Properties

Label 4032.2.p.k.1567.11
Level 4032
Weight 2
Character 4032.1567
Analytic conductor 32.196
Analytic rank 0
Dimension 12
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.p (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.11
Root \(0.385124 - 0.500000i\)
Character \(\chi\) = 4032.1567
Dual form 4032.2.p.k.1567.12

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+2.76873 q^{5}\) \(+(0.480901 - 2.60168i) q^{7}\) \(+O(q^{10})\) \(q\)\(+2.76873 q^{5}\) \(+(0.480901 - 2.60168i) q^{7}\) \(+5.75739 q^{11}\) \(+2.00000 q^{13}\) \(-6.71919i q^{17}\) \(+5.20336i q^{19}\) \(+4.43462i q^{23}\) \(+2.66589 q^{25}\) \(+1.54050i q^{29}\) \(+8.05068 q^{31}\) \(+(1.33149 - 7.20336i) q^{35}\) \(+4.42590i q^{37}\) \(+0.209011i q^{41}\) \(+10.5530 q^{43}\) \(-4.58658 q^{47}\) \(+(-6.53747 - 2.50230i) q^{49}\) \(-8.05068i q^{53}\) \(+15.9407 q^{55}\) \(+5.53747i q^{59}\) \(-10.8692 q^{61}\) \(+5.53747 q^{65}\) \(-4.04280 q^{67}\) \(-1.10284i q^{71}\) \(+9.59118i q^{73}\) \(+(2.76873 - 14.9789i) q^{77}\) \(+14.7408i q^{79}\) \(-8.86925i q^{83}\) \(-18.6037i q^{85}\) \(-13.6474i q^{89}\) \(+(0.961802 - 5.20336i) q^{91}\) \(+14.4067i q^{95}\) \(+4.58658i q^{97}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 24q^{13} \) \(\mathstrut +\mathstrut 36q^{25} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 72q^{61} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(5\) 2.76873 1.23822 0.619108 0.785306i \(-0.287494\pi\)
0.619108 + 0.785306i \(0.287494\pi\)
\(6\) 0 0
\(7\) 0.480901 2.60168i 0.181763 0.983342i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.75739 1.73592 0.867959 0.496635i \(-0.165431\pi\)
0.867959 + 0.496635i \(0.165431\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.71919i 1.62964i −0.579712 0.814822i \(-0.696835\pi\)
0.579712 0.814822i \(-0.303165\pi\)
\(18\) 0 0
\(19\) 5.20336i 1.19373i 0.802341 + 0.596866i \(0.203588\pi\)
−0.802341 + 0.596866i \(0.796412\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.43462i 0.924683i 0.886702 + 0.462342i \(0.152991\pi\)
−0.886702 + 0.462342i \(0.847009\pi\)
\(24\) 0 0
\(25\) 2.66589 0.533178
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.54050i 0.286063i 0.989718 + 0.143032i \(0.0456851\pi\)
−0.989718 + 0.143032i \(0.954315\pi\)
\(30\) 0 0
\(31\) 8.05068 1.44594 0.722972 0.690877i \(-0.242775\pi\)
0.722972 + 0.690877i \(0.242775\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.33149 7.20336i 0.225062 1.21759i
\(36\) 0 0
\(37\) 4.42590i 0.727614i 0.931474 + 0.363807i \(0.118523\pi\)
−0.931474 + 0.363807i \(0.881477\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.209011i 0.0326420i 0.999867 + 0.0163210i \(0.00519537\pi\)
−0.999867 + 0.0163210i \(0.994805\pi\)
\(42\) 0 0
\(43\) 10.5530 1.60931 0.804657 0.593740i \(-0.202349\pi\)
0.804657 + 0.593740i \(0.202349\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.58658 −0.669021 −0.334511 0.942392i \(-0.608571\pi\)
−0.334511 + 0.942392i \(0.608571\pi\)
\(48\) 0 0
\(49\) −6.53747 2.50230i −0.933924 0.357471i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.05068i 1.10585i −0.833232 0.552923i \(-0.813512\pi\)
0.833232 0.552923i \(-0.186488\pi\)
\(54\) 0 0
\(55\) 15.9407 2.14944
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.53747i 0.720917i 0.932775 + 0.360459i \(0.117380\pi\)
−0.932775 + 0.360459i \(0.882620\pi\)
\(60\) 0 0
\(61\) −10.8692 −1.39166 −0.695832 0.718204i \(-0.744964\pi\)
−0.695832 + 0.718204i \(0.744964\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.53747 0.686838
\(66\) 0 0
\(67\) −4.04280 −0.493906 −0.246953 0.969027i \(-0.579429\pi\)
−0.246953 + 0.969027i \(0.579429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.10284i 0.130884i −0.997856 0.0654418i \(-0.979154\pi\)
0.997856 0.0654418i \(-0.0208457\pi\)
\(72\) 0 0
\(73\) 9.59118i 1.12256i 0.827625 + 0.561281i \(0.189691\pi\)
−0.827625 + 0.561281i \(0.810309\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.76873 14.9789i 0.315527 1.70700i
\(78\) 0 0
\(79\) 14.7408i 1.65847i 0.558898 + 0.829236i \(0.311224\pi\)
−0.558898 + 0.829236i \(0.688776\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.86925i 0.973526i −0.873534 0.486763i \(-0.838177\pi\)
0.873534 0.486763i \(-0.161823\pi\)
\(84\) 0 0
\(85\) 18.6037i 2.01785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.6474i 1.44662i −0.690523 0.723311i \(-0.742620\pi\)
0.690523 0.723311i \(-0.257380\pi\)
\(90\) 0 0
\(91\) 0.961802 5.20336i 0.100824 0.545460i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 14.4067i 1.47810i
\(96\) 0 0
\(97\) 4.58658i 0.465696i 0.972513 + 0.232848i \(0.0748045\pi\)
−0.972513 + 0.232848i \(0.925195\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.76873 0.275499 0.137750 0.990467i \(-0.456013\pi\)
0.137750 + 0.990467i \(0.456013\pi\)
\(102\) 0 0
\(103\) −19.9835 −1.96903 −0.984515 0.175298i \(-0.943911\pi\)
−0.984515 + 0.175298i \(0.943911\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.752791 0.0727750 0.0363875 0.999338i \(-0.488415\pi\)
0.0363875 + 0.999338i \(0.488415\pi\)
\(108\) 0 0
\(109\) 8.85181i 0.847849i −0.905697 0.423925i \(-0.860652\pi\)
0.905697 0.423925i \(-0.139348\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.5375 −1.08535 −0.542677 0.839942i \(-0.682589\pi\)
−0.542677 + 0.839942i \(0.682589\pi\)
\(114\) 0 0
\(115\) 12.2783i 1.14496i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −17.4812 3.23127i −1.60250 0.296210i
\(120\) 0 0
\(121\) 22.1475 2.01341
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.46253 −0.578026
\(126\) 0 0
\(127\) 3.66589i 0.325295i −0.986684 0.162648i \(-0.947997\pi\)
0.986684 0.162648i \(-0.0520034\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.13075i 0.273535i 0.990603 + 0.136768i \(0.0436713\pi\)
−0.990603 + 0.136768i \(0.956329\pi\)
\(132\) 0 0
\(133\) 13.5375 + 2.50230i 1.17385 + 0.216977i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.4817 1.66443 0.832215 0.554453i \(-0.187073\pi\)
0.832215 + 0.554453i \(0.187073\pi\)
\(138\) 0 0
\(139\) 4.66822i 0.395953i 0.980207 + 0.197977i \(0.0634370\pi\)
−0.980207 + 0.197977i \(0.936563\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.5148 0.962914
\(144\) 0 0
\(145\) 4.26523i 0.354208i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.97428i 0.817125i −0.912730 0.408563i \(-0.866030\pi\)
0.912730 0.408563i \(-0.133970\pi\)
\(150\) 0 0
\(151\) 2.79664i 0.227587i −0.993504 0.113794i \(-0.963700\pi\)
0.993504 0.113794i \(-0.0363003\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 22.2902 1.79039
\(156\) 0 0
\(157\) −21.9442 −1.75134 −0.875668 0.482913i \(-0.839579\pi\)
−0.875668 + 0.482913i \(0.839579\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.5375 + 2.13261i 0.909280 + 0.168074i
\(162\) 0 0
\(163\) 7.89001 0.617993 0.308996 0.951063i \(-0.400007\pi\)
0.308996 + 0.951063i \(0.400007\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.00460 −0.387268 −0.193634 0.981074i \(-0.562027\pi\)
−0.193634 + 0.981074i \(0.562027\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.50723 −0.646793 −0.323396 0.946264i \(-0.604825\pi\)
−0.323396 + 0.946264i \(0.604825\pi\)
\(174\) 0 0
\(175\) 1.28203 6.93579i 0.0969123 0.524296i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.3440 0.773144 0.386572 0.922259i \(-0.373659\pi\)
0.386572 + 0.922259i \(0.373659\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.2542i 0.900943i
\(186\) 0 0
\(187\) 38.6850i 2.82893i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.50956i 0.253943i 0.991906 + 0.126971i \(0.0405257\pi\)
−0.991906 + 0.126971i \(0.959474\pi\)
\(192\) 0 0
\(193\) 13.4090 0.965204 0.482602 0.875840i \(-0.339692\pi\)
0.482602 + 0.875840i \(0.339692\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.70905i 0.406753i −0.979101 0.203377i \(-0.934808\pi\)
0.979101 0.203377i \(-0.0651916\pi\)
\(198\) 0 0
\(199\) −4.80901 −0.340902 −0.170451 0.985366i \(-0.554522\pi\)
−0.170451 + 0.985366i \(0.554522\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.00788 + 0.740827i 0.281298 + 0.0519959i
\(204\) 0 0
\(205\) 0.578696i 0.0404179i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.9578i 2.07222i
\(210\) 0 0
\(211\) 14.4002 0.991350 0.495675 0.868508i \(-0.334921\pi\)
0.495675 + 0.868508i \(0.334921\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 29.2184 1.99268
\(216\) 0 0
\(217\) 3.87158 20.9453i 0.262820 1.42186i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.4384i 0.903964i
\(222\) 0 0
\(223\) −12.0586 −0.807501 −0.403750 0.914869i \(-0.632294\pi\)
−0.403750 + 0.914869i \(0.632294\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.7385i 1.17735i −0.808372 0.588673i \(-0.799651\pi\)
0.808372 0.588673i \(-0.200349\pi\)
\(228\) 0 0
\(229\) 11.4817 0.758729 0.379365 0.925247i \(-0.376143\pi\)
0.379365 + 0.925247i \(0.376143\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.6077 1.35006 0.675029 0.737791i \(-0.264131\pi\)
0.675029 + 0.737791i \(0.264131\pi\)
\(234\) 0 0
\(235\) −12.6990 −0.828392
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.6403i 1.20574i 0.797839 + 0.602871i \(0.205977\pi\)
−0.797839 + 0.602871i \(0.794023\pi\)
\(240\) 0 0
\(241\) 14.5958i 0.940197i 0.882614 + 0.470098i \(0.155782\pi\)
−0.882614 + 0.470098i \(0.844218\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −18.1005 6.92820i −1.15640 0.442627i
\(246\) 0 0
\(247\) 10.4067i 0.662164i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.4817i 0.850954i 0.904969 + 0.425477i \(0.139894\pi\)
−0.904969 + 0.425477i \(0.860106\pi\)
\(252\) 0 0
\(253\) 25.5319i 1.60517i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.6566i 1.47566i −0.674988 0.737829i \(-0.735851\pi\)
0.674988 0.737829i \(-0.264149\pi\)
\(258\) 0 0
\(259\) 11.5148 + 2.12842i 0.715494 + 0.132254i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.8972i 0.671947i −0.941871 0.335974i \(-0.890935\pi\)
0.941871 0.335974i \(-0.109065\pi\)
\(264\) 0 0
\(265\) 22.2902i 1.36928i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.0447 −0.856320 −0.428160 0.903703i \(-0.640838\pi\)
−0.428160 + 0.903703i \(0.640838\pi\)
\(270\) 0 0
\(271\) −26.9117 −1.63477 −0.817384 0.576093i \(-0.804577\pi\)
−0.817384 + 0.576093i \(0.804577\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.3486 0.925553
\(276\) 0 0
\(277\) 2.50230i 0.150349i −0.997170 0.0751743i \(-0.976049\pi\)
0.997170 0.0751743i \(-0.0239513\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.3509 0.975413 0.487707 0.873008i \(-0.337834\pi\)
0.487707 + 0.873008i \(0.337834\pi\)
\(282\) 0 0
\(283\) 14.9419i 0.888201i −0.895977 0.444101i \(-0.853523\pi\)
0.895977 0.444101i \(-0.146477\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.543780 + 0.100514i 0.0320983 + 0.00593313i
\(288\) 0 0
\(289\) −28.1475 −1.65574
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.50723 −0.496998 −0.248499 0.968632i \(-0.579937\pi\)
−0.248499 + 0.968632i \(0.579937\pi\)
\(294\) 0 0
\(295\) 15.3318i 0.892651i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.86925i 0.512922i
\(300\) 0 0
\(301\) 5.07494 27.4555i 0.292515 1.58251i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −30.0941 −1.72318
\(306\) 0 0
\(307\) 12.5351i 0.715418i 0.933833 + 0.357709i \(0.116442\pi\)
−0.933833 + 0.357709i \(0.883558\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.5194 0.936728 0.468364 0.883536i \(-0.344844\pi\)
0.468364 + 0.883536i \(0.344844\pi\)
\(312\) 0 0
\(313\) 4.68325i 0.264713i 0.991202 + 0.132357i \(0.0422544\pi\)
−0.991202 + 0.132357i \(0.957746\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.97428i 0.560212i −0.959969 0.280106i \(-0.909630\pi\)
0.959969 0.280106i \(-0.0903695\pi\)
\(318\) 0 0
\(319\) 8.86925i 0.496583i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 34.9624 1.94536
\(324\) 0 0
\(325\) 5.33178 0.295754
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.20569 + 11.9328i −0.121604 + 0.657877i
\(330\) 0 0
\(331\) 3.62478 0.199236 0.0996178 0.995026i \(-0.468238\pi\)
0.0996178 + 0.995026i \(0.468238\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.1934 −0.611563
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 46.3509 2.51004
\(342\) 0 0
\(343\) −9.65406 + 15.8050i −0.521270 + 0.853392i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.7666 −0.846395 −0.423197 0.906037i \(-0.639092\pi\)
−0.423197 + 0.906037i \(0.639092\pi\)
\(348\) 0 0
\(349\) 17.1308 0.916988 0.458494 0.888697i \(-0.348389\pi\)
0.458494 + 0.888697i \(0.348389\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.6678i 1.41938i 0.704513 + 0.709691i \(0.251165\pi\)
−0.704513 + 0.709691i \(0.748835\pi\)
\(354\) 0 0
\(355\) 3.05348i 0.162062i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.1028i 1.32488i 0.749116 + 0.662439i \(0.230479\pi\)
−0.749116 + 0.662439i \(0.769521\pi\)
\(360\) 0 0
\(361\) −8.07494 −0.424997
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 26.5554i 1.38997i
\(366\) 0 0
\(367\) 22.0678 1.15193 0.575964 0.817475i \(-0.304627\pi\)
0.575964 + 0.817475i \(0.304627\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −20.9453 3.87158i −1.08742 0.201002i
\(372\) 0 0
\(373\) 19.9486i 1.03290i −0.856318 0.516449i \(-0.827254\pi\)
0.856318 0.516449i \(-0.172746\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.08100i 0.158679i
\(378\) 0 0
\(379\) −21.6497 −1.11207 −0.556036 0.831158i \(-0.687678\pi\)
−0.556036 + 0.831158i \(0.687678\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 29.2184 1.49299 0.746495 0.665391i \(-0.231735\pi\)
0.746495 + 0.665391i \(0.231735\pi\)
\(384\) 0 0
\(385\) 7.66589 41.4725i 0.390690 2.11364i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.70905i 0.289460i 0.989471 + 0.144730i \(0.0462314\pi\)
−0.989471 + 0.144730i \(0.953769\pi\)
\(390\) 0 0
\(391\) 29.7971 1.50690
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 40.8134i 2.05355i
\(396\) 0 0
\(397\) 13.2760 0.666302 0.333151 0.942874i \(-0.391888\pi\)
0.333151 + 0.942874i \(0.391888\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.1452 −1.30563 −0.652815 0.757518i \(-0.726412\pi\)
−0.652815 + 0.757518i \(0.726412\pi\)
\(402\) 0 0
\(403\) 16.1014 0.802066
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.4817i 1.26308i
\(408\) 0 0
\(409\) 14.5958i 0.721715i 0.932621 + 0.360857i \(0.117516\pi\)
−0.932621 + 0.360857i \(0.882484\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.4067 + 2.66297i 0.708908 + 0.131036i
\(414\) 0 0
\(415\) 24.5566i 1.20544i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.6124i 1.39781i −0.715216 0.698904i \(-0.753672\pi\)
0.715216 0.698904i \(-0.246328\pi\)
\(420\) 0 0
\(421\) 12.5115i 0.609773i 0.952389 + 0.304887i \(0.0986186\pi\)
−0.952389 + 0.304887i \(0.901381\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.9126i 0.868890i
\(426\) 0 0
\(427\) −5.22703 + 28.2783i −0.252954 + 1.36848i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 35.4537i 1.70775i −0.520481 0.853873i \(-0.674247\pi\)
0.520481 0.853873i \(-0.325753\pi\)
\(432\) 0 0
\(433\) 36.8860i 1.77263i −0.463086 0.886313i \(-0.653258\pi\)
0.463086 0.886313i \(-0.346742\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −23.0749 −1.10382
\(438\) 0 0
\(439\) −10.9710 −0.523617 −0.261809 0.965120i \(-0.584319\pi\)
−0.261809 + 0.965120i \(0.584319\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.5374 1.02327 0.511636 0.859202i \(-0.329039\pi\)
0.511636 + 0.859202i \(0.329039\pi\)
\(444\) 0 0
\(445\) 37.7860i 1.79123i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.2760 −0.815303 −0.407652 0.913138i \(-0.633652\pi\)
−0.407652 + 0.913138i \(0.633652\pi\)
\(450\) 0 0
\(451\) 1.20336i 0.0566639i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.66297 14.4067i 0.124842 0.675397i
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −37.8437 −1.76256 −0.881278 0.472599i \(-0.843316\pi\)
−0.881278 + 0.472599i \(0.843316\pi\)
\(462\) 0 0
\(463\) 2.99767i 0.139313i 0.997571 + 0.0696567i \(0.0221904\pi\)
−0.997571 + 0.0696567i \(0.977810\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.66822i 0.401117i 0.979682 + 0.200559i \(0.0642757\pi\)
−0.979682 + 0.200559i \(0.935724\pi\)
\(468\) 0 0
\(469\) −1.94419 + 10.5181i −0.0897741 + 0.485679i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 60.7576 2.79364
\(474\) 0 0
\(475\) 13.8716i 0.636472i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.26523 0.194883 0.0974417 0.995241i \(-0.468934\pi\)
0.0974417 + 0.995241i \(0.468934\pi\)
\(480\) 0 0
\(481\) 8.85181i 0.403608i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.6990i 0.576633i
\(486\) 0 0
\(487\) 22.9419i 1.03959i −0.854290 0.519797i \(-0.826007\pi\)
0.854290 0.519797i \(-0.173993\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.25641 0.417736 0.208868 0.977944i \(-0.433022\pi\)
0.208868 + 0.977944i \(0.433022\pi\)
\(492\) 0 0
\(493\) 10.3509 0.466181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.86925 0.530359i −0.128703 0.0237899i
\(498\) 0 0
\(499\) 5.54838 0.248380 0.124190 0.992258i \(-0.460367\pi\)
0.124190 + 0.992258i \(0.460367\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.26983 −0.413321 −0.206661 0.978413i \(-0.566260\pi\)
−0.206661 + 0.978413i \(0.566260\pi\)
\(504\) 0 0
\(505\) 7.66589 0.341128
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0447 1.15441 0.577205 0.816599i \(-0.304143\pi\)
0.577205 + 0.816599i \(0.304143\pi\)
\(510\) 0 0
\(511\) 24.9532 + 4.61241i 1.10386 + 0.204041i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −55.3290 −2.43808
\(516\) 0 0
\(517\) −26.4067 −1.16137
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 33.5960i 1.47187i −0.677054 0.735933i \(-0.736744\pi\)
0.677054 0.735933i \(-0.263256\pi\)
\(522\) 0 0
\(523\) 26.8181i 1.17267i −0.810067 0.586337i \(-0.800570\pi\)
0.810067 0.586337i \(-0.199430\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 54.0941i 2.35637i
\(528\) 0 0
\(529\) 3.33411 0.144961
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.418022i 0.0181065i
\(534\) 0 0
\(535\) 2.08428 0.0901112
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −37.6388 14.4067i −1.62122 0.620541i
\(540\) 0 0
\(541\) 28.2217i 1.21334i 0.794952 + 0.606672i \(0.207496\pi\)
−0.794952 + 0.606672i \(0.792504\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.5083i 1.04982i
\(546\) 0 0
\(547\) −1.28315 −0.0548635 −0.0274318 0.999624i \(-0.508733\pi\)
−0.0274318 + 0.999624i \(0.508733\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.01576 −0.341483
\(552\) 0 0
\(553\) 38.3509 + 7.08888i 1.63085 + 0.301450i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 40.7681i 1.72740i −0.504007 0.863700i \(-0.668141\pi\)
0.504007 0.863700i \(-0.331859\pi\)
\(558\) 0 0
\(559\) 21.1060 0.892687
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.9442i 0.840547i 0.907397 + 0.420274i \(0.138066\pi\)
−0.907397 + 0.420274i \(0.861934\pi\)
\(564\) 0 0
\(565\) −31.9442 −1.34390
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.8692 −1.12642 −0.563209 0.826315i \(-0.690433\pi\)
−0.563209 + 0.826315i \(0.690433\pi\)
\(570\) 0 0
\(571\) 29.7622 1.24551 0.622754 0.782418i \(-0.286014\pi\)
0.622754 + 0.782418i \(0.286014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.8222i 0.493021i
\(576\) 0 0
\(577\) 30.7240i 1.27906i 0.768768 + 0.639528i \(0.220870\pi\)
−0.768768 + 0.639528i \(0.779130\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −23.0749 4.26523i −0.957310 0.176952i
\(582\) 0 0
\(583\) 46.3509i 1.91966i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.33178i 0.137517i −0.997633 0.0687586i \(-0.978096\pi\)
0.997633 0.0687586i \(-0.0219038\pi\)
\(588\) 0 0
\(589\) 41.8906i 1.72607i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.29001i 0.135104i −0.997716 0.0675522i \(-0.978481\pi\)
0.997716 0.0675522i \(-0.0215189\pi\)
\(594\) 0 0
\(595\) −48.4007 8.94652i −1.98424 0.366771i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.71059i 0.151611i 0.997123 + 0.0758053i \(0.0241527\pi\)
−0.997123 + 0.0758053i \(0.975847\pi\)
\(600\) 0 0
\(601\) 13.3417i 0.544220i 0.962266 + 0.272110i \(0.0877214\pi\)
−0.962266 + 0.272110i \(0.912279\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 61.3207 2.49304
\(606\) 0 0
\(607\) −26.7510 −1.08579 −0.542895 0.839801i \(-0.682672\pi\)
−0.542895 + 0.839801i \(0.682672\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.17315 −0.371106
\(612\) 0 0
\(613\) 30.2791i 1.22296i 0.791259 + 0.611481i \(0.209426\pi\)
−0.791259 + 0.611481i \(0.790574\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.48165 0.301200 0.150600 0.988595i \(-0.451879\pi\)
0.150600 + 0.988595i \(0.451879\pi\)
\(618\) 0 0
\(619\) 16.1452i 0.648931i 0.945898 + 0.324465i \(0.105184\pi\)
−0.945898 + 0.324465i \(0.894816\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −35.5061 6.56305i −1.42252 0.262943i
\(624\) 0 0
\(625\) −31.2225 −1.24890
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 29.7385 1.18575
\(630\) 0 0
\(631\) 5.00233i 0.199140i 0.995031 + 0.0995698i \(0.0317467\pi\)
−0.995031 + 0.0995698i \(0.968253\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.1499i 0.402785i
\(636\) 0 0
\(637\) −13.0749 5.00460i −0.518048 0.198289i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28.5519 −1.12773 −0.563867 0.825866i \(-0.690687\pi\)
−0.563867 + 0.825866i \(0.690687\pi\)
\(642\) 0 0
\(643\) 8.27830i 0.326464i −0.986588 0.163232i \(-0.947808\pi\)
0.986588 0.163232i \(-0.0521919\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.9486 −0.784259 −0.392130 0.919910i \(-0.628261\pi\)
−0.392130 + 0.919910i \(0.628261\pi\)
\(648\) 0 0
\(649\) 31.8814i 1.25145i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.6825i 1.27896i 0.768806 + 0.639482i \(0.220851\pi\)
−0.768806 + 0.639482i \(0.779149\pi\)
\(654\) 0 0
\(655\) 8.66822i 0.338695i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.93046 0.153109 0.0765545 0.997065i \(-0.475608\pi\)
0.0765545 + 0.997065i \(0.475608\pi\)
\(660\) 0 0
\(661\) −11.6827 −0.454404 −0.227202 0.973848i \(-0.572958\pi\)
−0.227202 + 0.973848i \(0.572958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 37.4817 + 6.92820i 1.45348 + 0.268664i
\(666\) 0 0
\(667\) −6.83153 −0.264518
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −62.5785 −2.41582
\(672\) 0 0
\(673\) −17.6659 −0.680970 −0.340485 0.940250i \(-0.610591\pi\)
−0.340485 + 0.940250i \(0.610591\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.6571 −1.17825 −0.589124 0.808043i \(-0.700527\pi\)
−0.589124 + 0.808043i \(0.700527\pi\)
\(678\) 0 0
\(679\) 11.9328 + 2.20569i 0.457939 + 0.0846466i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23.8790 −0.913706 −0.456853 0.889542i \(-0.651023\pi\)
−0.456853 + 0.889542i \(0.651023\pi\)
\(684\) 0 0
\(685\) 53.9395 2.06092
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.1014i 0.613413i
\(690\) 0 0
\(691\) 44.5566i 1.69501i 0.530785 + 0.847506i \(0.321897\pi\)
−0.530785 + 0.847506i \(0.678103\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.9251i 0.490276i
\(696\) 0 0
\(697\) 1.40439 0.0531949
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.0940i 1.74095i 0.492215 + 0.870474i \(0.336188\pi\)
−0.492215 + 0.870474i \(0.663812\pi\)
\(702\) 0 0
\(703\) −23.0296 −0.868576
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.33149 7.20336i 0.0500757 0.270910i
\(708\) 0 0
\(709\) 23.8656i 0.896292i −0.893960 0.448146i \(-0.852085\pi\)
0.893960 0.448146i \(-0.147915\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 35.7017i 1.33704i
\(714\) 0 0
\(715\) 31.8814 1.19230
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.42919 −0.127887 −0.0639435 0.997954i \(-0.520368\pi\)
−0.0639435 + 0.997954i \(0.520368\pi\)
\(720\) 0 0
\(721\) −9.61007 + 51.9906i −0.357898 + 1.93623i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.10680i 0.152523i
\(726\) 0 0
\(727\) −13.0553 −0.484193 −0.242097 0.970252i \(-0.577835\pi\)
−0.242097 + 0.970252i \(0.577835\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 70.9075i 2.62261i
\(732\) 0 0
\(733\) −18.1452 −0.670209 −0.335104 0.942181i \(-0.608772\pi\)
−0.335104 + 0.942181i \(0.608772\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.2760 −0.857381
\(738\) 0 0
\(739\) −53.1131 −1.95380 −0.976898 0.213705i \(-0.931447\pi\)
−0.976898 + 0.213705i \(0.931447\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.2481i 0.779516i −0.920917 0.389758i \(-0.872559\pi\)
0.920917 0.389758i \(-0.127441\pi\)
\(744\) 0 0
\(745\) 27.6161i 1.01178i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.362018 1.95852i 0.0132278 0.0715628i
\(750\) 0 0
\(751\) 9.61474i 0.350847i 0.984493 + 0.175423i \(0.0561294\pi\)
−0.984493 + 0.175423i \(0.943871\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.74316i 0.281802i
\(756\) 0 0
\(757\) 16.8676i 0.613062i 0.951861 + 0.306531i \(0.0991684\pi\)
−0.951861 + 0.306531i \(0.900832\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.1576i 0.730712i −0.930868 0.365356i \(-0.880947\pi\)
0.930868 0.365356i \(-0.119053\pi\)
\(762\) 0 0
\(763\) −23.0296 4.25684i −0.833726 0.154108i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.0749i 0.399893i
\(768\) 0 0
\(769\) 35.7286i 1.28841i 0.764855 + 0.644203i \(0.222811\pi\)
−0.764855 + 0.644203i \(0.777189\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15.6938 −0.564467 −0.282233 0.959346i \(-0.591075\pi\)
−0.282233 + 0.959346i \(0.591075\pi\)
\(774\) 0 0
\(775\) 21.4622 0.770946
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.08756 −0.0389659
\(780\) 0 0
\(781\) 6.34951i 0.227203i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −60.7576 −2.16853
\(786\) 0 0
\(787\) 4.00000i 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.54838 + 30.0168i −0.197278 + 1.06727i
\(792\) 0 0
\(793\) −21.7385 −0.771957
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.4705 1.25643 0.628215 0.778039i \(-0.283786\pi\)
0.628215 + 0.778039i \(0.283786\pi\)
\(798\) 0 0
\(799\) 30.8181i 1.09027i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 55.2201i 1.94868i
\(804\) 0 0
\(805\) 31.9442 + 5.90464i 1.12588 + 0.208111i