Properties

Label 4032.2.b.p.3583.7
Level 4032
Weight 2
Character 4032.3583
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3583.7
Root \(-0.382683 + 0.923880i\)
Character \(\chi\) = 4032.3583
Dual form 4032.2.b.p.3583.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61313i q^{5} +(-2.61313 - 0.414214i) q^{7} +O(q^{10})\) \(q+2.61313i q^{5} +(-2.61313 - 0.414214i) q^{7} +2.00000i q^{11} -4.77791i q^{13} +3.06147i q^{17} +4.14386 q^{19} -7.65685i q^{23} -1.82843 q^{25} +3.65685 q^{29} +3.06147 q^{31} +(1.08239 - 6.82843i) q^{35} +7.65685 q^{37} +9.55582i q^{41} +3.65685i q^{43} -7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} -2.00000 q^{53} -5.22625 q^{55} +8.47343 q^{59} +2.61313i q^{61} +12.4853 q^{65} -15.6569i q^{67} +8.82843i q^{71} +12.6173i q^{73} +(0.828427 - 5.22625i) q^{77} +12.8284i q^{79} -11.5349 q^{83} -8.00000 q^{85} -2.16478i q^{89} +(-1.97908 + 12.4853i) q^{91} +10.8284i q^{95} +13.5140i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} - 16q^{29} + 16q^{37} + 8q^{49} - 16q^{53} + 32q^{65} - 16q^{77} - 64q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(5\) 2.61313i 1.16863i 0.811529 + 0.584313i \(0.198636\pi\)
−0.811529 + 0.584313i \(0.801364\pi\)
\(6\) 0 0
\(7\) −2.61313 0.414214i −0.987669 0.156558i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 4.77791i 1.32515i −0.748994 0.662577i \(-0.769463\pi\)
0.748994 0.662577i \(-0.230537\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.06147i 0.742515i 0.928530 + 0.371257i \(0.121073\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 4.14386 0.950667 0.475333 0.879806i \(-0.342327\pi\)
0.475333 + 0.879806i \(0.342327\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.65685i 1.59656i −0.602284 0.798282i \(-0.705742\pi\)
0.602284 0.798282i \(-0.294258\pi\)
\(24\) 0 0
\(25\) −1.82843 −0.365685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) 3.06147 0.549856 0.274928 0.961465i \(-0.411346\pi\)
0.274928 + 0.961465i \(0.411346\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.08239 6.82843i 0.182958 1.15421i
\(36\) 0 0
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.55582i 1.49237i 0.665740 + 0.746184i \(0.268116\pi\)
−0.665740 + 0.746184i \(0.731884\pi\)
\(42\) 0 0
\(43\) 3.65685i 0.557665i 0.960340 + 0.278833i \(0.0899474\pi\)
−0.960340 + 0.278833i \(0.910053\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.39104 −1.07809 −0.539047 0.842276i \(-0.681215\pi\)
−0.539047 + 0.842276i \(0.681215\pi\)
\(48\) 0 0
\(49\) 6.65685 + 2.16478i 0.950979 + 0.309255i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −5.22625 −0.704708
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.47343 1.10315 0.551573 0.834126i \(-0.314028\pi\)
0.551573 + 0.834126i \(0.314028\pi\)
\(60\) 0 0
\(61\) 2.61313i 0.334576i 0.985908 + 0.167288i \(0.0535011\pi\)
−0.985908 + 0.167288i \(0.946499\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.4853 1.54861
\(66\) 0 0
\(67\) 15.6569i 1.91279i −0.292078 0.956395i \(-0.594347\pi\)
0.292078 0.956395i \(-0.405653\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.82843i 1.04774i 0.851798 + 0.523871i \(0.175513\pi\)
−0.851798 + 0.523871i \(0.824487\pi\)
\(72\) 0 0
\(73\) 12.6173i 1.47674i 0.674395 + 0.738371i \(0.264405\pi\)
−0.674395 + 0.738371i \(0.735595\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.828427 5.22625i 0.0944080 0.595587i
\(78\) 0 0
\(79\) 12.8284i 1.44331i 0.692252 + 0.721655i \(0.256618\pi\)
−0.692252 + 0.721655i \(0.743382\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.5349 −1.26612 −0.633060 0.774103i \(-0.718201\pi\)
−0.633060 + 0.774103i \(0.718201\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.16478i 0.229467i −0.993396 0.114733i \(-0.963399\pi\)
0.993396 0.114733i \(-0.0366014\pi\)
\(90\) 0 0
\(91\) −1.97908 + 12.4853i −0.207463 + 1.30881i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.8284i 1.11097i
\(96\) 0 0
\(97\) 13.5140i 1.37214i 0.727537 + 0.686068i \(0.240665\pi\)
−0.727537 + 0.686068i \(0.759335\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.67459i 0.564643i 0.959320 + 0.282322i \(0.0911045\pi\)
−0.959320 + 0.282322i \(0.908895\pi\)
\(102\) 0 0
\(103\) −16.9469 −1.66982 −0.834912 0.550384i \(-0.814481\pi\)
−0.834912 + 0.550384i \(0.814481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.343146i 0.0331732i −0.999862 0.0165866i \(-0.994720\pi\)
0.999862 0.0165866i \(-0.00527991\pi\)
\(108\) 0 0
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.82843 −0.830509 −0.415254 0.909705i \(-0.636307\pi\)
−0.415254 + 0.909705i \(0.636307\pi\)
\(114\) 0 0
\(115\) 20.0083 1.86579
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.26810 8.00000i 0.116247 0.733359i
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.28772i 0.741276i
\(126\) 0 0
\(127\) 3.65685i 0.324493i −0.986750 0.162247i \(-0.948126\pi\)
0.986750 0.162247i \(-0.0518740\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.6997 1.19695 0.598473 0.801143i \(-0.295774\pi\)
0.598473 + 0.801143i \(0.295774\pi\)
\(132\) 0 0
\(133\) −10.8284 1.71644i −0.938944 0.148834i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 1.97908 0.167863 0.0839315 0.996472i \(-0.473252\pi\)
0.0839315 + 0.996472i \(0.473252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.55582 0.799098
\(144\) 0 0
\(145\) 9.55582i 0.793568i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.34315i 0.678956i −0.940614 0.339478i \(-0.889750\pi\)
0.940614 0.339478i \(-0.110250\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) 5.67459i 0.452882i 0.974025 + 0.226441i \(0.0727090\pi\)
−0.974025 + 0.226441i \(0.927291\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.17157 + 20.0083i −0.249955 + 1.57688i
\(162\) 0 0
\(163\) 17.3137i 1.35611i 0.735009 + 0.678057i \(0.237178\pi\)
−0.735009 + 0.678057i \(0.762822\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.49435 −0.502548 −0.251274 0.967916i \(-0.580850\pi\)
−0.251274 + 0.967916i \(0.580850\pi\)
\(168\) 0 0
\(169\) −9.82843 −0.756033
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.9008i 0.828776i 0.910100 + 0.414388i \(0.136004\pi\)
−0.910100 + 0.414388i \(0.863996\pi\)
\(174\) 0 0
\(175\) 4.77791 + 0.757359i 0.361176 + 0.0572510i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.00000i 0.149487i 0.997203 + 0.0747435i \(0.0238138\pi\)
−0.997203 + 0.0747435i \(0.976186\pi\)
\(180\) 0 0
\(181\) 1.71644i 0.127582i 0.997963 + 0.0637911i \(0.0203191\pi\)
−0.997963 + 0.0637911i \(0.979681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.0083i 1.47104i
\(186\) 0 0
\(187\) −6.12293 −0.447753
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.82843i 0.349373i −0.984624 0.174686i \(-0.944109\pi\)
0.984624 0.174686i \(-0.0558911\pi\)
\(192\) 0 0
\(193\) −10.4853 −0.754747 −0.377374 0.926061i \(-0.623173\pi\)
−0.377374 + 0.926061i \(0.623173\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.65685 0.260540 0.130270 0.991479i \(-0.458416\pi\)
0.130270 + 0.991479i \(0.458416\pi\)
\(198\) 0 0
\(199\) 12.6173 0.894416 0.447208 0.894430i \(-0.352418\pi\)
0.447208 + 0.894430i \(0.352418\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.55582 1.51472i −0.670687 0.106312i
\(204\) 0 0
\(205\) −24.9706 −1.74402
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.28772i 0.573274i
\(210\) 0 0
\(211\) 10.9706i 0.755245i −0.925960 0.377622i \(-0.876742\pi\)
0.925960 0.377622i \(-0.123258\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.55582 −0.651702
\(216\) 0 0
\(217\) −8.00000 1.26810i −0.543075 0.0860843i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.6274 0.983947
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.5545 1.23151 0.615753 0.787939i \(-0.288852\pi\)
0.615753 + 0.787939i \(0.288852\pi\)
\(228\) 0 0
\(229\) 2.98454i 0.197224i 0.995126 + 0.0986121i \(0.0314403\pi\)
−0.995126 + 0.0986121i \(0.968560\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 19.3137i 1.25989i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.34315i 0.539673i 0.962906 + 0.269837i \(0.0869697\pi\)
−0.962906 + 0.269837i \(0.913030\pi\)
\(240\) 0 0
\(241\) 23.9665i 1.54382i −0.635734 0.771908i \(-0.719302\pi\)
0.635734 0.771908i \(-0.280698\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.65685 + 17.3952i −0.361403 + 1.11134i
\(246\) 0 0
\(247\) 19.7990i 1.25978i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.08239 0.0683200 0.0341600 0.999416i \(-0.489124\pi\)
0.0341600 + 0.999416i \(0.489124\pi\)
\(252\) 0 0
\(253\) 15.3137 0.962765
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.12293i 0.381938i 0.981596 + 0.190969i \(0.0611630\pi\)
−0.981596 + 0.190969i \(0.938837\pi\)
\(258\) 0 0
\(259\) −20.0083 3.17157i −1.24326 0.197072i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.14214i 0.255415i 0.991812 + 0.127708i \(0.0407619\pi\)
−0.991812 + 0.127708i \(0.959238\pi\)
\(264\) 0 0
\(265\) 5.22625i 0.321046i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.4370i 0.819271i 0.912249 + 0.409636i \(0.134344\pi\)
−0.912249 + 0.409636i \(0.865656\pi\)
\(270\) 0 0
\(271\) −1.79337 −0.108939 −0.0544696 0.998515i \(-0.517347\pi\)
−0.0544696 + 0.998515i \(0.517347\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.65685i 0.220517i
\(276\) 0 0
\(277\) 26.9706 1.62050 0.810252 0.586082i \(-0.199330\pi\)
0.810252 + 0.586082i \(0.199330\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.6274 1.70777 0.853884 0.520463i \(-0.174241\pi\)
0.853884 + 0.520463i \(0.174241\pi\)
\(282\) 0 0
\(283\) −1.08239 −0.0643415 −0.0321708 0.999482i \(-0.510242\pi\)
−0.0321708 + 0.999482i \(0.510242\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.95815 24.9706i 0.233642 1.47397i
\(288\) 0 0
\(289\) 7.62742 0.448672
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.67459i 0.331513i 0.986167 + 0.165757i \(0.0530066\pi\)
−0.986167 + 0.165757i \(0.946993\pi\)
\(294\) 0 0
\(295\) 22.1421i 1.28916i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −36.5838 −2.11569
\(300\) 0 0
\(301\) 1.51472 9.55582i 0.0873069 0.550788i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.82843 −0.390995
\(306\) 0 0
\(307\) 11.5349 0.658331 0.329166 0.944272i \(-0.393233\pi\)
0.329166 + 0.944272i \(0.393233\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.0083 −1.13457 −0.567284 0.823522i \(-0.692006\pi\)
−0.567284 + 0.823522i \(0.692006\pi\)
\(312\) 0 0
\(313\) 15.6788i 0.886216i 0.896468 + 0.443108i \(0.146124\pi\)
−0.896468 + 0.443108i \(0.853876\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.6569 1.55336 0.776682 0.629893i \(-0.216901\pi\)
0.776682 + 0.629893i \(0.216901\pi\)
\(318\) 0 0
\(319\) 7.31371i 0.409489i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.6863i 0.705884i
\(324\) 0 0
\(325\) 8.73606i 0.484589i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.3137 + 3.06147i 1.06480 + 0.168784i
\(330\) 0 0
\(331\) 17.3137i 0.951647i 0.879541 + 0.475824i \(0.157850\pi\)
−0.879541 + 0.475824i \(0.842150\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 40.9133 2.23533
\(336\) 0 0
\(337\) 8.82843 0.480915 0.240458 0.970660i \(-0.422703\pi\)
0.240458 + 0.970660i \(0.422703\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.12293i 0.331576i
\(342\) 0 0
\(343\) −16.4985 8.41421i −0.890836 0.454325i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.6569i 1.05524i −0.849482 0.527618i \(-0.823085\pi\)
0.849482 0.527618i \(-0.176915\pi\)
\(348\) 0 0
\(349\) 8.73606i 0.467631i 0.972281 + 0.233815i \(0.0751211\pi\)
−0.972281 + 0.233815i \(0.924879\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.5641i 1.57354i −0.617246 0.786770i \(-0.711752\pi\)
0.617246 0.786770i \(-0.288248\pi\)
\(354\) 0 0
\(355\) −23.0698 −1.22442
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.9706i 0.790116i 0.918656 + 0.395058i \(0.129276\pi\)
−0.918656 + 0.395058i \(0.870724\pi\)
\(360\) 0 0
\(361\) −1.82843 −0.0962330
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −32.9706 −1.72576
\(366\) 0 0
\(367\) 10.4525 0.545616 0.272808 0.962068i \(-0.412048\pi\)
0.272808 + 0.962068i \(0.412048\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.22625 + 0.828427i 0.271333 + 0.0430098i
\(372\) 0 0
\(373\) −6.97056 −0.360922 −0.180461 0.983582i \(-0.557759\pi\)
−0.180461 + 0.983582i \(0.557759\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17.4721i 0.899860i
\(378\) 0 0
\(379\) 5.31371i 0.272947i −0.990644 0.136473i \(-0.956423\pi\)
0.990644 0.136473i \(-0.0435768\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.7206 0.598895 0.299447 0.954113i \(-0.403198\pi\)
0.299447 + 0.954113i \(0.403198\pi\)
\(384\) 0 0
\(385\) 13.6569 + 2.16478i 0.696018 + 0.110328i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.6569 0.996642 0.498321 0.866993i \(-0.333950\pi\)
0.498321 + 0.866993i \(0.333950\pi\)
\(390\) 0 0
\(391\) 23.4412 1.18547
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −33.5223 −1.68669
\(396\) 0 0
\(397\) 33.9706i 1.70494i 0.522778 + 0.852469i \(0.324896\pi\)
−0.522778 + 0.852469i \(0.675104\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.1716 0.757632 0.378816 0.925472i \(-0.376331\pi\)
0.378816 + 0.925472i \(0.376331\pi\)
\(402\) 0 0
\(403\) 14.6274i 0.728644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.3137i 0.759072i
\(408\) 0 0
\(409\) 0.896683i 0.0443381i 0.999754 + 0.0221691i \(0.00705721\pi\)
−0.999754 + 0.0221691i \(0.992943\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −22.1421 3.50981i −1.08954 0.172706i
\(414\) 0 0
\(415\) 30.1421i 1.47962i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 34.6047 1.69055 0.845275 0.534332i \(-0.179437\pi\)
0.845275 + 0.534332i \(0.179437\pi\)
\(420\) 0 0
\(421\) 21.3137 1.03877 0.519383 0.854541i \(-0.326162\pi\)
0.519383 + 0.854541i \(0.326162\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.59767i 0.271527i
\(426\) 0 0
\(427\) 1.08239 6.82843i 0.0523806 0.330451i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.3431i 0.594548i −0.954792 0.297274i \(-0.903922\pi\)
0.954792 0.297274i \(-0.0960775\pi\)
\(432\) 0 0
\(433\) 5.59767i 0.269007i −0.990913 0.134503i \(-0.957056\pi\)
0.990913 0.134503i \(-0.0429439\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.7289i 1.51780i
\(438\) 0 0
\(439\) 22.5445 1.07599 0.537996 0.842948i \(-0.319182\pi\)
0.537996 + 0.842948i \(0.319182\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0000i 0.475114i 0.971374 + 0.237557i \(0.0763467\pi\)
−0.971374 + 0.237557i \(0.923653\pi\)
\(444\) 0 0
\(445\) 5.65685 0.268161
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −21.3137 −1.00586 −0.502928 0.864328i \(-0.667744\pi\)
−0.502928 + 0.864328i \(0.667744\pi\)
\(450\) 0 0
\(451\) −19.1116 −0.899932
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −32.6256 5.17157i −1.52951 0.242447i
\(456\) 0 0
\(457\) 37.1127 1.73606 0.868029 0.496513i \(-0.165386\pi\)
0.868029 + 0.496513i \(0.165386\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.3756i 0.483239i 0.970371 + 0.241619i \(0.0776786\pi\)
−0.970371 + 0.241619i \(0.922321\pi\)
\(462\) 0 0
\(463\) 16.1421i 0.750189i 0.926987 + 0.375094i \(0.122390\pi\)
−0.926987 + 0.375094i \(0.877610\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.10201 0.374916 0.187458 0.982273i \(-0.439975\pi\)
0.187458 + 0.982273i \(0.439975\pi\)
\(468\) 0 0
\(469\) −6.48528 + 40.9133i −0.299462 + 1.88920i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.31371 −0.336285
\(474\) 0 0
\(475\) −7.57675 −0.347645
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.0894 1.37482 0.687410 0.726269i \(-0.258748\pi\)
0.687410 + 0.726269i \(0.258748\pi\)
\(480\) 0 0
\(481\) 36.5838i 1.66808i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −35.3137 −1.60351
\(486\) 0 0
\(487\) 30.2843i 1.37231i 0.727455 + 0.686156i \(0.240703\pi\)
−0.727455 + 0.686156i \(0.759297\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.2843i 1.36671i 0.730086 + 0.683355i \(0.239480\pi\)
−0.730086 + 0.683355i \(0.760520\pi\)
\(492\) 0 0
\(493\) 11.1953i 0.504213i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.65685 23.0698i 0.164032 1.03482i
\(498\) 0 0
\(499\) 17.3137i 0.775068i 0.921855 + 0.387534i \(0.126673\pi\)
−0.921855 + 0.387534i \(0.873327\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.4721 −0.779043 −0.389522 0.921017i \(-0.627360\pi\)
−0.389522 + 0.921017i \(0.627360\pi\)
\(504\) 0 0
\(505\) −14.8284 −0.659856
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0543i 1.15484i −0.816448 0.577419i \(-0.804060\pi\)
0.816448 0.577419i \(-0.195940\pi\)
\(510\) 0 0
\(511\) 5.22625 32.9706i 0.231196 1.45853i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 44.2843i 1.95140i
\(516\) 0 0
\(517\) 14.7821i 0.650115i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.896683i 0.0392844i 0.999807 + 0.0196422i \(0.00625271\pi\)
−0.999807 + 0.0196422i \(0.993747\pi\)
\(522\) 0 0
\(523\) −18.9259 −0.827573 −0.413787 0.910374i \(-0.635794\pi\)
−0.413787 + 0.910374i \(0.635794\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.37258i 0.408276i
\(528\) 0 0
\(529\) −35.6274 −1.54902
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 45.6569 1.97762
\(534\) 0 0
\(535\) 0.896683 0.0387670
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.32957 + 13.3137i −0.186488 + 0.573462i
\(540\) 0 0
\(541\) −25.3137 −1.08832 −0.544161 0.838981i \(-0.683152\pi\)
−0.544161 + 0.838981i \(0.683152\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.8854i 0.594785i
\(546\) 0 0
\(547\) 33.3137i 1.42439i 0.701981 + 0.712196i \(0.252299\pi\)
−0.701981 + 0.712196i \(0.747701\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.1535 0.645560
\(552\) 0 0
\(553\) 5.31371 33.5223i 0.225962 1.42551i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.31371 −0.225149 −0.112575 0.993643i \(-0.535910\pi\)
−0.112575 + 0.993643i \(0.535910\pi\)
\(558\) 0 0
\(559\) 17.4721 0.738992
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.51528 0.190296 0.0951481 0.995463i \(-0.469668\pi\)
0.0951481 + 0.995463i \(0.469668\pi\)
\(564\) 0 0
\(565\) 23.0698i 0.970553i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.85786 −0.329419 −0.164709 0.986342i \(-0.552669\pi\)
−0.164709 + 0.986342i \(0.552669\pi\)
\(570\) 0 0
\(571\) 8.34315i 0.349150i 0.984644 + 0.174575i \(0.0558551\pi\)
−0.984644 + 0.174575i \(0.944145\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) 35.6871i 1.48567i 0.669473 + 0.742836i \(0.266520\pi\)
−0.669473 + 0.742836i \(0.733480\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 30.1421 + 4.77791i 1.25051 + 0.198221i
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −29.0070 −1.19725 −0.598624 0.801030i \(-0.704286\pi\)
−0.598624 + 0.801030i \(0.704286\pi\)
\(588\) 0 0
\(589\) 12.6863 0.522730
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.79337i 0.0736447i −0.999322 0.0368224i \(-0.988276\pi\)
0.999322 0.0368224i \(-0.0117236\pi\)
\(594\) 0 0
\(595\) 20.9050 + 3.31371i 0.857022 + 0.135849i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.1716i 1.27364i −0.771014 0.636818i \(-0.780250\pi\)
0.771014 0.636818i \(-0.219750\pi\)
\(600\) 0 0
\(601\) 42.1814i 1.72062i −0.509774 0.860308i \(-0.670271\pi\)
0.509774 0.860308i \(-0.329729\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.2919i 0.743671i
\(606\) 0 0
\(607\) −17.3183 −0.702927 −0.351464 0.936202i \(-0.614316\pi\)
−0.351464 + 0.936202i \(0.614316\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.3137i 1.42864i
\(612\) 0 0
\(613\) 5.31371 0.214619 0.107309 0.994226i \(-0.465776\pi\)
0.107309 + 0.994226i \(0.465776\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.1127 −0.527897 −0.263949 0.964537i \(-0.585025\pi\)
−0.263949 + 0.964537i \(0.585025\pi\)
\(618\) 0 0
\(619\) −2.87576 −0.115586 −0.0577932 0.998329i \(-0.518406\pi\)
−0.0577932 + 0.998329i \(0.518406\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.896683 + 5.65685i −0.0359248 + 0.226637i
\(624\) 0 0
\(625\) −30.7990 −1.23196
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.4412i 0.934662i
\(630\) 0 0
\(631\) 26.4853i 1.05436i 0.849753 + 0.527181i \(0.176751\pi\)
−0.849753 + 0.527181i \(0.823249\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.55582 0.379211
\(636\) 0 0
\(637\) 10.3431 31.8059i 0.409810 1.26019i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28.1421 −1.11155 −0.555774 0.831334i \(-0.687578\pi\)
−0.555774 + 0.831334i \(0.687578\pi\)
\(642\) 0 0
\(643\) 17.1326 0.675642 0.337821 0.941210i \(-0.390310\pi\)
0.337821 + 0.941210i \(0.390310\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.70099 0.184815 0.0924074 0.995721i \(-0.470544\pi\)
0.0924074 + 0.995721i \(0.470544\pi\)
\(648\) 0 0
\(649\) 16.9469i 0.665222i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.68629 0.105123 0.0525614 0.998618i \(-0.483261\pi\)
0.0525614 + 0.998618i \(0.483261\pi\)
\(654\) 0 0
\(655\) 35.7990i 1.39878i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.0294i 0.663373i 0.943390 + 0.331686i \(0.107618\pi\)
−0.943390 + 0.331686i \(0.892382\pi\)
\(660\) 0 0
\(661\) 30.9092i 1.20223i −0.799164 0.601114i \(-0.794724\pi\)
0.799164 0.601114i \(-0.205276\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.48528 28.2960i 0.173932 1.09727i
\(666\) 0 0
\(667\) 28.0000i 1.08416i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.22625 −0.201757
\(672\) 0 0
\(673\) 6.68629 0.257738 0.128869 0.991662i \(-0.458865\pi\)
0.128869 + 0.991662i \(0.458865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.4148i 0.938338i −0.883109 0.469169i \(-0.844554\pi\)
0.883109 0.469169i \(-0.155446\pi\)
\(678\) 0 0
\(679\) 5.59767 35.3137i 0.214819 1.35522i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 43.6569i 1.67048i −0.549883 0.835242i \(-0.685328\pi\)
0.549883 0.835242i \(-0.314672\pi\)
\(684\) 0 0
\(685\) 5.22625i 0.199685i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.55582i 0.364048i
\(690\) 0 0
\(691\) 32.4399 1.23407 0.617036 0.786935i \(-0.288333\pi\)
0.617036 + 0.786935i \(0.288333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.17157i 0.196169i
\(696\) 0 0
\(697\) −29.2548 −1.10811
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.31371 0.351774 0.175887 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(702\) 0 0
\(703\) 31.7289 1.19668
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.35049 14.8284i 0.0883994 0.557680i
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 23.4412i 0.877880i
\(714\) 0 0
\(715\) 24.9706i 0.933846i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 34.4190 1.28361 0.641806 0.766867i \(-0.278186\pi\)
0.641806 + 0.766867i \(0.278186\pi\)
\(720\) 0 0
\(721\) 44.2843 + 7.01962i 1.64923 + 0.261424i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.68629 −0.248323
\(726\) 0 0
\(727\) −42.1814 −1.56442 −0.782211 0.623013i \(-0.785908\pi\)
−0.782211 + 0.623013i \(0.785908\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11.1953 −0.414075
\(732\) 0 0
\(733\) 24.4148i 0.901782i −0.892579 0.450891i \(-0.851106\pi\)
0.892579 0.450891i \(-0.148894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.3137 1.15346
\(738\) 0 0
\(739\) 16.6274i 0.611649i −0.952088 0.305825i \(-0.901068\pi\)
0.952088 0.305825i \(-0.0989322\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.3431i 0.893063i 0.894768 + 0.446532i \(0.147341\pi\)
−0.894768 + 0.446532i \(0.852659\pi\)
\(744\) 0 0
\(745\) 26.1313i 0.957375i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.142136 + 0.896683i −0.00519352 + 0.0327641i
\(750\) 0 0
\(751\) 14.9706i 0.546284i −0.961974 0.273142i \(-0.911937\pi\)
0.961974 0.273142i \(-0.0880628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.8017 0.793445
\(756\) 0 0
\(757\) −49.3137 −1.79234 −0.896169 0.443714i \(-0.853661\pi\)
−0.896169 + 0.443714i \(0.853661\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.76245i 0.281389i 0.990053 + 0.140694i \(0.0449335\pi\)
−0.990053 + 0.140694i \(0.955067\pi\)
\(762\) 0 0
\(763\) −13.8854 2.20101i −0.502685 0.0796819i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40.4853i 1.46184i
\(768\) 0 0
\(769\) 13.5140i 0.487326i −0.969860 0.243663i \(-0.921651\pi\)
0.969860 0.243663i \(-0.0783491\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 26.9510i 0.969361i −0.874691 0.484680i \(-0.838936\pi\)
0.874691 0.484680i \(-0.161064\pi\)
\(774\) 0 0
\(775\) −5.59767 −0.201074
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 39.5980i 1.41874i
\(780\) 0 0
\(781\) −17.6569 −0.631812
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −14.8284 −0.529249
\(786\) 0 0
\(787\) −4.51528 −0.160952 −0.0804761 0.996757i \(-0.525644\pi\)
−0.0804761 + 0.996757i \(0.525644\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 23.0698 + 3.65685i 0.820267 + 0.130023i
\(792\) 0 0
\(793\) 12.4853 0.443365
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 41.3617i 1.46511i −0.680710 0.732553i \(-0.738329\pi\)
0.680710 0.732553i \(-0.261671\pi\)
\(798\) 0 0
\(799\) 22.6274i 0.800500i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −25.2346 −0.890509
\(804\) 0 0
\(805\) −52.2843 8.28772i −1.84278 0.292104i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.5147 0.615785 0.307892 0.951421i \(-0.400376\pi\)
0.307892 + 0.951421i \(0.400376\pi\)