Properties

Label 804.2.ba.a.329.1
Level 804
Weight 2
Character 804.329
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.ba (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label 329.1
Root \(0.580057 + 0.814576i\)
Character \(\chi\) = 804.329
Dual form 804.2.ba.a.413.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.57553 - 0.719520i) q^{3} +(-0.842599 - 4.37182i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})\) \(q+(-1.57553 - 0.719520i) q^{3} +(-0.842599 - 4.37182i) q^{7} +(1.96458 + 2.26725i) q^{9} +(7.17337 + 0.341710i) q^{13} +(-6.35738 - 1.22528i) q^{19} +(-1.81807 + 7.49420i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(-1.46393 - 4.98567i) q^{27} +(-4.75556 + 0.226535i) q^{31} +(-6.07910 - 10.5293i) q^{37} +(-11.0560 - 5.69976i) q^{39} +(5.51453 - 0.792870i) q^{43} +(-11.9043 + 4.76575i) q^{49} +(9.13462 + 6.50473i) q^{57} +(5.09645 - 5.34500i) q^{61} +(8.25665 - 10.4992i) q^{63} +(-7.90436 + 2.12630i) q^{67} +(-8.03940 - 7.66555i) q^{73} +(4.68209 + 7.28547i) q^{75} +(-1.45319 - 2.81879i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(-4.55038 - 31.6486i) q^{91} +(7.65551 + 3.06480i) q^{93} +(-10.7932 + 6.23144i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{66}\right)\) \(1\)

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 \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(6\) 0 0
\(7\) −0.842599 4.37182i −0.318473 1.65239i −0.689881 0.723923i \(-0.742337\pi\)
0.371408 0.928470i \(-0.378875\pi\)
\(8\) 0 0
\(9\) 1.96458 + 2.26725i 0.654861 + 0.755750i
\(10\) 0 0
\(11\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(12\) 0 0
\(13\) 7.17337 + 0.341710i 1.98953 + 0.0947732i 0.999074 0.0430222i \(-0.0136986\pi\)
0.990461 + 0.137795i \(0.0440016\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(18\) 0 0
\(19\) −6.35738 1.22528i −1.45848 0.281099i −0.602506 0.798114i \(-0.705831\pi\)
−0.855976 + 0.517015i \(0.827043\pi\)
\(20\) 0 0
\(21\) −1.81807 + 7.49420i −0.396736 + 1.63537i
\(22\) 0 0
\(23\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(24\) 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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −4.75556 + 0.226535i −0.854123 + 0.0406869i −0.470075 0.882626i \(-0.655773\pi\)
−0.384048 + 0.923313i \(0.625470\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.07910 10.5293i −0.999398 1.73101i −0.529738 0.848162i \(-0.677710\pi\)
−0.469661 0.882847i \(-0.655624\pi\)
\(38\) 0 0
\(39\) −11.0560 5.69976i −1.77037 0.912691i
\(40\) 0 0
\(41\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(42\) 0 0
\(43\) 5.51453 0.792870i 0.840958 0.120911i 0.291643 0.956527i \(-0.405798\pi\)
0.549315 + 0.835616i \(0.314889\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(48\) 0 0
\(49\) −11.9043 + 4.76575i −1.70061 + 0.680821i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.13462 + 6.50473i 1.20991 + 0.861573i
\(58\) 0 0
\(59\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(60\) 0 0
\(61\) 5.09645 5.34500i 0.652533 0.684357i −0.311467 0.950257i \(-0.600820\pi\)
0.964001 + 0.265900i \(0.0856690\pi\)
\(62\) 0 0
\(63\) 8.25665 10.4992i 1.04024 1.32277i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.90436 + 2.12630i −0.965671 + 0.259769i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(72\) 0 0
\(73\) −8.03940 7.66555i −0.940941 0.897185i 0.0539089 0.998546i \(-0.482832\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 4.68209 + 7.28547i 0.540641 + 0.841254i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.45319 2.81879i −0.163496 0.317138i 0.792829 0.609445i \(-0.208608\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) −1.28083 + 8.90839i −0.142315 + 0.989821i
\(82\) 0 0
\(83\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(90\) 0 0
\(91\) −4.55038 31.6486i −0.477010 3.31768i
\(92\) 0 0
\(93\) 7.65551 + 3.06480i 0.793840 + 0.317805i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.7932 + 6.23144i −1.09588 + 0.632707i −0.935136 0.354289i \(-0.884723\pi\)
−0.160744 + 0.986996i \(0.551390\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(102\) 0 0
\(103\) 0.498621 + 10.4674i 0.0491306 + 1.03138i 0.880019 + 0.474939i \(0.157530\pi\)
−0.830888 + 0.556440i \(0.812167\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(108\) 0 0
\(109\) −1.59842 + 2.48719i −0.153101 + 0.238230i −0.909329 0.416079i \(-0.863404\pi\)
0.756227 + 0.654309i \(0.227040\pi\)
\(110\) 0 0
\(111\) 2.00175 + 20.9633i 0.189998 + 1.98975i
\(112\) 0 0
\(113\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 13.3179 + 16.9351i 1.23124 + 1.56565i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.523401 + 10.9875i −0.0475819 + 0.998867i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 22.1314 4.26548i 1.96385 0.378501i 0.981315 0.192410i \(-0.0616302\pi\)
0.982533 0.186091i \(-0.0595819\pi\)
\(128\) 0 0
\(129\) −9.25879 2.71862i −0.815191 0.239362i
\(130\) 0 0
\(131\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(132\) 0 0
\(133\) 28.8257i 2.49951i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(138\) 0 0
\(139\) 5.28614 18.0029i 0.448365 1.52699i −0.356939 0.934128i \(-0.616180\pi\)
0.805304 0.592862i \(-0.202002\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 22.1846 + 1.05678i 1.82975 + 0.0871618i
\(148\) 0 0
\(149\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(150\) 0 0
\(151\) 15.4290 12.1335i 1.25560 0.987411i 0.255809 0.966727i \(-0.417658\pi\)
0.999786 0.0206838i \(-0.00658434\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 24.7634 2.36462i 1.97633 0.188717i 0.976671 0.214742i \(-0.0688911\pi\)
0.999661 + 0.0260253i \(0.00828506\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.35057 + 7.53540i −0.340763 + 0.590218i −0.984575 0.174966i \(-0.944019\pi\)
0.643812 + 0.765184i \(0.277352\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(168\) 0 0
\(169\) 38.3993 + 3.66669i 2.95379 + 0.282053i
\(170\) 0 0
\(171\) −9.71157 16.8209i −0.742662 1.28633i
\(172\) 0 0
\(173\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(174\) 0 0
\(175\) −8.27372 + 20.6668i −0.625435 + 1.56226i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(180\) 0 0
\(181\) −14.6312 20.5466i −1.08753 1.52722i −0.827374 0.561651i \(-0.810166\pi\)
−0.260153 0.965567i \(-0.583773\pi\)
\(182\) 0 0
\(183\) −11.8754 + 4.75421i −0.877857 + 0.351441i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −20.5630 + 10.6009i −1.49574 + 0.771105i
\(190\) 0 0
\(191\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(192\) 0 0
\(193\) 21.4810 13.8050i 1.54624 0.993705i 0.559974 0.828510i \(-0.310811\pi\)
0.986261 0.165195i \(-0.0528253\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(198\) 0 0
\(199\) −8.08081 + 23.3480i −0.572834 + 1.65509i 0.169865 + 0.985467i \(0.445667\pi\)
−0.742698 + 0.669626i \(0.766454\pi\)
\(200\) 0 0
\(201\) 13.9835 + 2.33729i 0.986317 + 0.164860i
\(202\) 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
\(211\) 15.8572 22.2683i 1.09165 1.53301i 0.270146 0.962819i \(-0.412928\pi\)
0.821509 0.570195i \(-0.193132\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.99740 + 20.5996i 0.339246 + 1.39839i
\(218\) 0 0
\(219\) 7.15079 + 17.8618i 0.483206 + 1.20699i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 11.9893 + 26.2529i 0.802864 + 1.75803i 0.635452 + 0.772140i \(0.280814\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 0 0
\(225\) −2.13472 14.8473i −0.142315 0.989821i
\(226\) 0 0
\(227\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(228\) 0 0
\(229\) 10.5227 20.4113i 0.695362 1.34881i −0.231287 0.972886i \(-0.574293\pi\)
0.926649 0.375929i \(-0.122676\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.261363 + 5.48668i 0.0169773 + 0.356398i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) −27.1993 + 7.98643i −1.75206 + 0.514451i −0.990957 0.134179i \(-0.957160\pi\)
−0.761103 + 0.648630i \(0.775342\pi\)
\(242\) 0 0
\(243\) 8.42776 13.1138i 0.540641 0.841254i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −45.1851 10.9618i −2.87506 0.697482i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(258\) 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 \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(264\) 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
\(271\) −29.2219 13.3452i −1.77511 0.810664i −0.978524 0.206133i \(-0.933912\pi\)
−0.796582 0.604530i \(-0.793361\pi\)
\(272\) 0 0
\(273\) −15.6025 + 53.1374i −0.944309 + 3.21602i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.96986 5.73553i −0.298610 0.344614i 0.586540 0.809920i \(-0.300490\pi\)
−0.885150 + 0.465306i \(0.845944\pi\)
\(278\) 0 0
\(279\) −9.85629 10.3370i −0.590081 0.618859i
\(280\) 0 0
\(281\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(282\) 0 0
\(283\) 18.1894 20.9916i 1.08125 1.24782i 0.114135 0.993465i \(-0.463590\pi\)
0.967110 0.254358i \(-0.0818643\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.00790 16.5208i 0.235759 0.971812i
\(290\) 0 0
\(291\) 21.4886 2.05191i 1.25968 0.120285i
\(292\) 0 0
\(293\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −8.11282 23.4405i −0.467615 1.35109i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 23.5019 + 12.1160i 1.34132 + 0.691499i 0.971374 0.237553i \(-0.0763455\pi\)
0.369947 + 0.929053i \(0.379376\pi\)
\(308\) 0 0
\(309\) 6.74588 16.8504i 0.383759 0.958585i
\(310\) 0 0
\(311\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(312\) 0 0
\(313\) −31.6320 + 14.4459i −1.78795 + 0.816528i −0.817227 + 0.576316i \(0.804490\pi\)
−0.970720 + 0.240212i \(0.922783\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −29.2494 20.8284i −1.62246 1.15535i
\(326\) 0 0
\(327\) 4.30795 2.76855i 0.238230 0.153101i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.3238 28.3870i 1.22703 1.56029i 0.517317 0.855794i \(-0.326931\pi\)
0.709708 0.704496i \(-0.248827\pi\)
\(332\) 0 0
\(333\) 11.9297 34.4685i 0.653742 1.88886i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.3615 + 9.12381i 1.43600 + 0.497006i 0.930651 0.365907i \(-0.119241\pi\)
0.505352 + 0.862913i \(0.331363\pi\)
\(338\) 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 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(348\) 0 0
\(349\) 1.02341 7.11795i 0.0547817 0.381015i −0.943924 0.330162i \(-0.892897\pi\)
0.998706 0.0508535i \(-0.0161942\pi\)
\(350\) 0 0
\(351\) −8.79763 36.2643i −0.469582 1.93565i
\(352\) 0 0
\(353\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(360\) 0 0
\(361\) 21.2759 + 8.51760i 1.11979 + 0.448295i
\(362\) 0 0
\(363\) 8.73039 16.9346i 0.458227 0.888835i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.36828 + 35.2742i −0.175823 + 1.84130i 0.290730 + 0.956805i \(0.406102\pi\)
−0.466552 + 0.884494i \(0.654504\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 33.0614 + 19.0880i 1.71185 + 0.988340i 0.932060 + 0.362305i \(0.118010\pi\)
0.779795 + 0.626035i \(0.215323\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.90330 + 19.9323i 0.0977662 + 1.02385i 0.902637 + 0.430403i \(0.141629\pi\)
−0.804871 + 0.593450i \(0.797765\pi\)
\(380\) 0 0
\(381\) −37.9378 9.20361i −1.94361 0.471515i
\(382\) 0 0
\(383\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.6314 + 10.9452i 0.642089 + 0.556373i
\(388\) 0 0
\(389\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.8740 6.71641i −1.14801 0.337087i −0.348249 0.937402i \(-0.613224\pi\)
−0.799763 + 0.600315i \(0.795042\pi\)
\(398\) 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
\(403\) −34.1908 −1.70316
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.490026 + 2.54250i 0.0242302 + 0.125718i 0.992030 0.126004i \(-0.0402152\pi\)
−0.967800 + 0.251722i \(0.919003\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −21.2819 + 24.5607i −1.04218 + 1.20274i
\(418\) 0 0
\(419\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(420\) 0 0
\(421\) −34.7876 6.70476i −1.69544 0.326770i −0.752216 0.658917i \(-0.771015\pi\)
−0.943228 + 0.332147i \(0.892227\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −27.6616 17.7771i −1.33864 0.860292i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 36.0107 1.71540i 1.73056 0.0824369i 0.841511 0.540241i \(-0.181667\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 20.6409 + 35.7511i 0.985136 + 1.70631i 0.641331 + 0.767264i \(0.278382\pi\)
0.343805 + 0.939041i \(0.388284\pi\)
\(440\) 0 0
\(441\) −34.1920 17.6272i −1.62819 0.839392i
\(442\) 0 0
\(443\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −33.0392 + 8.01521i −1.55232 + 0.376588i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.7202 8.10434i 0.735361 0.379105i −0.0495022 0.998774i \(-0.515763\pi\)
0.784863 + 0.619669i \(0.212733\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(462\) 0 0
\(463\) −6.78938 + 7.12050i −0.315529 + 0.330918i −0.862331 0.506344i \(-0.830996\pi\)
0.546802 + 0.837262i \(0.315845\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(468\) 0 0
\(469\) 15.9560 + 32.7648i 0.736781 + 1.51294i
\(470\) 0 0
\(471\) −40.7168 14.0922i −1.87613 0.649335i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 23.4286 + 22.3392i 1.07498 + 1.02499i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(480\) 0 0
\(481\) −40.0097 77.6080i −1.82428 3.53862i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −9.09682 22.7228i −0.412216 1.02967i −0.978645 0.205556i \(-0.934100\pi\)
0.566429 0.824110i \(-0.308325\pi\)
\(488\) 0 0
\(489\) 12.2763 8.74193i 0.555154 0.395324i
\(490\) 0 0
\(491\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7.73117 4.46359i 0.346095 0.199818i −0.316869 0.948469i \(-0.602632\pi\)
0.662964 + 0.748651i \(0.269298\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −57.8610 33.4061i −2.56970 1.48362i
\(508\) 0 0
\(509\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(510\) 0 0
\(511\) −26.7384 + 41.6058i −1.18284 + 1.84053i
\(512\) 0 0
\(513\) 3.19786 + 33.4895i 0.141189 + 1.47860i
\(514\) 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 \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(522\) 0 0
\(523\) 1.86868 39.2283i 0.0817115 1.71533i −0.471143 0.882057i \(-0.656158\pi\)
0.552855 0.833278i \(-0.313538\pi\)
\(524\) 0 0
\(525\) 27.9056 26.6080i 1.21790 1.16127i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 22.5844 4.35278i 0.981929 0.189251i
\(530\) 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
\(541\) −9.54545 + 32.5088i −0.410391 + 1.39766i 0.452270 + 0.891881i \(0.350614\pi\)
−0.862661 + 0.505782i \(0.831204\pi\)
\(542\) 0 0
\(543\) 8.26815 + 42.8992i 0.354820 + 1.84098i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.90970 + 4.10038i 0.167167 + 0.175319i 0.801989 0.597339i \(-0.203775\pi\)
−0.634822 + 0.772658i \(0.718927\pi\)
\(548\) 0 0
\(549\) 22.1308 + 1.05422i 0.944521 + 0.0449931i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −11.0988 + 8.72818i −0.471968 + 0.371160i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(558\) 0 0
\(559\) 39.8287 3.80318i 1.68457 0.160857i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 40.0251 1.90663i 1.68090 0.0800710i
\(568\) 0 0
\(569\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(570\) 0 0
\(571\) 45.7558 + 4.36915i 1.91482 + 0.182843i 0.983444 0.181210i \(-0.0580014\pi\)
0.931378 + 0.364054i \(0.118607\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15.4004 38.4684i 0.641128 1.60146i −0.149222 0.988804i \(-0.547677\pi\)
0.790350 0.612656i \(-0.209899\pi\)
\(578\) 0 0
\(579\) −43.7769 + 6.29416i −1.81930 + 0.261577i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(588\) 0 0
\(589\) 30.5104 + 4.38674i 1.25716 + 0.180752i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.5309 30.9711i 1.20862 1.26756i
\(598\) 0 0
\(599\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(600\) 0 0
\(601\) 15.5368 44.8905i 0.633757 1.83112i 0.0828381 0.996563i \(-0.473602\pi\)
0.550919 0.834559i \(-0.314277\pi\)
\(602\) 0 0
\(603\) −20.3496 13.7438i −0.828700 0.559692i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28.3098 22.2631i −1.14906 0.903631i −0.152650 0.988280i \(-0.548781\pi\)
−0.996411 + 0.0846490i \(0.973023\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −15.8404 + 22.2447i −0.639788 + 0.898456i −0.999400 0.0346459i \(-0.988970\pi\)
0.359612 + 0.933102i \(0.382909\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(618\) 0 0
\(619\) −0.739352 3.04765i −0.0297171 0.122495i 0.955021 0.296538i \(-0.0958321\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 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
\(631\) −22.9558 + 44.5281i −0.913857 + 1.77263i −0.390068 + 0.920786i \(0.627548\pi\)
−0.523789 + 0.851848i \(0.675482\pi\)
\(632\) 0 0
\(633\) −41.0060 + 23.6748i −1.62984 + 0.940989i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −87.0222 + 30.1187i −3.44794 + 1.19335i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) −24.1475 + 7.09036i −0.952286 + 0.279616i −0.720738 0.693207i \(-0.756197\pi\)
−0.231548 + 0.972824i \(0.574379\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.94825 36.0509i 0.272323 1.41295i
\(652\) 0 0
\(653\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.58565 33.2869i 0.0618622 1.29865i
\(658\) 0 0
\(659\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(660\) 0 0
\(661\) 2.82623 2.44894i 0.109928 0.0952528i −0.598163 0.801375i \(-0.704102\pi\)
0.708090 + 0.706122i \(0.249557\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 49.9888i 1.93268i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 22.2638 + 10.1675i 0.858206 + 0.391930i 0.795392 0.606096i \(-0.207265\pi\)
0.0628142 + 0.998025i \(0.479992\pi\)
\(674\) 0 0
\(675\) −7.31963 + 24.9284i −0.281733 + 0.959493i
\(676\) 0 0
\(677\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(678\) 0 0
\(679\) 36.3371 + 41.9352i 1.39449 + 1.60932i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −31.2652 + 24.5872i −1.19284 + 0.938061i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.20973 + 33.8410i −0.312313 + 1.28737i 0.571914 + 0.820313i \(0.306201\pi\)
−0.884227 + 0.467057i \(0.845314\pi\)
\(692\) 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 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(702\) 0 0
\(703\) 25.7457 + 74.3875i 0.971019 + 2.80558i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 31.0876 + 16.0268i 1.16752 + 0.601898i 0.929305 0.369314i \(-0.120407\pi\)
0.238215 + 0.971212i \(0.423438\pi\)
\(710\) 0 0
\(711\) 3.53599 8.83247i 0.132610 0.331244i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(720\) 0 0
\(721\) 45.3412 10.9997i 1.68860 0.409649i
\(722\) 0 0
\(723\) 48.5997 + 6.98758i 1.80744 + 0.259871i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −36.8977 26.2747i −1.36846 0.974475i −0.998976 0.0452379i \(-0.985595\pi\)
−0.369483 0.929237i \(-0.620465\pi\)
\(728\) 0 0
\(729\) −22.7138 + 14.5973i −0.841254 + 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −31.7724 + 40.4020i −1.17354 + 1.49228i −0.339477 + 0.940614i \(0.610250\pi\)
−0.834065 + 0.551666i \(0.813992\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 39.9667 + 13.8326i 1.47020 + 0.508841i 0.940633 0.339426i \(-0.110233\pi\)
0.529568 + 0.848268i \(0.322354\pi\)
\(740\) 0 0
\(741\) 63.3032 + 49.7822i 2.32550 + 1.82880i
\(742\) 0 0
\(743\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(744\) 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 −0.948753 0.316017i \(-0.897654\pi\)
0.999355 0.0359215i \(-0.0114366\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 43.2062 30.7670i 1.57035 1.11824i 0.630748 0.775988i \(-0.282748\pi\)
0.939606 0.342257i \(-0.111191\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(762\) 0 0
\(763\) 12.2204 + 4.89231i 0.442408 + 0.177114i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −4.76998 + 49.9535i −0.172010 + 1.80137i 0.337943 + 0.941167i \(0.390269\pi\)
−0.509953 + 0.860202i \(0.670337\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(774\) 0 0
\(775\) 20.6155 + 11.9024i 0.740531 + 0.427546i
\(776\) 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
\(787\) −20.5167 26.0892i −0.731343 0.929978i 0.268099 0.963391i \(-0.413605\pi\)
−0.999442 + 0.0334133i \(0.989362\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 38.3851 36.6002i 1.36310 1.29971i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0