Properties

Label 3024.2.k.k.1889.4
Level $3024$
Weight $2$
Character 3024.1889
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 24 x^{14} + 230 x^{12} - 1052 x^{10} + 2139 x^{8} - 1244 x^{6} + 1134 x^{4} - 104 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.4
Root \(-0.415570 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.k.1889.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.70790 q^{5} +(0.946562 + 2.47063i) q^{7} +O(q^{10})\) \(q-2.70790 q^{5} +(0.946562 + 2.47063i) q^{7} +0.464391i q^{11} -3.03772i q^{13} -0.900912 q^{17} -0.831139i q^{19} -7.12979i q^{23} +2.33270 q^{25} +4.22582i q^{29} -1.30567i q^{31} +(-2.56319 - 6.69021i) q^{35} +10.3804 q^{37} -5.51235 q^{41} +11.8200 q^{43} -8.63862 q^{47} +(-5.20804 + 4.67721i) q^{49} +8.45164i q^{53} -1.25752i q^{55} +14.2475 q^{59} +3.27899i q^{61} +8.22582i q^{65} +3.40392 q^{67} -7.76143i q^{71} +4.83954i q^{73} +(-1.14734 + 0.439575i) q^{77} -5.04773 q^{79} +5.31407 q^{83} +2.43958 q^{85} +5.22294 q^{89} +(7.50508 - 2.87539i) q^{91} +2.25064i q^{95} +12.3087i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{7} + O(q^{10}) \) \( 16q + 2q^{7} - 12q^{25} - 8q^{37} - 8q^{43} + 2q^{49} + 28q^{67} + 44q^{79} + 16q^{85} - 18q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.70790 −1.21101 −0.605504 0.795842i \(-0.707028\pi\)
−0.605504 + 0.795842i \(0.707028\pi\)
\(6\) 0 0
\(7\) 0.946562 + 2.47063i 0.357767 + 0.933811i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.464391i 0.140019i 0.997546 + 0.0700096i \(0.0223030\pi\)
−0.997546 + 0.0700096i \(0.977697\pi\)
\(12\) 0 0
\(13\) 3.03772i 0.842511i −0.906942 0.421256i \(-0.861590\pi\)
0.906942 0.421256i \(-0.138410\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.900912 −0.218503 −0.109252 0.994014i \(-0.534845\pi\)
−0.109252 + 0.994014i \(0.534845\pi\)
\(18\) 0 0
\(19\) 0.831139i 0.190676i −0.995445 0.0953382i \(-0.969607\pi\)
0.995445 0.0953382i \(-0.0303932\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.12979i 1.48666i −0.668923 0.743332i \(-0.733244\pi\)
0.668923 0.743332i \(-0.266756\pi\)
\(24\) 0 0
\(25\) 2.33270 0.466540
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.22582i 0.784715i 0.919813 + 0.392358i \(0.128340\pi\)
−0.919813 + 0.392358i \(0.871660\pi\)
\(30\) 0 0
\(31\) 1.30567i 0.234505i −0.993102 0.117252i \(-0.962591\pi\)
0.993102 0.117252i \(-0.0374086\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.56319 6.69021i −0.433258 1.13085i
\(36\) 0 0
\(37\) 10.3804 1.70653 0.853266 0.521476i \(-0.174619\pi\)
0.853266 + 0.521476i \(0.174619\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.51235 −0.860885 −0.430442 0.902618i \(-0.641642\pi\)
−0.430442 + 0.902618i \(0.641642\pi\)
\(42\) 0 0
\(43\) 11.8200 1.80253 0.901267 0.433265i \(-0.142639\pi\)
0.901267 + 0.433265i \(0.142639\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.63862 −1.26007 −0.630036 0.776566i \(-0.716960\pi\)
−0.630036 + 0.776566i \(0.716960\pi\)
\(48\) 0 0
\(49\) −5.20804 + 4.67721i −0.744006 + 0.668173i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.45164i 1.16092i 0.814288 + 0.580461i \(0.197128\pi\)
−0.814288 + 0.580461i \(0.802872\pi\)
\(54\) 0 0
\(55\) 1.25752i 0.169564i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.2475 1.85487 0.927436 0.373982i \(-0.122008\pi\)
0.927436 + 0.373982i \(0.122008\pi\)
\(60\) 0 0
\(61\) 3.27899i 0.419831i 0.977720 + 0.209916i \(0.0673189\pi\)
−0.977720 + 0.209916i \(0.932681\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.22582i 1.02029i
\(66\) 0 0
\(67\) 3.40392 0.415854 0.207927 0.978144i \(-0.433328\pi\)
0.207927 + 0.978144i \(0.433328\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.76143i 0.921112i −0.887631 0.460556i \(-0.847650\pi\)
0.887631 0.460556i \(-0.152350\pi\)
\(72\) 0 0
\(73\) 4.83954i 0.566425i 0.959057 + 0.283213i \(0.0914002\pi\)
−0.959057 + 0.283213i \(0.908600\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.14734 + 0.439575i −0.130752 + 0.0500942i
\(78\) 0 0
\(79\) −5.04773 −0.567914 −0.283957 0.958837i \(-0.591647\pi\)
−0.283957 + 0.958837i \(0.591647\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.31407 0.583295 0.291647 0.956526i \(-0.405797\pi\)
0.291647 + 0.956526i \(0.405797\pi\)
\(84\) 0 0
\(85\) 2.43958 0.264609
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.22294 0.553631 0.276815 0.960923i \(-0.410721\pi\)
0.276815 + 0.960923i \(0.410721\pi\)
\(90\) 0 0
\(91\) 7.50508 2.87539i 0.786746 0.301422i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.25064i 0.230911i
\(96\) 0 0
\(97\) 12.3087i 1.24976i 0.780719 + 0.624882i \(0.214853\pi\)
−0.780719 + 0.624882i \(0.785147\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.82912 −0.779026 −0.389513 0.921021i \(-0.627357\pi\)
−0.389513 + 0.921021i \(0.627357\pi\)
\(102\) 0 0
\(103\) 6.24693i 0.615528i 0.951463 + 0.307764i \(0.0995808\pi\)
−0.951463 + 0.307764i \(0.900419\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.01207i 0.194514i −0.995259 0.0972570i \(-0.968993\pi\)
0.995259 0.0972570i \(-0.0310069\pi\)
\(108\) 0 0
\(109\) 6.19016 0.592910 0.296455 0.955047i \(-0.404196\pi\)
0.296455 + 0.955047i \(0.404196\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.879150i 0.0827035i 0.999145 + 0.0413517i \(0.0131664\pi\)
−0.999145 + 0.0413517i \(0.986834\pi\)
\(114\) 0 0
\(115\) 19.3067i 1.80036i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.852768 2.22582i −0.0781732 0.204041i
\(120\) 0 0
\(121\) 10.7843 0.980395
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.22277 0.646025
\(126\) 0 0
\(127\) 10.9409 0.970843 0.485422 0.874280i \(-0.338666\pi\)
0.485422 + 0.874280i \(0.338666\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.3491 1.42843 0.714214 0.699928i \(-0.246785\pi\)
0.714214 + 0.699928i \(0.246785\pi\)
\(132\) 0 0
\(133\) 2.05344 0.786724i 0.178056 0.0682176i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.5350i 1.58355i 0.610810 + 0.791777i \(0.290844\pi\)
−0.610810 + 0.791777i \(0.709156\pi\)
\(138\) 0 0
\(139\) 4.57103i 0.387710i 0.981030 + 0.193855i \(0.0620991\pi\)
−0.981030 + 0.193855i \(0.937901\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.41069 0.117968
\(144\) 0 0
\(145\) 11.4431i 0.950296i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.7609i 1.53695i −0.639881 0.768474i \(-0.721016\pi\)
0.639881 0.768474i \(-0.278984\pi\)
\(150\) 0 0
\(151\) −5.89312 −0.479576 −0.239788 0.970825i \(-0.577078\pi\)
−0.239788 + 0.970825i \(0.577078\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.53561i 0.283987i
\(156\) 0 0
\(157\) 10.1238i 0.807967i 0.914766 + 0.403983i \(0.132375\pi\)
−0.914766 + 0.403983i \(0.867625\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.6151 6.74878i 1.38826 0.531879i
\(162\) 0 0
\(163\) 7.19016 0.563177 0.281588 0.959535i \(-0.409139\pi\)
0.281588 + 0.959535i \(0.409139\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.93347 0.691293 0.345646 0.938365i \(-0.387660\pi\)
0.345646 + 0.938365i \(0.387660\pi\)
\(168\) 0 0
\(169\) 3.77227 0.290175
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.6634 1.19087 0.595433 0.803405i \(-0.296981\pi\)
0.595433 + 0.803405i \(0.296981\pi\)
\(174\) 0 0
\(175\) 2.20804 + 5.76324i 0.166912 + 0.435660i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.8912i 0.814048i 0.913418 + 0.407024i \(0.133433\pi\)
−0.913418 + 0.407024i \(0.866567\pi\)
\(180\) 0 0
\(181\) 23.4143i 1.74037i −0.492726 0.870185i \(-0.663999\pi\)
0.492726 0.870185i \(-0.336001\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −28.1091 −2.06662
\(186\) 0 0
\(187\) 0.418376i 0.0305947i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.2506i 0.741710i −0.928691 0.370855i \(-0.879065\pi\)
0.928691 0.370855i \(-0.120935\pi\)
\(192\) 0 0
\(193\) 14.4161 1.03769 0.518846 0.854868i \(-0.326362\pi\)
0.518846 + 0.854868i \(0.326362\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.4301i 1.24184i 0.783874 + 0.620920i \(0.213241\pi\)
−0.783874 + 0.620920i \(0.786759\pi\)
\(198\) 0 0
\(199\) 4.02673i 0.285447i −0.989763 0.142724i \(-0.954414\pi\)
0.989763 0.142724i \(-0.0455860\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.4404 + 4.00000i −0.732776 + 0.280745i
\(204\) 0 0
\(205\) 14.9269 1.04254
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.385974 0.0266984
\(210\) 0 0
\(211\) 15.2239 1.04806 0.524029 0.851701i \(-0.324428\pi\)
0.524029 + 0.851701i \(0.324428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −32.0073 −2.18288
\(216\) 0 0
\(217\) 3.22582 1.23589i 0.218983 0.0838979i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.73672i 0.184091i
\(222\) 0 0
\(223\) 24.3795i 1.63257i −0.577647 0.816287i \(-0.696029\pi\)
0.577647 0.816287i \(-0.303971\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.02482 0.200765 0.100382 0.994949i \(-0.467993\pi\)
0.100382 + 0.994949i \(0.467993\pi\)
\(228\) 0 0
\(229\) 15.9580i 1.05453i 0.849700 + 0.527266i \(0.176783\pi\)
−0.849700 + 0.527266i \(0.823217\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i −0.851658 0.524097i \(-0.824403\pi\)
0.851658 0.524097i \(-0.175597\pi\)
\(234\) 0 0
\(235\) 23.3925 1.52596
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.4637i 1.45306i 0.687137 + 0.726528i \(0.258867\pi\)
−0.687137 + 0.726528i \(0.741133\pi\)
\(240\) 0 0
\(241\) 9.86963i 0.635759i 0.948131 + 0.317880i \(0.102971\pi\)
−0.948131 + 0.317880i \(0.897029\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.1028 12.6654i 0.900997 0.809162i
\(246\) 0 0
\(247\) −2.52477 −0.160647
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.80709 −0.240302 −0.120151 0.992756i \(-0.538338\pi\)
−0.120151 + 0.992756i \(0.538338\pi\)
\(252\) 0 0
\(253\) 3.31101 0.208162
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.3660 1.27039 0.635197 0.772350i \(-0.280919\pi\)
0.635197 + 0.772350i \(0.280919\pi\)
\(258\) 0 0
\(259\) 9.82571 + 25.6462i 0.610540 + 1.59358i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.5102i 0.771413i 0.922622 + 0.385706i \(0.126042\pi\)
−0.922622 + 0.385706i \(0.873958\pi\)
\(264\) 0 0
\(265\) 22.8862i 1.40589i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.5669 −0.705246 −0.352623 0.935766i \(-0.614710\pi\)
−0.352623 + 0.935766i \(0.614710\pi\)
\(270\) 0 0
\(271\) 22.7010i 1.37899i −0.724290 0.689495i \(-0.757832\pi\)
0.724290 0.689495i \(-0.242168\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.08328i 0.0653245i
\(276\) 0 0
\(277\) −18.4873 −1.11079 −0.555397 0.831585i \(-0.687434\pi\)
−0.555397 + 0.831585i \(0.687434\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 27.4142i 1.63539i −0.575650 0.817696i \(-0.695251\pi\)
0.575650 0.817696i \(-0.304749\pi\)
\(282\) 0 0
\(283\) 1.75856i 0.104536i −0.998633 0.0522679i \(-0.983355\pi\)
0.998633 0.0522679i \(-0.0166450\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.21778 13.6190i −0.307996 0.803904i
\(288\) 0 0
\(289\) −16.1884 −0.952256
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.8168 −1.91718 −0.958590 0.284790i \(-0.908076\pi\)
−0.958590 + 0.284790i \(0.908076\pi\)
\(294\) 0 0
\(295\) −38.5809 −2.24626
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −21.6583 −1.25253
\(300\) 0 0
\(301\) 11.1884 + 29.2029i 0.644886 + 1.68323i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.87915i 0.508419i
\(306\) 0 0
\(307\) 21.2911i 1.21515i −0.794264 0.607573i \(-0.792143\pi\)
0.794264 0.607573i \(-0.207857\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.4950 1.27558 0.637788 0.770212i \(-0.279850\pi\)
0.637788 + 0.770212i \(0.279850\pi\)
\(312\) 0 0
\(313\) 10.7482i 0.607524i 0.952748 + 0.303762i \(0.0982427\pi\)
−0.952748 + 0.303762i \(0.901757\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.9867i 0.841735i −0.907122 0.420868i \(-0.861726\pi\)
0.907122 0.420868i \(-0.138274\pi\)
\(318\) 0 0
\(319\) −1.96243 −0.109875
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.748783i 0.0416634i
\(324\) 0 0
\(325\) 7.08608i 0.393065i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.17699 21.3429i −0.450812 1.17667i
\(330\) 0 0
\(331\) 29.5331 1.62329 0.811644 0.584153i \(-0.198573\pi\)
0.811644 + 0.584153i \(0.198573\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.21745 −0.503603
\(336\) 0 0
\(337\) −1.87915 −0.102364 −0.0511819 0.998689i \(-0.516299\pi\)
−0.0511819 + 0.998689i \(0.516299\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.606340 0.0328352
\(342\) 0 0
\(343\) −16.4854 8.43989i −0.890128 0.455711i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.8569i 1.06597i 0.846124 + 0.532987i \(0.178930\pi\)
−0.846124 + 0.532987i \(0.821070\pi\)
\(348\) 0 0
\(349\) 14.4103i 0.771366i 0.922631 + 0.385683i \(0.126034\pi\)
−0.922631 + 0.385683i \(0.873966\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.73009 −0.251757 −0.125879 0.992046i \(-0.540175\pi\)
−0.125879 + 0.992046i \(0.540175\pi\)
\(354\) 0 0
\(355\) 21.0171i 1.11547i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.8912i 1.63038i 0.579196 + 0.815188i \(0.303366\pi\)
−0.579196 + 0.815188i \(0.696634\pi\)
\(360\) 0 0
\(361\) 18.3092 0.963643
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.1050i 0.685946i
\(366\) 0 0
\(367\) 29.8945i 1.56048i 0.625481 + 0.780239i \(0.284903\pi\)
−0.625481 + 0.780239i \(0.715097\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −20.8809 + 8.00000i −1.08408 + 0.415339i
\(372\) 0 0
\(373\) −32.8441 −1.70060 −0.850302 0.526294i \(-0.823581\pi\)
−0.850302 + 0.526294i \(0.823581\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.8369 0.661131
\(378\) 0 0
\(379\) −7.11905 −0.365681 −0.182840 0.983143i \(-0.558529\pi\)
−0.182840 + 0.983143i \(0.558529\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.80676 0.398907 0.199453 0.979907i \(-0.436083\pi\)
0.199453 + 0.979907i \(0.436083\pi\)
\(384\) 0 0
\(385\) 3.10688 1.19032i 0.158341 0.0606645i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.6400i 1.40140i −0.713454 0.700702i \(-0.752870\pi\)
0.713454 0.700702i \(-0.247130\pi\)
\(390\) 0 0
\(391\) 6.42331i 0.324841i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.6687 0.687748
\(396\) 0 0
\(397\) 28.7948i 1.44517i −0.691282 0.722585i \(-0.742954\pi\)
0.691282 0.722585i \(-0.257046\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.2717i 1.46176i 0.682505 + 0.730881i \(0.260890\pi\)
−0.682505 + 0.730881i \(0.739110\pi\)
\(402\) 0 0
\(403\) −3.96625 −0.197573
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.82058i 0.238947i
\(408\) 0 0
\(409\) 18.6959i 0.924455i 0.886761 + 0.462228i \(0.152950\pi\)
−0.886761 + 0.462228i \(0.847050\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.4862 + 35.2004i 0.663611 + 1.73210i
\(414\) 0 0
\(415\) −14.3899 −0.706374
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.3439 −0.993863 −0.496932 0.867790i \(-0.665540\pi\)
−0.496932 + 0.867790i \(0.665540\pi\)
\(420\) 0 0
\(421\) −16.3804 −0.798333 −0.399167 0.916878i \(-0.630700\pi\)
−0.399167 + 0.916878i \(0.630700\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.10156 −0.101940
\(426\) 0 0
\(427\) −8.10117 + 3.10376i −0.392043 + 0.150202i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.74936i 0.276937i 0.990367 + 0.138469i \(0.0442180\pi\)
−0.990367 + 0.138469i \(0.955782\pi\)
\(432\) 0 0
\(433\) 27.6801i 1.33022i −0.746744 0.665111i \(-0.768384\pi\)
0.746744 0.665111i \(-0.231616\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.92584 −0.283472
\(438\) 0 0
\(439\) 1.17974i 0.0563060i 0.999604 + 0.0281530i \(0.00896256\pi\)
−0.999604 + 0.0281530i \(0.991037\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.8448i 1.41797i −0.705224 0.708985i \(-0.749153\pi\)
0.705224 0.708985i \(-0.250847\pi\)
\(444\) 0 0
\(445\) −14.1432 −0.670451
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.0241351i 0.00113900i −1.00000 0.000569502i \(-0.999819\pi\)
1.00000 0.000569502i \(-0.000181278\pi\)
\(450\) 0 0
\(451\) 2.55989i 0.120540i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −20.3230 + 7.78625i −0.952756 + 0.365025i
\(456\) 0 0
\(457\) −13.6540 −0.638706 −0.319353 0.947636i \(-0.603466\pi\)
−0.319353 + 0.947636i \(0.603466\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.7649 0.827396 0.413698 0.910414i \(-0.364237\pi\)
0.413698 + 0.910414i \(0.364237\pi\)
\(462\) 0 0
\(463\) −32.1272 −1.49308 −0.746539 0.665342i \(-0.768286\pi\)
−0.746539 + 0.665342i \(0.768286\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −35.3319 −1.63496 −0.817482 0.575954i \(-0.804631\pi\)
−0.817482 + 0.575954i \(0.804631\pi\)
\(468\) 0 0
\(469\) 3.22202 + 8.40982i 0.148779 + 0.388329i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.48911i 0.252389i
\(474\) 0 0
\(475\) 1.93880i 0.0889581i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 41.9652 1.91744 0.958720 0.284351i \(-0.0917780\pi\)
0.958720 + 0.284351i \(0.0917780\pi\)
\(480\) 0 0
\(481\) 31.5328i 1.43777i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 33.3308i 1.51347i
\(486\) 0 0
\(487\) −18.3785 −0.832810 −0.416405 0.909179i \(-0.636710\pi\)
−0.416405 + 0.909179i \(0.636710\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.35561i 0.151437i −0.997129 0.0757183i \(-0.975875\pi\)
0.997129 0.0757183i \(-0.0241250\pi\)
\(492\) 0 0
\(493\) 3.80709i 0.171463i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.1756 7.34667i 0.860145 0.329543i
\(498\) 0 0
\(499\) −2.91672 −0.130570 −0.0652851 0.997867i \(-0.520796\pi\)
−0.0652851 + 0.997867i \(0.520796\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.6880 1.36831 0.684156 0.729336i \(-0.260171\pi\)
0.684156 + 0.729336i \(0.260171\pi\)
\(504\) 0 0
\(505\) 21.2004 0.943407
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 43.4450 1.92566 0.962832 0.270100i \(-0.0870568\pi\)
0.962832 + 0.270100i \(0.0870568\pi\)
\(510\) 0 0
\(511\) −11.9567 + 4.58092i −0.528934 + 0.202648i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.9160i 0.745410i
\(516\) 0 0
\(517\) 4.01170i 0.176434i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.8539 −0.650760 −0.325380 0.945583i \(-0.605492\pi\)
−0.325380 + 0.945583i \(0.605492\pi\)
\(522\) 0 0
\(523\) 14.1296i 0.617845i −0.951087 0.308923i \(-0.900032\pi\)
0.951087 0.308923i \(-0.0999684\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.17629i 0.0512400i
\(528\) 0 0
\(529\) −27.8339 −1.21017
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16.7450i 0.725305i
\(534\) 0 0
\(535\) 5.44847i 0.235558i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.17206 2.41857i −0.0935571 0.104175i
\(540\) 0 0
\(541\) −4.80793 −0.206709 −0.103355 0.994645i \(-0.532958\pi\)
−0.103355 + 0.994645i \(0.532958\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.7623 −0.718019
\(546\) 0 0
\(547\) 2.10688 0.0900835 0.0450418 0.998985i \(-0.485658\pi\)
0.0450418 + 0.998985i \(0.485658\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.51224 0.149627
\(552\) 0 0
\(553\) −4.77798 12.4711i −0.203181 0.530324i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.43006i 0.399564i −0.979840 0.199782i \(-0.935977\pi\)
0.979840 0.199782i \(-0.0640235\pi\)
\(558\) 0 0
\(559\) 35.9058i 1.51865i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.9776 0.968389 0.484194 0.874960i \(-0.339113\pi\)
0.484194 + 0.874960i \(0.339113\pi\)
\(564\) 0 0
\(565\) 2.38065i 0.100155i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 33.8817i 1.42039i 0.704003 + 0.710197i \(0.251394\pi\)
−0.704003 + 0.710197i \(0.748606\pi\)
\(570\) 0 0
\(571\) 25.9606 1.08642 0.543209 0.839597i \(-0.317209\pi\)
0.543209 + 0.839597i \(0.317209\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.6316i 0.693587i
\(576\) 0 0
\(577\) 23.5811i 0.981692i 0.871246 + 0.490846i \(0.163312\pi\)
−0.871246 + 0.490846i \(0.836688\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.03009 + 13.1291i 0.208683 + 0.544687i
\(582\) 0 0
\(583\) −3.92487 −0.162551
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.4282 1.70993 0.854963 0.518689i \(-0.173580\pi\)
0.854963 + 0.518689i \(0.173580\pi\)
\(588\) 0 0
\(589\) −1.08519 −0.0447145
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.14846 0.211422 0.105711 0.994397i \(-0.466288\pi\)
0.105711 + 0.994397i \(0.466288\pi\)
\(594\) 0 0
\(595\) 2.30921 + 6.02729i 0.0946683 + 0.247095i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.37149i 0.0560375i −0.999607 0.0280187i \(-0.991080\pi\)
0.999607 0.0280187i \(-0.00891981\pi\)
\(600\) 0 0
\(601\) 17.8887i 0.729697i −0.931067 0.364849i \(-0.881121\pi\)
0.931067 0.364849i \(-0.118879\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −29.2029 −1.18727
\(606\) 0 0
\(607\) 41.4099i 1.68078i −0.541986 0.840388i \(-0.682327\pi\)
0.541986 0.840388i \(-0.317673\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 26.2417i 1.06163i
\(612\) 0 0
\(613\) −8.72073 −0.352227 −0.176114 0.984370i \(-0.556353\pi\)
−0.176114 + 0.984370i \(0.556353\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.0834i 0.405942i −0.979185 0.202971i \(-0.934940\pi\)
0.979185 0.202971i \(-0.0650597\pi\)
\(618\) 0 0
\(619\) 12.4673i 0.501105i 0.968103 + 0.250552i \(0.0806123\pi\)
−0.968103 + 0.250552i \(0.919388\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.94384 + 12.9040i 0.198071 + 0.516986i
\(624\) 0 0
\(625\) −31.2220 −1.24888
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.35185 −0.372883
\(630\) 0 0
\(631\) −4.06931 −0.161997 −0.0809984 0.996714i \(-0.525811\pi\)
−0.0809984 + 0.996714i \(0.525811\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −29.6267 −1.17570
\(636\) 0 0
\(637\) 14.2080 + 15.8206i 0.562943 + 0.626833i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.0675i 1.10860i 0.832317 + 0.554300i \(0.187014\pi\)
−0.832317 + 0.554300i \(0.812986\pi\)
\(642\) 0 0
\(643\) 31.9192i 1.25877i −0.777093 0.629386i \(-0.783307\pi\)
0.777093 0.629386i \(-0.216693\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.9723 0.824506 0.412253 0.911069i \(-0.364742\pi\)
0.412253 + 0.911069i \(0.364742\pi\)
\(648\) 0 0
\(649\) 6.61643i 0.259718i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16.5107i 0.646113i 0.946380 + 0.323057i \(0.104710\pi\)
−0.946380 + 0.323057i \(0.895290\pi\)
\(654\) 0 0
\(655\) −44.2716 −1.72984
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.9160i 0.814773i 0.913256 + 0.407387i \(0.133560\pi\)
−0.913256 + 0.407387i \(0.866440\pi\)
\(660\) 0 0
\(661\) 10.1694i 0.395542i 0.980248 + 0.197771i \(0.0633703\pi\)
−0.980248 + 0.197771i \(0.936630\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.56050 + 2.13037i −0.215627 + 0.0826121i
\(666\) 0 0
\(667\) 30.1292 1.16661
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.52273 −0.0587844
\(672\) 0 0
\(673\) −27.8180 −1.07230 −0.536152 0.844121i \(-0.680123\pi\)
−0.536152 + 0.844121i \(0.680123\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.5640 −0.790337 −0.395169 0.918609i \(-0.629314\pi\)
−0.395169 + 0.918609i \(0.629314\pi\)
\(678\) 0 0
\(679\) −30.4104 + 11.6510i −1.16704 + 0.447124i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.5064i 0.937711i −0.883275 0.468856i \(-0.844666\pi\)
0.883275 0.468856i \(-0.155334\pi\)
\(684\) 0 0
\(685\) 50.1909i 1.91770i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 25.6737 0.978090
\(690\) 0 0
\(691\) 6.68150i 0.254176i −0.991891 0.127088i \(-0.959437\pi\)
0.991891 0.127088i \(-0.0405631\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.3779i 0.469520i
\(696\) 0 0
\(697\) 4.96614 0.188106
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.0083i 1.73771i 0.495069 + 0.868854i \(0.335143\pi\)
−0.495069 + 0.868854i \(0.664857\pi\)
\(702\) 0 0
\(703\) 8.62758i 0.325395i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.41074 19.3429i −0.278710 0.727463i
\(708\) 0 0
\(709\) −10.1508 −0.381221 −0.190611 0.981666i \(-0.561047\pi\)
−0.190611 + 0.981666i \(0.561047\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.30913 −0.348629
\(714\) 0 0
\(715\) −3.82000 −0.142860
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.6071 1.06687 0.533433 0.845842i \(-0.320902\pi\)
0.533433 + 0.845842i \(0.320902\pi\)
\(720\) 0 0
\(721\) −15.4339 + 5.91310i −0.574787 + 0.220215i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.85757i 0.366101i
\(726\) 0 0
\(727\) 35.0803i 1.30106i −0.759482 0.650528i \(-0.774548\pi\)
0.759482 0.650528i \(-0.225452\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.6488 −0.393859
\(732\) 0 0
\(733\) 24.1836i 0.893243i 0.894723 + 0.446622i \(0.147373\pi\)
−0.894723 + 0.446622i \(0.852627\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.58075i 0.0582276i
\(738\) 0 0
\(739\) 9.26328 0.340755 0.170378 0.985379i \(-0.445501\pi\)
0.170378 + 0.985379i \(0.445501\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 42.5464i 1.56088i −0.625233 0.780438i \(-0.714996\pi\)
0.625233 0.780438i \(-0.285004\pi\)
\(744\) 0 0
\(745\) 50.8024i 1.86126i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.97108 1.90455i 0.181639 0.0695906i
\(750\) 0 0
\(751\) −36.9015 −1.34655 −0.673277 0.739390i \(-0.735114\pi\)
−0.673277 + 0.739390i \(0.735114\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.9580 0.580770
\(756\) 0 0
\(757\) −2.70095 −0.0981678 −0.0490839 0.998795i \(-0.515630\pi\)
−0.0490839 + 0.998795i \(0.515630\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.1887 0.804340 0.402170 0.915565i \(-0.368256\pi\)
0.402170 + 0.915565i \(0.368256\pi\)
\(762\) 0 0
\(763\) 5.85937 + 15.2936i 0.212123 + 0.553666i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 43.2800i 1.56275i
\(768\) 0 0
\(769\) 39.1470i 1.41168i −0.708373 0.705838i \(-0.750571\pi\)
0.708373 0.705838i \(-0.249429\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −46.3242 −1.66617 −0.833083 0.553149i \(-0.813426\pi\)
−0.833083 + 0.553149i \(0.813426\pi\)
\(774\) 0 0
\(775\) 3.04573i 0.109406i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.58153i 0.164150i
\(780\) 0 0
\(781\) 3.60434 0.128973
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 27.4142i 0.978454i
\(786\) 0 0
\(787\) 42.7292i 1.52313i 0.648088 + 0.761565i \(0.275569\pi\)
−0.648088 + 0.761565i \(0.724431\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.17206 + 0.832170i −0.0772294 + 0.0295885i
\(792\) 0 0
\(793\) 9.96063 0.353712
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.5081 0.903543 0.451772 0.892134i \(-0.350792\pi\)
0.451772 + 0.892134i \(0.350792\pi\)
\(798\) 0 0
\(799\) 7.78264 0.275330
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.24744 −0.0793105
\(804\) 0