\pi\)
−0.643761 + 0.765227i \(0.722627\pi\)
\(348\) 0 0
\(349\) 9.76108 0.522499 0.261249 0.965271i \(-0.415866\pi\)
0.261249 + 0.965271i \(0.415866\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.86629i 0.418680i −0.977843 0.209340i \(-0.932868\pi\)
0.977843 0.209340i \(-0.0671316\pi\)
\(354\) 0 0
\(355\) 5.60034i 0.297235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.1406i 0.693533i −0.937951 0.346767i \(-0.887280\pi\)
0.937951 0.346767i \(-0.112720\pi\)
\(360\) 0 0
\(361\) 13.6525 0.718551
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 36.1400i 1.89165i
\(366\) 0 0
\(367\) 9.12461 0.476301 0.238150 0.971228i \(-0.423459\pi\)
0.238150 + 0.971228i \(0.423459\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.89487 0.254129
\(372\) 0 0
\(373\) 0.343027i 0.0177613i −0.999961 0.00888063i \(-0.997173\pi\)
0.999961 0.00888063i \(-0.00282683\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.24264 + 10.9010i −0.115502 + 0.561432i
\(378\) 0 0
\(379\) 18.8101 0.966210 0.483105 0.875563i \(-0.339509\pi\)
0.483105 + 0.875563i \(0.339509\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.2241i 0.829010i 0.910047 + 0.414505i \(0.136045\pi\)
−0.910047 + 0.414505i \(0.863955\pi\)
\(384\) 0 0
\(385\) 48.6082i 2.47730i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.6182i 0.589067i 0.955641 + 0.294534i \(0.0951643\pi\)
−0.955641 + 0.294534i \(0.904836\pi\)
\(390\) 0 0
\(391\) −21.3050 −1.07744
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 29.4400 1.48129
\(396\) 0 0
\(397\) −24.5243 −1.23084 −0.615420 0.788199i \(-0.711014\pi\)
−0.615420 + 0.788199i \(0.711014\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.3438i 1.16574i −0.812567 0.582868i \(-0.801931\pi\)
0.812567 0.582868i \(-0.198069\pi\)
\(402\) 0 0
\(403\) −6.74474 + 32.7849i −0.335979 + 1.63313i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.7426 0.582059
\(408\) 0 0
\(409\) 15.3951i 0.761239i 0.924732 + 0.380620i \(0.124289\pi\)
−0.924732 + 0.380620i \(0.875711\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.53850i 0.469359i
\(414\) 0 0
\(415\) −26.1803 −1.28514
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.9833i 0.731981i −0.930619 0.365990i \(-0.880730\pi\)
0.930619 0.365990i \(-0.119270\pi\)
\(420\) 0 0
\(421\) 15.7938 0.769741 0.384870 0.922971i \(-0.374246\pi\)
0.384870 + 0.922971i \(0.374246\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −13.1286 −0.636832
\(426\) 0 0
\(427\) 29.2832 1.41711
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.8005i 1.00193i 0.865468 + 0.500963i \(0.167021\pi\)
−0.865468 + 0.500963i \(0.832979\pi\)
\(432\) 0 0
\(433\) 14.1803 0.681464 0.340732 0.940161i \(-0.389325\pi\)
0.340732 + 0.940161i \(0.389325\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7350 0.991891
\(438\) 0 0
\(439\) −24.1803 −1.15406 −0.577032 0.816721i \(-0.695789\pi\)
−0.577032 + 0.816721i \(0.695789\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.17902i 0.0560171i −0.999608 0.0280086i \(-0.991083\pi\)
0.999608 0.0280086i \(-0.00891656\pi\)
\(444\) 0 0
\(445\) 32.7849i 1.55416i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.3221i 1.57257i −0.617865 0.786284i \(-0.712002\pi\)
0.617865 0.786284i \(-0.287998\pi\)
\(450\) 0 0
\(451\) 48.3449i 2.27647i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.11393 39.4404i 0.380387 1.84899i
\(456\) 0 0
\(457\) 38.3448i 1.79369i 0.442342 + 0.896847i \(0.354148\pi\)
−0.442342 + 0.896847i \(0.645852\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.6951 1.15016 0.575082 0.818096i \(-0.304970\pi\)
0.575082 + 0.818096i \(0.304970\pi\)
\(462\) 0 0
\(463\) 4.15163i 0.192943i 0.995336 + 0.0964714i \(0.0307556\pi\)
−0.995336 + 0.0964714i \(0.969244\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.17345i 0.285673i 0.989746 + 0.142837i \(0.0456223\pi\)
−0.989746 + 0.142837i \(0.954378\pi\)
\(468\) 0 0
\(469\) 41.8465i 1.93229i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.5579i 0.761333i
\(474\) 0 0
\(475\) 12.7774 0.586268
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3122i 0.471174i −0.971853 0.235587i \(-0.924299\pi\)
0.971853 0.235587i \(-0.0757013\pi\)
\(480\) 0 0
\(481\) −9.52786 1.96014i −0.434433 0.0893745i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.8042i 0.626819i
\(486\) 0 0
\(487\) 17.5866i 0.796925i −0.917185 0.398463i \(-0.869544\pi\)
0.917185 0.398463i \(-0.130456\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.1511i 1.18018i −0.807336 0.590092i \(-0.799091\pi\)
0.807336 0.590092i \(-0.200909\pi\)
\(492\) 0 0
\(493\) 18.1231i 0.816222i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.64335 −0.387707
\(498\) 0 0
\(499\) −39.3628 −1.76212 −0.881059 0.473006i \(-0.843169\pi\)
−0.881059 + 0.473006i \(0.843169\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.9999 0.847164 0.423582 0.905858i \(-0.360772\pi\)
0.423582 + 0.905858i \(0.360772\pi\)
\(504\) 0 0
\(505\) 40.3049i 1.79355i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 27.5350 1.22047 0.610233 0.792222i \(-0.291076\pi\)
0.610233 + 0.792222i \(0.291076\pi\)
\(510\) 0 0
\(511\) 55.7771 2.46743
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.59506 0.202482
\(516\) 0 0
\(517\) 21.3723i 0.939951i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −24.8712 −1.08963 −0.544814 0.838557i \(-0.683400\pi\)
−0.544814 + 0.838557i \(0.683400\pi\)
\(522\) 0 0
\(523\) 21.9273i 0.958814i −0.877593 0.479407i \(-0.840852\pi\)
0.877593 0.479407i \(-0.159148\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 54.5052i 2.37429i
\(528\) 0 0
\(529\) −9.83282 −0.427514
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.06998 + 39.2267i −0.349550 + 1.69910i
\(534\) 0 0
\(535\) 13.4350i 0.580844i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 44.5525 1.91901
\(540\) 0 0
\(541\) 41.3486 1.77771 0.888857 0.458184i \(-0.151500\pi\)
0.888857 + 0.458184i \(0.151500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 23.4852 1.00600
\(546\) 0 0
\(547\) 0.212002i 0.00906456i 0.999990 + 0.00453228i \(0.00144267\pi\)
−0.999990 + 0.00453228i \(0.998557\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.6383i 0.751415i
\(552\) 0 0
\(553\) 45.4366i 1.93216i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.47492 0.147237 0.0736186 0.997286i \(-0.476545\pi\)
0.0736186 + 0.997286i \(0.476545\pi\)
\(558\) 0 0
\(559\) 2.76393 13.4350i 0.116902 0.568239i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 21.1567i 0.891649i 0.895120 + 0.445825i \(0.147090\pi\)
−0.895120 + 0.445825i \(0.852910\pi\)
\(564\) 0 0
\(565\) 15.7938 0.664448
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.48527 0.188032 0.0940162 0.995571i \(-0.470029\pi\)
0.0940162 + 0.995571i \(0.470029\pi\)
\(570\) 0 0
\(571\) 5.81234i 0.243239i −0.992577 0.121619i \(-0.961191\pi\)
0.992577 0.121619i \(-0.0388087\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.11393 0.338374
\(576\) 0 0
\(577\) 14.6464i 0.609738i −0.952394 0.304869i \(-0.901387\pi\)
0.952394 0.304869i \(-0.0986126\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.4057i 1.67631i
\(582\) 0 0
\(583\) 5.13170i 0.212533i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25.0875 −1.03547 −0.517736 0.855540i \(-0.673225\pi\)
−0.517736 + 0.855540i \(0.673225\pi\)
\(588\) 0 0
\(589\) 53.0472i 2.18577i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.1769i 0.705370i 0.935742 + 0.352685i \(0.114731\pi\)
−0.935742 + 0.352685i \(0.885269\pi\)
\(594\) 0 0
\(595\) 65.5699i 2.68810i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.9705 1.14284 0.571421 0.820657i \(-0.306392\pi\)
0.571421 + 0.820657i \(0.306392\pi\)
\(600\) 0 0
\(601\) −7.41641 −0.302522 −0.151261 0.988494i \(-0.548333\pi\)
−0.151261 + 0.988494i \(0.548333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.3700 −0.868816
\(606\) 0 0
\(607\) −2.29180 −0.0930211 −0.0465106 0.998918i \(-0.514810\pi\)
−0.0465106 + 0.998918i \(0.514810\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.56757 + 17.3413i −0.144329 + 0.701555i
\(612\) 0 0
\(613\) −34.2854 −1.38477 −0.692387 0.721526i \(-0.743441\pi\)
−0.692387 + 0.721526i \(0.743441\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.53089i 0.262924i 0.991321 + 0.131462i \(0.0419671\pi\)
−0.991321 + 0.131462i \(0.958033\pi\)
\(618\) 0 0
\(619\) 32.2996 1.29823 0.649115 0.760691i \(-0.275140\pi\)
0.649115 + 0.760691i \(0.275140\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 50.5990 2.02721
\(624\) 0 0
\(625\) −31.1803 −1.24721
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.8401 0.631588
\(630\) 0 0
\(631\) 7.32320i 0.291532i −0.989319 0.145766i \(-0.953435\pi\)
0.989319 0.145766i \(-0.0465647\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.1400 −0.640495
\(636\) 0 0
\(637\) −36.1496 7.43694i −1.43230 0.294662i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.6433 1.09184 0.545922 0.837836i \(-0.316180\pi\)
0.545922 + 0.837836i \(0.316180\pi\)
\(642\) 0 0
\(643\) −38.9691 −1.53679 −0.768396 0.639974i \(-0.778945\pi\)
−0.768396 + 0.639974i \(0.778945\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.2279 −0.637983 −0.318992 0.947758i \(-0.603344\pi\)
−0.318992 + 0.947758i \(0.603344\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.7694i 1.47803i 0.673689 + 0.739015i \(0.264709\pi\)
−0.673689 + 0.739015i \(0.735291\pi\)
\(654\) 0 0
\(655\) 23.6984i 0.925972i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.45735i 0.0567704i 0.999597 + 0.0283852i \(0.00903651\pi\)
−0.999597 + 0.0283852i \(0.990963\pi\)
\(660\) 0 0
\(661\) −23.4938 −0.913804 −0.456902 0.889517i \(-0.651041\pi\)
−0.456902 + 0.889517i \(0.651041\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 63.8158i 2.47467i
\(666\) 0 0
\(667\) 11.2007i 0.433692i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 30.7000i 1.18516i
\(672\) 0 0
\(673\) 18.6525 0.719000 0.359500 0.933145i \(-0.382947\pi\)
0.359500 + 0.933145i \(0.382947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.5036i 1.28765i −0.765173 0.643824i \(-0.777347\pi\)
0.765173 0.643824i \(-0.222653\pi\)
\(678\) 0 0
\(679\) 21.3050 0.817609
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25.0875 −0.959947 −0.479974 0.877283i \(-0.659354\pi\)
−0.479974 + 0.877283i \(0.659354\pi\)
\(684\) 0 0
\(685\) 28.0827i 1.07298i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.856611 4.16383i 0.0326343 0.158629i
\(690\) 0 0
\(691\) −1.34895 −0.0513164 −0.0256582 0.999671i \(-0.508168\pi\)
−0.0256582 + 0.999671i \(0.508168\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38.5176i 1.46106i
\(696\) 0 0
\(697\) 65.2147i 2.47018i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.44477i 0.205646i −0.994700 0.102823i \(-0.967212\pi\)
0.994700 0.102823i \(-0.0327876\pi\)
\(702\) 0 0
\(703\) −15.4164 −0.581441
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 62.2051 2.33946
\(708\) 0 0
\(709\) −15.7938 −0.593147 −0.296573 0.955010i \(-0.595844\pi\)
−0.296573 + 0.955010i \(0.595844\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33.6861i 1.26155i
\(714\) 0 0
\(715\) 41.3486 + 8.50651i 1.54635 + 0.318125i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27.1139 −1.01118 −0.505588 0.862775i \(-0.668724\pi\)
−0.505588 + 0.862775i \(0.668724\pi\)
\(720\) 0 0
\(721\) 7.09184i 0.264114i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.90212i 0.256338i
\(726\) 0 0
\(727\) −22.3607 −0.829312 −0.414656 0.909978i \(-0.636098\pi\)
−0.414656 + 0.909978i \(0.636098\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 22.3357i 0.826117i
\(732\) 0 0
\(733\) −24.9179 −0.920365 −0.460183 0.887824i \(-0.652216\pi\)
−0.460183 + 0.887824i \(0.652216\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 43.8711 1.61601
\(738\) 0 0
\(739\) −29.6017 −1.08892 −0.544458 0.838788i \(-0.683264\pi\)
−0.544458 + 0.838788i \(0.683264\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.31990i 0.121795i 0.998144 + 0.0608977i \(0.0193963\pi\)
−0.998144 + 0.0608977i \(0.980604\pi\)
\(744\) 0 0
\(745\) −1.70820 −0.0625837
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 20.7350 0.757641
\(750\) 0 0
\(751\) −12.5410 −0.457628 −0.228814 0.973470i \(-0.573485\pi\)
−0.228814 + 0.973470i \(0.573485\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.1679i 0.406440i
\(756\) 0 0
\(757\) 5.25731i 0.191080i 0.995426 + 0.0955401i \(0.0304578\pi\)
−0.995426 + 0.0955401i \(0.969542\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.2681i 0.480968i 0.970653 + 0.240484i \(0.0773062\pi\)
−0.970653 + 0.240484i \(0.922694\pi\)
\(762\) 0 0
\(763\) 36.2461i 1.31220i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.11393 1.66925i −0.292977 0.0602732i
\(768\) 0 0
\(769\) 46.6480i 1.68217i −0.540902 0.841086i \(-0.681917\pi\)
0.540902 0.841086i \(-0.318083\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −37.0249 −1.33169 −0.665846 0.746089i \(-0.731929\pi\)
−0.665846 + 0.746089i \(0.731929\pi\)
\(774\) 0 0
\(775\) 20.7582i 0.745656i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 63.4702i 2.27405i
\(780\) 0 0
\(781\) 9.06154i 0.324247i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 43.3491i 1.54720i
\(786\) 0 0
\(787\) −5.07735 −0.180988 −0.0904940 0.995897i \(-0.528845\pi\)
−0.0904940 + 0.995897i \(0.528845\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 24.3755i 0.866692i
\(792\) 0 0
\(793\) 5.12461 24.9098i 0.181980 0.884573i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.4336i 0.546687i 0.961917 + 0.273343i \(0.0881295\pi\)
−0.961917 + 0.273343i \(0.911870\pi\)
\(798\) 0 0
\(799\) 28.8301i 1.01993i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 58.4757i 2.06356i
\(804\) 0 0
\(805\) 40.5244i 1.42830i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.3859 0.716732 0.358366 0.933581i \(-0.383334\pi\)
0.358366 + 0.933581i \(0.383334\pi\)