Properties

Label 804.2.s.a.713.2
Level 804
Weight 2
Character 804.713
Analytic conductor 6.420
Analytic rank 0
Dimension 20
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.s (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label 713.2
Root \(-0.327068 + 0.945001i\)
Character \(\chi\) = 804.713
Dual form 804.2.s.a.53.2

$q$-expansion

\(f(q)\) \(=\) \(q+(0.936417 + 1.45709i) q^{3} +(4.81332 + 2.19817i) q^{7} +(-1.24625 + 2.72890i) q^{9} +O(q^{10})\) \(q+(0.936417 + 1.45709i) q^{3} +(4.81332 + 2.19817i) q^{7} +(-1.24625 + 2.72890i) q^{9} +(-0.298303 + 1.01593i) q^{13} +(-2.01775 - 4.41826i) q^{19} +(1.30434 + 9.07186i) q^{21} +(4.79746 + 1.40866i) q^{25} +(-5.14326 + 0.739490i) q^{27} +(-1.53214 - 5.21798i) q^{31} -7.64303 q^{37} +(-1.75964 + 0.516676i) q^{39} +(5.83248 + 5.05388i) q^{43} +(13.7521 + 15.8707i) q^{49} +(4.54836 - 7.07739i) q^{57} +(1.72288 - 0.247713i) q^{61} +(-11.9972 + 10.3956i) q^{63} +(-8.16392 - 0.591970i) q^{67} +(-1.74937 - 12.1671i) q^{73} +(2.43988 + 8.30945i) q^{75} +(-4.36459 + 14.8644i) q^{79} +(-5.89375 - 6.80175i) q^{81} +(-3.66900 + 4.23426i) q^{91} +(6.16836 - 7.11867i) q^{93} -19.2257i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{22}\right)\) \(1\)

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
\(3\) 0.936417 + 1.45709i 0.540641 + 0.841254i
\(4\) 0 0
\(5\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(6\) 0 0
\(7\) 4.81332 + 2.19817i 1.81926 + 0.830830i 0.909746 + 0.415166i \(0.136276\pi\)
0.909518 + 0.415664i \(0.136451\pi\)
\(8\) 0 0
\(9\) −1.24625 + 2.72890i −0.415415 + 0.909632i
\(10\) 0 0
\(11\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(12\) 0 0
\(13\) −0.298303 + 1.01593i −0.0827343 + 0.281767i −0.990461 0.137795i \(-0.955998\pi\)
0.907726 + 0.419563i \(0.137817\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(18\) 0 0
\(19\) −2.01775 4.41826i −0.462904 1.01362i −0.986816 0.161846i \(-0.948255\pi\)
0.523912 0.851772i \(-0.324472\pi\)
\(20\) 0 0
\(21\) 1.30434 + 9.07186i 0.284630 + 1.97964i
\(22\) 0 0
\(23\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(24\) 0 0
\(25\) 4.79746 + 1.40866i 0.959493 + 0.281733i
\(26\) 0 0
\(27\) −5.14326 + 0.739490i −0.989821 + 0.142315i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.53214 5.21798i −0.275180 0.937176i −0.974878 0.222738i \(-0.928501\pi\)
0.699699 0.714438i \(-0.253318\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.64303 −1.25651 −0.628253 0.778009i \(-0.716230\pi\)
−0.628253 + 0.778009i \(0.716230\pi\)
\(38\) 0 0
\(39\) −1.75964 + 0.516676i −0.281767 + 0.0827343i
\(40\) 0 0
\(41\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(42\) 0 0
\(43\) 5.83248 + 5.05388i 0.889446 + 0.770709i 0.974198 0.225693i \(-0.0724646\pi\)
−0.0847529 + 0.996402i \(0.527010\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(48\) 0 0
\(49\) 13.7521 + 15.8707i 1.96458 + 2.26725i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.54836 7.07739i 0.602445 0.937423i
\(58\) 0 0
\(59\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(60\) 0 0
\(61\) 1.72288 0.247713i 0.220592 0.0317164i −0.0311325 0.999515i \(-0.509911\pi\)
0.251725 + 0.967799i \(0.419002\pi\)
\(62\) 0 0
\(63\) −11.9972 + 10.3956i −1.51150 + 1.30972i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.16392 0.591970i −0.997381 0.0723206i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(72\) 0 0
\(73\) −1.74937 12.1671i −0.204748 1.42405i −0.789953 0.613167i \(-0.789895\pi\)
0.585206 0.810885i \(-0.301014\pi\)
\(74\) 0 0
\(75\) 2.43988 + 8.30945i 0.281733 + 0.959493i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.36459 + 14.8644i −0.491055 + 1.67238i 0.225018 + 0.974355i \(0.427756\pi\)
−0.716073 + 0.698026i \(0.754062\pi\)
\(80\) 0 0
\(81\) −5.89375 6.80175i −0.654861 0.755750i
\(82\) 0 0
\(83\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(90\) 0 0
\(91\) −3.66900 + 4.23426i −0.384616 + 0.443871i
\(92\) 0 0
\(93\) 6.16836 7.11867i 0.639629 0.738172i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 19.2257i 1.95208i −0.217599 0.976038i \(-0.569823\pi\)
0.217599 0.976038i \(-0.430177\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(102\) 0 0
\(103\) 4.94834 1.45296i 0.487574 0.143165i −0.0286999 0.999588i \(-0.509137\pi\)
0.516274 + 0.856423i \(0.327319\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(108\) 0 0
\(109\) 0.275623 0.938687i 0.0263999 0.0899099i −0.945239 0.326378i \(-0.894172\pi\)
0.971639 + 0.236468i \(0.0759899\pi\)
\(110\) 0 0
\(111\) −7.15707 11.1366i −0.679319 1.05704i
\(112\) 0 0
\(113\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.40060 2.08013i −0.221935 0.192308i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.5544 + 3.09906i 0.959493 + 0.281733i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.18803 20.1190i 0.815306 1.78527i 0.232631 0.972565i \(-0.425267\pi\)
0.582675 0.812705i \(-0.302006\pi\)
\(128\) 0 0
\(129\) −1.90233 + 13.2310i −0.167491 + 1.16493i
\(130\) 0 0
\(131\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(132\) 0 0
\(133\) 25.7019i 2.22863i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(138\) 0 0
\(139\) −5.86042 0.842601i −0.497074 0.0714685i −0.110782 0.993845i \(-0.535335\pi\)
−0.386293 + 0.922376i \(0.626245\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.2475 + 34.8997i −0.845198 + 2.87848i
\(148\) 0 0
\(149\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(150\) 0 0
\(151\) 15.3451 17.7092i 1.24877 1.44115i 0.396511 0.918030i \(-0.370221\pi\)
0.852256 0.523125i \(-0.175234\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.7976 13.3658i 1.65983 1.06671i 0.741448 0.671010i \(-0.234139\pi\)
0.918381 0.395698i \(-0.129497\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −23.5648 −1.84574 −0.922870 0.385111i \(-0.874163\pi\)
−0.922870 + 0.385111i \(0.874163\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(168\) 0 0
\(169\) 9.99317 + 6.42222i 0.768706 + 0.494017i
\(170\) 0 0
\(171\) 14.5716 1.11432
\(172\) 0 0
\(173\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(174\) 0 0
\(175\) 19.9953 + 17.3260i 1.51150 + 1.30972i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(180\) 0 0
\(181\) 21.5515 13.8503i 1.60191 1.02948i 0.635624 0.771998i \(-0.280743\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) 1.97427 + 2.27843i 0.145943 + 0.168427i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −26.3817 7.74636i −1.91899 0.563465i
\(190\) 0 0
\(191\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(192\) 0 0
\(193\) −24.9845 + 7.33612i −1.79842 + 0.528065i −0.997498 0.0706968i \(-0.977478\pi\)
−0.800927 + 0.598762i \(0.795660\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(198\) 0 0
\(199\) −6.52954 + 14.2977i −0.462867 + 1.01354i 0.523958 + 0.851744i \(0.324455\pi\)
−0.986825 + 0.161793i \(0.948272\pi\)
\(200\) 0 0
\(201\) −6.78228 12.4499i −0.478385 0.878150i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −22.1072 14.2075i −1.52193 0.978082i −0.991465 0.130372i \(-0.958383\pi\)
−0.530460 0.847710i \(-0.677981\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.09534 28.4837i 0.278010 1.93360i
\(218\) 0 0
\(219\) 16.0905 13.9425i 1.08729 0.942145i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −15.0892 9.69726i −1.01045 0.649376i −0.0729398 0.997336i \(-0.523238\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −9.82291 + 11.3362i −0.654861 + 0.755750i
\(226\) 0 0
\(227\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(228\) 0 0
\(229\) −7.36472 25.0819i −0.486674 1.65746i −0.726900 0.686743i \(-0.759040\pi\)
0.240226 0.970717i \(-0.422778\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −25.7460 + 7.55970i −1.67238 + 0.491055i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 3.02284 + 21.0243i 0.194718 + 1.35429i 0.819313 + 0.573346i \(0.194355\pi\)
−0.624595 + 0.780949i \(0.714736\pi\)
\(242\) 0 0
\(243\) 4.39178 14.9570i 0.281733 0.959493i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.09053 0.731907i 0.323902 0.0465701i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(258\) 0 0
\(259\) −36.7883 16.8007i −2.28592 1.04394i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 9.87994 + 15.3735i 0.600164 + 0.933873i 0.999852 + 0.0172215i \(0.00548205\pi\)
−0.399688 + 0.916651i \(0.630882\pi\)
\(272\) 0 0
\(273\) −9.60543 1.38105i −0.581347 0.0835851i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.5983 29.7762i 0.817045 1.78908i 0.243508 0.969899i \(-0.421702\pi\)
0.573537 0.819180i \(-0.305571\pi\)
\(278\) 0 0
\(279\) 16.1487 + 2.32184i 0.966799 + 0.139005i
\(280\) 0 0
\(281\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(282\) 0 0
\(283\) 2.38135 + 5.21442i 0.141556 + 0.309965i 0.967110 0.254358i \(-0.0818643\pi\)
−0.825554 + 0.564324i \(0.809137\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.41935 16.8270i −0.142315 0.989821i
\(290\) 0 0
\(291\) 28.0137 18.0033i 1.64219 1.05537i
\(292\) 0 0
\(293\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 16.9643 + 37.1467i 0.977808 + 2.14110i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −32.6609 + 9.59009i −1.86405 + 0.547336i −0.865100 + 0.501599i \(0.832745\pi\)
−0.998954 + 0.0457370i \(0.985436\pi\)
\(308\) 0 0
\(309\) 6.75081 + 5.84961i 0.384040 + 0.332773i
\(310\) 0 0
\(311\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(312\) 0 0
\(313\) −17.0433 + 26.5199i −0.963343 + 1.49899i −0.0996196 + 0.995026i \(0.531763\pi\)
−0.863724 + 0.503966i \(0.831874\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.86219 + 4.45366i −0.158766 + 0.247045i
\(326\) 0 0
\(327\) 1.62585 0.477394i 0.0899099 0.0263999i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −24.0868 + 20.8714i −1.32393 + 1.14719i −0.346008 + 0.938231i \(0.612463\pi\)
−0.977924 + 0.208962i \(0.932991\pi\)
\(332\) 0 0
\(333\) 9.52509 20.8570i 0.521972 1.14296i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.8070 + 9.50222i 1.13343 + 0.517619i 0.891656 0.452715i \(-0.149544\pi\)
0.241771 + 0.970333i \(0.422272\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.8710 + 71.0802i 1.12693 + 3.83797i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(348\) 0 0
\(349\) 24.4436 + 28.2094i 1.30844 + 1.51001i 0.687705 + 0.725991i \(0.258618\pi\)
0.620730 + 0.784024i \(0.286836\pi\)
\(350\) 0 0
\(351\) 0.782983 5.44577i 0.0417925 0.290674i
\(352\) 0 0
\(353\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(360\) 0 0
\(361\) −3.00734 + 3.47065i −0.158281 + 0.182666i
\(362\) 0 0
\(363\) 5.36773 + 18.2808i 0.281733 + 0.959493i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.791302 + 1.23129i −0.0413056 + 0.0642728i −0.861292 0.508110i \(-0.830344\pi\)
0.819987 + 0.572383i \(0.193981\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 31.6879i 1.64074i 0.571834 + 0.820370i \(0.306232\pi\)
−0.571834 + 0.820370i \(0.693768\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.62377 7.19472i −0.237507 0.369568i 0.701955 0.712221i \(-0.252311\pi\)
−0.939462 + 0.342653i \(0.888674\pi\)
\(380\) 0 0
\(381\) 37.9191 5.45194i 1.94265 0.279311i
\(382\) 0 0
\(383\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −21.0602 + 9.61787i −1.07055 + 0.488904i
\(388\) 0 0
\(389\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.933202 6.49056i 0.0468361 0.325752i −0.952911 0.303251i \(-0.901928\pi\)
0.999747 0.0225011i \(-0.00716292\pi\)
\(398\) 0 0
\(399\) 37.4500 24.0677i 1.87485 1.20489i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 5.75812 0.286832
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.4204 + 14.8059i 1.60309 + 0.732105i 0.997949 0.0640160i \(-0.0203909\pi\)
0.605137 + 0.796121i \(0.293118\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.26005 9.32821i −0.208616 0.456804i
\(418\) 0 0
\(419\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(420\) 0 0
\(421\) −12.8232 28.0788i −0.624963 1.36848i −0.911854 0.410515i \(-0.865349\pi\)
0.286890 0.957963i \(-0.407378\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.83728 + 2.59486i 0.427666 + 0.125574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 11.0783 + 37.7291i 0.532387 + 1.81314i 0.580444 + 0.814300i \(0.302879\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 5.22584 0.249416 0.124708 0.992194i \(-0.460201\pi\)
0.124708 + 0.992194i \(0.460201\pi\)
\(440\) 0 0
\(441\) −60.4481 + 17.7491i −2.87848 + 0.845198i
\(442\) 0 0
\(443\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 40.1734 + 5.77606i 1.88751 + 0.271383i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −32.2951 9.48271i −1.51070 0.443582i −0.581622 0.813459i \(-0.697582\pi\)
−0.929081 + 0.369877i \(0.879400\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(462\) 0 0
\(463\) −2.33915 + 0.336319i −0.108709 + 0.0156301i −0.196455 0.980513i \(-0.562943\pi\)
0.0877454 + 0.996143i \(0.472034\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(468\) 0 0
\(469\) −37.9943 20.7950i −1.75441 0.960225i
\(470\) 0 0
\(471\) 38.9505 + 17.7881i 1.79474 + 0.819632i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.45625 24.0388i −0.158584 1.10297i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(480\) 0 0
\(481\) 2.27994 7.76475i 0.103956 0.354042i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −26.8453 + 23.2616i −1.21648 + 1.05408i −0.219564 + 0.975598i \(0.570463\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) −22.0665 34.3362i −0.997883 1.55274i
\(490\) 0 0
\(491\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.9295i 1.74272i 0.490642 + 0.871361i \(0.336762\pi\)
−0.490642 + 0.871361i \(0.663238\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.5749i 0.913762i
\(508\) 0 0
\(509\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(510\) 0 0
\(511\) 18.3251 62.4096i 0.810655 2.76084i
\(512\) 0 0
\(513\) 13.6451 + 21.2322i 0.602445 + 0.937423i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(522\) 0 0
\(523\) 42.4931 + 12.4771i 1.85809 + 0.545586i 0.999455 + 0.0330064i \(0.0105082\pi\)
0.858640 + 0.512579i \(0.171310\pi\)
\(524\) 0 0
\(525\) −6.52168 + 45.3593i −0.284630 + 1.97964i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 9.55455 20.9215i 0.415415 0.909632i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 29.9198 + 4.30182i 1.28635 + 0.184950i 0.751352 0.659902i \(-0.229402\pi\)
0.535001 + 0.844851i \(0.320311\pi\)
\(542\) 0 0
\(543\) 40.3623 + 18.4328i 1.73211 + 0.791029i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −45.6402 6.56207i −1.95143 0.280574i −0.951777 0.306791i \(-0.900745\pi\)
−0.999656 + 0.0262168i \(0.991654\pi\)
\(548\) 0 0
\(549\) −1.47115 + 5.01027i −0.0627871 + 0.213833i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −53.6827 + 61.9532i −2.28282 + 2.63452i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(558\) 0 0
\(559\) −6.87421 + 4.41779i −0.290748 + 0.186853i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −13.4171 45.6944i −0.563465 1.91899i
\(568\) 0 0
\(569\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(570\) 0 0
\(571\) −31.8052 20.4400i −1.33101 0.855386i −0.334790 0.942293i \(-0.608665\pi\)
−0.996216 + 0.0869073i \(0.972302\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 32.9203 + 28.5256i 1.37049 + 1.18754i 0.961437 + 0.275027i \(0.0886866\pi\)
0.409054 + 0.912510i \(0.365859\pi\)
\(578\) 0 0
\(579\) −34.0853 29.5351i −1.41654 1.22744i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(588\) 0 0
\(589\) −19.9629 + 17.2980i −0.822557 + 0.712750i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26.9475 + 3.87446i −1.10289 + 0.158571i
\(598\) 0 0
\(599\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(600\) 0 0
\(601\) −15.5556 + 34.0620i −0.634526 + 1.38942i 0.269942 + 0.962877i \(0.412995\pi\)
−0.904468 + 0.426542i \(0.859732\pi\)
\(602\) 0 0
\(603\) 11.7897 21.5407i 0.480112 0.877207i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −32.2663 37.2373i −1.30965 1.51142i −0.662722 0.748866i \(-0.730599\pi\)
−0.646928 0.762551i \(-0.723947\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 36.3426 + 23.3560i 1.46786 + 0.943338i 0.998168 + 0.0605099i \(0.0192727\pi\)
0.469696 + 0.882828i \(0.344364\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(618\) 0 0
\(619\) 6.01584 41.8411i 0.241797 1.68173i −0.401297 0.915948i \(-0.631441\pi\)
0.643094 0.765787i \(-0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 21.0313 + 13.5160i 0.841254 + 0.540641i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.954758 + 3.25161i 0.0380083 + 0.129444i 0.976312 0.216369i \(-0.0694213\pi\)
−0.938303 + 0.345813i \(0.887603\pi\)
\(632\) 0 0
\(633\) 45.5164i 1.80912i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −20.2258 + 9.23681i −0.801375 + 0.365976i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0.374642 + 2.60569i 0.0147745 + 0.102759i 0.995874 0.0907437i \(-0.0289244\pi\)
−0.981100 + 0.193502i \(0.938015\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 45.3384 20.7053i 1.77695 0.811506i
\(652\) 0 0
\(653\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 35.3829 + 10.3894i 1.38042 + 0.405327i
\(658\) 0 0
\(659\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(660\) 0 0
\(661\) −46.7583 21.3538i −1.81869 0.830566i −0.919807 0.392370i \(-0.871655\pi\)
−0.898879 0.438196i \(-0.855618\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 31.0671i 1.20112i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −27.8230 43.2935i −1.07250 1.66884i −0.643255 0.765652i \(-0.722417\pi\)
−0.429243 0.903189i \(-0.641220\pi\)
\(674\) 0 0
\(675\) −25.7163 3.69745i −0.989821 0.142315i
\(676\) 0 0
\(677\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(678\) 0 0
\(679\) 42.2614 92.5396i 1.62184 3.55134i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 29.6503 34.2182i 1.13123 1.30551i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.990982 + 6.89243i 0.0376987 + 0.262201i 0.999950 0.00996078i \(-0.00317067\pi\)
−0.962252 + 0.272161i \(0.912262\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(702\) 0 0
\(703\) 15.4217 + 33.7689i 0.581642 + 1.27362i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −42.9155 + 12.6011i −1.61173 + 0.473245i −0.958778 0.284158i \(-0.908286\pi\)
−0.652948 + 0.757403i \(0.726468\pi\)
\(710\) 0 0
\(711\) −35.1242 30.4353i −1.31726 1.14141i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(720\) 0 0
\(721\) 27.0118 + 3.88371i 1.00597 + 0.144637i
\(722\) 0 0
\(723\) −27.8037 + 24.0921i −1.03403 + 0.895994i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 29.0159 45.1497i 1.07614 1.67451i 0.456140 0.889908i \(-0.349232\pi\)
0.620002 0.784600i \(-0.287132\pi\)
\(728\) 0 0
\(729\) 25.9063 7.60678i 0.959493 0.281733i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −40.8980 + 35.4383i −1.51060 + 1.30894i −0.732754 + 0.680494i \(0.761765\pi\)
−0.777849 + 0.628451i \(0.783689\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −19.9533 9.11236i −0.733994 0.335204i 0.0131073 0.999914i \(-0.495828\pi\)
−0.747101 + 0.664710i \(0.768555\pi\)
\(740\) 0 0
\(741\) 5.83331 + 6.73200i 0.214292 + 0.247306i
\(742\) 0 0
\(743\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −13.7277 15.8426i −0.500931 0.578105i 0.447822 0.894123i \(-0.352200\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.1821 + 15.8437i 0.370076 + 0.575849i 0.975485 0.220065i \(-0.0706269\pi\)
−0.605409 + 0.795914i \(0.706991\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(762\) 0 0
\(763\) 3.39006 3.91233i 0.122728 0.141636i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −29.0562 + 45.2123i −1.04779 + 1.63040i −0.316150 + 0.948709i \(0.602390\pi\)
−0.731643 + 0.681688i \(0.761246\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(774\) 0 0
\(775\) 27.1913i 0.976741i
\(776\) 0 0
\(777\) −9.96909 69.3365i −0.357639 2.48743i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.41153 + 8.15514i 0.335485 + 0.290699i 0.806269 0.591549i \(-0.201483\pi\)
−0.470784 + 0.882248i \(0.656029\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.262282 + 1.82421i −0.00931391 + 0.0647796i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\)