Properties

Label 4032.2.v.e.1583.16
Level 4032
Weight 2
Character 4032.1583
Analytic conductor 32.196
Analytic rank 0
Dimension 40
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.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1583.16
Character \(\chi\) = 4032.1583
Dual form 4032.2.v.e.3599.16

$q$-expansion

\(f(q)\) \(=\) \(q+(1.65702 + 1.65702i) q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+(1.65702 + 1.65702i) q^{5} -1.00000 q^{7} +(-0.993222 + 0.993222i) q^{11} +(2.62686 + 2.62686i) q^{13} +2.77728i q^{17} +(-1.56309 + 1.56309i) q^{19} -1.05272i q^{23} +0.491445i q^{25} +(-3.47823 + 3.47823i) q^{29} +1.06124i q^{31} +(-1.65702 - 1.65702i) q^{35} +(-0.0657126 + 0.0657126i) q^{37} -6.31313 q^{41} +(2.38841 + 2.38841i) q^{43} -1.47190 q^{47} +1.00000 q^{49} +(7.63807 + 7.63807i) q^{53} -3.29158 q^{55} +(-4.15490 + 4.15490i) q^{59} +(-7.78123 - 7.78123i) q^{61} +8.70553i q^{65} +(-1.98557 + 1.98557i) q^{67} -13.0855i q^{71} -9.50329i q^{73} +(0.993222 - 0.993222i) q^{77} +9.85051i q^{79} +(1.13002 + 1.13002i) q^{83} +(-4.60202 + 4.60202i) q^{85} -7.04453 q^{89} +(-2.62686 - 2.62686i) q^{91} -5.18016 q^{95} +5.35352 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{7} + O(q^{10}) \) \( 40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 1.65702 + 1.65702i 0.741043 + 0.741043i 0.972779 0.231736i \(-0.0744405\pi\)
−0.231736 + 0.972779i \(0.574441\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.993222 + 0.993222i −0.299468 + 0.299468i −0.840805 0.541338i \(-0.817918\pi\)
0.541338 + 0.840805i \(0.317918\pi\)
\(12\) 0 0
\(13\) 2.62686 + 2.62686i 0.728560 + 0.728560i 0.970333 0.241773i \(-0.0777290\pi\)
−0.241773 + 0.970333i \(0.577729\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.77728i 0.673589i 0.941578 + 0.336795i \(0.109343\pi\)
−0.941578 + 0.336795i \(0.890657\pi\)
\(18\) 0 0
\(19\) −1.56309 + 1.56309i −0.358598 + 0.358598i −0.863296 0.504698i \(-0.831604\pi\)
0.504698 + 0.863296i \(0.331604\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.05272i 0.219507i −0.993959 0.109754i \(-0.964994\pi\)
0.993959 0.109754i \(-0.0350062\pi\)
\(24\) 0 0
\(25\) 0.491445i 0.0982890i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.47823 + 3.47823i −0.645891 + 0.645891i −0.951997 0.306106i \(-0.900974\pi\)
0.306106 + 0.951997i \(0.400974\pi\)
\(30\) 0 0
\(31\) 1.06124i 0.190605i 0.995448 + 0.0953026i \(0.0303819\pi\)
−0.995448 + 0.0953026i \(0.969618\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.65702 1.65702i −0.280088 0.280088i
\(36\) 0 0
\(37\) −0.0657126 + 0.0657126i −0.0108031 + 0.0108031i −0.712488 0.701685i \(-0.752432\pi\)
0.701685 + 0.712488i \(0.252432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.31313 −0.985945 −0.492972 0.870045i \(-0.664090\pi\)
−0.492972 + 0.870045i \(0.664090\pi\)
\(42\) 0 0
\(43\) 2.38841 + 2.38841i 0.364229 + 0.364229i 0.865367 0.501138i \(-0.167085\pi\)
−0.501138 + 0.865367i \(0.667085\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.47190 −0.214699 −0.107349 0.994221i \(-0.534236\pi\)
−0.107349 + 0.994221i \(0.534236\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.63807 + 7.63807i 1.04917 + 1.04917i 0.998727 + 0.0504431i \(0.0160634\pi\)
0.0504431 + 0.998727i \(0.483937\pi\)
\(54\) 0 0
\(55\) −3.29158 −0.443837
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.15490 + 4.15490i −0.540921 + 0.540921i −0.923799 0.382878i \(-0.874933\pi\)
0.382878 + 0.923799i \(0.374933\pi\)
\(60\) 0 0
\(61\) −7.78123 7.78123i −0.996285 0.996285i 0.00370838 0.999993i \(-0.498820\pi\)
−0.999993 + 0.00370838i \(0.998820\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.70553i 1.07979i
\(66\) 0 0
\(67\) −1.98557 + 1.98557i −0.242576 + 0.242576i −0.817915 0.575339i \(-0.804870\pi\)
0.575339 + 0.817915i \(0.304870\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.0855i 1.55297i −0.630137 0.776484i \(-0.717001\pi\)
0.630137 0.776484i \(-0.282999\pi\)
\(72\) 0 0
\(73\) 9.50329i 1.11228i −0.831090 0.556138i \(-0.812283\pi\)
0.831090 0.556138i \(-0.187717\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.993222 0.993222i 0.113188 0.113188i
\(78\) 0 0
\(79\) 9.85051i 1.10827i 0.832427 + 0.554134i \(0.186951\pi\)
−0.832427 + 0.554134i \(0.813049\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.13002 + 1.13002i 0.124036 + 0.124036i 0.766400 0.642364i \(-0.222046\pi\)
−0.642364 + 0.766400i \(0.722046\pi\)
\(84\) 0 0
\(85\) −4.60202 + 4.60202i −0.499159 + 0.499159i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.04453 −0.746718 −0.373359 0.927687i \(-0.621794\pi\)
−0.373359 + 0.927687i \(0.621794\pi\)
\(90\) 0 0
\(91\) −2.62686 2.62686i −0.275370 0.275370i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.18016 −0.531473
\(96\) 0 0
\(97\) 5.35352 0.543568 0.271784 0.962358i \(-0.412386\pi\)
0.271784 + 0.962358i \(0.412386\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.34591 + 5.34591i 0.531938 + 0.531938i 0.921149 0.389211i \(-0.127252\pi\)
−0.389211 + 0.921149i \(0.627252\pi\)
\(102\) 0 0
\(103\) −3.41355 −0.336347 −0.168173 0.985757i \(-0.553787\pi\)
−0.168173 + 0.985757i \(0.553787\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.7004 + 10.7004i −1.03445 + 1.03445i −0.0350640 + 0.999385i \(0.511164\pi\)
−0.999385 + 0.0350640i \(0.988836\pi\)
\(108\) 0 0
\(109\) −5.36354 5.36354i −0.513734 0.513734i 0.401934 0.915668i \(-0.368338\pi\)
−0.915668 + 0.401934i \(0.868338\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.2463i 1.71647i 0.513256 + 0.858236i \(0.328439\pi\)
−0.513256 + 0.858236i \(0.671561\pi\)
\(114\) 0 0
\(115\) 1.74438 1.74438i 0.162664 0.162664i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.77728i 0.254593i
\(120\) 0 0
\(121\) 9.02702i 0.820638i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.47078 7.47078i 0.668206 0.668206i
\(126\) 0 0
\(127\) 0.0998053i 0.00885629i 0.999990 + 0.00442814i \(0.00140953\pi\)
−0.999990 + 0.00442814i \(0.998590\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.8436 + 12.8436i 1.12215 + 1.12215i 0.991417 + 0.130734i \(0.0417335\pi\)
0.130734 + 0.991417i \(0.458266\pi\)
\(132\) 0 0
\(133\) 1.56309 1.56309i 0.135537 0.135537i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.10918 0.265635 0.132817 0.991141i \(-0.457598\pi\)
0.132817 + 0.991141i \(0.457598\pi\)
\(138\) 0 0
\(139\) −10.5794 10.5794i −0.897330 0.897330i 0.0978696 0.995199i \(-0.468797\pi\)
−0.995199 + 0.0978696i \(0.968797\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.21811 −0.436360
\(144\) 0 0
\(145\) −11.5270 −0.957266
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.32230 + 6.32230i 0.517943 + 0.517943i 0.916948 0.399006i \(-0.130645\pi\)
−0.399006 + 0.916948i \(0.630645\pi\)
\(150\) 0 0
\(151\) −22.9393 −1.86677 −0.933385 0.358877i \(-0.883160\pi\)
−0.933385 + 0.358877i \(0.883160\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.75851 + 1.75851i −0.141247 + 0.141247i
\(156\) 0 0
\(157\) −12.7269 12.7269i −1.01571 1.01571i −0.999875 0.0158402i \(-0.994958\pi\)
−0.0158402 0.999875i \(-0.505042\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.05272i 0.0829660i
\(162\) 0 0
\(163\) −5.66445 + 5.66445i −0.443674 + 0.443674i −0.893245 0.449571i \(-0.851577\pi\)
0.449571 + 0.893245i \(0.351577\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.3445i 1.57431i 0.616756 + 0.787154i \(0.288446\pi\)
−0.616756 + 0.787154i \(0.711554\pi\)
\(168\) 0 0
\(169\) 0.800782i 0.0615986i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.94764 3.94764i 0.300133 0.300133i −0.540933 0.841066i \(-0.681929\pi\)
0.841066 + 0.540933i \(0.181929\pi\)
\(174\) 0 0
\(175\) 0.491445i 0.0371497i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.5761 + 12.5761i 0.939982 + 0.939982i 0.998298 0.0583166i \(-0.0185733\pi\)
−0.0583166 + 0.998298i \(0.518573\pi\)
\(180\) 0 0
\(181\) 16.0136 16.0136i 1.19028 1.19028i 0.213295 0.976988i \(-0.431580\pi\)
0.976988 0.213295i \(-0.0684196\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.217775 −0.0160111
\(186\) 0 0
\(187\) −2.75846 2.75846i −0.201718 0.201718i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.8336 −0.856251 −0.428126 0.903719i \(-0.640826\pi\)
−0.428126 + 0.903719i \(0.640826\pi\)
\(192\) 0 0
\(193\) 21.1021 1.51896 0.759481 0.650529i \(-0.225453\pi\)
0.759481 + 0.650529i \(0.225453\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.83308 5.83308i −0.415590 0.415590i 0.468091 0.883680i \(-0.344942\pi\)
−0.883680 + 0.468091i \(0.844942\pi\)
\(198\) 0 0
\(199\) −23.2865 −1.65074 −0.825369 0.564594i \(-0.809033\pi\)
−0.825369 + 0.564594i \(0.809033\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.47823 3.47823i 0.244124 0.244124i
\(204\) 0 0
\(205\) −10.4610 10.4610i −0.730627 0.730627i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.10500i 0.214777i
\(210\) 0 0
\(211\) 16.2720 16.2720i 1.12021 1.12021i 0.128501 0.991709i \(-0.458983\pi\)
0.991709 0.128501i \(-0.0410165\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.91529i 0.539818i
\(216\) 0 0
\(217\) 1.06124i 0.0720420i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.29553 + 7.29553i −0.490750 + 0.490750i
\(222\) 0 0
\(223\) 22.8221i 1.52828i 0.645051 + 0.764139i \(0.276836\pi\)
−0.645051 + 0.764139i \(0.723164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.09419 + 6.09419i 0.404486 + 0.404486i 0.879810 0.475325i \(-0.157669\pi\)
−0.475325 + 0.879810i \(0.657669\pi\)
\(228\) 0 0
\(229\) −8.57148 + 8.57148i −0.566420 + 0.566420i −0.931124 0.364704i \(-0.881170\pi\)
0.364704 + 0.931124i \(0.381170\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.90649 −0.321435 −0.160717 0.987000i \(-0.551381\pi\)
−0.160717 + 0.987000i \(0.551381\pi\)
\(234\) 0 0
\(235\) −2.43897 2.43897i −0.159101 0.159101i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.1659 0.722260 0.361130 0.932515i \(-0.382391\pi\)
0.361130 + 0.932515i \(0.382391\pi\)
\(240\) 0 0
\(241\) −17.8083 −1.14713 −0.573566 0.819159i \(-0.694440\pi\)
−0.573566 + 0.819159i \(0.694440\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.65702 + 1.65702i 0.105863 + 0.105863i
\(246\) 0 0
\(247\) −8.21206 −0.522521
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.7431 + 10.7431i −0.678099 + 0.678099i −0.959570 0.281471i \(-0.909178\pi\)
0.281471 + 0.959570i \(0.409178\pi\)
\(252\) 0 0
\(253\) 1.04559 + 1.04559i 0.0657354 + 0.0657354i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.20282i 0.0750300i 0.999296 + 0.0375150i \(0.0119442\pi\)
−0.999296 + 0.0375150i \(0.988056\pi\)
\(258\) 0 0
\(259\) 0.0657126 0.0657126i 0.00408318 0.00408318i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.9654i 0.922803i −0.887191 0.461402i \(-0.847347\pi\)
0.887191 0.461402i \(-0.152653\pi\)
\(264\) 0 0
\(265\) 25.3129i 1.55496i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.58780 6.58780i 0.401665 0.401665i −0.477154 0.878819i \(-0.658332\pi\)
0.878819 + 0.477154i \(0.158332\pi\)
\(270\) 0 0
\(271\) 27.1436i 1.64886i −0.565965 0.824429i \(-0.691496\pi\)
0.565965 0.824429i \(-0.308504\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.488114 0.488114i −0.0294344 0.0294344i
\(276\) 0 0
\(277\) 12.0118 12.0118i 0.721720 0.721720i −0.247235 0.968955i \(-0.579522\pi\)
0.968955 + 0.247235i \(0.0795221\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.46830 −0.505176 −0.252588 0.967574i \(-0.581282\pi\)
−0.252588 + 0.967574i \(0.581282\pi\)
\(282\) 0 0
\(283\) −8.77919 8.77919i −0.521868 0.521868i 0.396267 0.918135i \(-0.370305\pi\)
−0.918135 + 0.396267i \(0.870305\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.31313 0.372652
\(288\) 0 0
\(289\) 9.28671 0.546277
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.25965 9.25965i −0.540955 0.540955i 0.382854 0.923809i \(-0.374941\pi\)
−0.923809 + 0.382854i \(0.874941\pi\)
\(294\) 0 0
\(295\) −13.7695 −0.801692
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.76535 2.76535i 0.159924 0.159924i
\(300\) 0 0
\(301\) −2.38841 2.38841i −0.137665 0.137665i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 25.7873i 1.47658i
\(306\) 0 0
\(307\) 9.60065 9.60065i 0.547938 0.547938i −0.377906 0.925844i \(-0.623356\pi\)
0.925844 + 0.377906i \(0.123356\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.1840i 1.31464i 0.753611 + 0.657321i \(0.228310\pi\)
−0.753611 + 0.657321i \(0.771690\pi\)
\(312\) 0 0
\(313\) 27.9361i 1.57904i 0.613723 + 0.789522i \(0.289671\pi\)
−0.613723 + 0.789522i \(0.710329\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.5910 + 21.5910i −1.21267 + 1.21267i −0.242529 + 0.970144i \(0.577977\pi\)
−0.970144 + 0.242529i \(0.922023\pi\)
\(318\) 0 0
\(319\) 6.90931i 0.386847i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.34115 4.34115i −0.241548 0.241548i
\(324\) 0 0
\(325\) −1.29096 + 1.29096i −0.0716094 + 0.0716094i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.47190 0.0811485
\(330\) 0 0
\(331\) 18.4892 + 18.4892i 1.01626 + 1.01626i 0.999866 + 0.0163917i \(0.00521788\pi\)
0.0163917 + 0.999866i \(0.494782\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.58027 −0.359519
\(336\) 0 0
\(337\) −8.58189 −0.467486 −0.233743 0.972298i \(-0.575097\pi\)
−0.233743 + 0.972298i \(0.575097\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.05405 1.05405i −0.0570801 0.0570801i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.6238 19.6238i 1.05346 1.05346i 0.0549718 0.998488i \(-0.482493\pi\)
0.998488 0.0549718i \(-0.0175069\pi\)
\(348\) 0 0
\(349\) 7.67820 + 7.67820i 0.411004 + 0.411004i 0.882088 0.471084i \(-0.156137\pi\)
−0.471084 + 0.882088i \(0.656137\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 25.9405i 1.38067i 0.723489 + 0.690336i \(0.242537\pi\)
−0.723489 + 0.690336i \(0.757463\pi\)
\(354\) 0 0
\(355\) 21.6830 21.6830i 1.15082 1.15082i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.08367i 0.162750i 0.996684 + 0.0813750i \(0.0259312\pi\)
−0.996684 + 0.0813750i \(0.974069\pi\)
\(360\) 0 0
\(361\) 14.1135i 0.742814i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.7472 15.7472i 0.824244 0.824244i
\(366\) 0 0
\(367\) 18.1386i 0.946825i 0.880841 + 0.473412i \(0.156978\pi\)
−0.880841 + 0.473412i \(0.843022\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.63807 7.63807i −0.396549 0.396549i
\(372\) 0 0
\(373\) −1.70711 + 1.70711i −0.0883908 + 0.0883908i −0.749920 0.661529i \(-0.769908\pi\)
0.661529 + 0.749920i \(0.269908\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −18.2736 −0.941140
\(378\) 0 0
\(379\) 24.2888 + 24.2888i 1.24763 + 1.24763i 0.956762 + 0.290870i \(0.0939448\pi\)
0.290870 + 0.956762i \(0.406055\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.2655 −0.524544 −0.262272 0.964994i \(-0.584472\pi\)
−0.262272 + 0.964994i \(0.584472\pi\)
\(384\) 0 0
\(385\) 3.29158 0.167755
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.924351 0.924351i −0.0468664 0.0468664i 0.683285 0.730152i \(-0.260551\pi\)
−0.730152 + 0.683285i \(0.760551\pi\)
\(390\) 0 0
\(391\) 2.92370 0.147858
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.3225 + 16.3225i −0.821275 + 0.821275i
\(396\) 0 0
\(397\) −18.2727 18.2727i −0.917080 0.917080i 0.0797361 0.996816i \(-0.474592\pi\)
−0.996816 + 0.0797361i \(0.974592\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.9686i 1.84612i 0.384651 + 0.923062i \(0.374322\pi\)
−0.384651 + 0.923062i \(0.625678\pi\)
\(402\) 0 0
\(403\) −2.78774 + 2.78774i −0.138867 + 0.138867i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.130534i 0.00647035i
\(408\) 0 0
\(409\) 4.66305i 0.230573i 0.993332 + 0.115286i \(0.0367786\pi\)
−0.993332 + 0.115286i \(0.963221\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.15490 4.15490i 0.204449 0.204449i
\(414\) 0 0
\(415\) 3.74494i 0.183832i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.9392 + 20.9392i 1.02294 + 1.02294i 0.999731 + 0.0232142i \(0.00738996\pi\)
0.0232142 + 0.999731i \(0.492610\pi\)
\(420\) 0 0
\(421\) −18.0980 + 18.0980i −0.882042 + 0.882042i −0.993742 0.111700i \(-0.964370\pi\)
0.111700 + 0.993742i \(0.464370\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.36488 −0.0662064
\(426\) 0 0
\(427\) 7.78123 + 7.78123i 0.376560 + 0.376560i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.3329 −1.46108 −0.730541 0.682868i \(-0.760732\pi\)
−0.730541 + 0.682868i \(0.760732\pi\)
\(432\) 0 0
\(433\) −25.8726 −1.24336 −0.621679 0.783272i \(-0.713549\pi\)
−0.621679 + 0.783272i \(0.713549\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.64550 + 1.64550i 0.0787150 + 0.0787150i
\(438\) 0 0
\(439\) 18.1840 0.867874 0.433937 0.900943i \(-0.357124\pi\)
0.433937 + 0.900943i \(0.357124\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.0172574 0.0172574i 0.000819922 0.000819922i −0.706697 0.707517i \(-0.749815\pi\)
0.707517 + 0.706697i \(0.249815\pi\)
\(444\) 0 0
\(445\) −11.6729 11.6729i −0.553350 0.553350i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.83335i 0.416872i −0.978036 0.208436i \(-0.933163\pi\)
0.978036 0.208436i \(-0.0668372\pi\)
\(450\) 0 0
\(451\) 6.27034 6.27034i 0.295259 0.295259i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.70553i 0.408121i
\(456\) 0 0
\(457\) 3.53330i 0.165281i 0.996579 + 0.0826404i \(0.0263353\pi\)
−0.996579 + 0.0826404i \(0.973665\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13.9587 13.9587i 0.650123 0.650123i −0.302900 0.953022i \(-0.597955\pi\)
0.953022 + 0.302900i \(0.0979547\pi\)
\(462\) 0 0
\(463\) 10.6448i 0.494705i −0.968926 0.247353i \(-0.920439\pi\)
0.968926 0.247353i \(-0.0795606\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.4545 + 12.4545i 0.576325 + 0.576325i 0.933889 0.357564i \(-0.116393\pi\)
−0.357564 + 0.933889i \(0.616393\pi\)
\(468\) 0 0
\(469\) 1.98557 1.98557i 0.0916852 0.0916852i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.74444 −0.218149
\(474\) 0 0
\(475\) −0.768175 0.768175i −0.0352463 0.0352463i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 22.0204 1.00614 0.503068 0.864247i \(-0.332205\pi\)
0.503068 + 0.864247i \(0.332205\pi\)
\(480\) 0 0
\(481\) −0.345236 −0.0157414
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.87091 + 8.87091i 0.402807 + 0.402807i
\(486\) 0 0
\(487\) 23.2822 1.05502 0.527509 0.849549i \(-0.323126\pi\)
0.527509 + 0.849549i \(0.323126\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.129520 + 0.129520i −0.00584515 + 0.00584515i −0.710023 0.704178i \(-0.751316\pi\)
0.704178 + 0.710023i \(0.251316\pi\)
\(492\) 0 0
\(493\) −9.66002 9.66002i −0.435065 0.435065i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.0855i 0.586967i
\(498\) 0 0
\(499\) 10.7818 10.7818i 0.482661 0.482661i −0.423319 0.905980i \(-0.639135\pi\)
0.905980 + 0.423319i \(0.139135\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.04034i 0.269326i −0.990891 0.134663i \(-0.957005\pi\)
0.990891 0.134663i \(-0.0429951\pi\)
\(504\) 0 0
\(505\) 17.7166i 0.788378i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.74993 + 9.74993i −0.432158 + 0.432158i −0.889362 0.457204i \(-0.848851\pi\)
0.457204 + 0.889362i \(0.348851\pi\)
\(510\) 0 0
\(511\) 9.50329i 0.420401i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.65632 5.65632i −0.249247 0.249247i
\(516\) 0 0
\(517\) 1.46192 1.46192i 0.0642954 0.0642954i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.68047 −0.0736225 −0.0368113 0.999322i \(-0.511720\pi\)
−0.0368113 + 0.999322i \(0.511720\pi\)
\(522\) 0 0
\(523\) 16.9606 + 16.9606i 0.741635 + 0.741635i 0.972892 0.231258i \(-0.0742841\pi\)
−0.231258 + 0.972892i \(0.574284\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.94737 −0.128390
\(528\) 0 0
\(529\) 21.8918 0.951816
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.5837 16.5837i −0.718320 0.718320i
\(534\) 0 0
\(535\) −35.4617 −1.53314
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.993222 + 0.993222i −0.0427811 + 0.0427811i
\(540\) 0 0
\(541\) −5.82266 5.82266i −0.250336 0.250336i 0.570773 0.821108i \(-0.306644\pi\)
−0.821108 + 0.570773i \(0.806644\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.7750i 0.761398i
\(546\) 0 0
\(547\) 15.9661 15.9661i 0.682662 0.682662i −0.277937 0.960599i \(-0.589651\pi\)
0.960599 + 0.277937i \(0.0896507\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.8736i 0.463231i
\(552\) 0 0
\(553\) 9.85051i 0.418886i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1695 14.1695i 0.600381 0.600381i −0.340033 0.940414i \(-0.610438\pi\)
0.940414 + 0.340033i \(0.110438\pi\)
\(558\) 0 0
\(559\) 12.5480i 0.530725i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.96715 + 5.96715i 0.251485 + 0.251485i 0.821579 0.570094i \(-0.193093\pi\)
−0.570094 + 0.821579i \(0.693093\pi\)
\(564\) 0 0
\(565\) −30.2346 + 30.2346i −1.27198 + 1.27198i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.5770 −0.611099 −0.305550 0.952176i \(-0.598840\pi\)
−0.305550 + 0.952176i \(0.598840\pi\)
\(570\) 0 0
\(571\) 1.94424 + 1.94424i 0.0813641 + 0.0813641i 0.746618 0.665253i \(-0.231677\pi\)
−0.665253 + 0.746618i \(0.731677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.517354 0.0215752
\(576\) 0 0
\(577\) −5.60353 −0.233278 −0.116639 0.993174i \(-0.537212\pi\)
−0.116639 + 0.993174i \(0.537212\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.13002 1.13002i −0.0468812 0.0468812i
\(582\) 0 0
\(583\) −15.1726 −0.628385
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.8040 15.8040i 0.652300 0.652300i −0.301246 0.953546i \(-0.597403\pi\)
0.953546 + 0.301246i \(0.0974026\pi\)
\(588\) 0 0
\(589\) −1.65883 1.65883i −0.0683507 0.0683507i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.08911i 0.332180i −0.986111 0.166090i \(-0.946886\pi\)
0.986111 0.166090i \(-0.0531142\pi\)
\(594\) 0 0
\(595\) 4.60202 4.60202i 0.188664 0.188664i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25.5885i 1.04552i −0.852480 0.522759i \(-0.824903\pi\)
0.852480 0.522759i \(-0.175097\pi\)
\(600\) 0 0
\(601\) 26.8368i 1.09470i 0.836905 + 0.547348i \(0.184363\pi\)
−0.836905 + 0.547348i \(0.815637\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −14.9580 + 14.9580i −0.608128 + 0.608128i
\(606\) 0 0
\(607\) 17.9052i 0.726750i −0.931643 0.363375i \(-0.881624\pi\)
0.931643 0.363375i \(-0.118376\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.86648 3.86648i −0.156421 0.156421i
\(612\) 0 0
\(613\) 13.9653 13.9653i 0.564053 0.564053i −0.366403 0.930456i \(-0.619411\pi\)
0.930456 + 0.366403i \(0.119411\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.19844 0.249540 0.124770 0.992186i \(-0.460181\pi\)
0.124770 + 0.992186i \(0.460181\pi\)
\(618\) 0 0
\(619\) −26.7837 26.7837i −1.07653 1.07653i −0.996818 0.0797087i \(-0.974601\pi\)
−0.0797087 0.996818i \(-0.525399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.04453 0.282233
\(624\) 0 0
\(625\) 27.2157 1.08863
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.182502 0.182502i −0.00727685 0.00727685i
\(630\) 0 0
\(631\) 30.5763 1.21723 0.608613 0.793468i \(-0.291726\pi\)
0.608613 + 0.793468i \(0.291726\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.165380 + 0.165380i −0.00656289 + 0.00656289i
\(636\) 0 0
\(637\) 2.62686 + 2.62686i 0.104080 + 0.104080i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 43.7713i 1.72886i −0.502752 0.864431i \(-0.667679\pi\)
0.502752 0.864431i \(-0.332321\pi\)
\(642\) 0 0
\(643\) 24.5474 24.5474i 0.968053 0.968053i −0.0314518 0.999505i \(-0.510013\pi\)
0.999505 + 0.0314518i \(0.0100131\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.216780i 0.00852248i 0.999991 + 0.00426124i \(0.00135640\pi\)
−0.999991 + 0.00426124i \(0.998644\pi\)
\(648\) 0 0
\(649\) 8.25347i 0.323977i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.1867 12.1867i 0.476904 0.476904i −0.427236 0.904140i \(-0.640513\pi\)
0.904140 + 0.427236i \(0.140513\pi\)
\(654\) 0 0
\(655\) 42.5643i 1.66313i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.3745 + 14.3745i 0.559951 + 0.559951i 0.929293 0.369343i \(-0.120417\pi\)
−0.369343 + 0.929293i \(0.620417\pi\)
\(660\) 0 0
\(661\) 28.7815 28.7815i 1.11947 1.11947i 0.127651 0.991819i \(-0.459256\pi\)
0.991819 0.127651i \(-0.0407436\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.18016 0.200878
\(666\) 0 0
\(667\) 3.66160 + 3.66160i 0.141778 + 0.141778i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.4570 0.596710
\(672\) 0 0
\(673\) 39.5913 1.52613 0.763066 0.646320i \(-0.223693\pi\)
0.763066 + 0.646320i \(0.223693\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.7471 13.7471i −0.528342 0.528342i 0.391735 0.920078i \(-0.371875\pi\)
−0.920078 + 0.391735i \(0.871875\pi\)
\(678\) 0 0
\(679\) −5.35352 −0.205449
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.00325507 0.00325507i 0.000124552 0.000124552i −0.707045 0.707169i \(-0.749972\pi\)
0.707169 + 0.707045i \(0.249972\pi\)
\(684\) 0 0
\(685\) 5.15197 + 5.15197i 0.196847 + 0.196847i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 40.1283i 1.52877i
\(690\) 0 0
\(691\) 17.5751 17.5751i 0.668590 0.668590i −0.288800 0.957390i \(-0.593256\pi\)
0.957390 + 0.288800i \(0.0932562\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 35.0605i 1.32992i
\(696\) 0 0
\(697\) 17.5333i 0.664122i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.3683 29.3683i 1.10923 1.10923i 0.115975 0.993252i \(-0.463001\pi\)
0.993252 0.115975i \(-0.0369991\pi\)
\(702\) 0 0
\(703\) 0.205430i 0.00774794i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.34591 5.34591i −0.201054 0.201054i
\(708\) 0 0
\(709\) −21.5493 + 21.5493i −0.809300 + 0.809300i −0.984528 0.175228i \(-0.943934\pi\)
0.175228 + 0.984528i \(0.443934\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.11719 0.0418393
\(714\) 0 0
\(715\) −8.64652 8.64652i −0.323362 0.323362i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.1027 0.861585 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(720\) 0 0
\(721\) 3.41355 0.127127
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.70936 1.70936i −0.0634840 0.0634840i
\(726\) 0 0
\(727\) 10.8397 0.402024 0.201012 0.979589i \(-0.435577\pi\)
0.201012 + 0.979589i \(0.435577\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.63328 + 6.63328i −0.245341 + 0.245341i
\(732\) 0 0
\(733\) −5.15354 5.15354i −0.190350 0.190350i 0.605497 0.795848i \(-0.292974\pi\)
−0.795848 + 0.605497i \(0.792974\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.94423i 0.145287i
\(738\) 0 0
\(739\) −12.6864 + 12.6864i −0.466678 + 0.466678i −0.900837 0.434158i \(-0.857046\pi\)
0.434158 + 0.900837i \(0.357046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.3109i 0.891879i −0.895063 0.445940i \(-0.852870\pi\)
0.895063 0.445940i \(-0.147130\pi\)
\(744\) 0 0
\(745\) 20.9524i 0.767635i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.7004 10.7004i 0.390985 0.390985i
\(750\) 0 0
\(751\) 47.8161i 1.74483i −0.488762 0.872417i \(-0.662551\pi\)
0.488762 0.872417i \(-0.337449\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −38.0109 38.0109i −1.38336 1.38336i
\(756\) 0 0
\(757\) 20.6181 20.6181i 0.749377 0.749377i −0.224986 0.974362i \(-0.572234\pi\)
0.974362 + 0.224986i \(0.0722335\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.6635 −0.676553 −0.338276 0.941047i \(-0.609844\pi\)
−0.338276 + 0.941047i \(0.609844\pi\)
\(762\) 0 0
\(763\) 5.36354 + 5.36354i 0.194173 + 0.194173i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −21.8287 −0.788187
\(768\) 0 0
\(769\) −13.7015 −0.494090 −0.247045 0.969004i \(-0.579459\pi\)
−0.247045 + 0.969004i \(0.579459\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.1383 + 21.1383i 0.760294 + 0.760294i 0.976375 0.216082i \(-0.0693278\pi\)
−0.216082 + 0.976375i \(0.569328\pi\)
\(774\) 0 0
\(775\) −0.521543 −0.0187344
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.86801 9.86801i 0.353558 0.353558i
\(780\) 0 0
\(781\) 12.9969 + 12.9969i 0.465064 + 0.465064i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 42.1774i 1.50538i
\(786\) 0 0
\(787\) −25.7503 + 25.7503i −0.917899 + 0.917899i −0.996876 0.0789778i \(-0.974834\pi\)
0.0789778 + 0.996876i \(0.474834\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.2463i 0.648765i
\(792\) 0 0
\(793\) 40.8804i 1.45171i
\(794\) 0