Properties

Label 4032.2.v.e.1583.2
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 1583.2
Character \(\chi\) = 4032.1583
Dual form 4032.2.v.e.3599.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.62814 - 2.62814i) q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+(-2.62814 - 2.62814i) q^{5} -1.00000 q^{7} +(0.583140 - 0.583140i) q^{11} +(1.76590 + 1.76590i) q^{13} -3.63734i q^{17} +(-0.963353 + 0.963353i) q^{19} +3.77401i q^{23} +8.81423i q^{25} +(-4.76058 + 4.76058i) q^{29} +4.89207i q^{31} +(2.62814 + 2.62814i) q^{35} +(-4.66911 + 4.66911i) q^{37} -3.83668 q^{41} +(3.45278 + 3.45278i) q^{43} +11.6693 q^{47} +1.00000 q^{49} +(-5.48624 - 5.48624i) q^{53} -3.06515 q^{55} +(4.40688 - 4.40688i) q^{59} +(-7.99959 - 7.99959i) q^{61} -9.28208i q^{65} +(11.3524 - 11.3524i) q^{67} -5.85120i q^{71} +0.564936i q^{73} +(-0.583140 + 0.583140i) q^{77} +16.4692i q^{79} +(-6.92234 - 6.92234i) q^{83} +(-9.55945 + 9.55945i) q^{85} +10.7726 q^{89} +(-1.76590 - 1.76590i) q^{91} +5.06365 q^{95} +8.64969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{7} + O(q^{10}) \) \( 40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100}) \)

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
\(5\) −2.62814 2.62814i −1.17534 1.17534i −0.980918 0.194421i \(-0.937717\pi\)
−0.194421 0.980918i \(1.43772\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.583140 0.583140i 0.175823 0.175823i −0.613709 0.789532i \(-0.710323\pi\)
0.789532 + 0.613709i \(0.210323\pi\)
\(12\) 0 0
\(13\) 1.76590 + 1.76590i 0.489774 + 0.489774i 0.908235 0.418461i \(-0.137430\pi\)
−0.418461 + 0.908235i \(0.637430\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.63734i 0.882186i −0.897462 0.441093i \(-0.854591\pi\)
0.897462 0.441093i \(-0.145409\pi\)
\(18\) 0 0
\(19\) −0.963353 + 0.963353i −0.221008 + 0.221008i −0.808923 0.587915i \(-0.799949\pi\)
0.587915 + 0.808923i \(0.299949\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.77401i 0.786935i 0.919338 + 0.393468i \(0.128725\pi\)
−0.919338 + 0.393468i \(0.871275\pi\)
\(24\) 0 0
\(25\) 8.81423i 1.76285i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.76058 + 4.76058i −0.884018 + 0.884018i −0.993940 0.109922i \(-0.964940\pi\)
0.109922 + 0.993940i \(0.464940\pi\)
\(30\) 0 0
\(31\) 4.89207i 0.878641i 0.898330 + 0.439320i \(0.144781\pi\)
−0.898330 + 0.439320i \(0.855219\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.62814 + 2.62814i 0.444237 + 0.444237i
\(36\) 0 0
\(37\) −4.66911 + 4.66911i −0.767598 + 0.767598i −0.977683 0.210085i \(-0.932626\pi\)
0.210085 + 0.977683i \(0.432626\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.83668 −0.599188 −0.299594 0.954067i \(-0.596851\pi\)
−0.299594 + 0.954067i \(0.596851\pi\)
\(42\) 0 0
\(43\) 3.45278 + 3.45278i 0.526544 + 0.526544i 0.919540 0.392996i \(-0.128561\pi\)
−0.392996 + 0.919540i \(0.628561\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.6693 1.70214 0.851070 0.525052i \(-0.175954\pi\)
0.851070 + 0.525052i \(0.175954\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.48624 5.48624i −0.753593 0.753593i 0.221555 0.975148i \(-0.428887\pi\)
−0.975148 + 0.221555i \(0.928887\pi\)
\(54\) 0 0
\(55\) −3.06515 −0.413304
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.40688 4.40688i 0.573727 0.573727i −0.359441 0.933168i \(-0.617033\pi\)
0.933168 + 0.359441i \(0.117033\pi\)
\(60\) 0 0
\(61\) −7.99959 7.99959i −1.02424 1.02424i −0.999699 0.0245439i \(-0.992187\pi\)
−0.0245439 0.999699i \(-0.507813\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.28208i 1.15130i
\(66\) 0 0
\(67\) 11.3524 11.3524i 1.38692 1.38692i 0.555213 0.831708i \(-0.312637\pi\)
0.831708 0.555213i \(-0.187363\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.85120i 0.694409i −0.937789 0.347205i \(-0.887131\pi\)
0.937789 0.347205i \(-0.112869\pi\)
\(72\) 0 0
\(73\) 0.564936i 0.0661207i 0.999453 + 0.0330604i \(0.0105254\pi\)
−0.999453 + 0.0330604i \(0.989475\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.583140 + 0.583140i −0.0664550 + 0.0664550i
\(78\) 0 0
\(79\) 16.4692i 1.85293i 0.376382 + 0.926465i \(0.377168\pi\)
−0.376382 + 0.926465i \(0.622832\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.92234 6.92234i −0.759826 0.759826i 0.216465 0.976290i \(-0.430547\pi\)
−0.976290 + 0.216465i \(0.930547\pi\)
\(84\) 0 0
\(85\) −9.55945 + 9.55945i −1.03687 + 1.03687i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.7726 1.14189 0.570946 0.820988i \(-0.306577\pi\)
0.570946 + 0.820988i \(0.306577\pi\)
\(90\) 0 0
\(91\) −1.76590 1.76590i −0.185117 0.185117i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.06365 0.519520
\(96\) 0 0
\(97\) 8.64969 0.878243 0.439122 0.898428i \(-0.355290\pi\)
0.439122 + 0.898428i \(0.355290\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.9849 + 10.9849i 1.09303 + 1.09303i 0.995203 + 0.0978307i \(0.0311904\pi\)
0.0978307 + 0.995203i \(0.468810\pi\)
\(102\) 0 0
\(103\) 2.31765 0.228365 0.114183 0.993460i \(-0.463575\pi\)
0.114183 + 0.993460i \(0.463575\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.38720 4.38720i 0.424127 0.424127i −0.462495 0.886622i \(-0.653046\pi\)
0.886622 + 0.462495i \(0.153046\pi\)
\(108\) 0 0
\(109\) 1.97102 + 1.97102i 0.188790 + 0.188790i 0.795173 0.606383i \(-0.207380\pi\)
−0.606383 + 0.795173i \(0.707380\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.79490i 0.733283i 0.930362 + 0.366641i \(0.119492\pi\)
−0.930362 + 0.366641i \(0.880508\pi\)
\(114\) 0 0
\(115\) 9.91862 9.91862i 0.924916 0.924916i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.63734i 0.333435i
\(120\) 0 0
\(121\) 10.3199i 0.938172i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.0243 10.0243i 0.896603 0.896603i
\(126\) 0 0
\(127\) 5.11789i 0.454139i 0.973879 + 0.227070i \(0.0729145\pi\)
−0.973879 + 0.227070i \(0.927086\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.29337 + 6.29337i 0.549854 + 0.549854i 0.926399 0.376544i \(-0.122888\pi\)
−0.376544 + 0.926399i \(0.622888\pi\)
\(132\) 0 0
\(133\) 0.963353 0.963353i 0.0835333 0.0835333i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.2421 1.21678 0.608392 0.793636i \(-0.291815\pi\)
0.608392 + 0.793636i \(0.291815\pi\)
\(138\) 0 0
\(139\) −1.23142 1.23142i −0.104448 0.104448i 0.652952 0.757400i \(-0.273530\pi\)
−0.757400 + 0.652952i \(0.773530\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.05954 0.172227
\(144\) 0 0
\(145\) 25.0229 2.07804
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.2649 14.2649i −1.16863 1.16863i −0.982531 0.186098i \(-0.940416\pi\)
−0.186098 0.982531i \(1.44042\pi\)
\(150\) 0 0
\(151\) 15.2951 1.24470 0.622349 0.782740i \(-0.286178\pi\)
0.622349 + 0.782740i \(0.286178\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.8570 12.8570i 1.03270 1.03270i
\(156\) 0 0
\(157\) −4.79120 4.79120i −0.382379 0.382379i 0.489580 0.871959i \(-0.337150\pi\)
−0.871959 + 0.489580i \(0.837150\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.77401i 0.297434i
\(162\) 0 0
\(163\) −3.28056 + 3.28056i −0.256953 + 0.256953i −0.823814 0.566861i \(-0.808158\pi\)
0.566861 + 0.823814i \(0.308158\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.8772i 1.69291i 0.532461 + 0.846455i \(0.321267\pi\)
−0.532461 + 0.846455i \(0.678733\pi\)
\(168\) 0 0
\(169\) 6.76317i 0.520244i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.4598 12.4598i 0.947304 0.947304i −0.0513757 0.998679i \(-0.516361\pi\)
0.998679 + 0.0513757i \(0.0163606\pi\)
\(174\) 0 0
\(175\) 8.81423i 0.666293i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.24841 + 9.24841i 0.691259 + 0.691259i 0.962509 0.271250i \(-0.0874371\pi\)
−0.271250 + 0.962509i \(0.587437\pi\)
\(180\) 0 0
\(181\) −16.6449 + 16.6449i −1.23721 + 1.23721i −0.276072 + 0.961137i \(0.589033\pi\)
−0.961137 + 0.276072i \(0.910967\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 24.5422 1.80438
\(186\) 0 0
\(187\) −2.12108 2.12108i −0.155109 0.155109i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.93740 0.140185 0.0700927 0.997540i \(-0.477670\pi\)
0.0700927 + 0.997540i \(0.477670\pi\)
\(192\) 0 0
\(193\) −10.0684 −0.724742 −0.362371 0.932034i \(-0.618033\pi\)
−0.362371 + 0.932034i \(0.618033\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.45632 + 1.45632i 0.103758 + 0.103758i 0.757080 0.653322i \(-0.226625\pi\)
−0.653322 + 0.757080i \(0.726625\pi\)
\(198\) 0 0
\(199\) 16.3258 1.15730 0.578652 0.815574i \(-0.303579\pi\)
0.578652 + 0.815574i \(0.303579\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.76058 4.76058i 0.334127 0.334127i
\(204\) 0 0
\(205\) 10.0833 + 10.0833i 0.704250 + 0.704250i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.12354i 0.0777168i
\(210\) 0 0
\(211\) 6.81896 6.81896i 0.469437 0.469437i −0.432295 0.901732i \(-0.642296\pi\)
0.901732 + 0.432295i \(0.142296\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.1488i 1.23774i
\(216\) 0 0
\(217\) 4.89207i 0.332095i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.42320 6.42320i 0.432071 0.432071i
\(222\) 0 0
\(223\) 2.90066i 0.194243i −0.995273 0.0971213i \(-0.969037\pi\)
0.995273 0.0971213i \(-0.0309635\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.48182 + 9.48182i 0.629331 + 0.629331i 0.947900 0.318569i \(-0.103202\pi\)
−0.318569 + 0.947900i \(0.603202\pi\)
\(228\) 0 0
\(229\) 14.9933 14.9933i 0.990786 0.990786i −0.00917178 0.999958i \(-0.502920\pi\)
0.999958 + 0.00917178i \(0.00291951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.9359 −0.781945 −0.390973 0.920402i \(-0.627861\pi\)
−0.390973 + 0.920402i \(0.627861\pi\)
\(234\) 0 0
\(235\) −30.6685 30.6685i −2.00059 2.00059i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.4892 −0.937227 −0.468614 0.883403i \(-0.655246\pi\)
−0.468614 + 0.883403i \(0.655246\pi\)
\(240\) 0 0
\(241\) 12.5295 0.807094 0.403547 0.914959i \(-0.367777\pi\)
0.403547 + 0.914959i \(0.367777\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.62814 2.62814i −0.167906 0.167906i
\(246\) 0 0
\(247\) −3.40238 −0.216488
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.1384 12.1384i 0.766166 0.766166i −0.211263 0.977429i \(-0.567758\pi\)
0.977429 + 0.211263i \(0.0677576\pi\)
\(252\) 0 0
\(253\) 2.20078 + 2.20078i 0.138362 + 0.138362i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.8533i 0.801764i 0.916130 + 0.400882i \(0.131296\pi\)
−0.916130 + 0.400882i \(0.868704\pi\)
\(258\) 0 0
\(259\) 4.66911 4.66911i 0.290125 0.290125i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 25.0594i 1.54523i −0.634877 0.772613i \(-0.718949\pi\)
0.634877 0.772613i \(-0.281051\pi\)
\(264\) 0 0
\(265\) 28.8372i 1.77145i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.1085 21.1085i 1.28701 1.28701i 0.350409 0.936597i \(-0.386043\pi\)
0.936597 0.350409i \(-0.113957\pi\)
\(270\) 0 0
\(271\) 27.0517i 1.64327i 0.570013 + 0.821636i \(0.306938\pi\)
−0.570013 + 0.821636i \(0.693062\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.13993 + 5.13993i 0.309949 + 0.309949i
\(276\) 0 0
\(277\) 9.25373 9.25373i 0.556003 0.556003i −0.372164 0.928167i \(-0.621384\pi\)
0.928167 + 0.372164i \(0.121384\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.2367 −0.968600 −0.484300 0.874902i \(-0.660926\pi\)
−0.484300 + 0.874902i \(0.660926\pi\)
\(282\) 0 0
\(283\) 9.02844 + 9.02844i 0.536685 + 0.536685i 0.922554 0.385869i \(-0.126098\pi\)
−0.385869 + 0.922554i \(0.626098\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.83668 0.226472
\(288\) 0 0
\(289\) 3.76973 0.221749
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.6256 + 20.6256i 1.20496 + 1.20496i 0.972639 + 0.232324i \(0.0746328\pi\)
0.232324 + 0.972639i \(0.425367\pi\)
\(294\) 0 0
\(295\) −23.1638 −1.34865
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.66454 + 6.66454i −0.385420 + 0.385420i
\(300\) 0 0
\(301\) −3.45278 3.45278i −0.199015 0.199015i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 42.0481i 2.40767i
\(306\) 0 0
\(307\) −9.53094 + 9.53094i −0.543959 + 0.543959i −0.924687 0.380728i \(-0.875673\pi\)
0.380728 + 0.924687i \(0.375673\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.39638i 0.249296i 0.992201 + 0.124648i \(0.0397802\pi\)
−0.992201 + 0.124648i \(0.960220\pi\)
\(312\) 0 0
\(313\) 27.8579i 1.57462i −0.616558 0.787310i \(-0.711473\pi\)
0.616558 0.787310i \(-0.288527\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.5283 11.5283i 0.647491 0.647491i −0.304895 0.952386i \(-0.598621\pi\)
0.952386 + 0.304895i \(0.0986214\pi\)
\(318\) 0 0
\(319\) 5.55217i 0.310862i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.50405 + 3.50405i 0.194970 + 0.194970i
\(324\) 0 0
\(325\) −15.5651 + 15.5651i −0.863395 + 0.863395i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.6693 −0.643349
\(330\) 0 0
\(331\) 21.4888 + 21.4888i 1.18113 + 1.18113i 0.979451 + 0.201682i \(0.0646408\pi\)
0.201682 + 0.979451i \(0.435359\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −59.6716 −3.26021
\(336\) 0 0
\(337\) 6.64762 0.362119 0.181059 0.983472i \(-0.442047\pi\)
0.181059 + 0.983472i \(0.442047\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.85276 + 2.85276i 0.154486 + 0.154486i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.4459 + 19.4459i −1.04391 + 1.04391i −0.0449218 + 0.998991i \(0.514304\pi\)
−0.998991 + 0.0449218i \(0.985696\pi\)
\(348\) 0 0
\(349\) −6.38328 6.38328i −0.341689 0.341689i 0.515313 0.857002i \(-0.327676\pi\)
−0.857002 + 0.515313i \(0.827676\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.05143i 0.375310i −0.982235 0.187655i \(-0.939911\pi\)
0.982235 0.187655i \(-0.0600886\pi\)
\(354\) 0 0
\(355\) −15.3778 + 15.3778i −0.816167 + 0.816167i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.7598i 1.51788i −0.651159 0.758941i \(-0.725717\pi\)
0.651159 0.758941i \(-0.274283\pi\)
\(360\) 0 0
\(361\) 17.1439i 0.902311i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.48473 1.48473i 0.0777143 0.0777143i
\(366\) 0 0
\(367\) 9.89445i 0.516486i −0.966080 0.258243i \(-0.916856\pi\)
0.966080 0.258243i \(-0.0831435\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.48624 + 5.48624i 0.284831 + 0.284831i
\(372\) 0 0
\(373\) −24.6444 + 24.6444i −1.27604 + 1.27604i −0.333175 + 0.942865i \(0.608120\pi\)
−0.942865 + 0.333175i \(0.891880\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.8135 −0.865937
\(378\) 0 0
\(379\) −3.92064 3.92064i −0.201390 0.201390i 0.599206 0.800595i \(-0.295483\pi\)
−0.800595 + 0.599206i \(0.795483\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.5474 −0.947730 −0.473865 0.880598i \(-0.657142\pi\)
−0.473865 + 0.880598i \(0.657142\pi\)
\(384\) 0 0
\(385\) 3.06515 0.156214
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.7437 19.7437i −1.00104 1.00104i −0.999999 0.00104509i \(-0.999667\pi\)
−0.00104509 0.999999i \(1.49967\pi\)
\(390\) 0 0
\(391\) 13.7274 0.694223
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 43.2833 43.2833i 2.17782 2.17782i
\(396\) 0 0
\(397\) 13.4592 + 13.4592i 0.675497 + 0.675497i 0.958978 0.283481i \(-0.0914893\pi\)
−0.283481 + 0.958978i \(0.591489\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.4638i 1.47135i 0.677334 + 0.735676i \(0.263135\pi\)
−0.677334 + 0.735676i \(0.736865\pi\)
\(402\) 0 0
\(403\) −8.63891 + 8.63891i −0.430335 + 0.430335i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.44550i 0.269923i
\(408\) 0 0
\(409\) 15.9026i 0.786333i −0.919467 0.393167i \(-0.871380\pi\)
0.919467 0.393167i \(-0.128620\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.40688 + 4.40688i −0.216848 + 0.216848i
\(414\) 0 0
\(415\) 36.3858i 1.78611i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.45606 6.45606i −0.315399 0.315399i 0.531598 0.846997i \(-0.321592\pi\)
−0.846997 + 0.531598i \(0.821592\pi\)
\(420\) 0 0
\(421\) 11.8827 11.8827i 0.579126 0.579126i −0.355536 0.934663i \(-0.615702\pi\)
0.934663 + 0.355536i \(0.115702\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 32.0604 1.55516
\(426\) 0 0
\(427\) 7.99959 + 7.99959i 0.387127 + 0.387127i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.49791 0.264825 0.132412 0.991195i \(-0.457728\pi\)
0.132412 + 0.991195i \(0.457728\pi\)
\(432\) 0 0
\(433\) 20.0080 0.961522 0.480761 0.876852i \(-0.340360\pi\)
0.480761 + 0.876852i \(0.340360\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.63570 3.63570i −0.173919 0.173919i
\(438\) 0 0
\(439\) 0.00616772 0.000294369 0.000147185 1.00000i \(-0.499953\pi\)
0.000147185 1.00000i \(0.499953\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.02537 + 4.02537i −0.191251 + 0.191251i −0.796237 0.604985i \(-0.793179\pi\)
0.604985 + 0.796237i \(0.293179\pi\)
\(444\) 0 0
\(445\) −28.3118 28.3118i −1.34211 1.34211i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 19.6378i 0.926765i −0.886158 0.463382i \(-0.846636\pi\)
0.886158 0.463382i \(-0.153364\pi\)
\(450\) 0 0
\(451\) −2.23732 + 2.23732i −0.105351 + 0.105351i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.28208i 0.435151i
\(456\) 0 0
\(457\) 18.0292i 0.843372i 0.906742 + 0.421686i \(0.138562\pi\)
−0.906742 + 0.421686i \(0.861438\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.06055 2.06055i 0.0959694 0.0959694i −0.657492 0.753461i \(-0.728383\pi\)
0.753461 + 0.657492i \(0.228383\pi\)
\(462\) 0 0
\(463\) 38.8765i 1.80674i 0.428857 + 0.903372i \(0.358916\pi\)
−0.428857 + 0.903372i \(0.641084\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.88850 6.88850i −0.318762 0.318762i 0.529530 0.848291i \(-0.322368\pi\)
−0.848291 + 0.529530i \(0.822368\pi\)
\(468\) 0 0
\(469\) −11.3524 + 11.3524i −0.524207 + 0.524207i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.02691 0.185157
\(474\) 0 0
\(475\) −8.49121 8.49121i −0.389604 0.389604i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.25769 0.148848 0.0744239 0.997227i \(-0.476288\pi\)
0.0744239 + 0.997227i \(0.476288\pi\)
\(480\) 0 0
\(481\) −16.4904 −0.751898
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.7326 22.7326i −1.03223 1.03223i
\(486\) 0 0
\(487\) 27.6993 1.25517 0.627587 0.778547i \(-0.284043\pi\)
0.627587 + 0.778547i \(0.284043\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20.9676 + 20.9676i −0.946256 + 0.946256i −0.998628 0.0523720i \(-0.983322\pi\)
0.0523720 + 0.998628i \(0.483322\pi\)
\(492\) 0 0
\(493\) 17.3159 + 17.3159i 0.779868 + 0.779868i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.85120i 0.262462i
\(498\) 0 0
\(499\) −2.46939 + 2.46939i −0.110545 + 0.110545i −0.760216 0.649671i \(-0.774907\pi\)
0.649671 + 0.760216i \(0.274907\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19.9403i 0.889093i −0.895756 0.444547i \(-0.853365\pi\)
0.895756 0.444547i \(-0.146635\pi\)
\(504\) 0 0
\(505\) 57.7394i 2.56937i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.45882 + 7.45882i −0.330606 + 0.330606i −0.852817 0.522210i \(-0.825108\pi\)
0.522210 + 0.852817i \(0.325108\pi\)
\(510\) 0 0
\(511\) 0.564936i 0.0249913i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.09112 6.09112i −0.268407 0.268407i
\(516\) 0 0
\(517\) 6.80483 6.80483i 0.299276 0.299276i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.2354 −1.01796 −0.508981 0.860778i \(-0.669978\pi\)
−0.508981 + 0.860778i \(0.669978\pi\)
\(522\) 0 0
\(523\) −19.1847 19.1847i −0.838888 0.838888i 0.149824 0.988713i \(-0.452129\pi\)
−0.988713 + 0.149824i \(0.952129\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.7941 0.775124
\(528\) 0 0
\(529\) 8.75685 0.380733
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.77520 6.77520i −0.293467 0.293467i
\(534\) 0 0
\(535\) −23.0604 −0.996986
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.583140 0.583140i 0.0251176 0.0251176i
\(540\) 0 0
\(541\) −9.33342 9.33342i −0.401275 0.401275i 0.477407 0.878682i \(-0.341577\pi\)
−0.878682 + 0.477407i \(0.841577\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.3602i 0.443784i
\(546\) 0 0
\(547\) −12.2760 + 12.2760i −0.524883 + 0.524883i −0.919042 0.394159i \(-0.871036\pi\)
0.394159 + 0.919042i \(0.371036\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.17224i 0.390751i
\(552\) 0 0
\(553\) 16.4692i 0.700342i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.3475 + 14.3475i −0.607923 + 0.607923i −0.942403 0.334480i \(-0.891439\pi\)
0.334480 + 0.942403i \(0.391439\pi\)
\(558\) 0 0
\(559\) 12.1946i 0.515775i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.8013 + 14.8013i 0.623802 + 0.623802i 0.946501 0.322700i \(-0.104590\pi\)
−0.322700 + 0.946501i \(0.604590\pi\)
\(564\) 0 0
\(565\) 20.4861 20.4861i 0.861856 0.861856i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.30895 0.348329 0.174165 0.984717i \(-0.444277\pi\)
0.174165 + 0.984717i \(0.444277\pi\)
\(570\) 0 0
\(571\) −9.56981 9.56981i −0.400484 0.400484i 0.477920 0.878404i \(-0.341391\pi\)
−0.878404 + 0.477920i \(0.841391\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −33.2650 −1.38725
\(576\) 0 0
\(577\) 36.5616 1.52208 0.761039 0.648706i \(-0.224690\pi\)
0.761039 + 0.648706i \(0.224690\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.92234 + 6.92234i 0.287187 + 0.287187i
\(582\) 0 0
\(583\) −6.39849 −0.264998
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.86143 2.86143i 0.118104 0.118104i −0.645585 0.763689i \(-0.723386\pi\)
0.763689 + 0.645585i \(0.223386\pi\)
\(588\) 0 0
\(589\) −4.71279 4.71279i −0.194187 0.194187i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 40.5600i 1.66560i 0.553575 + 0.832799i \(0.313263\pi\)
−0.553575 + 0.832799i \(0.686737\pi\)
\(594\) 0 0
\(595\) 9.55945 9.55945i 0.391899 0.391899i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 41.0811i 1.67853i 0.543725 + 0.839263i \(0.317013\pi\)
−0.543725 + 0.839263i \(0.682987\pi\)
\(600\) 0 0
\(601\) 26.2778i 1.07190i −0.844251 0.535948i \(-0.819954\pi\)
0.844251 0.535948i \(-0.180046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 27.1221 27.1221i 1.10267 1.10267i
\(606\) 0 0
\(607\) 47.8974i 1.94410i 0.234784 + 0.972048i \(0.424562\pi\)
−0.234784 + 0.972048i \(0.575438\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.6068 + 20.6068i 0.833663 + 0.833663i
\(612\) 0 0
\(613\) 0.694342 0.694342i 0.0280442 0.0280442i −0.692946 0.720990i \(-0.743687\pi\)
0.720990 + 0.692946i \(0.243687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.60105 −0.185231 −0.0926156 0.995702i \(-0.529523\pi\)
−0.0926156 + 0.995702i \(0.529523\pi\)
\(618\) 0 0
\(619\) 6.50800 + 6.50800i 0.261579 + 0.261579i 0.825695 0.564117i \(-0.190783\pi\)
−0.564117 + 0.825695i \(0.690783\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.7726 −0.431594
\(624\) 0 0
\(625\) −8.61949 −0.344779
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9832 + 16.9832i 0.677164 + 0.677164i
\(630\) 0 0
\(631\) −30.8637 −1.22867 −0.614333 0.789047i \(-0.710575\pi\)
−0.614333 + 0.789047i \(0.710575\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.4505 13.4505i 0.533768 0.533768i
\(636\) 0 0
\(637\) 1.76590 + 1.76590i 0.0699676 + 0.0699676i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.57360i 0.0621536i −0.999517 0.0310768i \(-0.990106\pi\)
0.999517 0.0310768i \(-0.00989365\pi\)
\(642\) 0 0
\(643\) −0.203779 + 0.203779i −0.00803626 + 0.00803626i −0.711113 0.703077i \(-0.751809\pi\)
0.703077 + 0.711113i \(0.251809\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.73020i 0.303906i 0.988388 + 0.151953i \(0.0485562\pi\)
−0.988388 + 0.151953i \(0.951444\pi\)
\(648\) 0 0
\(649\) 5.13966i 0.201749i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.7507 + 25.7507i −1.00770 + 1.00770i −0.00773247 + 0.999970i \(0.502461\pi\)
−0.999970 + 0.00773247i \(0.997539\pi\)
\(654\) 0 0
\(655\) 33.0797i 1.29253i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.1443 + 14.1443i 0.550984 + 0.550984i 0.926725 0.375741i \(-0.122612\pi\)
−0.375741 + 0.926725i \(0.622612\pi\)
\(660\) 0 0
\(661\) −8.31619 + 8.31619i −0.323462 + 0.323462i −0.850094 0.526632i \(-0.823455\pi\)
0.526632 + 0.850094i \(0.323455\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.06365 −0.196360
\(666\) 0 0
\(667\) −17.9665 17.9665i −0.695665 0.695665i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.32976 −0.360172
\(672\) 0 0
\(673\) 34.9355 1.34667 0.673333 0.739339i \(-0.264862\pi\)
0.673333 + 0.739339i \(0.264862\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.44273 + 5.44273i 0.209181 + 0.209181i 0.803919 0.594738i \(-0.202744\pi\)
−0.594738 + 0.803919i \(0.702744\pi\)
\(678\) 0 0
\(679\) −8.64969 −0.331945
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.7753 + 33.7753i −1.29238 + 1.29238i −0.359063 + 0.933313i \(0.616904\pi\)
−0.933313 + 0.359063i \(0.883096\pi\)
\(684\) 0 0
\(685\) −37.4302 37.4302i −1.43014 1.43014i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.3763i 0.738179i
\(690\) 0 0
\(691\) 31.8468 31.8468i 1.21151 1.21151i 0.240979 0.970530i \(-0.422531\pi\)
0.970530 0.240979i \(-0.0774686\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.47269i 0.245523i
\(696\) 0 0
\(697\) 13.9553i 0.528595i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.0196 + 17.0196i −0.642822 + 0.642822i −0.951248 0.308426i \(-0.900198\pi\)
0.308426 + 0.951248i \(0.400198\pi\)
\(702\) 0 0
\(703\) 8.99601i 0.339291i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.9849 10.9849i −0.413128 0.413128i
\(708\) 0 0
\(709\) 4.45471 4.45471i 0.167300 0.167300i −0.618491 0.785792i \(-0.712256\pi\)
0.785792 + 0.618491i \(0.212256\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.4627 −0.691434
\(714\) 0 0
\(715\) −5.41275 5.41275i −0.202425 0.202425i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 17.3426 0.646771 0.323385 0.946267i \(-0.395179\pi\)
0.323385 + 0.946267i \(0.395179\pi\)
\(720\) 0 0
\(721\) −2.31765 −0.0863139
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −41.9609 41.9609i −1.55839 1.55839i
\(726\) 0 0
\(727\) −0.938724 −0.0348153 −0.0174077 0.999848i \(-0.505541\pi\)
−0.0174077 + 0.999848i \(0.505541\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.5590 12.5590i 0.464510 0.464510i
\(732\) 0 0
\(733\) −17.3212 17.3212i −0.639775 0.639775i 0.310725 0.950500i \(-0.399428\pi\)
−0.950500 + 0.310725i \(0.899428\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.2401i 0.487706i
\(738\) 0 0
\(739\) 30.7806 30.7806i 1.13228 1.13228i 0.142486 0.989797i \(-0.454490\pi\)
0.989797 0.142486i \(-0.0455096\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 41.3295i 1.51623i 0.652120 + 0.758116i \(0.273880\pi\)
−0.652120 + 0.758116i \(0.726120\pi\)
\(744\) 0 0
\(745\) 74.9805i 2.74707i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.38720 + 4.38720i −0.160305 + 0.160305i
\(750\) 0 0
\(751\) 50.1560i 1.83022i −0.403204 0.915110i \(-0.632104\pi\)
0.403204 0.915110i \(-0.367896\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −40.1977 40.1977i −1.46294 1.46294i
\(756\) 0 0
\(757\) −2.81463 + 2.81463i −0.102299 + 0.102299i −0.756404 0.654105i \(-0.773046\pi\)
0.654105 + 0.756404i \(0.273046\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.6910 0.786297 0.393149 0.919475i \(-0.371386\pi\)
0.393149 + 0.919475i \(0.371386\pi\)
\(762\) 0 0
\(763\) −1.97102 1.97102i −0.0713558 0.0713558i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.5642 0.561992
\(768\) 0 0
\(769\) −4.53040 −0.163371 −0.0816853 0.996658i \(-0.526030\pi\)
−0.0816853 + 0.996658i \(0.526030\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 23.6600 + 23.6600i 0.850990 + 0.850990i 0.990255 0.139265i \(-0.0444739\pi\)
−0.139265 + 0.990255i \(0.544474\pi\)
\(774\) 0 0
\(775\) −43.1198 −1.54891
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.69607 3.69607i 0.132426 0.132426i
\(780\) 0 0
\(781\) −3.41207 3.41207i −0.122093 0.122093i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.1839i 0.898850i
\(786\) 0 0
\(787\) 1.29221 1.29221i 0.0460623 0.0460623i −0.683700 0.729763i \(-0.739631\pi\)
0.729763 + 0.683700i \(0.239631\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.79490i 0.277155i
\(792\) 0 0
\(793\) 28.2530i 1.00329i
\(794\) 0 0