Properties

Label 804.2.s.a.137.2
Level 804
Weight 2
Character 804.137
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 137.2
Root \(-0.888835 + 0.458227i\)
Character \(\chi\) = 804.137
Dual form 804.2.s.a.581.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.71442 + 0.246497i) q^{3} +(0.211757 - 0.721178i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})\) \(q+(1.71442 + 0.246497i) q^{3} +(0.211757 - 0.721178i) q^{7} +(2.87848 + 0.845198i) q^{9} +(4.59805 + 3.98424i) q^{13} +(-4.38235 + 1.28677i) q^{19} +(0.540809 - 1.18421i) q^{21} +(3.27430 - 3.77875i) q^{25} +(4.72659 + 2.15856i) q^{27} +(6.01248 - 5.20984i) q^{31} -1.31262 q^{37} +(6.90090 + 7.96406i) q^{39} +(0.410625 + 0.638945i) q^{43} +(5.41352 + 3.47906i) q^{49} +(-7.83038 + 1.12584i) q^{57} +(-13.5021 - 6.16621i) q^{61} +(1.21908 - 1.89692i) q^{63} +(-6.54788 + 4.91174i) q^{67} +(-3.51029 + 7.68646i) q^{73} +(6.54498 - 5.67126i) q^{75} +(-6.28748 - 5.44813i) q^{79} +(7.57128 + 4.86577i) q^{81} +(3.84702 - 2.47233i) q^{91} +(11.5921 - 7.44981i) q^{93} +11.8855i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{22}\right)\) \(1\)

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
\(3\) 1.71442 + 0.246497i 0.989821 + 0.142315i
\(4\) 0 0
\(5\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(6\) 0 0
\(7\) 0.211757 0.721178i 0.0800367 0.272580i −0.909746 0.415166i \(-0.863724\pi\)
0.989782 + 0.142586i \(0.0455417\pi\)
\(8\) 0 0
\(9\) 2.87848 + 0.845198i 0.959493 + 0.281733i
\(10\) 0 0
\(11\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(12\) 0 0
\(13\) 4.59805 + 3.98424i 1.27527 + 1.10503i 0.989162 + 0.146831i \(0.0469074\pi\)
0.286109 + 0.958197i \(0.407638\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(18\) 0 0
\(19\) −4.38235 + 1.28677i −1.00538 + 0.295206i −0.742662 0.669666i \(-0.766437\pi\)
−0.262718 + 0.964873i \(0.584619\pi\)
\(20\) 0 0
\(21\) 0.540809 1.18421i 0.118014 0.258415i
\(22\) 0 0
\(23\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(24\) 0 0
\(25\) 3.27430 3.77875i 0.654861 0.755750i
\(26\) 0 0
\(27\) 4.72659 + 2.15856i 0.909632 + 0.415415i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 6.01248 5.20984i 1.07987 0.935715i 0.0817313 0.996654i \(-0.473955\pi\)
0.998141 + 0.0609393i \(0.0194096\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.31262 −0.215794 −0.107897 0.994162i \(-0.534412\pi\)
−0.107897 + 0.994162i \(0.534412\pi\)
\(38\) 0 0
\(39\) 6.90090 + 7.96406i 1.10503 + 1.27527i
\(40\) 0 0
\(41\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(42\) 0 0
\(43\) 0.410625 + 0.638945i 0.0626198 + 0.0974383i 0.871152 0.491014i \(-0.163374\pi\)
−0.808532 + 0.588453i \(0.799737\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(48\) 0 0
\(49\) 5.41352 + 3.47906i 0.773360 + 0.497008i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.83038 + 1.12584i −1.03716 + 0.149121i
\(58\) 0 0
\(59\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(60\) 0 0
\(61\) −13.5021 6.16621i −1.72877 0.789502i −0.993772 0.111430i \(-0.964457\pi\)
−0.734996 0.678072i \(-0.762816\pi\)
\(62\) 0 0
\(63\) 1.21908 1.89692i 0.153589 0.238989i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.54788 + 4.91174i −0.799951 + 0.600065i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(72\) 0 0
\(73\) −3.51029 + 7.68646i −0.410848 + 0.899633i 0.585206 + 0.810885i \(0.301014\pi\)
−0.996054 + 0.0887477i \(0.971714\pi\)
\(74\) 0 0
\(75\) 6.54498 5.67126i 0.755750 0.654861i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.28748 5.44813i −0.707396 0.612962i 0.225018 0.974355i \(-0.427756\pi\)
−0.932414 + 0.361392i \(0.882301\pi\)
\(80\) 0 0
\(81\) 7.57128 + 4.86577i 0.841254 + 0.540641i
\(82\) 0 0
\(83\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(90\) 0 0
\(91\) 3.84702 2.47233i 0.403277 0.259170i
\(92\) 0 0
\(93\) 11.5921 7.44981i 1.20205 0.772509i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.8855i 1.20679i 0.797442 + 0.603396i \(0.206186\pi\)
−0.797442 + 0.603396i \(0.793814\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(102\) 0 0
\(103\) −11.9276 13.7652i −1.17526 1.35632i −0.921180 0.389138i \(-0.872773\pi\)
−0.254079 0.967183i \(-0.581772\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(108\) 0 0
\(109\) −5.15041 4.46286i −0.493320 0.427464i 0.372340 0.928096i \(-0.378556\pi\)
−0.865660 + 0.500632i \(0.833101\pi\)
\(110\) 0 0
\(111\) −2.25038 0.323556i −0.213597 0.0307106i
\(112\) 0 0
\(113\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.86793 + 15.3548i 0.912291 + 1.41955i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.20347 8.31325i 0.654861 0.755750i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −20.1025 5.90263i −1.78381 0.523774i −0.788038 0.615627i \(-0.788903\pi\)
−0.995773 + 0.0918526i \(0.970721\pi\)
\(128\) 0 0
\(129\) 0.546487 + 1.19664i 0.0481155 + 0.105358i
\(130\) 0 0
\(131\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(132\) 0 0
\(133\) 3.43294i 0.297674i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(138\) 0 0
\(139\) −16.6482 + 7.60297i −1.41208 + 0.644876i −0.967963 0.251093i \(-0.919210\pi\)
−0.444118 + 0.895969i \(0.646483\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.42347 + 7.29898i 0.694756 + 0.602010i
\(148\) 0 0
\(149\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(150\) 0 0
\(151\) 17.1580 11.0268i 1.39630 0.897346i 0.396511 0.918030i \(-0.370221\pi\)
0.999786 + 0.0206838i \(0.00658434\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.74505 19.0923i 0.219079 1.52373i −0.522369 0.852719i \(-0.674952\pi\)
0.741448 0.671010i \(-0.234139\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.37981 −0.499706 −0.249853 0.968284i \(-0.580382\pi\)
−0.249853 + 0.968284i \(0.580382\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(168\) 0 0
\(169\) 3.41787 + 23.7718i 0.262913 + 1.82860i
\(170\) 0 0
\(171\) −13.7021 −1.04782
\(172\) 0 0
\(173\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(174\) 0 0
\(175\) −2.03179 3.16153i −0.153589 0.238989i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(180\) 0 0
\(181\) −0.449359 + 3.12536i −0.0334006 + 0.232306i −0.999683 0.0251785i \(-0.991985\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −21.6284 13.8997i −1.59881 1.02750i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.55759 2.95162i 0.186038 0.214699i
\(190\) 0 0
\(191\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(192\) 0 0
\(193\) 6.36956 + 7.35086i 0.458491 + 0.529127i 0.937175 0.348861i \(-0.113431\pi\)
−0.478684 + 0.877987i \(0.658886\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(198\) 0 0
\(199\) −24.3979 7.16388i −1.72952 0.507834i −0.742698 0.669626i \(-0.766454\pi\)
−0.986825 + 0.161793i \(0.948272\pi\)
\(200\) 0 0
\(201\) −12.4366 + 6.80677i −0.877207 + 0.480112i
\(202\) 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
\(211\) −4.08501 28.4119i −0.281224 1.95595i −0.293279 0.956027i \(-0.594746\pi\)
0.0120548 0.999927i \(-0.496163\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.48404 5.43929i −0.168628 0.369243i
\(218\) 0 0
\(219\) −7.91280 + 12.3126i −0.534698 + 0.832006i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.03102 14.1261i −0.136007 0.945952i −0.937509 0.347960i \(-0.886874\pi\)
0.801502 0.597992i \(-0.204035\pi\)
\(224\) 0 0
\(225\) 12.6188 8.10961i 0.841254 0.540641i
\(226\) 0 0
\(227\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(228\) 0 0
\(229\) −22.8520 + 19.8014i −1.51010 + 1.30851i −0.726900 + 0.686743i \(0.759040\pi\)
−0.783202 + 0.621768i \(0.786415\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −9.43644 10.8902i −0.612962 0.707396i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −12.5090 + 27.3908i −0.805774 + 1.76440i −0.181179 + 0.983450i \(0.557991\pi\)
−0.624595 + 0.780949i \(0.714736\pi\)
\(242\) 0 0
\(243\) 11.7810 + 10.2083i 0.755750 + 0.654861i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −25.2771 11.5437i −1.60834 0.734506i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(258\) 0 0
\(259\) −0.277957 + 0.946634i −0.0172714 + 0.0588210i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(264\) 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
\(271\) 8.64153 + 1.24246i 0.524936 + 0.0754744i 0.399688 0.916651i \(-0.369118\pi\)
0.125247 + 0.992126i \(0.460028\pi\)
\(272\) 0 0
\(273\) 7.20482 3.29033i 0.436056 0.199140i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.2048 + 2.99639i 0.613145 + 0.180036i 0.573537 0.819180i \(-0.305571\pi\)
0.0396081 + 0.999215i \(0.487389\pi\)
\(278\) 0 0
\(279\) 21.7101 9.91469i 1.29975 0.593577i
\(280\) 0 0
\(281\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(282\) 0 0
\(283\) −3.68456 + 1.08188i −0.219024 + 0.0643113i −0.389404 0.921067i \(-0.627319\pi\)
0.170379 + 0.985379i \(0.445501\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.06206 15.4637i 0.415415 0.909632i
\(290\) 0 0
\(291\) −2.92974 + 20.3768i −0.171744 + 1.19451i
\(292\) 0 0
\(293\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.547746 0.160833i 0.0315716 0.00927025i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.22355 2.56611i −0.126905 0.146456i 0.688741 0.725007i \(-0.258164\pi\)
−0.815646 + 0.578551i \(0.803618\pi\)
\(308\) 0 0
\(309\) −17.0558 26.5394i −0.970272 1.50977i
\(310\) 0 0
\(311\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(312\) 0 0
\(313\) 32.4546 4.66627i 1.83444 0.263753i 0.863724 0.503966i \(-0.168126\pi\)
0.970720 + 0.240212i \(0.0772171\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 30.1108 4.32929i 1.67025 0.240146i
\(326\) 0 0
\(327\) −7.72990 8.92078i −0.427464 0.493320i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.7918 + 24.5724i −0.867993 + 1.35062i 0.0676621 + 0.997708i \(0.478446\pi\)
−0.935655 + 0.352915i \(0.885190\pi\)
\(332\) 0 0
\(333\) −3.77835 1.10942i −0.207052 0.0607961i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −9.79570 + 33.3611i −0.533606 + 1.81729i 0.0413966 + 0.999143i \(0.486819\pi\)
−0.575002 + 0.818152i \(0.694999\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 7.63165 6.61286i 0.412070 0.357061i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(348\) 0 0
\(349\) 19.5107 + 12.5388i 1.04438 + 0.671184i 0.946067 0.323971i \(-0.105018\pi\)
0.0983163 + 0.995155i \(0.468654\pi\)
\(350\) 0 0
\(351\) 13.1329 + 28.7570i 0.700982 + 1.53494i
\(352\) 0 0
\(353\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(360\) 0 0
\(361\) 1.56539 1.00602i 0.0823890 0.0529482i
\(362\) 0 0
\(363\) 14.3990 12.4768i 0.755750 0.654861i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 12.0668 1.73495i 0.629883 0.0905635i 0.180023 0.983662i \(-0.442383\pi\)
0.449860 + 0.893099i \(0.351474\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.05827i 0.210130i 0.994465 + 0.105065i \(0.0335050\pi\)
−0.994465 + 0.105065i \(0.966495\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 33.9585 + 4.88250i 1.74433 + 0.250797i 0.939462 0.342653i \(-0.111326\pi\)
0.804871 + 0.593450i \(0.202235\pi\)
\(380\) 0 0
\(381\) −33.0092 15.0748i −1.69111 0.772305i
\(382\) 0 0
\(383\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.641941 + 2.18625i 0.0326317 + 0.111133i
\(388\) 0 0
\(389\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.5500 + 36.2394i 0.830620 + 1.81880i 0.435778 + 0.900054i \(0.356473\pi\)
0.394842 + 0.918749i \(0.370799\pi\)
\(398\) 0 0
\(399\) −0.846208 + 5.88550i −0.0423634 + 0.294644i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 48.4029 2.41112
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.11667 27.6429i 0.401344 1.36685i −0.472794 0.881173i \(-0.656755\pi\)
0.874138 0.485678i \(-0.161427\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −30.4161 + 8.93097i −1.48948 + 0.437352i
\(418\) 0 0
\(419\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(420\) 0 0
\(421\) 38.9428 11.4346i 1.89796 0.557290i 0.907384 0.420303i \(-0.138076\pi\)
0.990573 0.136988i \(-0.0437421\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.30610 + 8.43169i −0.353567 + 0.408038i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 19.4307 16.8368i 0.933782 0.809126i −0.0480569 0.998845i \(-0.515303\pi\)
0.981839 + 0.189718i \(0.0607574\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 40.4106 1.92869 0.964347 0.264642i \(-0.0852539\pi\)
0.964347 + 0.264642i \(0.0852539\pi\)
\(440\) 0 0
\(441\) 12.6422 + 14.5899i 0.602010 + 0.694756i
\(442\) 0 0
\(443\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 32.1341 14.6751i 1.50979 0.689498i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.38599 1.59952i 0.0648341 0.0748225i −0.722406 0.691469i \(-0.756964\pi\)
0.787240 + 0.616647i \(0.211509\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(462\) 0 0
\(463\) 28.1321 + 12.8475i 1.30741 + 0.597075i 0.942574 0.333997i \(-0.108397\pi\)
0.364837 + 0.931071i \(0.381125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(468\) 0 0
\(469\) 2.15568 + 5.76229i 0.0995402 + 0.266078i
\(470\) 0 0
\(471\) 9.41236 32.0555i 0.433699 1.47704i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −9.48675 + 20.7731i −0.435282 + 0.953135i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(480\) 0 0
\(481\) −6.03550 5.22979i −0.275195 0.238458i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −11.2853 + 17.5603i −0.511388 + 0.795735i −0.996915 0.0784867i \(-0.974991\pi\)
0.485528 + 0.874221i \(0.338628\pi\)
\(488\) 0 0
\(489\) −10.9377 1.57260i −0.494619 0.0711155i
\(490\) 0 0
\(491\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 44.6005i 1.99659i −0.0583392 0.998297i \(-0.518580\pi\)
0.0583392 0.998297i \(-0.481420\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 41.5973i 1.84740i
\(508\) 0 0
\(509\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(510\) 0 0
\(511\) 4.79998 + 4.15921i 0.212339 + 0.183993i
\(512\) 0 0
\(513\) −23.4911 3.37752i −1.03716 0.149121i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(522\) 0 0
\(523\) −13.3351 + 15.3896i −0.583104 + 0.672938i −0.968269 0.249910i \(-0.919599\pi\)
0.385165 + 0.922848i \(0.374145\pi\)
\(524\) 0 0
\(525\) −2.70404 5.92103i −0.118014 0.258415i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0683 6.47985i −0.959493 0.281733i
\(530\) 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
\(541\) −27.9237 + 12.7523i −1.20054 + 0.548266i −0.912391 0.409320i \(-0.865766\pi\)
−0.288145 + 0.957587i \(0.593039\pi\)
\(542\) 0 0
\(543\) −1.54078 + 5.24742i −0.0661212 + 0.225188i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.11550 0.509431i 0.0476953 0.0217817i −0.391425 0.920210i \(-0.628018\pi\)
0.439120 + 0.898428i \(0.355290\pi\)
\(548\) 0 0
\(549\) −33.6539 29.1612i −1.43631 1.24457i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −5.26049 + 3.38071i −0.223699 + 0.143762i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(558\) 0 0
\(559\) −0.657632 + 4.57393i −0.0278149 + 0.193457i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 5.11236 4.42988i 0.214699 0.186038i
\(568\) 0 0
\(569\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(570\) 0 0
\(571\) 6.01962 + 41.8674i 0.251913 + 1.75210i 0.586703 + 0.809802i \(0.300426\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19.5101 30.3582i −0.812214 1.26383i −0.961437 0.275027i \(-0.911313\pi\)
0.149222 0.988804i \(-0.452323\pi\)
\(578\) 0 0
\(579\) 9.10815 + 14.1726i 0.378522 + 0.588991i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(588\) 0 0
\(589\) −19.6449 + 30.5681i −0.809454 + 1.25953i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −40.0624 18.2959i −1.63965 0.748802i
\(598\) 0 0
\(599\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(600\) 0 0
\(601\) −25.9177 7.61013i −1.05721 0.310424i −0.293481 0.955965i \(-0.594814\pi\)
−0.763725 + 0.645541i \(0.776632\pi\)
\(602\) 0 0
\(603\) −22.9993 + 8.60410i −0.936605 + 0.350386i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −26.8168 17.2341i −1.08846 0.699511i −0.131964 0.991254i \(-0.542128\pi\)
−0.956497 + 0.291743i \(0.905765\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.53422 17.6259i −0.102356 0.711903i −0.974782 0.223157i \(-0.928364\pi\)
0.872426 0.488746i \(-0.162545\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(618\) 0 0
\(619\) 19.9212 + 43.6213i 0.800700 + 1.75329i 0.643094 + 0.765787i \(0.277650\pi\)
0.157606 + 0.987502i \(0.449622\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3.55787 24.7455i −0.142315 0.989821i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 35.5226 30.7805i 1.41413 1.22535i 0.475828 0.879538i \(-0.342149\pi\)
0.938303 0.345813i \(-0.112397\pi\)
\(632\) 0 0
\(633\) 49.7169i 1.97607i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 11.0303 + 37.5656i 0.437035 + 1.48840i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 20.9808 45.9416i 0.827402 1.81176i 0.331158 0.943575i \(-0.392561\pi\)
0.496245 0.868183i \(-0.334712\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.91792 9.93754i −0.114362 0.389483i
\(652\) 0 0
\(653\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −16.6009 + 19.1584i −0.647662 + 0.747442i
\(658\) 0 0
\(659\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(660\) 0 0
\(661\) −2.42220 + 8.24927i −0.0942128 + 0.320859i −0.993092 0.117337i \(-0.962564\pi\)
0.898879 + 0.438196i \(0.144382\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 24.7187i 0.955680i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 50.7162 + 7.29189i 1.95497 + 0.281082i 0.999884 0.0152551i \(-0.00485604\pi\)
0.955083 + 0.296337i \(0.0957651\pi\)
\(674\) 0 0
\(675\) 23.6329 10.7928i 0.909632 0.415415i
\(676\) 0 0
\(677\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(678\) 0 0
\(679\) 8.57159 + 2.51684i 0.328947 + 0.0965876i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −44.0589 + 28.3149i −1.68095 + 1.08028i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −14.1370 + 30.9557i −0.537797 + 1.17761i 0.424455 + 0.905449i \(0.360466\pi\)
−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 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(702\) 0 0
\(703\) 5.75236 1.68905i 0.216955 0.0637036i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −29.3855 33.9126i −1.10359 1.27362i −0.958778 0.284158i \(-0.908286\pi\)
−0.144817 0.989458i \(-0.546259\pi\)
\(710\) 0 0
\(711\) −13.4936 20.9965i −0.506050 0.787430i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(720\) 0 0
\(721\) −12.4529 + 5.68704i −0.463769 + 0.211796i
\(722\) 0 0
\(723\) −28.1974 + 43.8760i −1.04867 + 1.63177i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.41466 + 0.347176i −0.0895549 + 0.0128760i −0.186947 0.982370i \(-0.559859\pi\)
0.0973917 + 0.995246i \(0.468950\pi\)
\(728\) 0 0
\(729\) 17.6812 + 20.4052i 0.654861 + 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 18.3976 28.6273i 0.679532 1.05737i −0.314602 0.949224i \(-0.601871\pi\)
0.994135 0.108149i \(-0.0344924\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −14.7524 + 50.2420i −0.542675 + 1.84818i −0.0131073 + 0.999914i \(0.504172\pi\)
−0.529568 + 0.848268i \(0.677646\pi\)
\(740\) 0 0
\(741\) −40.4901 26.0214i −1.48744 0.955921i
\(742\) 0 0
\(743\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −44.6784 28.7131i −1.63034 1.04776i −0.948753 0.316017i \(-0.897654\pi\)
−0.681586 0.731738i \(-0.738709\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.46821 0.498653i −0.126054 0.0181239i 0.0789989 0.996875i \(-0.474828\pi\)
−0.205053 + 0.978751i \(0.565737\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(762\) 0 0
\(763\) −4.30915 + 2.76933i −0.156002 + 0.100256i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −37.4227 + 5.38058i −1.34950 + 0.194029i −0.778873 0.627182i \(-0.784208\pi\)
−0.570626 + 0.821210i \(0.693299\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(774\) 0 0
\(775\) 39.7782i 1.42888i
\(776\) 0 0
\(777\) −0.709877 + 1.55441i −0.0254667 + 0.0557643i
\(778\) 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
\(787\) 21.6545 + 33.6951i 0.771901 + 1.20110i 0.975057 + 0.221955i \(0.0712438\pi\)
−0.203156 + 0.979146i \(0.565120\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −37.5158 82.1482i −1.33223 2.91717i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\)