Properties

Label 804.2.s.a.5.1
Level 804
Weight 2
Character 804.5
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 5.1
Root \(-0.786053 + 0.618159i\)
Character \(\chi\) = 804.5
Dual form 804.2.s.a.161.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.57553 + 0.719520i) q^{3} +(-3.36481 - 2.91562i) q^{7} +(1.96458 - 2.26725i) q^{9} +O(q^{10})\) \(q+(-1.57553 + 0.719520i) q^{3} +(-3.36481 - 2.91562i) q^{7} +(1.96458 - 2.26725i) q^{9} +(1.63571 + 2.54522i) q^{13} +(4.23982 + 4.89301i) q^{19} +(7.39920 + 2.17260i) q^{21} +(-4.20627 + 2.70320i) q^{25} +(-1.46393 + 4.98567i) q^{27} +(-6.00018 + 9.33646i) q^{31} +12.1582 q^{37} +(-4.40845 - 2.83314i) q^{39} +(-12.9347 - 1.85973i) q^{43} +(1.82487 + 12.6923i) q^{49} +(-10.2006 - 4.65844i) q^{57} +(-4.39867 + 14.9805i) q^{61} +(-13.2209 + 1.90088i) q^{63} +(5.79361 - 5.78222i) q^{67} +(5.46060 + 1.60338i) q^{73} +(4.68209 - 7.28547i) q^{75} +(9.04681 + 14.0771i) q^{79} +(-1.28083 - 8.90839i) q^{81} +(1.91704 - 13.3333i) q^{91} +(2.73569 - 19.0271i) q^{93} +6.97869i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{22}\right)\) \(1\)

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
\(3\) −1.57553 + 0.719520i −0.909632 + 0.415415i
\(4\) 0 0
\(5\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(6\) 0 0
\(7\) −3.36481 2.91562i −1.27178 1.10200i −0.989782 0.142586i \(-0.954458\pi\)
−0.281995 0.959416i \(-0.590996\pi\)
\(8\) 0 0
\(9\) 1.96458 2.26725i 0.654861 0.755750i
\(10\) 0 0
\(11\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(12\) 0 0
\(13\) 1.63571 + 2.54522i 0.453665 + 0.705917i 0.990461 0.137795i \(-0.0440016\pi\)
−0.536795 + 0.843712i \(0.680365\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(18\) 0 0
\(19\) 4.23982 + 4.89301i 0.972680 + 1.12253i 0.992440 + 0.122728i \(0.0391644\pi\)
−0.0197599 + 0.999805i \(0.506290\pi\)
\(20\) 0 0
\(21\) 7.39920 + 2.17260i 1.61464 + 0.474100i
\(22\) 0 0
\(23\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(24\) 0 0
\(25\) −4.20627 + 2.70320i −0.841254 + 0.540641i
\(26\) 0 0
\(27\) −1.46393 + 4.98567i −0.281733 + 0.959493i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −6.00018 + 9.33646i −1.07766 + 1.67688i −0.470075 + 0.882626i \(0.655773\pi\)
−0.607589 + 0.794252i \(0.707863\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.1582 1.99880 0.999398 0.0346855i \(-0.0110430\pi\)
0.999398 + 0.0346855i \(0.0110430\pi\)
\(38\) 0 0
\(39\) −4.40845 2.83314i −0.705917 0.453665i
\(40\) 0 0
\(41\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(42\) 0 0
\(43\) −12.9347 1.85973i −1.97252 0.283606i −0.998322 0.0579125i \(-0.981556\pi\)
−0.974198 0.225693i \(-0.927535\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(48\) 0 0
\(49\) 1.82487 + 12.6923i 0.260696 + 1.81318i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.2006 4.65844i −1.35110 0.617026i
\(58\) 0 0
\(59\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(60\) 0 0
\(61\) −4.39867 + 14.9805i −0.563192 + 1.91806i −0.251725 + 0.967799i \(0.580998\pi\)
−0.311467 + 0.950257i \(0.600820\pi\)
\(62\) 0 0
\(63\) −13.2209 + 1.90088i −1.66567 + 0.239488i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.79361 5.78222i 0.707802 0.706411i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(72\) 0 0
\(73\) 5.46060 + 1.60338i 0.639115 + 0.187661i 0.585206 0.810885i \(-0.301014\pi\)
0.0539089 + 0.998546i \(0.482832\pi\)
\(74\) 0 0
\(75\) 4.68209 7.28547i 0.540641 0.841254i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.04681 + 14.0771i 1.01785 + 1.58380i 0.792829 + 0.609445i \(0.208608\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −1.28083 8.90839i −0.142315 0.989821i
\(82\) 0 0
\(83\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(90\) 0 0
\(91\) 1.91704 13.3333i 0.200960 1.39771i
\(92\) 0 0
\(93\) 2.73569 19.0271i 0.283677 1.97302i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.97869i 0.708578i 0.935136 + 0.354289i \(0.115277\pi\)
−0.935136 + 0.354289i \(0.884723\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(102\) 0 0
\(103\) 8.81568 + 5.66549i 0.868635 + 0.558238i 0.897335 0.441350i \(-0.145500\pi\)
−0.0286999 + 0.999588i \(0.509137\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(108\) 0 0
\(109\) 10.4772 + 16.3028i 1.00353 + 1.56153i 0.814999 + 0.579462i \(0.196737\pi\)
0.188534 + 0.982067i \(0.439626\pi\)
\(110\) 0 0
\(111\) −19.1556 + 8.74807i −1.81817 + 0.830330i
\(112\) 0 0
\(113\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.98414 + 1.29172i 0.830584 + 0.119420i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.25379 + 5.94705i −0.841254 + 0.540641i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.33874 8.46936i 0.651208 0.751534i −0.330107 0.943944i \(-0.607085\pi\)
0.981315 + 0.192410i \(0.0616302\pi\)
\(128\) 0 0
\(129\) 21.7171 6.37671i 1.91208 0.561438i
\(130\) 0 0
\(131\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(132\) 0 0
\(133\) 28.8257i 2.49951i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(138\) 0 0
\(139\) −6.12749 20.8683i −0.519727 1.77003i −0.630509 0.776182i \(-0.717154\pi\)
0.110782 0.993845i \(-0.464665\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −12.0075 18.6840i −0.990360 1.54103i
\(148\) 0 0
\(149\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(150\) 0 0
\(151\) 0.426045 2.96321i 0.0346711 0.241142i −0.965115 0.261826i \(-0.915675\pi\)
0.999786 + 0.0206838i \(0.00658434\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −10.3339 22.6280i −0.824732 1.80591i −0.522369 0.852719i \(-0.674952\pi\)
−0.302363 0.953193i \(-0.597776\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −25.1404 −1.96915 −0.984575 0.174966i \(-0.944019\pi\)
−0.984575 + 0.174966i \(0.944019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(168\) 0 0
\(169\) 1.59781 3.49871i 0.122908 0.269132i
\(170\) 0 0
\(171\) 19.4231 1.48532
\(172\) 0 0
\(173\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(174\) 0 0
\(175\) 22.0348 + 3.16813i 1.66567 + 0.239488i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(180\) 0 0
\(181\) 1.86882 + 4.09214i 0.138908 + 0.304166i 0.966282 0.257485i \(-0.0828937\pi\)
−0.827374 + 0.561651i \(0.810166\pi\)
\(182\) 0 0
\(183\) −3.84853 26.7671i −0.284492 1.97868i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 19.4622 12.5076i 1.41566 0.909792i
\(190\) 0 0
\(191\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(192\) 0 0
\(193\) −18.7210 12.0312i −1.34757 0.866028i −0.350068 0.936724i \(-0.613841\pi\)
−0.997498 + 0.0706968i \(0.977478\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(198\) 0 0
\(199\) 0.364085 0.420177i 0.0258093 0.0297855i −0.742698 0.669626i \(-0.766454\pi\)
0.768507 + 0.639841i \(0.221000\pi\)
\(200\) 0 0
\(201\) −4.96758 + 13.2787i −0.350386 + 0.936605i
\(202\) 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
\(211\) −9.21538 + 20.1789i −0.634413 + 1.38917i 0.270146 + 0.962819i \(0.412928\pi\)
−0.904558 + 0.426350i \(0.859799\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 47.4111 13.9211i 3.21847 0.945029i
\(218\) 0 0
\(219\) −9.75699 + 1.40284i −0.659316 + 0.0947954i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −8.75894 + 19.1794i −0.586542 + 1.28435i 0.350967 + 0.936388i \(0.385853\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −2.13472 + 14.8473i −0.142315 + 0.989821i
\(226\) 0 0
\(227\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(228\) 0 0
\(229\) 3.02274 4.70347i 0.199748 0.310814i −0.726900 0.686743i \(-0.759040\pi\)
0.926649 + 0.375929i \(0.122676\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −24.3823 15.6695i −1.58380 1.01785i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 3.07530 + 0.902989i 0.198097 + 0.0581666i 0.379276 0.925284i \(-0.376173\pi\)
−0.181179 + 0.983450i \(0.557991\pi\)
\(242\) 0 0
\(243\) 8.42776 + 13.1138i 0.540641 + 0.841254i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.51866 + 18.7948i −0.351144 + 1.19589i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(258\) 0 0
\(259\) −40.9100 35.4487i −2.54203 2.20268i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(264\) 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
\(271\) 8.93119 4.07874i 0.542531 0.247766i −0.125247 0.992126i \(-0.539972\pi\)
0.667779 + 0.744360i \(0.267245\pi\)
\(272\) 0 0
\(273\) 6.57322 + 22.3863i 0.397830 + 1.35488i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −15.8956 + 18.3445i −0.955073 + 1.10221i 0.0396081 + 0.999215i \(0.487389\pi\)
−0.994682 + 0.102998i \(0.967156\pi\)
\(278\) 0 0
\(279\) 9.38024 + 31.9461i 0.561580 + 1.91257i
\(280\) 0 0
\(281\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(282\) 0 0
\(283\) 18.1894 + 20.9916i 1.08125 + 1.24782i 0.967110 + 0.254358i \(0.0818643\pi\)
0.114135 + 0.993465i \(0.463590\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.3114 4.78945i −0.959493 0.281733i
\(290\) 0 0
\(291\) −5.02131 10.9951i −0.294354 0.644546i
\(292\) 0 0
\(293\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 38.1005 + 43.9703i 2.19607 + 2.53440i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 27.8763 + 17.9150i 1.59098 + 1.02246i 0.971374 + 0.237553i \(0.0763455\pi\)
0.619610 + 0.784910i \(0.287291\pi\)
\(308\) 0 0
\(309\) −17.9658 2.58309i −1.02204 0.146947i
\(310\) 0 0
\(311\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(312\) 0 0
\(313\) −31.6320 14.4459i −1.78795 0.816528i −0.970720 0.240212i \(-0.922783\pi\)
−0.817227 0.576316i \(-0.804490\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −13.7605 6.28421i −0.763295 0.348585i
\(326\) 0 0
\(327\) −28.2373 18.1470i −1.56153 1.00353i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 21.6900 3.11855i 1.19219 0.171411i 0.482481 0.875907i \(-0.339736\pi\)
0.709708 + 0.704496i \(0.248827\pi\)
\(332\) 0 0
\(333\) 23.8858 27.5657i 1.30893 1.51059i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5.08244 4.40396i −0.276858 0.239899i 0.505352 0.862913i \(-0.331363\pi\)
−0.782211 + 0.623014i \(0.785908\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 14.0159 21.8092i 0.756789 1.17759i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(348\) 0 0
\(349\) −5.03054 34.9882i −0.269279 1.87287i −0.455313 0.890332i \(-0.650472\pi\)
0.186034 0.982543i \(-0.440437\pi\)
\(350\) 0 0
\(351\) −15.0842 + 4.42912i −0.805135 + 0.236409i
\(352\) 0 0
\(353\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(360\) 0 0
\(361\) −3.26151 + 22.6843i −0.171658 + 1.19391i
\(362\) 0 0
\(363\) 10.3006 16.0280i 0.540641 0.841254i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 32.2325 + 14.7201i 1.68252 + 0.768382i 0.999270 + 0.0382006i \(0.0121626\pi\)
0.683253 + 0.730182i \(0.260565\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 24.1815i 1.25207i −0.779795 0.626035i \(-0.784677\pi\)
0.779795 0.626035i \(-0.215323\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −17.1989 + 7.85448i −0.883449 + 0.403458i −0.804871 0.593450i \(-0.797765\pi\)
−0.0785782 + 0.996908i \(0.525038\pi\)
\(380\) 0 0
\(381\) −5.46853 + 18.6241i −0.280161 + 0.954141i
\(382\) 0 0
\(383\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −29.6277 + 25.6726i −1.50606 + 1.30501i
\(388\) 0 0
\(389\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15.0970 + 4.43288i −0.757697 + 0.222480i −0.637690 0.770293i \(-0.720110\pi\)
−0.120007 + 0.992773i \(0.538292\pi\)
\(398\) 0 0
\(399\) 20.7407 + 45.4158i 1.03833 + 2.27363i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −33.5779 −1.67264
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.95685 + 1.69562i 0.0967601 + 0.0838431i 0.701898 0.712278i \(-0.252336\pi\)
−0.605137 + 0.796121i \(0.706882\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 24.6692 + 28.4698i 1.20806 + 1.39417i
\(418\) 0 0
\(419\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(420\) 0 0
\(421\) 0.144573 + 0.166847i 0.00704607 + 0.00813160i 0.759262 0.650785i \(-0.225560\pi\)
−0.752216 + 0.658917i \(0.771015\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 58.4781 37.5816i 2.82996 1.81870i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −19.4909 + 30.3285i −0.936674 + 1.45749i −0.0480569 + 0.998845i \(0.515303\pi\)
−0.888617 + 0.458649i \(0.848333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 26.8748 1.28266 0.641331 0.767264i \(-0.278382\pi\)
0.641331 + 0.767264i \(0.278382\pi\)
\(440\) 0 0
\(441\) 32.3616 + 20.7976i 1.54103 + 0.990360i
\(442\) 0 0
\(443\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.46084 + 4.97517i 0.0686363 + 0.233754i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −20.9197 + 13.4443i −0.978583 + 0.628897i −0.929081 0.369877i \(-0.879400\pi\)
−0.0495022 + 0.998774i \(0.515763\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(462\) 0 0
\(463\) −2.77184 + 9.44002i −0.128818 + 0.438715i −0.998491 0.0549137i \(-0.982512\pi\)
0.869673 + 0.493629i \(0.164330\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(468\) 0 0
\(469\) −36.3532 + 2.56408i −1.67863 + 0.118398i
\(470\) 0 0
\(471\) 32.5626 + 28.2157i 1.50041 + 1.30011i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −31.0606 9.12022i −1.42516 0.418464i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(480\) 0 0
\(481\) 19.8873 + 30.9453i 0.906785 + 1.41098i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −43.5968 + 6.26828i −1.97556 + 0.284043i −0.978645 + 0.205556i \(0.934100\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) 39.6094 18.0890i 1.79120 0.818014i
\(490\) 0 0
\(491\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.92718i 0.399636i −0.979833 0.199818i \(-0.935965\pi\)
0.979833 0.199818i \(-0.0640350\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.66198i 0.295869i
\(508\) 0 0
\(509\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(510\) 0 0
\(511\) −13.6990 21.3161i −0.606009 0.942969i
\(512\) 0 0
\(513\) −30.6017 + 13.9753i −1.35110 + 0.617026i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(522\) 0 0
\(523\) 33.0384 21.2325i 1.44467 0.928431i 0.445212 0.895425i \(-0.353128\pi\)
0.999455 0.0330064i \(-0.0105082\pi\)
\(524\) 0 0
\(525\) −36.9960 + 10.8630i −1.61464 + 0.474100i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0618 + 17.3822i −0.654861 + 0.755750i
\(530\) 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
\(541\) 12.5500 + 42.7414i 0.539567 + 1.83760i 0.546257 + 0.837618i \(0.316052\pi\)
−0.00668980 + 0.999978i \(0.502129\pi\)
\(542\) 0 0
\(543\) −5.88875 5.10263i −0.252710 0.218975i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12.1268 41.3002i −0.518506 1.76587i −0.634822 0.772658i \(-0.718927\pi\)
0.116316 0.993212i \(-0.462891\pi\)
\(548\) 0 0
\(549\) 25.3230 + 39.4033i 1.08076 + 1.68169i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 10.6028 73.7439i 0.450876 3.13591i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(558\) 0 0
\(559\) −16.4240 35.9636i −0.694662 1.52110i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −21.6638 + 33.7095i −0.909792 + 1.41566i
\(568\) 0 0
\(569\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(570\) 0 0
\(571\) 14.2558 31.2159i 0.596588 1.30635i −0.334790 0.942293i \(-0.608665\pi\)
0.931378 0.364054i \(-0.118607\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −0.332013 0.0477362i −0.0138219 0.00198729i 0.135400 0.990791i \(-0.456768\pi\)
−0.149222 + 0.988804i \(0.547677\pi\)
\(578\) 0 0
\(579\) 38.1522 + 5.48545i 1.58555 + 0.227968i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(588\) 0 0
\(589\) −71.1230 + 10.2259i −2.93057 + 0.421353i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.271301 + 0.923967i −0.0111036 + 0.0378155i
\(598\) 0 0
\(599\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(600\) 0 0
\(601\) −8.66735 + 10.0027i −0.353549 + 0.408017i −0.904468 0.426542i \(-0.859732\pi\)
0.550919 + 0.834559i \(0.314277\pi\)
\(602\) 0 0
\(603\) −1.72771 24.4952i −0.0703577 0.997522i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 6.70746 + 46.6514i 0.272247 + 1.89352i 0.424897 + 0.905242i \(0.360310\pi\)
−0.152650 + 0.988280i \(0.548781\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 20.5327 44.9603i 0.829308 1.81593i 0.359612 0.933102i \(-0.382909\pi\)
0.469696 0.882828i \(-0.344364\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(618\) 0 0
\(619\) 39.7606 11.6748i 1.59812 0.469249i 0.643094 0.765787i \(-0.277650\pi\)
0.955021 + 0.296538i \(0.0958321\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 10.3854 22.7408i 0.415415 0.909632i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −27.0845 + 42.1444i −1.07822 + 1.67774i −0.475828 + 0.879538i \(0.657851\pi\)
−0.602390 + 0.798202i \(0.705785\pi\)
\(632\) 0 0
\(633\) 38.4230i 1.52718i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −29.3197 + 25.4056i −1.16169 + 1.00661i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −24.1475 7.09036i −0.952286 0.279616i −0.231548 0.972824i \(-0.574379\pi\)
−0.720738 + 0.693207i \(0.756197\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −64.6809 + 56.0464i −2.53505 + 2.19663i
\(652\) 0 0
\(653\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.3630 9.23057i 0.560356 0.360119i
\(658\) 0 0
\(659\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(660\) 0 0
\(661\) 32.1520 + 27.8598i 1.25057 + 1.08362i 0.993092 + 0.117337i \(0.0374357\pi\)
0.257474 + 0.966285i \(0.417110\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 36.5199i 1.41194i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.9073 11.3748i 0.960107 0.438466i 0.127198 0.991877i \(-0.459401\pi\)
0.832908 + 0.553411i \(0.186674\pi\)
\(674\) 0 0
\(675\) −7.31963 24.9284i −0.281733 0.959493i
\(676\) 0 0
\(677\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(678\) 0 0
\(679\) 20.3472 23.4819i 0.780855 0.901154i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.37817 + 9.58538i −0.0525804 + 0.365705i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 33.4120 + 9.81065i 1.27105 + 0.373215i 0.846597 0.532234i \(-0.178648\pi\)
0.424455 + 0.905449i \(0.360466\pi\)
\(692\) 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 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(702\) 0 0
\(703\) 51.5485 + 59.4902i 1.94419 + 2.24372i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −29.4234 18.9093i −1.10502 0.710153i −0.144817 0.989458i \(-0.546259\pi\)
−0.960202 + 0.279306i \(0.909896\pi\)
\(710\) 0 0
\(711\) 49.6895 + 7.14428i 1.86350 + 0.267931i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(720\) 0 0
\(721\) −13.1446 44.7665i −0.489532 1.66719i
\(722\) 0 0
\(723\) −5.49494 + 0.790053i −0.204359 + 0.0293824i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −43.6524 19.9354i −1.61898 0.739362i −0.620002 0.784600i \(-0.712868\pi\)
−0.998976 + 0.0452379i \(0.985595\pi\)
\(728\) 0 0
\(729\) −22.7138 14.5973i −0.841254 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 50.8753 7.31477i 1.87912 0.270177i 0.894789 0.446489i \(-0.147326\pi\)
0.984334 + 0.176312i \(0.0564167\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −6.38022 5.52850i −0.234700 0.203369i 0.529568 0.848268i \(-0.322354\pi\)
−0.764268 + 0.644899i \(0.776900\pi\)
\(740\) 0 0
\(741\) −4.82845 33.5826i −0.177377 1.23369i
\(742\) 0 0
\(743\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.38669 + 9.64466i 0.0506011 + 0.351939i 0.999355 + 0.0359215i \(0.0114366\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 35.6648 16.2875i 1.29626 0.591981i 0.356651 0.934238i \(-0.383919\pi\)
0.939606 + 0.342257i \(0.111191\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(762\) 0 0
\(763\) 12.2792 85.4035i 0.444536 3.09181i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −41.4298 18.9204i −1.49400 0.682286i −0.509953 0.860202i \(-0.670337\pi\)
−0.984046 + 0.177916i \(0.943064\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(774\) 0 0
\(775\) 55.4914i 1.99331i
\(776\) 0 0
\(777\) 89.9610 + 26.4149i 3.22733 + 0.947630i
\(778\) 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
\(787\) 32.8522 + 4.72344i 1.17106 + 0.168372i 0.700272 0.713876i \(-0.253062\pi\)
0.470784 + 0.882248i \(0.343971\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −45.3236 + 13.3082i −1.60949 + 0.472589i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0