Properties

Label 1764.2.e.f.1079.1
Level $1764$
Weight $2$
Character 1764.1079
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1079.1
Root \(-0.581861 + 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1079
Dual form 1764.2.e.f.1079.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.28897 - 0.581861i) q^{2} +(1.32288 + 1.50000i) q^{4} +(-0.832353 - 2.70318i) q^{8} +O(q^{10})\) \(q+(-1.28897 - 0.581861i) q^{2} +(1.32288 + 1.50000i) q^{4} +(-0.832353 - 2.70318i) q^{8} -0.913230 q^{11} +(-0.500000 + 3.96863i) q^{16} +(1.17712 + 0.531373i) q^{22} -9.39851 q^{23} +5.00000 q^{25} -8.89753i q^{29} +(2.95367 - 4.82450i) q^{32} -10.5830 q^{37} +12.0000i q^{43} +(-1.20809 - 1.36985i) q^{44} +(12.1144 + 5.46863i) q^{46} +(-6.44484 - 2.90930i) q^{50} +0.412247i q^{53} +(-5.17712 + 11.4686i) q^{58} +(-6.61438 + 4.50000i) q^{64} -15.8745i q^{67} -7.57205 q^{71} +(13.6412 + 6.15784i) q^{74} -15.8745i q^{79} +(6.98233 - 15.4676i) q^{86} +(0.760130 + 2.46863i) q^{88} +(-12.4331 - 14.0978i) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{16} + 20q^{22} + 40q^{25} + 44q^{46} - 52q^{58} - 68q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28897 0.581861i −0.911438 0.411438i
\(3\) 0 0
\(4\) 1.32288 + 1.50000i 0.661438 + 0.750000i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.832353 2.70318i −0.294281 0.955719i
\(9\) 0 0
\(10\) 0 0
\(11\) −0.913230 −0.275349 −0.137675 0.990478i \(-0.543963\pi\)
−0.137675 + 0.990478i \(0.543963\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 3.96863i −0.125000 + 0.992157i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.17712 + 0.531373i 0.250964 + 0.113289i
\(23\) −9.39851 −1.95973 −0.979863 0.199673i \(-0.936012\pi\)
−0.979863 + 0.199673i \(0.936012\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.89753i 1.65223i −0.563502 0.826115i \(-0.690546\pi\)
0.563502 0.826115i \(-0.309454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 2.95367 4.82450i 0.522141 0.852859i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.5830 −1.73984 −0.869918 0.493197i \(-0.835828\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) −1.20809 1.36985i −0.182126 0.206512i
\(45\) 0 0
\(46\) 12.1144 + 5.46863i 1.78617 + 0.806305i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −6.44484 2.90930i −0.911438 0.411438i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.412247i 0.0566265i 0.999599 + 0.0283132i \(0.00901359\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −5.17712 + 11.4686i −0.679790 + 1.50590i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −6.61438 + 4.50000i −0.826797 + 0.562500i
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8745i 1.93938i −0.244339 0.969690i \(-0.578571\pi\)
0.244339 0.969690i \(-0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.57205 −0.898637 −0.449319 0.893372i \(-0.648333\pi\)
−0.449319 + 0.893372i \(0.648333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 13.6412 + 6.15784i 1.58575 + 0.715834i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.8745i 1.78602i −0.450035 0.893011i \(-0.648589\pi\)
0.450035 0.893011i \(-0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.98233 15.4676i 0.752924 1.66792i
\(87\) 0 0
\(88\) 0.760130 + 2.46863i 0.0810301 + 0.263157i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.4331 14.0978i −1.29624 1.46979i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 6.61438 + 7.50000i 0.661438 + 0.750000i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.239870 0.531373i 0.0232983 0.0516115i
\(107\) −17.8838 −1.72889 −0.864446 0.502726i \(-0.832330\pi\)
−0.864446 + 0.502726i \(0.832330\pi\)
\(108\) 0 0
\(109\) −10.5830 −1.01367 −0.506834 0.862044i \(-0.669184\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.5524i 1.27490i −0.770490 0.637452i \(-0.779988\pi\)
0.770490 0.637452i \(-0.220012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 13.3463 11.7703i 1.23917 1.09285i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.1660 −0.924183
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.8745i 1.40863i 0.709885 + 0.704317i \(0.248747\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 11.1441 1.95171i 0.985008 0.172508i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −9.23676 + 20.4617i −0.797934 + 1.76762i
\(135\) 0 0
\(136\) 0 0
\(137\) 22.0377i 1.88281i 0.337282 + 0.941404i \(0.390493\pi\)
−0.337282 + 0.941404i \(0.609507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.76013 + 4.40588i 0.819052 + 0.369733i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −14.0000 15.8745i −1.15079 1.30488i
\(149\) 8.07303i 0.661369i −0.943741 0.330684i \(-0.892720\pi\)
0.943741 0.330684i \(-0.107280\pi\)
\(150\) 0 0
\(151\) 24.0000i 1.95309i −0.215308 0.976546i \(-0.569076\pi\)
0.215308 0.976546i \(-0.430924\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −9.23676 + 20.4617i −0.734837 + 1.62785i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.8745i 1.24339i −0.783260 0.621694i \(-0.786445\pi\)
0.783260 0.621694i \(-0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −18.0000 + 15.8745i −1.37249 + 1.21042i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.456615 3.62427i 0.0344187 0.273190i
\(177\) 0 0
\(178\) 0 0
\(179\) −21.5367 −1.60973 −0.804865 0.593458i \(-0.797762\pi\)
−0.804865 + 0.593458i \(0.797762\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 7.82288 + 25.4059i 0.576710 + 1.87295i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.0514 −0.944369 −0.472184 0.881500i \(-0.656534\pi\)
−0.472184 + 0.881500i \(0.656534\pi\)
\(192\) 0 0
\(193\) −21.1660 −1.52356 −0.761781 0.647834i \(-0.775675\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.8681i 1.84303i 0.388348 + 0.921513i \(0.373046\pi\)
−0.388348 + 0.921513i \(0.626954\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −4.16176 13.5159i −0.294281 0.955719i
\(201\) 0 0
\(202\) 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
\(211\) 12.0000i 0.826114i −0.910705 0.413057i \(-0.864461\pi\)
0.910705 0.413057i \(-0.135539\pi\)
\(212\) −0.618370 + 0.545351i −0.0424699 + 0.0374549i
\(213\) 0 0
\(214\) 23.0516 + 10.4059i 1.57578 + 0.711331i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 13.6412 + 6.15784i 0.923895 + 0.417061i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7.88562 + 17.4686i −0.524544 + 1.16200i
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −24.0516 + 7.40588i −1.57907 + 0.486220i
\(233\) 30.5230i 1.99963i 0.0193169 + 0.999813i \(0.493851\pi\)
−0.0193169 + 0.999813i \(0.506149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.0220 1.94196 0.970981 0.239158i \(-0.0768713\pi\)
0.970981 + 0.239158i \(0.0768713\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 13.1037 + 5.91520i 0.842335 + 0.380244i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 8.58301 0.539609
\(254\) 9.23676 20.4617i 0.579566 1.28388i
\(255\) 0 0
\(256\) −15.5000 3.96863i −0.968750 0.248039i
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −26.3691 −1.62599 −0.812993 0.582273i \(-0.802164\pi\)
−0.812993 + 0.582273i \(0.802164\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 23.8118 21.0000i 1.45453 1.28278i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 12.8229 28.4059i 0.774658 1.71606i
\(275\) −4.56615 −0.275349
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.41815i 0.203910i −0.994789 0.101955i \(-0.967490\pi\)
0.994789 0.101955i \(-0.0325097\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −10.0169 11.3581i −0.594393 0.673978i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.80879 + 28.6078i 0.512001 + 1.66279i
\(297\) 0 0
\(298\) −4.69738 + 10.4059i −0.272112 + 0.602796i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −13.9647 + 30.9352i −0.803576 + 1.78012i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 23.8118 21.0000i 1.33952 1.18134i
\(317\) 16.5583i 0.930008i −0.885309 0.465004i \(-0.846053\pi\)
0.885309 0.465004i \(-0.153947\pi\)
\(318\) 0 0
\(319\) 8.12549i 0.454940i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −9.23676 + 20.4617i −0.511577 + 1.13327i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 36.0000i 1.97874i −0.145424 0.989369i \(-0.546455\pi\)
0.145424 0.989369i \(-0.453545\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 21.1660 1.15299 0.576493 0.817102i \(-0.304421\pi\)
0.576493 + 0.817102i \(0.304421\pi\)
\(338\) 16.7566 + 7.56419i 0.911438 + 0.411438i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 32.4382 9.98823i 1.74895 0.538529i
\(345\) 0 0
\(346\) 0 0
\(347\) 23.3632 1.25420 0.627100 0.778938i \(-0.284242\pi\)
0.627100 + 0.778938i \(0.284242\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.69738 + 4.40588i −0.143771 + 0.234834i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 27.7601 + 12.5314i 1.46717 + 0.662304i
\(359\) 31.8485 1.68090 0.840449 0.541891i \(-0.182292\pi\)
0.840449 + 0.541891i \(0.182292\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 4.69926 37.2992i 0.244966 1.94435i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.0000i 0.616399i −0.951322 0.308199i \(-0.900274\pi\)
0.951322 0.308199i \(-0.0997264\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.8229 + 7.59412i 0.860733 + 0.388549i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 27.2823 + 12.3157i 1.38863 + 0.626851i
\(387\) 0 0
\(388\) 0 0
\(389\) 34.3534i 1.74179i −0.491473 0.870893i \(-0.663542\pi\)
0.491473 0.870893i \(-0.336458\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 15.0516 33.3431i 0.758290 1.67980i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.50000 + 19.8431i −0.125000 + 0.992157i
\(401\) 39.0083i 1.94798i 0.226592 + 0.973990i \(0.427242\pi\)
−0.226592 + 0.973990i \(0.572758\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.66472 0.479062
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −6.98233 + 15.4676i −0.339895 + 0.752952i
\(423\) 0 0
\(424\) 1.11438 0.343135i 0.0541190 0.0166641i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −23.6580 26.8257i −1.14355 1.29667i
\(429\) 0 0
\(430\) 0 0
\(431\) 3.91913 0.188778 0.0943889 0.995535i \(-0.469910\pi\)
0.0943889 + 0.995535i \(0.469910\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 15.8745i −0.670478 0.760251i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 40.3337 1.91631 0.958157 0.286244i \(-0.0924067\pi\)
0.958157 + 0.286244i \(0.0924067\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.3475i 1.47938i 0.672948 + 0.739689i \(0.265028\pi\)
−0.672948 + 0.739689i \(0.734972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 20.3286 17.9282i 0.956178 0.843270i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 42.3320 1.98021 0.990104 0.140334i \(-0.0448177\pi\)
0.990104 + 0.140334i \(0.0448177\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 15.8745i 0.737751i −0.929479 0.368875i \(-0.879743\pi\)
0.929479 0.368875i \(-0.120257\pi\)
\(464\) 35.3110 + 4.44876i 1.63927 + 0.206529i
\(465\) 0 0
\(466\) 17.7601 39.3431i 0.822722 1.82254i
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.9588i 0.503884i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −38.6974 17.4686i −1.76998 0.798996i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −13.4484 15.2490i −0.611289 0.693137i
\(485\) 0 0
\(486\) 0 0
\(487\) 24.0000i 1.08754i 0.839233 + 0.543772i \(0.183004\pi\)
−0.839233 + 0.543772i \(0.816996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 34.8544 1.57296 0.786478 0.617619i \(-0.211903\pi\)
0.786478 + 0.617619i \(0.211903\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.0000i 1.61158i 0.592200 + 0.805791i \(0.298259\pi\)
−0.592200 + 0.805791i \(0.701741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −11.0632 4.99412i −0.491820 0.222016i
\(507\) 0 0
\(508\) −23.8118 + 21.0000i −1.05648 + 0.931724i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 17.6698 + 14.1343i 0.780903 + 0.624653i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 33.9889 + 15.3431i 1.48199 + 0.668992i
\(527\) 0 0
\(528\) 0 0
\(529\) 65.3320 2.84052
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −42.9117 + 13.2132i −1.85350 + 0.570723i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.8745i 0.678745i 0.940652 + 0.339372i \(0.110215\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −33.0565 + 29.1531i −1.41211 + 1.24536i
\(549\) 0 0
\(550\) 5.88562 + 2.65687i 0.250964 + 0.113289i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 12.8897 + 5.81861i 0.547630 + 0.247209i
\(555\) 0 0
\(556\) 0 0
\(557\) 25.0436i 1.06113i −0.847644 0.530566i \(-0.821980\pi\)
0.847644 0.530566i \(-0.178020\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.98889 + 4.40588i −0.0838961 + 0.185851i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 6.30262 + 20.4686i 0.264452 + 0.858845i
\(569\) 14.3769i 0.602711i −0.953512 0.301356i \(-0.902561\pi\)
0.953512 0.301356i \(-0.0974392\pi\)
\(570\) 0 0
\(571\) 47.6235i 1.99298i 0.0836974 + 0.996491i \(0.473327\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −46.9926 −1.95973
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −21.9125 9.89164i −0.911438 0.411438i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.376476i 0.0155921i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 5.29150 42.0000i 0.217479 1.72619i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.1096 10.6796i 0.496027 0.437454i
\(597\) 0 0
\(598\) 0 0
\(599\) 48.8190 1.99469 0.997346 0.0728143i \(-0.0231980\pi\)
0.997346 + 0.0728143i \(0.0231980\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 36.0000 31.7490i 1.46482 1.29185i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.3180i 1.94521i −0.232462 0.972605i \(-0.574678\pi\)
0.232462 0.972605i \(-0.425322\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 47.6235i 1.89586i −0.318475 0.947931i \(-0.603171\pi\)
0.318475 0.947931i \(-0.396829\pi\)
\(632\) −42.9117 + 13.2132i −1.70693 + 0.525592i
\(633\) 0 0
\(634\) −9.63464 + 21.3431i −0.382640 + 0.847644i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 4.72791 10.4735i 0.187180 0.414650i
\(639\) 0 0
\(640\) 0 0
\(641\) 47.4935i 1.87588i 0.346795 + 0.937941i \(0.387270\pi\)
−0.346795 + 0.937941i \(0.612730\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 23.8118 21.0000i 0.932541 0.822423i
\(653\) 42.8387i 1.67641i −0.545358 0.838203i \(-0.683606\pi\)
0.545358 0.838203i \(-0.316394\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.4044 −0.483207 −0.241604 0.970375i \(-0.577673\pi\)
−0.241604 + 0.970375i \(0.577673\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −20.9470 + 46.4028i −0.814128 + 1.80350i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 83.6235i 3.23792i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −42.3320 −1.63178 −0.815890 0.578208i \(-0.803752\pi\)
−0.815890 + 0.578208i \(0.803752\pi\)
\(674\) −27.2823 12.3157i −1.05088 0.474382i
\(675\) 0 0
\(676\) −17.1974 19.5000i −0.661438 0.750000i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.0279 −1.26378 −0.631889 0.775059i \(-0.717720\pi\)
−0.631889 + 0.775059i \(0.717720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −47.6235 6.00000i −1.81563 0.228748i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −30.1144 13.5941i −1.14313 0.516026i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0024i 1.35979i 0.733309 + 0.679895i \(0.237975\pi\)
−0.733309 + 0.679895i \(0.762025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.04045 4.10954i 0.227658 0.154884i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 52.9150 1.98727 0.993633 0.112667i \(-0.0359394\pi\)
0.993633 + 0.112667i \(0.0359394\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −28.4904 32.3051i −1.06474 1.20730i
\(717\) 0 0
\(718\) −41.0516 18.5314i −1.53203 0.691585i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −24.4904 11.0554i −0.911438 0.411438i
\(723\) 0 0
\(724\) 0 0
\(725\) 44.4876i 1.65223i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −27.7601 + 45.3431i −1.02325 + 1.67137i
\(737\) 14.4971i 0.534007i
\(738\) 0 0
\(739\) 15.8745i 0.583953i 0.956425 + 0.291977i \(0.0943129\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.09267 −0.0767726 −0.0383863 0.999263i \(-0.512222\pi\)
−0.0383863 + 0.999263i \(0.512222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 28.3573 + 12.8009i 1.03823 + 0.468676i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 48.0000i 1.75154i 0.482724 + 0.875772i \(0.339647\pi\)
−0.482724 + 0.875772i \(0.660353\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.5830 −0.384646 −0.192323 0.981332i \(-0.561602\pi\)
−0.192323 + 0.981332i \(0.561602\pi\)
\(758\) −6.98233 + 15.4676i −0.253610 + 0.561809i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −17.2654 19.5771i −0.624641 0.708276i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −28.0000 31.7490i −1.00774 1.14267i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −19.9889 + 44.2804i −0.716636 + 1.58753i
\(779\) 0 0
\(780\) 0 0
\(781\) 6.91503 0.247439
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −38.8021 + 34.2203i −1.38227 + 1.21905i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 14.7684 24.1225i 0.522141 0.852859i
\(801\) 0 0
\(802\) 22.6974 50.2804i 0.801472 1.77546i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.8033i 1.99710i 0.0538482 + 0.998549i \(0.482851\pi\)
−0.0538482 + 0.998549i \(0.517149\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\)